%----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, vPI: !>[X0: $tType]:(((X0 > $o) > $o))).
thf(func_def_130, type, db1: !>[X0: $tType]:(X0)).
thf(func_def_131, type, db0: !>[X0: $tType]:(X0)).
thf(func_def_132, type, db2: !>[X0: $tType]:(X0)).
thf(func_def_133, type, vEQ: !>[X0: $tType]:((X0 > X0 > $o))).
thf(func_def_134, type, vLAM: !>[X0: $tType, X1: $tType]:((X1) > (X0 > X1))).
thf(func_def_135, type, db4: !>[X0: $tType]:(X0)).
thf(func_def_136, type, db3: !>[X0: $tType]:(X0)).
thf(func_def_137, type, vIMP: ($o > $o > $o)).
thf(func_def_138, type, sP0: ((term > $o) > $o)).
thf(func_def_139, type, sP1: ((subst > term > subst > $o) > $o)).
thf(func_def_140, type, sP2: ((subst > term > subst > $o) > $o)).
thf(func_def_141, type, sP3: $o).
thf(func_def_142, type, sP4: ((subst > term > subst > $o) > $o)).
thf(func_def_143, type, sP5: $o).
thf(func_def_144, type, sP6: $o).
thf(func_def_145, type, sP7: ((subst > term > subst > $o) > $o)).
thf(func_def_146, type, sP8: $o).
thf(func_def_147, type, sP9: ((subst > term > subst > $o) > $o)).
thf(func_def_148, type, sP10: $o).
thf(func_def_149, type, sP11: $o).
thf(func_def_150, type, sP12: $o).
thf(func_def_151, type, sP13: ((term > $o) > $o)).
thf(func_def_152, type, sP14: $o).
thf(func_def_153, type, sP15: ((term > $o) > $o)).
thf(func_def_154, type, sP16: $o).
thf(func_def_155, type, sP17: ((subst > term > subst > $o) > $o)).
thf(func_def_156, type, sP18: $o).
thf(func_def_157, type, sP19: ((subst > term > subst > $o) > $o)).
thf(func_def_158, type, sP20: $o).
thf(func_def_159, type, sP21: ((subst > term > subst > $o) > $o)).
thf(func_def_160, type, sP22: $o).
thf(func_def_161, type, sP23: $o).
thf(func_def_162, type, sP24: ((subst > term > subst > $o) > $o)).
thf(func_def_163, type, sP25: ((term > $o) > $o)).
thf(func_def_164, type, sP26: ((term > $o) > $o)).
thf(func_def_165, type, sP27: $o).
thf(func_def_166, type, sP28: $o).
thf(func_def_167, type, sP29: $o).
thf(func_def_168, type, sP30: $o).
thf(func_def_169, type, sP31: ((term > $o) > $o)).
thf(func_def_170, type, sP32: ((subst > term > subst > $o) > $o)).
thf(func_def_171, type, sP33: $o).
thf(func_def_172, type, sP34: $o).
thf(func_def_173, type, sP35: $o).
thf(func_def_174, type, sP36: $o).
thf(func_def_175, type, sP37: ((subst > term > subst > $o) > $o)).
thf(func_def_176, type, sP38: ((subst > term > subst > $o) > $o)).
thf(func_def_177, type, sP39: ((term > $o) > $o)).
thf(func_def_178, type, sP40: ((subst > term > subst > $o) > $o)).
thf(func_def_179, type, sP41: ((subst > term > subst > $o) > $o)).
thf(func_def_180, type, sP42: $o).
thf(func_def_181, type, sP43: $o).
thf(func_def_182, type, sP44: $o).
thf(func_def_183, type, sP45: $o).
thf(func_def_184, type, sP46: $o).
thf(func_def_185, type, sP47: $o).
thf(func_def_186, type, sP48: ((term > $o) > $o)).
thf(func_def_187, type, sP49: ((term > $o) > $o)).
thf(func_def_188, type, sP50: $o).
thf(func_def_189, type, sP51: $o).
thf(func_def_190, type, sP52: $o).
thf(func_def_191, type, sP53: ((subst > term > subst > $o) > $o)).
thf(func_def_192, type, sP54: ((subst > term > subst > $o) > $o)).
thf(func_def_193, type, sP55: ((term > $o) > $o)).
thf(func_def_194, type, sP56: $o).
thf(func_def_195, type, sP57: $o).
thf(func_def_196, type, sP58: $o).
thf(func_def_197, type, sK59: ((term > $o) > term)).
thf(func_def_198, type, sK60: ((term > $o) > term)).
thf(func_def_199, type, sK61: ((subst > term > subst > $o) > term > $o)).
thf(func_def_200, type, sK62: term).
thf(func_def_201, type, sK63: (term > $o)).
thf(func_def_202, type, sK64: term).
thf(func_def_203, type, sK65: (term > term)).
thf(func_def_204, type, sK66: (subst > term > subst > $o)).
thf(func_def_205, type, sK67: ((term > $o) > term)).
thf(func_def_206, type, sK68: ((subst > term > subst > $o) > (term > $o) > term)).
thf(func_def_207, type, sK69: (term > (term > $o) > term)).
thf(func_def_208, type, sK70: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_209, type, sK71: (term > $o)).
thf(func_def_210, type, sK72: (subst > term > subst > $o)).
thf(func_def_211, type, sK73: term).
thf(func_def_212, type, sK74: ((subst > term > term) > term)).
thf(func_def_213, type, sK75: ((subst > term > term) > term)).
thf(func_def_214, type, sK76: ((subst > term > term) > subst)).
thf(func_def_215, type, sK77: ((subst > term > term) > subst)).
thf(func_def_216, type, sK78: (subst > term > subst > $o)).
thf(func_def_217, type, sK79: term).
thf(func_def_218, type, sK80: ((subst > term > subst > $o) > subst)).
thf(func_def_219, type, sK81: ((subst > term > subst > $o) > subst)).
thf(func_def_220, type, sK82: ((subst > term > subst > $o) > term)).
thf(func_def_221, type, sK83: ((subst > term > subst > $o) > subst)).
thf(func_def_222, type, sK84: ((subst > term > subst > $o) > term)).
thf(func_def_223, type, sK85: ((subst > term > subst > $o) > subst)).
thf(func_def_224, type, sK86: ((subst > term > subst > $o) > subst)).
thf(func_def_225, type, sK87: ((subst > term > subst > $o) > subst)).
thf(func_def_226, type, sK88: ((subst > term > subst > $o) > term)).
thf(func_def_227, type, sK89: ((term > $o) > term)).
thf(func_def_228, type, sK90: ((term > $o) > term)).
thf(func_def_229, type, sK91: ((subst > term > term) > subst)).
thf(func_def_230, type, sK92: ((subst > term > term) > subst)).
thf(func_def_231, type, sK93: ((subst > term > term) > term)).
thf(func_def_232, type, sK94: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_233, type, sK95: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_234, type, sK96: ((subst > term > subst > $o) > term)).
thf(func_def_235, type, sK97: ((subst > term > subst > $o) > term)).
thf(func_def_236, type, sK98: (subst > term > subst > $o)).
thf(func_def_237, type, sK99: term).
thf(func_def_238, type, sK100: ((subst > term > term) > term)).
thf(func_def_239, type, sK101: ((subst > term > term) > subst)).
thf(func_def_240, type, sK102: ((subst > term > term) > subst)).
thf(func_def_241, type, sK103: ((subst > term > term) > term)).
thf(func_def_242, type, sK104: (term > (subst > term > subst > $o) > term)).
thf(func_def_243, type, sK105: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_244, type, sK106: ((term > $o) > subst > term)).
thf(func_def_245, type, sK107: ((term > $o) > term)).
thf(func_def_246, type, sK108: term).
thf(func_def_247, type, sK109: subst).
thf(func_def_248, type, sK110: (term > $o)).
thf(func_def_249, type, sK111: (term > term)).
thf(func_def_250, type, sK112: term).
thf(func_def_251, type, sK113: term).
thf(func_def_252, type, sK114: (term > $o)).
thf(func_def_253, type, sK115: (term > term)).
thf(func_def_254, type, sK116: ((term > $o) > term)).
thf(func_def_255, type, sK117: ((term > $o) > term)).
thf(func_def_256, type, sK118: ((term > $o) > term)).
thf(func_def_257, type, sK119: ((term > $o) > term)).
thf(func_def_258, type, sK120: ((term > $o) > term)).
thf(func_def_259, type, sK121: ((term > $o) > term)).
thf(func_def_260, type, sK122: term).
thf(func_def_261, type, sK123: (subst > term > subst > $o)).
thf(func_def_262, type, sK124: ((subst > term > term) > subst)).
thf(func_def_263, type, sK125: ((subst > term > term) > term)).
thf(func_def_264, type, sK126: ((subst > term > term) > subst)).
thf(func_def_265, type, sK127: ((subst > term > term) > term)).
thf(func_def_266, type, sK128: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_267, type, sK129: ((subst > term > subst > $o) > term > term)).
thf(func_def_268, type, sK130: term).
thf(func_def_269, type, sK131: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_270, type, sK132: (term > (subst > term > subst > $o) > term)).
thf(func_def_271, type, sK133: (subst > term > subst > $o)).
thf(func_def_272, type, sK134: term).
thf(func_def_273, type, sK135: ((subst > term > term) > subst)).
thf(func_def_274, type, sK136: ((subst > term > term) > term)).
thf(func_def_275, type, sK137: ((subst > term > term) > subst)).
thf(func_def_276, type, sK138: ((subst > term > term) > term)).
thf(func_def_277, type, sK139: ((term > $o) > term)).
thf(func_def_278, type, sK140: ((term > $o) > term)).
thf(func_def_279, type, sK141: term).
thf(func_def_280, type, sK142: (term > $o)).
thf(func_def_281, type, sK143: (term > term)).
thf(func_def_282, type, sK144: ((subst > term > subst > $o) > subst)).
thf(func_def_283, type, sK145: ((subst > term > subst > $o) > term)).
thf(func_def_284, type, sK146: ((subst > term > subst > $o) > subst)).
thf(func_def_285, type, sK147: ((subst > term > subst > $o) > subst)).
thf(func_def_286, type, sK148: ((subst > term > subst > $o) > subst)).
thf(func_def_287, type, sK149: ((subst > term > subst > $o) > term)).
thf(func_def_288, type, sK150: ((subst > term > subst > $o) > subst)).
thf(func_def_289, type, sK151: ((subst > term > subst > $o) > subst)).
thf(func_def_290, type, sK152: term).
thf(func_def_291, type, sK153: (subst > term > subst > $o)).
thf(func_def_292, type, sK154: term).
thf(func_def_293, type, sK155: (subst > term > subst > $o)).
thf(func_def_294, type, sK156: term).
thf(func_def_295, type, sK157: ((subst > term > subst > $o) > subst)).
thf(func_def_296, type, sK158: ((subst > term > subst > $o) > subst)).
thf(func_def_297, type, sK159: ((subst > term > subst > $o) > term)).
thf(func_def_298, type, sK160: ((subst > term > subst > $o) > subst)).
thf(func_def_299, type, sK161: ((subst > term > subst > $o) > term)).
thf(func_def_300, type, sK162: ((subst > term > subst > $o) > subst)).
thf(func_def_301, type, sK163: ((subst > term > subst > $o) > subst)).
thf(func_def_302, type, sK164: ((subst > term > subst > $o) > term)).
thf(func_def_303, type, sK165: ((subst > term > subst > $o) > subst)).
thf(func_def_304, type, sK166: ((subst > term > subst > $o) > subst)).
thf(func_def_305, type, sK167: ((subst > term > subst > $o) > term)).
thf(func_def_306, type, sK168: ((subst > term > subst > $o) > subst)).
thf(func_def_307, type, sK169: ((subst > term > subst > $o) > subst)).
thf(func_def_308, type, sK170: ((subst > term > subst > $o) > term > term)).
thf(func_def_309, type, sK171: ((subst > term > subst > $o) > term)).
thf(func_def_310, type, sK172: ((subst > term > subst > $o) > subst)).
thf(func_def_311, type, sK173: ((subst > term > subst > $o) > subst)).
thf(func_def_312, type, sK174: ((subst > term > subst > $o) > subst)).
thf(func_def_313, type, sK175: term).
thf(func_def_314, type, sK176: (subst > term > subst > $o)).
thf(func_def_315, type, sK177: ((subst > term > subst > $o) > term)).
thf(func_def_316, type, sK178: ((subst > term > subst > $o) > term)).
thf(func_def_317, type, sK179: ((subst > term > term) > subst)).
thf(func_def_318, type, sK180: ((subst > term > term) > subst)).
thf(func_def_319, type, sK181: ((subst > term > term) > term)).
thf(func_def_320, type, sK182: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_321, type, sK183: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_322, type, sK184: ((term > $o) > term)).
thf(func_def_323, type, sK185: ((term > $o) > term)).
thf(func_def_324, type, sK186: ((subst > term > subst > $o) > term)).
thf(func_def_325, type, sK187: ((subst > term > subst > $o) > term)).
thf(func_def_326, type, sK188: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_327, type, sK189: ((subst > term > term) > subst)).
thf(func_def_328, type, sK190: ((subst > term > term) > subst)).
thf(func_def_329, type, sK191: ((subst > term > term) > term)).
thf(func_def_330, type, sK192: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_331, type, sK193: (term > subst > term > term)).
thf(func_def_332, type, sK194: (subst > term > subst > $o)).
thf(func_def_333, type, sK195: term).
thf(func_def_334, type, sK196: (term > $o)).
thf(func_def_335, type, sK197: ((subst > term > term) > subst)).
thf(func_def_336, type, sK198: ((subst > term > term) > term)).
thf(func_def_337, type, sK199: ((subst > term > term) > subst)).
thf(func_def_338, type, sK200: ((subst > term > term) > term)).
thf(func_def_339, type, sK201: term).
thf(func_def_340, type, sK202: ((subst > term > term) > subst)).
thf(func_def_341, type, sK203: ((subst > term > term) > term)).
thf(func_def_342, type, sK204: term).
thf(func_def_343, type, sK205: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_344, type, sK206: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_345, type, sK207: ((term > $o) > term > term)).
thf(func_def_346, type, sK208: (subst > term > subst > $o)).
thf(func_def_347, type, sK209: term).
thf(func_def_348, type, sK210: ((subst > term > subst > $o) > term > $o)).
thf(func_def_349, type, sK211: term).
thf(func_def_350, type, sK212: ((subst > term > subst > $o) > term)).
thf(func_def_351, type, sK213: ((subst > term > subst > $o) > term)).
thf(func_def_352, type, sK214: (subst > term > subst > $o)).
thf(func_def_353, type, sK215: ((term > $o) > term)).
thf(func_def_354, type, sK216: ((term > $o) > term)).
thf(func_def_355, type, sK217: ((term > $o) > term)).
thf(func_def_356, type, sK218: (term > $o)).
thf(func_def_357, type, sK219: term).
thf(func_def_358, type, sK220: (term > term)).
thf(func_def_359, type, sK221: (term > (term > $o) > term)).
thf(func_def_360, type, sK222: ((subst > term > subst > $o) > (term > $o) > term)).
thf(func_def_361, type, sK223: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_362, type, sK224: (term > $o)).
thf(func_def_363, type, sK225: term).
thf(func_def_364, type, sK226: (subst > term > subst > $o)).
thf(func_def_365, type, sK227: ((subst > term > term) > term)).
thf(func_def_366, type, sK228: ((subst > term > term) > subst)).
thf(func_def_367, type, sK229: ((subst > term > term) > subst)).
thf(func_def_368, type, sK230: ((subst > term > term) > term)).
thf(func_def_369, type, sK231: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_370, type, sK232: ((subst > term > term) > subst)).
thf(func_def_371, type, sK233: ((subst > term > term) > term)).
thf(func_def_372, type, sK234: ((subst > term > term) > subst)).
thf(func_def_373, type, sK235: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_374, type, sK236: ((term > $o) > term)).
thf(func_def_375, type, sK237: ((term > $o) > term)).
thf(func_def_376, type, sK238: subst).
thf(func_def_377, type, sK239: (subst > term > term)).
thf(func_def_378, type, sK240: term).
thf(func_def_379, type, sK241: (subst > term > term)).
thf(func_def_380, type, sK242: subst).
thf(func_def_381, type, sK243: term).
thf(func_def_382, type, sK244: ((subst > term > term) > term)).
thf(func_def_383, type, sK245: ((subst > term > term) > subst)).
thf(func_def_384, type, sK246: ((subst > term > term) > subst)).
thf(func_def_385, type, sK247: ((subst > term > term) > term)).
thf(func_def_386, type, sK248: ((subst > term > term) > subst)).
thf(func_def_387, type, sK249: ((subst > term > term) > subst)).
thf(func_def_388, type, sK250: subst).
thf(func_def_389, type, sK251: term).
thf(func_def_390, type, sK252: term).
thf(func_def_391, type, sK253: term).
thf(func_def_392, type, sK254: (term > $o)).
thf(func_def_393, type, sK255: term).
thf(func_def_394, type, sK256: (subst > term > subst > $o)).
thf(func_def_395, type, sK257: ((subst > term > term) > subst)).
thf(func_def_396, type, sK258: ((subst > term > term) > term)).
thf(func_def_397, type, sK259: ((subst > term > term) > subst)).
thf(func_def_398, type, sK260: ((subst > term > term) > term)).
thf(func_def_399, type, sK261: (term > (term > $o) > term)).
thf(func_def_400, type, sK262: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_401, type, sK263: ((subst > term > subst > $o) > (term > $o) > term)).
thf(func_def_402, type, sK264: ((term > $o) > term)).
thf(func_def_403, type, sK265: ((term > $o) > term)).
thf(func_def_404, type, sK266: term).
thf(func_def_405, type, sK267: (term > $o)).
thf(func_def_406, type, sK268: term).
thf(func_def_407, type, sK269: subst).
thf(func_def_408, type, sK270: subst).
thf(func_def_409, type, sK271: term).
thf(func_def_410, type, sK272: subst).
thf(func_def_411, type, sK273: term).
thf(func_def_412, type, sK274: (term > $o)).
thf(func_def_413, type, sK275: subst).
thf(func_def_414, type, sK276: (term > term)).
thf(func_def_415, type, sK277: (subst > (term > $o) > term)).
thf(func_def_416, type, sK278: ((term > $o) > term)).
thf(func_def_417, type, sK279: ((term > $o) > term)).
thf(func_def_418, type, sK280: ((term > $o) > term)).
thf(func_def_419, type, sK281: ((term > $o) > term)).
thf(func_def_420, type, sK282: ((term > $o) > term)).
thf(func_def_421, type, sK283: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_422, type, sK284: ((subst > term > term) > term)).
thf(func_def_423, type, sK285: ((subst > term > term) > subst)).
thf(func_def_424, type, sK286: ((subst > term > term) > subst)).
thf(func_def_425, type, sK287: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_426, type, sK288: (subst > term > subst > $o)).
thf(func_def_427, type, sK289: term).
thf(func_def_428, type, sK290: ((subst > term > subst > $o) > subst)).
thf(func_def_429, type, sK291: ((subst > term > subst > $o) > subst)).
thf(func_def_430, type, sK292: ((subst > term > subst > $o) > term)).
thf(func_def_431, type, sK293: ((subst > term > subst > $o) > subst)).
thf(func_def_432, type, sK294: ((subst > term > subst > $o) > subst)).
thf(func_def_433, type, sK295: ((subst > term > subst > $o) > term)).
thf(func_def_434, type, sK296: ((subst > term > subst > $o) > subst)).
thf(func_def_435, type, sK297: ((subst > term > subst > $o) > subst)).
thf(func_def_436, type, sK298: (term > (subst > term > subst > $o) > term)).
thf(func_def_437, type, sK299: term).
thf(func_def_438, type, sK300: ((subst > term > subst > $o) > term > term)).
thf(func_def_439, type, sK301: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_440, type, sK302: term).
thf(func_def_441, type, sK303: (subst > term > subst > $o)).
thf(func_def_442, type, sK304: ((subst > term > term) > term)).
thf(func_def_443, type, sK305: ((subst > term > term) > subst)).
thf(func_def_444, type, sK306: ((subst > term > term) > subst)).
thf(func_def_445, type, sK307: ((subst > term > term) > term)).
thf(func_def_446, type, sK308: ((subst > term > term) > term)).
thf(func_def_447, type, sK309: ((subst > term > term) > subst)).
thf(func_def_448, type, sK310: ((subst > term > term) > subst)).
thf(func_def_449, type, sK311: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_450, type, sK312: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_451, type, sK313: term).
thf(func_def_452, type, sK314: term).
thf(func_def_453, type, sK315: (term > $o)).
thf(func_def_454, type, sK316: (subst > term > subst > $o)).
thf(func_def_455, type, sK317: ((subst > term > subst > $o) > subst)).
thf(func_def_456, type, sK318: ((subst > term > subst > $o) > subst)).
thf(func_def_457, type, sK319: ((subst > term > subst > $o) > subst)).
thf(func_def_458, type, sK320: ((subst > term > subst > $o) > term)).
thf(func_def_459, type, sK321: ((subst > term > subst > $o) > (term > $o) > term)).
thf(func_def_460, type, sK322: ((subst > term > subst > $o) > subst)).
thf(func_def_461, type, sK323: ((subst > term > subst > $o) > term)).
thf(func_def_462, type, sK324: ((subst > term > subst > $o) > subst)).
thf(func_def_463, type, sK325: ((subst > term > subst > $o) > subst)).
thf(func_def_464, type, sK326: ((subst > term > subst > $o) > term)).
thf(func_def_465, type, sK327: ((subst > term > subst > $o) > term)).
thf(func_def_466, type, sK328: term).
thf(func_def_467, type, sK329: (subst > term > subst > $o)).
thf(func_def_468, type, sK330: ((subst > term > subst > $o) > term > term)).
thf(func_def_469, type, sK331: ((subst > term > subst > $o) > term)).
thf(func_def_470, type, sK332: ((subst > term > subst > $o) > subst)).
thf(func_def_471, type, sK333: ((subst > term > subst > $o) > subst)).
thf(func_def_472, type, sK334: ((subst > term > subst > $o) > subst)).
thf(func_def_473, type, sK335: ((subst > term > subst > $o) > subst)).
thf(func_def_474, type, sK336: ((subst > term > subst > $o) > subst)).
thf(func_def_475, type, sK337: ((subst > term > subst > $o) > subst)).
thf(func_def_476, type, sK338: ((subst > term > subst > $o) > term)).
thf(func_def_477, type, sK339: ((subst > term > term) > term)).
thf(func_def_478, type, sK340: ((subst > term > term) > subst)).
thf(func_def_479, type, sK341: ((subst > term > term) > subst)).
thf(func_def_480, type, sK342: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_481, type, sK343: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_482, type, sK344: term).
thf(func_def_483, type, sK345: (term > $o)).
thf(func_def_484, type, sK346: subst).
thf(func_def_485, type, sK347: (term > term)).
thf(func_def_486, type, sK348: (subst > (term > $o) > term)).
thf(func_def_487, type, sK349: ((term > $o) > term)).
thf(func_def_488, type, sK350: ((term > $o) > term)).
thf(func_def_489, type, sK351: ((term > $o) > term)).
thf(func_def_490, type, sK352: term).
thf(func_def_491, type, sK353: (term > $o)).
thf(func_def_492, type, sK354: (term > term)).
thf(func_def_493, type, sK355: ((term > $o) > term)).
thf(func_def_494, type, sK356: ((term > $o) > term)).
thf(func_def_495, type, sK357: ((term > $o) > term)).
thf(func_def_496, type, sK358: ((term > $o) > term)).
thf(func_def_497, type, sK359: (subst > term > subst > $o)).
thf(func_def_498, type, sK360: term).
thf(func_def_499, type, sK361: ((subst > term > term) > subst)).
thf(func_def_500, type, sK362: ((subst > term > term) > term)).
thf(func_def_501, type, sK363: ((subst > term > term) > subst)).
thf(func_def_502, type, sK364: ((subst > term > term) > term)).
thf(func_def_503, type, sK365: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_504, type, sK366: (term > (subst > term > subst > $o) > term)).
thf(func_def_505, type, sK367: term).
thf(func_def_506, type, sK368: (subst > term > subst > $o)).
thf(func_def_507, type, sK369: ((subst > term > term) > subst)).
thf(func_def_508, type, sK370: ((subst > term > term) > subst)).
thf(func_def_509, type, sK371: ((subst > term > term) > term)).
thf(func_def_510, type, sK372: ((subst > term > term) > term)).
thf(func_def_511, type, sK373: ((subst > term > subst > $o) > term > term)).
thf(func_def_512, type, sK374: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_513, type, sK375: (subst > term > subst > $o)).
thf(func_def_514, type, sK376: (term > $o)).
thf(func_def_515, type, sK377: term).
thf(func_def_516, type, sK378: term).
thf(func_def_517, type, sK379: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_518, type, sK380: ((subst > term > subst > $o) > term)).
thf(func_def_519, type, sK381: ((subst > term > subst > $o) > subst)).
thf(func_def_520, type, sK382: ((subst > term > subst > $o) > subst)).
thf(func_def_521, type, sK383: ((subst > term > subst > $o) > subst)).
thf(func_def_522, type, sK384: ((subst > term > subst > $o) > subst)).
thf(func_def_523, type, sK385: ((subst > term > subst > $o) > term)).
thf(func_def_524, type, sK386: ((subst > term > subst > $o) > subst)).
thf(func_def_525, type, sK387: ((subst > term > subst > $o) > subst)).
thf(func_def_526, type, sK388: (term > (term > $o) > term)).
thf(func_def_527, type, sK389: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_528, type, sK390: ((subst > term > term) > term)).
thf(func_def_529, type, sK391: ((subst > term > term) > subst)).
thf(func_def_530, type, sK392: ((subst > term > term) > subst)).
thf(func_def_531, type, sK393: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_532, type, sK394: term).
thf(func_def_533, type, sK395: (subst > term > subst > $o)).
thf(func_def_534, type, sK396: term).
thf(func_def_535, type, sK397: ((subst > term > subst > $o) > subst)).
thf(func_def_536, type, sK398: ((subst > term > subst > $o) > subst)).
thf(func_def_537, type, sK399: ((subst > term > subst > $o) > subst)).
thf(func_def_538, type, sK400: ((subst > term > subst > $o) > term)).
thf(func_def_539, type, sK401: ((subst > term > subst > $o) > subst)).
thf(func_def_540, type, sK402: ((subst > term > subst > $o) > term)).
thf(func_def_541, type, sK403: ((subst > term > subst > $o) > subst)).
thf(func_def_542, type, sK404: ((subst > term > subst > $o) > subst)).
thf(func_def_543, type, sK405: ((subst > term > subst > $o) > term)).
thf(func_def_544, type, sK406: ((subst > term > subst > $o) > term)).
thf(func_def_545, type, sK407: (subst > term > subst > $o)).
thf(func_def_546, type, sK408: term).
thf(func_def_547, type, sK409: ((subst > term > term) > term > term)).
thf(func_def_548, type, sK410: ((subst > term > term) > term > subst)).
thf(func_def_549, type, sK411: ((subst > term > term) > term > subst)).
thf(func_def_550, type, sK412: ((subst > term > term) > term)).
thf(func_def_551, type, sK413: ((subst > term > term) > term > subst)).
thf(func_def_552, type, sK414: ((subst > term > term) > term > term)).
thf(func_def_553, type, sK415: ((subst > term > term) > term > subst)).
thf(func_def_554, type, sK416: ((subst > term > subst > $o) > term)).
thf(func_def_555, type, sK417: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_556, type, sK418: (term > (subst > term > subst > $o) > term)).
thf(func_def_557, type, sK419: term).
thf(func_def_558, type, sK420: (term > $o)).
thf(func_def_559, type, sK421: (subst > term > subst > $o)).
thf(func_def_560, type, sK422: ((term > $o) > term > term)).
thf(func_def_561, type, sK423: ((subst > term > subst > $o) > (term > $o) > term)).
thf(func_def_562, type, sK424: ((subst > term > subst > $o) > subst)).
thf(func_def_563, type, sK425: ((subst > term > subst > $o) > subst)).
thf(func_def_564, type, sK426: ((subst > term > subst > $o) > subst)).
thf(func_def_565, type, sK427: ((subst > term > subst > $o) > term)).
thf(func_def_566, type, sK428: ((subst > term > subst > $o) > subst)).
thf(func_def_567, type, sK429: ((subst > term > subst > $o) > subst)).
thf(func_def_568, type, sK430: ((subst > term > subst > $o) > subst)).
thf(func_def_569, type, sK431: ((subst > term > subst > $o) > term)).
thf(func_def_570, type, sK432: ((subst > term > term) > subst)).
thf(func_def_571, type, sK433: ((subst > term > term) > term)).
thf(func_def_572, type, sK434: ((subst > term > term) > subst)).
thf(func_def_573, type, sK435: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_574, type, sK436: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_575, type, sK437: (subst > term > subst > $o)).
thf(func_def_576, type, sK438: term).
thf(func_def_577, type, sK439: ((subst > term > subst > $o) > subst)).
thf(func_def_578, type, sK440: ((subst > term > subst > $o) > term)).
thf(func_def_579, type, sK441: ((subst > term > subst > $o) > subst)).
thf(func_def_580, type, sK442: ((subst > term > subst > $o) > subst)).
thf(func_def_581, type, sK443: ((subst > term > subst > $o) > subst)).
thf(func_def_582, type, sK444: ((subst > term > subst > $o) > term)).
thf(func_def_583, type, sK445: ((subst > term > subst > $o) > subst)).
thf(func_def_584, type, sK446: ((subst > term > subst > $o) > subst)).
thf(func_def_585, type, sK447: ((subst > term > subst > $o) > term)).
thf(func_def_586, type, sK448: ((subst > term > subst > $o) > term)).
thf(func_def_587, type, sK449: ((subst > term > subst > $o) > term)).
thf(func_def_588, type, sK450: ((subst > term > term) > term)).
thf(func_def_589, type, sK451: ((subst > term > term) > subst)).
thf(func_def_590, type, sK452: ((subst > term > term) > subst)).
thf(func_def_591, type, sK453: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_592, type, sK454: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_593, type, sK455: ((term > $o) > term)).
thf(func_def_594, type, sK456: ((term > $o) > term)).
thf(func_def_595, type, sK457: ((term > $o) > term > (term > $o) > subst > term)).
thf(func_def_596, type, sK458: subst).
thf(func_def_597, type, sK459: (term > $o)).
thf(func_def_598, type, sK460: term).
thf(func_def_599, type, sK461: (term > $o)).
thf(func_def_600, type, sK462: term).
thf(func_def_601, type, sK463: ((subst > term > term) > subst)).
thf(func_def_602, type, sK464: ((subst > term > term) > term)).
thf(func_def_603, type, sK465: (term > subst > term > term)).
thf(func_def_604, type, sK466: term).
thf(func_def_605, type, sK467: term).
thf(func_def_606, type, sK468: term).
thf(func_def_607, type, sK469: term).
thf(func_def_608, type, sK470: term).
thf(func_def_609, type, sK471: term).
thf(func_def_610, type, sK472: term).
thf(func_def_611, type, sK473: term).
thf(func_def_612, type, sK474: term).
thf(func_def_613, type, sK475: term).
thf(func_def_614, type, sK476: term).
thf(func_def_615, type, sK477: term).
thf(func_def_616, type, sK478: term).
thf(func_def_617, type, sK479: term).
thf(func_def_618, type, sK480: term).
thf(func_def_619, type, sK481: term).
thf(func_def_620, type, sK482: term).
thf(func_def_621, type, sK483: term).
thf(func_def_622, type, sK484: term).
thf(func_def_623, type, sK485: subst).
thf(func_def_624, type, sK486: (subst > term > term)).
thf(func_def_625, type, sK487: term).
thf(func_def_626, type, sK488: (subst > term > term)).
thf(func_def_627, type, sK489: ((subst > term > term) > subst)).
thf(func_def_628, type, sK490: ((subst > term > term) > term)).
thf(func_def_629, type, sK491: ((subst > term > term) > subst)).
thf(func_def_630, type, sK492: ((subst > term > term) > subst)).
thf(func_def_631, type, sK493: ((subst > term > term) > term)).
thf(func_def_632, type, sK494: ((subst > term > term) > subst)).
thf(func_def_633, type, sK495: (subst > term > (term > $o) > term > $o)).
thf(func_def_634, type, sK496: (term > $o)).
thf(func_def_635, type, sK497: term).
thf(func_def_636, type, sK498: subst).
thf(func_def_637, type, sK499: ((term > $o) > term)).
thf(func_def_638, type, sK500: term).
thf(func_def_639, type, sK501: term).
thf(func_def_640, type, sK502: ((subst > term > subst > $o) > term)).
thf(func_def_641, type, sK503: ((subst > term > subst > $o) > term)).
thf(func_def_642, type, sK504: (subst > term > subst > $o)).
thf(func_def_643, type, sK505: term).
thf(func_def_644, type, sK506: subst).
thf(func_def_645, type, sK507: term).
thf(func_def_646, type, sK508: term).
thf(func_def_647, type, sK509: term).
thf(func_def_648, type, sK510: term).
thf(func_def_649, type, sK511: term).
thf(func_def_650, type, sK512: term).
thf(func_def_651, type, sK513: term).
thf(func_def_652, type, sK514: term).
thf(func_def_653, type, sK515: term).
thf(func_def_654, type, sK516: term).
thf(func_def_655, type, sK517: term).
thf(func_def_656, type, sK518: term).
thf(func_def_657, type, sK519: term).
thf(func_def_658, type, sK520: subst).
thf(func_def_659, type, sK521: term).
thf(func_def_660, type, sK522: term).
thf(func_def_661, type, sK523: (term > $o)).
thf(func_def_662, type, sK524: term).
thf(func_def_663, type, sK525: term).
thf(func_def_664, type, sK526: (term > $o)).
thf(func_def_665, type, sK527: subst).
thf(func_def_666, type, sK528: term).
thf(func_def_667, type, sK529: (term > subst > term > term)).
thf(func_def_668, type, sK530: subst).
thf(func_def_669, type, sK531: term).
thf(func_def_670, type, sK532: subst).
thf(func_def_671, type, sK533: ((term > $o) > subst > term > (term > $o) > term)).
thf(func_def_672, type, sK534: (term > $o)).
thf(func_def_673, type, sK535: term).
thf(func_def_674, type, sK536: subst).
thf(func_def_675, type, sK537: (term > $o)).
thf(func_def_676, type, sK538: ((subst > term > subst > $o) > term > $o)).
thf(func_def_677, type, sK539: (subst > term > subst > $o)).
thf(func_def_678, type, sK540: ((term > $o) > term)).
thf(func_def_679, type, sK541: subst).
thf(func_def_680, type, sK542: term).
thf(func_def_681, type, sK543: term).
thf(func_def_682, type, sK544: subst).
thf(func_def_683, type, sK545: term).
thf(func_def_684, type, sK546: subst).
thf(func_def_685, type, sK547: term).
thf(func_def_686, type, sK548: (term > $o)).
thf(func_def_687, type, sK549: (term > $o)).
thf(func_def_688, type, sK550: subst).
thf(func_def_689, type, sK551: subst).
thf(func_def_690, type, sK552: (subst > d_subst)).
thf(func_def_691, type, sK553: term).
thf(func_def_692, type, sK554: term).
thf(func_def_693, type, sK555: subst).
thf(func_def_694, type, sK556: subst).
thf(func_def_695, type, sK557: subst).
thf(func_def_696, type, sK558: subst).
thf(func_def_697, type, sK559: (term > $o)).
thf(func_def_698, type, sK560: subst).
thf(func_def_699, type, sK561: term).
thf(func_def_700, type, sK562: ((term > $o) > term)).
thf(func_def_701, type, sK563: (term > subst > (term > $o) > term > $o)).
thf(func_def_702, type, sK564: (term > subst > term > term)).
thf(func_def_703, type, sK565: term).
thf(func_def_704, type, sK566: ((subst > term > term) > subst)).
thf(func_def_705, type, sK567: ((subst > term > term) > term)).
thf(func_def_706, type, sK568: subst).
thf(func_def_707, type, sK569: term).
thf(func_def_708, type, sK570: (subst > term > term)).
thf(func_def_709, type, sK571: term).
thf(func_def_710, type, sK572: ((subst > term > term) > subst)).
thf(func_def_711, type, sK573: ((subst > term > term) > term)).
thf(func_def_712, type, sK574: ((subst > term > term) > subst)).
thf(func_def_713, type, sK575: (term > d_term)).
thf(func_def_714, type, sK576: subst).
thf(func_def_715, type, sK577: term).
thf(func_def_716, type, sK578: (subst > term > subst > $o)).
thf(func_def_717, type, sK579: ((term > $o) > term)).
thf(func_def_718, type, sK580: ((subst > term > subst > $o) > term > $o)).
thf(func_def_719, type, sK581: term).
thf(func_def_720, type, sK582: (term > subst > term > term)).
thf(func_def_721, type, sK583: term).
thf(func_def_722, type, sK584: (term > $o)).
thf(func_def_723, type, sK585: ((term > $o) > term)).
thf(func_def_724, type, sK586: ((term > $o) > term)).
thf(func_def_725, type, sK587: term).
thf(func_def_726, type, sK588: term).
thf(func_def_727, type, sK589: term).
thf(func_def_728, type, sK590: term).
thf(func_def_729, type, sF591: term).
thf(func_def_730, type, sF592: subst).
thf(func_def_731, type, sF593: $o).
thf(func_def_732, type, sK594: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_733, type, sK595: ((subst > term > subst > $o) > term > term)).
thf(func_def_734, type, sK596: ((subst > term > subst > $o) > term > term)).
thf(func_def_735, type, sK597: (subst > term > subst > $o)).
thf(func_def_736, type, sK598: ((subst > term > term) > term)).
thf(func_def_737, type, sK599: ((subst > term > term) > term)).
thf(func_def_738, type, sK600: ((subst > term > term) > subst)).
thf(func_def_739, type, sK601: ((subst > term > term) > subst)).
thf(func_def_740, type, sK602: ((subst > term > term) > subst)).
thf(func_def_741, type, sK603: ((subst > term > term) > subst)).
thf(func_def_742, type, sK604: term).
thf(f1,conjecture,(
  (! [X0 : (subst > term > subst > $o)] : (! [X1 : (subst > term > term)] : (! [X4 : subst,X3 : subst,X2 : term] : (((sub @ (X1 @ X3 @ X2) @ X4)) = ((X1 @ (comp @ X3 @ X4) @ (sub @ X2 @ X4)))) => (! [X2 : term] : ((X0 @ id @ X2 @ id) => (X0 @ id @ (X1 @ id @ X2) @ id)) => (X0 @ id @ (hoaslam @ id @ (^[X3 : subst, X2 : term] : ((X1 @ X3 @ X2)))) @ id))) => ! [X2 : term] : (! [X5 : term] : ((X0 @ id @ X5 @ id) => (X0 @ id @ (sub @ X2 @ (push @ X5 @ id)) @ id)) => (X0 @ id @ (lam @ (sub @ X2 @ (push @ one @ sh))) @ id))) = hoasinduction_lem3aa)),
  file('/export/starexec/sandbox/benchmark/theBenchmark.p',hoasinduction_lem3aa)).
thf(f2,negated_conjecture,(
  ~ (! [X0 : (subst > term > subst > $o)] : (! [X1 : (subst > term > term)] : (! [X4 : subst,X3 : subst,X2 : term] : (((sub @ (X1 @ X3 @ X2) @ X4)) = ((X1 @ (comp @ X3 @ X4) @ (sub @ X2 @ X4)))) => (! [X2 : term] : ((X0 @ id @ X2 @ id) => (X0 @ id @ (X1 @ id @ X2) @ id)) => (X0 @ id @ (hoaslam @ id @ (^[X3 : subst, X2 : term] : ((X1 @ X3 @ X2)))) @ id))) => ! [X2 : term] : (! [X5 : term] : ((X0 @ id @ X5 @ id) => (X0 @ id @ (sub @ X2 @ (push @ X5 @ id)) @ id)) => (X0 @ id @ (lam @ (sub @ X2 @ (push @ one @ sh))) @ id))) = hoasinduction_lem3aa)),
  inference(negated_conjecture,[status(cth)],[f1])).
