%----Output an unreadable proof
% SZS output start Proof for theBenchmark
thf(type_def_5, type, term: $tType).
thf(type_def_6, type, subst: $tType).
thf(type_def_7, type, sTfun: ($tType * $tType) > $tType).
thf(type_def_8, type, d_term: $tType).
thf(type_def_9, type, d_subst: $tType).
thf(func_def_0, type, one: term).
thf(func_def_1, type, ap: (term > term > term)).
thf(func_def_2, type, lam: (term > term)).
thf(func_def_3, type, sub: (term > subst > term)).
thf(func_def_4, type, id: subst).
thf(func_def_5, type, sh: subst).
thf(func_def_6, type, push: (term > subst > subst)).
thf(func_def_7, type, comp: (subst > subst > subst)).
thf(func_def_8, type, var: (term > $o)).
thf(func_def_9, type, pushprop_lem1v2: $o).
thf(func_def_10, type, pushprop_lem1_gthm: $o).
thf(func_def_11, type, axmap: $o).
thf(func_def_12, type, pushprop_lem0_gthm: $o).
thf(func_def_13, type, shinj: $o).
thf(func_def_14, type, hoasinduction_lem1v2: $o).
thf(func_def_15, type, hoasinduction_lem1v2_gthm: $o).
thf(func_def_16, type, hoasap: (subst > term > subst > term > term)).
thf(func_def_17, type, induction2lem: $o).
thf(func_def_18, type, hoasinduction_lem3v2_f: $o).
thf(func_def_19, type, axvarshift: $o).
thf(func_def_20, type, hoasapinj2: $o).
thf(func_def_21, type, hoasapnotvar_gthm: $o).
thf(func_def_22, type, hoasapinj1: $o).
thf(func_def_23, type, ulamvar1: $o).
thf(func_def_24, type, induction2lem_lthm: $o).
thf(func_def_25, type, hoasinduction_lem3v2_gthm: $o).
thf(func_def_26, type, apnotvar: $o).
thf(func_def_27, type, pushprop_lthm_orig: $o).
thf(func_def_28, type, hoasinduction_lem3v2_f_lthm: $o).
thf(func_def_29, type, hoasinduction_lthm: $o).
thf(func_def_30, type, hoasinduction_no_psi_cond_lthm: $o).
thf(func_def_31, type, hoaslaminj: $o).
thf(func_def_32, type, hoasinduction_lem3aaa: $o).
thf(func_def_33, type, induction2lem_gthm: $o).
thf(func_def_34, type, hoasinduction_lem3aa_lthm: $o).
thf(func_def_35, type, hoasinduction_lem3: $o).
thf(func_def_36, type, hoasinduction_lem2: $o).
thf(func_def_37, type, termmset_lthm: $o).
thf(func_def_38, type, hoasinduction_lem1: $o).
thf(func_def_39, type, hoaslamnotap_lthm: $o).
thf(func_def_40, type, pushprop_lem1v2_lthm: $o).
thf(func_def_41, type, hoasapnotvar: $o).
thf(func_def_42, type, hoasinduction_lem0: $o).
thf(func_def_43, type, hoasinduction: $o).
thf(func_def_44, type, hoasinduction_gthm: $o).
thf(func_def_45, type, axapp: $o).
thf(func_def_46, type, hoaslamnotvar_lthm: $o).
thf(func_def_47, type, pushprop_lem3v2_lthm: $o).
thf(func_def_48, type, hoasinduction_lem3b_lthm: $o).
thf(func_def_49, type, ulamvarind: $o).
thf(func_def_50, type, induction: $o).
thf(func_def_51, type, hoasinduction_lem3a_lthm: $o).
thf(func_def_52, type, termmset_gthm: $o).
thf(func_def_53, type, hoasinduction_lem3aa: $o).
thf(func_def_54, type, pushprop_lem1v2_gthm: $o).
thf(func_def_55, type, hoaslamnotap_gthm: $o).
thf(func_def_56, type, hoaslamnotvar_gthm: $o).
thf(func_def_57, type, hoasinduction_lem3b_gthm: $o).
thf(func_def_58, type, pushprop_lem2v2: $o).
thf(func_def_59, type, hoasinduction_lem3a_gthm: $o).
thf(func_def_60, type, axclos: $o).
thf(func_def_61, type, axassoc: $o).
thf(func_def_62, type, hoasinduction_lem2v2: $o).
thf(func_def_63, type, pushprop_lthm: $o).
thf(func_def_64, type, apinj2: $o).
thf(func_def_65, type, apinj1: $o).
thf(func_def_66, type, hoasapinj2_lthm: $o).
thf(func_def_67, type, hoasinduction_lem3v2a: $o).
thf(func_def_68, type, hoasapinj1_lthm: $o).
thf(func_def_69, type, hoaslaminj_lthm: $o).
thf(func_def_70, type, axvarcons: $o).
thf(func_def_71, type, hoaslam: (subst > (subst > term > term) > term)).
thf(func_def_72, type, axscons: $o).
thf(func_def_73, type, hoasinduction_lem2v2_gthm: $o).
thf(func_def_74, type, axidr: $o).
thf(func_def_75, type, pushprop_lem1: $o).
thf(func_def_76, type, laminj: $o).
thf(func_def_77, type, hoasinduction_lem3_lthm: $o).
thf(func_def_78, type, pushprop_lem0: $o).
thf(func_def_79, type, pushprop_gthm: $o).
thf(func_def_80, type, axabs: $o).
thf(func_def_81, type, hoasinduction_lem3v2a_lthm: $o).
thf(func_def_82, type, hoasinduction_lem2_lthm: $o).
thf(func_def_83, type, hoasapinj2_gthm: $o).
thf(func_def_84, type, hoasinduction_p_and_p_prime: ((subst > term > subst > $o) > (term > $o) > $o)).
thf(func_def_85, type, hoasinduction_lem1_lthm: $o).
thf(func_def_86, type, lamnotap: $o).
thf(func_def_87, type, hoasapinj1_gthm: $o).
thf(func_def_88, type, hoaslamnotvar: $o).
thf(func_def_89, type, axidl: $o).
thf(func_def_90, type, hoaslaminj_gthm: $o).
thf(func_def_91, type, induction2_lthm: $o).
thf(func_def_92, type, hoasinduction_lem0_lthm: $o).
thf(func_def_93, type, substmonoid_lthm: $o).
thf(func_def_94, type, pushprop: $o).
thf(func_def_95, type, hoasinduction_lem3_gthm: $o).
thf(func_def_96, type, hoasinduction_lem2_gthm: $o).
thf(func_def_97, type, hoasinduction_lem3b: $o).
thf(func_def_98, type, substmonoid: $o).
thf(func_def_99, type, lamnotvar: $o).
thf(func_def_100, type, hoasinduction_lem3a: $o).
thf(func_def_101, type, hoasinduction_lem1_gthm: $o).
thf(func_def_102, type, hoasinduction_no_psi_cond: $o).
thf(func_def_103, type, induction2_gthm: $o).
thf(func_def_104, type, pushprop_lem2v2_lthm: $o).
thf(func_def_105, type, hoasvar: (subst > term > subst > $o)).
thf(func_def_106, type, hoaslamnotap: $o).
thf(func_def_107, type, substmonoid_gthm: $o).
thf(func_def_108, type, ulamvarsh: $o).
thf(func_def_109, type, induction2: $o).
thf(func_def_110, type, pushprop_lem3v2: $o).
thf(func_def_111, type, pushprop_lem2v2_gthm: $o).
thf(func_def_112, type, pushprop_lem1_lthm: $o).
thf(func_def_113, type, hoasinduction_lem3v2: $o).
thf(func_def_114, type, axshiftcons: $o).
thf(func_def_115, type, termmset: $o).
thf(func_def_116, type, pushprop_lem0_lthm: $o).
thf(func_def_117, type, hoasapnotvar_lthm: $o).
thf(func_def_118, type, hoasinduction_lem3v2_lthm: $o).
thf(func_def_119, type, pushprop_p_and_p_prime: (term > subst > (term > $o) > (term > $o) > $o)).
thf(func_def_120, type, axvarid: $o).
thf(func_def_121, type, hoasinduction_lthm_3: $o).
thf(func_def_122, type, d2term: (d_term > term)).
thf(func_def_123, type, d2subst: (d_subst > subst)).
thf(func_def_124, type, d_one: d_term).
thf(func_def_125, type, d_id: d_subst).
thf(func_def_129, type, vLAM: !>[X0: $tType, X1: $tType]:((X1) > (X0 > X1))).
thf(func_def_130, type, vPI: !>[X0: $tType]:(((X0 > $o) > $o))).
thf(func_def_131, type, db2: !>[X0: $tType]:(X0)).
thf(func_def_132, type, db0: !>[X0: $tType]:(X0)).
thf(func_def_133, type, db4: !>[X0: $tType]:(X0)).
thf(func_def_134, type, db3: !>[X0: $tType]:(X0)).
thf(func_def_135, type, db1: !>[X0: $tType]:(X0)).
thf(func_def_136, type, vEQ: !>[X0: $tType]:((X0 > X0 > $o))).
thf(func_def_137, type, vIMP: ($o > $o > $o)).
thf(func_def_138, type, sP0: ((subst > term > subst > $o) > $o)).
thf(func_def_139, type, sP1: $o).
thf(func_def_140, type, sP2: $o).
thf(func_def_141, type, sP3: ((term > $o) > $o)).
thf(func_def_142, type, sP4: ((term > $o) > $o)).
thf(func_def_143, type, sP5: $o).
thf(func_def_144, type, sP6: $o).
thf(func_def_145, type, sP7: ((subst > term > subst > $o) > $o)).
thf(func_def_146, type, sP8: $o).
thf(func_def_147, type, sP9: $o).
thf(func_def_148, type, sP10: $o).
thf(func_def_149, type, sP11: $o).
thf(func_def_150, type, sP12: ((subst > term > subst > $o) > $o)).
thf(func_def_151, type, sP13: $o).
thf(func_def_152, type, sP14: ((subst > term > subst > $o) > $o)).
thf(func_def_153, type, sP15: ((subst > term > subst > $o) > $o)).
thf(func_def_154, type, sP16: ((term > $o) > $o)).
thf(func_def_155, type, sP17: ((subst > term > subst > $o) > $o)).
thf(func_def_156, type, sP18: ((subst > term > subst > $o) > $o)).
thf(func_def_157, type, sP19: $o).
thf(func_def_158, type, sP20: $o).
thf(func_def_159, type, sP21: $o).
thf(func_def_160, type, sP22: $o).
thf(func_def_161, type, sP23: $o).
thf(func_def_162, type, sP24: $o).
thf(func_def_163, type, sP25: ((subst > term > subst > $o) > $o)).
thf(func_def_164, type, sP26: $o).
thf(func_def_165, type, sP27: ((subst > term > subst > $o) > $o)).
thf(func_def_166, type, sP28: $o).
thf(func_def_167, type, sP29: $o).
thf(func_def_168, type, sP30: $o).
thf(func_def_169, type, sP31: ((subst > term > subst > $o) > $o)).
thf(func_def_170, type, sP32: ((subst > term > subst > $o) > $o)).
thf(func_def_171, type, sP33: $o).
thf(func_def_172, type, sP34: $o).
thf(func_def_173, type, sP35: ((term > $o) > $o)).
thf(func_def_174, type, sP36: ((term > $o) > $o)).
thf(func_def_175, type, sP37: $o).
thf(func_def_176, type, sP38: $o).
thf(func_def_177, type, sP39: ((term > $o) > $o)).
thf(func_def_178, type, sP40: $o).
thf(func_def_179, type, sP41: ((term > $o) > $o)).
thf(func_def_180, type, sP42: ((term > $o) > $o)).
thf(func_def_181, type, sP43: $o).
thf(func_def_182, type, sP44: $o).
thf(func_def_183, type, sP45: ((subst > term > subst > $o) > $o)).
thf(func_def_184, type, sP46: $o).
thf(func_def_185, type, sP47: ((subst > term > subst > $o) > $o)).
thf(func_def_186, type, sP48: ((term > $o) > $o)).
thf(func_def_187, type, sP49: $o).
thf(func_def_188, type, sP50: $o).
thf(func_def_189, type, sP51: $o).
thf(func_def_190, type, sP52: ((subst > term > subst > $o) > $o)).
thf(func_def_191, type, sP53: ((subst > term > subst > $o) > $o)).
thf(func_def_192, type, sP54: ((subst > term > subst > $o) > $o)).
thf(func_def_193, type, sP55: ((term > $o) > $o)).
thf(func_def_194, type, sP56: $o).
thf(func_def_195, type, sP57: $o).
thf(func_def_196, type, sP58: $o).
thf(func_def_197, type, sK59: ((subst > term > subst > $o) > term > $o)).
thf(func_def_198, type, sK60: ((term > $o) > term)).
thf(func_def_199, type, sK61: ((term > $o) > term)).
thf(func_def_200, type, sK62: term).
thf(func_def_201, type, sK63: (subst > term > subst > $o)).
thf(func_def_202, type, sK64: ((term > $o) > term)).
thf(func_def_203, type, sK65: term).
thf(func_def_204, type, sK66: (term > $o)).
thf(func_def_205, type, sK67: (term > term)).
thf(func_def_206, type, sK68: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_207, type, sK69: (term > (term > $o) > term)).
thf(func_def_208, type, sK70: ((subst > term > subst > $o) > (term > $o) > term)).
thf(func_def_209, type, sK71: (term > $o)).
thf(func_def_210, type, sK72: term).
thf(func_def_211, type, sK73: (subst > term > subst > $o)).
thf(func_def_212, type, sK74: ((subst > term > term) > term)).
thf(func_def_213, type, sK75: ((subst > term > term) > term)).
thf(func_def_214, type, sK76: ((subst > term > term) > subst)).
thf(func_def_215, type, sK77: ((subst > term > term) > subst)).
thf(func_def_216, type, sK78: term).
thf(func_def_217, type, sK79: (subst > term > subst > $o)).
thf(func_def_218, type, sK80: ((subst > term > subst > $o) > term)).
thf(func_def_219, type, sK81: ((subst > term > subst > $o) > subst)).
thf(func_def_220, type, sK82: ((subst > term > subst > $o) > subst)).
thf(func_def_221, type, sK83: ((subst > term > subst > $o) > term)).
thf(func_def_222, type, sK84: ((subst > term > subst > $o) > subst)).
thf(func_def_223, type, sK85: ((subst > term > subst > $o) > subst)).
thf(func_def_224, type, sK86: ((subst > term > subst > $o) > subst)).
thf(func_def_225, type, sK87: ((subst > term > subst > $o) > term)).
thf(func_def_226, type, sK88: ((subst > term > subst > $o) > subst)).
thf(func_def_227, type, sK89: ((term > $o) > term)).
thf(func_def_228, type, sK90: ((term > $o) > term)).
thf(func_def_229, type, sK91: ((subst > term > subst > $o) > term)).
thf(func_def_230, type, sK92: ((subst > term > subst > $o) > term)).
thf(func_def_231, type, sK93: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_232, type, sK94: ((subst > term > term) > subst)).
thf(func_def_233, type, sK95: ((subst > term > term) > subst)).
thf(func_def_234, type, sK96: ((subst > term > term) > term)).
thf(func_def_235, type, sK97: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_236, type, sK98: ((subst > term > term) > subst)).
thf(func_def_237, type, sK99: ((subst > term > term) > subst)).
thf(func_def_238, type, sK100: ((subst > term > term) > term)).
thf(func_def_239, type, sK101: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_240, type, sK102: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_241, type, sK103: ((term > $o) > term)).
thf(func_def_242, type, sK104: ((term > $o) > term)).
thf(func_def_243, type, sK105: ((subst > term > subst > $o) > term > $o)).
thf(func_def_244, type, sK106: term).
thf(func_def_245, type, sK107: (term > $o)).
thf(func_def_246, type, sK108: (term > term)).
thf(func_def_247, type, sK109: term).
thf(func_def_248, type, sK110: (subst > term > subst > $o)).
thf(func_def_249, type, sK111: ((term > $o) > term)).
thf(func_def_250, type, sK112: term).
thf(func_def_251, type, sK113: (subst > term > subst > $o)).
thf(func_def_252, type, sK114: ((subst > term > term) > subst)).
thf(func_def_253, type, sK115: ((subst > term > term) > term)).
thf(func_def_254, type, sK116: ((subst > term > term) > subst)).
thf(func_def_255, type, sK117: ((subst > term > term) > term)).
thf(func_def_256, type, sK118: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_257, type, sK119: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_258, type, sK120: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_259, type, sK121: (term > (term > $o) > term)).
thf(func_def_260, type, sK122: (subst > term > subst > $o)).
thf(func_def_261, type, sK123: (term > $o)).
thf(func_def_262, type, sK124: term).
thf(func_def_263, type, sK125: ((subst > term > term) > term)).
thf(func_def_264, type, sK126: ((subst > term > term) > subst)).
thf(func_def_265, type, sK127: ((subst > term > term) > subst)).
thf(func_def_266, type, sK128: ((subst > term > term) > term)).
thf(func_def_267, type, sK129: ((term > $o) > term)).
thf(func_def_268, type, sK130: ((term > $o) > term)).
thf(func_def_269, type, sK131: ((subst > term > subst > $o) > term)).
thf(func_def_270, type, sK132: ((subst > term > subst > $o) > term)).
thf(func_def_271, type, sK133: (subst > term > subst > $o)).
thf(func_def_272, type, sK134: (term > $o)).
thf(func_def_273, type, sK135: term).
thf(func_def_274, type, sK136: ((subst > term > subst > $o) > subst)).
thf(func_def_275, type, sK137: ((subst > term > subst > $o) > subst)).
thf(func_def_276, type, sK138: ((subst > term > subst > $o) > subst)).
thf(func_def_277, type, sK139: ((subst > term > subst > $o) > term)).
thf(func_def_278, type, sK140: (term > (term > $o) > term)).
thf(func_def_279, type, sK141: ((subst > term > subst > $o) > term)).
thf(func_def_280, type, sK142: ((subst > term > subst > $o) > subst)).
thf(func_def_281, type, sK143: ((subst > term > subst > $o) > subst)).
thf(func_def_282, type, sK144: ((subst > term > subst > $o) > subst)).
thf(func_def_283, type, sK145: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_284, type, sK146: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_285, type, sK147: ((subst > term > term) > subst)).
thf(func_def_286, type, sK148: ((subst > term > term) > subst)).
thf(func_def_287, type, sK149: ((subst > term > term) > term)).
thf(func_def_288, type, sK150: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_289, type, sK151: ((term > $o) > term)).
thf(func_def_290, type, sK152: ((term > $o) > term)).
thf(func_def_291, type, sK153: (term > $o)).
thf(func_def_292, type, sK154: term).
thf(func_def_293, type, sK155: subst).
thf(func_def_294, type, sK156: term).
thf(func_def_295, type, sK157: subst).
thf(func_def_296, type, sK158: term).
thf(func_def_297, type, sK159: subst).
thf(func_def_298, type, sK160: subst).
thf(func_def_299, type, sK161: term).
thf(func_def_300, type, sK162: (term > $o)).
thf(func_def_301, type, sK163: (term > term)).
thf(func_def_302, type, sK164: ((term > $o) > subst > term)).
thf(func_def_303, type, sK165: ((term > $o) > term)).
thf(func_def_304, type, sK166: ((term > $o) > term)).
thf(func_def_305, type, sK167: ((term > $o) > term)).
thf(func_def_306, type, sK168: ((term > $o) > term)).
thf(func_def_307, type, sK169: ((term > $o) > term)).
thf(func_def_308, type, sK170: (term > $o)).
thf(func_def_309, type, sK171: term).
thf(func_def_310, type, sK172: subst).
thf(func_def_311, type, sK173: (term > term)).
thf(func_def_312, type, sK174: (subst > (term > $o) > term)).
thf(func_def_313, type, sK175: ((term > $o) > term)).
thf(func_def_314, type, sK176: ((term > $o) > term)).
thf(func_def_315, type, sK177: ((term > $o) > term)).
thf(func_def_316, type, sK178: (subst > (term > $o) > term)).
thf(func_def_317, type, sK179: ((term > $o) > term)).
thf(func_def_318, type, sK180: term).
thf(func_def_319, type, sK181: term).
thf(func_def_320, type, sK182: (term > $o)).
thf(func_def_321, type, sK183: subst).
thf(func_def_322, type, sK184: (term > term)).
thf(func_def_323, type, sK185: (term > $o)).
thf(func_def_324, type, sK186: term).
thf(func_def_325, type, sK187: (term > term)).
thf(func_def_326, type, sK188: ((term > $o) > term)).
thf(func_def_327, type, sK189: ((term > $o) > term)).
thf(func_def_328, type, sK190: ((term > $o) > term)).
thf(func_def_329, type, sK191: ((term > $o) > term)).
thf(func_def_330, type, sK192: ((term > $o) > term)).
thf(func_def_331, type, sK193: ((term > $o) > term)).
thf(func_def_332, type, sK194: term).
thf(func_def_333, type, sK195: (subst > term > subst > $o)).
thf(func_def_334, type, sK196: ((subst > term > term) > subst)).
thf(func_def_335, type, sK197: ((subst > term > term) > subst)).
thf(func_def_336, type, sK198: ((subst > term > term) > term)).
thf(func_def_337, type, sK199: ((subst > term > term) > term)).
thf(func_def_338, type, sK200: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_339, type, sK201: ((subst > term > subst > $o) > term > term)).
thf(func_def_340, type, sK202: term).
thf(func_def_341, type, sK203: (subst > term > subst > $o)).
thf(func_def_342, type, sK204: ((subst > term > subst > $o) > subst)).
thf(func_def_343, type, sK205: ((subst > term > subst > $o) > term)).
thf(func_def_344, type, sK206: ((subst > term > subst > $o) > subst)).
thf(func_def_345, type, sK207: ((subst > term > subst > $o) > subst)).
thf(func_def_346, type, sK208: ((subst > term > subst > $o) > term)).
thf(func_def_347, type, sK209: ((subst > term > subst > $o) > subst)).
thf(func_def_348, type, sK210: ((subst > term > subst > $o) > subst)).
thf(func_def_349, type, sK211: ((subst > term > subst > $o) > term)).
thf(func_def_350, type, sK212: ((subst > term > subst > $o) > subst)).
thf(func_def_351, type, sK213: ((subst > term > subst > $o) > term)).
thf(func_def_352, type, sK214: ((subst > term > subst > $o) > term)).
thf(func_def_353, type, sK215: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_354, type, sK216: ((subst > term > term) > term)).
thf(func_def_355, type, sK217: ((subst > term > term) > subst)).
thf(func_def_356, type, sK218: ((subst > term > term) > subst)).
thf(func_def_357, type, sK219: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_358, type, sK220: term).
thf(func_def_359, type, sK221: (subst > term > term)).
thf(func_def_360, type, sK222: subst).
thf(func_def_361, type, sK223: (subst > term > term)).
thf(func_def_362, type, sK224: term).
thf(func_def_363, type, sK225: subst).
thf(func_def_364, type, sK226: ((subst > term > term) > subst)).
thf(func_def_365, type, sK227: ((subst > term > term) > term)).
thf(func_def_366, type, sK228: ((subst > term > term) > subst)).
thf(func_def_367, type, sK229: ((subst > term > term) > subst)).
thf(func_def_368, type, sK230: ((subst > term > term) > subst)).
thf(func_def_369, type, sK231: ((subst > term > term) > term)).
thf(func_def_370, type, sK232: subst).
thf(func_def_371, type, sK233: term).
thf(func_def_372, type, sK234: term).
thf(func_def_373, type, sK235: term).
thf(func_def_374, type, sK236: term).
thf(func_def_375, type, sK237: (subst > term > subst > $o)).
thf(func_def_376, type, sK238: ((subst > term > term) > term)).
thf(func_def_377, type, sK239: ((subst > term > term) > subst)).
thf(func_def_378, type, sK240: ((subst > term > term) > subst)).
thf(func_def_379, type, sK241: ((subst > term > term) > term)).
thf(func_def_380, type, sK242: ((subst > term > subst > $o) > term > term)).
thf(func_def_381, type, sK243: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_382, type, sK244: (subst > term > subst > $o)).
thf(func_def_383, type, sK245: term).
thf(func_def_384, type, sK246: term).
thf(func_def_385, type, sK247: (term > $o)).
thf(func_def_386, type, sK248: ((subst > term > subst > $o) > term)).
thf(func_def_387, type, sK249: ((subst > term > subst > $o) > subst)).
thf(func_def_388, type, sK250: ((subst > term > subst > $o) > subst)).
thf(func_def_389, type, sK251: ((subst > term > subst > $o) > subst)).
thf(func_def_390, type, sK252: ((subst > term > subst > $o) > subst)).
thf(func_def_391, type, sK253: ((subst > term > subst > $o) > subst)).
thf(func_def_392, type, sK254: ((subst > term > subst > $o) > term)).
thf(func_def_393, type, sK255: ((subst > term > subst > $o) > subst)).
thf(func_def_394, type, sK256: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_395, type, sK257: ((subst > term > subst > $o) > term)).
thf(func_def_396, type, sK258: ((subst > term > subst > $o) > term)).
thf(func_def_397, type, sK259: (subst > term > subst > $o)).
thf(func_def_398, type, sK260: term).
thf(func_def_399, type, sK261: ((subst > term > subst > $o) > term)).
thf(func_def_400, type, sK262: ((subst > term > subst > $o) > subst)).
thf(func_def_401, type, sK263: ((subst > term > subst > $o) > subst)).
thf(func_def_402, type, sK264: ((subst > term > subst > $o) > subst)).
thf(func_def_403, type, sK265: ((subst > term > subst > $o) > subst)).
thf(func_def_404, type, sK266: ((subst > term > subst > $o) > subst)).
thf(func_def_405, type, sK267: ((subst > term > subst > $o) > term)).
thf(func_def_406, type, sK268: ((subst > term > subst > $o) > subst)).
thf(func_def_407, type, sK269: (term > (subst > term > subst > $o) > term)).
thf(func_def_408, type, sK270: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_409, type, sK271: ((subst > term > term) > subst)).
thf(func_def_410, type, sK272: ((subst > term > term) > term)).
thf(func_def_411, type, sK273: ((subst > term > term) > subst)).
thf(func_def_412, type, sK274: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_413, type, sK275: (subst > term > subst > $o)).
thf(func_def_414, type, sK276: term).
thf(func_def_415, type, sK277: ((subst > term > term) > term)).
thf(func_def_416, type, sK278: ((subst > term > term) > subst)).
thf(func_def_417, type, sK279: ((subst > term > term) > subst)).
thf(func_def_418, type, sK280: ((subst > term > term) > term)).
thf(func_def_419, type, sK281: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_420, type, sK282: (term > (subst > term > subst > $o) > term)).
thf(func_def_421, type, sK283: (term > $o)).
thf(func_def_422, type, sK284: (subst > term > subst > $o)).
thf(func_def_423, type, sK285: term).
thf(func_def_424, type, sK286: ((subst > term > term) > term)).
thf(func_def_425, type, sK287: ((subst > term > term) > subst)).
thf(func_def_426, type, sK288: ((subst > term > term) > term)).
thf(func_def_427, type, sK289: ((subst > term > term) > subst)).
thf(func_def_428, type, sK290: (term > subst > term > term)).
thf(func_def_429, type, sK291: term).
thf(func_def_430, type, sK292: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_431, type, sK293: ((subst > term > subst > $o) > (term > $o) > term)).
thf(func_def_432, type, sK294: (term > (term > $o) > term)).
thf(func_def_433, type, sK295: term).
thf(func_def_434, type, sK296: ((subst > term > term) > term)).
thf(func_def_435, type, sK297: ((subst > term > term) > subst)).
thf(func_def_436, type, sK298: ((term > $o) > term)).
thf(func_def_437, type, sK299: ((term > $o) > term)).
thf(func_def_438, type, sK300: (term > $o)).
thf(func_def_439, type, sK301: term).
thf(func_def_440, type, sK302: (term > term)).
thf(func_def_441, type, sK303: ((subst > term > subst > $o) > subst)).
thf(func_def_442, type, sK304: ((subst > term > subst > $o) > subst)).
thf(func_def_443, type, sK305: ((subst > term > subst > $o) > subst)).
thf(func_def_444, type, sK306: ((subst > term > subst > $o) > term)).
thf(func_def_445, type, sK307: ((subst > term > subst > $o) > subst)).
thf(func_def_446, type, sK308: ((subst > term > subst > $o) > subst)).
thf(func_def_447, type, sK309: ((subst > term > subst > $o) > term)).
thf(func_def_448, type, sK310: ((subst > term > subst > $o) > subst)).
thf(func_def_449, type, sK311: term).
thf(func_def_450, type, sK312: term).
thf(func_def_451, type, sK313: (subst > term > subst > $o)).
thf(func_def_452, type, sK314: (subst > term > subst > $o)).
thf(func_def_453, type, sK315: term).
thf(func_def_454, type, sK316: ((subst > term > subst > $o) > term)).
thf(func_def_455, type, sK317: ((subst > term > subst > $o) > subst)).
thf(func_def_456, type, sK318: ((subst > term > subst > $o) > subst)).
thf(func_def_457, type, sK319: ((subst > term > subst > $o) > subst)).
thf(func_def_458, type, sK320: ((subst > term > subst > $o) > term)).
thf(func_def_459, type, sK321: ((subst > term > subst > $o) > term)).
thf(func_def_460, type, sK322: ((subst > term > subst > $o) > subst)).
thf(func_def_461, type, sK323: ((subst > term > subst > $o) > subst)).
thf(func_def_462, type, sK324: ((subst > term > subst > $o) > subst)).
thf(func_def_463, type, sK325: ((subst > term > subst > $o) > term > term)).
thf(func_def_464, type, sK326: ((subst > term > subst > $o) > subst)).
thf(func_def_465, type, sK327: ((subst > term > subst > $o) > subst)).
thf(func_def_466, type, sK328: ((subst > term > subst > $o) > term)).
thf(func_def_467, type, sK329: ((subst > term > subst > $o) > subst)).
thf(func_def_468, type, sK330: ((subst > term > subst > $o) > subst)).
thf(func_def_469, type, sK331: ((subst > term > subst > $o) > subst)).
thf(func_def_470, type, sK332: ((subst > term > subst > $o) > subst)).
thf(func_def_471, type, sK333: ((subst > term > subst > $o) > term)).
thf(func_def_472, type, sK334: term).
thf(func_def_473, type, sK335: (subst > term > subst > $o)).
thf(func_def_474, type, sK336: ((subst > term > subst > $o) > term)).
thf(func_def_475, type, sK337: ((subst > term > subst > $o) > term)).
thf(func_def_476, type, sK338: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_477, type, sK339: ((subst > term > term) > subst)).
thf(func_def_478, type, sK340: ((subst > term > term) > term)).
thf(func_def_479, type, sK341: ((subst > term > term) > subst)).
thf(func_def_480, type, sK342: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_481, type, sK343: ((term > $o) > term)).
thf(func_def_482, type, sK344: ((term > $o) > term)).
thf(func_def_483, type, sK345: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_484, type, sK346: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_485, type, sK347: ((subst > term > term) > subst)).
thf(func_def_486, type, sK348: ((subst > term > term) > subst)).
thf(func_def_487, type, sK349: ((subst > term > term) > term)).
thf(func_def_488, type, sK350: ((subst > term > subst > $o) > term)).
thf(func_def_489, type, sK351: ((subst > term > subst > $o) > term)).
thf(func_def_490, type, sK352: term).
thf(func_def_491, type, sK353: (term > $o)).
thf(func_def_492, type, sK354: (subst > term > subst > $o)).
thf(func_def_493, type, sK355: term).
thf(func_def_494, type, sK356: ((subst > term > subst > $o) > (term > $o) > term)).
thf(func_def_495, type, sK357: ((subst > term > subst > $o) > subst)).
thf(func_def_496, type, sK358: ((subst > term > subst > $o) > term)).
thf(func_def_497, type, sK359: ((subst > term > subst > $o) > subst)).
thf(func_def_498, type, sK360: ((subst > term > subst > $o) > subst)).
thf(func_def_499, type, sK361: ((term > $o) > term > term)).
thf(func_def_500, type, sK362: ((subst > term > subst > $o) > term)).
thf(func_def_501, type, sK363: ((subst > term > subst > $o) > subst)).
thf(func_def_502, type, sK364: ((subst > term > subst > $o) > subst)).
thf(func_def_503, type, sK365: ((subst > term > subst > $o) > subst)).
thf(func_def_504, type, sK366: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_505, type, sK367: ((subst > term > term) > subst)).
thf(func_def_506, type, sK368: ((subst > term > term) > subst)).
thf(func_def_507, type, sK369: ((subst > term > term) > term)).
thf(func_def_508, type, sK370: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_509, type, sK371: (subst > term > subst > $o)).
thf(func_def_510, type, sK372: term).
thf(func_def_511, type, sK373: ((subst > term > term) > subst)).
thf(func_def_512, type, sK374: ((subst > term > term) > term)).
thf(func_def_513, type, sK375: ((subst > term > term) > subst)).
thf(func_def_514, type, sK376: ((subst > term > term) > term)).
thf(func_def_515, type, sK377: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_516, type, sK378: (term > (subst > term > subst > $o) > term)).
thf(func_def_517, type, sK379: term).
thf(func_def_518, type, sK380: (term > (subst > term > subst > $o) > term)).
thf(func_def_519, type, sK381: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_520, type, sK382: (subst > term > subst > $o)).
thf(func_def_521, type, sK383: term).
thf(func_def_522, type, sK384: ((subst > term > term) > term)).
thf(func_def_523, type, sK385: ((subst > term > term) > subst)).
thf(func_def_524, type, sK386: ((subst > term > term) > subst)).
thf(func_def_525, type, sK387: ((subst > term > term) > term)).
thf(func_def_526, type, sK388: (subst > term > subst > $o)).
thf(func_def_527, type, sK389: term).
thf(func_def_528, type, sK390: ((subst > term > subst > $o) > term)).
thf(func_def_529, type, sK391: ((subst > term > subst > $o) > subst)).
thf(func_def_530, type, sK392: ((subst > term > subst > $o) > subst)).
thf(func_def_531, type, sK393: ((subst > term > subst > $o) > subst)).
thf(func_def_532, type, sK394: (term > (subst > term > subst > $o) > term)).
thf(func_def_533, type, sK395: ((subst > term > subst > $o) > subst)).
thf(func_def_534, type, sK396: ((subst > term > subst > $o) > term)).
thf(func_def_535, type, sK397: ((subst > term > subst > $o) > subst)).
thf(func_def_536, type, sK398: ((subst > term > subst > $o) > subst)).
thf(func_def_537, type, sK399: term).
thf(func_def_538, type, sK400: (term > (subst > term > subst > $o) > term)).
thf(func_def_539, type, sK401: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_540, type, sK402: (subst > term > subst > $o)).
thf(func_def_541, type, sK403: term).
thf(func_def_542, type, sK404: ((subst > term > term) > term)).
thf(func_def_543, type, sK405: ((subst > term > term) > subst)).
thf(func_def_544, type, sK406: ((subst > term > term) > term)).
thf(func_def_545, type, sK407: ((subst > term > term) > subst)).
thf(func_def_546, type, sK408: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_547, type, sK409: ((subst > term > term) > term)).
thf(func_def_548, type, sK410: ((subst > term > term) > subst)).
thf(func_def_549, type, sK411: ((subst > term > term) > subst)).
thf(func_def_550, type, sK412: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_551, type, sK413: term).
thf(func_def_552, type, sK414: (subst > term > subst > $o)).
thf(func_def_553, type, sK415: (term > $o)).
thf(func_def_554, type, sK416: ((subst > term > term) > term)).
thf(func_def_555, type, sK417: ((subst > term > term) > subst)).
thf(func_def_556, type, sK418: ((subst > term > term) > subst)).
thf(func_def_557, type, sK419: ((subst > term > term) > term)).
thf(func_def_558, type, sK420: ((term > $o) > term > term)).
thf(func_def_559, type, sK421: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_560, type, sK422: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_561, type, sK423: (term > $o)).
thf(func_def_562, type, sK424: term).
thf(func_def_563, type, sK425: (term > term)).
thf(func_def_564, type, sK426: ((term > $o) > term)).
thf(func_def_565, type, sK427: ((term > $o) > term)).
thf(func_def_566, type, sK428: ((term > $o) > term)).
thf(func_def_567, type, sK429: ((term > $o) > term)).
thf(func_def_568, type, sK430: ((term > $o) > term)).
thf(func_def_569, type, sK431: ((term > $o) > term)).
thf(func_def_570, type, sK432: term).
thf(func_def_571, type, sK433: (subst > term > subst > $o)).
thf(func_def_572, type, sK434: ((subst > term > term) > term > subst)).
thf(func_def_573, type, sK435: ((subst > term > term) > term > subst)).
thf(func_def_574, type, sK436: ((subst > term > term) > term > term)).
thf(func_def_575, type, sK437: ((subst > term > term) > term)).
thf(func_def_576, type, sK438: ((subst > term > term) > term > term)).
thf(func_def_577, type, sK439: ((subst > term > term) > term > subst)).
thf(func_def_578, type, sK440: ((subst > term > term) > term > subst)).
thf(func_def_579, type, sK441: ((subst > term > subst > $o) > term)).
thf(func_def_580, type, sK442: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_581, type, sK443: ((subst > term > subst > $o) > term > term)).
thf(func_def_582, type, sK444: term).
thf(func_def_583, type, sK445: term).
thf(func_def_584, type, sK446: (subst > term > subst > $o)).
thf(func_def_585, type, sK447: ((subst > term > subst > $o) > subst)).
thf(func_def_586, type, sK448: ((subst > term > subst > $o) > subst)).
thf(func_def_587, type, sK449: ((subst > term > subst > $o) > term)).
thf(func_def_588, type, sK450: ((subst > term > subst > $o) > subst)).
thf(func_def_589, type, sK451: ((subst > term > subst > $o) > subst)).
thf(func_def_590, type, sK452: ((subst > term > subst > $o) > term)).
thf(func_def_591, type, sK453: ((subst > term > subst > $o) > subst)).
thf(func_def_592, type, sK454: ((subst > term > subst > $o) > subst)).
thf(func_def_593, type, sK455: ((subst > term > subst > $o) > term)).
thf(func_def_594, type, sK456: ((subst > term > subst > $o) > term)).
thf(func_def_595, type, sK457: subst).
thf(func_def_596, type, sK458: subst).
thf(func_def_597, type, sK459: term).
thf(func_def_598, type, sK460: ((subst > term > term) > term)).
thf(func_def_599, type, sK461: ((subst > term > term) > subst)).
thf(func_def_600, type, sK462: (term > subst > term > term)).
thf(func_def_601, type, sK463: term).
thf(func_def_602, type, sK464: (subst > term > subst > $o)).
thf(func_def_603, type, sK465: ((subst > term > subst > $o) > term)).
thf(func_def_604, type, sK466: ((subst > term > subst > $o) > term)).
thf(func_def_605, type, sK467: (term > (term > $o) > subst > term > $o)).
thf(func_def_606, type, sK468: subst).
thf(func_def_607, type, sK469: (term > $o)).
thf(func_def_608, type, sK470: term).
thf(func_def_609, type, sK471: ((term > $o) > term)).
thf(func_def_610, type, sK472: term).
thf(func_def_611, type, sK473: term).
thf(func_def_612, type, sK474: subst).
thf(func_def_613, type, sK475: term).
thf(func_def_614, type, sK476: term).
thf(func_def_615, type, sK477: term).
thf(func_def_616, type, sK478: (term > $o)).
thf(func_def_617, type, sK479: subst).
thf(func_def_618, type, sK480: (term > $o)).
thf(func_def_619, type, sK481: subst).
thf(func_def_620, type, sK482: subst).
thf(func_def_621, type, sK483: term).
thf(func_def_622, type, sK484: term).
thf(func_def_623, type, sK485: term).
thf(func_def_624, type, sK486: term).
thf(func_def_625, type, sK487: term).
thf(func_def_626, type, sK488: term).
thf(func_def_627, type, sK489: term).
thf(func_def_628, type, sK490: term).
thf(func_def_629, type, sK491: term).
thf(func_def_630, type, sK492: subst).
thf(func_def_631, type, sK493: (term > $o)).
thf(func_def_632, type, sK494: (term > $o)).
thf(func_def_633, type, sK495: term).
thf(func_def_634, type, sK496: term).
thf(func_def_635, type, sK497: subst).
thf(func_def_636, type, sK498: term).
thf(func_def_637, type, sK499: term).
thf(func_def_638, type, sK500: subst).
thf(func_def_639, type, sK501: term).
thf(func_def_640, type, sK502: term).
thf(func_def_641, type, sK503: term).
thf(func_def_642, type, sK504: term).
thf(func_def_643, type, sK505: term).
thf(func_def_644, type, sK506: term).
thf(func_def_645, type, sK507: term).
thf(func_def_646, type, sK508: term).
thf(func_def_647, type, sK509: ((subst > term > term) > term)).
thf(func_def_648, type, sK510: ((subst > term > term) > subst)).
thf(func_def_649, type, sK511: ((subst > term > term) > subst)).
thf(func_def_650, type, sK512: term).
thf(func_def_651, type, sK513: term).
thf(func_def_652, type, sK514: (subst > term > term)).
thf(func_def_653, type, sK515: subst).
thf(func_def_654, type, sK516: term).
thf(func_def_655, type, sK517: term).
thf(func_def_656, type, sK518: term).
thf(func_def_657, type, sK519: term).
thf(func_def_658, type, sK520: term).
thf(func_def_659, type, sK521: (subst > (term > $o) > term > (term > $o) > term)).
thf(func_def_660, type, sK522: (term > $o)).
thf(func_def_661, type, sK523: term).
thf(func_def_662, type, sK524: (term > $o)).
thf(func_def_663, type, sK525: subst).
thf(func_def_664, type, sK526: ((term > $o) > (term > $o) > subst > term > term)).
thf(func_def_665, type, sK527: subst).
thf(func_def_666, type, sK528: term).
thf(func_def_667, type, sK529: term).
thf(func_def_668, type, sK530: subst).
thf(func_def_669, type, sK531: (term > $o)).
thf(func_def_670, type, sK532: (term > $o)).
thf(func_def_671, type, sK533: term).
thf(func_def_672, type, sK534: term).
thf(func_def_673, type, sK535: term).
thf(func_def_674, type, sK536: term).
thf(func_def_675, type, sK537: (subst > term > subst > $o)).
thf(func_def_676, type, sK538: ((term > $o) > term)).
thf(func_def_677, type, sK539: ((subst > term > subst > $o) > term > $o)).
thf(func_def_678, type, sK540: ((subst > term > term) > term)).
thf(func_def_679, type, sK541: ((subst > term > term) > subst)).
thf(func_def_680, type, sK542: ((subst > term > term) > subst)).
thf(func_def_681, type, sK543: ((subst > term > term) > subst)).
thf(func_def_682, type, sK544: ((subst > term > term) > term)).
thf(func_def_683, type, sK545: ((subst > term > term) > subst)).
thf(func_def_684, type, sK546: (subst > term > term)).
thf(func_def_685, type, sK547: subst).
thf(func_def_686, type, sK548: term).
thf(func_def_687, type, sK549: (subst > term > term)).
thf(func_def_688, type, sK550: subst).
thf(func_def_689, type, sK551: subst).
thf(func_def_690, type, sK552: subst).
thf(func_def_691, type, sK553: subst).
thf(func_def_692, type, sK554: term).
thf(func_def_693, type, sK555: term).
thf(func_def_694, type, sK556: term).
thf(func_def_695, type, sK557: term).
thf(func_def_696, type, sK558: term).
thf(func_def_697, type, sK559: (term > d_term)).
thf(func_def_698, type, sK560: term).
thf(func_def_699, type, sK561: (term > subst > term > term)).
thf(func_def_700, type, sK562: (subst > d_subst)).
thf(func_def_701, type, sK563: term).
thf(func_def_702, type, sK564: term).
thf(func_def_703, type, sK565: term).
thf(func_def_704, type, sK566: term).
thf(func_def_705, type, sK567: term).
thf(func_def_706, type, sK568: term).
thf(func_def_707, type, sK569: (term > subst > term > term)).
thf(func_def_708, type, sK570: term).
thf(func_def_709, type, sK571: ((subst > term > term) > subst)).
thf(func_def_710, type, sK572: ((subst > term > term) > term)).
thf(func_def_711, type, sK573: term).
thf(func_def_712, type, sK574: (subst > term > subst > $o)).
thf(func_def_713, type, sK575: ((term > $o) > term)).
thf(func_def_714, type, sK576: ((subst > term > subst > $o) > term > $o)).
thf(func_def_715, type, sK577: subst).
thf(func_def_716, type, sK578: term).
thf(func_def_717, type, sK579: term).
thf(func_def_718, type, sK580: term).
thf(func_def_719, type, sK581: (term > subst > term > term)).
thf(func_def_720, type, sK582: ((term > $o) > term)).
thf(func_def_721, type, sK583: ((term > $o) > term)).
thf(func_def_722, type, sK584: (term > $o)).
thf(func_def_723, type, sK585: term).
thf(func_def_724, type, sK586: (term > (term > $o) > subst > term > $o)).
thf(func_def_725, type, sK587: subst).
thf(func_def_726, type, sK588: (term > $o)).
thf(func_def_727, type, sK589: term).
thf(func_def_728, type, sK590: ((term > $o) > term)).
thf(func_def_729, type, vNOT: ($o > $o)).
thf(func_def_730, type, sK591: (subst > term > term)).
thf(func_def_731, type, sK592: (subst > term > term)).
thf(func_def_732, type, sK593: subst).
thf(func_def_733, type, sK594: term).
thf(func_def_734, type, sK595: ((subst > term > term) > subst)).
thf(func_def_735, type, sK596: ((subst > term > term) > subst)).
thf(func_def_736, type, sK597: ((subst > term > term) > term)).
thf(func_def_737, type, sK598: ((subst > term > term) > subst)).
thf(func_def_738, type, sK599: ((subst > term > term) > term)).
thf(func_def_739, type, sK600: ((subst > term > term) > subst)).
thf(f1,conjecture,(
  (! [X0 : (subst > term > term)] : (! [X2 : term,X1 : subst,X3 : subst] : (((sub @ (X0 @ X1 @ X2) @ X3)) = ((X0 @ (comp @ X1 @ X3) @ (sub @ X2 @ X3)))) => ! [X4 : (subst > term > term)] : (! [X3 : subst,X1 : subst,X2 : term] : (((sub @ (X4 @ X1 @ X2) @ X3)) = ((X4 @ (comp @ X1 @ X3) @ (sub @ X2 @ X3)))) => ((((hoaslam @ id @ (^[X1 : subst, X2 : term] : ((X0 @ X1 @ X2))))) = ((hoaslam @ id @ (^[X1 : subst, X2 : term] : ((X4 @ X1 @ X2)))))) => ! [X2 : term,X1 : subst] : (((X0 @ X1 @ X2)) = ((X4 @ X1 @ X2)))))) = hoaslaminj)),
  file('/export/starexec/sandbox/benchmark/theBenchmark.p',hoaslaminj)).
thf(f2,negated_conjecture,(
  ~ (! [X0 : (subst > term > term)] : (! [X2 : term,X1 : subst,X3 : subst] : (((sub @ (X0 @ X1 @ X2) @ X3)) = ((X0 @ (comp @ X1 @ X3) @ (sub @ X2 @ X3)))) => ! [X4 : (subst > term > term)] : (! [X3 : subst,X1 : subst,X2 : term] : (((sub @ (X4 @ X1 @ X2) @ X3)) = ((X4 @ (comp @ X1 @ X3) @ (sub @ X2 @ X3)))) => ((((hoaslam @ id @ (^[X1 : subst, X2 : term] : ((X0 @ X1 @ X2))))) = ((hoaslam @ id @ (^[X1 : subst, X2 : term] : ((X4 @ X1 @ X2)))))) => ! [X2 : term,X1 : subst] : (((X0 @ X1 @ X2)) = ((X4 @ X1 @ X2)))))) = hoaslaminj)),
  inference(negated_conjecture,[status(cth)],[f1])).
thf(f3,axiom,(
  (hoasinduction_lem3 <=> ! [X6 : (subst > term > subst > $o),X23 : term] : ((X6 @ id @ (lam @ X23) @ id) | ~ ! [X11 : term] : ((X6 @ id @ (sub @ X23 @ (push @ X11 @ id)) @ id) | ~(X6 @ id @ X11 @ id)) | ~ ! [X22 : (subst > term > term)] : (~ ! [X9 : term] : (~(X6 @ id @ X9 @ id) | (X6 @ id @ (X22 @ id @ X9) @ id)) | ~ ! [X19 : subst,X10 : subst,X9 : term] : (((sub @ (X22 @ X10 @ X9) @ X19)) = ((X22 @ (comp @ X10 @ X19) @ (sub @ X9 @ X19)))) | (X6 @ id @ (lam @ (X22 @ sh @ one)) @ id)) | ~ ! [X19 : subst,X10 : subst,X20 : subst,X9 : term] : (~(X6 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20) @ X19) | (X6 @ X10 @ X9 @ (comp @ X20 @ X19))) | ~ ! [X9 : term,X19 : subst,X20 : subst,X10 : subst] : (~(X6 @ X10 @ X9 @ (comp @ X20 @ X19)) | (X6 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20) @ X19)))) & (hoasinduction_lem3v2_lthm <=> (! [X9 : term] : (X9 = ((sub @ X9 @ id))) => ! [X40 : term,X6 : (subst > term > subst > $o),X7 : (term > $o)] : (~ ! [X8 : term] : (((X7 @ X8)) = ((X6 @ id @ X8 @ id))) | (X7 @ (lam @ X40)) | ~ ! [X19 : subst,X10 : subst,X20 : subst,X9 : term] : ((X6 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20) @ X19) | ~(X6 @ X10 @ X9 @ (comp @ X20 @ X19))) | ~ ! [X19 : subst,X9 : term,X20 : subst,X10 : subst] : ((X6 @ X10 @ X9 @ (comp @ X20 @ X19)) | ~(X6 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20) @ X19)) | ~ ! [X11 : term] : (~(X7 @ X11) | (X7 @ (sub @ X40 @ (push @ X11 @ id)))) | ~ ! [X22 : (subst > term > term)] : (~ ! [X19 : subst,X9 : term,X10 : subst] : (((sub @ (X22 @ X10 @ X9) @ X19)) = ((X22 @ (comp @ X10 @ X19) @ (sub @ X9 @ X19)))) | ~ ! [X9 : term] : (~(X6 @ id @ X9 @ id) | (X6 @ id @ (X22 @ id @ X9) @ id)) | (X6 @ id @ (lam @ (X22 @ sh @ one)) @ id))))) & hoasinduction_lem1_gthm & (hoasinduction_lem3v2a_lthm <=> (! [X11 : term] : ~ ! [X22 : (subst > term > term)] : ~ ! [X9 : term,X10 : subst] : (((sub @ X11 @ (push @ X9 @ X10))) = ((X22 @ X10 @ X9))) => (! [X9 : term] : (X9 = ((sub @ X9 @ id))) => ! [X7 : (term > $o),X6 : (subst > term > subst > $o),X26 : term] : (~ ! [X8 : term] : (((X7 @ X8)) = ((X6 @ id @ X8 @ id))) | (X7 @ (lam @ X26)) | ~ ! [X11 : term] : ((X7 @ (sub @ X26 @ (push @ X11 @ id))) | ~(X7 @ X11)) | ~ ! [X22 : (subst > term > term)] : ((X6 @ id @ (lam @ (X22 @ sh @ one)) @ id) | ~ ! [X9 : term] : (~(X6 @ id @ X9 @ id) | (X6 @ id @ (X22 @ id @ X9) @ id)) | ~ ! [X10 : subst,X19 : subst,X9 : term] : (((sub @ (X22 @ X10 @ X9) @ X19)) = ((X22 @ (comp @ X10 @ X19) @ (sub @ X9 @ X19))))))))) & hoaslamnotap_lthm & pushprop_lem1_gthm & axapp & (! [X41 : term,X42 : term,X22 : (subst > term > term)] : ((((lam @ (X22 @ sh @ one))) != ((ap @ (sub @ X41 @ id) @ X42))) | ~ ! [X10 : subst,X19 : subst,X9 : term] : (((sub @ (X22 @ X10 @ X9) @ X19)) = ((X22 @ (comp @ X10 @ X19) @ (sub @ X9 @ X19))))) <=> hoaslamnotap) & hoasapnotvar_lthm & (hoasinduction_lem3b <=> ! [X11 : term] : ~ ! [X22 : (subst > term > term)] : (((X22 @ sh @ one)) != ((sub @ X11 @ (push @ one @ sh))))) & ~lamnotap & hoasinduction_lem1 & hoaslamnotap_gthm & pushprop_lem1v2_gthm & pushprop_lem3v2_lthm & (! [X12 : term,X13 : subst,X6 : (term > $o)] : (~ ! [X9 : term] : (~ ! [X11 : term] : ((X6 @ (sub @ X9 @ (push @ X11 @ id))) | ~(X6 @ X11)) | (X6 @ (lam @ X9))) | ~ ! [X9 : term,X11 : term] : (~(X6 @ X9) | (X6 @ (ap @ X9 @ X11)) | ~(X6 @ X11)) | (X6 @ (sub @ X12 @ X13)) | ~ ! [X11 : term] : (~(var @ X11) | (X6 @ (sub @ X11 @ X13)))) <=> induction2lem) & hoasinduction_lem1_lthm & (hoasinduction_lem0_lthm <=> ! [X6 : (subst > term > subst > $o)] : ~ ! [X7 : (term > $o)] : ~ ! [X8 : term] : (((X7 @ X8)) = ((X6 @ id @ X8 @ id)))) & ! [X1 : d_term] : (X1 = d_one) & axabs & axassoc & ~ulamvar1 & hoasapnotvar_gthm & axclos & ((! [X10 : subst] : (X10 = ((comp @ X10 @ id))) & ! [X10 : subst] : (X10 = ((comp @ id @ X10)))) <=> substmonoid) & pushprop & ! [X47 : d_term,X48 : d_term] : ((((d2term @ X47)) = ((d2term @ X48))) => (X47 = X48)) & (hoasapinj2_lthm <=> (! [X14 : term,X15 : term,X11 : term,X9 : term] : ((((ap @ X9 @ X14)) != ((ap @ X11 @ X15))) | (X14 = X15)) => ! [X14 : term,X11 : term,X15 : term,X9 : term] : ((((ap @ (sub @ X9 @ id) @ X14)) != ((ap @ (sub @ X11 @ id) @ X15))) | (X14 = X15)))) & hoasinduction_lem2_gthm & pushprop_gthm & hoasinduction_lem3a_gthm & (! [X38 : term,X6 : (subst > term > subst > $o)] : (~ ! [X11 : term] : (~(X6 @ id @ X11 @ id) | (X6 @ id @ (sub @ X38 @ (push @ X11 @ id)) @ id)) | (X6 @ id @ (lam @ (sub @ X38 @ (push @ one @ sh))) @ id) | ~ ! [X22 : (subst > term > term)] : ((X6 @ id @ (lam @ (X22 @ sh @ one)) @ id) | ~ ! [X9 : term,X19 : subst,X10 : subst] : (((sub @ (X22 @ X10 @ X9) @ X19)) = ((X22 @ (comp @ X10 @ X19) @ (sub @ X9 @ X19)))) | ~ ! [X9 : term] : (~(X6 @ id @ X9 @ id) | (X6 @ id @ (X22 @ id @ X9) @ id)))) <=> hoasinduction_lem3aa) & pushprop_lthm & induction2lem_gthm & ((! [X6 : (term > $o),X24 : term] : (~ ! [X9 : term] : ((X6 @ (lam @ X9)) | ~ ! [X11 : term] : (~(X6 @ X11) | (X6 @ (sub @ X9 @ (push @ X11 @ id))))) | (X6 @ X24) | ~ ! [X9 : term,X11 : term] : (~(X6 @ X9) | ~(X6 @ X11) | (X6 @ (ap @ X9 @ X11))) | ~ ! [X9 : term] : ((X6 @ X9) | ~(var @ X9))) => (! [X18 : term,X6 : (subst > term > subst > $o),X17 : term] : ((X6 @ id @ (ap @ X17 @ X18) @ id) | ~ ! [X19 : subst,X9 : term,X10 : subst,X20 : subst] : (~(X6 @ X10 @ X9 @ (comp @ X20 @ X19)) | (X6 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20) @ X19)) | ~ ! [X20 : subst,X19 : subst,X10 : subst,X9 : term] : ((X6 @ X10 @ X9 @ (comp @ X20 @ X19)) | ~(X6 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20) @ X19)) | ~(X6 @ id @ X18 @ id) | ~ ! [X9 : term,X11 : term] : (~(X6 @ id @ X11 @ id) | ~(X6 @ id @ X9 @ id) | (X6 @ id @ (ap @ (sub @ X9 @ id) @ X11) @ id)) | ~(X6 @ id @ X17 @ id)) => (! [X6 : (subst > term > subst > $o),X23 : term] : (~ ! [X10 : subst,X19 : subst,X20 : subst,X9 : term] : (~(X6 @ X10 @ X9 @ (comp @ X20 @ X19)) | (X6 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20) @ X19)) | ~ ! [X19 : subst,X20 : subst,X9 : term,X10 : subst] : (~(X6 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20) @ X19) | (X6 @ X10 @ X9 @ (comp @ X20 @ X19))) | ~ ! [X22 : (subst > term > term)] : (~ ! [X9 : term] : (~(X6 @ id @ X9 @ id) | (X6 @ id @ (X22 @ id @ X9) @ id)) | ~ ! [X10 : subst,X19 : subst,X9 : term] : (((sub @ (X22 @ X10 @ X9) @ X19)) = ((X22 @ (comp @ X10 @ X19) @ (sub @ X9 @ X19)))) | (X6 @ id @ (lam @ (X22 @ sh @ one)) @ id)) | (X6 @ id @ (lam @ X23) @ id) | ~ ! [X11 : term] : (~(X6 @ id @ X11 @ id) | (X6 @ id @ (sub @ X23 @ (push @ X11 @ id)) @ id))) => ! [X6 : (subst > term > subst > $o),X21 : term] : (~ ! [X22 : (subst > term > term)] : (~ ! [X9 : term] : (~(X6 @ id @ X9 @ id) | (X6 @ id @ (X22 @ id @ X9) @ id)) | ~ ! [X9 : term,X10 : subst,X19 : subst] : (((sub @ (X22 @ X10 @ X9) @ X19)) = ((X22 @ (comp @ X10 @ X19) @ (sub @ X9 @ X19)))) | (X6 @ id @ (lam @ (X22 @ sh @ one)) @ id)) | (X6 @ id @ X21 @ id) | ~ ! [X20 : subst,X9 : term,X10 : subst,X19 : subst] : ((X6 @ X10 @ X9 @ (comp @ X20 @ X19)) | ~(X6 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20) @ X19)) | ~ ! [X19 : subst,X10 : subst,X9 : term,X20 : subst] : (~(X6 @ X10 @ X9 @ (comp @ X20 @ X19)) | (X6 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20) @ X19)) | ~ ! [X9 : term] : ((X6 @ id @ X9 @ id) | ~(var @ (sub @ X9 @ id))) | ~ ! [X9 : term,X11 : term] : (~(X6 @ id @ X9 @ id) | ~(X6 @ id @ X11 @ id) | (X6 @ id @ (ap @ (sub @ X9 @ id) @ X11) @ id)))))) <=> hoasinduction_lthm) & (! [X15 : term,X11 : term,X14 : term,X9 : term] : ((((ap @ (sub @ X9 @ id) @ X14)) != ((ap @ (sub @ X11 @ id) @ X15))) | (X9 = X11)) <=> hoasapinj1) & (hoasinduction_lem3b_lthm <=> ! [X11 : term] : ~ ! [X22 : (subst > term > term)] : (((X22 @ sh @ one)) != ((sub @ X11 @ (push @ one @ sh))))) & axmap & (((sub @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2term @ d_one))) & (axidr <=> ! [X10 : subst] : (X10 = ((comp @ X10 @ id)))) & (! [X9 : term] : (X9 = ((sub @ X9 @ id))) <=> termmset) & (hoasinduction_p_and_p_prime = (^[X6 : (subst > term > subst > $o), X7 : (term > $o)] : (! [X8 : term] : (((X7 @ X8)) = ((X6 @ id @ X8 @ id)))))) & pushprop_lem2v2_gthm & termmset_gthm & (axshiftcons <=> ! [X10 : subst,X9 : term] : (X10 = ((comp @ sh @ (push @ X9 @ X10))))) & ((! [X9 : term,X10 : subst] : (X9 = ((sub @ one @ (push @ X9 @ X10)))) => (! [X10 : subst,X9 : term] : (X10 = ((comp @ sh @ (push @ X9 @ X10)))) => (! [X10 : subst] : (X10 = ((comp @ X10 @ id))) => (! [X16 : term,X6 : (term > $o)] : ((X6 @ X16) | ~ ! [X9 : term] : (~(X6 @ X9) | (X6 @ (lam @ X9))) | ~ ! [X9 : term,X11 : term] : (~(X6 @ X11) | (X6 @ (ap @ X9 @ X11)) | ~(X6 @ X9)) | ~ ! [X9 : term] : (~(var @ X9) | (X6 @ X9))) => ! [X13 : subst,X6 : (term > $o),X12 : term] : (~ ! [X9 : term] : ((X6 @ (lam @ X9)) | ~ ! [X11 : term] : ((X6 @ (sub @ X9 @ (push @ X11 @ id))) | ~(X6 @ X11))) | (X6 @ (sub @ X12 @ X13)) | ~ ! [X9 : term,X11 : term] : (~(X6 @ X9) | (X6 @ (ap @ X9 @ X11)) | ~(X6 @ X11)) | ~ ! [X11 : term] : (~(var @ X11) | (X6 @ (sub @ X11 @ X13)))))))) <=> induction2lem_lthm) & (! [X25 : term,X6 : (subst > term > subst > $o)] : (~ ! [X11 : term,X9 : term] : (~(X6 @ id @ X9 @ id) | ~(X6 @ id @ X11 @ id) | (X6 @ id @ (ap @ (sub @ X9 @ id) @ X11) @ id)) | ~ ! [X22 : (subst > term > term)] : ((X6 @ id @ (lam @ (X22 @ sh @ one)) @ id) | ~ ! [X9 : term] : (~(X6 @ id @ X9 @ id) | (X6 @ id @ (X22 @ id @ X9) @ id)) | ~ ! [X10 : subst,X19 : subst,X9 : term] : (((sub @ (X22 @ X10 @ X9) @ X19)) = ((X22 @ (comp @ X10 @ X19) @ (sub @ X9 @ X19))))) | (X6 @ id @ X25 @ id)) <=> hoasinduction_no_psi_cond) & (sh = ((d2subst @ d_id))) & hoaslamnotvar & pushprop_lem2v2_lthm & hoasinduction_lem2_lthm & (((push @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & (! [X11 : term,X9 : term,X15 : term,X14 : term] : ((X14 = X15) | (((ap @ (sub @ X9 @ id) @ X14)) != ((ap @ (sub @ X11 @ id) @ X15)))) <=> hoasapinj2) & (pushprop_p_and_p_prime = (^[X9 : term, X10 : subst, X6 : (term > $o), X7 : (term > $o)] : (! [X8 : term] : (((X7 @ X8)) = ((X6 @ (sub @ X8 @ (push @ X9 @ X10)))))))) & ((! [X9 : term,X10 : subst] : (X9 = ((sub @ one @ (push @ X9 @ X10)))) => ! [X10 : subst,X7 : (term > $o),X9 : term,X6 : (term > $o)] : ((X7 @ one) | ~(X6 @ X9) | ~ ! [X8 : term] : (((X7 @ X8)) = ((X6 @ (sub @ X8 @ (push @ X9 @ X10))))))) <=> pushprop_lem1v2_lthm) & (hoaslam = (^[X4 : subst, X5 : (subst > term > term)] : ((d2term @ d_one)))) & hoasinduction_gthm & (hoasinduction_lem3v2a <=> ! [X7 : (term > $o),X26 : term,X6 : (subst > term > subst > $o)] : (~ ! [X22 : (subst > term > term)] : ((X6 @ id @ (lam @ (X22 @ sh @ one)) @ id) | ~ ! [X19 : subst,X10 : subst,X9 : term] : (((sub @ (X22 @ X10 @ X9) @ X19)) = ((X22 @ (comp @ X10 @ X19) @ (sub @ X9 @ X19)))) | ~ ! [X9 : term] : (~(X6 @ id @ X9 @ id) | (X6 @ id @ (X22 @ id @ X9) @ id))) | ~ ! [X11 : term] : (~(X7 @ X11) | (X7 @ (sub @ X26 @ (push @ X11 @ id)))) | (X7 @ (lam @ X26)) | ~ ! [X8 : term] : (((X7 @ X8)) = ((X6 @ id @ X8 @ id))))) & (hoasinduction_lem3a <=> ! [X39 : term,X6 : (subst > term > subst > $o)] : (~ ! [X22 : (subst > term > term)] : (~ ! [X10 : subst,X19 : subst,X9 : term] : (((sub @ (X22 @ X10 @ X9) @ X19)) = ((X22 @ (comp @ X10 @ X19) @ (sub @ X9 @ X19)))) | ~ ! [X9 : term] : ((X6 @ id @ (X22 @ id @ X9) @ id) | ~(X6 @ id @ X9 @ id)) | (X6 @ id @ (lam @ (X22 @ sh @ one)) @ id)) | (X6 @ id @ (lam @ X39) @ id) | ~ ! [X11 : term] : (~(X6 @ id @ X11 @ id) | (X6 @ id @ (sub @ X39 @ (push @ X11 @ id)) @ id)))) & (! [X11 : term] : ~ ! [X22 : (subst > term > term)] : ~ ! [X10 : subst,X9 : term] : (((sub @ X11 @ (push @ X9 @ X10))) = ((X22 @ X10 @ X9))) <=> hoasinduction_lem3v2_f) & axvarshift & (((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) = ((d2term @ d_one))) & ((! [X9 : term] : (X9 = ((sub @ X9 @ id))) => (! [X6 : (subst > term > subst > $o),X38 : term] : (~ ! [X11 : term] : (~(X6 @ id @ X11 @ id) | (X6 @ id @ (sub @ X38 @ (push @ X11 @ id)) @ id)) | (X6 @ id @ (lam @ (sub @ X38 @ (push @ one @ sh))) @ id) | ~ ! [X22 : (subst > term > term)] : (~ ! [X9 : term] : (~(X6 @ id @ X9 @ id) | (X6 @ id @ (X22 @ id @ X9) @ id)) | (X6 @ id @ (lam @ (X22 @ sh @ one)) @ id) | ~ ! [X19 : subst,X10 : subst,X9 : term] : (((sub @ (X22 @ X10 @ X9) @ X19)) = ((X22 @ (comp @ X10 @ X19) @ (sub @ X9 @ X19)))))) => ! [X39 : term,X6 : (subst > term > subst > $o)] : ((X6 @ id @ (lam @ X39) @ id) | ~ ! [X22 : (subst > term > term)] : (~ ! [X9 : term] : (~(X6 @ id @ X9 @ id) | (X6 @ id @ (X22 @ id @ X9) @ id)) | ~ ! [X10 : subst,X19 : subst,X9 : term] : (((sub @ (X22 @ X10 @ X9) @ X19)) = ((X22 @ (comp @ X10 @ X19) @ (sub @ X9 @ X19)))) | (X6 @ id @ (lam @ (X22 @ sh @ one)) @ id)) | ~ ! [X11 : term] : (~(X6 @ id @ X11 @ id) | (X6 @ id @ (sub @ X39 @ (push @ X11 @ id)) @ id))))) <=> hoasinduction_lem3a_lthm) & (! [X6 : (subst > term > subst > $o),X38 : term] : (~ ! [X22 : (subst > term > term)] : (~ ! [X9 : term] : (~(X6 @ id @ X9 @ id) | (X6 @ id @ (X22 @ id @ X9) @ id)) | ~ ! [X19 : subst,X9 : term,X10 : subst] : (((sub @ (X22 @ X10 @ X9) @ X19)) = ((X22 @ (comp @ X10 @ X19) @ (sub @ X9 @ X19)))) | (X6 @ id @ (lam @ (X22 @ sh @ one)) @ id)) | (X6 @ id @ (lam @ (sub @ X38 @ (push @ one @ sh))) @ id) | ~ ! [X11 : term] : (~(X6 @ id @ X11 @ id) | (X6 @ id @ (sub @ X38 @ (push @ X11 @ id)) @ id))) <=> hoasinduction_lem3aa_lthm) & (! [X16 : term,X6 : (term > $o)] : (~ ! [X9 : term] : (~(X6 @ X9) | (X6 @ (lam @ X9))) | (X6 @ X16) | ~ ! [X11 : term,X9 : term] : (~(X6 @ X9) | (X6 @ (ap @ X9 @ X11)) | ~(X6 @ X11)) | ~ ! [X9 : term] : ((X6 @ X9) | ~(var @ X9))) <=> induction) & hoasapinj2_gthm & hoasapnotvar & (! [X11 : term,X9 : term,X10 : subst,X6 : (term > $o),X20 : (term > $o)] : ((X20 @ (sub @ X9 @ (push @ X11 @ X10))) | ~(X6 @ X9)) <=> pushprop_lem1) & pushprop_lem3v2 & (! [X15 : term,X14 : term,X9 : term,X11 : term] : ((X14 = X15) | (((ap @ X9 @ X14)) != ((ap @ X11 @ X15)))) <=> apinj2) & (((ap @ (d2term @ d_one) @ (d2term @ d_one))) = ((d2term @ d_one))) & (induction2_lthm <=> (! [X9 : term] : (X9 = ((sub @ X9 @ id))) => (! [X6 : (term > $o),X12 : term,X13 : subst] : (~ ! [X11 : term] : (~(var @ X11) | (X6 @ (sub @ X11 @ X13))) | ~ ! [X11 : term,X9 : term] : (~(X6 @ X9) | ~(X6 @ X11) | (X6 @ (ap @ X9 @ X11))) | ~ ! [X9 : term] : (~ ! [X11 : term] : (~(X6 @ X11) | (X6 @ (sub @ X9 @ (push @ X11 @ id)))) | (X6 @ (lam @ X9))) | (X6 @ (sub @ X12 @ X13))) => ! [X6 : (term > $o),X24 : term] : (~ ! [X9 : term] : ((X6 @ (lam @ X9)) | ~ ! [X11 : term] : ((X6 @ (sub @ X9 @ (push @ X11 @ id))) | ~(X6 @ X11))) | ~ ! [X11 : term,X9 : term] : (~(X6 @ X11) | (X6 @ (ap @ X9 @ X11)) | ~(X6 @ X9)) | ~ ! [X9 : term] : (~(var @ X9) | (X6 @ X9)) | (X6 @ X24))))) & (hoasinduction_lem3_lthm <=> (! [X9 : term] : (X9 = ((sub @ X9 @ id))) => (! [X6 : (subst > term > subst > $o),X38 : term] : (~ ! [X11 : term] : ((X6 @ id @ (sub @ X38 @ (push @ X11 @ id)) @ id) | ~(X6 @ id @ X11 @ id)) | (X6 @ id @ (lam @ (sub @ X38 @ (push @ one @ sh))) @ id) | ~ ! [X22 : (subst > term > term)] : (~ ! [X9 : term] : (~(X6 @ id @ X9 @ id) | (X6 @ id @ (X22 @ id @ X9) @ id)) | (X6 @ id @ (lam @ (X22 @ sh @ one)) @ id) | ~ ! [X19 : subst,X9 : term,X10 : subst] : (((sub @ (X22 @ X10 @ X9) @ X19)) = ((X22 @ (comp @ X10 @ X19) @ (sub @ X9 @ X19)))))) => ! [X23 : term,X6 : (subst > term > subst > $o)] : (~ ! [X9 : term,X20 : subst,X10 : subst,X19 : subst] : (~(X6 @ X10 @ X9 @ (comp @ X20 @ X19)) | (X6 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20) @ X19)) | ~ ! [X11 : term] : ((X6 @ id @ (sub @ X23 @ (push @ X11 @ id)) @ id) | ~(X6 @ id @ X11 @ id)) | ~ ! [X19 : subst,X20 : subst,X9 : term,X10 : subst] : (~(X6 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20) @ X19) | (X6 @ X10 @ X9 @ (comp @ X20 @ X19))) | (X6 @ id @ (lam @ X23) @ id) | ~ ! [X22 : (subst > term > term)] : ((X6 @ id @ (lam @ (X22 @ sh @ one)) @ id) | ~ ! [X10 : subst,X19 : subst,X9 : term] : (((sub @ (X22 @ X10 @ X9) @ X19)) = ((X22 @ (comp @ X10 @ X19) @ (sub @ X9 @ X19)))) | ~ ! [X9 : term] : ((X6 @ id @ (X22 @ id @ X9) @ id) | ~(X6 @ id @ X9 @ id))))))) & (hoasapinj1_lthm <=> (! [X9 : term] : (X9 = ((sub @ X9 @ id))) => (! [X11 : term,X9 : term,X14 : term,X15 : term] : ((((ap @ X9 @ X14)) != ((ap @ X11 @ X15))) | (X9 = X11)) => ! [X9 : term,X15 : term,X11 : term,X14 : term] : ((((ap @ (sub @ X9 @ id) @ X14)) != ((ap @ (sub @ X11 @ id) @ X15))) | (X9 = X11))))) & hoasapinj1_gthm & ((! [X6 : (subst > term > subst > $o)] : ~ ! [X7 : (term > $o)] : ~ ! [X8 : term] : (((X7 @ X8)) = ((X6 @ id @ X8 @ id))) => (! [X6 : (term > $o),X24 : term] : (~ ! [X9 : term,X11 : term] : (~(X6 @ X9) | ~(X6 @ X11) | (X6 @ (ap @ X9 @ X11))) | (X6 @ X24) | ~ ! [X9 : term] : (~(var @ X9) | (X6 @ X9)) | ~ ! [X9 : term] : ((X6 @ (lam @ X9)) | ~ ! [X11 : term] : (~(X6 @ X11) | (X6 @ (sub @ X9 @ (push @ X11 @ id)))))) => (! [X9 : term] : (X9 = ((sub @ X9 @ id))) => (! [X7 : (term > $o),X6 : (subst > term > subst > $o),X26 : term] : (~ ! [X22 : (subst > term > term)] : ((X6 @ id @ (lam @ (X22 @ sh @ one)) @ id) | ~ ! [X19 : subst,X10 : subst,X9 : term] : (((sub @ (X22 @ X10 @ X9) @ X19)) = ((X22 @ (comp @ X10 @ X19) @ (sub @ X9 @ X19)))) | ~ ! [X9 : term] : (~(X6 @ id @ X9 @ id) | (X6 @ id @ (X22 @ id @ X9) @ id))) | ~ ! [X11 : term] : (~(X7 @ X11) | (X7 @ (sub @ X26 @ (push @ X11 @ id)))) | (X7 @ (lam @ X26)) | ~ ! [X8 : term] : (((X7 @ X8)) = ((X6 @ id @ X8 @ id)))) => ! [X6 : (subst > term > subst > $o),X21 : term] : (~ ! [X11 : term,X9 : term] : (~(X6 @ id @ X11 @ id) | ~(X6 @ id @ X9 @ id) | (X6 @ id @ (ap @ (sub @ X9 @ id) @ X11) @ id)) | ~ ! [X20 : subst,X10 : subst,X9 : term,X19 : subst] : ((X6 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20) @ X19) | ~(X6 @ X10 @ X9 @ (comp @ X20 @ X19))) | ~ ! [X22 : (subst > term > term)] : (~ ! [X9 : term] : (~(X6 @ id @ X9 @ id) | (X6 @ id @ (X22 @ id @ X9) @ id)) | ~ ! [X10 : subst,X9 : term,X19 : subst] : (((sub @ (X22 @ X10 @ X9) @ X19)) = ((X22 @ (comp @ X10 @ X19) @ (sub @ X9 @ X19)))) | (X6 @ id @ (lam @ (X22 @ sh @ one)) @ id)) | (X6 @ id @ X21 @ id) | ~ ! [X9 : term,X19 : subst,X20 : subst,X10 : subst] : ((X6 @ X10 @ X9 @ (comp @ X20 @ X19)) | ~(X6 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20) @ X19)) | ~ ! [X9 : term] : ((X6 @ id @ X9 @ id) | ~(var @ (sub @ X9 @ id)))))))) <=> hoasinduction_lthm_3) & ulamvarind & induction2_gthm & hoasinduction_lem3_gthm & ! [X45 : d_subst] : (X45 = d_id) & hoasinduction_lem3v2_gthm & (id = ((d2subst @ d_id))) & ((! [X10 : subst,X9 : term] : (X9 = ((sub @ one @ (push @ X9 @ X10)))) => (! [X9 : term,X10 : subst] : (X10 = ((comp @ sh @ (push @ X9 @ X10)))) => ! [X10 : subst,X11 : term,X9 : term,X20 : (term > $o),X6 : (term > $o)] : ((X20 @ (sub @ X9 @ (push @ X11 @ X10))) | ~(X6 @ X9)))) <=> pushprop_lem1_lthm) & hoasinduction_lem1v2 & ! [X46 : subst] : ? [X45 : d_subst] : (X46 = ((d2subst @ X45))) & (! [X10 : subst,X9 : term] : (X9 = ((sub @ one @ (push @ X9 @ X10)))) <=> axvarcons) & hoasinduction_lem1v2_gthm & pushprop_lem2v2 & (hoasinduction_lem0 <=> ! [X6 : (subst > term > subst > $o)] : ~ ! [X7 : (term > $o)] : ~ ! [X8 : term] : (((X7 @ X8)) = ((X6 @ id @ X8 @ id)))) & (hoaslaminj <=> ! [X27 : (subst > term > term),X29 : term,X28 : subst,X22 : (subst > term > term)] : (~ ! [X9 : term,X10 : subst,X19 : subst] : (((sub @ (X22 @ X10 @ X9) @ X19)) = ((X22 @ (comp @ X10 @ X19) @ (sub @ X9 @ X19)))) | (((X22 @ X28 @ X29)) = ((X27 @ X28 @ X29))) | (((lam @ (X22 @ sh @ one))) != ((lam @ (X27 @ sh @ one)))) | ~ ! [X19 : subst,X9 : term,X10 : subst] : (((sub @ (X27 @ X10 @ X9) @ X19)) = ((X27 @ (comp @ X10 @ X19) @ (sub @ X9 @ X19)))))) & pushprop_lthm_orig & (hoasinduction_no_psi_cond_lthm <=> (! [X6 : (subst > term > subst > $o)] : ~ ! [X7 : (term > $o)] : ~ ! [X8 : term] : (((X7 @ X8)) = ((X6 @ id @ X8 @ id))) => (! [X24 : term,X6 : (term > $o)] : (~ ! [X9 : term] : ((X6 @ X9) | ~(var @ X9)) | (X6 @ X24) | ~ ! [X9 : term] : ((X6 @ (lam @ X9)) | ~ ! [X11 : term] : ((X6 @ (sub @ X9 @ (push @ X11 @ id))) | ~(X6 @ X11))) | ~ ! [X11 : term,X9 : term] : (~(X6 @ X9) | (X6 @ (ap @ X9 @ X11)) | ~(X6 @ X11))) => (! [X9 : term] : (X9 = ((sub @ X9 @ id))) => (! [X6 : (subst > term > subst > $o),X26 : term,X7 : (term > $o)] : (~ ! [X11 : term] : (~(X7 @ X11) | (X7 @ (sub @ X26 @ (push @ X11 @ id)))) | ~ ! [X8 : term] : (((X7 @ X8)) = ((X6 @ id @ X8 @ id))) | (X7 @ (lam @ X26)) | ~ ! [X22 : (subst > term > term)] : (~ ! [X19 : subst,X9 : term,X10 : subst] : (((sub @ (X22 @ X10 @ X9) @ X19)) = ((X22 @ (comp @ X10 @ X19) @ (sub @ X9 @ X19)))) | (X6 @ id @ (lam @ (X22 @ sh @ one)) @ id) | ~ ! [X9 : term] : (~(X6 @ id @ X9 @ id) | (X6 @ id @ (X22 @ id @ X9) @ id)))) => ! [X6 : (subst > term > subst > $o),X25 : term] : ((X6 @ id @ X25 @ id) | ~ ! [X9 : term,X11 : term] : (~(X6 @ id @ X11 @ id) | (X6 @ id @ (ap @ (sub @ X9 @ id) @ X11) @ id) | ~(X6 @ id @ X9 @ id)) | ~ ! [X22 : (subst > term > term)] : ((X6 @ id @ (lam @ (X22 @ sh @ one)) @ id) | ~ ! [X9 : term] : ((X6 @ id @ (X22 @ id @ X9) @ id) | ~(X6 @ id @ X9 @ id)) | ~ ! [X9 : term,X10 : subst,X19 : subst] : (((sub @ (X22 @ X10 @ X9) @ X19)) = ((X22 @ (comp @ X10 @ X19) @ (sub @ X9 @ X19))))))))))) & (! [X9 : term,X10 : subst,X6 : (term > $o)] : ~ ! [X7 : (term > $o)] : ~ ! [X8 : term] : (((X7 @ X8)) = ((X6 @ (sub @ X8 @ (push @ X9 @ X10))))) <=> pushprop_lem0_lthm) & ! [X0 : term] : ? [X1 : d_term] : (X0 = ((d2term @ X1))) & (! [X9 : term,X6 : (term > $o),X10 : subst] : ~ ! [X7 : (term > $o)] : ~ ! [X8 : term] : (((X7 @ X8)) = ((X6 @ (sub @ X8 @ (push @ X9 @ X10))))) <=> pushprop_lem0) & (! [X11 : term,X9 : term] : ((X9 = X11) | (((lam @ X9)) != ((lam @ X11)))) <=> laminj) & pushprop_lem0_gthm & hoaslaminj_gthm & (! [X24 : term,X6 : (term > $o)] : (~ ! [X9 : term,X11 : term] : (~(X6 @ X9) | ~(X6 @ X11) | (X6 @ (ap @ X9 @ X11))) | ~ ! [X9 : term] : ((X6 @ (lam @ X9)) | ~ ! [X11 : term] : ((X6 @ (sub @ X9 @ (push @ X11 @ id))) | ~(X6 @ X11))) | (X6 @ X24) | ~ ! [X9 : term] : ((X6 @ X9) | ~(var @ X9))) <=> induction2) & (substmonoid_lthm <=> (! [X10 : subst] : (X10 = ((comp @ id @ X10))) => (! [X10 : subst] : (X10 = ((comp @ X10 @ id))) => (! [X10 : subst] : (X10 = ((comp @ id @ X10))) & ! [X10 : subst] : (X10 = ((comp @ X10 @ id))))))) & apnotvar & (one = ((d2term @ d_one))) & lamnotvar & ~(var @ (d2term @ d_one)) & (! [X10 : subst] : (X10 = ((push @ (sub @ one @ X10) @ (comp @ sh @ X10)))) <=> axscons) & ((! [X9 : term,X10 : subst] : (X9 = ((sub @ one @ (push @ X9 @ X10)))) => (! [X9 : term,X10 : subst] : (X10 = ((comp @ sh @ (push @ X9 @ X10)))) => (! [X9 : term,X11 : term] : ((((lam @ X9)) != ((lam @ X11))) | (X9 = X11)) => ! [X22 : (subst > term > term),X29 : term,X28 : subst,X27 : (subst > term > term)] : (~ ! [X19 : subst,X10 : subst,X9 : term] : (((sub @ (X27 @ X10 @ X9) @ X19)) = ((X27 @ (comp @ X10 @ X19) @ (sub @ X9 @ X19)))) | (((lam @ (X22 @ sh @ one))) != ((lam @ (X27 @ sh @ one)))) | ~ ! [X9 : term,X10 : subst,X19 : subst] : (((sub @ (X22 @ X10 @ X9) @ X19)) = ((X22 @ (comp @ X10 @ X19) @ (sub @ X9 @ X19)))) | (((X22 @ X28 @ X29)) = ((X27 @ X28 @ X29))))))) <=> hoaslaminj_lthm) & (! [X7 : (term > $o),X9 : term,X10 : subst,X6 : (term > $o)] : (~ ! [X8 : term] : (((X7 @ X8)) = ((X6 @ (sub @ X8 @ (push @ X9 @ X10))))) | (X7 @ one) | ~(X6 @ X9)) <=> pushprop_lem1v2) & (termmset_lthm <=> (! [X9 : term] : (X9 = ((sub @ X9 @ id))) => ! [X9 : term] : (X9 = ((sub @ X9 @ id))))) & hoaslamnotvar_lthm & (hoasinduction_lem3v2 <=> ! [X6 : (subst > term > subst > $o),X7 : (term > $o),X40 : term] : ((X7 @ (lam @ X40)) | ~ ! [X9 : term,X20 : subst,X19 : subst,X10 : subst] : ((X6 @ X10 @ X9 @ (comp @ X20 @ X19)) | ~(X6 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20) @ X19)) | ~ ! [X10 : subst,X19 : subst,X20 : subst,X9 : term] : ((X6 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20) @ X19) | ~(X6 @ X10 @ X9 @ (comp @ X20 @ X19))) | ~ ! [X11 : term] : ((X7 @ (sub @ X40 @ (push @ X11 @ id))) | ~(X7 @ X11)) | ~ ! [X8 : term] : (((X7 @ X8)) = ((X6 @ id @ X8 @ id))) | ~ ! [X22 : (subst > term > term)] : (~ ! [X19 : subst,X10 : subst,X9 : term] : (((sub @ (X22 @ X10 @ X9) @ X19)) = ((X22 @ (comp @ X10 @ X19) @ (sub @ X9 @ X19)))) | ~ ! [X9 : term] : (~(X6 @ id @ X9 @ id) | (X6 @ id @ (X22 @ id @ X9) @ id)) | (X6 @ id @ (lam @ (X22 @ sh @ one)) @ id)))) & (! [X9 : term,X11 : term] : ((X9 = X11) | (((sub @ X9 @ sh)) != ((sub @ X11 @ sh)))) <=> shinj) & hoasinduction_lem2v2_gthm & (axvarid <=> ! [X9 : term] : (X9 = ((sub @ X9 @ id)))) & (((comp @ (d2subst @ d_id) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & (! [X10 : subst] : (X10 = ((comp @ id @ X10))) <=> axidl) & (! [X9 : term,X15 : term,X14 : term,X11 : term] : ((((ap @ X9 @ X14)) != ((ap @ X11 @ X15))) | (X9 = X11)) <=> apinj1) & hoasinduction_lem3b_gthm & (((lam @ (d2term @ d_one))) = ((d2term @ d_one))) & ~(hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) & (hoasinduction_lem2 <=> ! [X17 : term,X6 : (subst > term > subst > $o),X18 : term] : ((X6 @ id @ (ap @ X17 @ X18) @ id) | ~ ! [X19 : subst,X10 : subst,X9 : term,X20 : subst] : ((X6 @ X10 @ X9 @ (comp @ X20 @ X19)) | ~(X6 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20) @ X19)) | ~(X6 @ id @ X18 @ id) | ~ ! [X9 : term,X11 : term] : (~(X6 @ id @ X11 @ id) | (X6 @ id @ (ap @ (sub @ X9 @ id) @ X11) @ id) | ~(X6 @ id @ X9 @ id)) | ~(X6 @ id @ X17 @ id) | ~ ! [X10 : subst,X9 : term,X19 : subst,X20 : subst] : ((X6 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20) @ X19) | ~(X6 @ X10 @ X9 @ (comp @ X20 @ X19))))) & (! [X21 : term,X6 : (subst > term > subst > $o)] : (~ ! [X11 : term,X9 : term] : (~(X6 @ id @ X11 @ id) | (X6 @ id @ (ap @ (sub @ X9 @ id) @ X11) @ id) | ~(X6 @ id @ X9 @ id)) | ~ ! [X22 : (subst > term > term)] : (~ ! [X10 : subst,X19 : subst,X9 : term] : (((sub @ (X22 @ X10 @ X9) @ X19)) = ((X22 @ (comp @ X10 @ X19) @ (sub @ X9 @ X19)))) | ~ ! [X9 : term] : (~(X6 @ id @ X9 @ id) | (X6 @ id @ (X22 @ id @ X9) @ id)) | (X6 @ id @ (lam @ (X22 @ sh @ one)) @ id)) | (X6 @ id @ X21 @ id) | ~ ! [X9 : term] : (~(var @ (sub @ X9 @ id)) | (X6 @ id @ X9 @ id)) | ~ ! [X10 : subst,X20 : subst,X19 : subst,X9 : term] : ((X6 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20) @ X19) | ~(X6 @ X10 @ X9 @ (comp @ X20 @ X19))) | ~ ! [X9 : term,X10 : subst,X19 : subst,X20 : subst] : ((X6 @ X10 @ X9 @ (comp @ X20 @ X19)) | ~(X6 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20) @ X19))) <=> hoasinduction) & (hoasinduction_lem3v2_f_lthm <=> ! [X11 : term] : ~ ! [X22 : (subst > term > term)] : ~ ! [X9 : term,X10 : subst] : (((sub @ X11 @ (push @ X9 @ X10))) = ((X22 @ X10 @ X9)))) & ! [X2 : d_subst,X3 : d_subst] : ((((d2subst @ X2)) = ((d2subst @ X3))) => (X2 = X3)) & (! [X6 : (subst > term > subst > $o),X44 : term,X43 : term,X7 : (term > $o)] : ((X7 @ (ap @ X43 @ X44)) | ~ ! [X9 : term,X11 : term] : (~(X6 @ id @ X11 @ id) | (X6 @ id @ (ap @ (sub @ X9 @ id) @ X11) @ id) | ~(X6 @ id @ X9 @ id)) | ~ ! [X8 : term] : (((X7 @ X8)) = ((X6 @ id @ X8 @ id))) | ~ ! [X10 : subst,X20 : subst,X19 : subst,X9 : term] : (~(X6 @ X10 @ X9 @ (comp @ X20 @ X19)) | (X6 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20) @ X19)) | ~(X7 @ X43) | ~(X7 @ X44) | ~ ! [X20 : subst,X19 : subst,X9 : term,X10 : subst] : (~(X6 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20) @ X19) | (X6 @ X10 @ X9 @ (comp @ X20 @ X19)))) <=> hoasinduction_lem2v2) & substmonoid_gthm & hoaslamnotvar_gthm & ulamvarsh & (hoasinduction_lem3aaa <=> ! [X6 : (subst > term > subst > $o),X30 : term] : (~ ! [X22 : (subst > term > term),X31 : term] : (~ ! [X34 : subst,X32 : subst,X33 : term] : (((sub @ (X22 @ X32 @ X33) @ X34)) = ((sub @ (sub @ X31 @ (push @ X33 @ X32)) @ X34))) | ~ ! [X9 : term] : (~(X6 @ id @ X9 @ id) | (X6 @ id @ (X22 @ id @ X9) @ id)) | ~ ! [X35 : subst,X36 : term,X37 : subst] : (((X22 @ (comp @ X35 @ X37) @ (sub @ X36 @ X37))) = ((sub @ X31 @ (push @ (sub @ X36 @ X37) @ (comp @ X35 @ X37))))) | (X6 @ id @ (lam @ (X22 @ sh @ one)) @ id)) | (X6 @ id @ (lam @ (sub @ X30 @ (push @ one @ sh))) @ id) | ~ ! [X11 : term] : (~(X6 @ id @ X11 @ id) | (X6 @ id @ (sub @ X30 @ (push @ X11 @ id)) @ id))))),
  file('/export/starexec/sandbox/benchmark/theBenchmark.p',alg444_1)).
thf(f4,plain,(
  (hoasinduction_lem3 <=> ! [X0 : (subst > term > subst > $o),X1 : term] : ((X0 @ id @ (lam @ X1) @ id) | ~ ! [X2 : term] : ((X0 @ id @ (sub @ X1 @ (push @ X2 @ id)) @ id) | ~(X0 @ id @ X2 @ id)) | ~ ! [X3 : (subst > term > term)] : (~ ! [X4 : term] : (~(X0 @ id @ X4 @ id) | (X0 @ id @ (X3 @ id @ X4) @ id)) | ~ ! [X5 : subst,X6 : subst,X7 : term] : (((X3 @ (comp @ X6 @ X5) @ (sub @ X7 @ X5))) = ((sub @ (X3 @ X6 @ X7) @ X5))) | (X0 @ id @ (lam @ (X3 @ sh @ one)) @ id)) | ~ ! [X8 : subst,X9 : subst,X10 : subst,X11 : term] : (~(X0 @ (comp @ X9 @ X10) @ (sub @ X11 @ X10) @ X8) | (X0 @ X9 @ X11 @ (comp @ X10 @ X8))) | ~ ! [X12 : term,X13 : subst,X14 : subst,X15 : subst] : (~(X0 @ X15 @ X12 @ (comp @ X14 @ X13)) | (X0 @ (comp @ X15 @ X14) @ (sub @ X12 @ X14) @ X13)))) & (hoasinduction_lem3v2_lthm <=> (! [X16 : term] : (((sub @ X16 @ id)) = X16) => ! [X17 : term,X18 : (subst > term > subst > $o),X19 : (term > $o)] : (~ ! [X20 : term] : (((X19 @ X20)) = ((X18 @ id @ X20 @ id))) | (X19 @ (lam @ X17)) | ~ ! [X21 : subst,X22 : subst,X23 : subst,X24 : term] : ((X18 @ (comp @ X22 @ X23) @ (sub @ X24 @ X23) @ X21) | ~(X18 @ X22 @ X24 @ (comp @ X23 @ X21))) | ~ ! [X25 : subst,X26 : term,X27 : subst,X28 : subst] : ((X18 @ X28 @ X26 @ (comp @ X27 @ X25)) | ~(X18 @ (comp @ X28 @ X27) @ (sub @ X26 @ X27) @ X25)) | ~ ! [X29 : term] : (~(X19 @ X29) | (X19 @ (sub @ X17 @ (push @ X29 @ id)))) | ~ ! [X30 : (subst > term > term)] : (~ ! [X31 : subst,X32 : term,X33 : subst] : (((sub @ (X30 @ X33 @ X32) @ X31)) = ((X30 @ (comp @ X33 @ X31) @ (sub @ X32 @ X31)))) | ~ ! [X34 : term] : (~(X18 @ id @ X34 @ id) | (X18 @ id @ (X30 @ id @ X34) @ id)) | (X18 @ id @ (lam @ (X30 @ sh @ one)) @ id))))) & hoasinduction_lem1_gthm & (hoasinduction_lem3v2a_lthm <=> (! [X35 : term] : ~ ! [X36 : (subst > term > term)] : ~ ! [X37 : term,X38 : subst] : (((X36 @ X38 @ X37)) = ((sub @ X35 @ (push @ X37 @ X38)))) => (! [X39 : term] : (((sub @ X39 @ id)) = X39) => ! [X40 : (term > $o),X41 : (subst > term > subst > $o),X42 : term] : (~ ! [X43 : term] : (((X41 @ id @ X43 @ id)) = ((X40 @ X43))) | (X40 @ (lam @ X42)) | ~ ! [X44 : term] : ((X40 @ (sub @ X42 @ (push @ X44 @ id))) | ~(X40 @ X44)) | ~ ! [X45 : (subst > term > term)] : ((X41 @ id @ (lam @ (X45 @ sh @ one)) @ id) | ~ ! [X46 : term] : (~(X41 @ id @ X46 @ id) | (X41 @ id @ (X45 @ id @ X46) @ id)) | ~ ! [X47 : subst,X48 : subst,X49 : term] : (((X45 @ (comp @ X47 @ X48) @ (sub @ X49 @ X48))) = ((sub @ (X45 @ X47 @ X49) @ X48)))))))) & hoaslamnotap_lthm & pushprop_lem1_gthm & axapp & (! [X50 : term,X51 : term,X52 : (subst > term > term)] : ((((lam @ (X52 @ sh @ one))) != ((ap @ (sub @ X50 @ id) @ X51))) | ~ ! [X53 : subst,X54 : subst,X55 : term] : (((sub @ (X52 @ X53 @ X55) @ X54)) = ((X52 @ (comp @ X53 @ X54) @ (sub @ X55 @ X54))))) <=> hoaslamnotap) & hoasapnotvar_lthm & (hoasinduction_lem3b <=> ! [X56 : term] : ~ ! [X57 : (subst > term > term)] : (((sub @ X56 @ (push @ one @ sh))) != ((X57 @ sh @ one)))) & ~lamnotap & hoasinduction_lem1 & hoaslamnotap_gthm & pushprop_lem1v2_gthm & pushprop_lem3v2_lthm & (! [X58 : term,X59 : subst,X60 : (term > $o)] : (~ ! [X61 : term] : (~ ! [X62 : term] : ((X60 @ (sub @ X61 @ (push @ X62 @ id))) | ~(X60 @ X62)) | (X60 @ (lam @ X61))) | ~ ! [X63 : term,X64 : term] : (~(X60 @ X63) | (X60 @ (ap @ X63 @ X64)) | ~(X60 @ X64)) | (X60 @ (sub @ X58 @ X59)) | ~ ! [X65 : term] : (~(var @ X65) | (X60 @ (sub @ X65 @ X59)))) <=> induction2lem) & hoasinduction_lem1_lthm & (hoasinduction_lem0_lthm <=> ! [X66 : (subst > term > subst > $o)] : ~ ! [X67 : (term > $o)] : ~ ! [X68 : term] : (((X67 @ X68)) = ((X66 @ id @ X68 @ id)))) & ! [X69 : d_term] : (d_one = X69) & axabs & axassoc & ~ulamvar1 & hoasapnotvar_gthm & axclos & ((! [X70 : subst] : (((comp @ X70 @ id)) = X70) & ! [X71 : subst] : (((comp @ id @ X71)) = X71)) <=> substmonoid) & pushprop & ! [X72 : d_term,X73 : d_term] : ((((d2term @ X72)) = ((d2term @ X73))) => (X72 = X73)) & (hoasapinj2_lthm <=> (! [X74 : term,X75 : term,X76 : term,X77 : term] : ((((ap @ X77 @ X74)) != ((ap @ X76 @ X75))) | (X74 = X75)) => ! [X78 : term,X79 : term,X80 : term,X81 : term] : ((((ap @ (sub @ X81 @ id) @ X78)) != ((ap @ (sub @ X79 @ id) @ X80))) | (X78 = X80)))) & hoasinduction_lem2_gthm & pushprop_gthm & hoasinduction_lem3a_gthm & (! [X82 : term,X83 : (subst > term > subst > $o)] : (~ ! [X84 : term] : (~(X83 @ id @ X84 @ id) | (X83 @ id @ (sub @ X82 @ (push @ X84 @ id)) @ id)) | (X83 @ id @ (lam @ (sub @ X82 @ (push @ one @ sh))) @ id) | ~ ! [X85 : (subst > term > term)] : ((X83 @ id @ (lam @ (X85 @ sh @ one)) @ id) | ~ ! [X86 : term,X87 : subst,X88 : subst] : (((X85 @ (comp @ X88 @ X87) @ (sub @ X86 @ X87))) = ((sub @ (X85 @ X88 @ X86) @ X87))) | ~ ! [X89 : term] : (~(X83 @ id @ X89 @ id) | (X83 @ id @ (X85 @ id @ X89) @ id)))) <=> hoasinduction_lem3aa) & pushprop_lthm & induction2lem_gthm & ((! [X90 : (term > $o),X91 : term] : (~ ! [X92 : term] : ((X90 @ (lam @ X92)) | ~ ! [X93 : term] : (~(X90 @ X93) | (X90 @ (sub @ X92 @ (push @ X93 @ id))))) | (X90 @ X91) | ~ ! [X94 : term,X95 : term] : (~(X90 @ X94) | ~(X90 @ X95) | (X90 @ (ap @ X94 @ X95))) | ~ ! [X96 : term] : ((X90 @ X96) | ~(var @ X96))) => (! [X97 : term,X98 : (subst > term > subst > $o),X99 : term] : ((X98 @ id @ (ap @ X99 @ X97) @ id) | ~ ! [X100 : subst,X101 : term,X102 : subst,X103 : subst] : (~(X98 @ X102 @ X101 @ (comp @ X103 @ X100)) | (X98 @ (comp @ X102 @ X103) @ (sub @ X101 @ X103) @ X100)) | ~ ! [X104 : subst,X105 : subst,X106 : subst,X107 : term] : ((X98 @ X106 @ X107 @ (comp @ X104 @ X105)) | ~(X98 @ (comp @ X106 @ X104) @ (sub @ X107 @ X104) @ X105)) | ~(X98 @ id @ X97 @ id) | ~ ! [X108 : term,X109 : term] : (~(X98 @ id @ X109 @ id) | ~(X98 @ id @ X108 @ id) | (X98 @ id @ (ap @ (sub @ X108 @ id) @ X109) @ id)) | ~(X98 @ id @ X99 @ id)) => (! [X110 : (subst > term > subst > $o),X111 : term] : (~ ! [X112 : subst,X113 : subst,X114 : subst,X115 : term] : (~(X110 @ X112 @ X115 @ (comp @ X114 @ X113)) | (X110 @ (comp @ X112 @ X114) @ (sub @ X115 @ X114) @ X113)) | ~ ! [X116 : subst,X117 : subst,X118 : term,X119 : subst] : (~(X110 @ (comp @ X119 @ X117) @ (sub @ X118 @ X117) @ X116) | (X110 @ X119 @ X118 @ (comp @ X117 @ X116))) | ~ ! [X120 : (subst > term > term)] : (~ ! [X121 : term] : (~(X110 @ id @ X121 @ id) | (X110 @ id @ (X120 @ id @ X121) @ id)) | ~ ! [X122 : subst,X123 : subst,X124 : term] : (((X120 @ (comp @ X122 @ X123) @ (sub @ X124 @ X123))) = ((sub @ (X120 @ X122 @ X124) @ X123))) | (X110 @ id @ (lam @ (X120 @ sh @ one)) @ id)) | (X110 @ id @ (lam @ X111) @ id) | ~ ! [X125 : term] : (~(X110 @ id @ X125 @ id) | (X110 @ id @ (sub @ X111 @ (push @ X125 @ id)) @ id))) => ! [X126 : (subst > term > subst > $o),X127 : term] : (~ ! [X128 : (subst > term > term)] : (~ ! [X129 : term] : (~(X126 @ id @ X129 @ id) | (X126 @ id @ (X128 @ id @ X129) @ id)) | ~ ! [X130 : term,X131 : subst,X132 : subst] : (((X128 @ (comp @ X131 @ X132) @ (sub @ X130 @ X132))) = ((sub @ (X128 @ X131 @ X130) @ X132))) | (X126 @ id @ (lam @ (X128 @ sh @ one)) @ id)) | (X126 @ id @ X127 @ id) | ~ ! [X133 : subst,X134 : term,X135 : subst,X136 : subst] : ((X126 @ X135 @ X134 @ (comp @ X133 @ X136)) | ~(X126 @ (comp @ X135 @ X133) @ (sub @ X134 @ X133) @ X136)) | ~ ! [X137 : subst,X138 : subst,X139 : term,X140 : subst] : (~(X126 @ X138 @ X139 @ (comp @ X140 @ X137)) | (X126 @ (comp @ X138 @ X140) @ (sub @ X139 @ X140) @ X137)) | ~ ! [X141 : term] : ((X126 @ id @ X141 @ id) | ~(var @ (sub @ X141 @ id))) | ~ ! [X142 : term,X143 : term] : (~(X126 @ id @ X142 @ id) | ~(X126 @ id @ X143 @ id) | (X126 @ id @ (ap @ (sub @ X142 @ id) @ X143) @ id)))))) <=> hoasinduction_lthm) & (! [X144 : term,X145 : term,X146 : term,X147 : term] : ((((ap @ (sub @ X147 @ id) @ X146)) != ((ap @ (sub @ X145 @ id) @ X144))) | (X145 = X147)) <=> hoasapinj1) & (hoasinduction_lem3b_lthm <=> ! [X148 : term] : ~ ! [X149 : (subst > term > term)] : (((sub @ X148 @ (push @ one @ sh))) != ((X149 @ sh @ one)))) & axmap & (((sub @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2term @ d_one))) & (axidr <=> ! [X150 : subst] : (((comp @ X150 @ id)) = X150)) & (! [X151 : term] : (((sub @ X151 @ id)) = X151) <=> termmset) & (hoasinduction_p_and_p_prime = (^[X152 : (subst > term > subst > $o), X153 : (term > $o)] : (! [X154 : term] : (((X153 @ X154)) = ((X152 @ id @ X154 @ id)))))) & pushprop_lem2v2_gthm & termmset_gthm & (axshiftcons <=> ! [X155 : subst,X156 : term] : (((comp @ sh @ (push @ X156 @ X155))) = X155)) & ((! [X157 : term,X158 : subst] : (((sub @ one @ (push @ X157 @ X158))) = X157) => (! [X159 : subst,X160 : term] : (((comp @ sh @ (push @ X160 @ X159))) = X159) => (! [X161 : subst] : (((comp @ X161 @ id)) = X161) => (! [X162 : term,X163 : (term > $o)] : ((X163 @ X162) | ~ ! [X164 : term] : (~(X163 @ X164) | (X163 @ (lam @ X164))) | ~ ! [X165 : term,X166 : term] : (~(X163 @ X166) | (X163 @ (ap @ X165 @ X166)) | ~(X163 @ X165)) | ~ ! [X167 : term] : (~(var @ X167) | (X163 @ X167))) => ! [X168 : subst,X169 : (term > $o),X170 : term] : (~ ! [X171 : term] : ((X169 @ (lam @ X171)) | ~ ! [X172 : term] : ((X169 @ (sub @ X171 @ (push @ X172 @ id))) | ~(X169 @ X172))) | (X169 @ (sub @ X170 @ X168)) | ~ ! [X173 : term,X174 : term] : (~(X169 @ X173) | (X169 @ (ap @ X173 @ X174)) | ~(X169 @ X174)) | ~ ! [X175 : term] : (~(var @ X175) | (X169 @ (sub @ X175 @ X168)))))))) <=> induction2lem_lthm) & (! [X176 : term,X177 : (subst > term > subst > $o)] : (~ ! [X178 : term,X179 : term] : (~(X177 @ id @ X179 @ id) | ~(X177 @ id @ X178 @ id) | (X177 @ id @ (ap @ (sub @ X179 @ id) @ X178) @ id)) | ~ ! [X180 : (subst > term > term)] : ((X177 @ id @ (lam @ (X180 @ sh @ one)) @ id) | ~ ! [X181 : term] : (~(X177 @ id @ X181 @ id) | (X177 @ id @ (X180 @ id @ X181) @ id)) | ~ ! [X182 : subst,X183 : subst,X184 : term] : (((X180 @ (comp @ X182 @ X183) @ (sub @ X184 @ X183))) = ((sub @ (X180 @ X182 @ X184) @ X183)))) | (X177 @ id @ X176 @ id)) <=> hoasinduction_no_psi_cond) & (sh = ((d2subst @ d_id))) & hoaslamnotvar & pushprop_lem2v2_lthm & hoasinduction_lem2_lthm & (((push @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & (! [X185 : term,X186 : term,X187 : term,X188 : term] : ((X187 = X188) | (((ap @ (sub @ X186 @ id) @ X188)) != ((ap @ (sub @ X185 @ id) @ X187)))) <=> hoasapinj2) & (pushprop_p_and_p_prime = (^[X189 : term, X190 : subst, X191 : (term > $o), X192 : (term > $o)] : (! [X193 : term] : (((X191 @ (sub @ X193 @ (push @ X189 @ X190)))) = ((X192 @ X193)))))) & ((! [X194 : term,X195 : subst] : (((sub @ one @ (push @ X194 @ X195))) = X194) => ! [X196 : subst,X197 : (term > $o),X198 : term,X199 : (term > $o)] : ((X197 @ one) | ~(X199 @ X198) | ~ ! [X200 : term] : (((X197 @ X200)) = ((X199 @ (sub @ X200 @ (push @ X198 @ X196))))))) <=> pushprop_lem1v2_lthm) & (hoaslam = (^[X201 : subst, X202 : (subst > term > term)] : ((d2term @ d_one)))) & hoasinduction_gthm & (hoasinduction_lem3v2a <=> ! [X203 : (term > $o),X204 : term,X205 : (subst > term > subst > $o)] : (~ ! [X206 : (subst > term > term)] : ((X205 @ id @ (lam @ (X206 @ sh @ one)) @ id) | ~ ! [X207 : subst,X208 : subst,X209 : term] : (((sub @ (X206 @ X208 @ X209) @ X207)) = ((X206 @ (comp @ X208 @ X207) @ (sub @ X209 @ X207)))) | ~ ! [X210 : term] : (~(X205 @ id @ X210 @ id) | (X205 @ id @ (X206 @ id @ X210) @ id))) | ~ ! [X211 : term] : (~(X203 @ X211) | (X203 @ (sub @ X204 @ (push @ X211 @ id)))) | (X203 @ (lam @ X204)) | ~ ! [X212 : term] : (((X203 @ X212)) = ((X205 @ id @ X212 @ id))))) & (hoasinduction_lem3a <=> ! [X213 : term,X214 : (subst > term > subst > $o)] : (~ ! [X215 : (subst > term > term)] : (~ ! [X216 : subst,X217 : subst,X218 : term] : (((sub @ (X215 @ X216 @ X218) @ X217)) = ((X215 @ (comp @ X216 @ X217) @ (sub @ X218 @ X217)))) | ~ ! [X219 : term] : ((X214 @ id @ (X215 @ id @ X219) @ id) | ~(X214 @ id @ X219 @ id)) | (X214 @ id @ (lam @ (X215 @ sh @ one)) @ id)) | (X214 @ id @ (lam @ X213) @ id) | ~ ! [X220 : term] : (~(X214 @ id @ X220 @ id) | (X214 @ id @ (sub @ X213 @ (push @ X220 @ id)) @ id)))) & (! [X221 : term] : ~ ! [X222 : (subst > term > term)] : ~ ! [X223 : subst,X224 : term] : (((X222 @ X223 @ X224)) = ((sub @ X221 @ (push @ X224 @ X223)))) <=> hoasinduction_lem3v2_f) & axvarshift & (((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) = ((d2term @ d_one))) & ((! [X225 : term] : (((sub @ X225 @ id)) = X225) => (! [X226 : (subst > term > subst > $o),X227 : term] : (~ ! [X228 : term] : (~(X226 @ id @ X228 @ id) | (X226 @ id @ (sub @ X227 @ (push @ X228 @ id)) @ id)) | (X226 @ id @ (lam @ (sub @ X227 @ (push @ one @ sh))) @ id) | ~ ! [X229 : (subst > term > term)] : (~ ! [X230 : term] : (~(X226 @ id @ X230 @ id) | (X226 @ id @ (X229 @ id @ X230) @ id)) | (X226 @ id @ (lam @ (X229 @ sh @ one)) @ id) | ~ ! [X231 : subst,X232 : subst,X233 : term] : (((X229 @ (comp @ X232 @ X231) @ (sub @ X233 @ X231))) = ((sub @ (X229 @ X232 @ X233) @ X231))))) => ! [X234 : term,X235 : (subst > term > subst > $o)] : ((X235 @ id @ (lam @ X234) @ id) | ~ ! [X236 : (subst > term > term)] : (~ ! [X237 : term] : (~(X235 @ id @ X237 @ id) | (X235 @ id @ (X236 @ id @ X237) @ id)) | ~ ! [X238 : subst,X239 : subst,X240 : term] : (((sub @ (X236 @ X238 @ X240) @ X239)) = ((X236 @ (comp @ X238 @ X239) @ (sub @ X240 @ X239)))) | (X235 @ id @ (lam @ (X236 @ sh @ one)) @ id)) | ~ ! [X241 : term] : (~(X235 @ id @ X241 @ id) | (X235 @ id @ (sub @ X234 @ (push @ X241 @ id)) @ id))))) <=> hoasinduction_lem3a_lthm) & (! [X242 : (subst > term > subst > $o),X243 : term] : (~ ! [X244 : (subst > term > term)] : (~ ! [X245 : term] : (~(X242 @ id @ X245 @ id) | (X242 @ id @ (X244 @ id @ X245) @ id)) | ~ ! [X246 : subst,X247 : term,X248 : subst] : (((X244 @ (comp @ X248 @ X246) @ (sub @ X247 @ X246))) = ((sub @ (X244 @ X248 @ X247) @ X246))) | (X242 @ id @ (lam @ (X244 @ sh @ one)) @ id)) | (X242 @ id @ (lam @ (sub @ X243 @ (push @ one @ sh))) @ id) | ~ ! [X249 : term] : (~(X242 @ id @ X249 @ id) | (X242 @ id @ (sub @ X243 @ (push @ X249 @ id)) @ id))) <=> hoasinduction_lem3aa_lthm) & (! [X250 : term,X251 : (term > $o)] : (~ ! [X252 : term] : (~(X251 @ X252) | (X251 @ (lam @ X252))) | (X251 @ X250) | ~ ! [X253 : term,X254 : term] : (~(X251 @ X254) | (X251 @ (ap @ X254 @ X253)) | ~(X251 @ X253)) | ~ ! [X255 : term] : ((X251 @ X255) | ~(var @ X255))) <=> induction) & hoasapinj2_gthm & hoasapnotvar & (! [X256 : term,X257 : term,X258 : subst,X259 : (term > $o),X260 : (term > $o)] : ((X260 @ (sub @ X257 @ (push @ X256 @ X258))) | ~(X259 @ X257)) <=> pushprop_lem1) & pushprop_lem3v2 & (! [X261 : term,X262 : term,X263 : term,X264 : term] : ((X261 = X262) | (((ap @ X263 @ X262)) != ((ap @ X264 @ X261)))) <=> apinj2) & (((ap @ (d2term @ d_one) @ (d2term @ d_one))) = ((d2term @ d_one))) & (induction2_lthm <=> (! [X265 : term] : (((sub @ X265 @ id)) = X265) => (! [X266 : (term > $o),X267 : term,X268 : subst] : (~ ! [X269 : term] : (~(var @ X269) | (X266 @ (sub @ X269 @ X268))) | ~ ! [X270 : term,X271 : term] : (~(X266 @ X271) | ~(X266 @ X270) | (X266 @ (ap @ X271 @ X270))) | ~ ! [X272 : term] : (~ ! [X273 : term] : (~(X266 @ X273) | (X266 @ (sub @ X272 @ (push @ X273 @ id)))) | (X266 @ (lam @ X272))) | (X266 @ (sub @ X267 @ X268))) => ! [X274 : (term > $o),X275 : term] : (~ ! [X276 : term] : ((X274 @ (lam @ X276)) | ~ ! [X277 : term] : ((X274 @ (sub @ X276 @ (push @ X277 @ id))) | ~(X274 @ X277))) | ~ ! [X278 : term,X279 : term] : (~(X274 @ X278) | (X274 @ (ap @ X279 @ X278)) | ~(X274 @ X279)) | ~ ! [X280 : term] : (~(var @ X280) | (X274 @ X280)) | (X274 @ X275))))) & (hoasinduction_lem3_lthm <=> (! [X281 : term] : (((sub @ X281 @ id)) = X281) => (! [X282 : (subst > term > subst > $o),X283 : term] : (~ ! [X284 : term] : ((X282 @ id @ (sub @ X283 @ (push @ X284 @ id)) @ id) | ~(X282 @ id @ X284 @ id)) | (X282 @ id @ (lam @ (sub @ X283 @ (push @ one @ sh))) @ id) | ~ ! [X285 : (subst > term > term)] : (~ ! [X286 : term] : (~(X282 @ id @ X286 @ id) | (X282 @ id @ (X285 @ id @ X286) @ id)) | (X282 @ id @ (lam @ (X285 @ sh @ one)) @ id) | ~ ! [X287 : subst,X288 : term,X289 : subst] : (((sub @ (X285 @ X289 @ X288) @ X287)) = ((X285 @ (comp @ X289 @ X287) @ (sub @ X288 @ X287)))))) => ! [X290 : term,X291 : (subst > term > subst > $o)] : (~ ! [X292 : term,X293 : subst,X294 : subst,X295 : subst] : (~(X291 @ X294 @ X292 @ (comp @ X293 @ X295)) | (X291 @ (comp @ X294 @ X293) @ (sub @ X292 @ X293) @ X295)) | ~ ! [X296 : term] : ((X291 @ id @ (sub @ X290 @ (push @ X296 @ id)) @ id) | ~(X291 @ id @ X296 @ id)) | ~ ! [X297 : subst,X298 : subst,X299 : term,X300 : subst] : (~(X291 @ (comp @ X300 @ X298) @ (sub @ X299 @ X298) @ X297) | (X291 @ X300 @ X299 @ (comp @ X298 @ X297))) | (X291 @ id @ (lam @ X290) @ id) | ~ ! [X301 : (subst > term > term)] : ((X291 @ id @ (lam @ (X301 @ sh @ one)) @ id) | ~ ! [X302 : subst,X303 : subst,X304 : term] : (((sub @ (X301 @ X302 @ X304) @ X303)) = ((X301 @ (comp @ X302 @ X303) @ (sub @ X304 @ X303)))) | ~ ! [X305 : term] : ((X291 @ id @ (X301 @ id @ X305) @ id) | ~(X291 @ id @ X305 @ id))))))) & (hoasapinj1_lthm <=> (! [X306 : term] : (((sub @ X306 @ id)) = X306) => (! [X307 : term,X308 : term,X309 : term,X310 : term] : ((((ap @ X307 @ X310)) != ((ap @ X308 @ X309))) | (X307 = X308)) => ! [X311 : term,X312 : term,X313 : term,X314 : term] : ((((ap @ (sub @ X313 @ id) @ X312)) != ((ap @ (sub @ X311 @ id) @ X314))) | (X311 = X313))))) & hoasapinj1_gthm & ((! [X315 : (subst > term > subst > $o)] : ~ ! [X316 : (term > $o)] : ~ ! [X317 : term] : (((X315 @ id @ X317 @ id)) = ((X316 @ X317))) => (! [X318 : (term > $o),X319 : term] : (~ ! [X320 : term,X321 : term] : (~(X318 @ X320) | ~(X318 @ X321) | (X318 @ (ap @ X320 @ X321))) | (X318 @ X319) | ~ ! [X322 : term] : (~(var @ X322) | (X318 @ X322)) | ~ ! [X323 : term] : ((X318 @ (lam @ X323)) | ~ ! [X324 : term] : (~(X318 @ X324) | (X318 @ (sub @ X323 @ (push @ X324 @ id)))))) => (! [X325 : term] : (((sub @ X325 @ id)) = X325) => (! [X326 : (term > $o),X327 : (subst > term > subst > $o),X328 : term] : (~ ! [X329 : (subst > term > term)] : ((X327 @ id @ (lam @ (X329 @ sh @ one)) @ id) | ~ ! [X330 : subst,X331 : subst,X332 : term] : (((sub @ (X329 @ X331 @ X332) @ X330)) = ((X329 @ (comp @ X331 @ X330) @ (sub @ X332 @ X330)))) | ~ ! [X333 : term] : (~(X327 @ id @ X333 @ id) | (X327 @ id @ (X329 @ id @ X333) @ id))) | ~ ! [X334 : term] : (~(X326 @ X334) | (X326 @ (sub @ X328 @ (push @ X334 @ id)))) | (X326 @ (lam @ X328)) | ~ ! [X335 : term] : (((X326 @ X335)) = ((X327 @ id @ X335 @ id)))) => ! [X336 : (subst > term > subst > $o),X337 : term] : (~ ! [X338 : term,X339 : term] : (~(X336 @ id @ X338 @ id) | ~(X336 @ id @ X339 @ id) | (X336 @ id @ (ap @ (sub @ X339 @ id) @ X338) @ id)) | ~ ! [X340 : subst,X341 : subst,X342 : term,X343 : subst] : ((X336 @ (comp @ X341 @ X340) @ (sub @ X342 @ X340) @ X343) | ~(X336 @ X341 @ X342 @ (comp @ X340 @ X343))) | ~ ! [X344 : (subst > term > term)] : (~ ! [X345 : term] : (~(X336 @ id @ X345 @ id) | (X336 @ id @ (X344 @ id @ X345) @ id)) | ~ ! [X346 : subst,X347 : term,X348 : subst] : (((X344 @ (comp @ X346 @ X348) @ (sub @ X347 @ X348))) = ((sub @ (X344 @ X346 @ X347) @ X348))) | (X336 @ id @ (lam @ (X344 @ sh @ one)) @ id)) | (X336 @ id @ X337 @ id) | ~ ! [X349 : term,X350 : subst,X351 : subst,X352 : subst] : ((X336 @ X352 @ X349 @ (comp @ X351 @ X350)) | ~(X336 @ (comp @ X352 @ X351) @ (sub @ X349 @ X351) @ X350)) | ~ ! [X353 : term] : ((X336 @ id @ X353 @ id) | ~(var @ (sub @ X353 @ id)))))))) <=> hoasinduction_lthm_3) & ulamvarind & induction2_gthm & hoasinduction_lem3_gthm & ! [X354 : d_subst] : (d_id = X354) & hoasinduction_lem3v2_gthm & (id = ((d2subst @ d_id))) & ((! [X355 : subst,X356 : term] : (((sub @ one @ (push @ X356 @ X355))) = X356) => (! [X357 : term,X358 : subst] : (((comp @ sh @ (push @ X357 @ X358))) = X358) => ! [X359 : subst,X360 : term,X361 : term,X362 : (term > $o),X363 : (term > $o)] : ((X362 @ (sub @ X361 @ (push @ X360 @ X359))) | ~(X363 @ X361)))) <=> pushprop_lem1_lthm) & hoasinduction_lem1v2 & ! [X364 : subst] : ? [X365 : d_subst] : (((d2subst @ X365)) = X364) & (! [X366 : subst,X367 : term] : (((sub @ one @ (push @ X367 @ X366))) = X367) <=> axvarcons) & hoasinduction_lem1v2_gthm & pushprop_lem2v2 & (hoasinduction_lem0 <=> ! [X368 : (subst > term > subst > $o)] : ~ ! [X369 : (term > $o)] : ~ ! [X370 : term] : (((X368 @ id @ X370 @ id)) = ((X369 @ X370)))) & (hoaslaminj <=> ! [X371 : (subst > term > term),X372 : term,X373 : subst,X374 : (subst > term > term)] : (~ ! [X375 : term,X376 : subst,X377 : subst] : (((X374 @ (comp @ X376 @ X377) @ (sub @ X375 @ X377))) = ((sub @ (X374 @ X376 @ X375) @ X377))) | (((X371 @ X373 @ X372)) = ((X374 @ X373 @ X372))) | (((lam @ (X374 @ sh @ one))) != ((lam @ (X371 @ sh @ one)))) | ~ ! [X378 : subst,X379 : term,X380 : subst] : (((sub @ (X371 @ X380 @ X379) @ X378)) = ((X371 @ (comp @ X380 @ X378) @ (sub @ X379 @ X378)))))) & pushprop_lthm_orig & (hoasinduction_no_psi_cond_lthm <=> (! [X381 : (subst > term > subst > $o)] : ~ ! [X382 : (term > $o)] : ~ ! [X383 : term] : (((X381 @ id @ X383 @ id)) = ((X382 @ X383))) => (! [X384 : term,X385 : (term > $o)] : (~ ! [X386 : term] : ((X385 @ X386) | ~(var @ X386)) | (X385 @ X384) | ~ ! [X387 : term] : ((X385 @ (lam @ X387)) | ~ ! [X388 : term] : ((X385 @ (sub @ X387 @ (push @ X388 @ id))) | ~(X385 @ X388))) | ~ ! [X389 : term,X390 : term] : (~(X385 @ X390) | (X385 @ (ap @ X390 @ X389)) | ~(X385 @ X389))) => (! [X391 : term] : (((sub @ X391 @ id)) = X391) => (! [X392 : (subst > term > subst > $o),X393 : term,X394 : (term > $o)] : (~ ! [X395 : term] : (~(X394 @ X395) | (X394 @ (sub @ X393 @ (push @ X395 @ id)))) | ~ ! [X396 : term] : (((X392 @ id @ X396 @ id)) = ((X394 @ X396))) | (X394 @ (lam @ X393)) | ~ ! [X397 : (subst > term > term)] : (~ ! [X398 : subst,X399 : term,X400 : subst] : (((sub @ (X397 @ X400 @ X399) @ X398)) = ((X397 @ (comp @ X400 @ X398) @ (sub @ X399 @ X398)))) | (X392 @ id @ (lam @ (X397 @ sh @ one)) @ id) | ~ ! [X401 : term] : (~(X392 @ id @ X401 @ id) | (X392 @ id @ (X397 @ id @ X401) @ id)))) => ! [X402 : (subst > term > subst > $o),X403 : term] : ((X402 @ id @ X403 @ id) | ~ ! [X404 : term,X405 : term] : (~(X402 @ id @ X405 @ id) | (X402 @ id @ (ap @ (sub @ X404 @ id) @ X405) @ id) | ~(X402 @ id @ X404 @ id)) | ~ ! [X406 : (subst > term > term)] : ((X402 @ id @ (lam @ (X406 @ sh @ one)) @ id) | ~ ! [X407 : term] : ((X402 @ id @ (X406 @ id @ X407) @ id) | ~(X402 @ id @ X407 @ id)) | ~ ! [X408 : term,X409 : subst,X410 : subst] : (((sub @ (X406 @ X409 @ X408) @ X410)) = ((X406 @ (comp @ X409 @ X410) @ (sub @ X408 @ X410))))))))))) & (! [X411 : term,X412 : subst,X413 : (term > $o)] : ~ ! [X414 : (term > $o)] : ~ ! [X415 : term] : (((X414 @ X415)) = ((X413 @ (sub @ X415 @ (push @ X411 @ X412))))) <=> pushprop_lem0_lthm) & ! [X416 : term] : ? [X417 : d_term] : (((d2term @ X417)) = X416) & (! [X418 : term,X419 : (term > $o),X420 : subst] : ~ ! [X421 : (term > $o)] : ~ ! [X422 : term] : (((X419 @ (sub @ X422 @ (push @ X418 @ X420)))) = ((X421 @ X422))) <=> pushprop_lem0) & (! [X423 : term,X424 : term] : ((X423 = X424) | (((lam @ X424)) != ((lam @ X423)))) <=> laminj) & pushprop_lem0_gthm & hoaslaminj_gthm & (! [X425 : term,X426 : (term > $o)] : (~ ! [X427 : term,X428 : term] : (~(X426 @ X427) | ~(X426 @ X428) | (X426 @ (ap @ X427 @ X428))) | ~ ! [X429 : term] : ((X426 @ (lam @ X429)) | ~ ! [X430 : term] : ((X426 @ (sub @ X429 @ (push @ X430 @ id))) | ~(X426 @ X430))) | (X426 @ X425) | ~ ! [X431 : term] : ((X426 @ X431) | ~(var @ X431))) <=> induction2) & (substmonoid_lthm <=> (! [X432 : subst] : (((comp @ id @ X432)) = X432) => (! [X433 : subst] : (((comp @ X433 @ id)) = X433) => (! [X434 : subst] : (((comp @ id @ X434)) = X434) & ! [X435 : subst] : (((comp @ X435 @ id)) = X435))))) & apnotvar & (one = ((d2term @ d_one))) & lamnotvar & ~(var @ (d2term @ d_one)) & (! [X436 : subst] : (((push @ (sub @ one @ X436) @ (comp @ sh @ X436))) = X436) <=> axscons) & ((! [X437 : term,X438 : subst] : (((sub @ one @ (push @ X437 @ X438))) = X437) => (! [X439 : term,X440 : subst] : (((comp @ sh @ (push @ X439 @ X440))) = X440) => (! [X441 : term,X442 : term] : ((((lam @ X442)) != ((lam @ X441))) | (X441 = X442)) => ! [X443 : (subst > term > term),X444 : term,X445 : subst,X446 : (subst > term > term)] : (~ ! [X447 : subst,X448 : subst,X449 : term] : (((sub @ (X446 @ X448 @ X449) @ X447)) = ((X446 @ (comp @ X448 @ X447) @ (sub @ X449 @ X447)))) | (((lam @ (X446 @ sh @ one))) != ((lam @ (X443 @ sh @ one)))) | ~ ! [X450 : term,X451 : subst,X452 : subst] : (((sub @ (X443 @ X451 @ X450) @ X452)) = ((X443 @ (comp @ X451 @ X452) @ (sub @ X450 @ X452)))) | (((X443 @ X445 @ X444)) = ((X446 @ X445 @ X444))))))) <=> hoaslaminj_lthm) & (! [X453 : (term > $o),X454 : term,X455 : subst,X456 : (term > $o)] : (~ ! [X457 : term] : (((X453 @ X457)) = ((X456 @ (sub @ X457 @ (push @ X454 @ X455))))) | (X453 @ one) | ~(X456 @ X454)) <=> pushprop_lem1v2) & (termmset_lthm <=> (! [X458 : term] : (((sub @ X458 @ id)) = X458) => ! [X459 : term] : (((sub @ X459 @ id)) = X459))) & hoaslamnotvar_lthm & (hoasinduction_lem3v2 <=> ! [X460 : (subst > term > subst > $o),X461 : (term > $o),X462 : term] : ((X461 @ (lam @ X462)) | ~ ! [X463 : term,X464 : subst,X465 : subst,X466 : subst] : ((X460 @ X466 @ X463 @ (comp @ X464 @ X465)) | ~(X460 @ (comp @ X466 @ X464) @ (sub @ X463 @ X464) @ X465)) | ~ ! [X467 : subst,X468 : subst,X469 : subst,X470 : term] : ((X460 @ (comp @ X467 @ X469) @ (sub @ X470 @ X469) @ X468) | ~(X460 @ X467 @ X470 @ (comp @ X469 @ X468))) | ~ ! [X471 : term] : ((X461 @ (sub @ X462 @ (push @ X471 @ id))) | ~(X461 @ X471)) | ~ ! [X472 : term] : (((X460 @ id @ X472 @ id)) = ((X461 @ X472))) | ~ ! [X473 : (subst > term > term)] : (~ ! [X474 : subst,X475 : subst,X476 : term] : (((sub @ (X473 @ X475 @ X476) @ X474)) = ((X473 @ (comp @ X475 @ X474) @ (sub @ X476 @ X474)))) | ~ ! [X477 : term] : (~(X460 @ id @ X477 @ id) | (X460 @ id @ (X473 @ id @ X477) @ id)) | (X460 @ id @ (lam @ (X473 @ sh @ one)) @ id)))) & (! [X478 : term,X479 : term] : ((X478 = X479) | (((sub @ X478 @ sh)) != ((sub @ X479 @ sh)))) <=> shinj) & hoasinduction_lem2v2_gthm & (axvarid <=> ! [X480 : term] : (((sub @ X480 @ id)) = X480)) & (((comp @ (d2subst @ d_id) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & (! [X481 : subst] : (((comp @ id @ X481)) = X481) <=> axidl) & (! [X482 : term,X483 : term,X484 : term,X485 : term] : ((((ap @ X482 @ X484)) != ((ap @ X485 @ X483))) | (X482 = X485)) <=> apinj1) & hoasinduction_lem3b_gthm & (((lam @ (d2term @ d_one))) = ((d2term @ d_one))) & ~(hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) & (hoasinduction_lem2 <=> ! [X486 : term,X487 : (subst > term > subst > $o),X488 : term] : ((X487 @ id @ (ap @ X486 @ X488) @ id) | ~ ! [X489 : subst,X490 : subst,X491 : term,X492 : subst] : ((X487 @ X490 @ X491 @ (comp @ X492 @ X489)) | ~(X487 @ (comp @ X490 @ X492) @ (sub @ X491 @ X492) @ X489)) | ~(X487 @ id @ X488 @ id) | ~ ! [X493 : term,X494 : term] : (~(X487 @ id @ X494 @ id) | (X487 @ id @ (ap @ (sub @ X493 @ id) @ X494) @ id) | ~(X487 @ id @ X493 @ id)) | ~(X487 @ id @ X486 @ id) | ~ ! [X495 : subst,X496 : term,X497 : subst,X498 : subst] : ((X487 @ (comp @ X495 @ X498) @ (sub @ X496 @ X498) @ X497) | ~(X487 @ X495 @ X496 @ (comp @ X498 @ X497))))) & (! [X499 : term,X500 : (subst > term > subst > $o)] : (~ ! [X501 : term,X502 : term] : (~(X500 @ id @ X501 @ id) | (X500 @ id @ (ap @ (sub @ X502 @ id) @ X501) @ id) | ~(X500 @ id @ X502 @ id)) | ~ ! [X503 : (subst > term > term)] : (~ ! [X504 : subst,X505 : subst,X506 : term] : (((X503 @ (comp @ X504 @ X505) @ (sub @ X506 @ X505))) = ((sub @ (X503 @ X504 @ X506) @ X505))) | ~ ! [X507 : term] : (~(X500 @ id @ X507 @ id) | (X500 @ id @ (X503 @ id @ X507) @ id)) | (X500 @ id @ (lam @ (X503 @ sh @ one)) @ id)) | (X500 @ id @ X499 @ id) | ~ ! [X508 : term] : (~(var @ (sub @ X508 @ id)) | (X500 @ id @ X508 @ id)) | ~ ! [X509 : subst,X510 : subst,X511 : subst,X512 : term] : ((X500 @ (comp @ X509 @ X510) @ (sub @ X512 @ X510) @ X511) | ~(X500 @ X509 @ X512 @ (comp @ X510 @ X511))) | ~ ! [X513 : term,X514 : subst,X515 : subst,X516 : subst] : ((X500 @ X514 @ X513 @ (comp @ X516 @ X515)) | ~(X500 @ (comp @ X514 @ X516) @ (sub @ X513 @ X516) @ X515))) <=> hoasinduction) & (hoasinduction_lem3v2_f_lthm <=> ! [X517 : term] : ~ ! [X518 : (subst > term > term)] : ~ ! [X519 : term,X520 : subst] : (((X518 @ X520 @ X519)) = ((sub @ X517 @ (push @ X519 @ X520))))) & ! [X521 : d_subst,X522 : d_subst] : ((((d2subst @ X522)) = ((d2subst @ X521))) => (X521 = X522)) & (! [X523 : (subst > term > subst > $o),X524 : term,X525 : term,X526 : (term > $o)] : ((X526 @ (ap @ X525 @ X524)) | ~ ! [X527 : term,X528 : term] : (~(X523 @ id @ X528 @ id) | (X523 @ id @ (ap @ (sub @ X527 @ id) @ X528) @ id) | ~(X523 @ id @ X527 @ id)) | ~ ! [X529 : term] : (((X523 @ id @ X529 @ id)) = ((X526 @ X529))) | ~ ! [X530 : subst,X531 : subst,X532 : subst,X533 : term] : (~(X523 @ X530 @ X533 @ (comp @ X531 @ X532)) | (X523 @ (comp @ X530 @ X531) @ (sub @ X533 @ X531) @ X532)) | ~(X526 @ X525) | ~(X526 @ X524) | ~ ! [X534 : subst,X535 : subst,X536 : term,X537 : subst] : (~(X523 @ (comp @ X537 @ X534) @ (sub @ X536 @ X534) @ X535) | (X523 @ X537 @ X536 @ (comp @ X534 @ X535)))) <=> hoasinduction_lem2v2) & substmonoid_gthm & hoaslamnotvar_gthm & ulamvarsh & (hoasinduction_lem3aaa <=> ! [X538 : (subst > term > subst > $o),X539 : term] : (~ ! [X540 : (subst > term > term),X541 : term] : (~ ! [X542 : subst,X543 : subst,X544 : term] : (((sub @ (X540 @ X543 @ X544) @ X542)) = ((sub @ (sub @ X541 @ (push @ X544 @ X543)) @ X542))) | ~ ! [X545 : term] : (~(X538 @ id @ X545 @ id) | (X538 @ id @ (X540 @ id @ X545) @ id)) | ~ ! [X546 : subst,X547 : term,X548 : subst] : (((X540 @ (comp @ X546 @ X548) @ (sub @ X547 @ X548))) = ((sub @ X541 @ (push @ (sub @ X547 @ X548) @ (comp @ X546 @ X548))))) | (X538 @ id @ (lam @ (X540 @ sh @ one)) @ id)) | (X538 @ id @ (lam @ (sub @ X539 @ (push @ one @ sh))) @ id) | ~ ! [X549 : term] : (~(X538 @ id @ X549 @ id) | (X538 @ id @ (sub @ X539 @ (push @ X549 @ id)) @ id))))),
  inference(rectify,[],[f3])).
