%----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, sP0: ((subst > term > subst > $o) > $o)).
thf(func_def_138, type, sP1: $o).
thf(func_def_139, type, sP2: $o).
thf(func_def_140, type, sP3: $o).
thf(func_def_141, type, sP4: ((term > $o) > $o)).
thf(func_def_142, type, sP5: ((term > $o) > $o)).
thf(func_def_143, type, sP6: $o).
thf(func_def_144, type, sP7: $o).
thf(func_def_145, type, sP8: $o).
thf(func_def_146, type, sP9: ((subst > term > subst > $o) > $o)).
thf(func_def_147, type, sP10: $o).
thf(func_def_148, type, sP11: ((term > $o) > $o)).
thf(func_def_149, type, sP12: ((term > $o) > $o)).
thf(func_def_150, type, sP13: $o).
thf(func_def_151, type, sP14: $o).
thf(func_def_152, type, sP15: $o).
thf(func_def_153, type, sP16: $o).
thf(func_def_154, type, sP17: ((subst > term > subst > $o) > $o)).
thf(func_def_155, type, sP18: $o).
thf(func_def_156, type, sP19: $o).
thf(func_def_157, type, sP20: ((subst > term > subst > $o) > $o)).
thf(func_def_158, type, sP21: ((subst > term > subst > $o) > $o)).
thf(func_def_159, type, sP22: ((subst > term > subst > $o) > $o)).
thf(func_def_160, type, sP23: ((term > $o) > $o)).
thf(func_def_161, type, sP24: ((subst > term > subst > $o) > $o)).
thf(func_def_162, type, sP25: $o).
thf(func_def_163, type, sP26: $o).
thf(func_def_164, type, sP27: $o).
thf(func_def_165, type, sP28: $o).
thf(func_def_166, type, sP29: $o).
thf(func_def_167, type, sP30: $o).
thf(func_def_168, type, sP31: ((subst > term > subst > $o) > $o)).
thf(func_def_169, type, sP32: ((subst > term > subst > $o) > $o)).
thf(func_def_170, type, sP33: ((term > $o) > $o)).
thf(func_def_171, type, sP34: $o).
thf(func_def_172, type, sP35: $o).
thf(func_def_173, type, sP36: $o).
thf(func_def_174, type, sP37: $o).
thf(func_def_175, type, sP38: ((subst > term > subst > $o) > $o)).
thf(func_def_176, type, sP39: $o).
thf(func_def_177, type, sP40: ((subst > term > subst > $o) > $o)).
thf(func_def_178, type, sP41: ((subst > term > subst > $o) > $o)).
thf(func_def_179, type, sP42: $o).
thf(func_def_180, type, sP43: ((subst > term > subst > $o) > $o)).
thf(func_def_181, type, sP44: $o).
thf(func_def_182, type, sP45: ((subst > term > subst > $o) > $o)).
thf(func_def_183, type, sP46: ((subst > term > subst > $o) > $o)).
thf(func_def_184, type, sP47: $o).
thf(func_def_185, type, sP48: ((subst > term > subst > $o) > $o)).
thf(func_def_186, type, sP49: ((term > $o) > $o)).
thf(func_def_187, type, sP50: $o).
thf(func_def_188, type, sP51: $o).
thf(func_def_189, type, sP52: $o).
thf(func_def_190, type, sP53: ((term > $o) > $o)).
thf(func_def_191, type, sP54: ((term > $o) > $o)).
thf(func_def_192, type, sP55: $o).
thf(func_def_193, type, sP56: ((term > $o) > $o)).
thf(func_def_194, type, sP57: $o).
thf(func_def_195, type, sP58: $o).
thf(func_def_196, type, sP59: ((subst > term > subst > $o) > $o)).
thf(func_def_197, type, sP60: $o).
thf(func_def_198, type, sK61: term).
thf(func_def_199, type, sK62: subst).
thf(func_def_200, type, sK63: (subst > term > term)).
thf(func_def_201, type, sK64: (subst > term > term)).
thf(func_def_202, type, sK65: ((subst > term > term) > subst)).
thf(func_def_203, type, sK66: ((subst > term > term) > subst)).
thf(func_def_204, type, sK67: ((subst > term > term) > term)).
thf(func_def_205, type, sK68: ((subst > term > term) > subst)).
thf(func_def_206, type, sK69: ((subst > term > term) > term)).
thf(func_def_207, type, sK70: ((subst > term > term) > subst)).
thf(func_def_208, type, sK71: term).
thf(func_def_209, type, sK72: subst).
thf(func_def_210, type, sK73: term).
thf(func_def_211, type, sK74: term).
thf(func_def_212, type, sK75: term).
thf(func_def_213, type, sK76: subst).
thf(func_def_214, type, sK77: (term > $o)).
thf(func_def_215, type, sK78: term).
thf(func_def_216, type, sK79: (term > term)).
thf(func_def_217, type, sK80: ((term > $o) > term)).
thf(func_def_218, type, sK81: ((term > $o) > term)).
thf(func_def_219, type, sK82: ((term > $o) > term)).
thf(func_def_220, type, sK83: ((term > $o) > term)).
thf(func_def_221, type, sK84: (term > $o)).
thf(func_def_222, type, sK85: term).
thf(func_def_223, type, sK86: subst).
thf(func_def_224, type, sK87: (term > term)).
thf(func_def_225, type, sK88: ((term > $o) > term)).
thf(func_def_226, type, sK89: (subst > (term > $o) > term)).
thf(func_def_227, type, sK90: ((term > $o) > term)).
thf(func_def_228, type, sK91: ((term > $o) > term)).
thf(func_def_229, type, sK92: ((term > $o) > term)).
thf(func_def_230, type, sK93: ((term > $o) > term)).
thf(func_def_231, type, sK94: ((term > $o) > term)).
thf(func_def_232, type, sK95: ((term > $o) > term)).
thf(func_def_233, type, sK96: ((subst > term > subst > $o) > term > $o)).
thf(func_def_234, type, sK97: (term > $o)).
thf(func_def_235, type, sK98: term).
thf(func_def_236, type, sK99: (term > term)).
thf(func_def_237, type, sK100: term).
thf(func_def_238, type, sK101: (subst > term > subst > $o)).
thf(func_def_239, type, sK102: ((term > $o) > term)).
thf(func_def_240, type, sK103: (subst > term > subst > $o)).
thf(func_def_241, type, sK104: term).
thf(func_def_242, type, sK105: ((subst > term > term) > subst)).
thf(func_def_243, type, sK106: ((subst > term > term) > subst)).
thf(func_def_244, type, sK107: ((subst > term > term) > term)).
thf(func_def_245, type, sK108: ((subst > term > term) > term)).
thf(func_def_246, type, sK109: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_247, type, sK110: (term > (term > $o) > term)).
thf(func_def_248, type, sK111: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_249, type, sK112: ((subst > term > subst > $o) > (term > $o) > term)).
thf(func_def_250, type, sK113: (term > $o)).
thf(func_def_251, type, sK114: term).
thf(func_def_252, type, sK115: (subst > term > subst > $o)).
thf(func_def_253, type, sK116: ((subst > term > term) > term)).
thf(func_def_254, type, sK117: ((subst > term > term) > subst)).
thf(func_def_255, type, sK118: ((subst > term > term) > subst)).
thf(func_def_256, type, sK119: ((subst > term > term) > term)).
thf(func_def_257, type, sK120: ((term > $o) > term)).
thf(func_def_258, type, sK121: ((term > $o) > term)).
thf(func_def_259, type, sK122: ((subst > term > subst > $o) > term)).
thf(func_def_260, type, sK123: ((subst > term > subst > $o) > term)).
thf(func_def_261, type, sK124: term).
thf(func_def_262, type, sK125: (subst > term > subst > $o)).
thf(func_def_263, type, sK126: term).
thf(func_def_264, type, sK127: ((subst > term > subst > $o) > term)).
thf(func_def_265, type, sK128: ((subst > term > subst > $o) > subst)).
thf(func_def_266, type, sK129: ((subst > term > subst > $o) > subst)).
thf(func_def_267, type, sK130: ((subst > term > subst > $o) > subst)).
thf(func_def_268, type, sK131: ((subst > term > subst > $o) > term)).
thf(func_def_269, type, sK132: ((subst > term > subst > $o) > subst)).
thf(func_def_270, type, sK133: ((subst > term > subst > $o) > subst)).
thf(func_def_271, type, sK134: ((subst > term > subst > $o) > subst)).
thf(func_def_272, type, sK135: ((subst > term > subst > $o) > term)).
thf(func_def_273, type, sK136: ((subst > term > subst > $o) > term)).
thf(func_def_274, type, sK137: ((subst > term > term) > subst)).
thf(func_def_275, type, sK138: ((subst > term > term) > term)).
thf(func_def_276, type, sK139: ((subst > term > term) > subst)).
thf(func_def_277, type, sK140: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_278, type, sK141: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_279, type, sK142: (subst > term > subst > $o)).
thf(func_def_280, type, sK143: (term > $o)).
thf(func_def_281, type, sK144: term).
thf(func_def_282, type, sK145: term).
thf(func_def_283, type, sK146: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_284, type, sK147: (term > (term > $o) > term)).
thf(func_def_285, type, sK148: ((subst > term > subst > $o) > subst)).
thf(func_def_286, type, sK149: ((subst > term > subst > $o) > subst)).
thf(func_def_287, type, sK150: ((subst > term > subst > $o) > term)).
thf(func_def_288, type, sK151: ((subst > term > subst > $o) > subst)).
thf(func_def_289, type, sK152: ((subst > term > subst > $o) > subst)).
thf(func_def_290, type, sK153: ((subst > term > subst > $o) > subst)).
thf(func_def_291, type, sK154: ((subst > term > subst > $o) > subst)).
thf(func_def_292, type, sK155: ((subst > term > subst > $o) > term)).
thf(func_def_293, type, sK156: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_294, type, sK157: ((subst > term > term) > subst)).
thf(func_def_295, type, sK158: ((subst > term > term) > subst)).
thf(func_def_296, type, sK159: ((subst > term > term) > term)).
thf(func_def_297, type, sK160: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_298, type, sK161: term).
thf(func_def_299, type, sK162: (subst > term > subst > $o)).
thf(func_def_300, type, sK163: ((subst > term > subst > $o) > term)).
thf(func_def_301, type, sK164: ((subst > term > subst > $o) > subst)).
thf(func_def_302, type, sK165: ((subst > term > subst > $o) > term)).
thf(func_def_303, type, sK166: ((subst > term > subst > $o) > subst)).
thf(func_def_304, type, sK167: ((subst > term > subst > $o) > subst)).
thf(func_def_305, type, sK168: ((subst > term > subst > $o) > subst)).
thf(func_def_306, type, sK169: ((subst > term > subst > $o) > subst)).
thf(func_def_307, type, sK170: ((subst > term > subst > $o) > subst)).
thf(func_def_308, type, sK171: ((subst > term > subst > $o) > term)).
thf(func_def_309, type, sK172: ((subst > term > subst > $o) > term)).
thf(func_def_310, type, sK173: ((subst > term > subst > $o) > term)).
thf(func_def_311, type, sK174: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_312, type, sK175: ((subst > term > term) > term)).
thf(func_def_313, type, sK176: ((subst > term > term) > subst)).
thf(func_def_314, type, sK177: ((subst > term > term) > subst)).
thf(func_def_315, type, sK178: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_316, type, sK179: term).
thf(func_def_317, type, sK180: term).
thf(func_def_318, type, sK181: (subst > term > subst > $o)).
thf(func_def_319, type, sK182: (term > $o)).
thf(func_def_320, type, sK183: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_321, type, sK184: ((subst > term > subst > $o) > subst)).
thf(func_def_322, type, sK185: ((subst > term > subst > $o) > term)).
thf(func_def_323, type, sK186: ((subst > term > subst > $o) > subst)).
thf(func_def_324, type, sK187: ((subst > term > subst > $o) > subst)).
thf(func_def_325, type, sK188: ((subst > term > subst > $o) > subst)).
thf(func_def_326, type, sK189: ((subst > term > subst > $o) > subst)).
thf(func_def_327, type, sK190: ((subst > term > subst > $o) > subst)).
thf(func_def_328, type, sK191: ((subst > term > subst > $o) > term)).
thf(func_def_329, type, sK192: ((subst > term > subst > $o) > term)).
thf(func_def_330, type, sK193: ((subst > term > subst > $o) > term)).
thf(func_def_331, type, sK194: (subst > term > subst > $o)).
thf(func_def_332, type, sK195: (term > $o)).
thf(func_def_333, type, sK196: term).
thf(func_def_334, type, sK197: ((subst > term > term) > term)).
thf(func_def_335, type, sK198: ((subst > term > term) > subst)).
thf(func_def_336, type, sK199: ((subst > term > term) > subst)).
thf(func_def_337, type, sK200: ((subst > term > term) > term)).
thf(func_def_338, type, sK201: (term > subst > term > term)).
thf(func_def_339, type, sK202: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_340, type, sK203: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_341, type, sK204: (term > (term > $o) > term)).
thf(func_def_342, type, sK205: term).
thf(func_def_343, type, sK206: ((subst > term > term) > term)).
thf(func_def_344, type, sK207: ((subst > term > term) > subst)).
thf(func_def_345, type, sK208: term).
thf(func_def_346, type, sK209: ((term > $o) > term)).
thf(func_def_347, type, sK210: ((term > $o) > term)).
thf(func_def_348, type, sK211: ((subst > term > subst > $o) > term > $o)).
thf(func_def_349, type, sK212: term).
thf(func_def_350, type, sK213: term).
thf(func_def_351, type, sK214: (term > $o)).
thf(func_def_352, type, sK215: (term > term)).
thf(func_def_353, type, sK216: (subst > term > subst > $o)).
thf(func_def_354, type, sK217: ((term > $o) > term)).
thf(func_def_355, type, sK218: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_356, type, sK219: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_357, type, sK220: ((term > $o) > term > term)).
thf(func_def_358, type, sK221: term).
thf(func_def_359, type, sK222: (subst > term > subst > $o)).
thf(func_def_360, type, sK223: (term > $o)).
thf(func_def_361, type, sK224: ((subst > term > term) > term)).
thf(func_def_362, type, sK225: ((subst > term > term) > subst)).
thf(func_def_363, type, sK226: ((subst > term > term) > subst)).
thf(func_def_364, type, sK227: ((subst > term > term) > term)).
thf(func_def_365, type, sK228: (subst > term > subst > $o)).
thf(func_def_366, type, sK229: term).
thf(func_def_367, type, sK230: ((subst > term > subst > $o) > term)).
thf(func_def_368, type, sK231: ((subst > term > subst > $o) > subst)).
thf(func_def_369, type, sK232: ((subst > term > subst > $o) > subst)).
thf(func_def_370, type, sK233: ((subst > term > subst > $o) > subst)).
thf(func_def_371, type, sK234: ((subst > term > subst > $o) > term)).
thf(func_def_372, type, sK235: ((subst > term > subst > $o) > subst)).
thf(func_def_373, type, sK236: ((subst > term > subst > $o) > subst)).
thf(func_def_374, type, sK237: ((subst > term > subst > $o) > subst)).
thf(func_def_375, type, sK238: ((subst > term > subst > $o) > term)).
thf(func_def_376, type, sK239: ((term > $o) > term)).
thf(func_def_377, type, sK240: ((term > $o) > term)).
thf(func_def_378, type, sK241: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_379, type, sK242: ((subst > term > term) > term)).
thf(func_def_380, type, sK243: ((subst > term > term) > subst)).
thf(func_def_381, type, sK244: ((subst > term > term) > subst)).
thf(func_def_382, type, sK245: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_383, type, sK246: ((subst > term > subst > $o) > term)).
thf(func_def_384, type, sK247: ((subst > term > subst > $o) > term)).
thf(func_def_385, type, sK248: (subst > term > subst > $o)).
thf(func_def_386, type, sK249: term).
thf(func_def_387, type, sK250: ((subst > term > term) > term)).
thf(func_def_388, type, sK251: ((subst > term > term) > subst)).
thf(func_def_389, type, sK252: ((subst > term > term) > subst)).
thf(func_def_390, type, sK253: ((subst > term > term) > term)).
thf(func_def_391, type, sK254: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_392, type, sK255: (term > (subst > term > subst > $o) > term)).
thf(func_def_393, type, sK256: (subst > term > subst > $o)).
thf(func_def_394, type, sK257: term).
thf(func_def_395, type, sK258: ((subst > term > term) > term)).
thf(func_def_396, type, sK259: ((subst > term > term) > subst)).
thf(func_def_397, type, sK260: ((subst > term > term) > subst)).
thf(func_def_398, type, sK261: ((subst > term > term) > term)).
thf(func_def_399, type, sK262: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_400, type, sK263: (term > (subst > term > subst > $o) > term)).
thf(func_def_401, type, sK264: ((subst > term > subst > $o) > term)).
thf(func_def_402, type, sK265: ((subst > term > subst > $o) > subst)).
thf(func_def_403, type, sK266: ((subst > term > subst > $o) > subst)).
thf(func_def_404, type, sK267: ((subst > term > subst > $o) > subst)).
thf(func_def_405, type, sK268: ((subst > term > subst > $o) > term)).
thf(func_def_406, type, sK269: ((subst > term > subst > $o) > subst)).
thf(func_def_407, type, sK270: ((subst > term > subst > $o) > subst)).
thf(func_def_408, type, sK271: ((subst > term > subst > $o) > subst)).
thf(func_def_409, type, sK272: (subst > term > subst > $o)).
thf(func_def_410, type, sK273: term).
thf(func_def_411, type, sK274: term).
thf(func_def_412, type, sK275: ((term > $o) > term)).
thf(func_def_413, type, sK276: ((term > $o) > term)).
thf(func_def_414, type, sK277: term).
thf(func_def_415, type, sK278: (term > $o)).
thf(func_def_416, type, sK279: (term > term)).
thf(func_def_417, type, sK280: term).
thf(func_def_418, type, sK281: (subst > term > subst > $o)).
thf(func_def_419, type, sK282: ((subst > term > subst > $o) > term)).
thf(func_def_420, type, sK283: ((subst > term > subst > $o) > subst)).
thf(func_def_421, type, sK284: ((subst > term > subst > $o) > subst)).
thf(func_def_422, type, sK285: ((subst > term > subst > $o) > term)).
thf(func_def_423, type, sK286: ((subst > term > subst > $o) > subst)).
thf(func_def_424, type, sK287: ((subst > term > subst > $o) > subst)).
thf(func_def_425, type, sK288: ((subst > term > subst > $o) > subst)).
thf(func_def_426, type, sK289: ((subst > term > subst > $o) > term)).
thf(func_def_427, type, sK290: ((subst > term > subst > $o) > subst)).
thf(func_def_428, type, sK291: ((subst > term > subst > $o) > subst)).
thf(func_def_429, type, sK292: ((subst > term > subst > $o) > subst)).
thf(func_def_430, type, sK293: ((subst > term > subst > $o) > term)).
thf(func_def_431, type, sK294: ((subst > term > subst > $o) > subst)).
thf(func_def_432, type, sK295: ((subst > term > subst > $o) > term > term)).
thf(func_def_433, type, sK296: ((subst > term > subst > $o) > subst)).
thf(func_def_434, type, sK297: ((subst > term > subst > $o) > subst)).
thf(func_def_435, type, sK298: ((subst > term > subst > $o) > term)).
thf(func_def_436, type, sK299: ((subst > term > subst > $o) > subst)).
thf(func_def_437, type, sK300: term).
thf(func_def_438, type, sK301: (subst > term > subst > $o)).
thf(func_def_439, type, sK302: ((subst > term > subst > $o) > term)).
thf(func_def_440, type, sK303: ((subst > term > subst > $o) > term)).
thf(func_def_441, type, sK304: ((term > $o) > term)).
thf(func_def_442, type, sK305: ((term > $o) > term)).
thf(func_def_443, type, sK306: ((subst > term > term) > subst)).
thf(func_def_444, type, sK307: ((subst > term > term) > subst)).
thf(func_def_445, type, sK308: ((subst > term > term) > term)).
thf(func_def_446, type, sK309: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_447, type, sK310: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_448, type, sK311: ((subst > term > subst > $o) > term)).
thf(func_def_449, type, sK312: ((subst > term > subst > $o) > term)).
thf(func_def_450, type, sK313: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_451, type, sK314: ((subst > term > term) > subst)).
thf(func_def_452, type, sK315: ((subst > term > term) > term)).
thf(func_def_453, type, sK316: ((subst > term > term) > subst)).
thf(func_def_454, type, sK317: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_455, type, sK318: term).
thf(func_def_456, type, sK319: (subst > term > subst > $o)).
thf(func_def_457, type, sK320: ((subst > term > subst > $o) > term)).
thf(func_def_458, type, sK321: ((subst > term > subst > $o) > subst)).
thf(func_def_459, type, sK322: ((subst > term > subst > $o) > subst)).
thf(func_def_460, type, sK323: ((subst > term > subst > $o) > subst)).
thf(func_def_461, type, sK324: ((subst > term > subst > $o) > term > term)).
thf(func_def_462, type, sK325: ((subst > term > subst > $o) > term)).
thf(func_def_463, type, sK326: ((subst > term > subst > $o) > subst)).
thf(func_def_464, type, sK327: ((subst > term > subst > $o) > subst)).
thf(func_def_465, type, sK328: ((subst > term > subst > $o) > subst)).
thf(func_def_466, type, sK329: term).
thf(func_def_467, type, sK330: (term > (subst > term > subst > $o) > term)).
thf(func_def_468, type, sK331: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_469, type, sK332: (subst > term > subst > $o)).
thf(func_def_470, type, sK333: term).
thf(func_def_471, type, sK334: ((subst > term > term) > term)).
thf(func_def_472, type, sK335: ((subst > term > term) > subst)).
thf(func_def_473, type, sK336: ((subst > term > term) > term)).
thf(func_def_474, type, sK337: ((subst > term > term) > subst)).
thf(func_def_475, type, sK338: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_476, type, sK339: ((subst > term > term) > term)).
thf(func_def_477, type, sK340: ((subst > term > term) > subst)).
thf(func_def_478, type, sK341: ((subst > term > term) > subst)).
thf(func_def_479, type, sK342: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_480, type, sK343: term).
thf(func_def_481, type, sK344: (subst > term > subst > $o)).
thf(func_def_482, type, sK345: ((subst > term > term) > term)).
thf(func_def_483, type, sK346: ((subst > term > term) > subst)).
thf(func_def_484, type, sK347: ((subst > term > term) > subst)).
thf(func_def_485, type, sK348: ((subst > term > term) > term)).
thf(func_def_486, type, sK349: ((subst > term > subst > $o) > term > term)).
thf(func_def_487, type, sK350: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_488, type, sK351: term).
thf(func_def_489, type, sK352: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_490, type, sK353: (term > (subst > term > subst > $o) > term)).
thf(func_def_491, type, sK354: (subst > term > subst > $o)).
thf(func_def_492, type, sK355: term).
thf(func_def_493, type, sK356: ((subst > term > term) > term)).
thf(func_def_494, type, sK357: ((subst > term > term) > subst)).
thf(func_def_495, type, sK358: ((subst > term > term) > subst)).
thf(func_def_496, type, sK359: ((subst > term > term) > term)).
thf(func_def_497, type, sK360: ((term > $o) > subst > term)).
thf(func_def_498, type, sK361: ((term > $o) > term)).
thf(func_def_499, type, sK362: term).
thf(func_def_500, type, sK363: subst).
thf(func_def_501, type, sK364: (term > $o)).
thf(func_def_502, type, sK365: term).
thf(func_def_503, type, sK366: (term > term)).
thf(func_def_504, type, sK367: term).
thf(func_def_505, type, sK368: (term > $o)).
thf(func_def_506, type, sK369: (term > term)).
thf(func_def_507, type, sK370: ((term > $o) > term)).
thf(func_def_508, type, sK371: ((term > $o) > term)).
thf(func_def_509, type, sK372: ((term > $o) > term)).
thf(func_def_510, type, sK373: ((term > $o) > term)).
thf(func_def_511, type, sK374: ((term > $o) > term)).
thf(func_def_512, type, sK375: ((term > $o) > term)).
thf(func_def_513, type, sK376: term).
thf(func_def_514, type, sK377: (subst > term > subst > $o)).
thf(func_def_515, type, sK378: ((subst > term > subst > $o) > subst)).
thf(func_def_516, type, sK379: ((subst > term > subst > $o) > subst)).
thf(func_def_517, type, sK380: ((subst > term > subst > $o) > term)).
thf(func_def_518, type, sK381: ((subst > term > subst > $o) > subst)).
thf(func_def_519, type, sK382: ((subst > term > subst > $o) > subst)).
thf(func_def_520, type, sK383: ((subst > term > subst > $o) > term)).
thf(func_def_521, type, sK384: ((subst > term > subst > $o) > subst)).
thf(func_def_522, type, sK385: ((subst > term > subst > $o) > subst)).
thf(func_def_523, type, sK386: ((subst > term > subst > $o) > term > term)).
thf(func_def_524, type, sK387: ((subst > term > term) > term)).
thf(func_def_525, type, sK388: ((subst > term > term) > subst)).
thf(func_def_526, type, sK389: ((subst > term > term) > subst)).
thf(func_def_527, type, sK390: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_528, type, sK391: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_529, type, sK392: term).
thf(func_def_530, type, sK393: (subst > term > subst > $o)).
thf(func_def_531, type, sK394: ((subst > term > term) > term)).
thf(func_def_532, type, sK395: (term > (subst > term > term) > term)).
thf(func_def_533, type, sK396: (term > (subst > term > term) > subst)).
thf(func_def_534, type, sK397: (term > (subst > term > term) > subst)).
thf(func_def_535, type, sK398: (term > (subst > term > term) > subst)).
thf(func_def_536, type, sK399: (term > (subst > term > term) > term)).
thf(func_def_537, type, sK400: (term > (subst > term > term) > subst)).
thf(func_def_538, type, sK401: ((subst > term > subst > $o) > term > term)).
thf(func_def_539, type, sK402: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_540, type, sK403: ((subst > term > subst > $o) > term)).
thf(func_def_541, type, sK404: ((term > $o) > term)).
thf(func_def_542, type, sK405: ((term > $o) > term)).
thf(func_def_543, type, sK406: term).
thf(func_def_544, type, sK407: subst).
thf(func_def_545, type, sK408: subst).
thf(func_def_546, type, sK409: (term > $o)).
thf(func_def_547, type, sK410: term).
thf(func_def_548, type, sK411: term).
thf(func_def_549, type, sK412: subst).
thf(func_def_550, type, sK413: subst).
thf(func_def_551, type, sK414: (term > $o)).
thf(func_def_552, type, sK415: term).
thf(func_def_553, type, sK416: (term > term)).
thf(func_def_554, type, sK417: ((term > $o) > subst > term)).
thf(func_def_555, type, sK418: ((term > $o) > term)).
thf(func_def_556, type, sK419: ((term > $o) > term)).
thf(func_def_557, type, sK420: ((term > $o) > term)).
thf(func_def_558, type, sK421: ((term > $o) > term)).
thf(func_def_559, type, sK422: ((term > $o) > term)).
thf(func_def_560, type, sK423: (term > $o)).
thf(func_def_561, type, sK424: (subst > term > subst > $o)).
thf(func_def_562, type, sK425: term).
thf(func_def_563, type, sK426: ((subst > term > term) > subst)).
thf(func_def_564, type, sK427: ((subst > term > term) > term)).
thf(func_def_565, type, sK428: ((subst > term > term) > subst)).
thf(func_def_566, type, sK429: ((subst > term > term) > term)).
thf(func_def_567, type, sK430: ((subst > term > subst > $o) > (term > $o) > term)).
thf(func_def_568, type, sK431: (term > (term > $o) > term)).
thf(func_def_569, type, sK432: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_570, type, sK433: term).
thf(func_def_571, type, sK434: (subst > term > subst > $o)).
thf(func_def_572, type, sK435: ((subst > term > term) > term)).
thf(func_def_573, type, sK436: ((subst > term > term) > term)).
thf(func_def_574, type, sK437: ((subst > term > term) > subst)).
thf(func_def_575, type, sK438: ((subst > term > term) > subst)).
thf(func_def_576, type, sK439: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_577, type, sK440: ((subst > term > subst > $o) > term > term)).
thf(func_def_578, type, sK441: (subst > term > subst > $o)).
thf(func_def_579, type, sK442: (term > $o)).
thf(func_def_580, type, sK443: term).
thf(func_def_581, type, sK444: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_582, type, sK445: (term > (term > $o) > term)).
thf(func_def_583, type, sK446: ((subst > term > subst > $o) > subst)).
thf(func_def_584, type, sK447: ((subst > term > subst > $o) > term)).
thf(func_def_585, type, sK448: ((subst > term > subst > $o) > subst)).
thf(func_def_586, type, sK449: ((subst > term > subst > $o) > subst)).
thf(func_def_587, type, sK450: ((subst > term > subst > $o) > subst)).
thf(func_def_588, type, sK451: ((subst > term > subst > $o) > subst)).
thf(func_def_589, type, sK452: ((subst > term > subst > $o) > term)).
thf(func_def_590, type, sK453: ((subst > term > subst > $o) > subst)).
thf(func_def_591, type, sK454: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_592, type, sK455: ((subst > term > term) > subst)).
thf(func_def_593, type, sK456: ((subst > term > term) > term)).
thf(func_def_594, type, sK457: ((subst > term > term) > subst)).
thf(func_def_595, type, sK458: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_596, type, sK459: subst).
thf(func_def_597, type, sK460: term).
thf(func_def_598, type, sK461: (term > d_term)).
thf(func_def_599, type, sK462: subst).
thf(func_def_600, type, sK463: (term > $o)).
thf(func_def_601, type, sK464: term).
thf(func_def_602, type, sK465: ((term > $o) > term)).
thf(func_def_603, type, sK466: (term > (term > $o) > subst > term > $o)).
thf(func_def_604, type, sK467: (term > subst > term > term)).
thf(func_def_605, type, sK468: term).
thf(func_def_606, type, sK469: ((subst > term > term) > term)).
thf(func_def_607, type, sK470: ((subst > term > term) > subst)).
thf(func_def_608, type, sK471: ((subst > term > subst > $o) > term > $o)).
thf(func_def_609, type, sK472: (subst > term > subst > $o)).
thf(func_def_610, type, sK473: ((term > $o) > term)).
thf(func_def_611, type, sK474: ((term > $o) > term)).
thf(func_def_612, type, sK475: ((term > $o) > term)).
thf(func_def_613, type, sK476: term).
thf(func_def_614, type, sK477: (term > $o)).
thf(func_def_615, type, sK478: term).
thf(func_def_616, type, sK479: term).
thf(func_def_617, type, sK480: (subst > term > term)).
thf(func_def_618, type, sK481: ((subst > term > term) > subst)).
thf(func_def_619, type, sK482: ((subst > term > term) > subst)).
thf(func_def_620, type, sK483: ((subst > term > term) > term)).
thf(func_def_621, type, sK484: term).
thf(func_def_622, type, sK485: term).
thf(func_def_623, type, sK486: term).
thf(func_def_624, type, sK487: term).
thf(func_def_625, type, sK488: term).
thf(func_def_626, type, sK489: term).
thf(func_def_627, type, sK490: term).
thf(func_def_628, type, sK491: term).
thf(func_def_629, type, sK492: term).
thf(func_def_630, type, sK493: ((subst > term > term) > term)).
thf(func_def_631, type, sK494: ((subst > term > term) > subst)).
thf(func_def_632, type, sK495: ((subst > term > term) > subst)).
thf(func_def_633, type, sK496: ((subst > term > term) > subst)).
thf(func_def_634, type, sK497: ((subst > term > term) > subst)).
thf(func_def_635, type, sK498: ((subst > term > term) > term)).
thf(func_def_636, type, sK499: term).
thf(func_def_637, type, sK500: subst).
thf(func_def_638, type, sK501: (subst > term > term)).
thf(func_def_639, type, sK502: (subst > term > term)).
thf(func_def_640, type, sK503: (subst > term > subst > $o)).
thf(func_def_641, type, sK504: term).
thf(func_def_642, type, sK505: ((subst > term > subst > $o) > term)).
thf(func_def_643, type, sK506: ((subst > term > subst > $o) > term)).
thf(func_def_644, type, sK507: ((subst > term > subst > $o) > term > $o)).
thf(func_def_645, type, sK508: (subst > term > subst > $o)).
thf(func_def_646, type, sK509: ((term > $o) > term)).
thf(func_def_647, type, sK510: (term > $o)).
thf(func_def_648, type, sK511: (term > $o)).
thf(func_def_649, type, sK512: term).
thf(func_def_650, type, sK513: subst).
thf(func_def_651, type, sK514: term).
thf(func_def_652, type, sK515: subst).
thf(func_def_653, type, sK516: (subst > term > (term > $o) > (term > $o) > term)).
thf(func_def_654, type, sK517: term).
thf(func_def_655, type, sK518: term).
thf(func_def_656, type, sK519: term).
thf(func_def_657, type, sK520: term).
thf(func_def_658, type, sK521: term).
thf(func_def_659, type, sK522: term).
thf(func_def_660, type, sK523: term).
thf(func_def_661, type, sK524: term).
thf(func_def_662, type, sK525: term).
thf(func_def_663, type, sK526: subst).
thf(func_def_664, type, sK527: (term > subst > term > term)).
thf(func_def_665, type, sK528: term).
thf(func_def_666, type, sK529: term).
thf(func_def_667, type, sK530: term).
thf(func_def_668, type, sK531: term).
thf(func_def_669, type, sK532: term).
thf(func_def_670, type, sK533: term).
thf(func_def_671, type, sK534: term).
thf(func_def_672, type, sK535: term).
thf(func_def_673, type, sK536: term).
thf(func_def_674, type, sK537: subst).
thf(func_def_675, type, sK538: (term > $o)).
thf(func_def_676, type, sK539: (term > $o)).
thf(func_def_677, type, sK540: subst).
thf(func_def_678, type, sK541: term).
thf(func_def_679, type, sK542: (term > subst > (term > $o) > (term > $o) > term)).
thf(func_def_680, type, sK543: ((term > $o) > term > subst > term > $o)).
thf(func_def_681, type, sK544: subst).
thf(func_def_682, type, sK545: term).
thf(func_def_683, type, sK546: (term > $o)).
thf(func_def_684, type, sK547: ((term > $o) > term)).
thf(func_def_685, type, sK548: term).
thf(func_def_686, type, sK549: term).
thf(func_def_687, type, sK550: (term > $o)).
thf(func_def_688, type, sK551: (term > $o)).
thf(func_def_689, type, sK552: subst).
thf(func_def_690, type, sK553: term).
thf(func_def_691, type, sK554: subst).
thf(func_def_692, type, sK555: term).
thf(func_def_693, type, sK556: term).
thf(func_def_694, type, sK557: subst).
thf(func_def_695, type, sK558: term).
thf(func_def_696, type, sK559: term).
thf(func_def_697, type, sK560: term).
thf(func_def_698, type, sK561: term).
thf(func_def_699, type, sK562: term).
thf(func_def_700, type, sK563: subst).
thf(func_def_701, type, sK564: term).
thf(func_def_702, type, sK565: term).
thf(func_def_703, type, sK566: term).
thf(func_def_704, type, sK567: term).
thf(func_def_705, type, sK568: (subst > d_subst)).
thf(func_def_706, type, sK569: term).
thf(func_def_707, type, sK570: term).
thf(func_def_708, type, sK571: subst).
thf(func_def_709, type, sK572: term).
thf(func_def_710, type, sK573: term).
thf(func_def_711, type, sK574: term).
thf(func_def_712, type, sK575: ((subst > term > term) > term)).
thf(func_def_713, type, sK576: ((subst > term > term) > subst)).
thf(func_def_714, type, sK577: (term > subst > term > term)).
thf(func_def_715, type, sK578: term).
thf(func_def_716, type, sK579: term).
thf(func_def_717, type, sK580: subst).
thf(func_def_718, type, sK581: subst).
thf(func_def_719, type, sK582: (term > $o)).
thf(func_def_720, type, sK583: (term > $o)).
thf(func_def_721, type, sK584: subst).
thf(func_def_722, type, sK585: term).
thf(func_def_723, type, sK586: term).
thf(func_def_724, type, sK587: subst).
thf(func_def_725, type, sK588: subst).
thf(func_def_726, type, sK589: subst).
thf(func_def_727, type, sK590: subst).
thf(func_def_728, type, sK591: (term > subst > term > term)).
thf(func_def_729, type, sK592: term).
thf(func_def_730, type, sK593: (term > $o)).
thf(func_def_731, type, sK594: (subst > term > subst > $o)).
thf(func_def_732, type, sK595: term).
thf(func_def_733, type, sK596: ((subst > term > subst > $o) > term)).
thf(func_def_734, type, sK597: ((subst > term > subst > $o) > subst)).
thf(func_def_735, type, sK598: ((subst > term > subst > $o) > subst)).
thf(func_def_736, type, sK599: ((subst > term > subst > $o) > subst)).
thf(func_def_737, type, sK600: (term > (term > $o) > term)).
thf(func_def_738, type, sK601: ((subst > term > subst > $o) > subst)).
thf(func_def_739, type, sK602: ((subst > term > subst > $o) > subst)).
thf(func_def_740, type, sK603: ((subst > term > subst > $o) > term)).
thf(func_def_741, type, sK604: ((subst > term > subst > $o) > subst)).
thf(func_def_742, type, sK605: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_743, type, sK606: ((subst > term > term) > subst)).
thf(func_def_744, type, sK607: ((subst > term > term) > subst)).
thf(func_def_745, type, sK608: ((subst > term > term) > term)).
thf(func_def_746, type, sK609: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_747, type, vNOT: ($o > $o)).
thf(func_def_748, type, sK610: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_749, type, sK611: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_750, type, sK612: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_751, type, sK613: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_752, type, sK614: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_753, type, sK615: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_754, type, sK616: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_755, type, sK617: ((term > $o) > (subst > term > subst > $o) > term)).
thf(f1,axiom,(
  axclos & pushprop_lem2v2_lthm & hoasinduction_lem1v2_gthm & hoasinduction_lem2v2_gthm & (hoasinduction_lem3v2_f <=> ! [X9 : term] : ~ ! [X10 : (subst > term > term)] : ~ ! [X4 : term,X5 : subst] : (((sub @ X9 @ (push @ X4 @ X5))) = ((X10 @ X5 @ X4)))) & ! [X46 : subst] : ? [X45 : d_subst] : (X46 = ((d2subst @ X45))) & (hoasinduction_lthm_3 <=> (! [X6 : (subst > term > subst > $o)] : ~ ! [X7 : (term > $o)] : ~ ! [X8 : term] : (((X7 @ X8)) = ((X6 @ id @ X8 @ id))) => (! [X6 : (term > $o),X35 : term] : (~ ! [X4 : term,X9 : term] : (~(X6 @ X4) | ~(X6 @ X9) | (X6 @ (ap @ X4 @ X9))) | ~ ! [X4 : term] : ((X6 @ X4) | ~(var @ X4)) | (X6 @ X35) | ~ ! [X4 : term] : (~ ! [X9 : term] : (~(X6 @ X9) | (X6 @ (sub @ X4 @ (push @ X9 @ id)))) | (X6 @ (lam @ X4)))) => (! [X4 : term] : (X4 = ((sub @ X4 @ id))) => (! [X34 : term,X7 : (term > $o),X6 : (subst > term > subst > $o)] : (~ ! [X10 : (subst > term > term)] : (~ ! [X4 : term] : (~(X6 @ id @ X4 @ id) | (X6 @ id @ (X10 @ id @ X4) @ id)) | (X6 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X4 : term,X5 : subst,X19 : subst] : (((sub @ (X10 @ X5 @ X4) @ X19)) = ((X10 @ (comp @ X5 @ X19) @ (sub @ X4 @ X19))))) | (X7 @ (lam @ X34)) | ~ ! [X9 : term] : ((X7 @ (sub @ X34 @ (push @ X9 @ id))) | ~(X7 @ X9)) | ~ ! [X8 : term] : (((X7 @ X8)) = ((X6 @ id @ X8 @ id)))) => ! [X38 : term,X6 : (subst > term > subst > $o)] : (~ ! [X9 : term,X4 : term] : (~(X6 @ id @ X4 @ id) | ~(X6 @ id @ X9 @ id) | (X6 @ id @ (ap @ (sub @ X4 @ id) @ X9) @ id)) | ~ ! [X10 : (subst > term > term)] : ((X6 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X4 : term] : (~(X6 @ id @ X4 @ id) | (X6 @ id @ (X10 @ id @ X4) @ id)) | ~ ! [X19 : subst,X4 : term,X5 : subst] : (((sub @ (X10 @ X5 @ X4) @ X19)) = ((X10 @ (comp @ X5 @ X19) @ (sub @ X4 @ X19))))) | ~ ! [X4 : term,X5 : subst,X19 : subst,X30 : subst] : (~(X6 @ (comp @ X5 @ X30) @ (sub @ X4 @ X30) @ X19) | (X6 @ X5 @ X4 @ (comp @ X30 @ X19))) | ~ ! [X30 : subst,X4 : term,X19 : subst,X5 : subst] : ((X6 @ (comp @ X5 @ X30) @ (sub @ X4 @ X30) @ X19) | ~(X6 @ X5 @ X4 @ (comp @ X30 @ X19))) | ~ ! [X4 : term] : (~(var @ (sub @ X4 @ id)) | (X6 @ id @ X4 @ id)) | (X6 @ id @ X38 @ id))))))) & hoaslamnotvar_gthm & ((^[X6 : (subst > term > subst > $o), X7 : (term > $o)] : (! [X8 : term] : (((X7 @ X8)) = ((X6 @ id @ X8 @ id))))) = hoasinduction_p_and_p_prime) & axmap & (! [X4 : term,X9 : term] : ((X4 = X9) | (((sub @ X4 @ sh)) != ((sub @ X9 @ sh)))) <=> shinj) & (! [X11 : term,X4 : term,X9 : term,X12 : term] : ((X11 = X12) | (((ap @ (sub @ X4 @ id) @ X11)) != ((ap @ (sub @ X9 @ id) @ X12)))) <=> hoasapinj2) & pushprop & pushprop_lem0_gthm & hoaslamnotvar & (! [X5 : subst,X4 : term] : (X5 = ((comp @ sh @ (push @ X4 @ X5)))) <=> axshiftcons) & substmonoid_gthm & pushprop_lem2v2_gthm & (hoasinduction_lem3v2 <=> ! [X39 : term,X6 : (subst > term > subst > $o),X7 : (term > $o)] : (~ ! [X10 : (subst > term > term)] : (~ ! [X4 : term] : (~(X6 @ id @ X4 @ id) | (X6 @ id @ (X10 @ id @ X4) @ id)) | ~ ! [X19 : subst,X5 : subst,X4 : term] : (((sub @ (X10 @ X5 @ X4) @ X19)) = ((X10 @ (comp @ X5 @ X19) @ (sub @ X4 @ X19)))) | (X6 @ id @ (lam @ (X10 @ sh @ one)) @ id)) | ~ ! [X5 : subst,X19 : subst,X30 : subst,X4 : term] : (~(X6 @ X5 @ X4 @ (comp @ X30 @ X19)) | (X6 @ (comp @ X5 @ X30) @ (sub @ X4 @ X30) @ X19)) | (X7 @ (lam @ X39)) | ~ ! [X4 : term,X19 : subst,X30 : subst,X5 : subst] : (~(X6 @ (comp @ X5 @ X30) @ (sub @ X4 @ X30) @ X19) | (X6 @ X5 @ X4 @ (comp @ X30 @ X19))) | ~ ! [X9 : term] : (~(X7 @ X9) | (X7 @ (sub @ X39 @ (push @ X9 @ id)))) | ~ ! [X8 : term] : (((X7 @ X8)) = ((X6 @ id @ X8 @ id))))) & (((lam @ (d2term @ d_one))) = ((d2term @ d_one))) & ulamvarsh & ((! [X4 : term] : (X4 = ((sub @ X4 @ id))) => ! [X39 : term,X6 : (subst > term > subst > $o),X7 : (term > $o)] : ((X7 @ (lam @ X39)) | ~ ! [X9 : term] : (~(X7 @ X9) | (X7 @ (sub @ X39 @ (push @ X9 @ id)))) | ~ ! [X4 : term,X5 : subst,X30 : subst,X19 : subst] : ((X6 @ (comp @ X5 @ X30) @ (sub @ X4 @ X30) @ X19) | ~(X6 @ X5 @ X4 @ (comp @ X30 @ X19))) | ~ ! [X10 : (subst > term > term)] : (~ ! [X4 : term] : (~(X6 @ id @ X4 @ id) | (X6 @ id @ (X10 @ id @ X4) @ id)) | ~ ! [X19 : subst,X4 : term,X5 : subst] : (((sub @ (X10 @ X5 @ X4) @ X19)) = ((X10 @ (comp @ X5 @ X19) @ (sub @ X4 @ X19)))) | (X6 @ id @ (lam @ (X10 @ sh @ one)) @ id)) | ~ ! [X5 : subst,X19 : subst,X30 : subst,X4 : term] : (~(X6 @ (comp @ X5 @ X30) @ (sub @ X4 @ X30) @ X19) | (X6 @ X5 @ X4 @ (comp @ X30 @ X19))) | ~ ! [X8 : term] : (((X7 @ X8)) = ((X6 @ id @ X8 @ id))))) <=> hoasinduction_lem3v2_lthm) & (hoasinduction_lem3v2_f_lthm <=> ! [X9 : term] : ~ ! [X10 : (subst > term > term)] : ~ ! [X4 : term,X5 : subst] : (((sub @ X9 @ (push @ X4 @ X5))) = ((X10 @ X5 @ X4)))) & ~ulamvar1 & ulamvarind & hoasinduction_lem1 & pushprop_lem2v2 & hoasinduction_lem3a_gthm & hoasapinj1_gthm & (hoaslaminj_lthm <=> (! [X4 : term,X5 : subst] : (X4 = ((sub @ one @ (push @ X4 @ X5)))) => (! [X4 : term,X5 : subst] : (X5 = ((comp @ sh @ (push @ X4 @ X5)))) => (! [X4 : term,X9 : term] : ((((lam @ X4)) != ((lam @ X9))) | (X4 = X9)) => ! [X18 : term,X10 : (subst > term > term),X16 : (subst > term > term),X17 : subst] : (~ ! [X4 : term,X5 : subst,X19 : subst] : (((sub @ (X16 @ X5 @ X4) @ X19)) = ((X16 @ (comp @ X5 @ X19) @ (sub @ X4 @ X19)))) | (((X10 @ X17 @ X18)) = ((X16 @ X17 @ X18))) | (((lam @ (X10 @ sh @ one))) != ((lam @ (X16 @ sh @ one)))) | ~ ! [X19 : subst,X5 : subst,X4 : term] : (((sub @ (X10 @ X5 @ X4) @ X19)) = ((X10 @ (comp @ X5 @ X19) @ (sub @ X4 @ X19))))))))) & ~(hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) & ((^[X4 : term, X5 : subst, X6 : (term > $o), X7 : (term > $o)] : (! [X8 : term] : (((X7 @ X8)) = ((X6 @ (sub @ X8 @ (push @ X4 @ X5))))))) = pushprop_p_and_p_prime) & hoaslamnotap_lthm & (! [X4 : term] : (X4 = ((sub @ X4 @ id))) <=> termmset) & pushprop_gthm & (id = ((d2subst @ d_id))) & ((! [X4 : term] : (X4 = ((sub @ X4 @ id))) => (! [X6 : (subst > term > subst > $o),X31 : term] : ((X6 @ id @ (lam @ (sub @ X31 @ (push @ one @ sh))) @ id) | ~ ! [X10 : (subst > term > term)] : (~ ! [X4 : term] : (~(X6 @ id @ X4 @ id) | (X6 @ id @ (X10 @ id @ X4) @ id)) | (X6 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X19 : subst,X4 : term,X5 : subst] : (((sub @ (X10 @ X5 @ X4) @ X19)) = ((X10 @ (comp @ X5 @ X19) @ (sub @ X4 @ X19))))) | ~ ! [X9 : term] : (~(X6 @ id @ X9 @ id) | (X6 @ id @ (sub @ X31 @ (push @ X9 @ id)) @ id))) => ! [X32 : term,X6 : (subst > term > subst > $o)] : (~ ! [X10 : (subst > term > term)] : (~ ! [X4 : term] : (~(X6 @ id @ X4 @ id) | (X6 @ id @ (X10 @ id @ X4) @ id)) | ~ ! [X4 : term,X5 : subst,X19 : subst] : (((sub @ (X10 @ X5 @ X4) @ X19)) = ((X10 @ (comp @ X5 @ X19) @ (sub @ X4 @ X19)))) | (X6 @ id @ (lam @ (X10 @ sh @ one)) @ id)) | ~ ! [X9 : term] : (~(X6 @ id @ X9 @ id) | (X6 @ id @ (sub @ X32 @ (push @ X9 @ id)) @ id)) | (X6 @ id @ (lam @ X32) @ id)))) <=> hoasinduction_lem3a_lthm) & (hoasinduction_lem3aaa <=> ! [X6 : (subst > term > subst > $o),X20 : term] : ((X6 @ id @ (lam @ (sub @ X20 @ (push @ one @ sh))) @ id) | ~ ! [X9 : term] : ((X6 @ id @ (sub @ X20 @ (push @ X9 @ id)) @ id) | ~(X6 @ id @ X9 @ id)) | ~ ! [X10 : (subst > term > term),X21 : term] : (~ ! [X27 : subst,X26 : term,X25 : subst] : (((sub @ (X10 @ X25 @ X26) @ X27)) = ((sub @ (sub @ X21 @ (push @ X26 @ X25)) @ X27))) | (X6 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X4 : term] : (~(X6 @ id @ X4 @ id) | (X6 @ id @ (X10 @ id @ X4) @ id)) | ~ ! [X24 : subst,X23 : term,X22 : subst] : (((X10 @ (comp @ X22 @ X24) @ (sub @ X23 @ X24))) = ((sub @ X21 @ (push @ (sub @ X23 @ X24) @ (comp @ X22 @ X24)))))))) & pushprop_lem3v2_lthm & axvarshift & hoasinduction_lem2_gthm & hoaslamnotvar_lthm & hoasinduction_gthm & (axidr <=> ! [X5 : subst] : (X5 = ((comp @ X5 @ id)))) & (axidl <=> ! [X5 : subst] : (X5 = ((comp @ id @ X5)))) & (! [X40 : term,X6 : (subst > term > subst > $o)] : (~ ! [X10 : (subst > term > term)] : (~ ! [X4 : term] : (~(X6 @ id @ X4 @ id) | (X6 @ id @ (X10 @ id @ X4) @ id)) | (X6 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X5 : subst,X19 : subst,X4 : term] : (((sub @ (X10 @ X5 @ X4) @ X19)) = ((X10 @ (comp @ X5 @ X19) @ (sub @ X4 @ X19))))) | (X6 @ id @ X40 @ id) | ~ ! [X4 : term,X9 : term] : (~(X6 @ id @ X4 @ id) | ~(X6 @ id @ X9 @ id) | (X6 @ id @ (ap @ (sub @ X4 @ id) @ X9) @ id))) <=> hoasinduction_no_psi_cond) & hoasapinj2_gthm & hoasinduction_lem2_lthm & (sh = ((d2subst @ d_id))) & ((! [X4 : term,X5 : subst] : (X4 = ((sub @ one @ (push @ X4 @ X5)))) => ! [X4 : term,X7 : (term > $o),X5 : subst,X6 : (term > $o)] : ((X7 @ one) | ~ ! [X8 : term] : (((X7 @ X8)) = ((X6 @ (sub @ X8 @ (push @ X4 @ X5))))) | ~(X6 @ X4))) <=> pushprop_lem1v2_lthm) & (induction2lem_lthm <=> (! [X5 : subst,X4 : term] : (X4 = ((sub @ one @ (push @ X4 @ X5)))) => (! [X5 : subst,X4 : term] : (X5 = ((comp @ sh @ (push @ X4 @ X5)))) => (! [X5 : subst] : (X5 = ((comp @ X5 @ id))) => (! [X15 : term,X6 : (term > $o)] : (~ ! [X4 : term,X9 : term] : (~(X6 @ X9) | ~(X6 @ X4) | (X6 @ (ap @ X4 @ X9))) | (X6 @ X15) | ~ ! [X4 : term] : (~(X6 @ X4) | (X6 @ (lam @ X4))) | ~ ! [X4 : term] : (~(var @ X4) | (X6 @ X4))) => ! [X13 : term,X14 : subst,X6 : (term > $o)] : (~ ! [X4 : term] : ((X6 @ (lam @ X4)) | ~ ! [X9 : term] : ((X6 @ (sub @ X4 @ (push @ X9 @ id))) | ~(X6 @ X9))) | ~ ! [X9 : term] : (~(var @ X9) | (X6 @ (sub @ X9 @ X14))) | (X6 @ (sub @ X13 @ X14)) | ~ ! [X4 : term,X9 : term] : ((X6 @ (ap @ X4 @ X9)) | ~(X6 @ X4) | ~(X6 @ X9)))))))) & hoasinduction_lem3b_gthm & (termmset_lthm <=> (! [X4 : term] : (X4 = ((sub @ X4 @ id))) => ! [X4 : term] : (X4 = ((sub @ X4 @ id))))) & (! [X14 : subst,X13 : term,X6 : (term > $o)] : (~ ! [X4 : term] : ((X6 @ (lam @ X4)) | ~ ! [X9 : term] : (~(X6 @ X9) | (X6 @ (sub @ X4 @ (push @ X9 @ id))))) | ~ ! [X9 : term,X4 : term] : (~(X6 @ X9) | ~(X6 @ X4) | (X6 @ (ap @ X4 @ X9))) | (X6 @ (sub @ X13 @ X14)) | ~ ! [X9 : term] : ((X6 @ (sub @ X9 @ X14)) | ~(var @ X9))) <=> induction2lem) & (((sub @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2term @ d_one))) & (hoasinduction_lem0 <=> ! [X6 : (subst > term > subst > $o)] : ~ ! [X7 : (term > $o)] : ~ ! [X8 : term] : (((X7 @ X8)) = ((X6 @ id @ X8 @ id)))) & hoasapnotvar_lthm & (! [X35 : term,X6 : (term > $o)] : (~ ! [X4 : term] : (~ ! [X9 : term] : ((X6 @ (sub @ X4 @ (push @ X9 @ id))) | ~(X6 @ X9)) | (X6 @ (lam @ X4))) | ~ ! [X9 : term,X4 : term] : (~(X6 @ X4) | (X6 @ (ap @ X4 @ X9)) | ~(X6 @ X9)) | ~ ! [X4 : term] : ((X6 @ X4) | ~(var @ X4)) | (X6 @ X35)) <=> induction2) & (induction <=> ! [X6 : (term > $o),X15 : term] : (~ ! [X4 : term] : ((X6 @ (lam @ X4)) | ~(X6 @ X4)) | (X6 @ X15) | ~ ! [X4 : term] : ((X6 @ X4) | ~(var @ X4)) | ~ ! [X4 : term,X9 : term] : (~(X6 @ X9) | ~(X6 @ X4) | (X6 @ (ap @ X4 @ X9))))) & ((! [X5 : subst,X4 : term] : (X4 = ((sub @ one @ (push @ X4 @ X5)))) => (! [X5 : subst,X4 : term] : (X5 = ((comp @ sh @ (push @ X4 @ X5)))) => ! [X9 : term,X4 : term,X6 : (term > $o),X30 : (term > $o),X5 : subst] : ((X30 @ (sub @ X4 @ (push @ X9 @ X5))) | ~(X6 @ X4)))) <=> pushprop_lem1_lthm) & axabs & (hoasinduction_lem2 <=> ! [X6 : (subst > term > subst > $o),X28 : term,X29 : term] : (~ ! [X5 : subst,X4 : term,X30 : subst,X19 : subst] : (~(X6 @ (comp @ X5 @ X30) @ (sub @ X4 @ X30) @ X19) | (X6 @ X5 @ X4 @ (comp @ X30 @ X19))) | ~ ! [X4 : term,X9 : term] : (~(X6 @ id @ X9 @ id) | (X6 @ id @ (ap @ (sub @ X4 @ id) @ X9) @ id) | ~(X6 @ id @ X4 @ id)) | ~(X6 @ id @ X29 @ id) | ~(X6 @ id @ X28 @ id) | (X6 @ id @ (ap @ X28 @ X29) @ id) | ~ ! [X30 : subst,X4 : term,X5 : subst,X19 : subst] : (~(X6 @ X5 @ X4 @ (comp @ X30 @ X19)) | (X6 @ (comp @ X5 @ X30) @ (sub @ X4 @ X30) @ X19)))) & pushprop_lthm & lamnotvar & (hoasinduction_lem3v2a_lthm <=> (! [X9 : term] : ~ ! [X10 : (subst > term > term)] : ~ ! [X4 : term,X5 : subst] : (((sub @ X9 @ (push @ X4 @ X5))) = ((X10 @ X5 @ X4))) => (! [X4 : term] : (X4 = ((sub @ X4 @ id))) => ! [X6 : (subst > term > subst > $o),X7 : (term > $o),X34 : term] : ((X7 @ (lam @ X34)) | ~ ! [X9 : term] : ((X7 @ (sub @ X34 @ (push @ X9 @ id))) | ~(X7 @ X9)) | ~ ! [X8 : term] : (((X7 @ X8)) = ((X6 @ id @ X8 @ id))) | ~ ! [X10 : (subst > term > term)] : ((X6 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X4 : term] : ((X6 @ id @ (X10 @ id @ X4) @ id) | ~(X6 @ id @ X4 @ id)) | ~ ! [X4 : term,X19 : subst,X5 : subst] : (((sub @ (X10 @ X5 @ X4) @ X19)) = ((X10 @ (comp @ X5 @ X19) @ (sub @ X4 @ X19))))))))) & ! [X44 : d_subst,X43 : d_subst] : ((((d2subst @ X43)) = ((d2subst @ X44))) => (X43 = X44)) & ! [X0 : term] : ? [X1 : d_term] : (X0 = ((d2term @ X1))) & (one = ((d2term @ d_one))) & ! [X45 : d_subst] : (X45 = d_id) & hoasinduction_lem3v2_gthm & (! [X12 : term,X11 : term,X4 : term,X9 : term] : ((X4 = X9) | (((ap @ (sub @ X4 @ id) @ X11)) != ((ap @ (sub @ X9 @ id) @ X12)))) <=> hoasapinj1) & axassoc & (((comp @ (d2subst @ d_id) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & ! [X1 : d_term] : (X1 = d_one) & termmset_gthm & (hoaslaminj <=> ! [X10 : (subst > term > term),X18 : term,X17 : subst,X16 : (subst > term > term)] : (~ ! [X4 : term,X5 : subst,X19 : subst] : (((sub @ (X16 @ X5 @ X4) @ X19)) = ((X16 @ (comp @ X5 @ X19) @ (sub @ X4 @ X19)))) | (((X10 @ X17 @ X18)) = ((X16 @ X17 @ X18))) | ~ ! [X19 : subst,X4 : term,X5 : subst] : (((sub @ (X10 @ X5 @ X4) @ X19)) = ((X10 @ (comp @ X5 @ X19) @ (sub @ X4 @ X19)))) | (((lam @ (X10 @ sh @ one))) != ((lam @ (X16 @ sh @ one)))))) & (hoaslamnotap <=> ! [X37 : term,X10 : (subst > term > term),X36 : term] : (~ ! [X4 : term,X5 : subst,X19 : subst] : (((sub @ (X10 @ X5 @ X4) @ X19)) = ((X10 @ (comp @ X5 @ X19) @ (sub @ X4 @ X19)))) | (((lam @ (X10 @ sh @ one))) != ((ap @ (sub @ X36 @ id) @ X37))))) & ~lamnotap & (hoasinduction_lem3v2a <=> ! [X6 : (subst > term > subst > $o),X34 : term,X7 : (term > $o)] : (~ ! [X9 : term] : ((X7 @ (sub @ X34 @ (push @ X9 @ id))) | ~(X7 @ X9)) | ~ ! [X10 : (subst > term > term)] : ((X6 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X4 : term] : (~(X6 @ id @ X4 @ id) | (X6 @ id @ (X10 @ id @ X4) @ id)) | ~ ! [X5 : subst,X4 : term,X19 : subst] : (((sub @ (X10 @ X5 @ X4) @ X19)) = ((X10 @ (comp @ X5 @ X19) @ (sub @ X4 @ X19))))) | (X7 @ (lam @ X34)) | ~ ! [X8 : term] : (((X7 @ X8)) = ((X6 @ id @ X8 @ id))))) & (pushprop_lem1v2 <=> ! [X7 : (term > $o),X5 : subst,X6 : (term > $o),X4 : term] : (~ ! [X8 : term] : (((X7 @ X8)) = ((X6 @ (sub @ X8 @ (push @ X4 @ X5))))) | ~(X6 @ X4) | (X7 @ one))) & (! [X5 : subst,X4 : term,X6 : (term > $o)] : ~ ! [X7 : (term > $o)] : ~ ! [X8 : term] : (((X7 @ X8)) = ((X6 @ (sub @ X8 @ (push @ X4 @ X5))))) <=> pushprop_lem0_lthm) & ((! [X9 : term,X12 : term,X11 : term,X4 : term] : ((X11 = X12) | (((ap @ X4 @ X11)) != ((ap @ X9 @ X12)))) => ! [X4 : term,X12 : term,X9 : term,X11 : term] : ((((ap @ (sub @ X4 @ id) @ X11)) != ((ap @ (sub @ X9 @ id) @ X12))) | (X11 = X12))) <=> hoasapinj2_lthm) & pushprop_lthm_orig & pushprop_lem1v2_gthm & (! [X4 : term,X5 : subst] : (X4 = ((sub @ one @ (push @ X4 @ X5)))) <=> axvarcons) & (! [X11 : term,X4 : term,X9 : term,X12 : term] : ((((ap @ X4 @ X11)) != ((ap @ X9 @ X12))) | (X4 = X9)) <=> apinj1) & (hoaslam = (^[X2 : subst, X3 : (subst > term > term)] : ((d2term @ d_one)))) & (pushprop_lem0 <=> ! [X4 : term,X5 : subst,X6 : (term > $o)] : ~ ! [X7 : (term > $o)] : ~ ! [X8 : term] : (((X7 @ X8)) = ((X6 @ (sub @ X8 @ (push @ X4 @ X5)))))) & induction2_gthm & (hoasinduction_lem3aa <=> ! [X6 : (subst > term > subst > $o),X31 : term] : (~ ! [X9 : term] : (~(X6 @ id @ X9 @ id) | (X6 @ id @ (sub @ X31 @ (push @ X9 @ id)) @ id)) | ~ ! [X10 : (subst > term > term)] : (~ ! [X4 : term] : (~(X6 @ id @ X4 @ id) | (X6 @ id @ (X10 @ id @ X4) @ id)) | (X6 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X19 : subst,X4 : term,X5 : subst] : (((sub @ (X10 @ X5 @ X4) @ X19)) = ((X10 @ (comp @ X5 @ X19) @ (sub @ X4 @ X19))))) | (X6 @ id @ (lam @ (sub @ X31 @ (push @ one @ sh))) @ id))) & (hoasinduction_no_psi_cond_lthm <=> (! [X6 : (subst > term > subst > $o)] : ~ ! [X7 : (term > $o)] : ~ ! [X8 : term] : (((X7 @ X8)) = ((X6 @ id @ X8 @ id))) => (! [X35 : term,X6 : (term > $o)] : (~ ! [X4 : term] : (~ ! [X9 : term] : ((X6 @ (sub @ X4 @ (push @ X9 @ id))) | ~(X6 @ X9)) | (X6 @ (lam @ X4))) | (X6 @ X35) | ~ ! [X4 : term,X9 : term] : ((X6 @ (ap @ X4 @ X9)) | ~(X6 @ X4) | ~(X6 @ X9)) | ~ ! [X4 : term] : (~(var @ X4) | (X6 @ X4))) => (! [X4 : term] : (X4 = ((sub @ X4 @ id))) => (! [X6 : (subst > term > subst > $o),X7 : (term > $o),X34 : term] : (~ ! [X10 : (subst > term > term)] : (~ ! [X4 : term] : (~(X6 @ id @ X4 @ id) | (X6 @ id @ (X10 @ id @ X4) @ id)) | ~ ! [X19 : subst,X4 : term,X5 : subst] : (((sub @ (X10 @ X5 @ X4) @ X19)) = ((X10 @ (comp @ X5 @ X19) @ (sub @ X4 @ X19)))) | (X6 @ id @ (lam @ (X10 @ sh @ one)) @ id)) | ~ ! [X9 : term] : ((X7 @ (sub @ X34 @ (push @ X9 @ id))) | ~(X7 @ X9)) | ~ ! [X8 : term] : (((X7 @ X8)) = ((X6 @ id @ X8 @ id))) | (X7 @ (lam @ X34))) => ! [X40 : term,X6 : (subst > term > subst > $o)] : (~ ! [X10 : (subst > term > term)] : (~ ! [X4 : term] : ((X6 @ id @ (X10 @ id @ X4) @ id) | ~(X6 @ id @ X4 @ id)) | (X6 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X4 : term,X19 : subst,X5 : subst] : (((sub @ (X10 @ X5 @ X4) @ X19)) = ((X10 @ (comp @ X5 @ X19) @ (sub @ X4 @ X19))))) | (X6 @ id @ X40 @ id) | ~ ! [X9 : term,X4 : term] : (~(X6 @ id @ X9 @ id) | (X6 @ id @ (ap @ (sub @ X4 @ id) @ X9) @ id) | ~(X6 @ id @ X4 @ id)))))))) & (induction2_lthm <=> (! [X4 : term] : (X4 = ((sub @ X4 @ id))) => (! [X13 : term,X14 : subst,X6 : (term > $o)] : (~ ! [X9 : term] : ((X6 @ (sub @ X9 @ X14)) | ~(var @ X9)) | (X6 @ (sub @ X13 @ X14)) | ~ ! [X9 : term,X4 : term] : (~(X6 @ X9) | ~(X6 @ X4) | (X6 @ (ap @ X4 @ X9))) | ~ ! [X4 : term] : (~ ! [X9 : term] : ((X6 @ (sub @ X4 @ (push @ X9 @ id))) | ~(X6 @ X9)) | (X6 @ (lam @ X4)))) => ! [X35 : term,X6 : (term > $o)] : (~ ! [X9 : term,X4 : term] : (~(X6 @ X9) | ~(X6 @ X4) | (X6 @ (ap @ X4 @ X9))) | (X6 @ X35) | ~ ! [X4 : term] : (~(var @ X4) | (X6 @ X4)) | ~ ! [X4 : term] : (~ ! [X9 : term] : (~(X6 @ X9) | (X6 @ (sub @ X4 @ (push @ X9 @ id)))) | (X6 @ (lam @ X4))))))) & (hoasapinj1_lthm <=> (! [X4 : term] : (X4 = ((sub @ X4 @ id))) => (! [X12 : term,X4 : term,X11 : term,X9 : term] : ((((ap @ X4 @ X11)) != ((ap @ X9 @ X12))) | (X4 = X9)) => ! [X11 : term,X9 : term,X12 : term,X4 : term] : ((X4 = X9) | (((ap @ (sub @ X4 @ id) @ X11)) != ((ap @ (sub @ X9 @ id) @ X12))))))) & (! [X42 : term,X6 : (subst > term > subst > $o),X7 : (term > $o),X41 : term] : (~ ! [X19 : subst,X4 : term,X30 : subst,X5 : subst] : ((X6 @ X5 @ X4 @ (comp @ X30 @ X19)) | ~(X6 @ (comp @ X5 @ X30) @ (sub @ X4 @ X30) @ X19)) | ~ ! [X5 : subst,X19 : subst,X4 : term,X30 : subst] : ((X6 @ (comp @ X5 @ X30) @ (sub @ X4 @ X30) @ X19) | ~(X6 @ X5 @ X4 @ (comp @ X30 @ X19))) | ~(X7 @ X42) | ~(X7 @ X41) | ~ ! [X4 : term,X9 : term] : ((X6 @ id @ (ap @ (sub @ X4 @ id) @ X9) @ id) | ~(X6 @ id @ X4 @ id) | ~(X6 @ id @ X9 @ id)) | (X7 @ (ap @ X41 @ X42)) | ~ ! [X8 : term] : (((X7 @ X8)) = ((X6 @ id @ X8 @ id)))) <=> hoasinduction_lem2v2) & (hoasinduction_lem3_lthm <=> (! [X4 : term] : (X4 = ((sub @ X4 @ id))) => (! [X6 : (subst > term > subst > $o),X31 : term] : ((X6 @ id @ (lam @ (sub @ X31 @ (push @ one @ sh))) @ id) | ~ ! [X10 : (subst > term > term)] : (~ ! [X5 : subst,X19 : subst,X4 : term] : (((sub @ (X10 @ X5 @ X4) @ X19)) = ((X10 @ (comp @ X5 @ X19) @ (sub @ X4 @ X19)))) | (X6 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X4 : term] : ((X6 @ id @ (X10 @ id @ X4) @ id) | ~(X6 @ id @ X4 @ id))) | ~ ! [X9 : term] : (~(X6 @ id @ X9 @ id) | (X6 @ id @ (sub @ X31 @ (push @ X9 @ id)) @ id))) => ! [X6 : (subst > term > subst > $o),X33 : term] : (~ ! [X10 : (subst > term > term)] : ((X6 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X4 : term] : ((X6 @ id @ (X10 @ id @ X4) @ id) | ~(X6 @ id @ X4 @ id)) | ~ ! [X5 : subst,X19 : subst,X4 : term] : (((sub @ (X10 @ X5 @ X4) @ X19)) = ((X10 @ (comp @ X5 @ X19) @ (sub @ X4 @ X19))))) | ~ ! [X9 : term] : ((X6 @ id @ (sub @ X33 @ (push @ X9 @ id)) @ id) | ~(X6 @ id @ X9 @ id)) | ~ ! [X4 : term,X30 : subst,X19 : subst,X5 : subst] : (~(X6 @ (comp @ X5 @ X30) @ (sub @ X4 @ X30) @ X19) | (X6 @ X5 @ X4 @ (comp @ X30 @ X19))) | ~ ! [X19 : subst,X4 : term,X30 : subst,X5 : subst] : (~(X6 @ X5 @ X4 @ (comp @ X30 @ X19)) | (X6 @ (comp @ X5 @ X30) @ (sub @ X4 @ X30) @ X19)) | (X6 @ id @ (lam @ X33) @ id))))) & hoasinduction_lem1_gthm & (pushprop_lem1 <=> ! [X5 : subst,X9 : term,X6 : (term > $o),X30 : (term > $o),X4 : term] : ((X30 @ (sub @ X4 @ (push @ X9 @ X5))) | ~(X6 @ X4))) & hoasapnotvar_gthm & (! [X33 : term,X6 : (subst > term > subst > $o)] : (~ ! [X5 : subst,X4 : term,X19 : subst,X30 : subst] : (~(X6 @ (comp @ X5 @ X30) @ (sub @ X4 @ X30) @ X19) | (X6 @ X5 @ X4 @ (comp @ X30 @ X19))) | ~ ! [X19 : subst,X5 : subst,X4 : term,X30 : subst] : ((X6 @ (comp @ X5 @ X30) @ (sub @ X4 @ X30) @ X19) | ~(X6 @ X5 @ X4 @ (comp @ X30 @ X19))) | ~ ! [X10 : (subst > term > term)] : (~ ! [X19 : subst,X4 : term,X5 : subst] : (((sub @ (X10 @ X5 @ X4) @ X19)) = ((X10 @ (comp @ X5 @ X19) @ (sub @ X4 @ X19)))) | (X6 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X4 : term] : (~(X6 @ id @ X4 @ id) | (X6 @ id @ (X10 @ id @ X4) @ id))) | (X6 @ id @ (lam @ X33) @ id) | ~ ! [X9 : term] : ((X6 @ id @ (sub @ X33 @ (push @ X9 @ id)) @ id) | ~(X6 @ id @ X9 @ id))) <=> hoasinduction_lem3) & (axscons <=> ! [X5 : subst] : (X5 = ((push @ (sub @ one @ X5) @ (comp @ sh @ X5))))) & (! [X9 : term] : ~ ! [X10 : (subst > term > term)] : (((X10 @ sh @ one)) != ((sub @ X9 @ (push @ one @ sh)))) <=> hoasinduction_lem3b) & apnotvar & (((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) = ((d2term @ d_one))) & ! [X48 : d_term,X47 : d_term] : ((((d2term @ X47)) = ((d2term @ X48))) => (X47 = X48)) & hoaslaminj_gthm & (! [X9 : term] : ~ ! [X10 : (subst > term > term)] : (((X10 @ sh @ one)) != ((sub @ X9 @ (push @ one @ sh)))) <=> hoasinduction_lem3b_lthm) & (substmonoid_lthm <=> (! [X5 : subst] : (X5 = ((comp @ id @ X5))) => (! [X5 : subst] : (X5 = ((comp @ X5 @ id))) => (! [X5 : subst] : (X5 = ((comp @ X5 @ id))) & ! [X5 : subst] : (X5 = ((comp @ id @ X5))))))) & axapp & (! [X6 : (subst > term > subst > $o),X32 : term] : (~ ! [X9 : term] : (~(X6 @ id @ X9 @ id) | (X6 @ id @ (sub @ X32 @ (push @ X9 @ id)) @ id)) | ~ ! [X10 : (subst > term > term)] : ((X6 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X4 : term] : (~(X6 @ id @ X4 @ id) | (X6 @ id @ (X10 @ id @ X4) @ id)) | ~ ! [X4 : term,X5 : subst,X19 : subst] : (((sub @ (X10 @ X5 @ X4) @ X19)) = ((X10 @ (comp @ X5 @ X19) @ (sub @ X4 @ X19))))) | (X6 @ id @ (lam @ X32) @ id)) <=> hoasinduction_lem3a) & (laminj <=> ! [X9 : term,X4 : term] : ((X4 = X9) | (((lam @ X4)) != ((lam @ X9))))) & hoasinduction_lem1_lthm & pushprop_lem3v2 & (((ap @ (d2term @ d_one) @ (d2term @ d_one))) = ((d2term @ d_one))) & ((! [X6 : (term > $o),X35 : term] : ((X6 @ X35) | ~ ! [X4 : term] : (~ ! [X9 : term] : ((X6 @ (sub @ X4 @ (push @ X9 @ id))) | ~(X6 @ X9)) | (X6 @ (lam @ X4))) | ~ ! [X4 : term] : (~(var @ X4) | (X6 @ X4)) | ~ ! [X4 : term,X9 : term] : (~(X6 @ X9) | ~(X6 @ X4) | (X6 @ (ap @ X4 @ X9)))) => (! [X6 : (subst > term > subst > $o),X29 : term,X28 : term] : (~(X6 @ id @ X29 @ id) | ~(X6 @ id @ X28 @ id) | ~ ! [X4 : term,X9 : term] : (~(X6 @ id @ X4 @ id) | ~(X6 @ id @ X9 @ id) | (X6 @ id @ (ap @ (sub @ X4 @ id) @ X9) @ id)) | (X6 @ id @ (ap @ X28 @ X29) @ id) | ~ ! [X4 : term,X30 : subst,X5 : subst,X19 : subst] : ((X6 @ (comp @ X5 @ X30) @ (sub @ X4 @ X30) @ X19) | ~(X6 @ X5 @ X4 @ (comp @ X30 @ X19))) | ~ ! [X5 : subst,X30 : subst,X4 : term,X19 : subst] : ((X6 @ X5 @ X4 @ (comp @ X30 @ X19)) | ~(X6 @ (comp @ X5 @ X30) @ (sub @ X4 @ X30) @ X19))) => (! [X33 : term,X6 : (subst > term > subst > $o)] : (~ ! [X19 : subst,X4 : term,X30 : subst,X5 : subst] : (~(X6 @ X5 @ X4 @ (comp @ X30 @ X19)) | (X6 @ (comp @ X5 @ X30) @ (sub @ X4 @ X30) @ X19)) | ~ ! [X30 : subst,X19 : subst,X4 : term,X5 : subst] : (~(X6 @ (comp @ X5 @ X30) @ (sub @ X4 @ X30) @ X19) | (X6 @ X5 @ X4 @ (comp @ X30 @ X19))) | ~ ! [X9 : term] : (~(X6 @ id @ X9 @ id) | (X6 @ id @ (sub @ X33 @ (push @ X9 @ id)) @ id)) | (X6 @ id @ (lam @ X33) @ id) | ~ ! [X10 : (subst > term > term)] : ((X6 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X4 : term,X19 : subst,X5 : subst] : (((sub @ (X10 @ X5 @ X4) @ X19)) = ((X10 @ (comp @ X5 @ X19) @ (sub @ X4 @ X19)))) | ~ ! [X4 : term] : (~(X6 @ id @ X4 @ id) | (X6 @ id @ (X10 @ id @ X4) @ id)))) => ! [X38 : term,X6 : (subst > term > subst > $o)] : (~ ! [X4 : term] : (~(var @ (sub @ X4 @ id)) | (X6 @ id @ X4 @ id)) | (X6 @ id @ X38 @ id) | ~ ! [X4 : term,X5 : subst,X30 : subst,X19 : subst] : ((X6 @ X5 @ X4 @ (comp @ X30 @ X19)) | ~(X6 @ (comp @ X5 @ X30) @ (sub @ X4 @ X30) @ X19)) | ~ ! [X5 : subst,X30 : subst,X4 : term,X19 : subst] : ((X6 @ (comp @ X5 @ X30) @ (sub @ X4 @ X30) @ X19) | ~(X6 @ X5 @ X4 @ (comp @ X30 @ X19))) | ~ ! [X4 : term,X9 : term] : ((X6 @ id @ (ap @ (sub @ X4 @ id) @ X9) @ id) | ~(X6 @ id @ X4 @ id) | ~(X6 @ id @ X9 @ id)) | ~ ! [X10 : (subst > term > term)] : (~ ! [X4 : term] : ((X6 @ id @ (X10 @ id @ X4) @ id) | ~(X6 @ id @ X4 @ id)) | ~ ! [X19 : subst,X5 : subst,X4 : term] : (((sub @ (X10 @ X5 @ X4) @ X19)) = ((X10 @ (comp @ X5 @ X19) @ (sub @ X4 @ X19)))) | (X6 @ id @ (lam @ (X10 @ sh @ one)) @ id)))))) <=> hoasinduction_lthm) & induction2lem_gthm & ((! [X5 : subst] : (X5 = ((comp @ X5 @ id))) & ! [X5 : subst] : (X5 = ((comp @ id @ X5)))) <=> substmonoid) & (! [X38 : term,X6 : (subst > term > subst > $o)] : (~ ! [X5 : subst,X30 : subst,X19 : subst,X4 : term] : ((X6 @ (comp @ X5 @ X30) @ (sub @ X4 @ X30) @ X19) | ~(X6 @ X5 @ X4 @ (comp @ X30 @ X19))) | ~ ! [X9 : term,X4 : term] : ((X6 @ id @ (ap @ (sub @ X4 @ id) @ X9) @ id) | ~(X6 @ id @ X9 @ id) | ~(X6 @ id @ X4 @ id)) | ~ ! [X5 : subst,X19 : subst,X30 : subst,X4 : term] : ((X6 @ X5 @ X4 @ (comp @ X30 @ X19)) | ~(X6 @ (comp @ X5 @ X30) @ (sub @ X4 @ X30) @ X19)) | ~ ! [X4 : term] : ((X6 @ id @ X4 @ id) | ~(var @ (sub @ X4 @ id))) | ~ ! [X10 : (subst > term > term)] : (~ ! [X4 : term] : ((X6 @ id @ (X10 @ id @ X4) @ id) | ~(X6 @ id @ X4 @ id)) | (X6 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X5 : subst,X19 : subst,X4 : term] : (((sub @ (X10 @ X5 @ X4) @ X19)) = ((X10 @ (comp @ X5 @ X19) @ (sub @ X4 @ X19))))) | (X6 @ id @ X38 @ id)) <=> hoasinduction) & hoaslamnotap_gthm & (((push @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & (hoasinduction_lem0_lthm <=> ! [X6 : (subst > term > subst > $o)] : ~ ! [X7 : (term > $o)] : ~ ! [X8 : term] : (((X7 @ X8)) = ((X6 @ id @ X8 @ id)))) & (! [X4 : term] : (X4 = ((sub @ X4 @ id))) <=> axvarid) & hoasinduction_lem3_gthm & (apinj2 <=> ! [X4 : term,X11 : term,X9 : term,X12 : term] : ((X11 = X12) | (((ap @ X4 @ X11)) != ((ap @ X9 @ X12))))) & hoasinduction_lem1v2 & ~(var @ (d2term @ d_one)) & (hoasinduction_lem3aa_lthm <=> ! [X31 : term,X6 : (subst > term > subst > $o)] : (~ ! [X9 : term] : (~(X6 @ id @ X9 @ id) | (X6 @ id @ (sub @ X31 @ (push @ X9 @ id)) @ id)) | ~ ! [X10 : (subst > term > term)] : ((X6 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X4 : term] : (~(X6 @ id @ X4 @ id) | (X6 @ id @ (X10 @ id @ X4) @ id)) | ~ ! [X19 : subst,X4 : term,X5 : subst] : (((sub @ (X10 @ X5 @ X4) @ X19)) = ((X10 @ (comp @ X5 @ X19) @ (sub @ X4 @ X19))))) | (X6 @ id @ (lam @ (sub @ X31 @ (push @ one @ sh))) @ id))) & pushprop_lem1_gthm & hoasapnotvar),
  file('/export/starexec/sandbox/benchmark/theBenchmark.p',alg444_1)).
