%----Output an unreadable proof
% SZS output start Proof for theBenchmark
thf(type_def_5, type, term: $tType).
thf(type_def_6, type, subst: $tType).
thf(type_def_7, type, sTfun: ($tType * $tType) > $tType).
thf(type_def_8, type, d_term: $tType).
thf(type_def_9, type, d_subst: $tType).
thf(func_def_0, type, one: term).
thf(func_def_1, type, ap: (term > term > term)).
thf(func_def_2, type, lam: (term > term)).
thf(func_def_3, type, sub: (term > subst > term)).
thf(func_def_4, type, id: subst).
thf(func_def_5, type, sh: subst).
thf(func_def_6, type, push: (term > subst > subst)).
thf(func_def_7, type, comp: (subst > subst > subst)).
thf(func_def_8, type, var: (term > $o)).
thf(func_def_9, type, pushprop_lem1v2: $o).
thf(func_def_10, type, pushprop_lem1_gthm: $o).
thf(func_def_11, type, axmap: $o).
thf(func_def_12, type, pushprop_lem0_gthm: $o).
thf(func_def_13, type, shinj: $o).
thf(func_def_14, type, hoasinduction_lem1v2: $o).
thf(func_def_15, type, hoasinduction_lem1v2_gthm: $o).
thf(func_def_16, type, hoasap: (subst > term > subst > term > term)).
thf(func_def_17, type, induction2lem: $o).
thf(func_def_18, type, hoasinduction_lem3v2_f: $o).
thf(func_def_19, type, axvarshift: $o).
thf(func_def_20, type, hoasapinj2: $o).
thf(func_def_21, type, hoasapnotvar_gthm: $o).
thf(func_def_22, type, hoasapinj1: $o).
thf(func_def_23, type, ulamvar1: $o).
thf(func_def_24, type, induction2lem_lthm: $o).
thf(func_def_25, type, hoasinduction_lem3v2_gthm: $o).
thf(func_def_26, type, apnotvar: $o).
thf(func_def_27, type, pushprop_lthm_orig: $o).
thf(func_def_28, type, hoasinduction_lem3v2_f_lthm: $o).
thf(func_def_29, type, hoasinduction_lthm: $o).
thf(func_def_30, type, hoasinduction_no_psi_cond_lthm: $o).
thf(func_def_31, type, hoaslaminj: $o).
thf(func_def_32, type, hoasinduction_lem3aaa: $o).
thf(func_def_33, type, induction2lem_gthm: $o).
thf(func_def_34, type, hoasinduction_lem3aa_lthm: $o).
thf(func_def_35, type, hoasinduction_lem3: $o).
thf(func_def_36, type, hoasinduction_lem2: $o).
thf(func_def_37, type, termmset_lthm: $o).
thf(func_def_38, type, hoasinduction_lem1: $o).
thf(func_def_39, type, hoaslamnotap_lthm: $o).
thf(func_def_40, type, pushprop_lem1v2_lthm: $o).
thf(func_def_41, type, hoasapnotvar: $o).
thf(func_def_42, type, hoasinduction_lem0: $o).
thf(func_def_43, type, hoasinduction: $o).
thf(func_def_44, type, hoasinduction_gthm: $o).
thf(func_def_45, type, axapp: $o).
thf(func_def_46, type, hoaslamnotvar_lthm: $o).
thf(func_def_47, type, pushprop_lem3v2_lthm: $o).
thf(func_def_48, type, hoasinduction_lem3b_lthm: $o).
thf(func_def_49, type, ulamvarind: $o).
thf(func_def_50, type, induction: $o).
thf(func_def_51, type, hoasinduction_lem3a_lthm: $o).
thf(func_def_52, type, termmset_gthm: $o).
thf(func_def_53, type, hoasinduction_lem3aa: $o).
thf(func_def_54, type, pushprop_lem1v2_gthm: $o).
thf(func_def_55, type, hoaslamnotap_gthm: $o).
thf(func_def_56, type, hoaslamnotvar_gthm: $o).
thf(func_def_57, type, hoasinduction_lem3b_gthm: $o).
thf(func_def_58, type, pushprop_lem2v2: $o).
thf(func_def_59, type, hoasinduction_lem3a_gthm: $o).
thf(func_def_60, type, axclos: $o).
thf(func_def_61, type, axassoc: $o).
thf(func_def_62, type, hoasinduction_lem2v2: $o).
thf(func_def_63, type, pushprop_lthm: $o).
thf(func_def_64, type, apinj2: $o).
thf(func_def_65, type, apinj1: $o).
thf(func_def_66, type, hoasapinj2_lthm: $o).
thf(func_def_67, type, hoasinduction_lem3v2a: $o).
thf(func_def_68, type, hoasapinj1_lthm: $o).
thf(func_def_69, type, hoaslaminj_lthm: $o).
thf(func_def_70, type, axvarcons: $o).
thf(func_def_71, type, hoaslam: (subst > (subst > term > term) > term)).
thf(func_def_72, type, axscons: $o).
thf(func_def_73, type, hoasinduction_lem2v2_gthm: $o).
thf(func_def_74, type, axidr: $o).
thf(func_def_75, type, pushprop_lem1: $o).
thf(func_def_76, type, laminj: $o).
thf(func_def_77, type, hoasinduction_lem3_lthm: $o).
thf(func_def_78, type, pushprop_lem0: $o).
thf(func_def_79, type, pushprop_gthm: $o).
thf(func_def_80, type, axabs: $o).
thf(func_def_81, type, hoasinduction_lem3v2a_lthm: $o).
thf(func_def_82, type, hoasinduction_lem2_lthm: $o).
thf(func_def_83, type, hoasapinj2_gthm: $o).
thf(func_def_84, type, hoasinduction_p_and_p_prime: ((subst > term > subst > $o) > (term > $o) > $o)).
thf(func_def_85, type, hoasinduction_lem1_lthm: $o).
thf(func_def_86, type, lamnotap: $o).
thf(func_def_87, type, hoasapinj1_gthm: $o).
thf(func_def_88, type, hoaslamnotvar: $o).
thf(func_def_89, type, axidl: $o).
thf(func_def_90, type, hoaslaminj_gthm: $o).
thf(func_def_91, type, induction2_lthm: $o).
thf(func_def_92, type, hoasinduction_lem0_lthm: $o).
thf(func_def_93, type, substmonoid_lthm: $o).
thf(func_def_94, type, pushprop: $o).
thf(func_def_95, type, hoasinduction_lem3_gthm: $o).
thf(func_def_96, type, hoasinduction_lem2_gthm: $o).
thf(func_def_97, type, hoasinduction_lem3b: $o).
thf(func_def_98, type, substmonoid: $o).
thf(func_def_99, type, lamnotvar: $o).
thf(func_def_100, type, hoasinduction_lem3a: $o).
thf(func_def_101, type, hoasinduction_lem1_gthm: $o).
thf(func_def_102, type, hoasinduction_no_psi_cond: $o).
thf(func_def_103, type, induction2_gthm: $o).
thf(func_def_104, type, pushprop_lem2v2_lthm: $o).
thf(func_def_105, type, hoasvar: (subst > term > subst > $o)).
thf(func_def_106, type, hoaslamnotap: $o).
thf(func_def_107, type, substmonoid_gthm: $o).
thf(func_def_108, type, ulamvarsh: $o).
thf(func_def_109, type, induction2: $o).
thf(func_def_110, type, pushprop_lem3v2: $o).
thf(func_def_111, type, pushprop_lem2v2_gthm: $o).
thf(func_def_112, type, pushprop_lem1_lthm: $o).
thf(func_def_113, type, hoasinduction_lem3v2: $o).
thf(func_def_114, type, axshiftcons: $o).
thf(func_def_115, type, termmset: $o).
thf(func_def_116, type, pushprop_lem0_lthm: $o).
thf(func_def_117, type, hoasapnotvar_lthm: $o).
thf(func_def_118, type, hoasinduction_lem3v2_lthm: $o).
thf(func_def_119, type, pushprop_p_and_p_prime: (term > subst > (term > $o) > (term > $o) > $o)).
thf(func_def_120, type, axvarid: $o).
thf(func_def_121, type, hoasinduction_lthm_3: $o).
thf(func_def_122, type, d2term: (d_term > term)).
thf(func_def_123, type, d2subst: (d_subst > subst)).
thf(func_def_124, type, d_one: d_term).
thf(func_def_125, type, d_id: d_subst).
thf(func_def_129, type, vLAM: !>[X0: $tType, X1: $tType]:((X1) > (X0 > X1))).
thf(func_def_130, type, vPI: !>[X0: $tType]:(((X0 > $o) > $o))).
thf(func_def_131, type, db1: !>[X0: $tType]:(X0)).
thf(func_def_132, type, db0: !>[X0: $tType]:(X0)).
thf(func_def_133, type, db2: !>[X0: $tType]:(X0)).
thf(func_def_134, type, db4: !>[X0: $tType]:(X0)).
thf(func_def_135, type, db3: !>[X0: $tType]:(X0)).
thf(func_def_136, type, vEQ: !>[X0: $tType]:((X0 > X0 > $o))).
thf(func_def_137, type, vSIGMA: !>[X0: $tType]:(((X0 > $o) > $o))).
thf(func_def_138, type, sP0: ((subst > term > subst > $o) > $o)).
thf(func_def_139, type, sP1: $o).
thf(func_def_140, type, sP2: ((term > $o) > $o)).
thf(func_def_141, type, sP3: ((subst > term > subst > $o) > $o)).
thf(func_def_142, type, sP4: $o).
thf(func_def_143, type, sP5: $o).
thf(func_def_144, type, sP6: $o).
thf(func_def_145, type, sP7: ((subst > term > subst > $o) > $o)).
thf(func_def_146, type, sP8: $o).
thf(func_def_147, type, sP9: ((subst > term > subst > $o) > $o)).
thf(func_def_148, type, sP10: ((subst > term > subst > $o) > $o)).
thf(func_def_149, type, sP11: ((subst > term > subst > $o) > $o)).
thf(func_def_150, type, sP12: ((subst > term > subst > $o) > $o)).
thf(func_def_151, type, sP13: ((term > $o) > $o)).
thf(func_def_152, type, sP14: $o).
thf(func_def_153, type, sP15: $o).
thf(func_def_154, type, sP16: $o).
thf(func_def_155, type, sP17: $o).
thf(func_def_156, type, sP18: $o).
thf(func_def_157, type, sP19: ((subst > term > subst > $o) > $o)).
thf(func_def_158, type, sP20: ((subst > term > subst > $o) > $o)).
thf(func_def_159, type, sP21: ((subst > term > subst > $o) > $o)).
thf(func_def_160, type, sP22: $o).
thf(func_def_161, type, sP23: ((term > $o) > $o)).
thf(func_def_162, type, sP24: ((subst > term > subst > $o) > $o)).
thf(func_def_163, type, sP25: $o).
thf(func_def_164, type, sP26: ((term > $o) > $o)).
thf(func_def_165, type, sP27: ((term > $o) > $o)).
thf(func_def_166, type, sP28: $o).
thf(func_def_167, type, sP29: $o).
thf(func_def_168, type, sP30: $o).
thf(func_def_169, type, sP31: $o).
thf(func_def_170, type, sP32: ((term > $o) > $o)).
thf(func_def_171, type, sP33: $o).
thf(func_def_172, type, sP34: ((term > $o) > $o)).
thf(func_def_173, type, sP35: ((term > $o) > $o)).
thf(func_def_174, type, sP36: $o).
thf(func_def_175, type, sP37: $o).
thf(func_def_176, type, sP38: $o).
thf(func_def_177, type, sP39: $o).
thf(func_def_178, type, sP40: ((subst > term > subst > $o) > $o)).
thf(func_def_179, type, sP41: $o).
thf(func_def_180, type, sP42: ((subst > term > subst > $o) > $o)).
thf(func_def_181, type, sP43: $o).
thf(func_def_182, type, sP44: $o).
thf(func_def_183, type, sP45: ((term > $o) > $o)).
thf(func_def_184, type, sP46: $o).
thf(func_def_185, type, sP47: $o).
thf(func_def_186, type, sP48: $o).
thf(func_def_187, type, sP49: $o).
thf(func_def_188, type, sP50: $o).
thf(func_def_189, type, sP51: ((subst > term > subst > $o) > $o)).
thf(func_def_190, type, sP52: $o).
thf(func_def_191, type, sP53: ((subst > term > subst > $o) > $o)).
thf(func_def_192, type, sP54: ((subst > term > subst > $o) > $o)).
thf(func_def_193, type, sP55: ((term > $o) > $o)).
thf(func_def_194, type, sP56: $o).
thf(func_def_195, type, sP57: $o).
thf(func_def_196, type, sP58: $o).
thf(func_def_197, type, sK59: ((term > $o) > term)).
thf(func_def_198, type, sK60: ((term > $o) > term)).
thf(func_def_199, type, sK61: ((subst > term > subst > $o) > term > $o)).
thf(func_def_200, type, sK62: term).
thf(func_def_201, type, sK63: term).
thf(func_def_202, type, sK64: (term > $o)).
thf(func_def_203, type, sK65: (term > term)).
thf(func_def_204, type, sK66: (subst > term > subst > $o)).
thf(func_def_205, type, sK67: ((term > $o) > term)).
thf(func_def_206, type, sK68: term).
thf(func_def_207, type, sK69: (subst > term > subst > $o)).
thf(func_def_208, type, sK70: ((subst > term > subst > $o) > subst)).
thf(func_def_209, type, sK71: ((subst > term > subst > $o) > subst)).
thf(func_def_210, type, sK72: ((subst > term > subst > $o) > subst)).
thf(func_def_211, type, sK73: ((subst > term > subst > $o) > term)).
thf(func_def_212, type, sK74: ((subst > term > subst > $o) > subst)).
thf(func_def_213, type, sK75: ((subst > term > subst > $o) > subst)).
thf(func_def_214, type, sK76: ((subst > term > subst > $o) > term)).
thf(func_def_215, type, sK77: ((subst > term > subst > $o) > subst)).
thf(func_def_216, type, sK78: ((subst > term > subst > $o) > term)).
thf(func_def_217, type, sK79: ((term > $o) > term > term)).
thf(func_def_218, type, sK80: ((subst > term > subst > $o) > (term > $o) > term)).
thf(func_def_219, type, sK81: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_220, type, sK82: term).
thf(func_def_221, type, sK83: (term > $o)).
thf(func_def_222, type, sK84: (subst > term > subst > $o)).
thf(func_def_223, type, sK85: ((subst > term > term) > subst)).
thf(func_def_224, type, sK86: ((subst > term > term) > term)).
thf(func_def_225, type, sK87: ((subst > term > term) > subst)).
thf(func_def_226, type, sK88: ((subst > term > term) > term)).
thf(func_def_227, type, sK89: ((term > $o) > term)).
thf(func_def_228, type, sK90: ((term > $o) > term)).
thf(func_def_229, type, sK91: ((subst > term > subst > $o) > term)).
thf(func_def_230, type, sK92: ((subst > term > subst > $o) > term)).
thf(func_def_231, type, sK93: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_232, type, sK94: ((subst > term > term) > subst)).
thf(func_def_233, type, sK95: ((subst > term > term) > subst)).
thf(func_def_234, type, sK96: ((subst > term > term) > term)).
thf(func_def_235, type, sK97: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_236, type, sK98: term).
thf(func_def_237, type, sK99: term).
thf(func_def_238, type, sK100: (subst > term > subst > $o)).
thf(func_def_239, type, sK101: ((subst > term > subst > $o) > subst)).
thf(func_def_240, type, sK102: ((subst > term > subst > $o) > term)).
thf(func_def_241, type, sK103: ((subst > term > subst > $o) > subst)).
thf(func_def_242, type, sK104: ((subst > term > subst > $o) > subst)).
thf(func_def_243, type, sK105: ((subst > term > subst > $o) > subst)).
thf(func_def_244, type, sK106: ((subst > term > subst > $o) > subst)).
thf(func_def_245, type, sK107: ((subst > term > subst > $o) > subst)).
thf(func_def_246, type, sK108: ((subst > term > subst > $o) > term)).
thf(func_def_247, type, sK109: ((subst > term > subst > $o) > term)).
thf(func_def_248, type, sK110: ((subst > term > subst > $o) > term)).
thf(func_def_249, type, sK111: (subst > term > subst > $o)).
thf(func_def_250, type, sK112: term).
thf(func_def_251, type, sK113: ((subst > term > term) > term)).
thf(func_def_252, type, sK114: ((subst > term > term) > subst)).
thf(func_def_253, type, sK115: ((subst > term > term) > term)).
thf(func_def_254, type, sK116: ((subst > term > term) > subst)).
thf(func_def_255, type, sK117: term).
thf(func_def_256, type, sK118: (term > (subst > term > subst > $o) > term)).
thf(func_def_257, type, sK119: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_258, type, sK120: (term > (subst > term > subst > $o) > term)).
thf(func_def_259, type, sK121: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_260, type, sK122: (subst > term > subst > $o)).
thf(func_def_261, type, sK123: term).
thf(func_def_262, type, sK124: ((subst > term > term) > term)).
thf(func_def_263, type, sK125: ((subst > term > term) > subst)).
thf(func_def_264, type, sK126: ((subst > term > term) > term)).
thf(func_def_265, type, sK127: ((subst > term > term) > subst)).
thf(func_def_266, type, sK128: (term > $o)).
thf(func_def_267, type, sK129: term).
thf(func_def_268, type, sK130: (subst > term > subst > $o)).
thf(func_def_269, type, sK131: ((subst > term > term) > term)).
thf(func_def_270, type, sK132: ((subst > term > term) > subst)).
thf(func_def_271, type, sK133: ((subst > term > term) > subst)).
thf(func_def_272, type, sK134: ((subst > term > term) > term)).
thf(func_def_273, type, sK135: ((subst > term > subst > $o) > (term > $o) > term)).
thf(func_def_274, type, sK136: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_275, type, sK137: (term > (term > $o) > term)).
thf(func_def_276, type, sK138: term).
thf(func_def_277, type, sK139: (subst > term > subst > $o)).
thf(func_def_278, type, sK140: ((subst > term > term) > term)).
thf(func_def_279, type, sK141: ((subst > term > term) > term)).
thf(func_def_280, type, sK142: ((subst > term > term) > subst)).
thf(func_def_281, type, sK143: ((subst > term > term) > subst)).
thf(func_def_282, type, sK144: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_283, type, sK145: ((subst > term > subst > $o) > term > term)).
thf(func_def_284, type, sK146: subst).
thf(func_def_285, type, sK147: term).
thf(func_def_286, type, sK148: (term > $o)).
thf(func_def_287, type, sK149: (term > term)).
thf(func_def_288, type, sK150: ((term > $o) > term)).
thf(func_def_289, type, sK151: ((term > $o) > subst > term)).
thf(func_def_290, type, sK152: ((term > $o) > term)).
thf(func_def_291, type, sK153: ((term > $o) > term)).
thf(func_def_292, type, sK154: term).
thf(func_def_293, type, sK155: (subst > term > subst > $o)).
thf(func_def_294, type, sK156: term).
thf(func_def_295, type, sK157: ((subst > term > subst > $o) > subst)).
thf(func_def_296, type, sK158: ((subst > term > subst > $o) > subst)).
thf(func_def_297, type, sK159: ((subst > term > subst > $o) > term)).
thf(func_def_298, type, sK160: ((subst > term > subst > $o) > subst)).
thf(func_def_299, type, sK161: ((subst > term > subst > $o) > subst)).
thf(func_def_300, type, sK162: ((subst > term > subst > $o) > subst)).
thf(func_def_301, type, sK163: ((subst > term > subst > $o) > subst)).
thf(func_def_302, type, sK164: ((subst > term > subst > $o) > term)).
thf(func_def_303, type, sK165: ((subst > term > subst > $o) > term > term)).
thf(func_def_304, type, sK166: ((subst > term > subst > $o) > term > term)).
thf(func_def_305, type, sK167: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_306, type, sK168: term).
thf(func_def_307, type, sK169: (subst > term > subst > $o)).
thf(func_def_308, type, sK170: ((subst > term > term) > subst)).
thf(func_def_309, type, sK171: ((subst > term > term) > term)).
thf(func_def_310, type, sK172: ((subst > term > term) > subst)).
thf(func_def_311, type, sK173: ((subst > term > term) > term)).
thf(func_def_312, type, sK174: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_313, type, sK175: ((subst > term > term) > subst)).
thf(func_def_314, type, sK176: ((subst > term > term) > subst)).
thf(func_def_315, type, sK177: ((subst > term > term) > term)).
thf(func_def_316, type, sK178: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_317, type, sK179: (subst > term > subst > $o)).
thf(func_def_318, type, sK180: term).
thf(func_def_319, type, sK181: ((subst > term > subst > $o) > term)).
thf(func_def_320, type, sK182: ((subst > term > subst > $o) > subst)).
thf(func_def_321, type, sK183: ((subst > term > subst > $o) > subst)).
thf(func_def_322, type, sK184: ((subst > term > subst > $o) > subst)).
thf(func_def_323, type, sK185: ((subst > term > subst > $o) > subst)).
thf(func_def_324, type, sK186: ((subst > term > subst > $o) > subst)).
thf(func_def_325, type, sK187: ((subst > term > subst > $o) > subst)).
thf(func_def_326, type, sK188: ((subst > term > subst > $o) > term)).
thf(func_def_327, type, sK189: (term > (subst > term > subst > $o) > term)).
thf(func_def_328, type, sK190: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_329, type, sK191: ((subst > term > term) > subst)).
thf(func_def_330, type, sK192: ((subst > term > term) > subst)).
thf(func_def_331, type, sK193: ((subst > term > term) > term)).
thf(func_def_332, type, sK194: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_333, type, sK195: (subst > term > subst > $o)).
thf(func_def_334, type, sK196: term).
thf(func_def_335, type, sK197: ((subst > term > term) > term)).
thf(func_def_336, type, sK198: ((subst > term > term) > term)).
thf(func_def_337, type, sK199: ((subst > term > term) > subst)).
thf(func_def_338, type, sK200: ((subst > term > term) > subst)).
thf(func_def_339, type, sK201: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_340, type, sK202: (term > (subst > term > subst > $o) > term)).
thf(func_def_341, type, sK203: term).
thf(func_def_342, type, sK204: (subst > term > subst > $o)).
thf(func_def_343, type, sK205: ((subst > term > term) > term)).
thf(func_def_344, type, sK206: ((subst > term > term) > subst)).
thf(func_def_345, type, sK207: ((subst > term > term) > subst)).
thf(func_def_346, type, sK208: ((subst > term > term) > term)).
thf(func_def_347, type, sK209: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_348, type, sK210: ((subst > term > subst > $o) > term > term)).
thf(func_def_349, type, sK211: ((term > $o) > term)).
thf(func_def_350, type, sK212: ((term > $o) > term)).
thf(func_def_351, type, sK213: (term > $o)).
thf(func_def_352, type, sK214: term).
thf(func_def_353, type, sK215: subst).
thf(func_def_354, type, sK216: term).
thf(func_def_355, type, sK217: subst).
thf(func_def_356, type, sK218: subst).
thf(func_def_357, type, sK219: term).
thf(func_def_358, type, sK220: subst).
thf(func_def_359, type, sK221: term).
thf(func_def_360, type, sK222: (term > $o)).
thf(func_def_361, type, sK223: (term > term)).
thf(func_def_362, type, sK224: ((term > $o) > subst > term)).
thf(func_def_363, type, sK225: ((term > $o) > term)).
thf(func_def_364, type, sK226: ((term > $o) > term)).
thf(func_def_365, type, sK227: ((term > $o) > term)).
thf(func_def_366, type, sK228: ((term > $o) > term)).
thf(func_def_367, type, sK229: ((term > $o) > term)).
thf(func_def_368, type, sK230: term).
thf(func_def_369, type, sK231: (term > $o)).
thf(func_def_370, type, sK232: (term > term)).
thf(func_def_371, type, sK233: ((term > $o) > term)).
thf(func_def_372, type, sK234: ((term > $o) > term)).
thf(func_def_373, type, sK235: ((term > $o) > term)).
thf(func_def_374, type, sK236: ((term > $o) > term)).
thf(func_def_375, type, sK237: (subst > term > term)).
thf(func_def_376, type, sK238: subst).
thf(func_def_377, type, sK239: (subst > term > term)).
thf(func_def_378, type, sK240: term).
thf(func_def_379, type, sK241: term).
thf(func_def_380, type, sK242: subst).
thf(func_def_381, type, sK243: ((subst > term > term) > subst)).
thf(func_def_382, type, sK244: ((subst > term > term) > term)).
thf(func_def_383, type, sK245: ((subst > term > term) > subst)).
thf(func_def_384, type, sK246: ((subst > term > term) > term)).
thf(func_def_385, type, sK247: ((subst > term > term) > subst)).
thf(func_def_386, type, sK248: ((subst > term > term) > subst)).
thf(func_def_387, type, sK249: subst).
thf(func_def_388, type, sK250: term).
thf(func_def_389, type, sK251: term).
thf(func_def_390, type, sK252: term).
thf(func_def_391, type, sK253: term).
thf(func_def_392, type, sK254: (subst > term > subst > $o)).
thf(func_def_393, type, sK255: (term > $o)).
thf(func_def_394, type, sK256: ((subst > term > term) > term)).
thf(func_def_395, type, sK257: ((subst > term > term) > subst)).
thf(func_def_396, type, sK258: ((subst > term > term) > subst)).
thf(func_def_397, type, sK259: ((subst > term > term) > term)).
thf(func_def_398, type, sK260: (term > subst > term > term)).
thf(func_def_399, type, sK261: term).
thf(func_def_400, type, sK262: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_401, type, sK263: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_402, type, sK264: ((term > $o) > term > term)).
thf(func_def_403, type, sK265: term).
thf(func_def_404, type, sK266: ((subst > term > term) > subst)).
thf(func_def_405, type, sK267: ((subst > term > term) > term)).
thf(func_def_406, type, sK268: (subst > (term > $o) > term)).
thf(func_def_407, type, sK269: ((term > $o) > term)).
thf(func_def_408, type, sK270: term).
thf(func_def_409, type, sK271: (term > $o)).
thf(func_def_410, type, sK272: subst).
thf(func_def_411, type, sK273: term).
thf(func_def_412, type, sK274: (term > term)).
thf(func_def_413, type, sK275: term).
thf(func_def_414, type, sK276: (term > $o)).
thf(func_def_415, type, sK277: (term > term)).
thf(func_def_416, type, sK278: ((term > $o) > term)).
thf(func_def_417, type, sK279: ((term > $o) > term)).
thf(func_def_418, type, sK280: ((term > $o) > term)).
thf(func_def_419, type, sK281: ((term > $o) > term)).
thf(func_def_420, type, sK282: ((term > $o) > term)).
thf(func_def_421, type, sK283: ((term > $o) > term)).
thf(func_def_422, type, sK284: (term > $o)).
thf(func_def_423, type, sK285: (subst > term > subst > $o)).
thf(func_def_424, type, sK286: term).
thf(func_def_425, type, sK287: (term > (term > $o) > term)).
thf(func_def_426, type, sK288: ((subst > term > subst > $o) > subst)).
thf(func_def_427, type, sK289: ((subst > term > subst > $o) > subst)).
thf(func_def_428, type, sK290: ((subst > term > subst > $o) > term)).
thf(func_def_429, type, sK291: ((subst > term > subst > $o) > subst)).
thf(func_def_430, type, sK292: ((subst > term > subst > $o) > (term > $o) > term)).
thf(func_def_431, type, sK293: ((subst > term > subst > $o) > subst)).
thf(func_def_432, type, sK294: ((subst > term > subst > $o) > subst)).
thf(func_def_433, type, sK295: ((subst > term > subst > $o) > subst)).
thf(func_def_434, type, sK296: ((subst > term > subst > $o) > term)).
thf(func_def_435, type, sK297: term).
thf(func_def_436, type, sK298: ((subst > term > term) > subst)).
thf(func_def_437, type, sK299: ((subst > term > term) > subst)).
thf(func_def_438, type, sK300: ((subst > term > term) > term)).
thf(func_def_439, type, sK301: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_440, type, sK302: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_441, type, sK303: ((term > $o) > term)).
thf(func_def_442, type, sK304: ((term > $o) > term)).
thf(func_def_443, type, sK305: (subst > term > subst > $o)).
thf(func_def_444, type, sK306: term).
thf(func_def_445, type, sK307: ((subst > term > subst > $o) > term)).
thf(func_def_446, type, sK308: ((subst > term > subst > $o) > term)).
thf(func_def_447, type, sK309: ((subst > term > subst > $o) > subst)).
thf(func_def_448, type, sK310: ((subst > term > subst > $o) > subst)).
thf(func_def_449, type, sK311: ((subst > term > subst > $o) > subst)).
thf(func_def_450, type, sK312: ((subst > term > subst > $o) > subst)).
thf(func_def_451, type, sK313: ((subst > term > subst > $o) > subst)).
thf(func_def_452, type, sK314: ((subst > term > subst > $o) > term)).
thf(func_def_453, type, sK315: ((subst > term > subst > $o) > subst)).
thf(func_def_454, type, sK316: ((subst > term > subst > $o) > term)).
thf(func_def_455, type, sK317: ((subst > term > subst > $o) > term)).
thf(func_def_456, type, sK318: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_457, type, sK319: ((subst > term > term) > term)).
thf(func_def_458, type, sK320: ((subst > term > term) > subst)).
thf(func_def_459, type, sK321: ((subst > term > term) > subst)).
thf(func_def_460, type, sK322: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_461, type, sK323: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_462, type, sK324: ((subst > term > term) > term)).
thf(func_def_463, type, sK325: ((subst > term > term) > subst)).
thf(func_def_464, type, sK326: ((subst > term > term) > subst)).
thf(func_def_465, type, sK327: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_466, type, sK328: (subst > term > subst > $o)).
thf(func_def_467, type, sK329: term).
thf(func_def_468, type, sK330: ((subst > term > term) > term > subst)).
thf(func_def_469, type, sK331: ((subst > term > term) > term > subst)).
thf(func_def_470, type, sK332: ((subst > term > term) > term > term)).
thf(func_def_471, type, sK333: ((subst > term > term) > term > subst)).
thf(func_def_472, type, sK334: ((subst > term > term) > term > subst)).
thf(func_def_473, type, sK335: ((subst > term > term) > term > term)).
thf(func_def_474, type, sK336: ((subst > term > term) > term)).
thf(func_def_475, type, sK337: ((subst > term > subst > $o) > term)).
thf(func_def_476, type, sK338: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_477, type, sK339: (term > (subst > term > subst > $o) > term)).
thf(func_def_478, type, sK340: ((term > $o) > term)).
thf(func_def_479, type, sK341: ((term > $o) > term)).
thf(func_def_480, type, sK342: (term > $o)).
thf(func_def_481, type, sK343: term).
thf(func_def_482, type, sK344: (term > term)).
thf(func_def_483, type, sK345: ((subst > term > subst > $o) > subst)).
thf(func_def_484, type, sK346: ((subst > term > subst > $o) > subst)).
thf(func_def_485, type, sK347: ((subst > term > subst > $o) > term)).
thf(func_def_486, type, sK348: ((subst > term > subst > $o) > subst)).
thf(func_def_487, type, sK349: ((subst > term > subst > $o) > subst)).
thf(func_def_488, type, sK350: ((subst > term > subst > $o) > term)).
thf(func_def_489, type, sK351: ((subst > term > subst > $o) > subst)).
thf(func_def_490, type, sK352: ((subst > term > subst > $o) > subst)).
thf(func_def_491, type, sK353: (subst > term > subst > $o)).
thf(func_def_492, type, sK354: term).
thf(func_def_493, type, sK355: term).
thf(func_def_494, type, sK356: (subst > term > subst > $o)).
thf(func_def_495, type, sK357: term).
thf(func_def_496, type, sK358: ((subst > term > subst > $o) > term)).
thf(func_def_497, type, sK359: ((subst > term > subst > $o) > subst)).
thf(func_def_498, type, sK360: ((subst > term > subst > $o) > subst)).
thf(func_def_499, type, sK361: ((subst > term > subst > $o) > subst)).
thf(func_def_500, type, sK362: ((subst > term > subst > $o) > subst)).
thf(func_def_501, type, sK363: ((subst > term > subst > $o) > subst)).
thf(func_def_502, type, sK364: ((subst > term > subst > $o) > subst)).
thf(func_def_503, type, sK365: ((subst > term > subst > $o) > term)).
thf(func_def_504, type, sK366: ((subst > term > subst > $o) > term)).
thf(func_def_505, type, sK367: ((subst > term > subst > $o) > subst)).
thf(func_def_506, type, sK368: ((subst > term > subst > $o) > subst)).
thf(func_def_507, type, sK369: ((subst > term > subst > $o) > term)).
thf(func_def_508, type, sK370: ((subst > term > subst > $o) > subst)).
thf(func_def_509, type, sK371: ((subst > term > subst > $o) > subst)).
thf(func_def_510, type, sK372: ((subst > term > subst > $o) > subst)).
thf(func_def_511, type, sK373: ((subst > term > subst > $o) > subst)).
thf(func_def_512, type, sK374: ((subst > term > subst > $o) > term)).
thf(func_def_513, type, sK375: ((subst > term > subst > $o) > term > term)).
thf(func_def_514, type, sK376: term).
thf(func_def_515, type, sK377: (subst > term > subst > $o)).
thf(func_def_516, type, sK378: ((term > $o) > term)).
thf(func_def_517, type, sK379: ((term > $o) > term)).
thf(func_def_518, type, sK380: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_519, type, sK381: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_520, type, sK382: ((subst > term > term) > term)).
thf(func_def_521, type, sK383: ((subst > term > term) > subst)).
thf(func_def_522, type, sK384: ((subst > term > term) > subst)).
thf(func_def_523, type, sK385: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_524, type, sK386: ((subst > term > term) > term)).
thf(func_def_525, type, sK387: ((subst > term > term) > subst)).
thf(func_def_526, type, sK388: ((subst > term > term) > subst)).
thf(func_def_527, type, sK389: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_528, type, sK390: ((subst > term > subst > $o) > term)).
thf(func_def_529, type, sK391: ((subst > term > subst > $o) > term)).
thf(func_def_530, type, sK392: ((subst > term > subst > $o) > term)).
thf(func_def_531, type, sK393: ((subst > term > subst > $o) > term)).
thf(func_def_532, type, sK394: (term > $o)).
thf(func_def_533, type, sK395: term).
thf(func_def_534, type, sK396: (subst > term > subst > $o)).
thf(func_def_535, type, sK397: ((subst > term > subst > $o) > (term > $o) > term)).
thf(func_def_536, type, sK398: ((subst > term > subst > $o) > subst)).
thf(func_def_537, type, sK399: ((subst > term > subst > $o) > subst)).
thf(func_def_538, type, sK400: ((subst > term > subst > $o) > subst)).
thf(func_def_539, type, sK401: ((subst > term > subst > $o) > term)).
thf(func_def_540, type, sK402: (term > (term > $o) > term)).
thf(func_def_541, type, sK403: ((subst > term > subst > $o) > subst)).
thf(func_def_542, type, sK404: ((subst > term > subst > $o) > subst)).
thf(func_def_543, type, sK405: ((subst > term > subst > $o) > term)).
thf(func_def_544, type, sK406: ((subst > term > subst > $o) > subst)).
thf(func_def_545, type, sK407: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_546, type, sK408: ((subst > term > term) > subst)).
thf(func_def_547, type, sK409: ((subst > term > term) > term)).
thf(func_def_548, type, sK410: ((subst > term > term) > subst)).
thf(func_def_549, type, sK411: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_550, type, sK412: term).
thf(func_def_551, type, sK413: (subst > term > subst > $o)).
thf(func_def_552, type, sK414: ((subst > term > term) > subst)).
thf(func_def_553, type, sK415: ((subst > term > term) > subst)).
thf(func_def_554, type, sK416: ((subst > term > term) > term)).
thf(func_def_555, type, sK417: ((subst > term > term) > term)).
thf(func_def_556, type, sK418: ((subst > term > subst > $o) > term > $o)).
thf(func_def_557, type, sK419: term).
thf(func_def_558, type, sK420: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_559, type, sK421: (subst > term > subst > $o)).
thf(func_def_560, type, sK422: ((term > $o) > term)).
thf(func_def_561, type, sK423: ((term > $o) > term)).
thf(func_def_562, type, sK424: ((term > $o) > term)).
thf(func_def_563, type, sK425: (term > $o)).
thf(func_def_564, type, sK426: term).
thf(func_def_565, type, sK427: (term > term)).
thf(func_def_566, type, sK428: (term > (term > $o) > term)).
thf(func_def_567, type, sK429: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_568, type, sK430: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_569, type, sK431: (subst > term > subst > $o)).
thf(func_def_570, type, sK432: (term > $o)).
thf(func_def_571, type, sK433: term).
thf(func_def_572, type, sK434: ((subst > term > term) > term)).
thf(func_def_573, type, sK435: ((subst > term > term) > subst)).
thf(func_def_574, type, sK436: ((subst > term > term) > term)).
thf(func_def_575, type, sK437: ((subst > term > term) > subst)).
thf(func_def_576, type, sK438: ((subst > term > subst > $o) > term)).
thf(func_def_577, type, sK439: ((subst > term > subst > $o) > term)).
thf(func_def_578, type, sK440: ((term > $o) > term)).
thf(func_def_579, type, sK441: ((term > $o) > term)).
thf(func_def_580, type, sK442: (term > $o)).
thf(func_def_581, type, sK443: term).
thf(func_def_582, type, sK444: term).
thf(func_def_583, type, sK445: (subst > term > subst > $o)).
thf(func_def_584, type, sK446: ((subst > term > subst > $o) > (term > $o) > term)).
thf(func_def_585, type, sK447: ((subst > term > subst > $o) > subst)).
thf(func_def_586, type, sK448: ((subst > term > subst > $o) > subst)).
thf(func_def_587, type, sK449: ((subst > term > subst > $o) > term)).
thf(func_def_588, type, sK450: ((subst > term > subst > $o) > subst)).
thf(func_def_589, type, sK451: ((subst > term > subst > $o) > term)).
thf(func_def_590, type, sK452: ((subst > term > subst > $o) > subst)).
thf(func_def_591, type, sK453: ((subst > term > subst > $o) > subst)).
thf(func_def_592, type, sK454: ((subst > term > subst > $o) > subst)).
thf(func_def_593, type, sK455: ((subst > term > subst > $o) > term)).
thf(func_def_594, type, sK456: ((subst > term > subst > $o) > term)).
thf(func_def_595, type, sK457: subst).
thf(func_def_596, type, sK458: (term > subst > (term > $o) > term > $o)).
thf(func_def_597, type, sK459: (term > $o)).
thf(func_def_598, type, sK460: subst).
thf(func_def_599, type, sK461: term).
thf(func_def_600, type, sK462: ((term > $o) > term)).
thf(func_def_601, type, sK463: (subst > term > term)).
thf(func_def_602, type, sK464: (subst > term > term)).
thf(func_def_603, type, sK465: term).
thf(func_def_604, type, sK466: subst).
thf(func_def_605, type, sK467: ((subst > term > term) > term)).
thf(func_def_606, type, sK468: ((subst > term > term) > subst)).
thf(func_def_607, type, sK469: ((subst > term > term) > subst)).
thf(func_def_608, type, sK470: ((subst > term > term) > subst)).
thf(func_def_609, type, sK471: ((subst > term > term) > term)).
thf(func_def_610, type, sK472: ((subst > term > term) > subst)).
thf(func_def_611, type, sK473: (term > subst > term > term)).
thf(func_def_612, type, sK474: term).
thf(func_def_613, type, sK475: term).
thf(func_def_614, type, sK476: term).
thf(func_def_615, type, sK477: term).
thf(func_def_616, type, sK478: term).
thf(func_def_617, type, sK479: (term > d_term)).
thf(func_def_618, type, sK480: (term > subst > term > term)).
thf(func_def_619, type, sK481: term).
thf(func_def_620, type, sK482: term).
thf(func_def_621, type, sK483: term).
thf(func_def_622, type, sK484: (subst > d_subst)).
thf(func_def_623, type, sK485: ((subst > term > subst > $o) > term > $o)).
thf(func_def_624, type, sK486: (subst > term > subst > $o)).
thf(func_def_625, type, sK487: ((term > $o) > term)).
thf(func_def_626, type, sK488: term).
thf(func_def_627, type, sK489: term).
thf(func_def_628, type, sK490: term).
thf(func_def_629, type, sK491: term).
thf(func_def_630, type, sK492: term).
thf(func_def_631, type, sK493: term).
thf(func_def_632, type, sK494: term).
thf(func_def_633, type, sK495: term).
thf(func_def_634, type, sK496: subst).
thf(func_def_635, type, sK497: term).
thf(func_def_636, type, sK498: subst).
thf(func_def_637, type, sK499: term).
thf(func_def_638, type, sK500: term).
thf(func_def_639, type, sK501: (term > $o)).
thf(func_def_640, type, sK502: subst).
thf(func_def_641, type, sK503: (term > $o)).
thf(func_def_642, type, sK504: term).
thf(func_def_643, type, sK505: term).
thf(func_def_644, type, sK506: term).
thf(func_def_645, type, sK507: ((subst > term > subst > $o) > term > $o)).
thf(func_def_646, type, sK508: (subst > term > subst > $o)).
thf(func_def_647, type, sK509: ((term > $o) > term)).
thf(func_def_648, type, sK510: ((subst > term > term) > term)).
thf(func_def_649, type, sK511: ((subst > term > term) > subst)).
thf(func_def_650, type, sK512: ((subst > term > term) > subst)).
thf(func_def_651, type, sK513: (subst > term > term)).
thf(func_def_652, type, sK514: term).
thf(func_def_653, type, sK515: term).
thf(func_def_654, type, sK516: term).
thf(func_def_655, type, sK517: subst).
thf(func_def_656, type, sK518: (term > (term > $o) > (term > $o) > subst > term)).
thf(func_def_657, type, sK519: term).
thf(func_def_658, type, sK520: subst).
thf(func_def_659, type, sK521: subst).
thf(func_def_660, type, sK522: (term > $o)).
thf(func_def_661, type, sK523: (term > $o)).
thf(func_def_662, type, sK524: term).
thf(func_def_663, type, sK525: term).
thf(func_def_664, type, sK526: subst).
thf(func_def_665, type, sK527: subst).
thf(func_def_666, type, sK528: subst).
thf(func_def_667, type, sK529: (term > $o)).
thf(func_def_668, type, sK530: subst).
thf(func_def_669, type, sK531: (term > $o)).
thf(func_def_670, type, sK532: term).
thf(func_def_671, type, sK533: term).
thf(func_def_672, type, sK534: term).
thf(func_def_673, type, sK535: term).
thf(func_def_674, type, sK536: term).
thf(func_def_675, type, sK537: subst).
thf(func_def_676, type, sK538: term).
thf(func_def_677, type, sK539: term).
thf(func_def_678, type, sK540: term).
thf(func_def_679, type, sK541: term).
thf(func_def_680, type, sK542: term).
thf(func_def_681, type, sK543: term).
thf(func_def_682, type, sK544: term).
thf(func_def_683, type, sK545: term).
thf(func_def_684, type, sK546: term).
thf(func_def_685, type, sK547: term).
thf(func_def_686, type, sK548: term).
thf(func_def_687, type, sK549: term).
thf(func_def_688, type, sK550: term).
thf(func_def_689, type, sK551: (term > $o)).
thf(func_def_690, type, sK552: term).
thf(func_def_691, type, sK553: ((term > $o) > term)).
thf(func_def_692, type, sK554: ((term > $o) > term)).
thf(func_def_693, type, sK555: term).
thf(func_def_694, type, sK556: term).
thf(func_def_695, type, sK557: term).
thf(func_def_696, type, sK558: term).
thf(func_def_697, type, sK559: subst).
thf(func_def_698, type, sK560: (term > subst > term > term)).
thf(func_def_699, type, sK561: term).
thf(func_def_700, type, sK562: ((subst > term > term) > term)).
thf(func_def_701, type, sK563: ((subst > term > term) > subst)).
thf(func_def_702, type, sK564: (subst > term > subst > $o)).
thf(func_def_703, type, sK565: term).
thf(func_def_704, type, sK566: ((subst > term > subst > $o) > term)).
thf(func_def_705, type, sK567: ((subst > term > subst > $o) > term)).
thf(func_def_706, type, sK568: term).
thf(func_def_707, type, sK569: (term > $o)).
thf(func_def_708, type, sK570: term).
thf(func_def_709, type, sK571: subst).
thf(func_def_710, type, sK572: (term > $o)).
thf(func_def_711, type, sK573: ((term > $o) > subst > term > (term > $o) > term)).
thf(func_def_712, type, sK574: (term > subst > term > term)).
thf(func_def_713, type, sK575: term).
thf(func_def_714, type, sK576: ((subst > term > term) > subst)).
thf(func_def_715, type, sK577: ((subst > term > term) > term)).
thf(func_def_716, type, sK578: subst).
thf(func_def_717, type, sK579: subst).
thf(func_def_718, type, sK580: subst).
thf(func_def_719, type, sK581: subst).
thf(func_def_720, type, sK582: term).
thf(func_def_721, type, sK583: term).
thf(func_def_722, type, sK584: term).
thf(func_def_723, type, sK585: term).
thf(func_def_724, type, sK586: (term > $o)).
thf(func_def_725, type, sK587: subst).
thf(func_def_726, type, sK588: term).
thf(func_def_727, type, sK589: ((term > $o) > term)).
thf(func_def_728, type, sK590: (term > subst > (term > $o) > term > $o)).
thf(func_def_729, type, vNOT: ($o > $o)).
thf(func_def_730, type, sK591: subst).
thf(func_def_731, type, db6: !>[X0: $tType]:(X0)).
thf(func_def_732, type, db5: !>[X0: $tType]:(X0)).
thf(func_def_733, type, sK592: term).
thf(func_def_734, type, sK593: (term > $o)).
thf(func_def_735, type, sK594: ((term > $o) > term)).
thf(func_def_736, type, sK595: ((term > $o) > subst > term > term > $o)).
thf(f1,conjecture,(
  (pushprop_lem0 = ! [X0 : (term > $o),X1 : term,X2 : subst] : ? [X3 : (term > $o)] : (pushprop_p_and_p_prime @ X1 @ X2 @ X0 @ X3))),
  file('/export/starexec/sandbox2/benchmark/theBenchmark.p',pushprop_lem0)).
thf(f2,negated_conjecture,(
  ~ (pushprop_lem0 = ! [X0 : (term > $o),X1 : term,X2 : subst] : ? [X3 : (term > $o)] : (pushprop_p_and_p_prime @ X1 @ X2 @ X0 @ X3))),
  inference(negated_conjecture,[status(cth)],[f1])).
thf(f3,axiom,(
  hoasinduction_lem1v2_gthm & hoaslamnotvar_lthm & ~lamnotap & hoasapnotvar_gthm & induction2_gthm & pushprop_lem1_gthm & (substmonoid_lthm <=> (! [X3 : subst] : (X3 = ((comp @ id @ X3))) => (! [X3 : subst] : (X3 = ((comp @ X3 @ id))) => (! [X3 : subst] : (X3 = ((comp @ id @ X3))) & ! [X3 : subst] : (X3 = ((comp @ X3 @ id))))))) & pushprop_lem2v2_lthm & hoasinduction_lem2_lthm & (hoasinduction_lthm <=> (! [X4 : (term > $o),X32 : term] : (~ ! [X7 : term,X2 : term] : (~(X4 @ X7) | (X4 @ (ap @ X2 @ X7)) | ~(X4 @ X2)) | (X4 @ X32) | ~ ! [X2 : term] : (~(var @ X2) | (X4 @ X2)) | ~ ! [X2 : term] : (~ ! [X7 : term] : ((X4 @ (sub @ X2 @ (push @ X7 @ id))) | ~(X4 @ X7)) | (X4 @ (lam @ X2)))) => (! [X39 : term,X4 : (subst > term > subst > $o),X40 : term] : (~(X4 @ id @ X39 @ id) | ~ ! [X2 : term,X23 : subst,X3 : subst,X25 : subst] : (~(X4 @ (comp @ X3 @ X25) @ (sub @ X2 @ X25) @ X23) | (X4 @ X3 @ X2 @ (comp @ X25 @ X23))) | (X4 @ id @ (ap @ X39 @ X40) @ id) | ~ ! [X7 : term,X2 : term] : (~(X4 @ id @ X2 @ id) | (X4 @ id @ (ap @ (sub @ X2 @ id) @ X7) @ id) | ~(X4 @ id @ X7 @ id)) | ~ ! [X3 : subst,X23 : subst,X2 : term,X25 : subst] : ((X4 @ (comp @ X3 @ X25) @ (sub @ X2 @ X25) @ X23) | ~(X4 @ X3 @ X2 @ (comp @ X25 @ X23))) | ~(X4 @ id @ X40 @ id)) => (! [X4 : (subst > term > subst > $o),X24 : term] : (~ ! [X25 : subst,X3 : subst,X23 : subst,X2 : term] : ((X4 @ (comp @ X3 @ X25) @ (sub @ X2 @ X25) @ X23) | ~(X4 @ X3 @ X2 @ (comp @ X25 @ X23))) | (X4 @ id @ (lam @ X24) @ id) | ~ ! [X8 : (subst > term > term)] : ((X4 @ id @ (lam @ (X8 @ sh @ one)) @ id) | ~ ! [X2 : term] : ((X4 @ id @ (X8 @ id @ X2) @ id) | ~(X4 @ id @ X2 @ id)) | ~ ! [X2 : term,X3 : subst,X23 : subst] : (((sub @ (X8 @ X3 @ X2) @ X23)) = ((X8 @ (comp @ X3 @ X23) @ (sub @ X2 @ X23))))) | ~ ! [X7 : term] : (~(X4 @ id @ X7 @ id) | (X4 @ id @ (sub @ X24 @ (push @ X7 @ id)) @ id)) | ~ ! [X23 : subst,X2 : term,X25 : subst,X3 : subst] : ((X4 @ X3 @ X2 @ (comp @ X25 @ X23)) | ~(X4 @ (comp @ X3 @ X25) @ (sub @ X2 @ X25) @ X23))) => ! [X34 : term,X4 : (subst > term > subst > $o)] : (~ ! [X7 : term,X2 : term] : (~(X4 @ id @ X2 @ id) | ~(X4 @ id @ X7 @ id) | (X4 @ id @ (ap @ (sub @ X2 @ id) @ X7) @ id)) | (X4 @ id @ X34 @ id) | ~ ! [X2 : term] : ((X4 @ id @ X2 @ id) | ~(var @ (sub @ X2 @ id))) | ~ ! [X8 : (subst > term > term)] : ((X4 @ id @ (lam @ (X8 @ sh @ one)) @ id) | ~ ! [X2 : term] : (~(X4 @ id @ X2 @ id) | (X4 @ id @ (X8 @ id @ X2) @ id)) | ~ ! [X23 : subst,X2 : term,X3 : subst] : (((sub @ (X8 @ X3 @ X2) @ X23)) = ((X8 @ (comp @ X3 @ X23) @ (sub @ X2 @ X23))))) | ~ ! [X25 : subst,X3 : subst,X2 : term,X23 : subst] : ((X4 @ X3 @ X2 @ (comp @ X25 @ X23)) | ~(X4 @ (comp @ X3 @ X25) @ (sub @ X2 @ X25) @ X23)) | ~ ! [X23 : subst,X25 : subst,X3 : subst,X2 : term] : (~(X4 @ X3 @ X2 @ (comp @ X25 @ X23)) | (X4 @ (comp @ X3 @ X25) @ (sub @ X2 @ X25) @ X23))))))) & (hoasinduction_lem3aaa <=> ! [X14 : term,X4 : (subst > term > subst > $o)] : (~ ! [X15 : term,X8 : (subst > term > term)] : (~ ! [X20 : term,X21 : subst,X19 : subst] : (((sub @ (X8 @ X19 @ X20) @ X21)) = ((sub @ (sub @ X15 @ (push @ X20 @ X19)) @ X21))) | ~ ! [X2 : term] : ((X4 @ id @ (X8 @ id @ X2) @ id) | ~(X4 @ id @ X2 @ id)) | (X4 @ id @ (lam @ (X8 @ sh @ one)) @ id) | ~ ! [X17 : term,X16 : subst,X18 : subst] : (((X8 @ (comp @ X16 @ X18) @ (sub @ X17 @ X18))) = ((sub @ X15 @ (push @ (sub @ X17 @ X18) @ (comp @ X16 @ X18)))))) | (X4 @ id @ (lam @ (sub @ X14 @ (push @ one @ sh))) @ id) | ~ ! [X7 : term] : ((X4 @ id @ (sub @ X14 @ (push @ X7 @ id)) @ id) | ~(X4 @ id @ X7 @ id)))) & (apinj1 <=> ! [X7 : term,X2 : term,X10 : term,X9 : term] : ((((ap @ X2 @ X9)) != ((ap @ X7 @ X10))) | (X2 = X7))) & ((! [X7 : term,X2 : term,X10 : term,X9 : term] : ((((ap @ X2 @ X9)) != ((ap @ X7 @ X10))) | (X9 = X10)) => ! [X7 : term,X9 : term,X2 : term,X10 : term] : ((X9 = X10) | (((ap @ (sub @ X2 @ id) @ X9)) != ((ap @ (sub @ X7 @ id) @ X10))))) <=> hoasapinj2_lthm) & (hoaslamnotap <=> ! [X8 : (subst > term > term),X30 : term,X31 : term] : (~ ! [X2 : term,X23 : subst,X3 : subst] : (((sub @ (X8 @ X3 @ X2) @ X23)) = ((X8 @ (comp @ X3 @ X23) @ (sub @ X2 @ X23)))) | (((lam @ (X8 @ sh @ one))) != ((ap @ (sub @ X30 @ id) @ X31))))) & (induction <=> ! [X11 : term,X4 : (term > $o)] : (~ ! [X2 : term,X7 : term] : (~(X4 @ X7) | ~(X4 @ X2) | (X4 @ (ap @ X2 @ X7))) | ~ ! [X2 : term] : ((X4 @ (lam @ X2)) | ~(X4 @ X2)) | (X4 @ X11) | ~ ! [X2 : term] : ((X4 @ X2) | ~(var @ X2)))) & (hoasinduction_lem2v2 <=> ! [X4 : (subst > term > subst > $o),X27 : term,X5 : (term > $o),X26 : term] : (~(X5 @ X27) | ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id))) | ~ ! [X2 : term,X7 : term] : (~(X4 @ id @ X7 @ id) | (X4 @ id @ (ap @ (sub @ X2 @ id) @ X7) @ id) | ~(X4 @ id @ X2 @ id)) | ~ ! [X25 : subst,X3 : subst,X2 : term,X23 : subst] : ((X4 @ (comp @ X3 @ X25) @ (sub @ X2 @ X25) @ X23) | ~(X4 @ X3 @ X2 @ (comp @ X25 @ X23))) | ~ ! [X25 : subst,X2 : term,X3 : subst,X23 : subst] : (~(X4 @ (comp @ X3 @ X25) @ (sub @ X2 @ X25) @ X23) | (X4 @ X3 @ X2 @ (comp @ X25 @ X23))) | ~(X5 @ X26) | (X5 @ (ap @ X26 @ X27)))) & ((! [X3 : subst,X2 : term] : (X2 = ((sub @ one @ (push @ X2 @ X3)))) => (! [X3 : subst,X2 : term] : (X3 = ((comp @ sh @ (push @ X2 @ X3)))) => (! [X7 : term,X2 : term] : ((((lam @ X2)) != ((lam @ X7))) | (X2 = X7)) => ! [X8 : (subst > term > term),X37 : subst,X36 : (subst > term > term),X38 : term] : (~ ! [X2 : term,X23 : subst,X3 : subst] : (((sub @ (X8 @ X3 @ X2) @ X23)) = ((X8 @ (comp @ X3 @ X23) @ (sub @ X2 @ X23)))) | (((lam @ (X8 @ sh @ one))) != ((lam @ (X36 @ sh @ one)))) | (((X8 @ X37 @ X38)) = ((X36 @ X37 @ X38))) | ~ ! [X23 : subst,X2 : term,X3 : subst] : (((sub @ (X36 @ X3 @ X2) @ X23)) = ((X36 @ (comp @ X3 @ X23) @ (sub @ X2 @ X23)))))))) <=> hoaslaminj_lthm) & apnotvar & (! [X9 : term,X2 : term,X7 : term,X10 : term] : ((X9 = X10) | (((ap @ (sub @ X2 @ id) @ X9)) != ((ap @ (sub @ X7 @ id) @ X10)))) <=> hoasapinj2) & (pushprop_lem1_lthm <=> (! [X2 : term,X3 : subst] : (X2 = ((sub @ one @ (push @ X2 @ X3)))) => (! [X2 : term,X3 : subst] : (X3 = ((comp @ sh @ (push @ X2 @ X3)))) => ! [X7 : term,X4 : (term > $o),X25 : (term > $o),X2 : term,X3 : subst] : (~(X4 @ X2) | (X25 @ (sub @ X2 @ (push @ X7 @ X3))))))) & pushprop & ((! [X2 : term] : (X2 = ((sub @ X2 @ id))) => ! [X2 : term] : (X2 = ((sub @ X2 @ id)))) <=> termmset_lthm) & ((! [X3 : subst] : (X3 = ((comp @ X3 @ id))) & ! [X3 : subst] : (X3 = ((comp @ id @ X3)))) <=> substmonoid) & (hoasinduction_p_and_p_prime = (^[X4 : (subst > term > subst > $o), X5 : (term > $o)] : (! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id)))))) & (hoasinduction_no_psi_cond <=> ! [X29 : term,X4 : (subst > term > subst > $o)] : (~ ! [X2 : term,X7 : term] : (~(X4 @ id @ X7 @ id) | (X4 @ id @ (ap @ (sub @ X2 @ id) @ X7) @ id) | ~(X4 @ id @ X2 @ id)) | (X4 @ id @ X29 @ id) | ~ ! [X8 : (subst > term > term)] : ((X4 @ id @ (lam @ (X8 @ sh @ one)) @ id) | ~ ! [X23 : subst,X2 : term,X3 : subst] : (((sub @ (X8 @ X3 @ X2) @ X23)) = ((X8 @ (comp @ X3 @ X23) @ (sub @ X2 @ X23)))) | ~ ! [X2 : term] : ((X4 @ id @ (X8 @ id @ X2) @ id) | ~(X4 @ id @ X2 @ id))))) & (laminj <=> ! [X2 : term,X7 : term] : ((X2 = X7) | (((lam @ X2)) != ((lam @ X7))))) & axclos & induction2lem_gthm & termmset_gthm & hoasinduction_gthm & (hoasinduction_lem3a_lthm <=> (! [X2 : term] : (X2 = ((sub @ X2 @ id))) => (! [X4 : (subst > term > subst > $o),X22 : term] : (~ ! [X8 : (subst > term > term)] : (~ ! [X2 : term] : ((X4 @ id @ (X8 @ id @ X2) @ id) | ~(X4 @ id @ X2 @ id)) | (X4 @ id @ (lam @ (X8 @ sh @ one)) @ id) | ~ ! [X2 : term,X3 : subst,X23 : subst] : (((sub @ (X8 @ X3 @ X2) @ X23)) = ((X8 @ (comp @ X3 @ X23) @ (sub @ X2 @ X23))))) | (X4 @ id @ (lam @ (sub @ X22 @ (push @ one @ sh))) @ id) | ~ ! [X7 : term] : (~(X4 @ id @ X7 @ id) | (X4 @ id @ (sub @ X22 @ (push @ X7 @ id)) @ id))) => ! [X4 : (subst > term > subst > $o),X35 : term] : (~ ! [X8 : (subst > term > term)] : (~ ! [X2 : term] : (~(X4 @ id @ X2 @ id) | (X4 @ id @ (X8 @ id @ X2) @ id)) | (X4 @ id @ (lam @ (X8 @ sh @ one)) @ id) | ~ ! [X23 : subst,X3 : subst,X2 : term] : (((sub @ (X8 @ X3 @ X2) @ X23)) = ((X8 @ (comp @ X3 @ X23) @ (sub @ X2 @ X23))))) | (X4 @ id @ (lam @ X35) @ id) | ~ ! [X7 : term] : (~(X4 @ id @ X7 @ id) | (X4 @ id @ (sub @ X35 @ (push @ X7 @ id)) @ id)))))) & pushprop_lthm & (induction2lem_lthm <=> (! [X3 : subst,X2 : term] : (X2 = ((sub @ one @ (push @ X2 @ X3)))) => (! [X3 : subst,X2 : term] : (X3 = ((comp @ sh @ (push @ X2 @ X3)))) => (! [X3 : subst] : (X3 = ((comp @ X3 @ id))) => (! [X11 : term,X4 : (term > $o)] : (~ ! [X2 : term] : (~(var @ X2) | (X4 @ X2)) | ~ ! [X2 : term] : (~(X4 @ X2) | (X4 @ (lam @ X2))) | (X4 @ X11) | ~ ! [X2 : term,X7 : term] : ((X4 @ (ap @ X2 @ X7)) | ~(X4 @ X2) | ~(X4 @ X7))) => ! [X4 : (term > $o),X12 : term,X13 : subst] : (~ ! [X7 : term,X2 : term] : (~(X4 @ X2) | (X4 @ (ap @ X2 @ X7)) | ~(X4 @ X7)) | (X4 @ (sub @ X12 @ X13)) | ~ ! [X7 : term] : ((X4 @ (sub @ X7 @ X13)) | ~(var @ X7)) | ~ ! [X2 : term] : (~ ! [X7 : term] : (~(X4 @ X7) | (X4 @ (sub @ X2 @ (push @ X7 @ id)))) | (X4 @ (lam @ X2))))))))) & axmap & axapp & hoasinduction_lem3b_gthm & (! [X4 : (subst > term > subst > $o),X33 : term,X5 : (term > $o)] : (~ ! [X8 : (subst > term > term)] : ((X4 @ id @ (lam @ (X8 @ sh @ one)) @ id) | ~ ! [X2 : term] : (~(X4 @ id @ X2 @ id) | (X4 @ id @ (X8 @ id @ X2) @ id)) | ~ ! [X23 : subst,X3 : subst,X2 : term] : (((sub @ (X8 @ X3 @ X2) @ X23)) = ((X8 @ (comp @ X3 @ X23) @ (sub @ X2 @ X23))))) | ~ ! [X7 : term] : ((X5 @ (sub @ X33 @ (push @ X7 @ id))) | ~(X5 @ X7)) | ~ ! [X23 : subst,X25 : subst,X3 : subst,X2 : term] : ((X4 @ X3 @ X2 @ (comp @ X25 @ X23)) | ~(X4 @ (comp @ X3 @ X25) @ (sub @ X2 @ X25) @ X23)) | (X5 @ (lam @ X33)) | ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id))) | ~ ! [X3 : subst,X23 : subst,X25 : subst,X2 : term] : (~(X4 @ X3 @ X2 @ (comp @ X25 @ X23)) | (X4 @ (comp @ X3 @ X25) @ (sub @ X2 @ X25) @ X23))) <=> hoasinduction_lem3v2) & (! [X5 : (term > $o),X28 : term,X4 : (subst > term > subst > $o)] : (~ ! [X7 : term] : (~(X5 @ X7) | (X5 @ (sub @ X28 @ (push @ X7 @ id)))) | (X5 @ (lam @ X28)) | ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id))) | ~ ! [X8 : (subst > term > term)] : ((X4 @ id @ (lam @ (X8 @ sh @ one)) @ id) | ~ ! [X2 : term] : (~(X4 @ id @ X2 @ id) | (X4 @ id @ (X8 @ id @ X2) @ id)) | ~ ! [X3 : subst,X23 : subst,X2 : term] : (((sub @ (X8 @ X3 @ X2) @ X23)) = ((X8 @ (comp @ X3 @ X23) @ (sub @ X2 @ X23)))))) <=> hoasinduction_lem3v2a) & pushprop_lem2v2_gthm & (axscons <=> ! [X3 : subst] : (X3 = ((push @ (sub @ one @ X3) @ (comp @ sh @ X3))))) & (hoasinduction_lem3aa <=> ! [X22 : term,X4 : (subst > term > subst > $o)] : ((X4 @ id @ (lam @ (sub @ X22 @ (push @ one @ sh))) @ id) | ~ ! [X7 : term] : ((X4 @ id @ (sub @ X22 @ (push @ X7 @ id)) @ id) | ~(X4 @ id @ X7 @ id)) | ~ ! [X8 : (subst > term > term)] : (~ ! [X3 : subst,X23 : subst,X2 : term] : (((sub @ (X8 @ X3 @ X2) @ X23)) = ((X8 @ (comp @ X3 @ X23) @ (sub @ X2 @ X23)))) | ~ ! [X2 : term] : ((X4 @ id @ (X8 @ id @ X2) @ id) | ~(X4 @ id @ X2 @ id)) | (X4 @ id @ (lam @ (X8 @ sh @ one)) @ id)))) & ! [X1 : d_subst] : (X1 = d_id) & pushprop_gthm & ulamvarind & (axvarcons <=> ! [X2 : term,X3 : subst] : (X2 = ((sub @ one @ (push @ X2 @ X3))))) & hoaslamnotvar & hoasinduction_lem1_gthm & (induction2lem <=> ! [X12 : term,X13 : subst,X4 : (term > $o)] : (~ ! [X7 : term] : (~(var @ X7) | (X4 @ (sub @ X7 @ X13))) | ~ ! [X2 : term,X7 : term] : (~(X4 @ X2) | ~(X4 @ X7) | (X4 @ (ap @ X2 @ X7))) | ~ ! [X2 : term] : ((X4 @ (lam @ X2)) | ~ ! [X7 : term] : (~(X4 @ X7) | (X4 @ (sub @ X2 @ (push @ X7 @ id))))) | (X4 @ (sub @ X12 @ X13)))) & (! [X2 : term,X7 : term] : ((((sub @ X2 @ sh)) != ((sub @ X7 @ sh))) | (X2 = X7)) <=> shinj) & (hoasinduction_lem0 <=> ! [X4 : (subst > term > subst > $o)] : ~ ! [X5 : (term > $o)] : ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id)))) & ! [X45 : d_term,X46 : d_term] : ((((d2term @ X45)) = ((d2term @ X46))) => (X45 = X46)) & axabs & (axidr <=> ! [X3 : subst] : (X3 = ((comp @ X3 @ id)))) & pushprop_lthm_orig & hoasinduction_lem3a_gthm & hoasinduction_lem2_gthm & hoaslamnotvar_gthm & (one = ((d2term @ d_one))) & (! [X24 : term,X4 : (subst > term > subst > $o)] : ((X4 @ id @ (lam @ X24) @ id) | ~ ! [X7 : term] : (~(X4 @ id @ X7 @ id) | (X4 @ id @ (sub @ X24 @ (push @ X7 @ id)) @ id)) | ~ ! [X23 : subst,X3 : subst,X25 : subst,X2 : term] : ((X4 @ X3 @ X2 @ (comp @ X25 @ X23)) | ~(X4 @ (comp @ X3 @ X25) @ (sub @ X2 @ X25) @ X23)) | ~ ! [X8 : (subst > term > term)] : (~ ! [X2 : term] : (~(X4 @ id @ X2 @ id) | (X4 @ id @ (X8 @ id @ X2) @ id)) | (X4 @ id @ (lam @ (X8 @ sh @ one)) @ id) | ~ ! [X3 : subst,X23 : subst,X2 : term] : (((sub @ (X8 @ X3 @ X2) @ X23)) = ((X8 @ (comp @ X3 @ X23) @ (sub @ X2 @ X23))))) | ~ ! [X3 : subst,X25 : subst,X2 : term,X23 : subst] : (~(X4 @ X3 @ X2 @ (comp @ X25 @ X23)) | (X4 @ (comp @ X3 @ X25) @ (sub @ X2 @ X25) @ X23))) <=> hoasinduction_lem3) & pushprop_lem2v2 & hoasinduction_lem3v2_gthm & (! [X40 : term,X4 : (subst > term > subst > $o),X39 : term] : (~(X4 @ id @ X39 @ id) | ~ ! [X2 : term,X25 : subst,X3 : subst,X23 : subst] : (~(X4 @ X3 @ X2 @ (comp @ X25 @ X23)) | (X4 @ (comp @ X3 @ X25) @ (sub @ X2 @ X25) @ X23)) | ~(X4 @ id @ X40 @ id) | ~ ! [X2 : term,X7 : term] : ((X4 @ id @ (ap @ (sub @ X2 @ id) @ X7) @ id) | ~(X4 @ id @ X7 @ id) | ~(X4 @ id @ X2 @ id)) | (X4 @ id @ (ap @ X39 @ X40) @ id) | ~ ! [X23 : subst,X3 : subst,X2 : term,X25 : subst] : (~(X4 @ (comp @ X3 @ X25) @ (sub @ X2 @ X25) @ X23) | (X4 @ X3 @ X2 @ (comp @ X25 @ X23)))) <=> hoasinduction_lem2) & pushprop_lem1v2_gthm & (((sub @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2term @ d_one))) & (! [X3 : subst,X5 : (term > $o),X2 : term,X4 : (term > $o)] : (~(X4 @ X2) | (X5 @ one) | ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ (sub @ X6 @ (push @ X2 @ X3)))))) <=> pushprop_lem1v2) & hoaslaminj_gthm & (! [X4 : (term > $o),X2 : term,X3 : subst] : ~ ! [X5 : (term > $o)] : ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ (sub @ X6 @ (push @ X2 @ X3))))) <=> pushprop_lem0_lthm) & hoasapinj2_gthm & (((ap @ (d2term @ d_one) @ (d2term @ d_one))) = ((d2term @ d_one))) & hoasapnotvar & (! [X35 : term,X4 : (subst > term > subst > $o)] : (~ ! [X7 : term] : ((X4 @ id @ (sub @ X35 @ (push @ X7 @ id)) @ id) | ~(X4 @ id @ X7 @ id)) | (X4 @ id @ (lam @ X35) @ id) | ~ ! [X8 : (subst > term > term)] : ((X4 @ id @ (lam @ (X8 @ sh @ one)) @ id) | ~ ! [X3 : subst,X23 : subst,X2 : term] : (((sub @ (X8 @ X3 @ X2) @ X23)) = ((X8 @ (comp @ X3 @ X23) @ (sub @ X2 @ X23)))) | ~ ! [X2 : term] : ((X4 @ id @ (X8 @ id @ X2) @ id) | ~(X4 @ id @ X2 @ id)))) <=> hoasinduction_lem3a) & ~(var @ (d2term @ d_one)) & (induction2 <=> ! [X4 : (term > $o),X32 : term] : (~ ! [X2 : term] : (~ ! [X7 : term] : (~(X4 @ X7) | (X4 @ (sub @ X2 @ (push @ X7 @ id)))) | (X4 @ (lam @ X2))) | (X4 @ X32) | ~ ! [X7 : term,X2 : term] : (~(X4 @ X7) | ~(X4 @ X2) | (X4 @ (ap @ X2 @ X7))) | ~ ! [X2 : term] : (~(var @ X2) | (X4 @ X2)))) & (! [X25 : (term > $o),X7 : term,X2 : term,X3 : subst,X4 : (term > $o)] : (~(X4 @ X2) | (X25 @ (sub @ X2 @ (push @ X7 @ X3)))) <=> pushprop_lem1) & (apinj2 <=> ! [X7 : term,X2 : term,X9 : term,X10 : term] : ((X9 = X10) | (((ap @ X2 @ X9)) != ((ap @ X7 @ X10))))) & (((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) = ((d2term @ d_one))) & ~(hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) & (hoasinduction_lem3v2_f_lthm <=> ! [X7 : term] : ~ ! [X8 : (subst > term > term)] : ~ ! [X2 : term,X3 : subst] : (((sub @ X7 @ (push @ X2 @ X3))) = ((X8 @ X3 @ X2)))) & (! [X7 : term] : ~ ! [X8 : (subst > term > term)] : (((X8 @ sh @ one)) != ((sub @ X7 @ (push @ one @ sh)))) <=> hoasinduction_lem3b) & ((! [X7 : term] : ~ ! [X8 : (subst > term > term)] : ~ ! [X2 : term,X3 : subst] : (((sub @ X7 @ (push @ X2 @ X3))) = ((X8 @ X3 @ X2))) => (! [X2 : term] : (X2 = ((sub @ X2 @ id))) => ! [X28 : term,X5 : (term > $o),X4 : (subst > term > subst > $o)] : (~ ! [X8 : (subst > term > term)] : (~ ! [X23 : subst,X3 : subst,X2 : term] : (((sub @ (X8 @ X3 @ X2) @ X23)) = ((X8 @ (comp @ X3 @ X23) @ (sub @ X2 @ X23)))) | (X4 @ id @ (lam @ (X8 @ sh @ one)) @ id) | ~ ! [X2 : term] : ((X4 @ id @ (X8 @ id @ X2) @ id) | ~(X4 @ id @ X2 @ id))) | ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id))) | ~ ! [X7 : term] : (~(X5 @ X7) | (X5 @ (sub @ X28 @ (push @ X7 @ id)))) | (X5 @ (lam @ X28))))) <=> hoasinduction_lem3v2a_lthm) & hoaslamnotap_lthm & hoasapnotvar_lthm & (! [X2 : term] : (X2 = ((sub @ X2 @ id))) <=> termmset) & (hoasinduction_lem3v2_lthm <=> (! [X2 : term] : (X2 = ((sub @ X2 @ id))) => ! [X33 : term,X4 : (subst > term > subst > $o),X5 : (term > $o)] : (~ ! [X7 : term] : (~(X5 @ X7) | (X5 @ (sub @ X33 @ (push @ X7 @ id)))) | ~ ! [X8 : (subst > term > term)] : (~ ! [X23 : subst,X2 : term,X3 : subst] : (((sub @ (X8 @ X3 @ X2) @ X23)) = ((X8 @ (comp @ X3 @ X23) @ (sub @ X2 @ X23)))) | ~ ! [X2 : term] : (~(X4 @ id @ X2 @ id) | (X4 @ id @ (X8 @ id @ X2) @ id)) | (X4 @ id @ (lam @ (X8 @ sh @ one)) @ id)) | (X5 @ (lam @ X33)) | ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id))) | ~ ! [X3 : subst,X23 : subst,X25 : subst,X2 : term] : ((X4 @ (comp @ X3 @ X25) @ (sub @ X2 @ X25) @ X23) | ~(X4 @ X3 @ X2 @ (comp @ X25 @ X23))) | ~ ! [X25 : subst,X2 : term,X3 : subst,X23 : subst] : (~(X4 @ (comp @ X3 @ X25) @ (sub @ X2 @ X25) @ X23) | (X4 @ X3 @ X2 @ (comp @ X25 @ X23)))))) & (hoasinduction_lem3b_lthm <=> ! [X7 : term] : ~ ! [X8 : (subst > term > term)] : (((X8 @ sh @ one)) != ((sub @ X7 @ (push @ one @ sh))))) & hoasinduction_lem3_gthm & pushprop_lem3v2_lthm & ! [X47 : d_term] : (X47 = d_one) & hoaslamnotap_gthm & (! [X2 : term] : (X2 = ((sub @ X2 @ id))) <=> axvarid) & ! [X0 : subst] : ? [X1 : d_subst] : (X0 = ((d2subst @ X1))) & (pushprop_lem1v2_lthm <=> (! [X2 : term,X3 : subst] : (X2 = ((sub @ one @ (push @ X2 @ X3)))) => ! [X5 : (term > $o),X4 : (term > $o),X2 : term,X3 : subst] : (~(X4 @ X2) | (X5 @ one) | ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ (sub @ X6 @ (push @ X2 @ X3)))))))) & axvarshift & (! [X4 : (subst > term > subst > $o)] : ~ ! [X5 : (term > $o)] : ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id))) <=> hoasinduction_lem0_lthm) & ((^[X2 : term, X3 : subst, X4 : (term > $o), X5 : (term > $o)] : (! [X6 : term] : (((X5 @ X6)) = ((X4 @ (sub @ X6 @ (push @ X2 @ X3))))))) = pushprop_p_and_p_prime) & (hoasinduction_lem3_lthm <=> (! [X2 : term] : (X2 = ((sub @ X2 @ id))) => (! [X4 : (subst > term > subst > $o),X22 : term] : ((X4 @ id @ (lam @ (sub @ X22 @ (push @ one @ sh))) @ id) | ~ ! [X7 : term] : (~(X4 @ id @ X7 @ id) | (X4 @ id @ (sub @ X22 @ (push @ X7 @ id)) @ id)) | ~ ! [X8 : (subst > term > term)] : (~ ! [X2 : term,X3 : subst,X23 : subst] : (((sub @ (X8 @ X3 @ X2) @ X23)) = ((X8 @ (comp @ X3 @ X23) @ (sub @ X2 @ X23)))) | ~ ! [X2 : term] : (~(X4 @ id @ X2 @ id) | (X4 @ id @ (X8 @ id @ X2) @ id)) | (X4 @ id @ (lam @ (X8 @ sh @ one)) @ id))) => ! [X4 : (subst > term > subst > $o),X24 : term] : (~ ! [X25 : subst,X23 : subst,X3 : subst,X2 : term] : ((X4 @ (comp @ X3 @ X25) @ (sub @ X2 @ X25) @ X23) | ~(X4 @ X3 @ X2 @ (comp @ X25 @ X23))) | (X4 @ id @ (lam @ X24) @ id) | ~ ! [X7 : term] : (~(X4 @ id @ X7 @ id) | (X4 @ id @ (sub @ X24 @ (push @ X7 @ id)) @ id)) | ~ ! [X23 : subst,X3 : subst,X25 : subst,X2 : term] : ((X4 @ X3 @ X2 @ (comp @ X25 @ X23)) | ~(X4 @ (comp @ X3 @ X25) @ (sub @ X2 @ X25) @ X23)) | ~ ! [X8 : (subst > term > term)] : (~ ! [X2 : term,X23 : subst,X3 : subst] : (((sub @ (X8 @ X3 @ X2) @ X23)) = ((X8 @ (comp @ X3 @ X23) @ (sub @ X2 @ X23)))) | ~ ! [X2 : term] : (~(X4 @ id @ X2 @ id) | (X4 @ id @ (X8 @ id @ X2) @ id)) | (X4 @ id @ (lam @ (X8 @ sh @ one)) @ id)))))) & (((push @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & (! [X3 : subst,X2 : term] : (X3 = ((comp @ sh @ (push @ X2 @ X3)))) <=> axshiftcons) & (hoasinduction <=> ! [X4 : (subst > term > subst > $o),X34 : term] : ((X4 @ id @ X34 @ id) | ~ ! [X25 : subst,X3 : subst,X2 : term,X23 : subst] : (~(X4 @ (comp @ X3 @ X25) @ (sub @ X2 @ X25) @ X23) | (X4 @ X3 @ X2 @ (comp @ X25 @ X23))) | ~ ! [X3 : subst,X2 : term,X25 : subst,X23 : subst] : ((X4 @ (comp @ X3 @ X25) @ (sub @ X2 @ X25) @ X23) | ~(X4 @ X3 @ X2 @ (comp @ X25 @ X23))) | ~ ! [X8 : (subst > term > term)] : (~ ! [X2 : term] : ((X4 @ id @ (X8 @ id @ X2) @ id) | ~(X4 @ id @ X2 @ id)) | ~ ! [X23 : subst,X2 : term,X3 : subst] : (((sub @ (X8 @ X3 @ X2) @ X23)) = ((X8 @ (comp @ X3 @ X23) @ (sub @ X2 @ X23)))) | (X4 @ id @ (lam @ (X8 @ sh @ one)) @ id)) | ~ ! [X7 : term,X2 : term] : ((X4 @ id @ (ap @ (sub @ X2 @ id) @ X7) @ id) | ~(X4 @ id @ X7 @ id) | ~(X4 @ id @ X2 @ id)) | ~ ! [X2 : term] : ((X4 @ id @ X2 @ id) | ~(var @ (sub @ X2 @ id))))) & pushprop_lem3v2 & (! [X7 : term] : ~ ! [X8 : (subst > term > term)] : ~ ! [X3 : subst,X2 : term] : (((sub @ X7 @ (push @ X2 @ X3))) = ((X8 @ X3 @ X2))) <=> hoasinduction_lem3v2_f) & hoasinduction_lem1v2 & ! [X43 : d_subst,X44 : d_subst] : ((((d2subst @ X43)) = ((d2subst @ X44))) => (X43 = X44)) & ((^[X41 : subst, X42 : (subst > term > term)] : ((d2term @ d_one))) = hoaslam) & ~ulamvar1 & (hoasapinj1_lthm <=> (! [X2 : term] : (X2 = ((sub @ X2 @ id))) => (! [X2 : term,X7 : term,X10 : term,X9 : term] : ((X2 = X7) | (((ap @ X2 @ X9)) != ((ap @ X7 @ X10)))) => ! [X7 : term,X10 : term,X9 : term,X2 : term] : ((X2 = X7) | (((ap @ (sub @ X2 @ id) @ X9)) != ((ap @ (sub @ X7 @ id) @ X10))))))) & ((! [X2 : term] : (X2 = ((sub @ X2 @ id))) => (! [X13 : subst,X12 : term,X4 : (term > $o)] : (~ ! [X2 : term] : (~ ! [X7 : term] : (~(X4 @ X7) | (X4 @ (sub @ X2 @ (push @ X7 @ id)))) | (X4 @ (lam @ X2))) | ~ ! [X7 : term] : ((X4 @ (sub @ X7 @ X13)) | ~(var @ X7)) | ~ ! [X7 : term,X2 : term] : ((X4 @ (ap @ X2 @ X7)) | ~(X4 @ X2) | ~(X4 @ X7)) | (X4 @ (sub @ X12 @ X13))) => ! [X4 : (term > $o),X32 : term] : (~ ! [X2 : term,X7 : term] : (~(X4 @ X2) | ~(X4 @ X7) | (X4 @ (ap @ X2 @ X7))) | (X4 @ X32) | ~ ! [X2 : term] : (~(var @ X2) | (X4 @ X2)) | ~ ! [X2 : term] : (~ ! [X7 : term] : ((X4 @ (sub @ X2 @ (push @ X7 @ id))) | ~(X4 @ X7)) | (X4 @ (lam @ X2)))))) <=> induction2_lthm) & (sh = ((d2subst @ d_id))) & axassoc & (! [X3 : subst] : (X3 = ((comp @ id @ X3))) <=> axidl) & pushprop_lem0_gthm & hoasinduction_lem2v2_gthm & (hoasinduction_lem3aa_lthm <=> ! [X4 : (subst > term > subst > $o),X22 : term] : (~ ! [X7 : term] : ((X4 @ id @ (sub @ X22 @ (push @ X7 @ id)) @ id) | ~(X4 @ id @ X7 @ id)) | ~ ! [X8 : (subst > term > term)] : (~ ! [X2 : term] : (~(X4 @ id @ X2 @ id) | (X4 @ id @ (X8 @ id @ X2) @ id)) | (X4 @ id @ (lam @ (X8 @ sh @ one)) @ id) | ~ ! [X23 : subst,X3 : subst,X2 : term] : (((sub @ (X8 @ X3 @ X2) @ X23)) = ((X8 @ (comp @ X3 @ X23) @ (sub @ X2 @ X23))))) | (X4 @ id @ (lam @ (sub @ X22 @ (push @ one @ sh))) @ id))) & hoasapinj1_gthm & hoasinduction_lem1_lthm & (id = ((d2subst @ d_id))) & (hoasinduction_no_psi_cond_lthm <=> (! [X4 : (subst > term > subst > $o)] : ~ ! [X5 : (term > $o)] : ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id))) => (! [X4 : (term > $o),X32 : term] : (~ ! [X2 : term] : ((X4 @ X2) | ~(var @ X2)) | ~ ! [X2 : term,X7 : term] : (~(X4 @ X7) | ~(X4 @ X2) | (X4 @ (ap @ X2 @ X7))) | ~ ! [X2 : term] : (~ ! [X7 : term] : ((X4 @ (sub @ X2 @ (push @ X7 @ id))) | ~(X4 @ X7)) | (X4 @ (lam @ X2))) | (X4 @ X32)) => (! [X2 : term] : (X2 = ((sub @ X2 @ id))) => (! [X28 : term,X5 : (term > $o),X4 : (subst > term > subst > $o)] : (~ ! [X7 : term] : (~(X5 @ X7) | (X5 @ (sub @ X28 @ (push @ X7 @ id)))) | ~ ! [X8 : (subst > term > term)] : (~ ! [X2 : term,X3 : subst,X23 : subst] : (((sub @ (X8 @ X3 @ X2) @ X23)) = ((X8 @ (comp @ X3 @ X23) @ (sub @ X2 @ X23)))) | (X4 @ id @ (lam @ (X8 @ sh @ one)) @ id) | ~ ! [X2 : term] : ((X4 @ id @ (X8 @ id @ X2) @ id) | ~(X4 @ id @ X2 @ id))) | (X5 @ (lam @ X28)) | ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id)))) => ! [X4 : (subst > term > subst > $o),X29 : term] : (~ ! [X8 : (subst > term > term)] : (~ ! [X2 : term] : (~(X4 @ id @ X2 @ id) | (X4 @ id @ (X8 @ id @ X2) @ id)) | ~ ! [X23 : subst,X3 : subst,X2 : term] : (((sub @ (X8 @ X3 @ X2) @ X23)) = ((X8 @ (comp @ X3 @ X23) @ (sub @ X2 @ X23)))) | (X4 @ id @ (lam @ (X8 @ sh @ one)) @ id)) | (X4 @ id @ X29 @ id) | ~ ! [X7 : term,X2 : term] : (~(X4 @ id @ X2 @ id) | (X4 @ id @ (ap @ (sub @ X2 @ id) @ X7) @ id) | ~(X4 @ id @ X7 @ id)))))))) & (((lam @ (d2term @ d_one))) = ((d2term @ d_one))) & (hoaslaminj <=> ! [X8 : (subst > term > term),X37 : subst,X38 : term,X36 : (subst > term > term)] : ((((X8 @ X37 @ X38)) = ((X36 @ X37 @ X38))) | ~ ! [X3 : subst,X23 : subst,X2 : term] : (((sub @ (X8 @ X3 @ X2) @ X23)) = ((X8 @ (comp @ X3 @ X23) @ (sub @ X2 @ X23)))) | (((lam @ (X8 @ sh @ one))) != ((lam @ (X36 @ sh @ one)))) | ~ ! [X3 : subst,X2 : term,X23 : subst] : (((sub @ (X36 @ X3 @ X2) @ X23)) = ((X36 @ (comp @ X3 @ X23) @ (sub @ X2 @ X23)))))) & (! [X7 : term,X10 : term,X2 : term,X9 : term] : ((X2 = X7) | (((ap @ (sub @ X2 @ id) @ X9)) != ((ap @ (sub @ X7 @ id) @ X10)))) <=> hoasapinj1) & lamnotvar & ulamvarsh & substmonoid_gthm & ! [X48 : term] : ? [X47 : d_term] : (X48 = ((d2term @ X47))) & (((comp @ (d2subst @ d_id) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & ((! [X4 : (subst > term > subst > $o)] : ~ ! [X5 : (term > $o)] : ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id))) => (! [X4 : (term > $o),X32 : term] : ((X4 @ X32) | ~ ! [X2 : term] : (~(var @ X2) | (X4 @ X2)) | ~ ! [X2 : term] : ((X4 @ (lam @ X2)) | ~ ! [X7 : term] : (~(X4 @ X7) | (X4 @ (sub @ X2 @ (push @ X7 @ id))))) | ~ ! [X2 : term,X7 : term] : (~(X4 @ X7) | (X4 @ (ap @ X2 @ X7)) | ~(X4 @ X2))) => (! [X2 : term] : (X2 = ((sub @ X2 @ id))) => (! [X5 : (term > $o),X28 : term,X4 : (subst > term > subst > $o)] : (~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id))) | (X5 @ (lam @ X28)) | ~ ! [X7 : term] : ((X5 @ (sub @ X28 @ (push @ X7 @ id))) | ~(X5 @ X7)) | ~ ! [X8 : (subst > term > term)] : (~ ! [X2 : term,X3 : subst,X23 : subst] : (((sub @ (X8 @ X3 @ X2) @ X23)) = ((X8 @ (comp @ X3 @ X23) @ (sub @ X2 @ X23)))) | (X4 @ id @ (lam @ (X8 @ sh @ one)) @ id) | ~ ! [X2 : term] : ((X4 @ id @ (X8 @ id @ X2) @ id) | ~(X4 @ id @ X2 @ id)))) => ! [X4 : (subst > term > subst > $o),X34 : term] : (~ ! [X2 : term,X7 : term] : (~(X4 @ id @ X7 @ id) | (X4 @ id @ (ap @ (sub @ X2 @ id) @ X7) @ id) | ~(X4 @ id @ X2 @ id)) | (X4 @ id @ X34 @ id) | ~ ! [X2 : term,X3 : subst,X23 : subst,X25 : subst] : (~(X4 @ (comp @ X3 @ X25) @ (sub @ X2 @ X25) @ X23) | (X4 @ X3 @ X2 @ (comp @ X25 @ X23))) | ~ ! [X8 : (subst > term > term)] : (~ ! [X2 : term,X23 : subst,X3 : subst] : (((sub @ (X8 @ X3 @ X2) @ X23)) = ((X8 @ (comp @ X3 @ X23) @ (sub @ X2 @ X23)))) | ~ ! [X2 : term] : ((X4 @ id @ (X8 @ id @ X2) @ id) | ~(X4 @ id @ X2 @ id)) | (X4 @ id @ (lam @ (X8 @ sh @ one)) @ id)) | ~ ! [X23 : subst,X2 : term,X3 : subst,X25 : subst] : ((X4 @ (comp @ X3 @ X25) @ (sub @ X2 @ X25) @ X23) | ~(X4 @ X3 @ X2 @ (comp @ X25 @ X23))) | ~ ! [X2 : term] : (~(var @ (sub @ X2 @ id)) | (X4 @ id @ X2 @ id))))))) <=> hoasinduction_lthm_3) & (pushprop_lem0 <=> ! [X4 : (term > $o),X2 : term,X3 : subst] : ~ ! [X5 : (term > $o)] : ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ (sub @ X6 @ (push @ X2 @ X3)))))) & hoasinduction_lem1),
  file('/export/starexec/sandbox2/benchmark/theBenchmark.p',alg444_1)).
thf(f4,plain,(
  hoasinduction_lem1v2_gthm & hoaslamnotvar_lthm & ~lamnotap & hoasapnotvar_gthm & induction2_gthm & pushprop_lem1_gthm & (substmonoid_lthm <=> (! [X0 : subst] : (((comp @ id @ X0)) = X0) => (! [X1 : subst] : (((comp @ X1 @ id)) = X1) => (! [X2 : subst] : (((comp @ id @ X2)) = X2) & ! [X3 : subst] : (X3 = ((comp @ X3 @ id))))))) & pushprop_lem2v2_lthm & hoasinduction_lem2_lthm & (hoasinduction_lthm <=> (! [X4 : (term > $o),X5 : term] : (~ ! [X6 : term,X7 : term] : (~(X4 @ X6) | (X4 @ (ap @ X7 @ X6)) | ~(X4 @ X7)) | (X4 @ X5) | ~ ! [X8 : term] : (~(var @ X8) | (X4 @ X8)) | ~ ! [X9 : term] : (~ ! [X10 : term] : ((X4 @ (sub @ X9 @ (push @ X10 @ id))) | ~(X4 @ X10)) | (X4 @ (lam @ X9)))) => (! [X11 : term,X12 : (subst > term > subst > $o),X13 : term] : (~(X12 @ id @ X11 @ id) | ~ ! [X14 : term,X15 : subst,X16 : subst,X17 : subst] : (~(X12 @ (comp @ X16 @ X17) @ (sub @ X14 @ X17) @ X15) | (X12 @ X16 @ X14 @ (comp @ X17 @ X15))) | (X12 @ id @ (ap @ X11 @ X13) @ id) | ~ ! [X18 : term,X19 : term] : (~(X12 @ id @ X19 @ id) | (X12 @ id @ (ap @ (sub @ X19 @ id) @ X18) @ id) | ~(X12 @ id @ X18 @ id)) | ~ ! [X20 : subst,X21 : subst,X22 : term,X23 : subst] : ((X12 @ (comp @ X20 @ X23) @ (sub @ X22 @ X23) @ X21) | ~(X12 @ X20 @ X22 @ (comp @ X23 @ X21))) | ~(X12 @ id @ X13 @ id)) => (! [X24 : (subst > term > subst > $o),X25 : term] : (~ ! [X26 : subst,X27 : subst,X28 : subst,X29 : term] : ((X24 @ (comp @ X27 @ X26) @ (sub @ X29 @ X26) @ X28) | ~(X24 @ X27 @ X29 @ (comp @ X26 @ X28))) | (X24 @ id @ (lam @ X25) @ id) | ~ ! [X30 : (subst > term > term)] : ((X24 @ id @ (lam @ (X30 @ sh @ one)) @ id) | ~ ! [X31 : term] : ((X24 @ id @ (X30 @ id @ X31) @ id) | ~(X24 @ id @ X31 @ id)) | ~ ! [X32 : term,X33 : subst,X34 : subst] : (((sub @ (X30 @ X33 @ X32) @ X34)) = ((X30 @ (comp @ X33 @ X34) @ (sub @ X32 @ X34))))) | ~ ! [X35 : term] : (~(X24 @ id @ X35 @ id) | (X24 @ id @ (sub @ X25 @ (push @ X35 @ id)) @ id)) | ~ ! [X36 : subst,X37 : term,X38 : subst,X39 : subst] : ((X24 @ X39 @ X37 @ (comp @ X38 @ X36)) | ~(X24 @ (comp @ X39 @ X38) @ (sub @ X37 @ X38) @ X36))) => ! [X40 : term,X41 : (subst > term > subst > $o)] : (~ ! [X42 : term,X43 : term] : (~(X41 @ id @ X43 @ id) | ~(X41 @ id @ X42 @ id) | (X41 @ id @ (ap @ (sub @ X43 @ id) @ X42) @ id)) | (X41 @ id @ X40 @ id) | ~ ! [X44 : term] : ((X41 @ id @ X44 @ id) | ~(var @ (sub @ X44 @ id))) | ~ ! [X45 : (subst > term > term)] : ((X41 @ id @ (lam @ (X45 @ sh @ one)) @ id) | ~ ! [X46 : term] : (~(X41 @ id @ X46 @ id) | (X41 @ id @ (X45 @ id @ X46) @ id)) | ~ ! [X47 : subst,X48 : term,X49 : subst] : (((sub @ (X45 @ X49 @ X48) @ X47)) = ((X45 @ (comp @ X49 @ X47) @ (sub @ X48 @ X47))))) | ~ ! [X50 : subst,X51 : subst,X52 : term,X53 : subst] : ((X41 @ X51 @ X52 @ (comp @ X50 @ X53)) | ~(X41 @ (comp @ X51 @ X50) @ (sub @ X52 @ X50) @ X53)) | ~ ! [X54 : subst,X55 : subst,X56 : subst,X57 : term] : (~(X41 @ X56 @ X57 @ (comp @ X55 @ X54)) | (X41 @ (comp @ X56 @ X55) @ (sub @ X57 @ X55) @ X54))))))) & (hoasinduction_lem3aaa <=> ! [X58 : term,X59 : (subst > term > subst > $o)] : (~ ! [X60 : term,X61 : (subst > term > term)] : (~ ! [X62 : term,X63 : subst,X64 : subst] : (((sub @ (X61 @ X64 @ X62) @ X63)) = ((sub @ (sub @ X60 @ (push @ X62 @ X64)) @ X63))) | ~ ! [X65 : term] : ((X59 @ id @ (X61 @ id @ X65) @ id) | ~(X59 @ id @ X65 @ id)) | (X59 @ id @ (lam @ (X61 @ sh @ one)) @ id) | ~ ! [X66 : term,X67 : subst,X68 : subst] : (((X61 @ (comp @ X67 @ X68) @ (sub @ X66 @ X68))) = ((sub @ X60 @ (push @ (sub @ X66 @ X68) @ (comp @ X67 @ X68)))))) | (X59 @ id @ (lam @ (sub @ X58 @ (push @ one @ sh))) @ id) | ~ ! [X69 : term] : ((X59 @ id @ (sub @ X58 @ (push @ X69 @ id)) @ id) | ~(X59 @ id @ X69 @ id)))) & (apinj1 <=> ! [X70 : term,X71 : term,X72 : term,X73 : term] : ((((ap @ X71 @ X73)) != ((ap @ X70 @ X72))) | (X70 = X71))) & ((! [X74 : term,X75 : term,X76 : term,X77 : term] : ((((ap @ X75 @ X77)) != ((ap @ X74 @ X76))) | (X76 = X77)) => ! [X78 : term,X79 : term,X80 : term,X81 : term] : ((X79 = X81) | (((ap @ (sub @ X80 @ id) @ X79)) != ((ap @ (sub @ X78 @ id) @ X81))))) <=> hoasapinj2_lthm) & (hoaslamnotap <=> ! [X82 : (subst > term > term),X83 : term,X84 : term] : (~ ! [X85 : term,X86 : subst,X87 : subst] : (((X82 @ (comp @ X87 @ X86) @ (sub @ X85 @ X86))) = ((sub @ (X82 @ X87 @ X85) @ X86))) | (((ap @ (sub @ X83 @ id) @ X84)) != ((lam @ (X82 @ sh @ one)))))) & (induction <=> ! [X88 : term,X89 : (term > $o)] : (~ ! [X90 : term,X91 : term] : (~(X89 @ X91) | ~(X89 @ X90) | (X89 @ (ap @ X90 @ X91))) | ~ ! [X92 : term] : ((X89 @ (lam @ X92)) | ~(X89 @ X92)) | (X89 @ X88) | ~ ! [X93 : term] : ((X89 @ X93) | ~(var @ X93)))) & (hoasinduction_lem2v2 <=> ! [X94 : (subst > term > subst > $o),X95 : term,X96 : (term > $o),X97 : term] : (~(X96 @ X95) | ~ ! [X98 : term] : (((X94 @ id @ X98 @ id)) = ((X96 @ X98))) | ~ ! [X99 : term,X100 : term] : (~(X94 @ id @ X100 @ id) | (X94 @ id @ (ap @ (sub @ X99 @ id) @ X100) @ id) | ~(X94 @ id @ X99 @ id)) | ~ ! [X101 : subst,X102 : subst,X103 : term,X104 : subst] : ((X94 @ (comp @ X102 @ X101) @ (sub @ X103 @ X101) @ X104) | ~(X94 @ X102 @ X103 @ (comp @ X101 @ X104))) | ~ ! [X105 : subst,X106 : term,X107 : subst,X108 : subst] : (~(X94 @ (comp @ X107 @ X105) @ (sub @ X106 @ X105) @ X108) | (X94 @ X107 @ X106 @ (comp @ X105 @ X108))) | ~(X96 @ X97) | (X96 @ (ap @ X97 @ X95)))) & ((! [X109 : subst,X110 : term] : (((sub @ one @ (push @ X110 @ X109))) = X110) => (! [X111 : subst,X112 : term] : (((comp @ sh @ (push @ X112 @ X111))) = X111) => (! [X113 : term,X114 : term] : ((((lam @ X113)) != ((lam @ X114))) | (X113 = X114)) => ! [X115 : (subst > term > term),X116 : subst,X117 : (subst > term > term),X118 : term] : (~ ! [X119 : term,X120 : subst,X121 : subst] : (((sub @ (X115 @ X121 @ X119) @ X120)) = ((X115 @ (comp @ X121 @ X120) @ (sub @ X119 @ X120)))) | (((lam @ (X115 @ sh @ one))) != ((lam @ (X117 @ sh @ one)))) | (((X117 @ X116 @ X118)) = ((X115 @ X116 @ X118))) | ~ ! [X122 : subst,X123 : term,X124 : subst] : (((X117 @ (comp @ X124 @ X122) @ (sub @ X123 @ X122))) = ((sub @ (X117 @ X124 @ X123) @ X122))))))) <=> hoaslaminj_lthm) & apnotvar & (! [X125 : term,X126 : term,X127 : term,X128 : term] : ((X125 = X128) | (((ap @ (sub @ X127 @ id) @ X128)) != ((ap @ (sub @ X126 @ id) @ X125)))) <=> hoasapinj2) & (pushprop_lem1_lthm <=> (! [X129 : term,X130 : subst] : (((sub @ one @ (push @ X129 @ X130))) = X129) => (! [X131 : term,X132 : subst] : (((comp @ sh @ (push @ X131 @ X132))) = X132) => ! [X133 : term,X134 : (term > $o),X135 : (term > $o),X136 : term,X137 : subst] : (~(X134 @ X136) | (X135 @ (sub @ X136 @ (push @ X133 @ X137))))))) & pushprop & ((! [X138 : term] : (((sub @ X138 @ id)) = X138) => ! [X139 : term] : (((sub @ X139 @ id)) = X139)) <=> termmset_lthm) & ((! [X140 : subst] : (((comp @ X140 @ id)) = X140) & ! [X141 : subst] : (((comp @ id @ X141)) = X141)) <=> substmonoid) & (hoasinduction_p_and_p_prime = (^[X142 : (subst > term > subst > $o), X143 : (term > $o)] : (! [X144 : term] : (((X142 @ id @ X144 @ id)) = ((X143 @ X144)))))) & (hoasinduction_no_psi_cond <=> ! [X145 : term,X146 : (subst > term > subst > $o)] : (~ ! [X147 : term,X148 : term] : (~(X146 @ id @ X148 @ id) | (X146 @ id @ (ap @ (sub @ X147 @ id) @ X148) @ id) | ~(X146 @ id @ X147 @ id)) | (X146 @ id @ X145 @ id) | ~ ! [X149 : (subst > term > term)] : ((X146 @ id @ (lam @ (X149 @ sh @ one)) @ id) | ~ ! [X150 : subst,X151 : term,X152 : subst] : (((sub @ (X149 @ X152 @ X151) @ X150)) = ((X149 @ (comp @ X152 @ X150) @ (sub @ X151 @ X150)))) | ~ ! [X153 : term] : ((X146 @ id @ (X149 @ id @ X153) @ id) | ~(X146 @ id @ X153 @ id))))) & (laminj <=> ! [X154 : term,X155 : term] : ((X154 = X155) | (((lam @ X154)) != ((lam @ X155))))) & axclos & induction2lem_gthm & termmset_gthm & hoasinduction_gthm & (hoasinduction_lem3a_lthm <=> (! [X156 : term] : (((sub @ X156 @ id)) = X156) => (! [X157 : (subst > term > subst > $o),X158 : term] : (~ ! [X159 : (subst > term > term)] : (~ ! [X160 : term] : ((X157 @ id @ (X159 @ id @ X160) @ id) | ~(X157 @ id @ X160 @ id)) | (X157 @ id @ (lam @ (X159 @ sh @ one)) @ id) | ~ ! [X161 : term,X162 : subst,X163 : subst] : (((sub @ (X159 @ X162 @ X161) @ X163)) = ((X159 @ (comp @ X162 @ X163) @ (sub @ X161 @ X163))))) | (X157 @ id @ (lam @ (sub @ X158 @ (push @ one @ sh))) @ id) | ~ ! [X164 : term] : (~(X157 @ id @ X164 @ id) | (X157 @ id @ (sub @ X158 @ (push @ X164 @ id)) @ id))) => ! [X165 : (subst > term > subst > $o),X166 : term] : (~ ! [X167 : (subst > term > term)] : (~ ! [X168 : term] : (~(X165 @ id @ X168 @ id) | (X165 @ id @ (X167 @ id @ X168) @ id)) | (X165 @ id @ (lam @ (X167 @ sh @ one)) @ id) | ~ ! [X169 : subst,X170 : subst,X171 : term] : (((X167 @ (comp @ X170 @ X169) @ (sub @ X171 @ X169))) = ((sub @ (X167 @ X170 @ X171) @ X169)))) | (X165 @ id @ (lam @ X166) @ id) | ~ ! [X172 : term] : (~(X165 @ id @ X172 @ id) | (X165 @ id @ (sub @ X166 @ (push @ X172 @ id)) @ id)))))) & pushprop_lthm & (induction2lem_lthm <=> (! [X173 : subst,X174 : term] : (((sub @ one @ (push @ X174 @ X173))) = X174) => (! [X175 : subst,X176 : term] : (((comp @ sh @ (push @ X176 @ X175))) = X175) => (! [X177 : subst] : (((comp @ X177 @ id)) = X177) => (! [X178 : term,X179 : (term > $o)] : (~ ! [X180 : term] : (~(var @ X180) | (X179 @ X180)) | ~ ! [X181 : term] : (~(X179 @ X181) | (X179 @ (lam @ X181))) | (X179 @ X178) | ~ ! [X182 : term,X183 : term] : ((X179 @ (ap @ X182 @ X183)) | ~(X179 @ X182) | ~(X179 @ X183))) => ! [X184 : (term > $o),X185 : term,X186 : subst] : (~ ! [X187 : term,X188 : term] : (~(X184 @ X188) | (X184 @ (ap @ X188 @ X187)) | ~(X184 @ X187)) | (X184 @ (sub @ X185 @ X186)) | ~ ! [X189 : term] : ((X184 @ (sub @ X189 @ X186)) | ~(var @ X189)) | ~ ! [X190 : term] : (~ ! [X191 : term] : (~(X184 @ X191) | (X184 @ (sub @ X190 @ (push @ X191 @ id)))) | (X184 @ (lam @ X190))))))))) & axmap & axapp & hoasinduction_lem3b_gthm & (! [X192 : (subst > term > subst > $o),X193 : term,X194 : (term > $o)] : (~ ! [X195 : (subst > term > term)] : ((X192 @ id @ (lam @ (X195 @ sh @ one)) @ id) | ~ ! [X196 : term] : (~(X192 @ id @ X196 @ id) | (X192 @ id @ (X195 @ id @ X196) @ id)) | ~ ! [X197 : subst,X198 : subst,X199 : term] : (((X195 @ (comp @ X198 @ X197) @ (sub @ X199 @ X197))) = ((sub @ (X195 @ X198 @ X199) @ X197)))) | ~ ! [X200 : term] : ((X194 @ (sub @ X193 @ (push @ X200 @ id))) | ~(X194 @ X200)) | ~ ! [X201 : subst,X202 : subst,X203 : subst,X204 : term] : ((X192 @ X203 @ X204 @ (comp @ X202 @ X201)) | ~(X192 @ (comp @ X203 @ X202) @ (sub @ X204 @ X202) @ X201)) | (X194 @ (lam @ X193)) | ~ ! [X205 : term] : (((X194 @ X205)) = ((X192 @ id @ X205 @ id))) | ~ ! [X206 : subst,X207 : subst,X208 : subst,X209 : term] : (~(X192 @ X206 @ X209 @ (comp @ X208 @ X207)) | (X192 @ (comp @ X206 @ X208) @ (sub @ X209 @ X208) @ X207))) <=> hoasinduction_lem3v2) & (! [X210 : (term > $o),X211 : term,X212 : (subst > term > subst > $o)] : (~ ! [X213 : term] : (~(X210 @ X213) | (X210 @ (sub @ X211 @ (push @ X213 @ id)))) | (X210 @ (lam @ X211)) | ~ ! [X214 : term] : (((X210 @ X214)) = ((X212 @ id @ X214 @ id))) | ~ ! [X215 : (subst > term > term)] : ((X212 @ id @ (lam @ (X215 @ sh @ one)) @ id) | ~ ! [X216 : term] : (~(X212 @ id @ X216 @ id) | (X212 @ id @ (X215 @ id @ X216) @ id)) | ~ ! [X217 : subst,X218 : subst,X219 : term] : (((X215 @ (comp @ X217 @ X218) @ (sub @ X219 @ X218))) = ((sub @ (X215 @ X217 @ X219) @ X218))))) <=> hoasinduction_lem3v2a) & pushprop_lem2v2_gthm & (axscons <=> ! [X220 : subst] : (((push @ (sub @ one @ X220) @ (comp @ sh @ X220))) = X220)) & (hoasinduction_lem3aa <=> ! [X221 : term,X222 : (subst > term > subst > $o)] : ((X222 @ id @ (lam @ (sub @ X221 @ (push @ one @ sh))) @ id) | ~ ! [X223 : term] : ((X222 @ id @ (sub @ X221 @ (push @ X223 @ id)) @ id) | ~(X222 @ id @ X223 @ id)) | ~ ! [X224 : (subst > term > term)] : (~ ! [X225 : subst,X226 : subst,X227 : term] : (((X224 @ (comp @ X225 @ X226) @ (sub @ X227 @ X226))) = ((sub @ (X224 @ X225 @ X227) @ X226))) | ~ ! [X228 : term] : ((X222 @ id @ (X224 @ id @ X228) @ id) | ~(X222 @ id @ X228 @ id)) | (X222 @ id @ (lam @ (X224 @ sh @ one)) @ id)))) & ! [X229 : d_subst] : (d_id = X229) & pushprop_gthm & ulamvarind & (axvarcons <=> ! [X230 : term,X231 : subst] : (((sub @ one @ (push @ X230 @ X231))) = X230)) & hoaslamnotvar & hoasinduction_lem1_gthm & (induction2lem <=> ! [X232 : term,X233 : subst,X234 : (term > $o)] : (~ ! [X235 : term] : (~(var @ X235) | (X234 @ (sub @ X235 @ X233))) | ~ ! [X236 : term,X237 : term] : (~(X234 @ X236) | ~(X234 @ X237) | (X234 @ (ap @ X236 @ X237))) | ~ ! [X238 : term] : ((X234 @ (lam @ X238)) | ~ ! [X239 : term] : (~(X234 @ X239) | (X234 @ (sub @ X238 @ (push @ X239 @ id))))) | (X234 @ (sub @ X232 @ X233)))) & (! [X240 : term,X241 : term] : ((((sub @ X240 @ sh)) != ((sub @ X241 @ sh))) | (X240 = X241)) <=> shinj) & (hoasinduction_lem0 <=> ! [X242 : (subst > term > subst > $o)] : ~ ! [X243 : (term > $o)] : ~ ! [X244 : term] : (((X243 @ X244)) = ((X242 @ id @ X244 @ id)))) & ! [X245 : d_term,X246 : d_term] : ((((d2term @ X245)) = ((d2term @ X246))) => (X245 = X246)) & axabs & (axidr <=> ! [X247 : subst] : (((comp @ X247 @ id)) = X247)) & pushprop_lthm_orig & hoasinduction_lem3a_gthm & hoasinduction_lem2_gthm & hoaslamnotvar_gthm & (one = ((d2term @ d_one))) & (! [X248 : term,X249 : (subst > term > subst > $o)] : ((X249 @ id @ (lam @ X248) @ id) | ~ ! [X250 : term] : (~(X249 @ id @ X250 @ id) | (X249 @ id @ (sub @ X248 @ (push @ X250 @ id)) @ id)) | ~ ! [X251 : subst,X252 : subst,X253 : subst,X254 : term] : ((X249 @ X252 @ X254 @ (comp @ X253 @ X251)) | ~(X249 @ (comp @ X252 @ X253) @ (sub @ X254 @ X253) @ X251)) | ~ ! [X255 : (subst > term > term)] : (~ ! [X256 : term] : (~(X249 @ id @ X256 @ id) | (X249 @ id @ (X255 @ id @ X256) @ id)) | (X249 @ id @ (lam @ (X255 @ sh @ one)) @ id) | ~ ! [X257 : subst,X258 : subst,X259 : term] : (((sub @ (X255 @ X257 @ X259) @ X258)) = ((X255 @ (comp @ X257 @ X258) @ (sub @ X259 @ X258))))) | ~ ! [X260 : subst,X261 : subst,X262 : term,X263 : subst] : (~(X249 @ X260 @ X262 @ (comp @ X261 @ X263)) | (X249 @ (comp @ X260 @ X261) @ (sub @ X262 @ X261) @ X263))) <=> hoasinduction_lem3) & pushprop_lem2v2 & hoasinduction_lem3v2_gthm & (! [X264 : term,X265 : (subst > term > subst > $o),X266 : term] : (~(X265 @ id @ X266 @ id) | ~ ! [X267 : term,X268 : subst,X269 : subst,X270 : subst] : (~(X265 @ X269 @ X267 @ (comp @ X268 @ X270)) | (X265 @ (comp @ X269 @ X268) @ (sub @ X267 @ X268) @ X270)) | ~(X265 @ id @ X264 @ id) | ~ ! [X271 : term,X272 : term] : ((X265 @ id @ (ap @ (sub @ X271 @ id) @ X272) @ id) | ~(X265 @ id @ X272 @ id) | ~(X265 @ id @ X271 @ id)) | (X265 @ id @ (ap @ X266 @ X264) @ id) | ~ ! [X273 : subst,X274 : subst,X275 : term,X276 : subst] : (~(X265 @ (comp @ X274 @ X276) @ (sub @ X275 @ X276) @ X273) | (X265 @ X274 @ X275 @ (comp @ X276 @ X273)))) <=> hoasinduction_lem2) & pushprop_lem1v2_gthm & (((sub @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2term @ d_one))) & (! [X277 : subst,X278 : (term > $o),X279 : term,X280 : (term > $o)] : (~(X280 @ X279) | (X278 @ one) | ~ ! [X281 : term] : (((X278 @ X281)) = ((X280 @ (sub @ X281 @ (push @ X279 @ X277)))))) <=> pushprop_lem1v2) & hoaslaminj_gthm & (! [X282 : (term > $o),X283 : term,X284 : subst] : ~ ! [X285 : (term > $o)] : ~ ! [X286 : term] : (((X285 @ X286)) = ((X282 @ (sub @ X286 @ (push @ X283 @ X284))))) <=> pushprop_lem0_lthm) & hoasapinj2_gthm & (((ap @ (d2term @ d_one) @ (d2term @ d_one))) = ((d2term @ d_one))) & hoasapnotvar & (! [X287 : term,X288 : (subst > term > subst > $o)] : (~ ! [X289 : term] : ((X288 @ id @ (sub @ X287 @ (push @ X289 @ id)) @ id) | ~(X288 @ id @ X289 @ id)) | (X288 @ id @ (lam @ X287) @ id) | ~ ! [X290 : (subst > term > term)] : ((X288 @ id @ (lam @ (X290 @ sh @ one)) @ id) | ~ ! [X291 : subst,X292 : subst,X293 : term] : (((X290 @ (comp @ X291 @ X292) @ (sub @ X293 @ X292))) = ((sub @ (X290 @ X291 @ X293) @ X292))) | ~ ! [X294 : term] : ((X288 @ id @ (X290 @ id @ X294) @ id) | ~(X288 @ id @ X294 @ id)))) <=> hoasinduction_lem3a) & ~(var @ (d2term @ d_one)) & (induction2 <=> ! [X295 : (term > $o),X296 : term] : (~ ! [X297 : term] : (~ ! [X298 : term] : (~(X295 @ X298) | (X295 @ (sub @ X297 @ (push @ X298 @ id)))) | (X295 @ (lam @ X297))) | (X295 @ X296) | ~ ! [X299 : term,X300 : term] : (~(X295 @ X299) | ~(X295 @ X300) | (X295 @ (ap @ X300 @ X299))) | ~ ! [X301 : term] : (~(var @ X301) | (X295 @ X301)))) & (! [X302 : (term > $o),X303 : term,X304 : term,X305 : subst,X306 : (term > $o)] : (~(X306 @ X304) | (X302 @ (sub @ X304 @ (push @ X303 @ X305)))) <=> pushprop_lem1) & (apinj2 <=> ! [X307 : term,X308 : term,X309 : term,X310 : term] : ((X309 = X310) | (((ap @ X308 @ X309)) != ((ap @ X307 @ X310))))) & (((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) = ((d2term @ d_one))) & ~(hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) & (hoasinduction_lem3v2_f_lthm <=> ! [X311 : term] : ~ ! [X312 : (subst > term > term)] : ~ ! [X313 : term,X314 : subst] : (((X312 @ X314 @ X313)) = ((sub @ X311 @ (push @ X313 @ X314))))) & (! [X315 : term] : ~ ! [X316 : (subst > term > term)] : (((sub @ X315 @ (push @ one @ sh))) != ((X316 @ sh @ one))) <=> hoasinduction_lem3b) & ((! [X317 : term] : ~ ! [X318 : (subst > term > term)] : ~ ! [X319 : term,X320 : subst] : (((sub @ X317 @ (push @ X319 @ X320))) = ((X318 @ X320 @ X319))) => (! [X321 : term] : (((sub @ X321 @ id)) = X321) => ! [X322 : term,X323 : (term > $o),X324 : (subst > term > subst > $o)] : (~ ! [X325 : (subst > term > term)] : (~ ! [X326 : subst,X327 : subst,X328 : term] : (((X325 @ (comp @ X327 @ X326) @ (sub @ X328 @ X326))) = ((sub @ (X325 @ X327 @ X328) @ X326))) | (X324 @ id @ (lam @ (X325 @ sh @ one)) @ id) | ~ ! [X329 : term] : ((X324 @ id @ (X325 @ id @ X329) @ id) | ~(X324 @ id @ X329 @ id))) | ~ ! [X330 : term] : (((X323 @ X330)) = ((X324 @ id @ X330 @ id))) | ~ ! [X331 : term] : (~(X323 @ X331) | (X323 @ (sub @ X322 @ (push @ X331 @ id)))) | (X323 @ (lam @ X322))))) <=> hoasinduction_lem3v2a_lthm) & hoaslamnotap_lthm & hoasapnotvar_lthm & (! [X332 : term] : (((sub @ X332 @ id)) = X332) <=> termmset) & (hoasinduction_lem3v2_lthm <=> (! [X333 : term] : (((sub @ X333 @ id)) = X333) => ! [X334 : term,X335 : (subst > term > subst > $o),X336 : (term > $o)] : (~ ! [X337 : term] : (~(X336 @ X337) | (X336 @ (sub @ X334 @ (push @ X337 @ id)))) | ~ ! [X338 : (subst > term > term)] : (~ ! [X339 : subst,X340 : term,X341 : subst] : (((X338 @ (comp @ X341 @ X339) @ (sub @ X340 @ X339))) = ((sub @ (X338 @ X341 @ X340) @ X339))) | ~ ! [X342 : term] : (~(X335 @ id @ X342 @ id) | (X335 @ id @ (X338 @ id @ X342) @ id)) | (X335 @ id @ (lam @ (X338 @ sh @ one)) @ id)) | (X336 @ (lam @ X334)) | ~ ! [X343 : term] : (((X336 @ X343)) = ((X335 @ id @ X343 @ id))) | ~ ! [X344 : subst,X345 : subst,X346 : subst,X347 : term] : ((X335 @ (comp @ X344 @ X346) @ (sub @ X347 @ X346) @ X345) | ~(X335 @ X344 @ X347 @ (comp @ X346 @ X345))) | ~ ! [X348 : subst,X349 : term,X350 : subst,X351 : subst] : (~(X335 @ (comp @ X350 @ X348) @ (sub @ X349 @ X348) @ X351) | (X335 @ X350 @ X349 @ (comp @ X348 @ X351)))))) & (hoasinduction_lem3b_lthm <=> ! [X352 : term] : ~ ! [X353 : (subst > term > term)] : (((sub @ X352 @ (push @ one @ sh))) != ((X353 @ sh @ one)))) & hoasinduction_lem3_gthm & pushprop_lem3v2_lthm & ! [X354 : d_term] : (d_one = X354) & hoaslamnotap_gthm & (! [X355 : term] : (((sub @ X355 @ id)) = X355) <=> axvarid) & ! [X356 : subst] : ? [X357 : d_subst] : (((d2subst @ X357)) = X356) & (pushprop_lem1v2_lthm <=> (! [X358 : term,X359 : subst] : (((sub @ one @ (push @ X358 @ X359))) = X358) => ! [X360 : (term > $o),X361 : (term > $o),X362 : term,X363 : subst] : (~(X361 @ X362) | (X360 @ one) | ~ ! [X364 : term] : (((X361 @ (sub @ X364 @ (push @ X362 @ X363)))) = ((X360 @ X364)))))) & axvarshift & (! [X365 : (subst > term > subst > $o)] : ~ ! [X366 : (term > $o)] : ~ ! [X367 : term] : (((X366 @ X367)) = ((X365 @ id @ X367 @ id))) <=> hoasinduction_lem0_lthm) & ((^[X368 : term, X369 : subst, X370 : (term > $o), X371 : (term > $o)] : (! [X372 : term] : (((X371 @ X372)) = ((X370 @ (sub @ X372 @ (push @ X368 @ X369))))))) = pushprop_p_and_p_prime) & (hoasinduction_lem3_lthm <=> (! [X373 : term] : (((sub @ X373 @ id)) = X373) => (! [X374 : (subst > term > subst > $o),X375 : term] : ((X374 @ id @ (lam @ (sub @ X375 @ (push @ one @ sh))) @ id) | ~ ! [X376 : term] : (~(X374 @ id @ X376 @ id) | (X374 @ id @ (sub @ X375 @ (push @ X376 @ id)) @ id)) | ~ ! [X377 : (subst > term > term)] : (~ ! [X378 : term,X379 : subst,X380 : subst] : (((X377 @ (comp @ X379 @ X380) @ (sub @ X378 @ X380))) = ((sub @ (X377 @ X379 @ X378) @ X380))) | ~ ! [X381 : term] : (~(X374 @ id @ X381 @ id) | (X374 @ id @ (X377 @ id @ X381) @ id)) | (X374 @ id @ (lam @ (X377 @ sh @ one)) @ id))) => ! [X382 : (subst > term > subst > $o),X383 : term] : (~ ! [X384 : subst,X385 : subst,X386 : subst,X387 : term] : ((X382 @ (comp @ X386 @ X384) @ (sub @ X387 @ X384) @ X385) | ~(X382 @ X386 @ X387 @ (comp @ X384 @ X385))) | (X382 @ id @ (lam @ X383) @ id) | ~ ! [X388 : term] : (~(X382 @ id @ X388 @ id) | (X382 @ id @ (sub @ X383 @ (push @ X388 @ id)) @ id)) | ~ ! [X389 : subst,X390 : subst,X391 : subst,X392 : term] : ((X382 @ X390 @ X392 @ (comp @ X391 @ X389)) | ~(X382 @ (comp @ X390 @ X391) @ (sub @ X392 @ X391) @ X389)) | ~ ! [X393 : (subst > term > term)] : (~ ! [X394 : term,X395 : subst,X396 : subst] : (((X393 @ (comp @ X396 @ X395) @ (sub @ X394 @ X395))) = ((sub @ (X393 @ X396 @ X394) @ X395))) | ~ ! [X397 : term] : (~(X382 @ id @ X397 @ id) | (X382 @ id @ (X393 @ id @ X397) @ id)) | (X382 @ id @ (lam @ (X393 @ sh @ one)) @ id)))))) & (((push @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & (! [X398 : subst,X399 : term] : (((comp @ sh @ (push @ X399 @ X398))) = X398) <=> axshiftcons) & (hoasinduction <=> ! [X400 : (subst > term > subst > $o),X401 : term] : ((X400 @ id @ X401 @ id) | ~ ! [X402 : subst,X403 : subst,X404 : term,X405 : subst] : (~(X400 @ (comp @ X403 @ X402) @ (sub @ X404 @ X402) @ X405) | (X400 @ X403 @ X404 @ (comp @ X402 @ X405))) | ~ ! [X406 : subst,X407 : term,X408 : subst,X409 : subst] : ((X400 @ (comp @ X406 @ X408) @ (sub @ X407 @ X408) @ X409) | ~(X400 @ X406 @ X407 @ (comp @ X408 @ X409))) | ~ ! [X410 : (subst > term > term)] : (~ ! [X411 : term] : ((X400 @ id @ (X410 @ id @ X411) @ id) | ~(X400 @ id @ X411 @ id)) | ~ ! [X412 : subst,X413 : term,X414 : subst] : (((sub @ (X410 @ X414 @ X413) @ X412)) = ((X410 @ (comp @ X414 @ X412) @ (sub @ X413 @ X412)))) | (X400 @ id @ (lam @ (X410 @ sh @ one)) @ id)) | ~ ! [X415 : term,X416 : term] : ((X400 @ id @ (ap @ (sub @ X416 @ id) @ X415) @ id) | ~(X400 @ id @ X415 @ id) | ~(X400 @ id @ X416 @ id)) | ~ ! [X417 : term] : ((X400 @ id @ X417 @ id) | ~(var @ (sub @ X417 @ id))))) & pushprop_lem3v2 & (! [X418 : term] : ~ ! [X419 : (subst > term > term)] : ~ ! [X420 : subst,X421 : term] : (((X419 @ X420 @ X421)) = ((sub @ X418 @ (push @ X421 @ X420)))) <=> hoasinduction_lem3v2_f) & hoasinduction_lem1v2 & ! [X422 : d_subst,X423 : d_subst] : ((((d2subst @ X423)) = ((d2subst @ X422))) => (X422 = X423)) & ((^[X424 : subst, X425 : (subst > term > term)] : ((d2term @ d_one))) = hoaslam) & ~ulamvar1 & (hoasapinj1_lthm <=> (! [X426 : term] : (((sub @ X426 @ id)) = X426) => (! [X427 : term,X428 : term,X429 : term,X430 : term] : ((X427 = X428) | (((ap @ X427 @ X430)) != ((ap @ X428 @ X429)))) => ! [X431 : term,X432 : term,X433 : term,X434 : term] : ((X431 = X434) | (((ap @ (sub @ X431 @ id) @ X432)) != ((ap @ (sub @ X434 @ id) @ X433))))))) & ((! [X435 : term] : (((sub @ X435 @ id)) = X435) => (! [X436 : subst,X437 : term,X438 : (term > $o)] : (~ ! [X439 : term] : (~ ! [X440 : term] : (~(X438 @ X440) | (X438 @ (sub @ X439 @ (push @ X440 @ id)))) | (X438 @ (lam @ X439))) | ~ ! [X441 : term] : ((X438 @ (sub @ X441 @ X436)) | ~(var @ X441)) | ~ ! [X442 : term,X443 : term] : ((X438 @ (ap @ X443 @ X442)) | ~(X438 @ X443) | ~(X438 @ X442)) | (X438 @ (sub @ X437 @ X436))) => ! [X444 : (term > $o),X445 : term] : (~ ! [X446 : term,X447 : term] : (~(X444 @ X446) | ~(X444 @ X447) | (X444 @ (ap @ X446 @ X447))) | (X444 @ X445) | ~ ! [X448 : term] : (~(var @ X448) | (X444 @ X448)) | ~ ! [X449 : term] : (~ ! [X450 : term] : ((X444 @ (sub @ X449 @ (push @ X450 @ id))) | ~(X444 @ X450)) | (X444 @ (lam @ X449)))))) <=> induction2_lthm) & (sh = ((d2subst @ d_id))) & axassoc & (! [X451 : subst] : (((comp @ id @ X451)) = X451) <=> axidl) & pushprop_lem0_gthm & hoasinduction_lem2v2_gthm & (hoasinduction_lem3aa_lthm <=> ! [X452 : (subst > term > subst > $o),X453 : term] : (~ ! [X454 : term] : ((X452 @ id @ (sub @ X453 @ (push @ X454 @ id)) @ id) | ~(X452 @ id @ X454 @ id)) | ~ ! [X455 : (subst > term > term)] : (~ ! [X456 : term] : (~(X452 @ id @ X456 @ id) | (X452 @ id @ (X455 @ id @ X456) @ id)) | (X452 @ id @ (lam @ (X455 @ sh @ one)) @ id) | ~ ! [X457 : subst,X458 : subst,X459 : term] : (((X455 @ (comp @ X458 @ X457) @ (sub @ X459 @ X457))) = ((sub @ (X455 @ X458 @ X459) @ X457)))) | (X452 @ id @ (lam @ (sub @ X453 @ (push @ one @ sh))) @ id))) & hoasapinj1_gthm & hoasinduction_lem1_lthm & (id = ((d2subst @ d_id))) & (hoasinduction_no_psi_cond_lthm <=> (! [X460 : (subst > term > subst > $o)] : ~ ! [X461 : (term > $o)] : ~ ! [X462 : term] : (((X461 @ X462)) = ((X460 @ id @ X462 @ id))) => (! [X463 : (term > $o),X464 : term] : (~ ! [X465 : term] : ((X463 @ X465) | ~(var @ X465)) | ~ ! [X466 : term,X467 : term] : (~(X463 @ X467) | ~(X463 @ X466) | (X463 @ (ap @ X466 @ X467))) | ~ ! [X468 : term] : (~ ! [X469 : term] : ((X463 @ (sub @ X468 @ (push @ X469 @ id))) | ~(X463 @ X469)) | (X463 @ (lam @ X468))) | (X463 @ X464)) => (! [X470 : term] : (((sub @ X470 @ id)) = X470) => (! [X471 : term,X472 : (term > $o),X473 : (subst > term > subst > $o)] : (~ ! [X474 : term] : (~(X472 @ X474) | (X472 @ (sub @ X471 @ (push @ X474 @ id)))) | ~ ! [X475 : (subst > term > term)] : (~ ! [X476 : term,X477 : subst,X478 : subst] : (((sub @ (X475 @ X477 @ X476) @ X478)) = ((X475 @ (comp @ X477 @ X478) @ (sub @ X476 @ X478)))) | (X473 @ id @ (lam @ (X475 @ sh @ one)) @ id) | ~ ! [X479 : term] : ((X473 @ id @ (X475 @ id @ X479) @ id) | ~(X473 @ id @ X479 @ id))) | (X472 @ (lam @ X471)) | ~ ! [X480 : term] : (((X473 @ id @ X480 @ id)) = ((X472 @ X480)))) => ! [X481 : (subst > term > subst > $o),X482 : term] : (~ ! [X483 : (subst > term > term)] : (~ ! [X484 : term] : (~(X481 @ id @ X484 @ id) | (X481 @ id @ (X483 @ id @ X484) @ id)) | ~ ! [X485 : subst,X486 : subst,X487 : term] : (((X483 @ (comp @ X486 @ X485) @ (sub @ X487 @ X485))) = ((sub @ (X483 @ X486 @ X487) @ X485))) | (X481 @ id @ (lam @ (X483 @ sh @ one)) @ id)) | (X481 @ id @ X482 @ id) | ~ ! [X488 : term,X489 : term] : (~(X481 @ id @ X489 @ id) | (X481 @ id @ (ap @ (sub @ X489 @ id) @ X488) @ id) | ~(X481 @ id @ X488 @ id)))))))) & (((lam @ (d2term @ d_one))) = ((d2term @ d_one))) & (hoaslaminj <=> ! [X490 : (subst > term > term),X491 : subst,X492 : term,X493 : (subst > term > term)] : ((((X493 @ X491 @ X492)) = ((X490 @ X491 @ X492))) | ~ ! [X494 : subst,X495 : subst,X496 : term] : (((X490 @ (comp @ X494 @ X495) @ (sub @ X496 @ X495))) = ((sub @ (X490 @ X494 @ X496) @ X495))) | (((lam @ (X490 @ sh @ one))) != ((lam @ (X493 @ sh @ one)))) | ~ ! [X497 : subst,X498 : term,X499 : subst] : (((sub @ (X493 @ X497 @ X498) @ X499)) = ((X493 @ (comp @ X497 @ X499) @ (sub @ X498 @ X499)))))) & (! [X500 : term,X501 : term,X502 : term,X503 : term] : ((X500 = X502) | (((ap @ (sub @ X502 @ id) @ X503)) != ((ap @ (sub @ X500 @ id) @ X501)))) <=> hoasapinj1) & lamnotvar & ulamvarsh & substmonoid_gthm & ! [X504 : term] : ? [X505 : d_term] : (((d2term @ X505)) = X504) & (((comp @ (d2subst @ d_id) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & ((! [X506 : (subst > term > subst > $o)] : ~ ! [X507 : (term > $o)] : ~ ! [X508 : term] : (((X506 @ id @ X508 @ id)) = ((X507 @ X508))) => (! [X509 : (term > $o),X510 : term] : ((X509 @ X510) | ~ ! [X511 : term] : (~(var @ X511) | (X509 @ X511)) | ~ ! [X512 : term] : ((X509 @ (lam @ X512)) | ~ ! [X513 : term] : (~(X509 @ X513) | (X509 @ (sub @ X512 @ (push @ X513 @ id))))) | ~ ! [X514 : term,X515 : term] : (~(X509 @ X515) | (X509 @ (ap @ X514 @ X515)) | ~(X509 @ X514))) => (! [X516 : term] : (((sub @ X516 @ id)) = X516) => (! [X517 : (term > $o),X518 : term,X519 : (subst > term > subst > $o)] : (~ ! [X520 : term] : (((X519 @ id @ X520 @ id)) = ((X517 @ X520))) | (X517 @ (lam @ X518)) | ~ ! [X521 : term] : ((X517 @ (sub @ X518 @ (push @ X521 @ id))) | ~(X517 @ X521)) | ~ ! [X522 : (subst > term > term)] : (~ ! [X523 : term,X524 : subst,X525 : subst] : (((sub @ (X522 @ X524 @ X523) @ X525)) = ((X522 @ (comp @ X524 @ X525) @ (sub @ X523 @ X525)))) | (X519 @ id @ (lam @ (X522 @ sh @ one)) @ id) | ~ ! [X526 : term] : ((X519 @ id @ (X522 @ id @ X526) @ id) | ~(X519 @ id @ X526 @ id)))) => ! [X527 : (subst > term > subst > $o),X528 : term] : (~ ! [X529 : term,X530 : term] : (~(X527 @ id @ X530 @ id) | (X527 @ id @ (ap @ (sub @ X529 @ id) @ X530) @ id) | ~(X527 @ id @ X529 @ id)) | (X527 @ id @ X528 @ id) | ~ ! [X531 : term,X532 : subst,X533 : subst,X534 : subst] : (~(X527 @ (comp @ X532 @ X534) @ (sub @ X531 @ X534) @ X533) | (X527 @ X532 @ X531 @ (comp @ X534 @ X533))) | ~ ! [X535 : (subst > term > term)] : (~ ! [X536 : term,X537 : subst,X538 : subst] : (((X535 @ (comp @ X538 @ X537) @ (sub @ X536 @ X537))) = ((sub @ (X535 @ X538 @ X536) @ X537))) | ~ ! [X539 : term] : ((X527 @ id @ (X535 @ id @ X539) @ id) | ~(X527 @ id @ X539 @ id)) | (X527 @ id @ (lam @ (X535 @ sh @ one)) @ id)) | ~ ! [X540 : subst,X541 : term,X542 : subst,X543 : subst] : ((X527 @ (comp @ X542 @ X543) @ (sub @ X541 @ X543) @ X540) | ~(X527 @ X542 @ X541 @ (comp @ X543 @ X540))) | ~ ! [X544 : term] : (~(var @ (sub @ X544 @ id)) | (X527 @ id @ X544 @ id))))))) <=> hoasinduction_lthm_3) & (pushprop_lem0 <=> ! [X545 : (term > $o),X546 : term,X547 : subst] : ~ ! [X548 : (term > $o)] : ~ ! [X549 : term] : (((X545 @ (sub @ X549 @ (push @ X546 @ X547)))) = ((X548 @ X549)))) & hoasinduction_lem1),
  inference(rectify,[],[f3])).