thf(f5,plain,(
  ((hoasinduction_lem3 = $true) <=> ! [X0 : (subst > term > subst > $o),X1 : term] : ((((X0 @ id @ (lam @ X1) @ id)) = $true) | ~ ! [X2 : term] : ((((X0 @ id @ (sub @ X1 @ (push @ X2 @ id)) @ id)) = $true) | ~ (((X0 @ id @ X2 @ id)) = $true)) | ~ ! [X3 : (subst > term > term)] : (~ ! [X4 : term] : (~ (((X0 @ id @ X4 @ id)) = $true) | (((X0 @ id @ (X3 @ id @ X4) @ id)) = $true)) | ~ ! [X5 : subst,X7 : term,X6 : subst] : (((X3 @ (comp @ X6 @ X5) @ (sub @ X7 @ X5))) = ((sub @ (X3 @ X6 @ X7) @ X5))) | (((X0 @ id @ (lam @ (X3 @ sh @ one)) @ id)) = $true)) | ~ ! [X8 : subst,X9 : subst,X10 : subst,X11 : term] : (~ (((X0 @ (comp @ X9 @ X10) @ (sub @ X11 @ X10) @ X8)) = $true) | ($true = ((X0 @ X9 @ X11 @ (comp @ X10 @ X8))))) | ~ ! [X12 : term,X13 : subst,X14 : subst,X15 : subst] : (~ (((X0 @ X15 @ X12 @ (comp @ X14 @ X13))) = $true) | (((X0 @ (comp @ X15 @ X14) @ (sub @ X12 @ X14) @ X13)) = $true)))) & ((hoasinduction_lem3v2_lthm = $true) <=> (! [X16 : term] : (((sub @ X16 @ id)) = X16) => ! [X17 : term,X18 : (subst > term > subst > $o),X19 : (term > $o)] : (~ ! [X20 : term] : (((X19 @ X20)) = ((X18 @ id @ X20 @ id))) | (((X19 @ (lam @ X17))) = $true) | ~ ! [X21 : subst,X22 : subst,X23 : subst,X24 : term] : ((((X18 @ (comp @ X22 @ X23) @ (sub @ X24 @ X23) @ X21)) = $true) | ~ (((X18 @ X22 @ X24 @ (comp @ X23 @ X21))) = $true)) | ~ ! [X25 : subst,X26 : term,X27 : subst,X28 : subst] : ((((X18 @ X28 @ X26 @ (comp @ X27 @ X25))) = $true) | ~ ($true = ((X18 @ (comp @ X28 @ X27) @ (sub @ X26 @ X27) @ X25)))) | ~ ! [X29 : term] : (~ (((X19 @ X29)) = $true) | (((X19 @ (sub @ X17 @ (push @ X29 @ id)))) = $true)) | ~ ! [X30 : (subst > term > term)] : (~ ! [X31 : subst,X33 : subst,X32 : term] : (((sub @ (X30 @ X33 @ X32) @ X31)) = ((X30 @ (comp @ X33 @ X31) @ (sub @ X32 @ X31)))) | ~ ! [X34 : term] : (~ (((X18 @ id @ X34 @ id)) = $true) | (((X18 @ id @ (X30 @ id @ X34) @ id)) = $true)) | (((X18 @ id @ (lam @ (X30 @ sh @ one)) @ id)) = $true))))) & (hoasinduction_lem1_gthm = $true) & ((hoasinduction_lem3v2a_lthm = $true) <=> (! [X35 : term] : ~ ! [X36 : (subst > term > term)] : ~ ! [X38 : subst,X37 : term] : (((X36 @ X38 @ X37)) = ((sub @ X35 @ (push @ X37 @ X38)))) => (! [X39 : term] : (((sub @ X39 @ id)) = X39) => ! [X40 : (term > $o),X41 : (subst > term > subst > $o),X42 : term] : (~ ! [X43 : term] : (((X41 @ id @ X43 @ id)) = ((X40 @ X43))) | (((X40 @ (lam @ X42))) = $true) | ~ ! [X44 : term] : ((((X40 @ (sub @ X42 @ (push @ X44 @ id)))) = $true) | ~ (((X40 @ X44)) = $true)) | ~ ! [X45 : (subst > term > term)] : ((((X41 @ id @ (lam @ (X45 @ sh @ one)) @ id)) = $true) | ~ ! [X46 : term] : (~ (((X41 @ id @ X46 @ id)) = $true) | (((X41 @ id @ (X45 @ id @ X46) @ id)) = $true)) | ~ ! [X47 : subst,X49 : term,X48 : subst] : (((X45 @ (comp @ X47 @ X48) @ (sub @ X49 @ X48))) = ((sub @ (X45 @ X47 @ X49) @ X48)))))))) & (hoaslamnotap_lthm = $true) & (pushprop_lem1_gthm = $true) & (axapp = $true) & ((hoaslamnotap = $true) <=> ! [X52 : (subst > term > term),X50 : term,X51 : term] : ((((lam @ (X52 @ sh @ one))) != ((ap @ (sub @ X50 @ id) @ X51))) | ~ ! [X55 : term,X54 : subst,X53 : subst] : (((sub @ (X52 @ X53 @ X55) @ X54)) = ((X52 @ (comp @ X53 @ X54) @ (sub @ X55 @ X54)))))) & (hoasapnotvar_lthm = $true) & ((hoasinduction_lem3b = $true) <=> ! [X56 : term] : ~ ! [X57 : (subst > term > term)] : (((sub @ X56 @ (push @ one @ sh))) != ((X57 @ sh @ one)))) & ~ (lamnotap = $true) & (hoasinduction_lem1 = $true) & (hoaslamnotap_gthm = $true) & (pushprop_lem1v2_gthm = $true) & (pushprop_lem3v2_lthm = $true) & (! [X58 : term,X59 : subst,X60 : (term > $o)] : (~ ! [X61 : term] : (~ ! [X62 : term] : ((((X60 @ (sub @ X61 @ (push @ X62 @ id)))) = $true) | ~ (((X60 @ X62)) = $true)) | (((X60 @ (lam @ X61))) = $true)) | ~ ! [X63 : term,X64 : term] : (~ (((X60 @ X63)) = $true) | (((X60 @ (ap @ X63 @ X64))) = $true) | ~ (((X60 @ X64)) = $true)) | (((X60 @ (sub @ X58 @ X59))) = $true) | ~ ! [X65 : term] : (~ (((var @ X65)) = $true) | (((X60 @ (sub @ X65 @ X59))) = $true))) <=> (induction2lem = $true)) & (hoasinduction_lem1_lthm = $true) & ((hoasinduction_lem0_lthm = $true) <=> ! [X66 : (subst > term > subst > $o)] : ~ ! [X67 : (term > $o)] : ~ ! [X68 : term] : (((X67 @ X68)) = ((X66 @ id @ X68 @ id)))) & ! [X69 : d_term] : (d_one = X69) & (axabs = $true) & (axassoc = $true) & ~ (ulamvar1 = $true) & (hoasapnotvar_gthm = $true) & (axclos = $true) & ((substmonoid = $true) <=> (! [X71 : subst] : (((comp @ id @ X71)) = X71) & ! [X70 : subst] : (((comp @ X70 @ id)) = X70))) & (pushprop = $true) & ! [X72 : d_term,X73 : d_term] : ((((d2term @ X72)) = ((d2term @ X73))) => (X72 = X73)) & ((hoasapinj2_lthm = $true) <=> (! [X77 : term,X76 : term,X74 : term,X75 : term] : ((X74 = X75) | (((ap @ X77 @ X74)) != ((ap @ X76 @ X75)))) => ! [X80 : term,X81 : term,X79 : term,X78 : term] : ((((ap @ (sub @ X81 @ id) @ X78)) != ((ap @ (sub @ X79 @ id) @ X80))) | (X78 = X80)))) & (hoasinduction_lem2_gthm = $true) & (pushprop_gthm = $true) & (hoasinduction_lem3a_gthm = $true) & (! [X82 : term,X83 : (subst > term > subst > $o)] : (~ ! [X84 : term] : (~ (((X83 @ id @ X84 @ id)) = $true) | (((X83 @ id @ (sub @ X82 @ (push @ X84 @ id)) @ id)) = $true)) | (((X83 @ id @ (lam @ (sub @ X82 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X85 : (subst > term > term)] : (($true = ((X83 @ id @ (lam @ (X85 @ sh @ one)) @ id))) | ~ ! [X88 : subst,X87 : subst,X86 : term] : (((X85 @ (comp @ X88 @ X87) @ (sub @ X86 @ X87))) = ((sub @ (X85 @ X88 @ X86) @ X87))) | ~ ! [X89 : term] : (~ (((X83 @ id @ X89 @ id)) = $true) | (((X83 @ id @ (X85 @ id @ X89) @ id)) = $true)))) <=> (hoasinduction_lem3aa = $true)) & (pushprop_lthm = $true) & (induction2lem_gthm = $true) & ((! [X90 : (term > $o),X91 : term] : (~ ! [X92 : term] : ((((X90 @ (lam @ X92))) = $true) | ~ ! [X93 : term] : (~ (((X90 @ X93)) = $true) | (((X90 @ (sub @ X92 @ (push @ X93 @ id)))) = $true))) | (((X90 @ X91)) = $true) | ~ ! [X94 : term,X95 : term] : (~ (((X90 @ X94)) = $true) | ~ (((X90 @ X95)) = $true) | ($true = ((X90 @ (ap @ X94 @ X95))))) | ~ ! [X96 : term] : ((((X90 @ X96)) = $true) | ~ (((var @ X96)) = $true))) => (! [X97 : term,X98 : (subst > term > subst > $o),X99 : term] : ((((X98 @ id @ (ap @ X99 @ X97) @ id)) = $true) | ~ ! [X100 : subst,X101 : term,X102 : subst,X103 : subst] : (~ (((X98 @ X102 @ X101 @ (comp @ X103 @ X100))) = $true) | (((X98 @ (comp @ X102 @ X103) @ (sub @ X101 @ X103) @ X100)) = $true)) | ~ ! [X104 : subst,X105 : subst,X106 : subst,X107 : term] : ((((X98 @ X106 @ X107 @ (comp @ X104 @ X105))) = $true) | ~ (((X98 @ (comp @ X106 @ X104) @ (sub @ X107 @ X104) @ X105)) = $true)) | ~ ($true = ((X98 @ id @ X97 @ id))) | ~ ! [X108 : term,X109 : term] : (~ (((X98 @ id @ X109 @ id)) = $true) | ~ (((X98 @ id @ X108 @ id)) = $true) | (((X98 @ id @ (ap @ (sub @ X108 @ id) @ X109) @ id)) = $true)) | ~ (((X98 @ id @ X99 @ id)) = $true)) => (! [X110 : (subst > term > subst > $o),X111 : term] : (~ ! [X112 : subst,X113 : subst,X114 : subst,X115 : term] : (~ (((X110 @ X112 @ X115 @ (comp @ X114 @ X113))) = $true) | (((X110 @ (comp @ X112 @ X114) @ (sub @ X115 @ X114) @ X113)) = $true)) | ~ ! [X116 : subst,X117 : subst,X118 : term,X119 : subst] : (~ (((X110 @ (comp @ X119 @ X117) @ (sub @ X118 @ X117) @ X116)) = $true) | (((X110 @ X119 @ X118 @ (comp @ X117 @ X116))) = $true)) | ~ ! [X120 : (subst > term > term)] : (~ ! [X121 : term] : (~ (((X110 @ id @ X121 @ id)) = $true) | (((X110 @ id @ (X120 @ id @ X121) @ id)) = $true)) | ~ ! [X123 : subst,X124 : term,X122 : subst] : (((X120 @ (comp @ X122 @ X123) @ (sub @ X124 @ X123))) = ((sub @ (X120 @ X122 @ X124) @ X123))) | (((X110 @ id @ (lam @ (X120 @ sh @ one)) @ id)) = $true)) | ($true = ((X110 @ id @ (lam @ X111) @ id))) | ~ ! [X125 : term] : (~ ($true = ((X110 @ id @ X125 @ id))) | (((X110 @ id @ (sub @ X111 @ (push @ X125 @ id)) @ id)) = $true))) => ! [X126 : (subst > term > subst > $o),X127 : term] : (~ ! [X128 : (subst > term > term)] : (~ ! [X129 : term] : (~ (((X126 @ id @ X129 @ id)) = $true) | (((X126 @ id @ (X128 @ id @ X129) @ id)) = $true)) | ~ ! [X131 : subst,X130 : term,X132 : subst] : (((X128 @ (comp @ X131 @ X132) @ (sub @ X130 @ X132))) = ((sub @ (X128 @ X131 @ X130) @ X132))) | (((X126 @ id @ (lam @ (X128 @ sh @ one)) @ id)) = $true)) | (((X126 @ id @ X127 @ id)) = $true) | ~ ! [X133 : subst,X134 : term,X135 : subst,X136 : subst] : ((((X126 @ X135 @ X134 @ (comp @ X133 @ X136))) = $true) | ~ (((X126 @ (comp @ X135 @ X133) @ (sub @ X134 @ X133) @ X136)) = $true)) | ~ ! [X137 : subst,X138 : subst,X139 : term,X140 : subst] : (~ (((X126 @ X138 @ X139 @ (comp @ X140 @ X137))) = $true) | (((X126 @ (comp @ X138 @ X140) @ (sub @ X139 @ X140) @ X137)) = $true)) | ~ ! [X141 : term] : ((((X126 @ id @ X141 @ id)) = $true) | ~ (((var @ (sub @ X141 @ id))) = $true)) | ~ ! [X142 : term,X143 : term] : (~ (((X126 @ id @ X142 @ id)) = $true) | ~ (((X126 @ id @ X143 @ id)) = $true) | (((X126 @ id @ (ap @ (sub @ X142 @ id) @ X143) @ id)) = $true)))))) <=> (hoasinduction_lthm = $true)) & ((hoasapinj1 = $true) <=> ! [X144 : term,X147 : term,X146 : term,X145 : term] : ((((ap @ (sub @ X147 @ id) @ X146)) != ((ap @ (sub @ X145 @ id) @ X144))) | (X145 = X147))) & (! [X148 : term] : ~ ! [X149 : (subst > term > term)] : (((sub @ X148 @ (push @ one @ sh))) != ((X149 @ sh @ one))) <=> (hoasinduction_lem3b_lthm = $true)) & (axmap = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((axidr = $true) <=> ! [X150 : subst] : (((comp @ X150 @ id)) = X150)) & ((termmset = $true) <=> ! [X151 : term] : (((sub @ X151 @ id)) = X151)) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (pushprop_lem2v2_gthm = $true) & (termmset_gthm = $true) & ((axshiftcons = $true) <=> ! [X155 : subst,X156 : term] : (((comp @ sh @ (push @ X156 @ X155))) = X155)) & ((! [X157 : term,X158 : subst] : (((sub @ one @ (push @ X157 @ X158))) = X157) => (! [X159 : subst,X160 : term] : (((comp @ sh @ (push @ X160 @ X159))) = X159) => (! [X161 : subst] : (((comp @ X161 @ id)) = X161) => (! [X162 : term,X163 : (term > $o)] : ((((X163 @ X162)) = $true) | ~ ! [X164 : term] : (~ (((X163 @ X164)) = $true) | (((X163 @ (lam @ X164))) = $true)) | ~ ! [X165 : term,X166 : term] : (~ (((X163 @ X166)) = $true) | (((X163 @ (ap @ X165 @ X166))) = $true) | ~ ($true = ((X163 @ X165)))) | ~ ! [X167 : term] : (~ (((var @ X167)) = $true) | (((X163 @ X167)) = $true))) => ! [X168 : subst,X169 : (term > $o),X170 : term] : (~ ! [X171 : term] : ((((X169 @ (lam @ X171))) = $true) | ~ ! [X172 : term] : ((((X169 @ (sub @ X171 @ (push @ X172 @ id)))) = $true) | ~ (((X169 @ X172)) = $true))) | (((X169 @ (sub @ X170 @ X168))) = $true) | ~ ! [X173 : term,X174 : term] : (~ (((X169 @ X173)) = $true) | (((X169 @ (ap @ X173 @ X174))) = $true) | ~ (((X169 @ X174)) = $true)) | ~ ! [X175 : term] : (~ (((var @ X175)) = $true) | (((X169 @ (sub @ X175 @ X168))) = $true))))))) <=> (induction2lem_lthm = $true)) & (! [X176 : term,X177 : (subst > term > subst > $o)] : (~ ! [X178 : term,X179 : term] : (~ (((X177 @ id @ X179 @ id)) = $true) | ~ (((X177 @ id @ X178 @ id)) = $true) | (((X177 @ id @ (ap @ (sub @ X179 @ id) @ X178) @ id)) = $true)) | ~ ! [X180 : (subst > term > term)] : ((((X177 @ id @ (lam @ (X180 @ sh @ one)) @ id)) = $true) | ~ ! [X181 : term] : (~ (((X177 @ id @ X181 @ id)) = $true) | (((X177 @ id @ (X180 @ id @ X181) @ id)) = $true)) | ~ ! [X182 : subst,X183 : subst,X184 : term] : (((X180 @ (comp @ X182 @ X183) @ (sub @ X184 @ X183))) = ((sub @ (X180 @ X182 @ X184) @ X183)))) | (((X177 @ id @ X176 @ id)) = $true)) <=> (hoasinduction_no_psi_cond = $true)) & (((d2subst @ d_id)) = sh) & (hoaslamnotvar = $true) & (pushprop_lem2v2_lthm = $true) & (hoasinduction_lem2_lthm = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (! [X185 : term,X186 : term,X187 : term,X188 : term] : ((X187 = X188) | (((ap @ (sub @ X186 @ id) @ X188)) != ((ap @ (sub @ X185 @ id) @ X187)))) <=> (hoasapinj2 = $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)))))))))))) & ((! [X194 : term,X195 : subst] : (((sub @ one @ (push @ X194 @ X195))) = X194) => ! [X196 : subst,X197 : (term > $o),X198 : term,X199 : (term > $o)] : ((((X197 @ one)) = $true) | ~ (((X199 @ X198)) = $true) | ~ ! [X200 : term] : (((X197 @ X200)) = ((X199 @ (sub @ X200 @ (push @ X198 @ X196))))))) <=> (pushprop_lem1v2_lthm = $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (hoasinduction_gthm = $true) & ((hoasinduction_lem3v2a = $true) <=> ! [X203 : (term > $o),X204 : term,X205 : (subst > term > subst > $o)] : (~ ! [X206 : (subst > term > term)] : ((((X205 @ id @ (lam @ (X206 @ sh @ one)) @ id)) = $true) | ~ ! [X207 : subst,X208 : subst,X209 : term] : (((sub @ (X206 @ X208 @ X209) @ X207)) = ((X206 @ (comp @ X208 @ X207) @ (sub @ X209 @ X207)))) | ~ ! [X210 : term] : (~ (((X205 @ id @ X210 @ id)) = $true) | (((X205 @ id @ (X206 @ id @ X210) @ id)) = $true))) | ~ ! [X211 : term] : (~ (((X203 @ X211)) = $true) | (((X203 @ (sub @ X204 @ (push @ X211 @ id)))) = $true)) | (((X203 @ (lam @ X204))) = $true) | ~ ! [X212 : term] : (((X203 @ X212)) = ((X205 @ id @ X212 @ id))))) & ((hoasinduction_lem3a = $true) <=> ! [X213 : term,X214 : (subst > term > subst > $o)] : (~ ! [X215 : (subst > term > term)] : (~ ! [X216 : subst,X217 : subst,X218 : term] : (((sub @ (X215 @ X216 @ X218) @ X217)) = ((X215 @ (comp @ X216 @ X217) @ (sub @ X218 @ X217)))) | ~ ! [X219 : term] : (($true = ((X214 @ id @ (X215 @ id @ X219) @ id))) | ~ (((X214 @ id @ X219 @ id)) = $true)) | (((X214 @ id @ (lam @ (X215 @ sh @ one)) @ id)) = $true)) | ($true = ((X214 @ id @ (lam @ X213) @ id))) | ~ ! [X220 : term] : (~ (((X214 @ id @ X220 @ id)) = $true) | (((X214 @ id @ (sub @ X213 @ (push @ X220 @ id)) @ id)) = $true)))) & (! [X221 : term] : ~ ! [X222 : (subst > term > term)] : ~ ! [X223 : subst,X224 : term] : (((X222 @ X223 @ X224)) = ((sub @ X221 @ (push @ X224 @ X223)))) <=> (hoasinduction_lem3v2_f = $true)) & (axvarshift = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ((! [X225 : term] : (((sub @ X225 @ id)) = X225) => (! [X226 : (subst > term > subst > $o),X227 : term] : (~ ! [X228 : term] : (~ (((X226 @ id @ X228 @ id)) = $true) | (((X226 @ id @ (sub @ X227 @ (push @ X228 @ id)) @ id)) = $true)) | (((X226 @ id @ (lam @ (sub @ X227 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X229 : (subst > term > term)] : (~ ! [X230 : term] : (~ ($true = ((X226 @ id @ X230 @ id))) | (((X226 @ id @ (X229 @ id @ X230) @ id)) = $true)) | (((X226 @ id @ (lam @ (X229 @ sh @ one)) @ id)) = $true) | ~ ! [X231 : subst,X232 : subst,X233 : term] : (((X229 @ (comp @ X232 @ X231) @ (sub @ X233 @ X231))) = ((sub @ (X229 @ X232 @ X233) @ X231))))) => ! [X234 : term,X235 : (subst > term > subst > $o)] : ((((X235 @ id @ (lam @ X234) @ id)) = $true) | ~ ! [X236 : (subst > term > term)] : (~ ! [X237 : term] : (~ (((X235 @ id @ X237 @ id)) = $true) | (((X235 @ id @ (X236 @ id @ X237) @ id)) = $true)) | ~ ! [X238 : subst,X239 : subst,X240 : term] : (((sub @ (X236 @ X238 @ X240) @ X239)) = ((X236 @ (comp @ X238 @ X239) @ (sub @ X240 @ X239)))) | (((X235 @ id @ (lam @ (X236 @ sh @ one)) @ id)) = $true)) | ~ ! [X241 : term] : (~ (((X235 @ id @ X241 @ id)) = $true) | (((X235 @ id @ (sub @ X234 @ (push @ X241 @ id)) @ id)) = $true))))) <=> (hoasinduction_lem3a_lthm = $true)) & (! [X242 : (subst > term > subst > $o),X243 : term] : (~ ! [X244 : (subst > term > term)] : (~ ! [X245 : term] : (~ (((X242 @ id @ X245 @ id)) = $true) | (((X242 @ id @ (X244 @ id @ X245) @ id)) = $true)) | ~ ! [X246 : subst,X247 : term,X248 : subst] : (((X244 @ (comp @ X248 @ X246) @ (sub @ X247 @ X246))) = ((sub @ (X244 @ X248 @ X247) @ X246))) | (((X242 @ id @ (lam @ (X244 @ sh @ one)) @ id)) = $true)) | (((X242 @ id @ (lam @ (sub @ X243 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X249 : term] : (~ (((X242 @ id @ X249 @ id)) = $true) | (((X242 @ id @ (sub @ X243 @ (push @ X249 @ id)) @ id)) = $true))) <=> (hoasinduction_lem3aa_lthm = $true)) & (! [X250 : term,X251 : (term > $o)] : (~ ! [X252 : term] : (~ (((X251 @ X252)) = $true) | ($true = ((X251 @ (lam @ X252))))) | (((X251 @ X250)) = $true) | ~ ! [X253 : term,X254 : term] : (~ ($true = ((X251 @ X254))) | (((X251 @ (ap @ X254 @ X253))) = $true) | ~ ($true = ((X251 @ X253)))) | ~ ! [X255 : term] : ((((X251 @ X255)) = $true) | ~ ($true = ((var @ X255))))) <=> (induction = $true)) & (hoasapinj2_gthm = $true) & (hoasapnotvar = $true) & (! [X256 : term,X257 : term,X258 : subst,X259 : (term > $o),X260 : (term > $o)] : ((((X260 @ (sub @ X257 @ (push @ X256 @ X258)))) = $true) | ~ (((X259 @ X257)) = $true)) <=> (pushprop_lem1 = $true)) & (pushprop_lem3v2 = $true) & (! [X261 : term,X262 : term,X263 : term,X264 : term] : ((X261 = X262) | (((ap @ X263 @ X262)) != ((ap @ X264 @ X261)))) <=> (apinj2 = $true)) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((induction2_lthm = $true) <=> (! [X265 : term] : (((sub @ X265 @ id)) = X265) => (! [X266 : (term > $o),X267 : term,X268 : subst] : (~ ! [X269 : term] : (~ (((var @ X269)) = $true) | (((X266 @ (sub @ X269 @ X268))) = $true)) | ~ ! [X270 : term,X271 : term] : (~ (((X266 @ X271)) = $true) | ~ (((X266 @ X270)) = $true) | ($true = ((X266 @ (ap @ X271 @ X270))))) | ~ ! [X272 : term] : (~ ! [X273 : term] : (~ (((X266 @ X273)) = $true) | ($true = ((X266 @ (sub @ X272 @ (push @ X273 @ id)))))) | (((X266 @ (lam @ X272))) = $true)) | (((X266 @ (sub @ X267 @ X268))) = $true)) => ! [X274 : (term > $o),X275 : term] : (~ ! [X276 : term] : ((((X274 @ (lam @ X276))) = $true) | ~ ! [X277 : term] : ((((X274 @ (sub @ X276 @ (push @ X277 @ id)))) = $true) | ~ ($true = ((X274 @ X277))))) | ~ ! [X278 : term,X279 : term] : (~ (((X274 @ X278)) = $true) | (((X274 @ (ap @ X279 @ X278))) = $true) | ~ (((X274 @ X279)) = $true)) | ~ ! [X280 : term] : (~ (((var @ X280)) = $true) | (((X274 @ X280)) = $true)) | (((X274 @ X275)) = $true))))) & ((hoasinduction_lem3_lthm = $true) <=> (! [X281 : term] : (((sub @ X281 @ id)) = X281) => (! [X282 : (subst > term > subst > $o),X283 : term] : (~ ! [X284 : term] : (($true = ((X282 @ id @ (sub @ X283 @ (push @ X284 @ id)) @ id))) | ~ (((X282 @ id @ X284 @ id)) = $true)) | (((X282 @ id @ (lam @ (sub @ X283 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X285 : (subst > term > term)] : (~ ! [X286 : term] : (~ (((X282 @ id @ X286 @ id)) = $true) | (((X282 @ id @ (X285 @ id @ X286) @ id)) = $true)) | (((X282 @ id @ (lam @ (X285 @ sh @ one)) @ id)) = $true) | ~ ! [X287 : subst,X288 : term,X289 : subst] : (((sub @ (X285 @ X289 @ X288) @ X287)) = ((X285 @ (comp @ X289 @ X287) @ (sub @ X288 @ X287)))))) => ! [X290 : term,X291 : (subst > term > subst > $o)] : (~ ! [X292 : term,X293 : subst,X294 : subst,X295 : subst] : (~ (((X291 @ X294 @ X292 @ (comp @ X293 @ X295))) = $true) | (((X291 @ (comp @ X294 @ X293) @ (sub @ X292 @ X293) @ X295)) = $true)) | ~ ! [X296 : term] : ((((X291 @ id @ (sub @ X290 @ (push @ X296 @ id)) @ id)) = $true) | ~ (((X291 @ id @ X296 @ id)) = $true)) | ~ ! [X297 : subst,X298 : subst,X299 : term,X300 : subst] : (~ (((X291 @ (comp @ X300 @ X298) @ (sub @ X299 @ X298) @ X297)) = $true) | (((X291 @ X300 @ X299 @ (comp @ X298 @ X297))) = $true)) | (((X291 @ id @ (lam @ X290) @ id)) = $true) | ~ ! [X301 : (subst > term > term)] : ((((X291 @ id @ (lam @ (X301 @ sh @ one)) @ id)) = $true) | ~ ! [X302 : subst,X303 : subst,X304 : term] : (((sub @ (X301 @ X302 @ X304) @ X303)) = ((X301 @ (comp @ X302 @ X303) @ (sub @ X304 @ X303)))) | ~ ! [X305 : term] : ((((X291 @ id @ (X301 @ id @ X305) @ id)) = $true) | ~ (((X291 @ id @ X305 @ id)) = $true))))))) & ((hoasapinj1_lthm = $true) <=> (! [X306 : term] : (((sub @ X306 @ id)) = X306) => (! [X307 : term,X308 : term,X309 : term,X310 : term] : ((((ap @ X307 @ X310)) != ((ap @ X308 @ X309))) | (X307 = X308)) => ! [X311 : term,X312 : term,X313 : term,X314 : term] : ((((ap @ (sub @ X313 @ id) @ X312)) != ((ap @ (sub @ X311 @ id) @ X314))) | (X311 = X313))))) & (hoasapinj1_gthm = $true) & ((! [X315 : (subst > term > subst > $o)] : ~ ! [X316 : (term > $o)] : ~ ! [X317 : term] : (((X315 @ id @ X317 @ id)) = ((X316 @ X317))) => (! [X318 : (term > $o),X319 : term] : (~ ! [X320 : term,X321 : term] : (~ (((X318 @ X320)) = $true) | ~ (((X318 @ X321)) = $true) | (((X318 @ (ap @ X320 @ X321))) = $true)) | (((X318 @ X319)) = $true) | ~ ! [X322 : term] : (~ (((var @ X322)) = $true) | (((X318 @ X322)) = $true)) | ~ ! [X323 : term] : ((((X318 @ (lam @ X323))) = $true) | ~ ! [X324 : term] : (~ (((X318 @ X324)) = $true) | (((X318 @ (sub @ X323 @ (push @ X324 @ id)))) = $true)))) => (! [X325 : term] : (((sub @ X325 @ id)) = X325) => (! [X326 : (term > $o),X327 : (subst > term > subst > $o),X328 : term] : (~ ! [X329 : (subst > term > term)] : ((((X327 @ id @ (lam @ (X329 @ sh @ one)) @ id)) = $true) | ~ ! [X330 : subst,X331 : subst,X332 : term] : (((sub @ (X329 @ X331 @ X332) @ X330)) = ((X329 @ (comp @ X331 @ X330) @ (sub @ X332 @ X330)))) | ~ ! [X333 : term] : (~ (((X327 @ id @ X333 @ id)) = $true) | (((X327 @ id @ (X329 @ id @ X333) @ id)) = $true))) | ~ ! [X334 : term] : (~ (((X326 @ X334)) = $true) | (((X326 @ (sub @ X328 @ (push @ X334 @ id)))) = $true)) | (((X326 @ (lam @ X328))) = $true) | ~ ! [X335 : term] : (((X326 @ X335)) = ((X327 @ id @ X335 @ id)))) => ! [X336 : (subst > term > subst > $o),X337 : term] : (~ ! [X338 : term,X339 : term] : (~ (((X336 @ id @ X338 @ id)) = $true) | ~ (((X336 @ id @ X339 @ id)) = $true) | (((X336 @ id @ (ap @ (sub @ X339 @ id) @ X338) @ id)) = $true)) | ~ ! [X340 : subst,X341 : subst,X342 : term,X343 : subst] : ((((X336 @ (comp @ X341 @ X340) @ (sub @ X342 @ X340) @ X343)) = $true) | ~ (((X336 @ X341 @ X342 @ (comp @ X340 @ X343))) = $true)) | ~ ! [X344 : (subst > term > term)] : (~ ! [X345 : term] : (~ (((X336 @ id @ X345 @ id)) = $true) | (((X336 @ id @ (X344 @ id @ X345) @ id)) = $true)) | ~ ! [X346 : subst,X347 : term,X348 : subst] : (((X344 @ (comp @ X346 @ X348) @ (sub @ X347 @ X348))) = ((sub @ (X344 @ X346 @ X347) @ X348))) | (((X336 @ id @ (lam @ (X344 @ sh @ one)) @ id)) = $true)) | (((X336 @ id @ X337 @ id)) = $true) | ~ ! [X349 : term,X350 : subst,X351 : subst,X352 : subst] : ((((X336 @ X352 @ X349 @ (comp @ X351 @ X350))) = $true) | ~ (((X336 @ (comp @ X352 @ X351) @ (sub @ X349 @ X351) @ X350)) = $true)) | ~ ! [X353 : term] : ((((X336 @ id @ X353 @ id)) = $true) | ~ (((var @ (sub @ X353 @ id))) = $true))))))) <=> (hoasinduction_lthm_3 = $true)) & (ulamvarind = $true) & (induction2_gthm = $true) & (hoasinduction_lem3_gthm = $true) & ! [X354 : d_subst] : (d_id = X354) & (hoasinduction_lem3v2_gthm = $true) & (id = ((d2subst @ d_id))) & ((! [X355 : subst,X356 : term] : (((sub @ one @ (push @ X356 @ X355))) = X356) => (! [X357 : term,X358 : subst] : (((comp @ sh @ (push @ X357 @ X358))) = X358) => ! [X359 : subst,X360 : term,X361 : term,X362 : (term > $o),X363 : (term > $o)] : ((((X362 @ (sub @ X361 @ (push @ X360 @ X359)))) = $true) | ~ (((X363 @ X361)) = $true)))) <=> (pushprop_lem1_lthm = $true)) & (hoasinduction_lem1v2 = $true) & ! [X364 : subst] : ? [X365 : d_subst] : (((d2subst @ X365)) = X364) & (! [X366 : subst,X367 : term] : (((sub @ one @ (push @ X367 @ X366))) = X367) <=> (axvarcons = $true)) & (hoasinduction_lem1v2_gthm = $true) & (pushprop_lem2v2 = $true) & ((hoasinduction_lem0 = $true) <=> ! [X368 : (subst > term > subst > $o)] : ~ ! [X369 : (term > $o)] : ~ ! [X370 : term] : (((X368 @ id @ X370 @ id)) = ((X369 @ X370)))) & ((hoaslaminj = $true) <=> ! [X371 : (subst > term > term),X372 : term,X373 : subst,X374 : (subst > term > term)] : (~ ! [X375 : term,X376 : subst,X377 : subst] : (((X374 @ (comp @ X376 @ X377) @ (sub @ X375 @ X377))) = ((sub @ (X374 @ X376 @ X375) @ X377))) | (((X371 @ X373 @ X372)) = ((X374 @ X373 @ X372))) | (((lam @ (X374 @ sh @ one))) != ((lam @ (X371 @ sh @ one)))) | ~ ! [X378 : subst,X379 : term,X380 : subst] : (((sub @ (X371 @ X380 @ X379) @ X378)) = ((X371 @ (comp @ X380 @ X378) @ (sub @ X379 @ X378)))))) & (pushprop_lthm_orig = $true) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (! [X381 : (subst > term > subst > $o)] : ~ ! [X382 : (term > $o)] : ~ ! [X383 : term] : (((X381 @ id @ X383 @ id)) = ((X382 @ X383))) => (! [X384 : term,X385 : (term > $o)] : (~ ! [X386 : term] : ((((X385 @ X386)) = $true) | ~ (((var @ X386)) = $true)) | (((X385 @ X384)) = $true) | ~ ! [X387 : term] : ((((X385 @ (lam @ X387))) = $true) | ~ ! [X388 : term] : ((((X385 @ (sub @ X387 @ (push @ X388 @ id)))) = $true) | ~ (((X385 @ X388)) = $true))) | ~ ! [X389 : term,X390 : term] : (~ (((X385 @ X390)) = $true) | (((X385 @ (ap @ X390 @ X389))) = $true) | ~ (((X385 @ X389)) = $true))) => (! [X391 : term] : (((sub @ X391 @ id)) = X391) => (! [X392 : (subst > term > subst > $o),X393 : term,X394 : (term > $o)] : (~ ! [X395 : term] : (~ (((X394 @ X395)) = $true) | (((X394 @ (sub @ X393 @ (push @ X395 @ id)))) = $true)) | ~ ! [X396 : term] : (((X392 @ id @ X396 @ id)) = ((X394 @ X396))) | (((X394 @ (lam @ X393))) = $true) | ~ ! [X397 : (subst > term > term)] : (~ ! [X398 : subst,X399 : term,X400 : subst] : (((sub @ (X397 @ X400 @ X399) @ X398)) = ((X397 @ (comp @ X400 @ X398) @ (sub @ X399 @ X398)))) | (((X392 @ id @ (lam @ (X397 @ sh @ one)) @ id)) = $true) | ~ ! [X401 : term] : (~ (((X392 @ id @ X401 @ id)) = $true) | (((X392 @ id @ (X397 @ id @ X401) @ id)) = $true)))) => ! [X402 : (subst > term > subst > $o),X403 : term] : ((((X402 @ id @ X403 @ id)) = $true) | ~ ! [X404 : term,X405 : term] : (~ (((X402 @ id @ X405 @ id)) = $true) | (((X402 @ id @ (ap @ (sub @ X404 @ id) @ X405) @ id)) = $true) | ~ (((X402 @ id @ X404 @ id)) = $true)) | ~ ! [X406 : (subst > term > term)] : ((((X402 @ id @ (lam @ (X406 @ sh @ one)) @ id)) = $true) | ~ ! [X407 : term] : ((((X402 @ id @ (X406 @ id @ X407) @ id)) = $true) | ~ (((X402 @ id @ X407 @ id)) = $true)) | ~ ! [X408 : term,X409 : subst,X410 : subst] : (((sub @ (X406 @ X409 @ X408) @ X410)) = ((X406 @ (comp @ X409 @ X410) @ (sub @ X408 @ X410))))))))))) & (! [X411 : term,X412 : subst,X413 : (term > $o)] : ~ ! [X414 : (term > $o)] : ~ ! [X415 : term] : (((X414 @ X415)) = ((X413 @ (sub @ X415 @ (push @ X411 @ X412))))) <=> (pushprop_lem0_lthm = $true)) & ! [X416 : term] : ? [X417 : d_term] : (((d2term @ X417)) = X416) & (! [X418 : term,X419 : (term > $o),X420 : subst] : ~ ! [X421 : (term > $o)] : ~ ! [X422 : term] : (((X419 @ (sub @ X422 @ (push @ X418 @ X420)))) = ((X421 @ X422))) <=> (pushprop_lem0 = $true)) & (! [X423 : term,X424 : term] : ((X423 = X424) | (((lam @ X424)) != ((lam @ X423)))) <=> (laminj = $true)) & (pushprop_lem0_gthm = $true) & (hoaslaminj_gthm = $true) & (! [X425 : term,X426 : (term > $o)] : (~ ! [X427 : term,X428 : term] : (~ (((X426 @ X427)) = $true) | ~ (((X426 @ X428)) = $true) | (((X426 @ (ap @ X427 @ X428))) = $true)) | ~ ! [X429 : term] : ((((X426 @ (lam @ X429))) = $true) | ~ ! [X430 : term] : ((((X426 @ (sub @ X429 @ (push @ X430 @ id)))) = $true) | ~ (((X426 @ X430)) = $true))) | (((X426 @ X425)) = $true) | ~ ! [X431 : term] : ((((X426 @ X431)) = $true) | ~ ($true = ((var @ X431))))) <=> (induction2 = $true)) & ((substmonoid_lthm = $true) <=> (! [X432 : subst] : (((comp @ id @ X432)) = X432) => (! [X433 : subst] : (((comp @ X433 @ id)) = X433) => (! [X434 : subst] : (((comp @ id @ X434)) = X434) & ! [X435 : subst] : (((comp @ X435 @ id)) = X435))))) & (apnotvar = $true) & (one = ((d2term @ d_one))) & (lamnotvar = $true) & ~ (((var @ (d2term @ d_one))) = $true) & (! [X436 : subst] : (((push @ (sub @ one @ X436) @ (comp @ sh @ X436))) = X436) <=> (axscons = $true)) & ((! [X437 : term,X438 : subst] : (((sub @ one @ (push @ X437 @ X438))) = X437) => (! [X439 : term,X440 : subst] : (((comp @ sh @ (push @ X439 @ X440))) = X440) => (! [X441 : term,X442 : term] : ((((lam @ X442)) != ((lam @ X441))) | (X441 = X442)) => ! [X443 : (subst > term > term),X444 : term,X445 : subst,X446 : (subst > term > term)] : (~ ! [X447 : subst,X448 : subst,X449 : term] : (((sub @ (X446 @ X448 @ X449) @ X447)) = ((X446 @ (comp @ X448 @ X447) @ (sub @ X449 @ X447)))) | (((lam @ (X446 @ sh @ one))) != ((lam @ (X443 @ sh @ one)))) | ~ ! [X450 : term,X451 : subst,X452 : subst] : (((sub @ (X443 @ X451 @ X450) @ X452)) = ((X443 @ (comp @ X451 @ X452) @ (sub @ X450 @ X452)))) | (((X443 @ X445 @ X444)) = ((X446 @ X445 @ X444))))))) <=> (hoaslaminj_lthm = $true)) & (! [X453 : (term > $o),X454 : term,X455 : subst,X456 : (term > $o)] : (~ ! [X457 : term] : (((X453 @ X457)) = ((X456 @ (sub @ X457 @ (push @ X454 @ X455))))) | (((X453 @ one)) = $true) | ~ (((X456 @ X454)) = $true)) <=> (pushprop_lem1v2 = $true)) & ((termmset_lthm = $true) <=> (! [X458 : term] : (((sub @ X458 @ id)) = X458) => ! [X459 : term] : (((sub @ X459 @ id)) = X459))) & (hoaslamnotvar_lthm = $true) & ((hoasinduction_lem3v2 = $true) <=> ! [X460 : (subst > term > subst > $o),X461 : (term > $o),X462 : term] : ((((X461 @ (lam @ X462))) = $true) | ~ ! [X463 : term,X464 : subst,X465 : subst,X466 : subst] : ((((X460 @ X466 @ X463 @ (comp @ X464 @ X465))) = $true) | ~ (((X460 @ (comp @ X466 @ X464) @ (sub @ X463 @ X464) @ X465)) = $true)) | ~ ! [X467 : subst,X468 : subst,X469 : subst,X470 : term] : ((((X460 @ (comp @ X467 @ X469) @ (sub @ X470 @ X469) @ X468)) = $true) | ~ (((X460 @ X467 @ X470 @ (comp @ X469 @ X468))) = $true)) | ~ ! [X471 : term] : ((((X461 @ (sub @ X462 @ (push @ X471 @ id)))) = $true) | ~ (((X461 @ X471)) = $true)) | ~ ! [X472 : term] : (((X460 @ id @ X472 @ id)) = ((X461 @ X472))) | ~ ! [X473 : (subst > term > term)] : (~ ! [X474 : subst,X475 : subst,X476 : term] : (((sub @ (X473 @ X475 @ X476) @ X474)) = ((X473 @ (comp @ X475 @ X474) @ (sub @ X476 @ X474)))) | ~ ! [X477 : term] : (~ (((X460 @ id @ X477 @ id)) = $true) | ($true = ((X460 @ id @ (X473 @ id @ X477) @ id)))) | (((X460 @ id @ (lam @ (X473 @ sh @ one)) @ id)) = $true)))) & (! [X478 : term,X479 : term] : ((X478 = X479) | (((sub @ X478 @ sh)) != ((sub @ X479 @ sh)))) <=> (shinj = $true)) & (hoasinduction_lem2v2_gthm = $true) & ((axvarid = $true) <=> ! [X480 : term] : (((sub @ X480 @ id)) = X480)) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (! [X481 : subst] : (((comp @ id @ X481)) = X481) <=> (axidl = $true)) & (! [X482 : term,X483 : term,X484 : term,X485 : term] : ((((ap @ X482 @ X484)) != ((ap @ X485 @ X483))) | (X482 = X485)) <=> (apinj1 = $true)) & (hoasinduction_lem3b_gthm = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ~ (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) = $true) & ((hoasinduction_lem2 = $true) <=> ! [X486 : term,X487 : (subst > term > subst > $o),X488 : term] : ((((X487 @ id @ (ap @ X486 @ X488) @ id)) = $true) | ~ ! [X489 : subst,X490 : subst,X491 : term,X492 : subst] : (($true = ((X487 @ X490 @ X491 @ (comp @ X492 @ X489)))) | ~ (((X487 @ (comp @ X490 @ X492) @ (sub @ X491 @ X492) @ X489)) = $true)) | ~ (((X487 @ id @ X488 @ id)) = $true) | ~ ! [X493 : term,X494 : term] : (~ (((X487 @ id @ X494 @ id)) = $true) | (((X487 @ id @ (ap @ (sub @ X493 @ id) @ X494) @ id)) = $true) | ~ (((X487 @ id @ X493 @ id)) = $true)) | ~ (((X487 @ id @ X486 @ id)) = $true) | ~ ! [X495 : subst,X496 : term,X497 : subst,X498 : subst] : ((((X487 @ (comp @ X495 @ X498) @ (sub @ X496 @ X498) @ X497)) = $true) | ~ (((X487 @ X495 @ X496 @ (comp @ X498 @ X497))) = $true)))) & (! [X499 : term,X500 : (subst > term > subst > $o)] : (~ ! [X501 : term,X502 : term] : (~ (((X500 @ id @ X501 @ id)) = $true) | (((X500 @ id @ (ap @ (sub @ X502 @ id) @ X501) @ id)) = $true) | ~ (((X500 @ id @ X502 @ id)) = $true)) | ~ ! [X503 : (subst > term > term)] : (~ ! [X504 : subst,X505 : subst,X506 : term] : (((X503 @ (comp @ X504 @ X505) @ (sub @ X506 @ X505))) = ((sub @ (X503 @ X504 @ X506) @ X505))) | ~ ! [X507 : term] : (~ (((X500 @ id @ X507 @ id)) = $true) | ($true = ((X500 @ id @ (X503 @ id @ X507) @ id)))) | (((X500 @ id @ (lam @ (X503 @ sh @ one)) @ id)) = $true)) | (((X500 @ id @ X499 @ id)) = $true) | ~ ! [X508 : term] : (~ ($true = ((var @ (sub @ X508 @ id)))) | (((X500 @ id @ X508 @ id)) = $true)) | ~ ! [X509 : subst,X510 : subst,X511 : subst,X512 : term] : ((((X500 @ (comp @ X509 @ X510) @ (sub @ X512 @ X510) @ X511)) = $true) | ~ ($true = ((X500 @ X509 @ X512 @ (comp @ X510 @ X511))))) | ~ ! [X513 : term,X514 : subst,X515 : subst,X516 : subst] : ((((X500 @ X514 @ X513 @ (comp @ X516 @ X515))) = $true) | ~ (((X500 @ (comp @ X514 @ X516) @ (sub @ X513 @ X516) @ X515)) = $true))) <=> (hoasinduction = $true)) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X517 : term] : ~ ! [X518 : (subst > term > term)] : ~ ! [X519 : term,X520 : subst] : (((X518 @ X520 @ X519)) = ((sub @ X517 @ (push @ X519 @ X520))))) & ! [X521 : d_subst,X522 : d_subst] : ((((d2subst @ X522)) = ((d2subst @ X521))) => (X521 = X522)) & (! [X523 : (subst > term > subst > $o),X524 : term,X525 : term,X526 : (term > $o)] : (($true = ((X526 @ (ap @ X525 @ X524)))) | ~ ! [X527 : term,X528 : term] : (~ (((X523 @ id @ X528 @ id)) = $true) | (((X523 @ id @ (ap @ (sub @ X527 @ id) @ X528) @ id)) = $true) | ~ (((X523 @ id @ X527 @ id)) = $true)) | ~ ! [X529 : term] : (((X523 @ id @ X529 @ id)) = ((X526 @ X529))) | ~ ! [X530 : subst,X531 : subst,X532 : subst,X533 : term] : (~ (((X523 @ X530 @ X533 @ (comp @ X531 @ X532))) = $true) | (((X523 @ (comp @ X530 @ X531) @ (sub @ X533 @ X531) @ X532)) = $true)) | ~ (((X526 @ X525)) = $true) | ~ (((X526 @ X524)) = $true) | ~ ! [X534 : subst,X535 : subst,X536 : term,X537 : subst] : (~ (((X523 @ (comp @ X537 @ X534) @ (sub @ X536 @ X534) @ X535)) = $true) | (((X523 @ X537 @ X536 @ (comp @ X534 @ X535))) = $true))) <=> (hoasinduction_lem2v2 = $true)) & (substmonoid_gthm = $true) & (hoaslamnotvar_gthm = $true) & (ulamvarsh = $true) & ((hoasinduction_lem3aaa = $true) <=> ! [X538 : (subst > term > subst > $o),X539 : term] : (~ ! [X540 : (subst > term > term),X541 : term] : (~ ! [X542 : subst,X543 : subst,X544 : term] : (((sub @ (X540 @ X543 @ X544) @ X542)) = ((sub @ (sub @ X541 @ (push @ X544 @ X543)) @ X542))) | ~ ! [X545 : term] : (~ (((X538 @ id @ X545 @ id)) = $true) | (((X538 @ id @ (X540 @ id @ X545) @ id)) = $true)) | ~ ! [X546 : subst,X547 : term,X548 : subst] : (((X540 @ (comp @ X546 @ X548) @ (sub @ X547 @ X548))) = ((sub @ X541 @ (push @ (sub @ X547 @ X548) @ (comp @ X546 @ X548))))) | (((X538 @ id @ (lam @ (X540 @ sh @ one)) @ id)) = $true)) | (((X538 @ id @ (lam @ (sub @ X539 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X549 : term] : (~ ($true = ((X538 @ id @ X549 @ id))) | (((X538 @ id @ (sub @ X539 @ (push @ X549 @ id)) @ id)) = $true))))),
  inference(fool_elimination,[],[f4])).
thf(f6,plain,(
  ~ (! [X0 : (subst > term > term)] : (! [X1 : term,X2 : subst,X3 : subst] : (((sub @ (X0 @ X2 @ X1) @ X3)) = ((X0 @ (comp @ X2 @ X3) @ (sub @ X1 @ X3)))) => ! [X4 : (subst > term > term)] : (! [X5 : subst,X6 : subst,X7 : term] : (((sub @ (X4 @ X6 @ X7) @ X5)) = ((X4 @ (comp @ X6 @ X5) @ (sub @ X7 @ X5)))) => ((((hoaslam @ id @ (^[X8 : subst, X9 : term] : ((X0 @ X8 @ X9))))) = ((hoaslam @ id @ (^[X10 : subst, X11 : term] : ((X4 @ X10 @ X11)))))) => ! [X12 : term,X13 : subst] : (((X0 @ X13 @ X12)) = ((X4 @ X13 @ X12)))))) = hoaslaminj)),
  inference(rectify,[],[f2])).
thf(f7,plain,(
  ~ (hoaslaminj = ((!! @ (subst > term > term) @ (^[Y0 : subst > term > term]: ((!! @ subst @ (^[Y1 : subst]: (!! @ subst @ (^[Y2 : subst]: (!! @ term @ (^[Y3 : term]: ((sub @ (Y0 @ Y2 @ Y3) @ Y1) = (Y0 @ (comp @ Y2 @ Y1) @ (sub @ Y3 @ Y1))))))))) => (!! @ (subst > term > term) @ (^[Y1 : subst > term > term]: ((!! @ term @ (^[Y2 : term]: (!! @ subst @ (^[Y3 : subst]: (!! @ subst @ (^[Y4 : subst]: ((sub @ (Y1 @ Y3 @ Y2) @ Y4) = (Y1 @ (comp @ Y3 @ Y4) @ (sub @ Y2 @ Y4))))))))) => (((hoaslam @ id @ (^[Y2 : subst]: ((^[Y3 : term]: (Y0 @ Y2 @ Y3))))) = (hoaslam @ id @ (^[Y2 : subst]: ((^[Y3 : term]: (Y1 @ Y2 @ Y3)))))) => (!! @ subst @ (^[Y2 : subst]: (!! @ term @ (^[Y3 : term]: ((Y0 @ Y2 @ Y3) = (Y1 @ Y2 @ Y3)))))))))))))))),
  inference(fool_elimination,[],[f6])).
thf(f8,plain,(
  (hoaslaminj != ((!! @ (subst > term > term) @ (^[Y0 : subst > term > term]: ((!! @ subst @ (^[Y1 : subst]: (!! @ subst @ (^[Y2 : subst]: (!! @ term @ (^[Y3 : term]: ((sub @ (Y0 @ Y2 @ Y3) @ Y1) = (Y0 @ (comp @ Y2 @ Y1) @ (sub @ Y3 @ Y1))))))))) => (!! @ (subst > term > term) @ (^[Y1 : subst > term > term]: ((!! @ term @ (^[Y2 : term]: (!! @ subst @ (^[Y3 : subst]: (!! @ subst @ (^[Y4 : subst]: ((sub @ (Y1 @ Y3 @ Y2) @ Y4) = (Y1 @ (comp @ Y3 @ Y4) @ (sub @ Y2 @ Y4))))))))) => (((hoaslam @ id @ (^[Y2 : subst]: ((^[Y3 : term]: (Y0 @ Y2 @ Y3))))) = (hoaslam @ id @ (^[Y2 : subst]: ((^[Y3 : term]: (Y1 @ Y2 @ Y3)))))) => (!! @ subst @ (^[Y2 : subst]: (!! @ term @ (^[Y3 : term]: ((Y0 @ Y2 @ Y3) = (Y1 @ Y2 @ Y3)))))))))))))))),
  inference(flattening,[],[f7])).