thf(f2,conjecture,(
  (hoasinduction_lem3v2 = ! [X0 : (subst > term > subst > $o),X1 : (term > $o)] : (! [X4 : subst,X5 : subst,X3 : term,X2 : subst] : ((X0 @ X2 @ X3 @ (comp @ X5 @ X4)) => (X0 @ (comp @ X2 @ X5) @ (sub @ X3 @ X5) @ X4)) => (! [X5 : subst,X3 : term,X4 : subst,X2 : subst] : ((X0 @ (comp @ X2 @ X5) @ (sub @ X3 @ X5) @ X4) => (X0 @ X2 @ X3 @ (comp @ X5 @ X4))) => (! [X6 : (subst > term > term)] : (! [X2 : subst,X3 : term,X4 : subst] : (((sub @ (X6 @ X2 @ X3) @ X4)) = ((X6 @ (comp @ X2 @ X4) @ (sub @ X3 @ X4)))) => (! [X3 : term] : ((X0 @ id @ X3 @ id) => (X0 @ id @ (X6 @ id @ X3) @ id)) => (X0 @ id @ (hoaslam @ id @ (^[X2 : subst, X3 : term] : ((X6 @ X2 @ X3)))) @ id))) => ((hoasinduction_p_and_p_prime @ X0 @ X1) => ! [X3 : term] : (! [X7 : term] : ((X1 @ X7) => (X1 @ (sub @ X3 @ (push @ X7 @ id)))) => (X1 @ (lam @ X3))))))))),
  file('/export/starexec/sandbox/benchmark/theBenchmark.p',hoasinduction_lem3v2)).
thf(f3,negated_conjecture,(
  ~ (hoasinduction_lem3v2 = ! [X0 : (subst > term > subst > $o),X1 : (term > $o)] : (! [X4 : subst,X5 : subst,X3 : term,X2 : subst] : ((X0 @ X2 @ X3 @ (comp @ X5 @ X4)) => (X0 @ (comp @ X2 @ X5) @ (sub @ X3 @ X5) @ X4)) => (! [X5 : subst,X3 : term,X4 : subst,X2 : subst] : ((X0 @ (comp @ X2 @ X5) @ (sub @ X3 @ X5) @ X4) => (X0 @ X2 @ X3 @ (comp @ X5 @ X4))) => (! [X6 : (subst > term > term)] : (! [X2 : subst,X3 : term,X4 : subst] : (((sub @ (X6 @ X2 @ X3) @ X4)) = ((X6 @ (comp @ X2 @ X4) @ (sub @ X3 @ X4)))) => (! [X3 : term] : ((X0 @ id @ X3 @ id) => (X0 @ id @ (X6 @ id @ X3) @ id)) => (X0 @ id @ (hoaslam @ id @ (^[X2 : subst, X3 : term] : ((X6 @ X2 @ X3)))) @ id))) => ((hoasinduction_p_and_p_prime @ X0 @ X1) => ! [X3 : term] : (! [X7 : term] : ((X1 @ X7) => (X1 @ (sub @ X3 @ (push @ X7 @ id)))) => (X1 @ (lam @ X3))))))))),
  inference(negated_conjecture,[status(cth)],[f2])).
thf(f4,plain,(
  axclos & pushprop_lem2v2_lthm & hoasinduction_lem1v2_gthm & hoasinduction_lem2v2_gthm & (hoasinduction_lem3v2_f <=> ! [X0 : term] : ~ ! [X1 : (subst > term > term)] : ~ ! [X2 : term,X3 : subst] : (((sub @ X0 @ (push @ X2 @ X3))) = ((X1 @ X3 @ X2)))) & ! [X4 : subst] : ? [X5 : d_subst] : (((d2subst @ X5)) = X4) & (hoasinduction_lthm_3 <=> (! [X6 : (subst > term > subst > $o)] : ~ ! [X7 : (term > $o)] : ~ ! [X8 : term] : (((X7 @ X8)) = ((X6 @ id @ X8 @ id))) => (! [X9 : (term > $o),X10 : term] : (~ ! [X11 : term,X12 : term] : (~(X9 @ X11) | ~(X9 @ X12) | (X9 @ (ap @ X11 @ X12))) | ~ ! [X13 : term] : ((X9 @ X13) | ~(var @ X13)) | (X9 @ X10) | ~ ! [X14 : term] : (~ ! [X15 : term] : (~(X9 @ X15) | (X9 @ (sub @ X14 @ (push @ X15 @ id)))) | (X9 @ (lam @ X14)))) => (! [X16 : term] : (((sub @ X16 @ id)) = X16) => (! [X17 : term,X18 : (term > $o),X19 : (subst > term > subst > $o)] : (~ ! [X20 : (subst > term > term)] : (~ ! [X21 : term] : (~(X19 @ id @ X21 @ id) | (X19 @ id @ (X20 @ id @ X21) @ id)) | (X19 @ id @ (lam @ (X20 @ sh @ one)) @ id) | ~ ! [X22 : term,X23 : subst,X24 : subst] : (((X20 @ (comp @ X23 @ X24) @ (sub @ X22 @ X24))) = ((sub @ (X20 @ X23 @ X22) @ X24)))) | (X18 @ (lam @ X17)) | ~ ! [X25 : term] : ((X18 @ (sub @ X17 @ (push @ X25 @ id))) | ~(X18 @ X25)) | ~ ! [X26 : term] : (((X18 @ X26)) = ((X19 @ id @ X26 @ id)))) => ! [X27 : term,X28 : (subst > term > subst > $o)] : (~ ! [X29 : term,X30 : term] : (~(X28 @ id @ X30 @ id) | ~(X28 @ id @ X29 @ id) | (X28 @ id @ (ap @ (sub @ X30 @ id) @ X29) @ id)) | ~ ! [X31 : (subst > term > term)] : ((X28 @ id @ (lam @ (X31 @ sh @ one)) @ id) | ~ ! [X32 : term] : (~(X28 @ id @ X32 @ id) | (X28 @ id @ (X31 @ id @ X32) @ id)) | ~ ! [X33 : subst,X34 : term,X35 : subst] : (((sub @ (X31 @ X35 @ X34) @ X33)) = ((X31 @ (comp @ X35 @ X33) @ (sub @ X34 @ X33))))) | ~ ! [X36 : term,X37 : subst,X38 : subst,X39 : subst] : (~(X28 @ (comp @ X37 @ X39) @ (sub @ X36 @ X39) @ X38) | (X28 @ X37 @ X36 @ (comp @ X39 @ X38))) | ~ ! [X40 : subst,X41 : term,X42 : subst,X43 : subst] : ((X28 @ (comp @ X43 @ X40) @ (sub @ X41 @ X40) @ X42) | ~(X28 @ X43 @ X41 @ (comp @ X40 @ X42))) | ~ ! [X44 : term] : (~(var @ (sub @ X44 @ id)) | (X28 @ id @ X44 @ id)) | (X28 @ id @ X27 @ id))))))) & hoaslamnotvar_gthm & ((^[X45 : (subst > term > subst > $o), X46 : (term > $o)] : (! [X47 : term] : (((X46 @ X47)) = ((X45 @ id @ X47 @ id))))) = hoasinduction_p_and_p_prime) & axmap & (! [X48 : term,X49 : term] : ((X48 = X49) | (((sub @ X49 @ sh)) != ((sub @ X48 @ sh)))) <=> shinj) & (! [X50 : term,X51 : term,X52 : term,X53 : term] : ((X50 = X53) | (((ap @ (sub @ X52 @ id) @ X53)) != ((ap @ (sub @ X51 @ id) @ X50)))) <=> hoasapinj2) & pushprop & pushprop_lem0_gthm & hoaslamnotvar & (! [X54 : subst,X55 : term] : (((comp @ sh @ (push @ X55 @ X54))) = X54) <=> axshiftcons) & substmonoid_gthm & pushprop_lem2v2_gthm & (hoasinduction_lem3v2 <=> ! [X56 : term,X57 : (subst > term > subst > $o),X58 : (term > $o)] : (~ ! [X59 : (subst > term > term)] : (~ ! [X60 : term] : (~(X57 @ id @ X60 @ id) | (X57 @ id @ (X59 @ id @ X60) @ id)) | ~ ! [X61 : subst,X62 : subst,X63 : term] : (((sub @ (X59 @ X62 @ X63) @ X61)) = ((X59 @ (comp @ X62 @ X61) @ (sub @ X63 @ X61)))) | (X57 @ id @ (lam @ (X59 @ sh @ one)) @ id)) | ~ ! [X64 : subst,X65 : subst,X66 : subst,X67 : term] : (~(X57 @ X64 @ X67 @ (comp @ X66 @ X65)) | (X57 @ (comp @ X64 @ X66) @ (sub @ X67 @ X66) @ X65)) | (X58 @ (lam @ X56)) | ~ ! [X68 : term,X69 : subst,X70 : subst,X71 : subst] : (~(X57 @ (comp @ X71 @ X70) @ (sub @ X68 @ X70) @ X69) | (X57 @ X71 @ X68 @ (comp @ X70 @ X69))) | ~ ! [X72 : term] : (~(X58 @ X72) | (X58 @ (sub @ X56 @ (push @ X72 @ id)))) | ~ ! [X73 : term] : (((X57 @ id @ X73 @ id)) = ((X58 @ X73))))) & (((lam @ (d2term @ d_one))) = ((d2term @ d_one))) & ulamvarsh & ((! [X74 : term] : (((sub @ X74 @ id)) = X74) => ! [X75 : term,X76 : (subst > term > subst > $o),X77 : (term > $o)] : ((X77 @ (lam @ X75)) | ~ ! [X78 : term] : (~(X77 @ X78) | (X77 @ (sub @ X75 @ (push @ X78 @ id)))) | ~ ! [X79 : term,X80 : subst,X81 : subst,X82 : subst] : ((X76 @ (comp @ X80 @ X81) @ (sub @ X79 @ X81) @ X82) | ~(X76 @ X80 @ X79 @ (comp @ X81 @ X82))) | ~ ! [X83 : (subst > term > term)] : (~ ! [X84 : term] : (~(X76 @ id @ X84 @ id) | (X76 @ id @ (X83 @ id @ X84) @ id)) | ~ ! [X85 : subst,X86 : term,X87 : subst] : (((X83 @ (comp @ X87 @ X85) @ (sub @ X86 @ X85))) = ((sub @ (X83 @ X87 @ X86) @ X85))) | (X76 @ id @ (lam @ (X83 @ sh @ one)) @ id)) | ~ ! [X88 : subst,X89 : subst,X90 : subst,X91 : term] : (~(X76 @ (comp @ X88 @ X90) @ (sub @ X91 @ X90) @ X89) | (X76 @ X88 @ X91 @ (comp @ X90 @ X89))) | ~ ! [X92 : term] : (((X77 @ X92)) = ((X76 @ id @ X92 @ id))))) <=> hoasinduction_lem3v2_lthm) & (hoasinduction_lem3v2_f_lthm <=> ! [X93 : term] : ~ ! [X94 : (subst > term > term)] : ~ ! [X95 : term,X96 : subst] : (((sub @ X93 @ (push @ X95 @ X96))) = ((X94 @ X96 @ X95)))) & ~ulamvar1 & ulamvarind & hoasinduction_lem1 & pushprop_lem2v2 & hoasinduction_lem3a_gthm & hoasapinj1_gthm & (hoaslaminj_lthm <=> (! [X97 : term,X98 : subst] : (((sub @ one @ (push @ X97 @ X98))) = X97) => (! [X99 : term,X100 : subst] : (((comp @ sh @ (push @ X99 @ X100))) = X100) => (! [X101 : term,X102 : term] : ((((lam @ X102)) != ((lam @ X101))) | (X101 = X102)) => ! [X103 : term,X104 : (subst > term > term),X105 : (subst > term > term),X106 : subst] : (~ ! [X107 : term,X108 : subst,X109 : subst] : (((sub @ (X105 @ X108 @ X107) @ X109)) = ((X105 @ (comp @ X108 @ X109) @ (sub @ X107 @ X109)))) | (((X104 @ X106 @ X103)) = ((X105 @ X106 @ X103))) | (((lam @ (X105 @ sh @ one))) != ((lam @ (X104 @ sh @ one)))) | ~ ! [X110 : subst,X111 : subst,X112 : term] : (((sub @ (X104 @ X111 @ X112) @ X110)) = ((X104 @ (comp @ X111 @ X110) @ (sub @ X112 @ X110))))))))) & ~(hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) & ((^[X113 : term, X114 : subst, X115 : (term > $o), X116 : (term > $o)] : (! [X117 : term] : (((X115 @ (sub @ X117 @ (push @ X113 @ X114)))) = ((X116 @ X117))))) = pushprop_p_and_p_prime) & hoaslamnotap_lthm & (! [X118 : term] : (((sub @ X118 @ id)) = X118) <=> termmset) & pushprop_gthm & (id = ((d2subst @ d_id))) & ((! [X119 : term] : (((sub @ X119 @ id)) = X119) => (! [X120 : (subst > term > subst > $o),X121 : term] : ((X120 @ id @ (lam @ (sub @ X121 @ (push @ one @ sh))) @ id) | ~ ! [X122 : (subst > term > term)] : (~ ! [X123 : term] : (~(X120 @ id @ X123 @ id) | (X120 @ id @ (X122 @ id @ X123) @ id)) | (X120 @ id @ (lam @ (X122 @ sh @ one)) @ id) | ~ ! [X124 : subst,X125 : term,X126 : subst] : (((X122 @ (comp @ X126 @ X124) @ (sub @ X125 @ X124))) = ((sub @ (X122 @ X126 @ X125) @ X124)))) | ~ ! [X127 : term] : (~(X120 @ id @ X127 @ id) | (X120 @ id @ (sub @ X121 @ (push @ X127 @ id)) @ id))) => ! [X128 : term,X129 : (subst > term > subst > $o)] : (~ ! [X130 : (subst > term > term)] : (~ ! [X131 : term] : (~(X129 @ id @ X131 @ id) | (X129 @ id @ (X130 @ id @ X131) @ id)) | ~ ! [X132 : term,X133 : subst,X134 : subst] : (((sub @ (X130 @ X133 @ X132) @ X134)) = ((X130 @ (comp @ X133 @ X134) @ (sub @ X132 @ X134)))) | (X129 @ id @ (lam @ (X130 @ sh @ one)) @ id)) | ~ ! [X135 : term] : (~(X129 @ id @ X135 @ id) | (X129 @ id @ (sub @ X128 @ (push @ X135 @ id)) @ id)) | (X129 @ id @ (lam @ X128) @ id)))) <=> hoasinduction_lem3a_lthm) & (hoasinduction_lem3aaa <=> ! [X136 : (subst > term > subst > $o),X137 : term] : ((X136 @ id @ (lam @ (sub @ X137 @ (push @ one @ sh))) @ id) | ~ ! [X138 : term] : ((X136 @ id @ (sub @ X137 @ (push @ X138 @ id)) @ id) | ~(X136 @ id @ X138 @ id)) | ~ ! [X139 : (subst > term > term),X140 : term] : (~ ! [X141 : subst,X142 : term,X143 : subst] : (((sub @ (X139 @ X143 @ X142) @ X141)) = ((sub @ (sub @ X140 @ (push @ X142 @ X143)) @ X141))) | (X136 @ id @ (lam @ (X139 @ sh @ one)) @ id) | ~ ! [X144 : term] : (~(X136 @ id @ X144 @ id) | (X136 @ id @ (X139 @ id @ X144) @ id)) | ~ ! [X145 : subst,X146 : term,X147 : subst] : (((sub @ X140 @ (push @ (sub @ X146 @ X145) @ (comp @ X147 @ X145)))) = ((X139 @ (comp @ X147 @ X145) @ (sub @ X146 @ X145))))))) & pushprop_lem3v2_lthm & axvarshift & hoasinduction_lem2_gthm & hoaslamnotvar_lthm & hoasinduction_gthm & (axidr <=> ! [X148 : subst] : (((comp @ X148 @ id)) = X148)) & (axidl <=> ! [X149 : subst] : (((comp @ id @ X149)) = X149)) & (! [X150 : term,X151 : (subst > term > subst > $o)] : (~ ! [X152 : (subst > term > term)] : (~ ! [X153 : term] : (~(X151 @ id @ X153 @ id) | (X151 @ id @ (X152 @ id @ X153) @ id)) | (X151 @ id @ (lam @ (X152 @ sh @ one)) @ id) | ~ ! [X154 : subst,X155 : subst,X156 : term] : (((sub @ (X152 @ X154 @ X156) @ X155)) = ((X152 @ (comp @ X154 @ X155) @ (sub @ X156 @ X155))))) | (X151 @ id @ X150 @ id) | ~ ! [X157 : term,X158 : term] : (~(X151 @ id @ X157 @ id) | ~(X151 @ id @ X158 @ id) | (X151 @ id @ (ap @ (sub @ X157 @ id) @ X158) @ id))) <=> hoasinduction_no_psi_cond) & hoasapinj2_gthm & hoasinduction_lem2_lthm & (sh = ((d2subst @ d_id))) & ((! [X159 : term,X160 : subst] : (((sub @ one @ (push @ X159 @ X160))) = X159) => ! [X161 : term,X162 : (term > $o),X163 : subst,X164 : (term > $o)] : ((X162 @ one) | ~ ! [X165 : term] : (((X164 @ (sub @ X165 @ (push @ X161 @ X163)))) = ((X162 @ X165))) | ~(X164 @ X161))) <=> pushprop_lem1v2_lthm) & (induction2lem_lthm <=> (! [X166 : subst,X167 : term] : (((sub @ one @ (push @ X167 @ X166))) = X167) => (! [X168 : subst,X169 : term] : (((comp @ sh @ (push @ X169 @ X168))) = X168) => (! [X170 : subst] : (((comp @ X170 @ id)) = X170) => (! [X171 : term,X172 : (term > $o)] : (~ ! [X173 : term,X174 : term] : (~(X172 @ X174) | ~(X172 @ X173) | (X172 @ (ap @ X173 @ X174))) | (X172 @ X171) | ~ ! [X175 : term] : (~(X172 @ X175) | (X172 @ (lam @ X175))) | ~ ! [X176 : term] : (~(var @ X176) | (X172 @ X176))) => ! [X177 : term,X178 : subst,X179 : (term > $o)] : (~ ! [X180 : term] : ((X179 @ (lam @ X180)) | ~ ! [X181 : term] : ((X179 @ (sub @ X180 @ (push @ X181 @ id))) | ~(X179 @ X181))) | ~ ! [X182 : term] : (~(var @ X182) | (X179 @ (sub @ X182 @ X178))) | (X179 @ (sub @ X177 @ X178)) | ~ ! [X183 : term,X184 : term] : ((X179 @ (ap @ X183 @ X184)) | ~(X179 @ X183) | ~(X179 @ X184)))))))) & hoasinduction_lem3b_gthm & (termmset_lthm <=> (! [X185 : term] : (((sub @ X185 @ id)) = X185) => ! [X186 : term] : (((sub @ X186 @ id)) = X186))) & (! [X187 : subst,X188 : term,X189 : (term > $o)] : (~ ! [X190 : term] : ((X189 @ (lam @ X190)) | ~ ! [X191 : term] : (~(X189 @ X191) | (X189 @ (sub @ X190 @ (push @ X191 @ id))))) | ~ ! [X192 : term,X193 : term] : (~(X189 @ X192) | ~(X189 @ X193) | (X189 @ (ap @ X193 @ X192))) | (X189 @ (sub @ X188 @ X187)) | ~ ! [X194 : term] : ((X189 @ (sub @ X194 @ X187)) | ~(var @ X194))) <=> induction2lem) & (((sub @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2term @ d_one))) & (hoasinduction_lem0 <=> ! [X195 : (subst > term > subst > $o)] : ~ ! [X196 : (term > $o)] : ~ ! [X197 : term] : (((X195 @ id @ X197 @ id)) = ((X196 @ X197)))) & hoasapnotvar_lthm & (! [X198 : term,X199 : (term > $o)] : (~ ! [X200 : term] : (~ ! [X201 : term] : ((X199 @ (sub @ X200 @ (push @ X201 @ id))) | ~(X199 @ X201)) | (X199 @ (lam @ X200))) | ~ ! [X202 : term,X203 : term] : (~(X199 @ X203) | (X199 @ (ap @ X203 @ X202)) | ~(X199 @ X202)) | ~ ! [X204 : term] : ((X199 @ X204) | ~(var @ X204)) | (X199 @ X198)) <=> induction2) & (induction <=> ! [X205 : (term > $o),X206 : term] : (~ ! [X207 : term] : ((X205 @ (lam @ X207)) | ~(X205 @ X207)) | (X205 @ X206) | ~ ! [X208 : term] : ((X205 @ X208) | ~(var @ X208)) | ~ ! [X209 : term,X210 : term] : (~(X205 @ X210) | ~(X205 @ X209) | (X205 @ (ap @ X209 @ X210))))) & ((! [X211 : subst,X212 : term] : (((sub @ one @ (push @ X212 @ X211))) = X212) => (! [X213 : subst,X214 : term] : (((comp @ sh @ (push @ X214 @ X213))) = X213) => ! [X215 : term,X216 : term,X217 : (term > $o),X218 : (term > $o),X219 : subst] : ((X218 @ (sub @ X216 @ (push @ X215 @ X219))) | ~(X217 @ X216)))) <=> pushprop_lem1_lthm) & axabs & (hoasinduction_lem2 <=> ! [X220 : (subst > term > subst > $o),X221 : term,X222 : term] : (~ ! [X223 : subst,X224 : term,X225 : subst,X226 : subst] : (~(X220 @ (comp @ X223 @ X225) @ (sub @ X224 @ X225) @ X226) | (X220 @ X223 @ X224 @ (comp @ X225 @ X226))) | ~ ! [X227 : term,X228 : term] : (~(X220 @ id @ X228 @ id) | (X220 @ id @ (ap @ (sub @ X227 @ id) @ X228) @ id) | ~(X220 @ id @ X227 @ id)) | ~(X220 @ id @ X222 @ id) | ~(X220 @ id @ X221 @ id) | (X220 @ id @ (ap @ X221 @ X222) @ id) | ~ ! [X229 : subst,X230 : term,X231 : subst,X232 : subst] : (~(X220 @ X231 @ X230 @ (comp @ X229 @ X232)) | (X220 @ (comp @ X231 @ X229) @ (sub @ X230 @ X229) @ X232)))) & pushprop_lthm & lamnotvar & (hoasinduction_lem3v2a_lthm <=> (! [X233 : term] : ~ ! [X234 : (subst > term > term)] : ~ ! [X235 : term,X236 : subst] : (((sub @ X233 @ (push @ X235 @ X236))) = ((X234 @ X236 @ X235))) => (! [X237 : term] : (((sub @ X237 @ id)) = X237) => ! [X238 : (subst > term > subst > $o),X239 : (term > $o),X240 : term] : ((X239 @ (lam @ X240)) | ~ ! [X241 : term] : ((X239 @ (sub @ X240 @ (push @ X241 @ id))) | ~(X239 @ X241)) | ~ ! [X242 : term] : (((X239 @ X242)) = ((X238 @ id @ X242 @ id))) | ~ ! [X243 : (subst > term > term)] : ((X238 @ id @ (lam @ (X243 @ sh @ one)) @ id) | ~ ! [X244 : term] : ((X238 @ id @ (X243 @ id @ X244) @ id) | ~(X238 @ id @ X244 @ id)) | ~ ! [X245 : term,X246 : subst,X247 : subst] : (((sub @ (X243 @ X247 @ X245) @ X246)) = ((X243 @ (comp @ X247 @ X246) @ (sub @ X245 @ X246))))))))) & ! [X248 : d_subst,X249 : d_subst] : ((((d2subst @ X249)) = ((d2subst @ X248))) => (X248 = X249)) & ! [X250 : term] : ? [X251 : d_term] : (((d2term @ X251)) = X250) & (one = ((d2term @ d_one))) & ! [X252 : d_subst] : (d_id = X252) & hoasinduction_lem3v2_gthm & (! [X253 : term,X254 : term,X255 : term,X256 : term] : ((X255 = X256) | (((ap @ (sub @ X256 @ id) @ X253)) != ((ap @ (sub @ X255 @ id) @ X254)))) <=> hoasapinj1) & axassoc & (((comp @ (d2subst @ d_id) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & ! [X257 : d_term] : (d_one = X257) & termmset_gthm & (hoaslaminj <=> ! [X258 : (subst > term > term),X259 : term,X260 : subst,X261 : (subst > term > term)] : (~ ! [X262 : term,X263 : subst,X264 : subst] : (((X261 @ (comp @ X263 @ X264) @ (sub @ X262 @ X264))) = ((sub @ (X261 @ X263 @ X262) @ X264))) | (((X261 @ X260 @ X259)) = ((X258 @ X260 @ X259))) | ~ ! [X265 : subst,X266 : term,X267 : subst] : (((X258 @ (comp @ X267 @ X265) @ (sub @ X266 @ X265))) = ((sub @ (X258 @ X267 @ X266) @ X265))) | (((lam @ (X258 @ sh @ one))) != ((lam @ (X261 @ sh @ one)))))) & (hoaslamnotap <=> ! [X268 : term,X269 : (subst > term > term),X270 : term] : (~ ! [X271 : term,X272 : subst,X273 : subst] : (((X269 @ (comp @ X272 @ X273) @ (sub @ X271 @ X273))) = ((sub @ (X269 @ X272 @ X271) @ X273))) | (((ap @ (sub @ X270 @ id) @ X268)) != ((lam @ (X269 @ sh @ one)))))) & ~lamnotap & (hoasinduction_lem3v2a <=> ! [X274 : (subst > term > subst > $o),X275 : term,X276 : (term > $o)] : (~ ! [X277 : term] : ((X276 @ (sub @ X275 @ (push @ X277 @ id))) | ~(X276 @ X277)) | ~ ! [X278 : (subst > term > term)] : ((X274 @ id @ (lam @ (X278 @ sh @ one)) @ id) | ~ ! [X279 : term] : (~(X274 @ id @ X279 @ id) | (X274 @ id @ (X278 @ id @ X279) @ id)) | ~ ! [X280 : subst,X281 : term,X282 : subst] : (((sub @ (X278 @ X280 @ X281) @ X282)) = ((X278 @ (comp @ X280 @ X282) @ (sub @ X281 @ X282))))) | (X276 @ (lam @ X275)) | ~ ! [X283 : term] : (((X274 @ id @ X283 @ id)) = ((X276 @ X283))))) & (pushprop_lem1v2 <=> ! [X284 : (term > $o),X285 : subst,X286 : (term > $o),X287 : term] : (~ ! [X288 : term] : (((X284 @ X288)) = ((X286 @ (sub @ X288 @ (push @ X287 @ X285))))) | ~(X286 @ X287) | (X284 @ one))) & (! [X289 : subst,X290 : term,X291 : (term > $o)] : ~ ! [X292 : (term > $o)] : ~ ! [X293 : term] : (((X291 @ (sub @ X293 @ (push @ X290 @ X289)))) = ((X292 @ X293))) <=> pushprop_lem0_lthm) & ((! [X294 : term,X295 : term,X296 : term,X297 : term] : ((X295 = X296) | (((ap @ X294 @ X295)) != ((ap @ X297 @ X296)))) => ! [X298 : term,X299 : term,X300 : term,X301 : term] : ((((ap @ (sub @ X298 @ id) @ X301)) != ((ap @ (sub @ X300 @ id) @ X299))) | (X299 = X301))) <=> hoasapinj2_lthm) & pushprop_lthm_orig & pushprop_lem1v2_gthm & (! [X302 : term,X303 : subst] : (((sub @ one @ (push @ X302 @ X303))) = X302) <=> axvarcons) & (! [X304 : term,X305 : term,X306 : term,X307 : term] : ((((ap @ X305 @ X304)) != ((ap @ X306 @ X307))) | (X305 = X306)) <=> apinj1) & (hoaslam = (^[X308 : subst, X309 : (subst > term > term)] : ((d2term @ d_one)))) & (pushprop_lem0 <=> ! [X310 : term,X311 : subst,X312 : (term > $o)] : ~ ! [X313 : (term > $o)] : ~ ! [X314 : term] : (((X313 @ X314)) = ((X312 @ (sub @ X314 @ (push @ X310 @ X311)))))) & induction2_gthm & (hoasinduction_lem3aa <=> ! [X315 : (subst > term > subst > $o),X316 : term] : (~ ! [X317 : term] : (~(X315 @ id @ X317 @ id) | (X315 @ id @ (sub @ X316 @ (push @ X317 @ id)) @ id)) | ~ ! [X318 : (subst > term > term)] : (~ ! [X319 : term] : (~(X315 @ id @ X319 @ id) | (X315 @ id @ (X318 @ id @ X319) @ id)) | (X315 @ id @ (lam @ (X318 @ sh @ one)) @ id) | ~ ! [X320 : subst,X321 : term,X322 : subst] : (((X318 @ (comp @ X322 @ X320) @ (sub @ X321 @ X320))) = ((sub @ (X318 @ X322 @ X321) @ X320)))) | (X315 @ id @ (lam @ (sub @ X316 @ (push @ one @ sh))) @ id))) & (hoasinduction_no_psi_cond_lthm <=> (! [X323 : (subst > term > subst > $o)] : ~ ! [X324 : (term > $o)] : ~ ! [X325 : term] : (((X323 @ id @ X325 @ id)) = ((X324 @ X325))) => (! [X326 : term,X327 : (term > $o)] : (~ ! [X328 : term] : (~ ! [X329 : term] : ((X327 @ (sub @ X328 @ (push @ X329 @ id))) | ~(X327 @ X329)) | (X327 @ (lam @ X328))) | (X327 @ X326) | ~ ! [X330 : term,X331 : term] : ((X327 @ (ap @ X330 @ X331)) | ~(X327 @ X330) | ~(X327 @ X331)) | ~ ! [X332 : term] : (~(var @ X332) | (X327 @ X332))) => (! [X333 : term] : (((sub @ X333 @ id)) = X333) => (! [X334 : (subst > term > subst > $o),X335 : (term > $o),X336 : term] : (~ ! [X337 : (subst > term > term)] : (~ ! [X338 : term] : (~(X334 @ id @ X338 @ id) | (X334 @ id @ (X337 @ id @ X338) @ id)) | ~ ! [X339 : subst,X340 : term,X341 : subst] : (((sub @ (X337 @ X341 @ X340) @ X339)) = ((X337 @ (comp @ X341 @ X339) @ (sub @ X340 @ X339)))) | (X334 @ id @ (lam @ (X337 @ sh @ one)) @ id)) | ~ ! [X342 : term] : ((X335 @ (sub @ X336 @ (push @ X342 @ id))) | ~(X335 @ X342)) | ~ ! [X343 : term] : (((X335 @ X343)) = ((X334 @ id @ X343 @ id))) | (X335 @ (lam @ X336))) => ! [X344 : term,X345 : (subst > term > subst > $o)] : (~ ! [X346 : (subst > term > term)] : (~ ! [X347 : term] : ((X345 @ id @ (X346 @ id @ X347) @ id) | ~(X345 @ id @ X347 @ id)) | (X345 @ id @ (lam @ (X346 @ sh @ one)) @ id) | ~ ! [X348 : term,X349 : subst,X350 : subst] : (((X346 @ (comp @ X350 @ X349) @ (sub @ X348 @ X349))) = ((sub @ (X346 @ X350 @ X348) @ X349)))) | (X345 @ id @ X344 @ id) | ~ ! [X351 : term,X352 : term] : (~(X345 @ id @ X351 @ id) | (X345 @ id @ (ap @ (sub @ X352 @ id) @ X351) @ id) | ~(X345 @ id @ X352 @ id)))))))) & (induction2_lthm <=> (! [X353 : term] : (((sub @ X353 @ id)) = X353) => (! [X354 : term,X355 : subst,X356 : (term > $o)] : (~ ! [X357 : term] : ((X356 @ (sub @ X357 @ X355)) | ~(var @ X357)) | (X356 @ (sub @ X354 @ X355)) | ~ ! [X358 : term,X359 : term] : (~(X356 @ X358) | ~(X356 @ X359) | (X356 @ (ap @ X359 @ X358))) | ~ ! [X360 : term] : (~ ! [X361 : term] : ((X356 @ (sub @ X360 @ (push @ X361 @ id))) | ~(X356 @ X361)) | (X356 @ (lam @ X360)))) => ! [X362 : term,X363 : (term > $o)] : (~ ! [X364 : term,X365 : term] : (~(X363 @ X364) | ~(X363 @ X365) | (X363 @ (ap @ X365 @ X364))) | (X363 @ X362) | ~ ! [X366 : term] : (~(var @ X366) | (X363 @ X366)) | ~ ! [X367 : term] : (~ ! [X368 : term] : (~(X363 @ X368) | (X363 @ (sub @ X367 @ (push @ X368 @ id)))) | (X363 @ (lam @ X367))))))) & (hoasapinj1_lthm <=> (! [X369 : term] : (((sub @ X369 @ id)) = X369) => (! [X370 : term,X371 : term,X372 : term,X373 : term] : ((((ap @ X371 @ X372)) != ((ap @ X373 @ X370))) | (X371 = X373)) => ! [X374 : term,X375 : term,X376 : term,X377 : term] : ((X375 = X377) | (((ap @ (sub @ X377 @ id) @ X374)) != ((ap @ (sub @ X375 @ id) @ X376))))))) & (! [X378 : term,X379 : (subst > term > subst > $o),X380 : (term > $o),X381 : term] : (~ ! [X382 : subst,X383 : term,X384 : subst,X385 : subst] : ((X379 @ X385 @ X383 @ (comp @ X384 @ X382)) | ~(X379 @ (comp @ X385 @ X384) @ (sub @ X383 @ X384) @ X382)) | ~ ! [X386 : subst,X387 : subst,X388 : term,X389 : subst] : ((X379 @ (comp @ X386 @ X389) @ (sub @ X388 @ X389) @ X387) | ~(X379 @ X386 @ X388 @ (comp @ X389 @ X387))) | ~(X380 @ X378) | ~(X380 @ X381) | ~ ! [X390 : term,X391 : term] : ((X379 @ id @ (ap @ (sub @ X390 @ id) @ X391) @ id) | ~(X379 @ id @ X390 @ id) | ~(X379 @ id @ X391 @ id)) | (X380 @ (ap @ X381 @ X378)) | ~ ! [X392 : term] : (((X379 @ id @ X392 @ id)) = ((X380 @ X392)))) <=> hoasinduction_lem2v2) & (hoasinduction_lem3_lthm <=> (! [X393 : term] : (((sub @ X393 @ id)) = X393) => (! [X394 : (subst > term > subst > $o),X395 : term] : ((X394 @ id @ (lam @ (sub @ X395 @ (push @ one @ sh))) @ id) | ~ ! [X396 : (subst > term > term)] : (~ ! [X397 : subst,X398 : subst,X399 : term] : (((X396 @ (comp @ X397 @ X398) @ (sub @ X399 @ X398))) = ((sub @ (X396 @ X397 @ X399) @ X398))) | (X394 @ id @ (lam @ (X396 @ sh @ one)) @ id) | ~ ! [X400 : term] : ((X394 @ id @ (X396 @ id @ X400) @ id) | ~(X394 @ id @ X400 @ id))) | ~ ! [X401 : term] : (~(X394 @ id @ X401 @ id) | (X394 @ id @ (sub @ X395 @ (push @ X401 @ id)) @ id))) => ! [X402 : (subst > term > subst > $o),X403 : term] : (~ ! [X404 : (subst > term > term)] : ((X402 @ id @ (lam @ (X404 @ sh @ one)) @ id) | ~ ! [X405 : term] : ((X402 @ id @ (X404 @ id @ X405) @ id) | ~(X402 @ id @ X405 @ id)) | ~ ! [X406 : subst,X407 : subst,X408 : term] : (((sub @ (X404 @ X406 @ X408) @ X407)) = ((X404 @ (comp @ X406 @ X407) @ (sub @ X408 @ X407))))) | ~ ! [X409 : term] : ((X402 @ id @ (sub @ X403 @ (push @ X409 @ id)) @ id) | ~(X402 @ id @ X409 @ id)) | ~ ! [X410 : term,X411 : subst,X412 : subst,X413 : subst] : (~(X402 @ (comp @ X413 @ X411) @ (sub @ X410 @ X411) @ X412) | (X402 @ X413 @ X410 @ (comp @ X411 @ X412))) | ~ ! [X414 : subst,X415 : term,X416 : subst,X417 : subst] : (~(X402 @ X417 @ X415 @ (comp @ X416 @ X414)) | (X402 @ (comp @ X417 @ X416) @ (sub @ X415 @ X416) @ X414)) | (X402 @ id @ (lam @ X403) @ id))))) & hoasinduction_lem1_gthm & (pushprop_lem1 <=> ! [X418 : subst,X419 : term,X420 : (term > $o),X421 : (term > $o),X422 : term] : ((X421 @ (sub @ X422 @ (push @ X419 @ X418))) | ~(X420 @ X422))) & hoasapnotvar_gthm & (! [X423 : term,X424 : (subst > term > subst > $o)] : (~ ! [X425 : subst,X426 : term,X427 : subst,X428 : subst] : (~(X424 @ (comp @ X425 @ X428) @ (sub @ X426 @ X428) @ X427) | (X424 @ X425 @ X426 @ (comp @ X428 @ X427))) | ~ ! [X429 : subst,X430 : subst,X431 : term,X432 : subst] : ((X424 @ (comp @ X430 @ X432) @ (sub @ X431 @ X432) @ X429) | ~(X424 @ X430 @ X431 @ (comp @ X432 @ X429))) | ~ ! [X433 : (subst > term > term)] : (~ ! [X434 : subst,X435 : term,X436 : subst] : (((sub @ (X433 @ X436 @ X435) @ X434)) = ((X433 @ (comp @ X436 @ X434) @ (sub @ X435 @ X434)))) | (X424 @ id @ (lam @ (X433 @ sh @ one)) @ id) | ~ ! [X437 : term] : (~(X424 @ id @ X437 @ id) | (X424 @ id @ (X433 @ id @ X437) @ id))) | (X424 @ id @ (lam @ X423) @ id) | ~ ! [X438 : term] : ((X424 @ id @ (sub @ X423 @ (push @ X438 @ id)) @ id) | ~(X424 @ id @ X438 @ id))) <=> hoasinduction_lem3) & (axscons <=> ! [X439 : subst] : (((push @ (sub @ one @ X439) @ (comp @ sh @ X439))) = X439)) & (! [X440 : term] : ~ ! [X441 : (subst > term > term)] : (((X441 @ sh @ one)) != ((sub @ X440 @ (push @ one @ sh)))) <=> hoasinduction_lem3b) & apnotvar & (((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) = ((d2term @ d_one))) & ! [X442 : d_term,X443 : d_term] : ((((d2term @ X443)) = ((d2term @ X442))) => (X442 = X443)) & hoaslaminj_gthm & (! [X444 : term] : ~ ! [X445 : (subst > term > term)] : (((X445 @ sh @ one)) != ((sub @ X444 @ (push @ one @ sh)))) <=> hoasinduction_lem3b_lthm) & (substmonoid_lthm <=> (! [X446 : subst] : (((comp @ id @ X446)) = X446) => (! [X447 : subst] : (((comp @ X447 @ id)) = X447) => (! [X448 : subst] : (((comp @ X448 @ id)) = X448) & ! [X449 : subst] : (((comp @ id @ X449)) = X449))))) & axapp & (! [X450 : (subst > term > subst > $o),X451 : term] : (~ ! [X452 : term] : (~(X450 @ id @ X452 @ id) | (X450 @ id @ (sub @ X451 @ (push @ X452 @ id)) @ id)) | ~ ! [X453 : (subst > term > term)] : ((X450 @ id @ (lam @ (X453 @ sh @ one)) @ id) | ~ ! [X454 : term] : (~(X450 @ id @ X454 @ id) | (X450 @ id @ (X453 @ id @ X454) @ id)) | ~ ! [X455 : term,X456 : subst,X457 : subst] : (((X453 @ (comp @ X456 @ X457) @ (sub @ X455 @ X457))) = ((sub @ (X453 @ X456 @ X455) @ X457)))) | (X450 @ id @ (lam @ X451) @ id)) <=> hoasinduction_lem3a) & (laminj <=> ! [X458 : term,X459 : term] : ((X458 = X459) | (((lam @ X459)) != ((lam @ X458))))) & hoasinduction_lem1_lthm & pushprop_lem3v2 & (((ap @ (d2term @ d_one) @ (d2term @ d_one))) = ((d2term @ d_one))) & ((! [X460 : (term > $o),X461 : term] : ((X460 @ X461) | ~ ! [X462 : term] : (~ ! [X463 : term] : ((X460 @ (sub @ X462 @ (push @ X463 @ id))) | ~(X460 @ X463)) | (X460 @ (lam @ X462))) | ~ ! [X464 : term] : (~(var @ X464) | (X460 @ X464)) | ~ ! [X465 : term,X466 : term] : (~(X460 @ X466) | ~(X460 @ X465) | (X460 @ (ap @ X465 @ X466)))) => (! [X467 : (subst > term > subst > $o),X468 : term,X469 : term] : (~(X467 @ id @ X468 @ id) | ~(X467 @ id @ X469 @ id) | ~ ! [X470 : term,X471 : term] : (~(X467 @ id @ X470 @ id) | ~(X467 @ id @ X471 @ id) | (X467 @ id @ (ap @ (sub @ X470 @ id) @ X471) @ id)) | (X467 @ id @ (ap @ X469 @ X468) @ id) | ~ ! [X472 : term,X473 : subst,X474 : subst,X475 : subst] : ((X467 @ (comp @ X474 @ X473) @ (sub @ X472 @ X473) @ X475) | ~(X467 @ X474 @ X472 @ (comp @ X473 @ X475))) | ~ ! [X476 : subst,X477 : subst,X478 : term,X479 : subst] : ((X467 @ X476 @ X478 @ (comp @ X477 @ X479)) | ~(X467 @ (comp @ X476 @ X477) @ (sub @ X478 @ X477) @ X479))) => (! [X480 : term,X481 : (subst > term > subst > $o)] : (~ ! [X482 : subst,X483 : term,X484 : subst,X485 : subst] : (~(X481 @ X485 @ X483 @ (comp @ X484 @ X482)) | (X481 @ (comp @ X485 @ X484) @ (sub @ X483 @ X484) @ X482)) | ~ ! [X486 : subst,X487 : subst,X488 : term,X489 : subst] : (~(X481 @ (comp @ X489 @ X486) @ (sub @ X488 @ X486) @ X487) | (X481 @ X489 @ X488 @ (comp @ X486 @ X487))) | ~ ! [X490 : term] : (~(X481 @ id @ X490 @ id) | (X481 @ id @ (sub @ X480 @ (push @ X490 @ id)) @ id)) | (X481 @ id @ (lam @ X480) @ id) | ~ ! [X491 : (subst > term > term)] : ((X481 @ id @ (lam @ (X491 @ sh @ one)) @ id) | ~ ! [X492 : term,X493 : subst,X494 : subst] : (((X491 @ (comp @ X494 @ X493) @ (sub @ X492 @ X493))) = ((sub @ (X491 @ X494 @ X492) @ X493))) | ~ ! [X495 : term] : (~(X481 @ id @ X495 @ id) | (X481 @ id @ (X491 @ id @ X495) @ id)))) => ! [X496 : term,X497 : (subst > term > subst > $o)] : (~ ! [X498 : term] : (~(var @ (sub @ X498 @ id)) | (X497 @ id @ X498 @ id)) | (X497 @ id @ X496 @ id) | ~ ! [X499 : term,X500 : subst,X501 : subst,X502 : subst] : ((X497 @ X500 @ X499 @ (comp @ X501 @ X502)) | ~(X497 @ (comp @ X500 @ X501) @ (sub @ X499 @ X501) @ X502)) | ~ ! [X503 : subst,X504 : subst,X505 : term,X506 : subst] : ((X497 @ (comp @ X503 @ X504) @ (sub @ X505 @ X504) @ X506) | ~(X497 @ X503 @ X505 @ (comp @ X504 @ X506))) | ~ ! [X507 : term,X508 : term] : ((X497 @ id @ (ap @ (sub @ X507 @ id) @ X508) @ id) | ~(X497 @ id @ X507 @ id) | ~(X497 @ id @ X508 @ id)) | ~ ! [X509 : (subst > term > term)] : (~ ! [X510 : term] : ((X497 @ id @ (X509 @ id @ X510) @ id) | ~(X497 @ id @ X510 @ id)) | ~ ! [X511 : subst,X512 : subst,X513 : term] : (((X509 @ (comp @ X512 @ X511) @ (sub @ X513 @ X511))) = ((sub @ (X509 @ X512 @ X513) @ X511))) | (X497 @ id @ (lam @ (X509 @ sh @ one)) @ id)))))) <=> hoasinduction_lthm) & induction2lem_gthm & ((! [X514 : subst] : (((comp @ X514 @ id)) = X514) & ! [X515 : subst] : (((comp @ id @ X515)) = X515)) <=> substmonoid) & (! [X516 : term,X517 : (subst > term > subst > $o)] : (~ ! [X518 : subst,X519 : subst,X520 : subst,X521 : term] : ((X517 @ (comp @ X518 @ X519) @ (sub @ X521 @ X519) @ X520) | ~(X517 @ X518 @ X521 @ (comp @ X519 @ X520))) | ~ ! [X522 : term,X523 : term] : ((X517 @ id @ (ap @ (sub @ X523 @ id) @ X522) @ id) | ~(X517 @ id @ X522 @ id) | ~(X517 @ id @ X523 @ id)) | ~ ! [X524 : subst,X525 : subst,X526 : subst,X527 : term] : ((X517 @ X524 @ X527 @ (comp @ X526 @ X525)) | ~(X517 @ (comp @ X524 @ X526) @ (sub @ X527 @ X526) @ X525)) | ~ ! [X528 : term] : ((X517 @ id @ X528 @ id) | ~(var @ (sub @ X528 @ id))) | ~ ! [X529 : (subst > term > term)] : (~ ! [X530 : term] : ((X517 @ id @ (X529 @ id @ X530) @ id) | ~(X517 @ id @ X530 @ id)) | (X517 @ id @ (lam @ (X529 @ sh @ one)) @ id) | ~ ! [X531 : subst,X532 : subst,X533 : term] : (((X529 @ (comp @ X531 @ X532) @ (sub @ X533 @ X532))) = ((sub @ (X529 @ X531 @ X533) @ X532)))) | (X517 @ id @ X516 @ id)) <=> hoasinduction) & hoaslamnotap_gthm & (((push @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & (hoasinduction_lem0_lthm <=> ! [X534 : (subst > term > subst > $o)] : ~ ! [X535 : (term > $o)] : ~ ! [X536 : term] : (((X534 @ id @ X536 @ id)) = ((X535 @ X536)))) & (! [X537 : term] : (((sub @ X537 @ id)) = X537) <=> axvarid) & hoasinduction_lem3_gthm & (apinj2 <=> ! [X538 : term,X539 : term,X540 : term,X541 : term] : ((X539 = X541) | (((ap @ X538 @ X539)) != ((ap @ X540 @ X541))))) & hoasinduction_lem1v2 & ~(var @ (d2term @ d_one)) & (hoasinduction_lem3aa_lthm <=> ! [X542 : term,X543 : (subst > term > subst > $o)] : (~ ! [X544 : term] : (~(X543 @ id @ X544 @ id) | (X543 @ id @ (sub @ X542 @ (push @ X544 @ id)) @ id)) | ~ ! [X545 : (subst > term > term)] : ((X543 @ id @ (lam @ (X545 @ sh @ one)) @ id) | ~ ! [X546 : term] : (~(X543 @ id @ X546 @ id) | (X543 @ id @ (X545 @ id @ X546) @ id)) | ~ ! [X547 : subst,X548 : term,X549 : subst] : (((X545 @ (comp @ X549 @ X547) @ (sub @ X548 @ X547))) = ((sub @ (X545 @ X549 @ X548) @ X547)))) | (X543 @ id @ (lam @ (sub @ X542 @ (push @ one @ sh))) @ id))) & pushprop_lem1_gthm & hoasapnotvar),
  inference(rectify,[],[f1])).