thf(f3,axiom,(
  hoaslamnotvar_lthm & (hoasapinj1 <=> ! [X15 : term,X16 : term,X11 : term,X17 : term] : ((X11 = X15) | (((ap @ (sub @ X11 @ id) @ X16)) != ((ap @ (sub @ X15 @ id) @ X17))))) & hoaslamnotvar_gthm & (hoasinduction_lem3aa_lthm <=> ! [X8 : (subst > term > subst > $o),X28 : term] : ((X8 @ id @ (lam @ (sub @ X28 @ (push @ one @ sh))) @ id) | ~ ! [X19 : (subst > term > term)] : (~ ! [X11 : term] : (~(X8 @ id @ X11 @ id) | (X8 @ id @ (X19 @ id @ X11) @ id)) | (X8 @ id @ (lam @ (X19 @ sh @ one)) @ id) | ~ ! [X29 : subst,X12 : subst,X11 : term] : (((sub @ (X19 @ X12 @ X11) @ X29)) = ((X19 @ (comp @ X12 @ X29) @ (sub @ X11 @ X29))))) | ~ ! [X15 : term] : (~(X8 @ id @ X15 @ id) | (X8 @ id @ (sub @ X28 @ (push @ X15 @ id)) @ id)))) & pushprop_lthm_orig & ((! [X16 : term,X15 : term,X17 : term,X11 : term] : ((X16 = X17) | (((ap @ X11 @ X16)) != ((ap @ X15 @ X17)))) => ! [X17 : term,X11 : term,X15 : term,X16 : term] : ((((ap @ (sub @ X11 @ id) @ X16)) != ((ap @ (sub @ X15 @ id) @ X17))) | (X16 = X17))) <=> hoasapinj2_lthm) & (pushprop_lem0 <=> ! [X11 : term,X12 : subst,X8 : (term > $o)] : ~ ! [X9 : (term > $o)] : ~ ! [X10 : term] : (((X9 @ X10)) = ((X8 @ (sub @ X10 @ (push @ X11 @ X12)))))) & ~(hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) & ~lamnotap & hoaslamnotap_lthm & (hoasinduction_lem3aaa <=> ! [X20 : term,X8 : (subst > term > subst > $o)] : ((X8 @ id @ (lam @ (sub @ X20 @ (push @ one @ sh))) @ id) | ~ ! [X15 : term] : ((X8 @ id @ (sub @ X20 @ (push @ X15 @ id)) @ id) | ~(X8 @ id @ X15 @ id)) | ~ ! [X21 : term,X19 : (subst > term > term)] : ((X8 @ id @ (lam @ (X19 @ sh @ one)) @ id) | ~ ! [X26 : term,X25 : subst,X27 : subst] : (((sub @ (X19 @ X25 @ X26) @ X27)) = ((sub @ (sub @ X21 @ (push @ X26 @ X25)) @ X27))) | ~ ! [X22 : subst,X24 : subst,X23 : term] : (((X19 @ (comp @ X22 @ X24) @ (sub @ X23 @ X24))) = ((sub @ X21 @ (push @ (sub @ X23 @ X24) @ (comp @ X22 @ X24))))) | ~ ! [X11 : term] : (~(X8 @ id @ X11 @ id) | (X8 @ id @ (X19 @ id @ X11) @ id))))) & (hoaslaminj_lthm <=> (! [X11 : term,X12 : subst] : (X11 = ((sub @ one @ (push @ X11 @ X12)))) => (! [X12 : subst,X11 : term] : (X12 = ((comp @ sh @ (push @ X11 @ X12)))) => (! [X15 : term,X11 : term] : ((((lam @ X11)) != ((lam @ X15))) | (X11 = X15)) => ! [X36 : term,X34 : (subst > term > term),X35 : subst,X19 : (subst > term > term)] : (~ ! [X11 : term,X12 : subst,X29 : subst] : (((sub @ (X34 @ X12 @ X11) @ X29)) = ((X34 @ (comp @ X12 @ X29) @ (sub @ X11 @ X29)))) | (((X19 @ X35 @ X36)) = ((X34 @ X35 @ X36))) | (((lam @ (X19 @ sh @ one))) != ((lam @ (X34 @ sh @ one)))) | ~ ! [X29 : subst,X11 : term,X12 : subst] : (((sub @ (X19 @ X12 @ X11) @ X29)) = ((X19 @ (comp @ X12 @ X29) @ (sub @ X11 @ X29))))))))) & (! [X11 : term] : (X11 = ((sub @ X11 @ id))) <=> axvarid) & axapp & (hoasinduction_lem3b <=> ! [X15 : term] : ~ ! [X19 : (subst > term > term)] : (((X19 @ sh @ one)) != ((sub @ X15 @ (push @ one @ sh))))) & hoasapinj2_gthm & axvarshift & axassoc & (((push @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & (hoasinduction <=> ! [X40 : term,X8 : (subst > term > subst > $o)] : (~ ! [X11 : term] : ((X8 @ id @ X11 @ id) | ~(var @ (sub @ X11 @ id))) | (X8 @ id @ X40 @ id) | ~ ! [X11 : term,X12 : subst,X31 : subst,X29 : subst] : ((X8 @ (comp @ X12 @ X31) @ (sub @ X11 @ X31) @ X29) | ~(X8 @ X12 @ X11 @ (comp @ X31 @ X29))) | ~ ! [X19 : (subst > term > term)] : (~ ! [X11 : term,X12 : subst,X29 : subst] : (((sub @ (X19 @ X12 @ X11) @ X29)) = ((X19 @ (comp @ X12 @ X29) @ (sub @ X11 @ X29)))) | ~ ! [X11 : term] : (~(X8 @ id @ X11 @ id) | (X8 @ id @ (X19 @ id @ X11) @ id)) | (X8 @ id @ (lam @ (X19 @ sh @ one)) @ id)) | ~ ! [X15 : term,X11 : term] : (~(X8 @ id @ X15 @ id) | ~(X8 @ id @ X11 @ id) | (X8 @ id @ (ap @ (sub @ X11 @ id) @ X15) @ id)) | ~ ! [X12 : subst,X11 : term,X29 : subst,X31 : subst] : ((X8 @ X12 @ X11 @ (comp @ X31 @ X29)) | ~(X8 @ (comp @ X12 @ X31) @ (sub @ X11 @ X31) @ X29)))) & ! [X3 : d_subst] : (X3 = d_id) & ulamvarsh & (induction <=> ! [X18 : term,X8 : (term > $o)] : (~ ! [X11 : term] : (~(var @ X11) | (X8 @ X11)) | ~ ! [X11 : term] : (~(X8 @ X11) | (X8 @ (lam @ X11))) | (X8 @ X18) | ~ ! [X11 : term,X15 : term] : (~(X8 @ X11) | ~(X8 @ X15) | (X8 @ (ap @ X11 @ X15))))) & (! [X15 : term] : ~ ! [X19 : (subst > term > term)] : ~ ! [X12 : subst,X11 : term] : (((sub @ X15 @ (push @ X11 @ X12))) = ((X19 @ X12 @ X11))) <=> hoasinduction_lem3v2_f_lthm) & hoasapinj1_gthm & ~(var @ (d2term @ d_one)) & (! [X15 : term] : ~ ! [X19 : (subst > term > term)] : (((X19 @ sh @ one)) != ((sub @ X15 @ (push @ one @ sh)))) <=> hoasinduction_lem3b_lthm) & ((! [X11 : term] : (X11 = ((sub @ X11 @ id))) => ! [X8 : (subst > term > subst > $o),X39 : term,X9 : (term > $o)] : (~ ! [X11 : term,X31 : subst,X12 : subst,X29 : subst] : (~(X8 @ X12 @ X11 @ (comp @ X31 @ X29)) | (X8 @ (comp @ X12 @ X31) @ (sub @ X11 @ X31) @ X29)) | (X9 @ (lam @ X39)) | ~ ! [X19 : (subst > term > term)] : (~ ! [X11 : term] : (~(X8 @ id @ X11 @ id) | (X8 @ id @ (X19 @ id @ X11) @ id)) | ~ ! [X12 : subst,X11 : term,X29 : subst] : (((sub @ (X19 @ X12 @ X11) @ X29)) = ((X19 @ (comp @ X12 @ X29) @ (sub @ X11 @ X29)))) | (X8 @ id @ (lam @ (X19 @ sh @ one)) @ id)) | ~ ! [X10 : term] : (((X9 @ X10)) = ((X8 @ id @ X10 @ id))) | ~ ! [X31 : subst,X12 : subst,X29 : subst,X11 : term] : ((X8 @ X12 @ X11 @ (comp @ X31 @ X29)) | ~(X8 @ (comp @ X12 @ X31) @ (sub @ X11 @ X31) @ X29)) | ~ ! [X15 : term] : ((X9 @ (sub @ X39 @ (push @ X15 @ id))) | ~(X9 @ X15)))) <=> hoasinduction_lem3v2_lthm) & hoasinduction_lem3b_gthm & hoaslamnotap_gthm & (pushprop_p_and_p_prime = (^[X11 : term, X12 : subst, X8 : (term > $o), X9 : (term > $o)] : (! [X10 : term] : (((X9 @ X10)) = ((X8 @ (sub @ X10 @ (push @ X11 @ X12)))))))) & (induction2 <=> ! [X8 : (term > $o),X37 : term] : ((X8 @ X37) | ~ ! [X11 : term] : (~ ! [X15 : term] : ((X8 @ (sub @ X11 @ (push @ X15 @ id))) | ~(X8 @ X15)) | (X8 @ (lam @ X11))) | ~ ! [X11 : term,X15 : term] : (~(X8 @ X15) | ~(X8 @ X11) | (X8 @ (ap @ X11 @ X15))) | ~ ! [X11 : term] : ((X8 @ X11) | ~(var @ X11)))) & (one = ((d2term @ d_one))) & ((! [X12 : subst] : (X12 = ((comp @ X12 @ id))) & ! [X12 : subst] : (X12 = ((comp @ id @ X12)))) <=> substmonoid) & (! [X8 : (term > $o),X14 : subst,X13 : term] : (~ ! [X15 : term] : ((X8 @ (sub @ X15 @ X14)) | ~(var @ X15)) | ~ ! [X11 : term] : (~ ! [X15 : term] : (~(X8 @ X15) | (X8 @ (sub @ X11 @ (push @ X15 @ id)))) | (X8 @ (lam @ X11))) | (X8 @ (sub @ X13 @ X14)) | ~ ! [X15 : term,X11 : term] : ((X8 @ (ap @ X11 @ X15)) | ~(X8 @ X11) | ~(X8 @ X15))) <=> induction2lem) & pushprop_lem3v2 & hoasinduction_lem1v2_gthm & (((comp @ (d2subst @ d_id) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & (! [X15 : term] : ~ ! [X19 : (subst > term > term)] : ~ ! [X12 : subst,X11 : term] : (((sub @ X15 @ (push @ X11 @ X12))) = ((X19 @ X12 @ X11))) <=> hoasinduction_lem3v2_f) & (hoasinduction_lem3v2a_lthm <=> (! [X15 : term] : ~ ! [X19 : (subst > term > term)] : ~ ! [X12 : subst,X11 : term] : (((sub @ X15 @ (push @ X11 @ X12))) = ((X19 @ X12 @ X11))) => (! [X11 : term] : (X11 = ((sub @ X11 @ id))) => ! [X8 : (subst > term > subst > $o),X9 : (term > $o),X41 : term] : (~ ! [X10 : term] : (((X9 @ X10)) = ((X8 @ id @ X10 @ id))) | ~ ! [X19 : (subst > term > term)] : (~ ! [X11 : term,X29 : subst,X12 : subst] : (((sub @ (X19 @ X12 @ X11) @ X29)) = ((X19 @ (comp @ X12 @ X29) @ (sub @ X11 @ X29)))) | (X8 @ id @ (lam @ (X19 @ sh @ one)) @ id) | ~ ! [X11 : term] : ((X8 @ id @ (X19 @ id @ X11) @ id) | ~(X8 @ id @ X11 @ id))) | (X9 @ (lam @ X41)) | ~ ! [X15 : term] : (~(X9 @ X15) | (X9 @ (sub @ X41 @ (push @ X15 @ id)))))))) & (! [X8 : (subst > term > subst > $o)] : ~ ! [X9 : (term > $o)] : ~ ! [X10 : term] : (((X9 @ X10)) = ((X8 @ id @ X10 @ id))) <=> hoasinduction_lem0_lthm) & hoasinduction_gthm & ~ulamvar1 & hoasinduction_lem2_gthm & pushprop_lem3v2_lthm & hoaslamnotvar & (((sub @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2term @ d_one))) & ((^[X6 : subst, X7 : (subst > term > term)] : ((d2term @ d_one))) = hoaslam) & (! [X8 : (subst > term > subst > $o),X9 : (term > $o),X45 : term,X46 : term] : (~ ! [X10 : term] : (((X9 @ X10)) = ((X8 @ id @ X10 @ id))) | ~ ! [X11 : term,X31 : subst,X29 : subst,X12 : subst] : ((X8 @ X12 @ X11 @ (comp @ X31 @ X29)) | ~(X8 @ (comp @ X12 @ X31) @ (sub @ X11 @ X31) @ X29)) | ~ ! [X11 : term,X15 : term] : (~(X8 @ id @ X11 @ id) | ~(X8 @ id @ X15 @ id) | (X8 @ id @ (ap @ (sub @ X11 @ id) @ X15) @ id)) | ~ ! [X11 : term,X12 : subst,X31 : subst,X29 : subst] : ((X8 @ (comp @ X12 @ X31) @ (sub @ X11 @ X31) @ X29) | ~(X8 @ X12 @ X11 @ (comp @ X31 @ X29))) | ~(X9 @ X45) | (X9 @ (ap @ X45 @ X46)) | ~(X9 @ X46)) <=> hoasinduction_lem2v2) & ! [X1 : d_term] : (X1 = d_one) & (! [X8 : (term > $o),X11 : term,X31 : (term > $o),X12 : subst,X15 : term] : ((X31 @ (sub @ X11 @ (push @ X15 @ X12))) | ~(X8 @ X11)) <=> pushprop_lem1) & hoasinduction_lem1_gthm & ! [X2 : subst] : ? [X3 : d_subst] : (X2 = ((d2subst @ X3))) & (hoasinduction_lem3v2a <=> ! [X41 : term,X9 : (term > $o),X8 : (subst > term > subst > $o)] : (~ ! [X15 : term] : ((X9 @ (sub @ X41 @ (push @ X15 @ id))) | ~(X9 @ X15)) | ~ ! [X19 : (subst > term > term)] : (~ ! [X12 : subst,X11 : term,X29 : subst] : (((sub @ (X19 @ X12 @ X11) @ X29)) = ((X19 @ (comp @ X12 @ X29) @ (sub @ X11 @ X29)))) | (X8 @ id @ (lam @ (X19 @ sh @ one)) @ id) | ~ ! [X11 : term] : ((X8 @ id @ (X19 @ id @ X11) @ id) | ~(X8 @ id @ X11 @ id))) | (X9 @ (lam @ X41)) | ~ ! [X10 : term] : (((X9 @ X10)) = ((X8 @ id @ X10 @ id))))) & ((! [X11 : term] : (X11 = ((sub @ X11 @ id))) => ! [X11 : term] : (X11 = ((sub @ X11 @ id)))) <=> termmset_lthm) & pushprop_lem2v2 & pushprop_lem2v2_lthm & (! [X44 : term,X8 : (subst > term > subst > $o)] : ((X8 @ id @ X44 @ id) | ~ ! [X19 : (subst > term > term)] : (~ ! [X11 : term] : (~(X8 @ id @ X11 @ id) | (X8 @ id @ (X19 @ id @ X11) @ id)) | ~ ! [X29 : subst,X12 : subst,X11 : term] : (((sub @ (X19 @ X12 @ X11) @ X29)) = ((X19 @ (comp @ X12 @ X29) @ (sub @ X11 @ X29)))) | (X8 @ id @ (lam @ (X19 @ sh @ one)) @ id)) | ~ ! [X11 : term,X15 : term] : (~(X8 @ id @ X15 @ id) | ~(X8 @ id @ X11 @ id) | (X8 @ id @ (ap @ (sub @ X11 @ id) @ X15) @ id))) <=> hoasinduction_no_psi_cond) & (hoasinduction_lem3a_lthm <=> (! [X11 : term] : (X11 = ((sub @ X11 @ id))) => (! [X28 : term,X8 : (subst > term > subst > $o)] : (~ ! [X19 : (subst > term > term)] : ((X8 @ id @ (lam @ (X19 @ sh @ one)) @ id) | ~ ! [X11 : term] : ((X8 @ id @ (X19 @ id @ X11) @ id) | ~(X8 @ id @ X11 @ id)) | ~ ! [X29 : subst,X12 : subst,X11 : term] : (((sub @ (X19 @ X12 @ X11) @ X29)) = ((X19 @ (comp @ X12 @ X29) @ (sub @ X11 @ X29))))) | ~ ! [X15 : term] : (~(X8 @ id @ X15 @ id) | (X8 @ id @ (sub @ X28 @ (push @ X15 @ id)) @ id)) | (X8 @ id @ (lam @ (sub @ X28 @ (push @ one @ sh))) @ id)) => ! [X8 : (subst > term > subst > $o),X38 : term] : (~ ! [X15 : term] : (~(X8 @ id @ X15 @ id) | (X8 @ id @ (sub @ X38 @ (push @ X15 @ id)) @ id)) | ~ ! [X19 : (subst > term > term)] : (~ ! [X29 : subst,X11 : term,X12 : subst] : (((sub @ (X19 @ X12 @ X11) @ X29)) = ((X19 @ (comp @ X12 @ X29) @ (sub @ X11 @ X29)))) | (X8 @ id @ (lam @ (X19 @ sh @ one)) @ id) | ~ ! [X11 : term] : ((X8 @ id @ (X19 @ id @ X11) @ id) | ~(X8 @ id @ X11 @ id))) | (X8 @ id @ (lam @ X38) @ id))))) & hoasinduction_lem1 & (id = ((d2subst @ d_id))) & pushprop_lem1_gthm & (sh = ((d2subst @ d_id))) & (induction2_lthm <=> (! [X11 : term] : (X11 = ((sub @ X11 @ id))) => (! [X14 : subst,X13 : term,X8 : (term > $o)] : (~ ! [X11 : term] : (~ ! [X15 : term] : ((X8 @ (sub @ X11 @ (push @ X15 @ id))) | ~(X8 @ X15)) | (X8 @ (lam @ X11))) | (X8 @ (sub @ X13 @ X14)) | ~ ! [X11 : term,X15 : term] : (~(X8 @ X11) | ~(X8 @ X15) | (X8 @ (ap @ X11 @ X15))) | ~ ! [X15 : term] : (~(var @ X15) | (X8 @ (sub @ X15 @ X14)))) => ! [X8 : (term > $o),X37 : term] : (~ ! [X15 : term,X11 : term] : ((X8 @ (ap @ X11 @ X15)) | ~(X8 @ X11) | ~(X8 @ X15)) | ~ ! [X11 : term] : (~(var @ X11) | (X8 @ X11)) | ~ ! [X11 : term] : ((X8 @ (lam @ X11)) | ~ ! [X15 : term] : (~(X8 @ X15) | (X8 @ (sub @ X11 @ (push @ X15 @ id))))) | (X8 @ X37))))) & (axidl <=> ! [X12 : subst] : (X12 = ((comp @ id @ X12)))) & (hoasinduction_lthm_3 <=> (! [X8 : (subst > term > subst > $o)] : ~ ! [X9 : (term > $o)] : ~ ! [X10 : term] : (((X9 @ X10)) = ((X8 @ id @ X10 @ id))) => (! [X8 : (term > $o),X37 : term] : (~ ! [X11 : term] : (~(var @ X11) | (X8 @ X11)) | ~ ! [X15 : term,X11 : term] : ((X8 @ (ap @ X11 @ X15)) | ~(X8 @ X11) | ~(X8 @ X15)) | ~ ! [X11 : term] : (~ ! [X15 : term] : ((X8 @ (sub @ X11 @ (push @ X15 @ id))) | ~(X8 @ X15)) | (X8 @ (lam @ X11))) | (X8 @ X37)) => (! [X11 : term] : (X11 = ((sub @ X11 @ id))) => (! [X9 : (term > $o),X41 : term,X8 : (subst > term > subst > $o)] : (~ ! [X10 : term] : (((X9 @ X10)) = ((X8 @ id @ X10 @ id))) | ~ ! [X19 : (subst > term > term)] : (~ ! [X11 : term] : ((X8 @ id @ (X19 @ id @ X11) @ id) | ~(X8 @ id @ X11 @ id)) | (X8 @ id @ (lam @ (X19 @ sh @ one)) @ id) | ~ ! [X29 : subst,X12 : subst,X11 : term] : (((sub @ (X19 @ X12 @ X11) @ X29)) = ((X19 @ (comp @ X12 @ X29) @ (sub @ X11 @ X29))))) | (X9 @ (lam @ X41)) | ~ ! [X15 : term] : ((X9 @ (sub @ X41 @ (push @ X15 @ id))) | ~(X9 @ X15))) => ! [X8 : (subst > term > subst > $o),X40 : term] : ((X8 @ id @ X40 @ id) | ~ ! [X12 : subst,X11 : term,X29 : subst,X31 : subst] : (~(X8 @ X12 @ X11 @ (comp @ X31 @ X29)) | (X8 @ (comp @ X12 @ X31) @ (sub @ X11 @ X31) @ X29)) | ~ ! [X29 : subst,X31 : subst,X11 : term,X12 : subst] : (~(X8 @ (comp @ X12 @ X31) @ (sub @ X11 @ X31) @ X29) | (X8 @ X12 @ X11 @ (comp @ X31 @ X29))) | ~ ! [X11 : term,X15 : term] : ((X8 @ id @ (ap @ (sub @ X11 @ id) @ X15) @ id) | ~(X8 @ id @ X11 @ id) | ~(X8 @ id @ X15 @ id)) | ~ ! [X11 : term] : ((X8 @ id @ X11 @ id) | ~(var @ (sub @ X11 @ id))) | ~ ! [X19 : (subst > term > term)] : (~ ! [X29 : subst,X11 : term,X12 : subst] : (((sub @ (X19 @ X12 @ X11) @ X29)) = ((X19 @ (comp @ X12 @ X29) @ (sub @ X11 @ X29)))) | ~ ! [X11 : term] : (~(X8 @ id @ X11 @ id) | (X8 @ id @ (X19 @ id @ X11) @ id)) | (X8 @ id @ (lam @ (X19 @ sh @ one)) @ id)))))))) & ((! [X12 : subst,X11 : term] : (X11 = ((sub @ one @ (push @ X11 @ X12)))) => (! [X12 : subst,X11 : term] : (X12 = ((comp @ sh @ (push @ X11 @ X12)))) => (! [X12 : subst] : (X12 = ((comp @ X12 @ id))) => (! [X18 : term,X8 : (term > $o)] : ((X8 @ X18) | ~ ! [X11 : term,X15 : term] : (~(X8 @ X15) | ~(X8 @ X11) | (X8 @ (ap @ X11 @ X15))) | ~ ! [X11 : term] : (~(var @ X11) | (X8 @ X11)) | ~ ! [X11 : term] : (~(X8 @ X11) | (X8 @ (lam @ X11)))) => ! [X8 : (term > $o),X13 : term,X14 : subst] : (~ ! [X15 : term] : ((X8 @ (sub @ X15 @ X14)) | ~(var @ X15)) | ~ ! [X11 : term] : ((X8 @ (lam @ X11)) | ~ ! [X15 : term] : ((X8 @ (sub @ X11 @ (push @ X15 @ id))) | ~(X8 @ X15))) | ~ ! [X11 : term,X15 : term] : (~(X8 @ X15) | (X8 @ (ap @ X11 @ X15)) | ~(X8 @ X11)) | (X8 @ (sub @ X13 @ X14))))))) <=> induction2lem_lthm) & ((! [X11 : term] : (X11 = ((sub @ X11 @ id))) => (! [X16 : term,X15 : term,X11 : term,X17 : term] : ((X11 = X15) | (((ap @ X11 @ X16)) != ((ap @ X15 @ X17)))) => ! [X15 : term,X17 : term,X11 : term,X16 : term] : ((((ap @ (sub @ X11 @ id) @ X16)) != ((ap @ (sub @ X15 @ id) @ X17))) | (X11 = X15)))) <=> hoasapinj1_lthm) & (termmset <=> ! [X11 : term] : (X11 = ((sub @ X11 @ id)))) & (pushprop_lem1_lthm <=> (! [X12 : subst,X11 : term] : (X11 = ((sub @ one @ (push @ X11 @ X12)))) => (! [X11 : term,X12 : subst] : (X12 = ((comp @ sh @ (push @ X11 @ X12)))) => ! [X12 : subst,X31 : (term > $o),X11 : term,X15 : term,X8 : (term > $o)] : (~(X8 @ X11) | (X31 @ (sub @ X11 @ (push @ X15 @ X12))))))) & pushprop_lem2v2_gthm & hoasinduction_lem1_lthm & pushprop_lem1v2_gthm & pushprop_gthm & (substmonoid_lthm <=> (! [X12 : subst] : (X12 = ((comp @ id @ X12))) => (! [X12 : subst] : (X12 = ((comp @ X12 @ id))) => (! [X12 : subst] : (X12 = ((comp @ X12 @ id))) & ! [X12 : subst] : (X12 = ((comp @ id @ X12))))))) & axmap & (! [X34 : (subst > term > term),X19 : (subst > term > term),X35 : subst,X36 : term] : ((((lam @ (X19 @ sh @ one))) != ((lam @ (X34 @ sh @ one)))) | (((X19 @ X35 @ X36)) = ((X34 @ X35 @ X36))) | ~ ! [X11 : term,X29 : subst,X12 : subst] : (((sub @ (X34 @ X12 @ X11) @ X29)) = ((X34 @ (comp @ X12 @ X29) @ (sub @ X11 @ X29)))) | ~ ! [X11 : term,X29 : subst,X12 : subst] : (((sub @ (X19 @ X12 @ X11) @ X29)) = ((X19 @ (comp @ X12 @ X29) @ (sub @ X11 @ X29))))) <=> hoaslaminj) & ulamvarind & hoasinduction_lem1v2 & substmonoid_gthm & (! [X8 : (term > $o),X12 : subst,X11 : term] : ~ ! [X9 : (term > $o)] : ~ ! [X10 : term] : (((X9 @ X10)) = ((X8 @ (sub @ X10 @ (push @ X11 @ X12))))) <=> pushprop_lem0_lthm) & hoasapnotvar & (hoasinduction_lem3 <=> ! [X8 : (subst > term > subst > $o),X30 : term] : (~ ! [X11 : term,X31 : subst,X29 : subst,X12 : subst] : ((X8 @ (comp @ X12 @ X31) @ (sub @ X11 @ X31) @ X29) | ~(X8 @ X12 @ X11 @ (comp @ X31 @ X29))) | (X8 @ id @ (lam @ X30) @ id) | ~ ! [X19 : (subst > term > term)] : (~ ! [X12 : subst,X29 : subst,X11 : term] : (((sub @ (X19 @ X12 @ X11) @ X29)) = ((X19 @ (comp @ X12 @ X29) @ (sub @ X11 @ X29)))) | (X8 @ id @ (lam @ (X19 @ sh @ one)) @ id) | ~ ! [X11 : term] : (~(X8 @ id @ X11 @ id) | (X8 @ id @ (X19 @ id @ X11) @ id))) | ~ ! [X15 : term] : ((X8 @ id @ (sub @ X30 @ (push @ X15 @ id)) @ id) | ~(X8 @ id @ X15 @ id)) | ~ ! [X29 : subst,X12 : subst,X31 : subst,X11 : term] : (~(X8 @ (comp @ X12 @ X31) @ (sub @ X11 @ X31) @ X29) | (X8 @ X12 @ X11 @ (comp @ X31 @ X29))))) & (hoasinduction_lem2 <=> ! [X8 : (subst > term > subst > $o),X33 : term,X32 : term] : ((X8 @ id @ (ap @ X32 @ X33) @ id) | ~ ! [X31 : subst,X12 : subst,X29 : subst,X11 : term] : ((X8 @ (comp @ X12 @ X31) @ (sub @ X11 @ X31) @ X29) | ~(X8 @ X12 @ X11 @ (comp @ X31 @ X29))) | ~(X8 @ id @ X32 @ id) | ~ ! [X29 : subst,X12 : subst,X11 : term,X31 : subst] : (~(X8 @ (comp @ X12 @ X31) @ (sub @ X11 @ X31) @ X29) | (X8 @ X12 @ X11 @ (comp @ X31 @ X29))) | ~ ! [X11 : term,X15 : term] : (~(X8 @ id @ X11 @ id) | (X8 @ id @ (ap @ (sub @ X11 @ id) @ X15) @ id) | ~(X8 @ id @ X15 @ id)) | ~(X8 @ id @ X33 @ id))) & ! [X0 : term] : ? [X1 : d_term] : (X0 = ((d2term @ X1))) & (((ap @ (d2term @ d_one) @ (d2term @ d_one))) = ((d2term @ d_one))) & axabs & (hoasinduction_lem3a <=> ! [X8 : (subst > term > subst > $o),X38 : term] : ((X8 @ id @ (lam @ X38) @ id) | ~ ! [X19 : (subst > term > term)] : (~ ! [X11 : term,X29 : subst,X12 : subst] : (((sub @ (X19 @ X12 @ X11) @ X29)) = ((X19 @ (comp @ X12 @ X29) @ (sub @ X11 @ X29)))) | ~ ! [X11 : term] : ((X8 @ id @ (X19 @ id @ X11) @ id) | ~(X8 @ id @ X11 @ id)) | (X8 @ id @ (lam @ (X19 @ sh @ one)) @ id)) | ~ ! [X15 : term] : (~(X8 @ id @ X15 @ id) | (X8 @ id @ (sub @ X38 @ (push @ X15 @ id)) @ id)))) & (apinj1 <=> ! [X17 : term,X11 : term,X15 : term,X16 : term] : ((X11 = X15) | (((ap @ X11 @ X16)) != ((ap @ X15 @ X17))))) & induction2lem_gthm & (hoasinduction_no_psi_cond_lthm <=> (! [X8 : (subst > term > subst > $o)] : ~ ! [X9 : (term > $o)] : ~ ! [X10 : term] : (((X9 @ X10)) = ((X8 @ id @ X10 @ id))) => (! [X8 : (term > $o),X37 : term] : ((X8 @ X37) | ~ ! [X11 : term] : (~(var @ X11) | (X8 @ X11)) | ~ ! [X15 : term,X11 : term] : (~(X8 @ X11) | (X8 @ (ap @ X11 @ X15)) | ~(X8 @ X15)) | ~ ! [X11 : term] : (~ ! [X15 : term] : (~(X8 @ X15) | (X8 @ (sub @ X11 @ (push @ X15 @ id)))) | (X8 @ (lam @ X11)))) => (! [X11 : term] : (X11 = ((sub @ X11 @ id))) => (! [X41 : term,X9 : (term > $o),X8 : (subst > term > subst > $o)] : (~ ! [X15 : term] : (~(X9 @ X15) | (X9 @ (sub @ X41 @ (push @ X15 @ id)))) | ~ ! [X19 : (subst > term > term)] : (~ ! [X29 : subst,X12 : subst,X11 : term] : (((sub @ (X19 @ X12 @ X11) @ X29)) = ((X19 @ (comp @ X12 @ X29) @ (sub @ X11 @ X29)))) | (X8 @ id @ (lam @ (X19 @ sh @ one)) @ id) | ~ ! [X11 : term] : ((X8 @ id @ (X19 @ id @ X11) @ id) | ~(X8 @ id @ X11 @ id))) | ~ ! [X10 : term] : (((X9 @ X10)) = ((X8 @ id @ X10 @ id))) | (X9 @ (lam @ X41))) => ! [X44 : term,X8 : (subst > term > subst > $o)] : ((X8 @ id @ X44 @ id) | ~ ! [X11 : term,X15 : term] : (~(X8 @ id @ X11 @ id) | (X8 @ id @ (ap @ (sub @ X11 @ id) @ X15) @ id) | ~(X8 @ id @ X15 @ id)) | ~ ! [X19 : (subst > term > term)] : (~ ! [X29 : subst,X12 : subst,X11 : term] : (((sub @ (X19 @ X12 @ X11) @ X29)) = ((X19 @ (comp @ X12 @ X29) @ (sub @ X11 @ X29)))) | ~ ! [X11 : term] : (~(X8 @ id @ X11 @ id) | (X8 @ id @ (X19 @ id @ X11) @ id)) | (X8 @ id @ (lam @ (X19 @ sh @ one)) @ id)))))))) & ((! [X12 : subst,X11 : term] : (X11 = ((sub @ one @ (push @ X11 @ X12)))) => ! [X9 : (term > $o),X8 : (term > $o),X12 : subst,X11 : term] : ((X9 @ one) | ~ ! [X10 : term] : (((X9 @ X10)) = ((X8 @ (sub @ X10 @ (push @ X11 @ X12))))) | ~(X8 @ X11))) <=> pushprop_lem1v2_lthm) & induction2_gthm & (pushprop_lem1v2 <=> ! [X12 : subst,X9 : (term > $o),X8 : (term > $o),X11 : term] : (~ ! [X10 : term] : (((X9 @ X10)) = ((X8 @ (sub @ X10 @ (push @ X11 @ X12))))) | ~(X8 @ X11) | (X9 @ one))) & hoasinduction_lem2v2_gthm & (apinj2 <=> ! [X15 : term,X11 : term,X16 : term,X17 : term] : ((X16 = X17) | (((ap @ X11 @ X16)) != ((ap @ X15 @ X17))))) & (! [X12 : subst,X11 : term] : (X12 = ((comp @ sh @ (push @ X11 @ X12)))) <=> axshiftcons) & apnotvar & (hoasinduction_lem0 <=> ! [X8 : (subst > term > subst > $o)] : ~ ! [X9 : (term > $o)] : ~ ! [X10 : term] : (((X9 @ X10)) = ((X8 @ id @ X10 @ id)))) & (! [X16 : term,X17 : term,X11 : term,X15 : term] : ((X16 = X17) | (((ap @ (sub @ X11 @ id) @ X16)) != ((ap @ (sub @ X15 @ id) @ X17)))) <=> hoasapinj2) & pushprop & (! [X19 : (subst > term > term),X43 : term,X42 : term] : ((((lam @ (X19 @ sh @ one))) != ((ap @ (sub @ X42 @ id) @ X43))) | ~ ! [X12 : subst,X29 : subst,X11 : term] : (((sub @ (X19 @ X12 @ X11) @ X29)) = ((X19 @ (comp @ X12 @ X29) @ (sub @ X11 @ X29))))) <=> hoaslamnotap) & pushprop_lem0_gthm & hoaslaminj_gthm & (axvarcons <=> ! [X12 : subst,X11 : term] : (X11 = ((sub @ one @ (push @ X11 @ X12))))) & (hoasinduction_lem3_lthm <=> (! [X11 : term] : (X11 = ((sub @ X11 @ id))) => (! [X8 : (subst > term > subst > $o),X28 : term] : ((X8 @ id @ (lam @ (sub @ X28 @ (push @ one @ sh))) @ id) | ~ ! [X19 : (subst > term > term)] : (~ ! [X11 : term] : (~(X8 @ id @ X11 @ id) | (X8 @ id @ (X19 @ id @ X11) @ id)) | (X8 @ id @ (lam @ (X19 @ sh @ one)) @ id) | ~ ! [X29 : subst,X11 : term,X12 : subst] : (((sub @ (X19 @ X12 @ X11) @ X29)) = ((X19 @ (comp @ X12 @ X29) @ (sub @ X11 @ X29))))) | ~ ! [X15 : term] : (~(X8 @ id @ X15 @ id) | (X8 @ id @ (sub @ X28 @ (push @ X15 @ id)) @ id))) => ! [X30 : term,X8 : (subst > term > subst > $o)] : ((X8 @ id @ (lam @ X30) @ id) | ~ ! [X11 : term,X29 : subst,X12 : subst,X31 : subst] : ((X8 @ (comp @ X12 @ X31) @ (sub @ X11 @ X31) @ X29) | ~(X8 @ X12 @ X11 @ (comp @ X31 @ X29))) | ~ ! [X15 : term] : (~(X8 @ id @ X15 @ id) | (X8 @ id @ (sub @ X30 @ (push @ X15 @ id)) @ id)) | ~ ! [X19 : (subst > term > term)] : (~ ! [X12 : subst,X11 : term,X29 : subst] : (((sub @ (X19 @ X12 @ X11) @ X29)) = ((X19 @ (comp @ X12 @ X29) @ (sub @ X11 @ X29)))) | (X8 @ id @ (lam @ (X19 @ sh @ one)) @ id) | ~ ! [X11 : term] : ((X8 @ id @ (X19 @ id @ X11) @ id) | ~(X8 @ id @ X11 @ id))) | ~ ! [X12 : subst,X29 : subst,X11 : term,X31 : subst] : (~(X8 @ (comp @ X12 @ X31) @ (sub @ X11 @ X31) @ X29) | (X8 @ X12 @ X11 @ (comp @ X31 @ X29))))))) & termmset_gthm & ! [X47 : d_term,X48 : d_term] : ((((d2term @ X47)) = ((d2term @ X48))) => (X47 = X48)) & ! [X5 : d_subst,X4 : d_subst] : ((((d2subst @ X4)) = ((d2subst @ X5))) => (X4 = X5)) & lamnotvar & (((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) = ((d2term @ d_one))) & (! [X12 : subst] : (X12 = ((push @ (sub @ one @ X12) @ (comp @ sh @ X12)))) <=> axscons) & (shinj <=> ! [X11 : term,X15 : term] : ((((sub @ X11 @ sh)) != ((sub @ X15 @ sh))) | (X11 = X15))) & pushprop_lthm & hoasapnotvar_lthm & hoasinduction_lem3_gthm & hoasinduction_lem3v2_gthm & ((^[X8 : (subst > term > subst > $o), X9 : (term > $o)] : (! [X10 : term] : (((X9 @ X10)) = ((X8 @ id @ X10 @ id))))) = hoasinduction_p_and_p_prime) & hoasinduction_lem2_lthm & axclos & (hoasinduction_lem3aa <=> ! [X8 : (subst > term > subst > $o),X28 : term] : (~ ! [X19 : (subst > term > term)] : ((X8 @ id @ (lam @ (X19 @ sh @ one)) @ id) | ~ ! [X11 : term] : ((X8 @ id @ (X19 @ id @ X11) @ id) | ~(X8 @ id @ X11 @ id)) | ~ ! [X11 : term,X29 : subst,X12 : subst] : (((sub @ (X19 @ X12 @ X11) @ X29)) = ((X19 @ (comp @ X12 @ X29) @ (sub @ X11 @ X29))))) | ~ ! [X15 : term] : (~(X8 @ id @ X15 @ id) | (X8 @ id @ (sub @ X28 @ (push @ X15 @ id)) @ id)) | (X8 @ id @ (lam @ (sub @ X28 @ (push @ one @ sh))) @ id))) & (hoasinduction_lthm <=> (! [X37 : term,X8 : (term > $o)] : (~ ! [X11 : term] : (~(var @ X11) | (X8 @ X11)) | (X8 @ X37) | ~ ! [X11 : term,X15 : term] : (~(X8 @ X15) | ~(X8 @ X11) | (X8 @ (ap @ X11 @ X15))) | ~ ! [X11 : term] : (~ ! [X15 : term] : (~(X8 @ X15) | (X8 @ (sub @ X11 @ (push @ X15 @ id)))) | (X8 @ (lam @ X11)))) => (! [X32 : term,X33 : term,X8 : (subst > term > subst > $o)] : ((X8 @ id @ (ap @ X32 @ X33) @ id) | ~(X8 @ id @ X32 @ id) | ~(X8 @ id @ X33 @ id) | ~ ! [X11 : term,X12 : subst,X31 : subst,X29 : subst] : ((X8 @ (comp @ X12 @ X31) @ (sub @ X11 @ X31) @ X29) | ~(X8 @ X12 @ X11 @ (comp @ X31 @ X29))) | ~ ! [X29 : subst,X12 : subst,X31 : subst,X11 : term] : (~(X8 @ (comp @ X12 @ X31) @ (sub @ X11 @ X31) @ X29) | (X8 @ X12 @ X11 @ (comp @ X31 @ X29))) | ~ ! [X11 : term,X15 : term] : (~(X8 @ id @ X11 @ id) | ~(X8 @ id @ X15 @ id) | (X8 @ id @ (ap @ (sub @ X11 @ id) @ X15) @ id))) => (! [X30 : term,X8 : (subst > term > subst > $o)] : (~ ! [X29 : subst,X11 : term,X12 : subst,X31 : subst] : ((X8 @ X12 @ X11 @ (comp @ X31 @ X29)) | ~(X8 @ (comp @ X12 @ X31) @ (sub @ X11 @ X31) @ X29)) | ~ ! [X19 : (subst > term > term)] : ((X8 @ id @ (lam @ (X19 @ sh @ one)) @ id) | ~ ! [X12 : subst,X29 : subst,X11 : term] : (((sub @ (X19 @ X12 @ X11) @ X29)) = ((X19 @ (comp @ X12 @ X29) @ (sub @ X11 @ X29)))) | ~ ! [X11 : term] : ((X8 @ id @ (X19 @ id @ X11) @ id) | ~(X8 @ id @ X11 @ id))) | (X8 @ id @ (lam @ X30) @ id) | ~ ! [X11 : term,X31 : subst,X29 : subst,X12 : subst] : (~(X8 @ X12 @ X11 @ (comp @ X31 @ X29)) | (X8 @ (comp @ X12 @ X31) @ (sub @ X11 @ X31) @ X29)) | ~ ! [X15 : term] : (~(X8 @ id @ X15 @ id) | (X8 @ id @ (sub @ X30 @ (push @ X15 @ id)) @ id))) => ! [X8 : (subst > term > subst > $o),X40 : term] : (~ ! [X15 : term,X11 : term] : (~(X8 @ id @ X15 @ id) | ~(X8 @ id @ X11 @ id) | (X8 @ id @ (ap @ (sub @ X11 @ id) @ X15) @ id)) | ~ ! [X11 : term] : (~(var @ (sub @ X11 @ id)) | (X8 @ id @ X11 @ id)) | (X8 @ id @ X40 @ id) | ~ ! [X19 : (subst > term > term)] : ((X8 @ id @ (lam @ (X19 @ sh @ one)) @ id) | ~ ! [X11 : term] : (~(X8 @ id @ X11 @ id) | (X8 @ id @ (X19 @ id @ X11) @ id)) | ~ ! [X11 : term,X29 : subst,X12 : subst] : (((sub @ (X19 @ X12 @ X11) @ X29)) = ((X19 @ (comp @ X12 @ X29) @ (sub @ X11 @ X29))))) | ~ ! [X29 : subst,X11 : term,X31 : subst,X12 : subst] : (~(X8 @ (comp @ X12 @ X31) @ (sub @ X11 @ X31) @ X29) | (X8 @ X12 @ X11 @ (comp @ X31 @ X29))) | ~ ! [X11 : term,X31 : subst,X12 : subst,X29 : subst] : (~(X8 @ X12 @ X11 @ (comp @ X31 @ X29)) | (X8 @ (comp @ X12 @ X31) @ (sub @ X11 @ X31) @ X29))))))) & hoasapnotvar_gthm & (! [X12 : subst] : (X12 = ((comp @ X12 @ id))) <=> axidr) & (laminj <=> ! [X11 : term,X15 : term] : ((X11 = X15) | (((lam @ X11)) != ((lam @ X15))))) & (((lam @ (d2term @ d_one))) = ((d2term @ d_one))) & (! [X9 : (term > $o),X39 : term,X8 : (subst > term > subst > $o)] : ((X9 @ (lam @ X39)) | ~ ! [X12 : subst,X31 : subst,X11 : term,X29 : subst] : ((X8 @ (comp @ X12 @ X31) @ (sub @ X11 @ X31) @ X29) | ~(X8 @ X12 @ X11 @ (comp @ X31 @ X29))) | ~ ! [X29 : subst,X31 : subst,X11 : term,X12 : subst] : (~(X8 @ (comp @ X12 @ X31) @ (sub @ X11 @ X31) @ X29) | (X8 @ X12 @ X11 @ (comp @ X31 @ X29))) | ~ ! [X15 : term] : (~(X9 @ X15) | (X9 @ (sub @ X39 @ (push @ X15 @ id)))) | ~ ! [X19 : (subst > term > term)] : (~ ! [X11 : term] : (~(X8 @ id @ X11 @ id) | (X8 @ id @ (X19 @ id @ X11) @ id)) | (X8 @ id @ (lam @ (X19 @ sh @ one)) @ id) | ~ ! [X12 : subst,X11 : term,X29 : subst] : (((sub @ (X19 @ X12 @ X11) @ X29)) = ((X19 @ (comp @ X12 @ X29) @ (sub @ X11 @ X29))))) | ~ ! [X10 : term] : (((X9 @ X10)) = ((X8 @ id @ X10 @ id)))) <=> hoasinduction_lem3v2) & hoasinduction_lem3a_gthm),
  file('/export/starexec/sandbox/benchmark/theBenchmark.p',alg444_1)).