thf(f5,plain,(
  (hoasinduction_lem1v2_gthm = $true) & (hoaslamnotvar_lthm = $true) & ~ (lamnotap = $true) & (hoasapnotvar_gthm = $true) & (induction2_gthm = $true) & (pushprop_lem1_gthm = $true) & ((substmonoid_lthm = $true) <=> (! [X0 : subst] : (((comp @ id @ X0)) = X0) => (! [X1 : subst] : (((comp @ X1 @ id)) = X1) => (! [X3 : subst] : (((comp @ X3 @ id)) = X3) & ! [X2 : subst] : (((comp @ id @ X2)) = X2))))) & (pushprop_lem2v2_lthm = $true) & (hoasinduction_lem2_lthm = $true) & ((hoasinduction_lthm = $true) <=> (! [X4 : (term > $o),X5 : term] : (~ ! [X6 : term,X7 : term] : (~ ($true = ((X4 @ X6))) | (((X4 @ (ap @ X7 @ X6))) = $true) | ~ (((X4 @ X7)) = $true)) | (((X4 @ X5)) = $true) | ~ ! [X8 : term] : (~ ($true = ((var @ X8))) | (((X4 @ X8)) = $true)) | ~ ! [X9 : term] : (~ ! [X10 : term] : (($true = ((X4 @ (sub @ X9 @ (push @ X10 @ id))))) | ~ (((X4 @ X10)) = $true)) | ($true = ((X4 @ (lam @ X9)))))) => (! [X11 : term,X12 : (subst > term > subst > $o),X13 : term] : (~ (((X12 @ id @ X11 @ id)) = $true) | ~ ! [X14 : term,X15 : subst,X16 : subst,X17 : subst] : (~ (((X12 @ (comp @ X16 @ X17) @ (sub @ X14 @ X17) @ X15)) = $true) | ($true = ((X12 @ X16 @ X14 @ (comp @ X17 @ X15))))) | (((X12 @ id @ (ap @ X11 @ X13) @ id)) = $true) | ~ ! [X18 : term,X19 : term] : (~ (((X12 @ id @ X19 @ id)) = $true) | (((X12 @ id @ (ap @ (sub @ X19 @ id) @ X18) @ id)) = $true) | ~ (((X12 @ id @ X18 @ id)) = $true)) | ~ ! [X20 : subst,X21 : subst,X22 : term,X23 : subst] : ((((X12 @ (comp @ X20 @ X23) @ (sub @ X22 @ X23) @ X21)) = $true) | ~ ($true = ((X12 @ X20 @ X22 @ (comp @ X23 @ X21))))) | ~ ($true = ((X12 @ id @ X13 @ id)))) => (! [X24 : (subst > term > subst > $o),X25 : term] : (~ ! [X26 : subst,X27 : subst,X28 : subst,X29 : term] : ((((X24 @ (comp @ X27 @ X26) @ (sub @ X29 @ X26) @ X28)) = $true) | ~ ($true = ((X24 @ X27 @ X29 @ (comp @ X26 @ X28))))) | (((X24 @ id @ (lam @ X25) @ id)) = $true) | ~ ! [X30 : (subst > term > term)] : ((((X24 @ id @ (lam @ (X30 @ sh @ one)) @ id)) = $true) | ~ ! [X31 : term] : ((((X24 @ id @ (X30 @ id @ X31) @ id)) = $true) | ~ (((X24 @ id @ X31 @ id)) = $true)) | ~ ! [X33 : subst,X34 : subst,X32 : term] : (((sub @ (X30 @ X33 @ X32) @ X34)) = ((X30 @ (comp @ X33 @ X34) @ (sub @ X32 @ X34))))) | ~ ! [X35 : term] : (~ (((X24 @ id @ X35 @ id)) = $true) | (((X24 @ id @ (sub @ X25 @ (push @ X35 @ id)) @ id)) = $true)) | ~ ! [X36 : subst,X37 : term,X38 : subst,X39 : subst] : (($true = ((X24 @ X39 @ X37 @ (comp @ X38 @ X36)))) | ~ (((X24 @ (comp @ X39 @ X38) @ (sub @ X37 @ X38) @ X36)) = $true))) => ! [X40 : term,X41 : (subst > term > subst > $o)] : (~ ! [X42 : term,X43 : term] : (~ ($true = ((X41 @ id @ X43 @ id))) | ~ ($true = ((X41 @ id @ X42 @ id))) | ($true = ((X41 @ id @ (ap @ (sub @ X43 @ id) @ X42) @ id)))) | (((X41 @ id @ X40 @ id)) = $true) | ~ ! [X44 : term] : ((((X41 @ id @ X44 @ id)) = $true) | ~ (((var @ (sub @ X44 @ id))) = $true)) | ~ ! [X45 : (subst > term > term)] : ((((X41 @ id @ (lam @ (X45 @ sh @ one)) @ id)) = $true) | ~ ! [X46 : term] : (~ (((X41 @ id @ X46 @ id)) = $true) | ($true = ((X41 @ id @ (X45 @ id @ X46) @ id)))) | ~ ! [X48 : term,X49 : subst,X47 : subst] : (((sub @ (X45 @ X49 @ X48) @ X47)) = ((X45 @ (comp @ X49 @ X47) @ (sub @ X48 @ X47))))) | ~ ! [X50 : subst,X51 : subst,X52 : term,X53 : subst] : (($true = ((X41 @ X51 @ X52 @ (comp @ X50 @ X53)))) | ~ (((X41 @ (comp @ X51 @ X50) @ (sub @ X52 @ X50) @ X53)) = $true)) | ~ ! [X54 : subst,X55 : subst,X56 : subst,X57 : term] : (~ ($true = ((X41 @ X56 @ X57 @ (comp @ X55 @ X54)))) | (((X41 @ (comp @ X56 @ X55) @ (sub @ X57 @ X55) @ X54)) = $true))))))) & ((hoasinduction_lem3aaa = $true) <=> ! [X58 : term,X59 : (subst > term > subst > $o)] : (~ ! [X60 : term,X61 : (subst > term > term)] : (~ ! [X64 : subst,X63 : subst,X62 : term] : (((sub @ (X61 @ X64 @ X62) @ X63)) = ((sub @ (sub @ X60 @ (push @ X62 @ X64)) @ X63))) | ~ ! [X65 : term] : (($true = ((X59 @ id @ (X61 @ id @ X65) @ id))) | ~ ($true = ((X59 @ id @ X65 @ id)))) | ($true = ((X59 @ id @ (lam @ (X61 @ sh @ one)) @ id))) | ~ ! [X68 : subst,X67 : subst,X66 : term] : (((X61 @ (comp @ X67 @ X68) @ (sub @ X66 @ X68))) = ((sub @ X60 @ (push @ (sub @ X66 @ X68) @ (comp @ X67 @ X68)))))) | (((X59 @ id @ (lam @ (sub @ X58 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X69 : term] : ((((X59 @ id @ (sub @ X58 @ (push @ X69 @ id)) @ id)) = $true) | ~ (((X59 @ id @ X69 @ id)) = $true)))) & (! [X72 : term,X71 : term,X70 : term,X73 : term] : ((((ap @ X71 @ X73)) != ((ap @ X70 @ X72))) | (X70 = X71)) <=> (apinj1 = $true)) & ((hoasapinj2_lthm = $true) <=> (! [X76 : term,X77 : term,X75 : term,X74 : term] : ((X76 = X77) | (((ap @ X75 @ X77)) != ((ap @ X74 @ X76)))) => ! [X80 : term,X81 : term,X78 : term,X79 : term] : ((X79 = X81) | (((ap @ (sub @ X80 @ id) @ X79)) != ((ap @ (sub @ X78 @ id) @ X81)))))) & (! [X84 : term,X82 : (subst > term > term),X83 : term] : ((((ap @ (sub @ X83 @ id) @ X84)) != ((lam @ (X82 @ sh @ one)))) | ~ ! [X85 : term,X86 : subst,X87 : subst] : (((X82 @ (comp @ X87 @ X86) @ (sub @ X85 @ X86))) = ((sub @ (X82 @ X87 @ X85) @ X86)))) <=> (hoaslamnotap = $true)) & ((induction = $true) <=> ! [X88 : term,X89 : (term > $o)] : (~ ! [X90 : term,X91 : term] : (~ ($true = ((X89 @ X91))) | ~ (((X89 @ X90)) = $true) | ($true = ((X89 @ (ap @ X90 @ X91))))) | ~ ! [X92 : term] : (($true = ((X89 @ (lam @ X92)))) | ~ ($true = ((X89 @ X92)))) | (((X89 @ X88)) = $true) | ~ ! [X93 : term] : ((((X89 @ X93)) = $true) | ~ (((var @ X93)) = $true)))) & ((hoasinduction_lem2v2 = $true) <=> ! [X94 : (subst > term > subst > $o),X95 : term,X96 : (term > $o),X97 : term] : (~ (((X96 @ X95)) = $true) | ~ ! [X98 : term] : (((X94 @ id @ X98 @ id)) = ((X96 @ X98))) | ~ ! [X99 : term,X100 : term] : (~ (((X94 @ id @ X100 @ id)) = $true) | (((X94 @ id @ (ap @ (sub @ X99 @ id) @ X100) @ id)) = $true) | ~ (((X94 @ id @ X99 @ id)) = $true)) | ~ ! [X101 : subst,X102 : subst,X103 : term,X104 : subst] : (($true = ((X94 @ (comp @ X102 @ X101) @ (sub @ X103 @ X101) @ X104))) | ~ (((X94 @ X102 @ X103 @ (comp @ X101 @ X104))) = $true)) | ~ ! [X105 : subst,X106 : term,X107 : subst,X108 : subst] : (~ (((X94 @ (comp @ X107 @ X105) @ (sub @ X106 @ X105) @ X108)) = $true) | (((X94 @ X107 @ X106 @ (comp @ X105 @ X108))) = $true)) | ~ (((X96 @ X97)) = $true) | ($true = ((X96 @ (ap @ X97 @ X95)))))) & ((hoaslaminj_lthm = $true) <=> (! [X109 : subst,X110 : term] : (((sub @ one @ (push @ X110 @ X109))) = X110) => (! [X112 : term,X111 : subst] : (((comp @ sh @ (push @ X112 @ X111))) = X111) => (! [X113 : term,X114 : term] : ((((lam @ X113)) != ((lam @ X114))) | (X113 = X114)) => ! [X117 : (subst > term > term),X118 : term,X115 : (subst > term > term),X116 : subst] : ((((lam @ (X115 @ sh @ one))) != ((lam @ (X117 @ sh @ one)))) | ~ ! [X122 : subst,X124 : subst,X123 : term] : (((X117 @ (comp @ X124 @ X122) @ (sub @ X123 @ X122))) = ((sub @ (X117 @ X124 @ X123) @ X122))) | ~ ! [X121 : subst,X119 : term,X120 : subst] : (((sub @ (X115 @ X121 @ X119) @ X120)) = ((X115 @ (comp @ X121 @ X120) @ (sub @ X119 @ X120)))) | (((X117 @ X116 @ X118)) = ((X115 @ X116 @ X118)))))))) & (apnotvar = $true) & (! [X128 : term,X125 : term,X127 : term,X126 : term] : ((((ap @ (sub @ X127 @ id) @ X128)) != ((ap @ (sub @ X126 @ id) @ X125))) | (X125 = X128)) <=> (hoasapinj2 = $true)) & ((pushprop_lem1_lthm = $true) <=> (! [X130 : subst,X129 : term] : (((sub @ one @ (push @ X129 @ X130))) = X129) => (! [X132 : subst,X131 : term] : (((comp @ sh @ (push @ X131 @ X132))) = X132) => ! [X133 : term,X134 : (term > $o),X135 : (term > $o),X136 : term,X137 : subst] : (~ ($true = ((X134 @ X136))) | (((X135 @ (sub @ X136 @ (push @ X133 @ X137)))) = $true))))) & (pushprop = $true) & ((! [X138 : term] : (((sub @ X138 @ id)) = X138) => ! [X139 : term] : (((sub @ X139 @ id)) = X139)) <=> (termmset_lthm = $true)) & ((! [X141 : subst] : (((comp @ id @ X141)) = X141) & ! [X140 : subst] : (((comp @ X140 @ id)) = X140)) <=> (substmonoid = $true)) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & ((hoasinduction_no_psi_cond = $true) <=> ! [X145 : term,X146 : (subst > term > subst > $o)] : (~ ! [X147 : term,X148 : term] : (~ (((X146 @ id @ X148 @ id)) = $true) | ($true = ((X146 @ id @ (ap @ (sub @ X147 @ id) @ X148) @ id))) | ~ ($true = ((X146 @ id @ X147 @ id)))) | (((X146 @ id @ X145 @ id)) = $true) | ~ ! [X149 : (subst > term > term)] : ((((X146 @ id @ (lam @ (X149 @ sh @ one)) @ id)) = $true) | ~ ! [X150 : subst,X151 : term,X152 : subst] : (((sub @ (X149 @ X152 @ X151) @ X150)) = ((X149 @ (comp @ X152 @ X150) @ (sub @ X151 @ X150)))) | ~ ! [X153 : term] : (($true = ((X146 @ id @ (X149 @ id @ X153) @ id))) | ~ (((X146 @ id @ X153 @ id)) = $true))))) & ((laminj = $true) <=> ! [X154 : term,X155 : term] : ((X154 = X155) | (((lam @ X154)) != ((lam @ X155))))) & (axclos = $true) & (induction2lem_gthm = $true) & (termmset_gthm = $true) & (hoasinduction_gthm = $true) & ((hoasinduction_lem3a_lthm = $true) <=> (! [X156 : term] : (((sub @ X156 @ id)) = X156) => (! [X157 : (subst > term > subst > $o),X158 : term] : (~ ! [X159 : (subst > term > term)] : (~ ! [X160 : term] : (($true = ((X157 @ id @ (X159 @ id @ X160) @ id))) | ~ (((X157 @ id @ X160 @ id)) = $true)) | (((X157 @ id @ (lam @ (X159 @ sh @ one)) @ id)) = $true) | ~ ! [X161 : term,X162 : subst,X163 : subst] : (((sub @ (X159 @ X162 @ X161) @ X163)) = ((X159 @ (comp @ X162 @ X163) @ (sub @ X161 @ X163))))) | (((X157 @ id @ (lam @ (sub @ X158 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X164 : term] : (~ (((X157 @ id @ X164 @ id)) = $true) | ($true = ((X157 @ id @ (sub @ X158 @ (push @ X164 @ id)) @ id))))) => ! [X165 : (subst > term > subst > $o),X166 : term] : (~ ! [X167 : (subst > term > term)] : (~ ! [X168 : term] : (~ ($true = ((X165 @ id @ X168 @ id))) | ($true = ((X165 @ id @ (X167 @ id @ X168) @ id)))) | ($true = ((X165 @ id @ (lam @ (X167 @ sh @ one)) @ id))) | ~ ! [X169 : subst,X170 : subst,X171 : term] : (((X167 @ (comp @ X170 @ X169) @ (sub @ X171 @ X169))) = ((sub @ (X167 @ X170 @ X171) @ X169)))) | (((X165 @ id @ (lam @ X166) @ id)) = $true) | ~ ! [X172 : term] : (~ (((X165 @ id @ X172 @ id)) = $true) | (((X165 @ id @ (sub @ X166 @ (push @ X172 @ id)) @ id)) = $true)))))) & (pushprop_lthm = $true) & ((induction2lem_lthm = $true) <=> (! [X173 : subst,X174 : term] : (((sub @ one @ (push @ X174 @ X173))) = X174) => (! [X175 : subst,X176 : term] : (((comp @ sh @ (push @ X176 @ X175))) = X175) => (! [X177 : subst] : (((comp @ X177 @ id)) = X177) => (! [X178 : term,X179 : (term > $o)] : (~ ! [X180 : term] : (~ (((var @ X180)) = $true) | ($true = ((X179 @ X180)))) | ~ ! [X181 : term] : (~ (((X179 @ X181)) = $true) | (((X179 @ (lam @ X181))) = $true)) | ($true = ((X179 @ X178))) | ~ ! [X182 : term,X183 : term] : ((((X179 @ (ap @ X182 @ X183))) = $true) | ~ ($true = ((X179 @ X182))) | ~ (((X179 @ X183)) = $true))) => ! [X184 : (term > $o),X185 : term,X186 : subst] : (~ ! [X187 : term,X188 : term] : (~ ($true = ((X184 @ X188))) | ($true = ((X184 @ (ap @ X188 @ X187)))) | ~ ($true = ((X184 @ X187)))) | (((X184 @ (sub @ X185 @ X186))) = $true) | ~ ! [X189 : term] : ((((X184 @ (sub @ X189 @ X186))) = $true) | ~ ($true = ((var @ X189)))) | ~ ! [X190 : term] : (~ ! [X191 : term] : (~ (((X184 @ X191)) = $true) | (((X184 @ (sub @ X190 @ (push @ X191 @ id)))) = $true)) | ($true = ((X184 @ (lam @ X190))))))))))) & (axmap = $true) & (axapp = $true) & (hoasinduction_lem3b_gthm = $true) & (! [X192 : (subst > term > subst > $o),X193 : term,X194 : (term > $o)] : (~ ! [X195 : (subst > term > term)] : (($true = ((X192 @ id @ (lam @ (X195 @ sh @ one)) @ id))) | ~ ! [X196 : term] : (~ (((X192 @ id @ X196 @ id)) = $true) | (((X192 @ id @ (X195 @ id @ X196) @ id)) = $true)) | ~ ! [X197 : subst,X198 : subst,X199 : term] : (((X195 @ (comp @ X198 @ X197) @ (sub @ X199 @ X197))) = ((sub @ (X195 @ X198 @ X199) @ X197)))) | ~ ! [X200 : term] : (($true = ((X194 @ (sub @ X193 @ (push @ X200 @ id))))) | ~ ($true = ((X194 @ X200)))) | ~ ! [X201 : subst,X202 : subst,X203 : subst,X204 : term] : ((((X192 @ X203 @ X204 @ (comp @ X202 @ X201))) = $true) | ~ (((X192 @ (comp @ X203 @ X202) @ (sub @ X204 @ X202) @ X201)) = $true)) | (((X194 @ (lam @ X193))) = $true) | ~ ! [X205 : term] : (((X194 @ X205)) = ((X192 @ id @ X205 @ id))) | ~ ! [X206 : subst,X207 : subst,X208 : subst,X209 : term] : (~ (((X192 @ X206 @ X209 @ (comp @ X208 @ X207))) = $true) | (((X192 @ (comp @ X206 @ X208) @ (sub @ X209 @ X208) @ X207)) = $true))) <=> (hoasinduction_lem3v2 = $true)) & (! [X210 : (term > $o),X211 : term,X212 : (subst > term > subst > $o)] : (~ ! [X213 : term] : (~ ($true = ((X210 @ X213))) | (((X210 @ (sub @ X211 @ (push @ X213 @ id)))) = $true)) | (((X210 @ (lam @ X211))) = $true) | ~ ! [X214 : term] : (((X210 @ X214)) = ((X212 @ id @ X214 @ id))) | ~ ! [X215 : (subst > term > term)] : ((((X212 @ id @ (lam @ (X215 @ sh @ one)) @ id)) = $true) | ~ ! [X216 : term] : (~ (((X212 @ id @ X216 @ id)) = $true) | (((X212 @ id @ (X215 @ id @ X216) @ id)) = $true)) | ~ ! [X217 : subst,X218 : subst,X219 : term] : (((X215 @ (comp @ X217 @ X218) @ (sub @ X219 @ X218))) = ((sub @ (X215 @ X217 @ X219) @ X218))))) <=> (hoasinduction_lem3v2a = $true)) & (pushprop_lem2v2_gthm = $true) & ((axscons = $true) <=> ! [X220 : subst] : (((push @ (sub @ one @ X220) @ (comp @ sh @ X220))) = X220)) & ((hoasinduction_lem3aa = $true) <=> ! [X221 : term,X222 : (subst > term > subst > $o)] : ((((X222 @ id @ (lam @ (sub @ X221 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X223 : term] : ((((X222 @ id @ (sub @ X221 @ (push @ X223 @ id)) @ id)) = $true) | ~ (((X222 @ id @ X223 @ id)) = $true)) | ~ ! [X224 : (subst > term > term)] : (~ ! [X225 : subst,X226 : subst,X227 : term] : (((X224 @ (comp @ X225 @ X226) @ (sub @ X227 @ X226))) = ((sub @ (X224 @ X225 @ X227) @ X226))) | ~ ! [X228 : term] : ((((X222 @ id @ (X224 @ id @ X228) @ id)) = $true) | ~ (((X222 @ id @ X228 @ id)) = $true)) | (((X222 @ id @ (lam @ (X224 @ sh @ one)) @ id)) = $true)))) & ! [X229 : d_subst] : (d_id = X229) & (pushprop_gthm = $true) & (ulamvarind = $true) & ((axvarcons = $true) <=> ! [X230 : term,X231 : subst] : (((sub @ one @ (push @ X230 @ X231))) = X230)) & (hoaslamnotvar = $true) & (hoasinduction_lem1_gthm = $true) & ((induction2lem = $true) <=> ! [X232 : term,X233 : subst,X234 : (term > $o)] : (~ ! [X235 : term] : (~ (((var @ X235)) = $true) | ($true = ((X234 @ (sub @ X235 @ X233))))) | ~ ! [X236 : term,X237 : term] : (~ ($true = ((X234 @ X236))) | ~ ($true = ((X234 @ X237))) | (((X234 @ (ap @ X236 @ X237))) = $true)) | ~ ! [X238 : term] : ((((X234 @ (lam @ X238))) = $true) | ~ ! [X239 : term] : (~ ($true = ((X234 @ X239))) | ($true = ((X234 @ (sub @ X238 @ (push @ X239 @ id))))))) | (((X234 @ (sub @ X232 @ X233))) = $true))) & (! [X240 : term,X241 : term] : ((((sub @ X240 @ sh)) != ((sub @ X241 @ sh))) | (X240 = X241)) <=> (shinj = $true)) & ((hoasinduction_lem0 = $true) <=> ! [X242 : (subst > term > subst > $o)] : ~ ! [X243 : (term > $o)] : ~ ! [X244 : term] : (((X243 @ X244)) = ((X242 @ id @ X244 @ id)))) & ! [X245 : d_term,X246 : d_term] : ((((d2term @ X245)) = ((d2term @ X246))) => (X245 = X246)) & (axabs = $true) & ((axidr = $true) <=> ! [X247 : subst] : (((comp @ X247 @ id)) = X247)) & (pushprop_lthm_orig = $true) & (hoasinduction_lem3a_gthm = $true) & (hoasinduction_lem2_gthm = $true) & (hoaslamnotvar_gthm = $true) & (one = ((d2term @ d_one))) & (! [X248 : term,X249 : (subst > term > subst > $o)] : (($true = ((X249 @ id @ (lam @ X248) @ id))) | ~ ! [X250 : term] : (~ ($true = ((X249 @ id @ X250 @ id))) | (((X249 @ id @ (sub @ X248 @ (push @ X250 @ id)) @ id)) = $true)) | ~ ! [X251 : subst,X252 : subst,X253 : subst,X254 : term] : ((((X249 @ X252 @ X254 @ (comp @ X253 @ X251))) = $true) | ~ ($true = ((X249 @ (comp @ X252 @ X253) @ (sub @ X254 @ X253) @ X251)))) | ~ ! [X255 : (subst > term > term)] : (~ ! [X256 : term] : (~ (((X249 @ id @ X256 @ id)) = $true) | ($true = ((X249 @ id @ (X255 @ id @ X256) @ id)))) | (((X249 @ id @ (lam @ (X255 @ sh @ one)) @ id)) = $true) | ~ ! [X257 : subst,X258 : subst,X259 : term] : (((sub @ (X255 @ X257 @ X259) @ X258)) = ((X255 @ (comp @ X257 @ X258) @ (sub @ X259 @ X258))))) | ~ ! [X260 : subst,X261 : subst,X262 : term,X263 : subst] : (~ (((X249 @ X260 @ X262 @ (comp @ X261 @ X263))) = $true) | ($true = ((X249 @ (comp @ X260 @ X261) @ (sub @ X262 @ X261) @ X263))))) <=> (hoasinduction_lem3 = $true)) & (pushprop_lem2v2 = $true) & (hoasinduction_lem3v2_gthm = $true) & (! [X264 : term,X265 : (subst > term > subst > $o),X266 : term] : (~ (((X265 @ id @ X266 @ id)) = $true) | ~ ! [X267 : term,X268 : subst,X269 : subst,X270 : subst] : (~ ($true = ((X265 @ X269 @ X267 @ (comp @ X268 @ X270)))) | (((X265 @ (comp @ X269 @ X268) @ (sub @ X267 @ X268) @ X270)) = $true)) | ~ (((X265 @ id @ X264 @ id)) = $true) | ~ ! [X271 : term,X272 : term] : (($true = ((X265 @ id @ (ap @ (sub @ X271 @ id) @ X272) @ id))) | ~ (((X265 @ id @ X272 @ id)) = $true) | ~ (((X265 @ id @ X271 @ id)) = $true)) | (((X265 @ id @ (ap @ X266 @ X264) @ id)) = $true) | ~ ! [X273 : subst,X274 : subst,X275 : term,X276 : subst] : (~ ($true = ((X265 @ (comp @ X274 @ X276) @ (sub @ X275 @ X276) @ X273))) | ($true = ((X265 @ X274 @ X275 @ (comp @ X276 @ X273)))))) <=> (hoasinduction_lem2 = $true)) & (pushprop_lem1v2_gthm = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (! [X277 : subst,X278 : (term > $o),X279 : term,X280 : (term > $o)] : (~ ($true = ((X280 @ X279))) | ($true = ((X278 @ one))) | ~ ! [X281 : term] : (((X278 @ X281)) = ((X280 @ (sub @ X281 @ (push @ X279 @ X277)))))) <=> (pushprop_lem1v2 = $true)) & (hoaslaminj_gthm = $true) & (! [X282 : (term > $o),X283 : term,X284 : subst] : ~ ! [X285 : (term > $o)] : ~ ! [X286 : term] : (((X285 @ X286)) = ((X282 @ (sub @ X286 @ (push @ X283 @ X284))))) <=> (pushprop_lem0_lthm = $true)) & (hoasapinj2_gthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (hoasapnotvar = $true) & (! [X287 : term,X288 : (subst > term > subst > $o)] : (~ ! [X289 : term] : ((((X288 @ id @ (sub @ X287 @ (push @ X289 @ id)) @ id)) = $true) | ~ (((X288 @ id @ X289 @ id)) = $true)) | ($true = ((X288 @ id @ (lam @ X287) @ id))) | ~ ! [X290 : (subst > term > term)] : ((((X288 @ id @ (lam @ (X290 @ sh @ one)) @ id)) = $true) | ~ ! [X291 : subst,X292 : subst,X293 : term] : (((X290 @ (comp @ X291 @ X292) @ (sub @ X293 @ X292))) = ((sub @ (X290 @ X291 @ X293) @ X292))) | ~ ! [X294 : term] : ((((X288 @ id @ (X290 @ id @ X294) @ id)) = $true) | ~ (((X288 @ id @ X294 @ id)) = $true)))) <=> (hoasinduction_lem3a = $true)) & ~ (((var @ (d2term @ d_one))) = $true) & ((induction2 = $true) <=> ! [X295 : (term > $o),X296 : term] : (~ ! [X297 : term] : (~ ! [X298 : term] : (~ ($true = ((X295 @ X298))) | ($true = ((X295 @ (sub @ X297 @ (push @ X298 @ id)))))) | ($true = ((X295 @ (lam @ X297))))) | (((X295 @ X296)) = $true) | ~ ! [X299 : term,X300 : term] : (~ (((X295 @ X299)) = $true) | ~ (((X295 @ X300)) = $true) | (((X295 @ (ap @ X300 @ X299))) = $true)) | ~ ! [X301 : term] : (~ (((var @ X301)) = $true) | (((X295 @ X301)) = $true)))) & (! [X302 : (term > $o),X303 : term,X304 : term,X305 : subst,X306 : (term > $o)] : (~ ($true = ((X306 @ X304))) | (((X302 @ (sub @ X304 @ (push @ X303 @ X305)))) = $true)) <=> (pushprop_lem1 = $true)) & ((apinj2 = $true) <=> ! [X307 : term,X308 : term,X309 : term,X310 : term] : ((X309 = X310) | (((ap @ X308 @ X309)) != ((ap @ X307 @ X310))))) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ~ (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) = $true) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X311 : term] : ~ ! [X312 : (subst > term > term)] : ~ ! [X313 : term,X314 : subst] : (((X312 @ X314 @ X313)) = ((sub @ X311 @ (push @ X313 @ X314))))) & (! [X315 : term] : ~ ! [X316 : (subst > term > term)] : (((sub @ X315 @ (push @ one @ sh))) != ((X316 @ sh @ one))) <=> (hoasinduction_lem3b = $true)) & ((! [X317 : term] : ~ ! [X318 : (subst > term > term)] : ~ ! [X319 : term,X320 : subst] : (((sub @ X317 @ (push @ X319 @ X320))) = ((X318 @ X320 @ X319))) => (! [X321 : term] : (((sub @ X321 @ id)) = X321) => ! [X322 : term,X323 : (term > $o),X324 : (subst > term > subst > $o)] : (~ ! [X325 : (subst > term > term)] : (~ ! [X326 : subst,X327 : subst,X328 : term] : (((X325 @ (comp @ X327 @ X326) @ (sub @ X328 @ X326))) = ((sub @ (X325 @ X327 @ X328) @ X326))) | (((X324 @ id @ (lam @ (X325 @ sh @ one)) @ id)) = $true) | ~ ! [X329 : term] : ((((X324 @ id @ (X325 @ id @ X329) @ id)) = $true) | ~ ($true = ((X324 @ id @ X329 @ id))))) | ~ ! [X330 : term] : (((X323 @ X330)) = ((X324 @ id @ X330 @ id))) | ~ ! [X331 : term] : (~ ($true = ((X323 @ X331))) | (((X323 @ (sub @ X322 @ (push @ X331 @ id)))) = $true)) | (((X323 @ (lam @ X322))) = $true)))) <=> (hoasinduction_lem3v2a_lthm = $true)) & (hoaslamnotap_lthm = $true) & (hoasapnotvar_lthm = $true) & (! [X332 : term] : (((sub @ X332 @ id)) = X332) <=> (termmset = $true)) & ((hoasinduction_lem3v2_lthm = $true) <=> (! [X333 : term] : (((sub @ X333 @ id)) = X333) => ! [X334 : term,X335 : (subst > term > subst > $o),X336 : (term > $o)] : (~ ! [X337 : term] : (~ ($true = ((X336 @ X337))) | (((X336 @ (sub @ X334 @ (push @ X337 @ id)))) = $true)) | ~ ! [X338 : (subst > term > term)] : (~ ! [X339 : subst,X340 : term,X341 : subst] : (((X338 @ (comp @ X341 @ X339) @ (sub @ X340 @ X339))) = ((sub @ (X338 @ X341 @ X340) @ X339))) | ~ ! [X342 : term] : (~ ($true = ((X335 @ id @ X342 @ id))) | (((X335 @ id @ (X338 @ id @ X342) @ id)) = $true)) | (((X335 @ id @ (lam @ (X338 @ sh @ one)) @ id)) = $true)) | (((X336 @ (lam @ X334))) = $true) | ~ ! [X343 : term] : (((X336 @ X343)) = ((X335 @ id @ X343 @ id))) | ~ ! [X344 : subst,X345 : subst,X346 : subst,X347 : term] : (($true = ((X335 @ (comp @ X344 @ X346) @ (sub @ X347 @ X346) @ X345))) | ~ (((X335 @ X344 @ X347 @ (comp @ X346 @ X345))) = $true)) | ~ ! [X348 : subst,X349 : term,X350 : subst,X351 : subst] : (~ (((X335 @ (comp @ X350 @ X348) @ (sub @ X349 @ X348) @ X351)) = $true) | ($true = ((X335 @ X350 @ X349 @ (comp @ X348 @ X351)))))))) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X352 : term] : ~ ! [X353 : (subst > term > term)] : (((sub @ X352 @ (push @ one @ sh))) != ((X353 @ sh @ one)))) & (hoasinduction_lem3_gthm = $true) & (pushprop_lem3v2_lthm = $true) & ! [X354 : d_term] : (d_one = X354) & (hoaslamnotap_gthm = $true) & (! [X355 : term] : (((sub @ X355 @ id)) = X355) <=> (axvarid = $true)) & ! [X356 : subst] : ? [X357 : d_subst] : (((d2subst @ X357)) = X356) & ((pushprop_lem1v2_lthm = $true) <=> (! [X358 : term,X359 : subst] : (((sub @ one @ (push @ X358 @ X359))) = X358) => ! [X360 : (term > $o),X361 : (term > $o),X362 : term,X363 : subst] : (~ ($true = ((X361 @ X362))) | (((X360 @ one)) = $true) | ~ ! [X364 : term] : (((X361 @ (sub @ X364 @ (push @ X362 @ X363)))) = ((X360 @ X364)))))) & (axvarshift = $true) & (! [X365 : (subst > term > subst > $o)] : ~ ! [X366 : (term > $o)] : ~ ! [X367 : term] : (((X366 @ X367)) = ((X365 @ id @ X367 @ id))) <=> (hoasinduction_lem0_lthm = $true)) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & ((hoasinduction_lem3_lthm = $true) <=> (! [X373 : term] : (((sub @ X373 @ id)) = X373) => (! [X374 : (subst > term > subst > $o),X375 : term] : (($true = ((X374 @ id @ (lam @ (sub @ X375 @ (push @ one @ sh))) @ id))) | ~ ! [X376 : term] : (~ (((X374 @ id @ X376 @ id)) = $true) | (((X374 @ id @ (sub @ X375 @ (push @ X376 @ id)) @ id)) = $true)) | ~ ! [X377 : (subst > term > term)] : (~ ! [X378 : term,X379 : subst,X380 : subst] : (((X377 @ (comp @ X379 @ X380) @ (sub @ X378 @ X380))) = ((sub @ (X377 @ X379 @ X378) @ X380))) | ~ ! [X381 : term] : (~ ($true = ((X374 @ id @ X381 @ id))) | (((X374 @ id @ (X377 @ id @ X381) @ id)) = $true)) | (((X374 @ id @ (lam @ (X377 @ sh @ one)) @ id)) = $true))) => ! [X382 : (subst > term > subst > $o),X383 : term] : (~ ! [X384 : subst,X385 : subst,X386 : subst,X387 : term] : (($true = ((X382 @ (comp @ X386 @ X384) @ (sub @ X387 @ X384) @ X385))) | ~ (((X382 @ X386 @ X387 @ (comp @ X384 @ X385))) = $true)) | (((X382 @ id @ (lam @ X383) @ id)) = $true) | ~ ! [X388 : term] : (~ (((X382 @ id @ X388 @ id)) = $true) | (((X382 @ id @ (sub @ X383 @ (push @ X388 @ id)) @ id)) = $true)) | ~ ! [X389 : subst,X390 : subst,X391 : subst,X392 : term] : (($true = ((X382 @ X390 @ X392 @ (comp @ X391 @ X389)))) | ~ (((X382 @ (comp @ X390 @ X391) @ (sub @ X392 @ X391) @ X389)) = $true)) | ~ ! [X393 : (subst > term > term)] : (~ ! [X394 : term,X395 : subst,X396 : subst] : (((X393 @ (comp @ X396 @ X395) @ (sub @ X394 @ X395))) = ((sub @ (X393 @ X396 @ X394) @ X395))) | ~ ! [X397 : term] : (~ (((X382 @ id @ X397 @ id)) = $true) | (((X382 @ id @ (X393 @ id @ X397) @ id)) = $true)) | (((X382 @ id @ (lam @ (X393 @ sh @ one)) @ id)) = $true)))))) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (! [X398 : subst,X399 : term] : (((comp @ sh @ (push @ X399 @ X398))) = X398) <=> (axshiftcons = $true)) & ((hoasinduction = $true) <=> ! [X400 : (subst > term > subst > $o),X401 : term] : ((((X400 @ id @ X401 @ id)) = $true) | ~ ! [X402 : subst,X403 : subst,X404 : term,X405 : subst] : (~ ($true = ((X400 @ (comp @ X403 @ X402) @ (sub @ X404 @ X402) @ X405))) | ($true = ((X400 @ X403 @ X404 @ (comp @ X402 @ X405))))) | ~ ! [X406 : subst,X407 : term,X408 : subst,X409 : subst] : (($true = ((X400 @ (comp @ X406 @ X408) @ (sub @ X407 @ X408) @ X409))) | ~ (((X400 @ X406 @ X407 @ (comp @ X408 @ X409))) = $true)) | ~ ! [X410 : (subst > term > term)] : (~ ! [X411 : term] : ((((X400 @ id @ (X410 @ id @ X411) @ id)) = $true) | ~ (((X400 @ id @ X411 @ id)) = $true)) | ~ ! [X412 : subst,X413 : term,X414 : subst] : (((sub @ (X410 @ X414 @ X413) @ X412)) = ((X410 @ (comp @ X414 @ X412) @ (sub @ X413 @ X412)))) | (((X400 @ id @ (lam @ (X410 @ sh @ one)) @ id)) = $true)) | ~ ! [X415 : term,X416 : term] : ((((X400 @ id @ (ap @ (sub @ X416 @ id) @ X415) @ id)) = $true) | ~ (((X400 @ id @ X415 @ id)) = $true) | ~ ($true = ((X400 @ id @ X416 @ id)))) | ~ ! [X417 : term] : ((((X400 @ id @ X417 @ id)) = $true) | ~ (((var @ (sub @ X417 @ id))) = $true)))) & (pushprop_lem3v2 = $true) & (! [X418 : term] : ~ ! [X419 : (subst > term > term)] : ~ ! [X420 : subst,X421 : term] : (((X419 @ X420 @ X421)) = ((sub @ X418 @ (push @ X421 @ X420)))) <=> (hoasinduction_lem3v2_f = $true)) & (hoasinduction_lem1v2 = $true) & ! [X422 : d_subst,X423 : d_subst] : ((((d2subst @ X423)) = ((d2subst @ X422))) => (X422 = X423)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ~ (ulamvar1 = $true) & ((hoasapinj1_lthm = $true) <=> (! [X426 : term] : (((sub @ X426 @ id)) = X426) => (! [X427 : term,X428 : term,X429 : term,X430 : term] : ((X427 = X428) | (((ap @ X427 @ X430)) != ((ap @ X428 @ X429)))) => ! [X431 : term,X432 : term,X433 : term,X434 : term] : ((X431 = X434) | (((ap @ (sub @ X431 @ id) @ X432)) != ((ap @ (sub @ X434 @ id) @ X433))))))) & ((! [X435 : term] : (((sub @ X435 @ id)) = X435) => (! [X436 : subst,X437 : term,X438 : (term > $o)] : (~ ! [X439 : term] : (~ ! [X440 : term] : (~ (((X438 @ X440)) = $true) | ($true = ((X438 @ (sub @ X439 @ (push @ X440 @ id)))))) | (((X438 @ (lam @ X439))) = $true)) | ~ ! [X441 : term] : (($true = ((X438 @ (sub @ X441 @ X436)))) | ~ (((var @ X441)) = $true)) | ~ ! [X442 : term,X443 : term] : (($true = ((X438 @ (ap @ X443 @ X442)))) | ~ (((X438 @ X443)) = $true) | ~ (((X438 @ X442)) = $true)) | ($true = ((X438 @ (sub @ X437 @ X436))))) => ! [X444 : (term > $o),X445 : term] : (~ ! [X446 : term,X447 : term] : (~ (((X444 @ X446)) = $true) | ~ ($true = ((X444 @ X447))) | (((X444 @ (ap @ X446 @ X447))) = $true)) | (((X444 @ X445)) = $true) | ~ ! [X448 : term] : (~ ($true = ((var @ X448))) | ($true = ((X444 @ X448)))) | ~ ! [X449 : term] : (~ ! [X450 : term] : ((((X444 @ (sub @ X449 @ (push @ X450 @ id)))) = $true) | ~ ($true = ((X444 @ X450)))) | ($true = ((X444 @ (lam @ X449)))))))) <=> (induction2_lthm = $true)) & (((d2subst @ d_id)) = sh) & (axassoc = $true) & (! [X451 : subst] : (((comp @ id @ X451)) = X451) <=> (axidl = $true)) & (pushprop_lem0_gthm = $true) & (hoasinduction_lem2v2_gthm = $true) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X452 : (subst > term > subst > $o),X453 : term] : (~ ! [X454 : term] : ((((X452 @ id @ (sub @ X453 @ (push @ X454 @ id)) @ id)) = $true) | ~ (((X452 @ id @ X454 @ id)) = $true)) | ~ ! [X455 : (subst > term > term)] : (~ ! [X456 : term] : (~ ($true = ((X452 @ id @ X456 @ id))) | ($true = ((X452 @ id @ (X455 @ id @ X456) @ id)))) | (((X452 @ id @ (lam @ (X455 @ sh @ one)) @ id)) = $true) | ~ ! [X457 : subst,X458 : subst,X459 : term] : (((X455 @ (comp @ X458 @ X457) @ (sub @ X459 @ X457))) = ((sub @ (X455 @ X458 @ X459) @ X457)))) | (((X452 @ id @ (lam @ (sub @ X453 @ (push @ one @ sh))) @ id)) = $true))) & (hoasapinj1_gthm = $true) & (hoasinduction_lem1_lthm = $true) & (((d2subst @ d_id)) = id) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (! [X460 : (subst > term > subst > $o)] : ~ ! [X461 : (term > $o)] : ~ ! [X462 : term] : (((X461 @ X462)) = ((X460 @ id @ X462 @ id))) => (! [X463 : (term > $o),X464 : term] : (~ ! [X465 : term] : ((((X463 @ X465)) = $true) | ~ (((var @ X465)) = $true)) | ~ ! [X466 : term,X467 : term] : (~ (((X463 @ X467)) = $true) | ~ (((X463 @ X466)) = $true) | ($true = ((X463 @ (ap @ X466 @ X467))))) | ~ ! [X468 : term] : (~ ! [X469 : term] : ((((X463 @ (sub @ X468 @ (push @ X469 @ id)))) = $true) | ~ ($true = ((X463 @ X469)))) | (((X463 @ (lam @ X468))) = $true)) | (((X463 @ X464)) = $true)) => (! [X470 : term] : (((sub @ X470 @ id)) = X470) => (! [X471 : term,X472 : (term > $o),X473 : (subst > term > subst > $o)] : (~ ! [X474 : term] : (~ (((X472 @ X474)) = $true) | (((X472 @ (sub @ X471 @ (push @ X474 @ id)))) = $true)) | ~ ! [X475 : (subst > term > term)] : (~ ! [X476 : term,X477 : subst,X478 : subst] : (((sub @ (X475 @ X477 @ X476) @ X478)) = ((X475 @ (comp @ X477 @ X478) @ (sub @ X476 @ X478)))) | (((X473 @ id @ (lam @ (X475 @ sh @ one)) @ id)) = $true) | ~ ! [X479 : term] : ((((X473 @ id @ (X475 @ id @ X479) @ id)) = $true) | ~ (((X473 @ id @ X479 @ id)) = $true))) | (((X472 @ (lam @ X471))) = $true) | ~ ! [X480 : term] : (((X473 @ id @ X480 @ id)) = ((X472 @ X480)))) => ! [X481 : (subst > term > subst > $o),X482 : term] : (~ ! [X483 : (subst > term > term)] : (~ ! [X484 : term] : (~ ($true = ((X481 @ id @ X484 @ id))) | (((X481 @ id @ (X483 @ id @ X484) @ id)) = $true)) | ~ ! [X485 : subst,X486 : subst,X487 : term] : (((X483 @ (comp @ X486 @ X485) @ (sub @ X487 @ X485))) = ((sub @ (X483 @ X486 @ X487) @ X485))) | (((X481 @ id @ (lam @ (X483 @ sh @ one)) @ id)) = $true)) | ($true = ((X481 @ id @ X482 @ id))) | ~ ! [X488 : term,X489 : term] : (~ (((X481 @ id @ X489 @ id)) = $true) | ($true = ((X481 @ id @ (ap @ (sub @ X489 @ id) @ X488) @ id))) | ~ (((X481 @ id @ X488 @ id)) = $true)))))))) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((hoaslaminj = $true) <=> ! [X490 : (subst > term > term),X491 : subst,X492 : term,X493 : (subst > term > term)] : ((((X493 @ X491 @ X492)) = ((X490 @ X491 @ X492))) | ~ ! [X494 : subst,X495 : subst,X496 : term] : (((X490 @ (comp @ X494 @ X495) @ (sub @ X496 @ X495))) = ((sub @ (X490 @ X494 @ X496) @ X495))) | (((lam @ (X490 @ sh @ one))) != ((lam @ (X493 @ sh @ one)))) | ~ ! [X497 : subst,X498 : term,X499 : subst] : (((sub @ (X493 @ X497 @ X498) @ X499)) = ((X493 @ (comp @ X497 @ X499) @ (sub @ X498 @ X499)))))) & (! [X500 : term,X501 : term,X502 : term,X503 : term] : ((X500 = X502) | (((ap @ (sub @ X502 @ id) @ X503)) != ((ap @ (sub @ X500 @ id) @ X501)))) <=> (hoasapinj1 = $true)) & (lamnotvar = $true) & (ulamvarsh = $true) & (substmonoid_gthm = $true) & ! [X504 : term] : ? [X505 : d_term] : (((d2term @ X505)) = X504) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ((! [X506 : (subst > term > subst > $o)] : ~ ! [X507 : (term > $o)] : ~ ! [X508 : term] : (((X506 @ id @ X508 @ id)) = ((X507 @ X508))) => (! [X509 : (term > $o),X510 : term] : (($true = ((X509 @ X510))) | ~ ! [X511 : term] : (~ (((var @ X511)) = $true) | (((X509 @ X511)) = $true)) | ~ ! [X512 : term] : ((((X509 @ (lam @ X512))) = $true) | ~ ! [X513 : term] : (~ (((X509 @ X513)) = $true) | ($true = ((X509 @ (sub @ X512 @ (push @ X513 @ id))))))) | ~ ! [X514 : term,X515 : term] : (~ (((X509 @ X515)) = $true) | ($true = ((X509 @ (ap @ X514 @ X515)))) | ~ ($true = ((X509 @ X514))))) => (! [X516 : term] : (((sub @ X516 @ id)) = X516) => (! [X517 : (term > $o),X518 : term,X519 : (subst > term > subst > $o)] : (~ ! [X520 : term] : (((X519 @ id @ X520 @ id)) = ((X517 @ X520))) | (((X517 @ (lam @ X518))) = $true) | ~ ! [X521 : term] : (($true = ((X517 @ (sub @ X518 @ (push @ X521 @ id))))) | ~ (((X517 @ X521)) = $true)) | ~ ! [X522 : (subst > term > term)] : (~ ! [X523 : term,X524 : subst,X525 : subst] : (((sub @ (X522 @ X524 @ X523) @ X525)) = ((X522 @ (comp @ X524 @ X525) @ (sub @ X523 @ X525)))) | (((X519 @ id @ (lam @ (X522 @ sh @ one)) @ id)) = $true) | ~ ! [X526 : term] : (($true = ((X519 @ id @ (X522 @ id @ X526) @ id))) | ~ (((X519 @ id @ X526 @ id)) = $true)))) => ! [X527 : (subst > term > subst > $o),X528 : term] : (~ ! [X529 : term,X530 : term] : (~ (((X527 @ id @ X530 @ id)) = $true) | (((X527 @ id @ (ap @ (sub @ X529 @ id) @ X530) @ id)) = $true) | ~ (((X527 @ id @ X529 @ id)) = $true)) | ($true = ((X527 @ id @ X528 @ id))) | ~ ! [X531 : term,X532 : subst,X533 : subst,X534 : subst] : (~ (((X527 @ (comp @ X532 @ X534) @ (sub @ X531 @ X534) @ X533)) = $true) | ($true = ((X527 @ X532 @ X531 @ (comp @ X534 @ X533))))) | ~ ! [X535 : (subst > term > term)] : (~ ! [X536 : term,X537 : subst,X538 : subst] : (((X535 @ (comp @ X538 @ X537) @ (sub @ X536 @ X537))) = ((sub @ (X535 @ X538 @ X536) @ X537))) | ~ ! [X539 : term] : ((((X527 @ id @ (X535 @ id @ X539) @ id)) = $true) | ~ (((X527 @ id @ X539 @ id)) = $true)) | ($true = ((X527 @ id @ (lam @ (X535 @ sh @ one)) @ id)))) | ~ ! [X540 : subst,X541 : term,X542 : subst,X543 : subst] : ((((X527 @ (comp @ X542 @ X543) @ (sub @ X541 @ X543) @ X540)) = $true) | ~ (((X527 @ X542 @ X541 @ (comp @ X543 @ X540))) = $true)) | ~ ! [X544 : term] : (~ (((var @ (sub @ X544 @ id))) = $true) | (((X527 @ id @ X544 @ id)) = $true))))))) <=> (hoasinduction_lthm_3 = $true)) & ((pushprop_lem0 = $true) <=> ! [X545 : (term > $o),X546 : term,X547 : subst] : ~ ! [X548 : (term > $o)] : ~ ! [X549 : term] : (((X545 @ (sub @ X549 @ (push @ X546 @ X547)))) = ((X548 @ X549)))) & (hoasinduction_lem1 = $true)),
  inference(fool_elimination,[],[f4])).
thf(f6,plain,(
  ~ (pushprop_lem0 = ! [X0 : (term > $o),X1 : term,X2 : subst] : ? [X3 : (term > $o)] : (pushprop_p_and_p_prime @ X1 @ X2 @ X0 @ X3))),
  inference(rectify,[],[f2])).
thf(f7,plain,(
  ~ (pushprop_lem0 = ((!! @ subst @ (^[Y0 : subst]: (!! @ term @ (^[Y1 : term]: (!! @ (term > $o) @ (^[Y2 : term > $o]: (?? @ (term > $o) @ (^[Y3 : term > $o]: (pushprop_p_and_p_prime @ Y1 @ Y0 @ Y2 @ Y3)))))))))))),
  inference(fool_elimination,[],[f6])).
thf(f8,plain,(
  (pushprop_lem0 != ((!! @ subst @ (^[Y0 : subst]: (!! @ term @ (^[Y1 : term]: (!! @ (term > $o) @ (^[Y2 : term > $o]: (?? @ (term > $o) @ (^[Y3 : term > $o]: (pushprop_p_and_p_prime @ Y1 @ Y0 @ Y2 @ Y3)))))))))))),
  inference(flattening,[],[f7])).
thf(f9,plain,(
  (hoasinduction_lem1v2_gthm = $true) & (hoaslamnotvar_lthm = $true) & ~ (lamnotap = $true) & (hoasapnotvar_gthm = $true) & (induction2_gthm = $true) & (pushprop_lem1_gthm = $true) & ((substmonoid_lthm = $true) <=> (! [X0 : subst] : (((comp @ id @ X0)) = X0) => (! [X1 : subst] : (((comp @ X1 @ id)) = X1) => (! [X3 : subst] : (((comp @ X3 @ id)) = X3) & ! [X2 : subst] : (((comp @ id @ X2)) = X2))))) & (pushprop_lem2v2_lthm = $true) & (hoasinduction_lem2_lthm = $true) & ((hoasinduction_lthm = $true) <=> (! [X4 : (term > $o),X5 : term] : (~ ! [X6 : term,X7 : term] : (~ ($true = ((X4 @ X6))) | (((X4 @ (ap @ X7 @ X6))) = $true) | ~ (((X4 @ X7)) = $true)) | (((X4 @ X5)) = $true) | ~ ! [X8 : term] : (~ ($true = ((var @ X8))) | (((X4 @ X8)) = $true)) | ~ ! [X9 : term] : (~ ! [X10 : term] : (($true = ((X4 @ (sub @ X9 @ (push @ X10 @ id))))) | ~ (((X4 @ X10)) = $true)) | ($true = ((X4 @ (lam @ X9)))))) => (! [X11 : term,X12 : (subst > term > subst > $o),X13 : term] : (~ (((X12 @ id @ X11 @ id)) = $true) | ~ ! [X14 : term,X15 : subst,X16 : subst,X17 : subst] : (~ (((X12 @ (comp @ X16 @ X17) @ (sub @ X14 @ X17) @ X15)) = $true) | ($true = ((X12 @ X16 @ X14 @ (comp @ X17 @ X15))))) | (((X12 @ id @ (ap @ X11 @ X13) @ id)) = $true) | ~ ! [X18 : term,X19 : term] : (~ (((X12 @ id @ X19 @ id)) = $true) | (((X12 @ id @ (ap @ (sub @ X19 @ id) @ X18) @ id)) = $true) | ~ (((X12 @ id @ X18 @ id)) = $true)) | ~ ! [X20 : subst,X21 : subst,X22 : term,X23 : subst] : ((((X12 @ (comp @ X20 @ X23) @ (sub @ X22 @ X23) @ X21)) = $true) | ~ ($true = ((X12 @ X20 @ X22 @ (comp @ X23 @ X21))))) | ~ ($true = ((X12 @ id @ X13 @ id)))) => (! [X24 : (subst > term > subst > $o),X25 : term] : (~ ! [X26 : subst,X27 : subst,X28 : subst,X29 : term] : ((((X24 @ (comp @ X27 @ X26) @ (sub @ X29 @ X26) @ X28)) = $true) | ~ ($true = ((X24 @ X27 @ X29 @ (comp @ X26 @ X28))))) | (((X24 @ id @ (lam @ X25) @ id)) = $true) | ~ ! [X30 : (subst > term > term)] : ((((X24 @ id @ (lam @ (X30 @ sh @ one)) @ id)) = $true) | ~ ! [X31 : term] : ((((X24 @ id @ (X30 @ id @ X31) @ id)) = $true) | ~ (((X24 @ id @ X31 @ id)) = $true)) | ~ ! [X33 : subst,X34 : subst,X32 : term] : (((sub @ (X30 @ X33 @ X32) @ X34)) = ((X30 @ (comp @ X33 @ X34) @ (sub @ X32 @ X34))))) | ~ ! [X35 : term] : (~ (((X24 @ id @ X35 @ id)) = $true) | (((X24 @ id @ (sub @ X25 @ (push @ X35 @ id)) @ id)) = $true)) | ~ ! [X36 : subst,X37 : term,X38 : subst,X39 : subst] : (($true = ((X24 @ X39 @ X37 @ (comp @ X38 @ X36)))) | ~ (((X24 @ (comp @ X39 @ X38) @ (sub @ X37 @ X38) @ X36)) = $true))) => ! [X40 : term,X41 : (subst > term > subst > $o)] : (~ ! [X42 : term,X43 : term] : (~ ($true = ((X41 @ id @ X43 @ id))) | ~ ($true = ((X41 @ id @ X42 @ id))) | ($true = ((X41 @ id @ (ap @ (sub @ X43 @ id) @ X42) @ id)))) | (((X41 @ id @ X40 @ id)) = $true) | ~ ! [X44 : term] : ((((X41 @ id @ X44 @ id)) = $true) | ~ (((var @ (sub @ X44 @ id))) = $true)) | ~ ! [X45 : (subst > term > term)] : ((((X41 @ id @ (lam @ (X45 @ sh @ one)) @ id)) = $true) | ~ ! [X46 : term] : (~ (((X41 @ id @ X46 @ id)) = $true) | ($true = ((X41 @ id @ (X45 @ id @ X46) @ id)))) | ~ ! [X48 : term,X49 : subst,X47 : subst] : (((sub @ (X45 @ X49 @ X48) @ X47)) = ((X45 @ (comp @ X49 @ X47) @ (sub @ X48 @ X47))))) | ~ ! [X50 : subst,X51 : subst,X52 : term,X53 : subst] : (($true = ((X41 @ X51 @ X52 @ (comp @ X50 @ X53)))) | ~ (((X41 @ (comp @ X51 @ X50) @ (sub @ X52 @ X50) @ X53)) = $true)) | ~ ! [X54 : subst,X55 : subst,X56 : subst,X57 : term] : (~ ($true = ((X41 @ X56 @ X57 @ (comp @ X55 @ X54)))) | (((X41 @ (comp @ X56 @ X55) @ (sub @ X57 @ X55) @ X54)) = $true))))))) & ((hoasinduction_lem3aaa = $true) <=> ! [X58 : term,X59 : (subst > term > subst > $o)] : (~ ! [X60 : term,X61 : (subst > term > term)] : (~ ! [X64 : subst,X63 : subst,X62 : term] : (((sub @ (X61 @ X64 @ X62) @ X63)) = ((sub @ (sub @ X60 @ (push @ X62 @ X64)) @ X63))) | ~ ! [X65 : term] : (($true = ((X59 @ id @ (X61 @ id @ X65) @ id))) | ~ ($true = ((X59 @ id @ X65 @ id)))) | ($true = ((X59 @ id @ (lam @ (X61 @ sh @ one)) @ id))) | ~ ! [X68 : subst,X67 : subst,X66 : term] : (((X61 @ (comp @ X67 @ X68) @ (sub @ X66 @ X68))) = ((sub @ X60 @ (push @ (sub @ X66 @ X68) @ (comp @ X67 @ X68)))))) | (((X59 @ id @ (lam @ (sub @ X58 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X69 : term] : ((((X59 @ id @ (sub @ X58 @ (push @ X69 @ id)) @ id)) = $true) | ~ (((X59 @ id @ X69 @ id)) = $true)))) & (! [X72 : term,X71 : term,X70 : term,X73 : term] : ((((ap @ X71 @ X73)) != ((ap @ X70 @ X72))) | (X70 = X71)) <=> (apinj1 = $true)) & ((hoasapinj2_lthm = $true) <=> (! [X76 : term,X77 : term,X75 : term,X74 : term] : ((X76 = X77) | (((ap @ X75 @ X77)) != ((ap @ X74 @ X76)))) => ! [X80 : term,X81 : term,X78 : term,X79 : term] : ((X79 = X81) | (((ap @ (sub @ X80 @ id) @ X79)) != ((ap @ (sub @ X78 @ id) @ X81)))))) & (! [X84 : term,X82 : (subst > term > term),X83 : term] : ((((ap @ (sub @ X83 @ id) @ X84)) != ((lam @ (X82 @ sh @ one)))) | ~ ! [X85 : term,X86 : subst,X87 : subst] : (((X82 @ (comp @ X87 @ X86) @ (sub @ X85 @ X86))) = ((sub @ (X82 @ X87 @ X85) @ X86)))) <=> (hoaslamnotap = $true)) & ((induction = $true) <=> ! [X88 : term,X89 : (term > $o)] : (~ ! [X90 : term,X91 : term] : (~ ($true = ((X89 @ X91))) | ~ (((X89 @ X90)) = $true) | ($true = ((X89 @ (ap @ X90 @ X91))))) | ~ ! [X92 : term] : (($true = ((X89 @ (lam @ X92)))) | ~ ($true = ((X89 @ X92)))) | (((X89 @ X88)) = $true) | ~ ! [X93 : term] : ((((X89 @ X93)) = $true) | ~ (((var @ X93)) = $true)))) & ((hoasinduction_lem2v2 = $true) <=> ! [X94 : (subst > term > subst > $o),X95 : term,X96 : (term > $o),X97 : term] : (~ (((X96 @ X95)) = $true) | ~ ! [X98 : term] : (((X94 @ id @ X98 @ id)) = ((X96 @ X98))) | ~ ! [X99 : term,X100 : term] : (~ (((X94 @ id @ X100 @ id)) = $true) | (((X94 @ id @ (ap @ (sub @ X99 @ id) @ X100) @ id)) = $true) | ~ (((X94 @ id @ X99 @ id)) = $true)) | ~ ! [X101 : subst,X102 : subst,X103 : term,X104 : subst] : (($true = ((X94 @ (comp @ X102 @ X101) @ (sub @ X103 @ X101) @ X104))) | ~ (((X94 @ X102 @ X103 @ (comp @ X101 @ X104))) = $true)) | ~ ! [X105 : subst,X106 : term,X107 : subst,X108 : subst] : (~ (((X94 @ (comp @ X107 @ X105) @ (sub @ X106 @ X105) @ X108)) = $true) | (((X94 @ X107 @ X106 @ (comp @ X105 @ X108))) = $true)) | ~ (((X96 @ X97)) = $true) | ($true = ((X96 @ (ap @ X97 @ X95)))))) & ((hoaslaminj_lthm = $true) <=> (! [X109 : subst,X110 : term] : (((sub @ one @ (push @ X110 @ X109))) = X110) => (! [X112 : term,X111 : subst] : (((comp @ sh @ (push @ X112 @ X111))) = X111) => (! [X113 : term,X114 : term] : ((((lam @ X113)) != ((lam @ X114))) | (X113 = X114)) => ! [X117 : (subst > term > term),X118 : term,X115 : (subst > term > term),X116 : subst] : ((((lam @ (X115 @ sh @ one))) != ((lam @ (X117 @ sh @ one)))) | ~ ! [X122 : subst,X124 : subst,X123 : term] : (((X117 @ (comp @ X124 @ X122) @ (sub @ X123 @ X122))) = ((sub @ (X117 @ X124 @ X123) @ X122))) | ~ ! [X121 : subst,X119 : term,X120 : subst] : (((sub @ (X115 @ X121 @ X119) @ X120)) = ((X115 @ (comp @ X121 @ X120) @ (sub @ X119 @ X120)))) | (((X117 @ X116 @ X118)) = ((X115 @ X116 @ X118)))))))) & (apnotvar = $true) & (! [X128 : term,X125 : term,X127 : term,X126 : term] : ((((ap @ (sub @ X127 @ id) @ X128)) != ((ap @ (sub @ X126 @ id) @ X125))) | (X125 = X128)) <=> (hoasapinj2 = $true)) & ((pushprop_lem1_lthm = $true) <=> (! [X130 : subst,X129 : term] : (((sub @ one @ (push @ X129 @ X130))) = X129) => (! [X132 : subst,X131 : term] : (((comp @ sh @ (push @ X131 @ X132))) = X132) => ! [X133 : term,X134 : (term > $o),X135 : (term > $o),X136 : term,X137 : subst] : (~ ($true = ((X134 @ X136))) | (((X135 @ (sub @ X136 @ (push @ X133 @ X137)))) = $true))))) & (pushprop = $true) & ((! [X138 : term] : (((sub @ X138 @ id)) = X138) => ! [X139 : term] : (((sub @ X139 @ id)) = X139)) <=> (termmset_lthm = $true)) & ((! [X141 : subst] : (((comp @ id @ X141)) = X141) & ! [X140 : subst] : (((comp @ X140 @ id)) = X140)) <=> (substmonoid = $true)) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & ((hoasinduction_no_psi_cond = $true) <=> ! [X142 : term,X143 : (subst > term > subst > $o)] : (~ ! [X144 : term,X145 : term] : (~ (((X143 @ id @ X145 @ id)) = $true) | (((X143 @ id @ (ap @ (sub @ X144 @ id) @ X145) @ id)) = $true) | ~ (((X143 @ id @ X144 @ id)) = $true)) | (((X143 @ id @ X142 @ id)) = $true) | ~ ! [X146 : (subst > term > term)] : ((((X143 @ id @ (lam @ (X146 @ sh @ one)) @ id)) = $true) | ~ ! [X148 : term,X147 : subst,X149 : subst] : (((sub @ (X146 @ X149 @ X148) @ X147)) = ((X146 @ (comp @ X149 @ X147) @ (sub @ X148 @ X147)))) | ~ ! [X150 : term] : ((((X143 @ id @ (X146 @ id @ X150) @ id)) = $true) | ~ ($true = ((X143 @ id @ X150 @ id))))))) & (! [X151 : term,X152 : term] : ((X151 = X152) | (((lam @ X151)) != ((lam @ X152)))) <=> (laminj = $true)) & (axclos = $true) & (induction2lem_gthm = $true) & (termmset_gthm = $true) & (hoasinduction_gthm = $true) & ((hoasinduction_lem3a_lthm = $true) <=> (! [X153 : term] : (((sub @ X153 @ id)) = X153) => (! [X154 : (subst > term > subst > $o),X155 : term] : (~ ! [X156 : (subst > term > term)] : (~ ! [X157 : term] : (($true = ((X154 @ id @ (X156 @ id @ X157) @ id))) | ~ ($true = ((X154 @ id @ X157 @ id)))) | (((X154 @ id @ (lam @ (X156 @ sh @ one)) @ id)) = $true) | ~ ! [X160 : subst,X158 : term,X159 : subst] : (((X156 @ (comp @ X159 @ X160) @ (sub @ X158 @ X160))) = ((sub @ (X156 @ X159 @ X158) @ X160)))) | (((X154 @ id @ (lam @ (sub @ X155 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X161 : term] : (~ (((X154 @ id @ X161 @ id)) = $true) | (((X154 @ id @ (sub @ X155 @ (push @ X161 @ id)) @ id)) = $true))) => ! [X162 : (subst > term > subst > $o),X163 : term] : (~ ! [X164 : (subst > term > term)] : (~ ! [X165 : term] : (~ ($true = ((X162 @ id @ X165 @ id))) | (((X162 @ id @ (X164 @ id @ X165) @ id)) = $true)) | (((X162 @ id @ (lam @ (X164 @ sh @ one)) @ id)) = $true) | ~ ! [X167 : subst,X168 : term,X166 : subst] : (((sub @ (X164 @ X167 @ X168) @ X166)) = ((X164 @ (comp @ X167 @ X166) @ (sub @ X168 @ X166))))) | (((X162 @ id @ (lam @ X163) @ id)) = $true) | ~ ! [X169 : term] : (~ ($true = ((X162 @ id @ X169 @ id))) | (((X162 @ id @ (sub @ X163 @ (push @ X169 @ id)) @ id)) = $true)))))) & (pushprop_lthm = $true) & ((induction2lem_lthm = $true) <=> (! [X170 : subst,X171 : term] : (((sub @ one @ (push @ X171 @ X170))) = X171) => (! [X172 : subst,X173 : term] : (((comp @ sh @ (push @ X173 @ X172))) = X172) => (! [X174 : subst] : (((comp @ X174 @ id)) = X174) => (! [X175 : term,X176 : (term > $o)] : (~ ! [X177 : term] : (~ (((var @ X177)) = $true) | (((X176 @ X177)) = $true)) | ~ ! [X178 : term] : (~ (((X176 @ X178)) = $true) | ($true = ((X176 @ (lam @ X178))))) | (((X176 @ X175)) = $true) | ~ ! [X179 : term,X180 : term] : ((((X176 @ (ap @ X179 @ X180))) = $true) | ~ ($true = ((X176 @ X179))) | ~ ($true = ((X176 @ X180))))) => ! [X181 : (term > $o),X182 : term,X183 : subst] : (~ ! [X184 : term,X185 : term] : (~ (((X181 @ X185)) = $true) | ($true = ((X181 @ (ap @ X185 @ X184)))) | ~ (((X181 @ X184)) = $true)) | ($true = ((X181 @ (sub @ X182 @ X183)))) | ~ ! [X186 : term] : ((((X181 @ (sub @ X186 @ X183))) = $true) | ~ (((var @ X186)) = $true)) | ~ ! [X187 : term] : (~ ! [X188 : term] : (~ (((X181 @ X188)) = $true) | (((X181 @ (sub @ X187 @ (push @ X188 @ id)))) = $true)) | (((X181 @ (lam @ X187))) = $true)))))))) & (axmap = $true) & (axapp = $true) & (hoasinduction_lem3b_gthm = $true) & (! [X189 : (subst > term > subst > $o),X190 : term,X191 : (term > $o)] : (~ ! [X192 : (subst > term > term)] : (($true = ((X189 @ id @ (lam @ (X192 @ sh @ one)) @ id))) | ~ ! [X193 : term] : (~ (((X189 @ id @ X193 @ id)) = $true) | ($true = ((X189 @ id @ (X192 @ id @ X193) @ id)))) | ~ ! [X195 : subst,X196 : term,X194 : subst] : (((X192 @ (comp @ X195 @ X194) @ (sub @ X196 @ X194))) = ((sub @ (X192 @ X195 @ X196) @ X194)))) | ~ ! [X197 : term] : (($true = ((X191 @ (sub @ X190 @ (push @ X197 @ id))))) | ~ ($true = ((X191 @ X197)))) | ~ ! [X198 : subst,X199 : subst,X200 : subst,X201 : term] : ((((X189 @ X200 @ X201 @ (comp @ X199 @ X198))) = $true) | ~ ($true = ((X189 @ (comp @ X200 @ X199) @ (sub @ X201 @ X199) @ X198)))) | (((X191 @ (lam @ X190))) = $true) | ~ ! [X202 : term] : (((X189 @ id @ X202 @ id)) = ((X191 @ X202))) | ~ ! [X203 : subst,X204 : subst,X205 : subst,X206 : term] : (~ (((X189 @ X203 @ X206 @ (comp @ X205 @ X204))) = $true) | (((X189 @ (comp @ X203 @ X205) @ (sub @ X206 @ X205) @ X204)) = $true))) <=> (hoasinduction_lem3v2 = $true)) & (! [X207 : (term > $o),X208 : term,X209 : (subst > term > subst > $o)] : (~ ! [X210 : term] : (~ (((X207 @ X210)) = $true) | (((X207 @ (sub @ X208 @ (push @ X210 @ id)))) = $true)) | (((X207 @ (lam @ X208))) = $true) | ~ ! [X211 : term] : (((X209 @ id @ X211 @ id)) = ((X207 @ X211))) | ~ ! [X212 : (subst > term > term)] : ((((X209 @ id @ (lam @ (X212 @ sh @ one)) @ id)) = $true) | ~ ! [X213 : term] : (~ (((X209 @ id @ X213 @ id)) = $true) | (((X209 @ id @ (X212 @ id @ X213) @ id)) = $true)) | ~ ! [X215 : subst,X214 : subst,X216 : term] : (((X212 @ (comp @ X214 @ X215) @ (sub @ X216 @ X215))) = ((sub @ (X212 @ X214 @ X216) @ X215))))) <=> (hoasinduction_lem3v2a = $true)) & (pushprop_lem2v2_gthm = $true) & ((axscons = $true) <=> ! [X217 : subst] : (((push @ (sub @ one @ X217) @ (comp @ sh @ X217))) = X217)) & ((hoasinduction_lem3aa = $true) <=> ! [X218 : term,X219 : (subst > term > subst > $o)] : ((((X219 @ id @ (lam @ (sub @ X218 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X220 : term] : ((((X219 @ id @ (sub @ X218 @ (push @ X220 @ id)) @ id)) = $true) | ~ ($true = ((X219 @ id @ X220 @ id)))) | ~ ! [X221 : (subst > term > term)] : (~ ! [X223 : subst,X222 : subst,X224 : term] : (((X221 @ (comp @ X222 @ X223) @ (sub @ X224 @ X223))) = ((sub @ (X221 @ X222 @ X224) @ X223))) | ~ ! [X225 : term] : ((((X219 @ id @ (X221 @ id @ X225) @ id)) = $true) | ~ ($true = ((X219 @ id @ X225 @ id)))) | (((X219 @ id @ (lam @ (X221 @ sh @ one)) @ id)) = $true)))) & ! [X226 : d_subst] : (d_id = X226) & (pushprop_gthm = $true) & (ulamvarind = $true) & (! [X227 : term,X228 : subst] : (((sub @ one @ (push @ X227 @ X228))) = X227) <=> (axvarcons = $true)) & (hoaslamnotvar = $true) & (hoasinduction_lem1_gthm = $true) & ((induction2lem = $true) <=> ! [X229 : term,X230 : subst,X231 : (term > $o)] : (~ ! [X232 : term] : (~ (((var @ X232)) = $true) | (((X231 @ (sub @ X232 @ X230))) = $true)) | ~ ! [X233 : term,X234 : term] : (~ ($true = ((X231 @ X233))) | ~ (((X231 @ X234)) = $true) | (((X231 @ (ap @ X233 @ X234))) = $true)) | ~ ! [X235 : term] : ((((X231 @ (lam @ X235))) = $true) | ~ ! [X236 : term] : (~ (((X231 @ X236)) = $true) | (((X231 @ (sub @ X235 @ (push @ X236 @ id)))) = $true))) | ($true = ((X231 @ (sub @ X229 @ X230)))))) & (! [X238 : term,X237 : term] : ((X237 = X238) | (((sub @ X237 @ sh)) != ((sub @ X238 @ sh)))) <=> (shinj = $true)) & (! [X239 : (subst > term > subst > $o)] : ~ ! [X240 : (term > $o)] : ~ ! [X241 : term] : (((X240 @ X241)) = ((X239 @ id @ X241 @ id))) <=> (hoasinduction_lem0 = $true)) & ! [X242 : d_term,X243 : d_term] : ((((d2term @ X242)) = ((d2term @ X243))) => (X242 = X243)) & (axabs = $true) & ((axidr = $true) <=> ! [X244 : subst] : (((comp @ X244 @ id)) = X244)) & (pushprop_lthm_orig = $true) & (hoasinduction_lem3a_gthm = $true) & (hoasinduction_lem2_gthm = $true) & (hoaslamnotvar_gthm = $true) & (one = ((d2term @ d_one))) & (! [X245 : term,X246 : (subst > term > subst > $o)] : ((((X246 @ id @ (lam @ X245) @ id)) = $true) | ~ ! [X247 : term] : (~ ($true = ((X246 @ id @ X247 @ id))) | ($true = ((X246 @ id @ (sub @ X245 @ (push @ X247 @ id)) @ id)))) | ~ ! [X248 : subst,X249 : subst,X250 : subst,X251 : term] : (($true = ((X246 @ X249 @ X251 @ (comp @ X250 @ X248)))) | ~ ($true = ((X246 @ (comp @ X249 @ X250) @ (sub @ X251 @ X250) @ X248)))) | ~ ! [X252 : (subst > term > term)] : (~ ! [X253 : term] : (~ (((X246 @ id @ X253 @ id)) = $true) | ($true = ((X246 @ id @ (X252 @ id @ X253) @ id)))) | ($true = ((X246 @ id @ (lam @ (X252 @ sh @ one)) @ id))) | ~ ! [X255 : subst,X254 : subst,X256 : term] : (((X252 @ (comp @ X254 @ X255) @ (sub @ X256 @ X255))) = ((sub @ (X252 @ X254 @ X256) @ X255)))) | ~ ! [X257 : subst,X258 : subst,X259 : term,X260 : subst] : (~ (((X246 @ X257 @ X259 @ (comp @ X258 @ X260))) = $true) | (((X246 @ (comp @ X257 @ X258) @ (sub @ X259 @ X258) @ X260)) = $true))) <=> (hoasinduction_lem3 = $true)) & (pushprop_lem2v2 = $true) & (hoasinduction_lem3v2_gthm = $true) & (! [X261 : term,X262 : (subst > term > subst > $o),X263 : term] : (~ ($true = ((X262 @ id @ X263 @ id))) | ~ ! [X264 : term,X265 : subst,X266 : subst,X267 : subst] : (~ (((X262 @ X266 @ X264 @ (comp @ X265 @ X267))) = $true) | (((X262 @ (comp @ X266 @ X265) @ (sub @ X264 @ X265) @ X267)) = $true)) | ~ (((X262 @ id @ X261 @ id)) = $true) | ~ ! [X268 : term,X269 : term] : (($true = ((X262 @ id @ (ap @ (sub @ X268 @ id) @ X269) @ id))) | ~ ($true = ((X262 @ id @ X269 @ id))) | ~ ($true = ((X262 @ id @ X268 @ id)))) | ($true = ((X262 @ id @ (ap @ X263 @ X261) @ id))) | ~ ! [X270 : subst,X271 : subst,X272 : term,X273 : subst] : (~ (((X262 @ (comp @ X271 @ X273) @ (sub @ X272 @ X273) @ X270)) = $true) | (((X262 @ X271 @ X272 @ (comp @ X273 @ X270))) = $true))) <=> (hoasinduction_lem2 = $true)) & (pushprop_lem1v2_gthm = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (! [X274 : subst,X275 : (term > $o),X276 : term,X277 : (term > $o)] : (~ (((X277 @ X276)) = $true) | (((X275 @ one)) = $true) | ~ ! [X278 : term] : (((X275 @ X278)) = ((X277 @ (sub @ X278 @ (push @ X276 @ X274)))))) <=> (pushprop_lem1v2 = $true)) & (hoaslaminj_gthm = $true) & (! [X281 : subst,X279 : (term > $o),X280 : term] : ~ ! [X282 : (term > $o)] : ~ ! [X283 : term] : (((X279 @ (sub @ X283 @ (push @ X280 @ X281)))) = ((X282 @ X283))) <=> (pushprop_lem0_lthm = $true)) & (hoasapinj2_gthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (hoasapnotvar = $true) & (! [X284 : term,X285 : (subst > term > subst > $o)] : (~ ! [X286 : term] : ((((X285 @ id @ (sub @ X284 @ (push @ X286 @ id)) @ id)) = $true) | ~ (((X285 @ id @ X286 @ id)) = $true)) | ($true = ((X285 @ id @ (lam @ X284) @ id))) | ~ ! [X287 : (subst > term > term)] : (($true = ((X285 @ id @ (lam @ (X287 @ sh @ one)) @ id))) | ~ ! [X290 : term,X288 : subst,X289 : subst] : (((X287 @ (comp @ X288 @ X289) @ (sub @ X290 @ X289))) = ((sub @ (X287 @ X288 @ X290) @ X289))) | ~ ! [X291 : term] : (($true = ((X285 @ id @ (X287 @ id @ X291) @ id))) | ~ (((X285 @ id @ X291 @ id)) = $true)))) <=> (hoasinduction_lem3a = $true)) & ~ (((var @ (d2term @ d_one))) = $true) & ((induction2 = $true) <=> ! [X292 : (term > $o),X293 : term] : (~ ! [X294 : term] : (~ ! [X295 : term] : (~ (((X292 @ X295)) = $true) | (((X292 @ (sub @ X294 @ (push @ X295 @ id)))) = $true)) | (((X292 @ (lam @ X294))) = $true)) | (((X292 @ X293)) = $true) | ~ ! [X296 : term,X297 : term] : (~ ($true = ((X292 @ X296))) | ~ (((X292 @ X297)) = $true) | ($true = ((X292 @ (ap @ X297 @ X296))))) | ~ ! [X298 : term] : (~ ($true = ((var @ X298))) | (((X292 @ X298)) = $true)))) & (! [X299 : (term > $o),X300 : term,X301 : term,X302 : subst,X303 : (term > $o)] : (~ (((X303 @ X301)) = $true) | (((X299 @ (sub @ X301 @ (push @ X300 @ X302)))) = $true)) <=> (pushprop_lem1 = $true)) & (! [X304 : term,X307 : term,X305 : term,X306 : term] : ((X306 = X307) | (((ap @ X304 @ X307)) != ((ap @ X305 @ X306)))) <=> (apinj2 = $true)) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ~ (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) = $true) & (! [X308 : term] : ~ ! [X309 : (subst > term > term)] : ~ ! [X311 : subst,X310 : term] : (((X309 @ X311 @ X310)) = ((sub @ X308 @ (push @ X310 @ X311)))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & (! [X312 : term] : ~ ! [X313 : (subst > term > term)] : (((X313 @ sh @ one)) != ((sub @ X312 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b = $true)) & ((! [X314 : term] : ~ ! [X315 : (subst > term > term)] : ~ ! [X317 : subst,X316 : term] : (((sub @ X314 @ (push @ X316 @ X317))) = ((X315 @ X317 @ X316))) => (! [X318 : term] : (((sub @ X318 @ id)) = X318) => ! [X319 : term,X320 : (term > $o),X321 : (subst > term > subst > $o)] : (~ ! [X322 : (subst > term > term)] : (~ ! [X323 : subst,X324 : subst,X325 : term] : (((sub @ (X322 @ X324 @ X325) @ X323)) = ((X322 @ (comp @ X324 @ X323) @ (sub @ X325 @ X323)))) | (((X321 @ id @ (lam @ (X322 @ sh @ one)) @ id)) = $true) | ~ ! [X326 : term] : (($true = ((X321 @ id @ (X322 @ id @ X326) @ id))) | ~ (((X321 @ id @ X326 @ id)) = $true))) | ~ ! [X327 : term] : (((X320 @ X327)) = ((X321 @ id @ X327 @ id))) | ~ ! [X328 : term] : (~ (((X320 @ X328)) = $true) | (((X320 @ (sub @ X319 @ (push @ X328 @ id)))) = $true)) | (((X320 @ (lam @ X319))) = $true)))) <=> (hoasinduction_lem3v2a_lthm = $true)) & (hoaslamnotap_lthm = $true) & (hoasapnotvar_lthm = $true) & ((termmset = $true) <=> ! [X329 : term] : (((sub @ X329 @ id)) = X329)) & ((hoasinduction_lem3v2_lthm = $true) <=> (! [X330 : term] : (((sub @ X330 @ id)) = X330) => ! [X331 : term,X332 : (subst > term > subst > $o),X333 : (term > $o)] : (~ ! [X334 : term] : (~ (((X333 @ X334)) = $true) | ($true = ((X333 @ (sub @ X331 @ (push @ X334 @ id)))))) | ~ ! [X335 : (subst > term > term)] : (~ ! [X338 : subst,X336 : subst,X337 : term] : (((sub @ (X335 @ X338 @ X337) @ X336)) = ((X335 @ (comp @ X338 @ X336) @ (sub @ X337 @ X336)))) | ~ ! [X339 : term] : (~ (((X332 @ id @ X339 @ id)) = $true) | (((X332 @ id @ (X335 @ id @ X339) @ id)) = $true)) | ($true = ((X332 @ id @ (lam @ (X335 @ sh @ one)) @ id)))) | (((X333 @ (lam @ X331))) = $true) | ~ ! [X340 : term] : (((X333 @ X340)) = ((X332 @ id @ X340 @ id))) | ~ ! [X341 : subst,X342 : subst,X343 : subst,X344 : term] : (($true = ((X332 @ (comp @ X341 @ X343) @ (sub @ X344 @ X343) @ X342))) | ~ (((X332 @ X341 @ X344 @ (comp @ X343 @ X342))) = $true)) | ~ ! [X345 : subst,X346 : term,X347 : subst,X348 : subst] : (~ (((X332 @ (comp @ X347 @ X345) @ (sub @ X346 @ X345) @ X348)) = $true) | ($true = ((X332 @ X347 @ X346 @ (comp @ X345 @ X348)))))))) & (! [X349 : term] : ~ ! [X350 : (subst > term > term)] : (((X350 @ sh @ one)) != ((sub @ X349 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b_lthm = $true)) & (hoasinduction_lem3_gthm = $true) & (pushprop_lem3v2_lthm = $true) & ! [X351 : d_term] : (d_one = X351) & (hoaslamnotap_gthm = $true) & (! [X352 : term] : (((sub @ X352 @ id)) = X352) <=> (axvarid = $true)) & ! [X353 : subst] : ? [X354 : d_subst] : (((d2subst @ X354)) = X353) & ((pushprop_lem1v2_lthm = $true) <=> (! [X355 : term,X356 : subst] : (((sub @ one @ (push @ X355 @ X356))) = X355) => ! [X357 : (term > $o),X358 : (term > $o),X359 : term,X360 : subst] : (~ (((X358 @ X359)) = $true) | (((X357 @ one)) = $true) | ~ ! [X361 : term] : (((X358 @ (sub @ X361 @ (push @ X359 @ X360)))) = ((X357 @ X361)))))) & (axvarshift = $true) & (! [X362 : (subst > term > subst > $o)] : ~ ! [X363 : (term > $o)] : ~ ! [X364 : term] : (((X362 @ id @ X364 @ id)) = ((X363 @ X364))) <=> (hoasinduction_lem0_lthm = $true)) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & ((hoasinduction_lem3_lthm = $true) <=> (! [X365 : term] : (((sub @ X365 @ id)) = X365) => (! [X366 : (subst > term > subst > $o),X367 : term] : ((((X366 @ id @ (lam @ (sub @ X367 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X368 : term] : (~ (((X366 @ id @ X368 @ id)) = $true) | ($true = ((X366 @ id @ (sub @ X367 @ (push @ X368 @ id)) @ id)))) | ~ ! [X369 : (subst > term > term)] : (~ ! [X371 : subst,X370 : term,X372 : subst] : (((X369 @ (comp @ X371 @ X372) @ (sub @ X370 @ X372))) = ((sub @ (X369 @ X371 @ X370) @ X372))) | ~ ! [X373 : term] : (~ ($true = ((X366 @ id @ X373 @ id))) | (((X366 @ id @ (X369 @ id @ X373) @ id)) = $true)) | (((X366 @ id @ (lam @ (X369 @ sh @ one)) @ id)) = $true))) => ! [X374 : (subst > term > subst > $o),X375 : term] : (~ ! [X376 : subst,X377 : subst,X378 : subst,X379 : term] : (($true = ((X374 @ (comp @ X378 @ X376) @ (sub @ X379 @ X376) @ X377))) | ~ (((X374 @ X378 @ X379 @ (comp @ X376 @ X377))) = $true)) | ($true = ((X374 @ id @ (lam @ X375) @ id))) | ~ ! [X380 : term] : (~ (((X374 @ id @ X380 @ id)) = $true) | (((X374 @ id @ (sub @ X375 @ (push @ X380 @ id)) @ id)) = $true)) | ~ ! [X381 : subst,X382 : subst,X383 : subst,X384 : term] : (($true = ((X374 @ X382 @ X384 @ (comp @ X383 @ X381)))) | ~ ($true = ((X374 @ (comp @ X382 @ X383) @ (sub @ X384 @ X383) @ X381)))) | ~ ! [X385 : (subst > term > term)] : (~ ! [X388 : subst,X387 : subst,X386 : term] : (((X385 @ (comp @ X388 @ X387) @ (sub @ X386 @ X387))) = ((sub @ (X385 @ X388 @ X386) @ X387))) | ~ ! [X389 : term] : (~ (((X374 @ id @ X389 @ id)) = $true) | (((X374 @ id @ (X385 @ id @ X389) @ id)) = $true)) | (((X374 @ id @ (lam @ (X385 @ sh @ one)) @ id)) = $true)))))) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (! [X391 : term,X390 : subst] : (((comp @ sh @ (push @ X391 @ X390))) = X390) <=> (axshiftcons = $true)) & ((hoasinduction = $true) <=> ! [X392 : (subst > term > subst > $o),X393 : term] : (($true = ((X392 @ id @ X393 @ id))) | ~ ! [X394 : subst,X395 : subst,X396 : term,X397 : subst] : (~ ($true = ((X392 @ (comp @ X395 @ X394) @ (sub @ X396 @ X394) @ X397))) | ($true = ((X392 @ X395 @ X396 @ (comp @ X394 @ X397))))) | ~ ! [X398 : subst,X399 : term,X400 : subst,X401 : subst] : ((((X392 @ (comp @ X398 @ X400) @ (sub @ X399 @ X400) @ X401)) = $true) | ~ (((X392 @ X398 @ X399 @ (comp @ X400 @ X401))) = $true)) | ~ ! [X402 : (subst > term > term)] : (~ ! [X403 : term] : ((((X392 @ id @ (X402 @ id @ X403) @ id)) = $true) | ~ ($true = ((X392 @ id @ X403 @ id)))) | ~ ! [X405 : term,X404 : subst,X406 : subst] : (((sub @ (X402 @ X406 @ X405) @ X404)) = ((X402 @ (comp @ X406 @ X404) @ (sub @ X405 @ X404)))) | (((X392 @ id @ (lam @ (X402 @ sh @ one)) @ id)) = $true)) | ~ ! [X407 : term,X408 : term] : ((((X392 @ id @ (ap @ (sub @ X408 @ id) @ X407) @ id)) = $true) | ~ (((X392 @ id @ X407 @ id)) = $true) | ~ (((X392 @ id @ X408 @ id)) = $true)) | ~ ! [X409 : term] : (($true = ((X392 @ id @ X409 @ id))) | ~ (((var @ (sub @ X409 @ id))) = $true)))) & (pushprop_lem3v2 = $true) & ((hoasinduction_lem3v2_f = $true) <=> ! [X410 : term] : ~ ! [X411 : (subst > term > term)] : ~ ! [X413 : term,X412 : subst] : (((sub @ X410 @ (push @ X413 @ X412))) = ((X411 @ X412 @ X413)))) & (hoasinduction_lem1v2 = $true) & ! [X415 : d_subst,X414 : d_subst] : ((((d2subst @ X415)) = ((d2subst @ X414))) => (X414 = X415)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ~ (ulamvar1 = $true) & ((hoasapinj1_lthm = $true) <=> (! [X416 : term] : (((sub @ X416 @ id)) = X416) => (! [X420 : term,X417 : term,X419 : term,X418 : term] : ((X417 = X418) | (((ap @ X418 @ X419)) != ((ap @ X417 @ X420)))) => ! [X423 : term,X424 : term,X422 : term,X421 : term] : ((((ap @ (sub @ X421 @ id) @ X422)) != ((ap @ (sub @ X424 @ id) @ X423))) | (X421 = X424))))) & ((! [X425 : term] : (((sub @ X425 @ id)) = X425) => (! [X426 : subst,X427 : term,X428 : (term > $o)] : (~ ! [X429 : term] : (~ ! [X430 : term] : (~ (((X428 @ X430)) = $true) | ($true = ((X428 @ (sub @ X429 @ (push @ X430 @ id)))))) | (((X428 @ (lam @ X429))) = $true)) | ~ ! [X431 : term] : ((((X428 @ (sub @ X431 @ X426))) = $true) | ~ ($true = ((var @ X431)))) | ~ ! [X432 : term,X433 : term] : (($true = ((X428 @ (ap @ X433 @ X432)))) | ~ (((X428 @ X433)) = $true) | ~ ($true = ((X428 @ X432)))) | ($true = ((X428 @ (sub @ X427 @ X426))))) => ! [X434 : (term > $o),X435 : term] : (~ ! [X436 : term,X437 : term] : (~ (((X434 @ X436)) = $true) | ~ (((X434 @ X437)) = $true) | (((X434 @ (ap @ X436 @ X437))) = $true)) | (((X434 @ X435)) = $true) | ~ ! [X438 : term] : (~ (((var @ X438)) = $true) | (((X434 @ X438)) = $true)) | ~ ! [X439 : term] : (~ ! [X440 : term] : ((((X434 @ (sub @ X439 @ (push @ X440 @ id)))) = $true) | ~ ($true = ((X434 @ X440)))) | (((X434 @ (lam @ X439))) = $true))))) <=> (induction2_lthm = $true)) & (((d2subst @ d_id)) = sh) & (axassoc = $true) & (! [X441 : subst] : (((comp @ id @ X441)) = X441) <=> (axidl = $true)) & (pushprop_lem0_gthm = $true) & (hoasinduction_lem2v2_gthm = $true) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X442 : (subst > term > subst > $o),X443 : term] : (~ ! [X444 : term] : (($true = ((X442 @ id @ (sub @ X443 @ (push @ X444 @ id)) @ id))) | ~ (((X442 @ id @ X444 @ id)) = $true)) | ~ ! [X445 : (subst > term > term)] : (~ ! [X446 : term] : (~ (((X442 @ id @ X446 @ id)) = $true) | (((X442 @ id @ (X445 @ id @ X446) @ id)) = $true)) | ($true = ((X442 @ id @ (lam @ (X445 @ sh @ one)) @ id))) | ~ ! [X449 : term,X448 : subst,X447 : subst] : (((sub @ (X445 @ X448 @ X449) @ X447)) = ((X445 @ (comp @ X448 @ X447) @ (sub @ X449 @ X447))))) | (((X442 @ id @ (lam @ (sub @ X443 @ (push @ one @ sh))) @ id)) = $true))) & (hoasapinj1_gthm = $true) & (hoasinduction_lem1_lthm = $true) & (((d2subst @ d_id)) = id) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (! [X450 : (subst > term > subst > $o)] : ~ ! [X451 : (term > $o)] : ~ ! [X452 : term] : (((X451 @ X452)) = ((X450 @ id @ X452 @ id))) => (! [X453 : (term > $o),X454 : term] : (~ ! [X455 : term] : (($true = ((X453 @ X455))) | ~ (((var @ X455)) = $true)) | ~ ! [X456 : term,X457 : term] : (~ (((X453 @ X457)) = $true) | ~ (((X453 @ X456)) = $true) | (((X453 @ (ap @ X456 @ X457))) = $true)) | ~ ! [X458 : term] : (~ ! [X459 : term] : ((((X453 @ (sub @ X458 @ (push @ X459 @ id)))) = $true) | ~ (((X453 @ X459)) = $true)) | (((X453 @ (lam @ X458))) = $true)) | ($true = ((X453 @ X454)))) => (! [X460 : term] : (((sub @ X460 @ id)) = X460) => (! [X461 : term,X462 : (term > $o),X463 : (subst > term > subst > $o)] : (~ ! [X464 : term] : (~ (((X462 @ X464)) = $true) | (((X462 @ (sub @ X461 @ (push @ X464 @ id)))) = $true)) | ~ ! [X465 : (subst > term > term)] : (~ ! [X467 : subst,X468 : subst,X466 : term] : (((sub @ (X465 @ X467 @ X466) @ X468)) = ((X465 @ (comp @ X467 @ X468) @ (sub @ X466 @ X468)))) | (((X463 @ id @ (lam @ (X465 @ sh @ one)) @ id)) = $true) | ~ ! [X469 : term] : (($true = ((X463 @ id @ (X465 @ id @ X469) @ id))) | ~ (((X463 @ id @ X469 @ id)) = $true))) | (((X462 @ (lam @ X461))) = $true) | ~ ! [X470 : term] : (((X462 @ X470)) = ((X463 @ id @ X470 @ id)))) => ! [X471 : (subst > term > subst > $o),X472 : term] : (~ ! [X473 : (subst > term > term)] : (~ ! [X474 : term] : (~ (((X471 @ id @ X474 @ id)) = $true) | ($true = ((X471 @ id @ (X473 @ id @ X474) @ id)))) | ~ ! [X475 : subst,X476 : subst,X477 : term] : (((X473 @ (comp @ X476 @ X475) @ (sub @ X477 @ X475))) = ((sub @ (X473 @ X476 @ X477) @ X475))) | (((X471 @ id @ (lam @ (X473 @ sh @ one)) @ id)) = $true)) | ($true = ((X471 @ id @ X472 @ id))) | ~ ! [X478 : term,X479 : term] : (~ ($true = ((X471 @ id @ X479 @ id))) | (((X471 @ id @ (ap @ (sub @ X479 @ id) @ X478) @ id)) = $true) | ~ ($true = ((X471 @ id @ X478 @ id)))))))))) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (! [X482 : term,X480 : (subst > term > term),X481 : subst,X483 : (subst > term > term)] : (~ ! [X486 : term,X484 : subst,X485 : subst] : (((sub @ (X480 @ X484 @ X486) @ X485)) = ((X480 @ (comp @ X484 @ X485) @ (sub @ X486 @ X485)))) | ~ ! [X488 : term,X487 : subst,X489 : subst] : (((X483 @ (comp @ X487 @ X489) @ (sub @ X488 @ X489))) = ((sub @ (X483 @ X487 @ X488) @ X489))) | (((X480 @ X481 @ X482)) = ((X483 @ X481 @ X482))) | (((lam @ (X483 @ sh @ one))) != ((lam @ (X480 @ sh @ one))))) <=> (hoaslaminj = $true)) & (! [X492 : term,X491 : term,X493 : term,X490 : term] : ((X490 = X492) | (((ap @ (sub @ X490 @ id) @ X491)) != ((ap @ (sub @ X492 @ id) @ X493)))) <=> (hoasapinj1 = $true)) & (lamnotvar = $true) & (ulamvarsh = $true) & (substmonoid_gthm = $true) & ! [X494 : term] : ? [X495 : d_term] : (((d2term @ X495)) = X494) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ((! [X496 : (subst > term > subst > $o)] : ~ ! [X497 : (term > $o)] : ~ ! [X498 : term] : (((X496 @ id @ X498 @ id)) = ((X497 @ X498))) => (! [X499 : (term > $o),X500 : term] : (($true = ((X499 @ X500))) | ~ ! [X501 : term] : (~ (((var @ X501)) = $true) | (((X499 @ X501)) = $true)) | ~ ! [X502 : term] : ((((X499 @ (lam @ X502))) = $true) | ~ ! [X503 : term] : (~ ($true = ((X499 @ X503))) | ($true = ((X499 @ (sub @ X502 @ (push @ X503 @ id))))))) | ~ ! [X504 : term,X505 : term] : (~ ($true = ((X499 @ X505))) | ($true = ((X499 @ (ap @ X504 @ X505)))) | ~ (((X499 @ X504)) = $true))) => (! [X506 : term] : (((sub @ X506 @ id)) = X506) => (! [X507 : (term > $o),X508 : term,X509 : (subst > term > subst > $o)] : (~ ! [X510 : term] : (((X507 @ X510)) = ((X509 @ id @ X510 @ id))) | (((X507 @ (lam @ X508))) = $true) | ~ ! [X511 : term] : (($true = ((X507 @ (sub @ X508 @ (push @ X511 @ id))))) | ~ (((X507 @ X511)) = $true)) | ~ ! [X512 : (subst > term > term)] : (~ ! [X513 : term,X515 : subst,X514 : subst] : (((sub @ (X512 @ X514 @ X513) @ X515)) = ((X512 @ (comp @ X514 @ X515) @ (sub @ X513 @ X515)))) | ($true = ((X509 @ id @ (lam @ (X512 @ sh @ one)) @ id))) | ~ ! [X516 : term] : ((((X509 @ id @ (X512 @ id @ X516) @ id)) = $true) | ~ (((X509 @ id @ X516 @ id)) = $true)))) => ! [X517 : (subst > term > subst > $o),X518 : term] : (~ ! [X519 : term,X520 : term] : (~ ($true = ((X517 @ id @ X520 @ id))) | (((X517 @ id @ (ap @ (sub @ X519 @ id) @ X520) @ id)) = $true) | ~ (((X517 @ id @ X519 @ id)) = $true)) | (((X517 @ id @ X518 @ id)) = $true) | ~ ! [X521 : term,X522 : subst,X523 : subst,X524 : subst] : (~ ($true = ((X517 @ (comp @ X522 @ X524) @ (sub @ X521 @ X524) @ X523))) | ($true = ((X517 @ X522 @ X521 @ (comp @ X524 @ X523))))) | ~ ! [X525 : (subst > term > term)] : (~ ! [X528 : subst,X527 : subst,X526 : term] : (((X525 @ (comp @ X528 @ X527) @ (sub @ X526 @ X527))) = ((sub @ (X525 @ X528 @ X526) @ X527))) | ~ ! [X529 : term] : ((((X517 @ id @ (X525 @ id @ X529) @ id)) = $true) | ~ ($true = ((X517 @ id @ X529 @ id)))) | ($true = ((X517 @ id @ (lam @ (X525 @ sh @ one)) @ id)))) | ~ ! [X530 : subst,X531 : term,X532 : subst,X533 : subst] : (($true = ((X517 @ (comp @ X532 @ X533) @ (sub @ X531 @ X533) @ X530))) | ~ (((X517 @ X532 @ X531 @ (comp @ X533 @ X530))) = $true)) | ~ ! [X534 : term] : (~ (((var @ (sub @ X534 @ id))) = $true) | ($true = ((X517 @ id @ X534 @ id))))))))) <=> (hoasinduction_lthm_3 = $true)) & (! [X537 : subst,X536 : term,X535 : (term > $o)] : ~ ! [X538 : (term > $o)] : ~ ! [X539 : term] : (((X538 @ X539)) = ((X535 @ (sub @ X539 @ (push @ X536 @ X537))))) <=> (pushprop_lem0 = $true)) & (hoasinduction_lem1 = $true)),
  inference(rectify,[],[f5])).
thf(f10,plain,(
  (pushprop_lem2v2 = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (one = ((d2term @ d_one))) & ((hoaslaminj_lthm = $true) <=> (! [X109 : subst,X110 : term] : (((sub @ one @ (push @ X110 @ X109))) = X110) => (! [X112 : term,X111 : subst] : (((comp @ sh @ (push @ X112 @ X111))) = X111) => (! [X113 : term,X114 : term] : ((((lam @ X113)) != ((lam @ X114))) | (X113 = X114)) => ! [X117 : (subst > term > term),X118 : term,X115 : (subst > term > term),X116 : subst] : ((((lam @ (X115 @ sh @ one))) != ((lam @ (X117 @ sh @ one)))) | ~ ! [X122 : subst,X124 : subst,X123 : term] : (((X117 @ (comp @ X124 @ X122) @ (sub @ X123 @ X122))) = ((sub @ (X117 @ X124 @ X123) @ X122))) | ~ ! [X121 : subst,X119 : term,X120 : subst] : (((sub @ (X115 @ X121 @ X119) @ X120)) = ((X115 @ (comp @ X121 @ X120) @ (sub @ X119 @ X120)))) | (((X117 @ X116 @ X118)) = ((X115 @ X116 @ X118)))))))) & (pushprop_lem1_gthm = $true) & (pushprop_lem2v2_gthm = $true) & (apnotvar = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (hoasinduction_lem1v2 = $true) & (! [X391 : term,X390 : subst] : (((comp @ sh @ (push @ X391 @ X390))) = X390) <=> (axshiftcons = $true)) & (hoasinduction_lem2_lthm = $true) & (hoaslamnotap_lthm = $true) & (! [X59 : (subst > term > subst > $o),X58 : term] : (~ ! [X69 : term] : ((((X59 @ id @ (sub @ X58 @ (push @ X69 @ id)) @ id)) = $true) | (((X59 @ id @ X69 @ id)) != $true)) | ~ ! [X60 : term,X61 : (subst > term > term)] : (~ ! [X65 : term] : (($true != ((X59 @ id @ X65 @ id))) | ($true = ((X59 @ id @ (X61 @ id @ X65) @ id)))) | ~ ! [X68 : subst,X67 : subst,X66 : term] : (((X61 @ (comp @ X67 @ X68) @ (sub @ X66 @ X68))) = ((sub @ X60 @ (push @ (sub @ X66 @ X68) @ (comp @ X67 @ X68))))) | ($true = ((X59 @ id @ (lam @ (X61 @ sh @ one)) @ id))) | ~ ! [X64 : subst,X63 : subst,X62 : term] : (((sub @ (X61 @ X64 @ X62) @ X63)) = ((sub @ (sub @ X60 @ (push @ X62 @ X64)) @ X63)))) | (((X59 @ id @ (lam @ (sub @ X58 @ (push @ one @ sh))) @ id)) = $true)) <=> (hoasinduction_lem3aaa = $true)) & (((d2subst @ d_id)) = sh) & ((! [X153 : term] : (((sub @ X153 @ id)) = X153) => (! [X155 : term,X154 : (subst > term > subst > $o)] : (~ ! [X161 : term] : ((((X154 @ id @ X161 @ id)) != $true) | (((X154 @ id @ (sub @ X155 @ (push @ X161 @ id)) @ id)) = $true)) | (((X154 @ id @ (lam @ (sub @ X155 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X156 : (subst > term > term)] : (~ ! [X157 : term] : (($true = ((X154 @ id @ (X156 @ id @ X157) @ id))) | ($true != ((X154 @ id @ X157 @ id)))) | ~ ! [X160 : subst,X158 : term,X159 : subst] : (((X156 @ (comp @ X159 @ X160) @ (sub @ X158 @ X160))) = ((sub @ (X156 @ X159 @ X158) @ X160))) | (((X154 @ id @ (lam @ (X156 @ sh @ one)) @ id)) = $true))) => ! [X163 : term,X162 : (subst > term > subst > $o)] : (~ ! [X169 : term] : ((((X162 @ id @ (sub @ X163 @ (push @ X169 @ id)) @ id)) = $true) | ($true != ((X162 @ id @ X169 @ id)))) | ~ ! [X164 : (subst > term > term)] : (~ ! [X165 : term] : (($true != ((X162 @ id @ X165 @ id))) | (((X162 @ id @ (X164 @ id @ X165) @ id)) = $true)) | (((X162 @ id @ (lam @ (X164 @ sh @ one)) @ id)) = $true) | ~ ! [X167 : subst,X168 : term,X166 : subst] : (((sub @ (X164 @ X167 @ X168) @ X166)) = ((X164 @ (comp @ X167 @ X166) @ (sub @ X168 @ X166))))) | (((X162 @ id @ (lam @ X163) @ id)) = $true)))) <=> (hoasinduction_lem3a_lthm = $true)) & ((pushprop_lem1v2 = $true) <=> ! [X276 : term,X275 : (term > $o),X277 : (term > $o),X274 : subst] : (~ ! [X278 : term] : (((X275 @ X278)) = ((X277 @ (sub @ X278 @ (push @ X276 @ X274))))) | (((X277 @ X276)) != $true) | (((X275 @ one)) = $true))) & ((substmonoid_lthm = $true) <=> (! [X0 : subst] : (((comp @ id @ X0)) = X0) => (! [X1 : subst] : (((comp @ X1 @ id)) = X1) => (! [X3 : subst] : (((comp @ X3 @ id)) = X3) & ! [X2 : subst] : (((comp @ id @ X2)) = X2))))) & (((d2subst @ d_id)) = id) & ((pushprop_lem1_lthm = $true) <=> (! [X130 : subst,X129 : term] : (((sub @ one @ (push @ X129 @ X130))) = X129) => (! [X132 : subst,X131 : term] : (((comp @ sh @ (push @ X131 @ X132))) = X132) => ! [X136 : term,X134 : (term > $o),X137 : subst,X135 : (term > $o),X133 : term] : ((((X135 @ (sub @ X136 @ (push @ X133 @ X137)))) = $true) | ($true != ((X134 @ X136))))))) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (! [X450 : (subst > term > subst > $o)] : ~ ! [X451 : (term > $o)] : ~ ! [X452 : term] : (((X451 @ X452)) = ((X450 @ id @ X452 @ id))) => (! [X453 : (term > $o),X454 : term] : (~ ! [X456 : term,X457 : term] : ((((X453 @ (ap @ X456 @ X457))) = $true) | (((X453 @ X456)) != $true) | (((X453 @ X457)) != $true)) | ~ ! [X458 : term] : ((((X453 @ (lam @ X458))) = $true) | ~ ! [X459 : term] : ((((X453 @ (sub @ X458 @ (push @ X459 @ id)))) = $true) | (((X453 @ X459)) != $true))) | ($true = ((X453 @ X454))) | ~ ! [X455 : term] : (($true = ((X453 @ X455))) | (((var @ X455)) != $true))) => (! [X460 : term] : (((sub @ X460 @ id)) = X460) => (! [X462 : (term > $o),X463 : (subst > term > subst > $o),X461 : term] : (~ ! [X464 : term] : ((((X462 @ X464)) != $true) | (((X462 @ (sub @ X461 @ (push @ X464 @ id)))) = $true)) | ~ ! [X470 : term] : (((X462 @ X470)) = ((X463 @ id @ X470 @ id))) | (((X462 @ (lam @ X461))) = $true) | ~ ! [X465 : (subst > term > term)] : ((((X463 @ id @ (lam @ (X465 @ sh @ one)) @ id)) = $true) | ~ ! [X467 : subst,X468 : subst,X466 : term] : (((sub @ (X465 @ X467 @ X466) @ X468)) = ((X465 @ (comp @ X467 @ X468) @ (sub @ X466 @ X468)))) | ~ ! [X469 : term] : (($true = ((X463 @ id @ (X465 @ id @ X469) @ id))) | (((X463 @ id @ X469 @ id)) != $true)))) => ! [X472 : term,X471 : (subst > term > subst > $o)] : (~ ! [X473 : (subst > term > term)] : (~ ! [X474 : term] : (($true = ((X471 @ id @ (X473 @ id @ X474) @ id))) | (((X471 @ id @ X474 @ id)) != $true)) | (((X471 @ id @ (lam @ (X473 @ sh @ one)) @ id)) = $true) | ~ ! [X475 : subst,X476 : subst,X477 : term] : (((X473 @ (comp @ X476 @ X475) @ (sub @ X477 @ X475))) = ((sub @ (X473 @ X476 @ X477) @ X475)))) | ($true = ((X471 @ id @ X472 @ id))) | ~ ! [X478 : term,X479 : term] : (($true != ((X471 @ id @ X479 @ id))) | (((X471 @ id @ (ap @ (sub @ X479 @ id) @ X478) @ id)) = $true) | ($true != ((X471 @ id @ X478 @ id)))))))))) & (! [X441 : subst] : (((comp @ id @ X441)) = X441) <=> (axidl = $true)) & (! [X443 : term,X442 : (subst > term > subst > $o)] : (~ ! [X444 : term] : (($true = ((X442 @ id @ (sub @ X443 @ (push @ X444 @ id)) @ id))) | (((X442 @ id @ X444 @ id)) != $true)) | ~ ! [X445 : (subst > term > term)] : (~ ! [X446 : term] : ((((X442 @ id @ X446 @ id)) != $true) | (((X442 @ id @ (X445 @ id @ X446) @ id)) = $true)) | ($true = ((X442 @ id @ (lam @ (X445 @ sh @ one)) @ id))) | ~ ! [X449 : term,X448 : subst,X447 : subst] : (((sub @ (X445 @ X448 @ X449) @ X447)) = ((X445 @ (comp @ X448 @ X447) @ (sub @ X449 @ X447))))) | (((X442 @ id @ (lam @ (sub @ X443 @ (push @ one @ sh))) @ id)) = $true)) <=> (hoasinduction_lem3aa_lthm = $true)) & ((pushprop_lem1v2_lthm = $true) <=> (! [X355 : term,X356 : subst] : (((sub @ one @ (push @ X355 @ X356))) = X355) => ! [X358 : (term > $o),X359 : term,X360 : subst,X357 : (term > $o)] : (~ ! [X361 : term] : (((X358 @ (sub @ X361 @ (push @ X359 @ X360)))) = ((X357 @ X361))) | (((X357 @ one)) = $true) | (((X358 @ X359)) != $true)))) & ((! [X141 : subst] : (((comp @ id @ X141)) = X141) & ! [X140 : subst] : (((comp @ X140 @ id)) = X140)) <=> (substmonoid = $true)) & (pushprop_lthm = $true) & ! [X351 : d_term] : (d_one = X351) & (hoasinduction_lem1_gthm = $true) & (axassoc = $true) & (hoasapinj2_gthm = $true) & (lamnotvar = $true) & ((induction2lem = $true) <=> ! [X231 : (term > $o),X230 : subst,X229 : term] : (~ ! [X234 : term,X233 : term] : ((((X231 @ X234)) != $true) | (((X231 @ (ap @ X233 @ X234))) = $true) | ($true != ((X231 @ X233)))) | ($true = ((X231 @ (sub @ X229 @ X230)))) | ~ ! [X232 : term] : ((((var @ X232)) != $true) | (((X231 @ (sub @ X232 @ X230))) = $true)) | ~ ! [X235 : term] : ((((X231 @ (lam @ X235))) = $true) | ~ ! [X236 : term] : ((((X231 @ (sub @ X235 @ (push @ X236 @ id)))) = $true) | (((X231 @ X236)) != $true))))) & ((induction2_lthm = $true) <=> (! [X425 : term] : (((sub @ X425 @ id)) = X425) => (! [X428 : (term > $o),X427 : term,X426 : subst] : (~ ! [X432 : term,X433 : term] : ((((X428 @ X433)) != $true) | ($true != ((X428 @ X432))) | ($true = ((X428 @ (ap @ X433 @ X432))))) | ($true = ((X428 @ (sub @ X427 @ X426)))) | ~ ! [X429 : term] : (~ ! [X430 : term] : (($true = ((X428 @ (sub @ X429 @ (push @ X430 @ id))))) | (((X428 @ X430)) != $true)) | (((X428 @ (lam @ X429))) = $true)) | ~ ! [X431 : term] : (($true != ((var @ X431))) | (((X428 @ (sub @ X431 @ X426))) = $true))) => ! [X434 : (term > $o),X435 : term] : (~ ! [X439 : term] : (~ ! [X440 : term] : (($true != ((X434 @ X440))) | (((X434 @ (sub @ X439 @ (push @ X440 @ id)))) = $true)) | (((X434 @ (lam @ X439))) = $true)) | ~ ! [X437 : term,X436 : term] : ((((X434 @ X437)) != $true) | (((X434 @ (ap @ X436 @ X437))) = $true) | (((X434 @ X436)) != $true)) | (((X434 @ X435)) = $true) | ~ ! [X438 : term] : ((((X434 @ X438)) = $true) | (((var @ X438)) != $true)))))) & ((! [X170 : subst,X171 : term] : (((sub @ one @ (push @ X171 @ X170))) = X171) => (! [X172 : subst,X173 : term] : (((comp @ sh @ (push @ X173 @ X172))) = X172) => (! [X174 : subst] : (((comp @ X174 @ id)) = X174) => (! [X176 : (term > $o),X175 : term] : (~ ! [X180 : term,X179 : term] : ((((X176 @ (ap @ X179 @ X180))) = $true) | ($true != ((X176 @ X179))) | ($true != ((X176 @ X180)))) | ~ ! [X177 : term] : ((((var @ X177)) != $true) | (((X176 @ X177)) = $true)) | (((X176 @ X175)) = $true) | ~ ! [X178 : term] : (($true = ((X176 @ (lam @ X178)))) | (((X176 @ X178)) != $true))) => ! [X183 : subst,X182 : term,X181 : (term > $o)] : (~ ! [X185 : term,X184 : term] : ((((X181 @ X184)) != $true) | ($true = ((X181 @ (ap @ X185 @ X184)))) | (((X181 @ X185)) != $true)) | ($true = ((X181 @ (sub @ X182 @ X183)))) | ~ ! [X187 : term] : ((((X181 @ (lam @ X187))) = $true) | ~ ! [X188 : term] : ((((X181 @ X188)) != $true) | (((X181 @ (sub @ X187 @ (push @ X188 @ id)))) = $true))) | ~ ! [X186 : term] : ((((var @ X186)) != $true) | (((X181 @ (sub @ X186 @ X183))) = $true))))))) <=> (induction2lem_lthm = $true)) & (hoaslamnotvar_lthm = $true) & (lamnotap != $true) & (hoasinduction_lem3a_gthm = $true) & ((hoasinduction = $true) <=> ! [X392 : (subst > term > subst > $o),X393 : term] : (~ ! [X409 : term] : (($true = ((X392 @ id @ X409 @ id))) | (((var @ (sub @ X409 @ id))) != $true)) | ~ ! [X408 : term,X407 : term] : ((((X392 @ id @ X407 @ id)) != $true) | (((X392 @ id @ (ap @ (sub @ X408 @ id) @ X407) @ id)) = $true) | (((X392 @ id @ X408 @ id)) != $true)) | ($true = ((X392 @ id @ X393 @ id))) | ~ ! [X402 : (subst > term > term)] : (~ ! [X403 : term] : ((((X392 @ id @ (X402 @ id @ X403) @ id)) = $true) | ($true != ((X392 @ id @ X403 @ id)))) | (((X392 @ id @ (lam @ (X402 @ sh @ one)) @ id)) = $true) | ~ ! [X405 : term,X404 : subst,X406 : subst] : (((sub @ (X402 @ X406 @ X405) @ X404)) = ((X402 @ (comp @ X406 @ X404) @ (sub @ X405 @ X404))))) | ~ ! [X401 : subst,X399 : term,X398 : subst,X400 : subst] : ((((X392 @ X398 @ X399 @ (comp @ X400 @ X401))) != $true) | (((X392 @ (comp @ X398 @ X400) @ (sub @ X399 @ X400) @ X401)) = $true)) | ~ ! [X394 : subst,X397 : subst,X395 : subst,X396 : term] : (($true != ((X392 @ (comp @ X395 @ X394) @ (sub @ X396 @ X394) @ X397))) | ($true = ((X392 @ X395 @ X396 @ (comp @ X394 @ X397))))))) & (ulamvarind = $true) & (hoasinduction_lem2v2_gthm = $true) & (hoasapnotvar = $true) & (! [X304 : term,X307 : term,X305 : term,X306 : term] : ((X306 = X307) | (((ap @ X304 @ X307)) != ((ap @ X305 @ X306)))) <=> (apinj2 = $true)) & ((termmset = $true) <=> ! [X329 : term] : (((sub @ X329 @ id)) = X329)) & (hoasinduction_lem2_gthm = $true) & (! [X262 : (subst > term > subst > $o),X263 : term,X261 : term] : (($true != ((X262 @ id @ X263 @ id))) | ($true = ((X262 @ id @ (ap @ X263 @ X261) @ id))) | (((X262 @ id @ X261 @ id)) != $true) | ~ ! [X268 : term,X269 : term] : (($true != ((X262 @ id @ X268 @ id))) | ($true != ((X262 @ id @ X269 @ id))) | ($true = ((X262 @ id @ (ap @ (sub @ X268 @ id) @ X269) @ id)))) | ~ ! [X265 : subst,X266 : subst,X264 : term,X267 : subst] : ((((X262 @ (comp @ X266 @ X265) @ (sub @ X264 @ X265) @ X267)) = $true) | (((X262 @ X266 @ X264 @ (comp @ X265 @ X267))) != $true)) | ~ ! [X273 : subst,X272 : term,X270 : subst,X271 : subst] : ((((X262 @ X271 @ X272 @ (comp @ X273 @ X270))) = $true) | (((X262 @ (comp @ X271 @ X273) @ (sub @ X272 @ X273) @ X270)) != $true))) <=> (hoasinduction_lem2 = $true)) & (pushprop_lem0_gthm = $true) & (axvarshift = $true) & (pushprop_lem3v2_lthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (hoaslamnotvar_gthm = $true) & ((axidr = $true) <=> ! [X244 : subst] : (((comp @ X244 @ id)) = X244)) & ((axscons = $true) <=> ! [X217 : subst] : (((push @ (sub @ one @ X217) @ (comp @ sh @ X217))) = X217)) & (hoaslamnotvar = $true) & (! [X84 : term,X82 : (subst > term > term),X83 : term] : ((((ap @ (sub @ X83 @ id) @ X84)) != ((lam @ (X82 @ sh @ one)))) | ~ ! [X85 : term,X86 : subst,X87 : subst] : (((X82 @ (comp @ X87 @ X86) @ (sub @ X85 @ X86))) = ((sub @ (X82 @ X87 @ X85) @ X86)))) <=> (hoaslamnotap = $true)) & (pushprop = $true) & (hoaslamnotap_gthm = $true) & (! [X281 : subst,X279 : (term > $o),X280 : term] : ~ ! [X282 : (term > $o)] : ~ ! [X283 : term] : (((X279 @ (sub @ X283 @ (push @ X280 @ X281)))) = ((X282 @ X283))) <=> (pushprop_lem0_lthm = $true)) & (! [X303 : (term > $o),X302 : subst,X299 : (term > $o),X300 : term,X301 : term] : ((((X303 @ X301)) != $true) | (((X299 @ (sub @ X301 @ (push @ X300 @ X302)))) = $true)) <=> (pushprop_lem1 = $true)) & (! [X352 : term] : (((sub @ X352 @ id)) = X352) <=> (axvarid = $true)) & (! [X492 : term,X491 : term,X493 : term,X490 : term] : ((X490 = X492) | (((ap @ (sub @ X490 @ id) @ X491)) != ((ap @ (sub @ X492 @ id) @ X493)))) <=> (hoasapinj1 = $true)) & (! [X312 : term] : ~ ! [X313 : (subst > term > term)] : (((X313 @ sh @ one)) != ((sub @ X312 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b = $true)) & (hoasinduction_lem3v2_gthm = $true) & (hoasinduction_lem1_lthm = $true) & (axclos = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (! [X362 : (subst > term > subst > $o)] : ~ ! [X363 : (term > $o)] : ~ ! [X364 : term] : (((X362 @ id @ X364 @ id)) = ((X363 @ X364))) <=> (hoasinduction_lem0_lthm = $true)) & ((hoasinduction_lthm_3 = $true) <=> (! [X496 : (subst > term > subst > $o)] : ~ ! [X497 : (term > $o)] : ~ ! [X498 : term] : (((X496 @ id @ X498 @ id)) = ((X497 @ X498))) => (! [X499 : (term > $o),X500 : term] : (($true = ((X499 @ X500))) | ~ ! [X501 : term] : ((((X499 @ X501)) = $true) | (((var @ X501)) != $true)) | ~ ! [X502 : term] : (~ ! [X503 : term] : (($true = ((X499 @ (sub @ X502 @ (push @ X503 @ id))))) | ($true != ((X499 @ X503)))) | (((X499 @ (lam @ X502))) = $true)) | ~ ! [X505 : term,X504 : term] : (($true != ((X499 @ X505))) | ($true = ((X499 @ (ap @ X504 @ X505)))) | (((X499 @ X504)) != $true))) => (! [X506 : term] : (((sub @ X506 @ id)) = X506) => (! [X508 : term,X507 : (term > $o),X509 : (subst > term > subst > $o)] : ((((X507 @ (lam @ X508))) = $true) | ~ ! [X511 : term] : (($true = ((X507 @ (sub @ X508 @ (push @ X511 @ id))))) | (((X507 @ X511)) != $true)) | ~ ! [X512 : (subst > term > term)] : (~ ! [X513 : term,X515 : subst,X514 : subst] : (((sub @ (X512 @ X514 @ X513) @ X515)) = ((X512 @ (comp @ X514 @ X515) @ (sub @ X513 @ X515)))) | ~ ! [X516 : term] : ((((X509 @ id @ (X512 @ id @ X516) @ id)) = $true) | (((X509 @ id @ X516 @ id)) != $true)) | ($true = ((X509 @ id @ (lam @ (X512 @ sh @ one)) @ id)))) | ~ ! [X510 : term] : (((X507 @ X510)) = ((X509 @ id @ X510 @ id)))) => ! [X518 : term,X517 : (subst > term > subst > $o)] : (~ ! [X525 : (subst > term > term)] : (~ ! [X528 : subst,X527 : subst,X526 : term] : (((X525 @ (comp @ X528 @ X527) @ (sub @ X526 @ X527))) = ((sub @ (X525 @ X528 @ X526) @ X527))) | ~ ! [X529 : term] : ((((X517 @ id @ (X525 @ id @ X529) @ id)) = $true) | ($true != ((X517 @ id @ X529 @ id)))) | ($true = ((X517 @ id @ (lam @ (X525 @ sh @ one)) @ id)))) | ~ ! [X523 : subst,X521 : term,X522 : subst,X524 : subst] : (($true != ((X517 @ (comp @ X522 @ X524) @ (sub @ X521 @ X524) @ X523))) | ($true = ((X517 @ X522 @ X521 @ (comp @ X524 @ X523))))) | ~ ! [X531 : term,X533 : subst,X532 : subst,X530 : subst] : (($true = ((X517 @ (comp @ X532 @ X533) @ (sub @ X531 @ X533) @ X530))) | (((X517 @ X532 @ X531 @ (comp @ X533 @ X530))) != $true)) | (((X517 @ id @ X518 @ id)) = $true) | ~ ! [X534 : term] : (($true = ((X517 @ id @ X534 @ id))) | (((var @ (sub @ X534 @ id))) != $true)) | ~ ! [X519 : term,X520 : term] : ((((X517 @ id @ X519 @ id)) != $true) | (((X517 @ id @ (ap @ (sub @ X519 @ id) @ X520) @ id)) = $true) | ($true != ((X517 @ id @ X520 @ id)))))))))) & ! [X494 : term] : ? [X495 : d_term] : (((d2term @ X495)) = X494) & (hoaslaminj_gthm = $true) & ((hoasinduction_lem3v2a = $true) <=> ! [X209 : (subst > term > subst > $o),X208 : term,X207 : (term > $o)] : ((((X207 @ (lam @ X208))) = $true) | ~ ! [X212 : (subst > term > term)] : ((((X209 @ id @ (lam @ (X212 @ sh @ one)) @ id)) = $true) | ~ ! [X215 : subst,X214 : subst,X216 : term] : (((X212 @ (comp @ X214 @ X215) @ (sub @ X216 @ X215))) = ((sub @ (X212 @ X214 @ X216) @ X215))) | ~ ! [X213 : term] : ((((X209 @ id @ X213 @ id)) != $true) | (((X209 @ id @ (X212 @ id @ X213) @ id)) = $true))) | ~ ! [X210 : term] : ((((X207 @ (sub @ X208 @ (push @ X210 @ id)))) = $true) | (((X207 @ X210)) != $true)) | ~ ! [X211 : term] : (((X209 @ id @ X211 @ id)) = ((X207 @ X211))))) & (hoasinduction_lem1 = $true) & ((hoasapinj2_lthm = $true) <=> (! [X76 : term,X77 : term,X75 : term,X74 : term] : ((X76 = X77) | (((ap @ X75 @ X77)) != ((ap @ X74 @ X76)))) => ! [X80 : term,X81 : term,X78 : term,X79 : term] : ((X79 = X81) | (((ap @ (sub @ X80 @ id) @ X79)) != ((ap @ (sub @ X78 @ id) @ X81)))))) & (pushprop_lem1v2_gthm = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ! [X415 : d_subst,X414 : d_subst] : ((((d2subst @ X415)) = ((d2subst @ X414))) => (X414 = X415)) & (! [X219 : (subst > term > subst > $o),X218 : term] : (~ ! [X221 : (subst > term > term)] : (~ ! [X223 : subst,X222 : subst,X224 : term] : (((X221 @ (comp @ X222 @ X223) @ (sub @ X224 @ X223))) = ((sub @ (X221 @ X222 @ X224) @ X223))) | ~ ! [X225 : term] : (($true != ((X219 @ id @ X225 @ id))) | (((X219 @ id @ (X221 @ id @ X225) @ id)) = $true)) | (((X219 @ id @ (lam @ (X221 @ sh @ one)) @ id)) = $true)) | (((X219 @ id @ (lam @ (sub @ X218 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X220 : term] : (($true != ((X219 @ id @ X220 @ id))) | (((X219 @ id @ (sub @ X218 @ (push @ X220 @ id)) @ id)) = $true))) <=> (hoasinduction_lem3aa = $true)) & (axmap = $true) & (! [X349 : term] : ~ ! [X350 : (subst > term > term)] : (((X350 @ sh @ one)) != ((sub @ X349 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b_lthm = $true)) & (! [X128 : term,X125 : term,X127 : term,X126 : term] : ((((ap @ (sub @ X127 @ id) @ X128)) != ((ap @ (sub @ X126 @ id) @ X125))) | (X125 = X128)) <=> (hoasapinj2 = $true)) & ((hoasinduction_lem3a = $true) <=> ! [X285 : (subst > term > subst > $o),X284 : term] : (~ ! [X286 : term] : ((((X285 @ id @ (sub @ X284 @ (push @ X286 @ id)) @ id)) = $true) | (((X285 @ id @ X286 @ id)) != $true)) | ~ ! [X287 : (subst > term > term)] : (~ ! [X291 : term] : (($true = ((X285 @ id @ (X287 @ id @ X291) @ id))) | (((X285 @ id @ X291 @ id)) != $true)) | ~ ! [X290 : term,X288 : subst,X289 : subst] : (((X287 @ (comp @ X288 @ X289) @ (sub @ X290 @ X289))) = ((sub @ (X287 @ X288 @ X290) @ X289))) | ($true = ((X285 @ id @ (lam @ (X287 @ sh @ one)) @ id)))) | ($true = ((X285 @ id @ (lam @ X284) @ id))))) & ((hoasinduction_lem2v2 = $true) <=> ! [X96 : (term > $o),X95 : term,X94 : (subst > term > subst > $o),X97 : term] : (~ ! [X100 : term,X99 : term] : ((((X94 @ id @ X99 @ id)) != $true) | (((X94 @ id @ X100 @ id)) != $true) | (((X94 @ id @ (ap @ (sub @ X99 @ id) @ X100) @ id)) = $true)) | ($true = ((X96 @ (ap @ X97 @ X95)))) | ~ ! [X106 : term,X105 : subst,X107 : subst,X108 : subst] : ((((X94 @ X107 @ X106 @ (comp @ X105 @ X108))) = $true) | (((X94 @ (comp @ X107 @ X105) @ (sub @ X106 @ X105) @ X108)) != $true)) | ~ ! [X104 : subst,X102 : subst,X103 : term,X101 : subst] : ((((X94 @ X102 @ X103 @ (comp @ X101 @ X104))) != $true) | ($true = ((X94 @ (comp @ X102 @ X101) @ (sub @ X103 @ X101) @ X104)))) | (((X96 @ X95)) != $true) | (((X96 @ X97)) != $true) | ~ ! [X98 : term] : (((X94 @ id @ X98 @ id)) = ((X96 @ X98))))) & (! [X238 : term,X237 : term] : ((X237 = X238) | (((sub @ X237 @ sh)) != ((sub @ X238 @ sh)))) <=> (shinj = $true)) & (ulamvar1 != $true) & (hoasapnotvar_lthm = $true) & (induction2lem_gthm = $true) & (axapp = $true) & (hoasinduction_lem1v2_gthm = $true) & ! [X242 : d_term,X243 : d_term] : ((((d2term @ X242)) = ((d2term @ X243))) => (X242 = X243)) & (ulamvarsh = $true) & ((hoasinduction_lem3v2 = $true) <=> ! [X190 : term,X189 : (subst > term > subst > $o),X191 : (term > $o)] : (~ ! [X192 : (subst > term > term)] : (~ ! [X195 : subst,X196 : term,X194 : subst] : (((X192 @ (comp @ X195 @ X194) @ (sub @ X196 @ X194))) = ((sub @ (X192 @ X195 @ X196) @ X194))) | ~ ! [X193 : term] : ((((X189 @ id @ X193 @ id)) != $true) | ($true = ((X189 @ id @ (X192 @ id @ X193) @ id)))) | ($true = ((X189 @ id @ (lam @ (X192 @ sh @ one)) @ id)))) | ~ ! [X200 : subst,X201 : term,X198 : subst,X199 : subst] : (($true != ((X189 @ (comp @ X200 @ X199) @ (sub @ X201 @ X199) @ X198))) | (((X189 @ X200 @ X201 @ (comp @ X199 @ X198))) = $true)) | ~ ! [X202 : term] : (((X189 @ id @ X202 @ id)) = ((X191 @ X202))) | ~ ! [X203 : subst,X206 : term,X204 : subst,X205 : subst] : ((((X189 @ (comp @ X203 @ X205) @ (sub @ X206 @ X205) @ X204)) = $true) | (((X189 @ X203 @ X206 @ (comp @ X205 @ X204))) != $true)) | ~ ! [X197 : term] : (($true = ((X191 @ (sub @ X190 @ (push @ X197 @ id))))) | ($true != ((X191 @ X197)))) | (((X191 @ (lam @ X190))) = $true))) & (! [X537 : subst,X536 : term,X535 : (term > $o)] : ~ ! [X538 : (term > $o)] : ~ ! [X539 : term] : (((X538 @ X539)) = ((X535 @ (sub @ X539 @ (push @ X536 @ X537))))) <=> (pushprop_lem0 = $true)) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (! [X482 : term,X480 : (subst > term > term),X481 : subst,X483 : (subst > term > term)] : (~ ! [X486 : term,X484 : subst,X485 : subst] : (((sub @ (X480 @ X484 @ X486) @ X485)) = ((X480 @ (comp @ X484 @ X485) @ (sub @ X486 @ X485)))) | ~ ! [X488 : term,X487 : subst,X489 : subst] : (((X483 @ (comp @ X487 @ X489) @ (sub @ X488 @ X489))) = ((sub @ (X483 @ X487 @ X488) @ X489))) | (((X480 @ X481 @ X482)) = ((X483 @ X481 @ X482))) | (((lam @ (X483 @ sh @ one))) != ((lam @ (X480 @ sh @ one))))) <=> (hoaslaminj = $true)) & (! [X227 : term,X228 : subst] : (((sub @ one @ (push @ X227 @ X228))) = X227) <=> (axvarcons = $true)) & ((hoasinduction_lthm = $true) <=> (! [X4 : (term > $o),X5 : term] : (~ ! [X7 : term,X6 : term] : ((((X4 @ (ap @ X7 @ X6))) = $true) | ($true != ((X4 @ X6))) | (((X4 @ X7)) != $true)) | ~ ! [X8 : term] : ((((X4 @ X8)) = $true) | ($true != ((var @ X8)))) | ~ ! [X9 : term] : (~ ! [X10 : term] : (($true = ((X4 @ (sub @ X9 @ (push @ X10 @ id))))) | (((X4 @ X10)) != $true)) | ($true = ((X4 @ (lam @ X9))))) | (((X4 @ X5)) = $true)) => (! [X11 : term,X12 : (subst > term > subst > $o),X13 : term] : (~ ! [X16 : subst,X14 : term,X15 : subst,X17 : subst] : (($true = ((X12 @ X16 @ X14 @ (comp @ X17 @ X15)))) | (((X12 @ (comp @ X16 @ X17) @ (sub @ X14 @ X17) @ X15)) != $true)) | (((X12 @ id @ (ap @ X11 @ X13) @ id)) = $true) | (((X12 @ id @ X11 @ id)) != $true) | ~ ! [X19 : term,X18 : term] : ((((X12 @ id @ X18 @ id)) != $true) | (((X12 @ id @ (ap @ (sub @ X19 @ id) @ X18) @ id)) = $true) | (((X12 @ id @ X19 @ id)) != $true)) | ~ ! [X20 : subst,X21 : subst,X22 : term,X23 : subst] : (($true != ((X12 @ X20 @ X22 @ (comp @ X23 @ X21)))) | (((X12 @ (comp @ X20 @ X23) @ (sub @ X22 @ X23) @ X21)) = $true)) | ($true != ((X12 @ id @ X13 @ id)))) => (! [X25 : term,X24 : (subst > term > subst > $o)] : (~ ! [X35 : term] : ((((X24 @ id @ X35 @ id)) != $true) | (((X24 @ id @ (sub @ X25 @ (push @ X35 @ id)) @ id)) = $true)) | ~ ! [X36 : subst,X39 : subst,X37 : term,X38 : subst] : (($true = ((X24 @ X39 @ X37 @ (comp @ X38 @ X36)))) | (((X24 @ (comp @ X39 @ X38) @ (sub @ X37 @ X38) @ X36)) != $true)) | ~ ! [X30 : (subst > term > term)] : ((((X24 @ id @ (lam @ (X30 @ sh @ one)) @ id)) = $true) | ~ ! [X31 : term] : ((((X24 @ id @ (X30 @ id @ X31) @ id)) = $true) | (((X24 @ id @ X31 @ id)) != $true)) | ~ ! [X33 : subst,X34 : subst,X32 : term] : (((sub @ (X30 @ X33 @ X32) @ X34)) = ((X30 @ (comp @ X33 @ X34) @ (sub @ X32 @ X34))))) | (((X24 @ id @ (lam @ X25) @ id)) = $true) | ~ ! [X27 : subst,X26 : subst,X28 : subst,X29 : term] : ((((X24 @ (comp @ X27 @ X26) @ (sub @ X29 @ X26) @ X28)) = $true) | ($true != ((X24 @ X27 @ X29 @ (comp @ X26 @ X28)))))) => ! [X41 : (subst > term > subst > $o),X40 : term] : (~ ! [X43 : term,X42 : term] : (($true != ((X41 @ id @ X43 @ id))) | ($true = ((X41 @ id @ (ap @ (sub @ X43 @ id) @ X42) @ id))) | ($true != ((X41 @ id @ X42 @ id)))) | ~ ! [X54 : subst,X56 : subst,X55 : subst,X57 : term] : ((((X41 @ (comp @ X56 @ X55) @ (sub @ X57 @ X55) @ X54)) = $true) | ($true != ((X41 @ X56 @ X57 @ (comp @ X55 @ X54))))) | ~ ! [X45 : (subst > term > term)] : (~ ! [X48 : term,X49 : subst,X47 : subst] : (((sub @ (X45 @ X49 @ X48) @ X47)) = ((X45 @ (comp @ X49 @ X47) @ (sub @ X48 @ X47)))) | (((X41 @ id @ (lam @ (X45 @ sh @ one)) @ id)) = $true) | ~ ! [X46 : term] : (($true = ((X41 @ id @ (X45 @ id @ X46) @ id))) | (((X41 @ id @ X46 @ id)) != $true))) | ~ ! [X52 : term,X51 : subst,X50 : subst,X53 : subst] : ((((X41 @ (comp @ X51 @ X50) @ (sub @ X52 @ X50) @ X53)) != $true) | ($true = ((X41 @ X51 @ X52 @ (comp @ X50 @ X53))))) | ~ ! [X44 : term] : ((((X41 @ id @ X44 @ id)) = $true) | (((var @ (sub @ X44 @ id))) != $true)) | (((X41 @ id @ X40 @ id)) = $true)))))) & (! [X239 : (subst > term > subst > $o)] : ~ ! [X240 : (term > $o)] : ~ ! [X241 : term] : (((X240 @ X241)) = ((X239 @ id @ X241 @ id))) <=> (hoasinduction_lem0 = $true)) & ! [X226 : d_subst] : (d_id = X226) & (termmset_gthm = $true) & (! [X245 : term,X246 : (subst > term > subst > $o)] : (~ ! [X247 : term] : (($true = ((X246 @ id @ (sub @ X245 @ (push @ X247 @ id)) @ id))) | ($true != ((X246 @ id @ X247 @ id)))) | (((X246 @ id @ (lam @ X245) @ id)) = $true) | ~ ! [X248 : subst,X249 : subst,X251 : term,X250 : subst] : (($true = ((X246 @ X249 @ X251 @ (comp @ X250 @ X248)))) | ($true != ((X246 @ (comp @ X249 @ X250) @ (sub @ X251 @ X250) @ X248)))) | ~ ! [X259 : term,X260 : subst,X257 : subst,X258 : subst] : ((((X246 @ X257 @ X259 @ (comp @ X258 @ X260))) != $true) | (((X246 @ (comp @ X257 @ X258) @ (sub @ X259 @ X258) @ X260)) = $true)) | ~ ! [X252 : (subst > term > term)] : (~ ! [X253 : term] : (($true = ((X246 @ id @ (X252 @ id @ X253) @ id))) | (((X246 @ id @ X253 @ id)) != $true)) | ($true = ((X246 @ id @ (lam @ (X252 @ sh @ one)) @ id))) | ~ ! [X255 : subst,X254 : subst,X256 : term] : (((X252 @ (comp @ X254 @ X255) @ (sub @ X256 @ X255))) = ((sub @ (X252 @ X254 @ X256) @ X255))))) <=> (hoasinduction_lem3 = $true)) & (! [X151 : term,X152 : term] : ((X151 = X152) | (((lam @ X151)) != ((lam @ X152)))) <=> (laminj = $true)) & (substmonoid_gthm = $true) & (hoasapinj1_gthm = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (! [X72 : term,X71 : term,X70 : term,X73 : term] : ((((ap @ X71 @ X73)) != ((ap @ X70 @ X72))) | (X70 = X71)) <=> (apinj1 = $true)) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (hoasinduction_gthm = $true) & ! [X353 : subst] : ? [X354 : d_subst] : (((d2subst @ X354)) = X353) & (hoasinduction_lem3_gthm = $true) & (hoasapnotvar_gthm = $true) & ((! [X365 : term] : (((sub @ X365 @ id)) = X365) => (! [X367 : term,X366 : (subst > term > subst > $o)] : (~ ! [X368 : term] : (($true = ((X366 @ id @ (sub @ X367 @ (push @ X368 @ id)) @ id))) | (((X366 @ id @ X368 @ id)) != $true)) | ~ ! [X369 : (subst > term > term)] : (~ ! [X373 : term] : (($true != ((X366 @ id @ X373 @ id))) | (((X366 @ id @ (X369 @ id @ X373) @ id)) = $true)) | ~ ! [X371 : subst,X370 : term,X372 : subst] : (((X369 @ (comp @ X371 @ X372) @ (sub @ X370 @ X372))) = ((sub @ (X369 @ X371 @ X370) @ X372))) | (((X366 @ id @ (lam @ (X369 @ sh @ one)) @ id)) = $true)) | (((X366 @ id @ (lam @ (sub @ X367 @ (push @ one @ sh))) @ id)) = $true)) => ! [X374 : (subst > term > subst > $o),X375 : term] : (~ ! [X384 : term,X382 : subst,X383 : subst,X381 : subst] : (($true != ((X374 @ (comp @ X382 @ X383) @ (sub @ X384 @ X383) @ X381))) | ($true = ((X374 @ X382 @ X384 @ (comp @ X383 @ X381))))) | ~ ! [X380 : term] : ((((X374 @ id @ X380 @ id)) != $true) | (((X374 @ id @ (sub @ X375 @ (push @ X380 @ id)) @ id)) = $true)) | ($true = ((X374 @ id @ (lam @ X375) @ id))) | ~ ! [X385 : (subst > term > term)] : (~ ! [X388 : subst,X387 : subst,X386 : term] : (((X385 @ (comp @ X388 @ X387) @ (sub @ X386 @ X387))) = ((sub @ (X385 @ X388 @ X386) @ X387))) | (((X374 @ id @ (lam @ (X385 @ sh @ one)) @ id)) = $true) | ~ ! [X389 : term] : ((((X374 @ id @ (X385 @ id @ X389) @ id)) = $true) | (((X374 @ id @ X389 @ id)) != $true))) | ~ ! [X379 : term,X377 : subst,X376 : subst,X378 : subst] : (($true = ((X374 @ (comp @ X378 @ X376) @ (sub @ X379 @ X376) @ X377))) | (((X374 @ X378 @ X379 @ (comp @ X376 @ X377))) != $true))))) <=> (hoasinduction_lem3_lthm = $true)) & ((! [X330 : term] : (((sub @ X330 @ id)) = X330) => ! [X333 : (term > $o),X331 : term,X332 : (subst > term > subst > $o)] : (~ ! [X335 : (subst > term > term)] : (~ ! [X338 : subst,X336 : subst,X337 : term] : (((sub @ (X335 @ X338 @ X337) @ X336)) = ((X335 @ (comp @ X338 @ X336) @ (sub @ X337 @ X336)))) | ($true = ((X332 @ id @ (lam @ (X335 @ sh @ one)) @ id))) | ~ ! [X339 : term] : ((((X332 @ id @ (X335 @ id @ X339) @ id)) = $true) | (((X332 @ id @ X339 @ id)) != $true))) | ~ ! [X344 : term,X343 : subst,X342 : subst,X341 : subst] : ((((X332 @ X341 @ X344 @ (comp @ X343 @ X342))) != $true) | ($true = ((X332 @ (comp @ X341 @ X343) @ (sub @ X344 @ X343) @ X342)))) | (((X333 @ (lam @ X331))) = $true) | ~ ! [X340 : term] : (((X333 @ X340)) = ((X332 @ id @ X340 @ id))) | ~ ! [X334 : term] : ((((X333 @ X334)) != $true) | ($true = ((X333 @ (sub @ X331 @ (push @ X334 @ id)))))) | ~ ! [X346 : term,X348 : subst,X345 : subst,X347 : subst] : ((((X332 @ (comp @ X347 @ X345) @ (sub @ X346 @ X345) @ X348)) != $true) | ($true = ((X332 @ X347 @ X346 @ (comp @ X345 @ X348))))))) <=> (hoasinduction_lem3v2_lthm = $true)) & (! [X308 : term] : ~ ! [X309 : (subst > term > term)] : ~ ! [X311 : subst,X310 : term] : (((X309 @ X311 @ X310)) = ((sub @ X308 @ (push @ X310 @ X311)))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & (axabs = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (! [X88 : term,X89 : (term > $o)] : (~ ! [X90 : term,X91 : term] : (($true != ((X89 @ X91))) | (((X89 @ X90)) != $true) | ($true = ((X89 @ (ap @ X90 @ X91))))) | (((X89 @ X88)) = $true) | ~ ! [X92 : term] : (($true != ((X89 @ X92))) | ($true = ((X89 @ (lam @ X92))))) | ~ ! [X93 : term] : ((((X89 @ X93)) = $true) | (((var @ X93)) != $true))) <=> (induction = $true)) & ((! [X314 : term] : ~ ! [X315 : (subst > term > term)] : ~ ! [X317 : subst,X316 : term] : (((sub @ X314 @ (push @ X316 @ X317))) = ((X315 @ X317 @ X316))) => (! [X318 : term] : (((sub @ X318 @ id)) = X318) => ! [X319 : term,X321 : (subst > term > subst > $o),X320 : (term > $o)] : (~ ! [X322 : (subst > term > term)] : (~ ! [X323 : subst,X324 : subst,X325 : term] : (((sub @ (X322 @ X324 @ X325) @ X323)) = ((X322 @ (comp @ X324 @ X323) @ (sub @ X325 @ X323)))) | ~ ! [X326 : term] : ((((X321 @ id @ X326 @ id)) != $true) | ($true = ((X321 @ id @ (X322 @ id @ X326) @ id)))) | (((X321 @ id @ (lam @ (X322 @ sh @ one)) @ id)) = $true)) | ~ ! [X327 : term] : (((X320 @ X327)) = ((X321 @ id @ X327 @ id))) | ~ ! [X328 : term] : ((((X320 @ X328)) != $true) | (((X320 @ (sub @ X319 @ (push @ X328 @ id)))) = $true)) | (((X320 @ (lam @ X319))) = $true)))) <=> (hoasinduction_lem3v2a_lthm = $true)) & (pushprop_lem2v2_lthm = $true) & (pushprop_lem3v2 = $true) & (((var @ (d2term @ d_one))) != $true) & ((hoasinduction_lem3v2_f = $true) <=> ! [X410 : term] : ~ ! [X411 : (subst > term > term)] : ~ ! [X413 : term,X412 : subst] : (((sub @ X410 @ (push @ X413 @ X412))) = ((X411 @ X412 @ X413)))) & ((! [X138 : term] : (((sub @ X138 @ id)) = X138) => ! [X139 : term] : (((sub @ X139 @ id)) = X139)) <=> (termmset_lthm = $true)) & (pushprop_lthm_orig = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (induction2_gthm = $true) & (pushprop_gthm = $true) & (! [X292 : (term > $o),X293 : term] : (~ ! [X298 : term] : (($true != ((var @ X298))) | (((X292 @ X298)) = $true)) | ~ ! [X296 : term,X297 : term] : ((((X292 @ X297)) != $true) | ($true = ((X292 @ (ap @ X297 @ X296)))) | ($true != ((X292 @ X296)))) | ~ ! [X294 : term] : (~ ! [X295 : term] : ((((X292 @ X295)) != $true) | (((X292 @ (sub @ X294 @ (push @ X295 @ id)))) = $true)) | (((X292 @ (lam @ X294))) = $true)) | (((X292 @ X293)) = $true)) <=> (induction2 = $true)) & (hoasinduction_lem3b_gthm = $true) & ((hoasinduction_no_psi_cond = $true) <=> ! [X142 : term,X143 : (subst > term > subst > $o)] : ((((X143 @ id @ X142 @ id)) = $true) | ~ ! [X145 : term,X144 : term] : ((((X143 @ id @ X144 @ id)) != $true) | (((X143 @ id @ X145 @ id)) != $true) | (((X143 @ id @ (ap @ (sub @ X144 @ id) @ X145) @ id)) = $true)) | ~ ! [X146 : (subst > term > term)] : (~ ! [X150 : term] : ((((X143 @ id @ (X146 @ id @ X150) @ id)) = $true) | ($true != ((X143 @ id @ X150 @ id)))) | (((X143 @ id @ (lam @ (X146 @ sh @ one)) @ id)) = $true) | ~ ! [X148 : term,X147 : subst,X149 : subst] : (((sub @ (X146 @ X149 @ X148) @ X147)) = ((X146 @ (comp @ X149 @ X147) @ (sub @ X148 @ X147))))))) & ((hoasapinj1_lthm = $true) <=> (! [X416 : term] : (((sub @ X416 @ id)) = X416) => (! [X420 : term,X417 : term,X419 : term,X418 : term] : ((X417 = X418) | (((ap @ X418 @ X419)) != ((ap @ X417 @ X420)))) => ! [X423 : term,X424 : term,X422 : term,X421 : term] : ((((ap @ (sub @ X421 @ id) @ X422)) != ((ap @ (sub @ X424 @ id) @ X423))) | (X421 = X424)))))),
  inference(flattening,[],[f9])).
