%----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) > $o))). thf(func_def_128, type, vIMP: ($o > $o > $o)). thf(func_def_129, type, db3: !>[X0: $tType]:(X0)). thf(func_def_130, type, db0: !>[X0: $tType]:(X0)). thf(func_def_131, type, db1: !>[X0: $tType]:(X0)). thf(func_def_132, type, db2: !>[X0: $tType]:(X0)). thf(func_def_133, type, vEQ: !>[X0: $tType]:((X0 > X0 > $o))). thf(func_def_134, type, vLAM: !>[X0: $tType, X1: $tType]:((X1) > (X0 > X1))). thf(func_def_137, type, db4: !>[X0: $tType]:(X0)). thf(func_def_138, type, sP0: $o). thf(func_def_139, type, sP1: ((term > $o) > $o)). thf(func_def_140, type, sP2: ((term > $o) > $o)). thf(func_def_141, type, sP3: $o). thf(func_def_142, type, sP4: $o). thf(func_def_143, type, sP5: ((subst > term > subst > $o) > $o)). thf(func_def_144, type, sP6: ((subst > term > subst > $o) > $o)). thf(func_def_145, type, sP7: $o). thf(func_def_146, type, sP8: ((subst > term > subst > $o) > $o)). thf(func_def_147, type, sP9: $o). thf(func_def_148, type, sP10: $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: $o). thf(func_def_155, type, sP17: ((subst > term > subst > $o) > $o)). thf(func_def_156, type, sP18: ((subst > term > subst > $o) > $o)). thf(func_def_157, type, sP19: ((subst > term > subst > $o) > $o)). thf(func_def_158, type, sP20: ((term > $o) > $o)). thf(func_def_159, type, sP21: ((subst > term > subst > $o) > $o)). thf(func_def_160, type, sP22: $o). thf(func_def_161, type, sP23: $o). thf(func_def_162, type, sP24: $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: ((subst > term > subst > $o) > $o)). thf(func_def_167, type, sP29: ((subst > term > subst > $o) > $o)). thf(func_def_168, type, sP30: $o). thf(func_def_169, type, sP31: ((subst > term > subst > $o) > $o)). thf(func_def_170, type, sP32: $o). thf(func_def_171, type, sP33: ((subst > term > subst > $o) > $o)). thf(func_def_172, type, sP34: $o). thf(func_def_173, type, sP35: $o). thf(func_def_174, type, sP36: ((subst > term > subst > $o) > $o)). thf(func_def_175, type, sP37: ((subst > term > subst > $o) > $o)). thf(func_def_176, type, sP38: ((term > $o) > $o)). thf(func_def_177, type, sP39: $o). thf(func_def_178, type, sP40: $o). thf(func_def_179, type, sP41: $o). thf(func_def_180, type, sP42: ((term > $o) > $o)). thf(func_def_181, type, sP43: ((term > $o) > $o)). thf(func_def_182, type, sP44: ((term > $o) > $o)). thf(func_def_183, type, sP45: $o). thf(func_def_184, type, sP46: $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: $o). thf(func_def_189, type, sP51: ((term > $o) > $o)). thf(func_def_190, type, sP52: $o). thf(func_def_191, type, sP53: ((term > $o) > $o)). thf(func_def_192, type, sP54: ((subst > term > subst > $o) > $o)). thf(func_def_193, type, sP55: $o). thf(func_def_194, type, sP56: $o). thf(func_def_195, type, sP57: $o). thf(func_def_196, type, sP58: $o). thf(func_def_197, type, sK59: term). thf(func_def_198, type, sK60: (subst > term > term)). thf(func_def_199, type, sK61: subst). thf(func_def_200, type, sK62: (subst > term > term)). thf(func_def_201, type, sK63: subst). thf(func_def_202, type, sK64: term). thf(func_def_203, type, sK65: term). thf(func_def_204, type, sK66: term). thf(func_def_205, type, sK67: term). thf(func_def_206, type, sK68: subst). thf(func_def_207, type, sK69: ((subst > term > term) > term)). thf(func_def_208, type, sK70: ((subst > term > term) > subst)). thf(func_def_209, type, sK71: ((subst > term > term) > subst)). thf(func_def_210, type, sK72: ((subst > term > term) > subst)). thf(func_def_211, type, sK73: ((subst > term > term) > term)). thf(func_def_212, type, sK74: ((subst > term > term) > subst)). thf(func_def_213, type, sK75: ((subst > term > subst > $o) > term > $o)). thf(func_def_214, type, sK76: term). thf(func_def_215, type, sK77: (subst > term > subst > $o)). thf(func_def_216, type, sK78: term). thf(func_def_217, type, sK79: (subst > term > subst > $o)). thf(func_def_218, type, sK80: ((term > $o) > term)). thf(func_def_219, type, sK81: ((subst > term > subst > $o) > term)). thf(func_def_220, type, sK82: ((subst > term > subst > $o) > term)). thf(func_def_221, type, sK83: ((term > $o) > term)). thf(func_def_222, type, sK84: ((term > $o) > term)). thf(func_def_223, type, sK85: (term > $o)). thf(func_def_224, type, sK86: term). thf(func_def_225, type, sK87: (term > term)). thf(func_def_226, type, sK88: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_227, type, sK89: ((term > $o) > term > term)). thf(func_def_228, type, sK90: ((term > $o) > (subst > term > subst > $o) > term)). thf(func_def_229, type, sK91: (subst > term > subst > $o)). thf(func_def_230, type, sK92: term). thf(func_def_231, type, sK93: (term > $o)). thf(func_def_232, type, sK94: ((subst > term > term) > subst)). thf(func_def_233, type, sK95: ((subst > term > term) > subst)). thf(func_def_234, type, sK96: ((subst > term > term) > term)). thf(func_def_235, type, sK97: ((subst > term > term) > term)). thf(func_def_236, type, sK98: ((subst > term > term) > (subst > term > subst > $o) > term)). thf(func_def_237, type, sK99: ((subst > term > term) > subst)). thf(func_def_238, type, sK100: ((subst > term > term) > term)). thf(func_def_239, type, sK101: ((subst > term > term) > subst)). thf(func_def_240, type, sK102: ((subst > term > subst > $o) > subst > term > 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). thf(func_def_244, type, sK106: (term > $o)). thf(func_def_245, type, sK107: (term > term)). thf(func_def_246, type, sK108: ((term > $o) > term)). thf(func_def_247, type, sK109: ((term > $o) > term)). thf(func_def_248, type, sK110: ((term > $o) > term)). thf(func_def_249, type, sK111: ((term > $o) > term)). thf(func_def_250, type, sK112: term). thf(func_def_251, type, sK113: (subst > term > subst > $o)). thf(func_def_252, type, sK114: ((subst > term > term) > term > term)). thf(func_def_253, type, sK115: ((subst > term > term) > term > subst)). thf(func_def_254, type, sK116: ((subst > term > term) > term > subst)). thf(func_def_255, type, sK117: ((subst > term > term) > term > term)). thf(func_def_256, type, sK118: ((subst > term > term) > term > subst)). thf(func_def_257, type, sK119: ((subst > term > term) > term > subst)). thf(func_def_258, type, sK120: ((subst > term > term) > term)). thf(func_def_259, type, sK121: ((subst > term > subst > $o) > term)). thf(func_def_260, type, sK122: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_261, type, sK123: ((subst > term > subst > $o) > term > term)). thf(func_def_262, type, sK124: term). thf(func_def_263, type, sK125: (term > $o)). thf(func_def_264, type, sK126: (subst > term > subst > $o)). thf(func_def_265, type, sK127: term). thf(func_def_266, type, sK128: ((subst > term > subst > $o) > subst)). thf(func_def_267, type, sK129: ((subst > term > subst > $o) > subst)). thf(func_def_268, type, sK130: ((subst > term > subst > $o) > term)). thf(func_def_269, type, sK131: ((subst > term > subst > $o) > subst)). thf(func_def_270, type, sK132: ((subst > term > subst > $o) > (term > $o) > term)). thf(func_def_271, type, sK133: ((subst > term > subst > $o) > term)). thf(func_def_272, type, sK134: ((subst > term > subst > $o) > subst)). thf(func_def_273, type, sK135: ((subst > term > subst > $o) > subst)). thf(func_def_274, type, sK136: ((subst > term > subst > $o) > subst)). thf(func_def_275, type, sK137: ((subst > term > subst > $o) > term)). thf(func_def_276, type, sK138: ((subst > term > subst > $o) > term)). thf(func_def_277, type, sK139: (term > subst > term > term)). thf(func_def_278, type, sK140: term). thf(func_def_279, type, sK141: (term > $o)). thf(func_def_280, type, sK142: (subst > term > subst > $o)). thf(func_def_281, type, sK143: ((subst > term > term) > term)). thf(func_def_282, type, sK144: ((subst > term > term) > subst)). thf(func_def_283, type, sK145: ((subst > term > term) > term)). thf(func_def_284, type, sK146: ((subst > term > term) > subst)). thf(func_def_285, type, sK147: term). thf(func_def_286, type, sK148: term). thf(func_def_287, type, sK149: ((subst > term > term) > term)). thf(func_def_288, type, sK150: ((subst > term > term) > subst)). thf(func_def_289, type, sK151: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_290, type, sK152: ((term > $o) > term > term)). thf(func_def_291, type, sK153: ((subst > term > subst > $o) > (term > $o) > term)). thf(func_def_292, type, sK154: (subst > (term > $o) > term)). thf(func_def_293, type, sK155: ((term > $o) > term)). thf(func_def_294, type, sK156: term). thf(func_def_295, type, sK157: (term > $o)). thf(func_def_296, type, sK158: term). thf(func_def_297, type, sK159: subst). thf(func_def_298, type, sK160: (term > term)). thf(func_def_299, type, sK161: (term > $o)). thf(func_def_300, type, sK162: term). thf(func_def_301, type, sK163: (term > term)). thf(func_def_302, type, sK164: ((term > $o) > term)). thf(func_def_303, type, sK165: ((term > $o) > term)). thf(func_def_304, type, sK166: ((term > $o) > term)). thf(func_def_305, type, sK167: ((term > $o) > term)). thf(func_def_306, type, sK168: ((term > $o) > term)). thf(func_def_307, type, sK169: ((term > $o) > term)). thf(func_def_308, type, sK170: ((term > $o) > term)). thf(func_def_309, type, sK171: ((term > $o) > term)). thf(func_def_310, type, sK172: ((subst > term > subst > $o) > term > $o)). thf(func_def_311, type, sK173: ((term > $o) > term)). thf(func_def_312, type, sK174: ((term > $o) > term)). thf(func_def_313, type, sK175: (subst > term > subst > $o)). thf(func_def_314, type, sK176: ((term > $o) > term)). thf(func_def_315, type, sK177: term). thf(func_def_316, type, sK178: (term > $o)). thf(func_def_317, type, sK179: term). thf(func_def_318, type, sK180: (term > term)). thf(func_def_319, type, sK181: ((subst > term > subst > $o) > (term > $o) > term)). thf(func_def_320, type, sK182: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_321, type, sK183: ((term > $o) > term > term)). thf(func_def_322, type, sK184: term). thf(func_def_323, type, sK185: (term > $o)). thf(func_def_324, type, sK186: (subst > term > subst > $o)). thf(func_def_325, type, sK187: ((subst > term > term) > term)). thf(func_def_326, type, sK188: ((subst > term > term) > subst)). thf(func_def_327, type, sK189: ((subst > term > term) > subst)). thf(func_def_328, type, sK190: ((subst > term > term) > term)). thf(func_def_329, type, sK191: (subst > term > subst > $o)). thf(func_def_330, type, sK192: term). thf(func_def_331, type, sK193: ((subst > term > subst > $o) > term)). thf(func_def_332, type, sK194: ((subst > term > subst > $o) > subst)). thf(func_def_333, type, sK195: ((subst > term > subst > $o) > subst)). thf(func_def_334, type, sK196: ((subst > term > subst > $o) > subst)). thf(func_def_335, type, sK197: ((subst > term > subst > $o) > term)). thf(func_def_336, type, sK198: ((subst > term > subst > $o) > subst)). thf(func_def_337, type, sK199: ((subst > term > subst > $o) > term)). thf(func_def_338, type, sK200: ((subst > term > subst > $o) > subst)). thf(func_def_339, type, sK201: ((subst > term > subst > $o) > subst)). thf(func_def_340, type, sK202: ((term > $o) > term)). thf(func_def_341, type, sK203: ((term > $o) > term)). thf(func_def_342, type, sK204: ((subst > term > term) > (subst > term > subst > $o) > term)). thf(func_def_343, type, sK205: ((subst > term > term) > term)). thf(func_def_344, type, sK206: ((subst > term > term) > subst)). thf(func_def_345, type, sK207: ((subst > term > term) > subst)). thf(func_def_346, type, sK208: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_347, type, sK209: ((subst > term > subst > $o) > term)). thf(func_def_348, type, sK210: ((subst > term > subst > $o) > term)). thf(func_def_349, type, sK211: 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 > subst > $o) > term > term)). thf(func_def_353, type, sK215: ((subst > term > subst > $o) > term)). thf(func_def_354, type, sK216: ((subst > term > subst > $o) > subst)). thf(func_def_355, type, sK217: ((subst > term > subst > $o) > subst)). thf(func_def_356, type, sK218: ((subst > term > subst > $o) > subst)). thf(func_def_357, type, sK219: ((subst > term > subst > $o) > term)). thf(func_def_358, type, sK220: ((subst > term > subst > $o) > subst)). thf(func_def_359, type, sK221: ((subst > term > subst > $o) > subst)). thf(func_def_360, type, sK222: ((subst > term > subst > $o) > subst)). thf(func_def_361, type, sK223: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_362, type, sK224: (term > (subst > term > subst > $o) > term)). thf(func_def_363, type, sK225: (subst > term > subst > $o)). thf(func_def_364, type, sK226: term). thf(func_def_365, type, sK227: ((subst > term > term) > term)). thf(func_def_366, type, sK228: ((subst > term > term) > subst)). thf(func_def_367, type, sK229: ((subst > term > term) > term)). thf(func_def_368, type, sK230: ((subst > term > term) > subst)). thf(func_def_369, type, sK231: ((subst > term > term) > (subst > term > subst > $o) > term)). thf(func_def_370, type, sK232: ((subst > term > term) > subst)). thf(func_def_371, type, sK233: ((subst > term > term) > term)). thf(func_def_372, type, sK234: ((subst > term > term) > subst)). thf(func_def_373, type, sK235: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_374, type, sK236: term). thf(func_def_375, type, sK237: (subst > term > subst > $o)). thf(func_def_376, type, sK238: ((subst > term > subst > $o) > term > term)). thf(func_def_377, type, sK239: ((subst > term > subst > $o) > subst)). thf(func_def_378, type, sK240: ((subst > term > subst > $o) > subst)). thf(func_def_379, type, sK241: ((subst > term > subst > $o) > term)). thf(func_def_380, type, sK242: ((subst > term > subst > $o) > subst)). thf(func_def_381, type, sK243: ((subst > term > subst > $o) > term)). thf(func_def_382, type, sK244: ((subst > term > subst > $o) > subst)). thf(func_def_383, type, sK245: ((subst > term > subst > $o) > subst)). thf(func_def_384, type, sK246: ((subst > term > subst > $o) > subst)). thf(func_def_385, type, sK247: ((subst > term > term) > (subst > term > subst > $o) > term)). thf(func_def_386, type, sK248: ((subst > term > term) > subst)). thf(func_def_387, type, sK249: ((subst > term > term) > subst)). thf(func_def_388, type, sK250: ((subst > term > term) > term)). thf(func_def_389, type, sK251: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_390, type, sK252: term). thf(func_def_391, type, sK253: (subst > term > subst > $o)). thf(func_def_392, type, sK254: ((subst > term > subst > $o) > subst)). thf(func_def_393, type, sK255: ((subst > term > subst > $o) > subst)). thf(func_def_394, type, sK256: ((subst > term > subst > $o) > term)). thf(func_def_395, type, sK257: ((subst > term > subst > $o) > subst)). thf(func_def_396, type, sK258: ((subst > term > subst > $o) > term)). thf(func_def_397, type, sK259: ((subst > term > subst > $o) > subst)). thf(func_def_398, type, sK260: ((subst > term > subst > $o) > subst)). thf(func_def_399, type, sK261: ((subst > term > subst > $o) > subst)). thf(func_def_400, type, sK262: ((subst > term > subst > $o) > term)). thf(func_def_401, type, sK263: ((subst > term > subst > $o) > term)). thf(func_def_402, type, sK264: ((subst > term > subst > $o) > term)). thf(func_def_403, type, sK265: ((subst > term > term) > subst)). thf(func_def_404, type, sK266: ((subst > term > term) > subst)). thf(func_def_405, type, sK267: ((subst > term > term) > term)). thf(func_def_406, type, sK268: ((subst > term > term) > (subst > term > subst > $o) > term)). thf(func_def_407, type, sK269: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_408, type, sK270: subst). thf(func_def_409, type, sK271: term). thf(func_def_410, type, sK272: (term > $o)). thf(func_def_411, type, sK273: (term > term)). thf(func_def_412, type, sK274: ((term > $o) > term)). thf(func_def_413, type, sK275: ((term > $o) > subst > term)). thf(func_def_414, type, sK276: ((term > $o) > term)). thf(func_def_415, type, sK277: ((term > $o) > term)). thf(func_def_416, type, sK278: ((term > $o) > term)). thf(func_def_417, type, sK279: ((term > $o) > term)). thf(func_def_418, type, sK280: term). thf(func_def_419, type, sK281: (term > $o)). thf(func_def_420, type, sK282: (term > term)). thf(func_def_421, type, sK283: ((subst > term > subst > $o) > subst)). thf(func_def_422, type, sK284: ((subst > term > subst > $o) > subst)). thf(func_def_423, type, sK285: ((subst > term > subst > $o) > subst)). thf(func_def_424, type, sK286: ((subst > term > subst > $o) > term)). thf(func_def_425, type, sK287: ((subst > term > subst > $o) > subst)). thf(func_def_426, type, sK288: ((subst > term > subst > $o) > subst)). thf(func_def_427, type, sK289: ((subst > term > subst > $o) > subst)). thf(func_def_428, type, sK290: ((subst > term > subst > $o) > term)). thf(func_def_429, type, sK291: term). thf(func_def_430, type, sK292: (subst > term > subst > $o)). thf(func_def_431, type, sK293: term). thf(func_def_432, type, sK294: term). thf(func_def_433, type, sK295: (subst > term > subst > $o)). thf(func_def_434, type, sK296: ((subst > term > subst > $o) > term)). thf(func_def_435, type, sK297: ((subst > term > subst > $o) > subst)). thf(func_def_436, type, sK298: ((subst > term > subst > $o) > subst)). thf(func_def_437, type, sK299: ((subst > term > subst > $o) > subst)). thf(func_def_438, type, sK300: ((subst > term > subst > $o) > term)). thf(func_def_439, type, sK301: ((subst > term > subst > $o) > subst)). thf(func_def_440, type, sK302: ((subst > term > subst > $o) > subst)). thf(func_def_441, type, sK303: ((subst > term > subst > $o) > subst)). thf(func_def_442, type, sK304: ((subst > term > subst > $o) > term)). thf(func_def_443, type, sK305: (term > (subst > term > subst > $o) > term)). thf(func_def_444, type, sK306: ((subst > term > subst > $o) > subst)). thf(func_def_445, type, sK307: ((subst > term > subst > $o) > term)). thf(func_def_446, type, sK308: ((subst > term > subst > $o) > subst)). thf(func_def_447, type, sK309: ((subst > term > subst > $o) > subst)). thf(func_def_448, type, sK310: ((subst > term > subst > $o) > subst)). thf(func_def_449, type, sK311: ((subst > term > subst > $o) > subst)). thf(func_def_450, type, sK312: ((subst > term > subst > $o) > subst)). thf(func_def_451, type, sK313: ((subst > term > subst > $o) > term)). thf(func_def_452, type, sK314: (subst > term > subst > $o)). thf(func_def_453, type, sK315: term). thf(func_def_454, type, sK316: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_455, type, sK317: ((subst > term > term) > (subst > term > subst > $o) > term)). thf(func_def_456, type, sK318: ((subst > term > term) > term)). thf(func_def_457, type, sK319: ((subst > term > term) > subst)). thf(func_def_458, type, sK320: ((subst > term > term) > subst)). thf(func_def_459, type, sK321: ((term > $o) > term)). thf(func_def_460, type, sK322: ((term > $o) > term)). thf(func_def_461, type, sK323: ((subst > term > subst > $o) > term)). thf(func_def_462, type, sK324: ((subst > term > subst > $o) > term)). thf(func_def_463, type, sK325: ((subst > term > term) > subst)). thf(func_def_464, type, sK326: ((subst > term > term) > term)). thf(func_def_465, type, sK327: ((subst > term > term) > subst)). thf(func_def_466, type, sK328: ((subst > term > term) > (subst > term > subst > $o) > term)). thf(func_def_467, type, sK329: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_468, type, sK330: ((subst > term > subst > $o) > term)). thf(func_def_469, type, sK331: ((subst > term > subst > $o) > term)). thf(func_def_470, type, sK332: term). thf(func_def_471, type, sK333: (subst > term > subst > $o)). thf(func_def_472, type, sK334: ((subst > term > term) > term)). thf(func_def_473, type, sK335: ((subst > term > term) > subst)). thf(func_def_474, type, sK336: ((subst > term > term) > subst)). thf(func_def_475, type, sK337: ((subst > term > term) > term)). thf(func_def_476, type, sK338: ((subst > term > subst > $o) > term > term)). thf(func_def_477, type, sK339: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_478, type, sK340: term). thf(func_def_479, type, sK341: (subst > term > subst > $o)). thf(func_def_480, type, sK342: ((subst > term > term) > term)). 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)). thf(func_def_484, type, sK346: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_485, type, sK347: ((subst > term > subst > $o) > term > term)). thf(func_def_486, type, sK348: term). thf(func_def_487, type, sK349: (subst > term > subst > $o)). thf(func_def_488, type, sK350: ((subst > term > term) > subst)). thf(func_def_489, type, sK351: ((subst > term > term) > term)). thf(func_def_490, type, sK352: ((subst > term > term) > subst)). thf(func_def_491, type, sK353: ((subst > term > term) > term)). thf(func_def_492, type, sK354: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_493, type, sK355: ((subst > term > subst > $o) > term > term)). thf(func_def_494, type, sK356: term). thf(func_def_495, type, sK357: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_496, type, sK358: (term > (subst > term > subst > $o) > term)). thf(func_def_497, type, sK359: (subst > term > subst > $o)). thf(func_def_498, type, sK360: term). thf(func_def_499, type, sK361: ((subst > term > term) > subst)). thf(func_def_500, type, sK362: ((subst > term > term) > subst)). thf(func_def_501, type, sK363: ((subst > term > term) > term)). thf(func_def_502, type, sK364: ((subst > term > term) > term)). thf(func_def_503, type, sK365: term). thf(func_def_504, type, sK366: term). thf(func_def_505, type, sK367: (subst > term > subst > $o)). thf(func_def_506, type, sK368: ((subst > term > subst > $o) > subst)). thf(func_def_507, type, sK369: ((subst > term > subst > $o) > term)). thf(func_def_508, type, sK370: ((subst > term > subst > $o) > subst)). thf(func_def_509, type, sK371: ((subst > term > subst > $o) > subst)). thf(func_def_510, type, sK372: ((subst > term > subst > $o) > subst)). thf(func_def_511, type, sK373: ((subst > term > subst > $o) > subst)). thf(func_def_512, type, sK374: ((subst > term > subst > $o) > subst)). thf(func_def_513, type, sK375: ((subst > term > subst > $o) > term)). thf(func_def_514, type, sK376: ((subst > term > subst > $o) > term)). thf(func_def_515, type, sK377: ((subst > term > subst > $o) > term)). thf(func_def_516, type, sK378: term). thf(func_def_517, type, sK379: (subst > term > subst > $o)). thf(func_def_518, type, sK380: ((subst > term > term) > subst)). thf(func_def_519, type, sK381: ((subst > term > term) > subst)). thf(func_def_520, type, sK382: ((subst > term > term) > term)). thf(func_def_521, type, sK383: ((subst > term > term) > term)). thf(func_def_522, type, sK384: ((subst > term > subst > $o) > term > term)). thf(func_def_523, type, sK385: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_524, type, sK386: term). thf(func_def_525, type, sK387: (subst > term > subst > $o)). thf(func_def_526, type, sK388: (term > $o)). thf(func_def_527, type, sK389: ((term > $o) > term > term)). thf(func_def_528, type, sK390: ((subst > term > subst > $o) > subst)). thf(func_def_529, type, sK391: ((subst > term > subst > $o) > subst)). thf(func_def_530, type, sK392: ((subst > term > subst > $o) > term)). thf(func_def_531, type, sK393: ((subst > term > subst > $o) > subst)). thf(func_def_532, type, sK394: ((subst > term > subst > $o) > subst)). thf(func_def_533, type, sK395: ((subst > term > subst > $o) > subst)). thf(func_def_534, type, sK396: ((subst > term > subst > $o) > subst)). thf(func_def_535, type, sK397: ((subst > term > subst > $o) > term)). thf(func_def_536, type, sK398: ((term > $o) > (subst > term > subst > $o) > term)). thf(func_def_537, type, sK399: ((subst > term > term) > subst)). thf(func_def_538, type, sK400: ((subst > term > term) > subst)). thf(func_def_539, type, sK401: ((subst > term > term) > term)). thf(func_def_540, type, sK402: ((subst > term > term) > (subst > term > subst > $o) > term)). thf(func_def_541, type, sK403: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_542, type, sK404: term). thf(func_def_543, type, sK405: (term > $o)). thf(func_def_544, type, sK406: (subst > term > subst > $o)). thf(func_def_545, type, sK407: ((subst > term > subst > $o) > (term > $o) > term)). thf(func_def_546, type, sK408: ((term > $o) > term > term)). thf(func_def_547, type, sK409: ((subst > term > subst > $o) > subst)). thf(func_def_548, type, sK410: ((subst > term > subst > $o) > subst)). thf(func_def_549, type, sK411: ((subst > term > subst > $o) > subst)). thf(func_def_550, type, sK412: ((subst > term > subst > $o) > term)). thf(func_def_551, type, sK413: ((subst > term > subst > $o) > subst)). thf(func_def_552, type, sK414: ((subst > term > subst > $o) > subst)). thf(func_def_553, type, sK415: ((subst > term > subst > $o) > term)). thf(func_def_554, type, sK416: ((subst > term > subst > $o) > subst)). thf(func_def_555, type, sK417: term). thf(func_def_556, type, sK418: ((subst > term > term) > (subst > term > subst > $o) > term)). thf(func_def_557, type, sK419: ((subst > term > term) > subst)). thf(func_def_558, type, sK420: ((subst > term > term) > term)). thf(func_def_559, type, sK421: ((subst > term > term) > subst)). thf(func_def_560, type, sK422: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_561, type, sK423: ((subst > term > term) > subst)). thf(func_def_562, type, sK424: ((subst > term > term) > subst)). thf(func_def_563, type, sK425: ((subst > term > term) > term)). thf(func_def_564, type, sK426: ((subst > term > term) > (subst > term > subst > $o) > term)). thf(func_def_565, type, sK427: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_566, type, sK428: ((term > $o) > term)). thf(func_def_567, type, sK429: ((term > $o) > term)). thf(func_def_568, type, sK430: subst). thf(func_def_569, type, sK431: subst). thf(func_def_570, type, sK432: term). thf(func_def_571, type, sK433: subst). thf(func_def_572, type, sK434: term). thf(func_def_573, type, sK435: (term > $o)). thf(func_def_574, type, sK436: term). thf(func_def_575, type, sK437: subst). thf(func_def_576, type, sK438: term). thf(func_def_577, type, sK439: (term > $o)). thf(func_def_578, type, sK440: (term > term)). thf(func_def_579, type, sK441: ((term > $o) > subst > term)). thf(func_def_580, type, sK442: ((term > $o) > term)). thf(func_def_581, type, sK443: ((term > $o) > term)). thf(func_def_582, type, sK444: ((term > $o) > term)). thf(func_def_583, type, sK445: ((term > $o) > term)). thf(func_def_584, type, sK446: ((term > $o) > term)). thf(func_def_585, type, sK447: (subst > term > subst > $o)). thf(func_def_586, type, sK448: term). thf(func_def_587, type, sK449: (term > $o)). thf(func_def_588, type, sK450: ((subst > term > term) > subst)). thf(func_def_589, type, sK451: ((subst > term > term) > term)). thf(func_def_590, type, sK452: ((subst > term > term) > subst)). thf(func_def_591, type, sK453: ((subst > term > term) > term)). thf(func_def_592, type, sK454: ((term > $o) > term > term)). thf(func_def_593, type, sK455: ((term > $o) > (subst > term > subst > $o) > term)). thf(func_def_594, type, sK456: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_595, type, sK457: term). thf(func_def_596, type, sK458: (term > subst > term > term)). thf(func_def_597, type, sK459: ((subst > term > subst > $o) > term > $o)). thf(func_def_598, type, sK460: (subst > term > subst > $o)). thf(func_def_599, type, sK461: ((term > $o) > term)). thf(func_def_600, type, sK462: term). thf(func_def_601, type, sK463: subst). thf(func_def_602, type, sK464: term). thf(func_def_603, type, sK465: term). thf(func_def_604, type, sK466: term). 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). thf(func_def_610, type, sK472: term). thf(func_def_611, type, sK473: term). thf(func_def_612, type, sK474: term). thf(func_def_613, type, sK475: term). thf(func_def_614, type, sK476: term). thf(func_def_615, type, sK477: term). thf(func_def_616, type, sK478: term). thf(func_def_617, type, sK479: term). thf(func_def_618, type, sK480: term). thf(func_def_619, type, sK481: term). thf(func_def_620, type, sK482: (term > $o)). thf(func_def_621, type, sK483: subst). thf(func_def_622, type, sK484: (term > $o)). thf(func_def_623, type, sK485: term). thf(func_def_624, type, sK486: subst). thf(func_def_625, type, sK487: ((term > $o) > subst > (term > $o) > term > term)). thf(func_def_626, type, sK488: term). thf(func_def_627, type, sK489: term). thf(func_def_628, type, sK490: (term > $o)). thf(func_def_629, type, sK491: subst). thf(func_def_630, type, sK492: term). thf(func_def_631, type, sK493: ((term > $o) > term)). thf(func_def_632, type, sK494: (term > subst > (term > $o) > term > $o)). thf(func_def_633, type, sK495: (term > d_term)). 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: ((subst > term > subst > $o) > term > $o)). thf(func_def_639, type, sK501: (subst > term > subst > $o)). thf(func_def_640, type, sK502: ((term > $o) > term)). thf(func_def_641, type, sK503: ((term > $o) > (term > $o) > term > subst > term)). thf(func_def_642, type, sK504: subst). thf(func_def_643, type, sK505: term). thf(func_def_644, type, sK506: (term > $o)). thf(func_def_645, type, sK507: (term > $o)). thf(func_def_646, type, sK508: term). thf(func_def_647, type, sK509: term). thf(func_def_648, type, sK510: term). thf(func_def_649, type, sK511: subst). thf(func_def_650, type, sK512: term). thf(func_def_651, type, sK513: (subst > term > term)). thf(func_def_652, type, sK514: term). thf(func_def_653, type, sK515: ((subst > term > term) > subst)). thf(func_def_654, type, sK516: ((subst > term > term) > term)). thf(func_def_655, type, sK517: ((subst > term > term) > subst)). 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: ((term > $o) > term)). thf(func_def_661, type, sK523: ((term > $o) > term)). thf(func_def_662, type, sK524: term). thf(func_def_663, type, sK525: (term > $o)). thf(func_def_664, type, sK526: subst). thf(func_def_665, type, sK527: term). thf(func_def_666, type, sK528: term). thf(func_def_667, type, sK529: term). thf(func_def_668, type, sK530: term). thf(func_def_669, type, sK531: subst). thf(func_def_670, type, sK532: subst). thf(func_def_671, type, sK533: term). thf(func_def_672, type, sK534: term). thf(func_def_673, type, sK535: subst). thf(func_def_674, type, sK536: ((subst > term > term) > term)). thf(func_def_675, type, sK537: ((subst > term > term) > subst)). thf(func_def_676, type, sK538: ((subst > term > term) > subst)). thf(func_def_677, type, sK539: ((subst > term > term) > subst)). thf(func_def_678, type, sK540: ((subst > term > term) > term)). thf(func_def_679, type, sK541: ((subst > term > term) > subst)). thf(func_def_680, type, sK542: term). thf(func_def_681, type, sK543: subst). thf(func_def_682, type, sK544: (subst > term > term)). thf(func_def_683, type, sK545: (subst > term > term)). thf(func_def_684, type, sK546: (term > subst > term > term)). thf(func_def_685, type, sK547: term). thf(func_def_686, type, sK548: ((subst > term > term) > term)). thf(func_def_687, type, sK549: ((subst > term > term) > subst)). thf(func_def_688, type, sK550: (term > $o)). thf(func_def_689, type, sK551: (term > $o)). thf(func_def_690, type, sK552: term). thf(func_def_691, type, sK553: term). thf(func_def_692, type, sK554: subst). thf(func_def_693, type, sK555: subst). thf(func_def_694, type, sK556: term). thf(func_def_695, type, sK557: subst). thf(func_def_696, type, sK558: term). thf(func_def_697, type, sK559: term). thf(func_def_698, type, sK560: subst). thf(func_def_699, type, sK561: term). thf(func_def_700, type, sK562: (term > $o)). thf(func_def_701, type, sK563: (term > $o)). thf(func_def_702, type, sK564: term). thf(func_def_703, type, sK565: subst). thf(func_def_704, type, sK566: subst). thf(func_def_705, type, sK567: subst). thf(func_def_706, type, sK568: subst). thf(func_def_707, type, sK569: subst). thf(func_def_708, type, sK570: (term > $o)). thf(func_def_709, type, sK571: term). thf(func_def_710, type, sK572: ((term > $o) > term)). thf(func_def_711, type, sK573: (term > (term > $o) > subst > term > $o)). thf(func_def_712, type, sK574: term). thf(func_def_713, type, sK575: subst). thf(func_def_714, type, sK576: (subst > d_subst)). thf(func_def_715, type, sK577: term). thf(func_def_716, type, sK578: ((subst > term > term) > term)). thf(func_def_717, type, sK579: ((subst > term > term) > subst)). thf(func_def_718, type, sK580: (term > subst > term > term)). thf(func_def_719, type, sK581: ((subst > term > subst > $o) > term)). thf(func_def_720, type, sK582: ((subst > term > subst > $o) > term)). thf(func_def_721, type, sK583: term). thf(func_def_722, type, sK584: (subst > term > subst > $o)). thf(func_def_723, type, sK585: (term > subst > term > term)). thf(func_def_724, type, sK586: term). thf(func_def_725, type, sK587: term). thf(func_def_726, type, sK588: term). thf(func_def_727, type, sK589: term). thf(func_def_728, type, sK590: term). thf(func_def_729, type, vNOT: ($o > $o)). thf(func_def_730, type, sK591: (subst > term > subst > $o)). thf(func_def_731, type, sK592: (term > $o)). thf(func_def_732, type, sK593: ((subst > term > term) > term)). thf(func_def_733, type, sK594: ((subst > term > term) > term)). thf(func_def_734, type, sK595: ((subst > term > term) > subst)). thf(func_def_735, type, sK596: ((subst > term > term) > subst)). thf(func_def_736, type, sK597: ((subst > term > term) > subst)). thf(func_def_737, type, sK598: term). thf(func_def_738, type, sK599: ((subst > term > subst > $o) > (term > $o) > term)). thf(func_def_739, type, sK600: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_740, type, sK601: (term > (term > $o) > term)). thf(func_def_741, type, sK602: (term > (term > $o) > term)). thf(func_def_742, type, sK603: (term > (term > $o) > term)). thf(func_def_743, type, sK604: (term > (term > $o) > term)). thf(f1,conjecture,( (hoasinduction_lem3v2a = ! [X1 : (term > $o),X0 : (subst > term > subst > $o)] : (! [X2 : (subst > term > term)] : (! [X4 : subst,X5 : subst,X3 : term] : (((sub @ (X2 @ X4 @ X3) @ X5)) = ((X2 @ (comp @ X4 @ X5) @ (sub @ X3 @ X5)))) => (! [X3 : term] : ((X0 @ id @ X3 @ id) => (X0 @ id @ (X2 @ id @ X3) @ id)) => (X0 @ id @ (hoaslam @ id @ (^[X4 : subst, X3 : term] : ((X2 @ X4 @ X3)))) @ id))) => ((hoasinduction_p_and_p_prime @ X0 @ X1) => ! [X3 : term] : (! [X6 : term] : ((X1 @ X6) => (X1 @ (sub @ X3 @ (push @ X6 @ id)))) => (X1 @ (lam @ X3))))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p',hoasinduction_lem3v2a)). thf(f2,negated_conjecture,( ~ (hoasinduction_lem3v2a = ! [X1 : (term > $o),X0 : (subst > term > subst > $o)] : (! [X2 : (subst > term > term)] : (! [X4 : subst,X5 : subst,X3 : term] : (((sub @ (X2 @ X4 @ X3) @ X5)) = ((X2 @ (comp @ X4 @ X5) @ (sub @ X3 @ X5)))) => (! [X3 : term] : ((X0 @ id @ X3 @ id) => (X0 @ id @ (X2 @ id @ X3) @ id)) => (X0 @ id @ (hoaslam @ id @ (^[X4 : subst, X3 : term] : ((X2 @ X4 @ X3)))) @ id))) => ((hoasinduction_p_and_p_prime @ X0 @ X1) => ! [X3 : term] : (! [X6 : term] : ((X1 @ X6) => (X1 @ (sub @ X3 @ (push @ X6 @ id)))) => (X1 @ (lam @ X3))))))), inference(negated_conjecture,[status(cth)],[f1])). thf(f3,axiom,( (((sub @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2term @ d_one))) & pushprop_gthm & (((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) = ((d2term @ d_one))) & (((ap @ (d2term @ d_one) @ (d2term @ d_one))) = ((d2term @ d_one))) & (! [X7 : term,X6 : term] : ((((sub @ X6 @ sh)) != ((sub @ X7 @ sh))) | (X6 = X7)) <=> shinj) & hoaslamnotvar_lthm & ((! [X6 : term,X15 : subst] : (X6 = ((sub @ one @ (push @ X6 @ X15)))) => (! [X6 : term,X15 : subst] : (X15 = ((comp @ sh @ (push @ X6 @ X15)))) => (! [X7 : term,X6 : term] : ((X6 = X7) | (((lam @ X6)) != ((lam @ X7)))) => ! [X37 : subst,X19 : (subst > term > term),X38 : term,X36 : (subst > term > term)] : (~ ! [X16 : subst,X6 : term,X15 : subst] : (((sub @ (X19 @ X15 @ X6) @ X16)) = ((X19 @ (comp @ X15 @ X16) @ (sub @ X6 @ X16)))) | (((X19 @ X37 @ X38)) = ((X36 @ X37 @ X38))) | (((lam @ (X19 @ sh @ one))) != ((lam @ (X36 @ sh @ one)))) | ~ ! [X6 : term,X15 : subst,X16 : subst] : (((sub @ (X36 @ X15 @ X6) @ X16)) = ((X36 @ (comp @ X15 @ X16) @ (sub @ X6 @ X16)))))))) <=> hoaslaminj_lthm) & hoasinduction_lem1v2 & hoaslamnotap_gthm & (! [X14 : term,X13 : term,X3 : (subst > term > subst > $o)] : ((X3 @ id @ (ap @ X13 @ X14) @ id) | ~(X3 @ id @ X13 @ id) | ~ ! [X15 : subst,X6 : term,X16 : subst,X17 : subst] : (~(X3 @ (comp @ X15 @ X17) @ (sub @ X6 @ X17) @ X16) | (X3 @ X15 @ X6 @ (comp @ X17 @ X16))) | ~ ! [X6 : term,X17 : subst,X16 : subst,X15 : subst] : (~(X3 @ X15 @ X6 @ (comp @ X17 @ X16)) | (X3 @ (comp @ X15 @ X17) @ (sub @ X6 @ X17) @ X16)) | ~(X3 @ id @ X14 @ id) | ~ ! [X6 : term,X7 : term] : (~(X3 @ id @ X6 @ id) | ~(X3 @ id @ X7 @ id) | (X3 @ id @ (ap @ (sub @ X6 @ id) @ X7) @ id))) <=> hoasinduction_lem2) & ! [X2 : d_subst] : (X2 = d_id) & (! [X6 : term,X11 : term,X7 : term,X10 : term] : ((((ap @ (sub @ X6 @ id) @ X10)) != ((ap @ (sub @ X7 @ id) @ X11))) | (X6 = X7)) <=> hoasapinj1) & ! [X0 : d_term] : (X0 = d_one) & pushprop_lem1v2_gthm & ((! [X6 : term] : (X6 = ((sub @ X6 @ id))) => (! [X3 : (subst > term > subst > $o),X31 : term] : ((X3 @ id @ (lam @ (sub @ X31 @ (push @ one @ sh))) @ id) | ~ ! [X19 : (subst > term > term)] : (~ ! [X6 : term,X15 : subst,X16 : subst] : (((sub @ (X19 @ X15 @ X6) @ X16)) = ((X19 @ (comp @ X15 @ X16) @ (sub @ X6 @ X16)))) | ~ ! [X6 : term] : ((X3 @ id @ (X19 @ id @ X6) @ id) | ~(X3 @ id @ X6 @ id)) | (X3 @ id @ (lam @ (X19 @ sh @ one)) @ id)) | ~ ! [X7 : term] : (~(X3 @ id @ X7 @ id) | (X3 @ id @ (sub @ X31 @ (push @ X7 @ id)) @ id))) => ! [X32 : term,X3 : (subst > term > subst > $o)] : (~ ! [X19 : (subst > term > term)] : (~ ! [X6 : term] : (~(X3 @ id @ X6 @ id) | (X3 @ id @ (X19 @ id @ X6) @ id)) | (X3 @ id @ (lam @ (X19 @ sh @ one)) @ id) | ~ ! [X15 : subst,X6 : term,X16 : subst] : (((sub @ (X19 @ X15 @ X6) @ X16)) = ((X19 @ (comp @ X15 @ X16) @ (sub @ X6 @ X16))))) | ~ ! [X7 : term] : ((X3 @ id @ (sub @ X32 @ (push @ X7 @ id)) @ id) | ~(X3 @ id @ X7 @ id)) | (X3 @ id @ (lam @ X32) @ id)))) <=> hoasinduction_lem3a_lthm) & pushprop_lthm & (hoasinduction_no_psi_cond_lthm <=> (! [X3 : (subst > term > subst > $o)] : ~ ! [X4 : (term > $o)] : ~ ! [X5 : term] : (((X4 @ X5)) = ((X3 @ id @ X5 @ id))) => (! [X3 : (term > $o),X12 : term] : (~ ! [X6 : term,X7 : term] : (~(X3 @ X6) | ~(X3 @ X7) | (X3 @ (ap @ X6 @ X7))) | (X3 @ X12) | ~ ! [X6 : term] : ((X3 @ X6) | ~(var @ X6)) | ~ ! [X6 : term] : ((X3 @ (lam @ X6)) | ~ ! [X7 : term] : ((X3 @ (sub @ X6 @ (push @ X7 @ id))) | ~(X3 @ X7)))) => (! [X6 : term] : (X6 = ((sub @ X6 @ id))) => (! [X22 : term,X3 : (subst > term > subst > $o),X4 : (term > $o)] : (~ ! [X7 : term] : ((X4 @ (sub @ X22 @ (push @ X7 @ id))) | ~(X4 @ X7)) | (X4 @ (lam @ X22)) | ~ ! [X19 : (subst > term > term)] : ((X3 @ id @ (lam @ (X19 @ sh @ one)) @ id) | ~ ! [X15 : subst,X16 : subst,X6 : term] : (((sub @ (X19 @ X15 @ X6) @ X16)) = ((X19 @ (comp @ X15 @ X16) @ (sub @ X6 @ X16)))) | ~ ! [X6 : term] : ((X3 @ id @ (X19 @ id @ X6) @ id) | ~(X3 @ id @ X6 @ id))) | ~ ! [X5 : term] : (((X4 @ X5)) = ((X3 @ id @ X5 @ id)))) => ! [X21 : term,X3 : (subst > term > subst > $o)] : ((X3 @ id @ X21 @ id) | ~ ! [X19 : (subst > term > term)] : (~ ! [X15 : subst,X16 : subst,X6 : term] : (((sub @ (X19 @ X15 @ X6) @ X16)) = ((X19 @ (comp @ X15 @ X16) @ (sub @ X6 @ X16)))) | ~ ! [X6 : term] : ((X3 @ id @ (X19 @ id @ X6) @ id) | ~(X3 @ id @ X6 @ id)) | (X3 @ id @ (lam @ (X19 @ sh @ one)) @ id)) | ~ ! [X6 : term,X7 : term] : ((X3 @ id @ (ap @ (sub @ X6 @ id) @ X7) @ id) | ~(X3 @ id @ X7 @ id) | ~(X3 @ id @ X6 @ id)))))))) & hoasinduction_lem2_gthm & apnotvar & (induction2lem <=> ! [X3 : (term > $o),X8 : term,X9 : subst] : ((X3 @ (sub @ X8 @ X9)) | ~ ! [X7 : term,X6 : term] : (~(X3 @ X7) | (X3 @ (ap @ X6 @ X7)) | ~(X3 @ X6)) | ~ ! [X7 : term] : ((X3 @ (sub @ X7 @ X9)) | ~(var @ X7)) | ~ ! [X6 : term] : (~ ! [X7 : term] : (~(X3 @ X7) | (X3 @ (sub @ X6 @ (push @ X7 @ id)))) | (X3 @ (lam @ X6))))) & (axvarid <=> ! [X6 : term] : (X6 = ((sub @ X6 @ id)))) & (hoasinduction_lthm_3 <=> (! [X3 : (subst > term > subst > $o)] : ~ ! [X4 : (term > $o)] : ~ ! [X5 : term] : (((X4 @ X5)) = ((X3 @ id @ X5 @ id))) => (! [X3 : (term > $o),X12 : term] : ((X3 @ X12) | ~ ! [X6 : term] : ((X3 @ X6) | ~(var @ X6)) | ~ ! [X6 : term] : (~ ! [X7 : term] : ((X3 @ (sub @ X6 @ (push @ X7 @ id))) | ~(X3 @ X7)) | (X3 @ (lam @ X6))) | ~ ! [X6 : term,X7 : term] : (~(X3 @ X7) | ~(X3 @ X6) | (X3 @ (ap @ X6 @ X7)))) => (! [X6 : term] : (X6 = ((sub @ X6 @ id))) => (! [X22 : term,X3 : (subst > term > subst > $o),X4 : (term > $o)] : (~ ! [X5 : term] : (((X4 @ X5)) = ((X3 @ id @ X5 @ id))) | ~ ! [X19 : (subst > term > term)] : ((X3 @ id @ (lam @ (X19 @ sh @ one)) @ id) | ~ ! [X6 : term] : ((X3 @ id @ (X19 @ id @ X6) @ id) | ~(X3 @ id @ X6 @ id)) | ~ ! [X15 : subst,X6 : term,X16 : subst] : (((sub @ (X19 @ X15 @ X6) @ X16)) = ((X19 @ (comp @ X15 @ X16) @ (sub @ X6 @ X16))))) | (X4 @ (lam @ X22)) | ~ ! [X7 : term] : (~(X4 @ X7) | (X4 @ (sub @ X22 @ (push @ X7 @ id))))) => ! [X3 : (subst > term > subst > $o),X18 : term] : (~ ! [X6 : term,X16 : subst,X15 : subst,X17 : subst] : ((X3 @ (comp @ X15 @ X17) @ (sub @ X6 @ X17) @ X16) | ~(X3 @ X15 @ X6 @ (comp @ X17 @ X16))) | ~ ! [X6 : term,X7 : term] : (~(X3 @ id @ X6 @ id) | (X3 @ id @ (ap @ (sub @ X6 @ id) @ X7) @ id) | ~(X3 @ id @ X7 @ id)) | (X3 @ id @ X18 @ id) | ~ ! [X6 : term,X17 : subst,X15 : subst,X16 : subst] : ((X3 @ X15 @ X6 @ (comp @ X17 @ X16)) | ~(X3 @ (comp @ X15 @ X17) @ (sub @ X6 @ X17) @ X16)) | ~ ! [X19 : (subst > term > term)] : (~ ! [X6 : term,X16 : subst,X15 : subst] : (((sub @ (X19 @ X15 @ X6) @ X16)) = ((X19 @ (comp @ X15 @ X16) @ (sub @ X6 @ X16)))) | (X3 @ id @ (lam @ (X19 @ sh @ one)) @ id) | ~ ! [X6 : term] : ((X3 @ id @ (X19 @ id @ X6) @ id) | ~(X3 @ id @ X6 @ id))) | ~ ! [X6 : term] : (~(var @ (sub @ X6 @ id)) | (X3 @ id @ X6 @ id)))))))) & (((comp @ (d2subst @ d_id) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & hoasinduction_gthm & (hoasinduction_lem3aa <=> ! [X3 : (subst > term > subst > $o),X31 : term] : (~ ! [X19 : (subst > term > term)] : (~ ! [X6 : term] : ((X3 @ id @ (X19 @ id @ X6) @ id) | ~(X3 @ id @ X6 @ id)) | ~ ! [X6 : term,X16 : subst,X15 : subst] : (((sub @ (X19 @ X15 @ X6) @ X16)) = ((X19 @ (comp @ X15 @ X16) @ (sub @ X6 @ X16)))) | (X3 @ id @ (lam @ (X19 @ sh @ one)) @ id)) | ~ ! [X7 : term] : (~(X3 @ id @ X7 @ id) | (X3 @ id @ (sub @ X31 @ (push @ X7 @ id)) @ id)) | (X3 @ id @ (lam @ (sub @ X31 @ (push @ one @ sh))) @ id))) & (hoasinduction_lem0 <=> ! [X3 : (subst > term > subst > $o)] : ~ ! [X4 : (term > $o)] : ~ ! [X5 : term] : (((X4 @ X5)) = ((X3 @ id @ X5 @ id)))) & pushprop_lthm_orig & pushprop_lem3v2_lthm & hoasinduction_lem3b_gthm & ! [X47 : d_term,X46 : d_term] : ((((d2term @ X46)) = ((d2term @ X47))) => (X46 = X47)) & termmset_gthm & hoasinduction_lem3_gthm & hoasapinj1_gthm & (hoasinduction_p_and_p_prime = (^[X3 : (subst > term > subst > $o), X4 : (term > $o)] : (! [X5 : term] : (((X4 @ X5)) = ((X3 @ id @ X5 @ id)))))) & ((! [X7 : term] : ~ ! [X19 : (subst > term > term)] : ~ ! [X6 : term,X15 : subst] : (((sub @ X7 @ (push @ X6 @ X15))) = ((X19 @ X15 @ X6))) => (! [X6 : term] : (X6 = ((sub @ X6 @ id))) => ! [X4 : (term > $o),X3 : (subst > term > subst > $o),X22 : term] : (~ ! [X19 : (subst > term > term)] : (~ ! [X6 : term,X16 : subst,X15 : subst] : (((sub @ (X19 @ X15 @ X6) @ X16)) = ((X19 @ (comp @ X15 @ X16) @ (sub @ X6 @ X16)))) | (X3 @ id @ (lam @ (X19 @ sh @ one)) @ id) | ~ ! [X6 : term] : ((X3 @ id @ (X19 @ id @ X6) @ id) | ~(X3 @ id @ X6 @ id))) | (X4 @ (lam @ X22)) | ~ ! [X5 : term] : (((X4 @ X5)) = ((X3 @ id @ X5 @ id))) | ~ ! [X7 : term] : ((X4 @ (sub @ X22 @ (push @ X7 @ id))) | ~(X4 @ X7))))) <=> hoasinduction_lem3v2a_lthm) & induction2_gthm & (((push @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & (! [X7 : term] : ~ ! [X19 : (subst > term > term)] : ~ ! [X15 : subst,X6 : term] : (((sub @ X7 @ (push @ X6 @ X15))) = ((X19 @ X15 @ X6))) <=> hoasinduction_lem3v2_f_lthm) & (axvarcons <=> ! [X6 : term,X15 : subst] : (X6 = ((sub @ one @ (push @ X6 @ X15))))) & (apinj1 <=> ! [X7 : term,X10 : term,X6 : term,X11 : term] : ((((ap @ X6 @ X10)) != ((ap @ X7 @ X11))) | (X6 = X7))) & pushprop_lem2v2 & (induction2_lthm <=> (! [X6 : term] : (X6 = ((sub @ X6 @ id))) => (! [X3 : (term > $o),X8 : term,X9 : subst] : (~ ! [X7 : term] : ((X3 @ (sub @ X7 @ X9)) | ~(var @ X7)) | (X3 @ (sub @ X8 @ X9)) | ~ ! [X6 : term] : ((X3 @ (lam @ X6)) | ~ ! [X7 : term] : ((X3 @ (sub @ X6 @ (push @ X7 @ id))) | ~(X3 @ X7))) | ~ ! [X7 : term,X6 : term] : ((X3 @ (ap @ X6 @ X7)) | ~(X3 @ X7) | ~(X3 @ X6))) => ! [X12 : term,X3 : (term > $o)] : ((X3 @ X12) | ~ ! [X7 : term,X6 : term] : (~(X3 @ X6) | ~(X3 @ X7) | (X3 @ (ap @ X6 @ X7))) | ~ ! [X6 : term] : ((X3 @ (lam @ X6)) | ~ ! [X7 : term] : (~(X3 @ X7) | (X3 @ (sub @ X6 @ (push @ X7 @ id))))) | ~ ! [X6 : term] : ((X3 @ X6) | ~(var @ X6)))))) & (! [X41 : term,X3 : (term > $o)] : (~ ! [X6 : term] : ((X3 @ (lam @ X6)) | ~(X3 @ X6)) | ~ ! [X6 : term] : ((X3 @ X6) | ~(var @ X6)) | (X3 @ X41) | ~ ! [X7 : term,X6 : term] : (~(X3 @ X7) | ~(X3 @ X6) | (X3 @ (ap @ X6 @ X7)))) <=> induction) & pushprop_lem3v2 & ~(hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) & (pushprop_lem1_lthm <=> (! [X15 : subst,X6 : term] : (X6 = ((sub @ one @ (push @ X6 @ X15)))) => (! [X15 : subst,X6 : term] : (X15 = ((comp @ sh @ (push @ X6 @ X15)))) => ! [X3 : (term > $o),X15 : subst,X7 : term,X6 : term,X17 : (term > $o)] : ((X17 @ (sub @ X6 @ (push @ X7 @ X15))) | ~(X3 @ X6))))) & axassoc & induction2lem_gthm & (induction2lem_lthm <=> (! [X6 : term,X15 : subst] : (X6 = ((sub @ one @ (push @ X6 @ X15)))) => (! [X6 : term,X15 : subst] : (X15 = ((comp @ sh @ (push @ X6 @ X15)))) => (! [X15 : subst] : (X15 = ((comp @ X15 @ id))) => (! [X3 : (term > $o),X41 : term] : ((X3 @ X41) | ~ ! [X6 : term] : (~(X3 @ X6) | (X3 @ (lam @ X6))) | ~ ! [X6 : term] : (~(var @ X6) | (X3 @ X6)) | ~ ! [X6 : term,X7 : term] : (~(X3 @ X7) | (X3 @ (ap @ X6 @ X7)) | ~(X3 @ X6))) => ! [X3 : (term > $o),X9 : subst,X8 : term] : (~ ! [X6 : term] : (~ ! [X7 : term] : (~(X3 @ X7) | (X3 @ (sub @ X6 @ (push @ X7 @ id)))) | (X3 @ (lam @ X6))) | ~ ! [X7 : term,X6 : term] : (~(X3 @ X7) | ~(X3 @ X6) | (X3 @ (ap @ X6 @ X7))) | (X3 @ (sub @ X8 @ X9)) | ~ ! [X7 : term] : ((X3 @ (sub @ X7 @ X9)) | ~(var @ X7)))))))) & ~ulamvar1 & (hoasinduction_no_psi_cond <=> ! [X21 : term,X3 : (subst > term > subst > $o)] : (~ ! [X19 : (subst > term > term)] : (~ ! [X6 : term,X15 : subst,X16 : subst] : (((sub @ (X19 @ X15 @ X6) @ X16)) = ((X19 @ (comp @ X15 @ X16) @ (sub @ X6 @ X16)))) | (X3 @ id @ (lam @ (X19 @ sh @ one)) @ id) | ~ ! [X6 : term] : (~(X3 @ id @ X6 @ id) | (X3 @ id @ (X19 @ id @ X6) @ id))) | ~ ! [X7 : term,X6 : term] : ((X3 @ id @ (ap @ (sub @ X6 @ id) @ X7) @ id) | ~(X3 @ id @ X6 @ id) | ~(X3 @ id @ X7 @ id)) | (X3 @ id @ X21 @ id))) & (! [X6 : term,X15 : subst,X3 : (term > $o)] : ~ ! [X4 : (term > $o)] : ~ ! [X5 : term] : (((X4 @ X5)) = ((X3 @ (sub @ X5 @ (push @ X6 @ X15))))) <=> pushprop_lem0_lthm) & ! [X45 : d_subst,X44 : d_subst] : ((((d2subst @ X44)) = ((d2subst @ X45))) => (X44 = X45)) & ~lamnotap & ((! [X12 : term,X3 : (term > $o)] : (~ ! [X6 : term] : ((X3 @ X6) | ~(var @ X6)) | ~ ! [X7 : term,X6 : term] : (~(X3 @ X7) | (X3 @ (ap @ X6 @ X7)) | ~(X3 @ X6)) | ~ ! [X6 : term] : (~ ! [X7 : term] : ((X3 @ (sub @ X6 @ (push @ X7 @ id))) | ~(X3 @ X7)) | (X3 @ (lam @ X6))) | (X3 @ X12)) => (! [X3 : (subst > term > subst > $o),X13 : term,X14 : term] : (~(X3 @ id @ X14 @ id) | (X3 @ id @ (ap @ X13 @ X14) @ id) | ~ ! [X6 : term,X7 : term] : ((X3 @ id @ (ap @ (sub @ X6 @ id) @ X7) @ id) | ~(X3 @ id @ X6 @ id) | ~(X3 @ id @ X7 @ id)) | ~(X3 @ id @ X13 @ id) | ~ ! [X6 : term,X15 : subst,X16 : subst,X17 : subst] : (~(X3 @ (comp @ X15 @ X17) @ (sub @ X6 @ X17) @ X16) | (X3 @ X15 @ X6 @ (comp @ X17 @ X16))) | ~ ! [X17 : subst,X16 : subst,X15 : subst,X6 : term] : (~(X3 @ X15 @ X6 @ (comp @ X17 @ X16)) | (X3 @ (comp @ X15 @ X17) @ (sub @ X6 @ X17) @ X16))) => (! [X3 : (subst > term > subst > $o),X20 : term] : (~ ! [X19 : (subst > term > term)] : (~ ! [X15 : subst,X6 : term,X16 : subst] : (((sub @ (X19 @ X15 @ X6) @ X16)) = ((X19 @ (comp @ X15 @ X16) @ (sub @ X6 @ X16)))) | (X3 @ id @ (lam @ (X19 @ sh @ one)) @ id) | ~ ! [X6 : term] : (~(X3 @ id @ X6 @ id) | (X3 @ id @ (X19 @ id @ X6) @ id))) | ~ ! [X15 : subst,X17 : subst,X6 : term,X16 : subst] : ((X3 @ X15 @ X6 @ (comp @ X17 @ X16)) | ~(X3 @ (comp @ X15 @ X17) @ (sub @ X6 @ X17) @ X16)) | ~ ! [X15 : subst,X6 : term,X17 : subst,X16 : subst] : (~(X3 @ X15 @ X6 @ (comp @ X17 @ X16)) | (X3 @ (comp @ X15 @ X17) @ (sub @ X6 @ X17) @ X16)) | (X3 @ id @ (lam @ X20) @ id) | ~ ! [X7 : term] : ((X3 @ id @ (sub @ X20 @ (push @ X7 @ id)) @ id) | ~(X3 @ id @ X7 @ id))) => ! [X18 : term,X3 : (subst > term > subst > $o)] : (~ ! [X19 : (subst > term > term)] : (~ ! [X6 : term] : ((X3 @ id @ (X19 @ id @ X6) @ id) | ~(X3 @ id @ X6 @ id)) | (X3 @ id @ (lam @ (X19 @ sh @ one)) @ id) | ~ ! [X15 : subst,X6 : term,X16 : subst] : (((sub @ (X19 @ X15 @ X6) @ X16)) = ((X19 @ (comp @ X15 @ X16) @ (sub @ X6 @ X16))))) | ~ ! [X6 : term,X17 : subst,X16 : subst,X15 : subst] : (~(X3 @ (comp @ X15 @ X17) @ (sub @ X6 @ X17) @ X16) | (X3 @ X15 @ X6 @ (comp @ X17 @ X16))) | ~ ! [X6 : term] : (~(var @ (sub @ X6 @ id)) | (X3 @ id @ X6 @ id)) | (X3 @ id @ X18 @ id) | ~ ! [X15 : subst,X17 : subst,X6 : term,X16 : subst] : (~(X3 @ X15 @ X6 @ (comp @ X17 @ X16)) | (X3 @ (comp @ X15 @ X17) @ (sub @ X6 @ X17) @ X16)) | ~ ! [X7 : term,X6 : term] : (~(X3 @ id @ X6 @ id) | (X3 @ id @ (ap @ (sub @ X6 @ id) @ X7) @ id) | ~(X3 @ id @ X7 @ id)))))) <=> hoasinduction_lthm) & (! [X6 : term,X7 : term] : ((X6 = X7) | (((lam @ X6)) != ((lam @ X7)))) <=> laminj) & (pushprop_lem1v2 <=> ! [X4 : (term > $o),X6 : term,X3 : (term > $o),X15 : subst] : (~ ! [X5 : term] : (((X4 @ X5)) = ((X3 @ (sub @ X5 @ (push @ X6 @ X15))))) | (X4 @ one) | ~(X3 @ X6))) & pushprop_lem1_gthm & ! [X48 : term] : ? [X0 : d_term] : (X48 = ((d2term @ X0))) & hoasinduction_lem1_lthm & (hoasinduction_lem3_lthm <=> (! [X6 : term] : (X6 = ((sub @ X6 @ id))) => (! [X3 : (subst > term > subst > $o),X31 : term] : (~ ! [X19 : (subst > term > term)] : (~ ! [X6 : term,X15 : subst,X16 : subst] : (((sub @ (X19 @ X15 @ X6) @ X16)) = ((X19 @ (comp @ X15 @ X16) @ (sub @ X6 @ X16)))) | (X3 @ id @ (lam @ (X19 @ sh @ one)) @ id) | ~ ! [X6 : term] : (~(X3 @ id @ X6 @ id) | (X3 @ id @ (X19 @ id @ X6) @ id))) | (X3 @ id @ (lam @ (sub @ X31 @ (push @ one @ sh))) @ id) | ~ ! [X7 : term] : (~(X3 @ id @ X7 @ id) | (X3 @ id @ (sub @ X31 @ (push @ X7 @ id)) @ id))) => ! [X3 : (subst > term > subst > $o),X20 : term] : (~ ! [X15 : subst,X16 : subst,X6 : term,X17 : subst] : ((X3 @ X15 @ X6 @ (comp @ X17 @ X16)) | ~(X3 @ (comp @ X15 @ X17) @ (sub @ X6 @ X17) @ X16)) | ~ ! [X6 : term,X15 : subst,X16 : subst,X17 : subst] : ((X3 @ (comp @ X15 @ X17) @ (sub @ X6 @ X17) @ X16) | ~(X3 @ X15 @ X6 @ (comp @ X17 @ X16))) | ~ ! [X19 : (subst > term > term)] : ((X3 @ id @ (lam @ (X19 @ sh @ one)) @ id) | ~ ! [X6 : term] : ((X3 @ id @ (X19 @ id @ X6) @ id) | ~(X3 @ id @ X6 @ id)) | ~ ! [X15 : subst,X6 : term,X16 : subst] : (((sub @ (X19 @ X15 @ X6) @ X16)) = ((X19 @ (comp @ X15 @ X16) @ (sub @ X6 @ X16))))) | ~ ! [X7 : term] : (~(X3 @ id @ X7 @ id) | (X3 @ id @ (sub @ X20 @ (push @ X7 @ id)) @ id)) | (X3 @ id @ (lam @ X20) @ id))))) & hoaslamnotvar_gthm & hoaslamnotvar & (! [X7 : term] : ~ ! [X19 : (subst > term > term)] : (((X19 @ sh @ one)) != ((sub @ X7 @ (push @ one @ sh)))) <=> hoasinduction_lem3b_lthm) & (! [X7 : term,X6 : term,X15 : subst,X17 : (term > $o),X3 : (term > $o)] : ((X17 @ (sub @ X6 @ (push @ X7 @ X15))) | ~(X3 @ X6)) <=> pushprop_lem1) & ((! [X15 : subst] : (X15 = ((comp @ id @ X15))) => (! [X15 : subst] : (X15 = ((comp @ X15 @ id))) => (! [X15 : subst] : (X15 = ((comp @ id @ X15))) & ! [X15 : subst] : (X15 = ((comp @ X15 @ id)))))) <=> substmonoid_lthm) & ((^[X42 : subst, X43 : (subst > term > term)] : ((d2term @ d_one))) = hoaslam) & (pushprop_lem0 <=> ! [X3 : (term > $o),X6 : term,X15 : subst] : ~ ! [X4 : (term > $o)] : ~ ! [X5 : term] : (((X4 @ X5)) = ((X3 @ (sub @ X5 @ (push @ X6 @ X15)))))) & (axidl <=> ! [X15 : subst] : (X15 = ((comp @ id @ X15)))) & pushprop & axabs & (! [X20 : term,X3 : (subst > term > subst > $o)] : (~ ! [X15 : subst,X16 : subst,X17 : subst,X6 : term] : (~(X3 @ X15 @ X6 @ (comp @ X17 @ X16)) | (X3 @ (comp @ X15 @ X17) @ (sub @ X6 @ X17) @ X16)) | ~ ! [X7 : term] : (~(X3 @ id @ X7 @ id) | (X3 @ id @ (sub @ X20 @ (push @ X7 @ id)) @ id)) | (X3 @ id @ (lam @ X20) @ id) | ~ ! [X6 : term,X16 : subst,X17 : subst,X15 : subst] : ((X3 @ X15 @ X6 @ (comp @ X17 @ X16)) | ~(X3 @ (comp @ X15 @ X17) @ (sub @ X6 @ X17) @ X16)) | ~ ! [X19 : (subst > term > term)] : (~ ! [X16 : subst,X6 : term,X15 : subst] : (((sub @ (X19 @ X15 @ X6) @ X16)) = ((X19 @ (comp @ X15 @ X16) @ (sub @ X6 @ X16)))) | ~ ! [X6 : term] : ((X3 @ id @ (X19 @ id @ X6) @ id) | ~(X3 @ id @ X6 @ id)) | (X3 @ id @ (lam @ (X19 @ sh @ one)) @ id))) <=> hoasinduction_lem3) & hoasinduction_lem1 & hoasapnotvar & (pushprop_p_and_p_prime = (^[X6 : term, X15 : subst, X3 : (term > $o), X4 : (term > $o)] : (! [X5 : term] : (((X4 @ X5)) = ((X3 @ (sub @ X5 @ (push @ X6 @ X15)))))))) & (! [X3 : (subst > term > subst > $o)] : ~ ! [X4 : (term > $o)] : ~ ! [X5 : term] : (((X4 @ X5)) = ((X3 @ id @ X5 @ id))) <=> hoasinduction_lem0_lthm) & ! [X1 : subst] : ? [X2 : d_subst] : (X1 = ((d2subst @ X2))) & pushprop_lem2v2_gthm & (! [X15 : subst,X6 : term] : (X15 = ((comp @ sh @ (push @ X6 @ X15)))) <=> axshiftcons) & (apinj2 <=> ! [X11 : term,X10 : term,X7 : term,X6 : term] : ((((ap @ X6 @ X10)) != ((ap @ X7 @ X11))) | (X10 = X11))) & (hoasinduction_lem3b <=> ! [X7 : term] : ~ ! [X19 : (subst > term > term)] : (((X19 @ sh @ one)) != ((sub @ X7 @ (push @ one @ sh))))) & (! [X7 : term] : ~ ! [X19 : (subst > term > term)] : ~ ! [X6 : term,X15 : subst] : (((sub @ X7 @ (push @ X6 @ X15))) = ((X19 @ X15 @ X6))) <=> hoasinduction_lem3v2_f) & hoasapnotvar_lthm & (! [X3 : (subst > term > subst > $o),X23 : term] : (~ ! [X7 : term] : ((X3 @ id @ (sub @ X23 @ (push @ X7 @ id)) @ id) | ~(X3 @ id @ X7 @ id)) | ~ ! [X24 : term,X19 : (subst > term > term)] : (~ ! [X29 : term,X30 : subst,X28 : subst] : (((X19 @ (comp @ X28 @ X30) @ (sub @ X29 @ X30))) = ((sub @ X24 @ (push @ (sub @ X29 @ X30) @ (comp @ X28 @ X30))))) | (X3 @ id @ (lam @ (X19 @ sh @ one)) @ id) | ~ ! [X26 : term,X27 : subst,X25 : subst] : (((sub @ (X19 @ X25 @ X26) @ X27)) = ((sub @ (sub @ X24 @ (push @ X26 @ X25)) @ X27))) | ~ ! [X6 : term] : ((X3 @ id @ (X19 @ id @ X6) @ id) | ~(X3 @ id @ X6 @ id))) | (X3 @ id @ (lam @ (sub @ X23 @ (push @ one @ sh))) @ id)) <=> hoasinduction_lem3aaa) & (! [X3 : (subst > term > subst > $o),X32 : term] : (~ ! [X19 : (subst > term > term)] : (~ ! [X16 : subst,X15 : subst,X6 : term] : (((sub @ (X19 @ X15 @ X6) @ X16)) = ((X19 @ (comp @ X15 @ X16) @ (sub @ X6 @ X16)))) | ~ ! [X6 : term] : (~(X3 @ id @ X6 @ id) | (X3 @ id @ (X19 @ id @ X6) @ id)) | (X3 @ id @ (lam @ (X19 @ sh @ one)) @ id)) | ~ ! [X7 : term] : ((X3 @ id @ (sub @ X32 @ (push @ X7 @ id)) @ id) | ~(X3 @ id @ X7 @ id)) | (X3 @ id @ (lam @ X32) @ id)) <=> hoasinduction_lem3a) & hoasinduction_lem3v2_gthm & (sh = ((d2subst @ d_id))) & hoasapnotvar_gthm & hoasapinj2_gthm & (axscons <=> ! [X15 : subst] : (X15 = ((push @ (sub @ one @ X15) @ (comp @ sh @ X15))))) & hoasinduction_lem1v2_gthm & ((! [X6 : term] : (X6 = ((sub @ X6 @ id))) => ! [X6 : term] : (X6 = ((sub @ X6 @ id)))) <=> termmset_lthm) & axvarshift & (hoasinduction <=> ! [X18 : term,X3 : (subst > term > subst > $o)] : (~ ! [X17 : subst,X16 : subst,X15 : subst,X6 : term] : (~(X3 @ (comp @ X15 @ X17) @ (sub @ X6 @ X17) @ X16) | (X3 @ X15 @ X6 @ (comp @ X17 @ X16))) | ~ ! [X19 : (subst > term > term)] : ((X3 @ id @ (lam @ (X19 @ sh @ one)) @ id) | ~ ! [X6 : term] : (~(X3 @ id @ X6 @ id) | (X3 @ id @ (X19 @ id @ X6) @ id)) | ~ ! [X16 : subst,X15 : subst,X6 : term] : (((sub @ (X19 @ X15 @ X6) @ X16)) = ((X19 @ (comp @ X15 @ X16) @ (sub @ X6 @ X16))))) | ~ ! [X6 : term,X7 : term] : (~(X3 @ id @ X6 @ id) | ~(X3 @ id @ X7 @ id) | (X3 @ id @ (ap @ (sub @ X6 @ id) @ X7) @ id)) | ~ ! [X6 : term] : ((X3 @ id @ X6 @ id) | ~(var @ (sub @ X6 @ id))) | (X3 @ id @ X18 @ id) | ~ ! [X15 : subst,X6 : term,X16 : subst,X17 : subst] : (~(X3 @ X15 @ X6 @ (comp @ X17 @ X16)) | (X3 @ (comp @ X15 @ X17) @ (sub @ X6 @ X17) @ X16)))) & (hoasinduction_lem3v2_lthm <=> (! [X6 : term] : (X6 = ((sub @ X6 @ id))) => ! [X3 : (subst > term > subst > $o),X35 : term,X4 : (term > $o)] : (~ ! [X17 : subst,X6 : term,X16 : subst,X15 : subst] : (~(X3 @ X15 @ X6 @ (comp @ X17 @ X16)) | (X3 @ (comp @ X15 @ X17) @ (sub @ X6 @ X17) @ X16)) | (X4 @ (lam @ X35)) | ~ ! [X7 : term] : ((X4 @ (sub @ X35 @ (push @ X7 @ id))) | ~(X4 @ X7)) | ~ ! [X15 : subst,X16 : subst,X17 : subst,X6 : term] : ((X3 @ X15 @ X6 @ (comp @ X17 @ X16)) | ~(X3 @ (comp @ X15 @ X17) @ (sub @ X6 @ X17) @ X16)) | ~ ! [X19 : (subst > term > term)] : ((X3 @ id @ (lam @ (X19 @ sh @ one)) @ id) | ~ ! [X6 : term,X16 : subst,X15 : subst] : (((sub @ (X19 @ X15 @ X6) @ X16)) = ((X19 @ (comp @ X15 @ X16) @ (sub @ X6 @ X16)))) | ~ ! [X6 : term] : (~(X3 @ id @ X6 @ id) | (X3 @ id @ (X19 @ id @ X6) @ id))) | ~ ! [X5 : term] : (((X4 @ X5)) = ((X3 @ id @ X5 @ id)))))) & lamnotvar & pushprop_lem0_gthm & (! [X3 : (subst > term > subst > $o),X22 : term,X4 : (term > $o)] : (~ ! [X5 : term] : (((X4 @ X5)) = ((X3 @ id @ X5 @ id))) | (X4 @ (lam @ X22)) | ~ ! [X19 : (subst > term > term)] : (~ ! [X15 : subst,X16 : subst,X6 : term] : (((sub @ (X19 @ X15 @ X6) @ X16)) = ((X19 @ (comp @ X15 @ X16) @ (sub @ X6 @ X16)))) | (X3 @ id @ (lam @ (X19 @ sh @ one)) @ id) | ~ ! [X6 : term] : ((X3 @ id @ (X19 @ id @ X6) @ id) | ~(X3 @ id @ X6 @ id))) | ~ ! [X7 : term] : (~(X4 @ X7) | (X4 @ (sub @ X22 @ (push @ X7 @ id))))) <=> hoasinduction_lem3v2a) & (! [X15 : subst] : (X15 = ((comp @ X15 @ id))) <=> axidr) & (id = ((d2subst @ d_id))) & (! [X33 : term,X34 : term,X19 : (subst > term > term)] : ((((lam @ (X19 @ sh @ one))) != ((ap @ (sub @ X33 @ id) @ X34))) | ~ ! [X15 : subst,X16 : subst,X6 : term] : (((sub @ (X19 @ X15 @ X6) @ X16)) = ((X19 @ (comp @ X15 @ X16) @ (sub @ X6 @ X16))))) <=> hoaslamnotap) & (hoasapinj1_lthm <=> (! [X6 : term] : (X6 = ((sub @ X6 @ id))) => (! [X6 : term,X10 : term,X11 : term,X7 : term] : ((((ap @ X6 @ X10)) != ((ap @ X7 @ X11))) | (X6 = X7)) => ! [X10 : term,X11 : term,X6 : term,X7 : term] : ((((ap @ (sub @ X6 @ id) @ X10)) != ((ap @ (sub @ X7 @ id) @ X11))) | (X6 = X7))))) & pushprop_lem2v2_lthm & ulamvarsh & axapp & substmonoid_gthm & (one = ((d2term @ d_one))) & hoasinduction_lem2v2_gthm & (! [X36 : (subst > term > term),X37 : subst,X19 : (subst > term > term),X38 : term] : ((((X19 @ X37 @ X38)) = ((X36 @ X37 @ X38))) | ~ ! [X16 : subst,X15 : subst,X6 : term] : (((sub @ (X36 @ X15 @ X6) @ X16)) = ((X36 @ (comp @ X15 @ X16) @ (sub @ X6 @ X16)))) | (((lam @ (X19 @ sh @ one))) != ((lam @ (X36 @ sh @ one)))) | ~ ! [X15 : subst,X16 : subst,X6 : term] : (((sub @ (X19 @ X15 @ X6) @ X16)) = ((X19 @ (comp @ X15 @ X16) @ (sub @ X6 @ X16))))) <=> hoaslaminj) & (((lam @ (d2term @ d_one))) = ((d2term @ d_one))) & (hoasinduction_lem2v2 <=> ! [X39 : term,X4 : (term > $o),X3 : (subst > term > subst > $o),X40 : term] : ((X4 @ (ap @ X39 @ X40)) | ~ ! [X5 : term] : (((X4 @ X5)) = ((X3 @ id @ X5 @ id))) | ~ ! [X7 : term,X6 : term] : (~(X3 @ id @ X6 @ id) | ~(X3 @ id @ X7 @ id) | (X3 @ id @ (ap @ (sub @ X6 @ id) @ X7) @ id)) | ~ ! [X16 : subst,X15 : subst,X6 : term,X17 : subst] : ((X3 @ X15 @ X6 @ (comp @ X17 @ X16)) | ~(X3 @ (comp @ X15 @ X17) @ (sub @ X6 @ X17) @ X16)) | ~(X4 @ X40) | ~(X4 @ X39) | ~ ! [X15 : subst,X16 : subst,X6 : term,X17 : subst] : (~(X3 @ X15 @ X6 @ (comp @ X17 @ X16)) | (X3 @ (comp @ X15 @ X17) @ (sub @ X6 @ X17) @ X16)))) & hoasinduction_lem3a_gthm & (! [X3 : (term > $o),X12 : term] : ((X3 @ X12) | ~ ! [X6 : term] : (~(var @ X6) | (X3 @ X6)) | ~ ! [X6 : term,X7 : term] : (~(X3 @ X6) | (X3 @ (ap @ X6 @ X7)) | ~(X3 @ X7)) | ~ ! [X6 : term] : ((X3 @ (lam @ X6)) | ~ ! [X7 : term] : ((X3 @ (sub @ X6 @ (push @ X7 @ id))) | ~(X3 @ X7)))) <=> induction2) & ulamvarind & ~(var @ (d2term @ d_one)) & (! [X35 : term,X3 : (subst > term > subst > $o),X4 : (term > $o)] : ((X4 @ (lam @ X35)) | ~ ! [X5 : term] : (((X4 @ X5)) = ((X3 @ id @ X5 @ id))) | ~ ! [X19 : (subst > term > term)] : (~ ! [X6 : term] : (~(X3 @ id @ X6 @ id) | (X3 @ id @ (X19 @ id @ X6) @ id)) | ~ ! [X15 : subst,X6 : term,X16 : subst] : (((sub @ (X19 @ X15 @ X6) @ X16)) = ((X19 @ (comp @ X15 @ X16) @ (sub @ X6 @ X16)))) | (X3 @ id @ (lam @ (X19 @ sh @ one)) @ id)) | ~ ! [X6 : term,X16 : subst,X17 : subst,X15 : subst] : (~(X3 @ X15 @ X6 @ (comp @ X17 @ X16)) | (X3 @ (comp @ X15 @ X17) @ (sub @ X6 @ X17) @ X16)) | ~ ! [X7 : term] : (~(X4 @ X7) | (X4 @ (sub @ X35 @ (push @ X7 @ id)))) | ~ ! [X17 : subst,X6 : term,X16 : subst,X15 : subst] : (~(X3 @ (comp @ X15 @ X17) @ (sub @ X6 @ X17) @ X16) | (X3 @ X15 @ X6 @ (comp @ X17 @ X16)))) <=> hoasinduction_lem3v2) & hoasinduction_lem1_gthm & (! [X6 : term] : (X6 = ((sub @ X6 @ id))) <=> termmset) & hoaslaminj_gthm & axclos & hoasinduction_lem2_lthm & ((! [X15 : subst] : (X15 = ((comp @ X15 @ id))) & ! [X15 : subst] : (X15 = ((comp @ id @ X15)))) <=> substmonoid) & axmap & (hoasapinj2_lthm <=> (! [X11 : term,X10 : term,X6 : term,X7 : term] : ((X10 = X11) | (((ap @ X6 @ X10)) != ((ap @ X7 @ X11)))) => ! [X10 : term,X7 : term,X6 : term,X11 : term] : ((X10 = X11) | (((ap @ (sub @ X6 @ id) @ X10)) != ((ap @ (sub @ X7 @ id) @ X11)))))) & (! [X11 : term,X6 : term,X7 : term,X10 : term] : ((X10 = X11) | (((ap @ (sub @ X6 @ id) @ X10)) != ((ap @ (sub @ X7 @ id) @ X11)))) <=> hoasapinj2) & (hoasinduction_lem3aa_lthm <=> ! [X31 : term,X3 : (subst > term > subst > $o)] : (~ ! [X7 : term] : ((X3 @ id @ (sub @ X31 @ (push @ X7 @ id)) @ id) | ~(X3 @ id @ X7 @ id)) | ~ ! [X19 : (subst > term > term)] : (~ ! [X6 : term,X15 : subst,X16 : subst] : (((sub @ (X19 @ X15 @ X6) @ X16)) = ((X19 @ (comp @ X15 @ X16) @ (sub @ X6 @ X16)))) | (X3 @ id @ (lam @ (X19 @ sh @ one)) @ id) | ~ ! [X6 : term] : ((X3 @ id @ (X19 @ id @ X6) @ id) | ~(X3 @ id @ X6 @ id))) | (X3 @ id @ (lam @ (sub @ X31 @ (push @ one @ sh))) @ id))) & ((! [X6 : term,X15 : subst] : (X6 = ((sub @ one @ (push @ X6 @ X15)))) => ! [X6 : term,X3 : (term > $o),X15 : subst,X4 : (term > $o)] : ((X4 @ one) | ~ ! [X5 : term] : (((X4 @ X5)) = ((X3 @ (sub @ X5 @ (push @ X6 @ X15))))) | ~(X3 @ X6))) <=> pushprop_lem1v2_lthm) & hoaslamnotap_lthm), file('/export/starexec/sandbox/benchmark/theBenchmark.p',alg444_1)). thf(f4,plain,( ~ (hoasinduction_lem3v2a = ! [X0 : (term > $o),X1 : (subst > term > subst > $o)] : (! [X2 : (subst > term > term)] : (! [X3 : subst,X4 : subst,X5 : term] : (((X2 @ (comp @ X3 @ X4) @ (sub @ X5 @ X4))) = ((sub @ (X2 @ X3 @ X5) @ X4))) => (! [X6 : term] : ((X1 @ id @ X6 @ id) => (X1 @ id @ (X2 @ id @ X6) @ id)) => (X1 @ id @ (hoaslam @ id @ (^[X7 : subst, X8 : term] : ((X2 @ X7 @ X8)))) @ id))) => ((hoasinduction_p_and_p_prime @ X1 @ X0) => ! [X9 : term] : (! [X10 : term] : ((X0 @ X10) => (X0 @ (sub @ X9 @ (push @ X10 @ id)))) => (X0 @ (lam @ X9))))))), inference(rectify,[],[f2])). thf(f5,plain,( ~ (hoasinduction_lem3v2a = ((!! @ (subst > term > subst > $o) @ (^[Y0 : subst > term > subst > $o]: (!! @ (term > $o) @ (^[Y1 : term > $o]: ((!! @ (subst > term > term) @ (^[Y2 : subst > term > term]: ((!! @ term @ (^[Y3 : term]: (!! @ subst @ (^[Y4 : subst]: (!! @ subst @ (^[Y5 : subst]: ((Y2 @ (comp @ Y5 @ Y4) @ (sub @ Y3 @ Y4)) = (sub @ (Y2 @ Y5 @ Y3) @ Y4)))))))) => ((!! @ term @ (^[Y3 : term]: ((Y0 @ id @ Y3 @ id) => (Y0 @ id @ (Y2 @ id @ Y3) @ id)))) => (Y0 @ id @ (hoaslam @ id @ (^[Y3 : subst]: ((^[Y4 : term]: (Y2 @ Y3 @ Y4))))) @ id))))) => ((hoasinduction_p_and_p_prime @ Y0 @ Y1) => (!! @ term @ (^[Y2 : term]: ((!! @ term @ (^[Y3 : term]: ((Y1 @ Y3) => (Y1 @ (sub @ Y2 @ (push @ Y3 @ id)))))) => (Y1 @ (lam @ Y2)))))))))))))), inference(fool_elimination,[],[f4])). thf(f6,plain,( (((sub @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2term @ d_one))) & pushprop_gthm & (((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) = ((d2term @ d_one))) & (((ap @ (d2term @ d_one) @ (d2term @ d_one))) = ((d2term @ d_one))) & (! [X0 : term,X1 : term] : ((((sub @ X0 @ sh)) != ((sub @ X1 @ sh))) | (X0 = X1)) <=> shinj) & hoaslamnotvar_lthm & ((! [X2 : term,X3 : subst] : (((sub @ one @ (push @ X2 @ X3))) = X2) => (! [X4 : term,X5 : subst] : (((comp @ sh @ (push @ X4 @ X5))) = X5) => (! [X6 : term,X7 : term] : ((X6 = X7) | (((lam @ X6)) != ((lam @ X7)))) => ! [X8 : subst,X9 : (subst > term > term),X10 : term,X11 : (subst > term > term)] : (~ ! [X12 : subst,X13 : term,X14 : subst] : (((X9 @ (comp @ X14 @ X12) @ (sub @ X13 @ X12))) = ((sub @ (X9 @ X14 @ X13) @ X12))) | (((X9 @ X8 @ X10)) = ((X11 @ X8 @ X10))) | (((lam @ (X9 @ sh @ one))) != ((lam @ (X11 @ sh @ one)))) | ~ ! [X15 : term,X16 : subst,X17 : subst] : (((sub @ (X11 @ X16 @ X15) @ X17)) = ((X11 @ (comp @ X16 @ X17) @ (sub @ X15 @ X17)))))))) <=> hoaslaminj_lthm) & hoasinduction_lem1v2 & hoaslamnotap_gthm & (! [X18 : term,X19 : term,X20 : (subst > term > subst > $o)] : ((X20 @ id @ (ap @ X19 @ X18) @ id) | ~(X20 @ id @ X19 @ id) | ~ ! [X21 : subst,X22 : term,X23 : subst,X24 : subst] : (~(X20 @ (comp @ X21 @ X24) @ (sub @ X22 @ X24) @ X23) | (X20 @ X21 @ X22 @ (comp @ X24 @ X23))) | ~ ! [X25 : term,X26 : subst,X27 : subst,X28 : subst] : (~(X20 @ X28 @ X25 @ (comp @ X26 @ X27)) | (X20 @ (comp @ X28 @ X26) @ (sub @ X25 @ X26) @ X27)) | ~(X20 @ id @ X18 @ id) | ~ ! [X29 : term,X30 : term] : (~(X20 @ id @ X29 @ id) | ~(X20 @ id @ X30 @ id) | (X20 @ id @ (ap @ (sub @ X29 @ id) @ X30) @ id))) <=> hoasinduction_lem2) & ! [X31 : d_subst] : (d_id = X31) & (! [X32 : term,X33 : term,X34 : term,X35 : term] : ((((ap @ (sub @ X34 @ id) @ X33)) != ((ap @ (sub @ X32 @ id) @ X35))) | (X32 = X34)) <=> hoasapinj1) & ! [X36 : d_term] : (d_one = X36) & pushprop_lem1v2_gthm & ((! [X37 : term] : (((sub @ X37 @ id)) = X37) => (! [X38 : (subst > term > subst > $o),X39 : term] : ((X38 @ id @ (lam @ (sub @ X39 @ (push @ one @ sh))) @ id) | ~ ! [X40 : (subst > term > term)] : (~ ! [X41 : term,X42 : subst,X43 : subst] : (((X40 @ (comp @ X42 @ X43) @ (sub @ X41 @ X43))) = ((sub @ (X40 @ X42 @ X41) @ X43))) | ~ ! [X44 : term] : ((X38 @ id @ (X40 @ id @ X44) @ id) | ~(X38 @ id @ X44 @ id)) | (X38 @ id @ (lam @ (X40 @ sh @ one)) @ id)) | ~ ! [X45 : term] : (~(X38 @ id @ X45 @ id) | (X38 @ id @ (sub @ X39 @ (push @ X45 @ id)) @ id))) => ! [X46 : term,X47 : (subst > term > subst > $o)] : (~ ! [X48 : (subst > term > term)] : (~ ! [X49 : term] : (~(X47 @ id @ X49 @ id) | (X47 @ id @ (X48 @ id @ X49) @ id)) | (X47 @ id @ (lam @ (X48 @ sh @ one)) @ id) | ~ ! [X50 : subst,X51 : term,X52 : subst] : (((sub @ (X48 @ X50 @ X51) @ X52)) = ((X48 @ (comp @ X50 @ X52) @ (sub @ X51 @ X52))))) | ~ ! [X53 : term] : ((X47 @ id @ (sub @ X46 @ (push @ X53 @ id)) @ id) | ~(X47 @ id @ X53 @ id)) | (X47 @ id @ (lam @ X46) @ id)))) <=> hoasinduction_lem3a_lthm) & pushprop_lthm & (hoasinduction_no_psi_cond_lthm <=> (! [X54 : (subst > term > subst > $o)] : ~ ! [X55 : (term > $o)] : ~ ! [X56 : term] : (((X54 @ id @ X56 @ id)) = ((X55 @ X56))) => (! [X57 : (term > $o),X58 : term] : (~ ! [X59 : term,X60 : term] : (~(X57 @ X59) | ~(X57 @ X60) | (X57 @ (ap @ X59 @ X60))) | (X57 @ X58) | ~ ! [X61 : term] : ((X57 @ X61) | ~(var @ X61)) | ~ ! [X62 : term] : ((X57 @ (lam @ X62)) | ~ ! [X63 : term] : ((X57 @ (sub @ X62 @ (push @ X63 @ id))) | ~(X57 @ X63)))) => (! [X64 : term] : (((sub @ X64 @ id)) = X64) => (! [X65 : term,X66 : (subst > term > subst > $o),X67 : (term > $o)] : (~ ! [X68 : term] : ((X67 @ (sub @ X65 @ (push @ X68 @ id))) | ~(X67 @ X68)) | (X67 @ (lam @ X65)) | ~ ! [X69 : (subst > term > term)] : ((X66 @ id @ (lam @ (X69 @ sh @ one)) @ id) | ~ ! [X70 : subst,X71 : subst,X72 : term] : (((sub @ (X69 @ X70 @ X72) @ X71)) = ((X69 @ (comp @ X70 @ X71) @ (sub @ X72 @ X71)))) | ~ ! [X73 : term] : ((X66 @ id @ (X69 @ id @ X73) @ id) | ~(X66 @ id @ X73 @ id))) | ~ ! [X74 : term] : (((X66 @ id @ X74 @ id)) = ((X67 @ X74)))) => ! [X75 : term,X76 : (subst > term > subst > $o)] : ((X76 @ id @ X75 @ id) | ~ ! [X77 : (subst > term > term)] : (~ ! [X78 : subst,X79 : subst,X80 : term] : (((sub @ (X77 @ X78 @ X80) @ X79)) = ((X77 @ (comp @ X78 @ X79) @ (sub @ X80 @ X79)))) | ~ ! [X81 : term] : ((X76 @ id @ (X77 @ id @ X81) @ id) | ~(X76 @ id @ X81 @ id)) | (X76 @ id @ (lam @ (X77 @ sh @ one)) @ id)) | ~ ! [X82 : term,X83 : term] : ((X76 @ id @ (ap @ (sub @ X82 @ id) @ X83) @ id) | ~(X76 @ id @ X83 @ id) | ~(X76 @ id @ X82 @ id)))))))) & hoasinduction_lem2_gthm & apnotvar & (induction2lem <=> ! [X84 : (term > $o),X85 : term,X86 : subst] : ((X84 @ (sub @ X85 @ X86)) | ~ ! [X87 : term,X88 : term] : (~(X84 @ X87) | (X84 @ (ap @ X88 @ X87)) | ~(X84 @ X88)) | ~ ! [X89 : term] : ((X84 @ (sub @ X89 @ X86)) | ~(var @ X89)) | ~ ! [X90 : term] : (~ ! [X91 : term] : (~(X84 @ X91) | (X84 @ (sub @ X90 @ (push @ X91 @ id)))) | (X84 @ (lam @ X90))))) & (axvarid <=> ! [X92 : term] : (((sub @ X92 @ id)) = X92)) & (hoasinduction_lthm_3 <=> (! [X93 : (subst > term > subst > $o)] : ~ ! [X94 : (term > $o)] : ~ ! [X95 : term] : (((X93 @ id @ X95 @ id)) = ((X94 @ X95))) => (! [X96 : (term > $o),X97 : term] : ((X96 @ X97) | ~ ! [X98 : term] : ((X96 @ X98) | ~(var @ X98)) | ~ ! [X99 : term] : (~ ! [X100 : term] : ((X96 @ (sub @ X99 @ (push @ X100 @ id))) | ~(X96 @ X100)) | (X96 @ (lam @ X99))) | ~ ! [X101 : term,X102 : term] : (~(X96 @ X102) | ~(X96 @ X101) | (X96 @ (ap @ X101 @ X102)))) => (! [X103 : term] : (((sub @ X103 @ id)) = X103) => (! [X104 : term,X105 : (subst > term > subst > $o),X106 : (term > $o)] : (~ ! [X107 : term] : (((X105 @ id @ X107 @ id)) = ((X106 @ X107))) | ~ ! [X108 : (subst > term > term)] : ((X105 @ id @ (lam @ (X108 @ sh @ one)) @ id) | ~ ! [X109 : term] : ((X105 @ id @ (X108 @ id @ X109) @ id) | ~(X105 @ id @ X109 @ id)) | ~ ! [X110 : subst,X111 : term,X112 : subst] : (((X108 @ (comp @ X110 @ X112) @ (sub @ X111 @ X112))) = ((sub @ (X108 @ X110 @ X111) @ X112)))) | (X106 @ (lam @ X104)) | ~ ! [X113 : term] : (~(X106 @ X113) | (X106 @ (sub @ X104 @ (push @ X113 @ id))))) => ! [X114 : (subst > term > subst > $o),X115 : term] : (~ ! [X116 : term,X117 : subst,X118 : subst,X119 : subst] : ((X114 @ (comp @ X118 @ X119) @ (sub @ X116 @ X119) @ X117) | ~(X114 @ X118 @ X116 @ (comp @ X119 @ X117))) | ~ ! [X120 : term,X121 : term] : (~(X114 @ id @ X120 @ id) | (X114 @ id @ (ap @ (sub @ X120 @ id) @ X121) @ id) | ~(X114 @ id @ X121 @ id)) | (X114 @ id @ X115 @ id) | ~ ! [X122 : term,X123 : subst,X124 : subst,X125 : subst] : ((X114 @ X124 @ X122 @ (comp @ X123 @ X125)) | ~(X114 @ (comp @ X124 @ X123) @ (sub @ X122 @ X123) @ X125)) | ~ ! [X126 : (subst > term > term)] : (~ ! [X127 : term,X128 : subst,X129 : subst] : (((X126 @ (comp @ X129 @ X128) @ (sub @ X127 @ X128))) = ((sub @ (X126 @ X129 @ X127) @ X128))) | (X114 @ id @ (lam @ (X126 @ sh @ one)) @ id) | ~ ! [X130 : term] : ((X114 @ id @ (X126 @ id @ X130) @ id) | ~(X114 @ id @ X130 @ id))) | ~ ! [X131 : term] : (~(var @ (sub @ X131 @ id)) | (X114 @ id @ X131 @ id)))))))) & (((comp @ (d2subst @ d_id) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & hoasinduction_gthm & (hoasinduction_lem3aa <=> ! [X132 : (subst > term > subst > $o),X133 : term] : (~ ! [X134 : (subst > term > term)] : (~ ! [X135 : term] : ((X132 @ id @ (X134 @ id @ X135) @ id) | ~(X132 @ id @ X135 @ id)) | ~ ! [X136 : term,X137 : subst,X138 : subst] : (((sub @ (X134 @ X138 @ X136) @ X137)) = ((X134 @ (comp @ X138 @ X137) @ (sub @ X136 @ X137)))) | (X132 @ id @ (lam @ (X134 @ sh @ one)) @ id)) | ~ ! [X139 : term] : (~(X132 @ id @ X139 @ id) | (X132 @ id @ (sub @ X133 @ (push @ X139 @ id)) @ id)) | (X132 @ id @ (lam @ (sub @ X133 @ (push @ one @ sh))) @ id))) & (hoasinduction_lem0 <=> ! [X140 : (subst > term > subst > $o)] : ~ ! [X141 : (term > $o)] : ~ ! [X142 : term] : (((X141 @ X142)) = ((X140 @ id @ X142 @ id)))) & pushprop_lthm_orig & pushprop_lem3v2_lthm & hoasinduction_lem3b_gthm & ! [X143 : d_term,X144 : d_term] : ((((d2term @ X144)) = ((d2term @ X143))) => (X143 = X144)) & termmset_gthm & hoasinduction_lem3_gthm & hoasapinj1_gthm & (hoasinduction_p_and_p_prime = (^[X145 : (subst > term > subst > $o), X146 : (term > $o)] : (! [X147 : term] : (((X145 @ id @ X147 @ id)) = ((X146 @ X147)))))) & ((! [X148 : term] : ~ ! [X149 : (subst > term > term)] : ~ ! [X150 : term,X151 : subst] : (((X149 @ X151 @ X150)) = ((sub @ X148 @ (push @ X150 @ X151)))) => (! [X152 : term] : (((sub @ X152 @ id)) = X152) => ! [X153 : (term > $o),X154 : (subst > term > subst > $o),X155 : term] : (~ ! [X156 : (subst > term > term)] : (~ ! [X157 : term,X158 : subst,X159 : subst] : (((sub @ (X156 @ X159 @ X157) @ X158)) = ((X156 @ (comp @ X159 @ X158) @ (sub @ X157 @ X158)))) | (X154 @ id @ (lam @ (X156 @ sh @ one)) @ id) | ~ ! [X160 : term] : ((X154 @ id @ (X156 @ id @ X160) @ id) | ~(X154 @ id @ X160 @ id))) | (X153 @ (lam @ X155)) | ~ ! [X161 : term] : (((X153 @ X161)) = ((X154 @ id @ X161 @ id))) | ~ ! [X162 : term] : ((X153 @ (sub @ X155 @ (push @ X162 @ id))) | ~(X153 @ X162))))) <=> hoasinduction_lem3v2a_lthm) & induction2_gthm & (((push @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & (! [X163 : term] : ~ ! [X164 : (subst > term > term)] : ~ ! [X165 : subst,X166 : term] : (((X164 @ X165 @ X166)) = ((sub @ X163 @ (push @ X166 @ X165)))) <=> hoasinduction_lem3v2_f_lthm) & (axvarcons <=> ! [X167 : term,X168 : subst] : (((sub @ one @ (push @ X167 @ X168))) = X167)) & (apinj1 <=> ! [X169 : term,X170 : term,X171 : term,X172 : term] : ((((ap @ X169 @ X172)) != ((ap @ X171 @ X170))) | (X169 = X171))) & pushprop_lem2v2 & (induction2_lthm <=> (! [X173 : term] : (((sub @ X173 @ id)) = X173) => (! [X174 : (term > $o),X175 : term,X176 : subst] : (~ ! [X177 : term] : ((X174 @ (sub @ X177 @ X176)) | ~(var @ X177)) | (X174 @ (sub @ X175 @ X176)) | ~ ! [X178 : term] : ((X174 @ (lam @ X178)) | ~ ! [X179 : term] : ((X174 @ (sub @ X178 @ (push @ X179 @ id))) | ~(X174 @ X179))) | ~ ! [X180 : term,X181 : term] : ((X174 @ (ap @ X181 @ X180)) | ~(X174 @ X180) | ~(X174 @ X181))) => ! [X182 : term,X183 : (term > $o)] : ((X183 @ X182) | ~ ! [X184 : term,X185 : term] : (~(X183 @ X185) | ~(X183 @ X184) | (X183 @ (ap @ X185 @ X184))) | ~ ! [X186 : term] : ((X183 @ (lam @ X186)) | ~ ! [X187 : term] : (~(X183 @ X187) | (X183 @ (sub @ X186 @ (push @ X187 @ id))))) | ~ ! [X188 : term] : ((X183 @ X188) | ~(var @ X188)))))) & (! [X189 : term,X190 : (term > $o)] : (~ ! [X191 : term] : ((X190 @ (lam @ X191)) | ~(X190 @ X191)) | ~ ! [X192 : term] : ((X190 @ X192) | ~(var @ X192)) | (X190 @ X189) | ~ ! [X193 : term,X194 : term] : (~(X190 @ X193) | ~(X190 @ X194) | (X190 @ (ap @ X194 @ X193)))) <=> induction) & pushprop_lem3v2 & ~(hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) & (pushprop_lem1_lthm <=> (! [X195 : subst,X196 : term] : (((sub @ one @ (push @ X196 @ X195))) = X196) => (! [X197 : subst,X198 : term] : (((comp @ sh @ (push @ X198 @ X197))) = X197) => ! [X199 : (term > $o),X200 : subst,X201 : term,X202 : term,X203 : (term > $o)] : ((X203 @ (sub @ X202 @ (push @ X201 @ X200))) | ~(X199 @ X202))))) & axassoc & induction2lem_gthm & (induction2lem_lthm <=> (! [X204 : term,X205 : subst] : (((sub @ one @ (push @ X204 @ X205))) = X204) => (! [X206 : term,X207 : subst] : (((comp @ sh @ (push @ X206 @ X207))) = X207) => (! [X208 : subst] : (((comp @ X208 @ id)) = X208) => (! [X209 : (term > $o),X210 : term] : ((X209 @ X210) | ~ ! [X211 : term] : (~(X209 @ X211) | (X209 @ (lam @ X211))) | ~ ! [X212 : term] : (~(var @ X212) | (X209 @ X212)) | ~ ! [X213 : term,X214 : term] : (~(X209 @ X214) | (X209 @ (ap @ X213 @ X214)) | ~(X209 @ X213))) => ! [X215 : (term > $o),X216 : subst,X217 : term] : (~ ! [X218 : term] : (~ ! [X219 : term] : (~(X215 @ X219) | (X215 @ (sub @ X218 @ (push @ X219 @ id)))) | (X215 @ (lam @ X218))) | ~ ! [X220 : term,X221 : term] : (~(X215 @ X220) | ~(X215 @ X221) | (X215 @ (ap @ X221 @ X220))) | (X215 @ (sub @ X217 @ X216)) | ~ ! [X222 : term] : ((X215 @ (sub @ X222 @ X216)) | ~(var @ X222)))))))) & ~ulamvar1 & (hoasinduction_no_psi_cond <=> ! [X223 : term,X224 : (subst > term > subst > $o)] : (~ ! [X225 : (subst > term > term)] : (~ ! [X226 : term,X227 : subst,X228 : subst] : (((sub @ (X225 @ X227 @ X226) @ X228)) = ((X225 @ (comp @ X227 @ X228) @ (sub @ X226 @ X228)))) | (X224 @ id @ (lam @ (X225 @ sh @ one)) @ id) | ~ ! [X229 : term] : (~(X224 @ id @ X229 @ id) | (X224 @ id @ (X225 @ id @ X229) @ id))) | ~ ! [X230 : term,X231 : term] : ((X224 @ id @ (ap @ (sub @ X231 @ id) @ X230) @ id) | ~(X224 @ id @ X231 @ id) | ~(X224 @ id @ X230 @ id)) | (X224 @ id @ X223 @ id))) & (! [X232 : term,X233 : subst,X234 : (term > $o)] : ~ ! [X235 : (term > $o)] : ~ ! [X236 : term] : (((X234 @ (sub @ X236 @ (push @ X232 @ X233)))) = ((X235 @ X236))) <=> pushprop_lem0_lthm) & ! [X237 : d_subst,X238 : d_subst] : ((((d2subst @ X237)) = ((d2subst @ X238))) => (X237 = X238)) & ~lamnotap & ((! [X239 : term,X240 : (term > $o)] : (~ ! [X241 : term] : ((X240 @ X241) | ~(var @ X241)) | ~ ! [X242 : term,X243 : term] : (~(X240 @ X242) | (X240 @ (ap @ X243 @ X242)) | ~(X240 @ X243)) | ~ ! [X244 : term] : (~ ! [X245 : term] : ((X240 @ (sub @ X244 @ (push @ X245 @ id))) | ~(X240 @ X245)) | (X240 @ (lam @ X244))) | (X240 @ X239)) => (! [X246 : (subst > term > subst > $o),X247 : term,X248 : term] : (~(X246 @ id @ X248 @ id) | (X246 @ id @ (ap @ X247 @ X248) @ id) | ~ ! [X249 : term,X250 : term] : ((X246 @ id @ (ap @ (sub @ X249 @ id) @ X250) @ id) | ~(X246 @ id @ X249 @ id) | ~(X246 @ id @ X250 @ id)) | ~(X246 @ id @ X247 @ id) | ~ ! [X251 : term,X252 : subst,X253 : subst,X254 : subst] : (~(X246 @ (comp @ X252 @ X254) @ (sub @ X251 @ X254) @ X253) | (X246 @ X252 @ X251 @ (comp @ X254 @ X253))) | ~ ! [X255 : subst,X256 : subst,X257 : subst,X258 : term] : (~(X246 @ X257 @ X258 @ (comp @ X255 @ X256)) | (X246 @ (comp @ X257 @ X255) @ (sub @ X258 @ X255) @ X256))) => (! [X259 : (subst > term > subst > $o),X260 : term] : (~ ! [X261 : (subst > term > term)] : (~ ! [X262 : subst,X263 : term,X264 : subst] : (((sub @ (X261 @ X262 @ X263) @ X264)) = ((X261 @ (comp @ X262 @ X264) @ (sub @ X263 @ X264)))) | (X259 @ id @ (lam @ (X261 @ sh @ one)) @ id) | ~ ! [X265 : term] : (~(X259 @ id @ X265 @ id) | (X259 @ id @ (X261 @ id @ X265) @ id))) | ~ ! [X266 : subst,X267 : subst,X268 : term,X269 : subst] : ((X259 @ X266 @ X268 @ (comp @ X267 @ X269)) | ~(X259 @ (comp @ X266 @ X267) @ (sub @ X268 @ X267) @ X269)) | ~ ! [X270 : subst,X271 : term,X272 : subst,X273 : subst] : (~(X259 @ X270 @ X271 @ (comp @ X272 @ X273)) | (X259 @ (comp @ X270 @ X272) @ (sub @ X271 @ X272) @ X273)) | (X259 @ id @ (lam @ X260) @ id) | ~ ! [X274 : term] : ((X259 @ id @ (sub @ X260 @ (push @ X274 @ id)) @ id) | ~(X259 @ id @ X274 @ id))) => ! [X275 : term,X276 : (subst > term > subst > $o)] : (~ ! [X277 : (subst > term > term)] : (~ ! [X278 : term] : ((X276 @ id @ (X277 @ id @ X278) @ id) | ~(X276 @ id @ X278 @ id)) | (X276 @ id @ (lam @ (X277 @ sh @ one)) @ id) | ~ ! [X279 : subst,X280 : term,X281 : subst] : (((sub @ (X277 @ X279 @ X280) @ X281)) = ((X277 @ (comp @ X279 @ X281) @ (sub @ X280 @ X281))))) | ~ ! [X282 : term,X283 : subst,X284 : subst,X285 : subst] : (~(X276 @ (comp @ X285 @ X283) @ (sub @ X282 @ X283) @ X284) | (X276 @ X285 @ X282 @ (comp @ X283 @ X284))) | ~ ! [X286 : term] : (~(var @ (sub @ X286 @ id)) | (X276 @ id @ X286 @ id)) | (X276 @ id @ X275 @ id) | ~ ! [X287 : subst,X288 : subst,X289 : term,X290 : subst] : (~(X276 @ X287 @ X289 @ (comp @ X288 @ X290)) | (X276 @ (comp @ X287 @ X288) @ (sub @ X289 @ X288) @ X290)) | ~ ! [X291 : term,X292 : term] : (~(X276 @ id @ X292 @ id) | (X276 @ id @ (ap @ (sub @ X292 @ id) @ X291) @ id) | ~(X276 @ id @ X291 @ id)))))) <=> hoasinduction_lthm) & (! [X293 : term,X294 : term] : ((X293 = X294) | (((lam @ X294)) != ((lam @ X293)))) <=> laminj) & (pushprop_lem1v2 <=> ! [X295 : (term > $o),X296 : term,X297 : (term > $o),X298 : subst] : (~ ! [X299 : term] : (((X295 @ X299)) = ((X297 @ (sub @ X299 @ (push @ X296 @ X298))))) | (X295 @ one) | ~(X297 @ X296))) & pushprop_lem1_gthm & ! [X300 : term] : ? [X301 : d_term] : (((d2term @ X301)) = X300) & hoasinduction_lem1_lthm & (hoasinduction_lem3_lthm <=> (! [X302 : term] : (((sub @ X302 @ id)) = X302) => (! [X303 : (subst > term > subst > $o),X304 : term] : (~ ! [X305 : (subst > term > term)] : (~ ! [X306 : term,X307 : subst,X308 : subst] : (((sub @ (X305 @ X307 @ X306) @ X308)) = ((X305 @ (comp @ X307 @ X308) @ (sub @ X306 @ X308)))) | (X303 @ id @ (lam @ (X305 @ sh @ one)) @ id) | ~ ! [X309 : term] : (~(X303 @ id @ X309 @ id) | (X303 @ id @ (X305 @ id @ X309) @ id))) | (X303 @ id @ (lam @ (sub @ X304 @ (push @ one @ sh))) @ id) | ~ ! [X310 : term] : (~(X303 @ id @ X310 @ id) | (X303 @ id @ (sub @ X304 @ (push @ X310 @ id)) @ id))) => ! [X311 : (subst > term > subst > $o),X312 : term] : (~ ! [X313 : subst,X314 : subst,X315 : term,X316 : subst] : ((X311 @ X313 @ X315 @ (comp @ X316 @ X314)) | ~(X311 @ (comp @ X313 @ X316) @ (sub @ X315 @ X316) @ X314)) | ~ ! [X317 : term,X318 : subst,X319 : subst,X320 : subst] : ((X311 @ (comp @ X318 @ X320) @ (sub @ X317 @ X320) @ X319) | ~(X311 @ X318 @ X317 @ (comp @ X320 @ X319))) | ~ ! [X321 : (subst > term > term)] : ((X311 @ id @ (lam @ (X321 @ sh @ one)) @ id) | ~ ! [X322 : term] : ((X311 @ id @ (X321 @ id @ X322) @ id) | ~(X311 @ id @ X322 @ id)) | ~ ! [X323 : subst,X324 : term,X325 : subst] : (((X321 @ (comp @ X323 @ X325) @ (sub @ X324 @ X325))) = ((sub @ (X321 @ X323 @ X324) @ X325)))) | ~ ! [X326 : term] : (~(X311 @ id @ X326 @ id) | (X311 @ id @ (sub @ X312 @ (push @ X326 @ id)) @ id)) | (X311 @ id @ (lam @ X312) @ id))))) & hoaslamnotvar_gthm & hoaslamnotvar & (! [X327 : term] : ~ ! [X328 : (subst > term > term)] : (((X328 @ sh @ one)) != ((sub @ X327 @ (push @ one @ sh)))) <=> hoasinduction_lem3b_lthm) & (! [X329 : term,X330 : term,X331 : subst,X332 : (term > $o),X333 : (term > $o)] : ((X332 @ (sub @ X330 @ (push @ X329 @ X331))) | ~(X333 @ X330)) <=> pushprop_lem1) & ((! [X334 : subst] : (((comp @ id @ X334)) = X334) => (! [X335 : subst] : (((comp @ X335 @ id)) = X335) => (! [X336 : subst] : (((comp @ id @ X336)) = X336) & ! [X337 : subst] : (((comp @ X337 @ id)) = X337)))) <=> substmonoid_lthm) & ((^[X338 : subst, X339 : (subst > term > term)] : ((d2term @ d_one))) = hoaslam) & (pushprop_lem0 <=> ! [X340 : (term > $o),X341 : term,X342 : subst] : ~ ! [X343 : (term > $o)] : ~ ! [X344 : term] : (((X343 @ X344)) = ((X340 @ (sub @ X344 @ (push @ X341 @ X342)))))) & (axidl <=> ! [X345 : subst] : (((comp @ id @ X345)) = X345)) & pushprop & axabs & (! [X346 : term,X347 : (subst > term > subst > $o)] : (~ ! [X348 : subst,X349 : subst,X350 : subst,X351 : term] : (~(X347 @ X348 @ X351 @ (comp @ X350 @ X349)) | (X347 @ (comp @ X348 @ X350) @ (sub @ X351 @ X350) @ X349)) | ~ ! [X352 : term] : (~(X347 @ id @ X352 @ id) | (X347 @ id @ (sub @ X346 @ (push @ X352 @ id)) @ id)) | (X347 @ id @ (lam @ X346) @ id) | ~ ! [X353 : term,X354 : subst,X355 : subst,X356 : subst] : ((X347 @ X356 @ X353 @ (comp @ X355 @ X354)) | ~(X347 @ (comp @ X356 @ X355) @ (sub @ X353 @ X355) @ X354)) | ~ ! [X357 : (subst > term > term)] : (~ ! [X358 : subst,X359 : term,X360 : subst] : (((sub @ (X357 @ X360 @ X359) @ X358)) = ((X357 @ (comp @ X360 @ X358) @ (sub @ X359 @ X358)))) | ~ ! [X361 : term] : ((X347 @ id @ (X357 @ id @ X361) @ id) | ~(X347 @ id @ X361 @ id)) | (X347 @ id @ (lam @ (X357 @ sh @ one)) @ id))) <=> hoasinduction_lem3) & hoasinduction_lem1 & hoasapnotvar & (pushprop_p_and_p_prime = (^[X362 : term, X363 : subst, X364 : (term > $o), X365 : (term > $o)] : (! [X366 : term] : (((X364 @ (sub @ X366 @ (push @ X362 @ X363)))) = ((X365 @ X366)))))) & (! [X367 : (subst > term > subst > $o)] : ~ ! [X368 : (term > $o)] : ~ ! [X369 : term] : (((X367 @ id @ X369 @ id)) = ((X368 @ X369))) <=> hoasinduction_lem0_lthm) & ! [X370 : subst] : ? [X371 : d_subst] : (((d2subst @ X371)) = X370) & pushprop_lem2v2_gthm & (! [X372 : subst,X373 : term] : (((comp @ sh @ (push @ X373 @ X372))) = X372) <=> axshiftcons) & (apinj2 <=> ! [X374 : term,X375 : term,X376 : term,X377 : term] : ((((ap @ X376 @ X374)) != ((ap @ X377 @ X375))) | (X374 = X375))) & (hoasinduction_lem3b <=> ! [X378 : term] : ~ ! [X379 : (subst > term > term)] : (((X379 @ sh @ one)) != ((sub @ X378 @ (push @ one @ sh))))) & (! [X380 : term] : ~ ! [X381 : (subst > term > term)] : ~ ! [X382 : term,X383 : subst] : (((X381 @ X383 @ X382)) = ((sub @ X380 @ (push @ X382 @ X383)))) <=> hoasinduction_lem3v2_f) & hoasapnotvar_lthm & (! [X384 : (subst > term > subst > $o),X385 : term] : (~ ! [X386 : term] : ((X384 @ id @ (sub @ X385 @ (push @ X386 @ id)) @ id) | ~(X384 @ id @ X386 @ id)) | ~ ! [X387 : term,X388 : (subst > term > term)] : (~ ! [X389 : term,X390 : subst,X391 : subst] : (((sub @ X387 @ (push @ (sub @ X389 @ X390) @ (comp @ X391 @ X390)))) = ((X388 @ (comp @ X391 @ X390) @ (sub @ X389 @ X390)))) | (X384 @ id @ (lam @ (X388 @ sh @ one)) @ id) | ~ ! [X392 : term,X393 : subst,X394 : subst] : (((sub @ (sub @ X387 @ (push @ X392 @ X394)) @ X393)) = ((sub @ (X388 @ X394 @ X392) @ X393))) | ~ ! [X395 : term] : ((X384 @ id @ (X388 @ id @ X395) @ id) | ~(X384 @ id @ X395 @ id))) | (X384 @ id @ (lam @ (sub @ X385 @ (push @ one @ sh))) @ id)) <=> hoasinduction_lem3aaa) & (! [X396 : (subst > term > subst > $o),X397 : term] : (~ ! [X398 : (subst > term > term)] : (~ ! [X399 : subst,X400 : subst,X401 : term] : (((sub @ (X398 @ X400 @ X401) @ X399)) = ((X398 @ (comp @ X400 @ X399) @ (sub @ X401 @ X399)))) | ~ ! [X402 : term] : (~(X396 @ id @ X402 @ id) | (X396 @ id @ (X398 @ id @ X402) @ id)) | (X396 @ id @ (lam @ (X398 @ sh @ one)) @ id)) | ~ ! [X403 : term] : ((X396 @ id @ (sub @ X397 @ (push @ X403 @ id)) @ id) | ~(X396 @ id @ X403 @ id)) | (X396 @ id @ (lam @ X397) @ id)) <=> hoasinduction_lem3a) & hoasinduction_lem3v2_gthm & (sh = ((d2subst @ d_id))) & hoasapnotvar_gthm & hoasapinj2_gthm & (axscons <=> ! [X404 : subst] : (((push @ (sub @ one @ X404) @ (comp @ sh @ X404))) = X404)) & hoasinduction_lem1v2_gthm & ((! [X405 : term] : (((sub @ X405 @ id)) = X405) => ! [X406 : term] : (((sub @ X406 @ id)) = X406)) <=> termmset_lthm) & axvarshift & (hoasinduction <=> ! [X407 : term,X408 : (subst > term > subst > $o)] : (~ ! [X409 : subst,X410 : subst,X411 : subst,X412 : term] : (~(X408 @ (comp @ X411 @ X409) @ (sub @ X412 @ X409) @ X410) | (X408 @ X411 @ X412 @ (comp @ X409 @ X410))) | ~ ! [X413 : (subst > term > term)] : ((X408 @ id @ (lam @ (X413 @ sh @ one)) @ id) | ~ ! [X414 : term] : (~(X408 @ id @ X414 @ id) | (X408 @ id @ (X413 @ id @ X414) @ id)) | ~ ! [X415 : subst,X416 : subst,X417 : term] : (((sub @ (X413 @ X416 @ X417) @ X415)) = ((X413 @ (comp @ X416 @ X415) @ (sub @ X417 @ X415))))) | ~ ! [X418 : term,X419 : term] : (~(X408 @ id @ X418 @ id) | ~(X408 @ id @ X419 @ id) | (X408 @ id @ (ap @ (sub @ X418 @ id) @ X419) @ id)) | ~ ! [X420 : term] : ((X408 @ id @ X420 @ id) | ~(var @ (sub @ X420 @ id))) | (X408 @ id @ X407 @ id) | ~ ! [X421 : subst,X422 : term,X423 : subst,X424 : subst] : (~(X408 @ X421 @ X422 @ (comp @ X424 @ X423)) | (X408 @ (comp @ X421 @ X424) @ (sub @ X422 @ X424) @ X423)))) & (hoasinduction_lem3v2_lthm <=> (! [X425 : term] : (((sub @ X425 @ id)) = X425) => ! [X426 : (subst > term > subst > $o),X427 : term,X428 : (term > $o)] : (~ ! [X429 : subst,X430 : term,X431 : subst,X432 : subst] : (~(X426 @ X432 @ X430 @ (comp @ X429 @ X431)) | (X426 @ (comp @ X432 @ X429) @ (sub @ X430 @ X429) @ X431)) | (X428 @ (lam @ X427)) | ~ ! [X433 : term] : ((X428 @ (sub @ X427 @ (push @ X433 @ id))) | ~(X428 @ X433)) | ~ ! [X434 : subst,X435 : subst,X436 : subst,X437 : term] : ((X426 @ X434 @ X437 @ (comp @ X436 @ X435)) | ~(X426 @ (comp @ X434 @ X436) @ (sub @ X437 @ X436) @ X435)) | ~ ! [X438 : (subst > term > term)] : ((X426 @ id @ (lam @ (X438 @ sh @ one)) @ id) | ~ ! [X439 : term,X440 : subst,X441 : subst] : (((sub @ (X438 @ X441 @ X439) @ X440)) = ((X438 @ (comp @ X441 @ X440) @ (sub @ X439 @ X440)))) | ~ ! [X442 : term] : (~(X426 @ id @ X442 @ id) | (X426 @ id @ (X438 @ id @ X442) @ id))) | ~ ! [X443 : term] : (((X428 @ X443)) = ((X426 @ id @ X443 @ id)))))) & lamnotvar & pushprop_lem0_gthm & (! [X444 : (subst > term > subst > $o),X445 : term,X446 : (term > $o)] : (~ ! [X447 : term] : (((X446 @ X447)) = ((X444 @ id @ X447 @ id))) | (X446 @ (lam @ X445)) | ~ ! [X448 : (subst > term > term)] : (~ ! [X449 : subst,X450 : subst,X451 : term] : (((X448 @ (comp @ X449 @ X450) @ (sub @ X451 @ X450))) = ((sub @ (X448 @ X449 @ X451) @ X450))) | (X444 @ id @ (lam @ (X448 @ sh @ one)) @ id) | ~ ! [X452 : term] : ((X444 @ id @ (X448 @ id @ X452) @ id) | ~(X444 @ id @ X452 @ id))) | ~ ! [X453 : term] : (~(X446 @ X453) | (X446 @ (sub @ X445 @ (push @ X453 @ id))))) <=> hoasinduction_lem3v2a) & (! [X454 : subst] : (((comp @ X454 @ id)) = X454) <=> axidr) & (id = ((d2subst @ d_id))) & (! [X455 : term,X456 : term,X457 : (subst > term > term)] : ((((lam @ (X457 @ sh @ one))) != ((ap @ (sub @ X455 @ id) @ X456))) | ~ ! [X458 : subst,X459 : subst,X460 : term] : (((sub @ (X457 @ X458 @ X460) @ X459)) = ((X457 @ (comp @ X458 @ X459) @ (sub @ X460 @ X459))))) <=> hoaslamnotap) & (hoasapinj1_lthm <=> (! [X461 : term] : (((sub @ X461 @ id)) = X461) => (! [X462 : term,X463 : term,X464 : term,X465 : term] : ((((ap @ X465 @ X464)) != ((ap @ X462 @ X463))) | (X462 = X465)) => ! [X466 : term,X467 : term,X468 : term,X469 : term] : ((((ap @ (sub @ X469 @ id) @ X467)) != ((ap @ (sub @ X468 @ id) @ X466))) | (X468 = X469))))) & pushprop_lem2v2_lthm & ulamvarsh & axapp & substmonoid_gthm & (one = ((d2term @ d_one))) & hoasinduction_lem2v2_gthm & (! [X470 : (subst > term > term),X471 : subst,X472 : (subst > term > term),X473 : term] : ((((X470 @ X471 @ X473)) = ((X472 @ X471 @ X473))) | ~ ! [X474 : subst,X475 : subst,X476 : term] : (((X470 @ (comp @ X475 @ X474) @ (sub @ X476 @ X474))) = ((sub @ (X470 @ X475 @ X476) @ X474))) | (((lam @ (X470 @ sh @ one))) != ((lam @ (X472 @ sh @ one)))) | ~ ! [X477 : subst,X478 : subst,X479 : term] : (((sub @ (X472 @ X477 @ X479) @ X478)) = ((X472 @ (comp @ X477 @ X478) @ (sub @ X479 @ X478))))) <=> hoaslaminj) & (((lam @ (d2term @ d_one))) = ((d2term @ d_one))) & (hoasinduction_lem2v2 <=> ! [X480 : term,X481 : (term > $o),X482 : (subst > term > subst > $o),X483 : term] : ((X481 @ (ap @ X480 @ X483)) | ~ ! [X484 : term] : (((X481 @ X484)) = ((X482 @ id @ X484 @ id))) | ~ ! [X485 : term,X486 : term] : (~(X482 @ id @ X486 @ id) | ~(X482 @ id @ X485 @ id) | (X482 @ id @ (ap @ (sub @ X486 @ id) @ X485) @ id)) | ~ ! [X487 : subst,X488 : subst,X489 : term,X490 : subst] : ((X482 @ X488 @ X489 @ (comp @ X490 @ X487)) | ~(X482 @ (comp @ X488 @ X490) @ (sub @ X489 @ X490) @ X487)) | ~(X481 @ X483) | ~(X481 @ X480) | ~ ! [X491 : subst,X492 : subst,X493 : term,X494 : subst] : (~(X482 @ X491 @ X493 @ (comp @ X494 @ X492)) | (X482 @ (comp @ X491 @ X494) @ (sub @ X493 @ X494) @ X492)))) & hoasinduction_lem3a_gthm & (! [X495 : (term > $o),X496 : term] : ((X495 @ X496) | ~ ! [X497 : term] : (~(var @ X497) | (X495 @ X497)) | ~ ! [X498 : term,X499 : term] : (~(X495 @ X498) | (X495 @ (ap @ X498 @ X499)) | ~(X495 @ X499)) | ~ ! [X500 : term] : ((X495 @ (lam @ X500)) | ~ ! [X501 : term] : ((X495 @ (sub @ X500 @ (push @ X501 @ id))) | ~(X495 @ X501)))) <=> induction2) & ulamvarind & ~(var @ (d2term @ d_one)) & (! [X502 : term,X503 : (subst > term > subst > $o),X504 : (term > $o)] : ((X504 @ (lam @ X502)) | ~ ! [X505 : term] : (((X503 @ id @ X505 @ id)) = ((X504 @ X505))) | ~ ! [X506 : (subst > term > term)] : (~ ! [X507 : term] : (~(X503 @ id @ X507 @ id) | (X503 @ id @ (X506 @ id @ X507) @ id)) | ~ ! [X508 : subst,X509 : term,X510 : subst] : (((X506 @ (comp @ X508 @ X510) @ (sub @ X509 @ X510))) = ((sub @ (X506 @ X508 @ X509) @ X510))) | (X503 @ id @ (lam @ (X506 @ sh @ one)) @ id)) | ~ ! [X511 : term,X512 : subst,X513 : subst,X514 : subst] : (~(X503 @ X514 @ X511 @ (comp @ X513 @ X512)) | (X503 @ (comp @ X514 @ X513) @ (sub @ X511 @ X513) @ X512)) | ~ ! [X515 : term] : (~(X504 @ X515) | (X504 @ (sub @ X502 @ (push @ X515 @ id)))) | ~ ! [X516 : subst,X517 : term,X518 : subst,X519 : subst] : (~(X503 @ (comp @ X519 @ X516) @ (sub @ X517 @ X516) @ X518) | (X503 @ X519 @ X517 @ (comp @ X516 @ X518)))) <=> hoasinduction_lem3v2) & hoasinduction_lem1_gthm & (! [X520 : term] : (((sub @ X520 @ id)) = X520) <=> termmset) & hoaslaminj_gthm & axclos & hoasinduction_lem2_lthm & ((! [X521 : subst] : (((comp @ X521 @ id)) = X521) & ! [X522 : subst] : (((comp @ id @ X522)) = X522)) <=> substmonoid) & axmap & (hoasapinj2_lthm <=> (! [X523 : term,X524 : term,X525 : term,X526 : term] : ((X523 = X524) | (((ap @ X525 @ X524)) != ((ap @ X526 @ X523)))) => ! [X527 : term,X528 : term,X529 : term,X530 : term] : ((X527 = X530) | (((ap @ (sub @ X529 @ id) @ X527)) != ((ap @ (sub @ X528 @ id) @ X530)))))) & (! [X531 : term,X532 : term,X533 : term,X534 : term] : ((X531 = X534) | (((ap @ (sub @ X532 @ id) @ X534)) != ((ap @ (sub @ X533 @ id) @ X531)))) <=> hoasapinj2) & (hoasinduction_lem3aa_lthm <=> ! [X535 : term,X536 : (subst > term > subst > $o)] : (~ ! [X537 : term] : ((X536 @ id @ (sub @ X535 @ (push @ X537 @ id)) @ id) | ~(X536 @ id @ X537 @ id)) | ~ ! [X538 : (subst > term > term)] : (~ ! [X539 : term,X540 : subst,X541 : subst] : (((X538 @ (comp @ X540 @ X541) @ (sub @ X539 @ X541))) = ((sub @ (X538 @ X540 @ X539) @ X541))) | (X536 @ id @ (lam @ (X538 @ sh @ one)) @ id) | ~ ! [X542 : term] : ((X536 @ id @ (X538 @ id @ X542) @ id) | ~(X536 @ id @ X542 @ id))) | (X536 @ id @ (lam @ (sub @ X535 @ (push @ one @ sh))) @ id))) & ((! [X543 : term,X544 : subst] : (((sub @ one @ (push @ X543 @ X544))) = X543) => ! [X545 : term,X546 : (term > $o),X547 : subst,X548 : (term > $o)] : ((X548 @ one) | ~ ! [X549 : term] : (((X548 @ X549)) = ((X546 @ (sub @ X549 @ (push @ X545 @ X547))))) | ~(X546 @ X545))) <=> pushprop_lem1v2_lthm) & hoaslamnotap_lthm), inference(rectify,[],[f3])). thf(f7,plain,( (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (pushprop_gthm = $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)))) & (! [X0 : term,X1 : term] : ((X0 = X1) | (((sub @ X0 @ sh)) != ((sub @ X1 @ sh)))) <=> (shinj = $true)) & (hoaslamnotvar_lthm = $true) & ((hoaslaminj_lthm = $true) <=> (! [X3 : subst,X2 : term] : (((sub @ one @ (push @ X2 @ X3))) = X2) => (! [X4 : term,X5 : subst] : (((comp @ sh @ (push @ X4 @ X5))) = X5) => (! [X6 : term,X7 : term] : ((((lam @ X6)) != ((lam @ X7))) | (X6 = X7)) => ! [X9 : (subst > term > term),X8 : subst,X10 : term,X11 : (subst > term > term)] : (~ ! [X12 : subst,X14 : subst,X13 : term] : (((X9 @ (comp @ X14 @ X12) @ (sub @ X13 @ X12))) = ((sub @ (X9 @ X14 @ X13) @ X12))) | (((X9 @ X8 @ X10)) = ((X11 @ X8 @ X10))) | (((lam @ (X9 @ sh @ one))) != ((lam @ (X11 @ sh @ one)))) | ~ ! [X15 : term,X17 : subst,X16 : subst] : (((sub @ (X11 @ X16 @ X15) @ X17)) = ((X11 @ (comp @ X16 @ X17) @ (sub @ X15 @ X17))))))))) & (hoasinduction_lem1v2 = $true) & (hoaslamnotap_gthm = $true) & (! [X18 : term,X19 : term,X20 : (subst > term > subst > $o)] : (($true = ((X20 @ id @ (ap @ X19 @ X18) @ id))) | ~ ($true = ((X20 @ id @ X19 @ id))) | ~ ! [X21 : subst,X22 : term,X23 : subst,X24 : subst] : (~ ($true = ((X20 @ (comp @ X21 @ X24) @ (sub @ X22 @ X24) @ X23))) | ($true = ((X20 @ X21 @ X22 @ (comp @ X24 @ X23))))) | ~ ! [X25 : term,X26 : subst,X27 : subst,X28 : subst] : (~ ($true = ((X20 @ X28 @ X25 @ (comp @ X26 @ X27)))) | ($true = ((X20 @ (comp @ X28 @ X26) @ (sub @ X25 @ X26) @ X27)))) | ~ (((X20 @ id @ X18 @ id)) = $true) | ~ ! [X29 : term,X30 : term] : (~ (((X20 @ id @ X29 @ id)) = $true) | ~ ($true = ((X20 @ id @ X30 @ id))) | ($true = ((X20 @ id @ (ap @ (sub @ X29 @ id) @ X30) @ id))))) <=> (hoasinduction_lem2 = $true)) & ! [X31 : d_subst] : (d_id = X31) & (! [X32 : term,X33 : term,X34 : term,X35 : term] : ((((ap @ (sub @ X34 @ id) @ X33)) != ((ap @ (sub @ X32 @ id) @ X35))) | (X32 = X34)) <=> (hoasapinj1 = $true)) & ! [X36 : d_term] : (d_one = X36) & (pushprop_lem1v2_gthm = $true) & ((! [X37 : term] : (((sub @ X37 @ id)) = X37) => (! [X38 : (subst > term > subst > $o),X39 : term] : (($true = ((X38 @ id @ (lam @ (sub @ X39 @ (push @ one @ sh))) @ id))) | ~ ! [X40 : (subst > term > term)] : (~ ! [X42 : subst,X43 : subst,X41 : term] : (((X40 @ (comp @ X42 @ X43) @ (sub @ X41 @ X43))) = ((sub @ (X40 @ X42 @ X41) @ X43))) | ~ ! [X44 : term] : ((((X38 @ id @ (X40 @ id @ X44) @ id)) = $true) | ~ ($true = ((X38 @ id @ X44 @ id)))) | ($true = ((X38 @ id @ (lam @ (X40 @ sh @ one)) @ id)))) | ~ ! [X45 : term] : (~ ($true = ((X38 @ id @ X45 @ id))) | ($true = ((X38 @ id @ (sub @ X39 @ (push @ X45 @ id)) @ id))))) => ! [X46 : term,X47 : (subst > term > subst > $o)] : (~ ! [X48 : (subst > term > term)] : (~ ! [X49 : term] : (~ ($true = ((X47 @ id @ X49 @ id))) | (((X47 @ id @ (X48 @ id @ X49) @ id)) = $true)) | ($true = ((X47 @ id @ (lam @ (X48 @ sh @ one)) @ id))) | ~ ! [X50 : subst,X51 : term,X52 : subst] : (((sub @ (X48 @ X50 @ X51) @ X52)) = ((X48 @ (comp @ X50 @ X52) @ (sub @ X51 @ X52))))) | ~ ! [X53 : term] : (($true = ((X47 @ id @ (sub @ X46 @ (push @ X53 @ id)) @ id))) | ~ ($true = ((X47 @ id @ X53 @ id)))) | (((X47 @ id @ (lam @ X46) @ id)) = $true)))) <=> (hoasinduction_lem3a_lthm = $true)) & (pushprop_lthm = $true) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (! [X54 : (subst > term > subst > $o)] : ~ ! [X55 : (term > $o)] : ~ ! [X56 : term] : (((X54 @ id @ X56 @ id)) = ((X55 @ X56))) => (! [X57 : (term > $o),X58 : term] : (~ ! [X59 : term,X60 : term] : (~ ($true = ((X57 @ X59))) | ~ ($true = ((X57 @ X60))) | (((X57 @ (ap @ X59 @ X60))) = $true)) | ($true = ((X57 @ X58))) | ~ ! [X61 : term] : ((((X57 @ X61)) = $true) | ~ ($true = ((var @ X61)))) | ~ ! [X62 : term] : (($true = ((X57 @ (lam @ X62)))) | ~ ! [X63 : term] : ((((X57 @ (sub @ X62 @ (push @ X63 @ id)))) = $true) | ~ ($true = ((X57 @ X63)))))) => (! [X64 : term] : (((sub @ X64 @ id)) = X64) => (! [X65 : term,X66 : (subst > term > subst > $o),X67 : (term > $o)] : (~ ! [X68 : term] : (($true = ((X67 @ (sub @ X65 @ (push @ X68 @ id))))) | ~ ($true = ((X67 @ X68)))) | ($true = ((X67 @ (lam @ X65)))) | ~ ! [X69 : (subst > term > term)] : (($true = ((X66 @ id @ (lam @ (X69 @ sh @ one)) @ id))) | ~ ! [X72 : term,X71 : subst,X70 : subst] : (((sub @ (X69 @ X70 @ X72) @ X71)) = ((X69 @ (comp @ X70 @ X71) @ (sub @ X72 @ X71)))) | ~ ! [X73 : term] : (($true = ((X66 @ id @ (X69 @ id @ X73) @ id))) | ~ ($true = ((X66 @ id @ X73 @ id))))) | ~ ! [X74 : term] : (((X66 @ id @ X74 @ id)) = ((X67 @ X74)))) => ! [X75 : term,X76 : (subst > term > subst > $o)] : (($true = ((X76 @ id @ X75 @ id))) | ~ ! [X77 : (subst > term > term)] : (~ ! [X79 : subst,X80 : term,X78 : subst] : (((sub @ (X77 @ X78 @ X80) @ X79)) = ((X77 @ (comp @ X78 @ X79) @ (sub @ X80 @ X79)))) | ~ ! [X81 : term] : (($true = ((X76 @ id @ (X77 @ id @ X81) @ id))) | ~ ($true = ((X76 @ id @ X81 @ id)))) | ($true = ((X76 @ id @ (lam @ (X77 @ sh @ one)) @ id)))) | ~ ! [X82 : term,X83 : term] : (($true = ((X76 @ id @ (ap @ (sub @ X82 @ id) @ X83) @ id))) | ~ ($true = ((X76 @ id @ X83 @ id))) | ~ ($true = ((X76 @ id @ X82 @ id)))))))))) & (hoasinduction_lem2_gthm = $true) & (apnotvar = $true) & ((induction2lem = $true) <=> ! [X84 : (term > $o),X85 : term,X86 : subst] : ((((X84 @ (sub @ X85 @ X86))) = $true) | ~ ! [X87 : term,X88 : term] : (~ ($true = ((X84 @ X87))) | ($true = ((X84 @ (ap @ X88 @ X87)))) | ~ ($true = ((X84 @ X88)))) | ~ ! [X89 : term] : (($true = ((X84 @ (sub @ X89 @ X86)))) | ~ ($true = ((var @ X89)))) | ~ ! [X90 : term] : (~ ! [X91 : term] : (~ ($true = ((X84 @ X91))) | ($true = ((X84 @ (sub @ X90 @ (push @ X91 @ id)))))) | ($true = ((X84 @ (lam @ X90))))))) & (! [X92 : term] : (((sub @ X92 @ id)) = X92) <=> (axvarid = $true)) & ((hoasinduction_lthm_3 = $true) <=> (! [X93 : (subst > term > subst > $o)] : ~ ! [X94 : (term > $o)] : ~ ! [X95 : term] : (((X93 @ id @ X95 @ id)) = ((X94 @ X95))) => (! [X96 : (term > $o),X97 : term] : (($true = ((X96 @ X97))) | ~ ! [X98 : term] : (($true = ((X96 @ X98))) | ~ ($true = ((var @ X98)))) | ~ ! [X99 : term] : (~ ! [X100 : term] : (($true = ((X96 @ (sub @ X99 @ (push @ X100 @ id))))) | ~ ($true = ((X96 @ X100)))) | ($true = ((X96 @ (lam @ X99))))) | ~ ! [X101 : term,X102 : term] : (~ (((X96 @ X102)) = $true) | ~ (((X96 @ X101)) = $true) | (((X96 @ (ap @ X101 @ X102))) = $true))) => (! [X103 : term] : (((sub @ X103 @ id)) = X103) => (! [X104 : term,X105 : (subst > term > subst > $o),X106 : (term > $o)] : (~ ! [X107 : term] : (((X105 @ id @ X107 @ id)) = ((X106 @ X107))) | ~ ! [X108 : (subst > term > term)] : (($true = ((X105 @ id @ (lam @ (X108 @ sh @ one)) @ id))) | ~ ! [X109 : term] : (($true = ((X105 @ id @ (X108 @ id @ X109) @ id))) | ~ ($true = ((X105 @ id @ X109 @ id)))) | ~ ! [X111 : term,X110 : subst,X112 : subst] : (((X108 @ (comp @ X110 @ X112) @ (sub @ X111 @ X112))) = ((sub @ (X108 @ X110 @ X111) @ X112)))) | ($true = ((X106 @ (lam @ X104)))) | ~ ! [X113 : term] : (~ (((X106 @ X113)) = $true) | ($true = ((X106 @ (sub @ X104 @ (push @ X113 @ id))))))) => ! [X114 : (subst > term > subst > $o),X115 : term] : (~ ! [X116 : term,X117 : subst,X118 : subst,X119 : subst] : (($true = ((X114 @ (comp @ X118 @ X119) @ (sub @ X116 @ X119) @ X117))) | ~ ($true = ((X114 @ X118 @ X116 @ (comp @ X119 @ X117))))) | ~ ! [X120 : term,X121 : term] : (~ ($true = ((X114 @ id @ X120 @ id))) | (((X114 @ id @ (ap @ (sub @ X120 @ id) @ X121) @ id)) = $true) | ~ ($true = ((X114 @ id @ X121 @ id)))) | ($true = ((X114 @ id @ X115 @ id))) | ~ ! [X122 : term,X123 : subst,X124 : subst,X125 : subst] : (($true = ((X114 @ X124 @ X122 @ (comp @ X123 @ X125)))) | ~ ($true = ((X114 @ (comp @ X124 @ X123) @ (sub @ X122 @ X123) @ X125)))) | ~ ! [X126 : (subst > term > term)] : (~ ! [X127 : term,X129 : subst,X128 : subst] : (((X126 @ (comp @ X129 @ X128) @ (sub @ X127 @ X128))) = ((sub @ (X126 @ X129 @ X127) @ X128))) | ($true = ((X114 @ id @ (lam @ (X126 @ sh @ one)) @ id))) | ~ ! [X130 : term] : ((((X114 @ id @ (X126 @ id @ X130) @ id)) = $true) | ~ (((X114 @ id @ X130 @ id)) = $true))) | ~ ! [X131 : term] : (~ ($true = ((var @ (sub @ X131 @ id)))) | ($true = ((X114 @ id @ X131 @ id)))))))))) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (hoasinduction_gthm = $true) & ((hoasinduction_lem3aa = $true) <=> ! [X132 : (subst > term > subst > $o),X133 : term] : (~ ! [X134 : (subst > term > term)] : (~ ! [X135 : term] : (($true = ((X132 @ id @ (X134 @ id @ X135) @ id))) | ~ ($true = ((X132 @ id @ X135 @ id)))) | ~ ! [X138 : subst,X137 : subst,X136 : term] : (((sub @ (X134 @ X138 @ X136) @ X137)) = ((X134 @ (comp @ X138 @ X137) @ (sub @ X136 @ X137)))) | ($true = ((X132 @ id @ (lam @ (X134 @ sh @ one)) @ id)))) | ~ ! [X139 : term] : (~ ($true = ((X132 @ id @ X139 @ id))) | ($true = ((X132 @ id @ (sub @ X133 @ (push @ X139 @ id)) @ id)))) | ($true = ((X132 @ id @ (lam @ (sub @ X133 @ (push @ one @ sh))) @ id))))) & (! [X140 : (subst > term > subst > $o)] : ~ ! [X141 : (term > $o)] : ~ ! [X142 : term] : (((X141 @ X142)) = ((X140 @ id @ X142 @ id))) <=> (hoasinduction_lem0 = $true)) & (pushprop_lthm_orig = $true) & (pushprop_lem3v2_lthm = $true) & (hoasinduction_lem3b_gthm = $true) & ! [X143 : d_term,X144 : d_term] : ((((d2term @ X144)) = ((d2term @ X143))) => (X143 = X144)) & (termmset_gthm = $true) & (hoasinduction_lem3_gthm = $true) & (hoasapinj1_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & ((! [X148 : term] : ~ ! [X149 : (subst > term > term)] : ~ ! [X150 : term,X151 : subst] : (((X149 @ X151 @ X150)) = ((sub @ X148 @ (push @ X150 @ X151)))) => (! [X152 : term] : (((sub @ X152 @ id)) = X152) => ! [X153 : (term > $o),X154 : (subst > term > subst > $o),X155 : term] : (~ ! [X156 : (subst > term > term)] : (~ ! [X157 : term,X158 : subst,X159 : subst] : (((sub @ (X156 @ X159 @ X157) @ X158)) = ((X156 @ (comp @ X159 @ X158) @ (sub @ X157 @ X158)))) | (((X154 @ id @ (lam @ (X156 @ sh @ one)) @ id)) = $true) | ~ ! [X160 : term] : (($true = ((X154 @ id @ (X156 @ id @ X160) @ id))) | ~ ($true = ((X154 @ id @ X160 @ id))))) | ($true = ((X153 @ (lam @ X155)))) | ~ ! [X161 : term] : (((X153 @ X161)) = ((X154 @ id @ X161 @ id))) | ~ ! [X162 : term] : (($true = ((X153 @ (sub @ X155 @ (push @ X162 @ id))))) | ~ ($true = ((X153 @ X162))))))) <=> (hoasinduction_lem3v2a_lthm = $true)) & (induction2_gthm = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (! [X163 : term] : ~ ! [X164 : (subst > term > term)] : ~ ! [X165 : subst,X166 : term] : (((X164 @ X165 @ X166)) = ((sub @ X163 @ (push @ X166 @ X165)))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & ((axvarcons = $true) <=> ! [X167 : term,X168 : subst] : (((sub @ one @ (push @ X167 @ X168))) = X167)) & ((apinj1 = $true) <=> ! [X169 : term,X170 : term,X171 : term,X172 : term] : ((((ap @ X169 @ X172)) != ((ap @ X171 @ X170))) | (X169 = X171))) & (pushprop_lem2v2 = $true) & ((induction2_lthm = $true) <=> (! [X173 : term] : (((sub @ X173 @ id)) = X173) => (! [X174 : (term > $o),X175 : term,X176 : subst] : (~ ! [X177 : term] : (($true = ((X174 @ (sub @ X177 @ X176)))) | ~ ($true = ((var @ X177)))) | ($true = ((X174 @ (sub @ X175 @ X176)))) | ~ ! [X178 : term] : (($true = ((X174 @ (lam @ X178)))) | ~ ! [X179 : term] : (($true = ((X174 @ (sub @ X178 @ (push @ X179 @ id))))) | ~ ($true = ((X174 @ X179))))) | ~ ! [X180 : term,X181 : term] : (($true = ((X174 @ (ap @ X181 @ X180)))) | ~ ($true = ((X174 @ X180))) | ~ ($true = ((X174 @ X181))))) => ! [X182 : term,X183 : (term > $o)] : (($true = ((X183 @ X182))) | ~ ! [X184 : term,X185 : term] : (~ ($true = ((X183 @ X185))) | ~ ($true = ((X183 @ X184))) | ($true = ((X183 @ (ap @ X185 @ X184))))) | ~ ! [X186 : term] : (($true = ((X183 @ (lam @ X186)))) | ~ ! [X187 : term] : (~ ($true = ((X183 @ X187))) | ($true = ((X183 @ (sub @ X186 @ (push @ X187 @ id))))))) | ~ ! [X188 : term] : (($true = ((X183 @ X188))) | ~ ($true = ((var @ X188)))))))) & (! [X189 : term,X190 : (term > $o)] : (~ ! [X191 : term] : (($true = ((X190 @ (lam @ X191)))) | ~ ($true = ((X190 @ X191)))) | ~ ! [X192 : term] : (($true = ((X190 @ X192))) | ~ ($true = ((var @ X192)))) | ($true = ((X190 @ X189))) | ~ ! [X193 : term,X194 : term] : (~ ($true = ((X190 @ X193))) | ~ (((X190 @ X194)) = $true) | ($true = ((X190 @ (ap @ X194 @ X193)))))) <=> (induction = $true)) & (pushprop_lem3v2 = $true) & ~ (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) = $true) & ((pushprop_lem1_lthm = $true) <=> (! [X195 : subst,X196 : term] : (((sub @ one @ (push @ X196 @ X195))) = X196) => (! [X197 : subst,X198 : term] : (((comp @ sh @ (push @ X198 @ X197))) = X197) => ! [X199 : (term > $o),X200 : subst,X201 : term,X202 : term,X203 : (term > $o)] : (($true = ((X203 @ (sub @ X202 @ (push @ X201 @ X200))))) | ~ ($true = ((X199 @ X202))))))) & (axassoc = $true) & (induction2lem_gthm = $true) & ((induction2lem_lthm = $true) <=> (! [X204 : term,X205 : subst] : (((sub @ one @ (push @ X204 @ X205))) = X204) => (! [X206 : term,X207 : subst] : (((comp @ sh @ (push @ X206 @ X207))) = X207) => (! [X208 : subst] : (((comp @ X208 @ id)) = X208) => (! [X209 : (term > $o),X210 : term] : (($true = ((X209 @ X210))) | ~ ! [X211 : term] : (~ ($true = ((X209 @ X211))) | ($true = ((X209 @ (lam @ X211))))) | ~ ! [X212 : term] : (~ ($true = ((var @ X212))) | ($true = ((X209 @ X212)))) | ~ ! [X213 : term,X214 : term] : (~ ($true = ((X209 @ X214))) | ($true = ((X209 @ (ap @ X213 @ X214)))) | ~ ($true = ((X209 @ X213))))) => ! [X215 : (term > $o),X216 : subst,X217 : term] : (~ ! [X218 : term] : (~ ! [X219 : term] : (~ ($true = ((X215 @ X219))) | ($true = ((X215 @ (sub @ X218 @ (push @ X219 @ id)))))) | ($true = ((X215 @ (lam @ X218))))) | ~ ! [X220 : term,X221 : term] : (~ ($true = ((X215 @ X220))) | ~ ($true = ((X215 @ X221))) | ($true = ((X215 @ (ap @ X221 @ X220))))) | ($true = ((X215 @ (sub @ X217 @ X216)))) | ~ ! [X222 : term] : (($true = ((X215 @ (sub @ X222 @ X216)))) | ~ ($true = ((var @ X222)))))))))) & ~ (ulamvar1 = $true) & ((hoasinduction_no_psi_cond = $true) <=> ! [X223 : term,X224 : (subst > term > subst > $o)] : (~ ! [X225 : (subst > term > term)] : (~ ! [X226 : term,X227 : subst,X228 : subst] : (((sub @ (X225 @ X227 @ X226) @ X228)) = ((X225 @ (comp @ X227 @ X228) @ (sub @ X226 @ X228)))) | ($true = ((X224 @ id @ (lam @ (X225 @ sh @ one)) @ id))) | ~ ! [X229 : term] : (~ ($true = ((X224 @ id @ X229 @ id))) | (((X224 @ id @ (X225 @ id @ X229) @ id)) = $true))) | ~ ! [X230 : term,X231 : term] : (($true = ((X224 @ id @ (ap @ (sub @ X231 @ id) @ X230) @ id))) | ~ ($true = ((X224 @ id @ X231 @ id))) | ~ ($true = ((X224 @ id @ X230 @ id)))) | ($true = ((X224 @ id @ X223 @ id))))) & (! [X232 : term,X233 : subst,X234 : (term > $o)] : ~ ! [X235 : (term > $o)] : ~ ! [X236 : term] : (((X234 @ (sub @ X236 @ (push @ X232 @ X233)))) = ((X235 @ X236))) <=> (pushprop_lem0_lthm = $true)) & ! [X237 : d_subst,X238 : d_subst] : ((((d2subst @ X237)) = ((d2subst @ X238))) => (X237 = X238)) & ~ (lamnotap = $true) & ((! [X239 : term,X240 : (term > $o)] : (~ ! [X241 : term] : (($true = ((X240 @ X241))) | ~ ($true = ((var @ X241)))) | ~ ! [X242 : term,X243 : term] : (~ ($true = ((X240 @ X242))) | ($true = ((X240 @ (ap @ X243 @ X242)))) | ~ ($true = ((X240 @ X243)))) | ~ ! [X244 : term] : (~ ! [X245 : term] : (($true = ((X240 @ (sub @ X244 @ (push @ X245 @ id))))) | ~ ($true = ((X240 @ X245)))) | ($true = ((X240 @ (lam @ X244))))) | ($true = ((X240 @ X239)))) => (! [X246 : (subst > term > subst > $o),X247 : term,X248 : term] : (~ ($true = ((X246 @ id @ X248 @ id))) | ($true = ((X246 @ id @ (ap @ X247 @ X248) @ id))) | ~ ! [X249 : term,X250 : term] : (($true = ((X246 @ id @ (ap @ (sub @ X249 @ id) @ X250) @ id))) | ~ ($true = ((X246 @ id @ X249 @ id))) | ~ ($true = ((X246 @ id @ X250 @ id)))) | ~ ($true = ((X246 @ id @ X247 @ id))) | ~ ! [X251 : term,X252 : subst,X253 : subst,X254 : subst] : (~ ($true = ((X246 @ (comp @ X252 @ X254) @ (sub @ X251 @ X254) @ X253))) | ($true = ((X246 @ X252 @ X251 @ (comp @ X254 @ X253))))) | ~ ! [X255 : subst,X256 : subst,X257 : subst,X258 : term] : (~ ($true = ((X246 @ X257 @ X258 @ (comp @ X255 @ X256)))) | ($true = ((X246 @ (comp @ X257 @ X255) @ (sub @ X258 @ X255) @ X256))))) => (! [X259 : (subst > term > subst > $o),X260 : term] : (~ ! [X261 : (subst > term > term)] : (~ ! [X262 : subst,X263 : term,X264 : subst] : (((sub @ (X261 @ X262 @ X263) @ X264)) = ((X261 @ (comp @ X262 @ X264) @ (sub @ X263 @ X264)))) | ($true = ((X259 @ id @ (lam @ (X261 @ sh @ one)) @ id))) | ~ ! [X265 : term] : (~ ($true = ((X259 @ id @ X265 @ id))) | ($true = ((X259 @ id @ (X261 @ id @ X265) @ id))))) | ~ ! [X266 : subst,X267 : subst,X268 : term,X269 : subst] : (($true = ((X259 @ X266 @ X268 @ (comp @ X267 @ X269)))) | ~ ($true = ((X259 @ (comp @ X266 @ X267) @ (sub @ X268 @ X267) @ X269)))) | ~ ! [X270 : subst,X271 : term,X272 : subst,X273 : subst] : (~ ($true = ((X259 @ X270 @ X271 @ (comp @ X272 @ X273)))) | (((X259 @ (comp @ X270 @ X272) @ (sub @ X271 @ X272) @ X273)) = $true)) | ($true = ((X259 @ id @ (lam @ X260) @ id))) | ~ ! [X274 : term] : (($true = ((X259 @ id @ (sub @ X260 @ (push @ X274 @ id)) @ id))) | ~ ($true = ((X259 @ id @ X274 @ id))))) => ! [X275 : term,X276 : (subst > term > subst > $o)] : (~ ! [X277 : (subst > term > term)] : (~ ! [X278 : term] : (($true = ((X276 @ id @ (X277 @ id @ X278) @ id))) | ~ ($true = ((X276 @ id @ X278 @ id)))) | ($true = ((X276 @ id @ (lam @ (X277 @ sh @ one)) @ id))) | ~ ! [X279 : subst,X280 : term,X281 : subst] : (((sub @ (X277 @ X279 @ X280) @ X281)) = ((X277 @ (comp @ X279 @ X281) @ (sub @ X280 @ X281))))) | ~ ! [X282 : term,X283 : subst,X284 : subst,X285 : subst] : (~ ($true = ((X276 @ (comp @ X285 @ X283) @ (sub @ X282 @ X283) @ X284))) | ($true = ((X276 @ X285 @ X282 @ (comp @ X283 @ X284))))) | ~ ! [X286 : term] : (~ ($true = ((var @ (sub @ X286 @ id)))) | (((X276 @ id @ X286 @ id)) = $true)) | ($true = ((X276 @ id @ X275 @ id))) | ~ ! [X287 : subst,X288 : subst,X289 : term,X290 : subst] : (~ ($true = ((X276 @ X287 @ X289 @ (comp @ X288 @ X290)))) | ($true = ((X276 @ (comp @ X287 @ X288) @ (sub @ X289 @ X288) @ X290)))) | ~ ! [X291 : term,X292 : term] : (~ ($true = ((X276 @ id @ X292 @ id))) | ($true = ((X276 @ id @ (ap @ (sub @ X292 @ id) @ X291) @ id))) | ~ ($true = ((X276 @ id @ X291 @ id)))))))) <=> (hoasinduction_lthm = $true)) & (! [X293 : term,X294 : term] : ((X293 = X294) | (((lam @ X294)) != ((lam @ X293)))) <=> (laminj = $true)) & ((pushprop_lem1v2 = $true) <=> ! [X295 : (term > $o),X296 : term,X297 : (term > $o),X298 : subst] : (~ ! [X299 : term] : (((X295 @ X299)) = ((X297 @ (sub @ X299 @ (push @ X296 @ X298))))) | ($true = ((X295 @ one))) | ~ ($true = ((X297 @ X296))))) & (pushprop_lem1_gthm = $true) & ! [X300 : term] : ? [X301 : d_term] : (((d2term @ X301)) = X300) & (hoasinduction_lem1_lthm = $true) & ((hoasinduction_lem3_lthm = $true) <=> (! [X302 : term] : (((sub @ X302 @ id)) = X302) => (! [X303 : (subst > term > subst > $o),X304 : term] : (~ ! [X305 : (subst > term > term)] : (~ ! [X306 : term,X307 : subst,X308 : subst] : (((sub @ (X305 @ X307 @ X306) @ X308)) = ((X305 @ (comp @ X307 @ X308) @ (sub @ X306 @ X308)))) | ($true = ((X303 @ id @ (lam @ (X305 @ sh @ one)) @ id))) | ~ ! [X309 : term] : (~ (((X303 @ id @ X309 @ id)) = $true) | ($true = ((X303 @ id @ (X305 @ id @ X309) @ id))))) | ($true = ((X303 @ id @ (lam @ (sub @ X304 @ (push @ one @ sh))) @ id))) | ~ ! [X310 : term] : (~ ($true = ((X303 @ id @ X310 @ id))) | ($true = ((X303 @ id @ (sub @ X304 @ (push @ X310 @ id)) @ id))))) => ! [X311 : (subst > term > subst > $o),X312 : term] : (~ ! [X313 : subst,X314 : subst,X315 : term,X316 : subst] : (($true = ((X311 @ X313 @ X315 @ (comp @ X316 @ X314)))) | ~ ($true = ((X311 @ (comp @ X313 @ X316) @ (sub @ X315 @ X316) @ X314)))) | ~ ! [X317 : term,X318 : subst,X319 : subst,X320 : subst] : (($true = ((X311 @ (comp @ X318 @ X320) @ (sub @ X317 @ X320) @ X319))) | ~ ($true = ((X311 @ X318 @ X317 @ (comp @ X320 @ X319))))) | ~ ! [X321 : (subst > term > term)] : (($true = ((X311 @ id @ (lam @ (X321 @ sh @ one)) @ id))) | ~ ! [X322 : term] : (($true = ((X311 @ id @ (X321 @ id @ X322) @ id))) | ~ ($true = ((X311 @ id @ X322 @ id)))) | ~ ! [X323 : subst,X324 : term,X325 : subst] : (((X321 @ (comp @ X323 @ X325) @ (sub @ X324 @ X325))) = ((sub @ (X321 @ X323 @ X324) @ X325)))) | ~ ! [X326 : term] : (~ ($true = ((X311 @ id @ X326 @ id))) | ($true = ((X311 @ id @ (sub @ X312 @ (push @ X326 @ id)) @ id)))) | (((X311 @ id @ (lam @ X312) @ id)) = $true))))) & (hoaslamnotvar_gthm = $true) & (hoaslamnotvar = $true) & (! [X327 : term] : ~ ! [X328 : (subst > term > term)] : (((X328 @ sh @ one)) != ((sub @ X327 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b_lthm = $true)) & (! [X329 : term,X330 : term,X331 : subst,X332 : (term > $o),X333 : (term > $o)] : (($true = ((X332 @ (sub @ X330 @ (push @ X329 @ X331))))) | ~ ($true = ((X333 @ X330)))) <=> (pushprop_lem1 = $true)) & ((! [X334 : subst] : (((comp @ id @ X334)) = X334) => (! [X335 : subst] : (((comp @ X335 @ id)) = X335) => (! [X336 : subst] : (((comp @ id @ X336)) = X336) & ! [X337 : subst] : (((comp @ X337 @ id)) = X337)))) <=> (substmonoid_lthm = $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((pushprop_lem0 = $true) <=> ! [X340 : (term > $o),X341 : term,X342 : subst] : ~ ! [X343 : (term > $o)] : ~ ! [X344 : term] : (((X343 @ X344)) = ((X340 @ (sub @ X344 @ (push @ X341 @ X342)))))) & ((axidl = $true) <=> ! [X345 : subst] : (((comp @ id @ X345)) = X345)) & (pushprop = $true) & (axabs = $true) & (! [X346 : term,X347 : (subst > term > subst > $o)] : (~ ! [X348 : subst,X349 : subst,X350 : subst,X351 : term] : (~ ($true = ((X347 @ X348 @ X351 @ (comp @ X350 @ X349)))) | ($true = ((X347 @ (comp @ X348 @ X350) @ (sub @ X351 @ X350) @ X349)))) | ~ ! [X352 : term] : (~ (((X347 @ id @ X352 @ id)) = $true) | ($true = ((X347 @ id @ (sub @ X346 @ (push @ X352 @ id)) @ id)))) | ($true = ((X347 @ id @ (lam @ X346) @ id))) | ~ ! [X353 : term,X354 : subst,X355 : subst,X356 : subst] : (($true = ((X347 @ X356 @ X353 @ (comp @ X355 @ X354)))) | ~ (((X347 @ (comp @ X356 @ X355) @ (sub @ X353 @ X355) @ X354)) = $true)) | ~ ! [X357 : (subst > term > term)] : (~ ! [X358 : subst,X359 : term,X360 : subst] : (((sub @ (X357 @ X360 @ X359) @ X358)) = ((X357 @ (comp @ X360 @ X358) @ (sub @ X359 @ X358)))) | ~ ! [X361 : term] : (($true = ((X347 @ id @ (X357 @ id @ X361) @ id))) | ~ ($true = ((X347 @ id @ X361 @ id)))) | ($true = ((X347 @ id @ (lam @ (X357 @ sh @ one)) @ id))))) <=> (hoasinduction_lem3 = $true)) & (hoasinduction_lem1 = $true) & (hoasapnotvar = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & (! [X367 : (subst > term > subst > $o)] : ~ ! [X368 : (term > $o)] : ~ ! [X369 : term] : (((X367 @ id @ X369 @ id)) = ((X368 @ X369))) <=> (hoasinduction_lem0_lthm = $true)) & ! [X370 : subst] : ? [X371 : d_subst] : (((d2subst @ X371)) = X370) & (pushprop_lem2v2_gthm = $true) & (! [X372 : subst,X373 : term] : (((comp @ sh @ (push @ X373 @ X372))) = X372) <=> (axshiftcons = $true)) & ((apinj2 = $true) <=> ! [X374 : term,X375 : term,X376 : term,X377 : term] : ((((ap @ X376 @ X374)) != ((ap @ X377 @ X375))) | (X374 = X375))) & ((hoasinduction_lem3b = $true) <=> ! [X378 : term] : ~ ! [X379 : (subst > term > term)] : (((X379 @ sh @ one)) != ((sub @ X378 @ (push @ one @ sh))))) & (! [X380 : term] : ~ ! [X381 : (subst > term > term)] : ~ ! [X382 : term,X383 : subst] : (((X381 @ X383 @ X382)) = ((sub @ X380 @ (push @ X382 @ X383)))) <=> (hoasinduction_lem3v2_f = $true)) & (hoasapnotvar_lthm = $true) & (! [X384 : (subst > term > subst > $o),X385 : term] : (~ ! [X386 : term] : (($true = ((X384 @ id @ (sub @ X385 @ (push @ X386 @ id)) @ id))) | ~ ($true = ((X384 @ id @ X386 @ id)))) | ~ ! [X387 : term,X388 : (subst > term > term)] : (~ ! [X389 : term,X390 : subst,X391 : subst] : (((sub @ X387 @ (push @ (sub @ X389 @ X390) @ (comp @ X391 @ X390)))) = ((X388 @ (comp @ X391 @ X390) @ (sub @ X389 @ X390)))) | ($true = ((X384 @ id @ (lam @ (X388 @ sh @ one)) @ id))) | ~ ! [X392 : term,X393 : subst,X394 : subst] : (((sub @ (sub @ X387 @ (push @ X392 @ X394)) @ X393)) = ((sub @ (X388 @ X394 @ X392) @ X393))) | ~ ! [X395 : term] : (($true = ((X384 @ id @ (X388 @ id @ X395) @ id))) | ~ ($true = ((X384 @ id @ X395 @ id))))) | ($true = ((X384 @ id @ (lam @ (sub @ X385 @ (push @ one @ sh))) @ id)))) <=> (hoasinduction_lem3aaa = $true)) & (! [X396 : (subst > term > subst > $o),X397 : term] : (~ ! [X398 : (subst > term > term)] : (~ ! [X399 : subst,X400 : subst,X401 : term] : (((sub @ (X398 @ X400 @ X401) @ X399)) = ((X398 @ (comp @ X400 @ X399) @ (sub @ X401 @ X399)))) | ~ ! [X402 : term] : (~ ($true = ((X396 @ id @ X402 @ id))) | ($true = ((X396 @ id @ (X398 @ id @ X402) @ id)))) | ($true = ((X396 @ id @ (lam @ (X398 @ sh @ one)) @ id)))) | ~ ! [X403 : term] : ((((X396 @ id @ (sub @ X397 @ (push @ X403 @ id)) @ id)) = $true) | ~ (((X396 @ id @ X403 @ id)) = $true)) | ($true = ((X396 @ id @ (lam @ X397) @ id)))) <=> (hoasinduction_lem3a = $true)) & (hoasinduction_lem3v2_gthm = $true) & (((d2subst @ d_id)) = sh) & (hoasapnotvar_gthm = $true) & (hoasapinj2_gthm = $true) & ((axscons = $true) <=> ! [X404 : subst] : (((push @ (sub @ one @ X404) @ (comp @ sh @ X404))) = X404)) & (hoasinduction_lem1v2_gthm = $true) & ((! [X405 : term] : (((sub @ X405 @ id)) = X405) => ! [X406 : term] : (((sub @ X406 @ id)) = X406)) <=> (termmset_lthm = $true)) & (axvarshift = $true) & ((hoasinduction = $true) <=> ! [X407 : term,X408 : (subst > term > subst > $o)] : (~ ! [X409 : subst,X410 : subst,X411 : subst,X412 : term] : (~ ($true = ((X408 @ (comp @ X411 @ X409) @ (sub @ X412 @ X409) @ X410))) | ($true = ((X408 @ X411 @ X412 @ (comp @ X409 @ X410))))) | ~ ! [X413 : (subst > term > term)] : (($true = ((X408 @ id @ (lam @ (X413 @ sh @ one)) @ id))) | ~ ! [X414 : term] : (~ ($true = ((X408 @ id @ X414 @ id))) | ($true = ((X408 @ id @ (X413 @ id @ X414) @ id)))) | ~ ! [X415 : subst,X416 : subst,X417 : term] : (((sub @ (X413 @ X416 @ X417) @ X415)) = ((X413 @ (comp @ X416 @ X415) @ (sub @ X417 @ X415))))) | ~ ! [X418 : term,X419 : term] : (~ ($true = ((X408 @ id @ X418 @ id))) | ~ ($true = ((X408 @ id @ X419 @ id))) | ($true = ((X408 @ id @ (ap @ (sub @ X418 @ id) @ X419) @ id)))) | ~ ! [X420 : term] : (($true = ((X408 @ id @ X420 @ id))) | ~ ($true = ((var @ (sub @ X420 @ id))))) | ($true = ((X408 @ id @ X407 @ id))) | ~ ! [X421 : subst,X422 : term,X423 : subst,X424 : subst] : (~ ($true = ((X408 @ X421 @ X422 @ (comp @ X424 @ X423)))) | ($true = ((X408 @ (comp @ X421 @ X424) @ (sub @ X422 @ X424) @ X423)))))) & ((hoasinduction_lem3v2_lthm = $true) <=> (! [X425 : term] : (((sub @ X425 @ id)) = X425) => ! [X426 : (subst > term > subst > $o),X427 : term,X428 : (term > $o)] : (~ ! [X429 : subst,X430 : term,X431 : subst,X432 : subst] : (~ ($true = ((X426 @ X432 @ X430 @ (comp @ X429 @ X431)))) | ($true = ((X426 @ (comp @ X432 @ X429) @ (sub @ X430 @ X429) @ X431)))) | ($true = ((X428 @ (lam @ X427)))) | ~ ! [X433 : term] : (($true = ((X428 @ (sub @ X427 @ (push @ X433 @ id))))) | ~ (((X428 @ X433)) = $true)) | ~ ! [X434 : subst,X435 : subst,X436 : subst,X437 : term] : (($true = ((X426 @ X434 @ X437 @ (comp @ X436 @ X435)))) | ~ ($true = ((X426 @ (comp @ X434 @ X436) @ (sub @ X437 @ X436) @ X435)))) | ~ ! [X438 : (subst > term > term)] : (($true = ((X426 @ id @ (lam @ (X438 @ sh @ one)) @ id))) | ~ ! [X439 : term,X440 : subst,X441 : subst] : (((sub @ (X438 @ X441 @ X439) @ X440)) = ((X438 @ (comp @ X441 @ X440) @ (sub @ X439 @ X440)))) | ~ ! [X442 : term] : (~ ($true = ((X426 @ id @ X442 @ id))) | ($true = ((X426 @ id @ (X438 @ id @ X442) @ id))))) | ~ ! [X443 : term] : (((X428 @ X443)) = ((X426 @ id @ X443 @ id)))))) & (lamnotvar = $true) & (pushprop_lem0_gthm = $true) & (! [X444 : (subst > term > subst > $o),X445 : term,X446 : (term > $o)] : (~ ! [X447 : term] : (((X446 @ X447)) = ((X444 @ id @ X447 @ id))) | ($true = ((X446 @ (lam @ X445)))) | ~ ! [X448 : (subst > term > term)] : (~ ! [X449 : subst,X450 : subst,X451 : term] : (((X448 @ (comp @ X449 @ X450) @ (sub @ X451 @ X450))) = ((sub @ (X448 @ X449 @ X451) @ X450))) | ($true = ((X444 @ id @ (lam @ (X448 @ sh @ one)) @ id))) | ~ ! [X452 : term] : (($true = ((X444 @ id @ (X448 @ id @ X452) @ id))) | ~ ($true = ((X444 @ id @ X452 @ id))))) | ~ ! [X453 : term] : (~ ($true = ((X446 @ X453))) | ($true = ((X446 @ (sub @ X445 @ (push @ X453 @ id))))))) <=> (hoasinduction_lem3v2a = $true)) & (! [X454 : subst] : (((comp @ X454 @ id)) = X454) <=> (axidr = $true)) & (id = ((d2subst @ d_id))) & (! [X455 : term,X456 : term,X457 : (subst > term > term)] : ((((lam @ (X457 @ sh @ one))) != ((ap @ (sub @ X455 @ id) @ X456))) | ~ ! [X458 : subst,X459 : subst,X460 : term] : (((sub @ (X457 @ X458 @ X460) @ X459)) = ((X457 @ (comp @ X458 @ X459) @ (sub @ X460 @ X459))))) <=> (hoaslamnotap = $true)) & ((hoasapinj1_lthm = $true) <=> (! [X461 : term] : (((sub @ X461 @ id)) = X461) => (! [X462 : term,X463 : term,X464 : term,X465 : term] : ((((ap @ X465 @ X464)) != ((ap @ X462 @ X463))) | (X462 = X465)) => ! [X466 : term,X467 : term,X468 : term,X469 : term] : ((((ap @ (sub @ X469 @ id) @ X467)) != ((ap @ (sub @ X468 @ id) @ X466))) | (X468 = X469))))) & (pushprop_lem2v2_lthm = $true) & (ulamvarsh = $true) & (axapp = $true) & (substmonoid_gthm = $true) & (((d2term @ d_one)) = one) & (hoasinduction_lem2v2_gthm = $true) & (! [X470 : (subst > term > term),X471 : subst,X472 : (subst > term > term),X473 : term] : ((((X470 @ X471 @ X473)) = ((X472 @ X471 @ X473))) | ~ ! [X474 : subst,X475 : subst,X476 : term] : (((X470 @ (comp @ X475 @ X474) @ (sub @ X476 @ X474))) = ((sub @ (X470 @ X475 @ X476) @ X474))) | (((lam @ (X470 @ sh @ one))) != ((lam @ (X472 @ sh @ one)))) | ~ ! [X477 : subst,X478 : subst,X479 : term] : (((sub @ (X472 @ X477 @ X479) @ X478)) = ((X472 @ (comp @ X477 @ X478) @ (sub @ X479 @ X478))))) <=> (hoaslaminj = $true)) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((hoasinduction_lem2v2 = $true) <=> ! [X480 : term,X481 : (term > $o),X482 : (subst > term > subst > $o),X483 : term] : (($true = ((X481 @ (ap @ X480 @ X483)))) | ~ ! [X484 : term] : (((X481 @ X484)) = ((X482 @ id @ X484 @ id))) | ~ ! [X485 : term,X486 : term] : (~ ($true = ((X482 @ id @ X486 @ id))) | ~ ($true = ((X482 @ id @ X485 @ id))) | ($true = ((X482 @ id @ (ap @ (sub @ X486 @ id) @ X485) @ id)))) | ~ ! [X487 : subst,X488 : subst,X489 : term,X490 : subst] : (($true = ((X482 @ X488 @ X489 @ (comp @ X490 @ X487)))) | ~ ($true = ((X482 @ (comp @ X488 @ X490) @ (sub @ X489 @ X490) @ X487)))) | ~ ($true = ((X481 @ X483))) | ~ ($true = ((X481 @ X480))) | ~ ! [X491 : subst,X492 : subst,X493 : term,X494 : subst] : (~ ($true = ((X482 @ X491 @ X493 @ (comp @ X494 @ X492)))) | ($true = ((X482 @ (comp @ X491 @ X494) @ (sub @ X493 @ X494) @ X492)))))) & (hoasinduction_lem3a_gthm = $true) & (! [X495 : (term > $o),X496 : term] : (($true = ((X495 @ X496))) | ~ ! [X497 : term] : (~ ($true = ((var @ X497))) | ($true = ((X495 @ X497)))) | ~ ! [X498 : term,X499 : term] : (~ ($true = ((X495 @ X498))) | ($true = ((X495 @ (ap @ X498 @ X499)))) | ~ ($true = ((X495 @ X499)))) | ~ ! [X500 : term] : (($true = ((X495 @ (lam @ X500)))) | ~ ! [X501 : term] : (($true = ((X495 @ (sub @ X500 @ (push @ X501 @ id))))) | ~ ($true = ((X495 @ X501)))))) <=> (induction2 = $true)) & (ulamvarind = $true) & ~ (((var @ (d2term @ d_one))) = $true) & (! [X502 : term,X503 : (subst > term > subst > $o),X504 : (term > $o)] : (($true = ((X504 @ (lam @ X502)))) | ~ ! [X505 : term] : (((X503 @ id @ X505 @ id)) = ((X504 @ X505))) | ~ ! [X506 : (subst > term > term)] : (~ ! [X507 : term] : (~ ($true = ((X503 @ id @ X507 @ id))) | ($true = ((X503 @ id @ (X506 @ id @ X507) @ id)))) | ~ ! [X508 : subst,X509 : term,X510 : subst] : (((X506 @ (comp @ X508 @ X510) @ (sub @ X509 @ X510))) = ((sub @ (X506 @ X508 @ X509) @ X510))) | (((X503 @ id @ (lam @ (X506 @ sh @ one)) @ id)) = $true)) | ~ ! [X511 : term,X512 : subst,X513 : subst,X514 : subst] : (~ ($true = ((X503 @ X514 @ X511 @ (comp @ X513 @ X512)))) | ($true = ((X503 @ (comp @ X514 @ X513) @ (sub @ X511 @ X513) @ X512)))) | ~ ! [X515 : term] : (~ ($true = ((X504 @ X515))) | ($true = ((X504 @ (sub @ X502 @ (push @ X515 @ id)))))) | ~ ! [X516 : subst,X517 : term,X518 : subst,X519 : subst] : (~ ($true = ((X503 @ (comp @ X519 @ X516) @ (sub @ X517 @ X516) @ X518))) | ($true = ((X503 @ X519 @ X517 @ (comp @ X516 @ X518)))))) <=> (hoasinduction_lem3v2 = $true)) & (hoasinduction_lem1_gthm = $true) & (! [X520 : term] : (((sub @ X520 @ id)) = X520) <=> (termmset = $true)) & (hoaslaminj_gthm = $true) & (axclos = $true) & (hoasinduction_lem2_lthm = $true) & ((! [X521 : subst] : (((comp @ X521 @ id)) = X521) & ! [X522 : subst] : (((comp @ id @ X522)) = X522)) <=> (substmonoid = $true)) & (axmap = $true) & ((hoasapinj2_lthm = $true) <=> (! [X523 : term,X524 : term,X525 : term,X526 : term] : ((X523 = X524) | (((ap @ X525 @ X524)) != ((ap @ X526 @ X523)))) => ! [X527 : term,X528 : term,X529 : term,X530 : term] : ((X527 = X530) | (((ap @ (sub @ X529 @ id) @ X527)) != ((ap @ (sub @ X528 @ id) @ X530)))))) & (! [X531 : term,X532 : term,X533 : term,X534 : term] : ((X531 = X534) | (((ap @ (sub @ X532 @ id) @ X534)) != ((ap @ (sub @ X533 @ id) @ X531)))) <=> (hoasapinj2 = $true)) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X535 : term,X536 : (subst > term > subst > $o)] : (~ ! [X537 : term] : (($true = ((X536 @ id @ (sub @ X535 @ (push @ X537 @ id)) @ id))) | ~ (((X536 @ id @ X537 @ id)) = $true)) | ~ ! [X538 : (subst > term > term)] : (~ ! [X539 : term,X540 : subst,X541 : subst] : (((X538 @ (comp @ X540 @ X541) @ (sub @ X539 @ X541))) = ((sub @ (X538 @ X540 @ X539) @ X541))) | ($true = ((X536 @ id @ (lam @ (X538 @ sh @ one)) @ id))) | ~ ! [X542 : term] : (($true = ((X536 @ id @ (X538 @ id @ X542) @ id))) | ~ ($true = ((X536 @ id @ X542 @ id))))) | ($true = ((X536 @ id @ (lam @ (sub @ X535 @ (push @ one @ sh))) @ id))))) & ((! [X543 : term,X544 : subst] : (((sub @ one @ (push @ X543 @ X544))) = X543) => ! [X545 : term,X546 : (term > $o),X547 : subst,X548 : (term > $o)] : (($true = ((X548 @ one))) | ~ ! [X549 : term] : (((X548 @ X549)) = ((X546 @ (sub @ X549 @ (push @ X545 @ X547))))) | ~ (((X546 @ X545)) = $true))) <=> (pushprop_lem1v2_lthm = $true)) & (hoaslamnotap_lthm = $true)), inference(fool_elimination,[],[f6])). thf(f8,plain,( (hoasinduction_lem3v2a != ((!! @ (subst > term > subst > $o) @ (^[Y0 : subst > term > subst > $o]: (!! @ (term > $o) @ (^[Y1 : term > $o]: ((!! @ (subst > term > term) @ (^[Y2 : subst > term > term]: ((!! @ term @ (^[Y3 : term]: (!! @ subst @ (^[Y4 : subst]: (!! @ subst @ (^[Y5 : subst]: ((Y2 @ (comp @ Y5 @ Y4) @ (sub @ Y3 @ Y4)) = (sub @ (Y2 @ Y5 @ Y3) @ Y4)))))))) => ((!! @ term @ (^[Y3 : term]: ((Y0 @ id @ Y3 @ id) => (Y0 @ id @ (Y2 @ id @ Y3) @ id)))) => (Y0 @ id @ (hoaslam @ id @ (^[Y3 : subst]: ((^[Y4 : term]: (Y2 @ Y3 @ Y4))))) @ id))))) => ((hoasinduction_p_and_p_prime @ Y0 @ Y1) => (!! @ term @ (^[Y2 : term]: ((!! @ term @ (^[Y3 : term]: ((Y1 @ Y3) => (Y1 @ (sub @ Y2 @ (push @ Y3 @ id)))))) => (Y1 @ (lam @ Y2)))))))))))))), inference(flattening,[],[f5])). thf(f9,plain,( (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (pushprop_gthm = $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)))) & (! [X0 : term,X1 : term] : ((X0 = X1) | (((sub @ X0 @ sh)) != ((sub @ X1 @ sh)))) <=> (shinj = $true)) & (hoaslamnotvar_lthm = $true) & ((hoaslaminj_lthm = $true) <=> (! [X3 : subst,X2 : term] : (((sub @ one @ (push @ X2 @ X3))) = X2) => (! [X4 : term,X5 : subst] : (((comp @ sh @ (push @ X4 @ X5))) = X5) => (! [X6 : term,X7 : term] : ((((lam @ X6)) != ((lam @ X7))) | (X6 = X7)) => ! [X9 : (subst > term > term),X8 : subst,X10 : term,X11 : (subst > term > term)] : (~ ! [X12 : subst,X14 : subst,X13 : term] : (((X9 @ (comp @ X14 @ X12) @ (sub @ X13 @ X12))) = ((sub @ (X9 @ X14 @ X13) @ X12))) | (((X9 @ X8 @ X10)) = ((X11 @ X8 @ X10))) | (((lam @ (X9 @ sh @ one))) != ((lam @ (X11 @ sh @ one)))) | ~ ! [X15 : term,X17 : subst,X16 : subst] : (((sub @ (X11 @ X16 @ X15) @ X17)) = ((X11 @ (comp @ X16 @ X17) @ (sub @ X15 @ X17))))))))) & (hoasinduction_lem1v2 = $true) & (hoaslamnotap_gthm = $true) & (! [X18 : term,X19 : term,X20 : (subst > term > subst > $o)] : (($true = ((X20 @ id @ (ap @ X19 @ X18) @ id))) | ~ ($true = ((X20 @ id @ X19 @ id))) | ~ ! [X21 : subst,X22 : term,X23 : subst,X24 : subst] : (~ ($true = ((X20 @ (comp @ X21 @ X24) @ (sub @ X22 @ X24) @ X23))) | ($true = ((X20 @ X21 @ X22 @ (comp @ X24 @ X23))))) | ~ ! [X25 : term,X26 : subst,X27 : subst,X28 : subst] : (~ ($true = ((X20 @ X28 @ X25 @ (comp @ X26 @ X27)))) | ($true = ((X20 @ (comp @ X28 @ X26) @ (sub @ X25 @ X26) @ X27)))) | ~ (((X20 @ id @ X18 @ id)) = $true) | ~ ! [X29 : term,X30 : term] : (~ (((X20 @ id @ X29 @ id)) = $true) | ~ ($true = ((X20 @ id @ X30 @ id))) | ($true = ((X20 @ id @ (ap @ (sub @ X29 @ id) @ X30) @ id))))) <=> (hoasinduction_lem2 = $true)) & ! [X31 : d_subst] : (d_id = X31) & (! [X32 : term,X33 : term,X34 : term,X35 : term] : ((((ap @ (sub @ X34 @ id) @ X33)) != ((ap @ (sub @ X32 @ id) @ X35))) | (X32 = X34)) <=> (hoasapinj1 = $true)) & ! [X36 : d_term] : (d_one = X36) & (pushprop_lem1v2_gthm = $true) & ((! [X37 : term] : (((sub @ X37 @ id)) = X37) => (! [X38 : (subst > term > subst > $o),X39 : term] : (($true = ((X38 @ id @ (lam @ (sub @ X39 @ (push @ one @ sh))) @ id))) | ~ ! [X40 : (subst > term > term)] : (~ ! [X42 : subst,X43 : subst,X41 : term] : (((X40 @ (comp @ X42 @ X43) @ (sub @ X41 @ X43))) = ((sub @ (X40 @ X42 @ X41) @ X43))) | ~ ! [X44 : term] : ((((X38 @ id @ (X40 @ id @ X44) @ id)) = $true) | ~ ($true = ((X38 @ id @ X44 @ id)))) | ($true = ((X38 @ id @ (lam @ (X40 @ sh @ one)) @ id)))) | ~ ! [X45 : term] : (~ ($true = ((X38 @ id @ X45 @ id))) | ($true = ((X38 @ id @ (sub @ X39 @ (push @ X45 @ id)) @ id))))) => ! [X46 : term,X47 : (subst > term > subst > $o)] : (~ ! [X48 : (subst > term > term)] : (~ ! [X49 : term] : (~ ($true = ((X47 @ id @ X49 @ id))) | (((X47 @ id @ (X48 @ id @ X49) @ id)) = $true)) | ($true = ((X47 @ id @ (lam @ (X48 @ sh @ one)) @ id))) | ~ ! [X50 : subst,X51 : term,X52 : subst] : (((sub @ (X48 @ X50 @ X51) @ X52)) = ((X48 @ (comp @ X50 @ X52) @ (sub @ X51 @ X52))))) | ~ ! [X53 : term] : (($true = ((X47 @ id @ (sub @ X46 @ (push @ X53 @ id)) @ id))) | ~ ($true = ((X47 @ id @ X53 @ id)))) | (((X47 @ id @ (lam @ X46) @ id)) = $true)))) <=> (hoasinduction_lem3a_lthm = $true)) & (pushprop_lthm = $true) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (! [X54 : (subst > term > subst > $o)] : ~ ! [X55 : (term > $o)] : ~ ! [X56 : term] : (((X54 @ id @ X56 @ id)) = ((X55 @ X56))) => (! [X57 : (term > $o),X58 : term] : (~ ! [X59 : term,X60 : term] : (~ ($true = ((X57 @ X59))) | ~ ($true = ((X57 @ X60))) | (((X57 @ (ap @ X59 @ X60))) = $true)) | ($true = ((X57 @ X58))) | ~ ! [X61 : term] : ((((X57 @ X61)) = $true) | ~ ($true = ((var @ X61)))) | ~ ! [X62 : term] : (($true = ((X57 @ (lam @ X62)))) | ~ ! [X63 : term] : ((((X57 @ (sub @ X62 @ (push @ X63 @ id)))) = $true) | ~ ($true = ((X57 @ X63)))))) => (! [X64 : term] : (((sub @ X64 @ id)) = X64) => (! [X65 : term,X66 : (subst > term > subst > $o),X67 : (term > $o)] : (~ ! [X68 : term] : (($true = ((X67 @ (sub @ X65 @ (push @ X68 @ id))))) | ~ ($true = ((X67 @ X68)))) | ($true = ((X67 @ (lam @ X65)))) | ~ ! [X69 : (subst > term > term)] : (($true = ((X66 @ id @ (lam @ (X69 @ sh @ one)) @ id))) | ~ ! [X72 : term,X71 : subst,X70 : subst] : (((sub @ (X69 @ X70 @ X72) @ X71)) = ((X69 @ (comp @ X70 @ X71) @ (sub @ X72 @ X71)))) | ~ ! [X73 : term] : (($true = ((X66 @ id @ (X69 @ id @ X73) @ id))) | ~ ($true = ((X66 @ id @ X73 @ id))))) | ~ ! [X74 : term] : (((X66 @ id @ X74 @ id)) = ((X67 @ X74)))) => ! [X75 : term,X76 : (subst > term > subst > $o)] : (($true = ((X76 @ id @ X75 @ id))) | ~ ! [X77 : (subst > term > term)] : (~ ! [X79 : subst,X80 : term,X78 : subst] : (((sub @ (X77 @ X78 @ X80) @ X79)) = ((X77 @ (comp @ X78 @ X79) @ (sub @ X80 @ X79)))) | ~ ! [X81 : term] : (($true = ((X76 @ id @ (X77 @ id @ X81) @ id))) | ~ ($true = ((X76 @ id @ X81 @ id)))) | ($true = ((X76 @ id @ (lam @ (X77 @ sh @ one)) @ id)))) | ~ ! [X82 : term,X83 : term] : (($true = ((X76 @ id @ (ap @ (sub @ X82 @ id) @ X83) @ id))) | ~ ($true = ((X76 @ id @ X83 @ id))) | ~ ($true = ((X76 @ id @ X82 @ id)))))))))) & (hoasinduction_lem2_gthm = $true) & (apnotvar = $true) & ((induction2lem = $true) <=> ! [X84 : (term > $o),X85 : term,X86 : subst] : ((((X84 @ (sub @ X85 @ X86))) = $true) | ~ ! [X87 : term,X88 : term] : (~ ($true = ((X84 @ X87))) | ($true = ((X84 @ (ap @ X88 @ X87)))) | ~ ($true = ((X84 @ X88)))) | ~ ! [X89 : term] : (($true = ((X84 @ (sub @ X89 @ X86)))) | ~ ($true = ((var @ X89)))) | ~ ! [X90 : term] : (~ ! [X91 : term] : (~ ($true = ((X84 @ X91))) | ($true = ((X84 @ (sub @ X90 @ (push @ X91 @ id)))))) | ($true = ((X84 @ (lam @ X90))))))) & (! [X92 : term] : (((sub @ X92 @ id)) = X92) <=> (axvarid = $true)) & ((hoasinduction_lthm_3 = $true) <=> (! [X93 : (subst > term > subst > $o)] : ~ ! [X94 : (term > $o)] : ~ ! [X95 : term] : (((X93 @ id @ X95 @ id)) = ((X94 @ X95))) => (! [X96 : (term > $o),X97 : term] : (($true = ((X96 @ X97))) | ~ ! [X98 : term] : (($true = ((X96 @ X98))) | ~ ($true = ((var @ X98)))) | ~ ! [X99 : term] : (~ ! [X100 : term] : (($true = ((X96 @ (sub @ X99 @ (push @ X100 @ id))))) | ~ ($true = ((X96 @ X100)))) | ($true = ((X96 @ (lam @ X99))))) | ~ ! [X101 : term,X102 : term] : (~ (((X96 @ X102)) = $true) | ~ (((X96 @ X101)) = $true) | (((X96 @ (ap @ X101 @ X102))) = $true))) => (! [X103 : term] : (((sub @ X103 @ id)) = X103) => (! [X104 : term,X105 : (subst > term > subst > $o),X106 : (term > $o)] : (~ ! [X107 : term] : (((X105 @ id @ X107 @ id)) = ((X106 @ X107))) | ~ ! [X108 : (subst > term > term)] : (($true = ((X105 @ id @ (lam @ (X108 @ sh @ one)) @ id))) | ~ ! [X109 : term] : (($true = ((X105 @ id @ (X108 @ id @ X109) @ id))) | ~ ($true = ((X105 @ id @ X109 @ id)))) | ~ ! [X111 : term,X110 : subst,X112 : subst] : (((X108 @ (comp @ X110 @ X112) @ (sub @ X111 @ X112))) = ((sub @ (X108 @ X110 @ X111) @ X112)))) | ($true = ((X106 @ (lam @ X104)))) | ~ ! [X113 : term] : (~ (((X106 @ X113)) = $true) | ($true = ((X106 @ (sub @ X104 @ (push @ X113 @ id))))))) => ! [X114 : (subst > term > subst > $o),X115 : term] : (~ ! [X116 : term,X117 : subst,X118 : subst,X119 : subst] : (($true = ((X114 @ (comp @ X118 @ X119) @ (sub @ X116 @ X119) @ X117))) | ~ ($true = ((X114 @ X118 @ X116 @ (comp @ X119 @ X117))))) | ~ ! [X120 : term,X121 : term] : (~ ($true = ((X114 @ id @ X120 @ id))) | (((X114 @ id @ (ap @ (sub @ X120 @ id) @ X121) @ id)) = $true) | ~ ($true = ((X114 @ id @ X121 @ id)))) | ($true = ((X114 @ id @ X115 @ id))) | ~ ! [X122 : term,X123 : subst,X124 : subst,X125 : subst] : (($true = ((X114 @ X124 @ X122 @ (comp @ X123 @ X125)))) | ~ ($true = ((X114 @ (comp @ X124 @ X123) @ (sub @ X122 @ X123) @ X125)))) | ~ ! [X126 : (subst > term > term)] : (~ ! [X127 : term,X129 : subst,X128 : subst] : (((X126 @ (comp @ X129 @ X128) @ (sub @ X127 @ X128))) = ((sub @ (X126 @ X129 @ X127) @ X128))) | ($true = ((X114 @ id @ (lam @ (X126 @ sh @ one)) @ id))) | ~ ! [X130 : term] : ((((X114 @ id @ (X126 @ id @ X130) @ id)) = $true) | ~ (((X114 @ id @ X130 @ id)) = $true))) | ~ ! [X131 : term] : (~ ($true = ((var @ (sub @ X131 @ id)))) | ($true = ((X114 @ id @ X131 @ id)))))))))) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (hoasinduction_gthm = $true) & ((hoasinduction_lem3aa = $true) <=> ! [X132 : (subst > term > subst > $o),X133 : term] : (~ ! [X134 : (subst > term > term)] : (~ ! [X135 : term] : (($true = ((X132 @ id @ (X134 @ id @ X135) @ id))) | ~ ($true = ((X132 @ id @ X135 @ id)))) | ~ ! [X138 : subst,X137 : subst,X136 : term] : (((sub @ (X134 @ X138 @ X136) @ X137)) = ((X134 @ (comp @ X138 @ X137) @ (sub @ X136 @ X137)))) | ($true = ((X132 @ id @ (lam @ (X134 @ sh @ one)) @ id)))) | ~ ! [X139 : term] : (~ ($true = ((X132 @ id @ X139 @ id))) | ($true = ((X132 @ id @ (sub @ X133 @ (push @ X139 @ id)) @ id)))) | ($true = ((X132 @ id @ (lam @ (sub @ X133 @ (push @ one @ sh))) @ id))))) & (! [X140 : (subst > term > subst > $o)] : ~ ! [X141 : (term > $o)] : ~ ! [X142 : term] : (((X141 @ X142)) = ((X140 @ id @ X142 @ id))) <=> (hoasinduction_lem0 = $true)) & (pushprop_lthm_orig = $true) & (pushprop_lem3v2_lthm = $true) & (hoasinduction_lem3b_gthm = $true) & ! [X143 : d_term,X144 : d_term] : ((((d2term @ X144)) = ((d2term @ X143))) => (X143 = X144)) & (termmset_gthm = $true) & (hoasinduction_lem3_gthm = $true) & (hoasapinj1_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & ((! [X145 : term] : ~ ! [X146 : (subst > term > term)] : ~ ! [X147 : term,X148 : subst] : (((sub @ X145 @ (push @ X147 @ X148))) = ((X146 @ X148 @ X147))) => (! [X149 : term] : (((sub @ X149 @ id)) = X149) => ! [X150 : (term > $o),X151 : (subst > term > subst > $o),X152 : term] : (~ ! [X153 : (subst > term > term)] : (~ ! [X156 : subst,X154 : term,X155 : subst] : (((sub @ (X153 @ X156 @ X154) @ X155)) = ((X153 @ (comp @ X156 @ X155) @ (sub @ X154 @ X155)))) | ($true = ((X151 @ id @ (lam @ (X153 @ sh @ one)) @ id))) | ~ ! [X157 : term] : (($true = ((X151 @ id @ (X153 @ id @ X157) @ id))) | ~ (((X151 @ id @ X157 @ id)) = $true))) | ($true = ((X150 @ (lam @ X152)))) | ~ ! [X158 : term] : (((X151 @ id @ X158 @ id)) = ((X150 @ X158))) | ~ ! [X159 : term] : (($true = ((X150 @ (sub @ X152 @ (push @ X159 @ id))))) | ~ ($true = ((X150 @ X159))))))) <=> (hoasinduction_lem3v2a_lthm = $true)) & (induction2_gthm = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X160 : term] : ~ ! [X161 : (subst > term > term)] : ~ ! [X163 : term,X162 : subst] : (((sub @ X160 @ (push @ X163 @ X162))) = ((X161 @ X162 @ X163)))) & (! [X164 : term,X165 : subst] : (((sub @ one @ (push @ X164 @ X165))) = X164) <=> (axvarcons = $true)) & (! [X168 : term,X166 : term,X167 : term,X169 : term] : ((((ap @ X168 @ X167)) != ((ap @ X166 @ X169))) | (X166 = X168)) <=> (apinj1 = $true)) & (pushprop_lem2v2 = $true) & ((induction2_lthm = $true) <=> (! [X170 : term] : (((sub @ X170 @ id)) = X170) => (! [X171 : (term > $o),X172 : term,X173 : subst] : (~ ! [X174 : term] : (($true = ((X171 @ (sub @ X174 @ X173)))) | ~ ($true = ((var @ X174)))) | ($true = ((X171 @ (sub @ X172 @ X173)))) | ~ ! [X175 : term] : (($true = ((X171 @ (lam @ X175)))) | ~ ! [X176 : term] : (($true = ((X171 @ (sub @ X175 @ (push @ X176 @ id))))) | ~ ($true = ((X171 @ X176))))) | ~ ! [X177 : term,X178 : term] : (($true = ((X171 @ (ap @ X178 @ X177)))) | ~ ($true = ((X171 @ X177))) | ~ ($true = ((X171 @ X178))))) => ! [X179 : term,X180 : (term > $o)] : ((((X180 @ X179)) = $true) | ~ ! [X181 : term,X182 : term] : (~ ($true = ((X180 @ X182))) | ~ ($true = ((X180 @ X181))) | ($true = ((X180 @ (ap @ X182 @ X181))))) | ~ ! [X183 : term] : (($true = ((X180 @ (lam @ X183)))) | ~ ! [X184 : term] : (~ ($true = ((X180 @ X184))) | (((X180 @ (sub @ X183 @ (push @ X184 @ id)))) = $true))) | ~ ! [X185 : term] : (($true = ((X180 @ X185))) | ~ ($true = ((var @ X185)))))))) & (! [X186 : term,X187 : (term > $o)] : (~ ! [X188 : term] : (($true = ((X187 @ (lam @ X188)))) | ~ ($true = ((X187 @ X188)))) | ~ ! [X189 : term] : (($true = ((X187 @ X189))) | ~ ($true = ((var @ X189)))) | ($true = ((X187 @ X186))) | ~ ! [X190 : term,X191 : term] : (~ ($true = ((X187 @ X190))) | ~ ($true = ((X187 @ X191))) | ($true = ((X187 @ (ap @ X191 @ X190)))))) <=> (induction = $true)) & (pushprop_lem3v2 = $true) & ~ (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) = $true) & ((pushprop_lem1_lthm = $true) <=> (! [X192 : subst,X193 : term] : (((sub @ one @ (push @ X193 @ X192))) = X193) => (! [X195 : term,X194 : subst] : (((comp @ sh @ (push @ X195 @ X194))) = X194) => ! [X196 : (term > $o),X197 : subst,X198 : term,X199 : term,X200 : (term > $o)] : ((((X200 @ (sub @ X199 @ (push @ X198 @ X197)))) = $true) | ~ ($true = ((X196 @ X199))))))) & (axassoc = $true) & (induction2lem_gthm = $true) & ((induction2lem_lthm = $true) <=> (! [X201 : term,X202 : subst] : (((sub @ one @ (push @ X201 @ X202))) = X201) => (! [X204 : subst,X203 : term] : (((comp @ sh @ (push @ X203 @ X204))) = X204) => (! [X205 : subst] : (((comp @ X205 @ id)) = X205) => (! [X206 : (term > $o),X207 : term] : (($true = ((X206 @ X207))) | ~ ! [X208 : term] : (~ ($true = ((X206 @ X208))) | ($true = ((X206 @ (lam @ X208))))) | ~ ! [X209 : term] : (~ ($true = ((var @ X209))) | ($true = ((X206 @ X209)))) | ~ ! [X210 : term,X211 : term] : (~ ($true = ((X206 @ X211))) | ($true = ((X206 @ (ap @ X210 @ X211)))) | ~ ($true = ((X206 @ X210))))) => ! [X212 : (term > $o),X213 : subst,X214 : term] : (~ ! [X215 : term] : (~ ! [X216 : term] : (~ ($true = ((X212 @ X216))) | (((X212 @ (sub @ X215 @ (push @ X216 @ id)))) = $true)) | ($true = ((X212 @ (lam @ X215))))) | ~ ! [X217 : term,X218 : term] : (~ ($true = ((X212 @ X217))) | ~ ($true = ((X212 @ X218))) | ($true = ((X212 @ (ap @ X218 @ X217))))) | ($true = ((X212 @ (sub @ X214 @ X213)))) | ~ ! [X219 : term] : (($true = ((X212 @ (sub @ X219 @ X213)))) | ~ ($true = ((var @ X219)))))))))) & ~ (ulamvar1 = $true) & ((hoasinduction_no_psi_cond = $true) <=> ! [X220 : term,X221 : (subst > term > subst > $o)] : (~ ! [X222 : (subst > term > term)] : (~ ! [X225 : subst,X224 : subst,X223 : term] : (((sub @ (X222 @ X224 @ X223) @ X225)) = ((X222 @ (comp @ X224 @ X225) @ (sub @ X223 @ X225)))) | ($true = ((X221 @ id @ (lam @ (X222 @ sh @ one)) @ id))) | ~ ! [X226 : term] : (~ ($true = ((X221 @ id @ X226 @ id))) | ($true = ((X221 @ id @ (X222 @ id @ X226) @ id))))) | ~ ! [X227 : term,X228 : term] : (($true = ((X221 @ id @ (ap @ (sub @ X228 @ id) @ X227) @ id))) | ~ ($true = ((X221 @ id @ X228 @ id))) | ~ ($true = ((X221 @ id @ X227 @ id)))) | (((X221 @ id @ X220 @ id)) = $true))) & ((pushprop_lem0_lthm = $true) <=> ! [X229 : term,X231 : (term > $o),X230 : subst] : ~ ! [X232 : (term > $o)] : ~ ! [X233 : term] : (((X232 @ X233)) = ((X231 @ (sub @ X233 @ (push @ X229 @ X230)))))) & ! [X234 : d_subst,X235 : d_subst] : ((((d2subst @ X235)) = ((d2subst @ X234))) => (X234 = X235)) & ~ (lamnotap = $true) & ((! [X236 : term,X237 : (term > $o)] : (~ ! [X238 : term] : (($true = ((X237 @ X238))) | ~ ($true = ((var @ X238)))) | ~ ! [X239 : term,X240 : term] : (~ ($true = ((X237 @ X239))) | ($true = ((X237 @ (ap @ X240 @ X239)))) | ~ ($true = ((X237 @ X240)))) | ~ ! [X241 : term] : (~ ! [X242 : term] : (($true = ((X237 @ (sub @ X241 @ (push @ X242 @ id))))) | ~ ($true = ((X237 @ X242)))) | ($true = ((X237 @ (lam @ X241))))) | ($true = ((X237 @ X236)))) => (! [X243 : (subst > term > subst > $o),X244 : term,X245 : term] : (~ ($true = ((X243 @ id @ X245 @ id))) | ($true = ((X243 @ id @ (ap @ X244 @ X245) @ id))) | ~ ! [X246 : term,X247 : term] : (($true = ((X243 @ id @ (ap @ (sub @ X246 @ id) @ X247) @ id))) | ~ ($true = ((X243 @ id @ X246 @ id))) | ~ ($true = ((X243 @ id @ X247 @ id)))) | ~ ($true = ((X243 @ id @ X244 @ id))) | ~ ! [X248 : term,X249 : subst,X250 : subst,X251 : subst] : (~ ($true = ((X243 @ (comp @ X249 @ X251) @ (sub @ X248 @ X251) @ X250))) | ($true = ((X243 @ X249 @ X248 @ (comp @ X251 @ X250))))) | ~ ! [X252 : subst,X253 : subst,X254 : subst,X255 : term] : (~ ($true = ((X243 @ X254 @ X255 @ (comp @ X252 @ X253)))) | ($true = ((X243 @ (comp @ X254 @ X252) @ (sub @ X255 @ X252) @ X253))))) => (! [X256 : (subst > term > subst > $o),X257 : term] : (~ ! [X258 : (subst > term > term)] : (~ ! [X259 : subst,X261 : subst,X260 : term] : (((X258 @ (comp @ X259 @ X261) @ (sub @ X260 @ X261))) = ((sub @ (X258 @ X259 @ X260) @ X261))) | ($true = ((X256 @ id @ (lam @ (X258 @ sh @ one)) @ id))) | ~ ! [X262 : term] : (~ ($true = ((X256 @ id @ X262 @ id))) | ($true = ((X256 @ id @ (X258 @ id @ X262) @ id))))) | ~ ! [X263 : subst,X264 : subst,X265 : term,X266 : subst] : (($true = ((X256 @ X263 @ X265 @ (comp @ X264 @ X266)))) | ~ ($true = ((X256 @ (comp @ X263 @ X264) @ (sub @ X265 @ X264) @ X266)))) | ~ ! [X267 : subst,X268 : term,X269 : subst,X270 : subst] : (~ ($true = ((X256 @ X267 @ X268 @ (comp @ X269 @ X270)))) | ($true = ((X256 @ (comp @ X267 @ X269) @ (sub @ X268 @ X269) @ X270)))) | ($true = ((X256 @ id @ (lam @ X257) @ id))) | ~ ! [X271 : term] : (($true = ((X256 @ id @ (sub @ X257 @ (push @ X271 @ id)) @ id))) | ~ ($true = ((X256 @ id @ X271 @ id))))) => ! [X272 : term,X273 : (subst > term > subst > $o)] : (~ ! [X274 : (subst > term > term)] : (~ ! [X275 : term] : (($true = ((X273 @ id @ (X274 @ id @ X275) @ id))) | ~ ($true = ((X273 @ id @ X275 @ id)))) | ($true = ((X273 @ id @ (lam @ (X274 @ sh @ one)) @ id))) | ~ ! [X276 : subst,X277 : term,X278 : subst] : (((X274 @ (comp @ X276 @ X278) @ (sub @ X277 @ X278))) = ((sub @ (X274 @ X276 @ X277) @ X278)))) | ~ ! [X279 : term,X280 : subst,X281 : subst,X282 : subst] : (~ ($true = ((X273 @ (comp @ X282 @ X280) @ (sub @ X279 @ X280) @ X281))) | ($true = ((X273 @ X282 @ X279 @ (comp @ X280 @ X281))))) | ~ ! [X283 : term] : (~ ($true = ((var @ (sub @ X283 @ id)))) | ($true = ((X273 @ id @ X283 @ id)))) | ($true = ((X273 @ id @ X272 @ id))) | ~ ! [X284 : subst,X285 : subst,X286 : term,X287 : subst] : (~ ($true = ((X273 @ X284 @ X286 @ (comp @ X285 @ X287)))) | ($true = ((X273 @ (comp @ X284 @ X285) @ (sub @ X286 @ X285) @ X287)))) | ~ ! [X288 : term,X289 : term] : (~ (((X273 @ id @ X289 @ id)) = $true) | ($true = ((X273 @ id @ (ap @ (sub @ X289 @ id) @ X288) @ id))) | ~ ($true = ((X273 @ id @ X288 @ id)))))))) <=> (hoasinduction_lthm = $true)) & ((laminj = $true) <=> ! [X290 : term,X291 : term] : ((((lam @ X291)) != ((lam @ X290))) | (X290 = X291))) & ((pushprop_lem1v2 = $true) <=> ! [X292 : (term > $o),X293 : term,X294 : (term > $o),X295 : subst] : (~ ! [X296 : term] : (((X294 @ (sub @ X296 @ (push @ X293 @ X295)))) = ((X292 @ X296))) | ($true = ((X292 @ one))) | ~ ($true = ((X294 @ X293))))) & (pushprop_lem1_gthm = $true) & ! [X297 : term] : ? [X298 : d_term] : (((d2term @ X298)) = X297) & (hoasinduction_lem1_lthm = $true) & ((hoasinduction_lem3_lthm = $true) <=> (! [X299 : term] : (((sub @ X299 @ id)) = X299) => (! [X300 : (subst > term > subst > $o),X301 : term] : (~ ! [X302 : (subst > term > term)] : (~ ! [X303 : term,X304 : subst,X305 : subst] : (((sub @ (X302 @ X304 @ X303) @ X305)) = ((X302 @ (comp @ X304 @ X305) @ (sub @ X303 @ X305)))) | ($true = ((X300 @ id @ (lam @ (X302 @ sh @ one)) @ id))) | ~ ! [X306 : term] : (~ ($true = ((X300 @ id @ X306 @ id))) | ($true = ((X300 @ id @ (X302 @ id @ X306) @ id))))) | ($true = ((X300 @ id @ (lam @ (sub @ X301 @ (push @ one @ sh))) @ id))) | ~ ! [X307 : term] : (~ ($true = ((X300 @ id @ X307 @ id))) | ($true = ((X300 @ id @ (sub @ X301 @ (push @ X307 @ id)) @ id))))) => ! [X308 : (subst > term > subst > $o),X309 : term] : (~ ! [X310 : subst,X311 : subst,X312 : term,X313 : subst] : (($true = ((X308 @ X310 @ X312 @ (comp @ X313 @ X311)))) | ~ ($true = ((X308 @ (comp @ X310 @ X313) @ (sub @ X312 @ X313) @ X311)))) | ~ ! [X314 : term,X315 : subst,X316 : subst,X317 : subst] : ((((X308 @ (comp @ X315 @ X317) @ (sub @ X314 @ X317) @ X316)) = $true) | ~ (((X308 @ X315 @ X314 @ (comp @ X317 @ X316))) = $true)) | ~ ! [X318 : (subst > term > term)] : (($true = ((X308 @ id @ (lam @ (X318 @ sh @ one)) @ id))) | ~ ! [X319 : term] : (($true = ((X308 @ id @ (X318 @ id @ X319) @ id))) | ~ ($true = ((X308 @ id @ X319 @ id)))) | ~ ! [X320 : subst,X321 : term,X322 : subst] : (((X318 @ (comp @ X320 @ X322) @ (sub @ X321 @ X322))) = ((sub @ (X318 @ X320 @ X321) @ X322)))) | ~ ! [X323 : term] : (~ ($true = ((X308 @ id @ X323 @ id))) | ($true = ((X308 @ id @ (sub @ X309 @ (push @ X323 @ id)) @ id)))) | ($true = ((X308 @ id @ (lam @ X309) @ id))))))) & (hoaslamnotvar_gthm = $true) & (hoaslamnotvar = $true) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X324 : term] : ~ ! [X325 : (subst > term > term)] : (((sub @ X324 @ (push @ one @ sh))) != ((X325 @ sh @ one)))) & (! [X326 : term,X327 : term,X328 : subst,X329 : (term > $o),X330 : (term > $o)] : (($true = ((X329 @ (sub @ X327 @ (push @ X326 @ X328))))) | ~ ($true = ((X330 @ X327)))) <=> (pushprop_lem1 = $true)) & ((! [X331 : subst] : (((comp @ id @ X331)) = X331) => (! [X332 : subst] : (((comp @ X332 @ id)) = X332) => (! [X333 : subst] : (((comp @ id @ X333)) = X333) & ! [X334 : subst] : (((comp @ X334 @ id)) = X334)))) <=> (substmonoid_lthm = $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (! [X337 : subst,X336 : term,X335 : (term > $o)] : ~ ! [X338 : (term > $o)] : ~ ! [X339 : term] : (((X335 @ (sub @ X339 @ (push @ X336 @ X337)))) = ((X338 @ X339))) <=> (pushprop_lem0 = $true)) & (! [X340 : subst] : (((comp @ id @ X340)) = X340) <=> (axidl = $true)) & (pushprop = $true) & (axabs = $true) & (! [X341 : term,X342 : (subst > term > subst > $o)] : (~ ! [X343 : subst,X344 : subst,X345 : subst,X346 : term] : (~ ($true = ((X342 @ X343 @ X346 @ (comp @ X345 @ X344)))) | ($true = ((X342 @ (comp @ X343 @ X345) @ (sub @ X346 @ X345) @ X344)))) | ~ ! [X347 : term] : (~ ($true = ((X342 @ id @ X347 @ id))) | (((X342 @ id @ (sub @ X341 @ (push @ X347 @ id)) @ id)) = $true)) | ($true = ((X342 @ id @ (lam @ X341) @ id))) | ~ ! [X348 : term,X349 : subst,X350 : subst,X351 : subst] : ((((X342 @ X351 @ X348 @ (comp @ X350 @ X349))) = $true) | ~ ($true = ((X342 @ (comp @ X351 @ X350) @ (sub @ X348 @ X350) @ X349)))) | ~ ! [X352 : (subst > term > term)] : (~ ! [X355 : subst,X353 : subst,X354 : term] : (((X352 @ (comp @ X355 @ X353) @ (sub @ X354 @ X353))) = ((sub @ (X352 @ X355 @ X354) @ X353))) | ~ ! [X356 : term] : (($true = ((X342 @ id @ (X352 @ id @ X356) @ id))) | ~ (((X342 @ id @ X356 @ id)) = $true)) | ($true = ((X342 @ id @ (lam @ (X352 @ sh @ one)) @ id))))) <=> (hoasinduction_lem3 = $true)) & (hoasinduction_lem1 = $true) & (hoasapnotvar = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & ((hoasinduction_lem0_lthm = $true) <=> ! [X357 : (subst > term > subst > $o)] : ~ ! [X358 : (term > $o)] : ~ ! [X359 : term] : (((X358 @ X359)) = ((X357 @ id @ X359 @ id)))) & ! [X360 : subst] : ? [X361 : d_subst] : (((d2subst @ X361)) = X360) & (pushprop_lem2v2_gthm = $true) & (! [X363 : term,X362 : subst] : (((comp @ sh @ (push @ X363 @ X362))) = X362) <=> (axshiftcons = $true)) & ((apinj2 = $true) <=> ! [X364 : term,X365 : term,X366 : term,X367 : term] : ((X364 = X365) | (((ap @ X366 @ X364)) != ((ap @ X367 @ X365))))) & ((hoasinduction_lem3b = $true) <=> ! [X368 : term] : ~ ! [X369 : (subst > term > term)] : (((X369 @ sh @ one)) != ((sub @ X368 @ (push @ one @ sh))))) & (! [X370 : term] : ~ ! [X371 : (subst > term > term)] : ~ ! [X372 : term,X373 : subst] : (((X371 @ X373 @ X372)) = ((sub @ X370 @ (push @ X372 @ X373)))) <=> (hoasinduction_lem3v2_f = $true)) & (hoasapnotvar_lthm = $true) & (! [X374 : (subst > term > subst > $o),X375 : term] : (~ ! [X376 : term] : (($true = ((X374 @ id @ (sub @ X375 @ (push @ X376 @ id)) @ id))) | ~ ($true = ((X374 @ id @ X376 @ id)))) | ~ ! [X377 : term,X378 : (subst > term > term)] : (~ ! [X379 : term,X380 : subst,X381 : subst] : (((X378 @ (comp @ X381 @ X380) @ (sub @ X379 @ X380))) = ((sub @ X377 @ (push @ (sub @ X379 @ X380) @ (comp @ X381 @ X380))))) | ($true = ((X374 @ id @ (lam @ (X378 @ sh @ one)) @ id))) | ~ ! [X382 : term,X384 : subst,X383 : subst] : (((sub @ (X378 @ X384 @ X382) @ X383)) = ((sub @ (sub @ X377 @ (push @ X382 @ X384)) @ X383))) | ~ ! [X385 : term] : (($true = ((X374 @ id @ (X378 @ id @ X385) @ id))) | ~ ($true = ((X374 @ id @ X385 @ id))))) | (((X374 @ id @ (lam @ (sub @ X375 @ (push @ one @ sh))) @ id)) = $true)) <=> (hoasinduction_lem3aaa = $true)) & (! [X386 : (subst > term > subst > $o),X387 : term] : (~ ! [X388 : (subst > term > term)] : (~ ! [X389 : subst,X390 : subst,X391 : term] : (((X388 @ (comp @ X390 @ X389) @ (sub @ X391 @ X389))) = ((sub @ (X388 @ X390 @ X391) @ X389))) | ~ ! [X392 : term] : (~ ($true = ((X386 @ id @ X392 @ id))) | ($true = ((X386 @ id @ (X388 @ id @ X392) @ id)))) | ($true = ((X386 @ id @ (lam @ (X388 @ sh @ one)) @ id)))) | ~ ! [X393 : term] : (($true = ((X386 @ id @ (sub @ X387 @ (push @ X393 @ id)) @ id))) | ~ ($true = ((X386 @ id @ X393 @ id)))) | (((X386 @ id @ (lam @ X387) @ id)) = $true)) <=> (hoasinduction_lem3a = $true)) & (hoasinduction_lem3v2_gthm = $true) & (((d2subst @ d_id)) = sh) & (hoasapnotvar_gthm = $true) & (hoasapinj2_gthm = $true) & ((axscons = $true) <=> ! [X394 : subst] : (((push @ (sub @ one @ X394) @ (comp @ sh @ X394))) = X394)) & (hoasinduction_lem1v2_gthm = $true) & ((! [X395 : term] : (((sub @ X395 @ id)) = X395) => ! [X396 : term] : (((sub @ X396 @ id)) = X396)) <=> (termmset_lthm = $true)) & (axvarshift = $true) & ((hoasinduction = $true) <=> ! [X397 : term,X398 : (subst > term > subst > $o)] : (~ ! [X399 : subst,X400 : subst,X401 : subst,X402 : term] : (~ ($true = ((X398 @ (comp @ X401 @ X399) @ (sub @ X402 @ X399) @ X400))) | ($true = ((X398 @ X401 @ X402 @ (comp @ X399 @ X400))))) | ~ ! [X403 : (subst > term > term)] : ((((X398 @ id @ (lam @ (X403 @ sh @ one)) @ id)) = $true) | ~ ! [X404 : term] : (~ (((X398 @ id @ X404 @ id)) = $true) | ($true = ((X398 @ id @ (X403 @ id @ X404) @ id)))) | ~ ! [X405 : subst,X406 : subst,X407 : term] : (((sub @ (X403 @ X406 @ X407) @ X405)) = ((X403 @ (comp @ X406 @ X405) @ (sub @ X407 @ X405))))) | ~ ! [X408 : term,X409 : term] : (~ ($true = ((X398 @ id @ X408 @ id))) | ~ (((X398 @ id @ X409 @ id)) = $true) | ($true = ((X398 @ id @ (ap @ (sub @ X408 @ id) @ X409) @ id)))) | ~ ! [X410 : term] : (($true = ((X398 @ id @ X410 @ id))) | ~ ($true = ((var @ (sub @ X410 @ id))))) | ($true = ((X398 @ id @ X397 @ id))) | ~ ! [X411 : subst,X412 : term,X413 : subst,X414 : subst] : (~ ($true = ((X398 @ X411 @ X412 @ (comp @ X414 @ X413)))) | ($true = ((X398 @ (comp @ X411 @ X414) @ (sub @ X412 @ X414) @ X413)))))) & ((hoasinduction_lem3v2_lthm = $true) <=> (! [X415 : term] : (((sub @ X415 @ id)) = X415) => ! [X416 : (subst > term > subst > $o),X417 : term,X418 : (term > $o)] : (~ ! [X419 : subst,X420 : term,X421 : subst,X422 : subst] : (~ ($true = ((X416 @ X422 @ X420 @ (comp @ X419 @ X421)))) | ($true = ((X416 @ (comp @ X422 @ X419) @ (sub @ X420 @ X419) @ X421)))) | ($true = ((X418 @ (lam @ X417)))) | ~ ! [X423 : term] : (($true = ((X418 @ (sub @ X417 @ (push @ X423 @ id))))) | ~ ($true = ((X418 @ X423)))) | ~ ! [X424 : subst,X425 : subst,X426 : subst,X427 : term] : (($true = ((X416 @ X424 @ X427 @ (comp @ X426 @ X425)))) | ~ ($true = ((X416 @ (comp @ X424 @ X426) @ (sub @ X427 @ X426) @ X425)))) | ~ ! [X428 : (subst > term > term)] : (($true = ((X416 @ id @ (lam @ (X428 @ sh @ one)) @ id))) | ~ ! [X430 : subst,X429 : term,X431 : subst] : (((sub @ (X428 @ X431 @ X429) @ X430)) = ((X428 @ (comp @ X431 @ X430) @ (sub @ X429 @ X430)))) | ~ ! [X432 : term] : (~ ($true = ((X416 @ id @ X432 @ id))) | ($true = ((X416 @ id @ (X428 @ id @ X432) @ id))))) | ~ ! [X433 : term] : (((X418 @ X433)) = ((X416 @ id @ X433 @ id)))))) & (lamnotvar = $true) & (pushprop_lem0_gthm = $true) & (! [X434 : (subst > term > subst > $o),X435 : term,X436 : (term > $o)] : (~ ! [X437 : term] : (((X436 @ X437)) = ((X434 @ id @ X437 @ id))) | ($true = ((X436 @ (lam @ X435)))) | ~ ! [X438 : (subst > term > term)] : (~ ! [X440 : subst,X441 : term,X439 : subst] : (((sub @ (X438 @ X439 @ X441) @ X440)) = ((X438 @ (comp @ X439 @ X440) @ (sub @ X441 @ X440)))) | ($true = ((X434 @ id @ (lam @ (X438 @ sh @ one)) @ id))) | ~ ! [X442 : term] : ((((X434 @ id @ (X438 @ id @ X442) @ id)) = $true) | ~ ($true = ((X434 @ id @ X442 @ id))))) | ~ ! [X443 : term] : (~ ($true = ((X436 @ X443))) | ($true = ((X436 @ (sub @ X435 @ (push @ X443 @ id))))))) <=> (hoasinduction_lem3v2a = $true)) & (! [X444 : subst] : (((comp @ X444 @ id)) = X444) <=> (axidr = $true)) & (id = ((d2subst @ d_id))) & (! [X446 : term,X445 : term,X447 : (subst > term > term)] : ((((ap @ (sub @ X445 @ id) @ X446)) != ((lam @ (X447 @ sh @ one)))) | ~ ! [X449 : subst,X448 : subst,X450 : term] : (((X447 @ (comp @ X448 @ X449) @ (sub @ X450 @ X449))) = ((sub @ (X447 @ X448 @ X450) @ X449)))) <=> (hoaslamnotap = $true)) & ((hoasapinj1_lthm = $true) <=> (! [X451 : term] : (((sub @ X451 @ id)) = X451) => (! [X453 : term,X455 : term,X454 : term,X452 : term] : ((X452 = X455) | (((ap @ X452 @ X453)) != ((ap @ X455 @ X454)))) => ! [X458 : term,X457 : term,X456 : term,X459 : term] : ((((ap @ (sub @ X458 @ id) @ X456)) != ((ap @ (sub @ X459 @ id) @ X457))) | (X458 = X459))))) & (pushprop_lem2v2_lthm = $true) & (ulamvarsh = $true) & (axapp = $true) & (substmonoid_gthm = $true) & (((d2term @ d_one)) = one) & (hoasinduction_lem2v2_gthm = $true) & ((hoaslaminj = $true) <=> ! [X461 : subst,X460 : (subst > term > term),X462 : (subst > term > term),X463 : term] : (~ ! [X464 : subst,X466 : term,X465 : subst] : (((sub @ (X460 @ X465 @ X466) @ X464)) = ((X460 @ (comp @ X465 @ X464) @ (sub @ X466 @ X464)))) | (((lam @ (X462 @ sh @ one))) != ((lam @ (X460 @ sh @ one)))) | (((X462 @ X461 @ X463)) = ((X460 @ X461 @ X463))) | ~ ! [X467 : subst,X469 : term,X468 : subst] : (((sub @ (X462 @ X467 @ X469) @ X468)) = ((X462 @ (comp @ X467 @ X468) @ (sub @ X469 @ X468)))))) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((hoasinduction_lem2v2 = $true) <=> ! [X470 : term,X471 : (term > $o),X472 : (subst > term > subst > $o),X473 : term] : (($true = ((X471 @ (ap @ X470 @ X473)))) | ~ ! [X474 : term] : (((X472 @ id @ X474 @ id)) = ((X471 @ X474))) | ~ ! [X475 : term,X476 : term] : (~ ($true = ((X472 @ id @ X476 @ id))) | ~ ($true = ((X472 @ id @ X475 @ id))) | ($true = ((X472 @ id @ (ap @ (sub @ X476 @ id) @ X475) @ id)))) | ~ ! [X477 : subst,X478 : subst,X479 : term,X480 : subst] : (($true = ((X472 @ X478 @ X479 @ (comp @ X480 @ X477)))) | ~ ($true = ((X472 @ (comp @ X478 @ X480) @ (sub @ X479 @ X480) @ X477)))) | ~ ($true = ((X471 @ X473))) | ~ ($true = ((X471 @ X470))) | ~ ! [X481 : subst,X482 : subst,X483 : term,X484 : subst] : (~ ($true = ((X472 @ X481 @ X483 @ (comp @ X484 @ X482)))) | ($true = ((X472 @ (comp @ X481 @ X484) @ (sub @ X483 @ X484) @ X482)))))) & (hoasinduction_lem3a_gthm = $true) & (! [X485 : (term > $o),X486 : term] : (($true = ((X485 @ X486))) | ~ ! [X487 : term] : (~ ($true = ((var @ X487))) | ($true = ((X485 @ X487)))) | ~ ! [X488 : term,X489 : term] : (~ (((X485 @ X488)) = $true) | ($true = ((X485 @ (ap @ X488 @ X489)))) | ~ ($true = ((X485 @ X489)))) | ~ ! [X490 : term] : (($true = ((X485 @ (lam @ X490)))) | ~ ! [X491 : term] : ((((X485 @ (sub @ X490 @ (push @ X491 @ id)))) = $true) | ~ ($true = ((X485 @ X491)))))) <=> (induction2 = $true)) & (ulamvarind = $true) & ~ (((var @ (d2term @ d_one))) = $true) & (! [X492 : term,X493 : (subst > term > subst > $o),X494 : (term > $o)] : (($true = ((X494 @ (lam @ X492)))) | ~ ! [X495 : term] : (((X494 @ X495)) = ((X493 @ id @ X495 @ id))) | ~ ! [X496 : (subst > term > term)] : (~ ! [X497 : term] : (~ ($true = ((X493 @ id @ X497 @ id))) | ($true = ((X493 @ id @ (X496 @ id @ X497) @ id)))) | ~ ! [X498 : subst,X500 : subst,X499 : term] : (((sub @ (X496 @ X498 @ X499) @ X500)) = ((X496 @ (comp @ X498 @ X500) @ (sub @ X499 @ X500)))) | ($true = ((X493 @ id @ (lam @ (X496 @ sh @ one)) @ id)))) | ~ ! [X501 : term,X502 : subst,X503 : subst,X504 : subst] : (~ ($true = ((X493 @ X504 @ X501 @ (comp @ X503 @ X502)))) | ($true = ((X493 @ (comp @ X504 @ X503) @ (sub @ X501 @ X503) @ X502)))) | ~ ! [X505 : term] : (~ ($true = ((X494 @ X505))) | ($true = ((X494 @ (sub @ X492 @ (push @ X505 @ id)))))) | ~ ! [X506 : subst,X507 : term,X508 : subst,X509 : subst] : (~ ($true = ((X493 @ (comp @ X509 @ X506) @ (sub @ X507 @ X506) @ X508))) | ($true = ((X493 @ X509 @ X507 @ (comp @ X506 @ X508)))))) <=> (hoasinduction_lem3v2 = $true)) & (hoasinduction_lem1_gthm = $true) & (! [X510 : term] : (((sub @ X510 @ id)) = X510) <=> (termmset = $true)) & (hoaslaminj_gthm = $true) & (axclos = $true) & (hoasinduction_lem2_lthm = $true) & ((substmonoid = $true) <=> (! [X511 : subst] : (((comp @ X511 @ id)) = X511) & ! [X512 : subst] : (((comp @ id @ X512)) = X512))) & (axmap = $true) & ((hoasapinj2_lthm = $true) <=> (! [X516 : term,X514 : term,X515 : term,X513 : term] : ((((ap @ X515 @ X514)) != ((ap @ X516 @ X513))) | (X513 = X514)) => ! [X520 : term,X519 : term,X518 : term,X517 : term] : ((X517 = X520) | (((ap @ (sub @ X519 @ id) @ X517)) != ((ap @ (sub @ X518 @ id) @ X520)))))) & (! [X523 : term,X522 : term,X521 : term,X524 : term] : ((((ap @ (sub @ X523 @ id) @ X521)) != ((ap @ (sub @ X522 @ id) @ X524))) | (X521 = X524)) <=> (hoasapinj2 = $true)) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X525 : term,X526 : (subst > term > subst > $o)] : (~ ! [X527 : term] : (($true = ((X526 @ id @ (sub @ X525 @ (push @ X527 @ id)) @ id))) | ~ ($true = ((X526 @ id @ X527 @ id)))) | ~ ! [X528 : (subst > term > term)] : (~ ! [X529 : term,X531 : subst,X530 : subst] : (((sub @ (X528 @ X530 @ X529) @ X531)) = ((X528 @ (comp @ X530 @ X531) @ (sub @ X529 @ X531)))) | ($true = ((X526 @ id @ (lam @ (X528 @ sh @ one)) @ id))) | ~ ! [X532 : term] : (($true = ((X526 @ id @ (X528 @ id @ X532) @ id))) | ~ ($true = ((X526 @ id @ X532 @ id))))) | ($true = ((X526 @ id @ (lam @ (sub @ X525 @ (push @ one @ sh))) @ id))))) & ((! [X533 : term,X534 : subst] : (((sub @ one @ (push @ X533 @ X534))) = X533) => ! [X535 : term,X536 : (term > $o),X537 : subst,X538 : (term > $o)] : (($true = ((X538 @ one))) | ~ ! [X539 : term] : (((X536 @ (sub @ X539 @ (push @ X535 @ X537)))) = ((X538 @ X539))) | ~ ($true = ((X536 @ X535))))) <=> (pushprop_lem1v2_lthm = $true)) & (hoaslamnotap_lthm = $true)), inference(rectify,[],[f7])). thf(f10,plain,( ((! [X170 : term] : (((sub @ X170 @ id)) = X170) => (! [X172 : term,X173 : subst,X171 : (term > $o)] : (($true = ((X171 @ (sub @ X172 @ X173)))) | ~ ! [X178 : term,X177 : term] : (($true = ((X171 @ (ap @ X178 @ X177)))) | ($true != ((X171 @ X177))) | ($true != ((X171 @ X178)))) | ~ ! [X174 : term] : (($true != ((var @ X174))) | ($true = ((X171 @ (sub @ X174 @ X173))))) | ~ ! [X175 : term] : (($true = ((X171 @ (lam @ X175)))) | ~ ! [X176 : term] : (($true != ((X171 @ X176))) | ($true = ((X171 @ (sub @ X175 @ (push @ X176 @ id)))))))) => ! [X179 : term,X180 : (term > $o)] : (~ ! [X185 : term] : (($true = ((X180 @ X185))) | ($true != ((var @ X185)))) | ~ ! [X181 : term,X182 : term] : (($true != ((X180 @ X181))) | ($true = ((X180 @ (ap @ X182 @ X181)))) | ($true != ((X180 @ X182)))) | (((X180 @ X179)) = $true) | ~ ! [X183 : term] : (~ ! [X184 : term] : (($true != ((X180 @ X184))) | (((X180 @ (sub @ X183 @ (push @ X184 @ id)))) = $true)) | ($true = ((X180 @ (lam @ X183)))))))) <=> (induction2_lthm = $true)) & ((apinj2 = $true) <=> ! [X364 : term,X365 : term,X366 : term,X367 : term] : ((X364 = X365) | (((ap @ X366 @ X364)) != ((ap @ X367 @ X365))))) & (hoasapnotvar_gthm = $true) & (((d2subst @ d_id)) = sh) & ((induction = $true) <=> ! [X186 : term,X187 : (term > $o)] : (~ ! [X188 : term] : (($true != ((X187 @ X188))) | ($true = ((X187 @ (lam @ X188))))) | ~ ! [X189 : term] : (($true = ((X187 @ X189))) | ($true != ((var @ X189)))) | ~ ! [X191 : term,X190 : term] : (($true != ((X187 @ X191))) | ($true != ((X187 @ X190))) | ($true = ((X187 @ (ap @ X191 @ X190))))) | ($true = ((X187 @ X186))))) & (hoaslaminj_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (! [X370 : term] : ~ ! [X371 : (subst > term > term)] : ~ ! [X372 : term,X373 : subst] : (((X371 @ X373 @ X372)) = ((sub @ X370 @ (push @ X372 @ X373)))) <=> (hoasinduction_lem3v2_f = $true)) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X160 : term] : ~ ! [X161 : (subst > term > term)] : ~ ! [X163 : term,X162 : subst] : (((sub @ X160 @ (push @ X163 @ X162))) = ((X161 @ X162 @ X163)))) & (! [X446 : term,X445 : term,X447 : (subst > term > term)] : ((((ap @ (sub @ X445 @ id) @ X446)) != ((lam @ (X447 @ sh @ one)))) | ~ ! [X449 : subst,X448 : subst,X450 : term] : (((X447 @ (comp @ X448 @ X449) @ (sub @ X450 @ X449))) = ((sub @ (X447 @ X448 @ X450) @ X449)))) <=> (hoaslamnotap = $true)) & ((! [X395 : term] : (((sub @ X395 @ id)) = X395) => ! [X396 : term] : (((sub @ X396 @ id)) = X396)) <=> (termmset_lthm = $true)) & (! [X84 : (term > $o),X86 : subst,X85 : term] : ((((X84 @ (sub @ X85 @ X86))) = $true) | ~ ! [X90 : term] : (~ ! [X91 : term] : (($true != ((X84 @ X91))) | ($true = ((X84 @ (sub @ X90 @ (push @ X91 @ id)))))) | ($true = ((X84 @ (lam @ X90))))) | ~ ! [X89 : term] : (($true = ((X84 @ (sub @ X89 @ X86)))) | ($true != ((var @ X89)))) | ~ ! [X87 : term,X88 : term] : (($true = ((X84 @ (ap @ X88 @ X87)))) | ($true != ((X84 @ X88))) | ($true != ((X84 @ X87))))) <=> (induction2lem = $true)) & (! [X164 : term,X165 : subst] : (((sub @ one @ (push @ X164 @ X165))) = X164) <=> (axvarcons = $true)) & (pushprop_gthm = $true) & (hoaslamnotvar_lthm = $true) & (induction2_gthm = $true) & (! [X472 : (subst > term > subst > $o),X471 : (term > $o),X470 : term,X473 : term] : (~ ! [X481 : subst,X482 : subst,X483 : term,X484 : subst] : (($true = ((X472 @ (comp @ X481 @ X484) @ (sub @ X483 @ X484) @ X482))) | ($true != ((X472 @ X481 @ X483 @ (comp @ X484 @ X482))))) | ($true != ((X471 @ X470))) | ~ ! [X475 : term,X476 : term] : (($true != ((X472 @ id @ X475 @ id))) | ($true != ((X472 @ id @ X476 @ id))) | ($true = ((X472 @ id @ (ap @ (sub @ X476 @ id) @ X475) @ id)))) | ~ ! [X479 : term,X480 : subst,X478 : subst,X477 : subst] : (($true = ((X472 @ X478 @ X479 @ (comp @ X480 @ X477)))) | ($true != ((X472 @ (comp @ X478 @ X480) @ (sub @ X479 @ X480) @ X477)))) | ~ ! [X474 : term] : (((X472 @ id @ X474 @ id)) = ((X471 @ X474))) | ($true != ((X471 @ X473))) | ($true = ((X471 @ (ap @ X470 @ X473))))) <=> (hoasinduction_lem2v2 = $true)) & (! [X397 : term,X398 : (subst > term > subst > $o)] : (~ ! [X400 : subst,X399 : subst,X401 : subst,X402 : term] : (($true != ((X398 @ (comp @ X401 @ X399) @ (sub @ X402 @ X399) @ X400))) | ($true = ((X398 @ X401 @ X402 @ (comp @ X399 @ X400))))) | ~ ! [X410 : term] : (($true != ((var @ (sub @ X410 @ id)))) | ($true = ((X398 @ id @ X410 @ id)))) | ~ ! [X408 : term,X409 : term] : (($true != ((X398 @ id @ X408 @ id))) | (((X398 @ id @ X409 @ id)) != $true) | ($true = ((X398 @ id @ (ap @ (sub @ X408 @ id) @ X409) @ id)))) | ($true = ((X398 @ id @ X397 @ id))) | ~ ! [X403 : (subst > term > term)] : (~ ! [X405 : subst,X406 : subst,X407 : term] : (((sub @ (X403 @ X406 @ X407) @ X405)) = ((X403 @ (comp @ X406 @ X405) @ (sub @ X407 @ X405)))) | ~ ! [X404 : term] : (($true = ((X398 @ id @ (X403 @ id @ X404) @ id))) | (((X398 @ id @ X404 @ id)) != $true)) | (((X398 @ id @ (lam @ (X403 @ sh @ one)) @ id)) = $true)) | ~ ! [X412 : term,X411 : subst,X413 : subst,X414 : subst] : (($true = ((X398 @ (comp @ X411 @ X414) @ (sub @ X412 @ X414) @ X413))) | ($true != ((X398 @ X411 @ X412 @ (comp @ X414 @ X413)))))) <=> (hoasinduction = $true)) & (! [X494 : (term > $o),X493 : (subst > term > subst > $o),X492 : term] : (~ ! [X508 : subst,X507 : term,X509 : subst,X506 : subst] : (($true != ((X493 @ (comp @ X509 @ X506) @ (sub @ X507 @ X506) @ X508))) | ($true = ((X493 @ X509 @ X507 @ (comp @ X506 @ X508))))) | ~ ! [X496 : (subst > term > term)] : (~ ! [X498 : subst,X500 : subst,X499 : term] : (((sub @ (X496 @ X498 @ X499) @ X500)) = ((X496 @ (comp @ X498 @ X500) @ (sub @ X499 @ X500)))) | ($true = ((X493 @ id @ (lam @ (X496 @ sh @ one)) @ id))) | ~ ! [X497 : term] : (($true = ((X493 @ id @ (X496 @ id @ X497) @ id))) | ($true != ((X493 @ id @ X497 @ id))))) | ($true = ((X494 @ (lam @ X492)))) | ~ ! [X503 : subst,X501 : term,X504 : subst,X502 : subst] : (($true = ((X493 @ (comp @ X504 @ X503) @ (sub @ X501 @ X503) @ X502))) | ($true != ((X493 @ X504 @ X501 @ (comp @ X503 @ X502))))) | ~ ! [X495 : term] : (((X494 @ X495)) = ((X493 @ id @ X495 @ id))) | ~ ! [X505 : term] : (($true != ((X494 @ X505))) | ($true = ((X494 @ (sub @ X492 @ (push @ X505 @ id))))))) <=> (hoasinduction_lem3v2 = $true)) & ((! [X145 : term] : ~ ! [X146 : (subst > term > term)] : ~ ! [X147 : term,X148 : subst] : (((sub @ X145 @ (push @ X147 @ X148))) = ((X146 @ X148 @ X147))) => (! [X149 : term] : (((sub @ X149 @ id)) = X149) => ! [X152 : term,X151 : (subst > term > subst > $o),X150 : (term > $o)] : (~ ! [X158 : term] : (((X151 @ id @ X158 @ id)) = ((X150 @ X158))) | ~ ! [X153 : (subst > term > term)] : (~ ! [X156 : subst,X154 : term,X155 : subst] : (((sub @ (X153 @ X156 @ X154) @ X155)) = ((X153 @ (comp @ X156 @ X155) @ (sub @ X154 @ X155)))) | ($true = ((X151 @ id @ (lam @ (X153 @ sh @ one)) @ id))) | ~ ! [X157 : term] : (($true = ((X151 @ id @ (X153 @ id @ X157) @ id))) | (((X151 @ id @ X157 @ id)) != $true))) | ($true = ((X150 @ (lam @ X152)))) | ~ ! [X159 : term] : (($true = ((X150 @ (sub @ X152 @ (push @ X159 @ id))))) | ($true != ((X150 @ X159))))))) <=> (hoasinduction_lem3v2a_lthm = $true)) & ((! [X236 : term,X237 : (term > $o)] : (~ ! [X241 : term] : (($true = ((X237 @ (lam @ X241)))) | ~ ! [X242 : term] : (($true = ((X237 @ (sub @ X241 @ (push @ X242 @ id))))) | ($true != ((X237 @ X242))))) | ($true = ((X237 @ X236))) | ~ ! [X238 : term] : (($true = ((X237 @ X238))) | ($true != ((var @ X238)))) | ~ ! [X240 : term,X239 : term] : (($true != ((X237 @ X240))) | ($true = ((X237 @ (ap @ X240 @ X239)))) | ($true != ((X237 @ X239))))) => (! [X245 : term,X244 : term,X243 : (subst > term > subst > $o)] : (~ ! [X247 : term,X246 : term] : (($true != ((X243 @ id @ X247 @ id))) | ($true = ((X243 @ id @ (ap @ (sub @ X246 @ id) @ X247) @ id))) | ($true != ((X243 @ id @ X246 @ id)))) | ~ ! [X251 : subst,X250 : subst,X249 : subst,X248 : term] : (($true = ((X243 @ X249 @ X248 @ (comp @ X251 @ X250)))) | ($true != ((X243 @ (comp @ X249 @ X251) @ (sub @ X248 @ X251) @ X250)))) | ($true != ((X243 @ id @ X245 @ id))) | ($true = ((X243 @ id @ (ap @ X244 @ X245) @ id))) | ~ ! [X254 : subst,X252 : subst,X253 : subst,X255 : term] : (($true = ((X243 @ (comp @ X254 @ X252) @ (sub @ X255 @ X252) @ X253))) | ($true != ((X243 @ X254 @ X255 @ (comp @ X252 @ X253))))) | ($true != ((X243 @ id @ X244 @ id)))) => (! [X257 : term,X256 : (subst > term > subst > $o)] : (~ ! [X264 : subst,X266 : subst,X263 : subst,X265 : term] : (($true = ((X256 @ X263 @ X265 @ (comp @ X264 @ X266)))) | ($true != ((X256 @ (comp @ X263 @ X264) @ (sub @ X265 @ X264) @ X266)))) | ($true = ((X256 @ id @ (lam @ X257) @ id))) | ~ ! [X258 : (subst > term > term)] : (($true = ((X256 @ id @ (lam @ (X258 @ sh @ one)) @ id))) | ~ ! [X262 : term] : (($true != ((X256 @ id @ X262 @ id))) | ($true = ((X256 @ id @ (X258 @ id @ X262) @ id)))) | ~ ! [X259 : subst,X261 : subst,X260 : term] : (((X258 @ (comp @ X259 @ X261) @ (sub @ X260 @ X261))) = ((sub @ (X258 @ X259 @ X260) @ X261)))) | ~ ! [X271 : term] : (($true = ((X256 @ id @ (sub @ X257 @ (push @ X271 @ id)) @ id))) | ($true != ((X256 @ id @ X271 @ id)))) | ~ ! [X267 : subst,X268 : term,X269 : subst,X270 : subst] : (($true = ((X256 @ (comp @ X267 @ X269) @ (sub @ X268 @ X269) @ X270))) | ($true != ((X256 @ X267 @ X268 @ (comp @ X269 @ X270)))))) => ! [X272 : term,X273 : (subst > term > subst > $o)] : (~ ! [X281 : subst,X279 : term,X282 : subst,X280 : subst] : (($true = ((X273 @ X282 @ X279 @ (comp @ X280 @ X281)))) | ($true != ((X273 @ (comp @ X282 @ X280) @ (sub @ X279 @ X280) @ X281)))) | ~ ! [X283 : term] : (($true = ((X273 @ id @ X283 @ id))) | ($true != ((var @ (sub @ X283 @ id))))) | ($true = ((X273 @ id @ X272 @ id))) | ~ ! [X289 : term,X288 : term] : (($true = ((X273 @ id @ (ap @ (sub @ X289 @ id) @ X288) @ id))) | (((X273 @ id @ X289 @ id)) != $true) | ($true != ((X273 @ id @ X288 @ id)))) | ~ ! [X274 : (subst > term > term)] : (~ ! [X275 : term] : (($true = ((X273 @ id @ (X274 @ id @ X275) @ id))) | ($true != ((X273 @ id @ X275 @ id)))) | ($true = ((X273 @ id @ (lam @ (X274 @ sh @ one)) @ id))) | ~ ! [X276 : subst,X277 : term,X278 : subst] : (((X274 @ (comp @ X276 @ X278) @ (sub @ X277 @ X278))) = ((sub @ (X274 @ X276 @ X277) @ X278)))) | ~ ! [X286 : term,X287 : subst,X284 : subst,X285 : subst] : (($true = ((X273 @ (comp @ X284 @ X285) @ (sub @ X286 @ X285) @ X287))) | ($true != ((X273 @ X284 @ X286 @ (comp @ X285 @ X287))))))))) <=> (hoasinduction_lthm = $true)) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (substmonoid_gthm = $true) & (hoaslamnotap_lthm = $true) & ! [X360 : subst] : ? [X361 : d_subst] : (((d2subst @ X361)) = X360) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X324 : term] : ~ ! [X325 : (subst > term > term)] : (((sub @ X324 @ (push @ one @ sh))) != ((X325 @ sh @ one)))) & (hoasinduction_lem3_gthm = $true) & (hoasapinj2_gthm = $true) & (pushprop_lthm = $true) & (hoasinduction_lem1v2_gthm = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoasinduction_gthm = $true) & (! [X337 : subst,X336 : term,X335 : (term > $o)] : ~ ! [X338 : (term > $o)] : ~ ! [X339 : term] : (((X335 @ (sub @ X339 @ (push @ X336 @ X337)))) = ((X338 @ X339))) <=> (pushprop_lem0 = $true)) & ((substmonoid = $true) <=> (! [X511 : subst] : (((comp @ X511 @ id)) = X511) & ! [X512 : subst] : (((comp @ id @ X512)) = X512))) & (hoasinduction_lem2_gthm = $true) & (apnotvar = $true) & (! [X374 : (subst > term > subst > $o),X375 : term] : (~ ! [X376 : term] : (($true = ((X374 @ id @ (sub @ X375 @ (push @ X376 @ id)) @ id))) | ($true != ((X374 @ id @ X376 @ id)))) | ~ ! [X378 : (subst > term > term),X377 : term] : (~ ! [X385 : term] : (($true = ((X374 @ id @ (X378 @ id @ X385) @ id))) | ($true != ((X374 @ id @ X385 @ id)))) | ~ ! [X382 : term,X384 : subst,X383 : subst] : (((sub @ (X378 @ X384 @ X382) @ X383)) = ((sub @ (sub @ X377 @ (push @ X382 @ X384)) @ X383))) | ~ ! [X379 : term,X380 : subst,X381 : subst] : (((X378 @ (comp @ X381 @ X380) @ (sub @ X379 @ X380))) = ((sub @ X377 @ (push @ (sub @ X379 @ X380) @ (comp @ X381 @ X380))))) | ($true = ((X374 @ id @ (lam @ (X378 @ sh @ one)) @ id)))) | (((X374 @ id @ (lam @ (sub @ X375 @ (push @ one @ sh))) @ id)) = $true)) <=> (hoasinduction_lem3aaa = $true)) & (axclos = $true) & (hoasinduction_lem1_gthm = $true) & (hoaslamnotvar_gthm = $true) & (axassoc = $true) & ((induction2 = $true) <=> ! [X485 : (term > $o),X486 : term] : (~ ! [X487 : term] : (($true != ((var @ X487))) | ($true = ((X485 @ X487)))) | ~ ! [X490 : term] : (($true = ((X485 @ (lam @ X490)))) | ~ ! [X491 : term] : (($true != ((X485 @ X491))) | (((X485 @ (sub @ X490 @ (push @ X491 @ id)))) = $true))) | ~ ! [X489 : term,X488 : term] : ((((X485 @ X488)) != $true) | ($true = ((X485 @ (ap @ X488 @ X489)))) | ($true != ((X485 @ X489)))) | ($true = ((X485 @ X486))))) & (axabs = $true) & (hoasapnotvar_lthm = $true) & (id = ((d2subst @ d_id))) & (hoasinduction_lem3b_gthm = $true) & (! [X525 : term,X526 : (subst > term > subst > $o)] : (~ ! [X528 : (subst > term > term)] : (~ ! [X529 : term,X531 : subst,X530 : subst] : (((sub @ (X528 @ X530 @ X529) @ X531)) = ((X528 @ (comp @ X530 @ X531) @ (sub @ X529 @ X531)))) | ~ ! [X532 : term] : (($true = ((X526 @ id @ (X528 @ id @ X532) @ id))) | ($true != ((X526 @ id @ X532 @ id)))) | ($true = ((X526 @ id @ (lam @ (X528 @ sh @ one)) @ id)))) | ~ ! [X527 : term] : (($true != ((X526 @ id @ X527 @ id))) | ($true = ((X526 @ id @ (sub @ X525 @ (push @ X527 @ id)) @ id)))) | ($true = ((X526 @ id @ (lam @ (sub @ X525 @ (push @ one @ sh))) @ id)))) <=> (hoasinduction_lem3aa_lthm = $true)) & ((hoasinduction_lem0_lthm = $true) <=> ! [X357 : (subst > term > subst > $o)] : ~ ! [X358 : (term > $o)] : ~ ! [X359 : term] : (((X358 @ X359)) = ((X357 @ id @ X359 @ id)))) & (ulamvar1 != $true) & (ulamvarsh = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (pushprop_lem0_gthm = $true) & (! [X510 : term] : (((sub @ X510 @ id)) = X510) <=> (termmset = $true)) & ((hoasapinj2_lthm = $true) <=> (! [X516 : term,X514 : term,X515 : term,X513 : term] : ((((ap @ X515 @ X514)) != ((ap @ X516 @ X513))) | (X513 = X514)) => ! [X520 : term,X519 : term,X518 : term,X517 : term] : ((X517 = X520) | (((ap @ (sub @ X519 @ id) @ X517)) != ((ap @ (sub @ X518 @ id) @ X520)))))) & (! [X168 : term,X166 : term,X167 : term,X169 : term] : ((((ap @ X168 @ X167)) != ((ap @ X166 @ X169))) | (X166 = X168)) <=> (apinj1 = $true)) & (axvarshift = $true) & (! [X342 : (subst > term > subst > $o),X341 : term] : (~ ! [X352 : (subst > term > term)] : (($true = ((X342 @ id @ (lam @ (X352 @ sh @ one)) @ id))) | ~ ! [X356 : term] : (($true = ((X342 @ id @ (X352 @ id @ X356) @ id))) | (((X342 @ id @ X356 @ id)) != $true)) | ~ ! [X355 : subst,X353 : subst,X354 : term] : (((X352 @ (comp @ X355 @ X353) @ (sub @ X354 @ X353))) = ((sub @ (X352 @ X355 @ X354) @ X353)))) | ~ ! [X347 : term] : (($true != ((X342 @ id @ X347 @ id))) | (((X342 @ id @ (sub @ X341 @ (push @ X347 @ id)) @ id)) = $true)) | ~ ! [X348 : term,X350 : subst,X351 : subst,X349 : subst] : ((((X342 @ X351 @ X348 @ (comp @ X350 @ X349))) = $true) | ($true != ((X342 @ (comp @ X351 @ X350) @ (sub @ X348 @ X350) @ X349)))) | ~ ! [X343 : subst,X346 : term,X344 : subst,X345 : subst] : (($true != ((X342 @ X343 @ X346 @ (comp @ X345 @ X344)))) | ($true = ((X342 @ (comp @ X343 @ X345) @ (sub @ X346 @ X345) @ X344)))) | ($true = ((X342 @ id @ (lam @ X341) @ id)))) <=> (hoasinduction_lem3 = $true)) & (pushprop_lem2v2_lthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & (! [X444 : subst] : (((comp @ X444 @ id)) = X444) <=> (axidr = $true)) & (induction2lem_gthm = $true) & (hoasinduction_lem1_lthm = $true) & ! [X234 : d_subst,X235 : d_subst] : ((((d2subst @ X235)) = ((d2subst @ X234))) => (X234 = X235)) & (((var @ (d2term @ d_one))) != $true) & ((hoaslaminj_lthm = $true) <=> (! [X3 : subst,X2 : term] : (((sub @ one @ (push @ X2 @ X3))) = X2) => (! [X4 : term,X5 : subst] : (((comp @ sh @ (push @ X4 @ X5))) = X5) => (! [X6 : term,X7 : term] : ((((lam @ X6)) != ((lam @ X7))) | (X6 = X7)) => ! [X9 : (subst > term > term),X8 : subst,X10 : term,X11 : (subst > term > term)] : (~ ! [X12 : subst,X14 : subst,X13 : term] : (((X9 @ (comp @ X14 @ X12) @ (sub @ X13 @ X12))) = ((sub @ (X9 @ X14 @ X13) @ X12))) | (((X9 @ X8 @ X10)) = ((X11 @ X8 @ X10))) | (((lam @ (X9 @ sh @ one))) != ((lam @ (X11 @ sh @ one)))) | ~ ! [X15 : term,X17 : subst,X16 : subst] : (((sub @ (X11 @ X16 @ X15) @ X17)) = ((X11 @ (comp @ X16 @ X17) @ (sub @ X15 @ X17))))))))) & (! [X292 : (term > $o),X294 : (term > $o),X293 : term,X295 : subst] : (~ ! [X296 : term] : (((X294 @ (sub @ X296 @ (push @ X293 @ X295)))) = ((X292 @ X296))) | ($true != ((X294 @ X293))) | ($true = ((X292 @ one)))) <=> (pushprop_lem1v2 = $true)) & ! [X297 : term] : ? [X298 : d_term] : (((d2term @ X298)) = X297) & ((hoasinduction_no_psi_cond = $true) <=> ! [X221 : (subst > term > subst > $o),X220 : term] : ((((X221 @ id @ X220 @ id)) = $true) | ~ ! [X227 : term,X228 : term] : (($true != ((X221 @ id @ X227 @ id))) | ($true = ((X221 @ id @ (ap @ (sub @ X228 @ id) @ X227) @ id))) | ($true != ((X221 @ id @ X228 @ id)))) | ~ ! [X222 : (subst > term > term)] : (~ ! [X225 : subst,X224 : subst,X223 : term] : (((sub @ (X222 @ X224 @ X223) @ X225)) = ((X222 @ (comp @ X224 @ X225) @ (sub @ X223 @ X225)))) | ~ ! [X226 : term] : (($true != ((X221 @ id @ X226 @ id))) | ($true = ((X221 @ id @ (X222 @ id @ X226) @ id)))) | ($true = ((X221 @ id @ (lam @ (X222 @ sh @ one)) @ id)))))) & (axapp = $true) & (axmap = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (pushprop_lem1v2_gthm = $true) & (! [X32 : term,X33 : term,X34 : term,X35 : term] : ((((ap @ (sub @ X34 @ id) @ X33)) != ((ap @ (sub @ X32 @ id) @ X35))) | (X32 = X34)) <=> (hoasapinj1 = $true)) & ((hoasinduction_lem3b = $true) <=> ! [X368 : term] : ~ ! [X369 : (subst > term > term)] : (((X369 @ sh @ one)) != ((sub @ X368 @ (push @ one @ sh))))) & ((! [X201 : term,X202 : subst] : (((sub @ one @ (push @ X201 @ X202))) = X201) => (! [X204 : subst,X203 : term] : (((comp @ sh @ (push @ X203 @ X204))) = X204) => (! [X205 : subst] : (((comp @ X205 @ id)) = X205) => (! [X207 : term,X206 : (term > $o)] : (~ ! [X209 : term] : (($true = ((X206 @ X209))) | ($true != ((var @ X209)))) | ~ ! [X210 : term,X211 : term] : (($true != ((X206 @ X211))) | ($true != ((X206 @ X210))) | ($true = ((X206 @ (ap @ X210 @ X211))))) | ($true = ((X206 @ X207))) | ~ ! [X208 : term] : (($true != ((X206 @ X208))) | ($true = ((X206 @ (lam @ X208)))))) => ! [X213 : subst,X212 : (term > $o),X214 : term] : (($true = ((X212 @ (sub @ X214 @ X213)))) | ~ ! [X219 : term] : (($true = ((X212 @ (sub @ X219 @ X213)))) | ($true != ((var @ X219)))) | ~ ! [X215 : term] : (($true = ((X212 @ (lam @ X215)))) | ~ ! [X216 : term] : (($true != ((X212 @ X216))) | (((X212 @ (sub @ X215 @ (push @ X216 @ id)))) = $true))) | ~ ! [X217 : term,X218 : term] : (($true = ((X212 @ (ap @ X218 @ X217)))) | ($true != ((X212 @ X217))) | ($true != ((X212 @ X218))))))))) <=> (induction2lem_lthm = $true)) & ((laminj = $true) <=> ! [X290 : term,X291 : term] : ((((lam @ X291)) != ((lam @ X290))) | (X290 = X291))) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((hoasinduction_lem2 = $true) <=> ! [X19 : term,X18 : term,X20 : (subst > term > subst > $o)] : ((((X20 @ id @ X18 @ id)) != $true) | ($true = ((X20 @ id @ (ap @ X19 @ X18) @ id))) | ~ ! [X29 : term,X30 : term] : ((((X20 @ id @ X29 @ id)) != $true) | ($true != ((X20 @ id @ X30 @ id))) | ($true = ((X20 @ id @ (ap @ (sub @ X29 @ id) @ X30) @ id)))) | ($true != ((X20 @ id @ X19 @ id))) | ~ ! [X21 : subst,X24 : subst,X23 : subst,X22 : term] : (($true = ((X20 @ X21 @ X22 @ (comp @ X24 @ X23)))) | ($true != ((X20 @ (comp @ X21 @ X24) @ (sub @ X22 @ X24) @ X23)))) | ~ ! [X26 : subst,X28 : subst,X25 : term,X27 : subst] : (($true = ((X20 @ (comp @ X28 @ X26) @ (sub @ X25 @ X26) @ X27))) | ($true != ((X20 @ X28 @ X25 @ (comp @ X26 @ X27))))))) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (! [X133 : term,X132 : (subst > term > subst > $o)] : (~ ! [X134 : (subst > term > term)] : (($true = ((X132 @ id @ (lam @ (X134 @ sh @ one)) @ id))) | ~ ! [X138 : subst,X137 : subst,X136 : term] : (((sub @ (X134 @ X138 @ X136) @ X137)) = ((X134 @ (comp @ X138 @ X137) @ (sub @ X136 @ X137)))) | ~ ! [X135 : term] : (($true = ((X132 @ id @ (X134 @ id @ X135) @ id))) | ($true != ((X132 @ id @ X135 @ id))))) | ($true = ((X132 @ id @ (lam @ (sub @ X133 @ (push @ one @ sh))) @ id))) | ~ ! [X139 : term] : (($true = ((X132 @ id @ (sub @ X133 @ (push @ X139 @ id)) @ id))) | ($true != ((X132 @ id @ X139 @ id))))) <=> (hoasinduction_lem3aa = $true)) & (hoasinduction_lem2_lthm = $true) & ((! [X415 : term] : (((sub @ X415 @ id)) = X415) => ! [X417 : term,X416 : (subst > term > subst > $o),X418 : (term > $o)] : (~ ! [X425 : subst,X424 : subst,X427 : term,X426 : subst] : (($true = ((X416 @ X424 @ X427 @ (comp @ X426 @ X425)))) | ($true != ((X416 @ (comp @ X424 @ X426) @ (sub @ X427 @ X426) @ X425)))) | ~ ! [X423 : term] : (($true = ((X418 @ (sub @ X417 @ (push @ X423 @ id))))) | ($true != ((X418 @ X423)))) | ~ ! [X420 : term,X419 : subst,X422 : subst,X421 : subst] : (($true != ((X416 @ X422 @ X420 @ (comp @ X419 @ X421)))) | ($true = ((X416 @ (comp @ X422 @ X419) @ (sub @ X420 @ X419) @ X421)))) | ($true = ((X418 @ (lam @ X417)))) | ~ ! [X433 : term] : (((X418 @ X433)) = ((X416 @ id @ X433 @ id))) | ~ ! [X428 : (subst > term > term)] : (~ ! [X430 : subst,X429 : term,X431 : subst] : (((sub @ (X428 @ X431 @ X429) @ X430)) = ((X428 @ (comp @ X431 @ X430) @ (sub @ X429 @ X430)))) | ($true = ((X416 @ id @ (lam @ (X428 @ sh @ one)) @ id))) | ~ ! [X432 : term] : (($true != ((X416 @ id @ X432 @ id))) | ($true = ((X416 @ id @ (X428 @ id @ X432) @ id))))))) <=> (hoasinduction_lem3v2_lthm = $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (hoasinduction_lem1 = $true) & ((! [X54 : (subst > term > subst > $o)] : ~ ! [X55 : (term > $o)] : ~ ! [X56 : term] : (((X54 @ id @ X56 @ id)) = ((X55 @ X56))) => (! [X58 : term,X57 : (term > $o)] : (($true = ((X57 @ X58))) | ~ ! [X62 : term] : (($true = ((X57 @ (lam @ X62)))) | ~ ! [X63 : term] : (($true != ((X57 @ X63))) | (((X57 @ (sub @ X62 @ (push @ X63 @ id)))) = $true))) | ~ ! [X60 : term,X59 : term] : (($true != ((X57 @ X59))) | ($true != ((X57 @ X60))) | (((X57 @ (ap @ X59 @ X60))) = $true)) | ~ ! [X61 : term] : (($true != ((var @ X61))) | (((X57 @ X61)) = $true))) => (! [X64 : term] : (((sub @ X64 @ id)) = X64) => (! [X65 : term,X67 : (term > $o),X66 : (subst > term > subst > $o)] : (~ ! [X68 : term] : (($true != ((X67 @ X68))) | ($true = ((X67 @ (sub @ X65 @ (push @ X68 @ id)))))) | ~ ! [X69 : (subst > term > term)] : (~ ! [X73 : term] : (($true = ((X66 @ id @ (X69 @ id @ X73) @ id))) | ($true != ((X66 @ id @ X73 @ id)))) | ($true = ((X66 @ id @ (lam @ (X69 @ sh @ one)) @ id))) | ~ ! [X72 : term,X71 : subst,X70 : subst] : (((sub @ (X69 @ X70 @ X72) @ X71)) = ((X69 @ (comp @ X70 @ X71) @ (sub @ X72 @ X71))))) | ~ ! [X74 : term] : (((X66 @ id @ X74 @ id)) = ((X67 @ X74))) | ($true = ((X67 @ (lam @ X65))))) => ! [X75 : term,X76 : (subst > term > subst > $o)] : (~ ! [X77 : (subst > term > term)] : (~ ! [X81 : term] : (($true != ((X76 @ id @ X81 @ id))) | ($true = ((X76 @ id @ (X77 @ id @ X81) @ id)))) | ~ ! [X79 : subst,X80 : term,X78 : subst] : (((sub @ (X77 @ X78 @ X80) @ X79)) = ((X77 @ (comp @ X78 @ X79) @ (sub @ X80 @ X79)))) | ($true = ((X76 @ id @ (lam @ (X77 @ sh @ one)) @ id)))) | ($true = ((X76 @ id @ X75 @ id))) | ~ ! [X83 : term,X82 : term] : (($true = ((X76 @ id @ (ap @ (sub @ X82 @ id) @ X83) @ id))) | ($true != ((X76 @ id @ X82 @ id))) | ($true != ((X76 @ id @ X83 @ id))))))))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (pushprop_lem3v2_lthm = $true) & ((hoasinduction_lthm_3 = $true) <=> (! [X93 : (subst > term > subst > $o)] : ~ ! [X94 : (term > $o)] : ~ ! [X95 : term] : (((X93 @ id @ X95 @ id)) = ((X94 @ X95))) => (! [X96 : (term > $o),X97 : term] : (~ ! [X99 : term] : (~ ! [X100 : term] : (($true = ((X96 @ (sub @ X99 @ (push @ X100 @ id))))) | ($true != ((X96 @ X100)))) | ($true = ((X96 @ (lam @ X99))))) | ~ ! [X102 : term,X101 : term] : ((((X96 @ X102)) != $true) | (((X96 @ (ap @ X101 @ X102))) = $true) | (((X96 @ X101)) != $true)) | ~ ! [X98 : term] : (($true = ((X96 @ X98))) | ($true != ((var @ X98)))) | ($true = ((X96 @ X97)))) => (! [X103 : term] : (((sub @ X103 @ id)) = X103) => (! [X106 : (term > $o),X104 : term,X105 : (subst > term > subst > $o)] : (~ ! [X113 : term] : (($true = ((X106 @ (sub @ X104 @ (push @ X113 @ id))))) | (((X106 @ X113)) != $true)) | ~ ! [X108 : (subst > term > term)] : (~ ! [X111 : term,X110 : subst,X112 : subst] : (((X108 @ (comp @ X110 @ X112) @ (sub @ X111 @ X112))) = ((sub @ (X108 @ X110 @ X111) @ X112))) | ($true = ((X105 @ id @ (lam @ (X108 @ sh @ one)) @ id))) | ~ ! [X109 : term] : (($true != ((X105 @ id @ X109 @ id))) | ($true = ((X105 @ id @ (X108 @ id @ X109) @ id))))) | ($true = ((X106 @ (lam @ X104)))) | ~ ! [X107 : term] : (((X105 @ id @ X107 @ id)) = ((X106 @ X107)))) => ! [X115 : term,X114 : (subst > term > subst > $o)] : (~ ! [X131 : term] : (($true != ((var @ (sub @ X131 @ id)))) | ($true = ((X114 @ id @ X131 @ id)))) | ~ ! [X119 : subst,X117 : subst,X118 : subst,X116 : term] : (($true = ((X114 @ (comp @ X118 @ X119) @ (sub @ X116 @ X119) @ X117))) | ($true != ((X114 @ X118 @ X116 @ (comp @ X119 @ X117))))) | ($true = ((X114 @ id @ X115 @ id))) | ~ ! [X120 : term,X121 : term] : (($true != ((X114 @ id @ X120 @ id))) | ($true != ((X114 @ id @ X121 @ id))) | (((X114 @ id @ (ap @ (sub @ X120 @ id) @ X121) @ id)) = $true)) | ~ ! [X125 : subst,X124 : subst,X122 : term,X123 : subst] : (($true = ((X114 @ X124 @ X122 @ (comp @ X123 @ X125)))) | ($true != ((X114 @ (comp @ X124 @ X123) @ (sub @ X122 @ X123) @ X125)))) | ~ ! [X126 : (subst > term > term)] : (($true = ((X114 @ id @ (lam @ (X126 @ sh @ one)) @ id))) | ~ ! [X130 : term] : ((((X114 @ id @ (X126 @ id @ X130) @ id)) = $true) | (((X114 @ id @ X130 @ id)) != $true)) | ~ ! [X127 : term,X129 : subst,X128 : subst] : (((X126 @ (comp @ X129 @ X128) @ (sub @ X127 @ X128))) = ((sub @ (X126 @ X129 @ X127) @ X128)))))))))) & (hoasinduction_lem1v2 = $true) & (hoasinduction_lem3a_gthm = $true) & (termmset_gthm = $true) & (pushprop = $true) & (! [X0 : term,X1 : term] : ((X0 = X1) | (((sub @ X0 @ sh)) != ((sub @ X1 @ sh)))) <=> (shinj = $true)) & (pushprop_lem2v2_gthm = $true) & (! [X435 : term,X434 : (subst > term > subst > $o),X436 : (term > $o)] : (~ ! [X437 : term] : (((X436 @ X437)) = ((X434 @ id @ X437 @ id))) | ~ ! [X438 : (subst > term > term)] : (~ ! [X442 : term] : ((((X434 @ id @ (X438 @ id @ X442) @ id)) = $true) | ($true != ((X434 @ id @ X442 @ id)))) | ($true = ((X434 @ id @ (lam @ (X438 @ sh @ one)) @ id))) | ~ ! [X440 : subst,X441 : term,X439 : subst] : (((sub @ (X438 @ X439 @ X441) @ X440)) = ((X438 @ (comp @ X439 @ X440) @ (sub @ X441 @ X440))))) | ($true = ((X436 @ (lam @ X435)))) | ~ ! [X443 : term] : (($true = ((X436 @ (sub @ X435 @ (push @ X443 @ id))))) | ($true != ((X436 @ X443))))) <=> (hoasinduction_lem3v2a = $true)) & ((hoasinduction_lem3a = $true) <=> ! [X386 : (subst > term > subst > $o),X387 : term] : (~ ! [X388 : (subst > term > term)] : (($true = ((X386 @ id @ (lam @ (X388 @ sh @ one)) @ id))) | ~ ! [X389 : subst,X390 : subst,X391 : term] : (((X388 @ (comp @ X390 @ X389) @ (sub @ X391 @ X389))) = ((sub @ (X388 @ X390 @ X391) @ X389))) | ~ ! [X392 : term] : (($true = ((X386 @ id @ (X388 @ id @ X392) @ id))) | ($true != ((X386 @ id @ X392 @ id))))) | ~ ! [X393 : term] : (($true != ((X386 @ id @ X393 @ id))) | ($true = ((X386 @ id @ (sub @ X387 @ (push @ X393 @ id)) @ id)))) | (((X386 @ id @ (lam @ X387) @ id)) = $true))) & (! [X329 : (term > $o),X330 : (term > $o),X327 : term,X326 : term,X328 : subst] : (($true != ((X330 @ X327))) | ($true = ((X329 @ (sub @ X327 @ (push @ X326 @ X328)))))) <=> (pushprop_lem1 = $true)) & (! [X340 : subst] : (((comp @ id @ X340)) = X340) <=> (axidl = $true)) & ! [X36 : d_term] : (d_one = X36) & (hoaslamnotap_gthm = $true) & (pushprop_lem2v2 = $true) & ((! [X331 : subst] : (((comp @ id @ X331)) = X331) => (! [X332 : subst] : (((comp @ X332 @ id)) = X332) => (! [X333 : subst] : (((comp @ id @ X333)) = X333) & ! [X334 : subst] : (((comp @ X334 @ id)) = X334)))) <=> (substmonoid_lthm = $true)) & (pushprop_lem3v2 = $true) & (! [X140 : (subst > term > subst > $o)] : ~ ! [X141 : (term > $o)] : ~ ! [X142 : term] : (((X141 @ X142)) = ((X140 @ id @ X142 @ id))) <=> (hoasinduction_lem0 = $true)) & ((hoasapinj1_lthm = $true) <=> (! [X451 : term] : (((sub @ X451 @ id)) = X451) => (! [X453 : term,X455 : term,X454 : term,X452 : term] : ((X452 = X455) | (((ap @ X452 @ X453)) != ((ap @ X455 @ X454)))) => ! [X458 : term,X457 : term,X456 : term,X459 : term] : ((((ap @ (sub @ X458 @ id) @ X456)) != ((ap @ (sub @ X459 @ id) @ X457))) | (X458 = X459))))) & ! [X143 : d_term,X144 : d_term] : ((((d2term @ X144)) = ((d2term @ X143))) => (X143 = X144)) & (! [X523 : term,X522 : term,X521 : term,X524 : term] : ((((ap @ (sub @ X523 @ id) @ X521)) != ((ap @ (sub @ X522 @ id) @ X524))) | (X521 = X524)) <=> (hoasapinj2 = $true)) & ((! [X533 : term,X534 : subst] : (((sub @ one @ (push @ X533 @ X534))) = X533) => ! [X536 : (term > $o),X538 : (term > $o),X537 : subst,X535 : term] : (~ ! [X539 : term] : (((X536 @ (sub @ X539 @ (push @ X535 @ X537)))) = ((X538 @ X539))) | ($true != ((X536 @ X535))) | ($true = ((X538 @ one))))) <=> (pushprop_lem1v2_lthm = $true)) & (pushprop_lem1_gthm = $true) & (hoasapinj1_gthm = $true) & (hoaslamnotvar = $true) & (lamnotvar = $true) & (lamnotap != $true) & (pushprop_lthm_orig = $true) & ((hoasinduction_lem3_lthm = $true) <=> (! [X299 : term] : (((sub @ X299 @ id)) = X299) => (! [X301 : term,X300 : (subst > term > subst > $o)] : (($true = ((X300 @ id @ (lam @ (sub @ X301 @ (push @ one @ sh))) @ id))) | ~ ! [X302 : (subst > term > term)] : (~ ! [X303 : term,X304 : subst,X305 : subst] : (((sub @ (X302 @ X304 @ X303) @ X305)) = ((X302 @ (comp @ X304 @ X305) @ (sub @ X303 @ X305)))) | ~ ! [X306 : term] : (($true != ((X300 @ id @ X306 @ id))) | ($true = ((X300 @ id @ (X302 @ id @ X306) @ id)))) | ($true = ((X300 @ id @ (lam @ (X302 @ sh @ one)) @ id)))) | ~ ! [X307 : term] : (($true = ((X300 @ id @ (sub @ X301 @ (push @ X307 @ id)) @ id))) | ($true != ((X300 @ id @ X307 @ id))))) => ! [X308 : (subst > term > subst > $o),X309 : term] : (~ ! [X323 : term] : (($true != ((X308 @ id @ X323 @ id))) | ($true = ((X308 @ id @ (sub @ X309 @ (push @ X323 @ id)) @ id)))) | ~ ! [X311 : subst,X312 : term,X310 : subst,X313 : subst] : (($true = ((X308 @ X310 @ X312 @ (comp @ X313 @ X311)))) | ($true != ((X308 @ (comp @ X310 @ X313) @ (sub @ X312 @ X313) @ X311)))) | ~ ! [X318 : (subst > term > term)] : (~ ! [X320 : subst,X321 : term,X322 : subst] : (((X318 @ (comp @ X320 @ X322) @ (sub @ X321 @ X322))) = ((sub @ (X318 @ X320 @ X321) @ X322))) | ~ ! [X319 : term] : (($true != ((X308 @ id @ X319 @ id))) | ($true = ((X308 @ id @ (X318 @ id @ X319) @ id)))) | ($true = ((X308 @ id @ (lam @ (X318 @ sh @ one)) @ id)))) | ($true = ((X308 @ id @ (lam @ X309) @ id))) | ~ ! [X316 : subst,X315 : subst,X314 : term,X317 : subst] : ((((X308 @ X315 @ X314 @ (comp @ X317 @ X316))) != $true) | (((X308 @ (comp @ X315 @ X317) @ (sub @ X314 @ X317) @ X316)) = $true)))))) & ((pushprop_lem0_lthm = $true) <=> ! [X229 : term,X231 : (term > $o),X230 : subst] : ~ ! [X232 : (term > $o)] : ~ ! [X233 : term] : (((X232 @ X233)) = ((X231 @ (sub @ X233 @ (push @ X229 @ X230)))))) & (! [X363 : term,X362 : subst] : (((comp @ sh @ (push @ X363 @ X362))) = X362) <=> (axshiftcons = $true)) & ((hoasinduction_lem3a_lthm = $true) <=> (! [X37 : term] : (((sub @ X37 @ id)) = X37) => (! [X39 : term,X38 : (subst > term > subst > $o)] : (~ ! [X40 : (subst > term > term)] : (($true = ((X38 @ id @ (lam @ (X40 @ sh @ one)) @ id))) | ~ ! [X44 : term] : (($true != ((X38 @ id @ X44 @ id))) | (((X38 @ id @ (X40 @ id @ X44) @ id)) = $true)) | ~ ! [X42 : subst,X43 : subst,X41 : term] : (((X40 @ (comp @ X42 @ X43) @ (sub @ X41 @ X43))) = ((sub @ (X40 @ X42 @ X41) @ X43)))) | ($true = ((X38 @ id @ (lam @ (sub @ X39 @ (push @ one @ sh))) @ id))) | ~ ! [X45 : term] : (($true != ((X38 @ id @ X45 @ id))) | ($true = ((X38 @ id @ (sub @ X39 @ (push @ X45 @ id)) @ id))))) => ! [X47 : (subst > term > subst > $o),X46 : term] : (~ ! [X48 : (subst > term > term)] : (~ ! [X50 : subst,X51 : term,X52 : subst] : (((sub @ (X48 @ X50 @ X51) @ X52)) = ((X48 @ (comp @ X50 @ X52) @ (sub @ X51 @ X52)))) | ($true = ((X47 @ id @ (lam @ (X48 @ sh @ one)) @ id))) | ~ ! [X49 : term] : (($true != ((X47 @ id @ X49 @ id))) | (((X47 @ id @ (X48 @ id @ X49) @ id)) = $true))) | ~ ! [X53 : term] : (($true != ((X47 @ id @ X53 @ id))) | ($true = ((X47 @ id @ (sub @ X46 @ (push @ X53 @ id)) @ id)))) | (((X47 @ id @ (lam @ X46) @ id)) = $true))))) & (! [X92 : term] : (((sub @ X92 @ id)) = X92) <=> (axvarid = $true)) & ((hoaslaminj = $true) <=> ! [X461 : subst,X460 : (subst > term > term),X462 : (subst > term > term),X463 : term] : (~ ! [X464 : subst,X466 : term,X465 : subst] : (((sub @ (X460 @ X465 @ X466) @ X464)) = ((X460 @ (comp @ X465 @ X464) @ (sub @ X466 @ X464)))) | (((lam @ (X462 @ sh @ one))) != ((lam @ (X460 @ sh @ one)))) | (((X462 @ X461 @ X463)) = ((X460 @ X461 @ X463))) | ~ ! [X467 : subst,X469 : term,X468 : subst] : (((sub @ (X462 @ X467 @ X469) @ X468)) = ((X462 @ (comp @ X467 @ X468) @ (sub @ X469 @ X468)))))) & (hoasapnotvar = $true) & (hoasinduction_lem2v2_gthm = $true) & ((axscons = $true) <=> ! [X394 : subst] : (((push @ (sub @ one @ X394) @ (comp @ sh @ X394))) = X394)) & ! [X31 : d_subst] : (d_id = X31) & (((d2term @ d_one)) = one) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((pushprop_lem1_lthm = $true) <=> (! [X192 : subst,X193 : term] : (((sub @ one @ (push @ X193 @ X192))) = X193) => (! [X195 : term,X194 : subst] : (((comp @ sh @ (push @ X195 @ X194))) = X194) => ! [X198 : term,X197 : subst,X199 : term,X200 : (term > $o),X196 : (term > $o)] : (($true != ((X196 @ X199))) | (((X200 @ (sub @ X199 @ (push @ X198 @ X197)))) = $true))))) & (ulamvarind = $true) & (hoasinduction_lem3v2_gthm = $true)), inference(flattening,[],[f9])). thf(f11,plain,( (((! [X180 : (term > $o),X179 : term] : ((((X180 @ X179)) = $true) | ? [X181 : term,X182 : term] : (($true != ((X180 @ (ap @ X182 @ X181)))) & ($true = ((X180 @ X181))) & ($true = ((X180 @ X182)))) | ? [X185 : term] : (($true = ((var @ X185))) & ($true != ((X180 @ X185)))) | ? [X183 : term] : (($true != ((X180 @ (lam @ X183)))) & ! [X184 : term] : (($true != ((X180 @ X184))) | (((X180 @ (sub @ X183 @ (push @ X184 @ id)))) = $true)))) | ? [X171 : (term > $o),X172 : term,X173 : subst] : (! [X174 : term] : (($true != ((var @ X174))) | ($true = ((X171 @ (sub @ X174 @ X173))))) & ($true != ((X171 @ (sub @ X172 @ X173)))) & ! [X178 : term,X177 : term] : (($true = ((X171 @ (ap @ X178 @ X177)))) | ($true != ((X171 @ X177))) | ($true != ((X171 @ X178)))) & ! [X175 : term] : (? [X176 : term] : (($true = ((X171 @ X176))) & ($true != ((X171 @ (sub @ X175 @ (push @ X176 @ id)))))) | ($true = ((X171 @ (lam @ X175))))))) | ? [X170 : term] : (((sub @ X170 @ id)) != X170)) <=> (induction2_lthm = $true)) & ((apinj2 = $true) <=> ! [X364 : term,X365 : term,X366 : term,X367 : term] : ((X364 = X365) | (((ap @ X366 @ X364)) != ((ap @ X367 @ X365))))) & (hoasapnotvar_gthm = $true) & (((d2subst @ d_id)) = sh) & (! [X186 : term,X187 : (term > $o)] : (? [X189 : term] : (($true != ((X187 @ X189))) & ($true = ((var @ X189)))) | ($true = ((X187 @ X186))) | ? [X188 : term] : (($true = ((X187 @ X188))) & ($true != ((X187 @ (lam @ X188))))) | ? [X190 : term,X191 : term] : (($true = ((X187 @ X191))) & ($true = ((X187 @ X190))) & ($true != ((X187 @ (ap @ X191 @ X190)))))) <=> (induction = $true)) & (hoaslaminj_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (! [X370 : term] : ? [X371 : (subst > term > term)] : ! [X372 : term,X373 : subst] : (((X371 @ X373 @ X372)) = ((sub @ X370 @ (push @ X372 @ X373)))) <=> (hoasinduction_lem3v2_f = $true)) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X160 : term] : ? [X161 : (subst > term > term)] : ! [X163 : term,X162 : subst] : (((sub @ X160 @ (push @ X163 @ X162))) = ((X161 @ X162 @ X163)))) & ((hoaslamnotap = $true) <=> ! [X446 : term,X447 : (subst > term > term),X445 : term] : (? [X448 : subst,X450 : term,X449 : subst] : (((X447 @ (comp @ X448 @ X449) @ (sub @ X450 @ X449))) != ((sub @ (X447 @ X448 @ X450) @ X449))) | (((ap @ (sub @ X445 @ id) @ X446)) != ((lam @ (X447 @ sh @ one)))))) & ((! [X396 : term] : (((sub @ X396 @ id)) = X396) | ? [X395 : term] : (((sub @ X395 @ id)) != X395)) <=> (termmset_lthm = $true)) & (! [X86 : subst,X85 : term,X84 : (term > $o)] : ((((X84 @ (sub @ X85 @ X86))) = $true) | ? [X90 : term] : (($true != ((X84 @ (lam @ X90)))) & ! [X91 : term] : (($true != ((X84 @ X91))) | ($true = ((X84 @ (sub @ X90 @ (push @ X91 @ id))))))) | ? [X89 : term] : (($true != ((X84 @ (sub @ X89 @ X86)))) & ($true = ((var @ X89)))) | ? [X87 : term,X88 : term] : (($true = ((X84 @ X87))) & ($true != ((X84 @ (ap @ X88 @ X87)))) & ($true = ((X84 @ X88))))) <=> (induction2lem = $true)) & (! [X164 : term,X165 : subst] : (((sub @ one @ (push @ X164 @ X165))) = X164) <=> (axvarcons = $true)) & (pushprop_gthm = $true) & (hoaslamnotvar_lthm = $true) & (induction2_gthm = $true) & (! [X470 : term,X471 : (term > $o),X472 : (subst > term > subst > $o),X473 : term] : (($true != ((X471 @ X473))) | ? [X484 : subst,X481 : subst,X483 : term,X482 : subst] : (($true != ((X472 @ (comp @ X481 @ X484) @ (sub @ X483 @ X484) @ X482))) & ($true = ((X472 @ X481 @ X483 @ (comp @ X484 @ X482))))) | ? [X474 : term] : (((X472 @ id @ X474 @ id)) != ((X471 @ X474))) | ? [X476 : term,X475 : term] : (($true = ((X472 @ id @ X475 @ id))) & ($true = ((X472 @ id @ X476 @ id))) & ($true != ((X472 @ id @ (ap @ (sub @ X476 @ id) @ X475) @ id)))) | ? [X479 : term,X480 : subst,X477 : subst,X478 : subst] : (($true = ((X472 @ (comp @ X478 @ X480) @ (sub @ X479 @ X480) @ X477))) & ($true != ((X472 @ X478 @ X479 @ (comp @ X480 @ X477))))) | ($true != ((X471 @ X470))) | ($true = ((X471 @ (ap @ X470 @ X473))))) <=> (hoasinduction_lem2v2 = $true)) & ((hoasinduction = $true) <=> ! [X397 : term,X398 : (subst > term > subst > $o)] : (? [X414 : subst,X411 : subst,X412 : term,X413 : subst] : (($true = ((X398 @ X411 @ X412 @ (comp @ X414 @ X413)))) & ($true != ((X398 @ (comp @ X411 @ X414) @ (sub @ X412 @ X414) @ X413)))) | ($true = ((X398 @ id @ X397 @ id))) | ? [X410 : term] : (($true = ((var @ (sub @ X410 @ id)))) & ($true != ((X398 @ id @ X410 @ id)))) | ? [X401 : subst,X399 : subst,X400 : subst,X402 : term] : (($true = ((X398 @ (comp @ X401 @ X399) @ (sub @ X402 @ X399) @ X400))) & ($true != ((X398 @ X401 @ X402 @ (comp @ X399 @ X400))))) | ? [X409 : term,X408 : term] : ((((X398 @ id @ X409 @ id)) = $true) & ($true != ((X398 @ id @ (ap @ (sub @ X408 @ id) @ X409) @ id))) & ($true = ((X398 @ id @ X408 @ id)))) | ? [X403 : (subst > term > term)] : ((((X398 @ id @ (lam @ (X403 @ sh @ one)) @ id)) != $true) & ! [X405 : subst,X406 : subst,X407 : term] : (((sub @ (X403 @ X406 @ X407) @ X405)) = ((X403 @ (comp @ X406 @ X405) @ (sub @ X407 @ X405)))) & ! [X404 : term] : (($true = ((X398 @ id @ (X403 @ id @ X404) @ id))) | (((X398 @ id @ X404 @ id)) != $true))))) & ((hoasinduction_lem3v2 = $true) <=> ! [X492 : term,X493 : (subst > term > subst > $o),X494 : (term > $o)] : (($true = ((X494 @ (lam @ X492)))) | ? [X496 : (subst > term > term)] : (($true != ((X493 @ id @ (lam @ (X496 @ sh @ one)) @ id))) & ! [X498 : subst,X500 : subst,X499 : term] : (((sub @ (X496 @ X498 @ X499) @ X500)) = ((X496 @ (comp @ X498 @ X500) @ (sub @ X499 @ X500)))) & ! [X497 : term] : (($true = ((X493 @ id @ (X496 @ id @ X497) @ id))) | ($true != ((X493 @ id @ X497 @ id))))) | ? [X505 : term] : (($true = ((X494 @ X505))) & ($true != ((X494 @ (sub @ X492 @ (push @ X505 @ id)))))) | ? [X502 : subst,X504 : subst,X501 : term,X503 : subst] : (($true != ((X493 @ (comp @ X504 @ X503) @ (sub @ X501 @ X503) @ X502))) & ($true = ((X493 @ X504 @ X501 @ (comp @ X503 @ X502))))) | ? [X506 : subst,X508 : subst,X509 : subst,X507 : term] : (($true = ((X493 @ (comp @ X509 @ X506) @ (sub @ X507 @ X506) @ X508))) & ($true != ((X493 @ X509 @ X507 @ (comp @ X506 @ X508))))) | ? [X495 : term] : (((X494 @ X495)) != ((X493 @ id @ X495 @ id))))) & (((! [X152 : term,X150 : (term > $o),X151 : (subst > term > subst > $o)] : (? [X153 : (subst > term > term)] : (($true != ((X151 @ id @ (lam @ (X153 @ sh @ one)) @ id))) & ! [X157 : term] : (($true = ((X151 @ id @ (X153 @ id @ X157) @ id))) | (((X151 @ id @ X157 @ id)) != $true)) & ! [X156 : subst,X154 : term,X155 : subst] : (((sub @ (X153 @ X156 @ X154) @ X155)) = ((X153 @ (comp @ X156 @ X155) @ (sub @ X154 @ X155))))) | ? [X159 : term] : (($true = ((X150 @ X159))) & ($true != ((X150 @ (sub @ X152 @ (push @ X159 @ id)))))) | ($true = ((X150 @ (lam @ X152)))) | ? [X158 : term] : (((X151 @ id @ X158 @ id)) != ((X150 @ X158)))) | ? [X149 : term] : (((sub @ X149 @ id)) != X149)) | ? [X145 : term] : ! [X146 : (subst > term > term)] : ? [X147 : term,X148 : subst] : (((sub @ X145 @ (push @ X147 @ X148))) != ((X146 @ X148 @ X147)))) <=> (hoasinduction_lem3v2a_lthm = $true)) & ((((! [X272 : term,X273 : (subst > term > subst > $o)] : (? [X286 : term,X287 : subst,X284 : subst,X285 : subst] : (($true != ((X273 @ (comp @ X284 @ X285) @ (sub @ X286 @ X285) @ X287))) & ($true = ((X273 @ X284 @ X286 @ (comp @ X285 @ X287))))) | ($true = ((X273 @ id @ X272 @ id))) | ? [X274 : (subst > term > term)] : (($true != ((X273 @ id @ (lam @ (X274 @ sh @ one)) @ id))) & ! [X276 : subst,X277 : term,X278 : subst] : (((X274 @ (comp @ X276 @ X278) @ (sub @ X277 @ X278))) = ((sub @ (X274 @ X276 @ X277) @ X278))) & ! [X275 : term] : (($true = ((X273 @ id @ (X274 @ id @ X275) @ id))) | ($true != ((X273 @ id @ X275 @ id))))) | ? [X283 : term] : (($true = ((var @ (sub @ X283 @ id)))) & ($true != ((X273 @ id @ X283 @ id)))) | ? [X288 : term,X289 : term] : (($true = ((X273 @ id @ X288 @ id))) & ($true != ((X273 @ id @ (ap @ (sub @ X289 @ id) @ X288) @ id))) & (((X273 @ id @ X289 @ id)) = $true)) | ? [X281 : subst,X280 : subst,X282 : subst,X279 : term] : (($true = ((X273 @ (comp @ X282 @ X280) @ (sub @ X279 @ X280) @ X281))) & ($true != ((X273 @ X282 @ X279 @ (comp @ X280 @ X281)))))) | ? [X256 : (subst > term > subst > $o),X257 : term] : (! [X258 : (subst > term > term)] : (? [X262 : term] : (($true != ((X256 @ id @ (X258 @ id @ X262) @ id))) & ($true = ((X256 @ id @ X262 @ id)))) | ? [X260 : term,X261 : subst,X259 : subst] : (((X258 @ (comp @ X259 @ X261) @ (sub @ X260 @ X261))) != ((sub @ (X258 @ X259 @ X260) @ X261))) | ($true = ((X256 @ id @ (lam @ (X258 @ sh @ one)) @ id)))) & ! [X271 : term] : (($true = ((X256 @ id @ (sub @ X257 @ (push @ X271 @ id)) @ id))) | ($true != ((X256 @ id @ X271 @ id)))) & ! [X267 : subst,X268 : term,X269 : subst,X270 : subst] : (($true = ((X256 @ (comp @ X267 @ X269) @ (sub @ X268 @ X269) @ X270))) | ($true != ((X256 @ X267 @ X268 @ (comp @ X269 @ X270))))) & ! [X264 : subst,X266 : subst,X263 : subst,X265 : term] : (($true = ((X256 @ X263 @ X265 @ (comp @ X264 @ X266)))) | ($true != ((X256 @ (comp @ X263 @ X264) @ (sub @ X265 @ X264) @ X266)))) & ($true != ((X256 @ id @ (lam @ X257) @ id))))) | ? [X244 : term,X243 : (subst > term > subst > $o),X245 : term] : (($true = ((X243 @ id @ X244 @ id))) & ! [X247 : term,X246 : term] : (($true != ((X243 @ id @ X247 @ id))) | ($true = ((X243 @ id @ (ap @ (sub @ X246 @ id) @ X247) @ id))) | ($true != ((X243 @ id @ X246 @ id)))) & ($true = ((X243 @ id @ X245 @ id))) & ! [X251 : subst,X250 : subst,X249 : subst,X248 : term] : (($true = ((X243 @ X249 @ X248 @ (comp @ X251 @ X250)))) | ($true != ((X243 @ (comp @ X249 @ X251) @ (sub @ X248 @ X251) @ X250)))) & ($true != ((X243 @ id @ (ap @ X244 @ X245) @ id))) & ! [X254 : subst,X252 : subst,X253 : subst,X255 : term] : (($true = ((X243 @ (comp @ X254 @ X252) @ (sub @ X255 @ X252) @ X253))) | ($true != ((X243 @ X254 @ X255 @ (comp @ X252 @ X253))))))) | ? [X236 : term,X237 : (term > $o)] : (! [X240 : term,X239 : term] : (($true != ((X237 @ X240))) | ($true = ((X237 @ (ap @ X240 @ X239)))) | ($true != ((X237 @ X239)))) & ($true != ((X237 @ X236))) & ! [X241 : term] : (($true = ((X237 @ (lam @ X241)))) | ? [X242 : term] : (($true != ((X237 @ (sub @ X241 @ (push @ X242 @ id))))) & ($true = ((X237 @ X242))))) & ! [X238 : term] : (($true = ((X237 @ X238))) | ($true != ((var @ X238)))))) <=> (hoasinduction_lthm = $true)) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (substmonoid_gthm = $true) & (hoaslamnotap_lthm = $true) & ! [X360 : subst] : ? [X361 : d_subst] : (((d2subst @ X361)) = X360) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X324 : term] : ? [X325 : (subst > term > term)] : (((sub @ X324 @ (push @ one @ sh))) = ((X325 @ sh @ one)))) & (hoasinduction_lem3_gthm = $true) & (hoasapinj2_gthm = $true) & (pushprop_lthm = $true) & (hoasinduction_lem1v2_gthm = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoasinduction_gthm = $true) & ((pushprop_lem0 = $true) <=> ! [X335 : (term > $o),X337 : subst,X336 : term] : ? [X338 : (term > $o)] : ! [X339 : term] : (((X335 @ (sub @ X339 @ (push @ X336 @ X337)))) = ((X338 @ X339)))) & ((substmonoid = $true) <=> (! [X511 : subst] : (((comp @ X511 @ id)) = X511) & ! [X512 : subst] : (((comp @ id @ X512)) = X512))) & (hoasinduction_lem2_gthm = $true) & (apnotvar = $true) & ((hoasinduction_lem3aaa = $true) <=> ! [X375 : term,X374 : (subst > term > subst > $o)] : (? [X377 : term,X378 : (subst > term > term)] : (! [X382 : term,X384 : subst,X383 : subst] : (((sub @ (X378 @ X384 @ X382) @ X383)) = ((sub @ (sub @ X377 @ (push @ X382 @ X384)) @ X383))) & ! [X379 : term,X380 : subst,X381 : subst] : (((X378 @ (comp @ X381 @ X380) @ (sub @ X379 @ X380))) = ((sub @ X377 @ (push @ (sub @ X379 @ X380) @ (comp @ X381 @ X380))))) & ! [X385 : term] : (($true = ((X374 @ id @ (X378 @ id @ X385) @ id))) | ($true != ((X374 @ id @ X385 @ id)))) & ($true != ((X374 @ id @ (lam @ (X378 @ sh @ one)) @ id)))) | (((X374 @ id @ (lam @ (sub @ X375 @ (push @ one @ sh))) @ id)) = $true) | ? [X376 : term] : (($true = ((X374 @ id @ X376 @ id))) & ($true != ((X374 @ id @ (sub @ X375 @ (push @ X376 @ id)) @ id)))))) & (axclos = $true) & (hoasinduction_lem1_gthm = $true) & (hoaslamnotvar_gthm = $true) & (axassoc = $true) & (! [X486 : term,X485 : (term > $o)] : (? [X489 : term,X488 : term] : (($true != ((X485 @ (ap @ X488 @ X489)))) & (((X485 @ X488)) = $true) & ($true = ((X485 @ X489)))) | ? [X487 : term] : (($true = ((var @ X487))) & ($true != ((X485 @ X487)))) | ? [X490 : term] : (! [X491 : term] : (($true != ((X485 @ X491))) | (((X485 @ (sub @ X490 @ (push @ X491 @ id)))) = $true)) & ($true != ((X485 @ (lam @ X490))))) | ($true = ((X485 @ X486)))) <=> (induction2 = $true)) & (axabs = $true) & (hoasapnotvar_lthm = $true) & (id = ((d2subst @ d_id))) & (hoasinduction_lem3b_gthm = $true) & (! [X525 : term,X526 : (subst > term > subst > $o)] : (? [X528 : (subst > term > term)] : (! [X532 : term] : (($true = ((X526 @ id @ (X528 @ id @ X532) @ id))) | ($true != ((X526 @ id @ X532 @ id)))) & ($true != ((X526 @ id @ (lam @ (X528 @ sh @ one)) @ id))) & ! [X529 : term,X531 : subst,X530 : subst] : (((sub @ (X528 @ X530 @ X529) @ X531)) = ((X528 @ (comp @ X530 @ X531) @ (sub @ X529 @ X531))))) | ($true = ((X526 @ id @ (lam @ (sub @ X525 @ (push @ one @ sh))) @ id))) | ? [X527 : term] : (($true = ((X526 @ id @ X527 @ id))) & ($true != ((X526 @ id @ (sub @ X525 @ (push @ X527 @ id)) @ id))))) <=> (hoasinduction_lem3aa_lthm = $true)) & (! [X357 : (subst > term > subst > $o)] : ? [X358 : (term > $o)] : ! [X359 : term] : (((X358 @ X359)) = ((X357 @ id @ X359 @ id))) <=> (hoasinduction_lem0_lthm = $true)) & (ulamvar1 != $true) & (ulamvarsh = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (pushprop_lem0_gthm = $true) & (! [X510 : term] : (((sub @ X510 @ id)) = X510) <=> (termmset = $true)) & ((hoasapinj2_lthm = $true) <=> (! [X520 : term,X519 : term,X518 : term,X517 : term] : ((X517 = X520) | (((ap @ (sub @ X519 @ id) @ X517)) != ((ap @ (sub @ X518 @ id) @ X520)))) | ? [X516 : term,X514 : term,X513 : term,X515 : term] : ((((ap @ X515 @ X514)) = ((ap @ X516 @ X513))) & (X513 != X514)))) & (! [X168 : term,X166 : term,X167 : term,X169 : term] : ((((ap @ X168 @ X167)) != ((ap @ X166 @ X169))) | (X166 = X168)) <=> (apinj1 = $true)) & (axvarshift = $true) & (! [X341 : term,X342 : (subst > term > subst > $o)] : (? [X347 : term] : ((((X342 @ id @ (sub @ X341 @ (push @ X347 @ id)) @ id)) != $true) & ($true = ((X342 @ id @ X347 @ id)))) | ? [X345 : subst,X344 : subst,X346 : term,X343 : subst] : (($true = ((X342 @ X343 @ X346 @ (comp @ X345 @ X344)))) & ($true != ((X342 @ (comp @ X343 @ X345) @ (sub @ X346 @ X345) @ X344)))) | ($true = ((X342 @ id @ (lam @ X341) @ id))) | ? [X348 : term,X349 : subst,X351 : subst,X350 : subst] : ((((X342 @ X351 @ X348 @ (comp @ X350 @ X349))) != $true) & ($true = ((X342 @ (comp @ X351 @ X350) @ (sub @ X348 @ X350) @ X349)))) | ? [X352 : (subst > term > term)] : (! [X356 : term] : (($true = ((X342 @ id @ (X352 @ id @ X356) @ id))) | (((X342 @ id @ X356 @ id)) != $true)) & ($true != ((X342 @ id @ (lam @ (X352 @ sh @ one)) @ id))) & ! [X355 : subst,X353 : subst,X354 : term] : (((X352 @ (comp @ X355 @ X353) @ (sub @ X354 @ X353))) = ((sub @ (X352 @ X355 @ X354) @ X353))))) <=> (hoasinduction_lem3 = $true)) & (pushprop_lem2v2_lthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & (! [X444 : subst] : (((comp @ X444 @ id)) = X444) <=> (axidr = $true)) & (induction2lem_gthm = $true) & (hoasinduction_lem1_lthm = $true) & ! [X234 : d_subst,X235 : d_subst] : ((X234 = X235) | (((d2subst @ X235)) != ((d2subst @ X234)))) & (((var @ (d2term @ d_one))) != $true) & ((hoaslaminj_lthm = $true) <=> (((! [X10 : term,X11 : (subst > term > term),X8 : subst,X9 : (subst > term > term)] : ((((lam @ (X9 @ sh @ one))) != ((lam @ (X11 @ sh @ one)))) | ? [X15 : term,X17 : subst,X16 : subst] : (((sub @ (X11 @ X16 @ X15) @ X17)) != ((X11 @ (comp @ X16 @ X17) @ (sub @ X15 @ X17)))) | (((X9 @ X8 @ X10)) = ((X11 @ X8 @ X10))) | ? [X14 : subst,X13 : term,X12 : subst] : (((X9 @ (comp @ X14 @ X12) @ (sub @ X13 @ X12))) != ((sub @ (X9 @ X14 @ X13) @ X12)))) | ? [X6 : term,X7 : term] : ((((lam @ X6)) = ((lam @ X7))) & (X6 != X7))) | ? [X4 : term,X5 : subst] : (((comp @ sh @ (push @ X4 @ X5))) != X5)) | ? [X3 : subst,X2 : term] : (((sub @ one @ (push @ X2 @ X3))) != X2))) & (! [X295 : subst,X293 : term,X292 : (term > $o),X294 : (term > $o)] : (($true != ((X294 @ X293))) | ($true = ((X292 @ one))) | ? [X296 : term] : (((X294 @ (sub @ X296 @ (push @ X293 @ X295)))) != ((X292 @ X296)))) <=> (pushprop_lem1v2 = $true)) & ! [X297 : term] : ? [X298 : d_term] : (((d2term @ X298)) = X297) & (! [X220 : term,X221 : (subst > term > subst > $o)] : ((((X221 @ id @ X220 @ id)) = $true) | ? [X228 : term,X227 : term] : (($true = ((X221 @ id @ X227 @ id))) & ($true != ((X221 @ id @ (ap @ (sub @ X228 @ id) @ X227) @ id))) & ($true = ((X221 @ id @ X228 @ id)))) | ? [X222 : (subst > term > term)] : (($true != ((X221 @ id @ (lam @ (X222 @ sh @ one)) @ id))) & ! [X225 : subst,X224 : subst,X223 : term] : (((sub @ (X222 @ X224 @ X223) @ X225)) = ((X222 @ (comp @ X224 @ X225) @ (sub @ X223 @ X225)))) & ! [X226 : term] : (($true != ((X221 @ id @ X226 @ id))) | ($true = ((X221 @ id @ (X222 @ id @ X226) @ id)))))) <=> (hoasinduction_no_psi_cond = $true)) & (axapp = $true) & (axmap = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (pushprop_lem1v2_gthm = $true) & (! [X32 : term,X33 : term,X34 : term,X35 : term] : ((((ap @ (sub @ X34 @ id) @ X33)) != ((ap @ (sub @ X32 @ id) @ X35))) | (X32 = X34)) <=> (hoasapinj1 = $true)) & (! [X368 : term] : ? [X369 : (subst > term > term)] : (((X369 @ sh @ one)) = ((sub @ X368 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b = $true)) & (((((! [X213 : subst,X214 : term,X212 : (term > $o)] : (? [X219 : term] : (($true = ((var @ X219))) & ($true != ((X212 @ (sub @ X219 @ X213))))) | ? [X217 : term,X218 : term] : (($true = ((X212 @ X218))) & ($true = ((X212 @ X217))) & ($true != ((X212 @ (ap @ X218 @ X217))))) | ($true = ((X212 @ (sub @ X214 @ X213)))) | ? [X215 : term] : (! [X216 : term] : (($true != ((X212 @ X216))) | (((X212 @ (sub @ X215 @ (push @ X216 @ id)))) = $true)) & ($true != ((X212 @ (lam @ X215)))))) | ? [X206 : (term > $o),X207 : term] : (! [X209 : term] : (($true = ((X206 @ X209))) | ($true != ((var @ X209)))) & ! [X208 : term] : (($true != ((X206 @ X208))) | ($true = ((X206 @ (lam @ X208))))) & ($true != ((X206 @ X207))) & ! [X210 : term,X211 : term] : (($true != ((X206 @ X211))) | ($true != ((X206 @ X210))) | ($true = ((X206 @ (ap @ X210 @ X211))))))) | ? [X205 : subst] : (((comp @ X205 @ id)) != X205)) | ? [X204 : subst,X203 : term] : (((comp @ sh @ (push @ X203 @ X204))) != X204)) | ? [X202 : subst,X201 : term] : (((sub @ one @ (push @ X201 @ X202))) != X201)) <=> (induction2lem_lthm = $true)) & ((laminj = $true) <=> ! [X290 : term,X291 : term] : ((((lam @ X291)) != ((lam @ X290))) | (X290 = X291))) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((hoasinduction_lem2 = $true) <=> ! [X18 : term,X19 : term,X20 : (subst > term > subst > $o)] : (($true != ((X20 @ id @ X19 @ id))) | ? [X21 : subst,X22 : term,X23 : subst,X24 : subst] : (($true = ((X20 @ (comp @ X21 @ X24) @ (sub @ X22 @ X24) @ X23))) & ($true != ((X20 @ X21 @ X22 @ (comp @ X24 @ X23))))) | ($true = ((X20 @ id @ (ap @ X19 @ X18) @ id))) | (((X20 @ id @ X18 @ id)) != $true) | ? [X30 : term,X29 : term] : (($true = ((X20 @ id @ X30 @ id))) & (((X20 @ id @ X29 @ id)) = $true) & ($true != ((X20 @ id @ (ap @ (sub @ X29 @ id) @ X30) @ id)))) | ? [X27 : subst,X28 : subst,X26 : subst,X25 : term] : (($true != ((X20 @ (comp @ X28 @ X26) @ (sub @ X25 @ X26) @ X27))) & ($true = ((X20 @ X28 @ X25 @ (comp @ X26 @ X27))))))) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ((hoasinduction_lem3aa = $true) <=> ! [X133 : term,X132 : (subst > term > subst > $o)] : (? [X139 : term] : (($true = ((X132 @ id @ X139 @ id))) & ($true != ((X132 @ id @ (sub @ X133 @ (push @ X139 @ id)) @ id)))) | ? [X134 : (subst > term > term)] : (! [X135 : term] : (($true = ((X132 @ id @ (X134 @ id @ X135) @ id))) | ($true != ((X132 @ id @ X135 @ id)))) & ($true != ((X132 @ id @ (lam @ (X134 @ sh @ one)) @ id))) & ! [X138 : subst,X137 : subst,X136 : term] : (((sub @ (X134 @ X138 @ X136) @ X137)) = ((X134 @ (comp @ X138 @ X137) @ (sub @ X136 @ X137))))) | ($true = ((X132 @ id @ (lam @ (sub @ X133 @ (push @ one @ sh))) @ id))))) & (hoasinduction_lem2_lthm = $true) & ((hoasinduction_lem3v2_lthm = $true) <=> (! [X417 : term,X418 : (term > $o),X416 : (subst > term > subst > $o)] : (($true = ((X418 @ (lam @ X417)))) | ? [X428 : (subst > term > term)] : (($true != ((X416 @ id @ (lam @ (X428 @ sh @ one)) @ id))) & ! [X432 : term] : (($true != ((X416 @ id @ X432 @ id))) | ($true = ((X416 @ id @ (X428 @ id @ X432) @ id)))) & ! [X430 : subst,X429 : term,X431 : subst] : (((sub @ (X428 @ X431 @ X429) @ X430)) = ((X428 @ (comp @ X431 @ X430) @ (sub @ X429 @ X430))))) | ? [X433 : term] : (((X418 @ X433)) != ((X416 @ id @ X433 @ id))) | ? [X423 : term] : (($true = ((X418 @ X423))) & ($true != ((X418 @ (sub @ X417 @ (push @ X423 @ id)))))) | ? [X421 : subst,X419 : subst,X422 : subst,X420 : term] : (($true = ((X416 @ X422 @ X420 @ (comp @ X419 @ X421)))) & ($true != ((X416 @ (comp @ X422 @ X419) @ (sub @ X420 @ X419) @ X421)))) | ? [X424 : subst,X425 : subst,X427 : term,X426 : subst] : (($true != ((X416 @ X424 @ X427 @ (comp @ X426 @ X425)))) & ($true = ((X416 @ (comp @ X424 @ X426) @ (sub @ X427 @ X426) @ X425))))) | ? [X415 : term] : (((sub @ X415 @ id)) != X415))) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (hoasinduction_lem1 = $true) & (((((! [X75 : term,X76 : (subst > term > subst > $o)] : (? [X83 : term,X82 : term] : (($true = ((X76 @ id @ X82 @ id))) & ($true = ((X76 @ id @ X83 @ id))) & ($true != ((X76 @ id @ (ap @ (sub @ X82 @ id) @ X83) @ id)))) | ($true = ((X76 @ id @ X75 @ id))) | ? [X77 : (subst > term > term)] : (! [X81 : term] : (($true != ((X76 @ id @ X81 @ id))) | ($true = ((X76 @ id @ (X77 @ id @ X81) @ id)))) & ($true != ((X76 @ id @ (lam @ (X77 @ sh @ one)) @ id))) & ! [X79 : subst,X80 : term,X78 : subst] : (((sub @ (X77 @ X78 @ X80) @ X79)) = ((X77 @ (comp @ X78 @ X79) @ (sub @ X80 @ X79)))))) | ? [X66 : (subst > term > subst > $o),X65 : term,X67 : (term > $o)] : (($true != ((X67 @ (lam @ X65)))) & ! [X69 : (subst > term > term)] : (? [X71 : subst,X70 : subst,X72 : term] : (((sub @ (X69 @ X70 @ X72) @ X71)) != ((X69 @ (comp @ X70 @ X71) @ (sub @ X72 @ X71)))) | ? [X73 : term] : (($true = ((X66 @ id @ X73 @ id))) & ($true != ((X66 @ id @ (X69 @ id @ X73) @ id)))) | ($true = ((X66 @ id @ (lam @ (X69 @ sh @ one)) @ id)))) & ! [X68 : term] : (($true != ((X67 @ X68))) | ($true = ((X67 @ (sub @ X65 @ (push @ X68 @ id)))))) & ! [X74 : term] : (((X66 @ id @ X74 @ id)) = ((X67 @ X74))))) | ? [X64 : term] : (((sub @ X64 @ id)) != X64)) | ? [X57 : (term > $o),X58 : term] : (! [X62 : term] : (($true = ((X57 @ (lam @ X62)))) | ? [X63 : term] : (($true = ((X57 @ X63))) & (((X57 @ (sub @ X62 @ (push @ X63 @ id)))) != $true))) & ! [X60 : term,X59 : term] : (($true != ((X57 @ X59))) | ($true != ((X57 @ X60))) | (((X57 @ (ap @ X59 @ X60))) = $true)) & ($true != ((X57 @ X58))) & ! [X61 : term] : (($true != ((var @ X61))) | (((X57 @ X61)) = $true)))) | ? [X54 : (subst > term > subst > $o)] : ! [X55 : (term > $o)] : ? [X56 : term] : (((X54 @ id @ X56 @ id)) != ((X55 @ X56)))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (pushprop_lem3v2_lthm = $true) & ((hoasinduction_lthm_3 = $true) <=> ((((! [X114 : (subst > term > subst > $o),X115 : term] : (? [X122 : term,X124 : subst,X125 : subst,X123 : subst] : (($true != ((X114 @ X124 @ X122 @ (comp @ X123 @ X125)))) & ($true = ((X114 @ (comp @ X124 @ X123) @ (sub @ X122 @ X123) @ X125)))) | ? [X126 : (subst > term > term)] : (! [X130 : term] : ((((X114 @ id @ (X126 @ id @ X130) @ id)) = $true) | (((X114 @ id @ X130 @ id)) != $true)) & ! [X127 : term,X129 : subst,X128 : subst] : (((X126 @ (comp @ X129 @ X128) @ (sub @ X127 @ X128))) = ((sub @ (X126 @ X129 @ X127) @ X128))) & ($true != ((X114 @ id @ (lam @ (X126 @ sh @ one)) @ id)))) | ? [X131 : term] : (($true != ((X114 @ id @ X131 @ id))) & ($true = ((var @ (sub @ X131 @ id))))) | ($true = ((X114 @ id @ X115 @ id))) | ? [X120 : term,X121 : term] : (($true = ((X114 @ id @ X121 @ id))) & (((X114 @ id @ (ap @ (sub @ X120 @ id) @ X121) @ id)) != $true) & ($true = ((X114 @ id @ X120 @ id)))) | ? [X117 : subst,X116 : term,X118 : subst,X119 : subst] : (($true != ((X114 @ (comp @ X118 @ X119) @ (sub @ X116 @ X119) @ X117))) & ($true = ((X114 @ X118 @ X116 @ (comp @ X119 @ X117)))))) | ? [X104 : term,X106 : (term > $o),X105 : (subst > term > subst > $o)] : (($true != ((X106 @ (lam @ X104)))) & ! [X107 : term] : (((X105 @ id @ X107 @ id)) = ((X106 @ X107))) & ! [X108 : (subst > term > term)] : (($true = ((X105 @ id @ (lam @ (X108 @ sh @ one)) @ id))) | ? [X111 : term,X110 : subst,X112 : subst] : (((X108 @ (comp @ X110 @ X112) @ (sub @ X111 @ X112))) != ((sub @ (X108 @ X110 @ X111) @ X112))) | ? [X109 : term] : (($true = ((X105 @ id @ X109 @ id))) & ($true != ((X105 @ id @ (X108 @ id @ X109) @ id))))) & ! [X113 : term] : (($true = ((X106 @ (sub @ X104 @ (push @ X113 @ id))))) | (((X106 @ X113)) != $true)))) | ? [X103 : term] : (((sub @ X103 @ id)) != X103)) | ? [X96 : (term > $o),X97 : term] : (! [X102 : term,X101 : term] : ((((X96 @ X102)) != $true) | (((X96 @ (ap @ X101 @ X102))) = $true) | (((X96 @ X101)) != $true)) & ! [X98 : term] : (($true = ((X96 @ X98))) | ($true != ((var @ X98)))) & ! [X99 : term] : (? [X100 : term] : (($true = ((X96 @ X100))) & ($true != ((X96 @ (sub @ X99 @ (push @ X100 @ id)))))) | ($true = ((X96 @ (lam @ X99))))) & ($true != ((X96 @ X97))))) | ? [X93 : (subst > term > subst > $o)] : ! [X94 : (term > $o)] : ? [X95 : term] : (((X93 @ id @ X95 @ id)) != ((X94 @ X95))))) & (hoasinduction_lem1v2 = $true) & (hoasinduction_lem3a_gthm = $true) & (termmset_gthm = $true) & (pushprop = $true) & (! [X0 : term,X1 : term] : ((X0 = X1) | (((sub @ X0 @ sh)) != ((sub @ X1 @ sh)))) <=> (shinj = $true)) & (pushprop_lem2v2_gthm = $true) & ((hoasinduction_lem3v2a = $true) <=> ! [X434 : (subst > term > subst > $o),X435 : term,X436 : (term > $o)] : (($true = ((X436 @ (lam @ X435)))) | ? [X443 : term] : (($true != ((X436 @ (sub @ X435 @ (push @ X443 @ id))))) & ($true = ((X436 @ X443)))) | ? [X437 : term] : (((X436 @ X437)) != ((X434 @ id @ X437 @ id))) | ? [X438 : (subst > term > term)] : (! [X440 : subst,X441 : term,X439 : subst] : (((sub @ (X438 @ X439 @ X441) @ X440)) = ((X438 @ (comp @ X439 @ X440) @ (sub @ X441 @ X440)))) & ($true != ((X434 @ id @ (lam @ (X438 @ sh @ one)) @ id))) & ! [X442 : term] : ((((X434 @ id @ (X438 @ id @ X442) @ id)) = $true) | ($true != ((X434 @ id @ X442 @ id))))))) & (! [X387 : term,X386 : (subst > term > subst > $o)] : (? [X393 : term] : (($true != ((X386 @ id @ (sub @ X387 @ (push @ X393 @ id)) @ id))) & ($true = ((X386 @ id @ X393 @ id)))) | (((X386 @ id @ (lam @ X387) @ id)) = $true) | ? [X388 : (subst > term > term)] : (! [X389 : subst,X390 : subst,X391 : term] : (((X388 @ (comp @ X390 @ X389) @ (sub @ X391 @ X389))) = ((sub @ (X388 @ X390 @ X391) @ X389))) & ($true != ((X386 @ id @ (lam @ (X388 @ sh @ one)) @ id))) & ! [X392 : term] : (($true = ((X386 @ id @ (X388 @ id @ X392) @ id))) | ($true != ((X386 @ id @ X392 @ id)))))) <=> (hoasinduction_lem3a = $true)) & (! [X329 : (term > $o),X330 : (term > $o),X327 : term,X326 : term,X328 : subst] : (($true != ((X330 @ X327))) | ($true = ((X329 @ (sub @ X327 @ (push @ X326 @ X328)))))) <=> (pushprop_lem1 = $true)) & (! [X340 : subst] : (((comp @ id @ X340)) = X340) <=> (axidl = $true)) & ! [X36 : d_term] : (d_one = X36) & (hoaslamnotap_gthm = $true) & (pushprop_lem2v2 = $true) & ((((! [X333 : subst] : (((comp @ id @ X333)) = X333) & ! [X334 : subst] : (((comp @ X334 @ id)) = X334)) | ? [X332 : subst] : (((comp @ X332 @ id)) != X332)) | ? [X331 : subst] : (((comp @ id @ X331)) != X331)) <=> (substmonoid_lthm = $true)) & (pushprop_lem3v2 = $true) & (! [X140 : (subst > term > subst > $o)] : ? [X141 : (term > $o)] : ! [X142 : term] : (((X141 @ X142)) = ((X140 @ id @ X142 @ id))) <=> (hoasinduction_lem0 = $true)) & ((hoasapinj1_lthm = $true) <=> ((! [X458 : term,X457 : term,X456 : term,X459 : term] : ((((ap @ (sub @ X458 @ id) @ X456)) != ((ap @ (sub @ X459 @ id) @ X457))) | (X458 = X459)) | ? [X453 : term,X455 : term,X452 : term,X454 : term] : ((X452 != X455) & (((ap @ X452 @ X453)) = ((ap @ X455 @ X454))))) | ? [X451 : term] : (((sub @ X451 @ id)) != X451))) & ! [X144 : d_term,X143 : d_term] : ((X143 = X144) | (((d2term @ X144)) != ((d2term @ X143)))) & (! [X523 : term,X522 : term,X521 : term,X524 : term] : ((((ap @ (sub @ X523 @ id) @ X521)) != ((ap @ (sub @ X522 @ id) @ X524))) | (X521 = X524)) <=> (hoasapinj2 = $true)) & ((pushprop_lem1v2_lthm = $true) <=> (? [X533 : term,X534 : subst] : (((sub @ one @ (push @ X533 @ X534))) != X533) | ! [X535 : term,X538 : (term > $o),X537 : subst,X536 : (term > $o)] : (? [X539 : term] : (((X536 @ (sub @ X539 @ (push @ X535 @ X537)))) != ((X538 @ X539))) | ($true != ((X536 @ X535))) | ($true = ((X538 @ one)))))) & (pushprop_lem1_gthm = $true) & (hoasapinj1_gthm = $true) & (hoaslamnotvar = $true) & (lamnotvar = $true) & (lamnotap != $true) & (pushprop_lthm_orig = $true) & ((hoasinduction_lem3_lthm = $true) <=> ((! [X309 : term,X308 : (subst > term > subst > $o)] : (($true = ((X308 @ id @ (lam @ X309) @ id))) | ? [X323 : term] : (($true != ((X308 @ id @ (sub @ X309 @ (push @ X323 @ id)) @ id))) & ($true = ((X308 @ id @ X323 @ id)))) | ? [X318 : (subst > term > term)] : (! [X319 : term] : (($true != ((X308 @ id @ X319 @ id))) | ($true = ((X308 @ id @ (X318 @ id @ X319) @ id)))) & ! [X320 : subst,X321 : term,X322 : subst] : (((X318 @ (comp @ X320 @ X322) @ (sub @ X321 @ X322))) = ((sub @ (X318 @ X320 @ X321) @ X322))) & ($true != ((X308 @ id @ (lam @ (X318 @ sh @ one)) @ id)))) | ? [X312 : term,X311 : subst,X310 : subst,X313 : subst] : (($true != ((X308 @ X310 @ X312 @ (comp @ X313 @ X311)))) & ($true = ((X308 @ (comp @ X310 @ X313) @ (sub @ X312 @ X313) @ X311)))) | ? [X314 : term,X315 : subst,X317 : subst,X316 : subst] : ((((X308 @ (comp @ X315 @ X317) @ (sub @ X314 @ X317) @ X316)) != $true) & (((X308 @ X315 @ X314 @ (comp @ X317 @ X316))) = $true))) | ? [X300 : (subst > term > subst > $o),X301 : term] : (! [X302 : (subst > term > term)] : (? [X306 : term] : (($true = ((X300 @ id @ X306 @ id))) & ($true != ((X300 @ id @ (X302 @ id @ X306) @ id)))) | ($true = ((X300 @ id @ (lam @ (X302 @ sh @ one)) @ id))) | ? [X305 : subst,X303 : term,X304 : subst] : (((sub @ (X302 @ X304 @ X303) @ X305)) != ((X302 @ (comp @ X304 @ X305) @ (sub @ X303 @ X305))))) & ! [X307 : term] : (($true = ((X300 @ id @ (sub @ X301 @ (push @ X307 @ id)) @ id))) | ($true != ((X300 @ id @ X307 @ id)))) & ($true != ((X300 @ id @ (lam @ (sub @ X301 @ (push @ one @ sh))) @ id))))) | ? [X299 : term] : (((sub @ X299 @ id)) != X299))) & ((pushprop_lem0_lthm = $true) <=> ! [X230 : subst,X231 : (term > $o),X229 : term] : ? [X232 : (term > $o)] : ! [X233 : term] : (((X232 @ X233)) = ((X231 @ (sub @ X233 @ (push @ X229 @ X230)))))) & (! [X363 : term,X362 : subst] : (((comp @ sh @ (push @ X363 @ X362))) = X362) <=> (axshiftcons = $true)) & ((hoasinduction_lem3a_lthm = $true) <=> ((! [X46 : term,X47 : (subst > term > subst > $o)] : (? [X48 : (subst > term > term)] : (! [X50 : subst,X51 : term,X52 : subst] : (((sub @ (X48 @ X50 @ X51) @ X52)) = ((X48 @ (comp @ X50 @ X52) @ (sub @ X51 @ X52)))) & ! [X49 : term] : (($true != ((X47 @ id @ X49 @ id))) | (((X47 @ id @ (X48 @ id @ X49) @ id)) = $true)) & ($true != ((X47 @ id @ (lam @ (X48 @ sh @ one)) @ id)))) | ? [X53 : term] : (($true != ((X47 @ id @ (sub @ X46 @ (push @ X53 @ id)) @ id))) & ($true = ((X47 @ id @ X53 @ id)))) | (((X47 @ id @ (lam @ X46) @ id)) = $true)) | ? [X38 : (subst > term > subst > $o),X39 : term] : (! [X40 : (subst > term > term)] : (? [X42 : subst,X43 : subst,X41 : term] : (((X40 @ (comp @ X42 @ X43) @ (sub @ X41 @ X43))) != ((sub @ (X40 @ X42 @ X41) @ X43))) | ? [X44 : term] : ((((X38 @ id @ (X40 @ id @ X44) @ id)) != $true) & ($true = ((X38 @ id @ X44 @ id)))) | ($true = ((X38 @ id @ (lam @ (X40 @ sh @ one)) @ id)))) & ! [X45 : term] : (($true != ((X38 @ id @ X45 @ id))) | ($true = ((X38 @ id @ (sub @ X39 @ (push @ X45 @ id)) @ id)))) & ($true != ((X38 @ id @ (lam @ (sub @ X39 @ (push @ one @ sh))) @ id))))) | ? [X37 : term] : (((sub @ X37 @ id)) != X37))) & (! [X92 : term] : (((sub @ X92 @ id)) = X92) <=> (axvarid = $true)) & (! [X463 : term,X461 : subst,X460 : (subst > term > term),X462 : (subst > term > term)] : ((((lam @ (X462 @ sh @ one))) != ((lam @ (X460 @ sh @ one)))) | (((X462 @ X461 @ X463)) = ((X460 @ X461 @ X463))) | ? [X466 : term,X465 : subst,X464 : subst] : (((sub @ (X460 @ X465 @ X466) @ X464)) != ((X460 @ (comp @ X465 @ X464) @ (sub @ X466 @ X464)))) | ? [X467 : subst,X469 : term,X468 : subst] : (((sub @ (X462 @ X467 @ X469) @ X468)) != ((X462 @ (comp @ X467 @ X468) @ (sub @ X469 @ X468))))) <=> (hoaslaminj = $true)) & (hoasapnotvar = $true) & (hoasinduction_lem2v2_gthm = $true) & ((axscons = $true) <=> ! [X394 : subst] : (((push @ (sub @ one @ X394) @ (comp @ sh @ X394))) = X394)) & ! [X31 : d_subst] : (d_id = X31) & (((d2term @ d_one)) = one) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((pushprop_lem1_lthm = $true) <=> ((! [X198 : term,X197 : subst,X199 : term,X200 : (term > $o),X196 : (term > $o)] : (($true != ((X196 @ X199))) | (((X200 @ (sub @ X199 @ (push @ X198 @ X197)))) = $true)) | ? [X194 : subst,X195 : term] : (((comp @ sh @ (push @ X195 @ X194))) != X194)) | ? [X192 : subst,X193 : term] : (((sub @ one @ (push @ X193 @ X192))) != X193))) & (ulamvarind = $true) & (hoasinduction_lem3v2_gthm = $true)), inference(ennf_transformation,[],[f10])). thf(f12,plain,( (pushprop_lthm = $true) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X324 : term] : ? [X325 : (subst > term > term)] : (((sub @ X324 @ (push @ one @ sh))) = ((X325 @ sh @ one)))) & (hoasinduction_gthm = $true) & (((d2subst @ d_id)) = sh) & (pushprop_lem1_gthm = $true) & (hoasinduction_lem3_gthm = $true) & (ulamvarsh = $true) & (pushprop_lem3v2 = $true) & (! [X357 : (subst > term > subst > $o)] : ? [X358 : (term > $o)] : ! [X359 : term] : (((X358 @ X359)) = ((X357 @ id @ X359 @ id))) <=> (hoasinduction_lem0_lthm = $true)) & (! [X164 : term,X165 : subst] : (((sub @ one @ (push @ X164 @ X165))) = X164) <=> (axvarcons = $true)) & (((var @ (d2term @ d_one))) != $true) & ((? [X3 : subst,X2 : term] : (((sub @ one @ (push @ X2 @ X3))) != X2) | ? [X6 : term,X7 : term] : ((((lam @ X6)) = ((lam @ X7))) & (X6 != X7)) | ? [X4 : term,X5 : subst] : (((comp @ sh @ (push @ X4 @ X5))) != X5) | ! [X10 : term,X11 : (subst > term > term),X8 : subst,X9 : (subst > term > term)] : ((((lam @ (X9 @ sh @ one))) != ((lam @ (X11 @ sh @ one)))) | ? [X15 : term,X17 : subst,X16 : subst] : (((sub @ (X11 @ X16 @ X15) @ X17)) != ((X11 @ (comp @ X16 @ X17) @ (sub @ X15 @ X17)))) | (((X9 @ X8 @ X10)) = ((X11 @ X8 @ X10))) | ? [X14 : subst,X13 : term,X12 : subst] : (((X9 @ (comp @ X14 @ X12) @ (sub @ X13 @ X12))) != ((sub @ (X9 @ X14 @ X13) @ X12))))) <=> (hoaslaminj_lthm = $true)) & ((? [X64 : term] : (((sub @ X64 @ id)) != X64) | ? [X54 : (subst > term > subst > $o)] : ! [X55 : (term > $o)] : ? [X56 : term] : (((X54 @ id @ X56 @ id)) != ((X55 @ X56))) | ! [X75 : term,X76 : (subst > term > subst > $o)] : (? [X83 : term,X82 : term] : (($true = ((X76 @ id @ X82 @ id))) & ($true = ((X76 @ id @ X83 @ id))) & ($true != ((X76 @ id @ (ap @ (sub @ X82 @ id) @ X83) @ id)))) | ($true = ((X76 @ id @ X75 @ id))) | ? [X77 : (subst > term > term)] : (! [X81 : term] : (($true != ((X76 @ id @ X81 @ id))) | ($true = ((X76 @ id @ (X77 @ id @ X81) @ id)))) & ($true != ((X76 @ id @ (lam @ (X77 @ sh @ one)) @ id))) & ! [X79 : subst,X80 : term,X78 : subst] : (((sub @ (X77 @ X78 @ X80) @ X79)) = ((X77 @ (comp @ X78 @ X79) @ (sub @ X80 @ X79)))))) | ? [X57 : (term > $o),X58 : term] : (! [X62 : term] : (($true = ((X57 @ (lam @ X62)))) | ? [X63 : term] : (($true = ((X57 @ X63))) & (((X57 @ (sub @ X62 @ (push @ X63 @ id)))) != $true))) & ! [X60 : term,X59 : term] : (($true != ((X57 @ X59))) | ($true != ((X57 @ X60))) | (((X57 @ (ap @ X59 @ X60))) = $true)) & ($true != ((X57 @ X58))) & ! [X61 : term] : (($true != ((var @ X61))) | (((X57 @ X61)) = $true))) | ? [X66 : (subst > term > subst > $o),X65 : term,X67 : (term > $o)] : (($true != ((X67 @ (lam @ X65)))) & ! [X69 : (subst > term > term)] : (? [X71 : subst,X70 : subst,X72 : term] : (((sub @ (X69 @ X70 @ X72) @ X71)) != ((X69 @ (comp @ X70 @ X71) @ (sub @ X72 @ X71)))) | ? [X73 : term] : (($true = ((X66 @ id @ X73 @ id))) & ($true != ((X66 @ id @ (X69 @ id @ X73) @ id)))) | ($true = ((X66 @ id @ (lam @ (X69 @ sh @ one)) @ id)))) & ! [X68 : term] : (($true != ((X67 @ X68))) | ($true = ((X67 @ (sub @ X65 @ (push @ X68 @ id)))))) & ! [X74 : term] : (((X66 @ id @ X74 @ id)) = ((X67 @ X74))))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & ((hoasapinj1_lthm = $true) <=> (? [X453 : term,X455 : term,X452 : term,X454 : term] : ((X452 != X455) & (((ap @ X452 @ X453)) = ((ap @ X455 @ X454)))) | ? [X451 : term] : (((sub @ X451 @ id)) != X451) | ! [X458 : term,X457 : term,X456 : term,X459 : term] : ((((ap @ (sub @ X458 @ id) @ X456)) != ((ap @ (sub @ X459 @ id) @ X457))) | (X458 = X459)))) & (hoaslamnotvar_lthm = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ((hoasapinj2_lthm = $true) <=> (! [X520 : term,X519 : term,X518 : term,X517 : term] : ((X517 = X520) | (((ap @ (sub @ X519 @ id) @ X517)) != ((ap @ (sub @ X518 @ id) @ X520)))) | ? [X516 : term,X514 : term,X513 : term,X515 : term] : ((((ap @ X515 @ X514)) = ((ap @ X516 @ X513))) & (X513 != X514)))) & ((pushprop_lem1v2_lthm = $true) <=> (? [X533 : term,X534 : subst] : (((sub @ one @ (push @ X533 @ X534))) != X533) | ! [X535 : term,X538 : (term > $o),X537 : subst,X536 : (term > $o)] : (? [X539 : term] : (((X536 @ (sub @ X539 @ (push @ X535 @ X537)))) != ((X538 @ X539))) | ($true != ((X536 @ X535))) | ($true = ((X538 @ one)))))) & (axabs = $true) & ((! [X396 : term] : (((sub @ X396 @ id)) = X396) | ? [X395 : term] : (((sub @ X395 @ id)) != X395)) <=> (termmset_lthm = $true)) & (axvarshift = $true) & (axmap = $true) & (! [X486 : term,X485 : (term > $o)] : (? [X489 : term,X488 : term] : (($true != ((X485 @ (ap @ X488 @ X489)))) & (((X485 @ X488)) = $true) & ($true = ((X485 @ X489)))) | ? [X487 : term] : (($true = ((var @ X487))) & ($true != ((X485 @ X487)))) | ? [X490 : term] : (! [X491 : term] : (($true != ((X485 @ X491))) | (((X485 @ (sub @ X490 @ (push @ X491 @ id)))) = $true)) & ($true != ((X485 @ (lam @ X490))))) | ($true = ((X485 @ X486)))) <=> (induction2 = $true)) & (substmonoid_gthm = $true) & ((hoasinduction_lem3aaa = $true) <=> ! [X375 : term,X374 : (subst > term > subst > $o)] : (? [X377 : term,X378 : (subst > term > term)] : (! [X382 : term,X384 : subst,X383 : subst] : (((sub @ (X378 @ X384 @ X382) @ X383)) = ((sub @ (sub @ X377 @ (push @ X382 @ X384)) @ X383))) & ! [X379 : term,X380 : subst,X381 : subst] : (((X378 @ (comp @ X381 @ X380) @ (sub @ X379 @ X380))) = ((sub @ X377 @ (push @ (sub @ X379 @ X380) @ (comp @ X381 @ X380))))) & ! [X385 : term] : (($true = ((X374 @ id @ (X378 @ id @ X385) @ id))) | ($true != ((X374 @ id @ X385 @ id)))) & ($true != ((X374 @ id @ (lam @ (X378 @ sh @ one)) @ id)))) | (((X374 @ id @ (lam @ (sub @ X375 @ (push @ one @ sh))) @ id)) = $true) | ? [X376 : term] : (($true = ((X374 @ id @ X376 @ id))) & ($true != ((X374 @ id @ (sub @ X375 @ (push @ X376 @ id)) @ id)))))) & ((pushprop_lem0 = $true) <=> ! [X335 : (term > $o),X337 : subst,X336 : term] : ? [X338 : (term > $o)] : ! [X339 : term] : (((X335 @ (sub @ X339 @ (push @ X336 @ X337)))) = ((X338 @ X339)))) & ! [X297 : term] : ? [X298 : d_term] : (((d2term @ X298)) = X297) & (! [X523 : term,X522 : term,X521 : term,X524 : term] : ((((ap @ (sub @ X523 @ id) @ X521)) != ((ap @ (sub @ X522 @ id) @ X524))) | (X521 = X524)) <=> (hoasapinj2 = $true)) & (! [X470 : term,X471 : (term > $o),X472 : (subst > term > subst > $o),X473 : term] : (($true != ((X471 @ X473))) | ? [X484 : subst,X481 : subst,X483 : term,X482 : subst] : (($true != ((X472 @ (comp @ X481 @ X484) @ (sub @ X483 @ X484) @ X482))) & ($true = ((X472 @ X481 @ X483 @ (comp @ X484 @ X482))))) | ? [X474 : term] : (((X472 @ id @ X474 @ id)) != ((X471 @ X474))) | ? [X476 : term,X475 : term] : (($true = ((X472 @ id @ X475 @ id))) & ($true = ((X472 @ id @ X476 @ id))) & ($true != ((X472 @ id @ (ap @ (sub @ X476 @ id) @ X475) @ id)))) | ? [X479 : term,X480 : subst,X477 : subst,X478 : subst] : (($true = ((X472 @ (comp @ X478 @ X480) @ (sub @ X479 @ X480) @ X477))) & ($true != ((X472 @ X478 @ X479 @ (comp @ X480 @ X477))))) | ($true != ((X471 @ X470))) | ($true = ((X471 @ (ap @ X470 @ X473))))) <=> (hoasinduction_lem2v2 = $true)) & ! [X234 : d_subst,X235 : d_subst] : ((X234 = X235) | (((d2subst @ X235)) != ((d2subst @ X234)))) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_lem1v2 = $true) & (! [X140 : (subst > term > subst > $o)] : ? [X141 : (term > $o)] : ! [X142 : term] : (((X141 @ X142)) = ((X140 @ id @ X142 @ id))) <=> (hoasinduction_lem0 = $true)) & (! [X295 : subst,X293 : term,X292 : (term > $o),X294 : (term > $o)] : (($true != ((X294 @ X293))) | ($true = ((X292 @ one))) | ? [X296 : term] : (((X294 @ (sub @ X296 @ (push @ X293 @ X295)))) != ((X292 @ X296)))) <=> (pushprop_lem1v2 = $true)) & (hoasinduction_lem2_gthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (! [X92 : term] : (((sub @ X92 @ id)) = X92) <=> (axvarid = $true)) & ((laminj = $true) <=> ! [X290 : term,X291 : term] : ((((lam @ X291)) != ((lam @ X290))) | (X290 = X291))) & (hoaslamnotvar = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (axapp = $true) & (! [X444 : subst] : (((comp @ X444 @ id)) = X444) <=> (axidr = $true)) & (hoaslamnotap_gthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((? [X149 : term] : (((sub @ X149 @ id)) != X149) | ? [X145 : term] : ! [X146 : (subst > term > term)] : ? [X147 : term,X148 : subst] : (((sub @ X145 @ (push @ X147 @ X148))) != ((X146 @ X148 @ X147))) | ! [X152 : term,X150 : (term > $o),X151 : (subst > term > subst > $o)] : (? [X153 : (subst > term > term)] : (($true != ((X151 @ id @ (lam @ (X153 @ sh @ one)) @ id))) & ! [X157 : term] : (($true = ((X151 @ id @ (X153 @ id @ X157) @ id))) | (((X151 @ id @ X157 @ id)) != $true)) & ! [X156 : subst,X154 : term,X155 : subst] : (((sub @ (X153 @ X156 @ X154) @ X155)) = ((X153 @ (comp @ X156 @ X155) @ (sub @ X154 @ X155))))) | ? [X159 : term] : (($true = ((X150 @ X159))) & ($true != ((X150 @ (sub @ X152 @ (push @ X159 @ id)))))) | ($true = ((X150 @ (lam @ X152)))) | ? [X158 : term] : (((X151 @ id @ X158 @ id)) != ((X150 @ X158))))) <=> (hoasinduction_lem3v2a_lthm = $true)) & ((hoaslamnotap = $true) <=> ! [X446 : term,X447 : (subst > term > term),X445 : term] : (? [X448 : subst,X450 : term,X449 : subst] : (((X447 @ (comp @ X448 @ X449) @ (sub @ X450 @ X449))) != ((sub @ (X447 @ X448 @ X450) @ X449))) | (((ap @ (sub @ X445 @ id) @ X446)) != ((lam @ (X447 @ sh @ one)))))) & ((apinj2 = $true) <=> ! [X364 : term,X365 : term,X366 : term,X367 : term] : ((X364 = X365) | (((ap @ X366 @ X364)) != ((ap @ X367 @ X365))))) & ((! [X180 : (term > $o),X179 : term] : ((((X180 @ X179)) = $true) | ? [X181 : term,X182 : term] : (($true != ((X180 @ (ap @ X182 @ X181)))) & ($true = ((X180 @ X181))) & ($true = ((X180 @ X182)))) | ? [X185 : term] : (($true = ((var @ X185))) & ($true != ((X180 @ X185)))) | ? [X183 : term] : (($true != ((X180 @ (lam @ X183)))) & ! [X184 : term] : (($true != ((X180 @ X184))) | (((X180 @ (sub @ X183 @ (push @ X184 @ id)))) = $true)))) | ? [X170 : term] : (((sub @ X170 @ id)) != X170) | ? [X171 : (term > $o),X172 : term,X173 : subst] : (! [X174 : term] : (($true != ((var @ X174))) | ($true = ((X171 @ (sub @ X174 @ X173))))) & ($true != ((X171 @ (sub @ X172 @ X173)))) & ! [X178 : term,X177 : term] : (($true = ((X171 @ (ap @ X178 @ X177)))) | ($true != ((X171 @ X177))) | ($true != ((X171 @ X178)))) & ! [X175 : term] : (? [X176 : term] : (($true = ((X171 @ X176))) & ($true != ((X171 @ (sub @ X175 @ (push @ X176 @ id)))))) | ($true = ((X171 @ (lam @ X175))))))) <=> (induction2_lthm = $true)) & (! [X186 : term,X187 : (term > $o)] : (? [X189 : term] : (($true != ((X187 @ X189))) & ($true = ((var @ X189)))) | ($true = ((X187 @ X186))) | ? [X188 : term] : (($true = ((X187 @ X188))) & ($true != ((X187 @ (lam @ X188))))) | ? [X190 : term,X191 : term] : (($true = ((X187 @ X191))) & ($true = ((X187 @ X190))) & ($true != ((X187 @ (ap @ X191 @ X190)))))) <=> (induction = $true)) & (hoasapinj1_gthm = $true) & ! [X36 : d_term] : (d_one = X36) & ((axscons = $true) <=> ! [X394 : subst] : (((push @ (sub @ one @ X394) @ (comp @ sh @ X394))) = X394)) & (! [X32 : term,X33 : term,X34 : term,X35 : term] : ((((ap @ (sub @ X34 @ id) @ X33)) != ((ap @ (sub @ X32 @ id) @ X35))) | (X32 = X34)) <=> (hoasapinj1 = $true)) & (pushprop_lem2v2_lthm = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((substmonoid = $true) <=> (! [X511 : subst] : (((comp @ X511 @ id)) = X511) & ! [X512 : subst] : (((comp @ id @ X512)) = X512))) & (hoasinduction_lem3v2_gthm = $true) & (pushprop_lem0_gthm = $true) & (ulamvar1 != $true) & ((hoasinduction_lthm_3 = $true) <=> (? [X93 : (subst > term > subst > $o)] : ! [X94 : (term > $o)] : ? [X95 : term] : (((X93 @ id @ X95 @ id)) != ((X94 @ X95))) | ? [X104 : term,X106 : (term > $o),X105 : (subst > term > subst > $o)] : (($true != ((X106 @ (lam @ X104)))) & ! [X107 : term] : (((X105 @ id @ X107 @ id)) = ((X106 @ X107))) & ! [X108 : (subst > term > term)] : (($true = ((X105 @ id @ (lam @ (X108 @ sh @ one)) @ id))) | ? [X111 : term,X110 : subst,X112 : subst] : (((X108 @ (comp @ X110 @ X112) @ (sub @ X111 @ X112))) != ((sub @ (X108 @ X110 @ X111) @ X112))) | ? [X109 : term] : (($true = ((X105 @ id @ X109 @ id))) & ($true != ((X105 @ id @ (X108 @ id @ X109) @ id))))) & ! [X113 : term] : (($true = ((X106 @ (sub @ X104 @ (push @ X113 @ id))))) | (((X106 @ X113)) != $true))) | ? [X103 : term] : (((sub @ X103 @ id)) != X103) | ? [X96 : (term > $o),X97 : term] : (! [X102 : term,X101 : term] : ((((X96 @ X102)) != $true) | (((X96 @ (ap @ X101 @ X102))) = $true) | (((X96 @ X101)) != $true)) & ! [X98 : term] : (($true = ((X96 @ X98))) | ($true != ((var @ X98)))) & ! [X99 : term] : (? [X100 : term] : (($true = ((X96 @ X100))) & ($true != ((X96 @ (sub @ X99 @ (push @ X100 @ id)))))) | ($true = ((X96 @ (lam @ X99))))) & ($true != ((X96 @ X97)))) | ! [X114 : (subst > term > subst > $o),X115 : term] : (? [X122 : term,X124 : subst,X125 : subst,X123 : subst] : (($true != ((X114 @ X124 @ X122 @ (comp @ X123 @ X125)))) & ($true = ((X114 @ (comp @ X124 @ X123) @ (sub @ X122 @ X123) @ X125)))) | ? [X126 : (subst > term > term)] : (! [X130 : term] : ((((X114 @ id @ (X126 @ id @ X130) @ id)) = $true) | (((X114 @ id @ X130 @ id)) != $true)) & ! [X127 : term,X129 : subst,X128 : subst] : (((X126 @ (comp @ X129 @ X128) @ (sub @ X127 @ X128))) = ((sub @ (X126 @ X129 @ X127) @ X128))) & ($true != ((X114 @ id @ (lam @ (X126 @ sh @ one)) @ id)))) | ? [X131 : term] : (($true != ((X114 @ id @ X131 @ id))) & ($true = ((var @ (sub @ X131 @ id))))) | ($true = ((X114 @ id @ X115 @ id))) | ? [X120 : term,X121 : term] : (($true = ((X114 @ id @ X121 @ id))) & (((X114 @ id @ (ap @ (sub @ X120 @ id) @ X121) @ id)) != $true) & ($true = ((X114 @ id @ X120 @ id)))) | ? [X117 : subst,X116 : term,X118 : subst,X119 : subst] : (($true != ((X114 @ (comp @ X118 @ X119) @ (sub @ X116 @ X119) @ X117))) & ($true = ((X114 @ X118 @ X116 @ (comp @ X119 @ X117)))))))) & (! [X0 : term,X1 : term] : ((X0 = X1) | (((sub @ X0 @ sh)) != ((sub @ X1 @ sh)))) <=> (shinj = $true)) & (! [X340 : subst] : (((comp @ id @ X340)) = X340) <=> (axidl = $true)) & (! [X463 : term,X461 : subst,X460 : (subst > term > term),X462 : (subst > term > term)] : ((((lam @ (X462 @ sh @ one))) != ((lam @ (X460 @ sh @ one)))) | (((X462 @ X461 @ X463)) = ((X460 @ X461 @ X463))) | ? [X466 : term,X465 : subst,X464 : subst] : (((sub @ (X460 @ X465 @ X466) @ X464)) != ((X460 @ (comp @ X465 @ X464) @ (sub @ X466 @ X464)))) | ? [X467 : subst,X469 : term,X468 : subst] : (((sub @ (X462 @ X467 @ X469) @ X468)) != ((X462 @ (comp @ X467 @ X468) @ (sub @ X469 @ X468))))) <=> (hoaslaminj = $true)) & (hoaslaminj_gthm = $true) & (induction2lem_gthm = $true) & (id = ((d2subst @ d_id))) & (hoasinduction_lem1v2_gthm = $true) & (hoasinduction_lem1_lthm = $true) & (! [X370 : term] : ? [X371 : (subst > term > term)] : ! [X372 : term,X373 : subst] : (((X371 @ X373 @ X372)) = ((sub @ X370 @ (push @ X372 @ X373)))) <=> (hoasinduction_lem3v2_f = $true)) & (hoasinduction_lem3b_gthm = $true) & (lamnotvar = $true) & (pushprop = $true) & (axassoc = $true) & ((? [X299 : term] : (((sub @ X299 @ id)) != X299) | ! [X309 : term,X308 : (subst > term > subst > $o)] : (($true = ((X308 @ id @ (lam @ X309) @ id))) | ? [X323 : term] : (($true != ((X308 @ id @ (sub @ X309 @ (push @ X323 @ id)) @ id))) & ($true = ((X308 @ id @ X323 @ id)))) | ? [X318 : (subst > term > term)] : (! [X319 : term] : (($true != ((X308 @ id @ X319 @ id))) | ($true = ((X308 @ id @ (X318 @ id @ X319) @ id)))) & ! [X320 : subst,X321 : term,X322 : subst] : (((X318 @ (comp @ X320 @ X322) @ (sub @ X321 @ X322))) = ((sub @ (X318 @ X320 @ X321) @ X322))) & ($true != ((X308 @ id @ (lam @ (X318 @ sh @ one)) @ id)))) | ? [X312 : term,X311 : subst,X310 : subst,X313 : subst] : (($true != ((X308 @ X310 @ X312 @ (comp @ X313 @ X311)))) & ($true = ((X308 @ (comp @ X310 @ X313) @ (sub @ X312 @ X313) @ X311)))) | ? [X314 : term,X315 : subst,X317 : subst,X316 : subst] : ((((X308 @ (comp @ X315 @ X317) @ (sub @ X314 @ X317) @ X316)) != $true) & (((X308 @ X315 @ X314 @ (comp @ X317 @ X316))) = $true))) | ? [X300 : (subst > term > subst > $o),X301 : term] : (! [X302 : (subst > term > term)] : (? [X306 : term] : (($true = ((X300 @ id @ X306 @ id))) & ($true != ((X300 @ id @ (X302 @ id @ X306) @ id)))) | ($true = ((X300 @ id @ (lam @ (X302 @ sh @ one)) @ id))) | ? [X305 : subst,X303 : term,X304 : subst] : (((sub @ (X302 @ X304 @ X303) @ X305)) != ((X302 @ (comp @ X304 @ X305) @ (sub @ X303 @ X305))))) & ! [X307 : term] : (($true = ((X300 @ id @ (sub @ X301 @ (push @ X307 @ id)) @ id))) | ($true != ((X300 @ id @ X307 @ id)))) & ($true != ((X300 @ id @ (lam @ (sub @ X301 @ (push @ one @ sh))) @ id))))) <=> (hoasinduction_lem3_lthm = $true)) & ! [X144 : d_term,X143 : d_term] : ((X143 = X144) | (((d2term @ X144)) != ((d2term @ X143)))) & (! [X341 : term,X342 : (subst > term > subst > $o)] : (? [X347 : term] : ((((X342 @ id @ (sub @ X341 @ (push @ X347 @ id)) @ id)) != $true) & ($true = ((X342 @ id @ X347 @ id)))) | ? [X345 : subst,X344 : subst,X346 : term,X343 : subst] : (($true = ((X342 @ X343 @ X346 @ (comp @ X345 @ X344)))) & ($true != ((X342 @ (comp @ X343 @ X345) @ (sub @ X346 @ X345) @ X344)))) | ($true = ((X342 @ id @ (lam @ X341) @ id))) | ? [X348 : term,X349 : subst,X351 : subst,X350 : subst] : ((((X342 @ X351 @ X348 @ (comp @ X350 @ X349))) != $true) & ($true = ((X342 @ (comp @ X351 @ X350) @ (sub @ X348 @ X350) @ X349)))) | ? [X352 : (subst > term > term)] : (! [X356 : term] : (($true = ((X342 @ id @ (X352 @ id @ X356) @ id))) | (((X342 @ id @ X356 @ id)) != $true)) & ($true != ((X342 @ id @ (lam @ (X352 @ sh @ one)) @ id))) & ! [X355 : subst,X353 : subst,X354 : term] : (((X352 @ (comp @ X355 @ X353) @ (sub @ X354 @ X353))) = ((sub @ (X352 @ X355 @ X354) @ X353))))) <=> (hoasinduction_lem3 = $true)) & (pushprop_lthm_orig = $true) & (hoasinduction_lem1_gthm = $true) & (! [X329 : (term > $o),X330 : (term > $o),X327 : term,X326 : term,X328 : subst] : (($true != ((X330 @ X327))) | ($true = ((X329 @ (sub @ X327 @ (push @ X326 @ X328)))))) <=> (pushprop_lem1 = $true)) & (hoasapnotvar = $true) & ((hoasinduction = $true) <=> ! [X397 : term,X398 : (subst > term > subst > $o)] : (? [X414 : subst,X411 : subst,X412 : term,X413 : subst] : (($true = ((X398 @ X411 @ X412 @ (comp @ X414 @ X413)))) & ($true != ((X398 @ (comp @ X411 @ X414) @ (sub @ X412 @ X414) @ X413)))) | ($true = ((X398 @ id @ X397 @ id))) | ? [X410 : term] : (($true = ((var @ (sub @ X410 @ id)))) & ($true != ((X398 @ id @ X410 @ id)))) | ? [X401 : subst,X399 : subst,X400 : subst,X402 : term] : (($true = ((X398 @ (comp @ X401 @ X399) @ (sub @ X402 @ X399) @ X400))) & ($true != ((X398 @ X401 @ X402 @ (comp @ X399 @ X400))))) | ? [X409 : term,X408 : term] : ((((X398 @ id @ X409 @ id)) = $true) & ($true != ((X398 @ id @ (ap @ (sub @ X408 @ id) @ X409) @ id))) & ($true = ((X398 @ id @ X408 @ id)))) | ? [X403 : (subst > term > term)] : ((((X398 @ id @ (lam @ (X403 @ sh @ one)) @ id)) != $true) & ! [X405 : subst,X406 : subst,X407 : term] : (((sub @ (X403 @ X406 @ X407) @ X405)) = ((X403 @ (comp @ X406 @ X405) @ (sub @ X407 @ X405)))) & ! [X404 : term] : (($true = ((X398 @ id @ (X403 @ id @ X404) @ id))) | (((X398 @ id @ X404 @ id)) != $true))))) & (hoasinduction_lem2_lthm = $true) & (hoaslamnotap_lthm = $true) & ((? [X192 : subst,X193 : term] : (((sub @ one @ (push @ X193 @ X192))) != X193) | ! [X198 : term,X197 : subst,X199 : term,X200 : (term > $o),X196 : (term > $o)] : (($true != ((X196 @ X199))) | (((X200 @ (sub @ X199 @ (push @ X198 @ X197)))) = $true)) | ? [X194 : subst,X195 : term] : (((comp @ sh @ (push @ X195 @ X194))) != X194)) <=> (pushprop_lem1_lthm = $true)) & (hoaslamnotvar_gthm = $true) & ! [X31 : d_subst] : (d_id = X31) & (hoasapnotvar_lthm = $true) & (hoasinduction_lem2v2_gthm = $true) & (! [X86 : subst,X85 : term,X84 : (term > $o)] : ((((X84 @ (sub @ X85 @ X86))) = $true) | ? [X90 : term] : (($true != ((X84 @ (lam @ X90)))) & ! [X91 : term] : (($true != ((X84 @ X91))) | ($true = ((X84 @ (sub @ X90 @ (push @ X91 @ id))))))) | ? [X89 : term] : (($true != ((X84 @ (sub @ X89 @ X86)))) & ($true = ((var @ X89)))) | ? [X87 : term,X88 : term] : (($true = ((X84 @ X87))) & ($true != ((X84 @ (ap @ X88 @ X87)))) & ($true = ((X84 @ X88))))) <=> (induction2lem = $true)) & (apnotvar = $true) & (ulamvarind = $true) & (pushprop_lem2v2 = $true) & (lamnotap != $true) & ((hoasinduction_lthm = $true) <=> (? [X256 : (subst > term > subst > $o),X257 : term] : (! [X258 : (subst > term > term)] : (? [X262 : term] : (($true != ((X256 @ id @ (X258 @ id @ X262) @ id))) & ($true = ((X256 @ id @ X262 @ id)))) | ? [X260 : term,X261 : subst,X259 : subst] : (((X258 @ (comp @ X259 @ X261) @ (sub @ X260 @ X261))) != ((sub @ (X258 @ X259 @ X260) @ X261))) | ($true = ((X256 @ id @ (lam @ (X258 @ sh @ one)) @ id)))) & ! [X271 : term] : (($true = ((X256 @ id @ (sub @ X257 @ (push @ X271 @ id)) @ id))) | ($true != ((X256 @ id @ X271 @ id)))) & ! [X267 : subst,X268 : term,X269 : subst,X270 : subst] : (($true = ((X256 @ (comp @ X267 @ X269) @ (sub @ X268 @ X269) @ X270))) | ($true != ((X256 @ X267 @ X268 @ (comp @ X269 @ X270))))) & ! [X264 : subst,X266 : subst,X263 : subst,X265 : term] : (($true = ((X256 @ X263 @ X265 @ (comp @ X264 @ X266)))) | ($true != ((X256 @ (comp @ X263 @ X264) @ (sub @ X265 @ X264) @ X266)))) & ($true != ((X256 @ id @ (lam @ X257) @ id)))) | ? [X236 : term,X237 : (term > $o)] : (! [X240 : term,X239 : term] : (($true != ((X237 @ X240))) | ($true = ((X237 @ (ap @ X240 @ X239)))) | ($true != ((X237 @ X239)))) & ($true != ((X237 @ X236))) & ! [X241 : term] : (($true = ((X237 @ (lam @ X241)))) | ? [X242 : term] : (($true != ((X237 @ (sub @ X241 @ (push @ X242 @ id))))) & ($true = ((X237 @ X242))))) & ! [X238 : term] : (($true = ((X237 @ X238))) | ($true != ((var @ X238))))) | ? [X244 : term,X243 : (subst > term > subst > $o),X245 : term] : (($true = ((X243 @ id @ X244 @ id))) & ! [X247 : term,X246 : term] : (($true != ((X243 @ id @ X247 @ id))) | ($true = ((X243 @ id @ (ap @ (sub @ X246 @ id) @ X247) @ id))) | ($true != ((X243 @ id @ X246 @ id)))) & ($true = ((X243 @ id @ X245 @ id))) & ! [X251 : subst,X250 : subst,X249 : subst,X248 : term] : (($true = ((X243 @ X249 @ X248 @ (comp @ X251 @ X250)))) | ($true != ((X243 @ (comp @ X249 @ X251) @ (sub @ X248 @ X251) @ X250)))) & ($true != ((X243 @ id @ (ap @ X244 @ X245) @ id))) & ! [X254 : subst,X252 : subst,X253 : subst,X255 : term] : (($true = ((X243 @ (comp @ X254 @ X252) @ (sub @ X255 @ X252) @ X253))) | ($true != ((X243 @ X254 @ X255 @ (comp @ X252 @ X253)))))) | ! [X272 : term,X273 : (subst > term > subst > $o)] : (? [X286 : term,X287 : subst,X284 : subst,X285 : subst] : (($true != ((X273 @ (comp @ X284 @ X285) @ (sub @ X286 @ X285) @ X287))) & ($true = ((X273 @ X284 @ X286 @ (comp @ X285 @ X287))))) | ($true = ((X273 @ id @ X272 @ id))) | ? [X274 : (subst > term > term)] : (($true != ((X273 @ id @ (lam @ (X274 @ sh @ one)) @ id))) & ! [X276 : subst,X277 : term,X278 : subst] : (((X274 @ (comp @ X276 @ X278) @ (sub @ X277 @ X278))) = ((sub @ (X274 @ X276 @ X277) @ X278))) & ! [X275 : term] : (($true = ((X273 @ id @ (X274 @ id @ X275) @ id))) | ($true != ((X273 @ id @ X275 @ id))))) | ? [X283 : term] : (($true = ((var @ (sub @ X283 @ id)))) & ($true != ((X273 @ id @ X283 @ id)))) | ? [X288 : term,X289 : term] : (($true = ((X273 @ id @ X288 @ id))) & ($true != ((X273 @ id @ (ap @ (sub @ X289 @ id) @ X288) @ id))) & (((X273 @ id @ X289 @ id)) = $true)) | ? [X281 : subst,X280 : subst,X282 : subst,X279 : term] : (($true = ((X273 @ (comp @ X282 @ X280) @ (sub @ X279 @ X280) @ X281))) & ($true != ((X273 @ X282 @ X279 @ (comp @ X280 @ X281)))))))) & ((hoasinduction_lem3aa = $true) <=> ! [X133 : term,X132 : (subst > term > subst > $o)] : (? [X139 : term] : (($true = ((X132 @ id @ X139 @ id))) & ($true != ((X132 @ id @ (sub @ X133 @ (push @ X139 @ id)) @ id)))) | ? [X134 : (subst > term > term)] : (! [X135 : term] : (($true = ((X132 @ id @ (X134 @ id @ X135) @ id))) | ($true != ((X132 @ id @ X135 @ id)))) & ($true != ((X132 @ id @ (lam @ (X134 @ sh @ one)) @ id))) & ! [X138 : subst,X137 : subst,X136 : term] : (((sub @ (X134 @ X138 @ X136) @ X137)) = ((X134 @ (comp @ X138 @ X137) @ (sub @ X136 @ X137))))) | ($true = ((X132 @ id @ (lam @ (sub @ X133 @ (push @ one @ sh))) @ id))))) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (! [X510 : term] : (((sub @ X510 @ id)) = X510) <=> (termmset = $true)) & (! [X525 : term,X526 : (subst > term > subst > $o)] : (? [X528 : (subst > term > term)] : (! [X532 : term] : (($true = ((X526 @ id @ (X528 @ id @ X532) @ id))) | ($true != ((X526 @ id @ X532 @ id)))) & ($true != ((X526 @ id @ (lam @ (X528 @ sh @ one)) @ id))) & ! [X529 : term,X531 : subst,X530 : subst] : (((sub @ (X528 @ X530 @ X529) @ X531)) = ((X528 @ (comp @ X530 @ X531) @ (sub @ X529 @ X531))))) | ($true = ((X526 @ id @ (lam @ (sub @ X525 @ (push @ one @ sh))) @ id))) | ? [X527 : term] : (($true = ((X526 @ id @ X527 @ id))) & ($true != ((X526 @ id @ (sub @ X525 @ (push @ X527 @ id)) @ id))))) <=> (hoasinduction_lem3aa_lthm = $true)) & ((? [X331 : subst] : (((comp @ id @ X331)) != X331) | ? [X332 : subst] : (((comp @ X332 @ id)) != X332) | (! [X333 : subst] : (((comp @ id @ X333)) = X333) & ! [X334 : subst] : (((comp @ X334 @ id)) = X334))) <=> (substmonoid_lthm = $true)) & ((? [X38 : (subst > term > subst > $o),X39 : term] : (! [X40 : (subst > term > term)] : (? [X42 : subst,X43 : subst,X41 : term] : (((X40 @ (comp @ X42 @ X43) @ (sub @ X41 @ X43))) != ((sub @ (X40 @ X42 @ X41) @ X43))) | ? [X44 : term] : ((((X38 @ id @ (X40 @ id @ X44) @ id)) != $true) & ($true = ((X38 @ id @ X44 @ id)))) | ($true = ((X38 @ id @ (lam @ (X40 @ sh @ one)) @ id)))) & ! [X45 : term] : (($true != ((X38 @ id @ X45 @ id))) | ($true = ((X38 @ id @ (sub @ X39 @ (push @ X45 @ id)) @ id)))) & ($true != ((X38 @ id @ (lam @ (sub @ X39 @ (push @ one @ sh))) @ id)))) | ! [X46 : term,X47 : (subst > term > subst > $o)] : (? [X48 : (subst > term > term)] : (! [X50 : subst,X51 : term,X52 : subst] : (((sub @ (X48 @ X50 @ X51) @ X52)) = ((X48 @ (comp @ X50 @ X52) @ (sub @ X51 @ X52)))) & ! [X49 : term] : (($true != ((X47 @ id @ X49 @ id))) | (((X47 @ id @ (X48 @ id @ X49) @ id)) = $true)) & ($true != ((X47 @ id @ (lam @ (X48 @ sh @ one)) @ id)))) | ? [X53 : term] : (($true != ((X47 @ id @ (sub @ X46 @ (push @ X53 @ id)) @ id))) & ($true = ((X47 @ id @ X53 @ id)))) | (((X47 @ id @ (lam @ X46) @ id)) = $true)) | ? [X37 : term] : (((sub @ X37 @ id)) != X37)) <=> (hoasinduction_lem3a_lthm = $true)) & ((hoasinduction_lem2 = $true) <=> ! [X18 : term,X19 : term,X20 : (subst > term > subst > $o)] : (($true != ((X20 @ id @ X19 @ id))) | ? [X21 : subst,X22 : term,X23 : subst,X24 : subst] : (($true = ((X20 @ (comp @ X21 @ X24) @ (sub @ X22 @ X24) @ X23))) & ($true != ((X20 @ X21 @ X22 @ (comp @ X24 @ X23))))) | ($true = ((X20 @ id @ (ap @ X19 @ X18) @ id))) | (((X20 @ id @ X18 @ id)) != $true) | ? [X30 : term,X29 : term] : (($true = ((X20 @ id @ X30 @ id))) & (((X20 @ id @ X29 @ id)) = $true) & ($true != ((X20 @ id @ (ap @ (sub @ X29 @ id) @ X30) @ id)))) | ? [X27 : subst,X28 : subst,X26 : subst,X25 : term] : (($true != ((X20 @ (comp @ X28 @ X26) @ (sub @ X25 @ X26) @ X27))) & ($true = ((X20 @ X28 @ X25 @ (comp @ X26 @ X27))))))) & (! [X387 : term,X386 : (subst > term > subst > $o)] : (? [X393 : term] : (($true != ((X386 @ id @ (sub @ X387 @ (push @ X393 @ id)) @ id))) & ($true = ((X386 @ id @ X393 @ id)))) | (((X386 @ id @ (lam @ X387) @ id)) = $true) | ? [X388 : (subst > term > term)] : (! [X389 : subst,X390 : subst,X391 : term] : (((X388 @ (comp @ X390 @ X389) @ (sub @ X391 @ X389))) = ((sub @ (X388 @ X390 @ X391) @ X389))) & ($true != ((X386 @ id @ (lam @ (X388 @ sh @ one)) @ id))) & ! [X392 : term] : (($true = ((X386 @ id @ (X388 @ id @ X392) @ id))) | ($true != ((X386 @ id @ X392 @ id)))))) <=> (hoasinduction_lem3a = $true)) & (((d2term @ d_one)) = one) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (hoasinduction_lem3a_gthm = $true) & (hoasapinj2_gthm = $true) & (pushprop_gthm = $true) & ((pushprop_lem0_lthm = $true) <=> ! [X230 : subst,X231 : (term > $o),X229 : term] : ? [X232 : (term > $o)] : ! [X233 : term] : (((X232 @ X233)) = ((X231 @ (sub @ X233 @ (push @ X229 @ X230)))))) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (! [X363 : term,X362 : subst] : (((comp @ sh @ (push @ X363 @ X362))) = X362) <=> (axshiftcons = $true)) & ! [X360 : subst] : ? [X361 : d_subst] : (((d2subst @ X361)) = X360) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X160 : term] : ? [X161 : (subst > term > term)] : ! [X163 : term,X162 : subst] : (((sub @ X160 @ (push @ X163 @ X162))) = ((X161 @ X162 @ X163)))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (termmset_gthm = $true) & (axclos = $true) & (hoasinduction_lem1 = $true) & ((hoasinduction_lem3v2 = $true) <=> ! [X492 : term,X493 : (subst > term > subst > $o),X494 : (term > $o)] : (($true = ((X494 @ (lam @ X492)))) | ? [X496 : (subst > term > term)] : (($true != ((X493 @ id @ (lam @ (X496 @ sh @ one)) @ id))) & ! [X498 : subst,X500 : subst,X499 : term] : (((sub @ (X496 @ X498 @ X499) @ X500)) = ((X496 @ (comp @ X498 @ X500) @ (sub @ X499 @ X500)))) & ! [X497 : term] : (($true = ((X493 @ id @ (X496 @ id @ X497) @ id))) | ($true != ((X493 @ id @ X497 @ id))))) | ? [X505 : term] : (($true = ((X494 @ X505))) & ($true != ((X494 @ (sub @ X492 @ (push @ X505 @ id)))))) | ? [X502 : subst,X504 : subst,X501 : term,X503 : subst] : (($true != ((X493 @ (comp @ X504 @ X503) @ (sub @ X501 @ X503) @ X502))) & ($true = ((X493 @ X504 @ X501 @ (comp @ X503 @ X502))))) | ? [X506 : subst,X508 : subst,X509 : subst,X507 : term] : (($true = ((X493 @ (comp @ X509 @ X506) @ (sub @ X507 @ X506) @ X508))) & ($true != ((X493 @ X509 @ X507 @ (comp @ X506 @ X508))))) | ? [X495 : term] : (((X494 @ X495)) != ((X493 @ id @ X495 @ id))))) & ((hoasinduction_lem3v2_lthm = $true) <=> (! [X417 : term,X418 : (term > $o),X416 : (subst > term > subst > $o)] : (($true = ((X418 @ (lam @ X417)))) | ? [X428 : (subst > term > term)] : (($true != ((X416 @ id @ (lam @ (X428 @ sh @ one)) @ id))) & ! [X432 : term] : (($true != ((X416 @ id @ X432 @ id))) | ($true = ((X416 @ id @ (X428 @ id @ X432) @ id)))) & ! [X430 : subst,X429 : term,X431 : subst] : (((sub @ (X428 @ X431 @ X429) @ X430)) = ((X428 @ (comp @ X431 @ X430) @ (sub @ X429 @ X430))))) | ? [X433 : term] : (((X418 @ X433)) != ((X416 @ id @ X433 @ id))) | ? [X423 : term] : (($true = ((X418 @ X423))) & ($true != ((X418 @ (sub @ X417 @ (push @ X423 @ id)))))) | ? [X421 : subst,X419 : subst,X422 : subst,X420 : term] : (($true = ((X416 @ X422 @ X420 @ (comp @ X419 @ X421)))) & ($true != ((X416 @ (comp @ X422 @ X419) @ (sub @ X420 @ X419) @ X421)))) | ? [X424 : subst,X425 : subst,X427 : term,X426 : subst] : (($true != ((X416 @ X424 @ X427 @ (comp @ X426 @ X425)))) & ($true = ((X416 @ (comp @ X424 @ X426) @ (sub @ X427 @ X426) @ X425))))) | ? [X415 : term] : (((sub @ X415 @ id)) != X415))) & (pushprop_lem2v2_gthm = $true) & (! [X220 : term,X221 : (subst > term > subst > $o)] : ((((X221 @ id @ X220 @ id)) = $true) | ? [X228 : term,X227 : term] : (($true = ((X221 @ id @ X227 @ id))) & ($true != ((X221 @ id @ (ap @ (sub @ X228 @ id) @ X227) @ id))) & ($true = ((X221 @ id @ X228 @ id)))) | ? [X222 : (subst > term > term)] : (($true != ((X221 @ id @ (lam @ (X222 @ sh @ one)) @ id))) & ! [X225 : subst,X224 : subst,X223 : term] : (((sub @ (X222 @ X224 @ X223) @ X225)) = ((X222 @ (comp @ X224 @ X225) @ (sub @ X223 @ X225)))) & ! [X226 : term] : (($true != ((X221 @ id @ X226 @ id))) | ($true = ((X221 @ id @ (X222 @ id @ X226) @ id)))))) <=> (hoasinduction_no_psi_cond = $true)) & (! [X368 : term] : ? [X369 : (subst > term > term)] : (((X369 @ sh @ one)) = ((sub @ X368 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b = $true)) & ((induction2lem_lthm = $true) <=> (? [X205 : subst] : (((comp @ X205 @ id)) != X205) | ? [X204 : subst,X203 : term] : (((comp @ sh @ (push @ X203 @ X204))) != X204) | ! [X213 : subst,X214 : term,X212 : (term > $o)] : (? [X219 : term] : (($true = ((var @ X219))) & ($true != ((X212 @ (sub @ X219 @ X213))))) | ? [X217 : term,X218 : term] : (($true = ((X212 @ X218))) & ($true = ((X212 @ X217))) & ($true != ((X212 @ (ap @ X218 @ X217))))) | ($true = ((X212 @ (sub @ X214 @ X213)))) | ? [X215 : term] : (! [X216 : term] : (($true != ((X212 @ X216))) | (((X212 @ (sub @ X215 @ (push @ X216 @ id)))) = $true)) & ($true != ((X212 @ (lam @ X215)))))) | ? [X202 : subst,X201 : term] : (((sub @ one @ (push @ X201 @ X202))) != X201) | ? [X206 : (term > $o),X207 : term] : (! [X209 : term] : (($true = ((X206 @ X209))) | ($true != ((var @ X209)))) & ! [X208 : term] : (($true != ((X206 @ X208))) | ($true = ((X206 @ (lam @ X208))))) & ($true != ((X206 @ X207))) & ! [X210 : term,X211 : term] : (($true != ((X206 @ X211))) | ($true != ((X206 @ X210))) | ($true = ((X206 @ (ap @ X210 @ X211)))))))) & ((hoasinduction_lem3v2a = $true) <=> ! [X434 : (subst > term > subst > $o),X435 : term,X436 : (term > $o)] : (($true = ((X436 @ (lam @ X435)))) | ? [X443 : term] : (($true != ((X436 @ (sub @ X435 @ (push @ X443 @ id))))) & ($true = ((X436 @ X443)))) | ? [X437 : term] : (((X436 @ X437)) != ((X434 @ id @ X437 @ id))) | ? [X438 : (subst > term > term)] : (! [X440 : subst,X441 : term,X439 : subst] : (((sub @ (X438 @ X439 @ X441) @ X440)) = ((X438 @ (comp @ X439 @ X440) @ (sub @ X441 @ X440)))) & ($true != ((X434 @ id @ (lam @ (X438 @ sh @ one)) @ id))) & ! [X442 : term] : ((((X434 @ id @ (X438 @ id @ X442) @ id)) = $true) | ($true != ((X434 @ id @ X442 @ id))))))) & (induction2_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & (! [X168 : term,X166 : term,X167 : term,X169 : term] : ((((ap @ X168 @ X167)) != ((ap @ X166 @ X169))) | (X166 = X168)) <=> (apinj1 = $true)) & (pushprop_lem3v2_lthm = $true) & (hoasapnotvar_gthm = $true)), inference(flattening,[],[f11])). thf(f13,definition,( ($true = sP0) <=> ! [X434 : (subst > term > subst > $o),X435 : term,X436 : (term > $o)] : (($true = ((X436 @ (lam @ X435)))) | ? [X443 : term] : (($true != ((X436 @ (sub @ X435 @ (push @ X443 @ id))))) & ($true = ((X436 @ X443)))) | ? [X437 : term] : (((X436 @ X437)) != ((X434 @ id @ X437 @ id))) | ? [X438 : (subst > term > term)] : (! [X440 : subst,X441 : term,X439 : subst] : (((sub @ (X438 @ X439 @ X441) @ X440)) = ((X438 @ (comp @ X439 @ X440) @ (sub @ X441 @ X440)))) & ($true != ((X434 @ id @ (lam @ (X438 @ sh @ one)) @ id))) & ! [X442 : term] : ((((X434 @ id @ (X438 @ id @ X442) @ id)) = $true) | ($true != ((X434 @ id @ X442 @ id))))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f14,definition,( ! [X206 : (term > $o)] : (($true = ((sP1 @ X206))) <=> ! [X210 : term,X211 : term] : (($true != ((X206 @ X211))) | ($true != ((X206 @ X210))) | ($true = ((X206 @ (ap @ X210 @ X211))))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f15,definition,( ! [X212 : (term > $o)] : (($true = ((sP2 @ X212))) <=> ? [X217 : term,X218 : term] : (($true = ((X212 @ X218))) & ($true = ((X212 @ X217))) & ($true != ((X212 @ (ap @ X218 @ X217))))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f16,definition,( ($true = sP3) <=> ! [X213 : subst,X214 : term,X212 : (term > $o)] : (? [X219 : term] : (($true = ((var @ X219))) & ($true != ((X212 @ (sub @ X219 @ X213))))) | ($true = ((sP2 @ X212))) | ($true = ((X212 @ (sub @ X214 @ X213)))) | ? [X215 : term] : (! [X216 : term] : (($true != ((X212 @ X216))) | (((X212 @ (sub @ X215 @ (push @ X216 @ id)))) = $true)) & ($true != ((X212 @ (lam @ X215))))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f17,definition,( ($true = sP4) <=> (? [X205 : subst] : (((comp @ X205 @ id)) != X205) | ? [X204 : subst,X203 : term] : (((comp @ sh @ (push @ X203 @ X204))) != X204) | ($true = sP3) | ? [X202 : subst,X201 : term] : (((sub @ one @ (push @ X201 @ X202))) != X201) | ? [X206 : (term > $o),X207 : term] : (! [X209 : term] : (($true = ((X206 @ X209))) | ($true != ((var @ X209)))) & ! [X208 : term] : (($true != ((X206 @ X208))) | ($true = ((X206 @ (lam @ X208))))) & ($true != ((X206 @ X207))) & ($true = ((sP1 @ X206)))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f18,definition,( ! [X221 : (subst > term > subst > $o)] : (($true = ((sP5 @ X221))) <=> ? [X222 : (subst > term > term)] : (($true != ((X221 @ id @ (lam @ (X222 @ sh @ one)) @ id))) & ! [X225 : subst,X224 : subst,X223 : term] : (((sub @ (X222 @ X224 @ X223) @ X225)) = ((X222 @ (comp @ X224 @ X225) @ (sub @ X223 @ X225)))) & ! [X226 : term] : (($true != ((X221 @ id @ X226 @ id))) | ($true = ((X221 @ id @ (X222 @ id @ X226) @ id))))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f19,definition,( ! [X416 : (subst > term > subst > $o)] : (($true = ((sP6 @ X416))) <=> ? [X428 : (subst > term > term)] : (($true != ((X416 @ id @ (lam @ (X428 @ sh @ one)) @ id))) & ! [X432 : term] : (($true != ((X416 @ id @ X432 @ id))) | ($true = ((X416 @ id @ (X428 @ id @ X432) @ id)))) & ! [X430 : subst,X429 : term,X431 : subst] : (((sub @ (X428 @ X431 @ X429) @ X430)) = ((X428 @ (comp @ X431 @ X430) @ (sub @ X429 @ X430))))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f20,definition,( (sP7 = $true) <=> (! [X417 : term,X418 : (term > $o),X416 : (subst > term > subst > $o)] : (($true = ((X418 @ (lam @ X417)))) | ($true = ((sP6 @ X416))) | ? [X433 : term] : (((X418 @ X433)) != ((X416 @ id @ X433 @ id))) | ? [X423 : term] : (($true = ((X418 @ X423))) & ($true != ((X418 @ (sub @ X417 @ (push @ X423 @ id)))))) | ? [X421 : subst,X419 : subst,X422 : subst,X420 : term] : (($true = ((X416 @ X422 @ X420 @ (comp @ X419 @ X421)))) & ($true != ((X416 @ (comp @ X422 @ X419) @ (sub @ X420 @ X419) @ X421)))) | ? [X424 : subst,X425 : subst,X427 : term,X426 : subst] : (($true != ((X416 @ X424 @ X427 @ (comp @ X426 @ X425)))) & ($true = ((X416 @ (comp @ X424 @ X426) @ (sub @ X427 @ X426) @ X425))))) | ? [X415 : term] : (((sub @ X415 @ id)) != X415))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f21,definition,( ! [X493 : (subst > term > subst > $o)] : (($true = ((sP8 @ X493))) <=> ? [X496 : (subst > term > term)] : (($true != ((X493 @ id @ (lam @ (X496 @ sh @ one)) @ id))) & ! [X498 : subst,X500 : subst,X499 : term] : (((sub @ (X496 @ X498 @ X499) @ X500)) = ((X496 @ (comp @ X498 @ X500) @ (sub @ X499 @ X500)))) & ! [X497 : term] : (($true = ((X493 @ id @ (X496 @ id @ X497) @ id))) | ($true != ((X493 @ id @ X497 @ id))))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f22,definition,( ($true = sP9) <=> ! [X492 : term,X493 : (subst > term > subst > $o),X494 : (term > $o)] : (($true = ((X494 @ (lam @ X492)))) | ($true = ((sP8 @ X493))) | ? [X505 : term] : (($true = ((X494 @ X505))) & ($true != ((X494 @ (sub @ X492 @ (push @ X505 @ id)))))) | ? [X502 : subst,X504 : subst,X501 : term,X503 : subst] : (($true != ((X493 @ (comp @ X504 @ X503) @ (sub @ X501 @ X503) @ X502))) & ($true = ((X493 @ X504 @ X501 @ (comp @ X503 @ X502))))) | ? [X506 : subst,X508 : subst,X509 : subst,X507 : term] : (($true = ((X493 @ (comp @ X509 @ X506) @ (sub @ X507 @ X506) @ X508))) & ($true != ((X493 @ X509 @ X507 @ (comp @ X506 @ X508))))) | ? [X495 : term] : (((X494 @ X495)) != ((X493 @ id @ X495 @ id))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f23,definition,( ($true = sP10) <=> ! [X387 : term,X386 : (subst > term > subst > $o)] : (? [X393 : term] : (($true != ((X386 @ id @ (sub @ X387 @ (push @ X393 @ id)) @ id))) & ($true = ((X386 @ id @ X393 @ id)))) | (((X386 @ id @ (lam @ X387) @ id)) = $true) | ? [X388 : (subst > term > term)] : (! [X389 : subst,X390 : subst,X391 : term] : (((X388 @ (comp @ X390 @ X389) @ (sub @ X391 @ X389))) = ((sub @ (X388 @ X390 @ X391) @ X389))) & ($true != ((X386 @ id @ (lam @ (X388 @ sh @ one)) @ id))) & ! [X392 : term] : (($true = ((X386 @ id @ (X388 @ id @ X392) @ id))) | ($true != ((X386 @ id @ X392 @ id))))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f24,definition,( ! [X20 : (subst > term > subst > $o)] : (($true = ((sP11 @ X20))) <=> ? [X30 : term,X29 : term] : (($true = ((X20 @ id @ X30 @ id))) & (((X20 @ id @ X29 @ id)) = $true) & ($true != ((X20 @ id @ (ap @ (sub @ X29 @ id) @ X30) @ id)))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f25,definition,( ($true = sP12) <=> ! [X18 : term,X19 : term,X20 : (subst > term > subst > $o)] : (($true != ((X20 @ id @ X19 @ id))) | ? [X21 : subst,X22 : term,X23 : subst,X24 : subst] : (($true = ((X20 @ (comp @ X21 @ X24) @ (sub @ X22 @ X24) @ X23))) & ($true != ((X20 @ X21 @ X22 @ (comp @ X24 @ X23))))) | ($true = ((X20 @ id @ (ap @ X19 @ X18) @ id))) | (((X20 @ id @ X18 @ id)) != $true) | ($true = ((sP11 @ X20))) | ? [X27 : subst,X28 : subst,X26 : subst,X25 : term] : (($true != ((X20 @ (comp @ X28 @ X26) @ (sub @ X25 @ X26) @ X27))) & ($true = ((X20 @ X28 @ X25 @ (comp @ X26 @ X27))))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f26,definition,( ($true = sP13) <=> ? [X38 : (subst > term > subst > $o),X39 : term] : (! [X40 : (subst > term > term)] : (? [X42 : subst,X43 : subst,X41 : term] : (((X40 @ (comp @ X42 @ X43) @ (sub @ X41 @ X43))) != ((sub @ (X40 @ X42 @ X41) @ X43))) | ? [X44 : term] : ((((X38 @ id @ (X40 @ id @ X44) @ id)) != $true) & ($true = ((X38 @ id @ X44 @ id)))) | ($true = ((X38 @ id @ (lam @ (X40 @ sh @ one)) @ id)))) & ! [X45 : term] : (($true != ((X38 @ id @ X45 @ id))) | ($true = ((X38 @ id @ (sub @ X39 @ (push @ X45 @ id)) @ id)))) & ($true != ((X38 @ id @ (lam @ (sub @ X39 @ (push @ one @ sh))) @ id))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f27,definition,( ($true = sP14) <=> (($true = sP13) | ! [X46 : term,X47 : (subst > term > subst > $o)] : (? [X48 : (subst > term > term)] : (! [X50 : subst,X51 : term,X52 : subst] : (((sub @ (X48 @ X50 @ X51) @ X52)) = ((X48 @ (comp @ X50 @ X52) @ (sub @ X51 @ X52)))) & ! [X49 : term] : (($true != ((X47 @ id @ X49 @ id))) | (((X47 @ id @ (X48 @ id @ X49) @ id)) = $true)) & ($true != ((X47 @ id @ (lam @ (X48 @ sh @ one)) @ id)))) | ? [X53 : term] : (($true != ((X47 @ id @ (sub @ X46 @ (push @ X53 @ id)) @ id))) & ($true = ((X47 @ id @ X53 @ id)))) | (((X47 @ id @ (lam @ X46) @ id)) = $true)) | ? [X37 : term] : (((sub @ X37 @ id)) != X37))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f28,definition,( ($true = sP15) <=> ! [X525 : term,X526 : (subst > term > subst > $o)] : (? [X528 : (subst > term > term)] : (! [X532 : term] : (($true = ((X526 @ id @ (X528 @ id @ X532) @ id))) | ($true != ((X526 @ id @ X532 @ id)))) & ($true != ((X526 @ id @ (lam @ (X528 @ sh @ one)) @ id))) & ! [X529 : term,X531 : subst,X530 : subst] : (((sub @ (X528 @ X530 @ X529) @ X531)) = ((X528 @ (comp @ X530 @ X531) @ (sub @ X529 @ X531))))) | ($true = ((X526 @ id @ (lam @ (sub @ X525 @ (push @ one @ sh))) @ id))) | ? [X527 : term] : (($true = ((X526 @ id @ X527 @ id))) & ($true != ((X526 @ id @ (sub @ X525 @ (push @ X527 @ id)) @ id)))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f29,definition,( ($true = sP16) <=> ! [X133 : term,X132 : (subst > term > subst > $o)] : (? [X139 : term] : (($true = ((X132 @ id @ X139 @ id))) & ($true != ((X132 @ id @ (sub @ X133 @ (push @ X139 @ id)) @ id)))) | ? [X134 : (subst > term > term)] : (! [X135 : term] : (($true = ((X132 @ id @ (X134 @ id @ X135) @ id))) | ($true != ((X132 @ id @ X135 @ id)))) & ($true != ((X132 @ id @ (lam @ (X134 @ sh @ one)) @ id))) & ! [X138 : subst,X137 : subst,X136 : term] : (((sub @ (X134 @ X138 @ X136) @ X137)) = ((X134 @ (comp @ X138 @ X137) @ (sub @ X136 @ X137))))) | ($true = ((X132 @ id @ (lam @ (sub @ X133 @ (push @ one @ sh))) @ id))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f30,definition,( ! [X273 : (subst > term > subst > $o)] : ((((sP17 @ X273)) = $true) <=> ? [X288 : term,X289 : term] : (($true = ((X273 @ id @ X288 @ id))) & ($true != ((X273 @ id @ (ap @ (sub @ X289 @ id) @ X288) @ id))) & (((X273 @ id @ X289 @ id)) = $true)))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f31,definition,( ! [X273 : (subst > term > subst > $o)] : (($true = ((sP18 @ X273))) <=> ? [X274 : (subst > term > term)] : (($true != ((X273 @ id @ (lam @ (X274 @ sh @ one)) @ id))) & ! [X276 : subst,X277 : term,X278 : subst] : (((X274 @ (comp @ X276 @ X278) @ (sub @ X277 @ X278))) = ((sub @ (X274 @ X276 @ X277) @ X278))) & ! [X275 : term] : (($true = ((X273 @ id @ (X274 @ id @ X275) @ id))) | ($true != ((X273 @ id @ X275 @ id))))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f32,definition,( ! [X243 : (subst > term > subst > $o)] : ((((sP19 @ X243)) = $true) <=> ! [X247 : term,X246 : term] : (($true != ((X243 @ id @ X247 @ id))) | ($true = ((X243 @ id @ (ap @ (sub @ X246 @ id) @ X247) @ id))) | ($true != ((X243 @ id @ X246 @ id)))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f33,definition,( ! [X237 : (term > $o)] : (($true = ((sP20 @ X237))) <=> ! [X240 : term,X239 : term] : (($true != ((X237 @ X240))) | ($true = ((X237 @ (ap @ X240 @ X239)))) | ($true != ((X237 @ X239)))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f34,definition,( ! [X256 : (subst > term > subst > $o)] : (($true = ((sP21 @ X256))) <=> ! [X258 : (subst > term > term)] : (? [X262 : term] : (($true != ((X256 @ id @ (X258 @ id @ X262) @ id))) & ($true = ((X256 @ id @ X262 @ id)))) | ? [X260 : term,X261 : subst,X259 : subst] : (((X258 @ (comp @ X259 @ X261) @ (sub @ X260 @ X261))) != ((sub @ (X258 @ X259 @ X260) @ X261))) | ($true = ((X256 @ id @ (lam @ (X258 @ sh @ one)) @ id)))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f35,definition,( (sP22 = $true) <=> ? [X256 : (subst > term > subst > $o),X257 : term] : (($true = ((sP21 @ X256))) & ! [X271 : term] : (($true = ((X256 @ id @ (sub @ X257 @ (push @ X271 @ id)) @ id))) | ($true != ((X256 @ id @ X271 @ id)))) & ! [X267 : subst,X268 : term,X269 : subst,X270 : subst] : (($true = ((X256 @ (comp @ X267 @ X269) @ (sub @ X268 @ X269) @ X270))) | ($true != ((X256 @ X267 @ X268 @ (comp @ X269 @ X270))))) & ! [X264 : subst,X266 : subst,X263 : subst,X265 : term] : (($true = ((X256 @ X263 @ X265 @ (comp @ X264 @ X266)))) | ($true != ((X256 @ (comp @ X263 @ X264) @ (sub @ X265 @ X264) @ X266)))) & ($true != ((X256 @ id @ (lam @ X257) @ id))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f36,definition,( ($true = sP23) <=> ! [X272 : term,X273 : (subst > term > subst > $o)] : (? [X286 : term,X287 : subst,X284 : subst,X285 : subst] : (($true != ((X273 @ (comp @ X284 @ X285) @ (sub @ X286 @ X285) @ X287))) & ($true = ((X273 @ X284 @ X286 @ (comp @ X285 @ X287))))) | ($true = ((X273 @ id @ X272 @ id))) | ($true = ((sP18 @ X273))) | ? [X283 : term] : (($true = ((var @ (sub @ X283 @ id)))) & ($true != ((X273 @ id @ X283 @ id)))) | (((sP17 @ X273)) = $true) | ? [X281 : subst,X280 : subst,X282 : subst,X279 : term] : (($true = ((X273 @ (comp @ X282 @ X280) @ (sub @ X279 @ X280) @ X281))) & ($true != ((X273 @ X282 @ X279 @ (comp @ X280 @ X281))))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f37,definition,( ($true = sP24) <=> ? [X244 : term,X243 : (subst > term > subst > $o),X245 : term] : (($true = ((X243 @ id @ X244 @ id))) & (((sP19 @ X243)) = $true) & ($true = ((X243 @ id @ X245 @ id))) & ! [X251 : subst,X250 : subst,X249 : subst,X248 : term] : (($true = ((X243 @ X249 @ X248 @ (comp @ X251 @ X250)))) | ($true != ((X243 @ (comp @ X249 @ X251) @ (sub @ X248 @ X251) @ X250)))) & ($true != ((X243 @ id @ (ap @ X244 @ X245) @ id))) & ! [X254 : subst,X252 : subst,X253 : subst,X255 : term] : (($true = ((X243 @ (comp @ X254 @ X252) @ (sub @ X255 @ X252) @ X253))) | ($true != ((X243 @ X254 @ X255 @ (comp @ X252 @ X253))))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f38,definition,( ($true = sP25) <=> ((sP22 = $true) | ? [X236 : term,X237 : (term > $o)] : (($true = ((sP20 @ X237))) & ($true != ((X237 @ X236))) & ! [X241 : term] : (($true = ((X237 @ (lam @ X241)))) | ? [X242 : term] : (($true != ((X237 @ (sub @ X241 @ (push @ X242 @ id))))) & ($true = ((X237 @ X242))))) & ! [X238 : term] : (($true = ((X237 @ X238))) | ($true != ((var @ X238))))) | ($true = sP24) | ($true = sP23))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f39,definition,( ! [X84 : (term > $o)] : (($true = ((sP26 @ X84))) <=> ? [X87 : term,X88 : term] : (($true = ((X84 @ X87))) & ($true != ((X84 @ (ap @ X88 @ X87)))) & ($true = ((X84 @ X88)))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f40,definition,( ($true = sP27) <=> ! [X86 : subst,X85 : term,X84 : (term > $o)] : ((((X84 @ (sub @ X85 @ X86))) = $true) | ? [X90 : term] : (($true != ((X84 @ (lam @ X90)))) & ! [X91 : term] : (($true != ((X84 @ X91))) | ($true = ((X84 @ (sub @ X90 @ (push @ X91 @ id))))))) | ? [X89 : term] : (($true != ((X84 @ (sub @ X89 @ X86)))) & ($true = ((var @ X89)))) | ($true = ((sP26 @ X84))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f41,definition,( ! [X398 : (subst > term > subst > $o)] : (($true = ((sP28 @ X398))) <=> ? [X403 : (subst > term > term)] : ((((X398 @ id @ (lam @ (X403 @ sh @ one)) @ id)) != $true) & ! [X405 : subst,X406 : subst,X407 : term] : (((sub @ (X403 @ X406 @ X407) @ X405)) = ((X403 @ (comp @ X406 @ X405) @ (sub @ X407 @ X405)))) & ! [X404 : term] : (($true = ((X398 @ id @ (X403 @ id @ X404) @ id))) | (((X398 @ id @ X404 @ id)) != $true))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f42,definition,( ! [X398 : (subst > term > subst > $o)] : (($true = ((sP29 @ X398))) <=> ? [X409 : term,X408 : term] : ((((X398 @ id @ X409 @ id)) = $true) & ($true != ((X398 @ id @ (ap @ (sub @ X408 @ id) @ X409) @ id))) & ($true = ((X398 @ id @ X408 @ id)))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f43,definition,( ($true = sP30) <=> ! [X397 : term,X398 : (subst > term > subst > $o)] : (? [X414 : subst,X411 : subst,X412 : term,X413 : subst] : (($true = ((X398 @ X411 @ X412 @ (comp @ X414 @ X413)))) & ($true != ((X398 @ (comp @ X411 @ X414) @ (sub @ X412 @ X414) @ X413)))) | ($true = ((X398 @ id @ X397 @ id))) | ? [X410 : term] : (($true = ((var @ (sub @ X410 @ id)))) & ($true != ((X398 @ id @ X410 @ id)))) | ? [X401 : subst,X399 : subst,X400 : subst,X402 : term] : (($true = ((X398 @ (comp @ X401 @ X399) @ (sub @ X402 @ X399) @ X400))) & ($true != ((X398 @ X401 @ X402 @ (comp @ X399 @ X400))))) | ($true = ((sP29 @ X398))) | ($true = ((sP28 @ X398))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f44,definition,( ! [X342 : (subst > term > subst > $o)] : (($true = ((sP31 @ X342))) <=> ? [X352 : (subst > term > term)] : (! [X356 : term] : (($true = ((X342 @ id @ (X352 @ id @ X356) @ id))) | (((X342 @ id @ X356 @ id)) != $true)) & ($true != ((X342 @ id @ (lam @ (X352 @ sh @ one)) @ id))) & ! [X355 : subst,X353 : subst,X354 : term] : (((X352 @ (comp @ X355 @ X353) @ (sub @ X354 @ X353))) = ((sub @ (X352 @ X355 @ X354) @ X353)))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f45,definition,( ($true = sP32) <=> ! [X341 : term,X342 : (subst > term > subst > $o)] : (? [X347 : term] : ((((X342 @ id @ (sub @ X341 @ (push @ X347 @ id)) @ id)) != $true) & ($true = ((X342 @ id @ X347 @ id)))) | ? [X345 : subst,X344 : subst,X346 : term,X343 : subst] : (($true = ((X342 @ X343 @ X346 @ (comp @ X345 @ X344)))) & ($true != ((X342 @ (comp @ X343 @ X345) @ (sub @ X346 @ X345) @ X344)))) | ($true = ((X342 @ id @ (lam @ X341) @ id))) | ? [X348 : term,X349 : subst,X351 : subst,X350 : subst] : ((((X342 @ X351 @ X348 @ (comp @ X350 @ X349))) != $true) & ($true = ((X342 @ (comp @ X351 @ X350) @ (sub @ X348 @ X350) @ X349)))) | ($true = ((sP31 @ X342))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f46,definition,( ! [X308 : (subst > term > subst > $o)] : (($true = ((sP33 @ X308))) <=> ? [X318 : (subst > term > term)] : (! [X319 : term] : (($true != ((X308 @ id @ X319 @ id))) | ($true = ((X308 @ id @ (X318 @ id @ X319) @ id)))) & ! [X320 : subst,X321 : term,X322 : subst] : (((X318 @ (comp @ X320 @ X322) @ (sub @ X321 @ X322))) = ((sub @ (X318 @ X320 @ X321) @ X322))) & ($true != ((X308 @ id @ (lam @ (X318 @ sh @ one)) @ id)))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f47,definition,( ($true = sP34) <=> ? [X300 : (subst > term > subst > $o),X301 : term] : (! [X302 : (subst > term > term)] : (? [X306 : term] : (($true = ((X300 @ id @ X306 @ id))) & ($true != ((X300 @ id @ (X302 @ id @ X306) @ id)))) | ($true = ((X300 @ id @ (lam @ (X302 @ sh @ one)) @ id))) | ? [X305 : subst,X303 : term,X304 : subst] : (((sub @ (X302 @ X304 @ X303) @ X305)) != ((X302 @ (comp @ X304 @ X305) @ (sub @ X303 @ X305))))) & ! [X307 : term] : (($true = ((X300 @ id @ (sub @ X301 @ (push @ X307 @ id)) @ id))) | ($true != ((X300 @ id @ X307 @ id)))) & ($true != ((X300 @ id @ (lam @ (sub @ X301 @ (push @ one @ sh))) @ id))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f48,definition,( ($true = sP35) <=> (? [X299 : term] : (((sub @ X299 @ id)) != X299) | ! [X309 : term,X308 : (subst > term > subst > $o)] : (($true = ((X308 @ id @ (lam @ X309) @ id))) | ? [X323 : term] : (($true != ((X308 @ id @ (sub @ X309 @ (push @ X323 @ id)) @ id))) & ($true = ((X308 @ id @ X323 @ id)))) | ($true = ((sP33 @ X308))) | ? [X312 : term,X311 : subst,X310 : subst,X313 : subst] : (($true != ((X308 @ X310 @ X312 @ (comp @ X313 @ X311)))) & ($true = ((X308 @ (comp @ X310 @ X313) @ (sub @ X312 @ X313) @ X311)))) | ? [X314 : term,X315 : subst,X317 : subst,X316 : subst] : ((((X308 @ (comp @ X315 @ X317) @ (sub @ X314 @ X317) @ X316)) != $true) & (((X308 @ X315 @ X314 @ (comp @ X317 @ X316))) = $true))) | ($true = sP34))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f49,definition,( ! [X114 : (subst > term > subst > $o)] : (($true = ((sP36 @ X114))) <=> ? [X120 : term,X121 : term] : (($true = ((X114 @ id @ X121 @ id))) & (((X114 @ id @ (ap @ (sub @ X120 @ id) @ X121) @ id)) != $true) & ($true = ((X114 @ id @ X120 @ id)))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f50,definition,( ! [X114 : (subst > term > subst > $o)] : (($true = ((sP37 @ X114))) <=> ? [X126 : (subst > term > term)] : (! [X130 : term] : ((((X114 @ id @ (X126 @ id @ X130) @ id)) = $true) | (((X114 @ id @ X130 @ id)) != $true)) & ! [X127 : term,X129 : subst,X128 : subst] : (((X126 @ (comp @ X129 @ X128) @ (sub @ X127 @ X128))) = ((sub @ (X126 @ X129 @ X127) @ X128))) & ($true != ((X114 @ id @ (lam @ (X126 @ sh @ one)) @ id)))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f51,definition,( ! [X96 : (term > $o)] : (($true = ((sP38 @ X96))) <=> ! [X102 : term,X101 : term] : ((((X96 @ X102)) != $true) | (((X96 @ (ap @ X101 @ X102))) = $true) | (((X96 @ X101)) != $true)))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f52,definition,( ($true = sP39) <=> ! [X114 : (subst > term > subst > $o),X115 : term] : (? [X122 : term,X124 : subst,X125 : subst,X123 : subst] : (($true != ((X114 @ X124 @ X122 @ (comp @ X123 @ X125)))) & ($true = ((X114 @ (comp @ X124 @ X123) @ (sub @ X122 @ X123) @ X125)))) | ($true = ((sP37 @ X114))) | ? [X131 : term] : (($true != ((X114 @ id @ X131 @ id))) & ($true = ((var @ (sub @ X131 @ id))))) | ($true = ((X114 @ id @ X115 @ id))) | ($true = ((sP36 @ X114))) | ? [X117 : subst,X116 : term,X118 : subst,X119 : subst] : (($true != ((X114 @ (comp @ X118 @ X119) @ (sub @ X116 @ X119) @ X117))) & ($true = ((X114 @ X118 @ X116 @ (comp @ X119 @ X117))))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f53,definition,( ($true = sP40) <=> ? [X104 : term,X106 : (term > $o),X105 : (subst > term > subst > $o)] : (($true != ((X106 @ (lam @ X104)))) & ! [X107 : term] : (((X105 @ id @ X107 @ id)) = ((X106 @ X107))) & ! [X108 : (subst > term > term)] : (($true = ((X105 @ id @ (lam @ (X108 @ sh @ one)) @ id))) | ? [X111 : term,X110 : subst,X112 : subst] : (((X108 @ (comp @ X110 @ X112) @ (sub @ X111 @ X112))) != ((sub @ (X108 @ X110 @ X111) @ X112))) | ? [X109 : term] : (($true = ((X105 @ id @ X109 @ id))) & ($true != ((X105 @ id @ (X108 @ id @ X109) @ id))))) & ! [X113 : term] : (($true = ((X106 @ (sub @ X104 @ (push @ X113 @ id))))) | (((X106 @ X113)) != $true)))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f54,definition,( ($true = sP41) <=> (? [X93 : (subst > term > subst > $o)] : ! [X94 : (term > $o)] : ? [X95 : term] : (((X93 @ id @ X95 @ id)) != ((X94 @ X95))) | ($true = sP40) | ? [X103 : term] : (((sub @ X103 @ id)) != X103) | ? [X96 : (term > $o),X97 : term] : (($true = ((sP38 @ X96))) & ! [X98 : term] : (($true = ((X96 @ X98))) | ($true != ((var @ X98)))) & ! [X99 : term] : (? [X100 : term] : (($true = ((X96 @ X100))) & ($true != ((X96 @ (sub @ X99 @ (push @ X100 @ id)))))) | ($true = ((X96 @ (lam @ X99))))) & ($true != ((X96 @ X97)))) | ($true = sP39))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f55,definition,( ! [X187 : (term > $o)] : ((((sP42 @ X187)) = $true) <=> ? [X190 : term,X191 : term] : (($true = ((X187 @ X191))) & ($true = ((X187 @ X190))) & ($true != ((X187 @ (ap @ X191 @ X190))))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f56,definition,( ! [X171 : (term > $o)] : (($true = ((sP43 @ X171))) <=> ! [X178 : term,X177 : term] : (($true = ((X171 @ (ap @ X178 @ X177)))) | ($true != ((X171 @ X177))) | ($true != ((X171 @ X178)))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f57,definition,( ! [X180 : (term > $o)] : (($true = ((sP44 @ X180))) <=> ? [X181 : term,X182 : term] : (($true != ((X180 @ (ap @ X182 @ X181)))) & ($true = ((X180 @ X181))) & ($true = ((X180 @ X182)))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f58,definition,( ($true = sP45) <=> ! [X180 : (term > $o),X179 : term] : ((((X180 @ X179)) = $true) | ($true = ((sP44 @ X180))) | ? [X185 : term] : (($true = ((var @ X185))) & ($true != ((X180 @ X185)))) | ? [X183 : term] : (($true != ((X180 @ (lam @ X183)))) & ! [X184 : term] : (($true != ((X180 @ X184))) | (((X180 @ (sub @ X183 @ (push @ X184 @ id)))) = $true))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f59,definition,( ($true = sP46) <=> (($true = sP45) | ? [X170 : term] : (((sub @ X170 @ id)) != X170) | ? [X171 : (term > $o),X172 : term,X173 : subst] : (! [X174 : term] : (($true != ((var @ X174))) | ($true = ((X171 @ (sub @ X174 @ X173))))) & ($true != ((X171 @ (sub @ X172 @ X173)))) & ($true = ((sP43 @ X171))) & ! [X175 : term] : (? [X176 : term] : (($true = ((X171 @ X176))) & ($true != ((X171 @ (sub @ X175 @ (push @ X176 @ id)))))) | ($true = ((X171 @ (lam @ X175)))))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f60,definition,( ($true = sP47) <=> (? [X149 : term] : (((sub @ X149 @ id)) != X149) | ? [X145 : term] : ! [X146 : (subst > term > term)] : ? [X147 : term,X148 : subst] : (((sub @ X145 @ (push @ X147 @ X148))) != ((X146 @ X148 @ X147))) | ! [X152 : term,X150 : (term > $o),X151 : (subst > term > subst > $o)] : (? [X153 : (subst > term > term)] : (($true != ((X151 @ id @ (lam @ (X153 @ sh @ one)) @ id))) & ! [X157 : term] : (($true = ((X151 @ id @ (X153 @ id @ X157) @ id))) | (((X151 @ id @ X157 @ id)) != $true)) & ! [X156 : subst,X154 : term,X155 : subst] : (((sub @ (X153 @ X156 @ X154) @ X155)) = ((X153 @ (comp @ X156 @ X155) @ (sub @ X154 @ X155))))) | ? [X159 : term] : (($true = ((X150 @ X159))) & ($true != ((X150 @ (sub @ X152 @ (push @ X159 @ id)))))) | ($true = ((X150 @ (lam @ X152)))) | ? [X158 : term] : (((X151 @ id @ X158 @ id)) != ((X150 @ X158)))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f61,definition,( ! [X472 : (subst > term > subst > $o)] : (($true = ((sP48 @ X472))) <=> ? [X476 : term,X475 : term] : (($true = ((X472 @ id @ X475 @ id))) & ($true = ((X472 @ id @ X476 @ id))) & ($true != ((X472 @ id @ (ap @ (sub @ X476 @ id) @ X475) @ id)))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f62,definition,( ($true = sP49) <=> ! [X470 : term,X471 : (term > $o),X472 : (subst > term > subst > $o),X473 : term] : (($true != ((X471 @ X473))) | ? [X484 : subst,X481 : subst,X483 : term,X482 : subst] : (($true != ((X472 @ (comp @ X481 @ X484) @ (sub @ X483 @ X484) @ X482))) & ($true = ((X472 @ X481 @ X483 @ (comp @ X484 @ X482))))) | ? [X474 : term] : (((X472 @ id @ X474 @ id)) != ((X471 @ X474))) | ($true = ((sP48 @ X472))) | ? [X479 : term,X480 : subst,X477 : subst,X478 : subst] : (($true = ((X472 @ (comp @ X478 @ X480) @ (sub @ X479 @ X480) @ X477))) & ($true != ((X472 @ X478 @ X479 @ (comp @ X480 @ X477))))) | ($true != ((X471 @ X470))) | ($true = ((X471 @ (ap @ X470 @ X473)))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f63,definition,( ($true = sP50) <=> ! [X375 : term,X374 : (subst > term > subst > $o)] : (? [X377 : term,X378 : (subst > term > term)] : (! [X382 : term,X384 : subst,X383 : subst] : (((sub @ (X378 @ X384 @ X382) @ X383)) = ((sub @ (sub @ X377 @ (push @ X382 @ X384)) @ X383))) & ! [X379 : term,X380 : subst,X381 : subst] : (((X378 @ (comp @ X381 @ X380) @ (sub @ X379 @ X380))) = ((sub @ X377 @ (push @ (sub @ X379 @ X380) @ (comp @ X381 @ X380))))) & ! [X385 : term] : (($true = ((X374 @ id @ (X378 @ id @ X385) @ id))) | ($true != ((X374 @ id @ X385 @ id)))) & ($true != ((X374 @ id @ (lam @ (X378 @ sh @ one)) @ id)))) | (((X374 @ id @ (lam @ (sub @ X375 @ (push @ one @ sh))) @ id)) = $true) | ? [X376 : term] : (($true = ((X374 @ id @ X376 @ id))) & ($true != ((X374 @ id @ (sub @ X375 @ (push @ X376 @ id)) @ id)))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f64,definition,( ! [X485 : (term > $o)] : (($true = ((sP51 @ X485))) <=> ? [X489 : term,X488 : term] : (($true != ((X485 @ (ap @ X488 @ X489)))) & (((X485 @ X488)) = $true) & ($true = ((X485 @ X489)))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f65,definition,( ($true = sP52) <=> ! [X486 : term,X485 : (term > $o)] : (($true = ((sP51 @ X485))) | ? [X487 : term] : (($true = ((var @ X487))) & ($true != ((X485 @ X487)))) | ? [X490 : term] : (! [X491 : term] : (($true != ((X485 @ X491))) | (((X485 @ (sub @ X490 @ (push @ X491 @ id)))) = $true)) & ($true != ((X485 @ (lam @ X490))))) | ($true = ((X485 @ X486))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f66,definition,( ! [X57 : (term > $o)] : (($true = ((sP53 @ X57))) <=> ! [X60 : term,X59 : term] : (($true != ((X57 @ X59))) | ($true != ((X57 @ X60))) | (((X57 @ (ap @ X59 @ X60))) = $true)))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f67,definition,( ! [X76 : (subst > term > subst > $o)] : ((((sP54 @ X76)) = $true) <=> ? [X77 : (subst > term > term)] : (! [X81 : term] : (($true != ((X76 @ id @ X81 @ id))) | ($true = ((X76 @ id @ (X77 @ id @ X81) @ id)))) & ($true != ((X76 @ id @ (lam @ (X77 @ sh @ one)) @ id))) & ! [X79 : subst,X80 : term,X78 : subst] : (((sub @ (X77 @ X78 @ X80) @ X79)) = ((X77 @ (comp @ X78 @ X79) @ (sub @ X80 @ X79))))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f68,definition,( ($true = sP55) <=> ? [X66 : (subst > term > subst > $o),X65 : term,X67 : (term > $o)] : (($true != ((X67 @ (lam @ X65)))) & ! [X69 : (subst > term > term)] : (? [X71 : subst,X70 : subst,X72 : term] : (((sub @ (X69 @ X70 @ X72) @ X71)) != ((X69 @ (comp @ X70 @ X71) @ (sub @ X72 @ X71)))) | ? [X73 : term] : (($true = ((X66 @ id @ X73 @ id))) & ($true != ((X66 @ id @ (X69 @ id @ X73) @ id)))) | ($true = ((X66 @ id @ (lam @ (X69 @ sh @ one)) @ id)))) & ! [X68 : term] : (($true != ((X67 @ X68))) | ($true = ((X67 @ (sub @ X65 @ (push @ X68 @ id)))))) & ! [X74 : term] : (((X66 @ id @ X74 @ id)) = ((X67 @ X74))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f69,definition,( ($true = sP56) <=> ? [X57 : (term > $o),X58 : term] : (! [X62 : term] : (($true = ((X57 @ (lam @ X62)))) | ? [X63 : term] : (($true = ((X57 @ X63))) & (((X57 @ (sub @ X62 @ (push @ X63 @ id)))) != $true))) & ($true = ((sP53 @ X57))) & ($true != ((X57 @ X58))) & ! [X61 : term] : (($true != ((var @ X61))) | (((X57 @ X61)) = $true)))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f70,definition,( ($true = sP57) <=> (? [X64 : term] : (((sub @ X64 @ id)) != X64) | ? [X54 : (subst > term > subst > $o)] : ! [X55 : (term > $o)] : ? [X56 : term] : (((X54 @ id @ X56 @ id)) != ((X55 @ X56))) | ! [X75 : term,X76 : (subst > term > subst > $o)] : (? [X83 : term,X82 : term] : (($true = ((X76 @ id @ X82 @ id))) & ($true = ((X76 @ id @ X83 @ id))) & ($true != ((X76 @ id @ (ap @ (sub @ X82 @ id) @ X83) @ id)))) | ($true = ((X76 @ id @ X75 @ id))) | (((sP54 @ X76)) = $true)) | ($true = sP56) | ($true = sP55))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f71,definition,( ($true = sP58) <=> (? [X3 : subst,X2 : term] : (((sub @ one @ (push @ X2 @ X3))) != X2) | ? [X6 : term,X7 : term] : ((((lam @ X6)) = ((lam @ X7))) & (X6 != X7)) | ? [X4 : term,X5 : subst] : (((comp @ sh @ (push @ X4 @ X5))) != X5) | ! [X10 : term,X11 : (subst > term > term),X8 : subst,X9 : (subst > term > term)] : ((((lam @ (X9 @ sh @ one))) != ((lam @ (X11 @ sh @ one)))) | ? [X15 : term,X17 : subst,X16 : subst] : (((sub @ (X11 @ X16 @ X15) @ X17)) != ((X11 @ (comp @ X16 @ X17) @ (sub @ X15 @ X17)))) | (((X9 @ X8 @ X10)) = ((X11 @ X8 @ X10))) | ? [X14 : subst,X13 : term,X12 : subst] : (((X9 @ (comp @ X14 @ X12) @ (sub @ X13 @ X12))) != ((sub @ (X9 @ X14 @ X13) @ X12)))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f72,plain,( (pushprop_lthm = $true) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X324 : term] : ? [X325 : (subst > term > term)] : (((sub @ X324 @ (push @ one @ sh))) = ((X325 @ sh @ one)))) & (hoasinduction_gthm = $true) & (((d2subst @ d_id)) = sh) & (pushprop_lem1_gthm = $true) & (hoasinduction_lem3_gthm = $true) & (ulamvarsh = $true) & (pushprop_lem3v2 = $true) & (! [X357 : (subst > term > subst > $o)] : ? [X358 : (term > $o)] : ! [X359 : term] : (((X358 @ X359)) = ((X357 @ id @ X359 @ id))) <=> (hoasinduction_lem0_lthm = $true)) & (! [X164 : term,X165 : subst] : (((sub @ one @ (push @ X164 @ X165))) = X164) <=> (axvarcons = $true)) & (((var @ (d2term @ d_one))) != $true) & (($true = sP58) <=> (hoaslaminj_lthm = $true)) & (($true = sP57) <=> (hoasinduction_no_psi_cond_lthm = $true)) & ((hoasapinj1_lthm = $true) <=> (? [X453 : term,X455 : term,X452 : term,X454 : term] : ((X452 != X455) & (((ap @ X452 @ X453)) = ((ap @ X455 @ X454)))) | ? [X451 : term] : (((sub @ X451 @ id)) != X451) | ! [X458 : term,X457 : term,X456 : term,X459 : term] : ((((ap @ (sub @ X458 @ id) @ X456)) != ((ap @ (sub @ X459 @ id) @ X457))) | (X458 = X459)))) & (hoaslamnotvar_lthm = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ((hoasapinj2_lthm = $true) <=> (! [X520 : term,X519 : term,X518 : term,X517 : term] : ((X517 = X520) | (((ap @ (sub @ X519 @ id) @ X517)) != ((ap @ (sub @ X518 @ id) @ X520)))) | ? [X516 : term,X514 : term,X513 : term,X515 : term] : ((((ap @ X515 @ X514)) = ((ap @ X516 @ X513))) & (X513 != X514)))) & ((pushprop_lem1v2_lthm = $true) <=> (? [X533 : term,X534 : subst] : (((sub @ one @ (push @ X533 @ X534))) != X533) | ! [X535 : term,X538 : (term > $o),X537 : subst,X536 : (term > $o)] : (? [X539 : term] : (((X536 @ (sub @ X539 @ (push @ X535 @ X537)))) != ((X538 @ X539))) | ($true != ((X536 @ X535))) | ($true = ((X538 @ one)))))) & (axabs = $true) & ((! [X396 : term] : (((sub @ X396 @ id)) = X396) | ? [X395 : term] : (((sub @ X395 @ id)) != X395)) <=> (termmset_lthm = $true)) & (axvarshift = $true) & (axmap = $true) & (($true = sP52) <=> (induction2 = $true)) & (substmonoid_gthm = $true) & ((hoasinduction_lem3aaa = $true) <=> ($true = sP50)) & ((pushprop_lem0 = $true) <=> ! [X335 : (term > $o),X337 : subst,X336 : term] : ? [X338 : (term > $o)] : ! [X339 : term] : (((X335 @ (sub @ X339 @ (push @ X336 @ X337)))) = ((X338 @ X339)))) & ! [X297 : term] : ? [X298 : d_term] : (((d2term @ X298)) = X297) & (! [X523 : term,X522 : term,X521 : term,X524 : term] : ((((ap @ (sub @ X523 @ id) @ X521)) != ((ap @ (sub @ X522 @ id) @ X524))) | (X521 = X524)) <=> (hoasapinj2 = $true)) & (($true = sP49) <=> (hoasinduction_lem2v2 = $true)) & ! [X234 : d_subst,X235 : d_subst] : ((X234 = X235) | (((d2subst @ X235)) != ((d2subst @ X234)))) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_lem1v2 = $true) & (! [X140 : (subst > term > subst > $o)] : ? [X141 : (term > $o)] : ! [X142 : term] : (((X141 @ X142)) = ((X140 @ id @ X142 @ id))) <=> (hoasinduction_lem0 = $true)) & (! [X295 : subst,X293 : term,X292 : (term > $o),X294 : (term > $o)] : (($true != ((X294 @ X293))) | ($true = ((X292 @ one))) | ? [X296 : term] : (((X294 @ (sub @ X296 @ (push @ X293 @ X295)))) != ((X292 @ X296)))) <=> (pushprop_lem1v2 = $true)) & (hoasinduction_lem2_gthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (! [X92 : term] : (((sub @ X92 @ id)) = X92) <=> (axvarid = $true)) & ((laminj = $true) <=> ! [X290 : term,X291 : term] : ((((lam @ X291)) != ((lam @ X290))) | (X290 = X291))) & (hoaslamnotvar = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (axapp = $true) & (! [X444 : subst] : (((comp @ X444 @ id)) = X444) <=> (axidr = $true)) & (hoaslamnotap_gthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (($true = sP47) <=> (hoasinduction_lem3v2a_lthm = $true)) & ((hoaslamnotap = $true) <=> ! [X446 : term,X447 : (subst > term > term),X445 : term] : (? [X448 : subst,X450 : term,X449 : subst] : (((X447 @ (comp @ X448 @ X449) @ (sub @ X450 @ X449))) != ((sub @ (X447 @ X448 @ X450) @ X449))) | (((ap @ (sub @ X445 @ id) @ X446)) != ((lam @ (X447 @ sh @ one)))))) & ((apinj2 = $true) <=> ! [X364 : term,X365 : term,X366 : term,X367 : term] : ((X364 = X365) | (((ap @ X366 @ X364)) != ((ap @ X367 @ X365))))) & (($true = sP46) <=> (induction2_lthm = $true)) & (! [X186 : term,X187 : (term > $o)] : (? [X189 : term] : (($true != ((X187 @ X189))) & ($true = ((var @ X189)))) | ($true = ((X187 @ X186))) | ? [X188 : term] : (($true = ((X187 @ X188))) & ($true != ((X187 @ (lam @ X188))))) | (((sP42 @ X187)) = $true)) <=> (induction = $true)) & (hoasapinj1_gthm = $true) & ! [X36 : d_term] : (d_one = X36) & ((axscons = $true) <=> ! [X394 : subst] : (((push @ (sub @ one @ X394) @ (comp @ sh @ X394))) = X394)) & (! [X32 : term,X33 : term,X34 : term,X35 : term] : ((((ap @ (sub @ X34 @ id) @ X33)) != ((ap @ (sub @ X32 @ id) @ X35))) | (X32 = X34)) <=> (hoasapinj1 = $true)) & (pushprop_lem2v2_lthm = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((substmonoid = $true) <=> (! [X511 : subst] : (((comp @ X511 @ id)) = X511) & ! [X512 : subst] : (((comp @ id @ X512)) = X512))) & (hoasinduction_lem3v2_gthm = $true) & (pushprop_lem0_gthm = $true) & (ulamvar1 != $true) & ((hoasinduction_lthm_3 = $true) <=> ($true = sP41)) & (! [X0 : term,X1 : term] : ((X0 = X1) | (((sub @ X0 @ sh)) != ((sub @ X1 @ sh)))) <=> (shinj = $true)) & (! [X340 : subst] : (((comp @ id @ X340)) = X340) <=> (axidl = $true)) & (! [X463 : term,X461 : subst,X460 : (subst > term > term),X462 : (subst > term > term)] : ((((lam @ (X462 @ sh @ one))) != ((lam @ (X460 @ sh @ one)))) | (((X462 @ X461 @ X463)) = ((X460 @ X461 @ X463))) | ? [X466 : term,X465 : subst,X464 : subst] : (((sub @ (X460 @ X465 @ X466) @ X464)) != ((X460 @ (comp @ X465 @ X464) @ (sub @ X466 @ X464)))) | ? [X467 : subst,X469 : term,X468 : subst] : (((sub @ (X462 @ X467 @ X469) @ X468)) != ((X462 @ (comp @ X467 @ X468) @ (sub @ X469 @ X468))))) <=> (hoaslaminj = $true)) & (hoaslaminj_gthm = $true) & (induction2lem_gthm = $true) & (id = ((d2subst @ d_id))) & (hoasinduction_lem1v2_gthm = $true) & (hoasinduction_lem1_lthm = $true) & (! [X370 : term] : ? [X371 : (subst > term > term)] : ! [X372 : term,X373 : subst] : (((X371 @ X373 @ X372)) = ((sub @ X370 @ (push @ X372 @ X373)))) <=> (hoasinduction_lem3v2_f = $true)) & (hoasinduction_lem3b_gthm = $true) & (lamnotvar = $true) & (pushprop = $true) & (axassoc = $true) & (($true = sP35) <=> (hoasinduction_lem3_lthm = $true)) & ! [X144 : d_term,X143 : d_term] : ((X143 = X144) | (((d2term @ X144)) != ((d2term @ X143)))) & (($true = sP32) <=> (hoasinduction_lem3 = $true)) & (pushprop_lthm_orig = $true) & (hoasinduction_lem1_gthm = $true) & (! [X329 : (term > $o),X330 : (term > $o),X327 : term,X326 : term,X328 : subst] : (($true != ((X330 @ X327))) | ($true = ((X329 @ (sub @ X327 @ (push @ X326 @ X328)))))) <=> (pushprop_lem1 = $true)) & (hoasapnotvar = $true) & ((hoasinduction = $true) <=> ($true = sP30)) & (hoasinduction_lem2_lthm = $true) & (hoaslamnotap_lthm = $true) & ((? [X192 : subst,X193 : term] : (((sub @ one @ (push @ X193 @ X192))) != X193) | ! [X198 : term,X197 : subst,X199 : term,X200 : (term > $o),X196 : (term > $o)] : (($true != ((X196 @ X199))) | (((X200 @ (sub @ X199 @ (push @ X198 @ X197)))) = $true)) | ? [X194 : subst,X195 : term] : (((comp @ sh @ (push @ X195 @ X194))) != X194)) <=> (pushprop_lem1_lthm = $true)) & (hoaslamnotvar_gthm = $true) & ! [X31 : d_subst] : (d_id = X31) & (hoasapnotvar_lthm = $true) & (hoasinduction_lem2v2_gthm = $true) & (($true = sP27) <=> (induction2lem = $true)) & (apnotvar = $true) & (ulamvarind = $true) & (pushprop_lem2v2 = $true) & (lamnotap != $true) & ((hoasinduction_lthm = $true) <=> ($true = sP25)) & ((hoasinduction_lem3aa = $true) <=> ($true = sP16)) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (! [X510 : term] : (((sub @ X510 @ id)) = X510) <=> (termmset = $true)) & (($true = sP15) <=> (hoasinduction_lem3aa_lthm = $true)) & ((? [X331 : subst] : (((comp @ id @ X331)) != X331) | ? [X332 : subst] : (((comp @ X332 @ id)) != X332) | (! [X333 : subst] : (((comp @ id @ X333)) = X333) & ! [X334 : subst] : (((comp @ X334 @ id)) = X334))) <=> (substmonoid_lthm = $true)) & (($true = sP14) <=> (hoasinduction_lem3a_lthm = $true)) & ((hoasinduction_lem2 = $true) <=> ($true = sP12)) & (($true = sP10) <=> (hoasinduction_lem3a = $true)) & (((d2term @ d_one)) = one) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (hoasinduction_lem3a_gthm = $true) & (hoasapinj2_gthm = $true) & (pushprop_gthm = $true) & ((pushprop_lem0_lthm = $true) <=> ! [X230 : subst,X231 : (term > $o),X229 : term] : ? [X232 : (term > $o)] : ! [X233 : term] : (((X232 @ X233)) = ((X231 @ (sub @ X233 @ (push @ X229 @ X230)))))) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (! [X363 : term,X362 : subst] : (((comp @ sh @ (push @ X363 @ X362))) = X362) <=> (axshiftcons = $true)) & ! [X360 : subst] : ? [X361 : d_subst] : (((d2subst @ X361)) = X360) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X160 : term] : ? [X161 : (subst > term > term)] : ! [X163 : term,X162 : subst] : (((sub @ X160 @ (push @ X163 @ X162))) = ((X161 @ X162 @ X163)))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (termmset_gthm = $true) & (axclos = $true) & (hoasinduction_lem1 = $true) & ((hoasinduction_lem3v2 = $true) <=> ($true = sP9)) & ((hoasinduction_lem3v2_lthm = $true) <=> (sP7 = $true)) & (pushprop_lem2v2_gthm = $true) & (! [X220 : term,X221 : (subst > term > subst > $o)] : ((((X221 @ id @ X220 @ id)) = $true) | ? [X228 : term,X227 : term] : (($true = ((X221 @ id @ X227 @ id))) & ($true != ((X221 @ id @ (ap @ (sub @ X228 @ id) @ X227) @ id))) & ($true = ((X221 @ id @ X228 @ id)))) | ($true = ((sP5 @ X221)))) <=> (hoasinduction_no_psi_cond = $true)) & (! [X368 : term] : ? [X369 : (subst > term > term)] : (((X369 @ sh @ one)) = ((sub @ X368 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b = $true)) & ((induction2lem_lthm = $true) <=> ($true = sP4)) & ((hoasinduction_lem3v2a = $true) <=> ($true = sP0)) & (induction2_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & (! [X168 : term,X166 : term,X167 : term,X169 : term] : ((((ap @ X168 @ X167)) != ((ap @ X166 @ X169))) | (X166 = X168)) <=> (apinj1 = $true)) & (pushprop_lem3v2_lthm = $true) & (hoasapnotvar_gthm = $true)), inference(definition_folding,[],[f12,f71,f70,f69,f68,f67,f66,f65,f64,f63,f62,f61,f60,f59,f58,f57,f56,f55,f54,f53,f52,f51,f50,f49,f48,f47,f46,f45,f44,f43,f42,f41,f40,f39,f38,f37,f36,f35,f34,f33,f32,f31,f30,f29,f28,f27,f26,f25,f24,f23,f22,f21,f20,f19,f18,f17,f16,f15,f14,f13])). thf(f257,plain,( (($true = sP0) | ? [X434 : (subst > term > subst > $o),X435 : term,X436 : (term > $o)] : (($true != ((X436 @ (lam @ X435)))) & ! [X443 : term] : (($true = ((X436 @ (sub @ X435 @ (push @ X443 @ id))))) | ($true != ((X436 @ X443)))) & ! [X437 : term] : (((X436 @ X437)) = ((X434 @ id @ X437 @ id))) & ! [X438 : (subst > term > term)] : (? [X440 : subst,X441 : term,X439 : subst] : (((sub @ (X438 @ X439 @ X441) @ X440)) != ((X438 @ (comp @ X439 @ X440) @ (sub @ X441 @ X440)))) | ($true = ((X434 @ id @ (lam @ (X438 @ sh @ one)) @ id))) | ? [X442 : term] : ((((X434 @ id @ (X438 @ id @ X442) @ id)) != $true) & ($true = ((X434 @ id @ X442 @ id))))))) & (! [X434 : (subst > term > subst > $o),X435 : term,X436 : (term > $o)] : (($true = ((X436 @ (lam @ X435)))) | ? [X443 : term] : (($true != ((X436 @ (sub @ X435 @ (push @ X443 @ id))))) & ($true = ((X436 @ X443)))) | ? [X437 : term] : (((X436 @ X437)) != ((X434 @ id @ X437 @ id))) | ? [X438 : (subst > term > term)] : (! [X440 : subst,X441 : term,X439 : subst] : (((sub @ (X438 @ X439 @ X441) @ X440)) = ((X438 @ (comp @ X439 @ X440) @ (sub @ X441 @ X440)))) & ($true != ((X434 @ id @ (lam @ (X438 @ sh @ one)) @ id))) & ! [X442 : term] : ((((X434 @ id @ (X438 @ id @ X442) @ id)) = $true) | ($true != ((X434 @ id @ X442 @ id)))))) | ($true != sP0))), inference(nnf_transformation,[],[f13])). thf(f258,plain,( (($true = sP0) | ? [X0 : (subst > term > subst > $o),X1 : term,X2 : (term > $o)] : (($true != ((X2 @ (lam @ X1)))) & ! [X3 : term] : (($true = ((X2 @ (sub @ X1 @ (push @ X3 @ id))))) | ($true != ((X2 @ X3)))) & ! [X4 : term] : (((X2 @ X4)) = ((X0 @ id @ X4 @ id))) & ! [X5 : (subst > term > term)] : (? [X6 : subst,X7 : term,X8 : subst] : (((sub @ (X5 @ X8 @ X7) @ X6)) != ((X5 @ (comp @ X8 @ X6) @ (sub @ X7 @ X6)))) | ($true = ((X0 @ id @ (lam @ (X5 @ sh @ one)) @ id))) | ? [X9 : term] : (($true != ((X0 @ id @ (X5 @ id @ X9) @ id))) & ($true = ((X0 @ id @ X9 @ id))))))) & (! [X10 : (subst > term > subst > $o),X11 : term,X12 : (term > $o)] : (($true = ((X12 @ (lam @ X11)))) | ? [X13 : term] : (($true != ((X12 @ (sub @ X11 @ (push @ X13 @ id))))) & ($true = ((X12 @ X13)))) | ? [X14 : term] : (((X12 @ X14)) != ((X10 @ id @ X14 @ id))) | ? [X15 : (subst > term > term)] : (! [X16 : subst,X17 : term,X18 : subst] : (((sub @ (X15 @ X18 @ X17) @ X16)) = ((X15 @ (comp @ X18 @ X16) @ (sub @ X17 @ X16)))) & ($true != ((X10 @ id @ (lam @ (X15 @ sh @ one)) @ id))) & ! [X19 : term] : (($true = ((X10 @ id @ (X15 @ id @ X19) @ id))) | ($true != ((X10 @ id @ X19 @ id)))))) | ($true != sP0))), inference(rectify,[],[f257])). thf(f259,plain,( (($true = sP0) | ((((sK449 @ (lam @ sK448))) != $true) & ! [X3 : term] : (($true = ((sK449 @ (sub @ sK448 @ (push @ X3 @ id))))) | ($true != ((sK449 @ X3)))) & ! [X4 : term] : (((sK447 @ id @ X4 @ id)) = ((sK449 @ X4))) & ! [X5 : (subst > term > term)] : ((((sub @ (X5 @ (sK452 @ X5) @ (sK451 @ X5)) @ (sK450 @ X5))) != ((X5 @ (comp @ (sK452 @ X5) @ (sK450 @ X5)) @ (sub @ (sK451 @ X5) @ (sK450 @ X5))))) | ($true = ((sK447 @ id @ (lam @ (X5 @ sh @ one)) @ id))) | (($true != ((sK447 @ id @ (X5 @ id @ (sK453 @ X5)) @ id))) & ($true = ((sK447 @ id @ (sK453 @ X5) @ id))))))) & (! [X10 : (subst > term > subst > $o),X11 : term,X12 : (term > $o)] : (($true = ((X12 @ (lam @ X11)))) | (($true != ((X12 @ (sub @ X11 @ (push @ (sK454 @ X12 @ X11) @ id))))) & ($true = ((X12 @ (sK454 @ X12 @ X11))))) | (((X12 @ (sK455 @ X12 @ X10))) != ((X10 @ id @ (sK455 @ X12 @ X10) @ id))) | (! [X16 : subst,X17 : term,X18 : subst] : (((sK456 @ X10 @ (comp @ X18 @ X16) @ (sub @ X17 @ X16))) = ((sub @ (sK456 @ X10 @ X18 @ X17) @ X16))) & ($true != ((X10 @ id @ (lam @ (sK456 @ X10 @ sh @ one)) @ id))) & ! [X19 : term] : (($true = ((X10 @ id @ (sK456 @ X10 @ id @ X19) @ id))) | ($true != ((X10 @ id @ X19 @ id)))))) | ($true != sP0))), inference(skolemize,[status(esa),new_symbols(skolem,[sK447,sK448,sK449,vAPP]),skolemize(X0,$thf(sK447)),skolemize(X1,$thf(sK448)),skolemize(X2,$thf(sK449)),skolemize(X31,$thf(sK74 @ X25)),skolemize(X31,$thf(sK74 @ X25)),skolemize(X31,$thf(sK74 @ X25)),skolemize(X31,$thf(sK74 @ X25)),skolemize(X31,$thf(sK74 @ X25)),skolemize(X31,$thf(sK74 @ X25)),skolemize(X31,$thf(sK74 @ X25))],[f258])). thf(f260,plain,( (pushprop_lthm = $true) & (((hoasinduction_lem3b_lthm = $true) | ? [X324 : term] : ! [X325 : (subst > term > term)] : (((sub @ X324 @ (push @ one @ sh))) != ((X325 @ sh @ one)))) & (! [X324 : term] : ? [X325 : (subst > term > term)] : (((sub @ X324 @ (push @ one @ sh))) = ((X325 @ sh @ one))) | (hoasinduction_lem3b_lthm != $true))) & (hoasinduction_gthm = $true) & (((d2subst @ d_id)) = sh) & (pushprop_lem1_gthm = $true) & (hoasinduction_lem3_gthm = $true) & (ulamvarsh = $true) & (pushprop_lem3v2 = $true) & ((! [X357 : (subst > term > subst > $o)] : ? [X358 : (term > $o)] : ! [X359 : term] : (((X358 @ X359)) = ((X357 @ id @ X359 @ id))) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ? [X357 : (subst > term > subst > $o)] : ! [X358 : (term > $o)] : ? [X359 : term] : (((X358 @ X359)) != ((X357 @ id @ X359 @ id))))) & ((! [X164 : term,X165 : subst] : (((sub @ one @ (push @ X164 @ X165))) = X164) | (axvarcons != $true)) & ((axvarcons = $true) | ? [X164 : term,X165 : subst] : (((sub @ one @ (push @ X164 @ X165))) != X164))) & (((var @ (d2term @ d_one))) != $true) & ((($true = sP58) | (hoaslaminj_lthm != $true)) & ((hoaslaminj_lthm = $true) | ($true != sP58))) & ((($true = sP57) | (hoasinduction_no_psi_cond_lthm != $true)) & ((hoasinduction_no_psi_cond_lthm = $true) | ($true != sP57))) & (((hoasapinj1_lthm = $true) | (! [X453 : term,X455 : term,X452 : term,X454 : term] : ((X452 = X455) | (((ap @ X452 @ X453)) != ((ap @ X455 @ X454)))) & ! [X451 : term] : (((sub @ X451 @ id)) = X451) & ? [X458 : term,X457 : term,X456 : term,X459 : term] : ((((ap @ (sub @ X458 @ id) @ X456)) = ((ap @ (sub @ X459 @ id) @ X457))) & (X458 != X459)))) & ((? [X453 : term,X455 : term,X452 : term,X454 : term] : ((X452 != X455) & (((ap @ X452 @ X453)) = ((ap @ X455 @ X454)))) | ? [X451 : term] : (((sub @ X451 @ id)) != X451) | ! [X458 : term,X457 : term,X456 : term,X459 : term] : ((((ap @ (sub @ X458 @ id) @ X456)) != ((ap @ (sub @ X459 @ id) @ X457))) | (X458 = X459))) | (hoasapinj1_lthm != $true))) & (hoaslamnotvar_lthm = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (((hoasapinj2_lthm = $true) | (? [X520 : term,X519 : term,X518 : term,X517 : term] : ((X517 != X520) & (((ap @ (sub @ X519 @ id) @ X517)) = ((ap @ (sub @ X518 @ id) @ X520)))) & ! [X516 : term,X514 : term,X513 : term,X515 : term] : ((((ap @ X515 @ X514)) != ((ap @ X516 @ X513))) | (X513 = X514)))) & ((! [X520 : term,X519 : term,X518 : term,X517 : term] : ((X517 = X520) | (((ap @ (sub @ X519 @ id) @ X517)) != ((ap @ (sub @ X518 @ id) @ X520)))) | ? [X516 : term,X514 : term,X513 : term,X515 : term] : ((((ap @ X515 @ X514)) = ((ap @ X516 @ X513))) & (X513 != X514))) | (hoasapinj2_lthm != $true))) & (((pushprop_lem1v2_lthm = $true) | (! [X533 : term,X534 : subst] : (((sub @ one @ (push @ X533 @ X534))) = X533) & ? [X535 : term,X538 : (term > $o),X537 : subst,X536 : (term > $o)] : (! [X539 : term] : (((X536 @ (sub @ X539 @ (push @ X535 @ X537)))) = ((X538 @ X539))) & ($true = ((X536 @ X535))) & ($true != ((X538 @ one)))))) & ((? [X533 : term,X534 : subst] : (((sub @ one @ (push @ X533 @ X534))) != X533) | ! [X535 : term,X538 : (term > $o),X537 : subst,X536 : (term > $o)] : (? [X539 : term] : (((X536 @ (sub @ X539 @ (push @ X535 @ X537)))) != ((X538 @ X539))) | ($true != ((X536 @ X535))) | ($true = ((X538 @ one))))) | (pushprop_lem1v2_lthm != $true))) & (axabs = $true) & (((! [X396 : term] : (((sub @ X396 @ id)) = X396) | ? [X395 : term] : (((sub @ X395 @ id)) != X395)) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | (? [X396 : term] : (((sub @ X396 @ id)) != X396) & ! [X395 : term] : (((sub @ X395 @ id)) = X395)))) & (axvarshift = $true) & (axmap = $true) & ((($true = sP52) | (induction2 != $true)) & ((induction2 = $true) | ($true != sP52))) & (substmonoid_gthm = $true) & (((hoasinduction_lem3aaa = $true) | ($true != sP50)) & (($true = sP50) | (hoasinduction_lem3aaa != $true))) & (((pushprop_lem0 = $true) | ? [X335 : (term > $o),X337 : subst,X336 : term] : ! [X338 : (term > $o)] : ? [X339 : term] : (((X335 @ (sub @ X339 @ (push @ X336 @ X337)))) != ((X338 @ X339)))) & (! [X335 : (term > $o),X337 : subst,X336 : term] : ? [X338 : (term > $o)] : ! [X339 : term] : (((X335 @ (sub @ X339 @ (push @ X336 @ X337)))) = ((X338 @ X339))) | (pushprop_lem0 != $true))) & ! [X297 : term] : ? [X298 : d_term] : (((d2term @ X298)) = X297) & ((! [X523 : term,X522 : term,X521 : term,X524 : term] : ((((ap @ (sub @ X523 @ id) @ X521)) != ((ap @ (sub @ X522 @ id) @ X524))) | (X521 = X524)) | (hoasapinj2 != $true)) & ((hoasapinj2 = $true) | ? [X523 : term,X522 : term,X521 : term,X524 : term] : ((((ap @ (sub @ X523 @ id) @ X521)) = ((ap @ (sub @ X522 @ id) @ X524))) & (X521 != X524)))) & ((($true = sP49) | (hoasinduction_lem2v2 != $true)) & ((hoasinduction_lem2v2 = $true) | ($true != sP49))) & ! [X234 : d_subst,X235 : d_subst] : ((X234 = X235) | (((d2subst @ X235)) != ((d2subst @ X234)))) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_lem1v2 = $true) & ((! [X140 : (subst > term > subst > $o)] : ? [X141 : (term > $o)] : ! [X142 : term] : (((X141 @ X142)) = ((X140 @ id @ X142 @ id))) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem0 = $true) | ? [X140 : (subst > term > subst > $o)] : ! [X141 : (term > $o)] : ? [X142 : term] : (((X141 @ X142)) != ((X140 @ id @ X142 @ id))))) & ((! [X295 : subst,X293 : term,X292 : (term > $o),X294 : (term > $o)] : (($true != ((X294 @ X293))) | ($true = ((X292 @ one))) | ? [X296 : term] : (((X294 @ (sub @ X296 @ (push @ X293 @ X295)))) != ((X292 @ X296)))) | (pushprop_lem1v2 != $true)) & ((pushprop_lem1v2 = $true) | ? [X295 : subst,X293 : term,X292 : (term > $o),X294 : (term > $o)] : (($true = ((X294 @ X293))) & ($true != ((X292 @ one))) & ! [X296 : term] : (((X294 @ (sub @ X296 @ (push @ X293 @ X295)))) = ((X292 @ X296)))))) & (hoasinduction_lem2_gthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & ((! [X92 : term] : (((sub @ X92 @ id)) = X92) | (axvarid != $true)) & ((axvarid = $true) | ? [X92 : term] : (((sub @ X92 @ id)) != X92))) & (((laminj = $true) | ? [X290 : term,X291 : term] : ((((lam @ X291)) = ((lam @ X290))) & (X290 != X291))) & (! [X290 : term,X291 : term] : ((((lam @ X291)) != ((lam @ X290))) | (X290 = X291)) | (laminj != $true))) & (hoaslamnotvar = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (axapp = $true) & ((! [X444 : subst] : (((comp @ X444 @ id)) = X444) | (axidr != $true)) & ((axidr = $true) | ? [X444 : subst] : (((comp @ X444 @ id)) != X444))) & (hoaslamnotap_gthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((($true = sP47) | (hoasinduction_lem3v2a_lthm != $true)) & ((hoasinduction_lem3v2a_lthm = $true) | ($true != sP47))) & (((hoaslamnotap = $true) | ? [X446 : term,X447 : (subst > term > term),X445 : term] : (! [X448 : subst,X450 : term,X449 : subst] : (((X447 @ (comp @ X448 @ X449) @ (sub @ X450 @ X449))) = ((sub @ (X447 @ X448 @ X450) @ X449))) & (((ap @ (sub @ X445 @ id) @ X446)) = ((lam @ (X447 @ sh @ one)))))) & (! [X446 : term,X447 : (subst > term > term),X445 : term] : (? [X448 : subst,X450 : term,X449 : subst] : (((X447 @ (comp @ X448 @ X449) @ (sub @ X450 @ X449))) != ((sub @ (X447 @ X448 @ X450) @ X449))) | (((ap @ (sub @ X445 @ id) @ X446)) != ((lam @ (X447 @ sh @ one))))) | (hoaslamnotap != $true))) & (((apinj2 = $true) | ? [X364 : term,X365 : term,X366 : term,X367 : term] : ((X364 != X365) & (((ap @ X366 @ X364)) = ((ap @ X367 @ X365))))) & (! [X364 : term,X365 : term,X366 : term,X367 : term] : ((X364 = X365) | (((ap @ X366 @ X364)) != ((ap @ X367 @ X365)))) | (apinj2 != $true))) & ((($true = sP46) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | ($true != sP46))) & ((! [X186 : term,X187 : (term > $o)] : (? [X189 : term] : (($true != ((X187 @ X189))) & ($true = ((var @ X189)))) | ($true = ((X187 @ X186))) | ? [X188 : term] : (($true = ((X187 @ X188))) & ($true != ((X187 @ (lam @ X188))))) | (((sP42 @ X187)) = $true)) | (induction != $true)) & ((induction = $true) | ? [X186 : term,X187 : (term > $o)] : (! [X189 : term] : (($true = ((X187 @ X189))) | ($true != ((var @ X189)))) & ($true != ((X187 @ X186))) & ! [X188 : term] : (($true != ((X187 @ X188))) | ($true = ((X187 @ (lam @ X188))))) & (((sP42 @ X187)) != $true)))) & (hoasapinj1_gthm = $true) & ! [X36 : d_term] : (d_one = X36) & (((axscons = $true) | ? [X394 : subst] : (((push @ (sub @ one @ X394) @ (comp @ sh @ X394))) != X394)) & (! [X394 : subst] : (((push @ (sub @ one @ X394) @ (comp @ sh @ X394))) = X394) | (axscons != $true))) & ((! [X32 : term,X33 : term,X34 : term,X35 : term] : ((((ap @ (sub @ X34 @ id) @ X33)) != ((ap @ (sub @ X32 @ id) @ X35))) | (X32 = X34)) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | ? [X32 : term,X33 : term,X34 : term,X35 : term] : ((((ap @ (sub @ X34 @ id) @ X33)) = ((ap @ (sub @ X32 @ id) @ X35))) & (X32 != X34)))) & (pushprop_lem2v2_lthm = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (((substmonoid = $true) | (? [X511 : subst] : (((comp @ X511 @ id)) != X511) | ? [X512 : subst] : (((comp @ id @ X512)) != X512))) & ((! [X511 : subst] : (((comp @ X511 @ id)) = X511) & ! [X512 : subst] : (((comp @ id @ X512)) = X512)) | (substmonoid != $true))) & (hoasinduction_lem3v2_gthm = $true) & (pushprop_lem0_gthm = $true) & (ulamvar1 != $true) & (((hoasinduction_lthm_3 = $true) | ($true != sP41)) & (($true = sP41) | (hoasinduction_lthm_3 != $true))) & ((! [X0 : term,X1 : term] : ((X0 = X1) | (((sub @ X0 @ sh)) != ((sub @ X1 @ sh)))) | (shinj != $true)) & ((shinj = $true) | ? [X0 : term,X1 : term] : ((X0 != X1) & (((sub @ X0 @ sh)) = ((sub @ X1 @ sh)))))) & ((! [X340 : subst] : (((comp @ id @ X340)) = X340) | (axidl != $true)) & ((axidl = $true) | ? [X340 : subst] : (((comp @ id @ X340)) != X340))) & ((! [X463 : term,X461 : subst,X460 : (subst > term > term),X462 : (subst > term > term)] : ((((lam @ (X462 @ sh @ one))) != ((lam @ (X460 @ sh @ one)))) | (((X462 @ X461 @ X463)) = ((X460 @ X461 @ X463))) | ? [X466 : term,X465 : subst,X464 : subst] : (((sub @ (X460 @ X465 @ X466) @ X464)) != ((X460 @ (comp @ X465 @ X464) @ (sub @ X466 @ X464)))) | ? [X467 : subst,X469 : term,X468 : subst] : (((sub @ (X462 @ X467 @ X469) @ X468)) != ((X462 @ (comp @ X467 @ X468) @ (sub @ X469 @ X468))))) | (hoaslaminj != $true)) & ((hoaslaminj = $true) | ? [X463 : term,X461 : subst,X460 : (subst > term > term),X462 : (subst > term > term)] : ((((lam @ (X462 @ sh @ one))) = ((lam @ (X460 @ sh @ one)))) & (((X462 @ X461 @ X463)) != ((X460 @ X461 @ X463))) & ! [X466 : term,X465 : subst,X464 : subst] : (((sub @ (X460 @ X465 @ X466) @ X464)) = ((X460 @ (comp @ X465 @ X464) @ (sub @ X466 @ X464)))) & ! [X467 : subst,X469 : term,X468 : subst] : (((sub @ (X462 @ X467 @ X469) @ X468)) = ((X462 @ (comp @ X467 @ X468) @ (sub @ X469 @ X468))))))) & (hoaslaminj_gthm = $true) & (induction2lem_gthm = $true) & (id = ((d2subst @ d_id))) & (hoasinduction_lem1v2_gthm = $true) & (hoasinduction_lem1_lthm = $true) & ((! [X370 : term] : ? [X371 : (subst > term > term)] : ! [X372 : term,X373 : subst] : (((X371 @ X373 @ X372)) = ((sub @ X370 @ (push @ X372 @ X373)))) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ? [X370 : term] : ! [X371 : (subst > term > term)] : ? [X372 : term,X373 : subst] : (((X371 @ X373 @ X372)) != ((sub @ X370 @ (push @ X372 @ X373)))))) & (hoasinduction_lem3b_gthm = $true) & (lamnotvar = $true) & (pushprop = $true) & (axassoc = $true) & ((($true = sP35) | (hoasinduction_lem3_lthm != $true)) & ((hoasinduction_lem3_lthm = $true) | ($true != sP35))) & ! [X144 : d_term,X143 : d_term] : ((X143 = X144) | (((d2term @ X144)) != ((d2term @ X143)))) & ((($true = sP32) | (hoasinduction_lem3 != $true)) & ((hoasinduction_lem3 = $true) | ($true != sP32))) & (pushprop_lthm_orig = $true) & (hoasinduction_lem1_gthm = $true) & ((! [X329 : (term > $o),X330 : (term > $o),X327 : term,X326 : term,X328 : subst] : (($true != ((X330 @ X327))) | ($true = ((X329 @ (sub @ X327 @ (push @ X326 @ X328)))))) | (pushprop_lem1 != $true)) & ((pushprop_lem1 = $true) | ? [X329 : (term > $o),X330 : (term > $o),X327 : term,X326 : term,X328 : subst] : (($true = ((X330 @ X327))) & ($true != ((X329 @ (sub @ X327 @ (push @ X326 @ X328)))))))) & (hoasapnotvar = $true) & (((hoasinduction = $true) | ($true != sP30)) & (($true = sP30) | (hoasinduction != $true))) & (hoasinduction_lem2_lthm = $true) & (hoaslamnotap_lthm = $true) & (((? [X192 : subst,X193 : term] : (((sub @ one @ (push @ X193 @ X192))) != X193) | ! [X198 : term,X197 : subst,X199 : term,X200 : (term > $o),X196 : (term > $o)] : (($true != ((X196 @ X199))) | (((X200 @ (sub @ X199 @ (push @ X198 @ X197)))) = $true)) | ? [X194 : subst,X195 : term] : (((comp @ sh @ (push @ X195 @ X194))) != X194)) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (! [X192 : subst,X193 : term] : (((sub @ one @ (push @ X193 @ X192))) = X193) & ? [X198 : term,X197 : subst,X199 : term,X200 : (term > $o),X196 : (term > $o)] : (($true = ((X196 @ X199))) & (((X200 @ (sub @ X199 @ (push @ X198 @ X197)))) != $true)) & ! [X194 : subst,X195 : term] : (((comp @ sh @ (push @ X195 @ X194))) = X194)))) & (hoaslamnotvar_gthm = $true) & ! [X31 : d_subst] : (d_id = X31) & (hoasapnotvar_lthm = $true) & (hoasinduction_lem2v2_gthm = $true) & ((($true = sP27) | (induction2lem != $true)) & ((induction2lem = $true) | ($true != sP27))) & (apnotvar = $true) & (ulamvarind = $true) & (pushprop_lem2v2 = $true) & (lamnotap != $true) & (((hoasinduction_lthm = $true) | ($true != sP25)) & (($true = sP25) | (hoasinduction_lthm != $true))) & (((hoasinduction_lem3aa = $true) | ($true != sP16)) & (($true = sP16) | (hoasinduction_lem3aa != $true))) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ((! [X510 : term] : (((sub @ X510 @ id)) = X510) | (termmset != $true)) & ((termmset = $true) | ? [X510 : term] : (((sub @ X510 @ id)) != X510))) & ((($true = sP15) | (hoasinduction_lem3aa_lthm != $true)) & ((hoasinduction_lem3aa_lthm = $true) | ($true != sP15))) & (((? [X331 : subst] : (((comp @ id @ X331)) != X331) | ? [X332 : subst] : (((comp @ X332 @ id)) != X332) | (! [X333 : subst] : (((comp @ id @ X333)) = X333) & ! [X334 : subst] : (((comp @ X334 @ id)) = X334))) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | (! [X331 : subst] : (((comp @ id @ X331)) = X331) & ! [X332 : subst] : (((comp @ X332 @ id)) = X332) & (? [X333 : subst] : (((comp @ id @ X333)) != X333) | ? [X334 : subst] : (((comp @ X334 @ id)) != X334))))) & ((($true = sP14) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | ($true != sP14))) & (((hoasinduction_lem2 = $true) | ($true != sP12)) & (($true = sP12) | (hoasinduction_lem2 != $true))) & ((($true = sP10) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | ($true != sP10))) & (((d2term @ d_one)) = one) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (hoasinduction_lem3a_gthm = $true) & (hoasapinj2_gthm = $true) & (pushprop_gthm = $true) & (((pushprop_lem0_lthm = $true) | ? [X230 : subst,X231 : (term > $o),X229 : term] : ! [X232 : (term > $o)] : ? [X233 : term] : (((X232 @ X233)) != ((X231 @ (sub @ X233 @ (push @ X229 @ X230)))))) & (! [X230 : subst,X231 : (term > $o),X229 : term] : ? [X232 : (term > $o)] : ! [X233 : term] : (((X232 @ X233)) = ((X231 @ (sub @ X233 @ (push @ X229 @ X230))))) | (pushprop_lem0_lthm != $true))) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((! [X363 : term,X362 : subst] : (((comp @ sh @ (push @ X363 @ X362))) = X362) | (axshiftcons != $true)) & ((axshiftcons = $true) | ? [X363 : term,X362 : subst] : (((comp @ sh @ (push @ X363 @ X362))) != X362))) & ! [X360 : subst] : ? [X361 : d_subst] : (((d2subst @ X361)) = X360) & (((hoasinduction_lem3v2_f_lthm = $true) | ? [X160 : term] : ! [X161 : (subst > term > term)] : ? [X163 : term,X162 : subst] : (((sub @ X160 @ (push @ X163 @ X162))) != ((X161 @ X162 @ X163)))) & (! [X160 : term] : ? [X161 : (subst > term > term)] : ! [X163 : term,X162 : subst] : (((sub @ X160 @ (push @ X163 @ X162))) = ((X161 @ X162 @ X163))) | (hoasinduction_lem3v2_f_lthm != $true))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (termmset_gthm = $true) & (axclos = $true) & (hoasinduction_lem1 = $true) & (((hoasinduction_lem3v2 = $true) | ($true != sP9)) & (($true = sP9) | (hoasinduction_lem3v2 != $true))) & (((hoasinduction_lem3v2_lthm = $true) | (sP7 != $true)) & ((sP7 = $true) | (hoasinduction_lem3v2_lthm != $true))) & (pushprop_lem2v2_gthm = $true) & ((! [X220 : term,X221 : (subst > term > subst > $o)] : ((((X221 @ id @ X220 @ id)) = $true) | ? [X228 : term,X227 : term] : (($true = ((X221 @ id @ X227 @ id))) & ($true != ((X221 @ id @ (ap @ (sub @ X228 @ id) @ X227) @ id))) & ($true = ((X221 @ id @ X228 @ id)))) | ($true = ((sP5 @ X221)))) | (hoasinduction_no_psi_cond != $true)) & ((hoasinduction_no_psi_cond = $true) | ? [X220 : term,X221 : (subst > term > subst > $o)] : ((((X221 @ id @ X220 @ id)) != $true) & ! [X228 : term,X227 : term] : (($true != ((X221 @ id @ X227 @ id))) | ($true = ((X221 @ id @ (ap @ (sub @ X228 @ id) @ X227) @ id))) | ($true != ((X221 @ id @ X228 @ id)))) & ($true != ((sP5 @ X221)))))) & ((! [X368 : term] : ? [X369 : (subst > term > term)] : (((X369 @ sh @ one)) = ((sub @ X368 @ (push @ one @ sh)))) | (hoasinduction_lem3b != $true)) & ((hoasinduction_lem3b = $true) | ? [X368 : term] : ! [X369 : (subst > term > term)] : (((X369 @ sh @ one)) != ((sub @ X368 @ (push @ one @ sh)))))) & (((induction2lem_lthm = $true) | ($true != sP4)) & (($true = sP4) | (induction2lem_lthm != $true))) & (((hoasinduction_lem3v2a = $true) | ($true != sP0)) & (($true = sP0) | (hoasinduction_lem3v2a != $true))) & (induction2_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & ((! [X168 : term,X166 : term,X167 : term,X169 : term] : ((((ap @ X168 @ X167)) != ((ap @ X166 @ X169))) | (X166 = X168)) | (apinj1 != $true)) & ((apinj1 = $true) | ? [X168 : term,X166 : term,X167 : term,X169 : term] : ((((ap @ X168 @ X167)) = ((ap @ X166 @ X169))) & (X166 != X168)))) & (pushprop_lem3v2_lthm = $true) & (hoasapnotvar_gthm = $true)), inference(nnf_transformation,[],[f72])). thf(f261,plain,( (pushprop_lthm = $true) & ((hoasinduction_lem3b_lthm = $true) | ? [X324 : term] : ! [X325 : (subst > term > term)] : (((sub @ X324 @ (push @ one @ sh))) != ((X325 @ sh @ one)))) & (! [X324 : term] : ? [X325 : (subst > term > term)] : (((sub @ X324 @ (push @ one @ sh))) = ((X325 @ sh @ one))) | (hoasinduction_lem3b_lthm != $true)) & (hoasinduction_gthm = $true) & (((d2subst @ d_id)) = sh) & (pushprop_lem1_gthm = $true) & (hoasinduction_lem3_gthm = $true) & (ulamvarsh = $true) & (pushprop_lem3v2 = $true) & (! [X357 : (subst > term > subst > $o)] : ? [X358 : (term > $o)] : ! [X359 : term] : (((X358 @ X359)) = ((X357 @ id @ X359 @ id))) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ? [X357 : (subst > term > subst > $o)] : ! [X358 : (term > $o)] : ? [X359 : term] : (((X358 @ X359)) != ((X357 @ id @ X359 @ id)))) & (! [X164 : term,X165 : subst] : (((sub @ one @ (push @ X164 @ X165))) = X164) | (axvarcons != $true)) & ((axvarcons = $true) | ? [X164 : term,X165 : subst] : (((sub @ one @ (push @ X164 @ X165))) != X164)) & (((var @ (d2term @ d_one))) != $true) & (($true = sP58) | (hoaslaminj_lthm != $true)) & ((hoaslaminj_lthm = $true) | ($true != sP58)) & (($true = sP57) | (hoasinduction_no_psi_cond_lthm != $true)) & ((hoasinduction_no_psi_cond_lthm = $true) | ($true != sP57)) & ((hoasapinj1_lthm = $true) | (! [X453 : term,X455 : term,X452 : term,X454 : term] : ((X452 = X455) | (((ap @ X452 @ X453)) != ((ap @ X455 @ X454)))) & ! [X451 : term] : (((sub @ X451 @ id)) = X451) & ? [X458 : term,X457 : term,X456 : term,X459 : term] : ((((ap @ (sub @ X458 @ id) @ X456)) = ((ap @ (sub @ X459 @ id) @ X457))) & (X458 != X459)))) & (? [X453 : term,X455 : term,X452 : term,X454 : term] : ((X452 != X455) & (((ap @ X452 @ X453)) = ((ap @ X455 @ X454)))) | ? [X451 : term] : (((sub @ X451 @ id)) != X451) | ! [X458 : term,X457 : term,X456 : term,X459 : term] : ((((ap @ (sub @ X458 @ id) @ X456)) != ((ap @ (sub @ X459 @ id) @ X457))) | (X458 = X459)) | (hoasapinj1_lthm != $true)) & (hoaslamnotvar_lthm = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ((hoasapinj2_lthm = $true) | (? [X520 : term,X519 : term,X518 : term,X517 : term] : ((X517 != X520) & (((ap @ (sub @ X519 @ id) @ X517)) = ((ap @ (sub @ X518 @ id) @ X520)))) & ! [X516 : term,X514 : term,X513 : term,X515 : term] : ((((ap @ X515 @ X514)) != ((ap @ X516 @ X513))) | (X513 = X514)))) & (! [X520 : term,X519 : term,X518 : term,X517 : term] : ((X517 = X520) | (((ap @ (sub @ X519 @ id) @ X517)) != ((ap @ (sub @ X518 @ id) @ X520)))) | ? [X516 : term,X514 : term,X513 : term,X515 : term] : ((((ap @ X515 @ X514)) = ((ap @ X516 @ X513))) & (X513 != X514)) | (hoasapinj2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (! [X533 : term,X534 : subst] : (((sub @ one @ (push @ X533 @ X534))) = X533) & ? [X535 : term,X538 : (term > $o),X537 : subst,X536 : (term > $o)] : (! [X539 : term] : (((X536 @ (sub @ X539 @ (push @ X535 @ X537)))) = ((X538 @ X539))) & ($true = ((X536 @ X535))) & ($true != ((X538 @ one)))))) & (? [X533 : term,X534 : subst] : (((sub @ one @ (push @ X533 @ X534))) != X533) | ! [X535 : term,X538 : (term > $o),X537 : subst,X536 : (term > $o)] : (? [X539 : term] : (((X536 @ (sub @ X539 @ (push @ X535 @ X537)))) != ((X538 @ X539))) | ($true != ((X536 @ X535))) | ($true = ((X538 @ one)))) | (pushprop_lem1v2_lthm != $true)) & (axabs = $true) & (! [X396 : term] : (((sub @ X396 @ id)) = X396) | ? [X395 : term] : (((sub @ X395 @ id)) != X395) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | (? [X396 : term] : (((sub @ X396 @ id)) != X396) & ! [X395 : term] : (((sub @ X395 @ id)) = X395))) & (axvarshift = $true) & (axmap = $true) & (($true = sP52) | (induction2 != $true)) & ((induction2 = $true) | ($true != sP52)) & (substmonoid_gthm = $true) & ((hoasinduction_lem3aaa = $true) | ($true != sP50)) & (($true = sP50) | (hoasinduction_lem3aaa != $true)) & ((pushprop_lem0 = $true) | ? [X335 : (term > $o),X337 : subst,X336 : term] : ! [X338 : (term > $o)] : ? [X339 : term] : (((X335 @ (sub @ X339 @ (push @ X336 @ X337)))) != ((X338 @ X339)))) & (! [X335 : (term > $o),X337 : subst,X336 : term] : ? [X338 : (term > $o)] : ! [X339 : term] : (((X335 @ (sub @ X339 @ (push @ X336 @ X337)))) = ((X338 @ X339))) | (pushprop_lem0 != $true)) & ! [X297 : term] : ? [X298 : d_term] : (((d2term @ X298)) = X297) & (! [X523 : term,X522 : term,X521 : term,X524 : term] : ((((ap @ (sub @ X523 @ id) @ X521)) != ((ap @ (sub @ X522 @ id) @ X524))) | (X521 = X524)) | (hoasapinj2 != $true)) & ((hoasapinj2 = $true) | ? [X523 : term,X522 : term,X521 : term,X524 : term] : ((((ap @ (sub @ X523 @ id) @ X521)) = ((ap @ (sub @ X522 @ id) @ X524))) & (X521 != X524))) & (($true = sP49) | (hoasinduction_lem2v2 != $true)) & ((hoasinduction_lem2v2 = $true) | ($true != sP49)) & ! [X234 : d_subst,X235 : d_subst] : ((X234 = X235) | (((d2subst @ X235)) != ((d2subst @ X234)))) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_lem1v2 = $true) & (! [X140 : (subst > term > subst > $o)] : ? [X141 : (term > $o)] : ! [X142 : term] : (((X141 @ X142)) = ((X140 @ id @ X142 @ id))) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem0 = $true) | ? [X140 : (subst > term > subst > $o)] : ! [X141 : (term > $o)] : ? [X142 : term] : (((X141 @ X142)) != ((X140 @ id @ X142 @ id)))) & (! [X295 : subst,X293 : term,X292 : (term > $o),X294 : (term > $o)] : (($true != ((X294 @ X293))) | ($true = ((X292 @ one))) | ? [X296 : term] : (((X294 @ (sub @ X296 @ (push @ X293 @ X295)))) != ((X292 @ X296)))) | (pushprop_lem1v2 != $true)) & ((pushprop_lem1v2 = $true) | ? [X295 : subst,X293 : term,X292 : (term > $o),X294 : (term > $o)] : (($true = ((X294 @ X293))) & ($true != ((X292 @ one))) & ! [X296 : term] : (((X294 @ (sub @ X296 @ (push @ X293 @ X295)))) = ((X292 @ X296))))) & (hoasinduction_lem2_gthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (! [X92 : term] : (((sub @ X92 @ id)) = X92) | (axvarid != $true)) & ((axvarid = $true) | ? [X92 : term] : (((sub @ X92 @ id)) != X92)) & ((laminj = $true) | ? [X290 : term,X291 : term] : ((((lam @ X291)) = ((lam @ X290))) & (X290 != X291))) & (! [X290 : term,X291 : term] : ((((lam @ X291)) != ((lam @ X290))) | (X290 = X291)) | (laminj != $true)) & (hoaslamnotvar = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (axapp = $true) & (! [X444 : subst] : (((comp @ X444 @ id)) = X444) | (axidr != $true)) & ((axidr = $true) | ? [X444 : subst] : (((comp @ X444 @ id)) != X444)) & (hoaslamnotap_gthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (($true = sP47) | (hoasinduction_lem3v2a_lthm != $true)) & ((hoasinduction_lem3v2a_lthm = $true) | ($true != sP47)) & ((hoaslamnotap = $true) | ? [X446 : term,X447 : (subst > term > term),X445 : term] : (! [X448 : subst,X450 : term,X449 : subst] : (((X447 @ (comp @ X448 @ X449) @ (sub @ X450 @ X449))) = ((sub @ (X447 @ X448 @ X450) @ X449))) & (((ap @ (sub @ X445 @ id) @ X446)) = ((lam @ (X447 @ sh @ one)))))) & (! [X446 : term,X447 : (subst > term > term),X445 : term] : (? [X448 : subst,X450 : term,X449 : subst] : (((X447 @ (comp @ X448 @ X449) @ (sub @ X450 @ X449))) != ((sub @ (X447 @ X448 @ X450) @ X449))) | (((ap @ (sub @ X445 @ id) @ X446)) != ((lam @ (X447 @ sh @ one))))) | (hoaslamnotap != $true)) & ((apinj2 = $true) | ? [X364 : term,X365 : term,X366 : term,X367 : term] : ((X364 != X365) & (((ap @ X366 @ X364)) = ((ap @ X367 @ X365))))) & (! [X364 : term,X365 : term,X366 : term,X367 : term] : ((X364 = X365) | (((ap @ X366 @ X364)) != ((ap @ X367 @ X365)))) | (apinj2 != $true)) & (($true = sP46) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | ($true != sP46)) & (! [X186 : term,X187 : (term > $o)] : (? [X189 : term] : (($true != ((X187 @ X189))) & ($true = ((var @ X189)))) | ($true = ((X187 @ X186))) | ? [X188 : term] : (($true = ((X187 @ X188))) & ($true != ((X187 @ (lam @ X188))))) | (((sP42 @ X187)) = $true)) | (induction != $true)) & ((induction = $true) | ? [X186 : term,X187 : (term > $o)] : (! [X189 : term] : (($true = ((X187 @ X189))) | ($true != ((var @ X189)))) & ($true != ((X187 @ X186))) & ! [X188 : term] : (($true != ((X187 @ X188))) | ($true = ((X187 @ (lam @ X188))))) & (((sP42 @ X187)) != $true))) & (hoasapinj1_gthm = $true) & ! [X36 : d_term] : (d_one = X36) & ((axscons = $true) | ? [X394 : subst] : (((push @ (sub @ one @ X394) @ (comp @ sh @ X394))) != X394)) & (! [X394 : subst] : (((push @ (sub @ one @ X394) @ (comp @ sh @ X394))) = X394) | (axscons != $true)) & (! [X32 : term,X33 : term,X34 : term,X35 : term] : ((((ap @ (sub @ X34 @ id) @ X33)) != ((ap @ (sub @ X32 @ id) @ X35))) | (X32 = X34)) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | ? [X32 : term,X33 : term,X34 : term,X35 : term] : ((((ap @ (sub @ X34 @ id) @ X33)) = ((ap @ (sub @ X32 @ id) @ X35))) & (X32 != X34))) & (pushprop_lem2v2_lthm = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((substmonoid = $true) | ? [X511 : subst] : (((comp @ X511 @ id)) != X511) | ? [X512 : subst] : (((comp @ id @ X512)) != X512)) & ((! [X511 : subst] : (((comp @ X511 @ id)) = X511) & ! [X512 : subst] : (((comp @ id @ X512)) = X512)) | (substmonoid != $true)) & (hoasinduction_lem3v2_gthm = $true) & (pushprop_lem0_gthm = $true) & (ulamvar1 != $true) & ((hoasinduction_lthm_3 = $true) | ($true != sP41)) & (($true = sP41) | (hoasinduction_lthm_3 != $true)) & (! [X0 : term,X1 : term] : ((X0 = X1) | (((sub @ X0 @ sh)) != ((sub @ X1 @ sh)))) | (shinj != $true)) & ((shinj = $true) | ? [X0 : term,X1 : term] : ((X0 != X1) & (((sub @ X0 @ sh)) = ((sub @ X1 @ sh))))) & (! [X340 : subst] : (((comp @ id @ X340)) = X340) | (axidl != $true)) & ((axidl = $true) | ? [X340 : subst] : (((comp @ id @ X340)) != X340)) & (! [X463 : term,X461 : subst,X460 : (subst > term > term),X462 : (subst > term > term)] : ((((lam @ (X462 @ sh @ one))) != ((lam @ (X460 @ sh @ one)))) | (((X462 @ X461 @ X463)) = ((X460 @ X461 @ X463))) | ? [X466 : term,X465 : subst,X464 : subst] : (((sub @ (X460 @ X465 @ X466) @ X464)) != ((X460 @ (comp @ X465 @ X464) @ (sub @ X466 @ X464)))) | ? [X467 : subst,X469 : term,X468 : subst] : (((sub @ (X462 @ X467 @ X469) @ X468)) != ((X462 @ (comp @ X467 @ X468) @ (sub @ X469 @ X468))))) | (hoaslaminj != $true)) & ((hoaslaminj = $true) | ? [X463 : term,X461 : subst,X460 : (subst > term > term),X462 : (subst > term > term)] : ((((lam @ (X462 @ sh @ one))) = ((lam @ (X460 @ sh @ one)))) & (((X462 @ X461 @ X463)) != ((X460 @ X461 @ X463))) & ! [X466 : term,X465 : subst,X464 : subst] : (((sub @ (X460 @ X465 @ X466) @ X464)) = ((X460 @ (comp @ X465 @ X464) @ (sub @ X466 @ X464)))) & ! [X467 : subst,X469 : term,X468 : subst] : (((sub @ (X462 @ X467 @ X469) @ X468)) = ((X462 @ (comp @ X467 @ X468) @ (sub @ X469 @ X468)))))) & (hoaslaminj_gthm = $true) & (induction2lem_gthm = $true) & (id = ((d2subst @ d_id))) & (hoasinduction_lem1v2_gthm = $true) & (hoasinduction_lem1_lthm = $true) & (! [X370 : term] : ? [X371 : (subst > term > term)] : ! [X372 : term,X373 : subst] : (((X371 @ X373 @ X372)) = ((sub @ X370 @ (push @ X372 @ X373)))) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ? [X370 : term] : ! [X371 : (subst > term > term)] : ? [X372 : term,X373 : subst] : (((X371 @ X373 @ X372)) != ((sub @ X370 @ (push @ X372 @ X373))))) & (hoasinduction_lem3b_gthm = $true) & (lamnotvar = $true) & (pushprop = $true) & (axassoc = $true) & (($true = sP35) | (hoasinduction_lem3_lthm != $true)) & ((hoasinduction_lem3_lthm = $true) | ($true != sP35)) & ! [X144 : d_term,X143 : d_term] : ((X143 = X144) | (((d2term @ X144)) != ((d2term @ X143)))) & (($true = sP32) | (hoasinduction_lem3 != $true)) & ((hoasinduction_lem3 = $true) | ($true != sP32)) & (pushprop_lthm_orig = $true) & (hoasinduction_lem1_gthm = $true) & (! [X329 : (term > $o),X330 : (term > $o),X327 : term,X326 : term,X328 : subst] : (($true != ((X330 @ X327))) | ($true = ((X329 @ (sub @ X327 @ (push @ X326 @ X328)))))) | (pushprop_lem1 != $true)) & ((pushprop_lem1 = $true) | ? [X329 : (term > $o),X330 : (term > $o),X327 : term,X326 : term,X328 : subst] : (($true = ((X330 @ X327))) & ($true != ((X329 @ (sub @ X327 @ (push @ X326 @ X328))))))) & (hoasapnotvar = $true) & ((hoasinduction = $true) | ($true != sP30)) & (($true = sP30) | (hoasinduction != $true)) & (hoasinduction_lem2_lthm = $true) & (hoaslamnotap_lthm = $true) & (? [X192 : subst,X193 : term] : (((sub @ one @ (push @ X193 @ X192))) != X193) | ! [X198 : term,X197 : subst,X199 : term,X200 : (term > $o),X196 : (term > $o)] : (($true != ((X196 @ X199))) | (((X200 @ (sub @ X199 @ (push @ X198 @ X197)))) = $true)) | ? [X194 : subst,X195 : term] : (((comp @ sh @ (push @ X195 @ X194))) != X194) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (! [X192 : subst,X193 : term] : (((sub @ one @ (push @ X193 @ X192))) = X193) & ? [X198 : term,X197 : subst,X199 : term,X200 : (term > $o),X196 : (term > $o)] : (($true = ((X196 @ X199))) & (((X200 @ (sub @ X199 @ (push @ X198 @ X197)))) != $true)) & ! [X194 : subst,X195 : term] : (((comp @ sh @ (push @ X195 @ X194))) = X194))) & (hoaslamnotvar_gthm = $true) & ! [X31 : d_subst] : (d_id = X31) & (hoasapnotvar_lthm = $true) & (hoasinduction_lem2v2_gthm = $true) & (($true = sP27) | (induction2lem != $true)) & ((induction2lem = $true) | ($true != sP27)) & (apnotvar = $true) & (ulamvarind = $true) & (pushprop_lem2v2 = $true) & (lamnotap != $true) & ((hoasinduction_lthm = $true) | ($true != sP25)) & (($true = sP25) | (hoasinduction_lthm != $true)) & ((hoasinduction_lem3aa = $true) | ($true != sP16)) & (($true = sP16) | (hoasinduction_lem3aa != $true)) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (! [X510 : term] : (((sub @ X510 @ id)) = X510) | (termmset != $true)) & ((termmset = $true) | ? [X510 : term] : (((sub @ X510 @ id)) != X510)) & (($true = sP15) | (hoasinduction_lem3aa_lthm != $true)) & ((hoasinduction_lem3aa_lthm = $true) | ($true != sP15)) & (? [X331 : subst] : (((comp @ id @ X331)) != X331) | ? [X332 : subst] : (((comp @ X332 @ id)) != X332) | (! [X333 : subst] : (((comp @ id @ X333)) = X333) & ! [X334 : subst] : (((comp @ X334 @ id)) = X334)) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | (! [X331 : subst] : (((comp @ id @ X331)) = X331) & ! [X332 : subst] : (((comp @ X332 @ id)) = X332) & (? [X333 : subst] : (((comp @ id @ X333)) != X333) | ? [X334 : subst] : (((comp @ X334 @ id)) != X334)))) & (($true = sP14) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | ($true != sP14)) & ((hoasinduction_lem2 = $true) | ($true != sP12)) & (($true = sP12) | (hoasinduction_lem2 != $true)) & (($true = sP10) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | ($true != sP10)) & (((d2term @ d_one)) = one) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (hoasinduction_lem3a_gthm = $true) & (hoasapinj2_gthm = $true) & (pushprop_gthm = $true) & ((pushprop_lem0_lthm = $true) | ? [X230 : subst,X231 : (term > $o),X229 : term] : ! [X232 : (term > $o)] : ? [X233 : term] : (((X232 @ X233)) != ((X231 @ (sub @ X233 @ (push @ X229 @ X230)))))) & (! [X230 : subst,X231 : (term > $o),X229 : term] : ? [X232 : (term > $o)] : ! [X233 : term] : (((X232 @ X233)) = ((X231 @ (sub @ X233 @ (push @ X229 @ X230))))) | (pushprop_lem0_lthm != $true)) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (! [X363 : term,X362 : subst] : (((comp @ sh @ (push @ X363 @ X362))) = X362) | (axshiftcons != $true)) & ((axshiftcons = $true) | ? [X363 : term,X362 : subst] : (((comp @ sh @ (push @ X363 @ X362))) != X362)) & ! [X360 : subst] : ? [X361 : d_subst] : (((d2subst @ X361)) = X360) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X160 : term] : ! [X161 : (subst > term > term)] : ? [X163 : term,X162 : subst] : (((sub @ X160 @ (push @ X163 @ X162))) != ((X161 @ X162 @ X163)))) & (! [X160 : term] : ? [X161 : (subst > term > term)] : ! [X163 : term,X162 : subst] : (((sub @ X160 @ (push @ X163 @ X162))) = ((X161 @ X162 @ X163))) | (hoasinduction_lem3v2_f_lthm != $true)) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (termmset_gthm = $true) & (axclos = $true) & (hoasinduction_lem1 = $true) & ((hoasinduction_lem3v2 = $true) | ($true != sP9)) & (($true = sP9) | (hoasinduction_lem3v2 != $true)) & ((hoasinduction_lem3v2_lthm = $true) | (sP7 != $true)) & ((sP7 = $true) | (hoasinduction_lem3v2_lthm != $true)) & (pushprop_lem2v2_gthm = $true) & (! [X220 : term,X221 : (subst > term > subst > $o)] : ((((X221 @ id @ X220 @ id)) = $true) | ? [X228 : term,X227 : term] : (($true = ((X221 @ id @ X227 @ id))) & ($true != ((X221 @ id @ (ap @ (sub @ X228 @ id) @ X227) @ id))) & ($true = ((X221 @ id @ X228 @ id)))) | ($true = ((sP5 @ X221)))) | (hoasinduction_no_psi_cond != $true)) & ((hoasinduction_no_psi_cond = $true) | ? [X220 : term,X221 : (subst > term > subst > $o)] : ((((X221 @ id @ X220 @ id)) != $true) & ! [X228 : term,X227 : term] : (($true != ((X221 @ id @ X227 @ id))) | ($true = ((X221 @ id @ (ap @ (sub @ X228 @ id) @ X227) @ id))) | ($true != ((X221 @ id @ X228 @ id)))) & ($true != ((sP5 @ X221))))) & (! [X368 : term] : ? [X369 : (subst > term > term)] : (((X369 @ sh @ one)) = ((sub @ X368 @ (push @ one @ sh)))) | (hoasinduction_lem3b != $true)) & ((hoasinduction_lem3b = $true) | ? [X368 : term] : ! [X369 : (subst > term > term)] : (((X369 @ sh @ one)) != ((sub @ X368 @ (push @ one @ sh))))) & ((induction2lem_lthm = $true) | ($true != sP4)) & (($true = sP4) | (induction2lem_lthm != $true)) & ((hoasinduction_lem3v2a = $true) | ($true != sP0)) & (($true = sP0) | (hoasinduction_lem3v2a != $true)) & (induction2_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & (! [X168 : term,X166 : term,X167 : term,X169 : term] : ((((ap @ X168 @ X167)) != ((ap @ X166 @ X169))) | (X166 = X168)) | (apinj1 != $true)) & ((apinj1 = $true) | ? [X168 : term,X166 : term,X167 : term,X169 : term] : ((((ap @ X168 @ X167)) = ((ap @ X166 @ X169))) & (X166 != X168))) & (pushprop_lem3v2_lthm = $true) & (hoasapnotvar_gthm = $true)), inference(flattening,[],[f260])). thf(f262,plain,( (pushprop_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)) & (hoasinduction_gthm = $true) & (((d2subst @ d_id)) = sh) & (pushprop_lem1_gthm = $true) & (hoasinduction_lem3_gthm = $true) & (ulamvarsh = $true) & (pushprop_lem3v2 = $true) & (! [X4 : (subst > term > subst > $o)] : ? [X5 : (term > $o)] : ! [X6 : term] : (((X4 @ id @ X6 @ id)) = ((X5 @ X6))) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ? [X7 : (subst > term > subst > $o)] : ! [X8 : (term > $o)] : ? [X9 : term] : (((X7 @ id @ X9 @ id)) != ((X8 @ X9)))) & (! [X10 : term,X11 : subst] : (((sub @ one @ (push @ X10 @ X11))) = X10) | (axvarcons != $true)) & ((axvarcons = $true) | ? [X12 : term,X13 : subst] : (((sub @ one @ (push @ X12 @ X13))) != X12)) & (((var @ (d2term @ d_one))) != $true) & (($true = sP58) | (hoaslaminj_lthm != $true)) & ((hoaslaminj_lthm = $true) | ($true != sP58)) & (($true = sP57) | (hoasinduction_no_psi_cond_lthm != $true)) & ((hoasinduction_no_psi_cond_lthm = $true) | ($true != sP57)) & ((hoasapinj1_lthm = $true) | (! [X14 : term,X15 : term,X16 : term,X17 : term] : ((X15 = X16) | (((ap @ X16 @ X14)) != ((ap @ X15 @ X17)))) & ! [X18 : term] : (((sub @ X18 @ id)) = X18) & ? [X19 : term,X20 : term,X21 : term,X22 : term] : ((((ap @ (sub @ X19 @ id) @ X21)) = ((ap @ (sub @ X22 @ id) @ X20))) & (X19 != X22)))) & (? [X23 : term,X24 : term,X25 : term,X26 : term] : ((X24 != X25) & (((ap @ X25 @ X23)) = ((ap @ X24 @ X26)))) | ? [X27 : term] : (((sub @ X27 @ id)) != X27) | ! [X28 : term,X29 : term,X30 : term,X31 : term] : ((((ap @ (sub @ X28 @ id) @ X30)) != ((ap @ (sub @ X31 @ id) @ X29))) | (X28 = X31)) | (hoasapinj1_lthm != $true)) & (hoaslamnotvar_lthm = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ((hoasapinj2_lthm = $true) | (? [X32 : term,X33 : term,X34 : term,X35 : term] : ((X32 != X35) & (((ap @ (sub @ X33 @ id) @ X35)) = ((ap @ (sub @ X34 @ id) @ X32)))) & ! [X36 : term,X37 : term,X38 : term,X39 : term] : ((((ap @ X36 @ X38)) != ((ap @ X39 @ X37))) | (X37 = X38)))) & (! [X40 : term,X41 : term,X42 : term,X43 : term] : ((X40 = X43) | (((ap @ (sub @ X42 @ id) @ X40)) != ((ap @ (sub @ X41 @ id) @ X43)))) | ? [X44 : term,X45 : term,X46 : term,X47 : term] : ((((ap @ X44 @ X46)) = ((ap @ X47 @ X45))) & (X45 != X46)) | (hoasapinj2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (! [X48 : term,X49 : subst] : (((sub @ one @ (push @ X48 @ X49))) = X48) & ? [X50 : term,X51 : (term > $o),X52 : subst,X53 : (term > $o)] : (! [X54 : term] : (((X51 @ X54)) = ((X53 @ (sub @ X54 @ (push @ X50 @ X52))))) & ($true = ((X53 @ X50))) & ($true != ((X51 @ one)))))) & (? [X55 : term,X56 : subst] : (((sub @ one @ (push @ X55 @ X56))) != X55) | ! [X57 : term,X58 : (term > $o),X59 : subst,X60 : (term > $o)] : (? [X61 : term] : (((X58 @ X61)) != ((X60 @ (sub @ X61 @ (push @ X57 @ X59))))) | ($true != ((X60 @ X57))) | (((X58 @ one)) = $true)) | (pushprop_lem1v2_lthm != $true)) & (axabs = $true) & (! [X62 : term] : (((sub @ X62 @ id)) = X62) | ? [X63 : term] : (((sub @ X63 @ id)) != X63) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | (? [X64 : term] : (((sub @ X64 @ id)) != X64) & ! [X65 : term] : (((sub @ X65 @ id)) = X65))) & (axvarshift = $true) & (axmap = $true) & (($true = sP52) | (induction2 != $true)) & ((induction2 = $true) | ($true != sP52)) & (substmonoid_gthm = $true) & ((hoasinduction_lem3aaa = $true) | ($true != sP50)) & (($true = sP50) | (hoasinduction_lem3aaa != $true)) & ((pushprop_lem0 = $true) | ? [X66 : (term > $o),X67 : subst,X68 : term] : ! [X69 : (term > $o)] : ? [X70 : term] : (((X69 @ X70)) != ((X66 @ (sub @ X70 @ (push @ X68 @ X67)))))) & (! [X71 : (term > $o),X72 : subst,X73 : term] : ? [X74 : (term > $o)] : ! [X75 : term] : (((X74 @ X75)) = ((X71 @ (sub @ X75 @ (push @ X73 @ X72))))) | (pushprop_lem0 != $true)) & ! [X76 : term] : ? [X77 : d_term] : (((d2term @ X77)) = X76) & (! [X78 : term,X79 : term,X80 : term,X81 : term] : ((((ap @ (sub @ X79 @ id) @ X81)) != ((ap @ (sub @ X78 @ id) @ X80))) | (X80 = X81)) | (hoasapinj2 != $true)) & ((hoasapinj2 = $true) | ? [X82 : term,X83 : term,X84 : term,X85 : term] : ((((ap @ (sub @ X82 @ id) @ X84)) = ((ap @ (sub @ X83 @ id) @ X85))) & (X84 != X85))) & (($true = sP49) | (hoasinduction_lem2v2 != $true)) & ((hoasinduction_lem2v2 = $true) | ($true != sP49)) & ! [X86 : d_subst,X87 : d_subst] : ((X86 = X87) | (((d2subst @ X86)) != ((d2subst @ X87)))) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_lem1v2 = $true) & (! [X88 : (subst > term > subst > $o)] : ? [X89 : (term > $o)] : ! [X90 : term] : (((X89 @ X90)) = ((X88 @ id @ X90 @ id))) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem0 = $true) | ? [X91 : (subst > term > subst > $o)] : ! [X92 : (term > $o)] : ? [X93 : term] : (((X92 @ X93)) != ((X91 @ id @ X93 @ id)))) & (! [X94 : subst,X95 : term,X96 : (term > $o),X97 : (term > $o)] : (($true != ((X97 @ X95))) | ($true = ((X96 @ one))) | ? [X98 : term] : (((X96 @ X98)) != ((X97 @ (sub @ X98 @ (push @ X95 @ X94)))))) | (pushprop_lem1v2 != $true)) & ((pushprop_lem1v2 = $true) | ? [X99 : subst,X100 : term,X101 : (term > $o),X102 : (term > $o)] : (($true = ((X102 @ X100))) & ($true != ((X101 @ one))) & ! [X103 : term] : (((X102 @ (sub @ X103 @ (push @ X100 @ X99)))) = ((X101 @ X103))))) & (hoasinduction_lem2_gthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (! [X104 : term] : (((sub @ X104 @ id)) = X104) | (axvarid != $true)) & ((axvarid = $true) | ? [X105 : term] : (((sub @ X105 @ id)) != X105)) & ((laminj = $true) | ? [X106 : term,X107 : term] : ((((lam @ X107)) = ((lam @ X106))) & (X106 != X107))) & (! [X108 : term,X109 : term] : ((((lam @ X108)) != ((lam @ X109))) | (X108 = X109)) | (laminj != $true)) & (hoaslamnotvar = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (axapp = $true) & (! [X110 : subst] : (((comp @ X110 @ id)) = X110) | (axidr != $true)) & ((axidr = $true) | ? [X111 : subst] : (((comp @ X111 @ id)) != X111)) & (hoaslamnotap_gthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (($true = sP47) | (hoasinduction_lem3v2a_lthm != $true)) & ((hoasinduction_lem3v2a_lthm = $true) | ($true != sP47)) & ((hoaslamnotap = $true) | ? [X112 : term,X113 : (subst > term > term),X114 : term] : (! [X115 : subst,X116 : term,X117 : subst] : (((X113 @ (comp @ X115 @ X117) @ (sub @ X116 @ X117))) = ((sub @ (X113 @ X115 @ X116) @ X117))) & (((lam @ (X113 @ sh @ one))) = ((ap @ (sub @ X114 @ id) @ X112))))) & (! [X118 : term,X119 : (subst > term > term),X120 : term] : (? [X121 : subst,X122 : term,X123 : subst] : (((X119 @ (comp @ X121 @ X123) @ (sub @ X122 @ X123))) != ((sub @ (X119 @ X121 @ X122) @ X123))) | (((ap @ (sub @ X120 @ id) @ X118)) != ((lam @ (X119 @ sh @ one))))) | (hoaslamnotap != $true)) & ((apinj2 = $true) | ? [X124 : term,X125 : term,X126 : term,X127 : term] : ((X124 != X125) & (((ap @ X126 @ X124)) = ((ap @ X127 @ X125))))) & (! [X128 : term,X129 : term,X130 : term,X131 : term] : ((X128 = X129) | (((ap @ X130 @ X128)) != ((ap @ X131 @ X129)))) | (apinj2 != $true)) & (($true = sP46) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | ($true != sP46)) & (! [X132 : term,X133 : (term > $o)] : (? [X134 : term] : (($true != ((X133 @ X134))) & (((var @ X134)) = $true)) | ($true = ((X133 @ X132))) | ? [X135 : term] : (($true = ((X133 @ X135))) & ($true != ((X133 @ (lam @ X135))))) | ($true = ((sP42 @ X133)))) | (induction != $true)) & ((induction = $true) | ? [X136 : term,X137 : (term > $o)] : (! [X138 : term] : (($true = ((X137 @ X138))) | ($true != ((var @ X138)))) & ($true != ((X137 @ X136))) & ! [X139 : term] : (($true != ((X137 @ X139))) | ($true = ((X137 @ (lam @ X139))))) & ($true != ((sP42 @ X137))))) & (hoasapinj1_gthm = $true) & ! [X140 : d_term] : (d_one = X140) & ((axscons = $true) | ? [X141 : subst] : (((push @ (sub @ one @ X141) @ (comp @ sh @ X141))) != X141)) & (! [X142 : subst] : (((push @ (sub @ one @ X142) @ (comp @ sh @ X142))) = X142) | (axscons != $true)) & (! [X143 : term,X144 : term,X145 : term,X146 : term] : ((((ap @ (sub @ X143 @ id) @ X146)) != ((ap @ (sub @ X145 @ id) @ X144))) | (X143 = X145)) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | ? [X147 : term,X148 : term,X149 : term,X150 : term] : ((((ap @ (sub @ X147 @ id) @ X150)) = ((ap @ (sub @ X149 @ id) @ X148))) & (X147 != X149))) & (pushprop_lem2v2_lthm = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((substmonoid = $true) | ? [X151 : subst] : (((comp @ X151 @ id)) != X151) | ? [X152 : subst] : (((comp @ id @ X152)) != X152)) & ((! [X153 : subst] : (((comp @ X153 @ id)) = X153) & ! [X154 : subst] : (((comp @ id @ X154)) = X154)) | (substmonoid != $true)) & (hoasinduction_lem3v2_gthm = $true) & (pushprop_lem0_gthm = $true) & (ulamvar1 != $true) & ((hoasinduction_lthm_3 = $true) | ($true != sP41)) & (($true = sP41) | (hoasinduction_lthm_3 != $true)) & (! [X155 : term,X156 : term] : ((X155 = X156) | (((sub @ X156 @ sh)) != ((sub @ X155 @ sh)))) | (shinj != $true)) & ((shinj = $true) | ? [X157 : term,X158 : term] : ((X157 != X158) & (((sub @ X158 @ sh)) = ((sub @ X157 @ sh))))) & (! [X159 : subst] : (((comp @ id @ X159)) = X159) | (axidl != $true)) & ((axidl = $true) | ? [X160 : subst] : (((comp @ id @ X160)) != X160)) & (! [X161 : term,X162 : subst,X163 : (subst > term > term),X164 : (subst > term > term)] : ((((lam @ (X164 @ sh @ one))) != ((lam @ (X163 @ sh @ one)))) | (((X163 @ X162 @ X161)) = ((X164 @ X162 @ X161))) | ? [X165 : term,X166 : subst,X167 : subst] : (((sub @ (X163 @ X166 @ X165) @ X167)) != ((X163 @ (comp @ X166 @ X167) @ (sub @ X165 @ X167)))) | ? [X168 : subst,X169 : term,X170 : subst] : (((sub @ (X164 @ X168 @ X169) @ X170)) != ((X164 @ (comp @ X168 @ X170) @ (sub @ X169 @ X170))))) | (hoaslaminj != $true)) & ((hoaslaminj = $true) | ? [X171 : term,X172 : subst,X173 : (subst > term > term),X174 : (subst > term > term)] : ((((lam @ (X173 @ sh @ one))) = ((lam @ (X174 @ sh @ one)))) & (((X174 @ X172 @ X171)) != ((X173 @ X172 @ X171))) & ! [X175 : term,X176 : subst,X177 : subst] : (((sub @ (X173 @ X176 @ X175) @ X177)) = ((X173 @ (comp @ X176 @ X177) @ (sub @ X175 @ X177)))) & ! [X178 : subst,X179 : term,X180 : subst] : (((X174 @ (comp @ X178 @ X180) @ (sub @ X179 @ X180))) = ((sub @ (X174 @ X178 @ X179) @ X180))))) & (hoaslaminj_gthm = $true) & (induction2lem_gthm = $true) & (id = ((d2subst @ d_id))) & (hoasinduction_lem1v2_gthm = $true) & (hoasinduction_lem1_lthm = $true) & (! [X181 : term] : ? [X182 : (subst > term > term)] : ! [X183 : term,X184 : subst] : (((sub @ X181 @ (push @ X183 @ X184))) = ((X182 @ X184 @ X183))) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ? [X185 : term] : ! [X186 : (subst > term > term)] : ? [X187 : term,X188 : subst] : (((sub @ X185 @ (push @ X187 @ X188))) != ((X186 @ X188 @ X187)))) & (hoasinduction_lem3b_gthm = $true) & (lamnotvar = $true) & (pushprop = $true) & (axassoc = $true) & (($true = sP35) | (hoasinduction_lem3_lthm != $true)) & ((hoasinduction_lem3_lthm = $true) | ($true != sP35)) & ! [X189 : d_term,X190 : d_term] : ((X189 = X190) | (((d2term @ X189)) != ((d2term @ X190)))) & (($true = sP32) | (hoasinduction_lem3 != $true)) & ((hoasinduction_lem3 = $true) | ($true != sP32)) & (pushprop_lthm_orig = $true) & (hoasinduction_lem1_gthm = $true) & (! [X191 : (term > $o),X192 : (term > $o),X193 : term,X194 : term,X195 : subst] : (($true != ((X192 @ X193))) | ($true = ((X191 @ (sub @ X193 @ (push @ X194 @ X195)))))) | (pushprop_lem1 != $true)) & ((pushprop_lem1 = $true) | ? [X196 : (term > $o),X197 : (term > $o),X198 : term,X199 : term,X200 : subst] : ((((X197 @ X198)) = $true) & ($true != ((X196 @ (sub @ X198 @ (push @ X199 @ X200))))))) & (hoasapnotvar = $true) & ((hoasinduction = $true) | ($true != sP30)) & (($true = sP30) | (hoasinduction != $true)) & (hoasinduction_lem2_lthm = $true) & (hoaslamnotap_lthm = $true) & (? [X201 : subst,X202 : term] : (((sub @ one @ (push @ X202 @ X201))) != X202) | ! [X203 : term,X204 : subst,X205 : term,X206 : (term > $o),X207 : (term > $o)] : (($true != ((X207 @ X205))) | ($true = ((X206 @ (sub @ X205 @ (push @ X203 @ X204)))))) | ? [X208 : subst,X209 : term] : (((comp @ sh @ (push @ X209 @ X208))) != X208) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (! [X210 : subst,X211 : term] : (((sub @ one @ (push @ X211 @ X210))) = X211) & ? [X212 : term,X213 : subst,X214 : term,X215 : (term > $o),X216 : (term > $o)] : (($true = ((X216 @ X214))) & ($true != ((X215 @ (sub @ X214 @ (push @ X212 @ X213)))))) & ! [X217 : subst,X218 : term] : (((comp @ sh @ (push @ X218 @ X217))) = X217))) & (hoaslamnotvar_gthm = $true) & ! [X219 : d_subst] : (d_id = X219) & (hoasapnotvar_lthm = $true) & (hoasinduction_lem2v2_gthm = $true) & (($true = sP27) | (induction2lem != $true)) & ((induction2lem = $true) | ($true != sP27)) & (apnotvar = $true) & (ulamvarind = $true) & (pushprop_lem2v2 = $true) & (lamnotap != $true) & ((hoasinduction_lthm = $true) | ($true != sP25)) & (($true = sP25) | (hoasinduction_lthm != $true)) & ((hoasinduction_lem3aa = $true) | ($true != sP16)) & (($true = sP16) | (hoasinduction_lem3aa != $true)) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (! [X220 : term] : (((sub @ X220 @ id)) = X220) | (termmset != $true)) & ((termmset = $true) | ? [X221 : term] : (((sub @ X221 @ id)) != X221)) & (($true = sP15) | (hoasinduction_lem3aa_lthm != $true)) & ((hoasinduction_lem3aa_lthm = $true) | ($true != sP15)) & (? [X222 : subst] : (((comp @ id @ X222)) != X222) | ? [X223 : subst] : (((comp @ X223 @ id)) != X223) | (! [X224 : subst] : (((comp @ id @ X224)) = X224) & ! [X225 : subst] : (((comp @ X225 @ id)) = X225)) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | (! [X226 : subst] : (((comp @ id @ X226)) = X226) & ! [X227 : subst] : (((comp @ X227 @ id)) = X227) & (? [X228 : subst] : (((comp @ id @ X228)) != X228) | ? [X229 : subst] : (((comp @ X229 @ id)) != X229)))) & (($true = sP14) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | ($true != sP14)) & ((hoasinduction_lem2 = $true) | ($true != sP12)) & (($true = sP12) | (hoasinduction_lem2 != $true)) & (($true = sP10) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | ($true != sP10)) & (((d2term @ d_one)) = one) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (hoasinduction_lem3a_gthm = $true) & (hoasapinj2_gthm = $true) & (pushprop_gthm = $true) & ((pushprop_lem0_lthm = $true) | ? [X230 : subst,X231 : (term > $o),X232 : term] : ! [X233 : (term > $o)] : ? [X234 : term] : (((X233 @ X234)) != ((X231 @ (sub @ X234 @ (push @ X232 @ X230)))))) & (! [X235 : subst,X236 : (term > $o),X237 : term] : ? [X238 : (term > $o)] : ! [X239 : term] : (((X236 @ (sub @ X239 @ (push @ X237 @ X235)))) = ((X238 @ X239))) | (pushprop_lem0_lthm != $true)) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (! [X240 : term,X241 : subst] : (((comp @ sh @ (push @ X240 @ X241))) = X241) | (axshiftcons != $true)) & ((axshiftcons = $true) | ? [X242 : term,X243 : subst] : (((comp @ sh @ (push @ X242 @ X243))) != X243)) & ! [X244 : subst] : ? [X245 : d_subst] : (((d2subst @ X245)) = X244) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X246 : term] : ! [X247 : (subst > term > term)] : ? [X248 : term,X249 : subst] : (((X247 @ X249 @ X248)) != ((sub @ X246 @ (push @ X248 @ X249))))) & (! [X250 : term] : ? [X251 : (subst > term > term)] : ! [X252 : term,X253 : subst] : (((sub @ X250 @ (push @ X252 @ X253))) = ((X251 @ X253 @ X252))) | (hoasinduction_lem3v2_f_lthm != $true)) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (termmset_gthm = $true) & (axclos = $true) & (hoasinduction_lem1 = $true) & ((hoasinduction_lem3v2 = $true) | ($true != sP9)) & (($true = sP9) | (hoasinduction_lem3v2 != $true)) & ((hoasinduction_lem3v2_lthm = $true) | (sP7 != $true)) & ((sP7 = $true) | (hoasinduction_lem3v2_lthm != $true)) & (pushprop_lem2v2_gthm = $true) & (! [X254 : term,X255 : (subst > term > subst > $o)] : (($true = ((X255 @ id @ X254 @ id))) | ? [X256 : term,X257 : term] : (($true = ((X255 @ id @ X257 @ id))) & ($true != ((X255 @ id @ (ap @ (sub @ X256 @ id) @ X257) @ id))) & ($true = ((X255 @ id @ X256 @ id)))) | ($true = ((sP5 @ X255)))) | (hoasinduction_no_psi_cond != $true)) & ((hoasinduction_no_psi_cond = $true) | ? [X258 : term,X259 : (subst > term > subst > $o)] : (($true != ((X259 @ id @ X258 @ id))) & ! [X260 : term,X261 : term] : (($true != ((X259 @ id @ X261 @ id))) | ($true = ((X259 @ id @ (ap @ (sub @ X260 @ id) @ X261) @ id))) | (((X259 @ id @ X260 @ id)) != $true)) & ($true != ((sP5 @ X259))))) & (! [X262 : term] : ? [X263 : (subst > term > term)] : (((sub @ X262 @ (push @ one @ sh))) = ((X263 @ sh @ one))) | (hoasinduction_lem3b != $true)) & ((hoasinduction_lem3b = $true) | ? [X264 : term] : ! [X265 : (subst > term > term)] : (((X265 @ sh @ one)) != ((sub @ X264 @ (push @ one @ sh))))) & ((induction2lem_lthm = $true) | ($true != sP4)) & (($true = sP4) | (induction2lem_lthm != $true)) & ((hoasinduction_lem3v2a = $true) | ($true != sP0)) & (($true = sP0) | (hoasinduction_lem3v2a != $true)) & (induction2_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & (! [X266 : term,X267 : term,X268 : term,X269 : term] : ((((ap @ X266 @ X268)) != ((ap @ X267 @ X269))) | (X266 = X267)) | (apinj1 != $true)) & ((apinj1 = $true) | ? [X270 : term,X271 : term,X272 : term,X273 : term] : ((((ap @ X271 @ X273)) = ((ap @ X270 @ X272))) & (X270 != X271))) & (pushprop_lem3v2_lthm = $true) & (hoasapnotvar_gthm = $true)), inference(rectify,[],[f261])). thf(f263,plain,( (pushprop_lthm = $true) & ((hoasinduction_lem3b_lthm = $true) | ! [X1 : (subst > term > term)] : (((sub @ sK457 @ (push @ one @ sh))) != ((X1 @ sh @ one)))) & (! [X2 : term] : (((sK458 @ X2 @ sh @ one)) = ((sub @ X2 @ (push @ one @ sh)))) | (hoasinduction_lem3b_lthm != $true)) & (hoasinduction_gthm = $true) & (((d2subst @ d_id)) = sh) & (pushprop_lem1_gthm = $true) & (hoasinduction_lem3_gthm = $true) & (ulamvarsh = $true) & (pushprop_lem3v2 = $true) & (! [X4 : (subst > term > subst > $o)] : ! [X6 : term] : (((X4 @ id @ X6 @ id)) = ((sK459 @ X4 @ X6))) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ! [X8 : (term > $o)] : (((X8 @ (sK461 @ X8))) != ((sK460 @ id @ (sK461 @ X8) @ id)))) & (! [X10 : term,X11 : subst] : (((sub @ one @ (push @ X10 @ X11))) = X10) | (axvarcons != $true)) & ((axvarcons = $true) | (sK462 != ((sub @ one @ (push @ sK462 @ sK463))))) & (((var @ (d2term @ d_one))) != $true) & (($true = sP58) | (hoaslaminj_lthm != $true)) & ((hoaslaminj_lthm = $true) | ($true != sP58)) & (($true = sP57) | (hoasinduction_no_psi_cond_lthm != $true)) & ((hoasinduction_no_psi_cond_lthm = $true) | ($true != sP57)) & ((hoasapinj1_lthm = $true) | (! [X14 : term,X15 : term,X16 : term,X17 : term] : ((X15 = X16) | (((ap @ X16 @ X14)) != ((ap @ X15 @ X17)))) & ! [X18 : term] : (((sub @ X18 @ id)) = X18) & ((((ap @ (sub @ sK467 @ id) @ sK465)) = ((ap @ (sub @ sK464 @ id) @ sK466))) & (sK467 != sK464)))) & (((sK469 != sK470) & (((ap @ sK469 @ sK471)) = ((ap @ sK470 @ sK468)))) | (sK472 != ((sub @ sK472 @ id))) | ! [X28 : term,X29 : term,X30 : term,X31 : term] : ((((ap @ (sub @ X28 @ id) @ X30)) != ((ap @ (sub @ X31 @ id) @ X29))) | (X28 = X31)) | (hoasapinj1_lthm != $true)) & (hoaslamnotvar_lthm = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ((hoasapinj2_lthm = $true) | (((sK476 != sK473) & (((ap @ (sub @ sK474 @ id) @ sK476)) = ((ap @ (sub @ sK475 @ id) @ sK473)))) & ! [X36 : term,X37 : term,X38 : term,X39 : term] : ((((ap @ X36 @ X38)) != ((ap @ X39 @ X37))) | (X37 = X38)))) & (! [X40 : term,X41 : term,X42 : term,X43 : term] : ((X40 = X43) | (((ap @ (sub @ X42 @ id) @ X40)) != ((ap @ (sub @ X41 @ id) @ X43)))) | ((((ap @ sK477 @ sK479)) = ((ap @ sK480 @ sK478))) & (sK478 != sK479)) | (hoasapinj2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (! [X48 : term,X49 : subst] : (((sub @ one @ (push @ X48 @ X49))) = X48) & (! [X54 : term] : (((sK482 @ X54)) = ((sK484 @ (sub @ X54 @ (push @ sK481 @ sK483))))) & ($true = ((sK484 @ sK481))) & ($true != ((sK482 @ one)))))) & ((((sub @ one @ (push @ sK485 @ sK486))) != sK485) | ! [X57 : term,X58 : (term > $o),X59 : subst,X60 : (term > $o)] : ((((X60 @ (sub @ (sK487 @ X60 @ X59 @ X58 @ X57) @ (push @ X57 @ X59)))) != ((X58 @ (sK487 @ X60 @ X59 @ X58 @ X57)))) | ($true != ((X60 @ X57))) | (((X58 @ one)) = $true)) | (pushprop_lem1v2_lthm != $true)) & (axabs = $true) & (! [X62 : term] : (((sub @ X62 @ id)) = X62) | (((sub @ sK488 @ id)) != sK488) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | ((sK489 != ((sub @ sK489 @ id))) & ! [X65 : term] : (((sub @ X65 @ id)) = X65))) & (axvarshift = $true) & (axmap = $true) & (($true = sP52) | (induction2 != $true)) & ((induction2 = $true) | ($true != sP52)) & (substmonoid_gthm = $true) & ((hoasinduction_lem3aaa = $true) | ($true != sP50)) & (($true = sP50) | (hoasinduction_lem3aaa != $true)) & ((pushprop_lem0 = $true) | ! [X69 : (term > $o)] : (((sK490 @ (sub @ (sK493 @ X69) @ (push @ sK492 @ sK491)))) != ((X69 @ (sK493 @ X69))))) & (! [X71 : (term > $o),X72 : subst,X73 : term] : ! [X75 : term] : (((sK494 @ X73 @ X72 @ X71 @ X75)) = ((X71 @ (sub @ X75 @ (push @ X73 @ X72))))) | (pushprop_lem0 != $true)) & ! [X76 : term] : (((d2term @ (sK495 @ X76))) = X76) & (! [X78 : term,X79 : term,X80 : term,X81 : term] : ((((ap @ (sub @ X79 @ id) @ X81)) != ((ap @ (sub @ X78 @ id) @ X80))) | (X80 = X81)) | (hoasapinj2 != $true)) & ((hoasapinj2 = $true) | ((((ap @ (sub @ sK497 @ id) @ sK499)) = ((ap @ (sub @ sK496 @ id) @ sK498))) & (sK498 != sK499))) & (($true = sP49) | (hoasinduction_lem2v2 != $true)) & ((hoasinduction_lem2v2 = $true) | ($true != sP49)) & ! [X86 : d_subst,X87 : d_subst] : ((X86 = X87) | (((d2subst @ X86)) != ((d2subst @ X87)))) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_lem1v2 = $true) & (! [X88 : (subst > term > subst > $o)] : ! [X90 : term] : (((sK500 @ X88 @ X90)) = ((X88 @ id @ X90 @ id))) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem0 = $true) | ! [X92 : (term > $o)] : (((sK501 @ id @ (sK502 @ X92) @ id)) != ((X92 @ (sK502 @ X92))))) & (! [X94 : subst,X95 : term,X96 : (term > $o),X97 : (term > $o)] : (($true != ((X97 @ X95))) | ($true = ((X96 @ one))) | (((X96 @ (sK503 @ X97 @ X96 @ X95 @ X94))) != ((X97 @ (sub @ (sK503 @ X97 @ X96 @ X95 @ X94) @ (push @ X95 @ X94)))))) | (pushprop_lem1v2 != $true)) & ((pushprop_lem1v2 = $true) | (($true = ((sK507 @ sK505))) & ($true != ((sK506 @ one))) & ! [X103 : term] : (((sK507 @ (sub @ X103 @ (push @ sK505 @ sK504)))) = ((sK506 @ X103))))) & (hoasinduction_lem2_gthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (! [X104 : term] : (((sub @ X104 @ id)) = X104) | (axvarid != $true)) & ((axvarid = $true) | (sK508 != ((sub @ sK508 @ id)))) & ((laminj = $true) | ((((lam @ sK510)) = ((lam @ sK509))) & (sK509 != sK510))) & (! [X108 : term,X109 : term] : ((((lam @ X108)) != ((lam @ X109))) | (X108 = X109)) | (laminj != $true)) & (hoaslamnotvar = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (axapp = $true) & (! [X110 : subst] : (((comp @ X110 @ id)) = X110) | (axidr != $true)) & ((axidr = $true) | (sK511 != ((comp @ sK511 @ id)))) & (hoaslamnotap_gthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (($true = sP47) | (hoasinduction_lem3v2a_lthm != $true)) & ((hoasinduction_lem3v2a_lthm = $true) | ($true != sP47)) & ((hoaslamnotap = $true) | (! [X115 : subst,X116 : term,X117 : subst] : (((sub @ (sK513 @ X115 @ X116) @ X117)) = ((sK513 @ (comp @ X115 @ X117) @ (sub @ X116 @ X117)))) & (((lam @ (sK513 @ sh @ one))) = ((ap @ (sub @ sK514 @ id) @ sK512))))) & (! [X118 : term,X119 : (subst > term > term),X120 : term] : ((((sub @ (X119 @ (sK515 @ X119) @ (sK516 @ X119)) @ (sK517 @ X119))) != ((X119 @ (comp @ (sK515 @ X119) @ (sK517 @ X119)) @ (sub @ (sK516 @ X119) @ (sK517 @ X119))))) | (((ap @ (sub @ X120 @ id) @ X118)) != ((lam @ (X119 @ sh @ one))))) | (hoaslamnotap != $true)) & ((apinj2 = $true) | ((sK518 != sK519) & (((ap @ sK521 @ sK519)) = ((ap @ sK520 @ sK518))))) & (! [X128 : term,X129 : term,X130 : term,X131 : term] : ((X128 = X129) | (((ap @ X130 @ X128)) != ((ap @ X131 @ X129)))) | (apinj2 != $true)) & (($true = sP46) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | ($true != sP46)) & (! [X132 : term,X133 : (term > $o)] : ((($true != ((X133 @ (sK522 @ X133)))) & ($true = ((var @ (sK522 @ X133))))) | ($true = ((X133 @ X132))) | ((((X133 @ (sK523 @ X133))) = $true) & ($true != ((X133 @ (lam @ (sK523 @ X133)))))) | ($true = ((sP42 @ X133)))) | (induction != $true)) & ((induction = $true) | (! [X138 : term] : (($true = ((sK525 @ X138))) | ($true != ((var @ X138)))) & ($true != ((sK525 @ sK524))) & ! [X139 : term] : ((((sK525 @ X139)) != $true) | ($true = ((sK525 @ (lam @ X139))))) & ($true != ((sP42 @ sK525))))) & (hoasapinj1_gthm = $true) & ! [X140 : d_term] : (d_one = X140) & ((axscons = $true) | (sK526 != ((push @ (sub @ one @ sK526) @ (comp @ sh @ sK526))))) & (! [X142 : subst] : (((push @ (sub @ one @ X142) @ (comp @ sh @ X142))) = X142) | (axscons != $true)) & (! [X143 : term,X144 : term,X145 : term,X146 : term] : ((((ap @ (sub @ X143 @ id) @ X146)) != ((ap @ (sub @ X145 @ id) @ X144))) | (X143 = X145)) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | ((((ap @ (sub @ sK529 @ id) @ sK528)) = ((ap @ (sub @ sK527 @ id) @ sK530))) & (sK527 != sK529))) & (pushprop_lem2v2_lthm = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((substmonoid = $true) | (((comp @ sK531 @ id)) != sK531) | (((comp @ id @ sK532)) != sK532)) & ((! [X153 : subst] : (((comp @ X153 @ id)) = X153) & ! [X154 : subst] : (((comp @ id @ X154)) = X154)) | (substmonoid != $true)) & (hoasinduction_lem3v2_gthm = $true) & (pushprop_lem0_gthm = $true) & (ulamvar1 != $true) & ((hoasinduction_lthm_3 = $true) | ($true != sP41)) & (($true = sP41) | (hoasinduction_lthm_3 != $true)) & (! [X155 : term,X156 : term] : ((X155 = X156) | (((sub @ X156 @ sh)) != ((sub @ X155 @ sh)))) | (shinj != $true)) & ((shinj = $true) | ((sK533 != sK534) & (((sub @ sK534 @ sh)) = ((sub @ sK533 @ sh))))) & (! [X159 : subst] : (((comp @ id @ X159)) = X159) | (axidl != $true)) & ((axidl = $true) | (((comp @ id @ sK535)) != sK535)) & (! [X161 : term,X162 : subst,X163 : (subst > term > term),X164 : (subst > term > term)] : ((((lam @ (X164 @ sh @ one))) != ((lam @ (X163 @ sh @ one)))) | (((X163 @ X162 @ X161)) = ((X164 @ X162 @ X161))) | (((X163 @ (comp @ (sK537 @ X163) @ (sK538 @ X163)) @ (sub @ (sK536 @ X163) @ (sK538 @ X163)))) != ((sub @ (X163 @ (sK537 @ X163) @ (sK536 @ X163)) @ (sK538 @ X163)))) | (((X164 @ (comp @ (sK539 @ X164) @ (sK541 @ X164)) @ (sub @ (sK540 @ X164) @ (sK541 @ X164)))) != ((sub @ (X164 @ (sK539 @ X164) @ (sK540 @ X164)) @ (sK541 @ X164))))) | (hoaslaminj != $true)) & ((hoaslaminj = $true) | ((((lam @ (sK545 @ sh @ one))) = ((lam @ (sK544 @ sh @ one)))) & (((sK544 @ sK543 @ sK542)) != ((sK545 @ sK543 @ sK542))) & ! [X175 : term,X176 : subst,X177 : subst] : (((sub @ (sK544 @ X176 @ X175) @ X177)) = ((sK544 @ (comp @ X176 @ X177) @ (sub @ X175 @ X177)))) & ! [X178 : subst,X179 : term,X180 : subst] : (((sub @ (sK545 @ X178 @ X179) @ X180)) = ((sK545 @ (comp @ X178 @ X180) @ (sub @ X179 @ X180)))))) & (hoaslaminj_gthm = $true) & (induction2lem_gthm = $true) & (id = ((d2subst @ d_id))) & (hoasinduction_lem1v2_gthm = $true) & (hoasinduction_lem1_lthm = $true) & (! [X181 : term] : ! [X183 : term,X184 : subst] : (((sub @ X181 @ (push @ X183 @ X184))) = ((sK546 @ X181 @ X184 @ X183))) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ! [X186 : (subst > term > term)] : (((X186 @ (sK549 @ X186) @ (sK548 @ X186))) != ((sub @ sK547 @ (push @ (sK548 @ X186) @ (sK549 @ X186)))))) & (hoasinduction_lem3b_gthm = $true) & (lamnotvar = $true) & (pushprop = $true) & (axassoc = $true) & (($true = sP35) | (hoasinduction_lem3_lthm != $true)) & ((hoasinduction_lem3_lthm = $true) | ($true != sP35)) & ! [X189 : d_term,X190 : d_term] : ((X189 = X190) | (((d2term @ X189)) != ((d2term @ X190)))) & (($true = sP32) | (hoasinduction_lem3 != $true)) & ((hoasinduction_lem3 = $true) | ($true != sP32)) & (pushprop_lthm_orig = $true) & (hoasinduction_lem1_gthm = $true) & (! [X191 : (term > $o),X192 : (term > $o),X193 : term,X194 : term,X195 : subst] : (($true != ((X192 @ X193))) | ($true = ((X191 @ (sub @ X193 @ (push @ X194 @ X195)))))) | (pushprop_lem1 != $true)) & ((pushprop_lem1 = $true) | (($true = ((sK551 @ sK552))) & ($true != ((sK550 @ (sub @ sK552 @ (push @ sK553 @ sK554))))))) & (hoasapnotvar = $true) & ((hoasinduction = $true) | ($true != sP30)) & (($true = sP30) | (hoasinduction != $true)) & (hoasinduction_lem2_lthm = $true) & (hoaslamnotap_lthm = $true) & ((((sub @ one @ (push @ sK556 @ sK555))) != sK556) | ! [X203 : term,X204 : subst,X205 : term,X206 : (term > $o),X207 : (term > $o)] : (($true != ((X207 @ X205))) | ($true = ((X206 @ (sub @ X205 @ (push @ X203 @ X204)))))) | (((comp @ sh @ (push @ sK558 @ sK557))) != sK557) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (! [X210 : subst,X211 : term] : (((sub @ one @ (push @ X211 @ X210))) = X211) & (($true = ((sK563 @ sK561))) & ($true != ((sK562 @ (sub @ sK561 @ (push @ sK559 @ sK560)))))) & ! [X217 : subst,X218 : term] : (((comp @ sh @ (push @ X218 @ X217))) = X217))) & (hoaslamnotvar_gthm = $true) & ! [X219 : d_subst] : (d_id = X219) & (hoasapnotvar_lthm = $true) & (hoasinduction_lem2v2_gthm = $true) & (($true = sP27) | (induction2lem != $true)) & ((induction2lem = $true) | ($true != sP27)) & (apnotvar = $true) & (ulamvarind = $true) & (pushprop_lem2v2 = $true) & (lamnotap != $true) & ((hoasinduction_lthm = $true) | ($true != sP25)) & (($true = sP25) | (hoasinduction_lthm != $true)) & ((hoasinduction_lem3aa = $true) | ($true != sP16)) & (($true = sP16) | (hoasinduction_lem3aa != $true)) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (! [X220 : term] : (((sub @ X220 @ id)) = X220) | (termmset != $true)) & ((termmset = $true) | (sK564 != ((sub @ sK564 @ id)))) & (($true = sP15) | (hoasinduction_lem3aa_lthm != $true)) & ((hoasinduction_lem3aa_lthm = $true) | ($true != sP15)) & ((((comp @ id @ sK565)) != sK565) | (sK566 != ((comp @ sK566 @ id))) | (! [X224 : subst] : (((comp @ id @ X224)) = X224) & ! [X225 : subst] : (((comp @ X225 @ id)) = X225)) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | (! [X226 : subst] : (((comp @ id @ X226)) = X226) & ! [X227 : subst] : (((comp @ X227 @ id)) = X227) & ((((comp @ id @ sK567)) != sK567) | (((comp @ sK568 @ id)) != sK568)))) & (($true = sP14) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | ($true != sP14)) & ((hoasinduction_lem2 = $true) | ($true != sP12)) & (($true = sP12) | (hoasinduction_lem2 != $true)) & (($true = sP10) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | ($true != sP10)) & (((d2term @ d_one)) = one) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (hoasinduction_lem3a_gthm = $true) & (hoasapinj2_gthm = $true) & (pushprop_gthm = $true) & ((pushprop_lem0_lthm = $true) | ! [X233 : (term > $o)] : (((X233 @ (sK572 @ X233))) != ((sK570 @ (sub @ (sK572 @ X233) @ (push @ sK571 @ sK569)))))) & (! [X235 : subst,X236 : (term > $o),X237 : term] : ! [X239 : term] : (((X236 @ (sub @ X239 @ (push @ X237 @ X235)))) = ((sK573 @ X237 @ X236 @ X235 @ X239))) | (pushprop_lem0_lthm != $true)) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (! [X240 : term,X241 : subst] : (((comp @ sh @ (push @ X240 @ X241))) = X241) | (axshiftcons != $true)) & ((axshiftcons = $true) | (((comp @ sh @ (push @ sK574 @ sK575))) != sK575)) & ! [X244 : subst] : (((d2subst @ (sK576 @ X244))) = X244) & ((hoasinduction_lem3v2_f_lthm = $true) | ! [X247 : (subst > term > term)] : (((sub @ sK577 @ (push @ (sK578 @ X247) @ (sK579 @ X247)))) != ((X247 @ (sK579 @ X247) @ (sK578 @ X247))))) & (! [X250 : term] : ! [X252 : term,X253 : subst] : (((sub @ X250 @ (push @ X252 @ X253))) = ((sK580 @ X250 @ X253 @ X252))) | (hoasinduction_lem3v2_f_lthm != $true)) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (termmset_gthm = $true) & (axclos = $true) & (hoasinduction_lem1 = $true) & ((hoasinduction_lem3v2 = $true) | ($true != sP9)) & (($true = sP9) | (hoasinduction_lem3v2 != $true)) & ((hoasinduction_lem3v2_lthm = $true) | (sP7 != $true)) & ((sP7 = $true) | (hoasinduction_lem3v2_lthm != $true)) & (pushprop_lem2v2_gthm = $true) & (! [X254 : term,X255 : (subst > term > subst > $o)] : (($true = ((X255 @ id @ X254 @ id))) | (($true = ((X255 @ id @ (sK582 @ X255) @ id))) & ($true != ((X255 @ id @ (ap @ (sub @ (sK581 @ X255) @ id) @ (sK582 @ X255)) @ id))) & ($true = ((X255 @ id @ (sK581 @ X255) @ id)))) | ($true = ((sP5 @ X255)))) | (hoasinduction_no_psi_cond != $true)) & ((hoasinduction_no_psi_cond = $true) | (($true != ((sK584 @ id @ sK583 @ id))) & ! [X260 : term,X261 : term] : (($true != ((sK584 @ id @ X261 @ id))) | ($true = ((sK584 @ id @ (ap @ (sub @ X260 @ id) @ X261) @ id))) | ($true != ((sK584 @ id @ X260 @ id)))) & ($true != ((sP5 @ sK584))))) & (! [X262 : term] : (((sub @ X262 @ (push @ one @ sh))) = ((sK585 @ X262 @ sh @ one))) | (hoasinduction_lem3b != $true)) & ((hoasinduction_lem3b = $true) | ! [X265 : (subst > term > term)] : (((X265 @ sh @ one)) != ((sub @ sK586 @ (push @ one @ sh))))) & ((induction2lem_lthm = $true) | ($true != sP4)) & (($true = sP4) | (induction2lem_lthm != $true)) & ((hoasinduction_lem3v2a = $true) | ($true != sP0)) & (($true = sP0) | (hoasinduction_lem3v2a != $true)) & (induction2_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & (! [X266 : term,X267 : term,X268 : term,X269 : term] : ((((ap @ X266 @ X268)) != ((ap @ X267 @ X269))) | (X266 = X267)) | (apinj1 != $true)) & ((apinj1 = $true) | ((((ap @ sK587 @ sK589)) = ((ap @ sK588 @ sK590))) & (sK588 != sK587))) & (pushprop_lem3v2_lthm = $true) & (hoasapnotvar_gthm = $true)), inference(skolemize,[status(esa),new_symbols(skolem,[sK457,vAPP,sK460,vAPP,sK462,sK463,sK464,sK465,sK466,sK467,sK468,sK469,sK470,sK471,sK472,sK473,sK474,sK475,sK476,sK477,sK478,sK479,sK480,sK481,sK482,sK483,sK484,sK485,sK486,vAPP,sK488,sK489,sK490,sK491,sK492,vAPP,sK496,sK497,sK498,sK499,vAPP,sK501,vAPP,sK504,sK505,sK506,sK507,sK508,sK509,sK510,sK511,sK512,sK513,sK514,vAPP,sK518,sK519,sK520,sK521,vAPP,sK524,sK525,sK526,sK527,sK528,sK529,sK530,sK531,sK532,sK533,sK534,sK535,vAPP,sK542,sK543,sK544,sK545,vAPP,sK547,vAPP,sK550,sK551,sK552,sK553,sK554,sK555,sK556,sK557,sK558,sK559,sK560,sK561,sK562,sK563,sK564,sK565,sK566,sK567,sK568,sK569,sK570,sK571,vAPP,sK574,sK575,vAPP,sK577,vAPP,sK583,sK584,vAPP,sK586,sK587,sK588,sK589,sK590]),skolemize(X0,$thf(sK457)),skolemize(X31,$thf(sK74 @ X25)),skolemize(X31,$thf(sK74 @ X25)),skolemize(X7,$thf(sK460)),skolemize(X31,$thf(sK74 @ X25)),skolemize(X12,$thf(sK462)),skolemize(X13,$thf(sK463)),skolemize(X19,$thf(sK464)),skolemize(X20,$thf(sK465)),skolemize(X21,$thf(sK466)),skolemize(X22,$thf(sK467)),skolemize(X23,$thf(sK468)),skolemize(X24,$thf(sK469)),skolemize(X25,$thf(sK470)),skolemize(X26,$thf(sK471)),skolemize(X27,$thf(sK472)),skolemize(X32,$thf(sK473)),skolemize(X33,$thf(sK474)),skolemize(X34,$thf(sK475)),skolemize(X35,$thf(sK476)),skolemize(X44,$thf(sK477)),skolemize(X45,$thf(sK478)),skolemize(X46,$thf(sK479)),skolemize(X47,$thf(sK480)),skolemize(X50,$thf(sK481)),skolemize(X51,$thf(sK482)),skolemize(X52,$thf(sK483)),skolemize(X53,$thf(sK484)),skolemize(X55,$thf(sK485)),skolemize(X56,$thf(sK486)),skolemize(X31,$thf(sK74 @ X25)),skolemize(X63,$thf(sK488)),skolemize(X64,$thf(sK489)),skolemize(X66,$thf(sK490)),skolemize(X67,$thf(sK491)),skolemize(X68,$thf(sK492)),skolemize(X31,$thf(sK74 @ X25)),skolemize(X31,$thf(sK74 @ X25)),skolemize(X31,$thf(sK74 @ X25)),skolemize(X82,$thf(sK496)),skolemize(X83,$thf(sK497)),skolemize(X84,$thf(sK498)),skolemize(X85,$thf(sK499)),skolemize(X31,$thf(sK74 @ X25)),skolemize(X91,$thf(sK501)),skolemize(X31,$thf(sK74 @ X25)),skolemize(X31,$thf(sK74 @ X25)),skolemize(X99,$thf(sK504)),skolemize(X100,$thf(sK505)),skolemize(X101,$thf(sK506)),skolemize(X102,$thf(sK507)),skolemize(X105,$thf(sK508)),skolemize(X106,$thf(sK509)),skolemize(X107,$thf(sK510)),skolemize(X111,$thf(sK511)),skolemize(X112,$thf(sK512)),skolemize(X113,$thf(sK513)),skolemize(X114,$thf(sK514)),skolemize(X31,$thf(sK74 @ X25)),skolemize(X31,$thf(sK74 @ X25)),skolemize(X31,$thf(sK74 @ X25)),skolemize(X124,$thf(sK518)),skolemize(X125,$thf(sK519)),skolemize(X126,$thf(sK520)),skolemize(X127,$thf(sK521)),skolemize(X31,$thf(sK74 @ X25)),skolemize(X31,$thf(sK74 @ X25)),skolemize(X136,$thf(sK524)),skolemize(X137,$thf(sK525)),skolemize(X141,$thf(sK526)),skolemize(X147,$thf(sK527)),skolemize(X148,$thf(sK528)),skolemize(X149,$thf(sK529)),skolemize(X150,$thf(sK530)),skolemize(X151,$thf(sK531)),skolemize(X152,$thf(sK532)),skolemize(X157,$thf(sK533)),skolemize(X158,$thf(sK534)),skolemize(X160,$thf(sK535)),skolemize(X31,$thf(sK74 @ X25)),skolemize(X31,$thf(sK74 @ X25)),skolemize(X31,$thf(sK74 @ X25)),skolemize(X31,$thf(sK74 @ X25)),skolemize(X31,$thf(sK74 @ X25)),skolemize(X31,$thf(sK74 @ X25)),skolemize(X171,$thf(sK542)),skolemize(X172,$thf(sK543)),skolemize(X173,$thf(sK544)),skolemize(X174,$thf(sK545)),skolemize(X31,$thf(sK74 @ X25)),skolemize(X185,$thf(sK547)),skolemize(X31,$thf(sK74 @ X25)),skolemize(X31,$thf(sK74 @ X25)),skolemize(X196,$thf(sK550)),skolemize(X197,$thf(sK551)),skolemize(X198,$thf(sK552)),skolemize(X199,$thf(sK553)),skolemize(X200,$thf(sK554)),skolemize(X201,$thf(sK555)),skolemize(X202,$thf(sK556)),skolemize(X208,$thf(sK557)),skolemize(X209,$thf(sK558)),skolemize(X212,$thf(sK559)),skolemize(X213,$thf(sK560)),skolemize(X214,$thf(sK561)),skolemize(X215,$thf(sK562)),skolemize(X216,$thf(sK563)),skolemize(X221,$thf(sK564)),skolemize(X222,$thf(sK565)),skolemize(X223,$thf(sK566)),skolemize(X228,$thf(sK567)),skolemize(X229,$thf(sK568)),skolemize(X230,$thf(sK569)),skolemize(X231,$thf(sK570)),skolemize(X232,$thf(sK571)),skolemize(X31,$thf(sK74 @ X25)),skolemize(X31,$thf(sK74 @ X25)),skolemize(X242,$thf(sK574)),skolemize(X243,$thf(sK575)),skolemize(X31,$thf(sK74 @ X25)),skolemize(X246,$thf(sK577)),skolemize(X31,$thf(sK74 @ X25)),skolemize(X31,$thf(sK74 @ X25)),skolemize(X31,$thf(sK74 @ X25)),skolemize(X31,$thf(sK74 @ X25)),skolemize(X31,$thf(sK74 @ X25)),skolemize(X258,$thf(sK583)),skolemize(X259,$thf(sK584)),skolemize(X31,$thf(sK74 @ X25)),skolemize(X264,$thf(sK586)),skolemize(X270,$thf(sK587)),skolemize(X271,$thf(sK588)),skolemize(X272,$thf(sK589)),skolemize(X273,$thf(sK590))],[f262])). thf(f264,plain,( (hoasinduction_lem3v2a != ((!! @ (subst > term > subst > $o) @ (^[Y0 : subst > term > subst > $o]: (!! @ (term > $o) @ (^[Y1 : term > $o]: ((!! @ (subst > term > term) @ (^[Y2 : subst > term > term]: ((!! @ term @ (^[Y3 : term]: (!! @ subst @ (^[Y4 : subst]: (!! @ subst @ (^[Y5 : subst]: ((Y2 @ (comp @ Y5 @ Y4) @ (sub @ Y3 @ Y4)) = (sub @ (Y2 @ Y5 @ Y3) @ Y4)))))))) => ((!! @ term @ (^[Y3 : term]: ((Y0 @ id @ Y3 @ id) => (Y0 @ id @ (Y2 @ id @ Y3) @ id)))) => (Y0 @ id @ (hoaslam @ id @ (^[Y3 : subst]: ((^[Y4 : term]: (Y2 @ Y3 @ Y4))))) @ id))))) => ((hoasinduction_p_and_p_prime @ Y0 @ Y1) => (!! @ term @ (^[Y2 : term]: ((!! @ term @ (^[Y3 : term]: ((Y1 @ Y3) => (Y1 @ (sub @ Y2 @ (push @ Y3 @ id)))))) => (Y1 @ (lam @ Y2)))))))))))))), inference(cnf_transformation,[],[f8])). thf(f749,plain,( ( ! [X5 : (subst > term > term)] : ((((sub @ (X5 @ (sK452 @ X5) @ (sK451 @ X5)) @ (sK450 @ X5))) != ((X5 @ (comp @ (sK452 @ X5) @ (sK450 @ X5)) @ (sub @ (sK451 @ X5) @ (sK450 @ X5))))) | ($true = ((sK447 @ id @ (lam @ (X5 @ sh @ one)) @ id))) | ($true = sP0) | ($true = ((sK447 @ id @ (sK453 @ X5) @ id)))) )), inference(cnf_transformation,[],[f259])). thf(f751,plain,( ( ! [X4 : term] : (($true = sP0) | (((sK447 @ id @ X4 @ id)) = ((sK449 @ X4)))) )), inference(cnf_transformation,[],[f259])). thf(f752,plain,( ( ! [X3 : term] : (($true = ((sK449 @ (sub @ sK448 @ (push @ X3 @ id))))) | ($true = sP0) | ($true != ((sK449 @ X3)))) )), inference(cnf_transformation,[],[f259])). thf(f753,plain,( (((sK449 @ (lam @ sK448))) != $true) | ($true = sP0)), inference(cnf_transformation,[],[f259])). thf(f762,plain,( ($true != sP0) | (hoasinduction_lem3v2a = $true)), inference(cnf_transformation,[],[f263])). thf(f781,plain,( (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2))))))))), inference(cnf_transformation,[],[f263])). thf(f793,plain,( (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one)))))), inference(cnf_transformation,[],[f263])). thf(f794,plain,( (((d2term @ d_one)) = one)), inference(cnf_transformation,[],[f263])). thf(f881,plain,( ( ! [X140 : d_term] : ((d_one = X140)) )), inference(cnf_transformation,[],[f263])). thf(f929,plain,( ( ! [X76 : term] : ((((d2term @ (sK495 @ X76))) = X76)) )), inference(cnf_transformation,[],[f263])). thf(f980,definition,( ( ! [X0 : $o] : (($true = X0) | ($false = X0)) )), introduced(theory,[fool_exhaustiveness_axiom])). thf(f981,plain,( (hoasinduction_lem3v2a != ((!! @ (subst > term > subst > $o) @ (^[Y0 : subst > term > subst > $o]: (!! @ (term > $o) @ (^[Y1 : term > $o]: ((!! @ (subst > term > term) @ (^[Y2 : subst > term > term]: ((!! @ term @ (^[Y3 : term]: (!! @ subst @ (^[Y4 : subst]: (!! @ subst @ (^[Y5 : subst]: ((Y2 @ (comp @ Y5 @ Y4) @ (sub @ Y3 @ Y4)) = (sub @ (Y2 @ Y5 @ Y3) @ Y4)))))))) => ((!! @ term @ (^[Y3 : term]: ((Y0 @ id @ Y3 @ id) => (Y0 @ id @ (Y2 @ id @ Y3) @ id)))) => (Y0 @ id @ ((^[Y3 : subst]: ((^[Y4 : subst > term > term]: (d2term @ d_one)))) @ id @ (^[Y3 : subst]: ((^[Y4 : term]: (Y2 @ Y3 @ Y4))))) @ id))))) => (((^[Y2 : subst > term > subst > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ id @ Y4 @ id) = (Y3 @ Y4))))))) @ Y0 @ Y1) => (!! @ term @ (^[Y2 : term]: ((!! @ term @ (^[Y3 : term]: ((Y1 @ Y3) => (Y1 @ (sub @ Y2 @ (push @ Y3 @ id)))))) => (Y1 @ (lam @ Y2)))))))))))))), inference(definition_unfolding,[],[f264,f793,f781])). thf(f995,plain,( (((!! @ (subst > term > subst > $o) @ (^[Y0 : subst > term > subst > $o]: (!! @ (term > $o) @ (^[Y1 : term > $o]: ((!! @ (subst > term > term) @ (^[Y2 : subst > term > term]: ((!! @ term @ (^[Y3 : term]: (!! @ subst @ (^[Y4 : subst]: (!! @ subst @ (^[Y5 : subst]: ((Y2 @ (comp @ Y5 @ Y4) @ (sub @ Y3 @ Y4)) = (sub @ (Y2 @ Y5 @ Y3) @ Y4)))))))) => ((!! @ term @ (^[Y3 : term]: ((Y0 @ id @ Y3 @ id) => (Y0 @ id @ (Y2 @ id @ Y3) @ id)))) => (Y0 @ id @ ((^[Y3 : subst]: ((^[Y4 : subst > term > term]: (d2term @ d_one)))) @ id @ (^[Y3 : subst]: ((^[Y4 : term]: (Y2 @ Y3 @ Y4))))) @ id))))) => (((^[Y2 : subst > term > subst > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ id @ Y4 @ id) = (Y3 @ Y4))))))) @ Y0 @ Y1) => (!! @ term @ (^[Y2 : term]: ((!! @ term @ (^[Y3 : term]: ((Y1 @ Y3) => (Y1 @ (sub @ Y2 @ (push @ Y3 @ id)))))) => (Y1 @ (lam @ Y2)))))))))))) = $false) | (hoasinduction_lem3v2a = $false)), inference(xor_proxy_clausification,[],[f981])). thf(f996,plain,( ($false = (((^[Y0 : subst > term > subst > $o]: (!! @ (term > $o) @ (^[Y1 : term > $o]: ((!! @ (subst > term > term) @ (^[Y2 : subst > term > term]: ((!! @ term @ (^[Y3 : term]: (!! @ subst @ (^[Y4 : subst]: (!! @ subst @ (^[Y5 : subst]: ((Y2 @ (comp @ Y5 @ Y4) @ (sub @ Y3 @ Y4)) = (sub @ (Y2 @ Y5 @ Y3) @ Y4)))))))) => ((!! @ term @ (^[Y3 : term]: ((Y0 @ id @ Y3 @ id) => (Y0 @ id @ (Y2 @ id @ Y3) @ id)))) => (Y0 @ id @ ((^[Y3 : subst]: ((^[Y4 : subst > term > term]: (d2term @ d_one)))) @ id @ (^[Y3 : subst]: ((^[Y4 : term]: (Y2 @ Y3 @ Y4))))) @ id))))) => (((^[Y2 : subst > term > subst > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ id @ Y4 @ id) = (Y3 @ Y4))))))) @ Y0 @ Y1) => (!! @ term @ (^[Y2 : term]: ((!! @ term @ (^[Y3 : term]: ((Y1 @ Y3) => (Y1 @ (sub @ Y2 @ (push @ Y3 @ id)))))) => (Y1 @ (lam @ Y2)))))))))) @ sK591))) | (hoasinduction_lem3v2a = $false)), inference(sigma_proxy_clausification,[],[f995])). thf(f997,plain,( (hoasinduction_lem3v2a = $false) | (((!! @ (term > $o) @ (^[Y0 : term > $o]: ((!! @ (subst > term > term) @ (^[Y1 : subst > term > term]: ((!! @ term @ (^[Y2 : term]: (!! @ subst @ (^[Y3 : subst]: (!! @ subst @ (^[Y4 : subst]: ((Y1 @ (comp @ Y4 @ Y3) @ (sub @ Y2 @ Y3)) = (sub @ (Y1 @ Y4 @ Y2) @ Y3)))))))) => ((!! @ term @ (^[Y2 : term]: ((sK591 @ id @ Y2 @ id) => (sK591 @ id @ (Y1 @ id @ Y2) @ id)))) => (sK591 @ id @ (d2term @ d_one) @ id))))) => ((!! @ term @ (^[Y1 : term]: ((sK591 @ id @ Y1 @ id) = (Y0 @ Y1)))) => (!! @ term @ (^[Y1 : term]: ((!! @ term @ (^[Y2 : term]: ((Y0 @ Y2) => (Y0 @ (sub @ Y1 @ (push @ Y2 @ id)))))) => (Y0 @ (lam @ Y1)))))))))) = $false)), inference(beta-eta_normalization,[],[f996])). thf(f998,plain,( ($false = (((^[Y0 : term > $o]: ((!! @ (subst > term > term) @ (^[Y1 : subst > term > term]: ((!! @ term @ (^[Y2 : term]: (!! @ subst @ (^[Y3 : subst]: (!! @ subst @ (^[Y4 : subst]: ((Y1 @ (comp @ Y4 @ Y3) @ (sub @ Y2 @ Y3)) = (sub @ (Y1 @ Y4 @ Y2) @ Y3)))))))) => ((!! @ term @ (^[Y2 : term]: ((sK591 @ id @ Y2 @ id) => (sK591 @ id @ (Y1 @ id @ Y2) @ id)))) => (sK591 @ id @ (d2term @ d_one) @ id))))) => ((!! @ term @ (^[Y1 : term]: ((sK591 @ id @ Y1 @ id) = (Y0 @ Y1)))) => (!! @ term @ (^[Y1 : term]: ((!! @ term @ (^[Y2 : term]: ((Y0 @ Y2) => (Y0 @ (sub @ Y1 @ (push @ Y2 @ id)))))) => (Y0 @ (lam @ Y1)))))))) @ sK592))) | (hoasinduction_lem3v2a = $false)), inference(sigma_proxy_clausification,[],[f997])). thf(f999,plain,( (hoasinduction_lem3v2a = $false) | ($false = (((!! @ (subst > term > term) @ (^[Y0 : subst > term > term]: ((!! @ term @ (^[Y1 : term]: (!! @ subst @ (^[Y2 : subst]: (!! @ subst @ (^[Y3 : subst]: ((Y0 @ (comp @ Y3 @ Y2) @ (sub @ Y1 @ Y2)) = (sub @ (Y0 @ Y3 @ Y1) @ Y2)))))))) => ((!! @ term @ (^[Y1 : term]: ((sK591 @ id @ Y1 @ id) => (sK591 @ id @ (Y0 @ id @ Y1) @ id)))) => (sK591 @ id @ (d2term @ d_one) @ id))))) => ((!! @ term @ (^[Y0 : term]: ((sK591 @ id @ Y0 @ id) = (sK592 @ Y0)))) => (!! @ term @ (^[Y0 : term]: ((!! @ term @ (^[Y1 : term]: ((sK592 @ Y1) => (sK592 @ (sub @ Y0 @ (push @ Y1 @ id)))))) => (sK592 @ (lam @ Y0)))))))))), inference(beta-eta_normalization,[],[f998])). thf(f1000,plain,( (hoasinduction_lem3v2a = $false) | ($false = (((!! @ term @ (^[Y0 : term]: ((sK591 @ id @ Y0 @ id) = (sK592 @ Y0)))) => (!! @ term @ (^[Y0 : term]: ((!! @ term @ (^[Y1 : term]: ((sK592 @ Y1) => (sK592 @ (sub @ Y0 @ (push @ Y1 @ id)))))) => (sK592 @ (lam @ Y0))))))))), inference(imp_proxy_clausification,[],[f999])). thf(f1001,plain,( ($true = ((!! @ (subst > term > term) @ (^[Y0 : subst > term > term]: ((!! @ term @ (^[Y1 : term]: (!! @ subst @ (^[Y2 : subst]: (!! @ subst @ (^[Y3 : subst]: ((Y0 @ (comp @ Y3 @ Y2) @ (sub @ Y1 @ Y2)) = (sub @ (Y0 @ Y3 @ Y1) @ Y2)))))))) => ((!! @ term @ (^[Y1 : term]: ((sK591 @ id @ Y1 @ id) => (sK591 @ id @ (Y0 @ id @ Y1) @ id)))) => (sK591 @ id @ (d2term @ d_one) @ id))))))) | (hoasinduction_lem3v2a = $false)), inference(imp_proxy_clausification,[],[f999])). thf(f1002,plain,( ( ! [X1 : (subst > term > term)] : (($true = (((^[Y0 : subst > term > term]: ((!! @ term @ (^[Y1 : term]: (!! @ subst @ (^[Y2 : subst]: (!! @ subst @ (^[Y3 : subst]: ((Y0 @ (comp @ Y3 @ Y2) @ (sub @ Y1 @ Y2)) = (sub @ (Y0 @ Y3 @ Y1) @ Y2)))))))) => ((!! @ term @ (^[Y1 : term]: ((sK591 @ id @ Y1 @ id) => (sK591 @ id @ (Y0 @ id @ Y1) @ id)))) => (sK591 @ id @ (d2term @ d_one) @ id)))) @ X1))) | (hoasinduction_lem3v2a = $false)) )), inference(pi_proxy_clausification,[],[f1001])). thf(f1003,plain,( ( ! [X1 : (subst > term > term)] : (($true = (((!! @ term @ (^[Y0 : term]: (!! @ subst @ (^[Y1 : subst]: (!! @ subst @ (^[Y2 : subst]: ((X1 @ (comp @ Y2 @ Y1) @ (sub @ Y0 @ Y1)) = (sub @ (X1 @ Y2 @ Y0) @ Y1)))))))) => ((!! @ term @ (^[Y0 : term]: ((sK591 @ id @ Y0 @ id) => (sK591 @ id @ (X1 @ id @ Y0) @ id)))) => (sK591 @ id @ (d2term @ d_one) @ id))))) | (hoasinduction_lem3v2a = $false)) )), inference(beta-eta_normalization,[],[f1002])). thf(f1004,plain,( ( ! [X1 : (subst > term > term)] : (($true = (((!! @ term @ (^[Y0 : term]: ((sK591 @ id @ Y0 @ id) => (sK591 @ id @ (X1 @ id @ Y0) @ id)))) => (sK591 @ id @ (d2term @ d_one) @ id)))) | ($false = ((!! @ term @ (^[Y0 : term]: (!! @ subst @ (^[Y1 : subst]: (!! @ subst @ (^[Y2 : subst]: ((X1 @ (comp @ Y2 @ Y1) @ (sub @ Y0 @ Y1)) = (sub @ (X1 @ Y2 @ Y0) @ Y1)))))))))) | (hoasinduction_lem3v2a = $false)) )), inference(imp_proxy_clausification,[],[f1003])). thf(f1005,plain,( ( ! [X1 : (subst > term > term)] : (($true = ((sK591 @ id @ (d2term @ d_one) @ id))) | ($false = ((!! @ term @ (^[Y0 : term]: (!! @ subst @ (^[Y1 : subst]: (!! @ subst @ (^[Y2 : subst]: ((X1 @ (comp @ Y2 @ Y1) @ (sub @ Y0 @ Y1)) = (sub @ (X1 @ Y2 @ Y0) @ Y1)))))))))) | ($false = ((!! @ term @ (^[Y0 : term]: ((sK591 @ id @ Y0 @ id) => (sK591 @ id @ (X1 @ id @ Y0) @ id)))))) | (hoasinduction_lem3v2a = $false)) )), inference(imp_proxy_clausification,[],[f1004])). thf(f1006,plain,( ( ! [X1 : (subst > term > term)] : (($false = ((!! @ term @ (^[Y0 : term]: ((sK591 @ id @ Y0 @ id) => (sK591 @ id @ (X1 @ id @ Y0) @ id)))))) | (hoasinduction_lem3v2a = $false) | ($false = (((^[Y0 : term]: (!! @ subst @ (^[Y1 : subst]: (!! @ subst @ (^[Y2 : subst]: ((X1 @ (comp @ Y2 @ Y1) @ (sub @ Y0 @ Y1)) = (sub @ (X1 @ Y2 @ Y0) @ Y1))))))) @ (sK593 @ X1)))) | ($true = ((sK591 @ id @ (d2term @ d_one) @ id)))) )), inference(sigma_proxy_clausification,[],[f1005])). thf(f1007,plain,( ( ! [X1 : (subst > term > term)] : ((hoasinduction_lem3v2a = $false) | ($false = (((^[Y0 : term]: (!! @ subst @ (^[Y1 : subst]: (!! @ subst @ (^[Y2 : subst]: ((X1 @ (comp @ Y2 @ Y1) @ (sub @ Y0 @ Y1)) = (sub @ (X1 @ Y2 @ Y0) @ Y1))))))) @ (sK593 @ X1)))) | ($false = (((^[Y0 : term]: ((sK591 @ id @ Y0 @ id) => (sK591 @ id @ (X1 @ id @ Y0) @ id))) @ (sK594 @ X1)))) | ($true = ((sK591 @ id @ (d2term @ d_one) @ id)))) )), inference(sigma_proxy_clausification,[],[f1006])). thf(f1008,plain,( ( ! [X1 : (subst > term > term)] : ((hoasinduction_lem3v2a = $false) | ($false = ((!! @ subst @ (^[Y0 : subst]: (!! @ subst @ (^[Y1 : subst]: ((X1 @ (comp @ Y1 @ Y0) @ (sub @ (sK593 @ X1) @ Y0)) = (sub @ (X1 @ Y1 @ (sK593 @ X1)) @ Y0)))))))) | ((((sK591 @ id @ (sK594 @ X1) @ id) => (sK591 @ id @ (X1 @ id @ (sK594 @ X1)) @ id))) = $false) | ($true = ((sK591 @ id @ (d2term @ d_one) @ id)))) )), inference(beta-eta_normalization,[],[f1007])). thf(f1009,plain,( ( ! [X1 : (subst > term > term)] : (($true = ((sK591 @ id @ (d2term @ d_one) @ id))) | ((((sK591 @ id @ (sK594 @ X1) @ id) => (sK591 @ id @ (X1 @ id @ (sK594 @ X1)) @ id))) = $false) | (hoasinduction_lem3v2a = $false) | ((((^[Y0 : subst]: (!! @ subst @ (^[Y1 : subst]: ((X1 @ (comp @ Y1 @ Y0) @ (sub @ (sK593 @ X1) @ Y0)) = (sub @ (X1 @ Y1 @ (sK593 @ X1)) @ Y0))))) @ (sK595 @ X1))) = $false)) )), inference(sigma_proxy_clausification,[],[f1008])). thf(f1011,plain,( ( ! [X1 : (subst > term > term)] : (((((^[Y0 : subst]: (!! @ subst @ (^[Y1 : subst]: ((X1 @ (comp @ Y1 @ Y0) @ (sub @ (sK593 @ X1) @ Y0)) = (sub @ (X1 @ Y1 @ (sK593 @ X1)) @ Y0))))) @ (sK595 @ X1))) = $false) | ($true = ((sK591 @ id @ (sK594 @ X1) @ id))) | ($true = ((sK591 @ id @ (d2term @ d_one) @ id))) | (hoasinduction_lem3v2a = $false)) )), inference(imp_proxy_clausification,[],[f1009])). thf(f1012,plain,( ( ! [X1 : (subst > term > term)] : ((hoasinduction_lem3v2a = $false) | ($true = ((sK591 @ id @ (sK594 @ X1) @ id))) | ($true = ((sK591 @ id @ (d2term @ d_one) @ id))) | (((!! @ subst @ (^[Y0 : subst]: ((X1 @ (comp @ Y0 @ (sK595 @ X1)) @ (sub @ (sK593 @ X1) @ (sK595 @ X1))) = (sub @ (X1 @ Y0 @ (sK593 @ X1)) @ (sK595 @ X1)))))) = $false)) )), inference(beta-eta_normalization,[],[f1011])). thf(f1013,plain,( ( ! [X1 : (subst > term > term)] : (($true = ((sK591 @ id @ (d2term @ d_one) @ id))) | (hoasinduction_lem3v2a = $false) | ($true = ((sK591 @ id @ (sK594 @ X1) @ id))) | ($false = (((^[Y0 : subst]: ((X1 @ (comp @ Y0 @ (sK595 @ X1)) @ (sub @ (sK593 @ X1) @ (sK595 @ X1))) = (sub @ (X1 @ Y0 @ (sK593 @ X1)) @ (sK595 @ X1)))) @ (sK596 @ X1))))) )), inference(sigma_proxy_clausification,[],[f1012])). thf(f1014,plain,( ( ! [X1 : (subst > term > term)] : (($true = ((sK591 @ id @ (sK594 @ X1) @ id))) | ($true = ((sK591 @ id @ (d2term @ d_one) @ id))) | ((((X1 @ (comp @ (sK596 @ X1) @ (sK595 @ X1)) @ (sub @ (sK593 @ X1) @ (sK595 @ X1))) = (sub @ (X1 @ (sK596 @ X1) @ (sK593 @ X1)) @ (sK595 @ X1)))) = $false) | (hoasinduction_lem3v2a = $false)) )), inference(beta-eta_normalization,[],[f1013])). thf(f1015,plain,( ( ! [X1 : (subst > term > term)] : (($true = ((sK591 @ id @ (d2term @ d_one) @ id))) | (hoasinduction_lem3v2a = $false) | ($true = ((sK591 @ id @ (sK594 @ X1) @ id))) | (((sub @ (X1 @ (sK596 @ X1) @ (sK593 @ X1)) @ (sK595 @ X1))) != ((X1 @ (comp @ (sK596 @ X1) @ (sK595 @ X1)) @ (sub @ (sK593 @ X1) @ (sK595 @ X1)))))) )), inference(equality_proxy_clausification,[],[f1014])). thf(f1020,plain,( (hoasinduction_lem3v2a = $false) | (((!! @ term @ (^[Y0 : term]: ((!! @ term @ (^[Y1 : term]: ((sK592 @ Y1) => (sK592 @ (sub @ Y0 @ (push @ Y1 @ id)))))) => (sK592 @ (lam @ Y0)))))) = $false)), inference(imp_proxy_clausification,[],[f1000])). thf(f1021,plain,( (hoasinduction_lem3v2a = $false) | ($true = ((!! @ term @ (^[Y0 : term]: ((sK591 @ id @ Y0 @ id) = (sK592 @ Y0))))))), inference(imp_proxy_clausification,[],[f1000])). thf(f1022,plain,( ( ! [X1 : term] : ((hoasinduction_lem3v2a = $false) | ($true = (((^[Y0 : term]: ((sK591 @ id @ Y0 @ id) = (sK592 @ Y0))) @ X1)))) )), inference(pi_proxy_clausification,[],[f1021])). thf(f1023,plain,( ( ! [X1 : term] : (($true = (((sK591 @ id @ X1 @ id) = (sK592 @ X1)))) | (hoasinduction_lem3v2a = $false)) )), inference(beta-eta_normalization,[],[f1022])). thf(f1024,plain,( ( ! [X1 : term] : ((((sK592 @ X1)) = ((sK591 @ id @ X1 @ id))) | (hoasinduction_lem3v2a = $false)) )), inference(equality_proxy_clausification,[],[f1023])). thf(f1025,plain,( ( ! [X1 : term] : (($true = ((sK592 @ X1))) | (((sK591 @ id @ X1 @ id)) = $false) | (hoasinduction_lem3v2a = $false)) )), inference(iff_proxy_clausification,[],[f1024])). thf(f1027,plain,( ($false = (((^[Y0 : term]: ((!! @ term @ (^[Y1 : term]: ((sK592 @ Y1) => (sK592 @ (sub @ Y0 @ (push @ Y1 @ id)))))) => (sK592 @ (lam @ Y0)))) @ sK598))) | (hoasinduction_lem3v2a = $false)), inference(sigma_proxy_clausification,[],[f1020])). thf(f1028,plain,( ($false = (((!! @ term @ (^[Y0 : term]: ((sK592 @ Y0) => (sK592 @ (sub @ sK598 @ (push @ Y0 @ id)))))) => (sK592 @ (lam @ sK598))))) | (hoasinduction_lem3v2a = $false)), inference(beta-eta_normalization,[],[f1027])). thf(f1029,plain,( ($false = ((sK592 @ (lam @ sK598)))) | (hoasinduction_lem3v2a = $false)), inference(imp_proxy_clausification,[],[f1028])). thf(f1030,plain,( ($true = ((!! @ term @ (^[Y0 : term]: ((sK592 @ Y0) => (sK592 @ (sub @ sK598 @ (push @ Y0 @ id)))))))) | (hoasinduction_lem3v2a = $false)), inference(imp_proxy_clausification,[],[f1028])). thf(f1031,plain,( ( ! [X1 : term] : (($true = (((^[Y0 : term]: ((sK592 @ Y0) => (sK592 @ (sub @ sK598 @ (push @ Y0 @ id))))) @ X1))) | (hoasinduction_lem3v2a = $false)) )), inference(pi_proxy_clausification,[],[f1030])). thf(f1032,plain,( ( ! [X1 : term] : ((hoasinduction_lem3v2a = $false) | ($true = (((sK592 @ X1) => (sK592 @ (sub @ sK598 @ (push @ X1 @ id))))))) )), inference(beta-eta_normalization,[],[f1031])). thf(f1033,plain,( ( ! [X1 : term] : (($true = ((sK592 @ (sub @ sK598 @ (push @ X1 @ id))))) | (((sK592 @ X1)) = $false) | (hoasinduction_lem3v2a = $false)) )), inference(imp_proxy_clausification,[],[f1032])). thf(f1231,plain,( ( ! [X4 : term] : (($false = ((sK447 @ id @ X4 @ id))) | ($true = ((sK449 @ X4))) | ($true = sP0)) )), inference(iff_proxy_clausification,[],[f751])). thf(f1549,definition,( spl605_74 <=> ($true = sP0)), introduced(definition,[new_symbols(definition,[spl605_74])],[avatar_definition])). thf(f1640,definition,( spl605_95 <=> (hoasinduction_lem3v2a = $false)), introduced(definition,[new_symbols(definition,[spl605_95])],[avatar_definition])). thf(f1642,plain,( (hoasinduction_lem3v2a = $false) | ~spl605_95), inference(avatar_component_clause,[],[f1640])). thf(f1644,definition,( spl605_96 <=> ($true = ((sK591 @ id @ (d2term @ d_one) @ id)))), introduced(definition,[new_symbols(definition,[spl605_96])],[avatar_definition])). thf(f1646,plain,( ($true = ((sK591 @ id @ (d2term @ d_one) @ id))) | ~spl605_96), inference(avatar_component_clause,[],[f1644])). thf(f1648,definition,( spl605_97 <=> ! [X1 : (subst > term > term)] : (($true = ((sK591 @ id @ (sK594 @ X1) @ id))) | (((sub @ (X1 @ (sK596 @ X1) @ (sK593 @ X1)) @ (sK595 @ X1))) != ((X1 @ (comp @ (sK596 @ X1) @ (sK595 @ X1)) @ (sub @ (sK593 @ X1) @ (sK595 @ X1))))))), introduced(definition,[new_symbols(definition,[spl605_97])],[avatar_definition])). thf(f1649,plain,( ( ! [X1 : (subst > term > term)] : ((((sub @ (X1 @ (sK596 @ X1) @ (sK593 @ X1)) @ (sK595 @ X1))) != ((X1 @ (comp @ (sK596 @ X1) @ (sK595 @ X1)) @ (sub @ (sK593 @ X1) @ (sK595 @ X1))))) | ($true = ((sK591 @ id @ (sK594 @ X1) @ id)))) ) | ~spl605_97), inference(avatar_component_clause,[],[f1648])). thf(f1650,plain,( spl605_95 | spl605_96 | spl605_97), inference(avatar_split_clause,[],[f1015,f1648,f1644,f1640])). thf(f1660,definition,( spl605_100 <=> ! [X1 : term] : (($true = ((sK592 @ X1))) | (((sK591 @ id @ X1 @ id)) = $false))), introduced(definition,[new_symbols(definition,[spl605_100])],[avatar_definition])). thf(f1661,plain,( ( ! [X1 : term] : ((((sK591 @ id @ X1 @ id)) = $false) | ($true = ((sK592 @ X1)))) ) | ~spl605_100), inference(avatar_component_clause,[],[f1660])). thf(f1662,plain,( spl605_95 | spl605_100), inference(avatar_split_clause,[],[f1025,f1660,f1640])). thf(f1664,definition,( spl605_101 <=> ! [X1 : term] : (($true = ((sK592 @ (sub @ sK598 @ (push @ X1 @ id))))) | (((sK592 @ X1)) = $false))), introduced(definition,[new_symbols(definition,[spl605_101])],[avatar_definition])). thf(f1665,plain,( ( ! [X1 : term] : ((((sK592 @ X1)) = $false) | ($true = ((sK592 @ (sub @ sK598 @ (push @ X1 @ id)))))) ) | ~spl605_101), inference(avatar_component_clause,[],[f1664])). thf(f1666,plain,( spl605_95 | spl605_101), inference(avatar_split_clause,[],[f1033,f1664,f1640])). thf(f1668,definition,( spl605_102 <=> ($false = ((sK592 @ (lam @ sK598))))), introduced(definition,[new_symbols(definition,[spl605_102])],[avatar_definition])). thf(f1670,plain,( ($false = ((sK592 @ (lam @ sK598)))) | ~spl605_102), inference(avatar_component_clause,[],[f1668])). thf(f1671,plain,( spl605_102 | spl605_95), inference(avatar_split_clause,[],[f1029,f1640,f1668])). thf(f1673,definition,( spl605_103 <=> (hoasinduction_lem3v2a = $true)), introduced(definition,[new_symbols(definition,[spl605_103])],[avatar_definition])). thf(f1675,plain,( (hoasinduction_lem3v2a = $true) | ~spl605_103), inference(avatar_component_clause,[],[f1673])). thf(f1768,plain,( ( ! [X76 : term] : ((((d2term @ d_one)) = X76)) )), inference(forward_demodulation,[],[f929,f881])). thf(f2786,definition,( spl605_363 <=> ! [X5 : (subst > term > term)] : ((((sub @ (X5 @ (sK452 @ X5) @ (sK451 @ X5)) @ (sK450 @ X5))) != ((X5 @ (comp @ (sK452 @ X5) @ (sK450 @ X5)) @ (sub @ (sK451 @ X5) @ (sK450 @ X5))))) | ($true = ((sK447 @ id @ (sK453 @ X5) @ id))) | ($true = ((sK447 @ id @ (lam @ (X5 @ sh @ one)) @ id))))), introduced(definition,[new_symbols(definition,[spl605_363])],[avatar_definition])). thf(f2787,plain,( ( ! [X5 : (subst > term > term)] : (($true = ((sK447 @ id @ (sK453 @ X5) @ id))) | (((sub @ (X5 @ (sK452 @ X5) @ (sK451 @ X5)) @ (sK450 @ X5))) != ((X5 @ (comp @ (sK452 @ X5) @ (sK450 @ X5)) @ (sub @ (sK451 @ X5) @ (sK450 @ X5))))) | ($true = ((sK447 @ id @ (lam @ (X5 @ sh @ one)) @ id)))) ) | ~spl605_363), inference(avatar_component_clause,[],[f2786])). thf(f2788,plain,( spl605_363 | spl605_74), inference(avatar_split_clause,[],[f749,f1549,f2786])). thf(f2970,definition,( spl605_405 <=> (((sK449 @ (lam @ sK448))) = $true)), introduced(definition,[new_symbols(definition,[spl605_405])],[avatar_definition])). thf(f2972,plain,( (((sK449 @ (lam @ sK448))) != $true) | spl605_405), inference(avatar_component_clause,[],[f2970])). thf(f2973,plain,( ~spl605_405 | spl605_74), inference(avatar_split_clause,[],[f753,f1549,f2970])). thf(f3090,definition,( spl605_433 <=> ! [X3 : term] : (($true = ((sK449 @ (sub @ sK448 @ (push @ X3 @ id))))) | ($true != ((sK449 @ X3))))), introduced(definition,[new_symbols(definition,[spl605_433])],[avatar_definition])). thf(f3091,plain,( ( ! [X3 : term] : (($true != ((sK449 @ X3))) | ($true = ((sK449 @ (sub @ sK448 @ (push @ X3 @ id)))))) ) | ~spl605_433), inference(avatar_component_clause,[],[f3090])). thf(f3092,plain,( spl605_433 | spl605_74), inference(avatar_split_clause,[],[f752,f1549,f3090])). thf(f3751,plain,( spl605_103 | ~spl605_74), inference(avatar_split_clause,[],[f762,f1549,f1673])). thf(f3853,definition,( spl605_605 <=> ! [X4 : term] : (($false = ((sK447 @ id @ X4 @ id))) | ($true = ((sK449 @ X4))))), introduced(definition,[new_symbols(definition,[spl605_605])],[avatar_definition])). thf(f3854,plain,( ( ! [X4 : term] : (($false = ((sK447 @ id @ X4 @ id))) | ($true = ((sK449 @ X4)))) ) | ~spl605_605), inference(avatar_component_clause,[],[f3853])). thf(f3855,plain,( spl605_74 | spl605_605), inference(avatar_split_clause,[],[f1231,f3853,f1549])). thf(f4220,plain,( ( ! [X5 : (subst > term > term)] : ((((sub @ (X5 @ (sK452 @ X5) @ (sK451 @ X5)) @ (sK450 @ X5))) != ((X5 @ (comp @ (sK452 @ X5) @ (sK450 @ X5)) @ (sub @ (sK451 @ X5) @ (sK450 @ X5))))) | ($true = ((sK447 @ id @ (d2term @ d_one) @ id))) | ($true = ((sK447 @ id @ (lam @ (X5 @ sh @ one)) @ id)))) ) | ~spl605_363), inference(forward_demodulation,[],[f2787,f1768])). thf(f4238,plain,( ($true != ((sK449 @ (d2term @ d_one)))) | spl605_405), inference(forward_demodulation,[],[f2972,f1768])). thf(f4307,plain,( ( ! [X3 : term] : (($true != ((sK449 @ X3))) | ($true = ((sK449 @ (d2term @ d_one))))) ) | ~spl605_433), inference(forward_demodulation,[],[f3091,f1768])). thf(f4383,plain,( ( ! [X5 : (subst > term > term)] : (($true = ((sK447 @ id @ (lam @ (X5 @ sh @ one)) @ id))) | (((d2term @ d_one)) != ((sub @ (X5 @ (sK452 @ X5) @ (sK451 @ X5)) @ (sK450 @ X5)))) | ($true = ((sK447 @ id @ (d2term @ d_one) @ id)))) ) | ~spl605_363), inference(forward_demodulation,[],[f4220,f1768])). thf(f4406,plain,( ($true != ((sK449 @ one))) | spl605_405), inference(forward_demodulation,[],[f4238,f794])). thf(f4488,plain,( ( ! [X3 : term] : (($true != ((sK449 @ X3))) | ($true = ((sK449 @ one)))) ) | ~spl605_433), inference(forward_demodulation,[],[f4307,f794])). thf(f4562,plain,( ( ! [X5 : (subst > term > term)] : (($true = ((sK447 @ id @ (lam @ (X5 @ sh @ one)) @ id))) | ($true = ((sK447 @ id @ (d2term @ d_one) @ id)))) ) | ~spl605_363), inference(forward_subsumption_resolution,[],[f4383,f1768])). thf(f4632,plain,( ( ! [X3 : term] : (($true != ((sK449 @ X3)))) ) | (spl605_405 | ~spl605_433)), inference(forward_subsumption_resolution,[],[f4488,f4406])). thf(f4698,plain,( ($true = ((sK447 @ id @ (d2term @ d_one) @ id))) | ($true = ((sK447 @ id @ (d2term @ d_one) @ id))) | ~spl605_363), inference(forward_demodulation,[],[f4562,f1768])). thf(f4699,plain,( ($true = ((sK447 @ id @ (d2term @ d_one) @ id))) | ~spl605_363), inference(duplicate_literal_removal,[],[f4698])). thf(f4783,plain,( ($true = ((sK447 @ id @ one @ id))) | ~spl605_363), inference(forward_demodulation,[],[f4699,f794])). thf(f5103,plain,( ($true = $false) | (~spl605_95 | ~spl605_103)), inference(forward_demodulation,[],[f1675,f1642])). thf(f5104,plain,( $false | (~spl605_95 | ~spl605_103)), inference(trivial_inequality_removal,[],[f5103])). thf(f5105,plain,( ~spl605_95 | ~spl605_103), inference(avatar_contradiction_clause,[],[f5104])). thf(f5866,plain,( ($false = ((sK592 @ (d2term @ d_one)))) | ~spl605_102), inference(forward_demodulation,[],[f1670,f1768])). thf(f5871,plain,( ($true = ((sK591 @ id @ one @ id))) | ~spl605_96), inference(forward_demodulation,[],[f1646,f794])). thf(f5875,plain,( ( ! [X1 : term] : (($true = ((sK592 @ (d2term @ d_one)))) | (((sK592 @ X1)) = $false)) ) | ~spl605_101), inference(forward_demodulation,[],[f1665,f1768])). thf(f5909,plain,( ($false = ((sK592 @ one))) | ~spl605_102), inference(forward_demodulation,[],[f5866,f794])). thf(f5913,plain,( ( ! [X1 : term] : ((((sK592 @ X1)) = $false) | ($true = ((sK592 @ one)))) ) | ~spl605_101), inference(forward_demodulation,[],[f5875,f794])). thf(f5939,plain,( ( ! [X1 : term] : ((((sK592 @ X1)) = $false) | ($true = $false)) ) | (~spl605_101 | ~spl605_102)), inference(forward_demodulation,[],[f5913,f5909])). thf(f5940,plain,( ( ! [X1 : term] : ((((sK592 @ X1)) = $false)) ) | (~spl605_101 | ~spl605_102)), inference(trivial_inequality_removal,[],[f5939])). thf(f5964,plain,( ( ! [X1 : term] : (($true = $false) | (((sK591 @ id @ X1 @ id)) = $false)) ) | (~spl605_100 | ~spl605_101 | ~spl605_102)), inference(backward_demodulation,[],[f1661,f5940])). thf(f5965,plain,( ( ! [X1 : term] : ((((sK591 @ id @ X1 @ id)) = $false)) ) | (~spl605_100 | ~spl605_101 | ~spl605_102)), inference(trivial_inequality_removal,[],[f5964])). thf(f5976,plain,( ($true = $false) | (~spl605_96 | ~spl605_100 | ~spl605_101 | ~spl605_102)), inference(backward_demodulation,[],[f5871,f5965])). thf(f5977,plain,( $false | (~spl605_96 | ~spl605_100 | ~spl605_101 | ~spl605_102)), inference(trivial_inequality_removal,[],[f5976])). thf(f5978,plain,( ~spl605_96 | ~spl605_100 | ~spl605_101 | ~spl605_102), inference(avatar_contradiction_clause,[],[f5977])). thf(f6136,plain,( ( ! [X1 : (subst > term > term)] : (($true = ((sK591 @ id @ (sK594 @ X1) @ id))) | (((d2term @ d_one)) != ((sub @ (X1 @ (sK596 @ X1) @ (sK593 @ X1)) @ (sK595 @ X1))))) ) | ~spl605_97), inference(forward_demodulation,[],[f1649,f1768])). thf(f6166,plain,( ( ! [X1 : (subst > term > term)] : (($true = ((sK591 @ id @ (sK594 @ X1) @ id)))) ) | ~spl605_97), inference(forward_subsumption_resolution,[],[f6136,f1768])). thf(f6174,plain,( ($true = $false) | (~spl605_97 | ~spl605_100 | ~spl605_101 | ~spl605_102)), inference(forward_demodulation,[],[f6166,f5965])). thf(f6175,plain,( $false | (~spl605_97 | ~spl605_100 | ~spl605_101 | ~spl605_102)), inference(trivial_inequality_removal,[],[f6174])). thf(f6176,plain,( ~spl605_97 | ~spl605_100 | ~spl605_101 | ~spl605_102), inference(avatar_contradiction_clause,[],[f6175])). thf(f6695,plain,( ( ! [X0 : term] : (($true != $true) | ($false = ((sK449 @ X0)))) ) | (spl605_405 | ~spl605_433)), inference(constrained_superposition,[],[f4632,f980])). thf(f6696,plain,( ( ! [X0 : term] : (($false = ((sK449 @ X0)))) ) | (spl605_405 | ~spl605_433)), inference(trivial_inequality_removal,[],[f6695])). thf(f6698,plain,( ( ! [X4 : term] : (($true = $false) | ($false = ((sK447 @ id @ X4 @ id)))) ) | (spl605_405 | ~spl605_433 | ~spl605_605)), inference(backward_demodulation,[],[f3854,f6696])). thf(f6699,plain,( ( ! [X4 : term] : (($false = ((sK447 @ id @ X4 @ id)))) ) | (spl605_405 | ~spl605_433 | ~spl605_605)), inference(trivial_inequality_removal,[],[f6698])). thf(f6700,plain,( ($true = $false) | (~spl605_363 | spl605_405 | ~spl605_433 | ~spl605_605)), inference(backward_demodulation,[],[f4783,f6699])). thf(f6701,plain,( $false | (~spl605_363 | spl605_405 | ~spl605_433 | ~spl605_605)), inference(trivial_inequality_removal,[],[f6700])). thf(f6702,plain,( ~spl605_363 | spl605_405 | ~spl605_433 | ~spl605_605), inference(avatar_contradiction_clause,[],[f6701])). cnf(s57, plain, spl605_95 | spl605_96 | spl605_97, inference(sat_conversion,[],[f1650])). cnf(s60, plain, spl605_95 | spl605_100, inference(sat_conversion,[],[f1662])). cnf(s61, plain, spl605_95 | spl605_101, inference(sat_conversion,[],[f1666])). cnf(s62, plain, spl605_95 | spl605_102, inference(sat_conversion,[],[f1671])). cnf(s300, plain, spl605_74 | spl605_363, inference(sat_conversion,[],[f2788])). cnf(s343, plain, spl605_74 | ~spl605_405, inference(sat_conversion,[],[f2973])). cnf(s371, plain, spl605_74 | spl605_433, inference(sat_conversion,[],[f3092])). cnf(s543, plain, ~spl605_74 | spl605_103, inference(sat_conversion,[],[f3751])). cnf(s570, plain, spl605_74 | spl605_605, inference(sat_conversion,[],[f3855])). cnf(s681, plain, ~spl605_95 | ~spl605_103, inference(sat_conversion,[],[f5105])). cnf(s712, plain, ~spl605_96 | ~spl605_100 | ~spl605_101 | ~spl605_102, inference(sat_conversion,[],[f5978])). cnf(s732, plain, ~spl605_97 | ~spl605_100 | ~spl605_101 | ~spl605_102, inference(sat_conversion,[],[f6176])). cnf(s776, plain, ~spl605_363 | spl605_405 | ~spl605_433 | ~spl605_605, inference(sat_conversion,[],[f6702])). cnf(s908, plain, spl605_74, inference(rat,[],[s776,s300,s343,s371,s570])). cnf(s913, plain, spl605_103, inference(rat,[],[s543,s908])). cnf(s922, plain, ~spl605_95, inference(rat,[],[s681,s913])). cnf(s923, plain, spl605_102, inference(rat,[],[s62,s922])). cnf(s924, plain, spl605_101, inference(rat,[],[s61,s922])). cnf(s925, plain, spl605_100, inference(rat,[],[s60,s922])). cnf(s927, plain, ~spl605_97, inference(rat,[],[s732,s923,s924,s925])). cnf(s928, plain, ~spl605_96, inference(rat,[],[s712,s923,s924,s925])). cnf(s929, plain, $false, inference(rat,[],[s57,s922,s927,s928])). thf(f6703,plain,( $false), inference(avatar_sat_refutation,[],[s929])). % SZS output end Proof for theBenchmark