thf(f9,plain,(
  ((hoasinduction_lem3 = $true) <=> ! [X0 : (subst > term > subst > $o),X1 : term] : ((((X0 @ id @ (lam @ X1) @ id)) = $true) | ~ ! [X2 : term] : ((((X0 @ id @ (sub @ X1 @ (push @ X2 @ id)) @ id)) = $true) | ~ (((X0 @ id @ X2 @ id)) = $true)) | ~ ! [X3 : (subst > term > term)] : (~ ! [X4 : term] : (~ (((X0 @ id @ X4 @ id)) = $true) | (((X0 @ id @ (X3 @ id @ X4) @ id)) = $true)) | ~ ! [X5 : subst,X7 : term,X6 : subst] : (((X3 @ (comp @ X6 @ X5) @ (sub @ X7 @ X5))) = ((sub @ (X3 @ X6 @ X7) @ X5))) | (((X0 @ id @ (lam @ (X3 @ sh @ one)) @ id)) = $true)) | ~ ! [X8 : subst,X9 : subst,X10 : subst,X11 : term] : (~ (((X0 @ (comp @ X9 @ X10) @ (sub @ X11 @ X10) @ X8)) = $true) | ($true = ((X0 @ X9 @ X11 @ (comp @ X10 @ X8))))) | ~ ! [X12 : term,X13 : subst,X14 : subst,X15 : subst] : (~ (((X0 @ X15 @ X12 @ (comp @ X14 @ X13))) = $true) | (((X0 @ (comp @ X15 @ X14) @ (sub @ X12 @ X14) @ X13)) = $true)))) & ((hoasinduction_lem3v2_lthm = $true) <=> (! [X16 : term] : (((sub @ X16 @ id)) = X16) => ! [X17 : term,X18 : (subst > term > subst > $o),X19 : (term > $o)] : (~ ! [X20 : term] : (((X19 @ X20)) = ((X18 @ id @ X20 @ id))) | (((X19 @ (lam @ X17))) = $true) | ~ ! [X21 : subst,X22 : subst,X23 : subst,X24 : term] : ((((X18 @ (comp @ X22 @ X23) @ (sub @ X24 @ X23) @ X21)) = $true) | ~ (((X18 @ X22 @ X24 @ (comp @ X23 @ X21))) = $true)) | ~ ! [X25 : subst,X26 : term,X27 : subst,X28 : subst] : ((((X18 @ X28 @ X26 @ (comp @ X27 @ X25))) = $true) | ~ ($true = ((X18 @ (comp @ X28 @ X27) @ (sub @ X26 @ X27) @ X25)))) | ~ ! [X29 : term] : (~ (((X19 @ X29)) = $true) | (((X19 @ (sub @ X17 @ (push @ X29 @ id)))) = $true)) | ~ ! [X30 : (subst > term > term)] : (~ ! [X31 : subst,X33 : subst,X32 : term] : (((sub @ (X30 @ X33 @ X32) @ X31)) = ((X30 @ (comp @ X33 @ X31) @ (sub @ X32 @ X31)))) | ~ ! [X34 : term] : (~ (((X18 @ id @ X34 @ id)) = $true) | (((X18 @ id @ (X30 @ id @ X34) @ id)) = $true)) | (((X18 @ id @ (lam @ (X30 @ sh @ one)) @ id)) = $true))))) & (hoasinduction_lem1_gthm = $true) & ((hoasinduction_lem3v2a_lthm = $true) <=> (! [X35 : term] : ~ ! [X36 : (subst > term > term)] : ~ ! [X38 : subst,X37 : term] : (((X36 @ X38 @ X37)) = ((sub @ X35 @ (push @ X37 @ X38)))) => (! [X39 : term] : (((sub @ X39 @ id)) = X39) => ! [X40 : (term > $o),X41 : (subst > term > subst > $o),X42 : term] : (~ ! [X43 : term] : (((X41 @ id @ X43 @ id)) = ((X40 @ X43))) | (((X40 @ (lam @ X42))) = $true) | ~ ! [X44 : term] : ((((X40 @ (sub @ X42 @ (push @ X44 @ id)))) = $true) | ~ (((X40 @ X44)) = $true)) | ~ ! [X45 : (subst > term > term)] : ((((X41 @ id @ (lam @ (X45 @ sh @ one)) @ id)) = $true) | ~ ! [X46 : term] : (~ (((X41 @ id @ X46 @ id)) = $true) | (((X41 @ id @ (X45 @ id @ X46) @ id)) = $true)) | ~ ! [X47 : subst,X49 : term,X48 : subst] : (((X45 @ (comp @ X47 @ X48) @ (sub @ X49 @ X48))) = ((sub @ (X45 @ X47 @ X49) @ X48)))))))) & (hoaslamnotap_lthm = $true) & (pushprop_lem1_gthm = $true) & (axapp = $true) & ((hoaslamnotap = $true) <=> ! [X52 : (subst > term > term),X50 : term,X51 : term] : ((((lam @ (X52 @ sh @ one))) != ((ap @ (sub @ X50 @ id) @ X51))) | ~ ! [X55 : term,X54 : subst,X53 : subst] : (((sub @ (X52 @ X53 @ X55) @ X54)) = ((X52 @ (comp @ X53 @ X54) @ (sub @ X55 @ X54)))))) & (hoasapnotvar_lthm = $true) & ((hoasinduction_lem3b = $true) <=> ! [X56 : term] : ~ ! [X57 : (subst > term > term)] : (((sub @ X56 @ (push @ one @ sh))) != ((X57 @ sh @ one)))) & ~ (lamnotap = $true) & (hoasinduction_lem1 = $true) & (hoaslamnotap_gthm = $true) & (pushprop_lem1v2_gthm = $true) & (pushprop_lem3v2_lthm = $true) & (! [X58 : term,X59 : subst,X60 : (term > $o)] : (~ ! [X61 : term] : (~ ! [X62 : term] : ((((X60 @ (sub @ X61 @ (push @ X62 @ id)))) = $true) | ~ (((X60 @ X62)) = $true)) | (((X60 @ (lam @ X61))) = $true)) | ~ ! [X63 : term,X64 : term] : (~ (((X60 @ X63)) = $true) | (((X60 @ (ap @ X63 @ X64))) = $true) | ~ (((X60 @ X64)) = $true)) | (((X60 @ (sub @ X58 @ X59))) = $true) | ~ ! [X65 : term] : (~ (((var @ X65)) = $true) | (((X60 @ (sub @ X65 @ X59))) = $true))) <=> (induction2lem = $true)) & (hoasinduction_lem1_lthm = $true) & ((hoasinduction_lem0_lthm = $true) <=> ! [X66 : (subst > term > subst > $o)] : ~ ! [X67 : (term > $o)] : ~ ! [X68 : term] : (((X67 @ X68)) = ((X66 @ id @ X68 @ id)))) & ! [X69 : d_term] : (d_one = X69) & (axabs = $true) & (axassoc = $true) & ~ (ulamvar1 = $true) & (hoasapnotvar_gthm = $true) & (axclos = $true) & ((substmonoid = $true) <=> (! [X71 : subst] : (((comp @ id @ X71)) = X71) & ! [X70 : subst] : (((comp @ X70 @ id)) = X70))) & (pushprop = $true) & ! [X72 : d_term,X73 : d_term] : ((((d2term @ X72)) = ((d2term @ X73))) => (X72 = X73)) & ((hoasapinj2_lthm = $true) <=> (! [X77 : term,X76 : term,X74 : term,X75 : term] : ((X74 = X75) | (((ap @ X77 @ X74)) != ((ap @ X76 @ X75)))) => ! [X80 : term,X81 : term,X79 : term,X78 : term] : ((((ap @ (sub @ X81 @ id) @ X78)) != ((ap @ (sub @ X79 @ id) @ X80))) | (X78 = X80)))) & (hoasinduction_lem2_gthm = $true) & (pushprop_gthm = $true) & (hoasinduction_lem3a_gthm = $true) & (! [X82 : term,X83 : (subst > term > subst > $o)] : (~ ! [X84 : term] : (~ (((X83 @ id @ X84 @ id)) = $true) | (((X83 @ id @ (sub @ X82 @ (push @ X84 @ id)) @ id)) = $true)) | (((X83 @ id @ (lam @ (sub @ X82 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X85 : (subst > term > term)] : (($true = ((X83 @ id @ (lam @ (X85 @ sh @ one)) @ id))) | ~ ! [X88 : subst,X87 : subst,X86 : term] : (((X85 @ (comp @ X88 @ X87) @ (sub @ X86 @ X87))) = ((sub @ (X85 @ X88 @ X86) @ X87))) | ~ ! [X89 : term] : (~ (((X83 @ id @ X89 @ id)) = $true) | (((X83 @ id @ (X85 @ id @ X89) @ id)) = $true)))) <=> (hoasinduction_lem3aa = $true)) & (pushprop_lthm = $true) & (induction2lem_gthm = $true) & ((! [X90 : (term > $o),X91 : term] : (~ ! [X92 : term] : ((((X90 @ (lam @ X92))) = $true) | ~ ! [X93 : term] : (~ (((X90 @ X93)) = $true) | (((X90 @ (sub @ X92 @ (push @ X93 @ id)))) = $true))) | (((X90 @ X91)) = $true) | ~ ! [X94 : term,X95 : term] : (~ (((X90 @ X94)) = $true) | ~ (((X90 @ X95)) = $true) | ($true = ((X90 @ (ap @ X94 @ X95))))) | ~ ! [X96 : term] : ((((X90 @ X96)) = $true) | ~ (((var @ X96)) = $true))) => (! [X97 : term,X98 : (subst > term > subst > $o),X99 : term] : ((((X98 @ id @ (ap @ X99 @ X97) @ id)) = $true) | ~ ! [X100 : subst,X101 : term,X102 : subst,X103 : subst] : (~ (((X98 @ X102 @ X101 @ (comp @ X103 @ X100))) = $true) | (((X98 @ (comp @ X102 @ X103) @ (sub @ X101 @ X103) @ X100)) = $true)) | ~ ! [X104 : subst,X105 : subst,X106 : subst,X107 : term] : ((((X98 @ X106 @ X107 @ (comp @ X104 @ X105))) = $true) | ~ (((X98 @ (comp @ X106 @ X104) @ (sub @ X107 @ X104) @ X105)) = $true)) | ~ ($true = ((X98 @ id @ X97 @ id))) | ~ ! [X108 : term,X109 : term] : (~ (((X98 @ id @ X109 @ id)) = $true) | ~ (((X98 @ id @ X108 @ id)) = $true) | (((X98 @ id @ (ap @ (sub @ X108 @ id) @ X109) @ id)) = $true)) | ~ (((X98 @ id @ X99 @ id)) = $true)) => (! [X110 : (subst > term > subst > $o),X111 : term] : (~ ! [X112 : subst,X113 : subst,X114 : subst,X115 : term] : (~ (((X110 @ X112 @ X115 @ (comp @ X114 @ X113))) = $true) | (((X110 @ (comp @ X112 @ X114) @ (sub @ X115 @ X114) @ X113)) = $true)) | ~ ! [X116 : subst,X117 : subst,X118 : term,X119 : subst] : (~ (((X110 @ (comp @ X119 @ X117) @ (sub @ X118 @ X117) @ X116)) = $true) | (((X110 @ X119 @ X118 @ (comp @ X117 @ X116))) = $true)) | ~ ! [X120 : (subst > term > term)] : (~ ! [X121 : term] : (~ (((X110 @ id @ X121 @ id)) = $true) | (((X110 @ id @ (X120 @ id @ X121) @ id)) = $true)) | ~ ! [X123 : subst,X124 : term,X122 : subst] : (((X120 @ (comp @ X122 @ X123) @ (sub @ X124 @ X123))) = ((sub @ (X120 @ X122 @ X124) @ X123))) | (((X110 @ id @ (lam @ (X120 @ sh @ one)) @ id)) = $true)) | ($true = ((X110 @ id @ (lam @ X111) @ id))) | ~ ! [X125 : term] : (~ ($true = ((X110 @ id @ X125 @ id))) | (((X110 @ id @ (sub @ X111 @ (push @ X125 @ id)) @ id)) = $true))) => ! [X126 : (subst > term > subst > $o),X127 : term] : (~ ! [X128 : (subst > term > term)] : (~ ! [X129 : term] : (~ (((X126 @ id @ X129 @ id)) = $true) | (((X126 @ id @ (X128 @ id @ X129) @ id)) = $true)) | ~ ! [X131 : subst,X130 : term,X132 : subst] : (((X128 @ (comp @ X131 @ X132) @ (sub @ X130 @ X132))) = ((sub @ (X128 @ X131 @ X130) @ X132))) | (((X126 @ id @ (lam @ (X128 @ sh @ one)) @ id)) = $true)) | (((X126 @ id @ X127 @ id)) = $true) | ~ ! [X133 : subst,X134 : term,X135 : subst,X136 : subst] : ((((X126 @ X135 @ X134 @ (comp @ X133 @ X136))) = $true) | ~ (((X126 @ (comp @ X135 @ X133) @ (sub @ X134 @ X133) @ X136)) = $true)) | ~ ! [X137 : subst,X138 : subst,X139 : term,X140 : subst] : (~ (((X126 @ X138 @ X139 @ (comp @ X140 @ X137))) = $true) | (((X126 @ (comp @ X138 @ X140) @ (sub @ X139 @ X140) @ X137)) = $true)) | ~ ! [X141 : term] : ((((X126 @ id @ X141 @ id)) = $true) | ~ (((var @ (sub @ X141 @ id))) = $true)) | ~ ! [X142 : term,X143 : term] : (~ (((X126 @ id @ X142 @ id)) = $true) | ~ (((X126 @ id @ X143 @ id)) = $true) | (((X126 @ id @ (ap @ (sub @ X142 @ id) @ X143) @ id)) = $true)))))) <=> (hoasinduction_lthm = $true)) & ((hoasapinj1 = $true) <=> ! [X144 : term,X147 : term,X146 : term,X145 : term] : ((((ap @ (sub @ X147 @ id) @ X146)) != ((ap @ (sub @ X145 @ id) @ X144))) | (X145 = X147))) & (! [X148 : term] : ~ ! [X149 : (subst > term > term)] : (((sub @ X148 @ (push @ one @ sh))) != ((X149 @ sh @ one))) <=> (hoasinduction_lem3b_lthm = $true)) & (axmap = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((axidr = $true) <=> ! [X150 : subst] : (((comp @ X150 @ id)) = X150)) & ((termmset = $true) <=> ! [X151 : term] : (((sub @ X151 @ id)) = X151)) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (pushprop_lem2v2_gthm = $true) & (termmset_gthm = $true) & (! [X152 : subst,X153 : term] : (((comp @ sh @ (push @ X153 @ X152))) = X152) <=> (axshiftcons = $true)) & ((! [X155 : subst,X154 : term] : (((sub @ one @ (push @ X154 @ X155))) = X154) => (! [X156 : subst,X157 : term] : (((comp @ sh @ (push @ X157 @ X156))) = X156) => (! [X158 : subst] : (((comp @ X158 @ id)) = X158) => (! [X159 : term,X160 : (term > $o)] : ((((X160 @ X159)) = $true) | ~ ! [X161 : term] : (~ (((X160 @ X161)) = $true) | (((X160 @ (lam @ X161))) = $true)) | ~ ! [X162 : term,X163 : term] : (~ (((X160 @ X163)) = $true) | (((X160 @ (ap @ X162 @ X163))) = $true) | ~ (((X160 @ X162)) = $true)) | ~ ! [X164 : term] : (~ (((var @ X164)) = $true) | ($true = ((X160 @ X164))))) => ! [X165 : subst,X166 : (term > $o),X167 : term] : (~ ! [X168 : term] : ((((X166 @ (lam @ X168))) = $true) | ~ ! [X169 : term] : ((((X166 @ (sub @ X168 @ (push @ X169 @ id)))) = $true) | ~ ($true = ((X166 @ X169))))) | (((X166 @ (sub @ X167 @ X165))) = $true) | ~ ! [X170 : term,X171 : term] : (~ (((X166 @ X170)) = $true) | (((X166 @ (ap @ X170 @ X171))) = $true) | ~ (((X166 @ X171)) = $true)) | ~ ! [X172 : term] : (~ (((var @ X172)) = $true) | (((X166 @ (sub @ X172 @ X165))) = $true))))))) <=> (induction2lem_lthm = $true)) & (! [X173 : term,X174 : (subst > term > subst > $o)] : (~ ! [X175 : term,X176 : term] : (~ (((X174 @ id @ X176 @ id)) = $true) | ~ (((X174 @ id @ X175 @ id)) = $true) | (((X174 @ id @ (ap @ (sub @ X176 @ id) @ X175) @ id)) = $true)) | ~ ! [X177 : (subst > term > term)] : ((((X174 @ id @ (lam @ (X177 @ sh @ one)) @ id)) = $true) | ~ ! [X178 : term] : (~ (((X174 @ id @ X178 @ id)) = $true) | (((X174 @ id @ (X177 @ id @ X178) @ id)) = $true)) | ~ ! [X179 : subst,X180 : subst,X181 : term] : (((sub @ (X177 @ X179 @ X181) @ X180)) = ((X177 @ (comp @ X179 @ X180) @ (sub @ X181 @ X180))))) | (((X174 @ id @ X173 @ id)) = $true)) <=> (hoasinduction_no_psi_cond = $true)) & (((d2subst @ d_id)) = sh) & (hoaslamnotvar = $true) & (pushprop_lem2v2_lthm = $true) & (hoasinduction_lem2_lthm = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoasapinj2 = $true) <=> ! [X185 : term,X183 : term,X184 : term,X182 : term] : ((((ap @ (sub @ X182 @ id) @ X184)) != ((ap @ (sub @ X183 @ id) @ X185))) | (X184 = X185))) & (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)))))))))))) & ((! [X187 : subst,X186 : term] : (((sub @ one @ (push @ X186 @ X187))) = X186) => ! [X188 : subst,X189 : (term > $o),X190 : term,X191 : (term > $o)] : ((((X189 @ one)) = $true) | ~ ($true = ((X191 @ X190))) | ~ ! [X192 : term] : (((X191 @ (sub @ X192 @ (push @ X190 @ X188)))) = ((X189 @ X192))))) <=> (pushprop_lem1v2_lthm = $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (hoasinduction_gthm = $true) & ((hoasinduction_lem3v2a = $true) <=> ! [X193 : (term > $o),X194 : term,X195 : (subst > term > subst > $o)] : (~ ! [X196 : (subst > term > term)] : ((((X195 @ id @ (lam @ (X196 @ sh @ one)) @ id)) = $true) | ~ ! [X199 : term,X197 : subst,X198 : subst] : (((X196 @ (comp @ X198 @ X197) @ (sub @ X199 @ X197))) = ((sub @ (X196 @ X198 @ X199) @ X197))) | ~ ! [X200 : term] : (~ (((X195 @ id @ X200 @ id)) = $true) | (((X195 @ id @ (X196 @ id @ X200) @ id)) = $true))) | ~ ! [X201 : term] : (~ (((X193 @ X201)) = $true) | (((X193 @ (sub @ X194 @ (push @ X201 @ id)))) = $true)) | (((X193 @ (lam @ X194))) = $true) | ~ ! [X202 : term] : (((X193 @ X202)) = ((X195 @ id @ X202 @ id))))) & ((hoasinduction_lem3a = $true) <=> ! [X203 : term,X204 : (subst > term > subst > $o)] : (~ ! [X205 : (subst > term > term)] : (~ ! [X208 : term,X207 : subst,X206 : subst] : (((X205 @ (comp @ X206 @ X207) @ (sub @ X208 @ X207))) = ((sub @ (X205 @ X206 @ X208) @ X207))) | ~ ! [X209 : term] : (($true = ((X204 @ id @ (X205 @ id @ X209) @ id))) | ~ (((X204 @ id @ X209 @ id)) = $true)) | (((X204 @ id @ (lam @ (X205 @ sh @ one)) @ id)) = $true)) | ($true = ((X204 @ id @ (lam @ X203) @ id))) | ~ ! [X210 : term] : (~ (((X204 @ id @ X210 @ id)) = $true) | (((X204 @ id @ (sub @ X203 @ (push @ X210 @ id)) @ id)) = $true)))) & ((hoasinduction_lem3v2_f = $true) <=> ! [X211 : term] : ~ ! [X212 : (subst > term > term)] : ~ ! [X213 : subst,X214 : term] : (((X212 @ X213 @ X214)) = ((sub @ X211 @ (push @ X214 @ X213))))) & (axvarshift = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ((! [X215 : term] : (((sub @ X215 @ id)) = X215) => (! [X216 : (subst > term > subst > $o),X217 : term] : (~ ! [X218 : term] : (~ (((X216 @ id @ X218 @ id)) = $true) | (((X216 @ id @ (sub @ X217 @ (push @ X218 @ id)) @ id)) = $true)) | (((X216 @ id @ (lam @ (sub @ X217 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X219 : (subst > term > term)] : (~ ! [X220 : term] : (~ (((X216 @ id @ X220 @ id)) = $true) | (((X216 @ id @ (X219 @ id @ X220) @ id)) = $true)) | (((X216 @ id @ (lam @ (X219 @ sh @ one)) @ id)) = $true) | ~ ! [X221 : subst,X223 : term,X222 : subst] : (((sub @ (X219 @ X222 @ X223) @ X221)) = ((X219 @ (comp @ X222 @ X221) @ (sub @ X223 @ X221)))))) => ! [X224 : term,X225 : (subst > term > subst > $o)] : (($true = ((X225 @ id @ (lam @ X224) @ id))) | ~ ! [X226 : (subst > term > term)] : (~ ! [X227 : term] : (~ (((X225 @ id @ X227 @ id)) = $true) | (((X225 @ id @ (X226 @ id @ X227) @ id)) = $true)) | ~ ! [X229 : subst,X230 : term,X228 : subst] : (((X226 @ (comp @ X228 @ X229) @ (sub @ X230 @ X229))) = ((sub @ (X226 @ X228 @ X230) @ X229))) | (((X225 @ id @ (lam @ (X226 @ sh @ one)) @ id)) = $true)) | ~ ! [X231 : term] : (~ (((X225 @ id @ X231 @ id)) = $true) | ($true = ((X225 @ id @ (sub @ X224 @ (push @ X231 @ id)) @ id))))))) <=> (hoasinduction_lem3a_lthm = $true)) & (! [X232 : (subst > term > subst > $o),X233 : term] : (~ ! [X234 : (subst > term > term)] : (~ ! [X235 : term] : (~ ($true = ((X232 @ id @ X235 @ id))) | (((X232 @ id @ (X234 @ id @ X235) @ id)) = $true)) | ~ ! [X238 : subst,X236 : subst,X237 : term] : (((sub @ (X234 @ X238 @ X237) @ X236)) = ((X234 @ (comp @ X238 @ X236) @ (sub @ X237 @ X236)))) | (((X232 @ id @ (lam @ (X234 @ sh @ one)) @ id)) = $true)) | (((X232 @ id @ (lam @ (sub @ X233 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X239 : term] : (~ (((X232 @ id @ X239 @ id)) = $true) | (((X232 @ id @ (sub @ X233 @ (push @ X239 @ id)) @ id)) = $true))) <=> (hoasinduction_lem3aa_lthm = $true)) & (! [X240 : term,X241 : (term > $o)] : (~ ! [X242 : term] : (~ (((X241 @ X242)) = $true) | (((X241 @ (lam @ X242))) = $true)) | (((X241 @ X240)) = $true) | ~ ! [X243 : term,X244 : term] : (~ (((X241 @ X244)) = $true) | (((X241 @ (ap @ X244 @ X243))) = $true) | ~ (((X241 @ X243)) = $true)) | ~ ! [X245 : term] : ((((X241 @ X245)) = $true) | ~ (((var @ X245)) = $true))) <=> (induction = $true)) & (hoasapinj2_gthm = $true) & (hoasapnotvar = $true) & (! [X246 : term,X247 : term,X248 : subst,X249 : (term > $o),X250 : (term > $o)] : ((((X250 @ (sub @ X247 @ (push @ X246 @ X248)))) = $true) | ~ (((X249 @ X247)) = $true)) <=> (pushprop_lem1 = $true)) & (pushprop_lem3v2 = $true) & (! [X251 : term,X252 : term,X254 : term,X253 : term] : ((((ap @ X254 @ X251)) != ((ap @ X253 @ X252))) | (X251 = X252)) <=> (apinj2 = $true)) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((induction2_lthm = $true) <=> (! [X255 : term] : (((sub @ X255 @ id)) = X255) => (! [X256 : (term > $o),X257 : term,X258 : subst] : (~ ! [X259 : term] : (~ ($true = ((var @ X259))) | (((X256 @ (sub @ X259 @ X258))) = $true)) | ~ ! [X260 : term,X261 : term] : (~ (((X256 @ X261)) = $true) | ~ (((X256 @ X260)) = $true) | (((X256 @ (ap @ X261 @ X260))) = $true)) | ~ ! [X262 : term] : (~ ! [X263 : term] : (~ (((X256 @ X263)) = $true) | (((X256 @ (sub @ X262 @ (push @ X263 @ id)))) = $true)) | (((X256 @ (lam @ X262))) = $true)) | (((X256 @ (sub @ X257 @ X258))) = $true)) => ! [X264 : (term > $o),X265 : term] : (~ ! [X266 : term] : ((((X264 @ (lam @ X266))) = $true) | ~ ! [X267 : term] : ((((X264 @ (sub @ X266 @ (push @ X267 @ id)))) = $true) | ~ (((X264 @ X267)) = $true))) | ~ ! [X268 : term,X269 : term] : (~ (((X264 @ X268)) = $true) | (((X264 @ (ap @ X269 @ X268))) = $true) | ~ (((X264 @ X269)) = $true)) | ~ ! [X270 : term] : (~ (((var @ X270)) = $true) | (((X264 @ X270)) = $true)) | (((X264 @ X265)) = $true))))) & ((hoasinduction_lem3_lthm = $true) <=> (! [X271 : term] : (((sub @ X271 @ id)) = X271) => (! [X272 : (subst > term > subst > $o),X273 : term] : (~ ! [X274 : term] : ((((X272 @ id @ (sub @ X273 @ (push @ X274 @ id)) @ id)) = $true) | ~ (((X272 @ id @ X274 @ id)) = $true)) | (((X272 @ id @ (lam @ (sub @ X273 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X275 : (subst > term > term)] : (~ ! [X276 : term] : (~ (((X272 @ id @ X276 @ id)) = $true) | ($true = ((X272 @ id @ (X275 @ id @ X276) @ id)))) | ($true = ((X272 @ id @ (lam @ (X275 @ sh @ one)) @ id))) | ~ ! [X277 : subst,X278 : term,X279 : subst] : (((X275 @ (comp @ X279 @ X277) @ (sub @ X278 @ X277))) = ((sub @ (X275 @ X279 @ X278) @ X277))))) => ! [X280 : term,X281 : (subst > term > subst > $o)] : (~ ! [X282 : term,X283 : subst,X284 : subst,X285 : subst] : (~ ($true = ((X281 @ X284 @ X282 @ (comp @ X283 @ X285)))) | (((X281 @ (comp @ X284 @ X283) @ (sub @ X282 @ X283) @ X285)) = $true)) | ~ ! [X286 : term] : ((((X281 @ id @ (sub @ X280 @ (push @ X286 @ id)) @ id)) = $true) | ~ (((X281 @ id @ X286 @ id)) = $true)) | ~ ! [X287 : subst,X288 : subst,X289 : term,X290 : subst] : (~ (((X281 @ (comp @ X290 @ X288) @ (sub @ X289 @ X288) @ X287)) = $true) | (((X281 @ X290 @ X289 @ (comp @ X288 @ X287))) = $true)) | (((X281 @ id @ (lam @ X280) @ id)) = $true) | ~ ! [X291 : (subst > term > term)] : ((((X281 @ id @ (lam @ (X291 @ sh @ one)) @ id)) = $true) | ~ ! [X294 : term,X293 : subst,X292 : subst] : (((sub @ (X291 @ X292 @ X294) @ X293)) = ((X291 @ (comp @ X292 @ X293) @ (sub @ X294 @ X293)))) | ~ ! [X295 : term] : ((((X281 @ id @ (X291 @ id @ X295) @ id)) = $true) | ~ (((X281 @ id @ X295 @ id)) = $true))))))) & ((! [X296 : term] : (((sub @ X296 @ id)) = X296) => (! [X299 : term,X298 : term,X297 : term,X300 : term] : ((((ap @ X298 @ X299)) != ((ap @ X297 @ X300))) | (X297 = X298)) => ! [X303 : term,X301 : term,X302 : term,X304 : term] : ((X301 = X303) | (((ap @ (sub @ X303 @ id) @ X302)) != ((ap @ (sub @ X301 @ id) @ X304)))))) <=> (hoasapinj1_lthm = $true)) & (hoasapinj1_gthm = $true) & ((! [X305 : (subst > term > subst > $o)] : ~ ! [X306 : (term > $o)] : ~ ! [X307 : term] : (((X306 @ X307)) = ((X305 @ id @ X307 @ id))) => (! [X308 : (term > $o),X309 : term] : (~ ! [X310 : term,X311 : term] : (~ (((X308 @ X310)) = $true) | ~ (((X308 @ X311)) = $true) | (((X308 @ (ap @ X310 @ X311))) = $true)) | (((X308 @ X309)) = $true) | ~ ! [X312 : term] : (~ (((var @ X312)) = $true) | (((X308 @ X312)) = $true)) | ~ ! [X313 : term] : ((((X308 @ (lam @ X313))) = $true) | ~ ! [X314 : term] : (~ (((X308 @ X314)) = $true) | (((X308 @ (sub @ X313 @ (push @ X314 @ id)))) = $true)))) => (! [X315 : term] : (((sub @ X315 @ id)) = X315) => (! [X316 : (term > $o),X317 : (subst > term > subst > $o),X318 : term] : (~ ! [X319 : (subst > term > term)] : ((((X317 @ id @ (lam @ (X319 @ sh @ one)) @ id)) = $true) | ~ ! [X321 : subst,X320 : subst,X322 : term] : (((X319 @ (comp @ X321 @ X320) @ (sub @ X322 @ X320))) = ((sub @ (X319 @ X321 @ X322) @ X320))) | ~ ! [X323 : term] : (~ (((X317 @ id @ X323 @ id)) = $true) | (((X317 @ id @ (X319 @ id @ X323) @ id)) = $true))) | ~ ! [X324 : term] : (~ (((X316 @ X324)) = $true) | (((X316 @ (sub @ X318 @ (push @ X324 @ id)))) = $true)) | (((X316 @ (lam @ X318))) = $true) | ~ ! [X325 : term] : (((X317 @ id @ X325 @ id)) = ((X316 @ X325)))) => ! [X326 : (subst > term > subst > $o),X327 : term] : (~ ! [X328 : term,X329 : term] : (~ (((X326 @ id @ X328 @ id)) = $true) | ~ (((X326 @ id @ X329 @ id)) = $true) | (((X326 @ id @ (ap @ (sub @ X329 @ id) @ X328) @ id)) = $true)) | ~ ! [X330 : subst,X331 : subst,X332 : term,X333 : subst] : ((((X326 @ (comp @ X331 @ X330) @ (sub @ X332 @ X330) @ X333)) = $true) | ~ (((X326 @ X331 @ X332 @ (comp @ X330 @ X333))) = $true)) | ~ ! [X334 : (subst > term > term)] : (~ ! [X335 : term] : (~ (((X326 @ id @ X335 @ id)) = $true) | (((X326 @ id @ (X334 @ id @ X335) @ id)) = $true)) | ~ ! [X336 : subst,X338 : subst,X337 : term] : (((sub @ (X334 @ X336 @ X337) @ X338)) = ((X334 @ (comp @ X336 @ X338) @ (sub @ X337 @ X338)))) | (((X326 @ id @ (lam @ (X334 @ sh @ one)) @ id)) = $true)) | ($true = ((X326 @ id @ X327 @ id))) | ~ ! [X339 : term,X340 : subst,X341 : subst,X342 : subst] : ((((X326 @ X342 @ X339 @ (comp @ X341 @ X340))) = $true) | ~ (((X326 @ (comp @ X342 @ X341) @ (sub @ X339 @ X341) @ X340)) = $true)) | ~ ! [X343 : term] : ((((X326 @ id @ X343 @ id)) = $true) | ~ (((var @ (sub @ X343 @ id))) = $true))))))) <=> (hoasinduction_lthm_3 = $true)) & (ulamvarind = $true) & (induction2_gthm = $true) & (hoasinduction_lem3_gthm = $true) & ! [X344 : d_subst] : (d_id = X344) & (hoasinduction_lem3v2_gthm = $true) & (id = ((d2subst @ d_id))) & ((! [X345 : subst,X346 : term] : (((sub @ one @ (push @ X346 @ X345))) = X346) => (! [X347 : term,X348 : subst] : (((comp @ sh @ (push @ X347 @ X348))) = X348) => ! [X349 : subst,X350 : term,X351 : term,X352 : (term > $o),X353 : (term > $o)] : ((((X352 @ (sub @ X351 @ (push @ X350 @ X349)))) = $true) | ~ (((X353 @ X351)) = $true)))) <=> (pushprop_lem1_lthm = $true)) & (hoasinduction_lem1v2 = $true) & ! [X354 : subst] : ? [X355 : d_subst] : (((d2subst @ X355)) = X354) & ((axvarcons = $true) <=> ! [X356 : subst,X357 : term] : (((sub @ one @ (push @ X357 @ X356))) = X357)) & (hoasinduction_lem1v2_gthm = $true) & (pushprop_lem2v2 = $true) & (! [X358 : (subst > term > subst > $o)] : ~ ! [X359 : (term > $o)] : ~ ! [X360 : term] : (((X358 @ id @ X360 @ id)) = ((X359 @ X360))) <=> (hoasinduction_lem0 = $true)) & ((hoaslaminj = $true) <=> ! [X361 : (subst > term > term),X364 : (subst > term > term),X362 : term,X363 : subst] : ((((X364 @ X363 @ X362)) = ((X361 @ X363 @ X362))) | (((lam @ (X361 @ sh @ one))) != ((lam @ (X364 @ sh @ one)))) | ~ ! [X369 : term,X368 : subst,X370 : subst] : (((sub @ (X361 @ X370 @ X369) @ X368)) = ((X361 @ (comp @ X370 @ X368) @ (sub @ X369 @ X368)))) | ~ ! [X367 : subst,X365 : term,X366 : subst] : (((X364 @ (comp @ X366 @ X367) @ (sub @ X365 @ X367))) = ((sub @ (X364 @ X366 @ X365) @ X367))))) & (pushprop_lthm_orig = $true) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (! [X371 : (subst > term > subst > $o)] : ~ ! [X372 : (term > $o)] : ~ ! [X373 : term] : (((X371 @ id @ X373 @ id)) = ((X372 @ X373))) => (! [X374 : term,X375 : (term > $o)] : (~ ! [X376 : term] : ((((X375 @ X376)) = $true) | ~ ($true = ((var @ X376)))) | (((X375 @ X374)) = $true) | ~ ! [X377 : term] : ((((X375 @ (lam @ X377))) = $true) | ~ ! [X378 : term] : ((((X375 @ (sub @ X377 @ (push @ X378 @ id)))) = $true) | ~ (((X375 @ X378)) = $true))) | ~ ! [X379 : term,X380 : term] : (~ (((X375 @ X380)) = $true) | (((X375 @ (ap @ X380 @ X379))) = $true) | ~ (((X375 @ X379)) = $true))) => (! [X381 : term] : (((sub @ X381 @ id)) = X381) => (! [X382 : (subst > term > subst > $o),X383 : term,X384 : (term > $o)] : (~ ! [X385 : term] : (~ (((X384 @ X385)) = $true) | (((X384 @ (sub @ X383 @ (push @ X385 @ id)))) = $true)) | ~ ! [X386 : term] : (((X382 @ id @ X386 @ id)) = ((X384 @ X386))) | (((X384 @ (lam @ X383))) = $true) | ~ ! [X387 : (subst > term > term)] : (~ ! [X388 : subst,X389 : term,X390 : subst] : (((X387 @ (comp @ X390 @ X388) @ (sub @ X389 @ X388))) = ((sub @ (X387 @ X390 @ X389) @ X388))) | (((X382 @ id @ (lam @ (X387 @ sh @ one)) @ id)) = $true) | ~ ! [X391 : term] : (~ (((X382 @ id @ X391 @ id)) = $true) | (((X382 @ id @ (X387 @ id @ X391) @ id)) = $true)))) => ! [X392 : (subst > term > subst > $o),X393 : term] : ((((X392 @ id @ X393 @ id)) = $true) | ~ ! [X394 : term,X395 : term] : (~ ($true = ((X392 @ id @ X395 @ id))) | (((X392 @ id @ (ap @ (sub @ X394 @ id) @ X395) @ id)) = $true) | ~ (((X392 @ id @ X394 @ id)) = $true)) | ~ ! [X396 : (subst > term > term)] : ((((X392 @ id @ (lam @ (X396 @ sh @ one)) @ id)) = $true) | ~ ! [X397 : term] : ((((X392 @ id @ (X396 @ id @ X397) @ id)) = $true) | ~ (((X392 @ id @ X397 @ id)) = $true)) | ~ ! [X400 : subst,X398 : term,X399 : subst] : (((X396 @ (comp @ X399 @ X400) @ (sub @ X398 @ X400))) = ((sub @ (X396 @ X399 @ X398) @ X400)))))))))) & ((pushprop_lem0_lthm = $true) <=> ! [X401 : term,X402 : subst,X403 : (term > $o)] : ~ ! [X404 : (term > $o)] : ~ ! [X405 : term] : (((X403 @ (sub @ X405 @ (push @ X401 @ X402)))) = ((X404 @ X405)))) & ! [X406 : term] : ? [X407 : d_term] : (((d2term @ X407)) = X406) & ((pushprop_lem0 = $true) <=> ! [X408 : term,X409 : (term > $o),X410 : subst] : ~ ! [X411 : (term > $o)] : ~ ! [X412 : term] : (((X409 @ (sub @ X412 @ (push @ X408 @ X410)))) = ((X411 @ X412)))) & ((laminj = $true) <=> ! [X414 : term,X413 : term] : ((X413 = X414) | (((lam @ X413)) != ((lam @ X414))))) & (pushprop_lem0_gthm = $true) & (hoaslaminj_gthm = $true) & (! [X415 : term,X416 : (term > $o)] : (~ ! [X417 : term,X418 : term] : (~ (((X416 @ X417)) = $true) | ~ (((X416 @ X418)) = $true) | (((X416 @ (ap @ X417 @ X418))) = $true)) | ~ ! [X419 : term] : (($true = ((X416 @ (lam @ X419)))) | ~ ! [X420 : term] : ((((X416 @ (sub @ X419 @ (push @ X420 @ id)))) = $true) | ~ (((X416 @ X420)) = $true))) | (((X416 @ X415)) = $true) | ~ ! [X421 : term] : ((((X416 @ X421)) = $true) | ~ (((var @ X421)) = $true))) <=> (induction2 = $true)) & ((substmonoid_lthm = $true) <=> (! [X422 : subst] : (((comp @ id @ X422)) = X422) => (! [X423 : subst] : (((comp @ X423 @ id)) = X423) => (! [X424 : subst] : (((comp @ id @ X424)) = X424) & ! [X425 : subst] : (((comp @ X425 @ id)) = X425))))) & (apnotvar = $true) & (one = ((d2term @ d_one))) & (lamnotvar = $true) & ~ (((var @ (d2term @ d_one))) = $true) & (! [X426 : subst] : (((push @ (sub @ one @ X426) @ (comp @ sh @ X426))) = X426) <=> (axscons = $true)) & ((hoaslaminj_lthm = $true) <=> (! [X427 : term,X428 : subst] : (((sub @ one @ (push @ X427 @ X428))) = X427) => (! [X430 : subst,X429 : term] : (((comp @ sh @ (push @ X429 @ X430))) = X430) => (! [X431 : term,X432 : term] : ((((lam @ X432)) != ((lam @ X431))) | (X431 = X432)) => ! [X436 : (subst > term > term),X435 : subst,X433 : (subst > term > term),X434 : term] : ((((X433 @ X435 @ X434)) = ((X436 @ X435 @ X434))) | (((lam @ (X433 @ sh @ one))) != ((lam @ (X436 @ sh @ one)))) | ~ ! [X437 : subst,X438 : subst,X439 : term] : (((sub @ (X436 @ X438 @ X439) @ X437)) = ((X436 @ (comp @ X438 @ X437) @ (sub @ X439 @ X437)))) | ~ ! [X440 : term,X441 : subst,X442 : subst] : (((X433 @ (comp @ X441 @ X442) @ (sub @ X440 @ X442))) = ((sub @ (X433 @ X441 @ X440) @ X442)))))))) & (! [X443 : (term > $o),X444 : term,X445 : subst,X446 : (term > $o)] : (~ ! [X447 : term] : (((X443 @ X447)) = ((X446 @ (sub @ X447 @ (push @ X444 @ X445))))) | (((X443 @ one)) = $true) | ~ (((X446 @ X444)) = $true)) <=> (pushprop_lem1v2 = $true)) & ((! [X448 : term] : (((sub @ X448 @ id)) = X448) => ! [X449 : term] : (((sub @ X449 @ id)) = X449)) <=> (termmset_lthm = $true)) & (hoaslamnotvar_lthm = $true) & ((hoasinduction_lem3v2 = $true) <=> ! [X450 : (subst > term > subst > $o),X451 : (term > $o),X452 : term] : ((((X451 @ (lam @ X452))) = $true) | ~ ! [X453 : term,X454 : subst,X455 : subst,X456 : subst] : ((((X450 @ X456 @ X453 @ (comp @ X454 @ X455))) = $true) | ~ (((X450 @ (comp @ X456 @ X454) @ (sub @ X453 @ X454) @ X455)) = $true)) | ~ ! [X457 : subst,X458 : subst,X459 : subst,X460 : term] : ((((X450 @ (comp @ X457 @ X459) @ (sub @ X460 @ X459) @ X458)) = $true) | ~ (((X450 @ X457 @ X460 @ (comp @ X459 @ X458))) = $true)) | ~ ! [X461 : term] : ((((X451 @ (sub @ X452 @ (push @ X461 @ id)))) = $true) | ~ (((X451 @ X461)) = $true)) | ~ ! [X462 : term] : (((X451 @ X462)) = ((X450 @ id @ X462 @ id))) | ~ ! [X463 : (subst > term > term)] : (~ ! [X465 : subst,X464 : subst,X466 : term] : (((X463 @ (comp @ X465 @ X464) @ (sub @ X466 @ X464))) = ((sub @ (X463 @ X465 @ X466) @ X464))) | ~ ! [X467 : term] : (~ (((X450 @ id @ X467 @ id)) = $true) | (((X450 @ id @ (X463 @ id @ X467) @ id)) = $true)) | (((X450 @ id @ (lam @ (X463 @ sh @ one)) @ id)) = $true)))) & (! [X469 : term,X468 : term] : ((X468 = X469) | (((sub @ X469 @ sh)) != ((sub @ X468 @ sh)))) <=> (shinj = $true)) & (hoasinduction_lem2v2_gthm = $true) & (! [X470 : term] : (((sub @ X470 @ id)) = X470) <=> (axvarid = $true)) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (! [X471 : subst] : (((comp @ id @ X471)) = X471) <=> (axidl = $true)) & (! [X472 : term,X473 : term,X474 : term,X475 : term] : ((((ap @ X475 @ X473)) != ((ap @ X472 @ X474))) | (X472 = X475)) <=> (apinj1 = $true)) & (hoasinduction_lem3b_gthm = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ~ (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) = $true) & ((hoasinduction_lem2 = $true) <=> ! [X476 : term,X477 : (subst > term > subst > $o),X478 : term] : ((((X477 @ id @ (ap @ X476 @ X478) @ id)) = $true) | ~ ! [X479 : subst,X480 : subst,X481 : term,X482 : subst] : ((((X477 @ X480 @ X481 @ (comp @ X482 @ X479))) = $true) | ~ (((X477 @ (comp @ X480 @ X482) @ (sub @ X481 @ X482) @ X479)) = $true)) | ~ (((X477 @ id @ X478 @ id)) = $true) | ~ ! [X483 : term,X484 : term] : (~ (((X477 @ id @ X484 @ id)) = $true) | (((X477 @ id @ (ap @ (sub @ X483 @ id) @ X484) @ id)) = $true) | ~ (((X477 @ id @ X483 @ id)) = $true)) | ~ (((X477 @ id @ X476 @ id)) = $true) | ~ ! [X485 : subst,X486 : term,X487 : subst,X488 : subst] : ((((X477 @ (comp @ X485 @ X488) @ (sub @ X486 @ X488) @ X487)) = $true) | ~ ($true = ((X477 @ X485 @ X486 @ (comp @ X488 @ X487))))))) & (! [X489 : term,X490 : (subst > term > subst > $o)] : (~ ! [X491 : term,X492 : term] : (~ (((X490 @ id @ X491 @ id)) = $true) | (((X490 @ id @ (ap @ (sub @ X492 @ id) @ X491) @ id)) = $true) | ~ (((X490 @ id @ X492 @ id)) = $true)) | ~ ! [X493 : (subst > term > term)] : (~ ! [X496 : term,X494 : subst,X495 : subst] : (((X493 @ (comp @ X494 @ X495) @ (sub @ X496 @ X495))) = ((sub @ (X493 @ X494 @ X496) @ X495))) | ~ ! [X497 : term] : (~ ($true = ((X490 @ id @ X497 @ id))) | (((X490 @ id @ (X493 @ id @ X497) @ id)) = $true)) | (((X490 @ id @ (lam @ (X493 @ sh @ one)) @ id)) = $true)) | (((X490 @ id @ X489 @ id)) = $true) | ~ ! [X498 : term] : (~ (((var @ (sub @ X498 @ id))) = $true) | (((X490 @ id @ X498 @ id)) = $true)) | ~ ! [X499 : subst,X500 : subst,X501 : subst,X502 : term] : (($true = ((X490 @ (comp @ X499 @ X500) @ (sub @ X502 @ X500) @ X501))) | ~ (((X490 @ X499 @ X502 @ (comp @ X500 @ X501))) = $true)) | ~ ! [X503 : term,X504 : subst,X505 : subst,X506 : subst] : ((((X490 @ X504 @ X503 @ (comp @ X506 @ X505))) = $true) | ~ (((X490 @ (comp @ X504 @ X506) @ (sub @ X503 @ X506) @ X505)) = $true))) <=> (hoasinduction = $true)) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X507 : term] : ~ ! [X508 : (subst > term > term)] : ~ ! [X509 : term,X510 : subst] : (((sub @ X507 @ (push @ X509 @ X510))) = ((X508 @ X510 @ X509)))) & ! [X511 : d_subst,X512 : d_subst] : ((((d2subst @ X511)) = ((d2subst @ X512))) => (X511 = X512)) & (! [X513 : (subst > term > subst > $o),X514 : term,X515 : term,X516 : (term > $o)] : ((((X516 @ (ap @ X515 @ X514))) = $true) | ~ ! [X517 : term,X518 : term] : (~ (((X513 @ id @ X518 @ id)) = $true) | (((X513 @ id @ (ap @ (sub @ X517 @ id) @ X518) @ id)) = $true) | ~ (((X513 @ id @ X517 @ id)) = $true)) | ~ ! [X519 : term] : (((X513 @ id @ X519 @ id)) = ((X516 @ X519))) | ~ ! [X520 : subst,X521 : subst,X522 : subst,X523 : term] : (~ (((X513 @ X520 @ X523 @ (comp @ X521 @ X522))) = $true) | (((X513 @ (comp @ X520 @ X521) @ (sub @ X523 @ X521) @ X522)) = $true)) | ~ ($true = ((X516 @ X515))) | ~ (((X516 @ X514)) = $true) | ~ ! [X524 : subst,X525 : subst,X526 : term,X527 : subst] : (~ (((X513 @ (comp @ X527 @ X524) @ (sub @ X526 @ X524) @ X525)) = $true) | (((X513 @ X527 @ X526 @ (comp @ X524 @ X525))) = $true))) <=> (hoasinduction_lem2v2 = $true)) & (substmonoid_gthm = $true) & (hoaslamnotvar_gthm = $true) & (ulamvarsh = $true) & ((hoasinduction_lem3aaa = $true) <=> ! [X528 : (subst > term > subst > $o),X529 : term] : (~ ! [X530 : (subst > term > term),X531 : term] : (~ ! [X534 : term,X532 : subst,X533 : subst] : (((sub @ (X530 @ X533 @ X534) @ X532)) = ((sub @ (sub @ X531 @ (push @ X534 @ X533)) @ X532))) | ~ ! [X535 : term] : (~ ($true = ((X528 @ id @ X535 @ id))) | (((X528 @ id @ (X530 @ id @ X535) @ id)) = $true)) | ~ ! [X538 : subst,X536 : subst,X537 : term] : (((sub @ X531 @ (push @ (sub @ X537 @ X538) @ (comp @ X536 @ X538)))) = ((X530 @ (comp @ X536 @ X538) @ (sub @ X537 @ X538)))) | (((X528 @ id @ (lam @ (X530 @ sh @ one)) @ id)) = $true)) | (((X528 @ id @ (lam @ (sub @ X529 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X539 : term] : (~ (((X528 @ id @ X539 @ id)) = $true) | (((X528 @ id @ (sub @ X529 @ (push @ X539 @ id)) @ id)) = $true))))),
  inference(rectify,[],[f5])).
thf(f10,plain,(
  (pushprop_lem0_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (hoasapnotvar = $true) & (hoasinduction_lem3a_gthm = $true) & ((substmonoid_lthm = $true) <=> (! [X422 : subst] : (((comp @ id @ X422)) = X422) => (! [X423 : subst] : (((comp @ X423 @ id)) = X423) => (! [X424 : subst] : (((comp @ id @ X424)) = X424) & ! [X425 : subst] : (((comp @ X425 @ id)) = X425))))) & ((! [X187 : subst,X186 : term] : (((sub @ one @ (push @ X186 @ X187))) = X186) => ! [X188 : subst,X191 : (term > $o),X190 : term,X189 : (term > $o)] : (~ ! [X192 : term] : (((X191 @ (sub @ X192 @ (push @ X190 @ X188)))) = ((X189 @ X192))) | (((X189 @ one)) = $true) | ($true != ((X191 @ X190))))) <=> (pushprop_lem1v2_lthm = $true)) & (((d2subst @ d_id)) = sh) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (induction2lem_gthm = $true) & (hoasinduction_lem1_gthm = $true) & (hoaslamnotvar_lthm = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (! [X60 : (term > $o),X58 : term,X59 : subst] : (~ ! [X65 : term] : ((((var @ X65)) != $true) | (((X60 @ (sub @ X65 @ X59))) = $true)) | ~ ! [X61 : term] : ((((X60 @ (lam @ X61))) = $true) | ~ ! [X62 : term] : ((((X60 @ (sub @ X61 @ (push @ X62 @ id)))) = $true) | (((X60 @ X62)) != $true))) | (((X60 @ (sub @ X58 @ X59))) = $true) | ~ ! [X64 : term,X63 : term] : ((((X60 @ (ap @ X63 @ X64))) = $true) | (((X60 @ X64)) != $true) | (((X60 @ X63)) != $true))) <=> (induction2lem = $true)) & (axmap = $true) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X507 : term] : ~ ! [X508 : (subst > term > term)] : ~ ! [X509 : term,X510 : subst] : (((sub @ X507 @ (push @ X509 @ X510))) = ((X508 @ X510 @ X509)))) & (hoasinduction_gthm = $true) & (! [X470 : term] : (((sub @ X470 @ id)) = X470) <=> (axvarid = $true)) & ((termmset = $true) <=> ! [X151 : term] : (((sub @ X151 @ id)) = X151)) & (! [X416 : (term > $o),X415 : term] : (~ ! [X418 : term,X417 : term] : ((((X416 @ X417)) != $true) | (((X416 @ X418)) != $true) | (((X416 @ (ap @ X417 @ X418))) = $true)) | (((X416 @ X415)) = $true) | ~ ! [X421 : term] : ((((var @ X421)) != $true) | (((X416 @ X421)) = $true)) | ~ ! [X419 : term] : (~ ! [X420 : term] : ((((X416 @ (sub @ X419 @ (push @ X420 @ id)))) = $true) | (((X416 @ X420)) != $true)) | ($true = ((X416 @ (lam @ X419)))))) <=> (induction2 = $true)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoasinduction_lem3v2_f = $true) <=> ! [X211 : term] : ~ ! [X212 : (subst > term > term)] : ~ ! [X213 : subst,X214 : term] : (((X212 @ X213 @ X214)) = ((sub @ X211 @ (push @ X214 @ X213))))) & (! [X82 : term,X83 : (subst > term > subst > $o)] : ((((X83 @ id @ (lam @ (sub @ X82 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X85 : (subst > term > term)] : (~ ! [X88 : subst,X87 : subst,X86 : term] : (((X85 @ (comp @ X88 @ X87) @ (sub @ X86 @ X87))) = ((sub @ (X85 @ X88 @ X86) @ X87))) | ($true = ((X83 @ id @ (lam @ (X85 @ sh @ one)) @ id))) | ~ ! [X89 : term] : ((((X83 @ id @ X89 @ id)) != $true) | (((X83 @ id @ (X85 @ id @ X89) @ id)) = $true))) | ~ ! [X84 : term] : ((((X83 @ id @ X84 @ id)) != $true) | (((X83 @ id @ (sub @ X82 @ (push @ X84 @ id)) @ id)) = $true))) <=> (hoasinduction_lem3aa = $true)) & ! [X69 : d_term] : (d_one = X69) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (pushprop_lthm_orig = $true) & (hoasinduction_lem3v2_gthm = $true) & (! [X358 : (subst > term > subst > $o)] : ~ ! [X359 : (term > $o)] : ~ ! [X360 : term] : (((X358 @ id @ X360 @ id)) = ((X359 @ X360))) <=> (hoasinduction_lem0 = $true)) & (hoaslamnotvar_gthm = $true) & ((pushprop_lem0_lthm = $true) <=> ! [X401 : term,X402 : subst,X403 : (term > $o)] : ~ ! [X404 : (term > $o)] : ~ ! [X405 : term] : (((X403 @ (sub @ X405 @ (push @ X401 @ X402)))) = ((X404 @ X405)))) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (substmonoid_gthm = $true) & ! [X511 : d_subst,X512 : d_subst] : ((((d2subst @ X511)) = ((d2subst @ X512))) => (X511 = X512)) & (hoaslamnotap_gthm = $true) & (! [X241 : (term > $o),X240 : term] : ((((X241 @ X240)) = $true) | ~ ! [X242 : term] : ((((X241 @ X242)) != $true) | (((X241 @ (lam @ X242))) = $true)) | ~ ! [X245 : term] : ((((var @ X245)) != $true) | (((X241 @ X245)) = $true)) | ~ ! [X244 : term,X243 : term] : ((((X241 @ (ap @ X244 @ X243))) = $true) | (((X241 @ X243)) != $true) | (((X241 @ X244)) != $true))) <=> (induction = $true)) & ((hoaslaminj = $true) <=> ! [X361 : (subst > term > term),X364 : (subst > term > term),X362 : term,X363 : subst] : ((((X364 @ X363 @ X362)) = ((X361 @ X363 @ X362))) | (((lam @ (X361 @ sh @ one))) != ((lam @ (X364 @ sh @ one)))) | ~ ! [X369 : term,X368 : subst,X370 : subst] : (((sub @ (X361 @ X370 @ X369) @ X368)) = ((X361 @ (comp @ X370 @ X368) @ (sub @ X369 @ X368)))) | ~ ! [X367 : subst,X365 : term,X366 : subst] : (((X364 @ (comp @ X366 @ X367) @ (sub @ X365 @ X367))) = ((sub @ (X364 @ X366 @ X365) @ X367))))) & ((pushprop_lem0 = $true) <=> ! [X408 : term,X409 : (term > $o),X410 : subst] : ~ ! [X411 : (term > $o)] : ~ ! [X412 : term] : (((X409 @ (sub @ X412 @ (push @ X408 @ X410)))) = ((X411 @ X412)))) & (hoasinduction_lem1 = $true) & ! [X344 : d_subst] : (d_id = X344) & ((! [X271 : term] : (((sub @ X271 @ id)) = X271) => (! [X273 : term,X272 : (subst > term > subst > $o)] : (~ ! [X275 : (subst > term > term)] : (($true = ((X272 @ id @ (lam @ (X275 @ sh @ one)) @ id))) | ~ ! [X277 : subst,X278 : term,X279 : subst] : (((X275 @ (comp @ X279 @ X277) @ (sub @ X278 @ X277))) = ((sub @ (X275 @ X279 @ X278) @ X277))) | ~ ! [X276 : term] : ((((X272 @ id @ X276 @ id)) != $true) | ($true = ((X272 @ id @ (X275 @ id @ X276) @ id))))) | (((X272 @ id @ (lam @ (sub @ X273 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X274 : term] : ((((X272 @ id @ X274 @ id)) != $true) | (((X272 @ id @ (sub @ X273 @ (push @ X274 @ id)) @ id)) = $true))) => ! [X281 : (subst > term > subst > $o),X280 : term] : (~ ! [X284 : subst,X283 : subst,X285 : subst,X282 : term] : ((((X281 @ (comp @ X284 @ X283) @ (sub @ X282 @ X283) @ X285)) = $true) | ($true != ((X281 @ X284 @ X282 @ (comp @ X283 @ X285))))) | ~ ! [X287 : subst,X289 : term,X290 : subst,X288 : subst] : ((((X281 @ X290 @ X289 @ (comp @ X288 @ X287))) = $true) | (((X281 @ (comp @ X290 @ X288) @ (sub @ X289 @ X288) @ X287)) != $true)) | (((X281 @ id @ (lam @ X280) @ id)) = $true) | ~ ! [X286 : term] : ((((X281 @ id @ (sub @ X280 @ (push @ X286 @ id)) @ id)) = $true) | (((X281 @ id @ X286 @ id)) != $true)) | ~ ! [X291 : (subst > term > term)] : ((((X281 @ id @ (lam @ (X291 @ sh @ one)) @ id)) = $true) | ~ ! [X295 : term] : ((((X281 @ id @ (X291 @ id @ X295) @ id)) = $true) | (((X281 @ id @ X295 @ id)) != $true)) | ~ ! [X294 : term,X293 : subst,X292 : subst] : (((sub @ (X291 @ X292 @ X294) @ X293)) = ((X291 @ (comp @ X292 @ X293) @ (sub @ X294 @ X293)))))))) <=> (hoasinduction_lem3_lthm = $true)) & (pushprop_lem2v2_lthm = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ((hoasapinj2 = $true) <=> ! [X185 : term,X183 : term,X184 : term,X182 : term] : ((((ap @ (sub @ X182 @ id) @ X184)) != ((ap @ (sub @ X183 @ id) @ X185))) | (X184 = X185))) & ((! [X155 : subst,X154 : term] : (((sub @ one @ (push @ X154 @ X155))) = X154) => (! [X156 : subst,X157 : term] : (((comp @ sh @ (push @ X157 @ X156))) = X156) => (! [X158 : subst] : (((comp @ X158 @ id)) = X158) => (! [X160 : (term > $o),X159 : term] : ((((X160 @ X159)) = $true) | ~ ! [X161 : term] : ((((X160 @ (lam @ X161))) = $true) | (((X160 @ X161)) != $true)) | ~ ! [X164 : term] : ((((var @ X164)) != $true) | ($true = ((X160 @ X164)))) | ~ ! [X162 : term,X163 : term] : ((((X160 @ X163)) != $true) | (((X160 @ X162)) != $true) | (((X160 @ (ap @ X162 @ X163))) = $true))) => ! [X167 : term,X166 : (term > $o),X165 : subst] : (~ ! [X172 : term] : ((((var @ X172)) != $true) | (((X166 @ (sub @ X172 @ X165))) = $true)) | ~ ! [X168 : term] : ((((X166 @ (lam @ X168))) = $true) | ~ ! [X169 : term] : (($true != ((X166 @ X169))) | (((X166 @ (sub @ X168 @ (push @ X169 @ id)))) = $true))) | ~ ! [X171 : term,X170 : term] : ((((X166 @ X170)) != $true) | (((X166 @ X171)) != $true) | (((X166 @ (ap @ X170 @ X171))) = $true)) | (((X166 @ (sub @ X167 @ X165))) = $true)))))) <=> (induction2lem_lthm = $true)) & (hoaslaminj_gthm = $true) & (hoasinduction_lem1v2 = $true) & ((axidr = $true) <=> ! [X150 : subst] : (((comp @ X150 @ id)) = X150)) & ((pushprop_lem1 = $true) <=> ! [X246 : term,X247 : term,X250 : (term > $o),X248 : subst,X249 : (term > $o)] : ((((X249 @ X247)) != $true) | (((X250 @ (sub @ X247 @ (push @ X246 @ X248)))) = $true))) & (hoasinduction_lem2_lthm = $true) & ((hoaslamnotap = $true) <=> ! [X52 : (subst > term > term),X50 : term,X51 : term] : ((((lam @ (X52 @ sh @ one))) != ((ap @ (sub @ X50 @ id) @ X51))) | ~ ! [X55 : term,X54 : subst,X53 : subst] : (((sub @ (X52 @ X53 @ X55) @ X54)) = ((X52 @ (comp @ X53 @ X54) @ (sub @ X55 @ X54)))))) & (hoasapinj2_gthm = $true) & ((! [X448 : term] : (((sub @ X448 @ id)) = X448) => ! [X449 : term] : (((sub @ X449 @ id)) = X449)) <=> (termmset_lthm = $true)) & (hoasinduction_lem2v2_gthm = $true) & (pushprop_lem3v2 = $true) & (hoaslamnotap_lthm = $true) & (pushprop_lem1v2_gthm = $true) & (! [X426 : subst] : (((push @ (sub @ one @ X426) @ (comp @ sh @ X426))) = X426) <=> (axscons = $true)) & (axvarshift = $true) & (lamnotvar = $true) & (axabs = $true) & (pushprop = $true) & (! [X469 : term,X468 : term] : ((X468 = X469) | (((sub @ X469 @ sh)) != ((sub @ X468 @ sh)))) <=> (shinj = $true)) & ((hoasinduction_lem3v2a = $true) <=> ! [X195 : (subst > term > subst > $o),X193 : (term > $o),X194 : term] : (~ ! [X196 : (subst > term > term)] : (~ ! [X200 : term] : ((((X195 @ id @ X200 @ id)) != $true) | (((X195 @ id @ (X196 @ id @ X200) @ id)) = $true)) | ~ ! [X199 : term,X197 : subst,X198 : subst] : (((X196 @ (comp @ X198 @ X197) @ (sub @ X199 @ X197))) = ((sub @ (X196 @ X198 @ X199) @ X197))) | (((X195 @ id @ (lam @ (X196 @ sh @ one)) @ id)) = $true)) | (((X193 @ (lam @ X194))) = $true) | ~ ! [X202 : term] : (((X193 @ X202)) = ((X195 @ id @ X202 @ id))) | ~ ! [X201 : term] : ((((X193 @ X201)) != $true) | (((X193 @ (sub @ X194 @ (push @ X201 @ id)))) = $true)))) & ((hoasinduction_lthm = $true) <=> (! [X90 : (term > $o),X91 : term] : (~ ! [X96 : term] : ((((var @ X96)) != $true) | (((X90 @ X96)) = $true)) | ~ ! [X92 : term] : (~ ! [X93 : term] : ((((X90 @ X93)) != $true) | (((X90 @ (sub @ X92 @ (push @ X93 @ id)))) = $true)) | (((X90 @ (lam @ X92))) = $true)) | ~ ! [X95 : term,X94 : term] : (($true = ((X90 @ (ap @ X94 @ X95)))) | (((X90 @ X94)) != $true) | (((X90 @ X95)) != $true)) | (((X90 @ X91)) = $true)) => (! [X98 : (subst > term > subst > $o),X97 : term,X99 : term] : ((((X98 @ id @ X99 @ id)) != $true) | ~ ! [X104 : subst,X105 : subst,X106 : subst,X107 : term] : ((((X98 @ (comp @ X106 @ X104) @ (sub @ X107 @ X104) @ X105)) != $true) | (((X98 @ X106 @ X107 @ (comp @ X104 @ X105))) = $true)) | (((X98 @ id @ (ap @ X99 @ X97) @ id)) = $true) | ~ ! [X108 : term,X109 : term] : ((((X98 @ id @ X108 @ id)) != $true) | (((X98 @ id @ X109 @ id)) != $true) | (((X98 @ id @ (ap @ (sub @ X108 @ id) @ X109) @ id)) = $true)) | ~ ! [X103 : subst,X100 : subst,X101 : term,X102 : subst] : ((((X98 @ (comp @ X102 @ X103) @ (sub @ X101 @ X103) @ X100)) = $true) | (((X98 @ X102 @ X101 @ (comp @ X103 @ X100))) != $true)) | ($true != ((X98 @ id @ X97 @ id)))) => (! [X110 : (subst > term > subst > $o),X111 : term] : (($true = ((X110 @ id @ (lam @ X111) @ id))) | ~ ! [X120 : (subst > term > term)] : (~ ! [X123 : subst,X124 : term,X122 : subst] : (((X120 @ (comp @ X122 @ X123) @ (sub @ X124 @ X123))) = ((sub @ (X120 @ X122 @ X124) @ X123))) | (((X110 @ id @ (lam @ (X120 @ sh @ one)) @ id)) = $true) | ~ ! [X121 : term] : ((((X110 @ id @ (X120 @ id @ X121) @ id)) = $true) | (((X110 @ id @ X121 @ id)) != $true))) | ~ ! [X119 : subst,X117 : subst,X116 : subst,X118 : term] : ((((X110 @ (comp @ X119 @ X117) @ (sub @ X118 @ X117) @ X116)) != $true) | (((X110 @ X119 @ X118 @ (comp @ X117 @ X116))) = $true)) | ~ ! [X114 : subst,X113 : subst,X115 : term,X112 : subst] : ((((X110 @ X112 @ X115 @ (comp @ X114 @ X113))) != $true) | (((X110 @ (comp @ X112 @ X114) @ (sub @ X115 @ X114) @ X113)) = $true)) | ~ ! [X125 : term] : (($true != ((X110 @ id @ X125 @ id))) | (((X110 @ id @ (sub @ X111 @ (push @ X125 @ id)) @ id)) = $true))) => ! [X126 : (subst > term > subst > $o),X127 : term] : ((((X126 @ id @ X127 @ id)) = $true) | ~ ! [X128 : (subst > term > term)] : (~ ! [X129 : term] : ((((X126 @ id @ X129 @ id)) != $true) | (((X126 @ id @ (X128 @ id @ X129) @ id)) = $true)) | ~ ! [X131 : subst,X130 : term,X132 : subst] : (((X128 @ (comp @ X131 @ X132) @ (sub @ X130 @ X132))) = ((sub @ (X128 @ X131 @ X130) @ X132))) | (((X126 @ id @ (lam @ (X128 @ sh @ one)) @ id)) = $true)) | ~ ! [X138 : subst,X140 : subst,X139 : term,X137 : subst] : ((((X126 @ X138 @ X139 @ (comp @ X140 @ X137))) != $true) | (((X126 @ (comp @ X138 @ X140) @ (sub @ X139 @ X140) @ X137)) = $true)) | ~ ! [X142 : term,X143 : term] : ((((X126 @ id @ (ap @ (sub @ X142 @ id) @ X143) @ id)) = $true) | (((X126 @ id @ X142 @ id)) != $true) | (((X126 @ id @ X143 @ id)) != $true)) | ~ ! [X141 : term] : ((((X126 @ id @ X141 @ id)) = $true) | (((var @ (sub @ X141 @ id))) != $true)) | ~ ! [X136 : subst,X133 : subst,X134 : term,X135 : subst] : ((((X126 @ X135 @ X134 @ (comp @ X133 @ X136))) = $true) | (((X126 @ (comp @ X135 @ X133) @ (sub @ X134 @ X133) @ X136)) != $true))))))) & ((hoasinduction_lem3a_lthm = $true) <=> (! [X215 : term] : (((sub @ X215 @ id)) = X215) => (! [X217 : term,X216 : (subst > term > subst > $o)] : ((((X216 @ id @ (lam @ (sub @ X217 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X218 : term] : ((((X216 @ id @ (sub @ X217 @ (push @ X218 @ id)) @ id)) = $true) | (((X216 @ id @ X218 @ id)) != $true)) | ~ ! [X219 : (subst > term > term)] : ((((X216 @ id @ (lam @ (X219 @ sh @ one)) @ id)) = $true) | ~ ! [X220 : term] : ((((X216 @ id @ X220 @ id)) != $true) | (((X216 @ id @ (X219 @ id @ X220) @ id)) = $true)) | ~ ! [X221 : subst,X223 : term,X222 : subst] : (((sub @ (X219 @ X222 @ X223) @ X221)) = ((X219 @ (comp @ X222 @ X221) @ (sub @ X223 @ X221)))))) => ! [X224 : term,X225 : (subst > term > subst > $o)] : (~ ! [X226 : (subst > term > term)] : (~ ! [X229 : subst,X230 : term,X228 : subst] : (((X226 @ (comp @ X228 @ X229) @ (sub @ X230 @ X229))) = ((sub @ (X226 @ X228 @ X230) @ X229))) | (((X225 @ id @ (lam @ (X226 @ sh @ one)) @ id)) = $true) | ~ ! [X227 : term] : ((((X225 @ id @ X227 @ id)) != $true) | (((X225 @ id @ (X226 @ id @ X227) @ id)) = $true))) | ~ ! [X231 : term] : ((((X225 @ id @ X231 @ id)) != $true) | ($true = ((X225 @ id @ (sub @ X224 @ (push @ X231 @ id)) @ id)))) | ($true = ((X225 @ id @ (lam @ X224) @ id))))))) & (! [X471 : subst] : (((comp @ id @ X471)) = X471) <=> (axidl = $true)) & ((hoasinduction_lem3v2_lthm = $true) <=> (! [X16 : term] : (((sub @ X16 @ id)) = X16) => ! [X17 : term,X18 : (subst > term > subst > $o),X19 : (term > $o)] : (~ ! [X30 : (subst > term > term)] : (~ ! [X34 : term] : ((((X18 @ id @ (X30 @ id @ X34) @ id)) = $true) | (((X18 @ id @ X34 @ id)) != $true)) | (((X18 @ id @ (lam @ (X30 @ sh @ one)) @ id)) = $true) | ~ ! [X31 : subst,X33 : subst,X32 : term] : (((sub @ (X30 @ X33 @ X32) @ X31)) = ((X30 @ (comp @ X33 @ X31) @ (sub @ X32 @ X31))))) | ~ ! [X20 : term] : (((X19 @ X20)) = ((X18 @ id @ X20 @ id))) | ~ ! [X29 : term] : ((((X19 @ X29)) != $true) | (((X19 @ (sub @ X17 @ (push @ X29 @ id)))) = $true)) | ~ ! [X26 : term,X28 : subst,X25 : subst,X27 : subst] : ((((X18 @ X28 @ X26 @ (comp @ X27 @ X25))) = $true) | ($true != ((X18 @ (comp @ X28 @ X27) @ (sub @ X26 @ X27) @ X25)))) | ~ ! [X24 : term,X21 : subst,X22 : subst,X23 : subst] : ((((X18 @ X22 @ X24 @ (comp @ X23 @ X21))) != $true) | (((X18 @ (comp @ X22 @ X23) @ (sub @ X24 @ X23) @ X21)) = $true)) | (((X19 @ (lam @ X17))) = $true)))) & (hoasapinj1_gthm = $true) & (hoasinduction_lem1v2_gthm = $true) & ! [X406 : term] : ? [X407 : d_term] : (((d2term @ X407)) = X406) & ((hoasinduction_lem2v2 = $true) <=> ! [X515 : term,X514 : term,X513 : (subst > term > subst > $o),X516 : (term > $o)] : (~ ! [X517 : term,X518 : term] : ((((X513 @ id @ X517 @ id)) != $true) | (((X513 @ id @ (ap @ (sub @ X517 @ id) @ X518) @ id)) = $true) | (((X513 @ id @ X518 @ id)) != $true)) | (((X516 @ (ap @ X515 @ X514))) = $true) | ~ ! [X519 : term] : (((X513 @ id @ X519 @ id)) = ((X516 @ X519))) | (((X516 @ X514)) != $true) | ($true != ((X516 @ X515))) | ~ ! [X520 : subst,X523 : term,X521 : subst,X522 : subst] : ((((X513 @ X520 @ X523 @ (comp @ X521 @ X522))) != $true) | (((X513 @ (comp @ X520 @ X521) @ (sub @ X523 @ X521) @ X522)) = $true)) | ~ ! [X524 : subst,X526 : term,X525 : subst,X527 : subst] : ((((X513 @ (comp @ X527 @ X524) @ (sub @ X526 @ X524) @ X525)) != $true) | (((X513 @ X527 @ X526 @ (comp @ X524 @ X525))) = $true)))) & (! [X445 : subst,X443 : (term > $o),X444 : term,X446 : (term > $o)] : (~ ! [X447 : term] : (((X443 @ X447)) = ((X446 @ (sub @ X447 @ (push @ X444 @ X445))))) | (((X446 @ X444)) != $true) | (((X443 @ one)) = $true)) <=> (pushprop_lem1v2 = $true)) & ((! [X345 : subst,X346 : term] : (((sub @ one @ (push @ X346 @ X345))) = X346) => (! [X347 : term,X348 : subst] : (((comp @ sh @ (push @ X347 @ X348))) = X348) => ! [X349 : subst,X353 : (term > $o),X352 : (term > $o),X350 : term,X351 : term] : ((((X352 @ (sub @ X351 @ (push @ X350 @ X349)))) = $true) | (((X353 @ X351)) != $true)))) <=> (pushprop_lem1_lthm = $true)) & (ulamvar1 != $true) & ((substmonoid = $true) <=> (! [X71 : subst] : (((comp @ id @ X71)) = X71) & ! [X70 : subst] : (((comp @ X70 @ id)) = X70))) & (! [X472 : term,X473 : term,X474 : term,X475 : term] : ((((ap @ X475 @ X473)) != ((ap @ X472 @ X474))) | (X472 = X475)) <=> (apinj1 = $true)) & (axclos = $true) & ((! [X35 : term] : ~ ! [X36 : (subst > term > term)] : ~ ! [X38 : subst,X37 : term] : (((X36 @ X38 @ X37)) = ((sub @ X35 @ (push @ X37 @ X38)))) => (! [X39 : term] : (((sub @ X39 @ id)) = X39) => ! [X40 : (term > $o),X41 : (subst > term > subst > $o),X42 : term] : (~ ! [X44 : term] : ((((X40 @ (sub @ X42 @ (push @ X44 @ id)))) = $true) | (((X40 @ X44)) != $true)) | ~ ! [X43 : term] : (((X41 @ id @ X43 @ id)) = ((X40 @ X43))) | ~ ! [X45 : (subst > term > term)] : (~ ! [X46 : term] : ((((X41 @ id @ X46 @ id)) != $true) | (((X41 @ id @ (X45 @ id @ X46) @ id)) = $true)) | (((X41 @ id @ (lam @ (X45 @ sh @ one)) @ id)) = $true) | ~ ! [X47 : subst,X49 : term,X48 : subst] : (((X45 @ (comp @ X47 @ X48) @ (sub @ X49 @ X48))) = ((sub @ (X45 @ X47 @ X49) @ X48)))) | (((X40 @ (lam @ X42))) = $true)))) <=> (hoasinduction_lem3v2a_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)))))))))))) & (one = ((d2term @ d_one))) & (! [X148 : term] : ~ ! [X149 : (subst > term > term)] : (((sub @ X148 @ (push @ one @ sh))) != ((X149 @ sh @ one))) <=> (hoasinduction_lem3b_lthm = $true)) & (pushprop_lthm = $true) & (! [X451 : (term > $o),X452 : term,X450 : (subst > term > subst > $o)] : ((((X451 @ (lam @ X452))) = $true) | ~ ! [X463 : (subst > term > term)] : (~ ! [X465 : subst,X464 : subst,X466 : term] : (((X463 @ (comp @ X465 @ X464) @ (sub @ X466 @ X464))) = ((sub @ (X463 @ X465 @ X466) @ X464))) | ~ ! [X467 : term] : ((((X450 @ id @ (X463 @ id @ X467) @ id)) = $true) | (((X450 @ id @ X467 @ id)) != $true)) | (((X450 @ id @ (lam @ (X463 @ sh @ one)) @ id)) = $true)) | ~ ! [X455 : subst,X454 : subst,X456 : subst,X453 : term] : ((((X450 @ (comp @ X456 @ X454) @ (sub @ X453 @ X454) @ X455)) != $true) | (((X450 @ X456 @ X453 @ (comp @ X454 @ X455))) = $true)) | ~ ! [X462 : term] : (((X451 @ X462)) = ((X450 @ id @ X462 @ id))) | ~ ! [X460 : term,X458 : subst,X457 : subst,X459 : subst] : ((((X450 @ (comp @ X457 @ X459) @ (sub @ X460 @ X459) @ X458)) = $true) | (((X450 @ X457 @ X460 @ (comp @ X459 @ X458))) != $true)) | ~ ! [X461 : term] : ((((X451 @ X461)) != $true) | (((X451 @ (sub @ X452 @ (push @ X461 @ id)))) = $true))) <=> (hoasinduction_lem3v2 = $true)) & ((! [X305 : (subst > term > subst > $o)] : ~ ! [X306 : (term > $o)] : ~ ! [X307 : term] : (((X306 @ X307)) = ((X305 @ id @ X307 @ id))) => (! [X309 : term,X308 : (term > $o)] : (~ ! [X310 : term,X311 : term] : ((((X308 @ X311)) != $true) | (((X308 @ (ap @ X310 @ X311))) = $true) | (((X308 @ X310)) != $true)) | (((X308 @ X309)) = $true) | ~ ! [X312 : term] : ((((X308 @ X312)) = $true) | (((var @ X312)) != $true)) | ~ ! [X313 : term] : (~ ! [X314 : term] : ((((X308 @ (sub @ X313 @ (push @ X314 @ id)))) = $true) | (((X308 @ X314)) != $true)) | (((X308 @ (lam @ X313))) = $true))) => (! [X315 : term] : (((sub @ X315 @ id)) = X315) => (! [X317 : (subst > term > subst > $o),X316 : (term > $o),X318 : term] : (~ ! [X319 : (subst > term > term)] : (~ ! [X321 : subst,X320 : subst,X322 : term] : (((X319 @ (comp @ X321 @ X320) @ (sub @ X322 @ X320))) = ((sub @ (X319 @ X321 @ X322) @ X320))) | ~ ! [X323 : term] : ((((X317 @ id @ X323 @ id)) != $true) | (((X317 @ id @ (X319 @ id @ X323) @ id)) = $true)) | (((X317 @ id @ (lam @ (X319 @ sh @ one)) @ id)) = $true)) | ~ ! [X324 : term] : ((((X316 @ (sub @ X318 @ (push @ X324 @ id)))) = $true) | (((X316 @ X324)) != $true)) | ~ ! [X325 : term] : (((X317 @ id @ X325 @ id)) = ((X316 @ X325))) | (((X316 @ (lam @ X318))) = $true)) => ! [X327 : term,X326 : (subst > term > subst > $o)] : (~ ! [X341 : subst,X342 : subst,X339 : term,X340 : subst] : ((((X326 @ (comp @ X342 @ X341) @ (sub @ X339 @ X341) @ X340)) != $true) | (((X326 @ X342 @ X339 @ (comp @ X341 @ X340))) = $true)) | ~ ! [X333 : subst,X332 : term,X331 : subst,X330 : subst] : ((((X326 @ X331 @ X332 @ (comp @ X330 @ X333))) != $true) | (((X326 @ (comp @ X331 @ X330) @ (sub @ X332 @ X330) @ X333)) = $true)) | ~ ! [X334 : (subst > term > term)] : (~ ! [X336 : subst,X338 : subst,X337 : term] : (((sub @ (X334 @ X336 @ X337) @ X338)) = ((X334 @ (comp @ X336 @ X338) @ (sub @ X337 @ X338)))) | (((X326 @ id @ (lam @ (X334 @ sh @ one)) @ id)) = $true) | ~ ! [X335 : term] : ((((X326 @ id @ X335 @ id)) != $true) | (((X326 @ id @ (X334 @ id @ X335) @ id)) = $true))) | ~ ! [X343 : term] : ((((var @ (sub @ X343 @ id))) != $true) | (((X326 @ id @ X343 @ id)) = $true)) | ~ ! [X328 : term,X329 : term] : ((((X326 @ id @ X328 @ id)) != $true) | (((X326 @ id @ (ap @ (sub @ X329 @ id) @ X328) @ id)) = $true) | (((X326 @ id @ X329 @ id)) != $true)) | ($true = ((X326 @ id @ X327 @ id)))))))) <=> (hoasinduction_lthm_3 = $true)) & ((axvarcons = $true) <=> ! [X356 : subst,X357 : term] : (((sub @ one @ (push @ X357 @ X356))) = X357)) & (pushprop_lem3v2_lthm = $true) & ! [X72 : d_term,X73 : d_term] : ((((d2term @ X72)) = ((d2term @ X73))) => (X72 = X73)) & ((hoaslaminj_lthm = $true) <=> (! [X427 : term,X428 : subst] : (((sub @ one @ (push @ X427 @ X428))) = X427) => (! [X430 : subst,X429 : term] : (((comp @ sh @ (push @ X429 @ X430))) = X430) => (! [X431 : term,X432 : term] : ((((lam @ X432)) != ((lam @ X431))) | (X431 = X432)) => ! [X436 : (subst > term > term),X435 : subst,X433 : (subst > term > term),X434 : term] : ((((X433 @ X435 @ X434)) = ((X436 @ X435 @ X434))) | (((lam @ (X433 @ sh @ one))) != ((lam @ (X436 @ sh @ one)))) | ~ ! [X437 : subst,X438 : subst,X439 : term] : (((sub @ (X436 @ X438 @ X439) @ X437)) = ((X436 @ (comp @ X438 @ X437) @ (sub @ X439 @ X437)))) | ~ ! [X440 : term,X441 : subst,X442 : subst] : (((X433 @ (comp @ X441 @ X442) @ (sub @ X440 @ X442))) = ((sub @ (X433 @ X441 @ X440) @ X442)))))))) & ((! [X296 : term] : (((sub @ X296 @ id)) = X296) => (! [X299 : term,X298 : term,X297 : term,X300 : term] : ((((ap @ X298 @ X299)) != ((ap @ X297 @ X300))) | (X297 = X298)) => ! [X303 : term,X301 : term,X302 : term,X304 : term] : ((X301 = X303) | (((ap @ (sub @ X303 @ id) @ X302)) != ((ap @ (sub @ X301 @ id) @ X304)))))) <=> (hoasapinj1_lthm = $true)) & (hoaslamnotvar = $true) & (induction2_gthm = $true) & (id = ((d2subst @ d_id))) & ((! [X255 : term] : (((sub @ X255 @ id)) = X255) => (! [X257 : term,X258 : subst,X256 : (term > $o)] : (~ ! [X259 : term] : ((((X256 @ (sub @ X259 @ X258))) = $true) | ($true != ((var @ X259)))) | ~ ! [X262 : term] : ((((X256 @ (lam @ X262))) = $true) | ~ ! [X263 : term] : ((((X256 @ (sub @ X262 @ (push @ X263 @ id)))) = $true) | (((X256 @ X263)) != $true))) | ~ ! [X260 : term,X261 : term] : ((((X256 @ X260)) != $true) | (((X256 @ X261)) != $true) | (((X256 @ (ap @ X261 @ X260))) = $true)) | (((X256 @ (sub @ X257 @ X258))) = $true)) => ! [X264 : (term > $o),X265 : term] : (~ ! [X266 : term] : ((((X264 @ (lam @ X266))) = $true) | ~ ! [X267 : term] : ((((X264 @ X267)) != $true) | (((X264 @ (sub @ X266 @ (push @ X267 @ id)))) = $true))) | ~ ! [X270 : term] : ((((var @ X270)) != $true) | (((X264 @ X270)) = $true)) | (((X264 @ X265)) = $true) | ~ ! [X268 : term,X269 : term] : ((((X264 @ X269)) != $true) | (((X264 @ X268)) != $true) | (((X264 @ (ap @ X269 @ X268))) = $true))))) <=> (induction2_lthm = $true)) & (pushprop_lem1_gthm = $true) & (lamnotap != $true) & (! [X152 : subst,X153 : term] : (((comp @ sh @ (push @ X153 @ X152))) = X152) <=> (axshiftcons = $true)) & (axassoc = $true) & ((laminj = $true) <=> ! [X414 : term,X413 : term] : ((X413 = X414) | (((lam @ X413)) != ((lam @ X414))))) & (! [X203 : term,X204 : (subst > term > subst > $o)] : (($true = ((X204 @ id @ (lam @ X203) @ id))) | ~ ! [X210 : term] : ((((X204 @ id @ X210 @ id)) != $true) | (((X204 @ id @ (sub @ X203 @ (push @ X210 @ id)) @ id)) = $true)) | ~ ! [X205 : (subst > term > term)] : ((((X204 @ id @ (lam @ (X205 @ sh @ one)) @ id)) = $true) | ~ ! [X208 : term,X207 : subst,X206 : subst] : (((X205 @ (comp @ X206 @ X207) @ (sub @ X208 @ X207))) = ((sub @ (X205 @ X206 @ X208) @ X207))) | ~ ! [X209 : term] : ((((X204 @ id @ X209 @ id)) != $true) | ($true = ((X204 @ id @ (X205 @ id @ X209) @ id)))))) <=> (hoasinduction_lem3a = $true)) & (hoasapnotvar_lthm = $true) & (axapp = $true) & (pushprop_lem2v2 = $true) & (! [X251 : term,X252 : term,X254 : term,X253 : term] : ((((ap @ X254 @ X251)) != ((ap @ X253 @ X252))) | (X251 = X252)) <=> (apinj2 = $true)) & ((hoasinduction_lem3b = $true) <=> ! [X56 : term] : ~ ! [X57 : (subst > term > term)] : (((sub @ X56 @ (push @ one @ sh))) != ((X57 @ sh @ one)))) & ((hoasapinj2_lthm = $true) <=> (! [X77 : term,X76 : term,X74 : term,X75 : term] : ((X74 = X75) | (((ap @ X77 @ X74)) != ((ap @ X76 @ X75)))) => ! [X80 : term,X81 : term,X79 : term,X78 : term] : ((((ap @ (sub @ X81 @ id) @ X78)) != ((ap @ (sub @ X79 @ id) @ X80))) | (X78 = X80)))) & ((hoasinduction_no_psi_cond = $true) <=> ! [X174 : (subst > term > subst > $o),X173 : term] : (~ ! [X177 : (subst > term > term)] : ((((X174 @ id @ (lam @ (X177 @ sh @ one)) @ id)) = $true) | ~ ! [X178 : term] : ((((X174 @ id @ (X177 @ id @ X178) @ id)) = $true) | (((X174 @ id @ X178 @ id)) != $true)) | ~ ! [X179 : subst,X180 : subst,X181 : term] : (((sub @ (X177 @ X179 @ X181) @ X180)) = ((X177 @ (comp @ X179 @ X180) @ (sub @ X181 @ X180))))) | ~ ! [X175 : term,X176 : term] : ((((X174 @ id @ (ap @ (sub @ X176 @ id) @ X175) @ id)) = $true) | (((X174 @ id @ X176 @ id)) != $true) | (((X174 @ id @ X175 @ id)) != $true)) | (((X174 @ id @ X173 @ id)) = $true))) & (! [X490 : (subst > term > subst > $o),X489 : term] : (~ ! [X502 : term,X499 : subst,X501 : subst,X500 : subst] : ((((X490 @ X499 @ X502 @ (comp @ X500 @ X501))) != $true) | ($true = ((X490 @ (comp @ X499 @ X500) @ (sub @ X502 @ X500) @ X501)))) | ~ ! [X504 : subst,X505 : subst,X506 : subst,X503 : term] : ((((X490 @ X504 @ X503 @ (comp @ X506 @ X505))) = $true) | (((X490 @ (comp @ X504 @ X506) @ (sub @ X503 @ X506) @ X505)) != $true)) | ~ ! [X493 : (subst > term > term)] : (~ ! [X497 : term] : (($true != ((X490 @ id @ X497 @ id))) | (((X490 @ id @ (X493 @ id @ X497) @ id)) = $true)) | (((X490 @ id @ (lam @ (X493 @ sh @ one)) @ id)) = $true) | ~ ! [X496 : term,X494 : subst,X495 : subst] : (((X493 @ (comp @ X494 @ X495) @ (sub @ X496 @ X495))) = ((sub @ (X493 @ X494 @ X496) @ X495)))) | ~ ! [X492 : term,X491 : term] : ((((X490 @ id @ X491 @ id)) != $true) | (((X490 @ id @ (ap @ (sub @ X492 @ id) @ X491) @ id)) = $true) | (((X490 @ id @ X492 @ id)) != $true)) | (((X490 @ id @ X489 @ id)) = $true) | ~ ! [X498 : term] : ((((var @ (sub @ X498 @ id))) != $true) | (((X490 @ id @ X498 @ id)) = $true))) <=> (hoasinduction = $true)) & (termmset_gthm = $true) & ! [X354 : subst] : ? [X355 : d_subst] : (((d2subst @ X355)) = X354) & (ulamvarind = $true) & (hoasinduction_lem1_lthm = $true) & ((! [X371 : (subst > term > subst > $o)] : ~ ! [X372 : (term > $o)] : ~ ! [X373 : term] : (((X371 @ id @ X373 @ id)) = ((X372 @ X373))) => (! [X374 : term,X375 : (term > $o)] : (~ ! [X379 : term,X380 : term] : ((((X375 @ X380)) != $true) | (((X375 @ X379)) != $true) | (((X375 @ (ap @ X380 @ X379))) = $true)) | ~ ! [X376 : term] : ((((X375 @ X376)) = $true) | ($true != ((var @ X376)))) | (((X375 @ X374)) = $true) | ~ ! [X377 : term] : (~ ! [X378 : term] : ((((X375 @ (sub @ X377 @ (push @ X378 @ id)))) = $true) | (((X375 @ X378)) != $true)) | (((X375 @ (lam @ X377))) = $true))) => (! [X381 : term] : (((sub @ X381 @ id)) = X381) => (! [X383 : term,X382 : (subst > term > subst > $o),X384 : (term > $o)] : (~ ! [X385 : term] : ((((X384 @ X385)) != $true) | (((X384 @ (sub @ X383 @ (push @ X385 @ id)))) = $true)) | ~ ! [X386 : term] : (((X382 @ id @ X386 @ id)) = ((X384 @ X386))) | ~ ! [X387 : (subst > term > term)] : ((((X382 @ id @ (lam @ (X387 @ sh @ one)) @ id)) = $true) | ~ ! [X391 : term] : ((((X382 @ id @ (X387 @ id @ X391) @ id)) = $true) | (((X382 @ id @ X391 @ id)) != $true)) | ~ ! [X388 : subst,X389 : term,X390 : subst] : (((X387 @ (comp @ X390 @ X388) @ (sub @ X389 @ X388))) = ((sub @ (X387 @ X390 @ X389) @ X388)))) | (((X384 @ (lam @ X383))) = $true)) => ! [X393 : term,X392 : (subst > term > subst > $o)] : ((((X392 @ id @ X393 @ id)) = $true) | ~ ! [X395 : term,X394 : term] : ((((X392 @ id @ (ap @ (sub @ X394 @ id) @ X395) @ id)) = $true) | ($true != ((X392 @ id @ X395 @ id))) | (((X392 @ id @ X394 @ id)) != $true)) | ~ ! [X396 : (subst > term > term)] : (~ ! [X397 : term] : ((((X392 @ id @ (X396 @ id @ X397) @ id)) = $true) | (((X392 @ id @ X397 @ id)) != $true)) | (((X392 @ id @ (lam @ (X396 @ sh @ one)) @ id)) = $true) | ~ ! [X400 : subst,X398 : term,X399 : subst] : (((X396 @ (comp @ X399 @ X400) @ (sub @ X398 @ X400))) = ((sub @ (X396 @ X399 @ X398) @ X400))))))))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (pushprop_lem2v2_gthm = $true) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem2_gthm = $true) & (((var @ (d2term @ d_one))) != $true) & (hoasinduction_lem3_gthm = $true) & ((hoasinduction_lem0_lthm = $true) <=> ! [X66 : (subst > term > subst > $o)] : ~ ! [X67 : (term > $o)] : ~ ! [X68 : term] : (((X67 @ X68)) = ((X66 @ id @ X68 @ id)))) & (! [X0 : (subst > term > subst > $o),X1 : term] : (~ ! [X3 : (subst > term > term)] : (~ ! [X5 : subst,X7 : term,X6 : subst] : (((X3 @ (comp @ X6 @ X5) @ (sub @ X7 @ X5))) = ((sub @ (X3 @ X6 @ X7) @ X5))) | (((X0 @ id @ (lam @ (X3 @ sh @ one)) @ id)) = $true) | ~ ! [X4 : term] : ((((X0 @ id @ (X3 @ id @ X4) @ id)) = $true) | (((X0 @ id @ X4 @ id)) != $true))) | ~ ! [X2 : term] : ((((X0 @ id @ (sub @ X1 @ (push @ X2 @ id)) @ id)) = $true) | (((X0 @ id @ X2 @ id)) != $true)) | (((X0 @ id @ (lam @ X1) @ id)) = $true) | ~ ! [X15 : subst,X13 : subst,X14 : subst,X12 : term] : ((((X0 @ X15 @ X12 @ (comp @ X14 @ X13))) != $true) | (((X0 @ (comp @ X15 @ X14) @ (sub @ X12 @ X14) @ X13)) = $true)) | ~ ! [X10 : subst,X11 : term,X8 : subst,X9 : subst] : ((((X0 @ (comp @ X9 @ X10) @ (sub @ X11 @ X10) @ X8)) != $true) | ($true = ((X0 @ X9 @ X11 @ (comp @ X10 @ X8)))))) <=> (hoasinduction_lem3 = $true)) & ((hoasinduction_lem2 = $true) <=> ! [X477 : (subst > term > subst > $o),X478 : term,X476 : term] : (~ ! [X480 : subst,X482 : subst,X479 : subst,X481 : term] : ((((X477 @ (comp @ X480 @ X482) @ (sub @ X481 @ X482) @ X479)) != $true) | (((X477 @ X480 @ X481 @ (comp @ X482 @ X479))) = $true)) | (((X477 @ id @ X476 @ id)) != $true) | ~ ! [X485 : subst,X488 : subst,X487 : subst,X486 : term] : ((((X477 @ (comp @ X485 @ X488) @ (sub @ X486 @ X488) @ X487)) = $true) | ($true != ((X477 @ X485 @ X486 @ (comp @ X488 @ X487))))) | (((X477 @ id @ X478 @ id)) != $true) | ~ ! [X484 : term,X483 : term] : ((((X477 @ id @ (ap @ (sub @ X483 @ id) @ X484) @ id)) = $true) | (((X477 @ id @ X483 @ id)) != $true) | (((X477 @ id @ X484 @ id)) != $true)) | (((X477 @ id @ (ap @ X476 @ X478) @ id)) = $true))) & (! [X529 : term,X528 : (subst > term > subst > $o)] : (~ ! [X530 : (subst > term > term),X531 : term] : (~ ! [X535 : term] : ((((X528 @ id @ (X530 @ id @ X535) @ id)) = $true) | ($true != ((X528 @ id @ X535 @ id)))) | (((X528 @ id @ (lam @ (X530 @ sh @ one)) @ id)) = $true) | ~ ! [X534 : term,X532 : subst,X533 : subst] : (((sub @ (X530 @ X533 @ X534) @ X532)) = ((sub @ (sub @ X531 @ (push @ X534 @ X533)) @ X532))) | ~ ! [X538 : subst,X536 : subst,X537 : term] : (((sub @ X531 @ (push @ (sub @ X537 @ X538) @ (comp @ X536 @ X538)))) = ((X530 @ (comp @ X536 @ X538) @ (sub @ X537 @ X538))))) | ~ ! [X539 : term] : ((((X528 @ id @ X539 @ id)) != $true) | (((X528 @ id @ (sub @ X529 @ (push @ X539 @ id)) @ id)) = $true)) | (((X528 @ id @ (lam @ (sub @ X529 @ (push @ one @ sh))) @ id)) = $true)) <=> (hoasinduction_lem3aaa = $true)) & (pushprop_gthm = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (ulamvarsh = $true) & ((hoasapinj1 = $true) <=> ! [X144 : term,X147 : term,X146 : term,X145 : term] : ((((ap @ (sub @ X147 @ id) @ X146)) != ((ap @ (sub @ X145 @ id) @ X144))) | (X145 = X147))) & (apnotvar = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (hoasinduction_lem3b_gthm = $true) & (! [X232 : (subst > term > subst > $o),X233 : term] : (~ ! [X239 : term] : ((((X232 @ id @ X239 @ id)) != $true) | (((X232 @ id @ (sub @ X233 @ (push @ X239 @ id)) @ id)) = $true)) | ~ ! [X234 : (subst > term > term)] : ((((X232 @ id @ (lam @ (X234 @ sh @ one)) @ id)) = $true) | ~ ! [X235 : term] : (($true != ((X232 @ id @ X235 @ id))) | (((X232 @ id @ (X234 @ id @ X235) @ id)) = $true)) | ~ ! [X238 : subst,X236 : subst,X237 : term] : (((sub @ (X234 @ X238 @ X237) @ X236)) = ((X234 @ (comp @ X238 @ X236) @ (sub @ X237 @ X236))))) | (((X232 @ id @ (lam @ (sub @ X233 @ (push @ one @ sh))) @ id)) = $true)) <=> (hoasinduction_lem3aa_lthm = $true))),
  inference(flattening,[],[f9])).