thf(f11,plain,(
  (pushprop_lem2v2 = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (one = ((d2term @ d_one))) & ((hoaslaminj_lthm = $true) <=> (((! [X115 : (subst > term > term),X116 : subst,X117 : (subst > term > term),X118 : term] : ((((X117 @ X116 @ X118)) = ((X115 @ X116 @ X118))) | ? [X122 : subst,X123 : term,X124 : subst] : (((X117 @ (comp @ X124 @ X122) @ (sub @ X123 @ X122))) != ((sub @ (X117 @ X124 @ X123) @ X122))) | ? [X119 : term,X121 : subst,X120 : subst] : (((sub @ (X115 @ X121 @ X119) @ X120)) != ((X115 @ (comp @ X121 @ X120) @ (sub @ X119 @ X120)))) | (((lam @ (X115 @ sh @ one))) != ((lam @ (X117 @ sh @ one))))) | ? [X114 : term,X113 : term] : ((((lam @ X113)) = ((lam @ X114))) & (X113 != X114))) | ? [X112 : term,X111 : subst] : (((comp @ sh @ (push @ X112 @ X111))) != X111)) | ? [X109 : subst,X110 : term] : (((sub @ one @ (push @ X110 @ X109))) != X110))) & (pushprop_lem1_gthm = $true) & (pushprop_lem2v2_gthm = $true) & (apnotvar = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (hoasinduction_lem1v2 = $true) & (! [X391 : term,X390 : subst] : (((comp @ sh @ (push @ X391 @ X390))) = X390) <=> (axshiftcons = $true)) & (hoasinduction_lem2_lthm = $true) & (hoaslamnotap_lthm = $true) & (! [X59 : (subst > term > subst > $o),X58 : term] : (? [X60 : term,X61 : (subst > term > term)] : (! [X64 : subst,X63 : subst,X62 : term] : (((sub @ (X61 @ X64 @ X62) @ X63)) = ((sub @ (sub @ X60 @ (push @ X62 @ X64)) @ X63))) & ! [X68 : subst,X67 : subst,X66 : term] : (((X61 @ (comp @ X67 @ X68) @ (sub @ X66 @ X68))) = ((sub @ X60 @ (push @ (sub @ X66 @ X68) @ (comp @ X67 @ X68))))) & ($true != ((X59 @ id @ (lam @ (X61 @ sh @ one)) @ id))) & ! [X65 : term] : (($true != ((X59 @ id @ X65 @ id))) | ($true = ((X59 @ id @ (X61 @ id @ X65) @ id))))) | (((X59 @ id @ (lam @ (sub @ X58 @ (push @ one @ sh))) @ id)) = $true) | ? [X69 : term] : ((((X59 @ id @ (sub @ X58 @ (push @ X69 @ id)) @ id)) != $true) & (((X59 @ id @ X69 @ id)) = $true))) <=> (hoasinduction_lem3aaa = $true)) & (((d2subst @ d_id)) = sh) & (((! [X162 : (subst > term > subst > $o),X163 : term] : ((((X162 @ id @ (lam @ X163) @ id)) = $true) | ? [X169 : term] : (($true = ((X162 @ id @ X169 @ id))) & (((X162 @ id @ (sub @ X163 @ (push @ X169 @ id)) @ id)) != $true)) | ? [X164 : (subst > term > term)] : (! [X165 : term] : (($true != ((X162 @ id @ X165 @ id))) | (((X162 @ id @ (X164 @ id @ X165) @ id)) = $true)) & ! [X167 : subst,X168 : term,X166 : subst] : (((sub @ (X164 @ X167 @ X168) @ X166)) = ((X164 @ (comp @ X167 @ X166) @ (sub @ X168 @ X166)))) & (((X162 @ id @ (lam @ (X164 @ sh @ one)) @ id)) != $true))) | ? [X154 : (subst > term > subst > $o),X155 : term] : (! [X161 : term] : ((((X154 @ id @ X161 @ id)) != $true) | (((X154 @ id @ (sub @ X155 @ (push @ X161 @ id)) @ id)) = $true)) & (((X154 @ id @ (lam @ (sub @ X155 @ (push @ one @ sh))) @ id)) != $true) & ! [X156 : (subst > term > term)] : ((((X154 @ id @ (lam @ (X156 @ sh @ one)) @ id)) = $true) | ? [X157 : term] : (($true != ((X154 @ id @ (X156 @ id @ X157) @ id))) & ($true = ((X154 @ id @ X157 @ id)))) | ? [X160 : subst,X158 : term,X159 : subst] : (((X156 @ (comp @ X159 @ X160) @ (sub @ X158 @ X160))) != ((sub @ (X156 @ X159 @ X158) @ X160)))))) | ? [X153 : term] : (((sub @ X153 @ id)) != X153)) <=> (hoasinduction_lem3a_lthm = $true)) & ((pushprop_lem1v2 = $true) <=> ! [X277 : (term > $o),X276 : term,X274 : subst,X275 : (term > $o)] : ((((X275 @ one)) = $true) | (((X277 @ X276)) != $true) | ? [X278 : term] : (((X275 @ X278)) != ((X277 @ (sub @ X278 @ (push @ X276 @ X274))))))) & ((substmonoid_lthm = $true) <=> (((! [X3 : subst] : (((comp @ X3 @ id)) = X3) & ! [X2 : subst] : (((comp @ id @ X2)) = X2)) | ? [X1 : subst] : (((comp @ X1 @ id)) != X1)) | ? [X0 : subst] : (((comp @ id @ X0)) != X0))) & (((d2subst @ d_id)) = id) & ((pushprop_lem1_lthm = $true) <=> ((! [X136 : term,X134 : (term > $o),X137 : subst,X135 : (term > $o),X133 : term] : ((((X135 @ (sub @ X136 @ (push @ X133 @ X137)))) = $true) | ($true != ((X134 @ X136)))) | ? [X132 : subst,X131 : term] : (((comp @ sh @ (push @ X131 @ X132))) != X132)) | ? [X130 : subst,X129 : term] : (((sub @ one @ (push @ X129 @ X130))) != X129))) & ((hoasinduction_no_psi_cond_lthm = $true) <=> ((((! [X472 : term,X471 : (subst > term > subst > $o)] : (($true = ((X471 @ id @ X472 @ id))) | ? [X473 : (subst > term > term)] : (! [X475 : subst,X476 : subst,X477 : term] : (((X473 @ (comp @ X476 @ X475) @ (sub @ X477 @ X475))) = ((sub @ (X473 @ X476 @ X477) @ X475))) & ! [X474 : term] : (($true = ((X471 @ id @ (X473 @ id @ X474) @ id))) | (((X471 @ id @ X474 @ id)) != $true)) & (((X471 @ id @ (lam @ (X473 @ sh @ one)) @ id)) != $true)) | ? [X478 : term,X479 : term] : (($true = ((X471 @ id @ X479 @ id))) & (((X471 @ id @ (ap @ (sub @ X479 @ id) @ X478) @ id)) != $true) & ($true = ((X471 @ id @ X478 @ id))))) | ? [X463 : (subst > term > subst > $o),X462 : (term > $o),X461 : term] : (! [X464 : term] : ((((X462 @ X464)) != $true) | (((X462 @ (sub @ X461 @ (push @ X464 @ id)))) = $true)) & ! [X470 : term] : (((X462 @ X470)) = ((X463 @ id @ X470 @ id))) & ! [X465 : (subst > term > term)] : (? [X469 : term] : ((((X463 @ id @ X469 @ id)) = $true) & ($true != ((X463 @ id @ (X465 @ id @ X469) @ id)))) | (((X463 @ id @ (lam @ (X465 @ sh @ one)) @ id)) = $true) | ? [X467 : subst,X466 : term,X468 : subst] : (((sub @ (X465 @ X467 @ X466) @ X468)) != ((X465 @ (comp @ X467 @ X468) @ (sub @ X466 @ X468))))) & (((X462 @ (lam @ X461))) != $true))) | ? [X460 : term] : (((sub @ X460 @ id)) != X460)) | ? [X453 : (term > $o),X454 : term] : (($true != ((X453 @ X454))) & ! [X455 : term] : (($true = ((X453 @ X455))) | (((var @ X455)) != $true)) & ! [X456 : term,X457 : term] : ((((X453 @ (ap @ X456 @ X457))) = $true) | (((X453 @ X456)) != $true) | (((X453 @ X457)) != $true)) & ! [X458 : term] : (? [X459 : term] : ((((X453 @ X459)) = $true) & (((X453 @ (sub @ X458 @ (push @ X459 @ id)))) != $true)) | (((X453 @ (lam @ X458))) = $true)))) | ? [X450 : (subst > term > subst > $o)] : ! [X451 : (term > $o)] : ? [X452 : term] : (((X451 @ X452)) != ((X450 @ id @ X452 @ id))))) & (! [X441 : subst] : (((comp @ id @ X441)) = X441) <=> (axidl = $true)) & (! [X443 : term,X442 : (subst > term > subst > $o)] : (? [X445 : (subst > term > term)] : (($true != ((X442 @ id @ (lam @ (X445 @ sh @ one)) @ id))) & ! [X446 : term] : ((((X442 @ id @ X446 @ id)) != $true) | (((X442 @ id @ (X445 @ id @ X446) @ id)) = $true)) & ! [X449 : term,X448 : subst,X447 : subst] : (((sub @ (X445 @ X448 @ X449) @ X447)) = ((X445 @ (comp @ X448 @ X447) @ (sub @ X449 @ X447))))) | (((X442 @ id @ (lam @ (sub @ X443 @ (push @ one @ sh))) @ id)) = $true) | ? [X444 : term] : ((((X442 @ id @ X444 @ id)) = $true) & ($true != ((X442 @ id @ (sub @ X443 @ (push @ X444 @ id)) @ id))))) <=> (hoasinduction_lem3aa_lthm = $true)) & ((! [X360 : subst,X357 : (term > $o),X358 : (term > $o),X359 : term] : ((((X358 @ X359)) != $true) | ? [X361 : term] : (((X358 @ (sub @ X361 @ (push @ X359 @ X360)))) != ((X357 @ X361))) | (((X357 @ one)) = $true)) | ? [X355 : term,X356 : subst] : (((sub @ one @ (push @ X355 @ X356))) != X355)) <=> (pushprop_lem1v2_lthm = $true)) & ((! [X141 : subst] : (((comp @ id @ X141)) = X141) & ! [X140 : subst] : (((comp @ X140 @ id)) = X140)) <=> (substmonoid = $true)) & (pushprop_lthm = $true) & ! [X351 : d_term] : (d_one = X351) & (hoasinduction_lem1_gthm = $true) & (axassoc = $true) & (hoasapinj2_gthm = $true) & (lamnotvar = $true) & ((induction2lem = $true) <=> ! [X230 : subst,X229 : term,X231 : (term > $o)] : (? [X235 : term] : (! [X236 : term] : ((((X231 @ (sub @ X235 @ (push @ X236 @ id)))) = $true) | (((X231 @ X236)) != $true)) & (((X231 @ (lam @ X235))) != $true)) | ? [X234 : term,X233 : term] : ((((X231 @ X234)) = $true) & ($true = ((X231 @ X233))) & (((X231 @ (ap @ X233 @ X234))) != $true)) | ($true = ((X231 @ (sub @ X229 @ X230)))) | ? [X232 : term] : ((((X231 @ (sub @ X232 @ X230))) != $true) & (((var @ X232)) = $true)))) & ((induction2_lthm = $true) <=> ((! [X435 : term,X434 : (term > $o)] : ((((X434 @ X435)) = $true) | ? [X438 : term] : ((((X434 @ X438)) != $true) & (((var @ X438)) = $true)) | ? [X436 : term,X437 : term] : ((((X434 @ X436)) = $true) & (((X434 @ (ap @ X436 @ X437))) != $true) & (((X434 @ X437)) = $true)) | ? [X439 : term] : ((((X434 @ (lam @ X439))) != $true) & ! [X440 : term] : (($true != ((X434 @ X440))) | (((X434 @ (sub @ X439 @ (push @ X440 @ id)))) = $true)))) | ? [X428 : (term > $o),X426 : subst,X427 : term] : (! [X431 : term] : (($true != ((var @ X431))) | (((X428 @ (sub @ X431 @ X426))) = $true)) & ! [X429 : term] : ((((X428 @ (lam @ X429))) = $true) | ? [X430 : term] : ((((X428 @ X430)) = $true) & ($true != ((X428 @ (sub @ X429 @ (push @ X430 @ id))))))) & ! [X432 : term,X433 : term] : ((((X428 @ X433)) != $true) | ($true != ((X428 @ X432))) | ($true = ((X428 @ (ap @ X433 @ X432))))) & ($true != ((X428 @ (sub @ X427 @ X426)))))) | ? [X425 : term] : (((sub @ X425 @ id)) != X425))) & (((((! [X183 : subst,X182 : term,X181 : (term > $o)] : (? [X186 : term] : ((((X181 @ (sub @ X186 @ X183))) != $true) & (((var @ X186)) = $true)) | ? [X187 : term] : (! [X188 : term] : ((((X181 @ X188)) != $true) | (((X181 @ (sub @ X187 @ (push @ X188 @ id)))) = $true)) & (((X181 @ (lam @ X187))) != $true)) | ? [X184 : term,X185 : term] : ((((X181 @ X185)) = $true) & (((X181 @ X184)) = $true) & ($true != ((X181 @ (ap @ X185 @ X184))))) | ($true = ((X181 @ (sub @ X182 @ X183))))) | ? [X176 : (term > $o),X175 : term] : (! [X180 : term,X179 : term] : ((((X176 @ (ap @ X179 @ X180))) = $true) | ($true != ((X176 @ X179))) | ($true != ((X176 @ X180)))) & ! [X178 : term] : (($true = ((X176 @ (lam @ X178)))) | (((X176 @ X178)) != $true)) & (((X176 @ X175)) != $true) & ! [X177 : term] : ((((var @ X177)) != $true) | (((X176 @ X177)) = $true)))) | ? [X174 : subst] : (((comp @ X174 @ id)) != X174)) | ? [X172 : subst,X173 : term] : (((comp @ sh @ (push @ X173 @ X172))) != X172)) | ? [X171 : term,X170 : subst] : (((sub @ one @ (push @ X171 @ X170))) != X171)) <=> (induction2lem_lthm = $true)) & (hoaslamnotvar_lthm = $true) & (lamnotap != $true) & (hoasinduction_lem3a_gthm = $true) & ((hoasinduction = $true) <=> ! [X392 : (subst > term > subst > $o),X393 : term] : (($true = ((X392 @ id @ X393 @ id))) | ? [X409 : term] : ((((var @ (sub @ X409 @ id))) = $true) & ($true != ((X392 @ id @ X409 @ id)))) | ? [X399 : term,X400 : subst,X398 : subst,X401 : subst] : ((((X392 @ (comp @ X398 @ X400) @ (sub @ X399 @ X400) @ X401)) != $true) & (((X392 @ X398 @ X399 @ (comp @ X400 @ X401))) = $true)) | ? [X408 : term,X407 : term] : ((((X392 @ id @ X407 @ id)) = $true) & (((X392 @ id @ (ap @ (sub @ X408 @ id) @ X407) @ id)) != $true) & (((X392 @ id @ X408 @ id)) = $true)) | ? [X402 : (subst > term > term)] : (! [X403 : term] : ((((X392 @ id @ (X402 @ id @ X403) @ id)) = $true) | ($true != ((X392 @ id @ X403 @ id)))) & ! [X405 : term,X404 : subst,X406 : subst] : (((sub @ (X402 @ X406 @ X405) @ X404)) = ((X402 @ (comp @ X406 @ X404) @ (sub @ X405 @ X404)))) & (((X392 @ id @ (lam @ (X402 @ sh @ one)) @ id)) != $true)) | ? [X394 : subst,X397 : subst,X396 : term,X395 : subst] : (($true = ((X392 @ (comp @ X395 @ X394) @ (sub @ X396 @ X394) @ X397))) & ($true != ((X392 @ X395 @ X396 @ (comp @ X394 @ X397))))))) & (ulamvarind = $true) & (hoasinduction_lem2v2_gthm = $true) & (hoasapnotvar = $true) & (! [X304 : term,X307 : term,X305 : term,X306 : term] : ((X306 = X307) | (((ap @ X304 @ X307)) != ((ap @ X305 @ X306)))) <=> (apinj2 = $true)) & ((termmset = $true) <=> ! [X329 : term] : (((sub @ X329 @ id)) = X329)) & (hoasinduction_lem2_gthm = $true) & ((hoasinduction_lem2 = $true) <=> ! [X263 : term,X261 : term,X262 : (subst > term > subst > $o)] : ((((X262 @ id @ X261 @ id)) != $true) | ? [X265 : subst,X264 : term,X267 : subst,X266 : subst] : ((((X262 @ X266 @ X264 @ (comp @ X265 @ X267))) = $true) & (((X262 @ (comp @ X266 @ X265) @ (sub @ X264 @ X265) @ X267)) != $true)) | ($true = ((X262 @ id @ (ap @ X263 @ X261) @ id))) | ? [X269 : term,X268 : term] : (($true != ((X262 @ id @ (ap @ (sub @ X268 @ id) @ X269) @ id))) & ($true = ((X262 @ id @ X268 @ id))) & ($true = ((X262 @ id @ X269 @ id)))) | ? [X273 : subst,X270 : subst,X271 : subst,X272 : term] : ((((X262 @ X271 @ X272 @ (comp @ X273 @ X270))) != $true) & (((X262 @ (comp @ X271 @ X273) @ (sub @ X272 @ X273) @ X270)) = $true)) | ($true != ((X262 @ id @ X263 @ id))))) & (pushprop_lem0_gthm = $true) & (axvarshift = $true) & (pushprop_lem3v2_lthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (hoaslamnotvar_gthm = $true) & ((axidr = $true) <=> ! [X244 : subst] : (((comp @ X244 @ id)) = X244)) & ((axscons = $true) <=> ! [X217 : subst] : (((push @ (sub @ one @ X217) @ (comp @ sh @ X217))) = X217)) & (hoaslamnotvar = $true) & (! [X82 : (subst > term > term),X83 : term,X84 : term] : ((((ap @ (sub @ X83 @ id) @ X84)) != ((lam @ (X82 @ sh @ one)))) | ? [X85 : term,X86 : subst,X87 : subst] : (((X82 @ (comp @ X87 @ X86) @ (sub @ X85 @ X86))) != ((sub @ (X82 @ X87 @ X85) @ X86)))) <=> (hoaslamnotap = $true)) & (pushprop = $true) & (hoaslamnotap_gthm = $true) & ((pushprop_lem0_lthm = $true) <=> ! [X279 : (term > $o),X281 : subst,X280 : term] : ? [X282 : (term > $o)] : ! [X283 : term] : (((X279 @ (sub @ X283 @ (push @ X280 @ X281)))) = ((X282 @ X283)))) & (! [X303 : (term > $o),X302 : subst,X299 : (term > $o),X300 : term,X301 : term] : ((((X303 @ X301)) != $true) | (((X299 @ (sub @ X301 @ (push @ X300 @ X302)))) = $true)) <=> (pushprop_lem1 = $true)) & (! [X352 : term] : (((sub @ X352 @ id)) = X352) <=> (axvarid = $true)) & (! [X492 : term,X491 : term,X493 : term,X490 : term] : ((X490 = X492) | (((ap @ (sub @ X490 @ id) @ X491)) != ((ap @ (sub @ X492 @ id) @ X493)))) <=> (hoasapinj1 = $true)) & (! [X312 : term] : ? [X313 : (subst > term > term)] : (((X313 @ sh @ one)) = ((sub @ X312 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b = $true)) & (hoasinduction_lem3v2_gthm = $true) & (hoasinduction_lem1_lthm = $true) & (axclos = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (! [X362 : (subst > term > subst > $o)] : ? [X363 : (term > $o)] : ! [X364 : term] : (((X362 @ id @ X364 @ id)) = ((X363 @ X364))) <=> (hoasinduction_lem0_lthm = $true)) & ((hoasinduction_lthm_3 = $true) <=> ((((! [X518 : term,X517 : (subst > term > subst > $o)] : (? [X523 : subst,X524 : subst,X522 : subst,X521 : term] : (($true = ((X517 @ (comp @ X522 @ X524) @ (sub @ X521 @ X524) @ X523))) & ($true != ((X517 @ X522 @ X521 @ (comp @ X524 @ X523))))) | ? [X520 : term,X519 : term] : ((((X517 @ id @ X519 @ id)) = $true) & ($true = ((X517 @ id @ X520 @ id))) & (((X517 @ id @ (ap @ (sub @ X519 @ id) @ X520) @ id)) != $true)) | ? [X525 : (subst > term > term)] : (! [X529 : term] : ((((X517 @ id @ (X525 @ id @ X529) @ id)) = $true) | ($true != ((X517 @ id @ X529 @ id)))) & ($true != ((X517 @ id @ (lam @ (X525 @ sh @ one)) @ id))) & ! [X528 : subst,X527 : subst,X526 : term] : (((X525 @ (comp @ X528 @ X527) @ (sub @ X526 @ X527))) = ((sub @ (X525 @ X528 @ X526) @ X527)))) | (((X517 @ id @ X518 @ id)) = $true) | ? [X530 : subst,X532 : subst,X531 : term,X533 : subst] : (($true != ((X517 @ (comp @ X532 @ X533) @ (sub @ X531 @ X533) @ X530))) & (((X517 @ X532 @ X531 @ (comp @ X533 @ X530))) = $true)) | ? [X534 : term] : ((((var @ (sub @ X534 @ id))) = $true) & ($true != ((X517 @ id @ X534 @ id))))) | ? [X508 : term,X507 : (term > $o),X509 : (subst > term > subst > $o)] : ((((X507 @ (lam @ X508))) != $true) & ! [X511 : term] : (($true = ((X507 @ (sub @ X508 @ (push @ X511 @ id))))) | (((X507 @ X511)) != $true)) & ! [X510 : term] : (((X507 @ X510)) = ((X509 @ id @ X510 @ id))) & ! [X512 : (subst > term > term)] : (? [X514 : subst,X513 : term,X515 : subst] : (((sub @ (X512 @ X514 @ X513) @ X515)) != ((X512 @ (comp @ X514 @ X515) @ (sub @ X513 @ X515)))) | ? [X516 : term] : ((((X509 @ id @ X516 @ id)) = $true) & (((X509 @ id @ (X512 @ id @ X516) @ id)) != $true)) | ($true = ((X509 @ id @ (lam @ (X512 @ sh @ one)) @ id)))))) | ? [X506 : term] : (((sub @ X506 @ id)) != X506)) | ? [X500 : term,X499 : (term > $o)] : (! [X502 : term] : ((((X499 @ (lam @ X502))) = $true) | ? [X503 : term] : (($true = ((X499 @ X503))) & ($true != ((X499 @ (sub @ X502 @ (push @ X503 @ id))))))) & ($true != ((X499 @ X500))) & ! [X505 : term,X504 : term] : (($true != ((X499 @ X505))) | ($true = ((X499 @ (ap @ X504 @ X505)))) | (((X499 @ X504)) != $true)) & ! [X501 : term] : ((((X499 @ X501)) = $true) | (((var @ X501)) != $true)))) | ? [X496 : (subst > term > subst > $o)] : ! [X497 : (term > $o)] : ? [X498 : term] : (((X496 @ id @ X498 @ id)) != ((X497 @ X498))))) & ! [X494 : term] : ? [X495 : d_term] : (((d2term @ X495)) = X494) & (hoaslaminj_gthm = $true) & ((hoasinduction_lem3v2a = $true) <=> ! [X207 : (term > $o),X208 : term,X209 : (subst > term > subst > $o)] : (? [X211 : term] : (((X209 @ id @ X211 @ id)) != ((X207 @ X211))) | ? [X212 : (subst > term > term)] : (! [X213 : term] : ((((X209 @ id @ X213 @ id)) != $true) | (((X209 @ id @ (X212 @ id @ X213) @ id)) = $true)) & (((X209 @ id @ (lam @ (X212 @ sh @ one)) @ id)) != $true) & ! [X215 : subst,X214 : subst,X216 : term] : (((X212 @ (comp @ X214 @ X215) @ (sub @ X216 @ X215))) = ((sub @ (X212 @ X214 @ X216) @ X215)))) | (((X207 @ (lam @ X208))) = $true) | ? [X210 : term] : ((((X207 @ X210)) = $true) & (((X207 @ (sub @ X208 @ (push @ X210 @ id)))) != $true)))) & (hoasinduction_lem1 = $true) & ((hoasapinj2_lthm = $true) <=> (? [X77 : term,X75 : term,X76 : term,X74 : term] : ((X76 != X77) & (((ap @ X75 @ X77)) = ((ap @ X74 @ X76)))) | ! [X80 : term,X81 : term,X78 : term,X79 : term] : ((X79 = X81) | (((ap @ (sub @ X80 @ id) @ X79)) != ((ap @ (sub @ X78 @ id) @ X81)))))) & (pushprop_lem1v2_gthm = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ! [X415 : d_subst,X414 : d_subst] : ((((d2subst @ X415)) != ((d2subst @ X414))) | (X414 = X415)) & ((hoasinduction_lem3aa = $true) <=> ! [X218 : term,X219 : (subst > term > subst > $o)] : (? [X221 : (subst > term > term)] : (! [X225 : term] : (($true != ((X219 @ id @ X225 @ id))) | (((X219 @ id @ (X221 @ id @ X225) @ id)) = $true)) & (((X219 @ id @ (lam @ (X221 @ sh @ one)) @ id)) != $true) & ! [X223 : subst,X222 : subst,X224 : term] : (((X221 @ (comp @ X222 @ X223) @ (sub @ X224 @ X223))) = ((sub @ (X221 @ X222 @ X224) @ X223)))) | (((X219 @ id @ (lam @ (sub @ X218 @ (push @ one @ sh))) @ id)) = $true) | ? [X220 : term] : (($true = ((X219 @ id @ X220 @ id))) & (((X219 @ id @ (sub @ X218 @ (push @ X220 @ id)) @ id)) != $true)))) & (axmap = $true) & (! [X349 : term] : ? [X350 : (subst > term > term)] : (((X350 @ sh @ one)) = ((sub @ X349 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b_lthm = $true)) & (! [X128 : term,X125 : term,X127 : term,X126 : term] : ((((ap @ (sub @ X127 @ id) @ X128)) != ((ap @ (sub @ X126 @ id) @ X125))) | (X125 = X128)) <=> (hoasapinj2 = $true)) & ((hoasinduction_lem3a = $true) <=> ! [X285 : (subst > term > subst > $o),X284 : term] : (($true = ((X285 @ id @ (lam @ X284) @ id))) | ? [X287 : (subst > term > term)] : (! [X291 : term] : (($true = ((X285 @ id @ (X287 @ id @ X291) @ id))) | (((X285 @ id @ X291 @ id)) != $true)) & ($true != ((X285 @ id @ (lam @ (X287 @ sh @ one)) @ id))) & ! [X290 : term,X288 : subst,X289 : subst] : (((X287 @ (comp @ X288 @ X289) @ (sub @ X290 @ X289))) = ((sub @ (X287 @ X288 @ X290) @ X289)))) | ? [X286 : term] : ((((X285 @ id @ X286 @ id)) = $true) & (((X285 @ id @ (sub @ X284 @ (push @ X286 @ id)) @ id)) != $true)))) & ((hoasinduction_lem2v2 = $true) <=> ! [X96 : (term > $o),X95 : term,X97 : term,X94 : (subst > term > subst > $o)] : (? [X98 : term] : (((X94 @ id @ X98 @ id)) != ((X96 @ X98))) | ? [X108 : subst,X107 : subst,X106 : term,X105 : subst] : ((((X94 @ (comp @ X107 @ X105) @ (sub @ X106 @ X105) @ X108)) = $true) & (((X94 @ X107 @ X106 @ (comp @ X105 @ X108))) != $true)) | (((X96 @ X97)) != $true) | (((X96 @ X95)) != $true) | ? [X99 : term,X100 : term] : ((((X94 @ id @ X100 @ id)) = $true) & (((X94 @ id @ X99 @ id)) = $true) & (((X94 @ id @ (ap @ (sub @ X99 @ id) @ X100) @ id)) != $true)) | ? [X103 : term,X102 : subst,X101 : subst,X104 : subst] : ((((X94 @ X102 @ X103 @ (comp @ X101 @ X104))) = $true) & ($true != ((X94 @ (comp @ X102 @ X101) @ (sub @ X103 @ X101) @ X104)))) | ($true = ((X96 @ (ap @ X97 @ X95)))))) & (! [X238 : term,X237 : term] : ((X237 = X238) | (((sub @ X237 @ sh)) != ((sub @ X238 @ sh)))) <=> (shinj = $true)) & (ulamvar1 != $true) & (hoasapnotvar_lthm = $true) & (induction2lem_gthm = $true) & (axapp = $true) & (hoasinduction_lem1v2_gthm = $true) & ! [X242 : d_term,X243 : d_term] : ((((d2term @ X242)) != ((d2term @ X243))) | (X242 = X243)) & (ulamvarsh = $true) & ((hoasinduction_lem3v2 = $true) <=> ! [X191 : (term > $o),X190 : term,X189 : (subst > term > subst > $o)] : (? [X202 : term] : (((X189 @ id @ X202 @ id)) != ((X191 @ X202))) | ? [X200 : subst,X199 : subst,X198 : subst,X201 : term] : (($true = ((X189 @ (comp @ X200 @ X199) @ (sub @ X201 @ X199) @ X198))) & (((X189 @ X200 @ X201 @ (comp @ X199 @ X198))) != $true)) | ? [X197 : term] : (($true != ((X191 @ (sub @ X190 @ (push @ X197 @ id))))) & ($true = ((X191 @ X197)))) | (((X191 @ (lam @ X190))) = $true) | ? [X204 : subst,X205 : subst,X206 : term,X203 : subst] : ((((X189 @ (comp @ X203 @ X205) @ (sub @ X206 @ X205) @ X204)) != $true) & (((X189 @ X203 @ X206 @ (comp @ X205 @ X204))) = $true)) | ? [X192 : (subst > term > term)] : (! [X193 : term] : ((((X189 @ id @ X193 @ id)) != $true) | ($true = ((X189 @ id @ (X192 @ id @ X193) @ id)))) & ! [X195 : subst,X196 : term,X194 : subst] : (((X192 @ (comp @ X195 @ X194) @ (sub @ X196 @ X194))) = ((sub @ (X192 @ X195 @ X196) @ X194))) & ($true != ((X189 @ id @ (lam @ (X192 @ sh @ one)) @ id)))))) & (! [X535 : (term > $o),X537 : subst,X536 : term] : ? [X538 : (term > $o)] : ! [X539 : term] : (((X538 @ X539)) = ((X535 @ (sub @ X539 @ (push @ X536 @ X537))))) <=> (pushprop_lem0 = $true)) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ((hoaslaminj = $true) <=> ! [X480 : (subst > term > term),X483 : (subst > term > term),X482 : term,X481 : subst] : (? [X488 : term,X487 : subst,X489 : subst] : (((X483 @ (comp @ X487 @ X489) @ (sub @ X488 @ X489))) != ((sub @ (X483 @ X487 @ X488) @ X489))) | (((X480 @ X481 @ X482)) = ((X483 @ X481 @ X482))) | ? [X485 : subst,X486 : term,X484 : subst] : (((sub @ (X480 @ X484 @ X486) @ X485)) != ((X480 @ (comp @ X484 @ X485) @ (sub @ X486 @ X485)))) | (((lam @ (X483 @ sh @ one))) != ((lam @ (X480 @ sh @ one)))))) & (! [X227 : term,X228 : subst] : (((sub @ one @ (push @ X227 @ X228))) = X227) <=> (axvarcons = $true)) & ((hoasinduction_lthm = $true) <=> (((! [X41 : (subst > term > subst > $o),X40 : term] : (? [X57 : term,X54 : subst,X56 : subst,X55 : subst] : ((((X41 @ (comp @ X56 @ X55) @ (sub @ X57 @ X55) @ X54)) != $true) & ($true = ((X41 @ X56 @ X57 @ (comp @ X55 @ X54))))) | ? [X50 : subst,X53 : subst,X51 : subst,X52 : term] : (($true != ((X41 @ X51 @ X52 @ (comp @ X50 @ X53)))) & (((X41 @ (comp @ X51 @ X50) @ (sub @ X52 @ X50) @ X53)) = $true)) | ? [X44 : term] : ((((X41 @ id @ X44 @ id)) != $true) & (((var @ (sub @ X44 @ id))) = $true)) | ? [X45 : (subst > term > term)] : ((((X41 @ id @ (lam @ (X45 @ sh @ one)) @ id)) != $true) & ! [X46 : term] : (($true = ((X41 @ id @ (X45 @ id @ X46) @ id))) | (((X41 @ id @ X46 @ id)) != $true)) & ! [X48 : term,X49 : subst,X47 : subst] : (((sub @ (X45 @ X49 @ X48) @ X47)) = ((X45 @ (comp @ X49 @ X47) @ (sub @ X48 @ X47))))) | ? [X42 : term,X43 : term] : (($true = ((X41 @ id @ X43 @ id))) & ($true = ((X41 @ id @ X42 @ id))) & ($true != ((X41 @ id @ (ap @ (sub @ X43 @ id) @ X42) @ id)))) | (((X41 @ id @ X40 @ id)) = $true)) | ? [X25 : term,X24 : (subst > term > subst > $o)] : (! [X36 : subst,X39 : subst,X37 : term,X38 : subst] : (($true = ((X24 @ X39 @ X37 @ (comp @ X38 @ X36)))) | (((X24 @ (comp @ X39 @ X38) @ (sub @ X37 @ X38) @ X36)) != $true)) & ! [X27 : subst,X26 : subst,X28 : subst,X29 : term] : ((((X24 @ (comp @ X27 @ X26) @ (sub @ X29 @ X26) @ X28)) = $true) | ($true != ((X24 @ X27 @ X29 @ (comp @ X26 @ X28))))) & ! [X35 : term] : ((((X24 @ id @ X35 @ id)) != $true) | (((X24 @ id @ (sub @ X25 @ (push @ X35 @ id)) @ id)) = $true)) & (((X24 @ id @ (lam @ X25) @ id)) != $true) & ! [X30 : (subst > term > term)] : (? [X31 : term] : ((((X24 @ id @ X31 @ id)) = $true) & (((X24 @ id @ (X30 @ id @ X31) @ id)) != $true)) | (((X24 @ id @ (lam @ (X30 @ sh @ one)) @ id)) = $true) | ? [X32 : term,X33 : subst,X34 : subst] : (((sub @ (X30 @ X33 @ X32) @ X34)) != ((X30 @ (comp @ X33 @ X34) @ (sub @ X32 @ X34))))))) | ? [X12 : (subst > term > subst > $o),X13 : term,X11 : term] : ((((X12 @ id @ X11 @ id)) = $true) & ($true = ((X12 @ id @ X13 @ id))) & ! [X20 : subst,X21 : subst,X22 : term,X23 : subst] : (($true != ((X12 @ X20 @ X22 @ (comp @ X23 @ X21)))) | (((X12 @ (comp @ X20 @ X23) @ (sub @ X22 @ X23) @ X21)) = $true)) & (((X12 @ id @ (ap @ X11 @ X13) @ id)) != $true) & ! [X16 : subst,X14 : term,X15 : subst,X17 : subst] : (($true = ((X12 @ X16 @ X14 @ (comp @ X17 @ X15)))) | (((X12 @ (comp @ X16 @ X17) @ (sub @ X14 @ X17) @ X15)) != $true)) & ! [X19 : term,X18 : term] : ((((X12 @ id @ X18 @ id)) != $true) | (((X12 @ id @ (ap @ (sub @ X19 @ id) @ X18) @ id)) = $true) | (((X12 @ id @ X19 @ id)) != $true)))) | ? [X4 : (term > $o),X5 : term] : ((((X4 @ X5)) != $true) & ! [X7 : term,X6 : term] : ((((X4 @ (ap @ X7 @ X6))) = $true) | ($true != ((X4 @ X6))) | (((X4 @ X7)) != $true)) & ! [X8 : term] : ((((X4 @ X8)) = $true) | ($true != ((var @ X8)))) & ! [X9 : term] : (? [X10 : term] : (($true != ((X4 @ (sub @ X9 @ (push @ X10 @ id))))) & (((X4 @ X10)) = $true)) | ($true = ((X4 @ (lam @ X9)))))))) & (! [X239 : (subst > term > subst > $o)] : ? [X240 : (term > $o)] : ! [X241 : term] : (((X240 @ X241)) = ((X239 @ id @ X241 @ id))) <=> (hoasinduction_lem0 = $true)) & ! [X226 : d_subst] : (d_id = X226) & (termmset_gthm = $true) & (! [X246 : (subst > term > subst > $o),X245 : term] : (? [X252 : (subst > term > term)] : (! [X253 : term] : (($true = ((X246 @ id @ (X252 @ id @ X253) @ id))) | (((X246 @ id @ X253 @ id)) != $true)) & ! [X255 : subst,X254 : subst,X256 : term] : (((X252 @ (comp @ X254 @ X255) @ (sub @ X256 @ X255))) = ((sub @ (X252 @ X254 @ X256) @ X255))) & ($true != ((X246 @ id @ (lam @ (X252 @ sh @ one)) @ id)))) | (((X246 @ id @ (lam @ X245) @ id)) = $true) | ? [X251 : term,X249 : subst,X248 : subst,X250 : subst] : (($true != ((X246 @ X249 @ X251 @ (comp @ X250 @ X248)))) & ($true = ((X246 @ (comp @ X249 @ X250) @ (sub @ X251 @ X250) @ X248)))) | ? [X257 : subst,X258 : subst,X260 : subst,X259 : term] : ((((X246 @ X257 @ X259 @ (comp @ X258 @ X260))) = $true) & (((X246 @ (comp @ X257 @ X258) @ (sub @ X259 @ X258) @ X260)) != $true)) | ? [X247 : term] : (($true = ((X246 @ id @ X247 @ id))) & ($true != ((X246 @ id @ (sub @ X245 @ (push @ X247 @ id)) @ id))))) <=> (hoasinduction_lem3 = $true)) & (! [X151 : term,X152 : term] : ((X151 = X152) | (((lam @ X151)) != ((lam @ X152)))) <=> (laminj = $true)) & (substmonoid_gthm = $true) & (hoasapinj1_gthm = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (! [X72 : term,X71 : term,X70 : term,X73 : term] : ((((ap @ X71 @ X73)) != ((ap @ X70 @ X72))) | (X70 = X71)) <=> (apinj1 = $true)) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (hoasinduction_gthm = $true) & ! [X353 : subst] : ? [X354 : d_subst] : (((d2subst @ X354)) = X353) & (hoasinduction_lem3_gthm = $true) & (hoasapnotvar_gthm = $true) & (((! [X375 : term,X374 : (subst > term > subst > $o)] : (? [X381 : subst,X382 : subst,X384 : term,X383 : subst] : (($true = ((X374 @ (comp @ X382 @ X383) @ (sub @ X384 @ X383) @ X381))) & ($true != ((X374 @ X382 @ X384 @ (comp @ X383 @ X381))))) | ($true = ((X374 @ id @ (lam @ X375) @ id))) | ? [X385 : (subst > term > term)] : (! [X389 : term] : ((((X374 @ id @ (X385 @ id @ X389) @ id)) = $true) | (((X374 @ id @ X389 @ id)) != $true)) & ! [X388 : subst,X387 : subst,X386 : term] : (((X385 @ (comp @ X388 @ X387) @ (sub @ X386 @ X387))) = ((sub @ (X385 @ X388 @ X386) @ X387))) & (((X374 @ id @ (lam @ (X385 @ sh @ one)) @ id)) != $true)) | ? [X376 : subst,X378 : subst,X377 : subst,X379 : term] : ((((X374 @ X378 @ X379 @ (comp @ X376 @ X377))) = $true) & ($true != ((X374 @ (comp @ X378 @ X376) @ (sub @ X379 @ X376) @ X377)))) | ? [X380 : term] : ((((X374 @ id @ X380 @ id)) = $true) & (((X374 @ id @ (sub @ X375 @ (push @ X380 @ id)) @ id)) != $true))) | ? [X367 : term,X366 : (subst > term > subst > $o)] : ((((X366 @ id @ (lam @ (sub @ X367 @ (push @ one @ sh))) @ id)) != $true) & ! [X368 : term] : (($true = ((X366 @ id @ (sub @ X367 @ (push @ X368 @ id)) @ id))) | (((X366 @ id @ X368 @ id)) != $true)) & ! [X369 : (subst > term > term)] : (? [X371 : subst,X370 : term,X372 : subst] : (((X369 @ (comp @ X371 @ X372) @ (sub @ X370 @ X372))) != ((sub @ (X369 @ X371 @ X370) @ X372))) | ? [X373 : term] : (($true = ((X366 @ id @ X373 @ id))) & (((X366 @ id @ (X369 @ id @ X373) @ id)) != $true)) | (((X366 @ id @ (lam @ (X369 @ sh @ one)) @ id)) = $true)))) | ? [X365 : term] : (((sub @ X365 @ id)) != X365)) <=> (hoasinduction_lem3_lthm = $true)) & ((hoasinduction_lem3v2_lthm = $true) <=> (! [X333 : (term > $o),X332 : (subst > term > subst > $o),X331 : term] : ((((X333 @ (lam @ X331))) = $true) | ? [X335 : (subst > term > term)] : (! [X338 : subst,X336 : subst,X337 : term] : (((sub @ (X335 @ X338 @ X337) @ X336)) = ((X335 @ (comp @ X338 @ X336) @ (sub @ X337 @ X336)))) & ($true != ((X332 @ id @ (lam @ (X335 @ sh @ one)) @ id))) & ! [X339 : term] : ((((X332 @ id @ (X335 @ id @ X339) @ id)) = $true) | (((X332 @ id @ X339 @ id)) != $true))) | ? [X334 : term] : ((((X333 @ X334)) = $true) & ($true != ((X333 @ (sub @ X331 @ (push @ X334 @ id)))))) | ? [X345 : subst,X348 : subst,X346 : term,X347 : subst] : ((((X332 @ (comp @ X347 @ X345) @ (sub @ X346 @ X345) @ X348)) = $true) & ($true != ((X332 @ X347 @ X346 @ (comp @ X345 @ X348))))) | ? [X340 : term] : (((X333 @ X340)) != ((X332 @ id @ X340 @ id))) | ? [X341 : subst,X342 : subst,X343 : subst,X344 : term] : (($true != ((X332 @ (comp @ X341 @ X343) @ (sub @ X344 @ X343) @ X342))) & (((X332 @ X341 @ X344 @ (comp @ X343 @ X342))) = $true))) | ? [X330 : term] : (((sub @ X330 @ id)) != X330))) & (! [X308 : term] : ? [X309 : (subst > term > term)] : ! [X311 : subst,X310 : term] : (((X309 @ X311 @ X310)) = ((sub @ X308 @ (push @ X310 @ X311)))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & (axabs = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((induction = $true) <=> ! [X89 : (term > $o),X88 : term] : (? [X92 : term] : (($true = ((X89 @ X92))) & ($true != ((X89 @ (lam @ X92))))) | ? [X90 : term,X91 : term] : (($true != ((X89 @ (ap @ X90 @ X91)))) & ($true = ((X89 @ X91))) & (((X89 @ X90)) = $true)) | (((X89 @ X88)) = $true) | ? [X93 : term] : ((((var @ X93)) = $true) & (((X89 @ X93)) != $true)))) & (((! [X319 : term,X321 : (subst > term > subst > $o),X320 : (term > $o)] : (? [X322 : (subst > term > term)] : ((((X321 @ id @ (lam @ (X322 @ sh @ one)) @ id)) != $true) & ! [X326 : term] : ((((X321 @ id @ X326 @ id)) != $true) | ($true = ((X321 @ id @ (X322 @ id @ X326) @ id)))) & ! [X323 : subst,X324 : subst,X325 : term] : (((sub @ (X322 @ X324 @ X325) @ X323)) = ((X322 @ (comp @ X324 @ X323) @ (sub @ X325 @ X323))))) | (((X320 @ (lam @ X319))) = $true) | ? [X327 : term] : (((X320 @ X327)) != ((X321 @ id @ X327 @ id))) | ? [X328 : term] : ((((X320 @ X328)) = $true) & (((X320 @ (sub @ X319 @ (push @ X328 @ id)))) != $true))) | ? [X318 : term] : (((sub @ X318 @ id)) != X318)) | ? [X314 : term] : ! [X315 : (subst > term > term)] : ? [X317 : subst,X316 : term] : (((sub @ X314 @ (push @ X316 @ X317))) != ((X315 @ X317 @ X316)))) <=> (hoasinduction_lem3v2a_lthm = $true)) & (pushprop_lem2v2_lthm = $true) & (pushprop_lem3v2 = $true) & (((var @ (d2term @ d_one))) != $true) & (! [X410 : term] : ? [X411 : (subst > term > term)] : ! [X413 : term,X412 : subst] : (((sub @ X410 @ (push @ X413 @ X412))) = ((X411 @ X412 @ X413))) <=> (hoasinduction_lem3v2_f = $true)) & ((termmset_lthm = $true) <=> (! [X139 : term] : (((sub @ X139 @ id)) = X139) | ? [X138 : term] : (((sub @ X138 @ id)) != X138))) & (pushprop_lthm_orig = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (induction2_gthm = $true) & (pushprop_gthm = $true) & (! [X293 : term,X292 : (term > $o)] : ((((X292 @ X293)) = $true) | ? [X298 : term] : (($true = ((var @ X298))) & (((X292 @ X298)) != $true)) | ? [X297 : term,X296 : term] : (($true = ((X292 @ X296))) & ($true != ((X292 @ (ap @ X297 @ X296)))) & (((X292 @ X297)) = $true)) | ? [X294 : term] : (! [X295 : term] : ((((X292 @ X295)) != $true) | (((X292 @ (sub @ X294 @ (push @ X295 @ id)))) = $true)) & (((X292 @ (lam @ X294))) != $true))) <=> (induction2 = $true)) & (hoasinduction_lem3b_gthm = $true) & ((hoasinduction_no_psi_cond = $true) <=> ! [X143 : (subst > term > subst > $o),X142 : term] : (? [X145 : term,X144 : term] : ((((X143 @ id @ (ap @ (sub @ X144 @ id) @ X145) @ id)) != $true) & (((X143 @ id @ X145 @ id)) = $true) & (((X143 @ id @ X144 @ id)) = $true)) | (((X143 @ id @ X142 @ id)) = $true) | ? [X146 : (subst > term > term)] : (! [X150 : term] : ((((X143 @ id @ (X146 @ id @ X150) @ id)) = $true) | ($true != ((X143 @ id @ X150 @ id)))) & ! [X148 : term,X147 : subst,X149 : subst] : (((sub @ (X146 @ X149 @ X148) @ X147)) = ((X146 @ (comp @ X149 @ X147) @ (sub @ X148 @ X147)))) & (((X143 @ id @ (lam @ (X146 @ sh @ one)) @ id)) != $true)))) & ((hoasapinj1_lthm = $true) <=> ((! [X423 : term,X424 : term,X422 : term,X421 : term] : ((((ap @ (sub @ X421 @ id) @ X422)) != ((ap @ (sub @ X424 @ id) @ X423))) | (X421 = X424)) | ? [X419 : term,X420 : term,X418 : term,X417 : term] : ((X417 != X418) & (((ap @ X418 @ X419)) = ((ap @ X417 @ X420))))) | ? [X416 : term] : (((sub @ X416 @ id)) != X416)))),
  inference(ennf_transformation,[],[f10])).