thf(f5,plain,(
  (axclos = $true) & (pushprop_lem2v2_lthm = $true) & (hoasinduction_lem1v2_gthm = $true) & (hoasinduction_lem2v2_gthm = $true) & (! [X0 : term] : ~ ! [X1 : (subst > term > term)] : ~ ! [X2 : term,X3 : subst] : (((sub @ X0 @ (push @ X2 @ X3))) = ((X1 @ X3 @ X2))) <=> (hoasinduction_lem3v2_f = $true)) & ! [X4 : subst] : ? [X5 : d_subst] : (((d2subst @ X5)) = X4) & ((hoasinduction_lthm_3 = $true) <=> (! [X6 : (subst > term > subst > $o)] : ~ ! [X7 : (term > $o)] : ~ ! [X8 : term] : (((X7 @ X8)) = ((X6 @ id @ X8 @ id))) => (! [X9 : (term > $o),X10 : term] : (~ ! [X11 : term,X12 : term] : (~ ($true = ((X9 @ X11))) | ~ (((X9 @ X12)) = $true) | (((X9 @ (ap @ X11 @ X12))) = $true)) | ~ ! [X13 : term] : (($true = ((X9 @ X13))) | ~ ($true = ((var @ X13)))) | ($true = ((X9 @ X10))) | ~ ! [X14 : term] : (~ ! [X15 : term] : (~ ($true = ((X9 @ X15))) | ($true = ((X9 @ (sub @ X14 @ (push @ X15 @ id)))))) | ($true = ((X9 @ (lam @ X14)))))) => (! [X16 : term] : (((sub @ X16 @ id)) = X16) => (! [X17 : term,X18 : (term > $o),X19 : (subst > term > subst > $o)] : (~ ! [X20 : (subst > term > term)] : (~ ! [X21 : term] : (~ (((X19 @ id @ X21 @ id)) = $true) | ($true = ((X19 @ id @ (X20 @ id @ X21) @ id)))) | (((X19 @ id @ (lam @ (X20 @ sh @ one)) @ id)) = $true) | ~ ! [X22 : term,X23 : subst,X24 : subst] : (((X20 @ (comp @ X23 @ X24) @ (sub @ X22 @ X24))) = ((sub @ (X20 @ X23 @ X22) @ X24)))) | ($true = ((X18 @ (lam @ X17)))) | ~ ! [X25 : term] : (($true = ((X18 @ (sub @ X17 @ (push @ X25 @ id))))) | ~ ($true = ((X18 @ X25)))) | ~ ! [X26 : term] : (((X18 @ X26)) = ((X19 @ id @ X26 @ id)))) => ! [X27 : term,X28 : (subst > term > subst > $o)] : (~ ! [X29 : term,X30 : term] : (~ ($true = ((X28 @ id @ X30 @ id))) | ~ ($true = ((X28 @ id @ X29 @ id))) | ($true = ((X28 @ id @ (ap @ (sub @ X30 @ id) @ X29) @ id)))) | ~ ! [X31 : (subst > term > term)] : ((((X28 @ id @ (lam @ (X31 @ sh @ one)) @ id)) = $true) | ~ ! [X32 : term] : (~ ($true = ((X28 @ id @ X32 @ id))) | ($true = ((X28 @ id @ (X31 @ id @ X32) @ id)))) | ~ ! [X34 : term,X35 : subst,X33 : subst] : (((sub @ (X31 @ X35 @ X34) @ X33)) = ((X31 @ (comp @ X35 @ X33) @ (sub @ X34 @ X33))))) | ~ ! [X36 : term,X37 : subst,X38 : subst,X39 : subst] : (~ ($true = ((X28 @ (comp @ X37 @ X39) @ (sub @ X36 @ X39) @ X38))) | ($true = ((X28 @ X37 @ X36 @ (comp @ X39 @ X38))))) | ~ ! [X40 : subst,X41 : term,X42 : subst,X43 : subst] : ((((X28 @ (comp @ X43 @ X40) @ (sub @ X41 @ X40) @ X42)) = $true) | ~ (((X28 @ X43 @ X41 @ (comp @ X40 @ X42))) = $true)) | ~ ! [X44 : term] : (~ ($true = ((var @ (sub @ X44 @ id)))) | ($true = ((X28 @ id @ X44 @ id)))) | (((X28 @ id @ X27 @ id)) = $true))))))) & (hoaslamnotvar_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (axmap = $true) & (! [X48 : term,X49 : term] : ((X48 = X49) | (((sub @ X49 @ sh)) != ((sub @ X48 @ sh)))) <=> (shinj = $true)) & (! [X50 : term,X51 : term,X52 : term,X53 : term] : ((X50 = X53) | (((ap @ (sub @ X52 @ id) @ X53)) != ((ap @ (sub @ X51 @ id) @ X50)))) <=> (hoasapinj2 = $true)) & (pushprop = $true) & (pushprop_lem0_gthm = $true) & (hoaslamnotvar = $true) & (! [X54 : subst,X55 : term] : (((comp @ sh @ (push @ X55 @ X54))) = X54) <=> (axshiftcons = $true)) & (substmonoid_gthm = $true) & (pushprop_lem2v2_gthm = $true) & ((hoasinduction_lem3v2 = $true) <=> ! [X56 : term,X57 : (subst > term > subst > $o),X58 : (term > $o)] : (~ ! [X59 : (subst > term > term)] : (~ ! [X60 : term] : (~ ($true = ((X57 @ id @ X60 @ id))) | (((X57 @ id @ (X59 @ id @ X60) @ id)) = $true)) | ~ ! [X61 : subst,X62 : subst,X63 : term] : (((sub @ (X59 @ X62 @ X63) @ X61)) = ((X59 @ (comp @ X62 @ X61) @ (sub @ X63 @ X61)))) | ($true = ((X57 @ id @ (lam @ (X59 @ sh @ one)) @ id)))) | ~ ! [X64 : subst,X65 : subst,X66 : subst,X67 : term] : (~ ($true = ((X57 @ X64 @ X67 @ (comp @ X66 @ X65)))) | ($true = ((X57 @ (comp @ X64 @ X66) @ (sub @ X67 @ X66) @ X65)))) | ($true = ((X58 @ (lam @ X56)))) | ~ ! [X68 : term,X69 : subst,X70 : subst,X71 : subst] : (~ ($true = ((X57 @ (comp @ X71 @ X70) @ (sub @ X68 @ X70) @ X69))) | (((X57 @ X71 @ X68 @ (comp @ X70 @ X69))) = $true)) | ~ ! [X72 : term] : (~ ($true = ((X58 @ X72))) | (((X58 @ (sub @ X56 @ (push @ X72 @ id)))) = $true)) | ~ ! [X73 : term] : (((X57 @ id @ X73 @ id)) = ((X58 @ X73))))) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (ulamvarsh = $true) & ((! [X74 : term] : (((sub @ X74 @ id)) = X74) => ! [X75 : term,X76 : (subst > term > subst > $o),X77 : (term > $o)] : (($true = ((X77 @ (lam @ X75)))) | ~ ! [X78 : term] : (~ ($true = ((X77 @ X78))) | ($true = ((X77 @ (sub @ X75 @ (push @ X78 @ id)))))) | ~ ! [X79 : term,X80 : subst,X81 : subst,X82 : subst] : (($true = ((X76 @ (comp @ X80 @ X81) @ (sub @ X79 @ X81) @ X82))) | ~ (((X76 @ X80 @ X79 @ (comp @ X81 @ X82))) = $true)) | ~ ! [X83 : (subst > term > term)] : (~ ! [X84 : term] : (~ ($true = ((X76 @ id @ X84 @ id))) | ($true = ((X76 @ id @ (X83 @ id @ X84) @ id)))) | ~ ! [X85 : subst,X86 : term,X87 : subst] : (((X83 @ (comp @ X87 @ X85) @ (sub @ X86 @ X85))) = ((sub @ (X83 @ X87 @ X86) @ X85))) | (((X76 @ id @ (lam @ (X83 @ sh @ one)) @ id)) = $true)) | ~ ! [X88 : subst,X89 : subst,X90 : subst,X91 : term] : (~ (((X76 @ (comp @ X88 @ X90) @ (sub @ X91 @ X90) @ X89)) = $true) | ($true = ((X76 @ X88 @ X91 @ (comp @ X90 @ X89))))) | ~ ! [X92 : term] : (((X77 @ X92)) = ((X76 @ id @ X92 @ id))))) <=> (hoasinduction_lem3v2_lthm = $true)) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X93 : term] : ~ ! [X94 : (subst > term > term)] : ~ ! [X95 : term,X96 : subst] : (((sub @ X93 @ (push @ X95 @ X96))) = ((X94 @ X96 @ X95)))) & ~ (ulamvar1 = $true) & (ulamvarind = $true) & (hoasinduction_lem1 = $true) & (pushprop_lem2v2 = $true) & (hoasinduction_lem3a_gthm = $true) & (hoasapinj1_gthm = $true) & ((hoaslaminj_lthm = $true) <=> (! [X97 : term,X98 : subst] : (((sub @ one @ (push @ X97 @ X98))) = X97) => (! [X99 : term,X100 : subst] : (((comp @ sh @ (push @ X99 @ X100))) = X100) => (! [X101 : term,X102 : term] : ((((lam @ X102)) != ((lam @ X101))) | (X101 = X102)) => ! [X103 : term,X104 : (subst > term > term),X105 : (subst > term > term),X106 : subst] : (~ ! [X107 : term,X108 : subst,X109 : subst] : (((sub @ (X105 @ X108 @ X107) @ X109)) = ((X105 @ (comp @ X108 @ X109) @ (sub @ X107 @ X109)))) | (((X104 @ X106 @ X103)) = ((X105 @ X106 @ X103))) | (((lam @ (X105 @ sh @ one))) != ((lam @ (X104 @ sh @ one)))) | ~ ! [X110 : subst,X111 : subst,X112 : term] : (((sub @ (X104 @ X111 @ X112) @ X110)) = ((X104 @ (comp @ X111 @ X110) @ (sub @ X112 @ X110))))))))) & ~ (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & (hoaslamnotap_lthm = $true) & (! [X118 : term] : (((sub @ X118 @ id)) = X118) <=> (termmset = $true)) & (pushprop_gthm = $true) & (((d2subst @ d_id)) = id) & ((! [X119 : term] : (((sub @ X119 @ id)) = X119) => (! [X120 : (subst > term > subst > $o),X121 : term] : (($true = ((X120 @ id @ (lam @ (sub @ X121 @ (push @ one @ sh))) @ id))) | ~ ! [X122 : (subst > term > term)] : (~ ! [X123 : term] : (~ ($true = ((X120 @ id @ X123 @ id))) | (((X120 @ id @ (X122 @ id @ X123) @ id)) = $true)) | ($true = ((X120 @ id @ (lam @ (X122 @ sh @ one)) @ id))) | ~ ! [X124 : subst,X125 : term,X126 : subst] : (((X122 @ (comp @ X126 @ X124) @ (sub @ X125 @ X124))) = ((sub @ (X122 @ X126 @ X125) @ X124)))) | ~ ! [X127 : term] : (~ (((X120 @ id @ X127 @ id)) = $true) | ($true = ((X120 @ id @ (sub @ X121 @ (push @ X127 @ id)) @ id))))) => ! [X128 : term,X129 : (subst > term > subst > $o)] : (~ ! [X130 : (subst > term > term)] : (~ ! [X131 : term] : (~ ($true = ((X129 @ id @ X131 @ id))) | ($true = ((X129 @ id @ (X130 @ id @ X131) @ id)))) | ~ ! [X132 : term,X133 : subst,X134 : subst] : (((sub @ (X130 @ X133 @ X132) @ X134)) = ((X130 @ (comp @ X133 @ X134) @ (sub @ X132 @ X134)))) | ($true = ((X129 @ id @ (lam @ (X130 @ sh @ one)) @ id)))) | ~ ! [X135 : term] : (~ (((X129 @ id @ X135 @ id)) = $true) | (((X129 @ id @ (sub @ X128 @ (push @ X135 @ id)) @ id)) = $true)) | ($true = ((X129 @ id @ (lam @ X128) @ id)))))) <=> (hoasinduction_lem3a_lthm = $true)) & ((hoasinduction_lem3aaa = $true) <=> ! [X136 : (subst > term > subst > $o),X137 : term] : (($true = ((X136 @ id @ (lam @ (sub @ X137 @ (push @ one @ sh))) @ id))) | ~ ! [X138 : term] : ((((X136 @ id @ (sub @ X137 @ (push @ X138 @ id)) @ id)) = $true) | ~ ($true = ((X136 @ id @ X138 @ id)))) | ~ ! [X139 : (subst > term > term),X140 : term] : (~ ! [X141 : subst,X142 : term,X143 : subst] : (((sub @ (X139 @ X143 @ X142) @ X141)) = ((sub @ (sub @ X140 @ (push @ X142 @ X143)) @ X141))) | (((X136 @ id @ (lam @ (X139 @ sh @ one)) @ id)) = $true) | ~ ! [X144 : term] : (~ (((X136 @ id @ X144 @ id)) = $true) | (((X136 @ id @ (X139 @ id @ X144) @ id)) = $true)) | ~ ! [X145 : subst,X146 : term,X147 : subst] : (((sub @ X140 @ (push @ (sub @ X146 @ X145) @ (comp @ X147 @ X145)))) = ((X139 @ (comp @ X147 @ X145) @ (sub @ X146 @ X145))))))) & (pushprop_lem3v2_lthm = $true) & (axvarshift = $true) & (hoasinduction_lem2_gthm = $true) & (hoaslamnotvar_lthm = $true) & (hoasinduction_gthm = $true) & ((axidr = $true) <=> ! [X148 : subst] : (((comp @ X148 @ id)) = X148)) & ((axidl = $true) <=> ! [X149 : subst] : (((comp @ id @ X149)) = X149)) & (! [X150 : term,X151 : (subst > term > subst > $o)] : (~ ! [X152 : (subst > term > term)] : (~ ! [X153 : term] : (~ ($true = ((X151 @ id @ X153 @ id))) | ($true = ((X151 @ id @ (X152 @ id @ X153) @ id)))) | (((X151 @ id @ (lam @ (X152 @ sh @ one)) @ id)) = $true) | ~ ! [X154 : subst,X155 : subst,X156 : term] : (((sub @ (X152 @ X154 @ X156) @ X155)) = ((X152 @ (comp @ X154 @ X155) @ (sub @ X156 @ X155))))) | ($true = ((X151 @ id @ X150 @ id))) | ~ ! [X157 : term,X158 : term] : (~ ($true = ((X151 @ id @ X157 @ id))) | ~ ($true = ((X151 @ id @ X158 @ id))) | ($true = ((X151 @ id @ (ap @ (sub @ X157 @ id) @ X158) @ id))))) <=> (hoasinduction_no_psi_cond = $true)) & (hoasapinj2_gthm = $true) & (hoasinduction_lem2_lthm = $true) & (((d2subst @ d_id)) = sh) & ((! [X159 : term,X160 : subst] : (((sub @ one @ (push @ X159 @ X160))) = X159) => ! [X161 : term,X162 : (term > $o),X163 : subst,X164 : (term > $o)] : ((((X162 @ one)) = $true) | ~ ! [X165 : term] : (((X164 @ (sub @ X165 @ (push @ X161 @ X163)))) = ((X162 @ X165))) | ~ ($true = ((X164 @ X161))))) <=> (pushprop_lem1v2_lthm = $true)) & ((induction2lem_lthm = $true) <=> (! [X166 : subst,X167 : term] : (((sub @ one @ (push @ X167 @ X166))) = X167) => (! [X168 : subst,X169 : term] : (((comp @ sh @ (push @ X169 @ X168))) = X168) => (! [X170 : subst] : (((comp @ X170 @ id)) = X170) => (! [X171 : term,X172 : (term > $o)] : (~ ! [X173 : term,X174 : term] : (~ ($true = ((X172 @ X174))) | ~ ($true = ((X172 @ X173))) | ($true = ((X172 @ (ap @ X173 @ X174))))) | ($true = ((X172 @ X171))) | ~ ! [X175 : term] : (~ ($true = ((X172 @ X175))) | (((X172 @ (lam @ X175))) = $true)) | ~ ! [X176 : term] : (~ ($true = ((var @ X176))) | (((X172 @ X176)) = $true))) => ! [X177 : term,X178 : subst,X179 : (term > $o)] : (~ ! [X180 : term] : ((((X179 @ (lam @ X180))) = $true) | ~ ! [X181 : term] : (($true = ((X179 @ (sub @ X180 @ (push @ X181 @ id))))) | ~ (((X179 @ X181)) = $true))) | ~ ! [X182 : term] : (~ ($true = ((var @ X182))) | ($true = ((X179 @ (sub @ X182 @ X178))))) | ($true = ((X179 @ (sub @ X177 @ X178)))) | ~ ! [X183 : term,X184 : term] : ((((X179 @ (ap @ X183 @ X184))) = $true) | ~ (((X179 @ X183)) = $true) | ~ ($true = ((X179 @ X184)))))))))) & (hoasinduction_lem3b_gthm = $true) & ((termmset_lthm = $true) <=> (! [X185 : term] : (((sub @ X185 @ id)) = X185) => ! [X186 : term] : (((sub @ X186 @ id)) = X186))) & (! [X187 : subst,X188 : term,X189 : (term > $o)] : (~ ! [X190 : term] : (($true = ((X189 @ (lam @ X190)))) | ~ ! [X191 : term] : (~ (((X189 @ X191)) = $true) | ($true = ((X189 @ (sub @ X190 @ (push @ X191 @ id))))))) | ~ ! [X192 : term,X193 : term] : (~ (((X189 @ X192)) = $true) | ~ ($true = ((X189 @ X193))) | (((X189 @ (ap @ X193 @ X192))) = $true)) | ($true = ((X189 @ (sub @ X188 @ X187)))) | ~ ! [X194 : term] : (($true = ((X189 @ (sub @ X194 @ X187)))) | ~ (((var @ X194)) = $true))) <=> (induction2lem = $true)) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoasinduction_lem0 = $true) <=> ! [X195 : (subst > term > subst > $o)] : ~ ! [X196 : (term > $o)] : ~ ! [X197 : term] : (((X195 @ id @ X197 @ id)) = ((X196 @ X197)))) & (hoasapnotvar_lthm = $true) & (! [X198 : term,X199 : (term > $o)] : (~ ! [X200 : term] : (~ ! [X201 : term] : (($true = ((X199 @ (sub @ X200 @ (push @ X201 @ id))))) | ~ ($true = ((X199 @ X201)))) | (((X199 @ (lam @ X200))) = $true)) | ~ ! [X202 : term,X203 : term] : (~ (((X199 @ X203)) = $true) | ($true = ((X199 @ (ap @ X203 @ X202)))) | ~ ($true = ((X199 @ X202)))) | ~ ! [X204 : term] : (($true = ((X199 @ X204))) | ~ ($true = ((var @ X204)))) | ($true = ((X199 @ X198)))) <=> (induction2 = $true)) & ((induction = $true) <=> ! [X205 : (term > $o),X206 : term] : (~ ! [X207 : term] : (($true = ((X205 @ (lam @ X207)))) | ~ ($true = ((X205 @ X207)))) | ($true = ((X205 @ X206))) | ~ ! [X208 : term] : (($true = ((X205 @ X208))) | ~ ($true = ((var @ X208)))) | ~ ! [X209 : term,X210 : term] : (~ (((X205 @ X210)) = $true) | ~ ($true = ((X205 @ X209))) | ($true = ((X205 @ (ap @ X209 @ X210))))))) & ((! [X211 : subst,X212 : term] : (((sub @ one @ (push @ X212 @ X211))) = X212) => (! [X213 : subst,X214 : term] : (((comp @ sh @ (push @ X214 @ X213))) = X213) => ! [X215 : term,X216 : term,X217 : (term > $o),X218 : (term > $o),X219 : subst] : ((((X218 @ (sub @ X216 @ (push @ X215 @ X219)))) = $true) | ~ (((X217 @ X216)) = $true)))) <=> (pushprop_lem1_lthm = $true)) & (axabs = $true) & ((hoasinduction_lem2 = $true) <=> ! [X220 : (subst > term > subst > $o),X221 : term,X222 : term] : (~ ! [X223 : subst,X224 : term,X225 : subst,X226 : subst] : (~ ($true = ((X220 @ (comp @ X223 @ X225) @ (sub @ X224 @ X225) @ X226))) | ($true = ((X220 @ X223 @ X224 @ (comp @ X225 @ X226))))) | ~ ! [X227 : term,X228 : term] : (~ (((X220 @ id @ X228 @ id)) = $true) | ($true = ((X220 @ id @ (ap @ (sub @ X227 @ id) @ X228) @ id))) | ~ ($true = ((X220 @ id @ X227 @ id)))) | ~ ($true = ((X220 @ id @ X222 @ id))) | ~ ($true = ((X220 @ id @ X221 @ id))) | (((X220 @ id @ (ap @ X221 @ X222) @ id)) = $true) | ~ ! [X229 : subst,X230 : term,X231 : subst,X232 : subst] : (~ ($true = ((X220 @ X231 @ X230 @ (comp @ X229 @ X232)))) | (((X220 @ (comp @ X231 @ X229) @ (sub @ X230 @ X229) @ X232)) = $true)))) & (pushprop_lthm = $true) & (lamnotvar = $true) & ((hoasinduction_lem3v2a_lthm = $true) <=> (! [X233 : term] : ~ ! [X234 : (subst > term > term)] : ~ ! [X235 : term,X236 : subst] : (((sub @ X233 @ (push @ X235 @ X236))) = ((X234 @ X236 @ X235))) => (! [X237 : term] : (((sub @ X237 @ id)) = X237) => ! [X238 : (subst > term > subst > $o),X239 : (term > $o),X240 : term] : (($true = ((X239 @ (lam @ X240)))) | ~ ! [X241 : term] : (($true = ((X239 @ (sub @ X240 @ (push @ X241 @ id))))) | ~ ($true = ((X239 @ X241)))) | ~ ! [X242 : term] : (((X239 @ X242)) = ((X238 @ id @ X242 @ id))) | ~ ! [X243 : (subst > term > term)] : (($true = ((X238 @ id @ (lam @ (X243 @ sh @ one)) @ id))) | ~ ! [X244 : term] : (($true = ((X238 @ id @ (X243 @ id @ X244) @ id))) | ~ (((X238 @ id @ X244 @ id)) = $true)) | ~ ! [X245 : term,X246 : subst,X247 : subst] : (((sub @ (X243 @ X247 @ X245) @ X246)) = ((X243 @ (comp @ X247 @ X246) @ (sub @ X245 @ X246))))))))) & ! [X248 : d_subst,X249 : d_subst] : ((((d2subst @ X249)) = ((d2subst @ X248))) => (X248 = X249)) & ! [X250 : term] : ? [X251 : d_term] : (((d2term @ X251)) = X250) & (((d2term @ d_one)) = one) & ! [X252 : d_subst] : (d_id = X252) & (hoasinduction_lem3v2_gthm = $true) & (! [X253 : term,X254 : term,X255 : term,X256 : term] : ((X255 = X256) | (((ap @ (sub @ X256 @ id) @ X253)) != ((ap @ (sub @ X255 @ id) @ X254)))) <=> (hoasapinj1 = $true)) & (axassoc = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ! [X257 : d_term] : (d_one = X257) & (termmset_gthm = $true) & ((hoaslaminj = $true) <=> ! [X258 : (subst > term > term),X259 : term,X260 : subst,X261 : (subst > term > term)] : (~ ! [X262 : term,X263 : subst,X264 : subst] : (((X261 @ (comp @ X263 @ X264) @ (sub @ X262 @ X264))) = ((sub @ (X261 @ X263 @ X262) @ X264))) | (((X261 @ X260 @ X259)) = ((X258 @ X260 @ X259))) | ~ ! [X265 : subst,X266 : term,X267 : subst] : (((X258 @ (comp @ X267 @ X265) @ (sub @ X266 @ X265))) = ((sub @ (X258 @ X267 @ X266) @ X265))) | (((lam @ (X258 @ sh @ one))) != ((lam @ (X261 @ sh @ one)))))) & ((hoaslamnotap = $true) <=> ! [X268 : term,X269 : (subst > term > term),X270 : term] : (~ ! [X271 : term,X272 : subst,X273 : subst] : (((X269 @ (comp @ X272 @ X273) @ (sub @ X271 @ X273))) = ((sub @ (X269 @ X272 @ X271) @ X273))) | (((ap @ (sub @ X270 @ id) @ X268)) != ((lam @ (X269 @ sh @ one)))))) & ~ (lamnotap = $true) & ((hoasinduction_lem3v2a = $true) <=> ! [X274 : (subst > term > subst > $o),X275 : term,X276 : (term > $o)] : (~ ! [X277 : term] : ((((X276 @ (sub @ X275 @ (push @ X277 @ id)))) = $true) | ~ ($true = ((X276 @ X277)))) | ~ ! [X278 : (subst > term > term)] : (($true = ((X274 @ id @ (lam @ (X278 @ sh @ one)) @ id))) | ~ ! [X279 : term] : (~ ($true = ((X274 @ id @ X279 @ id))) | (((X274 @ id @ (X278 @ id @ X279) @ id)) = $true)) | ~ ! [X280 : subst,X281 : term,X282 : subst] : (((sub @ (X278 @ X280 @ X281) @ X282)) = ((X278 @ (comp @ X280 @ X282) @ (sub @ X281 @ X282))))) | (((X276 @ (lam @ X275))) = $true) | ~ ! [X283 : term] : (((X274 @ id @ X283 @ id)) = ((X276 @ X283))))) & ((pushprop_lem1v2 = $true) <=> ! [X284 : (term > $o),X285 : subst,X286 : (term > $o),X287 : term] : (~ ! [X288 : term] : (((X284 @ X288)) = ((X286 @ (sub @ X288 @ (push @ X287 @ X285))))) | ~ ($true = ((X286 @ X287))) | ($true = ((X284 @ one))))) & (! [X289 : subst,X290 : term,X291 : (term > $o)] : ~ ! [X292 : (term > $o)] : ~ ! [X293 : term] : (((X291 @ (sub @ X293 @ (push @ X290 @ X289)))) = ((X292 @ X293))) <=> (pushprop_lem0_lthm = $true)) & ((! [X294 : term,X295 : term,X296 : term,X297 : term] : ((X295 = X296) | (((ap @ X294 @ X295)) != ((ap @ X297 @ X296)))) => ! [X298 : term,X299 : term,X300 : term,X301 : term] : ((((ap @ (sub @ X298 @ id) @ X301)) != ((ap @ (sub @ X300 @ id) @ X299))) | (X299 = X301))) <=> (hoasapinj2_lthm = $true)) & (pushprop_lthm_orig = $true) & (pushprop_lem1v2_gthm = $true) & (! [X302 : term,X303 : subst] : (((sub @ one @ (push @ X302 @ X303))) = X302) <=> (axvarcons = $true)) & (! [X304 : term,X305 : term,X306 : term,X307 : term] : ((((ap @ X305 @ X304)) != ((ap @ X306 @ X307))) | (X305 = X306)) <=> (apinj1 = $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((pushprop_lem0 = $true) <=> ! [X310 : term,X311 : subst,X312 : (term > $o)] : ~ ! [X313 : (term > $o)] : ~ ! [X314 : term] : (((X313 @ X314)) = ((X312 @ (sub @ X314 @ (push @ X310 @ X311)))))) & (induction2_gthm = $true) & ((hoasinduction_lem3aa = $true) <=> ! [X315 : (subst > term > subst > $o),X316 : term] : (~ ! [X317 : term] : (~ ($true = ((X315 @ id @ X317 @ id))) | ($true = ((X315 @ id @ (sub @ X316 @ (push @ X317 @ id)) @ id)))) | ~ ! [X318 : (subst > term > term)] : (~ ! [X319 : term] : (~ (((X315 @ id @ X319 @ id)) = $true) | ($true = ((X315 @ id @ (X318 @ id @ X319) @ id)))) | (((X315 @ id @ (lam @ (X318 @ sh @ one)) @ id)) = $true) | ~ ! [X320 : subst,X321 : term,X322 : subst] : (((X318 @ (comp @ X322 @ X320) @ (sub @ X321 @ X320))) = ((sub @ (X318 @ X322 @ X321) @ X320)))) | ($true = ((X315 @ id @ (lam @ (sub @ X316 @ (push @ one @ sh))) @ id))))) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (! [X323 : (subst > term > subst > $o)] : ~ ! [X324 : (term > $o)] : ~ ! [X325 : term] : (((X323 @ id @ X325 @ id)) = ((X324 @ X325))) => (! [X326 : term,X327 : (term > $o)] : (~ ! [X328 : term] : (~ ! [X329 : term] : (($true = ((X327 @ (sub @ X328 @ (push @ X329 @ id))))) | ~ ($true = ((X327 @ X329)))) | ($true = ((X327 @ (lam @ X328))))) | ($true = ((X327 @ X326))) | ~ ! [X330 : term,X331 : term] : ((((X327 @ (ap @ X330 @ X331))) = $true) | ~ (((X327 @ X330)) = $true) | ~ (((X327 @ X331)) = $true)) | ~ ! [X332 : term] : (~ ($true = ((var @ X332))) | ($true = ((X327 @ X332))))) => (! [X333 : term] : (((sub @ X333 @ id)) = X333) => (! [X334 : (subst > term > subst > $o),X335 : (term > $o),X336 : term] : (~ ! [X337 : (subst > term > term)] : (~ ! [X338 : term] : (~ ($true = ((X334 @ id @ X338 @ id))) | ($true = ((X334 @ id @ (X337 @ id @ X338) @ id)))) | ~ ! [X339 : subst,X340 : term,X341 : subst] : (((sub @ (X337 @ X341 @ X340) @ X339)) = ((X337 @ (comp @ X341 @ X339) @ (sub @ X340 @ X339)))) | (((X334 @ id @ (lam @ (X337 @ sh @ one)) @ id)) = $true)) | ~ ! [X342 : term] : (($true = ((X335 @ (sub @ X336 @ (push @ X342 @ id))))) | ~ ($true = ((X335 @ X342)))) | ~ ! [X343 : term] : (((X335 @ X343)) = ((X334 @ id @ X343 @ id))) | ($true = ((X335 @ (lam @ X336))))) => ! [X344 : term,X345 : (subst > term > subst > $o)] : (~ ! [X346 : (subst > term > term)] : (~ ! [X347 : term] : (($true = ((X345 @ id @ (X346 @ id @ X347) @ id))) | ~ ($true = ((X345 @ id @ X347 @ id)))) | ($true = ((X345 @ id @ (lam @ (X346 @ sh @ one)) @ id))) | ~ ! [X348 : term,X349 : subst,X350 : subst] : (((X346 @ (comp @ X350 @ X349) @ (sub @ X348 @ X349))) = ((sub @ (X346 @ X350 @ X348) @ X349)))) | (((X345 @ id @ X344 @ id)) = $true) | ~ ! [X351 : term,X352 : term] : (~ ($true = ((X345 @ id @ X351 @ id))) | ($true = ((X345 @ id @ (ap @ (sub @ X352 @ id) @ X351) @ id))) | ~ ($true = ((X345 @ id @ X352 @ id)))))))))) & ((induction2_lthm = $true) <=> (! [X353 : term] : (((sub @ X353 @ id)) = X353) => (! [X354 : term,X355 : subst,X356 : (term > $o)] : (~ ! [X357 : term] : (($true = ((X356 @ (sub @ X357 @ X355)))) | ~ ($true = ((var @ X357)))) | ($true = ((X356 @ (sub @ X354 @ X355)))) | ~ ! [X358 : term,X359 : term] : (~ ($true = ((X356 @ X358))) | ~ ($true = ((X356 @ X359))) | ($true = ((X356 @ (ap @ X359 @ X358))))) | ~ ! [X360 : term] : (~ ! [X361 : term] : ((((X356 @ (sub @ X360 @ (push @ X361 @ id)))) = $true) | ~ (((X356 @ X361)) = $true)) | ($true = ((X356 @ (lam @ X360)))))) => ! [X362 : term,X363 : (term > $o)] : (~ ! [X364 : term,X365 : term] : (~ ($true = ((X363 @ X364))) | ~ ($true = ((X363 @ X365))) | ($true = ((X363 @ (ap @ X365 @ X364))))) | (((X363 @ X362)) = $true) | ~ ! [X366 : term] : (~ ($true = ((var @ X366))) | ($true = ((X363 @ X366)))) | ~ ! [X367 : term] : (~ ! [X368 : term] : (~ (((X363 @ X368)) = $true) | ($true = ((X363 @ (sub @ X367 @ (push @ X368 @ id)))))) | ($true = ((X363 @ (lam @ X367))))))))) & ((hoasapinj1_lthm = $true) <=> (! [X369 : term] : (((sub @ X369 @ id)) = X369) => (! [X370 : term,X371 : term,X372 : term,X373 : term] : ((((ap @ X371 @ X372)) != ((ap @ X373 @ X370))) | (X371 = X373)) => ! [X374 : term,X375 : term,X376 : term,X377 : term] : ((X375 = X377) | (((ap @ (sub @ X377 @ id) @ X374)) != ((ap @ (sub @ X375 @ id) @ X376))))))) & (! [X378 : term,X379 : (subst > term > subst > $o),X380 : (term > $o),X381 : term] : (~ ! [X382 : subst,X383 : term,X384 : subst,X385 : subst] : (($true = ((X379 @ X385 @ X383 @ (comp @ X384 @ X382)))) | ~ ($true = ((X379 @ (comp @ X385 @ X384) @ (sub @ X383 @ X384) @ X382)))) | ~ ! [X386 : subst,X387 : subst,X388 : term,X389 : subst] : (($true = ((X379 @ (comp @ X386 @ X389) @ (sub @ X388 @ X389) @ X387))) | ~ ($true = ((X379 @ X386 @ X388 @ (comp @ X389 @ X387))))) | ~ ($true = ((X380 @ X378))) | ~ ($true = ((X380 @ X381))) | ~ ! [X390 : term,X391 : term] : (($true = ((X379 @ id @ (ap @ (sub @ X390 @ id) @ X391) @ id))) | ~ ($true = ((X379 @ id @ X390 @ id))) | ~ ($true = ((X379 @ id @ X391 @ id)))) | ($true = ((X380 @ (ap @ X381 @ X378)))) | ~ ! [X392 : term] : (((X379 @ id @ X392 @ id)) = ((X380 @ X392)))) <=> (hoasinduction_lem2v2 = $true)) & ((hoasinduction_lem3_lthm = $true) <=> (! [X393 : term] : (((sub @ X393 @ id)) = X393) => (! [X394 : (subst > term > subst > $o),X395 : term] : (($true = ((X394 @ id @ (lam @ (sub @ X395 @ (push @ one @ sh))) @ id))) | ~ ! [X396 : (subst > term > term)] : (~ ! [X397 : subst,X398 : subst,X399 : term] : (((X396 @ (comp @ X397 @ X398) @ (sub @ X399 @ X398))) = ((sub @ (X396 @ X397 @ X399) @ X398))) | ($true = ((X394 @ id @ (lam @ (X396 @ sh @ one)) @ id))) | ~ ! [X400 : term] : (($true = ((X394 @ id @ (X396 @ id @ X400) @ id))) | ~ ($true = ((X394 @ id @ X400 @ id))))) | ~ ! [X401 : term] : (~ ($true = ((X394 @ id @ X401 @ id))) | (((X394 @ id @ (sub @ X395 @ (push @ X401 @ id)) @ id)) = $true))) => ! [X402 : (subst > term > subst > $o),X403 : term] : (~ ! [X404 : (subst > term > term)] : (($true = ((X402 @ id @ (lam @ (X404 @ sh @ one)) @ id))) | ~ ! [X405 : term] : (($true = ((X402 @ id @ (X404 @ id @ X405) @ id))) | ~ (((X402 @ id @ X405 @ id)) = $true)) | ~ ! [X406 : subst,X407 : subst,X408 : term] : (((sub @ (X404 @ X406 @ X408) @ X407)) = ((X404 @ (comp @ X406 @ X407) @ (sub @ X408 @ X407))))) | ~ ! [X409 : term] : (($true = ((X402 @ id @ (sub @ X403 @ (push @ X409 @ id)) @ id))) | ~ ($true = ((X402 @ id @ X409 @ id)))) | ~ ! [X410 : term,X411 : subst,X412 : subst,X413 : subst] : (~ ($true = ((X402 @ (comp @ X413 @ X411) @ (sub @ X410 @ X411) @ X412))) | ($true = ((X402 @ X413 @ X410 @ (comp @ X411 @ X412))))) | ~ ! [X414 : subst,X415 : term,X416 : subst,X417 : subst] : (~ (((X402 @ X417 @ X415 @ (comp @ X416 @ X414))) = $true) | ($true = ((X402 @ (comp @ X417 @ X416) @ (sub @ X415 @ X416) @ X414)))) | ($true = ((X402 @ id @ (lam @ X403) @ id))))))) & (hoasinduction_lem1_gthm = $true) & ((pushprop_lem1 = $true) <=> ! [X418 : subst,X419 : term,X420 : (term > $o),X421 : (term > $o),X422 : term] : (($true = ((X421 @ (sub @ X422 @ (push @ X419 @ X418))))) | ~ ($true = ((X420 @ X422))))) & (hoasapnotvar_gthm = $true) & (! [X423 : term,X424 : (subst > term > subst > $o)] : (~ ! [X425 : subst,X426 : term,X427 : subst,X428 : subst] : (~ ($true = ((X424 @ (comp @ X425 @ X428) @ (sub @ X426 @ X428) @ X427))) | ($true = ((X424 @ X425 @ X426 @ (comp @ X428 @ X427))))) | ~ ! [X429 : subst,X430 : subst,X431 : term,X432 : subst] : ((((X424 @ (comp @ X430 @ X432) @ (sub @ X431 @ X432) @ X429)) = $true) | ~ ($true = ((X424 @ X430 @ X431 @ (comp @ X432 @ X429))))) | ~ ! [X433 : (subst > term > term)] : (~ ! [X434 : subst,X435 : term,X436 : subst] : (((sub @ (X433 @ X436 @ X435) @ X434)) = ((X433 @ (comp @ X436 @ X434) @ (sub @ X435 @ X434)))) | (((X424 @ id @ (lam @ (X433 @ sh @ one)) @ id)) = $true) | ~ ! [X437 : term] : (~ ($true = ((X424 @ id @ X437 @ id))) | ($true = ((X424 @ id @ (X433 @ id @ X437) @ id))))) | ($true = ((X424 @ id @ (lam @ X423) @ id))) | ~ ! [X438 : term] : ((((X424 @ id @ (sub @ X423 @ (push @ X438 @ id)) @ id)) = $true) | ~ ($true = ((X424 @ id @ X438 @ id))))) <=> (hoasinduction_lem3 = $true)) & ((axscons = $true) <=> ! [X439 : subst] : (((push @ (sub @ one @ X439) @ (comp @ sh @ X439))) = X439)) & (! [X440 : term] : ~ ! [X441 : (subst > term > term)] : (((X441 @ sh @ one)) != ((sub @ X440 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b = $true)) & (apnotvar = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ! [X442 : d_term,X443 : d_term] : ((((d2term @ X443)) = ((d2term @ X442))) => (X442 = X443)) & (hoaslaminj_gthm = $true) & (! [X444 : term] : ~ ! [X445 : (subst > term > term)] : (((X445 @ sh @ one)) != ((sub @ X444 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b_lthm = $true)) & ((substmonoid_lthm = $true) <=> (! [X446 : subst] : (((comp @ id @ X446)) = X446) => (! [X447 : subst] : (((comp @ X447 @ id)) = X447) => (! [X448 : subst] : (((comp @ X448 @ id)) = X448) & ! [X449 : subst] : (((comp @ id @ X449)) = X449))))) & (axapp = $true) & (! [X450 : (subst > term > subst > $o),X451 : term] : (~ ! [X452 : term] : (~ ($true = ((X450 @ id @ X452 @ id))) | (((X450 @ id @ (sub @ X451 @ (push @ X452 @ id)) @ id)) = $true)) | ~ ! [X453 : (subst > term > term)] : ((((X450 @ id @ (lam @ (X453 @ sh @ one)) @ id)) = $true) | ~ ! [X454 : term] : (~ ($true = ((X450 @ id @ X454 @ id))) | ($true = ((X450 @ id @ (X453 @ id @ X454) @ id)))) | ~ ! [X455 : term,X456 : subst,X457 : subst] : (((X453 @ (comp @ X456 @ X457) @ (sub @ X455 @ X457))) = ((sub @ (X453 @ X456 @ X455) @ X457)))) | ($true = ((X450 @ id @ (lam @ X451) @ id)))) <=> (hoasinduction_lem3a = $true)) & ((laminj = $true) <=> ! [X458 : term,X459 : term] : ((X458 = X459) | (((lam @ X459)) != ((lam @ X458))))) & (hoasinduction_lem1_lthm = $true) & (pushprop_lem3v2 = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((! [X460 : (term > $o),X461 : term] : (($true = ((X460 @ X461))) | ~ ! [X462 : term] : (~ ! [X463 : term] : ((((X460 @ (sub @ X462 @ (push @ X463 @ id)))) = $true) | ~ ($true = ((X460 @ X463)))) | ($true = ((X460 @ (lam @ X462))))) | ~ ! [X464 : term] : (~ ($true = ((var @ X464))) | (((X460 @ X464)) = $true)) | ~ ! [X465 : term,X466 : term] : (~ (((X460 @ X466)) = $true) | ~ (((X460 @ X465)) = $true) | ($true = ((X460 @ (ap @ X465 @ X466)))))) => (! [X467 : (subst > term > subst > $o),X468 : term,X469 : term] : (~ ($true = ((X467 @ id @ X468 @ id))) | ~ (((X467 @ id @ X469 @ id)) = $true) | ~ ! [X470 : term,X471 : term] : (~ (((X467 @ id @ X470 @ id)) = $true) | ~ (((X467 @ id @ X471 @ id)) = $true) | ($true = ((X467 @ id @ (ap @ (sub @ X470 @ id) @ X471) @ id)))) | ($true = ((X467 @ id @ (ap @ X469 @ X468) @ id))) | ~ ! [X472 : term,X473 : subst,X474 : subst,X475 : subst] : (($true = ((X467 @ (comp @ X474 @ X473) @ (sub @ X472 @ X473) @ X475))) | ~ (((X467 @ X474 @ X472 @ (comp @ X473 @ X475))) = $true)) | ~ ! [X476 : subst,X477 : subst,X478 : term,X479 : subst] : ((((X467 @ X476 @ X478 @ (comp @ X477 @ X479))) = $true) | ~ ($true = ((X467 @ (comp @ X476 @ X477) @ (sub @ X478 @ X477) @ X479))))) => (! [X480 : term,X481 : (subst > term > subst > $o)] : (~ ! [X482 : subst,X483 : term,X484 : subst,X485 : subst] : (~ ($true = ((X481 @ X485 @ X483 @ (comp @ X484 @ X482)))) | (((X481 @ (comp @ X485 @ X484) @ (sub @ X483 @ X484) @ X482)) = $true)) | ~ ! [X486 : subst,X487 : subst,X488 : term,X489 : subst] : (~ ($true = ((X481 @ (comp @ X489 @ X486) @ (sub @ X488 @ X486) @ X487))) | ($true = ((X481 @ X489 @ X488 @ (comp @ X486 @ X487))))) | ~ ! [X490 : term] : (~ ($true = ((X481 @ id @ X490 @ id))) | ($true = ((X481 @ id @ (sub @ X480 @ (push @ X490 @ id)) @ id)))) | ($true = ((X481 @ id @ (lam @ X480) @ id))) | ~ ! [X491 : (subst > term > term)] : (($true = ((X481 @ id @ (lam @ (X491 @ sh @ one)) @ id))) | ~ ! [X492 : term,X493 : subst,X494 : subst] : (((X491 @ (comp @ X494 @ X493) @ (sub @ X492 @ X493))) = ((sub @ (X491 @ X494 @ X492) @ X493))) | ~ ! [X495 : term] : (~ ($true = ((X481 @ id @ X495 @ id))) | ($true = ((X481 @ id @ (X491 @ id @ X495) @ id)))))) => ! [X496 : term,X497 : (subst > term > subst > $o)] : (~ ! [X498 : term] : (~ ($true = ((var @ (sub @ X498 @ id)))) | ($true = ((X497 @ id @ X498 @ id)))) | ($true = ((X497 @ id @ X496 @ id))) | ~ ! [X499 : term,X500 : subst,X501 : subst,X502 : subst] : (($true = ((X497 @ X500 @ X499 @ (comp @ X501 @ X502)))) | ~ (((X497 @ (comp @ X500 @ X501) @ (sub @ X499 @ X501) @ X502)) = $true)) | ~ ! [X503 : subst,X504 : subst,X505 : term,X506 : subst] : (($true = ((X497 @ (comp @ X503 @ X504) @ (sub @ X505 @ X504) @ X506))) | ~ ($true = ((X497 @ X503 @ X505 @ (comp @ X504 @ X506))))) | ~ ! [X507 : term,X508 : term] : (($true = ((X497 @ id @ (ap @ (sub @ X507 @ id) @ X508) @ id))) | ~ ($true = ((X497 @ id @ X507 @ id))) | ~ ($true = ((X497 @ id @ X508 @ id)))) | ~ ! [X509 : (subst > term > term)] : (~ ! [X510 : term] : ((((X497 @ id @ (X509 @ id @ X510) @ id)) = $true) | ~ ($true = ((X497 @ id @ X510 @ id)))) | ~ ! [X511 : subst,X512 : subst,X513 : term] : (((X509 @ (comp @ X512 @ X511) @ (sub @ X513 @ X511))) = ((sub @ (X509 @ X512 @ X513) @ X511))) | ($true = ((X497 @ id @ (lam @ (X509 @ sh @ one)) @ id)))))))) <=> (hoasinduction_lthm = $true)) & (induction2lem_gthm = $true) & ((! [X514 : subst] : (((comp @ X514 @ id)) = X514) & ! [X515 : subst] : (((comp @ id @ X515)) = X515)) <=> (substmonoid = $true)) & (! [X516 : term,X517 : (subst > term > subst > $o)] : (~ ! [X518 : subst,X519 : subst,X520 : subst,X521 : term] : (($true = ((X517 @ (comp @ X518 @ X519) @ (sub @ X521 @ X519) @ X520))) | ~ ($true = ((X517 @ X518 @ X521 @ (comp @ X519 @ X520))))) | ~ ! [X522 : term,X523 : term] : (($true = ((X517 @ id @ (ap @ (sub @ X523 @ id) @ X522) @ id))) | ~ (((X517 @ id @ X522 @ id)) = $true) | ~ (((X517 @ id @ X523 @ id)) = $true)) | ~ ! [X524 : subst,X525 : subst,X526 : subst,X527 : term] : (($true = ((X517 @ X524 @ X527 @ (comp @ X526 @ X525)))) | ~ ($true = ((X517 @ (comp @ X524 @ X526) @ (sub @ X527 @ X526) @ X525)))) | ~ ! [X528 : term] : (($true = ((X517 @ id @ X528 @ id))) | ~ ($true = ((var @ (sub @ X528 @ id))))) | ~ ! [X529 : (subst > term > term)] : (~ ! [X530 : term] : (($true = ((X517 @ id @ (X529 @ id @ X530) @ id))) | ~ ($true = ((X517 @ id @ X530 @ id)))) | (((X517 @ id @ (lam @ (X529 @ sh @ one)) @ id)) = $true) | ~ ! [X531 : subst,X532 : subst,X533 : term] : (((X529 @ (comp @ X531 @ X532) @ (sub @ X533 @ X532))) = ((sub @ (X529 @ X531 @ X533) @ X532)))) | ($true = ((X517 @ id @ X516 @ id)))) <=> (hoasinduction = $true)) & (hoaslamnotap_gthm = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoasinduction_lem0_lthm = $true) <=> ! [X534 : (subst > term > subst > $o)] : ~ ! [X535 : (term > $o)] : ~ ! [X536 : term] : (((X534 @ id @ X536 @ id)) = ((X535 @ X536)))) & (! [X537 : term] : (((sub @ X537 @ id)) = X537) <=> (axvarid = $true)) & (hoasinduction_lem3_gthm = $true) & ((apinj2 = $true) <=> ! [X538 : term,X539 : term,X540 : term,X541 : term] : ((X539 = X541) | (((ap @ X538 @ X539)) != ((ap @ X540 @ X541))))) & (hoasinduction_lem1v2 = $true) & ~ (((var @ (d2term @ d_one))) = $true) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X542 : term,X543 : (subst > term > subst > $o)] : (~ ! [X544 : term] : (~ ($true = ((X543 @ id @ X544 @ id))) | ($true = ((X543 @ id @ (sub @ X542 @ (push @ X544 @ id)) @ id)))) | ~ ! [X545 : (subst > term > term)] : (($true = ((X543 @ id @ (lam @ (X545 @ sh @ one)) @ id))) | ~ ! [X546 : term] : (~ ($true = ((X543 @ id @ X546 @ id))) | ($true = ((X543 @ id @ (X545 @ id @ X546) @ id)))) | ~ ! [X547 : subst,X548 : term,X549 : subst] : (((X545 @ (comp @ X549 @ X547) @ (sub @ X548 @ X547))) = ((sub @ (X545 @ X549 @ X548) @ X547)))) | ($true = ((X543 @ id @ (lam @ (sub @ X542 @ (push @ one @ sh))) @ id))))) & (pushprop_lem1_gthm = $true) & (hoasapnotvar = $true)),
  inference(fool_elimination,[],[f4])).
thf(f6,plain,(
  ~ (hoasinduction_lem3v2 = ! [X0 : (subst > term > subst > $o),X1 : (term > $o)] : (! [X2 : subst,X3 : subst,X4 : term,X5 : subst] : ((X0 @ X5 @ X4 @ (comp @ X3 @ X2)) => (X0 @ (comp @ X5 @ X3) @ (sub @ X4 @ X3) @ X2)) => (! [X6 : subst,X7 : term,X8 : subst,X9 : subst] : ((X0 @ (comp @ X9 @ X6) @ (sub @ X7 @ X6) @ X8) => (X0 @ X9 @ X7 @ (comp @ X6 @ X8))) => (! [X10 : (subst > term > term)] : (! [X11 : subst,X12 : term,X13 : subst] : (((X10 @ (comp @ X11 @ X13) @ (sub @ X12 @ X13))) = ((sub @ (X10 @ X11 @ X12) @ X13))) => (! [X14 : term] : ((X0 @ id @ X14 @ id) => (X0 @ id @ (X10 @ id @ X14) @ id)) => (X0 @ id @ (hoaslam @ id @ (^[X15 : subst, X16 : term] : ((X10 @ X15 @ X16)))) @ id))) => ((hoasinduction_p_and_p_prime @ X0 @ X1) => ! [X17 : term] : (! [X18 : term] : ((X1 @ X18) => (X1 @ (sub @ X17 @ (push @ X18 @ id)))) => (X1 @ (lam @ X17))))))))),
  inference(rectify,[],[f3])).
thf(f7,plain,(
  ~((hoasinduction_lem3v2 = $true) <=> ! [X0 : (subst > term > subst > $o),X1 : (term > $o)] : (! [X2 : subst,X4 : term,X3 : subst,X5 : subst] : (($true = ((X0 @ X5 @ X4 @ (comp @ X3 @ X2)))) => ($true = ((X0 @ (comp @ X5 @ X3) @ (sub @ X4 @ X3) @ X2)))) => (! [X9 : subst,X8 : subst,X7 : term,X6 : subst] : ((((X0 @ (comp @ X9 @ X6) @ (sub @ X7 @ X6) @ X8)) = $true) => (((X0 @ X9 @ X7 @ (comp @ X6 @ X8))) = $true)) => (! [X10 : (subst > term > term)] : (! [X11 : subst,X13 : subst,X12 : term] : (((X10 @ (comp @ X11 @ X13) @ (sub @ X12 @ X13))) = ((sub @ (X10 @ X11 @ X12) @ X13))) => (! [X14 : term] : ((((X0 @ id @ X14 @ id)) = $true) => (((X0 @ id @ (X10 @ id @ X14) @ id)) = $true)) => ($true = ((X0 @ id @ (hoaslam @ id @ (^[Y0 : subst]: ((^[Y1 : term]: (X10 @ Y0 @ Y1))))) @ id))))) => ((((hoasinduction_p_and_p_prime @ X0 @ X1)) = $true) => ! [X17 : term] : (! [X18 : term] : (($true = ((X1 @ X18))) => (((X1 @ (sub @ X17 @ (push @ X18 @ id)))) = $true)) => ($true = ((X1 @ (lam @ X17))))))))))),
  inference(fool_elimination,[],[f6])).
thf(f8,plain,(
  (axclos = $true) & (pushprop_lem2v2_lthm = $true) & (hoasinduction_lem1v2_gthm = $true) & (hoasinduction_lem2v2_gthm = $true) & (! [X0 : term] : ~ ! [X1 : (subst > term > term)] : ~ ! [X2 : term,X3 : subst] : (((sub @ X0 @ (push @ X2 @ X3))) = ((X1 @ X3 @ X2))) <=> (hoasinduction_lem3v2_f = $true)) & ! [X4 : subst] : ? [X5 : d_subst] : (((d2subst @ X5)) = X4) & ((hoasinduction_lthm_3 = $true) <=> (! [X6 : (subst > term > subst > $o)] : ~ ! [X7 : (term > $o)] : ~ ! [X8 : term] : (((X7 @ X8)) = ((X6 @ id @ X8 @ id))) => (! [X9 : (term > $o),X10 : term] : (~ ! [X11 : term,X12 : term] : (~ ($true = ((X9 @ X11))) | ~ (((X9 @ X12)) = $true) | (((X9 @ (ap @ X11 @ X12))) = $true)) | ~ ! [X13 : term] : (($true = ((X9 @ X13))) | ~ ($true = ((var @ X13)))) | ($true = ((X9 @ X10))) | ~ ! [X14 : term] : (~ ! [X15 : term] : (~ ($true = ((X9 @ X15))) | ($true = ((X9 @ (sub @ X14 @ (push @ X15 @ id)))))) | ($true = ((X9 @ (lam @ X14)))))) => (! [X16 : term] : (((sub @ X16 @ id)) = X16) => (! [X17 : term,X18 : (term > $o),X19 : (subst > term > subst > $o)] : (~ ! [X20 : (subst > term > term)] : (~ ! [X21 : term] : (~ (((X19 @ id @ X21 @ id)) = $true) | ($true = ((X19 @ id @ (X20 @ id @ X21) @ id)))) | (((X19 @ id @ (lam @ (X20 @ sh @ one)) @ id)) = $true) | ~ ! [X22 : term,X23 : subst,X24 : subst] : (((X20 @ (comp @ X23 @ X24) @ (sub @ X22 @ X24))) = ((sub @ (X20 @ X23 @ X22) @ X24)))) | ($true = ((X18 @ (lam @ X17)))) | ~ ! [X25 : term] : (($true = ((X18 @ (sub @ X17 @ (push @ X25 @ id))))) | ~ ($true = ((X18 @ X25)))) | ~ ! [X26 : term] : (((X18 @ X26)) = ((X19 @ id @ X26 @ id)))) => ! [X27 : term,X28 : (subst > term > subst > $o)] : (~ ! [X29 : term,X30 : term] : (~ ($true = ((X28 @ id @ X30 @ id))) | ~ ($true = ((X28 @ id @ X29 @ id))) | ($true = ((X28 @ id @ (ap @ (sub @ X30 @ id) @ X29) @ id)))) | ~ ! [X31 : (subst > term > term)] : ((((X28 @ id @ (lam @ (X31 @ sh @ one)) @ id)) = $true) | ~ ! [X32 : term] : (~ ($true = ((X28 @ id @ X32 @ id))) | ($true = ((X28 @ id @ (X31 @ id @ X32) @ id)))) | ~ ! [X34 : term,X35 : subst,X33 : subst] : (((sub @ (X31 @ X35 @ X34) @ X33)) = ((X31 @ (comp @ X35 @ X33) @ (sub @ X34 @ X33))))) | ~ ! [X36 : term,X37 : subst,X38 : subst,X39 : subst] : (~ ($true = ((X28 @ (comp @ X37 @ X39) @ (sub @ X36 @ X39) @ X38))) | ($true = ((X28 @ X37 @ X36 @ (comp @ X39 @ X38))))) | ~ ! [X40 : subst,X41 : term,X42 : subst,X43 : subst] : ((((X28 @ (comp @ X43 @ X40) @ (sub @ X41 @ X40) @ X42)) = $true) | ~ (((X28 @ X43 @ X41 @ (comp @ X40 @ X42))) = $true)) | ~ ! [X44 : term] : (~ ($true = ((var @ (sub @ X44 @ id)))) | ($true = ((X28 @ id @ X44 @ id)))) | (((X28 @ id @ X27 @ id)) = $true))))))) & (hoaslamnotvar_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (axmap = $true) & ((shinj = $true) <=> ! [X45 : term,X46 : term] : ((X45 = X46) | (((sub @ X46 @ sh)) != ((sub @ X45 @ sh))))) & ((hoasapinj2 = $true) <=> ! [X50 : term,X49 : term,X47 : term,X48 : term] : ((((ap @ (sub @ X48 @ id) @ X47)) != ((ap @ (sub @ X49 @ id) @ X50))) | (X47 = X50))) & (pushprop = $true) & (pushprop_lem0_gthm = $true) & (hoaslamnotvar = $true) & ((axshiftcons = $true) <=> ! [X52 : term,X51 : subst] : (((comp @ sh @ (push @ X52 @ X51))) = X51)) & (substmonoid_gthm = $true) & (pushprop_lem2v2_gthm = $true) & ((hoasinduction_lem3v2 = $true) <=> ! [X53 : term,X54 : (subst > term > subst > $o),X55 : (term > $o)] : (~ ! [X56 : (subst > term > term)] : (~ ! [X57 : term] : (~ ($true = ((X54 @ id @ X57 @ id))) | ($true = ((X54 @ id @ (X56 @ id @ X57) @ id)))) | ~ ! [X59 : subst,X60 : term,X58 : subst] : (((X56 @ (comp @ X59 @ X58) @ (sub @ X60 @ X58))) = ((sub @ (X56 @ X59 @ X60) @ X58))) | ($true = ((X54 @ id @ (lam @ (X56 @ sh @ one)) @ id)))) | ~ ! [X61 : subst,X62 : subst,X63 : subst,X64 : term] : (~ ($true = ((X54 @ X61 @ X64 @ (comp @ X63 @ X62)))) | (((X54 @ (comp @ X61 @ X63) @ (sub @ X64 @ X63) @ X62)) = $true)) | ($true = ((X55 @ (lam @ X53)))) | ~ ! [X65 : term,X66 : subst,X67 : subst,X68 : subst] : (~ ($true = ((X54 @ (comp @ X68 @ X67) @ (sub @ X65 @ X67) @ X66))) | (((X54 @ X68 @ X65 @ (comp @ X67 @ X66))) = $true)) | ~ ! [X69 : term] : (~ (((X55 @ X69)) = $true) | (((X55 @ (sub @ X53 @ (push @ X69 @ id)))) = $true)) | ~ ! [X70 : term] : (((X54 @ id @ X70 @ id)) = ((X55 @ X70))))) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (ulamvarsh = $true) & ((! [X71 : term] : (((sub @ X71 @ id)) = X71) => ! [X72 : term,X73 : (subst > term > subst > $o),X74 : (term > $o)] : (($true = ((X74 @ (lam @ X72)))) | ~ ! [X75 : term] : (~ ($true = ((X74 @ X75))) | ($true = ((X74 @ (sub @ X72 @ (push @ X75 @ id)))))) | ~ ! [X76 : term,X77 : subst,X78 : subst,X79 : subst] : (($true = ((X73 @ (comp @ X77 @ X78) @ (sub @ X76 @ X78) @ X79))) | ~ (((X73 @ X77 @ X76 @ (comp @ X78 @ X79))) = $true)) | ~ ! [X80 : (subst > term > term)] : (~ ! [X81 : term] : (~ ($true = ((X73 @ id @ X81 @ id))) | (((X73 @ id @ (X80 @ id @ X81) @ id)) = $true)) | ~ ! [X84 : subst,X82 : subst,X83 : term] : (((sub @ (X80 @ X84 @ X83) @ X82)) = ((X80 @ (comp @ X84 @ X82) @ (sub @ X83 @ X82)))) | ($true = ((X73 @ id @ (lam @ (X80 @ sh @ one)) @ id)))) | ~ ! [X85 : subst,X86 : subst,X87 : subst,X88 : term] : (~ (((X73 @ (comp @ X85 @ X87) @ (sub @ X88 @ X87) @ X86)) = $true) | ($true = ((X73 @ X85 @ X88 @ (comp @ X87 @ X86))))) | ~ ! [X89 : term] : (((X74 @ X89)) = ((X73 @ id @ X89 @ id))))) <=> (hoasinduction_lem3v2_lthm = $true)) & (! [X90 : term] : ~ ! [X91 : (subst > term > term)] : ~ ! [X92 : term,X93 : subst] : (((X91 @ X93 @ X92)) = ((sub @ X90 @ (push @ X92 @ X93)))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & ~ (ulamvar1 = $true) & (ulamvarind = $true) & (hoasinduction_lem1 = $true) & (pushprop_lem2v2 = $true) & (hoasinduction_lem3a_gthm = $true) & (hoasapinj1_gthm = $true) & ((! [X94 : term,X95 : subst] : (((sub @ one @ (push @ X94 @ X95))) = X94) => (! [X97 : subst,X96 : term] : (((comp @ sh @ (push @ X96 @ X97))) = X97) => (! [X98 : term,X99 : term] : ((((lam @ X99)) != ((lam @ X98))) | (X98 = X99)) => ! [X101 : (subst > term > term),X103 : subst,X100 : term,X102 : (subst > term > term)] : (~ ! [X104 : term,X105 : subst,X106 : subst] : (((X102 @ (comp @ X105 @ X106) @ (sub @ X104 @ X106))) = ((sub @ (X102 @ X105 @ X104) @ X106))) | (((lam @ (X102 @ sh @ one))) != ((lam @ (X101 @ sh @ one)))) | ~ ! [X108 : subst,X109 : term,X107 : subst] : (((X101 @ (comp @ X108 @ X107) @ (sub @ X109 @ X107))) = ((sub @ (X101 @ X108 @ X109) @ X107))) | (((X102 @ X103 @ X100)) = ((X101 @ X103 @ X100))))))) <=> (hoaslaminj_lthm = $true)) & ~ (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & (hoaslamnotap_lthm = $true) & ((termmset = $true) <=> ! [X110 : term] : (((sub @ X110 @ id)) = X110)) & (pushprop_gthm = $true) & (((d2subst @ d_id)) = id) & ((! [X111 : term] : (((sub @ X111 @ id)) = X111) => (! [X112 : (subst > term > subst > $o),X113 : term] : (($true = ((X112 @ id @ (lam @ (sub @ X113 @ (push @ one @ sh))) @ id))) | ~ ! [X114 : (subst > term > term)] : (~ ! [X115 : term] : (~ (((X112 @ id @ X115 @ id)) = $true) | (((X112 @ id @ (X114 @ id @ X115) @ id)) = $true)) | ($true = ((X112 @ id @ (lam @ (X114 @ sh @ one)) @ id))) | ~ ! [X116 : subst,X118 : subst,X117 : term] : (((X114 @ (comp @ X118 @ X116) @ (sub @ X117 @ X116))) = ((sub @ (X114 @ X118 @ X117) @ X116)))) | ~ ! [X119 : term] : (~ ($true = ((X112 @ id @ X119 @ id))) | ($true = ((X112 @ id @ (sub @ X113 @ (push @ X119 @ id)) @ id))))) => ! [X120 : term,X121 : (subst > term > subst > $o)] : (~ ! [X122 : (subst > term > term)] : (~ ! [X123 : term] : (~ ($true = ((X121 @ id @ X123 @ id))) | ($true = ((X121 @ id @ (X122 @ id @ X123) @ id)))) | ~ ! [X124 : term,X125 : subst,X126 : subst] : (((X122 @ (comp @ X125 @ X126) @ (sub @ X124 @ X126))) = ((sub @ (X122 @ X125 @ X124) @ X126))) | ($true = ((X121 @ id @ (lam @ (X122 @ sh @ one)) @ id)))) | ~ ! [X127 : term] : (~ ($true = ((X121 @ id @ X127 @ id))) | ($true = ((X121 @ id @ (sub @ X120 @ (push @ X127 @ id)) @ id)))) | ($true = ((X121 @ id @ (lam @ X120) @ id)))))) <=> (hoasinduction_lem3a_lthm = $true)) & ((hoasinduction_lem3aaa = $true) <=> ! [X128 : (subst > term > subst > $o),X129 : term] : (($true = ((X128 @ id @ (lam @ (sub @ X129 @ (push @ one @ sh))) @ id))) | ~ ! [X130 : term] : (($true = ((X128 @ id @ (sub @ X129 @ (push @ X130 @ id)) @ id))) | ~ ($true = ((X128 @ id @ X130 @ id)))) | ~ ! [X131 : (subst > term > term),X132 : term] : (~ ! [X134 : term,X135 : subst,X133 : subst] : (((sub @ (sub @ X132 @ (push @ X134 @ X135)) @ X133)) = ((sub @ (X131 @ X135 @ X134) @ X133))) | (((X128 @ id @ (lam @ (X131 @ sh @ one)) @ id)) = $true) | ~ ! [X136 : term] : (~ (((X128 @ id @ X136 @ id)) = $true) | ($true = ((X128 @ id @ (X131 @ id @ X136) @ id)))) | ~ ! [X139 : subst,X138 : term,X137 : subst] : (((sub @ X132 @ (push @ (sub @ X138 @ X137) @ (comp @ X139 @ X137)))) = ((X131 @ (comp @ X139 @ X137) @ (sub @ X138 @ X137))))))) & (pushprop_lem3v2_lthm = $true) & (axvarshift = $true) & (hoasinduction_lem2_gthm = $true) & (hoaslamnotvar_lthm = $true) & (hoasinduction_gthm = $true) & (! [X140 : subst] : (((comp @ X140 @ id)) = X140) <=> (axidr = $true)) & (! [X141 : subst] : (((comp @ id @ X141)) = X141) <=> (axidl = $true)) & (! [X142 : term,X143 : (subst > term > subst > $o)] : (~ ! [X144 : (subst > term > term)] : (~ ! [X145 : term] : (~ (((X143 @ id @ X145 @ id)) = $true) | ($true = ((X143 @ id @ (X144 @ id @ X145) @ id)))) | (((X143 @ id @ (lam @ (X144 @ sh @ one)) @ id)) = $true) | ~ ! [X147 : subst,X148 : term,X146 : subst] : (((X144 @ (comp @ X146 @ X147) @ (sub @ X148 @ X147))) = ((sub @ (X144 @ X146 @ X148) @ X147)))) | ($true = ((X143 @ id @ X142 @ id))) | ~ ! [X149 : term,X150 : term] : (~ (((X143 @ id @ X149 @ id)) = $true) | ~ ($true = ((X143 @ id @ X150 @ id))) | ($true = ((X143 @ id @ (ap @ (sub @ X149 @ id) @ X150) @ id))))) <=> (hoasinduction_no_psi_cond = $true)) & (hoasapinj2_gthm = $true) & (hoasinduction_lem2_lthm = $true) & (((d2subst @ d_id)) = sh) & ((! [X151 : term,X152 : subst] : (((sub @ one @ (push @ X151 @ X152))) = X151) => ! [X153 : term,X154 : (term > $o),X155 : subst,X156 : (term > $o)] : ((((X154 @ one)) = $true) | ~ ! [X157 : term] : (((X156 @ (sub @ X157 @ (push @ X153 @ X155)))) = ((X154 @ X157))) | ~ ($true = ((X156 @ X153))))) <=> (pushprop_lem1v2_lthm = $true)) & ((induction2lem_lthm = $true) <=> (! [X159 : term,X158 : subst] : (((sub @ one @ (push @ X159 @ X158))) = X159) => (! [X160 : subst,X161 : term] : (((comp @ sh @ (push @ X161 @ X160))) = X160) => (! [X162 : subst] : (((comp @ X162 @ id)) = X162) => (! [X163 : term,X164 : (term > $o)] : (~ ! [X165 : term,X166 : term] : (~ ($true = ((X164 @ X166))) | ~ ($true = ((X164 @ X165))) | ($true = ((X164 @ (ap @ X165 @ X166))))) | ($true = ((X164 @ X163))) | ~ ! [X167 : term] : (~ ($true = ((X164 @ X167))) | (((X164 @ (lam @ X167))) = $true)) | ~ ! [X168 : term] : (~ ($true = ((var @ X168))) | ($true = ((X164 @ X168))))) => ! [X169 : term,X170 : subst,X171 : (term > $o)] : (~ ! [X172 : term] : (($true = ((X171 @ (lam @ X172)))) | ~ ! [X173 : term] : (($true = ((X171 @ (sub @ X172 @ (push @ X173 @ id))))) | ~ ($true = ((X171 @ X173))))) | ~ ! [X174 : term] : (~ ($true = ((var @ X174))) | ($true = ((X171 @ (sub @ X174 @ X170))))) | ($true = ((X171 @ (sub @ X169 @ X170)))) | ~ ! [X175 : term,X176 : term] : (($true = ((X171 @ (ap @ X175 @ X176)))) | ~ ($true = ((X171 @ X175))) | ~ (((X171 @ X176)) = $true)))))))) & (hoasinduction_lem3b_gthm = $true) & ((! [X177 : term] : (((sub @ X177 @ id)) = X177) => ! [X178 : term] : (((sub @ X178 @ id)) = X178)) <=> (termmset_lthm = $true)) & (! [X179 : subst,X180 : term,X181 : (term > $o)] : (~ ! [X182 : term] : (($true = ((X181 @ (lam @ X182)))) | ~ ! [X183 : term] : (~ ($true = ((X181 @ X183))) | ($true = ((X181 @ (sub @ X182 @ (push @ X183 @ id))))))) | ~ ! [X184 : term,X185 : term] : (~ ($true = ((X181 @ X184))) | ~ ($true = ((X181 @ X185))) | ($true = ((X181 @ (ap @ X185 @ X184))))) | ($true = ((X181 @ (sub @ X180 @ X179)))) | ~ ! [X186 : term] : ((((X181 @ (sub @ X186 @ X179))) = $true) | ~ ($true = ((var @ X186))))) <=> (induction2lem = $true)) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoasinduction_lem0 = $true) <=> ! [X187 : (subst > term > subst > $o)] : ~ ! [X188 : (term > $o)] : ~ ! [X189 : term] : (((X187 @ id @ X189 @ id)) = ((X188 @ X189)))) & (hoasapnotvar_lthm = $true) & (! [X190 : term,X191 : (term > $o)] : (~ ! [X192 : term] : (~ ! [X193 : term] : (($true = ((X191 @ (sub @ X192 @ (push @ X193 @ id))))) | ~ ($true = ((X191 @ X193)))) | ($true = ((X191 @ (lam @ X192))))) | ~ ! [X194 : term,X195 : term] : (~ ($true = ((X191 @ X195))) | (((X191 @ (ap @ X195 @ X194))) = $true) | ~ ($true = ((X191 @ X194)))) | ~ ! [X196 : term] : (($true = ((X191 @ X196))) | ~ ($true = ((var @ X196)))) | ($true = ((X191 @ X190)))) <=> (induction2 = $true)) & ((induction = $true) <=> ! [X197 : (term > $o),X198 : term] : (~ ! [X199 : term] : (($true = ((X197 @ (lam @ X199)))) | ~ ($true = ((X197 @ X199)))) | ($true = ((X197 @ X198))) | ~ ! [X200 : term] : (($true = ((X197 @ X200))) | ~ ($true = ((var @ X200)))) | ~ ! [X201 : term,X202 : term] : (~ ($true = ((X197 @ X202))) | ~ (((X197 @ X201)) = $true) | (((X197 @ (ap @ X201 @ X202))) = $true)))) & ((! [X203 : subst,X204 : term] : (((sub @ one @ (push @ X204 @ X203))) = X204) => (! [X206 : term,X205 : subst] : (((comp @ sh @ (push @ X206 @ X205))) = X205) => ! [X207 : term,X208 : term,X209 : (term > $o),X210 : (term > $o),X211 : subst] : ((((X210 @ (sub @ X208 @ (push @ X207 @ X211)))) = $true) | ~ ($true = ((X209 @ X208)))))) <=> (pushprop_lem1_lthm = $true)) & (axabs = $true) & ((hoasinduction_lem2 = $true) <=> ! [X212 : (subst > term > subst > $o),X213 : term,X214 : term] : (~ ! [X215 : subst,X216 : term,X217 : subst,X218 : subst] : (~ ($true = ((X212 @ (comp @ X215 @ X217) @ (sub @ X216 @ X217) @ X218))) | ($true = ((X212 @ X215 @ X216 @ (comp @ X217 @ X218))))) | ~ ! [X219 : term,X220 : term] : (~ ($true = ((X212 @ id @ X220 @ id))) | ($true = ((X212 @ id @ (ap @ (sub @ X219 @ id) @ X220) @ id))) | ~ (((X212 @ id @ X219 @ id)) = $true)) | ~ ($true = ((X212 @ id @ X214 @ id))) | ~ ($true = ((X212 @ id @ X213 @ id))) | ($true = ((X212 @ id @ (ap @ X213 @ X214) @ id))) | ~ ! [X221 : subst,X222 : term,X223 : subst,X224 : subst] : (~ (((X212 @ X223 @ X222 @ (comp @ X221 @ X224))) = $true) | ($true = ((X212 @ (comp @ X223 @ X221) @ (sub @ X222 @ X221) @ X224)))))) & (pushprop_lthm = $true) & (lamnotvar = $true) & ((hoasinduction_lem3v2a_lthm = $true) <=> (! [X225 : term] : ~ ! [X226 : (subst > term > term)] : ~ ! [X228 : subst,X227 : term] : (((X226 @ X228 @ X227)) = ((sub @ X225 @ (push @ X227 @ X228)))) => (! [X229 : term] : (((sub @ X229 @ id)) = X229) => ! [X230 : (subst > term > subst > $o),X231 : (term > $o),X232 : term] : (($true = ((X231 @ (lam @ X232)))) | ~ ! [X233 : term] : (($true = ((X231 @ (sub @ X232 @ (push @ X233 @ id))))) | ~ ($true = ((X231 @ X233)))) | ~ ! [X234 : term] : (((X230 @ id @ X234 @ id)) = ((X231 @ X234))) | ~ ! [X235 : (subst > term > term)] : (($true = ((X230 @ id @ (lam @ (X235 @ sh @ one)) @ id))) | ~ ! [X236 : term] : (($true = ((X230 @ id @ (X235 @ id @ X236) @ id))) | ~ (((X230 @ id @ X236 @ id)) = $true)) | ~ ! [X239 : subst,X238 : subst,X237 : term] : (((sub @ (X235 @ X239 @ X237) @ X238)) = ((X235 @ (comp @ X239 @ X238) @ (sub @ X237 @ X238))))))))) & ! [X241 : d_subst,X240 : d_subst] : ((((d2subst @ X241)) = ((d2subst @ X240))) => (X240 = X241)) & ! [X242 : term] : ? [X243 : d_term] : (((d2term @ X243)) = X242) & (((d2term @ d_one)) = one) & ! [X244 : d_subst] : (d_id = X244) & (hoasinduction_lem3v2_gthm = $true) & (! [X246 : term,X247 : term,X245 : term,X248 : term] : ((X247 = X248) | (((ap @ (sub @ X247 @ id) @ X246)) != ((ap @ (sub @ X248 @ id) @ X245)))) <=> (hoasapinj1 = $true)) & (axassoc = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ! [X249 : d_term] : (d_one = X249) & (termmset_gthm = $true) & (! [X251 : term,X253 : (subst > term > term),X250 : (subst > term > term),X252 : subst] : (~ ! [X259 : subst,X257 : subst,X258 : term] : (((sub @ (X250 @ X259 @ X258) @ X257)) = ((X250 @ (comp @ X259 @ X257) @ (sub @ X258 @ X257)))) | (((X253 @ X252 @ X251)) = ((X250 @ X252 @ X251))) | (((lam @ (X250 @ sh @ one))) != ((lam @ (X253 @ sh @ one)))) | ~ ! [X256 : subst,X255 : subst,X254 : term] : (((sub @ (X253 @ X255 @ X254) @ X256)) = ((X253 @ (comp @ X255 @ X256) @ (sub @ X254 @ X256))))) <=> (hoaslaminj = $true)) & (! [X260 : term,X261 : (subst > term > term),X262 : term] : (~ ! [X264 : subst,X263 : term,X265 : subst] : (((sub @ (X261 @ X264 @ X263) @ X265)) = ((X261 @ (comp @ X264 @ X265) @ (sub @ X263 @ X265)))) | (((ap @ (sub @ X262 @ id) @ X260)) != ((lam @ (X261 @ sh @ one))))) <=> (hoaslamnotap = $true)) & ~ (lamnotap = $true) & ((hoasinduction_lem3v2a = $true) <=> ! [X266 : (subst > term > subst > $o),X267 : term,X268 : (term > $o)] : (~ ! [X269 : term] : (($true = ((X268 @ (sub @ X267 @ (push @ X269 @ id))))) | ~ ($true = ((X268 @ X269)))) | ~ ! [X270 : (subst > term > term)] : (($true = ((X266 @ id @ (lam @ (X270 @ sh @ one)) @ id))) | ~ ! [X271 : term] : (~ (((X266 @ id @ X271 @ id)) = $true) | (((X266 @ id @ (X270 @ id @ X271) @ id)) = $true)) | ~ ! [X272 : subst,X273 : term,X274 : subst] : (((sub @ (X270 @ X272 @ X273) @ X274)) = ((X270 @ (comp @ X272 @ X274) @ (sub @ X273 @ X274))))) | ($true = ((X268 @ (lam @ X267)))) | ~ ! [X275 : term] : (((X266 @ id @ X275 @ id)) = ((X268 @ X275))))) & ((pushprop_lem1v2 = $true) <=> ! [X276 : (term > $o),X277 : subst,X278 : (term > $o),X279 : term] : (~ ! [X280 : term] : (((X276 @ X280)) = ((X278 @ (sub @ X280 @ (push @ X279 @ X277))))) | ~ ($true = ((X278 @ X279))) | (((X276 @ one)) = $true))) & (! [X282 : term,X283 : (term > $o),X281 : subst] : ~ ! [X284 : (term > $o)] : ~ ! [X285 : term] : (((X283 @ (sub @ X285 @ (push @ X282 @ X281)))) = ((X284 @ X285))) <=> (pushprop_lem0_lthm = $true)) & ((! [X289 : term,X288 : term,X287 : term,X286 : term] : ((X287 = X288) | (((ap @ X286 @ X287)) != ((ap @ X289 @ X288)))) => ! [X291 : term,X290 : term,X293 : term,X292 : term] : ((((ap @ (sub @ X292 @ id) @ X291)) != ((ap @ (sub @ X290 @ id) @ X293))) | (X291 = X293))) <=> (hoasapinj2_lthm = $true)) & (pushprop_lthm_orig = $true) & (pushprop_lem1v2_gthm = $true) & (! [X295 : subst,X294 : term] : (((sub @ one @ (push @ X294 @ X295))) = X294) <=> (axvarcons = $true)) & (! [X296 : term,X298 : term,X299 : term,X297 : term] : ((X297 = X298) | (((ap @ X298 @ X299)) != ((ap @ X297 @ X296)))) <=> (apinj1 = $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (! [X300 : term,X301 : subst,X302 : (term > $o)] : ~ ! [X303 : (term > $o)] : ~ ! [X304 : term] : (((X302 @ (sub @ X304 @ (push @ X300 @ X301)))) = ((X303 @ X304))) <=> (pushprop_lem0 = $true)) & (induction2_gthm = $true) & ((hoasinduction_lem3aa = $true) <=> ! [X305 : (subst > term > subst > $o),X306 : term] : (~ ! [X307 : term] : (~ ($true = ((X305 @ id @ X307 @ id))) | ($true = ((X305 @ id @ (sub @ X306 @ (push @ X307 @ id)) @ id)))) | ~ ! [X308 : (subst > term > term)] : (~ ! [X309 : term] : (~ (((X305 @ id @ X309 @ id)) = $true) | ($true = ((X305 @ id @ (X308 @ id @ X309) @ id)))) | ($true = ((X305 @ id @ (lam @ (X308 @ sh @ one)) @ id))) | ~ ! [X312 : subst,X310 : subst,X311 : term] : (((sub @ (X308 @ X312 @ X311) @ X310)) = ((X308 @ (comp @ X312 @ X310) @ (sub @ X311 @ X310))))) | ($true = ((X305 @ id @ (lam @ (sub @ X306 @ (push @ one @ sh))) @ id))))) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (! [X313 : (subst > term > subst > $o)] : ~ ! [X314 : (term > $o)] : ~ ! [X315 : term] : (((X313 @ id @ X315 @ id)) = ((X314 @ X315))) => (! [X316 : term,X317 : (term > $o)] : (~ ! [X318 : term] : (~ ! [X319 : term] : ((((X317 @ (sub @ X318 @ (push @ X319 @ id)))) = $true) | ~ (((X317 @ X319)) = $true)) | ($true = ((X317 @ (lam @ X318))))) | ($true = ((X317 @ X316))) | ~ ! [X320 : term,X321 : term] : ((((X317 @ (ap @ X320 @ X321))) = $true) | ~ ($true = ((X317 @ X320))) | ~ (((X317 @ X321)) = $true)) | ~ ! [X322 : term] : (~ ($true = ((var @ X322))) | ($true = ((X317 @ X322))))) => (! [X323 : term] : (((sub @ X323 @ id)) = X323) => (! [X324 : (subst > term > subst > $o),X325 : (term > $o),X326 : term] : (~ ! [X327 : (subst > term > term)] : (~ ! [X328 : term] : (~ ($true = ((X324 @ id @ X328 @ id))) | ($true = ((X324 @ id @ (X327 @ id @ X328) @ id)))) | ~ ! [X329 : subst,X330 : term,X331 : subst] : (((X327 @ (comp @ X331 @ X329) @ (sub @ X330 @ X329))) = ((sub @ (X327 @ X331 @ X330) @ X329))) | (((X324 @ id @ (lam @ (X327 @ sh @ one)) @ id)) = $true)) | ~ ! [X332 : term] : (($true = ((X325 @ (sub @ X326 @ (push @ X332 @ id))))) | ~ ($true = ((X325 @ X332)))) | ~ ! [X333 : term] : (((X325 @ X333)) = ((X324 @ id @ X333 @ id))) | ($true = ((X325 @ (lam @ X326))))) => ! [X334 : term,X335 : (subst > term > subst > $o)] : (~ ! [X336 : (subst > term > term)] : (~ ! [X337 : term] : (($true = ((X335 @ id @ (X336 @ id @ X337) @ id))) | ~ (((X335 @ id @ X337 @ id)) = $true)) | (((X335 @ id @ (lam @ (X336 @ sh @ one)) @ id)) = $true) | ~ ! [X339 : subst,X340 : subst,X338 : term] : (((sub @ (X336 @ X340 @ X338) @ X339)) = ((X336 @ (comp @ X340 @ X339) @ (sub @ X338 @ X339))))) | ($true = ((X335 @ id @ X334 @ id))) | ~ ! [X341 : term,X342 : term] : (~ ($true = ((X335 @ id @ X341 @ id))) | ($true = ((X335 @ id @ (ap @ (sub @ X342 @ id) @ X341) @ id))) | ~ ($true = ((X335 @ id @ X342 @ id)))))))))) & ((induction2_lthm = $true) <=> (! [X343 : term] : (((sub @ X343 @ id)) = X343) => (! [X344 : term,X345 : subst,X346 : (term > $o)] : (~ ! [X347 : term] : (($true = ((X346 @ (sub @ X347 @ X345)))) | ~ (((var @ X347)) = $true)) | ($true = ((X346 @ (sub @ X344 @ X345)))) | ~ ! [X348 : term,X349 : term] : (~ ($true = ((X346 @ X348))) | ~ (((X346 @ X349)) = $true) | (((X346 @ (ap @ X349 @ X348))) = $true)) | ~ ! [X350 : term] : (~ ! [X351 : term] : (($true = ((X346 @ (sub @ X350 @ (push @ X351 @ id))))) | ~ ($true = ((X346 @ X351)))) | ($true = ((X346 @ (lam @ X350)))))) => ! [X352 : term,X353 : (term > $o)] : (~ ! [X354 : term,X355 : term] : (~ ($true = ((X353 @ X354))) | ~ ($true = ((X353 @ X355))) | ($true = ((X353 @ (ap @ X355 @ X354))))) | ($true = ((X353 @ X352))) | ~ ! [X356 : term] : (~ ($true = ((var @ X356))) | ($true = ((X353 @ X356)))) | ~ ! [X357 : term] : (~ ! [X358 : term] : (~ ($true = ((X353 @ X358))) | ($true = ((X353 @ (sub @ X357 @ (push @ X358 @ id)))))) | ($true = ((X353 @ (lam @ X357))))))))) & ((hoasapinj1_lthm = $true) <=> (! [X359 : term] : (((sub @ X359 @ id)) = X359) => (! [X360 : term,X361 : term,X363 : term,X362 : term] : ((X361 = X363) | (((ap @ X361 @ X362)) != ((ap @ X363 @ X360)))) => ! [X364 : term,X366 : term,X365 : term,X367 : term] : ((X365 = X367) | (((ap @ (sub @ X367 @ id) @ X364)) != ((ap @ (sub @ X365 @ id) @ X366))))))) & (! [X368 : term,X369 : (subst > term > subst > $o),X370 : (term > $o),X371 : term] : (~ ! [X372 : subst,X373 : term,X374 : subst,X375 : subst] : (($true = ((X369 @ X375 @ X373 @ (comp @ X374 @ X372)))) | ~ ($true = ((X369 @ (comp @ X375 @ X374) @ (sub @ X373 @ X374) @ X372)))) | ~ ! [X376 : subst,X377 : subst,X378 : term,X379 : subst] : (($true = ((X369 @ (comp @ X376 @ X379) @ (sub @ X378 @ X379) @ X377))) | ~ ($true = ((X369 @ X376 @ X378 @ (comp @ X379 @ X377))))) | ~ ($true = ((X370 @ X368))) | ~ ($true = ((X370 @ X371))) | ~ ! [X380 : term,X381 : term] : (($true = ((X369 @ id @ (ap @ (sub @ X380 @ id) @ X381) @ id))) | ~ ($true = ((X369 @ id @ X380 @ id))) | ~ (((X369 @ id @ X381 @ id)) = $true)) | ($true = ((X370 @ (ap @ X371 @ X368)))) | ~ ! [X382 : term] : (((X369 @ id @ X382 @ id)) = ((X370 @ X382)))) <=> (hoasinduction_lem2v2 = $true)) & ((hoasinduction_lem3_lthm = $true) <=> (! [X383 : term] : (((sub @ X383 @ id)) = X383) => (! [X384 : (subst > term > subst > $o),X385 : term] : (($true = ((X384 @ id @ (lam @ (sub @ X385 @ (push @ one @ sh))) @ id))) | ~ ! [X386 : (subst > term > term)] : (~ ! [X389 : term,X387 : subst,X388 : subst] : (((sub @ (X386 @ X387 @ X389) @ X388)) = ((X386 @ (comp @ X387 @ X388) @ (sub @ X389 @ X388)))) | ($true = ((X384 @ id @ (lam @ (X386 @ sh @ one)) @ id))) | ~ ! [X390 : term] : (($true = ((X384 @ id @ (X386 @ id @ X390) @ id))) | ~ ($true = ((X384 @ id @ X390 @ id))))) | ~ ! [X391 : term] : (~ (((X384 @ id @ X391 @ id)) = $true) | ($true = ((X384 @ id @ (sub @ X385 @ (push @ X391 @ id)) @ id))))) => ! [X392 : (subst > term > subst > $o),X393 : term] : (~ ! [X394 : (subst > term > term)] : (($true = ((X392 @ id @ (lam @ (X394 @ sh @ one)) @ id))) | ~ ! [X395 : term] : ((((X392 @ id @ (X394 @ id @ X395) @ id)) = $true) | ~ (((X392 @ id @ X395 @ id)) = $true)) | ~ ! [X398 : term,X396 : subst,X397 : subst] : (((sub @ (X394 @ X396 @ X398) @ X397)) = ((X394 @ (comp @ X396 @ X397) @ (sub @ X398 @ X397))))) | ~ ! [X399 : term] : (($true = ((X392 @ id @ (sub @ X393 @ (push @ X399 @ id)) @ id))) | ~ ($true = ((X392 @ id @ X399 @ id)))) | ~ ! [X400 : term,X401 : subst,X402 : subst,X403 : subst] : (~ (((X392 @ (comp @ X403 @ X401) @ (sub @ X400 @ X401) @ X402)) = $true) | ($true = ((X392 @ X403 @ X400 @ (comp @ X401 @ X402))))) | ~ ! [X404 : subst,X405 : term,X406 : subst,X407 : subst] : (~ (((X392 @ X407 @ X405 @ (comp @ X406 @ X404))) = $true) | (((X392 @ (comp @ X407 @ X406) @ (sub @ X405 @ X406) @ X404)) = $true)) | (((X392 @ id @ (lam @ X393) @ id)) = $true))))) & (hoasinduction_lem1_gthm = $true) & ((pushprop_lem1 = $true) <=> ! [X408 : subst,X409 : term,X410 : (term > $o),X411 : (term > $o),X412 : term] : (($true = ((X411 @ (sub @ X412 @ (push @ X409 @ X408))))) | ~ ($true = ((X410 @ X412))))) & (hoasapnotvar_gthm = $true) & (! [X413 : term,X414 : (subst > term > subst > $o)] : (~ ! [X415 : subst,X416 : term,X417 : subst,X418 : subst] : (~ (((X414 @ (comp @ X415 @ X418) @ (sub @ X416 @ X418) @ X417)) = $true) | ($true = ((X414 @ X415 @ X416 @ (comp @ X418 @ X417))))) | ~ ! [X419 : subst,X420 : subst,X421 : term,X422 : subst] : (($true = ((X414 @ (comp @ X420 @ X422) @ (sub @ X421 @ X422) @ X419))) | ~ ($true = ((X414 @ X420 @ X421 @ (comp @ X422 @ X419))))) | ~ ! [X423 : (subst > term > term)] : (~ ! [X425 : term,X426 : subst,X424 : subst] : (((X423 @ (comp @ X426 @ X424) @ (sub @ X425 @ X424))) = ((sub @ (X423 @ X426 @ X425) @ X424))) | (((X414 @ id @ (lam @ (X423 @ sh @ one)) @ id)) = $true) | ~ ! [X427 : term] : (~ ($true = ((X414 @ id @ X427 @ id))) | (((X414 @ id @ (X423 @ id @ X427) @ id)) = $true))) | ($true = ((X414 @ id @ (lam @ X413) @ id))) | ~ ! [X428 : term] : (($true = ((X414 @ id @ (sub @ X413 @ (push @ X428 @ id)) @ id))) | ~ ($true = ((X414 @ id @ X428 @ id))))) <=> (hoasinduction_lem3 = $true)) & ((axscons = $true) <=> ! [X429 : subst] : (((push @ (sub @ one @ X429) @ (comp @ sh @ X429))) = X429)) & (! [X430 : term] : ~ ! [X431 : (subst > term > term)] : (((X431 @ sh @ one)) != ((sub @ X430 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b = $true)) & (apnotvar = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ! [X433 : d_term,X432 : d_term] : ((((d2term @ X432)) = ((d2term @ X433))) => (X432 = X433)) & (hoaslaminj_gthm = $true) & (! [X434 : term] : ~ ! [X435 : (subst > term > term)] : (((X435 @ sh @ one)) != ((sub @ X434 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b_lthm = $true)) & ((substmonoid_lthm = $true) <=> (! [X436 : subst] : (((comp @ id @ X436)) = X436) => (! [X437 : subst] : (((comp @ X437 @ id)) = X437) => (! [X438 : subst] : (((comp @ X438 @ id)) = X438) & ! [X439 : subst] : (((comp @ id @ X439)) = X439))))) & (axapp = $true) & (! [X440 : (subst > term > subst > $o),X441 : term] : (~ ! [X442 : term] : (~ ($true = ((X440 @ id @ X442 @ id))) | ($true = ((X440 @ id @ (sub @ X441 @ (push @ X442 @ id)) @ id)))) | ~ ! [X443 : (subst > term > term)] : (($true = ((X440 @ id @ (lam @ (X443 @ sh @ one)) @ id))) | ~ ! [X444 : term] : (~ ($true = ((X440 @ id @ X444 @ id))) | ($true = ((X440 @ id @ (X443 @ id @ X444) @ id)))) | ~ ! [X445 : term,X446 : subst,X447 : subst] : (((X443 @ (comp @ X446 @ X447) @ (sub @ X445 @ X447))) = ((sub @ (X443 @ X446 @ X445) @ X447)))) | ($true = ((X440 @ id @ (lam @ X441) @ id)))) <=> (hoasinduction_lem3a = $true)) & ((laminj = $true) <=> ! [X448 : term,X449 : term] : ((X448 = X449) | (((lam @ X449)) != ((lam @ X448))))) & (hoasinduction_lem1_lthm = $true) & (pushprop_lem3v2 = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((! [X450 : (term > $o),X451 : term] : ((((X450 @ X451)) = $true) | ~ ! [X452 : term] : (~ ! [X453 : term] : (($true = ((X450 @ (sub @ X452 @ (push @ X453 @ id))))) | ~ ($true = ((X450 @ X453)))) | ($true = ((X450 @ (lam @ X452))))) | ~ ! [X454 : term] : (~ (((var @ X454)) = $true) | ($true = ((X450 @ X454)))) | ~ ! [X455 : term,X456 : term] : (~ ($true = ((X450 @ X456))) | ~ ($true = ((X450 @ X455))) | ($true = ((X450 @ (ap @ X455 @ X456)))))) => (! [X457 : (subst > term > subst > $o),X458 : term,X459 : term] : (~ (((X457 @ id @ X458 @ id)) = $true) | ~ ($true = ((X457 @ id @ X459 @ id))) | ~ ! [X460 : term,X461 : term] : (~ ($true = ((X457 @ id @ X460 @ id))) | ~ ($true = ((X457 @ id @ X461 @ id))) | ($true = ((X457 @ id @ (ap @ (sub @ X460 @ id) @ X461) @ id)))) | ($true = ((X457 @ id @ (ap @ X459 @ X458) @ id))) | ~ ! [X462 : term,X463 : subst,X464 : subst,X465 : subst] : (($true = ((X457 @ (comp @ X464 @ X463) @ (sub @ X462 @ X463) @ X465))) | ~ ($true = ((X457 @ X464 @ X462 @ (comp @ X463 @ X465))))) | ~ ! [X466 : subst,X467 : subst,X468 : term,X469 : subst] : ((((X457 @ X466 @ X468 @ (comp @ X467 @ X469))) = $true) | ~ ($true = ((X457 @ (comp @ X466 @ X467) @ (sub @ X468 @ X467) @ X469))))) => (! [X470 : term,X471 : (subst > term > subst > $o)] : (~ ! [X472 : subst,X473 : term,X474 : subst,X475 : subst] : (~ ($true = ((X471 @ X475 @ X473 @ (comp @ X474 @ X472)))) | ($true = ((X471 @ (comp @ X475 @ X474) @ (sub @ X473 @ X474) @ X472)))) | ~ ! [X476 : subst,X477 : subst,X478 : term,X479 : subst] : (~ ($true = ((X471 @ (comp @ X479 @ X476) @ (sub @ X478 @ X476) @ X477))) | ($true = ((X471 @ X479 @ X478 @ (comp @ X476 @ X477))))) | ~ ! [X480 : term] : (~ ($true = ((X471 @ id @ X480 @ id))) | ($true = ((X471 @ id @ (sub @ X470 @ (push @ X480 @ id)) @ id)))) | (((X471 @ id @ (lam @ X470) @ id)) = $true) | ~ ! [X481 : (subst > term > term)] : ((((X471 @ id @ (lam @ (X481 @ sh @ one)) @ id)) = $true) | ~ ! [X482 : term,X484 : subst,X483 : subst] : (((X481 @ (comp @ X484 @ X483) @ (sub @ X482 @ X483))) = ((sub @ (X481 @ X484 @ X482) @ X483))) | ~ ! [X485 : term] : (~ (((X471 @ id @ X485 @ id)) = $true) | (((X471 @ id @ (X481 @ id @ X485) @ id)) = $true)))) => ! [X486 : term,X487 : (subst > term > subst > $o)] : (~ ! [X488 : term] : (~ ($true = ((var @ (sub @ X488 @ id)))) | ($true = ((X487 @ id @ X488 @ id)))) | ($true = ((X487 @ id @ X486 @ id))) | ~ ! [X489 : term,X490 : subst,X491 : subst,X492 : subst] : ((((X487 @ X490 @ X489 @ (comp @ X491 @ X492))) = $true) | ~ ($true = ((X487 @ (comp @ X490 @ X491) @ (sub @ X489 @ X491) @ X492)))) | ~ ! [X493 : subst,X494 : subst,X495 : term,X496 : subst] : (($true = ((X487 @ (comp @ X493 @ X494) @ (sub @ X495 @ X494) @ X496))) | ~ ($true = ((X487 @ X493 @ X495 @ (comp @ X494 @ X496))))) | ~ ! [X497 : term,X498 : term] : (($true = ((X487 @ id @ (ap @ (sub @ X497 @ id) @ X498) @ id))) | ~ ($true = ((X487 @ id @ X497 @ id))) | ~ (((X487 @ id @ X498 @ id)) = $true)) | ~ ! [X499 : (subst > term > term)] : (~ ! [X500 : term] : (($true = ((X487 @ id @ (X499 @ id @ X500) @ id))) | ~ ($true = ((X487 @ id @ X500 @ id)))) | ~ ! [X501 : subst,X502 : subst,X503 : term] : (((X499 @ (comp @ X502 @ X501) @ (sub @ X503 @ X501))) = ((sub @ (X499 @ X502 @ X503) @ X501))) | (((X487 @ id @ (lam @ (X499 @ sh @ one)) @ id)) = $true)))))) <=> (hoasinduction_lthm = $true)) & (induction2lem_gthm = $true) & ((! [X505 : subst] : (((comp @ id @ X505)) = X505) & ! [X504 : subst] : (((comp @ X504 @ id)) = X504)) <=> (substmonoid = $true)) & (! [X506 : term,X507 : (subst > term > subst > $o)] : (~ ! [X508 : subst,X509 : subst,X510 : subst,X511 : term] : ((((X507 @ (comp @ X508 @ X509) @ (sub @ X511 @ X509) @ X510)) = $true) | ~ ($true = ((X507 @ X508 @ X511 @ (comp @ X509 @ X510))))) | ~ ! [X512 : term,X513 : term] : (($true = ((X507 @ id @ (ap @ (sub @ X513 @ id) @ X512) @ id))) | ~ ($true = ((X507 @ id @ X512 @ id))) | ~ ($true = ((X507 @ id @ X513 @ id)))) | ~ ! [X514 : subst,X515 : subst,X516 : subst,X517 : term] : (($true = ((X507 @ X514 @ X517 @ (comp @ X516 @ X515)))) | ~ (((X507 @ (comp @ X514 @ X516) @ (sub @ X517 @ X516) @ X515)) = $true)) | ~ ! [X518 : term] : (($true = ((X507 @ id @ X518 @ id))) | ~ ($true = ((var @ (sub @ X518 @ id))))) | ~ ! [X519 : (subst > term > term)] : (~ ! [X520 : term] : (($true = ((X507 @ id @ (X519 @ id @ X520) @ id))) | ~ ($true = ((X507 @ id @ X520 @ id)))) | (((X507 @ id @ (lam @ (X519 @ sh @ one)) @ id)) = $true) | ~ ! [X523 : term,X522 : subst,X521 : subst] : (((X519 @ (comp @ X521 @ X522) @ (sub @ X523 @ X522))) = ((sub @ (X519 @ X521 @ X523) @ X522)))) | ($true = ((X507 @ id @ X506 @ id)))) <=> (hoasinduction = $true)) & (hoaslamnotap_gthm = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (! [X524 : (subst > term > subst > $o)] : ~ ! [X525 : (term > $o)] : ~ ! [X526 : term] : (((X524 @ id @ X526 @ id)) = ((X525 @ X526))) <=> (hoasinduction_lem0_lthm = $true)) & (! [X527 : term] : (((sub @ X527 @ id)) = X527) <=> (axvarid = $true)) & (hoasinduction_lem3_gthm = $true) & ((apinj2 = $true) <=> ! [X529 : term,X528 : term,X531 : term,X530 : term] : ((X529 = X531) | (((ap @ X528 @ X529)) != ((ap @ X530 @ X531))))) & (hoasinduction_lem1v2 = $true) & ~ (((var @ (d2term @ d_one))) = $true) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X532 : term,X533 : (subst > term > subst > $o)] : (~ ! [X534 : term] : (~ ($true = ((X533 @ id @ X534 @ id))) | (((X533 @ id @ (sub @ X532 @ (push @ X534 @ id)) @ id)) = $true)) | ~ ! [X535 : (subst > term > term)] : ((((X533 @ id @ (lam @ (X535 @ sh @ one)) @ id)) = $true) | ~ ! [X536 : term] : (~ ($true = ((X533 @ id @ X536 @ id))) | ($true = ((X533 @ id @ (X535 @ id @ X536) @ id)))) | ~ ! [X537 : subst,X539 : subst,X538 : term] : (((sub @ (X535 @ X539 @ X538) @ X537)) = ((X535 @ (comp @ X539 @ X537) @ (sub @ X538 @ X537))))) | ($true = ((X533 @ id @ (lam @ (sub @ X532 @ (push @ one @ sh))) @ id))))) & (pushprop_lem1_gthm = $true) & (hoasapnotvar = $true)),
  inference(rectify,[],[f5])).