thf(f11,plain,(
  (pushprop_lem0_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (hoasapnotvar = $true) & (hoasinduction_lem3a_gthm = $true) & ((substmonoid_lthm = $true) <=> (((! [X424 : subst] : (((comp @ id @ X424)) = X424) & ! [X425 : subst] : (((comp @ X425 @ id)) = X425)) | ? [X423 : subst] : (((comp @ X423 @ id)) != X423)) | ? [X422 : subst] : (((comp @ id @ X422)) != X422))) & ((! [X190 : term,X188 : subst,X191 : (term > $o),X189 : (term > $o)] : (($true != ((X191 @ X190))) | ? [X192 : term] : (((X191 @ (sub @ X192 @ (push @ X190 @ X188)))) != ((X189 @ X192))) | (((X189 @ one)) = $true)) | ? [X187 : subst,X186 : term] : (((sub @ one @ (push @ X186 @ X187))) != X186)) <=> (pushprop_lem1v2_lthm = $true)) & (((d2subst @ d_id)) = sh) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (induction2lem_gthm = $true) & (hoasinduction_lem1_gthm = $true) & (hoaslamnotvar_lthm = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (! [X60 : (term > $o),X58 : term,X59 : subst] : (? [X65 : term] : ((((var @ X65)) = $true) & (((X60 @ (sub @ X65 @ X59))) != $true)) | ? [X64 : term,X63 : term] : ((((X60 @ X63)) = $true) & (((X60 @ (ap @ X63 @ X64))) != $true) & (((X60 @ X64)) = $true)) | ? [X61 : term] : ((((X60 @ (lam @ X61))) != $true) & ! [X62 : term] : ((((X60 @ (sub @ X61 @ (push @ X62 @ id)))) = $true) | (((X60 @ X62)) != $true))) | (((X60 @ (sub @ X58 @ X59))) = $true)) <=> (induction2lem = $true)) & (axmap = $true) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X507 : term] : ? [X508 : (subst > term > term)] : ! [X509 : term,X510 : subst] : (((sub @ X507 @ (push @ X509 @ X510))) = ((X508 @ X510 @ X509)))) & (hoasinduction_gthm = $true) & (! [X470 : term] : (((sub @ X470 @ id)) = X470) <=> (axvarid = $true)) & ((termmset = $true) <=> ! [X151 : term] : (((sub @ X151 @ id)) = X151)) & (! [X416 : (term > $o),X415 : term] : (? [X418 : term,X417 : term] : ((((X416 @ X417)) = $true) & (((X416 @ X418)) = $true) & (((X416 @ (ap @ X417 @ X418))) != $true)) | ? [X419 : term] : (($true != ((X416 @ (lam @ X419)))) & ! [X420 : term] : ((((X416 @ (sub @ X419 @ (push @ X420 @ id)))) = $true) | (((X416 @ X420)) != $true))) | ? [X421 : term] : ((((var @ X421)) = $true) & (((X416 @ X421)) != $true)) | (((X416 @ X415)) = $true)) <=> (induction2 = $true)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (! [X211 : term] : ? [X212 : (subst > term > term)] : ! [X213 : subst,X214 : term] : (((X212 @ X213 @ X214)) = ((sub @ X211 @ (push @ X214 @ X213)))) <=> (hoasinduction_lem3v2_f = $true)) & ((hoasinduction_lem3aa = $true) <=> ! [X83 : (subst > term > subst > $o),X82 : term] : ((((X83 @ id @ (lam @ (sub @ X82 @ (push @ one @ sh))) @ id)) = $true) | ? [X85 : (subst > term > term)] : (($true != ((X83 @ id @ (lam @ (X85 @ sh @ one)) @ id))) & ! [X89 : term] : ((((X83 @ id @ X89 @ id)) != $true) | (((X83 @ id @ (X85 @ id @ X89) @ id)) = $true)) & ! [X88 : subst,X87 : subst,X86 : term] : (((X85 @ (comp @ X88 @ X87) @ (sub @ X86 @ X87))) = ((sub @ (X85 @ X88 @ X86) @ X87)))) | ? [X84 : term] : ((((X83 @ id @ X84 @ id)) = $true) & (((X83 @ id @ (sub @ X82 @ (push @ X84 @ id)) @ id)) != $true)))) & ! [X69 : d_term] : (d_one = X69) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (pushprop_lthm_orig = $true) & (hoasinduction_lem3v2_gthm = $true) & ((hoasinduction_lem0 = $true) <=> ! [X358 : (subst > term > subst > $o)] : ? [X359 : (term > $o)] : ! [X360 : term] : (((X358 @ id @ X360 @ id)) = ((X359 @ X360)))) & (hoaslamnotvar_gthm = $true) & (! [X402 : subst,X403 : (term > $o),X401 : term] : ? [X404 : (term > $o)] : ! [X405 : term] : (((X403 @ (sub @ X405 @ (push @ X401 @ X402)))) = ((X404 @ X405))) <=> (pushprop_lem0_lthm = $true)) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (substmonoid_gthm = $true) & ! [X512 : d_subst,X511 : d_subst] : ((X511 = X512) | (((d2subst @ X511)) != ((d2subst @ X512)))) & (hoaslamnotap_gthm = $true) & (! [X241 : (term > $o),X240 : term] : (? [X242 : term] : ((((X241 @ X242)) = $true) & (((X241 @ (lam @ X242))) != $true)) | ? [X245 : term] : ((((var @ X245)) = $true) & (((X241 @ X245)) != $true)) | ? [X244 : term,X243 : term] : ((((X241 @ X243)) = $true) & (((X241 @ (ap @ X244 @ X243))) != $true) & (((X241 @ X244)) = $true)) | (((X241 @ X240)) = $true)) <=> (induction = $true)) & (! [X361 : (subst > term > term),X363 : subst,X362 : term,X364 : (subst > term > term)] : (? [X365 : term,X366 : subst,X367 : subst] : (((X364 @ (comp @ X366 @ X367) @ (sub @ X365 @ X367))) != ((sub @ (X364 @ X366 @ X365) @ X367))) | ? [X370 : subst,X369 : term,X368 : subst] : (((sub @ (X361 @ X370 @ X369) @ X368)) != ((X361 @ (comp @ X370 @ X368) @ (sub @ X369 @ X368)))) | (((X364 @ X363 @ X362)) = ((X361 @ X363 @ X362))) | (((lam @ (X361 @ sh @ one))) != ((lam @ (X364 @ sh @ one))))) <=> (hoaslaminj = $true)) & (! [X410 : subst,X409 : (term > $o),X408 : term] : ? [X411 : (term > $o)] : ! [X412 : term] : (((X409 @ (sub @ X412 @ (push @ X408 @ X410)))) = ((X411 @ X412))) <=> (pushprop_lem0 = $true)) & (hoasinduction_lem1 = $true) & ! [X344 : d_subst] : (d_id = X344) & (((! [X281 : (subst > term > subst > $o),X280 : term] : ((((X281 @ id @ (lam @ X280) @ id)) = $true) | ? [X291 : (subst > term > term)] : (! [X295 : term] : ((((X281 @ id @ (X291 @ id @ X295) @ id)) = $true) | (((X281 @ id @ X295 @ id)) != $true)) & (((X281 @ id @ (lam @ (X291 @ sh @ one)) @ id)) != $true) & ! [X294 : term,X293 : subst,X292 : subst] : (((sub @ (X291 @ X292 @ X294) @ X293)) = ((X291 @ (comp @ X292 @ X293) @ (sub @ X294 @ X293))))) | ? [X289 : term,X290 : subst,X288 : subst,X287 : subst] : ((((X281 @ X290 @ X289 @ (comp @ X288 @ X287))) != $true) & (((X281 @ (comp @ X290 @ X288) @ (sub @ X289 @ X288) @ X287)) = $true)) | ? [X286 : term] : ((((X281 @ id @ X286 @ id)) = $true) & (((X281 @ id @ (sub @ X280 @ (push @ X286 @ id)) @ id)) != $true)) | ? [X285 : subst,X282 : term,X284 : subst,X283 : subst] : ((((X281 @ (comp @ X284 @ X283) @ (sub @ X282 @ X283) @ X285)) != $true) & ($true = ((X281 @ X284 @ X282 @ (comp @ X283 @ X285)))))) | ? [X272 : (subst > term > subst > $o),X273 : term] : (! [X274 : term] : ((((X272 @ id @ X274 @ id)) != $true) | (((X272 @ id @ (sub @ X273 @ (push @ X274 @ id)) @ id)) = $true)) & (((X272 @ id @ (lam @ (sub @ X273 @ (push @ one @ sh))) @ id)) != $true) & ! [X275 : (subst > term > term)] : (($true = ((X272 @ id @ (lam @ (X275 @ sh @ one)) @ id))) | ? [X276 : term] : ((((X272 @ id @ X276 @ id)) = $true) & ($true != ((X272 @ id @ (X275 @ id @ X276) @ id)))) | ? [X277 : subst,X278 : term,X279 : subst] : (((X275 @ (comp @ X279 @ X277) @ (sub @ X278 @ X277))) != ((sub @ (X275 @ X279 @ X278) @ X277)))))) | ? [X271 : term] : (((sub @ X271 @ id)) != X271)) <=> (hoasinduction_lem3_lthm = $true)) & (pushprop_lem2v2_lthm = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ((hoasapinj2 = $true) <=> ! [X185 : term,X183 : term,X184 : term,X182 : term] : ((((ap @ (sub @ X182 @ id) @ X184)) != ((ap @ (sub @ X183 @ id) @ X185))) | (X184 = X185))) & (((((! [X165 : subst,X167 : term,X166 : (term > $o)] : (? [X172 : term] : ((((X166 @ (sub @ X172 @ X165))) != $true) & (((var @ X172)) = $true)) | ? [X171 : term,X170 : term] : ((((X166 @ X171)) = $true) & (((X166 @ X170)) = $true) & (((X166 @ (ap @ X170 @ X171))) != $true)) | ? [X168 : term] : ((((X166 @ (lam @ X168))) != $true) & ! [X169 : term] : (($true != ((X166 @ X169))) | (((X166 @ (sub @ X168 @ (push @ X169 @ id)))) = $true))) | (((X166 @ (sub @ X167 @ X165))) = $true)) | ? [X160 : (term > $o),X159 : term] : (! [X162 : term,X163 : term] : ((((X160 @ X163)) != $true) | (((X160 @ X162)) != $true) | (((X160 @ (ap @ X162 @ X163))) = $true)) & (((X160 @ X159)) != $true) & ! [X161 : term] : ((((X160 @ (lam @ X161))) = $true) | (((X160 @ X161)) != $true)) & ! [X164 : term] : ((((var @ X164)) != $true) | ($true = ((X160 @ X164)))))) | ? [X158 : subst] : (((comp @ X158 @ id)) != X158)) | ? [X157 : term,X156 : subst] : (((comp @ sh @ (push @ X157 @ X156))) != X156)) | ? [X154 : term,X155 : subst] : (((sub @ one @ (push @ X154 @ X155))) != X154)) <=> (induction2lem_lthm = $true)) & (hoaslaminj_gthm = $true) & (hoasinduction_lem1v2 = $true) & ((axidr = $true) <=> ! [X150 : subst] : (((comp @ X150 @ id)) = X150)) & ((pushprop_lem1 = $true) <=> ! [X246 : term,X247 : term,X250 : (term > $o),X248 : subst,X249 : (term > $o)] : ((((X249 @ X247)) != $true) | (((X250 @ (sub @ X247 @ (push @ X246 @ X248)))) = $true))) & (hoasinduction_lem2_lthm = $true) & (! [X51 : term,X50 : term,X52 : (subst > term > term)] : (? [X55 : term,X53 : subst,X54 : subst] : (((sub @ (X52 @ X53 @ X55) @ X54)) != ((X52 @ (comp @ X53 @ X54) @ (sub @ X55 @ X54)))) | (((lam @ (X52 @ sh @ one))) != ((ap @ (sub @ X50 @ id) @ X51)))) <=> (hoaslamnotap = $true)) & (hoasapinj2_gthm = $true) & ((termmset_lthm = $true) <=> (! [X449 : term] : (((sub @ X449 @ id)) = X449) | ? [X448 : term] : (((sub @ X448 @ id)) != X448))) & (hoasinduction_lem2v2_gthm = $true) & (pushprop_lem3v2 = $true) & (hoaslamnotap_lthm = $true) & (pushprop_lem1v2_gthm = $true) & (! [X426 : subst] : (((push @ (sub @ one @ X426) @ (comp @ sh @ X426))) = X426) <=> (axscons = $true)) & (axvarshift = $true) & (lamnotvar = $true) & (axabs = $true) & (pushprop = $true) & (! [X469 : term,X468 : term] : ((X468 = X469) | (((sub @ X469 @ sh)) != ((sub @ X468 @ sh)))) <=> (shinj = $true)) & (! [X194 : term,X195 : (subst > term > subst > $o),X193 : (term > $o)] : (? [X201 : term] : ((((X193 @ (sub @ X194 @ (push @ X201 @ id)))) != $true) & (((X193 @ X201)) = $true)) | ? [X202 : term] : (((X193 @ X202)) != ((X195 @ id @ X202 @ id))) | ? [X196 : (subst > term > term)] : (! [X199 : term,X197 : subst,X198 : subst] : (((X196 @ (comp @ X198 @ X197) @ (sub @ X199 @ X197))) = ((sub @ (X196 @ X198 @ X199) @ X197))) & ! [X200 : term] : ((((X195 @ id @ X200 @ id)) != $true) | (((X195 @ id @ (X196 @ id @ X200) @ id)) = $true)) & (((X195 @ id @ (lam @ (X196 @ sh @ one)) @ id)) != $true)) | (((X193 @ (lam @ X194))) = $true)) <=> (hoasinduction_lem3v2a = $true)) & ((hoasinduction_lthm = $true) <=> (((! [X126 : (subst > term > subst > $o),X127 : term] : ((((X126 @ id @ X127 @ id)) = $true) | ? [X142 : term,X143 : term] : ((((X126 @ id @ X143 @ id)) = $true) & (((X126 @ id @ (ap @ (sub @ X142 @ id) @ X143) @ id)) != $true) & (((X126 @ id @ X142 @ id)) = $true)) | ? [X139 : term,X137 : subst,X138 : subst,X140 : subst] : ((((X126 @ (comp @ X138 @ X140) @ (sub @ X139 @ X140) @ X137)) != $true) & (((X126 @ X138 @ X139 @ (comp @ X140 @ X137))) = $true)) | ? [X141 : term] : ((((var @ (sub @ X141 @ id))) = $true) & (((X126 @ id @ X141 @ id)) != $true)) | ? [X134 : term,X133 : subst,X136 : subst,X135 : subst] : ((((X126 @ X135 @ X134 @ (comp @ X133 @ X136))) != $true) & (((X126 @ (comp @ X135 @ X133) @ (sub @ X134 @ X133) @ X136)) = $true)) | ? [X128 : (subst > term > term)] : (! [X129 : term] : ((((X126 @ id @ X129 @ id)) != $true) | (((X126 @ id @ (X128 @ id @ X129) @ id)) = $true)) & (((X126 @ id @ (lam @ (X128 @ sh @ one)) @ id)) != $true) & ! [X131 : subst,X130 : term,X132 : subst] : (((X128 @ (comp @ X131 @ X132) @ (sub @ X130 @ X132))) = ((sub @ (X128 @ X131 @ X130) @ X132))))) | ? [X111 : term,X110 : (subst > term > subst > $o)] : (! [X125 : term] : (($true != ((X110 @ id @ X125 @ id))) | (((X110 @ id @ (sub @ X111 @ (push @ X125 @ id)) @ id)) = $true)) & ($true != ((X110 @ id @ (lam @ X111) @ id))) & ! [X114 : subst,X113 : subst,X115 : term,X112 : subst] : ((((X110 @ X112 @ X115 @ (comp @ X114 @ X113))) != $true) | (((X110 @ (comp @ X112 @ X114) @ (sub @ X115 @ X114) @ X113)) = $true)) & ! [X119 : subst,X117 : subst,X116 : subst,X118 : term] : ((((X110 @ (comp @ X119 @ X117) @ (sub @ X118 @ X117) @ X116)) != $true) | (((X110 @ X119 @ X118 @ (comp @ X117 @ X116))) = $true)) & ! [X120 : (subst > term > term)] : ((((X110 @ id @ (lam @ (X120 @ sh @ one)) @ id)) = $true) | ? [X121 : term] : ((((X110 @ id @ (X120 @ id @ X121) @ id)) != $true) & (((X110 @ id @ X121 @ id)) = $true)) | ? [X123 : subst,X122 : subst,X124 : term] : (((X120 @ (comp @ X122 @ X123) @ (sub @ X124 @ X123))) != ((sub @ (X120 @ X122 @ X124) @ X123)))))) | ? [X97 : term,X99 : term,X98 : (subst > term > subst > $o)] : (! [X104 : subst,X105 : subst,X106 : subst,X107 : term] : ((((X98 @ (comp @ X106 @ X104) @ (sub @ X107 @ X104) @ X105)) != $true) | (((X98 @ X106 @ X107 @ (comp @ X104 @ X105))) = $true)) & (((X98 @ id @ (ap @ X99 @ X97) @ id)) != $true) & ! [X108 : term,X109 : term] : ((((X98 @ id @ X108 @ id)) != $true) | (((X98 @ id @ X109 @ id)) != $true) | (((X98 @ id @ (ap @ (sub @ X108 @ id) @ X109) @ id)) = $true)) & (((X98 @ id @ X99 @ id)) = $true) & ! [X103 : subst,X100 : subst,X101 : term,X102 : subst] : ((((X98 @ (comp @ X102 @ X103) @ (sub @ X101 @ X103) @ X100)) = $true) | (((X98 @ X102 @ X101 @ (comp @ X103 @ X100))) != $true)) & ($true = ((X98 @ id @ X97 @ id))))) | ? [X90 : (term > $o),X91 : term] : (! [X92 : term] : ((((X90 @ (lam @ X92))) = $true) | ? [X93 : term] : ((((X90 @ X93)) = $true) & (((X90 @ (sub @ X92 @ (push @ X93 @ id)))) != $true))) & ! [X95 : term,X94 : term] : (($true = ((X90 @ (ap @ X94 @ X95)))) | (((X90 @ X94)) != $true) | (((X90 @ X95)) != $true)) & (((X90 @ X91)) != $true) & ! [X96 : term] : ((((var @ X96)) != $true) | (((X90 @ X96)) = $true))))) & ((hoasinduction_lem3a_lthm = $true) <=> ((! [X225 : (subst > term > subst > $o),X224 : term] : (? [X226 : (subst > term > term)] : ((((X225 @ id @ (lam @ (X226 @ sh @ one)) @ id)) != $true) & ! [X229 : subst,X230 : term,X228 : subst] : (((X226 @ (comp @ X228 @ X229) @ (sub @ X230 @ X229))) = ((sub @ (X226 @ X228 @ X230) @ X229))) & ! [X227 : term] : ((((X225 @ id @ X227 @ id)) != $true) | (((X225 @ id @ (X226 @ id @ X227) @ id)) = $true))) | ($true = ((X225 @ id @ (lam @ X224) @ id))) | ? [X231 : term] : (($true != ((X225 @ id @ (sub @ X224 @ (push @ X231 @ id)) @ id))) & (((X225 @ id @ X231 @ id)) = $true))) | ? [X216 : (subst > term > subst > $o),X217 : term] : (! [X218 : term] : ((((X216 @ id @ (sub @ X217 @ (push @ X218 @ id)) @ id)) = $true) | (((X216 @ id @ X218 @ id)) != $true)) & ! [X219 : (subst > term > term)] : ((((X216 @ id @ (lam @ (X219 @ sh @ one)) @ id)) = $true) | ? [X220 : term] : ((((X216 @ id @ (X219 @ id @ X220) @ id)) != $true) & (((X216 @ id @ X220 @ id)) = $true)) | ? [X222 : subst,X221 : subst,X223 : term] : (((sub @ (X219 @ X222 @ X223) @ X221)) != ((X219 @ (comp @ X222 @ X221) @ (sub @ X223 @ X221))))) & (((X216 @ id @ (lam @ (sub @ X217 @ (push @ one @ sh))) @ id)) != $true))) | ? [X215 : term] : (((sub @ X215 @ id)) != X215))) & (! [X471 : subst] : (((comp @ id @ X471)) = X471) <=> (axidl = $true)) & ((? [X16 : term] : (((sub @ X16 @ id)) != X16) | ! [X17 : term,X19 : (term > $o),X18 : (subst > term > subst > $o)] : (? [X20 : term] : (((X19 @ X20)) != ((X18 @ id @ X20 @ id))) | ? [X30 : (subst > term > term)] : (! [X34 : term] : ((((X18 @ id @ (X30 @ id @ X34) @ id)) = $true) | (((X18 @ id @ X34 @ id)) != $true)) & ! [X31 : subst,X33 : subst,X32 : term] : (((sub @ (X30 @ X33 @ X32) @ X31)) = ((X30 @ (comp @ X33 @ X31) @ (sub @ X32 @ X31)))) & (((X18 @ id @ (lam @ (X30 @ sh @ one)) @ id)) != $true)) | ? [X23 : subst,X24 : term,X22 : subst,X21 : subst] : ((((X18 @ (comp @ X22 @ X23) @ (sub @ X24 @ X23) @ X21)) != $true) & (((X18 @ X22 @ X24 @ (comp @ X23 @ X21))) = $true)) | (((X19 @ (lam @ X17))) = $true) | ? [X29 : term] : ((((X19 @ X29)) = $true) & (((X19 @ (sub @ X17 @ (push @ X29 @ id)))) != $true)) | ? [X26 : term,X25 : subst,X27 : subst,X28 : subst] : (($true = ((X18 @ (comp @ X28 @ X27) @ (sub @ X26 @ X27) @ X25))) & (((X18 @ X28 @ X26 @ (comp @ X27 @ X25))) != $true)))) <=> (hoasinduction_lem3v2_lthm = $true)) & (hoasapinj1_gthm = $true) & (hoasinduction_lem1v2_gthm = $true) & ! [X406 : term] : ? [X407 : d_term] : (((d2term @ X407)) = X406) & (! [X513 : (subst > term > subst > $o),X515 : term,X514 : term,X516 : (term > $o)] : ((((X516 @ (ap @ X515 @ X514))) = $true) | (((X516 @ X514)) != $true) | ? [X523 : term,X520 : subst,X521 : subst,X522 : subst] : ((((X513 @ X520 @ X523 @ (comp @ X521 @ X522))) = $true) & (((X513 @ (comp @ X520 @ X521) @ (sub @ X523 @ X521) @ X522)) != $true)) | ($true != ((X516 @ X515))) | ? [X518 : term,X517 : term] : ((((X513 @ id @ X517 @ id)) = $true) & (((X513 @ id @ (ap @ (sub @ X517 @ id) @ X518) @ id)) != $true) & (((X513 @ id @ X518 @ id)) = $true)) | ? [X527 : subst,X524 : subst,X526 : term,X525 : subst] : ((((X513 @ (comp @ X527 @ X524) @ (sub @ X526 @ X524) @ X525)) = $true) & (((X513 @ X527 @ X526 @ (comp @ X524 @ X525))) != $true)) | ? [X519 : term] : (((X513 @ id @ X519 @ id)) != ((X516 @ X519)))) <=> (hoasinduction_lem2v2 = $true)) & (! [X443 : (term > $o),X444 : term,X446 : (term > $o),X445 : subst] : (? [X447 : term] : (((X443 @ X447)) != ((X446 @ (sub @ X447 @ (push @ X444 @ X445))))) | (((X446 @ X444)) != $true) | (((X443 @ one)) = $true)) <=> (pushprop_lem1v2 = $true)) & (((! [X349 : subst,X353 : (term > $o),X352 : (term > $o),X350 : term,X351 : term] : ((((X352 @ (sub @ X351 @ (push @ X350 @ X349)))) = $true) | (((X353 @ X351)) != $true)) | ? [X347 : term,X348 : subst] : (((comp @ sh @ (push @ X347 @ X348))) != X348)) | ? [X345 : subst,X346 : term] : (((sub @ one @ (push @ X346 @ X345))) != X346)) <=> (pushprop_lem1_lthm = $true)) & (ulamvar1 != $true) & ((substmonoid = $true) <=> (! [X71 : subst] : (((comp @ id @ X71)) = X71) & ! [X70 : subst] : (((comp @ X70 @ id)) = X70))) & (! [X472 : term,X473 : term,X474 : term,X475 : term] : ((((ap @ X475 @ X473)) != ((ap @ X472 @ X474))) | (X472 = X475)) <=> (apinj1 = $true)) & (axclos = $true) & (((! [X40 : (term > $o),X41 : (subst > term > subst > $o),X42 : term] : (? [X45 : (subst > term > term)] : (! [X46 : term] : ((((X41 @ id @ X46 @ id)) != $true) | (((X41 @ id @ (X45 @ id @ X46) @ id)) = $true)) & ! [X47 : subst,X49 : term,X48 : subst] : (((X45 @ (comp @ X47 @ X48) @ (sub @ X49 @ X48))) = ((sub @ (X45 @ X47 @ X49) @ X48))) & (((X41 @ id @ (lam @ (X45 @ sh @ one)) @ id)) != $true)) | ? [X43 : term] : (((X41 @ id @ X43 @ id)) != ((X40 @ X43))) | (((X40 @ (lam @ X42))) = $true) | ? [X44 : term] : ((((X40 @ X44)) = $true) & (((X40 @ (sub @ X42 @ (push @ X44 @ id)))) != $true))) | ? [X39 : term] : (((sub @ X39 @ id)) != X39)) | ? [X35 : term] : ! [X36 : (subst > term > term)] : ? [X37 : term,X38 : subst] : (((X36 @ X38 @ X37)) != ((sub @ X35 @ (push @ X37 @ X38))))) <=> (hoasinduction_lem3v2a_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)))))))))))) & (one = ((d2term @ d_one))) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X148 : term] : ? [X149 : (subst > term > term)] : (((sub @ X148 @ (push @ one @ sh))) = ((X149 @ sh @ one)))) & (pushprop_lthm = $true) & ((hoasinduction_lem3v2 = $true) <=> ! [X450 : (subst > term > subst > $o),X451 : (term > $o),X452 : term] : ((((X451 @ (lam @ X452))) = $true) | ? [X458 : subst,X459 : subst,X457 : subst,X460 : term] : ((((X450 @ X457 @ X460 @ (comp @ X459 @ X458))) = $true) & (((X450 @ (comp @ X457 @ X459) @ (sub @ X460 @ X459) @ X458)) != $true)) | ? [X461 : term] : ((((X451 @ (sub @ X452 @ (push @ X461 @ id)))) != $true) & (((X451 @ X461)) = $true)) | ? [X453 : term,X455 : subst,X454 : subst,X456 : subst] : ((((X450 @ X456 @ X453 @ (comp @ X454 @ X455))) != $true) & (((X450 @ (comp @ X456 @ X454) @ (sub @ X453 @ X454) @ X455)) = $true)) | ? [X463 : (subst > term > term)] : (! [X467 : term] : ((((X450 @ id @ (X463 @ id @ X467) @ id)) = $true) | (((X450 @ id @ X467 @ id)) != $true)) & (((X450 @ id @ (lam @ (X463 @ sh @ one)) @ id)) != $true) & ! [X465 : subst,X464 : subst,X466 : term] : (((X463 @ (comp @ X465 @ X464) @ (sub @ X466 @ X464))) = ((sub @ (X463 @ X465 @ X466) @ X464)))) | ? [X462 : term] : (((X451 @ X462)) != ((X450 @ id @ X462 @ id))))) & (((((! [X327 : term,X326 : (subst > term > subst > $o)] : (? [X343 : term] : ((((var @ (sub @ X343 @ id))) = $true) & (((X326 @ id @ X343 @ id)) != $true)) | ? [X329 : term,X328 : term] : ((((X326 @ id @ X328 @ id)) = $true) & (((X326 @ id @ X329 @ id)) = $true) & (((X326 @ id @ (ap @ (sub @ X329 @ id) @ X328) @ id)) != $true)) | ? [X331 : subst,X333 : subst,X332 : term,X330 : subst] : ((((X326 @ X331 @ X332 @ (comp @ X330 @ X333))) = $true) & (((X326 @ (comp @ X331 @ X330) @ (sub @ X332 @ X330) @ X333)) != $true)) | ($true = ((X326 @ id @ X327 @ id))) | ? [X341 : subst,X342 : subst,X339 : term,X340 : subst] : ((((X326 @ (comp @ X342 @ X341) @ (sub @ X339 @ X341) @ X340)) = $true) & (((X326 @ X342 @ X339 @ (comp @ X341 @ X340))) != $true)) | ? [X334 : (subst > term > term)] : ((((X326 @ id @ (lam @ (X334 @ sh @ one)) @ id)) != $true) & ! [X335 : term] : ((((X326 @ id @ X335 @ id)) != $true) | (((X326 @ id @ (X334 @ id @ X335) @ id)) = $true)) & ! [X336 : subst,X338 : subst,X337 : term] : (((sub @ (X334 @ X336 @ X337) @ X338)) = ((X334 @ (comp @ X336 @ X338) @ (sub @ X337 @ X338)))))) | ? [X316 : (term > $o),X318 : term,X317 : (subst > term > subst > $o)] : (! [X319 : (subst > term > term)] : ((((X317 @ id @ (lam @ (X319 @ sh @ one)) @ id)) = $true) | ? [X323 : term] : ((((X317 @ id @ (X319 @ id @ X323) @ id)) != $true) & (((X317 @ id @ X323 @ id)) = $true)) | ? [X322 : term,X320 : subst,X321 : subst] : (((X319 @ (comp @ X321 @ X320) @ (sub @ X322 @ X320))) != ((sub @ (X319 @ X321 @ X322) @ X320)))) & ! [X324 : term] : ((((X316 @ (sub @ X318 @ (push @ X324 @ id)))) = $true) | (((X316 @ X324)) != $true)) & (((X316 @ (lam @ X318))) != $true) & ! [X325 : term] : (((X317 @ id @ X325 @ id)) = ((X316 @ X325))))) | ? [X315 : term] : (((sub @ X315 @ id)) != X315)) | ? [X309 : term,X308 : (term > $o)] : ((((X308 @ X309)) != $true) & ! [X312 : term] : ((((X308 @ X312)) = $true) | (((var @ X312)) != $true)) & ! [X313 : term] : (? [X314 : term] : ((((X308 @ (sub @ X313 @ (push @ X314 @ id)))) != $true) & (((X308 @ X314)) = $true)) | (((X308 @ (lam @ X313))) = $true)) & ! [X310 : term,X311 : term] : ((((X308 @ X311)) != $true) | (((X308 @ (ap @ X310 @ X311))) = $true) | (((X308 @ X310)) != $true)))) | ? [X305 : (subst > term > subst > $o)] : ! [X306 : (term > $o)] : ? [X307 : term] : (((X306 @ X307)) != ((X305 @ id @ X307 @ id)))) <=> (hoasinduction_lthm_3 = $true)) & ((axvarcons = $true) <=> ! [X356 : subst,X357 : term] : (((sub @ one @ (push @ X357 @ X356))) = X357)) & (pushprop_lem3v2_lthm = $true) & ! [X72 : d_term,X73 : d_term] : ((X72 = X73) | (((d2term @ X72)) != ((d2term @ X73)))) & ((hoaslaminj_lthm = $true) <=> (((! [X434 : term,X433 : (subst > term > term),X435 : subst,X436 : (subst > term > term)] : (? [X442 : subst,X440 : term,X441 : subst] : (((X433 @ (comp @ X441 @ X442) @ (sub @ X440 @ X442))) != ((sub @ (X433 @ X441 @ X440) @ X442))) | (((lam @ (X433 @ sh @ one))) != ((lam @ (X436 @ sh @ one)))) | ? [X437 : subst,X438 : subst,X439 : term] : (((sub @ (X436 @ X438 @ X439) @ X437)) != ((X436 @ (comp @ X438 @ X437) @ (sub @ X439 @ X437)))) | (((X433 @ X435 @ X434)) = ((X436 @ X435 @ X434)))) | ? [X432 : term,X431 : term] : ((((lam @ X432)) = ((lam @ X431))) & (X431 != X432))) | ? [X430 : subst,X429 : term] : (((comp @ sh @ (push @ X429 @ X430))) != X430)) | ? [X427 : term,X428 : subst] : (((sub @ one @ (push @ X427 @ X428))) != X427))) & (((! [X303 : term,X301 : term,X302 : term,X304 : term] : ((X301 = X303) | (((ap @ (sub @ X303 @ id) @ X302)) != ((ap @ (sub @ X301 @ id) @ X304)))) | ? [X297 : term,X300 : term,X298 : term,X299 : term] : ((X297 != X298) & (((ap @ X298 @ X299)) = ((ap @ X297 @ X300))))) | ? [X296 : term] : (((sub @ X296 @ id)) != X296)) <=> (hoasapinj1_lthm = $true)) & (hoaslamnotvar = $true) & (induction2_gthm = $true) & (id = ((d2subst @ d_id))) & (((! [X264 : (term > $o),X265 : term] : (? [X266 : term] : (! [X267 : term] : ((((X264 @ X267)) != $true) | (((X264 @ (sub @ X266 @ (push @ X267 @ id)))) = $true)) & (((X264 @ (lam @ X266))) != $true)) | ? [X269 : term,X268 : term] : ((((X264 @ X268)) = $true) & (((X264 @ X269)) = $true) & (((X264 @ (ap @ X269 @ X268))) != $true)) | (((X264 @ X265)) = $true) | ? [X270 : term] : ((((X264 @ X270)) != $true) & (((var @ X270)) = $true))) | ? [X257 : term,X256 : (term > $o),X258 : subst] : (! [X259 : term] : ((((X256 @ (sub @ X259 @ X258))) = $true) | ($true != ((var @ X259)))) & (((X256 @ (sub @ X257 @ X258))) != $true) & ! [X262 : term] : ((((X256 @ (lam @ X262))) = $true) | ? [X263 : term] : ((((X256 @ X263)) = $true) & (((X256 @ (sub @ X262 @ (push @ X263 @ id)))) != $true))) & ! [X260 : term,X261 : term] : ((((X256 @ X260)) != $true) | (((X256 @ X261)) != $true) | (((X256 @ (ap @ X261 @ X260))) = $true)))) | ? [X255 : term] : (((sub @ X255 @ id)) != X255)) <=> (induction2_lthm = $true)) & (pushprop_lem1_gthm = $true) & (lamnotap != $true) & (! [X152 : subst,X153 : term] : (((comp @ sh @ (push @ X153 @ X152))) = X152) <=> (axshiftcons = $true)) & (axassoc = $true) & ((laminj = $true) <=> ! [X414 : term,X413 : term] : ((X413 = X414) | (((lam @ X413)) != ((lam @ X414))))) & (! [X203 : term,X204 : (subst > term > subst > $o)] : (? [X210 : term] : ((((X204 @ id @ X210 @ id)) = $true) & (((X204 @ id @ (sub @ X203 @ (push @ X210 @ id)) @ id)) != $true)) | ? [X205 : (subst > term > term)] : (! [X208 : term,X207 : subst,X206 : subst] : (((X205 @ (comp @ X206 @ X207) @ (sub @ X208 @ X207))) = ((sub @ (X205 @ X206 @ X208) @ X207))) & (((X204 @ id @ (lam @ (X205 @ sh @ one)) @ id)) != $true) & ! [X209 : term] : ((((X204 @ id @ X209 @ id)) != $true) | ($true = ((X204 @ id @ (X205 @ id @ X209) @ id))))) | ($true = ((X204 @ id @ (lam @ X203) @ id)))) <=> (hoasinduction_lem3a = $true)) & (hoasapnotvar_lthm = $true) & (axapp = $true) & (pushprop_lem2v2 = $true) & (! [X251 : term,X252 : term,X254 : term,X253 : term] : ((((ap @ X254 @ X251)) != ((ap @ X253 @ X252))) | (X251 = X252)) <=> (apinj2 = $true)) & ((hoasinduction_lem3b = $true) <=> ! [X56 : term] : ? [X57 : (subst > term > term)] : (((sub @ X56 @ (push @ one @ sh))) = ((X57 @ sh @ one)))) & ((? [X77 : term,X74 : term,X75 : term,X76 : term] : ((((ap @ X77 @ X74)) = ((ap @ X76 @ X75))) & (X74 != X75)) | ! [X80 : term,X81 : term,X79 : term,X78 : term] : ((((ap @ (sub @ X81 @ id) @ X78)) != ((ap @ (sub @ X79 @ id) @ X80))) | (X78 = X80))) <=> (hoasapinj2_lthm = $true)) & ((hoasinduction_no_psi_cond = $true) <=> ! [X173 : term,X174 : (subst > term > subst > $o)] : (? [X176 : term,X175 : term] : ((((X174 @ id @ (ap @ (sub @ X176 @ id) @ X175) @ id)) != $true) & (((X174 @ id @ X175 @ id)) = $true) & (((X174 @ id @ X176 @ id)) = $true)) | ? [X177 : (subst > term > term)] : (! [X179 : subst,X180 : subst,X181 : term] : (((sub @ (X177 @ X179 @ X181) @ X180)) = ((X177 @ (comp @ X179 @ X180) @ (sub @ X181 @ X180)))) & ! [X178 : term] : ((((X174 @ id @ (X177 @ id @ X178) @ id)) = $true) | (((X174 @ id @ X178 @ id)) != $true)) & (((X174 @ id @ (lam @ (X177 @ sh @ one)) @ id)) != $true)) | (((X174 @ id @ X173 @ id)) = $true))) & ((hoasinduction = $true) <=> ! [X489 : term,X490 : (subst > term > subst > $o)] : (? [X504 : subst,X503 : term,X505 : subst,X506 : subst] : ((((X490 @ (comp @ X504 @ X506) @ (sub @ X503 @ X506) @ X505)) = $true) & (((X490 @ X504 @ X503 @ (comp @ X506 @ X505))) != $true)) | (((X490 @ id @ X489 @ id)) = $true) | ? [X498 : term] : ((((var @ (sub @ X498 @ id))) = $true) & (((X490 @ id @ X498 @ id)) != $true)) | ? [X501 : subst,X499 : subst,X502 : term,X500 : subst] : ((((X490 @ X499 @ X502 @ (comp @ X500 @ X501))) = $true) & ($true != ((X490 @ (comp @ X499 @ X500) @ (sub @ X502 @ X500) @ X501)))) | ? [X492 : term,X491 : term] : ((((X490 @ id @ X492 @ id)) = $true) & (((X490 @ id @ (ap @ (sub @ X492 @ id) @ X491) @ id)) != $true) & (((X490 @ id @ X491 @ id)) = $true)) | ? [X493 : (subst > term > term)] : (! [X497 : term] : (($true != ((X490 @ id @ X497 @ id))) | (((X490 @ id @ (X493 @ id @ X497) @ id)) = $true)) & (((X490 @ id @ (lam @ (X493 @ sh @ one)) @ id)) != $true) & ! [X496 : term,X494 : subst,X495 : subst] : (((X493 @ (comp @ X494 @ X495) @ (sub @ X496 @ X495))) = ((sub @ (X493 @ X494 @ X496) @ X495)))))) & (termmset_gthm = $true) & ! [X354 : subst] : ? [X355 : d_subst] : (((d2subst @ X355)) = X354) & (ulamvarind = $true) & (hoasinduction_lem1_lthm = $true) & (((((! [X393 : term,X392 : (subst > term > subst > $o)] : ((((X392 @ id @ X393 @ id)) = $true) | ? [X396 : (subst > term > term)] : ((((X392 @ id @ (lam @ (X396 @ sh @ one)) @ id)) != $true) & ! [X400 : subst,X398 : term,X399 : subst] : (((X396 @ (comp @ X399 @ X400) @ (sub @ X398 @ X400))) = ((sub @ (X396 @ X399 @ X398) @ X400))) & ! [X397 : term] : ((((X392 @ id @ (X396 @ id @ X397) @ id)) = $true) | (((X392 @ id @ X397 @ id)) != $true))) | ? [X395 : term,X394 : term] : ((((X392 @ id @ (ap @ (sub @ X394 @ id) @ X395) @ id)) != $true) & ($true = ((X392 @ id @ X395 @ id))) & (((X392 @ id @ X394 @ id)) = $true))) | ? [X382 : (subst > term > subst > $o),X384 : (term > $o),X383 : term] : ((((X384 @ (lam @ X383))) != $true) & ! [X386 : term] : (((X382 @ id @ X386 @ id)) = ((X384 @ X386))) & ! [X387 : (subst > term > term)] : (? [X391 : term] : ((((X382 @ id @ X391 @ id)) = $true) & (((X382 @ id @ (X387 @ id @ X391) @ id)) != $true)) | ? [X390 : subst,X388 : subst,X389 : term] : (((X387 @ (comp @ X390 @ X388) @ (sub @ X389 @ X388))) != ((sub @ (X387 @ X390 @ X389) @ X388))) | (((X382 @ id @ (lam @ (X387 @ sh @ one)) @ id)) = $true)) & ! [X385 : term] : ((((X384 @ X385)) != $true) | (((X384 @ (sub @ X383 @ (push @ X385 @ id)))) = $true)))) | ? [X381 : term] : (((sub @ X381 @ id)) != X381)) | ? [X374 : term,X375 : (term > $o)] : (! [X377 : term] : ((((X375 @ (lam @ X377))) = $true) | ? [X378 : term] : ((((X375 @ (sub @ X377 @ (push @ X378 @ id)))) != $true) & (((X375 @ X378)) = $true))) & ! [X376 : term] : ((((X375 @ X376)) = $true) | ($true != ((var @ X376)))) & ! [X379 : term,X380 : term] : ((((X375 @ X380)) != $true) | (((X375 @ X379)) != $true) | (((X375 @ (ap @ X380 @ X379))) = $true)) & (((X375 @ X374)) != $true))) | ? [X371 : (subst > term > subst > $o)] : ! [X372 : (term > $o)] : ? [X373 : term] : (((X371 @ id @ X373 @ id)) != ((X372 @ X373)))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (pushprop_lem2v2_gthm = $true) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem2_gthm = $true) & (((var @ (d2term @ d_one))) != $true) & (hoasinduction_lem3_gthm = $true) & ((hoasinduction_lem0_lthm = $true) <=> ! [X66 : (subst > term > subst > $o)] : ? [X67 : (term > $o)] : ! [X68 : term] : (((X67 @ X68)) = ((X66 @ id @ X68 @ id)))) & ((hoasinduction_lem3 = $true) <=> ! [X0 : (subst > term > subst > $o),X1 : term] : (? [X11 : term,X9 : subst,X8 : subst,X10 : subst] : ((((X0 @ (comp @ X9 @ X10) @ (sub @ X11 @ X10) @ X8)) = $true) & ($true != ((X0 @ X9 @ X11 @ (comp @ X10 @ X8))))) | (((X0 @ id @ (lam @ X1) @ id)) = $true) | ? [X15 : subst,X14 : subst,X12 : term,X13 : subst] : ((((X0 @ X15 @ X12 @ (comp @ X14 @ X13))) = $true) & (((X0 @ (comp @ X15 @ X14) @ (sub @ X12 @ X14) @ X13)) != $true)) | ? [X3 : (subst > term > term)] : (! [X4 : term] : ((((X0 @ id @ (X3 @ id @ X4) @ id)) = $true) | (((X0 @ id @ X4 @ id)) != $true)) & ! [X5 : subst,X7 : term,X6 : subst] : (((X3 @ (comp @ X6 @ X5) @ (sub @ X7 @ X5))) = ((sub @ (X3 @ X6 @ X7) @ X5))) & (((X0 @ id @ (lam @ (X3 @ sh @ one)) @ id)) != $true)) | ? [X2 : term] : ((((X0 @ id @ X2 @ id)) = $true) & (((X0 @ id @ (sub @ X1 @ (push @ X2 @ id)) @ id)) != $true)))) & (! [X478 : term,X476 : term,X477 : (subst > term > subst > $o)] : (? [X484 : term,X483 : term] : ((((X477 @ id @ (ap @ (sub @ X483 @ id) @ X484) @ id)) != $true) & (((X477 @ id @ X484 @ id)) = $true) & (((X477 @ id @ X483 @ id)) = $true)) | ? [X482 : subst,X479 : subst,X481 : term,X480 : subst] : ((((X477 @ (comp @ X480 @ X482) @ (sub @ X481 @ X482) @ X479)) = $true) & (((X477 @ X480 @ X481 @ (comp @ X482 @ X479))) != $true)) | (((X477 @ id @ (ap @ X476 @ X478) @ id)) = $true) | ? [X487 : subst,X486 : term,X488 : subst,X485 : subst] : (($true = ((X477 @ X485 @ X486 @ (comp @ X488 @ X487)))) & (((X477 @ (comp @ X485 @ X488) @ (sub @ X486 @ X488) @ X487)) != $true)) | (((X477 @ id @ X478 @ id)) != $true) | (((X477 @ id @ X476 @ id)) != $true)) <=> (hoasinduction_lem2 = $true)) & ((hoasinduction_lem3aaa = $true) <=> ! [X529 : term,X528 : (subst > term > subst > $o)] : ((((X528 @ id @ (lam @ (sub @ X529 @ (push @ one @ sh))) @ id)) = $true) | ? [X531 : term,X530 : (subst > term > term)] : ((((X528 @ id @ (lam @ (X530 @ sh @ one)) @ id)) != $true) & ! [X538 : subst,X536 : subst,X537 : term] : (((sub @ X531 @ (push @ (sub @ X537 @ X538) @ (comp @ X536 @ X538)))) = ((X530 @ (comp @ X536 @ X538) @ (sub @ X537 @ X538)))) & ! [X535 : term] : ((((X528 @ id @ (X530 @ id @ X535) @ id)) = $true) | ($true != ((X528 @ id @ X535 @ id)))) & ! [X534 : term,X532 : subst,X533 : subst] : (((sub @ (X530 @ X533 @ X534) @ X532)) = ((sub @ (sub @ X531 @ (push @ X534 @ X533)) @ X532)))) | ? [X539 : term] : ((((X528 @ id @ X539 @ id)) = $true) & (((X528 @ id @ (sub @ X529 @ (push @ X539 @ id)) @ id)) != $true)))) & (pushprop_gthm = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (ulamvarsh = $true) & ((hoasapinj1 = $true) <=> ! [X144 : term,X147 : term,X146 : term,X145 : term] : ((((ap @ (sub @ X147 @ id) @ X146)) != ((ap @ (sub @ X145 @ id) @ X144))) | (X145 = X147))) & (apnotvar = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (hoasinduction_lem3b_gthm = $true) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X233 : term,X232 : (subst > term > subst > $o)] : (? [X234 : (subst > term > term)] : (! [X238 : subst,X236 : subst,X237 : term] : (((sub @ (X234 @ X238 @ X237) @ X236)) = ((X234 @ (comp @ X238 @ X236) @ (sub @ X237 @ X236)))) & (((X232 @ id @ (lam @ (X234 @ sh @ one)) @ id)) != $true) & ! [X235 : term] : (($true != ((X232 @ id @ X235 @ id))) | (((X232 @ id @ (X234 @ id @ X235) @ id)) = $true))) | (((X232 @ id @ (lam @ (sub @ X233 @ (push @ one @ sh))) @ id)) = $true) | ? [X239 : term] : ((((X232 @ id @ (sub @ X233 @ (push @ X239 @ id)) @ id)) != $true) & (((X232 @ id @ X239 @ id)) = $true))))),
  inference(ennf_transformation,[],[f10])).
