%----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_127, type, vPI: !>[X0: $tType]:(((X0 > $o) > $o))).
thf(func_def_128, type, db1: !>[X0: $tType]:(X0)).
thf(func_def_129, type, db0: !>[X0: $tType]:(X0)).
thf(func_def_130, type, db2: !>[X0: $tType]:(X0)).
thf(func_def_131, type, vEQ: !>[X0: $tType]:((X0 > X0 > $o))).
thf(func_def_132, type, vLAM: !>[X0: $tType, X1: $tType]:((X1) > (X0 > X1))).
thf(func_def_135, type, db4: !>[X0: $tType]:(X0)).
thf(func_def_136, type, db3: !>[X0: $tType]:(X0)).
thf(func_def_137, type, vSIGMA: !>[X0: $tType]:(((X0 > $o) > $o))).
thf(func_def_138, type, sP0: $o).
thf(func_def_139, type, sP1: ((subst > term > subst > $o) > $o)).
thf(func_def_140, type, sP2: ((subst > term > subst > $o) > $o)).
thf(func_def_141, type, sP3: $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: ((subst > term > subst > $o) > $o)).
thf(func_def_147, type, sP9: ((subst > term > subst > $o) > $o)).
thf(func_def_148, type, sP10: ((term > $o) > $o)).
thf(func_def_149, type, sP11: ((subst > term > subst > $o) > $o)).
thf(func_def_150, type, sP12: $o).
thf(func_def_151, type, sP13: $o).
thf(func_def_152, type, sP14: $o).
thf(func_def_153, type, sP15: $o).
thf(func_def_154, type, sP16: ((subst > term > subst > $o) > $o)).
thf(func_def_155, type, sP17: $o).
thf(func_def_156, type, sP18: ((term > $o) > $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: $o).
thf(func_def_160, type, sP22: $o).
thf(func_def_161, type, sP23: $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: $o).
thf(func_def_166, type, sP28: $o).
thf(func_def_167, type, sP29: ((subst > term > subst > $o) > $o)).
thf(func_def_168, type, sP30: ((term > $o) > $o)).
thf(func_def_169, type, sP31: $o).
thf(func_def_170, type, sP32: $o).
thf(func_def_171, type, sP33: $o).
thf(func_def_172, type, sP34: ((subst > term > subst > $o) > $o)).
thf(func_def_173, type, sP35: $o).
thf(func_def_174, type, sP36: $o).
thf(func_def_175, type, sP37: ((subst > term > subst > $o) > $o)).
thf(func_def_176, type, sP38: $o).
thf(func_def_177, type, sP39: ((subst > term > subst > $o) > $o)).
thf(func_def_178, type, sP40: ((term > $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: $o).
thf(func_def_184, type, sP46: ((term > $o) > $o)).
thf(func_def_185, type, sP47: $o).
thf(func_def_186, type, sP48: ((subst > term > subst > $o) > $o)).
thf(func_def_187, type, sP49: $o).
thf(func_def_188, type, sP50: ((term > $o) > $o)).
thf(func_def_189, type, sP51: ((term > $o) > $o)).
thf(func_def_190, type, sP52: $o).
thf(func_def_191, type, sP53: $o).
thf(func_def_192, type, sP54: $o).
thf(func_def_193, type, sP55: ((term > $o) > $o)).
thf(func_def_194, type, sP56: ((term > $o) > $o)).
thf(func_def_195, type, sP57: $o).
thf(func_def_196, type, sP58: $o).
thf(func_def_197, type, sK59: (subst > (term > $o) > term)).
thf(func_def_198, type, sK60: ((term > $o) > term)).
thf(func_def_199, type, sK61: (term > $o)).
thf(func_def_200, type, sK62: subst).
thf(func_def_201, type, sK63: term).
thf(func_def_202, type, sK64: (term > term)).
thf(func_def_203, type, sK65: term).
thf(func_def_204, type, sK66: term).
thf(func_def_205, type, sK67: (term > $o)).
thf(func_def_206, type, sK68: (term > term)).
thf(func_def_207, type, sK69: ((term > $o) > term)).
thf(func_def_208, type, sK70: ((term > $o) > term)).
thf(func_def_209, type, sK71: ((term > $o) > term)).
thf(func_def_210, type, sK72: ((term > $o) > term)).
thf(func_def_211, type, sK73: ((term > $o) > term)).
thf(func_def_212, type, sK74: ((term > $o) > term)).
thf(func_def_213, type, sK75: (subst > term > subst > $o)).
thf(func_def_214, type, sK76: term).
thf(func_def_215, type, sK77: ((subst > term > term) > term)).
thf(func_def_216, type, sK78: ((subst > term > term) > term > subst)).
thf(func_def_217, type, sK79: ((subst > term > term) > term > term)).
thf(func_def_218, type, sK80: ((subst > term > term) > term > subst)).
thf(func_def_219, type, sK81: ((subst > term > term) > term > subst)).
thf(func_def_220, type, sK82: ((subst > term > term) > term > subst)).
thf(func_def_221, type, sK83: ((subst > term > term) > term > term)).
thf(func_def_222, type, sK84: ((subst > term > subst > $o) > term)).
thf(func_def_223, type, sK85: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_224, type, sK86: (term > (subst > term > subst > $o) > term)).
thf(func_def_225, type, sK87: ((term > $o) > term)).
thf(func_def_226, type, sK88: ((term > $o) > term)).
thf(func_def_227, type, sK89: term).
thf(func_def_228, type, sK90: subst).
thf(func_def_229, type, sK91: subst).
thf(func_def_230, type, sK92: (term > $o)).
thf(func_def_231, type, sK93: term).
thf(func_def_232, type, sK94: term).
thf(func_def_233, type, sK95: subst).
thf(func_def_234, type, sK96: subst).
thf(func_def_235, type, sK97: term).
thf(func_def_236, type, sK98: (term > $o)).
thf(func_def_237, type, sK99: (term > term)).
thf(func_def_238, type, sK100: ((term > $o) > term)).
thf(func_def_239, type, sK101: ((term > $o) > subst > term)).
thf(func_def_240, type, sK102: ((term > $o) > term)).
thf(func_def_241, type, sK103: ((term > $o) > term)).
thf(func_def_242, type, sK104: ((term > $o) > term)).
thf(func_def_243, type, sK105: ((term > $o) > term)).
thf(func_def_244, type, sK106: (subst > term > subst > $o)).
thf(func_def_245, type, sK107: term).
thf(func_def_246, type, sK108: (term > $o)).
thf(func_def_247, type, sK109: term).
thf(func_def_248, type, sK110: ((term > $o) > term > term)).
thf(func_def_249, type, sK111: ((subst > term > subst > $o) > subst)).
thf(func_def_250, type, sK112: ((subst > term > subst > $o) > term)).
thf(func_def_251, type, sK113: ((subst > term > subst > $o) > subst)).
thf(func_def_252, type, sK114: ((subst > term > subst > $o) > subst)).
thf(func_def_253, type, sK115: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_254, type, sK116: ((subst > term > subst > $o) > subst)).
thf(func_def_255, type, sK117: ((subst > term > subst > $o) > subst)).
thf(func_def_256, type, sK118: ((subst > term > subst > $o) > term)).
thf(func_def_257, type, sK119: ((subst > term > subst > $o) > subst)).
thf(func_def_258, type, sK120: ((subst > term > term) > subst)).
thf(func_def_259, type, sK121: ((subst > term > term) > subst)).
thf(func_def_260, type, sK122: ((subst > term > term) > term)).
thf(func_def_261, type, sK123: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_262, type, sK124: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_263, type, sK125: (subst > term > subst > $o)).
thf(func_def_264, type, sK126: term).
thf(func_def_265, type, sK127: ((subst > term > term) > subst)).
thf(func_def_266, type, sK128: ((subst > term > term) > subst)).
thf(func_def_267, type, sK129: ((subst > term > term) > term)).
thf(func_def_268, type, sK130: ((subst > term > term) > term)).
thf(func_def_269, type, sK131: (term > (subst > term > subst > $o) > term)).
thf(func_def_270, type, sK132: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_271, type, sK133: ((term > $o) > term)).
thf(func_def_272, type, sK134: ((term > $o) > term)).
thf(func_def_273, type, sK135: (subst > term > subst > $o)).
thf(func_def_274, type, sK136: term).
thf(func_def_275, type, sK137: ((subst > term > term) > term)).
thf(func_def_276, type, sK138: ((subst > term > term) > subst)).
thf(func_def_277, type, sK139: ((subst > term > term) > subst)).
thf(func_def_278, type, sK140: ((subst > term > term) > term)).
thf(func_def_279, type, sK141: (term > (subst > term > subst > $o) > term)).
thf(func_def_280, type, sK142: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_281, type, sK143: term).
thf(func_def_282, type, sK144: (subst > term > term)).
thf(func_def_283, type, sK145: subst).
thf(func_def_284, type, sK146: (subst > term > term)).
thf(func_def_285, type, sK147: term).
thf(func_def_286, type, sK148: subst).
thf(func_def_287, type, sK149: term).
thf(func_def_288, type, sK150: term).
thf(func_def_289, type, sK151: subst).
thf(func_def_290, type, sK152: term).
thf(func_def_291, type, sK153: ((subst > term > term) > subst)).
thf(func_def_292, type, sK154: ((subst > term > term) > term)).
thf(func_def_293, type, sK155: ((subst > term > term) > subst)).
thf(func_def_294, type, sK156: ((subst > term > term) > subst)).
thf(func_def_295, type, sK157: ((subst > term > term) > subst)).
thf(func_def_296, type, sK158: ((subst > term > term) > term)).
thf(func_def_297, type, sK159: (subst > term > subst > $o)).
thf(func_def_298, type, sK160: term).
thf(func_def_299, type, sK161: term).
thf(func_def_300, type, sK162: ((subst > term > subst > $o) > term)).
thf(func_def_301, type, sK163: ((subst > term > subst > $o) > subst)).
thf(func_def_302, type, sK164: ((subst > term > subst > $o) > subst)).
thf(func_def_303, type, sK165: ((subst > term > subst > $o) > subst)).
thf(func_def_304, type, sK166: ((subst > term > subst > $o) > term)).
thf(func_def_305, type, sK167: ((subst > term > subst > $o) > subst)).
thf(func_def_306, type, sK168: ((subst > term > subst > $o) > subst)).
thf(func_def_307, type, sK169: ((subst > term > subst > $o) > subst)).
thf(func_def_308, type, sK170: ((subst > term > subst > $o) > term)).
thf(func_def_309, type, sK171: ((subst > term > subst > $o) > term)).
thf(func_def_310, type, sK172: (term > $o)).
thf(func_def_311, type, sK173: term).
thf(func_def_312, type, sK174: subst).
thf(func_def_313, type, sK175: (term > term)).
thf(func_def_314, type, sK176: ((term > $o) > term)).
thf(func_def_315, type, sK177: (subst > (term > $o) > term)).
thf(func_def_316, type, sK178: ((term > $o) > term)).
thf(func_def_317, type, sK179: ((term > $o) > term)).
thf(func_def_318, type, sK180: ((subst > term > subst > $o) > term)).
thf(func_def_319, type, sK181: ((subst > term > subst > $o) > term)).
thf(func_def_320, type, sK182: (subst > term > subst > $o)).
thf(func_def_321, type, sK183: term).
thf(func_def_322, type, sK184: ((subst > term > subst > $o) > subst)).
thf(func_def_323, type, sK185: ((subst > term > subst > $o) > term)).
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: (term > (subst > term > subst > $o) > term)).
thf(func_def_327, type, sK189: ((subst > term > subst > $o) > subst)).
thf(func_def_328, type, sK190: ((subst > term > subst > $o) > subst)).
thf(func_def_329, type, sK191: ((subst > term > subst > $o) > term)).
thf(func_def_330, type, sK192: ((subst > term > subst > $o) > subst)).
thf(func_def_331, type, sK193: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_332, type, sK194: ((subst > term > term) > subst)).
thf(func_def_333, type, sK195: ((subst > term > term) > subst)).
thf(func_def_334, type, sK196: ((subst > term > term) > term)).
thf(func_def_335, type, sK197: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_336, type, sK198: term).
thf(func_def_337, type, sK199: (subst > term > subst > $o)).
thf(func_def_338, type, sK200: term).
thf(func_def_339, type, sK201: ((subst > term > subst > $o) > subst)).
thf(func_def_340, type, sK202: ((subst > term > subst > $o) > subst)).
thf(func_def_341, type, sK203: ((subst > term > subst > $o) > subst)).
thf(func_def_342, type, sK204: ((subst > term > subst > $o) > term)).
thf(func_def_343, type, sK205: ((subst > term > subst > $o) > term > term)).
thf(func_def_344, type, sK206: ((subst > term > subst > $o) > term)).
thf(func_def_345, type, sK207: ((subst > term > subst > $o) > subst)).
thf(func_def_346, type, sK208: ((subst > term > subst > $o) > subst)).
thf(func_def_347, type, sK209: ((subst > term > subst > $o) > subst)).
thf(func_def_348, type, sK210: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_349, type, sK211: (term > (subst > term > subst > $o) > term)).
thf(func_def_350, type, sK212: (subst > term > subst > $o)).
thf(func_def_351, type, sK213: term).
thf(func_def_352, type, sK214: ((subst > term > term) > term)).
thf(func_def_353, type, sK215: ((subst > term > term) > subst)).
thf(func_def_354, type, sK216: ((subst > term > term) > subst)).
thf(func_def_355, type, sK217: ((subst > term > term) > term)).
thf(func_def_356, type, sK218: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_357, type, sK219: ((subst > term > term) > subst)).
thf(func_def_358, type, sK220: ((subst > term > term) > subst)).
thf(func_def_359, type, sK221: ((subst > term > term) > term)).
thf(func_def_360, type, sK222: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_361, type, sK223: ((subst > term > subst > $o) > term > $o)).
thf(func_def_362, type, sK224: (subst > term > subst > $o)).
thf(func_def_363, type, sK225: term).
thf(func_def_364, type, sK226: (subst > term > subst > $o)).
thf(func_def_365, type, sK227: ((term > $o) > term)).
thf(func_def_366, type, sK228: term).
thf(func_def_367, type, sK229: ((subst > term > subst > $o) > term)).
thf(func_def_368, type, sK230: ((subst > term > subst > $o) > term)).
thf(func_def_369, type, sK231: ((term > $o) > term)).
thf(func_def_370, type, sK232: ((term > $o) > term)).
thf(func_def_371, type, sK233: term).
thf(func_def_372, type, sK234: (term > $o)).
thf(func_def_373, type, sK235: (term > term)).
thf(func_def_374, type, sK236: ((term > $o) > term > term)).
thf(func_def_375, type, sK237: ((subst > term > subst > $o) > (term > $o) > term)).
thf(func_def_376, type, sK238: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_377, type, sK239: term).
thf(func_def_378, type, sK240: (term > $o)).
thf(func_def_379, type, sK241: (subst > term > subst > $o)).
thf(func_def_380, type, sK242: ((subst > term > term) > term)).
thf(func_def_381, type, sK243: ((subst > term > term) > term)).
thf(func_def_382, type, sK244: ((subst > term > term) > subst)).
thf(func_def_383, type, sK245: ((subst > term > term) > subst)).
thf(func_def_384, type, sK246: ((term > $o) > term)).
thf(func_def_385, type, sK247: ((term > $o) > term)).
thf(func_def_386, type, sK248: ((subst > term > term) > term)).
thf(func_def_387, type, sK249: ((subst > term > term) > subst)).
thf(func_def_388, type, sK250: ((subst > term > term) > subst)).
thf(func_def_389, type, sK251: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_390, type, sK252: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_391, type, sK253: (term > subst > term > term)).
thf(func_def_392, type, sK254: (subst > term > subst > $o)).
thf(func_def_393, type, sK255: term).
thf(func_def_394, type, sK256: (term > $o)).
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: ((subst > term > term) > term)).
thf(func_def_399, type, sK261: term).
thf(func_def_400, type, sK262: ((subst > term > term) > subst)).
thf(func_def_401, type, sK263: ((subst > term > term) > term)).
thf(func_def_402, type, sK264: term).
thf(func_def_403, type, sK265: ((term > $o) > term > term)).
thf(func_def_404, type, sK266: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_405, type, sK267: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_406, type, sK268: term).
thf(func_def_407, type, sK269: (term > $o)).
thf(func_def_408, type, sK270: (term > term)).
thf(func_def_409, type, sK271: ((term > $o) > term)).
thf(func_def_410, type, sK272: ((term > $o) > term)).
thf(func_def_411, type, sK273: ((term > $o) > term)).
thf(func_def_412, type, sK274: ((term > $o) > term)).
thf(func_def_413, type, sK275: (term > $o)).
thf(func_def_414, type, sK276: (subst > term > subst > $o)).
thf(func_def_415, type, sK277: term).
thf(func_def_416, type, sK278: term).
thf(func_def_417, type, sK279: ((subst > term > subst > $o) > subst)).
thf(func_def_418, type, sK280: ((subst > term > subst > $o) > subst)).
thf(func_def_419, type, sK281: ((subst > term > subst > $o) > subst)).
thf(func_def_420, type, sK282: ((subst > term > subst > $o) > term)).
thf(func_def_421, type, sK283: ((subst > term > subst > $o) > subst)).
thf(func_def_422, type, sK284: ((subst > term > subst > $o) > term)).
thf(func_def_423, type, sK285: ((subst > term > subst > $o) > subst)).
thf(func_def_424, type, sK286: ((subst > term > subst > $o) > subst)).
thf(func_def_425, type, sK287: ((subst > term > subst > $o) > (term > $o) > term)).
thf(func_def_426, type, sK288: ((subst > term > subst > $o) > term)).
thf(func_def_427, type, sK289: ((subst > term > subst > $o) > term)).
thf(func_def_428, type, sK290: ((subst > term > subst > $o) > term > $o)).
thf(func_def_429, type, sK291: ((term > $o) > term)).
thf(func_def_430, type, sK292: ((term > $o) > term)).
thf(func_def_431, type, sK293: (subst > term > subst > $o)).
thf(func_def_432, type, sK294: ((term > $o) > term)).
thf(func_def_433, type, sK295: term).
thf(func_def_434, type, sK296: term).
thf(func_def_435, type, sK297: (term > $o)).
thf(func_def_436, type, sK298: (term > term)).
thf(func_def_437, type, sK299: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_438, type, sK300: ((term > $o) > term > term)).
thf(func_def_439, type, sK301: ((subst > term > subst > $o) > (term > $o) > term)).
thf(func_def_440, type, sK302: term).
thf(func_def_441, type, sK303: (term > $o)).
thf(func_def_442, type, sK304: (subst > term > subst > $o)).
thf(func_def_443, type, sK305: ((subst > term > term) > term)).
thf(func_def_444, type, sK306: ((subst > term > term) > subst)).
thf(func_def_445, type, sK307: ((subst > term > term) > subst)).
thf(func_def_446, type, sK308: ((subst > term > term) > term)).
thf(func_def_447, type, sK309: (subst > term > subst > $o)).
thf(func_def_448, type, sK310: term).
thf(func_def_449, type, sK311: ((subst > term > subst > $o) > term)).
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) > subst)).
thf(func_def_453, type, sK315: ((subst > term > subst > $o) > term)).
thf(func_def_454, type, sK316: ((subst > term > subst > $o) > term)).
thf(func_def_455, type, sK317: ((subst > term > subst > $o) > subst)).
thf(func_def_456, type, sK318: ((subst > term > subst > $o) > subst)).
thf(func_def_457, type, sK319: ((subst > term > subst > $o) > subst)).
thf(func_def_458, type, sK320: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_459, type, sK321: ((subst > term > term) > subst)).
thf(func_def_460, type, sK322: ((subst > term > term) > subst)).
thf(func_def_461, type, sK323: ((subst > term > term) > term)).
thf(func_def_462, type, sK324: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_463, type, sK325: ((subst > term > subst > $o) > term)).
thf(func_def_464, type, sK326: ((subst > term > subst > $o) > term)).
thf(func_def_465, type, sK327: ((term > $o) > term)).
thf(func_def_466, type, sK328: ((term > $o) > term)).
thf(func_def_467, type, sK329: (subst > term > subst > $o)).
thf(func_def_468, type, sK330: term).
thf(func_def_469, type, sK331: (term > $o)).
thf(func_def_470, type, sK332: ((term > $o) > term > term)).
thf(func_def_471, type, sK333: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_472, type, sK334: ((subst > term > subst > $o) > subst)).
thf(func_def_473, type, sK335: ((subst > term > subst > $o) > subst)).
thf(func_def_474, type, sK336: ((subst > term > subst > $o) > subst)).
thf(func_def_475, type, sK337: ((subst > term > subst > $o) > term)).
thf(func_def_476, type, sK338: ((subst > term > subst > $o) > subst)).
thf(func_def_477, type, sK339: ((subst > term > subst > $o) > subst)).
thf(func_def_478, type, sK340: ((subst > term > subst > $o) > term)).
thf(func_def_479, type, sK341: ((subst > term > subst > $o) > subst)).
thf(func_def_480, type, sK342: ((subst > term > term) > subst)).
thf(func_def_481, type, sK343: ((subst > term > term) > term)).
thf(func_def_482, type, sK344: ((subst > term > term) > subst)).
thf(func_def_483, type, sK345: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_484, type, sK346: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_485, type, sK347: ((subst > term > subst > $o) > subst)).
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) > term)).
thf(func_def_491, type, sK353: ((subst > term > subst > $o) > subst)).
thf(func_def_492, type, sK354: ((subst > term > subst > $o) > subst)).
thf(func_def_493, type, sK355: (subst > term > subst > $o)).
thf(func_def_494, type, sK356: term).
thf(func_def_495, type, sK357: term).
thf(func_def_496, type, sK358: ((term > $o) > term)).
thf(func_def_497, type, sK359: ((term > $o) > term)).
thf(func_def_498, type, sK360: term).
thf(func_def_499, type, sK361: (term > $o)).
thf(func_def_500, type, sK362: (term > term)).
thf(func_def_501, type, sK363: (subst > term > subst > $o)).
thf(func_def_502, type, sK364: term).
thf(func_def_503, type, sK365: ((subst > term > subst > $o) > subst)).
thf(func_def_504, type, sK366: ((subst > term > subst > $o) > subst)).
thf(func_def_505, type, sK367: ((subst > term > subst > $o) > term)).
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) > term)).
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: (term > (subst > term > subst > $o) > term)).
thf(func_def_513, type, sK375: ((subst > term > subst > $o) > subst)).
thf(func_def_514, type, sK376: ((subst > term > subst > $o) > term)).
thf(func_def_515, type, sK377: ((subst > term > subst > $o) > subst)).
thf(func_def_516, type, sK378: ((subst > term > subst > $o) > subst)).
thf(func_def_517, type, sK379: ((subst > term > subst > $o) > subst)).
thf(func_def_518, type, sK380: ((subst > term > subst > $o) > subst)).
thf(func_def_519, type, sK381: ((subst > term > subst > $o) > term)).
thf(func_def_520, type, sK382: ((subst > term > subst > $o) > subst)).
thf(func_def_521, type, sK383: (subst > term > subst > $o)).
thf(func_def_522, type, sK384: term).
thf(func_def_523, type, sK385: ((subst > term > subst > $o) > term)).
thf(func_def_524, type, sK386: ((subst > term > subst > $o) > term)).
thf(func_def_525, type, sK387: ((term > $o) > term)).
thf(func_def_526, type, sK388: ((term > $o) > term)).
thf(func_def_527, type, sK389: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_528, type, sK390: ((subst > term > term) > subst)).
thf(func_def_529, type, sK391: ((subst > term > term) > term)).
thf(func_def_530, type, sK392: ((subst > term > term) > subst)).
thf(func_def_531, type, sK393: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_532, type, sK394: ((subst > term > subst > $o) > term)).
thf(func_def_533, type, sK395: ((subst > term > subst > $o) > term)).
thf(func_def_534, type, sK396: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_535, type, sK397: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_536, type, sK398: ((subst > term > term) > subst)).
thf(func_def_537, type, sK399: ((subst > term > term) > subst)).
thf(func_def_538, type, sK400: ((subst > term > term) > term)).
thf(func_def_539, type, sK401: (term > $o)).
thf(func_def_540, type, sK402: term).
thf(func_def_541, type, sK403: (subst > term > subst > $o)).
thf(func_def_542, type, sK404: ((subst > term > term) > subst)).
thf(func_def_543, type, sK405: ((subst > term > term) > subst)).
thf(func_def_544, type, sK406: ((subst > term > term) > term)).
thf(func_def_545, type, sK407: ((subst > term > term) > term)).
thf(func_def_546, type, sK408: (term > (term > $o) > term)).
thf(func_def_547, type, sK409: ((subst > term > subst > $o) > (term > $o) > term)).
thf(func_def_548, type, sK410: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_549, type, sK411: (subst > term > subst > $o)).
thf(func_def_550, type, sK412: term).
thf(func_def_551, type, sK413: ((subst > term > term) > term)).
thf(func_def_552, type, sK414: ((subst > term > term) > term)).
thf(func_def_553, type, sK415: ((subst > term > term) > subst)).
thf(func_def_554, type, sK416: ((subst > term > term) > subst)).
thf(func_def_555, type, sK417: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_556, type, sK418: (term > (subst > term > subst > $o) > term)).
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: (term > (subst > term > subst > $o) > term)).
thf(func_def_560, type, sK422: (subst > term > subst > $o)).
thf(func_def_561, type, sK423: term).
thf(func_def_562, type, sK424: ((subst > term > term) > term)).
thf(func_def_563, type, sK425: ((subst > term > term) > subst)).
thf(func_def_564, type, sK426: ((subst > term > term) > term)).
thf(func_def_565, type, sK427: ((subst > term > term) > subst)).
thf(func_def_566, type, sK428: term).
thf(func_def_567, type, sK429: (subst > term > subst > $o)).
thf(func_def_568, type, sK430: ((subst > term > subst > $o) > subst)).
thf(func_def_569, type, sK431: ((subst > term > subst > $o) > subst)).
thf(func_def_570, type, sK432: ((subst > term > subst > $o) > term)).
thf(func_def_571, type, sK433: ((subst > term > subst > $o) > subst)).
thf(func_def_572, type, sK434: ((subst > term > subst > $o) > term)).
thf(func_def_573, type, sK435: ((subst > term > subst > $o) > subst)).
thf(func_def_574, type, sK436: ((subst > term > subst > $o) > subst)).
thf(func_def_575, type, sK437: ((subst > term > subst > $o) > subst)).
thf(func_def_576, type, sK438: ((subst > term > subst > $o) > term)).
thf(func_def_577, type, sK439: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_578, type, sK440: ((subst > term > term) > subst)).
thf(func_def_579, type, sK441: ((subst > term > term) > term)).
thf(func_def_580, type, sK442: ((subst > term > term) > subst)).
thf(func_def_581, type, sK443: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_582, type, sK444: ((subst > term > subst > $o) > term)).
thf(func_def_583, type, sK445: ((subst > term > subst > $o) > term)).
thf(func_def_584, type, sK446: (subst > term > subst > $o)).
thf(func_def_585, type, sK447: term).
thf(func_def_586, type, sK448: ((subst > term > term) > term)).
thf(func_def_587, type, sK449: ((subst > term > term) > subst)).
thf(func_def_588, type, sK450: ((subst > term > term) > term)).
thf(func_def_589, type, sK451: ((subst > term > term) > subst)).
thf(func_def_590, type, sK452: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_591, type, sK453: (term > (subst > term > subst > $o) > term)).
thf(func_def_592, type, sK454: term).
thf(func_def_593, type, sK455: (term > subst > term > term)).
thf(func_def_594, type, sK456: term).
thf(func_def_595, type, sK457: (term > $o)).
thf(func_def_596, type, sK458: subst).
thf(func_def_597, type, sK459: term).
thf(func_def_598, type, sK460: (term > $o)).
thf(func_def_599, type, sK461: subst).
thf(func_def_600, type, sK462: term).
thf(func_def_601, type, sK463: (term > subst > term > term)).
thf(func_def_602, type, sK464: term).
thf(func_def_603, type, sK465: ((subst > term > term) > term)).
thf(func_def_604, type, sK466: ((subst > term > term) > subst)).
thf(func_def_605, type, sK467: term).
thf(func_def_606, type, sK468: term).
thf(func_def_607, type, sK469: term).
thf(func_def_608, type, sK470: term).
thf(func_def_609, type, sK471: ((term > $o) > term)).
thf(func_def_610, type, sK472: ((term > $o) > term)).
thf(func_def_611, type, sK473: term).
thf(func_def_612, type, sK474: (term > $o)).
thf(func_def_613, type, sK475: (subst > term > subst > $o)).
thf(func_def_614, type, sK476: ((term > $o) > term)).
thf(func_def_615, type, sK477: ((subst > term > subst > $o) > term > $o)).
thf(func_def_616, type, sK478: (subst > d_subst)).
thf(func_def_617, type, sK479: term).
thf(func_def_618, type, sK480: term).
thf(func_def_619, type, sK481: term).
thf(func_def_620, type, sK482: term).
thf(func_def_621, type, sK483: subst).
thf(func_def_622, type, sK484: (subst > term > term)).
thf(func_def_623, type, sK485: term).
thf(func_def_624, type, sK486: (subst > term > term)).
thf(func_def_625, type, sK487: ((subst > term > term) > subst)).
thf(func_def_626, type, sK488: ((subst > term > term) > subst)).
thf(func_def_627, type, sK489: ((subst > term > term) > term)).
thf(func_def_628, type, sK490: ((subst > term > term) > term)).
thf(func_def_629, type, sK491: ((subst > term > term) > subst)).
thf(func_def_630, type, sK492: ((subst > term > term) > subst)).
thf(func_def_631, type, sK493: subst).
thf(func_def_632, type, sK494: term).
thf(func_def_633, type, sK495: subst).
thf(func_def_634, type, sK496: term).
thf(func_def_635, type, sK497: term).
thf(func_def_636, type, sK498: term).
thf(func_def_637, type, sK499: term).
thf(func_def_638, type, sK500: term).
thf(func_def_639, type, sK501: term).
thf(func_def_640, type, sK502: term).
thf(func_def_641, type, sK503: (subst > term > term)).
thf(func_def_642, type, sK504: ((subst > term > term) > subst)).
thf(func_def_643, type, sK505: ((subst > term > term) > term)).
thf(func_def_644, type, sK506: ((subst > term > term) > subst)).
thf(func_def_645, type, sK507: term).
thf(func_def_646, type, sK508: term).
thf(func_def_647, type, sK509: subst).
thf(func_def_648, type, sK510: term).
thf(func_def_649, type, sK511: term).
thf(func_def_650, type, sK512: subst).
thf(func_def_651, type, sK513: term).
thf(func_def_652, type, sK514: subst).
thf(func_def_653, type, sK515: term).
thf(func_def_654, type, sK516: (term > $o)).
thf(func_def_655, type, sK517: (term > $o)).
thf(func_def_656, type, sK518: term).
thf(func_def_657, type, sK519: term).
thf(func_def_658, type, sK520: term).
thf(func_def_659, type, sK521: term).
thf(func_def_660, type, sK522: subst).
thf(func_def_661, type, sK523: (term > $o)).
thf(func_def_662, type, sK524: (term > $o)).
thf(func_def_663, type, sK525: term).
thf(func_def_664, type, sK526: (term > (term > $o) > (term > $o) > subst > term)).
thf(func_def_665, type, sK527: term).
thf(func_def_666, type, sK528: (term > subst > term > term)).
thf(func_def_667, type, sK529: ((term > $o) > (term > $o) > term > subst > term)).
thf(func_def_668, type, sK530: subst).
thf(func_def_669, type, sK531: term).
thf(func_def_670, type, sK532: subst).
thf(func_def_671, type, sK533: term).
thf(func_def_672, type, sK534: (term > $o)).
thf(func_def_673, type, sK535: (term > $o)).
thf(func_def_674, type, sK536: term).
thf(func_def_675, type, sK537: ((subst > term > term) > term)).
thf(func_def_676, type, sK538: ((subst > term > term) > subst)).
thf(func_def_677, type, sK539: (term > subst > term > term)).
thf(func_def_678, type, sK540: (term > d_term)).
thf(func_def_679, type, sK541: (subst > term > subst > $o)).
thf(func_def_680, type, sK542: term).
thf(func_def_681, type, sK543: ((subst > term > term) > subst)).
thf(func_def_682, type, sK544: ((subst > term > term) > subst)).
thf(func_def_683, type, sK545: ((subst > term > term) > term)).
thf(func_def_684, type, sK546: ((subst > term > term) > term)).
thf(func_def_685, type, sK547: ((subst > term > subst > $o) > subst > term > 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).
thf(func_def_690, type, sK552: term).
thf(func_def_691, type, sK553: term).
thf(func_def_692, type, sK554: 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 > $o)).
thf(func_def_697, type, sK559: subst).
thf(func_def_698, type, sK560: term).
thf(func_def_699, type, sK561: ((term > $o) > term)).
thf(func_def_700, type, sK562: (term > subst > (term > $o) > term > $o)).
thf(func_def_701, type, sK563: subst).
thf(func_def_702, type, sK564: (term > (term > $o) > subst > term > $o)).
thf(func_def_703, type, sK565: subst).
thf(func_def_704, type, sK566: (term > $o)).
thf(func_def_705, type, sK567: term).
thf(func_def_706, type, sK568: ((term > $o) > term)).
thf(func_def_707, type, sK569: subst).
thf(func_def_708, type, sK570: subst).
thf(func_def_709, type, sK571: (subst > term > subst > $o)).
thf(func_def_710, type, sK572: ((term > $o) > term)).
thf(func_def_711, type, sK573: ((subst > term > subst > $o) > term > $o)).
thf(func_def_712, type, sK574: subst).
thf(func_def_713, type, sK575: term).
thf(func_def_714, type, sK576: term).
thf(func_def_715, type, sK577: term).
thf(func_def_716, type, sK578: term).
thf(func_def_717, type, sK579: term).
thf(func_def_718, type, sK580: term).
thf(func_def_719, type, sK581: term).
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).
thf(func_def_725, type, sK587: subst).
thf(func_def_726, type, sK588: subst).
thf(func_def_727, type, sK589: subst).
thf(func_def_728, type, sK590: subst).
thf(func_def_729, type, sF591: $o).
thf(func_def_730, type, sK592: (subst > term > subst > $o)).
thf(func_def_731, type, sK593: ((term > $o) > term)).
thf(func_def_732, type, sK594: ((subst > term > subst > $o) > term > $o)).
thf(f1,axiom,(
  (hoaslamnotap <=> ! [X10 : (subst > term > term),X28 : term,X27 : term] : (~ ! [X8 : subst,X7 : term,X14 : subst] : (((sub @ (X10 @ X8 @ X7) @ X14)) = ((X10 @ (comp @ X8 @ X14) @ (sub @ X7 @ X14)))) | (((lam @ (X10 @ sh @ one))) != ((ap @ (sub @ X27 @ id) @ X28))))) & (! [X4 : (subst > term > subst > $o),X20 : term] : ((X4 @ id @ (lam @ X20) @ id) | ~ ! [X8 : subst,X17 : subst,X7 : term,X14 : subst] : ((X4 @ (comp @ X8 @ X17) @ (sub @ X7 @ X17) @ X14) | ~(X4 @ X8 @ X7 @ (comp @ X17 @ X14))) | ~ ! [X9 : term] : ((X4 @ id @ (sub @ X20 @ (push @ X9 @ id)) @ id) | ~(X4 @ id @ X9 @ id)) | ~ ! [X17 : subst,X7 : term,X8 : subst,X14 : subst] : (~(X4 @ (comp @ X8 @ X17) @ (sub @ X7 @ X17) @ X14) | (X4 @ X8 @ X7 @ (comp @ X17 @ X14))) | ~ ! [X10 : (subst > term > term)] : (~ ! [X7 : term] : (~(X4 @ id @ X7 @ id) | (X4 @ id @ (X10 @ id @ X7) @ id)) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X7 : term,X14 : subst,X8 : subst] : (((sub @ (X10 @ X8 @ X7) @ X14)) = ((X10 @ (comp @ X8 @ X14) @ (sub @ X7 @ X14)))))) <=> hoasinduction_lem3) & ~(var @ (d2term @ d_one)) & ~(hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) & (hoasinduction_lem3v2 <=> ! [X5 : (term > $o),X4 : (subst > term > subst > $o),X25 : term] : (~ ! [X7 : term,X14 : subst,X17 : subst,X8 : subst] : (~(X4 @ X8 @ X7 @ (comp @ X17 @ X14)) | (X4 @ (comp @ X8 @ X17) @ (sub @ X7 @ X17) @ X14)) | ~ ! [X10 : (subst > term > term)] : ((X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X7 : term] : ((X4 @ id @ (X10 @ id @ X7) @ id) | ~(X4 @ id @ X7 @ id)) | ~ ! [X8 : subst,X14 : subst,X7 : term] : (((sub @ (X10 @ X8 @ X7) @ X14)) = ((X10 @ (comp @ X8 @ X14) @ (sub @ X7 @ X14))))) | ~ ! [X14 : subst,X17 : subst,X7 : term,X8 : subst] : (~(X4 @ (comp @ X8 @ X17) @ (sub @ X7 @ X17) @ X14) | (X4 @ X8 @ X7 @ (comp @ X17 @ X14))) | ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id))) | (X5 @ (lam @ X25)) | ~ ! [X9 : term] : (~(X5 @ X9) | (X5 @ (sub @ X25 @ (push @ X9 @ id)))))) & (! [X8 : subst,X7 : term,X5 : (term > $o),X4 : (term > $o)] : (~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ (sub @ X6 @ (push @ X7 @ X8))))) | (X5 @ one) | ~(X4 @ X7)) <=> pushprop_lem1v2) & pushprop & (axidl <=> ! [X8 : subst] : (X8 = ((comp @ id @ X8)))) & (! [X15 : term,X16 : term,X4 : (subst > term > subst > $o)] : (~(X4 @ id @ X16 @ id) | ~ ! [X8 : subst,X7 : term,X14 : subst,X17 : subst] : (~(X4 @ (comp @ X8 @ X17) @ (sub @ X7 @ X17) @ X14) | (X4 @ X8 @ X7 @ (comp @ X17 @ X14))) | ~ ! [X7 : term,X9 : term] : (~(X4 @ id @ X9 @ id) | ~(X4 @ id @ X7 @ id) | (X4 @ id @ (ap @ (sub @ X7 @ id) @ X9) @ id)) | ~ ! [X8 : subst,X7 : term,X17 : subst,X14 : subst] : (~(X4 @ X8 @ X7 @ (comp @ X17 @ X14)) | (X4 @ (comp @ X8 @ X17) @ (sub @ X7 @ X17) @ X14)) | (X4 @ id @ (ap @ X15 @ X16) @ id) | ~(X4 @ id @ X15 @ id)) <=> hoasinduction_lem2) & (! [X13 : term,X4 : (subst > term > subst > $o)] : (~ ! [X9 : term] : (~(X4 @ id @ X9 @ id) | (X4 @ id @ (sub @ X13 @ (push @ X9 @ id)) @ id)) | ~ ! [X10 : (subst > term > term)] : (~ ! [X14 : subst,X7 : term,X8 : subst] : (((sub @ (X10 @ X8 @ X7) @ X14)) = ((X10 @ (comp @ X8 @ X14) @ (sub @ X7 @ X14)))) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X7 : term] : ((X4 @ id @ (X10 @ id @ X7) @ id) | ~(X4 @ id @ X7 @ id))) | (X4 @ id @ (lam @ (sub @ X13 @ (push @ one @ sh))) @ id)) <=> hoasinduction_lem3aa) & ~lamnotap & (! [X9 : term,X7 : term,X11 : term,X12 : term] : ((X7 = X9) | (((ap @ (sub @ X7 @ id) @ X11)) != ((ap @ (sub @ X9 @ id) @ X12)))) <=> hoasapinj1) & axmap & hoasinduction_lem1_gthm & (apinj2 <=> ! [X12 : term,X11 : term,X7 : term,X9 : term] : ((X11 = X12) | (((ap @ X7 @ X11)) != ((ap @ X9 @ X12))))) & substmonoid_gthm & (induction2lem <=> ! [X30 : subst,X4 : (term > $o),X29 : term] : ((X4 @ (sub @ X29 @ X30)) | ~ ! [X9 : term] : ((X4 @ (sub @ X9 @ X30)) | ~(var @ X9)) | ~ ! [X9 : term,X7 : term] : (~(X4 @ X7) | ~(X4 @ X9) | (X4 @ (ap @ X7 @ X9))) | ~ ! [X7 : term] : ((X4 @ (lam @ X7)) | ~ ! [X9 : term] : (~(X4 @ X9) | (X4 @ (sub @ X7 @ (push @ X9 @ id))))))) & (one = ((d2term @ d_one))) & (hoasinduction_lem3v2_f_lthm <=> ! [X9 : term] : ~ ! [X10 : (subst > term > term)] : ~ ! [X7 : term,X8 : subst] : (((sub @ X9 @ (push @ X7 @ X8))) = ((X10 @ X8 @ X7)))) & (! [X8 : subst] : (X8 = ((push @ (sub @ one @ X8) @ (comp @ sh @ X8)))) <=> axscons) & hoasinduction_gthm & (! [X13 : term,X4 : (subst > term > subst > $o)] : ((X4 @ id @ (lam @ (sub @ X13 @ (push @ one @ sh))) @ id) | ~ ! [X10 : (subst > term > term)] : (~ ! [X7 : term] : ((X4 @ id @ (X10 @ id @ X7) @ id) | ~(X4 @ id @ X7 @ id)) | ~ ! [X7 : term,X8 : subst,X14 : subst] : (((sub @ (X10 @ X8 @ X7) @ X14)) = ((X10 @ (comp @ X8 @ X14) @ (sub @ X7 @ X14)))) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id)) | ~ ! [X9 : term] : (~(X4 @ id @ X9 @ id) | (X4 @ id @ (sub @ X13 @ (push @ X9 @ id)) @ id))) <=> hoasinduction_lem3aa_lthm) & axabs & hoasinduction_lem2_gthm & hoasinduction_lem3b_gthm & ! [X0 : d_term] : (X0 = d_one) & termmset_gthm & apnotvar & hoasapinj2_gthm & (pushprop_lem0 <=> ! [X7 : term,X8 : subst,X4 : (term > $o)] : ~ ! [X5 : (term > $o)] : ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ (sub @ X6 @ (push @ X7 @ X8)))))) & (! [X12 : term,X7 : term,X9 : term,X11 : term] : ((((ap @ (sub @ X7 @ id) @ X11)) != ((ap @ (sub @ X9 @ id) @ X12))) | (X11 = X12)) <=> hoasapinj2) & (! [X7 : term] : (X7 = ((sub @ X7 @ id))) <=> axvarid) & (! [X18 : term,X19 : term,X5 : (term > $o),X4 : (subst > term > subst > $o)] : ((X5 @ (ap @ X18 @ X19)) | ~ ! [X7 : term,X8 : subst,X14 : subst,X17 : subst] : ((X4 @ X8 @ X7 @ (comp @ X17 @ X14)) | ~(X4 @ (comp @ X8 @ X17) @ (sub @ X7 @ X17) @ X14)) | ~ ! [X8 : subst,X17 : subst,X7 : term,X14 : subst] : (~(X4 @ X8 @ X7 @ (comp @ X17 @ X14)) | (X4 @ (comp @ X8 @ X17) @ (sub @ X7 @ X17) @ X14)) | ~(X5 @ X19) | ~(X5 @ X18) | ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id))) | ~ ! [X7 : term,X9 : term] : ((X4 @ id @ (ap @ (sub @ X7 @ id) @ X9) @ id) | ~(X4 @ id @ X7 @ id) | ~(X4 @ id @ X9 @ id))) <=> hoasinduction_lem2v2) & (hoasinduction_lem3v2_lthm <=> (! [X7 : term] : (X7 = ((sub @ X7 @ id))) => ! [X25 : term,X5 : (term > $o),X4 : (subst > term > subst > $o)] : (~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id))) | ~ ! [X7 : term,X14 : subst,X8 : subst,X17 : subst] : (~(X4 @ (comp @ X8 @ X17) @ (sub @ X7 @ X17) @ X14) | (X4 @ X8 @ X7 @ (comp @ X17 @ X14))) | ~ ! [X10 : (subst > term > term)] : (~ ! [X7 : term] : ((X4 @ id @ (X10 @ id @ X7) @ id) | ~(X4 @ id @ X7 @ id)) | ~ ! [X7 : term,X8 : subst,X14 : subst] : (((sub @ (X10 @ X8 @ X7) @ X14)) = ((X10 @ (comp @ X8 @ X14) @ (sub @ X7 @ X14)))) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id)) | ~ ! [X14 : subst,X7 : term,X17 : subst,X8 : subst] : ((X4 @ (comp @ X8 @ X17) @ (sub @ X7 @ X17) @ X14) | ~(X4 @ X8 @ X7 @ (comp @ X17 @ X14))) | (X5 @ (lam @ X25)) | ~ ! [X9 : term] : ((X5 @ (sub @ X25 @ (push @ X9 @ id))) | ~(X5 @ X9))))) & pushprop_lem0_gthm & hoaslaminj_gthm & (((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))) <=> hoasinduction_lem0_lthm) & ulamvarind & pushprop_gthm & hoaslamnotvar & (hoaslaminj_lthm <=> (! [X7 : term,X8 : subst] : (X7 = ((sub @ one @ (push @ X7 @ X8)))) => (! [X7 : term,X8 : subst] : (X8 = ((comp @ sh @ (push @ X7 @ X8)))) => (! [X7 : term,X9 : term] : ((X7 = X9) | (((lam @ X7)) != ((lam @ X9)))) => ! [X32 : subst,X31 : (subst > term > term),X33 : term,X10 : (subst > term > term)] : ((((X10 @ X32 @ X33)) = ((X31 @ X32 @ X33))) | ~ ! [X7 : term,X8 : subst,X14 : subst] : (((sub @ (X10 @ X8 @ X7) @ X14)) = ((X10 @ (comp @ X8 @ X14) @ (sub @ X7 @ X14)))) | ~ ! [X8 : subst,X14 : subst,X7 : term] : (((sub @ (X31 @ X8 @ X7) @ X14)) = ((X31 @ (comp @ X8 @ X14) @ (sub @ X7 @ X14)))) | (((lam @ (X10 @ sh @ one))) != ((lam @ (X31 @ sh @ one))))))))) & (induction2_lthm <=> (! [X7 : term] : (X7 = ((sub @ X7 @ id))) => (! [X29 : term,X4 : (term > $o),X30 : subst] : (~ ! [X7 : term,X9 : term] : (~(X4 @ X9) | (X4 @ (ap @ X7 @ X9)) | ~(X4 @ X7)) | ~ ! [X7 : term] : ((X4 @ (lam @ X7)) | ~ ! [X9 : term] : ((X4 @ (sub @ X7 @ (push @ X9 @ id))) | ~(X4 @ X9))) | (X4 @ (sub @ X29 @ X30)) | ~ ! [X9 : term] : ((X4 @ (sub @ X9 @ X30)) | ~(var @ X9))) => ! [X24 : term,X4 : (term > $o)] : ((X4 @ X24) | ~ ! [X7 : term] : (~ ! [X9 : term] : ((X4 @ (sub @ X7 @ (push @ X9 @ id))) | ~(X4 @ X9)) | (X4 @ (lam @ X7))) | ~ ! [X9 : term,X7 : term] : (~(X4 @ X9) | ~(X4 @ X7) | (X4 @ (ap @ X7 @ X9))) | ~ ! [X7 : term] : (~(var @ X7) | (X4 @ X7)))))) & (hoaslaminj <=> ! [X32 : subst,X33 : term,X31 : (subst > term > term),X10 : (subst > term > term)] : ((((lam @ (X10 @ sh @ one))) != ((lam @ (X31 @ sh @ one)))) | ~ ! [X7 : term,X14 : subst,X8 : subst] : (((sub @ (X31 @ X8 @ X7) @ X14)) = ((X31 @ (comp @ X8 @ X14) @ (sub @ X7 @ X14)))) | ~ ! [X14 : subst,X8 : subst,X7 : term] : (((sub @ (X10 @ X8 @ X7) @ X14)) = ((X10 @ (comp @ X8 @ X14) @ (sub @ X7 @ X14)))) | (((X10 @ X32 @ X33)) = ((X31 @ X32 @ X33))))) & ((! [X4 : (subst > term > subst > $o)] : ~ ! [X5 : (term > $o)] : ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id))) => (! [X4 : (term > $o),X24 : term] : ((X4 @ X24) | ~ ! [X7 : term,X9 : term] : ((X4 @ (ap @ X7 @ X9)) | ~(X4 @ X9) | ~(X4 @ X7)) | ~ ! [X7 : term] : ((X4 @ (lam @ X7)) | ~ ! [X9 : term] : (~(X4 @ X9) | (X4 @ (sub @ X7 @ (push @ X9 @ id))))) | ~ ! [X7 : term] : (~(var @ X7) | (X4 @ X7))) => (! [X7 : term] : (X7 = ((sub @ X7 @ id))) => (! [X5 : (term > $o),X4 : (subst > term > subst > $o),X21 : term] : (~ ! [X10 : (subst > term > term)] : (~ ! [X7 : term,X14 : subst,X8 : subst] : (((sub @ (X10 @ X8 @ X7) @ X14)) = ((X10 @ (comp @ X8 @ X14) @ (sub @ X7 @ X14)))) | ~ ! [X7 : term] : ((X4 @ id @ (X10 @ id @ X7) @ id) | ~(X4 @ id @ X7 @ id)) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id)) | (X5 @ (lam @ X21)) | ~ ! [X9 : term] : (~(X5 @ X9) | (X5 @ (sub @ X21 @ (push @ X9 @ id)))) | ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id)))) => ! [X26 : term,X4 : (subst > term > subst > $o)] : (~ ! [X7 : term] : (~(var @ (sub @ X7 @ id)) | (X4 @ id @ X7 @ id)) | ~ ! [X9 : term,X7 : term] : (~(X4 @ id @ X9 @ id) | (X4 @ id @ (ap @ (sub @ X7 @ id) @ X9) @ id) | ~(X4 @ id @ X7 @ id)) | ~ ! [X17 : subst,X14 : subst,X7 : term,X8 : subst] : (~(X4 @ X8 @ X7 @ (comp @ X17 @ X14)) | (X4 @ (comp @ X8 @ X17) @ (sub @ X7 @ X17) @ X14)) | ~ ! [X10 : (subst > term > term)] : (~ ! [X8 : subst,X7 : term,X14 : subst] : (((sub @ (X10 @ X8 @ X7) @ X14)) = ((X10 @ (comp @ X8 @ X14) @ (sub @ X7 @ X14)))) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X7 : term] : ((X4 @ id @ (X10 @ id @ X7) @ id) | ~(X4 @ id @ X7 @ id))) | (X4 @ id @ X26 @ id) | ~ ! [X8 : subst,X7 : term,X14 : subst,X17 : subst] : (~(X4 @ (comp @ X8 @ X17) @ (sub @ X7 @ X17) @ X14) | (X4 @ X8 @ X7 @ (comp @ X17 @ X14)))))))) <=> hoasinduction_lthm_3) & (! [X4 : (subst > term > subst > $o),X23 : term] : (~ ! [X7 : term,X9 : term] : (~(X4 @ id @ X9 @ id) | (X4 @ id @ (ap @ (sub @ X7 @ id) @ X9) @ id) | ~(X4 @ id @ X7 @ id)) | ~ ! [X10 : (subst > term > term)] : ((X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X7 : term] : (~(X4 @ id @ X7 @ id) | (X4 @ id @ (X10 @ id @ X7) @ id)) | ~ ! [X8 : subst,X7 : term,X14 : subst] : (((sub @ (X10 @ X8 @ X7) @ X14)) = ((X10 @ (comp @ X8 @ X14) @ (sub @ X7 @ X14))))) | (X4 @ id @ X23 @ id)) <=> hoasinduction_no_psi_cond) & (! [X7 : term,X8 : subst] : (X7 = ((sub @ one @ (push @ X7 @ X8)))) <=> axvarcons) & pushprop_lem2v2_gthm & (hoasinduction_lem3a_lthm <=> (! [X7 : term] : (X7 = ((sub @ X7 @ id))) => (! [X4 : (subst > term > subst > $o),X13 : term] : (~ ! [X10 : (subst > term > term)] : ((X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X14 : subst,X8 : subst,X7 : term] : (((sub @ (X10 @ X8 @ X7) @ X14)) = ((X10 @ (comp @ X8 @ X14) @ (sub @ X7 @ X14)))) | ~ ! [X7 : term] : (~(X4 @ id @ X7 @ id) | (X4 @ id @ (X10 @ id @ X7) @ id))) | (X4 @ id @ (lam @ (sub @ X13 @ (push @ one @ sh))) @ id) | ~ ! [X9 : term] : ((X4 @ id @ (sub @ X13 @ (push @ X9 @ id)) @ id) | ~(X4 @ id @ X9 @ id))) => ! [X4 : (subst > term > subst > $o),X22 : term] : ((X4 @ id @ (lam @ X22) @ id) | ~ ! [X9 : term] : (~(X4 @ id @ X9 @ id) | (X4 @ id @ (sub @ X22 @ (push @ X9 @ id)) @ id)) | ~ ! [X10 : (subst > term > term)] : ((X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X14 : subst,X8 : subst,X7 : term] : (((sub @ (X10 @ X8 @ X7) @ X14)) = ((X10 @ (comp @ X8 @ X14) @ (sub @ X7 @ X14)))) | ~ ! [X7 : term] : (~(X4 @ id @ X7 @ id) | (X4 @ id @ (X10 @ id @ X7) @ id))))))) & (((push @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & (hoasinduction_lem3b <=> ! [X9 : term] : ~ ! [X10 : (subst > term > term)] : (((X10 @ sh @ one)) != ((sub @ X9 @ (push @ one @ sh))))) & (pushprop_lem0_lthm <=> ! [X8 : subst,X4 : (term > $o),X7 : term] : ~ ! [X5 : (term > $o)] : ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ (sub @ X6 @ (push @ X7 @ X8)))))) & axapp & (sh = ((d2subst @ d_id))) & (hoaslam = (^[X43 : subst, X44 : (subst > term > term)] : ((d2term @ d_one)))) & axassoc & (((sub @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2term @ d_one))) & (hoasinduction_lthm <=> (! [X4 : (term > $o),X24 : term] : ((X4 @ X24) | ~ ! [X7 : term] : ((X4 @ (lam @ X7)) | ~ ! [X9 : term] : (~(X4 @ X9) | (X4 @ (sub @ X7 @ (push @ X9 @ id))))) | ~ ! [X7 : term] : (~(var @ X7) | (X4 @ X7)) | ~ ! [X7 : term,X9 : term] : (~(X4 @ X7) | (X4 @ (ap @ X7 @ X9)) | ~(X4 @ X9))) => (! [X16 : term,X4 : (subst > term > subst > $o),X15 : term] : (~(X4 @ id @ X16 @ id) | ~(X4 @ id @ X15 @ id) | ~ ! [X8 : subst,X7 : term,X14 : subst,X17 : subst] : (~(X4 @ (comp @ X8 @ X17) @ (sub @ X7 @ X17) @ X14) | (X4 @ X8 @ X7 @ (comp @ X17 @ X14))) | ~ ! [X14 : subst,X8 : subst,X17 : subst,X7 : term] : (~(X4 @ X8 @ X7 @ (comp @ X17 @ X14)) | (X4 @ (comp @ X8 @ X17) @ (sub @ X7 @ X17) @ X14)) | (X4 @ id @ (ap @ X15 @ X16) @ id) | ~ ! [X7 : term,X9 : term] : (~(X4 @ id @ X9 @ id) | (X4 @ id @ (ap @ (sub @ X7 @ id) @ X9) @ id) | ~(X4 @ id @ X7 @ id))) => (! [X20 : term,X4 : (subst > term > subst > $o)] : ((X4 @ id @ (lam @ X20) @ id) | ~ ! [X8 : subst,X17 : subst,X14 : subst,X7 : term] : (~(X4 @ X8 @ X7 @ (comp @ X17 @ X14)) | (X4 @ (comp @ X8 @ X17) @ (sub @ X7 @ X17) @ X14)) | ~ ! [X9 : term] : (~(X4 @ id @ X9 @ id) | (X4 @ id @ (sub @ X20 @ (push @ X9 @ id)) @ id)) | ~ ! [X14 : subst,X8 : subst,X17 : subst,X7 : term] : (~(X4 @ (comp @ X8 @ X17) @ (sub @ X7 @ X17) @ X14) | (X4 @ X8 @ X7 @ (comp @ X17 @ X14))) | ~ ! [X10 : (subst > term > term)] : ((X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X14 : subst,X7 : term,X8 : subst] : (((sub @ (X10 @ X8 @ X7) @ X14)) = ((X10 @ (comp @ X8 @ X14) @ (sub @ X7 @ X14)))) | ~ ! [X7 : term] : ((X4 @ id @ (X10 @ id @ X7) @ id) | ~(X4 @ id @ X7 @ id)))) => ! [X26 : term,X4 : (subst > term > subst > $o)] : (~ ! [X8 : subst,X17 : subst,X14 : subst,X7 : term] : ((X4 @ X8 @ X7 @ (comp @ X17 @ X14)) | ~(X4 @ (comp @ X8 @ X17) @ (sub @ X7 @ X17) @ X14)) | ~ ! [X10 : (subst > term > term)] : ((X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X7 : term] : ((X4 @ id @ (X10 @ id @ X7) @ id) | ~(X4 @ id @ X7 @ id)) | ~ ! [X7 : term,X14 : subst,X8 : subst] : (((sub @ (X10 @ X8 @ X7) @ X14)) = ((X10 @ (comp @ X8 @ X14) @ (sub @ X7 @ X14))))) | ~ ! [X8 : subst,X17 : subst,X14 : subst,X7 : term] : (~(X4 @ X8 @ X7 @ (comp @ X17 @ X14)) | (X4 @ (comp @ X8 @ X17) @ (sub @ X7 @ X17) @ X14)) | ~ ! [X7 : term] : ((X4 @ id @ X7 @ id) | ~(var @ (sub @ X7 @ id))) | ~ ! [X7 : term,X9 : term] : (~(X4 @ id @ X7 @ id) | (X4 @ id @ (ap @ (sub @ X7 @ id) @ X9) @ id) | ~(X4 @ id @ X9 @ id)) | (X4 @ id @ X26 @ id)))))) & (((lam @ (d2term @ d_one))) = ((d2term @ d_one))) & (hoasinduction_lem3v2a_lthm <=> (! [X9 : term] : ~ ! [X10 : (subst > term > term)] : ~ ! [X8 : subst,X7 : term] : (((sub @ X9 @ (push @ X7 @ X8))) = ((X10 @ X8 @ X7))) => (! [X7 : term] : (X7 = ((sub @ X7 @ id))) => ! [X4 : (subst > term > subst > $o),X5 : (term > $o),X21 : term] : (~ ! [X10 : (subst > term > term)] : (~ ! [X8 : subst,X7 : term,X14 : subst] : (((sub @ (X10 @ X8 @ X7) @ X14)) = ((X10 @ (comp @ X8 @ X14) @ (sub @ X7 @ X14)))) | ~ ! [X7 : term] : (~(X4 @ id @ X7 @ id) | (X4 @ id @ (X10 @ id @ X7) @ id)) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id)) | (X5 @ (lam @ X21)) | ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id))) | ~ ! [X9 : term] : (~(X5 @ X9) | (X5 @ (sub @ X21 @ (push @ X9 @ id)))))))) & pushprop_lem1_gthm & pushprop_lem3v2_lthm & hoasinduction_lem1 & pushprop_lem3v2 & ~ulamvar1 & hoasinduction_lem1v2 & (substmonoid_lthm <=> (! [X8 : subst] : (X8 = ((comp @ id @ X8))) => (! [X8 : subst] : (X8 = ((comp @ X8 @ id))) => (! [X8 : subst] : (X8 = ((comp @ X8 @ id))) & ! [X8 : subst] : (X8 = ((comp @ id @ X8))))))) & induction2_gthm & (hoasinduction_lem3_lthm <=> (! [X7 : term] : (X7 = ((sub @ X7 @ id))) => (! [X13 : term,X4 : (subst > term > subst > $o)] : (~ ! [X9 : term] : (~(X4 @ id @ X9 @ id) | (X4 @ id @ (sub @ X13 @ (push @ X9 @ id)) @ id)) | ~ ! [X10 : (subst > term > term)] : (~ ! [X7 : term] : ((X4 @ id @ (X10 @ id @ X7) @ id) | ~(X4 @ id @ X7 @ id)) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X7 : term,X14 : subst,X8 : subst] : (((sub @ (X10 @ X8 @ X7) @ X14)) = ((X10 @ (comp @ X8 @ X14) @ (sub @ X7 @ X14))))) | (X4 @ id @ (lam @ (sub @ X13 @ (push @ one @ sh))) @ id)) => ! [X4 : (subst > term > subst > $o),X20 : term] : (~ ! [X9 : term] : (~(X4 @ id @ X9 @ id) | (X4 @ id @ (sub @ X20 @ (push @ X9 @ id)) @ id)) | (X4 @ id @ (lam @ X20) @ id) | ~ ! [X8 : subst,X7 : term,X17 : subst,X14 : subst] : ((X4 @ X8 @ X7 @ (comp @ X17 @ X14)) | ~(X4 @ (comp @ X8 @ X17) @ (sub @ X7 @ X17) @ X14)) | ~ ! [X10 : (subst > term > term)] : (~ ! [X7 : term] : ((X4 @ id @ (X10 @ id @ X7) @ id) | ~(X4 @ id @ X7 @ id)) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X7 : term,X14 : subst,X8 : subst] : (((sub @ (X10 @ X8 @ X7) @ X14)) = ((X10 @ (comp @ X8 @ X14) @ (sub @ X7 @ X14))))) | ~ ! [X14 : subst,X17 : subst,X8 : subst,X7 : term] : ((X4 @ (comp @ X8 @ X17) @ (sub @ X7 @ X17) @ X14) | ~(X4 @ X8 @ X7 @ (comp @ X17 @ X14))))))) & pushprop_lem1v2_gthm & lamnotvar & ! [X47 : d_term,X46 : d_term] : ((((d2term @ X46)) = ((d2term @ X47))) => (X46 = X47)) & axclos & (id = ((d2subst @ d_id))) & hoasinduction_lem2v2_gthm & (! [X7 : term] : (X7 = ((sub @ X7 @ id))) <=> termmset) & pushprop_lem2v2_lthm & ulamvarsh & hoasapinj1_gthm & ((! [X7 : term,X8 : subst] : (X7 = ((sub @ one @ (push @ X7 @ X8)))) => (! [X7 : term,X8 : subst] : (X8 = ((comp @ sh @ (push @ X7 @ X8)))) => ! [X4 : (term > $o),X8 : subst,X17 : (term > $o),X9 : term,X7 : term] : (~(X4 @ X7) | (X17 @ (sub @ X7 @ (push @ X9 @ X8)))))) <=> pushprop_lem1_lthm) & hoasinduction_lem3v2_gthm & hoasinduction_lem2_lthm & (((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) = ((d2term @ d_one))) & hoasinduction_lem1_lthm & (hoasinduction_lem3aaa <=> ! [X35 : term,X4 : (subst > term > subst > $o)] : ((X4 @ id @ (lam @ (sub @ X35 @ (push @ one @ sh))) @ id) | ~ ! [X10 : (subst > term > term),X36 : term] : (~ ! [X7 : term] : ((X4 @ id @ (X10 @ id @ X7) @ id) | ~(X4 @ id @ X7 @ id)) | ~ ! [X41 : term,X42 : subst,X40 : subst] : (((sub @ (X10 @ X40 @ X41) @ X42)) = ((sub @ (sub @ X36 @ (push @ X41 @ X40)) @ X42))) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X37 : subst,X39 : subst,X38 : term] : (((X10 @ (comp @ X37 @ X39) @ (sub @ X38 @ X39))) = ((sub @ X36 @ (push @ (sub @ X38 @ X39) @ (comp @ X37 @ X39)))))) | ~ ! [X9 : term] : (~(X4 @ id @ X9 @ id) | (X4 @ id @ (sub @ X35 @ (push @ X9 @ id)) @ id)))) & ((! [X4 : (subst > term > subst > $o)] : ~ ! [X5 : (term > $o)] : ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id))) => (! [X4 : (term > $o),X24 : term] : ((X4 @ X24) | ~ ! [X9 : term,X7 : term] : ((X4 @ (ap @ X7 @ X9)) | ~(X4 @ X9) | ~(X4 @ X7)) | ~ ! [X7 : term] : (~ ! [X9 : term] : ((X4 @ (sub @ X7 @ (push @ X9 @ id))) | ~(X4 @ X9)) | (X4 @ (lam @ X7))) | ~ ! [X7 : term] : ((X4 @ X7) | ~(var @ X7))) => (! [X7 : term] : (X7 = ((sub @ X7 @ id))) => (! [X4 : (subst > term > subst > $o),X21 : term,X5 : (term > $o)] : (~ ! [X10 : (subst > term > term)] : ((X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X7 : term] : ((X4 @ id @ (X10 @ id @ X7) @ id) | ~(X4 @ id @ X7 @ id)) | ~ ! [X8 : subst,X7 : term,X14 : subst] : (((sub @ (X10 @ X8 @ X7) @ X14)) = ((X10 @ (comp @ X8 @ X14) @ (sub @ X7 @ X14))))) | ~ ! [X9 : term] : ((X5 @ (sub @ X21 @ (push @ X9 @ id))) | ~(X5 @ X9)) | (X5 @ (lam @ X21)) | ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id)))) => ! [X4 : (subst > term > subst > $o),X23 : term] : ((X4 @ id @ X23 @ id) | ~ ! [X10 : (subst > term > term)] : (~ ! [X14 : subst,X8 : subst,X7 : term] : (((sub @ (X10 @ X8 @ X7) @ X14)) = ((X10 @ (comp @ X8 @ X14) @ (sub @ X7 @ X14)))) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X7 : term] : (~(X4 @ id @ X7 @ id) | (X4 @ id @ (X10 @ id @ X7) @ id))) | ~ ! [X9 : term,X7 : term] : (~(X4 @ id @ X7 @ id) | (X4 @ id @ (ap @ (sub @ X7 @ id) @ X9) @ id) | ~(X4 @ id @ X9 @ id))))))) <=> hoasinduction_no_psi_cond_lthm) & hoasinduction_lem3a_gthm & hoasinduction_lem1v2_gthm & ! [X3 : d_subst,X2 : d_subst] : ((((d2subst @ X2)) = ((d2subst @ X3))) => (X2 = X3)) & (! [X22 : term,X4 : (subst > term > subst > $o)] : ((X4 @ id @ (lam @ X22) @ id) | ~ ! [X9 : term] : ((X4 @ id @ (sub @ X22 @ (push @ X9 @ id)) @ id) | ~(X4 @ id @ X9 @ id)) | ~ ! [X10 : (subst > term > term)] : ((X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X7 : term,X14 : subst,X8 : subst] : (((sub @ (X10 @ X8 @ X7) @ X14)) = ((X10 @ (comp @ X8 @ X14) @ (sub @ X7 @ X14)))) | ~ ! [X7 : term] : (~(X4 @ id @ X7 @ id) | (X4 @ id @ (X10 @ id @ X7) @ id)))) <=> hoasinduction_lem3a) & ! [X48 : term] : ? [X0 : d_term] : (X48 = ((d2term @ X0))) & hoasapnotvar & axvarshift & (! [X8 : subst] : (X8 = ((comp @ X8 @ id))) <=> axidr) & hoaslamnotvar_gthm & (hoasinduction <=> ! [X26 : term,X4 : (subst > term > subst > $o)] : ((X4 @ id @ X26 @ id) | ~ ! [X9 : term,X7 : term] : (~(X4 @ id @ X7 @ id) | (X4 @ id @ (ap @ (sub @ X7 @ id) @ X9) @ id) | ~(X4 @ id @ X9 @ id)) | ~ ! [X10 : (subst > term > term)] : (~ ! [X7 : term,X14 : subst,X8 : subst] : (((sub @ (X10 @ X8 @ X7) @ X14)) = ((X10 @ (comp @ X8 @ X14) @ (sub @ X7 @ X14)))) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X7 : term] : (~(X4 @ id @ X7 @ id) | (X4 @ id @ (X10 @ id @ X7) @ id))) | ~ ! [X7 : term,X14 : subst,X17 : subst,X8 : subst] : (~(X4 @ (comp @ X8 @ X17) @ (sub @ X7 @ X17) @ X14) | (X4 @ X8 @ X7 @ (comp @ X17 @ X14))) | ~ ! [X7 : term,X17 : subst,X8 : subst,X14 : subst] : (~(X4 @ X8 @ X7 @ (comp @ X17 @ X14)) | (X4 @ (comp @ X8 @ X17) @ (sub @ X7 @ X17) @ X14)) | ~ ! [X7 : term] : (~(var @ (sub @ X7 @ id)) | (X4 @ id @ X7 @ id)))) & (((ap @ (d2term @ d_one) @ (d2term @ d_one))) = ((d2term @ d_one))) & (laminj <=> ! [X7 : term,X9 : term] : ((X7 = X9) | (((lam @ X7)) != ((lam @ X9))))) & (! [X4 : (subst > term > subst > $o)] : ~ ! [X5 : (term > $o)] : ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id))) <=> hoasinduction_lem0) & ((! [X12 : term,X7 : term,X11 : term,X9 : term] : ((X11 = X12) | (((ap @ X7 @ X11)) != ((ap @ X9 @ X12)))) => ! [X11 : term,X9 : term,X7 : term,X12 : term] : ((((ap @ (sub @ X7 @ id) @ X11)) != ((ap @ (sub @ X9 @ id) @ X12))) | (X11 = X12))) <=> hoasapinj2_lthm) & (! [X4 : (term > $o),X24 : term] : ((X4 @ X24) | ~ ! [X9 : term,X7 : term] : ((X4 @ (ap @ X7 @ X9)) | ~(X4 @ X9) | ~(X4 @ X7)) | ~ ! [X7 : term] : (~ ! [X9 : term] : (~(X4 @ X9) | (X4 @ (sub @ X7 @ (push @ X9 @ id)))) | (X4 @ (lam @ X7))) | ~ ! [X7 : term] : ((X4 @ X7) | ~(var @ X7))) <=> induction2) & hoasinduction_lem3_gthm & ((! [X8 : subst,X7 : term] : (X7 = ((sub @ one @ (push @ X7 @ X8)))) => (! [X7 : term,X8 : subst] : (X8 = ((comp @ sh @ (push @ X7 @ X8)))) => (! [X8 : subst] : (X8 = ((comp @ X8 @ id))) => (! [X4 : (term > $o),X34 : term] : (~ ! [X9 : term,X7 : term] : ((X4 @ (ap @ X7 @ X9)) | ~(X4 @ X7) | ~(X4 @ X9)) | (X4 @ X34) | ~ ! [X7 : term] : (~(X4 @ X7) | (X4 @ (lam @ X7))) | ~ ! [X7 : term] : (~(var @ X7) | (X4 @ X7))) => ! [X30 : subst,X29 : term,X4 : (term > $o)] : (~ ! [X7 : term] : (~ ! [X9 : term] : (~(X4 @ X9) | (X4 @ (sub @ X7 @ (push @ X9 @ id)))) | (X4 @ (lam @ X7))) | ~ ! [X9 : term] : ((X4 @ (sub @ X9 @ X30)) | ~(var @ X9)) | (X4 @ (sub @ X29 @ X30)) | ~ ! [X9 : term,X7 : term] : ((X4 @ (ap @ X7 @ X9)) | ~(X4 @ X9) | ~(X4 @ X7))))))) <=> induction2lem_lthm) & hoasapnotvar_gthm & (shinj <=> ! [X9 : term,X7 : term] : ((((sub @ X7 @ sh)) != ((sub @ X9 @ sh))) | (X7 = X9))) & induction2lem_gthm & (! [X4 : (subst > term > subst > $o),X5 : (term > $o),X21 : term] : (~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id))) | (X5 @ (lam @ X21)) | ~ ! [X9 : term] : ((X5 @ (sub @ X21 @ (push @ X9 @ id))) | ~(X5 @ X9)) | ~ ! [X10 : (subst > term > term)] : (~ ! [X14 : subst,X7 : term,X8 : subst] : (((sub @ (X10 @ X8 @ X7) @ X14)) = ((X10 @ (comp @ X8 @ X14) @ (sub @ X7 @ X14)))) | ~ ! [X7 : term] : (~(X4 @ id @ X7 @ id) | (X4 @ id @ (X10 @ id @ X7) @ id)) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id))) <=> hoasinduction_lem3v2a) & (! [X9 : term] : ~ ! [X10 : (subst > term > term)] : ~ ! [X7 : term,X8 : subst] : (((sub @ X9 @ (push @ X7 @ X8))) = ((X10 @ X8 @ X7))) <=> hoasinduction_lem3v2_f) & pushprop_lthm & pushprop_lthm_orig & hoaslamnotap_gthm & hoaslamnotvar_lthm & (! [X9 : term] : ~ ! [X10 : (subst > term > term)] : (((X10 @ sh @ one)) != ((sub @ X9 @ (push @ one @ sh)))) <=> hoasinduction_lem3b_lthm) & hoasapnotvar_lthm & (termmset_lthm <=> (! [X7 : term] : (X7 = ((sub @ X7 @ id))) => ! [X7 : term] : (X7 = ((sub @ X7 @ id))))) & ((! [X8 : subst] : (X8 = ((comp @ X8 @ id))) & ! [X8 : subst] : (X8 = ((comp @ id @ X8)))) <=> substmonoid) & (pushprop_p_and_p_prime = (^[X7 : term, X8 : subst, X4 : (term > $o), X5 : (term > $o)] : (! [X6 : term] : (((X5 @ X6)) = ((X4 @ (sub @ X6 @ (push @ X7 @ X8)))))))) & (! [X8 : subst,X7 : term,X9 : term,X4 : (term > $o),X17 : (term > $o)] : ((X17 @ (sub @ X7 @ (push @ X9 @ X8))) | ~(X4 @ X7)) <=> pushprop_lem1) & ((! [X7 : term] : (X7 = ((sub @ X7 @ id))) => (! [X9 : term,X12 : term,X7 : term,X11 : term] : ((((ap @ X7 @ X11)) != ((ap @ X9 @ X12))) | (X7 = X9)) => ! [X9 : term,X11 : term,X7 : term,X12 : term] : ((X7 = X9) | (((ap @ (sub @ X7 @ id) @ X11)) != ((ap @ (sub @ X9 @ id) @ X12)))))) <=> hoasapinj1_lthm) & (! [X7 : term,X12 : term,X11 : term,X9 : term] : ((((ap @ X7 @ X11)) != ((ap @ X9 @ X12))) | (X7 = X9)) <=> apinj1) & (pushprop_lem1v2_lthm <=> (! [X7 : term,X8 : subst] : (X7 = ((sub @ one @ (push @ X7 @ X8)))) => ! [X5 : (term > $o),X7 : term,X8 : subst,X4 : (term > $o)] : ((X5 @ one) | ~(X4 @ X7) | ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ (sub @ X6 @ (push @ X7 @ X8)))))))) & (axshiftcons <=> ! [X8 : subst,X7 : term] : (X8 = ((comp @ sh @ (push @ X7 @ X8))))) & pushprop_lem2v2 & hoaslamnotap_lthm & ! [X1 : d_subst] : (X1 = d_id) & ! [X45 : subst] : ? [X1 : d_subst] : (X45 = ((d2subst @ X1))) & (! [X4 : (term > $o),X34 : term] : ((X4 @ X34) | ~ ! [X7 : term] : ((X4 @ X7) | ~(var @ X7)) | ~ ! [X9 : term,X7 : term] : (~(X4 @ X7) | (X4 @ (ap @ X7 @ X9)) | ~(X4 @ X9)) | ~ ! [X7 : term] : (~(X4 @ X7) | (X4 @ (lam @ X7)))) <=> induction) & (hoasinduction_p_and_p_prime = (^[X4 : (subst > term > subst > $o), X5 : (term > $o)] : (! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id))))))),
  file('/export/starexec/sandbox/benchmark/theBenchmark.p',alg444_1)).