thf(f9,plain,(
  (hoasinduction_lem3_gthm = $true) & (((var @ (d2term @ d_one))) != $true) & ((induction = $true) <=> ! [X197 : (term > $o),X198 : term] : (~ ! [X199 : term] : (($true != ((X197 @ X199))) | ($true = ((X197 @ (lam @ X199))))) | ($true = ((X197 @ X198))) | ~ ! [X202 : term,X201 : term] : ((((X197 @ (ap @ X201 @ X202))) = $true) | (((X197 @ X201)) != $true) | ($true != ((X197 @ X202)))) | ~ ! [X200 : term] : (($true = ((X197 @ X200))) | ($true != ((var @ X200)))))) & ! [X244 : d_subst] : (d_id = X244) & (pushprop_gthm = $true) & (axapp = $true) & (hoaslamnotap_gthm = $true) & ((hoasinduction_lem3aaa = $true) <=> ! [X129 : term,X128 : (subst > term > subst > $o)] : (~ ! [X130 : term] : (($true != ((X128 @ id @ X130 @ id))) | ($true = ((X128 @ id @ (sub @ X129 @ (push @ X130 @ id)) @ id)))) | ~ ! [X131 : (subst > term > term),X132 : term] : (~ ! [X136 : term] : ((((X128 @ id @ X136 @ id)) != $true) | ($true = ((X128 @ id @ (X131 @ id @ X136) @ id)))) | ~ ! [X134 : term,X135 : subst,X133 : subst] : (((sub @ (sub @ X132 @ (push @ X134 @ X135)) @ X133)) = ((sub @ (X131 @ X135 @ X134) @ X133))) | (((X128 @ id @ (lam @ (X131 @ sh @ one)) @ id)) = $true) | ~ ! [X139 : subst,X138 : term,X137 : subst] : (((sub @ X132 @ (push @ (sub @ X138 @ X137) @ (comp @ X139 @ X137)))) = ((X131 @ (comp @ X139 @ X137) @ (sub @ X138 @ X137))))) | ($true = ((X128 @ id @ (lam @ (sub @ X129 @ (push @ one @ sh))) @ id))))) & (hoaslamnotvar_gthm = $true) & ((induction2lem = $true) <=> ! [X179 : subst,X180 : term,X181 : (term > $o)] : (($true = ((X181 @ (sub @ X180 @ X179)))) | ~ ! [X186 : term] : ((((X181 @ (sub @ X186 @ X179))) = $true) | ($true != ((var @ X186)))) | ~ ! [X184 : term,X185 : term] : (($true = ((X181 @ (ap @ X185 @ X184)))) | ($true != ((X181 @ X185))) | ($true != ((X181 @ X184)))) | ~ ! [X182 : term] : (($true = ((X181 @ (lam @ X182)))) | ~ ! [X183 : term] : (($true != ((X181 @ X183))) | ($true = ((X181 @ (sub @ X182 @ (push @ X183 @ id))))))))) & ! [X242 : term] : ? [X243 : d_term] : (((d2term @ X243)) = X242) & (hoasinduction_lem2_lthm = $true) & (! [X141 : subst] : (((comp @ id @ X141)) = X141) <=> (axidl = $true)) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoasinduction_lem3a = $true) <=> ! [X440 : (subst > term > subst > $o),X441 : term] : (($true = ((X440 @ id @ (lam @ X441) @ id))) | ~ ! [X442 : term] : (($true = ((X440 @ id @ (sub @ X441 @ (push @ X442 @ id)) @ id))) | ($true != ((X440 @ id @ X442 @ id)))) | ~ ! [X443 : (subst > term > term)] : (~ ! [X444 : term] : (($true = ((X440 @ id @ (X443 @ id @ X444) @ id))) | ($true != ((X440 @ id @ X444 @ id)))) | ~ ! [X445 : term,X446 : subst,X447 : subst] : (((X443 @ (comp @ X446 @ X447) @ (sub @ X445 @ X447))) = ((sub @ (X443 @ X446 @ X445) @ X447))) | ($true = ((X440 @ id @ (lam @ (X443 @ sh @ one)) @ id)))))) & (hoasinduction_gthm = $true) & (! [X300 : term,X301 : subst,X302 : (term > $o)] : ~ ! [X303 : (term > $o)] : ~ ! [X304 : term] : (((X302 @ (sub @ X304 @ (push @ X300 @ X301)))) = ((X303 @ X304))) <=> (pushprop_lem0 = $true)) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (ulamvarsh = $true) & ((apinj2 = $true) <=> ! [X529 : term,X528 : term,X531 : term,X530 : term] : ((X529 = X531) | (((ap @ X528 @ X529)) != ((ap @ X530 @ X531))))) & (hoasinduction_lem1_lthm = $true) & ! [X4 : subst] : ? [X5 : d_subst] : (((d2subst @ X5)) = X4) & (! [X430 : term] : ~ ! [X431 : (subst > term > term)] : (((X431 @ sh @ one)) != ((sub @ X430 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b = $true)) & ((shinj = $true) <=> ! [X45 : term,X46 : term] : ((X45 = X46) | (((sub @ X46 @ sh)) != ((sub @ X45 @ sh))))) & ((hoasinduction = $true) <=> ! [X506 : term,X507 : (subst > term > subst > $o)] : (~ ! [X515 : subst,X514 : subst,X516 : subst,X517 : term] : (($true = ((X507 @ X514 @ X517 @ (comp @ X516 @ X515)))) | (((X507 @ (comp @ X514 @ X516) @ (sub @ X517 @ X516) @ X515)) != $true)) | ~ ! [X508 : subst,X511 : term,X510 : subst,X509 : subst] : ((((X507 @ (comp @ X508 @ X509) @ (sub @ X511 @ X509) @ X510)) = $true) | ($true != ((X507 @ X508 @ X511 @ (comp @ X509 @ X510))))) | ~ ! [X519 : (subst > term > term)] : ((((X507 @ id @ (lam @ (X519 @ sh @ one)) @ id)) = $true) | ~ ! [X520 : term] : (($true != ((X507 @ id @ X520 @ id))) | ($true = ((X507 @ id @ (X519 @ id @ X520) @ id)))) | ~ ! [X523 : term,X522 : subst,X521 : subst] : (((X519 @ (comp @ X521 @ X522) @ (sub @ X523 @ X522))) = ((sub @ (X519 @ X521 @ X523) @ X522)))) | ~ ! [X513 : term,X512 : term] : (($true != ((X507 @ id @ X512 @ id))) | ($true = ((X507 @ id @ (ap @ (sub @ X513 @ id) @ X512) @ id))) | ($true != ((X507 @ id @ X513 @ id)))) | ($true = ((X507 @ id @ X506 @ id))) | ~ ! [X518 : term] : (($true != ((var @ (sub @ X518 @ id)))) | ($true = ((X507 @ id @ X518 @ id)))))) & (pushprop_lem3v2 = $true) & (hoasapinj1_gthm = $true) & (axabs = $true) & (pushprop_lthm = $true) & (lamnotap != $true) & (! [X140 : subst] : (((comp @ X140 @ id)) = X140) <=> (axidr = $true)) & (! [X54 : (subst > term > subst > $o),X55 : (term > $o),X53 : term] : (~ ! [X56 : (subst > term > term)] : (~ ! [X57 : term] : (($true = ((X54 @ id @ (X56 @ id @ X57) @ id))) | ($true != ((X54 @ id @ X57 @ id)))) | ~ ! [X59 : subst,X60 : term,X58 : subst] : (((X56 @ (comp @ X59 @ X58) @ (sub @ X60 @ X58))) = ((sub @ (X56 @ X59 @ X60) @ X58))) | ($true = ((X54 @ id @ (lam @ (X56 @ sh @ one)) @ id)))) | ~ ! [X69 : term] : ((((X55 @ X69)) != $true) | (((X55 @ (sub @ X53 @ (push @ X69 @ id)))) = $true)) | ($true = ((X55 @ (lam @ X53)))) | ~ ! [X63 : subst,X61 : subst,X64 : term,X62 : subst] : (($true != ((X54 @ X61 @ X64 @ (comp @ X63 @ X62)))) | (((X54 @ (comp @ X61 @ X63) @ (sub @ X64 @ X63) @ X62)) = $true)) | ~ ! [X70 : term] : (((X54 @ id @ X70 @ id)) = ((X55 @ X70))) | ~ ! [X67 : subst,X66 : subst,X68 : subst,X65 : term] : ((((X54 @ X68 @ X65 @ (comp @ X67 @ X66))) = $true) | ($true != ((X54 @ (comp @ X68 @ X67) @ (sub @ X65 @ X67) @ X66))))) <=> (hoasinduction_lem3v2 = $true)) & (apnotvar = $true) & (hoaslamnotap_lthm = $true) & (pushprop_lem0_gthm = $true) & (! [X282 : term,X283 : (term > $o),X281 : subst] : ~ ! [X284 : (term > $o)] : ~ ! [X285 : term] : (((X283 @ (sub @ X285 @ (push @ X282 @ X281)))) = ((X284 @ X285))) <=> (pushprop_lem0_lthm = $true)) & ((hoasinduction_lem0 = $true) <=> ! [X187 : (subst > term > subst > $o)] : ~ ! [X188 : (term > $o)] : ~ ! [X189 : term] : (((X187 @ id @ X189 @ id)) = ((X188 @ X189)))) & (pushprop_lem2v2_lthm = $true) & (pushprop_lem1v2_gthm = $true) & ((! [X111 : term] : (((sub @ X111 @ id)) = X111) => (! [X113 : term,X112 : (subst > term > subst > $o)] : (($true = ((X112 @ id @ (lam @ (sub @ X113 @ (push @ one @ sh))) @ id))) | ~ ! [X114 : (subst > term > term)] : (($true = ((X112 @ id @ (lam @ (X114 @ sh @ one)) @ id))) | ~ ! [X115 : term] : ((((X112 @ id @ (X114 @ id @ X115) @ id)) = $true) | (((X112 @ id @ X115 @ id)) != $true)) | ~ ! [X116 : subst,X118 : subst,X117 : term] : (((X114 @ (comp @ X118 @ X116) @ (sub @ X117 @ X116))) = ((sub @ (X114 @ X118 @ X117) @ X116)))) | ~ ! [X119 : term] : (($true = ((X112 @ id @ (sub @ X113 @ (push @ X119 @ id)) @ id))) | ($true != ((X112 @ id @ X119 @ id))))) => ! [X120 : term,X121 : (subst > term > subst > $o)] : (($true = ((X121 @ id @ (lam @ X120) @ id))) | ~ ! [X127 : term] : (($true = ((X121 @ id @ (sub @ X120 @ (push @ X127 @ id)) @ id))) | ($true != ((X121 @ id @ X127 @ id)))) | ~ ! [X122 : (subst > term > term)] : (~ ! [X123 : term] : (($true != ((X121 @ id @ X123 @ id))) | ($true = ((X121 @ id @ (X122 @ id @ X123) @ id)))) | ~ ! [X124 : term,X125 : subst,X126 : subst] : (((X122 @ (comp @ X125 @ X126) @ (sub @ X124 @ X126))) = ((sub @ (X122 @ X125 @ X124) @ X126))) | ($true = ((X121 @ id @ (lam @ (X122 @ sh @ one)) @ id))))))) <=> (hoasinduction_lem3a_lthm = $true)) & (axmap = $true) & (! [X266 : (subst > term > subst > $o),X268 : (term > $o),X267 : term] : (~ ! [X275 : term] : (((X266 @ id @ X275 @ id)) = ((X268 @ X275))) | ~ ! [X270 : (subst > term > term)] : (~ ! [X271 : term] : ((((X266 @ id @ (X270 @ id @ X271) @ id)) = $true) | (((X266 @ id @ X271 @ id)) != $true)) | ~ ! [X272 : subst,X273 : term,X274 : subst] : (((sub @ (X270 @ X272 @ X273) @ X274)) = ((X270 @ (comp @ X272 @ X274) @ (sub @ X273 @ X274)))) | ($true = ((X266 @ id @ (lam @ (X270 @ sh @ one)) @ id)))) | ~ ! [X269 : term] : (($true = ((X268 @ (sub @ X267 @ (push @ X269 @ id))))) | ($true != ((X268 @ X269)))) | ($true = ((X268 @ (lam @ X267))))) <=> (hoasinduction_lem3v2a = $true)) & (hoasinduction_lem3v2_gthm = $true) & (! [X214 : term,X213 : term,X212 : (subst > term > subst > $o)] : (~ ! [X219 : term,X220 : term] : ((((X212 @ id @ X219 @ id)) != $true) | ($true != ((X212 @ id @ X220 @ id))) | ($true = ((X212 @ id @ (ap @ (sub @ X219 @ id) @ X220) @ id)))) | ~ ! [X215 : subst,X218 : subst,X216 : term,X217 : subst] : (($true != ((X212 @ (comp @ X215 @ X217) @ (sub @ X216 @ X217) @ X218))) | ($true = ((X212 @ X215 @ X216 @ (comp @ X217 @ X218))))) | ($true = ((X212 @ id @ (ap @ X213 @ X214) @ id))) | ($true != ((X212 @ id @ X213 @ id))) | ($true != ((X212 @ id @ X214 @ id))) | ~ ! [X223 : subst,X221 : subst,X224 : subst,X222 : term] : ((((X212 @ X223 @ X222 @ (comp @ X221 @ X224))) != $true) | ($true = ((X212 @ (comp @ X223 @ X221) @ (sub @ X222 @ X221) @ X224))))) <=> (hoasinduction_lem2 = $true)) & (hoasinduction_lem1v2 = $true) & ((! [X71 : term] : (((sub @ X71 @ id)) = X71) => ! [X72 : term,X74 : (term > $o),X73 : (subst > term > subst > $o)] : (~ ! [X89 : term] : (((X74 @ X89)) = ((X73 @ id @ X89 @ id))) | ~ ! [X86 : subst,X88 : term,X85 : subst,X87 : subst] : (($true = ((X73 @ X85 @ X88 @ (comp @ X87 @ X86)))) | (((X73 @ (comp @ X85 @ X87) @ (sub @ X88 @ X87) @ X86)) != $true)) | ($true = ((X74 @ (lam @ X72)))) | ~ ! [X75 : term] : (($true != ((X74 @ X75))) | ($true = ((X74 @ (sub @ X72 @ (push @ X75 @ id)))))) | ~ ! [X77 : subst,X78 : subst,X76 : term,X79 : subst] : (($true = ((X73 @ (comp @ X77 @ X78) @ (sub @ X76 @ X78) @ X79))) | (((X73 @ X77 @ X76 @ (comp @ X78 @ X79))) != $true)) | ~ ! [X80 : (subst > term > term)] : (($true = ((X73 @ id @ (lam @ (X80 @ sh @ one)) @ id))) | ~ ! [X84 : subst,X82 : subst,X83 : term] : (((sub @ (X80 @ X84 @ X83) @ X82)) = ((X80 @ (comp @ X84 @ X82) @ (sub @ X83 @ X82)))) | ~ ! [X81 : term] : (($true != ((X73 @ id @ X81 @ id))) | (((X73 @ id @ (X80 @ id @ X81) @ id)) = $true))))) <=> (hoasinduction_lem3v2_lthm = $true)) & ((laminj = $true) <=> ! [X448 : term,X449 : term] : ((X448 = X449) | (((lam @ X449)) != ((lam @ X448))))) & ((! [X289 : term,X288 : term,X287 : term,X286 : term] : ((X287 = X288) | (((ap @ X286 @ X287)) != ((ap @ X289 @ X288)))) => ! [X291 : term,X290 : term,X293 : term,X292 : term] : ((((ap @ (sub @ X292 @ id) @ X291)) != ((ap @ (sub @ X290 @ id) @ X293))) | (X291 = X293))) <=> (hoasapinj2_lthm = $true)) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (pushprop_lthm_orig = $true) & (! [X246 : term,X247 : term,X245 : term,X248 : term] : ((X247 = X248) | (((ap @ (sub @ X247 @ id) @ X246)) != ((ap @ (sub @ X248 @ id) @ X245)))) <=> (hoasapinj1 = $true)) & (! [X533 : (subst > term > subst > $o),X532 : term] : (($true = ((X533 @ id @ (lam @ (sub @ X532 @ (push @ one @ sh))) @ id))) | ~ ! [X535 : (subst > term > term)] : ((((X533 @ id @ (lam @ (X535 @ sh @ one)) @ id)) = $true) | ~ ! [X537 : subst,X539 : subst,X538 : term] : (((sub @ (X535 @ X539 @ X538) @ X537)) = ((X535 @ (comp @ X539 @ X537) @ (sub @ X538 @ X537)))) | ~ ! [X536 : term] : (($true != ((X533 @ id @ X536 @ id))) | ($true = ((X533 @ id @ (X535 @ id @ X536) @ id))))) | ~ ! [X534 : term] : ((((X533 @ id @ (sub @ X532 @ (push @ X534 @ id)) @ id)) = $true) | ($true != ((X533 @ id @ X534 @ id))))) <=> (hoasinduction_lem3aa_lthm = $true)) & (axvarshift = $true) & (! [X434 : term] : ~ ! [X435 : (subst > term > term)] : (((X435 @ sh @ one)) != ((sub @ X434 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b_lthm = $true)) & ((termmset = $true) <=> ! [X110 : term] : (((sub @ X110 @ id)) = X110)) & (! [X295 : subst,X294 : term] : (((sub @ one @ (push @ X294 @ X295))) = X294) <=> (axvarcons = $true)) & (hoasinduction_lem3a_gthm = $true) & ((hoasapinj2 = $true) <=> ! [X50 : term,X49 : term,X47 : term,X48 : term] : ((((ap @ (sub @ X48 @ id) @ X47)) != ((ap @ (sub @ X49 @ id) @ X50))) | (X47 = X50))) & (! [X90 : term] : ~ ! [X91 : (subst > term > term)] : ~ ! [X92 : term,X93 : subst] : (((X91 @ X93 @ X92)) = ((sub @ X90 @ (push @ X92 @ X93)))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & (ulamvar1 != $true) & ((induction2 = $true) <=> ! [X190 : term,X191 : (term > $o)] : (~ ! [X195 : term,X194 : term] : (($true != ((X191 @ X195))) | (((X191 @ (ap @ X195 @ X194))) = $true) | ($true != ((X191 @ X194)))) | ~ ! [X192 : term] : (($true = ((X191 @ (lam @ X192)))) | ~ ! [X193 : term] : (($true != ((X191 @ X193))) | ($true = ((X191 @ (sub @ X192 @ (push @ X193 @ id))))))) | ~ ! [X196 : term] : (($true != ((var @ X196))) | ($true = ((X191 @ X196)))) | ($true = ((X191 @ X190))))) & (hoasinduction_lem2_gthm = $true) & (substmonoid_gthm = $true) & ((axscons = $true) <=> ! [X429 : subst] : (((push @ (sub @ one @ X429) @ (comp @ sh @ X429))) = X429)) & (hoaslamnotvar_lthm = $true) & ((! [X313 : (subst > term > subst > $o)] : ~ ! [X314 : (term > $o)] : ~ ! [X315 : term] : (((X313 @ id @ X315 @ id)) = ((X314 @ X315))) => (! [X317 : (term > $o),X316 : term] : (~ ! [X318 : term] : (~ ! [X319 : term] : ((((X317 @ X319)) != $true) | (((X317 @ (sub @ X318 @ (push @ X319 @ id)))) = $true)) | ($true = ((X317 @ (lam @ X318))))) | ~ ! [X320 : term,X321 : term] : ((((X317 @ (ap @ X320 @ X321))) = $true) | (((X317 @ X321)) != $true) | ($true != ((X317 @ X320)))) | ($true = ((X317 @ X316))) | ~ ! [X322 : term] : (($true != ((var @ X322))) | ($true = ((X317 @ X322))))) => (! [X323 : term] : (((sub @ X323 @ id)) = X323) => (! [X325 : (term > $o),X326 : term,X324 : (subst > term > subst > $o)] : (($true = ((X325 @ (lam @ X326)))) | ~ ! [X333 : term] : (((X325 @ X333)) = ((X324 @ id @ X333 @ id))) | ~ ! [X327 : (subst > term > term)] : ((((X324 @ id @ (lam @ (X327 @ sh @ one)) @ id)) = $true) | ~ ! [X329 : subst,X330 : term,X331 : subst] : (((X327 @ (comp @ X331 @ X329) @ (sub @ X330 @ X329))) = ((sub @ (X327 @ X331 @ X330) @ X329))) | ~ ! [X328 : term] : (($true = ((X324 @ id @ (X327 @ id @ X328) @ id))) | ($true != ((X324 @ id @ X328 @ id))))) | ~ ! [X332 : term] : (($true != ((X325 @ X332))) | ($true = ((X325 @ (sub @ X326 @ (push @ X332 @ id))))))) => ! [X335 : (subst > term > subst > $o),X334 : term] : (~ ! [X336 : (subst > term > term)] : (~ ! [X339 : subst,X340 : subst,X338 : term] : (((sub @ (X336 @ X340 @ X338) @ X339)) = ((X336 @ (comp @ X340 @ X339) @ (sub @ X338 @ X339)))) | ~ ! [X337 : term] : ((((X335 @ id @ X337 @ id)) != $true) | ($true = ((X335 @ id @ (X336 @ id @ X337) @ id)))) | (((X335 @ id @ (lam @ (X336 @ sh @ one)) @ id)) = $true)) | ~ ! [X341 : term,X342 : term] : (($true != ((X335 @ id @ X341 @ id))) | ($true != ((X335 @ id @ X342 @ id))) | ($true = ((X335 @ id @ (ap @ (sub @ X342 @ id) @ X341) @ id)))) | ($true = ((X335 @ id @ X334 @ id)))))))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (hoaslamnotvar = $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 = $true) & (! [X296 : term,X298 : term,X299 : term,X297 : term] : ((X297 = X298) | (((ap @ X298 @ X299)) != ((ap @ X297 @ X296)))) <=> (apinj1 = $true)) & (hoasinduction_lem1_gthm = $true) & ((hoasinduction_lem3v2a_lthm = $true) <=> (! [X225 : term] : ~ ! [X226 : (subst > term > term)] : ~ ! [X228 : subst,X227 : term] : (((X226 @ X228 @ X227)) = ((sub @ X225 @ (push @ X227 @ X228)))) => (! [X229 : term] : (((sub @ X229 @ id)) = X229) => ! [X230 : (subst > term > subst > $o),X232 : term,X231 : (term > $o)] : (~ ! [X235 : (subst > term > term)] : (($true = ((X230 @ id @ (lam @ (X235 @ sh @ one)) @ id))) | ~ ! [X239 : subst,X238 : subst,X237 : term] : (((sub @ (X235 @ X239 @ X237) @ X238)) = ((X235 @ (comp @ X239 @ X238) @ (sub @ X237 @ X238)))) | ~ ! [X236 : term] : (($true = ((X230 @ id @ (X235 @ id @ X236) @ id))) | (((X230 @ id @ X236 @ id)) != $true))) | ~ ! [X234 : term] : (((X230 @ id @ X234 @ id)) = ((X231 @ X234))) | ~ ! [X233 : term] : (($true = ((X231 @ (sub @ X232 @ (push @ X233 @ id))))) | ($true != ((X231 @ X233)))) | ($true = ((X231 @ (lam @ X232)))))))) & (induction2lem_gthm = $true) & (! [X0 : term] : ~ ! [X1 : (subst > term > term)] : ~ ! [X2 : term,X3 : subst] : (((sub @ X0 @ (push @ X2 @ X3))) = ((X1 @ X3 @ X2))) <=> (hoasinduction_lem3v2_f = $true)) & (! [X413 : term,X414 : (subst > term > subst > $o)] : (~ ! [X420 : subst,X422 : subst,X421 : term,X419 : subst] : (($true = ((X414 @ (comp @ X420 @ X422) @ (sub @ X421 @ X422) @ X419))) | ($true != ((X414 @ X420 @ X421 @ (comp @ X422 @ X419))))) | ($true = ((X414 @ id @ (lam @ X413) @ id))) | ~ ! [X415 : subst,X418 : subst,X416 : term,X417 : subst] : (($true = ((X414 @ X415 @ X416 @ (comp @ X418 @ X417)))) | (((X414 @ (comp @ X415 @ X418) @ (sub @ X416 @ X418) @ X417)) != $true)) | ~ ! [X423 : (subst > term > term)] : (~ ! [X425 : term,X426 : subst,X424 : subst] : (((X423 @ (comp @ X426 @ X424) @ (sub @ X425 @ X424))) = ((sub @ (X423 @ X426 @ X425) @ X424))) | ~ ! [X427 : term] : ((((X414 @ id @ (X423 @ id @ X427) @ id)) = $true) | ($true != ((X414 @ id @ X427 @ id)))) | (((X414 @ id @ (lam @ (X423 @ sh @ one)) @ id)) = $true)) | ~ ! [X428 : term] : (($true != ((X414 @ id @ X428 @ id))) | ($true = ((X414 @ id @ (sub @ X413 @ (push @ X428 @ id)) @ id))))) <=> (hoasinduction_lem3 = $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((induction2lem_lthm = $true) <=> (! [X159 : term,X158 : subst] : (((sub @ one @ (push @ X159 @ X158))) = X159) => (! [X160 : subst,X161 : term] : (((comp @ sh @ (push @ X161 @ X160))) = X160) => (! [X162 : subst] : (((comp @ X162 @ id)) = X162) => (! [X164 : (term > $o),X163 : term] : (~ ! [X167 : term] : ((((X164 @ (lam @ X167))) = $true) | ($true != ((X164 @ X167)))) | ($true = ((X164 @ X163))) | ~ ! [X165 : term,X166 : term] : (($true != ((X164 @ X165))) | ($true != ((X164 @ X166))) | ($true = ((X164 @ (ap @ X165 @ X166))))) | ~ ! [X168 : term] : (($true = ((X164 @ X168))) | ($true != ((var @ X168))))) => ! [X171 : (term > $o),X170 : subst,X169 : term] : (~ ! [X176 : term,X175 : term] : (($true = ((X171 @ (ap @ X175 @ X176)))) | (((X171 @ X176)) != $true) | ($true != ((X171 @ X175)))) | ~ ! [X174 : term] : (($true = ((X171 @ (sub @ X174 @ X170)))) | ($true != ((var @ X174)))) | ($true = ((X171 @ (sub @ X169 @ X170)))) | ~ ! [X172 : term] : (($true = ((X171 @ (lam @ X172)))) | ~ ! [X173 : term] : (($true = ((X171 @ (sub @ X172 @ (push @ X173 @ id))))) | ($true != ((X171 @ X173))))))))))) & (termmset_gthm = $true) & (axassoc = $true) & (hoasapinj2_gthm = $true) & ((! [X505 : subst] : (((comp @ id @ X505)) = X505) & ! [X504 : subst] : (((comp @ X504 @ id)) = X504)) <=> (substmonoid = $true)) & ! [X433 : d_term,X432 : d_term] : ((((d2term @ X432)) = ((d2term @ X433))) => (X432 = X433)) & (lamnotvar = $true) & (((d2subst @ d_id)) = sh) & ((substmonoid_lthm = $true) <=> (! [X436 : subst] : (((comp @ id @ X436)) = X436) => (! [X437 : subst] : (((comp @ X437 @ id)) = X437) => (! [X438 : subst] : (((comp @ X438 @ id)) = X438) & ! [X439 : subst] : (((comp @ id @ X439)) = X439))))) & ((! [X177 : term] : (((sub @ X177 @ id)) = X177) => ! [X178 : term] : (((sub @ X178 @ id)) = X178)) <=> (termmset_lthm = $true)) & ((! [X94 : term,X95 : subst] : (((sub @ one @ (push @ X94 @ X95))) = X94) => (! [X97 : subst,X96 : term] : (((comp @ sh @ (push @ X96 @ X97))) = X97) => (! [X98 : term,X99 : term] : ((((lam @ X99)) != ((lam @ X98))) | (X98 = X99)) => ! [X101 : (subst > term > term),X103 : subst,X100 : term,X102 : (subst > term > term)] : (~ ! [X104 : term,X105 : subst,X106 : subst] : (((X102 @ (comp @ X105 @ X106) @ (sub @ X104 @ X106))) = ((sub @ (X102 @ X105 @ X104) @ X106))) | (((lam @ (X102 @ sh @ one))) != ((lam @ (X101 @ sh @ one)))) | ~ ! [X108 : subst,X109 : term,X107 : subst] : (((X101 @ (comp @ X108 @ X107) @ (sub @ X109 @ X107))) = ((sub @ (X101 @ X108 @ X109) @ X107))) | (((X102 @ X103 @ X100)) = ((X101 @ X103 @ X100))))))) <=> (hoaslaminj_lthm = $true)) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (hoasinduction_lem3b_gthm = $true) & ((hoasapinj1_lthm = $true) <=> (! [X359 : term] : (((sub @ X359 @ id)) = X359) => (! [X360 : term,X361 : term,X363 : term,X362 : term] : ((X361 = X363) | (((ap @ X361 @ X362)) != ((ap @ X363 @ X360)))) => ! [X364 : term,X366 : term,X365 : term,X367 : term] : ((X365 = X367) | (((ap @ (sub @ X367 @ id) @ X364)) != ((ap @ (sub @ X365 @ id) @ X366))))))) & (! [X527 : term] : (((sub @ X527 @ id)) = X527) <=> (axvarid = $true)) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ! [X241 : d_subst,X240 : d_subst] : ((((d2subst @ X241)) = ((d2subst @ X240))) => (X240 = X241)) & ((! [X343 : term] : (((sub @ X343 @ id)) = X343) => (! [X346 : (term > $o),X345 : subst,X344 : term] : (~ ! [X350 : term] : (($true = ((X346 @ (lam @ X350)))) | ~ ! [X351 : term] : (($true = ((X346 @ (sub @ X350 @ (push @ X351 @ id))))) | ($true != ((X346 @ X351))))) | ~ ! [X347 : term] : ((((var @ X347)) != $true) | ($true = ((X346 @ (sub @ X347 @ X345))))) | ~ ! [X349 : term,X348 : term] : (($true != ((X346 @ X348))) | (((X346 @ (ap @ X349 @ X348))) = $true) | (((X346 @ X349)) != $true)) | ($true = ((X346 @ (sub @ X344 @ X345))))) => ! [X353 : (term > $o),X352 : term] : (($true = ((X353 @ X352))) | ~ ! [X356 : term] : (($true = ((X353 @ X356))) | ($true != ((var @ X356)))) | ~ ! [X355 : term,X354 : term] : (($true = ((X353 @ (ap @ X355 @ X354)))) | ($true != ((X353 @ X355))) | ($true != ((X353 @ X354)))) | ~ ! [X357 : term] : (~ ! [X358 : term] : (($true = ((X353 @ (sub @ X357 @ (push @ X358 @ id))))) | ($true != ((X353 @ X358)))) | ($true = ((X353 @ (lam @ X357)))))))) <=> (induction2_lthm = $true)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (! [X524 : (subst > term > subst > $o)] : ~ ! [X525 : (term > $o)] : ~ ! [X526 : term] : (((X524 @ id @ X526 @ id)) = ((X525 @ X526))) <=> (hoasinduction_lem0_lthm = $true)) & (! [X276 : (term > $o),X279 : term,X278 : (term > $o),X277 : subst] : (~ ! [X280 : term] : (((X276 @ X280)) = ((X278 @ (sub @ X280 @ (push @ X279 @ X277))))) | ($true != ((X278 @ X279))) | (((X276 @ one)) = $true)) <=> (pushprop_lem1v2 = $true)) & (hoasapnotvar = $true) & (hoasapnotvar_gthm = $true) & (hoaslaminj_gthm = $true) & (((d2term @ d_one)) = one) & (hoasinduction_lem1 = $true) & (axclos = $true) & ((! [X151 : term,X152 : subst] : (((sub @ one @ (push @ X151 @ X152))) = X151) => ! [X156 : (term > $o),X155 : subst,X153 : term,X154 : (term > $o)] : (~ ! [X157 : term] : (((X156 @ (sub @ X157 @ (push @ X153 @ X155)))) = ((X154 @ X157))) | ($true != ((X156 @ X153))) | (((X154 @ one)) = $true))) <=> (pushprop_lem1v2_lthm = $true)) & (ulamvarind = $true) & (((d2subst @ d_id)) = id) & (hoasinduction_lem2v2_gthm = $true) & ((hoasinduction_lthm = $true) <=> (! [X450 : (term > $o),X451 : term] : (~ ! [X454 : term] : ((((var @ X454)) != $true) | ($true = ((X450 @ X454)))) | (((X450 @ X451)) = $true) | ~ ! [X456 : term,X455 : term] : (($true != ((X450 @ X456))) | ($true != ((X450 @ X455))) | ($true = ((X450 @ (ap @ X455 @ X456))))) | ~ ! [X452 : term] : (~ ! [X453 : term] : (($true = ((X450 @ (sub @ X452 @ (push @ X453 @ id))))) | ($true != ((X450 @ X453)))) | ($true = ((X450 @ (lam @ X452)))))) => (! [X458 : term,X459 : term,X457 : (subst > term > subst > $o)] : ((((X457 @ id @ X458 @ id)) != $true) | ~ ! [X462 : term,X463 : subst,X465 : subst,X464 : subst] : (($true != ((X457 @ X464 @ X462 @ (comp @ X463 @ X465)))) | ($true = ((X457 @ (comp @ X464 @ X463) @ (sub @ X462 @ X463) @ X465)))) | ($true = ((X457 @ id @ (ap @ X459 @ X458) @ id))) | ($true != ((X457 @ id @ X459 @ id))) | ~ ! [X461 : term,X460 : term] : (($true = ((X457 @ id @ (ap @ (sub @ X460 @ id) @ X461) @ id))) | ($true != ((X457 @ id @ X460 @ id))) | ($true != ((X457 @ id @ X461 @ id)))) | ~ ! [X468 : term,X467 : subst,X469 : subst,X466 : subst] : ((((X457 @ X466 @ X468 @ (comp @ X467 @ X469))) = $true) | ($true != ((X457 @ (comp @ X466 @ X467) @ (sub @ X468 @ X467) @ X469))))) => (! [X471 : (subst > term > subst > $o),X470 : term] : (~ ! [X480 : term] : (($true = ((X471 @ id @ (sub @ X470 @ (push @ X480 @ id)) @ id))) | ($true != ((X471 @ id @ X480 @ id)))) | (((X471 @ id @ (lam @ X470) @ id)) = $true) | ~ ! [X476 : subst,X477 : subst,X478 : term,X479 : subst] : (($true != ((X471 @ (comp @ X479 @ X476) @ (sub @ X478 @ X476) @ X477))) | ($true = ((X471 @ X479 @ X478 @ (comp @ X476 @ X477))))) | ~ ! [X481 : (subst > term > term)] : (~ ! [X482 : term,X484 : subst,X483 : subst] : (((X481 @ (comp @ X484 @ X483) @ (sub @ X482 @ X483))) = ((sub @ (X481 @ X484 @ X482) @ X483))) | (((X471 @ id @ (lam @ (X481 @ sh @ one)) @ id)) = $true) | ~ ! [X485 : term] : ((((X471 @ id @ (X481 @ id @ X485) @ id)) = $true) | (((X471 @ id @ X485 @ id)) != $true))) | ~ ! [X475 : subst,X474 : subst,X473 : term,X472 : subst] : (($true != ((X471 @ X475 @ X473 @ (comp @ X474 @ X472)))) | ($true = ((X471 @ (comp @ X475 @ X474) @ (sub @ X473 @ X474) @ X472))))) => ! [X487 : (subst > term > subst > $o),X486 : term] : (~ ! [X488 : term] : (($true != ((var @ (sub @ X488 @ id)))) | ($true = ((X487 @ id @ X488 @ id)))) | ~ ! [X499 : (subst > term > term)] : ((((X487 @ id @ (lam @ (X499 @ sh @ one)) @ id)) = $true) | ~ ! [X501 : subst,X502 : subst,X503 : term] : (((X499 @ (comp @ X502 @ X501) @ (sub @ X503 @ X501))) = ((sub @ (X499 @ X502 @ X503) @ X501))) | ~ ! [X500 : term] : (($true != ((X487 @ id @ X500 @ id))) | ($true = ((X487 @ id @ (X499 @ id @ X500) @ id))))) | ($true = ((X487 @ id @ X486 @ id))) | ~ ! [X495 : term,X494 : subst,X493 : subst,X496 : subst] : (($true = ((X487 @ (comp @ X493 @ X494) @ (sub @ X495 @ X494) @ X496))) | ($true != ((X487 @ X493 @ X495 @ (comp @ X494 @ X496))))) | ~ ! [X490 : subst,X492 : subst,X491 : subst,X489 : term] : ((((X487 @ X490 @ X489 @ (comp @ X491 @ X492))) = $true) | ($true != ((X487 @ (comp @ X490 @ X491) @ (sub @ X489 @ X491) @ X492)))) | ~ ! [X498 : term,X497 : term] : (($true = ((X487 @ id @ (ap @ (sub @ X497 @ id) @ X498) @ id))) | ($true != ((X487 @ id @ X497 @ id))) | (((X487 @ id @ X498 @ id)) != $true))))))) & (hoasapnotvar_lthm = $true) & ((axshiftcons = $true) <=> ! [X52 : term,X51 : subst] : (((comp @ sh @ (push @ X52 @ X51))) = X51)) & (induction2_gthm = $true) & (! [X371 : term,X370 : (term > $o),X368 : term,X369 : (subst > term > subst > $o)] : (~ ! [X380 : term,X381 : term] : (($true != ((X369 @ id @ X380 @ id))) | ($true = ((X369 @ id @ (ap @ (sub @ X380 @ id) @ X381) @ id))) | (((X369 @ id @ X381 @ id)) != $true)) | ($true != ((X370 @ X368))) | ~ ! [X378 : term,X379 : subst,X376 : subst,X377 : subst] : (($true != ((X369 @ X376 @ X378 @ (comp @ X379 @ X377)))) | ($true = ((X369 @ (comp @ X376 @ X379) @ (sub @ X378 @ X379) @ X377)))) | ($true != ((X370 @ X371))) | ($true = ((X370 @ (ap @ X371 @ X368)))) | ~ ! [X372 : subst,X373 : term,X375 : subst,X374 : subst] : (($true != ((X369 @ (comp @ X375 @ X374) @ (sub @ X373 @ X374) @ X372))) | ($true = ((X369 @ X375 @ X373 @ (comp @ X374 @ X372))))) | ~ ! [X382 : term] : (((X369 @ id @ X382 @ id)) = ((X370 @ X382)))) <=> (hoasinduction_lem2v2 = $true)) & (! [X260 : term,X261 : (subst > term > term),X262 : term] : (~ ! [X264 : subst,X263 : term,X265 : subst] : (((sub @ (X261 @ X264 @ X263) @ X265)) = ((X261 @ (comp @ X264 @ X265) @ (sub @ X263 @ X265)))) | (((ap @ (sub @ X262 @ id) @ X260)) != ((lam @ (X261 @ sh @ one))))) <=> (hoaslamnotap = $true)) & ((! [X203 : subst,X204 : term] : (((sub @ one @ (push @ X204 @ X203))) = X204) => (! [X206 : term,X205 : subst] : (((comp @ sh @ (push @ X206 @ X205))) = X205) => ! [X207 : term,X210 : (term > $o),X209 : (term > $o),X211 : subst,X208 : term] : (($true != ((X209 @ X208))) | (((X210 @ (sub @ X208 @ (push @ X207 @ X211)))) = $true)))) <=> (pushprop_lem1_lthm = $true)) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (pushprop_lem3v2_lthm = $true) & ((hoasinduction_no_psi_cond = $true) <=> ! [X143 : (subst > term > subst > $o),X142 : term] : (~ ! [X149 : term,X150 : term] : (($true != ((X143 @ id @ X150 @ id))) | (((X143 @ id @ X149 @ id)) != $true) | ($true = ((X143 @ id @ (ap @ (sub @ X149 @ id) @ X150) @ id)))) | ~ ! [X144 : (subst > term > term)] : (~ ! [X147 : subst,X148 : term,X146 : subst] : (((X144 @ (comp @ X146 @ X147) @ (sub @ X148 @ X147))) = ((sub @ (X144 @ X146 @ X148) @ X147))) | ~ ! [X145 : term] : (($true = ((X143 @ id @ (X144 @ id @ X145) @ id))) | (((X143 @ id @ X145 @ id)) != $true)) | (((X143 @ id @ (lam @ (X144 @ sh @ one)) @ id)) = $true)) | ($true = ((X143 @ id @ X142 @ id))))) & ((hoasinduction_lthm_3 = $true) <=> (! [X6 : (subst > term > subst > $o)] : ~ ! [X7 : (term > $o)] : ~ ! [X8 : term] : (((X7 @ X8)) = ((X6 @ id @ X8 @ id))) => (! [X9 : (term > $o),X10 : term] : (~ ! [X13 : term] : (($true != ((var @ X13))) | ($true = ((X9 @ X13)))) | ~ ! [X14 : term] : (~ ! [X15 : term] : (($true != ((X9 @ X15))) | ($true = ((X9 @ (sub @ X14 @ (push @ X15 @ id)))))) | ($true = ((X9 @ (lam @ X14))))) | ~ ! [X11 : term,X12 : term] : (($true != ((X9 @ X11))) | (((X9 @ X12)) != $true) | (((X9 @ (ap @ X11 @ X12))) = $true)) | ($true = ((X9 @ X10)))) => (! [X16 : term] : (((sub @ X16 @ id)) = X16) => (! [X17 : term,X18 : (term > $o),X19 : (subst > term > subst > $o)] : (~ ! [X25 : term] : (($true != ((X18 @ X25))) | ($true = ((X18 @ (sub @ X17 @ (push @ X25 @ id)))))) | ~ ! [X26 : term] : (((X18 @ X26)) = ((X19 @ id @ X26 @ id))) | ($true = ((X18 @ (lam @ X17)))) | ~ ! [X20 : (subst > term > term)] : (~ ! [X21 : term] : (($true = ((X19 @ id @ (X20 @ id @ X21) @ id))) | (((X19 @ id @ X21 @ id)) != $true)) | (((X19 @ id @ (lam @ (X20 @ sh @ one)) @ id)) = $true) | ~ ! [X22 : term,X23 : subst,X24 : subst] : (((X20 @ (comp @ X23 @ X24) @ (sub @ X22 @ X24))) = ((sub @ (X20 @ X23 @ X22) @ X24))))) => ! [X27 : term,X28 : (subst > term > subst > $o)] : (~ ! [X37 : subst,X39 : subst,X38 : subst,X36 : term] : (($true = ((X28 @ X37 @ X36 @ (comp @ X39 @ X38)))) | ($true != ((X28 @ (comp @ X37 @ X39) @ (sub @ X36 @ X39) @ X38)))) | ~ ! [X31 : (subst > term > term)] : ((((X28 @ id @ (lam @ (X31 @ sh @ one)) @ id)) = $true) | ~ ! [X34 : term,X35 : subst,X33 : subst] : (((sub @ (X31 @ X35 @ X34) @ X33)) = ((X31 @ (comp @ X35 @ X33) @ (sub @ X34 @ X33)))) | ~ ! [X32 : term] : (($true = ((X28 @ id @ (X31 @ id @ X32) @ id))) | ($true != ((X28 @ id @ X32 @ id))))) | ~ ! [X44 : term] : (($true = ((X28 @ id @ X44 @ id))) | ($true != ((var @ (sub @ X44 @ id))))) | ~ ! [X29 : term,X30 : term] : (($true = ((X28 @ id @ (ap @ (sub @ X30 @ id) @ X29) @ id))) | ($true != ((X28 @ id @ X30 @ id))) | ($true != ((X28 @ id @ X29 @ id)))) | (((X28 @ id @ X27 @ id)) = $true) | ~ ! [X41 : term,X40 : subst,X42 : subst,X43 : subst] : ((((X28 @ (comp @ X43 @ X40) @ (sub @ X41 @ X40) @ X42)) = $true) | (((X28 @ X43 @ X41 @ (comp @ X40 @ X42))) != $true)))))))) & ! [X249 : d_term] : (d_one = X249) & (pushprop_lem2v2_gthm = $true) & ((hoasinduction_lem3aa = $true) <=> ! [X306 : term,X305 : (subst > term > subst > $o)] : (~ ! [X307 : term] : (($true != ((X305 @ id @ X307 @ id))) | ($true = ((X305 @ id @ (sub @ X306 @ (push @ X307 @ id)) @ id)))) | ($true = ((X305 @ id @ (lam @ (sub @ X306 @ (push @ one @ sh))) @ id))) | ~ ! [X308 : (subst > term > term)] : (~ ! [X309 : term] : ((((X305 @ id @ X309 @ id)) != $true) | ($true = ((X305 @ id @ (X308 @ id @ X309) @ id)))) | ~ ! [X312 : subst,X310 : subst,X311 : term] : (((sub @ (X308 @ X312 @ X311) @ X310)) = ((X308 @ (comp @ X312 @ X310) @ (sub @ X311 @ X310)))) | ($true = ((X305 @ id @ (lam @ (X308 @ sh @ one)) @ id)))))) & (! [X251 : term,X253 : (subst > term > term),X250 : (subst > term > term),X252 : subst] : (~ ! [X259 : subst,X257 : subst,X258 : term] : (((sub @ (X250 @ X259 @ X258) @ X257)) = ((X250 @ (comp @ X259 @ X257) @ (sub @ X258 @ X257)))) | (((X253 @ X252 @ X251)) = ((X250 @ X252 @ X251))) | (((lam @ (X250 @ sh @ one))) != ((lam @ (X253 @ sh @ one)))) | ~ ! [X256 : subst,X255 : subst,X254 : term] : (((sub @ (X253 @ X255 @ X254) @ X256)) = ((X253 @ (comp @ X255 @ X256) @ (sub @ X254 @ X256))))) <=> (hoaslaminj = $true)) & (pushprop_lem1_gthm = $true) & ((pushprop_lem1 = $true) <=> ! [X410 : (term > $o),X411 : (term > $o),X408 : subst,X412 : term,X409 : term] : (($true = ((X411 @ (sub @ X412 @ (push @ X409 @ X408))))) | ($true != ((X410 @ X412))))) & (hoasinduction_lem1v2_gthm = $true) & (pushprop_lem2v2 = $true) & ((hoasinduction_lem3_lthm = $true) <=> (! [X383 : term] : (((sub @ X383 @ id)) = X383) => (! [X384 : (subst > term > subst > $o),X385 : term] : (($true = ((X384 @ id @ (lam @ (sub @ X385 @ (push @ one @ sh))) @ id))) | ~ ! [X386 : (subst > term > term)] : (($true = ((X384 @ id @ (lam @ (X386 @ sh @ one)) @ id))) | ~ ! [X390 : term] : (($true != ((X384 @ id @ X390 @ id))) | ($true = ((X384 @ id @ (X386 @ id @ X390) @ id)))) | ~ ! [X389 : term,X387 : subst,X388 : subst] : (((sub @ (X386 @ X387 @ X389) @ X388)) = ((X386 @ (comp @ X387 @ X388) @ (sub @ X389 @ X388))))) | ~ ! [X391 : term] : (($true = ((X384 @ id @ (sub @ X385 @ (push @ X391 @ id)) @ id))) | (((X384 @ id @ X391 @ id)) != $true))) => ! [X392 : (subst > term > subst > $o),X393 : term] : (~ ! [X407 : subst,X405 : term,X406 : subst,X404 : subst] : ((((X392 @ X407 @ X405 @ (comp @ X406 @ X404))) != $true) | (((X392 @ (comp @ X407 @ X406) @ (sub @ X405 @ X406) @ X404)) = $true)) | ~ ! [X394 : (subst > term > term)] : (~ ! [X395 : term] : ((((X392 @ id @ (X394 @ id @ X395) @ id)) = $true) | (((X392 @ id @ X395 @ id)) != $true)) | ~ ! [X398 : term,X396 : subst,X397 : subst] : (((sub @ (X394 @ X396 @ X398) @ X397)) = ((X394 @ (comp @ X396 @ X397) @ (sub @ X398 @ X397)))) | ($true = ((X392 @ id @ (lam @ (X394 @ sh @ one)) @ id)))) | ~ ! [X399 : term] : (($true != ((X392 @ id @ X399 @ id))) | ($true = ((X392 @ id @ (sub @ X393 @ (push @ X399 @ id)) @ id)))) | ~ ! [X402 : subst,X403 : subst,X400 : term,X401 : subst] : (($true = ((X392 @ X403 @ X400 @ (comp @ X401 @ X402)))) | (((X392 @ (comp @ X403 @ X401) @ (sub @ X400 @ X401) @ X402)) != $true)) | (((X392 @ id @ (lam @ X393) @ id)) = $true)))))),
  inference(flattening,[],[f8])).
thf(f10,plain,(
  ~((hoasinduction_lem3v2 = $true) <=> ! [X0 : (subst > term > subst > $o),X1 : (term > $o)] : (! [X2 : subst,X4 : term,X3 : subst,X5 : subst] : (($true = ((X0 @ X5 @ X4 @ (comp @ X3 @ X2)))) => ($true = ((X0 @ (comp @ X5 @ X3) @ (sub @ X4 @ X3) @ X2)))) => (! [X9 : subst,X8 : subst,X7 : term,X6 : subst] : ((((X0 @ (comp @ X9 @ X6) @ (sub @ X7 @ X6) @ X8)) = $true) => (((X0 @ X9 @ X7 @ (comp @ X6 @ X8))) = $true)) => (! [X10 : (subst > term > term)] : (! [X11 : subst,X13 : subst,X12 : term] : (((X10 @ (comp @ X11 @ X13) @ (sub @ X12 @ X13))) = ((sub @ (X10 @ X11 @ X12) @ X13))) => (! [X14 : term] : ((((X0 @ id @ X14 @ id)) = $true) => (((X0 @ id @ (X10 @ id @ X14) @ id)) = $true)) => ($true = ((X0 @ id @ (hoaslam @ id @ (^[Y0 : subst]: ((^[Y1 : term]: (X10 @ Y0 @ Y1))))) @ id))))) => ((((hoasinduction_p_and_p_prime @ X0 @ X1)) = $true) => ! [X15 : term] : (! [X16 : term] : ((((X1 @ X16)) = $true) => ($true = ((X1 @ (sub @ X15 @ (push @ X16 @ id)))))) => ($true = ((X1 @ (lam @ X15))))))))))),
  inference(rectify,[],[f7])).
thf(f11,plain,(
  (hoasinduction_lem3v2 = $true) <~> ! [X0 : (subst > term > subst > $o),X1 : (term > $o)] : ((((! [X15 : term] : (? [X16 : term] : ((((X1 @ X16)) = $true) & ($true != ((X1 @ (sub @ X15 @ (push @ X16 @ id)))))) | ($true = ((X1 @ (lam @ X15))))) | (((hoasinduction_p_and_p_prime @ X0 @ X1)) != $true)) | ? [X10 : (subst > term > term)] : ((($true != ((X0 @ id @ (hoaslam @ id @ (^[Y0 : subst]: ((^[Y1 : term]: (X10 @ Y0 @ Y1))))) @ id))) & ! [X14 : term] : ((((X0 @ id @ X14 @ id)) != $true) | (((X0 @ id @ (X10 @ id @ X14) @ id)) = $true))) & ! [X11 : subst,X13 : subst,X12 : term] : (((X10 @ (comp @ X11 @ X13) @ (sub @ X12 @ X13))) = ((sub @ (X10 @ X11 @ X12) @ X13))))) | ? [X7 : term,X6 : subst,X9 : subst,X8 : subst] : ((((X0 @ (comp @ X9 @ X6) @ (sub @ X7 @ X6) @ X8)) = $true) & (((X0 @ X9 @ X7 @ (comp @ X6 @ X8))) != $true))) | ? [X5 : subst,X3 : subst,X4 : term,X2 : subst] : (($true = ((X0 @ X5 @ X4 @ (comp @ X3 @ X2)))) & ($true != ((X0 @ (comp @ X5 @ X3) @ (sub @ X4 @ X3) @ X2)))))),
  inference(ennf_transformation,[],[f10])).
thf(f12,plain,(
  (hoasinduction_lem3v2 = $true) <~> ! [X1 : (term > $o),X0 : (subst > term > subst > $o)] : (? [X7 : term,X6 : subst,X9 : subst,X8 : subst] : ((((X0 @ (comp @ X9 @ X6) @ (sub @ X7 @ X6) @ X8)) = $true) & (((X0 @ X9 @ X7 @ (comp @ X6 @ X8))) != $true)) | ! [X15 : term] : (? [X16 : term] : ((((X1 @ X16)) = $true) & ($true != ((X1 @ (sub @ X15 @ (push @ X16 @ id)))))) | ($true = ((X1 @ (lam @ X15))))) | (((hoasinduction_p_and_p_prime @ X0 @ X1)) != $true) | ? [X10 : (subst > term > term)] : (! [X14 : term] : ((((X0 @ id @ X14 @ id)) != $true) | (((X0 @ id @ (X10 @ id @ X14) @ id)) = $true)) & ! [X11 : subst,X13 : subst,X12 : term] : (((X10 @ (comp @ X11 @ X13) @ (sub @ X12 @ X13))) = ((sub @ (X10 @ X11 @ X12) @ X13))) & ($true != ((X0 @ id @ (hoaslam @ id @ (^[Y0 : subst]: ((^[Y1 : term]: (X10 @ Y0 @ Y1))))) @ id)))) | ? [X5 : subst,X3 : subst,X4 : term,X2 : subst] : (($true = ((X0 @ X5 @ X4 @ (comp @ X3 @ X2)))) & ($true != ((X0 @ (comp @ X5 @ X3) @ (sub @ X4 @ X3) @ X2)))))),
  inference(flattening,[],[f11])).