thf(f4,plain,(
  hoaslamnotvar_lthm & (hoasapinj1 <=> ! [X0 : term,X1 : term,X2 : term,X3 : term] : ((X0 = X2) | (((ap @ (sub @ X2 @ id) @ X1)) != ((ap @ (sub @ X0 @ id) @ X3))))) & hoaslamnotvar_gthm & (hoasinduction_lem3aa_lthm <=> ! [X4 : (subst > term > subst > $o),X5 : term] : ((X4 @ id @ (lam @ (sub @ X5 @ (push @ one @ sh))) @ id) | ~ ! [X6 : (subst > term > term)] : (~ ! [X7 : term] : (~(X4 @ id @ X7 @ id) | (X4 @ id @ (X6 @ id @ X7) @ id)) | (X4 @ id @ (lam @ (X6 @ sh @ one)) @ id) | ~ ! [X8 : subst,X9 : subst,X10 : term] : (((sub @ (X6 @ X9 @ X10) @ X8)) = ((X6 @ (comp @ X9 @ X8) @ (sub @ X10 @ X8))))) | ~ ! [X11 : term] : (~(X4 @ id @ X11 @ id) | (X4 @ id @ (sub @ X5 @ (push @ X11 @ id)) @ id)))) & pushprop_lthm_orig & ((! [X12 : term,X13 : term,X14 : term,X15 : term] : ((X12 = X14) | (((ap @ X13 @ X14)) != ((ap @ X15 @ X12)))) => ! [X16 : term,X17 : term,X18 : term,X19 : term] : ((((ap @ (sub @ X17 @ id) @ X19)) != ((ap @ (sub @ X18 @ id) @ X16))) | (X16 = X19))) <=> hoasapinj2_lthm) & (pushprop_lem0 <=> ! [X20 : term,X21 : subst,X22 : (term > $o)] : ~ ! [X23 : (term > $o)] : ~ ! [X24 : term] : (((X23 @ X24)) = ((X22 @ (sub @ X24 @ (push @ X20 @ X21)))))) & ~(hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) & ~lamnotap & hoaslamnotap_lthm & (hoasinduction_lem3aaa <=> ! [X25 : term,X26 : (subst > term > subst > $o)] : ((X26 @ id @ (lam @ (sub @ X25 @ (push @ one @ sh))) @ id) | ~ ! [X27 : term] : ((X26 @ id @ (sub @ X25 @ (push @ X27 @ id)) @ id) | ~(X26 @ id @ X27 @ id)) | ~ ! [X28 : term,X29 : (subst > term > term)] : ((X26 @ id @ (lam @ (X29 @ sh @ one)) @ id) | ~ ! [X30 : term,X31 : subst,X32 : subst] : (((sub @ (sub @ X28 @ (push @ X30 @ X31)) @ X32)) = ((sub @ (X29 @ X31 @ X30) @ X32))) | ~ ! [X33 : subst,X34 : subst,X35 : term] : (((X29 @ (comp @ X33 @ X34) @ (sub @ X35 @ X34))) = ((sub @ X28 @ (push @ (sub @ X35 @ X34) @ (comp @ X33 @ X34))))) | ~ ! [X36 : term] : (~(X26 @ id @ X36 @ id) | (X26 @ id @ (X29 @ id @ X36) @ id))))) & (hoaslaminj_lthm <=> (! [X37 : term,X38 : subst] : (((sub @ one @ (push @ X37 @ X38))) = X37) => (! [X39 : subst,X40 : term] : (((comp @ sh @ (push @ X40 @ X39))) = X39) => (! [X41 : term,X42 : term] : ((((lam @ X41)) != ((lam @ X42))) | (X41 = X42)) => ! [X43 : term,X44 : (subst > term > term),X45 : subst,X46 : (subst > term > term)] : (~ ! [X47 : term,X48 : subst,X49 : subst] : (((X44 @ (comp @ X48 @ X49) @ (sub @ X47 @ X49))) = ((sub @ (X44 @ X48 @ X47) @ X49))) | (((X44 @ X45 @ X43)) = ((X46 @ X45 @ X43))) | (((lam @ (X44 @ sh @ one))) != ((lam @ (X46 @ sh @ one)))) | ~ ! [X50 : subst,X51 : term,X52 : subst] : (((X46 @ (comp @ X52 @ X50) @ (sub @ X51 @ X50))) = ((sub @ (X46 @ X52 @ X51) @ X50)))))))) & (! [X53 : term] : (((sub @ X53 @ id)) = X53) <=> axvarid) & axapp & (hoasinduction_lem3b <=> ! [X54 : term] : ~ ! [X55 : (subst > term > term)] : (((X55 @ sh @ one)) != ((sub @ X54 @ (push @ one @ sh))))) & hoasapinj2_gthm & axvarshift & axassoc & (((push @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & (hoasinduction <=> ! [X56 : term,X57 : (subst > term > subst > $o)] : (~ ! [X58 : term] : ((X57 @ id @ X58 @ id) | ~(var @ (sub @ X58 @ id))) | (X57 @ id @ X56 @ id) | ~ ! [X59 : term,X60 : subst,X61 : subst,X62 : subst] : ((X57 @ (comp @ X60 @ X61) @ (sub @ X59 @ X61) @ X62) | ~(X57 @ X60 @ X59 @ (comp @ X61 @ X62))) | ~ ! [X63 : (subst > term > term)] : (~ ! [X64 : term,X65 : subst,X66 : subst] : (((X63 @ (comp @ X65 @ X66) @ (sub @ X64 @ X66))) = ((sub @ (X63 @ X65 @ X64) @ X66))) | ~ ! [X67 : term] : (~(X57 @ id @ X67 @ id) | (X57 @ id @ (X63 @ id @ X67) @ id)) | (X57 @ id @ (lam @ (X63 @ sh @ one)) @ id)) | ~ ! [X68 : term,X69 : term] : (~(X57 @ id @ X68 @ id) | ~(X57 @ id @ X69 @ id) | (X57 @ id @ (ap @ (sub @ X69 @ id) @ X68) @ id)) | ~ ! [X70 : subst,X71 : term,X72 : subst,X73 : subst] : ((X57 @ X70 @ X71 @ (comp @ X73 @ X72)) | ~(X57 @ (comp @ X70 @ X73) @ (sub @ X71 @ X73) @ X72)))) & ! [X74 : d_subst] : (d_id = X74) & ulamvarsh & (induction <=> ! [X75 : term,X76 : (term > $o)] : (~ ! [X77 : term] : (~(var @ X77) | (X76 @ X77)) | ~ ! [X78 : term] : (~(X76 @ X78) | (X76 @ (lam @ X78))) | (X76 @ X75) | ~ ! [X79 : term,X80 : term] : (~(X76 @ X79) | ~(X76 @ X80) | (X76 @ (ap @ X79 @ X80))))) & (! [X81 : term] : ~ ! [X82 : (subst > term > term)] : ~ ! [X83 : subst,X84 : term] : (((sub @ X81 @ (push @ X84 @ X83))) = ((X82 @ X83 @ X84))) <=> hoasinduction_lem3v2_f_lthm) & hoasapinj1_gthm & ~(var @ (d2term @ d_one)) & (! [X85 : term] : ~ ! [X86 : (subst > term > term)] : (((sub @ X85 @ (push @ one @ sh))) != ((X86 @ sh @ one))) <=> hoasinduction_lem3b_lthm) & ((! [X87 : term] : (((sub @ X87 @ id)) = X87) => ! [X88 : (subst > term > subst > $o),X89 : term,X90 : (term > $o)] : (~ ! [X91 : term,X92 : subst,X93 : subst,X94 : subst] : (~(X88 @ X93 @ X91 @ (comp @ X92 @ X94)) | (X88 @ (comp @ X93 @ X92) @ (sub @ X91 @ X92) @ X94)) | (X90 @ (lam @ X89)) | ~ ! [X95 : (subst > term > term)] : (~ ! [X96 : term] : (~(X88 @ id @ X96 @ id) | (X88 @ id @ (X95 @ id @ X96) @ id)) | ~ ! [X97 : subst,X98 : term,X99 : subst] : (((sub @ (X95 @ X97 @ X98) @ X99)) = ((X95 @ (comp @ X97 @ X99) @ (sub @ X98 @ X99)))) | (X88 @ id @ (lam @ (X95 @ sh @ one)) @ id)) | ~ ! [X100 : term] : (((X90 @ X100)) = ((X88 @ id @ X100 @ id))) | ~ ! [X101 : subst,X102 : subst,X103 : subst,X104 : term] : ((X88 @ X102 @ X104 @ (comp @ X101 @ X103)) | ~(X88 @ (comp @ X102 @ X101) @ (sub @ X104 @ X101) @ X103)) | ~ ! [X105 : term] : ((X90 @ (sub @ X89 @ (push @ X105 @ id))) | ~(X90 @ X105)))) <=> hoasinduction_lem3v2_lthm) & hoasinduction_lem3b_gthm & hoaslamnotap_gthm & (pushprop_p_and_p_prime = (^[X106 : term, X107 : subst, X108 : (term > $o), X109 : (term > $o)] : (! [X110 : term] : (((X108 @ (sub @ X110 @ (push @ X106 @ X107)))) = ((X109 @ X110)))))) & (induction2 <=> ! [X111 : (term > $o),X112 : term] : ((X111 @ X112) | ~ ! [X113 : term] : (~ ! [X114 : term] : ((X111 @ (sub @ X113 @ (push @ X114 @ id))) | ~(X111 @ X114)) | (X111 @ (lam @ X113))) | ~ ! [X115 : term,X116 : term] : (~(X111 @ X116) | ~(X111 @ X115) | (X111 @ (ap @ X115 @ X116))) | ~ ! [X117 : term] : ((X111 @ X117) | ~(var @ X117)))) & (one = ((d2term @ d_one))) & ((! [X118 : subst] : (((comp @ X118 @ id)) = X118) & ! [X119 : subst] : (((comp @ id @ X119)) = X119)) <=> substmonoid) & (! [X120 : (term > $o),X121 : subst,X122 : term] : (~ ! [X123 : term] : ((X120 @ (sub @ X123 @ X121)) | ~(var @ X123)) | ~ ! [X124 : term] : (~ ! [X125 : term] : (~(X120 @ X125) | (X120 @ (sub @ X124 @ (push @ X125 @ id)))) | (X120 @ (lam @ X124))) | (X120 @ (sub @ X122 @ X121)) | ~ ! [X126 : term,X127 : term] : ((X120 @ (ap @ X127 @ X126)) | ~(X120 @ X127) | ~(X120 @ X126))) <=> induction2lem) & pushprop_lem3v2 & hoasinduction_lem1v2_gthm & (((comp @ (d2subst @ d_id) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & (! [X128 : term] : ~ ! [X129 : (subst > term > term)] : ~ ! [X130 : subst,X131 : term] : (((sub @ X128 @ (push @ X131 @ X130))) = ((X129 @ X130 @ X131))) <=> hoasinduction_lem3v2_f) & (hoasinduction_lem3v2a_lthm <=> (! [X132 : term] : ~ ! [X133 : (subst > term > term)] : ~ ! [X134 : subst,X135 : term] : (((sub @ X132 @ (push @ X135 @ X134))) = ((X133 @ X134 @ X135))) => (! [X136 : term] : (((sub @ X136 @ id)) = X136) => ! [X137 : (subst > term > subst > $o),X138 : (term > $o),X139 : term] : (~ ! [X140 : term] : (((X138 @ X140)) = ((X137 @ id @ X140 @ id))) | ~ ! [X141 : (subst > term > term)] : (~ ! [X142 : term,X143 : subst,X144 : subst] : (((sub @ (X141 @ X144 @ X142) @ X143)) = ((X141 @ (comp @ X144 @ X143) @ (sub @ X142 @ X143)))) | (X137 @ id @ (lam @ (X141 @ sh @ one)) @ id) | ~ ! [X145 : term] : ((X137 @ id @ (X141 @ id @ X145) @ id) | ~(X137 @ id @ X145 @ id))) | (X138 @ (lam @ X139)) | ~ ! [X146 : term] : (~(X138 @ X146) | (X138 @ (sub @ X139 @ (push @ X146 @ id)))))))) & (! [X147 : (subst > term > subst > $o)] : ~ ! [X148 : (term > $o)] : ~ ! [X149 : term] : (((X147 @ id @ X149 @ id)) = ((X148 @ X149))) <=> hoasinduction_lem0_lthm) & hoasinduction_gthm & ~ulamvar1 & hoasinduction_lem2_gthm & pushprop_lem3v2_lthm & hoaslamnotvar & (((sub @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2term @ d_one))) & ((^[X150 : subst, X151 : (subst > term > term)] : ((d2term @ d_one))) = hoaslam) & (! [X152 : (subst > term > subst > $o),X153 : (term > $o),X154 : term,X155 : term] : (~ ! [X156 : term] : (((X153 @ X156)) = ((X152 @ id @ X156 @ id))) | ~ ! [X157 : term,X158 : subst,X159 : subst,X160 : subst] : ((X152 @ X160 @ X157 @ (comp @ X158 @ X159)) | ~(X152 @ (comp @ X160 @ X158) @ (sub @ X157 @ X158) @ X159)) | ~ ! [X161 : term,X162 : term] : (~(X152 @ id @ X161 @ id) | ~(X152 @ id @ X162 @ id) | (X152 @ id @ (ap @ (sub @ X161 @ id) @ X162) @ id)) | ~ ! [X163 : term,X164 : subst,X165 : subst,X166 : subst] : ((X152 @ (comp @ X164 @ X165) @ (sub @ X163 @ X165) @ X166) | ~(X152 @ X164 @ X163 @ (comp @ X165 @ X166))) | ~(X153 @ X154) | (X153 @ (ap @ X154 @ X155)) | ~(X153 @ X155)) <=> hoasinduction_lem2v2) & ! [X167 : d_term] : (d_one = X167) & (! [X168 : (term > $o),X169 : term,X170 : (term > $o),X171 : subst,X172 : term] : ((X170 @ (sub @ X169 @ (push @ X172 @ X171))) | ~(X168 @ X169)) <=> pushprop_lem1) & hoasinduction_lem1_gthm & ! [X173 : subst] : ? [X174 : d_subst] : (((d2subst @ X174)) = X173) & (hoasinduction_lem3v2a <=> ! [X175 : term,X176 : (term > $o),X177 : (subst > term > subst > $o)] : (~ ! [X178 : term] : ((X176 @ (sub @ X175 @ (push @ X178 @ id))) | ~(X176 @ X178)) | ~ ! [X179 : (subst > term > term)] : (~ ! [X180 : subst,X181 : term,X182 : subst] : (((sub @ (X179 @ X180 @ X181) @ X182)) = ((X179 @ (comp @ X180 @ X182) @ (sub @ X181 @ X182)))) | (X177 @ id @ (lam @ (X179 @ sh @ one)) @ id) | ~ ! [X183 : term] : ((X177 @ id @ (X179 @ id @ X183) @ id) | ~(X177 @ id @ X183 @ id))) | (X176 @ (lam @ X175)) | ~ ! [X184 : term] : (((X177 @ id @ X184 @ id)) = ((X176 @ X184))))) & ((! [X185 : term] : (((sub @ X185 @ id)) = X185) => ! [X186 : term] : (((sub @ X186 @ id)) = X186)) <=> termmset_lthm) & pushprop_lem2v2 & pushprop_lem2v2_lthm & (! [X187 : term,X188 : (subst > term > subst > $o)] : ((X188 @ id @ X187 @ id) | ~ ! [X189 : (subst > term > term)] : (~ ! [X190 : term] : (~(X188 @ id @ X190 @ id) | (X188 @ id @ (X189 @ id @ X190) @ id)) | ~ ! [X191 : subst,X192 : subst,X193 : term] : (((X189 @ (comp @ X192 @ X191) @ (sub @ X193 @ X191))) = ((sub @ (X189 @ X192 @ X193) @ X191))) | (X188 @ id @ (lam @ (X189 @ sh @ one)) @ id)) | ~ ! [X194 : term,X195 : term] : (~(X188 @ id @ X195 @ id) | ~(X188 @ id @ X194 @ id) | (X188 @ id @ (ap @ (sub @ X194 @ id) @ X195) @ id))) <=> hoasinduction_no_psi_cond) & (hoasinduction_lem3a_lthm <=> (! [X196 : term] : (((sub @ X196 @ id)) = X196) => (! [X197 : term,X198 : (subst > term > subst > $o)] : (~ ! [X199 : (subst > term > term)] : ((X198 @ id @ (lam @ (X199 @ sh @ one)) @ id) | ~ ! [X200 : term] : ((X198 @ id @ (X199 @ id @ X200) @ id) | ~(X198 @ id @ X200 @ id)) | ~ ! [X201 : subst,X202 : subst,X203 : term] : (((X199 @ (comp @ X202 @ X201) @ (sub @ X203 @ X201))) = ((sub @ (X199 @ X202 @ X203) @ X201)))) | ~ ! [X204 : term] : (~(X198 @ id @ X204 @ id) | (X198 @ id @ (sub @ X197 @ (push @ X204 @ id)) @ id)) | (X198 @ id @ (lam @ (sub @ X197 @ (push @ one @ sh))) @ id)) => ! [X205 : (subst > term > subst > $o),X206 : term] : (~ ! [X207 : term] : (~(X205 @ id @ X207 @ id) | (X205 @ id @ (sub @ X206 @ (push @ X207 @ id)) @ id)) | ~ ! [X208 : (subst > term > term)] : (~ ! [X209 : subst,X210 : term,X211 : subst] : (((sub @ (X208 @ X211 @ X210) @ X209)) = ((X208 @ (comp @ X211 @ X209) @ (sub @ X210 @ X209)))) | (X205 @ id @ (lam @ (X208 @ sh @ one)) @ id) | ~ ! [X212 : term] : ((X205 @ id @ (X208 @ id @ X212) @ id) | ~(X205 @ id @ X212 @ id))) | (X205 @ id @ (lam @ X206) @ id))))) & hoasinduction_lem1 & (id = ((d2subst @ d_id))) & pushprop_lem1_gthm & (sh = ((d2subst @ d_id))) & (induction2_lthm <=> (! [X213 : term] : (((sub @ X213 @ id)) = X213) => (! [X214 : subst,X215 : term,X216 : (term > $o)] : (~ ! [X217 : term] : (~ ! [X218 : term] : ((X216 @ (sub @ X217 @ (push @ X218 @ id))) | ~(X216 @ X218)) | (X216 @ (lam @ X217))) | (X216 @ (sub @ X215 @ X214)) | ~ ! [X219 : term,X220 : term] : (~(X216 @ X219) | ~(X216 @ X220) | (X216 @ (ap @ X219 @ X220))) | ~ ! [X221 : term] : (~(var @ X221) | (X216 @ (sub @ X221 @ X214)))) => ! [X222 : (term > $o),X223 : term] : (~ ! [X224 : term,X225 : term] : ((X222 @ (ap @ X225 @ X224)) | ~(X222 @ X225) | ~(X222 @ X224)) | ~ ! [X226 : term] : (~(var @ X226) | (X222 @ X226)) | ~ ! [X227 : term] : ((X222 @ (lam @ X227)) | ~ ! [X228 : term] : (~(X222 @ X228) | (X222 @ (sub @ X227 @ (push @ X228 @ id))))) | (X222 @ X223))))) & (axidl <=> ! [X229 : subst] : (((comp @ id @ X229)) = X229)) & (hoasinduction_lthm_3 <=> (! [X230 : (subst > term > subst > $o)] : ~ ! [X231 : (term > $o)] : ~ ! [X232 : term] : (((X230 @ id @ X232 @ id)) = ((X231 @ X232))) => (! [X233 : (term > $o),X234 : term] : (~ ! [X235 : term] : (~(var @ X235) | (X233 @ X235)) | ~ ! [X236 : term,X237 : term] : ((X233 @ (ap @ X237 @ X236)) | ~(X233 @ X237) | ~(X233 @ X236)) | ~ ! [X238 : term] : (~ ! [X239 : term] : ((X233 @ (sub @ X238 @ (push @ X239 @ id))) | ~(X233 @ X239)) | (X233 @ (lam @ X238))) | (X233 @ X234)) => (! [X240 : term] : (((sub @ X240 @ id)) = X240) => (! [X241 : (term > $o),X242 : term,X243 : (subst > term > subst > $o)] : (~ ! [X244 : term] : (((X243 @ id @ X244 @ id)) = ((X241 @ X244))) | ~ ! [X245 : (subst > term > term)] : (~ ! [X246 : term] : ((X243 @ id @ (X245 @ id @ X246) @ id) | ~(X243 @ id @ X246 @ id)) | (X243 @ id @ (lam @ (X245 @ sh @ one)) @ id) | ~ ! [X247 : subst,X248 : subst,X249 : term] : (((sub @ (X245 @ X248 @ X249) @ X247)) = ((X245 @ (comp @ X248 @ X247) @ (sub @ X249 @ X247))))) | (X241 @ (lam @ X242)) | ~ ! [X250 : term] : ((X241 @ (sub @ X242 @ (push @ X250 @ id))) | ~(X241 @ X250))) => ! [X251 : (subst > term > subst > $o),X252 : term] : ((X251 @ id @ X252 @ id) | ~ ! [X253 : subst,X254 : term,X255 : subst,X256 : subst] : (~(X251 @ X253 @ X254 @ (comp @ X256 @ X255)) | (X251 @ (comp @ X253 @ X256) @ (sub @ X254 @ X256) @ X255)) | ~ ! [X257 : subst,X258 : subst,X259 : term,X260 : subst] : (~(X251 @ (comp @ X260 @ X258) @ (sub @ X259 @ X258) @ X257) | (X251 @ X260 @ X259 @ (comp @ X258 @ X257))) | ~ ! [X261 : term,X262 : term] : ((X251 @ id @ (ap @ (sub @ X261 @ id) @ X262) @ id) | ~(X251 @ id @ X261 @ id) | ~(X251 @ id @ X262 @ id)) | ~ ! [X263 : term] : ((X251 @ id @ X263 @ id) | ~(var @ (sub @ X263 @ id))) | ~ ! [X264 : (subst > term > term)] : (~ ! [X265 : subst,X266 : term,X267 : subst] : (((sub @ (X264 @ X267 @ X266) @ X265)) = ((X264 @ (comp @ X267 @ X265) @ (sub @ X266 @ X265)))) | ~ ! [X268 : term] : (~(X251 @ id @ X268 @ id) | (X251 @ id @ (X264 @ id @ X268) @ id)) | (X251 @ id @ (lam @ (X264 @ sh @ one)) @ id)))))))) & ((! [X269 : subst,X270 : term] : (((sub @ one @ (push @ X270 @ X269))) = X270) => (! [X271 : subst,X272 : term] : (((comp @ sh @ (push @ X272 @ X271))) = X271) => (! [X273 : subst] : (((comp @ X273 @ id)) = X273) => (! [X274 : term,X275 : (term > $o)] : ((X275 @ X274) | ~ ! [X276 : term,X277 : term] : (~(X275 @ X277) | ~(X275 @ X276) | (X275 @ (ap @ X276 @ X277))) | ~ ! [X278 : term] : (~(var @ X278) | (X275 @ X278)) | ~ ! [X279 : term] : (~(X275 @ X279) | (X275 @ (lam @ X279)))) => ! [X280 : (term > $o),X281 : term,X282 : subst] : (~ ! [X283 : term] : ((X280 @ (sub @ X283 @ X282)) | ~(var @ X283)) | ~ ! [X284 : term] : ((X280 @ (lam @ X284)) | ~ ! [X285 : term] : ((X280 @ (sub @ X284 @ (push @ X285 @ id))) | ~(X280 @ X285))) | ~ ! [X286 : term,X287 : term] : (~(X280 @ X287) | (X280 @ (ap @ X286 @ X287)) | ~(X280 @ X286)) | (X280 @ (sub @ X281 @ X282))))))) <=> induction2lem_lthm) & ((! [X288 : term] : (((sub @ X288 @ id)) = X288) => (! [X289 : term,X290 : term,X291 : term,X292 : term] : ((X290 = X291) | (((ap @ X290 @ X292)) != ((ap @ X291 @ X289)))) => ! [X293 : term,X294 : term,X295 : term,X296 : term] : ((((ap @ (sub @ X295 @ id) @ X296)) != ((ap @ (sub @ X293 @ id) @ X294))) | (X293 = X295)))) <=> hoasapinj1_lthm) & (termmset <=> ! [X297 : term] : (((sub @ X297 @ id)) = X297)) & (pushprop_lem1_lthm <=> (! [X298 : subst,X299 : term] : (((sub @ one @ (push @ X299 @ X298))) = X299) => (! [X300 : term,X301 : subst] : (((comp @ sh @ (push @ X300 @ X301))) = X301) => ! [X302 : subst,X303 : (term > $o),X304 : term,X305 : term,X306 : (term > $o)] : (~(X306 @ X304) | (X303 @ (sub @ X304 @ (push @ X305 @ X302))))))) & pushprop_lem2v2_gthm & hoasinduction_lem1_lthm & pushprop_lem1v2_gthm & pushprop_gthm & (substmonoid_lthm <=> (! [X307 : subst] : (((comp @ id @ X307)) = X307) => (! [X308 : subst] : (((comp @ X308 @ id)) = X308) => (! [X309 : subst] : (((comp @ X309 @ id)) = X309) & ! [X310 : subst] : (((comp @ id @ X310)) = X310))))) & axmap & (! [X311 : (subst > term > term),X312 : (subst > term > term),X313 : subst,X314 : term] : ((((lam @ (X311 @ sh @ one))) != ((lam @ (X312 @ sh @ one)))) | (((X312 @ X313 @ X314)) = ((X311 @ X313 @ X314))) | ~ ! [X315 : term,X316 : subst,X317 : subst] : (((X311 @ (comp @ X317 @ X316) @ (sub @ X315 @ X316))) = ((sub @ (X311 @ X317 @ X315) @ X316))) | ~ ! [X318 : term,X319 : subst,X320 : subst] : (((sub @ (X312 @ X320 @ X318) @ X319)) = ((X312 @ (comp @ X320 @ X319) @ (sub @ X318 @ X319))))) <=> hoaslaminj) & ulamvarind & hoasinduction_lem1v2 & substmonoid_gthm & (! [X321 : (term > $o),X322 : subst,X323 : term] : ~ ! [X324 : (term > $o)] : ~ ! [X325 : term] : (((X324 @ X325)) = ((X321 @ (sub @ X325 @ (push @ X323 @ X322))))) <=> pushprop_lem0_lthm) & hoasapnotvar & (hoasinduction_lem3 <=> ! [X326 : (subst > term > subst > $o),X327 : term] : (~ ! [X328 : term,X329 : subst,X330 : subst,X331 : subst] : ((X326 @ (comp @ X331 @ X329) @ (sub @ X328 @ X329) @ X330) | ~(X326 @ X331 @ X328 @ (comp @ X329 @ X330))) | (X326 @ id @ (lam @ X327) @ id) | ~ ! [X332 : (subst > term > term)] : (~ ! [X333 : subst,X334 : subst,X335 : term] : (((sub @ (X332 @ X333 @ X335) @ X334)) = ((X332 @ (comp @ X333 @ X334) @ (sub @ X335 @ X334)))) | (X326 @ id @ (lam @ (X332 @ sh @ one)) @ id) | ~ ! [X336 : term] : (~(X326 @ id @ X336 @ id) | (X326 @ id @ (X332 @ id @ X336) @ id))) | ~ ! [X337 : term] : ((X326 @ id @ (sub @ X327 @ (push @ X337 @ id)) @ id) | ~(X326 @ id @ X337 @ id)) | ~ ! [X338 : subst,X339 : subst,X340 : subst,X341 : term] : (~(X326 @ (comp @ X339 @ X340) @ (sub @ X341 @ X340) @ X338) | (X326 @ X339 @ X341 @ (comp @ X340 @ X338))))) & (hoasinduction_lem2 <=> ! [X342 : (subst > term > subst > $o),X343 : term,X344 : term] : ((X342 @ id @ (ap @ X344 @ X343) @ id) | ~ ! [X345 : subst,X346 : subst,X347 : subst,X348 : term] : ((X342 @ (comp @ X346 @ X345) @ (sub @ X348 @ X345) @ X347) | ~(X342 @ X346 @ X348 @ (comp @ X345 @ X347))) | ~(X342 @ id @ X344 @ id) | ~ ! [X349 : subst,X350 : subst,X351 : term,X352 : subst] : (~(X342 @ (comp @ X350 @ X352) @ (sub @ X351 @ X352) @ X349) | (X342 @ X350 @ X351 @ (comp @ X352 @ X349))) | ~ ! [X353 : term,X354 : term] : (~(X342 @ id @ X353 @ id) | (X342 @ id @ (ap @ (sub @ X353 @ id) @ X354) @ id) | ~(X342 @ id @ X354 @ id)) | ~(X342 @ id @ X343 @ id))) & ! [X355 : term] : ? [X356 : d_term] : (((d2term @ X356)) = X355) & (((ap @ (d2term @ d_one) @ (d2term @ d_one))) = ((d2term @ d_one))) & axabs & (hoasinduction_lem3a <=> ! [X357 : (subst > term > subst > $o),X358 : term] : ((X357 @ id @ (lam @ X358) @ id) | ~ ! [X359 : (subst > term > term)] : (~ ! [X360 : term,X361 : subst,X362 : subst] : (((sub @ (X359 @ X362 @ X360) @ X361)) = ((X359 @ (comp @ X362 @ X361) @ (sub @ X360 @ X361)))) | ~ ! [X363 : term] : ((X357 @ id @ (X359 @ id @ X363) @ id) | ~(X357 @ id @ X363 @ id)) | (X357 @ id @ (lam @ (X359 @ sh @ one)) @ id)) | ~ ! [X364 : term] : (~(X357 @ id @ X364 @ id) | (X357 @ id @ (sub @ X358 @ (push @ X364 @ id)) @ id)))) & (apinj1 <=> ! [X365 : term,X366 : term,X367 : term,X368 : term] : ((X366 = X367) | (((ap @ X366 @ X368)) != ((ap @ X367 @ X365))))) & induction2lem_gthm & (hoasinduction_no_psi_cond_lthm <=> (! [X369 : (subst > term > subst > $o)] : ~ ! [X370 : (term > $o)] : ~ ! [X371 : term] : (((X369 @ id @ X371 @ id)) = ((X370 @ X371))) => (! [X372 : (term > $o),X373 : term] : ((X372 @ X373) | ~ ! [X374 : term] : (~(var @ X374) | (X372 @ X374)) | ~ ! [X375 : term,X376 : term] : (~(X372 @ X376) | (X372 @ (ap @ X376 @ X375)) | ~(X372 @ X375)) | ~ ! [X377 : term] : (~ ! [X378 : term] : (~(X372 @ X378) | (X372 @ (sub @ X377 @ (push @ X378 @ id)))) | (X372 @ (lam @ X377)))) => (! [X379 : term] : (((sub @ X379 @ id)) = X379) => (! [X380 : term,X381 : (term > $o),X382 : (subst > term > subst > $o)] : (~ ! [X383 : term] : (~(X381 @ X383) | (X381 @ (sub @ X380 @ (push @ X383 @ id)))) | ~ ! [X384 : (subst > term > term)] : (~ ! [X385 : subst,X386 : subst,X387 : term] : (((sub @ (X384 @ X386 @ X387) @ X385)) = ((X384 @ (comp @ X386 @ X385) @ (sub @ X387 @ X385)))) | (X382 @ id @ (lam @ (X384 @ sh @ one)) @ id) | ~ ! [X388 : term] : ((X382 @ id @ (X384 @ id @ X388) @ id) | ~(X382 @ id @ X388 @ id))) | ~ ! [X389 : term] : (((X381 @ X389)) = ((X382 @ id @ X389 @ id))) | (X381 @ (lam @ X380))) => ! [X390 : term,X391 : (subst > term > subst > $o)] : ((X391 @ id @ X390 @ id) | ~ ! [X392 : term,X393 : term] : (~(X391 @ id @ X392 @ id) | (X391 @ id @ (ap @ (sub @ X392 @ id) @ X393) @ id) | ~(X391 @ id @ X393 @ id)) | ~ ! [X394 : (subst > term > term)] : (~ ! [X395 : subst,X396 : subst,X397 : term] : (((X394 @ (comp @ X396 @ X395) @ (sub @ X397 @ X395))) = ((sub @ (X394 @ X396 @ X397) @ X395))) | ~ ! [X398 : term] : (~(X391 @ id @ X398 @ id) | (X391 @ id @ (X394 @ id @ X398) @ id)) | (X391 @ id @ (lam @ (X394 @ sh @ one)) @ id)))))))) & ((! [X399 : subst,X400 : term] : (((sub @ one @ (push @ X400 @ X399))) = X400) => ! [X401 : (term > $o),X402 : (term > $o),X403 : subst,X404 : term] : ((X401 @ one) | ~ ! [X405 : term] : (((X401 @ X405)) = ((X402 @ (sub @ X405 @ (push @ X404 @ X403))))) | ~(X402 @ X404))) <=> pushprop_lem1v2_lthm) & induction2_gthm & (pushprop_lem1v2 <=> ! [X406 : subst,X407 : (term > $o),X408 : (term > $o),X409 : term] : (~ ! [X410 : term] : (((X408 @ (sub @ X410 @ (push @ X409 @ X406)))) = ((X407 @ X410))) | ~(X408 @ X409) | (X407 @ one))) & hoasinduction_lem2v2_gthm & (apinj2 <=> ! [X411 : term,X412 : term,X413 : term,X414 : term] : ((X413 = X414) | (((ap @ X412 @ X413)) != ((ap @ X411 @ X414))))) & (! [X415 : subst,X416 : term] : (((comp @ sh @ (push @ X416 @ X415))) = X415) <=> axshiftcons) & apnotvar & (hoasinduction_lem0 <=> ! [X417 : (subst > term > subst > $o)] : ~ ! [X418 : (term > $o)] : ~ ! [X419 : term] : (((X418 @ X419)) = ((X417 @ id @ X419 @ id)))) & (! [X420 : term,X421 : term,X422 : term,X423 : term] : ((X420 = X421) | (((ap @ (sub @ X422 @ id) @ X420)) != ((ap @ (sub @ X423 @ id) @ X421)))) <=> hoasapinj2) & pushprop & (! [X424 : (subst > term > term),X425 : term,X426 : term] : ((((ap @ (sub @ X426 @ id) @ X425)) != ((lam @ (X424 @ sh @ one)))) | ~ ! [X427 : subst,X428 : subst,X429 : term] : (((X424 @ (comp @ X427 @ X428) @ (sub @ X429 @ X428))) = ((sub @ (X424 @ X427 @ X429) @ X428)))) <=> hoaslamnotap) & pushprop_lem0_gthm & hoaslaminj_gthm & (axvarcons <=> ! [X430 : subst,X431 : term] : (((sub @ one @ (push @ X431 @ X430))) = X431)) & (hoasinduction_lem3_lthm <=> (! [X432 : term] : (((sub @ X432 @ id)) = X432) => (! [X433 : (subst > term > subst > $o),X434 : term] : ((X433 @ id @ (lam @ (sub @ X434 @ (push @ one @ sh))) @ id) | ~ ! [X435 : (subst > term > term)] : (~ ! [X436 : term] : (~(X433 @ id @ X436 @ id) | (X433 @ id @ (X435 @ id @ X436) @ id)) | (X433 @ id @ (lam @ (X435 @ sh @ one)) @ id) | ~ ! [X437 : subst,X438 : term,X439 : subst] : (((X435 @ (comp @ X439 @ X437) @ (sub @ X438 @ X437))) = ((sub @ (X435 @ X439 @ X438) @ X437)))) | ~ ! [X440 : term] : (~(X433 @ id @ X440 @ id) | (X433 @ id @ (sub @ X434 @ (push @ X440 @ id)) @ id))) => ! [X441 : term,X442 : (subst > term > subst > $o)] : ((X442 @ id @ (lam @ X441) @ id) | ~ ! [X443 : term,X444 : subst,X445 : subst,X446 : subst] : ((X442 @ (comp @ X445 @ X446) @ (sub @ X443 @ X446) @ X444) | ~(X442 @ X445 @ X443 @ (comp @ X446 @ X444))) | ~ ! [X447 : term] : (~(X442 @ id @ X447 @ id) | (X442 @ id @ (sub @ X441 @ (push @ X447 @ id)) @ id)) | ~ ! [X448 : (subst > term > term)] : (~ ! [X449 : subst,X450 : term,X451 : subst] : (((X448 @ (comp @ X449 @ X451) @ (sub @ X450 @ X451))) = ((sub @ (X448 @ X449 @ X450) @ X451))) | (X442 @ id @ (lam @ (X448 @ sh @ one)) @ id) | ~ ! [X452 : term] : ((X442 @ id @ (X448 @ id @ X452) @ id) | ~(X442 @ id @ X452 @ id))) | ~ ! [X453 : subst,X454 : subst,X455 : term,X456 : subst] : (~(X442 @ (comp @ X453 @ X456) @ (sub @ X455 @ X456) @ X454) | (X442 @ X453 @ X455 @ (comp @ X456 @ X454))))))) & termmset_gthm & ! [X457 : d_term,X458 : d_term] : ((((d2term @ X458)) = ((d2term @ X457))) => (X457 = X458)) & ! [X459 : d_subst,X460 : d_subst] : ((((d2subst @ X459)) = ((d2subst @ X460))) => (X459 = X460)) & lamnotvar & (((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) = ((d2term @ d_one))) & (! [X461 : subst] : (((push @ (sub @ one @ X461) @ (comp @ sh @ X461))) = X461) <=> axscons) & (shinj <=> ! [X462 : term,X463 : term] : ((((sub @ X462 @ sh)) != ((sub @ X463 @ sh))) | (X462 = X463))) & pushprop_lthm & hoasapnotvar_lthm & hoasinduction_lem3_gthm & hoasinduction_lem3v2_gthm & ((^[X464 : (subst > term > subst > $o), X465 : (term > $o)] : (! [X466 : term] : (((X465 @ X466)) = ((X464 @ id @ X466 @ id))))) = hoasinduction_p_and_p_prime) & hoasinduction_lem2_lthm & axclos & (hoasinduction_lem3aa <=> ! [X467 : (subst > term > subst > $o),X468 : term] : (~ ! [X469 : (subst > term > term)] : ((X467 @ id @ (lam @ (X469 @ sh @ one)) @ id) | ~ ! [X470 : term] : ((X467 @ id @ (X469 @ id @ X470) @ id) | ~(X467 @ id @ X470 @ id)) | ~ ! [X471 : term,X472 : subst,X473 : subst] : (((sub @ (X469 @ X473 @ X471) @ X472)) = ((X469 @ (comp @ X473 @ X472) @ (sub @ X471 @ X472))))) | ~ ! [X474 : term] : (~(X467 @ id @ X474 @ id) | (X467 @ id @ (sub @ X468 @ (push @ X474 @ id)) @ id)) | (X467 @ id @ (lam @ (sub @ X468 @ (push @ one @ sh))) @ id))) & (hoasinduction_lthm <=> (! [X475 : term,X476 : (term > $o)] : (~ ! [X477 : term] : (~(var @ X477) | (X476 @ X477)) | (X476 @ X475) | ~ ! [X478 : term,X479 : term] : (~(X476 @ X479) | ~(X476 @ X478) | (X476 @ (ap @ X478 @ X479))) | ~ ! [X480 : term] : (~ ! [X481 : term] : (~(X476 @ X481) | (X476 @ (sub @ X480 @ (push @ X481 @ id)))) | (X476 @ (lam @ X480)))) => (! [X482 : term,X483 : term,X484 : (subst > term > subst > $o)] : ((X484 @ id @ (ap @ X482 @ X483) @ id) | ~(X484 @ id @ X482 @ id) | ~(X484 @ id @ X483 @ id) | ~ ! [X485 : term,X486 : subst,X487 : subst,X488 : subst] : ((X484 @ (comp @ X486 @ X487) @ (sub @ X485 @ X487) @ X488) | ~(X484 @ X486 @ X485 @ (comp @ X487 @ X488))) | ~ ! [X489 : subst,X490 : subst,X491 : subst,X492 : term] : (~(X484 @ (comp @ X490 @ X491) @ (sub @ X492 @ X491) @ X489) | (X484 @ X490 @ X492 @ (comp @ X491 @ X489))) | ~ ! [X493 : term,X494 : term] : (~(X484 @ id @ X493 @ id) | ~(X484 @ id @ X494 @ id) | (X484 @ id @ (ap @ (sub @ X493 @ id) @ X494) @ id))) => (! [X495 : term,X496 : (subst > term > subst > $o)] : (~ ! [X497 : subst,X498 : term,X499 : subst,X500 : subst] : ((X496 @ X499 @ X498 @ (comp @ X500 @ X497)) | ~(X496 @ (comp @ X499 @ X500) @ (sub @ X498 @ X500) @ X497)) | ~ ! [X501 : (subst > term > term)] : ((X496 @ id @ (lam @ (X501 @ sh @ one)) @ id) | ~ ! [X502 : subst,X503 : subst,X504 : term] : (((X501 @ (comp @ X502 @ X503) @ (sub @ X504 @ X503))) = ((sub @ (X501 @ X502 @ X504) @ X503))) | ~ ! [X505 : term] : ((X496 @ id @ (X501 @ id @ X505) @ id) | ~(X496 @ id @ X505 @ id))) | (X496 @ id @ (lam @ X495) @ id) | ~ ! [X506 : term,X507 : subst,X508 : subst,X509 : subst] : (~(X496 @ X509 @ X506 @ (comp @ X507 @ X508)) | (X496 @ (comp @ X509 @ X507) @ (sub @ X506 @ X507) @ X508)) | ~ ! [X510 : term] : (~(X496 @ id @ X510 @ id) | (X496 @ id @ (sub @ X495 @ (push @ X510 @ id)) @ id))) => ! [X511 : (subst > term > subst > $o),X512 : term] : (~ ! [X513 : term,X514 : term] : (~(X511 @ id @ X513 @ id) | ~(X511 @ id @ X514 @ id) | (X511 @ id @ (ap @ (sub @ X514 @ id) @ X513) @ id)) | ~ ! [X515 : term] : (~(var @ (sub @ X515 @ id)) | (X511 @ id @ X515 @ id)) | (X511 @ id @ X512 @ id) | ~ ! [X516 : (subst > term > term)] : ((X511 @ id @ (lam @ (X516 @ sh @ one)) @ id) | ~ ! [X517 : term] : (~(X511 @ id @ X517 @ id) | (X511 @ id @ (X516 @ id @ X517) @ id)) | ~ ! [X518 : term,X519 : subst,X520 : subst] : (((X516 @ (comp @ X520 @ X519) @ (sub @ X518 @ X519))) = ((sub @ (X516 @ X520 @ X518) @ X519)))) | ~ ! [X521 : subst,X522 : term,X523 : subst,X524 : subst] : (~(X511 @ (comp @ X524 @ X523) @ (sub @ X522 @ X523) @ X521) | (X511 @ X524 @ X522 @ (comp @ X523 @ X521))) | ~ ! [X525 : term,X526 : subst,X527 : subst,X528 : subst] : (~(X511 @ X527 @ X525 @ (comp @ X526 @ X528)) | (X511 @ (comp @ X527 @ X526) @ (sub @ X525 @ X526) @ X528))))))) & hoasapnotvar_gthm & (! [X529 : subst] : (((comp @ X529 @ id)) = X529) <=> axidr) & (laminj <=> ! [X530 : term,X531 : term] : ((X530 = X531) | (((lam @ X530)) != ((lam @ X531))))) & (((lam @ (d2term @ d_one))) = ((d2term @ d_one))) & (! [X532 : (term > $o),X533 : term,X534 : (subst > term > subst > $o)] : ((X532 @ (lam @ X533)) | ~ ! [X535 : subst,X536 : subst,X537 : term,X538 : subst] : ((X534 @ (comp @ X535 @ X536) @ (sub @ X537 @ X536) @ X538) | ~(X534 @ X535 @ X537 @ (comp @ X536 @ X538))) | ~ ! [X539 : subst,X540 : subst,X541 : term,X542 : subst] : (~(X534 @ (comp @ X542 @ X540) @ (sub @ X541 @ X540) @ X539) | (X534 @ X542 @ X541 @ (comp @ X540 @ X539))) | ~ ! [X543 : term] : (~(X532 @ X543) | (X532 @ (sub @ X533 @ (push @ X543 @ id)))) | ~ ! [X544 : (subst > term > term)] : (~ ! [X545 : term] : (~(X534 @ id @ X545 @ id) | (X534 @ id @ (X544 @ id @ X545) @ id)) | (X534 @ id @ (lam @ (X544 @ sh @ one)) @ id) | ~ ! [X546 : subst,X547 : term,X548 : subst] : (((sub @ (X544 @ X546 @ X547) @ X548)) = ((X544 @ (comp @ X546 @ X548) @ (sub @ X547 @ X548))))) | ~ ! [X549 : term] : (((X534 @ id @ X549 @ id)) = ((X532 @ X549)))) <=> hoasinduction_lem3v2) & hoasinduction_lem3a_gthm),
  inference(rectify,[],[f3])).