thf(f2,conjecture,(
  (hoasinduction_lem0 = ! [X0 : (subst > term > subst > $o)] : ? [X1 : (term > $o)] : (hoasinduction_p_and_p_prime @ X0 @ X1))),
  file('/export/starexec/sandbox/benchmark/theBenchmark.p',hoasinduction_lem0)).
thf(f3,negated_conjecture,(
  ~ (hoasinduction_lem0 = ! [X0 : (subst > term > subst > $o)] : ? [X1 : (term > $o)] : (hoasinduction_p_and_p_prime @ X0 @ X1))),
  inference(negated_conjecture,[status(cth)],[f2])).
thf(f4,plain,(
  (hoaslamnotap <=> ! [X0 : (subst > term > term),X1 : term,X2 : term] : (~ ! [X3 : subst,X4 : term,X5 : subst] : (((X0 @ (comp @ X3 @ X5) @ (sub @ X4 @ X5))) = ((sub @ (X0 @ X3 @ X4) @ X5))) | (((ap @ (sub @ X2 @ id) @ X1)) != ((lam @ (X0 @ sh @ one)))))) & (! [X6 : (subst > term > subst > $o),X7 : term] : ((X6 @ id @ (lam @ X7) @ id) | ~ ! [X8 : subst,X9 : subst,X10 : term,X11 : subst] : ((X6 @ (comp @ X8 @ X9) @ (sub @ X10 @ X9) @ X11) | ~(X6 @ X8 @ X10 @ (comp @ X9 @ X11))) | ~ ! [X12 : term] : ((X6 @ id @ (sub @ X7 @ (push @ X12 @ id)) @ id) | ~(X6 @ id @ X12 @ id)) | ~ ! [X13 : subst,X14 : term,X15 : subst,X16 : subst] : (~(X6 @ (comp @ X15 @ X13) @ (sub @ X14 @ X13) @ X16) | (X6 @ X15 @ X14 @ (comp @ X13 @ X16))) | ~ ! [X17 : (subst > term > term)] : (~ ! [X18 : term] : (~(X6 @ id @ X18 @ id) | (X6 @ id @ (X17 @ id @ X18) @ id)) | (X6 @ id @ (lam @ (X17 @ sh @ one)) @ id) | ~ ! [X19 : term,X20 : subst,X21 : subst] : (((X17 @ (comp @ X21 @ X20) @ (sub @ X19 @ X20))) = ((sub @ (X17 @ X21 @ X19) @ X20))))) <=> hoasinduction_lem3) & ~(var @ (d2term @ d_one)) & ~(hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) & (hoasinduction_lem3v2 <=> ! [X22 : (term > $o),X23 : (subst > term > subst > $o),X24 : term] : (~ ! [X25 : term,X26 : subst,X27 : subst,X28 : subst] : (~(X23 @ X28 @ X25 @ (comp @ X27 @ X26)) | (X23 @ (comp @ X28 @ X27) @ (sub @ X25 @ X27) @ X26)) | ~ ! [X29 : (subst > term > term)] : ((X23 @ id @ (lam @ (X29 @ sh @ one)) @ id) | ~ ! [X30 : term] : ((X23 @ id @ (X29 @ id @ X30) @ id) | ~(X23 @ id @ X30 @ id)) | ~ ! [X31 : subst,X32 : subst,X33 : term] : (((X29 @ (comp @ X31 @ X32) @ (sub @ X33 @ X32))) = ((sub @ (X29 @ X31 @ X33) @ X32)))) | ~ ! [X34 : subst,X35 : subst,X36 : term,X37 : subst] : (~(X23 @ (comp @ X37 @ X35) @ (sub @ X36 @ X35) @ X34) | (X23 @ X37 @ X36 @ (comp @ X35 @ X34))) | ~ ! [X38 : term] : (((X23 @ id @ X38 @ id)) = ((X22 @ X38))) | (X22 @ (lam @ X24)) | ~ ! [X39 : term] : (~(X22 @ X39) | (X22 @ (sub @ X24 @ (push @ X39 @ id)))))) & (! [X40 : subst,X41 : term,X42 : (term > $o),X43 : (term > $o)] : (~ ! [X44 : term] : (((X43 @ (sub @ X44 @ (push @ X41 @ X40)))) = ((X42 @ X44))) | (X42 @ one) | ~(X43 @ X41)) <=> pushprop_lem1v2) & pushprop & (axidl <=> ! [X45 : subst] : (((comp @ id @ X45)) = X45)) & (! [X46 : term,X47 : term,X48 : (subst > term > subst > $o)] : (~(X48 @ id @ X47 @ id) | ~ ! [X49 : subst,X50 : term,X51 : subst,X52 : subst] : (~(X48 @ (comp @ X49 @ X52) @ (sub @ X50 @ X52) @ X51) | (X48 @ X49 @ X50 @ (comp @ X52 @ X51))) | ~ ! [X53 : term,X54 : term] : (~(X48 @ id @ X54 @ id) | ~(X48 @ id @ X53 @ id) | (X48 @ id @ (ap @ (sub @ X53 @ id) @ X54) @ id)) | ~ ! [X55 : subst,X56 : term,X57 : subst,X58 : subst] : (~(X48 @ X55 @ X56 @ (comp @ X57 @ X58)) | (X48 @ (comp @ X55 @ X57) @ (sub @ X56 @ X57) @ X58)) | (X48 @ id @ (ap @ X46 @ X47) @ id) | ~(X48 @ id @ X46 @ id)) <=> hoasinduction_lem2) & (! [X59 : term,X60 : (subst > term > subst > $o)] : (~ ! [X61 : term] : (~(X60 @ id @ X61 @ id) | (X60 @ id @ (sub @ X59 @ (push @ X61 @ id)) @ id)) | ~ ! [X62 : (subst > term > term)] : (~ ! [X63 : subst,X64 : term,X65 : subst] : (((sub @ (X62 @ X65 @ X64) @ X63)) = ((X62 @ (comp @ X65 @ X63) @ (sub @ X64 @ X63)))) | (X60 @ id @ (lam @ (X62 @ sh @ one)) @ id) | ~ ! [X66 : term] : ((X60 @ id @ (X62 @ id @ X66) @ id) | ~(X60 @ id @ X66 @ id))) | (X60 @ id @ (lam @ (sub @ X59 @ (push @ one @ sh))) @ id)) <=> hoasinduction_lem3aa) & ~lamnotap & (! [X67 : term,X68 : term,X69 : term,X70 : term] : ((X67 = X68) | (((ap @ (sub @ X67 @ id) @ X70)) != ((ap @ (sub @ X68 @ id) @ X69)))) <=> hoasapinj1) & axmap & hoasinduction_lem1_gthm & (apinj2 <=> ! [X71 : term,X72 : term,X73 : term,X74 : term] : ((X71 = X72) | (((ap @ X73 @ X72)) != ((ap @ X74 @ X71))))) & substmonoid_gthm & (induction2lem <=> ! [X75 : subst,X76 : (term > $o),X77 : term] : ((X76 @ (sub @ X77 @ X75)) | ~ ! [X78 : term] : ((X76 @ (sub @ X78 @ X75)) | ~(var @ X78)) | ~ ! [X79 : term,X80 : term] : (~(X76 @ X80) | ~(X76 @ X79) | (X76 @ (ap @ X80 @ X79))) | ~ ! [X81 : term] : ((X76 @ (lam @ X81)) | ~ ! [X82 : term] : (~(X76 @ X82) | (X76 @ (sub @ X81 @ (push @ X82 @ id))))))) & (one = ((d2term @ d_one))) & (hoasinduction_lem3v2_f_lthm <=> ! [X83 : term] : ~ ! [X84 : (subst > term > term)] : ~ ! [X85 : term,X86 : subst] : (((X84 @ X86 @ X85)) = ((sub @ X83 @ (push @ X85 @ X86))))) & (! [X87 : subst] : (((push @ (sub @ one @ X87) @ (comp @ sh @ X87))) = X87) <=> axscons) & hoasinduction_gthm & (! [X88 : term,X89 : (subst > term > subst > $o)] : ((X89 @ id @ (lam @ (sub @ X88 @ (push @ one @ sh))) @ id) | ~ ! [X90 : (subst > term > term)] : (~ ! [X91 : term] : ((X89 @ id @ (X90 @ id @ X91) @ id) | ~(X89 @ id @ X91 @ id)) | ~ ! [X92 : term,X93 : subst,X94 : subst] : (((sub @ (X90 @ X93 @ X92) @ X94)) = ((X90 @ (comp @ X93 @ X94) @ (sub @ X92 @ X94)))) | (X89 @ id @ (lam @ (X90 @ sh @ one)) @ id)) | ~ ! [X95 : term] : (~(X89 @ id @ X95 @ id) | (X89 @ id @ (sub @ X88 @ (push @ X95 @ id)) @ id))) <=> hoasinduction_lem3aa_lthm) & axabs & hoasinduction_lem2_gthm & hoasinduction_lem3b_gthm & ! [X96 : d_term] : (d_one = X96) & termmset_gthm & apnotvar & hoasapinj2_gthm & (pushprop_lem0 <=> ! [X97 : term,X98 : subst,X99 : (term > $o)] : ~ ! [X100 : (term > $o)] : ~ ! [X101 : term] : (((X99 @ (sub @ X101 @ (push @ X97 @ X98)))) = ((X100 @ X101)))) & (! [X102 : term,X103 : term,X104 : term,X105 : term] : ((((ap @ (sub @ X104 @ id) @ X102)) != ((ap @ (sub @ X103 @ id) @ X105))) | (X102 = X105)) <=> hoasapinj2) & (! [X106 : term] : (((sub @ X106 @ id)) = X106) <=> axvarid) & (! [X107 : term,X108 : term,X109 : (term > $o),X110 : (subst > term > subst > $o)] : ((X109 @ (ap @ X107 @ X108)) | ~ ! [X111 : term,X112 : subst,X113 : subst,X114 : subst] : ((X110 @ X112 @ X111 @ (comp @ X114 @ X113)) | ~(X110 @ (comp @ X112 @ X114) @ (sub @ X111 @ X114) @ X113)) | ~ ! [X115 : subst,X116 : subst,X117 : term,X118 : subst] : (~(X110 @ X115 @ X117 @ (comp @ X116 @ X118)) | (X110 @ (comp @ X115 @ X116) @ (sub @ X117 @ X116) @ X118)) | ~(X109 @ X108) | ~(X109 @ X107) | ~ ! [X119 : term] : (((X109 @ X119)) = ((X110 @ id @ X119 @ id))) | ~ ! [X120 : term,X121 : term] : ((X110 @ id @ (ap @ (sub @ X120 @ id) @ X121) @ id) | ~(X110 @ id @ X120 @ id) | ~(X110 @ id @ X121 @ id))) <=> hoasinduction_lem2v2) & (hoasinduction_lem3v2_lthm <=> (! [X122 : term] : (((sub @ X122 @ id)) = X122) => ! [X123 : term,X124 : (term > $o),X125 : (subst > term > subst > $o)] : (~ ! [X126 : term] : (((X124 @ X126)) = ((X125 @ id @ X126 @ id))) | ~ ! [X127 : term,X128 : subst,X129 : subst,X130 : subst] : (~(X125 @ (comp @ X129 @ X130) @ (sub @ X127 @ X130) @ X128) | (X125 @ X129 @ X127 @ (comp @ X130 @ X128))) | ~ ! [X131 : (subst > term > term)] : (~ ! [X132 : term] : ((X125 @ id @ (X131 @ id @ X132) @ id) | ~(X125 @ id @ X132 @ id)) | ~ ! [X133 : term,X134 : subst,X135 : subst] : (((X131 @ (comp @ X134 @ X135) @ (sub @ X133 @ X135))) = ((sub @ (X131 @ X134 @ X133) @ X135))) | (X125 @ id @ (lam @ (X131 @ sh @ one)) @ id)) | ~ ! [X136 : subst,X137 : term,X138 : subst,X139 : subst] : ((X125 @ (comp @ X139 @ X138) @ (sub @ X137 @ X138) @ X136) | ~(X125 @ X139 @ X137 @ (comp @ X138 @ X136))) | (X124 @ (lam @ X123)) | ~ ! [X140 : term] : ((X124 @ (sub @ X123 @ (push @ X140 @ id))) | ~(X124 @ X140))))) & pushprop_lem0_gthm & hoaslaminj_gthm & (((comp @ (d2subst @ d_id) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & (! [X141 : (subst > term > subst > $o)] : ~ ! [X142 : (term > $o)] : ~ ! [X143 : term] : (((X141 @ id @ X143 @ id)) = ((X142 @ X143))) <=> hoasinduction_lem0_lthm) & ulamvarind & pushprop_gthm & hoaslamnotvar & (hoaslaminj_lthm <=> (! [X144 : term,X145 : subst] : (((sub @ one @ (push @ X144 @ X145))) = X144) => (! [X146 : term,X147 : subst] : (((comp @ sh @ (push @ X146 @ X147))) = X147) => (! [X148 : term,X149 : term] : ((X148 = X149) | (((lam @ X149)) != ((lam @ X148)))) => ! [X150 : subst,X151 : (subst > term > term),X152 : term,X153 : (subst > term > term)] : ((((X151 @ X150 @ X152)) = ((X153 @ X150 @ X152))) | ~ ! [X154 : term,X155 : subst,X156 : subst] : (((sub @ (X153 @ X155 @ X154) @ X156)) = ((X153 @ (comp @ X155 @ X156) @ (sub @ X154 @ X156)))) | ~ ! [X157 : subst,X158 : subst,X159 : term] : (((X151 @ (comp @ X157 @ X158) @ (sub @ X159 @ X158))) = ((sub @ (X151 @ X157 @ X159) @ X158))) | (((lam @ (X151 @ sh @ one))) != ((lam @ (X153 @ sh @ one))))))))) & (induction2_lthm <=> (! [X160 : term] : (((sub @ X160 @ id)) = X160) => (! [X161 : term,X162 : (term > $o),X163 : subst] : (~ ! [X164 : term,X165 : term] : (~(X162 @ X165) | (X162 @ (ap @ X164 @ X165)) | ~(X162 @ X164)) | ~ ! [X166 : term] : ((X162 @ (lam @ X166)) | ~ ! [X167 : term] : ((X162 @ (sub @ X166 @ (push @ X167 @ id))) | ~(X162 @ X167))) | (X162 @ (sub @ X161 @ X163)) | ~ ! [X168 : term] : ((X162 @ (sub @ X168 @ X163)) | ~(var @ X168))) => ! [X169 : term,X170 : (term > $o)] : ((X170 @ X169) | ~ ! [X171 : term] : (~ ! [X172 : term] : ((X170 @ (sub @ X171 @ (push @ X172 @ id))) | ~(X170 @ X172)) | (X170 @ (lam @ X171))) | ~ ! [X173 : term,X174 : term] : (~(X170 @ X173) | ~(X170 @ X174) | (X170 @ (ap @ X174 @ X173))) | ~ ! [X175 : term] : (~(var @ X175) | (X170 @ X175)))))) & (hoaslaminj <=> ! [X176 : subst,X177 : term,X178 : (subst > term > term),X179 : (subst > term > term)] : ((((lam @ (X178 @ sh @ one))) != ((lam @ (X179 @ sh @ one)))) | ~ ! [X180 : term,X181 : subst,X182 : subst] : (((X178 @ (comp @ X182 @ X181) @ (sub @ X180 @ X181))) = ((sub @ (X178 @ X182 @ X180) @ X181))) | ~ ! [X183 : subst,X184 : subst,X185 : term] : (((X179 @ (comp @ X184 @ X183) @ (sub @ X185 @ X183))) = ((sub @ (X179 @ X184 @ X185) @ X183))) | (((X179 @ X176 @ X177)) = ((X178 @ X176 @ X177))))) & ((! [X186 : (subst > term > subst > $o)] : ~ ! [X187 : (term > $o)] : ~ ! [X188 : term] : (((X186 @ id @ X188 @ id)) = ((X187 @ X188))) => (! [X189 : (term > $o),X190 : term] : ((X189 @ X190) | ~ ! [X191 : term,X192 : term] : ((X189 @ (ap @ X191 @ X192)) | ~(X189 @ X192) | ~(X189 @ X191)) | ~ ! [X193 : term] : ((X189 @ (lam @ X193)) | ~ ! [X194 : term] : (~(X189 @ X194) | (X189 @ (sub @ X193 @ (push @ X194 @ id))))) | ~ ! [X195 : term] : (~(var @ X195) | (X189 @ X195))) => (! [X196 : term] : (((sub @ X196 @ id)) = X196) => (! [X197 : (term > $o),X198 : (subst > term > subst > $o),X199 : term] : (~ ! [X200 : (subst > term > term)] : (~ ! [X201 : term,X202 : subst,X203 : subst] : (((X200 @ (comp @ X203 @ X202) @ (sub @ X201 @ X202))) = ((sub @ (X200 @ X203 @ X201) @ X202))) | ~ ! [X204 : term] : ((X198 @ id @ (X200 @ id @ X204) @ id) | ~(X198 @ id @ X204 @ id)) | (X198 @ id @ (lam @ (X200 @ sh @ one)) @ id)) | (X197 @ (lam @ X199)) | ~ ! [X205 : term] : (~(X197 @ X205) | (X197 @ (sub @ X199 @ (push @ X205 @ id)))) | ~ ! [X206 : term] : (((X198 @ id @ X206 @ id)) = ((X197 @ X206)))) => ! [X207 : term,X208 : (subst > term > subst > $o)] : (~ ! [X209 : term] : (~(var @ (sub @ X209 @ id)) | (X208 @ id @ X209 @ id)) | ~ ! [X210 : term,X211 : term] : (~(X208 @ id @ X210 @ id) | (X208 @ id @ (ap @ (sub @ X211 @ id) @ X210) @ id) | ~(X208 @ id @ X211 @ id)) | ~ ! [X212 : subst,X213 : subst,X214 : term,X215 : subst] : (~(X208 @ X215 @ X214 @ (comp @ X212 @ X213)) | (X208 @ (comp @ X215 @ X212) @ (sub @ X214 @ X212) @ X213)) | ~ ! [X216 : (subst > term > term)] : (~ ! [X217 : subst,X218 : term,X219 : subst] : (((X216 @ (comp @ X217 @ X219) @ (sub @ X218 @ X219))) = ((sub @ (X216 @ X217 @ X218) @ X219))) | (X208 @ id @ (lam @ (X216 @ sh @ one)) @ id) | ~ ! [X220 : term] : ((X208 @ id @ (X216 @ id @ X220) @ id) | ~(X208 @ id @ X220 @ id))) | (X208 @ id @ X207 @ id) | ~ ! [X221 : subst,X222 : term,X223 : subst,X224 : subst] : (~(X208 @ (comp @ X221 @ X224) @ (sub @ X222 @ X224) @ X223) | (X208 @ X221 @ X222 @ (comp @ X224 @ X223)))))))) <=> hoasinduction_lthm_3) & (! [X225 : (subst > term > subst > $o),X226 : term] : (~ ! [X227 : term,X228 : term] : (~(X225 @ id @ X228 @ id) | (X225 @ id @ (ap @ (sub @ X227 @ id) @ X228) @ id) | ~(X225 @ id @ X227 @ id)) | ~ ! [X229 : (subst > term > term)] : ((X225 @ id @ (lam @ (X229 @ sh @ one)) @ id) | ~ ! [X230 : term] : (~(X225 @ id @ X230 @ id) | (X225 @ id @ (X229 @ id @ X230) @ id)) | ~ ! [X231 : subst,X232 : term,X233 : subst] : (((sub @ (X229 @ X231 @ X232) @ X233)) = ((X229 @ (comp @ X231 @ X233) @ (sub @ X232 @ X233))))) | (X225 @ id @ X226 @ id)) <=> hoasinduction_no_psi_cond) & (! [X234 : term,X235 : subst] : (((sub @ one @ (push @ X234 @ X235))) = X234) <=> axvarcons) & pushprop_lem2v2_gthm & (hoasinduction_lem3a_lthm <=> (! [X236 : term] : (((sub @ X236 @ id)) = X236) => (! [X237 : (subst > term > subst > $o),X238 : term] : (~ ! [X239 : (subst > term > term)] : ((X237 @ id @ (lam @ (X239 @ sh @ one)) @ id) | ~ ! [X240 : subst,X241 : subst,X242 : term] : (((sub @ (X239 @ X241 @ X242) @ X240)) = ((X239 @ (comp @ X241 @ X240) @ (sub @ X242 @ X240)))) | ~ ! [X243 : term] : (~(X237 @ id @ X243 @ id) | (X237 @ id @ (X239 @ id @ X243) @ id))) | (X237 @ id @ (lam @ (sub @ X238 @ (push @ one @ sh))) @ id) | ~ ! [X244 : term] : ((X237 @ id @ (sub @ X238 @ (push @ X244 @ id)) @ id) | ~(X237 @ id @ X244 @ id))) => ! [X245 : (subst > term > subst > $o),X246 : term] : ((X245 @ id @ (lam @ X246) @ id) | ~ ! [X247 : term] : (~(X245 @ id @ X247 @ id) | (X245 @ id @ (sub @ X246 @ (push @ X247 @ id)) @ id)) | ~ ! [X248 : (subst > term > term)] : ((X245 @ id @ (lam @ (X248 @ sh @ one)) @ id) | ~ ! [X249 : subst,X250 : subst,X251 : term] : (((X248 @ (comp @ X250 @ X249) @ (sub @ X251 @ X249))) = ((sub @ (X248 @ X250 @ X251) @ X249))) | ~ ! [X252 : term] : (~(X245 @ id @ X252 @ id) | (X245 @ id @ (X248 @ id @ X252) @ id))))))) & (((push @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & (hoasinduction_lem3b <=> ! [X253 : term] : ~ ! [X254 : (subst > term > term)] : (((X254 @ sh @ one)) != ((sub @ X253 @ (push @ one @ sh))))) & (pushprop_lem0_lthm <=> ! [X255 : subst,X256 : (term > $o),X257 : term] : ~ ! [X258 : (term > $o)] : ~ ! [X259 : term] : (((X258 @ X259)) = ((X256 @ (sub @ X259 @ (push @ X257 @ X255)))))) & axapp & (sh = ((d2subst @ d_id))) & (hoaslam = (^[X260 : subst, X261 : (subst > term > term)] : ((d2term @ d_one)))) & axassoc & (((sub @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2term @ d_one))) & (hoasinduction_lthm <=> (! [X262 : (term > $o),X263 : term] : ((X262 @ X263) | ~ ! [X264 : term] : ((X262 @ (lam @ X264)) | ~ ! [X265 : term] : (~(X262 @ X265) | (X262 @ (sub @ X264 @ (push @ X265 @ id))))) | ~ ! [X266 : term] : (~(var @ X266) | (X262 @ X266)) | ~ ! [X267 : term,X268 : term] : (~(X262 @ X267) | (X262 @ (ap @ X267 @ X268)) | ~(X262 @ X268))) => (! [X269 : term,X270 : (subst > term > subst > $o),X271 : term] : (~(X270 @ id @ X269 @ id) | ~(X270 @ id @ X271 @ id) | ~ ! [X272 : subst,X273 : term,X274 : subst,X275 : subst] : (~(X270 @ (comp @ X272 @ X275) @ (sub @ X273 @ X275) @ X274) | (X270 @ X272 @ X273 @ (comp @ X275 @ X274))) | ~ ! [X276 : subst,X277 : subst,X278 : subst,X279 : term] : (~(X270 @ X277 @ X279 @ (comp @ X278 @ X276)) | (X270 @ (comp @ X277 @ X278) @ (sub @ X279 @ X278) @ X276)) | (X270 @ id @ (ap @ X271 @ X269) @ id) | ~ ! [X280 : term,X281 : term] : (~(X270 @ id @ X281 @ id) | (X270 @ id @ (ap @ (sub @ X280 @ id) @ X281) @ id) | ~(X270 @ id @ X280 @ id))) => (! [X282 : term,X283 : (subst > term > subst > $o)] : ((X283 @ id @ (lam @ X282) @ id) | ~ ! [X284 : subst,X285 : subst,X286 : subst,X287 : term] : (~(X283 @ X284 @ X287 @ (comp @ X285 @ X286)) | (X283 @ (comp @ X284 @ X285) @ (sub @ X287 @ X285) @ X286)) | ~ ! [X288 : term] : (~(X283 @ id @ X288 @ id) | (X283 @ id @ (sub @ X282 @ (push @ X288 @ id)) @ id)) | ~ ! [X289 : subst,X290 : subst,X291 : subst,X292 : term] : (~(X283 @ (comp @ X290 @ X291) @ (sub @ X292 @ X291) @ X289) | (X283 @ X290 @ X292 @ (comp @ X291 @ X289))) | ~ ! [X293 : (subst > term > term)] : ((X283 @ id @ (lam @ (X293 @ sh @ one)) @ id) | ~ ! [X294 : subst,X295 : term,X296 : subst] : (((X293 @ (comp @ X296 @ X294) @ (sub @ X295 @ X294))) = ((sub @ (X293 @ X296 @ X295) @ X294))) | ~ ! [X297 : term] : ((X283 @ id @ (X293 @ id @ X297) @ id) | ~(X283 @ id @ X297 @ id)))) => ! [X298 : term,X299 : (subst > term > subst > $o)] : (~ ! [X300 : subst,X301 : subst,X302 : subst,X303 : term] : ((X299 @ X300 @ X303 @ (comp @ X301 @ X302)) | ~(X299 @ (comp @ X300 @ X301) @ (sub @ X303 @ X301) @ X302)) | ~ ! [X304 : (subst > term > term)] : ((X299 @ id @ (lam @ (X304 @ sh @ one)) @ id) | ~ ! [X305 : term] : ((X299 @ id @ (X304 @ id @ X305) @ id) | ~(X299 @ id @ X305 @ id)) | ~ ! [X306 : term,X307 : subst,X308 : subst] : (((X304 @ (comp @ X308 @ X307) @ (sub @ X306 @ X307))) = ((sub @ (X304 @ X308 @ X306) @ X307)))) | ~ ! [X309 : subst,X310 : subst,X311 : subst,X312 : term] : (~(X299 @ X309 @ X312 @ (comp @ X310 @ X311)) | (X299 @ (comp @ X309 @ X310) @ (sub @ X312 @ X310) @ X311)) | ~ ! [X313 : term] : ((X299 @ id @ X313 @ id) | ~(var @ (sub @ X313 @ id))) | ~ ! [X314 : term,X315 : term] : (~(X299 @ id @ X314 @ id) | (X299 @ id @ (ap @ (sub @ X314 @ id) @ X315) @ id) | ~(X299 @ id @ X315 @ id)) | (X299 @ id @ X298 @ id)))))) & (((lam @ (d2term @ d_one))) = ((d2term @ d_one))) & (hoasinduction_lem3v2a_lthm <=> (! [X316 : term] : ~ ! [X317 : (subst > term > term)] : ~ ! [X318 : subst,X319 : term] : (((sub @ X316 @ (push @ X319 @ X318))) = ((X317 @ X318 @ X319))) => (! [X320 : term] : (((sub @ X320 @ id)) = X320) => ! [X321 : (subst > term > subst > $o),X322 : (term > $o),X323 : term] : (~ ! [X324 : (subst > term > term)] : (~ ! [X325 : subst,X326 : term,X327 : subst] : (((X324 @ (comp @ X325 @ X327) @ (sub @ X326 @ X327))) = ((sub @ (X324 @ X325 @ X326) @ X327))) | ~ ! [X328 : term] : (~(X321 @ id @ X328 @ id) | (X321 @ id @ (X324 @ id @ X328) @ id)) | (X321 @ id @ (lam @ (X324 @ sh @ one)) @ id)) | (X322 @ (lam @ X323)) | ~ ! [X329 : term] : (((X322 @ X329)) = ((X321 @ id @ X329 @ id))) | ~ ! [X330 : term] : (~(X322 @ X330) | (X322 @ (sub @ X323 @ (push @ X330 @ id)))))))) & pushprop_lem1_gthm & pushprop_lem3v2_lthm & hoasinduction_lem1 & pushprop_lem3v2 & ~ulamvar1 & hoasinduction_lem1v2 & (substmonoid_lthm <=> (! [X331 : subst] : (((comp @ id @ X331)) = X331) => (! [X332 : subst] : (((comp @ X332 @ id)) = X332) => (! [X333 : subst] : (((comp @ X333 @ id)) = X333) & ! [X334 : subst] : (((comp @ id @ X334)) = X334))))) & induction2_gthm & (hoasinduction_lem3_lthm <=> (! [X335 : term] : (((sub @ X335 @ id)) = X335) => (! [X336 : term,X337 : (subst > term > subst > $o)] : (~ ! [X338 : term] : (~(X337 @ id @ X338 @ id) | (X337 @ id @ (sub @ X336 @ (push @ X338 @ id)) @ id)) | ~ ! [X339 : (subst > term > term)] : (~ ! [X340 : term] : ((X337 @ id @ (X339 @ id @ X340) @ id) | ~(X337 @ id @ X340 @ id)) | (X337 @ id @ (lam @ (X339 @ sh @ one)) @ id) | ~ ! [X341 : term,X342 : subst,X343 : subst] : (((X339 @ (comp @ X343 @ X342) @ (sub @ X341 @ X342))) = ((sub @ (X339 @ X343 @ X341) @ X342)))) | (X337 @ id @ (lam @ (sub @ X336 @ (push @ one @ sh))) @ id)) => ! [X344 : (subst > term > subst > $o),X345 : term] : (~ ! [X346 : term] : (~(X344 @ id @ X346 @ id) | (X344 @ id @ (sub @ X345 @ (push @ X346 @ id)) @ id)) | (X344 @ id @ (lam @ X345) @ id) | ~ ! [X347 : subst,X348 : term,X349 : subst,X350 : subst] : ((X344 @ X347 @ X348 @ (comp @ X349 @ X350)) | ~(X344 @ (comp @ X347 @ X349) @ (sub @ X348 @ X349) @ X350)) | ~ ! [X351 : (subst > term > term)] : (~ ! [X352 : term] : ((X344 @ id @ (X351 @ id @ X352) @ id) | ~(X344 @ id @ X352 @ id)) | (X344 @ id @ (lam @ (X351 @ sh @ one)) @ id) | ~ ! [X353 : term,X354 : subst,X355 : subst] : (((sub @ (X351 @ X355 @ X353) @ X354)) = ((X351 @ (comp @ X355 @ X354) @ (sub @ X353 @ X354))))) | ~ ! [X356 : subst,X357 : subst,X358 : subst,X359 : term] : ((X344 @ (comp @ X358 @ X357) @ (sub @ X359 @ X357) @ X356) | ~(X344 @ X358 @ X359 @ (comp @ X357 @ X356))))))) & pushprop_lem1v2_gthm & lamnotvar & ! [X360 : d_term,X361 : d_term] : ((((d2term @ X361)) = ((d2term @ X360))) => (X360 = X361)) & axclos & (id = ((d2subst @ d_id))) & hoasinduction_lem2v2_gthm & (! [X362 : term] : (((sub @ X362 @ id)) = X362) <=> termmset) & pushprop_lem2v2_lthm & ulamvarsh & hoasapinj1_gthm & ((! [X363 : term,X364 : subst] : (((sub @ one @ (push @ X363 @ X364))) = X363) => (! [X365 : term,X366 : subst] : (((comp @ sh @ (push @ X365 @ X366))) = X366) => ! [X367 : (term > $o),X368 : subst,X369 : (term > $o),X370 : term,X371 : term] : (~(X367 @ X371) | (X369 @ (sub @ X371 @ (push @ X370 @ X368)))))) <=> pushprop_lem1_lthm) & hoasinduction_lem3v2_gthm & hoasinduction_lem2_lthm & (((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) = ((d2term @ d_one))) & hoasinduction_lem1_lthm & (hoasinduction_lem3aaa <=> ! [X372 : term,X373 : (subst > term > subst > $o)] : ((X373 @ id @ (lam @ (sub @ X372 @ (push @ one @ sh))) @ id) | ~ ! [X374 : (subst > term > term),X375 : term] : (~ ! [X376 : term] : ((X373 @ id @ (X374 @ id @ X376) @ id) | ~(X373 @ id @ X376 @ id)) | ~ ! [X377 : term,X378 : subst,X379 : subst] : (((sub @ (sub @ X375 @ (push @ X377 @ X379)) @ X378)) = ((sub @ (X374 @ X379 @ X377) @ X378))) | (X373 @ id @ (lam @ (X374 @ sh @ one)) @ id) | ~ ! [X380 : subst,X381 : subst,X382 : term] : (((sub @ X375 @ (push @ (sub @ X382 @ X381) @ (comp @ X380 @ X381)))) = ((X374 @ (comp @ X380 @ X381) @ (sub @ X382 @ X381))))) | ~ ! [X383 : term] : (~(X373 @ id @ X383 @ id) | (X373 @ id @ (sub @ X372 @ (push @ X383 @ id)) @ id)))) & ((! [X384 : (subst > term > subst > $o)] : ~ ! [X385 : (term > $o)] : ~ ! [X386 : term] : (((X384 @ id @ X386 @ id)) = ((X385 @ X386))) => (! [X387 : (term > $o),X388 : term] : ((X387 @ X388) | ~ ! [X389 : term,X390 : term] : ((X387 @ (ap @ X390 @ X389)) | ~(X387 @ X389) | ~(X387 @ X390)) | ~ ! [X391 : term] : (~ ! [X392 : term] : ((X387 @ (sub @ X391 @ (push @ X392 @ id))) | ~(X387 @ X392)) | (X387 @ (lam @ X391))) | ~ ! [X393 : term] : ((X387 @ X393) | ~(var @ X393))) => (! [X394 : term] : (((sub @ X394 @ id)) = X394) => (! [X395 : (subst > term > subst > $o),X396 : term,X397 : (term > $o)] : (~ ! [X398 : (subst > term > term)] : ((X395 @ id @ (lam @ (X398 @ sh @ one)) @ id) | ~ ! [X399 : term] : ((X395 @ id @ (X398 @ id @ X399) @ id) | ~(X395 @ id @ X399 @ id)) | ~ ! [X400 : subst,X401 : term,X402 : subst] : (((X398 @ (comp @ X400 @ X402) @ (sub @ X401 @ X402))) = ((sub @ (X398 @ X400 @ X401) @ X402)))) | ~ ! [X403 : term] : ((X397 @ (sub @ X396 @ (push @ X403 @ id))) | ~(X397 @ X403)) | (X397 @ (lam @ X396)) | ~ ! [X404 : term] : (((X395 @ id @ X404 @ id)) = ((X397 @ X404)))) => ! [X405 : (subst > term > subst > $o),X406 : term] : ((X405 @ id @ X406 @ id) | ~ ! [X407 : (subst > term > term)] : (~ ! [X408 : subst,X409 : subst,X410 : term] : (((X407 @ (comp @ X409 @ X408) @ (sub @ X410 @ X408))) = ((sub @ (X407 @ X409 @ X410) @ X408))) | (X405 @ id @ (lam @ (X407 @ sh @ one)) @ id) | ~ ! [X411 : term] : (~(X405 @ id @ X411 @ id) | (X405 @ id @ (X407 @ id @ X411) @ id))) | ~ ! [X412 : term,X413 : term] : (~(X405 @ id @ X413 @ id) | (X405 @ id @ (ap @ (sub @ X413 @ id) @ X412) @ id) | ~(X405 @ id @ X412 @ id))))))) <=> hoasinduction_no_psi_cond_lthm) & hoasinduction_lem3a_gthm & hoasinduction_lem1v2_gthm & ! [X414 : d_subst,X415 : d_subst] : ((((d2subst @ X414)) = ((d2subst @ X415))) => (X414 = X415)) & (! [X416 : term,X417 : (subst > term > subst > $o)] : ((X417 @ id @ (lam @ X416) @ id) | ~ ! [X418 : term] : ((X417 @ id @ (sub @ X416 @ (push @ X418 @ id)) @ id) | ~(X417 @ id @ X418 @ id)) | ~ ! [X419 : (subst > term > term)] : ((X417 @ id @ (lam @ (X419 @ sh @ one)) @ id) | ~ ! [X420 : term,X421 : subst,X422 : subst] : (((X419 @ (comp @ X422 @ X421) @ (sub @ X420 @ X421))) = ((sub @ (X419 @ X422 @ X420) @ X421))) | ~ ! [X423 : term] : (~(X417 @ id @ X423 @ id) | (X417 @ id @ (X419 @ id @ X423) @ id)))) <=> hoasinduction_lem3a) & ! [X424 : term] : ? [X425 : d_term] : (((d2term @ X425)) = X424) & hoasapnotvar & axvarshift & (! [X426 : subst] : (((comp @ X426 @ id)) = X426) <=> axidr) & hoaslamnotvar_gthm & (hoasinduction <=> ! [X427 : term,X428 : (subst > term > subst > $o)] : ((X428 @ id @ X427 @ id) | ~ ! [X429 : term,X430 : term] : (~(X428 @ id @ X430 @ id) | (X428 @ id @ (ap @ (sub @ X430 @ id) @ X429) @ id) | ~(X428 @ id @ X429 @ id)) | ~ ! [X431 : (subst > term > term)] : (~ ! [X432 : term,X433 : subst,X434 : subst] : (((X431 @ (comp @ X434 @ X433) @ (sub @ X432 @ X433))) = ((sub @ (X431 @ X434 @ X432) @ X433))) | (X428 @ id @ (lam @ (X431 @ sh @ one)) @ id) | ~ ! [X435 : term] : (~(X428 @ id @ X435 @ id) | (X428 @ id @ (X431 @ id @ X435) @ id))) | ~ ! [X436 : term,X437 : subst,X438 : subst,X439 : subst] : (~(X428 @ (comp @ X439 @ X438) @ (sub @ X436 @ X438) @ X437) | (X428 @ X439 @ X436 @ (comp @ X438 @ X437))) | ~ ! [X440 : term,X441 : subst,X442 : subst,X443 : subst] : (~(X428 @ X442 @ X440 @ (comp @ X441 @ X443)) | (X428 @ (comp @ X442 @ X441) @ (sub @ X440 @ X441) @ X443)) | ~ ! [X444 : term] : (~(var @ (sub @ X444 @ id)) | (X428 @ id @ X444 @ id)))) & (((ap @ (d2term @ d_one) @ (d2term @ d_one))) = ((d2term @ d_one))) & (laminj <=> ! [X445 : term,X446 : term] : ((X445 = X446) | (((lam @ X445)) != ((lam @ X446))))) & (! [X447 : (subst > term > subst > $o)] : ~ ! [X448 : (term > $o)] : ~ ! [X449 : term] : (((X448 @ X449)) = ((X447 @ id @ X449 @ id))) <=> hoasinduction_lem0) & ((! [X450 : term,X451 : term,X452 : term,X453 : term] : ((X450 = X452) | (((ap @ X451 @ X452)) != ((ap @ X453 @ X450)))) => ! [X454 : term,X455 : term,X456 : term,X457 : term] : ((((ap @ (sub @ X456 @ id) @ X454)) != ((ap @ (sub @ X455 @ id) @ X457))) | (X454 = X457))) <=> hoasapinj2_lthm) & (! [X458 : (term > $o),X459 : term] : ((X458 @ X459) | ~ ! [X460 : term,X461 : term] : ((X458 @ (ap @ X461 @ X460)) | ~(X458 @ X460) | ~(X458 @ X461)) | ~ ! [X462 : term] : (~ ! [X463 : term] : (~(X458 @ X463) | (X458 @ (sub @ X462 @ (push @ X463 @ id)))) | (X458 @ (lam @ X462))) | ~ ! [X464 : term] : ((X458 @ X464) | ~(var @ X464))) <=> induction2) & hoasinduction_lem3_gthm & ((! [X465 : subst,X466 : term] : (((sub @ one @ (push @ X466 @ X465))) = X466) => (! [X467 : term,X468 : subst] : (((comp @ sh @ (push @ X467 @ X468))) = X468) => (! [X469 : subst] : (((comp @ X469 @ id)) = X469) => (! [X470 : (term > $o),X471 : term] : (~ ! [X472 : term,X473 : term] : ((X470 @ (ap @ X473 @ X472)) | ~(X470 @ X473) | ~(X470 @ X472)) | (X470 @ X471) | ~ ! [X474 : term] : (~(X470 @ X474) | (X470 @ (lam @ X474))) | ~ ! [X475 : term] : (~(var @ X475) | (X470 @ X475))) => ! [X476 : subst,X477 : term,X478 : (term > $o)] : (~ ! [X479 : term] : (~ ! [X480 : term] : (~(X478 @ X480) | (X478 @ (sub @ X479 @ (push @ X480 @ id)))) | (X478 @ (lam @ X479))) | ~ ! [X481 : term] : ((X478 @ (sub @ X481 @ X476)) | ~(var @ X481)) | (X478 @ (sub @ X477 @ X476)) | ~ ! [X482 : term,X483 : term] : ((X478 @ (ap @ X483 @ X482)) | ~(X478 @ X482) | ~(X478 @ X483))))))) <=> induction2lem_lthm) & hoasapnotvar_gthm & (shinj <=> ! [X484 : term,X485 : term] : ((((sub @ X485 @ sh)) != ((sub @ X484 @ sh))) | (X484 = X485))) & induction2lem_gthm & (! [X486 : (subst > term > subst > $o),X487 : (term > $o),X488 : term] : (~ ! [X489 : term] : (((X486 @ id @ X489 @ id)) = ((X487 @ X489))) | (X487 @ (lam @ X488)) | ~ ! [X490 : term] : ((X487 @ (sub @ X488 @ (push @ X490 @ id))) | ~(X487 @ X490)) | ~ ! [X491 : (subst > term > term)] : (~ ! [X492 : subst,X493 : term,X494 : subst] : (((sub @ (X491 @ X494 @ X493) @ X492)) = ((X491 @ (comp @ X494 @ X492) @ (sub @ X493 @ X492)))) | ~ ! [X495 : term] : (~(X486 @ id @ X495 @ id) | (X486 @ id @ (X491 @ id @ X495) @ id)) | (X486 @ id @ (lam @ (X491 @ sh @ one)) @ id))) <=> hoasinduction_lem3v2a) & (! [X496 : term] : ~ ! [X497 : (subst > term > term)] : ~ ! [X498 : term,X499 : subst] : (((sub @ X496 @ (push @ X498 @ X499))) = ((X497 @ X499 @ X498))) <=> hoasinduction_lem3v2_f) & pushprop_lthm & pushprop_lthm_orig & hoaslamnotap_gthm & hoaslamnotvar_lthm & (! [X500 : term] : ~ ! [X501 : (subst > term > term)] : (((sub @ X500 @ (push @ one @ sh))) != ((X501 @ sh @ one))) <=> hoasinduction_lem3b_lthm) & hoasapnotvar_lthm & (termmset_lthm <=> (! [X502 : term] : (((sub @ X502 @ id)) = X502) => ! [X503 : term] : (((sub @ X503 @ id)) = X503))) & ((! [X504 : subst] : (((comp @ X504 @ id)) = X504) & ! [X505 : subst] : (((comp @ id @ X505)) = X505)) <=> substmonoid) & (pushprop_p_and_p_prime = (^[X506 : term, X507 : subst, X508 : (term > $o), X509 : (term > $o)] : (! [X510 : term] : (((X509 @ X510)) = ((X508 @ (sub @ X510 @ (push @ X506 @ X507)))))))) & (! [X511 : subst,X512 : term,X513 : term,X514 : (term > $o),X515 : (term > $o)] : ((X515 @ (sub @ X512 @ (push @ X513 @ X511))) | ~(X514 @ X512)) <=> pushprop_lem1) & ((! [X516 : term] : (((sub @ X516 @ id)) = X516) => (! [X517 : term,X518 : term,X519 : term,X520 : term] : ((((ap @ X519 @ X520)) != ((ap @ X517 @ X518))) | (X517 = X519)) => ! [X521 : term,X522 : term,X523 : term,X524 : term] : ((X521 = X523) | (((ap @ (sub @ X523 @ id) @ X522)) != ((ap @ (sub @ X521 @ id) @ X524)))))) <=> hoasapinj1_lthm) & (! [X525 : term,X526 : term,X527 : term,X528 : term] : ((((ap @ X525 @ X527)) != ((ap @ X528 @ X526))) | (X525 = X528)) <=> apinj1) & (pushprop_lem1v2_lthm <=> (! [X529 : term,X530 : subst] : (((sub @ one @ (push @ X529 @ X530))) = X529) => ! [X531 : (term > $o),X532 : term,X533 : subst,X534 : (term > $o)] : ((X531 @ one) | ~(X534 @ X532) | ~ ! [X535 : term] : (((X534 @ (sub @ X535 @ (push @ X532 @ X533)))) = ((X531 @ X535)))))) & (axshiftcons <=> ! [X536 : subst,X537 : term] : (((comp @ sh @ (push @ X537 @ X536))) = X536)) & pushprop_lem2v2 & hoaslamnotap_lthm & ! [X538 : d_subst] : (d_id = X538) & ! [X539 : subst] : ? [X540 : d_subst] : (((d2subst @ X540)) = X539) & (! [X541 : (term > $o),X542 : term] : ((X541 @ X542) | ~ ! [X543 : term] : ((X541 @ X543) | ~(var @ X543)) | ~ ! [X544 : term,X545 : term] : (~(X541 @ X545) | (X541 @ (ap @ X545 @ X544)) | ~(X541 @ X544)) | ~ ! [X546 : term] : (~(X541 @ X546) | (X541 @ (lam @ X546)))) <=> induction) & (hoasinduction_p_and_p_prime = (^[X547 : (subst > term > subst > $o), X548 : (term > $o)] : (! [X549 : term] : (((X548 @ X549)) = ((X547 @ id @ X549 @ id))))))),
  inference(rectify,[],[f1])).