thf(f13,plain,(
  (hoasinduction_lem3_gthm = $true) & (((var @ (d2term @ d_one))) != $true) & (! [X198 : term,X197 : (term > $o)] : (? [X201 : term,X202 : term] : ((((X197 @ (ap @ X201 @ X202))) != $true) & ($true = ((X197 @ X202))) & (((X197 @ X201)) = $true)) | ? [X200 : term] : (($true = ((var @ X200))) & ($true != ((X197 @ X200)))) | ($true = ((X197 @ X198))) | ? [X199 : term] : (($true = ((X197 @ X199))) & ($true != ((X197 @ (lam @ X199)))))) <=> (induction = $true)) & ! [X244 : d_subst] : (d_id = X244) & (pushprop_gthm = $true) & (axapp = $true) & (hoaslamnotap_gthm = $true) & (! [X129 : term,X128 : (subst > term > subst > $o)] : (? [X130 : term] : (($true = ((X128 @ id @ X130 @ id))) & ($true != ((X128 @ id @ (sub @ X129 @ (push @ X130 @ id)) @ id)))) | ? [X131 : (subst > term > term),X132 : term] : (! [X136 : term] : ((((X128 @ id @ X136 @ id)) != $true) | ($true = ((X128 @ id @ (X131 @ id @ X136) @ id)))) & ! [X134 : term,X135 : subst,X133 : subst] : (((sub @ (sub @ X132 @ (push @ X134 @ X135)) @ X133)) = ((sub @ (X131 @ X135 @ X134) @ X133))) & ! [X139 : subst,X138 : term,X137 : subst] : (((sub @ X132 @ (push @ (sub @ X138 @ X137) @ (comp @ X139 @ X137)))) = ((X131 @ (comp @ X139 @ X137) @ (sub @ X138 @ X137)))) & (((X128 @ id @ (lam @ (X131 @ sh @ one)) @ id)) != $true)) | ($true = ((X128 @ id @ (lam @ (sub @ X129 @ (push @ one @ sh))) @ id)))) <=> (hoasinduction_lem3aaa = $true)) & (hoaslamnotvar_gthm = $true) & ((induction2lem = $true) <=> ! [X181 : (term > $o),X180 : term,X179 : subst] : (? [X182 : term] : (! [X183 : term] : (($true != ((X181 @ X183))) | ($true = ((X181 @ (sub @ X182 @ (push @ X183 @ id)))))) & ($true != ((X181 @ (lam @ X182))))) | ? [X186 : term] : ((((X181 @ (sub @ X186 @ X179))) != $true) & ($true = ((var @ X186)))) | ($true = ((X181 @ (sub @ X180 @ X179)))) | ? [X185 : term,X184 : term] : (($true = ((X181 @ X185))) & ($true = ((X181 @ X184))) & ($true != ((X181 @ (ap @ X185 @ X184))))))) & ! [X242 : term] : ? [X243 : d_term] : (((d2term @ X243)) = X242) & (hoasinduction_lem2_lthm = $true) & (! [X141 : subst] : (((comp @ id @ X141)) = X141) <=> (axidl = $true)) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoasinduction_lem3a = $true) <=> ! [X441 : term,X440 : (subst > term > subst > $o)] : (? [X443 : (subst > term > term)] : (! [X444 : term] : (($true = ((X440 @ id @ (X443 @ id @ X444) @ id))) | ($true != ((X440 @ id @ X444 @ id)))) & ! [X445 : term,X446 : subst,X447 : subst] : (((X443 @ (comp @ X446 @ X447) @ (sub @ X445 @ X447))) = ((sub @ (X443 @ X446 @ X445) @ X447))) & ($true != ((X440 @ id @ (lam @ (X443 @ sh @ one)) @ id)))) | ($true = ((X440 @ id @ (lam @ X441) @ id))) | ? [X442 : term] : (($true != ((X440 @ id @ (sub @ X441 @ (push @ X442 @ id)) @ id))) & ($true = ((X440 @ id @ X442 @ id)))))) & (hoasinduction_gthm = $true) & ((pushprop_lem0 = $true) <=> ! [X301 : subst,X302 : (term > $o),X300 : term] : ? [X303 : (term > $o)] : ! [X304 : term] : (((X302 @ (sub @ X304 @ (push @ X300 @ X301)))) = ((X303 @ X304)))) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (ulamvarsh = $true) & ((apinj2 = $true) <=> ! [X529 : term,X528 : term,X531 : term,X530 : term] : ((X529 = X531) | (((ap @ X528 @ X529)) != ((ap @ X530 @ X531))))) & (hoasinduction_lem1_lthm = $true) & ! [X4 : subst] : ? [X5 : d_subst] : (((d2subst @ X5)) = X4) & (! [X430 : term] : ? [X431 : (subst > term > term)] : (((X431 @ sh @ one)) = ((sub @ X430 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b = $true)) & ((shinj = $true) <=> ! [X45 : term,X46 : term] : ((X45 = X46) | (((sub @ X46 @ sh)) != ((sub @ X45 @ sh))))) & ((hoasinduction = $true) <=> ! [X506 : term,X507 : (subst > term > subst > $o)] : (? [X513 : term,X512 : term] : (($true != ((X507 @ id @ (ap @ (sub @ X513 @ id) @ X512) @ id))) & ($true = ((X507 @ id @ X512 @ id))) & ($true = ((X507 @ id @ X513 @ id)))) | ? [X518 : term] : (($true = ((var @ (sub @ X518 @ id)))) & ($true != ((X507 @ id @ X518 @ id)))) | ? [X508 : subst,X511 : term,X510 : subst,X509 : subst] : ((((X507 @ (comp @ X508 @ X509) @ (sub @ X511 @ X509) @ X510)) != $true) & ($true = ((X507 @ X508 @ X511 @ (comp @ X509 @ X510))))) | ($true = ((X507 @ id @ X506 @ id))) | ? [X519 : (subst > term > term)] : ((((X507 @ id @ (lam @ (X519 @ sh @ one)) @ id)) != $true) & ! [X520 : term] : (($true != ((X507 @ id @ X520 @ id))) | ($true = ((X507 @ id @ (X519 @ id @ X520) @ id)))) & ! [X523 : term,X522 : subst,X521 : subst] : (((X519 @ (comp @ X521 @ X522) @ (sub @ X523 @ X522))) = ((sub @ (X519 @ X521 @ X523) @ X522)))) | ? [X514 : subst,X516 : subst,X515 : subst,X517 : term] : (($true != ((X507 @ X514 @ X517 @ (comp @ X516 @ X515)))) & (((X507 @ (comp @ X514 @ X516) @ (sub @ X517 @ X516) @ X515)) = $true)))) & (pushprop_lem3v2 = $true) & (hoasapinj1_gthm = $true) & (axabs = $true) & (pushprop_lthm = $true) & (lamnotap != $true) & (! [X140 : subst] : (((comp @ X140 @ id)) = X140) <=> (axidr = $true)) & ((hoasinduction_lem3v2 = $true) <=> ! [X54 : (subst > term > subst > $o),X55 : (term > $o),X53 : term] : (? [X70 : term] : (((X54 @ id @ X70 @ id)) != ((X55 @ X70))) | ? [X56 : (subst > term > term)] : (! [X57 : term] : (($true = ((X54 @ id @ (X56 @ id @ X57) @ id))) | ($true != ((X54 @ id @ X57 @ id)))) & ! [X59 : subst,X60 : term,X58 : subst] : (((X56 @ (comp @ X59 @ X58) @ (sub @ X60 @ X58))) = ((sub @ (X56 @ X59 @ X60) @ X58))) & ($true != ((X54 @ id @ (lam @ (X56 @ sh @ one)) @ id)))) | ? [X69 : term] : ((((X55 @ (sub @ X53 @ (push @ X69 @ id)))) != $true) & (((X55 @ X69)) = $true)) | ($true = ((X55 @ (lam @ X53)))) | ? [X68 : subst,X65 : term,X66 : subst,X67 : subst] : (($true = ((X54 @ (comp @ X68 @ X67) @ (sub @ X65 @ X67) @ X66))) & (((X54 @ X68 @ X65 @ (comp @ X67 @ X66))) != $true)) | ? [X63 : subst,X62 : subst,X64 : term,X61 : subst] : ((((X54 @ (comp @ X61 @ X63) @ (sub @ X64 @ X63) @ X62)) != $true) & ($true = ((X54 @ X61 @ X64 @ (comp @ X63 @ X62))))))) & (apnotvar = $true) & (hoaslamnotap_lthm = $true) & (pushprop_lem0_gthm = $true) & (! [X281 : subst,X282 : term,X283 : (term > $o)] : ? [X284 : (term > $o)] : ! [X285 : term] : (((X283 @ (sub @ X285 @ (push @ X282 @ X281)))) = ((X284 @ X285))) <=> (pushprop_lem0_lthm = $true)) & (! [X187 : (subst > term > subst > $o)] : ? [X188 : (term > $o)] : ! [X189 : term] : (((X187 @ id @ X189 @ id)) = ((X188 @ X189))) <=> (hoasinduction_lem0 = $true)) & (pushprop_lem2v2_lthm = $true) & (pushprop_lem1v2_gthm = $true) & (((! [X120 : term,X121 : (subst > term > subst > $o)] : (? [X127 : term] : (($true != ((X121 @ id @ (sub @ X120 @ (push @ X127 @ id)) @ id))) & ($true = ((X121 @ id @ X127 @ id)))) | ? [X122 : (subst > term > term)] : (! [X124 : term,X125 : subst,X126 : subst] : (((X122 @ (comp @ X125 @ X126) @ (sub @ X124 @ X126))) = ((sub @ (X122 @ X125 @ X124) @ X126))) & ($true != ((X121 @ id @ (lam @ (X122 @ sh @ one)) @ id))) & ! [X123 : term] : (($true != ((X121 @ id @ X123 @ id))) | ($true = ((X121 @ id @ (X122 @ id @ X123) @ id))))) | ($true = ((X121 @ id @ (lam @ X120) @ id)))) | ? [X112 : (subst > term > subst > $o),X113 : term] : (($true != ((X112 @ id @ (lam @ (sub @ X113 @ (push @ one @ sh))) @ id))) & ! [X114 : (subst > term > term)] : (? [X115 : term] : ((((X112 @ id @ X115 @ id)) = $true) & (((X112 @ id @ (X114 @ id @ X115) @ id)) != $true)) | ($true = ((X112 @ id @ (lam @ (X114 @ sh @ one)) @ id))) | ? [X116 : subst,X118 : subst,X117 : term] : (((X114 @ (comp @ X118 @ X116) @ (sub @ X117 @ X116))) != ((sub @ (X114 @ X118 @ X117) @ X116)))) & ! [X119 : term] : (($true = ((X112 @ id @ (sub @ X113 @ (push @ X119 @ id)) @ id))) | ($true != ((X112 @ id @ X119 @ id)))))) | ? [X111 : term] : (((sub @ X111 @ id)) != X111)) <=> (hoasinduction_lem3a_lthm = $true)) & (axmap = $true) & (! [X268 : (term > $o),X266 : (subst > term > subst > $o),X267 : term] : (? [X275 : term] : (((X266 @ id @ X275 @ id)) != ((X268 @ X275))) | ? [X269 : term] : (($true = ((X268 @ X269))) & ($true != ((X268 @ (sub @ X267 @ (push @ X269 @ id)))))) | ? [X270 : (subst > term > term)] : (! [X272 : subst,X273 : term,X274 : subst] : (((sub @ (X270 @ X272 @ X273) @ X274)) = ((X270 @ (comp @ X272 @ X274) @ (sub @ X273 @ X274)))) & ! [X271 : term] : ((((X266 @ id @ (X270 @ id @ X271) @ id)) = $true) | (((X266 @ id @ X271 @ id)) != $true)) & ($true != ((X266 @ id @ (lam @ (X270 @ sh @ one)) @ id)))) | ($true = ((X268 @ (lam @ X267))))) <=> (hoasinduction_lem3v2a = $true)) & (hoasinduction_lem3v2_gthm = $true) & (! [X214 : term,X212 : (subst > term > subst > $o),X213 : term] : (($true != ((X212 @ id @ X213 @ id))) | ($true != ((X212 @ id @ X214 @ id))) | ? [X222 : term,X223 : subst,X224 : subst,X221 : subst] : (($true != ((X212 @ (comp @ X223 @ X221) @ (sub @ X222 @ X221) @ X224))) & (((X212 @ X223 @ X222 @ (comp @ X221 @ X224))) = $true)) | ? [X216 : term,X217 : subst,X215 : subst,X218 : subst] : (($true = ((X212 @ (comp @ X215 @ X217) @ (sub @ X216 @ X217) @ X218))) & ($true != ((X212 @ X215 @ X216 @ (comp @ X217 @ X218))))) | ? [X219 : term,X220 : term] : ((((X212 @ id @ X219 @ id)) = $true) & ($true = ((X212 @ id @ X220 @ id))) & ($true != ((X212 @ id @ (ap @ (sub @ X219 @ id) @ X220) @ id)))) | ($true = ((X212 @ id @ (ap @ X213 @ X214) @ id)))) <=> (hoasinduction_lem2 = $true)) & (hoasinduction_lem1v2 = $true) & ((? [X71 : term] : (((sub @ X71 @ id)) != X71) | ! [X73 : (subst > term > subst > $o),X74 : (term > $o),X72 : term] : (? [X89 : term] : (((X74 @ X89)) != ((X73 @ id @ X89 @ id))) | ? [X75 : term] : (($true != ((X74 @ (sub @ X72 @ (push @ X75 @ id))))) & ($true = ((X74 @ X75)))) | ($true = ((X74 @ (lam @ X72)))) | ? [X79 : subst,X77 : subst,X76 : term,X78 : subst] : ((((X73 @ X77 @ X76 @ (comp @ X78 @ X79))) = $true) & ($true != ((X73 @ (comp @ X77 @ X78) @ (sub @ X76 @ X78) @ X79)))) | ? [X80 : (subst > term > term)] : (! [X81 : term] : (($true != ((X73 @ id @ X81 @ id))) | (((X73 @ id @ (X80 @ id @ X81) @ id)) = $true)) & ($true != ((X73 @ id @ (lam @ (X80 @ sh @ one)) @ id))) & ! [X84 : subst,X82 : subst,X83 : term] : (((sub @ (X80 @ X84 @ X83) @ X82)) = ((X80 @ (comp @ X84 @ X82) @ (sub @ X83 @ X82))))) | ? [X87 : subst,X86 : subst,X85 : subst,X88 : term] : (($true != ((X73 @ X85 @ X88 @ (comp @ X87 @ X86)))) & (((X73 @ (comp @ X85 @ X87) @ (sub @ X88 @ X87) @ X86)) = $true)))) <=> (hoasinduction_lem3v2_lthm = $true)) & ((laminj = $true) <=> ! [X448 : term,X449 : term] : ((X448 = X449) | (((lam @ X449)) != ((lam @ X448))))) & ((hoasapinj2_lthm = $true) <=> (! [X291 : term,X290 : term,X293 : term,X292 : term] : ((((ap @ (sub @ X292 @ id) @ X291)) != ((ap @ (sub @ X290 @ id) @ X293))) | (X291 = X293)) | ? [X289 : term,X288 : term,X287 : term,X286 : term] : ((((ap @ X286 @ X287)) = ((ap @ X289 @ X288))) & (X287 != X288)))) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (pushprop_lthm_orig = $true) & (! [X246 : term,X247 : term,X245 : term,X248 : term] : ((X247 = X248) | (((ap @ (sub @ X247 @ id) @ X246)) != ((ap @ (sub @ X248 @ id) @ X245)))) <=> (hoasapinj1 = $true)) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X533 : (subst > term > subst > $o),X532 : term] : (? [X535 : (subst > term > term)] : (! [X536 : term] : (($true != ((X533 @ id @ X536 @ id))) | ($true = ((X533 @ id @ (X535 @ id @ X536) @ id)))) & ! [X537 : subst,X539 : subst,X538 : term] : (((sub @ (X535 @ X539 @ X538) @ X537)) = ((X535 @ (comp @ X539 @ X537) @ (sub @ X538 @ X537)))) & (((X533 @ id @ (lam @ (X535 @ sh @ one)) @ id)) != $true)) | ? [X534 : term] : ((((X533 @ id @ (sub @ X532 @ (push @ X534 @ id)) @ id)) != $true) & ($true = ((X533 @ id @ X534 @ id)))) | ($true = ((X533 @ id @ (lam @ (sub @ X532 @ (push @ one @ sh))) @ id))))) & (axvarshift = $true) & (! [X434 : term] : ? [X435 : (subst > term > term)] : (((X435 @ sh @ one)) = ((sub @ X434 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b_lthm = $true)) & ((termmset = $true) <=> ! [X110 : term] : (((sub @ X110 @ id)) = X110)) & (! [X295 : subst,X294 : term] : (((sub @ one @ (push @ X294 @ X295))) = X294) <=> (axvarcons = $true)) & (hoasinduction_lem3a_gthm = $true) & ((hoasapinj2 = $true) <=> ! [X50 : term,X49 : term,X47 : term,X48 : term] : ((((ap @ (sub @ X48 @ id) @ X47)) != ((ap @ (sub @ X49 @ id) @ X50))) | (X47 = X50))) & (! [X90 : term] : ? [X91 : (subst > term > term)] : ! [X92 : term,X93 : subst] : (((X91 @ X93 @ X92)) = ((sub @ X90 @ (push @ X92 @ X93)))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & (ulamvar1 != $true) & (! [X191 : (term > $o),X190 : term] : (? [X192 : term] : (! [X193 : term] : (($true != ((X191 @ X193))) | ($true = ((X191 @ (sub @ X192 @ (push @ X193 @ id)))))) & ($true != ((X191 @ (lam @ X192))))) | ? [X194 : term,X195 : term] : (($true = ((X191 @ X195))) & (((X191 @ (ap @ X195 @ X194))) != $true) & ($true = ((X191 @ X194)))) | ? [X196 : term] : (($true != ((X191 @ X196))) & ($true = ((var @ X196)))) | ($true = ((X191 @ X190)))) <=> (induction2 = $true)) & (hoasinduction_lem2_gthm = $true) & (substmonoid_gthm = $true) & ((axscons = $true) <=> ! [X429 : subst] : (((push @ (sub @ one @ X429) @ (comp @ sh @ X429))) = X429)) & (hoaslamnotvar_lthm = $true) & (((((! [X335 : (subst > term > subst > $o),X334 : term] : (($true = ((X335 @ id @ X334 @ id))) | ? [X336 : (subst > term > term)] : (! [X339 : subst,X340 : subst,X338 : term] : (((sub @ (X336 @ X340 @ X338) @ X339)) = ((X336 @ (comp @ X340 @ X339) @ (sub @ X338 @ X339)))) & (((X335 @ id @ (lam @ (X336 @ sh @ one)) @ id)) != $true) & ! [X337 : term] : ((((X335 @ id @ X337 @ id)) != $true) | ($true = ((X335 @ id @ (X336 @ id @ X337) @ id))))) | ? [X342 : term,X341 : term] : (($true = ((X335 @ id @ X341 @ id))) & ($true != ((X335 @ id @ (ap @ (sub @ X342 @ id) @ X341) @ id))) & ($true = ((X335 @ id @ X342 @ id))))) | ? [X325 : (term > $o),X326 : term,X324 : (subst > term > subst > $o)] : (! [X332 : term] : (($true != ((X325 @ X332))) | ($true = ((X325 @ (sub @ X326 @ (push @ X332 @ id)))))) & ! [X327 : (subst > term > term)] : ((((X324 @ id @ (lam @ (X327 @ sh @ one)) @ id)) = $true) | ? [X330 : term,X331 : subst,X329 : subst] : (((X327 @ (comp @ X331 @ X329) @ (sub @ X330 @ X329))) != ((sub @ (X327 @ X331 @ X330) @ X329))) | ? [X328 : term] : (($true = ((X324 @ id @ X328 @ id))) & ($true != ((X324 @ id @ (X327 @ id @ X328) @ id))))) & ! [X333 : term] : (((X325 @ X333)) = ((X324 @ id @ X333 @ id))) & ($true != ((X325 @ (lam @ X326)))))) | ? [X323 : term] : (((sub @ X323 @ id)) != X323)) | ? [X317 : (term > $o),X316 : term] : (! [X318 : term] : (($true = ((X317 @ (lam @ X318)))) | ? [X319 : term] : ((((X317 @ (sub @ X318 @ (push @ X319 @ id)))) != $true) & (((X317 @ X319)) = $true))) & ! [X320 : term,X321 : term] : ((((X317 @ (ap @ X320 @ X321))) = $true) | (((X317 @ X321)) != $true) | ($true != ((X317 @ X320)))) & ! [X322 : term] : (($true != ((var @ X322))) | ($true = ((X317 @ X322)))) & ($true != ((X317 @ X316))))) | ? [X313 : (subst > term > subst > $o)] : ! [X314 : (term > $o)] : ? [X315 : term] : (((X313 @ id @ X315 @ id)) != ((X314 @ X315)))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (hoaslamnotvar = $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 = $true) & (! [X296 : term,X298 : term,X299 : term,X297 : term] : ((X297 = X298) | (((ap @ X298 @ X299)) != ((ap @ X297 @ X296)))) <=> (apinj1 = $true)) & (hoasinduction_lem1_gthm = $true) & ((hoasinduction_lem3v2a_lthm = $true) <=> ((! [X230 : (subst > term > subst > $o),X231 : (term > $o),X232 : term] : (($true = ((X231 @ (lam @ X232)))) | ? [X235 : (subst > term > term)] : (! [X236 : term] : (($true = ((X230 @ id @ (X235 @ id @ X236) @ id))) | (((X230 @ id @ X236 @ id)) != $true)) & ! [X239 : subst,X238 : subst,X237 : term] : (((sub @ (X235 @ X239 @ X237) @ X238)) = ((X235 @ (comp @ X239 @ X238) @ (sub @ X237 @ X238)))) & ($true != ((X230 @ id @ (lam @ (X235 @ sh @ one)) @ id)))) | ? [X234 : term] : (((X230 @ id @ X234 @ id)) != ((X231 @ X234))) | ? [X233 : term] : (($true != ((X231 @ (sub @ X232 @ (push @ X233 @ id))))) & ($true = ((X231 @ X233))))) | ? [X229 : term] : (((sub @ X229 @ id)) != X229)) | ? [X225 : term] : ! [X226 : (subst > term > term)] : ? [X227 : term,X228 : subst] : (((X226 @ X228 @ X227)) != ((sub @ X225 @ (push @ X227 @ X228)))))) & (induction2lem_gthm = $true) & ((hoasinduction_lem3v2_f = $true) <=> ! [X0 : term] : ? [X1 : (subst > term > term)] : ! [X2 : term,X3 : subst] : (((sub @ X0 @ (push @ X2 @ X3))) = ((X1 @ X3 @ X2)))) & (! [X413 : term,X414 : (subst > term > subst > $o)] : (? [X417 : subst,X415 : subst,X416 : term,X418 : subst] : ((((X414 @ (comp @ X415 @ X418) @ (sub @ X416 @ X418) @ X417)) = $true) & ($true != ((X414 @ X415 @ X416 @ (comp @ X418 @ X417))))) | ? [X419 : subst,X421 : term,X422 : subst,X420 : subst] : (($true != ((X414 @ (comp @ X420 @ X422) @ (sub @ X421 @ X422) @ X419))) & ($true = ((X414 @ X420 @ X421 @ (comp @ X422 @ X419))))) | ? [X428 : term] : (($true != ((X414 @ id @ (sub @ X413 @ (push @ X428 @ id)) @ id))) & ($true = ((X414 @ id @ X428 @ id)))) | ? [X423 : (subst > term > term)] : ((((X414 @ id @ (lam @ (X423 @ sh @ one)) @ id)) != $true) & ! [X425 : term,X426 : subst,X424 : subst] : (((X423 @ (comp @ X426 @ X424) @ (sub @ X425 @ X424))) = ((sub @ (X423 @ X426 @ X425) @ X424))) & ! [X427 : term] : ((((X414 @ id @ (X423 @ id @ X427) @ id)) = $true) | ($true != ((X414 @ id @ X427 @ id))))) | ($true = ((X414 @ id @ (lam @ X413) @ id)))) <=> (hoasinduction_lem3 = $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((induction2lem_lthm = $true) <=> ((((! [X170 : subst,X171 : (term > $o),X169 : term] : (($true = ((X171 @ (sub @ X169 @ X170)))) | ? [X176 : term,X175 : term] : (($true = ((X171 @ X175))) & ($true != ((X171 @ (ap @ X175 @ X176)))) & (((X171 @ X176)) = $true)) | ? [X174 : term] : (($true != ((X171 @ (sub @ X174 @ X170)))) & ($true = ((var @ X174)))) | ? [X172 : term] : (! [X173 : term] : (($true = ((X171 @ (sub @ X172 @ (push @ X173 @ id))))) | ($true != ((X171 @ X173)))) & ($true != ((X171 @ (lam @ X172)))))) | ? [X164 : (term > $o),X163 : term] : (! [X167 : term] : ((((X164 @ (lam @ X167))) = $true) | ($true != ((X164 @ X167)))) & ! [X165 : term,X166 : term] : (($true != ((X164 @ X165))) | ($true != ((X164 @ X166))) | ($true = ((X164 @ (ap @ X165 @ X166))))) & ($true != ((X164 @ X163))) & ! [X168 : term] : (($true = ((X164 @ X168))) | ($true != ((var @ X168)))))) | ? [X162 : subst] : (((comp @ X162 @ id)) != X162)) | ? [X161 : term,X160 : subst] : (((comp @ sh @ (push @ X161 @ X160))) != X160)) | ? [X159 : term,X158 : subst] : (((sub @ one @ (push @ X159 @ X158))) != X159))) & (termmset_gthm = $true) & (axassoc = $true) & (hoasapinj2_gthm = $true) & ((! [X505 : subst] : (((comp @ id @ X505)) = X505) & ! [X504 : subst] : (((comp @ X504 @ id)) = X504)) <=> (substmonoid = $true)) & ! [X433 : d_term,X432 : d_term] : ((((d2term @ X432)) != ((d2term @ X433))) | (X432 = X433)) & (lamnotvar = $true) & (((d2subst @ d_id)) = sh) & ((substmonoid_lthm = $true) <=> (((! [X438 : subst] : (((comp @ X438 @ id)) = X438) & ! [X439 : subst] : (((comp @ id @ X439)) = X439)) | ? [X437 : subst] : (((comp @ X437 @ id)) != X437)) | ? [X436 : subst] : (((comp @ id @ X436)) != X436))) & ((termmset_lthm = $true) <=> (? [X177 : term] : (((sub @ X177 @ id)) != X177) | ! [X178 : term] : (((sub @ X178 @ id)) = X178))) & ((((! [X100 : term,X103 : subst,X102 : (subst > term > term),X101 : (subst > term > term)] : ((((X102 @ X103 @ X100)) = ((X101 @ X103 @ X100))) | (((lam @ (X102 @ sh @ one))) != ((lam @ (X101 @ sh @ one)))) | ? [X105 : subst,X106 : subst,X104 : term] : (((X102 @ (comp @ X105 @ X106) @ (sub @ X104 @ X106))) != ((sub @ (X102 @ X105 @ X104) @ X106))) | ? [X107 : subst,X109 : term,X108 : subst] : (((X101 @ (comp @ X108 @ X107) @ (sub @ X109 @ X107))) != ((sub @ (X101 @ X108 @ X109) @ X107)))) | ? [X98 : term,X99 : term] : ((((lam @ X99)) = ((lam @ X98))) & (X98 != X99))) | ? [X96 : term,X97 : subst] : (((comp @ sh @ (push @ X96 @ X97))) != X97)) | ? [X94 : term,X95 : subst] : (((sub @ one @ (push @ X94 @ X95))) != X94)) <=> (hoaslaminj_lthm = $true)) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (hoasinduction_lem3b_gthm = $true) & ((hoasapinj1_lthm = $true) <=> ((! [X364 : term,X366 : term,X365 : term,X367 : term] : ((X365 = X367) | (((ap @ (sub @ X367 @ id) @ X364)) != ((ap @ (sub @ X365 @ id) @ X366)))) | ? [X360 : term,X361 : term,X362 : term,X363 : term] : ((((ap @ X361 @ X362)) = ((ap @ X363 @ X360))) & (X361 != X363))) | ? [X359 : term] : (((sub @ X359 @ id)) != X359))) & (! [X527 : term] : (((sub @ X527 @ id)) = X527) <=> (axvarid = $true)) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ! [X240 : d_subst,X241 : d_subst] : ((((d2subst @ X241)) != ((d2subst @ X240))) | (X240 = X241)) & (((! [X352 : term,X353 : (term > $o)] : (($true = ((X353 @ X352))) | ? [X356 : term] : (($true = ((var @ X356))) & ($true != ((X353 @ X356)))) | ? [X357 : term] : (($true != ((X353 @ (lam @ X357)))) & ! [X358 : term] : (($true = ((X353 @ (sub @ X357 @ (push @ X358 @ id))))) | ($true != ((X353 @ X358))))) | ? [X354 : term,X355 : term] : (($true = ((X353 @ X355))) & ($true != ((X353 @ (ap @ X355 @ X354)))) & ($true = ((X353 @ X354))))) | ? [X345 : subst,X346 : (term > $o),X344 : term] : (($true != ((X346 @ (sub @ X344 @ X345)))) & ! [X347 : term] : ((((var @ X347)) != $true) | ($true = ((X346 @ (sub @ X347 @ X345))))) & ! [X349 : term,X348 : term] : (($true != ((X346 @ X348))) | (((X346 @ (ap @ X349 @ X348))) = $true) | (((X346 @ X349)) != $true)) & ! [X350 : term] : (($true = ((X346 @ (lam @ X350)))) | ? [X351 : term] : (($true != ((X346 @ (sub @ X350 @ (push @ X351 @ id))))) & ($true = ((X346 @ X351))))))) | ? [X343 : term] : (((sub @ X343 @ id)) != X343)) <=> (induction2_lthm = $true)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (! [X524 : (subst > term > subst > $o)] : ? [X525 : (term > $o)] : ! [X526 : term] : (((X524 @ id @ X526 @ id)) = ((X525 @ X526))) <=> (hoasinduction_lem0_lthm = $true)) & ((pushprop_lem1v2 = $true) <=> ! [X278 : (term > $o),X276 : (term > $o),X277 : subst,X279 : term] : (($true != ((X278 @ X279))) | (((X276 @ one)) = $true) | ? [X280 : term] : (((X276 @ X280)) != ((X278 @ (sub @ X280 @ (push @ X279 @ X277))))))) & (hoasapnotvar = $true) & (hoasapnotvar_gthm = $true) & (hoaslaminj_gthm = $true) & (((d2term @ d_one)) = one) & (hoasinduction_lem1 = $true) & (axclos = $true) & ((pushprop_lem1v2_lthm = $true) <=> (? [X151 : term,X152 : subst] : (((sub @ one @ (push @ X151 @ X152))) != X151) | ! [X154 : (term > $o),X156 : (term > $o),X153 : term,X155 : subst] : ((((X154 @ one)) = $true) | ? [X157 : term] : (((X156 @ (sub @ X157 @ (push @ X153 @ X155)))) != ((X154 @ X157))) | ($true != ((X156 @ X153)))))) & (ulamvarind = $true) & (((d2subst @ d_id)) = id) & (hoasinduction_lem2v2_gthm = $true) & ((hoasinduction_lthm = $true) <=> (((! [X486 : term,X487 : (subst > term > subst > $o)] : (? [X499 : (subst > term > term)] : (! [X501 : subst,X502 : subst,X503 : term] : (((X499 @ (comp @ X502 @ X501) @ (sub @ X503 @ X501))) = ((sub @ (X499 @ X502 @ X503) @ X501))) & ! [X500 : term] : (($true != ((X487 @ id @ X500 @ id))) | ($true = ((X487 @ id @ (X499 @ id @ X500) @ id)))) & (((X487 @ id @ (lam @ (X499 @ sh @ one)) @ id)) != $true)) | ($true = ((X487 @ id @ X486 @ id))) | ? [X497 : term,X498 : term] : (($true != ((X487 @ id @ (ap @ (sub @ X497 @ id) @ X498) @ id))) & (((X487 @ id @ X498 @ id)) = $true) & ($true = ((X487 @ id @ X497 @ id)))) | ? [X488 : term] : (($true != ((X487 @ id @ X488 @ id))) & ($true = ((var @ (sub @ X488 @ id))))) | ? [X490 : subst,X491 : subst,X489 : term,X492 : subst] : (($true = ((X487 @ (comp @ X490 @ X491) @ (sub @ X489 @ X491) @ X492))) & (((X487 @ X490 @ X489 @ (comp @ X491 @ X492))) != $true)) | ? [X496 : subst,X494 : subst,X495 : term,X493 : subst] : (($true != ((X487 @ (comp @ X493 @ X494) @ (sub @ X495 @ X494) @ X496))) & ($true = ((X487 @ X493 @ X495 @ (comp @ X494 @ X496)))))) | ? [X470 : term,X471 : (subst > term > subst > $o)] : ((((X471 @ id @ (lam @ X470) @ id)) != $true) & ! [X481 : (subst > term > term)] : (? [X483 : subst,X482 : term,X484 : subst] : (((X481 @ (comp @ X484 @ X483) @ (sub @ X482 @ X483))) != ((sub @ (X481 @ X484 @ X482) @ X483))) | (((X471 @ id @ (lam @ (X481 @ sh @ one)) @ id)) = $true) | ? [X485 : term] : ((((X471 @ id @ X485 @ id)) = $true) & (((X471 @ id @ (X481 @ id @ X485) @ id)) != $true))) & ! [X476 : subst,X477 : subst,X478 : term,X479 : subst] : (($true != ((X471 @ (comp @ X479 @ X476) @ (sub @ X478 @ X476) @ X477))) | ($true = ((X471 @ X479 @ X478 @ (comp @ X476 @ X477))))) & ! [X480 : term] : (($true = ((X471 @ id @ (sub @ X470 @ (push @ X480 @ id)) @ id))) | ($true != ((X471 @ id @ X480 @ id)))) & ! [X475 : subst,X474 : subst,X473 : term,X472 : subst] : (($true != ((X471 @ X475 @ X473 @ (comp @ X474 @ X472)))) | ($true = ((X471 @ (comp @ X475 @ X474) @ (sub @ X473 @ X474) @ X472)))))) | ? [X457 : (subst > term > subst > $o),X458 : term,X459 : term] : (! [X462 : term,X463 : subst,X465 : subst,X464 : subst] : (($true != ((X457 @ X464 @ X462 @ (comp @ X463 @ X465)))) | ($true = ((X457 @ (comp @ X464 @ X463) @ (sub @ X462 @ X463) @ X465)))) & ($true != ((X457 @ id @ (ap @ X459 @ X458) @ id))) & ($true = ((X457 @ id @ X459 @ id))) & (((X457 @ id @ X458 @ id)) = $true) & ! [X468 : term,X467 : subst,X469 : subst,X466 : subst] : ((((X457 @ X466 @ X468 @ (comp @ X467 @ X469))) = $true) | ($true != ((X457 @ (comp @ X466 @ X467) @ (sub @ X468 @ X467) @ X469)))) & ! [X461 : term,X460 : term] : (($true = ((X457 @ id @ (ap @ (sub @ X460 @ id) @ X461) @ id))) | ($true != ((X457 @ id @ X460 @ id))) | ($true != ((X457 @ id @ X461 @ id)))))) | ? [X451 : term,X450 : (term > $o)] : ((((X450 @ X451)) != $true) & ! [X452 : term] : (($true = ((X450 @ (lam @ X452)))) | ? [X453 : term] : (($true != ((X450 @ (sub @ X452 @ (push @ X453 @ id))))) & ($true = ((X450 @ X453))))) & ! [X454 : term] : ((((var @ X454)) != $true) | ($true = ((X450 @ X454)))) & ! [X456 : term,X455 : term] : (($true != ((X450 @ X456))) | ($true != ((X450 @ X455))) | ($true = ((X450 @ (ap @ X455 @ X456)))))))) & (hoasapnotvar_lthm = $true) & ((axshiftcons = $true) <=> ! [X52 : term,X51 : subst] : (((comp @ sh @ (push @ X52 @ X51))) = X51)) & (induction2_gthm = $true) & (! [X368 : term,X371 : term,X369 : (subst > term > subst > $o),X370 : (term > $o)] : (($true != ((X370 @ X371))) | ? [X382 : term] : (((X369 @ id @ X382 @ id)) != ((X370 @ X382))) | ? [X376 : subst,X378 : term,X379 : subst,X377 : subst] : (($true != ((X369 @ (comp @ X376 @ X379) @ (sub @ X378 @ X379) @ X377))) & ($true = ((X369 @ X376 @ X378 @ (comp @ X379 @ X377))))) | ? [X375 : subst,X372 : subst,X374 : subst,X373 : term] : (($true != ((X369 @ X375 @ X373 @ (comp @ X374 @ X372)))) & ($true = ((X369 @ (comp @ X375 @ X374) @ (sub @ X373 @ X374) @ X372)))) | ($true != ((X370 @ X368))) | ? [X381 : term,X380 : term] : ((((X369 @ id @ X381 @ id)) = $true) & ($true = ((X369 @ id @ X380 @ id))) & ($true != ((X369 @ id @ (ap @ (sub @ X380 @ id) @ X381) @ id)))) | ($true = ((X370 @ (ap @ X371 @ X368))))) <=> (hoasinduction_lem2v2 = $true)) & ((hoaslamnotap = $true) <=> ! [X262 : term,X260 : term,X261 : (subst > term > term)] : ((((ap @ (sub @ X262 @ id) @ X260)) != ((lam @ (X261 @ sh @ one)))) | ? [X264 : subst,X265 : subst,X263 : term] : (((sub @ (X261 @ X264 @ X263) @ X265)) != ((X261 @ (comp @ X264 @ X265) @ (sub @ X263 @ X265)))))) & (((! [X207 : term,X210 : (term > $o),X209 : (term > $o),X211 : subst,X208 : term] : (($true != ((X209 @ X208))) | (((X210 @ (sub @ X208 @ (push @ X207 @ X211)))) = $true)) | ? [X205 : subst,X206 : term] : (((comp @ sh @ (push @ X206 @ X205))) != X205)) | ? [X204 : term,X203 : subst] : (((sub @ one @ (push @ X204 @ X203))) != X204)) <=> (pushprop_lem1_lthm = $true)) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (pushprop_lem3v2_lthm = $true) & ((hoasinduction_no_psi_cond = $true) <=> ! [X143 : (subst > term > subst > $o),X142 : term] : (($true = ((X143 @ id @ X142 @ id))) | ? [X150 : term,X149 : term] : ((((X143 @ id @ X149 @ id)) = $true) & ($true = ((X143 @ id @ X150 @ id))) & ($true != ((X143 @ id @ (ap @ (sub @ X149 @ id) @ X150) @ id)))) | ? [X144 : (subst > term > term)] : ((((X143 @ id @ (lam @ (X144 @ sh @ one)) @ id)) != $true) & ! [X147 : subst,X148 : term,X146 : subst] : (((X144 @ (comp @ X146 @ X147) @ (sub @ X148 @ X147))) = ((sub @ (X144 @ X146 @ X148) @ X147))) & ! [X145 : term] : (($true = ((X143 @ id @ (X144 @ id @ X145) @ id))) | (((X143 @ id @ X145 @ id)) != $true))))) & ((hoasinduction_lthm_3 = $true) <=> ((((! [X28 : (subst > term > subst > $o),X27 : term] : (? [X31 : (subst > term > term)] : (! [X32 : term] : (($true = ((X28 @ id @ (X31 @ id @ X32) @ id))) | ($true != ((X28 @ id @ X32 @ id)))) & (((X28 @ id @ (lam @ (X31 @ sh @ one)) @ id)) != $true) & ! [X34 : term,X35 : subst,X33 : subst] : (((sub @ (X31 @ X35 @ X34) @ X33)) = ((X31 @ (comp @ X35 @ X33) @ (sub @ X34 @ X33))))) | (((X28 @ id @ X27 @ id)) = $true) | ? [X41 : term,X40 : subst,X43 : subst,X42 : subst] : ((((X28 @ X43 @ X41 @ (comp @ X40 @ X42))) = $true) & (((X28 @ (comp @ X43 @ X40) @ (sub @ X41 @ X40) @ X42)) != $true)) | ? [X29 : term,X30 : term] : (($true = ((X28 @ id @ X29 @ id))) & ($true != ((X28 @ id @ (ap @ (sub @ X30 @ id) @ X29) @ id))) & ($true = ((X28 @ id @ X30 @ id)))) | ? [X44 : term] : (($true != ((X28 @ id @ X44 @ id))) & ($true = ((var @ (sub @ X44 @ id))))) | ? [X39 : subst,X37 : subst,X38 : subst,X36 : term] : (($true = ((X28 @ (comp @ X37 @ X39) @ (sub @ X36 @ X39) @ X38))) & ($true != ((X28 @ X37 @ X36 @ (comp @ X39 @ X38)))))) | ? [X17 : term,X19 : (subst > term > subst > $o),X18 : (term > $o)] : (! [X26 : term] : (((X18 @ X26)) = ((X19 @ id @ X26 @ id))) & ($true != ((X18 @ (lam @ X17)))) & ! [X20 : (subst > term > term)] : (? [X21 : term] : ((((X19 @ id @ X21 @ id)) = $true) & ($true != ((X19 @ id @ (X20 @ id @ X21) @ id)))) | ? [X24 : subst,X23 : subst,X22 : term] : (((X20 @ (comp @ X23 @ X24) @ (sub @ X22 @ X24))) != ((sub @ (X20 @ X23 @ X22) @ X24))) | (((X19 @ id @ (lam @ (X20 @ sh @ one)) @ id)) = $true)) & ! [X25 : term] : (($true != ((X18 @ X25))) | ($true = ((X18 @ (sub @ X17 @ (push @ X25 @ id)))))))) | ? [X16 : term] : (((sub @ X16 @ id)) != X16)) | ? [X10 : term,X9 : (term > $o)] : (($true != ((X9 @ X10))) & ! [X14 : term] : (? [X15 : term] : (($true != ((X9 @ (sub @ X14 @ (push @ X15 @ id))))) & ($true = ((X9 @ X15)))) | ($true = ((X9 @ (lam @ X14))))) & ! [X13 : term] : (($true != ((var @ X13))) | ($true = ((X9 @ X13)))) & ! [X11 : term,X12 : term] : (($true != ((X9 @ X11))) | (((X9 @ X12)) != $true) | (((X9 @ (ap @ X11 @ X12))) = $true)))) | ? [X6 : (subst > term > subst > $o)] : ! [X7 : (term > $o)] : ? [X8 : term] : (((X7 @ X8)) != ((X6 @ id @ X8 @ id))))) & ! [X249 : d_term] : (d_one = X249) & (pushprop_lem2v2_gthm = $true) & (! [X305 : (subst > term > subst > $o),X306 : term] : (($true = ((X305 @ id @ (lam @ (sub @ X306 @ (push @ one @ sh))) @ id))) | ? [X308 : (subst > term > term)] : (($true != ((X305 @ id @ (lam @ (X308 @ sh @ one)) @ id))) & ! [X309 : term] : ((((X305 @ id @ X309 @ id)) != $true) | ($true = ((X305 @ id @ (X308 @ id @ X309) @ id)))) & ! [X312 : subst,X310 : subst,X311 : term] : (((sub @ (X308 @ X312 @ X311) @ X310)) = ((X308 @ (comp @ X312 @ X310) @ (sub @ X311 @ X310))))) | ? [X307 : term] : (($true = ((X305 @ id @ X307 @ id))) & ($true != ((X305 @ id @ (sub @ X306 @ (push @ X307 @ id)) @ id))))) <=> (hoasinduction_lem3aa = $true)) & (! [X251 : term,X252 : subst,X250 : (subst > term > term),X253 : (subst > term > term)] : (? [X254 : term,X256 : subst,X255 : subst] : (((sub @ (X253 @ X255 @ X254) @ X256)) != ((X253 @ (comp @ X255 @ X256) @ (sub @ X254 @ X256)))) | ? [X259 : subst,X257 : subst,X258 : term] : (((sub @ (X250 @ X259 @ X258) @ X257)) != ((X250 @ (comp @ X259 @ X257) @ (sub @ X258 @ X257)))) | (((X253 @ X252 @ X251)) = ((X250 @ X252 @ X251))) | (((lam @ (X250 @ sh @ one))) != ((lam @ (X253 @ sh @ one))))) <=> (hoaslaminj = $true)) & (pushprop_lem1_gthm = $true) & ((pushprop_lem1 = $true) <=> ! [X410 : (term > $o),X411 : (term > $o),X408 : subst,X412 : term,X409 : term] : (($true = ((X411 @ (sub @ X412 @ (push @ X409 @ X408))))) | ($true != ((X410 @ X412))))) & (hoasinduction_lem1v2_gthm = $true) & (pushprop_lem2v2 = $true) & ((hoasinduction_lem3_lthm = $true) <=> ((! [X393 : term,X392 : (subst > term > subst > $o)] : (? [X405 : term,X404 : subst,X406 : subst,X407 : subst] : ((((X392 @ (comp @ X407 @ X406) @ (sub @ X405 @ X406) @ X404)) != $true) & (((X392 @ X407 @ X405 @ (comp @ X406 @ X404))) = $true)) | (((X392 @ id @ (lam @ X393) @ id)) = $true) | ? [X399 : term] : (($true = ((X392 @ id @ X399 @ id))) & ($true != ((X392 @ id @ (sub @ X393 @ (push @ X399 @ id)) @ id)))) | ? [X394 : (subst > term > term)] : (! [X395 : term] : ((((X392 @ id @ (X394 @ id @ X395) @ id)) = $true) | (((X392 @ id @ X395 @ id)) != $true)) & ($true != ((X392 @ id @ (lam @ (X394 @ sh @ one)) @ id))) & ! [X398 : term,X396 : subst,X397 : subst] : (((sub @ (X394 @ X396 @ X398) @ X397)) = ((X394 @ (comp @ X396 @ X397) @ (sub @ X398 @ X397))))) | ? [X400 : term,X401 : subst,X402 : subst,X403 : subst] : ((((X392 @ (comp @ X403 @ X401) @ (sub @ X400 @ X401) @ X402)) = $true) & ($true != ((X392 @ X403 @ X400 @ (comp @ X401 @ X402)))))) | ? [X384 : (subst > term > subst > $o),X385 : term] : (($true != ((X384 @ id @ (lam @ (sub @ X385 @ (push @ one @ sh))) @ id))) & ! [X391 : term] : (($true = ((X384 @ id @ (sub @ X385 @ (push @ X391 @ id)) @ id))) | (((X384 @ id @ X391 @ id)) != $true)) & ! [X386 : (subst > term > term)] : (? [X390 : term] : (($true = ((X384 @ id @ X390 @ id))) & ($true != ((X384 @ id @ (X386 @ id @ X390) @ id)))) | ($true = ((X384 @ id @ (lam @ (X386 @ sh @ one)) @ id))) | ? [X388 : subst,X389 : term,X387 : subst] : (((sub @ (X386 @ X387 @ X389) @ X388)) != ((X386 @ (comp @ X387 @ X388) @ (sub @ X389 @ X388))))))) | ? [X383 : term] : (((sub @ X383 @ id)) != X383)))),
  inference(ennf_transformation,[],[f9])).
thf(f14,plain,(
  (hoaslamnotap_lthm = $true) & (lamnotap != $true) & (pushprop_lem0_gthm = $true) & ((hoaslaminj_lthm = $true) <=> (! [X100 : term,X103 : subst,X102 : (subst > term > term),X101 : (subst > term > term)] : ((((X102 @ X103 @ X100)) = ((X101 @ X103 @ X100))) | (((lam @ (X102 @ sh @ one))) != ((lam @ (X101 @ sh @ one)))) | ? [X105 : subst,X106 : subst,X104 : term] : (((X102 @ (comp @ X105 @ X106) @ (sub @ X104 @ X106))) != ((sub @ (X102 @ X105 @ X104) @ X106))) | ? [X107 : subst,X109 : term,X108 : subst] : (((X101 @ (comp @ X108 @ X107) @ (sub @ X109 @ X107))) != ((sub @ (X101 @ X108 @ X109) @ X107)))) | ? [X96 : term,X97 : subst] : (((comp @ sh @ (push @ X96 @ X97))) != X97) | ? [X98 : term,X99 : term] : ((((lam @ X99)) = ((lam @ X98))) & (X98 != X99)) | ? [X94 : term,X95 : subst] : (((sub @ one @ (push @ X94 @ X95))) != X94))) & (! [X191 : (term > $o),X190 : term] : (? [X192 : term] : (! [X193 : term] : (($true != ((X191 @ X193))) | ($true = ((X191 @ (sub @ X192 @ (push @ X193 @ id)))))) & ($true != ((X191 @ (lam @ X192))))) | ? [X194 : term,X195 : term] : (($true = ((X191 @ X195))) & (((X191 @ (ap @ X195 @ X194))) != $true) & ($true = ((X191 @ X194)))) | ? [X196 : term] : (($true != ((X191 @ X196))) & ($true = ((var @ X196)))) | ($true = ((X191 @ X190)))) <=> (induction2 = $true)) & (hoasinduction_lem2_gthm = $true) & (! [X295 : subst,X294 : term] : (((sub @ one @ (push @ X294 @ X295))) = X294) <=> (axvarcons = $true)) & ! [X242 : term] : ? [X243 : d_term] : (((d2term @ X243)) = X242) & (hoasinduction_lem1_lthm = $true) & ((induction2lem = $true) <=> ! [X181 : (term > $o),X180 : term,X179 : subst] : (? [X182 : term] : (! [X183 : term] : (($true != ((X181 @ X183))) | ($true = ((X181 @ (sub @ X182 @ (push @ X183 @ id)))))) & ($true != ((X181 @ (lam @ X182))))) | ? [X186 : term] : ((((X181 @ (sub @ X186 @ X179))) != $true) & ($true = ((var @ X186)))) | ($true = ((X181 @ (sub @ X180 @ X179)))) | ? [X185 : term,X184 : term] : (($true = ((X181 @ X185))) & ($true = ((X181 @ X184))) & ($true != ((X181 @ (ap @ X185 @ X184))))))) & (pushprop_gthm = $true) & (hoasinduction_lem1_gthm = $true) & ((pushprop_lem0 = $true) <=> ! [X301 : subst,X302 : (term > $o),X300 : term] : ? [X303 : (term > $o)] : ! [X304 : term] : (((X302 @ (sub @ X304 @ (push @ X300 @ X301)))) = ((X303 @ X304)))) & (! [X90 : term] : ? [X91 : (subst > term > term)] : ! [X92 : term,X93 : subst] : (((X91 @ X93 @ X92)) = ((sub @ X90 @ (push @ X92 @ X93)))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & (pushprop_lem1v2_gthm = $true) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem1v2_gthm = $true) & (((d2subst @ d_id)) = sh) & (! [X187 : (subst > term > subst > $o)] : ? [X188 : (term > $o)] : ! [X189 : term] : (((X187 @ id @ X189 @ id)) = ((X188 @ X189))) <=> (hoasinduction_lem0 = $true)) & (((d2subst @ d_id)) = id) & (! [X198 : term,X197 : (term > $o)] : (? [X201 : term,X202 : term] : ((((X197 @ (ap @ X201 @ X202))) != $true) & ($true = ((X197 @ X202))) & (((X197 @ X201)) = $true)) | ? [X200 : term] : (($true = ((var @ X200))) & ($true != ((X197 @ X200)))) | ($true = ((X197 @ X198))) | ? [X199 : term] : (($true = ((X197 @ X199))) & ($true != ((X197 @ (lam @ X199)))))) <=> (induction = $true)) & (((var @ (d2term @ d_one))) != $true) & ((hoaslamnotap = $true) <=> ! [X262 : term,X260 : term,X261 : (subst > term > term)] : ((((ap @ (sub @ X262 @ id) @ X260)) != ((lam @ (X261 @ sh @ one)))) | ? [X264 : subst,X265 : subst,X263 : term] : (((sub @ (X261 @ X264 @ X263) @ X265)) != ((X261 @ (comp @ X264 @ X265) @ (sub @ X263 @ X265)))))) & (hoasinduction_lem3_gthm = $true) & (pushprop_lem2v2 = $true) & ((apinj2 = $true) <=> ! [X529 : term,X528 : term,X531 : term,X530 : term] : ((X529 = X531) | (((ap @ X528 @ X529)) != ((ap @ X530 @ X531))))) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (? [X317 : (term > $o),X316 : term] : (! [X318 : term] : (($true = ((X317 @ (lam @ X318)))) | ? [X319 : term] : ((((X317 @ (sub @ X318 @ (push @ X319 @ id)))) != $true) & (((X317 @ X319)) = $true))) & ! [X320 : term,X321 : term] : ((((X317 @ (ap @ X320 @ X321))) = $true) | (((X317 @ X321)) != $true) | ($true != ((X317 @ X320)))) & ! [X322 : term] : (($true != ((var @ X322))) | ($true = ((X317 @ X322)))) & ($true != ((X317 @ X316)))) | ! [X335 : (subst > term > subst > $o),X334 : term] : (($true = ((X335 @ id @ X334 @ id))) | ? [X336 : (subst > term > term)] : (! [X339 : subst,X340 : subst,X338 : term] : (((sub @ (X336 @ X340 @ X338) @ X339)) = ((X336 @ (comp @ X340 @ X339) @ (sub @ X338 @ X339)))) & (((X335 @ id @ (lam @ (X336 @ sh @ one)) @ id)) != $true) & ! [X337 : term] : ((((X335 @ id @ X337 @ id)) != $true) | ($true = ((X335 @ id @ (X336 @ id @ X337) @ id))))) | ? [X342 : term,X341 : term] : (($true = ((X335 @ id @ X341 @ id))) & ($true != ((X335 @ id @ (ap @ (sub @ X342 @ id) @ X341) @ id))) & ($true = ((X335 @ id @ X342 @ id))))) | ? [X323 : term] : (((sub @ X323 @ id)) != X323) | ? [X313 : (subst > term > subst > $o)] : ! [X314 : (term > $o)] : ? [X315 : term] : (((X313 @ id @ X315 @ id)) != ((X314 @ X315))) | ? [X325 : (term > $o),X326 : term,X324 : (subst > term > subst > $o)] : (! [X332 : term] : (($true != ((X325 @ X332))) | ($true = ((X325 @ (sub @ X326 @ (push @ X332 @ id)))))) & ! [X327 : (subst > term > term)] : ((((X324 @ id @ (lam @ (X327 @ sh @ one)) @ id)) = $true) | ? [X330 : term,X331 : subst,X329 : subst] : (((X327 @ (comp @ X331 @ X329) @ (sub @ X330 @ X329))) != ((sub @ (X327 @ X331 @ X330) @ X329))) | ? [X328 : term] : (($true = ((X324 @ id @ X328 @ id))) & ($true != ((X324 @ id @ (X327 @ id @ X328) @ id))))) & ! [X333 : term] : (((X325 @ X333)) = ((X324 @ id @ X333 @ id))) & ($true != ((X325 @ (lam @ X326))))))) & (hoaslamnotap_gthm = $true) & ((termmset = $true) <=> ! [X110 : term] : (((sub @ X110 @ id)) = X110)) & (! [X214 : term,X212 : (subst > term > subst > $o),X213 : term] : (($true != ((X212 @ id @ X213 @ id))) | ($true != ((X212 @ id @ X214 @ id))) | ? [X222 : term,X223 : subst,X224 : subst,X221 : subst] : (($true != ((X212 @ (comp @ X223 @ X221) @ (sub @ X222 @ X221) @ X224))) & (((X212 @ X223 @ X222 @ (comp @ X221 @ X224))) = $true)) | ? [X216 : term,X217 : subst,X215 : subst,X218 : subst] : (($true = ((X212 @ (comp @ X215 @ X217) @ (sub @ X216 @ X217) @ X218))) & ($true != ((X212 @ X215 @ X216 @ (comp @ X217 @ X218))))) | ? [X219 : term,X220 : term] : ((((X212 @ id @ X219 @ id)) = $true) & ($true = ((X212 @ id @ X220 @ id))) & ($true != ((X212 @ id @ (ap @ (sub @ X219 @ id) @ X220) @ id)))) | ($true = ((X212 @ id @ (ap @ X213 @ X214) @ id)))) <=> (hoasinduction_lem2 = $true)) & (! [X246 : term,X247 : term,X245 : term,X248 : term] : ((X247 = X248) | (((ap @ (sub @ X247 @ id) @ X246)) != ((ap @ (sub @ X248 @ id) @ X245)))) <=> (hoasapinj1 = $true)) & (hoasapnotvar = $true) & (! [X251 : term,X252 : subst,X250 : (subst > term > term),X253 : (subst > term > term)] : (? [X254 : term,X256 : subst,X255 : subst] : (((sub @ (X253 @ X255 @ X254) @ X256)) != ((X253 @ (comp @ X255 @ X256) @ (sub @ X254 @ X256)))) | ? [X259 : subst,X257 : subst,X258 : term] : (((sub @ (X250 @ X259 @ X258) @ X257)) != ((X250 @ (comp @ X259 @ X257) @ (sub @ X258 @ X257)))) | (((X253 @ X252 @ X251)) = ((X250 @ X252 @ X251))) | (((lam @ (X250 @ sh @ one))) != ((lam @ (X253 @ sh @ one))))) <=> (hoaslaminj = $true)) & (hoasapnotvar_lthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((hoasinduction_no_psi_cond = $true) <=> ! [X143 : (subst > term > subst > $o),X142 : term] : (($true = ((X143 @ id @ X142 @ id))) | ? [X150 : term,X149 : term] : ((((X143 @ id @ X149 @ id)) = $true) & ($true = ((X143 @ id @ X150 @ id))) & ($true != ((X143 @ id @ (ap @ (sub @ X149 @ id) @ X150) @ id)))) | ? [X144 : (subst > term > term)] : ((((X143 @ id @ (lam @ (X144 @ sh @ one)) @ id)) != $true) & ! [X147 : subst,X148 : term,X146 : subst] : (((X144 @ (comp @ X146 @ X147) @ (sub @ X148 @ X147))) = ((sub @ (X144 @ X146 @ X148) @ X147))) & ! [X145 : term] : (($true = ((X143 @ id @ (X144 @ id @ X145) @ id))) | (((X143 @ id @ X145 @ id)) != $true))))) & (pushprop_lem2v2_lthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (! [X524 : (subst > term > subst > $o)] : ? [X525 : (term > $o)] : ! [X526 : term] : (((X524 @ id @ X526 @ id)) = ((X525 @ X526))) <=> (hoasinduction_lem0_lthm = $true)) & (termmset_gthm = $true) & ((pushprop_lem1v2_lthm = $true) <=> (? [X151 : term,X152 : subst] : (((sub @ one @ (push @ X151 @ X152))) != X151) | ! [X154 : (term > $o),X156 : (term > $o),X153 : term,X155 : subst] : ((((X154 @ one)) = $true) | ? [X157 : term] : (((X156 @ (sub @ X157 @ (push @ X153 @ X155)))) != ((X154 @ X157))) | ($true != ((X156 @ X153)))))) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((hoasapinj1_lthm = $true) <=> (? [X359 : term] : (((sub @ X359 @ id)) != X359) | ! [X364 : term,X366 : term,X365 : term,X367 : term] : ((X365 = X367) | (((ap @ (sub @ X367 @ id) @ X364)) != ((ap @ (sub @ X365 @ id) @ X366)))) | ? [X360 : term,X361 : term,X362 : term,X363 : term] : ((((ap @ X361 @ X362)) = ((ap @ X363 @ X360))) & (X361 != X363)))) & (hoaslamnotvar = $true) & (axclos = $true) & (! [X141 : subst] : (((comp @ id @ X141)) = X141) <=> (axidl = $true)) & ! [X433 : d_term,X432 : d_term] : ((((d2term @ X432)) != ((d2term @ X433))) | (X432 = X433)) & ((? [X71 : term] : (((sub @ X71 @ id)) != X71) | ! [X73 : (subst > term > subst > $o),X74 : (term > $o),X72 : term] : (? [X89 : term] : (((X74 @ X89)) != ((X73 @ id @ X89 @ id))) | ? [X75 : term] : (($true != ((X74 @ (sub @ X72 @ (push @ X75 @ id))))) & ($true = ((X74 @ X75)))) | ($true = ((X74 @ (lam @ X72)))) | ? [X79 : subst,X77 : subst,X76 : term,X78 : subst] : ((((X73 @ X77 @ X76 @ (comp @ X78 @ X79))) = $true) & ($true != ((X73 @ (comp @ X77 @ X78) @ (sub @ X76 @ X78) @ X79)))) | ? [X80 : (subst > term > term)] : (! [X81 : term] : (($true != ((X73 @ id @ X81 @ id))) | (((X73 @ id @ (X80 @ id @ X81) @ id)) = $true)) & ($true != ((X73 @ id @ (lam @ (X80 @ sh @ one)) @ id))) & ! [X84 : subst,X82 : subst,X83 : term] : (((sub @ (X80 @ X84 @ X83) @ X82)) = ((X80 @ (comp @ X84 @ X82) @ (sub @ X83 @ X82))))) | ? [X87 : subst,X86 : subst,X85 : subst,X88 : term] : (($true != ((X73 @ X85 @ X88 @ (comp @ X87 @ X86)))) & (((X73 @ (comp @ X85 @ X87) @ (sub @ X88 @ X87) @ X86)) = $true)))) <=> (hoasinduction_lem3v2_lthm = $true)) & (! [X430 : term] : ? [X431 : (subst > term > term)] : (((X431 @ sh @ one)) = ((sub @ X430 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b = $true)) & ((hoasinduction = $true) <=> ! [X506 : term,X507 : (subst > term > subst > $o)] : (? [X513 : term,X512 : term] : (($true != ((X507 @ id @ (ap @ (sub @ X513 @ id) @ X512) @ id))) & ($true = ((X507 @ id @ X512 @ id))) & ($true = ((X507 @ id @ X513 @ id)))) | ? [X518 : term] : (($true = ((var @ (sub @ X518 @ id)))) & ($true != ((X507 @ id @ X518 @ id)))) | ? [X508 : subst,X511 : term,X510 : subst,X509 : subst] : ((((X507 @ (comp @ X508 @ X509) @ (sub @ X511 @ X509) @ X510)) != $true) & ($true = ((X507 @ X508 @ X511 @ (comp @ X509 @ X510))))) | ($true = ((X507 @ id @ X506 @ id))) | ? [X519 : (subst > term > term)] : ((((X507 @ id @ (lam @ (X519 @ sh @ one)) @ id)) != $true) & ! [X520 : term] : (($true != ((X507 @ id @ X520 @ id))) | ($true = ((X507 @ id @ (X519 @ id @ X520) @ id)))) & ! [X523 : term,X522 : subst,X521 : subst] : (((X519 @ (comp @ X521 @ X522) @ (sub @ X523 @ X522))) = ((sub @ (X519 @ X521 @ X523) @ X522)))) | ? [X514 : subst,X516 : subst,X515 : subst,X517 : term] : (($true != ((X507 @ X514 @ X517 @ (comp @ X516 @ X515)))) & (((X507 @ (comp @ X514 @ X516) @ (sub @ X517 @ X516) @ X515)) = $true)))) & (hoaslaminj_gthm = $true) & ! [X249 : d_term] : (d_one = X249) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ! [X240 : d_subst,X241 : d_subst] : ((((d2subst @ X241)) != ((d2subst @ X240))) | (X240 = X241)) & (induction2_gthm = $true) & ((hoasapinj2_lthm = $true) <=> (! [X291 : term,X290 : term,X293 : term,X292 : term] : ((((ap @ (sub @ X292 @ id) @ X291)) != ((ap @ (sub @ X290 @ id) @ X293))) | (X291 = X293)) | ? [X289 : term,X288 : term,X287 : term,X286 : term] : ((((ap @ X286 @ X287)) = ((ap @ X289 @ X288))) & (X287 != X288)))) & (hoasinduction_lem1v2 = $true) & (! [X368 : term,X371 : term,X369 : (subst > term > subst > $o),X370 : (term > $o)] : (($true != ((X370 @ X371))) | ? [X382 : term] : (((X369 @ id @ X382 @ id)) != ((X370 @ X382))) | ? [X376 : subst,X378 : term,X379 : subst,X377 : subst] : (($true != ((X369 @ (comp @ X376 @ X379) @ (sub @ X378 @ X379) @ X377))) & ($true = ((X369 @ X376 @ X378 @ (comp @ X379 @ X377))))) | ? [X375 : subst,X372 : subst,X374 : subst,X373 : term] : (($true != ((X369 @ X375 @ X373 @ (comp @ X374 @ X372)))) & ($true = ((X369 @ (comp @ X375 @ X374) @ (sub @ X373 @ X374) @ X372)))) | ($true != ((X370 @ X368))) | ? [X381 : term,X380 : term] : ((((X369 @ id @ X381 @ id)) = $true) & ($true = ((X369 @ id @ X380 @ id))) & ($true != ((X369 @ id @ (ap @ (sub @ X380 @ id) @ X381) @ id)))) | ($true = ((X370 @ (ap @ X371 @ X368))))) <=> (hoasinduction_lem2v2 = $true)) & (axassoc = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (pushprop_lthm = $true) & ((hoasinduction_lem3v2a_lthm = $true) <=> (! [X230 : (subst > term > subst > $o),X231 : (term > $o),X232 : term] : (($true = ((X231 @ (lam @ X232)))) | ? [X235 : (subst > term > term)] : (! [X236 : term] : (($true = ((X230 @ id @ (X235 @ id @ X236) @ id))) | (((X230 @ id @ X236 @ id)) != $true)) & ! [X239 : subst,X238 : subst,X237 : term] : (((sub @ (X235 @ X239 @ X237) @ X238)) = ((X235 @ (comp @ X239 @ X238) @ (sub @ X237 @ X238)))) & ($true != ((X230 @ id @ (lam @ (X235 @ sh @ one)) @ id)))) | ? [X234 : term] : (((X230 @ id @ X234 @ id)) != ((X231 @ X234))) | ? [X233 : term] : (($true != ((X231 @ (sub @ X232 @ (push @ X233 @ id))))) & ($true = ((X231 @ X233))))) | ? [X225 : term] : ! [X226 : (subst > term > term)] : ? [X227 : term,X228 : subst] : (((X226 @ X228 @ X227)) != ((sub @ X225 @ (push @ X227 @ X228)))) | ? [X229 : term] : (((sub @ X229 @ id)) != X229))) & (hoasapinj1_gthm = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (hoasinduction_lem2v2_gthm = $true) & (! [X140 : subst] : (((comp @ X140 @ id)) = X140) <=> (axidr = $true)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((pushprop_lem1v2 = $true) <=> ! [X278 : (term > $o),X276 : (term > $o),X277 : subst,X279 : term] : (($true != ((X278 @ X279))) | (((X276 @ one)) = $true) | ? [X280 : term] : (((X276 @ X280)) != ((X278 @ (sub @ X280 @ (push @ X279 @ X277))))))) & ((? [X16 : term] : (((sub @ X16 @ id)) != X16) | ? [X10 : term,X9 : (term > $o)] : (($true != ((X9 @ X10))) & ! [X14 : term] : (? [X15 : term] : (($true != ((X9 @ (sub @ X14 @ (push @ X15 @ id))))) & ($true = ((X9 @ X15)))) | ($true = ((X9 @ (lam @ X14))))) & ! [X13 : term] : (($true != ((var @ X13))) | ($true = ((X9 @ X13)))) & ! [X11 : term,X12 : term] : (($true != ((X9 @ X11))) | (((X9 @ X12)) != $true) | (((X9 @ (ap @ X11 @ X12))) = $true))) | ? [X17 : term,X19 : (subst > term > subst > $o),X18 : (term > $o)] : (! [X26 : term] : (((X18 @ X26)) = ((X19 @ id @ X26 @ id))) & ($true != ((X18 @ (lam @ X17)))) & ! [X20 : (subst > term > term)] : (? [X21 : term] : ((((X19 @ id @ X21 @ id)) = $true) & ($true != ((X19 @ id @ (X20 @ id @ X21) @ id)))) | ? [X24 : subst,X23 : subst,X22 : term] : (((X20 @ (comp @ X23 @ X24) @ (sub @ X22 @ X24))) != ((sub @ (X20 @ X23 @ X22) @ X24))) | (((X19 @ id @ (lam @ (X20 @ sh @ one)) @ id)) = $true)) & ! [X25 : term] : (($true != ((X18 @ X25))) | ($true = ((X18 @ (sub @ X17 @ (push @ X25 @ id))))))) | ? [X6 : (subst > term > subst > $o)] : ! [X7 : (term > $o)] : ? [X8 : term] : (((X7 @ X8)) != ((X6 @ id @ X8 @ id))) | ! [X28 : (subst > term > subst > $o),X27 : term] : (? [X31 : (subst > term > term)] : (! [X32 : term] : (($true = ((X28 @ id @ (X31 @ id @ X32) @ id))) | ($true != ((X28 @ id @ X32 @ id)))) & (((X28 @ id @ (lam @ (X31 @ sh @ one)) @ id)) != $true) & ! [X34 : term,X35 : subst,X33 : subst] : (((sub @ (X31 @ X35 @ X34) @ X33)) = ((X31 @ (comp @ X35 @ X33) @ (sub @ X34 @ X33))))) | (((X28 @ id @ X27 @ id)) = $true) | ? [X41 : term,X40 : subst,X43 : subst,X42 : subst] : ((((X28 @ X43 @ X41 @ (comp @ X40 @ X42))) = $true) & (((X28 @ (comp @ X43 @ X40) @ (sub @ X41 @ X40) @ X42)) != $true)) | ? [X29 : term,X30 : term] : (($true = ((X28 @ id @ X29 @ id))) & ($true != ((X28 @ id @ (ap @ (sub @ X30 @ id) @ X29) @ id))) & ($true = ((X28 @ id @ X30 @ id)))) | ? [X44 : term] : (($true != ((X28 @ id @ X44 @ id))) & ($true = ((var @ (sub @ X44 @ id))))) | ? [X39 : subst,X37 : subst,X38 : subst,X36 : term] : (($true = ((X28 @ (comp @ X37 @ X39) @ (sub @ X36 @ X39) @ X38))) & ($true != ((X28 @ X37 @ X36 @ (comp @ X39 @ X38))))))) <=> (hoasinduction_lthm_3 = $true)) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X533 : (subst > term > subst > $o),X532 : term] : (? [X535 : (subst > term > term)] : (! [X536 : term] : (($true != ((X533 @ id @ X536 @ id))) | ($true = ((X533 @ id @ (X535 @ id @ X536) @ id)))) & ! [X537 : subst,X539 : subst,X538 : term] : (((sub @ (X535 @ X539 @ X538) @ X537)) = ((X535 @ (comp @ X539 @ X537) @ (sub @ X538 @ X537)))) & (((X533 @ id @ (lam @ (X535 @ sh @ one)) @ id)) != $true)) | ? [X534 : term] : ((((X533 @ id @ (sub @ X532 @ (push @ X534 @ id)) @ id)) != $true) & ($true = ((X533 @ id @ X534 @ id)))) | ($true = ((X533 @ id @ (lam @ (sub @ X532 @ (push @ one @ sh))) @ id))))) & (! [X305 : (subst > term > subst > $o),X306 : term] : (($true = ((X305 @ id @ (lam @ (sub @ X306 @ (push @ one @ sh))) @ id))) | ? [X308 : (subst > term > term)] : (($true != ((X305 @ id @ (lam @ (X308 @ sh @ one)) @ id))) & ! [X309 : term] : ((((X305 @ id @ X309 @ id)) != $true) | ($true = ((X305 @ id @ (X308 @ id @ X309) @ id)))) & ! [X312 : subst,X310 : subst,X311 : term] : (((sub @ (X308 @ X312 @ X311) @ X310)) = ((X308 @ (comp @ X312 @ X310) @ (sub @ X311 @ X310))))) | ? [X307 : term] : (($true = ((X305 @ id @ X307 @ id))) & ($true != ((X305 @ id @ (sub @ X306 @ (push @ X307 @ id)) @ id))))) <=> (hoasinduction_lem3aa = $true)) & (pushprop = $true) & (ulamvar1 != $true) & (! [X281 : subst,X282 : term,X283 : (term > $o)] : ? [X284 : (term > $o)] : ! [X285 : term] : (((X283 @ (sub @ X285 @ (push @ X282 @ X281)))) = ((X284 @ X285))) <=> (pushprop_lem0_lthm = $true)) & (! [X527 : term] : (((sub @ X527 @ id)) = X527) <=> (axvarid = $true)) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoaslamnotvar_lthm = $true) & (hoasinduction_lem3v2_gthm = $true) & ((pushprop_lem1_lthm = $true) <=> (? [X205 : subst,X206 : term] : (((comp @ sh @ (push @ X206 @ X205))) != X205) | ! [X207 : term,X210 : (term > $o),X209 : (term > $o),X211 : subst,X208 : term] : (($true != ((X209 @ X208))) | (((X210 @ (sub @ X208 @ (push @ X207 @ X211)))) = $true)) | ? [X204 : term,X203 : subst] : (((sub @ one @ (push @ X204 @ X203))) != X204))) & (pushprop_lthm_orig = $true) & (hoasapinj2_gthm = $true) & (ulamvarsh = $true) & (apnotvar = $true) & (lamnotvar = $true) & ((hoasapinj2 = $true) <=> ! [X50 : term,X49 : term,X47 : term,X48 : term] : ((((ap @ (sub @ X48 @ id) @ X47)) != ((ap @ (sub @ X49 @ id) @ X50))) | (X47 = X50))) & (induction2lem_gthm = $true) & (axapp = $true) & ((axshiftcons = $true) <=> ! [X52 : term,X51 : subst] : (((comp @ sh @ (push @ X52 @ X51))) = X51)) & ((hoasinduction_lthm = $true) <=> (? [X457 : (subst > term > subst > $o),X458 : term,X459 : term] : (! [X462 : term,X463 : subst,X465 : subst,X464 : subst] : (($true != ((X457 @ X464 @ X462 @ (comp @ X463 @ X465)))) | ($true = ((X457 @ (comp @ X464 @ X463) @ (sub @ X462 @ X463) @ X465)))) & ($true != ((X457 @ id @ (ap @ X459 @ X458) @ id))) & ($true = ((X457 @ id @ X459 @ id))) & (((X457 @ id @ X458 @ id)) = $true) & ! [X468 : term,X467 : subst,X469 : subst,X466 : subst] : ((((X457 @ X466 @ X468 @ (comp @ X467 @ X469))) = $true) | ($true != ((X457 @ (comp @ X466 @ X467) @ (sub @ X468 @ X467) @ X469)))) & ! [X461 : term,X460 : term] : (($true = ((X457 @ id @ (ap @ (sub @ X460 @ id) @ X461) @ id))) | ($true != ((X457 @ id @ X460 @ id))) | ($true != ((X457 @ id @ X461 @ id))))) | ? [X451 : term,X450 : (term > $o)] : ((((X450 @ X451)) != $true) & ! [X452 : term] : (($true = ((X450 @ (lam @ X452)))) | ? [X453 : term] : (($true != ((X450 @ (sub @ X452 @ (push @ X453 @ id))))) & ($true = ((X450 @ X453))))) & ! [X454 : term] : ((((var @ X454)) != $true) | ($true = ((X450 @ X454)))) & ! [X456 : term,X455 : term] : (($true != ((X450 @ X456))) | ($true != ((X450 @ X455))) | ($true = ((X450 @ (ap @ X455 @ X456)))))) | ! [X486 : term,X487 : (subst > term > subst > $o)] : (? [X499 : (subst > term > term)] : (! [X501 : subst,X502 : subst,X503 : term] : (((X499 @ (comp @ X502 @ X501) @ (sub @ X503 @ X501))) = ((sub @ (X499 @ X502 @ X503) @ X501))) & ! [X500 : term] : (($true != ((X487 @ id @ X500 @ id))) | ($true = ((X487 @ id @ (X499 @ id @ X500) @ id)))) & (((X487 @ id @ (lam @ (X499 @ sh @ one)) @ id)) != $true)) | ($true = ((X487 @ id @ X486 @ id))) | ? [X497 : term,X498 : term] : (($true != ((X487 @ id @ (ap @ (sub @ X497 @ id) @ X498) @ id))) & (((X487 @ id @ X498 @ id)) = $true) & ($true = ((X487 @ id @ X497 @ id)))) | ? [X488 : term] : (($true != ((X487 @ id @ X488 @ id))) & ($true = ((var @ (sub @ X488 @ id))))) | ? [X490 : subst,X491 : subst,X489 : term,X492 : subst] : (($true = ((X487 @ (comp @ X490 @ X491) @ (sub @ X489 @ X491) @ X492))) & (((X487 @ X490 @ X489 @ (comp @ X491 @ X492))) != $true)) | ? [X496 : subst,X494 : subst,X495 : term,X493 : subst] : (($true != ((X487 @ (comp @ X493 @ X494) @ (sub @ X495 @ X494) @ X496))) & ($true = ((X487 @ X493 @ X495 @ (comp @ X494 @ X496)))))) | ? [X470 : term,X471 : (subst > term > subst > $o)] : ((((X471 @ id @ (lam @ X470) @ id)) != $true) & ! [X481 : (subst > term > term)] : (? [X483 : subst,X482 : term,X484 : subst] : (((X481 @ (comp @ X484 @ X483) @ (sub @ X482 @ X483))) != ((sub @ (X481 @ X484 @ X482) @ X483))) | (((X471 @ id @ (lam @ (X481 @ sh @ one)) @ id)) = $true) | ? [X485 : term] : ((((X471 @ id @ X485 @ id)) = $true) & (((X471 @ id @ (X481 @ id @ X485) @ id)) != $true))) & ! [X476 : subst,X477 : subst,X478 : term,X479 : subst] : (($true != ((X471 @ (comp @ X479 @ X476) @ (sub @ X478 @ X476) @ X477))) | ($true = ((X471 @ X479 @ X478 @ (comp @ X476 @ X477))))) & ! [X480 : term] : (($true = ((X471 @ id @ (sub @ X470 @ (push @ X480 @ id)) @ id))) | ($true != ((X471 @ id @ X480 @ id)))) & ! [X475 : subst,X474 : subst,X473 : term,X472 : subst] : (($true != ((X471 @ X475 @ X473 @ (comp @ X474 @ X472)))) | ($true = ((X471 @ (comp @ X475 @ X474) @ (sub @ X473 @ X474) @ X472))))))) & ((! [X393 : term,X392 : (subst > term > subst > $o)] : (? [X405 : term,X404 : subst,X406 : subst,X407 : subst] : ((((X392 @ (comp @ X407 @ X406) @ (sub @ X405 @ X406) @ X404)) != $true) & (((X392 @ X407 @ X405 @ (comp @ X406 @ X404))) = $true)) | (((X392 @ id @ (lam @ X393) @ id)) = $true) | ? [X399 : term] : (($true = ((X392 @ id @ X399 @ id))) & ($true != ((X392 @ id @ (sub @ X393 @ (push @ X399 @ id)) @ id)))) | ? [X394 : (subst > term > term)] : (! [X395 : term] : ((((X392 @ id @ (X394 @ id @ X395) @ id)) = $true) | (((X392 @ id @ X395 @ id)) != $true)) & ($true != ((X392 @ id @ (lam @ (X394 @ sh @ one)) @ id))) & ! [X398 : term,X396 : subst,X397 : subst] : (((sub @ (X394 @ X396 @ X398) @ X397)) = ((X394 @ (comp @ X396 @ X397) @ (sub @ X398 @ X397))))) | ? [X400 : term,X401 : subst,X402 : subst,X403 : subst] : ((((X392 @ (comp @ X403 @ X401) @ (sub @ X400 @ X401) @ X402)) = $true) & ($true != ((X392 @ X403 @ X400 @ (comp @ X401 @ X402)))))) | ? [X383 : term] : (((sub @ X383 @ id)) != X383) | ? [X384 : (subst > term > subst > $o),X385 : term] : (($true != ((X384 @ id @ (lam @ (sub @ X385 @ (push @ one @ sh))) @ id))) & ! [X391 : term] : (($true = ((X384 @ id @ (sub @ X385 @ (push @ X391 @ id)) @ id))) | (((X384 @ id @ X391 @ id)) != $true)) & ! [X386 : (subst > term > term)] : (? [X390 : term] : (($true = ((X384 @ id @ X390 @ id))) & ($true != ((X384 @ id @ (X386 @ id @ X390) @ id)))) | ($true = ((X384 @ id @ (lam @ (X386 @ sh @ one)) @ id))) | ? [X388 : subst,X389 : term,X387 : subst] : (((sub @ (X386 @ X387 @ X389) @ X388)) != ((X386 @ (comp @ X387 @ X388) @ (sub @ X389 @ X388))))))) <=> (hoasinduction_lem3_lthm = $true)) & (substmonoid_gthm = $true) & (axvarshift = $true) & (! [X296 : term,X298 : term,X299 : term,X297 : term] : ((X297 = X298) | (((ap @ X298 @ X299)) != ((ap @ X297 @ X296)))) <=> (apinj1 = $true)) & ((! [X120 : term,X121 : (subst > term > subst > $o)] : (? [X127 : term] : (($true != ((X121 @ id @ (sub @ X120 @ (push @ X127 @ id)) @ id))) & ($true = ((X121 @ id @ X127 @ id)))) | ? [X122 : (subst > term > term)] : (! [X124 : term,X125 : subst,X126 : subst] : (((X122 @ (comp @ X125 @ X126) @ (sub @ X124 @ X126))) = ((sub @ (X122 @ X125 @ X124) @ X126))) & ($true != ((X121 @ id @ (lam @ (X122 @ sh @ one)) @ id))) & ! [X123 : term] : (($true != ((X121 @ id @ X123 @ id))) | ($true = ((X121 @ id @ (X122 @ id @ X123) @ id))))) | ($true = ((X121 @ id @ (lam @ X120) @ id)))) | ? [X112 : (subst > term > subst > $o),X113 : term] : (($true != ((X112 @ id @ (lam @ (sub @ X113 @ (push @ one @ sh))) @ id))) & ! [X114 : (subst > term > term)] : (? [X115 : term] : ((((X112 @ id @ X115 @ id)) = $true) & (((X112 @ id @ (X114 @ id @ X115) @ id)) != $true)) | ($true = ((X112 @ id @ (lam @ (X114 @ sh @ one)) @ id))) | ? [X116 : subst,X118 : subst,X117 : term] : (((X114 @ (comp @ X118 @ X116) @ (sub @ X117 @ X116))) != ((sub @ (X114 @ X118 @ X117) @ X116)))) & ! [X119 : term] : (($true = ((X112 @ id @ (sub @ X113 @ (push @ X119 @ id)) @ id))) | ($true != ((X112 @ id @ X119 @ id))))) | ? [X111 : term] : (((sub @ X111 @ id)) != X111)) <=> (hoasinduction_lem3a_lthm = $true)) & ((! [X352 : term,X353 : (term > $o)] : (($true = ((X353 @ X352))) | ? [X356 : term] : (($true = ((var @ X356))) & ($true != ((X353 @ X356)))) | ? [X357 : term] : (($true != ((X353 @ (lam @ X357)))) & ! [X358 : term] : (($true = ((X353 @ (sub @ X357 @ (push @ X358 @ id))))) | ($true != ((X353 @ X358))))) | ? [X354 : term,X355 : term] : (($true = ((X353 @ X355))) & ($true != ((X353 @ (ap @ X355 @ X354)))) & ($true = ((X353 @ X354))))) | ? [X343 : term] : (((sub @ X343 @ id)) != X343) | ? [X345 : subst,X346 : (term > $o),X344 : term] : (($true != ((X346 @ (sub @ X344 @ X345)))) & ! [X347 : term] : ((((var @ X347)) != $true) | ($true = ((X346 @ (sub @ X347 @ X345))))) & ! [X349 : term,X348 : term] : (($true != ((X346 @ X348))) | (((X346 @ (ap @ X349 @ X348))) = $true) | (((X346 @ X349)) != $true)) & ! [X350 : term] : (($true = ((X346 @ (lam @ X350)))) | ? [X351 : term] : (($true != ((X346 @ (sub @ X350 @ (push @ X351 @ id))))) & ($true = ((X346 @ X351))))))) <=> (induction2_lthm = $true)) & (hoasinduction_lem3a_gthm = $true) & (pushprop_lem1_gthm = $true) & ! [X4 : subst] : ? [X5 : d_subst] : (((d2subst @ X5)) = X4) & ((laminj = $true) <=> ! [X448 : term,X449 : term] : ((X448 = X449) | (((lam @ X449)) != ((lam @ X448))))) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (! [X413 : term,X414 : (subst > term > subst > $o)] : (? [X417 : subst,X415 : subst,X416 : term,X418 : subst] : ((((X414 @ (comp @ X415 @ X418) @ (sub @ X416 @ X418) @ X417)) = $true) & ($true != ((X414 @ X415 @ X416 @ (comp @ X418 @ X417))))) | ? [X419 : subst,X421 : term,X422 : subst,X420 : subst] : (($true != ((X414 @ (comp @ X420 @ X422) @ (sub @ X421 @ X422) @ X419))) & ($true = ((X414 @ X420 @ X421 @ (comp @ X422 @ X419))))) | ? [X428 : term] : (($true != ((X414 @ id @ (sub @ X413 @ (push @ X428 @ id)) @ id))) & ($true = ((X414 @ id @ X428 @ id)))) | ? [X423 : (subst > term > term)] : ((((X414 @ id @ (lam @ (X423 @ sh @ one)) @ id)) != $true) & ! [X425 : term,X426 : subst,X424 : subst] : (((X423 @ (comp @ X426 @ X424) @ (sub @ X425 @ X424))) = ((sub @ (X423 @ X426 @ X425) @ X424))) & ! [X427 : term] : ((((X414 @ id @ (X423 @ id @ X427) @ id)) = $true) | ($true != ((X414 @ id @ X427 @ id))))) | ($true = ((X414 @ id @ (lam @ X413) @ id)))) <=> (hoasinduction_lem3 = $true)) & (axabs = $true) & (hoaslamnotvar_gthm = $true) & (pushprop_lem2v2_gthm = $true) & (pushprop_lem3v2 = $true) & ((axscons = $true) <=> ! [X429 : subst] : (((push @ (sub @ one @ X429) @ (comp @ sh @ X429))) = X429)) & ((termmset_lthm = $true) <=> (? [X177 : term] : (((sub @ X177 @ id)) != X177) | ! [X178 : term] : (((sub @ X178 @ id)) = X178))) & (ulamvarind = $true) & ! [X244 : d_subst] : (d_id = X244) & ((hoasinduction_lem3v2_f = $true) <=> ! [X0 : term] : ? [X1 : (subst > term > term)] : ! [X2 : term,X3 : subst] : (((sub @ X0 @ (push @ X2 @ X3))) = ((X1 @ X3 @ X2)))) & (hoasinduction_lem2_lthm = $true) & (hoasinduction_lem3b_gthm = $true) & ((shinj = $true) <=> ! [X45 : term,X46 : term] : ((X45 = X46) | (((sub @ X46 @ sh)) != ((sub @ X45 @ sh))))) & (axmap = $true) & (pushprop_lem3v2_lthm = $true) & (hoasinduction_lem1 = $true) & (! [X129 : term,X128 : (subst > term > subst > $o)] : (? [X130 : term] : (($true = ((X128 @ id @ X130 @ id))) & ($true != ((X128 @ id @ (sub @ X129 @ (push @ X130 @ id)) @ id)))) | ? [X131 : (subst > term > term),X132 : term] : (! [X136 : term] : ((((X128 @ id @ X136 @ id)) != $true) | ($true = ((X128 @ id @ (X131 @ id @ X136) @ id)))) & ! [X134 : term,X135 : subst,X133 : subst] : (((sub @ (sub @ X132 @ (push @ X134 @ X135)) @ X133)) = ((sub @ (X131 @ X135 @ X134) @ X133))) & ! [X139 : subst,X138 : term,X137 : subst] : (((sub @ X132 @ (push @ (sub @ X138 @ X137) @ (comp @ X139 @ X137)))) = ((X131 @ (comp @ X139 @ X137) @ (sub @ X138 @ X137)))) & (((X128 @ id @ (lam @ (X131 @ sh @ one)) @ id)) != $true)) | ($true = ((X128 @ id @ (lam @ (sub @ X129 @ (push @ one @ sh))) @ id)))) <=> (hoasinduction_lem3aaa = $true)) & ((! [X505 : subst] : (((comp @ id @ X505)) = X505) & ! [X504 : subst] : (((comp @ X504 @ id)) = X504)) <=> (substmonoid = $true)) & ((? [X161 : term,X160 : subst] : (((comp @ sh @ (push @ X161 @ X160))) != X160) | ? [X162 : subst] : (((comp @ X162 @ id)) != X162) | ? [X164 : (term > $o),X163 : term] : (! [X167 : term] : ((((X164 @ (lam @ X167))) = $true) | ($true != ((X164 @ X167)))) & ! [X165 : term,X166 : term] : (($true != ((X164 @ X165))) | ($true != ((X164 @ X166))) | ($true = ((X164 @ (ap @ X165 @ X166))))) & ($true != ((X164 @ X163))) & ! [X168 : term] : (($true = ((X164 @ X168))) | ($true != ((var @ X168))))) | ! [X170 : subst,X171 : (term > $o),X169 : term] : (($true = ((X171 @ (sub @ X169 @ X170)))) | ? [X176 : term,X175 : term] : (($true = ((X171 @ X175))) & ($true != ((X171 @ (ap @ X175 @ X176)))) & (((X171 @ X176)) = $true)) | ? [X174 : term] : (($true != ((X171 @ (sub @ X174 @ X170)))) & ($true = ((var @ X174)))) | ? [X172 : term] : (! [X173 : term] : (($true = ((X171 @ (sub @ X172 @ (push @ X173 @ id))))) | ($true != ((X171 @ X173)))) & ($true != ((X171 @ (lam @ X172)))))) | ? [X159 : term,X158 : subst] : (((sub @ one @ (push @ X159 @ X158))) != X159)) <=> (induction2lem_lthm = $true)) & ((pushprop_lem1 = $true) <=> ! [X410 : (term > $o),X411 : (term > $o),X408 : subst,X412 : term,X409 : term] : (($true = ((X411 @ (sub @ X412 @ (push @ X409 @ X408))))) | ($true != ((X410 @ X412))))) & (((d2term @ d_one)) = one) & (! [X268 : (term > $o),X266 : (subst > term > subst > $o),X267 : term] : (? [X275 : term] : (((X266 @ id @ X275 @ id)) != ((X268 @ X275))) | ? [X269 : term] : (($true = ((X268 @ X269))) & ($true != ((X268 @ (sub @ X267 @ (push @ X269 @ id)))))) | ? [X270 : (subst > term > term)] : (! [X272 : subst,X273 : term,X274 : subst] : (((sub @ (X270 @ X272 @ X273) @ X274)) = ((X270 @ (comp @ X272 @ X274) @ (sub @ X273 @ X274)))) & ! [X271 : term] : ((((X266 @ id @ (X270 @ id @ X271) @ id)) = $true) | (((X266 @ id @ X271 @ id)) != $true)) & ($true != ((X266 @ id @ (lam @ (X270 @ sh @ one)) @ id)))) | ($true = ((X268 @ (lam @ X267))))) <=> (hoasinduction_lem3v2a = $true)) & (hoasinduction_gthm = $true) & ((hoasinduction_lem3a = $true) <=> ! [X441 : term,X440 : (subst > term > subst > $o)] : (? [X443 : (subst > term > term)] : (! [X444 : term] : (($true = ((X440 @ id @ (X443 @ id @ X444) @ id))) | ($true != ((X440 @ id @ X444 @ id)))) & ! [X445 : term,X446 : subst,X447 : subst] : (((X443 @ (comp @ X446 @ X447) @ (sub @ X445 @ X447))) = ((sub @ (X443 @ X446 @ X445) @ X447))) & ($true != ((X440 @ id @ (lam @ (X443 @ sh @ one)) @ id)))) | ($true = ((X440 @ id @ (lam @ X441) @ id))) | ? [X442 : term] : (($true != ((X440 @ id @ (sub @ X441 @ (push @ X442 @ id)) @ id))) & ($true = ((X440 @ id @ X442 @ id)))))) & ((hoasinduction_lem3v2 = $true) <=> ! [X54 : (subst > term > subst > $o),X55 : (term > $o),X53 : term] : (? [X70 : term] : (((X54 @ id @ X70 @ id)) != ((X55 @ X70))) | ? [X56 : (subst > term > term)] : (! [X57 : term] : (($true = ((X54 @ id @ (X56 @ id @ X57) @ id))) | ($true != ((X54 @ id @ X57 @ id)))) & ! [X59 : subst,X60 : term,X58 : subst] : (((X56 @ (comp @ X59 @ X58) @ (sub @ X60 @ X58))) = ((sub @ (X56 @ X59 @ X60) @ X58))) & ($true != ((X54 @ id @ (lam @ (X56 @ sh @ one)) @ id)))) | ? [X69 : term] : ((((X55 @ (sub @ X53 @ (push @ X69 @ id)))) != $true) & (((X55 @ X69)) = $true)) | ($true = ((X55 @ (lam @ X53)))) | ? [X68 : subst,X65 : term,X66 : subst,X67 : subst] : (($true = ((X54 @ (comp @ X68 @ X67) @ (sub @ X65 @ X67) @ X66))) & (((X54 @ X68 @ X65 @ (comp @ X67 @ X66))) != $true)) | ? [X63 : subst,X62 : subst,X64 : term,X61 : subst] : ((((X54 @ (comp @ X61 @ X63) @ (sub @ X64 @ X63) @ X62)) != $true) & ($true = ((X54 @ X61 @ X64 @ (comp @ X63 @ X62))))))) & ((? [X436 : subst] : (((comp @ id @ X436)) != X436) | ? [X437 : subst] : (((comp @ X437 @ id)) != X437) | (! [X438 : subst] : (((comp @ X438 @ id)) = X438) & ! [X439 : subst] : (((comp @ id @ X439)) = X439))) <=> (substmonoid_lthm = $true)) & (! [X434 : term] : ? [X435 : (subst > term > term)] : (((X435 @ sh @ one)) = ((sub @ X434 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b_lthm = $true))),
  inference(flattening,[],[f13])).