thf(f12,plain,(
  ((hoasinduction_lthm_3 = $true) <=> (? [X315 : term] : (((sub @ X315 @ id)) != X315) | ? [X305 : (subst > term > subst > $o)] : ! [X306 : (term > $o)] : ? [X307 : term] : (((X306 @ X307)) != ((X305 @ id @ X307 @ id))) | ? [X316 : (term > $o),X318 : term,X317 : (subst > term > subst > $o)] : (! [X319 : (subst > term > term)] : ((((X317 @ id @ (lam @ (X319 @ sh @ one)) @ id)) = $true) | ? [X323 : term] : ((((X317 @ id @ (X319 @ id @ X323) @ id)) != $true) & (((X317 @ id @ X323 @ id)) = $true)) | ? [X322 : term,X320 : subst,X321 : subst] : (((X319 @ (comp @ X321 @ X320) @ (sub @ X322 @ X320))) != ((sub @ (X319 @ X321 @ X322) @ X320)))) & ! [X324 : term] : ((((X316 @ (sub @ X318 @ (push @ X324 @ id)))) = $true) | (((X316 @ X324)) != $true)) & (((X316 @ (lam @ X318))) != $true) & ! [X325 : term] : (((X317 @ id @ X325 @ id)) = ((X316 @ X325)))) | ? [X309 : term,X308 : (term > $o)] : ((((X308 @ X309)) != $true) & ! [X312 : term] : ((((X308 @ X312)) = $true) | (((var @ X312)) != $true)) & ! [X313 : term] : (? [X314 : term] : ((((X308 @ (sub @ X313 @ (push @ X314 @ id)))) != $true) & (((X308 @ X314)) = $true)) | (((X308 @ (lam @ X313))) = $true)) & ! [X310 : term,X311 : term] : ((((X308 @ X311)) != $true) | (((X308 @ (ap @ X310 @ X311))) = $true) | (((X308 @ X310)) != $true))) | ! [X327 : term,X326 : (subst > term > subst > $o)] : (? [X343 : term] : ((((var @ (sub @ X343 @ id))) = $true) & (((X326 @ id @ X343 @ id)) != $true)) | ? [X329 : term,X328 : term] : ((((X326 @ id @ X328 @ id)) = $true) & (((X326 @ id @ X329 @ id)) = $true) & (((X326 @ id @ (ap @ (sub @ X329 @ id) @ X328) @ id)) != $true)) | ? [X331 : subst,X333 : subst,X332 : term,X330 : subst] : ((((X326 @ X331 @ X332 @ (comp @ X330 @ X333))) = $true) & (((X326 @ (comp @ X331 @ X330) @ (sub @ X332 @ X330) @ X333)) != $true)) | ($true = ((X326 @ id @ X327 @ id))) | ? [X341 : subst,X342 : subst,X339 : term,X340 : subst] : ((((X326 @ (comp @ X342 @ X341) @ (sub @ X339 @ X341) @ X340)) = $true) & (((X326 @ X342 @ X339 @ (comp @ X341 @ X340))) != $true)) | ? [X334 : (subst > term > term)] : ((((X326 @ id @ (lam @ (X334 @ sh @ one)) @ id)) != $true) & ! [X335 : term] : ((((X326 @ id @ X335 @ id)) != $true) | (((X326 @ id @ (X334 @ id @ X335) @ id)) = $true)) & ! [X336 : subst,X338 : subst,X337 : term] : (((sub @ (X334 @ X336 @ X337) @ X338)) = ((X334 @ (comp @ X336 @ X338) @ (sub @ X337 @ X338)))))))) & (termmset_gthm = $true) & ((substmonoid = $true) <=> (! [X71 : subst] : (((comp @ id @ X71)) = X71) & ! [X70 : subst] : (((comp @ X70 @ id)) = X70))) & (pushprop_gthm = $true) & (pushprop_lem3v2_lthm = $true) & (id = ((d2subst @ d_id))) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (pushprop_lthm = $true) & (((d2subst @ d_id)) = sh) & ! [X69 : d_term] : (d_one = X69) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X507 : term] : ? [X508 : (subst > term > term)] : ! [X509 : term,X510 : subst] : (((sub @ X507 @ (push @ X509 @ X510))) = ((X508 @ X510 @ X509)))) & (hoaslamnotvar_lthm = $true) & ! [X344 : d_subst] : (d_id = X344) & (induction2_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & (pushprop_lthm_orig = $true) & ((hoasinduction_no_psi_cond = $true) <=> ! [X173 : term,X174 : (subst > term > subst > $o)] : (? [X176 : term,X175 : term] : ((((X174 @ id @ (ap @ (sub @ X176 @ id) @ X175) @ id)) != $true) & (((X174 @ id @ X175 @ id)) = $true) & (((X174 @ id @ X176 @ id)) = $true)) | ? [X177 : (subst > term > term)] : (! [X179 : subst,X180 : subst,X181 : term] : (((sub @ (X177 @ X179 @ X181) @ X180)) = ((X177 @ (comp @ X179 @ X180) @ (sub @ X181 @ X180)))) & ! [X178 : term] : ((((X174 @ id @ (X177 @ id @ X178) @ id)) = $true) | (((X174 @ id @ X178 @ id)) != $true)) & (((X174 @ id @ (lam @ (X177 @ sh @ one)) @ id)) != $true)) | (((X174 @ id @ X173 @ id)) = $true))) & (hoaslamnotap_lthm = $true) & (! [X410 : subst,X409 : (term > $o),X408 : term] : ? [X411 : (term > $o)] : ! [X412 : term] : (((X409 @ (sub @ X412 @ (push @ X408 @ X410)))) = ((X411 @ X412))) <=> (pushprop_lem0 = $true)) & (hoaslamnotvar_gthm = $true) & (axclos = $true) & ((? [X374 : term,X375 : (term > $o)] : (! [X377 : term] : ((((X375 @ (lam @ X377))) = $true) | ? [X378 : term] : ((((X375 @ (sub @ X377 @ (push @ X378 @ id)))) != $true) & (((X375 @ X378)) = $true))) & ! [X376 : term] : ((((X375 @ X376)) = $true) | ($true != ((var @ X376)))) & ! [X379 : term,X380 : term] : ((((X375 @ X380)) != $true) | (((X375 @ X379)) != $true) | (((X375 @ (ap @ X380 @ X379))) = $true)) & (((X375 @ X374)) != $true)) | ! [X393 : term,X392 : (subst > term > subst > $o)] : ((((X392 @ id @ X393 @ id)) = $true) | ? [X396 : (subst > term > term)] : ((((X392 @ id @ (lam @ (X396 @ sh @ one)) @ id)) != $true) & ! [X400 : subst,X398 : term,X399 : subst] : (((X396 @ (comp @ X399 @ X400) @ (sub @ X398 @ X400))) = ((sub @ (X396 @ X399 @ X398) @ X400))) & ! [X397 : term] : ((((X392 @ id @ (X396 @ id @ X397) @ id)) = $true) | (((X392 @ id @ X397 @ id)) != $true))) | ? [X395 : term,X394 : term] : ((((X392 @ id @ (ap @ (sub @ X394 @ id) @ X395) @ id)) != $true) & ($true = ((X392 @ id @ X395 @ id))) & (((X392 @ id @ X394 @ id)) = $true))) | ? [X382 : (subst > term > subst > $o),X384 : (term > $o),X383 : term] : ((((X384 @ (lam @ X383))) != $true) & ! [X386 : term] : (((X382 @ id @ X386 @ id)) = ((X384 @ X386))) & ! [X387 : (subst > term > term)] : (? [X391 : term] : ((((X382 @ id @ X391 @ id)) = $true) & (((X382 @ id @ (X387 @ id @ X391) @ id)) != $true)) | ? [X390 : subst,X388 : subst,X389 : term] : (((X387 @ (comp @ X390 @ X388) @ (sub @ X389 @ X388))) != ((sub @ (X387 @ X390 @ X389) @ X388))) | (((X382 @ id @ (lam @ (X387 @ sh @ one)) @ id)) = $true)) & ! [X385 : term] : ((((X384 @ X385)) != $true) | (((X384 @ (sub @ X383 @ (push @ X385 @ id)))) = $true))) | ? [X381 : term] : (((sub @ X381 @ id)) != X381) | ? [X371 : (subst > term > subst > $o)] : ! [X372 : (term > $o)] : ? [X373 : term] : (((X371 @ id @ X373 @ id)) != ((X372 @ X373)))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (((var @ (d2term @ d_one))) != $true) & ((hoasinduction_lem3v2 = $true) <=> ! [X450 : (subst > term > subst > $o),X451 : (term > $o),X452 : term] : ((((X451 @ (lam @ X452))) = $true) | ? [X458 : subst,X459 : subst,X457 : subst,X460 : term] : ((((X450 @ X457 @ X460 @ (comp @ X459 @ X458))) = $true) & (((X450 @ (comp @ X457 @ X459) @ (sub @ X460 @ X459) @ X458)) != $true)) | ? [X461 : term] : ((((X451 @ (sub @ X452 @ (push @ X461 @ id)))) != $true) & (((X451 @ X461)) = $true)) | ? [X453 : term,X455 : subst,X454 : subst,X456 : subst] : ((((X450 @ X456 @ X453 @ (comp @ X454 @ X455))) != $true) & (((X450 @ (comp @ X456 @ X454) @ (sub @ X453 @ X454) @ X455)) = $true)) | ? [X463 : (subst > term > term)] : (! [X467 : term] : ((((X450 @ id @ (X463 @ id @ X467) @ id)) = $true) | (((X450 @ id @ X467 @ id)) != $true)) & (((X450 @ id @ (lam @ (X463 @ sh @ one)) @ id)) != $true) & ! [X465 : subst,X464 : subst,X466 : term] : (((X463 @ (comp @ X465 @ X464) @ (sub @ X466 @ X464))) = ((sub @ (X463 @ X465 @ X466) @ X464)))) | ? [X462 : term] : (((X451 @ X462)) != ((X450 @ id @ X462 @ id))))) & ((? [X160 : (term > $o),X159 : term] : (! [X162 : term,X163 : term] : ((((X160 @ X163)) != $true) | (((X160 @ X162)) != $true) | (((X160 @ (ap @ X162 @ X163))) = $true)) & (((X160 @ X159)) != $true) & ! [X161 : term] : ((((X160 @ (lam @ X161))) = $true) | (((X160 @ X161)) != $true)) & ! [X164 : term] : ((((var @ X164)) != $true) | ($true = ((X160 @ X164))))) | ? [X158 : subst] : (((comp @ X158 @ id)) != X158) | ? [X154 : term,X155 : subst] : (((sub @ one @ (push @ X154 @ X155))) != X154) | ! [X165 : subst,X167 : term,X166 : (term > $o)] : (? [X172 : term] : ((((X166 @ (sub @ X172 @ X165))) != $true) & (((var @ X172)) = $true)) | ? [X171 : term,X170 : term] : ((((X166 @ X171)) = $true) & (((X166 @ X170)) = $true) & (((X166 @ (ap @ X170 @ X171))) != $true)) | ? [X168 : term] : ((((X166 @ (lam @ X168))) != $true) & ! [X169 : term] : (($true != ((X166 @ X169))) | (((X166 @ (sub @ X168 @ (push @ X169 @ id)))) = $true))) | (((X166 @ (sub @ X167 @ X165))) = $true)) | ? [X157 : term,X156 : subst] : (((comp @ sh @ (push @ X157 @ X156))) != X156)) <=> (induction2lem_lthm = $true)) & ((termmset_lthm = $true) <=> (! [X449 : term] : (((sub @ X449 @ id)) = X449) | ? [X448 : term] : (((sub @ X448 @ id)) != X448))) & (! [X152 : subst,X153 : term] : (((comp @ sh @ (push @ X153 @ X152))) = X152) <=> (axshiftcons = $true)) & (hoasinduction_lem2v2_gthm = $true) & (pushprop_lem0_gthm = $true) & (hoasapnotvar_lthm = $true) & ((pushprop_lem1 = $true) <=> ! [X246 : term,X247 : term,X250 : (term > $o),X248 : subst,X249 : (term > $o)] : ((((X249 @ X247)) != $true) | (((X250 @ (sub @ X247 @ (push @ X246 @ X248)))) = $true))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (pushprop_lem1v2_gthm = $true) & ((axidr = $true) <=> ! [X150 : subst] : (((comp @ X150 @ id)) = X150)) & (one = ((d2term @ d_one))) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (! [X60 : (term > $o),X58 : term,X59 : subst] : (? [X65 : term] : ((((var @ X65)) = $true) & (((X60 @ (sub @ X65 @ X59))) != $true)) | ? [X64 : term,X63 : term] : ((((X60 @ X63)) = $true) & (((X60 @ (ap @ X63 @ X64))) != $true) & (((X60 @ X64)) = $true)) | ? [X61 : term] : ((((X60 @ (lam @ X61))) != $true) & ! [X62 : term] : ((((X60 @ (sub @ X61 @ (push @ X62 @ id)))) = $true) | (((X60 @ X62)) != $true))) | (((X60 @ (sub @ X58 @ X59))) = $true)) <=> (induction2lem = $true)) & ! [X512 : d_subst,X511 : d_subst] : ((X511 = X512) | (((d2subst @ X511)) != ((d2subst @ X512)))) & ((! [X264 : (term > $o),X265 : term] : (? [X266 : term] : (! [X267 : term] : ((((X264 @ X267)) != $true) | (((X264 @ (sub @ X266 @ (push @ X267 @ id)))) = $true)) & (((X264 @ (lam @ X266))) != $true)) | ? [X269 : term,X268 : term] : ((((X264 @ X268)) = $true) & (((X264 @ X269)) = $true) & (((X264 @ (ap @ X269 @ X268))) != $true)) | (((X264 @ X265)) = $true) | ? [X270 : term] : ((((X264 @ X270)) != $true) & (((var @ X270)) = $true))) | ? [X255 : term] : (((sub @ X255 @ id)) != X255) | ? [X257 : term,X256 : (term > $o),X258 : subst] : (! [X259 : term] : ((((X256 @ (sub @ X259 @ X258))) = $true) | ($true != ((var @ X259)))) & (((X256 @ (sub @ X257 @ X258))) != $true) & ! [X262 : term] : ((((X256 @ (lam @ X262))) = $true) | ? [X263 : term] : ((((X256 @ X263)) = $true) & (((X256 @ (sub @ X262 @ (push @ X263 @ id)))) != $true))) & ! [X260 : term,X261 : term] : ((((X256 @ X260)) != $true) | (((X256 @ X261)) != $true) | (((X256 @ (ap @ X261 @ X260))) = $true)))) <=> (induction2_lthm = $true)) & (hoasapinj2_gthm = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (hoasinduction_lem3v2_gthm = $true) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X233 : term,X232 : (subst > term > subst > $o)] : (? [X234 : (subst > term > term)] : (! [X238 : subst,X236 : subst,X237 : term] : (((sub @ (X234 @ X238 @ X237) @ X236)) = ((X234 @ (comp @ X238 @ X236) @ (sub @ X237 @ X236)))) & (((X232 @ id @ (lam @ (X234 @ sh @ one)) @ id)) != $true) & ! [X235 : term] : (($true != ((X232 @ id @ X235 @ id))) | (((X232 @ id @ (X234 @ id @ X235) @ id)) = $true))) | (((X232 @ id @ (lam @ (sub @ X233 @ (push @ one @ sh))) @ id)) = $true) | ? [X239 : term] : ((((X232 @ id @ (sub @ X233 @ (push @ X239 @ id)) @ id)) != $true) & (((X232 @ id @ X239 @ id)) = $true)))) & (! [X471 : subst] : (((comp @ id @ X471)) = X471) <=> (axidl = $true)) & ((hoasinduction = $true) <=> ! [X489 : term,X490 : (subst > term > subst > $o)] : (? [X504 : subst,X503 : term,X505 : subst,X506 : subst] : ((((X490 @ (comp @ X504 @ X506) @ (sub @ X503 @ X506) @ X505)) = $true) & (((X490 @ X504 @ X503 @ (comp @ X506 @ X505))) != $true)) | (((X490 @ id @ X489 @ id)) = $true) | ? [X498 : term] : ((((var @ (sub @ X498 @ id))) = $true) & (((X490 @ id @ X498 @ id)) != $true)) | ? [X501 : subst,X499 : subst,X502 : term,X500 : subst] : ((((X490 @ X499 @ X502 @ (comp @ X500 @ X501))) = $true) & ($true != ((X490 @ (comp @ X499 @ X500) @ (sub @ X502 @ X500) @ X501)))) | ? [X492 : term,X491 : term] : ((((X490 @ id @ X492 @ id)) = $true) & (((X490 @ id @ (ap @ (sub @ X492 @ id) @ X491) @ id)) != $true) & (((X490 @ id @ X491 @ id)) = $true)) | ? [X493 : (subst > term > term)] : (! [X497 : term] : (($true != ((X490 @ id @ X497 @ id))) | (((X490 @ id @ (X493 @ id @ X497) @ id)) = $true)) & (((X490 @ id @ (lam @ (X493 @ sh @ one)) @ id)) != $true) & ! [X496 : term,X494 : subst,X495 : subst] : (((X493 @ (comp @ X494 @ X495) @ (sub @ X496 @ X495))) = ((sub @ (X493 @ X494 @ X496) @ X495)))))) & (substmonoid_gthm = $true) & (hoasinduction_lem2_gthm = $true) & (axabs = $true) & ((! [X303 : term,X301 : term,X302 : term,X304 : term] : ((X301 = X303) | (((ap @ (sub @ X303 @ id) @ X302)) != ((ap @ (sub @ X301 @ id) @ X304)))) | ? [X296 : term] : (((sub @ X296 @ id)) != X296) | ? [X297 : term,X300 : term,X298 : term,X299 : term] : ((X297 != X298) & (((ap @ X298 @ X299)) = ((ap @ X297 @ X300))))) <=> (hoasapinj1_lthm = $true)) & (ulamvarind = $true) & ((pushprop_lem1_lthm = $true) <=> (? [X345 : subst,X346 : term] : (((sub @ one @ (push @ X346 @ X345))) != X346) | ? [X347 : term,X348 : subst] : (((comp @ sh @ (push @ X347 @ X348))) != X348) | ! [X349 : subst,X353 : (term > $o),X352 : (term > $o),X350 : term,X351 : term] : ((((X352 @ (sub @ X351 @ (push @ X350 @ X349)))) = $true) | (((X353 @ X351)) != $true)))) & (hoasinduction_lem1_lthm = $true) & (hoaslamnotap_gthm = $true) & (axvarshift = $true) & ((? [X77 : term,X74 : term,X75 : term,X76 : term] : ((((ap @ X77 @ X74)) = ((ap @ X76 @ X75))) & (X74 != X75)) | ! [X80 : term,X81 : term,X79 : term,X78 : term] : ((((ap @ (sub @ X81 @ id) @ X78)) != ((ap @ (sub @ X79 @ id) @ X80))) | (X78 = X80))) <=> (hoasapinj2_lthm = $true)) & (hoasinduction_lem1v2 = $true) & (ulamvar1 != $true) & ! [X72 : d_term,X73 : d_term] : ((X72 = X73) | (((d2term @ X72)) != ((d2term @ X73)))) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (pushprop_lem1_gthm = $true) & (! [X51 : term,X50 : term,X52 : (subst > term > term)] : (? [X55 : term,X53 : subst,X54 : subst] : (((sub @ (X52 @ X53 @ X55) @ X54)) != ((X52 @ (comp @ X53 @ X54) @ (sub @ X55 @ X54)))) | (((lam @ (X52 @ sh @ one))) != ((ap @ (sub @ X50 @ id) @ X51)))) <=> (hoaslamnotap = $true)) & ((axvarcons = $true) <=> ! [X356 : subst,X357 : term] : (((sub @ one @ (push @ X357 @ X356))) = X357)) & ((hoasapinj1 = $true) <=> ! [X144 : term,X147 : term,X146 : term,X145 : term] : ((((ap @ (sub @ X147 @ id) @ X146)) != ((ap @ (sub @ X145 @ id) @ X144))) | (X145 = X147))) & (hoasinduction_lem3a_gthm = $true) & (! [X443 : (term > $o),X444 : term,X446 : (term > $o),X445 : subst] : (? [X447 : term] : (((X443 @ X447)) != ((X446 @ (sub @ X447 @ (push @ X444 @ X445))))) | (((X446 @ X444)) != $true) | (((X443 @ one)) = $true)) <=> (pushprop_lem1v2 = $true)) & ((? [X427 : term,X428 : subst] : (((sub @ one @ (push @ X427 @ X428))) != X427) | ! [X434 : term,X433 : (subst > term > term),X435 : subst,X436 : (subst > term > term)] : (? [X442 : subst,X440 : term,X441 : subst] : (((X433 @ (comp @ X441 @ X442) @ (sub @ X440 @ X442))) != ((sub @ (X433 @ X441 @ X440) @ X442))) | (((lam @ (X433 @ sh @ one))) != ((lam @ (X436 @ sh @ one)))) | ? [X437 : subst,X438 : subst,X439 : term] : (((sub @ (X436 @ X438 @ X439) @ X437)) != ((X436 @ (comp @ X438 @ X437) @ (sub @ X439 @ X437)))) | (((X433 @ X435 @ X434)) = ((X436 @ X435 @ X434)))) | ? [X430 : subst,X429 : term] : (((comp @ sh @ (push @ X429 @ X430))) != X430) | ? [X432 : term,X431 : term] : ((((lam @ X432)) = ((lam @ X431))) & (X431 != X432))) <=> (hoaslaminj_lthm = $true)) & (hoaslamnotvar = $true) & (hoasinduction_lem1_gthm = $true) & (! [X203 : term,X204 : (subst > term > subst > $o)] : (? [X210 : term] : ((((X204 @ id @ X210 @ id)) = $true) & (((X204 @ id @ (sub @ X203 @ (push @ X210 @ id)) @ id)) != $true)) | ? [X205 : (subst > term > term)] : (! [X208 : term,X207 : subst,X206 : subst] : (((X205 @ (comp @ X206 @ X207) @ (sub @ X208 @ X207))) = ((sub @ (X205 @ X206 @ X208) @ X207))) & (((X204 @ id @ (lam @ (X205 @ sh @ one)) @ id)) != $true) & ! [X209 : term] : ((((X204 @ id @ X209 @ id)) != $true) | ($true = ((X204 @ id @ (X205 @ id @ X209) @ id))))) | ($true = ((X204 @ id @ (lam @ X203) @ id)))) <=> (hoasinduction_lem3a = $true)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((! [X190 : term,X188 : subst,X191 : (term > $o),X189 : (term > $o)] : (($true != ((X191 @ X190))) | ? [X192 : term] : (((X191 @ (sub @ X192 @ (push @ X190 @ X188)))) != ((X189 @ X192))) | (((X189 @ one)) = $true)) | ? [X187 : subst,X186 : term] : (((sub @ one @ (push @ X186 @ X187))) != X186)) <=> (pushprop_lem1v2_lthm = $true)) & (hoaslaminj_gthm = $true) & (! [X472 : term,X473 : term,X474 : term,X475 : term] : ((((ap @ X475 @ X473)) != ((ap @ X472 @ X474))) | (X472 = X475)) <=> (apinj1 = $true)) & (! [X513 : (subst > term > subst > $o),X515 : term,X514 : term,X516 : (term > $o)] : ((((X516 @ (ap @ X515 @ X514))) = $true) | (((X516 @ X514)) != $true) | ? [X523 : term,X520 : subst,X521 : subst,X522 : subst] : ((((X513 @ X520 @ X523 @ (comp @ X521 @ X522))) = $true) & (((X513 @ (comp @ X520 @ X521) @ (sub @ X523 @ X521) @ X522)) != $true)) | ($true != ((X516 @ X515))) | ? [X518 : term,X517 : term] : ((((X513 @ id @ X517 @ id)) = $true) & (((X513 @ id @ (ap @ (sub @ X517 @ id) @ X518) @ id)) != $true) & (((X513 @ id @ X518 @ id)) = $true)) | ? [X527 : subst,X524 : subst,X526 : term,X525 : subst] : ((((X513 @ (comp @ X527 @ X524) @ (sub @ X526 @ X524) @ X525)) = $true) & (((X513 @ X527 @ X526 @ (comp @ X524 @ X525))) != $true)) | ? [X519 : term] : (((X513 @ id @ X519 @ id)) != ((X516 @ X519)))) <=> (hoasinduction_lem2v2 = $true)) & (hoasinduction_gthm = $true) & ((hoasinduction_lem0_lthm = $true) <=> ! [X66 : (subst > term > subst > $o)] : ? [X67 : (term > $o)] : ! [X68 : term] : (((X67 @ X68)) = ((X66 @ id @ X68 @ id)))) & (! [X361 : (subst > term > term),X363 : subst,X362 : term,X364 : (subst > term > term)] : (? [X365 : term,X366 : subst,X367 : subst] : (((X364 @ (comp @ X366 @ X367) @ (sub @ X365 @ X367))) != ((sub @ (X364 @ X366 @ X365) @ X367))) | ? [X370 : subst,X369 : term,X368 : subst] : (((sub @ (X361 @ X370 @ X369) @ X368)) != ((X361 @ (comp @ X370 @ X368) @ (sub @ X369 @ X368)))) | (((X364 @ X363 @ X362)) = ((X361 @ X363 @ X362))) | (((lam @ (X361 @ sh @ one))) != ((lam @ (X364 @ sh @ one))))) <=> (hoaslaminj = $true)) & (hoasinduction_lem1v2_gthm = $true) & ((substmonoid_lthm = $true) <=> ((! [X424 : subst] : (((comp @ id @ X424)) = X424) & ! [X425 : subst] : (((comp @ X425 @ id)) = X425)) | ? [X423 : subst] : (((comp @ X423 @ id)) != X423) | ? [X422 : subst] : (((comp @ id @ X422)) != X422))) & (apnotvar = $true) & (hoasinduction_lem1 = $true) & (axmap = $true) & (axapp = $true) & (hoasapnotvar = $true) & (! [X470 : term] : (((sub @ X470 @ id)) = X470) <=> (axvarid = $true)) & (! [X251 : term,X252 : term,X254 : term,X253 : term] : ((((ap @ X254 @ X251)) != ((ap @ X253 @ X252))) | (X251 = X252)) <=> (apinj2 = $true)) & ((hoasinduction_lem3 = $true) <=> ! [X0 : (subst > term > subst > $o),X1 : term] : (? [X11 : term,X9 : subst,X8 : subst,X10 : subst] : ((((X0 @ (comp @ X9 @ X10) @ (sub @ X11 @ X10) @ X8)) = $true) & ($true != ((X0 @ X9 @ X11 @ (comp @ X10 @ X8))))) | (((X0 @ id @ (lam @ X1) @ id)) = $true) | ? [X15 : subst,X14 : subst,X12 : term,X13 : subst] : ((((X0 @ X15 @ X12 @ (comp @ X14 @ X13))) = $true) & (((X0 @ (comp @ X15 @ X14) @ (sub @ X12 @ X14) @ X13)) != $true)) | ? [X3 : (subst > term > term)] : (! [X4 : term] : ((((X0 @ id @ (X3 @ id @ X4) @ id)) = $true) | (((X0 @ id @ X4 @ id)) != $true)) & ! [X5 : subst,X7 : term,X6 : subst] : (((X3 @ (comp @ X6 @ X5) @ (sub @ X7 @ X5))) = ((sub @ (X3 @ X6 @ X7) @ X5))) & (((X0 @ id @ (lam @ (X3 @ sh @ one)) @ id)) != $true)) | ? [X2 : term] : ((((X0 @ id @ X2 @ id)) = $true) & (((X0 @ id @ (sub @ X1 @ (push @ X2 @ id)) @ id)) != $true)))) & (hoasinduction_lem2_lthm = $true) & ((hoasinduction_lem3aa = $true) <=> ! [X83 : (subst > term > subst > $o),X82 : term] : ((((X83 @ id @ (lam @ (sub @ X82 @ (push @ one @ sh))) @ id)) = $true) | ? [X85 : (subst > term > term)] : (($true != ((X83 @ id @ (lam @ (X85 @ sh @ one)) @ id))) & ! [X89 : term] : ((((X83 @ id @ X89 @ id)) != $true) | (((X83 @ id @ (X85 @ id @ X89) @ id)) = $true)) & ! [X88 : subst,X87 : subst,X86 : term] : (((X85 @ (comp @ X88 @ X87) @ (sub @ X86 @ X87))) = ((sub @ (X85 @ X88 @ X86) @ X87)))) | ? [X84 : term] : ((((X83 @ id @ X84 @ id)) = $true) & (((X83 @ id @ (sub @ X82 @ (push @ X84 @ id)) @ id)) != $true)))) & (induction2lem_gthm = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ! [X406 : term] : ? [X407 : d_term] : (((d2term @ X407)) = X406) & ((hoasinduction_lem3v2a_lthm = $true) <=> (? [X39 : term] : (((sub @ X39 @ id)) != X39) | ! [X40 : (term > $o),X41 : (subst > term > subst > $o),X42 : term] : (? [X45 : (subst > term > term)] : (! [X46 : term] : ((((X41 @ id @ X46 @ id)) != $true) | (((X41 @ id @ (X45 @ id @ X46) @ id)) = $true)) & ! [X47 : subst,X49 : term,X48 : subst] : (((X45 @ (comp @ X47 @ X48) @ (sub @ X49 @ X48))) = ((sub @ (X45 @ X47 @ X49) @ X48))) & (((X41 @ id @ (lam @ (X45 @ sh @ one)) @ id)) != $true)) | ? [X43 : term] : (((X41 @ id @ X43 @ id)) != ((X40 @ X43))) | (((X40 @ (lam @ X42))) = $true) | ? [X44 : term] : ((((X40 @ X44)) = $true) & (((X40 @ (sub @ X42 @ (push @ X44 @ id)))) != $true))) | ? [X35 : term] : ! [X36 : (subst > term > term)] : ? [X37 : term,X38 : subst] : (((X36 @ X38 @ X37)) != ((sub @ X35 @ (push @ X37 @ X38)))))) & ((hoasinduction_lem3b = $true) <=> ! [X56 : term] : ? [X57 : (subst > term > term)] : (((sub @ X56 @ (push @ one @ sh))) = ((X57 @ sh @ one)))) & (pushprop_lem3v2 = $true) & ! [X354 : subst] : ? [X355 : d_subst] : (((d2subst @ X355)) = X354) & (hoasapnotvar_gthm = $true) & (pushprop = $true) & ((hoasinduction_lthm = $true) <=> (! [X126 : (subst > term > subst > $o),X127 : term] : ((((X126 @ id @ X127 @ id)) = $true) | ? [X142 : term,X143 : term] : ((((X126 @ id @ X143 @ id)) = $true) & (((X126 @ id @ (ap @ (sub @ X142 @ id) @ X143) @ id)) != $true) & (((X126 @ id @ X142 @ id)) = $true)) | ? [X139 : term,X137 : subst,X138 : subst,X140 : subst] : ((((X126 @ (comp @ X138 @ X140) @ (sub @ X139 @ X140) @ X137)) != $true) & (((X126 @ X138 @ X139 @ (comp @ X140 @ X137))) = $true)) | ? [X141 : term] : ((((var @ (sub @ X141 @ id))) = $true) & (((X126 @ id @ X141 @ id)) != $true)) | ? [X134 : term,X133 : subst,X136 : subst,X135 : subst] : ((((X126 @ X135 @ X134 @ (comp @ X133 @ X136))) != $true) & (((X126 @ (comp @ X135 @ X133) @ (sub @ X134 @ X133) @ X136)) = $true)) | ? [X128 : (subst > term > term)] : (! [X129 : term] : ((((X126 @ id @ X129 @ id)) != $true) | (((X126 @ id @ (X128 @ id @ X129) @ id)) = $true)) & (((X126 @ id @ (lam @ (X128 @ sh @ one)) @ id)) != $true) & ! [X131 : subst,X130 : term,X132 : subst] : (((X128 @ (comp @ X131 @ X132) @ (sub @ X130 @ X132))) = ((sub @ (X128 @ X131 @ X130) @ X132))))) | ? [X90 : (term > $o),X91 : term] : (! [X92 : term] : ((((X90 @ (lam @ X92))) = $true) | ? [X93 : term] : ((((X90 @ X93)) = $true) & (((X90 @ (sub @ X92 @ (push @ X93 @ id)))) != $true))) & ! [X95 : term,X94 : term] : (($true = ((X90 @ (ap @ X94 @ X95)))) | (((X90 @ X94)) != $true) | (((X90 @ X95)) != $true)) & (((X90 @ X91)) != $true) & ! [X96 : term] : ((((var @ X96)) != $true) | (((X90 @ X96)) = $true))) | ? [X111 : term,X110 : (subst > term > subst > $o)] : (! [X125 : term] : (($true != ((X110 @ id @ X125 @ id))) | (((X110 @ id @ (sub @ X111 @ (push @ X125 @ id)) @ id)) = $true)) & ($true != ((X110 @ id @ (lam @ X111) @ id))) & ! [X114 : subst,X113 : subst,X115 : term,X112 : subst] : ((((X110 @ X112 @ X115 @ (comp @ X114 @ X113))) != $true) | (((X110 @ (comp @ X112 @ X114) @ (sub @ X115 @ X114) @ X113)) = $true)) & ! [X119 : subst,X117 : subst,X116 : subst,X118 : term] : ((((X110 @ (comp @ X119 @ X117) @ (sub @ X118 @ X117) @ X116)) != $true) | (((X110 @ X119 @ X118 @ (comp @ X117 @ X116))) = $true)) & ! [X120 : (subst > term > term)] : ((((X110 @ id @ (lam @ (X120 @ sh @ one)) @ id)) = $true) | ? [X121 : term] : ((((X110 @ id @ (X120 @ id @ X121) @ id)) != $true) & (((X110 @ id @ X121 @ id)) = $true)) | ? [X123 : subst,X122 : subst,X124 : term] : (((X120 @ (comp @ X122 @ X123) @ (sub @ X124 @ X123))) != ((sub @ (X120 @ X122 @ X124) @ X123))))) | ? [X97 : term,X99 : term,X98 : (subst > term > subst > $o)] : (! [X104 : subst,X105 : subst,X106 : subst,X107 : term] : ((((X98 @ (comp @ X106 @ X104) @ (sub @ X107 @ X104) @ X105)) != $true) | (((X98 @ X106 @ X107 @ (comp @ X104 @ X105))) = $true)) & (((X98 @ id @ (ap @ X99 @ X97) @ id)) != $true) & ! [X108 : term,X109 : term] : ((((X98 @ id @ X108 @ id)) != $true) | (((X98 @ id @ X109 @ id)) != $true) | (((X98 @ id @ (ap @ (sub @ X108 @ id) @ X109) @ id)) = $true)) & (((X98 @ id @ X99 @ id)) = $true) & ! [X103 : subst,X100 : subst,X101 : term,X102 : subst] : ((((X98 @ (comp @ X102 @ X103) @ (sub @ X101 @ X103) @ X100)) = $true) | (((X98 @ X102 @ X101 @ (comp @ X103 @ X100))) != $true)) & ($true = ((X98 @ id @ X97 @ id)))))) & (hoasinduction_lem3b_gthm = $true) & ((? [X16 : term] : (((sub @ X16 @ id)) != X16) | ! [X17 : term,X19 : (term > $o),X18 : (subst > term > subst > $o)] : (? [X20 : term] : (((X19 @ X20)) != ((X18 @ id @ X20 @ id))) | ? [X30 : (subst > term > term)] : (! [X34 : term] : ((((X18 @ id @ (X30 @ id @ X34) @ id)) = $true) | (((X18 @ id @ X34 @ id)) != $true)) & ! [X31 : subst,X33 : subst,X32 : term] : (((sub @ (X30 @ X33 @ X32) @ X31)) = ((X30 @ (comp @ X33 @ X31) @ (sub @ X32 @ X31)))) & (((X18 @ id @ (lam @ (X30 @ sh @ one)) @ id)) != $true)) | ? [X23 : subst,X24 : term,X22 : subst,X21 : subst] : ((((X18 @ (comp @ X22 @ X23) @ (sub @ X24 @ X23) @ X21)) != $true) & (((X18 @ X22 @ X24 @ (comp @ X23 @ X21))) = $true)) | (((X19 @ (lam @ X17))) = $true) | ? [X29 : term] : ((((X19 @ X29)) = $true) & (((X19 @ (sub @ X17 @ (push @ X29 @ id)))) != $true)) | ? [X26 : term,X25 : subst,X27 : subst,X28 : subst] : (($true = ((X18 @ (comp @ X28 @ X27) @ (sub @ X26 @ X27) @ X25))) & (((X18 @ X28 @ X26 @ (comp @ X27 @ X25))) != $true)))) <=> (hoasinduction_lem3v2_lthm = $true)) & (lamnotap != $true) & (pushprop_lem2v2 = $true) & (lamnotvar = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((hoasapinj2 = $true) <=> ! [X185 : term,X183 : term,X184 : term,X182 : term] : ((((ap @ (sub @ X182 @ id) @ X184)) != ((ap @ (sub @ X183 @ id) @ X185))) | (X184 = X185))) & ((laminj = $true) <=> ! [X414 : term,X413 : term] : ((X413 = X414) | (((lam @ X413)) != ((lam @ X414))))) & (pushprop_lem2v2_gthm = $true) & ((hoasinduction_lem3a_lthm = $true) <=> (! [X225 : (subst > term > subst > $o),X224 : term] : (? [X226 : (subst > term > term)] : ((((X225 @ id @ (lam @ (X226 @ sh @ one)) @ id)) != $true) & ! [X229 : subst,X230 : term,X228 : subst] : (((X226 @ (comp @ X228 @ X229) @ (sub @ X230 @ X229))) = ((sub @ (X226 @ X228 @ X230) @ X229))) & ! [X227 : term] : ((((X225 @ id @ X227 @ id)) != $true) | (((X225 @ id @ (X226 @ id @ X227) @ id)) = $true))) | ($true = ((X225 @ id @ (lam @ X224) @ id))) | ? [X231 : term] : (($true != ((X225 @ id @ (sub @ X224 @ (push @ X231 @ id)) @ id))) & (((X225 @ id @ X231 @ id)) = $true))) | ? [X215 : term] : (((sub @ X215 @ id)) != X215) | ? [X216 : (subst > term > subst > $o),X217 : term] : (! [X218 : term] : ((((X216 @ id @ (sub @ X217 @ (push @ X218 @ id)) @ id)) = $true) | (((X216 @ id @ X218 @ id)) != $true)) & ! [X219 : (subst > term > term)] : ((((X216 @ id @ (lam @ (X219 @ sh @ one)) @ id)) = $true) | ? [X220 : term] : ((((X216 @ id @ (X219 @ id @ X220) @ id)) != $true) & (((X216 @ id @ X220 @ id)) = $true)) | ? [X222 : subst,X221 : subst,X223 : term] : (((sub @ (X219 @ X222 @ X223) @ X221)) != ((X219 @ (comp @ X222 @ X221) @ (sub @ X223 @ X221))))) & (((X216 @ id @ (lam @ (sub @ X217 @ (push @ one @ sh))) @ id)) != $true)))) & (! [X211 : term] : ? [X212 : (subst > term > term)] : ! [X213 : subst,X214 : term] : (((X212 @ X213 @ X214)) = ((sub @ X211 @ (push @ X214 @ X213)))) <=> (hoasinduction_lem3v2_f = $true)) & ((hoasinduction_lem3_lthm = $true) <=> (! [X281 : (subst > term > subst > $o),X280 : term] : ((((X281 @ id @ (lam @ X280) @ id)) = $true) | ? [X291 : (subst > term > term)] : (! [X295 : term] : ((((X281 @ id @ (X291 @ id @ X295) @ id)) = $true) | (((X281 @ id @ X295 @ id)) != $true)) & (((X281 @ id @ (lam @ (X291 @ sh @ one)) @ id)) != $true) & ! [X294 : term,X293 : subst,X292 : subst] : (((sub @ (X291 @ X292 @ X294) @ X293)) = ((X291 @ (comp @ X292 @ X293) @ (sub @ X294 @ X293))))) | ? [X289 : term,X290 : subst,X288 : subst,X287 : subst] : ((((X281 @ X290 @ X289 @ (comp @ X288 @ X287))) != $true) & (((X281 @ (comp @ X290 @ X288) @ (sub @ X289 @ X288) @ X287)) = $true)) | ? [X286 : term] : ((((X281 @ id @ X286 @ id)) = $true) & (((X281 @ id @ (sub @ X280 @ (push @ X286 @ id)) @ id)) != $true)) | ? [X285 : subst,X282 : term,X284 : subst,X283 : subst] : ((((X281 @ (comp @ X284 @ X283) @ (sub @ X282 @ X283) @ X285)) != $true) & ($true = ((X281 @ X284 @ X282 @ (comp @ X283 @ X285)))))) | ? [X271 : term] : (((sub @ X271 @ id)) != X271) | ? [X272 : (subst > term > subst > $o),X273 : term] : (! [X274 : term] : ((((X272 @ id @ X274 @ id)) != $true) | (((X272 @ id @ (sub @ X273 @ (push @ X274 @ id)) @ id)) = $true)) & (((X272 @ id @ (lam @ (sub @ X273 @ (push @ one @ sh))) @ id)) != $true) & ! [X275 : (subst > term > term)] : (($true = ((X272 @ id @ (lam @ (X275 @ sh @ one)) @ id))) | ? [X276 : term] : ((((X272 @ id @ X276 @ id)) = $true) & ($true != ((X272 @ id @ (X275 @ id @ X276) @ id)))) | ? [X277 : subst,X278 : term,X279 : subst] : (((X275 @ (comp @ X279 @ X277) @ (sub @ X278 @ X277))) != ((sub @ (X275 @ X279 @ X278) @ X277))))))) & ((termmset = $true) <=> ! [X151 : term] : (((sub @ X151 @ id)) = X151)) & (hoasinduction_lem3_gthm = $true) & (! [X194 : term,X195 : (subst > term > subst > $o),X193 : (term > $o)] : (? [X201 : term] : ((((X193 @ (sub @ X194 @ (push @ X201 @ id)))) != $true) & (((X193 @ X201)) = $true)) | ? [X202 : term] : (((X193 @ X202)) != ((X195 @ id @ X202 @ id))) | ? [X196 : (subst > term > term)] : (! [X199 : term,X197 : subst,X198 : subst] : (((X196 @ (comp @ X198 @ X197) @ (sub @ X199 @ X197))) = ((sub @ (X196 @ X198 @ X199) @ X197))) & ! [X200 : term] : ((((X195 @ id @ X200 @ id)) != $true) | (((X195 @ id @ (X196 @ id @ X200) @ id)) = $true)) & (((X195 @ id @ (lam @ (X196 @ sh @ one)) @ id)) != $true)) | (((X193 @ (lam @ X194))) = $true)) <=> (hoasinduction_lem3v2a = $true)) & (! [X416 : (term > $o),X415 : term] : (? [X418 : term,X417 : term] : ((((X416 @ X417)) = $true) & (((X416 @ X418)) = $true) & (((X416 @ (ap @ X417 @ X418))) != $true)) | ? [X419 : term] : (($true != ((X416 @ (lam @ X419)))) & ! [X420 : term] : ((((X416 @ (sub @ X419 @ (push @ X420 @ id)))) = $true) | (((X416 @ X420)) != $true))) | ? [X421 : term] : ((((var @ X421)) = $true) & (((X416 @ X421)) != $true)) | (((X416 @ X415)) = $true)) <=> (induction2 = $true)) & ((hoasinduction_lem0 = $true) <=> ! [X358 : (subst > term > subst > $o)] : ? [X359 : (term > $o)] : ! [X360 : term] : (((X358 @ id @ X360 @ id)) = ((X359 @ X360)))) & (! [X426 : subst] : (((push @ (sub @ one @ X426) @ (comp @ sh @ X426))) = X426) <=> (axscons = $true)) & (pushprop_lem2v2_lthm = $true) & (! [X469 : term,X468 : term] : ((X468 = X469) | (((sub @ X469 @ sh)) != ((sub @ X468 @ sh)))) <=> (shinj = $true)) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X148 : term] : ? [X149 : (subst > term > term)] : (((sub @ X148 @ (push @ one @ sh))) = ((X149 @ sh @ one)))) & (! [X241 : (term > $o),X240 : term] : (? [X242 : term] : ((((X241 @ X242)) = $true) & (((X241 @ (lam @ X242))) != $true)) | ? [X245 : term] : ((((var @ X245)) = $true) & (((X241 @ X245)) != $true)) | ? [X244 : term,X243 : term] : ((((X241 @ X243)) = $true) & (((X241 @ (ap @ X244 @ X243))) != $true) & (((X241 @ X244)) = $true)) | (((X241 @ X240)) = $true)) <=> (induction = $true)) & ((hoasinduction_lem3aaa = $true) <=> ! [X529 : term,X528 : (subst > term > subst > $o)] : ((((X528 @ id @ (lam @ (sub @ X529 @ (push @ one @ sh))) @ id)) = $true) | ? [X531 : term,X530 : (subst > term > term)] : ((((X528 @ id @ (lam @ (X530 @ sh @ one)) @ id)) != $true) & ! [X538 : subst,X536 : subst,X537 : term] : (((sub @ X531 @ (push @ (sub @ X537 @ X538) @ (comp @ X536 @ X538)))) = ((X530 @ (comp @ X536 @ X538) @ (sub @ X537 @ X538)))) & ! [X535 : term] : ((((X528 @ id @ (X530 @ id @ X535) @ id)) = $true) | ($true != ((X528 @ id @ X535 @ id)))) & ! [X534 : term,X532 : subst,X533 : subst] : (((sub @ (X530 @ X533 @ X534) @ X532)) = ((sub @ (sub @ X531 @ (push @ X534 @ X533)) @ X532)))) | ? [X539 : term] : ((((X528 @ id @ X539 @ id)) = $true) & (((X528 @ id @ (sub @ X529 @ (push @ X539 @ id)) @ id)) != $true)))) & (! [X402 : subst,X403 : (term > $o),X401 : term] : ? [X404 : (term > $o)] : ! [X405 : term] : (((X403 @ (sub @ X405 @ (push @ X401 @ X402)))) = ((X404 @ X405))) <=> (pushprop_lem0_lthm = $true)) & (hoasapinj1_gthm = $true) & (ulamvarsh = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (! [X478 : term,X476 : term,X477 : (subst > term > subst > $o)] : (? [X484 : term,X483 : term] : ((((X477 @ id @ (ap @ (sub @ X483 @ id) @ X484) @ id)) != $true) & (((X477 @ id @ X484 @ id)) = $true) & (((X477 @ id @ X483 @ id)) = $true)) | ? [X482 : subst,X479 : subst,X481 : term,X480 : subst] : ((((X477 @ (comp @ X480 @ X482) @ (sub @ X481 @ X482) @ X479)) = $true) & (((X477 @ X480 @ X481 @ (comp @ X482 @ X479))) != $true)) | (((X477 @ id @ (ap @ X476 @ X478) @ id)) = $true) | ? [X487 : subst,X486 : term,X488 : subst,X485 : subst] : (($true = ((X477 @ X485 @ X486 @ (comp @ X488 @ X487)))) & (((X477 @ (comp @ X485 @ X488) @ (sub @ X486 @ X488) @ X487)) != $true)) | (((X477 @ id @ X478 @ id)) != $true) | (((X477 @ id @ X476 @ id)) != $true)) <=> (hoasinduction_lem2 = $true)) & (axassoc = $true)),
  inference(flattening,[],[f11])).
thf(f13,definition,(
  ! [X477 : (subst > term > subst > $o)] : ((((sP0 @ X477)) = $true) <=> ? [X484 : term,X483 : term] : ((((X477 @ id @ (ap @ (sub @ X483 @ id) @ X484) @ id)) != $true) & (((X477 @ id @ X484 @ id)) = $true) & (((X477 @ id @ X483 @ id)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f14,definition,(
  (sP1 = $true) <=> ! [X478 : term,X476 : term,X477 : (subst > term > subst > $o)] : ((((sP0 @ X477)) = $true) | ? [X482 : subst,X479 : subst,X481 : term,X480 : subst] : ((((X477 @ (comp @ X480 @ X482) @ (sub @ X481 @ X482) @ X479)) = $true) & (((X477 @ X480 @ X481 @ (comp @ X482 @ X479))) != $true)) | (((X477 @ id @ (ap @ X476 @ X478) @ id)) = $true) | ? [X487 : subst,X486 : term,X488 : subst,X485 : subst] : (($true = ((X477 @ X485 @ X486 @ (comp @ X488 @ X487)))) & (((X477 @ (comp @ X485 @ X488) @ (sub @ X486 @ X488) @ X487)) != $true)) | (((X477 @ id @ X478 @ id)) != $true) | (((X477 @ id @ X476 @ id)) != $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f15,definition,(
  (sP2 = $true) <=> ! [X529 : term,X528 : (subst > term > subst > $o)] : ((((X528 @ id @ (lam @ (sub @ X529 @ (push @ one @ sh))) @ id)) = $true) | ? [X531 : term,X530 : (subst > term > term)] : ((((X528 @ id @ (lam @ (X530 @ sh @ one)) @ id)) != $true) & ! [X538 : subst,X536 : subst,X537 : term] : (((sub @ X531 @ (push @ (sub @ X537 @ X538) @ (comp @ X536 @ X538)))) = ((X530 @ (comp @ X536 @ X538) @ (sub @ X537 @ X538)))) & ! [X535 : term] : ((((X528 @ id @ (X530 @ id @ X535) @ id)) = $true) | ($true != ((X528 @ id @ X535 @ id)))) & ! [X534 : term,X532 : subst,X533 : subst] : (((sub @ (X530 @ X533 @ X534) @ X532)) = ((sub @ (sub @ X531 @ (push @ X534 @ X533)) @ X532)))) | ? [X539 : term] : ((((X528 @ id @ X539 @ id)) = $true) & (((X528 @ id @ (sub @ X529 @ (push @ X539 @ id)) @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f16,definition,(
  ! [X241 : (term > $o)] : ((((sP3 @ X241)) = $true) <=> ? [X244 : term,X243 : term] : ((((X241 @ X243)) = $true) & (((X241 @ (ap @ X244 @ X243))) != $true) & (((X241 @ X244)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f17,definition,(
  ! [X416 : (term > $o)] : ((((sP4 @ X416)) = $true) <=> ? [X418 : term,X417 : term] : ((((X416 @ X417)) = $true) & (((X416 @ X418)) = $true) & (((X416 @ (ap @ X417 @ X418))) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f18,definition,(
  (sP5 = $true) <=> ! [X416 : (term > $o),X415 : term] : ((((sP4 @ X416)) = $true) | ? [X419 : term] : (($true != ((X416 @ (lam @ X419)))) & ! [X420 : term] : ((((X416 @ (sub @ X419 @ (push @ X420 @ id)))) = $true) | (((X416 @ X420)) != $true))) | ? [X421 : term] : ((((var @ X421)) = $true) & (((X416 @ X421)) != $true)) | (((X416 @ X415)) = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f19,definition,(
  (sP6 = $true) <=> ! [X194 : term,X195 : (subst > term > subst > $o),X193 : (term > $o)] : (? [X201 : term] : ((((X193 @ (sub @ X194 @ (push @ X201 @ id)))) != $true) & (((X193 @ X201)) = $true)) | ? [X202 : term] : (((X193 @ X202)) != ((X195 @ id @ X202 @ id))) | ? [X196 : (subst > term > term)] : (! [X199 : term,X197 : subst,X198 : subst] : (((X196 @ (comp @ X198 @ X197) @ (sub @ X199 @ X197))) = ((sub @ (X196 @ X198 @ X199) @ X197))) & ! [X200 : term] : ((((X195 @ id @ X200 @ id)) != $true) | (((X195 @ id @ (X196 @ id @ X200) @ id)) = $true)) & (((X195 @ id @ (lam @ (X196 @ sh @ one)) @ id)) != $true)) | (((X193 @ (lam @ X194))) = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f20,definition,(
  ! [X281 : (subst > term > subst > $o)] : ((((sP7 @ X281)) = $true) <=> ? [X291 : (subst > term > term)] : (! [X295 : term] : ((((X281 @ id @ (X291 @ id @ X295) @ id)) = $true) | (((X281 @ id @ X295 @ id)) != $true)) & (((X281 @ id @ (lam @ (X291 @ sh @ one)) @ id)) != $true) & ! [X294 : term,X293 : subst,X292 : subst] : (((sub @ (X291 @ X292 @ X294) @ X293)) = ((X291 @ (comp @ X292 @ X293) @ (sub @ X294 @ X293))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f21,definition,(
  ($true = sP8) <=> ? [X272 : (subst > term > subst > $o),X273 : term] : (! [X274 : term] : ((((X272 @ id @ X274 @ id)) != $true) | (((X272 @ id @ (sub @ X273 @ (push @ X274 @ id)) @ id)) = $true)) & (((X272 @ id @ (lam @ (sub @ X273 @ (push @ one @ sh))) @ id)) != $true) & ! [X275 : (subst > term > term)] : (($true = ((X272 @ id @ (lam @ (X275 @ sh @ one)) @ id))) | ? [X276 : term] : ((((X272 @ id @ X276 @ id)) = $true) & ($true != ((X272 @ id @ (X275 @ id @ X276) @ id)))) | ? [X277 : subst,X278 : term,X279 : subst] : (((X275 @ (comp @ X279 @ X277) @ (sub @ X278 @ X277))) != ((sub @ (X275 @ X279 @ X278) @ X277)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f22,definition,(
  (sP9 = $true) <=> (! [X281 : (subst > term > subst > $o),X280 : term] : ((((X281 @ id @ (lam @ X280) @ id)) = $true) | (((sP7 @ X281)) = $true) | ? [X289 : term,X290 : subst,X288 : subst,X287 : subst] : ((((X281 @ X290 @ X289 @ (comp @ X288 @ X287))) != $true) & (((X281 @ (comp @ X290 @ X288) @ (sub @ X289 @ X288) @ X287)) = $true)) | ? [X286 : term] : ((((X281 @ id @ X286 @ id)) = $true) & (((X281 @ id @ (sub @ X280 @ (push @ X286 @ id)) @ id)) != $true)) | ? [X285 : subst,X282 : term,X284 : subst,X283 : subst] : ((((X281 @ (comp @ X284 @ X283) @ (sub @ X282 @ X283) @ X285)) != $true) & ($true = ((X281 @ X284 @ X282 @ (comp @ X283 @ X285)))))) | ? [X271 : term] : (((sub @ X271 @ id)) != X271) | ($true = sP8))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f23,definition,(
  (sP10 = $true) <=> ? [X216 : (subst > term > subst > $o),X217 : term] : (! [X218 : term] : ((((X216 @ id @ (sub @ X217 @ (push @ X218 @ id)) @ id)) = $true) | (((X216 @ id @ X218 @ id)) != $true)) & ! [X219 : (subst > term > term)] : ((((X216 @ id @ (lam @ (X219 @ sh @ one)) @ id)) = $true) | ? [X220 : term] : ((((X216 @ id @ (X219 @ id @ X220) @ id)) != $true) & (((X216 @ id @ X220 @ id)) = $true)) | ? [X222 : subst,X221 : subst,X223 : term] : (((sub @ (X219 @ X222 @ X223) @ X221)) != ((X219 @ (comp @ X222 @ X221) @ (sub @ X223 @ X221))))) & (((X216 @ id @ (lam @ (sub @ X217 @ (push @ one @ sh))) @ id)) != $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f24,definition,(
  ($true = sP11) <=> (! [X225 : (subst > term > subst > $o),X224 : term] : (? [X226 : (subst > term > term)] : ((((X225 @ id @ (lam @ (X226 @ sh @ one)) @ id)) != $true) & ! [X229 : subst,X230 : term,X228 : subst] : (((X226 @ (comp @ X228 @ X229) @ (sub @ X230 @ X229))) = ((sub @ (X226 @ X228 @ X230) @ X229))) & ! [X227 : term] : ((((X225 @ id @ X227 @ id)) != $true) | (((X225 @ id @ (X226 @ id @ X227) @ id)) = $true))) | ($true = ((X225 @ id @ (lam @ X224) @ id))) | ? [X231 : term] : (($true != ((X225 @ id @ (sub @ X224 @ (push @ X231 @ id)) @ id))) & (((X225 @ id @ X231 @ id)) = $true))) | ? [X215 : term] : (((sub @ X215 @ id)) != X215) | (sP10 = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f25,definition,(
  ! [X18 : (subst > term > subst > $o)] : ((((sP12 @ X18)) = $true) <=> ? [X30 : (subst > term > term)] : (! [X34 : term] : ((((X18 @ id @ (X30 @ id @ X34) @ id)) = $true) | (((X18 @ id @ X34 @ id)) != $true)) & ! [X31 : subst,X33 : subst,X32 : term] : (((sub @ (X30 @ X33 @ X32) @ X31)) = ((X30 @ (comp @ X33 @ X31) @ (sub @ X32 @ X31)))) & (((X18 @ id @ (lam @ (X30 @ sh @ one)) @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f26,definition,(
  (sP13 = $true) <=> (? [X16 : term] : (((sub @ X16 @ id)) != X16) | ! [X17 : term,X19 : (term > $o),X18 : (subst > term > subst > $o)] : (? [X20 : term] : (((X19 @ X20)) != ((X18 @ id @ X20 @ id))) | (((sP12 @ X18)) = $true) | ? [X23 : subst,X24 : term,X22 : subst,X21 : subst] : ((((X18 @ (comp @ X22 @ X23) @ (sub @ X24 @ X23) @ X21)) != $true) & (((X18 @ X22 @ X24 @ (comp @ X23 @ X21))) = $true)) | (((X19 @ (lam @ X17))) = $true) | ? [X29 : term] : ((((X19 @ X29)) = $true) & (((X19 @ (sub @ X17 @ (push @ X29 @ id)))) != $true)) | ? [X26 : term,X25 : subst,X27 : subst,X28 : subst] : (($true = ((X18 @ (comp @ X28 @ X27) @ (sub @ X26 @ X27) @ X25))) & (((X18 @ X28 @ X26 @ (comp @ X27 @ X25))) != $true))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f27,definition,(
  ! [X98 : (subst > term > subst > $o)] : ((((sP14 @ X98)) = $true) <=> ! [X108 : term,X109 : term] : ((((X98 @ id @ X108 @ id)) != $true) | (((X98 @ id @ X109 @ id)) != $true) | (((X98 @ id @ (ap @ (sub @ X108 @ id) @ X109) @ id)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f28,definition,(
  ! [X110 : (subst > term > subst > $o)] : ((((sP15 @ X110)) = $true) <=> ! [X120 : (subst > term > term)] : ((((X110 @ id @ (lam @ (X120 @ sh @ one)) @ id)) = $true) | ? [X121 : term] : ((((X110 @ id @ (X120 @ id @ X121) @ id)) != $true) & (((X110 @ id @ X121 @ id)) = $true)) | ? [X123 : subst,X122 : subst,X124 : term] : (((X120 @ (comp @ X122 @ X123) @ (sub @ X124 @ X123))) != ((sub @ (X120 @ X122 @ X124) @ X123)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f29,definition,(
  ! [X90 : (term > $o)] : ((((sP16 @ X90)) = $true) <=> ! [X95 : term,X94 : term] : (($true = ((X90 @ (ap @ X94 @ X95)))) | (((X90 @ X94)) != $true) | (((X90 @ X95)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f30,definition,(
  ! [X126 : (subst > term > subst > $o)] : ((((sP17 @ X126)) = $true) <=> ? [X128 : (subst > term > term)] : (! [X129 : term] : ((((X126 @ id @ X129 @ id)) != $true) | (((X126 @ id @ (X128 @ id @ X129) @ id)) = $true)) & (((X126 @ id @ (lam @ (X128 @ sh @ one)) @ id)) != $true) & ! [X131 : subst,X130 : term,X132 : subst] : (((X128 @ (comp @ X131 @ X132) @ (sub @ X130 @ X132))) = ((sub @ (X128 @ X131 @ X130) @ X132)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f31,definition,(
  ! [X126 : (subst > term > subst > $o)] : (($true = ((sP18 @ X126))) <=> ? [X142 : term,X143 : term] : ((((X126 @ id @ X143 @ id)) = $true) & (((X126 @ id @ (ap @ (sub @ X142 @ id) @ X143) @ id)) != $true) & (((X126 @ id @ X142 @ id)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f32,definition,(
  (sP19 = $true) <=> ? [X111 : term,X110 : (subst > term > subst > $o)] : (! [X125 : term] : (($true != ((X110 @ id @ X125 @ id))) | (((X110 @ id @ (sub @ X111 @ (push @ X125 @ id)) @ id)) = $true)) & ($true != ((X110 @ id @ (lam @ X111) @ id))) & ! [X114 : subst,X113 : subst,X115 : term,X112 : subst] : ((((X110 @ X112 @ X115 @ (comp @ X114 @ X113))) != $true) | (((X110 @ (comp @ X112 @ X114) @ (sub @ X115 @ X114) @ X113)) = $true)) & ! [X119 : subst,X117 : subst,X116 : subst,X118 : term] : ((((X110 @ (comp @ X119 @ X117) @ (sub @ X118 @ X117) @ X116)) != $true) | (((X110 @ X119 @ X118 @ (comp @ X117 @ X116))) = $true)) & (((sP15 @ X110)) = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f33,definition,(
  (sP20 = $true) <=> ! [X126 : (subst > term > subst > $o),X127 : term] : ((((X126 @ id @ X127 @ id)) = $true) | ($true = ((sP18 @ X126))) | ? [X139 : term,X137 : subst,X138 : subst,X140 : subst] : ((((X126 @ (comp @ X138 @ X140) @ (sub @ X139 @ X140) @ X137)) != $true) & (((X126 @ X138 @ X139 @ (comp @ X140 @ X137))) = $true)) | ? [X141 : term] : ((((var @ (sub @ X141 @ id))) = $true) & (((X126 @ id @ X141 @ id)) != $true)) | ? [X134 : term,X133 : subst,X136 : subst,X135 : subst] : ((((X126 @ X135 @ X134 @ (comp @ X133 @ X136))) != $true) & (((X126 @ (comp @ X135 @ X133) @ (sub @ X134 @ X133) @ X136)) = $true)) | (((sP17 @ X126)) = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f34,definition,(
  (sP21 = $true) <=> ? [X97 : term,X99 : term,X98 : (subst > term > subst > $o)] : (! [X104 : subst,X105 : subst,X106 : subst,X107 : term] : ((((X98 @ (comp @ X106 @ X104) @ (sub @ X107 @ X104) @ X105)) != $true) | (((X98 @ X106 @ X107 @ (comp @ X104 @ X105))) = $true)) & (((X98 @ id @ (ap @ X99 @ X97) @ id)) != $true) & (((sP14 @ X98)) = $true) & (((X98 @ id @ X99 @ id)) = $true) & ! [X103 : subst,X100 : subst,X101 : term,X102 : subst] : ((((X98 @ (comp @ X102 @ X103) @ (sub @ X101 @ X103) @ X100)) = $true) | (((X98 @ X102 @ X101 @ (comp @ X103 @ X100))) != $true)) & ($true = ((X98 @ id @ X97 @ id))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f35,definition,(
  (sP22 = $true) <=> ((sP20 = $true) | ? [X90 : (term > $o),X91 : term] : (! [X92 : term] : ((((X90 @ (lam @ X92))) = $true) | ? [X93 : term] : ((((X90 @ X93)) = $true) & (((X90 @ (sub @ X92 @ (push @ X93 @ id)))) != $true))) & (((sP16 @ X90)) = $true) & (((X90 @ X91)) != $true) & ! [X96 : term] : ((((var @ X96)) != $true) | (((X90 @ X96)) = $true))) | (sP19 = $true) | (sP21 = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f36,definition,(
  (sP23 = $true) <=> (? [X39 : term] : (((sub @ X39 @ id)) != X39) | ! [X40 : (term > $o),X41 : (subst > term > subst > $o),X42 : term] : (? [X45 : (subst > term > term)] : (! [X46 : term] : ((((X41 @ id @ X46 @ id)) != $true) | (((X41 @ id @ (X45 @ id @ X46) @ id)) = $true)) & ! [X47 : subst,X49 : term,X48 : subst] : (((X45 @ (comp @ X47 @ X48) @ (sub @ X49 @ X48))) = ((sub @ (X45 @ X47 @ X49) @ X48))) & (((X41 @ id @ (lam @ (X45 @ sh @ one)) @ id)) != $true)) | ? [X43 : term] : (((X41 @ id @ X43 @ id)) != ((X40 @ X43))) | (((X40 @ (lam @ X42))) = $true) | ? [X44 : term] : ((((X40 @ X44)) = $true) & (((X40 @ (sub @ X42 @ (push @ X44 @ id)))) != $true))) | ? [X35 : term] : ! [X36 : (subst > term > term)] : ? [X37 : term,X38 : subst] : (((X36 @ X38 @ X37)) != ((sub @ X35 @ (push @ X37 @ X38)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f37,definition,(
  (sP24 = $true) <=> ! [X83 : (subst > term > subst > $o),X82 : term] : ((((X83 @ id @ (lam @ (sub @ X82 @ (push @ one @ sh))) @ id)) = $true) | ? [X85 : (subst > term > term)] : (($true != ((X83 @ id @ (lam @ (X85 @ sh @ one)) @ id))) & ! [X89 : term] : ((((X83 @ id @ X89 @ id)) != $true) | (((X83 @ id @ (X85 @ id @ X89) @ id)) = $true)) & ! [X88 : subst,X87 : subst,X86 : term] : (((X85 @ (comp @ X88 @ X87) @ (sub @ X86 @ X87))) = ((sub @ (X85 @ X88 @ X86) @ X87)))) | ? [X84 : term] : ((((X83 @ id @ X84 @ id)) = $true) & (((X83 @ id @ (sub @ X82 @ (push @ X84 @ id)) @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f38,definition,(
  ! [X0 : (subst > term > subst > $o)] : ((((sP25 @ X0)) = $true) <=> ? [X3 : (subst > term > term)] : (! [X4 : term] : ((((X0 @ id @ (X3 @ id @ X4) @ id)) = $true) | (((X0 @ id @ X4 @ id)) != $true)) & ! [X5 : subst,X7 : term,X6 : subst] : (((X3 @ (comp @ X6 @ X5) @ (sub @ X7 @ X5))) = ((sub @ (X3 @ X6 @ X7) @ X5))) & (((X0 @ id @ (lam @ (X3 @ sh @ one)) @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f39,definition,(
  (sP26 = $true) <=> ! [X0 : (subst > term > subst > $o),X1 : term] : (? [X11 : term,X9 : subst,X8 : subst,X10 : subst] : ((((X0 @ (comp @ X9 @ X10) @ (sub @ X11 @ X10) @ X8)) = $true) & ($true != ((X0 @ X9 @ X11 @ (comp @ X10 @ X8))))) | (((X0 @ id @ (lam @ X1) @ id)) = $true) | ? [X15 : subst,X14 : subst,X12 : term,X13 : subst] : ((((X0 @ X15 @ X12 @ (comp @ X14 @ X13))) = $true) & (((X0 @ (comp @ X15 @ X14) @ (sub @ X12 @ X14) @ X13)) != $true)) | (((sP25 @ X0)) = $true) | ? [X2 : term] : ((((X0 @ id @ X2 @ id)) = $true) & (((X0 @ id @ (sub @ X1 @ (push @ X2 @ id)) @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f40,definition,(
  ! [X513 : (subst > term > subst > $o)] : (($true = ((sP27 @ X513))) <=> ? [X518 : term,X517 : term] : ((((X513 @ id @ X517 @ id)) = $true) & (((X513 @ id @ (ap @ (sub @ X517 @ id) @ X518) @ id)) != $true) & (((X513 @ id @ X518 @ id)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f41,definition,(
  (sP28 = $true) <=> ! [X513 : (subst > term > subst > $o),X515 : term,X514 : term,X516 : (term > $o)] : ((((X516 @ (ap @ X515 @ X514))) = $true) | (((X516 @ X514)) != $true) | ? [X523 : term,X520 : subst,X521 : subst,X522 : subst] : ((((X513 @ X520 @ X523 @ (comp @ X521 @ X522))) = $true) & (((X513 @ (comp @ X520 @ X521) @ (sub @ X523 @ X521) @ X522)) != $true)) | ($true != ((X516 @ X515))) | ($true = ((sP27 @ X513))) | ? [X527 : subst,X524 : subst,X526 : term,X525 : subst] : ((((X513 @ (comp @ X527 @ X524) @ (sub @ X526 @ X524) @ X525)) = $true) & (((X513 @ X527 @ X526 @ (comp @ X524 @ X525))) != $true)) | ? [X519 : term] : (((X513 @ id @ X519 @ id)) != ((X516 @ X519))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f42,definition,(
  (sP29 = $true) <=> ! [X203 : term,X204 : (subst > term > subst > $o)] : (? [X210 : term] : ((((X204 @ id @ X210 @ id)) = $true) & (((X204 @ id @ (sub @ X203 @ (push @ X210 @ id)) @ id)) != $true)) | ? [X205 : (subst > term > term)] : (! [X208 : term,X207 : subst,X206 : subst] : (((X205 @ (comp @ X206 @ X207) @ (sub @ X208 @ X207))) = ((sub @ (X205 @ X206 @ X208) @ X207))) & (((X204 @ id @ (lam @ (X205 @ sh @ one)) @ id)) != $true) & ! [X209 : term] : ((((X204 @ id @ X209 @ id)) != $true) | ($true = ((X204 @ id @ (X205 @ id @ X209) @ id))))) | ($true = ((X204 @ id @ (lam @ X203) @ id))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f43,definition,(
  (sP30 = $true) <=> (? [X427 : term,X428 : subst] : (((sub @ one @ (push @ X427 @ X428))) != X427) | ! [X434 : term,X433 : (subst > term > term),X435 : subst,X436 : (subst > term > term)] : (? [X442 : subst,X440 : term,X441 : subst] : (((X433 @ (comp @ X441 @ X442) @ (sub @ X440 @ X442))) != ((sub @ (X433 @ X441 @ X440) @ X442))) | (((lam @ (X433 @ sh @ one))) != ((lam @ (X436 @ sh @ one)))) | ? [X437 : subst,X438 : subst,X439 : term] : (((sub @ (X436 @ X438 @ X439) @ X437)) != ((X436 @ (comp @ X438 @ X437) @ (sub @ X439 @ X437)))) | (((X433 @ X435 @ X434)) = ((X436 @ X435 @ X434)))) | ? [X430 : subst,X429 : term] : (((comp @ sh @ (push @ X429 @ X430))) != X430) | ? [X432 : term,X431 : term] : ((((lam @ X432)) = ((lam @ X431))) & (X431 != X432)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f44,definition,(
  ! [X490 : (subst > term > subst > $o)] : ((((sP31 @ X490)) = $true) <=> ? [X493 : (subst > term > term)] : (! [X497 : term] : (($true != ((X490 @ id @ X497 @ id))) | (((X490 @ id @ (X493 @ id @ X497) @ id)) = $true)) & (((X490 @ id @ (lam @ (X493 @ sh @ one)) @ id)) != $true) & ! [X496 : term,X494 : subst,X495 : subst] : (((X493 @ (comp @ X494 @ X495) @ (sub @ X496 @ X495))) = ((sub @ (X493 @ X494 @ X496) @ X495)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f45,definition,(
  ! [X490 : (subst > term > subst > $o)] : ((((sP32 @ X490)) = $true) <=> ? [X492 : term,X491 : term] : ((((X490 @ id @ X492 @ id)) = $true) & (((X490 @ id @ (ap @ (sub @ X492 @ id) @ X491) @ id)) != $true) & (((X490 @ id @ X491 @ id)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f46,definition,(
  (sP33 = $true) <=> ! [X489 : term,X490 : (subst > term > subst > $o)] : (? [X504 : subst,X503 : term,X505 : subst,X506 : subst] : ((((X490 @ (comp @ X504 @ X506) @ (sub @ X503 @ X506) @ X505)) = $true) & (((X490 @ X504 @ X503 @ (comp @ X506 @ X505))) != $true)) | (((X490 @ id @ X489 @ id)) = $true) | ? [X498 : term] : ((((var @ (sub @ X498 @ id))) = $true) & (((X490 @ id @ X498 @ id)) != $true)) | ? [X501 : subst,X499 : subst,X502 : term,X500 : subst] : ((((X490 @ X499 @ X502 @ (comp @ X500 @ X501))) = $true) & ($true != ((X490 @ (comp @ X499 @ X500) @ (sub @ X502 @ X500) @ X501)))) | (((sP32 @ X490)) = $true) | (((sP31 @ X490)) = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f47,definition,(
  (sP34 = $true) <=> ! [X233 : term,X232 : (subst > term > subst > $o)] : (? [X234 : (subst > term > term)] : (! [X238 : subst,X236 : subst,X237 : term] : (((sub @ (X234 @ X238 @ X237) @ X236)) = ((X234 @ (comp @ X238 @ X236) @ (sub @ X237 @ X236)))) & (((X232 @ id @ (lam @ (X234 @ sh @ one)) @ id)) != $true) & ! [X235 : term] : (($true != ((X232 @ id @ X235 @ id))) | (((X232 @ id @ (X234 @ id @ X235) @ id)) = $true))) | (((X232 @ id @ (lam @ (sub @ X233 @ (push @ one @ sh))) @ id)) = $true) | ? [X239 : term] : ((((X232 @ id @ (sub @ X233 @ (push @ X239 @ id)) @ id)) != $true) & (((X232 @ id @ X239 @ id)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f48,definition,(
  ! [X256 : (term > $o)] : ((((sP35 @ X256)) = $true) <=> ! [X260 : term,X261 : term] : ((((X256 @ X260)) != $true) | (((X256 @ X261)) != $true) | (((X256 @ (ap @ X261 @ X260))) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f49,definition,(
  ! [X264 : (term > $o)] : ((((sP36 @ X264)) = $true) <=> ? [X269 : term,X268 : term] : ((((X264 @ X268)) = $true) & (((X264 @ X269)) = $true) & (((X264 @ (ap @ X269 @ X268))) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f50,definition,(
  (sP37 = $true) <=> ! [X264 : (term > $o),X265 : term] : (? [X266 : term] : (! [X267 : term] : ((((X264 @ X267)) != $true) | (((X264 @ (sub @ X266 @ (push @ X267 @ id)))) = $true)) & (((X264 @ (lam @ X266))) != $true)) | (((sP36 @ X264)) = $true) | (((X264 @ X265)) = $true) | ? [X270 : term] : ((((X264 @ X270)) != $true) & (((var @ X270)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f51,definition,(
  (sP38 = $true) <=> ((sP37 = $true) | ? [X255 : term] : (((sub @ X255 @ id)) != X255) | ? [X257 : term,X256 : (term > $o),X258 : subst] : (! [X259 : term] : ((((X256 @ (sub @ X259 @ X258))) = $true) | ($true != ((var @ X259)))) & (((X256 @ (sub @ X257 @ X258))) != $true) & ! [X262 : term] : ((((X256 @ (lam @ X262))) = $true) | ? [X263 : term] : ((((X256 @ X263)) = $true) & (((X256 @ (sub @ X262 @ (push @ X263 @ id)))) != $true))) & (((sP35 @ X256)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f52,definition,(
  ! [X60 : (term > $o)] : ((((sP39 @ X60)) = $true) <=> ? [X64 : term,X63 : term] : ((((X60 @ X63)) = $true) & (((X60 @ (ap @ X63 @ X64))) != $true) & (((X60 @ X64)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f53,definition,(