thf(f12,plain,(
  ((axidr = $true) <=> ! [X244 : subst] : (((comp @ X244 @ id)) = X244)) & ((? [X506 : term] : (((sub @ X506 @ id)) != X506) | ? [X500 : term,X499 : (term > $o)] : (! [X502 : term] : ((((X499 @ (lam @ X502))) = $true) | ? [X503 : term] : (($true = ((X499 @ X503))) & ($true != ((X499 @ (sub @ X502 @ (push @ X503 @ id))))))) & ($true != ((X499 @ X500))) & ! [X505 : term,X504 : term] : (($true != ((X499 @ X505))) | ($true = ((X499 @ (ap @ X504 @ X505)))) | (((X499 @ X504)) != $true)) & ! [X501 : term] : ((((X499 @ X501)) = $true) | (((var @ X501)) != $true))) | ! [X518 : term,X517 : (subst > term > subst > $o)] : (? [X523 : subst,X524 : subst,X522 : subst,X521 : term] : (($true = ((X517 @ (comp @ X522 @ X524) @ (sub @ X521 @ X524) @ X523))) & ($true != ((X517 @ X522 @ X521 @ (comp @ X524 @ X523))))) | ? [X520 : term,X519 : term] : ((((X517 @ id @ X519 @ id)) = $true) & ($true = ((X517 @ id @ X520 @ id))) & (((X517 @ id @ (ap @ (sub @ X519 @ id) @ X520) @ id)) != $true)) | ? [X525 : (subst > term > term)] : (! [X529 : term] : ((((X517 @ id @ (X525 @ id @ X529) @ id)) = $true) | ($true != ((X517 @ id @ X529 @ id)))) & ($true != ((X517 @ id @ (lam @ (X525 @ sh @ one)) @ id))) & ! [X528 : subst,X527 : subst,X526 : term] : (((X525 @ (comp @ X528 @ X527) @ (sub @ X526 @ X527))) = ((sub @ (X525 @ X528 @ X526) @ X527)))) | (((X517 @ id @ X518 @ id)) = $true) | ? [X530 : subst,X532 : subst,X531 : term,X533 : subst] : (($true != ((X517 @ (comp @ X532 @ X533) @ (sub @ X531 @ X533) @ X530))) & (((X517 @ X532 @ X531 @ (comp @ X533 @ X530))) = $true)) | ? [X534 : term] : ((((var @ (sub @ X534 @ id))) = $true) & ($true != ((X517 @ id @ X534 @ id))))) | ? [X508 : term,X507 : (term > $o),X509 : (subst > term > subst > $o)] : ((((X507 @ (lam @ X508))) != $true) & ! [X511 : term] : (($true = ((X507 @ (sub @ X508 @ (push @ X511 @ id))))) | (((X507 @ X511)) != $true)) & ! [X510 : term] : (((X507 @ X510)) = ((X509 @ id @ X510 @ id))) & ! [X512 : (subst > term > term)] : (? [X514 : subst,X513 : term,X515 : subst] : (((sub @ (X512 @ X514 @ X513) @ X515)) != ((X512 @ (comp @ X514 @ X515) @ (sub @ X513 @ X515)))) | ? [X516 : term] : ((((X509 @ id @ X516 @ id)) = $true) & (((X509 @ id @ (X512 @ id @ X516) @ id)) != $true)) | ($true = ((X509 @ id @ (lam @ (X512 @ sh @ one)) @ id))))) | ? [X496 : (subst > term > subst > $o)] : ! [X497 : (term > $o)] : ? [X498 : term] : (((X496 @ id @ X498 @ id)) != ((X497 @ X498)))) <=> (hoasinduction_lthm_3 = $true)) & (pushprop_lem1_gthm = $true) & (! [X535 : (term > $o),X537 : subst,X536 : term] : ? [X538 : (term > $o)] : ! [X539 : term] : (((X538 @ X539)) = ((X535 @ (sub @ X539 @ (push @ X536 @ X537))))) <=> (pushprop_lem0 = $true)) & ((hoasinduction_lem2 = $true) <=> ! [X263 : term,X261 : term,X262 : (subst > term > subst > $o)] : ((((X262 @ id @ X261 @ id)) != $true) | ? [X265 : subst,X264 : term,X267 : subst,X266 : subst] : ((((X262 @ X266 @ X264 @ (comp @ X265 @ X267))) = $true) & (((X262 @ (comp @ X266 @ X265) @ (sub @ X264 @ X265) @ X267)) != $true)) | ($true = ((X262 @ id @ (ap @ X263 @ X261) @ id))) | ? [X269 : term,X268 : term] : (($true != ((X262 @ id @ (ap @ (sub @ X268 @ id) @ X269) @ id))) & ($true = ((X262 @ id @ X268 @ id))) & ($true = ((X262 @ id @ X269 @ id)))) | ? [X273 : subst,X270 : subst,X271 : subst,X272 : term] : ((((X262 @ X271 @ X272 @ (comp @ X273 @ X270))) != $true) & (((X262 @ (comp @ X271 @ X273) @ (sub @ X272 @ X273) @ X270)) = $true)) | ($true != ((X262 @ id @ X263 @ id))))) & ((hoaslaminj = $true) <=> ! [X480 : (subst > term > term),X483 : (subst > term > term),X482 : term,X481 : subst] : (? [X488 : term,X487 : subst,X489 : subst] : (((X483 @ (comp @ X487 @ X489) @ (sub @ X488 @ X489))) != ((sub @ (X483 @ X487 @ X488) @ X489))) | (((X480 @ X481 @ X482)) = ((X483 @ X481 @ X482))) | ? [X485 : subst,X486 : term,X484 : subst] : (((sub @ (X480 @ X484 @ X486) @ X485)) != ((X480 @ (comp @ X484 @ X485) @ (sub @ X486 @ X485)))) | (((lam @ (X483 @ sh @ one))) != ((lam @ (X480 @ sh @ one)))))) & (! [X312 : term] : ? [X313 : (subst > term > term)] : (((X313 @ sh @ one)) = ((sub @ X312 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b = $true)) & ! [X415 : d_subst,X414 : d_subst] : ((((d2subst @ X415)) != ((d2subst @ X414))) | (X414 = X415)) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ((hoasinduction_lem3a_lthm = $true) <=> (? [X153 : term] : (((sub @ X153 @ id)) != X153) | ? [X154 : (subst > term > subst > $o),X155 : term] : (! [X161 : term] : ((((X154 @ id @ X161 @ id)) != $true) | (((X154 @ id @ (sub @ X155 @ (push @ X161 @ id)) @ id)) = $true)) & (((X154 @ id @ (lam @ (sub @ X155 @ (push @ one @ sh))) @ id)) != $true) & ! [X156 : (subst > term > term)] : ((((X154 @ id @ (lam @ (X156 @ sh @ one)) @ id)) = $true) | ? [X157 : term] : (($true != ((X154 @ id @ (X156 @ id @ X157) @ id))) & ($true = ((X154 @ id @ X157 @ id)))) | ? [X160 : subst,X158 : term,X159 : subst] : (((X156 @ (comp @ X159 @ X160) @ (sub @ X158 @ X160))) != ((sub @ (X156 @ X159 @ X158) @ X160))))) | ! [X162 : (subst > term > subst > $o),X163 : term] : ((((X162 @ id @ (lam @ X163) @ id)) = $true) | ? [X169 : term] : (($true = ((X162 @ id @ X169 @ id))) & (((X162 @ id @ (sub @ X163 @ (push @ X169 @ id)) @ id)) != $true)) | ? [X164 : (subst > term > term)] : (! [X165 : term] : (($true != ((X162 @ id @ X165 @ id))) | (((X162 @ id @ (X164 @ id @ X165) @ id)) = $true)) & ! [X167 : subst,X168 : term,X166 : subst] : (((sub @ (X164 @ X167 @ X168) @ X166)) = ((X164 @ (comp @ X167 @ X166) @ (sub @ X168 @ X166)))) & (((X162 @ id @ (lam @ (X164 @ sh @ one)) @ id)) != $true))))) & (pushprop_lem2v2 = $true) & ((hoasinduction_lem3v2a = $true) <=> ! [X207 : (term > $o),X208 : term,X209 : (subst > term > subst > $o)] : (? [X211 : term] : (((X209 @ id @ X211 @ id)) != ((X207 @ X211))) | ? [X212 : (subst > term > term)] : (! [X213 : term] : ((((X209 @ id @ X213 @ id)) != $true) | (((X209 @ id @ (X212 @ id @ X213) @ id)) = $true)) & (((X209 @ id @ (lam @ (X212 @ sh @ one)) @ id)) != $true) & ! [X215 : subst,X214 : subst,X216 : term] : (((X212 @ (comp @ X214 @ X215) @ (sub @ X216 @ X215))) = ((sub @ (X212 @ X214 @ X216) @ X215)))) | (((X207 @ (lam @ X208))) = $true) | ? [X210 : term] : ((((X207 @ X210)) = $true) & (((X207 @ (sub @ X208 @ (push @ X210 @ id)))) != $true)))) & (! [X443 : term,X442 : (subst > term > subst > $o)] : (? [X445 : (subst > term > term)] : (($true != ((X442 @ id @ (lam @ (X445 @ sh @ one)) @ id))) & ! [X446 : term] : ((((X442 @ id @ X446 @ id)) != $true) | (((X442 @ id @ (X445 @ id @ X446) @ id)) = $true)) & ! [X449 : term,X448 : subst,X447 : subst] : (((sub @ (X445 @ X448 @ X449) @ X447)) = ((X445 @ (comp @ X448 @ X447) @ (sub @ X449 @ X447))))) | (((X442 @ id @ (lam @ (sub @ X443 @ (push @ one @ sh))) @ id)) = $true) | ? [X444 : term] : ((((X442 @ id @ X444 @ id)) = $true) & ($true != ((X442 @ id @ (sub @ X443 @ (push @ X444 @ id)) @ id))))) <=> (hoasinduction_lem3aa_lthm = $true)) & (pushprop_lem1v2_gthm = $true) & ((induction2lem = $true) <=> ! [X230 : subst,X229 : term,X231 : (term > $o)] : (? [X235 : term] : (! [X236 : term] : ((((X231 @ (sub @ X235 @ (push @ X236 @ id)))) = $true) | (((X231 @ X236)) != $true)) & (((X231 @ (lam @ X235))) != $true)) | ? [X234 : term,X233 : term] : ((((X231 @ X234)) = $true) & ($true = ((X231 @ X233))) & (((X231 @ (ap @ X233 @ X234))) != $true)) | ($true = ((X231 @ (sub @ X229 @ X230)))) | ? [X232 : term] : ((((X231 @ (sub @ X232 @ X230))) != $true) & (((var @ X232)) = $true)))) & (axassoc = $true) & (pushprop_lem0_gthm = $true) & (! [X492 : term,X491 : term,X493 : term,X490 : term] : ((X490 = X492) | (((ap @ (sub @ X490 @ id) @ X491)) != ((ap @ (sub @ X492 @ id) @ X493)))) <=> (hoasapinj1 = $true)) & ((? [X367 : term,X366 : (subst > term > subst > $o)] : ((((X366 @ id @ (lam @ (sub @ X367 @ (push @ one @ sh))) @ id)) != $true) & ! [X368 : term] : (($true = ((X366 @ id @ (sub @ X367 @ (push @ X368 @ id)) @ id))) | (((X366 @ id @ X368 @ id)) != $true)) & ! [X369 : (subst > term > term)] : (? [X371 : subst,X370 : term,X372 : subst] : (((X369 @ (comp @ X371 @ X372) @ (sub @ X370 @ X372))) != ((sub @ (X369 @ X371 @ X370) @ X372))) | ? [X373 : term] : (($true = ((X366 @ id @ X373 @ id))) & (((X366 @ id @ (X369 @ id @ X373) @ id)) != $true)) | (((X366 @ id @ (lam @ (X369 @ sh @ one)) @ id)) = $true))) | ? [X365 : term] : (((sub @ X365 @ id)) != X365) | ! [X375 : term,X374 : (subst > term > subst > $o)] : (? [X381 : subst,X382 : subst,X384 : term,X383 : subst] : (($true = ((X374 @ (comp @ X382 @ X383) @ (sub @ X384 @ X383) @ X381))) & ($true != ((X374 @ X382 @ X384 @ (comp @ X383 @ X381))))) | ($true = ((X374 @ id @ (lam @ X375) @ id))) | ? [X385 : (subst > term > term)] : (! [X389 : term] : ((((X374 @ id @ (X385 @ id @ X389) @ id)) = $true) | (((X374 @ id @ X389 @ id)) != $true)) & ! [X388 : subst,X387 : subst,X386 : term] : (((X385 @ (comp @ X388 @ X387) @ (sub @ X386 @ X387))) = ((sub @ (X385 @ X388 @ X386) @ X387))) & (((X374 @ id @ (lam @ (X385 @ sh @ one)) @ id)) != $true)) | ? [X376 : subst,X378 : subst,X377 : subst,X379 : term] : ((((X374 @ X378 @ X379 @ (comp @ X376 @ X377))) = $true) & ($true != ((X374 @ (comp @ X378 @ X376) @ (sub @ X379 @ X376) @ X377)))) | ? [X380 : term] : ((((X374 @ id @ X380 @ id)) = $true) & (((X374 @ id @ (sub @ X375 @ (push @ X380 @ id)) @ id)) != $true)))) <=> (hoasinduction_lem3_lthm = $true)) & (hoasapinj2_gthm = $true) & (((var @ (d2term @ d_one))) != $true) & (! [X246 : (subst > term > subst > $o),X245 : term] : (? [X252 : (subst > term > term)] : (! [X253 : term] : (($true = ((X246 @ id @ (X252 @ id @ X253) @ id))) | (((X246 @ id @ X253 @ id)) != $true)) & ! [X255 : subst,X254 : subst,X256 : term] : (((X252 @ (comp @ X254 @ X255) @ (sub @ X256 @ X255))) = ((sub @ (X252 @ X254 @ X256) @ X255))) & ($true != ((X246 @ id @ (lam @ (X252 @ sh @ one)) @ id)))) | (((X246 @ id @ (lam @ X245) @ id)) = $true) | ? [X251 : term,X249 : subst,X248 : subst,X250 : subst] : (($true != ((X246 @ X249 @ X251 @ (comp @ X250 @ X248)))) & ($true = ((X246 @ (comp @ X249 @ X250) @ (sub @ X251 @ X250) @ X248)))) | ? [X257 : subst,X258 : subst,X260 : subst,X259 : term] : ((((X246 @ X257 @ X259 @ (comp @ X258 @ X260))) = $true) & (((X246 @ (comp @ X257 @ X258) @ (sub @ X259 @ X258) @ X260)) != $true)) | ? [X247 : term] : (($true = ((X246 @ id @ X247 @ id))) & ($true != ((X246 @ id @ (sub @ X245 @ (push @ X247 @ id)) @ id))))) <=> (hoasinduction_lem3 = $true)) & ! [X494 : term] : ? [X495 : d_term] : (((d2term @ X495)) = X494) & (axabs = $true) & (hoasinduction_lem2_lthm = $true) & (hoasapnotvar_lthm = $true) & (hoasinduction_lem1v2_gthm = $true) & (hoasinduction_lem1 = $true) & (pushprop_lem2v2_lthm = $true) & (! [X349 : term] : ? [X350 : (subst > term > term)] : (((X350 @ sh @ one)) = ((sub @ X349 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b_lthm = $true)) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((hoasinduction_lem3a = $true) <=> ! [X285 : (subst > term > subst > $o),X284 : term] : (($true = ((X285 @ id @ (lam @ X284) @ id))) | ? [X287 : (subst > term > term)] : (! [X291 : term] : (($true = ((X285 @ id @ (X287 @ id @ X291) @ id))) | (((X285 @ id @ X291 @ id)) != $true)) & ($true != ((X285 @ id @ (lam @ (X287 @ sh @ one)) @ id))) & ! [X290 : term,X288 : subst,X289 : subst] : (((X287 @ (comp @ X288 @ X289) @ (sub @ X290 @ X289))) = ((sub @ (X287 @ X288 @ X290) @ X289)))) | ? [X286 : term] : ((((X285 @ id @ X286 @ id)) = $true) & (((X285 @ id @ (sub @ X284 @ (push @ X286 @ id)) @ id)) != $true)))) & ! [X242 : d_term,X243 : d_term] : ((((d2term @ X242)) != ((d2term @ X243))) | (X242 = X243)) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((hoasinduction_lem3aa = $true) <=> ! [X218 : term,X219 : (subst > term > subst > $o)] : (? [X221 : (subst > term > term)] : (! [X225 : term] : (($true != ((X219 @ id @ X225 @ id))) | (((X219 @ id @ (X221 @ id @ X225) @ id)) = $true)) & (((X219 @ id @ (lam @ (X221 @ sh @ one)) @ id)) != $true) & ! [X223 : subst,X222 : subst,X224 : term] : (((X221 @ (comp @ X222 @ X223) @ (sub @ X224 @ X223))) = ((sub @ (X221 @ X222 @ X224) @ X223)))) | (((X219 @ id @ (lam @ (sub @ X218 @ (push @ one @ sh))) @ id)) = $true) | ? [X220 : term] : (($true = ((X219 @ id @ X220 @ id))) & (((X219 @ id @ (sub @ X218 @ (push @ X220 @ id)) @ id)) != $true)))) & (one = ((d2term @ d_one))) & (! [X151 : term,X152 : term] : ((X151 = X152) | (((lam @ X151)) != ((lam @ X152)))) <=> (laminj = $true)) & (hoaslamnotvar = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ! [X353 : subst] : ? [X354 : d_subst] : (((d2subst @ X354)) = X353) & (! [X362 : (subst > term > subst > $o)] : ? [X363 : (term > $o)] : ! [X364 : term] : (((X362 @ id @ X364 @ id)) = ((X363 @ X364))) <=> (hoasinduction_lem0_lthm = $true)) & (((d2subst @ d_id)) = id) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & ((hoasapinj2_lthm = $true) <=> (? [X77 : term,X75 : term,X76 : term,X74 : term] : ((X76 != X77) & (((ap @ X75 @ X77)) = ((ap @ X74 @ X76)))) | ! [X80 : term,X81 : term,X78 : term,X79 : term] : ((X79 = X81) | (((ap @ (sub @ X80 @ id) @ X79)) != ((ap @ (sub @ X78 @ id) @ X81)))))) & (pushprop_lem2v2_gthm = $true) & ((? [X132 : subst,X131 : term] : (((comp @ sh @ (push @ X131 @ X132))) != X132) | ? [X130 : subst,X129 : term] : (((sub @ one @ (push @ X129 @ X130))) != X129) | ! [X136 : term,X134 : (term > $o),X137 : subst,X135 : (term > $o),X133 : term] : ((((X135 @ (sub @ X136 @ (push @ X133 @ X137)))) = $true) | ($true != ((X134 @ X136))))) <=> (pushprop_lem1_lthm = $true)) & ((? [X176 : (term > $o),X175 : term] : (! [X180 : term,X179 : term] : ((((X176 @ (ap @ X179 @ X180))) = $true) | ($true != ((X176 @ X179))) | ($true != ((X176 @ X180)))) & ! [X178 : term] : (($true = ((X176 @ (lam @ X178)))) | (((X176 @ X178)) != $true)) & (((X176 @ X175)) != $true) & ! [X177 : term] : ((((var @ X177)) != $true) | (((X176 @ X177)) = $true))) | ! [X183 : subst,X182 : term,X181 : (term > $o)] : (? [X186 : term] : ((((X181 @ (sub @ X186 @ X183))) != $true) & (((var @ X186)) = $true)) | ? [X187 : term] : (! [X188 : term] : ((((X181 @ X188)) != $true) | (((X181 @ (sub @ X187 @ (push @ X188 @ id)))) = $true)) & (((X181 @ (lam @ X187))) != $true)) | ? [X184 : term,X185 : term] : ((((X181 @ X185)) = $true) & (((X181 @ X184)) = $true) & ($true != ((X181 @ (ap @ X185 @ X184))))) | ($true = ((X181 @ (sub @ X182 @ X183))))) | ? [X174 : subst] : (((comp @ X174 @ id)) != X174) | ? [X171 : term,X170 : subst] : (((sub @ one @ (push @ X171 @ X170))) != X171) | ? [X172 : subst,X173 : term] : (((comp @ sh @ (push @ X173 @ X172))) != X172)) <=> (induction2lem_lthm = $true)) & ((termmset_lthm = $true) <=> (! [X139 : term] : (((sub @ X139 @ id)) = X139) | ? [X138 : term] : (((sub @ X138 @ id)) != X138))) & (pushprop_lem3v2 = $true) & (! [X293 : term,X292 : (term > $o)] : ((((X292 @ X293)) = $true) | ? [X298 : term] : (($true = ((var @ X298))) & (((X292 @ X298)) != $true)) | ? [X297 : term,X296 : term] : (($true = ((X292 @ X296))) & ($true != ((X292 @ (ap @ X297 @ X296)))) & (((X292 @ X297)) = $true)) | ? [X294 : term] : (! [X295 : term] : ((((X292 @ X295)) != $true) | (((X292 @ (sub @ X294 @ (push @ X295 @ id)))) = $true)) & (((X292 @ (lam @ X294))) != $true))) <=> (induction2 = $true)) & (pushprop_lem3v2_lthm = $true) & (hoasinduction_lem3v2_gthm = $true) & (((d2subst @ d_id)) = sh) & (hoaslaminj_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & ((hoaslaminj_lthm = $true) <=> (? [X112 : term,X111 : subst] : (((comp @ sh @ (push @ X112 @ X111))) != X111) | ! [X115 : (subst > term > term),X116 : subst,X117 : (subst > term > term),X118 : term] : ((((X117 @ X116 @ X118)) = ((X115 @ X116 @ X118))) | ? [X122 : subst,X123 : term,X124 : subst] : (((X117 @ (comp @ X124 @ X122) @ (sub @ X123 @ X122))) != ((sub @ (X117 @ X124 @ X123) @ X122))) | ? [X119 : term,X121 : subst,X120 : subst] : (((sub @ (X115 @ X121 @ X119) @ X120)) != ((X115 @ (comp @ X121 @ X120) @ (sub @ X119 @ X120)))) | (((lam @ (X115 @ sh @ one))) != ((lam @ (X117 @ sh @ one))))) | ? [X109 : subst,X110 : term] : (((sub @ one @ (push @ X110 @ X109))) != X110) | ? [X114 : term,X113 : term] : ((((lam @ X113)) = ((lam @ X114))) & (X113 != X114)))) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & ((hoasinduction_lem3v2a_lthm = $true) <=> (? [X318 : term] : (((sub @ X318 @ id)) != X318) | ! [X319 : term,X321 : (subst > term > subst > $o),X320 : (term > $o)] : (? [X322 : (subst > term > term)] : ((((X321 @ id @ (lam @ (X322 @ sh @ one)) @ id)) != $true) & ! [X326 : term] : ((((X321 @ id @ X326 @ id)) != $true) | ($true = ((X321 @ id @ (X322 @ id @ X326) @ id)))) & ! [X323 : subst,X324 : subst,X325 : term] : (((sub @ (X322 @ X324 @ X325) @ X323)) = ((X322 @ (comp @ X324 @ X323) @ (sub @ X325 @ X323))))) | (((X320 @ (lam @ X319))) = $true) | ? [X327 : term] : (((X320 @ X327)) != ((X321 @ id @ X327 @ id))) | ? [X328 : term] : ((((X320 @ X328)) = $true) & (((X320 @ (sub @ X319 @ (push @ X328 @ id)))) != $true))) | ? [X314 : term] : ! [X315 : (subst > term > term)] : ? [X317 : subst,X316 : term] : (((sub @ X314 @ (push @ X316 @ X317))) != ((X315 @ X317 @ X316))))) & ((? [X425 : term] : (((sub @ X425 @ id)) != X425) | ! [X435 : term,X434 : (term > $o)] : ((((X434 @ X435)) = $true) | ? [X438 : term] : ((((X434 @ X438)) != $true) & (((var @ X438)) = $true)) | ? [X436 : term,X437 : term] : ((((X434 @ X436)) = $true) & (((X434 @ (ap @ X436 @ X437))) != $true) & (((X434 @ X437)) = $true)) | ? [X439 : term] : ((((X434 @ (lam @ X439))) != $true) & ! [X440 : term] : (($true != ((X434 @ X440))) | (((X434 @ (sub @ X439 @ (push @ X440 @ id)))) = $true)))) | ? [X428 : (term > $o),X426 : subst,X427 : term] : (! [X431 : term] : (($true != ((var @ X431))) | (((X428 @ (sub @ X431 @ X426))) = $true)) & ! [X429 : term] : ((((X428 @ (lam @ X429))) = $true) | ? [X430 : term] : ((((X428 @ X430)) = $true) & ($true != ((X428 @ (sub @ X429 @ (push @ X430 @ id))))))) & ! [X432 : term,X433 : term] : ((((X428 @ X433)) != $true) | ($true != ((X428 @ X432))) | ($true = ((X428 @ (ap @ X433 @ X432))))) & ($true != ((X428 @ (sub @ X427 @ X426)))))) <=> (induction2_lthm = $true)) & (! [X239 : (subst > term > subst > $o)] : ? [X240 : (term > $o)] : ! [X241 : term] : (((X240 @ X241)) = ((X239 @ id @ X241 @ id))) <=> (hoasinduction_lem0 = $true)) & (! [X82 : (subst > term > term),X83 : term,X84 : term] : ((((ap @ (sub @ X83 @ id) @ X84)) != ((lam @ (X82 @ sh @ one)))) | ? [X85 : term,X86 : subst,X87 : subst] : (((X82 @ (comp @ X87 @ X86) @ (sub @ X85 @ X86))) != ((sub @ (X82 @ X87 @ X85) @ X86)))) <=> (hoaslamnotap = $true)) & (! [X227 : term,X228 : subst] : (((sub @ one @ (push @ X227 @ X228))) = X227) <=> (axvarcons = $true)) & (hoasinduction_lem3b_gthm = $true) & (hoaslamnotvar_lthm = $true) & (termmset_gthm = $true) & ((! [X360 : subst,X357 : (term > $o),X358 : (term > $o),X359 : term] : ((((X358 @ X359)) != $true) | ? [X361 : term] : (((X358 @ (sub @ X361 @ (push @ X359 @ X360)))) != ((X357 @ X361))) | (((X357 @ one)) = $true)) | ? [X355 : term,X356 : subst] : (((sub @ one @ (push @ X355 @ X356))) != X355)) <=> (pushprop_lem1v2_lthm = $true)) & (! [X391 : term,X390 : subst] : (((comp @ sh @ (push @ X391 @ X390))) = X390) <=> (axshiftcons = $true)) & ! [X351 : d_term] : (d_one = X351) & (hoasinduction_lem3_gthm = $true) & ((! [X141 : subst] : (((comp @ id @ X141)) = X141) & ! [X140 : subst] : (((comp @ X140 @ id)) = X140)) <=> (substmonoid = $true)) & (hoasinduction_gthm = $true) & (hoaslamnotvar_gthm = $true) & (axvarshift = $true) & (hoasinduction_lem2_gthm = $true) & (axclos = $true) & (! [X303 : (term > $o),X302 : subst,X299 : (term > $o),X300 : term,X301 : term] : ((((X303 @ X301)) != $true) | (((X299 @ (sub @ X301 @ (push @ X300 @ X302)))) = $true)) <=> (pushprop_lem1 = $true)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (induction2_gthm = $true) & (hoasinduction_lem1_gthm = $true) & (ulamvarsh = $true) & ((termmset = $true) <=> ! [X329 : term] : (((sub @ X329 @ id)) = X329)) & (induction2lem_gthm = $true) & (axmap = $true) & (! [X238 : term,X237 : term] : ((X237 = X238) | (((sub @ X237 @ sh)) != ((sub @ X238 @ sh)))) <=> (shinj = $true)) & (ulamvarind = $true) & (! [X441 : subst] : (((comp @ id @ X441)) = X441) <=> (axidl = $true)) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoasinduction_lem3v2_lthm = $true) <=> (! [X333 : (term > $o),X332 : (subst > term > subst > $o),X331 : term] : ((((X333 @ (lam @ X331))) = $true) | ? [X335 : (subst > term > term)] : (! [X338 : subst,X336 : subst,X337 : term] : (((sub @ (X335 @ X338 @ X337) @ X336)) = ((X335 @ (comp @ X338 @ X336) @ (sub @ X337 @ X336)))) & ($true != ((X332 @ id @ (lam @ (X335 @ sh @ one)) @ id))) & ! [X339 : term] : ((((X332 @ id @ (X335 @ id @ X339) @ id)) = $true) | (((X332 @ id @ X339 @ id)) != $true))) | ? [X334 : term] : ((((X333 @ X334)) = $true) & ($true != ((X333 @ (sub @ X331 @ (push @ X334 @ id)))))) | ? [X345 : subst,X348 : subst,X346 : term,X347 : subst] : ((((X332 @ (comp @ X347 @ X345) @ (sub @ X346 @ X345) @ X348)) = $true) & ($true != ((X332 @ X347 @ X346 @ (comp @ X345 @ X348))))) | ? [X340 : term] : (((X333 @ X340)) != ((X332 @ id @ X340 @ id))) | ? [X341 : subst,X342 : subst,X343 : subst,X344 : term] : (($true != ((X332 @ (comp @ X341 @ X343) @ (sub @ X344 @ X343) @ X342))) & (((X332 @ X341 @ X344 @ (comp @ X343 @ X342))) = $true))) | ? [X330 : term] : (((sub @ X330 @ id)) != X330))) & (pushprop_gthm = $true) & (apnotvar = $true) & ((hoasapinj1_lthm = $true) <=> (? [X419 : term,X420 : term,X418 : term,X417 : term] : ((X417 != X418) & (((ap @ X418 @ X419)) = ((ap @ X417 @ X420)))) | ? [X416 : term] : (((sub @ X416 @ id)) != X416) | ! [X423 : term,X424 : term,X422 : term,X421 : term] : ((((ap @ (sub @ X421 @ id) @ X422)) != ((ap @ (sub @ X424 @ id) @ X423))) | (X421 = X424)))) & (! [X304 : term,X307 : term,X305 : term,X306 : term] : ((X306 = X307) | (((ap @ X304 @ X307)) != ((ap @ X305 @ X306)))) <=> (apinj2 = $true)) & ((induction = $true) <=> ! [X89 : (term > $o),X88 : term] : (? [X92 : term] : (($true = ((X89 @ X92))) & ($true != ((X89 @ (lam @ X92))))) | ? [X90 : term,X91 : term] : (($true != ((X89 @ (ap @ X90 @ X91)))) & ($true = ((X89 @ X91))) & (((X89 @ X90)) = $true)) | (((X89 @ X88)) = $true) | ? [X93 : term] : ((((var @ X93)) = $true) & (((X89 @ X93)) != $true)))) & (! [X72 : term,X71 : term,X70 : term,X73 : term] : ((((ap @ X71 @ X73)) != ((ap @ X70 @ X72))) | (X70 = X71)) <=> (apinj1 = $true)) & ((hoasinduction = $true) <=> ! [X392 : (subst > term > subst > $o),X393 : term] : (($true = ((X392 @ id @ X393 @ id))) | ? [X409 : term] : ((((var @ (sub @ X409 @ id))) = $true) & ($true != ((X392 @ id @ X409 @ id)))) | ? [X399 : term,X400 : subst,X398 : subst,X401 : subst] : ((((X392 @ (comp @ X398 @ X400) @ (sub @ X399 @ X400) @ X401)) != $true) & (((X392 @ X398 @ X399 @ (comp @ X400 @ X401))) = $true)) | ? [X408 : term,X407 : term] : ((((X392 @ id @ X407 @ id)) = $true) & (((X392 @ id @ (ap @ (sub @ X408 @ id) @ X407) @ id)) != $true) & (((X392 @ id @ X408 @ id)) = $true)) | ? [X402 : (subst > term > term)] : (! [X403 : term] : ((((X392 @ id @ (X402 @ id @ X403) @ id)) = $true) | ($true != ((X392 @ id @ X403 @ id)))) & ! [X405 : term,X404 : subst,X406 : subst] : (((sub @ (X402 @ X406 @ X405) @ X404)) = ((X402 @ (comp @ X406 @ X404) @ (sub @ X405 @ X404)))) & (((X392 @ id @ (lam @ (X402 @ sh @ one)) @ id)) != $true)) | ? [X394 : subst,X397 : subst,X396 : term,X395 : subst] : (($true = ((X392 @ (comp @ X395 @ X394) @ (sub @ X396 @ X394) @ X397))) & ($true != ((X392 @ X395 @ X396 @ (comp @ X394 @ X397))))))) & ((axscons = $true) <=> ! [X217 : subst] : (((push @ (sub @ one @ X217) @ (comp @ sh @ X217))) = X217)) & (hoasapinj1_gthm = $true) & (lamnotap != $true) & (! [X410 : term] : ? [X411 : (subst > term > term)] : ! [X413 : term,X412 : subst] : (((sub @ X410 @ (push @ X413 @ X412))) = ((X411 @ X412 @ X413))) <=> (hoasinduction_lem3v2_f = $true)) & (lamnotvar = $true) & ((hoasinduction_no_psi_cond = $true) <=> ! [X143 : (subst > term > subst > $o),X142 : term] : (? [X145 : term,X144 : term] : ((((X143 @ id @ (ap @ (sub @ X144 @ id) @ X145) @ id)) != $true) & (((X143 @ id @ X145 @ id)) = $true) & (((X143 @ id @ X144 @ id)) = $true)) | (((X143 @ id @ X142 @ id)) = $true) | ? [X146 : (subst > term > term)] : (! [X150 : term] : ((((X143 @ id @ (X146 @ id @ X150) @ id)) = $true) | ($true != ((X143 @ id @ X150 @ id)))) & ! [X148 : term,X147 : subst,X149 : subst] : (((sub @ (X146 @ X149 @ X148) @ X147)) = ((X146 @ (comp @ X149 @ X147) @ (sub @ X148 @ X147)))) & (((X143 @ id @ (lam @ (X146 @ sh @ one)) @ id)) != $true)))) & (hoasinduction_lem1v2 = $true) & (! [X59 : (subst > term > subst > $o),X58 : term] : (? [X60 : term,X61 : (subst > term > term)] : (! [X64 : subst,X63 : subst,X62 : term] : (((sub @ (X61 @ X64 @ X62) @ X63)) = ((sub @ (sub @ X60 @ (push @ X62 @ X64)) @ X63))) & ! [X68 : subst,X67 : subst,X66 : term] : (((X61 @ (comp @ X67 @ X68) @ (sub @ X66 @ X68))) = ((sub @ X60 @ (push @ (sub @ X66 @ X68) @ (comp @ X67 @ X68))))) & ($true != ((X59 @ id @ (lam @ (X61 @ sh @ one)) @ id))) & ! [X65 : term] : (($true != ((X59 @ id @ X65 @ id))) | ($true = ((X59 @ id @ (X61 @ id @ X65) @ id))))) | (((X59 @ id @ (lam @ (sub @ X58 @ (push @ one @ sh))) @ id)) = $true) | ? [X69 : term] : ((((X59 @ id @ (sub @ X58 @ (push @ X69 @ id)) @ id)) != $true) & (((X59 @ id @ X69 @ id)) = $true))) <=> (hoasinduction_lem3aaa = $true)) & (! [X352 : term] : (((sub @ X352 @ id)) = X352) <=> (axvarid = $true)) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ((pushprop_lem1v2 = $true) <=> ! [X277 : (term > $o),X276 : term,X274 : subst,X275 : (term > $o)] : ((((X275 @ one)) = $true) | (((X277 @ X276)) != $true) | ? [X278 : term] : (((X275 @ X278)) != ((X277 @ (sub @ X278 @ (push @ X276 @ X274))))))) & (! [X308 : term] : ? [X309 : (subst > term > term)] : ! [X311 : subst,X310 : term] : (((X309 @ X311 @ X310)) = ((sub @ X308 @ (push @ X310 @ X311)))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & ((? [X4 : (term > $o),X5 : term] : ((((X4 @ X5)) != $true) & ! [X7 : term,X6 : term] : ((((X4 @ (ap @ X7 @ X6))) = $true) | ($true != ((X4 @ X6))) | (((X4 @ X7)) != $true)) & ! [X8 : term] : ((((X4 @ X8)) = $true) | ($true != ((var @ X8)))) & ! [X9 : term] : (? [X10 : term] : (($true != ((X4 @ (sub @ X9 @ (push @ X10 @ id))))) & (((X4 @ X10)) = $true)) | ($true = ((X4 @ (lam @ X9)))))) | ? [X25 : term,X24 : (subst > term > subst > $o)] : (! [X36 : subst,X39 : subst,X37 : term,X38 : subst] : (($true = ((X24 @ X39 @ X37 @ (comp @ X38 @ X36)))) | (((X24 @ (comp @ X39 @ X38) @ (sub @ X37 @ X38) @ X36)) != $true)) & ! [X27 : subst,X26 : subst,X28 : subst,X29 : term] : ((((X24 @ (comp @ X27 @ X26) @ (sub @ X29 @ X26) @ X28)) = $true) | ($true != ((X24 @ X27 @ X29 @ (comp @ X26 @ X28))))) & ! [X35 : term] : ((((X24 @ id @ X35 @ id)) != $true) | (((X24 @ id @ (sub @ X25 @ (push @ X35 @ id)) @ id)) = $true)) & (((X24 @ id @ (lam @ X25) @ id)) != $true) & ! [X30 : (subst > term > term)] : (? [X31 : term] : ((((X24 @ id @ X31 @ id)) = $true) & (((X24 @ id @ (X30 @ id @ X31) @ id)) != $true)) | (((X24 @ id @ (lam @ (X30 @ sh @ one)) @ id)) = $true) | ? [X32 : term,X33 : subst,X34 : subst] : (((sub @ (X30 @ X33 @ X32) @ X34)) != ((X30 @ (comp @ X33 @ X34) @ (sub @ X32 @ X34)))))) | ! [X41 : (subst > term > subst > $o),X40 : term] : (? [X57 : term,X54 : subst,X56 : subst,X55 : subst] : ((((X41 @ (comp @ X56 @ X55) @ (sub @ X57 @ X55) @ X54)) != $true) & ($true = ((X41 @ X56 @ X57 @ (comp @ X55 @ X54))))) | ? [X50 : subst,X53 : subst,X51 : subst,X52 : term] : (($true != ((X41 @ X51 @ X52 @ (comp @ X50 @ X53)))) & (((X41 @ (comp @ X51 @ X50) @ (sub @ X52 @ X50) @ X53)) = $true)) | ? [X44 : term] : ((((X41 @ id @ X44 @ id)) != $true) & (((var @ (sub @ X44 @ id))) = $true)) | ? [X45 : (subst > term > term)] : ((((X41 @ id @ (lam @ (X45 @ sh @ one)) @ id)) != $true) & ! [X46 : term] : (($true = ((X41 @ id @ (X45 @ id @ X46) @ id))) | (((X41 @ id @ X46 @ id)) != $true)) & ! [X48 : term,X49 : subst,X47 : subst] : (((sub @ (X45 @ X49 @ X48) @ X47)) = ((X45 @ (comp @ X49 @ X47) @ (sub @ X48 @ X47))))) | ? [X42 : term,X43 : term] : (($true = ((X41 @ id @ X43 @ id))) & ($true = ((X41 @ id @ X42 @ id))) & ($true != ((X41 @ id @ (ap @ (sub @ X43 @ id) @ X42) @ id)))) | (((X41 @ id @ X40 @ id)) = $true)) | ? [X12 : (subst > term > subst > $o),X13 : term,X11 : term] : ((((X12 @ id @ X11 @ id)) = $true) & ($true = ((X12 @ id @ X13 @ id))) & ! [X20 : subst,X21 : subst,X22 : term,X23 : subst] : (($true != ((X12 @ X20 @ X22 @ (comp @ X23 @ X21)))) | (((X12 @ (comp @ X20 @ X23) @ (sub @ X22 @ X23) @ X21)) = $true)) & (((X12 @ id @ (ap @ X11 @ X13) @ id)) != $true) & ! [X16 : subst,X14 : term,X15 : subst,X17 : subst] : (($true = ((X12 @ X16 @ X14 @ (comp @ X17 @ X15)))) | (((X12 @ (comp @ X16 @ X17) @ (sub @ X14 @ X17) @ X15)) != $true)) & ! [X19 : term,X18 : term] : ((((X12 @ id @ X18 @ id)) != $true) | (((X12 @ id @ (ap @ (sub @ X19 @ id) @ X18) @ id)) = $true) | (((X12 @ id @ X19 @ id)) != $true)))) <=> (hoasinduction_lthm = $true)) & (substmonoid_gthm = $true) & (pushprop_lthm = $true) & ((hoasinduction_lem3v2 = $true) <=> ! [X191 : (term > $o),X190 : term,X189 : (subst > term > subst > $o)] : (? [X202 : term] : (((X189 @ id @ X202 @ id)) != ((X191 @ X202))) | ? [X200 : subst,X199 : subst,X198 : subst,X201 : term] : (($true = ((X189 @ (comp @ X200 @ X199) @ (sub @ X201 @ X199) @ X198))) & (((X189 @ X200 @ X201 @ (comp @ X199 @ X198))) != $true)) | ? [X197 : term] : (($true != ((X191 @ (sub @ X190 @ (push @ X197 @ id))))) & ($true = ((X191 @ X197)))) | (((X191 @ (lam @ X190))) = $true) | ? [X204 : subst,X205 : subst,X206 : term,X203 : subst] : ((((X189 @ (comp @ X203 @ X205) @ (sub @ X206 @ X205) @ X204)) != $true) & (((X189 @ X203 @ X206 @ (comp @ X205 @ X204))) = $true)) | ? [X192 : (subst > term > term)] : (! [X193 : term] : ((((X189 @ id @ X193 @ id)) != $true) | ($true = ((X189 @ id @ (X192 @ id @ X193) @ id)))) & ! [X195 : subst,X196 : term,X194 : subst] : (((X192 @ (comp @ X195 @ X194) @ (sub @ X196 @ X194))) = ((sub @ (X192 @ X195 @ X196) @ X194))) & ($true != ((X189 @ id @ (lam @ (X192 @ sh @ one)) @ id)))))) & (axapp = $true) & (hoasinduction_lem2v2_gthm = $true) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem1_lthm = $true) & ((substmonoid_lthm = $true) <=> (? [X0 : subst] : (((comp @ id @ X0)) != X0) | (! [X3 : subst] : (((comp @ X3 @ id)) = X3) & ! [X2 : subst] : (((comp @ id @ X2)) = X2)) | ? [X1 : subst] : (((comp @ X1 @ id)) != X1))) & (hoaslamnotap_lthm = $true) & (hoasinduction_lem3a_gthm = $true) & ! [X226 : d_subst] : (d_id = X226) & (pushprop = $true) & (! [X128 : term,X125 : term,X127 : term,X126 : term] : ((((ap @ (sub @ X127 @ id) @ X128)) != ((ap @ (sub @ X126 @ id) @ X125))) | (X125 = X128)) <=> (hoasapinj2 = $true)) & ((? [X460 : term] : (((sub @ X460 @ id)) != X460) | ! [X472 : term,X471 : (subst > term > subst > $o)] : (($true = ((X471 @ id @ X472 @ id))) | ? [X473 : (subst > term > term)] : (! [X475 : subst,X476 : subst,X477 : term] : (((X473 @ (comp @ X476 @ X475) @ (sub @ X477 @ X475))) = ((sub @ (X473 @ X476 @ X477) @ X475))) & ! [X474 : term] : (($true = ((X471 @ id @ (X473 @ id @ X474) @ id))) | (((X471 @ id @ X474 @ id)) != $true)) & (((X471 @ id @ (lam @ (X473 @ sh @ one)) @ id)) != $true)) | ? [X478 : term,X479 : term] : (($true = ((X471 @ id @ X479 @ id))) & (((X471 @ id @ (ap @ (sub @ X479 @ id) @ X478) @ id)) != $true) & ($true = ((X471 @ id @ X478 @ id))))) | ? [X450 : (subst > term > subst > $o)] : ! [X451 : (term > $o)] : ? [X452 : term] : (((X451 @ X452)) != ((X450 @ id @ X452 @ id))) | ? [X463 : (subst > term > subst > $o),X462 : (term > $o),X461 : term] : (! [X464 : term] : ((((X462 @ X464)) != $true) | (((X462 @ (sub @ X461 @ (push @ X464 @ id)))) = $true)) & ! [X470 : term] : (((X462 @ X470)) = ((X463 @ id @ X470 @ id))) & ! [X465 : (subst > term > term)] : (? [X469 : term] : ((((X463 @ id @ X469 @ id)) = $true) & ($true != ((X463 @ id @ (X465 @ id @ X469) @ id)))) | (((X463 @ id @ (lam @ (X465 @ sh @ one)) @ id)) = $true) | ? [X467 : subst,X466 : term,X468 : subst] : (((sub @ (X465 @ X467 @ X466) @ X468)) != ((X465 @ (comp @ X467 @ X468) @ (sub @ X466 @ X468))))) & (((X462 @ (lam @ X461))) != $true)) | ? [X453 : (term > $o),X454 : term] : (($true != ((X453 @ X454))) & ! [X455 : term] : (($true = ((X453 @ X455))) | (((var @ X455)) != $true)) & ! [X456 : term,X457 : term] : ((((X453 @ (ap @ X456 @ X457))) = $true) | (((X453 @ X456)) != $true) | (((X453 @ X457)) != $true)) & ! [X458 : term] : (? [X459 : term] : ((((X453 @ X459)) = $true) & (((X453 @ (sub @ X458 @ (push @ X459 @ id)))) != $true)) | (((X453 @ (lam @ X458))) = $true)))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & ((pushprop_lem0_lthm = $true) <=> ! [X279 : (term > $o),X281 : subst,X280 : term] : ? [X282 : (term > $o)] : ! [X283 : term] : (((X279 @ (sub @ X283 @ (push @ X280 @ X281)))) = ((X282 @ X283)))) & (hoasapnotvar = $true) & (ulamvar1 != $true) & (pushprop_lthm_orig = $true) & ((hoasinduction_lem2v2 = $true) <=> ! [X96 : (term > $o),X95 : term,X97 : term,X94 : (subst > term > subst > $o)] : (? [X98 : term] : (((X94 @ id @ X98 @ id)) != ((X96 @ X98))) | ? [X108 : subst,X107 : subst,X106 : term,X105 : subst] : ((((X94 @ (comp @ X107 @ X105) @ (sub @ X106 @ X105) @ X108)) = $true) & (((X94 @ X107 @ X106 @ (comp @ X105 @ X108))) != $true)) | (((X96 @ X97)) != $true) | (((X96 @ X95)) != $true) | ? [X99 : term,X100 : term] : ((((X94 @ id @ X100 @ id)) = $true) & (((X94 @ id @ X99 @ id)) = $true) & (((X94 @ id @ (ap @ (sub @ X99 @ id) @ X100) @ id)) != $true)) | ? [X103 : term,X102 : subst,X101 : subst,X104 : subst] : ((((X94 @ X102 @ X103 @ (comp @ X101 @ X104))) = $true) & ($true != ((X94 @ (comp @ X102 @ X101) @ (sub @ X103 @ X101) @ X104)))) | ($true = ((X96 @ (ap @ X97 @ X95)))))) & (hoaslamnotap_gthm = $true)),
  inference(flattening,[],[f11])).
thf(f13,definition,(
  ! [X94 : (subst > term > subst > $o)] : ((((sP0 @ X94)) = $true) <=> ? [X99 : term,X100 : term] : ((((X94 @ id @ X100 @ id)) = $true) & (((X94 @ id @ X99 @ id)) = $true) & (((X94 @ id @ (ap @ (sub @ X99 @ id) @ X100) @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f14,definition,(
  (sP1 = $true) <=> ! [X96 : (term > $o),X95 : term,X97 : term,X94 : (subst > term > subst > $o)] : (? [X98 : term] : (((X94 @ id @ X98 @ id)) != ((X96 @ X98))) | ? [X108 : subst,X107 : subst,X106 : term,X105 : subst] : ((((X94 @ (comp @ X107 @ X105) @ (sub @ X106 @ X105) @ X108)) = $true) & (((X94 @ X107 @ X106 @ (comp @ X105 @ X108))) != $true)) | (((X96 @ X97)) != $true) | (((X96 @ X95)) != $true) | (((sP0 @ X94)) = $true) | ? [X103 : term,X102 : subst,X101 : subst,X104 : subst] : ((((X94 @ X102 @ X103 @ (comp @ X101 @ X104))) = $true) & ($true != ((X94 @ (comp @ X102 @ X101) @ (sub @ X103 @ X101) @ X104)))) | ($true = ((X96 @ (ap @ X97 @ X95)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f15,definition,(
  ! [X453 : (term > $o)] : ((((sP2 @ X453)) = $true) <=> ! [X456 : term,X457 : term] : ((((X453 @ (ap @ X456 @ X457))) = $true) | (((X453 @ X456)) != $true) | (((X453 @ X457)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f16,definition,(
  ! [X471 : (subst > term > subst > $o)] : ((((sP3 @ X471)) = $true) <=> ? [X478 : term,X479 : term] : (($true = ((X471 @ id @ X479 @ id))) & (((X471 @ id @ (ap @ (sub @ X479 @ id) @ X478) @ id)) != $true) & ($true = ((X471 @ id @ X478 @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f17,definition,(
  (sP4 = $true) <=> ? [X463 : (subst > term > subst > $o),X462 : (term > $o),X461 : term] : (! [X464 : term] : ((((X462 @ X464)) != $true) | (((X462 @ (sub @ X461 @ (push @ X464 @ id)))) = $true)) & ! [X470 : term] : (((X462 @ X470)) = ((X463 @ id @ X470 @ id))) & ! [X465 : (subst > term > term)] : (? [X469 : term] : ((((X463 @ id @ X469 @ id)) = $true) & ($true != ((X463 @ id @ (X465 @ id @ X469) @ id)))) | (((X463 @ id @ (lam @ (X465 @ sh @ one)) @ id)) = $true) | ? [X467 : subst,X466 : term,X468 : subst] : (((sub @ (X465 @ X467 @ X466) @ X468)) != ((X465 @ (comp @ X467 @ X468) @ (sub @ X466 @ X468))))) & (((X462 @ (lam @ X461))) != $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f18,definition,(
  (sP5 = $true) <=> ? [X453 : (term > $o),X454 : term] : (($true != ((X453 @ X454))) & ! [X455 : term] : (($true = ((X453 @ X455))) | (((var @ X455)) != $true)) & (((sP2 @ X453)) = $true) & ! [X458 : term] : (? [X459 : term] : ((((X453 @ X459)) = $true) & (((X453 @ (sub @ X458 @ (push @ X459 @ id)))) != $true)) | (((X453 @ (lam @ X458))) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f19,definition,(
  ($true = sP6) <=> (? [X460 : term] : (((sub @ X460 @ id)) != X460) | ! [X472 : term,X471 : (subst > term > subst > $o)] : (($true = ((X471 @ id @ X472 @ id))) | ? [X473 : (subst > term > term)] : (! [X475 : subst,X476 : subst,X477 : term] : (((X473 @ (comp @ X476 @ X475) @ (sub @ X477 @ X475))) = ((sub @ (X473 @ X476 @ X477) @ X475))) & ! [X474 : term] : (($true = ((X471 @ id @ (X473 @ id @ X474) @ id))) | (((X471 @ id @ X474 @ id)) != $true)) & (((X471 @ id @ (lam @ (X473 @ sh @ one)) @ id)) != $true)) | (((sP3 @ X471)) = $true)) | ? [X450 : (subst > term > subst > $o)] : ! [X451 : (term > $o)] : ? [X452 : term] : (((X451 @ X452)) != ((X450 @ id @ X452 @ id))) | (sP4 = $true) | (sP5 = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f20,definition,(
  ! [X189 : (subst > term > subst > $o)] : ((((sP7 @ X189)) = $true) <=> ? [X192 : (subst > term > term)] : (! [X193 : term] : ((((X189 @ id @ X193 @ id)) != $true) | ($true = ((X189 @ id @ (X192 @ id @ X193) @ id)))) & ! [X195 : subst,X196 : term,X194 : subst] : (((X192 @ (comp @ X195 @ X194) @ (sub @ X196 @ X194))) = ((sub @ (X192 @ X195 @ X196) @ X194))) & ($true != ((X189 @ id @ (lam @ (X192 @ sh @ one)) @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f21,definition,(
  ($true = sP8) <=> ! [X191 : (term > $o),X190 : term,X189 : (subst > term > subst > $o)] : (? [X202 : term] : (((X189 @ id @ X202 @ id)) != ((X191 @ X202))) | ? [X200 : subst,X199 : subst,X198 : subst,X201 : term] : (($true = ((X189 @ (comp @ X200 @ X199) @ (sub @ X201 @ X199) @ X198))) & (((X189 @ X200 @ X201 @ (comp @ X199 @ X198))) != $true)) | ? [X197 : term] : (($true != ((X191 @ (sub @ X190 @ (push @ X197 @ id))))) & ($true = ((X191 @ X197)))) | (((X191 @ (lam @ X190))) = $true) | ? [X204 : subst,X205 : subst,X206 : term,X203 : subst] : ((((X189 @ (comp @ X203 @ X205) @ (sub @ X206 @ X205) @ X204)) != $true) & (((X189 @ X203 @ X206 @ (comp @ X205 @ X204))) = $true)) | (((sP7 @ X189)) = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f22,definition,(
  ! [X12 : (subst > term > subst > $o)] : (($true = ((sP9 @ X12))) <=> ! [X19 : term,X18 : term] : ((((X12 @ id @ X18 @ id)) != $true) | (((X12 @ id @ (ap @ (sub @ X19 @ id) @ X18) @ id)) = $true) | (((X12 @ id @ X19 @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f23,definition,(
  ! [X41 : (subst > term > subst > $o)] : (($true = ((sP10 @ X41))) <=> ? [X42 : term,X43 : term] : (($true = ((X41 @ id @ X43 @ id))) & ($true = ((X41 @ id @ X42 @ id))) & ($true != ((X41 @ id @ (ap @ (sub @ X43 @ id) @ X42) @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f24,definition,(
  ! [X41 : (subst > term > subst > $o)] : ((((sP11 @ X41)) = $true) <=> ? [X45 : (subst > term > term)] : ((((X41 @ id @ (lam @ (X45 @ sh @ one)) @ id)) != $true) & ! [X46 : term] : (($true = ((X41 @ id @ (X45 @ id @ X46) @ id))) | (((X41 @ id @ X46 @ id)) != $true)) & ! [X48 : term,X49 : subst,X47 : subst] : (((sub @ (X45 @ X49 @ X48) @ X47)) = ((X45 @ (comp @ X49 @ X47) @ (sub @ X48 @ X47))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f25,definition,(
  ! [X24 : (subst > term > subst > $o)] : (($true = ((sP12 @ X24))) <=> ! [X30 : (subst > term > term)] : (? [X31 : term] : ((((X24 @ id @ X31 @ id)) = $true) & (((X24 @ id @ (X30 @ id @ X31) @ id)) != $true)) | (((X24 @ id @ (lam @ (X30 @ sh @ one)) @ id)) = $true) | ? [X32 : term,X33 : subst,X34 : subst] : (((sub @ (X30 @ X33 @ X32) @ X34)) != ((X30 @ (comp @ X33 @ X34) @ (sub @ X32 @ X34))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f26,definition,(
  ! [X4 : (term > $o)] : ((((sP13 @ X4)) = $true) <=> ! [X7 : term,X6 : term] : ((((X4 @ (ap @ X7 @ X6))) = $true) | ($true != ((X4 @ X6))) | (((X4 @ X7)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f27,definition,(
  (sP14 = $true) <=> ? [X25 : term,X24 : (subst > term > subst > $o)] : (! [X36 : subst,X39 : subst,X37 : term,X38 : subst] : (($true = ((X24 @ X39 @ X37 @ (comp @ X38 @ X36)))) | (((X24 @ (comp @ X39 @ X38) @ (sub @ X37 @ X38) @ X36)) != $true)) & ! [X27 : subst,X26 : subst,X28 : subst,X29 : term] : ((((X24 @ (comp @ X27 @ X26) @ (sub @ X29 @ X26) @ X28)) = $true) | ($true != ((X24 @ X27 @ X29 @ (comp @ X26 @ X28))))) & ! [X35 : term] : ((((X24 @ id @ X35 @ id)) != $true) | (((X24 @ id @ (sub @ X25 @ (push @ X35 @ id)) @ id)) = $true)) & (((X24 @ id @ (lam @ X25) @ id)) != $true) & ($true = ((sP12 @ X24))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f28,definition,(
  (sP15 = $true) <=> ! [X41 : (subst > term > subst > $o),X40 : term] : (? [X57 : term,X54 : subst,X56 : subst,X55 : subst] : ((((X41 @ (comp @ X56 @ X55) @ (sub @ X57 @ X55) @ X54)) != $true) & ($true = ((X41 @ X56 @ X57 @ (comp @ X55 @ X54))))) | ? [X50 : subst,X53 : subst,X51 : subst,X52 : term] : (($true != ((X41 @ X51 @ X52 @ (comp @ X50 @ X53)))) & (((X41 @ (comp @ X51 @ X50) @ (sub @ X52 @ X50) @ X53)) = $true)) | ? [X44 : term] : ((((X41 @ id @ X44 @ id)) != $true) & (((var @ (sub @ X44 @ id))) = $true)) | (((sP11 @ X41)) = $true) | ($true = ((sP10 @ X41))) | (((X41 @ id @ X40 @ id)) = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f29,definition,(
  (sP16 = $true) <=> ? [X12 : (subst > term > subst > $o),X13 : term,X11 : term] : ((((X12 @ id @ X11 @ id)) = $true) & ($true = ((X12 @ id @ X13 @ id))) & ! [X20 : subst,X21 : subst,X22 : term,X23 : subst] : (($true != ((X12 @ X20 @ X22 @ (comp @ X23 @ X21)))) | (((X12 @ (comp @ X20 @ X23) @ (sub @ X22 @ X23) @ X21)) = $true)) & (((X12 @ id @ (ap @ X11 @ X13) @ id)) != $true) & ! [X16 : subst,X14 : term,X15 : subst,X17 : subst] : (($true = ((X12 @ X16 @ X14 @ (comp @ X17 @ X15)))) | (((X12 @ (comp @ X16 @ X17) @ (sub @ X14 @ X17) @ X15)) != $true)) & ($true = ((sP9 @ X12))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f30,definition,(
  ($true = sP17) <=> (? [X4 : (term > $o),X5 : term] : ((((X4 @ X5)) != $true) & (((sP13 @ X4)) = $true) & ! [X8 : term] : ((((X4 @ X8)) = $true) | ($true != ((var @ X8)))) & ! [X9 : term] : (? [X10 : term] : (($true != ((X4 @ (sub @ X9 @ (push @ X10 @ id))))) & (((X4 @ X10)) = $true)) | ($true = ((X4 @ (lam @ X9)))))) | (sP14 = $true) | (sP15 = $true) | (sP16 = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f31,definition,(
  (sP18 = $true) <=> ! [X59 : (subst > term > subst > $o),X58 : term] : (? [X60 : term,X61 : (subst > term > term)] : (! [X64 : subst,X63 : subst,X62 : term] : (((sub @ (X61 @ X64 @ X62) @ X63)) = ((sub @ (sub @ X60 @ (push @ X62 @ X64)) @ X63))) & ! [X68 : subst,X67 : subst,X66 : term] : (((X61 @ (comp @ X67 @ X68) @ (sub @ X66 @ X68))) = ((sub @ X60 @ (push @ (sub @ X66 @ X68) @ (comp @ X67 @ X68))))) & ($true != ((X59 @ id @ (lam @ (X61 @ sh @ one)) @ id))) & ! [X65 : term] : (($true != ((X59 @ id @ X65 @ id))) | ($true = ((X59 @ id @ (X61 @ id @ X65) @ id))))) | (((X59 @ id @ (lam @ (sub @ X58 @ (push @ one @ sh))) @ id)) = $true) | ? [X69 : term] : ((((X59 @ id @ (sub @ X58 @ (push @ X69 @ id)) @ id)) != $true) & (((X59 @ id @ X69 @ id)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f32,definition,(
  ! [X143 : (subst > term > subst > $o)] : (($true = ((sP19 @ X143))) <=> ? [X146 : (subst > term > term)] : (! [X150 : term] : ((((X143 @ id @ (X146 @ id @ X150) @ id)) = $true) | ($true != ((X143 @ id @ X150 @ id)))) & ! [X148 : term,X147 : subst,X149 : subst] : (((sub @ (X146 @ X149 @ X148) @ X147)) = ((X146 @ (comp @ X149 @ X147) @ (sub @ X148 @ X147)))) & (((X143 @ id @ (lam @ (X146 @ sh @ one)) @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f33,definition,(
  ! [X392 : (subst > term > subst > $o)] : (($true = ((sP20 @ X392))) <=> ? [X402 : (subst > term > term)] : (! [X403 : term] : ((((X392 @ id @ (X402 @ id @ X403) @ id)) = $true) | ($true != ((X392 @ id @ X403 @ id)))) & ! [X405 : term,X404 : subst,X406 : subst] : (((sub @ (X402 @ X406 @ X405) @ X404)) = ((X402 @ (comp @ X406 @ X404) @ (sub @ X405 @ X404)))) & (((X392 @ id @ (lam @ (X402 @ sh @ one)) @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f34,definition,(
  ! [X392 : (subst > term > subst > $o)] : ((((sP21 @ X392)) = $true) <=> ? [X408 : term,X407 : term] : ((((X392 @ id @ X407 @ id)) = $true) & (((X392 @ id @ (ap @ (sub @ X408 @ id) @ X407) @ id)) != $true) & (((X392 @ id @ X408 @ id)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f35,definition,(
  ($true = sP22) <=> ! [X392 : (subst > term > subst > $o),X393 : term] : (($true = ((X392 @ id @ X393 @ id))) | ? [X409 : term] : ((((var @ (sub @ X409 @ id))) = $true) & ($true != ((X392 @ id @ X409 @ id)))) | ? [X399 : term,X400 : subst,X398 : subst,X401 : subst] : ((((X392 @ (comp @ X398 @ X400) @ (sub @ X399 @ X400) @ X401)) != $true) & (((X392 @ X398 @ X399 @ (comp @ X400 @ X401))) = $true)) | (((sP21 @ X392)) = $true) | ($true = ((sP20 @ X392))) | ? [X394 : subst,X397 : subst,X396 : term,X395 : subst] : (($true = ((X392 @ (comp @ X395 @ X394) @ (sub @ X396 @ X394) @ X397))) & ($true != ((X392 @ X395 @ X396 @ (comp @ X394 @ X397))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f36,definition,(
  ! [X89 : (term > $o)] : (($true = ((sP23 @ X89))) <=> ? [X90 : term,X91 : term] : (($true != ((X89 @ (ap @ X90 @ X91)))) & ($true = ((X89 @ X91))) & (((X89 @ X90)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f37,definition,(
  ! [X332 : (subst > term > subst > $o)] : ((((sP24 @ X332)) = $true) <=> ? [X335 : (subst > term > term)] : (! [X338 : subst,X336 : subst,X337 : term] : (((sub @ (X335 @ X338 @ X337) @ X336)) = ((X335 @ (comp @ X338 @ X336) @ (sub @ X337 @ X336)))) & ($true != ((X332 @ id @ (lam @ (X335 @ sh @ one)) @ id))) & ! [X339 : term] : ((((X332 @ id @ (X335 @ id @ X339) @ id)) = $true) | (((X332 @ id @ X339 @ id)) != $true))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f38,definition,(
  (sP25 = $true) <=> (! [X333 : (term > $o),X332 : (subst > term > subst > $o),X331 : term] : ((((X333 @ (lam @ X331))) = $true) | (((sP24 @ X332)) = $true) | ? [X334 : term] : ((((X333 @ X334)) = $true) & ($true != ((X333 @ (sub @ X331 @ (push @ X334 @ id)))))) | ? [X345 : subst,X348 : subst,X346 : term,X347 : subst] : ((((X332 @ (comp @ X347 @ X345) @ (sub @ X346 @ X345) @ X348)) = $true) & ($true != ((X332 @ X347 @ X346 @ (comp @ X345 @ X348))))) | ? [X340 : term] : (((X333 @ X340)) != ((X332 @ id @ X340 @ id))) | ? [X341 : subst,X342 : subst,X343 : subst,X344 : term] : (($true != ((X332 @ (comp @ X341 @ X343) @ (sub @ X344 @ X343) @ X342))) & (((X332 @ X341 @ X344 @ (comp @ X343 @ X342))) = $true))) | ? [X330 : term] : (((sub @ X330 @ id)) != X330))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f39,definition,(
  ! [X428 : (term > $o)] : ((((sP26 @ X428)) = $true) <=> ! [X432 : term,X433 : term] : ((((X428 @ X433)) != $true) | ($true != ((X428 @ X432))) | ($true = ((X428 @ (ap @ X433 @ X432))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f40,definition,(
  ! [X434 : (term > $o)] : ((((sP27 @ X434)) = $true) <=> ? [X436 : term,X437 : term] : ((((X434 @ X436)) = $true) & (((X434 @ (ap @ X436 @ X437))) != $true) & (((X434 @ X437)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f41,definition,(
  ($true = sP28) <=> ! [X435 : term,X434 : (term > $o)] : ((((X434 @ X435)) = $true) | ? [X438 : term] : ((((X434 @ X438)) != $true) & (((var @ X438)) = $true)) | (((sP27 @ X434)) = $true) | ? [X439 : term] : ((((X434 @ (lam @ X439))) != $true) & ! [X440 : term] : (($true != ((X434 @ X440))) | (((X434 @ (sub @ X439 @ (push @ X440 @ id)))) = $true))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f42,definition,(
  (sP29 = $true) <=> (? [X425 : term] : (((sub @ X425 @ id)) != X425) | ($true = sP28) | ? [X428 : (term > $o),X426 : subst,X427 : term] : (! [X431 : term] : (($true != ((var @ X431))) | (((X428 @ (sub @ X431 @ X426))) = $true)) & ! [X429 : term] : ((((X428 @ (lam @ X429))) = $true) | ? [X430 : term] : ((((X428 @ X430)) = $true) & ($true != ((X428 @ (sub @ X429 @ (push @ X430 @ id))))))) & (((sP26 @ X428)) = $true) & ($true != ((X428 @ (sub @ X427 @ X426))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f43,definition,(
  (sP30 = $true) <=> (? [X318 : term] : (((sub @ X318 @ id)) != X318) | ! [X319 : term,X321 : (subst > term > subst > $o),X320 : (term > $o)] : (? [X322 : (subst > term > term)] : ((((X321 @ id @ (lam @ (X322 @ sh @ one)) @ id)) != $true) & ! [X326 : term] : ((((X321 @ id @ X326 @ id)) != $true) | ($true = ((X321 @ id @ (X322 @ id @ X326) @ id)))) & ! [X323 : subst,X324 : subst,X325 : term] : (((sub @ (X322 @ X324 @ X325) @ X323)) = ((X322 @ (comp @ X324 @ X323) @ (sub @ X325 @ X323))))) | (((X320 @ (lam @ X319))) = $true) | ? [X327 : term] : (((X320 @ X327)) != ((X321 @ id @ X327 @ id))) | ? [X328 : term] : ((((X320 @ X328)) = $true) & (((X320 @ (sub @ X319 @ (push @ X328 @ id)))) != $true))) | ? [X314 : term] : ! [X315 : (subst > term > term)] : ? [X317 : subst,X316 : term] : (((sub @ X314 @ (push @ X316 @ X317))) != ((X315 @ X317 @ X316))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f44,definition,(
  (sP31 = $true) <=> (? [X112 : term,X111 : subst] : (((comp @ sh @ (push @ X112 @ X111))) != X111) | ! [X115 : (subst > term > term),X116 : subst,X117 : (subst > term > term),X118 : term] : ((((X117 @ X116 @ X118)) = ((X115 @ X116 @ X118))) | ? [X122 : subst,X123 : term,X124 : subst] : (((X117 @ (comp @ X124 @ X122) @ (sub @ X123 @ X122))) != ((sub @ (X117 @ X124 @ X123) @ X122))) | ? [X119 : term,X121 : subst,X120 : subst] : (((sub @ (X115 @ X121 @ X119) @ X120)) != ((X115 @ (comp @ X121 @ X120) @ (sub @ X119 @ X120)))) | (((lam @ (X115 @ sh @ one))) != ((lam @ (X117 @ sh @ one))))) | ? [X109 : subst,X110 : term] : (((sub @ one @ (push @ X110 @ X109))) != X110) | ? [X114 : term,X113 : term] : ((((lam @ X113)) = ((lam @ X114))) & (X113 != X114)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f45,definition,(
  ! [X292 : (term > $o)] : (($true = ((sP32 @ X292))) <=> ? [X297 : term,X296 : term] : (($true = ((X292 @ X296))) & ($true != ((X292 @ (ap @ X297 @ X296)))) & (((X292 @ X297)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f46,definition,(
  (sP33 = $true) <=> ! [X293 : term,X292 : (term > $o)] : ((((X292 @ X293)) = $true) | ? [X298 : term] : (($true = ((var @ X298))) & (((X292 @ X298)) != $true)) | ($true = ((sP32 @ X292))) | ? [X294 : term] : (! [X295 : term] : ((((X292 @ X295)) != $true) | (((X292 @ (sub @ X294 @ (push @ X295 @ id)))) = $true)) & (((X292 @ (lam @ X294))) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f47,definition,(
  ! [X181 : (term > $o)] : ((((sP34 @ X181)) = $true) <=> ? [X184 : term,X185 : term] : ((((X181 @ X185)) = $true) & (((X181 @ X184)) = $true) & ($true != ((X181 @ (ap @ X185 @ X184))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f48,definition,(
  ! [X176 : (term > $o)] : ((((sP35 @ X176)) = $true) <=> ! [X180 : term,X179 : term] : ((((X176 @ (ap @ X179 @ X180))) = $true) | ($true != ((X176 @ X179))) | ($true != ((X176 @ X180)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f49,definition,(
  ($true = sP36) <=> ! [X183 : subst,X182 : term,X181 : (term > $o)] : (? [X186 : term] : ((((X181 @ (sub @ X186 @ X183))) != $true) & (((var @ X186)) = $true)) | ? [X187 : term] : (! [X188 : term] : ((((X181 @ X188)) != $true) | (((X181 @ (sub @ X187 @ (push @ X188 @ id)))) = $true)) & (((X181 @ (lam @ X187))) != $true)) | (((sP34 @ X181)) = $true) | ($true = ((X181 @ (sub @ X182 @ X183)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f50,definition,(
  ($true = sP37) <=> (? [X176 : (term > $o),X175 : term] : ((((sP35 @ X176)) = $true) & ! [X178 : term] : (($true = ((X176 @ (lam @ X178)))) | (((X176 @ X178)) != $true)) & (((X176 @ X175)) != $true) & ! [X177 : term] : ((((var @ X177)) != $true) | (((X176 @ X177)) = $true))) | ($true = sP36) | ? [X174 : subst] : (((comp @ X174 @ id)) != X174) | ? [X171 : term,X170 : subst] : (((sub @ one @ (push @ X171 @ X170))) != X171) | ? [X172 : subst,X173 : term] : (((comp @ sh @ (push @ X173 @ X172))) != X172))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f51,definition,(
  ($true = sP38) <=> ! [X218 : term,X219 : (subst > term > subst > $o)] : (? [X221 : (subst > term > term)] : (! [X225 : term] : (($true != ((X219 @ id @ X225 @ id))) | (((X219 @ id @ (X221 @ id @ X225) @ id)) = $true)) & (((X219 @ id @ (lam @ (X221 @ sh @ one)) @ id)) != $true) & ! [X223 : subst,X222 : subst,X224 : term] : (((X221 @ (comp @ X222 @ X223) @ (sub @ X224 @ X223))) = ((sub @ (X221 @ X222 @ X224) @ X223)))) | (((X219 @ id @ (lam @ (sub @ X218 @ (push @ one @ sh))) @ id)) = $true) | ? [X220 : term] : (($true = ((X219 @ id @ X220 @ id))) & (((X219 @ id @ (sub @ X218 @ (push @ X220 @ id)) @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f52,definition,(
  ($true = sP39) <=> ! [X285 : (subst > term > subst > $o),X284 : term] : (($true = ((X285 @ id @ (lam @ X284) @ id))) | ? [X287 : (subst > term > term)] : (! [X291 : term] : (($true = ((X285 @ id @ (X287 @ id @ X291) @ id))) | (((X285 @ id @ X291 @ id)) != $true)) & ($true != ((X285 @ id @ (lam @ (X287 @ sh @ one)) @ id))) & ! [X290 : term,X288 : subst,X289 : subst] : (((X287 @ (comp @ X288 @ X289) @ (sub @ X290 @ X289))) = ((sub @ (X287 @ X288 @ X290) @ X289)))) | ? [X286 : term] : ((((X285 @ id @ X286 @ id)) = $true) & (((X285 @ id @ (sub @ X284 @ (push @ X286 @ id)) @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f53,definition,(
  ! [X246 : (subst > term > subst > $o)] : (($true = ((sP40 @ X246))) <=> ? [X252 : (subst > term > term)] : (! [X253 : term] : (($true = ((X246 @ id @ (X252 @ id @ X253) @ id))) | (((X246 @ id @ X253 @ id)) != $true)) & ! [X255 : subst,X254 : subst,X256 : term] : (((X252 @ (comp @ X254 @ X255) @ (sub @ X256 @ X255))) = ((sub @ (X252 @ X254 @ X256) @ X255))) & ($true != ((X246 @ id @ (lam @ (X252 @ sh @ one)) @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f54,definition,(
  ($true = sP41) <=> ! [X246 : (subst > term > subst > $o),X245 : term] : (($true = ((sP40 @ X246))) | (((X246 @ id @ (lam @ X245) @ id)) = $true) | ? [X251 : term,X249 : subst,X248 : subst,X250 : subst] : (($true != ((X246 @ X249 @ X251 @ (comp @ X250 @ X248)))) & ($true = ((X246 @ (comp @ X249 @ X250) @ (sub @ X251 @ X250) @ X248)))) | ? [X257 : subst,X258 : subst,X260 : subst,X259 : term] : ((((X246 @ X257 @ X259 @ (comp @ X258 @ X260))) = $true) & (((X246 @ (comp @ X257 @ X258) @ (sub @ X259 @ X258) @ X260)) != $true)) | ? [X247 : term] : (($true = ((X246 @ id @ X247 @ id))) & ($true != ((X246 @ id @ (sub @ X245 @ (push @ X247 @ id)) @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f55,definition,(
  ! [X374 : (subst > term > subst > $o)] : ((((sP42 @ X374)) = $true) <=> ? [X385 : (subst > term > term)] : (! [X389 : term] : ((((X374 @ id @ (X385 @ id @ X389) @ id)) = $true) | (((X374 @ id @ X389 @ id)) != $true)) & ! [X388 : subst,X387 : subst,X386 : term] : (((X385 @ (comp @ X388 @ X387) @ (sub @ X386 @ X387))) = ((sub @ (X385 @ X388 @ X386) @ X387))) & (((X374 @ id @ (lam @ (X385 @ sh @ one)) @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f56,definition,(
  ($true = sP43) <=> ? [X367 : term,X366 : (subst > term > subst > $o)] : ((((X366 @ id @ (lam @ (sub @ X367 @ (push @ one @ sh))) @ id)) != $true) & ! [X368 : term] : (($true = ((X366 @ id @ (sub @ X367 @ (push @ X368 @ id)) @ id))) | (((X366 @ id @ X368 @ id)) != $true)) & ! [X369 : (subst > term > term)] : (? [X371 : subst,X370 : term,X372 : subst] : (((X369 @ (comp @ X371 @ X372) @ (sub @ X370 @ X372))) != ((sub @ (X369 @ X371 @ X370) @ X372))) | ? [X373 : term] : (($true = ((X366 @ id @ X373 @ id))) & (((X366 @ id @ (X369 @ id @ X373) @ id)) != $true)) | (((X366 @ id @ (lam @ (X369 @ sh @ one)) @ id)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f57,definition,(
  (sP44 = $true) <=> (($true = sP43) | ? [X365 : term] : (((sub @ X365 @ id)) != X365) | ! [X375 : term,X374 : (subst > term > subst > $o)] : (? [X381 : subst,X382 : subst,X384 : term,X383 : subst] : (($true = ((X374 @ (comp @ X382 @ X383) @ (sub @ X384 @ X383) @ X381))) & ($true != ((X374 @ X382 @ X384 @ (comp @ X383 @ X381))))) | ($true = ((X374 @ id @ (lam @ X375) @ id))) | (((sP42 @ X374)) = $true) | ? [X376 : subst,X378 : subst,X377 : subst,X379 : term] : ((((X374 @ X378 @ X379 @ (comp @ X376 @ X377))) = $true) & ($true != ((X374 @ (comp @ X378 @ X376) @ (sub @ X379 @ X376) @ X377)))) | ? [X380 : term] : ((((X374 @ id @ X380 @ id)) = $true) & (((X374 @ id @ (sub @ X375 @ (push @ X380 @ id)) @ id)) != $true))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f58,definition,(
  ! [X231 : (term > $o)] : ((((sP45 @ X231)) = $true) <=> ? [X234 : term,X233 : term] : ((((X231 @ X234)) = $true) & ($true = ((X231 @ X233))) & (((X231 @ (ap @ X233 @ X234))) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f59,definition,(
  ($true = sP46) <=> ! [X230 : subst,X229 : term,X231 : (term > $o)] : (? [X235 : term] : (! [X236 : term] : ((((X231 @ (sub @ X235 @ (push @ X236 @ id)))) = $true) | (((X231 @ X236)) != $true)) & (((X231 @ (lam @ X235))) != $true)) | (((sP45 @ X231)) = $true) | ($true = ((X231 @ (sub @ X229 @ X230)))) | ? [X232 : term] : ((((X231 @ (sub @ X232 @ X230))) != $true) & (((var @ X232)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f60,definition,(
  (sP47 = $true) <=> ! [X443 : term,X442 : (subst > term > subst > $o)] : (? [X445 : (subst > term > term)] : (($true != ((X442 @ id @ (lam @ (X445 @ sh @ one)) @ id))) & ! [X446 : term] : ((((X442 @ id @ X446 @ id)) != $true) | (((X442 @ id @ (X445 @ id @ X446) @ id)) = $true)) & ! [X449 : term,X448 : subst,X447 : subst] : (((sub @ (X445 @ X448 @ X449) @ X447)) = ((X445 @ (comp @ X448 @ X447) @ (sub @ X449 @ X447))))) | (((X442 @ id @ (lam @ (sub @ X443 @ (push @ one @ sh))) @ id)) = $true) | ? [X444 : term] : ((((X442 @ id @ X444 @ id)) = $true) & ($true != ((X442 @ id @ (sub @ X443 @ (push @ X444 @ id)) @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f61,definition,(
  (sP48 = $true) <=> ! [X207 : (term > $o),X208 : term,X209 : (subst > term > subst > $o)] : (? [X211 : term] : (((X209 @ id @ X211 @ id)) != ((X207 @ X211))) | ? [X212 : (subst > term > term)] : (! [X213 : term] : ((((X209 @ id @ X213 @ id)) != $true) | (((X209 @ id @ (X212 @ id @ X213) @ id)) = $true)) & (((X209 @ id @ (lam @ (X212 @ sh @ one)) @ id)) != $true) & ! [X215 : subst,X214 : subst,X216 : term] : (((X212 @ (comp @ X214 @ X215) @ (sub @ X216 @ X215))) = ((sub @ (X212 @ X214 @ X216) @ X215)))) | (((X207 @ (lam @ X208))) = $true) | ? [X210 : term] : ((((X207 @ X210)) = $true) & (((X207 @ (sub @ X208 @ (push @ X210 @ id)))) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f62,definition,(
  (sP49 = $true) <=> ? [X154 : (subst > term > subst > $o),X155 : term] : (! [X161 : term] : ((((X154 @ id @ X161 @ id)) != $true) | (((X154 @ id @ (sub @ X155 @ (push @ X161 @ id)) @ id)) = $true)) & (((X154 @ id @ (lam @ (sub @ X155 @ (push @ one @ sh))) @ id)) != $true) & ! [X156 : (subst > term > term)] : ((((X154 @ id @ (lam @ (X156 @ sh @ one)) @ id)) = $true) | ? [X157 : term] : (($true != ((X154 @ id @ (X156 @ id @ X157) @ id))) & ($true = ((X154 @ id @ X157 @ id)))) | ? [X160 : subst,X158 : term,X159 : subst] : (((X156 @ (comp @ X159 @ X160) @ (sub @ X158 @ X160))) != ((sub @ (X156 @ X159 @ X158) @ X160)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f63,definition,(
  (sP50 = $true) <=> (? [X153 : term] : (((sub @ X153 @ id)) != X153) | (sP49 = $true) | ! [X162 : (subst > term > subst > $o),X163 : term] : ((((X162 @ id @ (lam @ X163) @ id)) = $true) | ? [X169 : term] : (($true = ((X162 @ id @ X169 @ id))) & (((X162 @ id @ (sub @ X163 @ (push @ X169 @ id)) @ id)) != $true)) | ? [X164 : (subst > term > term)] : (! [X165 : term] : (($true != ((X162 @ id @ X165 @ id))) | (((X162 @ id @ (X164 @ id @ X165) @ id)) = $true)) & ! [X167 : subst,X168 : term,X166 : subst] : (((sub @ (X164 @ X167 @ X168) @ X166)) = ((X164 @ (comp @ X167 @ X166) @ (sub @ X168 @ X166)))) & (((X162 @ id @ (lam @ (X164 @ sh @ one)) @ id)) != $true))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f64,definition,(
  ! [X262 : (subst > term > subst > $o)] : (($true = ((sP51 @ X262))) <=> ? [X269 : term,X268 : term] : (($true != ((X262 @ id @ (ap @ (sub @ X268 @ id) @ X269) @ id))) & ($true = ((X262 @ id @ X268 @ id))) & ($true = ((X262 @ id @ X269 @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f65,definition,(
  ($true = sP52) <=> ! [X263 : term,X261 : term,X262 : (subst > term > subst > $o)] : ((((X262 @ id @ X261 @ id)) != $true) | ? [X265 : subst,X264 : term,X267 : subst,X266 : subst] : ((((X262 @ X266 @ X264 @ (comp @ X265 @ X267))) = $true) & (((X262 @ (comp @ X266 @ X265) @ (sub @ X264 @ X265) @ X267)) != $true)) | ($true = ((X262 @ id @ (ap @ X263 @ X261) @ id))) | ($true = ((sP51 @ X262))) | ? [X273 : subst,X270 : subst,X271 : subst,X272 : term] : ((((X262 @ X271 @ X272 @ (comp @ X273 @ X270))) != $true) & (((X262 @ (comp @ X271 @ X273) @ (sub @ X272 @ X273) @ X270)) = $true)) | ($true != ((X262 @ id @ X263 @ id))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f66,definition,(
  ! [X517 : (subst > term > subst > $o)] : ((((sP53 @ X517)) = $true) <=> ? [X525 : (subst > term > term)] : (! [X529 : term] : ((((X517 @ id @ (X525 @ id @ X529) @ id)) = $true) | ($true != ((X517 @ id @ X529 @ id)))) & ($true != ((X517 @ id @ (lam @ (X525 @ sh @ one)) @ id))) & ! [X528 : subst,X527 : subst,X526 : term] : (((X525 @ (comp @ X528 @ X527) @ (sub @ X526 @ X527))) = ((sub @ (X525 @ X528 @ X526) @ X527)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f67,definition,(
  ! [X517 : (subst > term > subst > $o)] : (($true = ((sP54 @ X517))) <=> ? [X520 : term,X519 : term] : ((((X517 @ id @ X519 @ id)) = $true) & ($true = ((X517 @ id @ X520 @ id))) & (((X517 @ id @ (ap @ (sub @ X519 @ id) @ X520) @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f68,definition,(
  ! [X499 : (term > $o)] : (($true = ((sP55 @ X499))) <=> ! [X505 : term,X504 : term] : (($true != ((X499 @ X505))) | ($true = ((X499 @ (ap @ X504 @ X505)))) | (((X499 @ X504)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f69,definition,(
  (sP56 = $true) <=> ? [X508 : term,X507 : (term > $o),X509 : (subst > term > subst > $o)] : ((((X507 @ (lam @ X508))) != $true) & ! [X511 : term] : (($true = ((X507 @ (sub @ X508 @ (push @ X511 @ id))))) | (((X507 @ X511)) != $true)) & ! [X510 : term] : (((X507 @ X510)) = ((X509 @ id @ X510 @ id))) & ! [X512 : (subst > term > term)] : (? [X514 : subst,X513 : term,X515 : subst] : (((sub @ (X512 @ X514 @ X513) @ X515)) != ((X512 @ (comp @ X514 @ X515) @ (sub @ X513 @ X515)))) | ? [X516 : term] : ((((X509 @ id @ X516 @ id)) = $true) & (((X509 @ id @ (X512 @ id @ X516) @ id)) != $true)) | ($true = ((X509 @ id @ (lam @ (X512 @ sh @ one)) @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f70,definition,(
  (sP57 = $true) <=> ! [X518 : term,X517 : (subst > term > subst > $o)] : (? [X523 : subst,X524 : subst,X522 : subst,X521 : term] : (($true = ((X517 @ (comp @ X522 @ X524) @ (sub @ X521 @ X524) @ X523))) & ($true != ((X517 @ X522 @ X521 @ (comp @ X524 @ X523))))) | ($true = ((sP54 @ X517))) | (((sP53 @ X517)) = $true) | (((X517 @ id @ X518 @ id)) = $true) | ? [X530 : subst,X532 : subst,X531 : term,X533 : subst] : (($true != ((X517 @ (comp @ X532 @ X533) @ (sub @ X531 @ X533) @ X530))) & (((X517 @ X532 @ X531 @ (comp @ X533 @ X530))) = $true)) | ? [X534 : term] : ((((var @ (sub @ X534 @ id))) = $true) & ($true != ((X517 @ id @ X534 @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f71,definition,(
  (sP58 = $true) <=> (? [X506 : term] : (((sub @ X506 @ id)) != X506) | ? [X500 : term,X499 : (term > $o)] : (! [X502 : term] : ((((X499 @ (lam @ X502))) = $true) | ? [X503 : term] : (($true = ((X499 @ X503))) & ($true != ((X499 @ (sub @ X502 @ (push @ X503 @ id))))))) & ($true != ((X499 @ X500))) & ($true = ((sP55 @ X499))) & ! [X501 : term] : ((((X499 @ X501)) = $true) | (((var @ X501)) != $true))) | (sP57 = $true) | (sP56 = $true) | ? [X496 : (subst > term > subst > $o)] : ! [X497 : (term > $o)] : ? [X498 : term] : (((X496 @ id @ X498 @ id)) != ((X497 @ X498))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f72,plain,(
  ((axidr = $true) <=> ! [X244 : subst] : (((comp @ X244 @ id)) = X244)) & ((sP58 = $true) <=> (hoasinduction_lthm_3 = $true)) & (pushprop_lem1_gthm = $true) & (! [X535 : (term > $o),X537 : subst,X536 : term] : ? [X538 : (term > $o)] : ! [X539 : term] : (((X538 @ X539)) = ((X535 @ (sub @ X539 @ (push @ X536 @ X537))))) <=> (pushprop_lem0 = $true)) & ((hoasinduction_lem2 = $true) <=> ($true = sP52)) & ((hoaslaminj = $true) <=> ! [X480 : (subst > term > term),X483 : (subst > term > term),X482 : term,X481 : subst] : (? [X488 : term,X487 : subst,X489 : subst] : (((X483 @ (comp @ X487 @ X489) @ (sub @ X488 @ X489))) != ((sub @ (X483 @ X487 @ X488) @ X489))) | (((X480 @ X481 @ X482)) = ((X483 @ X481 @ X482))) | ? [X485 : subst,X486 : term,X484 : subst] : (((sub @ (X480 @ X484 @ X486) @ X485)) != ((X480 @ (comp @ X484 @ X485) @ (sub @ X486 @ X485)))) | (((lam @ (X483 @ sh @ one))) != ((lam @ (X480 @ sh @ one)))))) & (! [X312 : term] : ? [X313 : (subst > term > term)] : (((X313 @ sh @ one)) = ((sub @ X312 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b = $true)) & ! [X415 : d_subst,X414 : d_subst] : ((((d2subst @ X415)) != ((d2subst @ X414))) | (X414 = X415)) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ((hoasinduction_lem3a_lthm = $true) <=> (sP50 = $true)) & (pushprop_lem2v2 = $true) & ((hoasinduction_lem3v2a = $true) <=> (sP48 = $true)) & ((sP47 = $true) <=> (hoasinduction_lem3aa_lthm = $true)) & (pushprop_lem1v2_gthm = $true) & ((induction2lem = $true) <=> ($true = sP46)) & (axassoc = $true) & (pushprop_lem0_gthm = $true) & (! [X492 : term,X491 : term,X493 : term,X490 : term] : ((X490 = X492) | (((ap @ (sub @ X490 @ id) @ X491)) != ((ap @ (sub @ X492 @ id) @ X493)))) <=> (hoasapinj1 = $true)) & ((sP44 = $true) <=> (hoasinduction_lem3_lthm = $true)) & (hoasapinj2_gthm = $true) & (((var @ (d2term @ d_one))) != $true) & (($true = sP41) <=> (hoasinduction_lem3 = $true)) & ! [X494 : term] : ? [X495 : d_term] : (((d2term @ X495)) = X494) & (axabs = $true) & (hoasinduction_lem2_lthm = $true) & (hoasapnotvar_lthm = $true) & (hoasinduction_lem1v2_gthm = $true) & (hoasinduction_lem1 = $true) & (pushprop_lem2v2_lthm = $true) & (! [X349 : term] : ? [X350 : (subst > term > term)] : (((X350 @ sh @ one)) = ((sub @ X349 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b_lthm = $true)) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((hoasinduction_lem3a = $true) <=> ($true = sP39)) & ! [X242 : d_term,X243 : d_term] : ((((d2term @ X242)) != ((d2term @ X243))) | (X242 = X243)) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((hoasinduction_lem3aa = $true) <=> ($true = sP38)) & (one = ((d2term @ d_one))) & (! [X151 : term,X152 : term] : ((X151 = X152) | (((lam @ X151)) != ((lam @ X152)))) <=> (laminj = $true)) & (hoaslamnotvar = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ! [X353 : subst] : ? [X354 : d_subst] : (((d2subst @ X354)) = X353) & (! [X362 : (subst > term > subst > $o)] : ? [X363 : (term > $o)] : ! [X364 : term] : (((X362 @ id @ X364 @ id)) = ((X363 @ X364))) <=> (hoasinduction_lem0_lthm = $true)) & (((d2subst @ d_id)) = id) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & ((hoasapinj2_lthm = $true) <=> (? [X77 : term,X75 : term,X76 : term,X74 : term] : ((X76 != X77) & (((ap @ X75 @ X77)) = ((ap @ X74 @ X76)))) | ! [X80 : term,X81 : term,X78 : term,X79 : term] : ((X79 = X81) | (((ap @ (sub @ X80 @ id) @ X79)) != ((ap @ (sub @ X78 @ id) @ X81)))))) & (pushprop_lem2v2_gthm = $true) & ((? [X132 : subst,X131 : term] : (((comp @ sh @ (push @ X131 @ X132))) != X132) | ? [X130 : subst,X129 : term] : (((sub @ one @ (push @ X129 @ X130))) != X129) | ! [X136 : term,X134 : (term > $o),X137 : subst,X135 : (term > $o),X133 : term] : ((((X135 @ (sub @ X136 @ (push @ X133 @ X137)))) = $true) | ($true != ((X134 @ X136))))) <=> (pushprop_lem1_lthm = $true)) & (($true = sP37) <=> (induction2lem_lthm = $true)) & ((termmset_lthm = $true) <=> (! [X139 : term] : (((sub @ X139 @ id)) = X139) | ? [X138 : term] : (((sub @ X138 @ id)) != X138))) & (pushprop_lem3v2 = $true) & ((sP33 = $true) <=> (induction2 = $true)) & (pushprop_lem3v2_lthm = $true) & (hoasinduction_lem3v2_gthm = $true) & (((d2subst @ d_id)) = sh) & (hoaslaminj_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & ((hoaslaminj_lthm = $true) <=> (sP31 = $true)) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & ((hoasinduction_lem3v2a_lthm = $true) <=> (sP30 = $true)) & ((sP29 = $true) <=> (induction2_lthm = $true)) & (! [X239 : (subst > term > subst > $o)] : ? [X240 : (term > $o)] : ! [X241 : term] : (((X240 @ X241)) = ((X239 @ id @ X241 @ id))) <=> (hoasinduction_lem0 = $true)) & (! [X82 : (subst > term > term),X83 : term,X84 : term] : ((((ap @ (sub @ X83 @ id) @ X84)) != ((lam @ (X82 @ sh @ one)))) | ? [X85 : term,X86 : subst,X87 : subst] : (((X82 @ (comp @ X87 @ X86) @ (sub @ X85 @ X86))) != ((sub @ (X82 @ X87 @ X85) @ X86)))) <=> (hoaslamnotap = $true)) & (! [X227 : term,X228 : subst] : (((sub @ one @ (push @ X227 @ X228))) = X227) <=> (axvarcons = $true)) & (hoasinduction_lem3b_gthm = $true) & (hoaslamnotvar_lthm = $true) & (termmset_gthm = $true) & ((! [X360 : subst,X357 : (term > $o),X358 : (term > $o),X359 : term] : ((((X358 @ X359)) != $true) | ? [X361 : term] : (((X358 @ (sub @ X361 @ (push @ X359 @ X360)))) != ((X357 @ X361))) | (((X357 @ one)) = $true)) | ? [X355 : term,X356 : subst] : (((sub @ one @ (push @ X355 @ X356))) != X355)) <=> (pushprop_lem1v2_lthm = $true)) & (! [X391 : term,X390 : subst] : (((comp @ sh @ (push @ X391 @ X390))) = X390) <=> (axshiftcons = $true)) & ! [X351 : d_term] : (d_one = X351) & (hoasinduction_lem3_gthm = $true) & ((! [X141 : subst] : (((comp @ id @ X141)) = X141) & ! [X140 : subst] : (((comp @ X140 @ id)) = X140)) <=> (substmonoid = $true)) & (hoasinduction_gthm = $true) & (hoaslamnotvar_gthm = $true) & (axvarshift = $true) & (hoasinduction_lem2_gthm = $true) & (axclos = $true) & (! [X303 : (term > $o),X302 : subst,X299 : (term > $o),X300 : term,X301 : term] : ((((X303 @ X301)) != $true) | (((X299 @ (sub @ X301 @ (push @ X300 @ X302)))) = $true)) <=> (pushprop_lem1 = $true)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (induction2_gthm = $true) & (hoasinduction_lem1_gthm = $true) & (ulamvarsh = $true) & ((termmset = $true) <=> ! [X329 : term] : (((sub @ X329 @ id)) = X329)) & (induction2lem_gthm = $true) & (axmap = $true) & (! [X238 : term,X237 : term] : ((X237 = X238) | (((sub @ X237 @ sh)) != ((sub @ X238 @ sh)))) <=> (shinj = $true)) & (ulamvarind = $true) & (! [X441 : subst] : (((comp @ id @ X441)) = X441) <=> (axidl = $true)) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoasinduction_lem3v2_lthm = $true) <=> (sP25 = $true)) & (pushprop_gthm = $true) & (apnotvar = $true) & ((hoasapinj1_lthm = $true) <=> (? [X419 : term,X420 : term,X418 : term,X417 : term] : ((X417 != X418) & (((ap @ X418 @ X419)) = ((ap @ X417 @ X420)))) | ? [X416 : term] : (((sub @ X416 @ id)) != X416) | ! [X423 : term,X424 : term,X422 : term,X421 : term] : ((((ap @ (sub @ X421 @ id) @ X422)) != ((ap @ (sub @ X424 @ id) @ X423))) | (X421 = X424)))) & (! [X304 : term,X307 : term,X305 : term,X306 : term] : ((X306 = X307) | (((ap @ X304 @ X307)) != ((ap @ X305 @ X306)))) <=> (apinj2 = $true)) & ((induction = $true) <=> ! [X89 : (term > $o),X88 : term] : (? [X92 : term] : (($true = ((X89 @ X92))) & ($true != ((X89 @ (lam @ X92))))) | ($true = ((sP23 @ X89))) | (((X89 @ X88)) = $true) | ? [X93 : term] : ((((var @ X93)) = $true) & (((X89 @ X93)) != $true)))) & (! [X72 : term,X71 : term,X70 : term,X73 : term] : ((((ap @ X71 @ X73)) != ((ap @ X70 @ X72))) | (X70 = X71)) <=> (apinj1 = $true)) & ((hoasinduction = $true) <=> ($true = sP22)) & ((axscons = $true) <=> ! [X217 : subst] : (((push @ (sub @ one @ X217) @ (comp @ sh @ X217))) = X217)) & (hoasapinj1_gthm = $true) & (lamnotap != $true) & (! [X410 : term] : ? [X411 : (subst > term > term)] : ! [X413 : term,X412 : subst] : (((sub @ X410 @ (push @ X413 @ X412))) = ((X411 @ X412 @ X413))) <=> (hoasinduction_lem3v2_f = $true)) & (lamnotvar = $true) & ((hoasinduction_no_psi_cond = $true) <=> ! [X143 : (subst > term > subst > $o),X142 : term] : (? [X145 : term,X144 : term] : ((((X143 @ id @ (ap @ (sub @ X144 @ id) @ X145) @ id)) != $true) & (((X143 @ id @ X145 @ id)) = $true) & (((X143 @ id @ X144 @ id)) = $true)) | (((X143 @ id @ X142 @ id)) = $true) | ($true = ((sP19 @ X143))))) & (hoasinduction_lem1v2 = $true) & ((sP18 = $true) <=> (hoasinduction_lem3aaa = $true)) & (! [X352 : term] : (((sub @ X352 @ id)) = X352) <=> (axvarid = $true)) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ((pushprop_lem1v2 = $true) <=> ! [X277 : (term > $o),X276 : term,X274 : subst,X275 : (term > $o)] : ((((X275 @ one)) = $true) | (((X277 @ X276)) != $true) | ? [X278 : term] : (((X275 @ X278)) != ((X277 @ (sub @ X278 @ (push @ X276 @ X274))))))) & (! [X308 : term] : ? [X309 : (subst > term > term)] : ! [X311 : subst,X310 : term] : (((X309 @ X311 @ X310)) = ((sub @ X308 @ (push @ X310 @ X311)))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & (($true = sP17) <=> (hoasinduction_lthm = $true)) & (substmonoid_gthm = $true) & (pushprop_lthm = $true) & ((hoasinduction_lem3v2 = $true) <=> ($true = sP8)) & (axapp = $true) & (hoasinduction_lem2v2_gthm = $true) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem1_lthm = $true) & ((substmonoid_lthm = $true) <=> (? [X0 : subst] : (((comp @ id @ X0)) != X0) | (! [X3 : subst] : (((comp @ X3 @ id)) = X3) & ! [X2 : subst] : (((comp @ id @ X2)) = X2)) | ? [X1 : subst] : (((comp @ X1 @ id)) != X1))) & (hoaslamnotap_lthm = $true) & (hoasinduction_lem3a_gthm = $true) & ! [X226 : d_subst] : (d_id = X226) & (pushprop = $true) & (! [X128 : term,X125 : term,X127 : term,X126 : term] : ((((ap @ (sub @ X127 @ id) @ X128)) != ((ap @ (sub @ X126 @ id) @ X125))) | (X125 = X128)) <=> (hoasapinj2 = $true)) & (($true = sP6) <=> (hoasinduction_no_psi_cond_lthm = $true)) & ((pushprop_lem0_lthm = $true) <=> ! [X279 : (term > $o),X281 : subst,X280 : term] : ? [X282 : (term > $o)] : ! [X283 : term] : (((X279 @ (sub @ X283 @ (push @ X280 @ X281)))) = ((X282 @ X283)))) & (hoasapnotvar = $true) & (ulamvar1 != $true) & (pushprop_lthm_orig = $true) & ((hoasinduction_lem2v2 = $true) <=> (sP1 = $true)) & (hoaslamnotap_gthm = $true)),
  inference(definition_folding,[],[f12,f71,f70,f69,f68,f67,f66,f65,f64,f63,f62,f61,f60,f59,f58,f57,f56,f55,f54,f53,f52,f51,f50,f49,f48,f47,f46,f45,f44,f43,f42,f41,f40,f39,f38,f37,f36,f35,f34,f33,f32,f31,f30,f29,f28,f27,f26,f25,f24,f23,f22,f21,f20,f19,f18,f17,f16,f15,f14,f13])).