thf(f15,definition,(
  ! [X54 : (subst > term > subst > $o)] : ((((sP0 @ X54)) = $true) <=> ? [X56 : (subst > term > term)] : (! [X57 : term] : (($true = ((X54 @ id @ (X56 @ id @ X57) @ id))) | ($true != ((X54 @ id @ X57 @ id)))) & ! [X59 : subst,X60 : term,X58 : subst] : (((X56 @ (comp @ X59 @ X58) @ (sub @ X60 @ X58))) = ((sub @ (X56 @ X59 @ X60) @ X58))) & ($true != ((X54 @ id @ (lam @ (X56 @ sh @ one)) @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f16,definition,(
  ($true = sP1) <=> ! [X54 : (subst > term > subst > $o),X55 : (term > $o),X53 : term] : (? [X70 : term] : (((X54 @ id @ X70 @ id)) != ((X55 @ X70))) | (((sP0 @ X54)) = $true) | ? [X69 : term] : ((((X55 @ (sub @ X53 @ (push @ X69 @ id)))) != $true) & (((X55 @ X69)) = $true)) | ($true = ((X55 @ (lam @ X53)))) | ? [X68 : subst,X65 : term,X66 : subst,X67 : subst] : (($true = ((X54 @ (comp @ X68 @ X67) @ (sub @ X65 @ X67) @ X66))) & (((X54 @ X68 @ X65 @ (comp @ X67 @ X66))) != $true)) | ? [X63 : subst,X62 : subst,X64 : term,X61 : subst] : ((((X54 @ (comp @ X61 @ X63) @ (sub @ X64 @ X63) @ X62)) != $true) & ($true = ((X54 @ X61 @ X64 @ (comp @ X63 @ X62))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f17,definition,(
  ($true = sP2) <=> ! [X441 : term,X440 : (subst > term > subst > $o)] : (? [X443 : (subst > term > term)] : (! [X444 : term] : (($true = ((X440 @ id @ (X443 @ id @ X444) @ id))) | ($true != ((X440 @ id @ X444 @ id)))) & ! [X445 : term,X446 : subst,X447 : subst] : (((X443 @ (comp @ X446 @ X447) @ (sub @ X445 @ X447))) = ((sub @ (X443 @ X446 @ X445) @ X447))) & ($true != ((X440 @ id @ (lam @ (X443 @ sh @ one)) @ id)))) | ($true = ((X440 @ id @ (lam @ X441) @ id))) | ? [X442 : term] : (($true != ((X440 @ id @ (sub @ X441 @ (push @ X442 @ id)) @ id))) & ($true = ((X440 @ id @ X442 @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f18,definition,(
  (sP3 = $true) <=> ! [X268 : (term > $o),X266 : (subst > term > subst > $o),X267 : term] : (? [X275 : term] : (((X266 @ id @ X275 @ id)) != ((X268 @ X275))) | ? [X269 : term] : (($true = ((X268 @ X269))) & ($true != ((X268 @ (sub @ X267 @ (push @ X269 @ id)))))) | ? [X270 : (subst > term > term)] : (! [X272 : subst,X273 : term,X274 : subst] : (((sub @ (X270 @ X272 @ X273) @ X274)) = ((X270 @ (comp @ X272 @ X274) @ (sub @ X273 @ X274)))) & ! [X271 : term] : ((((X266 @ id @ (X270 @ id @ X271) @ id)) = $true) | (((X266 @ id @ X271 @ id)) != $true)) & ($true != ((X266 @ id @ (lam @ (X270 @ sh @ one)) @ id)))) | ($true = ((X268 @ (lam @ X267)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f19,definition,(
  ! [X171 : (term > $o)] : (($true = ((sP4 @ X171))) <=> ? [X176 : term,X175 : term] : (($true = ((X171 @ X175))) & ($true != ((X171 @ (ap @ X175 @ X176)))) & (((X171 @ X176)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f20,definition,(
  ! [X164 : (term > $o)] : ((((sP5 @ X164)) = $true) <=> ! [X165 : term,X166 : term] : (($true != ((X164 @ X165))) | ($true != ((X164 @ X166))) | ($true = ((X164 @ (ap @ X165 @ X166))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f21,definition,(
  (sP6 = $true) <=> ! [X170 : subst,X171 : (term > $o),X169 : term] : (($true = ((X171 @ (sub @ X169 @ X170)))) | ($true = ((sP4 @ X171))) | ? [X174 : term] : (($true != ((X171 @ (sub @ X174 @ X170)))) & ($true = ((var @ X174)))) | ? [X172 : term] : (! [X173 : term] : (($true = ((X171 @ (sub @ X172 @ (push @ X173 @ id))))) | ($true != ((X171 @ X173)))) & ($true != ((X171 @ (lam @ X172))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f22,definition,(
  ($true = sP7) <=> (? [X161 : term,X160 : subst] : (((comp @ sh @ (push @ X161 @ X160))) != X160) | ? [X162 : subst] : (((comp @ X162 @ id)) != X162) | ? [X164 : (term > $o),X163 : term] : (! [X167 : term] : ((((X164 @ (lam @ X167))) = $true) | ($true != ((X164 @ X167)))) & (((sP5 @ X164)) = $true) & ($true != ((X164 @ X163))) & ! [X168 : term] : (($true = ((X164 @ X168))) | ($true != ((var @ X168))))) | (sP6 = $true) | ? [X159 : term,X158 : subst] : (((sub @ one @ (push @ X159 @ X158))) != X159))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f23,definition,(
  ($true = sP8) <=> ! [X129 : term,X128 : (subst > term > subst > $o)] : (? [X130 : term] : (($true = ((X128 @ id @ X130 @ id))) & ($true != ((X128 @ id @ (sub @ X129 @ (push @ X130 @ id)) @ id)))) | ? [X131 : (subst > term > term),X132 : term] : (! [X136 : term] : ((((X128 @ id @ X136 @ id)) != $true) | ($true = ((X128 @ id @ (X131 @ id @ X136) @ id)))) & ! [X134 : term,X135 : subst,X133 : subst] : (((sub @ (sub @ X132 @ (push @ X134 @ X135)) @ X133)) = ((sub @ (X131 @ X135 @ X134) @ X133))) & ! [X139 : subst,X138 : term,X137 : subst] : (((sub @ X132 @ (push @ (sub @ X138 @ X137) @ (comp @ X139 @ X137)))) = ((X131 @ (comp @ X139 @ X137) @ (sub @ X138 @ X137)))) & (((X128 @ id @ (lam @ (X131 @ sh @ one)) @ id)) != $true)) | ($true = ((X128 @ id @ (lam @ (sub @ X129 @ (push @ one @ sh))) @ id))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f24,definition,(
  ! [X414 : (subst > term > subst > $o)] : (($true = ((sP9 @ X414))) <=> ? [X423 : (subst > term > term)] : ((((X414 @ id @ (lam @ (X423 @ sh @ one)) @ id)) != $true) & ! [X425 : term,X426 : subst,X424 : subst] : (((X423 @ (comp @ X426 @ X424) @ (sub @ X425 @ X424))) = ((sub @ (X423 @ X426 @ X425) @ X424))) & ! [X427 : term] : ((((X414 @ id @ (X423 @ id @ X427) @ id)) = $true) | ($true != ((X414 @ id @ X427 @ id))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f25,definition,(
  ($true = sP10) <=> ! [X413 : term,X414 : (subst > term > subst > $o)] : (? [X417 : subst,X415 : subst,X416 : term,X418 : subst] : ((((X414 @ (comp @ X415 @ X418) @ (sub @ X416 @ X418) @ X417)) = $true) & ($true != ((X414 @ X415 @ X416 @ (comp @ X418 @ X417))))) | ? [X419 : subst,X421 : term,X422 : subst,X420 : subst] : (($true != ((X414 @ (comp @ X420 @ X422) @ (sub @ X421 @ X422) @ X419))) & ($true = ((X414 @ X420 @ X421 @ (comp @ X422 @ X419))))) | ? [X428 : term] : (($true != ((X414 @ id @ (sub @ X413 @ (push @ X428 @ id)) @ id))) & ($true = ((X414 @ id @ X428 @ id)))) | ($true = ((sP9 @ X414))) | ($true = ((X414 @ id @ (lam @ X413) @ id))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f26,definition,(
  ! [X346 : (term > $o)] : (($true = ((sP11 @ X346))) <=> ! [X349 : term,X348 : term] : (($true != ((X346 @ X348))) | (((X346 @ (ap @ X349 @ X348))) = $true) | (((X346 @ X349)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f27,definition,(
  ! [X353 : (term > $o)] : (($true = ((sP12 @ X353))) <=> ? [X354 : term,X355 : term] : (($true = ((X353 @ X355))) & ($true != ((X353 @ (ap @ X355 @ X354)))) & ($true = ((X353 @ X354)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f28,definition,(
  ($true = sP13) <=> ! [X352 : term,X353 : (term > $o)] : (($true = ((X353 @ X352))) | ? [X356 : term] : (($true = ((var @ X356))) & ($true != ((X353 @ X356)))) | ? [X357 : term] : (($true != ((X353 @ (lam @ X357)))) & ! [X358 : term] : (($true = ((X353 @ (sub @ X357 @ (push @ X358 @ id))))) | ($true != ((X353 @ X358))))) | ($true = ((sP12 @ X353))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f29,definition,(
  ($true = sP14) <=> (($true = sP13) | ? [X343 : term] : (((sub @ X343 @ id)) != X343) | ? [X345 : subst,X346 : (term > $o),X344 : term] : (($true != ((X346 @ (sub @ X344 @ X345)))) & ! [X347 : term] : ((((var @ X347)) != $true) | ($true = ((X346 @ (sub @ X347 @ X345))))) & ($true = ((sP11 @ X346))) & ! [X350 : term] : (($true = ((X346 @ (lam @ X350)))) | ? [X351 : term] : (($true != ((X346 @ (sub @ X350 @ (push @ X351 @ id))))) & ($true = ((X346 @ X351)))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f30,definition,(
  ($true = sP15) <=> ? [X112 : (subst > term > subst > $o),X113 : term] : (($true != ((X112 @ id @ (lam @ (sub @ X113 @ (push @ one @ sh))) @ id))) & ! [X114 : (subst > term > term)] : (? [X115 : term] : ((((X112 @ id @ X115 @ id)) = $true) & (((X112 @ id @ (X114 @ id @ X115) @ id)) != $true)) | ($true = ((X112 @ id @ (lam @ (X114 @ sh @ one)) @ id))) | ? [X116 : subst,X118 : subst,X117 : term] : (((X114 @ (comp @ X118 @ X116) @ (sub @ X117 @ X116))) != ((sub @ (X114 @ X118 @ X117) @ X116)))) & ! [X119 : term] : (($true = ((X112 @ id @ (sub @ X113 @ (push @ X119 @ id)) @ id))) | ($true != ((X112 @ id @ X119 @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f31,definition,(
  ($true = sP16) <=> (! [X120 : term,X121 : (subst > term > subst > $o)] : (? [X127 : term] : (($true != ((X121 @ id @ (sub @ X120 @ (push @ X127 @ id)) @ id))) & ($true = ((X121 @ id @ X127 @ id)))) | ? [X122 : (subst > term > term)] : (! [X124 : term,X125 : subst,X126 : subst] : (((X122 @ (comp @ X125 @ X126) @ (sub @ X124 @ X126))) = ((sub @ (X122 @ X125 @ X124) @ X126))) & ($true != ((X121 @ id @ (lam @ (X122 @ sh @ one)) @ id))) & ! [X123 : term] : (($true != ((X121 @ id @ X123 @ id))) | ($true = ((X121 @ id @ (X122 @ id @ X123) @ id))))) | ($true = ((X121 @ id @ (lam @ X120) @ id)))) | ($true = sP15) | ? [X111 : term] : (((sub @ X111 @ id)) != X111))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f32,definition,(
  ! [X392 : (subst > term > subst > $o)] : (($true = ((sP17 @ X392))) <=> ? [X394 : (subst > term > term)] : (! [X395 : term] : ((((X392 @ id @ (X394 @ id @ X395) @ id)) = $true) | (((X392 @ id @ X395 @ id)) != $true)) & ($true != ((X392 @ id @ (lam @ (X394 @ sh @ one)) @ id))) & ! [X398 : term,X396 : subst,X397 : subst] : (((sub @ (X394 @ X396 @ X398) @ X397)) = ((X394 @ (comp @ X396 @ X397) @ (sub @ X398 @ X397))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f33,definition,(
  (sP18 = $true) <=> ? [X384 : (subst > term > subst > $o),X385 : term] : (($true != ((X384 @ id @ (lam @ (sub @ X385 @ (push @ one @ sh))) @ id))) & ! [X391 : term] : (($true = ((X384 @ id @ (sub @ X385 @ (push @ X391 @ id)) @ id))) | (((X384 @ id @ X391 @ id)) != $true)) & ! [X386 : (subst > term > term)] : (? [X390 : term] : (($true = ((X384 @ id @ X390 @ id))) & ($true != ((X384 @ id @ (X386 @ id @ X390) @ id)))) | ($true = ((X384 @ id @ (lam @ (X386 @ sh @ one)) @ id))) | ? [X388 : subst,X389 : term,X387 : subst] : (((sub @ (X386 @ X387 @ X389) @ X388)) != ((X386 @ (comp @ X387 @ X388) @ (sub @ X389 @ X388))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f34,definition,(
  ($true = sP19) <=> (! [X393 : term,X392 : (subst > term > subst > $o)] : (? [X405 : term,X404 : subst,X406 : subst,X407 : subst] : ((((X392 @ (comp @ X407 @ X406) @ (sub @ X405 @ X406) @ X404)) != $true) & (((X392 @ X407 @ X405 @ (comp @ X406 @ X404))) = $true)) | (((X392 @ id @ (lam @ X393) @ id)) = $true) | ? [X399 : term] : (($true = ((X392 @ id @ X399 @ id))) & ($true != ((X392 @ id @ (sub @ X393 @ (push @ X399 @ id)) @ id)))) | ($true = ((sP17 @ X392))) | ? [X400 : term,X401 : subst,X402 : subst,X403 : subst] : ((((X392 @ (comp @ X403 @ X401) @ (sub @ X400 @ X401) @ X402)) = $true) & ($true != ((X392 @ X403 @ X400 @ (comp @ X401 @ X402)))))) | ? [X383 : term] : (((sub @ X383 @ id)) != X383) | (sP18 = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f35,definition,(
  ! [X471 : (subst > term > subst > $o)] : ((((sP20 @ X471)) = $true) <=> ! [X481 : (subst > term > term)] : (? [X483 : subst,X482 : term,X484 : subst] : (((X481 @ (comp @ X484 @ X483) @ (sub @ X482 @ X483))) != ((sub @ (X481 @ X484 @ X482) @ X483))) | (((X471 @ id @ (lam @ (X481 @ sh @ one)) @ id)) = $true) | ? [X485 : term] : ((((X471 @ id @ X485 @ id)) = $true) & (((X471 @ id @ (X481 @ id @ X485) @ id)) != $true))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f36,definition,(
  ! [X487 : (subst > term > subst > $o)] : (($true = ((sP21 @ X487))) <=> ? [X497 : term,X498 : term] : (($true != ((X487 @ id @ (ap @ (sub @ X497 @ id) @ X498) @ id))) & (((X487 @ id @ X498 @ id)) = $true) & ($true = ((X487 @ id @ X497 @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f37,definition,(
  ! [X487 : (subst > term > subst > $o)] : (($true = ((sP22 @ X487))) <=> ? [X499 : (subst > term > term)] : (! [X501 : subst,X502 : subst,X503 : term] : (((X499 @ (comp @ X502 @ X501) @ (sub @ X503 @ X501))) = ((sub @ (X499 @ X502 @ X503) @ X501))) & ! [X500 : term] : (($true != ((X487 @ id @ X500 @ id))) | ($true = ((X487 @ id @ (X499 @ id @ X500) @ id)))) & (((X487 @ id @ (lam @ (X499 @ sh @ one)) @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f38,definition,(
  ! [X450 : (term > $o)] : ((((sP23 @ X450)) = $true) <=> ! [X456 : term,X455 : term] : (($true != ((X450 @ X456))) | ($true != ((X450 @ X455))) | ($true = ((X450 @ (ap @ X455 @ X456))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f39,definition,(
  ! [X457 : (subst > term > subst > $o)] : (($true = ((sP24 @ X457))) <=> ! [X461 : term,X460 : term] : (($true = ((X457 @ id @ (ap @ (sub @ X460 @ id) @ X461) @ id))) | ($true != ((X457 @ id @ X460 @ id))) | ($true != ((X457 @ id @ X461 @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f40,definition,(
  (sP25 = $true) <=> ? [X470 : term,X471 : (subst > term > subst > $o)] : ((((X471 @ id @ (lam @ X470) @ id)) != $true) & (((sP20 @ X471)) = $true) & ! [X476 : subst,X477 : subst,X478 : term,X479 : subst] : (($true != ((X471 @ (comp @ X479 @ X476) @ (sub @ X478 @ X476) @ X477))) | ($true = ((X471 @ X479 @ X478 @ (comp @ X476 @ X477))))) & ! [X480 : term] : (($true = ((X471 @ id @ (sub @ X470 @ (push @ X480 @ id)) @ id))) | ($true != ((X471 @ id @ X480 @ id)))) & ! [X475 : subst,X474 : subst,X473 : term,X472 : subst] : (($true != ((X471 @ X475 @ X473 @ (comp @ X474 @ X472)))) | ($true = ((X471 @ (comp @ X475 @ X474) @ (sub @ X473 @ X474) @ X472)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f41,definition,(
  ($true = sP26) <=> ! [X486 : term,X487 : (subst > term > subst > $o)] : (($true = ((sP22 @ X487))) | ($true = ((X487 @ id @ X486 @ id))) | ($true = ((sP21 @ X487))) | ? [X488 : term] : (($true != ((X487 @ id @ X488 @ id))) & ($true = ((var @ (sub @ X488 @ id))))) | ? [X490 : subst,X491 : subst,X489 : term,X492 : subst] : (($true = ((X487 @ (comp @ X490 @ X491) @ (sub @ X489 @ X491) @ X492))) & (((X487 @ X490 @ X489 @ (comp @ X491 @ X492))) != $true)) | ? [X496 : subst,X494 : subst,X495 : term,X493 : subst] : (($true != ((X487 @ (comp @ X493 @ X494) @ (sub @ X495 @ X494) @ X496))) & ($true = ((X487 @ X493 @ X495 @ (comp @ X494 @ X496))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f42,definition,(
  ($true = sP27) <=> ? [X451 : term,X450 : (term > $o)] : ((((X450 @ X451)) != $true) & ! [X452 : term] : (($true = ((X450 @ (lam @ X452)))) | ? [X453 : term] : (($true != ((X450 @ (sub @ X452 @ (push @ X453 @ id))))) & ($true = ((X450 @ X453))))) & ! [X454 : term] : ((((var @ X454)) != $true) | ($true = ((X450 @ X454)))) & (((sP23 @ X450)) = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f43,definition,(
  ($true = sP28) <=> (? [X457 : (subst > term > subst > $o),X458 : term,X459 : term] : (! [X462 : term,X463 : subst,X465 : subst,X464 : subst] : (($true != ((X457 @ X464 @ X462 @ (comp @ X463 @ X465)))) | ($true = ((X457 @ (comp @ X464 @ X463) @ (sub @ X462 @ X463) @ X465)))) & ($true != ((X457 @ id @ (ap @ X459 @ X458) @ id))) & ($true = ((X457 @ id @ X459 @ id))) & (((X457 @ id @ X458 @ id)) = $true) & ! [X468 : term,X467 : subst,X469 : subst,X466 : subst] : ((((X457 @ X466 @ X468 @ (comp @ X467 @ X469))) = $true) | ($true != ((X457 @ (comp @ X466 @ X467) @ (sub @ X468 @ X467) @ X469)))) & ($true = ((sP24 @ X457)))) | ($true = sP27) | ($true = sP26) | (sP25 = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f44,definition,(
  ($true = sP29) <=> ! [X305 : (subst > term > subst > $o),X306 : term] : (($true = ((X305 @ id @ (lam @ (sub @ X306 @ (push @ one @ sh))) @ id))) | ? [X308 : (subst > term > term)] : (($true != ((X305 @ id @ (lam @ (X308 @ sh @ one)) @ id))) & ! [X309 : term] : ((((X305 @ id @ X309 @ id)) != $true) | ($true = ((X305 @ id @ (X308 @ id @ X309) @ id)))) & ! [X312 : subst,X310 : subst,X311 : term] : (((sub @ (X308 @ X312 @ X311) @ X310)) = ((X308 @ (comp @ X312 @ X310) @ (sub @ X311 @ X310))))) | ? [X307 : term] : (($true = ((X305 @ id @ X307 @ id))) & ($true != ((X305 @ id @ (sub @ X306 @ (push @ X307 @ id)) @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f45,definition,(
  (sP30 = $true) <=> ! [X533 : (subst > term > subst > $o),X532 : term] : (? [X535 : (subst > term > term)] : (! [X536 : term] : (($true != ((X533 @ id @ X536 @ id))) | ($true = ((X533 @ id @ (X535 @ id @ X536) @ id)))) & ! [X537 : subst,X539 : subst,X538 : term] : (((sub @ (X535 @ X539 @ X538) @ X537)) = ((X535 @ (comp @ X539 @ X537) @ (sub @ X538 @ X537)))) & (((X533 @ id @ (lam @ (X535 @ sh @ one)) @ id)) != $true)) | ? [X534 : term] : ((((X533 @ id @ (sub @ X532 @ (push @ X534 @ id)) @ id)) != $true) & ($true = ((X533 @ id @ X534 @ id)))) | ($true = ((X533 @ id @ (lam @ (sub @ X532 @ (push @ one @ sh))) @ id))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f46,definition,(
  ! [X28 : (subst > term > subst > $o)] : (($true = ((sP31 @ X28))) <=> ? [X29 : term,X30 : term] : (($true = ((X28 @ id @ X29 @ id))) & ($true != ((X28 @ id @ (ap @ (sub @ X30 @ id) @ X29) @ id))) & ($true = ((X28 @ id @ X30 @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f47,definition,(
  ! [X28 : (subst > term > subst > $o)] : (($true = ((sP32 @ X28))) <=> ? [X31 : (subst > term > term)] : (! [X32 : term] : (($true = ((X28 @ id @ (X31 @ id @ X32) @ id))) | ($true != ((X28 @ id @ X32 @ id)))) & (((X28 @ id @ (lam @ (X31 @ sh @ one)) @ id)) != $true) & ! [X34 : term,X35 : subst,X33 : subst] : (((sub @ (X31 @ X35 @ X34) @ X33)) = ((X31 @ (comp @ X35 @ X33) @ (sub @ X34 @ X33))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f48,definition,(
  ! [X9 : (term > $o)] : (($true = ((sP33 @ X9))) <=> ! [X11 : term,X12 : term] : (($true != ((X9 @ X11))) | (((X9 @ X12)) != $true) | (((X9 @ (ap @ X11 @ X12))) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f49,definition,(
  ($true = sP34) <=> ! [X28 : (subst > term > subst > $o),X27 : term] : (($true = ((sP32 @ X28))) | (((X28 @ id @ X27 @ id)) = $true) | ? [X41 : term,X40 : subst,X43 : subst,X42 : subst] : ((((X28 @ X43 @ X41 @ (comp @ X40 @ X42))) = $true) & (((X28 @ (comp @ X43 @ X40) @ (sub @ X41 @ X40) @ X42)) != $true)) | ($true = ((sP31 @ X28))) | ? [X44 : term] : (($true != ((X28 @ id @ X44 @ id))) & ($true = ((var @ (sub @ X44 @ id))))) | ? [X39 : subst,X37 : subst,X38 : subst,X36 : term] : (($true = ((X28 @ (comp @ X37 @ X39) @ (sub @ X36 @ X39) @ X38))) & ($true != ((X28 @ X37 @ X36 @ (comp @ X39 @ X38))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f50,definition,(
  ($true = sP35) <=> ? [X17 : term,X19 : (subst > term > subst > $o),X18 : (term > $o)] : (! [X26 : term] : (((X18 @ X26)) = ((X19 @ id @ X26 @ id))) & ($true != ((X18 @ (lam @ X17)))) & ! [X20 : (subst > term > term)] : (? [X21 : term] : ((((X19 @ id @ X21 @ id)) = $true) & ($true != ((X19 @ id @ (X20 @ id @ X21) @ id)))) | ? [X24 : subst,X23 : subst,X22 : term] : (((X20 @ (comp @ X23 @ X24) @ (sub @ X22 @ X24))) != ((sub @ (X20 @ X23 @ X22) @ X24))) | (((X19 @ id @ (lam @ (X20 @ sh @ one)) @ id)) = $true)) & ! [X25 : term] : (($true != ((X18 @ X25))) | ($true = ((X18 @ (sub @ X17 @ (push @ X25 @ id)))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f51,definition,(
  ($true = sP36) <=> (? [X16 : term] : (((sub @ X16 @ id)) != X16) | ? [X10 : term,X9 : (term > $o)] : (($true != ((X9 @ X10))) & ! [X14 : term] : (? [X15 : term] : (($true != ((X9 @ (sub @ X14 @ (push @ X15 @ id))))) & ($true = ((X9 @ X15)))) | ($true = ((X9 @ (lam @ X14))))) & ! [X13 : term] : (($true != ((var @ X13))) | ($true = ((X9 @ X13)))) & ($true = ((sP33 @ X9)))) | ($true = sP35) | ? [X6 : (subst > term > subst > $o)] : ! [X7 : (term > $o)] : ? [X8 : term] : (((X7 @ X8)) != ((X6 @ id @ X8 @ id))) | ($true = sP34))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f52,definition,(
  ($true = sP37) <=> (! [X230 : (subst > term > subst > $o),X231 : (term > $o),X232 : term] : (($true = ((X231 @ (lam @ X232)))) | ? [X235 : (subst > term > term)] : (! [X236 : term] : (($true = ((X230 @ id @ (X235 @ id @ X236) @ id))) | (((X230 @ id @ X236 @ id)) != $true)) & ! [X239 : subst,X238 : subst,X237 : term] : (((sub @ (X235 @ X239 @ X237) @ X238)) = ((X235 @ (comp @ X239 @ X238) @ (sub @ X237 @ X238)))) & ($true != ((X230 @ id @ (lam @ (X235 @ sh @ one)) @ id)))) | ? [X234 : term] : (((X230 @ id @ X234 @ id)) != ((X231 @ X234))) | ? [X233 : term] : (($true != ((X231 @ (sub @ X232 @ (push @ X233 @ id))))) & ($true = ((X231 @ X233))))) | ? [X225 : term] : ! [X226 : (subst > term > term)] : ? [X227 : term,X228 : subst] : (((X226 @ X228 @ X227)) != ((sub @ X225 @ (push @ X227 @ X228)))) | ? [X229 : term] : (((sub @ X229 @ id)) != X229))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f53,definition,(
  ! [X369 : (subst > term > subst > $o)] : (($true = ((sP38 @ X369))) <=> ? [X381 : term,X380 : term] : ((((X369 @ id @ X381 @ id)) = $true) & ($true = ((X369 @ id @ X380 @ id))) & ($true != ((X369 @ id @ (ap @ (sub @ X380 @ id) @ X381) @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f54,definition,(
  (sP39 = $true) <=> ! [X368 : term,X371 : term,X369 : (subst > term > subst > $o),X370 : (term > $o)] : (($true != ((X370 @ X371))) | ? [X382 : term] : (((X369 @ id @ X382 @ id)) != ((X370 @ X382))) | ? [X376 : subst,X378 : term,X379 : subst,X377 : subst] : (($true != ((X369 @ (comp @ X376 @ X379) @ (sub @ X378 @ X379) @ X377))) & ($true = ((X369 @ X376 @ X378 @ (comp @ X379 @ X377))))) | ? [X375 : subst,X372 : subst,X374 : subst,X373 : term] : (($true != ((X369 @ X375 @ X373 @ (comp @ X374 @ X372)))) & ($true = ((X369 @ (comp @ X375 @ X374) @ (sub @ X373 @ X374) @ X372)))) | ($true != ((X370 @ X368))) | ($true = ((sP38 @ X369))) | ($true = ((X370 @ (ap @ X371 @ X368)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f55,definition,(
  ! [X507 : (subst > term > subst > $o)] : (($true = ((sP40 @ X507))) <=> ? [X519 : (subst > term > term)] : ((((X507 @ id @ (lam @ (X519 @ sh @ one)) @ id)) != $true) & ! [X520 : term] : (($true != ((X507 @ id @ X520 @ id))) | ($true = ((X507 @ id @ (X519 @ id @ X520) @ id)))) & ! [X523 : term,X522 : subst,X521 : subst] : (((X519 @ (comp @ X521 @ X522) @ (sub @ X523 @ X522))) = ((sub @ (X519 @ X521 @ X523) @ X522)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f56,definition,(
  ! [X507 : (subst > term > subst > $o)] : ((((sP41 @ X507)) = $true) <=> ? [X513 : term,X512 : term] : (($true != ((X507 @ id @ (ap @ (sub @ X513 @ id) @ X512) @ id))) & ($true = ((X507 @ id @ X512 @ id))) & ($true = ((X507 @ id @ X513 @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f57,definition,(
  (sP42 = $true) <=> ! [X506 : term,X507 : (subst > term > subst > $o)] : ((((sP41 @ X507)) = $true) | ? [X518 : term] : (($true = ((var @ (sub @ X518 @ id)))) & ($true != ((X507 @ id @ X518 @ id)))) | ? [X508 : subst,X511 : term,X510 : subst,X509 : subst] : ((((X507 @ (comp @ X508 @ X509) @ (sub @ X511 @ X509) @ X510)) != $true) & ($true = ((X507 @ X508 @ X511 @ (comp @ X509 @ X510))))) | ($true = ((X507 @ id @ X506 @ id))) | ($true = ((sP40 @ X507))) | ? [X514 : subst,X516 : subst,X515 : subst,X517 : term] : (($true != ((X507 @ X514 @ X517 @ (comp @ X516 @ X515)))) & (((X507 @ (comp @ X514 @ X516) @ (sub @ X517 @ X516) @ X515)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f58,definition,(
  ! [X73 : (subst > term > subst > $o)] : ((((sP43 @ X73)) = $true) <=> ? [X80 : (subst > term > term)] : (! [X81 : term] : (($true != ((X73 @ id @ X81 @ id))) | (((X73 @ id @ (X80 @ id @ X81) @ id)) = $true)) & ($true != ((X73 @ id @ (lam @ (X80 @ sh @ one)) @ id))) & ! [X84 : subst,X82 : subst,X83 : term] : (((sub @ (X80 @ X84 @ X83) @ X82)) = ((X80 @ (comp @ X84 @ X82) @ (sub @ X83 @ X82))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f59,definition,(
  ($true = sP44) <=> (? [X71 : term] : (((sub @ X71 @ id)) != X71) | ! [X73 : (subst > term > subst > $o),X74 : (term > $o),X72 : term] : (? [X89 : term] : (((X74 @ X89)) != ((X73 @ id @ X89 @ id))) | ? [X75 : term] : (($true != ((X74 @ (sub @ X72 @ (push @ X75 @ id))))) & ($true = ((X74 @ X75)))) | ($true = ((X74 @ (lam @ X72)))) | ? [X79 : subst,X77 : subst,X76 : term,X78 : subst] : ((((X73 @ X77 @ X76 @ (comp @ X78 @ X79))) = $true) & ($true != ((X73 @ (comp @ X77 @ X78) @ (sub @ X76 @ X78) @ X79)))) | (((sP43 @ X73)) = $true) | ? [X87 : subst,X86 : subst,X85 : subst,X88 : term] : (($true != ((X73 @ X85 @ X88 @ (comp @ X87 @ X86)))) & (((X73 @ (comp @ X85 @ X87) @ (sub @ X88 @ X87) @ X86)) = $true))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f60,definition,(
  ! [X143 : (subst > term > subst > $o)] : (($true = ((sP45 @ X143))) <=> ? [X144 : (subst > term > term)] : ((((X143 @ id @ (lam @ (X144 @ sh @ one)) @ id)) != $true) & ! [X147 : subst,X148 : term,X146 : subst] : (((X144 @ (comp @ X146 @ X147) @ (sub @ X148 @ X147))) = ((sub @ (X144 @ X146 @ X148) @ X147))) & ! [X145 : term] : (($true = ((X143 @ id @ (X144 @ id @ X145) @ id))) | (((X143 @ id @ X145 @ id)) != $true))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f61,definition,(
  ! [X212 : (subst > term > subst > $o)] : (($true = ((sP46 @ X212))) <=> ? [X219 : term,X220 : term] : ((((X212 @ id @ X219 @ id)) = $true) & ($true = ((X212 @ id @ X220 @ id))) & ($true != ((X212 @ id @ (ap @ (sub @ X219 @ id) @ X220) @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f62,definition,(
  (sP47 = $true) <=> ! [X214 : term,X212 : (subst > term > subst > $o),X213 : term] : (($true != ((X212 @ id @ X213 @ id))) | ($true != ((X212 @ id @ X214 @ id))) | ? [X222 : term,X223 : subst,X224 : subst,X221 : subst] : (($true != ((X212 @ (comp @ X223 @ X221) @ (sub @ X222 @ X221) @ X224))) & (((X212 @ X223 @ X222 @ (comp @ X221 @ X224))) = $true)) | ? [X216 : term,X217 : subst,X215 : subst,X218 : subst] : (($true = ((X212 @ (comp @ X215 @ X217) @ (sub @ X216 @ X217) @ X218))) & ($true != ((X212 @ X215 @ X216 @ (comp @ X217 @ X218))))) | ($true = ((sP46 @ X212))) | ($true = ((X212 @ id @ (ap @ X213 @ X214) @ id))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f63,definition,(
  ! [X335 : (subst > term > subst > $o)] : (($true = ((sP48 @ X335))) <=> ? [X342 : term,X341 : term] : (($true = ((X335 @ id @ X341 @ id))) & ($true != ((X335 @ id @ (ap @ (sub @ X342 @ id) @ X341) @ id))) & ($true = ((X335 @ id @ X342 @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f64,definition,(
  ! [X317 : (term > $o)] : (($true = ((sP49 @ X317))) <=> ! [X320 : term,X321 : term] : ((((X317 @ (ap @ X320 @ X321))) = $true) | (((X317 @ X321)) != $true) | ($true != ((X317 @ X320)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f65,definition,(
  ($true = sP50) <=> ? [X325 : (term > $o),X326 : term,X324 : (subst > term > subst > $o)] : (! [X332 : term] : (($true != ((X325 @ X332))) | ($true = ((X325 @ (sub @ X326 @ (push @ X332 @ id)))))) & ! [X327 : (subst > term > term)] : ((((X324 @ id @ (lam @ (X327 @ sh @ one)) @ id)) = $true) | ? [X330 : term,X331 : subst,X329 : subst] : (((X327 @ (comp @ X331 @ X329) @ (sub @ X330 @ X329))) != ((sub @ (X327 @ X331 @ X330) @ X329))) | ? [X328 : term] : (($true = ((X324 @ id @ X328 @ id))) & ($true != ((X324 @ id @ (X327 @ id @ X328) @ id))))) & ! [X333 : term] : (((X325 @ X333)) = ((X324 @ id @ X333 @ id))) & ($true != ((X325 @ (lam @ X326)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f66,definition,(
  (sP51 = $true) <=> ! [X335 : (subst > term > subst > $o),X334 : term] : (($true = ((X335 @ id @ X334 @ id))) | ? [X336 : (subst > term > term)] : (! [X339 : subst,X340 : subst,X338 : term] : (((sub @ (X336 @ X340 @ X338) @ X339)) = ((X336 @ (comp @ X340 @ X339) @ (sub @ X338 @ X339)))) & (((X335 @ id @ (lam @ (X336 @ sh @ one)) @ id)) != $true) & ! [X337 : term] : ((((X335 @ id @ X337 @ id)) != $true) | ($true = ((X335 @ id @ (X336 @ id @ X337) @ id))))) | ($true = ((sP48 @ X335))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f67,definition,(
  ($true = sP52) <=> (? [X317 : (term > $o),X316 : term] : (! [X318 : term] : (($true = ((X317 @ (lam @ X318)))) | ? [X319 : term] : ((((X317 @ (sub @ X318 @ (push @ X319 @ id)))) != $true) & (((X317 @ X319)) = $true))) & ($true = ((sP49 @ X317))) & ! [X322 : term] : (($true != ((var @ X322))) | ($true = ((X317 @ X322)))) & ($true != ((X317 @ X316)))) | (sP51 = $true) | ? [X323 : term] : (((sub @ X323 @ id)) != X323) | ? [X313 : (subst > term > subst > $o)] : ! [X314 : (term > $o)] : ? [X315 : term] : (((X313 @ id @ X315 @ id)) != ((X314 @ X315))) | ($true = sP50))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f68,definition,(
  ! [X197 : (term > $o)] : ((((sP53 @ X197)) = $true) <=> ? [X201 : term,X202 : term] : ((((X197 @ (ap @ X201 @ X202))) != $true) & ($true = ((X197 @ X202))) & (((X197 @ X201)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f69,definition,(
  ! [X181 : (term > $o)] : (($true = ((sP54 @ X181))) <=> ? [X185 : term,X184 : term] : (($true = ((X181 @ X185))) & ($true = ((X181 @ X184))) & ($true != ((X181 @ (ap @ X185 @ X184))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f70,definition,(
  ($true = sP55) <=> ! [X181 : (term > $o),X180 : term,X179 : subst] : (? [X182 : term] : (! [X183 : term] : (($true != ((X181 @ X183))) | ($true = ((X181 @ (sub @ X182 @ (push @ X183 @ id)))))) & ($true != ((X181 @ (lam @ X182))))) | ? [X186 : term] : ((((X181 @ (sub @ X186 @ X179))) != $true) & ($true = ((var @ X186)))) | ($true = ((X181 @ (sub @ X180 @ X179)))) | ($true = ((sP54 @ X181))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f71,definition,(
  ! [X191 : (term > $o)] : (($true = ((sP56 @ X191))) <=> ? [X194 : term,X195 : term] : (($true = ((X191 @ X195))) & (((X191 @ (ap @ X195 @ X194))) != $true) & ($true = ((X191 @ X194)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f72,definition,(
  ($true = sP57) <=> ! [X191 : (term > $o),X190 : term] : (? [X192 : term] : (! [X193 : term] : (($true != ((X191 @ X193))) | ($true = ((X191 @ (sub @ X192 @ (push @ X193 @ id)))))) & ($true != ((X191 @ (lam @ X192))))) | ($true = ((sP56 @ X191))) | ? [X196 : term] : (($true != ((X191 @ X196))) & ($true = ((var @ X196)))) | ($true = ((X191 @ X190))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f73,definition,(
  (sP58 = $true) <=> (! [X100 : term,X103 : subst,X102 : (subst > term > term),X101 : (subst > term > term)] : ((((X102 @ X103 @ X100)) = ((X101 @ X103 @ X100))) | (((lam @ (X102 @ sh @ one))) != ((lam @ (X101 @ sh @ one)))) | ? [X105 : subst,X106 : subst,X104 : term] : (((X102 @ (comp @ X105 @ X106) @ (sub @ X104 @ X106))) != ((sub @ (X102 @ X105 @ X104) @ X106))) | ? [X107 : subst,X109 : term,X108 : subst] : (((X101 @ (comp @ X108 @ X107) @ (sub @ X109 @ X107))) != ((sub @ (X101 @ X108 @ X109) @ X107)))) | ? [X96 : term,X97 : subst] : (((comp @ sh @ (push @ X96 @ X97))) != X97) | ? [X98 : term,X99 : term] : ((((lam @ X99)) = ((lam @ X98))) & (X98 != X99)) | ? [X94 : term,X95 : subst] : (((sub @ one @ (push @ X94 @ X95))) != X94))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f74,plain,(
  (hoaslamnotap_lthm = $true) & (lamnotap != $true) & (pushprop_lem0_gthm = $true) & ((hoaslaminj_lthm = $true) <=> (sP58 = $true)) & (($true = sP57) <=> (induction2 = $true)) & (hoasinduction_lem2_gthm = $true) & (! [X295 : subst,X294 : term] : (((sub @ one @ (push @ X294 @ X295))) = X294) <=> (axvarcons = $true)) & ! [X242 : term] : ? [X243 : d_term] : (((d2term @ X243)) = X242) & (hoasinduction_lem1_lthm = $true) & ((induction2lem = $true) <=> ($true = sP55)) & (pushprop_gthm = $true) & (hoasinduction_lem1_gthm = $true) & ((pushprop_lem0 = $true) <=> ! [X301 : subst,X302 : (term > $o),X300 : term] : ? [X303 : (term > $o)] : ! [X304 : term] : (((X302 @ (sub @ X304 @ (push @ X300 @ X301)))) = ((X303 @ X304)))) & (! [X90 : term] : ? [X91 : (subst > term > term)] : ! [X92 : term,X93 : subst] : (((X91 @ X93 @ X92)) = ((sub @ X90 @ (push @ X92 @ X93)))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & (pushprop_lem1v2_gthm = $true) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem1v2_gthm = $true) & (((d2subst @ d_id)) = sh) & (! [X187 : (subst > term > subst > $o)] : ? [X188 : (term > $o)] : ! [X189 : term] : (((X187 @ id @ X189 @ id)) = ((X188 @ X189))) <=> (hoasinduction_lem0 = $true)) & (((d2subst @ d_id)) = id) & (! [X198 : term,X197 : (term > $o)] : ((((sP53 @ X197)) = $true) | ? [X200 : term] : (($true = ((var @ X200))) & ($true != ((X197 @ X200)))) | ($true = ((X197 @ X198))) | ? [X199 : term] : (($true = ((X197 @ X199))) & ($true != ((X197 @ (lam @ X199)))))) <=> (induction = $true)) & (((var @ (d2term @ d_one))) != $true) & ((hoaslamnotap = $true) <=> ! [X262 : term,X260 : term,X261 : (subst > term > term)] : ((((ap @ (sub @ X262 @ id) @ X260)) != ((lam @ (X261 @ sh @ one)))) | ? [X264 : subst,X265 : subst,X263 : term] : (((sub @ (X261 @ X264 @ X263) @ X265)) != ((X261 @ (comp @ X264 @ X265) @ (sub @ X263 @ X265)))))) & (hoasinduction_lem3_gthm = $true) & (pushprop_lem2v2 = $true) & ((apinj2 = $true) <=> ! [X529 : term,X528 : term,X531 : term,X530 : term] : ((X529 = X531) | (((ap @ X528 @ X529)) != ((ap @ X530 @ X531))))) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & ((hoasinduction_no_psi_cond_lthm = $true) <=> ($true = sP52)) & (hoaslamnotap_gthm = $true) & ((termmset = $true) <=> ! [X110 : term] : (((sub @ X110 @ id)) = X110)) & ((sP47 = $true) <=> (hoasinduction_lem2 = $true)) & (! [X246 : term,X247 : term,X245 : term,X248 : term] : ((X247 = X248) | (((ap @ (sub @ X247 @ id) @ X246)) != ((ap @ (sub @ X248 @ id) @ X245)))) <=> (hoasapinj1 = $true)) & (hoasapnotvar = $true) & (! [X251 : term,X252 : subst,X250 : (subst > term > term),X253 : (subst > term > term)] : (? [X254 : term,X256 : subst,X255 : subst] : (((sub @ (X253 @ X255 @ X254) @ X256)) != ((X253 @ (comp @ X255 @ X256) @ (sub @ X254 @ X256)))) | ? [X259 : subst,X257 : subst,X258 : term] : (((sub @ (X250 @ X259 @ X258) @ X257)) != ((X250 @ (comp @ X259 @ X257) @ (sub @ X258 @ X257)))) | (((X253 @ X252 @ X251)) = ((X250 @ X252 @ X251))) | (((lam @ (X250 @ sh @ one))) != ((lam @ (X253 @ sh @ one))))) <=> (hoaslaminj = $true)) & (hoasapnotvar_lthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((hoasinduction_no_psi_cond = $true) <=> ! [X143 : (subst > term > subst > $o),X142 : term] : (($true = ((X143 @ id @ X142 @ id))) | ? [X150 : term,X149 : term] : ((((X143 @ id @ X149 @ id)) = $true) & ($true = ((X143 @ id @ X150 @ id))) & ($true != ((X143 @ id @ (ap @ (sub @ X149 @ id) @ X150) @ id)))) | ($true = ((sP45 @ X143))))) & (pushprop_lem2v2_lthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (! [X524 : (subst > term > subst > $o)] : ? [X525 : (term > $o)] : ! [X526 : term] : (((X524 @ id @ X526 @ id)) = ((X525 @ X526))) <=> (hoasinduction_lem0_lthm = $true)) & (termmset_gthm = $true) & ((pushprop_lem1v2_lthm = $true) <=> (? [X151 : term,X152 : subst] : (((sub @ one @ (push @ X151 @ X152))) != X151) | ! [X154 : (term > $o),X156 : (term > $o),X153 : term,X155 : subst] : ((((X154 @ one)) = $true) | ? [X157 : term] : (((X156 @ (sub @ X157 @ (push @ X153 @ X155)))) != ((X154 @ X157))) | ($true != ((X156 @ X153)))))) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((hoasapinj1_lthm = $true) <=> (? [X359 : term] : (((sub @ X359 @ id)) != X359) | ! [X364 : term,X366 : term,X365 : term,X367 : term] : ((X365 = X367) | (((ap @ (sub @ X367 @ id) @ X364)) != ((ap @ (sub @ X365 @ id) @ X366)))) | ? [X360 : term,X361 : term,X362 : term,X363 : term] : ((((ap @ X361 @ X362)) = ((ap @ X363 @ X360))) & (X361 != X363)))) & (hoaslamnotvar = $true) & (axclos = $true) & (! [X141 : subst] : (((comp @ id @ X141)) = X141) <=> (axidl = $true)) & ! [X433 : d_term,X432 : d_term] : ((((d2term @ X432)) != ((d2term @ X433))) | (X432 = X433)) & (($true = sP44) <=> (hoasinduction_lem3v2_lthm = $true)) & (! [X430 : term] : ? [X431 : (subst > term > term)] : (((X431 @ sh @ one)) = ((sub @ X430 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b = $true)) & ((hoasinduction = $true) <=> (sP42 = $true)) & (hoaslaminj_gthm = $true) & ! [X249 : d_term] : (d_one = X249) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ! [X240 : d_subst,X241 : d_subst] : ((((d2subst @ X241)) != ((d2subst @ X240))) | (X240 = X241)) & (induction2_gthm = $true) & ((hoasapinj2_lthm = $true) <=> (! [X291 : term,X290 : term,X293 : term,X292 : term] : ((((ap @ (sub @ X292 @ id) @ X291)) != ((ap @ (sub @ X290 @ id) @ X293))) | (X291 = X293)) | ? [X289 : term,X288 : term,X287 : term,X286 : term] : ((((ap @ X286 @ X287)) = ((ap @ X289 @ X288))) & (X287 != X288)))) & (hoasinduction_lem1v2 = $true) & ((sP39 = $true) <=> (hoasinduction_lem2v2 = $true)) & (axassoc = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (pushprop_lthm = $true) & ((hoasinduction_lem3v2a_lthm = $true) <=> ($true = sP37)) & (hoasapinj1_gthm = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (hoasinduction_lem2v2_gthm = $true) & (! [X140 : subst] : (((comp @ X140 @ id)) = X140) <=> (axidr = $true)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((pushprop_lem1v2 = $true) <=> ! [X278 : (term > $o),X276 : (term > $o),X277 : subst,X279 : term] : (($true != ((X278 @ X279))) | (((X276 @ one)) = $true) | ? [X280 : term] : (((X276 @ X280)) != ((X278 @ (sub @ X280 @ (push @ X279 @ X277))))))) & (($true = sP36) <=> (hoasinduction_lthm_3 = $true)) & ((hoasinduction_lem3aa_lthm = $true) <=> (sP30 = $true)) & (($true = sP29) <=> (hoasinduction_lem3aa = $true)) & (pushprop = $true) & (ulamvar1 != $true) & (! [X281 : subst,X282 : term,X283 : (term > $o)] : ? [X284 : (term > $o)] : ! [X285 : term] : (((X283 @ (sub @ X285 @ (push @ X282 @ X281)))) = ((X284 @ X285))) <=> (pushprop_lem0_lthm = $true)) & (! [X527 : term] : (((sub @ X527 @ id)) = X527) <=> (axvarid = $true)) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoaslamnotvar_lthm = $true) & (hoasinduction_lem3v2_gthm = $true) & ((pushprop_lem1_lthm = $true) <=> (? [X205 : subst,X206 : term] : (((comp @ sh @ (push @ X206 @ X205))) != X205) | ! [X207 : term,X210 : (term > $o),X209 : (term > $o),X211 : subst,X208 : term] : (($true != ((X209 @ X208))) | (((X210 @ (sub @ X208 @ (push @ X207 @ X211)))) = $true)) | ? [X204 : term,X203 : subst] : (((sub @ one @ (push @ X204 @ X203))) != X204))) & (pushprop_lthm_orig = $true) & (hoasapinj2_gthm = $true) & (ulamvarsh = $true) & (apnotvar = $true) & (lamnotvar = $true) & ((hoasapinj2 = $true) <=> ! [X50 : term,X49 : term,X47 : term,X48 : term] : ((((ap @ (sub @ X48 @ id) @ X47)) != ((ap @ (sub @ X49 @ id) @ X50))) | (X47 = X50))) & (induction2lem_gthm = $true) & (axapp = $true) & ((axshiftcons = $true) <=> ! [X52 : term,X51 : subst] : (((comp @ sh @ (push @ X52 @ X51))) = X51)) & ((hoasinduction_lthm = $true) <=> ($true = sP28)) & (($true = sP19) <=> (hoasinduction_lem3_lthm = $true)) & (substmonoid_gthm = $true) & (axvarshift = $true) & (! [X296 : term,X298 : term,X299 : term,X297 : term] : ((X297 = X298) | (((ap @ X298 @ X299)) != ((ap @ X297 @ X296)))) <=> (apinj1 = $true)) & (($true = sP16) <=> (hoasinduction_lem3a_lthm = $true)) & (($true = sP14) <=> (induction2_lthm = $true)) & (hoasinduction_lem3a_gthm = $true) & (pushprop_lem1_gthm = $true) & ! [X4 : subst] : ? [X5 : d_subst] : (((d2subst @ X5)) = X4) & ((laminj = $true) <=> ! [X448 : term,X449 : term] : ((X448 = X449) | (((lam @ X449)) != ((lam @ X448))))) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (($true = sP10) <=> (hoasinduction_lem3 = $true)) & (axabs = $true) & (hoaslamnotvar_gthm = $true) & (pushprop_lem2v2_gthm = $true) & (pushprop_lem3v2 = $true) & ((axscons = $true) <=> ! [X429 : subst] : (((push @ (sub @ one @ X429) @ (comp @ sh @ X429))) = X429)) & ((termmset_lthm = $true) <=> (? [X177 : term] : (((sub @ X177 @ id)) != X177) | ! [X178 : term] : (((sub @ X178 @ id)) = X178))) & (ulamvarind = $true) & ! [X244 : d_subst] : (d_id = X244) & ((hoasinduction_lem3v2_f = $true) <=> ! [X0 : term] : ? [X1 : (subst > term > term)] : ! [X2 : term,X3 : subst] : (((sub @ X0 @ (push @ X2 @ X3))) = ((X1 @ X3 @ X2)))) & (hoasinduction_lem2_lthm = $true) & (hoasinduction_lem3b_gthm = $true) & ((shinj = $true) <=> ! [X45 : term,X46 : term] : ((X45 = X46) | (((sub @ X46 @ sh)) != ((sub @ X45 @ sh))))) & (axmap = $true) & (pushprop_lem3v2_lthm = $true) & (hoasinduction_lem1 = $true) & (($true = sP8) <=> (hoasinduction_lem3aaa = $true)) & ((! [X505 : subst] : (((comp @ id @ X505)) = X505) & ! [X504 : subst] : (((comp @ X504 @ id)) = X504)) <=> (substmonoid = $true)) & (($true = sP7) <=> (induction2lem_lthm = $true)) & ((pushprop_lem1 = $true) <=> ! [X410 : (term > $o),X411 : (term > $o),X408 : subst,X412 : term,X409 : term] : (($true = ((X411 @ (sub @ X412 @ (push @ X409 @ X408))))) | ($true != ((X410 @ X412))))) & (((d2term @ d_one)) = one) & ((sP3 = $true) <=> (hoasinduction_lem3v2a = $true)) & (hoasinduction_gthm = $true) & ((hoasinduction_lem3a = $true) <=> ($true = sP2)) & ((hoasinduction_lem3v2 = $true) <=> ($true = sP1)) & ((? [X436 : subst] : (((comp @ id @ X436)) != X436) | ? [X437 : subst] : (((comp @ X437 @ id)) != X437) | (! [X438 : subst] : (((comp @ X438 @ id)) = X438) & ! [X439 : subst] : (((comp @ id @ X439)) = X439))) <=> (substmonoid_lthm = $true)) & (! [X434 : term] : ? [X435 : (subst > term > term)] : (((X435 @ sh @ one)) = ((sub @ X434 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b_lthm = $true))),
  inference(definition_folding,[],[f14,f73,f72,f71,f70,f69,f68,f67,f66,f65,f64,f63,f62,f61,f60,f59,f58,f57,f56,f55,f54,f53,f52,f51,f50,f49,f48,f47,f46,f45,f44,f43,f42,f41,f40,f39,f38,f37,f36,f35,f34,f33,f32,f31,f30,f29,f28,f27,f26,f25,f24,f23,f22,f21,f20,f19,f18,f17,f16,f15])).
thf(f75,definition,(
  ! [X0 : (subst > term > subst > $o)] : (($true = ((sP59 @ X0))) <=> ? [X10 : (subst > term > term)] : (! [X14 : term] : ((((X0 @ id @ X14 @ id)) != $true) | (((X0 @ id @ (X10 @ id @ X14) @ id)) = $true)) & ! [X11 : subst,X13 : subst,X12 : term] : (((X10 @ (comp @ X11 @ X13) @ (sub @ X12 @ X13))) = ((sub @ (X10 @ X11 @ X12) @ X13))) & ($true != ((X0 @ id @ (hoaslam @ id @ (^[Y0 : subst]: ((^[Y1 : term]: (X10 @ Y0 @ Y1))))) @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f76,definition,(
  ($true = sP60) <=> ! [X1 : (term > $o),X0 : (subst > term > subst > $o)] : (? [X7 : term,X6 : subst,X9 : subst,X8 : subst] : ((((X0 @ (comp @ X9 @ X6) @ (sub @ X7 @ X6) @ X8)) = $true) & (((X0 @ X9 @ X7 @ (comp @ X6 @ X8))) != $true)) | ! [X15 : term] : (? [X16 : term] : ((((X1 @ X16)) = $true) & ($true != ((X1 @ (sub @ X15 @ (push @ X16 @ id)))))) | ($true = ((X1 @ (lam @ X15))))) | (((hoasinduction_p_and_p_prime @ X0 @ X1)) != $true) | ($true = ((sP59 @ X0))) | ? [X5 : subst,X3 : subst,X4 : term,X2 : subst] : (($true = ((X0 @ X5 @ X4 @ (comp @ X3 @ X2)))) & ($true != ((X0 @ (comp @ X5 @ X3) @ (sub @ X4 @ X3) @ X2)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f77,plain,(
  (hoasinduction_lem3v2 = $true) <~> ($true = sP60)),
  inference(definition_folding,[],[f12,f76,f75])).
thf(f259,plain,(
  (($true = sP1) | ? [X54 : (subst > term > subst > $o),X55 : (term > $o),X53 : term] : (! [X70 : term] : (((X54 @ id @ X70 @ id)) = ((X55 @ X70))) & (((sP0 @ X54)) != $true) & ! [X69 : term] : ((((X55 @ (sub @ X53 @ (push @ X69 @ id)))) = $true) | (((X55 @ X69)) != $true)) & ($true != ((X55 @ (lam @ X53)))) & ! [X68 : subst,X65 : term,X66 : subst,X67 : subst] : (($true != ((X54 @ (comp @ X68 @ X67) @ (sub @ X65 @ X67) @ X66))) | (((X54 @ X68 @ X65 @ (comp @ X67 @ X66))) = $true)) & ! [X63 : subst,X62 : subst,X64 : term,X61 : subst] : ((((X54 @ (comp @ X61 @ X63) @ (sub @ X64 @ X63) @ X62)) = $true) | ($true != ((X54 @ X61 @ X64 @ (comp @ X63 @ X62))))))) & (! [X54 : (subst > term > subst > $o),X55 : (term > $o),X53 : term] : (? [X70 : term] : (((X54 @ id @ X70 @ id)) != ((X55 @ X70))) | (((sP0 @ X54)) = $true) | ? [X69 : term] : ((((X55 @ (sub @ X53 @ (push @ X69 @ id)))) != $true) & (((X55 @ X69)) = $true)) | ($true = ((X55 @ (lam @ X53)))) | ? [X68 : subst,X65 : term,X66 : subst,X67 : subst] : (($true = ((X54 @ (comp @ X68 @ X67) @ (sub @ X65 @ X67) @ X66))) & (((X54 @ X68 @ X65 @ (comp @ X67 @ X66))) != $true)) | ? [X63 : subst,X62 : subst,X64 : term,X61 : subst] : ((((X54 @ (comp @ X61 @ X63) @ (sub @ X64 @ X63) @ X62)) != $true) & ($true = ((X54 @ X61 @ X64 @ (comp @ X63 @ X62)))))) | ($true != sP1))),
  inference(nnf_transformation,[],[f16])).
thf(f260,plain,(
  (($true = sP1) | ? [X0 : (subst > term > subst > $o),X1 : (term > $o),X2 : term] : (! [X3 : term] : (((X0 @ id @ X3 @ id)) = ((X1 @ X3))) & (((sP0 @ X0)) != $true) & ! [X4 : term] : (($true = ((X1 @ (sub @ X2 @ (push @ X4 @ id))))) | ($true != ((X1 @ X4)))) & ($true != ((X1 @ (lam @ X2)))) & ! [X5 : subst,X6 : term,X7 : subst,X8 : subst] : ((((X0 @ (comp @ X5 @ X8) @ (sub @ X6 @ X8) @ X7)) != $true) | ($true = ((X0 @ X5 @ X6 @ (comp @ X8 @ X7))))) & ! [X9 : subst,X10 : subst,X11 : term,X12 : subst] : ((((X0 @ (comp @ X12 @ X9) @ (sub @ X11 @ X9) @ X10)) = $true) | ($true != ((X0 @ X12 @ X11 @ (comp @ X9 @ X10))))))) & (! [X13 : (subst > term > subst > $o),X14 : (term > $o),X15 : term] : (? [X16 : term] : (((X14 @ X16)) != ((X13 @ id @ X16 @ id))) | ($true = ((sP0 @ X13))) | ? [X17 : term] : (($true != ((X14 @ (sub @ X15 @ (push @ X17 @ id))))) & (((X14 @ X17)) = $true)) | ($true = ((X14 @ (lam @ X15)))) | ? [X18 : subst,X19 : term,X20 : subst,X21 : subst] : (($true = ((X13 @ (comp @ X18 @ X21) @ (sub @ X19 @ X21) @ X20))) & ($true != ((X13 @ X18 @ X19 @ (comp @ X21 @ X20))))) | ? [X22 : subst,X23 : subst,X24 : term,X25 : subst] : ((((X13 @ (comp @ X25 @ X22) @ (sub @ X24 @ X22) @ X23)) != $true) & ($true = ((X13 @ X25 @ X24 @ (comp @ X22 @ X23)))))) | ($true != sP1))),
  inference(rectify,[],[f259])).