thf(f5,plain,(
  (! [X2 : term,X1 : term,X0 : (subst > term > term)] : ((((ap @ (sub @ X2 @ id) @ X1)) != ((lam @ (X0 @ sh @ one)))) | ~ ! [X3 : subst,X5 : subst,X4 : term] : (((X0 @ (comp @ X3 @ X5) @ (sub @ X4 @ X5))) = ((sub @ (X0 @ X3 @ X4) @ X5)))) <=> (hoaslamnotap = $true)) & (! [X6 : (subst > term > subst > $o),X7 : term] : ((((X6 @ id @ (lam @ X7) @ id)) = $true) | ~ ! [X8 : subst,X9 : subst,X10 : term,X11 : subst] : (($true = ((X6 @ (comp @ X8 @ X9) @ (sub @ X10 @ X9) @ X11))) | ~ (((X6 @ X8 @ X10 @ (comp @ X9 @ X11))) = $true)) | ~ ! [X12 : term] : (($true = ((X6 @ id @ (sub @ X7 @ (push @ X12 @ id)) @ id))) | ~ (((X6 @ id @ X12 @ id)) = $true)) | ~ ! [X13 : subst,X14 : term,X15 : subst,X16 : subst] : (~ ($true = ((X6 @ (comp @ X15 @ X13) @ (sub @ X14 @ X13) @ X16))) | ($true = ((X6 @ X15 @ X14 @ (comp @ X13 @ X16))))) | ~ ! [X17 : (subst > term > term)] : (~ ! [X18 : term] : (~ ($true = ((X6 @ id @ X18 @ id))) | ($true = ((X6 @ id @ (X17 @ id @ X18) @ id)))) | (((X6 @ id @ (lam @ (X17 @ sh @ one)) @ id)) = $true) | ~ ! [X20 : subst,X21 : subst,X19 : term] : (((X17 @ (comp @ X21 @ X20) @ (sub @ X19 @ X20))) = ((sub @ (X17 @ X21 @ X19) @ X20))))) <=> (hoasinduction_lem3 = $true)) & ~ (((var @ (d2term @ d_one))) = $true) & ~ (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) = $true) & ((hoasinduction_lem3v2 = $true) <=> ! [X22 : (term > $o),X23 : (subst > term > subst > $o),X24 : term] : (~ ! [X25 : term,X26 : subst,X27 : subst,X28 : subst] : (~ (((X23 @ X28 @ X25 @ (comp @ X27 @ X26))) = $true) | (((X23 @ (comp @ X28 @ X27) @ (sub @ X25 @ X27) @ X26)) = $true)) | ~ ! [X29 : (subst > term > term)] : (($true = ((X23 @ id @ (lam @ (X29 @ sh @ one)) @ id))) | ~ ! [X30 : term] : (($true = ((X23 @ id @ (X29 @ id @ X30) @ id))) | ~ ($true = ((X23 @ id @ X30 @ id)))) | ~ ! [X31 : subst,X33 : term,X32 : subst] : (((X29 @ (comp @ X31 @ X32) @ (sub @ X33 @ X32))) = ((sub @ (X29 @ X31 @ X33) @ X32)))) | ~ ! [X34 : subst,X35 : subst,X36 : term,X37 : subst] : (~ ($true = ((X23 @ (comp @ X37 @ X35) @ (sub @ X36 @ X35) @ X34))) | ($true = ((X23 @ X37 @ X36 @ (comp @ X35 @ X34))))) | ~ ! [X38 : term] : (((X23 @ id @ X38 @ id)) = ((X22 @ X38))) | ($true = ((X22 @ (lam @ X24)))) | ~ ! [X39 : term] : (~ ($true = ((X22 @ X39))) | (((X22 @ (sub @ X24 @ (push @ X39 @ id)))) = $true)))) & (! [X40 : subst,X41 : term,X42 : (term > $o),X43 : (term > $o)] : (~ ! [X44 : term] : (((X43 @ (sub @ X44 @ (push @ X41 @ X40)))) = ((X42 @ X44))) | ($true = ((X42 @ one))) | ~ ($true = ((X43 @ X41)))) <=> (pushprop_lem1v2 = $true)) & (pushprop = $true) & (! [X45 : subst] : (((comp @ id @ X45)) = X45) <=> (axidl = $true)) & (! [X46 : term,X47 : term,X48 : (subst > term > subst > $o)] : (~ (((X48 @ id @ X47 @ id)) = $true) | ~ ! [X49 : subst,X50 : term,X51 : subst,X52 : subst] : (~ (((X48 @ (comp @ X49 @ X52) @ (sub @ X50 @ X52) @ X51)) = $true) | ($true = ((X48 @ X49 @ X50 @ (comp @ X52 @ X51))))) | ~ ! [X53 : term,X54 : term] : (~ ($true = ((X48 @ id @ X54 @ id))) | ~ ($true = ((X48 @ id @ X53 @ id))) | ($true = ((X48 @ id @ (ap @ (sub @ X53 @ id) @ X54) @ id)))) | ~ ! [X55 : subst,X56 : term,X57 : subst,X58 : subst] : (~ ($true = ((X48 @ X55 @ X56 @ (comp @ X57 @ X58)))) | ($true = ((X48 @ (comp @ X55 @ X57) @ (sub @ X56 @ X57) @ X58)))) | ($true = ((X48 @ id @ (ap @ X46 @ X47) @ id))) | ~ ($true = ((X48 @ id @ X46 @ id)))) <=> (hoasinduction_lem2 = $true)) & (! [X59 : term,X60 : (subst > term > subst > $o)] : (~ ! [X61 : term] : (~ ($true = ((X60 @ id @ X61 @ id))) | ($true = ((X60 @ id @ (sub @ X59 @ (push @ X61 @ id)) @ id)))) | ~ ! [X62 : (subst > term > term)] : (~ ! [X65 : subst,X64 : term,X63 : subst] : (((sub @ (X62 @ X65 @ X64) @ X63)) = ((X62 @ (comp @ X65 @ X63) @ (sub @ X64 @ X63)))) | (((X60 @ id @ (lam @ (X62 @ sh @ one)) @ id)) = $true) | ~ ! [X66 : term] : (($true = ((X60 @ id @ (X62 @ id @ X66) @ id))) | ~ ($true = ((X60 @ id @ X66 @ id))))) | (((X60 @ id @ (lam @ (sub @ X59 @ (push @ one @ sh))) @ id)) = $true)) <=> (hoasinduction_lem3aa = $true)) & ~ (lamnotap = $true) & ((hoasapinj1 = $true) <=> ! [X67 : term,X70 : term,X69 : term,X68 : term] : ((X67 = X68) | (((ap @ (sub @ X67 @ id) @ X70)) != ((ap @ (sub @ X68 @ id) @ X69))))) & (axmap = $true) & (hoasinduction_lem1_gthm = $true) & ((apinj2 = $true) <=> ! [X71 : term,X73 : term,X74 : term,X72 : term] : ((X71 = X72) | (((ap @ X73 @ X72)) != ((ap @ X74 @ X71))))) & (substmonoid_gthm = $true) & ((induction2lem = $true) <=> ! [X75 : subst,X76 : (term > $o),X77 : term] : (($true = ((X76 @ (sub @ X77 @ X75)))) | ~ ! [X78 : term] : ((((X76 @ (sub @ X78 @ X75))) = $true) | ~ (((var @ X78)) = $true)) | ~ ! [X79 : term,X80 : term] : (~ (((X76 @ X80)) = $true) | ~ (((X76 @ X79)) = $true) | (((X76 @ (ap @ X80 @ X79))) = $true)) | ~ ! [X81 : term] : ((((X76 @ (lam @ X81))) = $true) | ~ ! [X82 : term] : (~ ($true = ((X76 @ X82))) | ($true = ((X76 @ (sub @ X81 @ (push @ X82 @ id))))))))) & (one = ((d2term @ d_one))) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X83 : term] : ~ ! [X84 : (subst > term > term)] : ~ ! [X85 : term,X86 : subst] : (((X84 @ X86 @ X85)) = ((sub @ X83 @ (push @ X85 @ X86))))) & ((axscons = $true) <=> ! [X87 : subst] : (((push @ (sub @ one @ X87) @ (comp @ sh @ X87))) = X87)) & (hoasinduction_gthm = $true) & (! [X88 : term,X89 : (subst > term > subst > $o)] : ((((X89 @ id @ (lam @ (sub @ X88 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X90 : (subst > term > term)] : (~ ! [X91 : term] : (($true = ((X89 @ id @ (X90 @ id @ X91) @ id))) | ~ (((X89 @ id @ X91 @ id)) = $true)) | ~ ! [X93 : subst,X94 : subst,X92 : term] : (((sub @ (X90 @ X93 @ X92) @ X94)) = ((X90 @ (comp @ X93 @ X94) @ (sub @ X92 @ X94)))) | ($true = ((X89 @ id @ (lam @ (X90 @ sh @ one)) @ id)))) | ~ ! [X95 : term] : (~ (((X89 @ id @ X95 @ id)) = $true) | ($true = ((X89 @ id @ (sub @ X88 @ (push @ X95 @ id)) @ id))))) <=> (hoasinduction_lem3aa_lthm = $true)) & (axabs = $true) & (hoasinduction_lem2_gthm = $true) & (hoasinduction_lem3b_gthm = $true) & ! [X96 : d_term] : (d_one = X96) & (termmset_gthm = $true) & (apnotvar = $true) & (hoasapinj2_gthm = $true) & ((pushprop_lem0 = $true) <=> ! [X98 : subst,X99 : (term > $o),X97 : term] : ~ ! [X100 : (term > $o)] : ~ ! [X101 : term] : (((X99 @ (sub @ X101 @ (push @ X97 @ X98)))) = ((X100 @ X101)))) & ((hoasapinj2 = $true) <=> ! [X105 : term,X102 : term,X103 : term,X104 : term] : ((X102 = X105) | (((ap @ (sub @ X104 @ id) @ X102)) != ((ap @ (sub @ X103 @ id) @ X105))))) & (! [X106 : term] : (((sub @ X106 @ id)) = X106) <=> (axvarid = $true)) & (! [X107 : term,X108 : term,X109 : (term > $o),X110 : (subst > term > subst > $o)] : (($true = ((X109 @ (ap @ X107 @ X108)))) | ~ ! [X111 : term,X112 : subst,X113 : subst,X114 : subst] : (($true = ((X110 @ X112 @ X111 @ (comp @ X114 @ X113)))) | ~ ($true = ((X110 @ (comp @ X112 @ X114) @ (sub @ X111 @ X114) @ X113)))) | ~ ! [X115 : subst,X116 : subst,X117 : term,X118 : subst] : (~ ($true = ((X110 @ X115 @ X117 @ (comp @ X116 @ X118)))) | (((X110 @ (comp @ X115 @ X116) @ (sub @ X117 @ X116) @ X118)) = $true)) | ~ (((X109 @ X108)) = $true) | ~ (((X109 @ X107)) = $true) | ~ ! [X119 : term] : (((X109 @ X119)) = ((X110 @ id @ X119 @ id))) | ~ ! [X120 : term,X121 : term] : (($true = ((X110 @ id @ (ap @ (sub @ X120 @ id) @ X121) @ id))) | ~ ($true = ((X110 @ id @ X120 @ id))) | ~ ($true = ((X110 @ id @ X121 @ id))))) <=> (hoasinduction_lem2v2 = $true)) & ((hoasinduction_lem3v2_lthm = $true) <=> (! [X122 : term] : (((sub @ X122 @ id)) = X122) => ! [X123 : term,X124 : (term > $o),X125 : (subst > term > subst > $o)] : (~ ! [X126 : term] : (((X124 @ X126)) = ((X125 @ id @ X126 @ id))) | ~ ! [X127 : term,X128 : subst,X129 : subst,X130 : subst] : (~ ($true = ((X125 @ (comp @ X129 @ X130) @ (sub @ X127 @ X130) @ X128))) | (((X125 @ X129 @ X127 @ (comp @ X130 @ X128))) = $true)) | ~ ! [X131 : (subst > term > term)] : (~ ! [X132 : term] : (($true = ((X125 @ id @ (X131 @ id @ X132) @ id))) | ~ (((X125 @ id @ X132 @ id)) = $true)) | ~ ! [X134 : subst,X135 : subst,X133 : term] : (((X131 @ (comp @ X134 @ X135) @ (sub @ X133 @ X135))) = ((sub @ (X131 @ X134 @ X133) @ X135))) | ($true = ((X125 @ id @ (lam @ (X131 @ sh @ one)) @ id)))) | ~ ! [X136 : subst,X137 : term,X138 : subst,X139 : subst] : (($true = ((X125 @ (comp @ X139 @ X138) @ (sub @ X137 @ X138) @ X136))) | ~ (((X125 @ X139 @ X137 @ (comp @ X138 @ X136))) = $true)) | (((X124 @ (lam @ X123))) = $true) | ~ ! [X140 : term] : (($true = ((X124 @ (sub @ X123 @ (push @ X140 @ id))))) | ~ (((X124 @ X140)) = $true))))) & (pushprop_lem0_gthm = $true) & (hoaslaminj_gthm = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ((hoasinduction_lem0_lthm = $true) <=> ! [X141 : (subst > term > subst > $o)] : ~ ! [X142 : (term > $o)] : ~ ! [X143 : term] : (((X141 @ id @ X143 @ id)) = ((X142 @ X143)))) & (ulamvarind = $true) & (pushprop_gthm = $true) & (hoaslamnotvar = $true) & ((hoaslaminj_lthm = $true) <=> (! [X144 : term,X145 : subst] : (((sub @ one @ (push @ X144 @ X145))) = X144) => (! [X147 : subst,X146 : term] : (((comp @ sh @ (push @ X146 @ X147))) = X147) => (! [X149 : term,X148 : term] : ((((lam @ X149)) != ((lam @ X148))) | (X148 = X149)) => ! [X152 : term,X151 : (subst > term > term),X150 : subst,X153 : (subst > term > term)] : ((((X151 @ X150 @ X152)) = ((X153 @ X150 @ X152))) | ~ ! [X154 : term,X155 : subst,X156 : subst] : (((sub @ (X153 @ X155 @ X154) @ X156)) = ((X153 @ (comp @ X155 @ X156) @ (sub @ X154 @ X156)))) | ~ ! [X158 : subst,X157 : subst,X159 : term] : (((X151 @ (comp @ X157 @ X158) @ (sub @ X159 @ X158))) = ((sub @ (X151 @ X157 @ X159) @ X158))) | (((lam @ (X151 @ sh @ one))) != ((lam @ (X153 @ sh @ one))))))))) & ((induction2_lthm = $true) <=> (! [X160 : term] : (((sub @ X160 @ id)) = X160) => (! [X161 : term,X162 : (term > $o),X163 : subst] : (~ ! [X164 : term,X165 : term] : (~ ($true = ((X162 @ X165))) | ($true = ((X162 @ (ap @ X164 @ X165)))) | ~ (((X162 @ X164)) = $true)) | ~ ! [X166 : term] : (($true = ((X162 @ (lam @ X166)))) | ~ ! [X167 : term] : (($true = ((X162 @ (sub @ X166 @ (push @ X167 @ id))))) | ~ ($true = ((X162 @ X167))))) | (((X162 @ (sub @ X161 @ X163))) = $true) | ~ ! [X168 : term] : (($true = ((X162 @ (sub @ X168 @ X163)))) | ~ (((var @ X168)) = $true))) => ! [X169 : term,X170 : (term > $o)] : ((((X170 @ X169)) = $true) | ~ ! [X171 : term] : (~ ! [X172 : term] : (($true = ((X170 @ (sub @ X171 @ (push @ X172 @ id))))) | ~ ($true = ((X170 @ X172)))) | ($true = ((X170 @ (lam @ X171))))) | ~ ! [X173 : term,X174 : term] : (~ (((X170 @ X173)) = $true) | ~ ($true = ((X170 @ X174))) | (((X170 @ (ap @ X174 @ X173))) = $true)) | ~ ! [X175 : term] : (~ ($true = ((var @ X175))) | ($true = ((X170 @ X175)))))))) & ((hoaslaminj = $true) <=> ! [X176 : subst,X179 : (subst > term > term),X178 : (subst > term > term),X177 : term] : (~ ! [X183 : subst,X184 : subst,X185 : term] : (((X179 @ (comp @ X184 @ X183) @ (sub @ X185 @ X183))) = ((sub @ (X179 @ X184 @ X185) @ X183))) | (((X179 @ X176 @ X177)) = ((X178 @ X176 @ X177))) | ~ ! [X180 : term,X181 : subst,X182 : subst] : (((X178 @ (comp @ X182 @ X181) @ (sub @ X180 @ X181))) = ((sub @ (X178 @ X182 @ X180) @ X181))) | (((lam @ (X178 @ sh @ one))) != ((lam @ (X179 @ sh @ one)))))) & ((! [X186 : (subst > term > subst > $o)] : ~ ! [X187 : (term > $o)] : ~ ! [X188 : term] : (((X186 @ id @ X188 @ id)) = ((X187 @ X188))) => (! [X189 : (term > $o),X190 : term] : (($true = ((X189 @ X190))) | ~ ! [X191 : term,X192 : term] : ((((X189 @ (ap @ X191 @ X192))) = $true) | ~ ($true = ((X189 @ X192))) | ~ (((X189 @ X191)) = $true)) | ~ ! [X193 : term] : (($true = ((X189 @ (lam @ X193)))) | ~ ! [X194 : term] : (~ (((X189 @ X194)) = $true) | ($true = ((X189 @ (sub @ X193 @ (push @ X194 @ id))))))) | ~ ! [X195 : term] : (~ ($true = ((var @ X195))) | ($true = ((X189 @ X195))))) => (! [X196 : term] : (((sub @ X196 @ id)) = X196) => (! [X197 : (term > $o),X198 : (subst > term > subst > $o),X199 : term] : (~ ! [X200 : (subst > term > term)] : (~ ! [X203 : subst,X201 : term,X202 : subst] : (((X200 @ (comp @ X203 @ X202) @ (sub @ X201 @ X202))) = ((sub @ (X200 @ X203 @ X201) @ X202))) | ~ ! [X204 : term] : (($true = ((X198 @ id @ (X200 @ id @ X204) @ id))) | ~ ($true = ((X198 @ id @ X204 @ id)))) | (((X198 @ id @ (lam @ (X200 @ sh @ one)) @ id)) = $true)) | ($true = ((X197 @ (lam @ X199)))) | ~ ! [X205 : term] : (~ (((X197 @ X205)) = $true) | (((X197 @ (sub @ X199 @ (push @ X205 @ id)))) = $true)) | ~ ! [X206 : term] : (((X198 @ id @ X206 @ id)) = ((X197 @ X206)))) => ! [X207 : term,X208 : (subst > term > subst > $o)] : (~ ! [X209 : term] : (~ ($true = ((var @ (sub @ X209 @ id)))) | ($true = ((X208 @ id @ X209 @ id)))) | ~ ! [X210 : term,X211 : term] : (~ ($true = ((X208 @ id @ X210 @ id))) | ($true = ((X208 @ id @ (ap @ (sub @ X211 @ id) @ X210) @ id))) | ~ ($true = ((X208 @ id @ X211 @ id)))) | ~ ! [X212 : subst,X213 : subst,X214 : term,X215 : subst] : (~ (((X208 @ X215 @ X214 @ (comp @ X212 @ X213))) = $true) | ($true = ((X208 @ (comp @ X215 @ X212) @ (sub @ X214 @ X212) @ X213)))) | ~ ! [X216 : (subst > term > term)] : (~ ! [X217 : subst,X219 : subst,X218 : term] : (((X216 @ (comp @ X217 @ X219) @ (sub @ X218 @ X219))) = ((sub @ (X216 @ X217 @ X218) @ X219))) | ($true = ((X208 @ id @ (lam @ (X216 @ sh @ one)) @ id))) | ~ ! [X220 : term] : (($true = ((X208 @ id @ (X216 @ id @ X220) @ id))) | ~ (((X208 @ id @ X220 @ id)) = $true))) | ($true = ((X208 @ id @ X207 @ id))) | ~ ! [X221 : subst,X222 : term,X223 : subst,X224 : subst] : (~ ($true = ((X208 @ (comp @ X221 @ X224) @ (sub @ X222 @ X224) @ X223))) | ($true = ((X208 @ X221 @ X222 @ (comp @ X224 @ X223)))))))))) <=> (hoasinduction_lthm_3 = $true)) & (! [X225 : (subst > term > subst > $o),X226 : term] : (~ ! [X227 : term,X228 : term] : (~ ($true = ((X225 @ id @ X228 @ id))) | ($true = ((X225 @ id @ (ap @ (sub @ X227 @ id) @ X228) @ id))) | ~ ($true = ((X225 @ id @ X227 @ id)))) | ~ ! [X229 : (subst > term > term)] : (($true = ((X225 @ id @ (lam @ (X229 @ sh @ one)) @ id))) | ~ ! [X230 : term] : (~ ($true = ((X225 @ id @ X230 @ id))) | (((X225 @ id @ (X229 @ id @ X230) @ id)) = $true)) | ~ ! [X233 : subst,X231 : subst,X232 : term] : (((sub @ (X229 @ X231 @ X232) @ X233)) = ((X229 @ (comp @ X231 @ X233) @ (sub @ X232 @ X233))))) | (((X225 @ id @ X226 @ id)) = $true)) <=> (hoasinduction_no_psi_cond = $true)) & (! [X235 : subst,X234 : term] : (((sub @ one @ (push @ X234 @ X235))) = X234) <=> (axvarcons = $true)) & (pushprop_lem2v2_gthm = $true) & ((hoasinduction_lem3a_lthm = $true) <=> (! [X236 : term] : (((sub @ X236 @ id)) = X236) => (! [X237 : (subst > term > subst > $o),X238 : term] : (~ ! [X239 : (subst > term > term)] : ((((X237 @ id @ (lam @ (X239 @ sh @ one)) @ id)) = $true) | ~ ! [X241 : subst,X242 : term,X240 : subst] : (((sub @ (X239 @ X241 @ X242) @ X240)) = ((X239 @ (comp @ X241 @ X240) @ (sub @ X242 @ X240)))) | ~ ! [X243 : term] : (~ ($true = ((X237 @ id @ X243 @ id))) | ($true = ((X237 @ id @ (X239 @ id @ X243) @ id))))) | (((X237 @ id @ (lam @ (sub @ X238 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X244 : term] : (($true = ((X237 @ id @ (sub @ X238 @ (push @ X244 @ id)) @ id))) | ~ (((X237 @ id @ X244 @ id)) = $true))) => ! [X245 : (subst > term > subst > $o),X246 : term] : (($true = ((X245 @ id @ (lam @ X246) @ id))) | ~ ! [X247 : term] : (~ ($true = ((X245 @ id @ X247 @ id))) | ($true = ((X245 @ id @ (sub @ X246 @ (push @ X247 @ id)) @ id)))) | ~ ! [X248 : (subst > term > term)] : ((((X245 @ id @ (lam @ (X248 @ sh @ one)) @ id)) = $true) | ~ ! [X251 : term,X249 : subst,X250 : subst] : (((X248 @ (comp @ X250 @ X249) @ (sub @ X251 @ X249))) = ((sub @ (X248 @ X250 @ X251) @ X249))) | ~ ! [X252 : term] : (~ (((X245 @ id @ X252 @ id)) = $true) | (((X245 @ id @ (X248 @ id @ X252) @ id)) = $true))))))) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoasinduction_lem3b = $true) <=> ! [X253 : term] : ~ ! [X254 : (subst > term > term)] : (((X254 @ sh @ one)) != ((sub @ X253 @ (push @ one @ sh))))) & ((pushprop_lem0_lthm = $true) <=> ! [X255 : subst,X256 : (term > $o),X257 : term] : ~ ! [X258 : (term > $o)] : ~ ! [X259 : term] : (((X258 @ X259)) = ((X256 @ (sub @ X259 @ (push @ X257 @ X255)))))) & (axapp = $true) & (((d2subst @ d_id)) = sh) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (axassoc = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoasinduction_lthm = $true) <=> (! [X262 : (term > $o),X263 : term] : (($true = ((X262 @ X263))) | ~ ! [X264 : term] : (($true = ((X262 @ (lam @ X264)))) | ~ ! [X265 : term] : (~ ($true = ((X262 @ X265))) | ($true = ((X262 @ (sub @ X264 @ (push @ X265 @ id))))))) | ~ ! [X266 : term] : (~ ($true = ((var @ X266))) | ($true = ((X262 @ X266)))) | ~ ! [X267 : term,X268 : term] : (~ (((X262 @ X267)) = $true) | ($true = ((X262 @ (ap @ X267 @ X268)))) | ~ (((X262 @ X268)) = $true))) => (! [X269 : term,X270 : (subst > term > subst > $o),X271 : term] : (~ (((X270 @ id @ X269 @ id)) = $true) | ~ ($true = ((X270 @ id @ X271 @ id))) | ~ ! [X272 : subst,X273 : term,X274 : subst,X275 : subst] : (~ ($true = ((X270 @ (comp @ X272 @ X275) @ (sub @ X273 @ X275) @ X274))) | ($true = ((X270 @ X272 @ X273 @ (comp @ X275 @ X274))))) | ~ ! [X276 : subst,X277 : subst,X278 : subst,X279 : term] : (~ (((X270 @ X277 @ X279 @ (comp @ X278 @ X276))) = $true) | ($true = ((X270 @ (comp @ X277 @ X278) @ (sub @ X279 @ X278) @ X276)))) | ($true = ((X270 @ id @ (ap @ X271 @ X269) @ id))) | ~ ! [X280 : term,X281 : term] : (~ ($true = ((X270 @ id @ X281 @ id))) | ($true = ((X270 @ id @ (ap @ (sub @ X280 @ id) @ X281) @ id))) | ~ ($true = ((X270 @ id @ X280 @ id))))) => (! [X282 : term,X283 : (subst > term > subst > $o)] : ((((X283 @ id @ (lam @ X282) @ id)) = $true) | ~ ! [X284 : subst,X285 : subst,X286 : subst,X287 : term] : (~ ($true = ((X283 @ X284 @ X287 @ (comp @ X285 @ X286)))) | ($true = ((X283 @ (comp @ X284 @ X285) @ (sub @ X287 @ X285) @ X286)))) | ~ ! [X288 : term] : (~ ($true = ((X283 @ id @ X288 @ id))) | (((X283 @ id @ (sub @ X282 @ (push @ X288 @ id)) @ id)) = $true)) | ~ ! [X289 : subst,X290 : subst,X291 : subst,X292 : term] : (~ ($true = ((X283 @ (comp @ X290 @ X291) @ (sub @ X292 @ X291) @ X289))) | (((X283 @ X290 @ X292 @ (comp @ X291 @ X289))) = $true)) | ~ ! [X293 : (subst > term > term)] : (($true = ((X283 @ id @ (lam @ (X293 @ sh @ one)) @ id))) | ~ ! [X294 : subst,X295 : term,X296 : subst] : (((X293 @ (comp @ X296 @ X294) @ (sub @ X295 @ X294))) = ((sub @ (X293 @ X296 @ X295) @ X294))) | ~ ! [X297 : term] : (($true = ((X283 @ id @ (X293 @ id @ X297) @ id))) | ~ ($true = ((X283 @ id @ X297 @ id)))))) => ! [X298 : term,X299 : (subst > term > subst > $o)] : (~ ! [X300 : subst,X301 : subst,X302 : subst,X303 : term] : (($true = ((X299 @ X300 @ X303 @ (comp @ X301 @ X302)))) | ~ (((X299 @ (comp @ X300 @ X301) @ (sub @ X303 @ X301) @ X302)) = $true)) | ~ ! [X304 : (subst > term > term)] : (($true = ((X299 @ id @ (lam @ (X304 @ sh @ one)) @ id))) | ~ ! [X305 : term] : ((((X299 @ id @ (X304 @ id @ X305) @ id)) = $true) | ~ (((X299 @ id @ X305 @ id)) = $true)) | ~ ! [X306 : term,X307 : subst,X308 : subst] : (((X304 @ (comp @ X308 @ X307) @ (sub @ X306 @ X307))) = ((sub @ (X304 @ X308 @ X306) @ X307)))) | ~ ! [X309 : subst,X310 : subst,X311 : subst,X312 : term] : (~ ($true = ((X299 @ X309 @ X312 @ (comp @ X310 @ X311)))) | ($true = ((X299 @ (comp @ X309 @ X310) @ (sub @ X312 @ X310) @ X311)))) | ~ ! [X313 : term] : (($true = ((X299 @ id @ X313 @ id))) | ~ ($true = ((var @ (sub @ X313 @ id))))) | ~ ! [X314 : term,X315 : term] : (~ ($true = ((X299 @ id @ X314 @ id))) | ($true = ((X299 @ id @ (ap @ (sub @ X314 @ id) @ X315) @ id))) | ~ ($true = ((X299 @ id @ X315 @ id)))) | (((X299 @ id @ X298 @ id)) = $true)))))) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((hoasinduction_lem3v2a_lthm = $true) <=> (! [X316 : term] : ~ ! [X317 : (subst > term > term)] : ~ ! [X318 : subst,X319 : term] : (((sub @ X316 @ (push @ X319 @ X318))) = ((X317 @ X318 @ X319))) => (! [X320 : term] : (((sub @ X320 @ id)) = X320) => ! [X321 : (subst > term > subst > $o),X322 : (term > $o),X323 : term] : (~ ! [X324 : (subst > term > term)] : (~ ! [X325 : subst,X326 : term,X327 : subst] : (((X324 @ (comp @ X325 @ X327) @ (sub @ X326 @ X327))) = ((sub @ (X324 @ X325 @ X326) @ X327))) | ~ ! [X328 : term] : (~ ($true = ((X321 @ id @ X328 @ id))) | ($true = ((X321 @ id @ (X324 @ id @ X328) @ id)))) | ($true = ((X321 @ id @ (lam @ (X324 @ sh @ one)) @ id)))) | ($true = ((X322 @ (lam @ X323)))) | ~ ! [X329 : term] : (((X322 @ X329)) = ((X321 @ id @ X329 @ id))) | ~ ! [X330 : term] : (~ ($true = ((X322 @ X330))) | ($true = ((X322 @ (sub @ X323 @ (push @ X330 @ id)))))))))) & (pushprop_lem1_gthm = $true) & (pushprop_lem3v2_lthm = $true) & (hoasinduction_lem1 = $true) & (pushprop_lem3v2 = $true) & ~ (ulamvar1 = $true) & (hoasinduction_lem1v2 = $true) & ((substmonoid_lthm = $true) <=> (! [X331 : subst] : (((comp @ id @ X331)) = X331) => (! [X332 : subst] : (((comp @ X332 @ id)) = X332) => (! [X333 : subst] : (((comp @ X333 @ id)) = X333) & ! [X334 : subst] : (((comp @ id @ X334)) = X334))))) & (induction2_gthm = $true) & ((hoasinduction_lem3_lthm = $true) <=> (! [X335 : term] : (((sub @ X335 @ id)) = X335) => (! [X336 : term,X337 : (subst > term > subst > $o)] : (~ ! [X338 : term] : (~ (((X337 @ id @ X338 @ id)) = $true) | ($true = ((X337 @ id @ (sub @ X336 @ (push @ X338 @ id)) @ id)))) | ~ ! [X339 : (subst > term > term)] : (~ ! [X340 : term] : (($true = ((X337 @ id @ (X339 @ id @ X340) @ id))) | ~ ($true = ((X337 @ id @ X340 @ id)))) | ($true = ((X337 @ id @ (lam @ (X339 @ sh @ one)) @ id))) | ~ ! [X341 : term,X342 : subst,X343 : subst] : (((X339 @ (comp @ X343 @ X342) @ (sub @ X341 @ X342))) = ((sub @ (X339 @ X343 @ X341) @ X342)))) | ($true = ((X337 @ id @ (lam @ (sub @ X336 @ (push @ one @ sh))) @ id)))) => ! [X344 : (subst > term > subst > $o),X345 : term] : (~ ! [X346 : term] : (~ ($true = ((X344 @ id @ X346 @ id))) | ($true = ((X344 @ id @ (sub @ X345 @ (push @ X346 @ id)) @ id)))) | (((X344 @ id @ (lam @ X345) @ id)) = $true) | ~ ! [X347 : subst,X348 : term,X349 : subst,X350 : subst] : (($true = ((X344 @ X347 @ X348 @ (comp @ X349 @ X350)))) | ~ ($true = ((X344 @ (comp @ X347 @ X349) @ (sub @ X348 @ X349) @ X350)))) | ~ ! [X351 : (subst > term > term)] : (~ ! [X352 : term] : (($true = ((X344 @ id @ (X351 @ id @ X352) @ id))) | ~ ($true = ((X344 @ id @ X352 @ id)))) | (((X344 @ id @ (lam @ (X351 @ sh @ one)) @ id)) = $true) | ~ ! [X353 : term,X354 : subst,X355 : subst] : (((sub @ (X351 @ X355 @ X353) @ X354)) = ((X351 @ (comp @ X355 @ X354) @ (sub @ X353 @ X354))))) | ~ ! [X356 : subst,X357 : subst,X358 : subst,X359 : term] : (($true = ((X344 @ (comp @ X358 @ X357) @ (sub @ X359 @ X357) @ X356))) | ~ ($true = ((X344 @ X358 @ X359 @ (comp @ X357 @ X356))))))))) & (pushprop_lem1v2_gthm = $true) & (lamnotvar = $true) & ! [X360 : d_term,X361 : d_term] : ((((d2term @ X361)) = ((d2term @ X360))) => (X360 = X361)) & (axclos = $true) & (id = ((d2subst @ d_id))) & (hoasinduction_lem2v2_gthm = $true) & (! [X362 : term] : (((sub @ X362 @ id)) = X362) <=> (termmset = $true)) & (pushprop_lem2v2_lthm = $true) & (ulamvarsh = $true) & (hoasapinj1_gthm = $true) & ((! [X363 : term,X364 : subst] : (((sub @ one @ (push @ X363 @ X364))) = X363) => (! [X365 : term,X366 : subst] : (((comp @ sh @ (push @ X365 @ X366))) = X366) => ! [X367 : (term > $o),X368 : subst,X369 : (term > $o),X370 : term,X371 : term] : (~ ($true = ((X367 @ X371))) | ($true = ((X369 @ (sub @ X371 @ (push @ X370 @ X368)))))))) <=> (pushprop_lem1_lthm = $true)) & (hoasinduction_lem3v2_gthm = $true) & (hoasinduction_lem2_lthm = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (hoasinduction_lem1_lthm = $true) & ((hoasinduction_lem3aaa = $true) <=> ! [X372 : term,X373 : (subst > term > subst > $o)] : (($true = ((X373 @ id @ (lam @ (sub @ X372 @ (push @ one @ sh))) @ id))) | ~ ! [X374 : (subst > term > term),X375 : term] : (~ ! [X376 : term] : ((((X373 @ id @ (X374 @ id @ X376) @ id)) = $true) | ~ ($true = ((X373 @ id @ X376 @ id)))) | ~ ! [X377 : term,X378 : subst,X379 : subst] : (((sub @ (sub @ X375 @ (push @ X377 @ X379)) @ X378)) = ((sub @ (X374 @ X379 @ X377) @ X378))) | ($true = ((X373 @ id @ (lam @ (X374 @ sh @ one)) @ id))) | ~ ! [X380 : subst,X381 : subst,X382 : term] : (((sub @ X375 @ (push @ (sub @ X382 @ X381) @ (comp @ X380 @ X381)))) = ((X374 @ (comp @ X380 @ X381) @ (sub @ X382 @ X381))))) | ~ ! [X383 : term] : (~ ($true = ((X373 @ id @ X383 @ id))) | ($true = ((X373 @ id @ (sub @ X372 @ (push @ X383 @ id)) @ id)))))) & ((! [X384 : (subst > term > subst > $o)] : ~ ! [X385 : (term > $o)] : ~ ! [X386 : term] : (((X384 @ id @ X386 @ id)) = ((X385 @ X386))) => (! [X387 : (term > $o),X388 : term] : (($true = ((X387 @ X388))) | ~ ! [X389 : term,X390 : term] : (($true = ((X387 @ (ap @ X390 @ X389)))) | ~ ($true = ((X387 @ X389))) | ~ ($true = ((X387 @ X390)))) | ~ ! [X391 : term] : (~ ! [X392 : term] : ((((X387 @ (sub @ X391 @ (push @ X392 @ id)))) = $true) | ~ ($true = ((X387 @ X392)))) | ($true = ((X387 @ (lam @ X391))))) | ~ ! [X393 : term] : ((((X387 @ X393)) = $true) | ~ ($true = ((var @ X393))))) => (! [X394 : term] : (((sub @ X394 @ id)) = X394) => (! [X395 : (subst > term > subst > $o),X396 : term,X397 : (term > $o)] : (~ ! [X398 : (subst > term > term)] : (($true = ((X395 @ id @ (lam @ (X398 @ sh @ one)) @ id))) | ~ ! [X399 : term] : ((((X395 @ id @ (X398 @ id @ X399) @ id)) = $true) | ~ (((X395 @ id @ X399 @ id)) = $true)) | ~ ! [X400 : subst,X401 : term,X402 : subst] : (((X398 @ (comp @ X400 @ X402) @ (sub @ X401 @ X402))) = ((sub @ (X398 @ X400 @ X401) @ X402)))) | ~ ! [X403 : term] : (($true = ((X397 @ (sub @ X396 @ (push @ X403 @ id))))) | ~ (((X397 @ X403)) = $true)) | (((X397 @ (lam @ X396))) = $true) | ~ ! [X404 : term] : (((X395 @ id @ X404 @ id)) = ((X397 @ X404)))) => ! [X405 : (subst > term > subst > $o),X406 : term] : (($true = ((X405 @ id @ X406 @ id))) | ~ ! [X407 : (subst > term > term)] : (~ ! [X408 : subst,X409 : subst,X410 : term] : (((X407 @ (comp @ X409 @ X408) @ (sub @ X410 @ X408))) = ((sub @ (X407 @ X409 @ X410) @ X408))) | (((X405 @ id @ (lam @ (X407 @ sh @ one)) @ id)) = $true) | ~ ! [X411 : term] : (~ ($true = ((X405 @ id @ X411 @ id))) | ($true = ((X405 @ id @ (X407 @ id @ X411) @ id))))) | ~ ! [X412 : term,X413 : term] : (~ (((X405 @ id @ X413 @ id)) = $true) | (((X405 @ id @ (ap @ (sub @ X413 @ id) @ X412) @ id)) = $true) | ~ ($true = ((X405 @ id @ X412 @ id))))))))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (hoasinduction_lem3a_gthm = $true) & (hoasinduction_lem1v2_gthm = $true) & ! [X414 : d_subst,X415 : d_subst] : ((((d2subst @ X414)) = ((d2subst @ X415))) => (X414 = X415)) & (! [X416 : term,X417 : (subst > term > subst > $o)] : (($true = ((X417 @ id @ (lam @ X416) @ id))) | ~ ! [X418 : term] : (($true = ((X417 @ id @ (sub @ X416 @ (push @ X418 @ id)) @ id))) | ~ ($true = ((X417 @ id @ X418 @ id)))) | ~ ! [X419 : (subst > term > term)] : ((((X417 @ id @ (lam @ (X419 @ sh @ one)) @ id)) = $true) | ~ ! [X420 : term,X421 : subst,X422 : subst] : (((X419 @ (comp @ X422 @ X421) @ (sub @ X420 @ X421))) = ((sub @ (X419 @ X422 @ X420) @ X421))) | ~ ! [X423 : term] : (~ ($true = ((X417 @ id @ X423 @ id))) | (((X417 @ id @ (X419 @ id @ X423) @ id)) = $true)))) <=> (hoasinduction_lem3a = $true)) & ! [X424 : term] : ? [X425 : d_term] : (((d2term @ X425)) = X424) & (hoasapnotvar = $true) & (axvarshift = $true) & (! [X426 : subst] : (((comp @ X426 @ id)) = X426) <=> (axidr = $true)) & (hoaslamnotvar_gthm = $true) & ((hoasinduction = $true) <=> ! [X427 : term,X428 : (subst > term > subst > $o)] : (($true = ((X428 @ id @ X427 @ id))) | ~ ! [X429 : term,X430 : term] : (~ ($true = ((X428 @ id @ X430 @ id))) | ($true = ((X428 @ id @ (ap @ (sub @ X430 @ id) @ X429) @ id))) | ~ ($true = ((X428 @ id @ X429 @ id)))) | ~ ! [X431 : (subst > term > term)] : (~ ! [X432 : term,X433 : subst,X434 : subst] : (((X431 @ (comp @ X434 @ X433) @ (sub @ X432 @ X433))) = ((sub @ (X431 @ X434 @ X432) @ X433))) | ($true = ((X428 @ id @ (lam @ (X431 @ sh @ one)) @ id))) | ~ ! [X435 : term] : (~ ($true = ((X428 @ id @ X435 @ id))) | ($true = ((X428 @ id @ (X431 @ id @ X435) @ id))))) | ~ ! [X436 : term,X437 : subst,X438 : subst,X439 : subst] : (~ ($true = ((X428 @ (comp @ X439 @ X438) @ (sub @ X436 @ X438) @ X437))) | ($true = ((X428 @ X439 @ X436 @ (comp @ X438 @ X437))))) | ~ ! [X440 : term,X441 : subst,X442 : subst,X443 : subst] : (~ (((X428 @ X442 @ X440 @ (comp @ X441 @ X443))) = $true) | ($true = ((X428 @ (comp @ X442 @ X441) @ (sub @ X440 @ X441) @ X443)))) | ~ ! [X444 : term] : (~ (((var @ (sub @ X444 @ id))) = $true) | (((X428 @ id @ X444 @ id)) = $true)))) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((laminj = $true) <=> ! [X445 : term,X446 : term] : ((X445 = X446) | (((lam @ X445)) != ((lam @ X446))))) & (! [X447 : (subst > term > subst > $o)] : ~ ! [X448 : (term > $o)] : ~ ! [X449 : term] : (((X448 @ X449)) = ((X447 @ id @ X449 @ id))) <=> (hoasinduction_lem0 = $true)) & ((! [X450 : term,X451 : term,X452 : term,X453 : term] : ((X450 = X452) | (((ap @ X451 @ X452)) != ((ap @ X453 @ X450)))) => ! [X454 : term,X455 : term,X456 : term,X457 : term] : ((((ap @ (sub @ X456 @ id) @ X454)) != ((ap @ (sub @ X455 @ id) @ X457))) | (X454 = X457))) <=> (hoasapinj2_lthm = $true)) & (! [X458 : (term > $o),X459 : term] : (($true = ((X458 @ X459))) | ~ ! [X460 : term,X461 : term] : ((((X458 @ (ap @ X461 @ X460))) = $true) | ~ ($true = ((X458 @ X460))) | ~ (((X458 @ X461)) = $true)) | ~ ! [X462 : term] : (~ ! [X463 : term] : (~ ($true = ((X458 @ X463))) | ($true = ((X458 @ (sub @ X462 @ (push @ X463 @ id)))))) | (((X458 @ (lam @ X462))) = $true)) | ~ ! [X464 : term] : (($true = ((X458 @ X464))) | ~ (((var @ X464)) = $true))) <=> (induction2 = $true)) & (hoasinduction_lem3_gthm = $true) & ((! [X465 : subst,X466 : term] : (((sub @ one @ (push @ X466 @ X465))) = X466) => (! [X467 : term,X468 : subst] : (((comp @ sh @ (push @ X467 @ X468))) = X468) => (! [X469 : subst] : (((comp @ X469 @ id)) = X469) => (! [X470 : (term > $o),X471 : term] : (~ ! [X472 : term,X473 : term] : (($true = ((X470 @ (ap @ X473 @ X472)))) | ~ (((X470 @ X473)) = $true) | ~ ($true = ((X470 @ X472)))) | ($true = ((X470 @ X471))) | ~ ! [X474 : term] : (~ (((X470 @ X474)) = $true) | ($true = ((X470 @ (lam @ X474))))) | ~ ! [X475 : term] : (~ (((var @ X475)) = $true) | (((X470 @ X475)) = $true))) => ! [X476 : subst,X477 : term,X478 : (term > $o)] : (~ ! [X479 : term] : (~ ! [X480 : term] : (~ ($true = ((X478 @ X480))) | ($true = ((X478 @ (sub @ X479 @ (push @ X480 @ id)))))) | ($true = ((X478 @ (lam @ X479))))) | ~ ! [X481 : term] : (($true = ((X478 @ (sub @ X481 @ X476)))) | ~ ($true = ((var @ X481)))) | ($true = ((X478 @ (sub @ X477 @ X476)))) | ~ ! [X482 : term,X483 : term] : (($true = ((X478 @ (ap @ X483 @ X482)))) | ~ ($true = ((X478 @ X482))) | ~ ($true = ((X478 @ X483))))))))) <=> (induction2lem_lthm = $true)) & (hoasapnotvar_gthm = $true) & ((shinj = $true) <=> ! [X484 : term,X485 : term] : ((((sub @ X485 @ sh)) != ((sub @ X484 @ sh))) | (X484 = X485))) & (induction2lem_gthm = $true) & (! [X486 : (subst > term > subst > $o),X487 : (term > $o),X488 : term] : (~ ! [X489 : term] : (((X486 @ id @ X489 @ id)) = ((X487 @ X489))) | ($true = ((X487 @ (lam @ X488)))) | ~ ! [X490 : term] : (($true = ((X487 @ (sub @ X488 @ (push @ X490 @ id))))) | ~ ($true = ((X487 @ X490)))) | ~ ! [X491 : (subst > term > term)] : (~ ! [X492 : subst,X493 : term,X494 : subst] : (((sub @ (X491 @ X494 @ X493) @ X492)) = ((X491 @ (comp @ X494 @ X492) @ (sub @ X493 @ X492)))) | ~ ! [X495 : term] : (~ ($true = ((X486 @ id @ X495 @ id))) | ($true = ((X486 @ id @ (X491 @ id @ X495) @ id)))) | (((X486 @ id @ (lam @ (X491 @ sh @ one)) @ id)) = $true))) <=> (hoasinduction_lem3v2a = $true)) & (! [X496 : term] : ~ ! [X497 : (subst > term > term)] : ~ ! [X498 : term,X499 : subst] : (((sub @ X496 @ (push @ X498 @ X499))) = ((X497 @ X499 @ X498))) <=> (hoasinduction_lem3v2_f = $true)) & (pushprop_lthm = $true) & (pushprop_lthm_orig = $true) & (hoaslamnotap_gthm = $true) & (hoaslamnotvar_lthm = $true) & (! [X500 : term] : ~ ! [X501 : (subst > term > term)] : (((sub @ X500 @ (push @ one @ sh))) != ((X501 @ sh @ one))) <=> (hoasinduction_lem3b_lthm = $true)) & (hoasapnotvar_lthm = $true) & ((termmset_lthm = $true) <=> (! [X502 : term] : (((sub @ X502 @ id)) = X502) => ! [X503 : term] : (((sub @ X503 @ id)) = X503))) & ((! [X504 : subst] : (((comp @ X504 @ id)) = X504) & ! [X505 : subst] : (((comp @ id @ X505)) = X505)) <=> (substmonoid = $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)))))))))))))) & (! [X511 : subst,X512 : term,X513 : term,X514 : (term > $o),X515 : (term > $o)] : (($true = ((X515 @ (sub @ X512 @ (push @ X513 @ X511))))) | ~ ($true = ((X514 @ X512)))) <=> (pushprop_lem1 = $true)) & ((! [X516 : term] : (((sub @ X516 @ id)) = X516) => (! [X517 : term,X518 : term,X519 : term,X520 : term] : ((((ap @ X519 @ X520)) != ((ap @ X517 @ X518))) | (X517 = X519)) => ! [X521 : term,X522 : term,X523 : term,X524 : term] : ((X521 = X523) | (((ap @ (sub @ X523 @ id) @ X522)) != ((ap @ (sub @ X521 @ id) @ X524)))))) <=> (hoasapinj1_lthm = $true)) & (! [X525 : term,X526 : term,X527 : term,X528 : term] : ((((ap @ X525 @ X527)) != ((ap @ X528 @ X526))) | (X525 = X528)) <=> (apinj1 = $true)) & ((pushprop_lem1v2_lthm = $true) <=> (! [X529 : term,X530 : subst] : (((sub @ one @ (push @ X529 @ X530))) = X529) => ! [X531 : (term > $o),X532 : term,X533 : subst,X534 : (term > $o)] : (($true = ((X531 @ one))) | ~ ($true = ((X534 @ X532))) | ~ ! [X535 : term] : (((X534 @ (sub @ X535 @ (push @ X532 @ X533)))) = ((X531 @ X535)))))) & ((axshiftcons = $true) <=> ! [X536 : subst,X537 : term] : (((comp @ sh @ (push @ X537 @ X536))) = X536)) & (pushprop_lem2v2 = $true) & (hoaslamnotap_lthm = $true) & ! [X538 : d_subst] : (d_id = X538) & ! [X539 : subst] : ? [X540 : d_subst] : (((d2subst @ X540)) = X539) & (! [X541 : (term > $o),X542 : term] : (($true = ((X541 @ X542))) | ~ ! [X543 : term] : (($true = ((X541 @ X543))) | ~ ($true = ((var @ X543)))) | ~ ! [X544 : term,X545 : term] : (~ ($true = ((X541 @ X545))) | (((X541 @ (ap @ X545 @ X544))) = $true) | ~ (((X541 @ X544)) = $true)) | ~ ! [X546 : term] : (~ ($true = ((X541 @ X546))) | ($true = ((X541 @ (lam @ X546)))))) <=> (induction = $true)) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id))))))))),
  inference(fool_elimination,[],[f4])).
thf(f6,plain,(
  ~ (hoasinduction_lem0 = ! [X0 : (subst > term > subst > $o)] : ? [X1 : (term > $o)] : (hoasinduction_p_and_p_prime @ X0 @ X1))),
  inference(rectify,[],[f3])).
thf(f7,plain,(
  ~ (hoasinduction_lem0 = ((!! @ (subst > term > subst > $o) @ (^[Y0 : subst > term > subst > $o]: (?? @ (term > $o) @ (^[Y1 : term > $o]: (hoasinduction_p_and_p_prime @ Y0 @ Y1)))))))),
  inference(fool_elimination,[],[f6])).
thf(f8,plain,(
  (! [X2 : term,X1 : term,X0 : (subst > term > term)] : ((((ap @ (sub @ X2 @ id) @ X1)) != ((lam @ (X0 @ sh @ one)))) | ~ ! [X3 : subst,X5 : subst,X4 : term] : (((X0 @ (comp @ X3 @ X5) @ (sub @ X4 @ X5))) = ((sub @ (X0 @ X3 @ X4) @ X5)))) <=> (hoaslamnotap = $true)) & (! [X6 : (subst > term > subst > $o),X7 : term] : ((((X6 @ id @ (lam @ X7) @ id)) = $true) | ~ ! [X8 : subst,X9 : subst,X10 : term,X11 : subst] : (($true = ((X6 @ (comp @ X8 @ X9) @ (sub @ X10 @ X9) @ X11))) | ~ (((X6 @ X8 @ X10 @ (comp @ X9 @ X11))) = $true)) | ~ ! [X12 : term] : (($true = ((X6 @ id @ (sub @ X7 @ (push @ X12 @ id)) @ id))) | ~ (((X6 @ id @ X12 @ id)) = $true)) | ~ ! [X13 : subst,X14 : term,X15 : subst,X16 : subst] : (~ ($true = ((X6 @ (comp @ X15 @ X13) @ (sub @ X14 @ X13) @ X16))) | ($true = ((X6 @ X15 @ X14 @ (comp @ X13 @ X16))))) | ~ ! [X17 : (subst > term > term)] : (~ ! [X18 : term] : (~ ($true = ((X6 @ id @ X18 @ id))) | ($true = ((X6 @ id @ (X17 @ id @ X18) @ id)))) | (((X6 @ id @ (lam @ (X17 @ sh @ one)) @ id)) = $true) | ~ ! [X20 : subst,X21 : subst,X19 : term] : (((X17 @ (comp @ X21 @ X20) @ (sub @ X19 @ X20))) = ((sub @ (X17 @ X21 @ X19) @ X20))))) <=> (hoasinduction_lem3 = $true)) & ~ (((var @ (d2term @ d_one))) = $true) & ~ (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) = $true) & ((hoasinduction_lem3v2 = $true) <=> ! [X22 : (term > $o),X23 : (subst > term > subst > $o),X24 : term] : (~ ! [X25 : term,X26 : subst,X27 : subst,X28 : subst] : (~ (((X23 @ X28 @ X25 @ (comp @ X27 @ X26))) = $true) | (((X23 @ (comp @ X28 @ X27) @ (sub @ X25 @ X27) @ X26)) = $true)) | ~ ! [X29 : (subst > term > term)] : (($true = ((X23 @ id @ (lam @ (X29 @ sh @ one)) @ id))) | ~ ! [X30 : term] : (($true = ((X23 @ id @ (X29 @ id @ X30) @ id))) | ~ ($true = ((X23 @ id @ X30 @ id)))) | ~ ! [X31 : subst,X33 : term,X32 : subst] : (((X29 @ (comp @ X31 @ X32) @ (sub @ X33 @ X32))) = ((sub @ (X29 @ X31 @ X33) @ X32)))) | ~ ! [X34 : subst,X35 : subst,X36 : term,X37 : subst] : (~ ($true = ((X23 @ (comp @ X37 @ X35) @ (sub @ X36 @ X35) @ X34))) | ($true = ((X23 @ X37 @ X36 @ (comp @ X35 @ X34))))) | ~ ! [X38 : term] : (((X23 @ id @ X38 @ id)) = ((X22 @ X38))) | ($true = ((X22 @ (lam @ X24)))) | ~ ! [X39 : term] : (~ ($true = ((X22 @ X39))) | (((X22 @ (sub @ X24 @ (push @ X39 @ id)))) = $true)))) & (! [X40 : subst,X41 : term,X42 : (term > $o),X43 : (term > $o)] : (~ ! [X44 : term] : (((X43 @ (sub @ X44 @ (push @ X41 @ X40)))) = ((X42 @ X44))) | ($true = ((X42 @ one))) | ~ ($true = ((X43 @ X41)))) <=> (pushprop_lem1v2 = $true)) & (pushprop = $true) & (! [X45 : subst] : (((comp @ id @ X45)) = X45) <=> (axidl = $true)) & (! [X46 : term,X47 : term,X48 : (subst > term > subst > $o)] : (~ (((X48 @ id @ X47 @ id)) = $true) | ~ ! [X49 : subst,X50 : term,X51 : subst,X52 : subst] : (~ (((X48 @ (comp @ X49 @ X52) @ (sub @ X50 @ X52) @ X51)) = $true) | ($true = ((X48 @ X49 @ X50 @ (comp @ X52 @ X51))))) | ~ ! [X53 : term,X54 : term] : (~ ($true = ((X48 @ id @ X54 @ id))) | ~ ($true = ((X48 @ id @ X53 @ id))) | ($true = ((X48 @ id @ (ap @ (sub @ X53 @ id) @ X54) @ id)))) | ~ ! [X55 : subst,X56 : term,X57 : subst,X58 : subst] : (~ ($true = ((X48 @ X55 @ X56 @ (comp @ X57 @ X58)))) | ($true = ((X48 @ (comp @ X55 @ X57) @ (sub @ X56 @ X57) @ X58)))) | ($true = ((X48 @ id @ (ap @ X46 @ X47) @ id))) | ~ ($true = ((X48 @ id @ X46 @ id)))) <=> (hoasinduction_lem2 = $true)) & (! [X59 : term,X60 : (subst > term > subst > $o)] : (~ ! [X61 : term] : (~ ($true = ((X60 @ id @ X61 @ id))) | ($true = ((X60 @ id @ (sub @ X59 @ (push @ X61 @ id)) @ id)))) | ~ ! [X62 : (subst > term > term)] : (~ ! [X65 : subst,X64 : term,X63 : subst] : (((sub @ (X62 @ X65 @ X64) @ X63)) = ((X62 @ (comp @ X65 @ X63) @ (sub @ X64 @ X63)))) | (((X60 @ id @ (lam @ (X62 @ sh @ one)) @ id)) = $true) | ~ ! [X66 : term] : (($true = ((X60 @ id @ (X62 @ id @ X66) @ id))) | ~ ($true = ((X60 @ id @ X66 @ id))))) | (((X60 @ id @ (lam @ (sub @ X59 @ (push @ one @ sh))) @ id)) = $true)) <=> (hoasinduction_lem3aa = $true)) & ~ (lamnotap = $true) & ((hoasapinj1 = $true) <=> ! [X67 : term,X70 : term,X69 : term,X68 : term] : ((X67 = X68) | (((ap @ (sub @ X67 @ id) @ X70)) != ((ap @ (sub @ X68 @ id) @ X69))))) & (axmap = $true) & (hoasinduction_lem1_gthm = $true) & ((apinj2 = $true) <=> ! [X71 : term,X73 : term,X74 : term,X72 : term] : ((X71 = X72) | (((ap @ X73 @ X72)) != ((ap @ X74 @ X71))))) & (substmonoid_gthm = $true) & ((induction2lem = $true) <=> ! [X75 : subst,X76 : (term > $o),X77 : term] : (($true = ((X76 @ (sub @ X77 @ X75)))) | ~ ! [X78 : term] : ((((X76 @ (sub @ X78 @ X75))) = $true) | ~ (((var @ X78)) = $true)) | ~ ! [X79 : term,X80 : term] : (~ (((X76 @ X80)) = $true) | ~ (((X76 @ X79)) = $true) | (((X76 @ (ap @ X80 @ X79))) = $true)) | ~ ! [X81 : term] : ((((X76 @ (lam @ X81))) = $true) | ~ ! [X82 : term] : (~ ($true = ((X76 @ X82))) | ($true = ((X76 @ (sub @ X81 @ (push @ X82 @ id))))))))) & (one = ((d2term @ d_one))) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X83 : term] : ~ ! [X84 : (subst > term > term)] : ~ ! [X85 : term,X86 : subst] : (((X84 @ X86 @ X85)) = ((sub @ X83 @ (push @ X85 @ X86))))) & ((axscons = $true) <=> ! [X87 : subst] : (((push @ (sub @ one @ X87) @ (comp @ sh @ X87))) = X87)) & (hoasinduction_gthm = $true) & (! [X88 : term,X89 : (subst > term > subst > $o)] : ((((X89 @ id @ (lam @ (sub @ X88 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X90 : (subst > term > term)] : (~ ! [X91 : term] : (($true = ((X89 @ id @ (X90 @ id @ X91) @ id))) | ~ (((X89 @ id @ X91 @ id)) = $true)) | ~ ! [X93 : subst,X94 : subst,X92 : term] : (((sub @ (X90 @ X93 @ X92) @ X94)) = ((X90 @ (comp @ X93 @ X94) @ (sub @ X92 @ X94)))) | ($true = ((X89 @ id @ (lam @ (X90 @ sh @ one)) @ id)))) | ~ ! [X95 : term] : (~ (((X89 @ id @ X95 @ id)) = $true) | ($true = ((X89 @ id @ (sub @ X88 @ (push @ X95 @ id)) @ id))))) <=> (hoasinduction_lem3aa_lthm = $true)) & (axabs = $true) & (hoasinduction_lem2_gthm = $true) & (hoasinduction_lem3b_gthm = $true) & ! [X96 : d_term] : (d_one = X96) & (termmset_gthm = $true) & (apnotvar = $true) & (hoasapinj2_gthm = $true) & ((pushprop_lem0 = $true) <=> ! [X98 : subst,X99 : (term > $o),X97 : term] : ~ ! [X100 : (term > $o)] : ~ ! [X101 : term] : (((X99 @ (sub @ X101 @ (push @ X97 @ X98)))) = ((X100 @ X101)))) & ((hoasapinj2 = $true) <=> ! [X105 : term,X102 : term,X103 : term,X104 : term] : ((X102 = X105) | (((ap @ (sub @ X104 @ id) @ X102)) != ((ap @ (sub @ X103 @ id) @ X105))))) & (! [X106 : term] : (((sub @ X106 @ id)) = X106) <=> (axvarid = $true)) & (! [X107 : term,X108 : term,X109 : (term > $o),X110 : (subst > term > subst > $o)] : (($true = ((X109 @ (ap @ X107 @ X108)))) | ~ ! [X111 : term,X112 : subst,X113 : subst,X114 : subst] : (($true = ((X110 @ X112 @ X111 @ (comp @ X114 @ X113)))) | ~ ($true = ((X110 @ (comp @ X112 @ X114) @ (sub @ X111 @ X114) @ X113)))) | ~ ! [X115 : subst,X116 : subst,X117 : term,X118 : subst] : (~ ($true = ((X110 @ X115 @ X117 @ (comp @ X116 @ X118)))) | (((X110 @ (comp @ X115 @ X116) @ (sub @ X117 @ X116) @ X118)) = $true)) | ~ (((X109 @ X108)) = $true) | ~ (((X109 @ X107)) = $true) | ~ ! [X119 : term] : (((X109 @ X119)) = ((X110 @ id @ X119 @ id))) | ~ ! [X120 : term,X121 : term] : (($true = ((X110 @ id @ (ap @ (sub @ X120 @ id) @ X121) @ id))) | ~ ($true = ((X110 @ id @ X120 @ id))) | ~ ($true = ((X110 @ id @ X121 @ id))))) <=> (hoasinduction_lem2v2 = $true)) & ((hoasinduction_lem3v2_lthm = $true) <=> (! [X122 : term] : (((sub @ X122 @ id)) = X122) => ! [X123 : term,X124 : (term > $o),X125 : (subst > term > subst > $o)] : (~ ! [X126 : term] : (((X124 @ X126)) = ((X125 @ id @ X126 @ id))) | ~ ! [X127 : term,X128 : subst,X129 : subst,X130 : subst] : (~ ($true = ((X125 @ (comp @ X129 @ X130) @ (sub @ X127 @ X130) @ X128))) | (((X125 @ X129 @ X127 @ (comp @ X130 @ X128))) = $true)) | ~ ! [X131 : (subst > term > term)] : (~ ! [X132 : term] : (($true = ((X125 @ id @ (X131 @ id @ X132) @ id))) | ~ (((X125 @ id @ X132 @ id)) = $true)) | ~ ! [X134 : subst,X135 : subst,X133 : term] : (((X131 @ (comp @ X134 @ X135) @ (sub @ X133 @ X135))) = ((sub @ (X131 @ X134 @ X133) @ X135))) | ($true = ((X125 @ id @ (lam @ (X131 @ sh @ one)) @ id)))) | ~ ! [X136 : subst,X137 : term,X138 : subst,X139 : subst] : (($true = ((X125 @ (comp @ X139 @ X138) @ (sub @ X137 @ X138) @ X136))) | ~ (((X125 @ X139 @ X137 @ (comp @ X138 @ X136))) = $true)) | (((X124 @ (lam @ X123))) = $true) | ~ ! [X140 : term] : (($true = ((X124 @ (sub @ X123 @ (push @ X140 @ id))))) | ~ (((X124 @ X140)) = $true))))) & (pushprop_lem0_gthm = $true) & (hoaslaminj_gthm = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ((hoasinduction_lem0_lthm = $true) <=> ! [X141 : (subst > term > subst > $o)] : ~ ! [X142 : (term > $o)] : ~ ! [X143 : term] : (((X141 @ id @ X143 @ id)) = ((X142 @ X143)))) & (ulamvarind = $true) & (pushprop_gthm = $true) & (hoaslamnotvar = $true) & ((hoaslaminj_lthm = $true) <=> (! [X144 : term,X145 : subst] : (((sub @ one @ (push @ X144 @ X145))) = X144) => (! [X147 : subst,X146 : term] : (((comp @ sh @ (push @ X146 @ X147))) = X147) => (! [X149 : term,X148 : term] : ((((lam @ X149)) != ((lam @ X148))) | (X148 = X149)) => ! [X152 : term,X151 : (subst > term > term),X150 : subst,X153 : (subst > term > term)] : ((((X151 @ X150 @ X152)) = ((X153 @ X150 @ X152))) | ~ ! [X154 : term,X155 : subst,X156 : subst] : (((sub @ (X153 @ X155 @ X154) @ X156)) = ((X153 @ (comp @ X155 @ X156) @ (sub @ X154 @ X156)))) | ~ ! [X158 : subst,X157 : subst,X159 : term] : (((X151 @ (comp @ X157 @ X158) @ (sub @ X159 @ X158))) = ((sub @ (X151 @ X157 @ X159) @ X158))) | (((lam @ (X151 @ sh @ one))) != ((lam @ (X153 @ sh @ one))))))))) & ((induction2_lthm = $true) <=> (! [X160 : term] : (((sub @ X160 @ id)) = X160) => (! [X161 : term,X162 : (term > $o),X163 : subst] : (~ ! [X164 : term,X165 : term] : (~ ($true = ((X162 @ X165))) | ($true = ((X162 @ (ap @ X164 @ X165)))) | ~ (((X162 @ X164)) = $true)) | ~ ! [X166 : term] : (($true = ((X162 @ (lam @ X166)))) | ~ ! [X167 : term] : (($true = ((X162 @ (sub @ X166 @ (push @ X167 @ id))))) | ~ ($true = ((X162 @ X167))))) | (((X162 @ (sub @ X161 @ X163))) = $true) | ~ ! [X168 : term] : (($true = ((X162 @ (sub @ X168 @ X163)))) | ~ (((var @ X168)) = $true))) => ! [X169 : term,X170 : (term > $o)] : ((((X170 @ X169)) = $true) | ~ ! [X171 : term] : (~ ! [X172 : term] : (($true = ((X170 @ (sub @ X171 @ (push @ X172 @ id))))) | ~ ($true = ((X170 @ X172)))) | ($true = ((X170 @ (lam @ X171))))) | ~ ! [X173 : term,X174 : term] : (~ (((X170 @ X173)) = $true) | ~ ($true = ((X170 @ X174))) | (((X170 @ (ap @ X174 @ X173))) = $true)) | ~ ! [X175 : term] : (~ ($true = ((var @ X175))) | ($true = ((X170 @ X175)))))))) & ((hoaslaminj = $true) <=> ! [X176 : subst,X179 : (subst > term > term),X178 : (subst > term > term),X177 : term] : (~ ! [X183 : subst,X184 : subst,X185 : term] : (((X179 @ (comp @ X184 @ X183) @ (sub @ X185 @ X183))) = ((sub @ (X179 @ X184 @ X185) @ X183))) | (((X179 @ X176 @ X177)) = ((X178 @ X176 @ X177))) | ~ ! [X180 : term,X181 : subst,X182 : subst] : (((X178 @ (comp @ X182 @ X181) @ (sub @ X180 @ X181))) = ((sub @ (X178 @ X182 @ X180) @ X181))) | (((lam @ (X178 @ sh @ one))) != ((lam @ (X179 @ sh @ one)))))) & ((! [X186 : (subst > term > subst > $o)] : ~ ! [X187 : (term > $o)] : ~ ! [X188 : term] : (((X186 @ id @ X188 @ id)) = ((X187 @ X188))) => (! [X189 : (term > $o),X190 : term] : (($true = ((X189 @ X190))) | ~ ! [X191 : term,X192 : term] : ((((X189 @ (ap @ X191 @ X192))) = $true) | ~ ($true = ((X189 @ X192))) | ~ (((X189 @ X191)) = $true)) | ~ ! [X193 : term] : (($true = ((X189 @ (lam @ X193)))) | ~ ! [X194 : term] : (~ (((X189 @ X194)) = $true) | ($true = ((X189 @ (sub @ X193 @ (push @ X194 @ id))))))) | ~ ! [X195 : term] : (~ ($true = ((var @ X195))) | ($true = ((X189 @ X195))))) => (! [X196 : term] : (((sub @ X196 @ id)) = X196) => (! [X197 : (term > $o),X198 : (subst > term > subst > $o),X199 : term] : (~ ! [X200 : (subst > term > term)] : (~ ! [X203 : subst,X201 : term,X202 : subst] : (((X200 @ (comp @ X203 @ X202) @ (sub @ X201 @ X202))) = ((sub @ (X200 @ X203 @ X201) @ X202))) | ~ ! [X204 : term] : (($true = ((X198 @ id @ (X200 @ id @ X204) @ id))) | ~ ($true = ((X198 @ id @ X204 @ id)))) | (((X198 @ id @ (lam @ (X200 @ sh @ one)) @ id)) = $true)) | ($true = ((X197 @ (lam @ X199)))) | ~ ! [X205 : term] : (~ (((X197 @ X205)) = $true) | (((X197 @ (sub @ X199 @ (push @ X205 @ id)))) = $true)) | ~ ! [X206 : term] : (((X198 @ id @ X206 @ id)) = ((X197 @ X206)))) => ! [X207 : term,X208 : (subst > term > subst > $o)] : (~ ! [X209 : term] : (~ ($true = ((var @ (sub @ X209 @ id)))) | ($true = ((X208 @ id @ X209 @ id)))) | ~ ! [X210 : term,X211 : term] : (~ ($true = ((X208 @ id @ X210 @ id))) | ($true = ((X208 @ id @ (ap @ (sub @ X211 @ id) @ X210) @ id))) | ~ ($true = ((X208 @ id @ X211 @ id)))) | ~ ! [X212 : subst,X213 : subst,X214 : term,X215 : subst] : (~ (((X208 @ X215 @ X214 @ (comp @ X212 @ X213))) = $true) | ($true = ((X208 @ (comp @ X215 @ X212) @ (sub @ X214 @ X212) @ X213)))) | ~ ! [X216 : (subst > term > term)] : (~ ! [X217 : subst,X219 : subst,X218 : term] : (((X216 @ (comp @ X217 @ X219) @ (sub @ X218 @ X219))) = ((sub @ (X216 @ X217 @ X218) @ X219))) | ($true = ((X208 @ id @ (lam @ (X216 @ sh @ one)) @ id))) | ~ ! [X220 : term] : (($true = ((X208 @ id @ (X216 @ id @ X220) @ id))) | ~ (((X208 @ id @ X220 @ id)) = $true))) | ($true = ((X208 @ id @ X207 @ id))) | ~ ! [X221 : subst,X222 : term,X223 : subst,X224 : subst] : (~ ($true = ((X208 @ (comp @ X221 @ X224) @ (sub @ X222 @ X224) @ X223))) | ($true = ((X208 @ X221 @ X222 @ (comp @ X224 @ X223)))))))))) <=> (hoasinduction_lthm_3 = $true)) & (! [X225 : (subst > term > subst > $o),X226 : term] : (~ ! [X227 : term,X228 : term] : (~ ($true = ((X225 @ id @ X228 @ id))) | ($true = ((X225 @ id @ (ap @ (sub @ X227 @ id) @ X228) @ id))) | ~ ($true = ((X225 @ id @ X227 @ id)))) | ~ ! [X229 : (subst > term > term)] : (($true = ((X225 @ id @ (lam @ (X229 @ sh @ one)) @ id))) | ~ ! [X230 : term] : (~ ($true = ((X225 @ id @ X230 @ id))) | (((X225 @ id @ (X229 @ id @ X230) @ id)) = $true)) | ~ ! [X233 : subst,X231 : subst,X232 : term] : (((sub @ (X229 @ X231 @ X232) @ X233)) = ((X229 @ (comp @ X231 @ X233) @ (sub @ X232 @ X233))))) | (((X225 @ id @ X226 @ id)) = $true)) <=> (hoasinduction_no_psi_cond = $true)) & (! [X235 : subst,X234 : term] : (((sub @ one @ (push @ X234 @ X235))) = X234) <=> (axvarcons = $true)) & (pushprop_lem2v2_gthm = $true) & ((hoasinduction_lem3a_lthm = $true) <=> (! [X236 : term] : (((sub @ X236 @ id)) = X236) => (! [X237 : (subst > term > subst > $o),X238 : term] : (~ ! [X239 : (subst > term > term)] : ((((X237 @ id @ (lam @ (X239 @ sh @ one)) @ id)) = $true) | ~ ! [X241 : subst,X242 : term,X240 : subst] : (((sub @ (X239 @ X241 @ X242) @ X240)) = ((X239 @ (comp @ X241 @ X240) @ (sub @ X242 @ X240)))) | ~ ! [X243 : term] : (~ ($true = ((X237 @ id @ X243 @ id))) | ($true = ((X237 @ id @ (X239 @ id @ X243) @ id))))) | (((X237 @ id @ (lam @ (sub @ X238 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X244 : term] : (($true = ((X237 @ id @ (sub @ X238 @ (push @ X244 @ id)) @ id))) | ~ (((X237 @ id @ X244 @ id)) = $true))) => ! [X245 : (subst > term > subst > $o),X246 : term] : (($true = ((X245 @ id @ (lam @ X246) @ id))) | ~ ! [X247 : term] : (~ ($true = ((X245 @ id @ X247 @ id))) | ($true = ((X245 @ id @ (sub @ X246 @ (push @ X247 @ id)) @ id)))) | ~ ! [X248 : (subst > term > term)] : ((((X245 @ id @ (lam @ (X248 @ sh @ one)) @ id)) = $true) | ~ ! [X251 : term,X249 : subst,X250 : subst] : (((X248 @ (comp @ X250 @ X249) @ (sub @ X251 @ X249))) = ((sub @ (X248 @ X250 @ X251) @ X249))) | ~ ! [X252 : term] : (~ (((X245 @ id @ X252 @ id)) = $true) | (((X245 @ id @ (X248 @ id @ X252) @ id)) = $true))))))) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoasinduction_lem3b = $true) <=> ! [X253 : term] : ~ ! [X254 : (subst > term > term)] : (((X254 @ sh @ one)) != ((sub @ X253 @ (push @ one @ sh))))) & ((pushprop_lem0_lthm = $true) <=> ! [X255 : subst,X256 : (term > $o),X257 : term] : ~ ! [X258 : (term > $o)] : ~ ! [X259 : term] : (((X258 @ X259)) = ((X256 @ (sub @ X259 @ (push @ X257 @ X255)))))) & (axapp = $true) & (((d2subst @ d_id)) = sh) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (axassoc = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoasinduction_lthm = $true) <=> (! [X260 : (term > $o),X261 : term] : ((((X260 @ X261)) = $true) | ~ ! [X262 : term] : ((((X260 @ (lam @ X262))) = $true) | ~ ! [X263 : term] : (~ ($true = ((X260 @ X263))) | ($true = ((X260 @ (sub @ X262 @ (push @ X263 @ id))))))) | ~ ! [X264 : term] : (~ (((var @ X264)) = $true) | ($true = ((X260 @ X264)))) | ~ ! [X265 : term,X266 : term] : (~ ($true = ((X260 @ X265))) | ($true = ((X260 @ (ap @ X265 @ X266)))) | ~ (((X260 @ X266)) = $true))) => (! [X267 : term,X268 : (subst > term > subst > $o),X269 : term] : (~ ($true = ((X268 @ id @ X267 @ id))) | ~ ($true = ((X268 @ id @ X269 @ id))) | ~ ! [X270 : subst,X271 : term,X272 : subst,X273 : subst] : (~ ($true = ((X268 @ (comp @ X270 @ X273) @ (sub @ X271 @ X273) @ X272))) | (((X268 @ X270 @ X271 @ (comp @ X273 @ X272))) = $true)) | ~ ! [X274 : subst,X275 : subst,X276 : subst,X277 : term] : (~ ($true = ((X268 @ X275 @ X277 @ (comp @ X276 @ X274)))) | (((X268 @ (comp @ X275 @ X276) @ (sub @ X277 @ X276) @ X274)) = $true)) | ($true = ((X268 @ id @ (ap @ X269 @ X267) @ id))) | ~ ! [X278 : term,X279 : term] : (~ ($true = ((X268 @ id @ X279 @ id))) | (((X268 @ id @ (ap @ (sub @ X278 @ id) @ X279) @ id)) = $true) | ~ (((X268 @ id @ X278 @ id)) = $true))) => (! [X280 : term,X281 : (subst > term > subst > $o)] : (($true = ((X281 @ id @ (lam @ X280) @ id))) | ~ ! [X282 : subst,X283 : subst,X284 : subst,X285 : term] : (~ (((X281 @ X282 @ X285 @ (comp @ X283 @ X284))) = $true) | ($true = ((X281 @ (comp @ X282 @ X283) @ (sub @ X285 @ X283) @ X284)))) | ~ ! [X286 : term] : (~ (((X281 @ id @ X286 @ id)) = $true) | (((X281 @ id @ (sub @ X280 @ (push @ X286 @ id)) @ id)) = $true)) | ~ ! [X287 : subst,X288 : subst,X289 : subst,X290 : term] : (~ (((X281 @ (comp @ X288 @ X289) @ (sub @ X290 @ X289) @ X287)) = $true) | ($true = ((X281 @ X288 @ X290 @ (comp @ X289 @ X287))))) | ~ ! [X291 : (subst > term > term)] : (($true = ((X281 @ id @ (lam @ (X291 @ sh @ one)) @ id))) | ~ ! [X293 : term,X292 : subst,X294 : subst] : (((X291 @ (comp @ X294 @ X292) @ (sub @ X293 @ X292))) = ((sub @ (X291 @ X294 @ X293) @ X292))) | ~ ! [X295 : term] : (($true = ((X281 @ id @ (X291 @ id @ X295) @ id))) | ~ ($true = ((X281 @ id @ X295 @ id)))))) => ! [X296 : term,X297 : (subst > term > subst > $o)] : (~ ! [X298 : subst,X299 : subst,X300 : subst,X301 : term] : (($true = ((X297 @ X298 @ X301 @ (comp @ X299 @ X300)))) | ~ (((X297 @ (comp @ X298 @ X299) @ (sub @ X301 @ X299) @ X300)) = $true)) | ~ ! [X302 : (subst > term > term)] : (($true = ((X297 @ id @ (lam @ (X302 @ sh @ one)) @ id))) | ~ ! [X303 : term] : ((((X297 @ id @ (X302 @ id @ X303) @ id)) = $true) | ~ ($true = ((X297 @ id @ X303 @ id)))) | ~ ! [X306 : subst,X304 : term,X305 : subst] : (((sub @ (X302 @ X306 @ X304) @ X305)) = ((X302 @ (comp @ X306 @ X305) @ (sub @ X304 @ X305))))) | ~ ! [X307 : subst,X308 : subst,X309 : subst,X310 : term] : (~ ($true = ((X297 @ X307 @ X310 @ (comp @ X308 @ X309)))) | ($true = ((X297 @ (comp @ X307 @ X308) @ (sub @ X310 @ X308) @ X309)))) | ~ ! [X311 : term] : (($true = ((X297 @ id @ X311 @ id))) | ~ ($true = ((var @ (sub @ X311 @ id))))) | ~ ! [X312 : term,X313 : term] : (~ ($true = ((X297 @ id @ X312 @ id))) | ($true = ((X297 @ id @ (ap @ (sub @ X312 @ id) @ X313) @ id))) | ~ (((X297 @ id @ X313 @ id)) = $true)) | ($true = ((X297 @ id @ X296 @ id)))))))) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((hoasinduction_lem3v2a_lthm = $true) <=> (! [X314 : term] : ~ ! [X315 : (subst > term > term)] : ~ ! [X317 : term,X316 : subst] : (((sub @ X314 @ (push @ X317 @ X316))) = ((X315 @ X316 @ X317))) => (! [X318 : term] : (((sub @ X318 @ id)) = X318) => ! [X319 : (subst > term > subst > $o),X320 : (term > $o),X321 : term] : (~ ! [X322 : (subst > term > term)] : (~ ! [X323 : subst,X325 : subst,X324 : term] : (((sub @ (X322 @ X323 @ X324) @ X325)) = ((X322 @ (comp @ X323 @ X325) @ (sub @ X324 @ X325)))) | ~ ! [X326 : term] : (~ ($true = ((X319 @ id @ X326 @ id))) | ($true = ((X319 @ id @ (X322 @ id @ X326) @ id)))) | (((X319 @ id @ (lam @ (X322 @ sh @ one)) @ id)) = $true)) | ($true = ((X320 @ (lam @ X321)))) | ~ ! [X327 : term] : (((X319 @ id @ X327 @ id)) = ((X320 @ X327))) | ~ ! [X328 : term] : (~ ($true = ((X320 @ X328))) | ($true = ((X320 @ (sub @ X321 @ (push @ X328 @ id)))))))))) & (pushprop_lem1_gthm = $true) & (pushprop_lem3v2_lthm = $true) & (hoasinduction_lem1 = $true) & (pushprop_lem3v2 = $true) & ~ (ulamvar1 = $true) & (hoasinduction_lem1v2 = $true) & ((! [X329 : subst] : (((comp @ id @ X329)) = X329) => (! [X330 : subst] : (((comp @ X330 @ id)) = X330) => (! [X332 : subst] : (((comp @ id @ X332)) = X332) & ! [X331 : subst] : (((comp @ X331 @ id)) = X331)))) <=> (substmonoid_lthm = $true)) & (induction2_gthm = $true) & ((hoasinduction_lem3_lthm = $true) <=> (! [X333 : term] : (((sub @ X333 @ id)) = X333) => (! [X334 : term,X335 : (subst > term > subst > $o)] : (~ ! [X336 : term] : (~ (((X335 @ id @ X336 @ id)) = $true) | (((X335 @ id @ (sub @ X334 @ (push @ X336 @ id)) @ id)) = $true)) | ~ ! [X337 : (subst > term > term)] : (~ ! [X338 : term] : (($true = ((X335 @ id @ (X337 @ id @ X338) @ id))) | ~ ($true = ((X335 @ id @ X338 @ id)))) | ($true = ((X335 @ id @ (lam @ (X337 @ sh @ one)) @ id))) | ~ ! [X339 : term,X341 : subst,X340 : subst] : (((sub @ (X337 @ X341 @ X339) @ X340)) = ((X337 @ (comp @ X341 @ X340) @ (sub @ X339 @ X340))))) | ($true = ((X335 @ id @ (lam @ (sub @ X334 @ (push @ one @ sh))) @ id)))) => ! [X342 : (subst > term > subst > $o),X343 : term] : (~ ! [X344 : term] : (~ ($true = ((X342 @ id @ X344 @ id))) | ($true = ((X342 @ id @ (sub @ X343 @ (push @ X344 @ id)) @ id)))) | ($true = ((X342 @ id @ (lam @ X343) @ id))) | ~ ! [X345 : subst,X346 : term,X347 : subst,X348 : subst] : (($true = ((X342 @ X345 @ X346 @ (comp @ X347 @ X348)))) | ~ (((X342 @ (comp @ X345 @ X347) @ (sub @ X346 @ X347) @ X348)) = $true)) | ~ ! [X349 : (subst > term > term)] : (~ ! [X350 : term] : (($true = ((X342 @ id @ (X349 @ id @ X350) @ id))) | ~ (((X342 @ id @ X350 @ id)) = $true)) | ($true = ((X342 @ id @ (lam @ (X349 @ sh @ one)) @ id))) | ~ ! [X353 : subst,X352 : subst,X351 : term] : (((X349 @ (comp @ X353 @ X352) @ (sub @ X351 @ X352))) = ((sub @ (X349 @ X353 @ X351) @ X352)))) | ~ ! [X354 : subst,X355 : subst,X356 : subst,X357 : term] : (($true = ((X342 @ (comp @ X356 @ X355) @ (sub @ X357 @ X355) @ X354))) | ~ ($true = ((X342 @ X356 @ X357 @ (comp @ X355 @ X354))))))))) & (pushprop_lem1v2_gthm = $true) & (lamnotvar = $true) & ! [X358 : d_term,X359 : d_term] : ((((d2term @ X359)) = ((d2term @ X358))) => (X358 = X359)) & (axclos = $true) & (id = ((d2subst @ d_id))) & (hoasinduction_lem2v2_gthm = $true) & ((termmset = $true) <=> ! [X360 : term] : (((sub @ X360 @ id)) = X360)) & (pushprop_lem2v2_lthm = $true) & (ulamvarsh = $true) & (hoasapinj1_gthm = $true) & ((! [X362 : subst,X361 : term] : (((sub @ one @ (push @ X361 @ X362))) = X361) => (! [X364 : subst,X363 : term] : (((comp @ sh @ (push @ X363 @ X364))) = X364) => ! [X365 : (term > $o),X366 : subst,X367 : (term > $o),X368 : term,X369 : term] : (~ ($true = ((X365 @ X369))) | ($true = ((X367 @ (sub @ X369 @ (push @ X368 @ X366)))))))) <=> (pushprop_lem1_lthm = $true)) & (hoasinduction_lem3v2_gthm = $true) & (hoasinduction_lem2_lthm = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (hoasinduction_lem1_lthm = $true) & ((hoasinduction_lem3aaa = $true) <=> ! [X370 : term,X371 : (subst > term > subst > $o)] : (($true = ((X371 @ id @ (lam @ (sub @ X370 @ (push @ one @ sh))) @ id))) | ~ ! [X372 : (subst > term > term),X373 : term] : (~ ! [X374 : term] : (($true = ((X371 @ id @ (X372 @ id @ X374) @ id))) | ~ ($true = ((X371 @ id @ X374 @ id)))) | ~ ! [X376 : subst,X375 : term,X377 : subst] : (((sub @ (sub @ X373 @ (push @ X375 @ X377)) @ X376)) = ((sub @ (X372 @ X377 @ X375) @ X376))) | (((X371 @ id @ (lam @ (X372 @ sh @ one)) @ id)) = $true) | ~ ! [X378 : subst,X379 : subst,X380 : term] : (((sub @ X373 @ (push @ (sub @ X380 @ X379) @ (comp @ X378 @ X379)))) = ((X372 @ (comp @ X378 @ X379) @ (sub @ X380 @ X379))))) | ~ ! [X381 : term] : (~ ($true = ((X371 @ id @ X381 @ id))) | (((X371 @ id @ (sub @ X370 @ (push @ X381 @ id)) @ id)) = $true)))) & ((! [X382 : (subst > term > subst > $o)] : ~ ! [X383 : (term > $o)] : ~ ! [X384 : term] : (((X382 @ id @ X384 @ id)) = ((X383 @ X384))) => (! [X385 : (term > $o),X386 : term] : (($true = ((X385 @ X386))) | ~ ! [X387 : term,X388 : term] : ((((X385 @ (ap @ X388 @ X387))) = $true) | ~ ($true = ((X385 @ X387))) | ~ ($true = ((X385 @ X388)))) | ~ ! [X389 : term] : (~ ! [X390 : term] : (($true = ((X385 @ (sub @ X389 @ (push @ X390 @ id))))) | ~ ($true = ((X385 @ X390)))) | ($true = ((X385 @ (lam @ X389))))) | ~ ! [X391 : term] : (($true = ((X385 @ X391))) | ~ (((var @ X391)) = $true))) => (! [X392 : term] : (((sub @ X392 @ id)) = X392) => (! [X393 : (subst > term > subst > $o),X394 : term,X395 : (term > $o)] : (~ ! [X396 : (subst > term > term)] : (($true = ((X393 @ id @ (lam @ (X396 @ sh @ one)) @ id))) | ~ ! [X397 : term] : (($true = ((X393 @ id @ (X396 @ id @ X397) @ id))) | ~ ($true = ((X393 @ id @ X397 @ id)))) | ~ ! [X399 : term,X398 : subst,X400 : subst] : (((sub @ (X396 @ X398 @ X399) @ X400)) = ((X396 @ (comp @ X398 @ X400) @ (sub @ X399 @ X400))))) | ~ ! [X401 : term] : (($true = ((X395 @ (sub @ X394 @ (push @ X401 @ id))))) | ~ ($true = ((X395 @ X401)))) | ($true = ((X395 @ (lam @ X394)))) | ~ ! [X402 : term] : (((X393 @ id @ X402 @ id)) = ((X395 @ X402)))) => ! [X403 : (subst > term > subst > $o),X404 : term] : (($true = ((X403 @ id @ X404 @ id))) | ~ ! [X405 : (subst > term > term)] : (~ ! [X408 : term,X406 : subst,X407 : subst] : (((sub @ (X405 @ X407 @ X408) @ X406)) = ((X405 @ (comp @ X407 @ X406) @ (sub @ X408 @ X406)))) | (((X403 @ id @ (lam @ (X405 @ sh @ one)) @ id)) = $true) | ~ ! [X409 : term] : (~ ($true = ((X403 @ id @ X409 @ id))) | (((X403 @ id @ (X405 @ id @ X409) @ id)) = $true))) | ~ ! [X410 : term,X411 : term] : (~ (((X403 @ id @ X411 @ id)) = $true) | ($true = ((X403 @ id @ (ap @ (sub @ X411 @ id) @ X410) @ id))) | ~ ($true = ((X403 @ id @ X410 @ id))))))))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (hoasinduction_lem3a_gthm = $true) & (hoasinduction_lem1v2_gthm = $true) & ! [X412 : d_subst,X413 : d_subst] : ((((d2subst @ X413)) = ((d2subst @ X412))) => (X412 = X413)) & (! [X414 : term,X415 : (subst > term > subst > $o)] : (($true = ((X415 @ id @ (lam @ X414) @ id))) | ~ ! [X416 : term] : ((((X415 @ id @ (sub @ X414 @ (push @ X416 @ id)) @ id)) = $true) | ~ ($true = ((X415 @ id @ X416 @ id)))) | ~ ! [X417 : (subst > term > term)] : (($true = ((X415 @ id @ (lam @ (X417 @ sh @ one)) @ id))) | ~ ! [X418 : term,X420 : subst,X419 : subst] : (((sub @ (X417 @ X420 @ X418) @ X419)) = ((X417 @ (comp @ X420 @ X419) @ (sub @ X418 @ X419)))) | ~ ! [X421 : term] : (~ ($true = ((X415 @ id @ X421 @ id))) | (((X415 @ id @ (X417 @ id @ X421) @ id)) = $true)))) <=> (hoasinduction_lem3a = $true)) & ! [X422 : term] : ? [X423 : d_term] : (((d2term @ X423)) = X422) & (hoasapnotvar = $true) & (axvarshift = $true) & ((axidr = $true) <=> ! [X424 : subst] : (((comp @ X424 @ id)) = X424)) & (hoaslamnotvar_gthm = $true) & ((hoasinduction = $true) <=> ! [X425 : term,X426 : (subst > term > subst > $o)] : ((((X426 @ id @ X425 @ id)) = $true) | ~ ! [X427 : term,X428 : term] : (~ ($true = ((X426 @ id @ X428 @ id))) | ($true = ((X426 @ id @ (ap @ (sub @ X428 @ id) @ X427) @ id))) | ~ ($true = ((X426 @ id @ X427 @ id)))) | ~ ! [X429 : (subst > term > term)] : (~ ! [X431 : subst,X430 : term,X432 : subst] : (((X429 @ (comp @ X432 @ X431) @ (sub @ X430 @ X431))) = ((sub @ (X429 @ X432 @ X430) @ X431))) | ($true = ((X426 @ id @ (lam @ (X429 @ sh @ one)) @ id))) | ~ ! [X433 : term] : (~ ($true = ((X426 @ id @ X433 @ id))) | ($true = ((X426 @ id @ (X429 @ id @ X433) @ id))))) | ~ ! [X434 : term,X435 : subst,X436 : subst,X437 : subst] : (~ (((X426 @ (comp @ X437 @ X436) @ (sub @ X434 @ X436) @ X435)) = $true) | ($true = ((X426 @ X437 @ X434 @ (comp @ X436 @ X435))))) | ~ ! [X438 : term,X439 : subst,X440 : subst,X441 : subst] : (~ ($true = ((X426 @ X440 @ X438 @ (comp @ X439 @ X441)))) | (((X426 @ (comp @ X440 @ X439) @ (sub @ X438 @ X439) @ X441)) = $true)) | ~ ! [X442 : term] : (~ ($true = ((var @ (sub @ X442 @ id)))) | ($true = ((X426 @ id @ X442 @ id)))))) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (! [X443 : term,X444 : term] : ((((lam @ X443)) != ((lam @ X444))) | (X443 = X444)) <=> (laminj = $true)) & (! [X445 : (subst > term > subst > $o)] : ~ ! [X446 : (term > $o)] : ~ ! [X447 : term] : (((X445 @ id @ X447 @ id)) = ((X446 @ X447))) <=> (hoasinduction_lem0 = $true)) & ((! [X451 : term,X448 : term,X449 : term,X450 : term] : ((((ap @ X451 @ X448)) != ((ap @ X449 @ X450))) | (X448 = X450)) => ! [X455 : term,X453 : term,X454 : term,X452 : term] : ((((ap @ (sub @ X454 @ id) @ X452)) != ((ap @ (sub @ X453 @ id) @ X455))) | (X452 = X455))) <=> (hoasapinj2_lthm = $true)) & (! [X456 : (term > $o),X457 : term] : (($true = ((X456 @ X457))) | ~ ! [X458 : term,X459 : term] : (($true = ((X456 @ (ap @ X459 @ X458)))) | ~ (((X456 @ X458)) = $true) | ~ (((X456 @ X459)) = $true)) | ~ ! [X460 : term] : (~ ! [X461 : term] : (~ ($true = ((X456 @ X461))) | (((X456 @ (sub @ X460 @ (push @ X461 @ id)))) = $true)) | (((X456 @ (lam @ X460))) = $true)) | ~ ! [X462 : term] : (($true = ((X456 @ X462))) | ~ ($true = ((var @ X462))))) <=> (induction2 = $true)) & (hoasinduction_lem3_gthm = $true) & ((! [X463 : subst,X464 : term] : (((sub @ one @ (push @ X464 @ X463))) = X464) => (! [X465 : term,X466 : subst] : (((comp @ sh @ (push @ X465 @ X466))) = X466) => (! [X467 : subst] : (((comp @ X467 @ id)) = X467) => (! [X468 : (term > $o),X469 : term] : (~ ! [X470 : term,X471 : term] : (($true = ((X468 @ (ap @ X471 @ X470)))) | ~ ($true = ((X468 @ X471))) | ~ ($true = ((X468 @ X470)))) | ($true = ((X468 @ X469))) | ~ ! [X472 : term] : (~ ($true = ((X468 @ X472))) | ($true = ((X468 @ (lam @ X472))))) | ~ ! [X473 : term] : (~ (((var @ X473)) = $true) | (((X468 @ X473)) = $true))) => ! [X474 : subst,X475 : term,X476 : (term > $o)] : (~ ! [X477 : term] : (~ ! [X478 : term] : (~ ($true = ((X476 @ X478))) | ($true = ((X476 @ (sub @ X477 @ (push @ X478 @ id)))))) | ($true = ((X476 @ (lam @ X477))))) | ~ ! [X479 : term] : (($true = ((X476 @ (sub @ X479 @ X474)))) | ~ (((var @ X479)) = $true)) | ($true = ((X476 @ (sub @ X475 @ X474)))) | ~ ! [X480 : term,X481 : term] : ((((X476 @ (ap @ X481 @ X480))) = $true) | ~ ($true = ((X476 @ X480))) | ~ ($true = ((X476 @ X481))))))))) <=> (induction2lem_lthm = $true)) & (hoasapnotvar_gthm = $true) & (! [X483 : term,X482 : term] : ((X482 = X483) | (((sub @ X483 @ sh)) != ((sub @ X482 @ sh)))) <=> (shinj = $true)) & (induction2lem_gthm = $true) & (! [X484 : (subst > term > subst > $o),X485 : (term > $o),X486 : term] : (~ ! [X487 : term] : (((X484 @ id @ X487 @ id)) = ((X485 @ X487))) | ($true = ((X485 @ (lam @ X486)))) | ~ ! [X488 : term] : (($true = ((X485 @ (sub @ X486 @ (push @ X488 @ id))))) | ~ ($true = ((X485 @ X488)))) | ~ ! [X489 : (subst > term > term)] : (~ ! [X490 : subst,X492 : subst,X491 : term] : (((X489 @ (comp @ X492 @ X490) @ (sub @ X491 @ X490))) = ((sub @ (X489 @ X492 @ X491) @ X490))) | ~ ! [X493 : term] : (~ ($true = ((X484 @ id @ X493 @ id))) | ($true = ((X484 @ id @ (X489 @ id @ X493) @ id)))) | ($true = ((X484 @ id @ (lam @ (X489 @ sh @ one)) @ id))))) <=> (hoasinduction_lem3v2a = $true)) & ((hoasinduction_lem3v2_f = $true) <=> ! [X494 : term] : ~ ! [X495 : (subst > term > term)] : ~ ! [X496 : term,X497 : subst] : (((sub @ X494 @ (push @ X496 @ X497))) = ((X495 @ X497 @ X496)))) & (pushprop_lthm = $true) & (pushprop_lthm_orig = $true) & (hoaslamnotap_gthm = $true) & (hoaslamnotvar_lthm = $true) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X498 : term] : ~ ! [X499 : (subst > term > term)] : (((X499 @ sh @ one)) != ((sub @ X498 @ (push @ one @ sh))))) & (hoasapnotvar_lthm = $true) & ((termmset_lthm = $true) <=> (! [X500 : term] : (((sub @ X500 @ id)) = X500) => ! [X501 : term] : (((sub @ X501 @ id)) = X501))) & ((substmonoid = $true) <=> (! [X502 : subst] : (((comp @ X502 @ id)) = X502) & ! [X503 : subst] : (((comp @ id @ X503)) = X503))) & (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)))))))))))))) & (! [X504 : subst,X505 : term,X506 : term,X507 : (term > $o),X508 : (term > $o)] : (($true = ((X508 @ (sub @ X505 @ (push @ X506 @ X504))))) | ~ ($true = ((X507 @ X505)))) <=> (pushprop_lem1 = $true)) & ((! [X509 : term] : (((sub @ X509 @ id)) = X509) => (! [X510 : term,X513 : term,X511 : term,X512 : term] : ((X510 = X512) | (((ap @ X512 @ X513)) != ((ap @ X510 @ X511)))) => ! [X517 : term,X515 : term,X514 : term,X516 : term] : ((X514 = X516) | (((ap @ (sub @ X514 @ id) @ X517)) != ((ap @ (sub @ X516 @ id) @ X515)))))) <=> (hoasapinj1_lthm = $true)) & ((apinj1 = $true) <=> ! [X518 : term,X519 : term,X520 : term,X521 : term] : ((((ap @ X518 @ X520)) != ((ap @ X521 @ X519))) | (X518 = X521))) & ((pushprop_lem1v2_lthm = $true) <=> (! [X522 : term,X523 : subst] : (((sub @ one @ (push @ X522 @ X523))) = X522) => ! [X524 : (term > $o),X525 : term,X526 : subst,X527 : (term > $o)] : ((((X524 @ one)) = $true) | ~ ($true = ((X527 @ X525))) | ~ ! [X528 : term] : (((X524 @ X528)) = ((X527 @ (sub @ X528 @ (push @ X525 @ X526)))))))) & (! [X530 : term,X529 : subst] : (((comp @ sh @ (push @ X530 @ X529))) = X529) <=> (axshiftcons = $true)) & (pushprop_lem2v2 = $true) & (hoaslamnotap_lthm = $true) & ! [X531 : d_subst] : (d_id = X531) & ! [X532 : subst] : ? [X533 : d_subst] : (((d2subst @ X533)) = X532) & (! [X534 : (term > $o),X535 : term] : (($true = ((X534 @ X535))) | ~ ! [X536 : term] : (($true = ((X534 @ X536))) | ~ (((var @ X536)) = $true)) | ~ ! [X537 : term,X538 : term] : (~ (((X534 @ X538)) = $true) | ($true = ((X534 @ (ap @ X538 @ X537)))) | ~ ($true = ((X534 @ X537)))) | ~ ! [X539 : term] : (~ ($true = ((X534 @ X539))) | ($true = ((X534 @ (lam @ X539)))))) <=> (induction = $true)) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id))))))))),
  inference(rectify,[],[f5])).