  (sP40 = $true) <=> ! [X60 : (term > $o),X58 : term,X59 : subst] : (? [X65 : term] : ((((var @ X65)) = $true) & (((X60 @ (sub @ X65 @ X59))) != $true)) | (((sP39 @ X60)) = $true) | ? [X61 : term] : ((((X60 @ (lam @ X61))) != $true) & ! [X62 : term] : ((((X60 @ (sub @ X61 @ (push @ X62 @ id)))) = $true) | (((X60 @ X62)) != $true))) | (((X60 @ (sub @ X58 @ X59))) = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f54,definition,(
  ! [X166 : (term > $o)] : ((((sP41 @ X166)) = $true) <=> ? [X171 : term,X170 : term] : ((((X166 @ X171)) = $true) & (((X166 @ X170)) = $true) & (((X166 @ (ap @ X170 @ X171))) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f55,definition,(
  ! [X160 : (term > $o)] : ((((sP42 @ X160)) = $true) <=> ! [X162 : term,X163 : term] : ((((X160 @ X163)) != $true) | (((X160 @ X162)) != $true) | (((X160 @ (ap @ X162 @ X163))) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f56,definition,(
  (sP43 = $true) <=> ! [X165 : subst,X167 : term,X166 : (term > $o)] : (? [X172 : term] : ((((X166 @ (sub @ X172 @ X165))) != $true) & (((var @ X172)) = $true)) | (((sP41 @ X166)) = $true) | ? [X168 : term] : ((((X166 @ (lam @ X168))) != $true) & ! [X169 : term] : (($true != ((X166 @ X169))) | (((X166 @ (sub @ X168 @ (push @ X169 @ id)))) = $true))) | (((X166 @ (sub @ X167 @ X165))) = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f57,definition,(
  (sP44 = $true) <=> (? [X160 : (term > $o),X159 : term] : ((((sP42 @ X160)) = $true) & (((X160 @ X159)) != $true) & ! [X161 : term] : ((((X160 @ (lam @ X161))) = $true) | (((X160 @ X161)) != $true)) & ! [X164 : term] : ((((var @ X164)) != $true) | ($true = ((X160 @ X164))))) | ? [X158 : subst] : (((comp @ X158 @ id)) != X158) | ? [X154 : term,X155 : subst] : (((sub @ one @ (push @ X154 @ X155))) != X154) | (sP43 = $true) | ? [X157 : term,X156 : subst] : (((comp @ sh @ (push @ X157 @ X156))) != X156))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f58,definition,(
  ! [X450 : (subst > term > subst > $o)] : ((((sP45 @ X450)) = $true) <=> ? [X463 : (subst > term > term)] : (! [X467 : term] : ((((X450 @ id @ (X463 @ id @ X467) @ id)) = $true) | (((X450 @ id @ X467 @ id)) != $true)) & (((X450 @ id @ (lam @ (X463 @ sh @ one)) @ id)) != $true) & ! [X465 : subst,X464 : subst,X466 : term] : (((X463 @ (comp @ X465 @ X464) @ (sub @ X466 @ X464))) = ((sub @ (X463 @ X465 @ X466) @ X464)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f59,definition,(
  (sP46 = $true) <=> ! [X450 : (subst > term > subst > $o),X451 : (term > $o),X452 : term] : ((((X451 @ (lam @ X452))) = $true) | ? [X458 : subst,X459 : subst,X457 : subst,X460 : term] : ((((X450 @ X457 @ X460 @ (comp @ X459 @ X458))) = $true) & (((X450 @ (comp @ X457 @ X459) @ (sub @ X460 @ X459) @ X458)) != $true)) | ? [X461 : term] : ((((X451 @ (sub @ X452 @ (push @ X461 @ id)))) != $true) & (((X451 @ X461)) = $true)) | ? [X453 : term,X455 : subst,X454 : subst,X456 : subst] : ((((X450 @ X456 @ X453 @ (comp @ X454 @ X455))) != $true) & (((X450 @ (comp @ X456 @ X454) @ (sub @ X453 @ X454) @ X455)) = $true)) | (((sP45 @ X450)) = $true) | ? [X462 : term] : (((X451 @ X462)) != ((X450 @ id @ X462 @ id))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f60,definition,(
  ! [X392 : (subst > term > subst > $o)] : ((((sP47 @ X392)) = $true) <=> ? [X395 : term,X394 : term] : ((((X392 @ id @ (ap @ (sub @ X394 @ id) @ X395) @ id)) != $true) & ($true = ((X392 @ id @ X395 @ id))) & (((X392 @ id @ X394 @ id)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f61,definition,(
  ! [X375 : (term > $o)] : ((((sP48 @ X375)) = $true) <=> ! [X379 : term,X380 : term] : ((((X375 @ X380)) != $true) | (((X375 @ X379)) != $true) | (((X375 @ (ap @ X380 @ X379))) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f62,definition,(
  (sP49 = $true) <=> ? [X382 : (subst > term > subst > $o),X384 : (term > $o),X383 : term] : ((((X384 @ (lam @ X383))) != $true) & ! [X386 : term] : (((X382 @ id @ X386 @ id)) = ((X384 @ X386))) & ! [X387 : (subst > term > term)] : (? [X391 : term] : ((((X382 @ id @ X391 @ id)) = $true) & (((X382 @ id @ (X387 @ id @ X391) @ id)) != $true)) | ? [X390 : subst,X388 : subst,X389 : term] : (((X387 @ (comp @ X390 @ X388) @ (sub @ X389 @ X388))) != ((sub @ (X387 @ X390 @ X389) @ X388))) | (((X382 @ id @ (lam @ (X387 @ sh @ one)) @ id)) = $true)) & ! [X385 : term] : ((((X384 @ X385)) != $true) | (((X384 @ (sub @ X383 @ (push @ X385 @ id)))) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f63,definition,(
  (sP50 = $true) <=> ! [X393 : term,X392 : (subst > term > subst > $o)] : ((((X392 @ id @ X393 @ id)) = $true) | ? [X396 : (subst > term > term)] : ((((X392 @ id @ (lam @ (X396 @ sh @ one)) @ id)) != $true) & ! [X400 : subst,X398 : term,X399 : subst] : (((X396 @ (comp @ X399 @ X400) @ (sub @ X398 @ X400))) = ((sub @ (X396 @ X399 @ X398) @ X400))) & ! [X397 : term] : ((((X392 @ id @ (X396 @ id @ X397) @ id)) = $true) | (((X392 @ id @ X397 @ id)) != $true))) | (((sP47 @ X392)) = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f64,definition,(
  (sP51 = $true) <=> (? [X374 : term,X375 : (term > $o)] : (! [X377 : term] : ((((X375 @ (lam @ X377))) = $true) | ? [X378 : term] : ((((X375 @ (sub @ X377 @ (push @ X378 @ id)))) != $true) & (((X375 @ X378)) = $true))) & ! [X376 : term] : ((((X375 @ X376)) = $true) | ($true != ((var @ X376)))) & (((sP48 @ X375)) = $true) & (((X375 @ X374)) != $true)) | (sP50 = $true) | (sP49 = $true) | ? [X381 : term] : (((sub @ X381 @ id)) != X381) | ? [X371 : (subst > term > subst > $o)] : ! [X372 : (term > $o)] : ? [X373 : term] : (((X371 @ id @ X373 @ id)) != ((X372 @ X373))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f65,definition,(
  ! [X174 : (subst > term > subst > $o)] : ((((sP52 @ X174)) = $true) <=> ? [X177 : (subst > term > term)] : (! [X179 : subst,X180 : subst,X181 : term] : (((sub @ (X177 @ X179 @ X181) @ X180)) = ((X177 @ (comp @ X179 @ X180) @ (sub @ X181 @ X180)))) & ! [X178 : term] : ((((X174 @ id @ (X177 @ id @ X178) @ id)) = $true) | (((X174 @ id @ X178 @ id)) != $true)) & (((X174 @ id @ (lam @ (X177 @ sh @ one)) @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f66,definition,(
  ! [X326 : (subst > term > subst > $o)] : ((((sP53 @ X326)) = $true) <=> ? [X334 : (subst > term > term)] : ((((X326 @ id @ (lam @ (X334 @ sh @ one)) @ id)) != $true) & ! [X335 : term] : ((((X326 @ id @ X335 @ id)) != $true) | (((X326 @ id @ (X334 @ id @ X335) @ id)) = $true)) & ! [X336 : subst,X338 : subst,X337 : term] : (((sub @ (X334 @ X336 @ X337) @ X338)) = ((X334 @ (comp @ X336 @ X338) @ (sub @ X337 @ X338))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f67,definition,(
  ! [X326 : (subst > term > subst > $o)] : ((((sP54 @ X326)) = $true) <=> ? [X329 : term,X328 : term] : ((((X326 @ id @ X328 @ id)) = $true) & (((X326 @ id @ X329 @ id)) = $true) & (((X326 @ id @ (ap @ (sub @ X329 @ id) @ X328) @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f68,definition,(
  ! [X308 : (term > $o)] : ((((sP55 @ X308)) = $true) <=> ! [X310 : term,X311 : term] : ((((X308 @ X311)) != $true) | (((X308 @ (ap @ X310 @ X311))) = $true) | (((X308 @ X310)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f69,definition,(
  ($true = sP56) <=> ! [X327 : term,X326 : (subst > term > subst > $o)] : (? [X343 : term] : ((((var @ (sub @ X343 @ id))) = $true) & (((X326 @ id @ X343 @ id)) != $true)) | (((sP54 @ X326)) = $true) | ? [X331 : subst,X333 : subst,X332 : term,X330 : subst] : ((((X326 @ X331 @ X332 @ (comp @ X330 @ X333))) = $true) & (((X326 @ (comp @ X331 @ X330) @ (sub @ X332 @ X330) @ X333)) != $true)) | ($true = ((X326 @ id @ X327 @ id))) | ? [X341 : subst,X342 : subst,X339 : term,X340 : subst] : ((((X326 @ (comp @ X342 @ X341) @ (sub @ X339 @ X341) @ X340)) = $true) & (((X326 @ X342 @ X339 @ (comp @ X341 @ X340))) != $true)) | (((sP53 @ X326)) = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f70,definition,(
  (sP57 = $true) <=> ? [X316 : (term > $o),X318 : term,X317 : (subst > term > subst > $o)] : (! [X319 : (subst > term > term)] : ((((X317 @ id @ (lam @ (X319 @ sh @ one)) @ id)) = $true) | ? [X323 : term] : ((((X317 @ id @ (X319 @ id @ X323) @ id)) != $true) & (((X317 @ id @ X323 @ id)) = $true)) | ? [X322 : term,X320 : subst,X321 : subst] : (((X319 @ (comp @ X321 @ X320) @ (sub @ X322 @ X320))) != ((sub @ (X319 @ X321 @ X322) @ X320)))) & ! [X324 : term] : ((((X316 @ (sub @ X318 @ (push @ X324 @ id)))) = $true) | (((X316 @ X324)) != $true)) & (((X316 @ (lam @ X318))) != $true) & ! [X325 : term] : (((X317 @ id @ X325 @ id)) = ((X316 @ X325))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f71,definition,(
  (sP58 = $true) <=> (? [X315 : term] : (((sub @ X315 @ id)) != X315) | ? [X305 : (subst > term > subst > $o)] : ! [X306 : (term > $o)] : ? [X307 : term] : (((X306 @ X307)) != ((X305 @ id @ X307 @ id))) | (sP57 = $true) | ? [X309 : term,X308 : (term > $o)] : ((((X308 @ X309)) != $true) & ! [X312 : term] : ((((X308 @ X312)) = $true) | (((var @ X312)) != $true)) & ! [X313 : term] : (? [X314 : term] : ((((X308 @ (sub @ X313 @ (push @ X314 @ id)))) != $true) & (((X308 @ X314)) = $true)) | (((X308 @ (lam @ X313))) = $true)) & (((sP55 @ X308)) = $true)) | ($true = sP56))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f72,plain,(
  ((hoasinduction_lthm_3 = $true) <=> (sP58 = $true)) & (termmset_gthm = $true) & ((substmonoid = $true) <=> (! [X71 : subst] : (((comp @ id @ X71)) = X71) & ! [X70 : subst] : (((comp @ X70 @ id)) = X70))) & (pushprop_gthm = $true) & (pushprop_lem3v2_lthm = $true) & (id = ((d2subst @ d_id))) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (pushprop_lthm = $true) & (((d2subst @ d_id)) = sh) & ! [X69 : d_term] : (d_one = X69) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X507 : term] : ? [X508 : (subst > term > term)] : ! [X509 : term,X510 : subst] : (((sub @ X507 @ (push @ X509 @ X510))) = ((X508 @ X510 @ X509)))) & (hoaslamnotvar_lthm = $true) & ! [X344 : d_subst] : (d_id = X344) & (induction2_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & (pushprop_lthm_orig = $true) & ((hoasinduction_no_psi_cond = $true) <=> ! [X173 : term,X174 : (subst > term > subst > $o)] : (? [X176 : term,X175 : term] : ((((X174 @ id @ (ap @ (sub @ X176 @ id) @ X175) @ id)) != $true) & (((X174 @ id @ X175 @ id)) = $true) & (((X174 @ id @ X176 @ id)) = $true)) | (((sP52 @ X174)) = $true) | (((X174 @ id @ X173 @ id)) = $true))) & (hoaslamnotap_lthm = $true) & (! [X410 : subst,X409 : (term > $o),X408 : term] : ? [X411 : (term > $o)] : ! [X412 : term] : (((X409 @ (sub @ X412 @ (push @ X408 @ X410)))) = ((X411 @ X412))) <=> (pushprop_lem0 = $true)) & (hoaslamnotvar_gthm = $true) & (axclos = $true) & ((sP51 = $true) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (((var @ (d2term @ d_one))) != $true) & ((hoasinduction_lem3v2 = $true) <=> (sP46 = $true)) & ((sP44 = $true) <=> (induction2lem_lthm = $true)) & ((termmset_lthm = $true) <=> (! [X449 : term] : (((sub @ X449 @ id)) = X449) | ? [X448 : term] : (((sub @ X448 @ id)) != X448))) & (! [X152 : subst,X153 : term] : (((comp @ sh @ (push @ X153 @ X152))) = X152) <=> (axshiftcons = $true)) & (hoasinduction_lem2v2_gthm = $true) & (pushprop_lem0_gthm = $true) & (hoasapnotvar_lthm = $true) & ((pushprop_lem1 = $true) <=> ! [X246 : term,X247 : term,X250 : (term > $o),X248 : subst,X249 : (term > $o)] : ((((X249 @ X247)) != $true) | (((X250 @ (sub @ X247 @ (push @ X246 @ X248)))) = $true))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (pushprop_lem1v2_gthm = $true) & ((axidr = $true) <=> ! [X150 : subst] : (((comp @ X150 @ id)) = X150)) & (one = ((d2term @ d_one))) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((sP40 = $true) <=> (induction2lem = $true)) & ! [X512 : d_subst,X511 : d_subst] : ((X511 = X512) | (((d2subst @ X511)) != ((d2subst @ X512)))) & ((sP38 = $true) <=> (induction2_lthm = $true)) & (hoasapinj2_gthm = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (hoasinduction_lem3v2_gthm = $true) & ((hoasinduction_lem3aa_lthm = $true) <=> (sP34 = $true)) & (! [X471 : subst] : (((comp @ id @ X471)) = X471) <=> (axidl = $true)) & ((hoasinduction = $true) <=> (sP33 = $true)) & (substmonoid_gthm = $true) & (hoasinduction_lem2_gthm = $true) & (axabs = $true) & ((! [X303 : term,X301 : term,X302 : term,X304 : term] : ((X301 = X303) | (((ap @ (sub @ X303 @ id) @ X302)) != ((ap @ (sub @ X301 @ id) @ X304)))) | ? [X296 : term] : (((sub @ X296 @ id)) != X296) | ? [X297 : term,X300 : term,X298 : term,X299 : term] : ((X297 != X298) & (((ap @ X298 @ X299)) = ((ap @ X297 @ X300))))) <=> (hoasapinj1_lthm = $true)) & (ulamvarind = $true) & ((pushprop_lem1_lthm = $true) <=> (? [X345 : subst,X346 : term] : (((sub @ one @ (push @ X346 @ X345))) != X346) | ? [X347 : term,X348 : subst] : (((comp @ sh @ (push @ X347 @ X348))) != X348) | ! [X349 : subst,X353 : (term > $o),X352 : (term > $o),X350 : term,X351 : term] : ((((X352 @ (sub @ X351 @ (push @ X350 @ X349)))) = $true) | (((X353 @ X351)) != $true)))) & (hoasinduction_lem1_lthm = $true) & (hoaslamnotap_gthm = $true) & (axvarshift = $true) & ((? [X77 : term,X74 : term,X75 : term,X76 : term] : ((((ap @ X77 @ X74)) = ((ap @ X76 @ X75))) & (X74 != X75)) | ! [X80 : term,X81 : term,X79 : term,X78 : term] : ((((ap @ (sub @ X81 @ id) @ X78)) != ((ap @ (sub @ X79 @ id) @ X80))) | (X78 = X80))) <=> (hoasapinj2_lthm = $true)) & (hoasinduction_lem1v2 = $true) & (ulamvar1 != $true) & ! [X72 : d_term,X73 : d_term] : ((X72 = X73) | (((d2term @ X72)) != ((d2term @ X73)))) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (pushprop_lem1_gthm = $true) & (! [X51 : term,X50 : term,X52 : (subst > term > term)] : (? [X55 : term,X53 : subst,X54 : subst] : (((sub @ (X52 @ X53 @ X55) @ X54)) != ((X52 @ (comp @ X53 @ X54) @ (sub @ X55 @ X54)))) | (((lam @ (X52 @ sh @ one))) != ((ap @ (sub @ X50 @ id) @ X51)))) <=> (hoaslamnotap = $true)) & ((axvarcons = $true) <=> ! [X356 : subst,X357 : term] : (((sub @ one @ (push @ X357 @ X356))) = X357)) & ((hoasapinj1 = $true) <=> ! [X144 : term,X147 : term,X146 : term,X145 : term] : ((((ap @ (sub @ X147 @ id) @ X146)) != ((ap @ (sub @ X145 @ id) @ X144))) | (X145 = X147))) & (hoasinduction_lem3a_gthm = $true) & (! [X443 : (term > $o),X444 : term,X446 : (term > $o),X445 : subst] : (? [X447 : term] : (((X443 @ X447)) != ((X446 @ (sub @ X447 @ (push @ X444 @ X445))))) | (((X446 @ X444)) != $true) | (((X443 @ one)) = $true)) <=> (pushprop_lem1v2 = $true)) & ((sP30 = $true) <=> (hoaslaminj_lthm = $true)) & (hoaslamnotvar = $true) & (hoasinduction_lem1_gthm = $true) & ((sP29 = $true) <=> (hoasinduction_lem3a = $true)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((! [X190 : term,X188 : subst,X191 : (term > $o),X189 : (term > $o)] : (($true != ((X191 @ X190))) | ? [X192 : term] : (((X191 @ (sub @ X192 @ (push @ X190 @ X188)))) != ((X189 @ X192))) | (((X189 @ one)) = $true)) | ? [X187 : subst,X186 : term] : (((sub @ one @ (push @ X186 @ X187))) != X186)) <=> (pushprop_lem1v2_lthm = $true)) & (hoaslaminj_gthm = $true) & (! [X472 : term,X473 : term,X474 : term,X475 : term] : ((((ap @ X475 @ X473)) != ((ap @ X472 @ X474))) | (X472 = X475)) <=> (apinj1 = $true)) & ((sP28 = $true) <=> (hoasinduction_lem2v2 = $true)) & (hoasinduction_gthm = $true) & ((hoasinduction_lem0_lthm = $true) <=> ! [X66 : (subst > term > subst > $o)] : ? [X67 : (term > $o)] : ! [X68 : term] : (((X67 @ X68)) = ((X66 @ id @ X68 @ id)))) & (! [X361 : (subst > term > term),X363 : subst,X362 : term,X364 : (subst > term > term)] : (? [X365 : term,X366 : subst,X367 : subst] : (((X364 @ (comp @ X366 @ X367) @ (sub @ X365 @ X367))) != ((sub @ (X364 @ X366 @ X365) @ X367))) | ? [X370 : subst,X369 : term,X368 : subst] : (((sub @ (X361 @ X370 @ X369) @ X368)) != ((X361 @ (comp @ X370 @ X368) @ (sub @ X369 @ X368)))) | (((X364 @ X363 @ X362)) = ((X361 @ X363 @ X362))) | (((lam @ (X361 @ sh @ one))) != ((lam @ (X364 @ sh @ one))))) <=> (hoaslaminj = $true)) & (hoasinduction_lem1v2_gthm = $true) & ((substmonoid_lthm = $true) <=> ((! [X424 : subst] : (((comp @ id @ X424)) = X424) & ! [X425 : subst] : (((comp @ X425 @ id)) = X425)) | ? [X423 : subst] : (((comp @ X423 @ id)) != X423) | ? [X422 : subst] : (((comp @ id @ X422)) != X422))) & (apnotvar = $true) & (hoasinduction_lem1 = $true) & (axmap = $true) & (axapp = $true) & (hoasapnotvar = $true) & (! [X470 : term] : (((sub @ X470 @ id)) = X470) <=> (axvarid = $true)) & (! [X251 : term,X252 : term,X254 : term,X253 : term] : ((((ap @ X254 @ X251)) != ((ap @ X253 @ X252))) | (X251 = X252)) <=> (apinj2 = $true)) & ((hoasinduction_lem3 = $true) <=> (sP26 = $true)) & (hoasinduction_lem2_lthm = $true) & ((hoasinduction_lem3aa = $true) <=> (sP24 = $true)) & (induction2lem_gthm = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ! [X406 : term] : ? [X407 : d_term] : (((d2term @ X407)) = X406) & ((hoasinduction_lem3v2a_lthm = $true) <=> (sP23 = $true)) & ((hoasinduction_lem3b = $true) <=> ! [X56 : term] : ? [X57 : (subst > term > term)] : (((sub @ X56 @ (push @ one @ sh))) = ((X57 @ sh @ one)))) & (pushprop_lem3v2 = $true) & ! [X354 : subst] : ? [X355 : d_subst] : (((d2subst @ X355)) = X354) & (hoasapnotvar_gthm = $true) & (pushprop = $true) & ((hoasinduction_lthm = $true) <=> (sP22 = $true)) & (hoasinduction_lem3b_gthm = $true) & ((sP13 = $true) <=> (hoasinduction_lem3v2_lthm = $true)) & (lamnotap != $true) & (pushprop_lem2v2 = $true) & (lamnotvar = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((hoasapinj2 = $true) <=> ! [X185 : term,X183 : term,X184 : term,X182 : term] : ((((ap @ (sub @ X182 @ id) @ X184)) != ((ap @ (sub @ X183 @ id) @ X185))) | (X184 = X185))) & ((laminj = $true) <=> ! [X414 : term,X413 : term] : ((X413 = X414) | (((lam @ X413)) != ((lam @ X414))))) & (pushprop_lem2v2_gthm = $true) & ((hoasinduction_lem3a_lthm = $true) <=> ($true = sP11)) & (! [X211 : term] : ? [X212 : (subst > term > term)] : ! [X213 : subst,X214 : term] : (((X212 @ X213 @ X214)) = ((sub @ X211 @ (push @ X214 @ X213)))) <=> (hoasinduction_lem3v2_f = $true)) & ((hoasinduction_lem3_lthm = $true) <=> (sP9 = $true)) & ((termmset = $true) <=> ! [X151 : term] : (((sub @ X151 @ id)) = X151)) & (hoasinduction_lem3_gthm = $true) & ((sP6 = $true) <=> (hoasinduction_lem3v2a = $true)) & ((sP5 = $true) <=> (induction2 = $true)) & ((hoasinduction_lem0 = $true) <=> ! [X358 : (subst > term > subst > $o)] : ? [X359 : (term > $o)] : ! [X360 : term] : (((X358 @ id @ X360 @ id)) = ((X359 @ X360)))) & (! [X426 : subst] : (((push @ (sub @ one @ X426) @ (comp @ sh @ X426))) = X426) <=> (axscons = $true)) & (pushprop_lem2v2_lthm = $true) & (! [X469 : term,X468 : term] : ((X468 = X469) | (((sub @ X469 @ sh)) != ((sub @ X468 @ sh)))) <=> (shinj = $true)) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X148 : term] : ? [X149 : (subst > term > term)] : (((sub @ X148 @ (push @ one @ sh))) = ((X149 @ sh @ one)))) & (! [X241 : (term > $o),X240 : term] : (? [X242 : term] : ((((X241 @ X242)) = $true) & (((X241 @ (lam @ X242))) != $true)) | ? [X245 : term] : ((((var @ X245)) = $true) & (((X241 @ X245)) != $true)) | (((sP3 @ X241)) = $true) | (((X241 @ X240)) = $true)) <=> (induction = $true)) & ((hoasinduction_lem3aaa = $true) <=> (sP2 = $true)) & (! [X402 : subst,X403 : (term > $o),X401 : term] : ? [X404 : (term > $o)] : ! [X405 : term] : (((X403 @ (sub @ X405 @ (push @ X401 @ X402)))) = ((X404 @ X405))) <=> (pushprop_lem0_lthm = $true)) & (hoasapinj1_gthm = $true) & (ulamvarsh = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & ((sP1 = $true) <=> (hoasinduction_lem2 = $true)) & (axassoc = $true)),
  inference(definition_folding,[],[f12,f71,f70,f69,f68,f67,f66,f65,f64,f63,f62,f61,f60,f59,f58,f57,f56,f55,f54,f53,f52,f51,f50,f49,f48,f47,f46,f45,f44,f43,f42,f41,f40,f39,f38,f37,f36,f35,f34,f33,f32,f31,f30,f29,f28,f27,f26,f25,f24,f23,f22,f21,f20,f19,f18,f17,f16,f15,f14,f13])).
thf(f260,plain,(
  (((hoasinduction_lthm_3 = $true) | (sP58 != $true)) & ((sP58 = $true) | (hoasinduction_lthm_3 != $true))) & (termmset_gthm = $true) & (((substmonoid = $true) | (? [X71 : subst] : (((comp @ id @ X71)) != X71) | ? [X70 : subst] : (((comp @ X70 @ id)) != X70))) & ((! [X71 : subst] : (((comp @ id @ X71)) = X71) & ! [X70 : subst] : (((comp @ X70 @ id)) = X70)) | (substmonoid != $true))) & (pushprop_gthm = $true) & (pushprop_lem3v2_lthm = $true) & (id = ((d2subst @ d_id))) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (pushprop_lthm = $true) & (((d2subst @ d_id)) = sh) & ! [X69 : d_term] : (d_one = X69) & (((hoasinduction_lem3v2_f_lthm = $true) | ? [X507 : term] : ! [X508 : (subst > term > term)] : ? [X509 : term,X510 : subst] : (((sub @ X507 @ (push @ X509 @ X510))) != ((X508 @ X510 @ X509)))) & (! [X507 : term] : ? [X508 : (subst > term > term)] : ! [X509 : term,X510 : subst] : (((sub @ X507 @ (push @ X509 @ X510))) = ((X508 @ X510 @ X509))) | (hoasinduction_lem3v2_f_lthm != $true))) & (hoaslamnotvar_lthm = $true) & ! [X344 : d_subst] : (d_id = X344) & (induction2_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & (pushprop_lthm_orig = $true) & (((hoasinduction_no_psi_cond = $true) | ? [X173 : term,X174 : (subst > term > subst > $o)] : (! [X176 : term,X175 : term] : ((((X174 @ id @ (ap @ (sub @ X176 @ id) @ X175) @ id)) = $true) | (((X174 @ id @ X175 @ id)) != $true) | (((X174 @ id @ X176 @ id)) != $true)) & (((sP52 @ X174)) != $true) & (((X174 @ id @ X173 @ id)) != $true))) & (! [X173 : term,X174 : (subst > term > subst > $o)] : (? [X176 : term,X175 : term] : ((((X174 @ id @ (ap @ (sub @ X176 @ id) @ X175) @ id)) != $true) & (((X174 @ id @ X175 @ id)) = $true) & (((X174 @ id @ X176 @ id)) = $true)) | (((sP52 @ X174)) = $true) | (((X174 @ id @ X173 @ id)) = $true)) | (hoasinduction_no_psi_cond != $true))) & (hoaslamnotap_lthm = $true) & ((! [X410 : subst,X409 : (term > $o),X408 : term] : ? [X411 : (term > $o)] : ! [X412 : term] : (((X409 @ (sub @ X412 @ (push @ X408 @ X410)))) = ((X411 @ X412))) | (pushprop_lem0 != $true)) & ((pushprop_lem0 = $true) | ? [X410 : subst,X409 : (term > $o),X408 : term] : ! [X411 : (term > $o)] : ? [X412 : term] : (((X409 @ (sub @ X412 @ (push @ X408 @ X410)))) != ((X411 @ X412))))) & (hoaslamnotvar_gthm = $true) & (axclos = $true) & (((sP51 = $true) | (hoasinduction_no_psi_cond_lthm != $true)) & ((hoasinduction_no_psi_cond_lthm = $true) | (sP51 != $true))) & (((var @ (d2term @ d_one))) != $true) & (((hoasinduction_lem3v2 = $true) | (sP46 != $true)) & ((sP46 = $true) | (hoasinduction_lem3v2 != $true))) & (((sP44 = $true) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | (sP44 != $true))) & (((termmset_lthm = $true) | (? [X449 : term] : (((sub @ X449 @ id)) != X449) & ! [X448 : term] : (((sub @ X448 @ id)) = X448))) & ((! [X449 : term] : (((sub @ X449 @ id)) = X449) | ? [X448 : term] : (((sub @ X448 @ id)) != X448)) | (termmset_lthm != $true))) & ((! [X152 : subst,X153 : term] : (((comp @ sh @ (push @ X153 @ X152))) = X152) | (axshiftcons != $true)) & ((axshiftcons = $true) | ? [X152 : subst,X153 : term] : (((comp @ sh @ (push @ X153 @ X152))) != X152))) & (hoasinduction_lem2v2_gthm = $true) & (pushprop_lem0_gthm = $true) & (hoasapnotvar_lthm = $true) & (((pushprop_lem1 = $true) | ? [X246 : term,X247 : term,X250 : (term > $o),X248 : subst,X249 : (term > $o)] : ((((X249 @ X247)) = $true) & (((X250 @ (sub @ X247 @ (push @ X246 @ X248)))) != $true))) & (! [X246 : term,X247 : term,X250 : (term > $o),X248 : subst,X249 : (term > $o)] : ((((X249 @ X247)) != $true) | (((X250 @ (sub @ X247 @ (push @ X246 @ X248)))) = $true)) | (pushprop_lem1 != $true))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (pushprop_lem1v2_gthm = $true) & (((axidr = $true) | ? [X150 : subst] : (((comp @ X150 @ id)) != X150)) & (! [X150 : subst] : (((comp @ X150 @ id)) = X150) | (axidr != $true))) & (one = ((d2term @ d_one))) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (((sP40 = $true) | (induction2lem != $true)) & ((induction2lem = $true) | (sP40 != $true))) & ! [X512 : d_subst,X511 : d_subst] : ((X511 = X512) | (((d2subst @ X511)) != ((d2subst @ X512)))) & (((sP38 = $true) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | (sP38 != $true))) & (hoasapinj2_gthm = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (hoasinduction_lem3v2_gthm = $true) & (((hoasinduction_lem3aa_lthm = $true) | (sP34 != $true)) & ((sP34 = $true) | (hoasinduction_lem3aa_lthm != $true))) & ((! [X471 : subst] : (((comp @ id @ X471)) = X471) | (axidl != $true)) & ((axidl = $true) | ? [X471 : subst] : (((comp @ id @ X471)) != X471))) & (((hoasinduction = $true) | (sP33 != $true)) & ((sP33 = $true) | (hoasinduction != $true))) & (substmonoid_gthm = $true) & (hoasinduction_lem2_gthm = $true) & (axabs = $true) & (((! [X303 : term,X301 : term,X302 : term,X304 : term] : ((X301 = X303) | (((ap @ (sub @ X303 @ id) @ X302)) != ((ap @ (sub @ X301 @ id) @ X304)))) | ? [X296 : term] : (((sub @ X296 @ id)) != X296) | ? [X297 : term,X300 : term,X298 : term,X299 : term] : ((X297 != X298) & (((ap @ X298 @ X299)) = ((ap @ X297 @ X300))))) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | (? [X303 : term,X301 : term,X302 : term,X304 : term] : ((X301 != X303) & (((ap @ (sub @ X303 @ id) @ X302)) = ((ap @ (sub @ X301 @ id) @ X304)))) & ! [X296 : term] : (((sub @ X296 @ id)) = X296) & ! [X297 : term,X300 : term,X298 : term,X299 : term] : ((X297 = X298) | (((ap @ X298 @ X299)) != ((ap @ X297 @ X300))))))) & (ulamvarind = $true) & (((pushprop_lem1_lthm = $true) | (! [X345 : subst,X346 : term] : (((sub @ one @ (push @ X346 @ X345))) = X346) & ! [X347 : term,X348 : subst] : (((comp @ sh @ (push @ X347 @ X348))) = X348) & ? [X349 : subst,X353 : (term > $o),X352 : (term > $o),X350 : term,X351 : term] : ((((X352 @ (sub @ X351 @ (push @ X350 @ X349)))) != $true) & (((X353 @ X351)) = $true)))) & ((? [X345 : subst,X346 : term] : (((sub @ one @ (push @ X346 @ X345))) != X346) | ? [X347 : term,X348 : subst] : (((comp @ sh @ (push @ X347 @ X348))) != X348) | ! [X349 : subst,X353 : (term > $o),X352 : (term > $o),X350 : term,X351 : term] : ((((X352 @ (sub @ X351 @ (push @ X350 @ X349)))) = $true) | (((X353 @ X351)) != $true))) | (pushprop_lem1_lthm != $true))) & (hoasinduction_lem1_lthm = $true) & (hoaslamnotap_gthm = $true) & (axvarshift = $true) & (((? [X77 : term,X74 : term,X75 : term,X76 : term] : ((((ap @ X77 @ X74)) = ((ap @ X76 @ X75))) & (X74 != X75)) | ! [X80 : term,X81 : term,X79 : term,X78 : term] : ((((ap @ (sub @ X81 @ id) @ X78)) != ((ap @ (sub @ X79 @ id) @ X80))) | (X78 = X80))) | (hoasapinj2_lthm != $true)) & ((hoasapinj2_lthm = $true) | (! [X77 : term,X74 : term,X75 : term,X76 : term] : ((((ap @ X77 @ X74)) != ((ap @ X76 @ X75))) | (X74 = X75)) & ? [X80 : term,X81 : term,X79 : term,X78 : term] : ((((ap @ (sub @ X81 @ id) @ X78)) = ((ap @ (sub @ X79 @ id) @ X80))) & (X78 != X80))))) & (hoasinduction_lem1v2 = $true) & (ulamvar1 != $true) & ! [X72 : d_term,X73 : d_term] : ((X72 = X73) | (((d2term @ X72)) != ((d2term @ X73)))) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (pushprop_lem1_gthm = $true) & ((! [X51 : term,X50 : term,X52 : (subst > term > term)] : (? [X55 : term,X53 : subst,X54 : subst] : (((sub @ (X52 @ X53 @ X55) @ X54)) != ((X52 @ (comp @ X53 @ X54) @ (sub @ X55 @ X54)))) | (((lam @ (X52 @ sh @ one))) != ((ap @ (sub @ X50 @ id) @ X51)))) | (hoaslamnotap != $true)) & ((hoaslamnotap = $true) | ? [X51 : term,X50 : term,X52 : (subst > term > term)] : (! [X55 : term,X53 : subst,X54 : subst] : (((sub @ (X52 @ X53 @ X55) @ X54)) = ((X52 @ (comp @ X53 @ X54) @ (sub @ X55 @ X54)))) & (((lam @ (X52 @ sh @ one))) = ((ap @ (sub @ X50 @ id) @ X51)))))) & (((axvarcons = $true) | ? [X356 : subst,X357 : term] : (((sub @ one @ (push @ X357 @ X356))) != X357)) & (! [X356 : subst,X357 : term] : (((sub @ one @ (push @ X357 @ X356))) = X357) | (axvarcons != $true))) & (((hoasapinj1 = $true) | ? [X144 : term,X147 : term,X146 : term,X145 : term] : ((((ap @ (sub @ X147 @ id) @ X146)) = ((ap @ (sub @ X145 @ id) @ X144))) & (X145 != X147))) & (! [X144 : term,X147 : term,X146 : term,X145 : term] : ((((ap @ (sub @ X147 @ id) @ X146)) != ((ap @ (sub @ X145 @ id) @ X144))) | (X145 = X147)) | (hoasapinj1 != $true))) & (hoasinduction_lem3a_gthm = $true) & ((! [X443 : (term > $o),X444 : term,X446 : (term > $o),X445 : subst] : (? [X447 : term] : (((X443 @ X447)) != ((X446 @ (sub @ X447 @ (push @ X444 @ X445))))) | (((X446 @ X444)) != $true) | (((X443 @ one)) = $true)) | (pushprop_lem1v2 != $true)) & ((pushprop_lem1v2 = $true) | ? [X443 : (term > $o),X444 : term,X446 : (term > $o),X445 : subst] : (! [X447 : term] : (((X443 @ X447)) = ((X446 @ (sub @ X447 @ (push @ X444 @ X445))))) & (((X446 @ X444)) = $true) & (((X443 @ one)) != $true)))) & (((sP30 = $true) | (hoaslaminj_lthm != $true)) & ((hoaslaminj_lthm = $true) | (sP30 != $true))) & (hoaslamnotvar = $true) & (hoasinduction_lem1_gthm = $true) & (((sP29 = $true) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | (sP29 != $true))) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (((! [X190 : term,X188 : subst,X191 : (term > $o),X189 : (term > $o)] : (($true != ((X191 @ X190))) | ? [X192 : term] : (((X191 @ (sub @ X192 @ (push @ X190 @ X188)))) != ((X189 @ X192))) | (((X189 @ one)) = $true)) | ? [X187 : subst,X186 : term] : (((sub @ one @ (push @ X186 @ X187))) != X186)) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (? [X190 : term,X188 : subst,X191 : (term > $o),X189 : (term > $o)] : (($true = ((X191 @ X190))) & ! [X192 : term] : (((X191 @ (sub @ X192 @ (push @ X190 @ X188)))) = ((X189 @ X192))) & (((X189 @ one)) != $true)) & ! [X187 : subst,X186 : term] : (((sub @ one @ (push @ X186 @ X187))) = X186)))) & (hoaslaminj_gthm = $true) & ((! [X472 : term,X473 : term,X474 : term,X475 : term] : ((((ap @ X475 @ X473)) != ((ap @ X472 @ X474))) | (X472 = X475)) | (apinj1 != $true)) & ((apinj1 = $true) | ? [X472 : term,X473 : term,X474 : term,X475 : term] : ((((ap @ X475 @ X473)) = ((ap @ X472 @ X474))) & (X472 != X475)))) & (((sP28 = $true) | (hoasinduction_lem2v2 != $true)) & ((hoasinduction_lem2v2 = $true) | (sP28 != $true))) & (hoasinduction_gthm = $true) & (((hoasinduction_lem0_lthm = $true) | ? [X66 : (subst > term > subst > $o)] : ! [X67 : (term > $o)] : ? [X68 : term] : (((X67 @ X68)) != ((X66 @ id @ X68 @ id)))) & (! [X66 : (subst > term > subst > $o)] : ? [X67 : (term > $o)] : ! [X68 : term] : (((X67 @ X68)) = ((X66 @ id @ X68 @ id))) | (hoasinduction_lem0_lthm != $true))) & ((! [X361 : (subst > term > term),X363 : subst,X362 : term,X364 : (subst > term > term)] : (? [X365 : term,X366 : subst,X367 : subst] : (((X364 @ (comp @ X366 @ X367) @ (sub @ X365 @ X367))) != ((sub @ (X364 @ X366 @ X365) @ X367))) | ? [X370 : subst,X369 : term,X368 : subst] : (((sub @ (X361 @ X370 @ X369) @ X368)) != ((X361 @ (comp @ X370 @ X368) @ (sub @ X369 @ X368)))) | (((X364 @ X363 @ X362)) = ((X361 @ X363 @ X362))) | (((lam @ (X361 @ sh @ one))) != ((lam @ (X364 @ sh @ one))))) | (hoaslaminj != $true)) & ((hoaslaminj = $true) | ? [X361 : (subst > term > term),X363 : subst,X362 : term,X364 : (subst > term > term)] : (! [X365 : term,X366 : subst,X367 : subst] : (((X364 @ (comp @ X366 @ X367) @ (sub @ X365 @ X367))) = ((sub @ (X364 @ X366 @ X365) @ X367))) & ! [X370 : subst,X369 : term,X368 : subst] : (((sub @ (X361 @ X370 @ X369) @ X368)) = ((X361 @ (comp @ X370 @ X368) @ (sub @ X369 @ X368)))) & (((X364 @ X363 @ X362)) != ((X361 @ X363 @ X362))) & (((lam @ (X361 @ sh @ one))) = ((lam @ (X364 @ sh @ one))))))) & (hoasinduction_lem1v2_gthm = $true) & (((substmonoid_lthm = $true) | ((? [X424 : subst] : (((comp @ id @ X424)) != X424) | ? [X425 : subst] : (((comp @ X425 @ id)) != X425)) & ! [X423 : subst] : (((comp @ X423 @ id)) = X423) & ! [X422 : subst] : (((comp @ id @ X422)) = X422))) & (((! [X424 : subst] : (((comp @ id @ X424)) = X424) & ! [X425 : subst] : (((comp @ X425 @ id)) = X425)) | ? [X423 : subst] : (((comp @ X423 @ id)) != X423) | ? [X422 : subst] : (((comp @ id @ X422)) != X422)) | (substmonoid_lthm != $true))) & (apnotvar = $true) & (hoasinduction_lem1 = $true) & (axmap = $true) & (axapp = $true) & (hoasapnotvar = $true) & ((! [X470 : term] : (((sub @ X470 @ id)) = X470) | (axvarid != $true)) & ((axvarid = $true) | ? [X470 : term] : (((sub @ X470 @ id)) != X470))) & ((! [X251 : term,X252 : term,X254 : term,X253 : term] : ((((ap @ X254 @ X251)) != ((ap @ X253 @ X252))) | (X251 = X252)) | (apinj2 != $true)) & ((apinj2 = $true) | ? [X251 : term,X252 : term,X254 : term,X253 : term] : ((((ap @ X254 @ X251)) = ((ap @ X253 @ X252))) & (X251 != X252)))) & (((hoasinduction_lem3 = $true) | (sP26 != $true)) & ((sP26 = $true) | (hoasinduction_lem3 != $true))) & (hoasinduction_lem2_lthm = $true) & (((hoasinduction_lem3aa = $true) | (sP24 != $true)) & ((sP24 = $true) | (hoasinduction_lem3aa != $true))) & (induction2lem_gthm = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ! [X406 : term] : ? [X407 : d_term] : (((d2term @ X407)) = X406) & (((hoasinduction_lem3v2a_lthm = $true) | (sP23 != $true)) & ((sP23 = $true) | (hoasinduction_lem3v2a_lthm != $true))) & (((hoasinduction_lem3b = $true) | ? [X56 : term] : ! [X57 : (subst > term > term)] : (((sub @ X56 @ (push @ one @ sh))) != ((X57 @ sh @ one)))) & (! [X56 : term] : ? [X57 : (subst > term > term)] : (((sub @ X56 @ (push @ one @ sh))) = ((X57 @ sh @ one))) | (hoasinduction_lem3b != $true))) & (pushprop_lem3v2 = $true) & ! [X354 : subst] : ? [X355 : d_subst] : (((d2subst @ X355)) = X354) & (hoasapnotvar_gthm = $true) & (pushprop = $true) & (((hoasinduction_lthm = $true) | (sP22 != $true)) & ((sP22 = $true) | (hoasinduction_lthm != $true))) & (hoasinduction_lem3b_gthm = $true) & (((sP13 = $true) | (hoasinduction_lem3v2_lthm != $true)) & ((hoasinduction_lem3v2_lthm = $true) | (sP13 != $true))) & (lamnotap != $true) & (pushprop_lem2v2 = $true) & (lamnotvar = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (((hoasapinj2 = $true) | ? [X185 : term,X183 : term,X184 : term,X182 : term] : ((((ap @ (sub @ X182 @ id) @ X184)) = ((ap @ (sub @ X183 @ id) @ X185))) & (X184 != X185))) & (! [X185 : term,X183 : term,X184 : term,X182 : term] : ((((ap @ (sub @ X182 @ id) @ X184)) != ((ap @ (sub @ X183 @ id) @ X185))) | (X184 = X185)) | (hoasapinj2 != $true))) & (((laminj = $true) | ? [X414 : term,X413 : term] : ((X413 != X414) & (((lam @ X413)) = ((lam @ X414))))) & (! [X414 : term,X413 : term] : ((X413 = X414) | (((lam @ X413)) != ((lam @ X414)))) | (laminj != $true))) & (pushprop_lem2v2_gthm = $true) & (((hoasinduction_lem3a_lthm = $true) | ($true != sP11)) & (($true = sP11) | (hoasinduction_lem3a_lthm != $true))) & ((! [X211 : term] : ? [X212 : (subst > term > term)] : ! [X213 : subst,X214 : term] : (((X212 @ X213 @ X214)) = ((sub @ X211 @ (push @ X214 @ X213)))) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ? [X211 : term] : ! [X212 : (subst > term > term)] : ? [X213 : subst,X214 : term] : (((X212 @ X213 @ X214)) != ((sub @ X211 @ (push @ X214 @ X213)))))) & (((hoasinduction_lem3_lthm = $true) | (sP9 != $true)) & ((sP9 = $true) | (hoasinduction_lem3_lthm != $true))) & (((termmset = $true) | ? [X151 : term] : (((sub @ X151 @ id)) != X151)) & (! [X151 : term] : (((sub @ X151 @ id)) = X151) | (termmset != $true))) & (hoasinduction_lem3_gthm = $true) & (((sP6 = $true) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | (sP6 != $true))) & (((sP5 = $true) | (induction2 != $true)) & ((induction2 = $true) | (sP5 != $true))) & (((hoasinduction_lem0 = $true) | ? [X358 : (subst > term > subst > $o)] : ! [X359 : (term > $o)] : ? [X360 : term] : (((X358 @ id @ X360 @ id)) != ((X359 @ X360)))) & (! [X358 : (subst > term > subst > $o)] : ? [X359 : (term > $o)] : ! [X360 : term] : (((X358 @ id @ X360 @ id)) = ((X359 @ X360))) | (hoasinduction_lem0 != $true))) & ((! [X426 : subst] : (((push @ (sub @ one @ X426) @ (comp @ sh @ X426))) = X426) | (axscons != $true)) & ((axscons = $true) | ? [X426 : subst] : (((push @ (sub @ one @ X426) @ (comp @ sh @ X426))) != X426))) & (pushprop_lem2v2_lthm = $true) & ((! [X469 : term,X468 : term] : ((X468 = X469) | (((sub @ X469 @ sh)) != ((sub @ X468 @ sh)))) | (shinj != $true)) & ((shinj = $true) | ? [X469 : term,X468 : term] : ((X468 != X469) & (((sub @ X469 @ sh)) = ((sub @ X468 @ sh)))))) & (((hoasinduction_lem3b_lthm = $true) | ? [X148 : term] : ! [X149 : (subst > term > term)] : (((sub @ X148 @ (push @ one @ sh))) != ((X149 @ sh @ one)))) & (! [X148 : term] : ? [X149 : (subst > term > term)] : (((sub @ X148 @ (push @ one @ sh))) = ((X149 @ sh @ one))) | (hoasinduction_lem3b_lthm != $true))) & ((! [X241 : (term > $o),X240 : term] : (? [X242 : term] : ((((X241 @ X242)) = $true) & (((X241 @ (lam @ X242))) != $true)) | ? [X245 : term] : ((((var @ X245)) = $true) & (((X241 @ X245)) != $true)) | (((sP3 @ X241)) = $true) | (((X241 @ X240)) = $true)) | (induction != $true)) & ((induction = $true) | ? [X241 : (term > $o),X240 : term] : (! [X242 : term] : ((((X241 @ X242)) != $true) | (((X241 @ (lam @ X242))) = $true)) & ! [X245 : term] : ((((var @ X245)) != $true) | (((X241 @ X245)) = $true)) & (((sP3 @ X241)) != $true) & (((X241 @ X240)) != $true)))) & (((hoasinduction_lem3aaa = $true) | (sP2 != $true)) & ((sP2 = $true) | (hoasinduction_lem3aaa != $true))) & ((! [X402 : subst,X403 : (term > $o),X401 : term] : ? [X404 : (term > $o)] : ! [X405 : term] : (((X403 @ (sub @ X405 @ (push @ X401 @ X402)))) = ((X404 @ X405))) | (pushprop_lem0_lthm != $true)) & ((pushprop_lem0_lthm = $true) | ? [X402 : subst,X403 : (term > $o),X401 : term] : ! [X404 : (term > $o)] : ? [X405 : term] : (((X403 @ (sub @ X405 @ (push @ X401 @ X402)))) != ((X404 @ X405))))) & (hoasapinj1_gthm = $true) & (ulamvarsh = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (((sP1 = $true) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | (sP1 != $true))) & (axassoc = $true)),
  inference(nnf_transformation,[],[f72])).
thf(f261,plain,(
  ((hoasinduction_lthm_3 = $true) | (sP58 != $true)) & ((sP58 = $true) | (hoasinduction_lthm_3 != $true)) & (termmset_gthm = $true) & ((substmonoid = $true) | ? [X71 : subst] : (((comp @ id @ X71)) != X71) | ? [X70 : subst] : (((comp @ X70 @ id)) != X70)) & ((! [X71 : subst] : (((comp @ id @ X71)) = X71) & ! [X70 : subst] : (((comp @ X70 @ id)) = X70)) | (substmonoid != $true)) & (pushprop_gthm = $true) & (pushprop_lem3v2_lthm = $true) & (id = ((d2subst @ d_id))) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (pushprop_lthm = $true) & (((d2subst @ d_id)) = sh) & ! [X69 : d_term] : (d_one = X69) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X507 : term] : ! [X508 : (subst > term > term)] : ? [X509 : term,X510 : subst] : (((sub @ X507 @ (push @ X509 @ X510))) != ((X508 @ X510 @ X509)))) & (! [X507 : term] : ? [X508 : (subst > term > term)] : ! [X509 : term,X510 : subst] : (((sub @ X507 @ (push @ X509 @ X510))) = ((X508 @ X510 @ X509))) | (hoasinduction_lem3v2_f_lthm != $true)) & (hoaslamnotvar_lthm = $true) & ! [X344 : d_subst] : (d_id = X344) & (induction2_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & (pushprop_lthm_orig = $true) & ((hoasinduction_no_psi_cond = $true) | ? [X173 : term,X174 : (subst > term > subst > $o)] : (! [X176 : term,X175 : term] : ((((X174 @ id @ (ap @ (sub @ X176 @ id) @ X175) @ id)) = $true) | (((X174 @ id @ X175 @ id)) != $true) | (((X174 @ id @ X176 @ id)) != $true)) & (((sP52 @ X174)) != $true) & (((X174 @ id @ X173 @ id)) != $true))) & (! [X173 : term,X174 : (subst > term > subst > $o)] : (? [X176 : term,X175 : term] : ((((X174 @ id @ (ap @ (sub @ X176 @ id) @ X175) @ id)) != $true) & (((X174 @ id @ X175 @ id)) = $true) & (((X174 @ id @ X176 @ id)) = $true)) | (((sP52 @ X174)) = $true) | (((X174 @ id @ X173 @ id)) = $true)) | (hoasinduction_no_psi_cond != $true)) & (hoaslamnotap_lthm = $true) & (! [X410 : subst,X409 : (term > $o),X408 : term] : ? [X411 : (term > $o)] : ! [X412 : term] : (((X409 @ (sub @ X412 @ (push @ X408 @ X410)))) = ((X411 @ X412))) | (pushprop_lem0 != $true)) & ((pushprop_lem0 = $true) | ? [X410 : subst,X409 : (term > $o),X408 : term] : ! [X411 : (term > $o)] : ? [X412 : term] : (((X409 @ (sub @ X412 @ (push @ X408 @ X410)))) != ((X411 @ X412)))) & (hoaslamnotvar_gthm = $true) & (axclos = $true) & ((sP51 = $true) | (hoasinduction_no_psi_cond_lthm != $true)) & ((hoasinduction_no_psi_cond_lthm = $true) | (sP51 != $true)) & (((var @ (d2term @ d_one))) != $true) & ((hoasinduction_lem3v2 = $true) | (sP46 != $true)) & ((sP46 = $true) | (hoasinduction_lem3v2 != $true)) & ((sP44 = $true) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | (sP44 != $true)) & ((termmset_lthm = $true) | (? [X449 : term] : (((sub @ X449 @ id)) != X449) & ! [X448 : term] : (((sub @ X448 @ id)) = X448))) & (! [X449 : term] : (((sub @ X449 @ id)) = X449) | ? [X448 : term] : (((sub @ X448 @ id)) != X448) | (termmset_lthm != $true)) & (! [X152 : subst,X153 : term] : (((comp @ sh @ (push @ X153 @ X152))) = X152) | (axshiftcons != $true)) & ((axshiftcons = $true) | ? [X152 : subst,X153 : term] : (((comp @ sh @ (push @ X153 @ X152))) != X152)) & (hoasinduction_lem2v2_gthm = $true) & (pushprop_lem0_gthm = $true) & (hoasapnotvar_lthm = $true) & ((pushprop_lem1 = $true) | ? [X246 : term,X247 : term,X250 : (term > $o),X248 : subst,X249 : (term > $o)] : ((((X249 @ X247)) = $true) & (((X250 @ (sub @ X247 @ (push @ X246 @ X248)))) != $true))) & (! [X246 : term,X247 : term,X250 : (term > $o),X248 : subst,X249 : (term > $o)] : ((((X249 @ X247)) != $true) | (((X250 @ (sub @ X247 @ (push @ X246 @ X248)))) = $true)) | (pushprop_lem1 != $true)) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (pushprop_lem1v2_gthm = $true) & ((axidr = $true) | ? [X150 : subst] : (((comp @ X150 @ id)) != X150)) & (! [X150 : subst] : (((comp @ X150 @ id)) = X150) | (axidr != $true)) & (one = ((d2term @ d_one))) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((sP40 = $true) | (induction2lem != $true)) & ((induction2lem = $true) | (sP40 != $true)) & ! [X512 : d_subst,X511 : d_subst] : ((X511 = X512) | (((d2subst @ X511)) != ((d2subst @ X512)))) & ((sP38 = $true) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | (sP38 != $true)) & (hoasapinj2_gthm = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (hoasinduction_lem3v2_gthm = $true) & ((hoasinduction_lem3aa_lthm = $true) | (sP34 != $true)) & ((sP34 = $true) | (hoasinduction_lem3aa_lthm != $true)) & (! [X471 : subst] : (((comp @ id @ X471)) = X471) | (axidl != $true)) & ((axidl = $true) | ? [X471 : subst] : (((comp @ id @ X471)) != X471)) & ((hoasinduction = $true) | (sP33 != $true)) & ((sP33 = $true) | (hoasinduction != $true)) & (substmonoid_gthm = $true) & (hoasinduction_lem2_gthm = $true) & (axabs = $true) & (! [X303 : term,X301 : term,X302 : term,X304 : term] : ((X301 = X303) | (((ap @ (sub @ X303 @ id) @ X302)) != ((ap @ (sub @ X301 @ id) @ X304)))) | ? [X296 : term] : (((sub @ X296 @ id)) != X296) | ? [X297 : term,X300 : term,X298 : term,X299 : term] : ((X297 != X298) & (((ap @ X298 @ X299)) = ((ap @ X297 @ X300)))) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | (? [X303 : term,X301 : term,X302 : term,X304 : term] : ((X301 != X303) & (((ap @ (sub @ X303 @ id) @ X302)) = ((ap @ (sub @ X301 @ id) @ X304)))) & ! [X296 : term] : (((sub @ X296 @ id)) = X296) & ! [X297 : term,X300 : term,X298 : term,X299 : term] : ((X297 = X298) | (((ap @ X298 @ X299)) != ((ap @ X297 @ X300)))))) & (ulamvarind = $true) & ((pushprop_lem1_lthm = $true) | (! [X345 : subst,X346 : term] : (((sub @ one @ (push @ X346 @ X345))) = X346) & ! [X347 : term,X348 : subst] : (((comp @ sh @ (push @ X347 @ X348))) = X348) & ? [X349 : subst,X353 : (term > $o),X352 : (term > $o),X350 : term,X351 : term] : ((((X352 @ (sub @ X351 @ (push @ X350 @ X349)))) != $true) & (((X353 @ X351)) = $true)))) & (? [X345 : subst,X346 : term] : (((sub @ one @ (push @ X346 @ X345))) != X346) | ? [X347 : term,X348 : subst] : (((comp @ sh @ (push @ X347 @ X348))) != X348) | ! [X349 : subst,X353 : (term > $o),X352 : (term > $o),X350 : term,X351 : term] : ((((X352 @ (sub @ X351 @ (push @ X350 @ X349)))) = $true) | (((X353 @ X351)) != $true)) | (pushprop_lem1_lthm != $true)) & (hoasinduction_lem1_lthm = $true) & (hoaslamnotap_gthm = $true) & (axvarshift = $true) & (? [X77 : term,X74 : term,X75 : term,X76 : term] : ((((ap @ X77 @ X74)) = ((ap @ X76 @ X75))) & (X74 != X75)) | ! [X80 : term,X81 : term,X79 : term,X78 : term] : ((((ap @ (sub @ X81 @ id) @ X78)) != ((ap @ (sub @ X79 @ id) @ X80))) | (X78 = X80)) | (hoasapinj2_lthm != $true)) & ((hoasapinj2_lthm = $true) | (! [X77 : term,X74 : term,X75 : term,X76 : term] : ((((ap @ X77 @ X74)) != ((ap @ X76 @ X75))) | (X74 = X75)) & ? [X80 : term,X81 : term,X79 : term,X78 : term] : ((((ap @ (sub @ X81 @ id) @ X78)) = ((ap @ (sub @ X79 @ id) @ X80))) & (X78 != X80)))) & (hoasinduction_lem1v2 = $true) & (ulamvar1 != $true) & ! [X72 : d_term,X73 : d_term] : ((X72 = X73) | (((d2term @ X72)) != ((d2term @ X73)))) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (pushprop_lem1_gthm = $true) & (! [X51 : term,X50 : term,X52 : (subst > term > term)] : (? [X55 : term,X53 : subst,X54 : subst] : (((sub @ (X52 @ X53 @ X55) @ X54)) != ((X52 @ (comp @ X53 @ X54) @ (sub @ X55 @ X54)))) | (((lam @ (X52 @ sh @ one))) != ((ap @ (sub @ X50 @ id) @ X51)))) | (hoaslamnotap != $true)) & ((hoaslamnotap = $true) | ? [X51 : term,X50 : term,X52 : (subst > term > term)] : (! [X55 : term,X53 : subst,X54 : subst] : (((sub @ (X52 @ X53 @ X55) @ X54)) = ((X52 @ (comp @ X53 @ X54) @ (sub @ X55 @ X54)))) & (((lam @ (X52 @ sh @ one))) = ((ap @ (sub @ X50 @ id) @ X51))))) & ((axvarcons = $true) | ? [X356 : subst,X357 : term] : (((sub @ one @ (push @ X357 @ X356))) != X357)) & (! [X356 : subst,X357 : term] : (((sub @ one @ (push @ X357 @ X356))) = X357) | (axvarcons != $true)) & ((hoasapinj1 = $true) | ? [X144 : term,X147 : term,X146 : term,X145 : term] : ((((ap @ (sub @ X147 @ id) @ X146)) = ((ap @ (sub @ X145 @ id) @ X144))) & (X145 != X147))) & (! [X144 : term,X147 : term,X146 : term,X145 : term] : ((((ap @ (sub @ X147 @ id) @ X146)) != ((ap @ (sub @ X145 @ id) @ X144))) | (X145 = X147)) | (hoasapinj1 != $true)) & (hoasinduction_lem3a_gthm = $true) & (! [X443 : (term > $o),X444 : term,X446 : (term > $o),X445 : subst] : (? [X447 : term] : (((X443 @ X447)) != ((X446 @ (sub @ X447 @ (push @ X444 @ X445))))) | (((X446 @ X444)) != $true) | (((X443 @ one)) = $true)) | (pushprop_lem1v2 != $true)) & ((pushprop_lem1v2 = $true) | ? [X443 : (term > $o),X444 : term,X446 : (term > $o),X445 : subst] : (! [X447 : term] : (((X443 @ X447)) = ((X446 @ (sub @ X447 @ (push @ X444 @ X445))))) & (((X446 @ X444)) = $true) & (((X443 @ one)) != $true))) & ((sP30 = $true) | (hoaslaminj_lthm != $true)) & ((hoaslaminj_lthm = $true) | (sP30 != $true)) & (hoaslamnotvar = $true) & (hoasinduction_lem1_gthm = $true) & ((sP29 = $true) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | (sP29 != $true)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (! [X190 : term,X188 : subst,X191 : (term > $o),X189 : (term > $o)] : (($true != ((X191 @ X190))) | ? [X192 : term] : (((X191 @ (sub @ X192 @ (push @ X190 @ X188)))) != ((X189 @ X192))) | (((X189 @ one)) = $true)) | ? [X187 : subst,X186 : term] : (((sub @ one @ (push @ X186 @ X187))) != X186) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (? [X190 : term,X188 : subst,X191 : (term > $o),X189 : (term > $o)] : (($true = ((X191 @ X190))) & ! [X192 : term] : (((X191 @ (sub @ X192 @ (push @ X190 @ X188)))) = ((X189 @ X192))) & (((X189 @ one)) != $true)) & ! [X187 : subst,X186 : term] : (((sub @ one @ (push @ X186 @ X187))) = X186))) & (hoaslaminj_gthm = $true) & (! [X472 : term,X473 : term,X474 : term,X475 : term] : ((((ap @ X475 @ X473)) != ((ap @ X472 @ X474))) | (X472 = X475)) | (apinj1 != $true)) & ((apinj1 = $true) | ? [X472 : term,X473 : term,X474 : term,X475 : term] : ((((ap @ X475 @ X473)) = ((ap @ X472 @ X474))) & (X472 != X475))) & ((sP28 = $true) | (hoasinduction_lem2v2 != $true)) & ((hoasinduction_lem2v2 = $true) | (sP28 != $true)) & (hoasinduction_gthm = $true) & ((hoasinduction_lem0_lthm = $true) | ? [X66 : (subst > term > subst > $o)] : ! [X67 : (term > $o)] : ? [X68 : term] : (((X67 @ X68)) != ((X66 @ id @ X68 @ id)))) & (! [X66 : (subst > term > subst > $o)] : ? [X67 : (term > $o)] : ! [X68 : term] : (((X67 @ X68)) = ((X66 @ id @ X68 @ id))) | (hoasinduction_lem0_lthm != $true)) & (! [X361 : (subst > term > term),X363 : subst,X362 : term,X364 : (subst > term > term)] : (? [X365 : term,X366 : subst,X367 : subst] : (((X364 @ (comp @ X366 @ X367) @ (sub @ X365 @ X367))) != ((sub @ (X364 @ X366 @ X365) @ X367))) | ? [X370 : subst,X369 : term,X368 : subst] : (((sub @ (X361 @ X370 @ X369) @ X368)) != ((X361 @ (comp @ X370 @ X368) @ (sub @ X369 @ X368)))) | (((X364 @ X363 @ X362)) = ((X361 @ X363 @ X362))) | (((lam @ (X361 @ sh @ one))) != ((lam @ (X364 @ sh @ one))))) | (hoaslaminj != $true)) & ((hoaslaminj = $true) | ? [X361 : (subst > term > term),X363 : subst,X362 : term,X364 : (subst > term > term)] : (! [X365 : term,X366 : subst,X367 : subst] : (((X364 @ (comp @ X366 @ X367) @ (sub @ X365 @ X367))) = ((sub @ (X364 @ X366 @ X365) @ X367))) & ! [X370 : subst,X369 : term,X368 : subst] : (((sub @ (X361 @ X370 @ X369) @ X368)) = ((X361 @ (comp @ X370 @ X368) @ (sub @ X369 @ X368)))) & (((X364 @ X363 @ X362)) != ((X361 @ X363 @ X362))) & (((lam @ (X361 @ sh @ one))) = ((lam @ (X364 @ sh @ one)))))) & (hoasinduction_lem1v2_gthm = $true) & ((substmonoid_lthm = $true) | ((? [X424 : subst] : (((comp @ id @ X424)) != X424) | ? [X425 : subst] : (((comp @ X425 @ id)) != X425)) & ! [X423 : subst] : (((comp @ X423 @ id)) = X423) & ! [X422 : subst] : (((comp @ id @ X422)) = X422))) & ((! [X424 : subst] : (((comp @ id @ X424)) = X424) & ! [X425 : subst] : (((comp @ X425 @ id)) = X425)) | ? [X423 : subst] : (((comp @ X423 @ id)) != X423) | ? [X422 : subst] : (((comp @ id @ X422)) != X422) | (substmonoid_lthm != $true)) & (apnotvar = $true) & (hoasinduction_lem1 = $true) & (axmap = $true) & (axapp = $true) & (hoasapnotvar = $true) & (! [X470 : term] : (((sub @ X470 @ id)) = X470) | (axvarid != $true)) & ((axvarid = $true) | ? [X470 : term] : (((sub @ X470 @ id)) != X470)) & (! [X251 : term,X252 : term,X254 : term,X253 : term] : ((((ap @ X254 @ X251)) != ((ap @ X253 @ X252))) | (X251 = X252)) | (apinj2 != $true)) & ((apinj2 = $true) | ? [X251 : term,X252 : term,X254 : term,X253 : term] : ((((ap @ X254 @ X251)) = ((ap @ X253 @ X252))) & (X251 != X252))) & ((hoasinduction_lem3 = $true) | (sP26 != $true)) & ((sP26 = $true) | (hoasinduction_lem3 != $true)) & (hoasinduction_lem2_lthm = $true) & ((hoasinduction_lem3aa = $true) | (sP24 != $true)) & ((sP24 = $true) | (hoasinduction_lem3aa != $true)) & (induction2lem_gthm = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ! [X406 : term] : ? [X407 : d_term] : (((d2term @ X407)) = X406) & ((hoasinduction_lem3v2a_lthm = $true) | (sP23 != $true)) & ((sP23 = $true) | (hoasinduction_lem3v2a_lthm != $true)) & ((hoasinduction_lem3b = $true) | ? [X56 : term] : ! [X57 : (subst > term > term)] : (((sub @ X56 @ (push @ one @ sh))) != ((X57 @ sh @ one)))) & (! [X56 : term] : ? [X57 : (subst > term > term)] : (((sub @ X56 @ (push @ one @ sh))) = ((X57 @ sh @ one))) | (hoasinduction_lem3b != $true)) & (pushprop_lem3v2 = $true) & ! [X354 : subst] : ? [X355 : d_subst] : (((d2subst @ X355)) = X354) & (hoasapnotvar_gthm = $true) & (pushprop = $true) & ((hoasinduction_lthm = $true) | (sP22 != $true)) & ((sP22 = $true) | (hoasinduction_lthm != $true)) & (hoasinduction_lem3b_gthm = $true) & ((sP13 = $true) | (hoasinduction_lem3v2_lthm != $true)) & ((hoasinduction_lem3v2_lthm = $true) | (sP13 != $true)) & (lamnotap != $true) & (pushprop_lem2v2 = $true) & (lamnotvar = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((hoasapinj2 = $true) | ? [X185 : term,X183 : term,X184 : term,X182 : term] : ((((ap @ (sub @ X182 @ id) @ X184)) = ((ap @ (sub @ X183 @ id) @ X185))) & (X184 != X185))) & (! [X185 : term,X183 : term,X184 : term,X182 : term] : ((((ap @ (sub @ X182 @ id) @ X184)) != ((ap @ (sub @ X183 @ id) @ X185))) | (X184 = X185)) | (hoasapinj2 != $true)) & ((laminj = $true) | ? [X414 : term,X413 : term] : ((X413 != X414) & (((lam @ X413)) = ((lam @ X414))))) & (! [X414 : term,X413 : term] : ((X413 = X414) | (((lam @ X413)) != ((lam @ X414)))) | (laminj != $true)) & (pushprop_lem2v2_gthm = $true) & ((hoasinduction_lem3a_lthm = $true) | ($true != sP11)) & (($true = sP11) | (hoasinduction_lem3a_lthm != $true)) & (! [X211 : term] : ? [X212 : (subst > term > term)] : ! [X213 : subst,X214 : term] : (((X212 @ X213 @ X214)) = ((sub @ X211 @ (push @ X214 @ X213)))) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ? [X211 : term] : ! [X212 : (subst > term > term)] : ? [X213 : subst,X214 : term] : (((X212 @ X213 @ X214)) != ((sub @ X211 @ (push @ X214 @ X213))))) & ((hoasinduction_lem3_lthm = $true) | (sP9 != $true)) & ((sP9 = $true) | (hoasinduction_lem3_lthm != $true)) & ((termmset = $true) | ? [X151 : term] : (((sub @ X151 @ id)) != X151)) & (! [X151 : term] : (((sub @ X151 @ id)) = X151) | (termmset != $true)) & (hoasinduction_lem3_gthm = $true) & ((sP6 = $true) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | (sP6 != $true)) & ((sP5 = $true) | (induction2 != $true)) & ((induction2 = $true) | (sP5 != $true)) & ((hoasinduction_lem0 = $true) | ? [X358 : (subst > term > subst > $o)] : ! [X359 : (term > $o)] : ? [X360 : term] : (((X358 @ id @ X360 @ id)) != ((X359 @ X360)))) & (! [X358 : (subst > term > subst > $o)] : ? [X359 : (term > $o)] : ! [X360 : term] : (((X358 @ id @ X360 @ id)) = ((X359 @ X360))) | (hoasinduction_lem0 != $true)) & (! [X426 : subst] : (((push @ (sub @ one @ X426) @ (comp @ sh @ X426))) = X426) | (axscons != $true)) & ((axscons = $true) | ? [X426 : subst] : (((push @ (sub @ one @ X426) @ (comp @ sh @ X426))) != X426)) & (pushprop_lem2v2_lthm = $true) & (! [X469 : term,X468 : term] : ((X468 = X469) | (((sub @ X469 @ sh)) != ((sub @ X468 @ sh)))) | (shinj != $true)) & ((shinj = $true) | ? [X469 : term,X468 : term] : ((X468 != X469) & (((sub @ X469 @ sh)) = ((sub @ X468 @ sh))))) & ((hoasinduction_lem3b_lthm = $true) | ? [X148 : term] : ! [X149 : (subst > term > term)] : (((sub @ X148 @ (push @ one @ sh))) != ((X149 @ sh @ one)))) & (! [X148 : term] : ? [X149 : (subst > term > term)] : (((sub @ X148 @ (push @ one @ sh))) = ((X149 @ sh @ one))) | (hoasinduction_lem3b_lthm != $true)) & (! [X241 : (term > $o),X240 : term] : (? [X242 : term] : ((((X241 @ X242)) = $true) & (((X241 @ (lam @ X242))) != $true)) | ? [X245 : term] : ((((var @ X245)) = $true) & (((X241 @ X245)) != $true)) | (((sP3 @ X241)) = $true) | (((X241 @ X240)) = $true)) | (induction != $true)) & ((induction = $true) | ? [X241 : (term > $o),X240 : term] : (! [X242 : term] : ((((X241 @ X242)) != $true) | (((X241 @ (lam @ X242))) = $true)) & ! [X245 : term] : ((((var @ X245)) != $true) | (((X241 @ X245)) = $true)) & (((sP3 @ X241)) != $true) & (((X241 @ X240)) != $true))) & ((hoasinduction_lem3aaa = $true) | (sP2 != $true)) & ((sP2 = $true) | (hoasinduction_lem3aaa != $true)) & (! [X402 : subst,X403 : (term > $o),X401 : term] : ? [X404 : (term > $o)] : ! [X405 : term] : (((X403 @ (sub @ X405 @ (push @ X401 @ X402)))) = ((X404 @ X405))) | (pushprop_lem0_lthm != $true)) & ((pushprop_lem0_lthm = $true) | ? [X402 : subst,X403 : (term > $o),X401 : term] : ! [X404 : (term > $o)] : ? [X405 : term] : (((X403 @ (sub @ X405 @ (push @ X401 @ X402)))) != ((X404 @ X405)))) & (hoasapinj1_gthm = $true) & (ulamvarsh = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & ((sP1 = $true) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | (sP1 != $true)) & (axassoc = $true)),
  inference(flattening,[],[f260])).
