%----Output an unreadable proof % SZS output start Proof for theBenchmark thf(type_def_5, type, term: $tType). thf(type_def_6, type, subst: $tType). thf(type_def_7, type, sTfun: ($tType * $tType) > $tType). thf(type_def_8, type, d_term: $tType). thf(type_def_9, type, d_subst: $tType). thf(func_def_0, type, one: term). thf(func_def_1, type, ap: (term > term > term)). thf(func_def_2, type, lam: (term > term)). thf(func_def_3, type, sub: (term > subst > term)). thf(func_def_4, type, id: subst). thf(func_def_5, type, sh: subst). thf(func_def_6, type, push: (term > subst > subst)). thf(func_def_7, type, comp: (subst > subst > subst)). thf(func_def_8, type, var: (term > $o)). thf(func_def_9, type, pushprop_lem1v2: $o). thf(func_def_10, type, pushprop_lem1_gthm: $o). thf(func_def_11, type, axmap: $o). thf(func_def_12, type, pushprop_lem0_gthm: $o). thf(func_def_13, type, shinj: $o). thf(func_def_14, type, hoasinduction_lem1v2: $o). thf(func_def_15, type, hoasinduction_lem1v2_gthm: $o). thf(func_def_16, type, hoasap: (subst > term > subst > term > term)). thf(func_def_17, type, induction2lem: $o). thf(func_def_18, type, hoasinduction_lem3v2_f: $o). thf(func_def_19, type, axvarshift: $o). thf(func_def_20, type, hoasapinj2: $o). thf(func_def_21, type, hoasapnotvar_gthm: $o). thf(func_def_22, type, hoasapinj1: $o). thf(func_def_23, type, ulamvar1: $o). thf(func_def_24, type, induction2lem_lthm: $o). thf(func_def_25, type, hoasinduction_lem3v2_gthm: $o). thf(func_def_26, type, apnotvar: $o). thf(func_def_27, type, pushprop_lthm_orig: $o). thf(func_def_28, type, hoasinduction_lem3v2_f_lthm: $o). thf(func_def_29, type, hoasinduction_lthm: $o). thf(func_def_30, type, hoasinduction_no_psi_cond_lthm: $o). thf(func_def_31, type, hoaslaminj: $o). thf(func_def_32, type, hoasinduction_lem3aaa: $o). thf(func_def_33, type, induction2lem_gthm: $o). thf(func_def_34, type, hoasinduction_lem3aa_lthm: $o). thf(func_def_35, type, hoasinduction_lem3: $o). thf(func_def_36, type, hoasinduction_lem2: $o). thf(func_def_37, type, termmset_lthm: $o). thf(func_def_38, type, hoasinduction_lem1: $o). thf(func_def_39, type, hoaslamnotap_lthm: $o). thf(func_def_40, type, pushprop_lem1v2_lthm: $o). thf(func_def_41, type, hoasapnotvar: $o). thf(func_def_42, type, hoasinduction_lem0: $o). thf(func_def_43, type, hoasinduction: $o). thf(func_def_44, type, hoasinduction_gthm: $o). thf(func_def_45, type, axapp: $o). thf(func_def_46, type, hoaslamnotvar_lthm: $o). thf(func_def_47, type, pushprop_lem3v2_lthm: $o). thf(func_def_48, type, hoasinduction_lem3b_lthm: $o). thf(func_def_49, type, ulamvarind: $o). thf(func_def_50, type, induction: $o). thf(func_def_51, type, hoasinduction_lem3a_lthm: $o). thf(func_def_52, type, termmset_gthm: $o). thf(func_def_53, type, hoasinduction_lem3aa: $o). thf(func_def_54, type, pushprop_lem1v2_gthm: $o). thf(func_def_55, type, hoaslamnotap_gthm: $o). thf(func_def_56, type, hoaslamnotvar_gthm: $o). thf(func_def_57, type, hoasinduction_lem3b_gthm: $o). thf(func_def_58, type, pushprop_lem2v2: $o). thf(func_def_59, type, hoasinduction_lem3a_gthm: $o). thf(func_def_60, type, axclos: $o). thf(func_def_61, type, axassoc: $o). thf(func_def_62, type, hoasinduction_lem2v2: $o). thf(func_def_63, type, pushprop_lthm: $o). thf(func_def_64, type, apinj2: $o). thf(func_def_65, type, apinj1: $o). thf(func_def_66, type, hoasapinj2_lthm: $o). thf(func_def_67, type, hoasinduction_lem3v2a: $o). thf(func_def_68, type, hoasapinj1_lthm: $o). thf(func_def_69, type, hoaslaminj_lthm: $o). thf(func_def_70, type, axvarcons: $o). thf(func_def_71, type, hoaslam: (subst > (subst > term > term) > term)). thf(func_def_72, type, axscons: $o). thf(func_def_73, type, hoasinduction_lem2v2_gthm: $o). thf(func_def_74, type, axidr: $o). thf(func_def_75, type, pushprop_lem1: $o). thf(func_def_76, type, laminj: $o). thf(func_def_77, type, hoasinduction_lem3_lthm: $o). thf(func_def_78, type, pushprop_lem0: $o). thf(func_def_79, type, pushprop_gthm: $o). thf(func_def_80, type, axabs: $o). thf(func_def_81, type, hoasinduction_lem3v2a_lthm: $o). thf(func_def_82, type, hoasinduction_lem2_lthm: $o). thf(func_def_83, type, hoasapinj2_gthm: $o). thf(func_def_84, type, hoasinduction_p_and_p_prime: ((subst > term > subst > $o) > (term > $o) > $o)). thf(func_def_85, type, hoasinduction_lem1_lthm: $o). thf(func_def_86, type, lamnotap: $o). thf(func_def_87, type, hoasapinj1_gthm: $o). thf(func_def_88, type, hoaslamnotvar: $o). thf(func_def_89, type, axidl: $o). thf(func_def_90, type, hoaslaminj_gthm: $o). thf(func_def_91, type, induction2_lthm: $o). thf(func_def_92, type, hoasinduction_lem0_lthm: $o). thf(func_def_93, type, substmonoid_lthm: $o). thf(func_def_94, type, pushprop: $o). thf(func_def_95, type, hoasinduction_lem3_gthm: $o). thf(func_def_96, type, hoasinduction_lem2_gthm: $o). thf(func_def_97, type, hoasinduction_lem3b: $o). thf(func_def_98, type, substmonoid: $o). thf(func_def_99, type, lamnotvar: $o). thf(func_def_100, type, hoasinduction_lem3a: $o). thf(func_def_101, type, hoasinduction_lem1_gthm: $o). thf(func_def_102, type, hoasinduction_no_psi_cond: $o). thf(func_def_103, type, induction2_gthm: $o). thf(func_def_104, type, pushprop_lem2v2_lthm: $o). thf(func_def_105, type, hoasvar: (subst > term > subst > $o)). thf(func_def_106, type, hoaslamnotap: $o). thf(func_def_107, type, substmonoid_gthm: $o). thf(func_def_108, type, ulamvarsh: $o). thf(func_def_109, type, induction2: $o). thf(func_def_110, type, pushprop_lem3v2: $o). thf(func_def_111, type, pushprop_lem2v2_gthm: $o). thf(func_def_112, type, pushprop_lem1_lthm: $o). thf(func_def_113, type, hoasinduction_lem3v2: $o). thf(func_def_114, type, axshiftcons: $o). thf(func_def_115, type, termmset: $o). thf(func_def_116, type, pushprop_lem0_lthm: $o). thf(func_def_117, type, hoasapnotvar_lthm: $o). thf(func_def_118, type, hoasinduction_lem3v2_lthm: $o). thf(func_def_119, type, pushprop_p_and_p_prime: (term > subst > (term > $o) > (term > $o) > $o)). thf(func_def_120, type, axvarid: $o). thf(func_def_121, type, hoasinduction_lthm_3: $o). thf(func_def_122, type, d2term: (d_term > term)). thf(func_def_123, type, d2subst: (d_subst > subst)). thf(func_def_124, type, d_one: d_term). thf(func_def_125, type, d_id: d_subst). thf(func_def_129, type, db1: !>[X0: $tType]:(X0)). thf(func_def_130, type, db0: !>[X0: $tType]:(X0)). thf(func_def_131, type, vLAM: !>[X0: $tType, X1: $tType]:((X1) > (X0 > X1))). thf(func_def_132, type, vPI: !>[X0: $tType]:(((X0 > $o) > $o))). thf(func_def_133, type, db2: !>[X0: $tType]:(X0)). thf(func_def_134, type, vEQ: !>[X0: $tType]:((X0 > X0 > $o))). thf(func_def_135, type, db4: !>[X0: $tType]:(X0)). thf(func_def_136, type, db3: !>[X0: $tType]:(X0)). thf(func_def_137, type, sP0: ((subst > term > subst > $o) > $o)). thf(func_def_138, type, sP1: $o). thf(func_def_139, type, sP2: $o). thf(func_def_140, type, sP3: $o). thf(func_def_141, type, sP4: ((term > $o) > $o)). thf(func_def_142, type, sP5: $o). thf(func_def_143, type, sP6: ((subst > term > subst > $o) > $o)). thf(func_def_144, type, sP7: $o). thf(func_def_145, type, sP8: ((subst > term > subst > $o) > $o)). thf(func_def_146, type, sP9: ((subst > term > subst > $o) > $o)). thf(func_def_147, type, sP10: $o). thf(func_def_148, type, sP11: ((term > $o) > $o)). thf(func_def_149, type, sP12: ((subst > term > subst > $o) > $o)). thf(func_def_150, type, sP13: ((subst > term > subst > $o) > $o)). thf(func_def_151, type, sP14: $o). thf(func_def_152, type, sP15: $o). thf(func_def_153, type, sP16: $o). thf(func_def_154, type, sP17: $o). thf(func_def_155, type, sP18: $o). thf(func_def_156, type, sP19: $o). thf(func_def_157, type, sP20: $o). thf(func_def_158, type, sP21: ((term > $o) > $o)). thf(func_def_159, type, sP22: ((term > $o) > $o)). thf(func_def_160, type, sP23: ((subst > term > subst > $o) > $o)). thf(func_def_161, type, sP24: ((subst > term > subst > $o) > $o)). thf(func_def_162, type, sP25: ((subst > term > subst > $o) > $o)). thf(func_def_163, type, sP26: ((subst > term > subst > $o) > $o)). thf(func_def_164, type, sP27: $o). thf(func_def_165, type, sP28: $o). thf(func_def_166, type, sP29: $o). thf(func_def_167, type, sP30: $o). thf(func_def_168, type, sP31: $o). thf(func_def_169, type, sP32: $o). thf(func_def_170, type, sP33: $o). thf(func_def_171, type, sP34: ((subst > term > subst > $o) > $o)). thf(func_def_172, type, sP35: $o). thf(func_def_173, type, sP36: ((subst > term > subst > $o) > $o)). thf(func_def_174, type, sP37: $o). thf(func_def_175, type, sP38: ((subst > term > subst > $o) > $o)). thf(func_def_176, type, sP39: $o). thf(func_def_177, type, sP40: ((subst > term > subst > $o) > $o)). thf(func_def_178, type, sP41: $o). thf(func_def_179, type, sP42: ((term > $o) > $o)). thf(func_def_180, type, sP43: $o). thf(func_def_181, type, sP44: ((term > $o) > $o)). thf(func_def_182, type, sP45: ((term > $o) > $o)). thf(func_def_183, type, sP46: $o). thf(func_def_184, type, sP47: $o). thf(func_def_185, type, sP48: ((subst > term > subst > $o) > $o)). thf(func_def_186, type, sP49: ((subst > term > subst > $o) > $o)). thf(func_def_187, type, sP50: $o). thf(func_def_188, type, sP51: $o). thf(func_def_189, type, sP52: ((subst > term > subst > $o) > $o)). thf(func_def_190, type, sP53: ((term > $o) > $o)). thf(func_def_191, type, sP54: $o). thf(func_def_192, type, sP55: $o). thf(func_def_193, type, sP56: $o). thf(func_def_194, type, sP57: ((term > $o) > $o)). thf(func_def_195, type, sP58: ((term > $o) > $o)). thf(func_def_196, type, sP59: $o). thf(func_def_197, type, sP60: $o). thf(func_def_198, type, sK61: (subst > term > subst > $o)). thf(func_def_199, type, sK62: term). thf(func_def_200, type, sK63: (term > (subst > term > subst > $o) > term)). thf(func_def_201, type, sK64: ((subst > term > subst > $o) > subst)). thf(func_def_202, type, sK65: ((subst > term > subst > $o) > subst)). thf(func_def_203, type, sK66: ((subst > term > subst > $o) > subst)). thf(func_def_204, type, sK67: ((subst > term > subst > $o) > term)). thf(func_def_205, type, sK68: ((subst > term > subst > $o) > subst)). thf(func_def_206, type, sK69: ((subst > term > subst > $o) > subst)). thf(func_def_207, type, sK70: ((subst > term > subst > $o) > term)). thf(func_def_208, type, sK71: ((subst > term > subst > $o) > subst)). thf(func_def_209, type, sK72: ((subst > term > term) > (subst > term > subst > $o) > term)). thf(func_def_210, type, sK73: ((subst > term > term) > subst)). thf(func_def_211, type, sK74: ((subst > term > term) > term)). thf(func_def_212, type, sK75: ((subst > term > term) > subst)). thf(func_def_213, type, sK76: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_214, type, sK77: ((term > $o) > term)). thf(func_def_215, type, sK78: (subst > (term > $o) > term)). thf(func_def_216, type, sK79: (term > $o)). thf(func_def_217, type, sK80: term). thf(func_def_218, type, sK81: subst). thf(func_def_219, type, sK82: (term > term)). thf(func_def_220, type, sK83: term). thf(func_def_221, type, sK84: (term > $o)). thf(func_def_222, type, sK85: term). thf(func_def_223, type, sK86: (term > term)). thf(func_def_224, type, sK87: ((term > $o) > term)). thf(func_def_225, type, sK88: ((term > $o) > term)). thf(func_def_226, type, sK89: ((term > $o) > term)). thf(func_def_227, type, sK90: ((term > $o) > term)). thf(func_def_228, type, sK91: ((term > $o) > term)). thf(func_def_229, type, sK92: ((term > $o) > term)). thf(func_def_230, type, sK93: ((term > $o) > term)). thf(func_def_231, type, sK94: ((term > $o) > term)). thf(func_def_232, type, sK95: ((subst > term > subst > $o) > term > $o)). thf(func_def_233, type, sK96: (term > $o)). thf(func_def_234, type, sK97: term). thf(func_def_235, type, sK98: (term > term)). thf(func_def_236, type, sK99: (subst > term > subst > $o)). thf(func_def_237, type, sK100: ((term > $o) > term)). thf(func_def_238, type, sK101: term). thf(func_def_239, type, sK102: term). thf(func_def_240, type, sK103: (subst > term > subst > $o)). thf(func_def_241, type, sK104: ((subst > term > term) > subst)). thf(func_def_242, type, sK105: ((subst > term > term) > term)). thf(func_def_243, type, sK106: ((subst > term > term) > subst)). thf(func_def_244, type, sK107: ((subst > term > term) > term)). thf(func_def_245, type, sK108: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_246, type, sK109: ((term > $o) > term > term)). thf(func_def_247, type, sK110: ((term > $o) > (subst > term > subst > $o) > term)). thf(func_def_248, type, sK111: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_249, type, sK112: (subst > term > subst > $o)). thf(func_def_250, type, sK113: term). thf(func_def_251, type, sK114: (term > $o)). thf(func_def_252, type, sK115: ((subst > term > term) > subst)). thf(func_def_253, type, sK116: ((subst > term > term) > subst)). thf(func_def_254, type, sK117: ((subst > term > term) > term)). thf(func_def_255, type, sK118: ((subst > term > term) > term)). thf(func_def_256, type, sK119: ((term > $o) > term)). thf(func_def_257, type, sK120: ((term > $o) > term)). thf(func_def_258, type, sK121: ((subst > term > subst > $o) > term)). thf(func_def_259, type, sK122: ((subst > term > subst > $o) > term)). thf(func_def_260, type, sK123: (subst > term > subst > $o)). thf(func_def_261, type, sK124: term). thf(func_def_262, type, sK125: ((subst > term > subst > $o) > subst)). thf(func_def_263, type, sK126: ((subst > term > subst > $o) > subst)). thf(func_def_264, type, sK127: ((subst > term > subst > $o) > term)). thf(func_def_265, type, sK128: ((subst > term > subst > $o) > subst)). thf(func_def_266, type, sK129: (term > (subst > term > subst > $o) > term)). thf(func_def_267, type, sK130: ((subst > term > subst > $o) > subst)). thf(func_def_268, type, sK131: ((subst > term > subst > $o) > subst)). thf(func_def_269, type, sK132: ((subst > term > subst > $o) > term)). thf(func_def_270, type, sK133: ((subst > term > subst > $o) > subst)). thf(func_def_271, type, sK134: term). thf(func_def_272, type, sK135: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_273, type, sK136: (term > (subst > term > subst > $o) > term)). thf(func_def_274, type, sK137: (subst > term > subst > $o)). thf(func_def_275, type, sK138: term). thf(func_def_276, type, sK139: ((subst > term > term) > term)). thf(func_def_277, type, sK140: ((subst > term > term) > subst)). thf(func_def_278, type, sK141: ((subst > term > term) > subst)). thf(func_def_279, type, sK142: ((subst > term > term) > term)). thf(func_def_280, type, sK143: ((subst > term > term) > subst)). thf(func_def_281, type, sK144: ((subst > term > term) > subst)). thf(func_def_282, type, sK145: ((subst > term > term) > term)). thf(func_def_283, type, sK146: ((subst > term > term) > (subst > term > subst > $o) > term)). thf(func_def_284, type, sK147: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_285, type, sK148: ((subst > term > term) > subst)). thf(func_def_286, type, sK149: ((subst > term > term) > term)). thf(func_def_287, type, sK150: ((subst > term > term) > subst)). thf(func_def_288, type, sK151: ((subst > term > term) > (subst > term > subst > $o) > term)). thf(func_def_289, type, sK152: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_290, type, sK153: ((term > $o) > term)). thf(func_def_291, type, sK154: ((term > $o) > term)). thf(func_def_292, type, sK155: term). thf(func_def_293, type, sK156: subst). thf(func_def_294, type, sK157: subst). thf(func_def_295, type, sK158: term). thf(func_def_296, type, sK159: term). thf(func_def_297, type, sK160: (term > $o)). thf(func_def_298, type, sK161: subst). thf(func_def_299, type, sK162: term). thf(func_def_300, type, sK163: (term > $o)). thf(func_def_301, type, sK164: subst). thf(func_def_302, type, sK165: (term > term)). thf(func_def_303, type, sK166: (subst > (term > $o) > term)). thf(func_def_304, type, sK167: ((term > $o) > term)). thf(func_def_305, type, sK168: ((term > $o) > term)). thf(func_def_306, type, sK169: ((term > $o) > term)). thf(func_def_307, type, sK170: ((term > $o) > term)). thf(func_def_308, type, sK171: ((term > $o) > term)). thf(func_def_309, type, sK172: (term > $o)). thf(func_def_310, type, sK173: term). thf(func_def_311, type, sK174: (term > term)). thf(func_def_312, type, sK175: ((term > $o) > term)). thf(func_def_313, type, sK176: ((term > $o) > term)). thf(func_def_314, type, sK177: ((term > $o) > term)). thf(func_def_315, type, sK178: ((term > $o) > term)). thf(func_def_316, type, sK179: (subst > term > subst > $o)). thf(func_def_317, type, sK180: term). thf(func_def_318, type, sK181: ((subst > term > subst > $o) > subst)). thf(func_def_319, type, sK182: ((subst > term > subst > $o) > subst)). thf(func_def_320, type, sK183: ((subst > term > subst > $o) > subst)). thf(func_def_321, type, sK184: ((subst > term > subst > $o) > term)). thf(func_def_322, type, sK185: (term > (subst > term > subst > $o) > term)). thf(func_def_323, type, sK186: ((subst > term > subst > $o) > subst)). thf(func_def_324, type, sK187: ((subst > term > subst > $o) > term)). thf(func_def_325, type, sK188: ((subst > term > subst > $o) > subst)). thf(func_def_326, type, sK189: ((subst > term > subst > $o) > subst)). thf(func_def_327, type, sK190: ((subst > term > term) > (subst > term > subst > $o) > term)). thf(func_def_328, type, sK191: ((subst > term > term) > subst)). thf(func_def_329, type, sK192: ((subst > term > term) > term)). thf(func_def_330, type, sK193: ((subst > term > term) > subst)). thf(func_def_331, type, sK194: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_332, type, sK195: (term > $o)). thf(func_def_333, type, sK196: term). thf(func_def_334, type, sK197: (subst > term > subst > $o)). thf(func_def_335, type, sK198: ((subst > term > subst > $o) > (term > $o) > term)). thf(func_def_336, type, sK199: (term > (term > $o) > term)). thf(func_def_337, type, sK200: ((subst > term > subst > $o) > subst)). thf(func_def_338, type, sK201: ((subst > term > subst > $o) > term)). thf(func_def_339, type, sK202: ((subst > term > subst > $o) > subst)). thf(func_def_340, type, sK203: ((subst > term > subst > $o) > subst)). thf(func_def_341, type, sK204: ((subst > term > subst > $o) > subst)). thf(func_def_342, type, sK205: ((subst > term > subst > $o) > subst)). thf(func_def_343, type, sK206: ((subst > term > subst > $o) > term)). thf(func_def_344, type, sK207: ((subst > term > subst > $o) > subst)). thf(func_def_345, type, sK208: ((subst > term > term) > (subst > term > subst > $o) > term)). thf(func_def_346, type, sK209: ((subst > term > term) > subst)). thf(func_def_347, type, sK210: ((subst > term > term) > term)). thf(func_def_348, type, sK211: ((subst > term > term) > subst)). thf(func_def_349, type, sK212: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_350, type, sK213: (term > $o)). thf(func_def_351, type, sK214: term). thf(func_def_352, type, sK215: (subst > term > subst > $o)). thf(func_def_353, type, sK216: term). thf(func_def_354, type, sK217: ((subst > term > subst > $o) > subst)). thf(func_def_355, type, sK218: ((subst > term > subst > $o) > subst)). thf(func_def_356, type, sK219: ((subst > term > subst > $o) > subst)). thf(func_def_357, type, sK220: ((subst > term > subst > $o) > term)). thf(func_def_358, type, sK221: ((subst > term > subst > $o) > (term > $o) > term)). thf(func_def_359, type, sK222: ((subst > term > subst > $o) > subst)). thf(func_def_360, type, sK223: ((subst > term > subst > $o) > term)). thf(func_def_361, type, sK224: ((subst > term > subst > $o) > subst)). thf(func_def_362, type, sK225: ((subst > term > subst > $o) > subst)). thf(func_def_363, type, sK226: ((subst > term > subst > $o) > term)). thf(func_def_364, type, sK227: ((subst > term > subst > $o) > term)). thf(func_def_365, type, sK228: term). thf(func_def_366, type, sK229: term). thf(func_def_367, type, sK230: (subst > term > subst > $o)). thf(func_def_368, type, sK231: ((subst > term > subst > $o) > subst)). thf(func_def_369, type, sK232: ((subst > term > subst > $o) > subst)). thf(func_def_370, type, sK233: ((subst > term > subst > $o) > subst)). thf(func_def_371, type, sK234: ((subst > term > subst > $o) > term)). thf(func_def_372, type, sK235: ((subst > term > subst > $o) > subst)). thf(func_def_373, type, sK236: ((subst > term > subst > $o) > subst)). thf(func_def_374, type, sK237: ((subst > term > subst > $o) > term)). thf(func_def_375, type, sK238: ((subst > term > subst > $o) > subst)). thf(func_def_376, type, sK239: ((subst > term > subst > $o) > term)). thf(func_def_377, type, sK240: ((subst > term > subst > $o) > term)). thf(func_def_378, type, sK241: (subst > term > subst > $o)). thf(func_def_379, type, sK242: term). thf(func_def_380, type, sK243: ((subst > term > term) > term)). thf(func_def_381, type, sK244: ((subst > term > term) > term)). thf(func_def_382, type, sK245: ((subst > term > term) > subst)). thf(func_def_383, type, sK246: ((subst > term > term) > subst)). thf(func_def_384, type, sK247: (term > (subst > term > subst > $o) > term)). thf(func_def_385, type, sK248: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_386, type, sK249: (subst > term > subst > $o)). thf(func_def_387, type, sK250: term). thf(func_def_388, type, sK251: ((subst > term > term) > term)). thf(func_def_389, type, sK252: ((subst > term > term) > term > term)). thf(func_def_390, type, sK253: ((subst > term > term) > term > subst)). thf(func_def_391, type, sK254: ((subst > term > term) > term > subst)). thf(func_def_392, type, sK255: ((subst > term > term) > term > term)). thf(func_def_393, type, sK256: ((subst > term > term) > term > subst)). thf(func_def_394, type, sK257: ((subst > term > term) > term > subst)). thf(func_def_395, type, sK258: (term > (subst > term > subst > $o) > term)). thf(func_def_396, type, sK259: ((subst > term > subst > $o) > term)). thf(func_def_397, type, sK260: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_398, type, sK261: (term > $o)). thf(func_def_399, type, sK262: (subst > term > subst > $o)). thf(func_def_400, type, sK263: term). thf(func_def_401, type, sK264: ((subst > term > term) > subst)). thf(func_def_402, type, sK265: ((subst > term > term) > term)). thf(func_def_403, type, sK266: ((subst > term > term) > subst)). thf(func_def_404, type, sK267: ((subst > term > term) > term)). thf(func_def_405, type, sK268: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_406, type, sK269: ((subst > term > subst > $o) > (term > $o) > term)). thf(func_def_407, type, sK270: (term > (term > $o) > term)). thf(func_def_408, type, sK271: ((subst > term > subst > $o) > term)). thf(func_def_409, type, sK272: ((subst > term > subst > $o) > subst)). thf(func_def_410, type, sK273: ((subst > term > subst > $o) > subst)). thf(func_def_411, type, sK274: ((subst > term > subst > $o) > subst)). thf(func_def_412, type, sK275: ((subst > term > subst > $o) > subst)). thf(func_def_413, type, sK276: ((subst > term > subst > $o) > subst)). thf(func_def_414, type, sK277: ((subst > term > subst > $o) > subst)). thf(func_def_415, type, sK278: ((subst > term > subst > $o) > term)). thf(func_def_416, type, sK279: term). thf(func_def_417, type, sK280: (subst > term > subst > $o)). thf(func_def_418, type, sK281: term). thf(func_def_419, type, sK282: ((term > $o) > term)). thf(func_def_420, type, sK283: ((term > $o) > term)). thf(func_def_421, type, sK284: term). thf(func_def_422, type, sK285: (term > $o)). thf(func_def_423, type, sK286: (term > term)). thf(func_def_424, type, sK287: term). thf(func_def_425, type, sK288: (subst > term > subst > $o)). thf(func_def_426, type, sK289: ((subst > term > subst > $o) > subst)). thf(func_def_427, type, sK290: ((subst > term > subst > $o) > term)). thf(func_def_428, type, sK291: ((subst > term > subst > $o) > subst)). thf(func_def_429, type, sK292: ((subst > term > subst > $o) > subst)). thf(func_def_430, type, sK293: ((subst > term > subst > $o) > term)). thf(func_def_431, type, sK294: ((subst > term > subst > $o) > subst)). thf(func_def_432, type, sK295: ((subst > term > subst > $o) > subst)). thf(func_def_433, type, sK296: ((subst > term > subst > $o) > subst)). thf(func_def_434, type, sK297: ((subst > term > subst > $o) > term)). thf(func_def_435, type, sK298: (term > (subst > term > subst > $o) > term)). thf(func_def_436, type, sK299: ((subst > term > subst > $o) > subst)). thf(func_def_437, type, sK300: ((subst > term > subst > $o) > subst)). thf(func_def_438, type, sK301: ((subst > term > subst > $o) > term)). thf(func_def_439, type, sK302: ((subst > term > subst > $o) > subst)). thf(func_def_440, type, sK303: ((subst > term > subst > $o) > subst)). thf(func_def_441, type, sK304: ((subst > term > subst > $o) > subst)). thf(func_def_442, type, sK305: ((subst > term > subst > $o) > term)). thf(func_def_443, type, sK306: ((subst > term > subst > $o) > subst)). thf(func_def_444, type, sK307: (subst > term > subst > $o)). thf(func_def_445, type, sK308: term). thf(func_def_446, type, sK309: ((subst > term > subst > $o) > term)). thf(func_def_447, type, sK310: ((subst > term > subst > $o) > term)). thf(func_def_448, type, sK311: ((subst > term > term) > (subst > term > subst > $o) > term)). thf(func_def_449, type, sK312: ((subst > term > term) > term)). thf(func_def_450, type, sK313: ((subst > term > term) > subst)). thf(func_def_451, type, sK314: ((subst > term > term) > subst)). thf(func_def_452, type, sK315: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_453, type, sK316: ((subst > term > subst > $o) > term)). thf(func_def_454, type, sK317: ((subst > term > subst > $o) > term)). thf(func_def_455, type, sK318: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_456, type, sK319: ((subst > term > term) > (subst > term > subst > $o) > term)). thf(func_def_457, type, sK320: ((subst > term > term) > subst)). thf(func_def_458, type, sK321: ((subst > term > term) > subst)). thf(func_def_459, type, sK322: ((subst > term > term) > term)). thf(func_def_460, type, sK323: ((term > $o) > term)). thf(func_def_461, type, sK324: ((term > $o) > term)). thf(func_def_462, type, sK325: ((term > $o) > term)). thf(func_def_463, type, sK326: ((term > $o) > term)). thf(func_def_464, type, sK327: (subst > term > subst > $o)). thf(func_def_465, type, sK328: term). thf(func_def_466, type, sK329: ((subst > term > term) > term)). thf(func_def_467, type, sK330: ((subst > term > term) > term)). thf(func_def_468, type, sK331: ((subst > term > term) > subst)). thf(func_def_469, type, sK332: ((subst > term > term) > subst)). thf(func_def_470, type, sK333: (term > (subst > term > subst > $o) > term)). thf(func_def_471, type, sK334: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_472, type, sK335: term). thf(func_def_473, type, sK336: (subst > term > subst > $o)). thf(func_def_474, type, sK337: ((subst > term > term) > term)). thf(func_def_475, type, sK338: ((subst > term > term) > term)). thf(func_def_476, type, sK339: ((subst > term > term) > subst)). thf(func_def_477, type, sK340: ((subst > term > term) > subst)). thf(func_def_478, type, sK341: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_479, type, sK342: ((subst > term > subst > $o) > term > term)). thf(func_def_480, type, sK343: term). thf(func_def_481, type, sK344: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_482, type, sK345: (term > (subst > term > subst > $o) > term)). thf(func_def_483, type, sK346: (subst > term > subst > $o)). thf(func_def_484, type, sK347: term). thf(func_def_485, type, sK348: ((subst > term > term) > subst)). thf(func_def_486, type, sK349: ((subst > term > term) > term)). thf(func_def_487, type, sK350: ((subst > term > term) > subst)). thf(func_def_488, type, sK351: ((subst > term > term) > term)). thf(func_def_489, type, sK352: subst). thf(func_def_490, type, sK353: (subst > term > term)). thf(func_def_491, type, sK354: (subst > term > term)). thf(func_def_492, type, sK355: term). thf(func_def_493, type, sK356: ((subst > term > term) > subst)). thf(func_def_494, type, sK357: ((subst > term > term) > term)). thf(func_def_495, type, sK358: ((subst > term > term) > subst)). thf(func_def_496, type, sK359: ((subst > term > term) > term)). thf(func_def_497, type, sK360: ((subst > term > term) > subst)). thf(func_def_498, type, sK361: ((subst > term > term) > subst)). thf(func_def_499, type, sK362: term). thf(func_def_500, type, sK363: term). thf(func_def_501, type, sK364: subst). thf(func_def_502, type, sK365: term). thf(func_def_503, type, sK366: term). thf(func_def_504, type, sK367: subst). thf(func_def_505, type, sK368: ((subst > term > subst > $o) > term > $o)). thf(func_def_506, type, sK369: ((term > $o) > term)). thf(func_def_507, type, sK370: ((term > $o) > term)). thf(func_def_508, type, sK371: (subst > term > subst > $o)). thf(func_def_509, type, sK372: ((term > $o) > term)). thf(func_def_510, type, sK373: term). thf(func_def_511, type, sK374: term). thf(func_def_512, type, sK375: (term > $o)). thf(func_def_513, type, sK376: (term > term)). thf(func_def_514, type, sK377: (term > (term > $o) > term)). thf(func_def_515, type, sK378: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_516, type, sK379: ((subst > term > subst > $o) > (term > $o) > term)). thf(func_def_517, type, sK380: (term > $o)). thf(func_def_518, type, sK381: (subst > term > subst > $o)). thf(func_def_519, type, sK382: term). thf(func_def_520, type, sK383: ((subst > term > term) > subst)). thf(func_def_521, type, sK384: ((subst > term > term) > subst)). thf(func_def_522, type, sK385: ((subst > term > term) > term)). thf(func_def_523, type, sK386: ((subst > term > term) > term)). thf(func_def_524, type, sK387: term). thf(func_def_525, type, sK388: (subst > term > subst > $o)). thf(func_def_526, type, sK389: ((subst > term > subst > $o) > term)). thf(func_def_527, type, sK390: ((subst > term > subst > $o) > subst)). thf(func_def_528, type, sK391: ((subst > term > subst > $o) > subst)). thf(func_def_529, type, sK392: ((subst > term > subst > $o) > subst)). thf(func_def_530, type, sK393: ((subst > term > subst > $o) > term)). thf(func_def_531, type, sK394: ((subst > term > subst > $o) > subst)). thf(func_def_532, type, sK395: ((subst > term > subst > $o) > subst)). thf(func_def_533, type, sK396: ((subst > term > subst > $o) > subst)). thf(func_def_534, type, sK397: ((subst > term > subst > $o) > term)). thf(func_def_535, type, sK398: ((subst > term > subst > $o) > term)). thf(func_def_536, type, sK399: ((subst > term > subst > $o) > term)). thf(func_def_537, type, sK400: ((subst > term > term) > (subst > term > subst > $o) > term)). thf(func_def_538, type, sK401: ((subst > term > term) > term)). thf(func_def_539, type, sK402: ((subst > term > term) > subst)). thf(func_def_540, type, sK403: ((subst > term > term) > subst)). thf(func_def_541, type, sK404: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_542, type, sK405: ((term > $o) > term)). thf(func_def_543, type, sK406: ((term > $o) > term)). thf(func_def_544, type, sK407: term). thf(func_def_545, type, sK408: (subst > term > subst > $o)). thf(func_def_546, type, sK409: ((subst > term > subst > $o) > subst)). thf(func_def_547, type, sK410: ((subst > term > subst > $o) > subst)). thf(func_def_548, type, sK411: ((subst > term > subst > $o) > term)). thf(func_def_549, type, sK412: ((subst > term > subst > $o) > subst)). thf(func_def_550, type, sK413: ((subst > term > subst > $o) > subst)). thf(func_def_551, type, sK414: ((subst > term > subst > $o) > subst)). thf(func_def_552, type, sK415: ((subst > term > subst > $o) > subst)). thf(func_def_553, type, sK416: ((subst > term > subst > $o) > term)). thf(func_def_554, type, sK417: ((subst > term > subst > $o) > term)). thf(func_def_555, type, sK418: ((subst > term > term) > subst)). thf(func_def_556, type, sK419: ((subst > term > term) > term)). thf(func_def_557, type, sK420: ((subst > term > term) > subst)). thf(func_def_558, type, sK421: ((subst > term > term) > (subst > term > subst > $o) > term)). thf(func_def_559, type, sK422: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_560, type, sK423: ((subst > term > subst > $o) > term)). thf(func_def_561, type, sK424: ((subst > term > subst > $o) > term)). thf(func_def_562, type, sK425: (term > $o)). thf(func_def_563, type, sK426: (subst > term > subst > $o)). thf(func_def_564, type, sK427: term). thf(func_def_565, type, sK428: term). thf(func_def_566, type, sK429: ((subst > term > subst > $o) > subst)). thf(func_def_567, type, sK430: ((subst > term > subst > $o) > term)). thf(func_def_568, type, sK431: ((subst > term > subst > $o) > subst)). thf(func_def_569, type, sK432: ((subst > term > subst > $o) > subst)). thf(func_def_570, type, sK433: ((subst > term > subst > $o) > (term > $o) > term)). thf(func_def_571, type, sK434: ((subst > term > subst > $o) > subst)). thf(func_def_572, type, sK435: ((subst > term > subst > $o) > term)). thf(func_def_573, type, sK436: ((subst > term > subst > $o) > subst)). thf(func_def_574, type, sK437: ((subst > term > subst > $o) > subst)). thf(func_def_575, type, sK438: (term > (term > $o) > term)). thf(func_def_576, type, sK439: ((subst > term > term) > (subst > term > subst > $o) > term)). thf(func_def_577, type, sK440: ((subst > term > term) > subst)). thf(func_def_578, type, sK441: ((subst > term > term) > term)). thf(func_def_579, type, sK442: ((subst > term > term) > subst)). thf(func_def_580, type, sK443: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_581, type, sK444: (term > $o)). thf(func_def_582, type, sK445: subst). thf(func_def_583, type, sK446: term). thf(func_def_584, type, sK447: (term > term)). thf(func_def_585, type, sK448: (subst > (term > $o) > term)). thf(func_def_586, type, sK449: ((term > $o) > term)). thf(func_def_587, type, sK450: ((term > $o) > term)). thf(func_def_588, type, sK451: ((term > $o) > term)). thf(func_def_589, type, sK452: term). thf(func_def_590, type, sK453: (subst > term > subst > $o)). thf(func_def_591, type, sK454: ((subst > term > term) > term)). thf(func_def_592, type, sK455: ((subst > term > term) > subst)). thf(func_def_593, type, sK456: ((subst > term > term) > subst)). thf(func_def_594, type, sK457: ((subst > term > term) > term)). thf(func_def_595, type, sK458: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_596, type, sK459: ((subst > term > subst > $o) > term > term)). thf(func_def_597, type, sK460: (term > subst > term > term)). thf(func_def_598, type, sK461: (subst > term > subst > $o)). thf(func_def_599, type, sK462: term). thf(func_def_600, type, sK463: (term > $o)). thf(func_def_601, type, sK464: ((subst > term > term) > subst)). thf(func_def_602, type, sK465: ((subst > term > term) > subst)). thf(func_def_603, type, sK466: ((subst > term > term) > term)). thf(func_def_604, type, sK467: ((subst > term > term) > term)). thf(func_def_605, type, sK468: term). thf(func_def_606, type, sK469: ((subst > term > term) > subst)). thf(func_def_607, type, sK470: ((subst > term > term) > term)). thf(func_def_608, type, sK471: ((subst > term > subst > $o) > subst > term > term)). thf(func_def_609, type, sK472: ((term > $o) > (subst > term > subst > $o) > term)). thf(func_def_610, type, sK473: ((term > $o) > term > term)). thf(func_def_611, type, sK474: term). thf(func_def_612, type, sK475: term). thf(func_def_613, type, sK476: term). thf(func_def_614, type, sK477: term). thf(func_def_615, type, sK478: term). thf(func_def_616, type, sK479: (subst > d_subst)). thf(func_def_617, type, sK480: (subst > term > subst > $o)). thf(func_def_618, type, sK481: term). thf(func_def_619, type, sK482: ((subst > term > subst > $o) > term)). thf(func_def_620, type, sK483: ((subst > term > subst > $o) > term)). thf(func_def_621, type, sK484: (subst > term > subst > $o)). thf(func_def_622, type, sK485: ((term > $o) > term)). thf(func_def_623, type, sK486: ((subst > term > subst > $o) > term > $o)). thf(func_def_624, type, sK487: subst). thf(func_def_625, type, sK488: subst). thf(func_def_626, type, sK489: subst). thf(func_def_627, type, sK490: (term > (term > $o) > subst > term > $o)). thf(func_def_628, type, sK491: subst). thf(func_def_629, type, sK492: (term > $o)). thf(func_def_630, type, sK493: term). thf(func_def_631, type, sK494: ((term > $o) > term)). thf(func_def_632, type, sK495: term). thf(func_def_633, type, sK496: subst). thf(func_def_634, type, sK497: (term > $o)). thf(func_def_635, type, sK498: term). thf(func_def_636, type, sK499: (term > $o)). thf(func_def_637, type, sK500: term). thf(func_def_638, type, sK501: (term > subst > term > term)). thf(func_def_639, type, sK502: term). thf(func_def_640, type, sK503: subst). thf(func_def_641, type, sK504: (term > $o)). thf(func_def_642, type, sK505: term). thf(func_def_643, type, sK506: ((term > $o) > term)). thf(func_def_644, type, sK507: (term > (term > $o) > subst > term > $o)). thf(func_def_645, type, sK508: term). thf(func_def_646, type, sK509: ((subst > term > term) > subst)). thf(func_def_647, type, sK510: ((subst > term > term) > term)). thf(func_def_648, type, sK511: (term > subst > term > term)). thf(func_def_649, type, sK512: ((subst > term > subst > $o) > term > $o)). thf(func_def_650, type, sK513: (subst > term > subst > $o)). thf(func_def_651, type, sK514: ((term > $o) > term)). thf(func_def_652, type, sK515: term). thf(func_def_653, type, sK516: term). thf(func_def_654, type, sK517: term). thf(func_def_655, type, sK518: (subst > term > term)). thf(func_def_656, type, sK519: (subst > term > term)). thf(func_def_657, type, sK520: subst). thf(func_def_658, type, sK521: ((subst > term > term) > term)). thf(func_def_659, type, sK522: ((subst > term > term) > subst)). thf(func_def_660, type, sK523: ((subst > term > term) > subst)). thf(func_def_661, type, sK524: ((subst > term > term) > subst)). thf(func_def_662, type, sK525: ((subst > term > term) > subst)). thf(func_def_663, type, sK526: ((subst > term > term) > term)). thf(func_def_664, type, sK527: subst). thf(func_def_665, type, sK528: term). thf(func_def_666, type, sK529: subst). thf(func_def_667, type, sK530: term). thf(func_def_668, type, sK531: term). thf(func_def_669, type, sK532: ((subst > term > term) > subst)). thf(func_def_670, type, sK533: ((subst > term > term) > term)). thf(func_def_671, type, sK534: ((subst > term > term) > subst)). thf(func_def_672, type, sK535: term). thf(func_def_673, type, sK536: term). thf(func_def_674, type, sK537: (subst > term > term)). thf(func_def_675, type, sK538: subst). thf(func_def_676, type, sK539: subst). thf(func_def_677, type, sK540: subst). thf(func_def_678, type, sK541: subst). thf(func_def_679, type, sK542: term). thf(func_def_680, type, sK543: term). thf(func_def_681, type, sK544: term). thf(func_def_682, type, sK545: term). thf(func_def_683, type, sK546: term). thf(func_def_684, type, sK547: ((subst > term > term) > subst)). thf(func_def_685, type, sK548: ((subst > term > term) > term)). thf(func_def_686, type, sK549: (term > subst > term > term)). thf(func_def_687, type, sK550: term). thf(func_def_688, type, sK551: term). thf(func_def_689, type, sK552: term). thf(func_def_690, type, sK553: term). thf(func_def_691, type, sK554: term). thf(func_def_692, type, sK555: term). thf(func_def_693, type, sK556: term). thf(func_def_694, type, sK557: term). thf(func_def_695, type, sK558: term). thf(func_def_696, type, sK559: term). thf(func_def_697, type, sK560: term). thf(func_def_698, type, sK561: term). thf(func_def_699, type, sK562: ((term > $o) > term > subst > (term > $o) > term)). thf(func_def_700, type, sK563: (term > $o)). thf(func_def_701, type, sK564: subst). thf(func_def_702, type, sK565: term). thf(func_def_703, type, sK566: (term > $o)). thf(func_def_704, type, sK567: term). thf(func_def_705, type, sK568: term). thf(func_def_706, type, sK569: term). thf(func_def_707, type, sK570: term). thf(func_def_708, type, sK571: term). thf(func_def_709, type, sK572: term). thf(func_def_710, type, sK573: term). thf(func_def_711, type, sK574: term). thf(func_def_712, type, sK575: term). thf(func_def_713, type, sK576: term). thf(func_def_714, type, sK577: ((term > $o) > term)). thf(func_def_715, type, sK578: ((term > $o) > term)). thf(func_def_716, type, sK579: term). thf(func_def_717, type, sK580: (term > $o)). thf(func_def_718, type, sK581: (term > subst > term > term)). thf(func_def_719, type, sK582: term). thf(func_def_720, type, sK583: term). thf(func_def_721, type, sK584: term). thf(func_def_722, type, sK585: term). thf(func_def_723, type, sK586: term). thf(func_def_724, type, sK587: term). thf(func_def_725, type, sK588: (term > $o)). thf(func_def_726, type, sK589: subst). thf(func_def_727, type, sK590: (term > $o)). thf(func_def_728, type, sK591: ((term > $o) > subst > (term > $o) > term > term)). thf(func_def_729, type, sK592: term). thf(func_def_730, type, sK593: subst). thf(func_def_731, type, sK594: (term > d_term)). thf(func_def_732, type, sK595: term). thf(func_def_733, type, sK596: term). thf(func_def_734, type, sK597: subst). thf(func_def_735, type, sK598: term). thf(func_def_736, type, sK599: subst). thf(func_def_737, type, sK600: term). thf(func_def_738, type, sK601: term). thf(func_def_739, type, sK602: (term > $o)). thf(func_def_740, type, sK603: term). thf(func_def_741, type, sK604: (term > $o)). thf(func_def_742, type, sK605: subst). thf(func_def_743, type, sK606: subst). thf(func_def_744, type, sK607: subst). thf(func_def_745, type, sK608: term). thf(f1,axiom,( (! [X9 : term,X7 : term,X12 : term,X11 : term] : ((((ap @ X7 @ X11)) != ((ap @ X9 @ X12))) | (X7 = X9)) <=> apinj1) & (id = ((d2subst @ d_id))) & pushprop_lem3v2 & (hoasinduction_lem3v2 <=> ! [X31 : term,X5 : (term > $o),X4 : (subst > term > subst > $o)] : (~ ! [X24 : subst,X7 : term,X18 : subst,X8 : subst] : ((X4 @ X8 @ X7 @ (comp @ X24 @ X18)) | ~(X4 @ (comp @ X8 @ X24) @ (sub @ X7 @ X24) @ X18)) | ~ ! [X9 : term] : (~(X5 @ X9) | (X5 @ (sub @ X31 @ (push @ X9 @ id)))) | (X5 @ (lam @ X31)) | ~ ! [X10 : (subst > term > term)] : (~ ! [X7 : term] : ((X4 @ id @ (X10 @ id @ X7) @ id) | ~(X4 @ id @ X7 @ id)) | ~ ! [X7 : term,X8 : subst,X18 : subst] : (((sub @ (X10 @ X8 @ X7) @ X18)) = ((X10 @ (comp @ X8 @ X18) @ (sub @ X7 @ X18)))) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id)) | ~ ! [X18 : subst,X8 : subst,X24 : subst,X7 : term] : (~(X4 @ X8 @ X7 @ (comp @ X24 @ X18)) | (X4 @ (comp @ X8 @ X24) @ (sub @ X7 @ X24) @ X18)) | ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id))))) & (pushprop_p_and_p_prime = (^[X7 : term, X8 : subst, X4 : (term > $o), X5 : (term > $o)] : (! [X6 : term] : (((X5 @ X6)) = ((X4 @ (sub @ X6 @ (push @ X7 @ X8)))))))) & hoasinduction_lem1_lthm & hoasinduction_gthm & hoaslamnotvar & (! [X23 : term,X4 : (subst > term > subst > $o)] : (~ ! [X7 : term,X9 : term] : (~(X4 @ id @ X7 @ id) | ~(X4 @ id @ X9 @ id) | (X4 @ id @ (ap @ (sub @ X7 @ id) @ X9) @ id)) | ~ ! [X24 : subst,X8 : subst,X7 : term,X18 : subst] : ((X4 @ (comp @ X8 @ X24) @ (sub @ X7 @ X24) @ X18) | ~(X4 @ X8 @ X7 @ (comp @ X24 @ X18))) | ~ ! [X8 : subst,X24 : subst,X18 : subst,X7 : term] : (~(X4 @ (comp @ X8 @ X24) @ (sub @ X7 @ X24) @ X18) | (X4 @ X8 @ X7 @ (comp @ X24 @ X18))) | ~ ! [X7 : term] : ((X4 @ id @ X7 @ id) | ~(var @ (sub @ X7 @ id))) | ~ ! [X10 : (subst > term > term)] : (~ ! [X18 : subst,X8 : subst,X7 : term] : (((sub @ (X10 @ X8 @ X7) @ X18)) = ((X10 @ (comp @ X8 @ X18) @ (sub @ X7 @ X18)))) | ~ ! [X7 : term] : ((X4 @ id @ (X10 @ id @ X7) @ id) | ~(X4 @ id @ X7 @ id)) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id)) | (X4 @ id @ X23 @ id)) <=> hoasinduction) & induction2_gthm & ! [X46 : d_subst,X45 : d_subst] : ((((d2subst @ X45)) = ((d2subst @ X46))) => (X45 = X46)) & ~ulamvar1 & (! [X7 : term,X8 : subst,X4 : (term > $o)] : ~ ! [X5 : (term > $o)] : ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ (sub @ X6 @ (push @ X7 @ X8))))) <=> pushprop_lem0_lthm) & axvarshift & (axidl <=> ! [X8 : subst] : (X8 = ((comp @ id @ X8)))) & hoaslamnotvar_gthm & hoasapinj2_gthm & hoaslamnotap_gthm & induction2lem_gthm & (hoasinduction_lem3a <=> ! [X4 : (subst > term > subst > $o),X32 : term] : ((X4 @ id @ (lam @ X32) @ id) | ~ ! [X9 : term] : (~(X4 @ id @ X9 @ id) | (X4 @ id @ (sub @ X32 @ (push @ X9 @ id)) @ id)) | ~ ! [X10 : (subst > term > term)] : (~ ! [X18 : subst,X7 : term,X8 : subst] : (((sub @ (X10 @ X8 @ X7) @ X18)) = ((X10 @ (comp @ X8 @ X18) @ (sub @ X7 @ X18)))) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X7 : term] : (~(X4 @ id @ X7 @ id) | (X4 @ id @ (X10 @ id @ X7) @ id))))) & (! [X13 : term,X14 : subst,X4 : (term > $o)] : (~ ! [X9 : term,X7 : term] : (~(X4 @ X9) | (X4 @ (ap @ X7 @ X9)) | ~(X4 @ X7)) | ~ ! [X9 : term] : (~(var @ X9) | (X4 @ (sub @ X9 @ X14))) | ~ ! [X7 : term] : ((X4 @ (lam @ X7)) | ~ ! [X9 : term] : ((X4 @ (sub @ X7 @ (push @ X9 @ id))) | ~(X4 @ X9))) | (X4 @ (sub @ X13 @ X14))) <=> induction2lem) & hoasinduction_lem3b_gthm & axmap & pushprop_lem2v2 & axabs & hoaslamnotvar_lthm & (((comp @ (d2subst @ d_id) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & termmset_gthm & pushprop_lem2v2_gthm & (induction <=> ! [X4 : (term > $o),X15 : term] : ((X4 @ X15) | ~ ! [X9 : term,X7 : term] : (~(X4 @ X7) | ~(X4 @ X9) | (X4 @ (ap @ X7 @ X9))) | ~ ! [X7 : term] : (~(X4 @ X7) | (X4 @ (lam @ X7))) | ~ ! [X7 : term] : ((X4 @ X7) | ~(var @ X7)))) & (hoasinduction_lem3aa_lthm <=> ! [X4 : (subst > term > subst > $o),X28 : term] : (~ ! [X9 : term] : ((X4 @ id @ (sub @ X28 @ (push @ X9 @ id)) @ id) | ~(X4 @ id @ X9 @ id)) | ~ ! [X10 : (subst > term > term)] : (~ ! [X7 : term] : ((X4 @ id @ (X10 @ id @ X7) @ id) | ~(X4 @ id @ X7 @ id)) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X8 : subst,X7 : term,X18 : subst] : (((sub @ (X10 @ X8 @ X7) @ X18)) = ((X10 @ (comp @ X8 @ X18) @ (sub @ X7 @ X18))))) | (X4 @ id @ (lam @ (sub @ X28 @ (push @ one @ sh))) @ id))) & hoasinduction_lem1v2 & ulamvarind & (hoasinduction_lem3v2_f <=> ! [X9 : term] : ~ ! [X10 : (subst > term > term)] : ~ ! [X7 : term,X8 : subst] : (((sub @ X9 @ (push @ X7 @ X8))) = ((X10 @ X8 @ X7)))) & (hoasinduction_lem3b_lthm <=> ! [X9 : term] : ~ ! [X10 : (subst > term > term)] : (((X10 @ sh @ one)) != ((sub @ X9 @ (push @ one @ sh))))) & (induction2_lthm <=> (! [X7 : term] : (X7 = ((sub @ X7 @ id))) => (! [X4 : (term > $o),X14 : subst,X13 : term] : (~ ! [X7 : term] : ((X4 @ (lam @ X7)) | ~ ! [X9 : term] : (~(X4 @ X9) | (X4 @ (sub @ X7 @ (push @ X9 @ id))))) | ~ ! [X9 : term,X7 : term] : ((X4 @ (ap @ X7 @ X9)) | ~(X4 @ X9) | ~(X4 @ X7)) | (X4 @ (sub @ X13 @ X14)) | ~ ! [X9 : term] : (~(var @ X9) | (X4 @ (sub @ X9 @ X14)))) => ! [X4 : (term > $o),X16 : term] : (~ ! [X9 : term,X7 : term] : (~(X4 @ X9) | (X4 @ (ap @ X7 @ X9)) | ~(X4 @ X7)) | (X4 @ X16) | ~ ! [X7 : term] : ((X4 @ X7) | ~(var @ X7)) | ~ ! [X7 : term] : (~ ! [X9 : term] : (~(X4 @ X9) | (X4 @ (sub @ X7 @ (push @ X9 @ id)))) | (X4 @ (lam @ X7))))))) & hoasapnotvar_gthm & ! [X1 : d_term,X2 : d_term] : ((((d2term @ X1)) = ((d2term @ X2))) => (X1 = X2)) & (substmonoid_lthm <=> (! [X8 : subst] : (X8 = ((comp @ id @ X8))) => (! [X8 : subst] : (X8 = ((comp @ X8 @ id))) => (! [X8 : subst] : (X8 = ((comp @ id @ X8))) & ! [X8 : subst] : (X8 = ((comp @ X8 @ id))))))) & ! [X48 : term] : ? [X0 : d_term] : (X48 = ((d2term @ X0))) & hoasinduction_lem2_lthm & (hoasinduction_lem3v2_lthm <=> (! [X7 : term] : (X7 = ((sub @ X7 @ id))) => ! [X4 : (subst > term > subst > $o),X5 : (term > $o),X31 : term] : ((X5 @ (lam @ X31)) | ~ ! [X10 : (subst > term > term)] : (~ ! [X7 : term] : ((X4 @ id @ (X10 @ id @ X7) @ id) | ~(X4 @ id @ X7 @ id)) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X18 : subst,X7 : term,X8 : subst] : (((sub @ (X10 @ X8 @ X7) @ X18)) = ((X10 @ (comp @ X8 @ X18) @ (sub @ X7 @ X18))))) | ~ ! [X18 : subst,X7 : term,X24 : subst,X8 : subst] : ((X4 @ (comp @ X8 @ X24) @ (sub @ X7 @ X24) @ X18) | ~(X4 @ X8 @ X7 @ (comp @ X24 @ X18))) | ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id))) | ~ ! [X9 : term] : ((X5 @ (sub @ X31 @ (push @ X9 @ id))) | ~(X5 @ X9)) | ~ ! [X18 : subst,X8 : subst,X7 : term,X24 : subst] : ((X4 @ X8 @ X7 @ (comp @ X24 @ X18)) | ~(X4 @ (comp @ X8 @ X24) @ (sub @ X7 @ X24) @ X18))))) & (hoasinduction_lem3aaa <=> ! [X35 : term,X4 : (subst > term > subst > $o)] : (~ ! [X9 : term] : ((X4 @ id @ (sub @ X35 @ (push @ X9 @ id)) @ id) | ~(X4 @ id @ X9 @ id)) | ~ ! [X36 : term,X10 : (subst > term > term)] : (~ ! [X38 : term,X37 : subst,X39 : subst] : (((sub @ (X10 @ X37 @ X38) @ X39)) = ((sub @ (sub @ X36 @ (push @ X38 @ X37)) @ X39))) | ~ ! [X42 : subst,X40 : subst,X41 : term] : (((X10 @ (comp @ X40 @ X42) @ (sub @ X41 @ X42))) = ((sub @ X36 @ (push @ (sub @ X41 @ X42) @ (comp @ X40 @ X42))))) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X7 : term] : ((X4 @ id @ (X10 @ id @ X7) @ id) | ~(X4 @ id @ X7 @ id))) | (X4 @ id @ (lam @ (sub @ X35 @ (push @ one @ sh))) @ id))) & (hoaslamnotap <=> ! [X30 : term,X29 : term,X10 : (subst > term > term)] : (~ ! [X7 : term,X8 : subst,X18 : subst] : (((sub @ (X10 @ X8 @ X7) @ X18)) = ((X10 @ (comp @ X8 @ X18) @ (sub @ X7 @ X18)))) | (((lam @ (X10 @ sh @ one))) != ((ap @ (sub @ X29 @ id) @ X30))))) & ((^[X43 : subst, X44 : (subst > term > term)] : ((d2term @ d_one))) = hoaslam) & apnotvar & hoaslaminj_gthm & (hoasapinj2_lthm <=> (! [X9 : term,X7 : term,X11 : term,X12 : term] : ((X11 = X12) | (((ap @ X7 @ X11)) != ((ap @ X9 @ X12)))) => ! [X7 : term,X11 : term,X12 : term,X9 : term] : ((((ap @ (sub @ X7 @ id) @ X11)) != ((ap @ (sub @ X9 @ id) @ X12))) | (X11 = X12)))) & ((! [X4 : (subst > term > subst > $o)] : ~ ! [X5 : (term > $o)] : ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id))) => (! [X16 : term,X4 : (term > $o)] : (~ ! [X7 : term] : (~ ! [X9 : term] : (~(X4 @ X9) | (X4 @ (sub @ X7 @ (push @ X9 @ id)))) | (X4 @ (lam @ X7))) | ~ ! [X9 : term,X7 : term] : (~(X4 @ X9) | ~(X4 @ X7) | (X4 @ (ap @ X7 @ X9))) | ~ ! [X7 : term] : ((X4 @ X7) | ~(var @ X7)) | (X4 @ X16)) => (! [X7 : term] : (X7 = ((sub @ X7 @ id))) => (! [X4 : (subst > term > subst > $o),X5 : (term > $o),X19 : term] : (~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id))) | ~ ! [X10 : (subst > term > term)] : ((X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X7 : term] : ((X4 @ id @ (X10 @ id @ X7) @ id) | ~(X4 @ id @ X7 @ id)) | ~ ! [X18 : subst,X7 : term,X8 : subst] : (((sub @ (X10 @ X8 @ X7) @ X18)) = ((X10 @ (comp @ X8 @ X18) @ (sub @ X7 @ X18))))) | (X5 @ (lam @ X19)) | ~ ! [X9 : term] : (~(X5 @ X9) | (X5 @ (sub @ X19 @ (push @ X9 @ id))))) => ! [X4 : (subst > term > subst > $o),X17 : term] : (~ ! [X10 : (subst > term > term)] : (~ ! [X8 : subst,X18 : subst,X7 : term] : (((sub @ (X10 @ X8 @ X7) @ X18)) = ((X10 @ (comp @ X8 @ X18) @ (sub @ X7 @ X18)))) | ~ ! [X7 : term] : (~(X4 @ id @ X7 @ id) | (X4 @ id @ (X10 @ id @ X7) @ id)) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id)) | ~ ! [X9 : term,X7 : term] : ((X4 @ id @ (ap @ (sub @ X7 @ id) @ X9) @ id) | ~(X4 @ id @ X7 @ id) | ~(X4 @ id @ X9 @ id)) | (X4 @ id @ X17 @ id)))))) <=> hoasinduction_no_psi_cond_lthm) & pushprop_lthm & (! [X8 : subst] : (X8 = ((comp @ X8 @ id))) <=> axidr) & (! [X9 : term,X7 : term,X11 : term,X12 : term] : ((((ap @ (sub @ X7 @ id) @ X11)) != ((ap @ (sub @ X9 @ id) @ X12))) | (X11 = X12)) <=> hoasapinj2) & pushprop_lthm_orig & hoasinduction_lem2_gthm & pushprop_lem1_gthm & (! [X9 : term] : ~ ! [X10 : (subst > term > term)] : ~ ! [X8 : subst,X7 : term] : (((sub @ X9 @ (push @ X7 @ X8))) = ((X10 @ X8 @ X7))) <=> hoasinduction_lem3v2_f_lthm) & (! [X4 : (subst > term > subst > $o)] : ~ ! [X5 : (term > $o)] : ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id))) <=> hoasinduction_lem0) & hoasinduction_lem1v2_gthm & (! [X9 : term,X7 : term] : ((X7 = X9) | (((sub @ X7 @ sh)) != ((sub @ X9 @ sh)))) <=> shinj) & hoasinduction_lem3v2_gthm & hoasapinj1_gthm & (induction2 <=> ! [X4 : (term > $o),X16 : term] : (~ ! [X9 : term,X7 : term] : (~(X4 @ X7) | (X4 @ (ap @ X7 @ X9)) | ~(X4 @ X9)) | (X4 @ X16) | ~ ! [X7 : term] : (~ ! [X9 : term] : ((X4 @ (sub @ X7 @ (push @ X9 @ id))) | ~(X4 @ X9)) | (X4 @ (lam @ X7))) | ~ ! [X7 : term] : ((X4 @ X7) | ~(var @ X7)))) & (axshiftcons <=> ! [X7 : term,X8 : subst] : (X8 = ((comp @ sh @ (push @ X7 @ X8))))) & ~(var @ (d2term @ d_one)) & ulamvarsh & ~(hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) & (! [X4 : (subst > term > subst > $o),X17 : term] : (~ ! [X7 : term,X9 : term] : ((X4 @ id @ (ap @ (sub @ X7 @ id) @ X9) @ id) | ~(X4 @ id @ X7 @ id) | ~(X4 @ id @ X9 @ id)) | (X4 @ id @ X17 @ id) | ~ ! [X10 : (subst > term > term)] : (~ ! [X7 : term] : ((X4 @ id @ (X10 @ id @ X7) @ id) | ~(X4 @ id @ X7 @ id)) | ~ ! [X18 : subst,X7 : term,X8 : subst] : (((sub @ (X10 @ X8 @ X7) @ X18)) = ((X10 @ (comp @ X8 @ X18) @ (sub @ X7 @ X18)))) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id))) <=> hoasinduction_no_psi_cond) & hoasinduction_lem2v2_gthm & hoasinduction_lem3_gthm & (hoasinduction_lem2 <=> ! [X4 : (subst > term > subst > $o),X33 : term,X34 : term] : (~(X4 @ id @ X33 @ id) | ~ ! [X24 : subst,X18 : subst,X8 : subst,X7 : term] : ((X4 @ (comp @ X8 @ X24) @ (sub @ X7 @ X24) @ X18) | ~(X4 @ X8 @ X7 @ (comp @ X24 @ X18))) | (X4 @ id @ (ap @ X33 @ X34) @ id) | ~ ! [X9 : term,X7 : term] : (~(X4 @ id @ X9 @ id) | ~(X4 @ id @ X7 @ id) | (X4 @ id @ (ap @ (sub @ X7 @ id) @ X9) @ id)) | ~ ! [X7 : term,X24 : subst,X18 : subst,X8 : subst] : (~(X4 @ (comp @ X8 @ X24) @ (sub @ X7 @ X24) @ X18) | (X4 @ X8 @ X7 @ (comp @ X24 @ X18))) | ~(X4 @ id @ X34 @ id))) & ! [X47 : subst] : ? [X3 : d_subst] : (X47 = ((d2subst @ X3))) & (((push @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & (substmonoid <=> (! [X8 : subst] : (X8 = ((comp @ id @ X8))) & ! [X8 : subst] : (X8 = ((comp @ X8 @ id))))) & (((lam @ (d2term @ d_one))) = ((d2term @ d_one))) & (! [X7 : term,X9 : term] : ((X7 = X9) | (((lam @ X7)) != ((lam @ X9)))) <=> laminj) & (! [X8 : subst,X7 : term] : (X7 = ((sub @ one @ (push @ X7 @ X8)))) <=> axvarcons) & pushprop & hoasapnotvar & (one = ((d2term @ d_one))) & (! [X11 : term,X7 : term,X9 : term,X12 : term] : ((((ap @ X7 @ X11)) != ((ap @ X9 @ X12))) | (X11 = X12)) <=> apinj2) & (hoasinduction_lem3b <=> ! [X9 : term] : ~ ! [X10 : (subst > term > term)] : (((X10 @ sh @ one)) != ((sub @ X9 @ (push @ one @ sh))))) & axclos & (pushprop_lem0 <=> ! [X7 : term,X4 : (term > $o),X8 : subst] : ~ ! [X5 : (term > $o)] : ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ (sub @ X6 @ (push @ X7 @ X8)))))) & (! [X8 : subst] : (X8 = ((push @ (sub @ one @ X8) @ (comp @ sh @ X8)))) <=> axscons) & (! [X7 : term] : (X7 = ((sub @ X7 @ id))) <=> termmset) & (! [X19 : term,X5 : (term > $o),X4 : (subst > term > subst > $o)] : ((X5 @ (lam @ X19)) | ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id))) | ~ ! [X10 : (subst > term > term)] : ((X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X8 : subst,X7 : term,X18 : subst] : (((sub @ (X10 @ X8 @ X7) @ X18)) = ((X10 @ (comp @ X8 @ X18) @ (sub @ X7 @ X18)))) | ~ ! [X7 : term] : (~(X4 @ id @ X7 @ id) | (X4 @ id @ (X10 @ id @ X7) @ id))) | ~ ! [X9 : term] : ((X5 @ (sub @ X19 @ (push @ X9 @ id))) | ~(X5 @ X9))) <=> hoasinduction_lem3v2a) & (((ap @ (d2term @ d_one) @ (d2term @ d_one))) = ((d2term @ d_one))) & ((! [X7 : term] : (X7 = ((sub @ X7 @ id))) => (! [X12 : term,X7 : term,X9 : term,X11 : term] : ((((ap @ X7 @ X11)) != ((ap @ X9 @ X12))) | (X7 = X9)) => ! [X11 : term,X12 : term,X7 : term,X9 : term] : ((((ap @ (sub @ X7 @ id) @ X11)) != ((ap @ (sub @ X9 @ id) @ X12))) | (X7 = X9)))) <=> hoasapinj1_lthm) & (pushprop_lem1 <=> ! [X9 : term,X4 : (term > $o),X7 : term,X24 : (term > $o),X8 : subst] : ((X24 @ (sub @ X7 @ (push @ X9 @ X8))) | ~(X4 @ X7))) & ((! [X7 : term,X8 : subst] : (X7 = ((sub @ one @ (push @ X7 @ X8)))) => (! [X7 : term,X8 : subst] : (X8 = ((comp @ sh @ (push @ X7 @ X8)))) => (! [X8 : subst] : (X8 = ((comp @ X8 @ id))) => (! [X4 : (term > $o),X15 : term] : ((X4 @ X15) | ~ ! [X7 : term] : (~(X4 @ X7) | (X4 @ (lam @ X7))) | ~ ! [X7 : term,X9 : term] : (~(X4 @ X9) | (X4 @ (ap @ X7 @ X9)) | ~(X4 @ X7)) | ~ ! [X7 : term] : ((X4 @ X7) | ~(var @ X7))) => ! [X4 : (term > $o),X14 : subst,X13 : term] : (~ ! [X9 : term,X7 : term] : (~(X4 @ X7) | (X4 @ (ap @ X7 @ X9)) | ~(X4 @ X9)) | (X4 @ (sub @ X13 @ X14)) | ~ ! [X9 : term] : ((X4 @ (sub @ X9 @ X14)) | ~(var @ X9)) | ~ ! [X7 : term] : ((X4 @ (lam @ X7)) | ~ ! [X9 : term] : (~(X4 @ X9) | (X4 @ (sub @ X7 @ (push @ X9 @ id)))))))))) <=> induction2lem_lthm) & ((! [X7 : term] : (X7 = ((sub @ X7 @ id))) => (! [X4 : (subst > term > subst > $o),X28 : term] : ((X4 @ id @ (lam @ (sub @ X28 @ (push @ one @ sh))) @ id) | ~ ! [X10 : (subst > term > term)] : (~ ! [X7 : term] : (~(X4 @ id @ X7 @ id) | (X4 @ id @ (X10 @ id @ X7) @ id)) | ~ ! [X7 : term,X18 : subst,X8 : subst] : (((sub @ (X10 @ X8 @ X7) @ X18)) = ((X10 @ (comp @ X8 @ X18) @ (sub @ X7 @ X18)))) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id)) | ~ ! [X9 : term] : ((X4 @ id @ (sub @ X28 @ (push @ X9 @ id)) @ id) | ~(X4 @ id @ X9 @ id))) => ! [X32 : term,X4 : (subst > term > subst > $o)] : (~ ! [X10 : (subst > term > term)] : ((X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X8 : subst,X7 : term,X18 : subst] : (((sub @ (X10 @ X8 @ X7) @ X18)) = ((X10 @ (comp @ X8 @ X18) @ (sub @ X7 @ X18)))) | ~ ! [X7 : term] : (~(X4 @ id @ X7 @ id) | (X4 @ id @ (X10 @ id @ X7) @ id))) | (X4 @ id @ (lam @ X32) @ id) | ~ ! [X9 : term] : ((X4 @ id @ (sub @ X32 @ (push @ X9 @ id)) @ id) | ~(X4 @ id @ X9 @ id))))) <=> hoasinduction_lem3a_lthm) & (hoasinduction_p_and_p_prime = (^[X4 : (subst > term > subst > $o), X5 : (term > $o)] : (! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id)))))) & (sh = ((d2subst @ d_id))) & pushprop_gthm & (! [X4 : (term > $o),X5 : (term > $o),X7 : term,X8 : subst] : ((X5 @ one) | ~(X4 @ X7) | ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ (sub @ X6 @ (push @ X7 @ X8)))))) <=> pushprop_lem1v2) & hoasinduction_lem1_gthm & lamnotvar & (! [X7 : term,X11 : term,X9 : term,X12 : term] : ((((ap @ (sub @ X7 @ id) @ X11)) != ((ap @ (sub @ X9 @ id) @ X12))) | (X7 = X9)) <=> hoasapinj1) & hoasinduction_lem1 & ! [X0 : d_term] : (X0 = d_one) & (hoasinduction_lem3v2a_lthm <=> (! [X9 : term] : ~ ! [X10 : (subst > term > term)] : ~ ! [X8 : subst,X7 : term] : (((sub @ X9 @ (push @ X7 @ X8))) = ((X10 @ X8 @ X7))) => (! [X7 : term] : (X7 = ((sub @ X7 @ id))) => ! [X5 : (term > $o),X4 : (subst > term > subst > $o),X19 : term] : (~ ! [X9 : term] : ((X5 @ (sub @ X19 @ (push @ X9 @ id))) | ~(X5 @ X9)) | ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id))) | ~ ! [X10 : (subst > term > term)] : (~ ! [X8 : subst,X7 : term,X18 : subst] : (((sub @ (X10 @ X8 @ X7) @ X18)) = ((X10 @ (comp @ X8 @ X18) @ (sub @ X7 @ X18)))) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X7 : term] : ((X4 @ id @ (X10 @ id @ X7) @ id) | ~(X4 @ id @ X7 @ id))) | (X5 @ (lam @ X19)))))) & (! [X4 : (subst > term > subst > $o),X28 : term] : (~ ! [X10 : (subst > term > term)] : (~ ! [X7 : term] : (~(X4 @ id @ X7 @ id) | (X4 @ id @ (X10 @ id @ X7) @ id)) | ~ ! [X8 : subst,X18 : subst,X7 : term] : (((sub @ (X10 @ X8 @ X7) @ X18)) = ((X10 @ (comp @ X8 @ X18) @ (sub @ X7 @ X18)))) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id)) | (X4 @ id @ (lam @ (sub @ X28 @ (push @ one @ sh))) @ id) | ~ ! [X9 : term] : (~(X4 @ id @ X9 @ id) | (X4 @ id @ (sub @ X28 @ (push @ X9 @ id)) @ id))) <=> hoasinduction_lem3aa) & hoaslamnotap_lthm & pushprop_lem1v2_gthm & pushprop_lem3v2_lthm & (hoasinduction_lthm <=> (! [X4 : (term > $o),X16 : term] : (~ ! [X7 : term] : (~ ! [X9 : term] : (~(X4 @ X9) | (X4 @ (sub @ X7 @ (push @ X9 @ id)))) | (X4 @ (lam @ X7))) | (X4 @ X16) | ~ ! [X7 : term,X9 : term] : (~(X4 @ X9) | (X4 @ (ap @ X7 @ X9)) | ~(X4 @ X7)) | ~ ! [X7 : term] : ((X4 @ X7) | ~(var @ X7))) => (! [X33 : term,X4 : (subst > term > subst > $o),X34 : term] : (~ ! [X8 : subst,X7 : term,X24 : subst,X18 : subst] : ((X4 @ X8 @ X7 @ (comp @ X24 @ X18)) | ~(X4 @ (comp @ X8 @ X24) @ (sub @ X7 @ X24) @ X18)) | ~(X4 @ id @ X34 @ id) | ~ ! [X7 : term,X8 : subst,X24 : subst,X18 : subst] : ((X4 @ (comp @ X8 @ X24) @ (sub @ X7 @ X24) @ X18) | ~(X4 @ X8 @ X7 @ (comp @ X24 @ X18))) | ~(X4 @ id @ X33 @ id) | (X4 @ id @ (ap @ X33 @ X34) @ id) | ~ ! [X7 : term,X9 : term] : ((X4 @ id @ (ap @ (sub @ X7 @ id) @ X9) @ id) | ~(X4 @ id @ X7 @ id) | ~(X4 @ id @ X9 @ id))) => (! [X27 : term,X4 : (subst > term > subst > $o)] : (~ ! [X9 : term] : (~(X4 @ id @ X9 @ id) | (X4 @ id @ (sub @ X27 @ (push @ X9 @ id)) @ id)) | ~ ! [X8 : subst,X18 : subst,X24 : subst,X7 : term] : (~(X4 @ X8 @ X7 @ (comp @ X24 @ X18)) | (X4 @ (comp @ X8 @ X24) @ (sub @ X7 @ X24) @ X18)) | (X4 @ id @ (lam @ X27) @ id) | ~ ! [X18 : subst,X8 : subst,X24 : subst,X7 : term] : (~(X4 @ (comp @ X8 @ X24) @ (sub @ X7 @ X24) @ X18) | (X4 @ X8 @ X7 @ (comp @ X24 @ X18))) | ~ ! [X10 : (subst > term > term)] : ((X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X7 : term] : (~(X4 @ id @ X7 @ id) | (X4 @ id @ (X10 @ id @ X7) @ id)) | ~ ! [X7 : term,X8 : subst,X18 : subst] : (((sub @ (X10 @ X8 @ X7) @ X18)) = ((X10 @ (comp @ X8 @ X18) @ (sub @ X7 @ X18)))))) => ! [X4 : (subst > term > subst > $o),X23 : term] : (~ ! [X7 : term,X8 : subst,X18 : subst,X24 : subst] : ((X4 @ (comp @ X8 @ X24) @ (sub @ X7 @ X24) @ X18) | ~(X4 @ X8 @ X7 @ (comp @ X24 @ X18))) | ~ ! [X10 : (subst > term > term)] : ((X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X7 : term] : (~(X4 @ id @ X7 @ id) | (X4 @ id @ (X10 @ id @ X7) @ id)) | ~ ! [X18 : subst,X7 : term,X8 : subst] : (((sub @ (X10 @ X8 @ X7) @ X18)) = ((X10 @ (comp @ X8 @ X18) @ (sub @ X7 @ X18))))) | ~ ! [X7 : term,X9 : term] : (~(X4 @ id @ X9 @ id) | ~(X4 @ id @ X7 @ id) | (X4 @ id @ (ap @ (sub @ X7 @ id) @ X9) @ id)) | ~ ! [X7 : term] : (~(var @ (sub @ X7 @ id)) | (X4 @ id @ X7 @ id)) | ~ ! [X24 : subst,X8 : subst,X18 : subst,X7 : term] : ((X4 @ X8 @ X7 @ (comp @ X24 @ X18)) | ~(X4 @ (comp @ X8 @ X24) @ (sub @ X7 @ X24) @ X18)) | (X4 @ id @ X23 @ id)))))) & hoasinduction_lem3a_gthm & (((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) = ((d2term @ d_one))) & (hoasinduction_lem3 <=> ! [X27 : term,X4 : (subst > term > subst > $o)] : ((X4 @ id @ (lam @ X27) @ id) | ~ ! [X10 : (subst > term > term)] : (~ ! [X7 : term] : (~(X4 @ id @ X7 @ id) | (X4 @ id @ (X10 @ id @ X7) @ id)) | ~ ! [X7 : term,X8 : subst,X18 : subst] : (((sub @ (X10 @ X8 @ X7) @ X18)) = ((X10 @ (comp @ X8 @ X18) @ (sub @ X7 @ X18)))) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id)) | ~ ! [X18 : subst,X8 : subst,X24 : subst,X7 : term] : (~(X4 @ X8 @ X7 @ (comp @ X24 @ X18)) | (X4 @ (comp @ X8 @ X24) @ (sub @ X7 @ X24) @ X18)) | ~ ! [X24 : subst,X18 : subst,X7 : term,X8 : subst] : ((X4 @ X8 @ X7 @ (comp @ X24 @ X18)) | ~(X4 @ (comp @ X8 @ X24) @ (sub @ X7 @ X24) @ X18)) | ~ ! [X9 : term] : (~(X4 @ id @ X9 @ id) | (X4 @ id @ (sub @ X27 @ (push @ X9 @ id)) @ id)))) & substmonoid_gthm & ((! [X7 : term,X8 : subst] : (X7 = ((sub @ one @ (push @ X7 @ X8)))) => ! [X4 : (term > $o),X5 : (term > $o),X8 : subst,X7 : term] : (~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ (sub @ X6 @ (push @ X7 @ X8))))) | (X5 @ one) | ~(X4 @ X7))) <=> pushprop_lem1v2_lthm) & (hoasinduction_lem3_lthm <=> (! [X7 : term] : (X7 = ((sub @ X7 @ id))) => (! [X4 : (subst > term > subst > $o),X28 : term] : (~ ! [X10 : (subst > term > term)] : (~ ! [X7 : term] : ((X4 @ id @ (X10 @ id @ X7) @ id) | ~(X4 @ id @ X7 @ id)) | ~ ! [X18 : subst,X8 : subst,X7 : term] : (((sub @ (X10 @ X8 @ X7) @ X18)) = ((X10 @ (comp @ X8 @ X18) @ (sub @ X7 @ X18)))) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id)) | (X4 @ id @ (lam @ (sub @ X28 @ (push @ one @ sh))) @ id) | ~ ! [X9 : term] : (~(X4 @ id @ X9 @ id) | (X4 @ id @ (sub @ X28 @ (push @ X9 @ id)) @ id))) => ! [X4 : (subst > term > subst > $o),X27 : term] : (~ ! [X18 : subst,X7 : term,X24 : subst,X8 : subst] : ((X4 @ (comp @ X8 @ X24) @ (sub @ X7 @ X24) @ X18) | ~(X4 @ X8 @ X7 @ (comp @ X24 @ X18))) | ~ ! [X9 : term] : ((X4 @ id @ (sub @ X27 @ (push @ X9 @ id)) @ id) | ~(X4 @ id @ X9 @ id)) | (X4 @ id @ (lam @ X27) @ id) | ~ ! [X10 : (subst > term > term)] : ((X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X7 : term] : ((X4 @ id @ (X10 @ id @ X7) @ id) | ~(X4 @ id @ X7 @ id)) | ~ ! [X7 : term,X8 : subst,X18 : subst] : (((sub @ (X10 @ X8 @ X7) @ X18)) = ((X10 @ (comp @ X8 @ X18) @ (sub @ X7 @ X18))))) | ~ ! [X7 : term,X18 : subst,X24 : subst,X8 : subst] : (~(X4 @ (comp @ X8 @ X24) @ (sub @ X7 @ X24) @ X18) | (X4 @ X8 @ X7 @ (comp @ X24 @ X18))))))) & pushprop_lem2v2_lthm & ((! [X7 : term] : (X7 = ((sub @ X7 @ id))) => ! [X7 : term] : (X7 = ((sub @ X7 @ id)))) <=> termmset_lthm) & pushprop_lem0_gthm & (((sub @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2term @ d_one))) & (hoasinduction_lthm_3 <=> (! [X4 : (subst > term > subst > $o)] : ~ ! [X5 : (term > $o)] : ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id))) => (! [X16 : term,X4 : (term > $o)] : (~ ! [X9 : term,X7 : term] : (~(X4 @ X7) | (X4 @ (ap @ X7 @ X9)) | ~(X4 @ X9)) | ~ ! [X7 : term] : ((X4 @ (lam @ X7)) | ~ ! [X9 : term] : ((X4 @ (sub @ X7 @ (push @ X9 @ id))) | ~(X4 @ X9))) | ~ ! [X7 : term] : ((X4 @ X7) | ~(var @ X7)) | (X4 @ X16)) => (! [X7 : term] : (X7 = ((sub @ X7 @ id))) => (! [X4 : (subst > term > subst > $o),X19 : term,X5 : (term > $o)] : (~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id))) | (X5 @ (lam @ X19)) | ~ ! [X9 : term] : ((X5 @ (sub @ X19 @ (push @ X9 @ id))) | ~(X5 @ X9)) | ~ ! [X10 : (subst > term > term)] : (~ ! [X8 : subst,X18 : subst,X7 : term] : (((sub @ (X10 @ X8 @ X7) @ X18)) = ((X10 @ (comp @ X8 @ X18) @ (sub @ X7 @ X18)))) | ~ ! [X7 : term] : (~(X4 @ id @ X7 @ id) | (X4 @ id @ (X10 @ id @ X7) @ id)) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id))) => ! [X4 : (subst > term > subst > $o),X23 : term] : (~ ! [X7 : term,X9 : term] : (~(X4 @ id @ X9 @ id) | (X4 @ id @ (ap @ (sub @ X7 @ id) @ X9) @ id) | ~(X4 @ id @ X7 @ id)) | ~ ! [X7 : term] : ((X4 @ id @ X7 @ id) | ~(var @ (sub @ X7 @ id))) | ~ ! [X7 : term,X24 : subst,X8 : subst,X18 : subst] : ((X4 @ (comp @ X8 @ X24) @ (sub @ X7 @ X24) @ X18) | ~(X4 @ X8 @ X7 @ (comp @ X24 @ X18))) | (X4 @ id @ X23 @ id) | ~ ! [X10 : (subst > term > term)] : (~ ! [X7 : term] : ((X4 @ id @ (X10 @ id @ X7) @ id) | ~(X4 @ id @ X7 @ id)) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X8 : subst,X7 : term,X18 : subst] : (((sub @ (X10 @ X8 @ X7) @ X18)) = ((X10 @ (comp @ X8 @ X18) @ (sub @ X7 @ X18))))) | ~ ! [X24 : subst,X7 : term,X18 : subst,X8 : subst] : ((X4 @ X8 @ X7 @ (comp @ X24 @ X18)) | ~(X4 @ (comp @ X8 @ X24) @ (sub @ X7 @ X24) @ X18)))))))) & ((! [X8 : subst,X7 : term] : (X7 = ((sub @ one @ (push @ X7 @ X8)))) => (! [X8 : subst,X7 : term] : (X8 = ((comp @ sh @ (push @ X7 @ X8)))) => ! [X9 : term,X7 : term,X24 : (term > $o),X8 : subst,X4 : (term > $o)] : (~(X4 @ X7) | (X24 @ (sub @ X7 @ (push @ X9 @ X8)))))) <=> pushprop_lem1_lthm) & (hoaslaminj_lthm <=> (! [X7 : term,X8 : subst] : (X7 = ((sub @ one @ (push @ X7 @ X8)))) => (! [X7 : term,X8 : subst] : (X8 = ((comp @ sh @ (push @ X7 @ X8)))) => (! [X9 : term,X7 : term] : ((((lam @ X7)) != ((lam @ X9))) | (X7 = X9)) => ! [X20 : (subst > term > term),X21 : subst,X10 : (subst > term > term),X22 : term] : ((((lam @ (X10 @ sh @ one))) != ((lam @ (X20 @ sh @ one)))) | (((X10 @ X21 @ X22)) = ((X20 @ X21 @ X22))) | ~ ! [X18 : subst,X8 : subst,X7 : term] : (((sub @ (X10 @ X8 @ X7) @ X18)) = ((X10 @ (comp @ X8 @ X18) @ (sub @ X7 @ X18)))) | ~ ! [X7 : term,X18 : subst,X8 : subst] : (((sub @ (X20 @ X8 @ X7) @ X18)) = ((X20 @ (comp @ X8 @ X18) @ (sub @ X7 @ X18))))))))) & (! [X21 : subst,X22 : term,X20 : (subst > term > term),X10 : (subst > term > term)] : ((((lam @ (X10 @ sh @ one))) != ((lam @ (X20 @ sh @ one)))) | ~ ! [X7 : term,X8 : subst,X18 : subst] : (((sub @ (X10 @ X8 @ X7) @ X18)) = ((X10 @ (comp @ X8 @ X18) @ (sub @ X7 @ X18)))) | (((X10 @ X21 @ X22)) = ((X20 @ X21 @ X22))) | ~ ! [X18 : subst,X7 : term,X8 : subst] : (((sub @ (X20 @ X8 @ X7) @ X18)) = ((X20 @ (comp @ X8 @ X18) @ (sub @ X7 @ X18))))) <=> hoaslaminj) & (axvarid <=> ! [X7 : term] : (X7 = ((sub @ X7 @ id)))) & hoasapnotvar_lthm & (! [X4 : (subst > term > subst > $o)] : ~ ! [X5 : (term > $o)] : ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id))) <=> hoasinduction_lem0_lthm) & axassoc & ! [X3 : d_subst] : (X3 = d_id) & (hoasinduction_lem2v2 <=> ! [X26 : term,X25 : term,X4 : (subst > term > subst > $o),X5 : (term > $o)] : (~ ! [X9 : term,X7 : term] : (~(X4 @ id @ X7 @ id) | ~(X4 @ id @ X9 @ id) | (X4 @ id @ (ap @ (sub @ X7 @ id) @ X9) @ id)) | ~ ! [X18 : subst,X8 : subst,X7 : term,X24 : subst] : ((X4 @ X8 @ X7 @ (comp @ X24 @ X18)) | ~(X4 @ (comp @ X8 @ X24) @ (sub @ X7 @ X24) @ X18)) | ~ ! [X8 : subst,X24 : subst,X18 : subst,X7 : term] : (~(X4 @ X8 @ X7 @ (comp @ X24 @ X18)) | (X4 @ (comp @ X8 @ X24) @ (sub @ X7 @ X24) @ X18)) | (X5 @ (ap @ X25 @ X26)) | ~(X5 @ X25) | ~(X5 @ X26) | ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id))))) & axapp & ~lamnotap), file('/export/starexec/sandbox2/benchmark/theBenchmark.p',alg444_1)). thf(f2,conjecture,( (! [X0 : (subst > term > subst > $o)] : (! [X4 : subst,X2 : term,X1 : subst,X3 : subst] : ((X0 @ X1 @ X2 @ (comp @ X4 @ X3)) => (X0 @ (comp @ X1 @ X4) @ (sub @ X2 @ X4) @ X3)) => (! [X1 : subst,X4 : subst,X2 : term,X3 : subst] : ((X0 @ (comp @ X1 @ X4) @ (sub @ X2 @ X4) @ X3) => (X0 @ X1 @ X2 @ (comp @ X4 @ X3))) => (! [X6 : (subst > term > term)] : (! [X2 : term,X3 : subst,X1 : subst] : (((sub @ (X6 @ X1 @ X2) @ X3)) = ((X6 @ (comp @ X1 @ X3) @ (sub @ X2 @ X3)))) => (! [X2 : term] : ((X0 @ id @ X2 @ id) => (X0 @ id @ (X6 @ id @ X2) @ id)) => (X0 @ id @ (hoaslam @ id @ (^[X1 : subst, X2 : term] : ((X6 @ X1 @ X2)))) @ id))) => ! [X2 : term] : (! [X5 : term] : ((X0 @ id @ X5 @ id) => (X0 @ id @ (sub @ X2 @ (push @ X5 @ id)) @ id)) => (X0 @ id @ (lam @ X2) @ id))))) = hoasinduction_lem3)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p',hoasinduction_lem3)). thf(f3,negated_conjecture,( ~ (! [X0 : (subst > term > subst > $o)] : (! [X4 : subst,X2 : term,X1 : subst,X3 : subst] : ((X0 @ X1 @ X2 @ (comp @ X4 @ X3)) => (X0 @ (comp @ X1 @ X4) @ (sub @ X2 @ X4) @ X3)) => (! [X1 : subst,X4 : subst,X2 : term,X3 : subst] : ((X0 @ (comp @ X1 @ X4) @ (sub @ X2 @ X4) @ X3) => (X0 @ X1 @ X2 @ (comp @ X4 @ X3))) => (! [X6 : (subst > term > term)] : (! [X2 : term,X3 : subst,X1 : subst] : (((sub @ (X6 @ X1 @ X2) @ X3)) = ((X6 @ (comp @ X1 @ X3) @ (sub @ X2 @ X3)))) => (! [X2 : term] : ((X0 @ id @ X2 @ id) => (X0 @ id @ (X6 @ id @ X2) @ id)) => (X0 @ id @ (hoaslam @ id @ (^[X1 : subst, X2 : term] : ((X6 @ X1 @ X2)))) @ id))) => ! [X2 : term] : (! [X5 : term] : ((X0 @ id @ X5 @ id) => (X0 @ id @ (sub @ X2 @ (push @ X5 @ id)) @ id)) => (X0 @ id @ (lam @ X2) @ id))))) = hoasinduction_lem3)), inference(negated_conjecture,[status(cth)],[f2])). thf(f4,plain,( ~ (! [X0 : (subst > term > subst > $o)] : (! [X1 : subst,X2 : term,X3 : subst,X4 : subst] : ((X0 @ X3 @ X2 @ (comp @ X1 @ X4)) => (X0 @ (comp @ X3 @ X1) @ (sub @ X2 @ X1) @ X4)) => (! [X5 : subst,X6 : subst,X7 : term,X8 : subst] : ((X0 @ (comp @ X5 @ X6) @ (sub @ X7 @ X6) @ X8) => (X0 @ X5 @ X7 @ (comp @ X6 @ X8))) => (! [X9 : (subst > term > term)] : (! [X10 : term,X11 : subst,X12 : subst] : (((X9 @ (comp @ X12 @ X11) @ (sub @ X10 @ X11))) = ((sub @ (X9 @ X12 @ X10) @ X11))) => (! [X13 : term] : ((X0 @ id @ X13 @ id) => (X0 @ id @ (X9 @ id @ X13) @ id)) => (X0 @ id @ (hoaslam @ id @ (^[X14 : subst, X15 : term] : ((X9 @ X14 @ X15)))) @ id))) => ! [X16 : term] : (! [X17 : term] : ((X0 @ id @ X17 @ id) => (X0 @ id @ (sub @ X16 @ (push @ X17 @ id)) @ id)) => (X0 @ id @ (lam @ X16) @ id))))) = hoasinduction_lem3)), inference(rectify,[],[f3])). thf(f5,plain,( ~(! [X0 : (subst > term > subst > $o)] : (! [X3 : subst,X4 : subst,X2 : term,X1 : subst] : (($true = ((X0 @ X3 @ X2 @ (comp @ X1 @ X4)))) => (((X0 @ (comp @ X3 @ X1) @ (sub @ X2 @ X1) @ X4)) = $true)) => (! [X5 : subst,X7 : term,X8 : subst,X6 : subst] : ((((X0 @ (comp @ X5 @ X6) @ (sub @ X7 @ X6) @ X8)) = $true) => (((X0 @ X5 @ X7 @ (comp @ X6 @ X8))) = $true)) => (! [X9 : (subst > term > term)] : (! [X12 : subst,X10 : term,X11 : subst] : (((X9 @ (comp @ X12 @ X11) @ (sub @ X10 @ X11))) = ((sub @ (X9 @ X12 @ X10) @ X11))) => (! [X13 : term] : ((((X0 @ id @ X13 @ id)) = $true) => ($true = ((X0 @ id @ (X9 @ id @ X13) @ id)))) => ($true = ((X0 @ id @ (hoaslam @ id @ (^[Y0 : subst]: ((^[Y1 : term]: (X9 @ Y0 @ Y1))))) @ id))))) => ! [X16 : term] : (! [X17 : term] : (($true = ((X0 @ id @ X17 @ id))) => (((X0 @ id @ (sub @ X16 @ (push @ X17 @ id)) @ id)) = $true)) => (((X0 @ id @ (lam @ X16) @ id)) = $true))))) <=> (hoasinduction_lem3 = $true))), inference(fool_elimination,[],[f4])). thf(f6,plain,( (! [X0 : term,X1 : term,X2 : term,X3 : term] : ((((ap @ X1 @ X3)) != ((ap @ X0 @ X2))) | (X0 = X1)) <=> apinj1) & (id = ((d2subst @ d_id))) & pushprop_lem3v2 & (hoasinduction_lem3v2 <=> ! [X4 : term,X5 : (term > $o),X6 : (subst > term > subst > $o)] : (~ ! [X7 : subst,X8 : term,X9 : subst,X10 : subst] : ((X6 @ X10 @ X8 @ (comp @ X7 @ X9)) | ~(X6 @ (comp @ X10 @ X7) @ (sub @ X8 @ X7) @ X9)) | ~ ! [X11 : term] : (~(X5 @ X11) | (X5 @ (sub @ X4 @ (push @ X11 @ id)))) | (X5 @ (lam @ X4)) | ~ ! [X12 : (subst > term > term)] : (~ ! [X13 : term] : ((X6 @ id @ (X12 @ id @ X13) @ id) | ~(X6 @ id @ X13 @ id)) | ~ ! [X14 : term,X15 : subst,X16 : subst] : (((sub @ (X12 @ X15 @ X14) @ X16)) = ((X12 @ (comp @ X15 @ X16) @ (sub @ X14 @ X16)))) | (X6 @ id @ (lam @ (X12 @ sh @ one)) @ id)) | ~ ! [X17 : subst,X18 : subst,X19 : subst,X20 : term] : (~(X6 @ X18 @ X20 @ (comp @ X19 @ X17)) | (X6 @ (comp @ X18 @ X19) @ (sub @ X20 @ X19) @ X17)) | ~ ! [X21 : term] : (((X6 @ id @ X21 @ id)) = ((X5 @ X21))))) & (pushprop_p_and_p_prime = (^[X22 : term, X23 : subst, X24 : (term > $o), X25 : (term > $o)] : (! [X26 : term] : (((X24 @ (sub @ X26 @ (push @ X22 @ X23)))) = ((X25 @ X26)))))) & hoasinduction_lem1_lthm & hoasinduction_gthm & hoaslamnotvar & (! [X27 : term,X28 : (subst > term > subst > $o)] : (~ ! [X29 : term,X30 : term] : (~(X28 @ id @ X29 @ id) | ~(X28 @ id @ X30 @ id) | (X28 @ id @ (ap @ (sub @ X29 @ id) @ X30) @ id)) | ~ ! [X31 : subst,X32 : subst,X33 : term,X34 : subst] : ((X28 @ (comp @ X32 @ X31) @ (sub @ X33 @ X31) @ X34) | ~(X28 @ X32 @ X33 @ (comp @ X31 @ X34))) | ~ ! [X35 : subst,X36 : subst,X37 : subst,X38 : term] : (~(X28 @ (comp @ X35 @ X36) @ (sub @ X38 @ X36) @ X37) | (X28 @ X35 @ X38 @ (comp @ X36 @ X37))) | ~ ! [X39 : term] : ((X28 @ id @ X39 @ id) | ~(var @ (sub @ X39 @ id))) | ~ ! [X40 : (subst > term > term)] : (~ ! [X41 : subst,X42 : subst,X43 : term] : (((X40 @ (comp @ X42 @ X41) @ (sub @ X43 @ X41))) = ((sub @ (X40 @ X42 @ X43) @ X41))) | ~ ! [X44 : term] : ((X28 @ id @ (X40 @ id @ X44) @ id) | ~(X28 @ id @ X44 @ id)) | (X28 @ id @ (lam @ (X40 @ sh @ one)) @ id)) | (X28 @ id @ X27 @ id)) <=> hoasinduction) & induction2_gthm & ! [X45 : d_subst,X46 : d_subst] : ((((d2subst @ X45)) = ((d2subst @ X46))) => (X45 = X46)) & ~ulamvar1 & (! [X47 : term,X48 : subst,X49 : (term > $o)] : ~ ! [X50 : (term > $o)] : ~ ! [X51 : term] : (((X50 @ X51)) = ((X49 @ (sub @ X51 @ (push @ X47 @ X48))))) <=> pushprop_lem0_lthm) & axvarshift & (axidl <=> ! [X52 : subst] : (((comp @ id @ X52)) = X52)) & hoaslamnotvar_gthm & hoasapinj2_gthm & hoaslamnotap_gthm & induction2lem_gthm & (hoasinduction_lem3a <=> ! [X53 : (subst > term > subst > $o),X54 : term] : ((X53 @ id @ (lam @ X54) @ id) | ~ ! [X55 : term] : (~(X53 @ id @ X55 @ id) | (X53 @ id @ (sub @ X54 @ (push @ X55 @ id)) @ id)) | ~ ! [X56 : (subst > term > term)] : (~ ! [X57 : subst,X58 : term,X59 : subst] : (((sub @ (X56 @ X59 @ X58) @ X57)) = ((X56 @ (comp @ X59 @ X57) @ (sub @ X58 @ X57)))) | (X53 @ id @ (lam @ (X56 @ sh @ one)) @ id) | ~ ! [X60 : term] : (~(X53 @ id @ X60 @ id) | (X53 @ id @ (X56 @ id @ X60) @ id))))) & (! [X61 : term,X62 : subst,X63 : (term > $o)] : (~ ! [X64 : term,X65 : term] : (~(X63 @ X64) | (X63 @ (ap @ X65 @ X64)) | ~(X63 @ X65)) | ~ ! [X66 : term] : (~(var @ X66) | (X63 @ (sub @ X66 @ X62))) | ~ ! [X67 : term] : ((X63 @ (lam @ X67)) | ~ ! [X68 : term] : ((X63 @ (sub @ X67 @ (push @ X68 @ id))) | ~(X63 @ X68))) | (X63 @ (sub @ X61 @ X62))) <=> induction2lem) & hoasinduction_lem3b_gthm & axmap & pushprop_lem2v2 & axabs & hoaslamnotvar_lthm & (((comp @ (d2subst @ d_id) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & termmset_gthm & pushprop_lem2v2_gthm & (induction <=> ! [X69 : (term > $o),X70 : term] : ((X69 @ X70) | ~ ! [X71 : term,X72 : term] : (~(X69 @ X72) | ~(X69 @ X71) | (X69 @ (ap @ X72 @ X71))) | ~ ! [X73 : term] : (~(X69 @ X73) | (X69 @ (lam @ X73))) | ~ ! [X74 : term] : ((X69 @ X74) | ~(var @ X74)))) & (hoasinduction_lem3aa_lthm <=> ! [X75 : (subst > term > subst > $o),X76 : term] : (~ ! [X77 : term] : ((X75 @ id @ (sub @ X76 @ (push @ X77 @ id)) @ id) | ~(X75 @ id @ X77 @ id)) | ~ ! [X78 : (subst > term > term)] : (~ ! [X79 : term] : ((X75 @ id @ (X78 @ id @ X79) @ id) | ~(X75 @ id @ X79 @ id)) | (X75 @ id @ (lam @ (X78 @ sh @ one)) @ id) | ~ ! [X80 : subst,X81 : term,X82 : subst] : (((X78 @ (comp @ X80 @ X82) @ (sub @ X81 @ X82))) = ((sub @ (X78 @ X80 @ X81) @ X82)))) | (X75 @ id @ (lam @ (sub @ X76 @ (push @ one @ sh))) @ id))) & hoasinduction_lem1v2 & ulamvarind & (hoasinduction_lem3v2_f <=> ! [X83 : term] : ~ ! [X84 : (subst > term > term)] : ~ ! [X85 : term,X86 : subst] : (((sub @ X83 @ (push @ X85 @ X86))) = ((X84 @ X86 @ X85)))) & (hoasinduction_lem3b_lthm <=> ! [X87 : term] : ~ ! [X88 : (subst > term > term)] : (((sub @ X87 @ (push @ one @ sh))) != ((X88 @ sh @ one)))) & (induction2_lthm <=> (! [X89 : term] : (((sub @ X89 @ id)) = X89) => (! [X90 : (term > $o),X91 : subst,X92 : term] : (~ ! [X93 : term] : ((X90 @ (lam @ X93)) | ~ ! [X94 : term] : (~(X90 @ X94) | (X90 @ (sub @ X93 @ (push @ X94 @ id))))) | ~ ! [X95 : term,X96 : term] : ((X90 @ (ap @ X96 @ X95)) | ~(X90 @ X95) | ~(X90 @ X96)) | (X90 @ (sub @ X92 @ X91)) | ~ ! [X97 : term] : (~(var @ X97) | (X90 @ (sub @ X97 @ X91)))) => ! [X98 : (term > $o),X99 : term] : (~ ! [X100 : term,X101 : term] : (~(X98 @ X100) | (X98 @ (ap @ X101 @ X100)) | ~(X98 @ X101)) | (X98 @ X99) | ~ ! [X102 : term] : ((X98 @ X102) | ~(var @ X102)) | ~ ! [X103 : term] : (~ ! [X104 : term] : (~(X98 @ X104) | (X98 @ (sub @ X103 @ (push @ X104 @ id)))) | (X98 @ (lam @ X103))))))) & hoasapnotvar_gthm & ! [X105 : d_term,X106 : d_term] : ((((d2term @ X105)) = ((d2term @ X106))) => (X105 = X106)) & (substmonoid_lthm <=> (! [X107 : subst] : (((comp @ id @ X107)) = X107) => (! [X108 : subst] : (((comp @ X108 @ id)) = X108) => (! [X109 : subst] : (((comp @ id @ X109)) = X109) & ! [X110 : subst] : (((comp @ X110 @ id)) = X110))))) & ! [X111 : term] : ? [X112 : d_term] : (((d2term @ X112)) = X111) & hoasinduction_lem2_lthm & (hoasinduction_lem3v2_lthm <=> (! [X113 : term] : (((sub @ X113 @ id)) = X113) => ! [X114 : (subst > term > subst > $o),X115 : (term > $o),X116 : term] : ((X115 @ (lam @ X116)) | ~ ! [X117 : (subst > term > term)] : (~ ! [X118 : term] : ((X114 @ id @ (X117 @ id @ X118) @ id) | ~(X114 @ id @ X118 @ id)) | (X114 @ id @ (lam @ (X117 @ sh @ one)) @ id) | ~ ! [X119 : subst,X120 : term,X121 : subst] : (((sub @ (X117 @ X121 @ X120) @ X119)) = ((X117 @ (comp @ X121 @ X119) @ (sub @ X120 @ X119))))) | ~ ! [X122 : subst,X123 : term,X124 : subst,X125 : subst] : ((X114 @ (comp @ X125 @ X124) @ (sub @ X123 @ X124) @ X122) | ~(X114 @ X125 @ X123 @ (comp @ X124 @ X122))) | ~ ! [X126 : term] : (((X115 @ X126)) = ((X114 @ id @ X126 @ id))) | ~ ! [X127 : term] : ((X115 @ (sub @ X116 @ (push @ X127 @ id))) | ~(X115 @ X127)) | ~ ! [X128 : subst,X129 : subst,X130 : term,X131 : subst] : ((X114 @ X129 @ X130 @ (comp @ X131 @ X128)) | ~(X114 @ (comp @ X129 @ X131) @ (sub @ X130 @ X131) @ X128))))) & (hoasinduction_lem3aaa <=> ! [X132 : term,X133 : (subst > term > subst > $o)] : (~ ! [X134 : term] : ((X133 @ id @ (sub @ X132 @ (push @ X134 @ id)) @ id) | ~(X133 @ id @ X134 @ id)) | ~ ! [X135 : term,X136 : (subst > term > term)] : (~ ! [X137 : term,X138 : subst,X139 : subst] : (((sub @ (X136 @ X138 @ X137) @ X139)) = ((sub @ (sub @ X135 @ (push @ X137 @ X138)) @ X139))) | ~ ! [X140 : subst,X141 : subst,X142 : term] : (((sub @ X135 @ (push @ (sub @ X142 @ X140) @ (comp @ X141 @ X140)))) = ((X136 @ (comp @ X141 @ X140) @ (sub @ X142 @ X140)))) | (X133 @ id @ (lam @ (X136 @ sh @ one)) @ id) | ~ ! [X143 : term] : ((X133 @ id @ (X136 @ id @ X143) @ id) | ~(X133 @ id @ X143 @ id))) | (X133 @ id @ (lam @ (sub @ X132 @ (push @ one @ sh))) @ id))) & (hoaslamnotap <=> ! [X144 : term,X145 : term,X146 : (subst > term > term)] : (~ ! [X147 : term,X148 : subst,X149 : subst] : (((X146 @ (comp @ X148 @ X149) @ (sub @ X147 @ X149))) = ((sub @ (X146 @ X148 @ X147) @ X149))) | (((lam @ (X146 @ sh @ one))) != ((ap @ (sub @ X145 @ id) @ X144))))) & ((^[X150 : subst, X151 : (subst > term > term)] : ((d2term @ d_one))) = hoaslam) & apnotvar & hoaslaminj_gthm & (hoasapinj2_lthm <=> (! [X152 : term,X153 : term,X154 : term,X155 : term] : ((X154 = X155) | (((ap @ X152 @ X155)) != ((ap @ X153 @ X154)))) => ! [X156 : term,X157 : term,X158 : term,X159 : term] : ((((ap @ (sub @ X156 @ id) @ X157)) != ((ap @ (sub @ X159 @ id) @ X158))) | (X157 = X158)))) & ((! [X160 : (subst > term > subst > $o)] : ~ ! [X161 : (term > $o)] : ~ ! [X162 : term] : (((X160 @ id @ X162 @ id)) = ((X161 @ X162))) => (! [X163 : term,X164 : (term > $o)] : (~ ! [X165 : term] : (~ ! [X166 : term] : (~(X164 @ X166) | (X164 @ (sub @ X165 @ (push @ X166 @ id)))) | (X164 @ (lam @ X165))) | ~ ! [X167 : term,X168 : term] : (~(X164 @ X167) | ~(X164 @ X168) | (X164 @ (ap @ X168 @ X167))) | ~ ! [X169 : term] : ((X164 @ X169) | ~(var @ X169)) | (X164 @ X163)) => (! [X170 : term] : (((sub @ X170 @ id)) = X170) => (! [X171 : (subst > term > subst > $o),X172 : (term > $o),X173 : term] : (~ ! [X174 : term] : (((X172 @ X174)) = ((X171 @ id @ X174 @ id))) | ~ ! [X175 : (subst > term > term)] : ((X171 @ id @ (lam @ (X175 @ sh @ one)) @ id) | ~ ! [X176 : term] : ((X171 @ id @ (X175 @ id @ X176) @ id) | ~(X171 @ id @ X176 @ id)) | ~ ! [X177 : subst,X178 : term,X179 : subst] : (((X175 @ (comp @ X179 @ X177) @ (sub @ X178 @ X177))) = ((sub @ (X175 @ X179 @ X178) @ X177)))) | (X172 @ (lam @ X173)) | ~ ! [X180 : term] : (~(X172 @ X180) | (X172 @ (sub @ X173 @ (push @ X180 @ id))))) => ! [X181 : (subst > term > subst > $o),X182 : term] : (~ ! [X183 : (subst > term > term)] : (~ ! [X184 : subst,X185 : subst,X186 : term] : (((X183 @ (comp @ X184 @ X185) @ (sub @ X186 @ X185))) = ((sub @ (X183 @ X184 @ X186) @ X185))) | ~ ! [X187 : term] : (~(X181 @ id @ X187 @ id) | (X181 @ id @ (X183 @ id @ X187) @ id)) | (X181 @ id @ (lam @ (X183 @ sh @ one)) @ id)) | ~ ! [X188 : term,X189 : term] : ((X181 @ id @ (ap @ (sub @ X189 @ id) @ X188) @ id) | ~(X181 @ id @ X189 @ id) | ~(X181 @ id @ X188 @ id)) | (X181 @ id @ X182 @ id)))))) <=> hoasinduction_no_psi_cond_lthm) & pushprop_lthm & (! [X190 : subst] : (((comp @ X190 @ id)) = X190) <=> axidr) & (! [X191 : term,X192 : term,X193 : term,X194 : term] : ((((ap @ (sub @ X191 @ id) @ X194)) != ((ap @ (sub @ X192 @ id) @ X193))) | (X193 = X194)) <=> hoasapinj2) & pushprop_lthm_orig & hoasinduction_lem2_gthm & pushprop_lem1_gthm & (! [X195 : term] : ~ ! [X196 : (subst > term > term)] : ~ ! [X197 : subst,X198 : term] : (((X196 @ X197 @ X198)) = ((sub @ X195 @ (push @ X198 @ X197)))) <=> hoasinduction_lem3v2_f_lthm) & (! [X199 : (subst > term > subst > $o)] : ~ ! [X200 : (term > $o)] : ~ ! [X201 : term] : (((X199 @ id @ X201 @ id)) = ((X200 @ X201))) <=> hoasinduction_lem0) & hoasinduction_lem1v2_gthm & (! [X202 : term,X203 : term] : ((X202 = X203) | (((sub @ X202 @ sh)) != ((sub @ X203 @ sh)))) <=> shinj) & hoasinduction_lem3v2_gthm & hoasapinj1_gthm & (induction2 <=> ! [X204 : (term > $o),X205 : term] : (~ ! [X206 : term,X207 : term] : (~(X204 @ X207) | (X204 @ (ap @ X207 @ X206)) | ~(X204 @ X206)) | (X204 @ X205) | ~ ! [X208 : term] : (~ ! [X209 : term] : ((X204 @ (sub @ X208 @ (push @ X209 @ id))) | ~(X204 @ X209)) | (X204 @ (lam @ X208))) | ~ ! [X210 : term] : ((X204 @ X210) | ~(var @ X210)))) & (axshiftcons <=> ! [X211 : term,X212 : subst] : (((comp @ sh @ (push @ X211 @ X212))) = X212)) & ~(var @ (d2term @ d_one)) & ulamvarsh & ~(hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) & (! [X213 : (subst > term > subst > $o),X214 : term] : (~ ! [X215 : term,X216 : term] : ((X213 @ id @ (ap @ (sub @ X215 @ id) @ X216) @ id) | ~(X213 @ id @ X215 @ id) | ~(X213 @ id @ X216 @ id)) | (X213 @ id @ X214 @ id) | ~ ! [X217 : (subst > term > term)] : (~ ! [X218 : term] : ((X213 @ id @ (X217 @ id @ X218) @ id) | ~(X213 @ id @ X218 @ id)) | ~ ! [X219 : subst,X220 : term,X221 : subst] : (((sub @ (X217 @ X221 @ X220) @ X219)) = ((X217 @ (comp @ X221 @ X219) @ (sub @ X220 @ X219)))) | (X213 @ id @ (lam @ (X217 @ sh @ one)) @ id))) <=> hoasinduction_no_psi_cond) & hoasinduction_lem2v2_gthm & hoasinduction_lem3_gthm & (hoasinduction_lem2 <=> ! [X222 : (subst > term > subst > $o),X223 : term,X224 : term] : (~(X222 @ id @ X223 @ id) | ~ ! [X225 : subst,X226 : subst,X227 : subst,X228 : term] : ((X222 @ (comp @ X227 @ X225) @ (sub @ X228 @ X225) @ X226) | ~(X222 @ X227 @ X228 @ (comp @ X225 @ X226))) | (X222 @ id @ (ap @ X223 @ X224) @ id) | ~ ! [X229 : term,X230 : term] : (~(X222 @ id @ X229 @ id) | ~(X222 @ id @ X230 @ id) | (X222 @ id @ (ap @ (sub @ X230 @ id) @ X229) @ id)) | ~ ! [X231 : term,X232 : subst,X233 : subst,X234 : subst] : (~(X222 @ (comp @ X234 @ X232) @ (sub @ X231 @ X232) @ X233) | (X222 @ X234 @ X231 @ (comp @ X232 @ X233))) | ~(X222 @ id @ X224 @ id))) & ! [X235 : subst] : ? [X236 : d_subst] : (((d2subst @ X236)) = X235) & (((push @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & (substmonoid <=> (! [X237 : subst] : (((comp @ id @ X237)) = X237) & ! [X238 : subst] : (((comp @ X238 @ id)) = X238))) & (((lam @ (d2term @ d_one))) = ((d2term @ d_one))) & (! [X239 : term,X240 : term] : ((X239 = X240) | (((lam @ X239)) != ((lam @ X240)))) <=> laminj) & (! [X241 : subst,X242 : term] : (((sub @ one @ (push @ X242 @ X241))) = X242) <=> axvarcons) & pushprop & hoasapnotvar & (one = ((d2term @ d_one))) & (! [X243 : term,X244 : term,X245 : term,X246 : term] : ((((ap @ X244 @ X243)) != ((ap @ X245 @ X246))) | (X243 = X246)) <=> apinj2) & (hoasinduction_lem3b <=> ! [X247 : term] : ~ ! [X248 : (subst > term > term)] : (((sub @ X247 @ (push @ one @ sh))) != ((X248 @ sh @ one)))) & axclos & (pushprop_lem0 <=> ! [X249 : term,X250 : (term > $o),X251 : subst] : ~ ! [X252 : (term > $o)] : ~ ! [X253 : term] : (((X250 @ (sub @ X253 @ (push @ X249 @ X251)))) = ((X252 @ X253)))) & (! [X254 : subst] : (((push @ (sub @ one @ X254) @ (comp @ sh @ X254))) = X254) <=> axscons) & (! [X255 : term] : (((sub @ X255 @ id)) = X255) <=> termmset) & (! [X256 : term,X257 : (term > $o),X258 : (subst > term > subst > $o)] : ((X257 @ (lam @ X256)) | ~ ! [X259 : term] : (((X258 @ id @ X259 @ id)) = ((X257 @ X259))) | ~ ! [X260 : (subst > term > term)] : ((X258 @ id @ (lam @ (X260 @ sh @ one)) @ id) | ~ ! [X261 : subst,X262 : term,X263 : subst] : (((X260 @ (comp @ X261 @ X263) @ (sub @ X262 @ X263))) = ((sub @ (X260 @ X261 @ X262) @ X263))) | ~ ! [X264 : term] : (~(X258 @ id @ X264 @ id) | (X258 @ id @ (X260 @ id @ X264) @ id))) | ~ ! [X265 : term] : ((X257 @ (sub @ X256 @ (push @ X265 @ id))) | ~(X257 @ X265))) <=> hoasinduction_lem3v2a) & (((ap @ (d2term @ d_one) @ (d2term @ d_one))) = ((d2term @ d_one))) & ((! [X266 : term] : (((sub @ X266 @ id)) = X266) => (! [X267 : term,X268 : term,X269 : term,X270 : term] : ((((ap @ X269 @ X267)) != ((ap @ X268 @ X270))) | (X268 = X269)) => ! [X271 : term,X272 : term,X273 : term,X274 : term] : ((((ap @ (sub @ X273 @ id) @ X271)) != ((ap @ (sub @ X274 @ id) @ X272))) | (X273 = X274)))) <=> hoasapinj1_lthm) & (pushprop_lem1 <=> ! [X275 : term,X276 : (term > $o),X277 : term,X278 : (term > $o),X279 : subst] : ((X278 @ (sub @ X277 @ (push @ X275 @ X279))) | ~(X276 @ X277))) & ((! [X280 : term,X281 : subst] : (((sub @ one @ (push @ X280 @ X281))) = X280) => (! [X282 : term,X283 : subst] : (((comp @ sh @ (push @ X282 @ X283))) = X283) => (! [X284 : subst] : (((comp @ X284 @ id)) = X284) => (! [X285 : (term > $o),X286 : term] : ((X285 @ X286) | ~ ! [X287 : term] : (~(X285 @ X287) | (X285 @ (lam @ X287))) | ~ ! [X288 : term,X289 : term] : (~(X285 @ X289) | (X285 @ (ap @ X288 @ X289)) | ~(X285 @ X288)) | ~ ! [X290 : term] : ((X285 @ X290) | ~(var @ X290))) => ! [X291 : (term > $o),X292 : subst,X293 : term] : (~ ! [X294 : term,X295 : term] : (~(X291 @ X295) | (X291 @ (ap @ X295 @ X294)) | ~(X291 @ X294)) | (X291 @ (sub @ X293 @ X292)) | ~ ! [X296 : term] : ((X291 @ (sub @ X296 @ X292)) | ~(var @ X296)) | ~ ! [X297 : term] : ((X291 @ (lam @ X297)) | ~ ! [X298 : term] : (~(X291 @ X298) | (X291 @ (sub @ X297 @ (push @ X298 @ id)))))))))) <=> induction2lem_lthm) & ((! [X299 : term] : (((sub @ X299 @ id)) = X299) => (! [X300 : (subst > term > subst > $o),X301 : term] : ((X300 @ id @ (lam @ (sub @ X301 @ (push @ one @ sh))) @ id) | ~ ! [X302 : (subst > term > term)] : (~ ! [X303 : term] : (~(X300 @ id @ X303 @ id) | (X300 @ id @ (X302 @ id @ X303) @ id)) | ~ ! [X304 : term,X305 : subst,X306 : subst] : (((sub @ (X302 @ X306 @ X304) @ X305)) = ((X302 @ (comp @ X306 @ X305) @ (sub @ X304 @ X305)))) | (X300 @ id @ (lam @ (X302 @ sh @ one)) @ id)) | ~ ! [X307 : term] : ((X300 @ id @ (sub @ X301 @ (push @ X307 @ id)) @ id) | ~(X300 @ id @ X307 @ id))) => ! [X308 : term,X309 : (subst > term > subst > $o)] : (~ ! [X310 : (subst > term > term)] : ((X309 @ id @ (lam @ (X310 @ sh @ one)) @ id) | ~ ! [X311 : subst,X312 : term,X313 : subst] : (((X310 @ (comp @ X311 @ X313) @ (sub @ X312 @ X313))) = ((sub @ (X310 @ X311 @ X312) @ X313))) | ~ ! [X314 : term] : (~(X309 @ id @ X314 @ id) | (X309 @ id @ (X310 @ id @ X314) @ id))) | (X309 @ id @ (lam @ X308) @ id) | ~ ! [X315 : term] : ((X309 @ id @ (sub @ X308 @ (push @ X315 @ id)) @ id) | ~(X309 @ id @ X315 @ id))))) <=> hoasinduction_lem3a_lthm) & (hoasinduction_p_and_p_prime = (^[X316 : (subst > term > subst > $o), X317 : (term > $o)] : (! [X318 : term] : (((X317 @ X318)) = ((X316 @ id @ X318 @ id)))))) & (sh = ((d2subst @ d_id))) & pushprop_gthm & (! [X319 : (term > $o),X320 : (term > $o),X321 : term,X322 : subst] : ((X320 @ one) | ~(X319 @ X321) | ~ ! [X323 : term] : (((X320 @ X323)) = ((X319 @ (sub @ X323 @ (push @ X321 @ X322)))))) <=> pushprop_lem1v2) & hoasinduction_lem1_gthm & lamnotvar & (! [X324 : term,X325 : term,X326 : term,X327 : term] : ((((ap @ (sub @ X324 @ id) @ X325)) != ((ap @ (sub @ X326 @ id) @ X327))) | (X324 = X326)) <=> hoasapinj1) & hoasinduction_lem1 & ! [X328 : d_term] : (d_one = X328) & (hoasinduction_lem3v2a_lthm <=> (! [X329 : term] : ~ ! [X330 : (subst > term > term)] : ~ ! [X331 : subst,X332 : term] : (((sub @ X329 @ (push @ X332 @ X331))) = ((X330 @ X331 @ X332))) => (! [X333 : term] : (((sub @ X333 @ id)) = X333) => ! [X334 : (term > $o),X335 : (subst > term > subst > $o),X336 : term] : (~ ! [X337 : term] : ((X334 @ (sub @ X336 @ (push @ X337 @ id))) | ~(X334 @ X337)) | ~ ! [X338 : term] : (((X335 @ id @ X338 @ id)) = ((X334 @ X338))) | ~ ! [X339 : (subst > term > term)] : (~ ! [X340 : subst,X341 : term,X342 : subst] : (((sub @ (X339 @ X340 @ X341) @ X342)) = ((X339 @ (comp @ X340 @ X342) @ (sub @ X341 @ X342)))) | (X335 @ id @ (lam @ (X339 @ sh @ one)) @ id) | ~ ! [X343 : term] : ((X335 @ id @ (X339 @ id @ X343) @ id) | ~(X335 @ id @ X343 @ id))) | (X334 @ (lam @ X336)))))) & (! [X344 : (subst > term > subst > $o),X345 : term] : (~ ! [X346 : (subst > term > term)] : (~ ! [X347 : term] : (~(X344 @ id @ X347 @ id) | (X344 @ id @ (X346 @ id @ X347) @ id)) | ~ ! [X348 : subst,X349 : subst,X350 : term] : (((sub @ (X346 @ X348 @ X350) @ X349)) = ((X346 @ (comp @ X348 @ X349) @ (sub @ X350 @ X349)))) | (X344 @ id @ (lam @ (X346 @ sh @ one)) @ id)) | (X344 @ id @ (lam @ (sub @ X345 @ (push @ one @ sh))) @ id) | ~ ! [X351 : term] : (~(X344 @ id @ X351 @ id) | (X344 @ id @ (sub @ X345 @ (push @ X351 @ id)) @ id))) <=> hoasinduction_lem3aa) & hoaslamnotap_lthm & pushprop_lem1v2_gthm & pushprop_lem3v2_lthm & (hoasinduction_lthm <=> (! [X352 : (term > $o),X353 : term] : (~ ! [X354 : term] : (~ ! [X355 : term] : (~(X352 @ X355) | (X352 @ (sub @ X354 @ (push @ X355 @ id)))) | (X352 @ (lam @ X354))) | (X352 @ X353) | ~ ! [X356 : term,X357 : term] : (~(X352 @ X357) | (X352 @ (ap @ X356 @ X357)) | ~(X352 @ X356)) | ~ ! [X358 : term] : ((X352 @ X358) | ~(var @ X358))) => (! [X359 : term,X360 : (subst > term > subst > $o),X361 : term] : (~ ! [X362 : subst,X363 : term,X364 : subst,X365 : subst] : ((X360 @ X362 @ X363 @ (comp @ X364 @ X365)) | ~(X360 @ (comp @ X362 @ X364) @ (sub @ X363 @ X364) @ X365)) | ~(X360 @ id @ X361 @ id) | ~ ! [X366 : term,X367 : subst,X368 : subst,X369 : subst] : ((X360 @ (comp @ X367 @ X368) @ (sub @ X366 @ X368) @ X369) | ~(X360 @ X367 @ X366 @ (comp @ X368 @ X369))) | ~(X360 @ id @ X359 @ id) | (X360 @ id @ (ap @ X359 @ X361) @ id) | ~ ! [X370 : term,X371 : term] : ((X360 @ id @ (ap @ (sub @ X370 @ id) @ X371) @ id) | ~(X360 @ id @ X370 @ id) | ~(X360 @ id @ X371 @ id))) => (! [X372 : term,X373 : (subst > term > subst > $o)] : (~ ! [X374 : term] : (~(X373 @ id @ X374 @ id) | (X373 @ id @ (sub @ X372 @ (push @ X374 @ id)) @ id)) | ~ ! [X375 : subst,X376 : subst,X377 : subst,X378 : term] : (~(X373 @ X375 @ X378 @ (comp @ X377 @ X376)) | (X373 @ (comp @ X375 @ X377) @ (sub @ X378 @ X377) @ X376)) | (X373 @ id @ (lam @ X372) @ id) | ~ ! [X379 : subst,X380 : subst,X381 : subst,X382 : term] : (~(X373 @ (comp @ X380 @ X381) @ (sub @ X382 @ X381) @ X379) | (X373 @ X380 @ X382 @ (comp @ X381 @ X379))) | ~ ! [X383 : (subst > term > term)] : ((X373 @ id @ (lam @ (X383 @ sh @ one)) @ id) | ~ ! [X384 : term] : (~(X373 @ id @ X384 @ id) | (X373 @ id @ (X383 @ id @ X384) @ id)) | ~ ! [X385 : term,X386 : subst,X387 : subst] : (((X383 @ (comp @ X386 @ X387) @ (sub @ X385 @ X387))) = ((sub @ (X383 @ X386 @ X385) @ X387))))) => ! [X388 : (subst > term > subst > $o),X389 : term] : (~ ! [X390 : term,X391 : subst,X392 : subst,X393 : subst] : ((X388 @ (comp @ X391 @ X393) @ (sub @ X390 @ X393) @ X392) | ~(X388 @ X391 @ X390 @ (comp @ X393 @ X392))) | ~ ! [X394 : (subst > term > term)] : ((X388 @ id @ (lam @ (X394 @ sh @ one)) @ id) | ~ ! [X395 : term] : (~(X388 @ id @ X395 @ id) | (X388 @ id @ (X394 @ id @ X395) @ id)) | ~ ! [X396 : subst,X397 : term,X398 : subst] : (((sub @ (X394 @ X398 @ X397) @ X396)) = ((X394 @ (comp @ X398 @ X396) @ (sub @ X397 @ X396))))) | ~ ! [X399 : term,X400 : term] : (~(X388 @ id @ X400 @ id) | ~(X388 @ id @ X399 @ id) | (X388 @ id @ (ap @ (sub @ X399 @ id) @ X400) @ id)) | ~ ! [X401 : term] : (~(var @ (sub @ X401 @ id)) | (X388 @ id @ X401 @ id)) | ~ ! [X402 : subst,X403 : subst,X404 : subst,X405 : term] : ((X388 @ X403 @ X405 @ (comp @ X402 @ X404)) | ~(X388 @ (comp @ X403 @ X402) @ (sub @ X405 @ X402) @ X404)) | (X388 @ id @ X389 @ id)))))) & hoasinduction_lem3a_gthm & (((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) = ((d2term @ d_one))) & (hoasinduction_lem3 <=> ! [X406 : term,X407 : (subst > term > subst > $o)] : ((X407 @ id @ (lam @ X406) @ id) | ~ ! [X408 : (subst > term > term)] : (~ ! [X409 : term] : (~(X407 @ id @ X409 @ id) | (X407 @ id @ (X408 @ id @ X409) @ id)) | ~ ! [X410 : term,X411 : subst,X412 : subst] : (((X408 @ (comp @ X411 @ X412) @ (sub @ X410 @ X412))) = ((sub @ (X408 @ X411 @ X410) @ X412))) | (X407 @ id @ (lam @ (X408 @ sh @ one)) @ id)) | ~ ! [X413 : subst,X414 : subst,X415 : subst,X416 : term] : (~(X407 @ X414 @ X416 @ (comp @ X415 @ X413)) | (X407 @ (comp @ X414 @ X415) @ (sub @ X416 @ X415) @ X413)) | ~ ! [X417 : subst,X418 : subst,X419 : term,X420 : subst] : ((X407 @ X420 @ X419 @ (comp @ X417 @ X418)) | ~(X407 @ (comp @ X420 @ X417) @ (sub @ X419 @ X417) @ X418)) | ~ ! [X421 : term] : (~(X407 @ id @ X421 @ id) | (X407 @ id @ (sub @ X406 @ (push @ X421 @ id)) @ id)))) & substmonoid_gthm & ((! [X422 : term,X423 : subst] : (((sub @ one @ (push @ X422 @ X423))) = X422) => ! [X424 : (term > $o),X425 : (term > $o),X426 : subst,X427 : term] : (~ ! [X428 : term] : (((X425 @ X428)) = ((X424 @ (sub @ X428 @ (push @ X427 @ X426))))) | (X425 @ one) | ~(X424 @ X427))) <=> pushprop_lem1v2_lthm) & (hoasinduction_lem3_lthm <=> (! [X429 : term] : (((sub @ X429 @ id)) = X429) => (! [X430 : (subst > term > subst > $o),X431 : term] : (~ ! [X432 : (subst > term > term)] : (~ ! [X433 : term] : ((X430 @ id @ (X432 @ id @ X433) @ id) | ~(X430 @ id @ X433 @ id)) | ~ ! [X434 : subst,X435 : subst,X436 : term] : (((sub @ (X432 @ X435 @ X436) @ X434)) = ((X432 @ (comp @ X435 @ X434) @ (sub @ X436 @ X434)))) | (X430 @ id @ (lam @ (X432 @ sh @ one)) @ id)) | (X430 @ id @ (lam @ (sub @ X431 @ (push @ one @ sh))) @ id) | ~ ! [X437 : term] : (~(X430 @ id @ X437 @ id) | (X430 @ id @ (sub @ X431 @ (push @ X437 @ id)) @ id))) => ! [X438 : (subst > term > subst > $o),X439 : term] : (~ ! [X440 : subst,X441 : term,X442 : subst,X443 : subst] : ((X438 @ (comp @ X443 @ X442) @ (sub @ X441 @ X442) @ X440) | ~(X438 @ X443 @ X441 @ (comp @ X442 @ X440))) | ~ ! [X444 : term] : ((X438 @ id @ (sub @ X439 @ (push @ X444 @ id)) @ id) | ~(X438 @ id @ X444 @ id)) | (X438 @ id @ (lam @ X439) @ id) | ~ ! [X445 : (subst > term > term)] : ((X438 @ id @ (lam @ (X445 @ sh @ one)) @ id) | ~ ! [X446 : term] : ((X438 @ id @ (X445 @ id @ X446) @ id) | ~(X438 @ id @ X446 @ id)) | ~ ! [X447 : term,X448 : subst,X449 : subst] : (((X445 @ (comp @ X448 @ X449) @ (sub @ X447 @ X449))) = ((sub @ (X445 @ X448 @ X447) @ X449)))) | ~ ! [X450 : term,X451 : subst,X452 : subst,X453 : subst] : (~(X438 @ (comp @ X453 @ X452) @ (sub @ X450 @ X452) @ X451) | (X438 @ X453 @ X450 @ (comp @ X452 @ X451))))))) & pushprop_lem2v2_lthm & ((! [X454 : term] : (((sub @ X454 @ id)) = X454) => ! [X455 : term] : (((sub @ X455 @ id)) = X455)) <=> termmset_lthm) & pushprop_lem0_gthm & (((sub @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2term @ d_one))) & (hoasinduction_lthm_3 <=> (! [X456 : (subst > term > subst > $o)] : ~ ! [X457 : (term > $o)] : ~ ! [X458 : term] : (((X457 @ X458)) = ((X456 @ id @ X458 @ id))) => (! [X459 : term,X460 : (term > $o)] : (~ ! [X461 : term,X462 : term] : (~(X460 @ X462) | (X460 @ (ap @ X462 @ X461)) | ~(X460 @ X461)) | ~ ! [X463 : term] : ((X460 @ (lam @ X463)) | ~ ! [X464 : term] : ((X460 @ (sub @ X463 @ (push @ X464 @ id))) | ~(X460 @ X464))) | ~ ! [X465 : term] : ((X460 @ X465) | ~(var @ X465)) | (X460 @ X459)) => (! [X466 : term] : (((sub @ X466 @ id)) = X466) => (! [X467 : (subst > term > subst > $o),X468 : term,X469 : (term > $o)] : (~ ! [X470 : term] : (((X467 @ id @ X470 @ id)) = ((X469 @ X470))) | (X469 @ (lam @ X468)) | ~ ! [X471 : term] : ((X469 @ (sub @ X468 @ (push @ X471 @ id))) | ~(X469 @ X471)) | ~ ! [X472 : (subst > term > term)] : (~ ! [X473 : subst,X474 : subst,X475 : term] : (((X472 @ (comp @ X473 @ X474) @ (sub @ X475 @ X474))) = ((sub @ (X472 @ X473 @ X475) @ X474))) | ~ ! [X476 : term] : (~(X467 @ id @ X476 @ id) | (X467 @ id @ (X472 @ id @ X476) @ id)) | (X467 @ id @ (lam @ (X472 @ sh @ one)) @ id))) => ! [X477 : (subst > term > subst > $o),X478 : term] : (~ ! [X479 : term,X480 : term] : (~(X477 @ id @ X480 @ id) | (X477 @ id @ (ap @ (sub @ X479 @ id) @ X480) @ id) | ~(X477 @ id @ X479 @ id)) | ~ ! [X481 : term] : ((X477 @ id @ X481 @ id) | ~(var @ (sub @ X481 @ id))) | ~ ! [X482 : term,X483 : subst,X484 : subst,X485 : subst] : ((X477 @ (comp @ X484 @ X483) @ (sub @ X482 @ X483) @ X485) | ~(X477 @ X484 @ X482 @ (comp @ X483 @ X485))) | (X477 @ id @ X478 @ id) | ~ ! [X486 : (subst > term > term)] : (~ ! [X487 : term] : ((X477 @ id @ (X486 @ id @ X487) @ id) | ~(X477 @ id @ X487 @ id)) | (X477 @ id @ (lam @ (X486 @ sh @ one)) @ id) | ~ ! [X488 : subst,X489 : term,X490 : subst] : (((sub @ (X486 @ X488 @ X489) @ X490)) = ((X486 @ (comp @ X488 @ X490) @ (sub @ X489 @ X490))))) | ~ ! [X491 : subst,X492 : term,X493 : subst,X494 : subst] : ((X477 @ X494 @ X492 @ (comp @ X491 @ X493)) | ~(X477 @ (comp @ X494 @ X491) @ (sub @ X492 @ X491) @ X493)))))))) & ((! [X495 : subst,X496 : term] : (((sub @ one @ (push @ X496 @ X495))) = X496) => (! [X497 : subst,X498 : term] : (((comp @ sh @ (push @ X498 @ X497))) = X497) => ! [X499 : term,X500 : term,X501 : (term > $o),X502 : subst,X503 : (term > $o)] : (~(X503 @ X500) | (X501 @ (sub @ X500 @ (push @ X499 @ X502)))))) <=> pushprop_lem1_lthm) & (hoaslaminj_lthm <=> (! [X504 : term,X505 : subst] : (((sub @ one @ (push @ X504 @ X505))) = X504) => (! [X506 : term,X507 : subst] : (((comp @ sh @ (push @ X506 @ X507))) = X507) => (! [X508 : term,X509 : term] : ((((lam @ X509)) != ((lam @ X508))) | (X508 = X509)) => ! [X510 : (subst > term > term),X511 : subst,X512 : (subst > term > term),X513 : term] : ((((lam @ (X510 @ sh @ one))) != ((lam @ (X512 @ sh @ one)))) | (((X510 @ X511 @ X513)) = ((X512 @ X511 @ X513))) | ~ ! [X514 : subst,X515 : subst,X516 : term] : (((X512 @ (comp @ X515 @ X514) @ (sub @ X516 @ X514))) = ((sub @ (X512 @ X515 @ X516) @ X514))) | ~ ! [X517 : term,X518 : subst,X519 : subst] : (((X510 @ (comp @ X519 @ X518) @ (sub @ X517 @ X518))) = ((sub @ (X510 @ X519 @ X517) @ X518)))))))) & (! [X520 : subst,X521 : term,X522 : (subst > term > term),X523 : (subst > term > term)] : ((((lam @ (X522 @ sh @ one))) != ((lam @ (X523 @ sh @ one)))) | ~ ! [X524 : term,X525 : subst,X526 : subst] : (((sub @ (X523 @ X525 @ X524) @ X526)) = ((X523 @ (comp @ X525 @ X526) @ (sub @ X524 @ X526)))) | (((X522 @ X520 @ X521)) = ((X523 @ X520 @ X521))) | ~ ! [X527 : subst,X528 : term,X529 : subst] : (((sub @ (X522 @ X529 @ X528) @ X527)) = ((X522 @ (comp @ X529 @ X527) @ (sub @ X528 @ X527))))) <=> hoaslaminj) & (axvarid <=> ! [X530 : term] : (((sub @ X530 @ id)) = X530)) & hoasapnotvar_lthm & (! [X531 : (subst > term > subst > $o)] : ~ ! [X532 : (term > $o)] : ~ ! [X533 : term] : (((X531 @ id @ X533 @ id)) = ((X532 @ X533))) <=> hoasinduction_lem0_lthm) & axassoc & ! [X534 : d_subst] : (d_id = X534) & (hoasinduction_lem2v2 <=> ! [X535 : term,X536 : term,X537 : (subst > term > subst > $o),X538 : (term > $o)] : (~ ! [X539 : term,X540 : term] : (~(X537 @ id @ X540 @ id) | ~(X537 @ id @ X539 @ id) | (X537 @ id @ (ap @ (sub @ X540 @ id) @ X539) @ id)) | ~ ! [X541 : subst,X542 : subst,X543 : term,X544 : subst] : ((X537 @ X542 @ X543 @ (comp @ X544 @ X541)) | ~(X537 @ (comp @ X542 @ X544) @ (sub @ X543 @ X544) @ X541)) | ~ ! [X545 : subst,X546 : subst,X547 : subst,X548 : term] : (~(X537 @ X545 @ X548 @ (comp @ X546 @ X547)) | (X537 @ (comp @ X545 @ X546) @ (sub @ X548 @ X546) @ X547)) | (X538 @ (ap @ X536 @ X535)) | ~(X538 @ X536) | ~(X538 @ X535) | ~ ! [X549 : term] : (((X538 @ X549)) = ((X537 @ id @ X549 @ id))))) & axapp & ~lamnotap), inference(rectify,[],[f1])). thf(f7,plain,( ((apinj1 = $true) <=> ! [X3 : term,X1 : term,X0 : term,X2 : term] : ((((ap @ X1 @ X3)) != ((ap @ X0 @ X2))) | (X0 = X1))) & (((d2subst @ d_id)) = id) & (pushprop_lem3v2 = $true) & ((hoasinduction_lem3v2 = $true) <=> ! [X4 : term,X5 : (term > $o),X6 : (subst > term > subst > $o)] : (~ ! [X7 : subst,X8 : term,X9 : subst,X10 : subst] : ((((X6 @ X10 @ X8 @ (comp @ X7 @ X9))) = $true) | ~ ($true = ((X6 @ (comp @ X10 @ X7) @ (sub @ X8 @ X7) @ X9)))) | ~ ! [X11 : term] : (~ ($true = ((X5 @ X11))) | ($true = ((X5 @ (sub @ X4 @ (push @ X11 @ id)))))) | (((X5 @ (lam @ X4))) = $true) | ~ ! [X12 : (subst > term > term)] : (~ ! [X13 : term] : ((((X6 @ id @ (X12 @ id @ X13) @ id)) = $true) | ~ (((X6 @ id @ X13 @ id)) = $true)) | ~ ! [X16 : subst,X14 : term,X15 : subst] : (((sub @ (X12 @ X15 @ X14) @ X16)) = ((X12 @ (comp @ X15 @ X16) @ (sub @ X14 @ X16)))) | (((X6 @ id @ (lam @ (X12 @ sh @ one)) @ id)) = $true)) | ~ ! [X17 : subst,X18 : subst,X19 : subst,X20 : term] : (~ (((X6 @ X18 @ X20 @ (comp @ X19 @ X17))) = $true) | ($true = ((X6 @ (comp @ X18 @ X19) @ (sub @ X20 @ X19) @ X17)))) | ~ ! [X21 : term] : (((X6 @ id @ X21 @ id)) = ((X5 @ X21))))) & (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_lem1_lthm = $true) & (hoasinduction_gthm = $true) & (hoaslamnotvar = $true) & (! [X27 : term,X28 : (subst > term > subst > $o)] : (~ ! [X29 : term,X30 : term] : (~ ($true = ((X28 @ id @ X29 @ id))) | ~ (((X28 @ id @ X30 @ id)) = $true) | (((X28 @ id @ (ap @ (sub @ X29 @ id) @ X30) @ id)) = $true)) | ~ ! [X31 : subst,X32 : subst,X33 : term,X34 : subst] : ((((X28 @ (comp @ X32 @ X31) @ (sub @ X33 @ X31) @ X34)) = $true) | ~ (((X28 @ X32 @ X33 @ (comp @ X31 @ X34))) = $true)) | ~ ! [X35 : subst,X36 : subst,X37 : subst,X38 : term] : (~ (((X28 @ (comp @ X35 @ X36) @ (sub @ X38 @ X36) @ X37)) = $true) | ($true = ((X28 @ X35 @ X38 @ (comp @ X36 @ X37))))) | ~ ! [X39 : term] : ((((X28 @ id @ X39 @ id)) = $true) | ~ (((var @ (sub @ X39 @ id))) = $true)) | ~ ! [X40 : (subst > term > term)] : (~ ! [X41 : subst,X42 : subst,X43 : term] : (((X40 @ (comp @ X42 @ X41) @ (sub @ X43 @ X41))) = ((sub @ (X40 @ X42 @ X43) @ X41))) | ~ ! [X44 : term] : (($true = ((X28 @ id @ (X40 @ id @ X44) @ id))) | ~ (((X28 @ id @ X44 @ id)) = $true)) | ($true = ((X28 @ id @ (lam @ (X40 @ sh @ one)) @ id)))) | (((X28 @ id @ X27 @ id)) = $true)) <=> (hoasinduction = $true)) & (induction2_gthm = $true) & ! [X45 : d_subst,X46 : d_subst] : ((((d2subst @ X45)) = ((d2subst @ X46))) => (X45 = X46)) & ~ (ulamvar1 = $true) & (! [X47 : term,X48 : subst,X49 : (term > $o)] : ~ ! [X50 : (term > $o)] : ~ ! [X51 : term] : (((X50 @ X51)) = ((X49 @ (sub @ X51 @ (push @ X47 @ X48))))) <=> (pushprop_lem0_lthm = $true)) & (axvarshift = $true) & ((axidl = $true) <=> ! [X52 : subst] : (((comp @ id @ X52)) = X52)) & (hoaslamnotvar_gthm = $true) & (hoasapinj2_gthm = $true) & (hoaslamnotap_gthm = $true) & (induction2lem_gthm = $true) & ((hoasinduction_lem3a = $true) <=> ! [X53 : (subst > term > subst > $o),X54 : term] : ((((X53 @ id @ (lam @ X54) @ id)) = $true) | ~ ! [X55 : term] : (~ ($true = ((X53 @ id @ X55 @ id))) | ($true = ((X53 @ id @ (sub @ X54 @ (push @ X55 @ id)) @ id)))) | ~ ! [X56 : (subst > term > term)] : (~ ! [X57 : subst,X58 : term,X59 : subst] : (((sub @ (X56 @ X59 @ X58) @ X57)) = ((X56 @ (comp @ X59 @ X57) @ (sub @ X58 @ X57)))) | (((X53 @ id @ (lam @ (X56 @ sh @ one)) @ id)) = $true) | ~ ! [X60 : term] : (~ (((X53 @ id @ X60 @ id)) = $true) | (((X53 @ id @ (X56 @ id @ X60) @ id)) = $true))))) & (! [X61 : term,X62 : subst,X63 : (term > $o)] : (~ ! [X64 : term,X65 : term] : (~ ($true = ((X63 @ X64))) | ($true = ((X63 @ (ap @ X65 @ X64)))) | ~ (((X63 @ X65)) = $true)) | ~ ! [X66 : term] : (~ (((var @ X66)) = $true) | (((X63 @ (sub @ X66 @ X62))) = $true)) | ~ ! [X67 : term] : ((((X63 @ (lam @ X67))) = $true) | ~ ! [X68 : term] : ((((X63 @ (sub @ X67 @ (push @ X68 @ id)))) = $true) | ~ ($true = ((X63 @ X68))))) | (((X63 @ (sub @ X61 @ X62))) = $true)) <=> (induction2lem = $true)) & (hoasinduction_lem3b_gthm = $true) & (axmap = $true) & (pushprop_lem2v2 = $true) & (axabs = $true) & (hoaslamnotvar_lthm = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (termmset_gthm = $true) & (pushprop_lem2v2_gthm = $true) & ((induction = $true) <=> ! [X69 : (term > $o),X70 : term] : ((((X69 @ X70)) = $true) | ~ ! [X71 : term,X72 : term] : (~ (((X69 @ X72)) = $true) | ~ (((X69 @ X71)) = $true) | ($true = ((X69 @ (ap @ X72 @ X71))))) | ~ ! [X73 : term] : (~ ($true = ((X69 @ X73))) | (((X69 @ (lam @ X73))) = $true)) | ~ ! [X74 : term] : ((((X69 @ X74)) = $true) | ~ ($true = ((var @ X74)))))) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X75 : (subst > term > subst > $o),X76 : term] : (~ ! [X77 : term] : (($true = ((X75 @ id @ (sub @ X76 @ (push @ X77 @ id)) @ id))) | ~ (((X75 @ id @ X77 @ id)) = $true)) | ~ ! [X78 : (subst > term > term)] : (~ ! [X79 : term] : (($true = ((X75 @ id @ (X78 @ id @ X79) @ id))) | ~ ($true = ((X75 @ id @ X79 @ id)))) | (((X75 @ id @ (lam @ (X78 @ sh @ one)) @ id)) = $true) | ~ ! [X80 : subst,X81 : term,X82 : subst] : (((X78 @ (comp @ X80 @ X82) @ (sub @ X81 @ X82))) = ((sub @ (X78 @ X80 @ X81) @ X82)))) | (((X75 @ id @ (lam @ (sub @ X76 @ (push @ one @ sh))) @ id)) = $true))) & (hoasinduction_lem1v2 = $true) & (ulamvarind = $true) & ((hoasinduction_lem3v2_f = $true) <=> ! [X83 : term] : ~ ! [X84 : (subst > term > term)] : ~ ! [X85 : term,X86 : subst] : (((sub @ X83 @ (push @ X85 @ X86))) = ((X84 @ X86 @ X85)))) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X87 : term] : ~ ! [X88 : (subst > term > term)] : (((sub @ X87 @ (push @ one @ sh))) != ((X88 @ sh @ one)))) & ((induction2_lthm = $true) <=> (! [X89 : term] : (((sub @ X89 @ id)) = X89) => (! [X90 : (term > $o),X91 : subst,X92 : term] : (~ ! [X93 : term] : ((((X90 @ (lam @ X93))) = $true) | ~ ! [X94 : term] : (~ (((X90 @ X94)) = $true) | (((X90 @ (sub @ X93 @ (push @ X94 @ id)))) = $true))) | ~ ! [X95 : term,X96 : term] : ((((X90 @ (ap @ X96 @ X95))) = $true) | ~ ($true = ((X90 @ X95))) | ~ ($true = ((X90 @ X96)))) | (((X90 @ (sub @ X92 @ X91))) = $true) | ~ ! [X97 : term] : (~ (((var @ X97)) = $true) | ($true = ((X90 @ (sub @ X97 @ X91)))))) => ! [X98 : (term > $o),X99 : term] : (~ ! [X100 : term,X101 : term] : (~ ($true = ((X98 @ X100))) | (((X98 @ (ap @ X101 @ X100))) = $true) | ~ (((X98 @ X101)) = $true)) | ($true = ((X98 @ X99))) | ~ ! [X102 : term] : ((((X98 @ X102)) = $true) | ~ ($true = ((var @ X102)))) | ~ ! [X103 : term] : (~ ! [X104 : term] : (~ (((X98 @ X104)) = $true) | ($true = ((X98 @ (sub @ X103 @ (push @ X104 @ id)))))) | ($true = ((X98 @ (lam @ X103))))))))) & (hoasapnotvar_gthm = $true) & ! [X105 : d_term,X106 : d_term] : ((((d2term @ X105)) = ((d2term @ X106))) => (X105 = X106)) & ((substmonoid_lthm = $true) <=> (! [X107 : subst] : (((comp @ id @ X107)) = X107) => (! [X108 : subst] : (((comp @ X108 @ id)) = X108) => (! [X109 : subst] : (((comp @ id @ X109)) = X109) & ! [X110 : subst] : (((comp @ X110 @ id)) = X110))))) & ! [X111 : term] : ? [X112 : d_term] : (((d2term @ X112)) = X111) & (hoasinduction_lem2_lthm = $true) & ((hoasinduction_lem3v2_lthm = $true) <=> (! [X113 : term] : (((sub @ X113 @ id)) = X113) => ! [X114 : (subst > term > subst > $o),X115 : (term > $o),X116 : term] : (($true = ((X115 @ (lam @ X116)))) | ~ ! [X117 : (subst > term > term)] : (~ ! [X118 : term] : ((((X114 @ id @ (X117 @ id @ X118) @ id)) = $true) | ~ ($true = ((X114 @ id @ X118 @ id)))) | ($true = ((X114 @ id @ (lam @ (X117 @ sh @ one)) @ id))) | ~ ! [X119 : subst,X120 : term,X121 : subst] : (((sub @ (X117 @ X121 @ X120) @ X119)) = ((X117 @ (comp @ X121 @ X119) @ (sub @ X120 @ X119))))) | ~ ! [X122 : subst,X123 : term,X124 : subst,X125 : subst] : ((((X114 @ (comp @ X125 @ X124) @ (sub @ X123 @ X124) @ X122)) = $true) | ~ (((X114 @ X125 @ X123 @ (comp @ X124 @ X122))) = $true)) | ~ ! [X126 : term] : (((X115 @ X126)) = ((X114 @ id @ X126 @ id))) | ~ ! [X127 : term] : ((((X115 @ (sub @ X116 @ (push @ X127 @ id)))) = $true) | ~ (((X115 @ X127)) = $true)) | ~ ! [X128 : subst,X129 : subst,X130 : term,X131 : subst] : (($true = ((X114 @ X129 @ X130 @ (comp @ X131 @ X128)))) | ~ (((X114 @ (comp @ X129 @ X131) @ (sub @ X130 @ X131) @ X128)) = $true))))) & ((hoasinduction_lem3aaa = $true) <=> ! [X132 : term,X133 : (subst > term > subst > $o)] : (~ ! [X134 : term] : (($true = ((X133 @ id @ (sub @ X132 @ (push @ X134 @ id)) @ id))) | ~ (((X133 @ id @ X134 @ id)) = $true)) | ~ ! [X135 : term,X136 : (subst > term > term)] : (~ ! [X137 : term,X138 : subst,X139 : subst] : (((sub @ (X136 @ X138 @ X137) @ X139)) = ((sub @ (sub @ X135 @ (push @ X137 @ X138)) @ X139))) | ~ ! [X140 : subst,X141 : subst,X142 : term] : (((sub @ X135 @ (push @ (sub @ X142 @ X140) @ (comp @ X141 @ X140)))) = ((X136 @ (comp @ X141 @ X140) @ (sub @ X142 @ X140)))) | (((X133 @ id @ (lam @ (X136 @ sh @ one)) @ id)) = $true) | ~ ! [X143 : term] : ((((X133 @ id @ (X136 @ id @ X143) @ id)) = $true) | ~ (((X133 @ id @ X143 @ id)) = $true))) | ($true = ((X133 @ id @ (lam @ (sub @ X132 @ (push @ one @ sh))) @ id))))) & ((hoaslamnotap = $true) <=> ! [X144 : term,X145 : term,X146 : (subst > term > term)] : (~ ! [X147 : term,X148 : subst,X149 : subst] : (((X146 @ (comp @ X148 @ X149) @ (sub @ X147 @ X149))) = ((sub @ (X146 @ X148 @ X147) @ X149))) | (((lam @ (X146 @ sh @ one))) != ((ap @ (sub @ X145 @ id) @ X144))))) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (apnotvar = $true) & (hoaslaminj_gthm = $true) & ((hoasapinj2_lthm = $true) <=> (! [X152 : term,X153 : term,X154 : term,X155 : term] : ((X154 = X155) | (((ap @ X152 @ X155)) != ((ap @ X153 @ X154)))) => ! [X156 : term,X157 : term,X158 : term,X159 : term] : ((((ap @ (sub @ X156 @ id) @ X157)) != ((ap @ (sub @ X159 @ id) @ X158))) | (X157 = X158)))) & ((! [X160 : (subst > term > subst > $o)] : ~ ! [X161 : (term > $o)] : ~ ! [X162 : term] : (((X160 @ id @ X162 @ id)) = ((X161 @ X162))) => (! [X163 : term,X164 : (term > $o)] : (~ ! [X165 : term] : (~ ! [X166 : term] : (~ ($true = ((X164 @ X166))) | (((X164 @ (sub @ X165 @ (push @ X166 @ id)))) = $true)) | ($true = ((X164 @ (lam @ X165))))) | ~ ! [X167 : term,X168 : term] : (~ ($true = ((X164 @ X167))) | ~ ($true = ((X164 @ X168))) | ($true = ((X164 @ (ap @ X168 @ X167))))) | ~ ! [X169 : term] : ((((X164 @ X169)) = $true) | ~ (((var @ X169)) = $true)) | (((X164 @ X163)) = $true)) => (! [X170 : term] : (((sub @ X170 @ id)) = X170) => (! [X171 : (subst > term > subst > $o),X172 : (term > $o),X173 : term] : (~ ! [X174 : term] : (((X172 @ X174)) = ((X171 @ id @ X174 @ id))) | ~ ! [X175 : (subst > term > term)] : (($true = ((X171 @ id @ (lam @ (X175 @ sh @ one)) @ id))) | ~ ! [X176 : term] : (($true = ((X171 @ id @ (X175 @ id @ X176) @ id))) | ~ (((X171 @ id @ X176 @ id)) = $true)) | ~ ! [X177 : subst,X178 : term,X179 : subst] : (((X175 @ (comp @ X179 @ X177) @ (sub @ X178 @ X177))) = ((sub @ (X175 @ X179 @ X178) @ X177)))) | (((X172 @ (lam @ X173))) = $true) | ~ ! [X180 : term] : (~ (((X172 @ X180)) = $true) | (((X172 @ (sub @ X173 @ (push @ X180 @ id)))) = $true))) => ! [X181 : (subst > term > subst > $o),X182 : term] : (~ ! [X183 : (subst > term > term)] : (~ ! [X184 : subst,X185 : subst,X186 : term] : (((X183 @ (comp @ X184 @ X185) @ (sub @ X186 @ X185))) = ((sub @ (X183 @ X184 @ X186) @ X185))) | ~ ! [X187 : term] : (~ ($true = ((X181 @ id @ X187 @ id))) | (((X181 @ id @ (X183 @ id @ X187) @ id)) = $true)) | (((X181 @ id @ (lam @ (X183 @ sh @ one)) @ id)) = $true)) | ~ ! [X188 : term,X189 : term] : ((((X181 @ id @ (ap @ (sub @ X189 @ id) @ X188) @ id)) = $true) | ~ ($true = ((X181 @ id @ X189 @ id))) | ~ ($true = ((X181 @ id @ X188 @ id)))) | (((X181 @ id @ X182 @ id)) = $true)))))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (pushprop_lthm = $true) & (! [X190 : subst] : (((comp @ X190 @ id)) = X190) <=> (axidr = $true)) & (! [X191 : term,X192 : term,X193 : term,X194 : term] : ((((ap @ (sub @ X191 @ id) @ X194)) != ((ap @ (sub @ X192 @ id) @ X193))) | (X193 = X194)) <=> (hoasapinj2 = $true)) & (pushprop_lthm_orig = $true) & (hoasinduction_lem2_gthm = $true) & (pushprop_lem1_gthm = $true) & (! [X195 : term] : ~ ! [X196 : (subst > term > term)] : ~ ! [X197 : subst,X198 : term] : (((X196 @ X197 @ X198)) = ((sub @ X195 @ (push @ X198 @ X197)))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & (! [X199 : (subst > term > subst > $o)] : ~ ! [X200 : (term > $o)] : ~ ! [X201 : term] : (((X199 @ id @ X201 @ id)) = ((X200 @ X201))) <=> (hoasinduction_lem0 = $true)) & (hoasinduction_lem1v2_gthm = $true) & (! [X202 : term,X203 : term] : ((X202 = X203) | (((sub @ X202 @ sh)) != ((sub @ X203 @ sh)))) <=> (shinj = $true)) & (hoasinduction_lem3v2_gthm = $true) & (hoasapinj1_gthm = $true) & ((induction2 = $true) <=> ! [X204 : (term > $o),X205 : term] : (~ ! [X206 : term,X207 : term] : (~ ($true = ((X204 @ X207))) | ($true = ((X204 @ (ap @ X207 @ X206)))) | ~ (((X204 @ X206)) = $true)) | (((X204 @ X205)) = $true) | ~ ! [X208 : term] : (~ ! [X209 : term] : (($true = ((X204 @ (sub @ X208 @ (push @ X209 @ id))))) | ~ ($true = ((X204 @ X209)))) | (((X204 @ (lam @ X208))) = $true)) | ~ ! [X210 : term] : ((((X204 @ X210)) = $true) | ~ (((var @ X210)) = $true)))) & ((axshiftcons = $true) <=> ! [X211 : term,X212 : subst] : (((comp @ sh @ (push @ X211 @ X212))) = X212)) & ~ (((var @ (d2term @ d_one))) = $true) & (ulamvarsh = $true) & ~ (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) = $true) & (! [X213 : (subst > term > subst > $o),X214 : term] : (~ ! [X215 : term,X216 : term] : ((((X213 @ id @ (ap @ (sub @ X215 @ id) @ X216) @ id)) = $true) | ~ (((X213 @ id @ X215 @ id)) = $true) | ~ (((X213 @ id @ X216 @ id)) = $true)) | ($true = ((X213 @ id @ X214 @ id))) | ~ ! [X217 : (subst > term > term)] : (~ ! [X218 : term] : (($true = ((X213 @ id @ (X217 @ id @ X218) @ id))) | ~ (((X213 @ id @ X218 @ id)) = $true)) | ~ ! [X219 : subst,X220 : term,X221 : subst] : (((sub @ (X217 @ X221 @ X220) @ X219)) = ((X217 @ (comp @ X221 @ X219) @ (sub @ X220 @ X219)))) | ($true = ((X213 @ id @ (lam @ (X217 @ sh @ one)) @ id))))) <=> (hoasinduction_no_psi_cond = $true)) & (hoasinduction_lem2v2_gthm = $true) & (hoasinduction_lem3_gthm = $true) & ((hoasinduction_lem2 = $true) <=> ! [X222 : (subst > term > subst > $o),X223 : term,X224 : term] : (~ ($true = ((X222 @ id @ X223 @ id))) | ~ ! [X225 : subst,X226 : subst,X227 : subst,X228 : term] : ((((X222 @ (comp @ X227 @ X225) @ (sub @ X228 @ X225) @ X226)) = $true) | ~ (((X222 @ X227 @ X228 @ (comp @ X225 @ X226))) = $true)) | (((X222 @ id @ (ap @ X223 @ X224) @ id)) = $true) | ~ ! [X229 : term,X230 : term] : (~ ($true = ((X222 @ id @ X229 @ id))) | ~ (((X222 @ id @ X230 @ id)) = $true) | ($true = ((X222 @ id @ (ap @ (sub @ X230 @ id) @ X229) @ id)))) | ~ ! [X231 : term,X232 : subst,X233 : subst,X234 : subst] : (~ (((X222 @ (comp @ X234 @ X232) @ (sub @ X231 @ X232) @ X233)) = $true) | (((X222 @ X234 @ X231 @ (comp @ X232 @ X233))) = $true)) | ~ (((X222 @ id @ X224 @ id)) = $true))) & ! [X235 : subst] : ? [X236 : d_subst] : (((d2subst @ X236)) = X235) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((substmonoid = $true) <=> (! [X237 : subst] : (((comp @ id @ X237)) = X237) & ! [X238 : subst] : (((comp @ X238 @ id)) = X238))) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (! [X239 : term,X240 : term] : ((X239 = X240) | (((lam @ X239)) != ((lam @ X240)))) <=> (laminj = $true)) & (! [X241 : subst,X242 : term] : (((sub @ one @ (push @ X242 @ X241))) = X242) <=> (axvarcons = $true)) & (pushprop = $true) & (hoasapnotvar = $true) & (((d2term @ d_one)) = one) & (! [X243 : term,X244 : term,X245 : term,X246 : term] : ((((ap @ X244 @ X243)) != ((ap @ X245 @ X246))) | (X243 = X246)) <=> (apinj2 = $true)) & ((hoasinduction_lem3b = $true) <=> ! [X247 : term] : ~ ! [X248 : (subst > term > term)] : (((sub @ X247 @ (push @ one @ sh))) != ((X248 @ sh @ one)))) & (axclos = $true) & ((pushprop_lem0 = $true) <=> ! [X249 : term,X250 : (term > $o),X251 : subst] : ~ ! [X252 : (term > $o)] : ~ ! [X253 : term] : (((X250 @ (sub @ X253 @ (push @ X249 @ X251)))) = ((X252 @ X253)))) & (! [X254 : subst] : (((push @ (sub @ one @ X254) @ (comp @ sh @ X254))) = X254) <=> (axscons = $true)) & (! [X255 : term] : (((sub @ X255 @ id)) = X255) <=> (termmset = $true)) & (! [X256 : term,X257 : (term > $o),X258 : (subst > term > subst > $o)] : (($true = ((X257 @ (lam @ X256)))) | ~ ! [X259 : term] : (((X258 @ id @ X259 @ id)) = ((X257 @ X259))) | ~ ! [X260 : (subst > term > term)] : (($true = ((X258 @ id @ (lam @ (X260 @ sh @ one)) @ id))) | ~ ! [X261 : subst,X262 : term,X263 : subst] : (((X260 @ (comp @ X261 @ X263) @ (sub @ X262 @ X263))) = ((sub @ (X260 @ X261 @ X262) @ X263))) | ~ ! [X264 : term] : (~ (((X258 @ id @ X264 @ id)) = $true) | (((X258 @ id @ (X260 @ id @ X264) @ id)) = $true))) | ~ ! [X265 : term] : ((((X257 @ (sub @ X256 @ (push @ X265 @ id)))) = $true) | ~ (((X257 @ X265)) = $true))) <=> (hoasinduction_lem3v2a = $true)) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((! [X266 : term] : (((sub @ X266 @ id)) = X266) => (! [X267 : term,X268 : term,X269 : term,X270 : term] : ((((ap @ X269 @ X267)) != ((ap @ X268 @ X270))) | (X268 = X269)) => ! [X271 : term,X272 : term,X273 : term,X274 : term] : ((((ap @ (sub @ X273 @ id) @ X271)) != ((ap @ (sub @ X274 @ id) @ X272))) | (X273 = X274)))) <=> (hoasapinj1_lthm = $true)) & ((pushprop_lem1 = $true) <=> ! [X275 : term,X276 : (term > $o),X277 : term,X278 : (term > $o),X279 : subst] : (($true = ((X278 @ (sub @ X277 @ (push @ X275 @ X279))))) | ~ (((X276 @ X277)) = $true))) & ((! [X280 : term,X281 : subst] : (((sub @ one @ (push @ X280 @ X281))) = X280) => (! [X282 : term,X283 : subst] : (((comp @ sh @ (push @ X282 @ X283))) = X283) => (! [X284 : subst] : (((comp @ X284 @ id)) = X284) => (! [X285 : (term > $o),X286 : term] : (($true = ((X285 @ X286))) | ~ ! [X287 : term] : (~ ($true = ((X285 @ X287))) | (((X285 @ (lam @ X287))) = $true)) | ~ ! [X288 : term,X289 : term] : (~ (((X285 @ X289)) = $true) | (((X285 @ (ap @ X288 @ X289))) = $true) | ~ ($true = ((X285 @ X288)))) | ~ ! [X290 : term] : ((((X285 @ X290)) = $true) | ~ ($true = ((var @ X290))))) => ! [X291 : (term > $o),X292 : subst,X293 : term] : (~ ! [X294 : term,X295 : term] : (~ (((X291 @ X295)) = $true) | (((X291 @ (ap @ X295 @ X294))) = $true) | ~ (((X291 @ X294)) = $true)) | ($true = ((X291 @ (sub @ X293 @ X292)))) | ~ ! [X296 : term] : (($true = ((X291 @ (sub @ X296 @ X292)))) | ~ (((var @ X296)) = $true)) | ~ ! [X297 : term] : ((((X291 @ (lam @ X297))) = $true) | ~ ! [X298 : term] : (~ (((X291 @ X298)) = $true) | ($true = ((X291 @ (sub @ X297 @ (push @ X298 @ id)))))))))))) <=> (induction2lem_lthm = $true)) & ((! [X299 : term] : (((sub @ X299 @ id)) = X299) => (! [X300 : (subst > term > subst > $o),X301 : term] : (($true = ((X300 @ id @ (lam @ (sub @ X301 @ (push @ one @ sh))) @ id))) | ~ ! [X302 : (subst > term > term)] : (~ ! [X303 : term] : (~ (((X300 @ id @ X303 @ id)) = $true) | (((X300 @ id @ (X302 @ id @ X303) @ id)) = $true)) | ~ ! [X304 : term,X305 : subst,X306 : subst] : (((sub @ (X302 @ X306 @ X304) @ X305)) = ((X302 @ (comp @ X306 @ X305) @ (sub @ X304 @ X305)))) | ($true = ((X300 @ id @ (lam @ (X302 @ sh @ one)) @ id)))) | ~ ! [X307 : term] : ((((X300 @ id @ (sub @ X301 @ (push @ X307 @ id)) @ id)) = $true) | ~ ($true = ((X300 @ id @ X307 @ id))))) => ! [X308 : term,X309 : (subst > term > subst > $o)] : (~ ! [X310 : (subst > term > term)] : ((((X309 @ id @ (lam @ (X310 @ sh @ one)) @ id)) = $true) | ~ ! [X311 : subst,X312 : term,X313 : subst] : (((X310 @ (comp @ X311 @ X313) @ (sub @ X312 @ X313))) = ((sub @ (X310 @ X311 @ X312) @ X313))) | ~ ! [X314 : term] : (~ (((X309 @ id @ X314 @ id)) = $true) | (((X309 @ id @ (X310 @ id @ X314) @ id)) = $true))) | (((X309 @ id @ (lam @ X308) @ id)) = $true) | ~ ! [X315 : term] : (($true = ((X309 @ id @ (sub @ X308 @ (push @ X315 @ id)) @ id))) | ~ (((X309 @ id @ X315 @ id)) = $true))))) <=> (hoasinduction_lem3a_lthm = $true)) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (((d2subst @ d_id)) = sh) & (pushprop_gthm = $true) & (! [X319 : (term > $o),X320 : (term > $o),X321 : term,X322 : subst] : (($true = ((X320 @ one))) | ~ ($true = ((X319 @ X321))) | ~ ! [X323 : term] : (((X320 @ X323)) = ((X319 @ (sub @ X323 @ (push @ X321 @ X322)))))) <=> (pushprop_lem1v2 = $true)) & (hoasinduction_lem1_gthm = $true) & (lamnotvar = $true) & (! [X324 : term,X325 : term,X326 : term,X327 : term] : ((((ap @ (sub @ X324 @ id) @ X325)) != ((ap @ (sub @ X326 @ id) @ X327))) | (X324 = X326)) <=> (hoasapinj1 = $true)) & (hoasinduction_lem1 = $true) & ! [X328 : d_term] : (d_one = X328) & ((hoasinduction_lem3v2a_lthm = $true) <=> (! [X329 : term] : ~ ! [X330 : (subst > term > term)] : ~ ! [X331 : subst,X332 : term] : (((sub @ X329 @ (push @ X332 @ X331))) = ((X330 @ X331 @ X332))) => (! [X333 : term] : (((sub @ X333 @ id)) = X333) => ! [X334 : (term > $o),X335 : (subst > term > subst > $o),X336 : term] : (~ ! [X337 : term] : (($true = ((X334 @ (sub @ X336 @ (push @ X337 @ id))))) | ~ (((X334 @ X337)) = $true)) | ~ ! [X338 : term] : (((X335 @ id @ X338 @ id)) = ((X334 @ X338))) | ~ ! [X339 : (subst > term > term)] : (~ ! [X340 : subst,X341 : term,X342 : subst] : (((sub @ (X339 @ X340 @ X341) @ X342)) = ((X339 @ (comp @ X340 @ X342) @ (sub @ X341 @ X342)))) | ($true = ((X335 @ id @ (lam @ (X339 @ sh @ one)) @ id))) | ~ ! [X343 : term] : (($true = ((X335 @ id @ (X339 @ id @ X343) @ id))) | ~ (((X335 @ id @ X343 @ id)) = $true))) | (((X334 @ (lam @ X336))) = $true))))) & (! [X344 : (subst > term > subst > $o),X345 : term] : (~ ! [X346 : (subst > term > term)] : (~ ! [X347 : term] : (~ ($true = ((X344 @ id @ X347 @ id))) | ($true = ((X344 @ id @ (X346 @ id @ X347) @ id)))) | ~ ! [X348 : subst,X349 : subst,X350 : term] : (((sub @ (X346 @ X348 @ X350) @ X349)) = ((X346 @ (comp @ X348 @ X349) @ (sub @ X350 @ X349)))) | (((X344 @ id @ (lam @ (X346 @ sh @ one)) @ id)) = $true)) | (((X344 @ id @ (lam @ (sub @ X345 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X351 : term] : (~ (((X344 @ id @ X351 @ id)) = $true) | (((X344 @ id @ (sub @ X345 @ (push @ X351 @ id)) @ id)) = $true))) <=> (hoasinduction_lem3aa = $true)) & (hoaslamnotap_lthm = $true) & (pushprop_lem1v2_gthm = $true) & (pushprop_lem3v2_lthm = $true) & ((hoasinduction_lthm = $true) <=> (! [X352 : (term > $o),X353 : term] : (~ ! [X354 : term] : (~ ! [X355 : term] : (~ (((X352 @ X355)) = $true) | (((X352 @ (sub @ X354 @ (push @ X355 @ id)))) = $true)) | (((X352 @ (lam @ X354))) = $true)) | (((X352 @ X353)) = $true) | ~ ! [X356 : term,X357 : term] : (~ (((X352 @ X357)) = $true) | (((X352 @ (ap @ X356 @ X357))) = $true) | ~ (((X352 @ X356)) = $true)) | ~ ! [X358 : term] : (($true = ((X352 @ X358))) | ~ (((var @ X358)) = $true))) => (! [X359 : term,X360 : (subst > term > subst > $o),X361 : term] : (~ ! [X362 : subst,X363 : term,X364 : subst,X365 : subst] : ((((X360 @ X362 @ X363 @ (comp @ X364 @ X365))) = $true) | ~ ($true = ((X360 @ (comp @ X362 @ X364) @ (sub @ X363 @ X364) @ X365)))) | ~ ($true = ((X360 @ id @ X361 @ id))) | ~ ! [X366 : term,X367 : subst,X368 : subst,X369 : subst] : ((((X360 @ (comp @ X367 @ X368) @ (sub @ X366 @ X368) @ X369)) = $true) | ~ (((X360 @ X367 @ X366 @ (comp @ X368 @ X369))) = $true)) | ~ (((X360 @ id @ X359 @ id)) = $true) | ($true = ((X360 @ id @ (ap @ X359 @ X361) @ id))) | ~ ! [X370 : term,X371 : term] : ((((X360 @ id @ (ap @ (sub @ X370 @ id) @ X371) @ id)) = $true) | ~ (((X360 @ id @ X370 @ id)) = $true) | ~ (((X360 @ id @ X371 @ id)) = $true))) => (! [X372 : term,X373 : (subst > term > subst > $o)] : (~ ! [X374 : term] : (~ ($true = ((X373 @ id @ X374 @ id))) | ($true = ((X373 @ id @ (sub @ X372 @ (push @ X374 @ id)) @ id)))) | ~ ! [X375 : subst,X376 : subst,X377 : subst,X378 : term] : (~ ($true = ((X373 @ X375 @ X378 @ (comp @ X377 @ X376)))) | ($true = ((X373 @ (comp @ X375 @ X377) @ (sub @ X378 @ X377) @ X376)))) | (((X373 @ id @ (lam @ X372) @ id)) = $true) | ~ ! [X379 : subst,X380 : subst,X381 : subst,X382 : term] : (~ (((X373 @ (comp @ X380 @ X381) @ (sub @ X382 @ X381) @ X379)) = $true) | ($true = ((X373 @ X380 @ X382 @ (comp @ X381 @ X379))))) | ~ ! [X383 : (subst > term > term)] : (($true = ((X373 @ id @ (lam @ (X383 @ sh @ one)) @ id))) | ~ ! [X384 : term] : (~ ($true = ((X373 @ id @ X384 @ id))) | ($true = ((X373 @ id @ (X383 @ id @ X384) @ id)))) | ~ ! [X385 : term,X386 : subst,X387 : subst] : (((X383 @ (comp @ X386 @ X387) @ (sub @ X385 @ X387))) = ((sub @ (X383 @ X386 @ X385) @ X387))))) => ! [X388 : (subst > term > subst > $o),X389 : term] : (~ ! [X390 : term,X391 : subst,X392 : subst,X393 : subst] : (($true = ((X388 @ (comp @ X391 @ X393) @ (sub @ X390 @ X393) @ X392))) | ~ ($true = ((X388 @ X391 @ X390 @ (comp @ X393 @ X392))))) | ~ ! [X394 : (subst > term > term)] : (($true = ((X388 @ id @ (lam @ (X394 @ sh @ one)) @ id))) | ~ ! [X395 : term] : (~ ($true = ((X388 @ id @ X395 @ id))) | ($true = ((X388 @ id @ (X394 @ id @ X395) @ id)))) | ~ ! [X396 : subst,X397 : term,X398 : subst] : (((sub @ (X394 @ X398 @ X397) @ X396)) = ((X394 @ (comp @ X398 @ X396) @ (sub @ X397 @ X396))))) | ~ ! [X399 : term,X400 : term] : (~ (((X388 @ id @ X400 @ id)) = $true) | ~ ($true = ((X388 @ id @ X399 @ id))) | ($true = ((X388 @ id @ (ap @ (sub @ X399 @ id) @ X400) @ id)))) | ~ ! [X401 : term] : (~ ($true = ((var @ (sub @ X401 @ id)))) | ($true = ((X388 @ id @ X401 @ id)))) | ~ ! [X402 : subst,X403 : subst,X404 : subst,X405 : term] : ((((X388 @ X403 @ X405 @ (comp @ X402 @ X404))) = $true) | ~ (((X388 @ (comp @ X403 @ X402) @ (sub @ X405 @ X402) @ X404)) = $true)) | ($true = ((X388 @ id @ X389 @ id)))))))) & (hoasinduction_lem3a_gthm = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ((hoasinduction_lem3 = $true) <=> ! [X406 : term,X407 : (subst > term > subst > $o)] : ((((X407 @ id @ (lam @ X406) @ id)) = $true) | ~ ! [X408 : (subst > term > term)] : (~ ! [X409 : term] : (~ (((X407 @ id @ X409 @ id)) = $true) | (((X407 @ id @ (X408 @ id @ X409) @ id)) = $true)) | ~ ! [X410 : term,X411 : subst,X412 : subst] : (((X408 @ (comp @ X411 @ X412) @ (sub @ X410 @ X412))) = ((sub @ (X408 @ X411 @ X410) @ X412))) | ($true = ((X407 @ id @ (lam @ (X408 @ sh @ one)) @ id)))) | ~ ! [X413 : subst,X414 : subst,X415 : subst,X416 : term] : (~ ($true = ((X407 @ X414 @ X416 @ (comp @ X415 @ X413)))) | (((X407 @ (comp @ X414 @ X415) @ (sub @ X416 @ X415) @ X413)) = $true)) | ~ ! [X417 : subst,X418 : subst,X419 : term,X420 : subst] : ((((X407 @ X420 @ X419 @ (comp @ X417 @ X418))) = $true) | ~ (((X407 @ (comp @ X420 @ X417) @ (sub @ X419 @ X417) @ X418)) = $true)) | ~ ! [X421 : term] : (~ (((X407 @ id @ X421 @ id)) = $true) | (((X407 @ id @ (sub @ X406 @ (push @ X421 @ id)) @ id)) = $true)))) & (substmonoid_gthm = $true) & ((! [X422 : term,X423 : subst] : (((sub @ one @ (push @ X422 @ X423))) = X422) => ! [X424 : (term > $o),X425 : (term > $o),X426 : subst,X427 : term] : (~ ! [X428 : term] : (((X425 @ X428)) = ((X424 @ (sub @ X428 @ (push @ X427 @ X426))))) | ($true = ((X425 @ one))) | ~ (((X424 @ X427)) = $true))) <=> (pushprop_lem1v2_lthm = $true)) & ((hoasinduction_lem3_lthm = $true) <=> (! [X429 : term] : (((sub @ X429 @ id)) = X429) => (! [X430 : (subst > term > subst > $o),X431 : term] : (~ ! [X432 : (subst > term > term)] : (~ ! [X433 : term] : (($true = ((X430 @ id @ (X432 @ id @ X433) @ id))) | ~ (((X430 @ id @ X433 @ id)) = $true)) | ~ ! [X434 : subst,X435 : subst,X436 : term] : (((sub @ (X432 @ X435 @ X436) @ X434)) = ((X432 @ (comp @ X435 @ X434) @ (sub @ X436 @ X434)))) | (((X430 @ id @ (lam @ (X432 @ sh @ one)) @ id)) = $true)) | (((X430 @ id @ (lam @ (sub @ X431 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X437 : term] : (~ ($true = ((X430 @ id @ X437 @ id))) | ($true = ((X430 @ id @ (sub @ X431 @ (push @ X437 @ id)) @ id))))) => ! [X438 : (subst > term > subst > $o),X439 : term] : (~ ! [X440 : subst,X441 : term,X442 : subst,X443 : subst] : ((((X438 @ (comp @ X443 @ X442) @ (sub @ X441 @ X442) @ X440)) = $true) | ~ (((X438 @ X443 @ X441 @ (comp @ X442 @ X440))) = $true)) | ~ ! [X444 : term] : ((((X438 @ id @ (sub @ X439 @ (push @ X444 @ id)) @ id)) = $true) | ~ ($true = ((X438 @ id @ X444 @ id)))) | (((X438 @ id @ (lam @ X439) @ id)) = $true) | ~ ! [X445 : (subst > term > term)] : ((((X438 @ id @ (lam @ (X445 @ sh @ one)) @ id)) = $true) | ~ ! [X446 : term] : (($true = ((X438 @ id @ (X445 @ id @ X446) @ id))) | ~ (((X438 @ id @ X446 @ id)) = $true)) | ~ ! [X447 : term,X448 : subst,X449 : subst] : (((X445 @ (comp @ X448 @ X449) @ (sub @ X447 @ X449))) = ((sub @ (X445 @ X448 @ X447) @ X449)))) | ~ ! [X450 : term,X451 : subst,X452 : subst,X453 : subst] : (~ (((X438 @ (comp @ X453 @ X452) @ (sub @ X450 @ X452) @ X451)) = $true) | (((X438 @ X453 @ X450 @ (comp @ X452 @ X451))) = $true)))))) & (pushprop_lem2v2_lthm = $true) & ((! [X454 : term] : (((sub @ X454 @ id)) = X454) => ! [X455 : term] : (((sub @ X455 @ id)) = X455)) <=> (termmset_lthm = $true)) & (pushprop_lem0_gthm = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoasinduction_lthm_3 = $true) <=> (! [X456 : (subst > term > subst > $o)] : ~ ! [X457 : (term > $o)] : ~ ! [X458 : term] : (((X457 @ X458)) = ((X456 @ id @ X458 @ id))) => (! [X459 : term,X460 : (term > $o)] : (~ ! [X461 : term,X462 : term] : (~ (((X460 @ X462)) = $true) | (((X460 @ (ap @ X462 @ X461))) = $true) | ~ ($true = ((X460 @ X461)))) | ~ ! [X463 : term] : (($true = ((X460 @ (lam @ X463)))) | ~ ! [X464 : term] : ((((X460 @ (sub @ X463 @ (push @ X464 @ id)))) = $true) | ~ (((X460 @ X464)) = $true))) | ~ ! [X465 : term] : ((((X460 @ X465)) = $true) | ~ ($true = ((var @ X465)))) | ($true = ((X460 @ X459)))) => (! [X466 : term] : (((sub @ X466 @ id)) = X466) => (! [X467 : (subst > term > subst > $o),X468 : term,X469 : (term > $o)] : (~ ! [X470 : term] : (((X467 @ id @ X470 @ id)) = ((X469 @ X470))) | (((X469 @ (lam @ X468))) = $true) | ~ ! [X471 : term] : (($true = ((X469 @ (sub @ X468 @ (push @ X471 @ id))))) | ~ (((X469 @ X471)) = $true)) | ~ ! [X472 : (subst > term > term)] : (~ ! [X473 : subst,X474 : subst,X475 : term] : (((X472 @ (comp @ X473 @ X474) @ (sub @ X475 @ X474))) = ((sub @ (X472 @ X473 @ X475) @ X474))) | ~ ! [X476 : term] : (~ (((X467 @ id @ X476 @ id)) = $true) | ($true = ((X467 @ id @ (X472 @ id @ X476) @ id)))) | (((X467 @ id @ (lam @ (X472 @ sh @ one)) @ id)) = $true))) => ! [X477 : (subst > term > subst > $o),X478 : term] : (~ ! [X479 : term,X480 : term] : (~ ($true = ((X477 @ id @ X480 @ id))) | (((X477 @ id @ (ap @ (sub @ X479 @ id) @ X480) @ id)) = $true) | ~ (((X477 @ id @ X479 @ id)) = $true)) | ~ ! [X481 : term] : ((((X477 @ id @ X481 @ id)) = $true) | ~ (((var @ (sub @ X481 @ id))) = $true)) | ~ ! [X482 : term,X483 : subst,X484 : subst,X485 : subst] : ((((X477 @ (comp @ X484 @ X483) @ (sub @ X482 @ X483) @ X485)) = $true) | ~ (((X477 @ X484 @ X482 @ (comp @ X483 @ X485))) = $true)) | (((X477 @ id @ X478 @ id)) = $true) | ~ ! [X486 : (subst > term > term)] : (~ ! [X487 : term] : ((((X477 @ id @ (X486 @ id @ X487) @ id)) = $true) | ~ ($true = ((X477 @ id @ X487 @ id)))) | ($true = ((X477 @ id @ (lam @ (X486 @ sh @ one)) @ id))) | ~ ! [X488 : subst,X489 : term,X490 : subst] : (((sub @ (X486 @ X488 @ X489) @ X490)) = ((X486 @ (comp @ X488 @ X490) @ (sub @ X489 @ X490))))) | ~ ! [X491 : subst,X492 : term,X493 : subst,X494 : subst] : (($true = ((X477 @ X494 @ X492 @ (comp @ X491 @ X493)))) | ~ ($true = ((X477 @ (comp @ X494 @ X491) @ (sub @ X492 @ X491) @ X493)))))))))) & ((! [X495 : subst,X496 : term] : (((sub @ one @ (push @ X496 @ X495))) = X496) => (! [X497 : subst,X498 : term] : (((comp @ sh @ (push @ X498 @ X497))) = X497) => ! [X499 : term,X500 : term,X501 : (term > $o),X502 : subst,X503 : (term > $o)] : (~ (((X503 @ X500)) = $true) | (((X501 @ (sub @ X500 @ (push @ X499 @ X502)))) = $true)))) <=> (pushprop_lem1_lthm = $true)) & ((hoaslaminj_lthm = $true) <=> (! [X504 : term,X505 : subst] : (((sub @ one @ (push @ X504 @ X505))) = X504) => (! [X506 : term,X507 : subst] : (((comp @ sh @ (push @ X506 @ X507))) = X507) => (! [X508 : term,X509 : term] : ((((lam @ X509)) != ((lam @ X508))) | (X508 = X509)) => ! [X510 : (subst > term > term),X511 : subst,X512 : (subst > term > term),X513 : term] : ((((lam @ (X510 @ sh @ one))) != ((lam @ (X512 @ sh @ one)))) | (((X510 @ X511 @ X513)) = ((X512 @ X511 @ X513))) | ~ ! [X514 : subst,X515 : subst,X516 : term] : (((X512 @ (comp @ X515 @ X514) @ (sub @ X516 @ X514))) = ((sub @ (X512 @ X515 @ X516) @ X514))) | ~ ! [X517 : term,X518 : subst,X519 : subst] : (((X510 @ (comp @ X519 @ X518) @ (sub @ X517 @ X518))) = ((sub @ (X510 @ X519 @ X517) @ X518)))))))) & (! [X520 : subst,X521 : term,X522 : (subst > term > term),X523 : (subst > term > term)] : ((((lam @ (X522 @ sh @ one))) != ((lam @ (X523 @ sh @ one)))) | ~ ! [X524 : term,X525 : subst,X526 : subst] : (((sub @ (X523 @ X525 @ X524) @ X526)) = ((X523 @ (comp @ X525 @ X526) @ (sub @ X524 @ X526)))) | (((X522 @ X520 @ X521)) = ((X523 @ X520 @ X521))) | ~ ! [X527 : subst,X528 : term,X529 : subst] : (((sub @ (X522 @ X529 @ X528) @ X527)) = ((X522 @ (comp @ X529 @ X527) @ (sub @ X528 @ X527))))) <=> (hoaslaminj = $true)) & ((axvarid = $true) <=> ! [X530 : term] : (((sub @ X530 @ id)) = X530)) & (hoasapnotvar_lthm = $true) & (! [X531 : (subst > term > subst > $o)] : ~ ! [X532 : (term > $o)] : ~ ! [X533 : term] : (((X531 @ id @ X533 @ id)) = ((X532 @ X533))) <=> (hoasinduction_lem0_lthm = $true)) & (axassoc = $true) & ! [X534 : d_subst] : (d_id = X534) & ((hoasinduction_lem2v2 = $true) <=> ! [X535 : term,X536 : term,X537 : (subst > term > subst > $o),X538 : (term > $o)] : (~ ! [X539 : term,X540 : term] : (~ ($true = ((X537 @ id @ X540 @ id))) | ~ (((X537 @ id @ X539 @ id)) = $true) | ($true = ((X537 @ id @ (ap @ (sub @ X540 @ id) @ X539) @ id)))) | ~ ! [X541 : subst,X542 : subst,X543 : term,X544 : subst] : ((((X537 @ X542 @ X543 @ (comp @ X544 @ X541))) = $true) | ~ ($true = ((X537 @ (comp @ X542 @ X544) @ (sub @ X543 @ X544) @ X541)))) | ~ ! [X545 : subst,X546 : subst,X547 : subst,X548 : term] : (~ ($true = ((X537 @ X545 @ X548 @ (comp @ X546 @ X547)))) | (((X537 @ (comp @ X545 @ X546) @ (sub @ X548 @ X546) @ X547)) = $true)) | ($true = ((X538 @ (ap @ X536 @ X535)))) | ~ (((X538 @ X536)) = $true) | ~ (((X538 @ X535)) = $true) | ~ ! [X549 : term] : (((X538 @ X549)) = ((X537 @ id @ X549 @ id))))) & (axapp = $true) & ~ (lamnotap = $true)), inference(fool_elimination,[],[f6])). thf(f8,plain,( ~((hoasinduction_lem3 = $true) <=> ! [X0 : (subst > term > subst > $o)] : (! [X3 : subst,X4 : subst,X2 : term,X1 : subst] : (($true = ((X0 @ X3 @ X2 @ (comp @ X1 @ X4)))) => (((X0 @ (comp @ X3 @ X1) @ (sub @ X2 @ X1) @ X4)) = $true)) => (! [X5 : subst,X7 : term,X8 : subst,X6 : subst] : ((((X0 @ (comp @ X5 @ X6) @ (sub @ X7 @ X6) @ X8)) = $true) => (((X0 @ X5 @ X7 @ (comp @ X6 @ X8))) = $true)) => (! [X9 : (subst > term > term)] : (! [X12 : subst,X10 : term,X11 : subst] : (((X9 @ (comp @ X12 @ X11) @ (sub @ X10 @ X11))) = ((sub @ (X9 @ X12 @ X10) @ X11))) => (! [X13 : term] : ((((X0 @ id @ X13 @ id)) = $true) => ($true = ((X0 @ id @ (X9 @ id @ X13) @ id)))) => ($true = ((X0 @ id @ (hoaslam @ id @ (^[Y0 : subst]: ((^[Y1 : term]: (X9 @ Y0 @ Y1))))) @ id))))) => ! [X14 : term] : (! [X15 : term] : ((((X0 @ id @ X15 @ id)) = $true) => ($true = ((X0 @ id @ (sub @ X14 @ (push @ X15 @ id)) @ id)))) => (((X0 @ id @ (lam @ X14) @ id)) = $true))))))), inference(rectify,[],[f5])). thf(f9,plain,( ((apinj1 = $true) <=> ! [X3 : term,X1 : term,X0 : term,X2 : term] : ((((ap @ X1 @ X3)) != ((ap @ X0 @ X2))) | (X0 = X1))) & (((d2subst @ d_id)) = id) & (pushprop_lem3v2 = $true) & ((hoasinduction_lem3v2 = $true) <=> ! [X4 : term,X5 : (term > $o),X6 : (subst > term > subst > $o)] : (~ ! [X7 : subst,X8 : term,X9 : subst,X10 : subst] : ((((X6 @ X10 @ X8 @ (comp @ X7 @ X9))) = $true) | ~ ($true = ((X6 @ (comp @ X10 @ X7) @ (sub @ X8 @ X7) @ X9)))) | ~ ! [X11 : term] : (~ ($true = ((X5 @ X11))) | ($true = ((X5 @ (sub @ X4 @ (push @ X11 @ id)))))) | (((X5 @ (lam @ X4))) = $true) | ~ ! [X12 : (subst > term > term)] : (~ ! [X13 : term] : ((((X6 @ id @ (X12 @ id @ X13) @ id)) = $true) | ~ (((X6 @ id @ X13 @ id)) = $true)) | ~ ! [X16 : subst,X14 : term,X15 : subst] : (((sub @ (X12 @ X15 @ X14) @ X16)) = ((X12 @ (comp @ X15 @ X16) @ (sub @ X14 @ X16)))) | (((X6 @ id @ (lam @ (X12 @ sh @ one)) @ id)) = $true)) | ~ ! [X17 : subst,X18 : subst,X19 : subst,X20 : term] : (~ (((X6 @ X18 @ X20 @ (comp @ X19 @ X17))) = $true) | ($true = ((X6 @ (comp @ X18 @ X19) @ (sub @ X20 @ X19) @ X17)))) | ~ ! [X21 : term] : (((X6 @ id @ X21 @ id)) = ((X5 @ X21))))) & (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_lem1_lthm = $true) & (hoasinduction_gthm = $true) & (hoaslamnotvar = $true) & (! [X22 : term,X23 : (subst > term > subst > $o)] : (~ ! [X24 : term,X25 : term] : (~ (((X23 @ id @ X24 @ id)) = $true) | ~ (((X23 @ id @ X25 @ id)) = $true) | (((X23 @ id @ (ap @ (sub @ X24 @ id) @ X25) @ id)) = $true)) | ~ ! [X26 : subst,X27 : subst,X28 : term,X29 : subst] : ((((X23 @ (comp @ X27 @ X26) @ (sub @ X28 @ X26) @ X29)) = $true) | ~ (((X23 @ X27 @ X28 @ (comp @ X26 @ X29))) = $true)) | ~ ! [X30 : subst,X31 : subst,X32 : subst,X33 : term] : (~ ($true = ((X23 @ (comp @ X30 @ X31) @ (sub @ X33 @ X31) @ X32))) | (((X23 @ X30 @ X33 @ (comp @ X31 @ X32))) = $true)) | ~ ! [X34 : term] : ((((X23 @ id @ X34 @ id)) = $true) | ~ ($true = ((var @ (sub @ X34 @ id))))) | ~ ! [X35 : (subst > term > term)] : (~ ! [X37 : subst,X38 : term,X36 : subst] : (((X35 @ (comp @ X37 @ X36) @ (sub @ X38 @ X36))) = ((sub @ (X35 @ X37 @ X38) @ X36))) | ~ ! [X39 : term] : (($true = ((X23 @ id @ (X35 @ id @ X39) @ id))) | ~ (((X23 @ id @ X39 @ id)) = $true)) | (((X23 @ id @ (lam @ (X35 @ sh @ one)) @ id)) = $true)) | (((X23 @ id @ X22 @ id)) = $true)) <=> (hoasinduction = $true)) & (induction2_gthm = $true) & ! [X41 : d_subst,X40 : d_subst] : ((((d2subst @ X40)) = ((d2subst @ X41))) => (X40 = X41)) & ~ (ulamvar1 = $true) & ((pushprop_lem0_lthm = $true) <=> ! [X44 : (term > $o),X43 : subst,X42 : term] : ~ ! [X45 : (term > $o)] : ~ ! [X46 : term] : (((X45 @ X46)) = ((X44 @ (sub @ X46 @ (push @ X42 @ X43)))))) & (axvarshift = $true) & ((axidl = $true) <=> ! [X47 : subst] : (((comp @ id @ X47)) = X47)) & (hoaslamnotvar_gthm = $true) & (hoasapinj2_gthm = $true) & (hoaslamnotap_gthm = $true) & (induction2lem_gthm = $true) & ((hoasinduction_lem3a = $true) <=> ! [X48 : (subst > term > subst > $o),X49 : term] : (($true = ((X48 @ id @ (lam @ X49) @ id))) | ~ ! [X50 : term] : (~ ($true = ((X48 @ id @ X50 @ id))) | ($true = ((X48 @ id @ (sub @ X49 @ (push @ X50 @ id)) @ id)))) | ~ ! [X51 : (subst > term > term)] : (~ ! [X53 : term,X54 : subst,X52 : subst] : (((X51 @ (comp @ X54 @ X52) @ (sub @ X53 @ X52))) = ((sub @ (X51 @ X54 @ X53) @ X52))) | (((X48 @ id @ (lam @ (X51 @ sh @ one)) @ id)) = $true) | ~ ! [X55 : term] : (~ ($true = ((X48 @ id @ X55 @ id))) | (((X48 @ id @ (X51 @ id @ X55) @ id)) = $true))))) & (! [X56 : term,X57 : subst,X58 : (term > $o)] : (~ ! [X59 : term,X60 : term] : (~ (((X58 @ X59)) = $true) | (((X58 @ (ap @ X60 @ X59))) = $true) | ~ (((X58 @ X60)) = $true)) | ~ ! [X61 : term] : (~ (((var @ X61)) = $true) | ($true = ((X58 @ (sub @ X61 @ X57))))) | ~ ! [X62 : term] : (($true = ((X58 @ (lam @ X62)))) | ~ ! [X63 : term] : ((((X58 @ (sub @ X62 @ (push @ X63 @ id)))) = $true) | ~ ($true = ((X58 @ X63))))) | (((X58 @ (sub @ X56 @ X57))) = $true)) <=> (induction2lem = $true)) & (hoasinduction_lem3b_gthm = $true) & (axmap = $true) & (pushprop_lem2v2 = $true) & (axabs = $true) & (hoaslamnotvar_lthm = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (termmset_gthm = $true) & (pushprop_lem2v2_gthm = $true) & ((induction = $true) <=> ! [X64 : (term > $o),X65 : term] : ((((X64 @ X65)) = $true) | ~ ! [X66 : term,X67 : term] : (~ (((X64 @ X67)) = $true) | ~ (((X64 @ X66)) = $true) | (((X64 @ (ap @ X67 @ X66))) = $true)) | ~ ! [X68 : term] : (~ (((X64 @ X68)) = $true) | ($true = ((X64 @ (lam @ X68))))) | ~ ! [X69 : term] : (($true = ((X64 @ X69))) | ~ (((var @ X69)) = $true)))) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X70 : (subst > term > subst > $o),X71 : term] : (~ ! [X72 : term] : (($true = ((X70 @ id @ (sub @ X71 @ (push @ X72 @ id)) @ id))) | ~ ($true = ((X70 @ id @ X72 @ id)))) | ~ ! [X73 : (subst > term > term)] : (~ ! [X74 : term] : (($true = ((X70 @ id @ (X73 @ id @ X74) @ id))) | ~ ($true = ((X70 @ id @ X74 @ id)))) | (((X70 @ id @ (lam @ (X73 @ sh @ one)) @ id)) = $true) | ~ ! [X76 : term,X77 : subst,X75 : subst] : (((sub @ (X73 @ X75 @ X76) @ X77)) = ((X73 @ (comp @ X75 @ X77) @ (sub @ X76 @ X77))))) | (((X70 @ id @ (lam @ (sub @ X71 @ (push @ one @ sh))) @ id)) = $true))) & (hoasinduction_lem1v2 = $true) & (ulamvarind = $true) & (! [X78 : term] : ~ ! [X79 : (subst > term > term)] : ~ ! [X81 : subst,X80 : term] : (((X79 @ X81 @ X80)) = ((sub @ X78 @ (push @ X80 @ X81)))) <=> (hoasinduction_lem3v2_f = $true)) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X82 : term] : ~ ! [X83 : (subst > term > term)] : (((sub @ X82 @ (push @ one @ sh))) != ((X83 @ sh @ one)))) & ((induction2_lthm = $true) <=> (! [X84 : term] : (((sub @ X84 @ id)) = X84) => (! [X85 : (term > $o),X86 : subst,X87 : term] : (~ ! [X88 : term] : ((((X85 @ (lam @ X88))) = $true) | ~ ! [X89 : term] : (~ (((X85 @ X89)) = $true) | (((X85 @ (sub @ X88 @ (push @ X89 @ id)))) = $true))) | ~ ! [X90 : term,X91 : term] : ((((X85 @ (ap @ X91 @ X90))) = $true) | ~ (((X85 @ X90)) = $true) | ~ (((X85 @ X91)) = $true)) | (((X85 @ (sub @ X87 @ X86))) = $true) | ~ ! [X92 : term] : (~ ($true = ((var @ X92))) | (((X85 @ (sub @ X92 @ X86))) = $true))) => ! [X93 : (term > $o),X94 : term] : (~ ! [X95 : term,X96 : term] : (~ ($true = ((X93 @ X95))) | ($true = ((X93 @ (ap @ X96 @ X95)))) | ~ (((X93 @ X96)) = $true)) | (((X93 @ X94)) = $true) | ~ ! [X97 : term] : ((((X93 @ X97)) = $true) | ~ (((var @ X97)) = $true)) | ~ ! [X98 : term] : (~ ! [X99 : term] : (~ (((X93 @ X99)) = $true) | ($true = ((X93 @ (sub @ X98 @ (push @ X99 @ id)))))) | ($true = ((X93 @ (lam @ X98))))))))) & (hoasapnotvar_gthm = $true) & ! [X101 : d_term,X100 : d_term] : ((((d2term @ X101)) = ((d2term @ X100))) => (X100 = X101)) & ((! [X102 : subst] : (((comp @ id @ X102)) = X102) => (! [X103 : subst] : (((comp @ X103 @ id)) = X103) => (! [X105 : subst] : (((comp @ X105 @ id)) = X105) & ! [X104 : subst] : (((comp @ id @ X104)) = X104)))) <=> (substmonoid_lthm = $true)) & ! [X106 : term] : ? [X107 : d_term] : (((d2term @ X107)) = X106) & (hoasinduction_lem2_lthm = $true) & ((hoasinduction_lem3v2_lthm = $true) <=> (! [X108 : term] : (((sub @ X108 @ id)) = X108) => ! [X109 : (subst > term > subst > $o),X110 : (term > $o),X111 : term] : (($true = ((X110 @ (lam @ X111)))) | ~ ! [X112 : (subst > term > term)] : (~ ! [X113 : term] : ((((X109 @ id @ (X112 @ id @ X113) @ id)) = $true) | ~ ($true = ((X109 @ id @ X113 @ id)))) | (((X109 @ id @ (lam @ (X112 @ sh @ one)) @ id)) = $true) | ~ ! [X114 : subst,X115 : term,X116 : subst] : (((sub @ (X112 @ X116 @ X115) @ X114)) = ((X112 @ (comp @ X116 @ X114) @ (sub @ X115 @ X114))))) | ~ ! [X117 : subst,X118 : term,X119 : subst,X120 : subst] : ((((X109 @ (comp @ X120 @ X119) @ (sub @ X118 @ X119) @ X117)) = $true) | ~ (((X109 @ X120 @ X118 @ (comp @ X119 @ X117))) = $true)) | ~ ! [X121 : term] : (((X110 @ X121)) = ((X109 @ id @ X121 @ id))) | ~ ! [X122 : term] : (($true = ((X110 @ (sub @ X111 @ (push @ X122 @ id))))) | ~ (((X110 @ X122)) = $true)) | ~ ! [X123 : subst,X124 : subst,X125 : term,X126 : subst] : (($true = ((X109 @ X124 @ X125 @ (comp @ X126 @ X123)))) | ~ ($true = ((X109 @ (comp @ X124 @ X126) @ (sub @ X125 @ X126) @ X123))))))) & ((hoasinduction_lem3aaa = $true) <=> ! [X127 : term,X128 : (subst > term > subst > $o)] : (~ ! [X129 : term] : (($true = ((X128 @ id @ (sub @ X127 @ (push @ X129 @ id)) @ id))) | ~ (((X128 @ id @ X129 @ id)) = $true)) | ~ ! [X130 : term,X131 : (subst > term > term)] : (~ ! [X132 : term,X133 : subst,X134 : subst] : (((sub @ (X131 @ X133 @ X132) @ X134)) = ((sub @ (sub @ X130 @ (push @ X132 @ X133)) @ X134))) | ~ ! [X137 : term,X136 : subst,X135 : subst] : (((X131 @ (comp @ X136 @ X135) @ (sub @ X137 @ X135))) = ((sub @ X130 @ (push @ (sub @ X137 @ X135) @ (comp @ X136 @ X135))))) | (((X128 @ id @ (lam @ (X131 @ sh @ one)) @ id)) = $true) | ~ ! [X138 : term] : ((((X128 @ id @ (X131 @ id @ X138) @ id)) = $true) | ~ ($true = ((X128 @ id @ X138 @ id))))) | (((X128 @ id @ (lam @ (sub @ X127 @ (push @ one @ sh))) @ id)) = $true))) & ((hoaslamnotap = $true) <=> ! [X141 : (subst > term > term),X140 : term,X139 : term] : (~ ! [X143 : subst,X142 : term,X144 : subst] : (((X141 @ (comp @ X143 @ X144) @ (sub @ X142 @ X144))) = ((sub @ (X141 @ X143 @ X142) @ X144))) | (((lam @ (X141 @ sh @ one))) != ((ap @ (sub @ X140 @ id) @ X139))))) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (apnotvar = $true) & (hoaslaminj_gthm = $true) & ((hoasapinj2_lthm = $true) <=> (! [X145 : term,X146 : term,X148 : term,X147 : term] : ((X147 = X148) | (((ap @ X145 @ X148)) != ((ap @ X146 @ X147)))) => ! [X149 : term,X152 : term,X151 : term,X150 : term] : ((X150 = X151) | (((ap @ (sub @ X149 @ id) @ X150)) != ((ap @ (sub @ X152 @ id) @ X151)))))) & ((! [X153 : (subst > term > subst > $o)] : ~ ! [X154 : (term > $o)] : ~ ! [X155 : term] : (((X154 @ X155)) = ((X153 @ id @ X155 @ id))) => (! [X156 : term,X157 : (term > $o)] : (~ ! [X158 : term] : (~ ! [X159 : term] : (~ ($true = ((X157 @ X159))) | (((X157 @ (sub @ X158 @ (push @ X159 @ id)))) = $true)) | (((X157 @ (lam @ X158))) = $true)) | ~ ! [X160 : term,X161 : term] : (~ (((X157 @ X160)) = $true) | ~ ($true = ((X157 @ X161))) | (((X157 @ (ap @ X161 @ X160))) = $true)) | ~ ! [X162 : term] : (($true = ((X157 @ X162))) | ~ (((var @ X162)) = $true)) | ($true = ((X157 @ X156)))) => (! [X163 : term] : (((sub @ X163 @ id)) = X163) => (! [X164 : (subst > term > subst > $o),X165 : (term > $o),X166 : term] : (~ ! [X167 : term] : (((X165 @ X167)) = ((X164 @ id @ X167 @ id))) | ~ ! [X168 : (subst > term > term)] : (($true = ((X164 @ id @ (lam @ (X168 @ sh @ one)) @ id))) | ~ ! [X169 : term] : ((((X164 @ id @ (X168 @ id @ X169) @ id)) = $true) | ~ ($true = ((X164 @ id @ X169 @ id)))) | ~ ! [X171 : term,X172 : subst,X170 : subst] : (((X168 @ (comp @ X172 @ X170) @ (sub @ X171 @ X170))) = ((sub @ (X168 @ X172 @ X171) @ X170)))) | ($true = ((X165 @ (lam @ X166)))) | ~ ! [X173 : term] : (~ ($true = ((X165 @ X173))) | (((X165 @ (sub @ X166 @ (push @ X173 @ id)))) = $true))) => ! [X174 : (subst > term > subst > $o),X175 : term] : (~ ! [X176 : (subst > term > term)] : (~ ! [X178 : subst,X179 : term,X177 : subst] : (((sub @ (X176 @ X177 @ X179) @ X178)) = ((X176 @ (comp @ X177 @ X178) @ (sub @ X179 @ X178)))) | ~ ! [X180 : term] : (~ ($true = ((X174 @ id @ X180 @ id))) | (((X174 @ id @ (X176 @ id @ X180) @ id)) = $true)) | (((X174 @ id @ (lam @ (X176 @ sh @ one)) @ id)) = $true)) | ~ ! [X181 : term,X182 : term] : ((((X174 @ id @ (ap @ (sub @ X182 @ id) @ X181) @ id)) = $true) | ~ ($true = ((X174 @ id @ X182 @ id))) | ~ ($true = ((X174 @ id @ X181 @ id)))) | ($true = ((X174 @ id @ X175 @ id)))))))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (pushprop_lthm = $true) & ((axidr = $true) <=> ! [X183 : subst] : (((comp @ X183 @ id)) = X183)) & (! [X187 : term,X184 : term,X186 : term,X185 : term] : ((X186 = X187) | (((ap @ (sub @ X185 @ id) @ X186)) != ((ap @ (sub @ X184 @ id) @ X187)))) <=> (hoasapinj2 = $true)) & (pushprop_lthm_orig = $true) & (hoasinduction_lem2_gthm = $true) & (pushprop_lem1_gthm = $true) & (! [X188 : term] : ~ ! [X189 : (subst > term > term)] : ~ ! [X190 : subst,X191 : term] : (((X189 @ X190 @ X191)) = ((sub @ X188 @ (push @ X191 @ X190)))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & (! [X192 : (subst > term > subst > $o)] : ~ ! [X193 : (term > $o)] : ~ ! [X194 : term] : (((X192 @ id @ X194 @ id)) = ((X193 @ X194))) <=> (hoasinduction_lem0 = $true)) & (hoasinduction_lem1v2_gthm = $true) & (! [X195 : term,X196 : term] : ((((sub @ X196 @ sh)) != ((sub @ X195 @ sh))) | (X195 = X196)) <=> (shinj = $true)) & (hoasinduction_lem3v2_gthm = $true) & (hoasapinj1_gthm = $true) & ((induction2 = $true) <=> ! [X197 : (term > $o),X198 : term] : (~ ! [X199 : term,X200 : term] : (~ ($true = ((X197 @ X200))) | (((X197 @ (ap @ X200 @ X199))) = $true) | ~ ($true = ((X197 @ X199)))) | (((X197 @ X198)) = $true) | ~ ! [X201 : term] : (~ ! [X202 : term] : ((((X197 @ (sub @ X201 @ (push @ X202 @ id)))) = $true) | ~ ($true = ((X197 @ X202)))) | (((X197 @ (lam @ X201))) = $true)) | ~ ! [X203 : term] : (($true = ((X197 @ X203))) | ~ (((var @ X203)) = $true)))) & (! [X204 : term,X205 : subst] : (((comp @ sh @ (push @ X204 @ X205))) = X205) <=> (axshiftcons = $true)) & ~ (((var @ (d2term @ d_one))) = $true) & (ulamvarsh = $true) & ~ (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) = $true) & (! [X206 : (subst > term > subst > $o),X207 : term] : (~ ! [X208 : term,X209 : term] : (($true = ((X206 @ id @ (ap @ (sub @ X208 @ id) @ X209) @ id))) | ~ (((X206 @ id @ X208 @ id)) = $true) | ~ ($true = ((X206 @ id @ X209 @ id)))) | (((X206 @ id @ X207 @ id)) = $true) | ~ ! [X210 : (subst > term > term)] : (~ ! [X211 : term] : (($true = ((X206 @ id @ (X210 @ id @ X211) @ id))) | ~ (((X206 @ id @ X211 @ id)) = $true)) | ~ ! [X212 : subst,X213 : term,X214 : subst] : (((X210 @ (comp @ X214 @ X212) @ (sub @ X213 @ X212))) = ((sub @ (X210 @ X214 @ X213) @ X212))) | ($true = ((X206 @ id @ (lam @ (X210 @ sh @ one)) @ id))))) <=> (hoasinduction_no_psi_cond = $true)) & (hoasinduction_lem2v2_gthm = $true) & (hoasinduction_lem3_gthm = $true) & ((hoasinduction_lem2 = $true) <=> ! [X215 : (subst > term > subst > $o),X216 : term,X217 : term] : (~ (((X215 @ id @ X216 @ id)) = $true) | ~ ! [X218 : subst,X219 : subst,X220 : subst,X221 : term] : (($true = ((X215 @ (comp @ X220 @ X218) @ (sub @ X221 @ X218) @ X219))) | ~ ($true = ((X215 @ X220 @ X221 @ (comp @ X218 @ X219))))) | ($true = ((X215 @ id @ (ap @ X216 @ X217) @ id))) | ~ ! [X222 : term,X223 : term] : (~ (((X215 @ id @ X222 @ id)) = $true) | ~ (((X215 @ id @ X223 @ id)) = $true) | ($true = ((X215 @ id @ (ap @ (sub @ X223 @ id) @ X222) @ id)))) | ~ ! [X224 : term,X225 : subst,X226 : subst,X227 : subst] : (~ ($true = ((X215 @ (comp @ X227 @ X225) @ (sub @ X224 @ X225) @ X226))) | ($true = ((X215 @ X227 @ X224 @ (comp @ X225 @ X226))))) | ~ (((X215 @ id @ X217 @ id)) = $true))) & ! [X228 : subst] : ? [X229 : d_subst] : (((d2subst @ X229)) = X228) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((! [X231 : subst] : (((comp @ X231 @ id)) = X231) & ! [X230 : subst] : (((comp @ id @ X230)) = X230)) <=> (substmonoid = $true)) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((laminj = $true) <=> ! [X233 : term,X232 : term] : ((((lam @ X232)) != ((lam @ X233))) | (X232 = X233))) & ((axvarcons = $true) <=> ! [X234 : subst,X235 : term] : (((sub @ one @ (push @ X235 @ X234))) = X235)) & (pushprop = $true) & (hoasapnotvar = $true) & (((d2term @ d_one)) = one) & ((apinj2 = $true) <=> ! [X236 : term,X238 : term,X237 : term,X239 : term] : ((X236 = X239) | (((ap @ X237 @ X236)) != ((ap @ X238 @ X239))))) & (! [X240 : term] : ~ ! [X241 : (subst > term > term)] : (((sub @ X240 @ (push @ one @ sh))) != ((X241 @ sh @ one))) <=> (hoasinduction_lem3b = $true)) & (axclos = $true) & (! [X244 : subst,X243 : (term > $o),X242 : term] : ~ ! [X245 : (term > $o)] : ~ ! [X246 : term] : (((X245 @ X246)) = ((X243 @ (sub @ X246 @ (push @ X242 @ X244))))) <=> (pushprop_lem0 = $true)) & ((axscons = $true) <=> ! [X247 : subst] : (((push @ (sub @ one @ X247) @ (comp @ sh @ X247))) = X247)) & (! [X248 : term] : (((sub @ X248 @ id)) = X248) <=> (termmset = $true)) & (! [X249 : term,X250 : (term > $o),X251 : (subst > term > subst > $o)] : ((((X250 @ (lam @ X249))) = $true) | ~ ! [X252 : term] : (((X250 @ X252)) = ((X251 @ id @ X252 @ id))) | ~ ! [X253 : (subst > term > term)] : (($true = ((X251 @ id @ (lam @ (X253 @ sh @ one)) @ id))) | ~ ! [X254 : subst,X255 : term,X256 : subst] : (((sub @ (X253 @ X254 @ X255) @ X256)) = ((X253 @ (comp @ X254 @ X256) @ (sub @ X255 @ X256)))) | ~ ! [X257 : term] : (~ ($true = ((X251 @ id @ X257 @ id))) | ($true = ((X251 @ id @ (X253 @ id @ X257) @ id))))) | ~ ! [X258 : term] : ((((X250 @ (sub @ X249 @ (push @ X258 @ id)))) = $true) | ~ (((X250 @ X258)) = $true))) <=> (hoasinduction_lem3v2a = $true)) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((! [X259 : term] : (((sub @ X259 @ id)) = X259) => (! [X262 : term,X263 : term,X260 : term,X261 : term] : ((((ap @ X261 @ X263)) != ((ap @ X262 @ X260))) | (X261 = X262)) => ! [X265 : term,X266 : term,X267 : term,X264 : term] : ((((ap @ (sub @ X267 @ id) @ X265)) != ((ap @ (sub @ X266 @ id) @ X264))) | (X266 = X267)))) <=> (hoasapinj1_lthm = $true)) & ((pushprop_lem1 = $true) <=> ! [X268 : term,X269 : (term > $o),X270 : term,X271 : (term > $o),X272 : subst] : ((((X271 @ (sub @ X270 @ (push @ X268 @ X272)))) = $true) | ~ (((X269 @ X270)) = $true))) & ((! [X274 : subst,X273 : term] : (((sub @ one @ (push @ X273 @ X274))) = X273) => (! [X276 : subst,X275 : term] : (((comp @ sh @ (push @ X275 @ X276))) = X276) => (! [X277 : subst] : (((comp @ X277 @ id)) = X277) => (! [X278 : (term > $o),X279 : term] : ((((X278 @ X279)) = $true) | ~ ! [X280 : term] : (~ ($true = ((X278 @ X280))) | (((X278 @ (lam @ X280))) = $true)) | ~ ! [X281 : term,X282 : term] : (~ (((X278 @ X282)) = $true) | (((X278 @ (ap @ X281 @ X282))) = $true) | ~ (((X278 @ X281)) = $true)) | ~ ! [X283 : term] : (($true = ((X278 @ X283))) | ~ ($true = ((var @ X283))))) => ! [X284 : (term > $o),X285 : subst,X286 : term] : (~ ! [X287 : term,X288 : term] : (~ (((X284 @ X288)) = $true) | (((X284 @ (ap @ X288 @ X287))) = $true) | ~ (((X284 @ X287)) = $true)) | (((X284 @ (sub @ X286 @ X285))) = $true) | ~ ! [X289 : term] : ((((X284 @ (sub @ X289 @ X285))) = $true) | ~ (((var @ X289)) = $true)) | ~ ! [X290 : term] : ((((X284 @ (lam @ X290))) = $true) | ~ ! [X291 : term] : (~ (((X284 @ X291)) = $true) | (((X284 @ (sub @ X290 @ (push @ X291 @ id)))) = $true)))))))) <=> (induction2lem_lthm = $true)) & ((! [X292 : term] : (((sub @ X292 @ id)) = X292) => (! [X293 : (subst > term > subst > $o),X294 : term] : (($true = ((X293 @ id @ (lam @ (sub @ X294 @ (push @ one @ sh))) @ id))) | ~ ! [X295 : (subst > term > term)] : (~ ! [X296 : term] : (~ ($true = ((X293 @ id @ X296 @ id))) | ($true = ((X293 @ id @ (X295 @ id @ X296) @ id)))) | ~ ! [X298 : subst,X297 : term,X299 : subst] : (((X295 @ (comp @ X299 @ X298) @ (sub @ X297 @ X298))) = ((sub @ (X295 @ X299 @ X297) @ X298))) | (((X293 @ id @ (lam @ (X295 @ sh @ one)) @ id)) = $true)) | ~ ! [X300 : term] : (($true = ((X293 @ id @ (sub @ X294 @ (push @ X300 @ id)) @ id))) | ~ (((X293 @ id @ X300 @ id)) = $true))) => ! [X301 : term,X302 : (subst > term > subst > $o)] : (~ ! [X303 : (subst > term > term)] : (($true = ((X302 @ id @ (lam @ (X303 @ sh @ one)) @ id))) | ~ ! [X305 : term,X304 : subst,X306 : subst] : (((sub @ (X303 @ X304 @ X305) @ X306)) = ((X303 @ (comp @ X304 @ X306) @ (sub @ X305 @ X306)))) | ~ ! [X307 : term] : (~ (((X302 @ id @ X307 @ id)) = $true) | ($true = ((X302 @ id @ (X303 @ id @ X307) @ id))))) | (((X302 @ id @ (lam @ X301) @ id)) = $true) | ~ ! [X308 : term] : (($true = ((X302 @ id @ (sub @ X301 @ (push @ X308 @ id)) @ id))) | ~ ($true = ((X302 @ id @ X308 @ id))))))) <=> (hoasinduction_lem3a_lthm = $true)) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (((d2subst @ d_id)) = sh) & (pushprop_gthm = $true) & (! [X309 : (term > $o),X310 : (term > $o),X311 : term,X312 : subst] : (($true = ((X310 @ one))) | ~ (((X309 @ X311)) = $true) | ~ ! [X313 : term] : (((X309 @ (sub @ X313 @ (push @ X311 @ X312)))) = ((X310 @ X313)))) <=> (pushprop_lem1v2 = $true)) & (hoasinduction_lem1_gthm = $true) & (lamnotvar = $true) & ((hoasapinj1 = $true) <=> ! [X314 : term,X317 : term,X316 : term,X315 : term] : ((X314 = X316) | (((ap @ (sub @ X316 @ id) @ X317)) != ((ap @ (sub @ X314 @ id) @ X315))))) & (hoasinduction_lem1 = $true) & ! [X318 : d_term] : (d_one = X318) & ((hoasinduction_lem3v2a_lthm = $true) <=> (! [X319 : term] : ~ ! [X320 : (subst > term > term)] : ~ ! [X322 : term,X321 : subst] : (((sub @ X319 @ (push @ X322 @ X321))) = ((X320 @ X321 @ X322))) => (! [X323 : term] : (((sub @ X323 @ id)) = X323) => ! [X324 : (term > $o),X325 : (subst > term > subst > $o),X326 : term] : (~ ! [X327 : term] : ((((X324 @ (sub @ X326 @ (push @ X327 @ id)))) = $true) | ~ ($true = ((X324 @ X327)))) | ~ ! [X328 : term] : (((X325 @ id @ X328 @ id)) = ((X324 @ X328))) | ~ ! [X329 : (subst > term > term)] : (~ ! [X332 : subst,X330 : subst,X331 : term] : (((sub @ (X329 @ X330 @ X331) @ X332)) = ((X329 @ (comp @ X330 @ X332) @ (sub @ X331 @ X332)))) | ($true = ((X325 @ id @ (lam @ (X329 @ sh @ one)) @ id))) | ~ ! [X333 : term] : (($true = ((X325 @ id @ (X329 @ id @ X333) @ id))) | ~ (((X325 @ id @ X333 @ id)) = $true))) | ($true = ((X324 @ (lam @ X326)))))))) & (! [X334 : (subst > term > subst > $o),X335 : term] : (~ ! [X336 : (subst > term > term)] : (~ ! [X337 : term] : (~ ($true = ((X334 @ id @ X337 @ id))) | (((X334 @ id @ (X336 @ id @ X337) @ id)) = $true)) | ~ ! [X340 : term,X339 : subst,X338 : subst] : (((sub @ (X336 @ X338 @ X340) @ X339)) = ((X336 @ (comp @ X338 @ X339) @ (sub @ X340 @ X339)))) | (((X334 @ id @ (lam @ (X336 @ sh @ one)) @ id)) = $true)) | ($true = ((X334 @ id @ (lam @ (sub @ X335 @ (push @ one @ sh))) @ id))) | ~ ! [X341 : term] : (~ (((X334 @ id @ X341 @ id)) = $true) | ($true = ((X334 @ id @ (sub @ X335 @ (push @ X341 @ id)) @ id))))) <=> (hoasinduction_lem3aa = $true)) & (hoaslamnotap_lthm = $true) & (pushprop_lem1v2_gthm = $true) & (pushprop_lem3v2_lthm = $true) & ((hoasinduction_lthm = $true) <=> (! [X342 : (term > $o),X343 : term] : (~ ! [X344 : term] : (~ ! [X345 : term] : (~ ($true = ((X342 @ X345))) | ($true = ((X342 @ (sub @ X344 @ (push @ X345 @ id)))))) | (((X342 @ (lam @ X344))) = $true)) | ($true = ((X342 @ X343))) | ~ ! [X346 : term,X347 : term] : (~ (((X342 @ X347)) = $true) | (((X342 @ (ap @ X346 @ X347))) = $true) | ~ ($true = ((X342 @ X346)))) | ~ ! [X348 : term] : ((((X342 @ X348)) = $true) | ~ ($true = ((var @ X348))))) => (! [X349 : term,X350 : (subst > term > subst > $o),X351 : term] : (~ ! [X352 : subst,X353 : term,X354 : subst,X355 : subst] : (($true = ((X350 @ X352 @ X353 @ (comp @ X354 @ X355)))) | ~ ($true = ((X350 @ (comp @ X352 @ X354) @ (sub @ X353 @ X354) @ X355)))) | ~ ($true = ((X350 @ id @ X351 @ id))) | ~ ! [X356 : term,X357 : subst,X358 : subst,X359 : subst] : ((((X350 @ (comp @ X357 @ X358) @ (sub @ X356 @ X358) @ X359)) = $true) | ~ (((X350 @ X357 @ X356 @ (comp @ X358 @ X359))) = $true)) | ~ (((X350 @ id @ X349 @ id)) = $true) | ($true = ((X350 @ id @ (ap @ X349 @ X351) @ id))) | ~ ! [X360 : term,X361 : term] : ((((X350 @ id @ (ap @ (sub @ X360 @ id) @ X361) @ id)) = $true) | ~ ($true = ((X350 @ id @ X360 @ id))) | ~ (((X350 @ id @ X361 @ id)) = $true))) => (! [X362 : term,X363 : (subst > term > subst > $o)] : (~ ! [X364 : term] : (~ (((X363 @ id @ X364 @ id)) = $true) | ($true = ((X363 @ id @ (sub @ X362 @ (push @ X364 @ id)) @ id)))) | ~ ! [X365 : subst,X366 : subst,X367 : subst,X368 : term] : (~ ($true = ((X363 @ X365 @ X368 @ (comp @ X367 @ X366)))) | (((X363 @ (comp @ X365 @ X367) @ (sub @ X368 @ X367) @ X366)) = $true)) | ($true = ((X363 @ id @ (lam @ X362) @ id))) | ~ ! [X369 : subst,X370 : subst,X371 : subst,X372 : term] : (~ (((X363 @ (comp @ X370 @ X371) @ (sub @ X372 @ X371) @ X369)) = $true) | ($true = ((X363 @ X370 @ X372 @ (comp @ X371 @ X369))))) | ~ ! [X373 : (subst > term > term)] : (($true = ((X363 @ id @ (lam @ (X373 @ sh @ one)) @ id))) | ~ ! [X374 : term] : (~ ($true = ((X363 @ id @ X374 @ id))) | ($true = ((X363 @ id @ (X373 @ id @ X374) @ id)))) | ~ ! [X377 : subst,X376 : subst,X375 : term] : (((sub @ (X373 @ X376 @ X375) @ X377)) = ((X373 @ (comp @ X376 @ X377) @ (sub @ X375 @ X377)))))) => ! [X378 : (subst > term > subst > $o),X379 : term] : (~ ! [X380 : term,X381 : subst,X382 : subst,X383 : subst] : ((((X378 @ (comp @ X381 @ X383) @ (sub @ X380 @ X383) @ X382)) = $true) | ~ (((X378 @ X381 @ X380 @ (comp @ X383 @ X382))) = $true)) | ~ ! [X384 : (subst > term > term)] : (($true = ((X378 @ id @ (lam @ (X384 @ sh @ one)) @ id))) | ~ ! [X385 : term] : (~ ($true = ((X378 @ id @ X385 @ id))) | (((X378 @ id @ (X384 @ id @ X385) @ id)) = $true)) | ~ ! [X387 : term,X388 : subst,X386 : subst] : (((X384 @ (comp @ X388 @ X386) @ (sub @ X387 @ X386))) = ((sub @ (X384 @ X388 @ X387) @ X386)))) | ~ ! [X389 : term,X390 : term] : (~ (((X378 @ id @ X390 @ id)) = $true) | ~ ($true = ((X378 @ id @ X389 @ id))) | (((X378 @ id @ (ap @ (sub @ X389 @ id) @ X390) @ id)) = $true)) | ~ ! [X391 : term] : (~ (((var @ (sub @ X391 @ id))) = $true) | (((X378 @ id @ X391 @ id)) = $true)) | ~ ! [X392 : subst,X393 : subst,X394 : subst,X395 : term] : ((((X378 @ X393 @ X395 @ (comp @ X392 @ X394))) = $true) | ~ ($true = ((X378 @ (comp @ X393 @ X392) @ (sub @ X395 @ X392) @ X394)))) | ($true = ((X378 @ id @ X379 @ id)))))))) & (hoasinduction_lem3a_gthm = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ((hoasinduction_lem3 = $true) <=> ! [X396 : term,X397 : (subst > term > subst > $o)] : ((((X397 @ id @ (lam @ X396) @ id)) = $true) | ~ ! [X398 : (subst > term > term)] : (~ ! [X399 : term] : (~ (((X397 @ id @ X399 @ id)) = $true) | ($true = ((X397 @ id @ (X398 @ id @ X399) @ id)))) | ~ ! [X401 : subst,X400 : term,X402 : subst] : (((X398 @ (comp @ X401 @ X402) @ (sub @ X400 @ X402))) = ((sub @ (X398 @ X401 @ X400) @ X402))) | ($true = ((X397 @ id @ (lam @ (X398 @ sh @ one)) @ id)))) | ~ ! [X403 : subst,X404 : subst,X405 : subst,X406 : term] : (~ (((X397 @ X404 @ X406 @ (comp @ X405 @ X403))) = $true) | (((X397 @ (comp @ X404 @ X405) @ (sub @ X406 @ X405) @ X403)) = $true)) | ~ ! [X407 : subst,X408 : subst,X409 : term,X410 : subst] : ((((X397 @ X410 @ X409 @ (comp @ X407 @ X408))) = $true) | ~ ($true = ((X397 @ (comp @ X410 @ X407) @ (sub @ X409 @ X407) @ X408)))) | ~ ! [X411 : term] : (~ (((X397 @ id @ X411 @ id)) = $true) | (((X397 @ id @ (sub @ X396 @ (push @ X411 @ id)) @ id)) = $true)))) & (substmonoid_gthm = $true) & ((! [X412 : term,X413 : subst] : (((sub @ one @ (push @ X412 @ X413))) = X412) => ! [X414 : (term > $o),X415 : (term > $o),X416 : subst,X417 : term] : (~ ! [X418 : term] : (((X415 @ X418)) = ((X414 @ (sub @ X418 @ (push @ X417 @ X416))))) | (((X415 @ one)) = $true) | ~ ($true = ((X414 @ X417))))) <=> (pushprop_lem1v2_lthm = $true)) & ((hoasinduction_lem3_lthm = $true) <=> (! [X419 : term] : (((sub @ X419 @ id)) = X419) => (! [X420 : (subst > term > subst > $o),X421 : term] : (~ ! [X422 : (subst > term > term)] : (~ ! [X423 : term] : (($true = ((X420 @ id @ (X422 @ id @ X423) @ id))) | ~ (((X420 @ id @ X423 @ id)) = $true)) | ~ ! [X424 : subst,X426 : term,X425 : subst] : (((sub @ (X422 @ X425 @ X426) @ X424)) = ((X422 @ (comp @ X425 @ X424) @ (sub @ X426 @ X424)))) | (((X420 @ id @ (lam @ (X422 @ sh @ one)) @ id)) = $true)) | ($true = ((X420 @ id @ (lam @ (sub @ X421 @ (push @ one @ sh))) @ id))) | ~ ! [X427 : term] : (~ (((X420 @ id @ X427 @ id)) = $true) | (((X420 @ id @ (sub @ X421 @ (push @ X427 @ id)) @ id)) = $true))) => ! [X428 : (subst > term > subst > $o),X429 : term] : (~ ! [X430 : subst,X431 : term,X432 : subst,X433 : subst] : ((((X428 @ (comp @ X433 @ X432) @ (sub @ X431 @ X432) @ X430)) = $true) | ~ (((X428 @ X433 @ X431 @ (comp @ X432 @ X430))) = $true)) | ~ ! [X434 : term] : (($true = ((X428 @ id @ (sub @ X429 @ (push @ X434 @ id)) @ id))) | ~ ($true = ((X428 @ id @ X434 @ id)))) | ($true = ((X428 @ id @ (lam @ X429) @ id))) | ~ ! [X435 : (subst > term > term)] : ((((X428 @ id @ (lam @ (X435 @ sh @ one)) @ id)) = $true) | ~ ! [X436 : term] : ((((X428 @ id @ (X435 @ id @ X436) @ id)) = $true) | ~ ($true = ((X428 @ id @ X436 @ id)))) | ~ ! [X438 : subst,X439 : subst,X437 : term] : (((X435 @ (comp @ X438 @ X439) @ (sub @ X437 @ X439))) = ((sub @ (X435 @ X438 @ X437) @ X439)))) | ~ ! [X440 : term,X441 : subst,X442 : subst,X443 : subst] : (~ (((X428 @ (comp @ X443 @ X442) @ (sub @ X440 @ X442) @ X441)) = $true) | (((X428 @ X443 @ X440 @ (comp @ X442 @ X441))) = $true)))))) & (pushprop_lem2v2_lthm = $true) & ((! [X444 : term] : (((sub @ X444 @ id)) = X444) => ! [X445 : term] : (((sub @ X445 @ id)) = X445)) <=> (termmset_lthm = $true)) & (pushprop_lem0_gthm = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoasinduction_lthm_3 = $true) <=> (! [X446 : (subst > term > subst > $o)] : ~ ! [X447 : (term > $o)] : ~ ! [X448 : term] : (((X447 @ X448)) = ((X446 @ id @ X448 @ id))) => (! [X449 : term,X450 : (term > $o)] : (~ ! [X451 : term,X452 : term] : (~ ($true = ((X450 @ X452))) | ($true = ((X450 @ (ap @ X452 @ X451)))) | ~ ($true = ((X450 @ X451)))) | ~ ! [X453 : term] : ((((X450 @ (lam @ X453))) = $true) | ~ ! [X454 : term] : ((((X450 @ (sub @ X453 @ (push @ X454 @ id)))) = $true) | ~ ($true = ((X450 @ X454))))) | ~ ! [X455 : term] : (($true = ((X450 @ X455))) | ~ (((var @ X455)) = $true)) | ($true = ((X450 @ X449)))) => (! [X456 : term] : (((sub @ X456 @ id)) = X456) => (! [X457 : (subst > term > subst > $o),X458 : term,X459 : (term > $o)] : (~ ! [X460 : term] : (((X459 @ X460)) = ((X457 @ id @ X460 @ id))) | (((X459 @ (lam @ X458))) = $true) | ~ ! [X461 : term] : (($true = ((X459 @ (sub @ X458 @ (push @ X461 @ id))))) | ~ ($true = ((X459 @ X461)))) | ~ ! [X462 : (subst > term > term)] : (~ ! [X465 : term,X464 : subst,X463 : subst] : (((X462 @ (comp @ X463 @ X464) @ (sub @ X465 @ X464))) = ((sub @ (X462 @ X463 @ X465) @ X464))) | ~ ! [X466 : term] : (~ (((X457 @ id @ X466 @ id)) = $true) | ($true = ((X457 @ id @ (X462 @ id @ X466) @ id)))) | (((X457 @ id @ (lam @ (X462 @ sh @ one)) @ id)) = $true))) => ! [X467 : (subst > term > subst > $o),X468 : term] : (~ ! [X469 : term,X470 : term] : (~ (((X467 @ id @ X470 @ id)) = $true) | (((X467 @ id @ (ap @ (sub @ X469 @ id) @ X470) @ id)) = $true) | ~ (((X467 @ id @ X469 @ id)) = $true)) | ~ ! [X471 : term] : ((((X467 @ id @ X471 @ id)) = $true) | ~ (((var @ (sub @ X471 @ id))) = $true)) | ~ ! [X472 : term,X473 : subst,X474 : subst,X475 : subst] : ((((X467 @ (comp @ X474 @ X473) @ (sub @ X472 @ X473) @ X475)) = $true) | ~ (((X467 @ X474 @ X472 @ (comp @ X473 @ X475))) = $true)) | (((X467 @ id @ X468 @ id)) = $true) | ~ ! [X476 : (subst > term > term)] : (~ ! [X477 : term] : ((((X467 @ id @ (X476 @ id @ X477) @ id)) = $true) | ~ ($true = ((X467 @ id @ X477 @ id)))) | (((X467 @ id @ (lam @ (X476 @ sh @ one)) @ id)) = $true) | ~ ! [X479 : term,X480 : subst,X478 : subst] : (((X476 @ (comp @ X478 @ X480) @ (sub @ X479 @ X480))) = ((sub @ (X476 @ X478 @ X479) @ X480)))) | ~ ! [X481 : subst,X482 : term,X483 : subst,X484 : subst] : (($true = ((X467 @ X484 @ X482 @ (comp @ X481 @ X483)))) | ~ (((X467 @ (comp @ X484 @ X481) @ (sub @ X482 @ X481) @ X483)) = $true)))))))) & ((! [X486 : term,X485 : subst] : (((sub @ one @ (push @ X486 @ X485))) = X486) => (! [X488 : term,X487 : subst] : (((comp @ sh @ (push @ X488 @ X487))) = X487) => ! [X489 : term,X490 : term,X491 : (term > $o),X492 : subst,X493 : (term > $o)] : (~ (((X493 @ X490)) = $true) | (((X491 @ (sub @ X490 @ (push @ X489 @ X492)))) = $true)))) <=> (pushprop_lem1_lthm = $true)) & ((hoaslaminj_lthm = $true) <=> (! [X495 : subst,X494 : term] : (((sub @ one @ (push @ X494 @ X495))) = X494) => (! [X497 : subst,X496 : term] : (((comp @ sh @ (push @ X496 @ X497))) = X497) => (! [X498 : term,X499 : term] : ((((lam @ X498)) != ((lam @ X499))) | (X498 = X499)) => ! [X501 : subst,X502 : (subst > term > term),X500 : (subst > term > term),X503 : term] : (~ ! [X505 : subst,X504 : subst,X506 : term] : (((sub @ (X502 @ X505 @ X506) @ X504)) = ((X502 @ (comp @ X505 @ X504) @ (sub @ X506 @ X504)))) | ~ ! [X507 : term,X509 : subst,X508 : subst] : (((X500 @ (comp @ X509 @ X508) @ (sub @ X507 @ X508))) = ((sub @ (X500 @ X509 @ X507) @ X508))) | (((lam @ (X502 @ sh @ one))) != ((lam @ (X500 @ sh @ one)))) | (((X502 @ X501 @ X503)) = ((X500 @ X501 @ X503)))))))) & ((hoaslaminj = $true) <=> ! [X511 : term,X510 : subst,X512 : (subst > term > term),X513 : (subst > term > term)] : ((((X512 @ X510 @ X511)) = ((X513 @ X510 @ X511))) | (((lam @ (X512 @ sh @ one))) != ((lam @ (X513 @ sh @ one)))) | ~ ! [X519 : subst,X518 : term,X517 : subst] : (((X512 @ (comp @ X519 @ X517) @ (sub @ X518 @ X517))) = ((sub @ (X512 @ X519 @ X518) @ X517))) | ~ ! [X514 : term,X515 : subst,X516 : subst] : (((sub @ (X513 @ X515 @ X514) @ X516)) = ((X513 @ (comp @ X515 @ X516) @ (sub @ X514 @ X516)))))) & ((axvarid = $true) <=> ! [X520 : term] : (((sub @ X520 @ id)) = X520)) & (hoasapnotvar_lthm = $true) & (! [X521 : (subst > term > subst > $o)] : ~ ! [X522 : (term > $o)] : ~ ! [X523 : term] : (((X521 @ id @ X523 @ id)) = ((X522 @ X523))) <=> (hoasinduction_lem0_lthm = $true)) & (axassoc = $true) & ! [X524 : d_subst] : (d_id = X524) & ((hoasinduction_lem2v2 = $true) <=> ! [X525 : term,X526 : term,X527 : (subst > term > subst > $o),X528 : (term > $o)] : (~ ! [X529 : term,X530 : term] : (~ (((X527 @ id @ X530 @ id)) = $true) | ~ ($true = ((X527 @ id @ X529 @ id))) | (((X527 @ id @ (ap @ (sub @ X530 @ id) @ X529) @ id)) = $true)) | ~ ! [X531 : subst,X532 : subst,X533 : term,X534 : subst] : (($true = ((X527 @ X532 @ X533 @ (comp @ X534 @ X531)))) | ~ (((X527 @ (comp @ X532 @ X534) @ (sub @ X533 @ X534) @ X531)) = $true)) | ~ ! [X535 : subst,X536 : subst,X537 : subst,X538 : term] : (~ ($true = ((X527 @ X535 @ X538 @ (comp @ X536 @ X537)))) | ($true = ((X527 @ (comp @ X535 @ X536) @ (sub @ X538 @ X536) @ X537)))) | ($true = ((X528 @ (ap @ X526 @ X525)))) | ~ ($true = ((X528 @ X526))) | ~ (((X528 @ X525)) = $true) | ~ ! [X539 : term] : (((X528 @ X539)) = ((X527 @ id @ X539 @ id))))) & (axapp = $true) & ~ (lamnotap = $true)), inference(rectify,[],[f7])). thf(f10,plain,( ((pushprop_lem0_lthm = $true) <=> ! [X44 : (term > $o),X43 : subst,X42 : term] : ~ ! [X45 : (term > $o)] : ~ ! [X46 : term] : (((X45 @ X46)) = ((X44 @ (sub @ X46 @ (push @ X42 @ X43)))))) & ((hoasinduction_lem3aa = $true) <=> ! [X335 : term,X334 : (subst > term > subst > $o)] : (($true = ((X334 @ id @ (lam @ (sub @ X335 @ (push @ one @ sh))) @ id))) | ~ ! [X336 : (subst > term > term)] : (~ ! [X340 : term,X339 : subst,X338 : subst] : (((sub @ (X336 @ X338 @ X340) @ X339)) = ((X336 @ (comp @ X338 @ X339) @ (sub @ X340 @ X339)))) | (((X334 @ id @ (lam @ (X336 @ sh @ one)) @ id)) = $true) | ~ ! [X337 : term] : (($true != ((X334 @ id @ X337 @ id))) | (((X334 @ id @ (X336 @ id @ X337) @ id)) = $true))) | ~ ! [X341 : term] : ((((X334 @ id @ X341 @ id)) != $true) | ($true = ((X334 @ id @ (sub @ X335 @ (push @ X341 @ id)) @ id)))))) & (hoasinduction_lem1v2_gthm = $true) & (pushprop_lthm_orig = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & ((hoasinduction_lem3a = $true) <=> ! [X49 : term,X48 : (subst > term > subst > $o)] : (~ ! [X51 : (subst > term > term)] : (~ ! [X55 : term] : (($true != ((X48 @ id @ X55 @ id))) | (((X48 @ id @ (X51 @ id @ X55) @ id)) = $true)) | (((X48 @ id @ (lam @ (X51 @ sh @ one)) @ id)) = $true) | ~ ! [X53 : term,X54 : subst,X52 : subst] : (((X51 @ (comp @ X54 @ X52) @ (sub @ X53 @ X52))) = ((sub @ (X51 @ X54 @ X53) @ X52)))) | ~ ! [X50 : term] : (($true != ((X48 @ id @ X50 @ id))) | ($true = ((X48 @ id @ (sub @ X49 @ (push @ X50 @ id)) @ id)))) | ($true = ((X48 @ id @ (lam @ X49) @ id))))) & (! [X187 : term,X184 : term,X186 : term,X185 : term] : ((X186 = X187) | (((ap @ (sub @ X185 @ id) @ X186)) != ((ap @ (sub @ X184 @ id) @ X187)))) <=> (hoasapinj2 = $true)) & (! [X525 : term,X527 : (subst > term > subst > $o),X528 : (term > $o),X526 : term] : (($true = ((X528 @ (ap @ X526 @ X525)))) | ~ ! [X539 : term] : (((X528 @ X539)) = ((X527 @ id @ X539 @ id))) | (((X528 @ X525)) != $true) | ~ ! [X532 : subst,X533 : term,X531 : subst,X534 : subst] : (($true = ((X527 @ X532 @ X533 @ (comp @ X534 @ X531)))) | (((X527 @ (comp @ X532 @ X534) @ (sub @ X533 @ X534) @ X531)) != $true)) | ($true != ((X528 @ X526))) | ~ ! [X529 : term,X530 : term] : ((((X527 @ id @ (ap @ (sub @ X530 @ id) @ X529) @ id)) = $true) | ($true != ((X527 @ id @ X529 @ id))) | (((X527 @ id @ X530 @ id)) != $true)) | ~ ! [X537 : subst,X538 : term,X535 : subst,X536 : subst] : (($true != ((X527 @ X535 @ X538 @ (comp @ X536 @ X537)))) | ($true = ((X527 @ (comp @ X535 @ X536) @ (sub @ X538 @ X536) @ X537))))) <=> (hoasinduction_lem2v2 = $true)) & (hoaslamnotap_gthm = $true) & (hoasinduction_lem1_lthm = $true) & (ulamvar1 != $true) & (! [X521 : (subst > term > subst > $o)] : ~ ! [X522 : (term > $o)] : ~ ! [X523 : term] : (((X521 @ id @ X523 @ id)) = ((X522 @ X523))) <=> (hoasinduction_lem0_lthm = $true)) & (! [X244 : subst,X243 : (term > $o),X242 : term] : ~ ! [X245 : (term > $o)] : ~ ! [X246 : term] : (((X245 @ X246)) = ((X243 @ (sub @ X246 @ (push @ X242 @ X244))))) <=> (pushprop_lem0 = $true)) & (axvarshift = $true) & ((hoaslaminj_lthm = $true) <=> (! [X495 : subst,X494 : term] : (((sub @ one @ (push @ X494 @ X495))) = X494) => (! [X497 : subst,X496 : term] : (((comp @ sh @ (push @ X496 @ X497))) = X497) => (! [X498 : term,X499 : term] : ((((lam @ X498)) != ((lam @ X499))) | (X498 = X499)) => ! [X501 : subst,X502 : (subst > term > term),X500 : (subst > term > term),X503 : term] : (~ ! [X505 : subst,X504 : subst,X506 : term] : (((sub @ (X502 @ X505 @ X506) @ X504)) = ((X502 @ (comp @ X505 @ X504) @ (sub @ X506 @ X504)))) | ~ ! [X507 : term,X509 : subst,X508 : subst] : (((X500 @ (comp @ X509 @ X508) @ (sub @ X507 @ X508))) = ((sub @ (X500 @ X509 @ X507) @ X508))) | (((lam @ (X502 @ sh @ one))) != ((lam @ (X500 @ sh @ one)))) | (((X502 @ X501 @ X503)) = ((X500 @ X501 @ X503)))))))) & (hoasapnotvar_lthm = $true) & ((hoaslaminj = $true) <=> ! [X511 : term,X510 : subst,X512 : (subst > term > term),X513 : (subst > term > term)] : ((((X512 @ X510 @ X511)) = ((X513 @ X510 @ X511))) | (((lam @ (X512 @ sh @ one))) != ((lam @ (X513 @ sh @ one)))) | ~ ! [X519 : subst,X518 : term,X517 : subst] : (((X512 @ (comp @ X519 @ X517) @ (sub @ X518 @ X517))) = ((sub @ (X512 @ X519 @ X518) @ X517))) | ~ ! [X514 : term,X515 : subst,X516 : subst] : (((sub @ (X513 @ X515 @ X514) @ X516)) = ((X513 @ (comp @ X515 @ X516) @ (sub @ X514 @ X516)))))) & (induction2_gthm = $true) & ((induction2 = $true) <=> ! [X197 : (term > $o),X198 : term] : (~ ! [X201 : term] : ((((X197 @ (lam @ X201))) = $true) | ~ ! [X202 : term] : (($true != ((X197 @ X202))) | (((X197 @ (sub @ X201 @ (push @ X202 @ id)))) = $true))) | (((X197 @ X198)) = $true) | ~ ! [X199 : term,X200 : term] : ((((X197 @ (ap @ X200 @ X199))) = $true) | ($true != ((X197 @ X199))) | ($true != ((X197 @ X200)))) | ~ ! [X203 : term] : ((((var @ X203)) != $true) | ($true = ((X197 @ X203)))))) & (lamnotap != $true) & (((d2subst @ d_id)) = id) & (! [X204 : term,X205 : subst] : (((comp @ sh @ (push @ X204 @ X205))) = X205) <=> (axshiftcons = $true)) & (hoasinduction_gthm = $true) & ((! [X342 : (term > $o),X343 : term] : (~ ! [X348 : term] : ((((X342 @ X348)) = $true) | ($true != ((var @ X348)))) | ($true = ((X342 @ X343))) | ~ ! [X344 : term] : (~ ! [X345 : term] : (($true != ((X342 @ X345))) | ($true = ((X342 @ (sub @ X344 @ (push @ X345 @ id)))))) | (((X342 @ (lam @ X344))) = $true)) | ~ ! [X347 : term,X346 : term] : ((((X342 @ (ap @ X346 @ X347))) = $true) | (((X342 @ X347)) != $true) | ($true != ((X342 @ X346))))) => (! [X349 : term,X350 : (subst > term > subst > $o),X351 : term] : (~ ! [X353 : term,X355 : subst,X354 : subst,X352 : subst] : (($true != ((X350 @ (comp @ X352 @ X354) @ (sub @ X353 @ X354) @ X355))) | ($true = ((X350 @ X352 @ X353 @ (comp @ X354 @ X355))))) | (((X350 @ id @ X349 @ id)) != $true) | ($true = ((X350 @ id @ (ap @ X349 @ X351) @ id))) | ($true != ((X350 @ id @ X351 @ id))) | ~ ! [X361 : term,X360 : term] : (($true != ((X350 @ id @ X360 @ id))) | (((X350 @ id @ X361 @ id)) != $true) | (((X350 @ id @ (ap @ (sub @ X360 @ id) @ X361) @ id)) = $true)) | ~ ! [X358 : subst,X357 : subst,X359 : subst,X356 : term] : ((((X350 @ (comp @ X357 @ X358) @ (sub @ X356 @ X358) @ X359)) = $true) | (((X350 @ X357 @ X356 @ (comp @ X358 @ X359))) != $true))) => (! [X362 : term,X363 : (subst > term > subst > $o)] : (~ ! [X365 : subst,X366 : subst,X368 : term,X367 : subst] : (($true != ((X363 @ X365 @ X368 @ (comp @ X367 @ X366)))) | (((X363 @ (comp @ X365 @ X367) @ (sub @ X368 @ X367) @ X366)) = $true)) | ($true = ((X363 @ id @ (lam @ X362) @ id))) | ~ ! [X370 : subst,X369 : subst,X372 : term,X371 : subst] : (($true = ((X363 @ X370 @ X372 @ (comp @ X371 @ X369)))) | (((X363 @ (comp @ X370 @ X371) @ (sub @ X372 @ X371) @ X369)) != $true)) | ~ ! [X364 : term] : (($true = ((X363 @ id @ (sub @ X362 @ (push @ X364 @ id)) @ id))) | (((X363 @ id @ X364 @ id)) != $true)) | ~ ! [X373 : (subst > term > term)] : (($true = ((X363 @ id @ (lam @ (X373 @ sh @ one)) @ id))) | ~ ! [X377 : subst,X376 : subst,X375 : term] : (((sub @ (X373 @ X376 @ X375) @ X377)) = ((X373 @ (comp @ X376 @ X377) @ (sub @ X375 @ X377)))) | ~ ! [X374 : term] : (($true != ((X363 @ id @ X374 @ id))) | ($true = ((X363 @ id @ (X373 @ id @ X374) @ id)))))) => ! [X378 : (subst > term > subst > $o),X379 : term] : (($true = ((X378 @ id @ X379 @ id))) | ~ ! [X384 : (subst > term > term)] : (~ ! [X387 : term,X388 : subst,X386 : subst] : (((X384 @ (comp @ X388 @ X386) @ (sub @ X387 @ X386))) = ((sub @ (X384 @ X388 @ X387) @ X386))) | ($true = ((X378 @ id @ (lam @ (X384 @ sh @ one)) @ id))) | ~ ! [X385 : term] : ((((X378 @ id @ (X384 @ id @ X385) @ id)) = $true) | ($true != ((X378 @ id @ X385 @ id))))) | ~ ! [X390 : term,X389 : term] : (($true != ((X378 @ id @ X389 @ id))) | (((X378 @ id @ X390 @ id)) != $true) | (((X378 @ id @ (ap @ (sub @ X389 @ id) @ X390) @ id)) = $true)) | ~ ! [X395 : term,X392 : subst,X394 : subst,X393 : subst] : (($true != ((X378 @ (comp @ X393 @ X392) @ (sub @ X395 @ X392) @ X394))) | (((X378 @ X393 @ X395 @ (comp @ X392 @ X394))) = $true)) | ~ ! [X391 : term] : ((((X378 @ id @ X391 @ id)) = $true) | (((var @ (sub @ X391 @ id))) != $true)) | ~ ! [X381 : subst,X380 : term,X383 : subst,X382 : subst] : ((((X378 @ X381 @ X380 @ (comp @ X383 @ X382))) != $true) | (((X378 @ (comp @ X381 @ X383) @ (sub @ X380 @ X383) @ X382)) = $true)))))) <=> (hoasinduction_lthm = $true)) & (hoasinduction_lem2_gthm = $true) & (hoasinduction_lem3b_gthm = $true) & (axassoc = $true) & ! [X524 : d_subst] : (d_id = X524) & (hoasapnotvar_gthm = $true) & (lamnotvar = $true) & ((hoasinduction_lem3 = $true) <=> ! [X396 : term,X397 : (subst > term > subst > $o)] : ((((X397 @ id @ (lam @ X396) @ id)) = $true) | ~ ! [X408 : subst,X409 : term,X410 : subst,X407 : subst] : (($true != ((X397 @ (comp @ X410 @ X407) @ (sub @ X409 @ X407) @ X408))) | (((X397 @ X410 @ X409 @ (comp @ X407 @ X408))) = $true)) | ~ ! [X411 : term] : ((((X397 @ id @ X411 @ id)) != $true) | (((X397 @ id @ (sub @ X396 @ (push @ X411 @ id)) @ id)) = $true)) | ~ ! [X403 : subst,X404 : subst,X405 : subst,X406 : term] : ((((X397 @ X404 @ X406 @ (comp @ X405 @ X403))) != $true) | (((X397 @ (comp @ X404 @ X405) @ (sub @ X406 @ X405) @ X403)) = $true)) | ~ ! [X398 : (subst > term > term)] : (~ ! [X399 : term] : ((((X397 @ id @ X399 @ id)) != $true) | ($true = ((X397 @ id @ (X398 @ id @ X399) @ id)))) | ($true = ((X397 @ id @ (lam @ (X398 @ sh @ one)) @ id))) | ~ ! [X401 : subst,X400 : term,X402 : subst] : (((X398 @ (comp @ X401 @ X402) @ (sub @ X400 @ X402))) = ((sub @ (X398 @ X401 @ X400) @ X402)))))) & (! [X248 : term] : (((sub @ X248 @ id)) = X248) <=> (termmset = $true)) & (apnotvar = $true) & (! [X206 : (subst > term > subst > $o),X207 : term] : (~ ! [X210 : (subst > term > term)] : (($true = ((X206 @ id @ (lam @ (X210 @ sh @ one)) @ id))) | ~ ! [X211 : term] : (($true = ((X206 @ id @ (X210 @ id @ X211) @ id))) | (((X206 @ id @ X211 @ id)) != $true)) | ~ ! [X212 : subst,X213 : term,X214 : subst] : (((X210 @ (comp @ X214 @ X212) @ (sub @ X213 @ X212))) = ((sub @ (X210 @ X214 @ X213) @ X212)))) | (((X206 @ id @ X207 @ id)) = $true) | ~ ! [X208 : term,X209 : term] : (($true != ((X206 @ id @ X209 @ id))) | (((X206 @ id @ X208 @ id)) != $true) | ($true = ((X206 @ id @ (ap @ (sub @ X208 @ id) @ X209) @ id))))) <=> (hoasinduction_no_psi_cond = $true)) & (substmonoid_gthm = $true) & ((hoasinduction_lem3a_lthm = $true) <=> (! [X292 : term] : (((sub @ X292 @ id)) = X292) => (! [X294 : term,X293 : (subst > term > subst > $o)] : (~ ! [X295 : (subst > term > term)] : (~ ! [X296 : term] : (($true = ((X293 @ id @ (X295 @ id @ X296) @ id))) | ($true != ((X293 @ id @ X296 @ id)))) | ~ ! [X298 : subst,X297 : term,X299 : subst] : (((X295 @ (comp @ X299 @ X298) @ (sub @ X297 @ X298))) = ((sub @ (X295 @ X299 @ X297) @ X298))) | (((X293 @ id @ (lam @ (X295 @ sh @ one)) @ id)) = $true)) | ($true = ((X293 @ id @ (lam @ (sub @ X294 @ (push @ one @ sh))) @ id))) | ~ ! [X300 : term] : (($true = ((X293 @ id @ (sub @ X294 @ (push @ X300 @ id)) @ id))) | (((X293 @ id @ X300 @ id)) != $true))) => ! [X302 : (subst > term > subst > $o),X301 : term] : (~ ! [X303 : (subst > term > term)] : (($true = ((X302 @ id @ (lam @ (X303 @ sh @ one)) @ id))) | ~ ! [X307 : term] : ((((X302 @ id @ X307 @ id)) != $true) | ($true = ((X302 @ id @ (X303 @ id @ X307) @ id)))) | ~ ! [X305 : term,X304 : subst,X306 : subst] : (((sub @ (X303 @ X304 @ X305) @ X306)) = ((X303 @ (comp @ X304 @ X306) @ (sub @ X305 @ X306))))) | (((X302 @ id @ (lam @ X301) @ id)) = $true) | ~ ! [X308 : term] : (($true != ((X302 @ id @ X308 @ id))) | ($true = ((X302 @ id @ (sub @ X301 @ (push @ X308 @ id)) @ id)))))))) & ((pushprop_lem1v2 = $true) <=> ! [X311 : term,X309 : (term > $o),X310 : (term > $o),X312 : subst] : (~ ! [X313 : term] : (((X309 @ (sub @ X313 @ (push @ X311 @ X312)))) = ((X310 @ X313))) | ($true = ((X310 @ one))) | (((X309 @ X311)) != $true))) & ((apinj1 = $true) <=> ! [X3 : term,X1 : term,X0 : term,X2 : term] : ((((ap @ X1 @ X3)) != ((ap @ X0 @ X2))) | (X0 = X1))) & (pushprop_lem1_gthm = $true) & ((induction = $true) <=> ! [X65 : term,X64 : (term > $o)] : (~ ! [X68 : term] : (($true = ((X64 @ (lam @ X68)))) | (((X64 @ X68)) != $true)) | ~ ! [X69 : term] : ((((var @ X69)) != $true) | ($true = ((X64 @ X69)))) | ~ ! [X66 : term,X67 : term] : ((((X64 @ X67)) != $true) | (((X64 @ X66)) != $true) | (((X64 @ (ap @ X67 @ X66))) = $true)) | (((X64 @ X65)) = $true))) & (hoaslamnotvar_gthm = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (pushprop_lem2v2 = $true) & (pushprop_lem1v2_gthm = $true) & ((induction2lem_lthm = $true) <=> (! [X274 : subst,X273 : term] : (((sub @ one @ (push @ X273 @ X274))) = X273) => (! [X276 : subst,X275 : term] : (((comp @ sh @ (push @ X275 @ X276))) = X276) => (! [X277 : subst] : (((comp @ X277 @ id)) = X277) => (! [X278 : (term > $o),X279 : term] : (~ ! [X283 : term] : (($true = ((X278 @ X283))) | ($true != ((var @ X283)))) | ~ ! [X280 : term] : ((((X278 @ (lam @ X280))) = $true) | ($true != ((X278 @ X280)))) | (((X278 @ X279)) = $true) | ~ ! [X282 : term,X281 : term] : ((((X278 @ X282)) != $true) | (((X278 @ X281)) != $true) | (((X278 @ (ap @ X281 @ X282))) = $true))) => ! [X284 : (term > $o),X286 : term,X285 : subst] : (~ ! [X289 : term] : ((((X284 @ (sub @ X289 @ X285))) = $true) | (((var @ X289)) != $true)) | ~ ! [X290 : term] : ((((X284 @ (lam @ X290))) = $true) | ~ ! [X291 : term] : ((((X284 @ (sub @ X290 @ (push @ X291 @ id)))) = $true) | (((X284 @ X291)) != $true))) | ~ ! [X287 : term,X288 : term] : ((((X284 @ X287)) != $true) | (((X284 @ (ap @ X288 @ X287))) = $true) | (((X284 @ X288)) != $true)) | (((X284 @ (sub @ X286 @ X285))) = $true))))))) & ! [X318 : d_term] : (d_one = X318) & ((hoasapinj2_lthm = $true) <=> (! [X145 : term,X146 : term,X148 : term,X147 : term] : ((X147 = X148) | (((ap @ X145 @ X148)) != ((ap @ X146 @ X147)))) => ! [X149 : term,X152 : term,X151 : term,X150 : term] : ((X150 = X151) | (((ap @ (sub @ X149 @ id) @ X150)) != ((ap @ (sub @ X152 @ id) @ X151)))))) & ((axidr = $true) <=> ! [X183 : subst] : (((comp @ X183 @ id)) = X183)) & (axclos = $true) & (pushprop_lem3v2 = $true) & ((pushprop_lem1_lthm = $true) <=> (! [X486 : term,X485 : subst] : (((sub @ one @ (push @ X486 @ X485))) = X486) => (! [X488 : term,X487 : subst] : (((comp @ sh @ (push @ X488 @ X487))) = X487) => ! [X489 : term,X493 : (term > $o),X490 : term,X491 : (term > $o),X492 : subst] : ((((X493 @ X490)) != $true) | (((X491 @ (sub @ X490 @ (push @ X489 @ X492)))) = $true))))) & (! [X188 : term] : ~ ! [X189 : (subst > term > term)] : ~ ! [X190 : subst,X191 : term] : (((X189 @ X190 @ X191)) = ((sub @ X188 @ (push @ X191 @ X190)))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & (hoasinduction_lem3_gthm = $true) & ! [X228 : subst] : ? [X229 : d_subst] : (((d2subst @ X229)) = X228) & (hoaslaminj_gthm = $true) & ((apinj2 = $true) <=> ! [X236 : term,X238 : term,X237 : term,X239 : term] : ((X236 = X239) | (((ap @ X237 @ X236)) != ((ap @ X238 @ X239))))) & ((! [X153 : (subst > term > subst > $o)] : ~ ! [X154 : (term > $o)] : ~ ! [X155 : term] : (((X154 @ X155)) = ((X153 @ id @ X155 @ id))) => (! [X156 : term,X157 : (term > $o)] : (~ ! [X160 : term,X161 : term] : ((((X157 @ (ap @ X161 @ X160))) = $true) | ($true != ((X157 @ X161))) | (((X157 @ X160)) != $true)) | ~ ! [X162 : term] : (($true = ((X157 @ X162))) | (((var @ X162)) != $true)) | ($true = ((X157 @ X156))) | ~ ! [X158 : term] : (~ ! [X159 : term] : ((((X157 @ (sub @ X158 @ (push @ X159 @ id)))) = $true) | ($true != ((X157 @ X159)))) | (((X157 @ (lam @ X158))) = $true))) => (! [X163 : term] : (((sub @ X163 @ id)) = X163) => (! [X166 : term,X165 : (term > $o),X164 : (subst > term > subst > $o)] : (~ ! [X173 : term] : (($true != ((X165 @ X173))) | (((X165 @ (sub @ X166 @ (push @ X173 @ id)))) = $true)) | ~ ! [X168 : (subst > term > term)] : (~ ! [X169 : term] : (($true != ((X164 @ id @ X169 @ id))) | (((X164 @ id @ (X168 @ id @ X169) @ id)) = $true)) | ($true = ((X164 @ id @ (lam @ (X168 @ sh @ one)) @ id))) | ~ ! [X171 : term,X172 : subst,X170 : subst] : (((X168 @ (comp @ X172 @ X170) @ (sub @ X171 @ X170))) = ((sub @ (X168 @ X172 @ X171) @ X170)))) | ($true = ((X165 @ (lam @ X166)))) | ~ ! [X167 : term] : (((X165 @ X167)) = ((X164 @ id @ X167 @ id)))) => ! [X175 : term,X174 : (subst > term > subst > $o)] : (~ ! [X181 : term,X182 : term] : (($true != ((X174 @ id @ X181 @ id))) | ($true != ((X174 @ id @ X182 @ id))) | (((X174 @ id @ (ap @ (sub @ X182 @ id) @ X181) @ id)) = $true)) | ($true = ((X174 @ id @ X175 @ id))) | ~ ! [X176 : (subst > term > term)] : (~ ! [X178 : subst,X179 : term,X177 : subst] : (((sub @ (X176 @ X177 @ X179) @ X178)) = ((X176 @ (comp @ X177 @ X178) @ (sub @ X179 @ X178)))) | (((X174 @ id @ (lam @ (X176 @ sh @ one)) @ id)) = $true) | ~ ! [X180 : term] : ((((X174 @ id @ (X176 @ id @ X180) @ id)) = $true) | ($true != ((X174 @ id @ X180 @ id)))))))))) <=> (hoasinduction_no_psi_cond_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)))))))))))) & ((hoasinduction = $true) <=> ! [X23 : (subst > term > subst > $o),X22 : term] : (~ ! [X35 : (subst > term > term)] : (~ ! [X37 : subst,X38 : term,X36 : subst] : (((X35 @ (comp @ X37 @ X36) @ (sub @ X38 @ X36))) = ((sub @ (X35 @ X37 @ X38) @ X36))) | ~ ! [X39 : term] : ((((X23 @ id @ X39 @ id)) != $true) | ($true = ((X23 @ id @ (X35 @ id @ X39) @ id)))) | (((X23 @ id @ (lam @ (X35 @ sh @ one)) @ id)) = $true)) | ~ ! [X26 : subst,X29 : subst,X27 : subst,X28 : term] : ((((X23 @ (comp @ X27 @ X26) @ (sub @ X28 @ X26) @ X29)) = $true) | (((X23 @ X27 @ X28 @ (comp @ X26 @ X29))) != $true)) | ~ ! [X25 : term,X24 : term] : ((((X23 @ id @ X24 @ id)) != $true) | (((X23 @ id @ X25 @ id)) != $true) | (((X23 @ id @ (ap @ (sub @ X24 @ id) @ X25) @ id)) = $true)) | (((X23 @ id @ X22 @ id)) = $true) | ~ ! [X34 : term] : ((((X23 @ id @ X34 @ id)) = $true) | ($true != ((var @ (sub @ X34 @ id))))) | ~ ! [X33 : term,X32 : subst,X31 : subst,X30 : subst] : ((((X23 @ X30 @ X33 @ (comp @ X31 @ X32))) = $true) | ($true != ((X23 @ (comp @ X30 @ X31) @ (sub @ X33 @ X31) @ X32)))))) & ((! [X412 : term,X413 : subst] : (((sub @ one @ (push @ X412 @ X413))) = X412) => ! [X416 : subst,X415 : (term > $o),X414 : (term > $o),X417 : term] : ((((X415 @ one)) = $true) | ($true != ((X414 @ X417))) | ~ ! [X418 : term] : (((X415 @ X418)) = ((X414 @ (sub @ X418 @ (push @ X417 @ X416))))))) <=> (pushprop_lem1v2_lthm = $true)) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (((var @ (d2term @ d_one))) != $true) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X71 : term,X70 : (subst > term > subst > $o)] : (~ ! [X73 : (subst > term > term)] : (~ ! [X76 : term,X77 : subst,X75 : subst] : (((sub @ (X73 @ X75 @ X76) @ X77)) = ((X73 @ (comp @ X75 @ X77) @ (sub @ X76 @ X77)))) | ~ ! [X74 : term] : (($true = ((X70 @ id @ (X73 @ id @ X74) @ id))) | ($true != ((X70 @ id @ X74 @ id)))) | (((X70 @ id @ (lam @ (X73 @ sh @ one)) @ id)) = $true)) | (((X70 @ id @ (lam @ (sub @ X71 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X72 : term] : (($true = ((X70 @ id @ (sub @ X71 @ (push @ X72 @ id)) @ id))) | ($true != ((X70 @ id @ X72 @ id)))))) & (hoasinduction_lem2_lthm = $true) & (hoasapinj2_gthm = $true) & ((hoasinduction_lem3v2 = $true) <=> ! [X4 : term,X5 : (term > $o),X6 : (subst > term > subst > $o)] : (~ ! [X11 : term] : (($true != ((X5 @ X11))) | ($true = ((X5 @ (sub @ X4 @ (push @ X11 @ id)))))) | ~ ! [X10 : subst,X7 : subst,X8 : term,X9 : subst] : (($true != ((X6 @ (comp @ X10 @ X7) @ (sub @ X8 @ X7) @ X9))) | (((X6 @ X10 @ X8 @ (comp @ X7 @ X9))) = $true)) | ~ ! [X12 : (subst > term > term)] : ((((X6 @ id @ (lam @ (X12 @ sh @ one)) @ id)) = $true) | ~ ! [X16 : subst,X14 : term,X15 : subst] : (((sub @ (X12 @ X15 @ X14) @ X16)) = ((X12 @ (comp @ X15 @ X16) @ (sub @ X14 @ X16)))) | ~ ! [X13 : term] : ((((X6 @ id @ X13 @ id)) != $true) | (((X6 @ id @ (X12 @ id @ X13) @ id)) = $true))) | (((X5 @ (lam @ X4))) = $true) | ~ ! [X18 : subst,X20 : term,X19 : subst,X17 : subst] : ((((X6 @ X18 @ X20 @ (comp @ X19 @ X17))) != $true) | ($true = ((X6 @ (comp @ X18 @ X19) @ (sub @ X20 @ X19) @ X17)))) | ~ ! [X21 : term] : (((X6 @ id @ X21 @ id)) = ((X5 @ X21))))) & (pushprop_lthm = $true) & (hoaslamnotvar_lthm = $true) & ((induction2_lthm = $true) <=> (! [X84 : term] : (((sub @ X84 @ id)) = X84) => (! [X86 : subst,X85 : (term > $o),X87 : term] : (~ ! [X91 : term,X90 : term] : ((((X85 @ X91)) != $true) | (((X85 @ X90)) != $true) | (((X85 @ (ap @ X91 @ X90))) = $true)) | ~ ! [X88 : term] : (~ ! [X89 : term] : ((((X85 @ (sub @ X88 @ (push @ X89 @ id)))) = $true) | (((X85 @ X89)) != $true)) | (((X85 @ (lam @ X88))) = $true)) | (((X85 @ (sub @ X87 @ X86))) = $true) | ~ ! [X92 : term] : (($true != ((var @ X92))) | (((X85 @ (sub @ X92 @ X86))) = $true))) => ! [X93 : (term > $o),X94 : term] : (~ ! [X98 : term] : (~ ! [X99 : term] : ((((X93 @ X99)) != $true) | ($true = ((X93 @ (sub @ X98 @ (push @ X99 @ id)))))) | ($true = ((X93 @ (lam @ X98))))) | ~ ! [X97 : term] : ((((X93 @ X97)) = $true) | (((var @ X97)) != $true)) | (((X93 @ X94)) = $true) | ~ ! [X96 : term,X95 : term] : (($true != ((X93 @ X95))) | ($true = ((X93 @ (ap @ X96 @ X95)))) | (((X93 @ X96)) != $true)))))) & ((pushprop_lem1 = $true) <=> ! [X270 : term,X272 : subst,X271 : (term > $o),X268 : term,X269 : (term > $o)] : ((((X271 @ (sub @ X270 @ (push @ X268 @ X272)))) = $true) | (((X269 @ X270)) != $true))) & ((hoasinduction_lem2 = $true) <=> ! [X215 : (subst > term > subst > $o),X216 : term,X217 : term] : (~ ! [X223 : term,X222 : term] : ((((X215 @ id @ X222 @ id)) != $true) | ($true = ((X215 @ id @ (ap @ (sub @ X223 @ id) @ X222) @ id))) | (((X215 @ id @ X223 @ id)) != $true)) | ~ ! [X224 : term,X225 : subst,X227 : subst,X226 : subst] : (($true = ((X215 @ X227 @ X224 @ (comp @ X225 @ X226)))) | ($true != ((X215 @ (comp @ X227 @ X225) @ (sub @ X224 @ X225) @ X226)))) | (((X215 @ id @ X217 @ id)) != $true) | ($true = ((X215 @ id @ (ap @ X216 @ X217) @ id))) | ~ ! [X218 : subst,X219 : subst,X221 : term,X220 : subst] : (($true = ((X215 @ (comp @ X220 @ X218) @ (sub @ X221 @ X218) @ X219))) | ($true != ((X215 @ X220 @ X221 @ (comp @ X218 @ X219))))) | (((X215 @ id @ X216 @ id)) != $true))) & (((d2subst @ d_id)) = sh) & (((d2term @ d_one)) = one) & (! [X192 : (subst > term > subst > $o)] : ~ ! [X193 : (term > $o)] : ~ ! [X194 : term] : (((X192 @ id @ X194 @ id)) = ((X193 @ X194))) <=> (hoasinduction_lem0 = $true)) & ((! [X102 : subst] : (((comp @ id @ X102)) = X102) => (! [X103 : subst] : (((comp @ X103 @ id)) = X103) => (! [X105 : subst] : (((comp @ X105 @ id)) = X105) & ! [X104 : subst] : (((comp @ id @ X104)) = X104)))) <=> (substmonoid_lthm = $true)) & (axabs = $true) & (! [X78 : term] : ~ ! [X79 : (subst > term > term)] : ~ ! [X81 : subst,X80 : term] : (((X79 @ X81 @ X80)) = ((sub @ X78 @ (push @ X80 @ X81)))) <=> (hoasinduction_lem3v2_f = $true)) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & ((axscons = $true) <=> ! [X247 : subst] : (((push @ (sub @ one @ X247) @ (comp @ sh @ X247))) = X247)) & ((! [X419 : term] : (((sub @ X419 @ id)) = X419) => (! [X421 : term,X420 : (subst > term > subst > $o)] : (~ ! [X422 : (subst > term > term)] : ((((X420 @ id @ (lam @ (X422 @ sh @ one)) @ id)) = $true) | ~ ! [X424 : subst,X426 : term,X425 : subst] : (((sub @ (X422 @ X425 @ X426) @ X424)) = ((X422 @ (comp @ X425 @ X424) @ (sub @ X426 @ X424)))) | ~ ! [X423 : term] : (($true = ((X420 @ id @ (X422 @ id @ X423) @ id))) | (((X420 @ id @ X423 @ id)) != $true))) | ~ ! [X427 : term] : ((((X420 @ id @ (sub @ X421 @ (push @ X427 @ id)) @ id)) = $true) | (((X420 @ id @ X427 @ id)) != $true)) | ($true = ((X420 @ id @ (lam @ (sub @ X421 @ (push @ one @ sh))) @ id)))) => ! [X428 : (subst > term > subst > $o),X429 : term] : (~ ! [X435 : (subst > term > term)] : (~ ! [X436 : term] : (($true != ((X428 @ id @ X436 @ id))) | (((X428 @ id @ (X435 @ id @ X436) @ id)) = $true)) | ~ ! [X438 : subst,X439 : subst,X437 : term] : (((X435 @ (comp @ X438 @ X439) @ (sub @ X437 @ X439))) = ((sub @ (X435 @ X438 @ X437) @ X439))) | (((X428 @ id @ (lam @ (X435 @ sh @ one)) @ id)) = $true)) | ($true = ((X428 @ id @ (lam @ X429) @ id))) | ~ ! [X434 : term] : (($true != ((X428 @ id @ X434 @ id))) | ($true = ((X428 @ id @ (sub @ X429 @ (push @ X434 @ id)) @ id)))) | ~ ! [X433 : subst,X431 : term,X432 : subst,X430 : subst] : ((((X428 @ (comp @ X433 @ X432) @ (sub @ X431 @ X432) @ X430)) = $true) | (((X428 @ X433 @ X431 @ (comp @ X432 @ X430))) != $true)) | ~ ! [X443 : subst,X441 : subst,X442 : subst,X440 : term] : ((((X428 @ (comp @ X443 @ X442) @ (sub @ X440 @ X442) @ X441)) != $true) | (((X428 @ X443 @ X440 @ (comp @ X442 @ X441))) = $true))))) <=> (hoasinduction_lem3_lthm = $true)) & ! [X41 : d_subst,X40 : d_subst] : ((((d2subst @ X40)) = ((d2subst @ X41))) => (X40 = X41)) & ((axvarid = $true) <=> ! [X520 : term] : (((sub @ X520 @ id)) = X520)) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (ulamvarsh = $true) & (hoasinduction_lem3a_gthm = $true) & ((! [X259 : term] : (((sub @ X259 @ id)) = X259) => (! [X262 : term,X263 : term,X260 : term,X261 : term] : ((((ap @ X261 @ X263)) != ((ap @ X262 @ X260))) | (X261 = X262)) => ! [X265 : term,X266 : term,X267 : term,X264 : term] : ((((ap @ (sub @ X267 @ id) @ X265)) != ((ap @ (sub @ X266 @ id) @ X264))) | (X266 = X267)))) <=> (hoasapinj1_lthm = $true)) & (termmset_gthm = $true) & (! [X250 : (term > $o),X251 : (subst > term > subst > $o),X249 : term] : (~ ! [X253 : (subst > term > term)] : (~ ! [X257 : term] : (($true = ((X251 @ id @ (X253 @ id @ X257) @ id))) | ($true != ((X251 @ id @ X257 @ id)))) | ~ ! [X254 : subst,X255 : term,X256 : subst] : (((sub @ (X253 @ X254 @ X255) @ X256)) = ((X253 @ (comp @ X254 @ X256) @ (sub @ X255 @ X256)))) | ($true = ((X251 @ id @ (lam @ (X253 @ sh @ one)) @ id)))) | (((X250 @ (lam @ X249))) = $true) | ~ ! [X252 : term] : (((X250 @ X252)) = ((X251 @ id @ X252 @ id))) | ~ ! [X258 : term] : ((((X250 @ X258)) != $true) | (((X250 @ (sub @ X249 @ (push @ X258 @ id)))) = $true))) <=> (hoasinduction_lem3v2a = $true)) & ((hoaslamnotap = $true) <=> ! [X141 : (subst > term > term),X140 : term,X139 : term] : (~ ! [X143 : subst,X142 : term,X144 : subst] : (((X141 @ (comp @ X143 @ X144) @ (sub @ X142 @ X144))) = ((sub @ (X141 @ X143 @ X142) @ X144))) | (((lam @ (X141 @ sh @ one))) != ((ap @ (sub @ X140 @ id) @ X139))))) & ! [X106 : term] : ? [X107 : d_term] : (((d2term @ X107)) = X106) & (! [X58 : (term > $o),X56 : term,X57 : subst] : (~ ! [X61 : term] : ((((var @ X61)) != $true) | ($true = ((X58 @ (sub @ X61 @ X57))))) | ~ ! [X62 : term] : (($true = ((X58 @ (lam @ X62)))) | ~ ! [X63 : term] : (($true != ((X58 @ X63))) | (((X58 @ (sub @ X62 @ (push @ X63 @ id)))) = $true))) | ~ ! [X60 : term,X59 : term] : ((((X58 @ X60)) != $true) | (((X58 @ (ap @ X60 @ X59))) = $true) | (((X58 @ X59)) != $true)) | (((X58 @ (sub @ X56 @ X57))) = $true)) <=> (induction2lem = $true)) & (pushprop_gthm = $true) & (hoasinduction_lem3v2_gthm = $true) & (hoasinduction_lem1v2 = $true) & ((! [X444 : term] : (((sub @ X444 @ id)) = X444) => ! [X445 : term] : (((sub @ X445 @ id)) = X445)) <=> (termmset_lthm = $true)) & (hoasinduction_lem1 = $true) & ((! [X231 : subst] : (((comp @ X231 @ id)) = X231) & ! [X230 : subst] : (((comp @ id @ X230)) = X230)) <=> (substmonoid = $true)) & (induction2lem_gthm = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (! [X240 : term] : ~ ! [X241 : (subst > term > term)] : (((sub @ X240 @ (push @ one @ sh))) != ((X241 @ sh @ one))) <=> (hoasinduction_lem3b = $true)) & (pushprop_lem0_gthm = $true) & (hoasinduction_lem1_gthm = $true) & (pushprop = $true) & ((! [X108 : term] : (((sub @ X108 @ id)) = X108) => ! [X110 : (term > $o),X111 : term,X109 : (subst > term > subst > $o)] : (~ ! [X122 : term] : (($true = ((X110 @ (sub @ X111 @ (push @ X122 @ id))))) | (((X110 @ X122)) != $true)) | ~ ! [X112 : (subst > term > term)] : ((((X109 @ id @ (lam @ (X112 @ sh @ one)) @ id)) = $true) | ~ ! [X114 : subst,X115 : term,X116 : subst] : (((sub @ (X112 @ X116 @ X115) @ X114)) = ((X112 @ (comp @ X116 @ X114) @ (sub @ X115 @ X114)))) | ~ ! [X113 : term] : (($true != ((X109 @ id @ X113 @ id))) | (((X109 @ id @ (X112 @ id @ X113) @ id)) = $true))) | ~ ! [X126 : subst,X124 : subst,X125 : term,X123 : subst] : (($true = ((X109 @ X124 @ X125 @ (comp @ X126 @ X123)))) | ($true != ((X109 @ (comp @ X124 @ X126) @ (sub @ X125 @ X126) @ X123)))) | ($true = ((X110 @ (lam @ X111)))) | ~ ! [X117 : subst,X118 : term,X119 : subst,X120 : subst] : ((((X109 @ (comp @ X120 @ X119) @ (sub @ X118 @ X119) @ X117)) = $true) | (((X109 @ X120 @ X118 @ (comp @ X119 @ X117))) != $true)) | ~ ! [X121 : term] : (((X110 @ X121)) = ((X109 @ id @ X121 @ id))))) <=> (hoasinduction_lem3v2_lthm = $true)) & ((hoasinduction_lthm_3 = $true) <=> (! [X446 : (subst > term > subst > $o)] : ~ ! [X447 : (term > $o)] : ~ ! [X448 : term] : (((X447 @ X448)) = ((X446 @ id @ X448 @ id))) => (! [X449 : term,X450 : (term > $o)] : (~ ! [X455 : term] : ((((var @ X455)) != $true) | ($true = ((X450 @ X455)))) | ~ ! [X452 : term,X451 : term] : (($true = ((X450 @ (ap @ X452 @ X451)))) | ($true != ((X450 @ X451))) | ($true != ((X450 @ X452)))) | ($true = ((X450 @ X449))) | ~ ! [X453 : term] : ((((X450 @ (lam @ X453))) = $true) | ~ ! [X454 : term] : (($true != ((X450 @ X454))) | (((X450 @ (sub @ X453 @ (push @ X454 @ id)))) = $true)))) => (! [X456 : term] : (((sub @ X456 @ id)) = X456) => (! [X457 : (subst > term > subst > $o),X458 : term,X459 : (term > $o)] : (~ ! [X462 : (subst > term > term)] : (~ ! [X465 : term,X464 : subst,X463 : subst] : (((X462 @ (comp @ X463 @ X464) @ (sub @ X465 @ X464))) = ((sub @ (X462 @ X463 @ X465) @ X464))) | ~ ! [X466 : term] : (($true = ((X457 @ id @ (X462 @ id @ X466) @ id))) | (((X457 @ id @ X466 @ id)) != $true)) | (((X457 @ id @ (lam @ (X462 @ sh @ one)) @ id)) = $true)) | (((X459 @ (lam @ X458))) = $true) | ~ ! [X461 : term] : (($true = ((X459 @ (sub @ X458 @ (push @ X461 @ id))))) | ($true != ((X459 @ X461)))) | ~ ! [X460 : term] : (((X459 @ X460)) = ((X457 @ id @ X460 @ id)))) => ! [X468 : term,X467 : (subst > term > subst > $o)] : (~ ! [X471 : term] : ((((X467 @ id @ X471 @ id)) = $true) | (((var @ (sub @ X471 @ id))) != $true)) | ~ ! [X476 : (subst > term > term)] : (~ ! [X477 : term] : ((((X467 @ id @ (X476 @ id @ X477) @ id)) = $true) | ($true != ((X467 @ id @ X477 @ id)))) | ~ ! [X479 : term,X480 : subst,X478 : subst] : (((X476 @ (comp @ X478 @ X480) @ (sub @ X479 @ X480))) = ((sub @ (X476 @ X478 @ X479) @ X480))) | (((X467 @ id @ (lam @ (X476 @ sh @ one)) @ id)) = $true)) | (((X467 @ id @ X468 @ id)) = $true) | ~ ! [X483 : subst,X481 : subst,X484 : subst,X482 : term] : (($true = ((X467 @ X484 @ X482 @ (comp @ X481 @ X483)))) | (((X467 @ (comp @ X484 @ X481) @ (sub @ X482 @ X481) @ X483)) != $true)) | ~ ! [X470 : term,X469 : term] : ((((X467 @ id @ X470 @ id)) != $true) | (((X467 @ id @ X469 @ id)) != $true) | (((X467 @ id @ (ap @ (sub @ X469 @ id) @ X470) @ id)) = $true)) | ~ ! [X473 : subst,X474 : subst,X475 : subst,X472 : term] : ((((X467 @ (comp @ X474 @ X473) @ (sub @ X472 @ X473) @ X475)) = $true) | (((X467 @ X474 @ X472 @ (comp @ X473 @ X475))) != $true)))))))) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X82 : term] : ~ ! [X83 : (subst > term > term)] : (((sub @ X82 @ (push @ one @ sh))) != ((X83 @ sh @ one)))) & (pushprop_lem2v2_lthm = $true) & (ulamvarind = $true) & (axmap = $true) & (hoasapnotvar = $true) & ((laminj = $true) <=> ! [X233 : term,X232 : term] : ((((lam @ X232)) != ((lam @ X233))) | (X232 = X233))) & ((axidl = $true) <=> ! [X47 : subst] : (((comp @ id @ X47)) = X47)) & ((! [X319 : term] : ~ ! [X320 : (subst > term > term)] : ~ ! [X322 : term,X321 : subst] : (((sub @ X319 @ (push @ X322 @ X321))) = ((X320 @ X321 @ X322))) => (! [X323 : term] : (((sub @ X323 @ id)) = X323) => ! [X326 : term,X324 : (term > $o),X325 : (subst > term > subst > $o)] : (~ ! [X329 : (subst > term > term)] : (($true = ((X325 @ id @ (lam @ (X329 @ sh @ one)) @ id))) | ~ ! [X333 : term] : (($true = ((X325 @ id @ (X329 @ id @ X333) @ id))) | (((X325 @ id @ X333 @ id)) != $true)) | ~ ! [X332 : subst,X330 : subst,X331 : term] : (((sub @ (X329 @ X330 @ X331) @ X332)) = ((X329 @ (comp @ X330 @ X332) @ (sub @ X331 @ X332))))) | ~ ! [X328 : term] : (((X325 @ id @ X328 @ id)) = ((X324 @ X328))) | ~ ! [X327 : term] : (($true != ((X324 @ X327))) | (((X324 @ (sub @ X326 @ (push @ X327 @ id)))) = $true)) | ($true = ((X324 @ (lam @ X326))))))) <=> (hoasinduction_lem3v2a_lthm = $true)) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (pushprop_lem2v2_gthm = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoasapinj1_gthm = $true) & ((axvarcons = $true) <=> ! [X234 : subst,X235 : term] : (((sub @ one @ (push @ X235 @ X234))) = X235)) & (! [X195 : term,X196 : term] : ((((sub @ X196 @ sh)) != ((sub @ X195 @ sh))) | (X195 = X196)) <=> (shinj = $true)) & (hoaslamnotap_lthm = $true) & ((hoasinduction_lem3aaa = $true) <=> ! [X127 : term,X128 : (subst > term > subst > $o)] : (~ ! [X131 : (subst > term > term),X130 : term] : (~ ! [X138 : term] : (($true != ((X128 @ id @ X138 @ id))) | (((X128 @ id @ (X131 @ id @ X138) @ id)) = $true)) | ~ ! [X132 : term,X133 : subst,X134 : subst] : (((sub @ (X131 @ X133 @ X132) @ X134)) = ((sub @ (sub @ X130 @ (push @ X132 @ X133)) @ X134))) | ~ ! [X137 : term,X136 : subst,X135 : subst] : (((X131 @ (comp @ X136 @ X135) @ (sub @ X137 @ X135))) = ((sub @ X130 @ (push @ (sub @ X137 @ X135) @ (comp @ X136 @ X135))))) | (((X128 @ id @ (lam @ (X131 @ sh @ one)) @ id)) = $true)) | (((X128 @ id @ (lam @ (sub @ X127 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X129 : term] : ((((X128 @ id @ X129 @ id)) != $true) | ($true = ((X128 @ id @ (sub @ X127 @ (push @ X129 @ id)) @ id)))))) & (hoasinduction_lem2v2_gthm = $true) & (axapp = $true) & (pushprop_lem3v2_lthm = $true) & (hoaslamnotvar = $true) & ((hoasapinj1 = $true) <=> ! [X314 : term,X317 : term,X316 : term,X315 : term] : ((X314 = X316) | (((ap @ (sub @ X316 @ id) @ X317)) != ((ap @ (sub @ X314 @ id) @ X315))))) & ! [X101 : d_term,X100 : d_term] : ((((d2term @ X101)) = ((d2term @ X100))) => (X100 = X101))), inference(flattening,[],[f9])). thf(f11,plain,( (hoasinduction_lem3 = $true) <~> ! [X0 : (subst > term > subst > $o)] : (((! [X14 : term] : ((((X0 @ id @ (lam @ X14) @ id)) = $true) | ? [X15 : term] : (($true != ((X0 @ id @ (sub @ X14 @ (push @ X15 @ id)) @ id))) & (((X0 @ id @ X15 @ id)) = $true))) | ? [X9 : (subst > term > term)] : ((($true != ((X0 @ id @ (hoaslam @ id @ (^[Y0 : subst]: ((^[Y1 : term]: (X9 @ Y0 @ Y1))))) @ id))) & ! [X13 : term] : ((((X0 @ id @ X13 @ id)) != $true) | ($true = ((X0 @ id @ (X9 @ id @ X13) @ id))))) & ! [X12 : subst,X10 : term,X11 : subst] : (((X9 @ (comp @ X12 @ X11) @ (sub @ X10 @ X11))) = ((sub @ (X9 @ X12 @ X10) @ X11))))) | ? [X6 : subst,X5 : subst,X8 : subst,X7 : term] : ((((X0 @ (comp @ X5 @ X6) @ (sub @ X7 @ X6) @ X8)) = $true) & (((X0 @ X5 @ X7 @ (comp @ X6 @ X8))) != $true))) | ? [X1 : subst,X3 : subst,X2 : term,X4 : subst] : ((((X0 @ (comp @ X3 @ X1) @ (sub @ X2 @ X1) @ X4)) != $true) & ($true = ((X0 @ X3 @ X2 @ (comp @ X1 @ X4))))))), inference(ennf_transformation,[],[f8])). thf(f12,plain,( ! [X0 : (subst > term > subst > $o)] : (! [X14 : term] : ((((X0 @ id @ (lam @ X14) @ id)) = $true) | ? [X15 : term] : (($true != ((X0 @ id @ (sub @ X14 @ (push @ X15 @ id)) @ id))) & (((X0 @ id @ X15 @ id)) = $true))) | ? [X6 : subst,X5 : subst,X8 : subst,X7 : term] : ((((X0 @ (comp @ X5 @ X6) @ (sub @ X7 @ X6) @ X8)) = $true) & (((X0 @ X5 @ X7 @ (comp @ X6 @ X8))) != $true)) | ? [X9 : (subst > term > term)] : (! [X13 : term] : ((((X0 @ id @ X13 @ id)) != $true) | ($true = ((X0 @ id @ (X9 @ id @ X13) @ id)))) & ($true != ((X0 @ id @ (hoaslam @ id @ (^[Y0 : subst]: ((^[Y1 : term]: (X9 @ Y0 @ Y1))))) @ id))) & ! [X12 : subst,X10 : term,X11 : subst] : (((X9 @ (comp @ X12 @ X11) @ (sub @ X10 @ X11))) = ((sub @ (X9 @ X12 @ X10) @ X11)))) | ? [X1 : subst,X3 : subst,X2 : term,X4 : subst] : ((((X0 @ (comp @ X3 @ X1) @ (sub @ X2 @ X1) @ X4)) != $true) & ($true = ((X0 @ X3 @ X2 @ (comp @ X1 @ X4)))))) <~> (hoasinduction_lem3 = $true)), inference(flattening,[],[f11])). thf(f13,plain,( ((pushprop_lem0_lthm = $true) <=> ! [X43 : subst,X44 : (term > $o),X42 : term] : ? [X45 : (term > $o)] : ! [X46 : term] : (((X45 @ X46)) = ((X44 @ (sub @ X46 @ (push @ X42 @ X43)))))) & (! [X334 : (subst > term > subst > $o),X335 : term] : (($true = ((X334 @ id @ (lam @ (sub @ X335 @ (push @ one @ sh))) @ id))) | ? [X341 : term] : (($true != ((X334 @ id @ (sub @ X335 @ (push @ X341 @ id)) @ id))) & (((X334 @ id @ X341 @ id)) = $true)) | ? [X336 : (subst > term > term)] : (! [X337 : term] : (($true != ((X334 @ id @ X337 @ id))) | (((X334 @ id @ (X336 @ id @ X337) @ id)) = $true)) & ! [X340 : term,X339 : subst,X338 : subst] : (((sub @ (X336 @ X338 @ X340) @ X339)) = ((X336 @ (comp @ X338 @ X339) @ (sub @ X340 @ X339)))) & (((X334 @ id @ (lam @ (X336 @ sh @ one)) @ id)) != $true))) <=> (hoasinduction_lem3aa = $true)) & (hoasinduction_lem1v2_gthm = $true) & (pushprop_lthm_orig = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (! [X49 : term,X48 : (subst > term > subst > $o)] : (($true = ((X48 @ id @ (lam @ X49) @ id))) | ? [X51 : (subst > term > term)] : ((((X48 @ id @ (lam @ (X51 @ sh @ one)) @ id)) != $true) & ! [X53 : term,X54 : subst,X52 : subst] : (((X51 @ (comp @ X54 @ X52) @ (sub @ X53 @ X52))) = ((sub @ (X51 @ X54 @ X53) @ X52))) & ! [X55 : term] : (($true != ((X48 @ id @ X55 @ id))) | (((X48 @ id @ (X51 @ id @ X55) @ id)) = $true))) | ? [X50 : term] : (($true != ((X48 @ id @ (sub @ X49 @ (push @ X50 @ id)) @ id))) & ($true = ((X48 @ id @ X50 @ id))))) <=> (hoasinduction_lem3a = $true)) & (! [X187 : term,X184 : term,X186 : term,X185 : term] : ((X186 = X187) | (((ap @ (sub @ X185 @ id) @ X186)) != ((ap @ (sub @ X184 @ id) @ X187)))) <=> (hoasapinj2 = $true)) & (! [X528 : (term > $o),X525 : term,X527 : (subst > term > subst > $o),X526 : term] : (? [X536 : subst,X537 : subst,X535 : subst,X538 : term] : (($true = ((X527 @ X535 @ X538 @ (comp @ X536 @ X537)))) & ($true != ((X527 @ (comp @ X535 @ X536) @ (sub @ X538 @ X536) @ X537)))) | ($true != ((X528 @ X526))) | ($true = ((X528 @ (ap @ X526 @ X525)))) | ? [X529 : term,X530 : term] : ((((X527 @ id @ (ap @ (sub @ X530 @ id) @ X529) @ id)) != $true) & (((X527 @ id @ X530 @ id)) = $true) & ($true = ((X527 @ id @ X529 @ id)))) | (((X528 @ X525)) != $true) | ? [X539 : term] : (((X528 @ X539)) != ((X527 @ id @ X539 @ id))) | ? [X532 : subst,X533 : term,X531 : subst,X534 : subst] : ((((X527 @ (comp @ X532 @ X534) @ (sub @ X533 @ X534) @ X531)) = $true) & ($true != ((X527 @ X532 @ X533 @ (comp @ X534 @ X531)))))) <=> (hoasinduction_lem2v2 = $true)) & (hoaslamnotap_gthm = $true) & (hoasinduction_lem1_lthm = $true) & (ulamvar1 != $true) & ((hoasinduction_lem0_lthm = $true) <=> ! [X521 : (subst > term > subst > $o)] : ? [X522 : (term > $o)] : ! [X523 : term] : (((X521 @ id @ X523 @ id)) = ((X522 @ X523)))) & (! [X244 : subst,X243 : (term > $o),X242 : term] : ? [X245 : (term > $o)] : ! [X246 : term] : (((X245 @ X246)) = ((X243 @ (sub @ X246 @ (push @ X242 @ X244))))) <=> (pushprop_lem0 = $true)) & (axvarshift = $true) & ((hoaslaminj_lthm = $true) <=> (((! [X501 : subst,X502 : (subst > term > term),X500 : (subst > term > term),X503 : term] : ((((X502 @ X501 @ X503)) = ((X500 @ X501 @ X503))) | (((lam @ (X502 @ sh @ one))) != ((lam @ (X500 @ sh @ one)))) | ? [X508 : subst,X507 : term,X509 : subst] : (((X500 @ (comp @ X509 @ X508) @ (sub @ X507 @ X508))) != ((sub @ (X500 @ X509 @ X507) @ X508))) | ? [X506 : term,X505 : subst,X504 : subst] : (((sub @ (X502 @ X505 @ X506) @ X504)) != ((X502 @ (comp @ X505 @ X504) @ (sub @ X506 @ X504))))) | ? [X499 : term,X498 : term] : ((((lam @ X498)) = ((lam @ X499))) & (X498 != X499))) | ? [X496 : term,X497 : subst] : (((comp @ sh @ (push @ X496 @ X497))) != X497)) | ? [X495 : subst,X494 : term] : (((sub @ one @ (push @ X494 @ X495))) != X494))) & (hoasapnotvar_lthm = $true) & ((hoaslaminj = $true) <=> ! [X511 : term,X512 : (subst > term > term),X513 : (subst > term > term),X510 : subst] : (? [X518 : term,X519 : subst,X517 : subst] : (((X512 @ (comp @ X519 @ X517) @ (sub @ X518 @ X517))) != ((sub @ (X512 @ X519 @ X518) @ X517))) | ? [X515 : subst,X516 : subst,X514 : term] : (((sub @ (X513 @ X515 @ X514) @ X516)) != ((X513 @ (comp @ X515 @ X516) @ (sub @ X514 @ X516)))) | (((lam @ (X512 @ sh @ one))) != ((lam @ (X513 @ sh @ one)))) | (((X512 @ X510 @ X511)) = ((X513 @ X510 @ X511))))) & (induction2_gthm = $true) & (! [X197 : (term > $o),X198 : term] : (? [X200 : term,X199 : term] : (($true = ((X197 @ X200))) & ($true = ((X197 @ X199))) & (((X197 @ (ap @ X200 @ X199))) != $true)) | (((X197 @ X198)) = $true) | ? [X201 : term] : (! [X202 : term] : (($true != ((X197 @ X202))) | (((X197 @ (sub @ X201 @ (push @ X202 @ id)))) = $true)) & (((X197 @ (lam @ X201))) != $true)) | ? [X203 : term] : (($true != ((X197 @ X203))) & (((var @ X203)) = $true))) <=> (induction2 = $true)) & (lamnotap != $true) & (((d2subst @ d_id)) = id) & (! [X204 : term,X205 : subst] : (((comp @ sh @ (push @ X204 @ X205))) = X205) <=> (axshiftcons = $true)) & (hoasinduction_gthm = $true) & ((((! [X379 : term,X378 : (subst > term > subst > $o)] : (? [X392 : subst,X395 : term,X394 : subst,X393 : subst] : (($true = ((X378 @ (comp @ X393 @ X392) @ (sub @ X395 @ X392) @ X394))) & (((X378 @ X393 @ X395 @ (comp @ X392 @ X394))) != $true)) | ($true = ((X378 @ id @ X379 @ id))) | ? [X384 : (subst > term > term)] : (($true != ((X378 @ id @ (lam @ (X384 @ sh @ one)) @ id))) & ! [X385 : term] : ((((X378 @ id @ (X384 @ id @ X385) @ id)) = $true) | ($true != ((X378 @ id @ X385 @ id)))) & ! [X387 : term,X388 : subst,X386 : subst] : (((X384 @ (comp @ X388 @ X386) @ (sub @ X387 @ X386))) = ((sub @ (X384 @ X388 @ X387) @ X386)))) | ? [X390 : term,X389 : term] : ((((X378 @ id @ X390 @ id)) = $true) & ($true = ((X378 @ id @ X389 @ id))) & (((X378 @ id @ (ap @ (sub @ X389 @ id) @ X390) @ id)) != $true)) | ? [X391 : term] : ((((var @ (sub @ X391 @ id))) = $true) & (((X378 @ id @ X391 @ id)) != $true)) | ? [X381 : subst,X383 : subst,X382 : subst,X380 : term] : ((((X378 @ X381 @ X380 @ (comp @ X383 @ X382))) = $true) & (((X378 @ (comp @ X381 @ X383) @ (sub @ X380 @ X383) @ X382)) != $true))) | ? [X363 : (subst > term > subst > $o),X362 : term] : (! [X364 : term] : (($true = ((X363 @ id @ (sub @ X362 @ (push @ X364 @ id)) @ id))) | (((X363 @ id @ X364 @ id)) != $true)) & ! [X370 : subst,X369 : subst,X372 : term,X371 : subst] : (($true = ((X363 @ X370 @ X372 @ (comp @ X371 @ X369)))) | (((X363 @ (comp @ X370 @ X371) @ (sub @ X372 @ X371) @ X369)) != $true)) & ($true != ((X363 @ id @ (lam @ X362) @ id))) & ! [X373 : (subst > term > term)] : (? [X374 : term] : (($true = ((X363 @ id @ X374 @ id))) & ($true != ((X363 @ id @ (X373 @ id @ X374) @ id)))) | ? [X377 : subst,X376 : subst,X375 : term] : (((sub @ (X373 @ X376 @ X375) @ X377)) != ((X373 @ (comp @ X376 @ X377) @ (sub @ X375 @ X377)))) | ($true = ((X363 @ id @ (lam @ (X373 @ sh @ one)) @ id)))) & ! [X365 : subst,X366 : subst,X368 : term,X367 : subst] : (($true != ((X363 @ X365 @ X368 @ (comp @ X367 @ X366)))) | (((X363 @ (comp @ X365 @ X367) @ (sub @ X368 @ X367) @ X366)) = $true)))) | ? [X351 : term,X350 : (subst > term > subst > $o),X349 : term] : (! [X361 : term,X360 : term] : (($true != ((X350 @ id @ X360 @ id))) | (((X350 @ id @ X361 @ id)) != $true) | (((X350 @ id @ (ap @ (sub @ X360 @ id) @ X361) @ id)) = $true)) & ! [X353 : term,X355 : subst,X354 : subst,X352 : subst] : (($true != ((X350 @ (comp @ X352 @ X354) @ (sub @ X353 @ X354) @ X355))) | ($true = ((X350 @ X352 @ X353 @ (comp @ X354 @ X355))))) & (((X350 @ id @ X349 @ id)) = $true) & ! [X358 : subst,X357 : subst,X359 : subst,X356 : term] : ((((X350 @ (comp @ X357 @ X358) @ (sub @ X356 @ X358) @ X359)) = $true) | (((X350 @ X357 @ X356 @ (comp @ X358 @ X359))) != $true)) & ($true = ((X350 @ id @ X351 @ id))) & ($true != ((X350 @ id @ (ap @ X349 @ X351) @ id))))) | ? [X343 : term,X342 : (term > $o)] : (($true != ((X342 @ X343))) & ! [X344 : term] : (? [X345 : term] : (($true = ((X342 @ X345))) & ($true != ((X342 @ (sub @ X344 @ (push @ X345 @ id)))))) | (((X342 @ (lam @ X344))) = $true)) & ! [X348 : term] : ((((X342 @ X348)) = $true) | ($true != ((var @ X348)))) & ! [X347 : term,X346 : term] : ((((X342 @ (ap @ X346 @ X347))) = $true) | (((X342 @ X347)) != $true) | ($true != ((X342 @ X346)))))) <=> (hoasinduction_lthm = $true)) & (hoasinduction_lem2_gthm = $true) & (hoasinduction_lem3b_gthm = $true) & (axassoc = $true) & ! [X524 : d_subst] : (d_id = X524) & (hoasapnotvar_gthm = $true) & (lamnotvar = $true) & (! [X397 : (subst > term > subst > $o),X396 : term] : (? [X403 : subst,X405 : subst,X404 : subst,X406 : term] : ((((X397 @ X404 @ X406 @ (comp @ X405 @ X403))) = $true) & (((X397 @ (comp @ X404 @ X405) @ (sub @ X406 @ X405) @ X403)) != $true)) | ? [X411 : term] : ((((X397 @ id @ (sub @ X396 @ (push @ X411 @ id)) @ id)) != $true) & (((X397 @ id @ X411 @ id)) = $true)) | ? [X408 : subst,X409 : term,X410 : subst,X407 : subst] : ((((X397 @ X410 @ X409 @ (comp @ X407 @ X408))) != $true) & ($true = ((X397 @ (comp @ X410 @ X407) @ (sub @ X409 @ X407) @ X408)))) | (((X397 @ id @ (lam @ X396) @ id)) = $true) | ? [X398 : (subst > term > term)] : (! [X399 : term] : ((((X397 @ id @ X399 @ id)) != $true) | ($true = ((X397 @ id @ (X398 @ id @ X399) @ id)))) & ($true != ((X397 @ id @ (lam @ (X398 @ sh @ one)) @ id))) & ! [X401 : subst,X400 : term,X402 : subst] : (((X398 @ (comp @ X401 @ X402) @ (sub @ X400 @ X402))) = ((sub @ (X398 @ X401 @ X400) @ X402))))) <=> (hoasinduction_lem3 = $true)) & (! [X248 : term] : (((sub @ X248 @ id)) = X248) <=> (termmset = $true)) & (apnotvar = $true) & ((hoasinduction_no_psi_cond = $true) <=> ! [X206 : (subst > term > subst > $o),X207 : term] : (? [X208 : term,X209 : term] : (($true = ((X206 @ id @ X209 @ id))) & ($true != ((X206 @ id @ (ap @ (sub @ X208 @ id) @ X209) @ id))) & (((X206 @ id @ X208 @ id)) = $true)) | ? [X210 : (subst > term > term)] : (! [X212 : subst,X213 : term,X214 : subst] : (((X210 @ (comp @ X214 @ X212) @ (sub @ X213 @ X212))) = ((sub @ (X210 @ X214 @ X213) @ X212))) & ($true != ((X206 @ id @ (lam @ (X210 @ sh @ one)) @ id))) & ! [X211 : term] : (($true = ((X206 @ id @ (X210 @ id @ X211) @ id))) | (((X206 @ id @ X211 @ id)) != $true))) | (((X206 @ id @ X207 @ id)) = $true))) & (substmonoid_gthm = $true) & ((hoasinduction_lem3a_lthm = $true) <=> ((! [X301 : term,X302 : (subst > term > subst > $o)] : ((((X302 @ id @ (lam @ X301) @ id)) = $true) | ? [X303 : (subst > term > term)] : (! [X307 : term] : ((((X302 @ id @ X307 @ id)) != $true) | ($true = ((X302 @ id @ (X303 @ id @ X307) @ id)))) & ($true != ((X302 @ id @ (lam @ (X303 @ sh @ one)) @ id))) & ! [X305 : term,X304 : subst,X306 : subst] : (((sub @ (X303 @ X304 @ X305) @ X306)) = ((X303 @ (comp @ X304 @ X306) @ (sub @ X305 @ X306))))) | ? [X308 : term] : (($true = ((X302 @ id @ X308 @ id))) & ($true != ((X302 @ id @ (sub @ X301 @ (push @ X308 @ id)) @ id))))) | ? [X293 : (subst > term > subst > $o),X294 : term] : (! [X295 : (subst > term > term)] : (? [X299 : subst,X297 : term,X298 : subst] : (((X295 @ (comp @ X299 @ X298) @ (sub @ X297 @ X298))) != ((sub @ (X295 @ X299 @ X297) @ X298))) | (((X293 @ id @ (lam @ (X295 @ sh @ one)) @ id)) = $true) | ? [X296 : term] : (($true != ((X293 @ id @ (X295 @ id @ X296) @ id))) & ($true = ((X293 @ id @ X296 @ id))))) & ! [X300 : term] : (($true = ((X293 @ id @ (sub @ X294 @ (push @ X300 @ id)) @ id))) | (((X293 @ id @ X300 @ id)) != $true)) & ($true != ((X293 @ id @ (lam @ (sub @ X294 @ (push @ one @ sh))) @ id))))) | ? [X292 : term] : (((sub @ X292 @ id)) != X292))) & (! [X309 : (term > $o),X312 : subst,X311 : term,X310 : (term > $o)] : ((((X309 @ X311)) != $true) | ? [X313 : term] : (((X309 @ (sub @ X313 @ (push @ X311 @ X312)))) != ((X310 @ X313))) | ($true = ((X310 @ one)))) <=> (pushprop_lem1v2 = $true)) & ((apinj1 = $true) <=> ! [X3 : term,X1 : term,X0 : term,X2 : term] : ((((ap @ X1 @ X3)) != ((ap @ X0 @ X2))) | (X0 = X1))) & (pushprop_lem1_gthm = $true) & (! [X65 : term,X64 : (term > $o)] : (? [X68 : term] : ((((X64 @ X68)) = $true) & ($true != ((X64 @ (lam @ X68))))) | ? [X66 : term,X67 : term] : ((((X64 @ X67)) = $true) & (((X64 @ X66)) = $true) & (((X64 @ (ap @ X67 @ X66))) != $true)) | ? [X69 : term] : ((((var @ X69)) = $true) & ($true != ((X64 @ X69)))) | (((X64 @ X65)) = $true)) <=> (induction = $true)) & (hoaslamnotvar_gthm = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (pushprop_lem2v2 = $true) & (pushprop_lem1v2_gthm = $true) & ((induction2lem_lthm = $true) <=> ((((! [X286 : term,X284 : (term > $o),X285 : subst] : (? [X289 : term] : ((((var @ X289)) = $true) & (((X284 @ (sub @ X289 @ X285))) != $true)) | ? [X290 : term] : ((((X284 @ (lam @ X290))) != $true) & ! [X291 : term] : ((((X284 @ (sub @ X290 @ (push @ X291 @ id)))) = $true) | (((X284 @ X291)) != $true))) | (((X284 @ (sub @ X286 @ X285))) = $true) | ? [X287 : term,X288 : term] : ((((X284 @ X288)) = $true) & (((X284 @ (ap @ X288 @ X287))) != $true) & (((X284 @ X287)) = $true))) | ? [X279 : term,X278 : (term > $o)] : (! [X282 : term,X281 : term] : ((((X278 @ X282)) != $true) | (((X278 @ X281)) != $true) | (((X278 @ (ap @ X281 @ X282))) = $true)) & ! [X283 : term] : (($true = ((X278 @ X283))) | ($true != ((var @ X283)))) & ! [X280 : term] : ((((X278 @ (lam @ X280))) = $true) | ($true != ((X278 @ X280)))) & (((X278 @ X279)) != $true))) | ? [X277 : subst] : (((comp @ X277 @ id)) != X277)) | ? [X275 : term,X276 : subst] : (((comp @ sh @ (push @ X275 @ X276))) != X276)) | ? [X274 : subst,X273 : term] : (((sub @ one @ (push @ X273 @ X274))) != X273))) & ! [X318 : d_term] : (d_one = X318) & ((hoasapinj2_lthm = $true) <=> (! [X149 : term,X152 : term,X151 : term,X150 : term] : ((X150 = X151) | (((ap @ (sub @ X149 @ id) @ X150)) != ((ap @ (sub @ X152 @ id) @ X151)))) | ? [X147 : term,X146 : term,X148 : term,X145 : term] : ((((ap @ X145 @ X148)) = ((ap @ X146 @ X147))) & (X147 != X148)))) & ((axidr = $true) <=> ! [X183 : subst] : (((comp @ X183 @ id)) = X183)) & (axclos = $true) & (pushprop_lem3v2 = $true) & ((pushprop_lem1_lthm = $true) <=> ((! [X489 : term,X493 : (term > $o),X490 : term,X491 : (term > $o),X492 : subst] : ((((X493 @ X490)) != $true) | (((X491 @ (sub @ X490 @ (push @ X489 @ X492)))) = $true)) | ? [X487 : subst,X488 : term] : (((comp @ sh @ (push @ X488 @ X487))) != X487)) | ? [X485 : subst,X486 : term] : (((sub @ one @ (push @ X486 @ X485))) != X486))) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X188 : term] : ? [X189 : (subst > term > term)] : ! [X190 : subst,X191 : term] : (((X189 @ X190 @ X191)) = ((sub @ X188 @ (push @ X191 @ X190))))) & (hoasinduction_lem3_gthm = $true) & ! [X228 : subst] : ? [X229 : d_subst] : (((d2subst @ X229)) = X228) & (hoaslaminj_gthm = $true) & ((apinj2 = $true) <=> ! [X236 : term,X238 : term,X237 : term,X239 : term] : ((X236 = X239) | (((ap @ X237 @ X236)) != ((ap @ X238 @ X239))))) & (((((! [X175 : term,X174 : (subst > term > subst > $o)] : (? [X176 : (subst > term > term)] : (! [X178 : subst,X179 : term,X177 : subst] : (((sub @ (X176 @ X177 @ X179) @ X178)) = ((X176 @ (comp @ X177 @ X178) @ (sub @ X179 @ X178)))) & ! [X180 : term] : ((((X174 @ id @ (X176 @ id @ X180) @ id)) = $true) | ($true != ((X174 @ id @ X180 @ id)))) & (((X174 @ id @ (lam @ (X176 @ sh @ one)) @ id)) != $true)) | ($true = ((X174 @ id @ X175 @ id))) | ? [X181 : term,X182 : term] : (($true = ((X174 @ id @ X182 @ id))) & (((X174 @ id @ (ap @ (sub @ X182 @ id) @ X181) @ id)) != $true) & ($true = ((X174 @ id @ X181 @ id))))) | ? [X164 : (subst > term > subst > $o),X166 : term,X165 : (term > $o)] : (! [X173 : term] : (($true != ((X165 @ X173))) | (((X165 @ (sub @ X166 @ (push @ X173 @ id)))) = $true)) & ! [X167 : term] : (((X165 @ X167)) = ((X164 @ id @ X167 @ id))) & ($true != ((X165 @ (lam @ X166)))) & ! [X168 : (subst > term > term)] : (($true = ((X164 @ id @ (lam @ (X168 @ sh @ one)) @ id))) | ? [X170 : subst,X172 : subst,X171 : term] : (((X168 @ (comp @ X172 @ X170) @ (sub @ X171 @ X170))) != ((sub @ (X168 @ X172 @ X171) @ X170))) | ? [X169 : term] : ((((X164 @ id @ (X168 @ id @ X169) @ id)) != $true) & ($true = ((X164 @ id @ X169 @ id))))))) | ? [X163 : term] : (((sub @ X163 @ id)) != X163)) | ? [X157 : (term > $o),X156 : term] : (($true != ((X157 @ X156))) & ! [X160 : term,X161 : term] : ((((X157 @ (ap @ X161 @ X160))) = $true) | ($true != ((X157 @ X161))) | (((X157 @ X160)) != $true)) & ! [X162 : term] : (($true = ((X157 @ X162))) | (((var @ X162)) != $true)) & ! [X158 : term] : (? [X159 : term] : ((((X157 @ (sub @ X158 @ (push @ X159 @ id)))) != $true) & ($true = ((X157 @ X159)))) | (((X157 @ (lam @ X158))) = $true)))) | ? [X153 : (subst > term > subst > $o)] : ! [X154 : (term > $o)] : ? [X155 : term] : (((X154 @ X155)) != ((X153 @ id @ X155 @ id)))) <=> (hoasinduction_no_psi_cond_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)))))))))))) & (! [X22 : term,X23 : (subst > term > subst > $o)] : (? [X26 : subst,X29 : subst,X28 : term,X27 : subst] : ((((X23 @ (comp @ X27 @ X26) @ (sub @ X28 @ X26) @ X29)) != $true) & (((X23 @ X27 @ X28 @ (comp @ X26 @ X29))) = $true)) | ? [X35 : (subst > term > term)] : ((((X23 @ id @ (lam @ (X35 @ sh @ one)) @ id)) != $true) & ! [X37 : subst,X38 : term,X36 : subst] : (((X35 @ (comp @ X37 @ X36) @ (sub @ X38 @ X36))) = ((sub @ (X35 @ X37 @ X38) @ X36))) & ! [X39 : term] : ((((X23 @ id @ X39 @ id)) != $true) | ($true = ((X23 @ id @ (X35 @ id @ X39) @ id))))) | ? [X25 : term,X24 : term] : ((((X23 @ id @ X24 @ id)) = $true) & (((X23 @ id @ (ap @ (sub @ X24 @ id) @ X25) @ id)) != $true) & (((X23 @ id @ X25 @ id)) = $true)) | ? [X31 : subst,X30 : subst,X32 : subst,X33 : term] : ((((X23 @ X30 @ X33 @ (comp @ X31 @ X32))) != $true) & ($true = ((X23 @ (comp @ X30 @ X31) @ (sub @ X33 @ X31) @ X32)))) | (((X23 @ id @ X22 @ id)) = $true) | ? [X34 : term] : (($true = ((var @ (sub @ X34 @ id)))) & (((X23 @ id @ X34 @ id)) != $true))) <=> (hoasinduction = $true)) & ((pushprop_lem1v2_lthm = $true) <=> (! [X417 : term,X414 : (term > $o),X416 : subst,X415 : (term > $o)] : ((((X415 @ one)) = $true) | ($true != ((X414 @ X417))) | ? [X418 : term] : (((X415 @ X418)) != ((X414 @ (sub @ X418 @ (push @ X417 @ X416)))))) | ? [X412 : term,X413 : subst] : (((sub @ one @ (push @ X412 @ X413))) != X412))) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (((var @ (d2term @ d_one))) != $true) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X70 : (subst > term > subst > $o),X71 : term] : (? [X72 : term] : (($true != ((X70 @ id @ (sub @ X71 @ (push @ X72 @ id)) @ id))) & ($true = ((X70 @ id @ X72 @ id)))) | (((X70 @ id @ (lam @ (sub @ X71 @ (push @ one @ sh))) @ id)) = $true) | ? [X73 : (subst > term > term)] : ((((X70 @ id @ (lam @ (X73 @ sh @ one)) @ id)) != $true) & ! [X74 : term] : (($true = ((X70 @ id @ (X73 @ id @ X74) @ id))) | ($true != ((X70 @ id @ X74 @ id)))) & ! [X76 : term,X77 : subst,X75 : subst] : (((sub @ (X73 @ X75 @ X76) @ X77)) = ((X73 @ (comp @ X75 @ X77) @ (sub @ X76 @ X77))))))) & (hoasinduction_lem2_lthm = $true) & (hoasapinj2_gthm = $true) & (! [X5 : (term > $o),X4 : term,X6 : (subst > term > subst > $o)] : ((((X5 @ (lam @ X4))) = $true) | ? [X21 : term] : (((X6 @ id @ X21 @ id)) != ((X5 @ X21))) | ? [X11 : term] : (($true != ((X5 @ (sub @ X4 @ (push @ X11 @ id))))) & ($true = ((X5 @ X11)))) | ? [X17 : subst,X20 : term,X18 : subst,X19 : subst] : (($true != ((X6 @ (comp @ X18 @ X19) @ (sub @ X20 @ X19) @ X17))) & (((X6 @ X18 @ X20 @ (comp @ X19 @ X17))) = $true)) | ? [X7 : subst,X9 : subst,X8 : term,X10 : subst] : (($true = ((X6 @ (comp @ X10 @ X7) @ (sub @ X8 @ X7) @ X9))) & (((X6 @ X10 @ X8 @ (comp @ X7 @ X9))) != $true)) | ? [X12 : (subst > term > term)] : (! [X13 : term] : ((((X6 @ id @ X13 @ id)) != $true) | (((X6 @ id @ (X12 @ id @ X13) @ id)) = $true)) & ! [X16 : subst,X14 : term,X15 : subst] : (((sub @ (X12 @ X15 @ X14) @ X16)) = ((X12 @ (comp @ X15 @ X16) @ (sub @ X14 @ X16)))) & (((X6 @ id @ (lam @ (X12 @ sh @ one)) @ id)) != $true))) <=> (hoasinduction_lem3v2 = $true)) & (pushprop_lthm = $true) & (hoaslamnotvar_lthm = $true) & ((induction2_lthm = $true) <=> ((! [X93 : (term > $o),X94 : term] : (? [X96 : term,X95 : term] : (($true = ((X93 @ X95))) & ($true != ((X93 @ (ap @ X96 @ X95)))) & (((X93 @ X96)) = $true)) | ? [X97 : term] : ((((var @ X97)) = $true) & (((X93 @ X97)) != $true)) | ? [X98 : term] : (! [X99 : term] : ((((X93 @ X99)) != $true) | ($true = ((X93 @ (sub @ X98 @ (push @ X99 @ id)))))) & ($true != ((X93 @ (lam @ X98))))) | (((X93 @ X94)) = $true)) | ? [X85 : (term > $o),X87 : term,X86 : subst] : ((((X85 @ (sub @ X87 @ X86))) != $true) & ! [X91 : term,X90 : term] : ((((X85 @ X91)) != $true) | (((X85 @ X90)) != $true) | (((X85 @ (ap @ X91 @ X90))) = $true)) & ! [X88 : term] : ((((X85 @ (lam @ X88))) = $true) | ? [X89 : term] : ((((X85 @ (sub @ X88 @ (push @ X89 @ id)))) != $true) & (((X85 @ X89)) = $true))) & ! [X92 : term] : (($true != ((var @ X92))) | (((X85 @ (sub @ X92 @ X86))) = $true)))) | ? [X84 : term] : (((sub @ X84 @ id)) != X84))) & ((pushprop_lem1 = $true) <=> ! [X270 : term,X272 : subst,X271 : (term > $o),X268 : term,X269 : (term > $o)] : ((((X271 @ (sub @ X270 @ (push @ X268 @ X272)))) = $true) | (((X269 @ X270)) != $true))) & (! [X216 : term,X217 : term,X215 : (subst > term > subst > $o)] : (? [X227 : subst,X226 : subst,X225 : subst,X224 : term] : (($true = ((X215 @ (comp @ X227 @ X225) @ (sub @ X224 @ X225) @ X226))) & ($true != ((X215 @ X227 @ X224 @ (comp @ X225 @ X226))))) | ? [X218 : subst,X220 : subst,X221 : term,X219 : subst] : (($true = ((X215 @ X220 @ X221 @ (comp @ X218 @ X219)))) & ($true != ((X215 @ (comp @ X220 @ X218) @ (sub @ X221 @ X218) @ X219)))) | (((X215 @ id @ X216 @ id)) != $true) | ($true = ((X215 @ id @ (ap @ X216 @ X217) @ id))) | ? [X223 : term,X222 : term] : ((((X215 @ id @ X222 @ id)) = $true) & (((X215 @ id @ X223 @ id)) = $true) & ($true != ((X215 @ id @ (ap @ (sub @ X223 @ id) @ X222) @ id)))) | (((X215 @ id @ X217 @ id)) != $true)) <=> (hoasinduction_lem2 = $true)) & (((d2subst @ d_id)) = sh) & (((d2term @ d_one)) = one) & (! [X192 : (subst > term > subst > $o)] : ? [X193 : (term > $o)] : ! [X194 : term] : (((X192 @ id @ X194 @ id)) = ((X193 @ X194))) <=> (hoasinduction_lem0 = $true)) & ((((! [X105 : subst] : (((comp @ X105 @ id)) = X105) & ! [X104 : subst] : (((comp @ id @ X104)) = X104)) | ? [X103 : subst] : (((comp @ X103 @ id)) != X103)) | ? [X102 : subst] : (((comp @ id @ X102)) != X102)) <=> (substmonoid_lthm = $true)) & (axabs = $true) & ((hoasinduction_lem3v2_f = $true) <=> ! [X78 : term] : ? [X79 : (subst > term > term)] : ! [X81 : subst,X80 : term] : (((X79 @ X81 @ X80)) = ((sub @ X78 @ (push @ X80 @ X81))))) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & ((axscons = $true) <=> ! [X247 : subst] : (((push @ (sub @ one @ X247) @ (comp @ sh @ X247))) = X247)) & (((! [X428 : (subst > term > subst > $o),X429 : term] : (? [X435 : (subst > term > term)] : ((((X428 @ id @ (lam @ (X435 @ sh @ one)) @ id)) != $true) & ! [X438 : subst,X439 : subst,X437 : term] : (((X435 @ (comp @ X438 @ X439) @ (sub @ X437 @ X439))) = ((sub @ (X435 @ X438 @ X437) @ X439))) & ! [X436 : term] : (($true != ((X428 @ id @ X436 @ id))) | (((X428 @ id @ (X435 @ id @ X436) @ id)) = $true))) | ? [X441 : subst,X442 : subst,X440 : term,X443 : subst] : ((((X428 @ X443 @ X440 @ (comp @ X442 @ X441))) != $true) & (((X428 @ (comp @ X443 @ X442) @ (sub @ X440 @ X442) @ X441)) = $true)) | ($true = ((X428 @ id @ (lam @ X429) @ id))) | ? [X434 : term] : (($true = ((X428 @ id @ X434 @ id))) & ($true != ((X428 @ id @ (sub @ X429 @ (push @ X434 @ id)) @ id)))) | ? [X432 : subst,X433 : subst,X431 : term,X430 : subst] : ((((X428 @ (comp @ X433 @ X432) @ (sub @ X431 @ X432) @ X430)) != $true) & (((X428 @ X433 @ X431 @ (comp @ X432 @ X430))) = $true))) | ? [X420 : (subst > term > subst > $o),X421 : term] : (($true != ((X420 @ id @ (lam @ (sub @ X421 @ (push @ one @ sh))) @ id))) & ! [X422 : (subst > term > term)] : (? [X426 : term,X425 : subst,X424 : subst] : (((sub @ (X422 @ X425 @ X426) @ X424)) != ((X422 @ (comp @ X425 @ X424) @ (sub @ X426 @ X424)))) | (((X420 @ id @ (lam @ (X422 @ sh @ one)) @ id)) = $true) | ? [X423 : term] : ((((X420 @ id @ X423 @ id)) = $true) & ($true != ((X420 @ id @ (X422 @ id @ X423) @ id))))) & ! [X427 : term] : ((((X420 @ id @ (sub @ X421 @ (push @ X427 @ id)) @ id)) = $true) | (((X420 @ id @ X427 @ id)) != $true)))) | ? [X419 : term] : (((sub @ X419 @ id)) != X419)) <=> (hoasinduction_lem3_lthm = $true)) & ! [X41 : d_subst,X40 : d_subst] : ((((d2subst @ X40)) != ((d2subst @ X41))) | (X40 = X41)) & ((axvarid = $true) <=> ! [X520 : term] : (((sub @ X520 @ id)) = X520)) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (ulamvarsh = $true) & (hoasinduction_lem3a_gthm = $true) & (((! [X265 : term,X266 : term,X267 : term,X264 : term] : ((((ap @ (sub @ X267 @ id) @ X265)) != ((ap @ (sub @ X266 @ id) @ X264))) | (X266 = X267)) | ? [X263 : term,X261 : term,X260 : term,X262 : term] : ((((ap @ X261 @ X263)) = ((ap @ X262 @ X260))) & (X261 != X262))) | ? [X259 : term] : (((sub @ X259 @ id)) != X259)) <=> (hoasapinj1_lthm = $true)) & (termmset_gthm = $true) & (! [X250 : (term > $o),X251 : (subst > term > subst > $o),X249 : term] : ((((X250 @ (lam @ X249))) = $true) | ? [X253 : (subst > term > term)] : (! [X254 : subst,X255 : term,X256 : subst] : (((sub @ (X253 @ X254 @ X255) @ X256)) = ((X253 @ (comp @ X254 @ X256) @ (sub @ X255 @ X256)))) & ! [X257 : term] : (($true = ((X251 @ id @ (X253 @ id @ X257) @ id))) | ($true != ((X251 @ id @ X257 @ id)))) & ($true != ((X251 @ id @ (lam @ (X253 @ sh @ one)) @ id)))) | ? [X252 : term] : (((X250 @ X252)) != ((X251 @ id @ X252 @ id))) | ? [X258 : term] : ((((X250 @ (sub @ X249 @ (push @ X258 @ id)))) != $true) & (((X250 @ X258)) = $true))) <=> (hoasinduction_lem3v2a = $true)) & (! [X140 : term,X139 : term,X141 : (subst > term > term)] : ((((lam @ (X141 @ sh @ one))) != ((ap @ (sub @ X140 @ id) @ X139))) | ? [X143 : subst,X142 : term,X144 : subst] : (((X141 @ (comp @ X143 @ X144) @ (sub @ X142 @ X144))) != ((sub @ (X141 @ X143 @ X142) @ X144)))) <=> (hoaslamnotap = $true)) & ! [X106 : term] : ? [X107 : d_term] : (((d2term @ X107)) = X106) & ((induction2lem = $true) <=> ! [X58 : (term > $o),X57 : subst,X56 : term] : (? [X61 : term] : (($true != ((X58 @ (sub @ X61 @ X57)))) & (((var @ X61)) = $true)) | ? [X60 : term,X59 : term] : ((((X58 @ X59)) = $true) & (((X58 @ X60)) = $true) & (((X58 @ (ap @ X60 @ X59))) != $true)) | ? [X62 : term] : (! [X63 : term] : (($true != ((X58 @ X63))) | (((X58 @ (sub @ X62 @ (push @ X63 @ id)))) = $true)) & ($true != ((X58 @ (lam @ X62))))) | (((X58 @ (sub @ X56 @ X57))) = $true))) & (pushprop_gthm = $true) & (hoasinduction_lem3v2_gthm = $true) & (hoasinduction_lem1v2 = $true) & ((? [X444 : term] : (((sub @ X444 @ id)) != X444) | ! [X445 : term] : (((sub @ X445 @ id)) = X445)) <=> (termmset_lthm = $true)) & (hoasinduction_lem1 = $true) & ((! [X231 : subst] : (((comp @ X231 @ id)) = X231) & ! [X230 : subst] : (((comp @ id @ X230)) = X230)) <=> (substmonoid = $true)) & (induction2lem_gthm = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ((hoasinduction_lem3b = $true) <=> ! [X240 : term] : ? [X241 : (subst > term > term)] : (((sub @ X240 @ (push @ one @ sh))) = ((X241 @ sh @ one)))) & (pushprop_lem0_gthm = $true) & (hoasinduction_lem1_gthm = $true) & (pushprop = $true) & ((? [X108 : term] : (((sub @ X108 @ id)) != X108) | ! [X110 : (term > $o),X109 : (subst > term > subst > $o),X111 : term] : (? [X123 : subst,X125 : term,X126 : subst,X124 : subst] : (($true = ((X109 @ (comp @ X124 @ X126) @ (sub @ X125 @ X126) @ X123))) & ($true != ((X109 @ X124 @ X125 @ (comp @ X126 @ X123))))) | ? [X121 : term] : (((X110 @ X121)) != ((X109 @ id @ X121 @ id))) | ($true = ((X110 @ (lam @ X111)))) | ? [X117 : subst,X118 : term,X120 : subst,X119 : subst] : ((((X109 @ X120 @ X118 @ (comp @ X119 @ X117))) = $true) & (((X109 @ (comp @ X120 @ X119) @ (sub @ X118 @ X119) @ X117)) != $true)) | ? [X112 : (subst > term > term)] : ((((X109 @ id @ (lam @ (X112 @ sh @ one)) @ id)) != $true) & ! [X113 : term] : (($true != ((X109 @ id @ X113 @ id))) | (((X109 @ id @ (X112 @ id @ X113) @ id)) = $true)) & ! [X114 : subst,X115 : term,X116 : subst] : (((sub @ (X112 @ X116 @ X115) @ X114)) = ((X112 @ (comp @ X116 @ X114) @ (sub @ X115 @ X114))))) | ? [X122 : term] : (($true != ((X110 @ (sub @ X111 @ (push @ X122 @ id))))) & (((X110 @ X122)) = $true)))) <=> (hoasinduction_lem3v2_lthm = $true)) & ((hoasinduction_lthm_3 = $true) <=> ((((! [X468 : term,X467 : (subst > term > subst > $o)] : ((((X467 @ id @ X468 @ id)) = $true) | ? [X482 : term,X481 : subst,X484 : subst,X483 : subst] : ((((X467 @ (comp @ X484 @ X481) @ (sub @ X482 @ X481) @ X483)) = $true) & ($true != ((X467 @ X484 @ X482 @ (comp @ X481 @ X483))))) | ? [X472 : term,X474 : subst,X475 : subst,X473 : subst] : ((((X467 @ (comp @ X474 @ X473) @ (sub @ X472 @ X473) @ X475)) != $true) & (((X467 @ X474 @ X472 @ (comp @ X473 @ X475))) = $true)) | ? [X471 : term] : ((((var @ (sub @ X471 @ id))) = $true) & (((X467 @ id @ X471 @ id)) != $true)) | ? [X470 : term,X469 : term] : ((((X467 @ id @ (ap @ (sub @ X469 @ id) @ X470) @ id)) != $true) & (((X467 @ id @ X469 @ id)) = $true) & (((X467 @ id @ X470 @ id)) = $true)) | ? [X476 : (subst > term > term)] : ((((X467 @ id @ (lam @ (X476 @ sh @ one)) @ id)) != $true) & ! [X477 : term] : ((((X467 @ id @ (X476 @ id @ X477) @ id)) = $true) | ($true != ((X467 @ id @ X477 @ id)))) & ! [X479 : term,X480 : subst,X478 : subst] : (((X476 @ (comp @ X478 @ X480) @ (sub @ X479 @ X480))) = ((sub @ (X476 @ X478 @ X479) @ X480))))) | ? [X459 : (term > $o),X457 : (subst > term > subst > $o),X458 : term] : (! [X461 : term] : (($true = ((X459 @ (sub @ X458 @ (push @ X461 @ id))))) | ($true != ((X459 @ X461)))) & (((X459 @ (lam @ X458))) != $true) & ! [X462 : (subst > term > term)] : (? [X464 : subst,X463 : subst,X465 : term] : (((X462 @ (comp @ X463 @ X464) @ (sub @ X465 @ X464))) != ((sub @ (X462 @ X463 @ X465) @ X464))) | ? [X466 : term] : ((((X457 @ id @ X466 @ id)) = $true) & ($true != ((X457 @ id @ (X462 @ id @ X466) @ id)))) | (((X457 @ id @ (lam @ (X462 @ sh @ one)) @ id)) = $true)) & ! [X460 : term] : (((X459 @ X460)) = ((X457 @ id @ X460 @ id))))) | ? [X456 : term] : (((sub @ X456 @ id)) != X456)) | ? [X449 : term,X450 : (term > $o)] : (! [X452 : term,X451 : term] : (($true = ((X450 @ (ap @ X452 @ X451)))) | ($true != ((X450 @ X451))) | ($true != ((X450 @ X452)))) & ! [X455 : term] : ((((var @ X455)) != $true) | ($true = ((X450 @ X455)))) & ($true != ((X450 @ X449))) & ! [X453 : term] : (? [X454 : term] : ((((X450 @ (sub @ X453 @ (push @ X454 @ id)))) != $true) & ($true = ((X450 @ X454)))) | (((X450 @ (lam @ X453))) = $true)))) | ? [X446 : (subst > term > subst > $o)] : ! [X447 : (term > $o)] : ? [X448 : term] : (((X447 @ X448)) != ((X446 @ id @ X448 @ id))))) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (! [X82 : term] : ? [X83 : (subst > term > term)] : (((sub @ X82 @ (push @ one @ sh))) = ((X83 @ sh @ one))) <=> (hoasinduction_lem3b_lthm = $true)) & (pushprop_lem2v2_lthm = $true) & (ulamvarind = $true) & (axmap = $true) & (hoasapnotvar = $true) & ((laminj = $true) <=> ! [X233 : term,X232 : term] : ((((lam @ X232)) != ((lam @ X233))) | (X232 = X233))) & ((axidl = $true) <=> ! [X47 : subst] : (((comp @ id @ X47)) = X47)) & (((! [X325 : (subst > term > subst > $o),X326 : term,X324 : (term > $o)] : (? [X329 : (subst > term > term)] : (! [X332 : subst,X330 : subst,X331 : term] : (((sub @ (X329 @ X330 @ X331) @ X332)) = ((X329 @ (comp @ X330 @ X332) @ (sub @ X331 @ X332)))) & ! [X333 : term] : (($true = ((X325 @ id @ (X329 @ id @ X333) @ id))) | (((X325 @ id @ X333 @ id)) != $true)) & ($true != ((X325 @ id @ (lam @ (X329 @ sh @ one)) @ id)))) | ? [X328 : term] : (((X325 @ id @ X328 @ id)) != ((X324 @ X328))) | ($true = ((X324 @ (lam @ X326)))) | ? [X327 : term] : ((((X324 @ (sub @ X326 @ (push @ X327 @ id)))) != $true) & ($true = ((X324 @ X327))))) | ? [X323 : term] : (((sub @ X323 @ id)) != X323)) | ? [X319 : term] : ! [X320 : (subst > term > term)] : ? [X321 : subst,X322 : term] : (((sub @ X319 @ (push @ X322 @ X321))) != ((X320 @ X321 @ X322)))) <=> (hoasinduction_lem3v2a_lthm = $true)) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (pushprop_lem2v2_gthm = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoasapinj1_gthm = $true) & ((axvarcons = $true) <=> ! [X234 : subst,X235 : term] : (((sub @ one @ (push @ X235 @ X234))) = X235)) & (! [X195 : term,X196 : term] : ((((sub @ X196 @ sh)) != ((sub @ X195 @ sh))) | (X195 = X196)) <=> (shinj = $true)) & (hoaslamnotap_lthm = $true) & ((hoasinduction_lem3aaa = $true) <=> ! [X128 : (subst > term > subst > $o),X127 : term] : (? [X129 : term] : ((((X128 @ id @ X129 @ id)) = $true) & ($true != ((X128 @ id @ (sub @ X127 @ (push @ X129 @ id)) @ id)))) | ? [X130 : term,X131 : (subst > term > term)] : (! [X138 : term] : (($true != ((X128 @ id @ X138 @ id))) | (((X128 @ id @ (X131 @ id @ X138) @ id)) = $true)) & ! [X132 : term,X133 : subst,X134 : subst] : (((sub @ (X131 @ X133 @ X132) @ X134)) = ((sub @ (sub @ X130 @ (push @ X132 @ X133)) @ X134))) & (((X128 @ id @ (lam @ (X131 @ sh @ one)) @ id)) != $true) & ! [X137 : term,X136 : subst,X135 : subst] : (((X131 @ (comp @ X136 @ X135) @ (sub @ X137 @ X135))) = ((sub @ X130 @ (push @ (sub @ X137 @ X135) @ (comp @ X136 @ X135)))))) | (((X128 @ id @ (lam @ (sub @ X127 @ (push @ one @ sh))) @ id)) = $true))) & (hoasinduction_lem2v2_gthm = $true) & (axapp = $true) & (pushprop_lem3v2_lthm = $true) & (hoaslamnotvar = $true) & ((hoasapinj1 = $true) <=> ! [X314 : term,X317 : term,X316 : term,X315 : term] : ((X314 = X316) | (((ap @ (sub @ X316 @ id) @ X317)) != ((ap @ (sub @ X314 @ id) @ X315))))) & ! [X101 : d_term,X100 : d_term] : ((X100 = X101) | (((d2term @ X101)) != ((d2term @ X100))))), inference(ennf_transformation,[],[f10])). thf(f14,plain,( ((apinj1 = $true) <=> ! [X3 : term,X1 : term,X0 : term,X2 : term] : ((((ap @ X1 @ X3)) != ((ap @ X0 @ X2))) | (X0 = X1))) & ! [X228 : subst] : ? [X229 : d_subst] : (((d2subst @ X229)) = X228) & (hoasapnotvar_lthm = $true) & ((induction2_lthm = $true) <=> (! [X93 : (term > $o),X94 : term] : (? [X96 : term,X95 : term] : (($true = ((X93 @ X95))) & ($true != ((X93 @ (ap @ X96 @ X95)))) & (((X93 @ X96)) = $true)) | ? [X97 : term] : ((((var @ X97)) = $true) & (((X93 @ X97)) != $true)) | ? [X98 : term] : (! [X99 : term] : ((((X93 @ X99)) != $true) | ($true = ((X93 @ (sub @ X98 @ (push @ X99 @ id)))))) & ($true != ((X93 @ (lam @ X98))))) | (((X93 @ X94)) = $true)) | ? [X85 : (term > $o),X87 : term,X86 : subst] : ((((X85 @ (sub @ X87 @ X86))) != $true) & ! [X91 : term,X90 : term] : ((((X85 @ X91)) != $true) | (((X85 @ X90)) != $true) | (((X85 @ (ap @ X91 @ X90))) = $true)) & ! [X88 : term] : ((((X85 @ (lam @ X88))) = $true) | ? [X89 : term] : ((((X85 @ (sub @ X88 @ (push @ X89 @ id)))) != $true) & (((X85 @ X89)) = $true))) & ! [X92 : term] : (($true != ((var @ X92))) | (((X85 @ (sub @ X92 @ X86))) = $true))) | ? [X84 : term] : (((sub @ X84 @ id)) != X84))) & (pushprop_gthm = $true) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (? [X164 : (subst > term > subst > $o),X166 : term,X165 : (term > $o)] : (! [X173 : term] : (($true != ((X165 @ X173))) | (((X165 @ (sub @ X166 @ (push @ X173 @ id)))) = $true)) & ! [X167 : term] : (((X165 @ X167)) = ((X164 @ id @ X167 @ id))) & ($true != ((X165 @ (lam @ X166)))) & ! [X168 : (subst > term > term)] : (($true = ((X164 @ id @ (lam @ (X168 @ sh @ one)) @ id))) | ? [X170 : subst,X172 : subst,X171 : term] : (((X168 @ (comp @ X172 @ X170) @ (sub @ X171 @ X170))) != ((sub @ (X168 @ X172 @ X171) @ X170))) | ? [X169 : term] : ((((X164 @ id @ (X168 @ id @ X169) @ id)) != $true) & ($true = ((X164 @ id @ X169 @ id)))))) | ? [X157 : (term > $o),X156 : term] : (($true != ((X157 @ X156))) & ! [X160 : term,X161 : term] : ((((X157 @ (ap @ X161 @ X160))) = $true) | ($true != ((X157 @ X161))) | (((X157 @ X160)) != $true)) & ! [X162 : term] : (($true = ((X157 @ X162))) | (((var @ X162)) != $true)) & ! [X158 : term] : (? [X159 : term] : ((((X157 @ (sub @ X158 @ (push @ X159 @ id)))) != $true) & ($true = ((X157 @ X159)))) | (((X157 @ (lam @ X158))) = $true))) | ! [X175 : term,X174 : (subst > term > subst > $o)] : (? [X176 : (subst > term > term)] : (! [X178 : subst,X179 : term,X177 : subst] : (((sub @ (X176 @ X177 @ X179) @ X178)) = ((X176 @ (comp @ X177 @ X178) @ (sub @ X179 @ X178)))) & ! [X180 : term] : ((((X174 @ id @ (X176 @ id @ X180) @ id)) = $true) | ($true != ((X174 @ id @ X180 @ id)))) & (((X174 @ id @ (lam @ (X176 @ sh @ one)) @ id)) != $true)) | ($true = ((X174 @ id @ X175 @ id))) | ? [X181 : term,X182 : term] : (($true = ((X174 @ id @ X182 @ id))) & (((X174 @ id @ (ap @ (sub @ X182 @ id) @ X181) @ id)) != $true) & ($true = ((X174 @ id @ X181 @ id))))) | ? [X153 : (subst > term > subst > $o)] : ! [X154 : (term > $o)] : ? [X155 : term] : (((X154 @ X155)) != ((X153 @ id @ X155 @ id))) | ? [X163 : term] : (((sub @ X163 @ id)) != X163))) & (induction2lem_gthm = $true) & ((! [X428 : (subst > term > subst > $o),X429 : term] : (? [X435 : (subst > term > term)] : ((((X428 @ id @ (lam @ (X435 @ sh @ one)) @ id)) != $true) & ! [X438 : subst,X439 : subst,X437 : term] : (((X435 @ (comp @ X438 @ X439) @ (sub @ X437 @ X439))) = ((sub @ (X435 @ X438 @ X437) @ X439))) & ! [X436 : term] : (($true != ((X428 @ id @ X436 @ id))) | (((X428 @ id @ (X435 @ id @ X436) @ id)) = $true))) | ? [X441 : subst,X442 : subst,X440 : term,X443 : subst] : ((((X428 @ X443 @ X440 @ (comp @ X442 @ X441))) != $true) & (((X428 @ (comp @ X443 @ X442) @ (sub @ X440 @ X442) @ X441)) = $true)) | ($true = ((X428 @ id @ (lam @ X429) @ id))) | ? [X434 : term] : (($true = ((X428 @ id @ X434 @ id))) & ($true != ((X428 @ id @ (sub @ X429 @ (push @ X434 @ id)) @ id)))) | ? [X432 : subst,X433 : subst,X431 : term,X430 : subst] : ((((X428 @ (comp @ X433 @ X432) @ (sub @ X431 @ X432) @ X430)) != $true) & (((X428 @ X433 @ X431 @ (comp @ X432 @ X430))) = $true))) | ? [X419 : term] : (((sub @ X419 @ id)) != X419) | ? [X420 : (subst > term > subst > $o),X421 : term] : (($true != ((X420 @ id @ (lam @ (sub @ X421 @ (push @ one @ sh))) @ id))) & ! [X422 : (subst > term > term)] : (? [X426 : term,X425 : subst,X424 : subst] : (((sub @ (X422 @ X425 @ X426) @ X424)) != ((X422 @ (comp @ X425 @ X424) @ (sub @ X426 @ X424)))) | (((X420 @ id @ (lam @ (X422 @ sh @ one)) @ id)) = $true) | ? [X423 : term] : ((((X420 @ id @ X423 @ id)) = $true) & ($true != ((X420 @ id @ (X422 @ id @ X423) @ id))))) & ! [X427 : term] : ((((X420 @ id @ (sub @ X421 @ (push @ X427 @ id)) @ id)) = $true) | (((X420 @ id @ X427 @ id)) != $true)))) <=> (hoasinduction_lem3_lthm = $true)) & ((hoasinduction_no_psi_cond = $true) <=> ! [X206 : (subst > term > subst > $o),X207 : term] : (? [X208 : term,X209 : term] : (($true = ((X206 @ id @ X209 @ id))) & ($true != ((X206 @ id @ (ap @ (sub @ X208 @ id) @ X209) @ id))) & (((X206 @ id @ X208 @ id)) = $true)) | ? [X210 : (subst > term > term)] : (! [X212 : subst,X213 : term,X214 : subst] : (((X210 @ (comp @ X214 @ X212) @ (sub @ X213 @ X212))) = ((sub @ (X210 @ X214 @ X213) @ X212))) & ($true != ((X206 @ id @ (lam @ (X210 @ sh @ one)) @ id))) & ! [X211 : term] : (($true = ((X206 @ id @ (X210 @ id @ X211) @ id))) | (((X206 @ id @ X211 @ id)) != $true))) | (((X206 @ id @ X207 @ id)) = $true))) & (pushprop_lthm_orig = $true) & (axassoc = $true) & (pushprop_lthm = $true) & (hoasinduction_lem1_lthm = $true) & (ulamvar1 != $true) & (substmonoid_gthm = $true) & (hoasapnotvar_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & ! [X524 : d_subst] : (d_id = X524) & (lamnotap != $true) & (((d2subst @ d_id)) = id) & ((? [X275 : term,X276 : subst] : (((comp @ sh @ (push @ X275 @ X276))) != X276) | ? [X274 : subst,X273 : term] : (((sub @ one @ (push @ X273 @ X274))) != X273) | ? [X279 : term,X278 : (term > $o)] : (! [X282 : term,X281 : term] : ((((X278 @ X282)) != $true) | (((X278 @ X281)) != $true) | (((X278 @ (ap @ X281 @ X282))) = $true)) & ! [X283 : term] : (($true = ((X278 @ X283))) | ($true != ((var @ X283)))) & ! [X280 : term] : ((((X278 @ (lam @ X280))) = $true) | ($true != ((X278 @ X280)))) & (((X278 @ X279)) != $true)) | ? [X277 : subst] : (((comp @ X277 @ id)) != X277) | ! [X286 : term,X284 : (term > $o),X285 : subst] : (? [X289 : term] : ((((var @ X289)) = $true) & (((X284 @ (sub @ X289 @ X285))) != $true)) | ? [X290 : term] : ((((X284 @ (lam @ X290))) != $true) & ! [X291 : term] : ((((X284 @ (sub @ X290 @ (push @ X291 @ id)))) = $true) | (((X284 @ X291)) != $true))) | (((X284 @ (sub @ X286 @ X285))) = $true) | ? [X287 : term,X288 : term] : ((((X284 @ X288)) = $true) & (((X284 @ (ap @ X288 @ X287))) != $true) & (((X284 @ X287)) = $true)))) <=> (induction2lem_lthm = $true)) & (hoasapnotvar = $true) & (axvarshift = $true) & ((hoasinduction_lem0_lthm = $true) <=> ! [X521 : (subst > term > subst > $o)] : ? [X522 : (term > $o)] : ! [X523 : term] : (((X521 @ id @ X523 @ id)) = ((X522 @ X523)))) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (! [X197 : (term > $o),X198 : term] : (? [X200 : term,X199 : term] : (($true = ((X197 @ X200))) & ($true = ((X197 @ X199))) & (((X197 @ (ap @ X200 @ X199))) != $true)) | (((X197 @ X198)) = $true) | ? [X201 : term] : (! [X202 : term] : (($true != ((X197 @ X202))) | (((X197 @ (sub @ X201 @ (push @ X202 @ id)))) = $true)) & (((X197 @ (lam @ X201))) != $true)) | ? [X203 : term] : (($true != ((X197 @ X203))) & (((var @ X203)) = $true))) <=> (induction2 = $true)) & (hoasinduction_lem1_gthm = $true) & ((axidr = $true) <=> ! [X183 : subst] : (((comp @ X183 @ id)) = X183)) & (! [X397 : (subst > term > subst > $o),X396 : term] : (? [X403 : subst,X405 : subst,X404 : subst,X406 : term] : ((((X397 @ X404 @ X406 @ (comp @ X405 @ X403))) = $true) & (((X397 @ (comp @ X404 @ X405) @ (sub @ X406 @ X405) @ X403)) != $true)) | ? [X411 : term] : ((((X397 @ id @ (sub @ X396 @ (push @ X411 @ id)) @ id)) != $true) & (((X397 @ id @ X411 @ id)) = $true)) | ? [X408 : subst,X409 : term,X410 : subst,X407 : subst] : ((((X397 @ X410 @ X409 @ (comp @ X407 @ X408))) != $true) & ($true = ((X397 @ (comp @ X410 @ X407) @ (sub @ X409 @ X407) @ X408)))) | (((X397 @ id @ (lam @ X396) @ id)) = $true) | ? [X398 : (subst > term > term)] : (! [X399 : term] : ((((X397 @ id @ X399 @ id)) != $true) | ($true = ((X397 @ id @ (X398 @ id @ X399) @ id)))) & ($true != ((X397 @ id @ (lam @ (X398 @ sh @ one)) @ id))) & ! [X401 : subst,X400 : term,X402 : subst] : (((X398 @ (comp @ X401 @ X402) @ (sub @ X400 @ X402))) = ((sub @ (X398 @ X401 @ X400) @ X402))))) <=> (hoasinduction_lem3 = $true)) & (! [X5 : (term > $o),X4 : term,X6 : (subst > term > subst > $o)] : ((((X5 @ (lam @ X4))) = $true) | ? [X21 : term] : (((X6 @ id @ X21 @ id)) != ((X5 @ X21))) | ? [X11 : term] : (($true != ((X5 @ (sub @ X4 @ (push @ X11 @ id))))) & ($true = ((X5 @ X11)))) | ? [X17 : subst,X20 : term,X18 : subst,X19 : subst] : (($true != ((X6 @ (comp @ X18 @ X19) @ (sub @ X20 @ X19) @ X17))) & (((X6 @ X18 @ X20 @ (comp @ X19 @ X17))) = $true)) | ? [X7 : subst,X9 : subst,X8 : term,X10 : subst] : (($true = ((X6 @ (comp @ X10 @ X7) @ (sub @ X8 @ X7) @ X9))) & (((X6 @ X10 @ X8 @ (comp @ X7 @ X9))) != $true)) | ? [X12 : (subst > term > term)] : (! [X13 : term] : ((((X6 @ id @ X13 @ id)) != $true) | (((X6 @ id @ (X12 @ id @ X13) @ id)) = $true)) & ! [X16 : subst,X14 : term,X15 : subst] : (((sub @ (X12 @ X15 @ X14) @ X16)) = ((X12 @ (comp @ X15 @ X16) @ (sub @ X14 @ X16)))) & (((X6 @ id @ (lam @ (X12 @ sh @ one)) @ id)) != $true))) <=> (hoasinduction_lem3v2 = $true)) & ((! [X231 : subst] : (((comp @ X231 @ id)) = X231) & ! [X230 : subst] : (((comp @ id @ X230)) = X230)) <=> (substmonoid = $true)) & (! [X528 : (term > $o),X525 : term,X527 : (subst > term > subst > $o),X526 : term] : (? [X536 : subst,X537 : subst,X535 : subst,X538 : term] : (($true = ((X527 @ X535 @ X538 @ (comp @ X536 @ X537)))) & ($true != ((X527 @ (comp @ X535 @ X536) @ (sub @ X538 @ X536) @ X537)))) | ($true != ((X528 @ X526))) | ($true = ((X528 @ (ap @ X526 @ X525)))) | ? [X529 : term,X530 : term] : ((((X527 @ id @ (ap @ (sub @ X530 @ id) @ X529) @ id)) != $true) & (((X527 @ id @ X530 @ id)) = $true) & ($true = ((X527 @ id @ X529 @ id)))) | (((X528 @ X525)) != $true) | ? [X539 : term] : (((X528 @ X539)) != ((X527 @ id @ X539 @ id))) | ? [X532 : subst,X533 : term,X531 : subst,X534 : subst] : ((((X527 @ (comp @ X532 @ X534) @ (sub @ X533 @ X534) @ X531)) = $true) & ($true != ((X527 @ X532 @ X533 @ (comp @ X534 @ X531)))))) <=> (hoasinduction_lem2v2 = $true)) & (hoaslamnotvar = $true) & (! [X216 : term,X217 : term,X215 : (subst > term > subst > $o)] : (? [X227 : subst,X226 : subst,X225 : subst,X224 : term] : (($true = ((X215 @ (comp @ X227 @ X225) @ (sub @ X224 @ X225) @ X226))) & ($true != ((X215 @ X227 @ X224 @ (comp @ X225 @ X226))))) | ? [X218 : subst,X220 : subst,X221 : term,X219 : subst] : (($true = ((X215 @ X220 @ X221 @ (comp @ X218 @ X219)))) & ($true != ((X215 @ (comp @ X220 @ X218) @ (sub @ X221 @ X218) @ X219)))) | (((X215 @ id @ X216 @ id)) != $true) | ($true = ((X215 @ id @ (ap @ X216 @ X217) @ id))) | ? [X223 : term,X222 : term] : ((((X215 @ id @ X222 @ id)) = $true) & (((X215 @ id @ X223 @ id)) = $true) & ($true != ((X215 @ id @ (ap @ (sub @ X223 @ id) @ X222) @ id)))) | (((X215 @ id @ X217 @ id)) != $true)) <=> (hoasinduction_lem2 = $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)))))))))))) & (hoaslamnotap_lthm = $true) & (! [X244 : subst,X243 : (term > $o),X242 : term] : ? [X245 : (term > $o)] : ! [X246 : term] : (((X245 @ X246)) = ((X243 @ (sub @ X246 @ (push @ X242 @ X244))))) <=> (pushprop_lem0 = $true)) & ((pushprop_lem1 = $true) <=> ! [X270 : term,X272 : subst,X271 : (term > $o),X268 : term,X269 : (term > $o)] : ((((X271 @ (sub @ X270 @ (push @ X268 @ X272)))) = $true) | (((X269 @ X270)) != $true))) & ((hoasinduction_lem3b = $true) <=> ! [X240 : term] : ? [X241 : (subst > term > term)] : (((sub @ X240 @ (push @ one @ sh))) = ((X241 @ sh @ one)))) & (hoasinduction_lem3a_gthm = $true) & (lamnotvar = $true) & (((d2subst @ d_id)) = sh) & ((axvarid = $true) <=> ! [X520 : term] : (((sub @ X520 @ id)) = X520)) & ((pushprop_lem0_lthm = $true) <=> ! [X43 : subst,X44 : (term > $o),X42 : term] : ? [X45 : (term > $o)] : ! [X46 : term] : (((X45 @ X46)) = ((X44 @ (sub @ X46 @ (push @ X42 @ X43)))))) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X70 : (subst > term > subst > $o),X71 : term] : (? [X72 : term] : (($true != ((X70 @ id @ (sub @ X71 @ (push @ X72 @ id)) @ id))) & ($true = ((X70 @ id @ X72 @ id)))) | (((X70 @ id @ (lam @ (sub @ X71 @ (push @ one @ sh))) @ id)) = $true) | ? [X73 : (subst > term > term)] : ((((X70 @ id @ (lam @ (X73 @ sh @ one)) @ id)) != $true) & ! [X74 : term] : (($true = ((X70 @ id @ (X73 @ id @ X74) @ id))) | ($true != ((X70 @ id @ X74 @ id)))) & ! [X76 : term,X77 : subst,X75 : subst] : (((sub @ (X73 @ X75 @ X76) @ X77)) = ((X73 @ (comp @ X75 @ X77) @ (sub @ X76 @ X77))))))) & (hoaslamnotvar_lthm = $true) & (pushprop_lem1v2_gthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & ((hoasinduction_lem3aaa = $true) <=> ! [X128 : (subst > term > subst > $o),X127 : term] : (? [X129 : term] : ((((X128 @ id @ X129 @ id)) = $true) & ($true != ((X128 @ id @ (sub @ X127 @ (push @ X129 @ id)) @ id)))) | ? [X130 : term,X131 : (subst > term > term)] : (! [X138 : term] : (($true != ((X128 @ id @ X138 @ id))) | (((X128 @ id @ (X131 @ id @ X138) @ id)) = $true)) & ! [X132 : term,X133 : subst,X134 : subst] : (((sub @ (X131 @ X133 @ X132) @ X134)) = ((sub @ (sub @ X130 @ (push @ X132 @ X133)) @ X134))) & (((X128 @ id @ (lam @ (X131 @ sh @ one)) @ id)) != $true) & ! [X137 : term,X136 : subst,X135 : subst] : (((X131 @ (comp @ X136 @ X135) @ (sub @ X137 @ X135))) = ((sub @ X130 @ (push @ (sub @ X137 @ X135) @ (comp @ X136 @ X135)))))) | (((X128 @ id @ (lam @ (sub @ X127 @ (push @ one @ sh))) @ id)) = $true))) & (! [X250 : (term > $o),X251 : (subst > term > subst > $o),X249 : term] : ((((X250 @ (lam @ X249))) = $true) | ? [X253 : (subst > term > term)] : (! [X254 : subst,X255 : term,X256 : subst] : (((sub @ (X253 @ X254 @ X255) @ X256)) = ((X253 @ (comp @ X254 @ X256) @ (sub @ X255 @ X256)))) & ! [X257 : term] : (($true = ((X251 @ id @ (X253 @ id @ X257) @ id))) | ($true != ((X251 @ id @ X257 @ id)))) & ($true != ((X251 @ id @ (lam @ (X253 @ sh @ one)) @ id)))) | ? [X252 : term] : (((X250 @ X252)) != ((X251 @ id @ X252 @ id))) | ? [X258 : term] : ((((X250 @ (sub @ X249 @ (push @ X258 @ id)))) != $true) & (((X250 @ X258)) = $true))) <=> (hoasinduction_lem3v2a = $true)) & (axabs = $true) & ((hoasinduction_lem3v2_f = $true) <=> ! [X78 : term] : ? [X79 : (subst > term > term)] : ! [X81 : subst,X80 : term] : (((X79 @ X81 @ X80)) = ((sub @ X78 @ (push @ X80 @ X81))))) & (hoasinduction_lem1 = $true) & ! [X101 : d_term,X100 : d_term] : ((X100 = X101) | (((d2term @ X101)) != ((d2term @ X100)))) & (! [X192 : (subst > term > subst > $o)] : ? [X193 : (term > $o)] : ! [X194 : term] : (((X192 @ id @ X194 @ id)) = ((X193 @ X194))) <=> (hoasinduction_lem0 = $true)) & ((hoasinduction_lthm = $true) <=> (? [X351 : term,X350 : (subst > term > subst > $o),X349 : term] : (! [X361 : term,X360 : term] : (($true != ((X350 @ id @ X360 @ id))) | (((X350 @ id @ X361 @ id)) != $true) | (((X350 @ id @ (ap @ (sub @ X360 @ id) @ X361) @ id)) = $true)) & ! [X353 : term,X355 : subst,X354 : subst,X352 : subst] : (($true != ((X350 @ (comp @ X352 @ X354) @ (sub @ X353 @ X354) @ X355))) | ($true = ((X350 @ X352 @ X353 @ (comp @ X354 @ X355))))) & (((X350 @ id @ X349 @ id)) = $true) & ! [X358 : subst,X357 : subst,X359 : subst,X356 : term] : ((((X350 @ (comp @ X357 @ X358) @ (sub @ X356 @ X358) @ X359)) = $true) | (((X350 @ X357 @ X356 @ (comp @ X358 @ X359))) != $true)) & ($true = ((X350 @ id @ X351 @ id))) & ($true != ((X350 @ id @ (ap @ X349 @ X351) @ id)))) | ! [X379 : term,X378 : (subst > term > subst > $o)] : (? [X392 : subst,X395 : term,X394 : subst,X393 : subst] : (($true = ((X378 @ (comp @ X393 @ X392) @ (sub @ X395 @ X392) @ X394))) & (((X378 @ X393 @ X395 @ (comp @ X392 @ X394))) != $true)) | ($true = ((X378 @ id @ X379 @ id))) | ? [X384 : (subst > term > term)] : (($true != ((X378 @ id @ (lam @ (X384 @ sh @ one)) @ id))) & ! [X385 : term] : ((((X378 @ id @ (X384 @ id @ X385) @ id)) = $true) | ($true != ((X378 @ id @ X385 @ id)))) & ! [X387 : term,X388 : subst,X386 : subst] : (((X384 @ (comp @ X388 @ X386) @ (sub @ X387 @ X386))) = ((sub @ (X384 @ X388 @ X387) @ X386)))) | ? [X390 : term,X389 : term] : ((((X378 @ id @ X390 @ id)) = $true) & ($true = ((X378 @ id @ X389 @ id))) & (((X378 @ id @ (ap @ (sub @ X389 @ id) @ X390) @ id)) != $true)) | ? [X391 : term] : ((((var @ (sub @ X391 @ id))) = $true) & (((X378 @ id @ X391 @ id)) != $true)) | ? [X381 : subst,X383 : subst,X382 : subst,X380 : term] : ((((X378 @ X381 @ X380 @ (comp @ X383 @ X382))) = $true) & (((X378 @ (comp @ X381 @ X383) @ (sub @ X380 @ X383) @ X382)) != $true))) | ? [X363 : (subst > term > subst > $o),X362 : term] : (! [X364 : term] : (($true = ((X363 @ id @ (sub @ X362 @ (push @ X364 @ id)) @ id))) | (((X363 @ id @ X364 @ id)) != $true)) & ! [X370 : subst,X369 : subst,X372 : term,X371 : subst] : (($true = ((X363 @ X370 @ X372 @ (comp @ X371 @ X369)))) | (((X363 @ (comp @ X370 @ X371) @ (sub @ X372 @ X371) @ X369)) != $true)) & ($true != ((X363 @ id @ (lam @ X362) @ id))) & ! [X373 : (subst > term > term)] : (? [X374 : term] : (($true = ((X363 @ id @ X374 @ id))) & ($true != ((X363 @ id @ (X373 @ id @ X374) @ id)))) | ? [X377 : subst,X376 : subst,X375 : term] : (((sub @ (X373 @ X376 @ X375) @ X377)) != ((X373 @ (comp @ X376 @ X377) @ (sub @ X375 @ X377)))) | ($true = ((X363 @ id @ (lam @ (X373 @ sh @ one)) @ id)))) & ! [X365 : subst,X366 : subst,X368 : term,X367 : subst] : (($true != ((X363 @ X365 @ X368 @ (comp @ X367 @ X366)))) | (((X363 @ (comp @ X365 @ X367) @ (sub @ X368 @ X367) @ X366)) = $true))) | ? [X343 : term,X342 : (term > $o)] : (($true != ((X342 @ X343))) & ! [X344 : term] : (? [X345 : term] : (($true = ((X342 @ X345))) & ($true != ((X342 @ (sub @ X344 @ (push @ X345 @ id)))))) | (((X342 @ (lam @ X344))) = $true)) & ! [X348 : term] : ((((X342 @ X348)) = $true) | ($true != ((var @ X348)))) & ! [X347 : term,X346 : term] : ((((X342 @ (ap @ X346 @ X347))) = $true) | (((X342 @ X347)) != $true) | ($true != ((X342 @ X346))))))) & ((? [X444 : term] : (((sub @ X444 @ id)) != X444) | ! [X445 : term] : (((sub @ X445 @ id)) = X445)) <=> (termmset_lthm = $true)) & ((hoaslaminj = $true) <=> ! [X511 : term,X512 : (subst > term > term),X513 : (subst > term > term),X510 : subst] : (? [X518 : term,X519 : subst,X517 : subst] : (((X512 @ (comp @ X519 @ X517) @ (sub @ X518 @ X517))) != ((sub @ (X512 @ X519 @ X518) @ X517))) | ? [X515 : subst,X516 : subst,X514 : term] : (((sub @ (X513 @ X515 @ X514) @ X516)) != ((X513 @ (comp @ X515 @ X516) @ (sub @ X514 @ X516)))) | (((lam @ (X512 @ sh @ one))) != ((lam @ (X513 @ sh @ one)))) | (((X512 @ X510 @ X511)) = ((X513 @ X510 @ X511))))) & (((d2term @ d_one)) = one) & (hoasinduction_lem1v2 = $true) & (pushprop_lem3v2_lthm = $true) & (((var @ (d2term @ d_one))) != $true) & (hoaslamnotap_gthm = $true) & ((axscons = $true) <=> ! [X247 : subst] : (((push @ (sub @ one @ X247) @ (comp @ sh @ X247))) = X247)) & (hoaslamnotvar_gthm = $true) & (hoaslaminj_gthm = $true) & (hoasinduction_lem3v2_gthm = $true) & (! [X204 : term,X205 : subst] : (((comp @ sh @ (push @ X204 @ X205))) = X205) <=> (axshiftcons = $true)) & (hoasinduction_lem3b_gthm = $true) & ((laminj = $true) <=> ! [X233 : term,X232 : term] : ((((lam @ X232)) != ((lam @ X233))) | (X232 = X233))) & (! [X140 : term,X139 : term,X141 : (subst > term > term)] : ((((lam @ (X141 @ sh @ one))) != ((ap @ (sub @ X140 @ id) @ X139))) | ? [X143 : subst,X142 : term,X144 : subst] : (((X141 @ (comp @ X143 @ X144) @ (sub @ X142 @ X144))) != ((sub @ (X141 @ X143 @ X142) @ X144)))) <=> (hoaslamnotap = $true)) & (pushprop_lem2v2 = $true) & ((? [X103 : subst] : (((comp @ X103 @ id)) != X103) | ? [X102 : subst] : (((comp @ id @ X102)) != X102) | (! [X105 : subst] : (((comp @ X105 @ id)) = X105) & ! [X104 : subst] : (((comp @ id @ X104)) = X104))) <=> (substmonoid_lthm = $true)) & (apnotvar = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((hoasapinj1 = $true) <=> ! [X314 : term,X317 : term,X316 : term,X315 : term] : ((X314 = X316) | (((ap @ (sub @ X316 @ id) @ X317)) != ((ap @ (sub @ X314 @ id) @ X315))))) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X188 : term] : ? [X189 : (subst > term > term)] : ! [X190 : subst,X191 : term] : (((X189 @ X190 @ X191)) = ((sub @ X188 @ (push @ X191 @ X190))))) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (hoasapinj2_gthm = $true) & ((hoasapinj2_lthm = $true) <=> (! [X149 : term,X152 : term,X151 : term,X150 : term] : ((X150 = X151) | (((ap @ (sub @ X149 @ id) @ X150)) != ((ap @ (sub @ X152 @ id) @ X151)))) | ? [X147 : term,X146 : term,X148 : term,X145 : term] : ((((ap @ X145 @ X148)) = ((ap @ X146 @ X147))) & (X147 != X148)))) & (hoasinduction_lem2_lthm = $true) & (! [X187 : term,X184 : term,X186 : term,X185 : term] : ((X186 = X187) | (((ap @ (sub @ X185 @ id) @ X186)) != ((ap @ (sub @ X184 @ id) @ X187)))) <=> (hoasapinj2 = $true)) & (! [X309 : (term > $o),X312 : subst,X311 : term,X310 : (term > $o)] : ((((X309 @ X311)) != $true) | ? [X313 : term] : (((X309 @ (sub @ X313 @ (push @ X311 @ X312)))) != ((X310 @ X313))) | ($true = ((X310 @ one)))) <=> (pushprop_lem1v2 = $true)) & (axclos = $true) & ((hoasapinj1_lthm = $true) <=> (? [X259 : term] : (((sub @ X259 @ id)) != X259) | ? [X263 : term,X261 : term,X260 : term,X262 : term] : ((((ap @ X261 @ X263)) = ((ap @ X262 @ X260))) & (X261 != X262)) | ! [X265 : term,X266 : term,X267 : term,X264 : term] : ((((ap @ (sub @ X267 @ id) @ X265)) != ((ap @ (sub @ X266 @ id) @ X264))) | (X266 = X267)))) & (pushprop_lem0_gthm = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (pushprop_lem3v2 = $true) & (! [X248 : term] : (((sub @ X248 @ id)) = X248) <=> (termmset = $true)) & (! [X65 : term,X64 : (term > $o)] : (? [X68 : term] : ((((X64 @ X68)) = $true) & ($true != ((X64 @ (lam @ X68))))) | ? [X66 : term,X67 : term] : ((((X64 @ X67)) = $true) & (((X64 @ X66)) = $true) & (((X64 @ (ap @ X67 @ X66))) != $true)) | ? [X69 : term] : ((((var @ X69)) = $true) & ($true != ((X64 @ X69)))) | (((X64 @ X65)) = $true)) <=> (induction = $true)) & (! [X334 : (subst > term > subst > $o),X335 : term] : (($true = ((X334 @ id @ (lam @ (sub @ X335 @ (push @ one @ sh))) @ id))) | ? [X341 : term] : (($true != ((X334 @ id @ (sub @ X335 @ (push @ X341 @ id)) @ id))) & (((X334 @ id @ X341 @ id)) = $true)) | ? [X336 : (subst > term > term)] : (! [X337 : term] : (($true != ((X334 @ id @ X337 @ id))) | (((X334 @ id @ (X336 @ id @ X337) @ id)) = $true)) & ! [X340 : term,X339 : subst,X338 : subst] : (((sub @ (X336 @ X338 @ X340) @ X339)) = ((X336 @ (comp @ X338 @ X339) @ (sub @ X340 @ X339)))) & (((X334 @ id @ (lam @ (X336 @ sh @ one)) @ id)) != $true))) <=> (hoasinduction_lem3aa = $true)) & (! [X82 : term] : ? [X83 : (subst > term > term)] : (((sub @ X82 @ (push @ one @ sh))) = ((X83 @ sh @ one))) <=> (hoasinduction_lem3b_lthm = $true)) & (pushprop_lem2v2_gthm = $true) & (hoasapinj1_gthm = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((! [X301 : term,X302 : (subst > term > subst > $o)] : ((((X302 @ id @ (lam @ X301) @ id)) = $true) | ? [X303 : (subst > term > term)] : (! [X307 : term] : ((((X302 @ id @ X307 @ id)) != $true) | ($true = ((X302 @ id @ (X303 @ id @ X307) @ id)))) & ($true != ((X302 @ id @ (lam @ (X303 @ sh @ one)) @ id))) & ! [X305 : term,X304 : subst,X306 : subst] : (((sub @ (X303 @ X304 @ X305) @ X306)) = ((X303 @ (comp @ X304 @ X306) @ (sub @ X305 @ X306))))) | ? [X308 : term] : (($true = ((X302 @ id @ X308 @ id))) & ($true != ((X302 @ id @ (sub @ X301 @ (push @ X308 @ id)) @ id))))) | ? [X292 : term] : (((sub @ X292 @ id)) != X292) | ? [X293 : (subst > term > subst > $o),X294 : term] : (! [X295 : (subst > term > term)] : (? [X299 : subst,X297 : term,X298 : subst] : (((X295 @ (comp @ X299 @ X298) @ (sub @ X297 @ X298))) != ((sub @ (X295 @ X299 @ X297) @ X298))) | (((X293 @ id @ (lam @ (X295 @ sh @ one)) @ id)) = $true) | ? [X296 : term] : (($true != ((X293 @ id @ (X295 @ id @ X296) @ id))) & ($true = ((X293 @ id @ X296 @ id))))) & ! [X300 : term] : (($true = ((X293 @ id @ (sub @ X294 @ (push @ X300 @ id)) @ id))) | (((X293 @ id @ X300 @ id)) != $true)) & ($true != ((X293 @ id @ (lam @ (sub @ X294 @ (push @ one @ sh))) @ id))))) <=> (hoasinduction_lem3a_lthm = $true)) & (pushprop_lem1_gthm = $true) & (axmap = $true) & ((hoaslaminj_lthm = $true) <=> (! [X501 : subst,X502 : (subst > term > term),X500 : (subst > term > term),X503 : term] : ((((X502 @ X501 @ X503)) = ((X500 @ X501 @ X503))) | (((lam @ (X502 @ sh @ one))) != ((lam @ (X500 @ sh @ one)))) | ? [X508 : subst,X507 : term,X509 : subst] : (((X500 @ (comp @ X509 @ X508) @ (sub @ X507 @ X508))) != ((sub @ (X500 @ X509 @ X507) @ X508))) | ? [X506 : term,X505 : subst,X504 : subst] : (((sub @ (X502 @ X505 @ X506) @ X504)) != ((X502 @ (comp @ X505 @ X504) @ (sub @ X506 @ X504))))) | ? [X499 : term,X498 : term] : ((((lam @ X498)) = ((lam @ X499))) & (X498 != X499)) | ? [X495 : subst,X494 : term] : (((sub @ one @ (push @ X494 @ X495))) != X494) | ? [X496 : term,X497 : subst] : (((comp @ sh @ (push @ X496 @ X497))) != X497))) & (hoasinduction_gthm = $true) & ((apinj2 = $true) <=> ! [X236 : term,X238 : term,X237 : term,X239 : term] : ((X236 = X239) | (((ap @ X237 @ X236)) != ((ap @ X238 @ X239))))) & (termmset_gthm = $true) & ((pushprop_lem1v2_lthm = $true) <=> (! [X417 : term,X414 : (term > $o),X416 : subst,X415 : (term > $o)] : ((((X415 @ one)) = $true) | ($true != ((X414 @ X417))) | ? [X418 : term] : (((X415 @ X418)) != ((X414 @ (sub @ X418 @ (push @ X417 @ X416)))))) | ? [X412 : term,X413 : subst] : (((sub @ one @ (push @ X412 @ X413))) != X412))) & ! [X106 : term] : ? [X107 : d_term] : (((d2term @ X107)) = X106) & ! [X318 : d_term] : (d_one = X318) & (hoasinduction_lem3_gthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (axapp = $true) & ((hoasinduction_lthm_3 = $true) <=> (? [X459 : (term > $o),X457 : (subst > term > subst > $o),X458 : term] : (! [X461 : term] : (($true = ((X459 @ (sub @ X458 @ (push @ X461 @ id))))) | ($true != ((X459 @ X461)))) & (((X459 @ (lam @ X458))) != $true) & ! [X462 : (subst > term > term)] : (? [X464 : subst,X463 : subst,X465 : term] : (((X462 @ (comp @ X463 @ X464) @ (sub @ X465 @ X464))) != ((sub @ (X462 @ X463 @ X465) @ X464))) | ? [X466 : term] : ((((X457 @ id @ X466 @ id)) = $true) & ($true != ((X457 @ id @ (X462 @ id @ X466) @ id)))) | (((X457 @ id @ (lam @ (X462 @ sh @ one)) @ id)) = $true)) & ! [X460 : term] : (((X459 @ X460)) = ((X457 @ id @ X460 @ id)))) | ? [X446 : (subst > term > subst > $o)] : ! [X447 : (term > $o)] : ? [X448 : term] : (((X447 @ X448)) != ((X446 @ id @ X448 @ id))) | ? [X456 : term] : (((sub @ X456 @ id)) != X456) | ! [X468 : term,X467 : (subst > term > subst > $o)] : ((((X467 @ id @ X468 @ id)) = $true) | ? [X482 : term,X481 : subst,X484 : subst,X483 : subst] : ((((X467 @ (comp @ X484 @ X481) @ (sub @ X482 @ X481) @ X483)) = $true) & ($true != ((X467 @ X484 @ X482 @ (comp @ X481 @ X483))))) | ? [X472 : term,X474 : subst,X475 : subst,X473 : subst] : ((((X467 @ (comp @ X474 @ X473) @ (sub @ X472 @ X473) @ X475)) != $true) & (((X467 @ X474 @ X472 @ (comp @ X473 @ X475))) = $true)) | ? [X471 : term] : ((((var @ (sub @ X471 @ id))) = $true) & (((X467 @ id @ X471 @ id)) != $true)) | ? [X470 : term,X469 : term] : ((((X467 @ id @ (ap @ (sub @ X469 @ id) @ X470) @ id)) != $true) & (((X467 @ id @ X469 @ id)) = $true) & (((X467 @ id @ X470 @ id)) = $true)) | ? [X476 : (subst > term > term)] : ((((X467 @ id @ (lam @ (X476 @ sh @ one)) @ id)) != $true) & ! [X477 : term] : ((((X467 @ id @ (X476 @ id @ X477) @ id)) = $true) | ($true != ((X467 @ id @ X477 @ id)))) & ! [X479 : term,X480 : subst,X478 : subst] : (((X476 @ (comp @ X478 @ X480) @ (sub @ X479 @ X480))) = ((sub @ (X476 @ X478 @ X479) @ X480))))) | ? [X449 : term,X450 : (term > $o)] : (! [X452 : term,X451 : term] : (($true = ((X450 @ (ap @ X452 @ X451)))) | ($true != ((X450 @ X451))) | ($true != ((X450 @ X452)))) & ! [X455 : term] : ((((var @ X455)) != $true) | ($true = ((X450 @ X455)))) & ($true != ((X450 @ X449))) & ! [X453 : term] : (? [X454 : term] : ((((X450 @ (sub @ X453 @ (push @ X454 @ id)))) != $true) & ($true = ((X450 @ X454)))) | (((X450 @ (lam @ X453))) = $true))))) & (hoasinduction_lem1v2_gthm = $true) & (! [X22 : term,X23 : (subst > term > subst > $o)] : (? [X26 : subst,X29 : subst,X28 : term,X27 : subst] : ((((X23 @ (comp @ X27 @ X26) @ (sub @ X28 @ X26) @ X29)) != $true) & (((X23 @ X27 @ X28 @ (comp @ X26 @ X29))) = $true)) | ? [X35 : (subst > term > term)] : ((((X23 @ id @ (lam @ (X35 @ sh @ one)) @ id)) != $true) & ! [X37 : subst,X38 : term,X36 : subst] : (((X35 @ (comp @ X37 @ X36) @ (sub @ X38 @ X36))) = ((sub @ (X35 @ X37 @ X38) @ X36))) & ! [X39 : term] : ((((X23 @ id @ X39 @ id)) != $true) | ($true = ((X23 @ id @ (X35 @ id @ X39) @ id))))) | ? [X25 : term,X24 : term] : ((((X23 @ id @ X24 @ id)) = $true) & (((X23 @ id @ (ap @ (sub @ X24 @ id) @ X25) @ id)) != $true) & (((X23 @ id @ X25 @ id)) = $true)) | ? [X31 : subst,X30 : subst,X32 : subst,X33 : term] : ((((X23 @ X30 @ X33 @ (comp @ X31 @ X32))) != $true) & ($true = ((X23 @ (comp @ X30 @ X31) @ (sub @ X33 @ X31) @ X32)))) | (((X23 @ id @ X22 @ id)) = $true) | ? [X34 : term] : (($true = ((var @ (sub @ X34 @ id)))) & (((X23 @ id @ X34 @ id)) != $true))) <=> (hoasinduction = $true)) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (! [X195 : term,X196 : term] : ((((sub @ X196 @ sh)) != ((sub @ X195 @ sh))) | (X195 = X196)) <=> (shinj = $true)) & ((? [X108 : term] : (((sub @ X108 @ id)) != X108) | ! [X110 : (term > $o),X109 : (subst > term > subst > $o),X111 : term] : (? [X123 : subst,X125 : term,X126 : subst,X124 : subst] : (($true = ((X109 @ (comp @ X124 @ X126) @ (sub @ X125 @ X126) @ X123))) & ($true != ((X109 @ X124 @ X125 @ (comp @ X126 @ X123))))) | ? [X121 : term] : (((X110 @ X121)) != ((X109 @ id @ X121 @ id))) | ($true = ((X110 @ (lam @ X111)))) | ? [X117 : subst,X118 : term,X120 : subst,X119 : subst] : ((((X109 @ X120 @ X118 @ (comp @ X119 @ X117))) = $true) & (((X109 @ (comp @ X120 @ X119) @ (sub @ X118 @ X119) @ X117)) != $true)) | ? [X112 : (subst > term > term)] : ((((X109 @ id @ (lam @ (X112 @ sh @ one)) @ id)) != $true) & ! [X113 : term] : (($true != ((X109 @ id @ X113 @ id))) | (((X109 @ id @ (X112 @ id @ X113) @ id)) = $true)) & ! [X114 : subst,X115 : term,X116 : subst] : (((sub @ (X112 @ X116 @ X115) @ X114)) = ((X112 @ (comp @ X116 @ X114) @ (sub @ X115 @ X114))))) | ? [X122 : term] : (($true != ((X110 @ (sub @ X111 @ (push @ X122 @ id))))) & (((X110 @ X122)) = $true)))) <=> (hoasinduction_lem3v2_lthm = $true)) & (hoasinduction_lem2_gthm = $true) & (hoasinduction_lem2v2_gthm = $true) & ((induction2lem = $true) <=> ! [X58 : (term > $o),X57 : subst,X56 : term] : (? [X61 : term] : (($true != ((X58 @ (sub @ X61 @ X57)))) & (((var @ X61)) = $true)) | ? [X60 : term,X59 : term] : ((((X58 @ X59)) = $true) & (((X58 @ X60)) = $true) & (((X58 @ (ap @ X60 @ X59))) != $true)) | ? [X62 : term] : (! [X63 : term] : (($true != ((X58 @ X63))) | (((X58 @ (sub @ X62 @ (push @ X63 @ id)))) = $true)) & ($true != ((X58 @ (lam @ X62))))) | (((X58 @ (sub @ X56 @ X57))) = $true))) & (pushprop_lem2v2_lthm = $true) & ((? [X485 : subst,X486 : term] : (((sub @ one @ (push @ X486 @ X485))) != X486) | ! [X489 : term,X493 : (term > $o),X490 : term,X491 : (term > $o),X492 : subst] : ((((X493 @ X490)) != $true) | (((X491 @ (sub @ X490 @ (push @ X489 @ X492)))) = $true)) | ? [X487 : subst,X488 : term] : (((comp @ sh @ (push @ X488 @ X487))) != X487)) <=> (pushprop_lem1_lthm = $true)) & ! [X41 : d_subst,X40 : d_subst] : ((((d2subst @ X40)) != ((d2subst @ X41))) | (X40 = X41)) & ((axidl = $true) <=> ! [X47 : subst] : (((comp @ id @ X47)) = X47)) & (! [X49 : term,X48 : (subst > term > subst > $o)] : (($true = ((X48 @ id @ (lam @ X49) @ id))) | ? [X51 : (subst > term > term)] : ((((X48 @ id @ (lam @ (X51 @ sh @ one)) @ id)) != $true) & ! [X53 : term,X54 : subst,X52 : subst] : (((X51 @ (comp @ X54 @ X52) @ (sub @ X53 @ X52))) = ((sub @ (X51 @ X54 @ X53) @ X52))) & ! [X55 : term] : (($true != ((X48 @ id @ X55 @ id))) | (((X48 @ id @ (X51 @ id @ X55) @ id)) = $true))) | ? [X50 : term] : (($true != ((X48 @ id @ (sub @ X49 @ (push @ X50 @ id)) @ id))) & ($true = ((X48 @ id @ X50 @ id))))) <=> (hoasinduction_lem3a = $true)) & (induction2_gthm = $true) & (ulamvarsh = $true) & ((hoasinduction_lem3v2a_lthm = $true) <=> (? [X319 : term] : ! [X320 : (subst > term > term)] : ? [X321 : subst,X322 : term] : (((sub @ X319 @ (push @ X322 @ X321))) != ((X320 @ X321 @ X322))) | ! [X325 : (subst > term > subst > $o),X326 : term,X324 : (term > $o)] : (? [X329 : (subst > term > term)] : (! [X332 : subst,X330 : subst,X331 : term] : (((sub @ (X329 @ X330 @ X331) @ X332)) = ((X329 @ (comp @ X330 @ X332) @ (sub @ X331 @ X332)))) & ! [X333 : term] : (($true = ((X325 @ id @ (X329 @ id @ X333) @ id))) | (((X325 @ id @ X333 @ id)) != $true)) & ($true != ((X325 @ id @ (lam @ (X329 @ sh @ one)) @ id)))) | ? [X328 : term] : (((X325 @ id @ X328 @ id)) != ((X324 @ X328))) | ($true = ((X324 @ (lam @ X326)))) | ? [X327 : term] : ((((X324 @ (sub @ X326 @ (push @ X327 @ id)))) != $true) & ($true = ((X324 @ X327))))) | ? [X323 : term] : (((sub @ X323 @ id)) != X323))) & (pushprop = $true) & ((axvarcons = $true) <=> ! [X234 : subst,X235 : term] : (((sub @ one @ (push @ X235 @ X234))) = X235)) & (ulamvarind = $true)), inference(flattening,[],[f13])). thf(f15,definition,( ! [X0 : (subst > term > subst > $o)] : (($true = ((sP0 @ X0))) <=> ? [X9 : (subst > term > term)] : (! [X13 : term] : ((((X0 @ id @ X13 @ id)) != $true) | ($true = ((X0 @ id @ (X9 @ id @ X13) @ id)))) & ($true != ((X0 @ id @ (hoaslam @ id @ (^[Y0 : subst]: ((^[Y1 : term]: (X9 @ Y0 @ Y1))))) @ id))) & ! [X12 : subst,X10 : term,X11 : subst] : (((X9 @ (comp @ X12 @ X11) @ (sub @ X10 @ X11))) = ((sub @ (X9 @ X12 @ X10) @ X11)))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f16,definition,( (sP1 = $true) <=> ! [X0 : (subst > term > subst > $o)] : (! [X14 : term] : ((((X0 @ id @ (lam @ X14) @ id)) = $true) | ? [X15 : term] : (($true != ((X0 @ id @ (sub @ X14 @ (push @ X15 @ id)) @ id))) & (((X0 @ id @ X15 @ id)) = $true))) | ? [X6 : subst,X5 : subst,X8 : subst,X7 : term] : ((((X0 @ (comp @ X5 @ X6) @ (sub @ X7 @ X6) @ X8)) = $true) & (((X0 @ X5 @ X7 @ (comp @ X6 @ X8))) != $true)) | ($true = ((sP0 @ X0))) | ? [X1 : subst,X3 : subst,X2 : term,X4 : subst] : ((((X0 @ (comp @ X3 @ X1) @ (sub @ X2 @ X1) @ X4)) != $true) & ($true = ((X0 @ X3 @ X2 @ (comp @ X1 @ X4))))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f17,plain,( (sP1 = $true) <~> (hoasinduction_lem3 = $true)), inference(definition_folding,[],[f12,f16,f15])). thf(f18,definition,( (sP2 = $true) <=> (? [X319 : term] : ! [X320 : (subst > term > term)] : ? [X321 : subst,X322 : term] : (((sub @ X319 @ (push @ X322 @ X321))) != ((X320 @ X321 @ X322))) | ! [X325 : (subst > term > subst > $o),X326 : term,X324 : (term > $o)] : (? [X329 : (subst > term > term)] : (! [X332 : subst,X330 : subst,X331 : term] : (((sub @ (X329 @ X330 @ X331) @ X332)) = ((X329 @ (comp @ X330 @ X332) @ (sub @ X331 @ X332)))) & ! [X333 : term] : (($true = ((X325 @ id @ (X329 @ id @ X333) @ id))) | (((X325 @ id @ X333 @ id)) != $true)) & ($true != ((X325 @ id @ (lam @ (X329 @ sh @ one)) @ id)))) | ? [X328 : term] : (((X325 @ id @ X328 @ id)) != ((X324 @ X328))) | ($true = ((X324 @ (lam @ X326)))) | ? [X327 : term] : ((((X324 @ (sub @ X326 @ (push @ X327 @ id)))) != $true) & ($true = ((X324 @ X327))))) | ? [X323 : term] : (((sub @ X323 @ id)) != X323))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f19,definition,( ($true = sP3) <=> ! [X49 : term,X48 : (subst > term > subst > $o)] : (($true = ((X48 @ id @ (lam @ X49) @ id))) | ? [X51 : (subst > term > term)] : ((((X48 @ id @ (lam @ (X51 @ sh @ one)) @ id)) != $true) & ! [X53 : term,X54 : subst,X52 : subst] : (((X51 @ (comp @ X54 @ X52) @ (sub @ X53 @ X52))) = ((sub @ (X51 @ X54 @ X53) @ X52))) & ! [X55 : term] : (($true != ((X48 @ id @ X55 @ id))) | (((X48 @ id @ (X51 @ id @ X55) @ id)) = $true))) | ? [X50 : term] : (($true != ((X48 @ id @ (sub @ X49 @ (push @ X50 @ id)) @ id))) & ($true = ((X48 @ id @ X50 @ id)))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f20,definition,( ! [X58 : (term > $o)] : ((((sP4 @ X58)) = $true) <=> ? [X60 : term,X59 : term] : ((((X58 @ X59)) = $true) & (((X58 @ X60)) = $true) & (((X58 @ (ap @ X60 @ X59))) != $true)))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f21,definition,( ($true = sP5) <=> ! [X58 : (term > $o),X57 : subst,X56 : term] : (? [X61 : term] : (($true != ((X58 @ (sub @ X61 @ X57)))) & (((var @ X61)) = $true)) | (((sP4 @ X58)) = $true) | ? [X62 : term] : (! [X63 : term] : (($true != ((X58 @ X63))) | (((X58 @ (sub @ X62 @ (push @ X63 @ id)))) = $true)) & ($true != ((X58 @ (lam @ X62))))) | (((X58 @ (sub @ X56 @ X57))) = $true))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f22,definition,( ! [X109 : (subst > term > subst > $o)] : ((((sP6 @ X109)) = $true) <=> ? [X112 : (subst > term > term)] : ((((X109 @ id @ (lam @ (X112 @ sh @ one)) @ id)) != $true) & ! [X113 : term] : (($true != ((X109 @ id @ X113 @ id))) | (((X109 @ id @ (X112 @ id @ X113) @ id)) = $true)) & ! [X114 : subst,X115 : term,X116 : subst] : (((sub @ (X112 @ X116 @ X115) @ X114)) = ((X112 @ (comp @ X116 @ X114) @ (sub @ X115 @ X114))))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f23,definition,( (sP7 = $true) <=> (? [X108 : term] : (((sub @ X108 @ id)) != X108) | ! [X110 : (term > $o),X109 : (subst > term > subst > $o),X111 : term] : (? [X123 : subst,X125 : term,X126 : subst,X124 : subst] : (($true = ((X109 @ (comp @ X124 @ X126) @ (sub @ X125 @ X126) @ X123))) & ($true != ((X109 @ X124 @ X125 @ (comp @ X126 @ X123))))) | ? [X121 : term] : (((X110 @ X121)) != ((X109 @ id @ X121 @ id))) | ($true = ((X110 @ (lam @ X111)))) | ? [X117 : subst,X118 : term,X120 : subst,X119 : subst] : ((((X109 @ X120 @ X118 @ (comp @ X119 @ X117))) = $true) & (((X109 @ (comp @ X120 @ X119) @ (sub @ X118 @ X119) @ X117)) != $true)) | (((sP6 @ X109)) = $true) | ? [X122 : term] : (($true != ((X110 @ (sub @ X111 @ (push @ X122 @ id))))) & (((X110 @ X122)) = $true))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f24,definition,( ! [X23 : (subst > term > subst > $o)] : ((((sP8 @ X23)) = $true) <=> ? [X25 : term,X24 : term] : ((((X23 @ id @ X24 @ id)) = $true) & (((X23 @ id @ (ap @ (sub @ X24 @ id) @ X25) @ id)) != $true) & (((X23 @ id @ X25 @ id)) = $true)))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f25,definition,( ! [X23 : (subst > term > subst > $o)] : ((((sP9 @ X23)) = $true) <=> ? [X35 : (subst > term > term)] : ((((X23 @ id @ (lam @ (X35 @ sh @ one)) @ id)) != $true) & ! [X37 : subst,X38 : term,X36 : subst] : (((X35 @ (comp @ X37 @ X36) @ (sub @ X38 @ X36))) = ((sub @ (X35 @ X37 @ X38) @ X36))) & ! [X39 : term] : ((((X23 @ id @ X39 @ id)) != $true) | ($true = ((X23 @ id @ (X35 @ id @ X39) @ id))))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f26,definition,( (sP10 = $true) <=> ! [X22 : term,X23 : (subst > term > subst > $o)] : (? [X26 : subst,X29 : subst,X28 : term,X27 : subst] : ((((X23 @ (comp @ X27 @ X26) @ (sub @ X28 @ X26) @ X29)) != $true) & (((X23 @ X27 @ X28 @ (comp @ X26 @ X29))) = $true)) | (((sP9 @ X23)) = $true) | (((sP8 @ X23)) = $true) | ? [X31 : subst,X30 : subst,X32 : subst,X33 : term] : ((((X23 @ X30 @ X33 @ (comp @ X31 @ X32))) != $true) & ($true = ((X23 @ (comp @ X30 @ X31) @ (sub @ X33 @ X31) @ X32)))) | (((X23 @ id @ X22 @ id)) = $true) | ? [X34 : term] : (($true = ((var @ (sub @ X34 @ id)))) & (((X23 @ id @ X34 @ id)) != $true)))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f27,definition,( ! [X450 : (term > $o)] : ((((sP11 @ X450)) = $true) <=> ! [X452 : term,X451 : term] : (($true = ((X450 @ (ap @ X452 @ X451)))) | ($true != ((X450 @ X451))) | ($true != ((X450 @ X452)))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f28,definition,( ! [X467 : (subst > term > subst > $o)] : (($true = ((sP12 @ X467))) <=> ? [X476 : (subst > term > term)] : ((((X467 @ id @ (lam @ (X476 @ sh @ one)) @ id)) != $true) & ! [X477 : term] : ((((X467 @ id @ (X476 @ id @ X477) @ id)) = $true) | ($true != ((X467 @ id @ X477 @ id)))) & ! [X479 : term,X480 : subst,X478 : subst] : (((X476 @ (comp @ X478 @ X480) @ (sub @ X479 @ X480))) = ((sub @ (X476 @ X478 @ X479) @ X480)))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f29,definition,( ! [X467 : (subst > term > subst > $o)] : ((((sP13 @ X467)) = $true) <=> ? [X470 : term,X469 : term] : ((((X467 @ id @ (ap @ (sub @ X469 @ id) @ X470) @ id)) != $true) & (((X467 @ id @ X469 @ id)) = $true) & (((X467 @ id @ X470 @ id)) = $true)))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f30,definition,( (sP14 = $true) <=> ! [X468 : term,X467 : (subst > term > subst > $o)] : ((((X467 @ id @ X468 @ id)) = $true) | ? [X482 : term,X481 : subst,X484 : subst,X483 : subst] : ((((X467 @ (comp @ X484 @ X481) @ (sub @ X482 @ X481) @ X483)) = $true) & ($true != ((X467 @ X484 @ X482 @ (comp @ X481 @ X483))))) | ? [X472 : term,X474 : subst,X475 : subst,X473 : subst] : ((((X467 @ (comp @ X474 @ X473) @ (sub @ X472 @ X473) @ X475)) != $true) & (((X467 @ X474 @ X472 @ (comp @ X473 @ X475))) = $true)) | ? [X471 : term] : ((((var @ (sub @ X471 @ id))) = $true) & (((X467 @ id @ X471 @ id)) != $true)) | (((sP13 @ X467)) = $true) | ($true = ((sP12 @ X467))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f31,definition,( ($true = sP15) <=> ? [X459 : (term > $o),X457 : (subst > term > subst > $o),X458 : term] : (! [X461 : term] : (($true = ((X459 @ (sub @ X458 @ (push @ X461 @ id))))) | ($true != ((X459 @ X461)))) & (((X459 @ (lam @ X458))) != $true) & ! [X462 : (subst > term > term)] : (? [X464 : subst,X463 : subst,X465 : term] : (((X462 @ (comp @ X463 @ X464) @ (sub @ X465 @ X464))) != ((sub @ (X462 @ X463 @ X465) @ X464))) | ? [X466 : term] : ((((X457 @ id @ X466 @ id)) = $true) & ($true != ((X457 @ id @ (X462 @ id @ X466) @ id)))) | (((X457 @ id @ (lam @ (X462 @ sh @ one)) @ id)) = $true)) & ! [X460 : term] : (((X459 @ X460)) = ((X457 @ id @ X460 @ id))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f32,definition,( ($true = sP16) <=> (($true = sP15) | ? [X446 : (subst > term > subst > $o)] : ! [X447 : (term > $o)] : ? [X448 : term] : (((X447 @ X448)) != ((X446 @ id @ X448 @ id))) | ? [X456 : term] : (((sub @ X456 @ id)) != X456) | (sP14 = $true) | ? [X449 : term,X450 : (term > $o)] : ((((sP11 @ X450)) = $true) & ! [X455 : term] : ((((var @ X455)) != $true) | ($true = ((X450 @ X455)))) & ($true != ((X450 @ X449))) & ! [X453 : term] : (? [X454 : term] : ((((X450 @ (sub @ X453 @ (push @ X454 @ id)))) != $true) & ($true = ((X450 @ X454)))) | (((X450 @ (lam @ X453))) = $true))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f33,definition,( ($true = sP17) <=> (! [X501 : subst,X502 : (subst > term > term),X500 : (subst > term > term),X503 : term] : ((((X502 @ X501 @ X503)) = ((X500 @ X501 @ X503))) | (((lam @ (X502 @ sh @ one))) != ((lam @ (X500 @ sh @ one)))) | ? [X508 : subst,X507 : term,X509 : subst] : (((X500 @ (comp @ X509 @ X508) @ (sub @ X507 @ X508))) != ((sub @ (X500 @ X509 @ X507) @ X508))) | ? [X506 : term,X505 : subst,X504 : subst] : (((sub @ (X502 @ X505 @ X506) @ X504)) != ((X502 @ (comp @ X505 @ X504) @ (sub @ X506 @ X504))))) | ? [X499 : term,X498 : term] : ((((lam @ X498)) = ((lam @ X499))) & (X498 != X499)) | ? [X495 : subst,X494 : term] : (((sub @ one @ (push @ X494 @ X495))) != X494) | ? [X496 : term,X497 : subst] : (((comp @ sh @ (push @ X496 @ X497))) != X497))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f34,definition,( (sP18 = $true) <=> ? [X293 : (subst > term > subst > $o),X294 : term] : (! [X295 : (subst > term > term)] : (? [X299 : subst,X297 : term,X298 : subst] : (((X295 @ (comp @ X299 @ X298) @ (sub @ X297 @ X298))) != ((sub @ (X295 @ X299 @ X297) @ X298))) | (((X293 @ id @ (lam @ (X295 @ sh @ one)) @ id)) = $true) | ? [X296 : term] : (($true != ((X293 @ id @ (X295 @ id @ X296) @ id))) & ($true = ((X293 @ id @ X296 @ id))))) & ! [X300 : term] : (($true = ((X293 @ id @ (sub @ X294 @ (push @ X300 @ id)) @ id))) | (((X293 @ id @ X300 @ id)) != $true)) & ($true != ((X293 @ id @ (lam @ (sub @ X294 @ (push @ one @ sh))) @ id))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f35,definition,( ($true = sP19) <=> (! [X301 : term,X302 : (subst > term > subst > $o)] : ((((X302 @ id @ (lam @ X301) @ id)) = $true) | ? [X303 : (subst > term > term)] : (! [X307 : term] : ((((X302 @ id @ X307 @ id)) != $true) | ($true = ((X302 @ id @ (X303 @ id @ X307) @ id)))) & ($true != ((X302 @ id @ (lam @ (X303 @ sh @ one)) @ id))) & ! [X305 : term,X304 : subst,X306 : subst] : (((sub @ (X303 @ X304 @ X305) @ X306)) = ((X303 @ (comp @ X304 @ X306) @ (sub @ X305 @ X306))))) | ? [X308 : term] : (($true = ((X302 @ id @ X308 @ id))) & ($true != ((X302 @ id @ (sub @ X301 @ (push @ X308 @ id)) @ id))))) | ? [X292 : term] : (((sub @ X292 @ id)) != X292) | (sP18 = $true))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f36,definition,( ($true = sP20) <=> ! [X334 : (subst > term > subst > $o),X335 : term] : (($true = ((X334 @ id @ (lam @ (sub @ X335 @ (push @ one @ sh))) @ id))) | ? [X341 : term] : (($true != ((X334 @ id @ (sub @ X335 @ (push @ X341 @ id)) @ id))) & (((X334 @ id @ X341 @ id)) = $true)) | ? [X336 : (subst > term > term)] : (! [X337 : term] : (($true != ((X334 @ id @ X337 @ id))) | (((X334 @ id @ (X336 @ id @ X337) @ id)) = $true)) & ! [X340 : term,X339 : subst,X338 : subst] : (((sub @ (X336 @ X338 @ X340) @ X339)) = ((X336 @ (comp @ X338 @ X339) @ (sub @ X340 @ X339)))) & (((X334 @ id @ (lam @ (X336 @ sh @ one)) @ id)) != $true)))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f37,definition,( ! [X64 : (term > $o)] : ((((sP21 @ X64)) = $true) <=> ? [X66 : term,X67 : term] : ((((X64 @ X67)) = $true) & (((X64 @ X66)) = $true) & (((X64 @ (ap @ X67 @ X66))) != $true)))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f38,definition,( ! [X342 : (term > $o)] : (($true = ((sP22 @ X342))) <=> ! [X347 : term,X346 : term] : ((((X342 @ (ap @ X346 @ X347))) = $true) | (((X342 @ X347)) != $true) | ($true != ((X342 @ X346)))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f39,definition,( ! [X363 : (subst > term > subst > $o)] : ((((sP23 @ X363)) = $true) <=> ! [X373 : (subst > term > term)] : (? [X374 : term] : (($true = ((X363 @ id @ X374 @ id))) & ($true != ((X363 @ id @ (X373 @ id @ X374) @ id)))) | ? [X377 : subst,X376 : subst,X375 : term] : (((sub @ (X373 @ X376 @ X375) @ X377)) != ((X373 @ (comp @ X376 @ X377) @ (sub @ X375 @ X377)))) | ($true = ((X363 @ id @ (lam @ (X373 @ sh @ one)) @ id)))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f40,definition,( ! [X378 : (subst > term > subst > $o)] : ((((sP24 @ X378)) = $true) <=> ? [X390 : term,X389 : term] : ((((X378 @ id @ X390 @ id)) = $true) & ($true = ((X378 @ id @ X389 @ id))) & (((X378 @ id @ (ap @ (sub @ X389 @ id) @ X390) @ id)) != $true)))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f41,definition,( ! [X378 : (subst > term > subst > $o)] : ((((sP25 @ X378)) = $true) <=> ? [X384 : (subst > term > term)] : (($true != ((X378 @ id @ (lam @ (X384 @ sh @ one)) @ id))) & ! [X385 : term] : ((((X378 @ id @ (X384 @ id @ X385) @ id)) = $true) | ($true != ((X378 @ id @ X385 @ id)))) & ! [X387 : term,X388 : subst,X386 : subst] : (((X384 @ (comp @ X388 @ X386) @ (sub @ X387 @ X386))) = ((sub @ (X384 @ X388 @ X387) @ X386)))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f42,definition,( ! [X350 : (subst > term > subst > $o)] : (($true = ((sP26 @ X350))) <=> ! [X361 : term,X360 : term] : (($true != ((X350 @ id @ X360 @ id))) | (((X350 @ id @ X361 @ id)) != $true) | (((X350 @ id @ (ap @ (sub @ X360 @ id) @ X361) @ id)) = $true)))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f43,definition,( (sP27 = $true) <=> ? [X363 : (subst > term > subst > $o),X362 : term] : (! [X364 : term] : (($true = ((X363 @ id @ (sub @ X362 @ (push @ X364 @ id)) @ id))) | (((X363 @ id @ X364 @ id)) != $true)) & ! [X370 : subst,X369 : subst,X372 : term,X371 : subst] : (($true = ((X363 @ X370 @ X372 @ (comp @ X371 @ X369)))) | (((X363 @ (comp @ X370 @ X371) @ (sub @ X372 @ X371) @ X369)) != $true)) & ($true != ((X363 @ id @ (lam @ X362) @ id))) & (((sP23 @ X363)) = $true) & ! [X365 : subst,X366 : subst,X368 : term,X367 : subst] : (($true != ((X363 @ X365 @ X368 @ (comp @ X367 @ X366)))) | (((X363 @ (comp @ X365 @ X367) @ (sub @ X368 @ X367) @ X366)) = $true)))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f44,definition,( ($true = sP28) <=> ! [X379 : term,X378 : (subst > term > subst > $o)] : (? [X392 : subst,X395 : term,X394 : subst,X393 : subst] : (($true = ((X378 @ (comp @ X393 @ X392) @ (sub @ X395 @ X392) @ X394))) & (((X378 @ X393 @ X395 @ (comp @ X392 @ X394))) != $true)) | ($true = ((X378 @ id @ X379 @ id))) | (((sP25 @ X378)) = $true) | (((sP24 @ X378)) = $true) | ? [X391 : term] : ((((var @ (sub @ X391 @ id))) = $true) & (((X378 @ id @ X391 @ id)) != $true)) | ? [X381 : subst,X383 : subst,X382 : subst,X380 : term] : ((((X378 @ X381 @ X380 @ (comp @ X383 @ X382))) = $true) & (((X378 @ (comp @ X381 @ X383) @ (sub @ X380 @ X383) @ X382)) != $true)))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f45,definition,( ($true = sP29) <=> ? [X343 : term,X342 : (term > $o)] : (($true != ((X342 @ X343))) & ! [X344 : term] : (? [X345 : term] : (($true = ((X342 @ X345))) & ($true != ((X342 @ (sub @ X344 @ (push @ X345 @ id)))))) | (((X342 @ (lam @ X344))) = $true)) & ! [X348 : term] : ((((X342 @ X348)) = $true) | ($true != ((var @ X348)))) & ($true = ((sP22 @ X342))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f46,definition,( ($true = sP30) <=> (? [X351 : term,X350 : (subst > term > subst > $o),X349 : term] : (($true = ((sP26 @ X350))) & ! [X353 : term,X355 : subst,X354 : subst,X352 : subst] : (($true != ((X350 @ (comp @ X352 @ X354) @ (sub @ X353 @ X354) @ X355))) | ($true = ((X350 @ X352 @ X353 @ (comp @ X354 @ X355))))) & (((X350 @ id @ X349 @ id)) = $true) & ! [X358 : subst,X357 : subst,X359 : subst,X356 : term] : ((((X350 @ (comp @ X357 @ X358) @ (sub @ X356 @ X358) @ X359)) = $true) | (((X350 @ X357 @ X356 @ (comp @ X358 @ X359))) != $true)) & ($true = ((X350 @ id @ X351 @ id))) & ($true != ((X350 @ id @ (ap @ X349 @ X351) @ id)))) | ($true = sP28) | (sP27 = $true) | ($true = sP29))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f47,definition,( (sP31 = $true) <=> ! [X250 : (term > $o),X251 : (subst > term > subst > $o),X249 : term] : ((((X250 @ (lam @ X249))) = $true) | ? [X253 : (subst > term > term)] : (! [X254 : subst,X255 : term,X256 : subst] : (((sub @ (X253 @ X254 @ X255) @ X256)) = ((X253 @ (comp @ X254 @ X256) @ (sub @ X255 @ X256)))) & ! [X257 : term] : (($true = ((X251 @ id @ (X253 @ id @ X257) @ id))) | ($true != ((X251 @ id @ X257 @ id)))) & ($true != ((X251 @ id @ (lam @ (X253 @ sh @ one)) @ id)))) | ? [X252 : term] : (((X250 @ X252)) != ((X251 @ id @ X252 @ id))) | ? [X258 : term] : ((((X250 @ (sub @ X249 @ (push @ X258 @ id)))) != $true) & (((X250 @ X258)) = $true)))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f48,definition,( ($true = sP32) <=> ! [X128 : (subst > term > subst > $o),X127 : term] : (? [X129 : term] : ((((X128 @ id @ X129 @ id)) = $true) & ($true != ((X128 @ id @ (sub @ X127 @ (push @ X129 @ id)) @ id)))) | ? [X130 : term,X131 : (subst > term > term)] : (! [X138 : term] : (($true != ((X128 @ id @ X138 @ id))) | (((X128 @ id @ (X131 @ id @ X138) @ id)) = $true)) & ! [X132 : term,X133 : subst,X134 : subst] : (((sub @ (X131 @ X133 @ X132) @ X134)) = ((sub @ (sub @ X130 @ (push @ X132 @ X133)) @ X134))) & (((X128 @ id @ (lam @ (X131 @ sh @ one)) @ id)) != $true) & ! [X137 : term,X136 : subst,X135 : subst] : (((X131 @ (comp @ X136 @ X135) @ (sub @ X137 @ X135))) = ((sub @ X130 @ (push @ (sub @ X137 @ X135) @ (comp @ X136 @ X135)))))) | (((X128 @ id @ (lam @ (sub @ X127 @ (push @ one @ sh))) @ id)) = $true))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f49,definition,( (sP33 = $true) <=> ! [X70 : (subst > term > subst > $o),X71 : term] : (? [X72 : term] : (($true != ((X70 @ id @ (sub @ X71 @ (push @ X72 @ id)) @ id))) & ($true = ((X70 @ id @ X72 @ id)))) | (((X70 @ id @ (lam @ (sub @ X71 @ (push @ one @ sh))) @ id)) = $true) | ? [X73 : (subst > term > term)] : ((((X70 @ id @ (lam @ (X73 @ sh @ one)) @ id)) != $true) & ! [X74 : term] : (($true = ((X70 @ id @ (X73 @ id @ X74) @ id))) | ($true != ((X70 @ id @ X74 @ id)))) & ! [X76 : term,X77 : subst,X75 : subst] : (((sub @ (X73 @ X75 @ X76) @ X77)) = ((X73 @ (comp @ X75 @ X77) @ (sub @ X76 @ X77))))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f50,definition,( ! [X215 : (subst > term > subst > $o)] : (($true = ((sP34 @ X215))) <=> ? [X223 : term,X222 : term] : ((((X215 @ id @ X222 @ id)) = $true) & (((X215 @ id @ X223 @ id)) = $true) & ($true != ((X215 @ id @ (ap @ (sub @ X223 @ id) @ X222) @ id)))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f51,definition,( (sP35 = $true) <=> ! [X216 : term,X217 : term,X215 : (subst > term > subst > $o)] : (? [X227 : subst,X226 : subst,X225 : subst,X224 : term] : (($true = ((X215 @ (comp @ X227 @ X225) @ (sub @ X224 @ X225) @ X226))) & ($true != ((X215 @ X227 @ X224 @ (comp @ X225 @ X226))))) | ? [X218 : subst,X220 : subst,X221 : term,X219 : subst] : (($true = ((X215 @ X220 @ X221 @ (comp @ X218 @ X219)))) & ($true != ((X215 @ (comp @ X220 @ X218) @ (sub @ X221 @ X218) @ X219)))) | (((X215 @ id @ X216 @ id)) != $true) | ($true = ((X215 @ id @ (ap @ X216 @ X217) @ id))) | ($true = ((sP34 @ X215))) | (((X215 @ id @ X217 @ id)) != $true))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f52,definition,( ! [X527 : (subst > term > subst > $o)] : ((((sP36 @ X527)) = $true) <=> ? [X529 : term,X530 : term] : ((((X527 @ id @ (ap @ (sub @ X530 @ id) @ X529) @ id)) != $true) & (((X527 @ id @ X530 @ id)) = $true) & ($true = ((X527 @ id @ X529 @ id)))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f53,definition,( ($true = sP37) <=> ! [X528 : (term > $o),X525 : term,X527 : (subst > term > subst > $o),X526 : term] : (? [X536 : subst,X537 : subst,X535 : subst,X538 : term] : (($true = ((X527 @ X535 @ X538 @ (comp @ X536 @ X537)))) & ($true != ((X527 @ (comp @ X535 @ X536) @ (sub @ X538 @ X536) @ X537)))) | ($true != ((X528 @ X526))) | ($true = ((X528 @ (ap @ X526 @ X525)))) | (((sP36 @ X527)) = $true) | (((X528 @ X525)) != $true) | ? [X539 : term] : (((X528 @ X539)) != ((X527 @ id @ X539 @ id))) | ? [X532 : subst,X533 : term,X531 : subst,X534 : subst] : ((((X527 @ (comp @ X532 @ X534) @ (sub @ X533 @ X534) @ X531)) = $true) & ($true != ((X527 @ X532 @ X533 @ (comp @ X534 @ X531))))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f54,definition,( ! [X6 : (subst > term > subst > $o)] : (($true = ((sP38 @ X6))) <=> ? [X12 : (subst > term > term)] : (! [X13 : term] : ((((X6 @ id @ X13 @ id)) != $true) | (((X6 @ id @ (X12 @ id @ X13) @ id)) = $true)) & ! [X16 : subst,X14 : term,X15 : subst] : (((sub @ (X12 @ X15 @ X14) @ X16)) = ((X12 @ (comp @ X15 @ X16) @ (sub @ X14 @ X16)))) & (((X6 @ id @ (lam @ (X12 @ sh @ one)) @ id)) != $true)))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f55,definition,( ($true = sP39) <=> ! [X5 : (term > $o),X4 : term,X6 : (subst > term > subst > $o)] : ((((X5 @ (lam @ X4))) = $true) | ? [X21 : term] : (((X6 @ id @ X21 @ id)) != ((X5 @ X21))) | ? [X11 : term] : (($true != ((X5 @ (sub @ X4 @ (push @ X11 @ id))))) & ($true = ((X5 @ X11)))) | ? [X17 : subst,X20 : term,X18 : subst,X19 : subst] : (($true != ((X6 @ (comp @ X18 @ X19) @ (sub @ X20 @ X19) @ X17))) & (((X6 @ X18 @ X20 @ (comp @ X19 @ X17))) = $true)) | ? [X7 : subst,X9 : subst,X8 : term,X10 : subst] : (($true = ((X6 @ (comp @ X10 @ X7) @ (sub @ X8 @ X7) @ X9))) & (((X6 @ X10 @ X8 @ (comp @ X7 @ X9))) != $true)) | ($true = ((sP38 @ X6))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f56,definition,( ! [X397 : (subst > term > subst > $o)] : ((((sP40 @ X397)) = $true) <=> ? [X398 : (subst > term > term)] : (! [X399 : term] : ((((X397 @ id @ X399 @ id)) != $true) | ($true = ((X397 @ id @ (X398 @ id @ X399) @ id)))) & ($true != ((X397 @ id @ (lam @ (X398 @ sh @ one)) @ id))) & ! [X401 : subst,X400 : term,X402 : subst] : (((X398 @ (comp @ X401 @ X402) @ (sub @ X400 @ X402))) = ((sub @ (X398 @ X401 @ X400) @ X402)))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f57,definition,( (sP41 = $true) <=> ! [X397 : (subst > term > subst > $o),X396 : term] : (? [X403 : subst,X405 : subst,X404 : subst,X406 : term] : ((((X397 @ X404 @ X406 @ (comp @ X405 @ X403))) = $true) & (((X397 @ (comp @ X404 @ X405) @ (sub @ X406 @ X405) @ X403)) != $true)) | ? [X411 : term] : ((((X397 @ id @ (sub @ X396 @ (push @ X411 @ id)) @ id)) != $true) & (((X397 @ id @ X411 @ id)) = $true)) | ? [X408 : subst,X409 : term,X410 : subst,X407 : subst] : ((((X397 @ X410 @ X409 @ (comp @ X407 @ X408))) != $true) & ($true = ((X397 @ (comp @ X410 @ X407) @ (sub @ X409 @ X407) @ X408)))) | (((X397 @ id @ (lam @ X396) @ id)) = $true) | (((sP40 @ X397)) = $true))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f58,definition,( ! [X197 : (term > $o)] : (($true = ((sP42 @ X197))) <=> ? [X200 : term,X199 : term] : (($true = ((X197 @ X200))) & ($true = ((X197 @ X199))) & (((X197 @ (ap @ X200 @ X199))) != $true)))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f59,definition,( (sP43 = $true) <=> ! [X197 : (term > $o),X198 : term] : (($true = ((sP42 @ X197))) | (((X197 @ X198)) = $true) | ? [X201 : term] : (! [X202 : term] : (($true != ((X197 @ X202))) | (((X197 @ (sub @ X201 @ (push @ X202 @ id)))) = $true)) & (((X197 @ (lam @ X201))) != $true)) | ? [X203 : term] : (($true != ((X197 @ X203))) & (((var @ X203)) = $true)))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f60,definition,( ! [X284 : (term > $o)] : (($true = ((sP44 @ X284))) <=> ? [X287 : term,X288 : term] : ((((X284 @ X288)) = $true) & (((X284 @ (ap @ X288 @ X287))) != $true) & (((X284 @ X287)) = $true)))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f61,definition,( ! [X278 : (term > $o)] : (($true = ((sP45 @ X278))) <=> ! [X282 : term,X281 : term] : ((((X278 @ X282)) != $true) | (((X278 @ X281)) != $true) | (((X278 @ (ap @ X281 @ X282))) = $true)))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f62,definition,( (sP46 = $true) <=> ! [X286 : term,X284 : (term > $o),X285 : subst] : (? [X289 : term] : ((((var @ X289)) = $true) & (((X284 @ (sub @ X289 @ X285))) != $true)) | ? [X290 : term] : ((((X284 @ (lam @ X290))) != $true) & ! [X291 : term] : ((((X284 @ (sub @ X290 @ (push @ X291 @ id)))) = $true) | (((X284 @ X291)) != $true))) | (((X284 @ (sub @ X286 @ X285))) = $true) | ($true = ((sP44 @ X284))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f63,definition,( ($true = sP47) <=> (? [X275 : term,X276 : subst] : (((comp @ sh @ (push @ X275 @ X276))) != X276) | ? [X274 : subst,X273 : term] : (((sub @ one @ (push @ X273 @ X274))) != X273) | ? [X279 : term,X278 : (term > $o)] : (($true = ((sP45 @ X278))) & ! [X283 : term] : (($true = ((X278 @ X283))) | ($true != ((var @ X283)))) & ! [X280 : term] : ((((X278 @ (lam @ X280))) = $true) | ($true != ((X278 @ X280)))) & (((X278 @ X279)) != $true)) | ? [X277 : subst] : (((comp @ X277 @ id)) != X277) | (sP46 = $true))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f64,definition,( ! [X206 : (subst > term > subst > $o)] : (($true = ((sP48 @ X206))) <=> ? [X210 : (subst > term > term)] : (! [X212 : subst,X213 : term,X214 : subst] : (((X210 @ (comp @ X214 @ X212) @ (sub @ X213 @ X212))) = ((sub @ (X210 @ X214 @ X213) @ X212))) & ($true != ((X206 @ id @ (lam @ (X210 @ sh @ one)) @ id))) & ! [X211 : term] : (($true = ((X206 @ id @ (X210 @ id @ X211) @ id))) | (((X206 @ id @ X211 @ id)) != $true))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f65,definition,( ! [X428 : (subst > term > subst > $o)] : (($true = ((sP49 @ X428))) <=> ? [X435 : (subst > term > term)] : ((((X428 @ id @ (lam @ (X435 @ sh @ one)) @ id)) != $true) & ! [X438 : subst,X439 : subst,X437 : term] : (((X435 @ (comp @ X438 @ X439) @ (sub @ X437 @ X439))) = ((sub @ (X435 @ X438 @ X437) @ X439))) & ! [X436 : term] : (($true != ((X428 @ id @ X436 @ id))) | (((X428 @ id @ (X435 @ id @ X436) @ id)) = $true))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f66,definition,( (sP50 = $true) <=> ? [X420 : (subst > term > subst > $o),X421 : term] : (($true != ((X420 @ id @ (lam @ (sub @ X421 @ (push @ one @ sh))) @ id))) & ! [X422 : (subst > term > term)] : (? [X426 : term,X425 : subst,X424 : subst] : (((sub @ (X422 @ X425 @ X426) @ X424)) != ((X422 @ (comp @ X425 @ X424) @ (sub @ X426 @ X424)))) | (((X420 @ id @ (lam @ (X422 @ sh @ one)) @ id)) = $true) | ? [X423 : term] : ((((X420 @ id @ X423 @ id)) = $true) & ($true != ((X420 @ id @ (X422 @ id @ X423) @ id))))) & ! [X427 : term] : ((((X420 @ id @ (sub @ X421 @ (push @ X427 @ id)) @ id)) = $true) | (((X420 @ id @ X427 @ id)) != $true)))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f67,definition,( (sP51 = $true) <=> (! [X428 : (subst > term > subst > $o),X429 : term] : (($true = ((sP49 @ X428))) | ? [X441 : subst,X442 : subst,X440 : term,X443 : subst] : ((((X428 @ X443 @ X440 @ (comp @ X442 @ X441))) != $true) & (((X428 @ (comp @ X443 @ X442) @ (sub @ X440 @ X442) @ X441)) = $true)) | ($true = ((X428 @ id @ (lam @ X429) @ id))) | ? [X434 : term] : (($true = ((X428 @ id @ X434 @ id))) & ($true != ((X428 @ id @ (sub @ X429 @ (push @ X434 @ id)) @ id)))) | ? [X432 : subst,X433 : subst,X431 : term,X430 : subst] : ((((X428 @ (comp @ X433 @ X432) @ (sub @ X431 @ X432) @ X430)) != $true) & (((X428 @ X433 @ X431 @ (comp @ X432 @ X430))) = $true))) | ? [X419 : term] : (((sub @ X419 @ id)) != X419) | (sP50 = $true))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f68,definition,( ! [X174 : (subst > term > subst > $o)] : ((((sP52 @ X174)) = $true) <=> ? [X181 : term,X182 : term] : (($true = ((X174 @ id @ X182 @ id))) & (((X174 @ id @ (ap @ (sub @ X182 @ id) @ X181) @ id)) != $true) & ($true = ((X174 @ id @ X181 @ id)))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f69,definition,( ! [X157 : (term > $o)] : ((((sP53 @ X157)) = $true) <=> ! [X160 : term,X161 : term] : ((((X157 @ (ap @ X161 @ X160))) = $true) | ($true != ((X157 @ X161))) | (((X157 @ X160)) != $true)))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f70,definition,( ($true = sP54) <=> ? [X164 : (subst > term > subst > $o),X166 : term,X165 : (term > $o)] : (! [X173 : term] : (($true != ((X165 @ X173))) | (((X165 @ (sub @ X166 @ (push @ X173 @ id)))) = $true)) & ! [X167 : term] : (((X165 @ X167)) = ((X164 @ id @ X167 @ id))) & ($true != ((X165 @ (lam @ X166)))) & ! [X168 : (subst > term > term)] : (($true = ((X164 @ id @ (lam @ (X168 @ sh @ one)) @ id))) | ? [X170 : subst,X172 : subst,X171 : term] : (((X168 @ (comp @ X172 @ X170) @ (sub @ X171 @ X170))) != ((sub @ (X168 @ X172 @ X171) @ X170))) | ? [X169 : term] : ((((X164 @ id @ (X168 @ id @ X169) @ id)) != $true) & ($true = ((X164 @ id @ X169 @ id))))))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f71,definition,( (sP55 = $true) <=> ! [X175 : term,X174 : (subst > term > subst > $o)] : (? [X176 : (subst > term > term)] : (! [X178 : subst,X179 : term,X177 : subst] : (((sub @ (X176 @ X177 @ X179) @ X178)) = ((X176 @ (comp @ X177 @ X178) @ (sub @ X179 @ X178)))) & ! [X180 : term] : ((((X174 @ id @ (X176 @ id @ X180) @ id)) = $true) | ($true != ((X174 @ id @ X180 @ id)))) & (((X174 @ id @ (lam @ (X176 @ sh @ one)) @ id)) != $true)) | ($true = ((X174 @ id @ X175 @ id))) | (((sP52 @ X174)) = $true))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f72,definition,( (sP56 = $true) <=> (($true = sP54) | ? [X157 : (term > $o),X156 : term] : (($true != ((X157 @ X156))) & (((sP53 @ X157)) = $true) & ! [X162 : term] : (($true = ((X157 @ X162))) | (((var @ X162)) != $true)) & ! [X158 : term] : (? [X159 : term] : ((((X157 @ (sub @ X158 @ (push @ X159 @ id)))) != $true) & ($true = ((X157 @ X159)))) | (((X157 @ (lam @ X158))) = $true))) | (sP55 = $true) | ? [X153 : (subst > term > subst > $o)] : ! [X154 : (term > $o)] : ? [X155 : term] : (((X154 @ X155)) != ((X153 @ id @ X155 @ id))) | ? [X163 : term] : (((sub @ X163 @ id)) != X163))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f73,definition,( ! [X85 : (term > $o)] : ((((sP57 @ X85)) = $true) <=> ! [X91 : term,X90 : term] : ((((X85 @ X91)) != $true) | (((X85 @ X90)) != $true) | (((X85 @ (ap @ X91 @ X90))) = $true)))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f74,definition,( ! [X93 : (term > $o)] : (($true = ((sP58 @ X93))) <=> ? [X96 : term,X95 : term] : (($true = ((X93 @ X95))) & ($true != ((X93 @ (ap @ X96 @ X95)))) & (((X93 @ X96)) = $true)))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f75,definition,( ($true = sP59) <=> ! [X93 : (term > $o),X94 : term] : (($true = ((sP58 @ X93))) | ? [X97 : term] : ((((var @ X97)) = $true) & (((X93 @ X97)) != $true)) | ? [X98 : term] : (! [X99 : term] : ((((X93 @ X99)) != $true) | ($true = ((X93 @ (sub @ X98 @ (push @ X99 @ id)))))) & ($true != ((X93 @ (lam @ X98))))) | (((X93 @ X94)) = $true))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f76,definition,( ($true = sP60) <=> (($true = sP59) | ? [X85 : (term > $o),X87 : term,X86 : subst] : ((((X85 @ (sub @ X87 @ X86))) != $true) & (((sP57 @ X85)) = $true) & ! [X88 : term] : ((((X85 @ (lam @ X88))) = $true) | ? [X89 : term] : ((((X85 @ (sub @ X88 @ (push @ X89 @ id)))) != $true) & (((X85 @ X89)) = $true))) & ! [X92 : term] : (($true != ((var @ X92))) | (((X85 @ (sub @ X92 @ X86))) = $true))) | ? [X84 : term] : (((sub @ X84 @ id)) != X84))), introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])). thf(f77,plain,( ((apinj1 = $true) <=> ! [X3 : term,X1 : term,X0 : term,X2 : term] : ((((ap @ X1 @ X3)) != ((ap @ X0 @ X2))) | (X0 = X1))) & ! [X228 : subst] : ? [X229 : d_subst] : (((d2subst @ X229)) = X228) & (hoasapnotvar_lthm = $true) & ((induction2_lthm = $true) <=> ($true = sP60)) & (pushprop_gthm = $true) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (sP56 = $true)) & (induction2lem_gthm = $true) & ((sP51 = $true) <=> (hoasinduction_lem3_lthm = $true)) & ((hoasinduction_no_psi_cond = $true) <=> ! [X206 : (subst > term > subst > $o),X207 : term] : (? [X208 : term,X209 : term] : (($true = ((X206 @ id @ X209 @ id))) & ($true != ((X206 @ id @ (ap @ (sub @ X208 @ id) @ X209) @ id))) & (((X206 @ id @ X208 @ id)) = $true)) | ($true = ((sP48 @ X206))) | (((X206 @ id @ X207 @ id)) = $true))) & (pushprop_lthm_orig = $true) & (axassoc = $true) & (pushprop_lthm = $true) & (hoasinduction_lem1_lthm = $true) & (ulamvar1 != $true) & (substmonoid_gthm = $true) & (hoasapnotvar_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & ! [X524 : d_subst] : (d_id = X524) & (lamnotap != $true) & (((d2subst @ d_id)) = id) & (($true = sP47) <=> (induction2lem_lthm = $true)) & (hoasapnotvar = $true) & (axvarshift = $true) & ((hoasinduction_lem0_lthm = $true) <=> ! [X521 : (subst > term > subst > $o)] : ? [X522 : (term > $o)] : ! [X523 : term] : (((X521 @ id @ X523 @ id)) = ((X522 @ X523)))) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((sP43 = $true) <=> (induction2 = $true)) & (hoasinduction_lem1_gthm = $true) & ((axidr = $true) <=> ! [X183 : subst] : (((comp @ X183 @ id)) = X183)) & ((sP41 = $true) <=> (hoasinduction_lem3 = $true)) & (($true = sP39) <=> (hoasinduction_lem3v2 = $true)) & ((! [X231 : subst] : (((comp @ X231 @ id)) = X231) & ! [X230 : subst] : (((comp @ id @ X230)) = X230)) <=> (substmonoid = $true)) & (($true = sP37) <=> (hoasinduction_lem2v2 = $true)) & (hoaslamnotvar = $true) & ((sP35 = $true) <=> (hoasinduction_lem2 = $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)))))))))))) & (hoaslamnotap_lthm = $true) & (! [X244 : subst,X243 : (term > $o),X242 : term] : ? [X245 : (term > $o)] : ! [X246 : term] : (((X245 @ X246)) = ((X243 @ (sub @ X246 @ (push @ X242 @ X244))))) <=> (pushprop_lem0 = $true)) & ((pushprop_lem1 = $true) <=> ! [X270 : term,X272 : subst,X271 : (term > $o),X268 : term,X269 : (term > $o)] : ((((X271 @ (sub @ X270 @ (push @ X268 @ X272)))) = $true) | (((X269 @ X270)) != $true))) & ((hoasinduction_lem3b = $true) <=> ! [X240 : term] : ? [X241 : (subst > term > term)] : (((sub @ X240 @ (push @ one @ sh))) = ((X241 @ sh @ one)))) & (hoasinduction_lem3a_gthm = $true) & (lamnotvar = $true) & (((d2subst @ d_id)) = sh) & ((axvarid = $true) <=> ! [X520 : term] : (((sub @ X520 @ id)) = X520)) & ((pushprop_lem0_lthm = $true) <=> ! [X43 : subst,X44 : (term > $o),X42 : term] : ? [X45 : (term > $o)] : ! [X46 : term] : (((X45 @ X46)) = ((X44 @ (sub @ X46 @ (push @ X42 @ X43)))))) & ((hoasinduction_lem3aa_lthm = $true) <=> (sP33 = $true)) & (hoaslamnotvar_lthm = $true) & (pushprop_lem1v2_gthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & ((hoasinduction_lem3aaa = $true) <=> ($true = sP32)) & ((sP31 = $true) <=> (hoasinduction_lem3v2a = $true)) & (axabs = $true) & ((hoasinduction_lem3v2_f = $true) <=> ! [X78 : term] : ? [X79 : (subst > term > term)] : ! [X81 : subst,X80 : term] : (((X79 @ X81 @ X80)) = ((sub @ X78 @ (push @ X80 @ X81))))) & (hoasinduction_lem1 = $true) & ! [X101 : d_term,X100 : d_term] : ((X100 = X101) | (((d2term @ X101)) != ((d2term @ X100)))) & (! [X192 : (subst > term > subst > $o)] : ? [X193 : (term > $o)] : ! [X194 : term] : (((X192 @ id @ X194 @ id)) = ((X193 @ X194))) <=> (hoasinduction_lem0 = $true)) & ((hoasinduction_lthm = $true) <=> ($true = sP30)) & ((? [X444 : term] : (((sub @ X444 @ id)) != X444) | ! [X445 : term] : (((sub @ X445 @ id)) = X445)) <=> (termmset_lthm = $true)) & ((hoaslaminj = $true) <=> ! [X511 : term,X512 : (subst > term > term),X513 : (subst > term > term),X510 : subst] : (? [X518 : term,X519 : subst,X517 : subst] : (((X512 @ (comp @ X519 @ X517) @ (sub @ X518 @ X517))) != ((sub @ (X512 @ X519 @ X518) @ X517))) | ? [X515 : subst,X516 : subst,X514 : term] : (((sub @ (X513 @ X515 @ X514) @ X516)) != ((X513 @ (comp @ X515 @ X516) @ (sub @ X514 @ X516)))) | (((lam @ (X512 @ sh @ one))) != ((lam @ (X513 @ sh @ one)))) | (((X512 @ X510 @ X511)) = ((X513 @ X510 @ X511))))) & (((d2term @ d_one)) = one) & (hoasinduction_lem1v2 = $true) & (pushprop_lem3v2_lthm = $true) & (((var @ (d2term @ d_one))) != $true) & (hoaslamnotap_gthm = $true) & ((axscons = $true) <=> ! [X247 : subst] : (((push @ (sub @ one @ X247) @ (comp @ sh @ X247))) = X247)) & (hoaslamnotvar_gthm = $true) & (hoaslaminj_gthm = $true) & (hoasinduction_lem3v2_gthm = $true) & (! [X204 : term,X205 : subst] : (((comp @ sh @ (push @ X204 @ X205))) = X205) <=> (axshiftcons = $true)) & (hoasinduction_lem3b_gthm = $true) & ((laminj = $true) <=> ! [X233 : term,X232 : term] : ((((lam @ X232)) != ((lam @ X233))) | (X232 = X233))) & (! [X140 : term,X139 : term,X141 : (subst > term > term)] : ((((lam @ (X141 @ sh @ one))) != ((ap @ (sub @ X140 @ id) @ X139))) | ? [X143 : subst,X142 : term,X144 : subst] : (((X141 @ (comp @ X143 @ X144) @ (sub @ X142 @ X144))) != ((sub @ (X141 @ X143 @ X142) @ X144)))) <=> (hoaslamnotap = $true)) & (pushprop_lem2v2 = $true) & ((? [X103 : subst] : (((comp @ X103 @ id)) != X103) | ? [X102 : subst] : (((comp @ id @ X102)) != X102) | (! [X105 : subst] : (((comp @ X105 @ id)) = X105) & ! [X104 : subst] : (((comp @ id @ X104)) = X104))) <=> (substmonoid_lthm = $true)) & (apnotvar = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((hoasapinj1 = $true) <=> ! [X314 : term,X317 : term,X316 : term,X315 : term] : ((X314 = X316) | (((ap @ (sub @ X316 @ id) @ X317)) != ((ap @ (sub @ X314 @ id) @ X315))))) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X188 : term] : ? [X189 : (subst > term > term)] : ! [X190 : subst,X191 : term] : (((X189 @ X190 @ X191)) = ((sub @ X188 @ (push @ X191 @ X190))))) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (hoasapinj2_gthm = $true) & ((hoasapinj2_lthm = $true) <=> (! [X149 : term,X152 : term,X151 : term,X150 : term] : ((X150 = X151) | (((ap @ (sub @ X149 @ id) @ X150)) != ((ap @ (sub @ X152 @ id) @ X151)))) | ? [X147 : term,X146 : term,X148 : term,X145 : term] : ((((ap @ X145 @ X148)) = ((ap @ X146 @ X147))) & (X147 != X148)))) & (hoasinduction_lem2_lthm = $true) & (! [X187 : term,X184 : term,X186 : term,X185 : term] : ((X186 = X187) | (((ap @ (sub @ X185 @ id) @ X186)) != ((ap @ (sub @ X184 @ id) @ X187)))) <=> (hoasapinj2 = $true)) & (! [X309 : (term > $o),X312 : subst,X311 : term,X310 : (term > $o)] : ((((X309 @ X311)) != $true) | ? [X313 : term] : (((X309 @ (sub @ X313 @ (push @ X311 @ X312)))) != ((X310 @ X313))) | ($true = ((X310 @ one)))) <=> (pushprop_lem1v2 = $true)) & (axclos = $true) & ((hoasapinj1_lthm = $true) <=> (? [X259 : term] : (((sub @ X259 @ id)) != X259) | ? [X263 : term,X261 : term,X260 : term,X262 : term] : ((((ap @ X261 @ X263)) = ((ap @ X262 @ X260))) & (X261 != X262)) | ! [X265 : term,X266 : term,X267 : term,X264 : term] : ((((ap @ (sub @ X267 @ id) @ X265)) != ((ap @ (sub @ X266 @ id) @ X264))) | (X266 = X267)))) & (pushprop_lem0_gthm = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (pushprop_lem3v2 = $true) & (! [X248 : term] : (((sub @ X248 @ id)) = X248) <=> (termmset = $true)) & (! [X65 : term,X64 : (term > $o)] : (? [X68 : term] : ((((X64 @ X68)) = $true) & ($true != ((X64 @ (lam @ X68))))) | (((sP21 @ X64)) = $true) | ? [X69 : term] : ((((var @ X69)) = $true) & ($true != ((X64 @ X69)))) | (((X64 @ X65)) = $true)) <=> (induction = $true)) & (($true = sP20) <=> (hoasinduction_lem3aa = $true)) & (! [X82 : term] : ? [X83 : (subst > term > term)] : (((sub @ X82 @ (push @ one @ sh))) = ((X83 @ sh @ one))) <=> (hoasinduction_lem3b_lthm = $true)) & (pushprop_lem2v2_gthm = $true) & (hoasapinj1_gthm = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (($true = sP19) <=> (hoasinduction_lem3a_lthm = $true)) & (pushprop_lem1_gthm = $true) & (axmap = $true) & ((hoaslaminj_lthm = $true) <=> ($true = sP17)) & (hoasinduction_gthm = $true) & ((apinj2 = $true) <=> ! [X236 : term,X238 : term,X237 : term,X239 : term] : ((X236 = X239) | (((ap @ X237 @ X236)) != ((ap @ X238 @ X239))))) & (termmset_gthm = $true) & ((pushprop_lem1v2_lthm = $true) <=> (! [X417 : term,X414 : (term > $o),X416 : subst,X415 : (term > $o)] : ((((X415 @ one)) = $true) | ($true != ((X414 @ X417))) | ? [X418 : term] : (((X415 @ X418)) != ((X414 @ (sub @ X418 @ (push @ X417 @ X416)))))) | ? [X412 : term,X413 : subst] : (((sub @ one @ (push @ X412 @ X413))) != X412))) & ! [X106 : term] : ? [X107 : d_term] : (((d2term @ X107)) = X106) & ! [X318 : d_term] : (d_one = X318) & (hoasinduction_lem3_gthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (axapp = $true) & ((hoasinduction_lthm_3 = $true) <=> ($true = sP16)) & (hoasinduction_lem1v2_gthm = $true) & ((sP10 = $true) <=> (hoasinduction = $true)) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (! [X195 : term,X196 : term] : ((((sub @ X196 @ sh)) != ((sub @ X195 @ sh))) | (X195 = X196)) <=> (shinj = $true)) & ((sP7 = $true) <=> (hoasinduction_lem3v2_lthm = $true)) & (hoasinduction_lem2_gthm = $true) & (hoasinduction_lem2v2_gthm = $true) & ((induction2lem = $true) <=> ($true = sP5)) & (pushprop_lem2v2_lthm = $true) & ((? [X485 : subst,X486 : term] : (((sub @ one @ (push @ X486 @ X485))) != X486) | ! [X489 : term,X493 : (term > $o),X490 : term,X491 : (term > $o),X492 : subst] : ((((X493 @ X490)) != $true) | (((X491 @ (sub @ X490 @ (push @ X489 @ X492)))) = $true)) | ? [X487 : subst,X488 : term] : (((comp @ sh @ (push @ X488 @ X487))) != X487)) <=> (pushprop_lem1_lthm = $true)) & ! [X41 : d_subst,X40 : d_subst] : ((((d2subst @ X40)) != ((d2subst @ X41))) | (X40 = X41)) & ((axidl = $true) <=> ! [X47 : subst] : (((comp @ id @ X47)) = X47)) & (($true = sP3) <=> (hoasinduction_lem3a = $true)) & (induction2_gthm = $true) & (ulamvarsh = $true) & ((hoasinduction_lem3v2a_lthm = $true) <=> (sP2 = $true)) & (pushprop = $true) & ((axvarcons = $true) <=> ! [X234 : subst,X235 : term] : (((sub @ one @ (push @ X235 @ X234))) = X235)) & (ulamvarind = $true)), inference(definition_folding,[],[f14,f76,f75,f74,f73,f72,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])). thf(f78,plain,( ((sP1 = $true) | ? [X0 : (subst > term > subst > $o)] : (? [X14 : term] : ((((X0 @ id @ (lam @ X14) @ id)) != $true) & ! [X15 : term] : (($true = ((X0 @ id @ (sub @ X14 @ (push @ X15 @ id)) @ id))) | (((X0 @ id @ X15 @ id)) != $true))) & ! [X6 : subst,X5 : subst,X8 : subst,X7 : term] : ((((X0 @ (comp @ X5 @ X6) @ (sub @ X7 @ X6) @ X8)) != $true) | (((X0 @ X5 @ X7 @ (comp @ X6 @ X8))) = $true)) & ($true != ((sP0 @ X0))) & ! [X1 : subst,X3 : subst,X2 : term,X4 : subst] : ((((X0 @ (comp @ X3 @ X1) @ (sub @ X2 @ X1) @ X4)) = $true) | ($true != ((X0 @ X3 @ X2 @ (comp @ X1 @ X4))))))) & (! [X0 : (subst > term > subst > $o)] : (! [X14 : term] : ((((X0 @ id @ (lam @ X14) @ id)) = $true) | ? [X15 : term] : (($true != ((X0 @ id @ (sub @ X14 @ (push @ X15 @ id)) @ id))) & (((X0 @ id @ X15 @ id)) = $true))) | ? [X6 : subst,X5 : subst,X8 : subst,X7 : term] : ((((X0 @ (comp @ X5 @ X6) @ (sub @ X7 @ X6) @ X8)) = $true) & (((X0 @ X5 @ X7 @ (comp @ X6 @ X8))) != $true)) | ($true = ((sP0 @ X0))) | ? [X1 : subst,X3 : subst,X2 : term,X4 : subst] : ((((X0 @ (comp @ X3 @ X1) @ (sub @ X2 @ X1) @ X4)) != $true) & ($true = ((X0 @ X3 @ X2 @ (comp @ X1 @ X4)))))) | (sP1 != $true))), inference(nnf_transformation,[],[f16])). thf(f79,plain,( ((sP1 = $true) | ? [X0 : (subst > term > subst > $o)] : (? [X1 : term] : (($true != ((X0 @ id @ (lam @ X1) @ id))) & ! [X2 : term] : (($true = ((X0 @ id @ (sub @ X1 @ (push @ X2 @ id)) @ id))) | (((X0 @ id @ X2 @ id)) != $true))) & ! [X3 : subst,X4 : subst,X5 : subst,X6 : term] : ((((X0 @ (comp @ X4 @ X3) @ (sub @ X6 @ X3) @ X5)) != $true) | (((X0 @ X4 @ X6 @ (comp @ X3 @ X5))) = $true)) & ($true != ((sP0 @ X0))) & ! [X7 : subst,X8 : subst,X9 : term,X10 : subst] : (($true = ((X0 @ (comp @ X8 @ X7) @ (sub @ X9 @ X7) @ X10))) | ($true != ((X0 @ X8 @ X9 @ (comp @ X7 @ X10))))))) & (! [X11 : (subst > term > subst > $o)] : (! [X12 : term] : ((((X11 @ id @ (lam @ X12) @ id)) = $true) | ? [X13 : term] : ((((X11 @ id @ (sub @ X12 @ (push @ X13 @ id)) @ id)) != $true) & (((X11 @ id @ X13 @ id)) = $true))) | ? [X14 : subst,X15 : subst,X16 : subst,X17 : term] : ((((X11 @ (comp @ X15 @ X14) @ (sub @ X17 @ X14) @ X16)) = $true) & (((X11 @ X15 @ X17 @ (comp @ X14 @ X16))) != $true)) | ($true = ((sP0 @ X11))) | ? [X18 : subst,X19 : subst,X20 : term,X21 : subst] : (($true != ((X11 @ (comp @ X19 @ X18) @ (sub @ X20 @ X18) @ X21))) & (((X11 @ X19 @ X20 @ (comp @ X18 @ X21))) = $true))) | (sP1 != $true))), inference(rectify,[],[f78])). thf(f80,plain,( ((sP1 = $true) | (((((sK61 @ id @ (lam @ sK62) @ id)) != $true) & ! [X2 : term] : (($true = ((sK61 @ id @ (sub @ sK62 @ (push @ X2 @ id)) @ id))) | ($true != ((sK61 @ id @ X2 @ id))))) & ! [X3 : subst,X4 : subst,X5 : subst,X6 : term] : ((((sK61 @ (comp @ X4 @ X3) @ (sub @ X6 @ X3) @ X5)) != $true) | (((sK61 @ X4 @ X6 @ (comp @ X3 @ X5))) = $true)) & (((sP0 @ sK61)) != $true) & ! [X7 : subst,X8 : subst,X9 : term,X10 : subst] : ((((sK61 @ (comp @ X8 @ X7) @ (sub @ X9 @ X7) @ X10)) = $true) | ($true != ((sK61 @ X8 @ X9 @ (comp @ X7 @ X10))))))) & (! [X11 : (subst > term > subst > $o)] : (! [X12 : term] : ((((X11 @ id @ (lam @ X12) @ id)) = $true) | ((((X11 @ id @ (sub @ X12 @ (push @ (sK63 @ X12 @ X11) @ id)) @ id)) != $true) & (((X11 @ id @ (sK63 @ X12 @ X11) @ id)) = $true))) | ((((X11 @ (comp @ (sK65 @ X11) @ (sK64 @ X11)) @ (sub @ (sK67 @ X11) @ (sK64 @ X11)) @ (sK66 @ X11))) = $true) & (((X11 @ (sK65 @ X11) @ (sK67 @ X11) @ (comp @ (sK64 @ X11) @ (sK66 @ X11)))) != $true)) | ($true = ((sP0 @ X11))) | ((((X11 @ (comp @ (sK69 @ X11) @ (sK68 @ X11)) @ (sub @ (sK70 @ X11) @ (sK68 @ X11)) @ (sK71 @ X11))) != $true) & ($true = ((X11 @ (sK69 @ X11) @ (sK70 @ X11) @ (comp @ (sK68 @ X11) @ (sK71 @ X11))))))) | (sP1 != $true))), inference(skolemize,[status(esa),new_symbols(skolem,[sK61,sK62,vAPP]),skolemize(X0,$thf(sK61)),skolemize(X1,$thf(sK62)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11))],[f79])). thf(f81,plain,( ! [X0 : (subst > term > subst > $o)] : ((($true = ((sP0 @ X0))) | ! [X9 : (subst > term > term)] : (? [X13 : term] : ((((X0 @ id @ X13 @ id)) = $true) & ($true != ((X0 @ id @ (X9 @ id @ X13) @ id)))) | ($true = ((X0 @ id @ (hoaslam @ id @ (^[Y0 : subst]: ((^[Y1 : term]: (X9 @ Y0 @ Y1))))) @ id))) | ? [X12 : subst,X10 : term,X11 : subst] : (((X9 @ (comp @ X12 @ X11) @ (sub @ X10 @ X11))) != ((sub @ (X9 @ X12 @ X10) @ X11))))) & (? [X9 : (subst > term > term)] : (! [X13 : term] : ((((X0 @ id @ X13 @ id)) != $true) | ($true = ((X0 @ id @ (X9 @ id @ X13) @ id)))) & ($true != ((X0 @ id @ (hoaslam @ id @ (^[Y0 : subst]: ((^[Y1 : term]: (X9 @ Y0 @ Y1))))) @ id))) & ! [X12 : subst,X10 : term,X11 : subst] : (((X9 @ (comp @ X12 @ X11) @ (sub @ X10 @ X11))) = ((sub @ (X9 @ X12 @ X10) @ X11)))) | ($true != ((sP0 @ X0)))))), inference(nnf_transformation,[],[f15])). thf(f82,plain,( ! [X0 : (subst > term > subst > $o)] : ((($true = ((sP0 @ X0))) | ! [X1 : (subst > term > term)] : (? [X2 : term] : ((((X0 @ id @ X2 @ id)) = $true) & (((X0 @ id @ (X1 @ id @ X2) @ id)) != $true)) | (((X0 @ id @ (hoaslam @ id @ (^[Y0 : subst]: ((^[Y1 : term]: (X1 @ Y0 @ Y1))))) @ id)) = $true) | ? [X3 : subst,X4 : term,X5 : subst] : (((X1 @ (comp @ X3 @ X5) @ (sub @ X4 @ X5))) != ((sub @ (X1 @ X3 @ X4) @ X5))))) & (? [X6 : (subst > term > term)] : (! [X7 : term] : ((((X0 @ id @ X7 @ id)) != $true) | (((X0 @ id @ (X6 @ id @ X7) @ id)) = $true)) & ($true != ((X0 @ id @ (hoaslam @ id @ (^[Y0 : subst]: ((^[Y1 : term]: (X6 @ Y0 @ Y1))))) @ id))) & ! [X8 : subst,X9 : term,X10 : subst] : (((sub @ (X6 @ X8 @ X9) @ X10)) = ((X6 @ (comp @ X8 @ X10) @ (sub @ X9 @ X10))))) | ($true != ((sP0 @ X0)))))), inference(rectify,[],[f81])). thf(f83,plain,( ! [X0 : (subst > term > subst > $o)] : ((($true = ((sP0 @ X0))) | ! [X1 : (subst > term > term)] : (((((X0 @ id @ (sK72 @ X1 @ X0) @ id)) = $true) & ($true != ((X0 @ id @ (X1 @ id @ (sK72 @ X1 @ X0)) @ id)))) | (((X0 @ id @ (hoaslam @ id @ (^[Y0 : subst]: ((^[Y1 : term]: (X1 @ Y0 @ Y1))))) @ id)) = $true) | (((sub @ (X1 @ (sK73 @ X1) @ (sK74 @ X1)) @ (sK75 @ X1))) != ((X1 @ (comp @ (sK73 @ X1) @ (sK75 @ X1)) @ (sub @ (sK74 @ X1) @ (sK75 @ X1))))))) & ((! [X7 : term] : ((((X0 @ id @ X7 @ id)) != $true) | (((X0 @ id @ (sK76 @ X0 @ id @ X7) @ id)) = $true)) & (((X0 @ id @ (hoaslam @ id @ (^[Y0 : subst]: ((^[Y1 : term]: (sK76 @ X0 @ Y0 @ Y1))))) @ id)) != $true) & ! [X8 : subst,X9 : term,X10 : subst] : (((sub @ (sK76 @ X0 @ X8 @ X9) @ X10)) = ((sK76 @ X0 @ (comp @ X8 @ X10) @ (sub @ X9 @ X10))))) | ($true != ((sP0 @ X0)))))), inference(skolemize,[status(esa),new_symbols(skolem,[vAPP]),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11))],[f82])). thf(f84,plain,( ((hoasinduction_lem3 != $true) | (sP1 != $true)) & ((hoasinduction_lem3 = $true) | (sP1 = $true))), inference(nnf_transformation,[],[f17])). thf(f111,plain,( ! [X174 : (subst > term > subst > $o)] : (((((sP52 @ X174)) = $true) | ! [X181 : term,X182 : term] : (($true != ((X174 @ id @ X182 @ id))) | (((X174 @ id @ (ap @ (sub @ X182 @ id) @ X181) @ id)) = $true) | ($true != ((X174 @ id @ X181 @ id))))) & (? [X181 : term,X182 : term] : (($true = ((X174 @ id @ X182 @ id))) & (((X174 @ id @ (ap @ (sub @ X182 @ id) @ X181) @ id)) != $true) & ($true = ((X174 @ id @ X181 @ id)))) | (((sP52 @ X174)) != $true)))), inference(nnf_transformation,[],[f68])). thf(f112,plain,( ! [X0 : (subst > term > subst > $o)] : (((((sP52 @ X0)) = $true) | ! [X1 : term,X2 : term] : ((((X0 @ id @ X2 @ id)) != $true) | (((X0 @ id @ (ap @ (sub @ X2 @ id) @ X1) @ id)) = $true) | (((X0 @ id @ X1 @ id)) != $true))) & (? [X3 : term,X4 : term] : (($true = ((X0 @ id @ X4 @ id))) & (((X0 @ id @ (ap @ (sub @ X4 @ id) @ X3) @ id)) != $true) & ($true = ((X0 @ id @ X3 @ id)))) | (((sP52 @ X0)) != $true)))), inference(rectify,[],[f111])). thf(f113,plain,( ! [X0 : (subst > term > subst > $o)] : (((((sP52 @ X0)) = $true) | ! [X1 : term,X2 : term] : ((((X0 @ id @ X2 @ id)) != $true) | (((X0 @ id @ (ap @ (sub @ X2 @ id) @ X1) @ id)) = $true) | (((X0 @ id @ X1 @ id)) != $true))) & ((($true = ((X0 @ id @ (sK122 @ X0) @ id))) & (((X0 @ id @ (ap @ (sub @ (sK122 @ X0) @ id) @ (sK121 @ X0)) @ id)) != $true) & (((X0 @ id @ (sK121 @ X0) @ id)) = $true)) | (((sP52 @ X0)) != $true)))), inference(skolemize,[status(esa),new_symbols(skolem,[vAPP]),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11))],[f112])). thf(f146,plain,( ((sP41 = $true) | ? [X397 : (subst > term > subst > $o),X396 : term] : (! [X403 : subst,X405 : subst,X404 : subst,X406 : term] : ((((X397 @ X404 @ X406 @ (comp @ X405 @ X403))) != $true) | (((X397 @ (comp @ X404 @ X405) @ (sub @ X406 @ X405) @ X403)) = $true)) & ! [X411 : term] : ((((X397 @ id @ (sub @ X396 @ (push @ X411 @ id)) @ id)) = $true) | (((X397 @ id @ X411 @ id)) != $true)) & ! [X408 : subst,X409 : term,X410 : subst,X407 : subst] : ((((X397 @ X410 @ X409 @ (comp @ X407 @ X408))) = $true) | ($true != ((X397 @ (comp @ X410 @ X407) @ (sub @ X409 @ X407) @ X408)))) & (((X397 @ id @ (lam @ X396) @ id)) != $true) & (((sP40 @ X397)) != $true))) & (! [X397 : (subst > term > subst > $o),X396 : term] : (? [X403 : subst,X405 : subst,X404 : subst,X406 : term] : ((((X397 @ X404 @ X406 @ (comp @ X405 @ X403))) = $true) & (((X397 @ (comp @ X404 @ X405) @ (sub @ X406 @ X405) @ X403)) != $true)) | ? [X411 : term] : ((((X397 @ id @ (sub @ X396 @ (push @ X411 @ id)) @ id)) != $true) & (((X397 @ id @ X411 @ id)) = $true)) | ? [X408 : subst,X409 : term,X410 : subst,X407 : subst] : ((((X397 @ X410 @ X409 @ (comp @ X407 @ X408))) != $true) & ($true = ((X397 @ (comp @ X410 @ X407) @ (sub @ X409 @ X407) @ X408)))) | (((X397 @ id @ (lam @ X396) @ id)) = $true) | (((sP40 @ X397)) = $true)) | (sP41 != $true))), inference(nnf_transformation,[],[f57])). thf(f147,plain,( ((sP41 = $true) | ? [X0 : (subst > term > subst > $o),X1 : term] : (! [X2 : subst,X3 : subst,X4 : subst,X5 : term] : ((((X0 @ X4 @ X5 @ (comp @ X3 @ X2))) != $true) | (((X0 @ (comp @ X4 @ X3) @ (sub @ X5 @ X3) @ X2)) = $true)) & ! [X6 : term] : (($true = ((X0 @ id @ (sub @ X1 @ (push @ X6 @ id)) @ id))) | (((X0 @ id @ X6 @ id)) != $true)) & ! [X7 : subst,X8 : term,X9 : subst,X10 : subst] : (($true = ((X0 @ X9 @ X8 @ (comp @ X10 @ X7)))) | ($true != ((X0 @ (comp @ X9 @ X10) @ (sub @ X8 @ X10) @ X7)))) & ($true != ((X0 @ id @ (lam @ X1) @ id))) & ($true != ((sP40 @ X0))))) & (! [X11 : (subst > term > subst > $o),X12 : term] : (? [X13 : subst,X14 : subst,X15 : subst,X16 : term] : (($true = ((X11 @ X15 @ X16 @ (comp @ X14 @ X13)))) & ($true != ((X11 @ (comp @ X15 @ X14) @ (sub @ X16 @ X14) @ X13)))) | ? [X17 : term] : ((((X11 @ id @ (sub @ X12 @ (push @ X17 @ id)) @ id)) != $true) & ($true = ((X11 @ id @ X17 @ id)))) | ? [X18 : subst,X19 : term,X20 : subst,X21 : subst] : ((((X11 @ X20 @ X19 @ (comp @ X21 @ X18))) != $true) & (((X11 @ (comp @ X20 @ X21) @ (sub @ X19 @ X21) @ X18)) = $true)) | (((X11 @ id @ (lam @ X12) @ id)) = $true) | ($true = ((sP40 @ X11)))) | (sP41 != $true))), inference(rectify,[],[f146])). thf(f148,plain,( ((sP41 = $true) | (! [X2 : subst,X3 : subst,X4 : subst,X5 : term] : (($true != ((sK179 @ X4 @ X5 @ (comp @ X3 @ X2)))) | (((sK179 @ (comp @ X4 @ X3) @ (sub @ X5 @ X3) @ X2)) = $true)) & ! [X6 : term] : (($true = ((sK179 @ id @ (sub @ sK180 @ (push @ X6 @ id)) @ id))) | ($true != ((sK179 @ id @ X6 @ id)))) & ! [X7 : subst,X8 : term,X9 : subst,X10 : subst] : (($true = ((sK179 @ X9 @ X8 @ (comp @ X10 @ X7)))) | ($true != ((sK179 @ (comp @ X9 @ X10) @ (sub @ X8 @ X10) @ X7)))) & (((sK179 @ id @ (lam @ sK180) @ id)) != $true) & ($true != ((sP40 @ sK179))))) & (! [X11 : (subst > term > subst > $o),X12 : term] : (((((X11 @ (sK183 @ X11) @ (sK184 @ X11) @ (comp @ (sK182 @ X11) @ (sK181 @ X11)))) = $true) & (((X11 @ (comp @ (sK183 @ X11) @ (sK182 @ X11)) @ (sub @ (sK184 @ X11) @ (sK182 @ X11)) @ (sK181 @ X11))) != $true)) | ((((X11 @ id @ (sub @ X12 @ (push @ (sK185 @ X12 @ X11) @ id)) @ id)) != $true) & (((X11 @ id @ (sK185 @ X12 @ X11) @ id)) = $true)) | (($true != ((X11 @ (sK188 @ X11) @ (sK187 @ X11) @ (comp @ (sK189 @ X11) @ (sK186 @ X11))))) & (((X11 @ (comp @ (sK188 @ X11) @ (sK189 @ X11)) @ (sub @ (sK187 @ X11) @ (sK189 @ X11)) @ (sK186 @ X11))) = $true)) | (((X11 @ id @ (lam @ X12) @ id)) = $true) | ($true = ((sP40 @ X11)))) | (sP41 != $true))), inference(skolemize,[status(esa),new_symbols(skolem,[sK179,sK180,vAPP]),skolemize(X0,$thf(sK179)),skolemize(X1,$thf(sK180)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11))],[f147])). thf(f149,plain,( ! [X397 : (subst > term > subst > $o)] : (((((sP40 @ X397)) = $true) | ! [X398 : (subst > term > term)] : (? [X399 : term] : ((((X397 @ id @ X399 @ id)) = $true) & ($true != ((X397 @ id @ (X398 @ id @ X399) @ id)))) | ($true = ((X397 @ id @ (lam @ (X398 @ sh @ one)) @ id))) | ? [X401 : subst,X400 : term,X402 : subst] : (((X398 @ (comp @ X401 @ X402) @ (sub @ X400 @ X402))) != ((sub @ (X398 @ X401 @ X400) @ X402))))) & (? [X398 : (subst > term > term)] : (! [X399 : term] : ((((X397 @ id @ X399 @ id)) != $true) | ($true = ((X397 @ id @ (X398 @ id @ X399) @ id)))) & ($true != ((X397 @ id @ (lam @ (X398 @ sh @ one)) @ id))) & ! [X401 : subst,X400 : term,X402 : subst] : (((X398 @ (comp @ X401 @ X402) @ (sub @ X400 @ X402))) = ((sub @ (X398 @ X401 @ X400) @ X402)))) | (((sP40 @ X397)) != $true)))), inference(nnf_transformation,[],[f56])). thf(f150,plain,( ! [X0 : (subst > term > subst > $o)] : ((($true = ((sP40 @ X0))) | ! [X1 : (subst > term > term)] : (? [X2 : term] : ((((X0 @ id @ X2 @ id)) = $true) & (((X0 @ id @ (X1 @ id @ X2) @ id)) != $true)) | (((X0 @ id @ (lam @ (X1 @ sh @ one)) @ id)) = $true) | ? [X3 : subst,X4 : term,X5 : subst] : (((X1 @ (comp @ X3 @ X5) @ (sub @ X4 @ X5))) != ((sub @ (X1 @ X3 @ X4) @ X5))))) & (? [X6 : (subst > term > term)] : (! [X7 : term] : ((((X0 @ id @ X7 @ id)) != $true) | (((X0 @ id @ (X6 @ id @ X7) @ id)) = $true)) & ($true != ((X0 @ id @ (lam @ (X6 @ sh @ one)) @ id))) & ! [X8 : subst,X9 : term,X10 : subst] : (((sub @ (X6 @ X8 @ X9) @ X10)) = ((X6 @ (comp @ X8 @ X10) @ (sub @ X9 @ X10))))) | ($true != ((sP40 @ X0)))))), inference(rectify,[],[f149])). thf(f151,plain,( ! [X0 : (subst > term > subst > $o)] : ((($true = ((sP40 @ X0))) | ! [X1 : (subst > term > term)] : ((($true = ((X0 @ id @ (sK190 @ X1 @ X0) @ id))) & ($true != ((X0 @ id @ (X1 @ id @ (sK190 @ X1 @ X0)) @ id)))) | (((X0 @ id @ (lam @ (X1 @ sh @ one)) @ id)) = $true) | (((X1 @ (comp @ (sK191 @ X1) @ (sK193 @ X1)) @ (sub @ (sK192 @ X1) @ (sK193 @ X1)))) != ((sub @ (X1 @ (sK191 @ X1) @ (sK192 @ X1)) @ (sK193 @ X1)))))) & ((! [X7 : term] : ((((X0 @ id @ X7 @ id)) != $true) | ($true = ((X0 @ id @ (sK194 @ X0 @ id @ X7) @ id)))) & ($true != ((X0 @ id @ (lam @ (sK194 @ X0 @ sh @ one)) @ id))) & ! [X8 : subst,X9 : term,X10 : subst] : (((sub @ (sK194 @ X0 @ X8 @ X9) @ X10)) = ((sK194 @ X0 @ (comp @ X8 @ X10) @ (sub @ X9 @ X10))))) | ($true != ((sP40 @ X0)))))), inference(skolemize,[status(esa),new_symbols(skolem,[vAPP]),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11))],[f150])). thf(f213,plain,( (($true = sP19) | (? [X301 : term,X302 : (subst > term > subst > $o)] : ((((X302 @ id @ (lam @ X301) @ id)) != $true) & ! [X303 : (subst > term > term)] : (? [X307 : term] : ((((X302 @ id @ X307 @ id)) = $true) & ($true != ((X302 @ id @ (X303 @ id @ X307) @ id)))) | ($true = ((X302 @ id @ (lam @ (X303 @ sh @ one)) @ id))) | ? [X305 : term,X304 : subst,X306 : subst] : (((sub @ (X303 @ X304 @ X305) @ X306)) != ((X303 @ (comp @ X304 @ X306) @ (sub @ X305 @ X306))))) & ! [X308 : term] : (($true != ((X302 @ id @ X308 @ id))) | ($true = ((X302 @ id @ (sub @ X301 @ (push @ X308 @ id)) @ id))))) & ! [X292 : term] : (((sub @ X292 @ id)) = X292) & (sP18 != $true))) & ((! [X301 : term,X302 : (subst > term > subst > $o)] : ((((X302 @ id @ (lam @ X301) @ id)) = $true) | ? [X303 : (subst > term > term)] : (! [X307 : term] : ((((X302 @ id @ X307 @ id)) != $true) | ($true = ((X302 @ id @ (X303 @ id @ X307) @ id)))) & ($true != ((X302 @ id @ (lam @ (X303 @ sh @ one)) @ id))) & ! [X305 : term,X304 : subst,X306 : subst] : (((sub @ (X303 @ X304 @ X305) @ X306)) = ((X303 @ (comp @ X304 @ X306) @ (sub @ X305 @ X306))))) | ? [X308 : term] : (($true = ((X302 @ id @ X308 @ id))) & ($true != ((X302 @ id @ (sub @ X301 @ (push @ X308 @ id)) @ id))))) | ? [X292 : term] : (((sub @ X292 @ id)) != X292) | (sP18 = $true)) | ($true != sP19))), inference(nnf_transformation,[],[f35])). thf(f214,plain,( (($true = sP19) | (? [X301 : term,X302 : (subst > term > subst > $o)] : ((((X302 @ id @ (lam @ X301) @ id)) != $true) & ! [X303 : (subst > term > term)] : (? [X307 : term] : ((((X302 @ id @ X307 @ id)) = $true) & ($true != ((X302 @ id @ (X303 @ id @ X307) @ id)))) | ($true = ((X302 @ id @ (lam @ (X303 @ sh @ one)) @ id))) | ? [X305 : term,X304 : subst,X306 : subst] : (((sub @ (X303 @ X304 @ X305) @ X306)) != ((X303 @ (comp @ X304 @ X306) @ (sub @ X305 @ X306))))) & ! [X308 : term] : (($true != ((X302 @ id @ X308 @ id))) | ($true = ((X302 @ id @ (sub @ X301 @ (push @ X308 @ id)) @ id))))) & ! [X292 : term] : (((sub @ X292 @ id)) = X292) & (sP18 != $true))) & (! [X301 : term,X302 : (subst > term > subst > $o)] : ((((X302 @ id @ (lam @ X301) @ id)) = $true) | ? [X303 : (subst > term > term)] : (! [X307 : term] : ((((X302 @ id @ X307 @ id)) != $true) | ($true = ((X302 @ id @ (X303 @ id @ X307) @ id)))) & ($true != ((X302 @ id @ (lam @ (X303 @ sh @ one)) @ id))) & ! [X305 : term,X304 : subst,X306 : subst] : (((sub @ (X303 @ X304 @ X305) @ X306)) = ((X303 @ (comp @ X304 @ X306) @ (sub @ X305 @ X306))))) | ? [X308 : term] : (($true = ((X302 @ id @ X308 @ id))) & ($true != ((X302 @ id @ (sub @ X301 @ (push @ X308 @ id)) @ id))))) | ? [X292 : term] : (((sub @ X292 @ id)) != X292) | (sP18 = $true) | ($true != sP19))), inference(flattening,[],[f213])). thf(f215,plain,( (($true = sP19) | (? [X0 : term,X1 : (subst > term > subst > $o)] : ((((X1 @ id @ (lam @ X0) @ id)) != $true) & ! [X2 : (subst > term > term)] : (? [X3 : term] : ((((X1 @ id @ X3 @ id)) = $true) & (((X1 @ id @ (X2 @ id @ X3) @ id)) != $true)) | ($true = ((X1 @ id @ (lam @ (X2 @ sh @ one)) @ id))) | ? [X4 : term,X5 : subst,X6 : subst] : (((X2 @ (comp @ X5 @ X6) @ (sub @ X4 @ X6))) != ((sub @ (X2 @ X5 @ X4) @ X6)))) & ! [X7 : term] : (($true != ((X1 @ id @ X7 @ id))) | ($true = ((X1 @ id @ (sub @ X0 @ (push @ X7 @ id)) @ id))))) & ! [X8 : term] : (((sub @ X8 @ id)) = X8) & (sP18 != $true))) & (! [X9 : term,X10 : (subst > term > subst > $o)] : (($true = ((X10 @ id @ (lam @ X9) @ id))) | ? [X11 : (subst > term > term)] : (! [X12 : term] : ((((X10 @ id @ X12 @ id)) != $true) | (((X10 @ id @ (X11 @ id @ X12) @ id)) = $true)) & ($true != ((X10 @ id @ (lam @ (X11 @ sh @ one)) @ id))) & ! [X13 : term,X14 : subst,X15 : subst] : (((sub @ (X11 @ X14 @ X13) @ X15)) = ((X11 @ (comp @ X14 @ X15) @ (sub @ X13 @ X15))))) | ? [X16 : term] : ((((X10 @ id @ X16 @ id)) = $true) & (((X10 @ id @ (sub @ X9 @ (push @ X16 @ id)) @ id)) != $true))) | ? [X17 : term] : (((sub @ X17 @ id)) != X17) | (sP18 = $true) | ($true != sP19))), inference(rectify,[],[f214])). thf(f216,plain,( (($true = sP19) | (((((sK336 @ id @ (lam @ sK335) @ id)) != $true) & ! [X2 : (subst > term > term)] : (((((sK336 @ id @ (sK337 @ X2) @ id)) = $true) & (((sK336 @ id @ (X2 @ id @ (sK337 @ X2)) @ id)) != $true)) | (((sK336 @ id @ (lam @ (X2 @ sh @ one)) @ id)) = $true) | (((sub @ (X2 @ (sK339 @ X2) @ (sK338 @ X2)) @ (sK340 @ X2))) != ((X2 @ (comp @ (sK339 @ X2) @ (sK340 @ X2)) @ (sub @ (sK338 @ X2) @ (sK340 @ X2)))))) & ! [X7 : term] : (($true != ((sK336 @ id @ X7 @ id))) | ($true = ((sK336 @ id @ (sub @ sK335 @ (push @ X7 @ id)) @ id))))) & ! [X8 : term] : (((sub @ X8 @ id)) = X8) & (sP18 != $true))) & (! [X9 : term,X10 : (subst > term > subst > $o)] : (($true = ((X10 @ id @ (lam @ X9) @ id))) | (! [X12 : term] : ((((X10 @ id @ X12 @ id)) != $true) | ($true = ((X10 @ id @ (sK341 @ X10 @ id @ X12) @ id)))) & (((X10 @ id @ (lam @ (sK341 @ X10 @ sh @ one)) @ id)) != $true) & ! [X13 : term,X14 : subst,X15 : subst] : (((sub @ (sK341 @ X10 @ X14 @ X13) @ X15)) = ((sK341 @ X10 @ (comp @ X14 @ X15) @ (sub @ X13 @ X15))))) | (($true = ((X10 @ id @ (sK342 @ X10 @ X9) @ id))) & (((X10 @ id @ (sub @ X9 @ (push @ (sK342 @ X10 @ X9) @ id)) @ id)) != $true))) | (((sub @ sK343 @ id)) != sK343) | (sP18 = $true) | ($true != sP19))), inference(skolemize,[status(esa),new_symbols(skolem,[sK335,sK336,vAPP,sK343]),skolemize(X0,$thf(sK335)),skolemize(X1,$thf(sK336)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X17,$thf(sK343))],[f215])). thf(f217,plain,( ((sP18 = $true) | ! [X293 : (subst > term > subst > $o),X294 : term] : (? [X295 : (subst > term > term)] : (! [X299 : subst,X297 : term,X298 : subst] : (((X295 @ (comp @ X299 @ X298) @ (sub @ X297 @ X298))) = ((sub @ (X295 @ X299 @ X297) @ X298))) & (((X293 @ id @ (lam @ (X295 @ sh @ one)) @ id)) != $true) & ! [X296 : term] : (($true = ((X293 @ id @ (X295 @ id @ X296) @ id))) | ($true != ((X293 @ id @ X296 @ id))))) | ? [X300 : term] : (($true != ((X293 @ id @ (sub @ X294 @ (push @ X300 @ id)) @ id))) & (((X293 @ id @ X300 @ id)) = $true)) | ($true = ((X293 @ id @ (lam @ (sub @ X294 @ (push @ one @ sh))) @ id))))) & (? [X293 : (subst > term > subst > $o),X294 : term] : (! [X295 : (subst > term > term)] : (? [X299 : subst,X297 : term,X298 : subst] : (((X295 @ (comp @ X299 @ X298) @ (sub @ X297 @ X298))) != ((sub @ (X295 @ X299 @ X297) @ X298))) | (((X293 @ id @ (lam @ (X295 @ sh @ one)) @ id)) = $true) | ? [X296 : term] : (($true != ((X293 @ id @ (X295 @ id @ X296) @ id))) & ($true = ((X293 @ id @ X296 @ id))))) & ! [X300 : term] : (($true = ((X293 @ id @ (sub @ X294 @ (push @ X300 @ id)) @ id))) | (((X293 @ id @ X300 @ id)) != $true)) & ($true != ((X293 @ id @ (lam @ (sub @ X294 @ (push @ one @ sh))) @ id)))) | (sP18 != $true))), inference(nnf_transformation,[],[f34])). thf(f218,plain,( ((sP18 = $true) | ! [X0 : (subst > term > subst > $o),X1 : term] : (? [X2 : (subst > term > term)] : (! [X3 : subst,X4 : term,X5 : subst] : (((X2 @ (comp @ X3 @ X5) @ (sub @ X4 @ X5))) = ((sub @ (X2 @ X3 @ X4) @ X5))) & ($true != ((X0 @ id @ (lam @ (X2 @ sh @ one)) @ id))) & ! [X6 : term] : ((((X0 @ id @ (X2 @ id @ X6) @ id)) = $true) | (((X0 @ id @ X6 @ id)) != $true))) | ? [X7 : term] : ((((X0 @ id @ (sub @ X1 @ (push @ X7 @ id)) @ id)) != $true) & (((X0 @ id @ X7 @ id)) = $true)) | (((X0 @ id @ (lam @ (sub @ X1 @ (push @ one @ sh))) @ id)) = $true))) & (? [X8 : (subst > term > subst > $o),X9 : term] : (! [X10 : (subst > term > term)] : (? [X11 : subst,X12 : term,X13 : subst] : (((sub @ (X10 @ X11 @ X12) @ X13)) != ((X10 @ (comp @ X11 @ X13) @ (sub @ X12 @ X13)))) | (((X8 @ id @ (lam @ (X10 @ sh @ one)) @ id)) = $true) | ? [X14 : term] : ((((X8 @ id @ (X10 @ id @ X14) @ id)) != $true) & (((X8 @ id @ X14 @ id)) = $true))) & ! [X15 : term] : ((((X8 @ id @ (sub @ X9 @ (push @ X15 @ id)) @ id)) = $true) | (((X8 @ id @ X15 @ id)) != $true)) & (((X8 @ id @ (lam @ (sub @ X9 @ (push @ one @ sh))) @ id)) != $true)) | (sP18 != $true))), inference(rectify,[],[f217])). thf(f219,plain,( ((sP18 = $true) | ! [X0 : (subst > term > subst > $o),X1 : term] : ((! [X3 : subst,X4 : term,X5 : subst] : (((sK344 @ X0 @ (comp @ X3 @ X5) @ (sub @ X4 @ X5))) = ((sub @ (sK344 @ X0 @ X3 @ X4) @ X5))) & (((X0 @ id @ (lam @ (sK344 @ X0 @ sh @ one)) @ id)) != $true) & ! [X6 : term] : ((((X0 @ id @ (sK344 @ X0 @ id @ X6) @ id)) = $true) | (((X0 @ id @ X6 @ id)) != $true))) | (($true != ((X0 @ id @ (sub @ X1 @ (push @ (sK345 @ X1 @ X0) @ id)) @ id))) & (((X0 @ id @ (sK345 @ X1 @ X0) @ id)) = $true)) | (((X0 @ id @ (lam @ (sub @ X1 @ (push @ one @ sh))) @ id)) = $true))) & ((! [X10 : (subst > term > term)] : ((((X10 @ (comp @ (sK348 @ X10) @ (sK350 @ X10)) @ (sub @ (sK349 @ X10) @ (sK350 @ X10)))) != ((sub @ (X10 @ (sK348 @ X10) @ (sK349 @ X10)) @ (sK350 @ X10)))) | ($true = ((sK346 @ id @ (lam @ (X10 @ sh @ one)) @ id))) | (($true != ((sK346 @ id @ (X10 @ id @ (sK351 @ X10)) @ id))) & (((sK346 @ id @ (sK351 @ X10) @ id)) = $true))) & ! [X15 : term] : (($true = ((sK346 @ id @ (sub @ sK347 @ (push @ X15 @ id)) @ id))) | ($true != ((sK346 @ id @ X15 @ id)))) & ($true != ((sK346 @ id @ (lam @ (sub @ sK347 @ (push @ one @ sh))) @ id)))) | (sP18 != $true))), inference(skolemize,[status(esa),new_symbols(skolem,[vAPP,sK346,sK347,vAPP]),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X8,$thf(sK346)),skolemize(X9,$thf(sK347)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11))],[f218])). thf(f272,plain,( (((apinj1 = $true) | ? [X3 : term,X1 : term,X0 : term,X2 : term] : ((((ap @ X1 @ X3)) = ((ap @ X0 @ X2))) & (X0 != X1))) & (! [X3 : term,X1 : term,X0 : term,X2 : term] : ((((ap @ X1 @ X3)) != ((ap @ X0 @ X2))) | (X0 = X1)) | (apinj1 != $true))) & ! [X228 : subst] : ? [X229 : d_subst] : (((d2subst @ X229)) = X228) & (hoasapnotvar_lthm = $true) & (((induction2_lthm = $true) | ($true != sP60)) & (($true = sP60) | (induction2_lthm != $true))) & (pushprop_gthm = $true) & (((hoasinduction_no_psi_cond_lthm = $true) | (sP56 != $true)) & ((sP56 = $true) | (hoasinduction_no_psi_cond_lthm != $true))) & (induction2lem_gthm = $true) & (((sP51 = $true) | (hoasinduction_lem3_lthm != $true)) & ((hoasinduction_lem3_lthm = $true) | (sP51 != $true))) & (((hoasinduction_no_psi_cond = $true) | ? [X206 : (subst > term > subst > $o),X207 : term] : (! [X208 : term,X209 : term] : (($true != ((X206 @ id @ X209 @ id))) | ($true = ((X206 @ id @ (ap @ (sub @ X208 @ id) @ X209) @ id))) | (((X206 @ id @ X208 @ id)) != $true)) & ($true != ((sP48 @ X206))) & (((X206 @ id @ X207 @ id)) != $true))) & (! [X206 : (subst > term > subst > $o),X207 : term] : (? [X208 : term,X209 : term] : (($true = ((X206 @ id @ X209 @ id))) & ($true != ((X206 @ id @ (ap @ (sub @ X208 @ id) @ X209) @ id))) & (((X206 @ id @ X208 @ id)) = $true)) | ($true = ((sP48 @ X206))) | (((X206 @ id @ X207 @ id)) = $true)) | (hoasinduction_no_psi_cond != $true))) & (pushprop_lthm_orig = $true) & (axassoc = $true) & (pushprop_lthm = $true) & (hoasinduction_lem1_lthm = $true) & (ulamvar1 != $true) & (substmonoid_gthm = $true) & (hoasapnotvar_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & ! [X524 : d_subst] : (d_id = X524) & (lamnotap != $true) & (((d2subst @ d_id)) = id) & ((($true = sP47) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | ($true != sP47))) & (hoasapnotvar = $true) & (axvarshift = $true) & (((hoasinduction_lem0_lthm = $true) | ? [X521 : (subst > term > subst > $o)] : ! [X522 : (term > $o)] : ? [X523 : term] : (((X521 @ id @ X523 @ id)) != ((X522 @ X523)))) & (! [X521 : (subst > term > subst > $o)] : ? [X522 : (term > $o)] : ! [X523 : term] : (((X521 @ id @ X523 @ id)) = ((X522 @ X523))) | (hoasinduction_lem0_lthm != $true))) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (((sP43 = $true) | (induction2 != $true)) & ((induction2 = $true) | (sP43 != $true))) & (hoasinduction_lem1_gthm = $true) & (((axidr = $true) | ? [X183 : subst] : (((comp @ X183 @ id)) != X183)) & (! [X183 : subst] : (((comp @ X183 @ id)) = X183) | (axidr != $true))) & (((sP41 = $true) | (hoasinduction_lem3 != $true)) & ((hoasinduction_lem3 = $true) | (sP41 != $true))) & ((($true = sP39) | (hoasinduction_lem3v2 != $true)) & ((hoasinduction_lem3v2 = $true) | ($true != sP39))) & (((! [X231 : subst] : (((comp @ X231 @ id)) = X231) & ! [X230 : subst] : (((comp @ id @ X230)) = X230)) | (substmonoid != $true)) & ((substmonoid = $true) | (? [X231 : subst] : (((comp @ X231 @ id)) != X231) | ? [X230 : subst] : (((comp @ id @ X230)) != X230)))) & ((($true = sP37) | (hoasinduction_lem2v2 != $true)) & ((hoasinduction_lem2v2 = $true) | ($true != sP37))) & (hoaslamnotvar = $true) & (((sP35 = $true) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | (sP35 != $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)))))))))))) & (hoaslamnotap_lthm = $true) & ((! [X244 : subst,X243 : (term > $o),X242 : term] : ? [X245 : (term > $o)] : ! [X246 : term] : (((X245 @ X246)) = ((X243 @ (sub @ X246 @ (push @ X242 @ X244))))) | (pushprop_lem0 != $true)) & ((pushprop_lem0 = $true) | ? [X244 : subst,X243 : (term > $o),X242 : term] : ! [X245 : (term > $o)] : ? [X246 : term] : (((X245 @ X246)) != ((X243 @ (sub @ X246 @ (push @ X242 @ X244))))))) & (((pushprop_lem1 = $true) | ? [X270 : term,X272 : subst,X271 : (term > $o),X268 : term,X269 : (term > $o)] : ((((X271 @ (sub @ X270 @ (push @ X268 @ X272)))) != $true) & (((X269 @ X270)) = $true))) & (! [X270 : term,X272 : subst,X271 : (term > $o),X268 : term,X269 : (term > $o)] : ((((X271 @ (sub @ X270 @ (push @ X268 @ X272)))) = $true) | (((X269 @ X270)) != $true)) | (pushprop_lem1 != $true))) & (((hoasinduction_lem3b = $true) | ? [X240 : term] : ! [X241 : (subst > term > term)] : (((sub @ X240 @ (push @ one @ sh))) != ((X241 @ sh @ one)))) & (! [X240 : term] : ? [X241 : (subst > term > term)] : (((sub @ X240 @ (push @ one @ sh))) = ((X241 @ sh @ one))) | (hoasinduction_lem3b != $true))) & (hoasinduction_lem3a_gthm = $true) & (lamnotvar = $true) & (((d2subst @ d_id)) = sh) & (((axvarid = $true) | ? [X520 : term] : (((sub @ X520 @ id)) != X520)) & (! [X520 : term] : (((sub @ X520 @ id)) = X520) | (axvarid != $true))) & (((pushprop_lem0_lthm = $true) | ? [X43 : subst,X44 : (term > $o),X42 : term] : ! [X45 : (term > $o)] : ? [X46 : term] : (((X45 @ X46)) != ((X44 @ (sub @ X46 @ (push @ X42 @ X43)))))) & (! [X43 : subst,X44 : (term > $o),X42 : term] : ? [X45 : (term > $o)] : ! [X46 : term] : (((X45 @ X46)) = ((X44 @ (sub @ X46 @ (push @ X42 @ X43))))) | (pushprop_lem0_lthm != $true))) & (((hoasinduction_lem3aa_lthm = $true) | (sP33 != $true)) & ((sP33 = $true) | (hoasinduction_lem3aa_lthm != $true))) & (hoaslamnotvar_lthm = $true) & (pushprop_lem1v2_gthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (((hoasinduction_lem3aaa = $true) | ($true != sP32)) & (($true = sP32) | (hoasinduction_lem3aaa != $true))) & (((sP31 = $true) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | (sP31 != $true))) & (axabs = $true) & (((hoasinduction_lem3v2_f = $true) | ? [X78 : term] : ! [X79 : (subst > term > term)] : ? [X81 : subst,X80 : term] : (((X79 @ X81 @ X80)) != ((sub @ X78 @ (push @ X80 @ X81))))) & (! [X78 : term] : ? [X79 : (subst > term > term)] : ! [X81 : subst,X80 : term] : (((X79 @ X81 @ X80)) = ((sub @ X78 @ (push @ X80 @ X81)))) | (hoasinduction_lem3v2_f != $true))) & (hoasinduction_lem1 = $true) & ! [X101 : d_term,X100 : d_term] : ((X100 = X101) | (((d2term @ X101)) != ((d2term @ X100)))) & ((! [X192 : (subst > term > subst > $o)] : ? [X193 : (term > $o)] : ! [X194 : term] : (((X192 @ id @ X194 @ id)) = ((X193 @ X194))) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem0 = $true) | ? [X192 : (subst > term > subst > $o)] : ! [X193 : (term > $o)] : ? [X194 : term] : (((X192 @ id @ X194 @ id)) != ((X193 @ X194))))) & (((hoasinduction_lthm = $true) | ($true != sP30)) & (($true = sP30) | (hoasinduction_lthm != $true))) & (((? [X444 : term] : (((sub @ X444 @ id)) != X444) | ! [X445 : term] : (((sub @ X445 @ id)) = X445)) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | (! [X444 : term] : (((sub @ X444 @ id)) = X444) & ? [X445 : term] : (((sub @ X445 @ id)) != X445)))) & (((hoaslaminj = $true) | ? [X511 : term,X512 : (subst > term > term),X513 : (subst > term > term),X510 : subst] : (! [X518 : term,X519 : subst,X517 : subst] : (((X512 @ (comp @ X519 @ X517) @ (sub @ X518 @ X517))) = ((sub @ (X512 @ X519 @ X518) @ X517))) & ! [X515 : subst,X516 : subst,X514 : term] : (((sub @ (X513 @ X515 @ X514) @ X516)) = ((X513 @ (comp @ X515 @ X516) @ (sub @ X514 @ X516)))) & (((lam @ (X512 @ sh @ one))) = ((lam @ (X513 @ sh @ one)))) & (((X512 @ X510 @ X511)) != ((X513 @ X510 @ X511))))) & (! [X511 : term,X512 : (subst > term > term),X513 : (subst > term > term),X510 : subst] : (? [X518 : term,X519 : subst,X517 : subst] : (((X512 @ (comp @ X519 @ X517) @ (sub @ X518 @ X517))) != ((sub @ (X512 @ X519 @ X518) @ X517))) | ? [X515 : subst,X516 : subst,X514 : term] : (((sub @ (X513 @ X515 @ X514) @ X516)) != ((X513 @ (comp @ X515 @ X516) @ (sub @ X514 @ X516)))) | (((lam @ (X512 @ sh @ one))) != ((lam @ (X513 @ sh @ one)))) | (((X512 @ X510 @ X511)) = ((X513 @ X510 @ X511)))) | (hoaslaminj != $true))) & (((d2term @ d_one)) = one) & (hoasinduction_lem1v2 = $true) & (pushprop_lem3v2_lthm = $true) & (((var @ (d2term @ d_one))) != $true) & (hoaslamnotap_gthm = $true) & (((axscons = $true) | ? [X247 : subst] : (((push @ (sub @ one @ X247) @ (comp @ sh @ X247))) != X247)) & (! [X247 : subst] : (((push @ (sub @ one @ X247) @ (comp @ sh @ X247))) = X247) | (axscons != $true))) & (hoaslamnotvar_gthm = $true) & (hoaslaminj_gthm = $true) & (hoasinduction_lem3v2_gthm = $true) & ((! [X204 : term,X205 : subst] : (((comp @ sh @ (push @ X204 @ X205))) = X205) | (axshiftcons != $true)) & ((axshiftcons = $true) | ? [X204 : term,X205 : subst] : (((comp @ sh @ (push @ X204 @ X205))) != X205))) & (hoasinduction_lem3b_gthm = $true) & (((laminj = $true) | ? [X233 : term,X232 : term] : ((((lam @ X232)) = ((lam @ X233))) & (X232 != X233))) & (! [X233 : term,X232 : term] : ((((lam @ X232)) != ((lam @ X233))) | (X232 = X233)) | (laminj != $true))) & ((! [X140 : term,X139 : term,X141 : (subst > term > term)] : ((((lam @ (X141 @ sh @ one))) != ((ap @ (sub @ X140 @ id) @ X139))) | ? [X143 : subst,X142 : term,X144 : subst] : (((X141 @ (comp @ X143 @ X144) @ (sub @ X142 @ X144))) != ((sub @ (X141 @ X143 @ X142) @ X144)))) | (hoaslamnotap != $true)) & ((hoaslamnotap = $true) | ? [X140 : term,X139 : term,X141 : (subst > term > term)] : ((((lam @ (X141 @ sh @ one))) = ((ap @ (sub @ X140 @ id) @ X139))) & ! [X143 : subst,X142 : term,X144 : subst] : (((X141 @ (comp @ X143 @ X144) @ (sub @ X142 @ X144))) = ((sub @ (X141 @ X143 @ X142) @ X144)))))) & (pushprop_lem2v2 = $true) & (((? [X103 : subst] : (((comp @ X103 @ id)) != X103) | ? [X102 : subst] : (((comp @ id @ X102)) != X102) | (! [X105 : subst] : (((comp @ X105 @ id)) = X105) & ! [X104 : subst] : (((comp @ id @ X104)) = X104))) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | (! [X103 : subst] : (((comp @ X103 @ id)) = X103) & ! [X102 : subst] : (((comp @ id @ X102)) = X102) & (? [X105 : subst] : (((comp @ X105 @ id)) != X105) | ? [X104 : subst] : (((comp @ id @ X104)) != X104))))) & (apnotvar = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (((hoasapinj1 = $true) | ? [X314 : term,X317 : term,X316 : term,X315 : term] : ((X314 != X316) & (((ap @ (sub @ X316 @ id) @ X317)) = ((ap @ (sub @ X314 @ id) @ X315))))) & (! [X314 : term,X317 : term,X316 : term,X315 : term] : ((X314 = X316) | (((ap @ (sub @ X316 @ id) @ X317)) != ((ap @ (sub @ X314 @ id) @ X315)))) | (hoasapinj1 != $true))) & (((hoasinduction_lem3v2_f_lthm = $true) | ? [X188 : term] : ! [X189 : (subst > term > term)] : ? [X190 : subst,X191 : term] : (((X189 @ X190 @ X191)) != ((sub @ X188 @ (push @ X191 @ X190))))) & (! [X188 : term] : ? [X189 : (subst > term > term)] : ! [X190 : subst,X191 : term] : (((X189 @ X190 @ X191)) = ((sub @ X188 @ (push @ X191 @ X190)))) | (hoasinduction_lem3v2_f_lthm != $true))) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (hoasapinj2_gthm = $true) & (((hoasapinj2_lthm = $true) | (? [X149 : term,X152 : term,X151 : term,X150 : term] : ((X150 != X151) & (((ap @ (sub @ X149 @ id) @ X150)) = ((ap @ (sub @ X152 @ id) @ X151)))) & ! [X147 : term,X146 : term,X148 : term,X145 : term] : ((((ap @ X145 @ X148)) != ((ap @ X146 @ X147))) | (X147 = X148)))) & ((! [X149 : term,X152 : term,X151 : term,X150 : term] : ((X150 = X151) | (((ap @ (sub @ X149 @ id) @ X150)) != ((ap @ (sub @ X152 @ id) @ X151)))) | ? [X147 : term,X146 : term,X148 : term,X145 : term] : ((((ap @ X145 @ X148)) = ((ap @ X146 @ X147))) & (X147 != X148))) | (hoasapinj2_lthm != $true))) & (hoasinduction_lem2_lthm = $true) & ((! [X187 : term,X184 : term,X186 : term,X185 : term] : ((X186 = X187) | (((ap @ (sub @ X185 @ id) @ X186)) != ((ap @ (sub @ X184 @ id) @ X187)))) | (hoasapinj2 != $true)) & ((hoasapinj2 = $true) | ? [X187 : term,X184 : term,X186 : term,X185 : term] : ((X186 != X187) & (((ap @ (sub @ X185 @ id) @ X186)) = ((ap @ (sub @ X184 @ id) @ X187)))))) & ((! [X309 : (term > $o),X312 : subst,X311 : term,X310 : (term > $o)] : ((((X309 @ X311)) != $true) | ? [X313 : term] : (((X309 @ (sub @ X313 @ (push @ X311 @ X312)))) != ((X310 @ X313))) | ($true = ((X310 @ one)))) | (pushprop_lem1v2 != $true)) & ((pushprop_lem1v2 = $true) | ? [X309 : (term > $o),X312 : subst,X311 : term,X310 : (term > $o)] : ((((X309 @ X311)) = $true) & ! [X313 : term] : (((X309 @ (sub @ X313 @ (push @ X311 @ X312)))) = ((X310 @ X313))) & ($true != ((X310 @ one)))))) & (axclos = $true) & (((hoasapinj1_lthm = $true) | (! [X259 : term] : (((sub @ X259 @ id)) = X259) & ! [X263 : term,X261 : term,X260 : term,X262 : term] : ((((ap @ X261 @ X263)) != ((ap @ X262 @ X260))) | (X261 = X262)) & ? [X265 : term,X266 : term,X267 : term,X264 : term] : ((((ap @ (sub @ X267 @ id) @ X265)) = ((ap @ (sub @ X266 @ id) @ X264))) & (X266 != X267)))) & ((? [X259 : term] : (((sub @ X259 @ id)) != X259) | ? [X263 : term,X261 : term,X260 : term,X262 : term] : ((((ap @ X261 @ X263)) = ((ap @ X262 @ X260))) & (X261 != X262)) | ! [X265 : term,X266 : term,X267 : term,X264 : term] : ((((ap @ (sub @ X267 @ id) @ X265)) != ((ap @ (sub @ X266 @ id) @ X264))) | (X266 = X267))) | (hoasapinj1_lthm != $true))) & (pushprop_lem0_gthm = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (pushprop_lem3v2 = $true) & ((! [X248 : term] : (((sub @ X248 @ id)) = X248) | (termmset != $true)) & ((termmset = $true) | ? [X248 : term] : (((sub @ X248 @ id)) != X248))) & ((! [X65 : term,X64 : (term > $o)] : (? [X68 : term] : ((((X64 @ X68)) = $true) & ($true != ((X64 @ (lam @ X68))))) | (((sP21 @ X64)) = $true) | ? [X69 : term] : ((((var @ X69)) = $true) & ($true != ((X64 @ X69)))) | (((X64 @ X65)) = $true)) | (induction != $true)) & ((induction = $true) | ? [X65 : term,X64 : (term > $o)] : (! [X68 : term] : ((((X64 @ X68)) != $true) | ($true = ((X64 @ (lam @ X68))))) & (((sP21 @ X64)) != $true) & ! [X69 : term] : ((((var @ X69)) != $true) | ($true = ((X64 @ X69)))) & (((X64 @ X65)) != $true)))) & ((($true = sP20) | (hoasinduction_lem3aa != $true)) & ((hoasinduction_lem3aa = $true) | ($true != sP20))) & ((! [X82 : term] : ? [X83 : (subst > term > term)] : (((sub @ X82 @ (push @ one @ sh))) = ((X83 @ sh @ one))) | (hoasinduction_lem3b_lthm != $true)) & ((hoasinduction_lem3b_lthm = $true) | ? [X82 : term] : ! [X83 : (subst > term > term)] : (((sub @ X82 @ (push @ one @ sh))) != ((X83 @ sh @ one))))) & (pushprop_lem2v2_gthm = $true) & (hoasapinj1_gthm = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((($true = sP19) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | ($true != sP19))) & (pushprop_lem1_gthm = $true) & (axmap = $true) & (((hoaslaminj_lthm = $true) | ($true != sP17)) & (($true = sP17) | (hoaslaminj_lthm != $true))) & (hoasinduction_gthm = $true) & (((apinj2 = $true) | ? [X236 : term,X238 : term,X237 : term,X239 : term] : ((X236 != X239) & (((ap @ X237 @ X236)) = ((ap @ X238 @ X239))))) & (! [X236 : term,X238 : term,X237 : term,X239 : term] : ((X236 = X239) | (((ap @ X237 @ X236)) != ((ap @ X238 @ X239)))) | (apinj2 != $true))) & (termmset_gthm = $true) & (((pushprop_lem1v2_lthm = $true) | (? [X417 : term,X414 : (term > $o),X416 : subst,X415 : (term > $o)] : ((((X415 @ one)) != $true) & ($true = ((X414 @ X417))) & ! [X418 : term] : (((X415 @ X418)) = ((X414 @ (sub @ X418 @ (push @ X417 @ X416)))))) & ! [X412 : term,X413 : subst] : (((sub @ one @ (push @ X412 @ X413))) = X412))) & ((! [X417 : term,X414 : (term > $o),X416 : subst,X415 : (term > $o)] : ((((X415 @ one)) = $true) | ($true != ((X414 @ X417))) | ? [X418 : term] : (((X415 @ X418)) != ((X414 @ (sub @ X418 @ (push @ X417 @ X416)))))) | ? [X412 : term,X413 : subst] : (((sub @ one @ (push @ X412 @ X413))) != X412)) | (pushprop_lem1v2_lthm != $true))) & ! [X106 : term] : ? [X107 : d_term] : (((d2term @ X107)) = X106) & ! [X318 : d_term] : (d_one = X318) & (hoasinduction_lem3_gthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (axapp = $true) & (((hoasinduction_lthm_3 = $true) | ($true != sP16)) & (($true = sP16) | (hoasinduction_lthm_3 != $true))) & (hoasinduction_lem1v2_gthm = $true) & (((sP10 = $true) | (hoasinduction != $true)) & ((hoasinduction = $true) | (sP10 != $true))) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ((! [X195 : term,X196 : term] : ((((sub @ X196 @ sh)) != ((sub @ X195 @ sh))) | (X195 = X196)) | (shinj != $true)) & ((shinj = $true) | ? [X195 : term,X196 : term] : ((((sub @ X196 @ sh)) = ((sub @ X195 @ sh))) & (X195 != X196)))) & (((sP7 = $true) | (hoasinduction_lem3v2_lthm != $true)) & ((hoasinduction_lem3v2_lthm = $true) | (sP7 != $true))) & (hoasinduction_lem2_gthm = $true) & (hoasinduction_lem2v2_gthm = $true) & (((induction2lem = $true) | ($true != sP5)) & (($true = sP5) | (induction2lem != $true))) & (pushprop_lem2v2_lthm = $true) & (((? [X485 : subst,X486 : term] : (((sub @ one @ (push @ X486 @ X485))) != X486) | ! [X489 : term,X493 : (term > $o),X490 : term,X491 : (term > $o),X492 : subst] : ((((X493 @ X490)) != $true) | (((X491 @ (sub @ X490 @ (push @ X489 @ X492)))) = $true)) | ? [X487 : subst,X488 : term] : (((comp @ sh @ (push @ X488 @ X487))) != X487)) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (! [X485 : subst,X486 : term] : (((sub @ one @ (push @ X486 @ X485))) = X486) & ? [X489 : term,X493 : (term > $o),X490 : term,X491 : (term > $o),X492 : subst] : ((((X493 @ X490)) = $true) & (((X491 @ (sub @ X490 @ (push @ X489 @ X492)))) != $true)) & ! [X487 : subst,X488 : term] : (((comp @ sh @ (push @ X488 @ X487))) = X487)))) & ! [X41 : d_subst,X40 : d_subst] : ((((d2subst @ X40)) != ((d2subst @ X41))) | (X40 = X41)) & (((axidl = $true) | ? [X47 : subst] : (((comp @ id @ X47)) != X47)) & (! [X47 : subst] : (((comp @ id @ X47)) = X47) | (axidl != $true))) & ((($true = sP3) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | ($true != sP3))) & (induction2_gthm = $true) & (ulamvarsh = $true) & (((hoasinduction_lem3v2a_lthm = $true) | (sP2 != $true)) & ((sP2 = $true) | (hoasinduction_lem3v2a_lthm != $true))) & (pushprop = $true) & (((axvarcons = $true) | ? [X234 : subst,X235 : term] : (((sub @ one @ (push @ X235 @ X234))) != X235)) & (! [X234 : subst,X235 : term] : (((sub @ one @ (push @ X235 @ X234))) = X235) | (axvarcons != $true))) & (ulamvarind = $true)), inference(nnf_transformation,[],[f77])). thf(f273,plain,( ((apinj1 = $true) | ? [X3 : term,X1 : term,X0 : term,X2 : term] : ((((ap @ X1 @ X3)) = ((ap @ X0 @ X2))) & (X0 != X1))) & (! [X3 : term,X1 : term,X0 : term,X2 : term] : ((((ap @ X1 @ X3)) != ((ap @ X0 @ X2))) | (X0 = X1)) | (apinj1 != $true)) & ! [X228 : subst] : ? [X229 : d_subst] : (((d2subst @ X229)) = X228) & (hoasapnotvar_lthm = $true) & ((induction2_lthm = $true) | ($true != sP60)) & (($true = sP60) | (induction2_lthm != $true)) & (pushprop_gthm = $true) & ((hoasinduction_no_psi_cond_lthm = $true) | (sP56 != $true)) & ((sP56 = $true) | (hoasinduction_no_psi_cond_lthm != $true)) & (induction2lem_gthm = $true) & ((sP51 = $true) | (hoasinduction_lem3_lthm != $true)) & ((hoasinduction_lem3_lthm = $true) | (sP51 != $true)) & ((hoasinduction_no_psi_cond = $true) | ? [X206 : (subst > term > subst > $o),X207 : term] : (! [X208 : term,X209 : term] : (($true != ((X206 @ id @ X209 @ id))) | ($true = ((X206 @ id @ (ap @ (sub @ X208 @ id) @ X209) @ id))) | (((X206 @ id @ X208 @ id)) != $true)) & ($true != ((sP48 @ X206))) & (((X206 @ id @ X207 @ id)) != $true))) & (! [X206 : (subst > term > subst > $o),X207 : term] : (? [X208 : term,X209 : term] : (($true = ((X206 @ id @ X209 @ id))) & ($true != ((X206 @ id @ (ap @ (sub @ X208 @ id) @ X209) @ id))) & (((X206 @ id @ X208 @ id)) = $true)) | ($true = ((sP48 @ X206))) | (((X206 @ id @ X207 @ id)) = $true)) | (hoasinduction_no_psi_cond != $true)) & (pushprop_lthm_orig = $true) & (axassoc = $true) & (pushprop_lthm = $true) & (hoasinduction_lem1_lthm = $true) & (ulamvar1 != $true) & (substmonoid_gthm = $true) & (hoasapnotvar_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & ! [X524 : d_subst] : (d_id = X524) & (lamnotap != $true) & (((d2subst @ d_id)) = id) & (($true = sP47) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | ($true != sP47)) & (hoasapnotvar = $true) & (axvarshift = $true) & ((hoasinduction_lem0_lthm = $true) | ? [X521 : (subst > term > subst > $o)] : ! [X522 : (term > $o)] : ? [X523 : term] : (((X521 @ id @ X523 @ id)) != ((X522 @ X523)))) & (! [X521 : (subst > term > subst > $o)] : ? [X522 : (term > $o)] : ! [X523 : term] : (((X521 @ id @ X523 @ id)) = ((X522 @ X523))) | (hoasinduction_lem0_lthm != $true)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((sP43 = $true) | (induction2 != $true)) & ((induction2 = $true) | (sP43 != $true)) & (hoasinduction_lem1_gthm = $true) & ((axidr = $true) | ? [X183 : subst] : (((comp @ X183 @ id)) != X183)) & (! [X183 : subst] : (((comp @ X183 @ id)) = X183) | (axidr != $true)) & ((sP41 = $true) | (hoasinduction_lem3 != $true)) & ((hoasinduction_lem3 = $true) | (sP41 != $true)) & (($true = sP39) | (hoasinduction_lem3v2 != $true)) & ((hoasinduction_lem3v2 = $true) | ($true != sP39)) & ((! [X231 : subst] : (((comp @ X231 @ id)) = X231) & ! [X230 : subst] : (((comp @ id @ X230)) = X230)) | (substmonoid != $true)) & ((substmonoid = $true) | ? [X231 : subst] : (((comp @ X231 @ id)) != X231) | ? [X230 : subst] : (((comp @ id @ X230)) != X230)) & (($true = sP37) | (hoasinduction_lem2v2 != $true)) & ((hoasinduction_lem2v2 = $true) | ($true != sP37)) & (hoaslamnotvar = $true) & ((sP35 = $true) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | (sP35 != $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)))))))))))) & (hoaslamnotap_lthm = $true) & (! [X244 : subst,X243 : (term > $o),X242 : term] : ? [X245 : (term > $o)] : ! [X246 : term] : (((X245 @ X246)) = ((X243 @ (sub @ X246 @ (push @ X242 @ X244))))) | (pushprop_lem0 != $true)) & ((pushprop_lem0 = $true) | ? [X244 : subst,X243 : (term > $o),X242 : term] : ! [X245 : (term > $o)] : ? [X246 : term] : (((X245 @ X246)) != ((X243 @ (sub @ X246 @ (push @ X242 @ X244)))))) & ((pushprop_lem1 = $true) | ? [X270 : term,X272 : subst,X271 : (term > $o),X268 : term,X269 : (term > $o)] : ((((X271 @ (sub @ X270 @ (push @ X268 @ X272)))) != $true) & (((X269 @ X270)) = $true))) & (! [X270 : term,X272 : subst,X271 : (term > $o),X268 : term,X269 : (term > $o)] : ((((X271 @ (sub @ X270 @ (push @ X268 @ X272)))) = $true) | (((X269 @ X270)) != $true)) | (pushprop_lem1 != $true)) & ((hoasinduction_lem3b = $true) | ? [X240 : term] : ! [X241 : (subst > term > term)] : (((sub @ X240 @ (push @ one @ sh))) != ((X241 @ sh @ one)))) & (! [X240 : term] : ? [X241 : (subst > term > term)] : (((sub @ X240 @ (push @ one @ sh))) = ((X241 @ sh @ one))) | (hoasinduction_lem3b != $true)) & (hoasinduction_lem3a_gthm = $true) & (lamnotvar = $true) & (((d2subst @ d_id)) = sh) & ((axvarid = $true) | ? [X520 : term] : (((sub @ X520 @ id)) != X520)) & (! [X520 : term] : (((sub @ X520 @ id)) = X520) | (axvarid != $true)) & ((pushprop_lem0_lthm = $true) | ? [X43 : subst,X44 : (term > $o),X42 : term] : ! [X45 : (term > $o)] : ? [X46 : term] : (((X45 @ X46)) != ((X44 @ (sub @ X46 @ (push @ X42 @ X43)))))) & (! [X43 : subst,X44 : (term > $o),X42 : term] : ? [X45 : (term > $o)] : ! [X46 : term] : (((X45 @ X46)) = ((X44 @ (sub @ X46 @ (push @ X42 @ X43))))) | (pushprop_lem0_lthm != $true)) & ((hoasinduction_lem3aa_lthm = $true) | (sP33 != $true)) & ((sP33 = $true) | (hoasinduction_lem3aa_lthm != $true)) & (hoaslamnotvar_lthm = $true) & (pushprop_lem1v2_gthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & ((hoasinduction_lem3aaa = $true) | ($true != sP32)) & (($true = sP32) | (hoasinduction_lem3aaa != $true)) & ((sP31 = $true) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | (sP31 != $true)) & (axabs = $true) & ((hoasinduction_lem3v2_f = $true) | ? [X78 : term] : ! [X79 : (subst > term > term)] : ? [X81 : subst,X80 : term] : (((X79 @ X81 @ X80)) != ((sub @ X78 @ (push @ X80 @ X81))))) & (! [X78 : term] : ? [X79 : (subst > term > term)] : ! [X81 : subst,X80 : term] : (((X79 @ X81 @ X80)) = ((sub @ X78 @ (push @ X80 @ X81)))) | (hoasinduction_lem3v2_f != $true)) & (hoasinduction_lem1 = $true) & ! [X101 : d_term,X100 : d_term] : ((X100 = X101) | (((d2term @ X101)) != ((d2term @ X100)))) & (! [X192 : (subst > term > subst > $o)] : ? [X193 : (term > $o)] : ! [X194 : term] : (((X192 @ id @ X194 @ id)) = ((X193 @ X194))) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem0 = $true) | ? [X192 : (subst > term > subst > $o)] : ! [X193 : (term > $o)] : ? [X194 : term] : (((X192 @ id @ X194 @ id)) != ((X193 @ X194)))) & ((hoasinduction_lthm = $true) | ($true != sP30)) & (($true = sP30) | (hoasinduction_lthm != $true)) & (? [X444 : term] : (((sub @ X444 @ id)) != X444) | ! [X445 : term] : (((sub @ X445 @ id)) = X445) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | (! [X444 : term] : (((sub @ X444 @ id)) = X444) & ? [X445 : term] : (((sub @ X445 @ id)) != X445))) & ((hoaslaminj = $true) | ? [X511 : term,X512 : (subst > term > term),X513 : (subst > term > term),X510 : subst] : (! [X518 : term,X519 : subst,X517 : subst] : (((X512 @ (comp @ X519 @ X517) @ (sub @ X518 @ X517))) = ((sub @ (X512 @ X519 @ X518) @ X517))) & ! [X515 : subst,X516 : subst,X514 : term] : (((sub @ (X513 @ X515 @ X514) @ X516)) = ((X513 @ (comp @ X515 @ X516) @ (sub @ X514 @ X516)))) & (((lam @ (X512 @ sh @ one))) = ((lam @ (X513 @ sh @ one)))) & (((X512 @ X510 @ X511)) != ((X513 @ X510 @ X511))))) & (! [X511 : term,X512 : (subst > term > term),X513 : (subst > term > term),X510 : subst] : (? [X518 : term,X519 : subst,X517 : subst] : (((X512 @ (comp @ X519 @ X517) @ (sub @ X518 @ X517))) != ((sub @ (X512 @ X519 @ X518) @ X517))) | ? [X515 : subst,X516 : subst,X514 : term] : (((sub @ (X513 @ X515 @ X514) @ X516)) != ((X513 @ (comp @ X515 @ X516) @ (sub @ X514 @ X516)))) | (((lam @ (X512 @ sh @ one))) != ((lam @ (X513 @ sh @ one)))) | (((X512 @ X510 @ X511)) = ((X513 @ X510 @ X511)))) | (hoaslaminj != $true)) & (((d2term @ d_one)) = one) & (hoasinduction_lem1v2 = $true) & (pushprop_lem3v2_lthm = $true) & (((var @ (d2term @ d_one))) != $true) & (hoaslamnotap_gthm = $true) & ((axscons = $true) | ? [X247 : subst] : (((push @ (sub @ one @ X247) @ (comp @ sh @ X247))) != X247)) & (! [X247 : subst] : (((push @ (sub @ one @ X247) @ (comp @ sh @ X247))) = X247) | (axscons != $true)) & (hoaslamnotvar_gthm = $true) & (hoaslaminj_gthm = $true) & (hoasinduction_lem3v2_gthm = $true) & (! [X204 : term,X205 : subst] : (((comp @ sh @ (push @ X204 @ X205))) = X205) | (axshiftcons != $true)) & ((axshiftcons = $true) | ? [X204 : term,X205 : subst] : (((comp @ sh @ (push @ X204 @ X205))) != X205)) & (hoasinduction_lem3b_gthm = $true) & ((laminj = $true) | ? [X233 : term,X232 : term] : ((((lam @ X232)) = ((lam @ X233))) & (X232 != X233))) & (! [X233 : term,X232 : term] : ((((lam @ X232)) != ((lam @ X233))) | (X232 = X233)) | (laminj != $true)) & (! [X140 : term,X139 : term,X141 : (subst > term > term)] : ((((lam @ (X141 @ sh @ one))) != ((ap @ (sub @ X140 @ id) @ X139))) | ? [X143 : subst,X142 : term,X144 : subst] : (((X141 @ (comp @ X143 @ X144) @ (sub @ X142 @ X144))) != ((sub @ (X141 @ X143 @ X142) @ X144)))) | (hoaslamnotap != $true)) & ((hoaslamnotap = $true) | ? [X140 : term,X139 : term,X141 : (subst > term > term)] : ((((lam @ (X141 @ sh @ one))) = ((ap @ (sub @ X140 @ id) @ X139))) & ! [X143 : subst,X142 : term,X144 : subst] : (((X141 @ (comp @ X143 @ X144) @ (sub @ X142 @ X144))) = ((sub @ (X141 @ X143 @ X142) @ X144))))) & (pushprop_lem2v2 = $true) & (? [X103 : subst] : (((comp @ X103 @ id)) != X103) | ? [X102 : subst] : (((comp @ id @ X102)) != X102) | (! [X105 : subst] : (((comp @ X105 @ id)) = X105) & ! [X104 : subst] : (((comp @ id @ X104)) = X104)) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | (! [X103 : subst] : (((comp @ X103 @ id)) = X103) & ! [X102 : subst] : (((comp @ id @ X102)) = X102) & (? [X105 : subst] : (((comp @ X105 @ id)) != X105) | ? [X104 : subst] : (((comp @ id @ X104)) != X104)))) & (apnotvar = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((hoasapinj1 = $true) | ? [X314 : term,X317 : term,X316 : term,X315 : term] : ((X314 != X316) & (((ap @ (sub @ X316 @ id) @ X317)) = ((ap @ (sub @ X314 @ id) @ X315))))) & (! [X314 : term,X317 : term,X316 : term,X315 : term] : ((X314 = X316) | (((ap @ (sub @ X316 @ id) @ X317)) != ((ap @ (sub @ X314 @ id) @ X315)))) | (hoasapinj1 != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X188 : term] : ! [X189 : (subst > term > term)] : ? [X190 : subst,X191 : term] : (((X189 @ X190 @ X191)) != ((sub @ X188 @ (push @ X191 @ X190))))) & (! [X188 : term] : ? [X189 : (subst > term > term)] : ! [X190 : subst,X191 : term] : (((X189 @ X190 @ X191)) = ((sub @ X188 @ (push @ X191 @ X190)))) | (hoasinduction_lem3v2_f_lthm != $true)) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (hoasapinj2_gthm = $true) & ((hoasapinj2_lthm = $true) | (? [X149 : term,X152 : term,X151 : term,X150 : term] : ((X150 != X151) & (((ap @ (sub @ X149 @ id) @ X150)) = ((ap @ (sub @ X152 @ id) @ X151)))) & ! [X147 : term,X146 : term,X148 : term,X145 : term] : ((((ap @ X145 @ X148)) != ((ap @ X146 @ X147))) | (X147 = X148)))) & (! [X149 : term,X152 : term,X151 : term,X150 : term] : ((X150 = X151) | (((ap @ (sub @ X149 @ id) @ X150)) != ((ap @ (sub @ X152 @ id) @ X151)))) | ? [X147 : term,X146 : term,X148 : term,X145 : term] : ((((ap @ X145 @ X148)) = ((ap @ X146 @ X147))) & (X147 != X148)) | (hoasapinj2_lthm != $true)) & (hoasinduction_lem2_lthm = $true) & (! [X187 : term,X184 : term,X186 : term,X185 : term] : ((X186 = X187) | (((ap @ (sub @ X185 @ id) @ X186)) != ((ap @ (sub @ X184 @ id) @ X187)))) | (hoasapinj2 != $true)) & ((hoasapinj2 = $true) | ? [X187 : term,X184 : term,X186 : term,X185 : term] : ((X186 != X187) & (((ap @ (sub @ X185 @ id) @ X186)) = ((ap @ (sub @ X184 @ id) @ X187))))) & (! [X309 : (term > $o),X312 : subst,X311 : term,X310 : (term > $o)] : ((((X309 @ X311)) != $true) | ? [X313 : term] : (((X309 @ (sub @ X313 @ (push @ X311 @ X312)))) != ((X310 @ X313))) | ($true = ((X310 @ one)))) | (pushprop_lem1v2 != $true)) & ((pushprop_lem1v2 = $true) | ? [X309 : (term > $o),X312 : subst,X311 : term,X310 : (term > $o)] : ((((X309 @ X311)) = $true) & ! [X313 : term] : (((X309 @ (sub @ X313 @ (push @ X311 @ X312)))) = ((X310 @ X313))) & ($true != ((X310 @ one))))) & (axclos = $true) & ((hoasapinj1_lthm = $true) | (! [X259 : term] : (((sub @ X259 @ id)) = X259) & ! [X263 : term,X261 : term,X260 : term,X262 : term] : ((((ap @ X261 @ X263)) != ((ap @ X262 @ X260))) | (X261 = X262)) & ? [X265 : term,X266 : term,X267 : term,X264 : term] : ((((ap @ (sub @ X267 @ id) @ X265)) = ((ap @ (sub @ X266 @ id) @ X264))) & (X266 != X267)))) & (? [X259 : term] : (((sub @ X259 @ id)) != X259) | ? [X263 : term,X261 : term,X260 : term,X262 : term] : ((((ap @ X261 @ X263)) = ((ap @ X262 @ X260))) & (X261 != X262)) | ! [X265 : term,X266 : term,X267 : term,X264 : term] : ((((ap @ (sub @ X267 @ id) @ X265)) != ((ap @ (sub @ X266 @ id) @ X264))) | (X266 = X267)) | (hoasapinj1_lthm != $true)) & (pushprop_lem0_gthm = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (pushprop_lem3v2 = $true) & (! [X248 : term] : (((sub @ X248 @ id)) = X248) | (termmset != $true)) & ((termmset = $true) | ? [X248 : term] : (((sub @ X248 @ id)) != X248)) & (! [X65 : term,X64 : (term > $o)] : (? [X68 : term] : ((((X64 @ X68)) = $true) & ($true != ((X64 @ (lam @ X68))))) | (((sP21 @ X64)) = $true) | ? [X69 : term] : ((((var @ X69)) = $true) & ($true != ((X64 @ X69)))) | (((X64 @ X65)) = $true)) | (induction != $true)) & ((induction = $true) | ? [X65 : term,X64 : (term > $o)] : (! [X68 : term] : ((((X64 @ X68)) != $true) | ($true = ((X64 @ (lam @ X68))))) & (((sP21 @ X64)) != $true) & ! [X69 : term] : ((((var @ X69)) != $true) | ($true = ((X64 @ X69)))) & (((X64 @ X65)) != $true))) & (($true = sP20) | (hoasinduction_lem3aa != $true)) & ((hoasinduction_lem3aa = $true) | ($true != sP20)) & (! [X82 : term] : ? [X83 : (subst > term > term)] : (((sub @ X82 @ (push @ one @ sh))) = ((X83 @ sh @ one))) | (hoasinduction_lem3b_lthm != $true)) & ((hoasinduction_lem3b_lthm = $true) | ? [X82 : term] : ! [X83 : (subst > term > term)] : (((sub @ X82 @ (push @ one @ sh))) != ((X83 @ sh @ one)))) & (pushprop_lem2v2_gthm = $true) & (hoasapinj1_gthm = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (($true = sP19) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | ($true != sP19)) & (pushprop_lem1_gthm = $true) & (axmap = $true) & ((hoaslaminj_lthm = $true) | ($true != sP17)) & (($true = sP17) | (hoaslaminj_lthm != $true)) & (hoasinduction_gthm = $true) & ((apinj2 = $true) | ? [X236 : term,X238 : term,X237 : term,X239 : term] : ((X236 != X239) & (((ap @ X237 @ X236)) = ((ap @ X238 @ X239))))) & (! [X236 : term,X238 : term,X237 : term,X239 : term] : ((X236 = X239) | (((ap @ X237 @ X236)) != ((ap @ X238 @ X239)))) | (apinj2 != $true)) & (termmset_gthm = $true) & ((pushprop_lem1v2_lthm = $true) | (? [X417 : term,X414 : (term > $o),X416 : subst,X415 : (term > $o)] : ((((X415 @ one)) != $true) & ($true = ((X414 @ X417))) & ! [X418 : term] : (((X415 @ X418)) = ((X414 @ (sub @ X418 @ (push @ X417 @ X416)))))) & ! [X412 : term,X413 : subst] : (((sub @ one @ (push @ X412 @ X413))) = X412))) & (! [X417 : term,X414 : (term > $o),X416 : subst,X415 : (term > $o)] : ((((X415 @ one)) = $true) | ($true != ((X414 @ X417))) | ? [X418 : term] : (((X415 @ X418)) != ((X414 @ (sub @ X418 @ (push @ X417 @ X416)))))) | ? [X412 : term,X413 : subst] : (((sub @ one @ (push @ X412 @ X413))) != X412) | (pushprop_lem1v2_lthm != $true)) & ! [X106 : term] : ? [X107 : d_term] : (((d2term @ X107)) = X106) & ! [X318 : d_term] : (d_one = X318) & (hoasinduction_lem3_gthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (axapp = $true) & ((hoasinduction_lthm_3 = $true) | ($true != sP16)) & (($true = sP16) | (hoasinduction_lthm_3 != $true)) & (hoasinduction_lem1v2_gthm = $true) & ((sP10 = $true) | (hoasinduction != $true)) & ((hoasinduction = $true) | (sP10 != $true)) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (! [X195 : term,X196 : term] : ((((sub @ X196 @ sh)) != ((sub @ X195 @ sh))) | (X195 = X196)) | (shinj != $true)) & ((shinj = $true) | ? [X195 : term,X196 : term] : ((((sub @ X196 @ sh)) = ((sub @ X195 @ sh))) & (X195 != X196))) & ((sP7 = $true) | (hoasinduction_lem3v2_lthm != $true)) & ((hoasinduction_lem3v2_lthm = $true) | (sP7 != $true)) & (hoasinduction_lem2_gthm = $true) & (hoasinduction_lem2v2_gthm = $true) & ((induction2lem = $true) | ($true != sP5)) & (($true = sP5) | (induction2lem != $true)) & (pushprop_lem2v2_lthm = $true) & (? [X485 : subst,X486 : term] : (((sub @ one @ (push @ X486 @ X485))) != X486) | ! [X489 : term,X493 : (term > $o),X490 : term,X491 : (term > $o),X492 : subst] : ((((X493 @ X490)) != $true) | (((X491 @ (sub @ X490 @ (push @ X489 @ X492)))) = $true)) | ? [X487 : subst,X488 : term] : (((comp @ sh @ (push @ X488 @ X487))) != X487) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (! [X485 : subst,X486 : term] : (((sub @ one @ (push @ X486 @ X485))) = X486) & ? [X489 : term,X493 : (term > $o),X490 : term,X491 : (term > $o),X492 : subst] : ((((X493 @ X490)) = $true) & (((X491 @ (sub @ X490 @ (push @ X489 @ X492)))) != $true)) & ! [X487 : subst,X488 : term] : (((comp @ sh @ (push @ X488 @ X487))) = X487))) & ! [X41 : d_subst,X40 : d_subst] : ((((d2subst @ X40)) != ((d2subst @ X41))) | (X40 = X41)) & ((axidl = $true) | ? [X47 : subst] : (((comp @ id @ X47)) != X47)) & (! [X47 : subst] : (((comp @ id @ X47)) = X47) | (axidl != $true)) & (($true = sP3) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | ($true != sP3)) & (induction2_gthm = $true) & (ulamvarsh = $true) & ((hoasinduction_lem3v2a_lthm = $true) | (sP2 != $true)) & ((sP2 = $true) | (hoasinduction_lem3v2a_lthm != $true)) & (pushprop = $true) & ((axvarcons = $true) | ? [X234 : subst,X235 : term] : (((sub @ one @ (push @ X235 @ X234))) != X235)) & (! [X234 : subst,X235 : term] : (((sub @ one @ (push @ X235 @ X234))) = X235) | (axvarcons != $true)) & (ulamvarind = $true)), inference(flattening,[],[f272])). thf(f274,plain,( ((apinj1 = $true) | ? [X0 : term,X1 : term,X2 : term,X3 : term] : ((((ap @ X2 @ X3)) = ((ap @ X1 @ X0))) & (X1 != X2))) & (! [X4 : term,X5 : term,X6 : term,X7 : term] : ((((ap @ X5 @ X4)) != ((ap @ X6 @ X7))) | (X5 = X6)) | (apinj1 != $true)) & ! [X8 : subst] : ? [X9 : d_subst] : (((d2subst @ X9)) = X8) & (hoasapnotvar_lthm = $true) & ((induction2_lthm = $true) | ($true != sP60)) & (($true = sP60) | (induction2_lthm != $true)) & (pushprop_gthm = $true) & ((hoasinduction_no_psi_cond_lthm = $true) | (sP56 != $true)) & ((sP56 = $true) | (hoasinduction_no_psi_cond_lthm != $true)) & (induction2lem_gthm = $true) & ((sP51 = $true) | (hoasinduction_lem3_lthm != $true)) & ((hoasinduction_lem3_lthm = $true) | (sP51 != $true)) & ((hoasinduction_no_psi_cond = $true) | ? [X10 : (subst > term > subst > $o),X11 : term] : (! [X12 : term,X13 : term] : ((((X10 @ id @ X13 @ id)) != $true) | (((X10 @ id @ (ap @ (sub @ X12 @ id) @ X13) @ id)) = $true) | (((X10 @ id @ X12 @ id)) != $true)) & (((sP48 @ X10)) != $true) & (((X10 @ id @ X11 @ id)) != $true))) & (! [X14 : (subst > term > subst > $o),X15 : term] : (? [X16 : term,X17 : term] : ((((X14 @ id @ X17 @ id)) = $true) & (((X14 @ id @ (ap @ (sub @ X16 @ id) @ X17) @ id)) != $true) & (((X14 @ id @ X16 @ id)) = $true)) | (((sP48 @ X14)) = $true) | (((X14 @ id @ X15 @ id)) = $true)) | (hoasinduction_no_psi_cond != $true)) & (pushprop_lthm_orig = $true) & (axassoc = $true) & (pushprop_lthm = $true) & (hoasinduction_lem1_lthm = $true) & (ulamvar1 != $true) & (substmonoid_gthm = $true) & (hoasapnotvar_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & ! [X18 : d_subst] : (d_id = X18) & (lamnotap != $true) & (((d2subst @ d_id)) = id) & (($true = sP47) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | ($true != sP47)) & (hoasapnotvar = $true) & (axvarshift = $true) & ((hoasinduction_lem0_lthm = $true) | ? [X19 : (subst > term > subst > $o)] : ! [X20 : (term > $o)] : ? [X21 : term] : (((X20 @ X21)) != ((X19 @ id @ X21 @ id)))) & (! [X22 : (subst > term > subst > $o)] : ? [X23 : (term > $o)] : ! [X24 : term] : (((X22 @ id @ X24 @ id)) = ((X23 @ X24))) | (hoasinduction_lem0_lthm != $true)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((sP43 = $true) | (induction2 != $true)) & ((induction2 = $true) | (sP43 != $true)) & (hoasinduction_lem1_gthm = $true) & ((axidr = $true) | ? [X25 : subst] : (((comp @ X25 @ id)) != X25)) & (! [X26 : subst] : (((comp @ X26 @ id)) = X26) | (axidr != $true)) & ((sP41 = $true) | (hoasinduction_lem3 != $true)) & ((hoasinduction_lem3 = $true) | (sP41 != $true)) & (($true = sP39) | (hoasinduction_lem3v2 != $true)) & ((hoasinduction_lem3v2 = $true) | ($true != sP39)) & ((! [X27 : subst] : (((comp @ X27 @ id)) = X27) & ! [X28 : subst] : (((comp @ id @ X28)) = X28)) | (substmonoid != $true)) & ((substmonoid = $true) | ? [X29 : subst] : (((comp @ X29 @ id)) != X29) | ? [X30 : subst] : (((comp @ id @ X30)) != X30)) & (($true = sP37) | (hoasinduction_lem2v2 != $true)) & ((hoasinduction_lem2v2 = $true) | ($true != sP37)) & (hoaslamnotvar = $true) & ((sP35 = $true) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | (sP35 != $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)))))))))))) & (hoaslamnotap_lthm = $true) & (! [X31 : subst,X32 : (term > $o),X33 : term] : ? [X34 : (term > $o)] : ! [X35 : term] : (((X32 @ (sub @ X35 @ (push @ X33 @ X31)))) = ((X34 @ X35))) | (pushprop_lem0 != $true)) & ((pushprop_lem0 = $true) | ? [X36 : subst,X37 : (term > $o),X38 : term] : ! [X39 : (term > $o)] : ? [X40 : term] : (((X37 @ (sub @ X40 @ (push @ X38 @ X36)))) != ((X39 @ X40)))) & ((pushprop_lem1 = $true) | ? [X41 : term,X42 : subst,X43 : (term > $o),X44 : term,X45 : (term > $o)] : ((((X43 @ (sub @ X41 @ (push @ X44 @ X42)))) != $true) & (((X45 @ X41)) = $true))) & (! [X46 : term,X47 : subst,X48 : (term > $o),X49 : term,X50 : (term > $o)] : ((((X48 @ (sub @ X46 @ (push @ X49 @ X47)))) = $true) | (((X50 @ X46)) != $true)) | (pushprop_lem1 != $true)) & ((hoasinduction_lem3b = $true) | ? [X51 : term] : ! [X52 : (subst > term > term)] : (((X52 @ sh @ one)) != ((sub @ X51 @ (push @ one @ sh))))) & (! [X53 : term] : ? [X54 : (subst > term > term)] : (((X54 @ sh @ one)) = ((sub @ X53 @ (push @ one @ sh)))) | (hoasinduction_lem3b != $true)) & (hoasinduction_lem3a_gthm = $true) & (lamnotvar = $true) & (((d2subst @ d_id)) = sh) & ((axvarid = $true) | ? [X55 : term] : (((sub @ X55 @ id)) != X55)) & (! [X56 : term] : (((sub @ X56 @ id)) = X56) | (axvarid != $true)) & ((pushprop_lem0_lthm = $true) | ? [X57 : subst,X58 : (term > $o),X59 : term] : ! [X60 : (term > $o)] : ? [X61 : term] : (((X60 @ X61)) != ((X58 @ (sub @ X61 @ (push @ X59 @ X57)))))) & (! [X62 : subst,X63 : (term > $o),X64 : term] : ? [X65 : (term > $o)] : ! [X66 : term] : (((X63 @ (sub @ X66 @ (push @ X64 @ X62)))) = ((X65 @ X66))) | (pushprop_lem0_lthm != $true)) & ((hoasinduction_lem3aa_lthm = $true) | (sP33 != $true)) & ((sP33 = $true) | (hoasinduction_lem3aa_lthm != $true)) & (hoaslamnotvar_lthm = $true) & (pushprop_lem1v2_gthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & ((hoasinduction_lem3aaa = $true) | ($true != sP32)) & (($true = sP32) | (hoasinduction_lem3aaa != $true)) & ((sP31 = $true) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | (sP31 != $true)) & (axabs = $true) & ((hoasinduction_lem3v2_f = $true) | ? [X67 : term] : ! [X68 : (subst > term > term)] : ? [X69 : subst,X70 : term] : (((X68 @ X69 @ X70)) != ((sub @ X67 @ (push @ X70 @ X69))))) & (! [X71 : term] : ? [X72 : (subst > term > term)] : ! [X73 : subst,X74 : term] : (((X72 @ X73 @ X74)) = ((sub @ X71 @ (push @ X74 @ X73)))) | (hoasinduction_lem3v2_f != $true)) & (hoasinduction_lem1 = $true) & ! [X75 : d_term,X76 : d_term] : ((X75 = X76) | (((d2term @ X75)) != ((d2term @ X76)))) & (! [X77 : (subst > term > subst > $o)] : ? [X78 : (term > $o)] : ! [X79 : term] : (((X78 @ X79)) = ((X77 @ id @ X79 @ id))) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem0 = $true) | ? [X80 : (subst > term > subst > $o)] : ! [X81 : (term > $o)] : ? [X82 : term] : (((X81 @ X82)) != ((X80 @ id @ X82 @ id)))) & ((hoasinduction_lthm = $true) | ($true != sP30)) & (($true = sP30) | (hoasinduction_lthm != $true)) & (? [X83 : term] : (((sub @ X83 @ id)) != X83) | ! [X84 : term] : (((sub @ X84 @ id)) = X84) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | (! [X85 : term] : (((sub @ X85 @ id)) = X85) & ? [X86 : term] : (((sub @ X86 @ id)) != X86))) & ((hoaslaminj = $true) | ? [X87 : term,X88 : (subst > term > term),X89 : (subst > term > term),X90 : subst] : (! [X91 : term,X92 : subst,X93 : subst] : (((X88 @ (comp @ X92 @ X93) @ (sub @ X91 @ X93))) = ((sub @ (X88 @ X92 @ X91) @ X93))) & ! [X94 : subst,X95 : subst,X96 : term] : (((X89 @ (comp @ X94 @ X95) @ (sub @ X96 @ X95))) = ((sub @ (X89 @ X94 @ X96) @ X95))) & (((lam @ (X88 @ sh @ one))) = ((lam @ (X89 @ sh @ one)))) & (((X88 @ X90 @ X87)) != ((X89 @ X90 @ X87))))) & (! [X97 : term,X98 : (subst > term > term),X99 : (subst > term > term),X100 : subst] : (? [X101 : term,X102 : subst,X103 : subst] : (((X98 @ (comp @ X102 @ X103) @ (sub @ X101 @ X103))) != ((sub @ (X98 @ X102 @ X101) @ X103))) | ? [X104 : subst,X105 : subst,X106 : term] : (((sub @ (X99 @ X104 @ X106) @ X105)) != ((X99 @ (comp @ X104 @ X105) @ (sub @ X106 @ X105)))) | (((lam @ (X99 @ sh @ one))) != ((lam @ (X98 @ sh @ one)))) | (((X99 @ X100 @ X97)) = ((X98 @ X100 @ X97)))) | (hoaslaminj != $true)) & (((d2term @ d_one)) = one) & (hoasinduction_lem1v2 = $true) & (pushprop_lem3v2_lthm = $true) & (((var @ (d2term @ d_one))) != $true) & (hoaslamnotap_gthm = $true) & ((axscons = $true) | ? [X107 : subst] : (((push @ (sub @ one @ X107) @ (comp @ sh @ X107))) != X107)) & (! [X108 : subst] : (((push @ (sub @ one @ X108) @ (comp @ sh @ X108))) = X108) | (axscons != $true)) & (hoaslamnotvar_gthm = $true) & (hoaslaminj_gthm = $true) & (hoasinduction_lem3v2_gthm = $true) & (! [X109 : term,X110 : subst] : (((comp @ sh @ (push @ X109 @ X110))) = X110) | (axshiftcons != $true)) & ((axshiftcons = $true) | ? [X111 : term,X112 : subst] : (((comp @ sh @ (push @ X111 @ X112))) != X112)) & (hoasinduction_lem3b_gthm = $true) & ((laminj = $true) | ? [X113 : term,X114 : term] : ((((lam @ X113)) = ((lam @ X114))) & (X113 != X114))) & (! [X115 : term,X116 : term] : ((((lam @ X115)) != ((lam @ X116))) | (X115 = X116)) | (laminj != $true)) & (! [X117 : term,X118 : term,X119 : (subst > term > term)] : ((((lam @ (X119 @ sh @ one))) != ((ap @ (sub @ X117 @ id) @ X118))) | ? [X120 : subst,X121 : term,X122 : subst] : (((X119 @ (comp @ X120 @ X122) @ (sub @ X121 @ X122))) != ((sub @ (X119 @ X120 @ X121) @ X122)))) | (hoaslamnotap != $true)) & ((hoaslamnotap = $true) | ? [X123 : term,X124 : term,X125 : (subst > term > term)] : ((((lam @ (X125 @ sh @ one))) = ((ap @ (sub @ X123 @ id) @ X124))) & ! [X126 : subst,X127 : term,X128 : subst] : (((sub @ (X125 @ X126 @ X127) @ X128)) = ((X125 @ (comp @ X126 @ X128) @ (sub @ X127 @ X128)))))) & (pushprop_lem2v2 = $true) & (? [X129 : subst] : (((comp @ X129 @ id)) != X129) | ? [X130 : subst] : (((comp @ id @ X130)) != X130) | (! [X131 : subst] : (((comp @ X131 @ id)) = X131) & ! [X132 : subst] : (((comp @ id @ X132)) = X132)) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | (! [X133 : subst] : (((comp @ X133 @ id)) = X133) & ! [X134 : subst] : (((comp @ id @ X134)) = X134) & (? [X135 : subst] : (((comp @ X135 @ id)) != X135) | ? [X136 : subst] : (((comp @ id @ X136)) != X136)))) & (apnotvar = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((hoasapinj1 = $true) | ? [X137 : term,X138 : term,X139 : term,X140 : term] : ((X137 != X139) & (((ap @ (sub @ X139 @ id) @ X138)) = ((ap @ (sub @ X137 @ id) @ X140))))) & (! [X141 : term,X142 : term,X143 : term,X144 : term] : ((X141 = X143) | (((ap @ (sub @ X141 @ id) @ X144)) != ((ap @ (sub @ X143 @ id) @ X142)))) | (hoasapinj1 != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X145 : term] : ! [X146 : (subst > term > term)] : ? [X147 : subst,X148 : term] : (((sub @ X145 @ (push @ X148 @ X147))) != ((X146 @ X147 @ X148)))) & (! [X149 : term] : ? [X150 : (subst > term > term)] : ! [X151 : subst,X152 : term] : (((sub @ X149 @ (push @ X152 @ X151))) = ((X150 @ X151 @ X152))) | (hoasinduction_lem3v2_f_lthm != $true)) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (hoasapinj2_gthm = $true) & ((hoasapinj2_lthm = $true) | (? [X153 : term,X154 : term,X155 : term,X156 : term] : ((X155 != X156) & (((ap @ (sub @ X153 @ id) @ X156)) = ((ap @ (sub @ X154 @ id) @ X155)))) & ! [X157 : term,X158 : term,X159 : term,X160 : term] : ((((ap @ X160 @ X159)) != ((ap @ X158 @ X157))) | (X157 = X159)))) & (! [X161 : term,X162 : term,X163 : term,X164 : term] : ((X163 = X164) | (((ap @ (sub @ X162 @ id) @ X163)) != ((ap @ (sub @ X161 @ id) @ X164)))) | ? [X165 : term,X166 : term,X167 : term,X168 : term] : ((((ap @ X168 @ X167)) = ((ap @ X166 @ X165))) & (X165 != X167)) | (hoasapinj2_lthm != $true)) & (hoasinduction_lem2_lthm = $true) & (! [X169 : term,X170 : term,X171 : term,X172 : term] : ((X169 = X171) | (((ap @ (sub @ X172 @ id) @ X171)) != ((ap @ (sub @ X170 @ id) @ X169)))) | (hoasapinj2 != $true)) & ((hoasapinj2 = $true) | ? [X173 : term,X174 : term,X175 : term,X176 : term] : ((X173 != X175) & (((ap @ (sub @ X174 @ id) @ X173)) = ((ap @ (sub @ X176 @ id) @ X175))))) & (! [X177 : (term > $o),X178 : subst,X179 : term,X180 : (term > $o)] : (($true != ((X177 @ X179))) | ? [X181 : term] : (((X177 @ (sub @ X181 @ (push @ X179 @ X178)))) != ((X180 @ X181))) | (((X180 @ one)) = $true)) | (pushprop_lem1v2 != $true)) & ((pushprop_lem1v2 = $true) | ? [X182 : (term > $o),X183 : subst,X184 : term,X185 : (term > $o)] : (($true = ((X182 @ X184))) & ! [X186 : term] : (((X185 @ X186)) = ((X182 @ (sub @ X186 @ (push @ X184 @ X183))))) & ($true != ((X185 @ one))))) & (axclos = $true) & ((hoasapinj1_lthm = $true) | (! [X187 : term] : (((sub @ X187 @ id)) = X187) & ! [X188 : term,X189 : term,X190 : term,X191 : term] : ((((ap @ X191 @ X190)) != ((ap @ X189 @ X188))) | (X189 = X191)) & ? [X192 : term,X193 : term,X194 : term,X195 : term] : ((((ap @ (sub @ X194 @ id) @ X192)) = ((ap @ (sub @ X193 @ id) @ X195))) & (X193 != X194)))) & (? [X196 : term] : (((sub @ X196 @ id)) != X196) | ? [X197 : term,X198 : term,X199 : term,X200 : term] : ((((ap @ X200 @ X199)) = ((ap @ X198 @ X197))) & (X198 != X200)) | ! [X201 : term,X202 : term,X203 : term,X204 : term] : ((((ap @ (sub @ X203 @ id) @ X201)) != ((ap @ (sub @ X202 @ id) @ X204))) | (X202 = X203)) | (hoasapinj1_lthm != $true)) & (pushprop_lem0_gthm = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (pushprop_lem3v2 = $true) & (! [X205 : term] : (((sub @ X205 @ id)) = X205) | (termmset != $true)) & ((termmset = $true) | ? [X206 : term] : (((sub @ X206 @ id)) != X206)) & (! [X207 : term,X208 : (term > $o)] : (? [X209 : term] : (($true = ((X208 @ X209))) & (((X208 @ (lam @ X209))) != $true)) | ($true = ((sP21 @ X208))) | ? [X210 : term] : ((((var @ X210)) = $true) & ($true != ((X208 @ X210)))) | (((X208 @ X207)) = $true)) | (induction != $true)) & ((induction = $true) | ? [X211 : term,X212 : (term > $o)] : (! [X213 : term] : (($true != ((X212 @ X213))) | ($true = ((X212 @ (lam @ X213))))) & ($true != ((sP21 @ X212))) & ! [X214 : term] : (($true != ((var @ X214))) | (((X212 @ X214)) = $true)) & (((X212 @ X211)) != $true))) & (($true = sP20) | (hoasinduction_lem3aa != $true)) & ((hoasinduction_lem3aa = $true) | ($true != sP20)) & (! [X215 : term] : ? [X216 : (subst > term > term)] : (((sub @ X215 @ (push @ one @ sh))) = ((X216 @ sh @ one))) | (hoasinduction_lem3b_lthm != $true)) & ((hoasinduction_lem3b_lthm = $true) | ? [X217 : term] : ! [X218 : (subst > term > term)] : (((sub @ X217 @ (push @ one @ sh))) != ((X218 @ sh @ one)))) & (pushprop_lem2v2_gthm = $true) & (hoasapinj1_gthm = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (($true = sP19) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | ($true != sP19)) & (pushprop_lem1_gthm = $true) & (axmap = $true) & ((hoaslaminj_lthm = $true) | ($true != sP17)) & (($true = sP17) | (hoaslaminj_lthm != $true)) & (hoasinduction_gthm = $true) & ((apinj2 = $true) | ? [X219 : term,X220 : term,X221 : term,X222 : term] : ((X219 != X222) & (((ap @ X220 @ X222)) = ((ap @ X221 @ X219))))) & (! [X223 : term,X224 : term,X225 : term,X226 : term] : ((X223 = X226) | (((ap @ X224 @ X226)) != ((ap @ X225 @ X223)))) | (apinj2 != $true)) & (termmset_gthm = $true) & ((pushprop_lem1v2_lthm = $true) | (? [X227 : term,X228 : (term > $o),X229 : subst,X230 : (term > $o)] : ((((X230 @ one)) != $true) & (((X228 @ X227)) = $true) & ! [X231 : term] : (((X230 @ X231)) = ((X228 @ (sub @ X231 @ (push @ X227 @ X229)))))) & ! [X232 : term,X233 : subst] : (((sub @ one @ (push @ X232 @ X233))) = X232))) & (! [X234 : term,X235 : (term > $o),X236 : subst,X237 : (term > $o)] : ((((X237 @ one)) = $true) | (((X235 @ X234)) != $true) | ? [X238 : term] : (((X235 @ (sub @ X238 @ (push @ X234 @ X236)))) != ((X237 @ X238)))) | ? [X239 : term,X240 : subst] : (((sub @ one @ (push @ X239 @ X240))) != X239) | (pushprop_lem1v2_lthm != $true)) & ! [X241 : term] : ? [X242 : d_term] : (((d2term @ X242)) = X241) & ! [X243 : d_term] : (d_one = X243) & (hoasinduction_lem3_gthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (axapp = $true) & ((hoasinduction_lthm_3 = $true) | ($true != sP16)) & (($true = sP16) | (hoasinduction_lthm_3 != $true)) & (hoasinduction_lem1v2_gthm = $true) & ((sP10 = $true) | (hoasinduction != $true)) & ((hoasinduction = $true) | (sP10 != $true)) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (! [X244 : term,X245 : term] : ((((sub @ X244 @ sh)) != ((sub @ X245 @ sh))) | (X244 = X245)) | (shinj != $true)) & ((shinj = $true) | ? [X246 : term,X247 : term] : ((((sub @ X247 @ sh)) = ((sub @ X246 @ sh))) & (X246 != X247))) & ((sP7 = $true) | (hoasinduction_lem3v2_lthm != $true)) & ((hoasinduction_lem3v2_lthm = $true) | (sP7 != $true)) & (hoasinduction_lem2_gthm = $true) & (hoasinduction_lem2v2_gthm = $true) & ((induction2lem = $true) | ($true != sP5)) & (($true = sP5) | (induction2lem != $true)) & (pushprop_lem2v2_lthm = $true) & (? [X248 : subst,X249 : term] : (((sub @ one @ (push @ X249 @ X248))) != X249) | ! [X250 : term,X251 : (term > $o),X252 : term,X253 : (term > $o),X254 : subst] : ((((X251 @ X252)) != $true) | ($true = ((X253 @ (sub @ X252 @ (push @ X250 @ X254)))))) | ? [X255 : subst,X256 : term] : (((comp @ sh @ (push @ X256 @ X255))) != X255) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (! [X257 : subst,X258 : term] : (((sub @ one @ (push @ X258 @ X257))) = X258) & ? [X259 : term,X260 : (term > $o),X261 : term,X262 : (term > $o),X263 : subst] : (($true = ((X260 @ X261))) & ($true != ((X262 @ (sub @ X261 @ (push @ X259 @ X263)))))) & ! [X264 : subst,X265 : term] : (((comp @ sh @ (push @ X265 @ X264))) = X264))) & ! [X266 : d_subst,X267 : d_subst] : ((((d2subst @ X266)) != ((d2subst @ X267))) | (X266 = X267)) & ((axidl = $true) | ? [X268 : subst] : (((comp @ id @ X268)) != X268)) & (! [X269 : subst] : (((comp @ id @ X269)) = X269) | (axidl != $true)) & (($true = sP3) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | ($true != sP3)) & (induction2_gthm = $true) & (ulamvarsh = $true) & ((hoasinduction_lem3v2a_lthm = $true) | (sP2 != $true)) & ((sP2 = $true) | (hoasinduction_lem3v2a_lthm != $true)) & (pushprop = $true) & ((axvarcons = $true) | ? [X270 : subst,X271 : term] : (((sub @ one @ (push @ X271 @ X270))) != X271)) & (! [X272 : subst,X273 : term] : (((sub @ one @ (push @ X273 @ X272))) = X273) | (axvarcons != $true)) & (ulamvarind = $true)), inference(rectify,[],[f273])). thf(f275,plain,( ((apinj1 = $true) | ((((ap @ sK476 @ sK475)) = ((ap @ sK477 @ sK478))) & (sK477 != sK476))) & (! [X4 : term,X5 : term,X6 : term,X7 : term] : ((((ap @ X5 @ X4)) != ((ap @ X6 @ X7))) | (X5 = X6)) | (apinj1 != $true)) & ! [X8 : subst] : (((d2subst @ (sK479 @ X8))) = X8) & (hoasapnotvar_lthm = $true) & ((induction2_lthm = $true) | ($true != sP60)) & (($true = sP60) | (induction2_lthm != $true)) & (pushprop_gthm = $true) & ((hoasinduction_no_psi_cond_lthm = $true) | (sP56 != $true)) & ((sP56 = $true) | (hoasinduction_no_psi_cond_lthm != $true)) & (induction2lem_gthm = $true) & ((sP51 = $true) | (hoasinduction_lem3_lthm != $true)) & ((hoasinduction_lem3_lthm = $true) | (sP51 != $true)) & ((hoasinduction_no_psi_cond = $true) | (! [X12 : term,X13 : term] : ((((sK480 @ id @ X13 @ id)) != $true) | ($true = ((sK480 @ id @ (ap @ (sub @ X12 @ id) @ X13) @ id))) | (((sK480 @ id @ X12 @ id)) != $true)) & (((sP48 @ sK480)) != $true) & (((sK480 @ id @ sK481 @ id)) != $true))) & (! [X14 : (subst > term > subst > $o),X15 : term] : ((($true = ((X14 @ id @ (sK483 @ X14) @ id))) & (((X14 @ id @ (ap @ (sub @ (sK482 @ X14) @ id) @ (sK483 @ X14)) @ id)) != $true) & ($true = ((X14 @ id @ (sK482 @ X14) @ id)))) | (((sP48 @ X14)) = $true) | (((X14 @ id @ X15 @ id)) = $true)) | (hoasinduction_no_psi_cond != $true)) & (pushprop_lthm_orig = $true) & (axassoc = $true) & (pushprop_lthm = $true) & (hoasinduction_lem1_lthm = $true) & (ulamvar1 != $true) & (substmonoid_gthm = $true) & (hoasapnotvar_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & ! [X18 : d_subst] : (d_id = X18) & (lamnotap != $true) & (((d2subst @ d_id)) = id) & (($true = sP47) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | ($true != sP47)) & (hoasapnotvar = $true) & (axvarshift = $true) & ((hoasinduction_lem0_lthm = $true) | ! [X20 : (term > $o)] : (((sK484 @ id @ (sK485 @ X20) @ id)) != ((X20 @ (sK485 @ X20))))) & (! [X22 : (subst > term > subst > $o)] : ! [X24 : term] : (((sK486 @ X22 @ X24)) = ((X22 @ id @ X24 @ id))) | (hoasinduction_lem0_lthm != $true)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((sP43 = $true) | (induction2 != $true)) & ((induction2 = $true) | (sP43 != $true)) & (hoasinduction_lem1_gthm = $true) & ((axidr = $true) | (sK487 != ((comp @ sK487 @ id)))) & (! [X26 : subst] : (((comp @ X26 @ id)) = X26) | (axidr != $true)) & ((sP41 = $true) | (hoasinduction_lem3 != $true)) & ((hoasinduction_lem3 = $true) | (sP41 != $true)) & (($true = sP39) | (hoasinduction_lem3v2 != $true)) & ((hoasinduction_lem3v2 = $true) | ($true != sP39)) & ((! [X27 : subst] : (((comp @ X27 @ id)) = X27) & ! [X28 : subst] : (((comp @ id @ X28)) = X28)) | (substmonoid != $true)) & ((substmonoid = $true) | (((comp @ sK488 @ id)) != sK488) | (((comp @ id @ sK489)) != sK489)) & (($true = sP37) | (hoasinduction_lem2v2 != $true)) & ((hoasinduction_lem2v2 = $true) | ($true != sP37)) & (hoaslamnotvar = $true) & ((sP35 = $true) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | (sP35 != $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)))))))))))) & (hoaslamnotap_lthm = $true) & (! [X31 : subst,X32 : (term > $o),X33 : term] : ! [X35 : term] : (((sK490 @ X33 @ X32 @ X31 @ X35)) = ((X32 @ (sub @ X35 @ (push @ X33 @ X31))))) | (pushprop_lem0 != $true)) & ((pushprop_lem0 = $true) | ! [X39 : (term > $o)] : (((sK492 @ (sub @ (sK494 @ X39) @ (push @ sK493 @ sK491)))) != ((X39 @ (sK494 @ X39))))) & ((pushprop_lem1 = $true) | ((((sK497 @ (sub @ sK495 @ (push @ sK498 @ sK496)))) != $true) & (((sK499 @ sK495)) = $true))) & (! [X46 : term,X47 : subst,X48 : (term > $o),X49 : term,X50 : (term > $o)] : ((((X48 @ (sub @ X46 @ (push @ X49 @ X47)))) = $true) | (((X50 @ X46)) != $true)) | (pushprop_lem1 != $true)) & ((hoasinduction_lem3b = $true) | ! [X52 : (subst > term > term)] : (((X52 @ sh @ one)) != ((sub @ sK500 @ (push @ one @ sh))))) & (! [X53 : term] : (((sK501 @ X53 @ sh @ one)) = ((sub @ X53 @ (push @ one @ sh)))) | (hoasinduction_lem3b != $true)) & (hoasinduction_lem3a_gthm = $true) & (lamnotvar = $true) & (((d2subst @ d_id)) = sh) & ((axvarid = $true) | (sK502 != ((sub @ sK502 @ id)))) & (! [X56 : term] : (((sub @ X56 @ id)) = X56) | (axvarid != $true)) & ((pushprop_lem0_lthm = $true) | ! [X60 : (term > $o)] : (((X60 @ (sK506 @ X60))) != ((sK504 @ (sub @ (sK506 @ X60) @ (push @ sK505 @ sK503)))))) & (! [X62 : subst,X63 : (term > $o),X64 : term] : ! [X66 : term] : (((sK507 @ X64 @ X63 @ X62 @ X66)) = ((X63 @ (sub @ X66 @ (push @ X64 @ X62))))) | (pushprop_lem0_lthm != $true)) & ((hoasinduction_lem3aa_lthm = $true) | (sP33 != $true)) & ((sP33 = $true) | (hoasinduction_lem3aa_lthm != $true)) & (hoaslamnotvar_lthm = $true) & (pushprop_lem1v2_gthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & ((hoasinduction_lem3aaa = $true) | ($true != sP32)) & (($true = sP32) | (hoasinduction_lem3aaa != $true)) & ((sP31 = $true) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | (sP31 != $true)) & (axabs = $true) & ((hoasinduction_lem3v2_f = $true) | ! [X68 : (subst > term > term)] : (((sub @ sK508 @ (push @ (sK510 @ X68) @ (sK509 @ X68)))) != ((X68 @ (sK509 @ X68) @ (sK510 @ X68))))) & (! [X71 : term] : ! [X73 : subst,X74 : term] : (((sK511 @ X71 @ X73 @ X74)) = ((sub @ X71 @ (push @ X74 @ X73)))) | (hoasinduction_lem3v2_f != $true)) & (hoasinduction_lem1 = $true) & ! [X75 : d_term,X76 : d_term] : ((X75 = X76) | (((d2term @ X75)) != ((d2term @ X76)))) & (! [X77 : (subst > term > subst > $o)] : ! [X79 : term] : (((sK512 @ X77 @ X79)) = ((X77 @ id @ X79 @ id))) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem0 = $true) | ! [X81 : (term > $o)] : (((sK513 @ id @ (sK514 @ X81) @ id)) != ((X81 @ (sK514 @ X81))))) & ((hoasinduction_lthm = $true) | ($true != sP30)) & (($true = sP30) | (hoasinduction_lthm != $true)) & ((sK515 != ((sub @ sK515 @ id))) | ! [X84 : term] : (((sub @ X84 @ id)) = X84) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | (! [X85 : term] : (((sub @ X85 @ id)) = X85) & (sK516 != ((sub @ sK516 @ id))))) & ((hoaslaminj = $true) | (! [X91 : term,X92 : subst,X93 : subst] : (((sub @ (sK518 @ X92 @ X91) @ X93)) = ((sK518 @ (comp @ X92 @ X93) @ (sub @ X91 @ X93)))) & ! [X94 : subst,X95 : subst,X96 : term] : (((sK519 @ (comp @ X94 @ X95) @ (sub @ X96 @ X95))) = ((sub @ (sK519 @ X94 @ X96) @ X95))) & (((lam @ (sK518 @ sh @ one))) = ((lam @ (sK519 @ sh @ one)))) & (((sK519 @ sK520 @ sK517)) != ((sK518 @ sK520 @ sK517))))) & (! [X97 : term,X98 : (subst > term > term),X99 : (subst > term > term),X100 : subst] : ((((sub @ (X98 @ (sK522 @ X98) @ (sK521 @ X98)) @ (sK523 @ X98))) != ((X98 @ (comp @ (sK522 @ X98) @ (sK523 @ X98)) @ (sub @ (sK521 @ X98) @ (sK523 @ X98))))) | (((X99 @ (comp @ (sK524 @ X99) @ (sK525 @ X99)) @ (sub @ (sK526 @ X99) @ (sK525 @ X99)))) != ((sub @ (X99 @ (sK524 @ X99) @ (sK526 @ X99)) @ (sK525 @ X99)))) | (((lam @ (X99 @ sh @ one))) != ((lam @ (X98 @ sh @ one)))) | (((X99 @ X100 @ X97)) = ((X98 @ X100 @ X97)))) | (hoaslaminj != $true)) & (((d2term @ d_one)) = one) & (hoasinduction_lem1v2 = $true) & (pushprop_lem3v2_lthm = $true) & (((var @ (d2term @ d_one))) != $true) & (hoaslamnotap_gthm = $true) & ((axscons = $true) | (((push @ (sub @ one @ sK527) @ (comp @ sh @ sK527))) != sK527)) & (! [X108 : subst] : (((push @ (sub @ one @ X108) @ (comp @ sh @ X108))) = X108) | (axscons != $true)) & (hoaslamnotvar_gthm = $true) & (hoaslaminj_gthm = $true) & (hoasinduction_lem3v2_gthm = $true) & (! [X109 : term,X110 : subst] : (((comp @ sh @ (push @ X109 @ X110))) = X110) | (axshiftcons != $true)) & ((axshiftcons = $true) | (((comp @ sh @ (push @ sK528 @ sK529))) != sK529)) & (hoasinduction_lem3b_gthm = $true) & ((laminj = $true) | ((((lam @ sK531)) = ((lam @ sK530))) & (sK530 != sK531))) & (! [X115 : term,X116 : term] : ((((lam @ X115)) != ((lam @ X116))) | (X115 = X116)) | (laminj != $true)) & (! [X117 : term,X118 : term,X119 : (subst > term > term)] : ((((lam @ (X119 @ sh @ one))) != ((ap @ (sub @ X117 @ id) @ X118))) | (((X119 @ (comp @ (sK532 @ X119) @ (sK534 @ X119)) @ (sub @ (sK533 @ X119) @ (sK534 @ X119)))) != ((sub @ (X119 @ (sK532 @ X119) @ (sK533 @ X119)) @ (sK534 @ X119))))) | (hoaslamnotap != $true)) & ((hoaslamnotap = $true) | ((((lam @ (sK537 @ sh @ one))) = ((ap @ (sub @ sK535 @ id) @ sK536))) & ! [X126 : subst,X127 : term,X128 : subst] : (((sK537 @ (comp @ X126 @ X128) @ (sub @ X127 @ X128))) = ((sub @ (sK537 @ X126 @ X127) @ X128))))) & (pushprop_lem2v2 = $true) & ((((comp @ sK538 @ id)) != sK538) | (((comp @ id @ sK539)) != sK539) | (! [X131 : subst] : (((comp @ X131 @ id)) = X131) & ! [X132 : subst] : (((comp @ id @ X132)) = X132)) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | (! [X133 : subst] : (((comp @ X133 @ id)) = X133) & ! [X134 : subst] : (((comp @ id @ X134)) = X134) & ((sK540 != ((comp @ sK540 @ id))) | (((comp @ id @ sK541)) != sK541)))) & (apnotvar = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((hoasapinj1 = $true) | ((sK544 != sK542) & (((ap @ (sub @ sK544 @ id) @ sK543)) = ((ap @ (sub @ sK542 @ id) @ sK545))))) & (! [X141 : term,X142 : term,X143 : term,X144 : term] : ((X141 = X143) | (((ap @ (sub @ X141 @ id) @ X144)) != ((ap @ (sub @ X143 @ id) @ X142)))) | (hoasapinj1 != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ! [X146 : (subst > term > term)] : (((sub @ sK546 @ (push @ (sK548 @ X146) @ (sK547 @ X146)))) != ((X146 @ (sK547 @ X146) @ (sK548 @ X146))))) & (! [X149 : term] : ! [X151 : subst,X152 : term] : (((sK549 @ X149 @ X151 @ X152)) = ((sub @ X149 @ (push @ X152 @ X151)))) | (hoasinduction_lem3v2_f_lthm != $true)) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (hoasapinj2_gthm = $true) & ((hoasapinj2_lthm = $true) | (((sK552 != sK553) & (((ap @ (sub @ sK550 @ id) @ sK553)) = ((ap @ (sub @ sK551 @ id) @ sK552)))) & ! [X157 : term,X158 : term,X159 : term,X160 : term] : ((((ap @ X160 @ X159)) != ((ap @ X158 @ X157))) | (X157 = X159)))) & (! [X161 : term,X162 : term,X163 : term,X164 : term] : ((X163 = X164) | (((ap @ (sub @ X162 @ id) @ X163)) != ((ap @ (sub @ X161 @ id) @ X164)))) | ((((ap @ sK557 @ sK556)) = ((ap @ sK555 @ sK554))) & (sK554 != sK556)) | (hoasapinj2_lthm != $true)) & (hoasinduction_lem2_lthm = $true) & (! [X169 : term,X170 : term,X171 : term,X172 : term] : ((X169 = X171) | (((ap @ (sub @ X172 @ id) @ X171)) != ((ap @ (sub @ X170 @ id) @ X169)))) | (hoasapinj2 != $true)) & ((hoasapinj2 = $true) | ((sK560 != sK558) & (((ap @ (sub @ sK559 @ id) @ sK558)) = ((ap @ (sub @ sK561 @ id) @ sK560))))) & (! [X177 : (term > $o),X178 : subst,X179 : term,X180 : (term > $o)] : (($true != ((X177 @ X179))) | (((X177 @ (sub @ (sK562 @ X180 @ X179 @ X178 @ X177) @ (push @ X179 @ X178)))) != ((X180 @ (sK562 @ X180 @ X179 @ X178 @ X177)))) | (((X180 @ one)) = $true)) | (pushprop_lem1v2 != $true)) & ((pushprop_lem1v2 = $true) | ((((sK563 @ sK565)) = $true) & ! [X186 : term] : (((sK566 @ X186)) = ((sK563 @ (sub @ X186 @ (push @ sK565 @ sK564))))) & ($true != ((sK566 @ one))))) & (axclos = $true) & ((hoasapinj1_lthm = $true) | (! [X187 : term] : (((sub @ X187 @ id)) = X187) & ! [X188 : term,X189 : term,X190 : term,X191 : term] : ((((ap @ X191 @ X190)) != ((ap @ X189 @ X188))) | (X189 = X191)) & ((((ap @ (sub @ sK569 @ id) @ sK567)) = ((ap @ (sub @ sK568 @ id) @ sK570))) & (sK568 != sK569)))) & ((sK571 != ((sub @ sK571 @ id))) | ((((ap @ sK575 @ sK574)) = ((ap @ sK573 @ sK572))) & (sK575 != sK573)) | ! [X201 : term,X202 : term,X203 : term,X204 : term] : ((((ap @ (sub @ X203 @ id) @ X201)) != ((ap @ (sub @ X202 @ id) @ X204))) | (X202 = X203)) | (hoasapinj1_lthm != $true)) & (pushprop_lem0_gthm = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (pushprop_lem3v2 = $true) & (! [X205 : term] : (((sub @ X205 @ id)) = X205) | (termmset != $true)) & ((termmset = $true) | (((sub @ sK576 @ id)) != sK576)) & (! [X207 : term,X208 : (term > $o)] : ((($true = ((X208 @ (sK577 @ X208)))) & ($true != ((X208 @ (lam @ (sK577 @ X208)))))) | ($true = ((sP21 @ X208))) | ((((var @ (sK578 @ X208))) = $true) & ($true != ((X208 @ (sK578 @ X208))))) | (((X208 @ X207)) = $true)) | (induction != $true)) & ((induction = $true) | (! [X213 : term] : (($true != ((sK580 @ X213))) | (((sK580 @ (lam @ X213))) = $true)) & ($true != ((sP21 @ sK580))) & ! [X214 : term] : (($true != ((var @ X214))) | ($true = ((sK580 @ X214)))) & (((sK580 @ sK579)) != $true))) & (($true = sP20) | (hoasinduction_lem3aa != $true)) & ((hoasinduction_lem3aa = $true) | ($true != sP20)) & (! [X215 : term] : (((sK581 @ X215 @ sh @ one)) = ((sub @ X215 @ (push @ one @ sh)))) | (hoasinduction_lem3b_lthm != $true)) & ((hoasinduction_lem3b_lthm = $true) | ! [X218 : (subst > term > term)] : (((X218 @ sh @ one)) != ((sub @ sK582 @ (push @ one @ sh))))) & (pushprop_lem2v2_gthm = $true) & (hoasapinj1_gthm = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (($true = sP19) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | ($true != sP19)) & (pushprop_lem1_gthm = $true) & (axmap = $true) & ((hoaslaminj_lthm = $true) | ($true != sP17)) & (($true = sP17) | (hoaslaminj_lthm != $true)) & (hoasinduction_gthm = $true) & ((apinj2 = $true) | ((sK583 != sK586) & (((ap @ sK585 @ sK583)) = ((ap @ sK584 @ sK586))))) & (! [X223 : term,X224 : term,X225 : term,X226 : term] : ((X223 = X226) | (((ap @ X224 @ X226)) != ((ap @ X225 @ X223)))) | (apinj2 != $true)) & (termmset_gthm = $true) & ((pushprop_lem1v2_lthm = $true) | (((((sK590 @ one)) != $true) & ($true = ((sK588 @ sK587))) & ! [X231 : term] : (((sK590 @ X231)) = ((sK588 @ (sub @ X231 @ (push @ sK587 @ sK589)))))) & ! [X232 : term,X233 : subst] : (((sub @ one @ (push @ X232 @ X233))) = X232))) & (! [X234 : term,X235 : (term > $o),X236 : subst,X237 : (term > $o)] : ((((X237 @ one)) = $true) | (((X235 @ X234)) != $true) | (((X235 @ (sub @ (sK591 @ X237 @ X236 @ X235 @ X234) @ (push @ X234 @ X236)))) != ((X237 @ (sK591 @ X237 @ X236 @ X235 @ X234))))) | (sK592 != ((sub @ one @ (push @ sK592 @ sK593)))) | (pushprop_lem1v2_lthm != $true)) & ! [X241 : term] : (((d2term @ (sK594 @ X241))) = X241) & ! [X243 : d_term] : (d_one = X243) & (hoasinduction_lem3_gthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (axapp = $true) & ((hoasinduction_lthm_3 = $true) | ($true != sP16)) & (($true = sP16) | (hoasinduction_lthm_3 != $true)) & (hoasinduction_lem1v2_gthm = $true) & ((sP10 = $true) | (hoasinduction != $true)) & ((hoasinduction = $true) | (sP10 != $true)) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (! [X244 : term,X245 : term] : ((((sub @ X244 @ sh)) != ((sub @ X245 @ sh))) | (X244 = X245)) | (shinj != $true)) & ((shinj = $true) | ((((sub @ sK596 @ sh)) = ((sub @ sK595 @ sh))) & (sK595 != sK596))) & ((sP7 = $true) | (hoasinduction_lem3v2_lthm != $true)) & ((hoasinduction_lem3v2_lthm = $true) | (sP7 != $true)) & (hoasinduction_lem2_gthm = $true) & (hoasinduction_lem2v2_gthm = $true) & ((induction2lem = $true) | ($true != sP5)) & (($true = sP5) | (induction2lem != $true)) & (pushprop_lem2v2_lthm = $true) & ((sK598 != ((sub @ one @ (push @ sK598 @ sK597)))) | ! [X250 : term,X251 : (term > $o),X252 : term,X253 : (term > $o),X254 : subst] : ((((X251 @ X252)) != $true) | ($true = ((X253 @ (sub @ X252 @ (push @ X250 @ X254)))))) | (((comp @ sh @ (push @ sK600 @ sK599))) != sK599) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (! [X257 : subst,X258 : term] : (((sub @ one @ (push @ X258 @ X257))) = X258) & ((((sK602 @ sK603)) = $true) & (((sK604 @ (sub @ sK603 @ (push @ sK601 @ sK605)))) != $true)) & ! [X264 : subst,X265 : term] : (((comp @ sh @ (push @ X265 @ X264))) = X264))) & ! [X266 : d_subst,X267 : d_subst] : ((((d2subst @ X266)) != ((d2subst @ X267))) | (X266 = X267)) & ((axidl = $true) | (sK606 != ((comp @ id @ sK606)))) & (! [X269 : subst] : (((comp @ id @ X269)) = X269) | (axidl != $true)) & (($true = sP3) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | ($true != sP3)) & (induction2_gthm = $true) & (ulamvarsh = $true) & ((hoasinduction_lem3v2a_lthm = $true) | (sP2 != $true)) & ((sP2 = $true) | (hoasinduction_lem3v2a_lthm != $true)) & (pushprop = $true) & ((axvarcons = $true) | (sK608 != ((sub @ one @ (push @ sK608 @ sK607))))) & (! [X272 : subst,X273 : term] : (((sub @ one @ (push @ X273 @ X272))) = X273) | (axvarcons != $true)) & (ulamvarind = $true)), inference(skolemize,[status(esa),new_symbols(skolem,[sK475,sK476,sK477,sK478,vAPP,sK480,sK481,vAPP,sK484,vAPP,sK487,sK488,sK489,vAPP,sK491,sK492,sK493,vAPP,sK495,sK496,sK497,sK498,sK499,sK500,vAPP,sK502,sK503,sK504,sK505,vAPP,sK508,vAPP,sK513,vAPP,sK515,sK516,sK517,sK518,sK519,sK520,vAPP,sK527,sK528,sK529,sK530,sK531,vAPP,sK535,sK536,sK537,sK538,sK539,sK540,sK541,sK542,sK543,sK544,sK545,sK546,vAPP,sK550,sK551,sK552,sK553,sK554,sK555,sK556,sK557,sK558,sK559,sK560,sK561,vAPP,sK563,sK564,sK565,sK566,sK567,sK568,sK569,sK570,sK571,sK572,sK573,sK574,sK575,sK576,vAPP,sK579,sK580,vAPP,sK582,sK583,sK584,sK585,sK586,sK587,sK588,sK589,sK590,vAPP,sK592,sK593,vAPP,sK595,sK596,sK597,sK598,sK599,sK600,sK601,sK602,sK603,sK604,sK605,sK606,sK607,sK608]),skolemize(X0,$thf(sK475)),skolemize(X1,$thf(sK476)),skolemize(X2,$thf(sK477)),skolemize(X3,$thf(sK478)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X10,$thf(sK480)),skolemize(X11,$thf(sK481)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X19,$thf(sK484)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X25,$thf(sK487)),skolemize(X29,$thf(sK488)),skolemize(X30,$thf(sK489)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X36,$thf(sK491)),skolemize(X37,$thf(sK492)),skolemize(X38,$thf(sK493)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X41,$thf(sK495)),skolemize(X42,$thf(sK496)),skolemize(X43,$thf(sK497)),skolemize(X44,$thf(sK498)),skolemize(X45,$thf(sK499)),skolemize(X51,$thf(sK500)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X55,$thf(sK502)),skolemize(X57,$thf(sK503)),skolemize(X58,$thf(sK504)),skolemize(X59,$thf(sK505)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X67,$thf(sK508)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X80,$thf(sK513)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X83,$thf(sK515)),skolemize(X86,$thf(sK516)),skolemize(X87,$thf(sK517)),skolemize(X88,$thf(sK518)),skolemize(X89,$thf(sK519)),skolemize(X90,$thf(sK520)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X107,$thf(sK527)),skolemize(X111,$thf(sK528)),skolemize(X112,$thf(sK529)),skolemize(X113,$thf(sK530)),skolemize(X114,$thf(sK531)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X123,$thf(sK535)),skolemize(X124,$thf(sK536)),skolemize(X125,$thf(sK537)),skolemize(X129,$thf(sK538)),skolemize(X130,$thf(sK539)),skolemize(X135,$thf(sK540)),skolemize(X136,$thf(sK541)),skolemize(X137,$thf(sK542)),skolemize(X138,$thf(sK543)),skolemize(X139,$thf(sK544)),skolemize(X140,$thf(sK545)),skolemize(X145,$thf(sK546)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X153,$thf(sK550)),skolemize(X154,$thf(sK551)),skolemize(X155,$thf(sK552)),skolemize(X156,$thf(sK553)),skolemize(X165,$thf(sK554)),skolemize(X166,$thf(sK555)),skolemize(X167,$thf(sK556)),skolemize(X168,$thf(sK557)),skolemize(X173,$thf(sK558)),skolemize(X174,$thf(sK559)),skolemize(X175,$thf(sK560)),skolemize(X176,$thf(sK561)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X182,$thf(sK563)),skolemize(X183,$thf(sK564)),skolemize(X184,$thf(sK565)),skolemize(X185,$thf(sK566)),skolemize(X192,$thf(sK567)),skolemize(X193,$thf(sK568)),skolemize(X194,$thf(sK569)),skolemize(X195,$thf(sK570)),skolemize(X196,$thf(sK571)),skolemize(X197,$thf(sK572)),skolemize(X198,$thf(sK573)),skolemize(X199,$thf(sK574)),skolemize(X200,$thf(sK575)),skolemize(X206,$thf(sK576)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X211,$thf(sK579)),skolemize(X212,$thf(sK580)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X217,$thf(sK582)),skolemize(X219,$thf(sK583)),skolemize(X220,$thf(sK584)),skolemize(X221,$thf(sK585)),skolemize(X222,$thf(sK586)),skolemize(X227,$thf(sK587)),skolemize(X228,$thf(sK588)),skolemize(X229,$thf(sK589)),skolemize(X230,$thf(sK590)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X239,$thf(sK592)),skolemize(X240,$thf(sK593)),skolemize(X21,$thf(sK71 @ X11)),skolemize(X246,$thf(sK595)),skolemize(X247,$thf(sK596)),skolemize(X248,$thf(sK597)),skolemize(X249,$thf(sK598)),skolemize(X255,$thf(sK599)),skolemize(X256,$thf(sK600)),skolemize(X259,$thf(sK601)),skolemize(X260,$thf(sK602)),skolemize(X261,$thf(sK603)),skolemize(X262,$thf(sK604)),skolemize(X263,$thf(sK605)),skolemize(X268,$thf(sK606)),skolemize(X270,$thf(sK607)),skolemize(X271,$thf(sK608))],[f274])). thf(f285,plain,( (((sP0 @ sK61)) != $true) | (sP1 = $true)), inference(cnf_transformation,[],[f80])). thf(f288,plain,( (sP1 = $true) | (((sK61 @ id @ (lam @ sK62) @ id)) != $true)), inference(cnf_transformation,[],[f80])). thf(f293,plain,( ( ! [X0 : (subst > term > subst > $o),X1 : (subst > term > term)] : (($true = ((sP0 @ X0))) | (((X0 @ id @ (sK72 @ X1 @ X0) @ id)) = $true) | (((sub @ (X1 @ (sK73 @ X1) @ (sK74 @ X1)) @ (sK75 @ X1))) != ((X1 @ (comp @ (sK73 @ X1) @ (sK75 @ X1)) @ (sub @ (sK74 @ X1) @ (sK75 @ X1))))) | (((X0 @ id @ (hoaslam @ id @ (^[Y0 : subst]: ((^[Y1 : term]: (X1 @ Y0 @ Y1))))) @ id)) = $true)) )), inference(cnf_transformation,[],[f83])). thf(f295,plain,( (sP1 != $true) | (hoasinduction_lem3 != $true)), inference(cnf_transformation,[],[f84])). thf(f359,plain,( ( ! [X0 : (subst > term > subst > $o)] : ((((sP52 @ X0)) != $true) | (((X0 @ id @ (sK121 @ X0) @ id)) = $true)) )), inference(cnf_transformation,[],[f113])). thf(f360,plain,( ( ! [X0 : (subst > term > subst > $o)] : ((((sP52 @ X0)) != $true) | (((X0 @ id @ (ap @ (sub @ (sK122 @ X0) @ id) @ (sK121 @ X0)) @ id)) != $true)) )), inference(cnf_transformation,[],[f113])). thf(f362,plain,( ( ! [X2 : term,X0 : (subst > term > subst > $o),X1 : term] : ((((sP52 @ X0)) = $true) | (((X0 @ id @ X2 @ id)) != $true) | (((X0 @ id @ X1 @ id)) != $true) | (((X0 @ id @ (ap @ (sub @ X2 @ id) @ X1) @ id)) = $true)) )), inference(cnf_transformation,[],[f113])). thf(f448,plain,( (sP41 = $true) | ($true != ((sP40 @ sK179)))), inference(cnf_transformation,[],[f148])). thf(f449,plain,( (sP41 = $true) | (((sK179 @ id @ (lam @ sK180) @ id)) != $true)), inference(cnf_transformation,[],[f148])). thf(f457,plain,( ( ! [X0 : (subst > term > subst > $o),X1 : (subst > term > term)] : ((((X1 @ (comp @ (sK191 @ X1) @ (sK193 @ X1)) @ (sub @ (sK192 @ X1) @ (sK193 @ X1)))) != ((sub @ (X1 @ (sK191 @ X1) @ (sK192 @ X1)) @ (sK193 @ X1)))) | ($true = ((sP40 @ X0))) | (((X0 @ id @ (lam @ (X1 @ sh @ one)) @ id)) = $true) | ($true = ((X0 @ id @ (sK190 @ X1 @ X0) @ id)))) )), inference(cnf_transformation,[],[f151])). thf(f626,plain,( ( ! [X10 : (subst > term > subst > $o),X9 : term] : (($true != sP19) | (((sub @ sK343 @ id)) != sK343) | ($true = ((X10 @ id @ (lam @ X9) @ id))) | (sP18 = $true) | (((X10 @ id @ (lam @ (sK341 @ X10 @ sh @ one)) @ id)) != $true) | (((X10 @ id @ (sub @ X9 @ (push @ (sK342 @ X10 @ X9) @ id)) @ id)) != $true)) )), inference(cnf_transformation,[],[f216])). thf(f629,plain,( ( ! [X10 : (subst > term > subst > $o),X9 : term,X12 : term] : (($true = ((X10 @ id @ (sK342 @ X10 @ X9) @ id))) | ($true != sP19) | (sP18 = $true) | (((sub @ sK343 @ id)) != sK343) | ($true = ((X10 @ id @ (lam @ X9) @ id))) | (((X10 @ id @ X12 @ id)) != $true) | ($true = ((X10 @ id @ (sK341 @ X10 @ id @ X12) @ id)))) )), inference(cnf_transformation,[],[f216])). thf(f634,plain,( ( ! [X2 : (subst > term > term)] : ((((sK336 @ id @ (sK337 @ X2) @ id)) = $true) | (((sub @ (X2 @ (sK339 @ X2) @ (sK338 @ X2)) @ (sK340 @ X2))) != ((X2 @ (comp @ (sK339 @ X2) @ (sK340 @ X2)) @ (sub @ (sK338 @ X2) @ (sK340 @ X2))))) | ($true = sP19) | (((sK336 @ id @ (lam @ (X2 @ sh @ one)) @ id)) = $true)) )), inference(cnf_transformation,[],[f216])). thf(f635,plain,( (((sK336 @ id @ (lam @ sK335) @ id)) != $true) | ($true = sP19)), inference(cnf_transformation,[],[f216])). thf(f636,plain,( (sP18 != $true) | ($true != ((sK346 @ id @ (lam @ (sub @ sK347 @ (push @ one @ sh))) @ id)))), inference(cnf_transformation,[],[f219])). thf(f638,plain,( ( ! [X10 : (subst > term > term)] : (($true = ((sK346 @ id @ (lam @ (X10 @ sh @ one)) @ id))) | (((sK346 @ id @ (sK351 @ X10) @ id)) = $true) | (((X10 @ (comp @ (sK348 @ X10) @ (sK350 @ X10)) @ (sub @ (sK349 @ X10) @ (sK350 @ X10)))) != ((sub @ (X10 @ (sK348 @ X10) @ (sK349 @ X10)) @ (sK350 @ X10)))) | (sP18 != $true)) )), inference(cnf_transformation,[],[f219])). thf(f640,plain,( ( ! [X0 : (subst > term > subst > $o),X1 : term,X6 : term] : ((sP18 = $true) | (((X0 @ id @ (lam @ (sub @ X1 @ (push @ one @ sh))) @ id)) = $true) | (((X0 @ id @ X6 @ id)) != $true) | (((X0 @ id @ (sK345 @ X1 @ X0) @ id)) = $true) | (((X0 @ id @ (sK344 @ X0 @ id @ X6) @ id)) = $true)) )), inference(cnf_transformation,[],[f219])). thf(f810,plain,( (shinj = $true) | (sK595 != sK596)), inference(cnf_transformation,[],[f275])). thf(f812,plain,( ( ! [X244 : term,X245 : term] : ((X244 = X245) | (((sub @ X244 @ sh)) != ((sub @ X245 @ sh))) | (shinj != $true)) )), inference(cnf_transformation,[],[f275])). thf(f822,plain,( ( ! [X243 : d_term] : ((d_one = X243)) )), inference(cnf_transformation,[],[f275])). thf(f823,plain,( ( ! [X241 : term] : ((((d2term @ (sK594 @ X241))) = X241)) )), inference(cnf_transformation,[],[f275])). thf(f913,plain,( (((d2term @ d_one)) = one)), inference(cnf_transformation,[],[f275])). thf(f966,plain,( (sP41 != $true) | (hoasinduction_lem3 = $true)), inference(cnf_transformation,[],[f275])). thf(f1100,plain,( ( ! [X0 : (subst > term > subst > $o),X1 : (subst > term > term)] : ((((X0 @ id @ (sK72 @ X1 @ X0) @ id)) = $true) | ($true = ((sP0 @ X0))) | (((X0 @ id @ (hoaslam @ id @ X1) @ id)) = $true) | (((sub @ (X1 @ (sK73 @ X1) @ (sK74 @ X1)) @ (sK75 @ X1))) != ((X1 @ (comp @ (sK73 @ X1) @ (sK75 @ X1)) @ (sub @ (sK74 @ X1) @ (sK75 @ X1)))))) )), inference(beta-eta_normalization,[],[f293])). thf(f1337,definition,( spl609_42 <=> (sP18 = $true)), introduced(definition,[new_symbols(definition,[spl609_42])],[avatar_definition])). thf(f1350,definition,( spl609_45 <=> ! [X10 : (subst > term > term)] : (($true = ((sK346 @ id @ (lam @ (X10 @ sh @ one)) @ id))) | (((X10 @ (comp @ (sK348 @ X10) @ (sK350 @ X10)) @ (sub @ (sK349 @ X10) @ (sK350 @ X10)))) != ((sub @ (X10 @ (sK348 @ X10) @ (sK349 @ X10)) @ (sK350 @ X10)))) | (((sK346 @ id @ (sK351 @ X10) @ id)) = $true))), introduced(definition,[new_symbols(definition,[spl609_45])],[avatar_definition])). thf(f1351,plain,( ( ! [X10 : (subst > term > term)] : (($true = ((sK346 @ id @ (lam @ (X10 @ sh @ one)) @ id))) | (((sK346 @ id @ (sK351 @ X10) @ id)) = $true) | (((X10 @ (comp @ (sK348 @ X10) @ (sK350 @ X10)) @ (sub @ (sK349 @ X10) @ (sK350 @ X10)))) != ((sub @ (X10 @ (sK348 @ X10) @ (sK349 @ X10)) @ (sK350 @ X10))))) ) | ~spl609_45), inference(avatar_component_clause,[],[f1350])). thf(f1352,plain,( spl609_45 | ~spl609_42), inference(avatar_split_clause,[],[f638,f1337,f1350])). thf(f1363,definition,( spl609_48 <=> ($true = sP19)), introduced(definition,[new_symbols(definition,[spl609_48])],[avatar_definition])). thf(f1460,definition,( spl609_71 <=> (sP41 = $true)), introduced(definition,[new_symbols(definition,[spl609_71])],[avatar_definition])). thf(f1464,definition,( spl609_72 <=> (hoasinduction_lem3 = $true)), introduced(definition,[new_symbols(definition,[spl609_72])],[avatar_definition])). thf(f1502,definition,( spl609_81 <=> (((sub @ sK343 @ id)) = sK343)), introduced(definition,[new_symbols(definition,[spl609_81])],[avatar_definition])). thf(f1504,plain,( (((sub @ sK343 @ id)) != sK343) | spl609_81), inference(avatar_component_clause,[],[f1502])). thf(f1507,definition,( spl609_82 <=> (shinj = $true)), introduced(definition,[new_symbols(definition,[spl609_82])],[avatar_definition])). thf(f1525,definition,( spl609_86 <=> ! [X2 : (subst > term > term)] : ((((sK336 @ id @ (sK337 @ X2) @ id)) = $true) | (((sK336 @ id @ (lam @ (X2 @ sh @ one)) @ id)) = $true) | (((sub @ (X2 @ (sK339 @ X2) @ (sK338 @ X2)) @ (sK340 @ X2))) != ((X2 @ (comp @ (sK339 @ X2) @ (sK340 @ X2)) @ (sub @ (sK338 @ X2) @ (sK340 @ X2))))))), introduced(definition,[new_symbols(definition,[spl609_86])],[avatar_definition])). thf(f1526,plain,( ( ! [X2 : (subst > term > term)] : ((((sub @ (X2 @ (sK339 @ X2) @ (sK338 @ X2)) @ (sK340 @ X2))) != ((X2 @ (comp @ (sK339 @ X2) @ (sK340 @ X2)) @ (sub @ (sK338 @ X2) @ (sK340 @ X2))))) | (((sK336 @ id @ (sK337 @ X2) @ id)) = $true) | (((sK336 @ id @ (lam @ (X2 @ sh @ one)) @ id)) = $true)) ) | ~spl609_86), inference(avatar_component_clause,[],[f1525])). thf(f1527,plain,( spl609_48 | spl609_86), inference(avatar_split_clause,[],[f634,f1525,f1363])). thf(f1560,plain,( ~spl609_71 | spl609_72), inference(avatar_split_clause,[],[f966,f1464,f1460])). thf(f1562,definition,( spl609_95 <=> ! [X9 : term,X10 : (subst > term > subst > $o)] : (($true = ((X10 @ id @ (lam @ X9) @ id))) | (((X10 @ id @ (sub @ X9 @ (push @ (sK342 @ X10 @ X9) @ id)) @ id)) != $true) | (((X10 @ id @ (lam @ (sK341 @ X10 @ sh @ one)) @ id)) != $true))), introduced(definition,[new_symbols(definition,[spl609_95])],[avatar_definition])). thf(f1563,plain,( ( ! [X10 : (subst > term > subst > $o),X9 : term] : (($true = ((X10 @ id @ (lam @ X9) @ id))) | (((X10 @ id @ (sub @ X9 @ (push @ (sK342 @ X10 @ X9) @ id)) @ id)) != $true) | (((X10 @ id @ (lam @ (sK341 @ X10 @ sh @ one)) @ id)) != $true)) ) | ~spl609_95), inference(avatar_component_clause,[],[f1562])). thf(f1564,plain,( ~spl609_48 | spl609_42 | spl609_95 | ~spl609_81), inference(avatar_split_clause,[],[f626,f1502,f1562,f1337,f1363])). thf(f1586,definition,( spl609_101 <=> (sP1 = $true)), introduced(definition,[new_symbols(definition,[spl609_101])],[avatar_definition])). thf(f1589,plain,( ~spl609_101 | ~spl609_72), inference(avatar_split_clause,[],[f295,f1464,f1586])). thf(f1700,definition,( spl609_128 <=> (((sP0 @ sK61)) = $true)), introduced(definition,[new_symbols(definition,[spl609_128])],[avatar_definition])). thf(f1702,plain,( (((sP0 @ sK61)) != $true) | spl609_128), inference(avatar_component_clause,[],[f1700])). thf(f1703,plain,( spl609_101 | ~spl609_128), inference(avatar_split_clause,[],[f285,f1700,f1586])). thf(f1819,definition,( spl609_155 <=> ! [X245 : term,X244 : term] : ((X244 = X245) | (((sub @ X244 @ sh)) != ((sub @ X245 @ sh))))), introduced(definition,[new_symbols(definition,[spl609_155])],[avatar_definition])). thf(f1820,plain,( ( ! [X244 : term,X245 : term] : ((X244 = X245) | (((sub @ X244 @ sh)) != ((sub @ X245 @ sh)))) ) | ~spl609_155), inference(avatar_component_clause,[],[f1819])). thf(f1821,plain,( spl609_155 | ~spl609_82), inference(avatar_split_clause,[],[f812,f1507,f1819])). thf(f1902,definition,( spl609_175 <=> ! [X9 : term,X12 : term,X10 : (subst > term > subst > $o)] : (($true = ((X10 @ id @ (sK342 @ X10 @ X9) @ id))) | ($true = ((X10 @ id @ (sK341 @ X10 @ id @ X12) @ id))) | (((X10 @ id @ X12 @ id)) != $true) | ($true = ((X10 @ id @ (lam @ X9) @ id))))), introduced(definition,[new_symbols(definition,[spl609_175])],[avatar_definition])). thf(f1903,plain,( ( ! [X10 : (subst > term > subst > $o),X9 : term,X12 : term] : (($true = ((X10 @ id @ (lam @ X9) @ id))) | (((X10 @ id @ X12 @ id)) != $true) | ($true = ((X10 @ id @ (sK341 @ X10 @ id @ X12) @ id))) | ($true = ((X10 @ id @ (sK342 @ X10 @ X9) @ id)))) ) | ~spl609_175), inference(avatar_component_clause,[],[f1902])). thf(f1904,plain,( ~spl609_48 | ~spl609_81 | spl609_175 | spl609_42), inference(avatar_split_clause,[],[f629,f1337,f1902,f1502,f1363])). thf(f2184,definition,( spl609_241 <=> (((sK179 @ id @ (lam @ sK180) @ id)) = $true)), introduced(definition,[new_symbols(definition,[spl609_241])],[avatar_definition])). thf(f2186,plain,( (((sK179 @ id @ (lam @ sK180) @ id)) != $true) | spl609_241), inference(avatar_component_clause,[],[f2184])). thf(f2187,plain,( ~spl609_241 | spl609_71), inference(avatar_split_clause,[],[f449,f1460,f2184])). thf(f2236,definition,( spl609_253 <=> ($true = ((sP40 @ sK179)))), introduced(definition,[new_symbols(definition,[spl609_253])],[avatar_definition])). thf(f2238,plain,( ($true != ((sP40 @ sK179))) | spl609_253), inference(avatar_component_clause,[],[f2236])). thf(f2239,plain,( ~spl609_253 | spl609_71), inference(avatar_split_clause,[],[f448,f1460,f2236])). thf(f2383,definition,( spl609_287 <=> ! [X6 : term,X0 : (subst > term > subst > $o),X1 : term] : ((((X0 @ id @ (lam @ (sub @ X1 @ (push @ one @ sh))) @ id)) = $true) | (((X0 @ id @ (sK344 @ X0 @ id @ X6) @ id)) = $true) | (((X0 @ id @ (sK345 @ X1 @ X0) @ id)) = $true) | (((X0 @ id @ X6 @ id)) != $true))), introduced(definition,[new_symbols(definition,[spl609_287])],[avatar_definition])). thf(f2384,plain,( ( ! [X0 : (subst > term > subst > $o),X1 : term,X6 : term] : ((((X0 @ id @ (sK344 @ X0 @ id @ X6) @ id)) = $true) | (((X0 @ id @ (lam @ (sub @ X1 @ (push @ one @ sh))) @ id)) = $true) | (((X0 @ id @ X6 @ id)) != $true) | (((X0 @ id @ (sK345 @ X1 @ X0) @ id)) = $true)) ) | ~spl609_287), inference(avatar_component_clause,[],[f2383])). thf(f2385,plain,( spl609_42 | spl609_287), inference(avatar_split_clause,[],[f640,f2383,f1337])). thf(f2820,definition,( spl609_387 <=> (((sK336 @ id @ (lam @ sK335) @ id)) = $true)), introduced(definition,[new_symbols(definition,[spl609_387])],[avatar_definition])). thf(f2822,plain,( (((sK336 @ id @ (lam @ sK335) @ id)) != $true) | spl609_387), inference(avatar_component_clause,[],[f2820])). thf(f2823,plain,( spl609_48 | ~spl609_387), inference(avatar_split_clause,[],[f635,f2820,f1363])). thf(f2966,definition,( spl609_420 <=> (((sK61 @ id @ (lam @ sK62) @ id)) = $true)), introduced(definition,[new_symbols(definition,[spl609_420])],[avatar_definition])). thf(f2968,plain,( (((sK61 @ id @ (lam @ sK62) @ id)) != $true) | spl609_420), inference(avatar_component_clause,[],[f2966])). thf(f2969,plain,( spl609_101 | ~spl609_420), inference(avatar_split_clause,[],[f288,f2966,f1586])). thf(f2991,definition,( spl609_426 <=> ($true = ((sK346 @ id @ (lam @ (sub @ sK347 @ (push @ one @ sh))) @ id)))), introduced(definition,[new_symbols(definition,[spl609_426])],[avatar_definition])). thf(f2993,plain,( ($true != ((sK346 @ id @ (lam @ (sub @ sK347 @ (push @ one @ sh))) @ id))) | spl609_426), inference(avatar_component_clause,[],[f2991])). thf(f2994,plain,( ~spl609_426 | ~spl609_42), inference(avatar_split_clause,[],[f636,f1337,f2991])). thf(f3789,definition,( spl609_609 <=> (sK595 = sK596)), introduced(definition,[new_symbols(definition,[spl609_609])],[avatar_definition])). thf(f3791,plain,( (sK595 != sK596) | spl609_609), inference(avatar_component_clause,[],[f3789])). thf(f3792,plain,( spl609_82 | ~spl609_609), inference(avatar_split_clause,[],[f810,f3789,f1507])). thf(f3930,plain,( ( ! [X0 : d_term] : ((one = ((d2term @ X0)))) )), inference(superposition,[],[f913,f822])). thf(f3945,plain,( ( ! [X241 : term] : ((one = X241)) )), inference(forward_demodulation,[],[f823,f3930])). thf(f3952,plain,( (one != sK595) | spl609_609), inference(superposition,[],[f3791,f3945])). thf(f3954,plain,( $false | spl609_609), inference(forward_subsumption_resolution,[],[f3952,f3945])). thf(f3955,plain,( spl609_609), inference(avatar_contradiction_clause,[],[f3954])). thf(f3959,plain,( ( ! [X244 : term,X245 : term] : ((X244 = X245) | (one != ((sub @ X244 @ sh)))) ) | ~spl609_155), inference(forward_demodulation,[],[f1820,f3945])). thf(f3961,plain,( ( ! [X244 : term,X245 : term] : ((X244 = X245)) ) | ~spl609_155), inference(forward_subsumption_resolution,[],[f3959,f3945])). thf(f4018,plain,( $false | (spl609_81 | ~spl609_155)), inference(forward_subsumption_resolution,[],[f1504,f3961])). thf(f4019,plain,( spl609_81 | ~spl609_155), inference(avatar_contradiction_clause,[],[f4018])). thf(f4021,plain,( (((sK336 @ id @ one @ id)) != $true) | spl609_387), inference(forward_demodulation,[],[f2822,f3945])). thf(f4023,plain,( ( ! [X2 : (subst > term > term)] : ((((sK336 @ id @ (sK337 @ X2) @ id)) = $true) | (((sK336 @ id @ (lam @ (X2 @ sh @ one)) @ id)) = $true)) ) | (~spl609_86 | ~spl609_155)), inference(forward_subsumption_resolution,[],[f1526,f3961])). thf(f4025,definition,( spl609_641 <=> (((sK336 @ id @ one @ id)) = $true)), introduced(definition,[new_symbols(definition,[spl609_641])],[avatar_definition])). thf(f4032,plain,( ~spl609_641 | spl609_387), inference(avatar_split_clause,[],[f4021,f2820,f4025])). thf(f4034,plain,( ( ! [X2 : (subst > term > term)] : ((((sK336 @ id @ (lam @ (X2 @ sh @ one)) @ id)) = $true) | (((sK336 @ id @ one @ id)) = $true)) ) | (~spl609_86 | ~spl609_155)), inference(forward_demodulation,[],[f4023,f3945])). thf(f4036,plain,( (((sK336 @ id @ one @ id)) = $true) | (((sK336 @ id @ one @ id)) = $true) | (~spl609_86 | ~spl609_155)), inference(forward_demodulation,[],[f4034,f3945])). thf(f4037,plain,( (((sK336 @ id @ one @ id)) = $true) | (~spl609_86 | ~spl609_155)), inference(duplicate_literal_removal,[],[f4036])). thf(f4038,plain,( spl609_641 | ~spl609_86 | ~spl609_155), inference(avatar_split_clause,[],[f4037,f1819,f1525,f4025])). thf(f4040,plain,( ( ! [X10 : (subst > term > subst > $o),X9 : term,X12 : term] : (($true = ((X10 @ id @ (lam @ X9) @ id))) | (((X10 @ id @ one @ id)) = $true) | ($true = ((X10 @ id @ (sK342 @ X10 @ X9) @ id))) | (((X10 @ id @ X12 @ id)) != $true)) ) | ~spl609_175), inference(forward_demodulation,[],[f1903,f3945])). thf(f4042,plain,( ( ! [X10 : (subst > term > subst > $o),X9 : term] : ((((X10 @ id @ one @ id)) != $true) | (((X10 @ id @ (lam @ (sK341 @ X10 @ sh @ one)) @ id)) != $true) | ($true = ((X10 @ id @ (lam @ X9) @ id)))) ) | ~spl609_95), inference(forward_demodulation,[],[f1563,f3945])). thf(f4044,plain,( ( ! [X10 : (subst > term > subst > $o),X9 : term,X12 : term] : ((((X10 @ id @ one @ id)) = $true) | ($true = ((X10 @ id @ (lam @ X9) @ id))) | (((X10 @ id @ one @ id)) = $true) | (((X10 @ id @ X12 @ id)) != $true)) ) | ~spl609_175), inference(forward_demodulation,[],[f4040,f3945])). thf(f4045,plain,( ( ! [X10 : (subst > term > subst > $o),X9 : term,X12 : term] : ((((X10 @ id @ one @ id)) = $true) | (((X10 @ id @ X12 @ id)) != $true) | ($true = ((X10 @ id @ (lam @ X9) @ id)))) ) | ~spl609_175), inference(duplicate_literal_removal,[],[f4044])). thf(f4047,plain,( ( ! [X10 : (subst > term > subst > $o),X9 : term] : (($true = ((X10 @ id @ (lam @ X9) @ id))) | (((X10 @ id @ one @ id)) != $true) | (((X10 @ id @ one @ id)) != $true)) ) | ~spl609_95), inference(forward_demodulation,[],[f4042,f3945])). thf(f4048,plain,( ( ! [X10 : (subst > term > subst > $o),X9 : term] : (($true = ((X10 @ id @ (lam @ X9) @ id))) | (((X10 @ id @ one @ id)) != $true)) ) | ~spl609_95), inference(duplicate_literal_removal,[],[f4047])). thf(f4347,plain,( ($true != ((sK179 @ id @ one @ id))) | spl609_241), inference(forward_demodulation,[],[f2186,f3945])). thf(f4348,plain,( ( ! [X0 : term] : (($true != ((sK179 @ id @ X0 @ id)))) ) | spl609_241), inference(superposition,[],[f4347,f3945])). thf(f4842,plain,( ( ! [X0 : (subst > term > subst > $o)] : ((((X0 @ id @ one @ id)) = $true) | (((sP52 @ X0)) != $true)) )), inference(forward_demodulation,[],[f359,f3945])). thf(f5974,plain,( ( ! [X10 : (subst > term > subst > $o),X9 : term,X12 : term] : (($true = ((X10 @ id @ (lam @ X9) @ id))) | (((X10 @ id @ X12 @ id)) != $true)) ) | (~spl609_95 | ~spl609_175)), inference(forward_subsumption_resolution,[],[f4045,f4048])). thf(f6023,plain,( ( ! [X0 : (subst > term > subst > $o)] : ((((X0 @ id @ one @ id)) != $true) | (((sP52 @ X0)) != $true)) )), inference(forward_demodulation,[],[f360,f3945])). thf(f6024,plain,( ( ! [X0 : (subst > term > subst > $o)] : ((((sP52 @ X0)) != $true)) )), inference(forward_subsumption_resolution,[],[f6023,f4842])). thf(f6497,plain,( ( ! [X2 : term,X0 : (subst > term > subst > $o),X1 : term] : ((((X0 @ id @ X1 @ id)) != $true) | (((X0 @ id @ (ap @ (sub @ X2 @ id) @ X1) @ id)) = $true) | (((X0 @ id @ X2 @ id)) != $true)) )), inference(forward_subsumption_resolution,[],[f362,f6024])). thf(f6498,plain,( ( ! [X2 : term,X0 : (subst > term > subst > $o),X1 : term] : ((((X0 @ id @ one @ id)) = $true) | (((X0 @ id @ X1 @ id)) != $true) | (((X0 @ id @ X2 @ id)) != $true)) )), inference(forward_demodulation,[],[f6497,f3945])). thf(f6641,plain,( ( ! [X10 : (subst > term > term)] : ((((sK346 @ id @ (sK351 @ X10) @ id)) = $true) | ($true = ((sK346 @ id @ (lam @ (X10 @ sh @ one)) @ id)))) ) | (~spl609_45 | ~spl609_155)), inference(forward_subsumption_resolution,[],[f1351,f3961])). thf(f6642,plain,( ( ! [X0 : (subst > term > subst > $o),X1 : term,X6 : term] : ((((X0 @ id @ X6 @ id)) != $true) | (((X0 @ id @ (sK345 @ X1 @ X0) @ id)) = $true) | (((X0 @ id @ one @ id)) = $true) | (((X0 @ id @ (lam @ (sub @ X1 @ (push @ one @ sh))) @ id)) = $true)) ) | ~spl609_287), inference(forward_demodulation,[],[f2384,f3945])). thf(f6648,plain,( (((sK346 @ id @ one @ id)) != $true) | spl609_426), inference(forward_demodulation,[],[f2993,f3945])). thf(f6651,plain,( ( ! [X10 : (subst > term > term)] : ((((sK346 @ id @ one @ id)) = $true) | ($true = ((sK346 @ id @ (lam @ (X10 @ sh @ one)) @ id)))) ) | (~spl609_45 | ~spl609_155)), inference(forward_demodulation,[],[f6641,f3945])). thf(f6652,plain,( ( ! [X0 : (subst > term > subst > $o),X1 : term,X6 : term] : ((((X0 @ id @ (lam @ (sub @ X1 @ (push @ one @ sh))) @ id)) = $true) | (((X0 @ id @ (sK345 @ X1 @ X0) @ id)) = $true) | (((X0 @ id @ X6 @ id)) != $true)) ) | (~spl609_95 | ~spl609_287)), inference(forward_subsumption_resolution,[],[f6642,f4048])). thf(f6656,definition,( spl609_673 <=> (((sK346 @ id @ one @ id)) = $true)), introduced(definition,[new_symbols(definition,[spl609_673])],[avatar_definition])). thf(f6663,plain,( ~spl609_673 | spl609_426), inference(avatar_split_clause,[],[f6648,f2991,f6656])). thf(f6668,plain,( ( ! [X0 : (subst > term > subst > $o),X1 : term,X6 : term] : ((((X0 @ id @ one @ id)) = $true) | (((X0 @ id @ (sK345 @ X1 @ X0) @ id)) = $true) | (((X0 @ id @ X6 @ id)) != $true)) ) | (~spl609_95 | ~spl609_287)), inference(forward_demodulation,[],[f6652,f3945])). thf(f6672,plain,( ( ! [X0 : (subst > term > subst > $o),X6 : term] : ((((X0 @ id @ one @ id)) = $true) | (((X0 @ id @ X6 @ id)) != $true)) ) | (~spl609_95 | ~spl609_287)), inference(forward_subsumption_resolution,[],[f6668,f6498])). thf(f7801,plain,( ( ! [X0 : (subst > term > subst > $o),X1 : (subst > term > term)] : ((((X0 @ id @ (lam @ (X1 @ sh @ one)) @ id)) = $true) | (((X1 @ (comp @ (sK191 @ X1) @ (sK193 @ X1)) @ (sub @ (sK192 @ X1) @ (sK193 @ X1)))) != ((sub @ (X1 @ (sK191 @ X1) @ (sK192 @ X1)) @ (sK193 @ X1)))) | ($true = ((sP40 @ X0)))) ) | (~spl609_95 | ~spl609_175)), inference(forward_subsumption_resolution,[],[f457,f5974])). thf(f7802,plain,( ( ! [X0 : (subst > term > subst > $o),X1 : (subst > term > term)] : (($true = ((sP40 @ X0))) | (((X0 @ id @ (lam @ (X1 @ sh @ one)) @ id)) = $true)) ) | (~spl609_95 | ~spl609_155 | ~spl609_175)), inference(forward_subsumption_resolution,[],[f7801,f3961])). thf(f7803,plain,( ( ! [X0 : (subst > term > subst > $o)] : ((((X0 @ id @ one @ id)) = $true) | ($true = ((sP40 @ X0)))) ) | (~spl609_95 | ~spl609_155 | ~spl609_175)), inference(forward_demodulation,[],[f7802,f3945])). thf(f7833,plain,( ($true != $true) | ($true = ((sP40 @ sK179))) | (~spl609_95 | ~spl609_155 | ~spl609_175 | spl609_241)), inference(superposition,[],[f4348,f7803])). thf(f7893,plain,( ($true = ((sP40 @ sK179))) | (~spl609_95 | ~spl609_155 | ~spl609_175 | spl609_241)), inference(trivial_inequality_removal,[],[f7833])). thf(f7914,plain,( $false | (~spl609_95 | ~spl609_155 | ~spl609_175 | spl609_241 | spl609_253)), inference(forward_subsumption_resolution,[],[f7893,f2238])). thf(f7915,plain,( ~spl609_95 | ~spl609_155 | ~spl609_175 | spl609_241 | spl609_253), inference(avatar_contradiction_clause,[],[f7914])). thf(f7959,plain,( (((sK61 @ id @ one @ id)) != $true) | spl609_420), inference(forward_demodulation,[],[f2968,f3945])). thf(f9103,plain,( ( ! [X0 : (subst > term > subst > $o),X1 : (subst > term > term)] : ((((X0 @ id @ (sK72 @ X1 @ X0) @ id)) = $true) | (((X0 @ id @ (hoaslam @ id @ X1) @ id)) = $true) | ($true = ((sP0 @ X0)))) ) | ~spl609_155), inference(forward_subsumption_resolution,[],[f1100,f3961])). thf(f9104,plain,( ( ! [X0 : (subst > term > subst > $o),X1 : (subst > term > term)] : (($true = ((sP0 @ X0))) | (((X0 @ id @ one @ id)) = $true) | (((X0 @ id @ (hoaslam @ id @ X1) @ id)) = $true)) ) | ~spl609_155), inference(forward_demodulation,[],[f9103,f3945])). thf(f9105,plain,( ( ! [X0 : (subst > term > subst > $o)] : ((((X0 @ id @ one @ id)) = $true) | ($true = ((sP0 @ X0)))) ) | (~spl609_95 | ~spl609_155 | ~spl609_287)), inference(forward_subsumption_resolution,[],[f9104,f6672])). thf(f9143,plain,( (((sP0 @ sK61)) = $true) | ($true != $true) | (~spl609_95 | ~spl609_155 | ~spl609_287 | spl609_420)), inference(superposition,[],[f7959,f9105])). thf(f9187,plain,( (((sP0 @ sK61)) = $true) | (~spl609_95 | ~spl609_155 | ~spl609_287 | spl609_420)), inference(trivial_inequality_removal,[],[f9143])). thf(f9202,plain,( $false | (~spl609_95 | spl609_128 | ~spl609_155 | ~spl609_287 | spl609_420)), inference(forward_subsumption_resolution,[],[f9187,f1702])). thf(f9203,plain,( ~spl609_95 | spl609_128 | ~spl609_155 | ~spl609_287 | spl609_420), inference(avatar_contradiction_clause,[],[f9202])). thf(f9442,plain,( (((sK346 @ id @ one @ id)) = $true) | (((sK346 @ id @ one @ id)) = $true) | (~spl609_45 | ~spl609_155)), inference(forward_demodulation,[],[f6651,f3945])). thf(f9443,plain,( (((sK346 @ id @ one @ id)) = $true) | (~spl609_45 | ~spl609_155)), inference(duplicate_literal_removal,[],[f9442])). thf(f9481,plain,( spl609_673 | ~spl609_45 | ~spl609_155), inference(avatar_split_clause,[],[f9443,f1819,f1350,f6656])). cnf(s23, plain, ~spl609_42 | spl609_45, inference(sat_conversion,[],[f1352])). cnf(s50, plain, spl609_48 | spl609_86, inference(sat_conversion,[],[f1527])). cnf(s56, plain, ~spl609_71 | spl609_72, inference(sat_conversion,[],[f1560])). cnf(s57, plain, spl609_42 | ~spl609_48 | ~spl609_81 | spl609_95, inference(sat_conversion,[],[f1564])). cnf(s63, plain, ~spl609_72 | ~spl609_101, inference(sat_conversion,[],[f1589])). cnf(s82, plain, spl609_101 | ~spl609_128, inference(sat_conversion,[],[f1703])). cnf(s103, plain, ~spl609_82 | spl609_155, inference(sat_conversion,[],[f1821])). cnf(s121, plain, spl609_42 | ~spl609_48 | ~spl609_81 | spl609_175, inference(sat_conversion,[],[f1904])). cnf(s182, plain, spl609_71 | ~spl609_241, inference(sat_conversion,[],[f2187])). cnf(s190, plain, spl609_71 | ~spl609_253, inference(sat_conversion,[],[f2239])). cnf(s221, plain, spl609_42 | spl609_287, inference(sat_conversion,[],[f2385])). cnf(s329, plain, spl609_48 | ~spl609_387, inference(sat_conversion,[],[f2823])). cnf(s368, plain, spl609_101 | ~spl609_420, inference(sat_conversion,[],[f2969])). cnf(s374, plain, ~spl609_42 | ~spl609_426, inference(sat_conversion,[],[f2994])). cnf(s584, plain, spl609_82 | ~spl609_609, inference(sat_conversion,[],[f3792])). cnf(s620, plain, spl609_609, inference(sat_conversion,[],[f3955])). cnf(s624, plain, spl609_81 | ~spl609_155, inference(sat_conversion,[],[f4019])). cnf(s626, plain, spl609_387 | ~spl609_641, inference(sat_conversion,[],[f4032])). cnf(s627, plain, ~spl609_86 | ~spl609_155 | spl609_641, inference(sat_conversion,[],[f4038])). cnf(s722, plain, spl609_426 | ~spl609_673, inference(sat_conversion,[],[f6663])). cnf(s765, plain, ~spl609_95 | ~spl609_155 | ~spl609_175 | spl609_241 | spl609_253, inference(sat_conversion,[],[f7915])). cnf(s859, plain, ~spl609_95 | spl609_128 | ~spl609_155 | ~spl609_287 | spl609_420, inference(sat_conversion,[],[f9203])). cnf(s865, plain, ~spl609_45 | ~spl609_155 | spl609_673, inference(sat_conversion,[],[f9481])). cnf(s869, plain, spl609_82, inference(rat,[],[s584,s620])). cnf(s883, plain, spl609_155, inference(rat,[],[s103,s869])). cnf(s896, plain, spl609_81, inference(rat,[],[s624,s883])). cnf(s909, plain, spl609_42 | ~spl609_48 | spl609_95, inference(rat,[],[s57,s896])). cnf(s928, plain, ~spl609_42, inference(rat,[],[s865,s722,s23,s374,s883])). cnf(s932, plain, spl609_287, inference(rat,[],[s221,s928])). cnf(s943, plain, spl609_48, inference(rat,[],[s627,s626,s50,s329,s883])). cnf(s948, plain, spl609_175, inference(rat,[],[s121,s928,s896,s943])). cnf(s949, plain, spl609_95, inference(rat,[],[s909,s928,s943])). cnf(s974, plain, spl609_71, inference(rat,[],[s765,s182,s190,s883,s949,s948])). cnf(s983, plain, spl609_72, inference(rat,[],[s56,s974])). cnf(s984, plain, ~spl609_101, inference(rat,[],[s63,s983])). cnf(s988, plain, ~spl609_420, inference(rat,[],[s368,s984])). cnf(s989, plain, ~spl609_128, inference(rat,[],[s82,s984])). cnf(s1000, plain, $false, inference(rat,[],[s859,s949,s932,s883,s988,s989])). thf(f9549,plain,( $false), inference(avatar_sat_refutation,[],[s1000])). % SZS output end Proof for theBenchmark