thf(f5,plain,(
  (hoaslamnotvar_lthm = $true) & (! [X0 : term,X3 : term,X1 : term,X2 : term] : ((X0 = X2) | (((ap @ (sub @ X2 @ id) @ X1)) != ((ap @ (sub @ X0 @ id) @ X3)))) <=> (hoasapinj1 = $true)) & (hoaslamnotvar_gthm = $true) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X4 : (subst > term > subst > $o),X5 : term] : (($true = ((X4 @ id @ (lam @ (sub @ X5 @ (push @ one @ sh))) @ id))) | ~ ! [X6 : (subst > term > term)] : (~ ! [X7 : term] : (~ (((X4 @ id @ X7 @ id)) = $true) | ($true = ((X4 @ id @ (X6 @ id @ X7) @ id)))) | (((X4 @ id @ (lam @ (X6 @ sh @ one)) @ id)) = $true) | ~ ! [X10 : term,X8 : subst,X9 : subst] : (((sub @ (X6 @ X9 @ X10) @ X8)) = ((X6 @ (comp @ X9 @ X8) @ (sub @ X10 @ X8))))) | ~ ! [X11 : term] : (~ (((X4 @ id @ X11 @ id)) = $true) | (((X4 @ id @ (sub @ X5 @ (push @ X11 @ id)) @ id)) = $true)))) & (pushprop_lthm_orig = $true) & ((! [X13 : term,X15 : term,X12 : term,X14 : term] : ((X12 = X14) | (((ap @ X13 @ X14)) != ((ap @ X15 @ X12)))) => ! [X19 : term,X17 : term,X18 : term,X16 : term] : ((X16 = X19) | (((ap @ (sub @ X17 @ id) @ X19)) != ((ap @ (sub @ X18 @ id) @ X16))))) <=> (hoasapinj2_lthm = $true)) & (! [X22 : (term > $o),X20 : term,X21 : subst] : ~ ! [X23 : (term > $o)] : ~ ! [X24 : term] : (((X23 @ X24)) = ((X22 @ (sub @ X24 @ (push @ X20 @ X21))))) <=> (pushprop_lem0 = $true)) & ~ (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) = $true) & ~ (lamnotap = $true) & (hoaslamnotap_lthm = $true) & ((hoasinduction_lem3aaa = $true) <=> ! [X25 : term,X26 : (subst > term > subst > $o)] : ((((X26 @ id @ (lam @ (sub @ X25 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X27 : term] : (($true = ((X26 @ id @ (sub @ X25 @ (push @ X27 @ id)) @ id))) | ~ (((X26 @ id @ X27 @ id)) = $true)) | ~ ! [X28 : term,X29 : (subst > term > term)] : ((((X26 @ id @ (lam @ (X29 @ sh @ one)) @ id)) = $true) | ~ ! [X30 : term,X31 : subst,X32 : subst] : (((sub @ (sub @ X28 @ (push @ X30 @ X31)) @ X32)) = ((sub @ (X29 @ X31 @ X30) @ X32))) | ~ ! [X33 : subst,X35 : term,X34 : subst] : (((X29 @ (comp @ X33 @ X34) @ (sub @ X35 @ X34))) = ((sub @ X28 @ (push @ (sub @ X35 @ X34) @ (comp @ X33 @ X34))))) | ~ ! [X36 : term] : (~ (((X26 @ id @ X36 @ id)) = $true) | (((X26 @ id @ (X29 @ id @ X36) @ id)) = $true))))) & ((! [X37 : term,X38 : subst] : (((sub @ one @ (push @ X37 @ X38))) = X37) => (! [X39 : subst,X40 : term] : (((comp @ sh @ (push @ X40 @ X39))) = X39) => (! [X41 : term,X42 : term] : ((X41 = X42) | (((lam @ X41)) != ((lam @ X42)))) => ! [X44 : (subst > term > term),X46 : (subst > term > term),X43 : term,X45 : subst] : ((((X44 @ X45 @ X43)) = ((X46 @ X45 @ X43))) | (((lam @ (X44 @ sh @ one))) != ((lam @ (X46 @ sh @ one)))) | ~ ! [X51 : term,X52 : subst,X50 : subst] : (((X46 @ (comp @ X52 @ X50) @ (sub @ X51 @ X50))) = ((sub @ (X46 @ X52 @ X51) @ X50))) | ~ ! [X47 : term,X49 : subst,X48 : subst] : (((X44 @ (comp @ X48 @ X49) @ (sub @ X47 @ X49))) = ((sub @ (X44 @ X48 @ X47) @ X49))))))) <=> (hoaslaminj_lthm = $true)) & ((axvarid = $true) <=> ! [X53 : term] : (((sub @ X53 @ id)) = X53)) & (axapp = $true) & ((hoasinduction_lem3b = $true) <=> ! [X54 : term] : ~ ! [X55 : (subst > term > term)] : (((X55 @ sh @ one)) != ((sub @ X54 @ (push @ one @ sh))))) & (hoasapinj2_gthm = $true) & (axvarshift = $true) & (axassoc = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoasinduction = $true) <=> ! [X56 : term,X57 : (subst > term > subst > $o)] : (~ ! [X58 : term] : ((((X57 @ id @ X58 @ id)) = $true) | ~ (((var @ (sub @ X58 @ id))) = $true)) | (((X57 @ id @ X56 @ id)) = $true) | ~ ! [X59 : term,X60 : subst,X61 : subst,X62 : subst] : ((((X57 @ (comp @ X60 @ X61) @ (sub @ X59 @ X61) @ X62)) = $true) | ~ (((X57 @ X60 @ X59 @ (comp @ X61 @ X62))) = $true)) | ~ ! [X63 : (subst > term > term)] : (~ ! [X64 : term,X66 : subst,X65 : subst] : (((X63 @ (comp @ X65 @ X66) @ (sub @ X64 @ X66))) = ((sub @ (X63 @ X65 @ X64) @ X66))) | ~ ! [X67 : term] : (~ (((X57 @ id @ X67 @ id)) = $true) | (((X57 @ id @ (X63 @ id @ X67) @ id)) = $true)) | ($true = ((X57 @ id @ (lam @ (X63 @ sh @ one)) @ id)))) | ~ ! [X68 : term,X69 : term] : (~ ($true = ((X57 @ id @ X68 @ id))) | ~ (((X57 @ id @ X69 @ id)) = $true) | (((X57 @ id @ (ap @ (sub @ X69 @ id) @ X68) @ id)) = $true)) | ~ ! [X70 : subst,X71 : term,X72 : subst,X73 : subst] : (($true = ((X57 @ X70 @ X71 @ (comp @ X73 @ X72)))) | ~ (((X57 @ (comp @ X70 @ X73) @ (sub @ X71 @ X73) @ X72)) = $true)))) & ! [X74 : d_subst] : (d_id = X74) & (ulamvarsh = $true) & ((induction = $true) <=> ! [X75 : term,X76 : (term > $o)] : (~ ! [X77 : term] : (~ (((var @ X77)) = $true) | (((X76 @ X77)) = $true)) | ~ ! [X78 : term] : (~ ($true = ((X76 @ X78))) | (((X76 @ (lam @ X78))) = $true)) | (((X76 @ X75)) = $true) | ~ ! [X79 : term,X80 : term] : (~ (((X76 @ X79)) = $true) | ~ ($true = ((X76 @ X80))) | (((X76 @ (ap @ X79 @ X80))) = $true)))) & (! [X81 : term] : ~ ! [X82 : (subst > term > term)] : ~ ! [X83 : subst,X84 : term] : (((sub @ X81 @ (push @ X84 @ X83))) = ((X82 @ X83 @ X84))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & (hoasapinj1_gthm = $true) & ~ (((var @ (d2term @ d_one))) = $true) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X85 : term] : ~ ! [X86 : (subst > term > term)] : (((sub @ X85 @ (push @ one @ sh))) != ((X86 @ sh @ one)))) & ((! [X87 : term] : (((sub @ X87 @ id)) = X87) => ! [X88 : (subst > term > subst > $o),X89 : term,X90 : (term > $o)] : (~ ! [X91 : term,X92 : subst,X93 : subst,X94 : subst] : (~ (((X88 @ X93 @ X91 @ (comp @ X92 @ X94))) = $true) | ($true = ((X88 @ (comp @ X93 @ X92) @ (sub @ X91 @ X92) @ X94)))) | (((X90 @ (lam @ X89))) = $true) | ~ ! [X95 : (subst > term > term)] : (~ ! [X96 : term] : (~ (((X88 @ id @ X96 @ id)) = $true) | ($true = ((X88 @ id @ (X95 @ id @ X96) @ id)))) | ~ ! [X98 : term,X99 : subst,X97 : subst] : (((sub @ (X95 @ X97 @ X98) @ X99)) = ((X95 @ (comp @ X97 @ X99) @ (sub @ X98 @ X99)))) | (((X88 @ id @ (lam @ (X95 @ sh @ one)) @ id)) = $true)) | ~ ! [X100 : term] : (((X90 @ X100)) = ((X88 @ id @ X100 @ id))) | ~ ! [X101 : subst,X102 : subst,X103 : subst,X104 : term] : ((((X88 @ X102 @ X104 @ (comp @ X101 @ X103))) = $true) | ~ (((X88 @ (comp @ X102 @ X101) @ (sub @ X104 @ X101) @ X103)) = $true)) | ~ ! [X105 : term] : ((((X90 @ (sub @ X89 @ (push @ X105 @ id)))) = $true) | ~ (((X90 @ X105)) = $true)))) <=> (hoasinduction_lem3v2_lthm = $true)) & (hoasinduction_lem3b_gthm = $true) & (hoaslamnotap_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & ((induction2 = $true) <=> ! [X111 : (term > $o),X112 : term] : (($true = ((X111 @ X112))) | ~ ! [X113 : term] : (~ ! [X114 : term] : ((((X111 @ (sub @ X113 @ (push @ X114 @ id)))) = $true) | ~ ($true = ((X111 @ X114)))) | ($true = ((X111 @ (lam @ X113))))) | ~ ! [X115 : term,X116 : term] : (~ ($true = ((X111 @ X116))) | ~ ($true = ((X111 @ X115))) | (((X111 @ (ap @ X115 @ X116))) = $true)) | ~ ! [X117 : term] : ((((X111 @ X117)) = $true) | ~ (((var @ X117)) = $true)))) & (one = ((d2term @ d_one))) & ((! [X118 : subst] : (((comp @ X118 @ id)) = X118) & ! [X119 : subst] : (((comp @ id @ X119)) = X119)) <=> (substmonoid = $true)) & (! [X120 : (term > $o),X121 : subst,X122 : term] : (~ ! [X123 : term] : ((((X120 @ (sub @ X123 @ X121))) = $true) | ~ (((var @ X123)) = $true)) | ~ ! [X124 : term] : (~ ! [X125 : term] : (~ (((X120 @ X125)) = $true) | (((X120 @ (sub @ X124 @ (push @ X125 @ id)))) = $true)) | ($true = ((X120 @ (lam @ X124))))) | (((X120 @ (sub @ X122 @ X121))) = $true) | ~ ! [X126 : term,X127 : term] : ((((X120 @ (ap @ X127 @ X126))) = $true) | ~ (((X120 @ X127)) = $true) | ~ ($true = ((X120 @ X126))))) <=> (induction2lem = $true)) & (pushprop_lem3v2 = $true) & (hoasinduction_lem1v2_gthm = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (! [X128 : term] : ~ ! [X129 : (subst > term > term)] : ~ ! [X130 : subst,X131 : term] : (((sub @ X128 @ (push @ X131 @ X130))) = ((X129 @ X130 @ X131))) <=> (hoasinduction_lem3v2_f = $true)) & ((hoasinduction_lem3v2a_lthm = $true) <=> (! [X132 : term] : ~ ! [X133 : (subst > term > term)] : ~ ! [X134 : subst,X135 : term] : (((sub @ X132 @ (push @ X135 @ X134))) = ((X133 @ X134 @ X135))) => (! [X136 : term] : (((sub @ X136 @ id)) = X136) => ! [X137 : (subst > term > subst > $o),X138 : (term > $o),X139 : term] : (~ ! [X140 : term] : (((X138 @ X140)) = ((X137 @ id @ X140 @ id))) | ~ ! [X141 : (subst > term > term)] : (~ ! [X142 : term,X143 : subst,X144 : subst] : (((sub @ (X141 @ X144 @ X142) @ X143)) = ((X141 @ (comp @ X144 @ X143) @ (sub @ X142 @ X143)))) | (((X137 @ id @ (lam @ (X141 @ sh @ one)) @ id)) = $true) | ~ ! [X145 : term] : (($true = ((X137 @ id @ (X141 @ id @ X145) @ id))) | ~ ($true = ((X137 @ id @ X145 @ id))))) | (((X138 @ (lam @ X139))) = $true) | ~ ! [X146 : term] : (~ (((X138 @ X146)) = $true) | (((X138 @ (sub @ X139 @ (push @ X146 @ id)))) = $true)))))) & (! [X147 : (subst > term > subst > $o)] : ~ ! [X148 : (term > $o)] : ~ ! [X149 : term] : (((X147 @ id @ X149 @ id)) = ((X148 @ X149))) <=> (hoasinduction_lem0_lthm = $true)) & (hoasinduction_gthm = $true) & ~ (ulamvar1 = $true) & (hoasinduction_lem2_gthm = $true) & (pushprop_lem3v2_lthm = $true) & (hoaslamnotvar = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (! [X152 : (subst > term > subst > $o),X153 : (term > $o),X154 : term,X155 : term] : (~ ! [X156 : term] : (((X153 @ X156)) = ((X152 @ id @ X156 @ id))) | ~ ! [X157 : term,X158 : subst,X159 : subst,X160 : subst] : (($true = ((X152 @ X160 @ X157 @ (comp @ X158 @ X159)))) | ~ (((X152 @ (comp @ X160 @ X158) @ (sub @ X157 @ X158) @ X159)) = $true)) | ~ ! [X161 : term,X162 : term] : (~ (((X152 @ id @ X161 @ id)) = $true) | ~ (((X152 @ id @ X162 @ id)) = $true) | (((X152 @ id @ (ap @ (sub @ X161 @ id) @ X162) @ id)) = $true)) | ~ ! [X163 : term,X164 : subst,X165 : subst,X166 : subst] : ((((X152 @ (comp @ X164 @ X165) @ (sub @ X163 @ X165) @ X166)) = $true) | ~ ($true = ((X152 @ X164 @ X163 @ (comp @ X165 @ X166))))) | ~ (((X153 @ X154)) = $true) | (((X153 @ (ap @ X154 @ X155))) = $true) | ~ (((X153 @ X155)) = $true)) <=> (hoasinduction_lem2v2 = $true)) & ! [X167 : d_term] : (d_one = X167) & (! [X168 : (term > $o),X169 : term,X170 : (term > $o),X171 : subst,X172 : term] : ((((X170 @ (sub @ X169 @ (push @ X172 @ X171)))) = $true) | ~ (((X168 @ X169)) = $true)) <=> (pushprop_lem1 = $true)) & (hoasinduction_lem1_gthm = $true) & ! [X173 : subst] : ? [X174 : d_subst] : (((d2subst @ X174)) = X173) & ((hoasinduction_lem3v2a = $true) <=> ! [X175 : term,X176 : (term > $o),X177 : (subst > term > subst > $o)] : (~ ! [X178 : term] : (($true = ((X176 @ (sub @ X175 @ (push @ X178 @ id))))) | ~ (((X176 @ X178)) = $true)) | ~ ! [X179 : (subst > term > term)] : (~ ! [X180 : subst,X181 : term,X182 : subst] : (((sub @ (X179 @ X180 @ X181) @ X182)) = ((X179 @ (comp @ X180 @ X182) @ (sub @ X181 @ X182)))) | (((X177 @ id @ (lam @ (X179 @ sh @ one)) @ id)) = $true) | ~ ! [X183 : term] : ((((X177 @ id @ (X179 @ id @ X183) @ id)) = $true) | ~ (((X177 @ id @ X183 @ id)) = $true))) | ($true = ((X176 @ (lam @ X175)))) | ~ ! [X184 : term] : (((X177 @ id @ X184 @ id)) = ((X176 @ X184))))) & ((! [X185 : term] : (((sub @ X185 @ id)) = X185) => ! [X186 : term] : (((sub @ X186 @ id)) = X186)) <=> (termmset_lthm = $true)) & (pushprop_lem2v2 = $true) & (pushprop_lem2v2_lthm = $true) & (! [X187 : term,X188 : (subst > term > subst > $o)] : ((((X188 @ id @ X187 @ id)) = $true) | ~ ! [X189 : (subst > term > term)] : (~ ! [X190 : term] : (~ (((X188 @ id @ X190 @ id)) = $true) | (((X188 @ id @ (X189 @ id @ X190) @ id)) = $true)) | ~ ! [X191 : subst,X192 : subst,X193 : term] : (((X189 @ (comp @ X192 @ X191) @ (sub @ X193 @ X191))) = ((sub @ (X189 @ X192 @ X193) @ X191))) | (((X188 @ id @ (lam @ (X189 @ sh @ one)) @ id)) = $true)) | ~ ! [X194 : term,X195 : term] : (~ (((X188 @ id @ X195 @ id)) = $true) | ~ (((X188 @ id @ X194 @ id)) = $true) | (((X188 @ id @ (ap @ (sub @ X194 @ id) @ X195) @ id)) = $true))) <=> (hoasinduction_no_psi_cond = $true)) & ((hoasinduction_lem3a_lthm = $true) <=> (! [X196 : term] : (((sub @ X196 @ id)) = X196) => (! [X197 : term,X198 : (subst > term > subst > $o)] : (~ ! [X199 : (subst > term > term)] : (($true = ((X198 @ id @ (lam @ (X199 @ sh @ one)) @ id))) | ~ ! [X200 : term] : (($true = ((X198 @ id @ (X199 @ id @ X200) @ id))) | ~ (((X198 @ id @ X200 @ id)) = $true)) | ~ ! [X201 : subst,X202 : subst,X203 : term] : (((X199 @ (comp @ X202 @ X201) @ (sub @ X203 @ X201))) = ((sub @ (X199 @ X202 @ X203) @ X201)))) | ~ ! [X204 : term] : (~ ($true = ((X198 @ id @ X204 @ id))) | ($true = ((X198 @ id @ (sub @ X197 @ (push @ X204 @ id)) @ id)))) | ($true = ((X198 @ id @ (lam @ (sub @ X197 @ (push @ one @ sh))) @ id)))) => ! [X205 : (subst > term > subst > $o),X206 : term] : (~ ! [X207 : term] : (~ (((X205 @ id @ X207 @ id)) = $true) | ($true = ((X205 @ id @ (sub @ X206 @ (push @ X207 @ id)) @ id)))) | ~ ! [X208 : (subst > term > term)] : (~ ! [X209 : subst,X210 : term,X211 : subst] : (((sub @ (X208 @ X211 @ X210) @ X209)) = ((X208 @ (comp @ X211 @ X209) @ (sub @ X210 @ X209)))) | ($true = ((X205 @ id @ (lam @ (X208 @ sh @ one)) @ id))) | ~ ! [X212 : term] : (($true = ((X205 @ id @ (X208 @ id @ X212) @ id))) | ~ (((X205 @ id @ X212 @ id)) = $true))) | ($true = ((X205 @ id @ (lam @ X206) @ id))))))) & (hoasinduction_lem1 = $true) & (id = ((d2subst @ d_id))) & (pushprop_lem1_gthm = $true) & (sh = ((d2subst @ d_id))) & ((induction2_lthm = $true) <=> (! [X213 : term] : (((sub @ X213 @ id)) = X213) => (! [X214 : subst,X215 : term,X216 : (term > $o)] : (~ ! [X217 : term] : (~ ! [X218 : term] : ((((X216 @ (sub @ X217 @ (push @ X218 @ id)))) = $true) | ~ (((X216 @ X218)) = $true)) | (((X216 @ (lam @ X217))) = $true)) | (((X216 @ (sub @ X215 @ X214))) = $true) | ~ ! [X219 : term,X220 : term] : (~ (((X216 @ X219)) = $true) | ~ (((X216 @ X220)) = $true) | (((X216 @ (ap @ X219 @ X220))) = $true)) | ~ ! [X221 : term] : (~ (((var @ X221)) = $true) | (((X216 @ (sub @ X221 @ X214))) = $true))) => ! [X222 : (term > $o),X223 : term] : (~ ! [X224 : term,X225 : term] : (($true = ((X222 @ (ap @ X225 @ X224)))) | ~ (((X222 @ X225)) = $true) | ~ (((X222 @ X224)) = $true)) | ~ ! [X226 : term] : (~ (((var @ X226)) = $true) | (((X222 @ X226)) = $true)) | ~ ! [X227 : term] : ((((X222 @ (lam @ X227))) = $true) | ~ ! [X228 : term] : (~ ($true = ((X222 @ X228))) | (((X222 @ (sub @ X227 @ (push @ X228 @ id)))) = $true))) | ($true = ((X222 @ X223))))))) & ((axidl = $true) <=> ! [X229 : subst] : (((comp @ id @ X229)) = X229)) & ((hoasinduction_lthm_3 = $true) <=> (! [X230 : (subst > term > subst > $o)] : ~ ! [X231 : (term > $o)] : ~ ! [X232 : term] : (((X230 @ id @ X232 @ id)) = ((X231 @ X232))) => (! [X233 : (term > $o),X234 : term] : (~ ! [X235 : term] : (~ (((var @ X235)) = $true) | ($true = ((X233 @ X235)))) | ~ ! [X236 : term,X237 : term] : ((((X233 @ (ap @ X237 @ X236))) = $true) | ~ (((X233 @ X237)) = $true) | ~ (((X233 @ X236)) = $true)) | ~ ! [X238 : term] : (~ ! [X239 : term] : ((((X233 @ (sub @ X238 @ (push @ X239 @ id)))) = $true) | ~ (((X233 @ X239)) = $true)) | (((X233 @ (lam @ X238))) = $true)) | (((X233 @ X234)) = $true)) => (! [X240 : term] : (((sub @ X240 @ id)) = X240) => (! [X241 : (term > $o),X242 : term,X243 : (subst > term > subst > $o)] : (~ ! [X244 : term] : (((X243 @ id @ X244 @ id)) = ((X241 @ X244))) | ~ ! [X245 : (subst > term > term)] : (~ ! [X246 : term] : ((((X243 @ id @ (X245 @ id @ X246) @ id)) = $true) | ~ (((X243 @ id @ X246 @ id)) = $true)) | (((X243 @ id @ (lam @ (X245 @ sh @ one)) @ id)) = $true) | ~ ! [X247 : subst,X248 : subst,X249 : term] : (((sub @ (X245 @ X248 @ X249) @ X247)) = ((X245 @ (comp @ X248 @ X247) @ (sub @ X249 @ X247))))) | (((X241 @ (lam @ X242))) = $true) | ~ ! [X250 : term] : ((((X241 @ (sub @ X242 @ (push @ X250 @ id)))) = $true) | ~ ($true = ((X241 @ X250))))) => ! [X251 : (subst > term > subst > $o),X252 : term] : ((((X251 @ id @ X252 @ id)) = $true) | ~ ! [X253 : subst,X254 : term,X255 : subst,X256 : subst] : (~ (((X251 @ X253 @ X254 @ (comp @ X256 @ X255))) = $true) | ($true = ((X251 @ (comp @ X253 @ X256) @ (sub @ X254 @ X256) @ X255)))) | ~ ! [X257 : subst,X258 : subst,X259 : term,X260 : subst] : (~ (((X251 @ (comp @ X260 @ X258) @ (sub @ X259 @ X258) @ X257)) = $true) | (((X251 @ X260 @ X259 @ (comp @ X258 @ X257))) = $true)) | ~ ! [X261 : term,X262 : term] : (($true = ((X251 @ id @ (ap @ (sub @ X261 @ id) @ X262) @ id))) | ~ ($true = ((X251 @ id @ X261 @ id))) | ~ (((X251 @ id @ X262 @ id)) = $true)) | ~ ! [X263 : term] : ((((X251 @ id @ X263 @ id)) = $true) | ~ ($true = ((var @ (sub @ X263 @ id))))) | ~ ! [X264 : (subst > term > term)] : (~ ! [X265 : subst,X266 : term,X267 : subst] : (((sub @ (X264 @ X267 @ X266) @ X265)) = ((X264 @ (comp @ X267 @ X265) @ (sub @ X266 @ X265)))) | ~ ! [X268 : term] : (~ (((X251 @ id @ X268 @ id)) = $true) | ($true = ((X251 @ id @ (X264 @ id @ X268) @ id)))) | (((X251 @ id @ (lam @ (X264 @ sh @ one)) @ id)) = $true)))))))) & ((! [X269 : subst,X270 : term] : (((sub @ one @ (push @ X270 @ X269))) = X270) => (! [X271 : subst,X272 : term] : (((comp @ sh @ (push @ X272 @ X271))) = X271) => (! [X273 : subst] : (((comp @ X273 @ id)) = X273) => (! [X274 : term,X275 : (term > $o)] : ((((X275 @ X274)) = $true) | ~ ! [X276 : term,X277 : term] : (~ (((X275 @ X277)) = $true) | ~ (((X275 @ X276)) = $true) | (((X275 @ (ap @ X276 @ X277))) = $true)) | ~ ! [X278 : term] : (~ ($true = ((var @ X278))) | (((X275 @ X278)) = $true)) | ~ ! [X279 : term] : (~ ($true = ((X275 @ X279))) | (((X275 @ (lam @ X279))) = $true))) => ! [X280 : (term > $o),X281 : term,X282 : subst] : (~ ! [X283 : term] : ((((X280 @ (sub @ X283 @ X282))) = $true) | ~ (((var @ X283)) = $true)) | ~ ! [X284 : term] : ((((X280 @ (lam @ X284))) = $true) | ~ ! [X285 : term] : (($true = ((X280 @ (sub @ X284 @ (push @ X285 @ id))))) | ~ (((X280 @ X285)) = $true))) | ~ ! [X286 : term,X287 : term] : (~ (((X280 @ X287)) = $true) | ($true = ((X280 @ (ap @ X286 @ X287)))) | ~ (((X280 @ X286)) = $true)) | (((X280 @ (sub @ X281 @ X282))) = $true)))))) <=> (induction2lem_lthm = $true)) & ((! [X288 : term] : (((sub @ X288 @ id)) = X288) => (! [X289 : term,X290 : term,X291 : term,X292 : term] : ((X290 = X291) | (((ap @ X290 @ X292)) != ((ap @ X291 @ X289)))) => ! [X293 : term,X294 : term,X295 : term,X296 : term] : ((((ap @ (sub @ X295 @ id) @ X296)) != ((ap @ (sub @ X293 @ id) @ X294))) | (X293 = X295)))) <=> (hoasapinj1_lthm = $true)) & ((termmset = $true) <=> ! [X297 : term] : (((sub @ X297 @ id)) = X297)) & ((pushprop_lem1_lthm = $true) <=> (! [X298 : subst,X299 : term] : (((sub @ one @ (push @ X299 @ X298))) = X299) => (! [X300 : term,X301 : subst] : (((comp @ sh @ (push @ X300 @ X301))) = X301) => ! [X302 : subst,X303 : (term > $o),X304 : term,X305 : term,X306 : (term > $o)] : (~ ($true = ((X306 @ X304))) | (((X303 @ (sub @ X304 @ (push @ X305 @ X302)))) = $true))))) & (pushprop_lem2v2_gthm = $true) & (hoasinduction_lem1_lthm = $true) & (pushprop_lem1v2_gthm = $true) & (pushprop_gthm = $true) & ((substmonoid_lthm = $true) <=> (! [X307 : subst] : (((comp @ id @ X307)) = X307) => (! [X308 : subst] : (((comp @ X308 @ id)) = X308) => (! [X309 : subst] : (((comp @ X309 @ id)) = X309) & ! [X310 : subst] : (((comp @ id @ X310)) = X310))))) & (axmap = $true) & (! [X311 : (subst > term > term),X312 : (subst > term > term),X313 : subst,X314 : term] : ((((lam @ (X311 @ sh @ one))) != ((lam @ (X312 @ sh @ one)))) | (((X312 @ X313 @ X314)) = ((X311 @ X313 @ X314))) | ~ ! [X315 : term,X316 : subst,X317 : subst] : (((X311 @ (comp @ X317 @ X316) @ (sub @ X315 @ X316))) = ((sub @ (X311 @ X317 @ X315) @ X316))) | ~ ! [X318 : term,X319 : subst,X320 : subst] : (((sub @ (X312 @ X320 @ X318) @ X319)) = ((X312 @ (comp @ X320 @ X319) @ (sub @ X318 @ X319))))) <=> (hoaslaminj = $true)) & (ulamvarind = $true) & (hoasinduction_lem1v2 = $true) & (substmonoid_gthm = $true) & (! [X321 : (term > $o),X322 : subst,X323 : term] : ~ ! [X324 : (term > $o)] : ~ ! [X325 : term] : (((X324 @ X325)) = ((X321 @ (sub @ X325 @ (push @ X323 @ X322))))) <=> (pushprop_lem0_lthm = $true)) & (hoasapnotvar = $true) & ((hoasinduction_lem3 = $true) <=> ! [X326 : (subst > term > subst > $o),X327 : term] : (~ ! [X328 : term,X329 : subst,X330 : subst,X331 : subst] : ((((X326 @ (comp @ X331 @ X329) @ (sub @ X328 @ X329) @ X330)) = $true) | ~ (((X326 @ X331 @ X328 @ (comp @ X329 @ X330))) = $true)) | ($true = ((X326 @ id @ (lam @ X327) @ id))) | ~ ! [X332 : (subst > term > term)] : (~ ! [X333 : subst,X334 : subst,X335 : term] : (((sub @ (X332 @ X333 @ X335) @ X334)) = ((X332 @ (comp @ X333 @ X334) @ (sub @ X335 @ X334)))) | (((X326 @ id @ (lam @ (X332 @ sh @ one)) @ id)) = $true) | ~ ! [X336 : term] : (~ (((X326 @ id @ X336 @ id)) = $true) | (((X326 @ id @ (X332 @ id @ X336) @ id)) = $true))) | ~ ! [X337 : term] : ((((X326 @ id @ (sub @ X327 @ (push @ X337 @ id)) @ id)) = $true) | ~ (((X326 @ id @ X337 @ id)) = $true)) | ~ ! [X338 : subst,X339 : subst,X340 : subst,X341 : term] : (~ (((X326 @ (comp @ X339 @ X340) @ (sub @ X341 @ X340) @ X338)) = $true) | ($true = ((X326 @ X339 @ X341 @ (comp @ X340 @ X338))))))) & ((hoasinduction_lem2 = $true) <=> ! [X342 : (subst > term > subst > $o),X343 : term,X344 : term] : (($true = ((X342 @ id @ (ap @ X344 @ X343) @ id))) | ~ ! [X345 : subst,X346 : subst,X347 : subst,X348 : term] : ((((X342 @ (comp @ X346 @ X345) @ (sub @ X348 @ X345) @ X347)) = $true) | ~ (((X342 @ X346 @ X348 @ (comp @ X345 @ X347))) = $true)) | ~ (((X342 @ id @ X344 @ id)) = $true) | ~ ! [X349 : subst,X350 : subst,X351 : term,X352 : subst] : (~ (((X342 @ (comp @ X350 @ X352) @ (sub @ X351 @ X352) @ X349)) = $true) | ($true = ((X342 @ X350 @ X351 @ (comp @ X352 @ X349))))) | ~ ! [X353 : term,X354 : term] : (~ (((X342 @ id @ X353 @ id)) = $true) | (((X342 @ id @ (ap @ (sub @ X353 @ id) @ X354) @ id)) = $true) | ~ ($true = ((X342 @ id @ X354 @ id)))) | ~ (((X342 @ id @ X343 @ id)) = $true))) & ! [X355 : term] : ? [X356 : d_term] : (((d2term @ X356)) = X355) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (axabs = $true) & ((hoasinduction_lem3a = $true) <=> ! [X357 : (subst > term > subst > $o),X358 : term] : ((((X357 @ id @ (lam @ X358) @ id)) = $true) | ~ ! [X359 : (subst > term > term)] : (~ ! [X360 : term,X361 : subst,X362 : subst] : (((sub @ (X359 @ X362 @ X360) @ X361)) = ((X359 @ (comp @ X362 @ X361) @ (sub @ X360 @ X361)))) | ~ ! [X363 : term] : (($true = ((X357 @ id @ (X359 @ id @ X363) @ id))) | ~ ($true = ((X357 @ id @ X363 @ id)))) | (((X357 @ id @ (lam @ (X359 @ sh @ one)) @ id)) = $true)) | ~ ! [X364 : term] : (~ (((X357 @ id @ X364 @ id)) = $true) | ($true = ((X357 @ id @ (sub @ X358 @ (push @ X364 @ id)) @ id)))))) & ((apinj1 = $true) <=> ! [X365 : term,X366 : term,X367 : term,X368 : term] : ((X366 = X367) | (((ap @ X366 @ X368)) != ((ap @ X367 @ X365))))) & (induction2lem_gthm = $true) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (! [X369 : (subst > term > subst > $o)] : ~ ! [X370 : (term > $o)] : ~ ! [X371 : term] : (((X369 @ id @ X371 @ id)) = ((X370 @ X371))) => (! [X372 : (term > $o),X373 : term] : ((((X372 @ X373)) = $true) | ~ ! [X374 : term] : (~ (((var @ X374)) = $true) | (((X372 @ X374)) = $true)) | ~ ! [X375 : term,X376 : term] : (~ (((X372 @ X376)) = $true) | (((X372 @ (ap @ X376 @ X375))) = $true) | ~ ($true = ((X372 @ X375)))) | ~ ! [X377 : term] : (~ ! [X378 : term] : (~ (((X372 @ X378)) = $true) | (((X372 @ (sub @ X377 @ (push @ X378 @ id)))) = $true)) | ($true = ((X372 @ (lam @ X377)))))) => (! [X379 : term] : (((sub @ X379 @ id)) = X379) => (! [X380 : term,X381 : (term > $o),X382 : (subst > term > subst > $o)] : (~ ! [X383 : term] : (~ (((X381 @ X383)) = $true) | ($true = ((X381 @ (sub @ X380 @ (push @ X383 @ id)))))) | ~ ! [X384 : (subst > term > term)] : (~ ! [X385 : subst,X386 : subst,X387 : term] : (((sub @ (X384 @ X386 @ X387) @ X385)) = ((X384 @ (comp @ X386 @ X385) @ (sub @ X387 @ X385)))) | (((X382 @ id @ (lam @ (X384 @ sh @ one)) @ id)) = $true) | ~ ! [X388 : term] : ((((X382 @ id @ (X384 @ id @ X388) @ id)) = $true) | ~ (((X382 @ id @ X388 @ id)) = $true))) | ~ ! [X389 : term] : (((X381 @ X389)) = ((X382 @ id @ X389 @ id))) | (((X381 @ (lam @ X380))) = $true)) => ! [X390 : term,X391 : (subst > term > subst > $o)] : (($true = ((X391 @ id @ X390 @ id))) | ~ ! [X392 : term,X393 : term] : (~ ($true = ((X391 @ id @ X392 @ id))) | ($true = ((X391 @ id @ (ap @ (sub @ X392 @ id) @ X393) @ id))) | ~ (((X391 @ id @ X393 @ id)) = $true)) | ~ ! [X394 : (subst > term > term)] : (~ ! [X395 : subst,X396 : subst,X397 : term] : (((X394 @ (comp @ X396 @ X395) @ (sub @ X397 @ X395))) = ((sub @ (X394 @ X396 @ X397) @ X395))) | ~ ! [X398 : term] : (~ (((X391 @ id @ X398 @ id)) = $true) | (((X391 @ id @ (X394 @ id @ X398) @ id)) = $true)) | ($true = ((X391 @ id @ (lam @ (X394 @ sh @ one)) @ id)))))))))) & ((! [X399 : subst,X400 : term] : (((sub @ one @ (push @ X400 @ X399))) = X400) => ! [X401 : (term > $o),X402 : (term > $o),X403 : subst,X404 : term] : ((((X401 @ one)) = $true) | ~ ! [X405 : term] : (((X401 @ X405)) = ((X402 @ (sub @ X405 @ (push @ X404 @ X403))))) | ~ (((X402 @ X404)) = $true))) <=> (pushprop_lem1v2_lthm = $true)) & (induction2_gthm = $true) & ((pushprop_lem1v2 = $true) <=> ! [X406 : subst,X407 : (term > $o),X408 : (term > $o),X409 : term] : (~ ! [X410 : term] : (((X408 @ (sub @ X410 @ (push @ X409 @ X406)))) = ((X407 @ X410))) | ~ ($true = ((X408 @ X409))) | ($true = ((X407 @ one))))) & (hoasinduction_lem2v2_gthm = $true) & ((apinj2 = $true) <=> ! [X411 : term,X412 : term,X413 : term,X414 : term] : ((X413 = X414) | (((ap @ X412 @ X413)) != ((ap @ X411 @ X414))))) & (! [X415 : subst,X416 : term] : (((comp @ sh @ (push @ X416 @ X415))) = X415) <=> (axshiftcons = $true)) & (apnotvar = $true) & ((hoasinduction_lem0 = $true) <=> ! [X417 : (subst > term > subst > $o)] : ~ ! [X418 : (term > $o)] : ~ ! [X419 : term] : (((X418 @ X419)) = ((X417 @ id @ X419 @ id)))) & (! [X420 : term,X421 : term,X422 : term,X423 : term] : ((X420 = X421) | (((ap @ (sub @ X422 @ id) @ X420)) != ((ap @ (sub @ X423 @ id) @ X421)))) <=> (hoasapinj2 = $true)) & (pushprop = $true) & (! [X424 : (subst > term > term),X425 : term,X426 : term] : ((((ap @ (sub @ X426 @ id) @ X425)) != ((lam @ (X424 @ sh @ one)))) | ~ ! [X427 : subst,X428 : subst,X429 : term] : (((X424 @ (comp @ X427 @ X428) @ (sub @ X429 @ X428))) = ((sub @ (X424 @ X427 @ X429) @ X428)))) <=> (hoaslamnotap = $true)) & (pushprop_lem0_gthm = $true) & (hoaslaminj_gthm = $true) & ((axvarcons = $true) <=> ! [X430 : subst,X431 : term] : (((sub @ one @ (push @ X431 @ X430))) = X431)) & ((hoasinduction_lem3_lthm = $true) <=> (! [X432 : term] : (((sub @ X432 @ id)) = X432) => (! [X433 : (subst > term > subst > $o),X434 : term] : (($true = ((X433 @ id @ (lam @ (sub @ X434 @ (push @ one @ sh))) @ id))) | ~ ! [X435 : (subst > term > term)] : (~ ! [X436 : term] : (~ ($true = ((X433 @ id @ X436 @ id))) | ($true = ((X433 @ id @ (X435 @ id @ X436) @ id)))) | (((X433 @ id @ (lam @ (X435 @ sh @ one)) @ id)) = $true) | ~ ! [X437 : subst,X438 : term,X439 : subst] : (((X435 @ (comp @ X439 @ X437) @ (sub @ X438 @ X437))) = ((sub @ (X435 @ X439 @ X438) @ X437)))) | ~ ! [X440 : term] : (~ ($true = ((X433 @ id @ X440 @ id))) | (((X433 @ id @ (sub @ X434 @ (push @ X440 @ id)) @ id)) = $true))) => ! [X441 : term,X442 : (subst > term > subst > $o)] : ((((X442 @ id @ (lam @ X441) @ id)) = $true) | ~ ! [X443 : term,X444 : subst,X445 : subst,X446 : subst] : ((((X442 @ (comp @ X445 @ X446) @ (sub @ X443 @ X446) @ X444)) = $true) | ~ (((X442 @ X445 @ X443 @ (comp @ X446 @ X444))) = $true)) | ~ ! [X447 : term] : (~ (((X442 @ id @ X447 @ id)) = $true) | (((X442 @ id @ (sub @ X441 @ (push @ X447 @ id)) @ id)) = $true)) | ~ ! [X448 : (subst > term > term)] : (~ ! [X449 : subst,X450 : term,X451 : subst] : (((X448 @ (comp @ X449 @ X451) @ (sub @ X450 @ X451))) = ((sub @ (X448 @ X449 @ X450) @ X451))) | (((X442 @ id @ (lam @ (X448 @ sh @ one)) @ id)) = $true) | ~ ! [X452 : term] : ((((X442 @ id @ (X448 @ id @ X452) @ id)) = $true) | ~ ($true = ((X442 @ id @ X452 @ id))))) | ~ ! [X453 : subst,X454 : subst,X455 : term,X456 : subst] : (~ (((X442 @ (comp @ X453 @ X456) @ (sub @ X455 @ X456) @ X454)) = $true) | (((X442 @ X453 @ X455 @ (comp @ X456 @ X454))) = $true)))))) & (termmset_gthm = $true) & ! [X457 : d_term,X458 : d_term] : ((((d2term @ X458)) = ((d2term @ X457))) => (X457 = X458)) & ! [X459 : d_subst,X460 : d_subst] : ((((d2subst @ X459)) = ((d2subst @ X460))) => (X459 = X460)) & (lamnotvar = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (! [X461 : subst] : (((push @ (sub @ one @ X461) @ (comp @ sh @ X461))) = X461) <=> (axscons = $true)) & ((shinj = $true) <=> ! [X462 : term,X463 : term] : ((((sub @ X462 @ sh)) != ((sub @ X463 @ sh))) | (X462 = X463))) & (pushprop_lthm = $true) & (hoasapnotvar_lthm = $true) & (hoasinduction_lem3_gthm = $true) & (hoasinduction_lem3v2_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (hoasinduction_lem2_lthm = $true) & (axclos = $true) & ((hoasinduction_lem3aa = $true) <=> ! [X467 : (subst > term > subst > $o),X468 : term] : (~ ! [X469 : (subst > term > term)] : ((((X467 @ id @ (lam @ (X469 @ sh @ one)) @ id)) = $true) | ~ ! [X470 : term] : ((((X467 @ id @ (X469 @ id @ X470) @ id)) = $true) | ~ ($true = ((X467 @ id @ X470 @ id)))) | ~ ! [X471 : term,X472 : subst,X473 : subst] : (((sub @ (X469 @ X473 @ X471) @ X472)) = ((X469 @ (comp @ X473 @ X472) @ (sub @ X471 @ X472))))) | ~ ! [X474 : term] : (~ (((X467 @ id @ X474 @ id)) = $true) | (((X467 @ id @ (sub @ X468 @ (push @ X474 @ id)) @ id)) = $true)) | (((X467 @ id @ (lam @ (sub @ X468 @ (push @ one @ sh))) @ id)) = $true))) & ((hoasinduction_lthm = $true) <=> (! [X475 : term,X476 : (term > $o)] : (~ ! [X477 : term] : (~ (((var @ X477)) = $true) | ($true = ((X476 @ X477)))) | (((X476 @ X475)) = $true) | ~ ! [X478 : term,X479 : term] : (~ (((X476 @ X479)) = $true) | ~ ($true = ((X476 @ X478))) | ($true = ((X476 @ (ap @ X478 @ X479))))) | ~ ! [X480 : term] : (~ ! [X481 : term] : (~ ($true = ((X476 @ X481))) | ($true = ((X476 @ (sub @ X480 @ (push @ X481 @ id)))))) | (((X476 @ (lam @ X480))) = $true))) => (! [X482 : term,X483 : term,X484 : (subst > term > subst > $o)] : ((((X484 @ id @ (ap @ X482 @ X483) @ id)) = $true) | ~ (((X484 @ id @ X482 @ id)) = $true) | ~ (((X484 @ id @ X483 @ id)) = $true) | ~ ! [X485 : term,X486 : subst,X487 : subst,X488 : subst] : ((((X484 @ (comp @ X486 @ X487) @ (sub @ X485 @ X487) @ X488)) = $true) | ~ (((X484 @ X486 @ X485 @ (comp @ X487 @ X488))) = $true)) | ~ ! [X489 : subst,X490 : subst,X491 : subst,X492 : term] : (~ ($true = ((X484 @ (comp @ X490 @ X491) @ (sub @ X492 @ X491) @ X489))) | (((X484 @ X490 @ X492 @ (comp @ X491 @ X489))) = $true)) | ~ ! [X493 : term,X494 : term] : (~ ($true = ((X484 @ id @ X493 @ id))) | ~ ($true = ((X484 @ id @ X494 @ id))) | (((X484 @ id @ (ap @ (sub @ X493 @ id) @ X494) @ id)) = $true))) => (! [X495 : term,X496 : (subst > term > subst > $o)] : (~ ! [X497 : subst,X498 : term,X499 : subst,X500 : subst] : (($true = ((X496 @ X499 @ X498 @ (comp @ X500 @ X497)))) | ~ (((X496 @ (comp @ X499 @ X500) @ (sub @ X498 @ X500) @ X497)) = $true)) | ~ ! [X501 : (subst > term > term)] : ((((X496 @ id @ (lam @ (X501 @ sh @ one)) @ id)) = $true) | ~ ! [X502 : subst,X503 : subst,X504 : term] : (((X501 @ (comp @ X502 @ X503) @ (sub @ X504 @ X503))) = ((sub @ (X501 @ X502 @ X504) @ X503))) | ~ ! [X505 : term] : ((((X496 @ id @ (X501 @ id @ X505) @ id)) = $true) | ~ (((X496 @ id @ X505 @ id)) = $true))) | (((X496 @ id @ (lam @ X495) @ id)) = $true) | ~ ! [X506 : term,X507 : subst,X508 : subst,X509 : subst] : (~ (((X496 @ X509 @ X506 @ (comp @ X507 @ X508))) = $true) | (((X496 @ (comp @ X509 @ X507) @ (sub @ X506 @ X507) @ X508)) = $true)) | ~ ! [X510 : term] : (~ (((X496 @ id @ X510 @ id)) = $true) | ($true = ((X496 @ id @ (sub @ X495 @ (push @ X510 @ id)) @ id))))) => ! [X511 : (subst > term > subst > $o),X512 : term] : (~ ! [X513 : term,X514 : term] : (~ ($true = ((X511 @ id @ X513 @ id))) | ~ (((X511 @ id @ X514 @ id)) = $true) | (((X511 @ id @ (ap @ (sub @ X514 @ id) @ X513) @ id)) = $true)) | ~ ! [X515 : term] : (~ ($true = ((var @ (sub @ X515 @ id)))) | (((X511 @ id @ X515 @ id)) = $true)) | (((X511 @ id @ X512 @ id)) = $true) | ~ ! [X516 : (subst > term > term)] : ((((X511 @ id @ (lam @ (X516 @ sh @ one)) @ id)) = $true) | ~ ! [X517 : term] : (~ (((X511 @ id @ X517 @ id)) = $true) | ($true = ((X511 @ id @ (X516 @ id @ X517) @ id)))) | ~ ! [X518 : term,X519 : subst,X520 : subst] : (((X516 @ (comp @ X520 @ X519) @ (sub @ X518 @ X519))) = ((sub @ (X516 @ X520 @ X518) @ X519)))) | ~ ! [X521 : subst,X522 : term,X523 : subst,X524 : subst] : (~ ($true = ((X511 @ (comp @ X524 @ X523) @ (sub @ X522 @ X523) @ X521))) | (((X511 @ X524 @ X522 @ (comp @ X523 @ X521))) = $true)) | ~ ! [X525 : term,X526 : subst,X527 : subst,X528 : subst] : (~ ($true = ((X511 @ X527 @ X525 @ (comp @ X526 @ X528)))) | (((X511 @ (comp @ X527 @ X526) @ (sub @ X525 @ X526) @ X528)) = $true))))))) & (hoasapnotvar_gthm = $true) & (! [X529 : subst] : (((comp @ X529 @ id)) = X529) <=> (axidr = $true)) & ((laminj = $true) <=> ! [X530 : term,X531 : term] : ((X530 = X531) | (((lam @ X530)) != ((lam @ X531))))) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (! [X532 : (term > $o),X533 : term,X534 : (subst > term > subst > $o)] : (($true = ((X532 @ (lam @ X533)))) | ~ ! [X535 : subst,X536 : subst,X537 : term,X538 : subst] : ((((X534 @ (comp @ X535 @ X536) @ (sub @ X537 @ X536) @ X538)) = $true) | ~ ($true = ((X534 @ X535 @ X537 @ (comp @ X536 @ X538))))) | ~ ! [X539 : subst,X540 : subst,X541 : term,X542 : subst] : (~ (((X534 @ (comp @ X542 @ X540) @ (sub @ X541 @ X540) @ X539)) = $true) | (((X534 @ X542 @ X541 @ (comp @ X540 @ X539))) = $true)) | ~ ! [X543 : term] : (~ (((X532 @ X543)) = $true) | (((X532 @ (sub @ X533 @ (push @ X543 @ id)))) = $true)) | ~ ! [X544 : (subst > term > term)] : (~ ! [X545 : term] : (~ (((X534 @ id @ X545 @ id)) = $true) | (((X534 @ id @ (X544 @ id @ X545) @ id)) = $true)) | (((X534 @ id @ (lam @ (X544 @ sh @ one)) @ id)) = $true) | ~ ! [X546 : subst,X547 : term,X548 : subst] : (((sub @ (X544 @ X546 @ X547) @ X548)) = ((X544 @ (comp @ X546 @ X548) @ (sub @ X547 @ X548))))) | ~ ! [X549 : term] : (((X534 @ id @ X549 @ id)) = ((X532 @ X549)))) <=> (hoasinduction_lem3v2 = $true)) & (hoasinduction_lem3a_gthm = $true)),
  inference(fool_elimination,[],[f4])).