thf(f260,plain,(
  (((axidr = $true) | ? [X244 : subst] : (((comp @ X244 @ id)) != X244)) & (! [X244 : subst] : (((comp @ X244 @ id)) = X244) | (axidr != $true))) & (((sP58 = $true) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | (sP58 != $true))) & (pushprop_lem1_gthm = $true) & ((! [X535 : (term > $o),X537 : subst,X536 : term] : ? [X538 : (term > $o)] : ! [X539 : term] : (((X538 @ X539)) = ((X535 @ (sub @ X539 @ (push @ X536 @ X537))))) | (pushprop_lem0 != $true)) & ((pushprop_lem0 = $true) | ? [X535 : (term > $o),X537 : subst,X536 : term] : ! [X538 : (term > $o)] : ? [X539 : term] : (((X538 @ X539)) != ((X535 @ (sub @ X539 @ (push @ X536 @ X537))))))) & (((hoasinduction_lem2 = $true) | ($true != sP52)) & (($true = sP52) | (hoasinduction_lem2 != $true))) & (((hoaslaminj = $true) | ? [X480 : (subst > term > term),X483 : (subst > term > term),X482 : term,X481 : subst] : (! [X488 : term,X487 : subst,X489 : subst] : (((X483 @ (comp @ X487 @ X489) @ (sub @ X488 @ X489))) = ((sub @ (X483 @ X487 @ X488) @ X489))) & (((X480 @ X481 @ X482)) != ((X483 @ X481 @ X482))) & ! [X485 : subst,X486 : term,X484 : subst] : (((sub @ (X480 @ X484 @ X486) @ X485)) = ((X480 @ (comp @ X484 @ X485) @ (sub @ X486 @ X485)))) & (((lam @ (X483 @ sh @ one))) = ((lam @ (X480 @ sh @ one)))))) & (! [X480 : (subst > term > term),X483 : (subst > term > term),X482 : term,X481 : subst] : (? [X488 : term,X487 : subst,X489 : subst] : (((X483 @ (comp @ X487 @ X489) @ (sub @ X488 @ X489))) != ((sub @ (X483 @ X487 @ X488) @ X489))) | (((X480 @ X481 @ X482)) = ((X483 @ X481 @ X482))) | ? [X485 : subst,X486 : term,X484 : subst] : (((sub @ (X480 @ X484 @ X486) @ X485)) != ((X480 @ (comp @ X484 @ X485) @ (sub @ X486 @ X485)))) | (((lam @ (X483 @ sh @ one))) != ((lam @ (X480 @ sh @ one))))) | (hoaslaminj != $true))) & ((! [X312 : term] : ? [X313 : (subst > term > term)] : (((X313 @ sh @ one)) = ((sub @ X312 @ (push @ one @ sh)))) | (hoasinduction_lem3b != $true)) & ((hoasinduction_lem3b = $true) | ? [X312 : term] : ! [X313 : (subst > term > term)] : (((X313 @ sh @ one)) != ((sub @ X312 @ (push @ one @ sh)))))) & ! [X415 : d_subst,X414 : d_subst] : ((((d2subst @ X415)) != ((d2subst @ X414))) | (X414 = X415)) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (((hoasinduction_lem3a_lthm = $true) | (sP50 != $true)) & ((sP50 = $true) | (hoasinduction_lem3a_lthm != $true))) & (pushprop_lem2v2 = $true) & (((hoasinduction_lem3v2a = $true) | (sP48 != $true)) & ((sP48 = $true) | (hoasinduction_lem3v2a != $true))) & (((sP47 = $true) | (hoasinduction_lem3aa_lthm != $true)) & ((hoasinduction_lem3aa_lthm = $true) | (sP47 != $true))) & (pushprop_lem1v2_gthm = $true) & (((induction2lem = $true) | ($true != sP46)) & (($true = sP46) | (induction2lem != $true))) & (axassoc = $true) & (pushprop_lem0_gthm = $true) & ((! [X492 : term,X491 : term,X493 : term,X490 : term] : ((X490 = X492) | (((ap @ (sub @ X490 @ id) @ X491)) != ((ap @ (sub @ X492 @ id) @ X493)))) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | ? [X492 : term,X491 : term,X493 : term,X490 : term] : ((X490 != X492) & (((ap @ (sub @ X490 @ id) @ X491)) = ((ap @ (sub @ X492 @ id) @ X493)))))) & (((sP44 = $true) | (hoasinduction_lem3_lthm != $true)) & ((hoasinduction_lem3_lthm = $true) | (sP44 != $true))) & (hoasapinj2_gthm = $true) & (((var @ (d2term @ d_one))) != $true) & ((($true = sP41) | (hoasinduction_lem3 != $true)) & ((hoasinduction_lem3 = $true) | ($true != sP41))) & ! [X494 : term] : ? [X495 : d_term] : (((d2term @ X495)) = X494) & (axabs = $true) & (hoasinduction_lem2_lthm = $true) & (hoasapnotvar_lthm = $true) & (hoasinduction_lem1v2_gthm = $true) & (hoasinduction_lem1 = $true) & (pushprop_lem2v2_lthm = $true) & ((! [X349 : term] : ? [X350 : (subst > term > term)] : (((X350 @ sh @ one)) = ((sub @ X349 @ (push @ one @ sh)))) | (hoasinduction_lem3b_lthm != $true)) & ((hoasinduction_lem3b_lthm = $true) | ? [X349 : term] : ! [X350 : (subst > term > term)] : (((X350 @ sh @ one)) != ((sub @ X349 @ (push @ one @ sh)))))) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (((hoasinduction_lem3a = $true) | ($true != sP39)) & (($true = sP39) | (hoasinduction_lem3a != $true))) & ! [X242 : d_term,X243 : d_term] : ((((d2term @ X242)) != ((d2term @ X243))) | (X242 = X243)) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (((hoasinduction_lem3aa = $true) | ($true != sP38)) & (($true = sP38) | (hoasinduction_lem3aa != $true))) & (one = ((d2term @ d_one))) & ((! [X151 : term,X152 : term] : ((X151 = X152) | (((lam @ X151)) != ((lam @ X152)))) | (laminj != $true)) & ((laminj = $true) | ? [X151 : term,X152 : term] : ((X151 != X152) & (((lam @ X151)) = ((lam @ X152)))))) & (hoaslamnotvar = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ! [X353 : subst] : ? [X354 : d_subst] : (((d2subst @ X354)) = X353) & ((! [X362 : (subst > term > subst > $o)] : ? [X363 : (term > $o)] : ! [X364 : term] : (((X362 @ id @ X364 @ id)) = ((X363 @ X364))) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ? [X362 : (subst > term > subst > $o)] : ! [X363 : (term > $o)] : ? [X364 : term] : (((X362 @ id @ X364 @ id)) != ((X363 @ X364))))) & (((d2subst @ d_id)) = id) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (((hoasapinj2_lthm = $true) | (! [X77 : term,X75 : term,X76 : term,X74 : term] : ((X76 = X77) | (((ap @ X75 @ X77)) != ((ap @ X74 @ X76)))) & ? [X80 : term,X81 : term,X78 : term,X79 : term] : ((X79 != X81) & (((ap @ (sub @ X80 @ id) @ X79)) = ((ap @ (sub @ X78 @ id) @ X81)))))) & ((? [X77 : term,X75 : term,X76 : term,X74 : term] : ((X76 != X77) & (((ap @ X75 @ X77)) = ((ap @ X74 @ X76)))) | ! [X80 : term,X81 : term,X78 : term,X79 : term] : ((X79 = X81) | (((ap @ (sub @ X80 @ id) @ X79)) != ((ap @ (sub @ X78 @ id) @ X81))))) | (hoasapinj2_lthm != $true))) & (pushprop_lem2v2_gthm = $true) & (((? [X132 : subst,X131 : term] : (((comp @ sh @ (push @ X131 @ X132))) != X132) | ? [X130 : subst,X129 : term] : (((sub @ one @ (push @ X129 @ X130))) != X129) | ! [X136 : term,X134 : (term > $o),X137 : subst,X135 : (term > $o),X133 : term] : ((((X135 @ (sub @ X136 @ (push @ X133 @ X137)))) = $true) | ($true != ((X134 @ X136))))) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (! [X132 : subst,X131 : term] : (((comp @ sh @ (push @ X131 @ X132))) = X132) & ! [X130 : subst,X129 : term] : (((sub @ one @ (push @ X129 @ X130))) = X129) & ? [X136 : term,X134 : (term > $o),X137 : subst,X135 : (term > $o),X133 : term] : ((((X135 @ (sub @ X136 @ (push @ X133 @ X137)))) != $true) & ($true = ((X134 @ X136))))))) & ((($true = sP37) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | ($true != sP37))) & (((termmset_lthm = $true) | (? [X139 : term] : (((sub @ X139 @ id)) != X139) & ! [X138 : term] : (((sub @ X138 @ id)) = X138))) & ((! [X139 : term] : (((sub @ X139 @ id)) = X139) | ? [X138 : term] : (((sub @ X138 @ id)) != X138)) | (termmset_lthm != $true))) & (pushprop_lem3v2 = $true) & (((sP33 = $true) | (induction2 != $true)) & ((induction2 = $true) | (sP33 != $true))) & (pushprop_lem3v2_lthm = $true) & (hoasinduction_lem3v2_gthm = $true) & (((d2subst @ d_id)) = sh) & (hoaslaminj_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (((hoaslaminj_lthm = $true) | (sP31 != $true)) & ((sP31 = $true) | (hoaslaminj_lthm != $true))) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (((hoasinduction_lem3v2a_lthm = $true) | (sP30 != $true)) & ((sP30 = $true) | (hoasinduction_lem3v2a_lthm != $true))) & (((sP29 = $true) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | (sP29 != $true))) & ((! [X239 : (subst > term > subst > $o)] : ? [X240 : (term > $o)] : ! [X241 : term] : (((X240 @ X241)) = ((X239 @ id @ X241 @ id))) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem0 = $true) | ? [X239 : (subst > term > subst > $o)] : ! [X240 : (term > $o)] : ? [X241 : term] : (((X240 @ X241)) != ((X239 @ id @ X241 @ id))))) & ((! [X82 : (subst > term > term),X83 : term,X84 : term] : ((((ap @ (sub @ X83 @ id) @ X84)) != ((lam @ (X82 @ sh @ one)))) | ? [X85 : term,X86 : subst,X87 : subst] : (((X82 @ (comp @ X87 @ X86) @ (sub @ X85 @ X86))) != ((sub @ (X82 @ X87 @ X85) @ X86)))) | (hoaslamnotap != $true)) & ((hoaslamnotap = $true) | ? [X82 : (subst > term > term),X83 : term,X84 : term] : ((((ap @ (sub @ X83 @ id) @ X84)) = ((lam @ (X82 @ sh @ one)))) & ! [X85 : term,X86 : subst,X87 : subst] : (((X82 @ (comp @ X87 @ X86) @ (sub @ X85 @ X86))) = ((sub @ (X82 @ X87 @ X85) @ X86)))))) & ((! [X227 : term,X228 : subst] : (((sub @ one @ (push @ X227 @ X228))) = X227) | (axvarcons != $true)) & ((axvarcons = $true) | ? [X227 : term,X228 : subst] : (((sub @ one @ (push @ X227 @ X228))) != X227))) & (hoasinduction_lem3b_gthm = $true) & (hoaslamnotvar_lthm = $true) & (termmset_gthm = $true) & (((! [X360 : subst,X357 : (term > $o),X358 : (term > $o),X359 : term] : ((((X358 @ X359)) != $true) | ? [X361 : term] : (((X358 @ (sub @ X361 @ (push @ X359 @ X360)))) != ((X357 @ X361))) | (((X357 @ one)) = $true)) | ? [X355 : term,X356 : subst] : (((sub @ one @ (push @ X355 @ X356))) != X355)) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (? [X360 : subst,X357 : (term > $o),X358 : (term > $o),X359 : term] : ((((X358 @ X359)) = $true) & ! [X361 : term] : (((X358 @ (sub @ X361 @ (push @ X359 @ X360)))) = ((X357 @ X361))) & (((X357 @ one)) != $true)) & ! [X355 : term,X356 : subst] : (((sub @ one @ (push @ X355 @ X356))) = X355)))) & ((! [X391 : term,X390 : subst] : (((comp @ sh @ (push @ X391 @ X390))) = X390) | (axshiftcons != $true)) & ((axshiftcons = $true) | ? [X391 : term,X390 : subst] : (((comp @ sh @ (push @ X391 @ X390))) != X390))) & ! [X351 : d_term] : (d_one = X351) & (hoasinduction_lem3_gthm = $true) & (((! [X141 : subst] : (((comp @ id @ X141)) = X141) & ! [X140 : subst] : (((comp @ X140 @ id)) = X140)) | (substmonoid != $true)) & ((substmonoid = $true) | (? [X141 : subst] : (((comp @ id @ X141)) != X141) | ? [X140 : subst] : (((comp @ X140 @ id)) != X140)))) & (hoasinduction_gthm = $true) & (hoaslamnotvar_gthm = $true) & (axvarshift = $true) & (hoasinduction_lem2_gthm = $true) & (axclos = $true) & ((! [X303 : (term > $o),X302 : subst,X299 : (term > $o),X300 : term,X301 : term] : ((((X303 @ X301)) != $true) | (((X299 @ (sub @ X301 @ (push @ X300 @ X302)))) = $true)) | (pushprop_lem1 != $true)) & ((pushprop_lem1 = $true) | ? [X303 : (term > $o),X302 : subst,X299 : (term > $o),X300 : term,X301 : term] : ((((X303 @ X301)) = $true) & (((X299 @ (sub @ X301 @ (push @ X300 @ X302)))) != $true)))) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (induction2_gthm = $true) & (hoasinduction_lem1_gthm = $true) & (ulamvarsh = $true) & (((termmset = $true) | ? [X329 : term] : (((sub @ X329 @ id)) != X329)) & (! [X329 : term] : (((sub @ X329 @ id)) = X329) | (termmset != $true))) & (induction2lem_gthm = $true) & (axmap = $true) & ((! [X238 : term,X237 : term] : ((X237 = X238) | (((sub @ X237 @ sh)) != ((sub @ X238 @ sh)))) | (shinj != $true)) & ((shinj = $true) | ? [X238 : term,X237 : term] : ((X237 != X238) & (((sub @ X237 @ sh)) = ((sub @ X238 @ sh)))))) & (ulamvarind = $true) & ((! [X441 : subst] : (((comp @ id @ X441)) = X441) | (axidl != $true)) & ((axidl = $true) | ? [X441 : subst] : (((comp @ id @ X441)) != X441))) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (((hoasinduction_lem3v2_lthm = $true) | (sP25 != $true)) & ((sP25 = $true) | (hoasinduction_lem3v2_lthm != $true))) & (pushprop_gthm = $true) & (apnotvar = $true) & (((hoasapinj1_lthm = $true) | (! [X419 : term,X420 : term,X418 : term,X417 : term] : ((X417 = X418) | (((ap @ X418 @ X419)) != ((ap @ X417 @ X420)))) & ! [X416 : term] : (((sub @ X416 @ id)) = X416) & ? [X423 : term,X424 : term,X422 : term,X421 : term] : ((((ap @ (sub @ X421 @ id) @ X422)) = ((ap @ (sub @ X424 @ id) @ X423))) & (X421 != X424)))) & ((? [X419 : term,X420 : term,X418 : term,X417 : term] : ((X417 != X418) & (((ap @ X418 @ X419)) = ((ap @ X417 @ X420)))) | ? [X416 : term] : (((sub @ X416 @ id)) != X416) | ! [X423 : term,X424 : term,X422 : term,X421 : term] : ((((ap @ (sub @ X421 @ id) @ X422)) != ((ap @ (sub @ X424 @ id) @ X423))) | (X421 = X424))) | (hoasapinj1_lthm != $true))) & ((! [X304 : term,X307 : term,X305 : term,X306 : term] : ((X306 = X307) | (((ap @ X304 @ X307)) != ((ap @ X305 @ X306)))) | (apinj2 != $true)) & ((apinj2 = $true) | ? [X304 : term,X307 : term,X305 : term,X306 : term] : ((X306 != X307) & (((ap @ X304 @ X307)) = ((ap @ X305 @ X306)))))) & (((induction = $true) | ? [X89 : (term > $o),X88 : term] : (! [X92 : term] : (($true != ((X89 @ X92))) | ($true = ((X89 @ (lam @ X92))))) & ($true != ((sP23 @ X89))) & (((X89 @ X88)) != $true) & ! [X93 : term] : ((((var @ X93)) != $true) | (((X89 @ X93)) = $true)))) & (! [X89 : (term > $o),X88 : term] : (? [X92 : term] : (($true = ((X89 @ X92))) & ($true != ((X89 @ (lam @ X92))))) | ($true = ((sP23 @ X89))) | (((X89 @ X88)) = $true) | ? [X93 : term] : ((((var @ X93)) = $true) & (((X89 @ X93)) != $true))) | (induction != $true))) & ((! [X72 : term,X71 : term,X70 : term,X73 : term] : ((((ap @ X71 @ X73)) != ((ap @ X70 @ X72))) | (X70 = X71)) | (apinj1 != $true)) & ((apinj1 = $true) | ? [X72 : term,X71 : term,X70 : term,X73 : term] : ((((ap @ X71 @ X73)) = ((ap @ X70 @ X72))) & (X70 != X71)))) & (((hoasinduction = $true) | ($true != sP22)) & (($true = sP22) | (hoasinduction != $true))) & (((axscons = $true) | ? [X217 : subst] : (((push @ (sub @ one @ X217) @ (comp @ sh @ X217))) != X217)) & (! [X217 : subst] : (((push @ (sub @ one @ X217) @ (comp @ sh @ X217))) = X217) | (axscons != $true))) & (hoasapinj1_gthm = $true) & (lamnotap != $true) & ((! [X410 : term] : ? [X411 : (subst > term > term)] : ! [X413 : term,X412 : subst] : (((sub @ X410 @ (push @ X413 @ X412))) = ((X411 @ X412 @ X413))) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ? [X410 : term] : ! [X411 : (subst > term > term)] : ? [X413 : term,X412 : subst] : (((sub @ X410 @ (push @ X413 @ X412))) != ((X411 @ X412 @ X413))))) & (lamnotvar = $true) & (((hoasinduction_no_psi_cond = $true) | ? [X143 : (subst > term > subst > $o),X142 : term] : (! [X145 : term,X144 : term] : ((((X143 @ id @ (ap @ (sub @ X144 @ id) @ X145) @ id)) = $true) | (((X143 @ id @ X145 @ id)) != $true) | (((X143 @ id @ X144 @ id)) != $true)) & (((X143 @ id @ X142 @ id)) != $true) & ($true != ((sP19 @ X143))))) & (! [X143 : (subst > term > subst > $o),X142 : term] : (? [X145 : term,X144 : term] : ((((X143 @ id @ (ap @ (sub @ X144 @ id) @ X145) @ id)) != $true) & (((X143 @ id @ X145 @ id)) = $true) & (((X143 @ id @ X144 @ id)) = $true)) | (((X143 @ id @ X142 @ id)) = $true) | ($true = ((sP19 @ X143)))) | (hoasinduction_no_psi_cond != $true))) & (hoasinduction_lem1v2 = $true) & (((sP18 = $true) | (hoasinduction_lem3aaa != $true)) & ((hoasinduction_lem3aaa = $true) | (sP18 != $true))) & ((! [X352 : term] : (((sub @ X352 @ id)) = X352) | (axvarid != $true)) & ((axvarid = $true) | ? [X352 : term] : (((sub @ X352 @ id)) != X352))) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (((pushprop_lem1v2 = $true) | ? [X277 : (term > $o),X276 : term,X274 : subst,X275 : (term > $o)] : ((((X275 @ one)) != $true) & (((X277 @ X276)) = $true) & ! [X278 : term] : (((X275 @ X278)) = ((X277 @ (sub @ X278 @ (push @ X276 @ X274))))))) & (! [X277 : (term > $o),X276 : term,X274 : subst,X275 : (term > $o)] : ((((X275 @ one)) = $true) | (((X277 @ X276)) != $true) | ? [X278 : term] : (((X275 @ X278)) != ((X277 @ (sub @ X278 @ (push @ X276 @ X274)))))) | (pushprop_lem1v2 != $true))) & ((! [X308 : term] : ? [X309 : (subst > term > term)] : ! [X311 : subst,X310 : term] : (((X309 @ X311 @ X310)) = ((sub @ X308 @ (push @ X310 @ X311)))) | (hoasinduction_lem3v2_f_lthm != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X308 : term] : ! [X309 : (subst > term > term)] : ? [X311 : subst,X310 : term] : (((X309 @ X311 @ X310)) != ((sub @ X308 @ (push @ X310 @ X311)))))) & ((($true = sP17) | (hoasinduction_lthm != $true)) & ((hoasinduction_lthm = $true) | ($true != sP17))) & (substmonoid_gthm = $true) & (pushprop_lthm = $true) & (((hoasinduction_lem3v2 = $true) | ($true != sP8)) & (($true = sP8) | (hoasinduction_lem3v2 != $true))) & (axapp = $true) & (hoasinduction_lem2v2_gthm = $true) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem1_lthm = $true) & (((substmonoid_lthm = $true) | (! [X0 : subst] : (((comp @ id @ X0)) = X0) & (? [X3 : subst] : (((comp @ X3 @ id)) != X3) | ? [X2 : subst] : (((comp @ id @ X2)) != X2)) & ! [X1 : subst] : (((comp @ X1 @ id)) = X1))) & ((? [X0 : subst] : (((comp @ id @ X0)) != X0) | (! [X3 : subst] : (((comp @ X3 @ id)) = X3) & ! [X2 : subst] : (((comp @ id @ X2)) = X2)) | ? [X1 : subst] : (((comp @ X1 @ id)) != X1)) | (substmonoid_lthm != $true))) & (hoaslamnotap_lthm = $true) & (hoasinduction_lem3a_gthm = $true) & ! [X226 : d_subst] : (d_id = X226) & (pushprop = $true) & ((! [X128 : term,X125 : term,X127 : term,X126 : term] : ((((ap @ (sub @ X127 @ id) @ X128)) != ((ap @ (sub @ X126 @ id) @ X125))) | (X125 = X128)) | (hoasapinj2 != $true)) & ((hoasapinj2 = $true) | ? [X128 : term,X125 : term,X127 : term,X126 : term] : ((((ap @ (sub @ X127 @ id) @ X128)) = ((ap @ (sub @ X126 @ id) @ X125))) & (X125 != X128)))) & ((($true = sP6) | (hoasinduction_no_psi_cond_lthm != $true)) & ((hoasinduction_no_psi_cond_lthm = $true) | ($true != sP6))) & (((pushprop_lem0_lthm = $true) | ? [X279 : (term > $o),X281 : subst,X280 : term] : ! [X282 : (term > $o)] : ? [X283 : term] : (((X279 @ (sub @ X283 @ (push @ X280 @ X281)))) != ((X282 @ X283)))) & (! [X279 : (term > $o),X281 : subst,X280 : term] : ? [X282 : (term > $o)] : ! [X283 : term] : (((X279 @ (sub @ X283 @ (push @ X280 @ X281)))) = ((X282 @ X283))) | (pushprop_lem0_lthm != $true))) & (hoasapnotvar = $true) & (ulamvar1 != $true) & (pushprop_lthm_orig = $true) & (((hoasinduction_lem2v2 = $true) | (sP1 != $true)) & ((sP1 = $true) | (hoasinduction_lem2v2 != $true))) & (hoaslamnotap_gthm = $true)),
  inference(nnf_transformation,[],[f72])).
thf(f261,plain,(
  ((axidr = $true) | ? [X244 : subst] : (((comp @ X244 @ id)) != X244)) & (! [X244 : subst] : (((comp @ X244 @ id)) = X244) | (axidr != $true)) & ((sP58 = $true) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | (sP58 != $true)) & (pushprop_lem1_gthm = $true) & (! [X535 : (term > $o),X537 : subst,X536 : term] : ? [X538 : (term > $o)] : ! [X539 : term] : (((X538 @ X539)) = ((X535 @ (sub @ X539 @ (push @ X536 @ X537))))) | (pushprop_lem0 != $true)) & ((pushprop_lem0 = $true) | ? [X535 : (term > $o),X537 : subst,X536 : term] : ! [X538 : (term > $o)] : ? [X539 : term] : (((X538 @ X539)) != ((X535 @ (sub @ X539 @ (push @ X536 @ X537)))))) & ((hoasinduction_lem2 = $true) | ($true != sP52)) & (($true = sP52) | (hoasinduction_lem2 != $true)) & ((hoaslaminj = $true) | ? [X480 : (subst > term > term),X483 : (subst > term > term),X482 : term,X481 : subst] : (! [X488 : term,X487 : subst,X489 : subst] : (((X483 @ (comp @ X487 @ X489) @ (sub @ X488 @ X489))) = ((sub @ (X483 @ X487 @ X488) @ X489))) & (((X480 @ X481 @ X482)) != ((X483 @ X481 @ X482))) & ! [X485 : subst,X486 : term,X484 : subst] : (((sub @ (X480 @ X484 @ X486) @ X485)) = ((X480 @ (comp @ X484 @ X485) @ (sub @ X486 @ X485)))) & (((lam @ (X483 @ sh @ one))) = ((lam @ (X480 @ sh @ one)))))) & (! [X480 : (subst > term > term),X483 : (subst > term > term),X482 : term,X481 : subst] : (? [X488 : term,X487 : subst,X489 : subst] : (((X483 @ (comp @ X487 @ X489) @ (sub @ X488 @ X489))) != ((sub @ (X483 @ X487 @ X488) @ X489))) | (((X480 @ X481 @ X482)) = ((X483 @ X481 @ X482))) | ? [X485 : subst,X486 : term,X484 : subst] : (((sub @ (X480 @ X484 @ X486) @ X485)) != ((X480 @ (comp @ X484 @ X485) @ (sub @ X486 @ X485)))) | (((lam @ (X483 @ sh @ one))) != ((lam @ (X480 @ sh @ one))))) | (hoaslaminj != $true)) & (! [X312 : term] : ? [X313 : (subst > term > term)] : (((X313 @ sh @ one)) = ((sub @ X312 @ (push @ one @ sh)))) | (hoasinduction_lem3b != $true)) & ((hoasinduction_lem3b = $true) | ? [X312 : term] : ! [X313 : (subst > term > term)] : (((X313 @ sh @ one)) != ((sub @ X312 @ (push @ one @ sh))))) & ! [X415 : d_subst,X414 : d_subst] : ((((d2subst @ X415)) != ((d2subst @ X414))) | (X414 = X415)) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ((hoasinduction_lem3a_lthm = $true) | (sP50 != $true)) & ((sP50 = $true) | (hoasinduction_lem3a_lthm != $true)) & (pushprop_lem2v2 = $true) & ((hoasinduction_lem3v2a = $true) | (sP48 != $true)) & ((sP48 = $true) | (hoasinduction_lem3v2a != $true)) & ((sP47 = $true) | (hoasinduction_lem3aa_lthm != $true)) & ((hoasinduction_lem3aa_lthm = $true) | (sP47 != $true)) & (pushprop_lem1v2_gthm = $true) & ((induction2lem = $true) | ($true != sP46)) & (($true = sP46) | (induction2lem != $true)) & (axassoc = $true) & (pushprop_lem0_gthm = $true) & (! [X492 : term,X491 : term,X493 : term,X490 : term] : ((X490 = X492) | (((ap @ (sub @ X490 @ id) @ X491)) != ((ap @ (sub @ X492 @ id) @ X493)))) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | ? [X492 : term,X491 : term,X493 : term,X490 : term] : ((X490 != X492) & (((ap @ (sub @ X490 @ id) @ X491)) = ((ap @ (sub @ X492 @ id) @ X493))))) & ((sP44 = $true) | (hoasinduction_lem3_lthm != $true)) & ((hoasinduction_lem3_lthm = $true) | (sP44 != $true)) & (hoasapinj2_gthm = $true) & (((var @ (d2term @ d_one))) != $true) & (($true = sP41) | (hoasinduction_lem3 != $true)) & ((hoasinduction_lem3 = $true) | ($true != sP41)) & ! [X494 : term] : ? [X495 : d_term] : (((d2term @ X495)) = X494) & (axabs = $true) & (hoasinduction_lem2_lthm = $true) & (hoasapnotvar_lthm = $true) & (hoasinduction_lem1v2_gthm = $true) & (hoasinduction_lem1 = $true) & (pushprop_lem2v2_lthm = $true) & (! [X349 : term] : ? [X350 : (subst > term > term)] : (((X350 @ sh @ one)) = ((sub @ X349 @ (push @ one @ sh)))) | (hoasinduction_lem3b_lthm != $true)) & ((hoasinduction_lem3b_lthm = $true) | ? [X349 : term] : ! [X350 : (subst > term > term)] : (((X350 @ sh @ one)) != ((sub @ X349 @ (push @ one @ sh))))) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((hoasinduction_lem3a = $true) | ($true != sP39)) & (($true = sP39) | (hoasinduction_lem3a != $true)) & ! [X242 : d_term,X243 : d_term] : ((((d2term @ X242)) != ((d2term @ X243))) | (X242 = X243)) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((hoasinduction_lem3aa = $true) | ($true != sP38)) & (($true = sP38) | (hoasinduction_lem3aa != $true)) & (one = ((d2term @ d_one))) & (! [X151 : term,X152 : term] : ((X151 = X152) | (((lam @ X151)) != ((lam @ X152)))) | (laminj != $true)) & ((laminj = $true) | ? [X151 : term,X152 : term] : ((X151 != X152) & (((lam @ X151)) = ((lam @ X152))))) & (hoaslamnotvar = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ! [X353 : subst] : ? [X354 : d_subst] : (((d2subst @ X354)) = X353) & (! [X362 : (subst > term > subst > $o)] : ? [X363 : (term > $o)] : ! [X364 : term] : (((X362 @ id @ X364 @ id)) = ((X363 @ X364))) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ? [X362 : (subst > term > subst > $o)] : ! [X363 : (term > $o)] : ? [X364 : term] : (((X362 @ id @ X364 @ id)) != ((X363 @ X364)))) & (((d2subst @ d_id)) = id) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & ((hoasapinj2_lthm = $true) | (! [X77 : term,X75 : term,X76 : term,X74 : term] : ((X76 = X77) | (((ap @ X75 @ X77)) != ((ap @ X74 @ X76)))) & ? [X80 : term,X81 : term,X78 : term,X79 : term] : ((X79 != X81) & (((ap @ (sub @ X80 @ id) @ X79)) = ((ap @ (sub @ X78 @ id) @ X81)))))) & (? [X77 : term,X75 : term,X76 : term,X74 : term] : ((X76 != X77) & (((ap @ X75 @ X77)) = ((ap @ X74 @ X76)))) | ! [X80 : term,X81 : term,X78 : term,X79 : term] : ((X79 = X81) | (((ap @ (sub @ X80 @ id) @ X79)) != ((ap @ (sub @ X78 @ id) @ X81)))) | (hoasapinj2_lthm != $true)) & (pushprop_lem2v2_gthm = $true) & (? [X132 : subst,X131 : term] : (((comp @ sh @ (push @ X131 @ X132))) != X132) | ? [X130 : subst,X129 : term] : (((sub @ one @ (push @ X129 @ X130))) != X129) | ! [X136 : term,X134 : (term > $o),X137 : subst,X135 : (term > $o),X133 : term] : ((((X135 @ (sub @ X136 @ (push @ X133 @ X137)))) = $true) | ($true != ((X134 @ X136)))) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (! [X132 : subst,X131 : term] : (((comp @ sh @ (push @ X131 @ X132))) = X132) & ! [X130 : subst,X129 : term] : (((sub @ one @ (push @ X129 @ X130))) = X129) & ? [X136 : term,X134 : (term > $o),X137 : subst,X135 : (term > $o),X133 : term] : ((((X135 @ (sub @ X136 @ (push @ X133 @ X137)))) != $true) & ($true = ((X134 @ X136)))))) & (($true = sP37) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | ($true != sP37)) & ((termmset_lthm = $true) | (? [X139 : term] : (((sub @ X139 @ id)) != X139) & ! [X138 : term] : (((sub @ X138 @ id)) = X138))) & (! [X139 : term] : (((sub @ X139 @ id)) = X139) | ? [X138 : term] : (((sub @ X138 @ id)) != X138) | (termmset_lthm != $true)) & (pushprop_lem3v2 = $true) & ((sP33 = $true) | (induction2 != $true)) & ((induction2 = $true) | (sP33 != $true)) & (pushprop_lem3v2_lthm = $true) & (hoasinduction_lem3v2_gthm = $true) & (((d2subst @ d_id)) = sh) & (hoaslaminj_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & ((hoaslaminj_lthm = $true) | (sP31 != $true)) & ((sP31 = $true) | (hoaslaminj_lthm != $true)) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & ((hoasinduction_lem3v2a_lthm = $true) | (sP30 != $true)) & ((sP30 = $true) | (hoasinduction_lem3v2a_lthm != $true)) & ((sP29 = $true) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | (sP29 != $true)) & (! [X239 : (subst > term > subst > $o)] : ? [X240 : (term > $o)] : ! [X241 : term] : (((X240 @ X241)) = ((X239 @ id @ X241 @ id))) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem0 = $true) | ? [X239 : (subst > term > subst > $o)] : ! [X240 : (term > $o)] : ? [X241 : term] : (((X240 @ X241)) != ((X239 @ id @ X241 @ id)))) & (! [X82 : (subst > term > term),X83 : term,X84 : term] : ((((ap @ (sub @ X83 @ id) @ X84)) != ((lam @ (X82 @ sh @ one)))) | ? [X85 : term,X86 : subst,X87 : subst] : (((X82 @ (comp @ X87 @ X86) @ (sub @ X85 @ X86))) != ((sub @ (X82 @ X87 @ X85) @ X86)))) | (hoaslamnotap != $true)) & ((hoaslamnotap = $true) | ? [X82 : (subst > term > term),X83 : term,X84 : term] : ((((ap @ (sub @ X83 @ id) @ X84)) = ((lam @ (X82 @ sh @ one)))) & ! [X85 : term,X86 : subst,X87 : subst] : (((X82 @ (comp @ X87 @ X86) @ (sub @ X85 @ X86))) = ((sub @ (X82 @ X87 @ X85) @ X86))))) & (! [X227 : term,X228 : subst] : (((sub @ one @ (push @ X227 @ X228))) = X227) | (axvarcons != $true)) & ((axvarcons = $true) | ? [X227 : term,X228 : subst] : (((sub @ one @ (push @ X227 @ X228))) != X227)) & (hoasinduction_lem3b_gthm = $true) & (hoaslamnotvar_lthm = $true) & (termmset_gthm = $true) & (! [X360 : subst,X357 : (term > $o),X358 : (term > $o),X359 : term] : ((((X358 @ X359)) != $true) | ? [X361 : term] : (((X358 @ (sub @ X361 @ (push @ X359 @ X360)))) != ((X357 @ X361))) | (((X357 @ one)) = $true)) | ? [X355 : term,X356 : subst] : (((sub @ one @ (push @ X355 @ X356))) != X355) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (? [X360 : subst,X357 : (term > $o),X358 : (term > $o),X359 : term] : ((((X358 @ X359)) = $true) & ! [X361 : term] : (((X358 @ (sub @ X361 @ (push @ X359 @ X360)))) = ((X357 @ X361))) & (((X357 @ one)) != $true)) & ! [X355 : term,X356 : subst] : (((sub @ one @ (push @ X355 @ X356))) = X355))) & (! [X391 : term,X390 : subst] : (((comp @ sh @ (push @ X391 @ X390))) = X390) | (axshiftcons != $true)) & ((axshiftcons = $true) | ? [X391 : term,X390 : subst] : (((comp @ sh @ (push @ X391 @ X390))) != X390)) & ! [X351 : d_term] : (d_one = X351) & (hoasinduction_lem3_gthm = $true) & ((! [X141 : subst] : (((comp @ id @ X141)) = X141) & ! [X140 : subst] : (((comp @ X140 @ id)) = X140)) | (substmonoid != $true)) & ((substmonoid = $true) | ? [X141 : subst] : (((comp @ id @ X141)) != X141) | ? [X140 : subst] : (((comp @ X140 @ id)) != X140)) & (hoasinduction_gthm = $true) & (hoaslamnotvar_gthm = $true) & (axvarshift = $true) & (hoasinduction_lem2_gthm = $true) & (axclos = $true) & (! [X303 : (term > $o),X302 : subst,X299 : (term > $o),X300 : term,X301 : term] : ((((X303 @ X301)) != $true) | (((X299 @ (sub @ X301 @ (push @ X300 @ X302)))) = $true)) | (pushprop_lem1 != $true)) & ((pushprop_lem1 = $true) | ? [X303 : (term > $o),X302 : subst,X299 : (term > $o),X300 : term,X301 : term] : ((((X303 @ X301)) = $true) & (((X299 @ (sub @ X301 @ (push @ X300 @ X302)))) != $true))) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (induction2_gthm = $true) & (hoasinduction_lem1_gthm = $true) & (ulamvarsh = $true) & ((termmset = $true) | ? [X329 : term] : (((sub @ X329 @ id)) != X329)) & (! [X329 : term] : (((sub @ X329 @ id)) = X329) | (termmset != $true)) & (induction2lem_gthm = $true) & (axmap = $true) & (! [X238 : term,X237 : term] : ((X237 = X238) | (((sub @ X237 @ sh)) != ((sub @ X238 @ sh)))) | (shinj != $true)) & ((shinj = $true) | ? [X238 : term,X237 : term] : ((X237 != X238) & (((sub @ X237 @ sh)) = ((sub @ X238 @ sh))))) & (ulamvarind = $true) & (! [X441 : subst] : (((comp @ id @ X441)) = X441) | (axidl != $true)) & ((axidl = $true) | ? [X441 : subst] : (((comp @ id @ X441)) != X441)) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoasinduction_lem3v2_lthm = $true) | (sP25 != $true)) & ((sP25 = $true) | (hoasinduction_lem3v2_lthm != $true)) & (pushprop_gthm = $true) & (apnotvar = $true) & ((hoasapinj1_lthm = $true) | (! [X419 : term,X420 : term,X418 : term,X417 : term] : ((X417 = X418) | (((ap @ X418 @ X419)) != ((ap @ X417 @ X420)))) & ! [X416 : term] : (((sub @ X416 @ id)) = X416) & ? [X423 : term,X424 : term,X422 : term,X421 : term] : ((((ap @ (sub @ X421 @ id) @ X422)) = ((ap @ (sub @ X424 @ id) @ X423))) & (X421 != X424)))) & (? [X419 : term,X420 : term,X418 : term,X417 : term] : ((X417 != X418) & (((ap @ X418 @ X419)) = ((ap @ X417 @ X420)))) | ? [X416 : term] : (((sub @ X416 @ id)) != X416) | ! [X423 : term,X424 : term,X422 : term,X421 : term] : ((((ap @ (sub @ X421 @ id) @ X422)) != ((ap @ (sub @ X424 @ id) @ X423))) | (X421 = X424)) | (hoasapinj1_lthm != $true)) & (! [X304 : term,X307 : term,X305 : term,X306 : term] : ((X306 = X307) | (((ap @ X304 @ X307)) != ((ap @ X305 @ X306)))) | (apinj2 != $true)) & ((apinj2 = $true) | ? [X304 : term,X307 : term,X305 : term,X306 : term] : ((X306 != X307) & (((ap @ X304 @ X307)) = ((ap @ X305 @ X306))))) & ((induction = $true) | ? [X89 : (term > $o),X88 : term] : (! [X92 : term] : (($true != ((X89 @ X92))) | ($true = ((X89 @ (lam @ X92))))) & ($true != ((sP23 @ X89))) & (((X89 @ X88)) != $true) & ! [X93 : term] : ((((var @ X93)) != $true) | (((X89 @ X93)) = $true)))) & (! [X89 : (term > $o),X88 : term] : (? [X92 : term] : (($true = ((X89 @ X92))) & ($true != ((X89 @ (lam @ X92))))) | ($true = ((sP23 @ X89))) | (((X89 @ X88)) = $true) | ? [X93 : term] : ((((var @ X93)) = $true) & (((X89 @ X93)) != $true))) | (induction != $true)) & (! [X72 : term,X71 : term,X70 : term,X73 : term] : ((((ap @ X71 @ X73)) != ((ap @ X70 @ X72))) | (X70 = X71)) | (apinj1 != $true)) & ((apinj1 = $true) | ? [X72 : term,X71 : term,X70 : term,X73 : term] : ((((ap @ X71 @ X73)) = ((ap @ X70 @ X72))) & (X70 != X71))) & ((hoasinduction = $true) | ($true != sP22)) & (($true = sP22) | (hoasinduction != $true)) & ((axscons = $true) | ? [X217 : subst] : (((push @ (sub @ one @ X217) @ (comp @ sh @ X217))) != X217)) & (! [X217 : subst] : (((push @ (sub @ one @ X217) @ (comp @ sh @ X217))) = X217) | (axscons != $true)) & (hoasapinj1_gthm = $true) & (lamnotap != $true) & (! [X410 : term] : ? [X411 : (subst > term > term)] : ! [X413 : term,X412 : subst] : (((sub @ X410 @ (push @ X413 @ X412))) = ((X411 @ X412 @ X413))) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ? [X410 : term] : ! [X411 : (subst > term > term)] : ? [X413 : term,X412 : subst] : (((sub @ X410 @ (push @ X413 @ X412))) != ((X411 @ X412 @ X413)))) & (lamnotvar = $true) & ((hoasinduction_no_psi_cond = $true) | ? [X143 : (subst > term > subst > $o),X142 : term] : (! [X145 : term,X144 : term] : ((((X143 @ id @ (ap @ (sub @ X144 @ id) @ X145) @ id)) = $true) | (((X143 @ id @ X145 @ id)) != $true) | (((X143 @ id @ X144 @ id)) != $true)) & (((X143 @ id @ X142 @ id)) != $true) & ($true != ((sP19 @ X143))))) & (! [X143 : (subst > term > subst > $o),X142 : term] : (? [X145 : term,X144 : term] : ((((X143 @ id @ (ap @ (sub @ X144 @ id) @ X145) @ id)) != $true) & (((X143 @ id @ X145 @ id)) = $true) & (((X143 @ id @ X144 @ id)) = $true)) | (((X143 @ id @ X142 @ id)) = $true) | ($true = ((sP19 @ X143)))) | (hoasinduction_no_psi_cond != $true)) & (hoasinduction_lem1v2 = $true) & ((sP18 = $true) | (hoasinduction_lem3aaa != $true)) & ((hoasinduction_lem3aaa = $true) | (sP18 != $true)) & (! [X352 : term] : (((sub @ X352 @ id)) = X352) | (axvarid != $true)) & ((axvarid = $true) | ? [X352 : term] : (((sub @ X352 @ id)) != X352)) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ((pushprop_lem1v2 = $true) | ? [X277 : (term > $o),X276 : term,X274 : subst,X275 : (term > $o)] : ((((X275 @ one)) != $true) & (((X277 @ X276)) = $true) & ! [X278 : term] : (((X275 @ X278)) = ((X277 @ (sub @ X278 @ (push @ X276 @ X274))))))) & (! [X277 : (term > $o),X276 : term,X274 : subst,X275 : (term > $o)] : ((((X275 @ one)) = $true) | (((X277 @ X276)) != $true) | ? [X278 : term] : (((X275 @ X278)) != ((X277 @ (sub @ X278 @ (push @ X276 @ X274)))))) | (pushprop_lem1v2 != $true)) & (! [X308 : term] : ? [X309 : (subst > term > term)] : ! [X311 : subst,X310 : term] : (((X309 @ X311 @ X310)) = ((sub @ X308 @ (push @ X310 @ X311)))) | (hoasinduction_lem3v2_f_lthm != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X308 : term] : ! [X309 : (subst > term > term)] : ? [X311 : subst,X310 : term] : (((X309 @ X311 @ X310)) != ((sub @ X308 @ (push @ X310 @ X311))))) & (($true = sP17) | (hoasinduction_lthm != $true)) & ((hoasinduction_lthm = $true) | ($true != sP17)) & (substmonoid_gthm = $true) & (pushprop_lthm = $true) & ((hoasinduction_lem3v2 = $true) | ($true != sP8)) & (($true = sP8) | (hoasinduction_lem3v2 != $true)) & (axapp = $true) & (hoasinduction_lem2v2_gthm = $true) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem1_lthm = $true) & ((substmonoid_lthm = $true) | (! [X0 : subst] : (((comp @ id @ X0)) = X0) & (? [X3 : subst] : (((comp @ X3 @ id)) != X3) | ? [X2 : subst] : (((comp @ id @ X2)) != X2)) & ! [X1 : subst] : (((comp @ X1 @ id)) = X1))) & (? [X0 : subst] : (((comp @ id @ X0)) != X0) | (! [X3 : subst] : (((comp @ X3 @ id)) = X3) & ! [X2 : subst] : (((comp @ id @ X2)) = X2)) | ? [X1 : subst] : (((comp @ X1 @ id)) != X1) | (substmonoid_lthm != $true)) & (hoaslamnotap_lthm = $true) & (hoasinduction_lem3a_gthm = $true) & ! [X226 : d_subst] : (d_id = X226) & (pushprop = $true) & (! [X128 : term,X125 : term,X127 : term,X126 : term] : ((((ap @ (sub @ X127 @ id) @ X128)) != ((ap @ (sub @ X126 @ id) @ X125))) | (X125 = X128)) | (hoasapinj2 != $true)) & ((hoasapinj2 = $true) | ? [X128 : term,X125 : term,X127 : term,X126 : term] : ((((ap @ (sub @ X127 @ id) @ X128)) = ((ap @ (sub @ X126 @ id) @ X125))) & (X125 != X128))) & (($true = sP6) | (hoasinduction_no_psi_cond_lthm != $true)) & ((hoasinduction_no_psi_cond_lthm = $true) | ($true != sP6)) & ((pushprop_lem0_lthm = $true) | ? [X279 : (term > $o),X281 : subst,X280 : term] : ! [X282 : (term > $o)] : ? [X283 : term] : (((X279 @ (sub @ X283 @ (push @ X280 @ X281)))) != ((X282 @ X283)))) & (! [X279 : (term > $o),X281 : subst,X280 : term] : ? [X282 : (term > $o)] : ! [X283 : term] : (((X279 @ (sub @ X283 @ (push @ X280 @ X281)))) = ((X282 @ X283))) | (pushprop_lem0_lthm != $true)) & (hoasapnotvar = $true) & (ulamvar1 != $true) & (pushprop_lthm_orig = $true) & ((hoasinduction_lem2v2 = $true) | (sP1 != $true)) & ((sP1 = $true) | (hoasinduction_lem2v2 != $true)) & (hoaslamnotap_gthm = $true)),
  inference(flattening,[],[f260])).