thf(f9,plain,(
  ((induction2lem = $true) <=> ! [X76 : (term > $o),X75 : subst,X77 : term] : (~ ! [X81 : term] : ((((X76 @ (lam @ X81))) = $true) | ~ ! [X82 : term] : (($true = ((X76 @ (sub @ X81 @ (push @ X82 @ id))))) | ($true != ((X76 @ X82))))) | ~ ! [X78 : term] : ((((var @ X78)) != $true) | (((X76 @ (sub @ X78 @ X75))) = $true)) | ($true = ((X76 @ (sub @ X77 @ X75)))) | ~ ! [X79 : term,X80 : term] : ((((X76 @ X79)) != $true) | (((X76 @ (ap @ X80 @ X79))) = $true) | (((X76 @ X80)) != $true)))) & (pushprop_lem2v2_gthm = $true) & (hoasapnotvar_lthm = $true) & (! [X2 : term,X1 : term,X0 : (subst > term > term)] : ((((ap @ (sub @ X2 @ id) @ X1)) != ((lam @ (X0 @ sh @ one)))) | ~ ! [X3 : subst,X5 : subst,X4 : term] : (((X0 @ (comp @ X3 @ X5) @ (sub @ X4 @ X5))) = ((sub @ (X0 @ X3 @ X4) @ X5)))) <=> (hoaslamnotap = $true)) & (pushprop_gthm = $true) & ((! [X451 : term,X448 : term,X449 : term,X450 : term] : ((((ap @ X451 @ X448)) != ((ap @ X449 @ X450))) | (X448 = X450)) => ! [X455 : term,X453 : term,X454 : term,X452 : term] : ((((ap @ (sub @ X454 @ id) @ X452)) != ((ap @ (sub @ X453 @ id) @ X455))) | (X452 = X455))) <=> (hoasapinj2_lthm = $true)) & ((hoasinduction_lem3_lthm = $true) <=> (! [X333 : term] : (((sub @ X333 @ id)) = X333) => (! [X334 : term,X335 : (subst > term > subst > $o)] : (~ ! [X336 : term] : ((((X335 @ id @ (sub @ X334 @ (push @ X336 @ id)) @ id)) = $true) | (((X335 @ id @ X336 @ id)) != $true)) | ($true = ((X335 @ id @ (lam @ (sub @ X334 @ (push @ one @ sh))) @ id))) | ~ ! [X337 : (subst > term > term)] : (~ ! [X338 : term] : (($true != ((X335 @ id @ X338 @ id))) | ($true = ((X335 @ id @ (X337 @ id @ X338) @ id)))) | ~ ! [X339 : term,X341 : subst,X340 : subst] : (((sub @ (X337 @ X341 @ X339) @ X340)) = ((X337 @ (comp @ X341 @ X340) @ (sub @ X339 @ X340)))) | ($true = ((X335 @ id @ (lam @ (X337 @ sh @ one)) @ id))))) => ! [X343 : term,X342 : (subst > term > subst > $o)] : (($true = ((X342 @ id @ (lam @ X343) @ id))) | ~ ! [X349 : (subst > term > term)] : (~ ! [X353 : subst,X352 : subst,X351 : term] : (((X349 @ (comp @ X353 @ X352) @ (sub @ X351 @ X352))) = ((sub @ (X349 @ X353 @ X351) @ X352))) | ($true = ((X342 @ id @ (lam @ (X349 @ sh @ one)) @ id))) | ~ ! [X350 : term] : ((((X342 @ id @ X350 @ id)) != $true) | ($true = ((X342 @ id @ (X349 @ id @ X350) @ id))))) | ~ ! [X344 : term] : (($true = ((X342 @ id @ (sub @ X343 @ (push @ X344 @ id)) @ id))) | ($true != ((X342 @ id @ X344 @ id)))) | ~ ! [X347 : subst,X345 : subst,X346 : term,X348 : subst] : (($true = ((X342 @ X345 @ X346 @ (comp @ X347 @ X348)))) | (((X342 @ (comp @ X345 @ X347) @ (sub @ X346 @ X347) @ X348)) != $true)) | ~ ! [X354 : subst,X356 : subst,X355 : subst,X357 : term] : (($true = ((X342 @ (comp @ X356 @ X355) @ (sub @ X357 @ X355) @ X354))) | ($true != ((X342 @ X356 @ X357 @ (comp @ X355 @ X354))))))))) & ((hoasinduction_lem3v2_lthm = $true) <=> (! [X122 : term] : (((sub @ X122 @ id)) = X122) => ! [X123 : term,X125 : (subst > term > subst > $o),X124 : (term > $o)] : (~ ! [X136 : subst,X139 : subst,X138 : subst,X137 : term] : (($true = ((X125 @ (comp @ X139 @ X138) @ (sub @ X137 @ X138) @ X136))) | (((X125 @ X139 @ X137 @ (comp @ X138 @ X136))) != $true)) | ~ ! [X127 : term,X130 : subst,X128 : subst,X129 : subst] : (($true != ((X125 @ (comp @ X129 @ X130) @ (sub @ X127 @ X130) @ X128))) | (((X125 @ X129 @ X127 @ (comp @ X130 @ X128))) = $true)) | ~ ! [X131 : (subst > term > term)] : (($true = ((X125 @ id @ (lam @ (X131 @ sh @ one)) @ id))) | ~ ! [X132 : term] : (($true = ((X125 @ id @ (X131 @ id @ X132) @ id))) | (((X125 @ id @ X132 @ id)) != $true)) | ~ ! [X134 : subst,X135 : subst,X133 : term] : (((X131 @ (comp @ X134 @ X135) @ (sub @ X133 @ X135))) = ((sub @ (X131 @ X134 @ X133) @ X135)))) | (((X124 @ (lam @ X123))) = $true) | ~ ! [X126 : term] : (((X124 @ X126)) = ((X125 @ id @ X126 @ id))) | ~ ! [X140 : term] : (($true = ((X124 @ (sub @ X123 @ (push @ X140 @ id))))) | (((X124 @ X140)) != $true))))) & ((pushprop_lem0_lthm = $true) <=> ! [X255 : subst,X256 : (term > $o),X257 : term] : ~ ! [X258 : (term > $o)] : ~ ! [X259 : term] : (((X258 @ X259)) = ((X256 @ (sub @ X259 @ (push @ X257 @ X255)))))) & (! [X45 : subst] : (((comp @ id @ X45)) = X45) <=> (axidl = $true)) & (hoasinduction_lem3a_gthm = $true) & (induction2lem_gthm = $true) & (axclos = $true) & (hoasinduction_lem1v2 = $true) & (hoasinduction_gthm = $true) & ((hoasinduction_lem0_lthm = $true) <=> ! [X141 : (subst > term > subst > $o)] : ~ ! [X142 : (term > $o)] : ~ ! [X143 : term] : (((X141 @ id @ X143 @ id)) = ((X142 @ X143)))) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (! [X382 : (subst > term > subst > $o)] : ~ ! [X383 : (term > $o)] : ~ ! [X384 : term] : (((X382 @ id @ X384 @ id)) = ((X383 @ X384))) => (! [X386 : term,X385 : (term > $o)] : (~ ! [X391 : term] : (($true = ((X385 @ X391))) | (((var @ X391)) != $true)) | ~ ! [X389 : term] : (~ ! [X390 : term] : (($true != ((X385 @ X390))) | ($true = ((X385 @ (sub @ X389 @ (push @ X390 @ id)))))) | ($true = ((X385 @ (lam @ X389))))) | ~ ! [X387 : term,X388 : term] : ((((X385 @ (ap @ X388 @ X387))) = $true) | ($true != ((X385 @ X387))) | ($true != ((X385 @ X388)))) | ($true = ((X385 @ X386)))) => (! [X392 : term] : (((sub @ X392 @ id)) = X392) => (! [X393 : (subst > term > subst > $o),X394 : term,X395 : (term > $o)] : (~ ! [X402 : term] : (((X393 @ id @ X402 @ id)) = ((X395 @ X402))) | ($true = ((X395 @ (lam @ X394)))) | ~ ! [X401 : term] : (($true = ((X395 @ (sub @ X394 @ (push @ X401 @ id))))) | ($true != ((X395 @ X401)))) | ~ ! [X396 : (subst > term > term)] : (($true = ((X393 @ id @ (lam @ (X396 @ sh @ one)) @ id))) | ~ ! [X397 : term] : (($true = ((X393 @ id @ (X396 @ id @ X397) @ id))) | ($true != ((X393 @ id @ X397 @ id)))) | ~ ! [X399 : term,X398 : subst,X400 : subst] : (((sub @ (X396 @ X398 @ X399) @ X400)) = ((X396 @ (comp @ X398 @ X400) @ (sub @ X399 @ X400)))))) => ! [X403 : (subst > term > subst > $o),X404 : term] : (~ ! [X411 : term,X410 : term] : ((((X403 @ id @ X411 @ id)) != $true) | ($true = ((X403 @ id @ (ap @ (sub @ X411 @ id) @ X410) @ id))) | ($true != ((X403 @ id @ X410 @ id)))) | ~ ! [X405 : (subst > term > term)] : (~ ! [X409 : term] : (($true != ((X403 @ id @ X409 @ id))) | (((X403 @ id @ (X405 @ id @ X409) @ id)) = $true)) | ~ ! [X408 : term,X406 : subst,X407 : subst] : (((sub @ (X405 @ X407 @ X408) @ X406)) = ((X405 @ (comp @ X407 @ X406) @ (sub @ X408 @ X406)))) | (((X403 @ id @ (lam @ (X405 @ sh @ one)) @ id)) = $true)) | ($true = ((X403 @ id @ X404 @ id))))))))) & (one = ((d2term @ d_one))) & (hoaslamnotap_lthm = $true) & (hoasinduction_lem3v2_gthm = $true) & (induction2_gthm = $true) & (hoasinduction_lem3_gthm = $true) & (pushprop_lem2v2_lthm = $true) & (pushprop_lem1v2_gthm = $true) & (! [X43 : (term > $o),X40 : subst,X41 : term,X42 : (term > $o)] : (~ ! [X44 : term] : (((X43 @ (sub @ X44 @ (push @ X41 @ X40)))) = ((X42 @ X44))) | ($true = ((X42 @ one))) | ($true != ((X43 @ X41)))) <=> (pushprop_lem1v2 = $true)) & (hoasinduction_lem1_gthm = $true) & (pushprop_lthm = $true) & ((hoasinduction_lthm = $true) <=> (! [X261 : term,X260 : (term > $o)] : (~ ! [X264 : term] : ((((var @ X264)) != $true) | ($true = ((X260 @ X264)))) | (((X260 @ X261)) = $true) | ~ ! [X262 : term] : (~ ! [X263 : term] : (($true = ((X260 @ (sub @ X262 @ (push @ X263 @ id))))) | ($true != ((X260 @ X263)))) | (((X260 @ (lam @ X262))) = $true)) | ~ ! [X266 : term,X265 : term] : (($true = ((X260 @ (ap @ X265 @ X266)))) | (((X260 @ X266)) != $true) | ($true != ((X260 @ X265))))) => (! [X267 : term,X268 : (subst > term > subst > $o),X269 : term] : (~ ! [X279 : term,X278 : term] : (($true != ((X268 @ id @ X279 @ id))) | (((X268 @ id @ (ap @ (sub @ X278 @ id) @ X279) @ id)) = $true) | (((X268 @ id @ X278 @ id)) != $true)) | ~ ! [X276 : subst,X277 : term,X275 : subst,X274 : subst] : (($true != ((X268 @ X275 @ X277 @ (comp @ X276 @ X274)))) | (((X268 @ (comp @ X275 @ X276) @ (sub @ X277 @ X276) @ X274)) = $true)) | ($true = ((X268 @ id @ (ap @ X269 @ X267) @ id))) | ($true != ((X268 @ id @ X267 @ id))) | ~ ! [X270 : subst,X273 : subst,X272 : subst,X271 : term] : (($true != ((X268 @ (comp @ X270 @ X273) @ (sub @ X271 @ X273) @ X272))) | (((X268 @ X270 @ X271 @ (comp @ X273 @ X272))) = $true)) | ($true != ((X268 @ id @ X269 @ id)))) => (! [X281 : (subst > term > subst > $o),X280 : term] : (($true = ((X281 @ id @ (lam @ X280) @ id))) | ~ ! [X291 : (subst > term > term)] : (($true = ((X281 @ id @ (lam @ (X291 @ sh @ one)) @ id))) | ~ ! [X293 : term,X292 : subst,X294 : subst] : (((X291 @ (comp @ X294 @ X292) @ (sub @ X293 @ X292))) = ((sub @ (X291 @ X294 @ X293) @ X292))) | ~ ! [X295 : term] : (($true != ((X281 @ id @ X295 @ id))) | ($true = ((X281 @ id @ (X291 @ id @ X295) @ id))))) | ~ ! [X287 : subst,X290 : term,X288 : subst,X289 : subst] : (($true = ((X281 @ X288 @ X290 @ (comp @ X289 @ X287)))) | (((X281 @ (comp @ X288 @ X289) @ (sub @ X290 @ X289) @ X287)) != $true)) | ~ ! [X283 : subst,X284 : subst,X285 : term,X282 : subst] : (($true = ((X281 @ (comp @ X282 @ X283) @ (sub @ X285 @ X283) @ X284))) | (((X281 @ X282 @ X285 @ (comp @ X283 @ X284))) != $true)) | ~ ! [X286 : term] : ((((X281 @ id @ (sub @ X280 @ (push @ X286 @ id)) @ id)) = $true) | (((X281 @ id @ X286 @ id)) != $true))) => ! [X297 : (subst > term > subst > $o),X296 : term] : (~ ! [X313 : term,X312 : term] : ((((X297 @ id @ X313 @ id)) != $true) | ($true != ((X297 @ id @ X312 @ id))) | ($true = ((X297 @ id @ (ap @ (sub @ X312 @ id) @ X313) @ id)))) | ~ ! [X302 : (subst > term > term)] : (~ ! [X306 : subst,X304 : term,X305 : subst] : (((sub @ (X302 @ X306 @ X304) @ X305)) = ((X302 @ (comp @ X306 @ X305) @ (sub @ X304 @ X305)))) | ~ ! [X303 : term] : ((((X297 @ id @ (X302 @ id @ X303) @ id)) = $true) | ($true != ((X297 @ id @ X303 @ id)))) | ($true = ((X297 @ id @ (lam @ (X302 @ sh @ one)) @ id)))) | ~ ! [X311 : term] : (($true != ((var @ (sub @ X311 @ id)))) | ($true = ((X297 @ id @ X311 @ id)))) | ($true = ((X297 @ id @ X296 @ id))) | ~ ! [X309 : subst,X310 : term,X307 : subst,X308 : subst] : (($true != ((X297 @ X307 @ X310 @ (comp @ X308 @ X309)))) | ($true = ((X297 @ (comp @ X307 @ X308) @ (sub @ X310 @ X308) @ X309)))) | ~ ! [X298 : subst,X301 : term,X300 : subst,X299 : subst] : (($true = ((X297 @ X298 @ X301 @ (comp @ X299 @ X300)))) | (((X297 @ (comp @ X298 @ X299) @ (sub @ X301 @ X299) @ X300)) != $true))))))) & ((termmset = $true) <=> ! [X360 : term] : (((sub @ X360 @ id)) = X360)) & ((axscons = $true) <=> ! [X87 : subst] : (((push @ (sub @ one @ X87) @ (comp @ sh @ X87))) = X87)) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & ((hoasinduction_lem3aaa = $true) <=> ! [X371 : (subst > term > subst > $o),X370 : term] : (($true = ((X371 @ id @ (lam @ (sub @ X370 @ (push @ one @ sh))) @ id))) | ~ ! [X381 : term] : (($true != ((X371 @ id @ X381 @ id))) | (((X371 @ id @ (sub @ X370 @ (push @ X381 @ id)) @ id)) = $true)) | ~ ! [X373 : term,X372 : (subst > term > term)] : ((((X371 @ id @ (lam @ (X372 @ sh @ one)) @ id)) = $true) | ~ ! [X376 : subst,X375 : term,X377 : subst] : (((sub @ (sub @ X373 @ (push @ X375 @ X377)) @ X376)) = ((sub @ (X372 @ X377 @ X375) @ X376))) | ~ ! [X374 : term] : (($true = ((X371 @ id @ (X372 @ id @ X374) @ id))) | ($true != ((X371 @ id @ X374 @ id)))) | ~ ! [X378 : subst,X379 : subst,X380 : term] : (((sub @ X373 @ (push @ (sub @ X380 @ X379) @ (comp @ X378 @ X379)))) = ((X372 @ (comp @ X378 @ X379) @ (sub @ X380 @ X379))))))) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X498 : term] : ~ ! [X499 : (subst > term > term)] : (((X499 @ sh @ one)) != ((sub @ X498 @ (push @ one @ sh))))) & (! [X235 : subst,X234 : term] : (((sub @ one @ (push @ X234 @ X235))) = X234) <=> (axvarcons = $true)) & ((hoasapinj2 = $true) <=> ! [X105 : term,X102 : term,X103 : term,X104 : term] : ((X102 = X105) | (((ap @ (sub @ X104 @ id) @ X102)) != ((ap @ (sub @ X103 @ id) @ X105))))) & ((hoasapinj1 = $true) <=> ! [X67 : term,X70 : term,X69 : term,X68 : term] : ((X67 = X68) | (((ap @ (sub @ X67 @ id) @ X70)) != ((ap @ (sub @ X68 @ id) @ X69))))) & ((hoasinduction_lem2 = $true) <=> ! [X46 : term,X47 : term,X48 : (subst > term > subst > $o)] : ((((X48 @ id @ X47 @ id)) != $true) | ($true = ((X48 @ id @ (ap @ X46 @ X47) @ id))) | ($true != ((X48 @ id @ X46 @ id))) | ~ ! [X56 : term,X55 : subst,X57 : subst,X58 : subst] : (($true != ((X48 @ X55 @ X56 @ (comp @ X57 @ X58)))) | ($true = ((X48 @ (comp @ X55 @ X57) @ (sub @ X56 @ X57) @ X58)))) | ~ ! [X52 : subst,X50 : term,X49 : subst,X51 : subst] : (($true = ((X48 @ X49 @ X50 @ (comp @ X52 @ X51)))) | (((X48 @ (comp @ X49 @ X52) @ (sub @ X50 @ X52) @ X51)) != $true)) | ~ ! [X54 : term,X53 : term] : (($true != ((X48 @ id @ X53 @ id))) | ($true != ((X48 @ id @ X54 @ id))) | ($true = ((X48 @ id @ (ap @ (sub @ X53 @ id) @ X54) @ id)))))) & (! [X486 : term,X485 : (term > $o),X484 : (subst > term > subst > $o)] : (($true = ((X485 @ (lam @ X486)))) | ~ ! [X487 : term] : (((X484 @ id @ X487 @ id)) = ((X485 @ X487))) | ~ ! [X489 : (subst > term > term)] : (($true = ((X484 @ id @ (lam @ (X489 @ sh @ one)) @ id))) | ~ ! [X490 : subst,X492 : subst,X491 : term] : (((X489 @ (comp @ X492 @ X490) @ (sub @ X491 @ X490))) = ((sub @ (X489 @ X492 @ X491) @ X490))) | ~ ! [X493 : term] : (($true != ((X484 @ id @ X493 @ id))) | ($true = ((X484 @ id @ (X489 @ id @ X493) @ id))))) | ~ ! [X488 : term] : (($true != ((X485 @ X488))) | ($true = ((X485 @ (sub @ X486 @ (push @ X488 @ id))))))) <=> (hoasinduction_lem3v2a = $true)) & ! [X532 : subst] : ? [X533 : d_subst] : (((d2subst @ X533)) = X532) & (! [X530 : term,X529 : subst] : (((comp @ sh @ (push @ X530 @ X529))) = X529) <=> (axshiftcons = $true)) & (hoaslamnotvar = $true) & (! [X425 : term,X426 : (subst > term > subst > $o)] : (~ ! [X429 : (subst > term > term)] : (~ ! [X431 : subst,X430 : term,X432 : subst] : (((X429 @ (comp @ X432 @ X431) @ (sub @ X430 @ X431))) = ((sub @ (X429 @ X432 @ X430) @ X431))) | ~ ! [X433 : term] : (($true = ((X426 @ id @ (X429 @ id @ X433) @ id))) | ($true != ((X426 @ id @ X433 @ id)))) | ($true = ((X426 @ id @ (lam @ (X429 @ sh @ one)) @ id)))) | (((X426 @ id @ X425 @ id)) = $true) | ~ ! [X440 : subst,X441 : subst,X438 : term,X439 : subst] : ((((X426 @ (comp @ X440 @ X439) @ (sub @ X438 @ X439) @ X441)) = $true) | ($true != ((X426 @ X440 @ X438 @ (comp @ X439 @ X441))))) | ~ ! [X442 : term] : (($true = ((X426 @ id @ X442 @ id))) | ($true != ((var @ (sub @ X442 @ id))))) | ~ ! [X436 : subst,X435 : subst,X437 : subst,X434 : term] : (($true = ((X426 @ X437 @ X434 @ (comp @ X436 @ X435)))) | (((X426 @ (comp @ X437 @ X436) @ (sub @ X434 @ X436) @ X435)) != $true)) | ~ ! [X427 : term,X428 : term] : (($true != ((X426 @ id @ X428 @ id))) | ($true != ((X426 @ id @ X427 @ id))) | ($true = ((X426 @ id @ (ap @ (sub @ X428 @ id) @ X427) @ id))))) <=> (hoasinduction = $true)) & ((apinj2 = $true) <=> ! [X71 : term,X73 : term,X74 : term,X72 : term] : ((X71 = X72) | (((ap @ X73 @ X72)) != ((ap @ X74 @ X71))))) & (! [X106 : term] : (((sub @ X106 @ id)) = X106) <=> (axvarid = $true)) & (lamnotap != $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (hoasinduction_lem1_lthm = $true) & ((pushprop_lem1v2_lthm = $true) <=> (! [X522 : term,X523 : subst] : (((sub @ one @ (push @ X522 @ X523))) = X522) => ! [X525 : term,X527 : (term > $o),X524 : (term > $o),X526 : subst] : (($true != ((X527 @ X525))) | (((X524 @ one)) = $true) | ~ ! [X528 : term] : (((X524 @ X528)) = ((X527 @ (sub @ X528 @ (push @ X525 @ X526)))))))) & ((hoasinduction_lem3v2_f = $true) <=> ! [X494 : term] : ~ ! [X495 : (subst > term > term)] : ~ ! [X496 : term,X497 : subst] : (((sub @ X494 @ (push @ X496 @ X497))) = ((X495 @ X497 @ X496)))) & (hoasapnotvar_gthm = $true) & ((! [X236 : term] : (((sub @ X236 @ id)) = X236) => (! [X238 : term,X237 : (subst > term > subst > $o)] : (~ ! [X244 : term] : (($true = ((X237 @ id @ (sub @ X238 @ (push @ X244 @ id)) @ id))) | (((X237 @ id @ X244 @ id)) != $true)) | (((X237 @ id @ (lam @ (sub @ X238 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X239 : (subst > term > term)] : (~ ! [X241 : subst,X242 : term,X240 : subst] : (((sub @ (X239 @ X241 @ X242) @ X240)) = ((X239 @ (comp @ X241 @ X240) @ (sub @ X242 @ X240)))) | ~ ! [X243 : term] : (($true != ((X237 @ id @ X243 @ id))) | ($true = ((X237 @ id @ (X239 @ id @ X243) @ id)))) | (((X237 @ id @ (lam @ (X239 @ sh @ one)) @ id)) = $true))) => ! [X245 : (subst > term > subst > $o),X246 : term] : (~ ! [X247 : term] : (($true = ((X245 @ id @ (sub @ X246 @ (push @ X247 @ id)) @ id))) | ($true != ((X245 @ id @ X247 @ id)))) | ($true = ((X245 @ id @ (lam @ X246) @ id))) | ~ ! [X248 : (subst > term > term)] : (~ ! [X251 : term,X249 : subst,X250 : subst] : (((X248 @ (comp @ X250 @ X249) @ (sub @ X251 @ X249))) = ((sub @ (X248 @ X250 @ X251) @ X249))) | ~ ! [X252 : term] : ((((X245 @ id @ X252 @ id)) != $true) | (((X245 @ id @ (X248 @ id @ X252) @ id)) = $true)) | (((X245 @ id @ (lam @ (X248 @ sh @ one)) @ id)) = $true))))) <=> (hoasinduction_lem3a_lthm = $true)) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (pushprop_lem0_gthm = $true) & ! [X422 : term] : ? [X423 : d_term] : (((d2term @ X423)) = X422) & (pushprop_lem2v2 = $true) & (! [X60 : (subst > term > subst > $o),X59 : term] : (~ ! [X62 : (subst > term > term)] : ((((X60 @ id @ (lam @ (X62 @ sh @ one)) @ id)) = $true) | ~ ! [X65 : subst,X64 : term,X63 : subst] : (((sub @ (X62 @ X65 @ X64) @ X63)) = ((X62 @ (comp @ X65 @ X63) @ (sub @ X64 @ X63)))) | ~ ! [X66 : term] : (($true != ((X60 @ id @ X66 @ id))) | ($true = ((X60 @ id @ (X62 @ id @ X66) @ id))))) | (((X60 @ id @ (lam @ (sub @ X59 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X61 : term] : (($true != ((X60 @ id @ X61 @ id))) | ($true = ((X60 @ id @ (sub @ X59 @ (push @ X61 @ id)) @ id))))) <=> (hoasinduction_lem3aa = $true)) & (! [X506 : term,X508 : (term > $o),X504 : subst,X505 : term,X507 : (term > $o)] : (($true != ((X507 @ X505))) | ($true = ((X508 @ (sub @ X505 @ (push @ X506 @ X504)))))) <=> (pushprop_lem1 = $true)) & ((hoasinduction_no_psi_cond = $true) <=> ! [X226 : term,X225 : (subst > term > subst > $o)] : (~ ! [X229 : (subst > term > term)] : (($true = ((X225 @ id @ (lam @ (X229 @ sh @ one)) @ id))) | ~ ! [X230 : term] : (($true != ((X225 @ id @ X230 @ id))) | (((X225 @ id @ (X229 @ id @ X230) @ id)) = $true)) | ~ ! [X233 : subst,X231 : subst,X232 : term] : (((sub @ (X229 @ X231 @ X232) @ X233)) = ((X229 @ (comp @ X231 @ X233) @ (sub @ X232 @ X233))))) | ~ ! [X227 : term,X228 : term] : (($true = ((X225 @ id @ (ap @ (sub @ X227 @ id) @ X228) @ id))) | ($true != ((X225 @ id @ X227 @ id))) | ($true != ((X225 @ id @ X228 @ id)))) | (((X225 @ id @ X226 @ id)) = $true))) & (axassoc = $true) & ((! [X362 : subst,X361 : term] : (((sub @ one @ (push @ X361 @ X362))) = X361) => (! [X364 : subst,X363 : term] : (((comp @ sh @ (push @ X363 @ X364))) = X364) => ! [X369 : term,X366 : subst,X368 : term,X367 : (term > $o),X365 : (term > $o)] : (($true != ((X365 @ X369))) | ($true = ((X367 @ (sub @ X369 @ (push @ X368 @ X366)))))))) <=> (pushprop_lem1_lthm = $true)) & ! [X412 : d_subst,X413 : d_subst] : ((((d2subst @ X413)) = ((d2subst @ X412))) => (X412 = X413)) & (hoasapnotvar = $true) & ! [X358 : d_term,X359 : d_term] : ((((d2term @ X359)) = ((d2term @ X358))) => (X358 = X359)) & ((axidr = $true) <=> ! [X424 : subst] : (((comp @ X424 @ id)) = X424)) & (! [X445 : (subst > term > subst > $o)] : ~ ! [X446 : (term > $o)] : ~ ! [X447 : term] : (((X445 @ id @ X447 @ id)) = ((X446 @ X447))) <=> (hoasinduction_lem0 = $true)) & ((! [X314 : term] : ~ ! [X315 : (subst > term > term)] : ~ ! [X317 : term,X316 : subst] : (((sub @ X314 @ (push @ X317 @ X316))) = ((X315 @ X316 @ X317))) => (! [X318 : term] : (((sub @ X318 @ id)) = X318) => ! [X319 : (subst > term > subst > $o),X320 : (term > $o),X321 : term] : (~ ! [X327 : term] : (((X319 @ id @ X327 @ id)) = ((X320 @ X327))) | ~ ! [X328 : term] : (($true != ((X320 @ X328))) | ($true = ((X320 @ (sub @ X321 @ (push @ X328 @ id)))))) | ~ ! [X322 : (subst > term > term)] : (~ ! [X323 : subst,X325 : subst,X324 : term] : (((sub @ (X322 @ X323 @ X324) @ X325)) = ((X322 @ (comp @ X323 @ X325) @ (sub @ X324 @ X325)))) | (((X319 @ id @ (lam @ (X322 @ sh @ one)) @ id)) = $true) | ~ ! [X326 : term] : (($true = ((X319 @ id @ (X322 @ id @ X326) @ id))) | ($true != ((X319 @ id @ X326 @ id))))) | ($true = ((X320 @ (lam @ X321))))))) <=> (hoasinduction_lem3v2a_lthm = $true)) & (axapp = $true) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X89 : (subst > term > subst > $o),X88 : term] : (~ ! [X95 : term] : ((((X89 @ id @ X95 @ id)) != $true) | ($true = ((X89 @ id @ (sub @ X88 @ (push @ X95 @ id)) @ id)))) | (((X89 @ id @ (lam @ (sub @ X88 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X90 : (subst > term > term)] : (($true = ((X89 @ id @ (lam @ (X90 @ sh @ one)) @ id))) | ~ ! [X93 : subst,X94 : subst,X92 : term] : (((sub @ (X90 @ X93 @ X92) @ X94)) = ((X90 @ (comp @ X93 @ X94) @ (sub @ X92 @ X94)))) | ~ ! [X91 : term] : (($true = ((X89 @ id @ (X90 @ id @ X91) @ id))) | (((X89 @ id @ X91 @ id)) != $true))))) & (hoasinduction_lem2v2_gthm = $true) & (hoasapinj2_gthm = $true) & (pushprop_lthm_orig = $true) & (hoasinduction_lem2_gthm = $true) & ((hoaslaminj_lthm = $true) <=> (! [X144 : term,X145 : subst] : (((sub @ one @ (push @ X144 @ X145))) = X144) => (! [X147 : subst,X146 : term] : (((comp @ sh @ (push @ X146 @ X147))) = X147) => (! [X149 : term,X148 : term] : ((((lam @ X149)) != ((lam @ X148))) | (X148 = X149)) => ! [X152 : term,X151 : (subst > term > term),X150 : subst,X153 : (subst > term > term)] : ((((X151 @ X150 @ X152)) = ((X153 @ X150 @ X152))) | ~ ! [X154 : term,X155 : subst,X156 : subst] : (((sub @ (X153 @ X155 @ X154) @ X156)) = ((X153 @ (comp @ X155 @ X156) @ (sub @ X154 @ X156)))) | ~ ! [X158 : subst,X157 : subst,X159 : term] : (((X151 @ (comp @ X157 @ X158) @ (sub @ X159 @ X158))) = ((sub @ (X151 @ X157 @ X159) @ X158))) | (((lam @ (X151 @ sh @ one))) != ((lam @ (X153 @ sh @ one))))))))) & (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_gthm = $true) & (((d2subst @ d_id)) = sh) & (pushprop = $true) & (hoasinduction_lem1v2_gthm = $true) & (ulamvar1 != $true) & (hoasinduction_lem1 = $true) & (axmap = $true) & (apnotvar = $true) & (! [X415 : (subst > term > subst > $o),X414 : term] : (($true = ((X415 @ id @ (lam @ X414) @ id))) | ~ ! [X417 : (subst > term > term)] : (~ ! [X418 : term,X420 : subst,X419 : subst] : (((sub @ (X417 @ X420 @ X418) @ X419)) = ((X417 @ (comp @ X420 @ X419) @ (sub @ X418 @ X419)))) | ~ ! [X421 : term] : (($true != ((X415 @ id @ X421 @ id))) | (((X415 @ id @ (X417 @ id @ X421) @ id)) = $true)) | ($true = ((X415 @ id @ (lam @ (X417 @ sh @ one)) @ id)))) | ~ ! [X416 : term] : (($true != ((X415 @ id @ X416 @ id))) | (((X415 @ id @ (sub @ X414 @ (push @ X416 @ id)) @ id)) = $true))) <=> (hoasinduction_lem3a = $true)) & ((substmonoid = $true) <=> (! [X502 : subst] : (((comp @ X502 @ id)) = X502) & ! [X503 : subst] : (((comp @ id @ X503)) = X503))) & ((! [X186 : (subst > term > subst > $o)] : ~ ! [X187 : (term > $o)] : ~ ! [X188 : term] : (((X186 @ id @ X188 @ id)) = ((X187 @ X188))) => (! [X190 : term,X189 : (term > $o)] : (~ ! [X192 : term,X191 : term] : (($true != ((X189 @ X192))) | (((X189 @ X191)) != $true) | (((X189 @ (ap @ X191 @ X192))) = $true)) | ~ ! [X193 : term] : (($true = ((X189 @ (lam @ X193)))) | ~ ! [X194 : term] : ((((X189 @ X194)) != $true) | ($true = ((X189 @ (sub @ X193 @ (push @ X194 @ id))))))) | ~ ! [X195 : term] : (($true != ((var @ X195))) | ($true = ((X189 @ X195)))) | ($true = ((X189 @ X190)))) => (! [X196 : term] : (((sub @ X196 @ id)) = X196) => (! [X197 : (term > $o),X198 : (subst > term > subst > $o),X199 : term] : (~ ! [X200 : (subst > term > term)] : (~ ! [X204 : term] : (($true != ((X198 @ id @ X204 @ id))) | ($true = ((X198 @ id @ (X200 @ id @ X204) @ id)))) | (((X198 @ id @ (lam @ (X200 @ sh @ one)) @ id)) = $true) | ~ ! [X203 : subst,X201 : term,X202 : subst] : (((X200 @ (comp @ X203 @ X202) @ (sub @ X201 @ X202))) = ((sub @ (X200 @ X203 @ X201) @ X202)))) | ~ ! [X205 : term] : ((((X197 @ (sub @ X199 @ (push @ X205 @ id)))) = $true) | (((X197 @ X205)) != $true)) | ($true = ((X197 @ (lam @ X199)))) | ~ ! [X206 : term] : (((X198 @ id @ X206 @ id)) = ((X197 @ X206)))) => ! [X208 : (subst > term > subst > $o),X207 : term] : (~ ! [X210 : term,X211 : term] : (($true != ((X208 @ id @ X211 @ id))) | ($true = ((X208 @ id @ (ap @ (sub @ X211 @ id) @ X210) @ id))) | ($true != ((X208 @ id @ X210 @ id)))) | ~ ! [X209 : term] : (($true != ((var @ (sub @ X209 @ id)))) | ($true = ((X208 @ id @ X209 @ id)))) | ~ ! [X216 : (subst > term > term)] : (~ ! [X220 : term] : ((((X208 @ id @ X220 @ id)) != $true) | ($true = ((X208 @ id @ (X216 @ id @ X220) @ id)))) | ($true = ((X208 @ id @ (lam @ (X216 @ sh @ one)) @ id))) | ~ ! [X217 : subst,X219 : subst,X218 : term] : (((X216 @ (comp @ X217 @ X219) @ (sub @ X218 @ X219))) = ((sub @ (X216 @ X217 @ X218) @ X219)))) | ~ ! [X212 : subst,X213 : subst,X214 : term,X215 : subst] : (($true = ((X208 @ (comp @ X215 @ X212) @ (sub @ X214 @ X212) @ X213))) | (((X208 @ X215 @ X214 @ (comp @ X212 @ X213))) != $true)) | ~ ! [X221 : subst,X222 : term,X223 : subst,X224 : subst] : (($true = ((X208 @ X221 @ X222 @ (comp @ X224 @ X223)))) | ($true != ((X208 @ (comp @ X221 @ X224) @ (sub @ X222 @ X224) @ X223)))) | ($true = ((X208 @ id @ X207 @ id)))))))) <=> (hoasinduction_lthm_3 = $true)) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (substmonoid_gthm = $true) & (ulamvarind = $true) & ((termmset_lthm = $true) <=> (! [X500 : term] : (((sub @ X500 @ id)) = X500) => ! [X501 : term] : (((sub @ X501 @ id)) = X501))) & (! [X483 : term,X482 : term] : ((X482 = X483) | (((sub @ X483 @ sh)) != ((sub @ X482 @ sh)))) <=> (shinj = $true)) & (((var @ (d2term @ d_one))) != $true) & ((induction = $true) <=> ! [X534 : (term > $o),X535 : term] : (($true = ((X534 @ X535))) | ~ ! [X539 : term] : (($true = ((X534 @ (lam @ X539)))) | ($true != ((X534 @ X539)))) | ~ ! [X537 : term,X538 : term] : (($true = ((X534 @ (ap @ X538 @ X537)))) | ($true != ((X534 @ X537))) | (((X534 @ X538)) != $true)) | ~ ! [X536 : term] : ((((var @ X536)) != $true) | ($true = ((X534 @ X536)))))) & (hoaslamnotap_gthm = $true) & (! [X443 : term,X444 : term] : ((((lam @ X443)) != ((lam @ X444))) | (X443 = X444)) <=> (laminj = $true)) & ((! [X329 : subst] : (((comp @ id @ X329)) = X329) => (! [X330 : subst] : (((comp @ X330 @ id)) = X330) => (! [X332 : subst] : (((comp @ id @ X332)) = X332) & ! [X331 : subst] : (((comp @ X331 @ id)) = X331)))) <=> (substmonoid_lthm = $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (pushprop_lem3v2 = $true) & ((induction2_lthm = $true) <=> (! [X160 : term] : (((sub @ X160 @ id)) = X160) => (! [X162 : (term > $o),X161 : term,X163 : subst] : (~ ! [X166 : term] : (~ ! [X167 : term] : (($true != ((X162 @ X167))) | ($true = ((X162 @ (sub @ X166 @ (push @ X167 @ id)))))) | ($true = ((X162 @ (lam @ X166))))) | (((X162 @ (sub @ X161 @ X163))) = $true) | ~ ! [X164 : term,X165 : term] : (($true = ((X162 @ (ap @ X164 @ X165)))) | (((X162 @ X164)) != $true) | ($true != ((X162 @ X165)))) | ~ ! [X168 : term] : (($true = ((X162 @ (sub @ X168 @ X163)))) | (((var @ X168)) != $true))) => ! [X170 : (term > $o),X169 : term] : (~ ! [X171 : term] : (($true = ((X170 @ (lam @ X171)))) | ~ ! [X172 : term] : (($true != ((X170 @ X172))) | ($true = ((X170 @ (sub @ X171 @ (push @ X172 @ id))))))) | (((X170 @ X169)) = $true) | ~ ! [X175 : term] : (($true != ((var @ X175))) | ($true = ((X170 @ X175)))) | ~ ! [X173 : term,X174 : term] : ((((X170 @ (ap @ X174 @ X173))) = $true) | ($true != ((X170 @ X174))) | (((X170 @ X173)) != $true)))))) & (hoaslamnotvar_lthm = $true) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X83 : term] : ~ ! [X84 : (subst > term > term)] : ~ ! [X85 : term,X86 : subst] : (((X84 @ X86 @ X85)) = ((sub @ X83 @ (push @ X85 @ X86))))) & (! [X456 : (term > $o),X457 : term] : (($true = ((X456 @ X457))) | ~ ! [X459 : term,X458 : term] : ((((X456 @ X458)) != $true) | (((X456 @ X459)) != $true) | ($true = ((X456 @ (ap @ X459 @ X458))))) | ~ ! [X460 : term] : (~ ! [X461 : term] : (($true != ((X456 @ X461))) | (((X456 @ (sub @ X460 @ (push @ X461 @ id)))) = $true)) | (((X456 @ (lam @ X460))) = $true)) | ~ ! [X462 : term] : (($true != ((var @ X462))) | ($true = ((X456 @ X462))))) <=> (induction2 = $true)) & ! [X96 : d_term] : (d_one = X96) & (axvarshift = $true) & (pushprop_lem3v2_lthm = $true) & ((hoaslaminj = $true) <=> ! [X176 : subst,X179 : (subst > term > term),X178 : (subst > term > term),X177 : term] : (~ ! [X183 : subst,X184 : subst,X185 : term] : (((X179 @ (comp @ X184 @ X183) @ (sub @ X185 @ X183))) = ((sub @ (X179 @ X184 @ X185) @ X183))) | (((X179 @ X176 @ X177)) = ((X178 @ X176 @ X177))) | ~ ! [X180 : term,X181 : subst,X182 : subst] : (((X178 @ (comp @ X182 @ X181) @ (sub @ X180 @ X181))) = ((sub @ (X178 @ X182 @ X180) @ X181))) | (((lam @ (X178 @ sh @ one))) != ((lam @ (X179 @ sh @ one)))))) & ((hoasinduction_lem2v2 = $true) <=> ! [X110 : (subst > term > subst > $o),X107 : term,X109 : (term > $o),X108 : term] : (($true = ((X109 @ (ap @ X107 @ X108)))) | ~ ! [X111 : term,X112 : subst,X114 : subst,X113 : subst] : (($true = ((X110 @ X112 @ X111 @ (comp @ X114 @ X113)))) | ($true != ((X110 @ (comp @ X112 @ X114) @ (sub @ X111 @ X114) @ X113)))) | (((X109 @ X108)) != $true) | ~ ! [X121 : term,X120 : term] : (($true = ((X110 @ id @ (ap @ (sub @ X120 @ id) @ X121) @ id))) | ($true != ((X110 @ id @ X120 @ id))) | ($true != ((X110 @ id @ X121 @ id)))) | ~ ! [X116 : subst,X117 : term,X115 : subst,X118 : subst] : ((((X110 @ (comp @ X115 @ X116) @ (sub @ X117 @ X116) @ X118)) = $true) | ($true != ((X110 @ X115 @ X117 @ (comp @ X116 @ X118))))) | (((X109 @ X107)) != $true) | ~ ! [X119 : term] : (((X109 @ X119)) = ((X110 @ id @ X119 @ id))))) & (ulamvarsh = $true) & ((hoasinduction_lem3b = $true) <=> ! [X253 : term] : ~ ! [X254 : (subst > term > term)] : (((X254 @ sh @ one)) != ((sub @ X253 @ (push @ one @ sh))))) & ((pushprop_lem0 = $true) <=> ! [X98 : subst,X99 : (term > $o),X97 : term] : ~ ! [X100 : (term > $o)] : ~ ! [X101 : term] : (((X99 @ (sub @ X101 @ (push @ X97 @ X98)))) = ((X100 @ X101)))) & ((hoasinduction_lem3v2 = $true) <=> ! [X22 : (term > $o),X23 : (subst > term > subst > $o),X24 : term] : (~ ! [X34 : subst,X36 : term,X35 : subst,X37 : subst] : (($true = ((X23 @ X37 @ X36 @ (comp @ X35 @ X34)))) | ($true != ((X23 @ (comp @ X37 @ X35) @ (sub @ X36 @ X35) @ X34)))) | ($true = ((X22 @ (lam @ X24)))) | ~ ! [X39 : term] : (($true != ((X22 @ X39))) | (((X22 @ (sub @ X24 @ (push @ X39 @ id)))) = $true)) | ~ ! [X29 : (subst > term > term)] : (~ ! [X31 : subst,X33 : term,X32 : subst] : (((X29 @ (comp @ X31 @ X32) @ (sub @ X33 @ X32))) = ((sub @ (X29 @ X31 @ X33) @ X32))) | ~ ! [X30 : term] : (($true = ((X23 @ id @ (X29 @ id @ X30) @ id))) | ($true != ((X23 @ id @ X30 @ id)))) | ($true = ((X23 @ id @ (lam @ (X29 @ sh @ one)) @ id)))) | ~ ! [X28 : subst,X26 : subst,X27 : subst,X25 : term] : ((((X23 @ (comp @ X28 @ X27) @ (sub @ X25 @ X27) @ X26)) = $true) | (((X23 @ X28 @ X25 @ (comp @ X27 @ X26))) != $true)) | ~ ! [X38 : term] : (((X23 @ id @ X38 @ id)) = ((X22 @ X38))))) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (termmset_gthm = $true) & (hoaslamnotvar_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (hoasinduction_lem3b_gthm = $true) & ((apinj1 = $true) <=> ! [X518 : term,X519 : term,X520 : term,X521 : term] : ((((ap @ X518 @ X520)) != ((ap @ X521 @ X519))) | (X518 = X521))) & ((induction2lem_lthm = $true) <=> (! [X463 : subst,X464 : term] : (((sub @ one @ (push @ X464 @ X463))) = X464) => (! [X465 : term,X466 : subst] : (((comp @ sh @ (push @ X465 @ X466))) = X466) => (! [X467 : subst] : (((comp @ X467 @ id)) = X467) => (! [X469 : term,X468 : (term > $o)] : (~ ! [X473 : term] : ((((X468 @ X473)) = $true) | (((var @ X473)) != $true)) | ~ ! [X470 : term,X471 : term] : (($true = ((X468 @ (ap @ X471 @ X470)))) | ($true != ((X468 @ X471))) | ($true != ((X468 @ X470)))) | ($true = ((X468 @ X469))) | ~ ! [X472 : term] : (($true != ((X468 @ X472))) | ($true = ((X468 @ (lam @ X472)))))) => ! [X476 : (term > $o),X475 : term,X474 : subst] : (($true = ((X476 @ (sub @ X475 @ X474)))) | ~ ! [X477 : term] : (~ ! [X478 : term] : (($true != ((X476 @ X478))) | ($true = ((X476 @ (sub @ X477 @ (push @ X478 @ id)))))) | ($true = ((X476 @ (lam @ X477))))) | ~ ! [X481 : term,X480 : term] : ((((X476 @ (ap @ X481 @ X480))) = $true) | ($true != ((X476 @ X480))) | ($true != ((X476 @ X481)))) | ~ ! [X479 : term] : ((((var @ X479)) != $true) | ($true = ((X476 @ (sub @ X479 @ X474))))))))))) & ((! [X509 : term] : (((sub @ X509 @ id)) = X509) => (! [X510 : term,X513 : term,X511 : term,X512 : term] : ((X510 = X512) | (((ap @ X512 @ X513)) != ((ap @ X510 @ X511)))) => ! [X517 : term,X515 : term,X514 : term,X516 : term] : ((X514 = X516) | (((ap @ (sub @ X514 @ id) @ X517)) != ((ap @ (sub @ X516 @ id) @ X515)))))) <=> (hoasapinj1_lthm = $true)) & (! [X6 : (subst > term > subst > $o),X7 : term] : ((((X6 @ id @ (lam @ X7) @ id)) = $true) | ~ ! [X17 : (subst > term > term)] : (~ ! [X18 : term] : (($true != ((X6 @ id @ X18 @ id))) | ($true = ((X6 @ id @ (X17 @ id @ X18) @ id)))) | (((X6 @ id @ (lam @ (X17 @ sh @ one)) @ id)) = $true) | ~ ! [X20 : subst,X21 : subst,X19 : term] : (((X17 @ (comp @ X21 @ X20) @ (sub @ X19 @ X20))) = ((sub @ (X17 @ X21 @ X19) @ X20)))) | ~ ! [X12 : term] : ((((X6 @ id @ X12 @ id)) != $true) | ($true = ((X6 @ id @ (sub @ X7 @ (push @ X12 @ id)) @ id)))) | ~ ! [X15 : subst,X14 : term,X13 : subst,X16 : subst] : (($true != ((X6 @ (comp @ X15 @ X13) @ (sub @ X14 @ X13) @ X16))) | ($true = ((X6 @ X15 @ X14 @ (comp @ X13 @ X16))))) | ~ ! [X9 : subst,X8 : subst,X10 : term,X11 : subst] : (($true = ((X6 @ (comp @ X8 @ X9) @ (sub @ X10 @ X9) @ X11))) | (((X6 @ X8 @ X10 @ (comp @ X9 @ X11))) != $true))) <=> (hoasinduction_lem3 = $true)) & (id = ((d2subst @ d_id))) & ! [X531 : d_subst] : (d_id = X531) & (axabs = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoasapinj1_gthm = $true) & (lamnotvar = $true) & (pushprop_lem1_gthm = $true) & (hoasinduction_lem2_lthm = $true)),
  inference(flattening,[],[f8])).
thf(f10,plain,(
  (hoasinduction_lem0 != ((!! @ (subst > term > subst > $o) @ (^[Y0 : subst > term > subst > $o]: (?? @ (term > $o) @ (^[Y1 : term > $o]: (hoasinduction_p_and_p_prime @ Y0 @ Y1)))))))),
  inference(flattening,[],[f7])).