thf(f262,plain,(
  ((hoasinduction_lthm_3 = $true) | (sP58 != $true)) & ((sP58 = $true) | (hoasinduction_lthm_3 != $true)) & (termmset_gthm = $true) & ((substmonoid = $true) | ? [X0 : subst] : (((comp @ id @ X0)) != X0) | ? [X1 : subst] : (((comp @ X1 @ id)) != X1)) & ((! [X2 : subst] : (((comp @ id @ X2)) = X2) & ! [X3 : subst] : (((comp @ X3 @ id)) = X3)) | (substmonoid != $true)) & (pushprop_gthm = $true) & (pushprop_lem3v2_lthm = $true) & (id = ((d2subst @ d_id))) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (pushprop_lthm = $true) & (((d2subst @ d_id)) = sh) & ! [X4 : d_term] : (d_one = X4) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X5 : term] : ! [X6 : (subst > term > term)] : ? [X7 : term,X8 : subst] : (((sub @ X5 @ (push @ X7 @ X8))) != ((X6 @ X8 @ X7)))) & (! [X9 : term] : ? [X10 : (subst > term > term)] : ! [X11 : term,X12 : subst] : (((X10 @ X12 @ X11)) = ((sub @ X9 @ (push @ X11 @ X12)))) | (hoasinduction_lem3v2_f_lthm != $true)) & (hoaslamnotvar_lthm = $true) & ! [X13 : d_subst] : (d_id = X13) & (induction2_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & (pushprop_lthm_orig = $true) & ((hoasinduction_no_psi_cond = $true) | ? [X14 : term,X15 : (subst > term > subst > $o)] : (! [X16 : term,X17 : term] : ((((X15 @ id @ (ap @ (sub @ X16 @ id) @ X17) @ id)) = $true) | (((X15 @ id @ X17 @ id)) != $true) | (((X15 @ id @ X16 @ id)) != $true)) & (((sP52 @ X15)) != $true) & (((X15 @ id @ X14 @ id)) != $true))) & (! [X18 : term,X19 : (subst > term > subst > $o)] : (? [X20 : term,X21 : term] : ((((X19 @ id @ (ap @ (sub @ X20 @ id) @ X21) @ id)) != $true) & (((X19 @ id @ X21 @ id)) = $true) & (((X19 @ id @ X20 @ id)) = $true)) | ($true = ((sP52 @ X19))) | (((X19 @ id @ X18 @ id)) = $true)) | (hoasinduction_no_psi_cond != $true)) & (hoaslamnotap_lthm = $true) & (! [X22 : subst,X23 : (term > $o),X24 : term] : ? [X25 : (term > $o)] : ! [X26 : term] : (((X25 @ X26)) = ((X23 @ (sub @ X26 @ (push @ X24 @ X22))))) | (pushprop_lem0 != $true)) & ((pushprop_lem0 = $true) | ? [X27 : subst,X28 : (term > $o),X29 : term] : ! [X30 : (term > $o)] : ? [X31 : term] : (((X28 @ (sub @ X31 @ (push @ X29 @ X27)))) != ((X30 @ X31)))) & (hoaslamnotvar_gthm = $true) & (axclos = $true) & ((sP51 = $true) | (hoasinduction_no_psi_cond_lthm != $true)) & ((hoasinduction_no_psi_cond_lthm = $true) | (sP51 != $true)) & (((var @ (d2term @ d_one))) != $true) & ((hoasinduction_lem3v2 = $true) | (sP46 != $true)) & ((sP46 = $true) | (hoasinduction_lem3v2 != $true)) & ((sP44 = $true) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | (sP44 != $true)) & ((termmset_lthm = $true) | (? [X32 : term] : (((sub @ X32 @ id)) != X32) & ! [X33 : term] : (((sub @ X33 @ id)) = X33))) & (! [X34 : term] : (((sub @ X34 @ id)) = X34) | ? [X35 : term] : (((sub @ X35 @ id)) != X35) | (termmset_lthm != $true)) & (! [X36 : subst,X37 : term] : (((comp @ sh @ (push @ X37 @ X36))) = X36) | (axshiftcons != $true)) & ((axshiftcons = $true) | ? [X38 : subst,X39 : term] : (((comp @ sh @ (push @ X39 @ X38))) != X38)) & (hoasinduction_lem2v2_gthm = $true) & (pushprop_lem0_gthm = $true) & (hoasapnotvar_lthm = $true) & ((pushprop_lem1 = $true) | ? [X40 : term,X41 : term,X42 : (term > $o),X43 : subst,X44 : (term > $o)] : ((((X44 @ X41)) = $true) & (((X42 @ (sub @ X41 @ (push @ X40 @ X43)))) != $true))) & (! [X45 : term,X46 : term,X47 : (term > $o),X48 : subst,X49 : (term > $o)] : ((((X49 @ X46)) != $true) | (((X47 @ (sub @ X46 @ (push @ X45 @ X48)))) = $true)) | (pushprop_lem1 != $true)) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (pushprop_lem1v2_gthm = $true) & ((axidr = $true) | ? [X50 : subst] : (((comp @ X50 @ id)) != X50)) & (! [X51 : subst] : (((comp @ X51 @ id)) = X51) | (axidr != $true)) & (one = ((d2term @ d_one))) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((sP40 = $true) | (induction2lem != $true)) & ((induction2lem = $true) | (sP40 != $true)) & ! [X52 : d_subst,X53 : d_subst] : ((X52 = X53) | (((d2subst @ X52)) != ((d2subst @ X53)))) & ((sP38 = $true) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | (sP38 != $true)) & (hoasapinj2_gthm = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (hoasinduction_lem3v2_gthm = $true) & ((hoasinduction_lem3aa_lthm = $true) | (sP34 != $true)) & ((sP34 = $true) | (hoasinduction_lem3aa_lthm != $true)) & (! [X54 : subst] : (((comp @ id @ X54)) = X54) | (axidl != $true)) & ((axidl = $true) | ? [X55 : subst] : (((comp @ id @ X55)) != X55)) & ((hoasinduction = $true) | (sP33 != $true)) & ((sP33 = $true) | (hoasinduction != $true)) & (substmonoid_gthm = $true) & (hoasinduction_lem2_gthm = $true) & (axabs = $true) & (! [X56 : term,X57 : term,X58 : term,X59 : term] : ((X56 = X57) | (((ap @ (sub @ X57 @ id) @ X59)) != ((ap @ (sub @ X56 @ id) @ X58)))) | ? [X60 : term] : (((sub @ X60 @ id)) != X60) | ? [X61 : term,X62 : term,X63 : term,X64 : term] : ((X61 != X63) & (((ap @ X61 @ X62)) = ((ap @ X63 @ X64)))) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | (? [X65 : term,X66 : term,X67 : term,X68 : term] : ((X65 != X66) & (((ap @ (sub @ X65 @ id) @ X67)) = ((ap @ (sub @ X66 @ id) @ X68)))) & ! [X69 : term] : (((sub @ X69 @ id)) = X69) & ! [X70 : term,X71 : term,X72 : term,X73 : term] : ((X70 = X72) | (((ap @ X70 @ X71)) != ((ap @ X72 @ X73)))))) & (ulamvarind = $true) & ((pushprop_lem1_lthm = $true) | (! [X74 : subst,X75 : term] : (((sub @ one @ (push @ X75 @ X74))) = X75) & ! [X76 : term,X77 : subst] : (((comp @ sh @ (push @ X76 @ X77))) = X77) & ? [X78 : subst,X79 : (term > $o),X80 : (term > $o),X81 : term,X82 : term] : ((((X80 @ (sub @ X82 @ (push @ X81 @ X78)))) != $true) & ($true = ((X79 @ X82)))))) & (? [X83 : subst,X84 : term] : (((sub @ one @ (push @ X84 @ X83))) != X84) | ? [X85 : term,X86 : subst] : (((comp @ sh @ (push @ X85 @ X86))) != X86) | ! [X87 : subst,X88 : (term > $o),X89 : (term > $o),X90 : term,X91 : term] : ((((X89 @ (sub @ X91 @ (push @ X90 @ X87)))) = $true) | (((X88 @ X91)) != $true)) | (pushprop_lem1_lthm != $true)) & (hoasinduction_lem1_lthm = $true) & (hoaslamnotap_gthm = $true) & (axvarshift = $true) & (? [X92 : term,X93 : term,X94 : term,X95 : term] : ((((ap @ X92 @ X93)) = ((ap @ X95 @ X94))) & (X93 != X94)) | ! [X96 : term,X97 : term,X98 : term,X99 : term] : ((((ap @ (sub @ X98 @ id) @ X96)) != ((ap @ (sub @ X97 @ id) @ X99))) | (X96 = X99)) | (hoasapinj2_lthm != $true)) & ((hoasapinj2_lthm = $true) | (! [X100 : term,X101 : term,X102 : term,X103 : term] : ((((ap @ X103 @ X102)) != ((ap @ X100 @ X101))) | (X101 = X102)) & ? [X104 : term,X105 : term,X106 : term,X107 : term] : ((((ap @ (sub @ X106 @ id) @ X104)) = ((ap @ (sub @ X105 @ id) @ X107))) & (X104 != X107)))) & (hoasinduction_lem1v2 = $true) & (ulamvar1 != $true) & ! [X108 : d_term,X109 : d_term] : ((X108 = X109) | (((d2term @ X108)) != ((d2term @ X109)))) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (pushprop_lem1_gthm = $true) & (! [X110 : term,X111 : term,X112 : (subst > term > term)] : (? [X113 : term,X114 : subst,X115 : subst] : (((sub @ (X112 @ X114 @ X113) @ X115)) != ((X112 @ (comp @ X114 @ X115) @ (sub @ X113 @ X115)))) | (((lam @ (X112 @ sh @ one))) != ((ap @ (sub @ X111 @ id) @ X110)))) | (hoaslamnotap != $true)) & ((hoaslamnotap = $true) | ? [X116 : term,X117 : term,X118 : (subst > term > term)] : (! [X119 : term,X120 : subst,X121 : subst] : (((sub @ (X118 @ X120 @ X119) @ X121)) = ((X118 @ (comp @ X120 @ X121) @ (sub @ X119 @ X121)))) & (((lam @ (X118 @ sh @ one))) = ((ap @ (sub @ X117 @ id) @ X116))))) & ((axvarcons = $true) | ? [X122 : subst,X123 : term] : (((sub @ one @ (push @ X123 @ X122))) != X123)) & (! [X124 : subst,X125 : term] : (((sub @ one @ (push @ X125 @ X124))) = X125) | (axvarcons != $true)) & ((hoasapinj1 = $true) | ? [X126 : term,X127 : term,X128 : term,X129 : term] : ((((ap @ (sub @ X129 @ id) @ X126)) = ((ap @ (sub @ X127 @ id) @ X128))) & (X127 != X129))) & (! [X130 : term,X131 : term,X132 : term,X133 : term] : ((((ap @ (sub @ X133 @ id) @ X130)) != ((ap @ (sub @ X131 @ id) @ X132))) | (X131 = X133)) | (hoasapinj1 != $true)) & (hoasinduction_lem3a_gthm = $true) & (! [X134 : (term > $o),X135 : term,X136 : (term > $o),X137 : subst] : (? [X138 : term] : (((X136 @ (sub @ X138 @ (push @ X135 @ X137)))) != ((X134 @ X138))) | (((X136 @ X135)) != $true) | (((X134 @ one)) = $true)) | (pushprop_lem1v2 != $true)) & ((pushprop_lem1v2 = $true) | ? [X139 : (term > $o),X140 : term,X141 : (term > $o),X142 : subst] : (! [X143 : term] : (((X141 @ (sub @ X143 @ (push @ X140 @ X142)))) = ((X139 @ X143))) & (((X141 @ X140)) = $true) & (((X139 @ one)) != $true))) & ((sP30 = $true) | (hoaslaminj_lthm != $true)) & ((hoaslaminj_lthm = $true) | (sP30 != $true)) & (hoaslamnotvar = $true) & (hoasinduction_lem1_gthm = $true) & ((sP29 = $true) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | (sP29 != $true)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (! [X144 : term,X145 : subst,X146 : (term > $o),X147 : (term > $o)] : ((((X146 @ X144)) != $true) | ? [X148 : term] : (((X147 @ X148)) != ((X146 @ (sub @ X148 @ (push @ X144 @ X145))))) | (((X147 @ one)) = $true)) | ? [X149 : subst,X150 : term] : (((sub @ one @ (push @ X150 @ X149))) != X150) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (? [X151 : term,X152 : subst,X153 : (term > $o),X154 : (term > $o)] : ((((X153 @ X151)) = $true) & ! [X155 : term] : (((X154 @ X155)) = ((X153 @ (sub @ X155 @ (push @ X151 @ X152))))) & (((X154 @ one)) != $true)) & ! [X156 : subst,X157 : term] : (((sub @ one @ (push @ X157 @ X156))) = X157))) & (hoaslaminj_gthm = $true) & (! [X158 : term,X159 : term,X160 : term,X161 : term] : ((((ap @ X161 @ X159)) != ((ap @ X158 @ X160))) | (X158 = X161)) | (apinj1 != $true)) & ((apinj1 = $true) | ? [X162 : term,X163 : term,X164 : term,X165 : term] : ((((ap @ X165 @ X163)) = ((ap @ X162 @ X164))) & (X162 != X165))) & ((sP28 = $true) | (hoasinduction_lem2v2 != $true)) & ((hoasinduction_lem2v2 = $true) | (sP28 != $true)) & (hoasinduction_gthm = $true) & ((hoasinduction_lem0_lthm = $true) | ? [X166 : (subst > term > subst > $o)] : ! [X167 : (term > $o)] : ? [X168 : term] : (((X167 @ X168)) != ((X166 @ id @ X168 @ id)))) & (! [X169 : (subst > term > subst > $o)] : ? [X170 : (term > $o)] : ! [X171 : term] : (((X169 @ id @ X171 @ id)) = ((X170 @ X171))) | (hoasinduction_lem0_lthm != $true)) & (! [X172 : (subst > term > term),X173 : subst,X174 : term,X175 : (subst > term > term)] : (? [X176 : term,X177 : subst,X178 : subst] : (((X175 @ (comp @ X177 @ X178) @ (sub @ X176 @ X178))) != ((sub @ (X175 @ X177 @ X176) @ X178))) | ? [X179 : subst,X180 : term,X181 : subst] : (((X172 @ (comp @ X179 @ X181) @ (sub @ X180 @ X181))) != ((sub @ (X172 @ X179 @ X180) @ X181))) | (((X175 @ X173 @ X174)) = ((X172 @ X173 @ X174))) | (((lam @ (X172 @ sh @ one))) != ((lam @ (X175 @ sh @ one))))) | (hoaslaminj != $true)) & ((hoaslaminj = $true) | ? [X182 : (subst > term > term),X183 : subst,X184 : term,X185 : (subst > term > term)] : (! [X186 : term,X187 : subst,X188 : subst] : (((sub @ (X185 @ X187 @ X186) @ X188)) = ((X185 @ (comp @ X187 @ X188) @ (sub @ X186 @ X188)))) & ! [X189 : subst,X190 : term,X191 : subst] : (((sub @ (X182 @ X189 @ X190) @ X191)) = ((X182 @ (comp @ X189 @ X191) @ (sub @ X190 @ X191)))) & (((X182 @ X183 @ X184)) != ((X185 @ X183 @ X184))) & (((lam @ (X185 @ sh @ one))) = ((lam @ (X182 @ sh @ one)))))) & (hoasinduction_lem1v2_gthm = $true) & ((substmonoid_lthm = $true) | ((? [X192 : subst] : (((comp @ id @ X192)) != X192) | ? [X193 : subst] : (((comp @ X193 @ id)) != X193)) & ! [X194 : subst] : (((comp @ X194 @ id)) = X194) & ! [X195 : subst] : (((comp @ id @ X195)) = X195))) & ((! [X196 : subst] : (((comp @ id @ X196)) = X196) & ! [X197 : subst] : (((comp @ X197 @ id)) = X197)) | ? [X198 : subst] : (((comp @ X198 @ id)) != X198) | ? [X199 : subst] : (((comp @ id @ X199)) != X199) | (substmonoid_lthm != $true)) & (apnotvar = $true) & (hoasinduction_lem1 = $true) & (axmap = $true) & (axapp = $true) & (hoasapnotvar = $true) & (! [X200 : term] : (((sub @ X200 @ id)) = X200) | (axvarid != $true)) & ((axvarid = $true) | ? [X201 : term] : (((sub @ X201 @ id)) != X201)) & (! [X202 : term,X203 : term,X204 : term,X205 : term] : ((((ap @ X204 @ X202)) != ((ap @ X205 @ X203))) | (X202 = X203)) | (apinj2 != $true)) & ((apinj2 = $true) | ? [X206 : term,X207 : term,X208 : term,X209 : term] : ((((ap @ X209 @ X207)) = ((ap @ X208 @ X206))) & (X206 != X207))) & ((hoasinduction_lem3 = $true) | (sP26 != $true)) & ((sP26 = $true) | (hoasinduction_lem3 != $true)) & (hoasinduction_lem2_lthm = $true) & ((hoasinduction_lem3aa = $true) | (sP24 != $true)) & ((sP24 = $true) | (hoasinduction_lem3aa != $true)) & (induction2lem_gthm = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ! [X210 : term] : ? [X211 : d_term] : (((d2term @ X211)) = X210) & ((hoasinduction_lem3v2a_lthm = $true) | (sP23 != $true)) & ((sP23 = $true) | (hoasinduction_lem3v2a_lthm != $true)) & ((hoasinduction_lem3b = $true) | ? [X212 : term] : ! [X213 : (subst > term > term)] : (((sub @ X212 @ (push @ one @ sh))) != ((X213 @ sh @ one)))) & (! [X214 : term] : ? [X215 : (subst > term > term)] : (((X215 @ sh @ one)) = ((sub @ X214 @ (push @ one @ sh)))) | (hoasinduction_lem3b != $true)) & (pushprop_lem3v2 = $true) & ! [X216 : subst] : ? [X217 : d_subst] : (((d2subst @ X217)) = X216) & (hoasapnotvar_gthm = $true) & (pushprop = $true) & ((hoasinduction_lthm = $true) | (sP22 != $true)) & ((sP22 = $true) | (hoasinduction_lthm != $true)) & (hoasinduction_lem3b_gthm = $true) & ((sP13 = $true) | (hoasinduction_lem3v2_lthm != $true)) & ((hoasinduction_lem3v2_lthm = $true) | (sP13 != $true)) & (lamnotap != $true) & (pushprop_lem2v2 = $true) & (lamnotvar = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((hoasapinj2 = $true) | ? [X218 : term,X219 : term,X220 : term,X221 : term] : ((((ap @ (sub @ X221 @ id) @ X220)) = ((ap @ (sub @ X219 @ id) @ X218))) & (X218 != X220))) & (! [X222 : term,X223 : term,X224 : term,X225 : term] : ((((ap @ (sub @ X225 @ id) @ X224)) != ((ap @ (sub @ X223 @ id) @ X222))) | (X222 = X224)) | (hoasapinj2 != $true)) & ((laminj = $true) | ? [X226 : term,X227 : term] : ((X226 != X227) & (((lam @ X227)) = ((lam @ X226))))) & (! [X228 : term,X229 : term] : ((X228 = X229) | (((lam @ X228)) != ((lam @ X229)))) | (laminj != $true)) & (pushprop_lem2v2_gthm = $true) & ((hoasinduction_lem3a_lthm = $true) | ($true != sP11)) & (($true = sP11) | (hoasinduction_lem3a_lthm != $true)) & (! [X230 : term] : ? [X231 : (subst > term > term)] : ! [X232 : subst,X233 : term] : (((X231 @ X232 @ X233)) = ((sub @ X230 @ (push @ X233 @ X232)))) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ? [X234 : term] : ! [X235 : (subst > term > term)] : ? [X236 : subst,X237 : term] : (((sub @ X234 @ (push @ X237 @ X236))) != ((X235 @ X236 @ X237)))) & ((hoasinduction_lem3_lthm = $true) | (sP9 != $true)) & ((sP9 = $true) | (hoasinduction_lem3_lthm != $true)) & ((termmset = $true) | ? [X238 : term] : (((sub @ X238 @ id)) != X238)) & (! [X239 : term] : (((sub @ X239 @ id)) = X239) | (termmset != $true)) & (hoasinduction_lem3_gthm = $true) & ((sP6 = $true) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | (sP6 != $true)) & ((sP5 = $true) | (induction2 != $true)) & ((induction2 = $true) | (sP5 != $true)) & ((hoasinduction_lem0 = $true) | ? [X240 : (subst > term > subst > $o)] : ! [X241 : (term > $o)] : ? [X242 : term] : (((X240 @ id @ X242 @ id)) != ((X241 @ X242)))) & (! [X243 : (subst > term > subst > $o)] : ? [X244 : (term > $o)] : ! [X245 : term] : (((X244 @ X245)) = ((X243 @ id @ X245 @ id))) | (hoasinduction_lem0 != $true)) & (! [X246 : subst] : (((push @ (sub @ one @ X246) @ (comp @ sh @ X246))) = X246) | (axscons != $true)) & ((axscons = $true) | ? [X247 : subst] : (((push @ (sub @ one @ X247) @ (comp @ sh @ X247))) != X247)) & (pushprop_lem2v2_lthm = $true) & (! [X248 : term,X249 : term] : ((X248 = X249) | (((sub @ X249 @ sh)) != ((sub @ X248 @ sh)))) | (shinj != $true)) & ((shinj = $true) | ? [X250 : term,X251 : term] : ((X250 != X251) & (((sub @ X250 @ sh)) = ((sub @ X251 @ sh))))) & ((hoasinduction_lem3b_lthm = $true) | ? [X252 : term] : ! [X253 : (subst > term > term)] : (((X253 @ sh @ one)) != ((sub @ X252 @ (push @ one @ sh))))) & (! [X254 : term] : ? [X255 : (subst > term > term)] : (((X255 @ sh @ one)) = ((sub @ X254 @ (push @ one @ sh)))) | (hoasinduction_lem3b_lthm != $true)) & (! [X256 : (term > $o),X257 : term] : (? [X258 : term] : ((((X256 @ X258)) = $true) & (((X256 @ (lam @ X258))) != $true)) | ? [X259 : term] : (($true = ((var @ X259))) & (((X256 @ X259)) != $true)) | (((sP3 @ X256)) = $true) | ($true = ((X256 @ X257)))) | (induction != $true)) & ((induction = $true) | ? [X260 : (term > $o),X261 : term] : (! [X262 : term] : ((((X260 @ X262)) != $true) | ($true = ((X260 @ (lam @ X262))))) & ! [X263 : term] : ((((var @ X263)) != $true) | (((X260 @ X263)) = $true)) & (((sP3 @ X260)) != $true) & (((X260 @ X261)) != $true))) & ((hoasinduction_lem3aaa = $true) | (sP2 != $true)) & ((sP2 = $true) | (hoasinduction_lem3aaa != $true)) & (! [X264 : subst,X265 : (term > $o),X266 : term] : ? [X267 : (term > $o)] : ! [X268 : term] : (((X267 @ X268)) = ((X265 @ (sub @ X268 @ (push @ X266 @ X264))))) | (pushprop_lem0_lthm != $true)) & ((pushprop_lem0_lthm = $true) | ? [X269 : subst,X270 : (term > $o),X271 : term] : ! [X272 : (term > $o)] : ? [X273 : term] : (((X272 @ X273)) != ((X270 @ (sub @ X273 @ (push @ X271 @ X269)))))) & (hoasapinj1_gthm = $true) & (ulamvarsh = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & ((sP1 = $true) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | (sP1 != $true)) & (axassoc = $true)),
  inference(rectify,[],[f261])).
thf(f263,plain,(
  ((hoasinduction_lthm_3 = $true) | (sP58 != $true)) & ((sP58 = $true) | (hoasinduction_lthm_3 != $true)) & (termmset_gthm = $true) & ((substmonoid = $true) | (((comp @ id @ sK457)) != sK457) | (sK458 != ((comp @ sK458 @ id)))) & ((! [X2 : subst] : (((comp @ id @ X2)) = X2) & ! [X3 : subst] : (((comp @ X3 @ id)) = X3)) | (substmonoid != $true)) & (pushprop_gthm = $true) & (pushprop_lem3v2_lthm = $true) & (id = ((d2subst @ d_id))) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (pushprop_lthm = $true) & (((d2subst @ d_id)) = sh) & ! [X4 : d_term] : (d_one = X4) & ((hoasinduction_lem3v2_f_lthm = $true) | ! [X6 : (subst > term > term)] : (((sub @ sK459 @ (push @ (sK460 @ X6) @ (sK461 @ X6)))) != ((X6 @ (sK461 @ X6) @ (sK460 @ X6))))) & (! [X9 : term] : ! [X11 : term,X12 : subst] : (((sK462 @ X9 @ X12 @ X11)) = ((sub @ X9 @ (push @ X11 @ X12)))) | (hoasinduction_lem3v2_f_lthm != $true)) & (hoaslamnotvar_lthm = $true) & ! [X13 : d_subst] : (d_id = X13) & (induction2_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & (pushprop_lthm_orig = $true) & ((hoasinduction_no_psi_cond = $true) | (! [X16 : term,X17 : term] : ((((sK464 @ id @ (ap @ (sub @ X16 @ id) @ X17) @ id)) = $true) | (((sK464 @ id @ X17 @ id)) != $true) | (((sK464 @ id @ X16 @ id)) != $true)) & (((sP52 @ sK464)) != $true) & (((sK464 @ id @ sK463 @ id)) != $true))) & (! [X18 : term,X19 : (subst > term > subst > $o)] : (((((X19 @ id @ (ap @ (sub @ (sK465 @ X19) @ id) @ (sK466 @ X19)) @ id)) != $true) & (((X19 @ id @ (sK466 @ X19) @ id)) = $true) & (((X19 @ id @ (sK465 @ X19) @ id)) = $true)) | ($true = ((sP52 @ X19))) | (((X19 @ id @ X18 @ id)) = $true)) | (hoasinduction_no_psi_cond != $true)) & (hoaslamnotap_lthm = $true) & (! [X22 : subst,X23 : (term > $o),X24 : term] : ! [X26 : term] : (((X23 @ (sub @ X26 @ (push @ X24 @ X22)))) = ((sK467 @ X24 @ X23 @ X22 @ X26))) | (pushprop_lem0 != $true)) & ((pushprop_lem0 = $true) | ! [X30 : (term > $o)] : (((sK469 @ (sub @ (sK471 @ X30) @ (push @ sK470 @ sK468)))) != ((X30 @ (sK471 @ X30))))) & (hoaslamnotvar_gthm = $true) & (axclos = $true) & ((sP51 = $true) | (hoasinduction_no_psi_cond_lthm != $true)) & ((hoasinduction_no_psi_cond_lthm = $true) | (sP51 != $true)) & (((var @ (d2term @ d_one))) != $true) & ((hoasinduction_lem3v2 = $true) | (sP46 != $true)) & ((sP46 = $true) | (hoasinduction_lem3v2 != $true)) & ((sP44 = $true) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | (sP44 != $true)) & ((termmset_lthm = $true) | ((((sub @ sK472 @ id)) != sK472) & ! [X33 : term] : (((sub @ X33 @ id)) = X33))) & (! [X34 : term] : (((sub @ X34 @ id)) = X34) | (((sub @ sK473 @ id)) != sK473) | (termmset_lthm != $true)) & (! [X36 : subst,X37 : term] : (((comp @ sh @ (push @ X37 @ X36))) = X36) | (axshiftcons != $true)) & ((axshiftcons = $true) | (((comp @ sh @ (push @ sK475 @ sK474))) != sK474)) & (hoasinduction_lem2v2_gthm = $true) & (pushprop_lem0_gthm = $true) & (hoasapnotvar_lthm = $true) & ((pushprop_lem1 = $true) | ((((sK480 @ sK477)) = $true) & (((sK478 @ (sub @ sK477 @ (push @ sK476 @ sK479)))) != $true))) & (! [X45 : term,X46 : term,X47 : (term > $o),X48 : subst,X49 : (term > $o)] : ((((X49 @ X46)) != $true) | (((X47 @ (sub @ X46 @ (push @ X45 @ X48)))) = $true)) | (pushprop_lem1 != $true)) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (pushprop_lem1v2_gthm = $true) & ((axidr = $true) | (((comp @ sK481 @ id)) != sK481)) & (! [X51 : subst] : (((comp @ X51 @ id)) = X51) | (axidr != $true)) & (one = ((d2term @ d_one))) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((sP40 = $true) | (induction2lem != $true)) & ((induction2lem = $true) | (sP40 != $true)) & ! [X52 : d_subst,X53 : d_subst] : ((X52 = X53) | (((d2subst @ X52)) != ((d2subst @ X53)))) & ((sP38 = $true) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | (sP38 != $true)) & (hoasapinj2_gthm = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (hoasinduction_lem3v2_gthm = $true) & ((hoasinduction_lem3aa_lthm = $true) | (sP34 != $true)) & ((sP34 = $true) | (hoasinduction_lem3aa_lthm != $true)) & (! [X54 : subst] : (((comp @ id @ X54)) = X54) | (axidl != $true)) & ((axidl = $true) | (sK482 != ((comp @ id @ sK482)))) & ((hoasinduction = $true) | (sP33 != $true)) & ((sP33 = $true) | (hoasinduction != $true)) & (substmonoid_gthm = $true) & (hoasinduction_lem2_gthm = $true) & (axabs = $true) & (! [X56 : term,X57 : term,X58 : term,X59 : term] : ((X56 = X57) | (((ap @ (sub @ X57 @ id) @ X59)) != ((ap @ (sub @ X56 @ id) @ X58)))) | (sK483 != ((sub @ sK483 @ id))) | ((sK486 != sK484) & (((ap @ sK484 @ sK485)) = ((ap @ sK486 @ sK487)))) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | (((sK488 != sK489) & (((ap @ (sub @ sK488 @ id) @ sK490)) = ((ap @ (sub @ sK489 @ id) @ sK491)))) & ! [X69 : term] : (((sub @ X69 @ id)) = X69) & ! [X70 : term,X71 : term,X72 : term,X73 : term] : ((X70 = X72) | (((ap @ X70 @ X71)) != ((ap @ X72 @ X73)))))) & (ulamvarind = $true) & ((pushprop_lem1_lthm = $true) | (! [X74 : subst,X75 : term] : (((sub @ one @ (push @ X75 @ X74))) = X75) & ! [X76 : term,X77 : subst] : (((comp @ sh @ (push @ X76 @ X77))) = X77) & (($true != ((sK494 @ (sub @ sK496 @ (push @ sK495 @ sK492))))) & (((sK493 @ sK496)) = $true)))) & ((sK498 != ((sub @ one @ (push @ sK498 @ sK497)))) | (sK500 != ((comp @ sh @ (push @ sK499 @ sK500)))) | ! [X87 : subst,X88 : (term > $o),X89 : (term > $o),X90 : term,X91 : term] : ((((X89 @ (sub @ X91 @ (push @ X90 @ X87)))) = $true) | (((X88 @ X91)) != $true)) | (pushprop_lem1_lthm != $true)) & (hoasinduction_lem1_lthm = $true) & (hoaslamnotap_gthm = $true) & (axvarshift = $true) & (((((ap @ sK504 @ sK503)) = ((ap @ sK501 @ sK502))) & (sK502 != sK503)) | ! [X96 : term,X97 : term,X98 : term,X99 : term] : ((((ap @ (sub @ X98 @ id) @ X96)) != ((ap @ (sub @ X97 @ id) @ X99))) | (X96 = X99)) | (hoasapinj2_lthm != $true)) & ((hoasapinj2_lthm = $true) | (! [X100 : term,X101 : term,X102 : term,X103 : term] : ((((ap @ X103 @ X102)) != ((ap @ X100 @ X101))) | (X101 = X102)) & ((((ap @ (sub @ sK507 @ id) @ sK505)) = ((ap @ (sub @ sK506 @ id) @ sK508))) & (sK508 != sK505)))) & (hoasinduction_lem1v2 = $true) & (ulamvar1 != $true) & ! [X108 : d_term,X109 : d_term] : ((X108 = X109) | (((d2term @ X108)) != ((d2term @ X109)))) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (pushprop_lem1_gthm = $true) & (! [X110 : term,X111 : term,X112 : (subst > term > term)] : ((((sub @ (X112 @ (sK510 @ X112) @ (sK509 @ X112)) @ (sK511 @ X112))) != ((X112 @ (comp @ (sK510 @ X112) @ (sK511 @ X112)) @ (sub @ (sK509 @ X112) @ (sK511 @ X112))))) | (((lam @ (X112 @ sh @ one))) != ((ap @ (sub @ X111 @ id) @ X110)))) | (hoaslamnotap != $true)) & ((hoaslamnotap = $true) | (! [X119 : term,X120 : subst,X121 : subst] : (((sK514 @ (comp @ X120 @ X121) @ (sub @ X119 @ X121))) = ((sub @ (sK514 @ X120 @ X119) @ X121))) & (((lam @ (sK514 @ sh @ one))) = ((ap @ (sub @ sK513 @ id) @ sK512))))) & ((axvarcons = $true) | (sK516 != ((sub @ one @ (push @ sK516 @ sK515))))) & (! [X124 : subst,X125 : term] : (((sub @ one @ (push @ X125 @ X124))) = X125) | (axvarcons != $true)) & ((hoasapinj1 = $true) | ((((ap @ (sub @ sK520 @ id) @ sK517)) = ((ap @ (sub @ sK518 @ id) @ sK519))) & (sK520 != sK518))) & (! [X130 : term,X131 : term,X132 : term,X133 : term] : ((((ap @ (sub @ X133 @ id) @ X130)) != ((ap @ (sub @ X131 @ id) @ X132))) | (X131 = X133)) | (hoasapinj1 != $true)) & (hoasinduction_lem3a_gthm = $true) & (! [X134 : (term > $o),X135 : term,X136 : (term > $o),X137 : subst] : ((((X136 @ (sub @ (sK521 @ X137 @ X136 @ X135 @ X134) @ (push @ X135 @ X137)))) != ((X134 @ (sK521 @ X137 @ X136 @ X135 @ X134)))) | (((X136 @ X135)) != $true) | (((X134 @ one)) = $true)) | (pushprop_lem1v2 != $true)) & ((pushprop_lem1v2 = $true) | (! [X143 : term] : (((sK522 @ X143)) = ((sK524 @ (sub @ X143 @ (push @ sK523 @ sK525))))) & (((sK524 @ sK523)) = $true) & (((sK522 @ one)) != $true))) & ((sP30 = $true) | (hoaslaminj_lthm != $true)) & ((hoaslaminj_lthm = $true) | (sP30 != $true)) & (hoaslamnotvar = $true) & (hoasinduction_lem1_gthm = $true) & ((sP29 = $true) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | (sP29 != $true)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (! [X144 : term,X145 : subst,X146 : (term > $o),X147 : (term > $o)] : ((((X146 @ X144)) != $true) | (((X146 @ (sub @ (sK526 @ X147 @ X146 @ X145 @ X144) @ (push @ X144 @ X145)))) != ((X147 @ (sK526 @ X147 @ X146 @ X145 @ X144)))) | (((X147 @ one)) = $true)) | (sK528 != ((sub @ one @ (push @ sK528 @ sK527)))) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (((((sK531 @ sK529)) = $true) & ! [X155 : term] : (((sK532 @ X155)) = ((sK531 @ (sub @ X155 @ (push @ sK529 @ sK530))))) & (((sK532 @ one)) != $true)) & ! [X156 : subst,X157 : term] : (((sub @ one @ (push @ X157 @ X156))) = X157))) & (hoaslaminj_gthm = $true) & (! [X158 : term,X159 : term,X160 : term,X161 : term] : ((((ap @ X161 @ X159)) != ((ap @ X158 @ X160))) | (X158 = X161)) | (apinj1 != $true)) & ((apinj1 = $true) | ((((ap @ sK536 @ sK534)) = ((ap @ sK533 @ sK535))) & (sK536 != sK533))) & ((sP28 = $true) | (hoasinduction_lem2v2 != $true)) & ((hoasinduction_lem2v2 = $true) | (sP28 != $true)) & (hoasinduction_gthm = $true) & ((hoasinduction_lem0_lthm = $true) | ! [X167 : (term > $o)] : (((sK537 @ id @ (sK538 @ X167) @ id)) != ((X167 @ (sK538 @ X167))))) & (! [X169 : (subst > term > subst > $o)] : ! [X171 : term] : (((sK539 @ X169 @ X171)) = ((X169 @ id @ X171 @ id))) | (hoasinduction_lem0_lthm != $true)) & (! [X172 : (subst > term > term),X173 : subst,X174 : term,X175 : (subst > term > term)] : ((((sub @ (X175 @ (sK541 @ X175) @ (sK540 @ X175)) @ (sK542 @ X175))) != ((X175 @ (comp @ (sK541 @ X175) @ (sK542 @ X175)) @ (sub @ (sK540 @ X175) @ (sK542 @ X175))))) | (((sub @ (X172 @ (sK543 @ X172) @ (sK544 @ X172)) @ (sK545 @ X172))) != ((X172 @ (comp @ (sK543 @ X172) @ (sK545 @ X172)) @ (sub @ (sK544 @ X172) @ (sK545 @ X172))))) | (((X175 @ X173 @ X174)) = ((X172 @ X173 @ X174))) | (((lam @ (X172 @ sh @ one))) != ((lam @ (X175 @ sh @ one))))) | (hoaslaminj != $true)) & ((hoaslaminj = $true) | (! [X186 : term,X187 : subst,X188 : subst] : (((sK549 @ (comp @ X187 @ X188) @ (sub @ X186 @ X188))) = ((sub @ (sK549 @ X187 @ X186) @ X188))) & ! [X189 : subst,X190 : term,X191 : subst] : (((sub @ (sK546 @ X189 @ X190) @ X191)) = ((sK546 @ (comp @ X189 @ X191) @ (sub @ X190 @ X191)))) & (((sK546 @ sK547 @ sK548)) != ((sK549 @ sK547 @ sK548))) & (((lam @ (sK546 @ sh @ one))) = ((lam @ (sK549 @ sh @ one)))))) & (hoasinduction_lem1v2_gthm = $true) & ((substmonoid_lthm = $true) | (((((comp @ id @ sK550)) != sK550) | (sK551 != ((comp @ sK551 @ id)))) & ! [X194 : subst] : (((comp @ X194 @ id)) = X194) & ! [X195 : subst] : (((comp @ id @ X195)) = X195))) & ((! [X196 : subst] : (((comp @ id @ X196)) = X196) & ! [X197 : subst] : (((comp @ X197 @ id)) = X197)) | (sK552 != ((comp @ sK552 @ id))) | (((comp @ id @ sK553)) != sK553) | (substmonoid_lthm != $true)) & (apnotvar = $true) & (hoasinduction_lem1 = $true) & (axmap = $true) & (axapp = $true) & (hoasapnotvar = $true) & (! [X200 : term] : (((sub @ X200 @ id)) = X200) | (axvarid != $true)) & ((axvarid = $true) | (((sub @ sK554 @ id)) != sK554)) & (! [X202 : term,X203 : term,X204 : term,X205 : term] : ((((ap @ X204 @ X202)) != ((ap @ X205 @ X203))) | (X202 = X203)) | (apinj2 != $true)) & ((apinj2 = $true) | ((((ap @ sK557 @ sK555)) = ((ap @ sK558 @ sK556))) & (sK556 != sK555))) & ((hoasinduction_lem3 = $true) | (sP26 != $true)) & ((sP26 = $true) | (hoasinduction_lem3 != $true)) & (hoasinduction_lem2_lthm = $true) & ((hoasinduction_lem3aa = $true) | (sP24 != $true)) & ((sP24 = $true) | (hoasinduction_lem3aa != $true)) & (induction2lem_gthm = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ! [X210 : term] : (((d2term @ (sK559 @ X210))) = X210) & ((hoasinduction_lem3v2a_lthm = $true) | (sP23 != $true)) & ((sP23 = $true) | (hoasinduction_lem3v2a_lthm != $true)) & ((hoasinduction_lem3b = $true) | ! [X213 : (subst > term > term)] : (((X213 @ sh @ one)) != ((sub @ sK560 @ (push @ one @ sh))))) & (! [X214 : term] : (((sK561 @ X214 @ sh @ one)) = ((sub @ X214 @ (push @ one @ sh)))) | (hoasinduction_lem3b != $true)) & (pushprop_lem3v2 = $true) & ! [X216 : subst] : (((d2subst @ (sK562 @ X216))) = X216) & (hoasapnotvar_gthm = $true) & (pushprop = $true) & ((hoasinduction_lthm = $true) | (sP22 != $true)) & ((sP22 = $true) | (hoasinduction_lthm != $true)) & (hoasinduction_lem3b_gthm = $true) & ((sP13 = $true) | (hoasinduction_lem3v2_lthm != $true)) & ((hoasinduction_lem3v2_lthm = $true) | (sP13 != $true)) & (lamnotap != $true) & (pushprop_lem2v2 = $true) & (lamnotvar = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((hoasapinj2 = $true) | ((((ap @ (sub @ sK564 @ id) @ sK563)) = ((ap @ (sub @ sK566 @ id) @ sK565))) & (sK565 != sK563))) & (! [X222 : term,X223 : term,X224 : term,X225 : term] : ((((ap @ (sub @ X225 @ id) @ X224)) != ((ap @ (sub @ X223 @ id) @ X222))) | (X222 = X224)) | (hoasapinj2 != $true)) & ((laminj = $true) | ((sK568 != sK567) & (((lam @ sK568)) = ((lam @ sK567))))) & (! [X228 : term,X229 : term] : ((X228 = X229) | (((lam @ X228)) != ((lam @ X229)))) | (laminj != $true)) & (pushprop_lem2v2_gthm = $true) & ((hoasinduction_lem3a_lthm = $true) | ($true != sP11)) & (($true = sP11) | (hoasinduction_lem3a_lthm != $true)) & (! [X230 : term] : ! [X232 : subst,X233 : term] : (((sK569 @ X230 @ X232 @ X233)) = ((sub @ X230 @ (push @ X233 @ X232)))) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ! [X235 : (subst > term > term)] : (((sub @ sK570 @ (push @ (sK572 @ X235) @ (sK571 @ X235)))) != ((X235 @ (sK571 @ X235) @ (sK572 @ X235))))) & ((hoasinduction_lem3_lthm = $true) | (sP9 != $true)) & ((sP9 = $true) | (hoasinduction_lem3_lthm != $true)) & ((termmset = $true) | (sK573 != ((sub @ sK573 @ id)))) & (! [X239 : term] : (((sub @ X239 @ id)) = X239) | (termmset != $true)) & (hoasinduction_lem3_gthm = $true) & ((sP6 = $true) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | (sP6 != $true)) & ((sP5 = $true) | (induction2 != $true)) & ((induction2 = $true) | (sP5 != $true)) & ((hoasinduction_lem0 = $true) | ! [X241 : (term > $o)] : (((sK574 @ id @ (sK575 @ X241) @ id)) != ((X241 @ (sK575 @ X241))))) & (! [X243 : (subst > term > subst > $o)] : ! [X245 : term] : (((X243 @ id @ X245 @ id)) = ((sK576 @ X243 @ X245))) | (hoasinduction_lem0 != $true)) & (! [X246 : subst] : (((push @ (sub @ one @ X246) @ (comp @ sh @ X246))) = X246) | (axscons != $true)) & ((axscons = $true) | (((push @ (sub @ one @ sK577) @ (comp @ sh @ sK577))) != sK577)) & (pushprop_lem2v2_lthm = $true) & (! [X248 : term,X249 : term] : ((X248 = X249) | (((sub @ X249 @ sh)) != ((sub @ X248 @ sh)))) | (shinj != $true)) & ((shinj = $true) | ((sK579 != sK578) & (((sub @ sK579 @ sh)) = ((sub @ sK578 @ sh))))) & ((hoasinduction_lem3b_lthm = $true) | ! [X253 : (subst > term > term)] : (((sub @ sK580 @ (push @ one @ sh))) != ((X253 @ sh @ one)))) & (! [X254 : term] : (((sK581 @ X254 @ sh @ one)) = ((sub @ X254 @ (push @ one @ sh)))) | (hoasinduction_lem3b_lthm != $true)) & (! [X256 : (term > $o),X257 : term] : (((((X256 @ (sK582 @ X256))) = $true) & (((X256 @ (lam @ (sK582 @ X256)))) != $true)) | ((((var @ (sK583 @ X256))) = $true) & (((X256 @ (sK583 @ X256))) != $true)) | (((sP3 @ X256)) = $true) | ($true = ((X256 @ X257)))) | (induction != $true)) & ((induction = $true) | (! [X262 : term] : ((((sK584 @ X262)) != $true) | (((sK584 @ (lam @ X262))) = $true)) & ! [X263 : term] : ((((var @ X263)) != $true) | (((sK584 @ X263)) = $true)) & (((sP3 @ sK584)) != $true) & (((sK584 @ sK585)) != $true))) & ((hoasinduction_lem3aaa = $true) | (sP2 != $true)) & ((sP2 = $true) | (hoasinduction_lem3aaa != $true)) & (! [X264 : subst,X265 : (term > $o),X266 : term] : ! [X268 : term] : (((X265 @ (sub @ X268 @ (push @ X266 @ X264)))) = ((sK586 @ X266 @ X265 @ X264 @ X268))) | (pushprop_lem0_lthm != $true)) & ((pushprop_lem0_lthm = $true) | ! [X272 : (term > $o)] : (((X272 @ (sK590 @ X272))) != ((sK588 @ (sub @ (sK590 @ X272) @ (push @ sK589 @ sK587)))))) & (hoasapinj1_gthm = $true) & (ulamvarsh = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & ((sP1 = $true) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | (sP1 != $true)) & (axassoc = $true)),
  inference(skolemize,[status(esa),new_symbols(skolem,[sK457,sK458,sK459,vAPP,sK463,sK464,vAPP,sK468,sK469,sK470,vAPP,sK472,sK473,sK474,sK475,sK476,sK477,sK478,sK479,sK480,sK481,sK482,sK483,sK484,sK485,sK486,sK487,sK488,sK489,sK490,sK491,sK492,sK493,sK494,sK495,sK496,sK497,sK498,sK499,sK500,sK501,sK502,sK503,sK504,sK505,sK506,sK507,sK508,vAPP,sK512,sK513,sK514,sK515,sK516,sK517,sK518,sK519,sK520,vAPP,sK522,sK523,sK524,sK525,vAPP,sK527,sK528,sK529,sK530,sK531,sK532,sK533,sK534,sK535,sK536,sK537,vAPP,sK546,sK547,sK548,sK549,sK550,sK551,sK552,sK553,sK554,sK555,sK556,sK557,sK558,vAPP,sK560,vAPP,sK563,sK564,sK565,sK566,sK567,sK568,vAPP,sK570,vAPP,sK573,sK574,vAPP,sK577,sK578,sK579,sK580,vAPP,sK584,sK585,vAPP,sK587,sK588,sK589,vAPP]),skolemize(X0,$thf(sK457)),skolemize(X1,$thf(sK458)),skolemize(X5,$thf(sK459)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X14,$thf(sK463)),skolemize(X15,$thf(sK464)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X27,$thf(sK468)),skolemize(X28,$thf(sK469)),skolemize(X29,$thf(sK470)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X32,$thf(sK472)),skolemize(X35,$thf(sK473)),skolemize(X38,$thf(sK474)),skolemize(X39,$thf(sK475)),skolemize(X40,$thf(sK476)),skolemize(X41,$thf(sK477)),skolemize(X42,$thf(sK478)),skolemize(X43,$thf(sK479)),skolemize(X44,$thf(sK480)),skolemize(X50,$thf(sK481)),skolemize(X55,$thf(sK482)),skolemize(X60,$thf(sK483)),skolemize(X61,$thf(sK484)),skolemize(X62,$thf(sK485)),skolemize(X63,$thf(sK486)),skolemize(X64,$thf(sK487)),skolemize(X65,$thf(sK488)),skolemize(X66,$thf(sK489)),skolemize(X67,$thf(sK490)),skolemize(X68,$thf(sK491)),skolemize(X78,$thf(sK492)),skolemize(X79,$thf(sK493)),skolemize(X80,$thf(sK494)),skolemize(X81,$thf(sK495)),skolemize(X82,$thf(sK496)),skolemize(X83,$thf(sK497)),skolemize(X84,$thf(sK498)),skolemize(X85,$thf(sK499)),skolemize(X86,$thf(sK500)),skolemize(X92,$thf(sK501)),skolemize(X93,$thf(sK502)),skolemize(X94,$thf(sK503)),skolemize(X95,$thf(sK504)),skolemize(X104,$thf(sK505)),skolemize(X105,$thf(sK506)),skolemize(X106,$thf(sK507)),skolemize(X107,$thf(sK508)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X116,$thf(sK512)),skolemize(X117,$thf(sK513)),skolemize(X118,$thf(sK514)),skolemize(X122,$thf(sK515)),skolemize(X123,$thf(sK516)),skolemize(X126,$thf(sK517)),skolemize(X127,$thf(sK518)),skolemize(X128,$thf(sK519)),skolemize(X129,$thf(sK520)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X139,$thf(sK522)),skolemize(X140,$thf(sK523)),skolemize(X141,$thf(sK524)),skolemize(X142,$thf(sK525)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X149,$thf(sK527)),skolemize(X150,$thf(sK528)),skolemize(X151,$thf(sK529)),skolemize(X152,$thf(sK530)),skolemize(X153,$thf(sK531)),skolemize(X154,$thf(sK532)),skolemize(X162,$thf(sK533)),skolemize(X163,$thf(sK534)),skolemize(X164,$thf(sK535)),skolemize(X165,$thf(sK536)),skolemize(X166,$thf(sK537)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X182,$thf(sK546)),skolemize(X183,$thf(sK547)),skolemize(X184,$thf(sK548)),skolemize(X185,$thf(sK549)),skolemize(X192,$thf(sK550)),skolemize(X193,$thf(sK551)),skolemize(X198,$thf(sK552)),skolemize(X199,$thf(sK553)),skolemize(X201,$thf(sK554)),skolemize(X206,$thf(sK555)),skolemize(X207,$thf(sK556)),skolemize(X208,$thf(sK557)),skolemize(X209,$thf(sK558)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X212,$thf(sK560)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X218,$thf(sK563)),skolemize(X219,$thf(sK564)),skolemize(X220,$thf(sK565)),skolemize(X221,$thf(sK566)),skolemize(X226,$thf(sK567)),skolemize(X227,$thf(sK568)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X234,$thf(sK570)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X238,$thf(sK573)),skolemize(X240,$thf(sK574)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X247,$thf(sK577)),skolemize(X250,$thf(sK578)),skolemize(X251,$thf(sK579)),skolemize(X252,$thf(sK580)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X260,$thf(sK584)),skolemize(X261,$thf(sK585)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X269,$thf(sK587)),skolemize(X270,$thf(sK588)),skolemize(X271,$thf(sK589)),skolemize(X17,$thf(sK67 @ X16))],[f262])).
thf(f818,plain,(
  ( ! [X210 : term] : ((((d2term @ (sK559 @ X210))) = X210)) )),
  inference(cnf_transformation,[],[f263])).
thf(f819,plain,(
  (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one)))))),
  inference(cnf_transformation,[],[f263])).
thf(f843,plain,(
  (hoaslaminj = $true) | (((sK546 @ sK547 @ sK548)) != ((sK549 @ sK547 @ sK548)))),
  inference(cnf_transformation,[],[f263])).
thf(f965,plain,(
  ( ! [X4 : d_term] : ((d_one = X4)) )),
  inference(cnf_transformation,[],[f263])).
thf(f978,plain,(
  (hoaslaminj != ((!! @ (subst > term > term) @ (^[Y0 : subst > term > term]: ((!! @ subst @ (^[Y1 : subst]: (!! @ subst @ (^[Y2 : subst]: (!! @ term @ (^[Y3 : term]: ((sub @ (Y0 @ Y2 @ Y3) @ Y1) = (Y0 @ (comp @ Y2 @ Y1) @ (sub @ Y3 @ Y1))))))))) => (!! @ (subst > term > term) @ (^[Y1 : subst > term > term]: ((!! @ term @ (^[Y2 : term]: (!! @ subst @ (^[Y3 : subst]: (!! @ subst @ (^[Y4 : subst]: ((sub @ (Y1 @ Y3 @ Y2) @ Y4) = (Y1 @ (comp @ Y3 @ Y4) @ (sub @ Y2 @ Y4))))))))) => (((hoaslam @ id @ (^[Y2 : subst]: ((^[Y3 : term]: (Y0 @ Y2 @ Y3))))) = (hoaslam @ id @ (^[Y2 : subst]: ((^[Y3 : term]: (Y1 @ Y2 @ Y3)))))) => (!! @ subst @ (^[Y2 : subst]: (!! @ term @ (^[Y3 : term]: ((Y0 @ Y2 @ Y3) = (Y1 @ Y2 @ Y3)))))))))))))))),
  inference(cnf_transformation,[],[f8])).
thf(f981,plain,(
  (hoaslaminj != ((!! @ (subst > term > term) @ (^[Y0 : subst > term > term]: ((!! @ subst @ (^[Y1 : subst]: (!! @ subst @ (^[Y2 : subst]: (!! @ term @ (^[Y3 : term]: ((sub @ (Y0 @ Y2 @ Y3) @ Y1) = (Y0 @ (comp @ Y2 @ Y1) @ (sub @ Y3 @ Y1))))))))) => (!! @ (subst > term > term) @ (^[Y1 : subst > term > term]: ((!! @ term @ (^[Y2 : term]: (!! @ subst @ (^[Y3 : subst]: (!! @ subst @ (^[Y4 : subst]: ((sub @ (Y1 @ Y3 @ Y2) @ Y4) = (Y1 @ (comp @ Y3 @ Y4) @ (sub @ Y2 @ Y4))))))))) => ((((^[Y2 : subst]: ((^[Y3 : subst > term > term]: (d2term @ d_one)))) @ id @ (^[Y2 : subst]: ((^[Y3 : term]: (Y0 @ Y2 @ Y3))))) = ((^[Y2 : subst]: ((^[Y3 : subst > term > term]: (d2term @ d_one)))) @ id @ (^[Y2 : subst]: ((^[Y3 : term]: (Y1 @ Y2 @ Y3)))))) => (!! @ subst @ (^[Y2 : subst]: (!! @ term @ (^[Y3 : term]: ((Y0 @ Y2 @ Y3) = (Y1 @ Y2 @ Y3)))))))))))))))),
  inference(definition_unfolding,[],[f978,f819,f819])).
thf(f1123,plain,(
  (hoaslaminj = $false) | (((!! @ (subst > term > term) @ (^[Y0 : subst > term > term]: ((!! @ subst @ (^[Y1 : subst]: (!! @ subst @ (^[Y2 : subst]: (!! @ term @ (^[Y3 : term]: ((sub @ (Y0 @ Y2 @ Y3) @ Y1) = (Y0 @ (comp @ Y2 @ Y1) @ (sub @ Y3 @ Y1))))))))) => (!! @ (subst > term > term) @ (^[Y1 : subst > term > term]: ((!! @ term @ (^[Y2 : term]: (!! @ subst @ (^[Y3 : subst]: (!! @ subst @ (^[Y4 : subst]: ((sub @ (Y1 @ Y3 @ Y2) @ Y4) = (Y1 @ (comp @ Y3 @ Y4) @ (sub @ Y2 @ Y4))))))))) => ((((^[Y2 : subst]: ((^[Y3 : subst > term > term]: (d2term @ d_one)))) @ id @ (^[Y2 : subst]: ((^[Y3 : term]: (Y0 @ Y2 @ Y3))))) = ((^[Y2 : subst]: ((^[Y3 : subst > term > term]: (d2term @ d_one)))) @ id @ (^[Y2 : subst]: ((^[Y3 : term]: (Y1 @ Y2 @ Y3)))))) => (!! @ subst @ (^[Y2 : subst]: (!! @ term @ (^[Y3 : term]: ((Y0 @ Y2 @ Y3) = (Y1 @ Y2 @ Y3)))))))))))))) = $false)),
  inference(xor_proxy_clausification,[],[f981])).
thf(f1124,plain,(
  (hoaslaminj = $false) | ((((^[Y0 : subst > term > term]: ((!! @ subst @ (^[Y1 : subst]: (!! @ subst @ (^[Y2 : subst]: (!! @ term @ (^[Y3 : term]: ((sub @ (Y0 @ Y2 @ Y3) @ Y1) = (Y0 @ (comp @ Y2 @ Y1) @ (sub @ Y3 @ Y1))))))))) => (!! @ (subst > term > term) @ (^[Y1 : subst > term > term]: ((!! @ term @ (^[Y2 : term]: (!! @ subst @ (^[Y3 : subst]: (!! @ subst @ (^[Y4 : subst]: ((sub @ (Y1 @ Y3 @ Y2) @ Y4) = (Y1 @ (comp @ Y3 @ Y4) @ (sub @ Y2 @ Y4))))))))) => ((((^[Y2 : subst]: ((^[Y3 : subst > term > term]: (d2term @ d_one)))) @ id @ (^[Y2 : subst]: ((^[Y3 : term]: (Y0 @ Y2 @ Y3))))) = ((^[Y2 : subst]: ((^[Y3 : subst > term > term]: (d2term @ d_one)))) @ id @ (^[Y2 : subst]: ((^[Y3 : term]: (Y1 @ Y2 @ Y3)))))) => (!! @ subst @ (^[Y2 : subst]: (!! @ term @ (^[Y3 : term]: ((Y0 @ Y2 @ Y3) = (Y1 @ Y2 @ Y3)))))))))))) @ sK591)) = $false)),
  inference(sigma_proxy_clausification,[],[f1123])).
thf(f1125,plain,(
  ((((!! @ subst @ (^[Y0 : subst]: (!! @ subst @ (^[Y1 : subst]: (!! @ term @ (^[Y2 : term]: ((sub @ (sK591 @ Y1 @ Y2) @ Y0) = (sK591 @ (comp @ Y1 @ Y0) @ (sub @ Y2 @ Y0))))))))) => (!! @ (subst > term > term) @ (^[Y0 : subst > term > term]: ((!! @ term @ (^[Y1 : term]: (!! @ subst @ (^[Y2 : subst]: (!! @ subst @ (^[Y3 : subst]: ((sub @ (Y0 @ Y2 @ Y1) @ Y3) = (Y0 @ (comp @ Y2 @ Y3) @ (sub @ Y1 @ Y3))))))))) => (((d2term @ d_one) = (d2term @ d_one)) => (!! @ subst @ (^[Y1 : subst]: (!! @ term @ (^[Y2 : term]: ((sK591 @ Y1 @ Y2) = (Y0 @ Y1 @ Y2)))))))))))) = $false) | (hoaslaminj = $false)),
  inference(beta-eta_normalization,[],[f1124])).
thf(f1126,plain,(
  (((!! @ (subst > term > term) @ (^[Y0 : subst > term > term]: ((!! @ term @ (^[Y1 : term]: (!! @ subst @ (^[Y2 : subst]: (!! @ subst @ (^[Y3 : subst]: ((sub @ (Y0 @ Y2 @ Y1) @ Y3) = (Y0 @ (comp @ Y2 @ Y3) @ (sub @ Y1 @ Y3))))))))) => (((d2term @ d_one) = (d2term @ d_one)) => (!! @ subst @ (^[Y1 : subst]: (!! @ term @ (^[Y2 : term]: ((sK591 @ Y1 @ Y2) = (Y0 @ Y1 @ Y2))))))))))) = $false) | (hoaslaminj = $false)),
  inference(imp_proxy_clausification,[],[f1125])).
thf(f1135,plain,(
  (hoaslaminj = $false) | ((((^[Y0 : subst > term > term]: ((!! @ term @ (^[Y1 : term]: (!! @ subst @ (^[Y2 : subst]: (!! @ subst @ (^[Y3 : subst]: ((sub @ (Y0 @ Y2 @ Y1) @ Y3) = (Y0 @ (comp @ Y2 @ Y3) @ (sub @ Y1 @ Y3))))))))) => (((d2term @ d_one) = (d2term @ d_one)) => (!! @ subst @ (^[Y1 : subst]: (!! @ term @ (^[Y2 : term]: ((sK591 @ Y1 @ Y2) = (Y0 @ Y1 @ Y2))))))))) @ sK592)) = $false)),
  inference(sigma_proxy_clausification,[],[f1126])).
thf(f1136,plain,(
  (hoaslaminj = $false) | ((((!! @ term @ (^[Y0 : term]: (!! @ subst @ (^[Y1 : subst]: (!! @ subst @ (^[Y2 : subst]: ((sub @ (sK592 @ Y1 @ Y0) @ Y2) = (sK592 @ (comp @ Y1 @ Y2) @ (sub @ Y0 @ Y2))))))))) => (((d2term @ d_one) = (d2term @ d_one)) => (!! @ subst @ (^[Y0 : subst]: (!! @ term @ (^[Y1 : term]: ((sK591 @ Y0 @ Y1) = (sK592 @ Y0 @ Y1))))))))) = $false)),
  inference(beta-eta_normalization,[],[f1135])).
thf(f1137,plain,(
  (((((d2term @ d_one) = (d2term @ d_one)) => (!! @ subst @ (^[Y0 : subst]: (!! @ term @ (^[Y1 : term]: ((sK591 @ Y0 @ Y1) = (sK592 @ Y0 @ Y1)))))))) = $false) | (hoaslaminj = $false)),
  inference(imp_proxy_clausification,[],[f1136])).
thf(f1146,plain,(
  (hoaslaminj = $false) | (((!! @ subst @ (^[Y0 : subst]: (!! @ term @ (^[Y1 : term]: ((sK591 @ Y0 @ Y1) = (sK592 @ Y0 @ Y1))))))) = $false)),
  inference(imp_proxy_clausification,[],[f1137])).
thf(f1149,plain,(
  (hoaslaminj = $false) | ($false = (((^[Y0 : subst]: (!! @ term @ (^[Y1 : term]: ((sK591 @ Y0 @ Y1) = (sK592 @ Y0 @ Y1))))) @ sK593)))),
  inference(sigma_proxy_clausification,[],[f1146])).
thf(f1150,plain,(
  (hoaslaminj = $false) | (((!! @ term @ (^[Y0 : term]: ((sK591 @ sK593 @ Y0) = (sK592 @ sK593 @ Y0))))) = $false)),
  inference(beta-eta_normalization,[],[f1149])).
thf(f1151,plain,(
  ((((^[Y0 : term]: ((sK591 @ sK593 @ Y0) = (sK592 @ sK593 @ Y0))) @ sK594)) = $false) | (hoaslaminj = $false)),
  inference(sigma_proxy_clausification,[],[f1150])).
thf(f1152,plain,(
  ((((sK591 @ sK593 @ sK594) = (sK592 @ sK593 @ sK594))) = $false) | (hoaslaminj = $false)),
  inference(beta-eta_normalization,[],[f1151])).
thf(f1153,plain,(
  (((sK592 @ sK593 @ sK594)) != ((sK591 @ sK593 @ sK594))) | (hoaslaminj = $false)),
  inference(equality_proxy_clausification,[],[f1152])).
thf(f2346,definition,(
  spl601_274 <=> (hoaslaminj = $true)),
  introduced(definition,[new_symbols(definition,[spl601_274])],[avatar_definition])).
thf(f2348,plain,(
  (hoaslaminj = $true) | ~spl601_274),
  inference(avatar_component_clause,[],[f2346])).
thf(f3216,definition,(
  spl601_474 <=> (((sK546 @ sK547 @ sK548)) = ((sK549 @ sK547 @ sK548)))),
  introduced(definition,[new_symbols(definition,[spl601_474])],[avatar_definition])).
thf(f3218,plain,(
  (((sK546 @ sK547 @ sK548)) != ((sK549 @ sK547 @ sK548))) | spl601_474),
  inference(avatar_component_clause,[],[f3216])).
thf(f3219,plain,(
  ~spl601_474 | spl601_274),
  inference(avatar_split_clause,[],[f843,f2346,f3216])).
thf(f3679,definition,(
  spl601_581 <=> (hoaslaminj = $false)),
  introduced(definition,[new_symbols(definition,[spl601_581])],[avatar_definition])).
thf(f3681,plain,(
  (hoaslaminj = $false) | ~spl601_581),
  inference(avatar_component_clause,[],[f3679])).
thf(f3688,definition,(
  spl601_583 <=> (((sK592 @ sK593 @ sK594)) = ((sK591 @ sK593 @ sK594)))),
  introduced(definition,[new_symbols(definition,[spl601_583])],[avatar_definition])).
thf(f3690,plain,(
  (((sK592 @ sK593 @ sK594)) != ((sK591 @ sK593 @ sK594))) | spl601_583),
  inference(avatar_component_clause,[],[f3688])).
thf(f3691,plain,(
  ~spl601_583 | spl601_581),
  inference(avatar_split_clause,[],[f1153,f3679,f3688])).
thf(f3913,plain,(
  ($false = $true) | (~spl601_274 | ~spl601_581)),
  inference(forward_demodulation,[],[f2348,f3681])).
thf(f3914,plain,(
  $false | (~spl601_274 | ~spl601_581)),
  inference(trivial_inequality_removal,[],[f3913])).
thf(f3915,plain,(
  ~spl601_274 | ~spl601_581),
  inference(avatar_contradiction_clause,[],[f3914])).
thf(f4041,plain,(
  ( ! [X210 : term] : ((((d2term @ d_one)) = X210)) )),
  inference(forward_demodulation,[],[f818,f965])).
thf(f4052,plain,(
  (((d2term @ d_one)) != ((sK591 @ sK593 @ sK594))) | spl601_583),
  inference(superposition,[],[f3690,f4041])).
thf(f4058,plain,(
  $false | spl601_583),
  inference(forward_subsumption_resolution,[],[f4052,f4041])).
thf(f4059,plain,(
  spl601_583),
  inference(avatar_contradiction_clause,[],[f4058])).
thf(f4062,plain,(
  (((d2term @ d_one)) != ((sK546 @ sK547 @ sK548))) | spl601_474),
  inference(forward_demodulation,[],[f3218,f4041])).
thf(f4067,plain,(
  $false | spl601_474),
  inference(forward_subsumption_resolution,[],[f4062,f4041])).
thf(f4068,plain,(
  spl601_474),
  inference(avatar_contradiction_clause,[],[f4067])).
cnf(s426, plain, spl601_274 | ~spl601_474, inference(sat_conversion,[],[f3219])).
cnf(s551, plain, spl601_581 | ~spl601_583, inference(sat_conversion,[],[f3691])).
cnf(s609, plain, ~spl601_274 | ~spl601_581, inference(sat_conversion,[],[f3915])).
cnf(s610, plain, spl601_583, inference(sat_conversion,[],[f4059])).
cnf(s611, plain, spl601_474, inference(sat_conversion,[],[f4068])).
cnf(s612, plain, spl601_581, inference(rat,[],[s551,s610])).
cnf(s613, plain, ~spl601_274, inference(rat,[],[s609,s612])).
cnf(s615, plain, $false, inference(rat,[],[s426,s611,s613])).
thf(f4073,plain,(
  $false),
  inference(avatar_sat_refutation,[],[s615])).
% SZS output end Proof for theBenchmark