thf(f261,plain,(
  (($true = sP1) | (! [X3 : term] : (((sK442 @ X3)) = ((sK441 @ id @ X3 @ id))) & ($true != ((sP0 @ sK441))) & ! [X4 : term] : (($true = ((sK442 @ (sub @ sK443 @ (push @ X4 @ id))))) | ($true != ((sK442 @ X4)))) & ($true != ((sK442 @ (lam @ sK443)))) & ! [X5 : subst,X6 : term,X7 : subst,X8 : subst] : (($true != ((sK441 @ (comp @ X5 @ X8) @ (sub @ X6 @ X8) @ X7))) | ($true = ((sK441 @ X5 @ X6 @ (comp @ X8 @ X7))))) & ! [X9 : subst,X10 : subst,X11 : term,X12 : subst] : (($true = ((sK441 @ (comp @ X12 @ X9) @ (sub @ X11 @ X9) @ X10))) | (((sK441 @ X12 @ X11 @ (comp @ X9 @ X10))) != $true)))) & (! [X13 : (subst > term > subst > $o),X14 : (term > $o),X15 : term] : ((((X14 @ (sK444 @ X14 @ X13))) != ((X13 @ id @ (sK444 @ X14 @ X13) @ id))) | ($true = ((sP0 @ X13))) | (($true != ((X14 @ (sub @ X15 @ (push @ (sK445 @ X15 @ X14) @ id))))) & ($true = ((X14 @ (sK445 @ X15 @ X14))))) | ($true = ((X14 @ (lam @ X15)))) | (($true = ((X13 @ (comp @ (sK446 @ X13) @ (sK449 @ X13)) @ (sub @ (sK447 @ X13) @ (sK449 @ X13)) @ (sK448 @ X13)))) & (((X13 @ (sK446 @ X13) @ (sK447 @ X13) @ (comp @ (sK449 @ X13) @ (sK448 @ X13)))) != $true)) | (($true != ((X13 @ (comp @ (sK453 @ X13) @ (sK450 @ X13)) @ (sub @ (sK452 @ X13) @ (sK450 @ X13)) @ (sK451 @ X13)))) & (((X13 @ (sK453 @ X13) @ (sK452 @ X13) @ (comp @ (sK450 @ X13) @ (sK451 @ X13)))) = $true))) | ($true != sP1))),
  inference(skolemize,[status(esa),new_symbols(skolem,[sK441,sK442,sK443,vAPP]),skolemize(X0,$thf(sK441)),skolemize(X1,$thf(sK442)),skolemize(X2,$thf(sK443)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19))],[f260])).
thf(f262,plain,(
  ! [X54 : (subst > term > subst > $o)] : (((((sP0 @ X54)) = $true) | ! [X56 : (subst > term > term)] : (? [X57 : term] : (($true != ((X54 @ id @ (X56 @ id @ X57) @ id))) & ($true = ((X54 @ id @ X57 @ id)))) | ? [X59 : subst,X60 : term,X58 : subst] : (((X56 @ (comp @ X59 @ X58) @ (sub @ X60 @ X58))) != ((sub @ (X56 @ X59 @ X60) @ X58))) | ($true = ((X54 @ id @ (lam @ (X56 @ sh @ one)) @ id))))) & (? [X56 : (subst > term > term)] : (! [X57 : term] : (($true = ((X54 @ id @ (X56 @ id @ X57) @ id))) | ($true != ((X54 @ id @ X57 @ id)))) & ! [X59 : subst,X60 : term,X58 : subst] : (((X56 @ (comp @ X59 @ X58) @ (sub @ X60 @ X58))) = ((sub @ (X56 @ X59 @ X60) @ X58))) & ($true != ((X54 @ id @ (lam @ (X56 @ sh @ one)) @ id)))) | (((sP0 @ X54)) != $true)))),
  inference(nnf_transformation,[],[f15])).
thf(f263,plain,(
  ! [X0 : (subst > term > subst > $o)] : (((((sP0 @ X0)) = $true) | ! [X1 : (subst > term > term)] : (? [X2 : term] : (($true != ((X0 @ id @ (X1 @ id @ X2) @ id))) & ($true = ((X0 @ id @ X2 @ id)))) | ? [X3 : subst,X4 : term,X5 : subst] : (((sub @ (X1 @ X3 @ X4) @ X5)) != ((X1 @ (comp @ X3 @ X5) @ (sub @ X4 @ X5)))) | ($true = ((X0 @ id @ (lam @ (X1 @ sh @ one)) @ id))))) & (? [X6 : (subst > term > term)] : (! [X7 : term] : (($true = ((X0 @ id @ (X6 @ id @ X7) @ id))) | ($true != ((X0 @ id @ X7 @ id)))) & ! [X8 : subst,X9 : term,X10 : subst] : (((sub @ (X6 @ X8 @ X9) @ X10)) = ((X6 @ (comp @ X8 @ X10) @ (sub @ X9 @ X10)))) & (((X0 @ id @ (lam @ (X6 @ sh @ one)) @ id)) != $true)) | (((sP0 @ X0)) != $true)))),
  inference(rectify,[],[f262])).
thf(f264,plain,(
  ! [X0 : (subst > term > subst > $o)] : (((((sP0 @ X0)) = $true) | ! [X1 : (subst > term > term)] : ((($true != ((X0 @ id @ (X1 @ id @ (sK454 @ X1 @ X0)) @ id))) & ($true = ((X0 @ id @ (sK454 @ X1 @ X0) @ id)))) | (((X1 @ (comp @ (sK455 @ X1) @ (sK457 @ X1)) @ (sub @ (sK456 @ X1) @ (sK457 @ X1)))) != ((sub @ (X1 @ (sK455 @ X1) @ (sK456 @ X1)) @ (sK457 @ X1)))) | ($true = ((X0 @ id @ (lam @ (X1 @ sh @ one)) @ id))))) & ((! [X7 : term] : (($true = ((X0 @ id @ (sK458 @ X0 @ id @ X7) @ id))) | ($true != ((X0 @ id @ X7 @ id)))) & ! [X8 : subst,X9 : term,X10 : subst] : (((sK458 @ X0 @ (comp @ X8 @ X10) @ (sub @ X9 @ X10))) = ((sub @ (sK458 @ X0 @ X8 @ X9) @ X10))) & ($true != ((X0 @ id @ (lam @ (sK458 @ X0 @ sh @ one)) @ id)))) | (((sP0 @ X0)) != $true)))),
  inference(skolemize,[status(esa),new_symbols(skolem,[vAPP]),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19))],[f263])).
thf(f265,plain,(
  (hoaslamnotap_lthm = $true) & (lamnotap != $true) & (pushprop_lem0_gthm = $true) & (((hoaslaminj_lthm = $true) | (sP58 != $true)) & ((sP58 = $true) | (hoaslaminj_lthm != $true))) & ((($true = sP57) | (induction2 != $true)) & ((induction2 = $true) | ($true != sP57))) & (hoasinduction_lem2_gthm = $true) & ((! [X295 : subst,X294 : term] : (((sub @ one @ (push @ X294 @ X295))) = X294) | (axvarcons != $true)) & ((axvarcons = $true) | ? [X295 : subst,X294 : term] : (((sub @ one @ (push @ X294 @ X295))) != X294))) & ! [X242 : term] : ? [X243 : d_term] : (((d2term @ X243)) = X242) & (hoasinduction_lem1_lthm = $true) & (((induction2lem = $true) | ($true != sP55)) & (($true = sP55) | (induction2lem != $true))) & (pushprop_gthm = $true) & (hoasinduction_lem1_gthm = $true) & (((pushprop_lem0 = $true) | ? [X301 : subst,X302 : (term > $o),X300 : term] : ! [X303 : (term > $o)] : ? [X304 : term] : (((X302 @ (sub @ X304 @ (push @ X300 @ X301)))) != ((X303 @ X304)))) & (! [X301 : subst,X302 : (term > $o),X300 : term] : ? [X303 : (term > $o)] : ! [X304 : term] : (((X302 @ (sub @ X304 @ (push @ X300 @ X301)))) = ((X303 @ X304))) | (pushprop_lem0 != $true))) & ((! [X90 : term] : ? [X91 : (subst > term > term)] : ! [X92 : term,X93 : subst] : (((X91 @ X93 @ X92)) = ((sub @ X90 @ (push @ X92 @ X93)))) | (hoasinduction_lem3v2_f_lthm != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X90 : term] : ! [X91 : (subst > term > term)] : ? [X92 : term,X93 : subst] : (((X91 @ X93 @ X92)) != ((sub @ X90 @ (push @ X92 @ X93)))))) & (pushprop_lem1v2_gthm = $true) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem1v2_gthm = $true) & (((d2subst @ d_id)) = sh) & ((! [X187 : (subst > term > subst > $o)] : ? [X188 : (term > $o)] : ! [X189 : term] : (((X187 @ id @ X189 @ id)) = ((X188 @ X189))) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem0 = $true) | ? [X187 : (subst > term > subst > $o)] : ! [X188 : (term > $o)] : ? [X189 : term] : (((X187 @ id @ X189 @ id)) != ((X188 @ X189))))) & (((d2subst @ d_id)) = id) & ((! [X198 : term,X197 : (term > $o)] : ((((sP53 @ X197)) = $true) | ? [X200 : term] : (($true = ((var @ X200))) & ($true != ((X197 @ X200)))) | ($true = ((X197 @ X198))) | ? [X199 : term] : (($true = ((X197 @ X199))) & ($true != ((X197 @ (lam @ X199)))))) | (induction != $true)) & ((induction = $true) | ? [X198 : term,X197 : (term > $o)] : ((((sP53 @ X197)) != $true) & ! [X200 : term] : (($true != ((var @ X200))) | ($true = ((X197 @ X200)))) & ($true != ((X197 @ X198))) & ! [X199 : term] : (($true != ((X197 @ X199))) | ($true = ((X197 @ (lam @ X199)))))))) & (((var @ (d2term @ d_one))) != $true) & (((hoaslamnotap = $true) | ? [X262 : term,X260 : term,X261 : (subst > term > term)] : ((((ap @ (sub @ X262 @ id) @ X260)) = ((lam @ (X261 @ sh @ one)))) & ! [X264 : subst,X265 : subst,X263 : term] : (((sub @ (X261 @ X264 @ X263) @ X265)) = ((X261 @ (comp @ X264 @ X265) @ (sub @ X263 @ X265)))))) & (! [X262 : term,X260 : term,X261 : (subst > term > term)] : ((((ap @ (sub @ X262 @ id) @ X260)) != ((lam @ (X261 @ sh @ one)))) | ? [X264 : subst,X265 : subst,X263 : term] : (((sub @ (X261 @ X264 @ X263) @ X265)) != ((X261 @ (comp @ X264 @ X265) @ (sub @ X263 @ X265))))) | (hoaslamnotap != $true))) & (hoasinduction_lem3_gthm = $true) & (pushprop_lem2v2 = $true) & (((apinj2 = $true) | ? [X529 : term,X528 : term,X531 : term,X530 : term] : ((X529 != X531) & (((ap @ X528 @ X529)) = ((ap @ X530 @ X531))))) & (! [X529 : term,X528 : term,X531 : term,X530 : term] : ((X529 = X531) | (((ap @ X528 @ X529)) != ((ap @ X530 @ X531)))) | (apinj2 != $true))) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & (((hoasinduction_no_psi_cond_lthm = $true) | ($true != sP52)) & (($true = sP52) | (hoasinduction_no_psi_cond_lthm != $true))) & (hoaslamnotap_gthm = $true) & (((termmset = $true) | ? [X110 : term] : (((sub @ X110 @ id)) != X110)) & (! [X110 : term] : (((sub @ X110 @ id)) = X110) | (termmset != $true))) & (((sP47 = $true) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | (sP47 != $true))) & ((! [X246 : term,X247 : term,X245 : term,X248 : term] : ((X247 = X248) | (((ap @ (sub @ X247 @ id) @ X246)) != ((ap @ (sub @ X248 @ id) @ X245)))) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | ? [X246 : term,X247 : term,X245 : term,X248 : term] : ((X247 != X248) & (((ap @ (sub @ X247 @ id) @ X246)) = ((ap @ (sub @ X248 @ id) @ X245)))))) & (hoasapnotvar = $true) & ((! [X251 : term,X252 : subst,X250 : (subst > term > term),X253 : (subst > term > term)] : (? [X254 : term,X256 : subst,X255 : subst] : (((sub @ (X253 @ X255 @ X254) @ X256)) != ((X253 @ (comp @ X255 @ X256) @ (sub @ X254 @ X256)))) | ? [X259 : subst,X257 : subst,X258 : term] : (((sub @ (X250 @ X259 @ X258) @ X257)) != ((X250 @ (comp @ X259 @ X257) @ (sub @ X258 @ X257)))) | (((X253 @ X252 @ X251)) = ((X250 @ X252 @ X251))) | (((lam @ (X250 @ sh @ one))) != ((lam @ (X253 @ sh @ one))))) | (hoaslaminj != $true)) & ((hoaslaminj = $true) | ? [X251 : term,X252 : subst,X250 : (subst > term > term),X253 : (subst > term > term)] : (! [X254 : term,X256 : subst,X255 : subst] : (((sub @ (X253 @ X255 @ X254) @ X256)) = ((X253 @ (comp @ X255 @ X256) @ (sub @ X254 @ X256)))) & ! [X259 : subst,X257 : subst,X258 : term] : (((sub @ (X250 @ X259 @ X258) @ X257)) = ((X250 @ (comp @ X259 @ X257) @ (sub @ X258 @ X257)))) & (((X253 @ X252 @ X251)) != ((X250 @ X252 @ X251))) & (((lam @ (X250 @ sh @ one))) = ((lam @ (X253 @ sh @ one))))))) & (hoasapnotvar_lthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (((hoasinduction_no_psi_cond = $true) | ? [X143 : (subst > term > subst > $o),X142 : term] : (($true != ((X143 @ id @ X142 @ id))) & ! [X150 : term,X149 : term] : ((((X143 @ id @ X149 @ id)) != $true) | ($true != ((X143 @ id @ X150 @ id))) | ($true = ((X143 @ id @ (ap @ (sub @ X149 @ id) @ X150) @ id)))) & ($true != ((sP45 @ X143))))) & (! [X143 : (subst > term > subst > $o),X142 : term] : (($true = ((X143 @ id @ X142 @ id))) | ? [X150 : term,X149 : term] : ((((X143 @ id @ X149 @ id)) = $true) & ($true = ((X143 @ id @ X150 @ id))) & ($true != ((X143 @ id @ (ap @ (sub @ X149 @ id) @ X150) @ id)))) | ($true = ((sP45 @ X143)))) | (hoasinduction_no_psi_cond != $true))) & (pushprop_lem2v2_lthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & ((! [X524 : (subst > term > subst > $o)] : ? [X525 : (term > $o)] : ! [X526 : term] : (((X524 @ id @ X526 @ id)) = ((X525 @ X526))) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ? [X524 : (subst > term > subst > $o)] : ! [X525 : (term > $o)] : ? [X526 : term] : (((X524 @ id @ X526 @ id)) != ((X525 @ X526))))) & (termmset_gthm = $true) & (((pushprop_lem1v2_lthm = $true) | (! [X151 : term,X152 : subst] : (((sub @ one @ (push @ X151 @ X152))) = X151) & ? [X154 : (term > $o),X156 : (term > $o),X153 : term,X155 : subst] : ((((X154 @ one)) != $true) & ! [X157 : term] : (((X156 @ (sub @ X157 @ (push @ X153 @ X155)))) = ((X154 @ X157))) & ($true = ((X156 @ X153)))))) & ((? [X151 : term,X152 : subst] : (((sub @ one @ (push @ X151 @ X152))) != X151) | ! [X154 : (term > $o),X156 : (term > $o),X153 : term,X155 : subst] : ((((X154 @ one)) = $true) | ? [X157 : term] : (((X156 @ (sub @ X157 @ (push @ X153 @ X155)))) != ((X154 @ X157))) | ($true != ((X156 @ X153))))) | (pushprop_lem1v2_lthm != $true))) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (((hoasapinj1_lthm = $true) | (! [X359 : term] : (((sub @ X359 @ id)) = X359) & ? [X364 : term,X366 : term,X365 : term,X367 : term] : ((X365 != X367) & (((ap @ (sub @ X367 @ id) @ X364)) = ((ap @ (sub @ X365 @ id) @ X366)))) & ! [X360 : term,X361 : term,X362 : term,X363 : term] : ((((ap @ X361 @ X362)) != ((ap @ X363 @ X360))) | (X361 = X363)))) & ((? [X359 : term] : (((sub @ X359 @ id)) != X359) | ! [X364 : term,X366 : term,X365 : term,X367 : term] : ((X365 = X367) | (((ap @ (sub @ X367 @ id) @ X364)) != ((ap @ (sub @ X365 @ id) @ X366)))) | ? [X360 : term,X361 : term,X362 : term,X363 : term] : ((((ap @ X361 @ X362)) = ((ap @ X363 @ X360))) & (X361 != X363))) | (hoasapinj1_lthm != $true))) & (hoaslamnotvar = $true) & (axclos = $true) & ((! [X141 : subst] : (((comp @ id @ X141)) = X141) | (axidl != $true)) & ((axidl = $true) | ? [X141 : subst] : (((comp @ id @ X141)) != X141))) & ! [X433 : d_term,X432 : d_term] : ((((d2term @ X432)) != ((d2term @ X433))) | (X432 = X433)) & ((($true = sP44) | (hoasinduction_lem3v2_lthm != $true)) & ((hoasinduction_lem3v2_lthm = $true) | ($true != sP44))) & ((! [X430 : term] : ? [X431 : (subst > term > term)] : (((X431 @ sh @ one)) = ((sub @ X430 @ (push @ one @ sh)))) | (hoasinduction_lem3b != $true)) & ((hoasinduction_lem3b = $true) | ? [X430 : term] : ! [X431 : (subst > term > term)] : (((X431 @ sh @ one)) != ((sub @ X430 @ (push @ one @ sh)))))) & (((hoasinduction = $true) | (sP42 != $true)) & ((sP42 = $true) | (hoasinduction != $true))) & (hoaslaminj_gthm = $true) & ! [X249 : d_term] : (d_one = X249) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ! [X240 : d_subst,X241 : d_subst] : ((((d2subst @ X241)) != ((d2subst @ X240))) | (X240 = X241)) & (induction2_gthm = $true) & (((hoasapinj2_lthm = $true) | (? [X291 : term,X290 : term,X293 : term,X292 : term] : ((((ap @ (sub @ X292 @ id) @ X291)) = ((ap @ (sub @ X290 @ id) @ X293))) & (X291 != X293)) & ! [X289 : term,X288 : term,X287 : term,X286 : term] : ((((ap @ X286 @ X287)) != ((ap @ X289 @ X288))) | (X287 = X288)))) & ((! [X291 : term,X290 : term,X293 : term,X292 : term] : ((((ap @ (sub @ X292 @ id) @ X291)) != ((ap @ (sub @ X290 @ id) @ X293))) | (X291 = X293)) | ? [X289 : term,X288 : term,X287 : term,X286 : term] : ((((ap @ X286 @ X287)) = ((ap @ X289 @ X288))) & (X287 != X288))) | (hoasapinj2_lthm != $true))) & (hoasinduction_lem1v2 = $true) & (((sP39 = $true) | (hoasinduction_lem2v2 != $true)) & ((hoasinduction_lem2v2 = $true) | (sP39 != $true))) & (axassoc = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (pushprop_lthm = $true) & (((hoasinduction_lem3v2a_lthm = $true) | ($true != sP37)) & (($true = sP37) | (hoasinduction_lem3v2a_lthm != $true))) & (hoasapinj1_gthm = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (hoasinduction_lem2v2_gthm = $true) & ((! [X140 : subst] : (((comp @ X140 @ id)) = X140) | (axidr != $true)) & ((axidr = $true) | ? [X140 : subst] : (((comp @ X140 @ id)) != X140))) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (((pushprop_lem1v2 = $true) | ? [X278 : (term > $o),X276 : (term > $o),X277 : subst,X279 : term] : (($true = ((X278 @ X279))) & (((X276 @ one)) != $true) & ! [X280 : term] : (((X276 @ X280)) = ((X278 @ (sub @ X280 @ (push @ X279 @ X277))))))) & (! [X278 : (term > $o),X276 : (term > $o),X277 : subst,X279 : term] : (($true != ((X278 @ X279))) | (((X276 @ one)) = $true) | ? [X280 : term] : (((X276 @ X280)) != ((X278 @ (sub @ X280 @ (push @ X279 @ X277)))))) | (pushprop_lem1v2 != $true))) & ((($true = sP36) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | ($true != sP36))) & (((hoasinduction_lem3aa_lthm = $true) | (sP30 != $true)) & ((sP30 = $true) | (hoasinduction_lem3aa_lthm != $true))) & ((($true = sP29) | (hoasinduction_lem3aa != $true)) & ((hoasinduction_lem3aa = $true) | ($true != sP29))) & (pushprop = $true) & (ulamvar1 != $true) & ((! [X281 : subst,X282 : term,X283 : (term > $o)] : ? [X284 : (term > $o)] : ! [X285 : term] : (((X283 @ (sub @ X285 @ (push @ X282 @ X281)))) = ((X284 @ X285))) | (pushprop_lem0_lthm != $true)) & ((pushprop_lem0_lthm = $true) | ? [X281 : subst,X282 : term,X283 : (term > $o)] : ! [X284 : (term > $o)] : ? [X285 : term] : (((X283 @ (sub @ X285 @ (push @ X282 @ X281)))) != ((X284 @ X285))))) & ((! [X527 : term] : (((sub @ X527 @ id)) = X527) | (axvarid != $true)) & ((axvarid = $true) | ? [X527 : term] : (((sub @ X527 @ id)) != X527))) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoaslamnotvar_lthm = $true) & (hoasinduction_lem3v2_gthm = $true) & (((pushprop_lem1_lthm = $true) | (! [X205 : subst,X206 : term] : (((comp @ sh @ (push @ X206 @ X205))) = X205) & ? [X207 : term,X210 : (term > $o),X209 : (term > $o),X211 : subst,X208 : term] : (($true = ((X209 @ X208))) & (((X210 @ (sub @ X208 @ (push @ X207 @ X211)))) != $true)) & ! [X204 : term,X203 : subst] : (((sub @ one @ (push @ X204 @ X203))) = X204))) & ((? [X205 : subst,X206 : term] : (((comp @ sh @ (push @ X206 @ X205))) != X205) | ! [X207 : term,X210 : (term > $o),X209 : (term > $o),X211 : subst,X208 : term] : (($true != ((X209 @ X208))) | (((X210 @ (sub @ X208 @ (push @ X207 @ X211)))) = $true)) | ? [X204 : term,X203 : subst] : (((sub @ one @ (push @ X204 @ X203))) != X204)) | (pushprop_lem1_lthm != $true))) & (pushprop_lthm_orig = $true) & (hoasapinj2_gthm = $true) & (ulamvarsh = $true) & (apnotvar = $true) & (lamnotvar = $true) & (((hoasapinj2 = $true) | ? [X50 : term,X49 : term,X47 : term,X48 : term] : ((((ap @ (sub @ X48 @ id) @ X47)) = ((ap @ (sub @ X49 @ id) @ X50))) & (X47 != X50))) & (! [X50 : term,X49 : term,X47 : term,X48 : term] : ((((ap @ (sub @ X48 @ id) @ X47)) != ((ap @ (sub @ X49 @ id) @ X50))) | (X47 = X50)) | (hoasapinj2 != $true))) & (induction2lem_gthm = $true) & (axapp = $true) & (((axshiftcons = $true) | ? [X52 : term,X51 : subst] : (((comp @ sh @ (push @ X52 @ X51))) != X51)) & (! [X52 : term,X51 : subst] : (((comp @ sh @ (push @ X52 @ X51))) = X51) | (axshiftcons != $true))) & (((hoasinduction_lthm = $true) | ($true != sP28)) & (($true = sP28) | (hoasinduction_lthm != $true))) & ((($true = sP19) | (hoasinduction_lem3_lthm != $true)) & ((hoasinduction_lem3_lthm = $true) | ($true != sP19))) & (substmonoid_gthm = $true) & (axvarshift = $true) & ((! [X296 : term,X298 : term,X299 : term,X297 : term] : ((X297 = X298) | (((ap @ X298 @ X299)) != ((ap @ X297 @ X296)))) | (apinj1 != $true)) & ((apinj1 = $true) | ? [X296 : term,X298 : term,X299 : term,X297 : term] : ((X297 != X298) & (((ap @ X298 @ X299)) = ((ap @ X297 @ X296)))))) & ((($true = sP16) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | ($true != sP16))) & ((($true = sP14) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | ($true != sP14))) & (hoasinduction_lem3a_gthm = $true) & (pushprop_lem1_gthm = $true) & ! [X4 : subst] : ? [X5 : d_subst] : (((d2subst @ X5)) = X4) & (((laminj = $true) | ? [X448 : term,X449 : term] : ((X448 != X449) & (((lam @ X449)) = ((lam @ X448))))) & (! [X448 : term,X449 : term] : ((X448 = X449) | (((lam @ X449)) != ((lam @ X448)))) | (laminj != $true))) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((($true = sP10) | (hoasinduction_lem3 != $true)) & ((hoasinduction_lem3 = $true) | ($true != sP10))) & (axabs = $true) & (hoaslamnotvar_gthm = $true) & (pushprop_lem2v2_gthm = $true) & (pushprop_lem3v2 = $true) & (((axscons = $true) | ? [X429 : subst] : (((push @ (sub @ one @ X429) @ (comp @ sh @ X429))) != X429)) & (! [X429 : subst] : (((push @ (sub @ one @ X429) @ (comp @ sh @ X429))) = X429) | (axscons != $true))) & (((termmset_lthm = $true) | (! [X177 : term] : (((sub @ X177 @ id)) = X177) & ? [X178 : term] : (((sub @ X178 @ id)) != X178))) & ((? [X177 : term] : (((sub @ X177 @ id)) != X177) | ! [X178 : term] : (((sub @ X178 @ id)) = X178)) | (termmset_lthm != $true))) & (ulamvarind = $true) & ! [X244 : d_subst] : (d_id = X244) & (((hoasinduction_lem3v2_f = $true) | ? [X0 : term] : ! [X1 : (subst > term > term)] : ? [X2 : term,X3 : subst] : (((sub @ X0 @ (push @ X2 @ X3))) != ((X1 @ X3 @ X2)))) & (! [X0 : term] : ? [X1 : (subst > term > term)] : ! [X2 : term,X3 : subst] : (((sub @ X0 @ (push @ X2 @ X3))) = ((X1 @ X3 @ X2))) | (hoasinduction_lem3v2_f != $true))) & (hoasinduction_lem2_lthm = $true) & (hoasinduction_lem3b_gthm = $true) & (((shinj = $true) | ? [X45 : term,X46 : term] : ((X45 != X46) & (((sub @ X46 @ sh)) = ((sub @ X45 @ sh))))) & (! [X45 : term,X46 : term] : ((X45 = X46) | (((sub @ X46 @ sh)) != ((sub @ X45 @ sh)))) | (shinj != $true))) & (axmap = $true) & (pushprop_lem3v2_lthm = $true) & (hoasinduction_lem1 = $true) & ((($true = sP8) | (hoasinduction_lem3aaa != $true)) & ((hoasinduction_lem3aaa = $true) | ($true != sP8))) & (((! [X505 : subst] : (((comp @ id @ X505)) = X505) & ! [X504 : subst] : (((comp @ X504 @ id)) = X504)) | (substmonoid != $true)) & ((substmonoid = $true) | (? [X505 : subst] : (((comp @ id @ X505)) != X505) | ? [X504 : subst] : (((comp @ X504 @ id)) != X504)))) & ((($true = sP7) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | ($true != sP7))) & (((pushprop_lem1 = $true) | ? [X410 : (term > $o),X411 : (term > $o),X408 : subst,X412 : term,X409 : term] : (($true != ((X411 @ (sub @ X412 @ (push @ X409 @ X408))))) & ($true = ((X410 @ X412))))) & (! [X410 : (term > $o),X411 : (term > $o),X408 : subst,X412 : term,X409 : term] : (($true = ((X411 @ (sub @ X412 @ (push @ X409 @ X408))))) | ($true != ((X410 @ X412)))) | (pushprop_lem1 != $true))) & (((d2term @ d_one)) = one) & (((sP3 = $true) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | (sP3 != $true))) & (hoasinduction_gthm = $true) & (((hoasinduction_lem3a = $true) | ($true != sP2)) & (($true = sP2) | (hoasinduction_lem3a != $true))) & (((hoasinduction_lem3v2 = $true) | ($true != sP1)) & (($true = sP1) | (hoasinduction_lem3v2 != $true))) & (((? [X436 : subst] : (((comp @ id @ X436)) != X436) | ? [X437 : subst] : (((comp @ X437 @ id)) != X437) | (! [X438 : subst] : (((comp @ X438 @ id)) = X438) & ! [X439 : subst] : (((comp @ id @ X439)) = X439))) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | (! [X436 : subst] : (((comp @ id @ X436)) = X436) & ! [X437 : subst] : (((comp @ X437 @ id)) = X437) & (? [X438 : subst] : (((comp @ X438 @ id)) != X438) | ? [X439 : subst] : (((comp @ id @ X439)) != X439))))) & ((! [X434 : term] : ? [X435 : (subst > term > term)] : (((X435 @ sh @ one)) = ((sub @ X434 @ (push @ one @ sh)))) | (hoasinduction_lem3b_lthm != $true)) & ((hoasinduction_lem3b_lthm = $true) | ? [X434 : term] : ! [X435 : (subst > term > term)] : (((X435 @ sh @ one)) != ((sub @ X434 @ (push @ one @ sh))))))),
  inference(nnf_transformation,[],[f74])).
thf(f266,plain,(
  (hoaslamnotap_lthm = $true) & (lamnotap != $true) & (pushprop_lem0_gthm = $true) & ((hoaslaminj_lthm = $true) | (sP58 != $true)) & ((sP58 = $true) | (hoaslaminj_lthm != $true)) & (($true = sP57) | (induction2 != $true)) & ((induction2 = $true) | ($true != sP57)) & (hoasinduction_lem2_gthm = $true) & (! [X295 : subst,X294 : term] : (((sub @ one @ (push @ X294 @ X295))) = X294) | (axvarcons != $true)) & ((axvarcons = $true) | ? [X295 : subst,X294 : term] : (((sub @ one @ (push @ X294 @ X295))) != X294)) & ! [X242 : term] : ? [X243 : d_term] : (((d2term @ X243)) = X242) & (hoasinduction_lem1_lthm = $true) & ((induction2lem = $true) | ($true != sP55)) & (($true = sP55) | (induction2lem != $true)) & (pushprop_gthm = $true) & (hoasinduction_lem1_gthm = $true) & ((pushprop_lem0 = $true) | ? [X301 : subst,X302 : (term > $o),X300 : term] : ! [X303 : (term > $o)] : ? [X304 : term] : (((X302 @ (sub @ X304 @ (push @ X300 @ X301)))) != ((X303 @ X304)))) & (! [X301 : subst,X302 : (term > $o),X300 : term] : ? [X303 : (term > $o)] : ! [X304 : term] : (((X302 @ (sub @ X304 @ (push @ X300 @ X301)))) = ((X303 @ X304))) | (pushprop_lem0 != $true)) & (! [X90 : term] : ? [X91 : (subst > term > term)] : ! [X92 : term,X93 : subst] : (((X91 @ X93 @ X92)) = ((sub @ X90 @ (push @ X92 @ X93)))) | (hoasinduction_lem3v2_f_lthm != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X90 : term] : ! [X91 : (subst > term > term)] : ? [X92 : term,X93 : subst] : (((X91 @ X93 @ X92)) != ((sub @ X90 @ (push @ X92 @ X93))))) & (pushprop_lem1v2_gthm = $true) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem1v2_gthm = $true) & (((d2subst @ d_id)) = sh) & (! [X187 : (subst > term > subst > $o)] : ? [X188 : (term > $o)] : ! [X189 : term] : (((X187 @ id @ X189 @ id)) = ((X188 @ X189))) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem0 = $true) | ? [X187 : (subst > term > subst > $o)] : ! [X188 : (term > $o)] : ? [X189 : term] : (((X187 @ id @ X189 @ id)) != ((X188 @ X189)))) & (((d2subst @ d_id)) = id) & (! [X198 : term,X197 : (term > $o)] : ((((sP53 @ X197)) = $true) | ? [X200 : term] : (($true = ((var @ X200))) & ($true != ((X197 @ X200)))) | ($true = ((X197 @ X198))) | ? [X199 : term] : (($true = ((X197 @ X199))) & ($true != ((X197 @ (lam @ X199)))))) | (induction != $true)) & ((induction = $true) | ? [X198 : term,X197 : (term > $o)] : ((((sP53 @ X197)) != $true) & ! [X200 : term] : (($true != ((var @ X200))) | ($true = ((X197 @ X200)))) & ($true != ((X197 @ X198))) & ! [X199 : term] : (($true != ((X197 @ X199))) | ($true = ((X197 @ (lam @ X199))))))) & (((var @ (d2term @ d_one))) != $true) & ((hoaslamnotap = $true) | ? [X262 : term,X260 : term,X261 : (subst > term > term)] : ((((ap @ (sub @ X262 @ id) @ X260)) = ((lam @ (X261 @ sh @ one)))) & ! [X264 : subst,X265 : subst,X263 : term] : (((sub @ (X261 @ X264 @ X263) @ X265)) = ((X261 @ (comp @ X264 @ X265) @ (sub @ X263 @ X265)))))) & (! [X262 : term,X260 : term,X261 : (subst > term > term)] : ((((ap @ (sub @ X262 @ id) @ X260)) != ((lam @ (X261 @ sh @ one)))) | ? [X264 : subst,X265 : subst,X263 : term] : (((sub @ (X261 @ X264 @ X263) @ X265)) != ((X261 @ (comp @ X264 @ X265) @ (sub @ X263 @ X265))))) | (hoaslamnotap != $true)) & (hoasinduction_lem3_gthm = $true) & (pushprop_lem2v2 = $true) & ((apinj2 = $true) | ? [X529 : term,X528 : term,X531 : term,X530 : term] : ((X529 != X531) & (((ap @ X528 @ X529)) = ((ap @ X530 @ X531))))) & (! [X529 : term,X528 : term,X531 : term,X530 : term] : ((X529 = X531) | (((ap @ X528 @ X529)) != ((ap @ X530 @ X531)))) | (apinj2 != $true)) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & ((hoasinduction_no_psi_cond_lthm = $true) | ($true != sP52)) & (($true = sP52) | (hoasinduction_no_psi_cond_lthm != $true)) & (hoaslamnotap_gthm = $true) & ((termmset = $true) | ? [X110 : term] : (((sub @ X110 @ id)) != X110)) & (! [X110 : term] : (((sub @ X110 @ id)) = X110) | (termmset != $true)) & ((sP47 = $true) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | (sP47 != $true)) & (! [X246 : term,X247 : term,X245 : term,X248 : term] : ((X247 = X248) | (((ap @ (sub @ X247 @ id) @ X246)) != ((ap @ (sub @ X248 @ id) @ X245)))) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | ? [X246 : term,X247 : term,X245 : term,X248 : term] : ((X247 != X248) & (((ap @ (sub @ X247 @ id) @ X246)) = ((ap @ (sub @ X248 @ id) @ X245))))) & (hoasapnotvar = $true) & (! [X251 : term,X252 : subst,X250 : (subst > term > term),X253 : (subst > term > term)] : (? [X254 : term,X256 : subst,X255 : subst] : (((sub @ (X253 @ X255 @ X254) @ X256)) != ((X253 @ (comp @ X255 @ X256) @ (sub @ X254 @ X256)))) | ? [X259 : subst,X257 : subst,X258 : term] : (((sub @ (X250 @ X259 @ X258) @ X257)) != ((X250 @ (comp @ X259 @ X257) @ (sub @ X258 @ X257)))) | (((X253 @ X252 @ X251)) = ((X250 @ X252 @ X251))) | (((lam @ (X250 @ sh @ one))) != ((lam @ (X253 @ sh @ one))))) | (hoaslaminj != $true)) & ((hoaslaminj = $true) | ? [X251 : term,X252 : subst,X250 : (subst > term > term),X253 : (subst > term > term)] : (! [X254 : term,X256 : subst,X255 : subst] : (((sub @ (X253 @ X255 @ X254) @ X256)) = ((X253 @ (comp @ X255 @ X256) @ (sub @ X254 @ X256)))) & ! [X259 : subst,X257 : subst,X258 : term] : (((sub @ (X250 @ X259 @ X258) @ X257)) = ((X250 @ (comp @ X259 @ X257) @ (sub @ X258 @ X257)))) & (((X253 @ X252 @ X251)) != ((X250 @ X252 @ X251))) & (((lam @ (X250 @ sh @ one))) = ((lam @ (X253 @ sh @ one)))))) & (hoasapnotvar_lthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((hoasinduction_no_psi_cond = $true) | ? [X143 : (subst > term > subst > $o),X142 : term] : (($true != ((X143 @ id @ X142 @ id))) & ! [X150 : term,X149 : term] : ((((X143 @ id @ X149 @ id)) != $true) | ($true != ((X143 @ id @ X150 @ id))) | ($true = ((X143 @ id @ (ap @ (sub @ X149 @ id) @ X150) @ id)))) & ($true != ((sP45 @ X143))))) & (! [X143 : (subst > term > subst > $o),X142 : term] : (($true = ((X143 @ id @ X142 @ id))) | ? [X150 : term,X149 : term] : ((((X143 @ id @ X149 @ id)) = $true) & ($true = ((X143 @ id @ X150 @ id))) & ($true != ((X143 @ id @ (ap @ (sub @ X149 @ id) @ X150) @ id)))) | ($true = ((sP45 @ X143)))) | (hoasinduction_no_psi_cond != $true)) & (pushprop_lem2v2_lthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (! [X524 : (subst > term > subst > $o)] : ? [X525 : (term > $o)] : ! [X526 : term] : (((X524 @ id @ X526 @ id)) = ((X525 @ X526))) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ? [X524 : (subst > term > subst > $o)] : ! [X525 : (term > $o)] : ? [X526 : term] : (((X524 @ id @ X526 @ id)) != ((X525 @ X526)))) & (termmset_gthm = $true) & ((pushprop_lem1v2_lthm = $true) | (! [X151 : term,X152 : subst] : (((sub @ one @ (push @ X151 @ X152))) = X151) & ? [X154 : (term > $o),X156 : (term > $o),X153 : term,X155 : subst] : ((((X154 @ one)) != $true) & ! [X157 : term] : (((X156 @ (sub @ X157 @ (push @ X153 @ X155)))) = ((X154 @ X157))) & ($true = ((X156 @ X153)))))) & (? [X151 : term,X152 : subst] : (((sub @ one @ (push @ X151 @ X152))) != X151) | ! [X154 : (term > $o),X156 : (term > $o),X153 : term,X155 : subst] : ((((X154 @ one)) = $true) | ? [X157 : term] : (((X156 @ (sub @ X157 @ (push @ X153 @ X155)))) != ((X154 @ X157))) | ($true != ((X156 @ X153)))) | (pushprop_lem1v2_lthm != $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((hoasapinj1_lthm = $true) | (! [X359 : term] : (((sub @ X359 @ id)) = X359) & ? [X364 : term,X366 : term,X365 : term,X367 : term] : ((X365 != X367) & (((ap @ (sub @ X367 @ id) @ X364)) = ((ap @ (sub @ X365 @ id) @ X366)))) & ! [X360 : term,X361 : term,X362 : term,X363 : term] : ((((ap @ X361 @ X362)) != ((ap @ X363 @ X360))) | (X361 = X363)))) & (? [X359 : term] : (((sub @ X359 @ id)) != X359) | ! [X364 : term,X366 : term,X365 : term,X367 : term] : ((X365 = X367) | (((ap @ (sub @ X367 @ id) @ X364)) != ((ap @ (sub @ X365 @ id) @ X366)))) | ? [X360 : term,X361 : term,X362 : term,X363 : term] : ((((ap @ X361 @ X362)) = ((ap @ X363 @ X360))) & (X361 != X363)) | (hoasapinj1_lthm != $true)) & (hoaslamnotvar = $true) & (axclos = $true) & (! [X141 : subst] : (((comp @ id @ X141)) = X141) | (axidl != $true)) & ((axidl = $true) | ? [X141 : subst] : (((comp @ id @ X141)) != X141)) & ! [X433 : d_term,X432 : d_term] : ((((d2term @ X432)) != ((d2term @ X433))) | (X432 = X433)) & (($true = sP44) | (hoasinduction_lem3v2_lthm != $true)) & ((hoasinduction_lem3v2_lthm = $true) | ($true != sP44)) & (! [X430 : term] : ? [X431 : (subst > term > term)] : (((X431 @ sh @ one)) = ((sub @ X430 @ (push @ one @ sh)))) | (hoasinduction_lem3b != $true)) & ((hoasinduction_lem3b = $true) | ? [X430 : term] : ! [X431 : (subst > term > term)] : (((X431 @ sh @ one)) != ((sub @ X430 @ (push @ one @ sh))))) & ((hoasinduction = $true) | (sP42 != $true)) & ((sP42 = $true) | (hoasinduction != $true)) & (hoaslaminj_gthm = $true) & ! [X249 : d_term] : (d_one = X249) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ! [X240 : d_subst,X241 : d_subst] : ((((d2subst @ X241)) != ((d2subst @ X240))) | (X240 = X241)) & (induction2_gthm = $true) & ((hoasapinj2_lthm = $true) | (? [X291 : term,X290 : term,X293 : term,X292 : term] : ((((ap @ (sub @ X292 @ id) @ X291)) = ((ap @ (sub @ X290 @ id) @ X293))) & (X291 != X293)) & ! [X289 : term,X288 : term,X287 : term,X286 : term] : ((((ap @ X286 @ X287)) != ((ap @ X289 @ X288))) | (X287 = X288)))) & (! [X291 : term,X290 : term,X293 : term,X292 : term] : ((((ap @ (sub @ X292 @ id) @ X291)) != ((ap @ (sub @ X290 @ id) @ X293))) | (X291 = X293)) | ? [X289 : term,X288 : term,X287 : term,X286 : term] : ((((ap @ X286 @ X287)) = ((ap @ X289 @ X288))) & (X287 != X288)) | (hoasapinj2_lthm != $true)) & (hoasinduction_lem1v2 = $true) & ((sP39 = $true) | (hoasinduction_lem2v2 != $true)) & ((hoasinduction_lem2v2 = $true) | (sP39 != $true)) & (axassoc = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (pushprop_lthm = $true) & ((hoasinduction_lem3v2a_lthm = $true) | ($true != sP37)) & (($true = sP37) | (hoasinduction_lem3v2a_lthm != $true)) & (hoasapinj1_gthm = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (hoasinduction_lem2v2_gthm = $true) & (! [X140 : subst] : (((comp @ X140 @ id)) = X140) | (axidr != $true)) & ((axidr = $true) | ? [X140 : subst] : (((comp @ X140 @ id)) != X140)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((pushprop_lem1v2 = $true) | ? [X278 : (term > $o),X276 : (term > $o),X277 : subst,X279 : term] : (($true = ((X278 @ X279))) & (((X276 @ one)) != $true) & ! [X280 : term] : (((X276 @ X280)) = ((X278 @ (sub @ X280 @ (push @ X279 @ X277))))))) & (! [X278 : (term > $o),X276 : (term > $o),X277 : subst,X279 : term] : (($true != ((X278 @ X279))) | (((X276 @ one)) = $true) | ? [X280 : term] : (((X276 @ X280)) != ((X278 @ (sub @ X280 @ (push @ X279 @ X277)))))) | (pushprop_lem1v2 != $true)) & (($true = sP36) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | ($true != sP36)) & ((hoasinduction_lem3aa_lthm = $true) | (sP30 != $true)) & ((sP30 = $true) | (hoasinduction_lem3aa_lthm != $true)) & (($true = sP29) | (hoasinduction_lem3aa != $true)) & ((hoasinduction_lem3aa = $true) | ($true != sP29)) & (pushprop = $true) & (ulamvar1 != $true) & (! [X281 : subst,X282 : term,X283 : (term > $o)] : ? [X284 : (term > $o)] : ! [X285 : term] : (((X283 @ (sub @ X285 @ (push @ X282 @ X281)))) = ((X284 @ X285))) | (pushprop_lem0_lthm != $true)) & ((pushprop_lem0_lthm = $true) | ? [X281 : subst,X282 : term,X283 : (term > $o)] : ! [X284 : (term > $o)] : ? [X285 : term] : (((X283 @ (sub @ X285 @ (push @ X282 @ X281)))) != ((X284 @ X285)))) & (! [X527 : term] : (((sub @ X527 @ id)) = X527) | (axvarid != $true)) & ((axvarid = $true) | ? [X527 : term] : (((sub @ X527 @ id)) != X527)) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoaslamnotvar_lthm = $true) & (hoasinduction_lem3v2_gthm = $true) & ((pushprop_lem1_lthm = $true) | (! [X205 : subst,X206 : term] : (((comp @ sh @ (push @ X206 @ X205))) = X205) & ? [X207 : term,X210 : (term > $o),X209 : (term > $o),X211 : subst,X208 : term] : (($true = ((X209 @ X208))) & (((X210 @ (sub @ X208 @ (push @ X207 @ X211)))) != $true)) & ! [X204 : term,X203 : subst] : (((sub @ one @ (push @ X204 @ X203))) = X204))) & (? [X205 : subst,X206 : term] : (((comp @ sh @ (push @ X206 @ X205))) != X205) | ! [X207 : term,X210 : (term > $o),X209 : (term > $o),X211 : subst,X208 : term] : (($true != ((X209 @ X208))) | (((X210 @ (sub @ X208 @ (push @ X207 @ X211)))) = $true)) | ? [X204 : term,X203 : subst] : (((sub @ one @ (push @ X204 @ X203))) != X204) | (pushprop_lem1_lthm != $true)) & (pushprop_lthm_orig = $true) & (hoasapinj2_gthm = $true) & (ulamvarsh = $true) & (apnotvar = $true) & (lamnotvar = $true) & ((hoasapinj2 = $true) | ? [X50 : term,X49 : term,X47 : term,X48 : term] : ((((ap @ (sub @ X48 @ id) @ X47)) = ((ap @ (sub @ X49 @ id) @ X50))) & (X47 != X50))) & (! [X50 : term,X49 : term,X47 : term,X48 : term] : ((((ap @ (sub @ X48 @ id) @ X47)) != ((ap @ (sub @ X49 @ id) @ X50))) | (X47 = X50)) | (hoasapinj2 != $true)) & (induction2lem_gthm = $true) & (axapp = $true) & ((axshiftcons = $true) | ? [X52 : term,X51 : subst] : (((comp @ sh @ (push @ X52 @ X51))) != X51)) & (! [X52 : term,X51 : subst] : (((comp @ sh @ (push @ X52 @ X51))) = X51) | (axshiftcons != $true)) & ((hoasinduction_lthm = $true) | ($true != sP28)) & (($true = sP28) | (hoasinduction_lthm != $true)) & (($true = sP19) | (hoasinduction_lem3_lthm != $true)) & ((hoasinduction_lem3_lthm = $true) | ($true != sP19)) & (substmonoid_gthm = $true) & (axvarshift = $true) & (! [X296 : term,X298 : term,X299 : term,X297 : term] : ((X297 = X298) | (((ap @ X298 @ X299)) != ((ap @ X297 @ X296)))) | (apinj1 != $true)) & ((apinj1 = $true) | ? [X296 : term,X298 : term,X299 : term,X297 : term] : ((X297 != X298) & (((ap @ X298 @ X299)) = ((ap @ X297 @ X296))))) & (($true = sP16) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | ($true != sP16)) & (($true = sP14) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | ($true != sP14)) & (hoasinduction_lem3a_gthm = $true) & (pushprop_lem1_gthm = $true) & ! [X4 : subst] : ? [X5 : d_subst] : (((d2subst @ X5)) = X4) & ((laminj = $true) | ? [X448 : term,X449 : term] : ((X448 != X449) & (((lam @ X449)) = ((lam @ X448))))) & (! [X448 : term,X449 : term] : ((X448 = X449) | (((lam @ X449)) != ((lam @ X448)))) | (laminj != $true)) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (($true = sP10) | (hoasinduction_lem3 != $true)) & ((hoasinduction_lem3 = $true) | ($true != sP10)) & (axabs = $true) & (hoaslamnotvar_gthm = $true) & (pushprop_lem2v2_gthm = $true) & (pushprop_lem3v2 = $true) & ((axscons = $true) | ? [X429 : subst] : (((push @ (sub @ one @ X429) @ (comp @ sh @ X429))) != X429)) & (! [X429 : subst] : (((push @ (sub @ one @ X429) @ (comp @ sh @ X429))) = X429) | (axscons != $true)) & ((termmset_lthm = $true) | (! [X177 : term] : (((sub @ X177 @ id)) = X177) & ? [X178 : term] : (((sub @ X178 @ id)) != X178))) & (? [X177 : term] : (((sub @ X177 @ id)) != X177) | ! [X178 : term] : (((sub @ X178 @ id)) = X178) | (termmset_lthm != $true)) & (ulamvarind = $true) & ! [X244 : d_subst] : (d_id = X244) & ((hoasinduction_lem3v2_f = $true) | ? [X0 : term] : ! [X1 : (subst > term > term)] : ? [X2 : term,X3 : subst] : (((sub @ X0 @ (push @ X2 @ X3))) != ((X1 @ X3 @ X2)))) & (! [X0 : term] : ? [X1 : (subst > term > term)] : ! [X2 : term,X3 : subst] : (((sub @ X0 @ (push @ X2 @ X3))) = ((X1 @ X3 @ X2))) | (hoasinduction_lem3v2_f != $true)) & (hoasinduction_lem2_lthm = $true) & (hoasinduction_lem3b_gthm = $true) & ((shinj = $true) | ? [X45 : term,X46 : term] : ((X45 != X46) & (((sub @ X46 @ sh)) = ((sub @ X45 @ sh))))) & (! [X45 : term,X46 : term] : ((X45 = X46) | (((sub @ X46 @ sh)) != ((sub @ X45 @ sh)))) | (shinj != $true)) & (axmap = $true) & (pushprop_lem3v2_lthm = $true) & (hoasinduction_lem1 = $true) & (($true = sP8) | (hoasinduction_lem3aaa != $true)) & ((hoasinduction_lem3aaa = $true) | ($true != sP8)) & ((! [X505 : subst] : (((comp @ id @ X505)) = X505) & ! [X504 : subst] : (((comp @ X504 @ id)) = X504)) | (substmonoid != $true)) & ((substmonoid = $true) | ? [X505 : subst] : (((comp @ id @ X505)) != X505) | ? [X504 : subst] : (((comp @ X504 @ id)) != X504)) & (($true = sP7) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | ($true != sP7)) & ((pushprop_lem1 = $true) | ? [X410 : (term > $o),X411 : (term > $o),X408 : subst,X412 : term,X409 : term] : (($true != ((X411 @ (sub @ X412 @ (push @ X409 @ X408))))) & ($true = ((X410 @ X412))))) & (! [X410 : (term > $o),X411 : (term > $o),X408 : subst,X412 : term,X409 : term] : (($true = ((X411 @ (sub @ X412 @ (push @ X409 @ X408))))) | ($true != ((X410 @ X412)))) | (pushprop_lem1 != $true)) & (((d2term @ d_one)) = one) & ((sP3 = $true) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | (sP3 != $true)) & (hoasinduction_gthm = $true) & ((hoasinduction_lem3a = $true) | ($true != sP2)) & (($true = sP2) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3v2 = $true) | ($true != sP1)) & (($true = sP1) | (hoasinduction_lem3v2 != $true)) & (? [X436 : subst] : (((comp @ id @ X436)) != X436) | ? [X437 : subst] : (((comp @ X437 @ id)) != X437) | (! [X438 : subst] : (((comp @ X438 @ id)) = X438) & ! [X439 : subst] : (((comp @ id @ X439)) = X439)) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | (! [X436 : subst] : (((comp @ id @ X436)) = X436) & ! [X437 : subst] : (((comp @ X437 @ id)) = X437) & (? [X438 : subst] : (((comp @ X438 @ id)) != X438) | ? [X439 : subst] : (((comp @ id @ X439)) != X439)))) & (! [X434 : term] : ? [X435 : (subst > term > term)] : (((X435 @ sh @ one)) = ((sub @ X434 @ (push @ one @ sh)))) | (hoasinduction_lem3b_lthm != $true)) & ((hoasinduction_lem3b_lthm = $true) | ? [X434 : term] : ! [X435 : (subst > term > term)] : (((X435 @ sh @ one)) != ((sub @ X434 @ (push @ one @ sh)))))),
  inference(flattening,[],[f265])).
thf(f267,plain,(
  (hoaslamnotap_lthm = $true) & (lamnotap != $true) & (pushprop_lem0_gthm = $true) & ((hoaslaminj_lthm = $true) | (sP58 != $true)) & ((sP58 = $true) | (hoaslaminj_lthm != $true)) & (($true = sP57) | (induction2 != $true)) & ((induction2 = $true) | ($true != sP57)) & (hoasinduction_lem2_gthm = $true) & (! [X0 : subst,X1 : term] : (((sub @ one @ (push @ X1 @ X0))) = X1) | (axvarcons != $true)) & ((axvarcons = $true) | ? [X2 : subst,X3 : term] : (((sub @ one @ (push @ X3 @ X2))) != X3)) & ! [X4 : term] : ? [X5 : d_term] : (((d2term @ X5)) = X4) & (hoasinduction_lem1_lthm = $true) & ((induction2lem = $true) | ($true != sP55)) & (($true = sP55) | (induction2lem != $true)) & (pushprop_gthm = $true) & (hoasinduction_lem1_gthm = $true) & ((pushprop_lem0 = $true) | ? [X6 : subst,X7 : (term > $o),X8 : term] : ! [X9 : (term > $o)] : ? [X10 : term] : (((X9 @ X10)) != ((X7 @ (sub @ X10 @ (push @ X8 @ X6)))))) & (! [X11 : subst,X12 : (term > $o),X13 : term] : ? [X14 : (term > $o)] : ! [X15 : term] : (((X14 @ X15)) = ((X12 @ (sub @ X15 @ (push @ X13 @ X11))))) | (pushprop_lem0 != $true)) & (! [X16 : term] : ? [X17 : (subst > term > term)] : ! [X18 : term,X19 : subst] : (((X17 @ X19 @ X18)) = ((sub @ X16 @ (push @ X18 @ X19)))) | (hoasinduction_lem3v2_f_lthm != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X20 : term] : ! [X21 : (subst > term > term)] : ? [X22 : term,X23 : subst] : (((X21 @ X23 @ X22)) != ((sub @ X20 @ (push @ X22 @ X23))))) & (pushprop_lem1v2_gthm = $true) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem1v2_gthm = $true) & (((d2subst @ d_id)) = sh) & (! [X24 : (subst > term > subst > $o)] : ? [X25 : (term > $o)] : ! [X26 : term] : (((X24 @ id @ X26 @ id)) = ((X25 @ X26))) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem0 = $true) | ? [X27 : (subst > term > subst > $o)] : ! [X28 : (term > $o)] : ? [X29 : term] : (((X27 @ id @ X29 @ id)) != ((X28 @ X29)))) & (((d2subst @ d_id)) = id) & (! [X30 : term,X31 : (term > $o)] : (($true = ((sP53 @ X31))) | ? [X32 : term] : (($true = ((var @ X32))) & ($true != ((X31 @ X32)))) | ($true = ((X31 @ X30))) | ? [X33 : term] : (($true = ((X31 @ X33))) & ($true != ((X31 @ (lam @ X33)))))) | (induction != $true)) & ((induction = $true) | ? [X34 : term,X35 : (term > $o)] : ((((sP53 @ X35)) != $true) & ! [X36 : term] : (($true != ((var @ X36))) | ($true = ((X35 @ X36)))) & ($true != ((X35 @ X34))) & ! [X37 : term] : (($true != ((X35 @ X37))) | ($true = ((X35 @ (lam @ X37))))))) & (((var @ (d2term @ d_one))) != $true) & ((hoaslamnotap = $true) | ? [X38 : term,X39 : term,X40 : (subst > term > term)] : ((((lam @ (X40 @ sh @ one))) = ((ap @ (sub @ X38 @ id) @ X39))) & ! [X41 : subst,X42 : subst,X43 : term] : (((X40 @ (comp @ X41 @ X42) @ (sub @ X43 @ X42))) = ((sub @ (X40 @ X41 @ X43) @ X42))))) & (! [X44 : term,X45 : term,X46 : (subst > term > term)] : ((((ap @ (sub @ X44 @ id) @ X45)) != ((lam @ (X46 @ sh @ one)))) | ? [X47 : subst,X48 : subst,X49 : term] : (((X46 @ (comp @ X47 @ X48) @ (sub @ X49 @ X48))) != ((sub @ (X46 @ X47 @ X49) @ X48)))) | (hoaslamnotap != $true)) & (hoasinduction_lem3_gthm = $true) & (pushprop_lem2v2 = $true) & ((apinj2 = $true) | ? [X50 : term,X51 : term,X52 : term,X53 : term] : ((X50 != X52) & (((ap @ X51 @ X50)) = ((ap @ X53 @ X52))))) & (! [X54 : term,X55 : term,X56 : term,X57 : term] : ((X54 = X56) | (((ap @ X57 @ X56)) != ((ap @ X55 @ X54)))) | (apinj2 != $true)) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & ((hoasinduction_no_psi_cond_lthm = $true) | ($true != sP52)) & (($true = sP52) | (hoasinduction_no_psi_cond_lthm != $true)) & (hoaslamnotap_gthm = $true) & ((termmset = $true) | ? [X58 : term] : (((sub @ X58 @ id)) != X58)) & (! [X59 : term] : (((sub @ X59 @ id)) = X59) | (termmset != $true)) & ((sP47 = $true) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | (sP47 != $true)) & (! [X60 : term,X61 : term,X62 : term,X63 : term] : ((X61 = X63) | (((ap @ (sub @ X63 @ id) @ X62)) != ((ap @ (sub @ X61 @ id) @ X60)))) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | ? [X64 : term,X65 : term,X66 : term,X67 : term] : ((X65 != X67) & (((ap @ (sub @ X67 @ id) @ X66)) = ((ap @ (sub @ X65 @ id) @ X64))))) & (hoasapnotvar = $true) & (! [X68 : term,X69 : subst,X70 : (subst > term > term),X71 : (subst > term > term)] : (? [X72 : term,X73 : subst,X74 : subst] : (((X71 @ (comp @ X74 @ X73) @ (sub @ X72 @ X73))) != ((sub @ (X71 @ X74 @ X72) @ X73))) | ? [X75 : subst,X76 : subst,X77 : term] : (((X70 @ (comp @ X75 @ X76) @ (sub @ X77 @ X76))) != ((sub @ (X70 @ X75 @ X77) @ X76))) | (((X70 @ X69 @ X68)) = ((X71 @ X69 @ X68))) | (((lam @ (X71 @ sh @ one))) != ((lam @ (X70 @ sh @ one))))) | (hoaslaminj != $true)) & ((hoaslaminj = $true) | ? [X78 : term,X79 : subst,X80 : (subst > term > term),X81 : (subst > term > term)] : (! [X82 : term,X83 : subst,X84 : subst] : (((sub @ (X81 @ X84 @ X82) @ X83)) = ((X81 @ (comp @ X84 @ X83) @ (sub @ X82 @ X83)))) & ! [X85 : subst,X86 : subst,X87 : term] : (((sub @ (X80 @ X85 @ X87) @ X86)) = ((X80 @ (comp @ X85 @ X86) @ (sub @ X87 @ X86)))) & (((X81 @ X79 @ X78)) != ((X80 @ X79 @ X78))) & (((lam @ (X80 @ sh @ one))) = ((lam @ (X81 @ sh @ one)))))) & (hoasapnotvar_lthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((hoasinduction_no_psi_cond = $true) | ? [X88 : (subst > term > subst > $o),X89 : term] : (($true != ((X88 @ id @ X89 @ id))) & ! [X90 : term,X91 : term] : (($true != ((X88 @ id @ X91 @ id))) | (((X88 @ id @ X90 @ id)) != $true) | ($true = ((X88 @ id @ (ap @ (sub @ X91 @ id) @ X90) @ id)))) & (((sP45 @ X88)) != $true))) & (! [X92 : (subst > term > subst > $o),X93 : term] : (($true = ((X92 @ id @ X93 @ id))) | ? [X94 : term,X95 : term] : (($true = ((X92 @ id @ X95 @ id))) & (((X92 @ id @ X94 @ id)) = $true) & ($true != ((X92 @ id @ (ap @ (sub @ X95 @ id) @ X94) @ id)))) | ($true = ((sP45 @ X92)))) | (hoasinduction_no_psi_cond != $true)) & (pushprop_lem2v2_lthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (! [X96 : (subst > term > subst > $o)] : ? [X97 : (term > $o)] : ! [X98 : term] : (((X96 @ id @ X98 @ id)) = ((X97 @ X98))) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ? [X99 : (subst > term > subst > $o)] : ! [X100 : (term > $o)] : ? [X101 : term] : (((X99 @ id @ X101 @ id)) != ((X100 @ X101)))) & (termmset_gthm = $true) & ((pushprop_lem1v2_lthm = $true) | (! [X102 : term,X103 : subst] : (((sub @ one @ (push @ X102 @ X103))) = X102) & ? [X104 : (term > $o),X105 : (term > $o),X106 : term,X107 : subst] : (($true != ((X104 @ one))) & ! [X108 : term] : (((X104 @ X108)) = ((X105 @ (sub @ X108 @ (push @ X106 @ X107))))) & ($true = ((X105 @ X106)))))) & (? [X109 : term,X110 : subst] : (((sub @ one @ (push @ X109 @ X110))) != X109) | ! [X111 : (term > $o),X112 : (term > $o),X113 : term,X114 : subst] : (($true = ((X111 @ one))) | ? [X115 : term] : (((X111 @ X115)) != ((X112 @ (sub @ X115 @ (push @ X113 @ X114))))) | ($true != ((X112 @ X113)))) | (pushprop_lem1v2_lthm != $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((hoasapinj1_lthm = $true) | (! [X116 : term] : (((sub @ X116 @ id)) = X116) & ? [X117 : term,X118 : term,X119 : term,X120 : term] : ((X119 != X120) & (((ap @ (sub @ X120 @ id) @ X117)) = ((ap @ (sub @ X119 @ id) @ X118)))) & ! [X121 : term,X122 : term,X123 : term,X124 : term] : ((((ap @ X122 @ X123)) != ((ap @ X124 @ X121))) | (X122 = X124)))) & (? [X125 : term] : (((sub @ X125 @ id)) != X125) | ! [X126 : term,X127 : term,X128 : term,X129 : term] : ((X128 = X129) | (((ap @ (sub @ X128 @ id) @ X127)) != ((ap @ (sub @ X129 @ id) @ X126)))) | ? [X130 : term,X131 : term,X132 : term,X133 : term] : ((((ap @ X133 @ X130)) = ((ap @ X131 @ X132))) & (X131 != X133)) | (hoasapinj1_lthm != $true)) & (hoaslamnotvar = $true) & (axclos = $true) & (! [X134 : subst] : (((comp @ id @ X134)) = X134) | (axidl != $true)) & ((axidl = $true) | ? [X135 : subst] : (((comp @ id @ X135)) != X135)) & ! [X136 : d_term,X137 : d_term] : ((((d2term @ X137)) != ((d2term @ X136))) | (X136 = X137)) & (($true = sP44) | (hoasinduction_lem3v2_lthm != $true)) & ((hoasinduction_lem3v2_lthm = $true) | ($true != sP44)) & (! [X138 : term] : ? [X139 : (subst > term > term)] : (((sub @ X138 @ (push @ one @ sh))) = ((X139 @ sh @ one))) | (hoasinduction_lem3b != $true)) & ((hoasinduction_lem3b = $true) | ? [X140 : term] : ! [X141 : (subst > term > term)] : (((X141 @ sh @ one)) != ((sub @ X140 @ (push @ one @ sh))))) & ((hoasinduction = $true) | (sP42 != $true)) & ((sP42 = $true) | (hoasinduction != $true)) & (hoaslaminj_gthm = $true) & ! [X142 : d_term] : (d_one = X142) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ! [X143 : d_subst,X144 : d_subst] : ((((d2subst @ X143)) != ((d2subst @ X144))) | (X143 = X144)) & (induction2_gthm = $true) & ((hoasapinj2_lthm = $true) | (? [X145 : term,X146 : term,X147 : term,X148 : term] : ((((ap @ (sub @ X148 @ id) @ X145)) = ((ap @ (sub @ X146 @ id) @ X147))) & (X145 != X147)) & ! [X149 : term,X150 : term,X151 : term,X152 : term] : ((((ap @ X152 @ X151)) != ((ap @ X149 @ X150))) | (X150 = X151)))) & (! [X153 : term,X154 : term,X155 : term,X156 : term] : ((((ap @ (sub @ X154 @ id) @ X155)) != ((ap @ (sub @ X156 @ id) @ X153))) | (X153 = X155)) | ? [X157 : term,X158 : term,X159 : term,X160 : term] : ((((ap @ X157 @ X158)) = ((ap @ X160 @ X159))) & (X158 != X159)) | (hoasapinj2_lthm != $true)) & (hoasinduction_lem1v2 = $true) & ((sP39 = $true) | (hoasinduction_lem2v2 != $true)) & ((hoasinduction_lem2v2 = $true) | (sP39 != $true)) & (axassoc = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (pushprop_lthm = $true) & ((hoasinduction_lem3v2a_lthm = $true) | ($true != sP37)) & (($true = sP37) | (hoasinduction_lem3v2a_lthm != $true)) & (hoasapinj1_gthm = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (hoasinduction_lem2v2_gthm = $true) & (! [X161 : subst] : (((comp @ X161 @ id)) = X161) | (axidr != $true)) & ((axidr = $true) | ? [X162 : subst] : (((comp @ X162 @ id)) != X162)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((pushprop_lem1v2 = $true) | ? [X163 : (term > $o),X164 : (term > $o),X165 : subst,X166 : term] : (($true = ((X163 @ X166))) & (((X164 @ one)) != $true) & ! [X167 : term] : (((X164 @ X167)) = ((X163 @ (sub @ X167 @ (push @ X166 @ X165))))))) & (! [X168 : (term > $o),X169 : (term > $o),X170 : subst,X171 : term] : (($true != ((X168 @ X171))) | ($true = ((X169 @ one))) | ? [X172 : term] : (((X169 @ X172)) != ((X168 @ (sub @ X172 @ (push @ X171 @ X170)))))) | (pushprop_lem1v2 != $true)) & (($true = sP36) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | ($true != sP36)) & ((hoasinduction_lem3aa_lthm = $true) | (sP30 != $true)) & ((sP30 = $true) | (hoasinduction_lem3aa_lthm != $true)) & (($true = sP29) | (hoasinduction_lem3aa != $true)) & ((hoasinduction_lem3aa = $true) | ($true != sP29)) & (pushprop = $true) & (ulamvar1 != $true) & (! [X173 : subst,X174 : term,X175 : (term > $o)] : ? [X176 : (term > $o)] : ! [X177 : term] : (((X176 @ X177)) = ((X175 @ (sub @ X177 @ (push @ X174 @ X173))))) | (pushprop_lem0_lthm != $true)) & ((pushprop_lem0_lthm = $true) | ? [X178 : subst,X179 : term,X180 : (term > $o)] : ! [X181 : (term > $o)] : ? [X182 : term] : (((X180 @ (sub @ X182 @ (push @ X179 @ X178)))) != ((X181 @ X182)))) & (! [X183 : term] : (((sub @ X183 @ id)) = X183) | (axvarid != $true)) & ((axvarid = $true) | ? [X184 : term] : (((sub @ X184 @ id)) != X184)) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoaslamnotvar_lthm = $true) & (hoasinduction_lem3v2_gthm = $true) & ((pushprop_lem1_lthm = $true) | (! [X185 : subst,X186 : term] : (((comp @ sh @ (push @ X186 @ X185))) = X185) & ? [X187 : term,X188 : (term > $o),X189 : (term > $o),X190 : subst,X191 : term] : ((((X189 @ X191)) = $true) & ($true != ((X188 @ (sub @ X191 @ (push @ X187 @ X190)))))) & ! [X192 : term,X193 : subst] : (((sub @ one @ (push @ X192 @ X193))) = X192))) & (? [X194 : subst,X195 : term] : (((comp @ sh @ (push @ X195 @ X194))) != X194) | ! [X196 : term,X197 : (term > $o),X198 : (term > $o),X199 : subst,X200 : term] : (($true != ((X198 @ X200))) | ($true = ((X197 @ (sub @ X200 @ (push @ X196 @ X199)))))) | ? [X201 : term,X202 : subst] : (((sub @ one @ (push @ X201 @ X202))) != X201) | (pushprop_lem1_lthm != $true)) & (pushprop_lthm_orig = $true) & (hoasapinj2_gthm = $true) & (ulamvarsh = $true) & (apnotvar = $true) & (lamnotvar = $true) & ((hoasapinj2 = $true) | ? [X203 : term,X204 : term,X205 : term,X206 : term] : ((((ap @ (sub @ X206 @ id) @ X205)) = ((ap @ (sub @ X204 @ id) @ X203))) & (X203 != X205))) & (! [X207 : term,X208 : term,X209 : term,X210 : term] : ((((ap @ (sub @ X210 @ id) @ X209)) != ((ap @ (sub @ X208 @ id) @ X207))) | (X207 = X209)) | (hoasapinj2 != $true)) & (induction2lem_gthm = $true) & (axapp = $true) & ((axshiftcons = $true) | ? [X211 : term,X212 : subst] : (((comp @ sh @ (push @ X211 @ X212))) != X212)) & (! [X213 : term,X214 : subst] : (((comp @ sh @ (push @ X213 @ X214))) = X214) | (axshiftcons != $true)) & ((hoasinduction_lthm = $true) | ($true != sP28)) & (($true = sP28) | (hoasinduction_lthm != $true)) & (($true = sP19) | (hoasinduction_lem3_lthm != $true)) & ((hoasinduction_lem3_lthm = $true) | ($true != sP19)) & (substmonoid_gthm = $true) & (axvarshift = $true) & (! [X215 : term,X216 : term,X217 : term,X218 : term] : ((X216 = X218) | (((ap @ X216 @ X217)) != ((ap @ X218 @ X215)))) | (apinj1 != $true)) & ((apinj1 = $true) | ? [X219 : term,X220 : term,X221 : term,X222 : term] : ((X220 != X222) & (((ap @ X220 @ X221)) = ((ap @ X222 @ X219))))) & (($true = sP16) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | ($true != sP16)) & (($true = sP14) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | ($true != sP14)) & (hoasinduction_lem3a_gthm = $true) & (pushprop_lem1_gthm = $true) & ! [X223 : subst] : ? [X224 : d_subst] : (((d2subst @ X224)) = X223) & ((laminj = $true) | ? [X225 : term,X226 : term] : ((X225 != X226) & (((lam @ X225)) = ((lam @ X226))))) & (! [X227 : term,X228 : term] : ((X227 = X228) | (((lam @ X228)) != ((lam @ X227)))) | (laminj != $true)) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (($true = sP10) | (hoasinduction_lem3 != $true)) & ((hoasinduction_lem3 = $true) | ($true != sP10)) & (axabs = $true) & (hoaslamnotvar_gthm = $true) & (pushprop_lem2v2_gthm = $true) & (pushprop_lem3v2 = $true) & ((axscons = $true) | ? [X229 : subst] : (((push @ (sub @ one @ X229) @ (comp @ sh @ X229))) != X229)) & (! [X230 : subst] : (((push @ (sub @ one @ X230) @ (comp @ sh @ X230))) = X230) | (axscons != $true)) & ((termmset_lthm = $true) | (! [X231 : term] : (((sub @ X231 @ id)) = X231) & ? [X232 : term] : (((sub @ X232 @ id)) != X232))) & (? [X233 : term] : (((sub @ X233 @ id)) != X233) | ! [X234 : term] : (((sub @ X234 @ id)) = X234) | (termmset_lthm != $true)) & (ulamvarind = $true) & ! [X235 : d_subst] : (d_id = X235) & ((hoasinduction_lem3v2_f = $true) | ? [X236 : term] : ! [X237 : (subst > term > term)] : ? [X238 : term,X239 : subst] : (((sub @ X236 @ (push @ X238 @ X239))) != ((X237 @ X239 @ X238)))) & (! [X240 : term] : ? [X241 : (subst > term > term)] : ! [X242 : term,X243 : subst] : (((sub @ X240 @ (push @ X242 @ X243))) = ((X241 @ X243 @ X242))) | (hoasinduction_lem3v2_f != $true)) & (hoasinduction_lem2_lthm = $true) & (hoasinduction_lem3b_gthm = $true) & ((shinj = $true) | ? [X244 : term,X245 : term] : ((X244 != X245) & (((sub @ X245 @ sh)) = ((sub @ X244 @ sh))))) & (! [X246 : term,X247 : term] : ((X246 = X247) | (((sub @ X246 @ sh)) != ((sub @ X247 @ sh)))) | (shinj != $true)) & (axmap = $true) & (pushprop_lem3v2_lthm = $true) & (hoasinduction_lem1 = $true) & (($true = sP8) | (hoasinduction_lem3aaa != $true)) & ((hoasinduction_lem3aaa = $true) | ($true != sP8)) & ((! [X248 : subst] : (((comp @ id @ X248)) = X248) & ! [X249 : subst] : (((comp @ X249 @ id)) = X249)) | (substmonoid != $true)) & ((substmonoid = $true) | ? [X250 : subst] : (((comp @ id @ X250)) != X250) | ? [X251 : subst] : (((comp @ X251 @ id)) != X251)) & (($true = sP7) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | ($true != sP7)) & ((pushprop_lem1 = $true) | ? [X252 : (term > $o),X253 : (term > $o),X254 : subst,X255 : term,X256 : term] : (($true != ((X253 @ (sub @ X255 @ (push @ X256 @ X254))))) & (((X252 @ X255)) = $true))) & (! [X257 : (term > $o),X258 : (term > $o),X259 : subst,X260 : term,X261 : term] : (($true = ((X258 @ (sub @ X260 @ (push @ X261 @ X259))))) | (((X257 @ X260)) != $true)) | (pushprop_lem1 != $true)) & (((d2term @ d_one)) = one) & ((sP3 = $true) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | (sP3 != $true)) & (hoasinduction_gthm = $true) & ((hoasinduction_lem3a = $true) | ($true != sP2)) & (($true = sP2) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3v2 = $true) | ($true != sP1)) & (($true = sP1) | (hoasinduction_lem3v2 != $true)) & (? [X262 : subst] : (((comp @ id @ X262)) != X262) | ? [X263 : subst] : (((comp @ X263 @ id)) != X263) | (! [X264 : subst] : (((comp @ X264 @ id)) = X264) & ! [X265 : subst] : (((comp @ id @ X265)) = X265)) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | (! [X266 : subst] : (((comp @ id @ X266)) = X266) & ! [X267 : subst] : (((comp @ X267 @ id)) = X267) & (? [X268 : subst] : (((comp @ X268 @ id)) != X268) | ? [X269 : subst] : (((comp @ id @ X269)) != X269)))) & (! [X270 : term] : ? [X271 : (subst > term > term)] : (((sub @ X270 @ (push @ one @ sh))) = ((X271 @ sh @ one))) | (hoasinduction_lem3b_lthm != $true)) & ((hoasinduction_lem3b_lthm = $true) | ? [X272 : term] : ! [X273 : (subst > term > term)] : (((X273 @ sh @ one)) != ((sub @ X272 @ (push @ one @ sh)))))),
  inference(rectify,[],[f266])).