thf(f262,plain,(
  ((axidr = $true) | ? [X0 : subst] : (((comp @ X0 @ id)) != X0)) & (! [X1 : subst] : (((comp @ X1 @ id)) = X1) | (axidr != $true)) & ((sP58 = $true) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | (sP58 != $true)) & (pushprop_lem1_gthm = $true) & (! [X2 : (term > $o),X3 : subst,X4 : term] : ? [X5 : (term > $o)] : ! [X6 : term] : (((X5 @ X6)) = ((X2 @ (sub @ X6 @ (push @ X4 @ X3))))) | (pushprop_lem0 != $true)) & ((pushprop_lem0 = $true) | ? [X7 : (term > $o),X8 : subst,X9 : term] : ! [X10 : (term > $o)] : ? [X11 : term] : (((X10 @ X11)) != ((X7 @ (sub @ X11 @ (push @ X9 @ X8)))))) & ((hoasinduction_lem2 = $true) | ($true != sP52)) & (($true = sP52) | (hoasinduction_lem2 != $true)) & ((hoaslaminj = $true) | ? [X12 : (subst > term > term),X13 : (subst > term > term),X14 : term,X15 : subst] : (! [X16 : term,X17 : subst,X18 : subst] : (((X13 @ (comp @ X17 @ X18) @ (sub @ X16 @ X18))) = ((sub @ (X13 @ X17 @ X16) @ X18))) & (((X12 @ X15 @ X14)) != ((X13 @ X15 @ X14))) & ! [X19 : subst,X20 : term,X21 : subst] : (((sub @ (X12 @ X21 @ X20) @ X19)) = ((X12 @ (comp @ X21 @ X19) @ (sub @ X20 @ X19)))) & (((lam @ (X12 @ sh @ one))) = ((lam @ (X13 @ sh @ one)))))) & (! [X22 : (subst > term > term),X23 : (subst > term > term),X24 : term,X25 : subst] : (? [X26 : term,X27 : subst,X28 : subst] : (((sub @ (X23 @ X27 @ X26) @ X28)) != ((X23 @ (comp @ X27 @ X28) @ (sub @ X26 @ X28)))) | (((X23 @ X25 @ X24)) = ((X22 @ X25 @ X24))) | ? [X29 : subst,X30 : term,X31 : subst] : (((X22 @ (comp @ X31 @ X29) @ (sub @ X30 @ X29))) != ((sub @ (X22 @ X31 @ X30) @ X29))) | (((lam @ (X22 @ sh @ one))) != ((lam @ (X23 @ sh @ one))))) | (hoaslaminj != $true)) & (! [X32 : term] : ? [X33 : (subst > term > term)] : (((sub @ X32 @ (push @ one @ sh))) = ((X33 @ sh @ one))) | (hoasinduction_lem3b != $true)) & ((hoasinduction_lem3b = $true) | ? [X34 : term] : ! [X35 : (subst > term > term)] : (((X35 @ sh @ one)) != ((sub @ X34 @ (push @ one @ sh))))) & ! [X36 : d_subst,X37 : d_subst] : ((((d2subst @ X36)) != ((d2subst @ X37))) | (X36 = X37)) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ((hoasinduction_lem3a_lthm = $true) | (sP50 != $true)) & ((sP50 = $true) | (hoasinduction_lem3a_lthm != $true)) & (pushprop_lem2v2 = $true) & ((hoasinduction_lem3v2a = $true) | (sP48 != $true)) & ((sP48 = $true) | (hoasinduction_lem3v2a != $true)) & ((sP47 = $true) | (hoasinduction_lem3aa_lthm != $true)) & ((hoasinduction_lem3aa_lthm = $true) | (sP47 != $true)) & (pushprop_lem1v2_gthm = $true) & ((induction2lem = $true) | ($true != sP46)) & (($true = sP46) | (induction2lem != $true)) & (axassoc = $true) & (pushprop_lem0_gthm = $true) & (! [X38 : term,X39 : term,X40 : term,X41 : term] : ((X38 = X41) | (((ap @ (sub @ X41 @ id) @ X39)) != ((ap @ (sub @ X38 @ id) @ X40)))) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | ? [X42 : term,X43 : term,X44 : term,X45 : term] : ((X42 != X45) & (((ap @ (sub @ X42 @ id) @ X44)) = ((ap @ (sub @ X45 @ id) @ X43))))) & ((sP44 = $true) | (hoasinduction_lem3_lthm != $true)) & ((hoasinduction_lem3_lthm = $true) | (sP44 != $true)) & (hoasapinj2_gthm = $true) & (((var @ (d2term @ d_one))) != $true) & (($true = sP41) | (hoasinduction_lem3 != $true)) & ((hoasinduction_lem3 = $true) | ($true != sP41)) & ! [X46 : term] : ? [X47 : d_term] : (((d2term @ X47)) = X46) & (axabs = $true) & (hoasinduction_lem2_lthm = $true) & (hoasapnotvar_lthm = $true) & (hoasinduction_lem1v2_gthm = $true) & (hoasinduction_lem1 = $true) & (pushprop_lem2v2_lthm = $true) & (! [X48 : term] : ? [X49 : (subst > term > term)] : (((sub @ X48 @ (push @ one @ sh))) = ((X49 @ sh @ one))) | (hoasinduction_lem3b_lthm != $true)) & ((hoasinduction_lem3b_lthm = $true) | ? [X50 : term] : ! [X51 : (subst > term > term)] : (((sub @ X50 @ (push @ one @ sh))) != ((X51 @ sh @ one)))) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((hoasinduction_lem3a = $true) | ($true != sP39)) & (($true = sP39) | (hoasinduction_lem3a != $true)) & ! [X52 : d_term,X53 : d_term] : ((((d2term @ X52)) != ((d2term @ X53))) | (X52 = X53)) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((hoasinduction_lem3aa = $true) | ($true != sP38)) & (($true = sP38) | (hoasinduction_lem3aa != $true)) & (one = ((d2term @ d_one))) & (! [X54 : term,X55 : term] : ((X54 = X55) | (((lam @ X55)) != ((lam @ X54)))) | (laminj != $true)) & ((laminj = $true) | ? [X56 : term,X57 : term] : ((X56 != X57) & (((lam @ X56)) = ((lam @ X57))))) & (hoaslamnotvar = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ! [X58 : subst] : ? [X59 : d_subst] : (((d2subst @ X59)) = X58) & (! [X60 : (subst > term > subst > $o)] : ? [X61 : (term > $o)] : ! [X62 : term] : (((X60 @ id @ X62 @ id)) = ((X61 @ X62))) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ? [X63 : (subst > term > subst > $o)] : ! [X64 : (term > $o)] : ? [X65 : term] : (((X64 @ X65)) != ((X63 @ id @ X65 @ id)))) & (((d2subst @ d_id)) = id) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & ((hoasapinj2_lthm = $true) | (! [X66 : term,X67 : term,X68 : term,X69 : term] : ((X66 = X68) | (((ap @ X67 @ X66)) != ((ap @ X69 @ X68)))) & ? [X70 : term,X71 : term,X72 : term,X73 : term] : ((X71 != X73) & (((ap @ (sub @ X72 @ id) @ X71)) = ((ap @ (sub @ X70 @ id) @ X73)))))) & (? [X74 : term,X75 : term,X76 : term,X77 : term] : ((X74 != X76) & (((ap @ X75 @ X74)) = ((ap @ X77 @ X76)))) | ! [X78 : term,X79 : term,X80 : term,X81 : term] : ((X79 = X81) | (((ap @ (sub @ X80 @ id) @ X79)) != ((ap @ (sub @ X78 @ id) @ X81)))) | (hoasapinj2_lthm != $true)) & (pushprop_lem2v2_gthm = $true) & (? [X82 : subst,X83 : term] : (((comp @ sh @ (push @ X83 @ X82))) != X82) | ? [X84 : subst,X85 : term] : (((sub @ one @ (push @ X85 @ X84))) != X85) | ! [X86 : term,X87 : (term > $o),X88 : subst,X89 : (term > $o),X90 : term] : (($true = ((X89 @ (sub @ X86 @ (push @ X90 @ X88))))) | ($true != ((X87 @ X86)))) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (! [X91 : subst,X92 : term] : (((comp @ sh @ (push @ X92 @ X91))) = X91) & ! [X93 : subst,X94 : term] : (((sub @ one @ (push @ X94 @ X93))) = X94) & ? [X95 : term,X96 : (term > $o),X97 : subst,X98 : (term > $o),X99 : term] : (($true != ((X98 @ (sub @ X95 @ (push @ X99 @ X97))))) & (((X96 @ X95)) = $true)))) & (($true = sP37) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | ($true != sP37)) & ((termmset_lthm = $true) | (? [X100 : term] : (((sub @ X100 @ id)) != X100) & ! [X101 : term] : (((sub @ X101 @ id)) = X101))) & (! [X102 : term] : (((sub @ X102 @ id)) = X102) | ? [X103 : term] : (((sub @ X103 @ id)) != X103) | (termmset_lthm != $true)) & (pushprop_lem3v2 = $true) & ((sP33 = $true) | (induction2 != $true)) & ((induction2 = $true) | (sP33 != $true)) & (pushprop_lem3v2_lthm = $true) & (hoasinduction_lem3v2_gthm = $true) & (((d2subst @ d_id)) = sh) & (hoaslaminj_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & ((hoaslaminj_lthm = $true) | (sP31 != $true)) & ((sP31 = $true) | (hoaslaminj_lthm != $true)) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & ((hoasinduction_lem3v2a_lthm = $true) | (sP30 != $true)) & ((sP30 = $true) | (hoasinduction_lem3v2a_lthm != $true)) & ((sP29 = $true) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | (sP29 != $true)) & (! [X104 : (subst > term > subst > $o)] : ? [X105 : (term > $o)] : ! [X106 : term] : (((X105 @ X106)) = ((X104 @ id @ X106 @ id))) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem0 = $true) | ? [X107 : (subst > term > subst > $o)] : ! [X108 : (term > $o)] : ? [X109 : term] : (((X107 @ id @ X109 @ id)) != ((X108 @ X109)))) & (! [X110 : (subst > term > term),X111 : term,X112 : term] : ((((ap @ (sub @ X111 @ id) @ X112)) != ((lam @ (X110 @ sh @ one)))) | ? [X113 : term,X114 : subst,X115 : subst] : (((X110 @ (comp @ X115 @ X114) @ (sub @ X113 @ X114))) != ((sub @ (X110 @ X115 @ X113) @ X114)))) | (hoaslamnotap != $true)) & ((hoaslamnotap = $true) | ? [X116 : (subst > term > term),X117 : term,X118 : term] : ((((lam @ (X116 @ sh @ one))) = ((ap @ (sub @ X117 @ id) @ X118))) & ! [X119 : term,X120 : subst,X121 : subst] : (((sub @ (X116 @ X121 @ X119) @ X120)) = ((X116 @ (comp @ X121 @ X120) @ (sub @ X119 @ X120)))))) & (! [X122 : term,X123 : subst] : (((sub @ one @ (push @ X122 @ X123))) = X122) | (axvarcons != $true)) & ((axvarcons = $true) | ? [X124 : term,X125 : subst] : (((sub @ one @ (push @ X124 @ X125))) != X124)) & (hoasinduction_lem3b_gthm = $true) & (hoaslamnotvar_lthm = $true) & (termmset_gthm = $true) & (! [X126 : subst,X127 : (term > $o),X128 : (term > $o),X129 : term] : (($true != ((X128 @ X129))) | ? [X130 : term] : (((X128 @ (sub @ X130 @ (push @ X129 @ X126)))) != ((X127 @ X130))) | (((X127 @ one)) = $true)) | ? [X131 : term,X132 : subst] : (((sub @ one @ (push @ X131 @ X132))) != X131) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (? [X133 : subst,X134 : (term > $o),X135 : (term > $o),X136 : term] : (($true = ((X135 @ X136))) & ! [X137 : term] : (((X134 @ X137)) = ((X135 @ (sub @ X137 @ (push @ X136 @ X133))))) & (((X134 @ one)) != $true)) & ! [X138 : term,X139 : subst] : (((sub @ one @ (push @ X138 @ X139))) = X138))) & (! [X140 : term,X141 : subst] : (((comp @ sh @ (push @ X140 @ X141))) = X141) | (axshiftcons != $true)) & ((axshiftcons = $true) | ? [X142 : term,X143 : subst] : (((comp @ sh @ (push @ X142 @ X143))) != X143)) & ! [X144 : d_term] : (d_one = X144) & (hoasinduction_lem3_gthm = $true) & ((! [X145 : subst] : (((comp @ id @ X145)) = X145) & ! [X146 : subst] : (((comp @ X146 @ id)) = X146)) | (substmonoid != $true)) & ((substmonoid = $true) | ? [X147 : subst] : (((comp @ id @ X147)) != X147) | ? [X148 : subst] : (((comp @ X148 @ id)) != X148)) & (hoasinduction_gthm = $true) & (hoaslamnotvar_gthm = $true) & (axvarshift = $true) & (hoasinduction_lem2_gthm = $true) & (axclos = $true) & (! [X149 : (term > $o),X150 : subst,X151 : (term > $o),X152 : term,X153 : term] : ((((X149 @ X153)) != $true) | ($true = ((X151 @ (sub @ X153 @ (push @ X152 @ X150)))))) | (pushprop_lem1 != $true)) & ((pushprop_lem1 = $true) | ? [X154 : (term > $o),X155 : subst,X156 : (term > $o),X157 : term,X158 : term] : ((((X154 @ X158)) = $true) & ($true != ((X156 @ (sub @ X158 @ (push @ X157 @ X155))))))) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (induction2_gthm = $true) & (hoasinduction_lem1_gthm = $true) & (ulamvarsh = $true) & ((termmset = $true) | ? [X159 : term] : (((sub @ X159 @ id)) != X159)) & (! [X160 : term] : (((sub @ X160 @ id)) = X160) | (termmset != $true)) & (induction2lem_gthm = $true) & (axmap = $true) & (! [X161 : term,X162 : term] : ((X161 = X162) | (((sub @ X161 @ sh)) != ((sub @ X162 @ sh)))) | (shinj != $true)) & ((shinj = $true) | ? [X163 : term,X164 : term] : ((X163 != X164) & (((sub @ X163 @ sh)) = ((sub @ X164 @ sh))))) & (ulamvarind = $true) & (! [X165 : subst] : (((comp @ id @ X165)) = X165) | (axidl != $true)) & ((axidl = $true) | ? [X166 : subst] : (((comp @ id @ X166)) != X166)) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoasinduction_lem3v2_lthm = $true) | (sP25 != $true)) & ((sP25 = $true) | (hoasinduction_lem3v2_lthm != $true)) & (pushprop_gthm = $true) & (apnotvar = $true) & ((hoasapinj1_lthm = $true) | (! [X167 : term,X168 : term,X169 : term,X170 : term] : ((X169 = X170) | (((ap @ X170 @ X168)) != ((ap @ X169 @ X167)))) & ! [X171 : term] : (((sub @ X171 @ id)) = X171) & ? [X172 : term,X173 : term,X174 : term,X175 : term] : ((((ap @ (sub @ X175 @ id) @ X174)) = ((ap @ (sub @ X173 @ id) @ X172))) & (X173 != X175)))) & (? [X176 : term,X177 : term,X178 : term,X179 : term] : ((X178 != X179) & (((ap @ X178 @ X176)) = ((ap @ X179 @ X177)))) | ? [X180 : term] : (((sub @ X180 @ id)) != X180) | ! [X181 : term,X182 : term,X183 : term,X184 : term] : ((((ap @ (sub @ X182 @ id) @ X181)) != ((ap @ (sub @ X184 @ id) @ X183))) | (X182 = X184)) | (hoasapinj1_lthm != $true)) & (! [X185 : term,X186 : term,X187 : term,X188 : term] : ((X186 = X188) | (((ap @ X185 @ X186)) != ((ap @ X187 @ X188)))) | (apinj2 != $true)) & ((apinj2 = $true) | ? [X189 : term,X190 : term,X191 : term,X192 : term] : ((X190 != X192) & (((ap @ X189 @ X190)) = ((ap @ X191 @ X192))))) & ((induction = $true) | ? [X193 : (term > $o),X194 : term] : (! [X195 : term] : ((((X193 @ X195)) != $true) | (((X193 @ (lam @ X195))) = $true)) & (((sP23 @ X193)) != $true) & ($true != ((X193 @ X194))) & ! [X196 : term] : (($true != ((var @ X196))) | ($true = ((X193 @ X196)))))) & (! [X197 : (term > $o),X198 : term] : (? [X199 : term] : ((((X197 @ X199)) = $true) & (((X197 @ (lam @ X199))) != $true)) | (((sP23 @ X197)) = $true) | ($true = ((X197 @ X198))) | ? [X200 : term] : (($true = ((var @ X200))) & ($true != ((X197 @ X200))))) | (induction != $true)) & (! [X201 : term,X202 : term,X203 : term,X204 : term] : ((((ap @ X203 @ X201)) != ((ap @ X202 @ X204))) | (X202 = X203)) | (apinj1 != $true)) & ((apinj1 = $true) | ? [X205 : term,X206 : term,X207 : term,X208 : term] : ((((ap @ X207 @ X205)) = ((ap @ X206 @ X208))) & (X206 != X207))) & ((hoasinduction = $true) | ($true != sP22)) & (($true = sP22) | (hoasinduction != $true)) & ((axscons = $true) | ? [X209 : subst] : (((push @ (sub @ one @ X209) @ (comp @ sh @ X209))) != X209)) & (! [X210 : subst] : (((push @ (sub @ one @ X210) @ (comp @ sh @ X210))) = X210) | (axscons != $true)) & (hoasapinj1_gthm = $true) & (lamnotap != $true) & (! [X211 : term] : ? [X212 : (subst > term > term)] : ! [X213 : term,X214 : subst] : (((X212 @ X214 @ X213)) = ((sub @ X211 @ (push @ X213 @ X214)))) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ? [X215 : term] : ! [X216 : (subst > term > term)] : ? [X217 : term,X218 : subst] : (((sub @ X215 @ (push @ X217 @ X218))) != ((X216 @ X218 @ X217)))) & (lamnotvar = $true) & ((hoasinduction_no_psi_cond = $true) | ? [X219 : (subst > term > subst > $o),X220 : term] : (! [X221 : term,X222 : term] : (($true = ((X219 @ id @ (ap @ (sub @ X222 @ id) @ X221) @ id))) | (((X219 @ id @ X221 @ id)) != $true) | ($true != ((X219 @ id @ X222 @ id)))) & ($true != ((X219 @ id @ X220 @ id))) & (((sP19 @ X219)) != $true))) & (! [X223 : (subst > term > subst > $o),X224 : term] : (? [X225 : term,X226 : term] : ((((X223 @ id @ (ap @ (sub @ X226 @ id) @ X225) @ id)) != $true) & ($true = ((X223 @ id @ X225 @ id))) & (((X223 @ id @ X226 @ id)) = $true)) | ($true = ((X223 @ id @ X224 @ id))) | (((sP19 @ X223)) = $true)) | (hoasinduction_no_psi_cond != $true)) & (hoasinduction_lem1v2 = $true) & ((sP18 = $true) | (hoasinduction_lem3aaa != $true)) & ((hoasinduction_lem3aaa = $true) | (sP18 != $true)) & (! [X227 : term] : (((sub @ X227 @ id)) = X227) | (axvarid != $true)) & ((axvarid = $true) | ? [X228 : term] : (((sub @ X228 @ id)) != X228)) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ((pushprop_lem1v2 = $true) | ? [X229 : (term > $o),X230 : term,X231 : subst,X232 : (term > $o)] : ((((X232 @ one)) != $true) & (((X229 @ X230)) = $true) & ! [X233 : term] : (((X232 @ X233)) = ((X229 @ (sub @ X233 @ (push @ X230 @ X231))))))) & (! [X234 : (term > $o),X235 : term,X236 : subst,X237 : (term > $o)] : (($true = ((X237 @ one))) | (((X234 @ X235)) != $true) | ? [X238 : term] : (((X237 @ X238)) != ((X234 @ (sub @ X238 @ (push @ X235 @ X236)))))) | (pushprop_lem1v2 != $true)) & (! [X239 : term] : ? [X240 : (subst > term > term)] : ! [X241 : subst,X242 : term] : (((X240 @ X241 @ X242)) = ((sub @ X239 @ (push @ X242 @ X241)))) | (hoasinduction_lem3v2_f_lthm != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X243 : term] : ! [X244 : (subst > term > term)] : ? [X245 : subst,X246 : term] : (((sub @ X243 @ (push @ X246 @ X245))) != ((X244 @ X245 @ X246)))) & (($true = sP17) | (hoasinduction_lthm != $true)) & ((hoasinduction_lthm = $true) | ($true != sP17)) & (substmonoid_gthm = $true) & (pushprop_lthm = $true) & ((hoasinduction_lem3v2 = $true) | ($true != sP8)) & (($true = sP8) | (hoasinduction_lem3v2 != $true)) & (axapp = $true) & (hoasinduction_lem2v2_gthm = $true) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem1_lthm = $true) & ((substmonoid_lthm = $true) | (! [X247 : subst] : (((comp @ id @ X247)) = X247) & (? [X248 : subst] : (((comp @ X248 @ id)) != X248) | ? [X249 : subst] : (((comp @ id @ X249)) != X249)) & ! [X250 : subst] : (((comp @ X250 @ id)) = X250))) & (? [X251 : subst] : (((comp @ id @ X251)) != X251) | (! [X252 : subst] : (((comp @ X252 @ id)) = X252) & ! [X253 : subst] : (((comp @ id @ X253)) = X253)) | ? [X254 : subst] : (((comp @ X254 @ id)) != X254) | (substmonoid_lthm != $true)) & (hoaslamnotap_lthm = $true) & (hoasinduction_lem3a_gthm = $true) & ! [X255 : d_subst] : (d_id = X255) & (pushprop = $true) & (! [X256 : term,X257 : term,X258 : term,X259 : term] : ((((ap @ (sub @ X259 @ id) @ X257)) != ((ap @ (sub @ X258 @ id) @ X256))) | (X256 = X257)) | (hoasapinj2 != $true)) & ((hoasapinj2 = $true) | ? [X260 : term,X261 : term,X262 : term,X263 : term] : ((((ap @ (sub @ X263 @ id) @ X261)) = ((ap @ (sub @ X262 @ id) @ X260))) & (X260 != X261))) & (($true = sP6) | (hoasinduction_no_psi_cond_lthm != $true)) & ((hoasinduction_no_psi_cond_lthm = $true) | ($true != sP6)) & ((pushprop_lem0_lthm = $true) | ? [X264 : (term > $o),X265 : subst,X266 : term] : ! [X267 : (term > $o)] : ? [X268 : term] : (((X267 @ X268)) != ((X264 @ (sub @ X268 @ (push @ X266 @ X265)))))) & (! [X269 : (term > $o),X270 : subst,X271 : term] : ? [X272 : (term > $o)] : ! [X273 : term] : (((X272 @ X273)) = ((X269 @ (sub @ X273 @ (push @ X271 @ X270))))) | (pushprop_lem0_lthm != $true)) & (hoasapnotvar = $true) & (ulamvar1 != $true) & (pushprop_lthm_orig = $true) & ((hoasinduction_lem2v2 = $true) | (sP1 != $true)) & ((sP1 = $true) | (hoasinduction_lem2v2 != $true)) & (hoaslamnotap_gthm = $true)),
  inference(rectify,[],[f261])).
thf(f263,plain,(
  ((axidr = $true) | (((comp @ sK457 @ id)) != sK457)) & (! [X1 : subst] : (((comp @ X1 @ id)) = X1) | (axidr != $true)) & ((sP58 = $true) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | (sP58 != $true)) & (pushprop_lem1_gthm = $true) & (! [X2 : (term > $o),X3 : subst,X4 : term] : ! [X6 : term] : (((sK458 @ X4 @ X3 @ X2 @ X6)) = ((X2 @ (sub @ X6 @ (push @ X4 @ X3))))) | (pushprop_lem0 != $true)) & ((pushprop_lem0 = $true) | ! [X10 : (term > $o)] : (((X10 @ (sK462 @ X10))) != ((sK459 @ (sub @ (sK462 @ X10) @ (push @ sK461 @ sK460)))))) & ((hoasinduction_lem2 = $true) | ($true != sP52)) & (($true = sP52) | (hoasinduction_lem2 != $true)) & ((hoaslaminj = $true) | (! [X16 : term,X17 : subst,X18 : subst] : (((sub @ (sK464 @ X17 @ X16) @ X18)) = ((sK464 @ (comp @ X17 @ X18) @ (sub @ X16 @ X18)))) & (((sK463 @ sK466 @ sK465)) != ((sK464 @ sK466 @ sK465))) & ! [X19 : subst,X20 : term,X21 : subst] : (((sub @ (sK463 @ X21 @ X20) @ X19)) = ((sK463 @ (comp @ X21 @ X19) @ (sub @ X20 @ X19)))) & (((lam @ (sK464 @ sh @ one))) = ((lam @ (sK463 @ sh @ one)))))) & (! [X22 : (subst > term > term),X23 : (subst > term > term),X24 : term,X25 : subst] : ((((sub @ (X23 @ (sK468 @ X23) @ (sK467 @ X23)) @ (sK469 @ X23))) != ((X23 @ (comp @ (sK468 @ X23) @ (sK469 @ X23)) @ (sub @ (sK467 @ X23) @ (sK469 @ X23))))) | (((X23 @ X25 @ X24)) = ((X22 @ X25 @ X24))) | (((sub @ (X22 @ (sK472 @ X22) @ (sK471 @ X22)) @ (sK470 @ X22))) != ((X22 @ (comp @ (sK472 @ X22) @ (sK470 @ X22)) @ (sub @ (sK471 @ X22) @ (sK470 @ X22))))) | (((lam @ (X22 @ sh @ one))) != ((lam @ (X23 @ sh @ one))))) | (hoaslaminj != $true)) & (! [X32 : term] : (((sub @ X32 @ (push @ one @ sh))) = ((sK473 @ X32 @ sh @ one))) | (hoasinduction_lem3b != $true)) & ((hoasinduction_lem3b = $true) | ! [X35 : (subst > term > term)] : (((X35 @ sh @ one)) != ((sub @ sK474 @ (push @ one @ sh))))) & ! [X36 : d_subst,X37 : d_subst] : ((((d2subst @ X36)) != ((d2subst @ X37))) | (X36 = X37)) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ((hoasinduction_lem3a_lthm = $true) | (sP50 != $true)) & ((sP50 = $true) | (hoasinduction_lem3a_lthm != $true)) & (pushprop_lem2v2 = $true) & ((hoasinduction_lem3v2a = $true) | (sP48 != $true)) & ((sP48 = $true) | (hoasinduction_lem3v2a != $true)) & ((sP47 = $true) | (hoasinduction_lem3aa_lthm != $true)) & ((hoasinduction_lem3aa_lthm = $true) | (sP47 != $true)) & (pushprop_lem1v2_gthm = $true) & ((induction2lem = $true) | ($true != sP46)) & (($true = sP46) | (induction2lem != $true)) & (axassoc = $true) & (pushprop_lem0_gthm = $true) & (! [X38 : term,X39 : term,X40 : term,X41 : term] : ((X38 = X41) | (((ap @ (sub @ X41 @ id) @ X39)) != ((ap @ (sub @ X38 @ id) @ X40)))) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | ((sK478 != sK475) & (((ap @ (sub @ sK478 @ id) @ sK476)) = ((ap @ (sub @ sK475 @ id) @ sK477))))) & ((sP44 = $true) | (hoasinduction_lem3_lthm != $true)) & ((hoasinduction_lem3_lthm = $true) | (sP44 != $true)) & (hoasapinj2_gthm = $true) & (((var @ (d2term @ d_one))) != $true) & (($true = sP41) | (hoasinduction_lem3 != $true)) & ((hoasinduction_lem3 = $true) | ($true != sP41)) & ! [X46 : term] : (((d2term @ (sK479 @ X46))) = X46) & (axabs = $true) & (hoasinduction_lem2_lthm = $true) & (hoasapnotvar_lthm = $true) & (hoasinduction_lem1v2_gthm = $true) & (hoasinduction_lem1 = $true) & (pushprop_lem2v2_lthm = $true) & (! [X48 : term] : (((sub @ X48 @ (push @ one @ sh))) = ((sK480 @ X48 @ sh @ one))) | (hoasinduction_lem3b_lthm != $true)) & ((hoasinduction_lem3b_lthm = $true) | ! [X51 : (subst > term > term)] : (((X51 @ sh @ one)) != ((sub @ sK481 @ (push @ one @ sh))))) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((hoasinduction_lem3a = $true) | ($true != sP39)) & (($true = sP39) | (hoasinduction_lem3a != $true)) & ! [X52 : d_term,X53 : d_term] : ((((d2term @ X52)) != ((d2term @ X53))) | (X52 = X53)) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((hoasinduction_lem3aa = $true) | ($true != sP38)) & (($true = sP38) | (hoasinduction_lem3aa != $true)) & (one = ((d2term @ d_one))) & (! [X54 : term,X55 : term] : ((X54 = X55) | (((lam @ X55)) != ((lam @ X54)))) | (laminj != $true)) & ((laminj = $true) | ((sK483 != sK482) & (((lam @ sK483)) = ((lam @ sK482))))) & (hoaslamnotvar = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ! [X58 : subst] : (((d2subst @ (sK484 @ X58))) = X58) & (! [X60 : (subst > term > subst > $o)] : ! [X62 : term] : (((X60 @ id @ X62 @ id)) = ((sK485 @ X60 @ X62))) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ! [X64 : (term > $o)] : (((X64 @ (sK487 @ X64))) != ((sK486 @ id @ (sK487 @ X64) @ id)))) & (((d2subst @ d_id)) = id) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & ((hoasapinj2_lthm = $true) | (! [X66 : term,X67 : term,X68 : term,X69 : term] : ((X66 = X68) | (((ap @ X67 @ X66)) != ((ap @ X69 @ X68)))) & ((sK489 != sK491) & (((ap @ (sub @ sK490 @ id) @ sK489)) = ((ap @ (sub @ sK488 @ id) @ sK491)))))) & (((sK494 != sK492) & (((ap @ sK495 @ sK494)) = ((ap @ sK493 @ sK492)))) | ! [X78 : term,X79 : term,X80 : term,X81 : term] : ((X79 = X81) | (((ap @ (sub @ X80 @ id) @ X79)) != ((ap @ (sub @ X78 @ id) @ X81)))) | (hoasapinj2_lthm != $true)) & (pushprop_lem2v2_gthm = $true) & ((((comp @ sh @ (push @ sK497 @ sK496))) != sK496) | (sK499 != ((sub @ one @ (push @ sK499 @ sK498)))) | ! [X86 : term,X87 : (term > $o),X88 : subst,X89 : (term > $o),X90 : term] : (($true = ((X89 @ (sub @ X86 @ (push @ X90 @ X88))))) | ($true != ((X87 @ X86)))) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (! [X91 : subst,X92 : term] : (((comp @ sh @ (push @ X92 @ X91))) = X91) & ! [X93 : subst,X94 : term] : (((sub @ one @ (push @ X94 @ X93))) = X94) & ((((sK503 @ (sub @ sK500 @ (push @ sK504 @ sK502)))) != $true) & (((sK501 @ sK500)) = $true)))) & (($true = sP37) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | ($true != sP37)) & ((termmset_lthm = $true) | ((sK505 != ((sub @ sK505 @ id))) & ! [X101 : term] : (((sub @ X101 @ id)) = X101))) & (! [X102 : term] : (((sub @ X102 @ id)) = X102) | (sK506 != ((sub @ sK506 @ id))) | (termmset_lthm != $true)) & (pushprop_lem3v2 = $true) & ((sP33 = $true) | (induction2 != $true)) & ((induction2 = $true) | (sP33 != $true)) & (pushprop_lem3v2_lthm = $true) & (hoasinduction_lem3v2_gthm = $true) & (((d2subst @ d_id)) = sh) & (hoaslaminj_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & ((hoaslaminj_lthm = $true) | (sP31 != $true)) & ((sP31 = $true) | (hoaslaminj_lthm != $true)) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & ((hoasinduction_lem3v2a_lthm = $true) | (sP30 != $true)) & ((sP30 = $true) | (hoasinduction_lem3v2a_lthm != $true)) & ((sP29 = $true) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | (sP29 != $true)) & (! [X104 : (subst > term > subst > $o)] : ! [X106 : term] : (((X104 @ id @ X106 @ id)) = ((sK507 @ X104 @ X106))) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem0 = $true) | ! [X108 : (term > $o)] : (((sK508 @ id @ (sK509 @ X108) @ id)) != ((X108 @ (sK509 @ X108))))) & (! [X110 : (subst > term > term),X111 : term,X112 : term] : ((((ap @ (sub @ X111 @ id) @ X112)) != ((lam @ (X110 @ sh @ one)))) | (((X110 @ (comp @ (sK512 @ X110) @ (sK511 @ X110)) @ (sub @ (sK510 @ X110) @ (sK511 @ X110)))) != ((sub @ (X110 @ (sK512 @ X110) @ (sK510 @ X110)) @ (sK511 @ X110))))) | (hoaslamnotap != $true)) & ((hoaslamnotap = $true) | ((((lam @ (sK513 @ sh @ one))) = ((ap @ (sub @ sK514 @ id) @ sK515))) & ! [X119 : term,X120 : subst,X121 : subst] : (((sK513 @ (comp @ X121 @ X120) @ (sub @ X119 @ X120))) = ((sub @ (sK513 @ X121 @ X119) @ X120))))) & (! [X122 : term,X123 : subst] : (((sub @ one @ (push @ X122 @ X123))) = X122) | (axvarcons != $true)) & ((axvarcons = $true) | (sK516 != ((sub @ one @ (push @ sK516 @ sK517))))) & (hoasinduction_lem3b_gthm = $true) & (hoaslamnotvar_lthm = $true) & (termmset_gthm = $true) & (! [X126 : subst,X127 : (term > $o),X128 : (term > $o),X129 : term] : (($true != ((X128 @ X129))) | (((X128 @ (sub @ (sK518 @ X129 @ X128 @ X127 @ X126) @ (push @ X129 @ X126)))) != ((X127 @ (sK518 @ X129 @ X128 @ X127 @ X126)))) | (((X127 @ one)) = $true)) | (((sub @ one @ (push @ sK519 @ sK520))) != sK519) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (((((sK523 @ sK524)) = $true) & ! [X137 : term] : (((sK522 @ X137)) = ((sK523 @ (sub @ X137 @ (push @ sK524 @ sK521))))) & (((sK522 @ one)) != $true)) & ! [X138 : term,X139 : subst] : (((sub @ one @ (push @ X138 @ X139))) = X138))) & (! [X140 : term,X141 : subst] : (((comp @ sh @ (push @ X140 @ X141))) = X141) | (axshiftcons != $true)) & ((axshiftcons = $true) | (sK526 != ((comp @ sh @ (push @ sK525 @ sK526))))) & ! [X144 : d_term] : (d_one = X144) & (hoasinduction_lem3_gthm = $true) & ((! [X145 : subst] : (((comp @ id @ X145)) = X145) & ! [X146 : subst] : (((comp @ X146 @ id)) = X146)) | (substmonoid != $true)) & ((substmonoid = $true) | (sK527 != ((comp @ id @ sK527))) | (sK528 != ((comp @ sK528 @ id)))) & (hoasinduction_gthm = $true) & (hoaslamnotvar_gthm = $true) & (axvarshift = $true) & (hoasinduction_lem2_gthm = $true) & (axclos = $true) & (! [X149 : (term > $o),X150 : subst,X151 : (term > $o),X152 : term,X153 : term] : ((((X149 @ X153)) != $true) | ($true = ((X151 @ (sub @ X153 @ (push @ X152 @ X150)))))) | (pushprop_lem1 != $true)) & ((pushprop_lem1 = $true) | ((((sK529 @ sK533)) = $true) & ($true != ((sK531 @ (sub @ sK533 @ (push @ sK532 @ sK530))))))) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (induction2_gthm = $true) & (hoasinduction_lem1_gthm = $true) & (ulamvarsh = $true) & ((termmset = $true) | (((sub @ sK534 @ id)) != sK534)) & (! [X160 : term] : (((sub @ X160 @ id)) = X160) | (termmset != $true)) & (induction2lem_gthm = $true) & (axmap = $true) & (! [X161 : term,X162 : term] : ((X161 = X162) | (((sub @ X161 @ sh)) != ((sub @ X162 @ sh)))) | (shinj != $true)) & ((shinj = $true) | ((sK535 != sK536) & (((sub @ sK536 @ sh)) = ((sub @ sK535 @ sh))))) & (ulamvarind = $true) & (! [X165 : subst] : (((comp @ id @ X165)) = X165) | (axidl != $true)) & ((axidl = $true) | (((comp @ id @ sK537)) != sK537)) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoasinduction_lem3v2_lthm = $true) | (sP25 != $true)) & ((sP25 = $true) | (hoasinduction_lem3v2_lthm != $true)) & (pushprop_gthm = $true) & (apnotvar = $true) & ((hoasapinj1_lthm = $true) | (! [X167 : term,X168 : term,X169 : term,X170 : term] : ((X169 = X170) | (((ap @ X170 @ X168)) != ((ap @ X169 @ X167)))) & ! [X171 : term] : (((sub @ X171 @ id)) = X171) & ((((ap @ (sub @ sK541 @ id) @ sK540)) = ((ap @ (sub @ sK539 @ id) @ sK538))) & (sK541 != sK539)))) & (((sK544 != sK545) & (((ap @ sK545 @ sK543)) = ((ap @ sK544 @ sK542)))) | (((sub @ sK546 @ id)) != sK546) | ! [X181 : term,X182 : term,X183 : term,X184 : term] : ((((ap @ (sub @ X182 @ id) @ X181)) != ((ap @ (sub @ X184 @ id) @ X183))) | (X182 = X184)) | (hoasapinj1_lthm != $true)) & (! [X185 : term,X186 : term,X187 : term,X188 : term] : ((X186 = X188) | (((ap @ X185 @ X186)) != ((ap @ X187 @ X188)))) | (apinj2 != $true)) & ((apinj2 = $true) | ((sK548 != sK550) & (((ap @ sK547 @ sK548)) = ((ap @ sK549 @ sK550))))) & ((induction = $true) | (! [X195 : term] : ((((sK551 @ X195)) != $true) | ($true = ((sK551 @ (lam @ X195))))) & ($true != ((sP23 @ sK551))) & (((sK551 @ sK552)) != $true) & ! [X196 : term] : (($true != ((var @ X196))) | (((sK551 @ X196)) = $true)))) & (! [X197 : (term > $o),X198 : term] : (((((X197 @ (sK553 @ X197))) = $true) & ($true != ((X197 @ (lam @ (sK553 @ X197)))))) | (((sP23 @ X197)) = $true) | ($true = ((X197 @ X198))) | ((((var @ (sK554 @ X197))) = $true) & ($true != ((X197 @ (sK554 @ X197)))))) | (induction != $true)) & (! [X201 : term,X202 : term,X203 : term,X204 : term] : ((((ap @ X203 @ X201)) != ((ap @ X202 @ X204))) | (X202 = X203)) | (apinj1 != $true)) & ((apinj1 = $true) | ((((ap @ sK557 @ sK555)) = ((ap @ sK556 @ sK558))) & (sK557 != sK556))) & ((hoasinduction = $true) | ($true != sP22)) & (($true = sP22) | (hoasinduction != $true)) & ((axscons = $true) | (sK559 != ((push @ (sub @ one @ sK559) @ (comp @ sh @ sK559))))) & (! [X210 : subst] : (((push @ (sub @ one @ X210) @ (comp @ sh @ X210))) = X210) | (axscons != $true)) & (hoasapinj1_gthm = $true) & (lamnotap != $true) & (! [X211 : term] : ! [X213 : term,X214 : subst] : (((sK560 @ X211 @ X214 @ X213)) = ((sub @ X211 @ (push @ X213 @ X214)))) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ! [X216 : (subst > term > term)] : (((X216 @ (sK563 @ X216) @ (sK562 @ X216))) != ((sub @ sK561 @ (push @ (sK562 @ X216) @ (sK563 @ X216)))))) & (lamnotvar = $true) & ((hoasinduction_no_psi_cond = $true) | (! [X221 : term,X222 : term] : (($true = ((sK564 @ id @ (ap @ (sub @ X222 @ id) @ X221) @ id))) | ($true != ((sK564 @ id @ X221 @ id))) | ($true != ((sK564 @ id @ X222 @ id)))) & (((sK564 @ id @ sK565 @ id)) != $true) & ($true != ((sP19 @ sK564))))) & (! [X223 : (subst > term > subst > $o),X224 : term] : (((((X223 @ id @ (ap @ (sub @ (sK567 @ X223) @ id) @ (sK566 @ X223)) @ id)) != $true) & (((X223 @ id @ (sK566 @ X223) @ id)) = $true) & (((X223 @ id @ (sK567 @ X223) @ id)) = $true)) | ($true = ((X223 @ id @ X224 @ id))) | (((sP19 @ X223)) = $true)) | (hoasinduction_no_psi_cond != $true)) & (hoasinduction_lem1v2 = $true) & ((sP18 = $true) | (hoasinduction_lem3aaa != $true)) & ((hoasinduction_lem3aaa = $true) | (sP18 != $true)) & (! [X227 : term] : (((sub @ X227 @ id)) = X227) | (axvarid != $true)) & ((axvarid = $true) | (sK568 != ((sub @ sK568 @ id)))) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ((pushprop_lem1v2 = $true) | ((((sK572 @ one)) != $true) & (((sK569 @ sK570)) = $true) & ! [X233 : term] : (((sK572 @ X233)) = ((sK569 @ (sub @ X233 @ (push @ sK570 @ sK571))))))) & (! [X234 : (term > $o),X235 : term,X236 : subst,X237 : (term > $o)] : (($true = ((X237 @ one))) | (((X234 @ X235)) != $true) | (((X237 @ (sK573 @ X237 @ X236 @ X235 @ X234))) != ((X234 @ (sub @ (sK573 @ X237 @ X236 @ X235 @ X234) @ (push @ X235 @ X236)))))) | (pushprop_lem1v2 != $true)) & (! [X239 : term] : ! [X241 : subst,X242 : term] : (((sK574 @ X239 @ X241 @ X242)) = ((sub @ X239 @ (push @ X242 @ X241)))) | (hoasinduction_lem3v2_f_lthm != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ! [X244 : (subst > term > term)] : (((sub @ sK575 @ (push @ (sK577 @ X244) @ (sK576 @ X244)))) != ((X244 @ (sK576 @ X244) @ (sK577 @ X244))))) & (($true = sP17) | (hoasinduction_lthm != $true)) & ((hoasinduction_lthm = $true) | ($true != sP17)) & (substmonoid_gthm = $true) & (pushprop_lthm = $true) & ((hoasinduction_lem3v2 = $true) | ($true != sP8)) & (($true = sP8) | (hoasinduction_lem3v2 != $true)) & (axapp = $true) & (hoasinduction_lem2v2_gthm = $true) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem1_lthm = $true) & ((substmonoid_lthm = $true) | (! [X247 : subst] : (((comp @ id @ X247)) = X247) & ((((comp @ sK578 @ id)) != sK578) | (sK579 != ((comp @ id @ sK579)))) & ! [X250 : subst] : (((comp @ X250 @ id)) = X250))) & ((((comp @ id @ sK580)) != sK580) | (! [X252 : subst] : (((comp @ X252 @ id)) = X252) & ! [X253 : subst] : (((comp @ id @ X253)) = X253)) | (((comp @ sK581 @ id)) != sK581) | (substmonoid_lthm != $true)) & (hoaslamnotap_lthm = $true) & (hoasinduction_lem3a_gthm = $true) & ! [X255 : d_subst] : (d_id = X255) & (pushprop = $true) & (! [X256 : term,X257 : term,X258 : term,X259 : term] : ((((ap @ (sub @ X259 @ id) @ X257)) != ((ap @ (sub @ X258 @ id) @ X256))) | (X256 = X257)) | (hoasapinj2 != $true)) & ((hoasapinj2 = $true) | ((((ap @ (sub @ sK584 @ id) @ sK582)) = ((ap @ (sub @ sK585 @ id) @ sK583))) & (sK582 != sK583))) & (($true = sP6) | (hoasinduction_no_psi_cond_lthm != $true)) & ((hoasinduction_no_psi_cond_lthm = $true) | ($true != sP6)) & ((pushprop_lem0_lthm = $true) | ! [X267 : (term > $o)] : (((X267 @ (sK589 @ X267))) != ((sK586 @ (sub @ (sK589 @ X267) @ (push @ sK588 @ sK587)))))) & (! [X269 : (term > $o),X270 : subst,X271 : term] : ! [X273 : term] : (((sK590 @ X271 @ X270 @ X269 @ X273)) = ((X269 @ (sub @ X273 @ (push @ X271 @ X270))))) | (pushprop_lem0_lthm != $true)) & (hoasapnotvar = $true) & (ulamvar1 != $true) & (pushprop_lthm_orig = $true) & ((hoasinduction_lem2v2 = $true) | (sP1 != $true)) & ((sP1 = $true) | (hoasinduction_lem2v2 != $true)) & (hoaslamnotap_gthm = $true)),
  inference(skolemize,[status(esa),new_symbols(skolem,[sK457,vAPP,sK459,sK460,sK461,vAPP,sK463,sK464,sK465,sK466,vAPP,sK474,sK475,sK476,sK477,sK478,vAPP,sK481,sK482,sK483,vAPP,sK486,vAPP,sK488,sK489,sK490,sK491,sK492,sK493,sK494,sK495,sK496,sK497,sK498,sK499,sK500,sK501,sK502,sK503,sK504,sK505,sK506,vAPP,sK508,vAPP,sK513,sK514,sK515,sK516,sK517,vAPP,sK519,sK520,sK521,sK522,sK523,sK524,sK525,sK526,sK527,sK528,sK529,sK530,sK531,sK532,sK533,sK534,sK535,sK536,sK537,sK538,sK539,sK540,sK541,sK542,sK543,sK544,sK545,sK546,sK547,sK548,sK549,sK550,sK551,sK552,vAPP,sK555,sK556,sK557,sK558,sK559,vAPP,sK561,vAPP,sK564,sK565,vAPP,sK568,sK569,sK570,sK571,sK572,vAPP,sK575,vAPP,sK578,sK579,sK580,sK581,sK582,sK583,sK584,sK585,sK586,sK587,sK588,vAPP]),skolemize(X0,$thf(sK457)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X7,$thf(sK459)),skolemize(X8,$thf(sK460)),skolemize(X9,$thf(sK461)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X12,$thf(sK463)),skolemize(X13,$thf(sK464)),skolemize(X14,$thf(sK465)),skolemize(X15,$thf(sK466)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X34,$thf(sK474)),skolemize(X42,$thf(sK475)),skolemize(X43,$thf(sK476)),skolemize(X44,$thf(sK477)),skolemize(X45,$thf(sK478)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X50,$thf(sK481)),skolemize(X56,$thf(sK482)),skolemize(X57,$thf(sK483)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X63,$thf(sK486)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X70,$thf(sK488)),skolemize(X71,$thf(sK489)),skolemize(X72,$thf(sK490)),skolemize(X73,$thf(sK491)),skolemize(X74,$thf(sK492)),skolemize(X75,$thf(sK493)),skolemize(X76,$thf(sK494)),skolemize(X77,$thf(sK495)),skolemize(X82,$thf(sK496)),skolemize(X83,$thf(sK497)),skolemize(X84,$thf(sK498)),skolemize(X85,$thf(sK499)),skolemize(X95,$thf(sK500)),skolemize(X96,$thf(sK501)),skolemize(X97,$thf(sK502)),skolemize(X98,$thf(sK503)),skolemize(X99,$thf(sK504)),skolemize(X100,$thf(sK505)),skolemize(X103,$thf(sK506)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X107,$thf(sK508)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X116,$thf(sK513)),skolemize(X117,$thf(sK514)),skolemize(X118,$thf(sK515)),skolemize(X124,$thf(sK516)),skolemize(X125,$thf(sK517)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X131,$thf(sK519)),skolemize(X132,$thf(sK520)),skolemize(X133,$thf(sK521)),skolemize(X134,$thf(sK522)),skolemize(X135,$thf(sK523)),skolemize(X136,$thf(sK524)),skolemize(X142,$thf(sK525)),skolemize(X143,$thf(sK526)),skolemize(X147,$thf(sK527)),skolemize(X148,$thf(sK528)),skolemize(X154,$thf(sK529)),skolemize(X155,$thf(sK530)),skolemize(X156,$thf(sK531)),skolemize(X157,$thf(sK532)),skolemize(X158,$thf(sK533)),skolemize(X159,$thf(sK534)),skolemize(X163,$thf(sK535)),skolemize(X164,$thf(sK536)),skolemize(X166,$thf(sK537)),skolemize(X172,$thf(sK538)),skolemize(X173,$thf(sK539)),skolemize(X174,$thf(sK540)),skolemize(X175,$thf(sK541)),skolemize(X176,$thf(sK542)),skolemize(X177,$thf(sK543)),skolemize(X178,$thf(sK544)),skolemize(X179,$thf(sK545)),skolemize(X180,$thf(sK546)),skolemize(X189,$thf(sK547)),skolemize(X190,$thf(sK548)),skolemize(X191,$thf(sK549)),skolemize(X192,$thf(sK550)),skolemize(X193,$thf(sK551)),skolemize(X194,$thf(sK552)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X205,$thf(sK555)),skolemize(X206,$thf(sK556)),skolemize(X207,$thf(sK557)),skolemize(X208,$thf(sK558)),skolemize(X209,$thf(sK559)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X215,$thf(sK561)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X219,$thf(sK564)),skolemize(X220,$thf(sK565)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X228,$thf(sK568)),skolemize(X229,$thf(sK569)),skolemize(X230,$thf(sK570)),skolemize(X231,$thf(sK571)),skolemize(X232,$thf(sK572)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X243,$thf(sK575)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X248,$thf(sK578)),skolemize(X249,$thf(sK579)),skolemize(X251,$thf(sK580)),skolemize(X254,$thf(sK581)),skolemize(X260,$thf(sK582)),skolemize(X261,$thf(sK583)),skolemize(X262,$thf(sK584)),skolemize(X263,$thf(sK585)),skolemize(X264,$thf(sK586)),skolemize(X265,$thf(sK587)),skolemize(X266,$thf(sK588)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16))],[f262])).
thf(f767,plain,(
  ( ! [X255 : d_subst] : ((d_id = X255)) )),
  inference(cnf_transformation,[],[f263])).
thf(f863,plain,(
  ( ! [X144 : d_term] : ((d_one = X144)) )),
  inference(cnf_transformation,[],[f263])).
thf(f888,plain,(
  (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))))))))))))))),
  inference(cnf_transformation,[],[f263])).
thf(f890,plain,(
  (((d2subst @ d_id)) = sh)),
  inference(cnf_transformation,[],[f263])).
thf(f916,plain,(
  ( ! [X58 : subst] : ((((d2subst @ (sK484 @ X58))) = X58)) )),
  inference(cnf_transformation,[],[f263])).
thf(f922,plain,(
  (one = ((d2term @ d_one)))),
  inference(cnf_transformation,[],[f263])).
thf(f938,plain,(
  ( ! [X46 : term] : ((((d2term @ (sK479 @ X46))) = X46)) )),
  inference(cnf_transformation,[],[f263])).
thf(f971,plain,(
  ( ! [X10 : (term > $o)] : ((((X10 @ (sK462 @ X10))) != ((sK459 @ (sub @ (sK462 @ X10) @ (push @ sK461 @ sK460))))) | (pushprop_lem0 = $true)) )),
  inference(cnf_transformation,[],[f263])).
thf(f978,plain,(
  (pushprop_lem0 != ((!! @ subst @ (^[Y0 : subst]: (!! @ term @ (^[Y1 : term]: (!! @ (term > $o) @ (^[Y2 : term > $o]: (?? @ (term > $o) @ (^[Y3 : term > $o]: (pushprop_p_and_p_prime @ Y1 @ Y0 @ Y2 @ Y3)))))))))))),
  inference(cnf_transformation,[],[f8])).
thf(f981,plain,(
  (pushprop_lem0 != ((!! @ subst @ (^[Y0 : subst]: (!! @ term @ (^[Y1 : term]: (!! @ (term > $o) @ (^[Y2 : term > $o]: (?? @ (term > $o) @ (^[Y3 : term > $o]: ((^[Y4 : term]: ((^[Y5 : subst]: ((^[Y6 : term > $o]: ((^[Y7 : term > $o]: (!! @ term @ (^[Y8 : term]: ((Y7 @ Y8) = (Y6 @ (sub @ Y8 @ (push @ Y4 @ Y5))))))))))))) @ Y1 @ Y0 @ Y2 @ Y3)))))))))))),
  inference(definition_unfolding,[],[f978,f888])).
thf(f1019,plain,(
  (pushprop_lem0 = $false) | (((!! @ subst @ (^[Y0 : subst]: (!! @ term @ (^[Y1 : term]: (!! @ (term > $o) @ (^[Y2 : term > $o]: (?? @ (term > $o) @ (^[Y3 : term > $o]: ((^[Y4 : term]: ((^[Y5 : subst]: ((^[Y6 : term > $o]: ((^[Y7 : term > $o]: (!! @ term @ (^[Y8 : term]: ((Y7 @ Y8) = (Y6 @ (sub @ Y8 @ (push @ Y4 @ Y5))))))))))))) @ Y1 @ Y0 @ Y2 @ Y3)))))))))) = $false)),
  inference(xor_proxy_clausification,[],[f981])).
thf(f1020,plain,(
  ((((^[Y0 : subst]: (!! @ term @ (^[Y1 : term]: (!! @ (term > $o) @ (^[Y2 : term > $o]: (?? @ (term > $o) @ (^[Y3 : term > $o]: ((^[Y4 : term]: ((^[Y5 : subst]: ((^[Y6 : term > $o]: ((^[Y7 : term > $o]: (!! @ term @ (^[Y8 : term]: ((Y7 @ Y8) = (Y6 @ (sub @ Y8 @ (push @ Y4 @ Y5))))))))))))) @ Y1 @ Y0 @ Y2 @ Y3)))))))) @ sK591)) = $false) | (pushprop_lem0 = $false)),
  inference(sigma_proxy_clausification,[],[f1019])).
thf(f1021,plain,(
  ($false = ((!! @ term @ (^[Y0 : term]: (!! @ (term > $o) @ (^[Y1 : term > $o]: (?? @ (term > $o) @ (^[Y2 : term > $o]: (!! @ term @ (^[Y3 : term]: ((Y2 @ Y3) = (Y1 @ (sub @ Y3 @ (push @ Y0 @ sK591)))))))))))))) | (pushprop_lem0 = $false)),
  inference(beta-eta_normalization,[],[f1020])).
thf(f1022,plain,(
  (pushprop_lem0 = $false) | ($false = (((^[Y0 : term]: (!! @ (term > $o) @ (^[Y1 : term > $o]: (?? @ (term > $o) @ (^[Y2 : term > $o]: (!! @ term @ (^[Y3 : term]: ((Y2 @ Y3) = (Y1 @ (sub @ Y3 @ (push @ Y0 @ sK591))))))))))) @ sK592)))),
  inference(sigma_proxy_clausification,[],[f1021])).
thf(f1023,plain,(
  ($false = ((!! @ (term > $o) @ (^[Y0 : term > $o]: (?? @ (term > $o) @ (^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ (sub @ Y2 @ (push @ sK592 @ sK591)))))))))))) | (pushprop_lem0 = $false)),
  inference(beta-eta_normalization,[],[f1022])).
thf(f1024,plain,(
  (pushprop_lem0 = $false) | ($false = (((^[Y0 : term > $o]: (?? @ (term > $o) @ (^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ (sub @ Y2 @ (push @ sK592 @ sK591))))))))) @ sK593)))),
  inference(sigma_proxy_clausification,[],[f1023])).
thf(f1025,plain,(
  (pushprop_lem0 = $false) | ($false = ((?? @ (term > $o) @ (^[Y0 : term > $o]: (!! @ term @ (^[Y1 : term]: ((Y0 @ Y1) = (sK593 @ (sub @ Y1 @ (push @ sK592 @ sK591))))))))))),
  inference(beta-eta_normalization,[],[f1024])).
thf(f1026,plain,(
  ( ! [X1 : (term > $o)] : (($false = (((^[Y0 : term > $o]: (!! @ term @ (^[Y1 : term]: ((Y0 @ Y1) = (sK593 @ (sub @ Y1 @ (push @ sK592 @ sK591))))))) @ X1))) | (pushprop_lem0 = $false)) )),
  inference(pi_proxy_clausification,[],[f1025])).
thf(f1027,plain,(
  ( ! [X1 : (term > $o)] : (($false = ((!! @ term @ (^[Y0 : term]: ((X1 @ Y0) = (sK593 @ (sub @ Y0 @ (push @ sK592 @ sK591)))))))) | (pushprop_lem0 = $false)) )),
  inference(beta-eta_normalization,[],[f1026])).
thf(f1028,plain,(
  ( ! [X1 : (term > $o)] : ((pushprop_lem0 = $false) | ($false = (((^[Y0 : term]: ((X1 @ Y0) = (sK593 @ (sub @ Y0 @ (push @ sK592 @ sK591))))) @ (sK594 @ X1))))) )),
  inference(sigma_proxy_clausification,[],[f1027])).
thf(f1029,plain,(
  ( ! [X1 : (term > $o)] : ((pushprop_lem0 = $false) | ($false = (((X1 @ (sK594 @ X1)) = (sK593 @ (sub @ (sK594 @ X1) @ (push @ sK592 @ sK591))))))) )),
  inference(beta-eta_normalization,[],[f1028])).
thf(f1030,plain,(
  ( ! [X1 : (term > $o)] : ((pushprop_lem0 = $false) | (((X1 @ (sK594 @ X1))) != ((sK593 @ (sub @ (sK594 @ X1) @ (push @ sK592 @ sK591)))))) )),
  inference(equality_proxy_clausification,[],[f1029])).
thf(f1031,plain,(
  ( ! [X1 : (term > $o)] : ((pushprop_lem0 = $false) | (((X1 @ (sK594 @ X1))) = $true) | (((sK593 @ (sub @ (sK594 @ X1) @ (push @ sK592 @ sK591)))) = $true)) )),
  inference(xor_proxy_clausification,[],[f1030])).
thf(f1032,plain,(
  ( ! [X1 : (term > $o)] : ((pushprop_lem0 = $false) | ($false = ((X1 @ (sK594 @ X1)))) | ($false = ((sK593 @ (sub @ (sK594 @ X1) @ (push @ sK592 @ sK591)))))) )),
  inference(xor_proxy_clausification,[],[f1030])).
thf(f1110,plain,(
  ( ! [X10 : (term > $o)] : ((((sK459 @ (sub @ (sK462 @ X10) @ (push @ sK461 @ sK460)))) = $true) | (((X10 @ (sK462 @ X10))) = $true) | (pushprop_lem0 = $true)) )),
  inference(xor_proxy_clausification,[],[f971])).
thf(f1111,plain,(
  ( ! [X10 : (term > $o)] : ((pushprop_lem0 = $true) | ($false = ((sK459 @ (sub @ (sK462 @ X10) @ (push @ sK461 @ sK460))))) | (((X10 @ (sK462 @ X10))) = $false)) )),
  inference(xor_proxy_clausification,[],[f971])).
thf(f1982,definition,(
  spl596_195 <=> (pushprop_lem0 = $false)),
  introduced(definition,[new_symbols(definition,[spl596_195])],[avatar_definition])).
thf(f1984,plain,(
  (pushprop_lem0 = $false) | ~spl596_195),
  inference(avatar_component_clause,[],[f1982])).
thf(f1986,definition,(
  spl596_196 <=> ! [X1 : (term > $o)] : (($false = ((X1 @ (sK594 @ X1)))) | ($false = ((sK593 @ (sub @ (sK594 @ X1) @ (push @ sK592 @ sK591))))))),
  introduced(definition,[new_symbols(definition,[spl596_196])],[avatar_definition])).
thf(f1987,plain,(
  ( ! [X1 : (term > $o)] : (($false = ((sK593 @ (sub @ (sK594 @ X1) @ (push @ sK592 @ sK591))))) | ($false = ((X1 @ (sK594 @ X1))))) ) | ~spl596_196),
  inference(avatar_component_clause,[],[f1986])).
thf(f1988,plain,(
  spl596_195 | spl596_196),
  inference(avatar_split_clause,[],[f1032,f1986,f1982])).
thf(f1990,definition,(
  spl596_197 <=> ! [X1 : (term > $o)] : ((((X1 @ (sK594 @ X1))) = $true) | (((sK593 @ (sub @ (sK594 @ X1) @ (push @ sK592 @ sK591)))) = $true))),
  introduced(definition,[new_symbols(definition,[spl596_197])],[avatar_definition])).
thf(f1991,plain,(
  ( ! [X1 : (term > $o)] : ((((sK593 @ (sub @ (sK594 @ X1) @ (push @ sK592 @ sK591)))) = $true) | (((X1 @ (sK594 @ X1))) = $true)) ) | ~spl596_197),
  inference(avatar_component_clause,[],[f1990])).
thf(f1992,plain,(
  spl596_197 | spl596_195),
  inference(avatar_split_clause,[],[f1031,f1982,f1990])).
thf(f1994,definition,(
  spl596_198 <=> (pushprop_lem0 = $true)),
  introduced(definition,[new_symbols(definition,[spl596_198])],[avatar_definition])).
thf(f1996,plain,(
  (pushprop_lem0 = $true) | ~spl596_198),
  inference(avatar_component_clause,[],[f1994])).
thf(f3017,definition,(
  spl596_436 <=> ! [X10 : (term > $o)] : (($false = ((sK459 @ (sub @ (sK462 @ X10) @ (push @ sK461 @ sK460))))) | (((X10 @ (sK462 @ X10))) = $false))),
  introduced(definition,[new_symbols(definition,[spl596_436])],[avatar_definition])).
thf(f3018,plain,(
  ( ! [X10 : (term > $o)] : ((((X10 @ (sK462 @ X10))) = $false) | ($false = ((sK459 @ (sub @ (sK462 @ X10) @ (push @ sK461 @ sK460)))))) ) | ~spl596_436),
  inference(avatar_component_clause,[],[f3017])).
thf(f3019,plain,(
  spl596_436 | spl596_198),
  inference(avatar_split_clause,[],[f1111,f1994,f3017])).
thf(f3021,definition,(
  spl596_437 <=> ! [X10 : (term > $o)] : ((((sK459 @ (sub @ (sK462 @ X10) @ (push @ sK461 @ sK460)))) = $true) | (((X10 @ (sK462 @ X10))) = $true))),
  introduced(definition,[new_symbols(definition,[spl596_437])],[avatar_definition])).
thf(f3022,plain,(
  ( ! [X10 : (term > $o)] : ((((sK459 @ (sub @ (sK462 @ X10) @ (push @ sK461 @ sK460)))) = $true) | (((X10 @ (sK462 @ X10))) = $true)) ) | ~spl596_437),
  inference(avatar_component_clause,[],[f3021])).
thf(f3023,plain,(
  spl596_198 | spl596_437),
  inference(avatar_split_clause,[],[f1110,f3021,f1994])).
thf(f3877,plain,(
  ($false = $true) | (~spl596_195 | ~spl596_198)),
  inference(forward_demodulation,[],[f1996,f1984])).
thf(f3878,plain,(
  $false | (~spl596_195 | ~spl596_198)),
  inference(trivial_inequality_removal,[],[f3877])).
thf(f3879,plain,(
  ~spl596_195 | ~spl596_198),
  inference(avatar_contradiction_clause,[],[f3878])).
thf(f4024,plain,(
  ( ! [X58 : subst] : ((((d2subst @ d_id)) = X58)) )),
  inference(forward_demodulation,[],[f916,f767])).
thf(f4036,plain,(
  ( ! [X0 : (term > $o)] : ((((X0 @ (sK594 @ X0))) = $true) | (((sK593 @ (sub @ (sK594 @ X0) @ (d2subst @ d_id)))) = $true)) ) | ~spl596_197),
  inference(superposition,[],[f1991,f4024])).
thf(f4038,plain,(
  ( ! [X0 : (term > $o)] : ((((sK593 @ (sub @ (sK594 @ X0) @ sh))) = $true) | (((X0 @ (sK594 @ X0))) = $true)) ) | ~spl596_197),
  inference(forward_demodulation,[],[f4036,f890])).
thf(f4043,plain,(
  ( ! [X46 : term] : ((((d2term @ d_one)) = X46)) )),
  inference(forward_demodulation,[],[f938,f863])).
thf(f4072,definition,(
  spl596_632 <=> ! [X0 : (term > $o)] : ($true = ((X0 @ one)))),
  introduced(definition,[new_symbols(definition,[spl596_632])],[avatar_definition])).
thf(f4073,plain,(
  ( ! [X0 : (term > $o)] : (($true = ((X0 @ one)))) ) | ~spl596_632),
  inference(avatar_component_clause,[],[f4072])).
thf(f4075,definition,(
  spl596_633 <=> (((sK593 @ one)) = $true)),
  introduced(definition,[new_symbols(definition,[spl596_633])],[avatar_definition])).
thf(f4077,plain,(
  (((sK593 @ one)) = $true) | ~spl596_633),
  inference(avatar_component_clause,[],[f4075])).
thf(f4080,definition,(
  spl596_634 <=> ! [X0 : (term > $o)] : ($false = ((X0 @ one)))),
  introduced(definition,[new_symbols(definition,[spl596_634])],[avatar_definition])).
thf(f4081,plain,(
  ( ! [X0 : (term > $o)] : (($false = ((X0 @ one)))) ) | ~spl596_634),
  inference(avatar_component_clause,[],[f4080])).
thf(f4083,definition,(
  spl596_635 <=> ($false = ((sK593 @ one)))),
  introduced(definition,[new_symbols(definition,[spl596_635])],[avatar_definition])).
thf(f4085,plain,(
  ($false = ((sK593 @ one))) | ~spl596_635),
  inference(avatar_component_clause,[],[f4083])).
thf(f4090,plain,(
  ( ! [X10 : (term > $o)] : (($false = ((X10 @ (d2term @ d_one)))) | ($false = ((sK459 @ (sub @ (sK462 @ X10) @ (push @ sK461 @ sK460)))))) ) | ~spl596_436),
  inference(forward_demodulation,[],[f3018,f4043])).
thf(f4099,plain,(
  ( ! [X10 : (term > $o)] : (($true = ((sK459 @ (d2term @ d_one)))) | (((X10 @ (sK462 @ X10))) = $true)) ) | ~spl596_437),
  inference(forward_demodulation,[],[f3022,f4043])).
thf(f4104,plain,(
  ( ! [X10 : (term > $o)] : (($false = ((X10 @ one))) | ($false = ((sK459 @ (sub @ (sK462 @ X10) @ (push @ sK461 @ sK460)))))) ) | ~spl596_436),
  inference(forward_demodulation,[],[f4090,f922])).
thf(f4105,plain,(
  ( ! [X10 : (term > $o)] : (($true = ((sK459 @ one))) | (((X10 @ (sK462 @ X10))) = $true)) ) | ~spl596_437),
  inference(forward_demodulation,[],[f4099,f922])).
thf(f4107,plain,(
  ( ! [X10 : (term > $o)] : (($false = ((X10 @ one))) | ($false = ((sK459 @ (d2term @ d_one))))) ) | ~spl596_436),
  inference(forward_demodulation,[],[f4104,f4043])).
thf(f4108,plain,(
  ( ! [X10 : (term > $o)] : ((((X10 @ (d2term @ d_one))) = $true) | ($true = ((sK459 @ one)))) ) | ~spl596_437),
  inference(forward_demodulation,[],[f4105,f4043])).
thf(f4109,plain,(
  ( ! [X10 : (term > $o)] : (($false = ((sK459 @ one))) | ($false = ((X10 @ one)))) ) | ~spl596_436),
  inference(forward_demodulation,[],[f4107,f922])).
thf(f4110,plain,(
  ( ! [X10 : (term > $o)] : ((((X10 @ one)) = $true) | ($true = ((sK459 @ one)))) ) | ~spl596_437),
  inference(forward_demodulation,[],[f4108,f922])).
thf(f4112,definition,(
  spl596_638 <=> ($false = ((sK459 @ one)))),
  introduced(definition,[new_symbols(definition,[spl596_638])],[avatar_definition])).
thf(f4114,plain,(
  ($false = ((sK459 @ one))) | ~spl596_638),
  inference(avatar_component_clause,[],[f4112])).
thf(f4115,plain,(
  spl596_634 | spl596_638 | ~spl596_436),
  inference(avatar_split_clause,[],[f4109,f3017,f4112,f4080])).
thf(f4117,definition,(
  spl596_639 <=> ($true = ((sK459 @ one)))),
  introduced(definition,[new_symbols(definition,[spl596_639])],[avatar_definition])).
thf(f4119,plain,(
  ($true = ((sK459 @ one))) | ~spl596_639),
  inference(avatar_component_clause,[],[f4117])).
thf(f4120,plain,(
  spl596_639 | spl596_632 | ~spl596_437),
  inference(avatar_split_clause,[],[f4110,f3021,f4072,f4117])).
thf(f4127,plain,(
  ( ! [X1 : (term > $o)] : (((((^[Y0 : term]: (~ (X1 @ Y0))) @ one)) = $true)) ) | ~spl596_632),
  inference(primitive_instantiation,[],[f4073])).
thf(f4132,plain,(
  ( ! [X1 : (term > $o)] : ((((~ (X1 @ one))) = $true)) ) | ~spl596_632),
  inference(beta-eta_normalization,[],[f4127])).
thf(f4133,plain,(
  ( ! [X1 : (term > $o)] : (($false = ((X1 @ one)))) ) | ~spl596_632),
  inference(not_proxy_clausification,[],[f4132])).
thf(f4134,plain,(
  spl596_634 | ~spl596_632),
  inference(avatar_split_clause,[],[f4133,f4072,f4080])).
thf(f4137,plain,(
  ( ! [X1 : (term > $o)] : (((((^[Y0 : term]: (~ (X1 @ Y0))) @ one)) = $false)) ) | ~spl596_634),
  inference(primitive_instantiation,[],[f4081])).
thf(f4147,plain,(
  ( ! [X1 : (term > $o)] : (($false = ((~ (X1 @ one))))) ) | ~spl596_634),
  inference(beta-eta_normalization,[],[f4137])).
thf(f4148,plain,(
  ( ! [X1 : (term > $o)] : ((((X1 @ one)) = $true)) ) | ~spl596_634),
  inference(not_proxy_clausification,[],[f4147])).
thf(f4150,plain,(
  ($false = $true) | ~spl596_634),
  inference(forward_demodulation,[],[f4148,f4081])).
thf(f4151,plain,(
  $false | ~spl596_634),
  inference(trivial_inequality_removal,[],[f4150])).
thf(f4152,plain,(
  ~spl596_634),
  inference(avatar_contradiction_clause,[],[f4151])).
thf(f4591,plain,(
  ($false = $true) | (~spl596_638 | ~spl596_639)),
  inference(forward_demodulation,[],[f4119,f4114])).
thf(f4592,plain,(
  $false | (~spl596_638 | ~spl596_639)),
  inference(trivial_inequality_removal,[],[f4591])).
thf(f4593,plain,(
  ~spl596_638 | ~spl596_639),
  inference(avatar_contradiction_clause,[],[f4592])).
thf(f4608,plain,(
  ( ! [X0 : (term > $o)] : ((((X0 @ (sK594 @ X0))) = $true) | (((sK593 @ (d2term @ d_one))) = $true)) ) | ~spl596_197),
  inference(forward_demodulation,[],[f4038,f4043])).
thf(f4609,plain,(
  ( ! [X1 : (term > $o)] : (($false = ((sK593 @ (d2term @ d_one)))) | ($false = ((X1 @ (sK594 @ X1))))) ) | ~spl596_196),
  inference(forward_demodulation,[],[f1987,f4043])).
thf(f4612,plain,(
  ( ! [X0 : (term > $o)] : ((((sK593 @ (d2term @ d_one))) = $true) | ($true = ((X0 @ (d2term @ d_one))))) ) | ~spl596_197),
  inference(forward_demodulation,[],[f4608,f4043])).
thf(f4613,plain,(
  ( ! [X1 : (term > $o)] : (($false = ((sK593 @ one))) | ($false = ((X1 @ (sK594 @ X1))))) ) | ~spl596_196),
  inference(forward_demodulation,[],[f4609,f922])).
thf(f4616,plain,(
  ( ! [X0 : (term > $o)] : ((((sK593 @ one)) = $true) | ($true = ((X0 @ (d2term @ d_one))))) ) | ~spl596_197),
  inference(forward_demodulation,[],[f4612,f922])).
thf(f4617,plain,(
  ( ! [X1 : (term > $o)] : (($false = ((sK593 @ one))) | ($false = ((X1 @ (d2term @ d_one))))) ) | ~spl596_196),
  inference(forward_demodulation,[],[f4613,f4043])).
thf(f4620,plain,(
  ( ! [X0 : (term > $o)] : ((((sK593 @ one)) = $true) | ($true = ((X0 @ one)))) ) | ~spl596_197),
  inference(forward_demodulation,[],[f4616,f922])).
thf(f4621,plain,(
  ( ! [X1 : (term > $o)] : (($false = ((sK593 @ one))) | ($false = ((X1 @ one)))) ) | ~spl596_196),
  inference(forward_demodulation,[],[f4617,f922])).
thf(f4624,plain,(
  spl596_632 | spl596_633 | ~spl596_197),
  inference(avatar_split_clause,[],[f4620,f1990,f4075,f4072])).
thf(f4625,plain,(
  spl596_634 | spl596_635 | ~spl596_196),
  inference(avatar_split_clause,[],[f4621,f1986,f4083,f4080])).
thf(f4636,plain,(
  ($false = $true) | (~spl596_633 | ~spl596_635)),
  inference(superposition,[],[f4077,f4085])).
thf(f4637,plain,(
  $false | (~spl596_633 | ~spl596_635)),
  inference(trivial_inequality_removal,[],[f4636])).
thf(f4638,plain,(
  ~spl596_633 | ~spl596_635),
  inference(avatar_contradiction_clause,[],[f4637])).
cnf(s144, plain, spl596_195 | spl596_196, inference(sat_conversion,[],[f1988])).
cnf(s145, plain, spl596_195 | spl596_197, inference(sat_conversion,[],[f1992])).
cnf(s384, plain, spl596_198 | spl596_436, inference(sat_conversion,[],[f3019])).
cnf(s385, plain, spl596_198 | spl596_437, inference(sat_conversion,[],[f3023])).
cnf(s607, plain, ~spl596_195 | ~spl596_198, inference(sat_conversion,[],[f3879])).
cnf(s615, plain, ~spl596_436 | spl596_634 | spl596_638, inference(sat_conversion,[],[f4115])).
cnf(s616, plain, ~spl596_437 | spl596_632 | spl596_639, inference(sat_conversion,[],[f4120])).
cnf(s619, plain, ~spl596_632 | spl596_634, inference(sat_conversion,[],[f4134])).
cnf(s621, plain, ~spl596_634, inference(sat_conversion,[],[f4152])).
cnf(s676, plain, ~spl596_638 | ~spl596_639, inference(sat_conversion,[],[f4593])).
cnf(s683, plain, ~spl596_197 | spl596_632 | spl596_633, inference(sat_conversion,[],[f4624])).
cnf(s684, plain, ~spl596_196 | spl596_634 | spl596_635, inference(sat_conversion,[],[f4625])).
cnf(s685, plain, ~spl596_633 | ~spl596_635, inference(sat_conversion,[],[f4638])).
cnf(s687, plain, ~spl596_632, inference(rat,[],[s619,s621])).
cnf(s688, plain, ~spl596_437 | spl596_639, inference(rat,[],[s616,s687])).
cnf(s689, plain, ~spl596_436 | spl596_638, inference(rat,[],[s615,s621])).
cnf(s721, plain, spl596_195, inference(rat,[],[s685,s684,s683,s144,s145,s621,s687])).
cnf(s722, plain, ~spl596_198, inference(rat,[],[s607,s721])).
cnf(s723, plain, spl596_437, inference(rat,[],[s385,s722])).
cnf(s724, plain, spl596_436, inference(rat,[],[s384,s722])).
cnf(s727, plain, spl596_639, inference(rat,[],[s688,s723])).
cnf(s728, plain, spl596_638, inference(rat,[],[s689,s724])).
cnf(s729, plain, $false, inference(rat,[],[s676,s727,s728])).
thf(f4642,plain,(
  $false),
  inference(avatar_sat_refutation,[],[s729])).
% SZS output end Proof for theBenchmark