thf(f11,plain,(
  (! [X76 : (term > $o),X77 : term,X75 : subst] : (? [X81 : term] : ((((X76 @ (lam @ X81))) != $true) & ! [X82 : term] : (($true = ((X76 @ (sub @ X81 @ (push @ X82 @ id))))) | ($true != ((X76 @ X82))))) | ? [X79 : term,X80 : term] : ((((X76 @ X80)) = $true) & (((X76 @ X79)) = $true) & (((X76 @ (ap @ X80 @ X79))) != $true)) | ? [X78 : term] : ((((X76 @ (sub @ X78 @ X75))) != $true) & (((var @ X78)) = $true)) | ($true = ((X76 @ (sub @ X77 @ X75))))) <=> (induction2lem = $true)) & (pushprop_lem2v2_gthm = $true) & (hoasapnotvar_lthm = $true) & ((hoaslamnotap = $true) <=> ! [X1 : term,X2 : term,X0 : (subst > term > term)] : (? [X3 : subst,X4 : term,X5 : subst] : (((X0 @ (comp @ X3 @ X5) @ (sub @ X4 @ X5))) != ((sub @ (X0 @ X3 @ X4) @ X5))) | (((ap @ (sub @ X2 @ id) @ X1)) != ((lam @ (X0 @ sh @ one)))))) & (pushprop_gthm = $true) & ((hoasapinj2_lthm = $true) <=> (! [X455 : term,X453 : term,X454 : term,X452 : term] : ((((ap @ (sub @ X454 @ id) @ X452)) != ((ap @ (sub @ X453 @ id) @ X455))) | (X452 = X455)) | ? [X449 : term,X450 : term,X451 : term,X448 : term] : ((X448 != X450) & (((ap @ X451 @ X448)) = ((ap @ X449 @ X450)))))) & ((hoasinduction_lem3_lthm = $true) <=> ((! [X343 : term,X342 : (subst > term > subst > $o)] : (($true = ((X342 @ id @ (lam @ X343) @ id))) | ? [X354 : subst,X355 : subst,X356 : subst,X357 : term] : (($true != ((X342 @ (comp @ X356 @ X355) @ (sub @ X357 @ X355) @ X354))) & ($true = ((X342 @ X356 @ X357 @ (comp @ X355 @ X354))))) | ? [X349 : (subst > term > term)] : (! [X350 : term] : ((((X342 @ id @ X350 @ id)) != $true) | ($true = ((X342 @ id @ (X349 @ id @ X350) @ id)))) & ($true != ((X342 @ id @ (lam @ (X349 @ sh @ one)) @ id))) & ! [X353 : subst,X352 : subst,X351 : term] : (((X349 @ (comp @ X353 @ X352) @ (sub @ X351 @ X352))) = ((sub @ (X349 @ X353 @ X351) @ X352)))) | ? [X344 : term] : (($true = ((X342 @ id @ X344 @ id))) & ($true != ((X342 @ id @ (sub @ X343 @ (push @ X344 @ id)) @ id)))) | ? [X346 : term,X345 : subst,X347 : subst,X348 : subst] : ((((X342 @ (comp @ X345 @ X347) @ (sub @ X346 @ X347) @ X348)) = $true) & ($true != ((X342 @ X345 @ X346 @ (comp @ X347 @ X348)))))) | ? [X335 : (subst > term > subst > $o),X334 : term] : (! [X337 : (subst > term > term)] : (? [X339 : term,X340 : subst,X341 : subst] : (((sub @ (X337 @ X341 @ X339) @ X340)) != ((X337 @ (comp @ X341 @ X340) @ (sub @ X339 @ X340)))) | ($true = ((X335 @ id @ (lam @ (X337 @ sh @ one)) @ id))) | ? [X338 : term] : (($true != ((X335 @ id @ (X337 @ id @ X338) @ id))) & ($true = ((X335 @ id @ X338 @ id))))) & ($true != ((X335 @ id @ (lam @ (sub @ X334 @ (push @ one @ sh))) @ id))) & ! [X336 : term] : ((((X335 @ id @ (sub @ X334 @ (push @ X336 @ id)) @ id)) = $true) | (((X335 @ id @ X336 @ id)) != $true)))) | ? [X333 : term] : (((sub @ X333 @ id)) != X333))) & ((hoasinduction_lem3v2_lthm = $true) <=> (? [X122 : term] : (((sub @ X122 @ id)) != X122) | ! [X125 : (subst > term > subst > $o),X123 : term,X124 : (term > $o)] : (? [X140 : term] : (($true != ((X124 @ (sub @ X123 @ (push @ X140 @ id))))) & (((X124 @ X140)) = $true)) | ? [X139 : subst,X137 : term,X136 : subst,X138 : subst] : ((((X125 @ X139 @ X137 @ (comp @ X138 @ X136))) = $true) & ($true != ((X125 @ (comp @ X139 @ X138) @ (sub @ X137 @ X138) @ X136)))) | ? [X126 : term] : (((X124 @ X126)) != ((X125 @ id @ X126 @ id))) | (((X124 @ (lam @ X123))) = $true) | ? [X131 : (subst > term > term)] : (($true != ((X125 @ id @ (lam @ (X131 @ sh @ one)) @ id))) & ! [X134 : subst,X135 : subst,X133 : term] : (((X131 @ (comp @ X134 @ X135) @ (sub @ X133 @ X135))) = ((sub @ (X131 @ X134 @ X133) @ X135))) & ! [X132 : term] : (($true = ((X125 @ id @ (X131 @ id @ X132) @ id))) | (((X125 @ id @ X132 @ id)) != $true))) | ? [X130 : subst,X129 : subst,X127 : term,X128 : subst] : ((((X125 @ X129 @ X127 @ (comp @ X130 @ X128))) != $true) & ($true = ((X125 @ (comp @ X129 @ X130) @ (sub @ X127 @ X130) @ X128))))))) & ((pushprop_lem0_lthm = $true) <=> ! [X256 : (term > $o),X255 : subst,X257 : term] : ? [X258 : (term > $o)] : ! [X259 : term] : (((X258 @ X259)) = ((X256 @ (sub @ X259 @ (push @ X257 @ X255)))))) & (! [X45 : subst] : (((comp @ id @ X45)) = X45) <=> (axidl = $true)) & (hoasinduction_lem3a_gthm = $true) & (induction2lem_gthm = $true) & (axclos = $true) & (hoasinduction_lem1v2 = $true) & (hoasinduction_gthm = $true) & ((hoasinduction_lem0_lthm = $true) <=> ! [X141 : (subst > term > subst > $o)] : ? [X142 : (term > $o)] : ! [X143 : term] : (((X141 @ id @ X143 @ id)) = ((X142 @ X143)))) & ((hoasinduction_no_psi_cond_lthm = $true) <=> ((((! [X403 : (subst > term > subst > $o),X404 : term] : (? [X410 : term,X411 : term] : (($true = ((X403 @ id @ X410 @ id))) & ($true != ((X403 @ id @ (ap @ (sub @ X411 @ id) @ X410) @ id))) & (((X403 @ id @ X411 @ id)) = $true)) | ($true = ((X403 @ id @ X404 @ id))) | ? [X405 : (subst > term > term)] : ((((X403 @ id @ (lam @ (X405 @ sh @ one)) @ id)) != $true) & ! [X408 : term,X406 : subst,X407 : subst] : (((sub @ (X405 @ X407 @ X408) @ X406)) = ((X405 @ (comp @ X407 @ X406) @ (sub @ X408 @ X406)))) & ! [X409 : term] : (($true != ((X403 @ id @ X409 @ id))) | (((X403 @ id @ (X405 @ id @ X409) @ id)) = $true)))) | ? [X394 : term,X395 : (term > $o),X393 : (subst > term > subst > $o)] : (! [X401 : term] : (($true = ((X395 @ (sub @ X394 @ (push @ X401 @ id))))) | ($true != ((X395 @ X401)))) & ! [X402 : term] : (((X393 @ id @ X402 @ id)) = ((X395 @ X402))) & ($true != ((X395 @ (lam @ X394)))) & ! [X396 : (subst > term > term)] : (? [X397 : term] : (($true = ((X393 @ id @ X397 @ id))) & ($true != ((X393 @ id @ (X396 @ id @ X397) @ id)))) | ($true = ((X393 @ id @ (lam @ (X396 @ sh @ one)) @ id))) | ? [X399 : term,X400 : subst,X398 : subst] : (((sub @ (X396 @ X398 @ X399) @ X400)) != ((X396 @ (comp @ X398 @ X400) @ (sub @ X399 @ X400))))))) | ? [X392 : term] : (((sub @ X392 @ id)) != X392)) | ? [X386 : term,X385 : (term > $o)] : (! [X391 : term] : (($true = ((X385 @ X391))) | (((var @ X391)) != $true)) & ($true != ((X385 @ X386))) & ! [X389 : term] : (? [X390 : term] : (($true = ((X385 @ X390))) & ($true != ((X385 @ (sub @ X389 @ (push @ X390 @ id)))))) | ($true = ((X385 @ (lam @ X389))))) & ! [X387 : term,X388 : term] : ((((X385 @ (ap @ X388 @ X387))) = $true) | ($true != ((X385 @ X387))) | ($true != ((X385 @ X388)))))) | ? [X382 : (subst > term > subst > $o)] : ! [X383 : (term > $o)] : ? [X384 : term] : (((X382 @ id @ X384 @ id)) != ((X383 @ X384))))) & (one = ((d2term @ d_one))) & (hoaslamnotap_lthm = $true) & (hoasinduction_lem3v2_gthm = $true) & (induction2_gthm = $true) & (hoasinduction_lem3_gthm = $true) & (pushprop_lem2v2_lthm = $true) & (pushprop_lem1v2_gthm = $true) & ((pushprop_lem1v2 = $true) <=> ! [X40 : subst,X43 : (term > $o),X42 : (term > $o),X41 : term] : (? [X44 : term] : (((X43 @ (sub @ X44 @ (push @ X41 @ X40)))) != ((X42 @ X44))) | ($true = ((X42 @ one))) | ($true != ((X43 @ X41))))) & (hoasinduction_lem1_gthm = $true) & (pushprop_lthm = $true) & ((hoasinduction_lthm = $true) <=> (((! [X297 : (subst > term > subst > $o),X296 : term] : (? [X302 : (subst > term > term)] : (($true != ((X297 @ id @ (lam @ (X302 @ sh @ one)) @ id))) & ! [X303 : term] : ((((X297 @ id @ (X302 @ id @ X303) @ id)) = $true) | ($true != ((X297 @ id @ X303 @ id)))) & ! [X306 : subst,X304 : term,X305 : subst] : (((sub @ (X302 @ X306 @ X304) @ X305)) = ((X302 @ (comp @ X306 @ X305) @ (sub @ X304 @ X305))))) | ? [X307 : subst,X309 : subst,X310 : term,X308 : subst] : (($true != ((X297 @ (comp @ X307 @ X308) @ (sub @ X310 @ X308) @ X309))) & ($true = ((X297 @ X307 @ X310 @ (comp @ X308 @ X309))))) | ? [X311 : term] : (($true = ((var @ (sub @ X311 @ id)))) & ($true != ((X297 @ id @ X311 @ id)))) | ? [X313 : term,X312 : term] : ((((X297 @ id @ X313 @ id)) = $true) & ($true = ((X297 @ id @ X312 @ id))) & ($true != ((X297 @ id @ (ap @ (sub @ X312 @ id) @ X313) @ id)))) | ? [X298 : subst,X301 : term,X300 : subst,X299 : subst] : (($true != ((X297 @ X298 @ X301 @ (comp @ X299 @ X300)))) & (((X297 @ (comp @ X298 @ X299) @ (sub @ X301 @ X299) @ X300)) = $true)) | ($true = ((X297 @ id @ X296 @ id)))) | ? [X281 : (subst > term > subst > $o),X280 : term] : (! [X286 : term] : ((((X281 @ id @ (sub @ X280 @ (push @ X286 @ id)) @ id)) = $true) | (((X281 @ id @ X286 @ id)) != $true)) & ($true != ((X281 @ id @ (lam @ X280) @ id))) & ! [X287 : subst,X290 : term,X288 : subst,X289 : subst] : (($true = ((X281 @ X288 @ X290 @ (comp @ X289 @ X287)))) | (((X281 @ (comp @ X288 @ X289) @ (sub @ X290 @ X289) @ X287)) != $true)) & ! [X283 : subst,X284 : subst,X285 : term,X282 : subst] : (($true = ((X281 @ (comp @ X282 @ X283) @ (sub @ X285 @ X283) @ X284))) | (((X281 @ X282 @ X285 @ (comp @ X283 @ X284))) != $true)) & ! [X291 : (subst > term > term)] : (? [X295 : term] : (($true != ((X281 @ id @ (X291 @ id @ X295) @ id))) & ($true = ((X281 @ id @ X295 @ id)))) | ($true = ((X281 @ id @ (lam @ (X291 @ sh @ one)) @ id))) | ? [X292 : subst,X294 : subst,X293 : term] : (((X291 @ (comp @ X294 @ X292) @ (sub @ X293 @ X292))) != ((sub @ (X291 @ X294 @ X293) @ X292)))))) | ? [X268 : (subst > term > subst > $o),X269 : term,X267 : term] : (($true != ((X268 @ id @ (ap @ X269 @ X267) @ id))) & ($true = ((X268 @ id @ X269 @ id))) & ! [X270 : subst,X273 : subst,X272 : subst,X271 : term] : (($true != ((X268 @ (comp @ X270 @ X273) @ (sub @ X271 @ X273) @ X272))) | (((X268 @ X270 @ X271 @ (comp @ X273 @ X272))) = $true)) & ($true = ((X268 @ id @ X267 @ id))) & ! [X279 : term,X278 : term] : (($true != ((X268 @ id @ X279 @ id))) | (((X268 @ id @ (ap @ (sub @ X278 @ id) @ X279) @ id)) = $true) | (((X268 @ id @ X278 @ id)) != $true)) & ! [X276 : subst,X277 : term,X275 : subst,X274 : subst] : (($true != ((X268 @ X275 @ X277 @ (comp @ X276 @ X274)))) | (((X268 @ (comp @ X275 @ X276) @ (sub @ X277 @ X276) @ X274)) = $true)))) | ? [X261 : term,X260 : (term > $o)] : (! [X266 : term,X265 : term] : (($true = ((X260 @ (ap @ X265 @ X266)))) | (((X260 @ X266)) != $true) | ($true != ((X260 @ X265)))) & ! [X262 : term] : (? [X263 : term] : (($true != ((X260 @ (sub @ X262 @ (push @ X263 @ id))))) & ($true = ((X260 @ X263)))) | (((X260 @ (lam @ X262))) = $true)) & ! [X264 : term] : ((((var @ X264)) != $true) | ($true = ((X260 @ X264)))) & (((X260 @ X261)) != $true)))) & ((termmset = $true) <=> ! [X360 : term] : (((sub @ X360 @ id)) = X360)) & ((axscons = $true) <=> ! [X87 : subst] : (((push @ (sub @ one @ X87) @ (comp @ sh @ X87))) = X87)) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & ((hoasinduction_lem3aaa = $true) <=> ! [X371 : (subst > term > subst > $o),X370 : term] : (? [X373 : term,X372 : (subst > term > term)] : (! [X374 : term] : (($true = ((X371 @ id @ (X372 @ id @ X374) @ id))) | ($true != ((X371 @ id @ X374 @ id)))) & (((X371 @ id @ (lam @ (X372 @ sh @ one)) @ id)) != $true) & ! [X376 : subst,X375 : term,X377 : subst] : (((sub @ (sub @ X373 @ (push @ X375 @ X377)) @ X376)) = ((sub @ (X372 @ X377 @ X375) @ X376))) & ! [X378 : subst,X379 : subst,X380 : term] : (((sub @ X373 @ (push @ (sub @ X380 @ X379) @ (comp @ X378 @ X379)))) = ((X372 @ (comp @ X378 @ X379) @ (sub @ X380 @ X379))))) | ($true = ((X371 @ id @ (lam @ (sub @ X370 @ (push @ one @ sh))) @ id))) | ? [X381 : term] : ((((X371 @ id @ (sub @ X370 @ (push @ X381 @ id)) @ id)) != $true) & ($true = ((X371 @ id @ X381 @ id)))))) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X498 : term] : ? [X499 : (subst > term > term)] : (((X499 @ sh @ one)) = ((sub @ X498 @ (push @ one @ sh))))) & (! [X235 : subst,X234 : term] : (((sub @ one @ (push @ X234 @ X235))) = X234) <=> (axvarcons = $true)) & ((hoasapinj2 = $true) <=> ! [X105 : term,X102 : term,X103 : term,X104 : term] : ((X102 = X105) | (((ap @ (sub @ X104 @ id) @ X102)) != ((ap @ (sub @ X103 @ id) @ X105))))) & ((hoasapinj1 = $true) <=> ! [X67 : term,X70 : term,X69 : term,X68 : term] : ((X67 = X68) | (((ap @ (sub @ X67 @ id) @ X70)) != ((ap @ (sub @ X68 @ id) @ X69))))) & (! [X48 : (subst > term > subst > $o),X47 : term,X46 : term] : (? [X53 : term,X54 : term] : (($true = ((X48 @ id @ X53 @ id))) & ($true != ((X48 @ id @ (ap @ (sub @ X53 @ id) @ X54) @ id))) & ($true = ((X48 @ id @ X54 @ id)))) | (((X48 @ id @ X47 @ id)) != $true) | ? [X56 : term,X58 : subst,X55 : subst,X57 : subst] : (($true = ((X48 @ X55 @ X56 @ (comp @ X57 @ X58)))) & ($true != ((X48 @ (comp @ X55 @ X57) @ (sub @ X56 @ X57) @ X58)))) | ($true != ((X48 @ id @ X46 @ id))) | ($true = ((X48 @ id @ (ap @ X46 @ X47) @ id))) | ? [X50 : term,X52 : subst,X51 : subst,X49 : subst] : (($true != ((X48 @ X49 @ X50 @ (comp @ X52 @ X51)))) & (((X48 @ (comp @ X49 @ X52) @ (sub @ X50 @ X52) @ X51)) = $true))) <=> (hoasinduction_lem2 = $true)) & (! [X485 : (term > $o),X486 : term,X484 : (subst > term > subst > $o)] : (? [X488 : term] : (($true = ((X485 @ X488))) & ($true != ((X485 @ (sub @ X486 @ (push @ X488 @ id)))))) | ? [X487 : term] : (((X484 @ id @ X487 @ id)) != ((X485 @ X487))) | ? [X489 : (subst > term > term)] : (($true != ((X484 @ id @ (lam @ (X489 @ sh @ one)) @ id))) & ! [X490 : subst,X492 : subst,X491 : term] : (((X489 @ (comp @ X492 @ X490) @ (sub @ X491 @ X490))) = ((sub @ (X489 @ X492 @ X491) @ X490))) & ! [X493 : term] : (($true != ((X484 @ id @ X493 @ id))) | ($true = ((X484 @ id @ (X489 @ id @ X493) @ id))))) | ($true = ((X485 @ (lam @ X486))))) <=> (hoasinduction_lem3v2a = $true)) & ! [X532 : subst] : ? [X533 : d_subst] : (((d2subst @ X533)) = X532) & (! [X530 : term,X529 : subst] : (((comp @ sh @ (push @ X530 @ X529))) = X529) <=> (axshiftcons = $true)) & (hoaslamnotvar = $true) & (! [X425 : term,X426 : (subst > term > subst > $o)] : (? [X429 : (subst > term > term)] : (! [X433 : term] : (($true = ((X426 @ id @ (X429 @ id @ X433) @ id))) | ($true != ((X426 @ id @ X433 @ id)))) & ! [X431 : subst,X430 : term,X432 : subst] : (((X429 @ (comp @ X432 @ X431) @ (sub @ X430 @ X431))) = ((sub @ (X429 @ X432 @ X430) @ X431))) & ($true != ((X426 @ id @ (lam @ (X429 @ sh @ one)) @ id)))) | ? [X439 : subst,X441 : subst,X438 : term,X440 : subst] : (($true = ((X426 @ X440 @ X438 @ (comp @ X439 @ X441)))) & (((X426 @ (comp @ X440 @ X439) @ (sub @ X438 @ X439) @ X441)) != $true)) | ? [X434 : term,X435 : subst,X437 : subst,X436 : subst] : (($true != ((X426 @ X437 @ X434 @ (comp @ X436 @ X435)))) & (((X426 @ (comp @ X437 @ X436) @ (sub @ X434 @ X436) @ X435)) = $true)) | ? [X442 : term] : (($true != ((X426 @ id @ X442 @ id))) & ($true = ((var @ (sub @ X442 @ id))))) | (((X426 @ id @ X425 @ id)) = $true) | ? [X428 : term,X427 : term] : (($true = ((X426 @ id @ X428 @ id))) & ($true != ((X426 @ id @ (ap @ (sub @ X428 @ id) @ X427) @ id))) & ($true = ((X426 @ id @ X427 @ id))))) <=> (hoasinduction = $true)) & ((apinj2 = $true) <=> ! [X71 : term,X73 : term,X74 : term,X72 : term] : ((X71 = X72) | (((ap @ X73 @ X72)) != ((ap @ X74 @ X71))))) & (! [X106 : term] : (((sub @ X106 @ id)) = X106) <=> (axvarid = $true)) & (lamnotap != $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (hoasinduction_lem1_lthm = $true) & ((! [X526 : subst,X525 : term,X527 : (term > $o),X524 : (term > $o)] : ((((X524 @ one)) = $true) | ? [X528 : term] : (((X524 @ X528)) != ((X527 @ (sub @ X528 @ (push @ X525 @ X526))))) | ($true != ((X527 @ X525)))) | ? [X523 : subst,X522 : term] : (((sub @ one @ (push @ X522 @ X523))) != X522)) <=> (pushprop_lem1v2_lthm = $true)) & (! [X494 : term] : ? [X495 : (subst > term > term)] : ! [X496 : term,X497 : subst] : (((sub @ X494 @ (push @ X496 @ X497))) = ((X495 @ X497 @ X496))) <=> (hoasinduction_lem3v2_f = $true)) & (hoasapnotvar_gthm = $true) & (((! [X245 : (subst > term > subst > $o),X246 : term] : (? [X248 : (subst > term > term)] : ((((X245 @ id @ (lam @ (X248 @ sh @ one)) @ id)) != $true) & ! [X252 : term] : ((((X245 @ id @ X252 @ id)) != $true) | (((X245 @ id @ (X248 @ id @ X252) @ id)) = $true)) & ! [X251 : term,X249 : subst,X250 : subst] : (((X248 @ (comp @ X250 @ X249) @ (sub @ X251 @ X249))) = ((sub @ (X248 @ X250 @ X251) @ X249)))) | ? [X247 : term] : (($true != ((X245 @ id @ (sub @ X246 @ (push @ X247 @ id)) @ id))) & ($true = ((X245 @ id @ X247 @ id)))) | ($true = ((X245 @ id @ (lam @ X246) @ id)))) | ? [X237 : (subst > term > subst > $o),X238 : term] : (! [X239 : (subst > term > term)] : (? [X243 : term] : (($true = ((X237 @ id @ X243 @ id))) & ($true != ((X237 @ id @ (X239 @ id @ X243) @ id)))) | ? [X240 : subst,X242 : term,X241 : subst] : (((sub @ (X239 @ X241 @ X242) @ X240)) != ((X239 @ (comp @ X241 @ X240) @ (sub @ X242 @ X240)))) | (((X237 @ id @ (lam @ (X239 @ sh @ one)) @ id)) = $true)) & ! [X244 : term] : (($true = ((X237 @ id @ (sub @ X238 @ (push @ X244 @ id)) @ id))) | (((X237 @ id @ X244 @ id)) != $true)) & (((X237 @ id @ (lam @ (sub @ X238 @ (push @ one @ sh))) @ id)) != $true))) | ? [X236 : term] : (((sub @ X236 @ id)) != X236)) <=> (hoasinduction_lem3a_lthm = $true)) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (pushprop_lem0_gthm = $true) & ! [X422 : term] : ? [X423 : d_term] : (((d2term @ X423)) = X422) & (pushprop_lem2v2 = $true) & ((hoasinduction_lem3aa = $true) <=> ! [X60 : (subst > term > subst > $o),X59 : term] : (? [X62 : (subst > term > term)] : (! [X66 : term] : (($true != ((X60 @ id @ X66 @ id))) | ($true = ((X60 @ id @ (X62 @ id @ X66) @ id)))) & ! [X65 : subst,X64 : term,X63 : subst] : (((sub @ (X62 @ X65 @ X64) @ X63)) = ((X62 @ (comp @ X65 @ X63) @ (sub @ X64 @ X63)))) & (((X60 @ id @ (lam @ (X62 @ sh @ one)) @ id)) != $true)) | ? [X61 : term] : (($true = ((X60 @ id @ X61 @ id))) & ($true != ((X60 @ id @ (sub @ X59 @ (push @ X61 @ id)) @ id)))) | (((X60 @ id @ (lam @ (sub @ X59 @ (push @ one @ sh))) @ id)) = $true))) & (! [X506 : term,X508 : (term > $o),X504 : subst,X505 : term,X507 : (term > $o)] : (($true != ((X507 @ X505))) | ($true = ((X508 @ (sub @ X505 @ (push @ X506 @ X504)))))) <=> (pushprop_lem1 = $true)) & ((hoasinduction_no_psi_cond = $true) <=> ! [X225 : (subst > term > subst > $o),X226 : term] : ((((X225 @ id @ X226 @ id)) = $true) | ? [X229 : (subst > term > term)] : (($true != ((X225 @ id @ (lam @ (X229 @ sh @ one)) @ id))) & ! [X233 : subst,X231 : subst,X232 : term] : (((sub @ (X229 @ X231 @ X232) @ X233)) = ((X229 @ (comp @ X231 @ X233) @ (sub @ X232 @ X233)))) & ! [X230 : term] : (($true != ((X225 @ id @ X230 @ id))) | (((X225 @ id @ (X229 @ id @ X230) @ id)) = $true))) | ? [X228 : term,X227 : term] : (($true = ((X225 @ id @ X227 @ id))) & ($true = ((X225 @ id @ X228 @ id))) & ($true != ((X225 @ id @ (ap @ (sub @ X227 @ id) @ X228) @ id)))))) & (axassoc = $true) & (((! [X369 : term,X366 : subst,X368 : term,X367 : (term > $o),X365 : (term > $o)] : (($true != ((X365 @ X369))) | ($true = ((X367 @ (sub @ X369 @ (push @ X368 @ X366)))))) | ? [X364 : subst,X363 : term] : (((comp @ sh @ (push @ X363 @ X364))) != X364)) | ? [X361 : term,X362 : subst] : (((sub @ one @ (push @ X361 @ X362))) != X361)) <=> (pushprop_lem1_lthm = $true)) & ! [X413 : d_subst,X412 : d_subst] : ((((d2subst @ X413)) != ((d2subst @ X412))) | (X412 = X413)) & (hoasapnotvar = $true) & ! [X358 : d_term,X359 : d_term] : ((((d2term @ X359)) != ((d2term @ X358))) | (X358 = X359)) & ((axidr = $true) <=> ! [X424 : subst] : (((comp @ X424 @ id)) = X424)) & ((hoasinduction_lem0 = $true) <=> ! [X445 : (subst > term > subst > $o)] : ? [X446 : (term > $o)] : ! [X447 : term] : (((X445 @ id @ X447 @ id)) = ((X446 @ X447)))) & (((! [X319 : (subst > term > subst > $o),X321 : term,X320 : (term > $o)] : (? [X328 : term] : (($true = ((X320 @ X328))) & ($true != ((X320 @ (sub @ X321 @ (push @ X328 @ id)))))) | ? [X327 : term] : (((X319 @ id @ X327 @ id)) != ((X320 @ X327))) | ($true = ((X320 @ (lam @ X321)))) | ? [X322 : (subst > term > term)] : (! [X323 : subst,X325 : subst,X324 : term] : (((sub @ (X322 @ X323 @ X324) @ X325)) = ((X322 @ (comp @ X323 @ X325) @ (sub @ X324 @ X325)))) & ! [X326 : term] : (($true = ((X319 @ id @ (X322 @ id @ X326) @ id))) | ($true != ((X319 @ id @ X326 @ id)))) & (((X319 @ id @ (lam @ (X322 @ sh @ one)) @ id)) != $true))) | ? [X318 : term] : (((sub @ X318 @ id)) != X318)) | ? [X314 : term] : ! [X315 : (subst > term > term)] : ? [X316 : subst,X317 : term] : (((sub @ X314 @ (push @ X317 @ X316))) != ((X315 @ X316 @ X317)))) <=> (hoasinduction_lem3v2a_lthm = $true)) & (axapp = $true) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X89 : (subst > term > subst > $o),X88 : term] : (? [X95 : term] : (($true != ((X89 @ id @ (sub @ X88 @ (push @ X95 @ id)) @ id))) & (((X89 @ id @ X95 @ id)) = $true)) | (((X89 @ id @ (lam @ (sub @ X88 @ (push @ one @ sh))) @ id)) = $true) | ? [X90 : (subst > term > term)] : (($true != ((X89 @ id @ (lam @ (X90 @ sh @ one)) @ id))) & ! [X93 : subst,X94 : subst,X92 : term] : (((sub @ (X90 @ X93 @ X92) @ X94)) = ((X90 @ (comp @ X93 @ X94) @ (sub @ X92 @ X94)))) & ! [X91 : term] : (($true = ((X89 @ id @ (X90 @ id @ X91) @ id))) | (((X89 @ id @ X91 @ id)) != $true))))) & (hoasinduction_lem2v2_gthm = $true) & (hoasapinj2_gthm = $true) & (pushprop_lthm_orig = $true) & (hoasinduction_lem2_gthm = $true) & ((hoaslaminj_lthm = $true) <=> (((! [X152 : term,X151 : (subst > term > term),X150 : subst,X153 : (subst > term > term)] : (? [X155 : subst,X154 : term,X156 : subst] : (((sub @ (X153 @ X155 @ X154) @ X156)) != ((X153 @ (comp @ X155 @ X156) @ (sub @ X154 @ X156)))) | ? [X158 : subst,X157 : subst,X159 : term] : (((X151 @ (comp @ X157 @ X158) @ (sub @ X159 @ X158))) != ((sub @ (X151 @ X157 @ X159) @ X158))) | (((lam @ (X151 @ sh @ one))) != ((lam @ (X153 @ sh @ one)))) | (((X151 @ X150 @ X152)) = ((X153 @ X150 @ X152)))) | ? [X148 : term,X149 : term] : ((X148 != X149) & (((lam @ X149)) = ((lam @ X148))))) | ? [X147 : subst,X146 : term] : (((comp @ sh @ (push @ X146 @ X147))) != X147)) | ? [X144 : term,X145 : subst] : (((sub @ one @ (push @ X144 @ X145))) != X144))) & (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_gthm = $true) & (((d2subst @ d_id)) = sh) & (pushprop = $true) & (hoasinduction_lem1v2_gthm = $true) & (ulamvar1 != $true) & (hoasinduction_lem1 = $true) & (axmap = $true) & (apnotvar = $true) & (! [X415 : (subst > term > subst > $o),X414 : term] : (($true = ((X415 @ id @ (lam @ X414) @ id))) | ? [X416 : term] : ((((X415 @ id @ (sub @ X414 @ (push @ X416 @ id)) @ id)) != $true) & ($true = ((X415 @ id @ X416 @ id)))) | ? [X417 : (subst > term > term)] : (($true != ((X415 @ id @ (lam @ (X417 @ sh @ one)) @ id))) & ! [X418 : term,X420 : subst,X419 : subst] : (((sub @ (X417 @ X420 @ X418) @ X419)) = ((X417 @ (comp @ X420 @ X419) @ (sub @ X418 @ X419)))) & ! [X421 : term] : (($true != ((X415 @ id @ X421 @ id))) | (((X415 @ id @ (X417 @ id @ X421) @ id)) = $true)))) <=> (hoasinduction_lem3a = $true)) & ((substmonoid = $true) <=> (! [X502 : subst] : (((comp @ X502 @ id)) = X502) & ! [X503 : subst] : (((comp @ id @ X503)) = X503))) & (((((! [X208 : (subst > term > subst > $o),X207 : term] : (($true = ((X208 @ id @ X207 @ id))) | ? [X216 : (subst > term > term)] : (! [X220 : term] : ((((X208 @ id @ X220 @ id)) != $true) | ($true = ((X208 @ id @ (X216 @ id @ X220) @ id)))) & ! [X217 : subst,X219 : subst,X218 : term] : (((X216 @ (comp @ X217 @ X219) @ (sub @ X218 @ X219))) = ((sub @ (X216 @ X217 @ X218) @ X219))) & ($true != ((X208 @ id @ (lam @ (X216 @ sh @ one)) @ id)))) | ? [X214 : term,X215 : subst,X212 : subst,X213 : subst] : (($true != ((X208 @ (comp @ X215 @ X212) @ (sub @ X214 @ X212) @ X213))) & (((X208 @ X215 @ X214 @ (comp @ X212 @ X213))) = $true)) | ? [X210 : term,X211 : term] : (($true = ((X208 @ id @ X210 @ id))) & ($true != ((X208 @ id @ (ap @ (sub @ X211 @ id) @ X210) @ id))) & ($true = ((X208 @ id @ X211 @ id)))) | ? [X209 : term] : (($true != ((X208 @ id @ X209 @ id))) & ($true = ((var @ (sub @ X209 @ id))))) | ? [X222 : term,X221 : subst,X223 : subst,X224 : subst] : (($true = ((X208 @ (comp @ X221 @ X224) @ (sub @ X222 @ X224) @ X223))) & ($true != ((X208 @ X221 @ X222 @ (comp @ X224 @ X223)))))) | ? [X199 : term,X197 : (term > $o),X198 : (subst > term > subst > $o)] : (! [X200 : (subst > term > term)] : (? [X204 : term] : (($true != ((X198 @ id @ (X200 @ id @ X204) @ id))) & ($true = ((X198 @ id @ X204 @ id)))) | (((X198 @ id @ (lam @ (X200 @ sh @ one)) @ id)) = $true) | ? [X202 : subst,X203 : subst,X201 : term] : (((X200 @ (comp @ X203 @ X202) @ (sub @ X201 @ X202))) != ((sub @ (X200 @ X203 @ X201) @ X202)))) & ! [X205 : term] : ((((X197 @ (sub @ X199 @ (push @ X205 @ id)))) = $true) | (((X197 @ X205)) != $true)) & ! [X206 : term] : (((X198 @ id @ X206 @ id)) = ((X197 @ X206))) & ($true != ((X197 @ (lam @ X199)))))) | ? [X196 : term] : (((sub @ X196 @ id)) != X196)) | ? [X190 : term,X189 : (term > $o)] : (! [X195 : term] : (($true != ((var @ X195))) | ($true = ((X189 @ X195)))) & ! [X192 : term,X191 : term] : (($true != ((X189 @ X192))) | (((X189 @ X191)) != $true) | (((X189 @ (ap @ X191 @ X192))) = $true)) & ($true != ((X189 @ X190))) & ! [X193 : term] : (? [X194 : term] : ((((X189 @ X194)) = $true) & ($true != ((X189 @ (sub @ X193 @ (push @ X194 @ id)))))) | ($true = ((X189 @ (lam @ X193))))))) | ? [X186 : (subst > term > subst > $o)] : ! [X187 : (term > $o)] : ? [X188 : term] : (((X186 @ id @ X188 @ id)) != ((X187 @ X188)))) <=> (hoasinduction_lthm_3 = $true)) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (substmonoid_gthm = $true) & (ulamvarind = $true) & ((? [X500 : term] : (((sub @ X500 @ id)) != X500) | ! [X501 : term] : (((sub @ X501 @ id)) = X501)) <=> (termmset_lthm = $true)) & (! [X483 : term,X482 : term] : ((X482 = X483) | (((sub @ X483 @ sh)) != ((sub @ X482 @ sh)))) <=> (shinj = $true)) & (((var @ (d2term @ d_one))) != $true) & (! [X535 : term,X534 : (term > $o)] : (? [X539 : term] : (($true != ((X534 @ (lam @ X539)))) & ($true = ((X534 @ X539)))) | ? [X537 : term,X538 : term] : (($true = ((X534 @ X537))) & ($true != ((X534 @ (ap @ X538 @ X537)))) & (((X534 @ X538)) = $true)) | ($true = ((X534 @ X535))) | ? [X536 : term] : ((((var @ X536)) = $true) & ($true != ((X534 @ X536))))) <=> (induction = $true)) & (hoaslamnotap_gthm = $true) & (! [X443 : term,X444 : term] : ((((lam @ X443)) != ((lam @ X444))) | (X443 = X444)) <=> (laminj = $true)) & ((((! [X332 : subst] : (((comp @ id @ X332)) = X332) & ! [X331 : subst] : (((comp @ X331 @ id)) = X331)) | ? [X330 : subst] : (((comp @ X330 @ id)) != X330)) | ? [X329 : subst] : (((comp @ id @ X329)) != X329)) <=> (substmonoid_lthm = $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (pushprop_lem3v2 = $true) & ((induction2_lthm = $true) <=> ((! [X169 : term,X170 : (term > $o)] : (? [X174 : term,X173 : term] : ((((X170 @ X173)) = $true) & ($true = ((X170 @ X174))) & (((X170 @ (ap @ X174 @ X173))) != $true)) | ? [X175 : term] : (($true != ((X170 @ X175))) & ($true = ((var @ X175)))) | (((X170 @ X169)) = $true) | ? [X171 : term] : (! [X172 : term] : (($true != ((X170 @ X172))) | ($true = ((X170 @ (sub @ X171 @ (push @ X172 @ id)))))) & ($true != ((X170 @ (lam @ X171)))))) | ? [X162 : (term > $o),X163 : subst,X161 : term] : (! [X164 : term,X165 : term] : (($true = ((X162 @ (ap @ X164 @ X165)))) | (((X162 @ X164)) != $true) | ($true != ((X162 @ X165)))) & ! [X168 : term] : (($true = ((X162 @ (sub @ X168 @ X163)))) | (((var @ X168)) != $true)) & ! [X166 : term] : (($true = ((X162 @ (lam @ X166)))) | ? [X167 : term] : (($true = ((X162 @ X167))) & ($true != ((X162 @ (sub @ X166 @ (push @ X167 @ id))))))) & (((X162 @ (sub @ X161 @ X163))) != $true))) | ? [X160 : term] : (((sub @ X160 @ id)) != X160))) & (hoaslamnotvar_lthm = $true) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X83 : term] : ? [X84 : (subst > term > term)] : ! [X85 : term,X86 : subst] : (((X84 @ X86 @ X85)) = ((sub @ X83 @ (push @ X85 @ X86))))) & (! [X457 : term,X456 : (term > $o)] : (? [X459 : term,X458 : term] : ((((X456 @ X459)) = $true) & (((X456 @ X458)) = $true) & ($true != ((X456 @ (ap @ X459 @ X458))))) | ? [X460 : term] : (! [X461 : term] : (($true != ((X456 @ X461))) | (((X456 @ (sub @ X460 @ (push @ X461 @ id)))) = $true)) & (((X456 @ (lam @ X460))) != $true)) | ? [X462 : term] : (($true != ((X456 @ X462))) & ($true = ((var @ X462)))) | ($true = ((X456 @ X457)))) <=> (induction2 = $true)) & ! [X96 : d_term] : (d_one = X96) & (axvarshift = $true) & (pushprop_lem3v2_lthm = $true) & ((hoaslaminj = $true) <=> ! [X176 : subst,X178 : (subst > term > term),X177 : term,X179 : (subst > term > term)] : (? [X183 : subst,X184 : subst,X185 : term] : (((X179 @ (comp @ X184 @ X183) @ (sub @ X185 @ X183))) != ((sub @ (X179 @ X184 @ X185) @ X183))) | ? [X180 : term,X181 : subst,X182 : subst] : (((X178 @ (comp @ X182 @ X181) @ (sub @ X180 @ X181))) != ((sub @ (X178 @ X182 @ X180) @ X181))) | (((X179 @ X176 @ X177)) = ((X178 @ X176 @ X177))) | (((lam @ (X178 @ sh @ one))) != ((lam @ (X179 @ sh @ one)))))) & ((hoasinduction_lem2v2 = $true) <=> ! [X109 : (term > $o),X110 : (subst > term > subst > $o),X107 : term,X108 : term] : (? [X112 : subst,X113 : subst,X114 : subst,X111 : term] : (($true = ((X110 @ (comp @ X112 @ X114) @ (sub @ X111 @ X114) @ X113))) & ($true != ((X110 @ X112 @ X111 @ (comp @ X114 @ X113))))) | ? [X115 : subst,X117 : term,X116 : subst,X118 : subst] : (($true = ((X110 @ X115 @ X117 @ (comp @ X116 @ X118)))) & (((X110 @ (comp @ X115 @ X116) @ (sub @ X117 @ X116) @ X118)) != $true)) | ? [X119 : term] : (((X109 @ X119)) != ((X110 @ id @ X119 @ id))) | ($true = ((X109 @ (ap @ X107 @ X108)))) | (((X109 @ X108)) != $true) | (((X109 @ X107)) != $true) | ? [X120 : term,X121 : term] : (($true = ((X110 @ id @ X120 @ id))) & ($true != ((X110 @ id @ (ap @ (sub @ X120 @ id) @ X121) @ id))) & ($true = ((X110 @ id @ X121 @ id)))))) & (ulamvarsh = $true) & ((hoasinduction_lem3b = $true) <=> ! [X253 : term] : ? [X254 : (subst > term > term)] : (((X254 @ sh @ one)) = ((sub @ X253 @ (push @ one @ sh))))) & (! [X98 : subst,X99 : (term > $o),X97 : term] : ? [X100 : (term > $o)] : ! [X101 : term] : (((X99 @ (sub @ X101 @ (push @ X97 @ X98)))) = ((X100 @ X101))) <=> (pushprop_lem0 = $true)) & (! [X23 : (subst > term > subst > $o),X24 : term,X22 : (term > $o)] : (? [X39 : term] : (($true = ((X22 @ X39))) & (((X22 @ (sub @ X24 @ (push @ X39 @ id)))) != $true)) | ? [X29 : (subst > term > term)] : (! [X31 : subst,X33 : term,X32 : subst] : (((X29 @ (comp @ X31 @ X32) @ (sub @ X33 @ X32))) = ((sub @ (X29 @ X31 @ X33) @ X32))) & ! [X30 : term] : (($true = ((X23 @ id @ (X29 @ id @ X30) @ id))) | ($true != ((X23 @ id @ X30 @ id)))) & ($true != ((X23 @ id @ (lam @ (X29 @ sh @ one)) @ id)))) | ? [X38 : term] : (((X23 @ id @ X38 @ id)) != ((X22 @ X38))) | ? [X37 : subst,X35 : subst,X34 : subst,X36 : term] : (($true = ((X23 @ (comp @ X37 @ X35) @ (sub @ X36 @ X35) @ X34))) & ($true != ((X23 @ X37 @ X36 @ (comp @ X35 @ X34))))) | ($true = ((X22 @ (lam @ X24)))) | ? [X27 : subst,X26 : subst,X25 : term,X28 : subst] : ((((X23 @ X28 @ X25 @ (comp @ X27 @ X26))) = $true) & (((X23 @ (comp @ X28 @ X27) @ (sub @ X25 @ X27) @ X26)) != $true))) <=> (hoasinduction_lem3v2 = $true)) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (termmset_gthm = $true) & (hoaslamnotvar_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (hoasinduction_lem3b_gthm = $true) & ((apinj1 = $true) <=> ! [X518 : term,X519 : term,X520 : term,X521 : term] : ((((ap @ X518 @ X520)) != ((ap @ X521 @ X519))) | (X518 = X521))) & ((induction2lem_lthm = $true) <=> ((((! [X474 : subst,X475 : term,X476 : (term > $o)] : (($true = ((X476 @ (sub @ X475 @ X474)))) | ? [X477 : term] : (! [X478 : term] : (($true != ((X476 @ X478))) | ($true = ((X476 @ (sub @ X477 @ (push @ X478 @ id)))))) & ($true != ((X476 @ (lam @ X477))))) | ? [X481 : term,X480 : term] : ((((X476 @ (ap @ X481 @ X480))) != $true) & ($true = ((X476 @ X480))) & ($true = ((X476 @ X481)))) | ? [X479 : term] : ((((var @ X479)) = $true) & ($true != ((X476 @ (sub @ X479 @ X474)))))) | ? [X468 : (term > $o),X469 : term] : (! [X472 : term] : (($true != ((X468 @ X472))) | ($true = ((X468 @ (lam @ X472))))) & ! [X473 : term] : ((((X468 @ X473)) = $true) | (((var @ X473)) != $true)) & ! [X470 : term,X471 : term] : (($true = ((X468 @ (ap @ X471 @ X470)))) | ($true != ((X468 @ X471))) | ($true != ((X468 @ X470)))) & ($true != ((X468 @ X469))))) | ? [X467 : subst] : (((comp @ X467 @ id)) != X467)) | ? [X465 : term,X466 : subst] : (((comp @ sh @ (push @ X465 @ X466))) != X466)) | ? [X464 : term,X463 : subst] : (((sub @ one @ (push @ X464 @ X463))) != X464))) & (((! [X517 : term,X515 : term,X514 : term,X516 : term] : ((X514 = X516) | (((ap @ (sub @ X514 @ id) @ X517)) != ((ap @ (sub @ X516 @ id) @ X515)))) | ? [X511 : term,X512 : term,X510 : term,X513 : term] : ((((ap @ X512 @ X513)) = ((ap @ X510 @ X511))) & (X510 != X512))) | ? [X509 : term] : (((sub @ X509 @ id)) != X509)) <=> (hoasapinj1_lthm = $true)) & ((hoasinduction_lem3 = $true) <=> ! [X6 : (subst > term > subst > $o),X7 : term] : ((((X6 @ id @ (lam @ X7) @ id)) = $true) | ? [X9 : subst,X10 : term,X8 : subst,X11 : subst] : (($true != ((X6 @ (comp @ X8 @ X9) @ (sub @ X10 @ X9) @ X11))) & (((X6 @ X8 @ X10 @ (comp @ X9 @ X11))) = $true)) | ? [X12 : term] : (($true != ((X6 @ id @ (sub @ X7 @ (push @ X12 @ id)) @ id))) & (((X6 @ id @ X12 @ id)) = $true)) | ? [X16 : subst,X13 : subst,X14 : term,X15 : subst] : (($true != ((X6 @ X15 @ X14 @ (comp @ X13 @ X16)))) & ($true = ((X6 @ (comp @ X15 @ X13) @ (sub @ X14 @ X13) @ X16)))) | ? [X17 : (subst > term > term)] : (! [X18 : term] : (($true != ((X6 @ id @ X18 @ id))) | ($true = ((X6 @ id @ (X17 @ id @ X18) @ id)))) & ! [X20 : subst,X21 : subst,X19 : term] : (((X17 @ (comp @ X21 @ X20) @ (sub @ X19 @ X20))) = ((sub @ (X17 @ X21 @ X19) @ X20))) & (((X6 @ id @ (lam @ (X17 @ sh @ one)) @ id)) != $true)))) & (id = ((d2subst @ d_id))) & ! [X531 : d_subst] : (d_id = X531) & (axabs = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoasapinj1_gthm = $true) & (lamnotvar = $true) & (pushprop_lem1_gthm = $true) & (hoasinduction_lem2_lthm = $true)),
  inference(ennf_transformation,[],[f9])).
thf(f12,plain,(
  ((! [X169 : term,X170 : (term > $o)] : (? [X174 : term,X173 : term] : ((((X170 @ X173)) = $true) & ($true = ((X170 @ X174))) & (((X170 @ (ap @ X174 @ X173))) != $true)) | ? [X175 : term] : (($true != ((X170 @ X175))) & ($true = ((var @ X175)))) | (((X170 @ X169)) = $true) | ? [X171 : term] : (! [X172 : term] : (($true != ((X170 @ X172))) | ($true = ((X170 @ (sub @ X171 @ (push @ X172 @ id)))))) & ($true != ((X170 @ (lam @ X171)))))) | ? [X162 : (term > $o),X163 : subst,X161 : term] : (! [X164 : term,X165 : term] : (($true = ((X162 @ (ap @ X164 @ X165)))) | (((X162 @ X164)) != $true) | ($true != ((X162 @ X165)))) & ! [X168 : term] : (($true = ((X162 @ (sub @ X168 @ X163)))) | (((var @ X168)) != $true)) & ! [X166 : term] : (($true = ((X162 @ (lam @ X166)))) | ? [X167 : term] : (($true = ((X162 @ X167))) & ($true != ((X162 @ (sub @ X166 @ (push @ X167 @ id))))))) & (((X162 @ (sub @ X161 @ X163))) != $true)) | ? [X160 : term] : (((sub @ X160 @ id)) != X160)) <=> (induction2_lthm = $true)) & ((hoasinduction_lem3aaa = $true) <=> ! [X371 : (subst > term > subst > $o),X370 : term] : (? [X373 : term,X372 : (subst > term > term)] : (! [X374 : term] : (($true = ((X371 @ id @ (X372 @ id @ X374) @ id))) | ($true != ((X371 @ id @ X374 @ id)))) & (((X371 @ id @ (lam @ (X372 @ sh @ one)) @ id)) != $true) & ! [X376 : subst,X375 : term,X377 : subst] : (((sub @ (sub @ X373 @ (push @ X375 @ X377)) @ X376)) = ((sub @ (X372 @ X377 @ X375) @ X376))) & ! [X378 : subst,X379 : subst,X380 : term] : (((sub @ X373 @ (push @ (sub @ X380 @ X379) @ (comp @ X378 @ X379)))) = ((X372 @ (comp @ X378 @ X379) @ (sub @ X380 @ X379))))) | ($true = ((X371 @ id @ (lam @ (sub @ X370 @ (push @ one @ sh))) @ id))) | ? [X381 : term] : ((((X371 @ id @ (sub @ X370 @ (push @ X381 @ id)) @ id)) != $true) & ($true = ((X371 @ id @ X381 @ id)))))) & (ulamvarind = $true) & (pushprop_lem3v2_lthm = $true) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X498 : term] : ? [X499 : (subst > term > term)] : (((X499 @ sh @ one)) = ((sub @ X498 @ (push @ one @ sh))))) & (substmonoid_gthm = $true) & ((! [X474 : subst,X475 : term,X476 : (term > $o)] : (($true = ((X476 @ (sub @ X475 @ X474)))) | ? [X477 : term] : (! [X478 : term] : (($true != ((X476 @ X478))) | ($true = ((X476 @ (sub @ X477 @ (push @ X478 @ id)))))) & ($true != ((X476 @ (lam @ X477))))) | ? [X481 : term,X480 : term] : ((((X476 @ (ap @ X481 @ X480))) != $true) & ($true = ((X476 @ X480))) & ($true = ((X476 @ X481)))) | ? [X479 : term] : ((((var @ X479)) = $true) & ($true != ((X476 @ (sub @ X479 @ X474)))))) | ? [X464 : term,X463 : subst] : (((sub @ one @ (push @ X464 @ X463))) != X464) | ? [X467 : subst] : (((comp @ X467 @ id)) != X467) | ? [X468 : (term > $o),X469 : term] : (! [X472 : term] : (($true != ((X468 @ X472))) | ($true = ((X468 @ (lam @ X472))))) & ! [X473 : term] : ((((X468 @ X473)) = $true) | (((var @ X473)) != $true)) & ! [X470 : term,X471 : term] : (($true = ((X468 @ (ap @ X471 @ X470)))) | ($true != ((X468 @ X471))) | ($true != ((X468 @ X470)))) & ($true != ((X468 @ X469)))) | ? [X465 : term,X466 : subst] : (((comp @ sh @ (push @ X465 @ X466))) != X466)) <=> (induction2lem_lthm = $true)) & (! [X506 : term,X508 : (term > $o),X504 : subst,X505 : term,X507 : (term > $o)] : (($true != ((X507 @ X505))) | ($true = ((X508 @ (sub @ X505 @ (push @ X506 @ X504)))))) <=> (pushprop_lem1 = $true)) & ((hoasinduction_lem3v2_lthm = $true) <=> (? [X122 : term] : (((sub @ X122 @ id)) != X122) | ! [X125 : (subst > term > subst > $o),X123 : term,X124 : (term > $o)] : (? [X140 : term] : (($true != ((X124 @ (sub @ X123 @ (push @ X140 @ id))))) & (((X124 @ X140)) = $true)) | ? [X139 : subst,X137 : term,X136 : subst,X138 : subst] : ((((X125 @ X139 @ X137 @ (comp @ X138 @ X136))) = $true) & ($true != ((X125 @ (comp @ X139 @ X138) @ (sub @ X137 @ X138) @ X136)))) | ? [X126 : term] : (((X124 @ X126)) != ((X125 @ id @ X126 @ id))) | (((X124 @ (lam @ X123))) = $true) | ? [X131 : (subst > term > term)] : (($true != ((X125 @ id @ (lam @ (X131 @ sh @ one)) @ id))) & ! [X134 : subst,X135 : subst,X133 : term] : (((X131 @ (comp @ X134 @ X135) @ (sub @ X133 @ X135))) = ((sub @ (X131 @ X134 @ X133) @ X135))) & ! [X132 : term] : (($true = ((X125 @ id @ (X131 @ id @ X132) @ id))) | (((X125 @ id @ X132 @ id)) != $true))) | ? [X130 : subst,X129 : subst,X127 : term,X128 : subst] : ((((X125 @ X129 @ X127 @ (comp @ X130 @ X128))) != $true) & ($true = ((X125 @ (comp @ X129 @ X130) @ (sub @ X127 @ X130) @ X128))))))) & (pushprop_lem3v2 = $true) & (hoaslamnotvar_gthm = $true) & (! [X235 : subst,X234 : term] : (((sub @ one @ (push @ X234 @ X235))) = X234) <=> (axvarcons = $true)) & (pushprop_lem2v2 = $true) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X89 : (subst > term > subst > $o),X88 : term] : (? [X95 : term] : (($true != ((X89 @ id @ (sub @ X88 @ (push @ X95 @ id)) @ id))) & (((X89 @ id @ X95 @ id)) = $true)) | (((X89 @ id @ (lam @ (sub @ X88 @ (push @ one @ sh))) @ id)) = $true) | ? [X90 : (subst > term > term)] : (($true != ((X89 @ id @ (lam @ (X90 @ sh @ one)) @ id))) & ! [X93 : subst,X94 : subst,X92 : term] : (((sub @ (X90 @ X93 @ X92) @ X94)) = ((X90 @ (comp @ X93 @ X94) @ (sub @ X92 @ X94)))) & ! [X91 : term] : (($true = ((X89 @ id @ (X90 @ id @ X91) @ id))) | (((X89 @ id @ X91 @ id)) != $true))))) & (hoasinduction_lem3_gthm = $true) & (! [X494 : term] : ? [X495 : (subst > term > term)] : ! [X496 : term,X497 : subst] : (((sub @ X494 @ (push @ X496 @ X497))) = ((X495 @ X497 @ X496))) <=> (hoasinduction_lem3v2_f = $true)) & (pushprop_lthm_orig = $true) & (lamnotvar = $true) & ((apinj1 = $true) <=> ! [X518 : term,X519 : term,X520 : term,X521 : term] : ((((ap @ X518 @ X520)) != ((ap @ X521 @ X519))) | (X518 = X521))) & (! [X535 : term,X534 : (term > $o)] : (? [X539 : term] : (($true != ((X534 @ (lam @ X539)))) & ($true = ((X534 @ X539)))) | ? [X537 : term,X538 : term] : (($true = ((X534 @ X537))) & ($true != ((X534 @ (ap @ X538 @ X537)))) & (((X534 @ X538)) = $true)) | ($true = ((X534 @ X535))) | ? [X536 : term] : ((((var @ X536)) = $true) & ($true != ((X534 @ X536))))) <=> (induction = $true)) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem1v2 = $true) & ! [X413 : d_subst,X412 : d_subst] : ((((d2subst @ X413)) != ((d2subst @ X412))) | (X412 = X413)) & ((hoasinduction_lem0 = $true) <=> ! [X445 : (subst > term > subst > $o)] : ? [X446 : (term > $o)] : ! [X447 : term] : (((X445 @ id @ X447 @ id)) = ((X446 @ X447)))) & (induction2_gthm = $true) & ! [X532 : subst] : ? [X533 : d_subst] : (((d2subst @ X533)) = X532) & (hoaslamnotap_gthm = $true) & (induction2lem_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (! [X415 : (subst > term > subst > $o),X414 : term] : (($true = ((X415 @ id @ (lam @ X414) @ id))) | ? [X416 : term] : ((((X415 @ id @ (sub @ X414 @ (push @ X416 @ id)) @ id)) != $true) & ($true = ((X415 @ id @ X416 @ id)))) | ? [X417 : (subst > term > term)] : (($true != ((X415 @ id @ (lam @ (X417 @ sh @ one)) @ id))) & ! [X418 : term,X420 : subst,X419 : subst] : (((sub @ (X417 @ X420 @ X418) @ X419)) = ((X417 @ (comp @ X420 @ X419) @ (sub @ X418 @ X419)))) & ! [X421 : term] : (($true != ((X415 @ id @ X421 @ id))) | (((X415 @ id @ (X417 @ id @ X421) @ id)) = $true)))) <=> (hoasinduction_lem3a = $true)) & (id = ((d2subst @ d_id))) & (hoasapinj1_gthm = $true) & ((apinj2 = $true) <=> ! [X71 : term,X73 : term,X74 : term,X72 : term] : ((X71 = X72) | (((ap @ X73 @ X72)) != ((ap @ X74 @ X71))))) & (((d2subst @ d_id)) = sh) & (hoasinduction_lem2_lthm = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoaslaminj = $true) <=> ! [X176 : subst,X178 : (subst > term > term),X177 : term,X179 : (subst > term > term)] : (? [X183 : subst,X184 : subst,X185 : term] : (((X179 @ (comp @ X184 @ X183) @ (sub @ X185 @ X183))) != ((sub @ (X179 @ X184 @ X185) @ X183))) | ? [X180 : term,X181 : subst,X182 : subst] : (((X178 @ (comp @ X182 @ X181) @ (sub @ X180 @ X181))) != ((sub @ (X178 @ X182 @ X180) @ X181))) | (((X179 @ X176 @ X177)) = ((X178 @ X176 @ X177))) | (((lam @ (X178 @ sh @ one))) != ((lam @ (X179 @ sh @ one)))))) & (hoasinduction_lem3b_gthm = $true) & ((axidr = $true) <=> ! [X424 : subst] : (((comp @ X424 @ id)) = X424)) & (termmset_gthm = $true) & (hoasinduction_lem1 = $true) & (pushprop = $true) & (apnotvar = $true) & ! [X96 : d_term] : (d_one = X96) & (hoaslamnotvar = $true) & (! [X530 : term,X529 : subst] : (((comp @ sh @ (push @ X530 @ X529))) = X529) <=> (axshiftcons = $true)) & (axapp = $true) & (hoasinduction_lem3a_gthm = $true) & (! [X106 : term] : (((sub @ X106 @ id)) = X106) <=> (axvarid = $true)) & (axmap = $true) & ((? [X144 : term,X145 : subst] : (((sub @ one @ (push @ X144 @ X145))) != X144) | ? [X148 : term,X149 : term] : ((X148 != X149) & (((lam @ X149)) = ((lam @ X148)))) | ? [X147 : subst,X146 : term] : (((comp @ sh @ (push @ X146 @ X147))) != X147) | ! [X152 : term,X151 : (subst > term > term),X150 : subst,X153 : (subst > term > term)] : (? [X155 : subst,X154 : term,X156 : subst] : (((sub @ (X153 @ X155 @ X154) @ X156)) != ((X153 @ (comp @ X155 @ X156) @ (sub @ X154 @ X156)))) | ? [X158 : subst,X157 : subst,X159 : term] : (((X151 @ (comp @ X157 @ X158) @ (sub @ X159 @ X158))) != ((sub @ (X151 @ X157 @ X159) @ X158))) | (((lam @ (X151 @ sh @ one))) != ((lam @ (X153 @ sh @ one)))) | (((X151 @ X150 @ X152)) = ((X153 @ X150 @ X152))))) <=> (hoaslaminj_lthm = $true)) & (axabs = $true) & ((hoasapinj2 = $true) <=> ! [X105 : term,X102 : term,X103 : term,X104 : term] : ((X102 = X105) | (((ap @ (sub @ X104 @ id) @ X102)) != ((ap @ (sub @ X103 @ id) @ X105))))) & ((hoaslamnotap = $true) <=> ! [X1 : term,X2 : term,X0 : (subst > term > term)] : (? [X3 : subst,X4 : term,X5 : subst] : (((X0 @ (comp @ X3 @ X5) @ (sub @ X4 @ X5))) != ((sub @ (X0 @ X3 @ X4) @ X5))) | (((ap @ (sub @ X2 @ id) @ X1)) != ((lam @ (X0 @ sh @ one)))))) & (hoaslamnotvar_lthm = $true) & (ulamvarsh = $true) & ((? [X500 : term] : (((sub @ X500 @ id)) != X500) | ! [X501 : term] : (((sub @ X501 @ id)) = X501)) <=> (termmset_lthm = $true)) & (! [X48 : (subst > term > subst > $o),X47 : term,X46 : term] : (? [X53 : term,X54 : term] : (($true = ((X48 @ id @ X53 @ id))) & ($true != ((X48 @ id @ (ap @ (sub @ X53 @ id) @ X54) @ id))) & ($true = ((X48 @ id @ X54 @ id)))) | (((X48 @ id @ X47 @ id)) != $true) | ? [X56 : term,X58 : subst,X55 : subst,X57 : subst] : (($true = ((X48 @ X55 @ X56 @ (comp @ X57 @ X58)))) & ($true != ((X48 @ (comp @ X55 @ X57) @ (sub @ X56 @ X57) @ X58)))) | ($true != ((X48 @ id @ X46 @ id))) | ($true = ((X48 @ id @ (ap @ X46 @ X47) @ id))) | ? [X50 : term,X52 : subst,X51 : subst,X49 : subst] : (($true != ((X48 @ X49 @ X50 @ (comp @ X52 @ X51)))) & (((X48 @ (comp @ X49 @ X52) @ (sub @ X50 @ X52) @ X51)) = $true))) <=> (hoasinduction_lem2 = $true)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoasapnotvar = $true) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_lem3v2_gthm = $true) & ((? [X364 : subst,X363 : term] : (((comp @ sh @ (push @ X363 @ X364))) != X364) | ! [X369 : term,X366 : subst,X368 : term,X367 : (term > $o),X365 : (term > $o)] : (($true != ((X365 @ X369))) | ($true = ((X367 @ (sub @ X369 @ (push @ X368 @ X366)))))) | ? [X361 : term,X362 : subst] : (((sub @ one @ (push @ X361 @ X362))) != X361)) <=> (pushprop_lem1_lthm = $true)) & ((hoasapinj1 = $true) <=> ! [X67 : term,X70 : term,X69 : term,X68 : term] : ((X67 = X68) | (((ap @ (sub @ X67 @ id) @ X70)) != ((ap @ (sub @ X68 @ id) @ X69))))) & (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)))))))))))))) & ((pushprop_lem1v2 = $true) <=> ! [X40 : subst,X43 : (term > $o),X42 : (term > $o),X41 : term] : (? [X44 : term] : (((X43 @ (sub @ X44 @ (push @ X41 @ X40)))) != ((X42 @ X44))) | ($true = ((X42 @ one))) | ($true != ((X43 @ X41))))) & (axclos = $true) & ((hoasinduction_lem3b = $true) <=> ! [X253 : term] : ? [X254 : (subst > term > term)] : (((X254 @ sh @ one)) = ((sub @ X253 @ (push @ one @ sh))))) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (lamnotap != $true) & (! [X76 : (term > $o),X77 : term,X75 : subst] : (? [X81 : term] : ((((X76 @ (lam @ X81))) != $true) & ! [X82 : term] : (($true = ((X76 @ (sub @ X81 @ (push @ X82 @ id))))) | ($true != ((X76 @ X82))))) | ? [X79 : term,X80 : term] : ((((X76 @ X80)) = $true) & (((X76 @ X79)) = $true) & (((X76 @ (ap @ X80 @ X79))) != $true)) | ? [X78 : term] : ((((X76 @ (sub @ X78 @ X75))) != $true) & (((var @ X78)) = $true)) | ($true = ((X76 @ (sub @ X77 @ X75))))) <=> (induction2lem = $true)) & ((! [X526 : subst,X525 : term,X527 : (term > $o),X524 : (term > $o)] : ((((X524 @ one)) = $true) | ? [X528 : term] : (((X524 @ X528)) != ((X527 @ (sub @ X528 @ (push @ X525 @ X526))))) | ($true != ((X527 @ X525)))) | ? [X523 : subst,X522 : term] : (((sub @ one @ (push @ X522 @ X523))) != X522)) <=> (pushprop_lem1v2_lthm = $true)) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X83 : term] : ? [X84 : (subst > term > term)] : ! [X85 : term,X86 : subst] : (((X84 @ X86 @ X85)) = ((sub @ X83 @ (push @ X85 @ X86))))) & ! [X422 : term] : ? [X423 : d_term] : (((d2term @ X423)) = X422) & ((hoasinduction_no_psi_cond = $true) <=> ! [X225 : (subst > term > subst > $o),X226 : term] : ((((X225 @ id @ X226 @ id)) = $true) | ? [X229 : (subst > term > term)] : (($true != ((X225 @ id @ (lam @ (X229 @ sh @ one)) @ id))) & ! [X233 : subst,X231 : subst,X232 : term] : (((sub @ (X229 @ X231 @ X232) @ X233)) = ((X229 @ (comp @ X231 @ X233) @ (sub @ X232 @ X233)))) & ! [X230 : term] : (($true != ((X225 @ id @ X230 @ id))) | (((X225 @ id @ (X229 @ id @ X230) @ id)) = $true))) | ? [X228 : term,X227 : term] : (($true = ((X225 @ id @ X227 @ id))) & ($true = ((X225 @ id @ X228 @ id))) & ($true != ((X225 @ id @ (ap @ (sub @ X227 @ id) @ X228) @ id)))))) & (! [X443 : term,X444 : term] : ((((lam @ X443)) != ((lam @ X444))) | (X443 = X444)) <=> (laminj = $true)) & (pushprop_lthm = $true) & ((hoasapinj2_lthm = $true) <=> (! [X455 : term,X453 : term,X454 : term,X452 : term] : ((((ap @ (sub @ X454 @ id) @ X452)) != ((ap @ (sub @ X453 @ id) @ X455))) | (X452 = X455)) | ? [X449 : term,X450 : term,X451 : term,X448 : term] : ((X448 != X450) & (((ap @ X451 @ X448)) = ((ap @ X449 @ X450)))))) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((hoasinduction_lem3 = $true) <=> ! [X6 : (subst > term > subst > $o),X7 : term] : ((((X6 @ id @ (lam @ X7) @ id)) = $true) | ? [X9 : subst,X10 : term,X8 : subst,X11 : subst] : (($true != ((X6 @ (comp @ X8 @ X9) @ (sub @ X10 @ X9) @ X11))) & (((X6 @ X8 @ X10 @ (comp @ X9 @ X11))) = $true)) | ? [X12 : term] : (($true != ((X6 @ id @ (sub @ X7 @ (push @ X12 @ id)) @ id))) & (((X6 @ id @ X12 @ id)) = $true)) | ? [X16 : subst,X13 : subst,X14 : term,X15 : subst] : (($true != ((X6 @ X15 @ X14 @ (comp @ X13 @ X16)))) & ($true = ((X6 @ (comp @ X15 @ X13) @ (sub @ X14 @ X13) @ X16)))) | ? [X17 : (subst > term > term)] : (! [X18 : term] : (($true != ((X6 @ id @ X18 @ id))) | ($true = ((X6 @ id @ (X17 @ id @ X18) @ id)))) & ! [X20 : subst,X21 : subst,X19 : term] : (((X17 @ (comp @ X21 @ X20) @ (sub @ X19 @ X20))) = ((sub @ (X17 @ X21 @ X19) @ X20))) & (((X6 @ id @ (lam @ (X17 @ sh @ one)) @ id)) != $true)))) & ((pushprop_lem0_lthm = $true) <=> ! [X256 : (term > $o),X255 : subst,X257 : term] : ? [X258 : (term > $o)] : ! [X259 : term] : (((X258 @ X259)) = ((X256 @ (sub @ X259 @ (push @ X257 @ X255)))))) & (! [X45 : subst] : (((comp @ id @ X45)) = X45) <=> (axidl = $true)) & (! [X98 : subst,X99 : (term > $o),X97 : term] : ? [X100 : (term > $o)] : ! [X101 : term] : (((X99 @ (sub @ X101 @ (push @ X97 @ X98)))) = ((X100 @ X101))) <=> (pushprop_lem0 = $true)) & (pushprop_lem0_gthm = $true) & ((hoasinduction_lem3_lthm = $true) <=> (? [X333 : term] : (((sub @ X333 @ id)) != X333) | ? [X335 : (subst > term > subst > $o),X334 : term] : (! [X337 : (subst > term > term)] : (? [X339 : term,X340 : subst,X341 : subst] : (((sub @ (X337 @ X341 @ X339) @ X340)) != ((X337 @ (comp @ X341 @ X340) @ (sub @ X339 @ X340)))) | ($true = ((X335 @ id @ (lam @ (X337 @ sh @ one)) @ id))) | ? [X338 : term] : (($true != ((X335 @ id @ (X337 @ id @ X338) @ id))) & ($true = ((X335 @ id @ X338 @ id))))) & ($true != ((X335 @ id @ (lam @ (sub @ X334 @ (push @ one @ sh))) @ id))) & ! [X336 : term] : ((((X335 @ id @ (sub @ X334 @ (push @ X336 @ id)) @ id)) = $true) | (((X335 @ id @ X336 @ id)) != $true))) | ! [X343 : term,X342 : (subst > term > subst > $o)] : (($true = ((X342 @ id @ (lam @ X343) @ id))) | ? [X354 : subst,X355 : subst,X356 : subst,X357 : term] : (($true != ((X342 @ (comp @ X356 @ X355) @ (sub @ X357 @ X355) @ X354))) & ($true = ((X342 @ X356 @ X357 @ (comp @ X355 @ X354))))) | ? [X349 : (subst > term > term)] : (! [X350 : term] : ((((X342 @ id @ X350 @ id)) != $true) | ($true = ((X342 @ id @ (X349 @ id @ X350) @ id)))) & ($true != ((X342 @ id @ (lam @ (X349 @ sh @ one)) @ id))) & ! [X353 : subst,X352 : subst,X351 : term] : (((X349 @ (comp @ X353 @ X352) @ (sub @ X351 @ X352))) = ((sub @ (X349 @ X353 @ X351) @ X352)))) | ? [X344 : term] : (($true = ((X342 @ id @ X344 @ id))) & ($true != ((X342 @ id @ (sub @ X343 @ (push @ X344 @ id)) @ id)))) | ? [X346 : term,X345 : subst,X347 : subst,X348 : subst] : ((((X342 @ (comp @ X345 @ X347) @ (sub @ X346 @ X347) @ X348)) = $true) & ($true != ((X342 @ X345 @ X346 @ (comp @ X347 @ X348)))))))) & ((? [X394 : term,X395 : (term > $o),X393 : (subst > term > subst > $o)] : (! [X401 : term] : (($true = ((X395 @ (sub @ X394 @ (push @ X401 @ id))))) | ($true != ((X395 @ X401)))) & ! [X402 : term] : (((X393 @ id @ X402 @ id)) = ((X395 @ X402))) & ($true != ((X395 @ (lam @ X394)))) & ! [X396 : (subst > term > term)] : (? [X397 : term] : (($true = ((X393 @ id @ X397 @ id))) & ($true != ((X393 @ id @ (X396 @ id @ X397) @ id)))) | ($true = ((X393 @ id @ (lam @ (X396 @ sh @ one)) @ id))) | ? [X399 : term,X400 : subst,X398 : subst] : (((sub @ (X396 @ X398 @ X399) @ X400)) != ((X396 @ (comp @ X398 @ X400) @ (sub @ X399 @ X400)))))) | ? [X382 : (subst > term > subst > $o)] : ! [X383 : (term > $o)] : ? [X384 : term] : (((X382 @ id @ X384 @ id)) != ((X383 @ X384))) | ? [X392 : term] : (((sub @ X392 @ id)) != X392) | ? [X386 : term,X385 : (term > $o)] : (! [X391 : term] : (($true = ((X385 @ X391))) | (((var @ X391)) != $true)) & ($true != ((X385 @ X386))) & ! [X389 : term] : (? [X390 : term] : (($true = ((X385 @ X390))) & ($true != ((X385 @ (sub @ X389 @ (push @ X390 @ id)))))) | ($true = ((X385 @ (lam @ X389))))) & ! [X387 : term,X388 : term] : ((((X385 @ (ap @ X388 @ X387))) = $true) | ($true != ((X385 @ X387))) | ($true != ((X385 @ X388))))) | ! [X403 : (subst > term > subst > $o),X404 : term] : (? [X410 : term,X411 : term] : (($true = ((X403 @ id @ X410 @ id))) & ($true != ((X403 @ id @ (ap @ (sub @ X411 @ id) @ X410) @ id))) & (((X403 @ id @ X411 @ id)) = $true)) | ($true = ((X403 @ id @ X404 @ id))) | ? [X405 : (subst > term > term)] : ((((X403 @ id @ (lam @ (X405 @ sh @ one)) @ id)) != $true) & ! [X408 : term,X406 : subst,X407 : subst] : (((sub @ (X405 @ X407 @ X408) @ X406)) = ((X405 @ (comp @ X407 @ X406) @ (sub @ X408 @ X406)))) & ! [X409 : term] : (($true != ((X403 @ id @ X409 @ id))) | (((X403 @ id @ (X405 @ id @ X409) @ id)) = $true))))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (hoasapinj2_gthm = $true) & (one = ((d2term @ d_one))) & (pushprop_gthm = $true) & (hoasinduction_lem1_lthm = $true) & ((hoasinduction_lem3v2a_lthm = $true) <=> (? [X314 : term] : ! [X315 : (subst > term > term)] : ? [X316 : subst,X317 : term] : (((sub @ X314 @ (push @ X317 @ X316))) != ((X315 @ X316 @ X317))) | ? [X318 : term] : (((sub @ X318 @ id)) != X318) | ! [X319 : (subst > term > subst > $o),X321 : term,X320 : (term > $o)] : (? [X328 : term] : (($true = ((X320 @ X328))) & ($true != ((X320 @ (sub @ X321 @ (push @ X328 @ id)))))) | ? [X327 : term] : (((X319 @ id @ X327 @ id)) != ((X320 @ X327))) | ($true = ((X320 @ (lam @ X321)))) | ? [X322 : (subst > term > term)] : (! [X323 : subst,X325 : subst,X324 : term] : (((sub @ (X322 @ X323 @ X324) @ X325)) = ((X322 @ (comp @ X323 @ X325) @ (sub @ X324 @ X325)))) & ! [X326 : term] : (($true = ((X319 @ id @ (X322 @ id @ X326) @ id))) | ($true != ((X319 @ id @ X326 @ id)))) & (((X319 @ id @ (lam @ (X322 @ sh @ one)) @ id)) != $true))))) & ((substmonoid = $true) <=> (! [X502 : subst] : (((comp @ X502 @ id)) = X502) & ! [X503 : subst] : (((comp @ id @ X503)) = X503))) & (hoasapnotvar_lthm = $true) & (pushprop_lem1_gthm = $true) & (hoaslaminj_gthm = $true) & (pushprop_lem2v2_gthm = $true) & (hoasinduction_lem2v2_gthm = $true) & (! [X457 : term,X456 : (term > $o)] : (? [X459 : term,X458 : term] : ((((X456 @ X459)) = $true) & (((X456 @ X458)) = $true) & ($true != ((X456 @ (ap @ X459 @ X458))))) | ? [X460 : term] : (! [X461 : term] : (($true != ((X456 @ X461))) | (((X456 @ (sub @ X460 @ (push @ X461 @ id)))) = $true)) & (((X456 @ (lam @ X460))) != $true)) | ? [X462 : term] : (($true != ((X456 @ X462))) & ($true = ((var @ X462)))) | ($true = ((X456 @ X457)))) <=> (induction2 = $true)) & ((hoasinduction_lem0_lthm = $true) <=> ! [X141 : (subst > term > subst > $o)] : ? [X142 : (term > $o)] : ! [X143 : term] : (((X141 @ id @ X143 @ id)) = ((X142 @ X143)))) & (hoasinduction_lem2_gthm = $true) & ((hoasinduction_lem2v2 = $true) <=> ! [X109 : (term > $o),X110 : (subst > term > subst > $o),X107 : term,X108 : term] : (? [X112 : subst,X113 : subst,X114 : subst,X111 : term] : (($true = ((X110 @ (comp @ X112 @ X114) @ (sub @ X111 @ X114) @ X113))) & ($true != ((X110 @ X112 @ X111 @ (comp @ X114 @ X113))))) | ? [X115 : subst,X117 : term,X116 : subst,X118 : subst] : (($true = ((X110 @ X115 @ X117 @ (comp @ X116 @ X118)))) & (((X110 @ (comp @ X115 @ X116) @ (sub @ X117 @ X116) @ X118)) != $true)) | ? [X119 : term] : (((X109 @ X119)) != ((X110 @ id @ X119 @ id))) | ($true = ((X109 @ (ap @ X107 @ X108)))) | (((X109 @ X108)) != $true) | (((X109 @ X107)) != $true) | ? [X120 : term,X121 : term] : (($true = ((X110 @ id @ X120 @ id))) & ($true != ((X110 @ id @ (ap @ (sub @ X120 @ id) @ X121) @ id))) & ($true = ((X110 @ id @ X121 @ id)))))) & ((hoasinduction_lthm_3 = $true) <=> (? [X199 : term,X197 : (term > $o),X198 : (subst > term > subst > $o)] : (! [X200 : (subst > term > term)] : (? [X204 : term] : (($true != ((X198 @ id @ (X200 @ id @ X204) @ id))) & ($true = ((X198 @ id @ X204 @ id)))) | (((X198 @ id @ (lam @ (X200 @ sh @ one)) @ id)) = $true) | ? [X202 : subst,X203 : subst,X201 : term] : (((X200 @ (comp @ X203 @ X202) @ (sub @ X201 @ X202))) != ((sub @ (X200 @ X203 @ X201) @ X202)))) & ! [X205 : term] : ((((X197 @ (sub @ X199 @ (push @ X205 @ id)))) = $true) | (((X197 @ X205)) != $true)) & ! [X206 : term] : (((X198 @ id @ X206 @ id)) = ((X197 @ X206))) & ($true != ((X197 @ (lam @ X199))))) | ! [X208 : (subst > term > subst > $o),X207 : term] : (($true = ((X208 @ id @ X207 @ id))) | ? [X216 : (subst > term > term)] : (! [X220 : term] : ((((X208 @ id @ X220 @ id)) != $true) | ($true = ((X208 @ id @ (X216 @ id @ X220) @ id)))) & ! [X217 : subst,X219 : subst,X218 : term] : (((X216 @ (comp @ X217 @ X219) @ (sub @ X218 @ X219))) = ((sub @ (X216 @ X217 @ X218) @ X219))) & ($true != ((X208 @ id @ (lam @ (X216 @ sh @ one)) @ id)))) | ? [X214 : term,X215 : subst,X212 : subst,X213 : subst] : (($true != ((X208 @ (comp @ X215 @ X212) @ (sub @ X214 @ X212) @ X213))) & (((X208 @ X215 @ X214 @ (comp @ X212 @ X213))) = $true)) | ? [X210 : term,X211 : term] : (($true = ((X208 @ id @ X210 @ id))) & ($true != ((X208 @ id @ (ap @ (sub @ X211 @ id) @ X210) @ id))) & ($true = ((X208 @ id @ X211 @ id)))) | ? [X209 : term] : (($true != ((X208 @ id @ X209 @ id))) & ($true = ((var @ (sub @ X209 @ id))))) | ? [X222 : term,X221 : subst,X223 : subst,X224 : subst] : (($true = ((X208 @ (comp @ X221 @ X224) @ (sub @ X222 @ X224) @ X223))) & ($true != ((X208 @ X221 @ X222 @ (comp @ X224 @ X223)))))) | ? [X186 : (subst > term > subst > $o)] : ! [X187 : (term > $o)] : ? [X188 : term] : (((X186 @ id @ X188 @ id)) != ((X187 @ X188))) | ? [X196 : term] : (((sub @ X196 @ id)) != X196) | ? [X190 : term,X189 : (term > $o)] : (! [X195 : term] : (($true != ((var @ X195))) | ($true = ((X189 @ X195)))) & ! [X192 : term,X191 : term] : (($true != ((X189 @ X192))) | (((X189 @ X191)) != $true) | (((X189 @ (ap @ X191 @ X192))) = $true)) & ($true != ((X189 @ X190))) & ! [X193 : term] : (? [X194 : term] : ((((X189 @ X194)) = $true) & ($true != ((X189 @ (sub @ X193 @ (push @ X194 @ id)))))) | ($true = ((X189 @ (lam @ X193)))))))) & (axvarshift = $true) & (! [X23 : (subst > term > subst > $o),X24 : term,X22 : (term > $o)] : (? [X39 : term] : (($true = ((X22 @ X39))) & (((X22 @ (sub @ X24 @ (push @ X39 @ id)))) != $true)) | ? [X29 : (subst > term > term)] : (! [X31 : subst,X33 : term,X32 : subst] : (((X29 @ (comp @ X31 @ X32) @ (sub @ X33 @ X32))) = ((sub @ (X29 @ X31 @ X33) @ X32))) & ! [X30 : term] : (($true = ((X23 @ id @ (X29 @ id @ X30) @ id))) | ($true != ((X23 @ id @ X30 @ id)))) & ($true != ((X23 @ id @ (lam @ (X29 @ sh @ one)) @ id)))) | ? [X38 : term] : (((X23 @ id @ X38 @ id)) != ((X22 @ X38))) | ? [X37 : subst,X35 : subst,X34 : subst,X36 : term] : (($true = ((X23 @ (comp @ X37 @ X35) @ (sub @ X36 @ X35) @ X34))) & ($true != ((X23 @ X37 @ X36 @ (comp @ X35 @ X34))))) | ($true = ((X22 @ (lam @ X24)))) | ? [X27 : subst,X26 : subst,X25 : term,X28 : subst] : ((((X23 @ X28 @ X25 @ (comp @ X27 @ X26))) = $true) & (((X23 @ (comp @ X28 @ X27) @ (sub @ X25 @ X27) @ X26)) != $true))) <=> (hoasinduction_lem3v2 = $true)) & ((axscons = $true) <=> ! [X87 : subst] : (((push @ (sub @ one @ X87) @ (comp @ sh @ X87))) = X87)) & ((? [X268 : (subst > term > subst > $o),X269 : term,X267 : term] : (($true != ((X268 @ id @ (ap @ X269 @ X267) @ id))) & ($true = ((X268 @ id @ X269 @ id))) & ! [X270 : subst,X273 : subst,X272 : subst,X271 : term] : (($true != ((X268 @ (comp @ X270 @ X273) @ (sub @ X271 @ X273) @ X272))) | (((X268 @ X270 @ X271 @ (comp @ X273 @ X272))) = $true)) & ($true = ((X268 @ id @ X267 @ id))) & ! [X279 : term,X278 : term] : (($true != ((X268 @ id @ X279 @ id))) | (((X268 @ id @ (ap @ (sub @ X278 @ id) @ X279) @ id)) = $true) | (((X268 @ id @ X278 @ id)) != $true)) & ! [X276 : subst,X277 : term,X275 : subst,X274 : subst] : (($true != ((X268 @ X275 @ X277 @ (comp @ X276 @ X274)))) | (((X268 @ (comp @ X275 @ X276) @ (sub @ X277 @ X276) @ X274)) = $true))) | ? [X261 : term,X260 : (term > $o)] : (! [X266 : term,X265 : term] : (($true = ((X260 @ (ap @ X265 @ X266)))) | (((X260 @ X266)) != $true) | ($true != ((X260 @ X265)))) & ! [X262 : term] : (? [X263 : term] : (($true != ((X260 @ (sub @ X262 @ (push @ X263 @ id))))) & ($true = ((X260 @ X263)))) | (((X260 @ (lam @ X262))) = $true)) & ! [X264 : term] : ((((var @ X264)) != $true) | ($true = ((X260 @ X264)))) & (((X260 @ X261)) != $true)) | ! [X297 : (subst > term > subst > $o),X296 : term] : (? [X302 : (subst > term > term)] : (($true != ((X297 @ id @ (lam @ (X302 @ sh @ one)) @ id))) & ! [X303 : term] : ((((X297 @ id @ (X302 @ id @ X303) @ id)) = $true) | ($true != ((X297 @ id @ X303 @ id)))) & ! [X306 : subst,X304 : term,X305 : subst] : (((sub @ (X302 @ X306 @ X304) @ X305)) = ((X302 @ (comp @ X306 @ X305) @ (sub @ X304 @ X305))))) | ? [X307 : subst,X309 : subst,X310 : term,X308 : subst] : (($true != ((X297 @ (comp @ X307 @ X308) @ (sub @ X310 @ X308) @ X309))) & ($true = ((X297 @ X307 @ X310 @ (comp @ X308 @ X309))))) | ? [X311 : term] : (($true = ((var @ (sub @ X311 @ id)))) & ($true != ((X297 @ id @ X311 @ id)))) | ? [X313 : term,X312 : term] : ((((X297 @ id @ X313 @ id)) = $true) & ($true = ((X297 @ id @ X312 @ id))) & ($true != ((X297 @ id @ (ap @ (sub @ X312 @ id) @ X313) @ id)))) | ? [X298 : subst,X301 : term,X300 : subst,X299 : subst] : (($true != ((X297 @ X298 @ X301 @ (comp @ X299 @ X300)))) & (((X297 @ (comp @ X298 @ X299) @ (sub @ X301 @ X299) @ X300)) = $true)) | ($true = ((X297 @ id @ X296 @ id)))) | ? [X281 : (subst > term > subst > $o),X280 : term] : (! [X286 : term] : ((((X281 @ id @ (sub @ X280 @ (push @ X286 @ id)) @ id)) = $true) | (((X281 @ id @ X286 @ id)) != $true)) & ($true != ((X281 @ id @ (lam @ X280) @ id))) & ! [X287 : subst,X290 : term,X288 : subst,X289 : subst] : (($true = ((X281 @ X288 @ X290 @ (comp @ X289 @ X287)))) | (((X281 @ (comp @ X288 @ X289) @ (sub @ X290 @ X289) @ X287)) != $true)) & ! [X283 : subst,X284 : subst,X285 : term,X282 : subst] : (($true = ((X281 @ (comp @ X282 @ X283) @ (sub @ X285 @ X283) @ X284))) | (((X281 @ X282 @ X285 @ (comp @ X283 @ X284))) != $true)) & ! [X291 : (subst > term > term)] : (? [X295 : term] : (($true != ((X281 @ id @ (X291 @ id @ X295) @ id))) & ($true = ((X281 @ id @ X295 @ id)))) | ($true = ((X281 @ id @ (lam @ (X291 @ sh @ one)) @ id))) | ? [X292 : subst,X294 : subst,X293 : term] : (((X291 @ (comp @ X294 @ X292) @ (sub @ X293 @ X292))) != ((sub @ (X291 @ X294 @ X293) @ X292)))))) <=> (hoasinduction_lthm = $true)) & (hoasinduction_gthm = $true) & ((termmset = $true) <=> ! [X360 : term] : (((sub @ X360 @ id)) = X360)) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (! [X485 : (term > $o),X486 : term,X484 : (subst > term > subst > $o)] : (? [X488 : term] : (($true = ((X485 @ X488))) & ($true != ((X485 @ (sub @ X486 @ (push @ X488 @ id)))))) | ? [X487 : term] : (((X484 @ id @ X487 @ id)) != ((X485 @ X487))) | ? [X489 : (subst > term > term)] : (($true != ((X484 @ id @ (lam @ (X489 @ sh @ one)) @ id))) & ! [X490 : subst,X492 : subst,X491 : term] : (((X489 @ (comp @ X492 @ X490) @ (sub @ X491 @ X490))) = ((sub @ (X489 @ X492 @ X491) @ X490))) & ! [X493 : term] : (($true != ((X484 @ id @ X493 @ id))) | ($true = ((X484 @ id @ (X489 @ id @ X493) @ id))))) | ($true = ((X485 @ (lam @ X486))))) <=> (hoasinduction_lem3v2a = $true)) & ((? [X511 : term,X512 : term,X510 : term,X513 : term] : ((((ap @ X512 @ X513)) = ((ap @ X510 @ X511))) & (X510 != X512)) | ? [X509 : term] : (((sub @ X509 @ id)) != X509) | ! [X517 : term,X515 : term,X514 : term,X516 : term] : ((X514 = X516) | (((ap @ (sub @ X514 @ id) @ X517)) != ((ap @ (sub @ X516 @ id) @ X515))))) <=> (hoasapinj1_lthm = $true)) & (ulamvar1 != $true) & ((hoasinduction_lem3a_lthm = $true) <=> (! [X245 : (subst > term > subst > $o),X246 : term] : (? [X248 : (subst > term > term)] : ((((X245 @ id @ (lam @ (X248 @ sh @ one)) @ id)) != $true) & ! [X252 : term] : ((((X245 @ id @ X252 @ id)) != $true) | (((X245 @ id @ (X248 @ id @ X252) @ id)) = $true)) & ! [X251 : term,X249 : subst,X250 : subst] : (((X248 @ (comp @ X250 @ X249) @ (sub @ X251 @ X249))) = ((sub @ (X248 @ X250 @ X251) @ X249)))) | ? [X247 : term] : (($true != ((X245 @ id @ (sub @ X246 @ (push @ X247 @ id)) @ id))) & ($true = ((X245 @ id @ X247 @ id)))) | ($true = ((X245 @ id @ (lam @ X246) @ id)))) | ? [X236 : term] : (((sub @ X236 @ id)) != X236) | ? [X237 : (subst > term > subst > $o),X238 : term] : (! [X239 : (subst > term > term)] : (? [X243 : term] : (($true = ((X237 @ id @ X243 @ id))) & ($true != ((X237 @ id @ (X239 @ id @ X243) @ id)))) | ? [X240 : subst,X242 : term,X241 : subst] : (((sub @ (X239 @ X241 @ X242) @ X240)) != ((X239 @ (comp @ X241 @ X240) @ (sub @ X242 @ X240)))) | (((X237 @ id @ (lam @ (X239 @ sh @ one)) @ id)) = $true)) & ! [X244 : term] : (($true = ((X237 @ id @ (sub @ X238 @ (push @ X244 @ id)) @ id))) | (((X237 @ id @ X244 @ id)) != $true)) & (((X237 @ id @ (lam @ (sub @ X238 @ (push @ one @ sh))) @ id)) != $true)))) & (axassoc = $true) & (! [X483 : term,X482 : term] : ((X482 = X483) | (((sub @ X483 @ sh)) != ((sub @ X482 @ sh)))) <=> (shinj = $true)) & (hoaslamnotap_lthm = $true) & (pushprop_lem2v2_lthm = $true) & (! [X425 : term,X426 : (subst > term > subst > $o)] : (? [X429 : (subst > term > term)] : (! [X433 : term] : (($true = ((X426 @ id @ (X429 @ id @ X433) @ id))) | ($true != ((X426 @ id @ X433 @ id)))) & ! [X431 : subst,X430 : term,X432 : subst] : (((X429 @ (comp @ X432 @ X431) @ (sub @ X430 @ X431))) = ((sub @ (X429 @ X432 @ X430) @ X431))) & ($true != ((X426 @ id @ (lam @ (X429 @ sh @ one)) @ id)))) | ? [X439 : subst,X441 : subst,X438 : term,X440 : subst] : (($true = ((X426 @ X440 @ X438 @ (comp @ X439 @ X441)))) & (((X426 @ (comp @ X440 @ X439) @ (sub @ X438 @ X439) @ X441)) != $true)) | ? [X434 : term,X435 : subst,X437 : subst,X436 : subst] : (($true != ((X426 @ X437 @ X434 @ (comp @ X436 @ X435)))) & (((X426 @ (comp @ X437 @ X436) @ (sub @ X434 @ X436) @ X435)) = $true)) | ? [X442 : term] : (($true != ((X426 @ id @ X442 @ id))) & ($true = ((var @ (sub @ X442 @ id))))) | (((X426 @ id @ X425 @ id)) = $true) | ? [X428 : term,X427 : term] : (($true = ((X426 @ id @ X428 @ id))) & ($true != ((X426 @ id @ (ap @ (sub @ X428 @ id) @ X427) @ id))) & ($true = ((X426 @ id @ X427 @ id))))) <=> (hoasinduction = $true)) & ! [X358 : d_term,X359 : d_term] : ((((d2term @ X359)) != ((d2term @ X358))) | (X358 = X359)) & (hoasinduction_lem1_gthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (((! [X332 : subst] : (((comp @ id @ X332)) = X332) & ! [X331 : subst] : (((comp @ X331 @ id)) = X331)) | ? [X330 : subst] : (((comp @ X330 @ id)) != X330) | ? [X329 : subst] : (((comp @ id @ X329)) != X329)) <=> (substmonoid_lthm = $true)) & ! [X531 : d_subst] : (d_id = X531) & (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_lem3aa = $true) <=> ! [X60 : (subst > term > subst > $o),X59 : term] : (? [X62 : (subst > term > term)] : (! [X66 : term] : (($true != ((X60 @ id @ X66 @ id))) | ($true = ((X60 @ id @ (X62 @ id @ X66) @ id)))) & ! [X65 : subst,X64 : term,X63 : subst] : (((sub @ (X62 @ X65 @ X64) @ X63)) = ((X62 @ (comp @ X65 @ X63) @ (sub @ X64 @ X63)))) & (((X60 @ id @ (lam @ (X62 @ sh @ one)) @ id)) != $true)) | ? [X61 : term] : (($true = ((X60 @ id @ X61 @ id))) & ($true != ((X60 @ id @ (sub @ X59 @ (push @ X61 @ id)) @ id)))) | (((X60 @ id @ (lam @ (sub @ X59 @ (push @ one @ sh))) @ id)) = $true))) & (((var @ (d2term @ d_one))) != $true)),
  inference(flattening,[],[f11])).
thf(f13,definition,(
  ($true = sP0) <=> ! [X60 : (subst > term > subst > $o),X59 : term] : (? [X62 : (subst > term > term)] : (! [X66 : term] : (($true != ((X60 @ id @ X66 @ id))) | ($true = ((X60 @ id @ (X62 @ id @ X66) @ id)))) & ! [X65 : subst,X64 : term,X63 : subst] : (((sub @ (X62 @ X65 @ X64) @ X63)) = ((X62 @ (comp @ X65 @ X63) @ (sub @ X64 @ X63)))) & (((X60 @ id @ (lam @ (X62 @ sh @ one)) @ id)) != $true)) | ? [X61 : term] : (($true = ((X60 @ id @ X61 @ id))) & ($true != ((X60 @ id @ (sub @ X59 @ (push @ X61 @ id)) @ id)))) | (((X60 @ id @ (lam @ (sub @ X59 @ (push @ one @ sh))) @ id)) = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f14,definition,(
  ! [X426 : (subst > term > subst > $o)] : (($true = ((sP1 @ X426))) <=> ? [X428 : term,X427 : term] : (($true = ((X426 @ id @ X428 @ id))) & ($true != ((X426 @ id @ (ap @ (sub @ X428 @ id) @ X427) @ id))) & ($true = ((X426 @ id @ X427 @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f15,definition,(
  ! [X426 : (subst > term > subst > $o)] : (($true = ((sP2 @ X426))) <=> ? [X429 : (subst > term > term)] : (! [X433 : term] : (($true = ((X426 @ id @ (X429 @ id @ X433) @ id))) | ($true != ((X426 @ id @ X433 @ id)))) & ! [X431 : subst,X430 : term,X432 : subst] : (((X429 @ (comp @ X432 @ X431) @ (sub @ X430 @ X431))) = ((sub @ (X429 @ X432 @ X430) @ X431))) & ($true != ((X426 @ id @ (lam @ (X429 @ sh @ one)) @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f16,definition,(
  (sP3 = $true) <=> ! [X425 : term,X426 : (subst > term > subst > $o)] : (($true = ((sP2 @ X426))) | ? [X439 : subst,X441 : subst,X438 : term,X440 : subst] : (($true = ((X426 @ X440 @ X438 @ (comp @ X439 @ X441)))) & (((X426 @ (comp @ X440 @ X439) @ (sub @ X438 @ X439) @ X441)) != $true)) | ? [X434 : term,X435 : subst,X437 : subst,X436 : subst] : (($true != ((X426 @ X437 @ X434 @ (comp @ X436 @ X435)))) & (((X426 @ (comp @ X437 @ X436) @ (sub @ X434 @ X436) @ X435)) = $true)) | ? [X442 : term] : (($true != ((X426 @ id @ X442 @ id))) & ($true = ((var @ (sub @ X442 @ id))))) | (((X426 @ id @ X425 @ id)) = $true) | ($true = ((sP1 @ X426))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f17,definition,(
  ($true = sP4) <=> ? [X237 : (subst > term > subst > $o),X238 : term] : (! [X239 : (subst > term > term)] : (? [X243 : term] : (($true = ((X237 @ id @ X243 @ id))) & ($true != ((X237 @ id @ (X239 @ id @ X243) @ id)))) | ? [X240 : subst,X242 : term,X241 : subst] : (((sub @ (X239 @ X241 @ X242) @ X240)) != ((X239 @ (comp @ X241 @ X240) @ (sub @ X242 @ X240)))) | (((X237 @ id @ (lam @ (X239 @ sh @ one)) @ id)) = $true)) & ! [X244 : term] : (($true = ((X237 @ id @ (sub @ X238 @ (push @ X244 @ id)) @ id))) | (((X237 @ id @ X244 @ id)) != $true)) & (((X237 @ id @ (lam @ (sub @ X238 @ (push @ one @ sh))) @ id)) != $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f18,definition,(
  (sP5 = $true) <=> (! [X245 : (subst > term > subst > $o),X246 : term] : (? [X248 : (subst > term > term)] : ((((X245 @ id @ (lam @ (X248 @ sh @ one)) @ id)) != $true) & ! [X252 : term] : ((((X245 @ id @ X252 @ id)) != $true) | (((X245 @ id @ (X248 @ id @ X252) @ id)) = $true)) & ! [X251 : term,X249 : subst,X250 : subst] : (((X248 @ (comp @ X250 @ X249) @ (sub @ X251 @ X249))) = ((sub @ (X248 @ X250 @ X251) @ X249)))) | ? [X247 : term] : (($true != ((X245 @ id @ (sub @ X246 @ (push @ X247 @ id)) @ id))) & ($true = ((X245 @ id @ X247 @ id)))) | ($true = ((X245 @ id @ (lam @ X246) @ id)))) | ? [X236 : term] : (((sub @ X236 @ id)) != X236) | ($true = sP4))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f19,definition,(
  ($true = sP6) <=> ! [X485 : (term > $o),X486 : term,X484 : (subst > term > subst > $o)] : (? [X488 : term] : (($true = ((X485 @ X488))) & ($true != ((X485 @ (sub @ X486 @ (push @ X488 @ id)))))) | ? [X487 : term] : (((X484 @ id @ X487 @ id)) != ((X485 @ X487))) | ? [X489 : (subst > term > term)] : (($true != ((X484 @ id @ (lam @ (X489 @ sh @ one)) @ id))) & ! [X490 : subst,X492 : subst,X491 : term] : (((X489 @ (comp @ X492 @ X490) @ (sub @ X491 @ X490))) = ((sub @ (X489 @ X492 @ X491) @ X490))) & ! [X493 : term] : (($true != ((X484 @ id @ X493 @ id))) | ($true = ((X484 @ id @ (X489 @ id @ X493) @ id))))) | ($true = ((X485 @ (lam @ X486)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f20,definition,(
  ! [X281 : (subst > term > subst > $o)] : (($true = ((sP7 @ X281))) <=> ! [X291 : (subst > term > term)] : (? [X295 : term] : (($true != ((X281 @ id @ (X291 @ id @ X295) @ id))) & ($true = ((X281 @ id @ X295 @ id)))) | ($true = ((X281 @ id @ (lam @ (X291 @ sh @ one)) @ id))) | ? [X292 : subst,X294 : subst,X293 : term] : (((X291 @ (comp @ X294 @ X292) @ (sub @ X293 @ X292))) != ((sub @ (X291 @ X294 @ X293) @ X292)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f21,definition,(
  ! [X297 : (subst > term > subst > $o)] : (($true = ((sP8 @ X297))) <=> ? [X313 : term,X312 : term] : ((((X297 @ id @ X313 @ id)) = $true) & ($true = ((X297 @ id @ X312 @ id))) & ($true != ((X297 @ id @ (ap @ (sub @ X312 @ id) @ X313) @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f22,definition,(
  ! [X297 : (subst > term > subst > $o)] : (($true = ((sP9 @ X297))) <=> ? [X302 : (subst > term > term)] : (($true != ((X297 @ id @ (lam @ (X302 @ sh @ one)) @ id))) & ! [X303 : term] : ((((X297 @ id @ (X302 @ id @ X303) @ id)) = $true) | ($true != ((X297 @ id @ X303 @ id)))) & ! [X306 : subst,X304 : term,X305 : subst] : (((sub @ (X302 @ X306 @ X304) @ X305)) = ((X302 @ (comp @ X306 @ X305) @ (sub @ X304 @ X305))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f23,definition,(
  ! [X260 : (term > $o)] : (($true = ((sP10 @ X260))) <=> ! [X266 : term,X265 : term] : (($true = ((X260 @ (ap @ X265 @ X266)))) | (((X260 @ X266)) != $true) | ($true != ((X260 @ X265)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f24,definition,(
  ! [X268 : (subst > term > subst > $o)] : (($true = ((sP11 @ X268))) <=> ! [X279 : term,X278 : term] : (($true != ((X268 @ id @ X279 @ id))) | (((X268 @ id @ (ap @ (sub @ X278 @ id) @ X279) @ id)) = $true) | (((X268 @ id @ X278 @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f25,definition,(
  ($true = sP12) <=> ? [X281 : (subst > term > subst > $o),X280 : term] : (! [X286 : term] : ((((X281 @ id @ (sub @ X280 @ (push @ X286 @ id)) @ id)) = $true) | (((X281 @ id @ X286 @ id)) != $true)) & ($true != ((X281 @ id @ (lam @ X280) @ id))) & ! [X287 : subst,X290 : term,X288 : subst,X289 : subst] : (($true = ((X281 @ X288 @ X290 @ (comp @ X289 @ X287)))) | (((X281 @ (comp @ X288 @ X289) @ (sub @ X290 @ X289) @ X287)) != $true)) & ! [X283 : subst,X284 : subst,X285 : term,X282 : subst] : (($true = ((X281 @ (comp @ X282 @ X283) @ (sub @ X285 @ X283) @ X284))) | (((X281 @ X282 @ X285 @ (comp @ X283 @ X284))) != $true)) & ($true = ((sP7 @ X281))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f26,definition,(
  (sP13 = $true) <=> ! [X297 : (subst > term > subst > $o),X296 : term] : (($true = ((sP9 @ X297))) | ? [X307 : subst,X309 : subst,X310 : term,X308 : subst] : (($true != ((X297 @ (comp @ X307 @ X308) @ (sub @ X310 @ X308) @ X309))) & ($true = ((X297 @ X307 @ X310 @ (comp @ X308 @ X309))))) | ? [X311 : term] : (($true = ((var @ (sub @ X311 @ id)))) & ($true != ((X297 @ id @ X311 @ id)))) | ($true = ((sP8 @ X297))) | ? [X298 : subst,X301 : term,X300 : subst,X299 : subst] : (($true != ((X297 @ X298 @ X301 @ (comp @ X299 @ X300)))) & (((X297 @ (comp @ X298 @ X299) @ (sub @ X301 @ X299) @ X300)) = $true)) | ($true = ((X297 @ id @ X296 @ id))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f27,definition,(
  (sP14 = $true) <=> ? [X261 : term,X260 : (term > $o)] : (($true = ((sP10 @ X260))) & ! [X262 : term] : (? [X263 : term] : (($true != ((X260 @ (sub @ X262 @ (push @ X263 @ id))))) & ($true = ((X260 @ X263)))) | (((X260 @ (lam @ X262))) = $true)) & ! [X264 : term] : ((((var @ X264)) != $true) | ($true = ((X260 @ X264)))) & (((X260 @ X261)) != $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f28,definition,(
  (sP15 = $true) <=> (? [X268 : (subst > term > subst > $o),X269 : term,X267 : term] : (($true != ((X268 @ id @ (ap @ X269 @ X267) @ id))) & ($true = ((X268 @ id @ X269 @ id))) & ! [X270 : subst,X273 : subst,X272 : subst,X271 : term] : (($true != ((X268 @ (comp @ X270 @ X273) @ (sub @ X271 @ X273) @ X272))) | (((X268 @ X270 @ X271 @ (comp @ X273 @ X272))) = $true)) & ($true = ((X268 @ id @ X267 @ id))) & ($true = ((sP11 @ X268))) & ! [X276 : subst,X277 : term,X275 : subst,X274 : subst] : (($true != ((X268 @ X275 @ X277 @ (comp @ X276 @ X274)))) | (((X268 @ (comp @ X275 @ X276) @ (sub @ X277 @ X276) @ X274)) = $true))) | (sP14 = $true) | (sP13 = $true) | ($true = sP12))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f29,definition,(
  ! [X23 : (subst > term > subst > $o)] : (($true = ((sP16 @ X23))) <=> ? [X29 : (subst > term > term)] : (! [X31 : subst,X33 : term,X32 : subst] : (((X29 @ (comp @ X31 @ X32) @ (sub @ X33 @ X32))) = ((sub @ (X29 @ X31 @ X33) @ X32))) & ! [X30 : term] : (($true = ((X23 @ id @ (X29 @ id @ X30) @ id))) | ($true != ((X23 @ id @ X30 @ id)))) & ($true != ((X23 @ id @ (lam @ (X29 @ sh @ one)) @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f30,definition,(
  (sP17 = $true) <=> ! [X23 : (subst > term > subst > $o),X24 : term,X22 : (term > $o)] : (? [X39 : term] : (($true = ((X22 @ X39))) & (((X22 @ (sub @ X24 @ (push @ X39 @ id)))) != $true)) | ($true = ((sP16 @ X23))) | ? [X38 : term] : (((X23 @ id @ X38 @ id)) != ((X22 @ X38))) | ? [X37 : subst,X35 : subst,X34 : subst,X36 : term] : (($true = ((X23 @ (comp @ X37 @ X35) @ (sub @ X36 @ X35) @ X34))) & ($true != ((X23 @ X37 @ X36 @ (comp @ X35 @ X34))))) | ($true = ((X22 @ (lam @ X24)))) | ? [X27 : subst,X26 : subst,X25 : term,X28 : subst] : ((((X23 @ X28 @ X25 @ (comp @ X27 @ X26))) = $true) & (((X23 @ (comp @ X28 @ X27) @ (sub @ X25 @ X27) @ X26)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f31,definition,(
  ! [X189 : (term > $o)] : (($true = ((sP18 @ X189))) <=> ! [X192 : term,X191 : term] : (($true != ((X189 @ X192))) | (((X189 @ X191)) != $true) | (((X189 @ (ap @ X191 @ X192))) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f32,definition,(
  ! [X208 : (subst > term > subst > $o)] : (($true = ((sP19 @ X208))) <=> ? [X210 : term,X211 : term] : (($true = ((X208 @ id @ X210 @ id))) & ($true != ((X208 @ id @ (ap @ (sub @ X211 @ id) @ X210) @ id))) & ($true = ((X208 @ id @ X211 @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f33,definition,(
  ! [X208 : (subst > term > subst > $o)] : ((((sP20 @ X208)) = $true) <=> ? [X216 : (subst > term > term)] : (! [X220 : term] : ((((X208 @ id @ X220 @ id)) != $true) | ($true = ((X208 @ id @ (X216 @ id @ X220) @ id)))) & ! [X217 : subst,X219 : subst,X218 : term] : (((X216 @ (comp @ X217 @ X219) @ (sub @ X218 @ X219))) = ((sub @ (X216 @ X217 @ X218) @ X219))) & ($true != ((X208 @ id @ (lam @ (X216 @ sh @ one)) @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f34,definition,(
  ($true = sP21) <=> ! [X208 : (subst > term > subst > $o),X207 : term] : (($true = ((X208 @ id @ X207 @ id))) | (((sP20 @ X208)) = $true) | ? [X214 : term,X215 : subst,X212 : subst,X213 : subst] : (($true != ((X208 @ (comp @ X215 @ X212) @ (sub @ X214 @ X212) @ X213))) & (((X208 @ X215 @ X214 @ (comp @ X212 @ X213))) = $true)) | ($true = ((sP19 @ X208))) | ? [X209 : term] : (($true != ((X208 @ id @ X209 @ id))) & ($true = ((var @ (sub @ X209 @ id))))) | ? [X222 : term,X221 : subst,X223 : subst,X224 : subst] : (($true = ((X208 @ (comp @ X221 @ X224) @ (sub @ X222 @ X224) @ X223))) & ($true != ((X208 @ X221 @ X222 @ (comp @ X224 @ X223))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f35,definition,(
  ($true = sP22) <=> ? [X199 : term,X197 : (term > $o),X198 : (subst > term > subst > $o)] : (! [X200 : (subst > term > term)] : (? [X204 : term] : (($true != ((X198 @ id @ (X200 @ id @ X204) @ id))) & ($true = ((X198 @ id @ X204 @ id)))) | (((X198 @ id @ (lam @ (X200 @ sh @ one)) @ id)) = $true) | ? [X202 : subst,X203 : subst,X201 : term] : (((X200 @ (comp @ X203 @ X202) @ (sub @ X201 @ X202))) != ((sub @ (X200 @ X203 @ X201) @ X202)))) & ! [X205 : term] : ((((X197 @ (sub @ X199 @ (push @ X205 @ id)))) = $true) | (((X197 @ X205)) != $true)) & ! [X206 : term] : (((X198 @ id @ X206 @ id)) = ((X197 @ X206))) & ($true != ((X197 @ (lam @ X199)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f36,definition,(
  (sP23 = $true) <=> (($true = sP22) | ($true = sP21) | ? [X186 : (subst > term > subst > $o)] : ! [X187 : (term > $o)] : ? [X188 : term] : (((X186 @ id @ X188 @ id)) != ((X187 @ X188))) | ? [X196 : term] : (((sub @ X196 @ id)) != X196) | ? [X190 : term,X189 : (term > $o)] : (! [X195 : term] : (($true != ((var @ X195))) | ($true = ((X189 @ X195)))) & ($true = ((sP18 @ X189))) & ($true != ((X189 @ X190))) & ! [X193 : term] : (? [X194 : term] : ((((X189 @ X194)) = $true) & ($true != ((X189 @ (sub @ X193 @ (push @ X194 @ id)))))) | ($true = ((X189 @ (lam @ X193)))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f37,definition,(
  ! [X110 : (subst > term > subst > $o)] : (($true = ((sP24 @ X110))) <=> ? [X120 : term,X121 : term] : (($true = ((X110 @ id @ X120 @ id))) & ($true != ((X110 @ id @ (ap @ (sub @ X120 @ id) @ X121) @ id))) & ($true = ((X110 @ id @ X121 @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f38,definition,(
  ($true = sP25) <=> ! [X109 : (term > $o),X110 : (subst > term > subst > $o),X107 : term,X108 : term] : (? [X112 : subst,X113 : subst,X114 : subst,X111 : term] : (($true = ((X110 @ (comp @ X112 @ X114) @ (sub @ X111 @ X114) @ X113))) & ($true != ((X110 @ X112 @ X111 @ (comp @ X114 @ X113))))) | ? [X115 : subst,X117 : term,X116 : subst,X118 : subst] : (($true = ((X110 @ X115 @ X117 @ (comp @ X116 @ X118)))) & (((X110 @ (comp @ X115 @ X116) @ (sub @ X117 @ X116) @ X118)) != $true)) | ? [X119 : term] : (((X109 @ X119)) != ((X110 @ id @ X119 @ id))) | ($true = ((X109 @ (ap @ X107 @ X108)))) | (((X109 @ X108)) != $true) | (((X109 @ X107)) != $true) | ($true = ((sP24 @ X110))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f39,definition,(
  ! [X456 : (term > $o)] : (($true = ((sP26 @ X456))) <=> ? [X459 : term,X458 : term] : ((((X456 @ X459)) = $true) & (((X456 @ X458)) = $true) & ($true != ((X456 @ (ap @ X459 @ X458))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f40,definition,(
  ($true = sP27) <=> ! [X457 : term,X456 : (term > $o)] : (($true = ((sP26 @ X456))) | ? [X460 : term] : (! [X461 : term] : (($true != ((X456 @ X461))) | (((X456 @ (sub @ X460 @ (push @ X461 @ id)))) = $true)) & (((X456 @ (lam @ X460))) != $true)) | ? [X462 : term] : (($true != ((X456 @ X462))) & ($true = ((var @ X462)))) | ($true = ((X456 @ X457))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f41,definition,(
  (sP28 = $true) <=> (? [X314 : term] : ! [X315 : (subst > term > term)] : ? [X316 : subst,X317 : term] : (((sub @ X314 @ (push @ X317 @ X316))) != ((X315 @ X316 @ X317))) | ? [X318 : term] : (((sub @ X318 @ id)) != X318) | ! [X319 : (subst > term > subst > $o),X321 : term,X320 : (term > $o)] : (? [X328 : term] : (($true = ((X320 @ X328))) & ($true != ((X320 @ (sub @ X321 @ (push @ X328 @ id)))))) | ? [X327 : term] : (((X319 @ id @ X327 @ id)) != ((X320 @ X327))) | ($true = ((X320 @ (lam @ X321)))) | ? [X322 : (subst > term > term)] : (! [X323 : subst,X325 : subst,X324 : term] : (((sub @ (X322 @ X323 @ X324) @ X325)) = ((X322 @ (comp @ X323 @ X325) @ (sub @ X324 @ X325)))) & ! [X326 : term] : (($true = ((X319 @ id @ (X322 @ id @ X326) @ id))) | ($true != ((X319 @ id @ X326 @ id)))) & (((X319 @ id @ (lam @ (X322 @ sh @ one)) @ id)) != $true))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f42,definition,(
  ! [X403 : (subst > term > subst > $o)] : (($true = ((sP29 @ X403))) <=> ? [X405 : (subst > term > term)] : ((((X403 @ id @ (lam @ (X405 @ sh @ one)) @ id)) != $true) & ! [X408 : term,X406 : subst,X407 : subst] : (((sub @ (X405 @ X407 @ X408) @ X406)) = ((X405 @ (comp @ X407 @ X406) @ (sub @ X408 @ X406)))) & ! [X409 : term] : (($true != ((X403 @ id @ X409 @ id))) | (((X403 @ id @ (X405 @ id @ X409) @ id)) = $true))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f43,definition,(
  ! [X385 : (term > $o)] : (($true = ((sP30 @ X385))) <=> ! [X387 : term,X388 : term] : ((((X385 @ (ap @ X388 @ X387))) = $true) | ($true != ((X385 @ X387))) | ($true != ((X385 @ X388)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f44,definition,(
  (sP31 = $true) <=> ? [X394 : term,X395 : (term > $o),X393 : (subst > term > subst > $o)] : (! [X401 : term] : (($true = ((X395 @ (sub @ X394 @ (push @ X401 @ id))))) | ($true != ((X395 @ X401)))) & ! [X402 : term] : (((X393 @ id @ X402 @ id)) = ((X395 @ X402))) & ($true != ((X395 @ (lam @ X394)))) & ! [X396 : (subst > term > term)] : (? [X397 : term] : (($true = ((X393 @ id @ X397 @ id))) & ($true != ((X393 @ id @ (X396 @ id @ X397) @ id)))) | ($true = ((X393 @ id @ (lam @ (X396 @ sh @ one)) @ id))) | ? [X399 : term,X400 : subst,X398 : subst] : (((sub @ (X396 @ X398 @ X399) @ X400)) != ((X396 @ (comp @ X398 @ X400) @ (sub @ X399 @ X400))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f45,definition,(
  ($true = sP32) <=> ? [X386 : term,X385 : (term > $o)] : (! [X391 : term] : (($true = ((X385 @ X391))) | (((var @ X391)) != $true)) & ($true != ((X385 @ X386))) & ! [X389 : term] : (? [X390 : term] : (($true = ((X385 @ X390))) & ($true != ((X385 @ (sub @ X389 @ (push @ X390 @ id)))))) | ($true = ((X385 @ (lam @ X389))))) & ($true = ((sP30 @ X385))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f46,definition,(
  ($true = sP33) <=> ((sP31 = $true) | ? [X382 : (subst > term > subst > $o)] : ! [X383 : (term > $o)] : ? [X384 : term] : (((X382 @ id @ X384 @ id)) != ((X383 @ X384))) | ? [X392 : term] : (((sub @ X392 @ id)) != X392) | ($true = sP32) | ! [X403 : (subst > term > subst > $o),X404 : term] : (? [X410 : term,X411 : term] : (($true = ((X403 @ id @ X410 @ id))) & ($true != ((X403 @ id @ (ap @ (sub @ X411 @ id) @ X410) @ id))) & (((X403 @ id @ X411 @ id)) = $true)) | ($true = ((X403 @ id @ X404 @ id))) | ($true = ((sP29 @ X403)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f47,definition,(
  ! [X342 : (subst > term > subst > $o)] : (($true = ((sP34 @ X342))) <=> ? [X349 : (subst > term > term)] : (! [X350 : term] : ((((X342 @ id @ X350 @ id)) != $true) | ($true = ((X342 @ id @ (X349 @ id @ X350) @ id)))) & ($true != ((X342 @ id @ (lam @ (X349 @ sh @ one)) @ id))) & ! [X353 : subst,X352 : subst,X351 : term] : (((X349 @ (comp @ X353 @ X352) @ (sub @ X351 @ X352))) = ((sub @ (X349 @ X353 @ X351) @ X352)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f48,definition,(
  ($true = sP35) <=> ? [X335 : (subst > term > subst > $o),X334 : term] : (! [X337 : (subst > term > term)] : (? [X339 : term,X340 : subst,X341 : subst] : (((sub @ (X337 @ X341 @ X339) @ X340)) != ((X337 @ (comp @ X341 @ X340) @ (sub @ X339 @ X340)))) | ($true = ((X335 @ id @ (lam @ (X337 @ sh @ one)) @ id))) | ? [X338 : term] : (($true != ((X335 @ id @ (X337 @ id @ X338) @ id))) & ($true = ((X335 @ id @ X338 @ id))))) & ($true != ((X335 @ id @ (lam @ (sub @ X334 @ (push @ one @ sh))) @ id))) & ! [X336 : term] : ((((X335 @ id @ (sub @ X334 @ (push @ X336 @ id)) @ id)) = $true) | (((X335 @ id @ X336 @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f49,definition,(
  ($true = sP36) <=> (? [X333 : term] : (((sub @ X333 @ id)) != X333) | ($true = sP35) | ! [X343 : term,X342 : (subst > term > subst > $o)] : (($true = ((X342 @ id @ (lam @ X343) @ id))) | ? [X354 : subst,X355 : subst,X356 : subst,X357 : term] : (($true != ((X342 @ (comp @ X356 @ X355) @ (sub @ X357 @ X355) @ X354))) & ($true = ((X342 @ X356 @ X357 @ (comp @ X355 @ X354))))) | ($true = ((sP34 @ X342))) | ? [X344 : term] : (($true = ((X342 @ id @ X344 @ id))) & ($true != ((X342 @ id @ (sub @ X343 @ (push @ X344 @ id)) @ id)))) | ? [X346 : term,X345 : subst,X347 : subst,X348 : subst] : ((((X342 @ (comp @ X345 @ X347) @ (sub @ X346 @ X347) @ X348)) = $true) & ($true != ((X342 @ X345 @ X346 @ (comp @ X347 @ X348)))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f50,definition,(
  ! [X6 : (subst > term > subst > $o)] : (($true = ((sP37 @ X6))) <=> ? [X17 : (subst > term > term)] : (! [X18 : term] : (($true != ((X6 @ id @ X18 @ id))) | ($true = ((X6 @ id @ (X17 @ id @ X18) @ id)))) & ! [X20 : subst,X21 : subst,X19 : term] : (((X17 @ (comp @ X21 @ X20) @ (sub @ X19 @ X20))) = ((sub @ (X17 @ X21 @ X19) @ X20))) & (((X6 @ id @ (lam @ (X17 @ sh @ one)) @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f51,definition,(
  ($true = sP38) <=> ! [X6 : (subst > term > subst > $o),X7 : term] : ((((X6 @ id @ (lam @ X7) @ id)) = $true) | ? [X9 : subst,X10 : term,X8 : subst,X11 : subst] : (($true != ((X6 @ (comp @ X8 @ X9) @ (sub @ X10 @ X9) @ X11))) & (((X6 @ X8 @ X10 @ (comp @ X9 @ X11))) = $true)) | ? [X12 : term] : (($true != ((X6 @ id @ (sub @ X7 @ (push @ X12 @ id)) @ id))) & (((X6 @ id @ X12 @ id)) = $true)) | ? [X16 : subst,X13 : subst,X14 : term,X15 : subst] : (($true != ((X6 @ X15 @ X14 @ (comp @ X13 @ X16)))) & ($true = ((X6 @ (comp @ X15 @ X13) @ (sub @ X14 @ X13) @ X16)))) | ($true = ((sP37 @ X6))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f52,definition,(
  ! [X225 : (subst > term > subst > $o)] : ((((sP39 @ X225)) = $true) <=> ? [X228 : term,X227 : term] : (($true = ((X225 @ id @ X227 @ id))) & ($true = ((X225 @ id @ X228 @ id))) & ($true != ((X225 @ id @ (ap @ (sub @ X227 @ id) @ X228) @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f53,definition,(
  ! [X76 : (term > $o)] : ((((sP40 @ X76)) = $true) <=> ? [X79 : term,X80 : term] : ((((X76 @ X80)) = $true) & (((X76 @ X79)) = $true) & (((X76 @ (ap @ X80 @ X79))) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f54,definition,(
  ($true = sP41) <=> ! [X76 : (term > $o),X77 : term,X75 : subst] : (? [X81 : term] : ((((X76 @ (lam @ X81))) != $true) & ! [X82 : term] : (($true = ((X76 @ (sub @ X81 @ (push @ X82 @ id))))) | ($true != ((X76 @ X82))))) | (((sP40 @ X76)) = $true) | ? [X78 : term] : ((((X76 @ (sub @ X78 @ X75))) != $true) & (((var @ X78)) = $true)) | ($true = ((X76 @ (sub @ X77 @ X75)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f55,definition,(
  ! [X48 : (subst > term > subst > $o)] : (($true = ((sP42 @ X48))) <=> ? [X53 : term,X54 : term] : (($true = ((X48 @ id @ X53 @ id))) & ($true != ((X48 @ id @ (ap @ (sub @ X53 @ id) @ X54) @ id))) & ($true = ((X48 @ id @ X54 @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f56,definition,(
  ($true = sP43) <=> ! [X48 : (subst > term > subst > $o),X47 : term,X46 : term] : (($true = ((sP42 @ X48))) | (((X48 @ id @ X47 @ id)) != $true) | ? [X56 : term,X58 : subst,X55 : subst,X57 : subst] : (($true = ((X48 @ X55 @ X56 @ (comp @ X57 @ X58)))) & ($true != ((X48 @ (comp @ X55 @ X57) @ (sub @ X56 @ X57) @ X58)))) | ($true != ((X48 @ id @ X46 @ id))) | ($true = ((X48 @ id @ (ap @ X46 @ X47) @ id))) | ? [X50 : term,X52 : subst,X51 : subst,X49 : subst] : (($true != ((X48 @ X49 @ X50 @ (comp @ X52 @ X51)))) & (((X48 @ (comp @ X49 @ X52) @ (sub @ X50 @ X52) @ X51)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f57,definition,(
  ($true = sP44) <=> (? [X144 : term,X145 : subst] : (((sub @ one @ (push @ X144 @ X145))) != X144) | ? [X148 : term,X149 : term] : ((X148 != X149) & (((lam @ X149)) = ((lam @ X148)))) | ? [X147 : subst,X146 : term] : (((comp @ sh @ (push @ X146 @ X147))) != X147) | ! [X152 : term,X151 : (subst > term > term),X150 : subst,X153 : (subst > term > term)] : (? [X155 : subst,X154 : term,X156 : subst] : (((sub @ (X153 @ X155 @ X154) @ X156)) != ((X153 @ (comp @ X155 @ X156) @ (sub @ X154 @ X156)))) | ? [X158 : subst,X157 : subst,X159 : term] : (((X151 @ (comp @ X157 @ X158) @ (sub @ X159 @ X158))) != ((sub @ (X151 @ X157 @ X159) @ X158))) | (((lam @ (X151 @ sh @ one))) != ((lam @ (X153 @ sh @ one)))) | (((X151 @ X150 @ X152)) = ((X153 @ X150 @ X152)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f58,definition,(
  ($true = sP45) <=> ! [X415 : (subst > term > subst > $o),X414 : term] : (($true = ((X415 @ id @ (lam @ X414) @ id))) | ? [X416 : term] : ((((X415 @ id @ (sub @ X414 @ (push @ X416 @ id)) @ id)) != $true) & ($true = ((X415 @ id @ X416 @ id)))) | ? [X417 : (subst > term > term)] : (($true != ((X415 @ id @ (lam @ (X417 @ sh @ one)) @ id))) & ! [X418 : term,X420 : subst,X419 : subst] : (((sub @ (X417 @ X420 @ X418) @ X419)) = ((X417 @ (comp @ X420 @ X419) @ (sub @ X418 @ X419)))) & ! [X421 : term] : (($true != ((X415 @ id @ X421 @ id))) | (((X415 @ id @ (X417 @ id @ X421) @ id)) = $true))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f59,definition,(
  ! [X534 : (term > $o)] : ((((sP46 @ X534)) = $true) <=> ? [X537 : term,X538 : term] : (($true = ((X534 @ X537))) & ($true != ((X534 @ (ap @ X538 @ X537)))) & (((X534 @ X538)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f60,definition,(
  (sP47 = $true) <=> ! [X89 : (subst > term > subst > $o),X88 : term] : (? [X95 : term] : (($true != ((X89 @ id @ (sub @ X88 @ (push @ X95 @ id)) @ id))) & (((X89 @ id @ X95 @ id)) = $true)) | (((X89 @ id @ (lam @ (sub @ X88 @ (push @ one @ sh))) @ id)) = $true) | ? [X90 : (subst > term > term)] : (($true != ((X89 @ id @ (lam @ (X90 @ sh @ one)) @ id))) & ! [X93 : subst,X94 : subst,X92 : term] : (((sub @ (X90 @ X93 @ X92) @ X94)) = ((X90 @ (comp @ X93 @ X94) @ (sub @ X92 @ X94)))) & ! [X91 : term] : (($true = ((X89 @ id @ (X90 @ id @ X91) @ id))) | (((X89 @ id @ X91 @ id)) != $true))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f61,definition,(
  ! [X125 : (subst > term > subst > $o)] : (($true = ((sP48 @ X125))) <=> ? [X131 : (subst > term > term)] : (($true != ((X125 @ id @ (lam @ (X131 @ sh @ one)) @ id))) & ! [X134 : subst,X135 : subst,X133 : term] : (((X131 @ (comp @ X134 @ X135) @ (sub @ X133 @ X135))) = ((sub @ (X131 @ X134 @ X133) @ X135))) & ! [X132 : term] : (($true = ((X125 @ id @ (X131 @ id @ X132) @ id))) | (((X125 @ id @ X132 @ id)) != $true))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f62,definition,(
  ($true = sP49) <=> (? [X122 : term] : (((sub @ X122 @ id)) != X122) | ! [X125 : (subst > term > subst > $o),X123 : term,X124 : (term > $o)] : (? [X140 : term] : (($true != ((X124 @ (sub @ X123 @ (push @ X140 @ id))))) & (((X124 @ X140)) = $true)) | ? [X139 : subst,X137 : term,X136 : subst,X138 : subst] : ((((X125 @ X139 @ X137 @ (comp @ X138 @ X136))) = $true) & ($true != ((X125 @ (comp @ X139 @ X138) @ (sub @ X137 @ X138) @ X136)))) | ? [X126 : term] : (((X124 @ X126)) != ((X125 @ id @ X126 @ id))) | (((X124 @ (lam @ X123))) = $true) | ($true = ((sP48 @ X125))) | ? [X130 : subst,X129 : subst,X127 : term,X128 : subst] : ((((X125 @ X129 @ X127 @ (comp @ X130 @ X128))) != $true) & ($true = ((X125 @ (comp @ X129 @ X130) @ (sub @ X127 @ X130) @ X128))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f63,definition,(
  ! [X468 : (term > $o)] : ((((sP50 @ X468)) = $true) <=> ! [X470 : term,X471 : term] : (($true = ((X468 @ (ap @ X471 @ X470)))) | ($true != ((X468 @ X471))) | ($true != ((X468 @ X470)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f64,definition,(
  ! [X476 : (term > $o)] : (($true = ((sP51 @ X476))) <=> ? [X481 : term,X480 : term] : ((((X476 @ (ap @ X481 @ X480))) != $true) & ($true = ((X476 @ X480))) & ($true = ((X476 @ X481)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f65,definition,(
  ($true = sP52) <=> ! [X474 : subst,X475 : term,X476 : (term > $o)] : (($true = ((X476 @ (sub @ X475 @ X474)))) | ? [X477 : term] : (! [X478 : term] : (($true != ((X476 @ X478))) | ($true = ((X476 @ (sub @ X477 @ (push @ X478 @ id)))))) & ($true != ((X476 @ (lam @ X477))))) | ($true = ((sP51 @ X476))) | ? [X479 : term] : ((((var @ X479)) = $true) & ($true != ((X476 @ (sub @ X479 @ X474))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f66,definition,(
  ($true = sP53) <=> (($true = sP52) | ? [X464 : term,X463 : subst] : (((sub @ one @ (push @ X464 @ X463))) != X464) | ? [X467 : subst] : (((comp @ X467 @ id)) != X467) | ? [X468 : (term > $o),X469 : term] : (! [X472 : term] : (($true != ((X468 @ X472))) | ($true = ((X468 @ (lam @ X472))))) & ! [X473 : term] : ((((X468 @ X473)) = $true) | (((var @ X473)) != $true)) & (((sP50 @ X468)) = $true) & ($true != ((X468 @ X469)))) | ? [X465 : term,X466 : subst] : (((comp @ sh @ (push @ X465 @ X466))) != X466))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f67,definition,(
  (sP54 = $true) <=> ! [X371 : (subst > term > subst > $o),X370 : term] : (? [X373 : term,X372 : (subst > term > term)] : (! [X374 : term] : (($true = ((X371 @ id @ (X372 @ id @ X374) @ id))) | ($true != ((X371 @ id @ X374 @ id)))) & (((X371 @ id @ (lam @ (X372 @ sh @ one)) @ id)) != $true) & ! [X376 : subst,X375 : term,X377 : subst] : (((sub @ (sub @ X373 @ (push @ X375 @ X377)) @ X376)) = ((sub @ (X372 @ X377 @ X375) @ X376))) & ! [X378 : subst,X379 : subst,X380 : term] : (((sub @ X373 @ (push @ (sub @ X380 @ X379) @ (comp @ X378 @ X379)))) = ((X372 @ (comp @ X378 @ X379) @ (sub @ X380 @ X379))))) | ($true = ((X371 @ id @ (lam @ (sub @ X370 @ (push @ one @ sh))) @ id))) | ? [X381 : term] : ((((X371 @ id @ (sub @ X370 @ (push @ X381 @ id)) @ id)) != $true) & ($true = ((X371 @ id @ X381 @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f68,definition,(
  ! [X162 : (term > $o)] : (($true = ((sP55 @ X162))) <=> ! [X164 : term,X165 : term] : (($true = ((X162 @ (ap @ X164 @ X165)))) | (((X162 @ X164)) != $true) | ($true != ((X162 @ X165)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f69,definition,(
  ! [X170 : (term > $o)] : ((((sP56 @ X170)) = $true) <=> ? [X174 : term,X173 : term] : ((((X170 @ X173)) = $true) & ($true = ((X170 @ X174))) & (((X170 @ (ap @ X174 @ X173))) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f70,definition,(
  ($true = sP57) <=> ! [X169 : term,X170 : (term > $o)] : ((((sP56 @ X170)) = $true) | ? [X175 : term] : (($true != ((X170 @ X175))) & ($true = ((var @ X175)))) | (((X170 @ X169)) = $true) | ? [X171 : term] : (! [X172 : term] : (($true != ((X170 @ X172))) | ($true = ((X170 @ (sub @ X171 @ (push @ X172 @ id)))))) & ($true != ((X170 @ (lam @ X171))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f71,definition,(
  ($true = sP58) <=> (($true = sP57) | ? [X162 : (term > $o),X163 : subst,X161 : term] : (($true = ((sP55 @ X162))) & ! [X168 : term] : (($true = ((X162 @ (sub @ X168 @ X163)))) | (((var @ X168)) != $true)) & ! [X166 : term] : (($true = ((X162 @ (lam @ X166)))) | ? [X167 : term] : (($true = ((X162 @ X167))) & ($true != ((X162 @ (sub @ X166 @ (push @ X167 @ id))))))) & (((X162 @ (sub @ X161 @ X163))) != $true)) | ? [X160 : term] : (((sub @ X160 @ id)) != X160))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f72,plain,(
  (($true = sP58) <=> (induction2_lthm = $true)) & ((hoasinduction_lem3aaa = $true) <=> (sP54 = $true)) & (ulamvarind = $true) & (pushprop_lem3v2_lthm = $true) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X498 : term] : ? [X499 : (subst > term > term)] : (((X499 @ sh @ one)) = ((sub @ X498 @ (push @ one @ sh))))) & (substmonoid_gthm = $true) & (($true = sP53) <=> (induction2lem_lthm = $true)) & (! [X506 : term,X508 : (term > $o),X504 : subst,X505 : term,X507 : (term > $o)] : (($true != ((X507 @ X505))) | ($true = ((X508 @ (sub @ X505 @ (push @ X506 @ X504)))))) <=> (pushprop_lem1 = $true)) & ((hoasinduction_lem3v2_lthm = $true) <=> ($true = sP49)) & (pushprop_lem3v2 = $true) & (hoaslamnotvar_gthm = $true) & (! [X235 : subst,X234 : term] : (((sub @ one @ (push @ X234 @ X235))) = X234) <=> (axvarcons = $true)) & (pushprop_lem2v2 = $true) & ((hoasinduction_lem3aa_lthm = $true) <=> (sP47 = $true)) & (hoasinduction_lem3_gthm = $true) & (! [X494 : term] : ? [X495 : (subst > term > term)] : ! [X496 : term,X497 : subst] : (((sub @ X494 @ (push @ X496 @ X497))) = ((X495 @ X497 @ X496))) <=> (hoasinduction_lem3v2_f = $true)) & (pushprop_lthm_orig = $true) & (lamnotvar = $true) & ((apinj1 = $true) <=> ! [X518 : term,X519 : term,X520 : term,X521 : term] : ((((ap @ X518 @ X520)) != ((ap @ X521 @ X519))) | (X518 = X521))) & (! [X535 : term,X534 : (term > $o)] : (? [X539 : term] : (($true != ((X534 @ (lam @ X539)))) & ($true = ((X534 @ X539)))) | (((sP46 @ X534)) = $true) | ($true = ((X534 @ X535))) | ? [X536 : term] : ((((var @ X536)) = $true) & ($true != ((X534 @ X536))))) <=> (induction = $true)) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem1v2 = $true) & ! [X413 : d_subst,X412 : d_subst] : ((((d2subst @ X413)) != ((d2subst @ X412))) | (X412 = X413)) & ((hoasinduction_lem0 = $true) <=> ! [X445 : (subst > term > subst > $o)] : ? [X446 : (term > $o)] : ! [X447 : term] : (((X445 @ id @ X447 @ id)) = ((X446 @ X447)))) & (induction2_gthm = $true) & ! [X532 : subst] : ? [X533 : d_subst] : (((d2subst @ X533)) = X532) & (hoaslamnotap_gthm = $true) & (induction2lem_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (($true = sP45) <=> (hoasinduction_lem3a = $true)) & (id = ((d2subst @ d_id))) & (hoasapinj1_gthm = $true) & ((apinj2 = $true) <=> ! [X71 : term,X73 : term,X74 : term,X72 : term] : ((X71 = X72) | (((ap @ X73 @ X72)) != ((ap @ X74 @ X71))))) & (((d2subst @ d_id)) = sh) & (hoasinduction_lem2_lthm = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoaslaminj = $true) <=> ! [X176 : subst,X178 : (subst > term > term),X177 : term,X179 : (subst > term > term)] : (? [X183 : subst,X184 : subst,X185 : term] : (((X179 @ (comp @ X184 @ X183) @ (sub @ X185 @ X183))) != ((sub @ (X179 @ X184 @ X185) @ X183))) | ? [X180 : term,X181 : subst,X182 : subst] : (((X178 @ (comp @ X182 @ X181) @ (sub @ X180 @ X181))) != ((sub @ (X178 @ X182 @ X180) @ X181))) | (((X179 @ X176 @ X177)) = ((X178 @ X176 @ X177))) | (((lam @ (X178 @ sh @ one))) != ((lam @ (X179 @ sh @ one)))))) & (hoasinduction_lem3b_gthm = $true) & ((axidr = $true) <=> ! [X424 : subst] : (((comp @ X424 @ id)) = X424)) & (termmset_gthm = $true) & (hoasinduction_lem1 = $true) & (pushprop = $true) & (apnotvar = $true) & ! [X96 : d_term] : (d_one = X96) & (hoaslamnotvar = $true) & (! [X530 : term,X529 : subst] : (((comp @ sh @ (push @ X530 @ X529))) = X529) <=> (axshiftcons = $true)) & (axapp = $true) & (hoasinduction_lem3a_gthm = $true) & (! [X106 : term] : (((sub @ X106 @ id)) = X106) <=> (axvarid = $true)) & (axmap = $true) & (($true = sP44) <=> (hoaslaminj_lthm = $true)) & (axabs = $true) & ((hoasapinj2 = $true) <=> ! [X105 : term,X102 : term,X103 : term,X104 : term] : ((X102 = X105) | (((ap @ (sub @ X104 @ id) @ X102)) != ((ap @ (sub @ X103 @ id) @ X105))))) & ((hoaslamnotap = $true) <=> ! [X1 : term,X2 : term,X0 : (subst > term > term)] : (? [X3 : subst,X4 : term,X5 : subst] : (((X0 @ (comp @ X3 @ X5) @ (sub @ X4 @ X5))) != ((sub @ (X0 @ X3 @ X4) @ X5))) | (((ap @ (sub @ X2 @ id) @ X1)) != ((lam @ (X0 @ sh @ one)))))) & (hoaslamnotvar_lthm = $true) & (ulamvarsh = $true) & ((? [X500 : term] : (((sub @ X500 @ id)) != X500) | ! [X501 : term] : (((sub @ X501 @ id)) = X501)) <=> (termmset_lthm = $true)) & (($true = sP43) <=> (hoasinduction_lem2 = $true)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoasapnotvar = $true) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_lem3v2_gthm = $true) & ((? [X364 : subst,X363 : term] : (((comp @ sh @ (push @ X363 @ X364))) != X364) | ! [X369 : term,X366 : subst,X368 : term,X367 : (term > $o),X365 : (term > $o)] : (($true != ((X365 @ X369))) | ($true = ((X367 @ (sub @ X369 @ (push @ X368 @ X366)))))) | ? [X361 : term,X362 : subst] : (((sub @ one @ (push @ X361 @ X362))) != X361)) <=> (pushprop_lem1_lthm = $true)) & ((hoasapinj1 = $true) <=> ! [X67 : term,X70 : term,X69 : term,X68 : term] : ((X67 = X68) | (((ap @ (sub @ X67 @ id) @ X70)) != ((ap @ (sub @ X68 @ id) @ X69))))) & (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)))))))))))))) & ((pushprop_lem1v2 = $true) <=> ! [X40 : subst,X43 : (term > $o),X42 : (term > $o),X41 : term] : (? [X44 : term] : (((X43 @ (sub @ X44 @ (push @ X41 @ X40)))) != ((X42 @ X44))) | ($true = ((X42 @ one))) | ($true != ((X43 @ X41))))) & (axclos = $true) & ((hoasinduction_lem3b = $true) <=> ! [X253 : term] : ? [X254 : (subst > term > term)] : (((X254 @ sh @ one)) = ((sub @ X253 @ (push @ one @ sh))))) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (lamnotap != $true) & (($true = sP41) <=> (induction2lem = $true)) & ((! [X526 : subst,X525 : term,X527 : (term > $o),X524 : (term > $o)] : ((((X524 @ one)) = $true) | ? [X528 : term] : (((X524 @ X528)) != ((X527 @ (sub @ X528 @ (push @ X525 @ X526))))) | ($true != ((X527 @ X525)))) | ? [X523 : subst,X522 : term] : (((sub @ one @ (push @ X522 @ X523))) != X522)) <=> (pushprop_lem1v2_lthm = $true)) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X83 : term] : ? [X84 : (subst > term > term)] : ! [X85 : term,X86 : subst] : (((X84 @ X86 @ X85)) = ((sub @ X83 @ (push @ X85 @ X86))))) & ! [X422 : term] : ? [X423 : d_term] : (((d2term @ X423)) = X422) & ((hoasinduction_no_psi_cond = $true) <=> ! [X225 : (subst > term > subst > $o),X226 : term] : ((((X225 @ id @ X226 @ id)) = $true) | ? [X229 : (subst > term > term)] : (($true != ((X225 @ id @ (lam @ (X229 @ sh @ one)) @ id))) & ! [X233 : subst,X231 : subst,X232 : term] : (((sub @ (X229 @ X231 @ X232) @ X233)) = ((X229 @ (comp @ X231 @ X233) @ (sub @ X232 @ X233)))) & ! [X230 : term] : (($true != ((X225 @ id @ X230 @ id))) | (((X225 @ id @ (X229 @ id @ X230) @ id)) = $true))) | (((sP39 @ X225)) = $true))) & (! [X443 : term,X444 : term] : ((((lam @ X443)) != ((lam @ X444))) | (X443 = X444)) <=> (laminj = $true)) & (pushprop_lthm = $true) & ((hoasapinj2_lthm = $true) <=> (! [X455 : term,X453 : term,X454 : term,X452 : term] : ((((ap @ (sub @ X454 @ id) @ X452)) != ((ap @ (sub @ X453 @ id) @ X455))) | (X452 = X455)) | ? [X449 : term,X450 : term,X451 : term,X448 : term] : ((X448 != X450) & (((ap @ X451 @ X448)) = ((ap @ X449 @ X450)))))) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((hoasinduction_lem3 = $true) <=> ($true = sP38)) & ((pushprop_lem0_lthm = $true) <=> ! [X256 : (term > $o),X255 : subst,X257 : term] : ? [X258 : (term > $o)] : ! [X259 : term] : (((X258 @ X259)) = ((X256 @ (sub @ X259 @ (push @ X257 @ X255)))))) & (! [X45 : subst] : (((comp @ id @ X45)) = X45) <=> (axidl = $true)) & (! [X98 : subst,X99 : (term > $o),X97 : term] : ? [X100 : (term > $o)] : ! [X101 : term] : (((X99 @ (sub @ X101 @ (push @ X97 @ X98)))) = ((X100 @ X101))) <=> (pushprop_lem0 = $true)) & (pushprop_lem0_gthm = $true) & ((hoasinduction_lem3_lthm = $true) <=> ($true = sP36)) & (($true = sP33) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (hoasapinj2_gthm = $true) & (one = ((d2term @ d_one))) & (pushprop_gthm = $true) & (hoasinduction_lem1_lthm = $true) & ((hoasinduction_lem3v2a_lthm = $true) <=> (sP28 = $true)) & ((substmonoid = $true) <=> (! [X502 : subst] : (((comp @ X502 @ id)) = X502) & ! [X503 : subst] : (((comp @ id @ X503)) = X503))) & (hoasapnotvar_lthm = $true) & (pushprop_lem1_gthm = $true) & (hoaslaminj_gthm = $true) & (pushprop_lem2v2_gthm = $true) & (hoasinduction_lem2v2_gthm = $true) & (($true = sP27) <=> (induction2 = $true)) & ((hoasinduction_lem0_lthm = $true) <=> ! [X141 : (subst > term > subst > $o)] : ? [X142 : (term > $o)] : ! [X143 : term] : (((X141 @ id @ X143 @ id)) = ((X142 @ X143)))) & (hoasinduction_lem2_gthm = $true) & ((hoasinduction_lem2v2 = $true) <=> ($true = sP25)) & ((hoasinduction_lthm_3 = $true) <=> (sP23 = $true)) & (axvarshift = $true) & ((sP17 = $true) <=> (hoasinduction_lem3v2 = $true)) & ((axscons = $true) <=> ! [X87 : subst] : (((push @ (sub @ one @ X87) @ (comp @ sh @ X87))) = X87)) & ((sP15 = $true) <=> (hoasinduction_lthm = $true)) & (hoasinduction_gthm = $true) & ((termmset = $true) <=> ! [X360 : term] : (((sub @ X360 @ id)) = X360)) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (($true = sP6) <=> (hoasinduction_lem3v2a = $true)) & ((? [X511 : term,X512 : term,X510 : term,X513 : term] : ((((ap @ X512 @ X513)) = ((ap @ X510 @ X511))) & (X510 != X512)) | ? [X509 : term] : (((sub @ X509 @ id)) != X509) | ! [X517 : term,X515 : term,X514 : term,X516 : term] : ((X514 = X516) | (((ap @ (sub @ X514 @ id) @ X517)) != ((ap @ (sub @ X516 @ id) @ X515))))) <=> (hoasapinj1_lthm = $true)) & (ulamvar1 != $true) & ((hoasinduction_lem3a_lthm = $true) <=> (sP5 = $true)) & (axassoc = $true) & (! [X483 : term,X482 : term] : ((X482 = X483) | (((sub @ X483 @ sh)) != ((sub @ X482 @ sh)))) <=> (shinj = $true)) & (hoaslamnotap_lthm = $true) & (pushprop_lem2v2_lthm = $true) & ((sP3 = $true) <=> (hoasinduction = $true)) & ! [X358 : d_term,X359 : d_term] : ((((d2term @ X359)) != ((d2term @ X358))) | (X358 = X359)) & (hoasinduction_lem1_gthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (((! [X332 : subst] : (((comp @ id @ X332)) = X332) & ! [X331 : subst] : (((comp @ X331 @ id)) = X331)) | ? [X330 : subst] : (((comp @ X330 @ id)) != X330) | ? [X329 : subst] : (((comp @ id @ X329)) != X329)) <=> (substmonoid_lthm = $true)) & ! [X531 : d_subst] : (d_id = X531) & (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_lem3aa = $true) <=> ($true = sP0)) & (((var @ (d2term @ d_one))) != $true)),
  inference(definition_folding,[],[f12,f71,f70,f69,f68,f67,f66,f65,f64,f63,f62,f61,f60,f59,f58,f57,f56,f55,f54,f53,f52,f51,f50,f49,f48,f47,f46,f45,f44,f43,f42,f41,f40,f39,f38,f37,f36,f35,f34,f33,f32,f31,f30,f29,f28,f27,f26,f25,f24,f23,f22,f21,f20,f19,f18,f17,f16,f15,f14,f13])).