thf(f268,plain,(
  (hoaslamnotap_lthm = $true) & (lamnotap != $true) & (pushprop_lem0_gthm = $true) & ((hoaslaminj_lthm = $true) | (sP58 != $true)) & ((sP58 = $true) | (hoaslaminj_lthm != $true)) & (($true = sP57) | (induction2 != $true)) & ((induction2 = $true) | ($true != sP57)) & (hoasinduction_lem2_gthm = $true) & (! [X0 : subst,X1 : term] : (((sub @ one @ (push @ X1 @ X0))) = X1) | (axvarcons != $true)) & ((axvarcons = $true) | (((sub @ one @ (push @ sK460 @ sK459))) != sK460)) & ! [X4 : term] : (((d2term @ (sK461 @ X4))) = X4) & (hoasinduction_lem1_lthm = $true) & ((induction2lem = $true) | ($true != sP55)) & (($true = sP55) | (induction2lem != $true)) & (pushprop_gthm = $true) & (hoasinduction_lem1_gthm = $true) & ((pushprop_lem0 = $true) | ! [X9 : (term > $o)] : (((X9 @ (sK465 @ X9))) != ((sK463 @ (sub @ (sK465 @ X9) @ (push @ sK464 @ sK462)))))) & (! [X11 : subst,X12 : (term > $o),X13 : term] : ! [X15 : term] : (((X12 @ (sub @ X15 @ (push @ X13 @ X11)))) = ((sK466 @ X13 @ X12 @ X11 @ X15))) | (pushprop_lem0 != $true)) & (! [X16 : term] : ! [X18 : term,X19 : subst] : (((sK467 @ X16 @ X19 @ X18)) = ((sub @ X16 @ (push @ X18 @ X19)))) | (hoasinduction_lem3v2_f_lthm != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ! [X21 : (subst > term > term)] : (((sub @ sK468 @ (push @ (sK469 @ X21) @ (sK470 @ X21)))) != ((X21 @ (sK470 @ X21) @ (sK469 @ X21))))) & (pushprop_lem1v2_gthm = $true) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem1v2_gthm = $true) & (((d2subst @ d_id)) = sh) & (! [X24 : (subst > term > subst > $o)] : ! [X26 : term] : (((X24 @ id @ X26 @ id)) = ((sK471 @ X24 @ X26))) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem0 = $true) | ! [X28 : (term > $o)] : (((sK472 @ id @ (sK473 @ X28) @ id)) != ((X28 @ (sK473 @ X28))))) & (((d2subst @ d_id)) = id) & (! [X30 : term,X31 : (term > $o)] : (($true = ((sP53 @ X31))) | (($true = ((var @ (sK474 @ X31)))) & ($true != ((X31 @ (sK474 @ X31))))) | ($true = ((X31 @ X30))) | (($true = ((X31 @ (sK475 @ X31)))) & ($true != ((X31 @ (lam @ (sK475 @ X31))))))) | (induction != $true)) & ((induction = $true) | (($true != ((sP53 @ sK477))) & ! [X36 : term] : (($true != ((var @ X36))) | ($true = ((sK477 @ X36)))) & ($true != ((sK477 @ sK476))) & ! [X37 : term] : (($true != ((sK477 @ X37))) | ($true = ((sK477 @ (lam @ X37))))))) & (((var @ (d2term @ d_one))) != $true) & ((hoaslamnotap = $true) | ((((lam @ (sK480 @ sh @ one))) = ((ap @ (sub @ sK478 @ id) @ sK479))) & ! [X41 : subst,X42 : subst,X43 : term] : (((sub @ (sK480 @ X41 @ X43) @ X42)) = ((sK480 @ (comp @ X41 @ X42) @ (sub @ X43 @ X42)))))) & (! [X44 : term,X45 : term,X46 : (subst > term > term)] : ((((ap @ (sub @ X44 @ id) @ X45)) != ((lam @ (X46 @ sh @ one)))) | (((X46 @ (comp @ (sK481 @ X46) @ (sK482 @ X46)) @ (sub @ (sK483 @ X46) @ (sK482 @ X46)))) != ((sub @ (X46 @ (sK481 @ X46) @ (sK483 @ X46)) @ (sK482 @ X46))))) | (hoaslamnotap != $true)) & (hoasinduction_lem3_gthm = $true) & (pushprop_lem2v2 = $true) & ((apinj2 = $true) | ((sK484 != sK486) & (((ap @ sK485 @ sK484)) = ((ap @ sK487 @ sK486))))) & (! [X54 : term,X55 : term,X56 : term,X57 : term] : ((X54 = X56) | (((ap @ X57 @ X56)) != ((ap @ X55 @ X54)))) | (apinj2 != $true)) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & ((hoasinduction_no_psi_cond_lthm = $true) | ($true != sP52)) & (($true = sP52) | (hoasinduction_no_psi_cond_lthm != $true)) & (hoaslamnotap_gthm = $true) & ((termmset = $true) | (sK488 != ((sub @ sK488 @ id)))) & (! [X59 : term] : (((sub @ X59 @ id)) = X59) | (termmset != $true)) & ((sP47 = $true) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | (sP47 != $true)) & (! [X60 : term,X61 : term,X62 : term,X63 : term] : ((X61 = X63) | (((ap @ (sub @ X63 @ id) @ X62)) != ((ap @ (sub @ X61 @ id) @ X60)))) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | ((sK490 != sK492) & (((ap @ (sub @ sK492 @ id) @ sK491)) = ((ap @ (sub @ sK490 @ id) @ sK489))))) & (hoasapnotvar = $true) & (! [X68 : term,X69 : subst,X70 : (subst > term > term),X71 : (subst > term > term)] : ((((sub @ (X71 @ (sK495 @ X71) @ (sK493 @ X71)) @ (sK494 @ X71))) != ((X71 @ (comp @ (sK495 @ X71) @ (sK494 @ X71)) @ (sub @ (sK493 @ X71) @ (sK494 @ X71))))) | (((X70 @ (comp @ (sK496 @ X70) @ (sK497 @ X70)) @ (sub @ (sK498 @ X70) @ (sK497 @ X70)))) != ((sub @ (X70 @ (sK496 @ X70) @ (sK498 @ X70)) @ (sK497 @ X70)))) | (((X70 @ X69 @ X68)) = ((X71 @ X69 @ X68))) | (((lam @ (X71 @ sh @ one))) != ((lam @ (X70 @ sh @ one))))) | (hoaslaminj != $true)) & ((hoaslaminj = $true) | (! [X82 : term,X83 : subst,X84 : subst] : (((sub @ (sK502 @ X84 @ X82) @ X83)) = ((sK502 @ (comp @ X84 @ X83) @ (sub @ X82 @ X83)))) & ! [X85 : subst,X86 : subst,X87 : term] : (((sub @ (sK501 @ X85 @ X87) @ X86)) = ((sK501 @ (comp @ X85 @ X86) @ (sub @ X87 @ X86)))) & (((sK501 @ sK500 @ sK499)) != ((sK502 @ sK500 @ sK499))) & (((lam @ (sK502 @ sh @ one))) = ((lam @ (sK501 @ sh @ one)))))) & (hoasapnotvar_lthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((hoasinduction_no_psi_cond = $true) | (($true != ((sK503 @ id @ sK504 @ id))) & ! [X90 : term,X91 : term] : (($true != ((sK503 @ id @ X91 @ id))) | ($true != ((sK503 @ id @ X90 @ id))) | ($true = ((sK503 @ id @ (ap @ (sub @ X91 @ id) @ X90) @ id)))) & ($true != ((sP45 @ sK503))))) & (! [X92 : (subst > term > subst > $o),X93 : term] : (($true = ((X92 @ id @ X93 @ id))) | (($true = ((X92 @ id @ (sK506 @ X92) @ id))) & ($true = ((X92 @ id @ (sK505 @ X92) @ id))) & ($true != ((X92 @ id @ (ap @ (sub @ (sK506 @ X92) @ id) @ (sK505 @ X92)) @ id)))) | ($true = ((sP45 @ X92)))) | (hoasinduction_no_psi_cond != $true)) & (pushprop_lem2v2_lthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (! [X96 : (subst > term > subst > $o)] : ! [X98 : term] : (((X96 @ id @ X98 @ id)) = ((sK507 @ X96 @ X98))) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ! [X100 : (term > $o)] : (((sK508 @ id @ (sK509 @ X100) @ id)) != ((X100 @ (sK509 @ X100))))) & (termmset_gthm = $true) & ((pushprop_lem1v2_lthm = $true) | (! [X102 : term,X103 : subst] : (((sub @ one @ (push @ X102 @ X103))) = X102) & ((((sK510 @ one)) != $true) & ! [X108 : term] : (((sK511 @ (sub @ X108 @ (push @ sK512 @ sK513)))) = ((sK510 @ X108))) & ($true = ((sK511 @ sK512)))))) & ((sK514 != ((sub @ one @ (push @ sK514 @ sK515)))) | ! [X111 : (term > $o),X112 : (term > $o),X113 : term,X114 : subst] : (($true = ((X111 @ one))) | (((X111 @ (sK516 @ X114 @ X113 @ X112 @ X111))) != ((X112 @ (sub @ (sK516 @ X114 @ X113 @ X112 @ X111) @ (push @ X113 @ X114))))) | ($true != ((X112 @ X113)))) | (pushprop_lem1v2_lthm != $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((hoasapinj1_lthm = $true) | (! [X116 : term] : (((sub @ X116 @ id)) = X116) & ((sK519 != sK520) & (((ap @ (sub @ sK520 @ id) @ sK517)) = ((ap @ (sub @ sK519 @ id) @ sK518)))) & ! [X121 : term,X122 : term,X123 : term,X124 : term] : ((((ap @ X122 @ X123)) != ((ap @ X124 @ X121))) | (X122 = X124)))) & ((sK521 != ((sub @ sK521 @ id))) | ! [X126 : term,X127 : term,X128 : term,X129 : term] : ((X128 = X129) | (((ap @ (sub @ X128 @ id) @ X127)) != ((ap @ (sub @ X129 @ id) @ X126)))) | ((((ap @ sK523 @ sK524)) = ((ap @ sK525 @ sK522))) & (sK525 != sK523)) | (hoasapinj1_lthm != $true)) & (hoaslamnotvar = $true) & (axclos = $true) & (! [X134 : subst] : (((comp @ id @ X134)) = X134) | (axidl != $true)) & ((axidl = $true) | (sK526 != ((comp @ id @ sK526)))) & ! [X136 : d_term,X137 : d_term] : ((((d2term @ X137)) != ((d2term @ X136))) | (X136 = X137)) & (($true = sP44) | (hoasinduction_lem3v2_lthm != $true)) & ((hoasinduction_lem3v2_lthm = $true) | ($true != sP44)) & (! [X138 : term] : (((sub @ X138 @ (push @ one @ sh))) = ((sK527 @ X138 @ sh @ one))) | (hoasinduction_lem3b != $true)) & ((hoasinduction_lem3b = $true) | ! [X141 : (subst > term > term)] : (((X141 @ sh @ one)) != ((sub @ sK528 @ (push @ one @ sh))))) & ((hoasinduction = $true) | (sP42 != $true)) & ((sP42 = $true) | (hoasinduction != $true)) & (hoaslaminj_gthm = $true) & ! [X142 : d_term] : (d_one = X142) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ! [X143 : d_subst,X144 : d_subst] : ((((d2subst @ X143)) != ((d2subst @ X144))) | (X143 = X144)) & (induction2_gthm = $true) & ((hoasapinj2_lthm = $true) | (((((ap @ (sub @ sK532 @ id) @ sK529)) = ((ap @ (sub @ sK530 @ id) @ sK531))) & (sK531 != sK529)) & ! [X149 : term,X150 : term,X151 : term,X152 : term] : ((((ap @ X152 @ X151)) != ((ap @ X149 @ X150))) | (X150 = X151)))) & (! [X153 : term,X154 : term,X155 : term,X156 : term] : ((((ap @ (sub @ X154 @ id) @ X155)) != ((ap @ (sub @ X156 @ id) @ X153))) | (X153 = X155)) | ((((ap @ sK536 @ sK535)) = ((ap @ sK533 @ sK534))) & (sK534 != sK535)) | (hoasapinj2_lthm != $true)) & (hoasinduction_lem1v2 = $true) & ((sP39 = $true) | (hoasinduction_lem2v2 != $true)) & ((hoasinduction_lem2v2 = $true) | (sP39 != $true)) & (axassoc = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (pushprop_lthm = $true) & ((hoasinduction_lem3v2a_lthm = $true) | ($true != sP37)) & (($true = sP37) | (hoasinduction_lem3v2a_lthm != $true)) & (hoasapinj1_gthm = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (hoasinduction_lem2v2_gthm = $true) & (! [X161 : subst] : (((comp @ X161 @ id)) = X161) | (axidr != $true)) & ((axidr = $true) | (((comp @ sK537 @ id)) != sK537)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((pushprop_lem1v2 = $true) | (($true = ((sK538 @ sK541))) & (((sK539 @ one)) != $true) & ! [X167 : term] : (((sK539 @ X167)) = ((sK538 @ (sub @ X167 @ (push @ sK541 @ sK540))))))) & (! [X168 : (term > $o),X169 : (term > $o),X170 : subst,X171 : term] : (($true != ((X168 @ X171))) | ($true = ((X169 @ one))) | (((X168 @ (sub @ (sK542 @ X171 @ X170 @ X169 @ X168) @ (push @ X171 @ X170)))) != ((X169 @ (sK542 @ X171 @ X170 @ X169 @ X168))))) | (pushprop_lem1v2 != $true)) & (($true = sP36) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | ($true != sP36)) & ((hoasinduction_lem3aa_lthm = $true) | (sP30 != $true)) & ((sP30 = $true) | (hoasinduction_lem3aa_lthm != $true)) & (($true = sP29) | (hoasinduction_lem3aa != $true)) & ((hoasinduction_lem3aa = $true) | ($true != sP29)) & (pushprop = $true) & (ulamvar1 != $true) & (! [X173 : subst,X174 : term,X175 : (term > $o)] : ! [X177 : term] : (((sK543 @ X175 @ X174 @ X173 @ X177)) = ((X175 @ (sub @ X177 @ (push @ X174 @ X173))))) | (pushprop_lem0_lthm != $true)) & ((pushprop_lem0_lthm = $true) | ! [X181 : (term > $o)] : (((sK546 @ (sub @ (sK547 @ X181) @ (push @ sK545 @ sK544)))) != ((X181 @ (sK547 @ X181))))) & (! [X183 : term] : (((sub @ X183 @ id)) = X183) | (axvarid != $true)) & ((axvarid = $true) | (sK548 != ((sub @ sK548 @ id)))) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoaslamnotvar_lthm = $true) & (hoasinduction_lem3v2_gthm = $true) & ((pushprop_lem1_lthm = $true) | (! [X185 : subst,X186 : term] : (((comp @ sh @ (push @ X186 @ X185))) = X185) & (($true = ((sK551 @ sK553))) & (((sK550 @ (sub @ sK553 @ (push @ sK549 @ sK552)))) != $true)) & ! [X192 : term,X193 : subst] : (((sub @ one @ (push @ X192 @ X193))) = X192))) & ((sK554 != ((comp @ sh @ (push @ sK555 @ sK554)))) | ! [X196 : term,X197 : (term > $o),X198 : (term > $o),X199 : subst,X200 : term] : (($true != ((X198 @ X200))) | ($true = ((X197 @ (sub @ X200 @ (push @ X196 @ X199)))))) | (sK556 != ((sub @ one @ (push @ sK556 @ sK557)))) | (pushprop_lem1_lthm != $true)) & (pushprop_lthm_orig = $true) & (hoasapinj2_gthm = $true) & (ulamvarsh = $true) & (apnotvar = $true) & (lamnotvar = $true) & ((hoasapinj2 = $true) | ((((ap @ (sub @ sK561 @ id) @ sK560)) = ((ap @ (sub @ sK559 @ id) @ sK558))) & (sK558 != sK560))) & (! [X207 : term,X208 : term,X209 : term,X210 : term] : ((((ap @ (sub @ X210 @ id) @ X209)) != ((ap @ (sub @ X208 @ id) @ X207))) | (X207 = X209)) | (hoasapinj2 != $true)) & (induction2lem_gthm = $true) & (axapp = $true) & ((axshiftcons = $true) | (sK563 != ((comp @ sh @ (push @ sK562 @ sK563))))) & (! [X213 : term,X214 : subst] : (((comp @ sh @ (push @ X213 @ X214))) = X214) | (axshiftcons != $true)) & ((hoasinduction_lthm = $true) | ($true != sP28)) & (($true = sP28) | (hoasinduction_lthm != $true)) & (($true = sP19) | (hoasinduction_lem3_lthm != $true)) & ((hoasinduction_lem3_lthm = $true) | ($true != sP19)) & (substmonoid_gthm = $true) & (axvarshift = $true) & (! [X215 : term,X216 : term,X217 : term,X218 : term] : ((X216 = X218) | (((ap @ X216 @ X217)) != ((ap @ X218 @ X215)))) | (apinj1 != $true)) & ((apinj1 = $true) | ((sK565 != sK567) & (((ap @ sK565 @ sK566)) = ((ap @ sK567 @ sK564))))) & (($true = sP16) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | ($true != sP16)) & (($true = sP14) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | ($true != sP14)) & (hoasinduction_lem3a_gthm = $true) & (pushprop_lem1_gthm = $true) & ! [X223 : subst] : (((d2subst @ (sK568 @ X223))) = X223) & ((laminj = $true) | ((sK569 != sK570) & (((lam @ sK570)) = ((lam @ sK569))))) & (! [X227 : term,X228 : term] : ((X227 = X228) | (((lam @ X228)) != ((lam @ X227)))) | (laminj != $true)) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (($true = sP10) | (hoasinduction_lem3 != $true)) & ((hoasinduction_lem3 = $true) | ($true != sP10)) & (axabs = $true) & (hoaslamnotvar_gthm = $true) & (pushprop_lem2v2_gthm = $true) & (pushprop_lem3v2 = $true) & ((axscons = $true) | (((push @ (sub @ one @ sK571) @ (comp @ sh @ sK571))) != sK571)) & (! [X230 : subst] : (((push @ (sub @ one @ X230) @ (comp @ sh @ X230))) = X230) | (axscons != $true)) & ((termmset_lthm = $true) | (! [X231 : term] : (((sub @ X231 @ id)) = X231) & (((sub @ sK572 @ id)) != sK572))) & ((((sub @ sK573 @ id)) != sK573) | ! [X234 : term] : (((sub @ X234 @ id)) = X234) | (termmset_lthm != $true)) & (ulamvarind = $true) & ! [X235 : d_subst] : (d_id = X235) & ((hoasinduction_lem3v2_f = $true) | ! [X237 : (subst > term > term)] : (((sub @ sK574 @ (push @ (sK575 @ X237) @ (sK576 @ X237)))) != ((X237 @ (sK576 @ X237) @ (sK575 @ X237))))) & (! [X240 : term] : ! [X242 : term,X243 : subst] : (((sub @ X240 @ (push @ X242 @ X243))) = ((sK577 @ X240 @ X243 @ X242))) | (hoasinduction_lem3v2_f != $true)) & (hoasinduction_lem2_lthm = $true) & (hoasinduction_lem3b_gthm = $true) & ((shinj = $true) | ((sK579 != sK578) & (((sub @ sK578 @ sh)) = ((sub @ sK579 @ sh))))) & (! [X246 : term,X247 : term] : ((X246 = X247) | (((sub @ X246 @ sh)) != ((sub @ X247 @ sh)))) | (shinj != $true)) & (axmap = $true) & (pushprop_lem3v2_lthm = $true) & (hoasinduction_lem1 = $true) & (($true = sP8) | (hoasinduction_lem3aaa != $true)) & ((hoasinduction_lem3aaa = $true) | ($true != sP8)) & ((! [X248 : subst] : (((comp @ id @ X248)) = X248) & ! [X249 : subst] : (((comp @ X249 @ id)) = X249)) | (substmonoid != $true)) & ((substmonoid = $true) | (sK580 != ((comp @ id @ sK580))) | (sK581 != ((comp @ sK581 @ id)))) & (($true = sP7) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | ($true != sP7)) & ((pushprop_lem1 = $true) | (($true != ((sK583 @ (sub @ sK585 @ (push @ sK586 @ sK584))))) & ($true = ((sK582 @ sK585))))) & (! [X257 : (term > $o),X258 : (term > $o),X259 : subst,X260 : term,X261 : term] : (($true = ((X258 @ (sub @ X260 @ (push @ X261 @ X259))))) | (((X257 @ X260)) != $true)) | (pushprop_lem1 != $true)) & (((d2term @ d_one)) = one) & ((sP3 = $true) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | (sP3 != $true)) & (hoasinduction_gthm = $true) & ((hoasinduction_lem3a = $true) | ($true != sP2)) & (($true = sP2) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3v2 = $true) | ($true != sP1)) & (($true = sP1) | (hoasinduction_lem3v2 != $true)) & ((((comp @ id @ sK587)) != sK587) | (sK588 != ((comp @ sK588 @ id))) | (! [X264 : subst] : (((comp @ X264 @ id)) = X264) & ! [X265 : subst] : (((comp @ id @ X265)) = X265)) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | (! [X266 : subst] : (((comp @ id @ X266)) = X266) & ! [X267 : subst] : (((comp @ X267 @ id)) = X267) & ((sK589 != ((comp @ sK589 @ id))) | (((comp @ id @ sK590)) != sK590)))) & (! [X270 : term] : (((sub @ X270 @ (push @ one @ sh))) = ((sK591 @ X270 @ sh @ one))) | (hoasinduction_lem3b_lthm != $true)) & ((hoasinduction_lem3b_lthm = $true) | ! [X273 : (subst > term > term)] : (((X273 @ sh @ one)) != ((sub @ sK592 @ (push @ one @ sh)))))),
  inference(skolemize,[status(esa),new_symbols(skolem,[sK459,sK460,vAPP,sK462,sK463,sK464,vAPP,sK468,vAPP,sK472,vAPP,sK476,sK477,sK478,sK479,sK480,vAPP,sK484,sK485,sK486,sK487,sK488,sK489,sK490,sK491,sK492,vAPP,sK499,sK500,sK501,sK502,sK503,sK504,vAPP,sK508,vAPP,sK510,sK511,sK512,sK513,sK514,sK515,vAPP,sK517,sK518,sK519,sK520,sK521,sK522,sK523,sK524,sK525,sK526,vAPP,sK528,sK529,sK530,sK531,sK532,sK533,sK534,sK535,sK536,sK537,sK538,sK539,sK540,sK541,vAPP,sK544,sK545,sK546,vAPP,sK548,sK549,sK550,sK551,sK552,sK553,sK554,sK555,sK556,sK557,sK558,sK559,sK560,sK561,sK562,sK563,sK564,sK565,sK566,sK567,vAPP,sK569,sK570,sK571,sK572,sK573,sK574,vAPP,sK578,sK579,sK580,sK581,sK582,sK583,sK584,sK585,sK586,sK587,sK588,sK589,sK590,vAPP,sK592]),skolemize(X2,$thf(sK459)),skolemize(X3,$thf(sK460)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X6,$thf(sK462)),skolemize(X7,$thf(sK463)),skolemize(X8,$thf(sK464)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X20,$thf(sK468)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X27,$thf(sK472)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X34,$thf(sK476)),skolemize(X35,$thf(sK477)),skolemize(X38,$thf(sK478)),skolemize(X39,$thf(sK479)),skolemize(X40,$thf(sK480)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X50,$thf(sK484)),skolemize(X51,$thf(sK485)),skolemize(X52,$thf(sK486)),skolemize(X53,$thf(sK487)),skolemize(X58,$thf(sK488)),skolemize(X64,$thf(sK489)),skolemize(X65,$thf(sK490)),skolemize(X66,$thf(sK491)),skolemize(X67,$thf(sK492)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X78,$thf(sK499)),skolemize(X79,$thf(sK500)),skolemize(X80,$thf(sK501)),skolemize(X81,$thf(sK502)),skolemize(X88,$thf(sK503)),skolemize(X89,$thf(sK504)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X99,$thf(sK508)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X104,$thf(sK510)),skolemize(X105,$thf(sK511)),skolemize(X106,$thf(sK512)),skolemize(X107,$thf(sK513)),skolemize(X109,$thf(sK514)),skolemize(X110,$thf(sK515)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X117,$thf(sK517)),skolemize(X118,$thf(sK518)),skolemize(X119,$thf(sK519)),skolemize(X120,$thf(sK520)),skolemize(X125,$thf(sK521)),skolemize(X130,$thf(sK522)),skolemize(X131,$thf(sK523)),skolemize(X132,$thf(sK524)),skolemize(X133,$thf(sK525)),skolemize(X135,$thf(sK526)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X140,$thf(sK528)),skolemize(X145,$thf(sK529)),skolemize(X146,$thf(sK530)),skolemize(X147,$thf(sK531)),skolemize(X148,$thf(sK532)),skolemize(X157,$thf(sK533)),skolemize(X158,$thf(sK534)),skolemize(X159,$thf(sK535)),skolemize(X160,$thf(sK536)),skolemize(X162,$thf(sK537)),skolemize(X163,$thf(sK538)),skolemize(X164,$thf(sK539)),skolemize(X165,$thf(sK540)),skolemize(X166,$thf(sK541)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X178,$thf(sK544)),skolemize(X179,$thf(sK545)),skolemize(X180,$thf(sK546)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X184,$thf(sK548)),skolemize(X187,$thf(sK549)),skolemize(X188,$thf(sK550)),skolemize(X189,$thf(sK551)),skolemize(X190,$thf(sK552)),skolemize(X191,$thf(sK553)),skolemize(X194,$thf(sK554)),skolemize(X195,$thf(sK555)),skolemize(X201,$thf(sK556)),skolemize(X202,$thf(sK557)),skolemize(X203,$thf(sK558)),skolemize(X204,$thf(sK559)),skolemize(X205,$thf(sK560)),skolemize(X206,$thf(sK561)),skolemize(X211,$thf(sK562)),skolemize(X212,$thf(sK563)),skolemize(X219,$thf(sK564)),skolemize(X220,$thf(sK565)),skolemize(X221,$thf(sK566)),skolemize(X222,$thf(sK567)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X225,$thf(sK569)),skolemize(X226,$thf(sK570)),skolemize(X229,$thf(sK571)),skolemize(X232,$thf(sK572)),skolemize(X233,$thf(sK573)),skolemize(X236,$thf(sK574)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X244,$thf(sK578)),skolemize(X245,$thf(sK579)),skolemize(X250,$thf(sK580)),skolemize(X251,$thf(sK581)),skolemize(X252,$thf(sK582)),skolemize(X253,$thf(sK583)),skolemize(X254,$thf(sK584)),skolemize(X255,$thf(sK585)),skolemize(X256,$thf(sK586)),skolemize(X262,$thf(sK587)),skolemize(X263,$thf(sK588)),skolemize(X268,$thf(sK589)),skolemize(X269,$thf(sK590)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X272,$thf(sK592))],[f267])).
thf(f269,plain,(
  (($true = sP60) | ? [X1 : (term > $o),X0 : (subst > term > subst > $o)] : (! [X7 : term,X6 : subst,X9 : subst,X8 : subst] : ((((X0 @ (comp @ X9 @ X6) @ (sub @ X7 @ X6) @ X8)) != $true) | (((X0 @ X9 @ X7 @ (comp @ X6 @ X8))) = $true)) & ? [X15 : term] : (! [X16 : term] : ((((X1 @ X16)) != $true) | ($true = ((X1 @ (sub @ X15 @ (push @ X16 @ id)))))) & ($true != ((X1 @ (lam @ X15))))) & (((hoasinduction_p_and_p_prime @ X0 @ X1)) = $true) & ($true != ((sP59 @ X0))) & ! [X5 : subst,X3 : subst,X4 : term,X2 : subst] : (($true != ((X0 @ X5 @ X4 @ (comp @ X3 @ X2)))) | ($true = ((X0 @ (comp @ X5 @ X3) @ (sub @ X4 @ X3) @ X2)))))) & (! [X1 : (term > $o),X0 : (subst > term > subst > $o)] : (? [X7 : term,X6 : subst,X9 : subst,X8 : subst] : ((((X0 @ (comp @ X9 @ X6) @ (sub @ X7 @ X6) @ X8)) = $true) & (((X0 @ X9 @ X7 @ (comp @ X6 @ X8))) != $true)) | ! [X15 : term] : (? [X16 : term] : ((((X1 @ X16)) = $true) & ($true != ((X1 @ (sub @ X15 @ (push @ X16 @ id)))))) | ($true = ((X1 @ (lam @ X15))))) | (((hoasinduction_p_and_p_prime @ X0 @ X1)) != $true) | ($true = ((sP59 @ X0))) | ? [X5 : subst,X3 : subst,X4 : term,X2 : subst] : (($true = ((X0 @ X5 @ X4 @ (comp @ X3 @ X2)))) & ($true != ((X0 @ (comp @ X5 @ X3) @ (sub @ X4 @ X3) @ X2))))) | ($true != sP60))),
  inference(nnf_transformation,[],[f76])).
thf(f270,plain,(
  (($true = sP60) | ? [X0 : (term > $o),X1 : (subst > term > subst > $o)] : (! [X2 : term,X3 : subst,X4 : subst,X5 : subst] : (($true != ((X1 @ (comp @ X4 @ X3) @ (sub @ X2 @ X3) @ X5))) | ($true = ((X1 @ X4 @ X2 @ (comp @ X3 @ X5))))) & ? [X6 : term] : (! [X7 : term] : ((((X0 @ X7)) != $true) | ($true = ((X0 @ (sub @ X6 @ (push @ X7 @ id)))))) & ($true != ((X0 @ (lam @ X6))))) & ($true = ((hoasinduction_p_and_p_prime @ X1 @ X0))) & ($true != ((sP59 @ X1))) & ! [X8 : subst,X9 : subst,X10 : term,X11 : subst] : ((((X1 @ X8 @ X10 @ (comp @ X9 @ X11))) != $true) | ($true = ((X1 @ (comp @ X8 @ X9) @ (sub @ X10 @ X9) @ X11)))))) & (! [X12 : (term > $o),X13 : (subst > term > subst > $o)] : (? [X14 : term,X15 : subst,X16 : subst,X17 : subst] : (($true = ((X13 @ (comp @ X16 @ X15) @ (sub @ X14 @ X15) @ X17))) & ($true != ((X13 @ X16 @ X14 @ (comp @ X15 @ X17))))) | ! [X18 : term] : (? [X19 : term] : (($true = ((X12 @ X19))) & (((X12 @ (sub @ X18 @ (push @ X19 @ id)))) != $true)) | ($true = ((X12 @ (lam @ X18))))) | ($true != ((hoasinduction_p_and_p_prime @ X13 @ X12))) | ($true = ((sP59 @ X13))) | ? [X20 : subst,X21 : subst,X22 : term,X23 : subst] : (($true = ((X13 @ X20 @ X22 @ (comp @ X21 @ X23)))) & ($true != ((X13 @ (comp @ X20 @ X21) @ (sub @ X22 @ X21) @ X23))))) | ($true != sP60))),
  inference(rectify,[],[f269])).
thf(f271,plain,(
  (($true = sP60) | (! [X2 : term,X3 : subst,X4 : subst,X5 : subst] : ((((sK594 @ (comp @ X4 @ X3) @ (sub @ X2 @ X3) @ X5)) != $true) | ($true = ((sK594 @ X4 @ X2 @ (comp @ X3 @ X5))))) & (! [X7 : term] : (($true != ((sK593 @ X7))) | (((sK593 @ (sub @ sK595 @ (push @ X7 @ id)))) = $true)) & ($true != ((sK593 @ (lam @ sK595))))) & ($true = ((hoasinduction_p_and_p_prime @ sK594 @ sK593))) & ($true != ((sP59 @ sK594))) & ! [X8 : subst,X9 : subst,X10 : term,X11 : subst] : (($true != ((sK594 @ X8 @ X10 @ (comp @ X9 @ X11)))) | ($true = ((sK594 @ (comp @ X8 @ X9) @ (sub @ X10 @ X9) @ X11)))))) & (! [X12 : (term > $o),X13 : (subst > term > subst > $o)] : ((($true = ((X13 @ (comp @ (sK598 @ X13) @ (sK597 @ X13)) @ (sub @ (sK596 @ X13) @ (sK597 @ X13)) @ (sK599 @ X13)))) & ($true != ((X13 @ (sK598 @ X13) @ (sK596 @ X13) @ (comp @ (sK597 @ X13) @ (sK599 @ X13)))))) | ! [X18 : term] : (((((X12 @ (sK600 @ X18 @ X12))) = $true) & (((X12 @ (sub @ X18 @ (push @ (sK600 @ X18 @ X12) @ id)))) != $true)) | ($true = ((X12 @ (lam @ X18))))) | ($true != ((hoasinduction_p_and_p_prime @ X13 @ X12))) | ($true = ((sP59 @ X13))) | (($true = ((X13 @ (sK601 @ X13) @ (sK603 @ X13) @ (comp @ (sK602 @ X13) @ (sK604 @ X13))))) & ($true != ((X13 @ (comp @ (sK601 @ X13) @ (sK602 @ X13)) @ (sub @ (sK603 @ X13) @ (sK602 @ X13)) @ (sK604 @ X13)))))) | ($true != sP60))),
  inference(skolemize,[status(esa),new_symbols(skolem,[sK593,sK594,sK595,vAPP]),skolemize(X0,$thf(sK593)),skolemize(X1,$thf(sK594)),skolemize(X6,$thf(sK595)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19))],[f270])).
thf(f272,plain,(
  ! [X0 : (subst > term > subst > $o)] : ((($true = ((sP59 @ X0))) | ! [X10 : (subst > term > term)] : (? [X14 : term] : ((((X0 @ id @ X14 @ id)) = $true) & (((X0 @ id @ (X10 @ id @ X14) @ id)) != $true)) | ? [X11 : subst,X13 : subst,X12 : term] : (((X10 @ (comp @ X11 @ X13) @ (sub @ X12 @ X13))) != ((sub @ (X10 @ X11 @ X12) @ X13))) | ($true = ((X0 @ id @ (hoaslam @ id @ (^[Y0 : subst]: ((^[Y1 : term]: (X10 @ Y0 @ Y1))))) @ id))))) & (? [X10 : (subst > term > term)] : (! [X14 : term] : ((((X0 @ id @ X14 @ id)) != $true) | (((X0 @ id @ (X10 @ id @ X14) @ id)) = $true)) & ! [X11 : subst,X13 : subst,X12 : term] : (((X10 @ (comp @ X11 @ X13) @ (sub @ X12 @ X13))) = ((sub @ (X10 @ X11 @ X12) @ X13))) & ($true != ((X0 @ id @ (hoaslam @ id @ (^[Y0 : subst]: ((^[Y1 : term]: (X10 @ Y0 @ Y1))))) @ id)))) | ($true != ((sP59 @ X0)))))),
  inference(nnf_transformation,[],[f75])).
thf(f273,plain,(
  ! [X0 : (subst > term > subst > $o)] : ((($true = ((sP59 @ X0))) | ! [X1 : (subst > term > term)] : (? [X2 : term] : (($true = ((X0 @ id @ X2 @ id))) & ($true != ((X0 @ id @ (X1 @ id @ X2) @ id)))) | ? [X3 : subst,X4 : subst,X5 : term] : (((X1 @ (comp @ X3 @ X4) @ (sub @ X5 @ X4))) != ((sub @ (X1 @ X3 @ X5) @ X4))) | (((X0 @ id @ (hoaslam @ id @ (^[Y0 : subst]: ((^[Y1 : term]: (X1 @ Y0 @ Y1))))) @ id)) = $true))) & (? [X6 : (subst > term > term)] : (! [X7 : term] : (($true != ((X0 @ id @ X7 @ id))) | ($true = ((X0 @ id @ (X6 @ id @ X7) @ id)))) & ! [X8 : subst,X9 : subst,X10 : term] : (((X6 @ (comp @ X8 @ X9) @ (sub @ X10 @ X9))) = ((sub @ (X6 @ X8 @ X10) @ X9))) & ($true != ((X0 @ id @ (hoaslam @ id @ (^[Y0 : subst]: ((^[Y1 : term]: (X6 @ Y0 @ Y1))))) @ id)))) | ($true != ((sP59 @ X0)))))),
  inference(rectify,[],[f272])).
thf(f274,plain,(
  ! [X0 : (subst > term > subst > $o)] : ((($true = ((sP59 @ X0))) | ! [X1 : (subst > term > term)] : ((($true = ((X0 @ id @ (sK605 @ X1 @ X0) @ id))) & ($true != ((X0 @ id @ (X1 @ id @ (sK605 @ X1 @ X0)) @ id)))) | (((X1 @ (comp @ (sK606 @ X1) @ (sK607 @ X1)) @ (sub @ (sK608 @ X1) @ (sK607 @ X1)))) != ((sub @ (X1 @ (sK606 @ X1) @ (sK608 @ X1)) @ (sK607 @ X1)))) | (((X0 @ id @ (hoaslam @ id @ (^[Y0 : subst]: ((^[Y1 : term]: (X1 @ Y0 @ Y1))))) @ id)) = $true))) & ((! [X7 : term] : (($true != ((X0 @ id @ X7 @ id))) | ($true = ((X0 @ id @ (sK609 @ X0 @ id @ X7) @ id)))) & ! [X8 : subst,X9 : subst,X10 : term] : (((sK609 @ X0 @ (comp @ X8 @ X9) @ (sub @ X10 @ X9))) = ((sub @ (sK609 @ X0 @ X8 @ X10) @ X9))) & (((X0 @ id @ (hoaslam @ id @ (^[Y0 : subst]: ((^[Y1 : term]: (sK609 @ X0 @ Y0 @ Y1))))) @ id)) != $true)) | ($true != ((sP59 @ X0)))))),
  inference(skolemize,[status(esa),new_symbols(skolem,[vAPP]),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19)),skolemize(X25,$thf(sK70 @ X19))],[f273])).
thf(f275,plain,(
  (($true != sP60) | (hoasinduction_lem3v2 != $true)) & (($true = sP60) | (hoasinduction_lem3v2 = $true))),
  inference(nnf_transformation,[],[f77])).
thf(f755,plain,(
  ( ! [X8 : subst,X6 : term,X7 : subst,X5 : subst] : (($true = ((sK441 @ X5 @ X6 @ (comp @ X8 @ X7)))) | ($true != ((sK441 @ (comp @ X5 @ X8) @ (sub @ X6 @ X8) @ X7))) | ($true = sP1)) )),
  inference(cnf_transformation,[],[f261])).
thf(f756,plain,(
  ($true = sP1) | ($true != ((sK442 @ (lam @ sK443))))),
  inference(cnf_transformation,[],[f261])).
thf(f758,plain,(
  ($true != ((sP0 @ sK441))) | ($true = sP1)),
  inference(cnf_transformation,[],[f261])).
thf(f759,plain,(
  ( ! [X3 : term] : (($true = sP1) | (((sK442 @ X3)) = ((sK441 @ id @ X3 @ id)))) )),
  inference(cnf_transformation,[],[f261])).
thf(f763,plain,(
  ( ! [X0 : (subst > term > subst > $o),X1 : (subst > term > term)] : ((((X1 @ (comp @ (sK455 @ X1) @ (sK457 @ X1)) @ (sub @ (sK456 @ X1) @ (sK457 @ X1)))) != ((sub @ (X1 @ (sK455 @ X1) @ (sK456 @ X1)) @ (sK457 @ X1)))) | ($true = ((X0 @ id @ (lam @ (X1 @ sh @ one)) @ id))) | (((sP0 @ X0)) = $true) | ($true = ((X0 @ id @ (sK454 @ X1 @ X0) @ id)))) )),
  inference(cnf_transformation,[],[f264])).
thf(f773,plain,(
  (hoasinduction_lem3v2 = $true) | ($true != sP1)),
  inference(cnf_transformation,[],[f268])).
thf(f779,plain,(
  (((d2term @ d_one)) = one)),
  inference(cnf_transformation,[],[f268])).
thf(f800,plain,(
  ( ! [X235 : d_subst] : ((d_id = X235)) )),
  inference(cnf_transformation,[],[f268])).
thf(f817,plain,(
  ( ! [X223 : subst] : ((((d2subst @ (sK568 @ X223))) = X223)) )),
  inference(cnf_transformation,[],[f268])).
thf(f878,plain,(
  (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id))))))))),
  inference(cnf_transformation,[],[f268])).
thf(f891,plain,(
  ( ! [X142 : d_term] : ((d_one = X142)) )),
  inference(cnf_transformation,[],[f268])).
thf(f910,plain,(
  (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one)))))),
  inference(cnf_transformation,[],[f268])).
thf(f963,plain,(
  (((d2subst @ d_id)) = id)),
  inference(cnf_transformation,[],[f268])).
thf(f966,plain,(
  (((d2subst @ d_id)) = sh)),
  inference(cnf_transformation,[],[f268])).
thf(f979,plain,(
  ( ! [X4 : term] : ((((d2term @ (sK461 @ X4))) = X4)) )),
  inference(cnf_transformation,[],[f268])).
thf(f999,plain,(
  ($true = sP60) | ($true != ((sP59 @ sK594)))),
  inference(cnf_transformation,[],[f271])).
thf(f1000,plain,(
  ($true = sP60) | ($true = ((hoasinduction_p_and_p_prime @ sK594 @ sK593)))),
  inference(cnf_transformation,[],[f271])).
thf(f1001,plain,(
  ($true = sP60) | ($true != ((sK593 @ (lam @ sK595))))),
  inference(cnf_transformation,[],[f271])).
thf(f1002,plain,(
  ( ! [X7 : term] : (($true = sP60) | (((sK593 @ (sub @ sK595 @ (push @ X7 @ id)))) = $true) | ($true != ((sK593 @ X7)))) )),
  inference(cnf_transformation,[],[f271])).
thf(f1008,plain,(
  ( ! [X0 : (subst > term > subst > $o),X1 : (subst > term > term)] : (($true = ((sP59 @ X0))) | ($true = ((X0 @ id @ (sK605 @ X1 @ X0) @ id))) | (((X1 @ (comp @ (sK606 @ X1) @ (sK607 @ X1)) @ (sub @ (sK608 @ X1) @ (sK607 @ X1)))) != ((sub @ (X1 @ (sK606 @ X1) @ (sK608 @ X1)) @ (sK607 @ X1)))) | (((X0 @ id @ (hoaslam @ id @ (^[Y0 : subst]: ((^[Y1 : term]: (X1 @ Y0 @ Y1))))) @ id)) = $true)) )),
  inference(cnf_transformation,[],[f274])).
thf(f1010,plain,(
  (hoasinduction_lem3v2 != $true) | ($true != sP60)),
  inference(cnf_transformation,[],[f275])).
thf(f1012,plain,(
  ($true = (((^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id))))))) @ sK594 @ sK593))) | ($true = sP60)),
  inference(definition_unfolding,[],[f1000,f878])).
thf(f1021,plain,(
  ( ! [X0 : (subst > term > subst > $o),X1 : (subst > term > term)] : ((((X1 @ (comp @ (sK606 @ X1) @ (sK607 @ X1)) @ (sub @ (sK608 @ X1) @ (sK607 @ X1)))) != ((sub @ (X1 @ (sK606 @ X1) @ (sK608 @ X1)) @ (sK607 @ X1)))) | ($true = ((X0 @ id @ (sK605 @ X1 @ X0) @ id))) | ($true = ((sP59 @ X0))) | (((X0 @ id @ ((^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one)))) @ id @ (^[Y0 : subst]: ((^[Y1 : term]: (X1 @ Y0 @ Y1))))) @ id)) = $true)) )),
  inference(definition_unfolding,[],[f1008,f910])).
thf(f1044,plain,(
  ( ! [X3 : term] : (($true = sP1) | ($false = ((sK441 @ id @ X3 @ id))) | ($true = ((sK442 @ X3)))) )),
  inference(iff_proxy_clausification,[],[f759])).
thf(f1085,plain,(
  ($true = ((!! @ term @ (^[Y0 : term]: ((sK593 @ Y0) = (sK594 @ id @ Y0 @ id)))))) | ($true = sP60)),
  inference(beta-eta_normalization,[],[f1012])).
thf(f1086,plain,(
  ( ! [X1 : term] : (($true = sP60) | ($true = (((^[Y0 : term]: ((sK593 @ Y0) = (sK594 @ id @ Y0 @ id))) @ X1)))) )),
  inference(pi_proxy_clausification,[],[f1085])).
thf(f1087,plain,(
  ( ! [X1 : term] : (($true = (((sK593 @ X1) = (sK594 @ id @ X1 @ id)))) | ($true = sP60)) )),
  inference(beta-eta_normalization,[],[f1086])).
thf(f1088,plain,(
  ( ! [X1 : term] : ((((sK594 @ id @ X1 @ id)) = ((sK593 @ X1))) | ($true = sP60)) )),
  inference(equality_proxy_clausification,[],[f1087])).
thf(f1090,plain,(
  ( ! [X1 : term] : (($true = sP60) | ($false = ((sK594 @ id @ X1 @ id))) | ($true = ((sK593 @ X1)))) )),
  inference(iff_proxy_clausification,[],[f1088])).
thf(f1113,plain,(
  ( ! [X0 : (subst > term > subst > $o),X1 : (subst > term > term)] : (($true = ((X0 @ id @ (sK605 @ X1 @ X0) @ id))) | ($true = ((X0 @ id @ (d2term @ d_one) @ id))) | ($true = ((sP59 @ X0))) | (((X1 @ (comp @ (sK606 @ X1) @ (sK607 @ X1)) @ (sub @ (sK608 @ X1) @ (sK607 @ X1)))) != ((sub @ (X1 @ (sK606 @ X1) @ (sK608 @ X1)) @ (sK607 @ X1))))) )),
  inference(beta-eta_normalization,[],[f1021])).
thf(f1441,definition,(
  spl618_52 <=> ($true = ((sP0 @ sK441)))),
  introduced(definition,[new_symbols(definition,[spl618_52])],[avatar_definition])).
thf(f1443,plain,(
  ($true != ((sP0 @ sK441))) | spl618_52),
  inference(avatar_component_clause,[],[f1441])).
thf(f1445,definition,(
  spl618_53 <=> ($true = sP1)),
  introduced(definition,[new_symbols(definition,[spl618_53])],[avatar_definition])).
thf(f1448,plain,(
  ~spl618_52 | spl618_53),
  inference(avatar_split_clause,[],[f758,f1445,f1441])).
thf(f1471,definition,(
  spl618_59 <=> (((d2subst @ d_id)) = sh)),
  introduced(definition,[new_symbols(definition,[spl618_59])],[avatar_definition])).
thf(f1473,plain,(
  (((d2subst @ d_id)) = sh) | ~spl618_59),
  inference(avatar_component_clause,[],[f1471])).
thf(f1474,plain,(
  spl618_59),
  inference(avatar_split_clause,[],[f966,f1471])).
thf(f1537,definition,(
  spl618_75 <=> ($true = sP60)),
  introduced(definition,[new_symbols(definition,[spl618_75])],[avatar_definition])).
thf(f1621,definition,(
  spl618_95 <=> (hoasinduction_lem3v2 = $true)),
  introduced(definition,[new_symbols(definition,[spl618_95])],[avatar_definition])).
thf(f1690,definition,(
  spl618_111 <=> ! [X3 : term] : (($false = ((sK441 @ id @ X3 @ id))) | ($true = ((sK442 @ X3))))),
  introduced(definition,[new_symbols(definition,[spl618_111])],[avatar_definition])).
thf(f1691,plain,(
  ( ! [X3 : term] : (($true = ((sK442 @ X3))) | ($false = ((sK441 @ id @ X3 @ id)))) ) | ~spl618_111),
  inference(avatar_component_clause,[],[f1690])).
thf(f1692,plain,(
  spl618_111 | spl618_53),
  inference(avatar_split_clause,[],[f1044,f1445,f1690])).
thf(f1852,definition,(
  spl618_150 <=> (((d2subst @ d_id)) = id)),
  introduced(definition,[new_symbols(definition,[spl618_150])],[avatar_definition])).
thf(f1854,plain,(
  (((d2subst @ d_id)) = id) | ~spl618_150),
  inference(avatar_component_clause,[],[f1852])).
thf(f1855,plain,(
  spl618_150),
  inference(avatar_split_clause,[],[f963,f1852])).
thf(f1888,plain,(
  spl618_95 | ~spl618_53),
  inference(avatar_split_clause,[],[f773,f1445,f1621])).
thf(f2210,plain,(
  ~spl618_75 | ~spl618_95),
  inference(avatar_split_clause,[],[f1010,f1621,f1537])).
thf(f2497,definition,(
  spl618_299 <=> ! [X7 : term] : ((((sK593 @ (sub @ sK595 @ (push @ X7 @ id)))) = $true) | ($true != ((sK593 @ X7))))),
  introduced(definition,[new_symbols(definition,[spl618_299])],[avatar_definition])).
thf(f2498,plain,(
  ( ! [X7 : term] : (($true != ((sK593 @ X7))) | (((sK593 @ (sub @ sK595 @ (push @ X7 @ id)))) = $true)) ) | ~spl618_299),
  inference(avatar_component_clause,[],[f2497])).
thf(f2499,plain,(
  spl618_75 | spl618_299),
  inference(avatar_split_clause,[],[f1002,f2497,f1537])).
thf(f2550,definition,(
  spl618_311 <=> ! [X1 : term] : (($false = ((sK594 @ id @ X1 @ id))) | ($true = ((sK593 @ X1))))),
  introduced(definition,[new_symbols(definition,[spl618_311])],[avatar_definition])).
thf(f2551,plain,(
  ( ! [X1 : term] : (($true = ((sK593 @ X1))) | ($false = ((sK594 @ id @ X1 @ id)))) ) | ~spl618_311),
  inference(avatar_component_clause,[],[f2550])).
thf(f2552,plain,(
  spl618_311 | spl618_75),
  inference(avatar_split_clause,[],[f1090,f1537,f2550])).
thf(f3062,definition,(
  spl618_428 <=> (((d2term @ d_one)) = one)),
  introduced(definition,[new_symbols(definition,[spl618_428])],[avatar_definition])).
thf(f3064,plain,(
  (((d2term @ d_one)) = one) | ~spl618_428),
  inference(avatar_component_clause,[],[f3062])).
thf(f3065,plain,(
  spl618_428),
  inference(avatar_split_clause,[],[f779,f3062])).
thf(f3348,definition,(
  spl618_494 <=> ($true = ((sK442 @ (lam @ sK443))))),
  introduced(definition,[new_symbols(definition,[spl618_494])],[avatar_definition])).
thf(f3350,plain,(
  ($true != ((sK442 @ (lam @ sK443)))) | spl618_494),
  inference(avatar_component_clause,[],[f3348])).
thf(f3351,plain,(
  spl618_53 | ~spl618_494),
  inference(avatar_split_clause,[],[f756,f3348,f1445])).
thf(f3883,definition,(
  spl618_615 <=> ($true = ((sP59 @ sK594)))),
  introduced(definition,[new_symbols(definition,[spl618_615])],[avatar_definition])).
thf(f3885,plain,(
  ($true != ((sP59 @ sK594))) | spl618_615),
  inference(avatar_component_clause,[],[f3883])).
thf(f3886,plain,(
  spl618_75 | ~spl618_615),
  inference(avatar_split_clause,[],[f999,f3883,f1537])).
thf(f3901,definition,(
  spl618_619 <=> ($true = ((sK593 @ (lam @ sK595))))),
  introduced(definition,[new_symbols(definition,[spl618_619])],[avatar_definition])).
thf(f3903,plain,(
  ($true != ((sK593 @ (lam @ sK595)))) | spl618_619),
  inference(avatar_component_clause,[],[f3901])).
thf(f3904,plain,(
  spl618_75 | ~spl618_619),
  inference(avatar_split_clause,[],[f1001,f3901,f1537])).
thf(f3997,definition,(
  spl618_641 <=> ! [X5 : subst,X7 : subst,X6 : term,X8 : subst] : (($true = ((sK441 @ X5 @ X6 @ (comp @ X8 @ X7)))) | ($true != ((sK441 @ (comp @ X5 @ X8) @ (sub @ X6 @ X8) @ X7))))),
  introduced(definition,[new_symbols(definition,[spl618_641])],[avatar_definition])).
thf(f3998,plain,(
  ( ! [X8 : subst,X6 : term,X7 : subst,X5 : subst] : (($true = ((sK441 @ X5 @ X6 @ (comp @ X8 @ X7)))) | ($true != ((sK441 @ (comp @ X5 @ X8) @ (sub @ X6 @ X8) @ X7)))) ) | ~spl618_641),
  inference(avatar_component_clause,[],[f3997])).
thf(f3999,plain,(
  spl618_641 | spl618_53),
  inference(avatar_split_clause,[],[f755,f1445,f3997])).
thf(f4396,plain,(
  (id = sh) | (~spl618_59 | ~spl618_150)),
  inference(superposition,[],[f1854,f1473])).
thf(f4399,definition,(
  spl618_692 <=> (id = sh)),
  introduced(definition,[new_symbols(definition,[spl618_692])],[avatar_definition])).
thf(f4401,plain,(
  (id = sh) | ~spl618_692),
  inference(avatar_component_clause,[],[f4399])).
thf(f4402,plain,(
  spl618_692 | ~spl618_59 | ~spl618_150),
  inference(avatar_split_clause,[],[f4396,f1852,f1471,f4399])).
thf(f4420,plain,(
  ( ! [X223 : subst] : ((((d2subst @ d_id)) = X223)) )),
  inference(forward_demodulation,[],[f817,f800])).
thf(f4586,plain,(
  ( ! [X4 : term] : ((((d2term @ d_one)) = X4)) )),
  inference(forward_demodulation,[],[f979,f891])).
thf(f4609,plain,(
  ( ! [X0 : term,X1 : term] : ((X0 = X1)) )),
  inference(superposition,[],[f4586,f4586])).
thf(f4939,plain,(
  ( ! [X8 : subst,X6 : term,X7 : subst,X5 : subst] : (($true = ((sK441 @ X5 @ X6 @ (d2subst @ d_id)))) | ($true != ((sK441 @ (comp @ X5 @ X8) @ (sub @ X6 @ X8) @ X7)))) ) | ~spl618_641),
  inference(forward_demodulation,[],[f3998,f4420])).
thf(f4940,plain,(
  ( ! [X8 : subst,X6 : term,X7 : subst,X5 : subst] : (($true != ((sK441 @ (comp @ X5 @ X8) @ (sub @ X6 @ X8) @ X7))) | ($true = ((sK441 @ X5 @ X6 @ sh)))) ) | (~spl618_59 | ~spl618_641)),
  inference(forward_demodulation,[],[f4939,f1473])).
thf(f4941,plain,(
  ( ! [X8 : subst,X6 : term,X7 : subst,X5 : subst] : (($true != ((sK441 @ (d2subst @ d_id) @ (sub @ X6 @ X8) @ X7))) | ($true = ((sK441 @ X5 @ X6 @ sh)))) ) | (~spl618_59 | ~spl618_641)),
  inference(forward_demodulation,[],[f4940,f4420])).
thf(f4942,plain,(
  ( ! [X8 : subst,X6 : term,X7 : subst,X5 : subst] : (($true != ((sK441 @ sh @ (sub @ X6 @ X8) @ X7))) | ($true = ((sK441 @ X5 @ X6 @ sh)))) ) | (~spl618_59 | ~spl618_641)),
  inference(forward_demodulation,[],[f4941,f1473])).
thf(f4943,plain,(
  ( ! [X8 : subst,X6 : term,X7 : subst,X5 : subst] : (($true = ((sK441 @ X5 @ X6 @ sh))) | ($true != ((sK441 @ id @ (sub @ X6 @ X8) @ X7)))) ) | (~spl618_59 | ~spl618_641 | ~spl618_692)),
  inference(forward_demodulation,[],[f4942,f4401])).
thf(f4944,plain,(
  ( ! [X8 : subst,X6 : term,X7 : subst,X5 : subst] : (($true != ((sK441 @ id @ (sub @ X6 @ X8) @ X7))) | (((sK441 @ X5 @ X6 @ id)) = $true)) ) | (~spl618_59 | ~spl618_641 | ~spl618_692)),
  inference(forward_demodulation,[],[f4943,f4401])).
thf(f4945,plain,(
  ( ! [X6 : term,X7 : subst,X5 : subst] : ((((sK441 @ X5 @ X6 @ id)) = $true) | ($true != ((sK441 @ id @ (d2term @ d_one) @ X7)))) ) | (~spl618_59 | ~spl618_641 | ~spl618_692)),
  inference(forward_demodulation,[],[f4944,f4586])).
thf(f4946,plain,(
  ( ! [X6 : term,X7 : subst,X5 : subst] : ((((sK441 @ X5 @ X6 @ id)) = $true) | ($true != ((sK441 @ id @ one @ X7)))) ) | (~spl618_59 | ~spl618_428 | ~spl618_641 | ~spl618_692)),
  inference(forward_demodulation,[],[f4945,f3064])).
thf(f4948,definition,(
  spl618_749 <=> ! [X6 : term,X5 : subst] : (((sK441 @ X5 @ X6 @ id)) = $true)),
  introduced(definition,[new_symbols(definition,[spl618_749])],[avatar_definition])).
thf(f4949,plain,(
  ( ! [X6 : term,X5 : subst] : ((((sK441 @ X5 @ X6 @ id)) = $true)) ) | ~spl618_749),
  inference(avatar_component_clause,[],[f4948])).
thf(f4951,definition,(
  spl618_750 <=> ! [X7 : subst] : ($true != ((sK441 @ id @ one @ X7)))),
  introduced(definition,[new_symbols(definition,[spl618_750])],[avatar_definition])).
thf(f4952,plain,(
  ( ! [X7 : subst] : (($true != ((sK441 @ id @ one @ X7)))) ) | ~spl618_750),
  inference(avatar_component_clause,[],[f4951])).
thf(f4953,plain,(
  spl618_749 | spl618_750 | ~spl618_59 | ~spl618_428 | ~spl618_641 | ~spl618_692),
  inference(avatar_split_clause,[],[f4946,f4399,f3997,f3062,f1471,f4951,f4948])).
thf(f4996,plain,(
  (((sK442 @ (d2term @ d_one))) != $true) | spl618_494),
  inference(forward_demodulation,[],[f3350,f4586])).
thf(f4997,plain,(
  ($true != ((sK442 @ one))) | (~spl618_428 | spl618_494)),
  inference(forward_demodulation,[],[f4996,f3064])).
thf(f4999,definition,(
  spl618_757 <=> ($true = ((sK442 @ one)))),
  introduced(definition,[new_symbols(definition,[spl618_757])],[avatar_definition])).
thf(f5001,plain,(
  ($true != ((sK442 @ one))) | spl618_757),
  inference(avatar_component_clause,[],[f4999])).
thf(f5002,plain,(
  ~spl618_757 | ~spl618_428 | spl618_494),
  inference(avatar_split_clause,[],[f4997,f3348,f3062,f4999])).
thf(f5013,plain,(
  ($true != ((sK593 @ (d2term @ d_one)))) | spl618_619),
  inference(forward_demodulation,[],[f3903,f4586])).
thf(f5014,plain,(
  (((sK593 @ one)) != $true) | (~spl618_428 | spl618_619)),
  inference(forward_demodulation,[],[f5013,f3064])).
thf(f5016,definition,(
  spl618_759 <=> (((sK593 @ one)) = $true)),
  introduced(definition,[new_symbols(definition,[spl618_759])],[avatar_definition])).
thf(f5019,plain,(
  ~spl618_759 | ~spl618_428 | spl618_619),
  inference(avatar_split_clause,[],[f5014,f3901,f3062,f5016])).
thf(f5364,plain,(
  ( ! [X0 : term] : (($true != $true) | (((sK593 @ (sub @ sK595 @ (push @ X0 @ id)))) = $true) | ($false = ((sK594 @ id @ X0 @ id)))) ) | (~spl618_299 | ~spl618_311)),
  inference(superposition,[],[f2498,f2551])).
thf(f5365,plain,(
  ( ! [X0 : term] : (($false = ((sK594 @ id @ X0 @ id))) | (((sK593 @ (sub @ sK595 @ (push @ X0 @ id)))) = $true)) ) | (~spl618_299 | ~spl618_311)),
  inference(trivial_inequality_removal,[],[f5364])).
thf(f5366,plain,(
  ( ! [X0 : term] : (($true = ((sK593 @ (d2term @ d_one)))) | ($false = ((sK594 @ id @ X0 @ id)))) ) | (~spl618_299 | ~spl618_311)),
  inference(forward_demodulation,[],[f5365,f4586])).
thf(f5367,plain,(
  ( ! [X0 : term] : (($false = ((sK594 @ id @ X0 @ id))) | (((sK593 @ one)) = $true)) ) | (~spl618_299 | ~spl618_311 | ~spl618_428)),
  inference(forward_demodulation,[],[f5366,f3064])).
thf(f5369,definition,(
  spl618_767 <=> ! [X0 : term] : ($false = ((sK594 @ id @ X0 @ id)))),
  introduced(definition,[new_symbols(definition,[spl618_767])],[avatar_definition])).
thf(f5370,plain,(
  ( ! [X0 : term] : (($false = ((sK594 @ id @ X0 @ id)))) ) | ~spl618_767),
  inference(avatar_component_clause,[],[f5369])).
thf(f5371,plain,(
  spl618_767 | spl618_759 | ~spl618_299 | ~spl618_311 | ~spl618_428),
  inference(avatar_split_clause,[],[f5367,f3062,f2550,f2497,f5016,f5369])).
thf(f5965,plain,(
  ( ! [X3 : term] : (($true = ((sK442 @ X3))) | ($false = $true)) ) | (~spl618_111 | ~spl618_749)),
  inference(forward_demodulation,[],[f1691,f4949])).
thf(f5966,plain,(
  ( ! [X3 : term] : (($true = ((sK442 @ X3)))) ) | (~spl618_111 | ~spl618_749)),
  inference(trivial_inequality_removal,[],[f5965])).
thf(f5997,plain,(
  ($true != $true) | (~spl618_111 | ~spl618_749 | spl618_757)),
  inference(superposition,[],[f5001,f5966])).
thf(f6000,plain,(
  $false | (~spl618_111 | ~spl618_749 | spl618_757)),
  inference(trivial_inequality_removal,[],[f5997])).
thf(f6001,plain,(
  ~spl618_111 | ~spl618_749 | spl618_757),
  inference(avatar_contradiction_clause,[],[f6000])).
thf(f6864,plain,(
  ( ! [X0 : (subst > term > subst > $o),X1 : (subst > term > term)] : (($true = ((X0 @ id @ (lam @ (X1 @ sh @ one)) @ id))) | (((sP0 @ X0)) = $true) | ($true = ((X0 @ id @ (sK454 @ X1 @ X0) @ id)))) )),
  inference(forward_subsumption_resolution,[],[f763,f4609])).
thf(f6865,plain,(
  ( ! [X0 : (subst > term > subst > $o),X1 : (subst > term > term)] : ((((sP0 @ X0)) = $true) | ($true = ((X0 @ id @ (d2term @ d_one) @ id))) | ($true = ((X0 @ id @ (sK454 @ X1 @ X0) @ id)))) )),
  inference(forward_demodulation,[],[f6864,f4586])).
thf(f6866,plain,(
  ( ! [X0 : (subst > term > subst > $o),X1 : (subst > term > term)] : (($true = ((X0 @ id @ one @ id))) | (((sP0 @ X0)) = $true) | ($true = ((X0 @ id @ (sK454 @ X1 @ X0) @ id)))) ) | ~spl618_428),
  inference(forward_demodulation,[],[f6865,f3064])).
thf(f6867,plain,(
  ( ! [X0 : (subst > term > subst > $o)] : ((((sP0 @ X0)) = $true) | ($true = ((X0 @ id @ (d2term @ d_one) @ id))) | ($true = ((X0 @ id @ one @ id)))) ) | ~spl618_428),
  inference(forward_demodulation,[],[f6866,f4586])).
thf(f6868,plain,(
  ( ! [X0 : (subst > term > subst > $o)] : (($true = ((X0 @ id @ one @ id))) | ($true = ((X0 @ id @ one @ id))) | (((sP0 @ X0)) = $true)) ) | ~spl618_428),
  inference(forward_demodulation,[],[f6867,f3064])).
thf(f6869,plain,(
  ( ! [X0 : (subst > term > subst > $o)] : ((((sP0 @ X0)) = $true) | ($true = ((X0 @ id @ one @ id)))) ) | ~spl618_428),
  inference(duplicate_literal_removal,[],[f6868])).
thf(f8650,definition,(
  spl618_860 <=> ! [X13 : (subst > term > subst > $o)] : (($true = ((sP0 @ X13))) | (((X13 @ id @ one @ id)) = $true))),
  introduced(definition,[new_symbols(definition,[spl618_860])],[avatar_definition])).
thf(f8651,plain,(
  ( ! [X13 : (subst > term > subst > $o)] : (($true = ((sP0 @ X13))) | (((X13 @ id @ one @ id)) = $true)) ) | ~spl618_860),
  inference(avatar_component_clause,[],[f8650])).
thf(f10031,plain,(
  ( ! [X0 : (subst > term > subst > $o),X1 : (subst > term > term)] : (($true = ((X0 @ id @ (sK605 @ X1 @ X0) @ id))) | ($true = ((sP59 @ X0))) | ($true = ((X0 @ id @ (d2term @ d_one) @ id)))) )),
  inference(forward_subsumption_resolution,[],[f1113,f4609])).
thf(f10032,plain,(
  ( ! [X0 : (subst > term > subst > $o)] : (($true = ((X0 @ id @ (d2term @ d_one) @ id))) | ($true = ((sP59 @ X0))) | ($true = ((X0 @ id @ (d2term @ d_one) @ id)))) )),
  inference(forward_demodulation,[],[f10031,f4586])).
thf(f10033,plain,(
  ( ! [X0 : (subst > term > subst > $o)] : (($true = ((X0 @ id @ (d2term @ d_one) @ id))) | ($true = ((sP59 @ X0)))) )),
  inference(duplicate_literal_removal,[],[f10032])).
thf(f10034,plain,(
  ( ! [X0 : (subst > term > subst > $o)] : (($true = ((sP59 @ X0))) | ($true = ((X0 @ id @ one @ id)))) ) | ~spl618_428),
  inference(forward_demodulation,[],[f10033,f3064])).
thf(f10204,plain,(
  ($true = ((sK594 @ id @ one @ id))) | (~spl618_428 | spl618_615)),
  inference(unit_resulting_resolution,[],[f10034,f3885])).
thf(f10225,plain,(
  ($false = $true) | (~spl618_428 | spl618_615 | ~spl618_767)),
  inference(forward_demodulation,[],[f10204,f5370])).
thf(f10226,plain,(
  $false | (~spl618_428 | spl618_615 | ~spl618_767)),
  inference(trivial_inequality_removal,[],[f10225])).
thf(f10227,plain,(
  ~spl618_428 | spl618_615 | ~spl618_767),
  inference(avatar_contradiction_clause,[],[f10226])).
thf(f39467,plain,(
  spl618_860 | ~spl618_428),
  inference(avatar_split_clause,[],[f6869,f3062,f8650])).
thf(f40872,plain,(
  ($true = ((sP0 @ sK441))) | (~spl618_750 | ~spl618_860)),
  inference(unit_resulting_resolution,[],[f8651,f4952])).
thf(f40931,plain,(
  $false | (spl618_52 | ~spl618_750 | ~spl618_860)),
  inference(forward_subsumption_resolution,[],[f40872,f1443])).
thf(f40932,plain,(
  spl618_52 | ~spl618_750 | ~spl618_860),
  inference(avatar_contradiction_clause,[],[f40931])).
cnf(s26, plain, ~spl618_52 | spl618_53, inference(sat_conversion,[],[f1448])).
cnf(s30, plain, spl618_59, inference(sat_conversion,[],[f1474])).
cnf(s69, plain, spl618_53 | spl618_111, inference(sat_conversion,[],[f1692])).
cnf(s99, plain, spl618_150, inference(sat_conversion,[],[f1855])).
cnf(s106, plain, ~spl618_53 | spl618_95, inference(sat_conversion,[],[f1888])).
cnf(s178, plain, ~spl618_75 | ~spl618_95, inference(sat_conversion,[],[f2210])).
cnf(s242, plain, spl618_75 | spl618_299, inference(sat_conversion,[],[f2499])).
cnf(s252, plain, spl618_75 | spl618_311, inference(sat_conversion,[],[f2552])).
cnf(s376, plain, spl618_428, inference(sat_conversion,[],[f3065])).
cnf(s447, plain, spl618_53 | ~spl618_494, inference(sat_conversion,[],[f3351])).
cnf(s587, plain, spl618_75 | ~spl618_615, inference(sat_conversion,[],[f3886])).
cnf(s591, plain, spl618_75 | ~spl618_619, inference(sat_conversion,[],[f3904])).
cnf(s614, plain, spl618_53 | spl618_641, inference(sat_conversion,[],[f3999])).
cnf(s672, plain, ~spl618_59 | ~spl618_150 | spl618_692, inference(sat_conversion,[],[f4402])).
cnf(s722, plain, ~spl618_59 | ~spl618_428 | ~spl618_641 | ~spl618_692 | spl618_749 | spl618_750, inference(sat_conversion,[],[f4953])).
cnf(s728, plain, ~spl618_428 | spl618_494 | ~spl618_757, inference(sat_conversion,[],[f5002])).
cnf(s731, plain, ~spl618_428 | spl618_619 | ~spl618_759, inference(sat_conversion,[],[f5019])).
cnf(s767, plain, ~spl618_299 | ~spl618_311 | ~spl618_428 | spl618_759 | spl618_767, inference(sat_conversion,[],[f5371])).
cnf(s819, plain, ~spl618_111 | ~spl618_749 | spl618_757, inference(sat_conversion,[],[f6001])).
cnf(s1365, plain, ~spl618_428 | spl618_615 | ~spl618_767, inference(sat_conversion,[],[f10227])).
cnf(s3147, plain, ~spl618_428 | spl618_860, inference(sat_conversion,[],[f39467])).
cnf(s3182, plain, spl618_52 | ~spl618_750 | ~spl618_860, inference(sat_conversion,[],[f40932])).
cnf(s3254, plain, spl618_860, inference(rat,[],[s3147,s376])).
cnf(s3357, plain, spl618_692, inference(rat,[],[s672,s99,s30])).
cnf(s3486, plain, spl618_53 | spl618_750, inference(rat,[],[s819,s728,s722,s69,s447,s614,s376,s30,s3357])).
cnf(s3487, plain, spl618_75, inference(rat,[],[s767,s1365,s731,s242,s252,s587,s591,s376])).
cnf(s3499, plain, ~spl618_95, inference(rat,[],[s178,s3487])).
cnf(s3507, plain, ~spl618_53, inference(rat,[],[s106,s3499])).
cnf(s3515, plain, ~spl618_52, inference(rat,[],[s26,s3507])).
cnf(s3516, plain, spl618_750, inference(rat,[],[s3486,s3507])).
cnf(s3522, plain, $false, inference(rat,[],[s3182,s3254,s3516,s3515])).
thf(f40982,plain,(
  $false),
  inference(avatar_sat_refutation,[],[s3522])).
% SZS output end Proof for theBenchmark