thf(f260,plain,(
  ((($true = sP58) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | ($true != sP58))) & (((hoasinduction_lem3aaa = $true) | (sP54 != $true)) & ((sP54 = $true) | (hoasinduction_lem3aaa != $true))) & (ulamvarind = $true) & (pushprop_lem3v2_lthm = $true) & (((hoasinduction_lem3b_lthm = $true) | ? [X498 : term] : ! [X499 : (subst > term > term)] : (((X499 @ sh @ one)) != ((sub @ X498 @ (push @ one @ sh))))) & (! [X498 : term] : ? [X499 : (subst > term > term)] : (((X499 @ sh @ one)) = ((sub @ X498 @ (push @ one @ sh)))) | (hoasinduction_lem3b_lthm != $true))) & (substmonoid_gthm = $true) & ((($true = sP53) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | ($true != sP53))) & ((! [X506 : term,X508 : (term > $o),X504 : subst,X505 : term,X507 : (term > $o)] : (($true != ((X507 @ X505))) | ($true = ((X508 @ (sub @ X505 @ (push @ X506 @ X504)))))) | (pushprop_lem1 != $true)) & ((pushprop_lem1 = $true) | ? [X506 : term,X508 : (term > $o),X504 : subst,X505 : term,X507 : (term > $o)] : (($true = ((X507 @ X505))) & ($true != ((X508 @ (sub @ X505 @ (push @ X506 @ X504)))))))) & (((hoasinduction_lem3v2_lthm = $true) | ($true != sP49)) & (($true = sP49) | (hoasinduction_lem3v2_lthm != $true))) & (pushprop_lem3v2 = $true) & (hoaslamnotvar_gthm = $true) & ((! [X235 : subst,X234 : term] : (((sub @ one @ (push @ X234 @ X235))) = X234) | (axvarcons != $true)) & ((axvarcons = $true) | ? [X235 : subst,X234 : term] : (((sub @ one @ (push @ X234 @ X235))) != X234))) & (pushprop_lem2v2 = $true) & (((hoasinduction_lem3aa_lthm = $true) | (sP47 != $true)) & ((sP47 = $true) | (hoasinduction_lem3aa_lthm != $true))) & (hoasinduction_lem3_gthm = $true) & ((! [X494 : term] : ? [X495 : (subst > term > term)] : ! [X496 : term,X497 : subst] : (((sub @ X494 @ (push @ X496 @ X497))) = ((X495 @ X497 @ X496))) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ? [X494 : term] : ! [X495 : (subst > term > term)] : ? [X496 : term,X497 : subst] : (((sub @ X494 @ (push @ X496 @ X497))) != ((X495 @ X497 @ X496))))) & (pushprop_lthm_orig = $true) & (lamnotvar = $true) & (((apinj1 = $true) | ? [X518 : term,X519 : term,X520 : term,X521 : term] : ((((ap @ X518 @ X520)) = ((ap @ X521 @ X519))) & (X518 != X521))) & (! [X518 : term,X519 : term,X520 : term,X521 : term] : ((((ap @ X518 @ X520)) != ((ap @ X521 @ X519))) | (X518 = X521)) | (apinj1 != $true))) & ((! [X535 : term,X534 : (term > $o)] : (? [X539 : term] : (($true != ((X534 @ (lam @ X539)))) & ($true = ((X534 @ X539)))) | (((sP46 @ X534)) = $true) | ($true = ((X534 @ X535))) | ? [X536 : term] : ((((var @ X536)) = $true) & ($true != ((X534 @ X536))))) | (induction != $true)) & ((induction = $true) | ? [X535 : term,X534 : (term > $o)] : (! [X539 : term] : (($true = ((X534 @ (lam @ X539)))) | ($true != ((X534 @ X539)))) & (((sP46 @ X534)) != $true) & ($true != ((X534 @ X535))) & ! [X536 : term] : ((((var @ X536)) != $true) | ($true = ((X534 @ X536))))))) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem1v2 = $true) & ! [X413 : d_subst,X412 : d_subst] : ((((d2subst @ X413)) != ((d2subst @ X412))) | (X412 = X413)) & (((hoasinduction_lem0 = $true) | ? [X445 : (subst > term > subst > $o)] : ! [X446 : (term > $o)] : ? [X447 : term] : (((X445 @ id @ X447 @ id)) != ((X446 @ X447)))) & (! [X445 : (subst > term > subst > $o)] : ? [X446 : (term > $o)] : ! [X447 : term] : (((X445 @ id @ X447 @ id)) = ((X446 @ X447))) | (hoasinduction_lem0 != $true))) & (induction2_gthm = $true) & ! [X532 : subst] : ? [X533 : d_subst] : (((d2subst @ X533)) = X532) & (hoaslamnotap_gthm = $true) & (induction2lem_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & ((($true = sP45) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | ($true != sP45))) & (id = ((d2subst @ d_id))) & (hoasapinj1_gthm = $true) & (((apinj2 = $true) | ? [X71 : term,X73 : term,X74 : term,X72 : term] : ((X71 != X72) & (((ap @ X73 @ X72)) = ((ap @ X74 @ X71))))) & (! [X71 : term,X73 : term,X74 : term,X72 : term] : ((X71 = X72) | (((ap @ X73 @ X72)) != ((ap @ X74 @ X71)))) | (apinj2 != $true))) & (((d2subst @ d_id)) = sh) & (hoasinduction_lem2_lthm = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (((hoaslaminj = $true) | ? [X176 : subst,X178 : (subst > term > term),X177 : term,X179 : (subst > term > term)] : (! [X183 : subst,X184 : subst,X185 : term] : (((X179 @ (comp @ X184 @ X183) @ (sub @ X185 @ X183))) = ((sub @ (X179 @ X184 @ X185) @ X183))) & ! [X180 : term,X181 : subst,X182 : subst] : (((X178 @ (comp @ X182 @ X181) @ (sub @ X180 @ X181))) = ((sub @ (X178 @ X182 @ X180) @ X181))) & (((X179 @ X176 @ X177)) != ((X178 @ X176 @ X177))) & (((lam @ (X178 @ sh @ one))) = ((lam @ (X179 @ sh @ one)))))) & (! [X176 : subst,X178 : (subst > term > term),X177 : term,X179 : (subst > term > term)] : (? [X183 : subst,X184 : subst,X185 : term] : (((X179 @ (comp @ X184 @ X183) @ (sub @ X185 @ X183))) != ((sub @ (X179 @ X184 @ X185) @ X183))) | ? [X180 : term,X181 : subst,X182 : subst] : (((X178 @ (comp @ X182 @ X181) @ (sub @ X180 @ X181))) != ((sub @ (X178 @ X182 @ X180) @ X181))) | (((X179 @ X176 @ X177)) = ((X178 @ X176 @ X177))) | (((lam @ (X178 @ sh @ one))) != ((lam @ (X179 @ sh @ one))))) | (hoaslaminj != $true))) & (hoasinduction_lem3b_gthm = $true) & (((axidr = $true) | ? [X424 : subst] : (((comp @ X424 @ id)) != X424)) & (! [X424 : subst] : (((comp @ X424 @ id)) = X424) | (axidr != $true))) & (termmset_gthm = $true) & (hoasinduction_lem1 = $true) & (pushprop = $true) & (apnotvar = $true) & ! [X96 : d_term] : (d_one = X96) & (hoaslamnotvar = $true) & ((! [X530 : term,X529 : subst] : (((comp @ sh @ (push @ X530 @ X529))) = X529) | (axshiftcons != $true)) & ((axshiftcons = $true) | ? [X530 : term,X529 : subst] : (((comp @ sh @ (push @ X530 @ X529))) != X529))) & (axapp = $true) & (hoasinduction_lem3a_gthm = $true) & ((! [X106 : term] : (((sub @ X106 @ id)) = X106) | (axvarid != $true)) & ((axvarid = $true) | ? [X106 : term] : (((sub @ X106 @ id)) != X106))) & (axmap = $true) & ((($true = sP44) | (hoaslaminj_lthm != $true)) & ((hoaslaminj_lthm = $true) | ($true != sP44))) & (axabs = $true) & (((hoasapinj2 = $true) | ? [X105 : term,X102 : term,X103 : term,X104 : term] : ((X102 != X105) & (((ap @ (sub @ X104 @ id) @ X102)) = ((ap @ (sub @ X103 @ id) @ X105))))) & (! [X105 : term,X102 : term,X103 : term,X104 : term] : ((X102 = X105) | (((ap @ (sub @ X104 @ id) @ X102)) != ((ap @ (sub @ X103 @ id) @ X105)))) | (hoasapinj2 != $true))) & (((hoaslamnotap = $true) | ? [X1 : term,X2 : term,X0 : (subst > term > term)] : (! [X3 : subst,X4 : term,X5 : subst] : (((X0 @ (comp @ X3 @ X5) @ (sub @ X4 @ X5))) = ((sub @ (X0 @ X3 @ X4) @ X5))) & (((ap @ (sub @ X2 @ id) @ X1)) = ((lam @ (X0 @ sh @ one)))))) & (! [X1 : term,X2 : term,X0 : (subst > term > term)] : (? [X3 : subst,X4 : term,X5 : subst] : (((X0 @ (comp @ X3 @ X5) @ (sub @ X4 @ X5))) != ((sub @ (X0 @ X3 @ X4) @ X5))) | (((ap @ (sub @ X2 @ id) @ X1)) != ((lam @ (X0 @ sh @ one))))) | (hoaslamnotap != $true))) & (hoaslamnotvar_lthm = $true) & (ulamvarsh = $true) & (((? [X500 : term] : (((sub @ X500 @ id)) != X500) | ! [X501 : term] : (((sub @ X501 @ id)) = X501)) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | (! [X500 : term] : (((sub @ X500 @ id)) = X500) & ? [X501 : term] : (((sub @ X501 @ id)) != X501)))) & ((($true = sP43) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | ($true != sP43))) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoasapnotvar = $true) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_lem3v2_gthm = $true) & (((? [X364 : subst,X363 : term] : (((comp @ sh @ (push @ X363 @ X364))) != X364) | ! [X369 : term,X366 : subst,X368 : term,X367 : (term > $o),X365 : (term > $o)] : (($true != ((X365 @ X369))) | ($true = ((X367 @ (sub @ X369 @ (push @ X368 @ X366)))))) | ? [X361 : term,X362 : subst] : (((sub @ one @ (push @ X361 @ X362))) != X361)) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (! [X364 : subst,X363 : term] : (((comp @ sh @ (push @ X363 @ X364))) = X364) & ? [X369 : term,X366 : subst,X368 : term,X367 : (term > $o),X365 : (term > $o)] : (($true = ((X365 @ X369))) & ($true != ((X367 @ (sub @ X369 @ (push @ X368 @ X366)))))) & ! [X361 : term,X362 : subst] : (((sub @ one @ (push @ X361 @ X362))) = X361)))) & (((hoasapinj1 = $true) | ? [X67 : term,X70 : term,X69 : term,X68 : term] : ((X67 != X68) & (((ap @ (sub @ X67 @ id) @ X70)) = ((ap @ (sub @ X68 @ id) @ X69))))) & (! [X67 : term,X70 : term,X69 : term,X68 : term] : ((X67 = X68) | (((ap @ (sub @ X67 @ id) @ X70)) != ((ap @ (sub @ X68 @ id) @ X69)))) | (hoasapinj1 != $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)))))))))))))) & (((pushprop_lem1v2 = $true) | ? [X40 : subst,X43 : (term > $o),X42 : (term > $o),X41 : term] : (! [X44 : term] : (((X43 @ (sub @ X44 @ (push @ X41 @ X40)))) = ((X42 @ X44))) & ($true != ((X42 @ one))) & ($true = ((X43 @ X41))))) & (! [X40 : subst,X43 : (term > $o),X42 : (term > $o),X41 : term] : (? [X44 : term] : (((X43 @ (sub @ X44 @ (push @ X41 @ X40)))) != ((X42 @ X44))) | ($true = ((X42 @ one))) | ($true != ((X43 @ X41)))) | (pushprop_lem1v2 != $true))) & (axclos = $true) & (((hoasinduction_lem3b = $true) | ? [X253 : term] : ! [X254 : (subst > term > term)] : (((X254 @ sh @ one)) != ((sub @ X253 @ (push @ one @ sh))))) & (! [X253 : term] : ? [X254 : (subst > term > term)] : (((X254 @ sh @ one)) = ((sub @ X253 @ (push @ one @ sh)))) | (hoasinduction_lem3b != $true))) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (lamnotap != $true) & ((($true = sP41) | (induction2lem != $true)) & ((induction2lem = $true) | ($true != sP41))) & (((! [X526 : subst,X525 : term,X527 : (term > $o),X524 : (term > $o)] : ((((X524 @ one)) = $true) | ? [X528 : term] : (((X524 @ X528)) != ((X527 @ (sub @ X528 @ (push @ X525 @ X526))))) | ($true != ((X527 @ X525)))) | ? [X523 : subst,X522 : term] : (((sub @ one @ (push @ X522 @ X523))) != X522)) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (? [X526 : subst,X525 : term,X527 : (term > $o),X524 : (term > $o)] : ((((X524 @ one)) != $true) & ! [X528 : term] : (((X524 @ X528)) = ((X527 @ (sub @ X528 @ (push @ X525 @ X526))))) & ($true = ((X527 @ X525)))) & ! [X523 : subst,X522 : term] : (((sub @ one @ (push @ X522 @ X523))) = X522)))) & (((hoasinduction_lem3v2_f_lthm = $true) | ? [X83 : term] : ! [X84 : (subst > term > term)] : ? [X85 : term,X86 : subst] : (((X84 @ X86 @ X85)) != ((sub @ X83 @ (push @ X85 @ X86))))) & (! [X83 : term] : ? [X84 : (subst > term > term)] : ! [X85 : term,X86 : subst] : (((X84 @ X86 @ X85)) = ((sub @ X83 @ (push @ X85 @ X86)))) | (hoasinduction_lem3v2_f_lthm != $true))) & ! [X422 : term] : ? [X423 : d_term] : (((d2term @ X423)) = X422) & (((hoasinduction_no_psi_cond = $true) | ? [X225 : (subst > term > subst > $o),X226 : term] : ((((X225 @ id @ X226 @ id)) != $true) & ! [X229 : (subst > term > term)] : (($true = ((X225 @ id @ (lam @ (X229 @ sh @ one)) @ id))) | ? [X233 : subst,X231 : subst,X232 : term] : (((sub @ (X229 @ X231 @ X232) @ X233)) != ((X229 @ (comp @ X231 @ X233) @ (sub @ X232 @ X233)))) | ? [X230 : term] : (($true = ((X225 @ id @ X230 @ id))) & (((X225 @ id @ (X229 @ id @ X230) @ id)) != $true))) & (((sP39 @ X225)) != $true))) & (! [X225 : (subst > term > subst > $o),X226 : term] : ((((X225 @ id @ X226 @ id)) = $true) | ? [X229 : (subst > term > term)] : (($true != ((X225 @ id @ (lam @ (X229 @ sh @ one)) @ id))) & ! [X233 : subst,X231 : subst,X232 : term] : (((sub @ (X229 @ X231 @ X232) @ X233)) = ((X229 @ (comp @ X231 @ X233) @ (sub @ X232 @ X233)))) & ! [X230 : term] : (($true != ((X225 @ id @ X230 @ id))) | (((X225 @ id @ (X229 @ id @ X230) @ id)) = $true))) | (((sP39 @ X225)) = $true)) | (hoasinduction_no_psi_cond != $true))) & ((! [X443 : term,X444 : term] : ((((lam @ X443)) != ((lam @ X444))) | (X443 = X444)) | (laminj != $true)) & ((laminj = $true) | ? [X443 : term,X444 : term] : ((((lam @ X443)) = ((lam @ X444))) & (X443 != X444)))) & (pushprop_lthm = $true) & (((hoasapinj2_lthm = $true) | (? [X455 : term,X453 : term,X454 : term,X452 : term] : ((((ap @ (sub @ X454 @ id) @ X452)) = ((ap @ (sub @ X453 @ id) @ X455))) & (X452 != X455)) & ! [X449 : term,X450 : term,X451 : term,X448 : term] : ((X448 = X450) | (((ap @ X451 @ X448)) != ((ap @ X449 @ X450)))))) & ((! [X455 : term,X453 : term,X454 : term,X452 : term] : ((((ap @ (sub @ X454 @ id) @ X452)) != ((ap @ (sub @ X453 @ id) @ X455))) | (X452 = X455)) | ? [X449 : term,X450 : term,X451 : term,X448 : term] : ((X448 != X450) & (((ap @ X451 @ X448)) = ((ap @ X449 @ X450))))) | (hoasapinj2_lthm != $true))) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (((hoasinduction_lem3 = $true) | ($true != sP38)) & (($true = sP38) | (hoasinduction_lem3 != $true))) & (((pushprop_lem0_lthm = $true) | ? [X256 : (term > $o),X255 : subst,X257 : term] : ! [X258 : (term > $o)] : ? [X259 : term] : (((X258 @ X259)) != ((X256 @ (sub @ X259 @ (push @ X257 @ X255)))))) & (! [X256 : (term > $o),X255 : subst,X257 : term] : ? [X258 : (term > $o)] : ! [X259 : term] : (((X258 @ X259)) = ((X256 @ (sub @ X259 @ (push @ X257 @ X255))))) | (pushprop_lem0_lthm != $true))) & ((! [X45 : subst] : (((comp @ id @ X45)) = X45) | (axidl != $true)) & ((axidl = $true) | ? [X45 : subst] : (((comp @ id @ X45)) != X45))) & ((! [X98 : subst,X99 : (term > $o),X97 : term] : ? [X100 : (term > $o)] : ! [X101 : term] : (((X99 @ (sub @ X101 @ (push @ X97 @ X98)))) = ((X100 @ X101))) | (pushprop_lem0 != $true)) & ((pushprop_lem0 = $true) | ? [X98 : subst,X99 : (term > $o),X97 : term] : ! [X100 : (term > $o)] : ? [X101 : term] : (((X99 @ (sub @ X101 @ (push @ X97 @ X98)))) != ((X100 @ X101))))) & (pushprop_lem0_gthm = $true) & (((hoasinduction_lem3_lthm = $true) | ($true != sP36)) & (($true = sP36) | (hoasinduction_lem3_lthm != $true))) & ((($true = sP33) | (hoasinduction_no_psi_cond_lthm != $true)) & ((hoasinduction_no_psi_cond_lthm = $true) | ($true != sP33))) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (hoasapinj2_gthm = $true) & (one = ((d2term @ d_one))) & (pushprop_gthm = $true) & (hoasinduction_lem1_lthm = $true) & (((hoasinduction_lem3v2a_lthm = $true) | (sP28 != $true)) & ((sP28 = $true) | (hoasinduction_lem3v2a_lthm != $true))) & (((substmonoid = $true) | (? [X502 : subst] : (((comp @ X502 @ id)) != X502) | ? [X503 : subst] : (((comp @ id @ X503)) != X503))) & ((! [X502 : subst] : (((comp @ X502 @ id)) = X502) & ! [X503 : subst] : (((comp @ id @ X503)) = X503)) | (substmonoid != $true))) & (hoasapnotvar_lthm = $true) & (pushprop_lem1_gthm = $true) & (hoaslaminj_gthm = $true) & (pushprop_lem2v2_gthm = $true) & (hoasinduction_lem2v2_gthm = $true) & ((($true = sP27) | (induction2 != $true)) & ((induction2 = $true) | ($true != sP27))) & (((hoasinduction_lem0_lthm = $true) | ? [X141 : (subst > term > subst > $o)] : ! [X142 : (term > $o)] : ? [X143 : term] : (((X141 @ id @ X143 @ id)) != ((X142 @ X143)))) & (! [X141 : (subst > term > subst > $o)] : ? [X142 : (term > $o)] : ! [X143 : term] : (((X141 @ id @ X143 @ id)) = ((X142 @ X143))) | (hoasinduction_lem0_lthm != $true))) & (hoasinduction_lem2_gthm = $true) & (((hoasinduction_lem2v2 = $true) | ($true != sP25)) & (($true = sP25) | (hoasinduction_lem2v2 != $true))) & (((hoasinduction_lthm_3 = $true) | (sP23 != $true)) & ((sP23 = $true) | (hoasinduction_lthm_3 != $true))) & (axvarshift = $true) & (((sP17 = $true) | (hoasinduction_lem3v2 != $true)) & ((hoasinduction_lem3v2 = $true) | (sP17 != $true))) & (((axscons = $true) | ? [X87 : subst] : (((push @ (sub @ one @ X87) @ (comp @ sh @ X87))) != X87)) & (! [X87 : subst] : (((push @ (sub @ one @ X87) @ (comp @ sh @ X87))) = X87) | (axscons != $true))) & (((sP15 = $true) | (hoasinduction_lthm != $true)) & ((hoasinduction_lthm = $true) | (sP15 != $true))) & (hoasinduction_gthm = $true) & (((termmset = $true) | ? [X360 : term] : (((sub @ X360 @ id)) != X360)) & (! [X360 : term] : (((sub @ X360 @ id)) = X360) | (termmset != $true))) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((($true = sP6) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | ($true != sP6))) & (((? [X511 : term,X512 : term,X510 : term,X513 : term] : ((((ap @ X512 @ X513)) = ((ap @ X510 @ X511))) & (X510 != X512)) | ? [X509 : term] : (((sub @ X509 @ id)) != X509) | ! [X517 : term,X515 : term,X514 : term,X516 : term] : ((X514 = X516) | (((ap @ (sub @ X514 @ id) @ X517)) != ((ap @ (sub @ X516 @ id) @ X515))))) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | (! [X511 : term,X512 : term,X510 : term,X513 : term] : ((((ap @ X512 @ X513)) != ((ap @ X510 @ X511))) | (X510 = X512)) & ! [X509 : term] : (((sub @ X509 @ id)) = X509) & ? [X517 : term,X515 : term,X514 : term,X516 : term] : ((X514 != X516) & (((ap @ (sub @ X514 @ id) @ X517)) = ((ap @ (sub @ X516 @ id) @ X515))))))) & (ulamvar1 != $true) & (((hoasinduction_lem3a_lthm = $true) | (sP5 != $true)) & ((sP5 = $true) | (hoasinduction_lem3a_lthm != $true))) & (axassoc = $true) & ((! [X483 : term,X482 : term] : ((X482 = X483) | (((sub @ X483 @ sh)) != ((sub @ X482 @ sh)))) | (shinj != $true)) & ((shinj = $true) | ? [X483 : term,X482 : term] : ((X482 != X483) & (((sub @ X483 @ sh)) = ((sub @ X482 @ sh)))))) & (hoaslamnotap_lthm = $true) & (pushprop_lem2v2_lthm = $true) & (((sP3 = $true) | (hoasinduction != $true)) & ((hoasinduction = $true) | (sP3 != $true))) & ! [X358 : d_term,X359 : d_term] : ((((d2term @ X359)) != ((d2term @ X358))) | (X358 = X359)) & (hoasinduction_lem1_gthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & ((((! [X332 : subst] : (((comp @ id @ X332)) = X332) & ! [X331 : subst] : (((comp @ X331 @ id)) = X331)) | ? [X330 : subst] : (((comp @ X330 @ id)) != X330) | ? [X329 : subst] : (((comp @ id @ X329)) != X329)) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | ((? [X332 : subst] : (((comp @ id @ X332)) != X332) | ? [X331 : subst] : (((comp @ X331 @ id)) != X331)) & ! [X330 : subst] : (((comp @ X330 @ id)) = X330) & ! [X329 : subst] : (((comp @ id @ X329)) = X329)))) & ! [X531 : d_subst] : (d_id = X531) & (hoasinduction_lem1v2_gthm = $true) & (((hoasinduction_lem3aa = $true) | ($true != sP0)) & (($true = sP0) | (hoasinduction_lem3aa != $true))) & (((var @ (d2term @ d_one))) != $true)),
  inference(nnf_transformation,[],[f72])).
thf(f261,plain,(
  (($true = sP58) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | ($true != sP58)) & ((hoasinduction_lem3aaa = $true) | (sP54 != $true)) & ((sP54 = $true) | (hoasinduction_lem3aaa != $true)) & (ulamvarind = $true) & (pushprop_lem3v2_lthm = $true) & ((hoasinduction_lem3b_lthm = $true) | ? [X498 : term] : ! [X499 : (subst > term > term)] : (((X499 @ sh @ one)) != ((sub @ X498 @ (push @ one @ sh))))) & (! [X498 : term] : ? [X499 : (subst > term > term)] : (((X499 @ sh @ one)) = ((sub @ X498 @ (push @ one @ sh)))) | (hoasinduction_lem3b_lthm != $true)) & (substmonoid_gthm = $true) & (($true = sP53) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | ($true != sP53)) & (! [X506 : term,X508 : (term > $o),X504 : subst,X505 : term,X507 : (term > $o)] : (($true != ((X507 @ X505))) | ($true = ((X508 @ (sub @ X505 @ (push @ X506 @ X504)))))) | (pushprop_lem1 != $true)) & ((pushprop_lem1 = $true) | ? [X506 : term,X508 : (term > $o),X504 : subst,X505 : term,X507 : (term > $o)] : (($true = ((X507 @ X505))) & ($true != ((X508 @ (sub @ X505 @ (push @ X506 @ X504))))))) & ((hoasinduction_lem3v2_lthm = $true) | ($true != sP49)) & (($true = sP49) | (hoasinduction_lem3v2_lthm != $true)) & (pushprop_lem3v2 = $true) & (hoaslamnotvar_gthm = $true) & (! [X235 : subst,X234 : term] : (((sub @ one @ (push @ X234 @ X235))) = X234) | (axvarcons != $true)) & ((axvarcons = $true) | ? [X235 : subst,X234 : term] : (((sub @ one @ (push @ X234 @ X235))) != X234)) & (pushprop_lem2v2 = $true) & ((hoasinduction_lem3aa_lthm = $true) | (sP47 != $true)) & ((sP47 = $true) | (hoasinduction_lem3aa_lthm != $true)) & (hoasinduction_lem3_gthm = $true) & (! [X494 : term] : ? [X495 : (subst > term > term)] : ! [X496 : term,X497 : subst] : (((sub @ X494 @ (push @ X496 @ X497))) = ((X495 @ X497 @ X496))) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ? [X494 : term] : ! [X495 : (subst > term > term)] : ? [X496 : term,X497 : subst] : (((sub @ X494 @ (push @ X496 @ X497))) != ((X495 @ X497 @ X496)))) & (pushprop_lthm_orig = $true) & (lamnotvar = $true) & ((apinj1 = $true) | ? [X518 : term,X519 : term,X520 : term,X521 : term] : ((((ap @ X518 @ X520)) = ((ap @ X521 @ X519))) & (X518 != X521))) & (! [X518 : term,X519 : term,X520 : term,X521 : term] : ((((ap @ X518 @ X520)) != ((ap @ X521 @ X519))) | (X518 = X521)) | (apinj1 != $true)) & (! [X535 : term,X534 : (term > $o)] : (? [X539 : term] : (($true != ((X534 @ (lam @ X539)))) & ($true = ((X534 @ X539)))) | (((sP46 @ X534)) = $true) | ($true = ((X534 @ X535))) | ? [X536 : term] : ((((var @ X536)) = $true) & ($true != ((X534 @ X536))))) | (induction != $true)) & ((induction = $true) | ? [X535 : term,X534 : (term > $o)] : (! [X539 : term] : (($true = ((X534 @ (lam @ X539)))) | ($true != ((X534 @ X539)))) & (((sP46 @ X534)) != $true) & ($true != ((X534 @ X535))) & ! [X536 : term] : ((((var @ X536)) != $true) | ($true = ((X534 @ X536)))))) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem1v2 = $true) & ! [X413 : d_subst,X412 : d_subst] : ((((d2subst @ X413)) != ((d2subst @ X412))) | (X412 = X413)) & ((hoasinduction_lem0 = $true) | ? [X445 : (subst > term > subst > $o)] : ! [X446 : (term > $o)] : ? [X447 : term] : (((X445 @ id @ X447 @ id)) != ((X446 @ X447)))) & (! [X445 : (subst > term > subst > $o)] : ? [X446 : (term > $o)] : ! [X447 : term] : (((X445 @ id @ X447 @ id)) = ((X446 @ X447))) | (hoasinduction_lem0 != $true)) & (induction2_gthm = $true) & ! [X532 : subst] : ? [X533 : d_subst] : (((d2subst @ X533)) = X532) & (hoaslamnotap_gthm = $true) & (induction2lem_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (($true = sP45) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | ($true != sP45)) & (id = ((d2subst @ d_id))) & (hoasapinj1_gthm = $true) & ((apinj2 = $true) | ? [X71 : term,X73 : term,X74 : term,X72 : term] : ((X71 != X72) & (((ap @ X73 @ X72)) = ((ap @ X74 @ X71))))) & (! [X71 : term,X73 : term,X74 : term,X72 : term] : ((X71 = X72) | (((ap @ X73 @ X72)) != ((ap @ X74 @ X71)))) | (apinj2 != $true)) & (((d2subst @ d_id)) = sh) & (hoasinduction_lem2_lthm = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoaslaminj = $true) | ? [X176 : subst,X178 : (subst > term > term),X177 : term,X179 : (subst > term > term)] : (! [X183 : subst,X184 : subst,X185 : term] : (((X179 @ (comp @ X184 @ X183) @ (sub @ X185 @ X183))) = ((sub @ (X179 @ X184 @ X185) @ X183))) & ! [X180 : term,X181 : subst,X182 : subst] : (((X178 @ (comp @ X182 @ X181) @ (sub @ X180 @ X181))) = ((sub @ (X178 @ X182 @ X180) @ X181))) & (((X179 @ X176 @ X177)) != ((X178 @ X176 @ X177))) & (((lam @ (X178 @ sh @ one))) = ((lam @ (X179 @ sh @ one)))))) & (! [X176 : subst,X178 : (subst > term > term),X177 : term,X179 : (subst > term > term)] : (? [X183 : subst,X184 : subst,X185 : term] : (((X179 @ (comp @ X184 @ X183) @ (sub @ X185 @ X183))) != ((sub @ (X179 @ X184 @ X185) @ X183))) | ? [X180 : term,X181 : subst,X182 : subst] : (((X178 @ (comp @ X182 @ X181) @ (sub @ X180 @ X181))) != ((sub @ (X178 @ X182 @ X180) @ X181))) | (((X179 @ X176 @ X177)) = ((X178 @ X176 @ X177))) | (((lam @ (X178 @ sh @ one))) != ((lam @ (X179 @ sh @ one))))) | (hoaslaminj != $true)) & (hoasinduction_lem3b_gthm = $true) & ((axidr = $true) | ? [X424 : subst] : (((comp @ X424 @ id)) != X424)) & (! [X424 : subst] : (((comp @ X424 @ id)) = X424) | (axidr != $true)) & (termmset_gthm = $true) & (hoasinduction_lem1 = $true) & (pushprop = $true) & (apnotvar = $true) & ! [X96 : d_term] : (d_one = X96) & (hoaslamnotvar = $true) & (! [X530 : term,X529 : subst] : (((comp @ sh @ (push @ X530 @ X529))) = X529) | (axshiftcons != $true)) & ((axshiftcons = $true) | ? [X530 : term,X529 : subst] : (((comp @ sh @ (push @ X530 @ X529))) != X529)) & (axapp = $true) & (hoasinduction_lem3a_gthm = $true) & (! [X106 : term] : (((sub @ X106 @ id)) = X106) | (axvarid != $true)) & ((axvarid = $true) | ? [X106 : term] : (((sub @ X106 @ id)) != X106)) & (axmap = $true) & (($true = sP44) | (hoaslaminj_lthm != $true)) & ((hoaslaminj_lthm = $true) | ($true != sP44)) & (axabs = $true) & ((hoasapinj2 = $true) | ? [X105 : term,X102 : term,X103 : term,X104 : term] : ((X102 != X105) & (((ap @ (sub @ X104 @ id) @ X102)) = ((ap @ (sub @ X103 @ id) @ X105))))) & (! [X105 : term,X102 : term,X103 : term,X104 : term] : ((X102 = X105) | (((ap @ (sub @ X104 @ id) @ X102)) != ((ap @ (sub @ X103 @ id) @ X105)))) | (hoasapinj2 != $true)) & ((hoaslamnotap = $true) | ? [X1 : term,X2 : term,X0 : (subst > term > term)] : (! [X3 : subst,X4 : term,X5 : subst] : (((X0 @ (comp @ X3 @ X5) @ (sub @ X4 @ X5))) = ((sub @ (X0 @ X3 @ X4) @ X5))) & (((ap @ (sub @ X2 @ id) @ X1)) = ((lam @ (X0 @ sh @ one)))))) & (! [X1 : term,X2 : term,X0 : (subst > term > term)] : (? [X3 : subst,X4 : term,X5 : subst] : (((X0 @ (comp @ X3 @ X5) @ (sub @ X4 @ X5))) != ((sub @ (X0 @ X3 @ X4) @ X5))) | (((ap @ (sub @ X2 @ id) @ X1)) != ((lam @ (X0 @ sh @ one))))) | (hoaslamnotap != $true)) & (hoaslamnotvar_lthm = $true) & (ulamvarsh = $true) & (? [X500 : term] : (((sub @ X500 @ id)) != X500) | ! [X501 : term] : (((sub @ X501 @ id)) = X501) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | (! [X500 : term] : (((sub @ X500 @ id)) = X500) & ? [X501 : term] : (((sub @ X501 @ id)) != X501))) & (($true = sP43) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | ($true != sP43)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoasapnotvar = $true) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_lem3v2_gthm = $true) & (? [X364 : subst,X363 : term] : (((comp @ sh @ (push @ X363 @ X364))) != X364) | ! [X369 : term,X366 : subst,X368 : term,X367 : (term > $o),X365 : (term > $o)] : (($true != ((X365 @ X369))) | ($true = ((X367 @ (sub @ X369 @ (push @ X368 @ X366)))))) | ? [X361 : term,X362 : subst] : (((sub @ one @ (push @ X361 @ X362))) != X361) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (! [X364 : subst,X363 : term] : (((comp @ sh @ (push @ X363 @ X364))) = X364) & ? [X369 : term,X366 : subst,X368 : term,X367 : (term > $o),X365 : (term > $o)] : (($true = ((X365 @ X369))) & ($true != ((X367 @ (sub @ X369 @ (push @ X368 @ X366)))))) & ! [X361 : term,X362 : subst] : (((sub @ one @ (push @ X361 @ X362))) = X361))) & ((hoasapinj1 = $true) | ? [X67 : term,X70 : term,X69 : term,X68 : term] : ((X67 != X68) & (((ap @ (sub @ X67 @ id) @ X70)) = ((ap @ (sub @ X68 @ id) @ X69))))) & (! [X67 : term,X70 : term,X69 : term,X68 : term] : ((X67 = X68) | (((ap @ (sub @ X67 @ id) @ X70)) != ((ap @ (sub @ X68 @ id) @ X69)))) | (hoasapinj1 != $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)))))))))))))) & ((pushprop_lem1v2 = $true) | ? [X40 : subst,X43 : (term > $o),X42 : (term > $o),X41 : term] : (! [X44 : term] : (((X43 @ (sub @ X44 @ (push @ X41 @ X40)))) = ((X42 @ X44))) & ($true != ((X42 @ one))) & ($true = ((X43 @ X41))))) & (! [X40 : subst,X43 : (term > $o),X42 : (term > $o),X41 : term] : (? [X44 : term] : (((X43 @ (sub @ X44 @ (push @ X41 @ X40)))) != ((X42 @ X44))) | ($true = ((X42 @ one))) | ($true != ((X43 @ X41)))) | (pushprop_lem1v2 != $true)) & (axclos = $true) & ((hoasinduction_lem3b = $true) | ? [X253 : term] : ! [X254 : (subst > term > term)] : (((X254 @ sh @ one)) != ((sub @ X253 @ (push @ one @ sh))))) & (! [X253 : term] : ? [X254 : (subst > term > term)] : (((X254 @ sh @ one)) = ((sub @ X253 @ (push @ one @ sh)))) | (hoasinduction_lem3b != $true)) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (lamnotap != $true) & (($true = sP41) | (induction2lem != $true)) & ((induction2lem = $true) | ($true != sP41)) & (! [X526 : subst,X525 : term,X527 : (term > $o),X524 : (term > $o)] : ((((X524 @ one)) = $true) | ? [X528 : term] : (((X524 @ X528)) != ((X527 @ (sub @ X528 @ (push @ X525 @ X526))))) | ($true != ((X527 @ X525)))) | ? [X523 : subst,X522 : term] : (((sub @ one @ (push @ X522 @ X523))) != X522) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (? [X526 : subst,X525 : term,X527 : (term > $o),X524 : (term > $o)] : ((((X524 @ one)) != $true) & ! [X528 : term] : (((X524 @ X528)) = ((X527 @ (sub @ X528 @ (push @ X525 @ X526))))) & ($true = ((X527 @ X525)))) & ! [X523 : subst,X522 : term] : (((sub @ one @ (push @ X522 @ X523))) = X522))) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X83 : term] : ! [X84 : (subst > term > term)] : ? [X85 : term,X86 : subst] : (((X84 @ X86 @ X85)) != ((sub @ X83 @ (push @ X85 @ X86))))) & (! [X83 : term] : ? [X84 : (subst > term > term)] : ! [X85 : term,X86 : subst] : (((X84 @ X86 @ X85)) = ((sub @ X83 @ (push @ X85 @ X86)))) | (hoasinduction_lem3v2_f_lthm != $true)) & ! [X422 : term] : ? [X423 : d_term] : (((d2term @ X423)) = X422) & ((hoasinduction_no_psi_cond = $true) | ? [X225 : (subst > term > subst > $o),X226 : term] : ((((X225 @ id @ X226 @ id)) != $true) & ! [X229 : (subst > term > term)] : (($true = ((X225 @ id @ (lam @ (X229 @ sh @ one)) @ id))) | ? [X233 : subst,X231 : subst,X232 : term] : (((sub @ (X229 @ X231 @ X232) @ X233)) != ((X229 @ (comp @ X231 @ X233) @ (sub @ X232 @ X233)))) | ? [X230 : term] : (($true = ((X225 @ id @ X230 @ id))) & (((X225 @ id @ (X229 @ id @ X230) @ id)) != $true))) & (((sP39 @ X225)) != $true))) & (! [X225 : (subst > term > subst > $o),X226 : term] : ((((X225 @ id @ X226 @ id)) = $true) | ? [X229 : (subst > term > term)] : (($true != ((X225 @ id @ (lam @ (X229 @ sh @ one)) @ id))) & ! [X233 : subst,X231 : subst,X232 : term] : (((sub @ (X229 @ X231 @ X232) @ X233)) = ((X229 @ (comp @ X231 @ X233) @ (sub @ X232 @ X233)))) & ! [X230 : term] : (($true != ((X225 @ id @ X230 @ id))) | (((X225 @ id @ (X229 @ id @ X230) @ id)) = $true))) | (((sP39 @ X225)) = $true)) | (hoasinduction_no_psi_cond != $true)) & (! [X443 : term,X444 : term] : ((((lam @ X443)) != ((lam @ X444))) | (X443 = X444)) | (laminj != $true)) & ((laminj = $true) | ? [X443 : term,X444 : term] : ((((lam @ X443)) = ((lam @ X444))) & (X443 != X444))) & (pushprop_lthm = $true) & ((hoasapinj2_lthm = $true) | (? [X455 : term,X453 : term,X454 : term,X452 : term] : ((((ap @ (sub @ X454 @ id) @ X452)) = ((ap @ (sub @ X453 @ id) @ X455))) & (X452 != X455)) & ! [X449 : term,X450 : term,X451 : term,X448 : term] : ((X448 = X450) | (((ap @ X451 @ X448)) != ((ap @ X449 @ X450)))))) & (! [X455 : term,X453 : term,X454 : term,X452 : term] : ((((ap @ (sub @ X454 @ id) @ X452)) != ((ap @ (sub @ X453 @ id) @ X455))) | (X452 = X455)) | ? [X449 : term,X450 : term,X451 : term,X448 : term] : ((X448 != X450) & (((ap @ X451 @ X448)) = ((ap @ X449 @ X450)))) | (hoasapinj2_lthm != $true)) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((hoasinduction_lem3 = $true) | ($true != sP38)) & (($true = sP38) | (hoasinduction_lem3 != $true)) & ((pushprop_lem0_lthm = $true) | ? [X256 : (term > $o),X255 : subst,X257 : term] : ! [X258 : (term > $o)] : ? [X259 : term] : (((X258 @ X259)) != ((X256 @ (sub @ X259 @ (push @ X257 @ X255)))))) & (! [X256 : (term > $o),X255 : subst,X257 : term] : ? [X258 : (term > $o)] : ! [X259 : term] : (((X258 @ X259)) = ((X256 @ (sub @ X259 @ (push @ X257 @ X255))))) | (pushprop_lem0_lthm != $true)) & (! [X45 : subst] : (((comp @ id @ X45)) = X45) | (axidl != $true)) & ((axidl = $true) | ? [X45 : subst] : (((comp @ id @ X45)) != X45)) & (! [X98 : subst,X99 : (term > $o),X97 : term] : ? [X100 : (term > $o)] : ! [X101 : term] : (((X99 @ (sub @ X101 @ (push @ X97 @ X98)))) = ((X100 @ X101))) | (pushprop_lem0 != $true)) & ((pushprop_lem0 = $true) | ? [X98 : subst,X99 : (term > $o),X97 : term] : ! [X100 : (term > $o)] : ? [X101 : term] : (((X99 @ (sub @ X101 @ (push @ X97 @ X98)))) != ((X100 @ X101)))) & (pushprop_lem0_gthm = $true) & ((hoasinduction_lem3_lthm = $true) | ($true != sP36)) & (($true = sP36) | (hoasinduction_lem3_lthm != $true)) & (($true = sP33) | (hoasinduction_no_psi_cond_lthm != $true)) & ((hoasinduction_no_psi_cond_lthm = $true) | ($true != sP33)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (hoasapinj2_gthm = $true) & (one = ((d2term @ d_one))) & (pushprop_gthm = $true) & (hoasinduction_lem1_lthm = $true) & ((hoasinduction_lem3v2a_lthm = $true) | (sP28 != $true)) & ((sP28 = $true) | (hoasinduction_lem3v2a_lthm != $true)) & ((substmonoid = $true) | ? [X502 : subst] : (((comp @ X502 @ id)) != X502) | ? [X503 : subst] : (((comp @ id @ X503)) != X503)) & ((! [X502 : subst] : (((comp @ X502 @ id)) = X502) & ! [X503 : subst] : (((comp @ id @ X503)) = X503)) | (substmonoid != $true)) & (hoasapnotvar_lthm = $true) & (pushprop_lem1_gthm = $true) & (hoaslaminj_gthm = $true) & (pushprop_lem2v2_gthm = $true) & (hoasinduction_lem2v2_gthm = $true) & (($true = sP27) | (induction2 != $true)) & ((induction2 = $true) | ($true != sP27)) & ((hoasinduction_lem0_lthm = $true) | ? [X141 : (subst > term > subst > $o)] : ! [X142 : (term > $o)] : ? [X143 : term] : (((X141 @ id @ X143 @ id)) != ((X142 @ X143)))) & (! [X141 : (subst > term > subst > $o)] : ? [X142 : (term > $o)] : ! [X143 : term] : (((X141 @ id @ X143 @ id)) = ((X142 @ X143))) | (hoasinduction_lem0_lthm != $true)) & (hoasinduction_lem2_gthm = $true) & ((hoasinduction_lem2v2 = $true) | ($true != sP25)) & (($true = sP25) | (hoasinduction_lem2v2 != $true)) & ((hoasinduction_lthm_3 = $true) | (sP23 != $true)) & ((sP23 = $true) | (hoasinduction_lthm_3 != $true)) & (axvarshift = $true) & ((sP17 = $true) | (hoasinduction_lem3v2 != $true)) & ((hoasinduction_lem3v2 = $true) | (sP17 != $true)) & ((axscons = $true) | ? [X87 : subst] : (((push @ (sub @ one @ X87) @ (comp @ sh @ X87))) != X87)) & (! [X87 : subst] : (((push @ (sub @ one @ X87) @ (comp @ sh @ X87))) = X87) | (axscons != $true)) & ((sP15 = $true) | (hoasinduction_lthm != $true)) & ((hoasinduction_lthm = $true) | (sP15 != $true)) & (hoasinduction_gthm = $true) & ((termmset = $true) | ? [X360 : term] : (((sub @ X360 @ id)) != X360)) & (! [X360 : term] : (((sub @ X360 @ id)) = X360) | (termmset != $true)) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (($true = sP6) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | ($true != sP6)) & (? [X511 : term,X512 : term,X510 : term,X513 : term] : ((((ap @ X512 @ X513)) = ((ap @ X510 @ X511))) & (X510 != X512)) | ? [X509 : term] : (((sub @ X509 @ id)) != X509) | ! [X517 : term,X515 : term,X514 : term,X516 : term] : ((X514 = X516) | (((ap @ (sub @ X514 @ id) @ X517)) != ((ap @ (sub @ X516 @ id) @ X515)))) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | (! [X511 : term,X512 : term,X510 : term,X513 : term] : ((((ap @ X512 @ X513)) != ((ap @ X510 @ X511))) | (X510 = X512)) & ! [X509 : term] : (((sub @ X509 @ id)) = X509) & ? [X517 : term,X515 : term,X514 : term,X516 : term] : ((X514 != X516) & (((ap @ (sub @ X514 @ id) @ X517)) = ((ap @ (sub @ X516 @ id) @ X515)))))) & (ulamvar1 != $true) & ((hoasinduction_lem3a_lthm = $true) | (sP5 != $true)) & ((sP5 = $true) | (hoasinduction_lem3a_lthm != $true)) & (axassoc = $true) & (! [X483 : term,X482 : term] : ((X482 = X483) | (((sub @ X483 @ sh)) != ((sub @ X482 @ sh)))) | (shinj != $true)) & ((shinj = $true) | ? [X483 : term,X482 : term] : ((X482 != X483) & (((sub @ X483 @ sh)) = ((sub @ X482 @ sh))))) & (hoaslamnotap_lthm = $true) & (pushprop_lem2v2_lthm = $true) & ((sP3 = $true) | (hoasinduction != $true)) & ((hoasinduction = $true) | (sP3 != $true)) & ! [X358 : d_term,X359 : d_term] : ((((d2term @ X359)) != ((d2term @ X358))) | (X358 = X359)) & (hoasinduction_lem1_gthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & ((! [X332 : subst] : (((comp @ id @ X332)) = X332) & ! [X331 : subst] : (((comp @ X331 @ id)) = X331)) | ? [X330 : subst] : (((comp @ X330 @ id)) != X330) | ? [X329 : subst] : (((comp @ id @ X329)) != X329) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | ((? [X332 : subst] : (((comp @ id @ X332)) != X332) | ? [X331 : subst] : (((comp @ X331 @ id)) != X331)) & ! [X330 : subst] : (((comp @ X330 @ id)) = X330) & ! [X329 : subst] : (((comp @ id @ X329)) = X329))) & ! [X531 : d_subst] : (d_id = X531) & (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_lem3aa = $true) | ($true != sP0)) & (($true = sP0) | (hoasinduction_lem3aa != $true)) & (((var @ (d2term @ d_one))) != $true)),
  inference(flattening,[],[f260])).
thf(f262,plain,(
  (($true = sP58) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | ($true != sP58)) & ((hoasinduction_lem3aaa = $true) | (sP54 != $true)) & ((sP54 = $true) | (hoasinduction_lem3aaa != $true)) & (ulamvarind = $true) & (pushprop_lem3v2_lthm = $true) & ((hoasinduction_lem3b_lthm = $true) | ? [X0 : term] : ! [X1 : (subst > term > term)] : (((sub @ X0 @ (push @ one @ sh))) != ((X1 @ sh @ one)))) & (! [X2 : term] : ? [X3 : (subst > term > term)] : (((X3 @ sh @ one)) = ((sub @ X2 @ (push @ one @ sh)))) | (hoasinduction_lem3b_lthm != $true)) & (substmonoid_gthm = $true) & (($true = sP53) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | ($true != sP53)) & (! [X4 : term,X5 : (term > $o),X6 : subst,X7 : term,X8 : (term > $o)] : (($true != ((X8 @ X7))) | ($true = ((X5 @ (sub @ X7 @ (push @ X4 @ X6)))))) | (pushprop_lem1 != $true)) & ((pushprop_lem1 = $true) | ? [X9 : term,X10 : (term > $o),X11 : subst,X12 : term,X13 : (term > $o)] : ((((X13 @ X12)) = $true) & ($true != ((X10 @ (sub @ X12 @ (push @ X9 @ X11))))))) & ((hoasinduction_lem3v2_lthm = $true) | ($true != sP49)) & (($true = sP49) | (hoasinduction_lem3v2_lthm != $true)) & (pushprop_lem3v2 = $true) & (hoaslamnotvar_gthm = $true) & (! [X14 : subst,X15 : term] : (((sub @ one @ (push @ X15 @ X14))) = X15) | (axvarcons != $true)) & ((axvarcons = $true) | ? [X16 : subst,X17 : term] : (((sub @ one @ (push @ X17 @ X16))) != X17)) & (pushprop_lem2v2 = $true) & ((hoasinduction_lem3aa_lthm = $true) | (sP47 != $true)) & ((sP47 = $true) | (hoasinduction_lem3aa_lthm != $true)) & (hoasinduction_lem3_gthm = $true) & (! [X18 : term] : ? [X19 : (subst > term > term)] : ! [X20 : term,X21 : subst] : (((X19 @ X21 @ X20)) = ((sub @ X18 @ (push @ X20 @ X21)))) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ? [X22 : term] : ! [X23 : (subst > term > term)] : ? [X24 : term,X25 : subst] : (((X23 @ X25 @ X24)) != ((sub @ X22 @ (push @ X24 @ X25))))) & (pushprop_lthm_orig = $true) & (lamnotvar = $true) & ((apinj1 = $true) | ? [X26 : term,X27 : term,X28 : term,X29 : term] : ((((ap @ X26 @ X28)) = ((ap @ X29 @ X27))) & (X26 != X29))) & (! [X30 : term,X31 : term,X32 : term,X33 : term] : ((((ap @ X33 @ X31)) != ((ap @ X30 @ X32))) | (X30 = X33)) | (apinj1 != $true)) & (! [X34 : term,X35 : (term > $o)] : (? [X36 : term] : ((((X35 @ (lam @ X36))) != $true) & ($true = ((X35 @ X36)))) | ($true = ((sP46 @ X35))) | ($true = ((X35 @ X34))) | ? [X37 : term] : (($true = ((var @ X37))) & ($true != ((X35 @ X37))))) | (induction != $true)) & ((induction = $true) | ? [X38 : term,X39 : (term > $o)] : (! [X40 : term] : (($true = ((X39 @ (lam @ X40)))) | ($true != ((X39 @ X40)))) & ($true != ((sP46 @ X39))) & ($true != ((X39 @ X38))) & ! [X41 : term] : (($true != ((var @ X41))) | ($true = ((X39 @ X41)))))) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem1v2 = $true) & ! [X42 : d_subst,X43 : d_subst] : ((((d2subst @ X42)) != ((d2subst @ X43))) | (X42 = X43)) & ((hoasinduction_lem0 = $true) | ? [X44 : (subst > term > subst > $o)] : ! [X45 : (term > $o)] : ? [X46 : term] : (((X44 @ id @ X46 @ id)) != ((X45 @ X46)))) & (! [X47 : (subst > term > subst > $o)] : ? [X48 : (term > $o)] : ! [X49 : term] : (((X47 @ id @ X49 @ id)) = ((X48 @ X49))) | (hoasinduction_lem0 != $true)) & (induction2_gthm = $true) & ! [X50 : subst] : ? [X51 : d_subst] : (((d2subst @ X51)) = X50) & (hoaslamnotap_gthm = $true) & (induction2lem_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (($true = sP45) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | ($true != sP45)) & (id = ((d2subst @ d_id))) & (hoasapinj1_gthm = $true) & ((apinj2 = $true) | ? [X52 : term,X53 : term,X54 : term,X55 : term] : ((X52 != X55) & (((ap @ X54 @ X52)) = ((ap @ X53 @ X55))))) & (! [X56 : term,X57 : term,X58 : term,X59 : term] : ((X56 = X59) | (((ap @ X57 @ X59)) != ((ap @ X58 @ X56)))) | (apinj2 != $true)) & (((d2subst @ d_id)) = sh) & (hoasinduction_lem2_lthm = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoaslaminj = $true) | ? [X60 : subst,X61 : (subst > term > term),X62 : term,X63 : (subst > term > term)] : (! [X64 : subst,X65 : subst,X66 : term] : (((sub @ (X63 @ X65 @ X66) @ X64)) = ((X63 @ (comp @ X65 @ X64) @ (sub @ X66 @ X64)))) & ! [X67 : term,X68 : subst,X69 : subst] : (((sub @ (X61 @ X69 @ X67) @ X68)) = ((X61 @ (comp @ X69 @ X68) @ (sub @ X67 @ X68)))) & (((X63 @ X60 @ X62)) != ((X61 @ X60 @ X62))) & (((lam @ (X61 @ sh @ one))) = ((lam @ (X63 @ sh @ one)))))) & (! [X70 : subst,X71 : (subst > term > term),X72 : term,X73 : (subst > term > term)] : (? [X74 : subst,X75 : subst,X76 : term] : (((sub @ (X73 @ X75 @ X76) @ X74)) != ((X73 @ (comp @ X75 @ X74) @ (sub @ X76 @ X74)))) | ? [X77 : term,X78 : subst,X79 : subst] : (((X71 @ (comp @ X79 @ X78) @ (sub @ X77 @ X78))) != ((sub @ (X71 @ X79 @ X77) @ X78))) | (((X71 @ X70 @ X72)) = ((X73 @ X70 @ X72))) | (((lam @ (X73 @ sh @ one))) != ((lam @ (X71 @ sh @ one))))) | (hoaslaminj != $true)) & (hoasinduction_lem3b_gthm = $true) & ((axidr = $true) | ? [X80 : subst] : (((comp @ X80 @ id)) != X80)) & (! [X81 : subst] : (((comp @ X81 @ id)) = X81) | (axidr != $true)) & (termmset_gthm = $true) & (hoasinduction_lem1 = $true) & (pushprop = $true) & (apnotvar = $true) & ! [X82 : d_term] : (d_one = X82) & (hoaslamnotvar = $true) & (! [X83 : term,X84 : subst] : (((comp @ sh @ (push @ X83 @ X84))) = X84) | (axshiftcons != $true)) & ((axshiftcons = $true) | ? [X85 : term,X86 : subst] : (((comp @ sh @ (push @ X85 @ X86))) != X86)) & (axapp = $true) & (hoasinduction_lem3a_gthm = $true) & (! [X87 : term] : (((sub @ X87 @ id)) = X87) | (axvarid != $true)) & ((axvarid = $true) | ? [X88 : term] : (((sub @ X88 @ id)) != X88)) & (axmap = $true) & (($true = sP44) | (hoaslaminj_lthm != $true)) & ((hoaslaminj_lthm = $true) | ($true != sP44)) & (axabs = $true) & ((hoasapinj2 = $true) | ? [X89 : term,X90 : term,X91 : term,X92 : term] : ((X89 != X90) & (((ap @ (sub @ X92 @ id) @ X90)) = ((ap @ (sub @ X91 @ id) @ X89))))) & (! [X93 : term,X94 : term,X95 : term,X96 : term] : ((X93 = X94) | (((ap @ (sub @ X95 @ id) @ X93)) != ((ap @ (sub @ X96 @ id) @ X94)))) | (hoasapinj2 != $true)) & ((hoaslamnotap = $true) | ? [X97 : term,X98 : term,X99 : (subst > term > term)] : (! [X100 : subst,X101 : term,X102 : subst] : (((sub @ (X99 @ X100 @ X101) @ X102)) = ((X99 @ (comp @ X100 @ X102) @ (sub @ X101 @ X102)))) & (((lam @ (X99 @ sh @ one))) = ((ap @ (sub @ X98 @ id) @ X97))))) & (! [X103 : term,X104 : term,X105 : (subst > term > term)] : (? [X106 : subst,X107 : term,X108 : subst] : (((sub @ (X105 @ X106 @ X107) @ X108)) != ((X105 @ (comp @ X106 @ X108) @ (sub @ X107 @ X108)))) | (((ap @ (sub @ X104 @ id) @ X103)) != ((lam @ (X105 @ sh @ one))))) | (hoaslamnotap != $true)) & (hoaslamnotvar_lthm = $true) & (ulamvarsh = $true) & (? [X109 : term] : (((sub @ X109 @ id)) != X109) | ! [X110 : term] : (((sub @ X110 @ id)) = X110) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | (! [X111 : term] : (((sub @ X111 @ id)) = X111) & ? [X112 : term] : (((sub @ X112 @ id)) != X112))) & (($true = sP43) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | ($true != sP43)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoasapnotvar = $true) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_lem3v2_gthm = $true) & (? [X113 : subst,X114 : term] : (((comp @ sh @ (push @ X114 @ X113))) != X113) | ! [X115 : term,X116 : subst,X117 : term,X118 : (term > $o),X119 : (term > $o)] : (($true != ((X119 @ X115))) | (((X118 @ (sub @ X115 @ (push @ X117 @ X116)))) = $true)) | ? [X120 : term,X121 : subst] : (((sub @ one @ (push @ X120 @ X121))) != X120) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (! [X122 : subst,X123 : term] : (((comp @ sh @ (push @ X123 @ X122))) = X122) & ? [X124 : term,X125 : subst,X126 : term,X127 : (term > $o),X128 : (term > $o)] : ((((X128 @ X124)) = $true) & ($true != ((X127 @ (sub @ X124 @ (push @ X126 @ X125)))))) & ! [X129 : term,X130 : subst] : (((sub @ one @ (push @ X129 @ X130))) = X129))) & ((hoasapinj1 = $true) | ? [X131 : term,X132 : term,X133 : term,X134 : term] : ((X131 != X134) & (((ap @ (sub @ X131 @ id) @ X132)) = ((ap @ (sub @ X134 @ id) @ X133))))) & (! [X135 : term,X136 : term,X137 : term,X138 : term] : ((X135 = X138) | (((ap @ (sub @ X135 @ id) @ X136)) != ((ap @ (sub @ X138 @ id) @ X137)))) | (hoasapinj1 != $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)))))))))))))) & ((pushprop_lem1v2 = $true) | ? [X139 : subst,X140 : (term > $o),X141 : (term > $o),X142 : term] : (! [X143 : term] : (((X140 @ (sub @ X143 @ (push @ X142 @ X139)))) = ((X141 @ X143))) & (((X141 @ one)) != $true) & (((X140 @ X142)) = $true))) & (! [X144 : subst,X145 : (term > $o),X146 : (term > $o),X147 : term] : (? [X148 : term] : (((X145 @ (sub @ X148 @ (push @ X147 @ X144)))) != ((X146 @ X148))) | (((X146 @ one)) = $true) | ($true != ((X145 @ X147)))) | (pushprop_lem1v2 != $true)) & (axclos = $true) & ((hoasinduction_lem3b = $true) | ? [X149 : term] : ! [X150 : (subst > term > term)] : (((X150 @ sh @ one)) != ((sub @ X149 @ (push @ one @ sh))))) & (! [X151 : term] : ? [X152 : (subst > term > term)] : (((sub @ X151 @ (push @ one @ sh))) = ((X152 @ sh @ one))) | (hoasinduction_lem3b != $true)) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (lamnotap != $true) & (($true = sP41) | (induction2lem != $true)) & ((induction2lem = $true) | ($true != sP41)) & (! [X153 : subst,X154 : term,X155 : (term > $o),X156 : (term > $o)] : (($true = ((X156 @ one))) | ? [X157 : term] : (((X155 @ (sub @ X157 @ (push @ X154 @ X153)))) != ((X156 @ X157))) | (((X155 @ X154)) != $true)) | ? [X158 : subst,X159 : term] : (((sub @ one @ (push @ X159 @ X158))) != X159) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (? [X160 : subst,X161 : term,X162 : (term > $o),X163 : (term > $o)] : (($true != ((X163 @ one))) & ! [X164 : term] : (((X163 @ X164)) = ((X162 @ (sub @ X164 @ (push @ X161 @ X160))))) & ($true = ((X162 @ X161)))) & ! [X165 : subst,X166 : term] : (((sub @ one @ (push @ X166 @ X165))) = X166))) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X167 : term] : ! [X168 : (subst > term > term)] : ? [X169 : term,X170 : subst] : (((X168 @ X170 @ X169)) != ((sub @ X167 @ (push @ X169 @ X170))))) & (! [X171 : term] : ? [X172 : (subst > term > term)] : ! [X173 : term,X174 : subst] : (((X172 @ X174 @ X173)) = ((sub @ X171 @ (push @ X173 @ X174)))) | (hoasinduction_lem3v2_f_lthm != $true)) & ! [X175 : term] : ? [X176 : d_term] : (((d2term @ X176)) = X175) & ((hoasinduction_no_psi_cond = $true) | ? [X177 : (subst > term > subst > $o),X178 : term] : (($true != ((X177 @ id @ X178 @ id))) & ! [X179 : (subst > term > term)] : (($true = ((X177 @ id @ (lam @ (X179 @ sh @ one)) @ id))) | ? [X180 : subst,X181 : subst,X182 : term] : (((X179 @ (comp @ X181 @ X180) @ (sub @ X182 @ X180))) != ((sub @ (X179 @ X181 @ X182) @ X180))) | ? [X183 : term] : ((((X177 @ id @ X183 @ id)) = $true) & ($true != ((X177 @ id @ (X179 @ id @ X183) @ id))))) & ($true != ((sP39 @ X177))))) & (! [X184 : (subst > term > subst > $o),X185 : term] : ((((X184 @ id @ X185 @ id)) = $true) | ? [X186 : (subst > term > term)] : ((((X184 @ id @ (lam @ (X186 @ sh @ one)) @ id)) != $true) & ! [X187 : subst,X188 : subst,X189 : term] : (((sub @ (X186 @ X188 @ X189) @ X187)) = ((X186 @ (comp @ X188 @ X187) @ (sub @ X189 @ X187)))) & ! [X190 : term] : (($true != ((X184 @ id @ X190 @ id))) | (((X184 @ id @ (X186 @ id @ X190) @ id)) = $true))) | ($true = ((sP39 @ X184)))) | (hoasinduction_no_psi_cond != $true)) & (! [X191 : term,X192 : term] : ((((lam @ X192)) != ((lam @ X191))) | (X191 = X192)) | (laminj != $true)) & ((laminj = $true) | ? [X193 : term,X194 : term] : ((((lam @ X193)) = ((lam @ X194))) & (X193 != X194))) & (pushprop_lthm = $true) & ((hoasapinj2_lthm = $true) | (? [X195 : term,X196 : term,X197 : term,X198 : term] : ((((ap @ (sub @ X196 @ id) @ X195)) = ((ap @ (sub @ X197 @ id) @ X198))) & (X195 != X198)) & ! [X199 : term,X200 : term,X201 : term,X202 : term] : ((X200 = X202) | (((ap @ X201 @ X202)) != ((ap @ X199 @ X200)))))) & (! [X203 : term,X204 : term,X205 : term,X206 : term] : ((((ap @ (sub @ X205 @ id) @ X206)) != ((ap @ (sub @ X204 @ id) @ X203))) | (X203 = X206)) | ? [X207 : term,X208 : term,X209 : term,X210 : term] : ((X208 != X210) & (((ap @ X207 @ X208)) = ((ap @ X209 @ X210)))) | (hoasapinj2_lthm != $true)) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((hoasinduction_lem3 = $true) | ($true != sP38)) & (($true = sP38) | (hoasinduction_lem3 != $true)) & ((pushprop_lem0_lthm = $true) | ? [X211 : (term > $o),X212 : subst,X213 : term] : ! [X214 : (term > $o)] : ? [X215 : term] : (((X214 @ X215)) != ((X211 @ (sub @ X215 @ (push @ X213 @ X212)))))) & (! [X216 : (term > $o),X217 : subst,X218 : term] : ? [X219 : (term > $o)] : ! [X220 : term] : (((X216 @ (sub @ X220 @ (push @ X218 @ X217)))) = ((X219 @ X220))) | (pushprop_lem0_lthm != $true)) & (! [X221 : subst] : (((comp @ id @ X221)) = X221) | (axidl != $true)) & ((axidl = $true) | ? [X222 : subst] : (((comp @ id @ X222)) != X222)) & (! [X223 : subst,X224 : (term > $o),X225 : term] : ? [X226 : (term > $o)] : ! [X227 : term] : (((X226 @ X227)) = ((X224 @ (sub @ X227 @ (push @ X225 @ X223))))) | (pushprop_lem0 != $true)) & ((pushprop_lem0 = $true) | ? [X228 : subst,X229 : (term > $o),X230 : term] : ! [X231 : (term > $o)] : ? [X232 : term] : (((X229 @ (sub @ X232 @ (push @ X230 @ X228)))) != ((X231 @ X232)))) & (pushprop_lem0_gthm = $true) & ((hoasinduction_lem3_lthm = $true) | ($true != sP36)) & (($true = sP36) | (hoasinduction_lem3_lthm != $true)) & (($true = sP33) | (hoasinduction_no_psi_cond_lthm != $true)) & ((hoasinduction_no_psi_cond_lthm = $true) | ($true != sP33)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (hoasapinj2_gthm = $true) & (one = ((d2term @ d_one))) & (pushprop_gthm = $true) & (hoasinduction_lem1_lthm = $true) & ((hoasinduction_lem3v2a_lthm = $true) | (sP28 != $true)) & ((sP28 = $true) | (hoasinduction_lem3v2a_lthm != $true)) & ((substmonoid = $true) | ? [X233 : subst] : (((comp @ X233 @ id)) != X233) | ? [X234 : subst] : (((comp @ id @ X234)) != X234)) & ((! [X235 : subst] : (((comp @ X235 @ id)) = X235) & ! [X236 : subst] : (((comp @ id @ X236)) = X236)) | (substmonoid != $true)) & (hoasapnotvar_lthm = $true) & (pushprop_lem1_gthm = $true) & (hoaslaminj_gthm = $true) & (pushprop_lem2v2_gthm = $true) & (hoasinduction_lem2v2_gthm = $true) & (($true = sP27) | (induction2 != $true)) & ((induction2 = $true) | ($true != sP27)) & ((hoasinduction_lem0_lthm = $true) | ? [X237 : (subst > term > subst > $o)] : ! [X238 : (term > $o)] : ? [X239 : term] : (((X238 @ X239)) != ((X237 @ id @ X239 @ id)))) & (! [X240 : (subst > term > subst > $o)] : ? [X241 : (term > $o)] : ! [X242 : term] : (((X240 @ id @ X242 @ id)) = ((X241 @ X242))) | (hoasinduction_lem0_lthm != $true)) & (hoasinduction_lem2_gthm = $true) & ((hoasinduction_lem2v2 = $true) | ($true != sP25)) & (($true = sP25) | (hoasinduction_lem2v2 != $true)) & ((hoasinduction_lthm_3 = $true) | (sP23 != $true)) & ((sP23 = $true) | (hoasinduction_lthm_3 != $true)) & (axvarshift = $true) & ((sP17 = $true) | (hoasinduction_lem3v2 != $true)) & ((hoasinduction_lem3v2 = $true) | (sP17 != $true)) & ((axscons = $true) | ? [X243 : subst] : (((push @ (sub @ one @ X243) @ (comp @ sh @ X243))) != X243)) & (! [X244 : subst] : (((push @ (sub @ one @ X244) @ (comp @ sh @ X244))) = X244) | (axscons != $true)) & ((sP15 = $true) | (hoasinduction_lthm != $true)) & ((hoasinduction_lthm = $true) | (sP15 != $true)) & (hoasinduction_gthm = $true) & ((termmset = $true) | ? [X245 : term] : (((sub @ X245 @ id)) != X245)) & (! [X246 : term] : (((sub @ X246 @ id)) = X246) | (termmset != $true)) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (($true = sP6) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | ($true != sP6)) & (? [X247 : term,X248 : term,X249 : term,X250 : term] : ((((ap @ X248 @ X250)) = ((ap @ X249 @ X247))) & (X248 != X249)) | ? [X251 : term] : (((sub @ X251 @ id)) != X251) | ! [X252 : term,X253 : term,X254 : term,X255 : term] : ((X254 = X255) | (((ap @ (sub @ X254 @ id) @ X252)) != ((ap @ (sub @ X255 @ id) @ X253)))) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | (! [X256 : term,X257 : term,X258 : term,X259 : term] : ((((ap @ X257 @ X259)) != ((ap @ X258 @ X256))) | (X257 = X258)) & ! [X260 : term] : (((sub @ X260 @ id)) = X260) & ? [X261 : term,X262 : term,X263 : term,X264 : term] : ((X263 != X264) & (((ap @ (sub @ X263 @ id) @ X261)) = ((ap @ (sub @ X264 @ id) @ X262)))))) & (ulamvar1 != $true) & ((hoasinduction_lem3a_lthm = $true) | (sP5 != $true)) & ((sP5 = $true) | (hoasinduction_lem3a_lthm != $true)) & (axassoc = $true) & (! [X265 : term,X266 : term] : ((X265 = X266) | (((sub @ X265 @ sh)) != ((sub @ X266 @ sh)))) | (shinj != $true)) & ((shinj = $true) | ? [X267 : term,X268 : term] : ((X267 != X268) & (((sub @ X268 @ sh)) = ((sub @ X267 @ sh))))) & (hoaslamnotap_lthm = $true) & (pushprop_lem2v2_lthm = $true) & ((sP3 = $true) | (hoasinduction != $true)) & ((hoasinduction = $true) | (sP3 != $true)) & ! [X269 : d_term,X270 : d_term] : ((((d2term @ X270)) != ((d2term @ X269))) | (X269 = X270)) & (hoasinduction_lem1_gthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & ((! [X271 : subst] : (((comp @ id @ X271)) = X271) & ! [X272 : subst] : (((comp @ X272 @ id)) = X272)) | ? [X273 : subst] : (((comp @ X273 @ id)) != X273) | ? [X274 : subst] : (((comp @ id @ X274)) != X274) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | ((? [X275 : subst] : (((comp @ id @ X275)) != X275) | ? [X276 : subst] : (((comp @ X276 @ id)) != X276)) & ! [X277 : subst] : (((comp @ X277 @ id)) = X277) & ! [X278 : subst] : (((comp @ id @ X278)) = X278))) & ! [X279 : d_subst] : (d_id = X279) & (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_lem3aa = $true) | ($true != sP0)) & (($true = sP0) | (hoasinduction_lem3aa != $true)) & (((var @ (d2term @ d_one))) != $true)),
  inference(rectify,[],[f261])).
thf(f263,plain,(
  (($true = sP58) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | ($true != sP58)) & ((hoasinduction_lem3aaa = $true) | (sP54 != $true)) & ((sP54 = $true) | (hoasinduction_lem3aaa != $true)) & (ulamvarind = $true) & (pushprop_lem3v2_lthm = $true) & ((hoasinduction_lem3b_lthm = $true) | ! [X1 : (subst > term > term)] : (((sub @ sK454 @ (push @ one @ sh))) != ((X1 @ sh @ one)))) & (! [X2 : term] : (((sK455 @ X2 @ sh @ one)) = ((sub @ X2 @ (push @ one @ sh)))) | (hoasinduction_lem3b_lthm != $true)) & (substmonoid_gthm = $true) & (($true = sP53) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | ($true != sP53)) & (! [X4 : term,X5 : (term > $o),X6 : subst,X7 : term,X8 : (term > $o)] : (($true != ((X8 @ X7))) | ($true = ((X5 @ (sub @ X7 @ (push @ X4 @ X6)))))) | (pushprop_lem1 != $true)) & ((pushprop_lem1 = $true) | ((((sK460 @ sK459)) = $true) & ($true != ((sK457 @ (sub @ sK459 @ (push @ sK456 @ sK458))))))) & ((hoasinduction_lem3v2_lthm = $true) | ($true != sP49)) & (($true = sP49) | (hoasinduction_lem3v2_lthm != $true)) & (pushprop_lem3v2 = $true) & (hoaslamnotvar_gthm = $true) & (! [X14 : subst,X15 : term] : (((sub @ one @ (push @ X15 @ X14))) = X15) | (axvarcons != $true)) & ((axvarcons = $true) | (((sub @ one @ (push @ sK462 @ sK461))) != sK462)) & (pushprop_lem2v2 = $true) & ((hoasinduction_lem3aa_lthm = $true) | (sP47 != $true)) & ((sP47 = $true) | (hoasinduction_lem3aa_lthm != $true)) & (hoasinduction_lem3_gthm = $true) & (! [X18 : term] : ! [X20 : term,X21 : subst] : (((sub @ X18 @ (push @ X20 @ X21))) = ((sK463 @ X18 @ X21 @ X20))) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ! [X23 : (subst > term > term)] : (((sub @ sK464 @ (push @ (sK465 @ X23) @ (sK466 @ X23)))) != ((X23 @ (sK466 @ X23) @ (sK465 @ X23))))) & (pushprop_lthm_orig = $true) & (lamnotvar = $true) & ((apinj1 = $true) | ((((ap @ sK467 @ sK469)) = ((ap @ sK470 @ sK468))) & (sK470 != sK467))) & (! [X30 : term,X31 : term,X32 : term,X33 : term] : ((((ap @ X33 @ X31)) != ((ap @ X30 @ X32))) | (X30 = X33)) | (apinj1 != $true)) & (! [X34 : term,X35 : (term > $o)] : (((((X35 @ (lam @ (sK471 @ X35)))) != $true) & ($true = ((X35 @ (sK471 @ X35))))) | ($true = ((sP46 @ X35))) | ($true = ((X35 @ X34))) | (($true = ((var @ (sK472 @ X35)))) & ($true != ((X35 @ (sK472 @ X35)))))) | (induction != $true)) & ((induction = $true) | (! [X40 : term] : (($true = ((sK474 @ (lam @ X40)))) | (((sK474 @ X40)) != $true)) & (((sP46 @ sK474)) != $true) & (((sK474 @ sK473)) != $true) & ! [X41 : term] : (($true != ((var @ X41))) | ($true = ((sK474 @ X41)))))) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem1v2 = $true) & ! [X42 : d_subst,X43 : d_subst] : ((((d2subst @ X42)) != ((d2subst @ X43))) | (X42 = X43)) & ((hoasinduction_lem0 = $true) | ! [X45 : (term > $o)] : (((X45 @ (sK476 @ X45))) != ((sK475 @ id @ (sK476 @ X45) @ id)))) & (! [X47 : (subst > term > subst > $o)] : ! [X49 : term] : (((X47 @ id @ X49 @ id)) = ((sK477 @ X47 @ X49))) | (hoasinduction_lem0 != $true)) & (induction2_gthm = $true) & ! [X50 : subst] : (((d2subst @ (sK478 @ X50))) = X50) & (hoaslamnotap_gthm = $true) & (induction2lem_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (($true = sP45) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | ($true != sP45)) & (id = ((d2subst @ d_id))) & (hoasapinj1_gthm = $true) & ((apinj2 = $true) | ((sK479 != sK482) & (((ap @ sK480 @ sK482)) = ((ap @ sK481 @ sK479))))) & (! [X56 : term,X57 : term,X58 : term,X59 : term] : ((X56 = X59) | (((ap @ X57 @ X59)) != ((ap @ X58 @ X56)))) | (apinj2 != $true)) & (((d2subst @ d_id)) = sh) & (hoasinduction_lem2_lthm = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoaslaminj = $true) | (! [X64 : subst,X65 : subst,X66 : term] : (((sK486 @ (comp @ X65 @ X64) @ (sub @ X66 @ X64))) = ((sub @ (sK486 @ X65 @ X66) @ X64))) & ! [X67 : term,X68 : subst,X69 : subst] : (((sub @ (sK484 @ X69 @ X67) @ X68)) = ((sK484 @ (comp @ X69 @ X68) @ (sub @ X67 @ X68)))) & (((sK486 @ sK483 @ sK485)) != ((sK484 @ sK483 @ sK485))) & (((lam @ (sK486 @ sh @ one))) = ((lam @ (sK484 @ sh @ one)))))) & (! [X70 : subst,X71 : (subst > term > term),X72 : term,X73 : (subst > term > term)] : ((((sub @ (X73 @ (sK488 @ X73) @ (sK489 @ X73)) @ (sK487 @ X73))) != ((X73 @ (comp @ (sK488 @ X73) @ (sK487 @ X73)) @ (sub @ (sK489 @ X73) @ (sK487 @ X73))))) | (((sub @ (X71 @ (sK492 @ X71) @ (sK490 @ X71)) @ (sK491 @ X71))) != ((X71 @ (comp @ (sK492 @ X71) @ (sK491 @ X71)) @ (sub @ (sK490 @ X71) @ (sK491 @ X71))))) | (((X71 @ X70 @ X72)) = ((X73 @ X70 @ X72))) | (((lam @ (X73 @ sh @ one))) != ((lam @ (X71 @ sh @ one))))) | (hoaslaminj != $true)) & (hoasinduction_lem3b_gthm = $true) & ((axidr = $true) | (sK493 != ((comp @ sK493 @ id)))) & (! [X81 : subst] : (((comp @ X81 @ id)) = X81) | (axidr != $true)) & (termmset_gthm = $true) & (hoasinduction_lem1 = $true) & (pushprop = $true) & (apnotvar = $true) & ! [X82 : d_term] : (d_one = X82) & (hoaslamnotvar = $true) & (! [X83 : term,X84 : subst] : (((comp @ sh @ (push @ X83 @ X84))) = X84) | (axshiftcons != $true)) & ((axshiftcons = $true) | (sK495 != ((comp @ sh @ (push @ sK494 @ sK495))))) & (axapp = $true) & (hoasinduction_lem3a_gthm = $true) & (! [X87 : term] : (((sub @ X87 @ id)) = X87) | (axvarid != $true)) & ((axvarid = $true) | (((sub @ sK496 @ id)) != sK496)) & (axmap = $true) & (($true = sP44) | (hoaslaminj_lthm != $true)) & ((hoaslaminj_lthm = $true) | ($true != sP44)) & (axabs = $true) & ((hoasapinj2 = $true) | ((sK497 != sK498) & (((ap @ (sub @ sK500 @ id) @ sK498)) = ((ap @ (sub @ sK499 @ id) @ sK497))))) & (! [X93 : term,X94 : term,X95 : term,X96 : term] : ((X93 = X94) | (((ap @ (sub @ X95 @ id) @ X93)) != ((ap @ (sub @ X96 @ id) @ X94)))) | (hoasapinj2 != $true)) & ((hoaslamnotap = $true) | (! [X100 : subst,X101 : term,X102 : subst] : (((sK503 @ (comp @ X100 @ X102) @ (sub @ X101 @ X102))) = ((sub @ (sK503 @ X100 @ X101) @ X102))) & (((ap @ (sub @ sK502 @ id) @ sK501)) = ((lam @ (sK503 @ sh @ one)))))) & (! [X103 : term,X104 : term,X105 : (subst > term > term)] : ((((sub @ (X105 @ (sK504 @ X105) @ (sK505 @ X105)) @ (sK506 @ X105))) != ((X105 @ (comp @ (sK504 @ X105) @ (sK506 @ X105)) @ (sub @ (sK505 @ X105) @ (sK506 @ X105))))) | (((ap @ (sub @ X104 @ id) @ X103)) != ((lam @ (X105 @ sh @ one))))) | (hoaslamnotap != $true)) & (hoaslamnotvar_lthm = $true) & (ulamvarsh = $true) & ((sK507 != ((sub @ sK507 @ id))) | ! [X110 : term] : (((sub @ X110 @ id)) = X110) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | (! [X111 : term] : (((sub @ X111 @ id)) = X111) & (((sub @ sK508 @ id)) != sK508))) & (($true = sP43) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | ($true != sP43)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoasapnotvar = $true) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_lem3v2_gthm = $true) & ((sK509 != ((comp @ sh @ (push @ sK510 @ sK509)))) | ! [X115 : term,X116 : subst,X117 : term,X118 : (term > $o),X119 : (term > $o)] : (($true != ((X119 @ X115))) | (((X118 @ (sub @ X115 @ (push @ X117 @ X116)))) = $true)) | (sK511 != ((sub @ one @ (push @ sK511 @ sK512)))) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (! [X122 : subst,X123 : term] : (((comp @ sh @ (push @ X123 @ X122))) = X122) & ((((sK517 @ sK513)) = $true) & ($true != ((sK516 @ (sub @ sK513 @ (push @ sK515 @ sK514)))))) & ! [X129 : term,X130 : subst] : (((sub @ one @ (push @ X129 @ X130))) = X129))) & ((hoasapinj1 = $true) | ((sK521 != sK518) & (((ap @ (sub @ sK518 @ id) @ sK519)) = ((ap @ (sub @ sK521 @ id) @ sK520))))) & (! [X135 : term,X136 : term,X137 : term,X138 : term] : ((X135 = X138) | (((ap @ (sub @ X135 @ id) @ X136)) != ((ap @ (sub @ X138 @ id) @ X137)))) | (hoasapinj1 != $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)))))))))))))) & ((pushprop_lem1v2 = $true) | (! [X143 : term] : (((sK523 @ (sub @ X143 @ (push @ sK525 @ sK522)))) = ((sK524 @ X143))) & ($true != ((sK524 @ one))) & ($true = ((sK523 @ sK525))))) & (! [X144 : subst,X145 : (term > $o),X146 : (term > $o),X147 : term] : ((((X146 @ (sK526 @ X147 @ X146 @ X145 @ X144))) != ((X145 @ (sub @ (sK526 @ X147 @ X146 @ X145 @ X144) @ (push @ X147 @ X144))))) | (((X146 @ one)) = $true) | ($true != ((X145 @ X147)))) | (pushprop_lem1v2 != $true)) & (axclos = $true) & ((hoasinduction_lem3b = $true) | ! [X150 : (subst > term > term)] : (((X150 @ sh @ one)) != ((sub @ sK527 @ (push @ one @ sh))))) & (! [X151 : term] : (((sK528 @ X151 @ sh @ one)) = ((sub @ X151 @ (push @ one @ sh)))) | (hoasinduction_lem3b != $true)) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (lamnotap != $true) & (($true = sP41) | (induction2lem != $true)) & ((induction2lem = $true) | ($true != sP41)) & (! [X153 : subst,X154 : term,X155 : (term > $o),X156 : (term > $o)] : (($true = ((X156 @ one))) | (((X155 @ (sub @ (sK529 @ X156 @ X155 @ X154 @ X153) @ (push @ X154 @ X153)))) != ((X156 @ (sK529 @ X156 @ X155 @ X154 @ X153)))) | (((X155 @ X154)) != $true)) | (sK531 != ((sub @ one @ (push @ sK531 @ sK530)))) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (((((sK535 @ one)) != $true) & ! [X164 : term] : (((sK534 @ (sub @ X164 @ (push @ sK533 @ sK532)))) = ((sK535 @ X164))) & ($true = ((sK534 @ sK533)))) & ! [X165 : subst,X166 : term] : (((sub @ one @ (push @ X166 @ X165))) = X166))) & ((hoasinduction_lem3v2_f_lthm = $true) | ! [X168 : (subst > term > term)] : (((sub @ sK536 @ (push @ (sK537 @ X168) @ (sK538 @ X168)))) != ((X168 @ (sK538 @ X168) @ (sK537 @ X168))))) & (! [X171 : term] : ! [X173 : term,X174 : subst] : (((sK539 @ X171 @ X174 @ X173)) = ((sub @ X171 @ (push @ X173 @ X174)))) | (hoasinduction_lem3v2_f_lthm != $true)) & ! [X175 : term] : (((d2term @ (sK540 @ X175))) = X175) & ((hoasinduction_no_psi_cond = $true) | (($true != ((sK541 @ id @ sK542 @ id))) & ! [X179 : (subst > term > term)] : ((((sK541 @ id @ (lam @ (X179 @ sh @ one)) @ id)) = $true) | (((X179 @ (comp @ (sK544 @ X179) @ (sK543 @ X179)) @ (sub @ (sK545 @ X179) @ (sK543 @ X179)))) != ((sub @ (X179 @ (sK544 @ X179) @ (sK545 @ X179)) @ (sK543 @ X179)))) | (($true = ((sK541 @ id @ (sK546 @ X179) @ id))) & ($true != ((sK541 @ id @ (X179 @ id @ (sK546 @ X179)) @ id))))) & ($true != ((sP39 @ sK541))))) & (! [X184 : (subst > term > subst > $o),X185 : term] : ((((X184 @ id @ X185 @ id)) = $true) | (($true != ((X184 @ id @ (lam @ (sK547 @ X184 @ sh @ one)) @ id))) & ! [X187 : subst,X188 : subst,X189 : term] : (((sK547 @ X184 @ (comp @ X188 @ X187) @ (sub @ X189 @ X187))) = ((sub @ (sK547 @ X184 @ X188 @ X189) @ X187))) & ! [X190 : term] : (($true != ((X184 @ id @ X190 @ id))) | ($true = ((X184 @ id @ (sK547 @ X184 @ id @ X190) @ id))))) | ($true = ((sP39 @ X184)))) | (hoasinduction_no_psi_cond != $true)) & (! [X191 : term,X192 : term] : ((((lam @ X192)) != ((lam @ X191))) | (X191 = X192)) | (laminj != $true)) & ((laminj = $true) | ((((lam @ sK548)) = ((lam @ sK549))) & (sK548 != sK549))) & (pushprop_lthm = $true) & ((hoasapinj2_lthm = $true) | (((((ap @ (sub @ sK552 @ id) @ sK553)) = ((ap @ (sub @ sK551 @ id) @ sK550))) & (sK553 != sK550)) & ! [X199 : term,X200 : term,X201 : term,X202 : term] : ((X200 = X202) | (((ap @ X201 @ X202)) != ((ap @ X199 @ X200)))))) & (! [X203 : term,X204 : term,X205 : term,X206 : term] : ((((ap @ (sub @ X205 @ id) @ X206)) != ((ap @ (sub @ X204 @ id) @ X203))) | (X203 = X206)) | ((sK555 != sK557) & (((ap @ sK554 @ sK555)) = ((ap @ sK556 @ sK557)))) | (hoasapinj2_lthm != $true)) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((hoasinduction_lem3 = $true) | ($true != sP38)) & (($true = sP38) | (hoasinduction_lem3 != $true)) & ((pushprop_lem0_lthm = $true) | ! [X214 : (term > $o)] : (((sK558 @ (sub @ (sK561 @ X214) @ (push @ sK560 @ sK559)))) != ((X214 @ (sK561 @ X214))))) & (! [X216 : (term > $o),X217 : subst,X218 : term] : ! [X220 : term] : (((X216 @ (sub @ X220 @ (push @ X218 @ X217)))) = ((sK562 @ X218 @ X217 @ X216 @ X220))) | (pushprop_lem0_lthm != $true)) & (! [X221 : subst] : (((comp @ id @ X221)) = X221) | (axidl != $true)) & ((axidl = $true) | (((comp @ id @ sK563)) != sK563)) & (! [X223 : subst,X224 : (term > $o),X225 : term] : ! [X227 : term] : (((X224 @ (sub @ X227 @ (push @ X225 @ X223)))) = ((sK564 @ X225 @ X224 @ X223 @ X227))) | (pushprop_lem0 != $true)) & ((pushprop_lem0 = $true) | ! [X231 : (term > $o)] : (((sK566 @ (sub @ (sK568 @ X231) @ (push @ sK567 @ sK565)))) != ((X231 @ (sK568 @ X231))))) & (pushprop_lem0_gthm = $true) & ((hoasinduction_lem3_lthm = $true) | ($true != sP36)) & (($true = sP36) | (hoasinduction_lem3_lthm != $true)) & (($true = sP33) | (hoasinduction_no_psi_cond_lthm != $true)) & ((hoasinduction_no_psi_cond_lthm = $true) | ($true != sP33)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (hoasapinj2_gthm = $true) & (one = ((d2term @ d_one))) & (pushprop_gthm = $true) & (hoasinduction_lem1_lthm = $true) & ((hoasinduction_lem3v2a_lthm = $true) | (sP28 != $true)) & ((sP28 = $true) | (hoasinduction_lem3v2a_lthm != $true)) & ((substmonoid = $true) | (((comp @ sK569 @ id)) != sK569) | (sK570 != ((comp @ id @ sK570)))) & ((! [X235 : subst] : (((comp @ X235 @ id)) = X235) & ! [X236 : subst] : (((comp @ id @ X236)) = X236)) | (substmonoid != $true)) & (hoasapnotvar_lthm = $true) & (pushprop_lem1_gthm = $true) & (hoaslaminj_gthm = $true) & (pushprop_lem2v2_gthm = $true) & (hoasinduction_lem2v2_gthm = $true) & (($true = sP27) | (induction2 != $true)) & ((induction2 = $true) | ($true != sP27)) & ((hoasinduction_lem0_lthm = $true) | ! [X238 : (term > $o)] : (((X238 @ (sK572 @ X238))) != ((sK571 @ id @ (sK572 @ X238) @ id)))) & (! [X240 : (subst > term > subst > $o)] : ! [X242 : term] : (((X240 @ id @ X242 @ id)) = ((sK573 @ X240 @ X242))) | (hoasinduction_lem0_lthm != $true)) & (hoasinduction_lem2_gthm = $true) & ((hoasinduction_lem2v2 = $true) | ($true != sP25)) & (($true = sP25) | (hoasinduction_lem2v2 != $true)) & ((hoasinduction_lthm_3 = $true) | (sP23 != $true)) & ((sP23 = $true) | (hoasinduction_lthm_3 != $true)) & (axvarshift = $true) & ((sP17 = $true) | (hoasinduction_lem3v2 != $true)) & ((hoasinduction_lem3v2 = $true) | (sP17 != $true)) & ((axscons = $true) | (((push @ (sub @ one @ sK574) @ (comp @ sh @ sK574))) != sK574)) & (! [X244 : subst] : (((push @ (sub @ one @ X244) @ (comp @ sh @ X244))) = X244) | (axscons != $true)) & ((sP15 = $true) | (hoasinduction_lthm != $true)) & ((hoasinduction_lthm = $true) | (sP15 != $true)) & (hoasinduction_gthm = $true) & ((termmset = $true) | (((sub @ sK575 @ id)) != sK575)) & (! [X246 : term] : (((sub @ X246 @ id)) = X246) | (termmset != $true)) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (($true = sP6) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | ($true != sP6)) & (((((ap @ sK577 @ sK579)) = ((ap @ sK578 @ sK576))) & (sK578 != sK577)) | (((sub @ sK580 @ id)) != sK580) | ! [X252 : term,X253 : term,X254 : term,X255 : term] : ((X254 = X255) | (((ap @ (sub @ X254 @ id) @ X252)) != ((ap @ (sub @ X255 @ id) @ X253)))) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | (! [X256 : term,X257 : term,X258 : term,X259 : term] : ((((ap @ X257 @ X259)) != ((ap @ X258 @ X256))) | (X257 = X258)) & ! [X260 : term] : (((sub @ X260 @ id)) = X260) & ((sK583 != sK584) & (((ap @ (sub @ sK583 @ id) @ sK581)) = ((ap @ (sub @ sK584 @ id) @ sK582)))))) & (ulamvar1 != $true) & ((hoasinduction_lem3a_lthm = $true) | (sP5 != $true)) & ((sP5 = $true) | (hoasinduction_lem3a_lthm != $true)) & (axassoc = $true) & (! [X265 : term,X266 : term] : ((X265 = X266) | (((sub @ X265 @ sh)) != ((sub @ X266 @ sh)))) | (shinj != $true)) & ((shinj = $true) | ((sK585 != sK586) & (((sub @ sK586 @ sh)) = ((sub @ sK585 @ sh))))) & (hoaslamnotap_lthm = $true) & (pushprop_lem2v2_lthm = $true) & ((sP3 = $true) | (hoasinduction != $true)) & ((hoasinduction = $true) | (sP3 != $true)) & ! [X269 : d_term,X270 : d_term] : ((((d2term @ X270)) != ((d2term @ X269))) | (X269 = X270)) & (hoasinduction_lem1_gthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & ((! [X271 : subst] : (((comp @ id @ X271)) = X271) & ! [X272 : subst] : (((comp @ X272 @ id)) = X272)) | (sK587 != ((comp @ sK587 @ id))) | (sK588 != ((comp @ id @ sK588))) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | (((((comp @ id @ sK589)) != sK589) | (sK590 != ((comp @ sK590 @ id)))) & ! [X277 : subst] : (((comp @ X277 @ id)) = X277) & ! [X278 : subst] : (((comp @ id @ X278)) = X278))) & ! [X279 : d_subst] : (d_id = X279) & (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_lem3aa = $true) | ($true != sP0)) & (($true = sP0) | (hoasinduction_lem3aa != $true)) & (((var @ (d2term @ d_one))) != $true)),
  inference(skolemize,[status(esa),new_symbols(skolem,[sK454,vAPP,sK456,sK457,sK458,sK459,sK460,sK461,sK462,vAPP,sK464,vAPP,sK467,sK468,sK469,sK470,vAPP,sK473,sK474,sK475,vAPP,sK479,sK480,sK481,sK482,sK483,sK484,sK485,sK486,vAPP,sK493,sK494,sK495,sK496,sK497,sK498,sK499,sK500,sK501,sK502,sK503,vAPP,sK507,sK508,sK509,sK510,sK511,sK512,sK513,sK514,sK515,sK516,sK517,sK518,sK519,sK520,sK521,sK522,sK523,sK524,sK525,vAPP,sK527,vAPP,sK530,sK531,sK532,sK533,sK534,sK535,sK536,vAPP,sK541,sK542,vAPP,sK548,sK549,sK550,sK551,sK552,sK553,sK554,sK555,sK556,sK557,sK558,sK559,sK560,vAPP,sK563,vAPP,sK565,sK566,sK567,vAPP,sK569,sK570,sK571,vAPP,sK574,sK575,sK576,sK577,sK578,sK579,sK580,sK581,sK582,sK583,sK584,sK585,sK586,sK587,sK588,sK589,sK590]),skolemize(X0,$thf(sK454)),skolemize(X12,$thf(sK64 @ X11)),skolemize(X9,$thf(sK456)),skolemize(X10,$thf(sK457)),skolemize(X11,$thf(sK458)),skolemize(X12,$thf(sK459)),skolemize(X13,$thf(sK460)),skolemize(X16,$thf(sK461)),skolemize(X17,$thf(sK462)),skolemize(X12,$thf(sK64 @ X11)),skolemize(X22,$thf(sK464)),skolemize(X12,$thf(sK64 @ X11)),skolemize(X12,$thf(sK64 @ X11)),skolemize(X26,$thf(sK467)),skolemize(X27,$thf(sK468)),skolemize(X28,$thf(sK469)),skolemize(X29,$thf(sK470)),skolemize(X12,$thf(sK64 @ X11)),skolemize(X12,$thf(sK64 @ X11)),skolemize(X38,$thf(sK473)),skolemize(X39,$thf(sK474)),skolemize(X44,$thf(sK475)),skolemize(X12,$thf(sK64 @ X11)),skolemize(X12,$thf(sK64 @ X11)),skolemize(X12,$thf(sK64 @ X11)),skolemize(X52,$thf(sK479)),skolemize(X53,$thf(sK480)),skolemize(X54,$thf(sK481)),skolemize(X55,$thf(sK482)),skolemize(X60,$thf(sK483)),skolemize(X61,$thf(sK484)),skolemize(X62,$thf(sK485)),skolemize(X63,$thf(sK486)),skolemize(X12,$thf(sK64 @ X11)),skolemize(X12,$thf(sK64 @ X11)),skolemize(X12,$thf(sK64 @ X11)),skolemize(X12,$thf(sK64 @ X11)),skolemize(X12,$thf(sK64 @ X11)),skolemize(X12,$thf(sK64 @ X11)),skolemize(X80,$thf(sK493)),skolemize(X85,$thf(sK494)),skolemize(X86,$thf(sK495)),skolemize(X88,$thf(sK496)),skolemize(X89,$thf(sK497)),skolemize(X90,$thf(sK498)),skolemize(X91,$thf(sK499)),skolemize(X92,$thf(sK500)),skolemize(X97,$thf(sK501)),skolemize(X98,$thf(sK502)),skolemize(X99,$thf(sK503)),skolemize(X12,$thf(sK64 @ X11)),skolemize(X12,$thf(sK64 @ X11)),skolemize(X12,$thf(sK64 @ X11)),skolemize(X109,$thf(sK507)),skolemize(X112,$thf(sK508)),skolemize(X113,$thf(sK509)),skolemize(X114,$thf(sK510)),skolemize(X120,$thf(sK511)),skolemize(X121,$thf(sK512)),skolemize(X124,$thf(sK513)),skolemize(X125,$thf(sK514)),skolemize(X126,$thf(sK515)),skolemize(X127,$thf(sK516)),skolemize(X128,$thf(sK517)),skolemize(X131,$thf(sK518)),skolemize(X132,$thf(sK519)),skolemize(X133,$thf(sK520)),skolemize(X134,$thf(sK521)),skolemize(X139,$thf(sK522)),skolemize(X140,$thf(sK523)),skolemize(X141,$thf(sK524)),skolemize(X142,$thf(sK525)),skolemize(X12,$thf(sK64 @ X11)),skolemize(X149,$thf(sK527)),skolemize(X12,$thf(sK64 @ X11)),skolemize(X12,$thf(sK64 @ X11)),skolemize(X158,$thf(sK530)),skolemize(X159,$thf(sK531)),skolemize(X160,$thf(sK532)),skolemize(X161,$thf(sK533)),skolemize(X162,$thf(sK534)),skolemize(X163,$thf(sK535)),skolemize(X167,$thf(sK536)),skolemize(X12,$thf(sK64 @ X11)),skolemize(X12,$thf(sK64 @ X11)),skolemize(X12,$thf(sK64 @ X11)),skolemize(X12,$thf(sK64 @ X11)),skolemize(X177,$thf(sK541)),skolemize(X178,$thf(sK542)),skolemize(X12,$thf(sK64 @ X11)),skolemize(X12,$thf(sK64 @ X11)),skolemize(X12,$thf(sK64 @ X11)),skolemize(X12,$thf(sK64 @ X11)),skolemize(X12,$thf(sK64 @ X11)),skolemize(X193,$thf(sK548)),skolemize(X194,$thf(sK549)),skolemize(X195,$thf(sK550)),skolemize(X196,$thf(sK551)),skolemize(X197,$thf(sK552)),skolemize(X198,$thf(sK553)),skolemize(X207,$thf(sK554)),skolemize(X208,$thf(sK555)),skolemize(X209,$thf(sK556)),skolemize(X210,$thf(sK557)),skolemize(X211,$thf(sK558)),skolemize(X212,$thf(sK559)),skolemize(X213,$thf(sK560)),skolemize(X12,$thf(sK64 @ X11)),skolemize(X12,$thf(sK64 @ X11)),skolemize(X222,$thf(sK563)),skolemize(X12,$thf(sK64 @ X11)),skolemize(X228,$thf(sK565)),skolemize(X229,$thf(sK566)),skolemize(X230,$thf(sK567)),skolemize(X12,$thf(sK64 @ X11)),skolemize(X233,$thf(sK569)),skolemize(X234,$thf(sK570)),skolemize(X237,$thf(sK571)),skolemize(X12,$thf(sK64 @ X11)),skolemize(X12,$thf(sK64 @ X11)),skolemize(X243,$thf(sK574)),skolemize(X245,$thf(sK575)),skolemize(X247,$thf(sK576)),skolemize(X248,$thf(sK577)),skolemize(X249,$thf(sK578)),skolemize(X250,$thf(sK579)),skolemize(X251,$thf(sK580)),skolemize(X261,$thf(sK581)),skolemize(X262,$thf(sK582)),skolemize(X263,$thf(sK583)),skolemize(X264,$thf(sK584)),skolemize(X267,$thf(sK585)),skolemize(X268,$thf(sK586)),skolemize(X273,$thf(sK587)),skolemize(X274,$thf(sK588)),skolemize(X275,$thf(sK589)),skolemize(X276,$thf(sK590))],[f262])).
thf(f752,plain,(
  (((var @ (d2term @ d_one))) != $true)),
  inference(cnf_transformation,[],[f263])).
thf(f756,plain,(
  ( ! [X279 : d_subst] : ((d_id = X279)) )),
  inference(cnf_transformation,[],[f263])).
thf(f816,plain,(
  (one = ((d2term @ d_one)))),
  inference(cnf_transformation,[],[f263])).
thf(f850,plain,(
  ( ! [X175 : term] : ((((d2term @ (sK540 @ X175))) = X175)) )),
  inference(cnf_transformation,[],[f263])).
thf(f906,plain,(
  ( ! [X82 : d_term] : ((d_one = X82)) )),
  inference(cnf_transformation,[],[f263])).
thf(f921,plain,(
  (((d2subst @ d_id)) = sh)),
  inference(cnf_transformation,[],[f263])).
thf(f929,plain,(
  (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id))))))))),
  inference(cnf_transformation,[],[f263])).
thf(f932,plain,(
  ( ! [X50 : subst] : ((((d2subst @ (sK478 @ X50))) = X50)) )),
  inference(cnf_transformation,[],[f263])).
thf(f935,plain,(
  ( ! [X45 : (term > $o)] : ((hoasinduction_lem0 = $true) | (((X45 @ (sK476 @ X45))) != ((sK475 @ id @ (sK476 @ X45) @ id)))) )),
  inference(cnf_transformation,[],[f263])).
thf(f965,plain,(
  (((sK460 @ sK459)) = $true) | (pushprop_lem1 = $true)),
  inference(cnf_transformation,[],[f263])).
thf(f966,plain,(
  ( ! [X8 : (term > $o),X6 : subst,X7 : term,X4 : term,X5 : (term > $o)] : (($true = ((X5 @ (sub @ X7 @ (push @ X4 @ X6))))) | (pushprop_lem1 != $true) | ($true != ((X8 @ X7)))) )),
  inference(cnf_transformation,[],[f263])).
thf(f978,plain,(
  (hoasinduction_lem0 != ((!! @ (subst > term > subst > $o) @ (^[Y0 : subst > term > subst > $o]: (?? @ (term > $o) @ (^[Y1 : term > $o]: (hoasinduction_p_and_p_prime @ Y0 @ Y1)))))))),
  inference(cnf_transformation,[],[f10])).
thf(f981,plain,(
  (hoasinduction_lem0 != ((!! @ (subst > term > subst > $o) @ (^[Y0 : subst > term > subst > $o]: (?? @ (term > $o) @ (^[Y1 : term > $o]: ((^[Y2 : subst > term > subst > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ id @ Y4 @ id))))))) @ Y0 @ Y1)))))))),
  inference(definition_unfolding,[],[f978,f929])).
thf(f982,definition,(
  (sF591 = ((!! @ (subst > term > subst > $o) @ (^[Y0 : subst > term > subst > $o]: (?? @ (term > $o) @ (^[Y1 : term > $o]: ((^[Y2 : subst > term > subst > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ id @ Y4 @ id))))))) @ Y0 @ Y1)))))))),
  introduced(definition,[new_symbols(definition,[sF591])],[function_definition])).
thf(f983,plain,(
  (((!! @ (subst > term > subst > $o) @ (^[Y0 : subst > term > subst > $o]: (?? @ (term > $o) @ (^[Y1 : term > $o]: ((^[Y2 : subst > term > subst > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ id @ Y4 @ id))))))) @ Y0 @ Y1)))))) = sF591)),
  inference(reorient_equations,[],[f982])).
thf(f984,plain,(
  (hoasinduction_lem0 != sF591)),
  inference(definition_folding,[],[f981,f983])).
thf(f993,plain,(
  ( ! [X45 : (term > $o)] : ((((X45 @ (sK476 @ X45))) = $true) | ($true = ((sK475 @ id @ (sK476 @ X45) @ id))) | (hoasinduction_lem0 = $true)) )),
  inference(xor_proxy_clausification,[],[f935])).
thf(f994,plain,(
  ( ! [X45 : (term > $o)] : ((((X45 @ (sK476 @ X45))) = $false) | ($false = ((sK475 @ id @ (sK476 @ X45) @ id))) | (hoasinduction_lem0 = $true)) )),
  inference(xor_proxy_clausification,[],[f935])).
thf(f1002,plain,(
  ($true = sF591) | (((!! @ (subst > term > subst > $o) @ (^[Y0 : subst > term > subst > $o]: (?? @ (term > $o) @ (^[Y1 : term > $o]: ((^[Y2 : subst > term > subst > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ id @ Y4 @ id))))))) @ Y0 @ Y1)))))) = $false)),
  inference(iff_proxy_clausification,[],[f983])).
thf(f1003,plain,(
  ((((^[Y0 : subst > term > subst > $o]: (?? @ (term > $o) @ (^[Y1 : term > $o]: ((^[Y2 : subst > term > subst > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ id @ Y4 @ id))))))) @ Y0 @ Y1)))) @ sK592)) = $false) | ($true = sF591)),
  inference(sigma_proxy_clausification,[],[f1002])).
thf(f1004,plain,(
  (((?? @ (term > $o) @ (^[Y0 : term > $o]: (!! @ term @ (^[Y1 : term]: ((Y0 @ Y1) = (sK592 @ id @ Y1 @ id))))))) = $false) | ($true = sF591)),
  inference(beta-eta_normalization,[],[f1003])).
thf(f1005,plain,(
  ( ! [X1 : (term > $o)] : (($true = sF591) | ((((^[Y0 : term > $o]: (!! @ term @ (^[Y1 : term]: ((Y0 @ Y1) = (sK592 @ id @ Y1 @ id))))) @ X1)) = $false)) )),
  inference(pi_proxy_clausification,[],[f1004])).
thf(f1006,plain,(
  ( ! [X1 : (term > $o)] : ((((!! @ term @ (^[Y0 : term]: ((X1 @ Y0) = (sK592 @ id @ Y0 @ id))))) = $false) | ($true = sF591)) )),
  inference(beta-eta_normalization,[],[f1005])).
thf(f1007,plain,(
  ( ! [X1 : (term > $o)] : (($true = sF591) | ((((^[Y0 : term]: ((X1 @ Y0) = (sK592 @ id @ Y0 @ id))) @ (sK593 @ X1))) = $false)) )),
  inference(sigma_proxy_clausification,[],[f1006])).
thf(f1008,plain,(
  ( ! [X1 : (term > $o)] : (($true = sF591) | ((((X1 @ (sK593 @ X1)) = (sK592 @ id @ (sK593 @ X1) @ id))) = $false)) )),
  inference(beta-eta_normalization,[],[f1007])).
thf(f1009,plain,(
  ( ! [X1 : (term > $o)] : ((((sK592 @ id @ (sK593 @ X1) @ id)) != ((X1 @ (sK593 @ X1)))) | ($true = sF591)) )),
  inference(equality_proxy_clausification,[],[f1008])).
thf(f1010,plain,(
  ( ! [X1 : (term > $o)] : (($true = ((sK592 @ id @ (sK593 @ X1) @ id))) | ($true = sF591) | ($true = ((X1 @ (sK593 @ X1))))) )),
  inference(xor_proxy_clausification,[],[f1009])).
thf(f1011,plain,(
  ( ! [X1 : (term > $o)] : ((((X1 @ (sK593 @ X1))) = $false) | ($true = sF591) | (((sK592 @ id @ (sK593 @ X1) @ id)) = $false)) )),
  inference(xor_proxy_clausification,[],[f1009])).
thf(f1062,plain,(
  (sF591 = $false) | (hoasinduction_lem0 = $false)),
  inference(xor_proxy_clausification,[],[f984])).
thf(f1249,definition,(
  spl595_25 <=> (hoasinduction_lem0 = $true)),
  introduced(definition,[new_symbols(definition,[spl595_25])],[avatar_definition])).
thf(f1251,plain,(
  (hoasinduction_lem0 = $true) | ~spl595_25),
  inference(avatar_component_clause,[],[f1249])).
thf(f1253,definition,(
  spl595_26 <=> ! [X45 : (term > $o)] : ((((X45 @ (sK476 @ X45))) = $false) | ($false = ((sK475 @ id @ (sK476 @ X45) @ id))))),
  introduced(definition,[new_symbols(definition,[spl595_26])],[avatar_definition])).
thf(f1254,plain,(
  ( ! [X45 : (term > $o)] : (($false = ((sK475 @ id @ (sK476 @ X45) @ id))) | (((X45 @ (sK476 @ X45))) = $false)) ) | ~spl595_26),
  inference(avatar_component_clause,[],[f1253])).
thf(f1255,plain,(
  spl595_25 | spl595_26),
  inference(avatar_split_clause,[],[f994,f1253,f1249])).
thf(f1257,definition,(
  spl595_27 <=> ! [X45 : (term > $o)] : ((((X45 @ (sK476 @ X45))) = $true) | ($true = ((sK475 @ id @ (sK476 @ X45) @ id))))),
  introduced(definition,[new_symbols(definition,[spl595_27])],[avatar_definition])).
thf(f1258,plain,(
  ( ! [X45 : (term > $o)] : (($true = ((sK475 @ id @ (sK476 @ X45) @ id))) | (((X45 @ (sK476 @ X45))) = $true)) ) | ~spl595_27),
  inference(avatar_component_clause,[],[f1257])).
thf(f1259,plain,(
  spl595_27 | spl595_25),
  inference(avatar_split_clause,[],[f993,f1249,f1257])).
thf(f1396,definition,(
  spl595_60 <=> ($true = sF591)),
  introduced(definition,[new_symbols(definition,[spl595_60])],[avatar_definition])).
thf(f1398,plain,(
  ($true = sF591) | ~spl595_60),
  inference(avatar_component_clause,[],[f1396])).
thf(f1400,definition,(
  spl595_61 <=> ! [X1 : (term > $o)] : ((((X1 @ (sK593 @ X1))) = $false) | (((sK592 @ id @ (sK593 @ X1) @ id)) = $false))),
  introduced(definition,[new_symbols(definition,[spl595_61])],[avatar_definition])).
thf(f1401,plain,(
  ( ! [X1 : (term > $o)] : ((((sK592 @ id @ (sK593 @ X1) @ id)) = $false) | (((X1 @ (sK593 @ X1))) = $false)) ) | ~spl595_61),
  inference(avatar_component_clause,[],[f1400])).
thf(f1402,plain,(
  spl595_60 | spl595_61),
  inference(avatar_split_clause,[],[f1011,f1400,f1396])).
thf(f1404,definition,(
  spl595_62 <=> ! [X1 : (term > $o)] : (($true = ((sK592 @ id @ (sK593 @ X1) @ id))) | ($true = ((X1 @ (sK593 @ X1)))))),
  introduced(definition,[new_symbols(definition,[spl595_62])],[avatar_definition])).
thf(f1405,plain,(
  ( ! [X1 : (term > $o)] : (($true = ((sK592 @ id @ (sK593 @ X1) @ id))) | ($true = ((X1 @ (sK593 @ X1))))) ) | ~spl595_62),
  inference(avatar_component_clause,[],[f1404])).
thf(f1406,plain,(
  spl595_60 | spl595_62),
  inference(avatar_split_clause,[],[f1010,f1404,f1396])).
thf(f1408,definition,(
  spl595_63 <=> (sF591 = $false)),
  introduced(definition,[new_symbols(definition,[spl595_63])],[avatar_definition])).
thf(f1410,plain,(
  (sF591 = $false) | ~spl595_63),
  inference(avatar_component_clause,[],[f1408])).
thf(f2339,definition,(
  spl595_279 <=> (hoasinduction_lem0 = $false)),
  introduced(definition,[new_symbols(definition,[spl595_279])],[avatar_definition])).
thf(f2341,plain,(
  (hoasinduction_lem0 = $false) | ~spl595_279),
  inference(avatar_component_clause,[],[f2339])).
thf(f2342,plain,(
  spl595_63 | spl595_279),
  inference(avatar_split_clause,[],[f1062,f2339,f1408])).
thf(f2699,definition,(
  spl595_362 <=> (pushprop_lem1 = $true)),
  introduced(definition,[new_symbols(definition,[spl595_362])],[avatar_definition])).
thf(f3065,definition,(
  spl595_447 <=> ! [X119 : (term > $o),X115 : term,X118 : (term > $o),X117 : term,X116 : subst] : ((((X118 @ (sub @ X115 @ (push @ X117 @ X116)))) = $true) | ($true != ((X119 @ X115))))),
  introduced(definition,[new_symbols(definition,[spl595_447])],[avatar_definition])).
thf(f3066,plain,(
  ( ! [X118 : (term > $o),X119 : (term > $o),X116 : subst,X117 : term,X115 : term] : ((((X118 @ (sub @ X115 @ (push @ X117 @ X116)))) = $true) | ($true != ((X119 @ X115)))) ) | ~spl595_447),
  inference(avatar_component_clause,[],[f3065])).
thf(f3095,plain,(
  ~spl595_362 | spl595_447),
  inference(avatar_split_clause,[],[f966,f3065,f2699])).
thf(f3427,definition,(
  spl595_529 <=> (((sK460 @ sK459)) = $true)),
  introduced(definition,[new_symbols(definition,[spl595_529])],[avatar_definition])).
thf(f3429,plain,(
  (((sK460 @ sK459)) = $true) | ~spl595_529),
  inference(avatar_component_clause,[],[f3427])).
thf(f3430,plain,(
  spl595_362 | spl595_529),
  inference(avatar_split_clause,[],[f965,f3427,f2699])).
thf(f3886,plain,(
  ( ! [X0 : d_term] : ((one = ((d2term @ X0)))) )),
  inference(superposition,[],[f816,f906])).
thf(f3887,plain,(
  ( ! [X0 : d_subst] : ((sh = ((d2subst @ X0)))) )),
  inference(superposition,[],[f921,f756])).
thf(f3896,plain,(
  ($true = $false) | (~spl595_60 | ~spl595_63)),
  inference(superposition,[],[f1410,f1398])).
thf(f3900,plain,(
  $false | (~spl595_60 | ~spl595_63)),
  inference(trivial_inequality_removal,[],[f3896])).
thf(f3901,plain,(
  ~spl595_60 | ~spl595_63),
  inference(avatar_contradiction_clause,[],[f3900])).
thf(f3905,plain,(
  ($true != ((var @ one)))),
  inference(forward_demodulation,[],[f752,f816])).
thf(f3906,plain,(
  ( ! [X175 : term] : ((one = X175)) )),
  inference(forward_demodulation,[],[f850,f3886])).
thf(f3907,plain,(
  ( ! [X0 : term,X1 : term] : ((X0 = X1)) )),
  inference(superposition,[],[f3906,f3906])).
thf(f3940,plain,(
  ( ! [X50 : subst] : ((sh = X50)) )),
  inference(forward_demodulation,[],[f932,f3887])).
thf(f4085,definition,(
  spl595_637 <=> ! [X22 : (term > $o)] : ($true = ((X22 @ one)))),
  introduced(definition,[new_symbols(definition,[spl595_637])],[avatar_definition])).
thf(f4086,plain,(
  ( ! [X22 : (term > $o)] : (($true = ((X22 @ one)))) ) | ~spl595_637),
  inference(avatar_component_clause,[],[f4085])).
thf(f4345,plain,(
  ($true != $true) | ~spl595_637),
  inference(superposition,[],[f3905,f4086])).
thf(f4389,plain,(
  $false | ~spl595_637),
  inference(trivial_inequality_removal,[],[f4345])).
thf(f4390,plain,(
  ~spl595_637),
  inference(avatar_contradiction_clause,[],[f4389])).
thf(f4901,plain,(
  ( ! [X118 : (term > $o),X119 : (term > $o),X115 : term] : (($true != ((X119 @ X115))) | (((X118 @ one)) = $true)) ) | ~spl595_447),
  inference(forward_demodulation,[],[f3066,f3906])).
thf(f4903,definition,(
  spl595_649 <=> ! [X119 : (term > $o),X115 : term] : ($true != ((X119 @ X115)))),
  introduced(definition,[new_symbols(definition,[spl595_649])],[avatar_definition])).
thf(f4904,plain,(
  ( ! [X119 : (term > $o),X115 : term] : (($true != ((X119 @ X115)))) ) | ~spl595_649),
  inference(avatar_component_clause,[],[f4903])).
thf(f4905,plain,(
  spl595_649 | spl595_637 | ~spl595_447),
  inference(avatar_split_clause,[],[f4901,f3065,f4085,f4903])).
thf(f4906,plain,(
  ( ! [X115 : term] : (($true != (((^[Y0 : term]: ($true)) @ X115)))) ) | ~spl595_649),
  inference(imitation,[],[f4904])).
thf(f4924,plain,(
  ($true != $true) | ~spl595_649),
  inference(beta-eta_normalization,[],[f4906])).
thf(f4925,plain,(
  $false | ~spl595_649),
  inference(trivial_inequality_removal,[],[f4924])).
thf(f4926,plain,(
  ~spl595_649),
  inference(avatar_contradiction_clause,[],[f4925])).
thf(f4966,plain,(
  ( ! [X0 : term] : (($true = ((sK460 @ X0)))) ) | ~spl595_529),
  inference(superposition,[],[f3429,f3907])).
thf(f4974,definition,(
  spl595_653 <=> ! [X0 : term] : ($true = ((sK460 @ X0)))),
  introduced(definition,[new_symbols(definition,[spl595_653])],[avatar_definition])).
thf(f4975,plain,(
  ( ! [X0 : term] : (($true = ((sK460 @ X0)))) ) | ~spl595_653),
  inference(avatar_component_clause,[],[f4974])).
thf(f4986,plain,(
  spl595_653 | ~spl595_529),
  inference(avatar_split_clause,[],[f4966,f3427,f4974])).
thf(f5791,plain,(
  ( ! [X45 : (term > $o)] : ((((sK475 @ sh @ (sK476 @ X45) @ sh)) = $false) | (((X45 @ (sK476 @ X45))) = $false)) ) | ~spl595_26),
  inference(forward_demodulation,[],[f1254,f3940])).
thf(f5792,plain,(
  ( ! [X45 : (term > $o)] : ((((sK475 @ sh @ one @ sh)) = $false) | (((X45 @ (sK476 @ X45))) = $false)) ) | ~spl595_26),
  inference(forward_demodulation,[],[f5791,f3906])).
thf(f5793,plain,(
  ( ! [X45 : (term > $o)] : ((((X45 @ one)) = $false) | (((sK475 @ sh @ one @ sh)) = $false)) ) | ~spl595_26),
  inference(forward_demodulation,[],[f5792,f3906])).
thf(f5795,definition,(
  spl595_656 <=> ! [X45 : (term > $o)] : (((X45 @ one)) = $false)),
  introduced(definition,[new_symbols(definition,[spl595_656])],[avatar_definition])).
thf(f5796,plain,(
  ( ! [X45 : (term > $o)] : ((((X45 @ one)) = $false)) ) | ~spl595_656),
  inference(avatar_component_clause,[],[f5795])).
thf(f5798,definition,(
  spl595_657 <=> (((sK475 @ sh @ one @ sh)) = $false)),
  introduced(definition,[new_symbols(definition,[spl595_657])],[avatar_definition])).
thf(f5800,plain,(
  (((sK475 @ sh @ one @ sh)) = $false) | ~spl595_657),
  inference(avatar_component_clause,[],[f5798])).
thf(f5801,plain,(
  spl595_656 | spl595_657 | ~spl595_26),
  inference(avatar_split_clause,[],[f5793,f1253,f5798,f5795])).
thf(f5811,plain,(
  ($true = $false) | (~spl595_653 | ~spl595_656)),
  inference(superposition,[],[f5796,f4975])).
thf(f5875,plain,(
  $false | (~spl595_653 | ~spl595_656)),
  inference(trivial_inequality_removal,[],[f5811])).
thf(f5876,plain,(
  ~spl595_653 | ~spl595_656),
  inference(avatar_contradiction_clause,[],[f5875])).
thf(f5891,plain,(
  ( ! [X0 : term] : ((((sK475 @ sh @ X0 @ sh)) = $false)) ) | ~spl595_657),
  inference(superposition,[],[f5800,f3906])).
thf(f5929,plain,(
  ( ! [X0 : subst,X1 : term] : ((((sK475 @ X0 @ X1 @ X0)) = $false)) ) | ~spl595_657),
  inference(superposition,[],[f5891,f3940])).
thf(f6031,plain,(
  ( ! [X45 : (term > $o)] : ((((X45 @ (sK476 @ X45))) = $true) | ($true = $false)) ) | (~spl595_27 | ~spl595_657)),
  inference(forward_demodulation,[],[f1258,f5929])).
thf(f6032,plain,(
  ( ! [X45 : (term > $o)] : ((((X45 @ (sK476 @ X45))) = $true)) ) | (~spl595_27 | ~spl595_657)),
  inference(trivial_inequality_removal,[],[f6031])).
thf(f6033,plain,(
  ( ! [X45 : (term > $o)] : ((((X45 @ one)) = $true)) ) | (~spl595_27 | ~spl595_657)),
  inference(forward_demodulation,[],[f6032,f3906])).
thf(f6034,plain,(
  spl595_637 | ~spl595_27 | ~spl595_657),
  inference(avatar_split_clause,[],[f6033,f5798,f1257,f4085])).
thf(f6038,plain,(
  ($true = $false) | (~spl595_25 | ~spl595_279)),
  inference(forward_demodulation,[],[f1251,f2341])).
thf(f6039,plain,(
  $false | (~spl595_25 | ~spl595_279)),
  inference(trivial_inequality_removal,[],[f6038])).
thf(f6040,plain,(
  ~spl595_25 | ~spl595_279),
  inference(avatar_contradiction_clause,[],[f6039])).
thf(f6048,plain,(
  ( ! [X1 : (term > $o)] : ((((X1 @ (sK593 @ X1))) = $false) | (((sK592 @ sh @ (sK593 @ X1) @ sh)) = $false)) ) | ~spl595_61),
  inference(forward_demodulation,[],[f1401,f3940])).
thf(f6049,plain,(
  ( ! [X1 : (term > $o)] : (($true = ((X1 @ (sK593 @ X1)))) | (((sK592 @ sh @ (sK593 @ X1) @ sh)) = $true)) ) | ~spl595_62),
  inference(forward_demodulation,[],[f1405,f3940])).
thf(f6050,plain,(
  ( ! [X1 : (term > $o)] : ((((sK592 @ sh @ (sK593 @ X1) @ sh)) = $false) | (((X1 @ one)) = $false)) ) | ~spl595_61),
  inference(forward_demodulation,[],[f6048,f3906])).
thf(f6051,plain,(
  ( ! [X1 : (term > $o)] : (($true = ((X1 @ one))) | (((sK592 @ sh @ (sK593 @ X1) @ sh)) = $true)) ) | ~spl595_62),
  inference(forward_demodulation,[],[f6049,f3906])).
thf(f6052,plain,(
  ( ! [X1 : (term > $o)] : ((((sK592 @ sh @ one @ sh)) = $false) | (((X1 @ one)) = $false)) ) | ~spl595_61),
  inference(forward_demodulation,[],[f6050,f3906])).
thf(f6053,plain,(
  ( ! [X1 : (term > $o)] : (($true = ((sK592 @ sh @ one @ sh))) | ($true = ((X1 @ one)))) ) | ~spl595_62),
  inference(forward_demodulation,[],[f6051,f3906])).
thf(f6055,definition,(
  spl595_659 <=> (((sK592 @ sh @ one @ sh)) = $false)),
  introduced(definition,[new_symbols(definition,[spl595_659])],[avatar_definition])).
thf(f6057,plain,(
  (((sK592 @ sh @ one @ sh)) = $false) | ~spl595_659),
  inference(avatar_component_clause,[],[f6055])).
thf(f6058,plain,(
  spl595_656 | spl595_659 | ~spl595_61),
  inference(avatar_split_clause,[],[f6052,f1400,f6055,f5795])).
thf(f6060,definition,(
  spl595_660 <=> ($true = ((sK592 @ sh @ one @ sh)))),
  introduced(definition,[new_symbols(definition,[spl595_660])],[avatar_definition])).
thf(f6062,plain,(
  ($true = ((sK592 @ sh @ one @ sh))) | ~spl595_660),
  inference(avatar_component_clause,[],[f6060])).
thf(f6063,plain,(
  spl595_660 | spl595_637 | ~spl595_62),
  inference(avatar_split_clause,[],[f6053,f1404,f4085,f6060])).
thf(f6125,plain,(
  ($true = $false) | (~spl595_659 | ~spl595_660)),
  inference(forward_demodulation,[],[f6062,f6057])).
thf(f6126,plain,(
  $false | (~spl595_659 | ~spl595_660)),
  inference(trivial_inequality_removal,[],[f6125])).
thf(f6127,plain,(
  ~spl595_659 | ~spl595_660),
  inference(avatar_contradiction_clause,[],[f6126])).
cnf(s14, plain, spl595_25 | spl595_26, inference(sat_conversion,[],[f1255])).
cnf(s15, plain, spl595_25 | spl595_27, inference(sat_conversion,[],[f1259])).
cnf(s36, plain, spl595_60 | spl595_61, inference(sat_conversion,[],[f1402])).
cnf(s37, plain, spl595_60 | spl595_62, inference(sat_conversion,[],[f1406])).
cnf(s221, plain, spl595_63 | spl595_279, inference(sat_conversion,[],[f2342])).
cnf(s404, plain, ~spl595_362 | spl595_447, inference(sat_conversion,[],[f3095])).
cnf(s486, plain, spl595_362 | spl595_529, inference(sat_conversion,[],[f3430])).
cnf(s610, plain, ~spl595_60 | ~spl595_63, inference(sat_conversion,[],[f3901])).
cnf(s670, plain, ~spl595_637, inference(sat_conversion,[],[f4390])).
cnf(s695, plain, ~spl595_447 | spl595_637 | spl595_649, inference(sat_conversion,[],[f4905])).
cnf(s700, plain, ~spl595_649, inference(sat_conversion,[],[f4926])).
cnf(s714, plain, ~spl595_529 | spl595_653, inference(sat_conversion,[],[f4986])).
cnf(s736, plain, ~spl595_26 | spl595_656 | spl595_657, inference(sat_conversion,[],[f5801])).
cnf(s747, plain, ~spl595_653 | ~spl595_656, inference(sat_conversion,[],[f5876])).
cnf(s755, plain, ~spl595_27 | spl595_637 | ~spl595_657, inference(sat_conversion,[],[f6034])).
cnf(s756, plain, ~spl595_25 | ~spl595_279, inference(sat_conversion,[],[f6040])).
cnf(s758, plain, ~spl595_61 | spl595_656 | spl595_659, inference(sat_conversion,[],[f6058])).
cnf(s759, plain, ~spl595_62 | spl595_637 | spl595_660, inference(sat_conversion,[],[f6063])).
cnf(s760, plain, ~spl595_659 | ~spl595_660, inference(sat_conversion,[],[f6127])).
cnf(s761, plain, ~spl595_447 | spl595_637, inference(rat,[],[s695,s700])).
cnf(s762, plain, ~spl595_447, inference(rat,[],[s761,s670])).
cnf(s789, plain, ~spl595_362, inference(rat,[],[s404,s762])).
cnf(s790, plain, spl595_529, inference(rat,[],[s486,s789])).
cnf(s791, plain, spl595_653, inference(rat,[],[s714,s790])).
cnf(s792, plain, ~spl595_656, inference(rat,[],[s747,s791])).
cnf(s861, plain, spl595_25, inference(rat,[],[s736,s755,s14,s15,s792,s670])).
cnf(s862, plain, ~spl595_279, inference(rat,[],[s756,s861])).
cnf(s865, plain, spl595_63, inference(rat,[],[s221,s862])).
cnf(s866, plain, ~spl595_60, inference(rat,[],[s610,s865])).
cnf(s867, plain, spl595_62, inference(rat,[],[s37,s866])).
cnf(s868, plain, spl595_61, inference(rat,[],[s36,s866])).
cnf(s869, plain, spl595_660, inference(rat,[],[s759,s670,s867])).
cnf(s870, plain, spl595_659, inference(rat,[],[s758,s792,s868])).
cnf(s871, plain, $false, inference(rat,[],[s760,s869,s870])).
thf(f6128,plain,(
  $false),
  inference(avatar_sat_refutation,[],[s871])).
% SZS output end Proof for theBenchmark