thf(f6,plain,(
  ~ (! [X0 : (subst > term > subst > $o)] : (! [X1 : (subst > term > term)] : (! [X2 : subst,X3 : subst,X4 : term] : (((X1 @ (comp @ X3 @ X2) @ (sub @ X4 @ X2))) = ((sub @ (X1 @ X3 @ X4) @ X2))) => (! [X5 : term] : ((X0 @ id @ X5 @ id) => (X0 @ id @ (X1 @ id @ X5) @ id)) => (X0 @ id @ (hoaslam @ id @ (^[X6 : subst, X7 : term] : ((X1 @ X6 @ X7)))) @ id))) => ! [X8 : term] : (! [X9 : term] : ((X0 @ id @ X9 @ id) => (X0 @ id @ (sub @ X8 @ (push @ X9 @ id)) @ id)) => (X0 @ id @ (lam @ (sub @ X8 @ (push @ one @ sh))) @ id))) = hoasinduction_lem3aa)),
  inference(rectify,[],[f2])).
thf(f7,plain,(
  ~ (hoasinduction_lem3aa = ((!! @ (subst > term > subst > $o) @ (^[Y0 : subst > term > subst > $o]: ((!! @ (subst > term > term) @ (^[Y1 : subst > term > term]: ((!! @ term @ (^[Y2 : term]: (!! @ subst @ (^[Y3 : subst]: (!! @ subst @ (^[Y4 : subst]: ((Y1 @ (comp @ Y3 @ Y4) @ (sub @ Y2 @ Y4)) = (sub @ (Y1 @ Y3 @ Y2) @ Y4)))))))) => ((!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) => (Y0 @ id @ (Y1 @ id @ Y2) @ id)))) => (Y0 @ id @ (hoaslam @ id @ (^[Y2 : subst]: ((^[Y3 : term]: (Y1 @ Y2 @ Y3))))) @ id))))) => (!! @ term @ (^[Y1 : term]: ((!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) => (Y0 @ id @ (sub @ Y1 @ (push @ Y2 @ id)) @ id)))) => (Y0 @ id @ (lam @ (sub @ Y1 @ (push @ one @ sh))) @ id)))))))))),
  inference(fool_elimination,[],[f6])).
thf(f8,plain,(
  (hoaslamnotvar_lthm = $true) & (! [X0 : term,X3 : term,X1 : term,X2 : term] : ((X0 = X2) | (((ap @ (sub @ X2 @ id) @ X1)) != ((ap @ (sub @ X0 @ id) @ X3)))) <=> (hoasapinj1 = $true)) & (hoaslamnotvar_gthm = $true) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X4 : (subst > term > subst > $o),X5 : term] : (($true = ((X4 @ id @ (lam @ (sub @ X5 @ (push @ one @ sh))) @ id))) | ~ ! [X6 : (subst > term > term)] : (~ ! [X7 : term] : (~ (((X4 @ id @ X7 @ id)) = $true) | ($true = ((X4 @ id @ (X6 @ id @ X7) @ id)))) | (((X4 @ id @ (lam @ (X6 @ sh @ one)) @ id)) = $true) | ~ ! [X10 : term,X8 : subst,X9 : subst] : (((sub @ (X6 @ X9 @ X10) @ X8)) = ((X6 @ (comp @ X9 @ X8) @ (sub @ X10 @ X8))))) | ~ ! [X11 : term] : (~ (((X4 @ id @ X11 @ id)) = $true) | (((X4 @ id @ (sub @ X5 @ (push @ X11 @ id)) @ id)) = $true)))) & (pushprop_lthm_orig = $true) & ((! [X13 : term,X15 : term,X12 : term,X14 : term] : ((X12 = X14) | (((ap @ X13 @ X14)) != ((ap @ X15 @ X12)))) => ! [X19 : term,X17 : term,X18 : term,X16 : term] : ((X16 = X19) | (((ap @ (sub @ X17 @ id) @ X19)) != ((ap @ (sub @ X18 @ id) @ X16))))) <=> (hoasapinj2_lthm = $true)) & (! [X22 : (term > $o),X20 : term,X21 : subst] : ~ ! [X23 : (term > $o)] : ~ ! [X24 : term] : (((X23 @ X24)) = ((X22 @ (sub @ X24 @ (push @ X20 @ X21))))) <=> (pushprop_lem0 = $true)) & ~ (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) = $true) & ~ (lamnotap = $true) & (hoaslamnotap_lthm = $true) & ((hoasinduction_lem3aaa = $true) <=> ! [X25 : term,X26 : (subst > term > subst > $o)] : ((((X26 @ id @ (lam @ (sub @ X25 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X27 : term] : (($true = ((X26 @ id @ (sub @ X25 @ (push @ X27 @ id)) @ id))) | ~ (((X26 @ id @ X27 @ id)) = $true)) | ~ ! [X28 : term,X29 : (subst > term > term)] : ((((X26 @ id @ (lam @ (X29 @ sh @ one)) @ id)) = $true) | ~ ! [X30 : term,X31 : subst,X32 : subst] : (((sub @ (sub @ X28 @ (push @ X30 @ X31)) @ X32)) = ((sub @ (X29 @ X31 @ X30) @ X32))) | ~ ! [X33 : subst,X35 : term,X34 : subst] : (((X29 @ (comp @ X33 @ X34) @ (sub @ X35 @ X34))) = ((sub @ X28 @ (push @ (sub @ X35 @ X34) @ (comp @ X33 @ X34))))) | ~ ! [X36 : term] : (~ (((X26 @ id @ X36 @ id)) = $true) | (((X26 @ id @ (X29 @ id @ X36) @ id)) = $true))))) & ((! [X37 : term,X38 : subst] : (((sub @ one @ (push @ X37 @ X38))) = X37) => (! [X39 : subst,X40 : term] : (((comp @ sh @ (push @ X40 @ X39))) = X39) => (! [X41 : term,X42 : term] : ((X41 = X42) | (((lam @ X41)) != ((lam @ X42)))) => ! [X44 : (subst > term > term),X46 : (subst > term > term),X43 : term,X45 : subst] : ((((X44 @ X45 @ X43)) = ((X46 @ X45 @ X43))) | (((lam @ (X44 @ sh @ one))) != ((lam @ (X46 @ sh @ one)))) | ~ ! [X51 : term,X52 : subst,X50 : subst] : (((X46 @ (comp @ X52 @ X50) @ (sub @ X51 @ X50))) = ((sub @ (X46 @ X52 @ X51) @ X50))) | ~ ! [X47 : term,X49 : subst,X48 : subst] : (((X44 @ (comp @ X48 @ X49) @ (sub @ X47 @ X49))) = ((sub @ (X44 @ X48 @ X47) @ X49))))))) <=> (hoaslaminj_lthm = $true)) & ((axvarid = $true) <=> ! [X53 : term] : (((sub @ X53 @ id)) = X53)) & (axapp = $true) & ((hoasinduction_lem3b = $true) <=> ! [X54 : term] : ~ ! [X55 : (subst > term > term)] : (((X55 @ sh @ one)) != ((sub @ X54 @ (push @ one @ sh))))) & (hoasapinj2_gthm = $true) & (axvarshift = $true) & (axassoc = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoasinduction = $true) <=> ! [X56 : term,X57 : (subst > term > subst > $o)] : (~ ! [X58 : term] : ((((X57 @ id @ X58 @ id)) = $true) | ~ (((var @ (sub @ X58 @ id))) = $true)) | (((X57 @ id @ X56 @ id)) = $true) | ~ ! [X59 : term,X60 : subst,X61 : subst,X62 : subst] : ((((X57 @ (comp @ X60 @ X61) @ (sub @ X59 @ X61) @ X62)) = $true) | ~ (((X57 @ X60 @ X59 @ (comp @ X61 @ X62))) = $true)) | ~ ! [X63 : (subst > term > term)] : (~ ! [X64 : term,X66 : subst,X65 : subst] : (((X63 @ (comp @ X65 @ X66) @ (sub @ X64 @ X66))) = ((sub @ (X63 @ X65 @ X64) @ X66))) | ~ ! [X67 : term] : (~ (((X57 @ id @ X67 @ id)) = $true) | (((X57 @ id @ (X63 @ id @ X67) @ id)) = $true)) | ($true = ((X57 @ id @ (lam @ (X63 @ sh @ one)) @ id)))) | ~ ! [X68 : term,X69 : term] : (~ ($true = ((X57 @ id @ X68 @ id))) | ~ (((X57 @ id @ X69 @ id)) = $true) | (((X57 @ id @ (ap @ (sub @ X69 @ id) @ X68) @ id)) = $true)) | ~ ! [X70 : subst,X71 : term,X72 : subst,X73 : subst] : (($true = ((X57 @ X70 @ X71 @ (comp @ X73 @ X72)))) | ~ (((X57 @ (comp @ X70 @ X73) @ (sub @ X71 @ X73) @ X72)) = $true)))) & ! [X74 : d_subst] : (d_id = X74) & (ulamvarsh = $true) & ((induction = $true) <=> ! [X75 : term,X76 : (term > $o)] : (~ ! [X77 : term] : (~ (((var @ X77)) = $true) | (((X76 @ X77)) = $true)) | ~ ! [X78 : term] : (~ ($true = ((X76 @ X78))) | (((X76 @ (lam @ X78))) = $true)) | (((X76 @ X75)) = $true) | ~ ! [X79 : term,X80 : term] : (~ (((X76 @ X79)) = $true) | ~ ($true = ((X76 @ X80))) | (((X76 @ (ap @ X79 @ X80))) = $true)))) & (! [X81 : term] : ~ ! [X82 : (subst > term > term)] : ~ ! [X83 : subst,X84 : term] : (((sub @ X81 @ (push @ X84 @ X83))) = ((X82 @ X83 @ X84))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & (hoasapinj1_gthm = $true) & ~ (((var @ (d2term @ d_one))) = $true) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X85 : term] : ~ ! [X86 : (subst > term > term)] : (((sub @ X85 @ (push @ one @ sh))) != ((X86 @ sh @ one)))) & ((! [X87 : term] : (((sub @ X87 @ id)) = X87) => ! [X88 : (subst > term > subst > $o),X89 : term,X90 : (term > $o)] : (~ ! [X91 : term,X92 : subst,X93 : subst,X94 : subst] : (~ (((X88 @ X93 @ X91 @ (comp @ X92 @ X94))) = $true) | ($true = ((X88 @ (comp @ X93 @ X92) @ (sub @ X91 @ X92) @ X94)))) | (((X90 @ (lam @ X89))) = $true) | ~ ! [X95 : (subst > term > term)] : (~ ! [X96 : term] : (~ (((X88 @ id @ X96 @ id)) = $true) | ($true = ((X88 @ id @ (X95 @ id @ X96) @ id)))) | ~ ! [X98 : term,X99 : subst,X97 : subst] : (((sub @ (X95 @ X97 @ X98) @ X99)) = ((X95 @ (comp @ X97 @ X99) @ (sub @ X98 @ X99)))) | (((X88 @ id @ (lam @ (X95 @ sh @ one)) @ id)) = $true)) | ~ ! [X100 : term] : (((X90 @ X100)) = ((X88 @ id @ X100 @ id))) | ~ ! [X101 : subst,X102 : subst,X103 : subst,X104 : term] : ((((X88 @ X102 @ X104 @ (comp @ X101 @ X103))) = $true) | ~ (((X88 @ (comp @ X102 @ X101) @ (sub @ X104 @ X101) @ X103)) = $true)) | ~ ! [X105 : term] : ((((X90 @ (sub @ X89 @ (push @ X105 @ id)))) = $true) | ~ (((X90 @ X105)) = $true)))) <=> (hoasinduction_lem3v2_lthm = $true)) & (hoasinduction_lem3b_gthm = $true) & (hoaslamnotap_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & ((induction2 = $true) <=> ! [X106 : (term > $o),X107 : term] : (($true = ((X106 @ X107))) | ~ ! [X108 : term] : (~ ! [X109 : term] : ((((X106 @ (sub @ X108 @ (push @ X109 @ id)))) = $true) | ~ (((X106 @ X109)) = $true)) | ($true = ((X106 @ (lam @ X108))))) | ~ ! [X110 : term,X111 : term] : (~ (((X106 @ X111)) = $true) | ~ (((X106 @ X110)) = $true) | (((X106 @ (ap @ X110 @ X111))) = $true)) | ~ ! [X112 : term] : ((((X106 @ X112)) = $true) | ~ (((var @ X112)) = $true)))) & (one = ((d2term @ d_one))) & ((! [X113 : subst] : (((comp @ X113 @ id)) = X113) & ! [X114 : subst] : (((comp @ id @ X114)) = X114)) <=> (substmonoid = $true)) & (! [X115 : (term > $o),X116 : subst,X117 : term] : (~ ! [X118 : term] : ((((X115 @ (sub @ X118 @ X116))) = $true) | ~ ($true = ((var @ X118)))) | ~ ! [X119 : term] : (~ ! [X120 : term] : (~ ($true = ((X115 @ X120))) | (((X115 @ (sub @ X119 @ (push @ X120 @ id)))) = $true)) | ($true = ((X115 @ (lam @ X119))))) | (((X115 @ (sub @ X117 @ X116))) = $true) | ~ ! [X121 : term,X122 : term] : ((((X115 @ (ap @ X122 @ X121))) = $true) | ~ (((X115 @ X122)) = $true) | ~ (((X115 @ X121)) = $true))) <=> (induction2lem = $true)) & (pushprop_lem3v2 = $true) & (hoasinduction_lem1v2_gthm = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (! [X123 : term] : ~ ! [X124 : (subst > term > term)] : ~ ! [X125 : subst,X126 : term] : (((sub @ X123 @ (push @ X126 @ X125))) = ((X124 @ X125 @ X126))) <=> (hoasinduction_lem3v2_f = $true)) & ((hoasinduction_lem3v2a_lthm = $true) <=> (! [X127 : term] : ~ ! [X128 : (subst > term > term)] : ~ ! [X129 : subst,X130 : term] : (((X128 @ X129 @ X130)) = ((sub @ X127 @ (push @ X130 @ X129)))) => (! [X131 : term] : (((sub @ X131 @ id)) = X131) => ! [X132 : (subst > term > subst > $o),X133 : (term > $o),X134 : term] : (~ ! [X135 : term] : (((X133 @ X135)) = ((X132 @ id @ X135 @ id))) | ~ ! [X136 : (subst > term > term)] : (~ ! [X139 : subst,X137 : term,X138 : subst] : (((X136 @ (comp @ X139 @ X138) @ (sub @ X137 @ X138))) = ((sub @ (X136 @ X139 @ X137) @ X138))) | (((X132 @ id @ (lam @ (X136 @ sh @ one)) @ id)) = $true) | ~ ! [X140 : term] : ((((X132 @ id @ (X136 @ id @ X140) @ id)) = $true) | ~ ($true = ((X132 @ id @ X140 @ id))))) | (((X133 @ (lam @ X134))) = $true) | ~ ! [X141 : term] : (~ (((X133 @ X141)) = $true) | ($true = ((X133 @ (sub @ X134 @ (push @ X141 @ id)))))))))) & (! [X142 : (subst > term > subst > $o)] : ~ ! [X143 : (term > $o)] : ~ ! [X144 : term] : (((X142 @ id @ X144 @ id)) = ((X143 @ X144))) <=> (hoasinduction_lem0_lthm = $true)) & (hoasinduction_gthm = $true) & ~ (ulamvar1 = $true) & (hoasinduction_lem2_gthm = $true) & (pushprop_lem3v2_lthm = $true) & (hoaslamnotvar = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (! [X145 : (subst > term > subst > $o),X146 : (term > $o),X147 : term,X148 : term] : (~ ! [X149 : term] : (((X145 @ id @ X149 @ id)) = ((X146 @ X149))) | ~ ! [X150 : term,X151 : subst,X152 : subst,X153 : subst] : ((((X145 @ X153 @ X150 @ (comp @ X151 @ X152))) = $true) | ~ ($true = ((X145 @ (comp @ X153 @ X151) @ (sub @ X150 @ X151) @ X152)))) | ~ ! [X154 : term,X155 : term] : (~ (((X145 @ id @ X154 @ id)) = $true) | ~ (((X145 @ id @ X155 @ id)) = $true) | ($true = ((X145 @ id @ (ap @ (sub @ X154 @ id) @ X155) @ id)))) | ~ ! [X156 : term,X157 : subst,X158 : subst,X159 : subst] : ((((X145 @ (comp @ X157 @ X158) @ (sub @ X156 @ X158) @ X159)) = $true) | ~ (((X145 @ X157 @ X156 @ (comp @ X158 @ X159))) = $true)) | ~ (((X146 @ X147)) = $true) | (((X146 @ (ap @ X147 @ X148))) = $true) | ~ (((X146 @ X148)) = $true)) <=> (hoasinduction_lem2v2 = $true)) & ! [X160 : d_term] : (d_one = X160) & (! [X161 : (term > $o),X162 : term,X163 : (term > $o),X164 : subst,X165 : term] : ((((X163 @ (sub @ X162 @ (push @ X165 @ X164)))) = $true) | ~ (((X161 @ X162)) = $true)) <=> (pushprop_lem1 = $true)) & (hoasinduction_lem1_gthm = $true) & ! [X166 : subst] : ? [X167 : d_subst] : (((d2subst @ X167)) = X166) & ((hoasinduction_lem3v2a = $true) <=> ! [X168 : term,X169 : (term > $o),X170 : (subst > term > subst > $o)] : (~ ! [X171 : term] : (($true = ((X169 @ (sub @ X168 @ (push @ X171 @ id))))) | ~ ($true = ((X169 @ X171)))) | ~ ! [X172 : (subst > term > term)] : (~ ! [X173 : subst,X174 : term,X175 : subst] : (((sub @ (X172 @ X173 @ X174) @ X175)) = ((X172 @ (comp @ X173 @ X175) @ (sub @ X174 @ X175)))) | (((X170 @ id @ (lam @ (X172 @ sh @ one)) @ id)) = $true) | ~ ! [X176 : term] : ((((X170 @ id @ (X172 @ id @ X176) @ id)) = $true) | ~ (((X170 @ id @ X176 @ id)) = $true))) | (((X169 @ (lam @ X168))) = $true) | ~ ! [X177 : term] : (((X169 @ X177)) = ((X170 @ id @ X177 @ id))))) & ((! [X178 : term] : (((sub @ X178 @ id)) = X178) => ! [X179 : term] : (((sub @ X179 @ id)) = X179)) <=> (termmset_lthm = $true)) & (pushprop_lem2v2 = $true) & (pushprop_lem2v2_lthm = $true) & (! [X180 : term,X181 : (subst > term > subst > $o)] : ((((X181 @ id @ X180 @ id)) = $true) | ~ ! [X182 : (subst > term > term)] : (~ ! [X183 : term] : (~ (((X181 @ id @ X183 @ id)) = $true) | (((X181 @ id @ (X182 @ id @ X183) @ id)) = $true)) | ~ ! [X186 : term,X185 : subst,X184 : subst] : (((X182 @ (comp @ X185 @ X184) @ (sub @ X186 @ X184))) = ((sub @ (X182 @ X185 @ X186) @ X184))) | ($true = ((X181 @ id @ (lam @ (X182 @ sh @ one)) @ id)))) | ~ ! [X187 : term,X188 : term] : (~ ($true = ((X181 @ id @ X188 @ id))) | ~ ($true = ((X181 @ id @ X187 @ id))) | (((X181 @ id @ (ap @ (sub @ X187 @ id) @ X188) @ id)) = $true))) <=> (hoasinduction_no_psi_cond = $true)) & ((hoasinduction_lem3a_lthm = $true) <=> (! [X189 : term] : (((sub @ X189 @ id)) = X189) => (! [X190 : term,X191 : (subst > term > subst > $o)] : (~ ! [X192 : (subst > term > term)] : (($true = ((X191 @ id @ (lam @ (X192 @ sh @ one)) @ id))) | ~ ! [X193 : term] : ((((X191 @ id @ (X192 @ id @ X193) @ id)) = $true) | ~ (((X191 @ id @ X193 @ id)) = $true)) | ~ ! [X196 : term,X194 : subst,X195 : subst] : (((X192 @ (comp @ X195 @ X194) @ (sub @ X196 @ X194))) = ((sub @ (X192 @ X195 @ X196) @ X194)))) | ~ ! [X197 : term] : (~ ($true = ((X191 @ id @ X197 @ id))) | (((X191 @ id @ (sub @ X190 @ (push @ X197 @ id)) @ id)) = $true)) | (((X191 @ id @ (lam @ (sub @ X190 @ (push @ one @ sh))) @ id)) = $true)) => ! [X198 : (subst > term > subst > $o),X199 : term] : (~ ! [X200 : term] : (~ (((X198 @ id @ X200 @ id)) = $true) | (((X198 @ id @ (sub @ X199 @ (push @ X200 @ id)) @ id)) = $true)) | ~ ! [X201 : (subst > term > term)] : (~ ! [X202 : subst,X203 : term,X204 : subst] : (((sub @ (X201 @ X204 @ X203) @ X202)) = ((X201 @ (comp @ X204 @ X202) @ (sub @ X203 @ X202)))) | ($true = ((X198 @ id @ (lam @ (X201 @ sh @ one)) @ id))) | ~ ! [X205 : term] : ((((X198 @ id @ (X201 @ id @ X205) @ id)) = $true) | ~ (((X198 @ id @ X205 @ id)) = $true))) | ($true = ((X198 @ id @ (lam @ X199) @ id))))))) & (hoasinduction_lem1 = $true) & (id = ((d2subst @ d_id))) & (pushprop_lem1_gthm = $true) & (sh = ((d2subst @ d_id))) & ((induction2_lthm = $true) <=> (! [X206 : term] : (((sub @ X206 @ id)) = X206) => (! [X207 : subst,X208 : term,X209 : (term > $o)] : (~ ! [X210 : term] : (~ ! [X211 : term] : (($true = ((X209 @ (sub @ X210 @ (push @ X211 @ id))))) | ~ (((X209 @ X211)) = $true)) | (((X209 @ (lam @ X210))) = $true)) | (((X209 @ (sub @ X208 @ X207))) = $true) | ~ ! [X212 : term,X213 : term] : (~ (((X209 @ X212)) = $true) | ~ (((X209 @ X213)) = $true) | (((X209 @ (ap @ X212 @ X213))) = $true)) | ~ ! [X214 : term] : (~ (((var @ X214)) = $true) | (((X209 @ (sub @ X214 @ X207))) = $true))) => ! [X215 : (term > $o),X216 : term] : (~ ! [X217 : term,X218 : term] : ((((X215 @ (ap @ X218 @ X217))) = $true) | ~ ($true = ((X215 @ X218))) | ~ (((X215 @ X217)) = $true)) | ~ ! [X219 : term] : (~ (((var @ X219)) = $true) | (((X215 @ X219)) = $true)) | ~ ! [X220 : term] : ((((X215 @ (lam @ X220))) = $true) | ~ ! [X221 : term] : (~ (((X215 @ X221)) = $true) | (((X215 @ (sub @ X220 @ (push @ X221 @ id)))) = $true))) | (((X215 @ X216)) = $true))))) & (! [X222 : subst] : (((comp @ id @ X222)) = X222) <=> (axidl = $true)) & ((hoasinduction_lthm_3 = $true) <=> (! [X223 : (subst > term > subst > $o)] : ~ ! [X224 : (term > $o)] : ~ ! [X225 : term] : (((X224 @ X225)) = ((X223 @ id @ X225 @ id))) => (! [X226 : (term > $o),X227 : term] : (~ ! [X228 : term] : (~ (((var @ X228)) = $true) | (((X226 @ X228)) = $true)) | ~ ! [X229 : term,X230 : term] : ((((X226 @ (ap @ X230 @ X229))) = $true) | ~ (((X226 @ X230)) = $true) | ~ (((X226 @ X229)) = $true)) | ~ ! [X231 : term] : (~ ! [X232 : term] : ((((X226 @ (sub @ X231 @ (push @ X232 @ id)))) = $true) | ~ (((X226 @ X232)) = $true)) | (((X226 @ (lam @ X231))) = $true)) | (((X226 @ X227)) = $true)) => (! [X233 : term] : (((sub @ X233 @ id)) = X233) => (! [X234 : (term > $o),X235 : term,X236 : (subst > term > subst > $o)] : (~ ! [X237 : term] : (((X234 @ X237)) = ((X236 @ id @ X237 @ id))) | ~ ! [X238 : (subst > term > term)] : (~ ! [X239 : term] : ((((X236 @ id @ (X238 @ id @ X239) @ id)) = $true) | ~ (((X236 @ id @ X239 @ id)) = $true)) | ($true = ((X236 @ id @ (lam @ (X238 @ sh @ one)) @ id))) | ~ ! [X240 : subst,X242 : term,X241 : subst] : (((sub @ (X238 @ X241 @ X242) @ X240)) = ((X238 @ (comp @ X241 @ X240) @ (sub @ X242 @ X240))))) | (((X234 @ (lam @ X235))) = $true) | ~ ! [X243 : term] : (($true = ((X234 @ (sub @ X235 @ (push @ X243 @ id))))) | ~ (((X234 @ X243)) = $true))) => ! [X244 : (subst > term > subst > $o),X245 : term] : (($true = ((X244 @ id @ X245 @ id))) | ~ ! [X246 : subst,X247 : term,X248 : subst,X249 : subst] : (~ (((X244 @ X246 @ X247 @ (comp @ X249 @ X248))) = $true) | (((X244 @ (comp @ X246 @ X249) @ (sub @ X247 @ X249) @ X248)) = $true)) | ~ ! [X250 : subst,X251 : subst,X252 : term,X253 : subst] : (~ ($true = ((X244 @ (comp @ X253 @ X251) @ (sub @ X252 @ X251) @ X250))) | ($true = ((X244 @ X253 @ X252 @ (comp @ X251 @ X250))))) | ~ ! [X254 : term,X255 : term] : ((((X244 @ id @ (ap @ (sub @ X254 @ id) @ X255) @ id)) = $true) | ~ (((X244 @ id @ X254 @ id)) = $true) | ~ (((X244 @ id @ X255 @ id)) = $true)) | ~ ! [X256 : term] : ((((X244 @ id @ X256 @ id)) = $true) | ~ (((var @ (sub @ X256 @ id))) = $true)) | ~ ! [X257 : (subst > term > term)] : (~ ! [X258 : subst,X260 : subst,X259 : term] : (((sub @ (X257 @ X260 @ X259) @ X258)) = ((X257 @ (comp @ X260 @ X258) @ (sub @ X259 @ X258)))) | ~ ! [X261 : term] : (~ (((X244 @ id @ X261 @ id)) = $true) | (((X244 @ id @ (X257 @ id @ X261) @ id)) = $true)) | (((X244 @ id @ (lam @ (X257 @ sh @ one)) @ id)) = $true)))))))) & ((! [X263 : term,X262 : subst] : (((sub @ one @ (push @ X263 @ X262))) = X263) => (! [X265 : term,X264 : subst] : (((comp @ sh @ (push @ X265 @ X264))) = X264) => (! [X266 : subst] : (((comp @ X266 @ id)) = X266) => (! [X267 : term,X268 : (term > $o)] : ((((X268 @ X267)) = $true) | ~ ! [X269 : term,X270 : term] : (~ (((X268 @ X270)) = $true) | ~ (((X268 @ X269)) = $true) | (((X268 @ (ap @ X269 @ X270))) = $true)) | ~ ! [X271 : term] : (~ ($true = ((var @ X271))) | (((X268 @ X271)) = $true)) | ~ ! [X272 : term] : (~ (((X268 @ X272)) = $true) | (((X268 @ (lam @ X272))) = $true))) => ! [X273 : (term > $o),X274 : term,X275 : subst] : (~ ! [X276 : term] : ((((X273 @ (sub @ X276 @ X275))) = $true) | ~ (((var @ X276)) = $true)) | ~ ! [X277 : term] : ((((X273 @ (lam @ X277))) = $true) | ~ ! [X278 : term] : ((((X273 @ (sub @ X277 @ (push @ X278 @ id)))) = $true) | ~ ($true = ((X273 @ X278))))) | ~ ! [X279 : term,X280 : term] : (~ (((X273 @ X280)) = $true) | ($true = ((X273 @ (ap @ X279 @ X280)))) | ~ (((X273 @ X279)) = $true)) | ($true = ((X273 @ (sub @ X274 @ X275))))))))) <=> (induction2lem_lthm = $true)) & ((! [X281 : term] : (((sub @ X281 @ id)) = X281) => (! [X283 : term,X285 : term,X284 : term,X282 : term] : ((((ap @ X283 @ X285)) != ((ap @ X284 @ X282))) | (X283 = X284)) => ! [X286 : term,X289 : term,X288 : term,X287 : term] : ((X286 = X288) | (((ap @ (sub @ X286 @ id) @ X287)) != ((ap @ (sub @ X288 @ id) @ X289)))))) <=> (hoasapinj1_lthm = $true)) & ((termmset = $true) <=> ! [X290 : term] : (((sub @ X290 @ id)) = X290)) & ((pushprop_lem1_lthm = $true) <=> (! [X291 : subst,X292 : term] : (((sub @ one @ (push @ X292 @ X291))) = X292) => (! [X294 : subst,X293 : term] : (((comp @ sh @ (push @ X293 @ X294))) = X294) => ! [X295 : subst,X296 : (term > $o),X297 : term,X298 : term,X299 : (term > $o)] : (~ (((X299 @ X297)) = $true) | (((X296 @ (sub @ X297 @ (push @ X298 @ X295)))) = $true))))) & (pushprop_lem2v2_gthm = $true) & (hoasinduction_lem1_lthm = $true) & (pushprop_lem1v2_gthm = $true) & (pushprop_gthm = $true) & ((! [X300 : subst] : (((comp @ id @ X300)) = X300) => (! [X301 : subst] : (((comp @ X301 @ id)) = X301) => (! [X302 : subst] : (((comp @ X302 @ id)) = X302) & ! [X303 : subst] : (((comp @ id @ X303)) = X303)))) <=> (substmonoid_lthm = $true)) & (axmap = $true) & (! [X305 : (subst > term > term),X304 : (subst > term > term),X307 : term,X306 : subst] : (~ ! [X312 : subst,X311 : term,X313 : subst] : (((sub @ (X305 @ X313 @ X311) @ X312)) = ((X305 @ (comp @ X313 @ X312) @ (sub @ X311 @ X312)))) | (((lam @ (X305 @ sh @ one))) != ((lam @ (X304 @ sh @ one)))) | (((X305 @ X306 @ X307)) = ((X304 @ X306 @ X307))) | ~ ! [X309 : subst,X310 : subst,X308 : term] : (((X304 @ (comp @ X310 @ X309) @ (sub @ X308 @ X309))) = ((sub @ (X304 @ X310 @ X308) @ X309)))) <=> (hoaslaminj = $true)) & (ulamvarind = $true) & (hoasinduction_lem1v2 = $true) & (substmonoid_gthm = $true) & (! [X315 : subst,X314 : (term > $o),X316 : term] : ~ ! [X317 : (term > $o)] : ~ ! [X318 : term] : (((X314 @ (sub @ X318 @ (push @ X316 @ X315)))) = ((X317 @ X318))) <=> (pushprop_lem0_lthm = $true)) & (hoasapnotvar = $true) & ((hoasinduction_lem3 = $true) <=> ! [X319 : (subst > term > subst > $o),X320 : term] : (~ ! [X321 : term,X322 : subst,X323 : subst,X324 : subst] : (($true = ((X319 @ (comp @ X324 @ X322) @ (sub @ X321 @ X322) @ X323))) | ~ (((X319 @ X324 @ X321 @ (comp @ X322 @ X323))) = $true)) | (((X319 @ id @ (lam @ X320) @ id)) = $true) | ~ ! [X325 : (subst > term > term)] : (~ ! [X328 : term,X326 : subst,X327 : subst] : (((sub @ (X325 @ X326 @ X328) @ X327)) = ((X325 @ (comp @ X326 @ X327) @ (sub @ X328 @ X327)))) | (((X319 @ id @ (lam @ (X325 @ sh @ one)) @ id)) = $true) | ~ ! [X329 : term] : (~ (((X319 @ id @ X329 @ id)) = $true) | ($true = ((X319 @ id @ (X325 @ id @ X329) @ id))))) | ~ ! [X330 : term] : ((((X319 @ id @ (sub @ X320 @ (push @ X330 @ id)) @ id)) = $true) | ~ (((X319 @ id @ X330 @ id)) = $true)) | ~ ! [X331 : subst,X332 : subst,X333 : subst,X334 : term] : (~ (((X319 @ (comp @ X332 @ X333) @ (sub @ X334 @ X333) @ X331)) = $true) | (((X319 @ X332 @ X334 @ (comp @ X333 @ X331))) = $true)))) & ((hoasinduction_lem2 = $true) <=> ! [X335 : (subst > term > subst > $o),X336 : term,X337 : term] : ((((X335 @ id @ (ap @ X337 @ X336) @ id)) = $true) | ~ ! [X338 : subst,X339 : subst,X340 : subst,X341 : term] : ((((X335 @ (comp @ X339 @ X338) @ (sub @ X341 @ X338) @ X340)) = $true) | ~ ($true = ((X335 @ X339 @ X341 @ (comp @ X338 @ X340))))) | ~ ($true = ((X335 @ id @ X337 @ id))) | ~ ! [X342 : subst,X343 : subst,X344 : term,X345 : subst] : (~ ($true = ((X335 @ (comp @ X343 @ X345) @ (sub @ X344 @ X345) @ X342))) | ($true = ((X335 @ X343 @ X344 @ (comp @ X345 @ X342))))) | ~ ! [X346 : term,X347 : term] : (~ ($true = ((X335 @ id @ X346 @ id))) | (((X335 @ id @ (ap @ (sub @ X346 @ id) @ X347) @ id)) = $true) | ~ (((X335 @ id @ X347 @ id)) = $true)) | ~ ($true = ((X335 @ id @ X336 @ id))))) & ! [X348 : term] : ? [X349 : d_term] : (((d2term @ X349)) = X348) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (axabs = $true) & ((hoasinduction_lem3a = $true) <=> ! [X350 : (subst > term > subst > $o),X351 : term] : ((((X350 @ id @ (lam @ X351) @ id)) = $true) | ~ ! [X352 : (subst > term > term)] : (~ ! [X355 : subst,X354 : subst,X353 : term] : (((X352 @ (comp @ X355 @ X354) @ (sub @ X353 @ X354))) = ((sub @ (X352 @ X355 @ X353) @ X354))) | ~ ! [X356 : term] : ((((X350 @ id @ (X352 @ id @ X356) @ id)) = $true) | ~ (((X350 @ id @ X356 @ id)) = $true)) | (((X350 @ id @ (lam @ (X352 @ sh @ one)) @ id)) = $true)) | ~ ! [X357 : term] : (~ (((X350 @ id @ X357 @ id)) = $true) | (((X350 @ id @ (sub @ X351 @ (push @ X357 @ id)) @ id)) = $true)))) & (! [X358 : term,X361 : term,X359 : term,X360 : term] : ((X359 = X360) | (((ap @ X359 @ X361)) != ((ap @ X360 @ X358)))) <=> (apinj1 = $true)) & (induction2lem_gthm = $true) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (! [X362 : (subst > term > subst > $o)] : ~ ! [X363 : (term > $o)] : ~ ! [X364 : term] : (((X362 @ id @ X364 @ id)) = ((X363 @ X364))) => (! [X365 : (term > $o),X366 : term] : ((((X365 @ X366)) = $true) | ~ ! [X367 : term] : (~ ($true = ((var @ X367))) | (((X365 @ X367)) = $true)) | ~ ! [X368 : term,X369 : term] : (~ (((X365 @ X369)) = $true) | (((X365 @ (ap @ X369 @ X368))) = $true) | ~ (((X365 @ X368)) = $true)) | ~ ! [X370 : term] : (~ ! [X371 : term] : (~ (((X365 @ X371)) = $true) | (((X365 @ (sub @ X370 @ (push @ X371 @ id)))) = $true)) | (((X365 @ (lam @ X370))) = $true))) => (! [X372 : term] : (((sub @ X372 @ id)) = X372) => (! [X373 : term,X374 : (term > $o),X375 : (subst > term > subst > $o)] : (~ ! [X376 : term] : (~ ($true = ((X374 @ X376))) | ($true = ((X374 @ (sub @ X373 @ (push @ X376 @ id)))))) | ~ ! [X377 : (subst > term > term)] : (~ ! [X379 : subst,X380 : term,X378 : subst] : (((sub @ (X377 @ X379 @ X380) @ X378)) = ((X377 @ (comp @ X379 @ X378) @ (sub @ X380 @ X378)))) | (((X375 @ id @ (lam @ (X377 @ sh @ one)) @ id)) = $true) | ~ ! [X381 : term] : (($true = ((X375 @ id @ (X377 @ id @ X381) @ id))) | ~ (((X375 @ id @ X381 @ id)) = $true))) | ~ ! [X382 : term] : (((X375 @ id @ X382 @ id)) = ((X374 @ X382))) | ($true = ((X374 @ (lam @ X373))))) => ! [X383 : term,X384 : (subst > term > subst > $o)] : (($true = ((X384 @ id @ X383 @ id))) | ~ ! [X385 : term,X386 : term] : (~ ($true = ((X384 @ id @ X385 @ id))) | (((X384 @ id @ (ap @ (sub @ X385 @ id) @ X386) @ id)) = $true) | ~ (((X384 @ id @ X386 @ id)) = $true)) | ~ ! [X387 : (subst > term > term)] : (~ ! [X389 : subst,X390 : term,X388 : subst] : (((X387 @ (comp @ X389 @ X388) @ (sub @ X390 @ X388))) = ((sub @ (X387 @ X389 @ X390) @ X388))) | ~ ! [X391 : term] : (~ ($true = ((X384 @ id @ X391 @ id))) | (((X384 @ id @ (X387 @ id @ X391) @ id)) = $true)) | (((X384 @ id @ (lam @ (X387 @ sh @ one)) @ id)) = $true)))))))) & ((! [X393 : term,X392 : subst] : (((sub @ one @ (push @ X393 @ X392))) = X393) => ! [X394 : (term > $o),X395 : (term > $o),X396 : subst,X397 : term] : (($true = ((X394 @ one))) | ~ ! [X398 : term] : (((X395 @ (sub @ X398 @ (push @ X397 @ X396)))) = ((X394 @ X398))) | ~ ($true = ((X395 @ X397))))) <=> (pushprop_lem1v2_lthm = $true)) & (induction2_gthm = $true) & ((pushprop_lem1v2 = $true) <=> ! [X399 : subst,X400 : (term > $o),X401 : (term > $o),X402 : term] : (~ ! [X403 : term] : (((X400 @ X403)) = ((X401 @ (sub @ X403 @ (push @ X402 @ X399))))) | ~ (((X401 @ X402)) = $true) | ($true = ((X400 @ one))))) & (hoasinduction_lem2v2_gthm = $true) & ((apinj2 = $true) <=> ! [X405 : term,X404 : term,X406 : term,X407 : term] : ((((ap @ X404 @ X407)) != ((ap @ X405 @ X406))) | (X406 = X407))) & (! [X409 : term,X408 : subst] : (((comp @ sh @ (push @ X409 @ X408))) = X408) <=> (axshiftcons = $true)) & (apnotvar = $true) & ((hoasinduction_lem0 = $true) <=> ! [X410 : (subst > term > subst > $o)] : ~ ! [X411 : (term > $o)] : ~ ! [X412 : term] : (((X410 @ id @ X412 @ id)) = ((X411 @ X412)))) & ((hoasapinj2 = $true) <=> ! [X415 : term,X413 : term,X414 : term,X416 : term] : ((X413 = X414) | (((ap @ (sub @ X415 @ id) @ X413)) != ((ap @ (sub @ X416 @ id) @ X414))))) & (pushprop = $true) & (! [X419 : term,X417 : (subst > term > term),X418 : term] : (~ ! [X420 : subst,X421 : subst,X422 : term] : (((X417 @ (comp @ X420 @ X421) @ (sub @ X422 @ X421))) = ((sub @ (X417 @ X420 @ X422) @ X421))) | (((lam @ (X417 @ sh @ one))) != ((ap @ (sub @ X419 @ id) @ X418)))) <=> (hoaslamnotap = $true)) & (pushprop_lem0_gthm = $true) & (hoaslaminj_gthm = $true) & (! [X423 : subst,X424 : term] : (((sub @ one @ (push @ X424 @ X423))) = X424) <=> (axvarcons = $true)) & ((hoasinduction_lem3_lthm = $true) <=> (! [X425 : term] : (((sub @ X425 @ id)) = X425) => (! [X426 : (subst > term > subst > $o),X427 : term] : (($true = ((X426 @ id @ (lam @ (sub @ X427 @ (push @ one @ sh))) @ id))) | ~ ! [X428 : (subst > term > term)] : (~ ! [X429 : term] : (~ ($true = ((X426 @ id @ X429 @ id))) | (((X426 @ id @ (X428 @ id @ X429) @ id)) = $true)) | ($true = ((X426 @ id @ (lam @ (X428 @ sh @ one)) @ id))) | ~ ! [X430 : subst,X432 : subst,X431 : term] : (((sub @ (X428 @ X432 @ X431) @ X430)) = ((X428 @ (comp @ X432 @ X430) @ (sub @ X431 @ X430))))) | ~ ! [X433 : term] : (~ ($true = ((X426 @ id @ X433 @ id))) | (((X426 @ id @ (sub @ X427 @ (push @ X433 @ id)) @ id)) = $true))) => ! [X434 : term,X435 : (subst > term > subst > $o)] : ((((X435 @ id @ (lam @ X434) @ id)) = $true) | ~ ! [X436 : term,X437 : subst,X438 : subst,X439 : subst] : (($true = ((X435 @ (comp @ X438 @ X439) @ (sub @ X436 @ X439) @ X437))) | ~ ($true = ((X435 @ X438 @ X436 @ (comp @ X439 @ X437))))) | ~ ! [X440 : term] : (~ (((X435 @ id @ X440 @ id)) = $true) | ($true = ((X435 @ id @ (sub @ X434 @ (push @ X440 @ id)) @ id)))) | ~ ! [X441 : (subst > term > term)] : (~ ! [X443 : term,X444 : subst,X442 : subst] : (((X441 @ (comp @ X442 @ X444) @ (sub @ X443 @ X444))) = ((sub @ (X441 @ X442 @ X443) @ X444))) | ($true = ((X435 @ id @ (lam @ (X441 @ sh @ one)) @ id))) | ~ ! [X445 : term] : ((((X435 @ id @ (X441 @ id @ X445) @ id)) = $true) | ~ (((X435 @ id @ X445 @ id)) = $true))) | ~ ! [X446 : subst,X447 : subst,X448 : term,X449 : subst] : (~ (((X435 @ (comp @ X446 @ X449) @ (sub @ X448 @ X449) @ X447)) = $true) | (((X435 @ X446 @ X448 @ (comp @ X449 @ X447))) = $true)))))) & (termmset_gthm = $true) & ! [X450 : d_term,X451 : d_term] : ((((d2term @ X450)) = ((d2term @ X451))) => (X450 = X451)) & ! [X453 : d_subst,X452 : d_subst] : ((((d2subst @ X453)) = ((d2subst @ X452))) => (X452 = X453)) & (lamnotvar = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ((axscons = $true) <=> ! [X454 : subst] : (((push @ (sub @ one @ X454) @ (comp @ sh @ X454))) = X454)) & ((shinj = $true) <=> ! [X455 : term,X456 : term] : ((((sub @ X455 @ sh)) != ((sub @ X456 @ sh))) | (X455 = X456))) & (pushprop_lthm = $true) & (hoasapnotvar_lthm = $true) & (hoasinduction_lem3_gthm = $true) & (hoasinduction_lem3v2_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (hoasinduction_lem2_lthm = $true) & (axclos = $true) & ((hoasinduction_lem3aa = $true) <=> ! [X457 : (subst > term > subst > $o),X458 : term] : (~ ! [X459 : (subst > term > term)] : ((((X457 @ id @ (lam @ (X459 @ sh @ one)) @ id)) = $true) | ~ ! [X460 : term] : ((((X457 @ id @ (X459 @ id @ X460) @ id)) = $true) | ~ ($true = ((X457 @ id @ X460 @ id)))) | ~ ! [X463 : subst,X461 : term,X462 : subst] : (((X459 @ (comp @ X463 @ X462) @ (sub @ X461 @ X462))) = ((sub @ (X459 @ X463 @ X461) @ X462)))) | ~ ! [X464 : term] : (~ (((X457 @ id @ X464 @ id)) = $true) | (((X457 @ id @ (sub @ X458 @ (push @ X464 @ id)) @ id)) = $true)) | (((X457 @ id @ (lam @ (sub @ X458 @ (push @ one @ sh))) @ id)) = $true))) & ((hoasinduction_lthm = $true) <=> (! [X465 : term,X466 : (term > $o)] : (~ ! [X467 : term] : (~ (((var @ X467)) = $true) | ($true = ((X466 @ X467)))) | ($true = ((X466 @ X465))) | ~ ! [X468 : term,X469 : term] : (~ ($true = ((X466 @ X469))) | ~ (((X466 @ X468)) = $true) | ($true = ((X466 @ (ap @ X468 @ X469))))) | ~ ! [X470 : term] : (~ ! [X471 : term] : (~ (((X466 @ X471)) = $true) | (((X466 @ (sub @ X470 @ (push @ X471 @ id)))) = $true)) | (((X466 @ (lam @ X470))) = $true))) => (! [X472 : term,X473 : term,X474 : (subst > term > subst > $o)] : ((((X474 @ id @ (ap @ X472 @ X473) @ id)) = $true) | ~ (((X474 @ id @ X472 @ id)) = $true) | ~ (((X474 @ id @ X473 @ id)) = $true) | ~ ! [X475 : term,X476 : subst,X477 : subst,X478 : subst] : ((((X474 @ (comp @ X476 @ X477) @ (sub @ X475 @ X477) @ X478)) = $true) | ~ ($true = ((X474 @ X476 @ X475 @ (comp @ X477 @ X478))))) | ~ ! [X479 : subst,X480 : subst,X481 : subst,X482 : term] : (~ ($true = ((X474 @ (comp @ X480 @ X481) @ (sub @ X482 @ X481) @ X479))) | (((X474 @ X480 @ X482 @ (comp @ X481 @ X479))) = $true)) | ~ ! [X483 : term,X484 : term] : (~ (((X474 @ id @ X483 @ id)) = $true) | ~ (((X474 @ id @ X484 @ id)) = $true) | (((X474 @ id @ (ap @ (sub @ X483 @ id) @ X484) @ id)) = $true))) => (! [X485 : term,X486 : (subst > term > subst > $o)] : (~ ! [X487 : subst,X488 : term,X489 : subst,X490 : subst] : (($true = ((X486 @ X489 @ X488 @ (comp @ X490 @ X487)))) | ~ ($true = ((X486 @ (comp @ X489 @ X490) @ (sub @ X488 @ X490) @ X487)))) | ~ ! [X491 : (subst > term > term)] : ((((X486 @ id @ (lam @ (X491 @ sh @ one)) @ id)) = $true) | ~ ! [X492 : subst,X493 : subst,X494 : term] : (((sub @ (X491 @ X492 @ X494) @ X493)) = ((X491 @ (comp @ X492 @ X493) @ (sub @ X494 @ X493)))) | ~ ! [X495 : term] : ((((X486 @ id @ (X491 @ id @ X495) @ id)) = $true) | ~ (((X486 @ id @ X495 @ id)) = $true))) | (((X486 @ id @ (lam @ X485) @ id)) = $true) | ~ ! [X496 : term,X497 : subst,X498 : subst,X499 : subst] : (~ (((X486 @ X499 @ X496 @ (comp @ X497 @ X498))) = $true) | ($true = ((X486 @ (comp @ X499 @ X497) @ (sub @ X496 @ X497) @ X498)))) | ~ ! [X500 : term] : (~ (((X486 @ id @ X500 @ id)) = $true) | (((X486 @ id @ (sub @ X485 @ (push @ X500 @ id)) @ id)) = $true))) => ! [X501 : (subst > term > subst > $o),X502 : term] : (~ ! [X503 : term,X504 : term] : (~ (((X501 @ id @ X503 @ id)) = $true) | ~ (((X501 @ id @ X504 @ id)) = $true) | (((X501 @ id @ (ap @ (sub @ X504 @ id) @ X503) @ id)) = $true)) | ~ ! [X505 : term] : (~ ($true = ((var @ (sub @ X505 @ id)))) | (((X501 @ id @ X505 @ id)) = $true)) | (((X501 @ id @ X502 @ id)) = $true) | ~ ! [X506 : (subst > term > term)] : ((((X501 @ id @ (lam @ (X506 @ sh @ one)) @ id)) = $true) | ~ ! [X507 : term] : (~ ($true = ((X501 @ id @ X507 @ id))) | (((X501 @ id @ (X506 @ id @ X507) @ id)) = $true)) | ~ ! [X509 : subst,X510 : subst,X508 : term] : (((X506 @ (comp @ X510 @ X509) @ (sub @ X508 @ X509))) = ((sub @ (X506 @ X510 @ X508) @ X509)))) | ~ ! [X511 : subst,X512 : term,X513 : subst,X514 : subst] : (~ (((X501 @ (comp @ X514 @ X513) @ (sub @ X512 @ X513) @ X511)) = $true) | (((X501 @ X514 @ X512 @ (comp @ X513 @ X511))) = $true)) | ~ ! [X515 : term,X516 : subst,X517 : subst,X518 : subst] : (~ (((X501 @ X517 @ X515 @ (comp @ X516 @ X518))) = $true) | (((X501 @ (comp @ X517 @ X516) @ (sub @ X515 @ X516) @ X518)) = $true))))))) & (hoasapnotvar_gthm = $true) & ((axidr = $true) <=> ! [X519 : subst] : (((comp @ X519 @ id)) = X519)) & (! [X521 : term,X520 : term] : ((((lam @ X521)) != ((lam @ X520))) | (X520 = X521)) <=> (laminj = $true)) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (! [X522 : (term > $o),X523 : term,X524 : (subst > term > subst > $o)] : ((((X522 @ (lam @ X523))) = $true) | ~ ! [X525 : subst,X526 : subst,X527 : term,X528 : subst] : ((((X524 @ (comp @ X525 @ X526) @ (sub @ X527 @ X526) @ X528)) = $true) | ~ ($true = ((X524 @ X525 @ X527 @ (comp @ X526 @ X528))))) | ~ ! [X529 : subst,X530 : subst,X531 : term,X532 : subst] : (~ (((X524 @ (comp @ X532 @ X530) @ (sub @ X531 @ X530) @ X529)) = $true) | (((X524 @ X532 @ X531 @ (comp @ X530 @ X529))) = $true)) | ~ ! [X533 : term] : (~ ($true = ((X522 @ X533))) | (((X522 @ (sub @ X523 @ (push @ X533 @ id)))) = $true)) | ~ ! [X534 : (subst > term > term)] : (~ ! [X535 : term] : (~ ($true = ((X524 @ id @ X535 @ id))) | (((X524 @ id @ (X534 @ id @ X535) @ id)) = $true)) | (((X524 @ id @ (lam @ (X534 @ sh @ one)) @ id)) = $true) | ~ ! [X538 : subst,X537 : term,X536 : subst] : (((X534 @ (comp @ X536 @ X538) @ (sub @ X537 @ X538))) = ((sub @ (X534 @ X536 @ X537) @ X538)))) | ~ ! [X539 : term] : (((X524 @ id @ X539 @ id)) = ((X522 @ X539)))) <=> (hoasinduction_lem3v2 = $true)) & (hoasinduction_lem3a_gthm = $true)),
  inference(rectify,[],[f5])).
thf(f9,plain,(
  (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X85 : term] : ~ ! [X86 : (subst > term > term)] : (((sub @ X85 @ (push @ one @ sh))) != ((X86 @ sh @ one)))) & ! [X453 : d_subst,X452 : d_subst] : ((((d2subst @ X453)) = ((d2subst @ X452))) => (X452 = X453)) & (hoasinduction_lem3a_gthm = $true) & ((axscons = $true) <=> ! [X454 : subst] : (((push @ (sub @ one @ X454) @ (comp @ sh @ X454))) = X454)) & ((pushprop_lem1_lthm = $true) <=> (! [X291 : subst,X292 : term] : (((sub @ one @ (push @ X292 @ X291))) = X292) => (! [X294 : subst,X293 : term] : (((comp @ sh @ (push @ X293 @ X294))) = X294) => ! [X298 : term,X295 : subst,X297 : term,X299 : (term > $o),X296 : (term > $o)] : ((((X299 @ X297)) != $true) | (((X296 @ (sub @ X297 @ (push @ X298 @ X295)))) = $true))))) & (pushprop_lem0_gthm = $true) & (hoaslamnotap_gthm = $true) & (! [X57 : (subst > term > subst > $o),X56 : term] : (~ ! [X63 : (subst > term > term)] : (~ ! [X67 : term] : ((((X57 @ id @ X67 @ id)) != $true) | (((X57 @ id @ (X63 @ id @ X67) @ id)) = $true)) | ($true = ((X57 @ id @ (lam @ (X63 @ sh @ one)) @ id))) | ~ ! [X64 : term,X66 : subst,X65 : subst] : (((X63 @ (comp @ X65 @ X66) @ (sub @ X64 @ X66))) = ((sub @ (X63 @ X65 @ X64) @ X66)))) | ~ ! [X70 : subst,X73 : subst,X72 : subst,X71 : term] : (($true = ((X57 @ X70 @ X71 @ (comp @ X73 @ X72)))) | (((X57 @ (comp @ X70 @ X73) @ (sub @ X71 @ X73) @ X72)) != $true)) | ~ ! [X62 : subst,X60 : subst,X59 : term,X61 : subst] : ((((X57 @ (comp @ X60 @ X61) @ (sub @ X59 @ X61) @ X62)) = $true) | (((X57 @ X60 @ X59 @ (comp @ X61 @ X62))) != $true)) | ~ ! [X68 : term,X69 : term] : ((((X57 @ id @ (ap @ (sub @ X69 @ id) @ X68) @ id)) = $true) | (((X57 @ id @ X69 @ id)) != $true) | ($true != ((X57 @ id @ X68 @ id)))) | (((X57 @ id @ X56 @ id)) = $true) | ~ ! [X58 : term] : ((((X57 @ id @ X58 @ id)) = $true) | (((var @ (sub @ X58 @ id))) != $true))) <=> (hoasinduction = $true)) & (axabs = $true) & (pushprop_lem2v2_lthm = $true) & (pushprop_gthm = $true) & ((! [X223 : (subst > term > subst > $o)] : ~ ! [X224 : (term > $o)] : ~ ! [X225 : term] : (((X224 @ X225)) = ((X223 @ id @ X225 @ id))) => (! [X226 : (term > $o),X227 : term] : (~ ! [X231 : term] : ((((X226 @ (lam @ X231))) = $true) | ~ ! [X232 : term] : ((((X226 @ X232)) != $true) | (((X226 @ (sub @ X231 @ (push @ X232 @ id)))) = $true))) | ~ ! [X230 : term,X229 : term] : ((((X226 @ (ap @ X230 @ X229))) = $true) | (((X226 @ X229)) != $true) | (((X226 @ X230)) != $true)) | ~ ! [X228 : term] : ((((var @ X228)) != $true) | (((X226 @ X228)) = $true)) | (((X226 @ X227)) = $true)) => (! [X233 : term] : (((sub @ X233 @ id)) = X233) => (! [X234 : (term > $o),X236 : (subst > term > subst > $o),X235 : term] : (~ ! [X237 : term] : (((X234 @ X237)) = ((X236 @ id @ X237 @ id))) | ~ ! [X243 : term] : ((((X234 @ X243)) != $true) | ($true = ((X234 @ (sub @ X235 @ (push @ X243 @ id)))))) | ~ ! [X238 : (subst > term > term)] : (~ ! [X239 : term] : ((((X236 @ id @ X239 @ id)) != $true) | (((X236 @ id @ (X238 @ id @ X239) @ id)) = $true)) | ~ ! [X240 : subst,X242 : term,X241 : subst] : (((sub @ (X238 @ X241 @ X242) @ X240)) = ((X238 @ (comp @ X241 @ X240) @ (sub @ X242 @ X240)))) | ($true = ((X236 @ id @ (lam @ (X238 @ sh @ one)) @ id)))) | (((X234 @ (lam @ X235))) = $true)) => ! [X244 : (subst > term > subst > $o),X245 : term] : (($true = ((X244 @ id @ X245 @ id))) | ~ ! [X257 : (subst > term > term)] : ((((X244 @ id @ (lam @ (X257 @ sh @ one)) @ id)) = $true) | ~ ! [X261 : term] : ((((X244 @ id @ (X257 @ id @ X261) @ id)) = $true) | (((X244 @ id @ X261 @ id)) != $true)) | ~ ! [X258 : subst,X260 : subst,X259 : term] : (((sub @ (X257 @ X260 @ X259) @ X258)) = ((X257 @ (comp @ X260 @ X258) @ (sub @ X259 @ X258))))) | ~ ! [X253 : subst,X250 : subst,X252 : term,X251 : subst] : (($true != ((X244 @ (comp @ X253 @ X251) @ (sub @ X252 @ X251) @ X250))) | ($true = ((X244 @ X253 @ X252 @ (comp @ X251 @ X250))))) | ~ ! [X248 : subst,X246 : subst,X249 : subst,X247 : term] : ((((X244 @ (comp @ X246 @ X249) @ (sub @ X247 @ X249) @ X248)) = $true) | (((X244 @ X246 @ X247 @ (comp @ X249 @ X248))) != $true)) | ~ ! [X256 : term] : ((((var @ (sub @ X256 @ id))) != $true) | (((X244 @ id @ X256 @ id)) = $true)) | ~ ! [X255 : term,X254 : term] : ((((X244 @ id @ X254 @ id)) != $true) | (((X244 @ id @ (ap @ (sub @ X254 @ id) @ X255) @ id)) = $true) | (((X244 @ id @ X255 @ id)) != $true))))))) <=> (hoasinduction_lthm_3 = $true)) & (hoasinduction_lem1_lthm = $true) & ! [X166 : subst] : ? [X167 : d_subst] : (((d2subst @ X167)) = X166) & (axassoc = $true) & ((pushprop_lem1v2_lthm = $true) <=> (! [X393 : term,X392 : subst] : (((sub @ one @ (push @ X393 @ X392))) = X393) => ! [X395 : (term > $o),X397 : term,X396 : subst,X394 : (term > $o)] : (($true != ((X395 @ X397))) | ($true = ((X394 @ one))) | ~ ! [X398 : term] : (((X395 @ (sub @ X398 @ (push @ X397 @ X396)))) = ((X394 @ X398)))))) & (hoasinduction_lem1_gthm = $true) & ((hoasinduction_lem3v2_lthm = $true) <=> (! [X87 : term] : (((sub @ X87 @ id)) = X87) => ! [X88 : (subst > term > subst > $o),X90 : (term > $o),X89 : term] : (~ ! [X95 : (subst > term > term)] : (~ ! [X98 : term,X99 : subst,X97 : subst] : (((sub @ (X95 @ X97 @ X98) @ X99)) = ((X95 @ (comp @ X97 @ X99) @ (sub @ X98 @ X99)))) | (((X88 @ id @ (lam @ (X95 @ sh @ one)) @ id)) = $true) | ~ ! [X96 : term] : (($true = ((X88 @ id @ (X95 @ id @ X96) @ id))) | (((X88 @ id @ X96 @ id)) != $true))) | (((X90 @ (lam @ X89))) = $true) | ~ ! [X100 : term] : (((X90 @ X100)) = ((X88 @ id @ X100 @ id))) | ~ ! [X94 : subst,X91 : term,X93 : subst,X92 : subst] : (($true = ((X88 @ (comp @ X93 @ X92) @ (sub @ X91 @ X92) @ X94))) | (((X88 @ X93 @ X91 @ (comp @ X92 @ X94))) != $true)) | ~ ! [X105 : term] : ((((X90 @ (sub @ X89 @ (push @ X105 @ id)))) = $true) | (((X90 @ X105)) != $true)) | ~ ! [X104 : term,X101 : subst,X102 : subst,X103 : subst] : ((((X88 @ (comp @ X102 @ X101) @ (sub @ X104 @ X101) @ X103)) != $true) | (((X88 @ X102 @ X104 @ (comp @ X101 @ X103))) = $true))))) & ((! [X113 : subst] : (((comp @ X113 @ id)) = X113) & ! [X114 : subst] : (((comp @ id @ X114)) = X114)) <=> (substmonoid = $true)) & ((hoasinduction_lem3b = $true) <=> ! [X54 : term] : ~ ! [X55 : (subst > term > term)] : (((X55 @ sh @ one)) != ((sub @ X54 @ (push @ one @ sh))))) & (induction2lem_gthm = $true) & (axapp = $true) & (! [X0 : term,X3 : term,X1 : term,X2 : term] : ((X0 = X2) | (((ap @ (sub @ X2 @ id) @ X1)) != ((ap @ (sub @ X0 @ id) @ X3)))) <=> (hoasapinj1 = $true)) & ((induction2lem_lthm = $true) <=> (! [X263 : term,X262 : subst] : (((sub @ one @ (push @ X263 @ X262))) = X263) => (! [X265 : term,X264 : subst] : (((comp @ sh @ (push @ X265 @ X264))) = X264) => (! [X266 : subst] : (((comp @ X266 @ id)) = X266) => (! [X267 : term,X268 : (term > $o)] : ((((X268 @ X267)) = $true) | ~ ! [X272 : term] : ((((X268 @ (lam @ X272))) = $true) | (((X268 @ X272)) != $true)) | ~ ! [X271 : term] : (($true != ((var @ X271))) | (((X268 @ X271)) = $true)) | ~ ! [X270 : term,X269 : term] : ((((X268 @ X269)) != $true) | (((X268 @ (ap @ X269 @ X270))) = $true) | (((X268 @ X270)) != $true))) => ! [X275 : subst,X274 : term,X273 : (term > $o)] : (~ ! [X276 : term] : ((((X273 @ (sub @ X276 @ X275))) = $true) | (((var @ X276)) != $true)) | ($true = ((X273 @ (sub @ X274 @ X275)))) | ~ ! [X277 : term] : ((((X273 @ (lam @ X277))) = $true) | ~ ! [X278 : term] : ((((X273 @ (sub @ X277 @ (push @ X278 @ id)))) = $true) | ($true != ((X273 @ X278))))) | ~ ! [X280 : term,X279 : term] : ((((X273 @ X279)) != $true) | (((X273 @ X280)) != $true) | ($true = ((X273 @ (ap @ X279 @ X280))))))))))) & (pushprop = $true) & ((hoasapinj2 = $true) <=> ! [X415 : term,X413 : term,X414 : term,X416 : term] : ((X413 = X414) | (((ap @ (sub @ X415 @ id) @ X413)) != ((ap @ (sub @ X416 @ id) @ X414))))) & ((! [X13 : term,X15 : term,X12 : term,X14 : term] : ((X12 = X14) | (((ap @ X13 @ X14)) != ((ap @ X15 @ X12)))) => ! [X19 : term,X17 : term,X18 : term,X16 : term] : ((X16 = X19) | (((ap @ (sub @ X17 @ id) @ X19)) != ((ap @ (sub @ X18 @ id) @ X16))))) <=> (hoasapinj2_lthm = $true)) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem2_lthm = $true) & ((axidr = $true) <=> ! [X519 : subst] : (((comp @ X519 @ id)) = X519)) & (! [X523 : term,X522 : (term > $o),X524 : (subst > term > subst > $o)] : (~ ! [X532 : subst,X531 : term,X530 : subst,X529 : subst] : ((((X524 @ (comp @ X532 @ X530) @ (sub @ X531 @ X530) @ X529)) != $true) | (((X524 @ X532 @ X531 @ (comp @ X530 @ X529))) = $true)) | ~ ! [X539 : term] : (((X524 @ id @ X539 @ id)) = ((X522 @ X539))) | (((X522 @ (lam @ X523))) = $true) | ~ ! [X534 : (subst > term > term)] : (~ ! [X535 : term] : ((((X524 @ id @ (X534 @ id @ X535) @ id)) = $true) | ($true != ((X524 @ id @ X535 @ id)))) | ~ ! [X538 : subst,X537 : term,X536 : subst] : (((X534 @ (comp @ X536 @ X538) @ (sub @ X537 @ X538))) = ((sub @ (X534 @ X536 @ X537) @ X538))) | (((X524 @ id @ (lam @ (X534 @ sh @ one)) @ id)) = $true)) | ~ ! [X527 : term,X525 : subst,X526 : subst,X528 : subst] : (($true != ((X524 @ X525 @ X527 @ (comp @ X526 @ X528)))) | (((X524 @ (comp @ X525 @ X526) @ (sub @ X527 @ X526) @ X528)) = $true)) | ~ ! [X533 : term] : ((((X522 @ (sub @ X523 @ (push @ X533 @ id)))) = $true) | ($true != ((X522 @ X533))))) <=> (hoasinduction_lem3v2 = $true)) & (induction2_gthm = $true) & (! [X399 : subst,X401 : (term > $o),X402 : term,X400 : (term > $o)] : ((((X401 @ X402)) != $true) | ($true = ((X400 @ one))) | ~ ! [X403 : term] : (((X400 @ X403)) = ((X401 @ (sub @ X403 @ (push @ X402 @ X399)))))) <=> (pushprop_lem1v2 = $true)) & (! [X409 : term,X408 : subst] : (((comp @ sh @ (push @ X409 @ X408))) = X408) <=> (axshiftcons = $true)) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((pushprop_lem1 = $true) <=> ! [X163 : (term > $o),X162 : term,X165 : term,X161 : (term > $o),X164 : subst] : ((((X161 @ X162)) != $true) | (((X163 @ (sub @ X162 @ (push @ X165 @ X164)))) = $true))) & ((hoasinduction_lem3v2a_lthm = $true) <=> (! [X127 : term] : ~ ! [X128 : (subst > term > term)] : ~ ! [X129 : subst,X130 : term] : (((X128 @ X129 @ X130)) = ((sub @ X127 @ (push @ X130 @ X129)))) => (! [X131 : term] : (((sub @ X131 @ id)) = X131) => ! [X134 : term,X133 : (term > $o),X132 : (subst > term > subst > $o)] : (~ ! [X136 : (subst > term > term)] : (~ ! [X140 : term] : ((((X132 @ id @ (X136 @ id @ X140) @ id)) = $true) | ($true != ((X132 @ id @ X140 @ id)))) | ~ ! [X139 : subst,X137 : term,X138 : subst] : (((X136 @ (comp @ X139 @ X138) @ (sub @ X137 @ X138))) = ((sub @ (X136 @ X139 @ X137) @ X138))) | (((X132 @ id @ (lam @ (X136 @ sh @ one)) @ id)) = $true)) | (((X133 @ (lam @ X134))) = $true) | ~ ! [X135 : term] : (((X133 @ X135)) = ((X132 @ id @ X135 @ id))) | ~ ! [X141 : term] : (($true = ((X133 @ (sub @ X134 @ (push @ X141 @ id))))) | (((X133 @ X141)) != $true)))))) & (one = ((d2term @ d_one))) & (hoasapnotvar = $true) & (hoasinduction_lem1v2 = $true) & (hoasinduction_lem3b_gthm = $true) & (! [X521 : term,X520 : term] : ((((lam @ X521)) != ((lam @ X520))) | (X520 = X521)) <=> (laminj = $true)) & (! [X75 : term,X76 : (term > $o)] : (~ ! [X80 : term,X79 : term] : ((((X76 @ (ap @ X79 @ X80))) = $true) | ($true != ((X76 @ X80))) | (((X76 @ X79)) != $true)) | (((X76 @ X75)) = $true) | ~ ! [X77 : term] : ((((X76 @ X77)) = $true) | (((var @ X77)) != $true)) | ~ ! [X78 : term] : ((((X76 @ (lam @ X78))) = $true) | ($true != ((X76 @ X78))))) <=> (induction = $true)) & (! [X358 : term,X361 : term,X359 : term,X360 : term] : ((X359 = X360) | (((ap @ X359 @ X361)) != ((ap @ X360 @ X358)))) <=> (apinj1 = $true)) & (apnotvar = $true) & ((hoasinduction_lem3aa = $true) <=> ! [X458 : term,X457 : (subst > term > subst > $o)] : (~ ! [X459 : (subst > term > term)] : ((((X457 @ id @ (lam @ (X459 @ sh @ one)) @ id)) = $true) | ~ ! [X460 : term] : (($true != ((X457 @ id @ X460 @ id))) | (((X457 @ id @ (X459 @ id @ X460) @ id)) = $true)) | ~ ! [X463 : subst,X461 : term,X462 : subst] : (((X459 @ (comp @ X463 @ X462) @ (sub @ X461 @ X462))) = ((sub @ (X459 @ X463 @ X461) @ X462)))) | (((X457 @ id @ (lam @ (sub @ X458 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X464 : term] : ((((X457 @ id @ X464 @ id)) != $true) | (((X457 @ id @ (sub @ X458 @ (push @ X464 @ id)) @ id)) = $true)))) & (pushprop_lem1v2_gthm = $true) & ((hoasinduction_lem2 = $true) <=> ! [X335 : (subst > term > subst > $o),X337 : term,X336 : term] : (~ ! [X341 : term,X339 : subst,X338 : subst,X340 : subst] : ((((X335 @ (comp @ X339 @ X338) @ (sub @ X341 @ X338) @ X340)) = $true) | ($true != ((X335 @ X339 @ X341 @ (comp @ X338 @ X340))))) | ~ ! [X346 : term,X347 : term] : ((((X335 @ id @ X347 @ id)) != $true) | (((X335 @ id @ (ap @ (sub @ X346 @ id) @ X347) @ id)) = $true) | ($true != ((X335 @ id @ X346 @ id)))) | ($true != ((X335 @ id @ X336 @ id))) | ($true != ((X335 @ id @ X337 @ id))) | ~ ! [X344 : term,X343 : subst,X345 : subst,X342 : subst] : (($true != ((X335 @ (comp @ X343 @ X345) @ (sub @ X344 @ X345) @ X342))) | ($true = ((X335 @ X343 @ X344 @ (comp @ X345 @ X342))))) | (((X335 @ id @ (ap @ X337 @ X336) @ id)) = $true))) & ((axvarid = $true) <=> ! [X53 : term] : (((sub @ X53 @ id)) = X53)) & ((hoasinduction_lem3_lthm = $true) <=> (! [X425 : term] : (((sub @ X425 @ id)) = X425) => (! [X426 : (subst > term > subst > $o),X427 : term] : (($true = ((X426 @ id @ (lam @ (sub @ X427 @ (push @ one @ sh))) @ id))) | ~ ! [X428 : (subst > term > term)] : (($true = ((X426 @ id @ (lam @ (X428 @ sh @ one)) @ id))) | ~ ! [X429 : term] : (($true != ((X426 @ id @ X429 @ id))) | (((X426 @ id @ (X428 @ id @ X429) @ id)) = $true)) | ~ ! [X430 : subst,X432 : subst,X431 : term] : (((sub @ (X428 @ X432 @ X431) @ X430)) = ((X428 @ (comp @ X432 @ X430) @ (sub @ X431 @ X430))))) | ~ ! [X433 : term] : (($true != ((X426 @ id @ X433 @ id))) | (((X426 @ id @ (sub @ X427 @ (push @ X433 @ id)) @ id)) = $true))) => ! [X434 : term,X435 : (subst > term > subst > $o)] : ((((X435 @ id @ (lam @ X434) @ id)) = $true) | ~ ! [X447 : subst,X446 : subst,X448 : term,X449 : subst] : ((((X435 @ (comp @ X446 @ X449) @ (sub @ X448 @ X449) @ X447)) != $true) | (((X435 @ X446 @ X448 @ (comp @ X449 @ X447))) = $true)) | ~ ! [X441 : (subst > term > term)] : (~ ! [X445 : term] : ((((X435 @ id @ X445 @ id)) != $true) | (((X435 @ id @ (X441 @ id @ X445) @ id)) = $true)) | ~ ! [X443 : term,X444 : subst,X442 : subst] : (((X441 @ (comp @ X442 @ X444) @ (sub @ X443 @ X444))) = ((sub @ (X441 @ X442 @ X443) @ X444))) | ($true = ((X435 @ id @ (lam @ (X441 @ sh @ one)) @ id)))) | ~ ! [X439 : subst,X437 : subst,X438 : subst,X436 : term] : (($true != ((X435 @ X438 @ X436 @ (comp @ X439 @ X437)))) | ($true = ((X435 @ (comp @ X438 @ X439) @ (sub @ X436 @ X439) @ X437)))) | ~ ! [X440 : term] : (($true = ((X435 @ id @ (sub @ X434 @ (push @ X440 @ id)) @ id))) | (((X435 @ id @ X440 @ id)) != $true)))))) & (! [X22 : (term > $o),X20 : term,X21 : subst] : ~ ! [X23 : (term > $o)] : ~ ! [X24 : term] : (((X23 @ X24)) = ((X22 @ (sub @ X24 @ (push @ X20 @ X21))))) <=> (pushprop_lem0 = $true)) & (ulamvarsh = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((hoasinduction_lem3a_lthm = $true) <=> (! [X189 : term] : (((sub @ X189 @ id)) = X189) => (! [X191 : (subst > term > subst > $o),X190 : term] : ((((X191 @ id @ (lam @ (sub @ X190 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X197 : term] : ((((X191 @ id @ (sub @ X190 @ (push @ X197 @ id)) @ id)) = $true) | ($true != ((X191 @ id @ X197 @ id)))) | ~ ! [X192 : (subst > term > term)] : (($true = ((X191 @ id @ (lam @ (X192 @ sh @ one)) @ id))) | ~ ! [X196 : term,X194 : subst,X195 : subst] : (((X192 @ (comp @ X195 @ X194) @ (sub @ X196 @ X194))) = ((sub @ (X192 @ X195 @ X196) @ X194))) | ~ ! [X193 : term] : ((((X191 @ id @ X193 @ id)) != $true) | (((X191 @ id @ (X192 @ id @ X193) @ id)) = $true)))) => ! [X198 : (subst > term > subst > $o),X199 : term] : (($true = ((X198 @ id @ (lam @ X199) @ id))) | ~ ! [X200 : term] : ((((X198 @ id @ X200 @ id)) != $true) | (((X198 @ id @ (sub @ X199 @ (push @ X200 @ id)) @ id)) = $true)) | ~ ! [X201 : (subst > term > term)] : (~ ! [X205 : term] : ((((X198 @ id @ (X201 @ id @ X205) @ id)) = $true) | (((X198 @ id @ X205 @ id)) != $true)) | ~ ! [X202 : subst,X203 : term,X204 : subst] : (((sub @ (X201 @ X204 @ X203) @ X202)) = ((X201 @ (comp @ X204 @ X202) @ (sub @ X203 @ X202)))) | ($true = ((X198 @ id @ (lam @ (X201 @ sh @ one)) @ id)))))))) & (((var @ (d2term @ d_one))) != $true) & ! [X74 : d_subst] : (d_id = X74) & ((termmset = $true) <=> ! [X290 : term] : (((sub @ X290 @ id)) = X290)) & ((! [X178 : term] : (((sub @ X178 @ id)) = X178) => ! [X179 : term] : (((sub @ X179 @ id)) = X179)) <=> (termmset_lthm = $true)) & (lamnotap != $true) & ((hoasinduction_lem3 = $true) <=> ! [X320 : term,X319 : (subst > term > subst > $o)] : (~ ! [X325 : (subst > term > term)] : (~ ! [X328 : term,X326 : subst,X327 : subst] : (((sub @ (X325 @ X326 @ X328) @ X327)) = ((X325 @ (comp @ X326 @ X327) @ (sub @ X328 @ X327)))) | (((X319 @ id @ (lam @ (X325 @ sh @ one)) @ id)) = $true) | ~ ! [X329 : term] : (($true = ((X319 @ id @ (X325 @ id @ X329) @ id))) | (((X319 @ id @ X329 @ id)) != $true))) | ~ ! [X331 : subst,X333 : subst,X334 : term,X332 : subst] : ((((X319 @ (comp @ X332 @ X333) @ (sub @ X334 @ X333) @ X331)) != $true) | (((X319 @ X332 @ X334 @ (comp @ X333 @ X331))) = $true)) | (((X319 @ id @ (lam @ X320) @ id)) = $true) | ~ ! [X330 : term] : ((((X319 @ id @ X330 @ id)) != $true) | (((X319 @ id @ (sub @ X320 @ (push @ X330 @ id)) @ id)) = $true)) | ~ ! [X323 : subst,X322 : subst,X321 : term,X324 : subst] : ((((X319 @ X324 @ X321 @ (comp @ X322 @ X323))) != $true) | ($true = ((X319 @ (comp @ X324 @ X322) @ (sub @ X321 @ X322) @ X323)))))) & (pushprop_lem2v2 = $true) & (pushprop_lem1_gthm = $true) & (! [X350 : (subst > term > subst > $o),X351 : term] : (~ ! [X357 : term] : ((((X350 @ id @ X357 @ id)) != $true) | (((X350 @ id @ (sub @ X351 @ (push @ X357 @ id)) @ id)) = $true)) | (((X350 @ id @ (lam @ X351) @ id)) = $true) | ~ ! [X352 : (subst > term > term)] : (~ ! [X356 : term] : ((((X350 @ id @ X356 @ id)) != $true) | (((X350 @ id @ (X352 @ id @ X356) @ id)) = $true)) | (((X350 @ id @ (lam @ (X352 @ sh @ one)) @ id)) = $true) | ~ ! [X355 : subst,X354 : subst,X353 : term] : (((X352 @ (comp @ X355 @ X354) @ (sub @ X353 @ X354))) = ((sub @ (X352 @ X355 @ X353) @ X354))))) <=> (hoasinduction_lem3a = $true)) & (! [X423 : subst,X424 : term] : (((sub @ one @ (push @ X424 @ X423))) = X424) <=> (axvarcons = $true)) & (hoaslamnotvar_gthm = $true) & (! [X142 : (subst > term > subst > $o)] : ~ ! [X143 : (term > $o)] : ~ ! [X144 : term] : (((X142 @ id @ X144 @ id)) = ((X143 @ X144))) <=> (hoasinduction_lem0_lthm = $true)) & ((shinj = $true) <=> ! [X455 : term,X456 : term] : ((((sub @ X455 @ sh)) != ((sub @ X456 @ sh))) | (X455 = X456))) & (hoasinduction_lem2v2_gthm = $true) & (hoasapnotvar_lthm = $true) & (termmset_gthm = $true) & (pushprop_lthm = $true) & ((! [X37 : term,X38 : subst] : (((sub @ one @ (push @ X37 @ X38))) = X37) => (! [X39 : subst,X40 : term] : (((comp @ sh @ (push @ X40 @ X39))) = X39) => (! [X41 : term,X42 : term] : ((X41 = X42) | (((lam @ X41)) != ((lam @ X42)))) => ! [X44 : (subst > term > term),X46 : (subst > term > term),X43 : term,X45 : subst] : ((((X44 @ X45 @ X43)) = ((X46 @ X45 @ X43))) | (((lam @ (X44 @ sh @ one))) != ((lam @ (X46 @ sh @ one)))) | ~ ! [X51 : term,X52 : subst,X50 : subst] : (((X46 @ (comp @ X52 @ X50) @ (sub @ X51 @ X50))) = ((sub @ (X46 @ X52 @ X51) @ X50))) | ~ ! [X47 : term,X49 : subst,X48 : subst] : (((X44 @ (comp @ X48 @ X49) @ (sub @ X47 @ X49))) = ((sub @ (X44 @ X48 @ X47) @ X49))))))) <=> (hoaslaminj_lthm = $true)) & ((induction2_lthm = $true) <=> (! [X206 : term] : (((sub @ X206 @ id)) = X206) => (! [X207 : subst,X209 : (term > $o),X208 : term] : ((((X209 @ (sub @ X208 @ X207))) = $true) | ~ ! [X213 : term,X212 : term] : ((((X209 @ X212)) != $true) | (((X209 @ X213)) != $true) | (((X209 @ (ap @ X212 @ X213))) = $true)) | ~ ! [X214 : term] : ((((X209 @ (sub @ X214 @ X207))) = $true) | (((var @ X214)) != $true)) | ~ ! [X210 : term] : (~ ! [X211 : term] : (($true = ((X209 @ (sub @ X210 @ (push @ X211 @ id))))) | (((X209 @ X211)) != $true)) | (((X209 @ (lam @ X210))) = $true))) => ! [X216 : term,X215 : (term > $o)] : (~ ! [X217 : term,X218 : term] : (($true != ((X215 @ X218))) | (((X215 @ (ap @ X218 @ X217))) = $true) | (((X215 @ X217)) != $true)) | ~ ! [X219 : term] : ((((X215 @ X219)) = $true) | (((var @ X219)) != $true)) | (((X215 @ X216)) = $true) | ~ ! [X220 : term] : (~ ! [X221 : term] : ((((X215 @ X221)) != $true) | (((X215 @ (sub @ X220 @ (push @ X221 @ id)))) = $true)) | (((X215 @ (lam @ X220))) = $true)))))) & (axvarshift = $true) & (substmonoid_gthm = $true) & (! [X315 : subst,X314 : (term > $o),X316 : term] : ~ ! [X317 : (term > $o)] : ~ ! [X318 : term] : (((X314 @ (sub @ X318 @ (push @ X316 @ X315)))) = ((X317 @ X318))) <=> (pushprop_lem0_lthm = $true)) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (! [X116 : subst,X117 : term,X115 : (term > $o)] : (~ ! [X118 : term] : ((((X115 @ (sub @ X118 @ X116))) = $true) | ($true != ((var @ X118)))) | (((X115 @ (sub @ X117 @ X116))) = $true) | ~ ! [X121 : term,X122 : term] : ((((X115 @ X121)) != $true) | (((X115 @ (ap @ X122 @ X121))) = $true) | (((X115 @ X122)) != $true)) | ~ ! [X119 : term] : (~ ! [X120 : term] : (($true != ((X115 @ X120))) | (((X115 @ (sub @ X119 @ (push @ X120 @ id)))) = $true)) | ($true = ((X115 @ (lam @ X119)))))) <=> (induction2lem = $true)) & (! [X305 : (subst > term > term),X304 : (subst > term > term),X307 : term,X306 : subst] : (~ ! [X312 : subst,X311 : term,X313 : subst] : (((sub @ (X305 @ X313 @ X311) @ X312)) = ((X305 @ (comp @ X313 @ X312) @ (sub @ X311 @ X312)))) | (((lam @ (X305 @ sh @ one))) != ((lam @ (X304 @ sh @ one)))) | (((X305 @ X306 @ X307)) = ((X304 @ X306 @ X307))) | ~ ! [X309 : subst,X310 : subst,X308 : term] : (((X304 @ (comp @ X310 @ X309) @ (sub @ X308 @ X309))) = ((sub @ (X304 @ X310 @ X308) @ X309)))) <=> (hoaslaminj = $true)) & (pushprop_lthm_orig = $true) & (sh = ((d2subst @ d_id))) & (lamnotvar = $true) & ((hoasinduction_lem2v2 = $true) <=> ! [X147 : term,X148 : term,X145 : (subst > term > subst > $o),X146 : (term > $o)] : ((((X146 @ (ap @ X147 @ X148))) = $true) | (((X146 @ X147)) != $true) | ~ ! [X150 : term,X151 : subst,X153 : subst,X152 : subst] : (($true != ((X145 @ (comp @ X153 @ X151) @ (sub @ X150 @ X151) @ X152))) | (((X145 @ X153 @ X150 @ (comp @ X151 @ X152))) = $true)) | (((X146 @ X148)) != $true) | ~ ! [X157 : subst,X156 : term,X159 : subst,X158 : subst] : ((((X145 @ (comp @ X157 @ X158) @ (sub @ X156 @ X158) @ X159)) = $true) | (((X145 @ X157 @ X156 @ (comp @ X158 @ X159))) != $true)) | ~ ! [X149 : term] : (((X145 @ id @ X149 @ id)) = ((X146 @ X149))) | ~ ! [X154 : term,X155 : term] : (($true = ((X145 @ id @ (ap @ (sub @ X154 @ id) @ X155) @ id))) | (((X145 @ id @ X154 @ id)) != $true) | (((X145 @ id @ X155 @ id)) != $true)))) & (! [X81 : term] : ~ ! [X82 : (subst > term > term)] : ~ ! [X83 : subst,X84 : term] : (((sub @ X81 @ (push @ X84 @ X83))) = ((X82 @ X83 @ X84))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & (ulamvar1 != $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoasinduction_lem3v2_gthm = $true) & (hoaslamnotvar_lthm = $true) & ((apinj2 = $true) <=> ! [X405 : term,X404 : term,X406 : term,X407 : term] : ((((ap @ X404 @ X407)) != ((ap @ X405 @ X406))) | (X406 = X407))) & ((! [X281 : term] : (((sub @ X281 @ id)) = X281) => (! [X283 : term,X285 : term,X284 : term,X282 : term] : ((((ap @ X283 @ X285)) != ((ap @ X284 @ X282))) | (X283 = X284)) => ! [X286 : term,X289 : term,X288 : term,X287 : term] : ((X286 = X288) | (((ap @ (sub @ X286 @ id) @ X287)) != ((ap @ (sub @ X288 @ id) @ X289)))))) <=> (hoasapinj1_lthm = $true)) & ((hoasinduction_lem3aaa = $true) <=> ! [X25 : term,X26 : (subst > term > subst > $o)] : (~ ! [X29 : (subst > term > term),X28 : term] : ((((X26 @ id @ (lam @ (X29 @ sh @ one)) @ id)) = $true) | ~ ! [X36 : term] : ((((X26 @ id @ (X29 @ id @ X36) @ id)) = $true) | (((X26 @ id @ X36 @ id)) != $true)) | ~ ! [X30 : term,X31 : subst,X32 : subst] : (((sub @ (sub @ X28 @ (push @ X30 @ X31)) @ X32)) = ((sub @ (X29 @ X31 @ X30) @ X32))) | ~ ! [X33 : subst,X35 : term,X34 : subst] : (((X29 @ (comp @ X33 @ X34) @ (sub @ X35 @ X34))) = ((sub @ X28 @ (push @ (sub @ X35 @ X34) @ (comp @ X33 @ X34)))))) | (((X26 @ id @ (lam @ (sub @ X25 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X27 : term] : (($true = ((X26 @ id @ (sub @ X25 @ (push @ X27 @ id)) @ id))) | (((X26 @ id @ X27 @ id)) != $true)))) & ((! [X362 : (subst > term > subst > $o)] : ~ ! [X363 : (term > $o)] : ~ ! [X364 : term] : (((X362 @ id @ X364 @ id)) = ((X363 @ X364))) => (! [X365 : (term > $o),X366 : term] : ((((X365 @ X366)) = $true) | ~ ! [X367 : term] : (($true != ((var @ X367))) | (((X365 @ X367)) = $true)) | ~ ! [X369 : term,X368 : term] : ((((X365 @ X368)) != $true) | (((X365 @ X369)) != $true) | (((X365 @ (ap @ X369 @ X368))) = $true)) | ~ ! [X370 : term] : (~ ! [X371 : term] : ((((X365 @ X371)) != $true) | (((X365 @ (sub @ X370 @ (push @ X371 @ id)))) = $true)) | (((X365 @ (lam @ X370))) = $true))) => (! [X372 : term] : (((sub @ X372 @ id)) = X372) => (! [X375 : (subst > term > subst > $o),X374 : (term > $o),X373 : term] : (~ ! [X377 : (subst > term > term)] : ((((X375 @ id @ (lam @ (X377 @ sh @ one)) @ id)) = $true) | ~ ! [X379 : subst,X380 : term,X378 : subst] : (((sub @ (X377 @ X379 @ X380) @ X378)) = ((X377 @ (comp @ X379 @ X378) @ (sub @ X380 @ X378)))) | ~ ! [X381 : term] : (($true = ((X375 @ id @ (X377 @ id @ X381) @ id))) | (((X375 @ id @ X381 @ id)) != $true))) | ($true = ((X374 @ (lam @ X373)))) | ~ ! [X376 : term] : (($true = ((X374 @ (sub @ X373 @ (push @ X376 @ id))))) | ($true != ((X374 @ X376)))) | ~ ! [X382 : term] : (((X375 @ id @ X382 @ id)) = ((X374 @ X382)))) => ! [X383 : term,X384 : (subst > term > subst > $o)] : (($true = ((X384 @ id @ X383 @ id))) | ~ ! [X387 : (subst > term > term)] : ((((X384 @ id @ (lam @ (X387 @ sh @ one)) @ id)) = $true) | ~ ! [X391 : term] : (($true != ((X384 @ id @ X391 @ id))) | (((X384 @ id @ (X387 @ id @ X391) @ id)) = $true)) | ~ ! [X389 : subst,X390 : term,X388 : subst] : (((X387 @ (comp @ X389 @ X388) @ (sub @ X390 @ X388))) = ((sub @ (X387 @ X389 @ X390) @ X388)))) | ~ ! [X386 : term,X385 : term] : (($true != ((X384 @ id @ X385 @ id))) | (((X384 @ id @ (ap @ (sub @ X385 @ id) @ X386) @ id)) = $true) | (((X384 @ id @ X386 @ id)) != $true))))))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (hoaslamnotvar = $true) & ((! [X300 : subst] : (((comp @ id @ X300)) = X300) => (! [X301 : subst] : (((comp @ X301 @ id)) = X301) => (! [X302 : subst] : (((comp @ X302 @ id)) = X302) & ! [X303 : subst] : (((comp @ id @ X303)) = X303)))) <=> (substmonoid_lthm = $true)) & (axmap = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (! [X4 : (subst > term > subst > $o),X5 : term] : (~ ! [X6 : (subst > term > term)] : (~ ! [X7 : term] : (($true = ((X4 @ id @ (X6 @ id @ X7) @ id))) | (((X4 @ id @ X7 @ id)) != $true)) | (((X4 @ id @ (lam @ (X6 @ sh @ one)) @ id)) = $true) | ~ ! [X10 : term,X8 : subst,X9 : subst] : (((sub @ (X6 @ X9 @ X10) @ X8)) = ((X6 @ (comp @ X9 @ X8) @ (sub @ X10 @ X8))))) | ($true = ((X4 @ id @ (lam @ (sub @ X5 @ (push @ one @ sh))) @ id))) | ~ ! [X11 : term] : ((((X4 @ id @ (sub @ X5 @ (push @ X11 @ id)) @ id)) = $true) | (((X4 @ id @ X11 @ id)) != $true))) <=> (hoasinduction_lem3aa_lthm = $true)) & (hoasinduction_lem1 = $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)))))))))))) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ! [X160 : d_term] : (d_one = X160) & (hoasapinj2_gthm = $true) & (pushprop_lem3v2 = $true) & (axclos = $true) & (hoasapinj1_gthm = $true) & ((induction2 = $true) <=> ! [X106 : (term > $o),X107 : term] : (~ ! [X108 : term] : (($true = ((X106 @ (lam @ X108)))) | ~ ! [X109 : term] : ((((X106 @ (sub @ X108 @ (push @ X109 @ id)))) = $true) | (((X106 @ X109)) != $true))) | ~ ! [X111 : term,X110 : term] : ((((X106 @ (ap @ X110 @ X111))) = $true) | (((X106 @ X110)) != $true) | (((X106 @ X111)) != $true)) | ($true = ((X106 @ X107))) | ~ ! [X112 : term] : ((((var @ X112)) != $true) | (((X106 @ X112)) = $true)))) & (pushprop_lem2v2_gthm = $true) & (pushprop_lem3v2_lthm = $true) & (hoaslaminj_gthm = $true) & (hoasinduction_lem2_gthm = $true) & (ulamvarind = $true) & (! [X419 : term,X417 : (subst > term > term),X418 : term] : (~ ! [X420 : subst,X421 : subst,X422 : term] : (((X417 @ (comp @ X420 @ X421) @ (sub @ X422 @ X421))) = ((sub @ (X417 @ X420 @ X422) @ X421))) | (((lam @ (X417 @ sh @ one))) != ((ap @ (sub @ X419 @ id) @ X418)))) <=> (hoaslamnotap = $true)) & ((! [X466 : (term > $o),X465 : term] : (~ ! [X470 : term] : (~ ! [X471 : term] : ((((X466 @ X471)) != $true) | (((X466 @ (sub @ X470 @ (push @ X471 @ id)))) = $true)) | (((X466 @ (lam @ X470))) = $true)) | ($true = ((X466 @ X465))) | ~ ! [X467 : term] : (($true = ((X466 @ X467))) | (((var @ X467)) != $true)) | ~ ! [X469 : term,X468 : term] : (($true = ((X466 @ (ap @ X468 @ X469)))) | (((X466 @ X468)) != $true) | ($true != ((X466 @ X469))))) => (! [X473 : term,X474 : (subst > term > subst > $o),X472 : term] : (~ ! [X477 : subst,X475 : term,X476 : subst,X478 : subst] : ((((X474 @ (comp @ X476 @ X477) @ (sub @ X475 @ X477) @ X478)) = $true) | ($true != ((X474 @ X476 @ X475 @ (comp @ X477 @ X478))))) | (((X474 @ id @ X472 @ id)) != $true) | (((X474 @ id @ X473 @ id)) != $true) | (((X474 @ id @ (ap @ X472 @ X473) @ id)) = $true) | ~ ! [X484 : term,X483 : term] : ((((X474 @ id @ X483 @ id)) != $true) | (((X474 @ id @ X484 @ id)) != $true) | (((X474 @ id @ (ap @ (sub @ X483 @ id) @ X484) @ id)) = $true)) | ~ ! [X480 : subst,X482 : term,X479 : subst,X481 : subst] : ((((X474 @ X480 @ X482 @ (comp @ X481 @ X479))) = $true) | ($true != ((X474 @ (comp @ X480 @ X481) @ (sub @ X482 @ X481) @ X479))))) => (! [X486 : (subst > term > subst > $o),X485 : term] : (~ ! [X500 : term] : ((((X486 @ id @ X500 @ id)) != $true) | (((X486 @ id @ (sub @ X485 @ (push @ X500 @ id)) @ id)) = $true)) | ~ ! [X491 : (subst > term > term)] : (~ ! [X492 : subst,X493 : subst,X494 : term] : (((sub @ (X491 @ X492 @ X494) @ X493)) = ((X491 @ (comp @ X492 @ X493) @ (sub @ X494 @ X493)))) | ~ ! [X495 : term] : ((((X486 @ id @ X495 @ id)) != $true) | (((X486 @ id @ (X491 @ id @ X495) @ id)) = $true)) | (((X486 @ id @ (lam @ (X491 @ sh @ one)) @ id)) = $true)) | ~ ! [X499 : subst,X496 : term,X498 : subst,X497 : subst] : (($true = ((X486 @ (comp @ X499 @ X497) @ (sub @ X496 @ X497) @ X498))) | (((X486 @ X499 @ X496 @ (comp @ X497 @ X498))) != $true)) | (((X486 @ id @ (lam @ X485) @ id)) = $true) | ~ ! [X488 : term,X490 : subst,X487 : subst,X489 : subst] : (($true = ((X486 @ X489 @ X488 @ (comp @ X490 @ X487)))) | ($true != ((X486 @ (comp @ X489 @ X490) @ (sub @ X488 @ X490) @ X487))))) => ! [X501 : (subst > term > subst > $o),X502 : term] : (~ ! [X514 : subst,X513 : subst,X511 : subst,X512 : term] : ((((X501 @ X514 @ X512 @ (comp @ X513 @ X511))) = $true) | (((X501 @ (comp @ X514 @ X513) @ (sub @ X512 @ X513) @ X511)) != $true)) | ~ ! [X504 : term,X503 : term] : ((((X501 @ id @ (ap @ (sub @ X504 @ id) @ X503) @ id)) = $true) | (((X501 @ id @ X503 @ id)) != $true) | (((X501 @ id @ X504 @ id)) != $true)) | ~ ! [X516 : subst,X517 : subst,X515 : term,X518 : subst] : ((((X501 @ X517 @ X515 @ (comp @ X516 @ X518))) != $true) | (((X501 @ (comp @ X517 @ X516) @ (sub @ X515 @ X516) @ X518)) = $true)) | ~ ! [X505 : term] : ((((X501 @ id @ X505 @ id)) = $true) | ($true != ((var @ (sub @ X505 @ id))))) | ~ ! [X506 : (subst > term > term)] : ((((X501 @ id @ (lam @ (X506 @ sh @ one)) @ id)) = $true) | ~ ! [X509 : subst,X510 : subst,X508 : term] : (((X506 @ (comp @ X510 @ X509) @ (sub @ X508 @ X509))) = ((sub @ (X506 @ X510 @ X508) @ X509))) | ~ ! [X507 : term] : ((((X501 @ id @ (X506 @ id @ X507) @ id)) = $true) | ($true != ((X501 @ id @ X507 @ id))))) | (((X501 @ id @ X502 @ id)) = $true))))) <=> (hoasinduction_lthm = $true)) & ((hoasinduction_lem3v2a = $true) <=> ! [X168 : term,X169 : (term > $o),X170 : (subst > term > subst > $o)] : (~ ! [X177 : term] : (((X169 @ X177)) = ((X170 @ id @ X177 @ id))) | ~ ! [X171 : term] : (($true != ((X169 @ X171))) | ($true = ((X169 @ (sub @ X168 @ (push @ X171 @ id)))))) | (((X169 @ (lam @ X168))) = $true) | ~ ! [X172 : (subst > term > term)] : (~ ! [X173 : subst,X174 : term,X175 : subst] : (((sub @ (X172 @ X173 @ X174) @ X175)) = ((X172 @ (comp @ X173 @ X175) @ (sub @ X174 @ X175)))) | (((X170 @ id @ (lam @ (X172 @ sh @ one)) @ id)) = $true) | ~ ! [X176 : term] : ((((X170 @ id @ (X172 @ id @ X176) @ id)) = $true) | (((X170 @ id @ X176 @ id)) != $true))))) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ! [X348 : term] : ? [X349 : d_term] : (((d2term @ X349)) = X348) & (! [X180 : term,X181 : (subst > term > subst > $o)] : (~ ! [X188 : term,X187 : term] : (($true != ((X181 @ id @ X187 @ id))) | (((X181 @ id @ (ap @ (sub @ X187 @ id) @ X188) @ id)) = $true) | ($true != ((X181 @ id @ X188 @ id)))) | ~ ! [X182 : (subst > term > term)] : (~ ! [X186 : term,X185 : subst,X184 : subst] : (((X182 @ (comp @ X185 @ X184) @ (sub @ X186 @ X184))) = ((sub @ (X182 @ X185 @ X186) @ X184))) | ~ ! [X183 : term] : ((((X181 @ id @ (X182 @ id @ X183) @ id)) = $true) | (((X181 @ id @ X183 @ id)) != $true)) | ($true = ((X181 @ id @ (lam @ (X182 @ sh @ one)) @ id)))) | (((X181 @ id @ X180 @ id)) = $true)) <=> (hoasinduction_no_psi_cond = $true)) & (! [X123 : term] : ~ ! [X124 : (subst > term > term)] : ~ ! [X125 : subst,X126 : term] : (((sub @ X123 @ (push @ X126 @ X125))) = ((X124 @ X125 @ X126))) <=> (hoasinduction_lem3v2_f = $true)) & ((hoasinduction_lem0 = $true) <=> ! [X410 : (subst > term > subst > $o)] : ~ ! [X411 : (term > $o)] : ~ ! [X412 : term] : (((X410 @ id @ X412 @ id)) = ((X411 @ X412)))) & (hoasinduction_gthm = $true) & (hoaslamnotap_lthm = $true) & (hoasinduction_lem3_gthm = $true) & (! [X222 : subst] : (((comp @ id @ X222)) = X222) <=> (axidl = $true)) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ! [X450 : d_term,X451 : d_term] : ((((d2term @ X450)) = ((d2term @ X451))) => (X450 = X451)) & (id = ((d2subst @ d_id)))),
  inference(flattening,[],[f8])).
thf(f10,plain,(
  (hoasinduction_lem3aa != ((!! @ (subst > term > subst > $o) @ (^[Y0 : subst > term > subst > $o]: ((!! @ (subst > term > term) @ (^[Y1 : subst > term > term]: ((!! @ term @ (^[Y2 : term]: (!! @ subst @ (^[Y3 : subst]: (!! @ subst @ (^[Y4 : subst]: ((Y1 @ (comp @ Y3 @ Y4) @ (sub @ Y2 @ Y4)) = (sub @ (Y1 @ Y3 @ Y2) @ Y4)))))))) => ((!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) => (Y0 @ id @ (Y1 @ id @ Y2) @ id)))) => (Y0 @ id @ (hoaslam @ id @ (^[Y2 : subst]: ((^[Y3 : term]: (Y1 @ Y2 @ Y3))))) @ id))))) => (!! @ term @ (^[Y1 : term]: ((!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) => (Y0 @ id @ (sub @ Y1 @ (push @ Y2 @ id)) @ id)))) => (Y0 @ id @ (lam @ (sub @ Y1 @ (push @ one @ sh))) @ id)))))))))),
  inference(flattening,[],[f7])).
thf(f11,plain,(
  (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X85 : term] : ? [X86 : (subst > term > term)] : (((sub @ X85 @ (push @ one @ sh))) = ((X86 @ sh @ one)))) & ! [X453 : d_subst,X452 : d_subst] : ((X452 = X453) | (((d2subst @ X453)) != ((d2subst @ X452)))) & (hoasinduction_lem3a_gthm = $true) & ((axscons = $true) <=> ! [X454 : subst] : (((push @ (sub @ one @ X454) @ (comp @ sh @ X454))) = X454)) & ((pushprop_lem1_lthm = $true) <=> ((! [X298 : term,X295 : subst,X297 : term,X299 : (term > $o),X296 : (term > $o)] : ((((X299 @ X297)) != $true) | (((X296 @ (sub @ X297 @ (push @ X298 @ X295)))) = $true)) | ? [X294 : subst,X293 : term] : (((comp @ sh @ (push @ X293 @ X294))) != X294)) | ? [X292 : term,X291 : subst] : (((sub @ one @ (push @ X292 @ X291))) != X292))) & (pushprop_lem0_gthm = $true) & (hoaslamnotap_gthm = $true) & ((hoasinduction = $true) <=> ! [X57 : (subst > term > subst > $o),X56 : term] : (? [X69 : term,X68 : term] : ((((X57 @ id @ (ap @ (sub @ X69 @ id) @ X68) @ id)) != $true) & ($true = ((X57 @ id @ X68 @ id))) & (((X57 @ id @ X69 @ id)) = $true)) | ? [X60 : subst,X59 : term,X61 : subst,X62 : subst] : ((((X57 @ (comp @ X60 @ X61) @ (sub @ X59 @ X61) @ X62)) != $true) & (((X57 @ X60 @ X59 @ (comp @ X61 @ X62))) = $true)) | ? [X72 : subst,X71 : term,X73 : subst,X70 : subst] : (($true != ((X57 @ X70 @ X71 @ (comp @ X73 @ X72)))) & (((X57 @ (comp @ X70 @ X73) @ (sub @ X71 @ X73) @ X72)) = $true)) | ? [X58 : term] : ((((var @ (sub @ X58 @ id))) = $true) & (((X57 @ id @ X58 @ id)) != $true)) | (((X57 @ id @ X56 @ id)) = $true) | ? [X63 : (subst > term > term)] : (($true != ((X57 @ id @ (lam @ (X63 @ sh @ one)) @ id))) & ! [X64 : term,X66 : subst,X65 : subst] : (((X63 @ (comp @ X65 @ X66) @ (sub @ X64 @ X66))) = ((sub @ (X63 @ X65 @ X64) @ X66))) & ! [X67 : term] : ((((X57 @ id @ X67 @ id)) != $true) | (((X57 @ id @ (X63 @ id @ X67) @ id)) = $true))))) & (axabs = $true) & (pushprop_lem2v2_lthm = $true) & (pushprop_gthm = $true) & (((((! [X244 : (subst > term > subst > $o),X245 : term] : (? [X257 : (subst > term > term)] : (! [X258 : subst,X260 : subst,X259 : term] : (((sub @ (X257 @ X260 @ X259) @ X258)) = ((X257 @ (comp @ X260 @ X258) @ (sub @ X259 @ X258)))) & ! [X261 : term] : ((((X244 @ id @ (X257 @ id @ X261) @ id)) = $true) | (((X244 @ id @ X261 @ id)) != $true)) & (((X244 @ id @ (lam @ (X257 @ sh @ one)) @ id)) != $true)) | ? [X250 : subst,X253 : subst,X252 : term,X251 : subst] : (($true != ((X244 @ X253 @ X252 @ (comp @ X251 @ X250)))) & ($true = ((X244 @ (comp @ X253 @ X251) @ (sub @ X252 @ X251) @ X250)))) | ? [X247 : term,X249 : subst,X248 : subst,X246 : subst] : ((((X244 @ X246 @ X247 @ (comp @ X249 @ X248))) = $true) & (((X244 @ (comp @ X246 @ X249) @ (sub @ X247 @ X249) @ X248)) != $true)) | ? [X255 : term,X254 : term] : ((((X244 @ id @ (ap @ (sub @ X254 @ id) @ X255) @ id)) != $true) & (((X244 @ id @ X255 @ id)) = $true) & (((X244 @ id @ X254 @ id)) = $true)) | ? [X256 : term] : ((((X244 @ id @ X256 @ id)) != $true) & (((var @ (sub @ X256 @ id))) = $true)) | ($true = ((X244 @ id @ X245 @ id)))) | ? [X234 : (term > $o),X236 : (subst > term > subst > $o),X235 : term] : ((((X234 @ (lam @ X235))) != $true) & ! [X237 : term] : (((X234 @ X237)) = ((X236 @ id @ X237 @ id))) & ! [X243 : term] : ((((X234 @ X243)) != $true) | ($true = ((X234 @ (sub @ X235 @ (push @ X243 @ id)))))) & ! [X238 : (subst > term > term)] : (? [X239 : term] : ((((X236 @ id @ (X238 @ id @ X239) @ id)) != $true) & (((X236 @ id @ X239 @ id)) = $true)) | ? [X242 : term,X241 : subst,X240 : subst] : (((sub @ (X238 @ X241 @ X242) @ X240)) != ((X238 @ (comp @ X241 @ X240) @ (sub @ X242 @ X240)))) | ($true = ((X236 @ id @ (lam @ (X238 @ sh @ one)) @ id)))))) | ? [X233 : term] : (((sub @ X233 @ id)) != X233)) | ? [X226 : (term > $o),X227 : term] : (! [X230 : term,X229 : term] : ((((X226 @ (ap @ X230 @ X229))) = $true) | (((X226 @ X229)) != $true) | (((X226 @ X230)) != $true)) & ! [X228 : term] : ((((var @ X228)) != $true) | (((X226 @ X228)) = $true)) & ! [X231 : term] : ((((X226 @ (lam @ X231))) = $true) | ? [X232 : term] : ((((X226 @ (sub @ X231 @ (push @ X232 @ id)))) != $true) & (((X226 @ X232)) = $true))) & (((X226 @ X227)) != $true))) | ? [X223 : (subst > term > subst > $o)] : ! [X224 : (term > $o)] : ? [X225 : term] : (((X224 @ X225)) != ((X223 @ id @ X225 @ id)))) <=> (hoasinduction_lthm_3 = $true)) & (hoasinduction_lem1_lthm = $true) & ! [X166 : subst] : ? [X167 : d_subst] : (((d2subst @ X167)) = X166) & (axassoc = $true) & ((? [X393 : term,X392 : subst] : (((sub @ one @ (push @ X393 @ X392))) != X393) | ! [X395 : (term > $o),X397 : term,X396 : subst,X394 : (term > $o)] : (? [X398 : term] : (((X395 @ (sub @ X398 @ (push @ X397 @ X396)))) != ((X394 @ X398))) | ($true != ((X395 @ X397))) | ($true = ((X394 @ one))))) <=> (pushprop_lem1v2_lthm = $true)) & (hoasinduction_lem1_gthm = $true) & ((? [X87 : term] : (((sub @ X87 @ id)) != X87) | ! [X88 : (subst > term > subst > $o),X90 : (term > $o),X89 : term] : (? [X100 : term] : (((X90 @ X100)) != ((X88 @ id @ X100 @ id))) | ? [X91 : term,X94 : subst,X93 : subst,X92 : subst] : (($true != ((X88 @ (comp @ X93 @ X92) @ (sub @ X91 @ X92) @ X94))) & (((X88 @ X93 @ X91 @ (comp @ X92 @ X94))) = $true)) | ? [X103 : subst,X104 : term,X101 : subst,X102 : subst] : ((((X88 @ (comp @ X102 @ X101) @ (sub @ X104 @ X101) @ X103)) = $true) & (((X88 @ X102 @ X104 @ (comp @ X101 @ X103))) != $true)) | ? [X105 : term] : ((((X90 @ X105)) = $true) & (((X90 @ (sub @ X89 @ (push @ X105 @ id)))) != $true)) | ? [X95 : (subst > term > term)] : (! [X96 : term] : (($true = ((X88 @ id @ (X95 @ id @ X96) @ id))) | (((X88 @ id @ X96 @ id)) != $true)) & (((X88 @ id @ (lam @ (X95 @ sh @ one)) @ id)) != $true) & ! [X98 : term,X99 : subst,X97 : subst] : (((sub @ (X95 @ X97 @ X98) @ X99)) = ((X95 @ (comp @ X97 @ X99) @ (sub @ X98 @ X99))))) | (((X90 @ (lam @ X89))) = $true))) <=> (hoasinduction_lem3v2_lthm = $true)) & ((! [X113 : subst] : (((comp @ X113 @ id)) = X113) & ! [X114 : subst] : (((comp @ id @ X114)) = X114)) <=> (substmonoid = $true)) & ((hoasinduction_lem3b = $true) <=> ! [X54 : term] : ? [X55 : (subst > term > term)] : (((X55 @ sh @ one)) = ((sub @ X54 @ (push @ one @ sh))))) & (induction2lem_gthm = $true) & (axapp = $true) & (! [X0 : term,X3 : term,X1 : term,X2 : term] : ((X0 = X2) | (((ap @ (sub @ X2 @ id) @ X1)) != ((ap @ (sub @ X0 @ id) @ X3)))) <=> (hoasapinj1 = $true)) & ((induction2lem_lthm = $true) <=> ((((! [X274 : term,X273 : (term > $o),X275 : subst] : (($true = ((X273 @ (sub @ X274 @ X275)))) | ? [X276 : term] : ((((var @ X276)) = $true) & (((X273 @ (sub @ X276 @ X275))) != $true)) | ? [X280 : term,X279 : term] : ((((X273 @ X279)) = $true) & (((X273 @ X280)) = $true) & ($true != ((X273 @ (ap @ X279 @ X280))))) | ? [X277 : term] : ((((X273 @ (lam @ X277))) != $true) & ! [X278 : term] : ((((X273 @ (sub @ X277 @ (push @ X278 @ id)))) = $true) | ($true != ((X273 @ X278)))))) | ? [X267 : term,X268 : (term > $o)] : (! [X270 : term,X269 : term] : ((((X268 @ X269)) != $true) | (((X268 @ (ap @ X269 @ X270))) = $true) | (((X268 @ X270)) != $true)) & ! [X272 : term] : ((((X268 @ (lam @ X272))) = $true) | (((X268 @ X272)) != $true)) & ! [X271 : term] : (($true != ((var @ X271))) | (((X268 @ X271)) = $true)) & (((X268 @ X267)) != $true))) | ? [X266 : subst] : (((comp @ X266 @ id)) != X266)) | ? [X265 : term,X264 : subst] : (((comp @ sh @ (push @ X265 @ X264))) != X264)) | ? [X262 : subst,X263 : term] : (((sub @ one @ (push @ X263 @ X262))) != X263))) & (pushprop = $true) & ((hoasapinj2 = $true) <=> ! [X415 : term,X413 : term,X414 : term,X416 : term] : ((X413 = X414) | (((ap @ (sub @ X415 @ id) @ X413)) != ((ap @ (sub @ X416 @ id) @ X414))))) & ((! [X19 : term,X17 : term,X18 : term,X16 : term] : ((X16 = X19) | (((ap @ (sub @ X17 @ id) @ X19)) != ((ap @ (sub @ X18 @ id) @ X16)))) | ? [X13 : term,X12 : term,X15 : term,X14 : term] : ((((ap @ X13 @ X14)) = ((ap @ X15 @ X12))) & (X12 != X14))) <=> (hoasapinj2_lthm = $true)) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem2_lthm = $true) & ((axidr = $true) <=> ! [X519 : subst] : (((comp @ X519 @ id)) = X519)) & (! [X523 : term,X522 : (term > $o),X524 : (subst > term > subst > $o)] : (? [X533 : term] : ((((X522 @ (sub @ X523 @ (push @ X533 @ id)))) != $true) & ($true = ((X522 @ X533)))) | ? [X539 : term] : (((X524 @ id @ X539 @ id)) != ((X522 @ X539))) | ? [X530 : subst,X529 : subst,X532 : subst,X531 : term] : ((((X524 @ X532 @ X531 @ (comp @ X530 @ X529))) != $true) & (((X524 @ (comp @ X532 @ X530) @ (sub @ X531 @ X530) @ X529)) = $true)) | ? [X534 : (subst > term > term)] : (! [X538 : subst,X537 : term,X536 : subst] : (((X534 @ (comp @ X536 @ X538) @ (sub @ X537 @ X538))) = ((sub @ (X534 @ X536 @ X537) @ X538))) & ! [X535 : term] : ((((X524 @ id @ (X534 @ id @ X535) @ id)) = $true) | ($true != ((X524 @ id @ X535 @ id)))) & (((X524 @ id @ (lam @ (X534 @ sh @ one)) @ id)) != $true)) | ? [X528 : subst,X526 : subst,X525 : subst,X527 : term] : ((((X524 @ (comp @ X525 @ X526) @ (sub @ X527 @ X526) @ X528)) != $true) & ($true = ((X524 @ X525 @ X527 @ (comp @ X526 @ X528))))) | (((X522 @ (lam @ X523))) = $true)) <=> (hoasinduction_lem3v2 = $true)) & (induction2_gthm = $true) & (! [X399 : subst,X400 : (term > $o),X402 : term,X401 : (term > $o)] : (? [X403 : term] : (((X400 @ X403)) != ((X401 @ (sub @ X403 @ (push @ X402 @ X399))))) | ($true = ((X400 @ one))) | (((X401 @ X402)) != $true)) <=> (pushprop_lem1v2 = $true)) & (! [X409 : term,X408 : subst] : (((comp @ sh @ (push @ X409 @ X408))) = X408) <=> (axshiftcons = $true)) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((pushprop_lem1 = $true) <=> ! [X163 : (term > $o),X162 : term,X165 : term,X161 : (term > $o),X164 : subst] : ((((X161 @ X162)) != $true) | (((X163 @ (sub @ X162 @ (push @ X165 @ X164)))) = $true))) & ((hoasinduction_lem3v2a_lthm = $true) <=> ((! [X132 : (subst > term > subst > $o),X134 : term,X133 : (term > $o)] : (? [X136 : (subst > term > term)] : ((((X132 @ id @ (lam @ (X136 @ sh @ one)) @ id)) != $true) & ! [X139 : subst,X137 : term,X138 : subst] : (((X136 @ (comp @ X139 @ X138) @ (sub @ X137 @ X138))) = ((sub @ (X136 @ X139 @ X137) @ X138))) & ! [X140 : term] : ((((X132 @ id @ (X136 @ id @ X140) @ id)) = $true) | ($true != ((X132 @ id @ X140 @ id))))) | ? [X135 : term] : (((X133 @ X135)) != ((X132 @ id @ X135 @ id))) | (((X133 @ (lam @ X134))) = $true) | ? [X141 : term] : ((((X133 @ X141)) = $true) & ($true != ((X133 @ (sub @ X134 @ (push @ X141 @ id))))))) | ? [X131 : term] : (((sub @ X131 @ id)) != X131)) | ? [X127 : term] : ! [X128 : (subst > term > term)] : ? [X129 : subst,X130 : term] : (((X128 @ X129 @ X130)) != ((sub @ X127 @ (push @ X130 @ X129)))))) & (one = ((d2term @ d_one))) & (hoasapnotvar = $true) & (hoasinduction_lem1v2 = $true) & (hoasinduction_lem3b_gthm = $true) & (! [X521 : term,X520 : term] : ((((lam @ X521)) != ((lam @ X520))) | (X520 = X521)) <=> (laminj = $true)) & ((induction = $true) <=> ! [X75 : term,X76 : (term > $o)] : ((((X76 @ X75)) = $true) | ? [X78 : term] : (($true = ((X76 @ X78))) & (((X76 @ (lam @ X78))) != $true)) | ? [X79 : term,X80 : term] : (($true = ((X76 @ X80))) & (((X76 @ X79)) = $true) & (((X76 @ (ap @ X79 @ X80))) != $true)) | ? [X77 : term] : ((((var @ X77)) = $true) & (((X76 @ X77)) != $true)))) & (! [X358 : term,X361 : term,X359 : term,X360 : term] : ((X359 = X360) | (((ap @ X359 @ X361)) != ((ap @ X360 @ X358)))) <=> (apinj1 = $true)) & (apnotvar = $true) & ((hoasinduction_lem3aa = $true) <=> ! [X457 : (subst > term > subst > $o),X458 : term] : (? [X459 : (subst > term > term)] : (! [X463 : subst,X461 : term,X462 : subst] : (((X459 @ (comp @ X463 @ X462) @ (sub @ X461 @ X462))) = ((sub @ (X459 @ X463 @ X461) @ X462))) & (((X457 @ id @ (lam @ (X459 @ sh @ one)) @ id)) != $true) & ! [X460 : term] : (($true != ((X457 @ id @ X460 @ id))) | (((X457 @ id @ (X459 @ id @ X460) @ id)) = $true))) | (((X457 @ id @ (lam @ (sub @ X458 @ (push @ one @ sh))) @ id)) = $true) | ? [X464 : term] : ((((X457 @ id @ X464 @ id)) = $true) & (((X457 @ id @ (sub @ X458 @ (push @ X464 @ id)) @ id)) != $true)))) & (pushprop_lem1v2_gthm = $true) & (! [X337 : term,X335 : (subst > term > subst > $o),X336 : term] : ((((X335 @ id @ (ap @ X337 @ X336) @ id)) = $true) | ($true != ((X335 @ id @ X337 @ id))) | ? [X345 : subst,X343 : subst,X342 : subst,X344 : term] : (($true = ((X335 @ (comp @ X343 @ X345) @ (sub @ X344 @ X345) @ X342))) & ($true != ((X335 @ X343 @ X344 @ (comp @ X345 @ X342))))) | ? [X340 : subst,X341 : term,X339 : subst,X338 : subst] : ((((X335 @ (comp @ X339 @ X338) @ (sub @ X341 @ X338) @ X340)) != $true) & ($true = ((X335 @ X339 @ X341 @ (comp @ X338 @ X340))))) | ? [X346 : term,X347 : term] : ((((X335 @ id @ X347 @ id)) = $true) & (((X335 @ id @ (ap @ (sub @ X346 @ id) @ X347) @ id)) != $true) & ($true = ((X335 @ id @ X346 @ id)))) | ($true != ((X335 @ id @ X336 @ id)))) <=> (hoasinduction_lem2 = $true)) & ((axvarid = $true) <=> ! [X53 : term] : (((sub @ X53 @ id)) = X53)) & ((hoasinduction_lem3_lthm = $true) <=> ((! [X435 : (subst > term > subst > $o),X434 : term] : ((((X435 @ id @ (lam @ X434) @ id)) = $true) | ? [X447 : subst,X446 : subst,X448 : term,X449 : subst] : ((((X435 @ X446 @ X448 @ (comp @ X449 @ X447))) != $true) & (((X435 @ (comp @ X446 @ X449) @ (sub @ X448 @ X449) @ X447)) = $true)) | ? [X438 : subst,X436 : term,X439 : subst,X437 : subst] : (($true != ((X435 @ (comp @ X438 @ X439) @ (sub @ X436 @ X439) @ X437))) & ($true = ((X435 @ X438 @ X436 @ (comp @ X439 @ X437))))) | ? [X440 : term] : ((((X435 @ id @ X440 @ id)) = $true) & ($true != ((X435 @ id @ (sub @ X434 @ (push @ X440 @ id)) @ id)))) | ? [X441 : (subst > term > term)] : (! [X443 : term,X444 : subst,X442 : subst] : (((X441 @ (comp @ X442 @ X444) @ (sub @ X443 @ X444))) = ((sub @ (X441 @ X442 @ X443) @ X444))) & ($true != ((X435 @ id @ (lam @ (X441 @ sh @ one)) @ id))) & ! [X445 : term] : ((((X435 @ id @ X445 @ id)) != $true) | (((X435 @ id @ (X441 @ id @ X445) @ id)) = $true)))) | ? [X427 : term,X426 : (subst > term > subst > $o)] : (! [X433 : term] : (($true != ((X426 @ id @ X433 @ id))) | (((X426 @ id @ (sub @ X427 @ (push @ X433 @ id)) @ id)) = $true)) & ($true != ((X426 @ id @ (lam @ (sub @ X427 @ (push @ one @ sh))) @ id))) & ! [X428 : (subst > term > term)] : (? [X431 : term,X432 : subst,X430 : subst] : (((sub @ (X428 @ X432 @ X431) @ X430)) != ((X428 @ (comp @ X432 @ X430) @ (sub @ X431 @ X430)))) | ($true = ((X426 @ id @ (lam @ (X428 @ sh @ one)) @ id))) | ? [X429 : term] : (($true = ((X426 @ id @ X429 @ id))) & (((X426 @ id @ (X428 @ id @ X429) @ id)) != $true))))) | ? [X425 : term] : (((sub @ X425 @ id)) != X425))) & ((pushprop_lem0 = $true) <=> ! [X22 : (term > $o),X21 : subst,X20 : term] : ? [X23 : (term > $o)] : ! [X24 : term] : (((X23 @ X24)) = ((X22 @ (sub @ X24 @ (push @ X20 @ X21)))))) & (ulamvarsh = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((hoasinduction_lem3a_lthm = $true) <=> ((! [X199 : term,X198 : (subst > term > subst > $o)] : (($true = ((X198 @ id @ (lam @ X199) @ id))) | ? [X201 : (subst > term > term)] : (($true != ((X198 @ id @ (lam @ (X201 @ sh @ one)) @ id))) & ! [X202 : subst,X203 : term,X204 : subst] : (((sub @ (X201 @ X204 @ X203) @ X202)) = ((X201 @ (comp @ X204 @ X202) @ (sub @ X203 @ X202)))) & ! [X205 : term] : ((((X198 @ id @ (X201 @ id @ X205) @ id)) = $true) | (((X198 @ id @ X205 @ id)) != $true))) | ? [X200 : term] : ((((X198 @ id @ (sub @ X199 @ (push @ X200 @ id)) @ id)) != $true) & (((X198 @ id @ X200 @ id)) = $true))) | ? [X191 : (subst > term > subst > $o),X190 : term] : (! [X192 : (subst > term > term)] : (? [X195 : subst,X196 : term,X194 : subst] : (((X192 @ (comp @ X195 @ X194) @ (sub @ X196 @ X194))) != ((sub @ (X192 @ X195 @ X196) @ X194))) | ? [X193 : term] : ((((X191 @ id @ (X192 @ id @ X193) @ id)) != $true) & (((X191 @ id @ X193 @ id)) = $true)) | ($true = ((X191 @ id @ (lam @ (X192 @ sh @ one)) @ id)))) & (((X191 @ id @ (lam @ (sub @ X190 @ (push @ one @ sh))) @ id)) != $true) & ! [X197 : term] : ((((X191 @ id @ (sub @ X190 @ (push @ X197 @ id)) @ id)) = $true) | ($true != ((X191 @ id @ X197 @ id)))))) | ? [X189 : term] : (((sub @ X189 @ id)) != X189))) & (((var @ (d2term @ d_one))) != $true) & ! [X74 : d_subst] : (d_id = X74) & ((termmset = $true) <=> ! [X290 : term] : (((sub @ X290 @ id)) = X290)) & ((! [X179 : term] : (((sub @ X179 @ id)) = X179) | ? [X178 : term] : (((sub @ X178 @ id)) != X178)) <=> (termmset_lthm = $true)) & (lamnotap != $true) & (! [X320 : term,X319 : (subst > term > subst > $o)] : (? [X330 : term] : ((((X319 @ id @ X330 @ id)) = $true) & (((X319 @ id @ (sub @ X320 @ (push @ X330 @ id)) @ id)) != $true)) | (((X319 @ id @ (lam @ X320) @ id)) = $true) | ? [X321 : term,X324 : subst,X322 : subst,X323 : subst] : ((((X319 @ X324 @ X321 @ (comp @ X322 @ X323))) = $true) & ($true != ((X319 @ (comp @ X324 @ X322) @ (sub @ X321 @ X322) @ X323)))) | ? [X333 : subst,X331 : subst,X332 : subst,X334 : term] : ((((X319 @ X332 @ X334 @ (comp @ X333 @ X331))) != $true) & (((X319 @ (comp @ X332 @ X333) @ (sub @ X334 @ X333) @ X331)) = $true)) | ? [X325 : (subst > term > term)] : ((((X319 @ id @ (lam @ (X325 @ sh @ one)) @ id)) != $true) & ! [X328 : term,X326 : subst,X327 : subst] : (((sub @ (X325 @ X326 @ X328) @ X327)) = ((X325 @ (comp @ X326 @ X327) @ (sub @ X328 @ X327)))) & ! [X329 : term] : (($true = ((X319 @ id @ (X325 @ id @ X329) @ id))) | (((X319 @ id @ X329 @ id)) != $true)))) <=> (hoasinduction_lem3 = $true)) & (pushprop_lem2v2 = $true) & (pushprop_lem1_gthm = $true) & ((hoasinduction_lem3a = $true) <=> ! [X351 : term,X350 : (subst > term > subst > $o)] : (? [X357 : term] : ((((X350 @ id @ X357 @ id)) = $true) & (((X350 @ id @ (sub @ X351 @ (push @ X357 @ id)) @ id)) != $true)) | (((X350 @ id @ (lam @ X351) @ id)) = $true) | ? [X352 : (subst > term > term)] : (! [X355 : subst,X354 : subst,X353 : term] : (((X352 @ (comp @ X355 @ X354) @ (sub @ X353 @ X354))) = ((sub @ (X352 @ X355 @ X353) @ X354))) & (((X350 @ id @ (lam @ (X352 @ sh @ one)) @ id)) != $true) & ! [X356 : term] : ((((X350 @ id @ X356 @ id)) != $true) | (((X350 @ id @ (X352 @ id @ X356) @ id)) = $true))))) & (! [X423 : subst,X424 : term] : (((sub @ one @ (push @ X424 @ X423))) = X424) <=> (axvarcons = $true)) & (hoaslamnotvar_gthm = $true) & (! [X142 : (subst > term > subst > $o)] : ? [X143 : (term > $o)] : ! [X144 : term] : (((X142 @ id @ X144 @ id)) = ((X143 @ X144))) <=> (hoasinduction_lem0_lthm = $true)) & ((shinj = $true) <=> ! [X455 : term,X456 : term] : ((((sub @ X455 @ sh)) != ((sub @ X456 @ sh))) | (X455 = X456))) & (hoasinduction_lem2v2_gthm = $true) & (hoasapnotvar_lthm = $true) & (termmset_gthm = $true) & (pushprop_lthm = $true) & ((((! [X45 : subst,X46 : (subst > term > term),X43 : term,X44 : (subst > term > term)] : ((((lam @ (X44 @ sh @ one))) != ((lam @ (X46 @ sh @ one)))) | ? [X51 : term,X50 : subst,X52 : subst] : (((X46 @ (comp @ X52 @ X50) @ (sub @ X51 @ X50))) != ((sub @ (X46 @ X52 @ X51) @ X50))) | (((X44 @ X45 @ X43)) = ((X46 @ X45 @ X43))) | ? [X47 : term,X49 : subst,X48 : subst] : (((X44 @ (comp @ X48 @ X49) @ (sub @ X47 @ X49))) != ((sub @ (X44 @ X48 @ X47) @ X49)))) | ? [X42 : term,X41 : term] : ((X41 != X42) & (((lam @ X41)) = ((lam @ X42))))) | ? [X39 : subst,X40 : term] : (((comp @ sh @ (push @ X40 @ X39))) != X39)) | ? [X38 : subst,X37 : term] : (((sub @ one @ (push @ X37 @ X38))) != X37)) <=> (hoaslaminj_lthm = $true)) & ((induction2_lthm = $true) <=> ((! [X216 : term,X215 : (term > $o)] : ((((X215 @ X216)) = $true) | ? [X219 : term] : ((((var @ X219)) = $true) & (((X215 @ X219)) != $true)) | ? [X217 : term,X218 : term] : ((((X215 @ X217)) = $true) & ($true = ((X215 @ X218))) & (((X215 @ (ap @ X218 @ X217))) != $true)) | ? [X220 : term] : ((((X215 @ (lam @ X220))) != $true) & ! [X221 : term] : ((((X215 @ X221)) != $true) | (((X215 @ (sub @ X220 @ (push @ X221 @ id)))) = $true)))) | ? [X208 : term,X207 : subst,X209 : (term > $o)] : ((((X209 @ (sub @ X208 @ X207))) != $true) & ! [X214 : term] : ((((X209 @ (sub @ X214 @ X207))) = $true) | (((var @ X214)) != $true)) & ! [X213 : term,X212 : term] : ((((X209 @ X212)) != $true) | (((X209 @ X213)) != $true) | (((X209 @ (ap @ X212 @ X213))) = $true)) & ! [X210 : term] : (? [X211 : term] : ((((X209 @ X211)) = $true) & ($true != ((X209 @ (sub @ X210 @ (push @ X211 @ id)))))) | (((X209 @ (lam @ X210))) = $true)))) | ? [X206 : term] : (((sub @ X206 @ id)) != X206))) & (axvarshift = $true) & (substmonoid_gthm = $true) & (! [X314 : (term > $o),X316 : term,X315 : subst] : ? [X317 : (term > $o)] : ! [X318 : term] : (((X314 @ (sub @ X318 @ (push @ X316 @ X315)))) = ((X317 @ X318))) <=> (pushprop_lem0_lthm = $true)) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (! [X117 : term,X115 : (term > $o),X116 : subst] : (? [X118 : term] : (($true = ((var @ X118))) & (((X115 @ (sub @ X118 @ X116))) != $true)) | (((X115 @ (sub @ X117 @ X116))) = $true) | ? [X122 : term,X121 : term] : ((((X115 @ (ap @ X122 @ X121))) != $true) & (((X115 @ X122)) = $true) & (((X115 @ X121)) = $true)) | ? [X119 : term] : (($true != ((X115 @ (lam @ X119)))) & ! [X120 : term] : (($true != ((X115 @ X120))) | (((X115 @ (sub @ X119 @ (push @ X120 @ id)))) = $true)))) <=> (induction2lem = $true)) & ((hoaslaminj = $true) <=> ! [X306 : subst,X304 : (subst > term > term),X307 : term,X305 : (subst > term > term)] : (? [X310 : subst,X308 : term,X309 : subst] : (((X304 @ (comp @ X310 @ X309) @ (sub @ X308 @ X309))) != ((sub @ (X304 @ X310 @ X308) @ X309))) | (((lam @ (X305 @ sh @ one))) != ((lam @ (X304 @ sh @ one)))) | (((X305 @ X306 @ X307)) = ((X304 @ X306 @ X307))) | ? [X312 : subst,X311 : term,X313 : subst] : (((sub @ (X305 @ X313 @ X311) @ X312)) != ((X305 @ (comp @ X313 @ X312) @ (sub @ X311 @ X312)))))) & (pushprop_lthm_orig = $true) & (sh = ((d2subst @ d_id))) & (lamnotvar = $true) & ((hoasinduction_lem2v2 = $true) <=> ! [X148 : term,X147 : term,X146 : (term > $o),X145 : (subst > term > subst > $o)] : ((((X146 @ X147)) != $true) | ? [X157 : subst,X159 : subst,X158 : subst,X156 : term] : ((((X145 @ X157 @ X156 @ (comp @ X158 @ X159))) = $true) & (((X145 @ (comp @ X157 @ X158) @ (sub @ X156 @ X158) @ X159)) != $true)) | (((X146 @ (ap @ X147 @ X148))) = $true) | ? [X149 : term] : (((X145 @ id @ X149 @ id)) != ((X146 @ X149))) | (((X146 @ X148)) != $true) | ? [X154 : term,X155 : term] : ((((X145 @ id @ X155 @ id)) = $true) & ($true != ((X145 @ id @ (ap @ (sub @ X154 @ id) @ X155) @ id))) & (((X145 @ id @ X154 @ id)) = $true)) | ? [X152 : subst,X150 : term,X153 : subst,X151 : subst] : ((((X145 @ X153 @ X150 @ (comp @ X151 @ X152))) != $true) & ($true = ((X145 @ (comp @ X153 @ X151) @ (sub @ X150 @ X151) @ X152)))))) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X81 : term] : ? [X82 : (subst > term > term)] : ! [X83 : subst,X84 : term] : (((sub @ X81 @ (push @ X84 @ X83))) = ((X82 @ X83 @ X84)))) & (ulamvar1 != $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoasinduction_lem3v2_gthm = $true) & (hoaslamnotvar_lthm = $true) & ((apinj2 = $true) <=> ! [X405 : term,X404 : term,X406 : term,X407 : term] : ((((ap @ X404 @ X407)) != ((ap @ X405 @ X406))) | (X406 = X407))) & (((! [X286 : term,X289 : term,X288 : term,X287 : term] : ((X286 = X288) | (((ap @ (sub @ X286 @ id) @ X287)) != ((ap @ (sub @ X288 @ id) @ X289)))) | ? [X282 : term,X283 : term,X284 : term,X285 : term] : ((X283 != X284) & (((ap @ X283 @ X285)) = ((ap @ X284 @ X282))))) | ? [X281 : term] : (((sub @ X281 @ id)) != X281)) <=> (hoasapinj1_lthm = $true)) & ((hoasinduction_lem3aaa = $true) <=> ! [X26 : (subst > term > subst > $o),X25 : term] : (? [X28 : term,X29 : (subst > term > term)] : (! [X30 : term,X31 : subst,X32 : subst] : (((sub @ (sub @ X28 @ (push @ X30 @ X31)) @ X32)) = ((sub @ (X29 @ X31 @ X30) @ X32))) & ! [X36 : term] : ((((X26 @ id @ (X29 @ id @ X36) @ id)) = $true) | (((X26 @ id @ X36 @ id)) != $true)) & ! [X33 : subst,X35 : term,X34 : subst] : (((X29 @ (comp @ X33 @ X34) @ (sub @ X35 @ X34))) = ((sub @ X28 @ (push @ (sub @ X35 @ X34) @ (comp @ X33 @ X34))))) & (((X26 @ id @ (lam @ (X29 @ sh @ one)) @ id)) != $true)) | ? [X27 : term] : ((((X26 @ id @ X27 @ id)) = $true) & ($true != ((X26 @ id @ (sub @ X25 @ (push @ X27 @ id)) @ id)))) | (((X26 @ id @ (lam @ (sub @ X25 @ (push @ one @ sh))) @ id)) = $true))) & (((((! [X384 : (subst > term > subst > $o),X383 : term] : (($true = ((X384 @ id @ X383 @ id))) | ? [X385 : term,X386 : term] : ((((X384 @ id @ (ap @ (sub @ X385 @ id) @ X386) @ id)) != $true) & ($true = ((X384 @ id @ X385 @ id))) & (((X384 @ id @ X386 @ id)) = $true)) | ? [X387 : (subst > term > term)] : (! [X391 : term] : (($true != ((X384 @ id @ X391 @ id))) | (((X384 @ id @ (X387 @ id @ X391) @ id)) = $true)) & (((X384 @ id @ (lam @ (X387 @ sh @ one)) @ id)) != $true) & ! [X389 : subst,X390 : term,X388 : subst] : (((X387 @ (comp @ X389 @ X388) @ (sub @ X390 @ X388))) = ((sub @ (X387 @ X389 @ X390) @ X388))))) | ? [X374 : (term > $o),X373 : term,X375 : (subst > term > subst > $o)] : (($true != ((X374 @ (lam @ X373)))) & ! [X376 : term] : (($true = ((X374 @ (sub @ X373 @ (push @ X376 @ id))))) | ($true != ((X374 @ X376)))) & ! [X382 : term] : (((X375 @ id @ X382 @ id)) = ((X374 @ X382))) & ! [X377 : (subst > term > term)] : (? [X380 : term,X379 : subst,X378 : subst] : (((sub @ (X377 @ X379 @ X380) @ X378)) != ((X377 @ (comp @ X379 @ X378) @ (sub @ X380 @ X378)))) | ? [X381 : term] : ((((X375 @ id @ X381 @ id)) = $true) & ($true != ((X375 @ id @ (X377 @ id @ X381) @ id)))) | (((X375 @ id @ (lam @ (X377 @ sh @ one)) @ id)) = $true)))) | ? [X372 : term] : (((sub @ X372 @ id)) != X372)) | ? [X365 : (term > $o),X366 : term] : (! [X367 : term] : (($true != ((var @ X367))) | (((X365 @ X367)) = $true)) & ! [X370 : term] : (? [X371 : term] : ((((X365 @ (sub @ X370 @ (push @ X371 @ id)))) != $true) & (((X365 @ X371)) = $true)) | (((X365 @ (lam @ X370))) = $true)) & (((X365 @ X366)) != $true) & ! [X369 : term,X368 : term] : ((((X365 @ X368)) != $true) | (((X365 @ X369)) != $true) | (((X365 @ (ap @ X369 @ X368))) = $true)))) | ? [X362 : (subst > term > subst > $o)] : ! [X363 : (term > $o)] : ? [X364 : term] : (((X362 @ id @ X364 @ id)) != ((X363 @ X364)))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (hoaslamnotvar = $true) & ((((! [X302 : subst] : (((comp @ X302 @ id)) = X302) & ! [X303 : subst] : (((comp @ id @ X303)) = X303)) | ? [X301 : subst] : (((comp @ X301 @ id)) != X301)) | ? [X300 : subst] : (((comp @ id @ X300)) != X300)) <=> (substmonoid_lthm = $true)) & (axmap = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X4 : (subst > term > subst > $o),X5 : term] : (($true = ((X4 @ id @ (lam @ (sub @ X5 @ (push @ one @ sh))) @ id))) | ? [X11 : term] : ((((X4 @ id @ X11 @ id)) = $true) & (((X4 @ id @ (sub @ X5 @ (push @ X11 @ id)) @ id)) != $true)) | ? [X6 : (subst > term > term)] : (! [X10 : term,X8 : subst,X9 : subst] : (((sub @ (X6 @ X9 @ X10) @ X8)) = ((X6 @ (comp @ X9 @ X8) @ (sub @ X10 @ X8)))) & ! [X7 : term] : (($true = ((X4 @ id @ (X6 @ id @ X7) @ id))) | (((X4 @ id @ X7 @ id)) != $true)) & (((X4 @ id @ (lam @ (X6 @ sh @ one)) @ id)) != $true)))) & (hoasinduction_lem1 = $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)))))))))))) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ! [X160 : d_term] : (d_one = X160) & (hoasapinj2_gthm = $true) & (pushprop_lem3v2 = $true) & (axclos = $true) & (hoasapinj1_gthm = $true) & (! [X107 : term,X106 : (term > $o)] : (? [X112 : term] : ((((X106 @ X112)) != $true) & (((var @ X112)) = $true)) | ($true = ((X106 @ X107))) | ? [X110 : term,X111 : term] : ((((X106 @ X110)) = $true) & (((X106 @ (ap @ X110 @ X111))) != $true) & (((X106 @ X111)) = $true)) | ? [X108 : term] : (($true != ((X106 @ (lam @ X108)))) & ! [X109 : term] : ((((X106 @ (sub @ X108 @ (push @ X109 @ id)))) = $true) | (((X106 @ X109)) != $true)))) <=> (induction2 = $true)) & (pushprop_lem2v2_gthm = $true) & (pushprop_lem3v2_lthm = $true) & (hoaslaminj_gthm = $true) & (hoasinduction_lem2_gthm = $true) & (ulamvarind = $true) & ((hoaslamnotap = $true) <=> ! [X419 : term,X417 : (subst > term > term),X418 : term] : (? [X421 : subst,X422 : term,X420 : subst] : (((X417 @ (comp @ X420 @ X421) @ (sub @ X422 @ X421))) != ((sub @ (X417 @ X420 @ X422) @ X421))) | (((lam @ (X417 @ sh @ one))) != ((ap @ (sub @ X419 @ id) @ X418))))) & ((((! [X501 : (subst > term > subst > $o),X502 : term] : (? [X514 : subst,X513 : subst,X512 : term,X511 : subst] : ((((X501 @ (comp @ X514 @ X513) @ (sub @ X512 @ X513) @ X511)) = $true) & (((X501 @ X514 @ X512 @ (comp @ X513 @ X511))) != $true)) | (((X501 @ id @ X502 @ id)) = $true) | ? [X505 : term] : ((((X501 @ id @ X505 @ id)) != $true) & ($true = ((var @ (sub @ X505 @ id))))) | ? [X504 : term,X503 : term] : ((((X501 @ id @ X504 @ id)) = $true) & (((X501 @ id @ X503 @ id)) = $true) & (((X501 @ id @ (ap @ (sub @ X504 @ id) @ X503) @ id)) != $true)) | ? [X506 : (subst > term > term)] : (! [X509 : subst,X510 : subst,X508 : term] : (((X506 @ (comp @ X510 @ X509) @ (sub @ X508 @ X509))) = ((sub @ (X506 @ X510 @ X508) @ X509))) & (((X501 @ id @ (lam @ (X506 @ sh @ one)) @ id)) != $true) & ! [X507 : term] : ((((X501 @ id @ (X506 @ id @ X507) @ id)) = $true) | ($true != ((X501 @ id @ X507 @ id))))) | ? [X517 : subst,X516 : subst,X515 : term,X518 : subst] : ((((X501 @ (comp @ X517 @ X516) @ (sub @ X515 @ X516) @ X518)) != $true) & (((X501 @ X517 @ X515 @ (comp @ X516 @ X518))) = $true))) | ? [X485 : term,X486 : (subst > term > subst > $o)] : ((((X486 @ id @ (lam @ X485) @ id)) != $true) & ! [X499 : subst,X496 : term,X498 : subst,X497 : subst] : (($true = ((X486 @ (comp @ X499 @ X497) @ (sub @ X496 @ X497) @ X498))) | (((X486 @ X499 @ X496 @ (comp @ X497 @ X498))) != $true)) & ! [X500 : term] : ((((X486 @ id @ X500 @ id)) != $true) | (((X486 @ id @ (sub @ X485 @ (push @ X500 @ id)) @ id)) = $true)) & ! [X488 : term,X490 : subst,X487 : subst,X489 : subst] : (($true = ((X486 @ X489 @ X488 @ (comp @ X490 @ X487)))) | ($true != ((X486 @ (comp @ X489 @ X490) @ (sub @ X488 @ X490) @ X487)))) & ! [X491 : (subst > term > term)] : (? [X493 : subst,X492 : subst,X494 : term] : (((sub @ (X491 @ X492 @ X494) @ X493)) != ((X491 @ (comp @ X492 @ X493) @ (sub @ X494 @ X493)))) | (((X486 @ id @ (lam @ (X491 @ sh @ one)) @ id)) = $true) | ? [X495 : term] : ((((X486 @ id @ X495 @ id)) = $true) & (((X486 @ id @ (X491 @ id @ X495) @ id)) != $true))))) | ? [X472 : term,X474 : (subst > term > subst > $o),X473 : term] : (! [X477 : subst,X475 : term,X476 : subst,X478 : subst] : ((((X474 @ (comp @ X476 @ X477) @ (sub @ X475 @ X477) @ X478)) = $true) | ($true != ((X474 @ X476 @ X475 @ (comp @ X477 @ X478))))) & (((X474 @ id @ (ap @ X472 @ X473) @ id)) != $true) & (((X474 @ id @ X473 @ id)) = $true) & ! [X480 : subst,X482 : term,X479 : subst,X481 : subst] : ((((X474 @ X480 @ X482 @ (comp @ X481 @ X479))) = $true) | ($true != ((X474 @ (comp @ X480 @ X481) @ (sub @ X482 @ X481) @ X479)))) & (((X474 @ id @ X472 @ id)) = $true) & ! [X484 : term,X483 : term] : ((((X474 @ id @ X483 @ id)) != $true) | (((X474 @ id @ X484 @ id)) != $true) | (((X474 @ id @ (ap @ (sub @ X483 @ id) @ X484) @ id)) = $true)))) | ? [X465 : term,X466 : (term > $o)] : (! [X469 : term,X468 : term] : (($true = ((X466 @ (ap @ X468 @ X469)))) | (((X466 @ X468)) != $true) | ($true != ((X466 @ X469)))) & ! [X467 : term] : (($true = ((X466 @ X467))) | (((var @ X467)) != $true)) & ! [X470 : term] : ((((X466 @ (lam @ X470))) = $true) | ? [X471 : term] : ((((X466 @ (sub @ X470 @ (push @ X471 @ id)))) != $true) & (((X466 @ X471)) = $true))) & ($true != ((X466 @ X465))))) <=> (hoasinduction_lthm = $true)) & ((hoasinduction_lem3v2a = $true) <=> ! [X169 : (term > $o),X168 : term,X170 : (subst > term > subst > $o)] : (? [X171 : term] : (($true = ((X169 @ X171))) & ($true != ((X169 @ (sub @ X168 @ (push @ X171 @ id)))))) | ? [X172 : (subst > term > term)] : ((((X170 @ id @ (lam @ (X172 @ sh @ one)) @ id)) != $true) & ! [X173 : subst,X174 : term,X175 : subst] : (((sub @ (X172 @ X173 @ X174) @ X175)) = ((X172 @ (comp @ X173 @ X175) @ (sub @ X174 @ X175)))) & ! [X176 : term] : ((((X170 @ id @ (X172 @ id @ X176) @ id)) = $true) | (((X170 @ id @ X176 @ id)) != $true))) | (((X169 @ (lam @ X168))) = $true) | ? [X177 : term] : (((X169 @ X177)) != ((X170 @ id @ X177 @ id))))) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ! [X348 : term] : ? [X349 : d_term] : (((d2term @ X349)) = X348) & (! [X181 : (subst > term > subst > $o),X180 : term] : ((((X181 @ id @ X180 @ id)) = $true) | ? [X188 : term,X187 : term] : (($true = ((X181 @ id @ X188 @ id))) & (((X181 @ id @ (ap @ (sub @ X187 @ id) @ X188) @ id)) != $true) & ($true = ((X181 @ id @ X187 @ id)))) | ? [X182 : (subst > term > term)] : (! [X183 : term] : ((((X181 @ id @ (X182 @ id @ X183) @ id)) = $true) | (((X181 @ id @ X183 @ id)) != $true)) & ($true != ((X181 @ id @ (lam @ (X182 @ sh @ one)) @ id))) & ! [X186 : term,X185 : subst,X184 : subst] : (((X182 @ (comp @ X185 @ X184) @ (sub @ X186 @ X184))) = ((sub @ (X182 @ X185 @ X186) @ X184))))) <=> (hoasinduction_no_psi_cond = $true)) & (! [X123 : term] : ? [X124 : (subst > term > term)] : ! [X125 : subst,X126 : term] : (((sub @ X123 @ (push @ X126 @ X125))) = ((X124 @ X125 @ X126))) <=> (hoasinduction_lem3v2_f = $true)) & ((hoasinduction_lem0 = $true) <=> ! [X410 : (subst > term > subst > $o)] : ? [X411 : (term > $o)] : ! [X412 : term] : (((X410 @ id @ X412 @ id)) = ((X411 @ X412)))) & (hoasinduction_gthm = $true) & (hoaslamnotap_lthm = $true) & (hoasinduction_lem3_gthm = $true) & (! [X222 : subst] : (((comp @ id @ X222)) = X222) <=> (axidl = $true)) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ! [X450 : d_term,X451 : d_term] : ((((d2term @ X450)) != ((d2term @ X451))) | (X450 = X451)) & (id = ((d2subst @ d_id)))),
  inference(ennf_transformation,[],[f9])).
thf(f12,plain,(
  ((? [X233 : term] : (((sub @ X233 @ id)) != X233) | ? [X226 : (term > $o),X227 : term] : (! [X230 : term,X229 : term] : ((((X226 @ (ap @ X230 @ X229))) = $true) | (((X226 @ X229)) != $true) | (((X226 @ X230)) != $true)) & ! [X228 : term] : ((((var @ X228)) != $true) | (((X226 @ X228)) = $true)) & ! [X231 : term] : ((((X226 @ (lam @ X231))) = $true) | ? [X232 : term] : ((((X226 @ (sub @ X231 @ (push @ X232 @ id)))) != $true) & (((X226 @ X232)) = $true))) & (((X226 @ X227)) != $true)) | ? [X234 : (term > $o),X236 : (subst > term > subst > $o),X235 : term] : ((((X234 @ (lam @ X235))) != $true) & ! [X237 : term] : (((X234 @ X237)) = ((X236 @ id @ X237 @ id))) & ! [X243 : term] : ((((X234 @ X243)) != $true) | ($true = ((X234 @ (sub @ X235 @ (push @ X243 @ id)))))) & ! [X238 : (subst > term > term)] : (? [X239 : term] : ((((X236 @ id @ (X238 @ id @ X239) @ id)) != $true) & (((X236 @ id @ X239 @ id)) = $true)) | ? [X242 : term,X241 : subst,X240 : subst] : (((sub @ (X238 @ X241 @ X242) @ X240)) != ((X238 @ (comp @ X241 @ X240) @ (sub @ X242 @ X240)))) | ($true = ((X236 @ id @ (lam @ (X238 @ sh @ one)) @ id))))) | ? [X223 : (subst > term > subst > $o)] : ! [X224 : (term > $o)] : ? [X225 : term] : (((X224 @ X225)) != ((X223 @ id @ X225 @ id))) | ! [X244 : (subst > term > subst > $o),X245 : term] : (? [X257 : (subst > term > term)] : (! [X258 : subst,X260 : subst,X259 : term] : (((sub @ (X257 @ X260 @ X259) @ X258)) = ((X257 @ (comp @ X260 @ X258) @ (sub @ X259 @ X258)))) & ! [X261 : term] : ((((X244 @ id @ (X257 @ id @ X261) @ id)) = $true) | (((X244 @ id @ X261 @ id)) != $true)) & (((X244 @ id @ (lam @ (X257 @ sh @ one)) @ id)) != $true)) | ? [X250 : subst,X253 : subst,X252 : term,X251 : subst] : (($true != ((X244 @ X253 @ X252 @ (comp @ X251 @ X250)))) & ($true = ((X244 @ (comp @ X253 @ X251) @ (sub @ X252 @ X251) @ X250)))) | ? [X247 : term,X249 : subst,X248 : subst,X246 : subst] : ((((X244 @ X246 @ X247 @ (comp @ X249 @ X248))) = $true) & (((X244 @ (comp @ X246 @ X249) @ (sub @ X247 @ X249) @ X248)) != $true)) | ? [X255 : term,X254 : term] : ((((X244 @ id @ (ap @ (sub @ X254 @ id) @ X255) @ id)) != $true) & (((X244 @ id @ X255 @ id)) = $true) & (((X244 @ id @ X254 @ id)) = $true)) | ? [X256 : term] : ((((X244 @ id @ X256 @ id)) != $true) & (((var @ (sub @ X256 @ id))) = $true)) | ($true = ((X244 @ id @ X245 @ id))))) <=> (hoasinduction_lthm_3 = $true)) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (induction2_gthm = $true) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X4 : (subst > term > subst > $o),X5 : term] : (($true = ((X4 @ id @ (lam @ (sub @ X5 @ (push @ one @ sh))) @ id))) | ? [X11 : term] : ((((X4 @ id @ X11 @ id)) = $true) & (((X4 @ id @ (sub @ X5 @ (push @ X11 @ id)) @ id)) != $true)) | ? [X6 : (subst > term > term)] : (! [X10 : term,X8 : subst,X9 : subst] : (((sub @ (X6 @ X9 @ X10) @ X8)) = ((X6 @ (comp @ X9 @ X8) @ (sub @ X10 @ X8)))) & ! [X7 : term] : (($true = ((X4 @ id @ (X6 @ id @ X7) @ id))) | (((X4 @ id @ X7 @ id)) != $true)) & (((X4 @ id @ (lam @ (X6 @ sh @ one)) @ id)) != $true)))) & ((? [X208 : term,X207 : subst,X209 : (term > $o)] : ((((X209 @ (sub @ X208 @ X207))) != $true) & ! [X214 : term] : ((((X209 @ (sub @ X214 @ X207))) = $true) | (((var @ X214)) != $true)) & ! [X213 : term,X212 : term] : ((((X209 @ X212)) != $true) | (((X209 @ X213)) != $true) | (((X209 @ (ap @ X212 @ X213))) = $true)) & ! [X210 : term] : (? [X211 : term] : ((((X209 @ X211)) = $true) & ($true != ((X209 @ (sub @ X210 @ (push @ X211 @ id)))))) | (((X209 @ (lam @ X210))) = $true))) | ? [X206 : term] : (((sub @ X206 @ id)) != X206) | ! [X216 : term,X215 : (term > $o)] : ((((X215 @ X216)) = $true) | ? [X219 : term] : ((((var @ X219)) = $true) & (((X215 @ X219)) != $true)) | ? [X217 : term,X218 : term] : ((((X215 @ X217)) = $true) & ($true = ((X215 @ X218))) & (((X215 @ (ap @ X218 @ X217))) != $true)) | ? [X220 : term] : ((((X215 @ (lam @ X220))) != $true) & ! [X221 : term] : ((((X215 @ X221)) != $true) | (((X215 @ (sub @ X220 @ (push @ X221 @ id)))) = $true))))) <=> (induction2_lthm = $true)) & (hoasinduction_lem2_gthm = $true) & (pushprop_lem2v2_gthm = $true) & ! [X453 : d_subst,X452 : d_subst] : ((X452 = X453) | (((d2subst @ X453)) != ((d2subst @ X452)))) & ((? [X191 : (subst > term > subst > $o),X190 : term] : (! [X192 : (subst > term > term)] : (? [X195 : subst,X196 : term,X194 : subst] : (((X192 @ (comp @ X195 @ X194) @ (sub @ X196 @ X194))) != ((sub @ (X192 @ X195 @ X196) @ X194))) | ? [X193 : term] : ((((X191 @ id @ (X192 @ id @ X193) @ id)) != $true) & (((X191 @ id @ X193 @ id)) = $true)) | ($true = ((X191 @ id @ (lam @ (X192 @ sh @ one)) @ id)))) & (((X191 @ id @ (lam @ (sub @ X190 @ (push @ one @ sh))) @ id)) != $true) & ! [X197 : term] : ((((X191 @ id @ (sub @ X190 @ (push @ X197 @ id)) @ id)) = $true) | ($true != ((X191 @ id @ X197 @ id))))) | ! [X199 : term,X198 : (subst > term > subst > $o)] : (($true = ((X198 @ id @ (lam @ X199) @ id))) | ? [X201 : (subst > term > term)] : (($true != ((X198 @ id @ (lam @ (X201 @ sh @ one)) @ id))) & ! [X202 : subst,X203 : term,X204 : subst] : (((sub @ (X201 @ X204 @ X203) @ X202)) = ((X201 @ (comp @ X204 @ X202) @ (sub @ X203 @ X202)))) & ! [X205 : term] : ((((X198 @ id @ (X201 @ id @ X205) @ id)) = $true) | (((X198 @ id @ X205 @ id)) != $true))) | ? [X200 : term] : ((((X198 @ id @ (sub @ X199 @ (push @ X200 @ id)) @ id)) != $true) & (((X198 @ id @ X200 @ id)) = $true))) | ? [X189 : term] : (((sub @ X189 @ id)) != X189)) <=> (hoasinduction_lem3a_lthm = $true)) & ((hoasinduction_lthm = $true) <=> (! [X501 : (subst > term > subst > $o),X502 : term] : (? [X514 : subst,X513 : subst,X512 : term,X511 : subst] : ((((X501 @ (comp @ X514 @ X513) @ (sub @ X512 @ X513) @ X511)) = $true) & (((X501 @ X514 @ X512 @ (comp @ X513 @ X511))) != $true)) | (((X501 @ id @ X502 @ id)) = $true) | ? [X505 : term] : ((((X501 @ id @ X505 @ id)) != $true) & ($true = ((var @ (sub @ X505 @ id))))) | ? [X504 : term,X503 : term] : ((((X501 @ id @ X504 @ id)) = $true) & (((X501 @ id @ X503 @ id)) = $true) & (((X501 @ id @ (ap @ (sub @ X504 @ id) @ X503) @ id)) != $true)) | ? [X506 : (subst > term > term)] : (! [X509 : subst,X510 : subst,X508 : term] : (((X506 @ (comp @ X510 @ X509) @ (sub @ X508 @ X509))) = ((sub @ (X506 @ X510 @ X508) @ X509))) & (((X501 @ id @ (lam @ (X506 @ sh @ one)) @ id)) != $true) & ! [X507 : term] : ((((X501 @ id @ (X506 @ id @ X507) @ id)) = $true) | ($true != ((X501 @ id @ X507 @ id))))) | ? [X517 : subst,X516 : subst,X515 : term,X518 : subst] : ((((X501 @ (comp @ X517 @ X516) @ (sub @ X515 @ X516) @ X518)) != $true) & (((X501 @ X517 @ X515 @ (comp @ X516 @ X518))) = $true))) | ? [X465 : term,X466 : (term > $o)] : (! [X469 : term,X468 : term] : (($true = ((X466 @ (ap @ X468 @ X469)))) | (((X466 @ X468)) != $true) | ($true != ((X466 @ X469)))) & ! [X467 : term] : (($true = ((X466 @ X467))) | (((var @ X467)) != $true)) & ! [X470 : term] : ((((X466 @ (lam @ X470))) = $true) | ? [X471 : term] : ((((X466 @ (sub @ X470 @ (push @ X471 @ id)))) != $true) & (((X466 @ X471)) = $true))) & ($true != ((X466 @ X465)))) | ? [X472 : term,X474 : (subst > term > subst > $o),X473 : term] : (! [X477 : subst,X475 : term,X476 : subst,X478 : subst] : ((((X474 @ (comp @ X476 @ X477) @ (sub @ X475 @ X477) @ X478)) = $true) | ($true != ((X474 @ X476 @ X475 @ (comp @ X477 @ X478))))) & (((X474 @ id @ (ap @ X472 @ X473) @ id)) != $true) & (((X474 @ id @ X473 @ id)) = $true) & ! [X480 : subst,X482 : term,X479 : subst,X481 : subst] : ((((X474 @ X480 @ X482 @ (comp @ X481 @ X479))) = $true) | ($true != ((X474 @ (comp @ X480 @ X481) @ (sub @ X482 @ X481) @ X479)))) & (((X474 @ id @ X472 @ id)) = $true) & ! [X484 : term,X483 : term] : ((((X474 @ id @ X483 @ id)) != $true) | (((X474 @ id @ X484 @ id)) != $true) | (((X474 @ id @ (ap @ (sub @ X483 @ id) @ X484) @ id)) = $true))) | ? [X485 : term,X486 : (subst > term > subst > $o)] : ((((X486 @ id @ (lam @ X485) @ id)) != $true) & ! [X499 : subst,X496 : term,X498 : subst,X497 : subst] : (($true = ((X486 @ (comp @ X499 @ X497) @ (sub @ X496 @ X497) @ X498))) | (((X486 @ X499 @ X496 @ (comp @ X497 @ X498))) != $true)) & ! [X500 : term] : ((((X486 @ id @ X500 @ id)) != $true) | (((X486 @ id @ (sub @ X485 @ (push @ X500 @ id)) @ id)) = $true)) & ! [X488 : term,X490 : subst,X487 : subst,X489 : subst] : (($true = ((X486 @ X489 @ X488 @ (comp @ X490 @ X487)))) | ($true != ((X486 @ (comp @ X489 @ X490) @ (sub @ X488 @ X490) @ X487)))) & ! [X491 : (subst > term > term)] : (? [X493 : subst,X492 : subst,X494 : term] : (((sub @ (X491 @ X492 @ X494) @ X493)) != ((X491 @ (comp @ X492 @ X493) @ (sub @ X494 @ X493)))) | (((X486 @ id @ (lam @ (X491 @ sh @ one)) @ id)) = $true) | ? [X495 : term] : ((((X486 @ id @ X495 @ id)) = $true) & (((X486 @ id @ (X491 @ id @ X495) @ id)) != $true)))))) & (axassoc = $true) & (hoasinduction_gthm = $true) & (! [X399 : subst,X400 : (term > $o),X402 : term,X401 : (term > $o)] : (? [X403 : term] : (((X400 @ X403)) != ((X401 @ (sub @ X403 @ (push @ X402 @ X399))))) | ($true = ((X400 @ one))) | (((X401 @ X402)) != $true)) <=> (pushprop_lem1v2 = $true)) & (hoasapinj1_gthm = $true) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X81 : term] : ? [X82 : (subst > term > term)] : ! [X83 : subst,X84 : term] : (((sub @ X81 @ (push @ X84 @ X83))) = ((X82 @ X83 @ X84)))) & (pushprop_gthm = $true) & ((? [X127 : term] : ! [X128 : (subst > term > term)] : ? [X129 : subst,X130 : term] : (((X128 @ X129 @ X130)) != ((sub @ X127 @ (push @ X130 @ X129)))) | ? [X131 : term] : (((sub @ X131 @ id)) != X131) | ! [X132 : (subst > term > subst > $o),X134 : term,X133 : (term > $o)] : (? [X136 : (subst > term > term)] : ((((X132 @ id @ (lam @ (X136 @ sh @ one)) @ id)) != $true) & ! [X139 : subst,X137 : term,X138 : subst] : (((X136 @ (comp @ X139 @ X138) @ (sub @ X137 @ X138))) = ((sub @ (X136 @ X139 @ X137) @ X138))) & ! [X140 : term] : ((((X132 @ id @ (X136 @ id @ X140) @ id)) = $true) | ($true != ((X132 @ id @ X140 @ id))))) | ? [X135 : term] : (((X133 @ X135)) != ((X132 @ id @ X135 @ id))) | (((X133 @ (lam @ X134))) = $true) | ? [X141 : term] : ((((X133 @ X141)) = $true) & ($true != ((X133 @ (sub @ X134 @ (push @ X141 @ id)))))))) <=> (hoasinduction_lem3v2a_lthm = $true)) & ((? [X282 : term,X283 : term,X284 : term,X285 : term] : ((X283 != X284) & (((ap @ X283 @ X285)) = ((ap @ X284 @ X282)))) | ? [X281 : term] : (((sub @ X281 @ id)) != X281) | ! [X286 : term,X289 : term,X288 : term,X287 : term] : ((X286 = X288) | (((ap @ (sub @ X286 @ id) @ X287)) != ((ap @ (sub @ X288 @ id) @ X289))))) <=> (hoasapinj1_lthm = $true)) & (ulamvar1 != $true) & (hoasinduction_lem1v2 = $true) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (? [X372 : term] : (((sub @ X372 @ id)) != X372) | ! [X384 : (subst > term > subst > $o),X383 : term] : (($true = ((X384 @ id @ X383 @ id))) | ? [X385 : term,X386 : term] : ((((X384 @ id @ (ap @ (sub @ X385 @ id) @ X386) @ id)) != $true) & ($true = ((X384 @ id @ X385 @ id))) & (((X384 @ id @ X386 @ id)) = $true)) | ? [X387 : (subst > term > term)] : (! [X391 : term] : (($true != ((X384 @ id @ X391 @ id))) | (((X384 @ id @ (X387 @ id @ X391) @ id)) = $true)) & (((X384 @ id @ (lam @ (X387 @ sh @ one)) @ id)) != $true) & ! [X389 : subst,X390 : term,X388 : subst] : (((X387 @ (comp @ X389 @ X388) @ (sub @ X390 @ X388))) = ((sub @ (X387 @ X389 @ X390) @ X388))))) | ? [X362 : (subst > term > subst > $o)] : ! [X363 : (term > $o)] : ? [X364 : term] : (((X362 @ id @ X364 @ id)) != ((X363 @ X364))) | ? [X365 : (term > $o),X366 : term] : (! [X367 : term] : (($true != ((var @ X367))) | (((X365 @ X367)) = $true)) & ! [X370 : term] : (? [X371 : term] : ((((X365 @ (sub @ X370 @ (push @ X371 @ id)))) != $true) & (((X365 @ X371)) = $true)) | (((X365 @ (lam @ X370))) = $true)) & (((X365 @ X366)) != $true) & ! [X369 : term,X368 : term] : ((((X365 @ X368)) != $true) | (((X365 @ X369)) != $true) | (((X365 @ (ap @ X369 @ X368))) = $true))) | ? [X374 : (term > $o),X373 : term,X375 : (subst > term > subst > $o)] : (($true != ((X374 @ (lam @ X373)))) & ! [X376 : term] : (($true = ((X374 @ (sub @ X373 @ (push @ X376 @ id))))) | ($true != ((X374 @ X376)))) & ! [X382 : term] : (((X375 @ id @ X382 @ id)) = ((X374 @ X382))) & ! [X377 : (subst > term > term)] : (? [X380 : term,X379 : subst,X378 : subst] : (((sub @ (X377 @ X379 @ X380) @ X378)) != ((X377 @ (comp @ X379 @ X378) @ (sub @ X380 @ X378)))) | ? [X381 : term] : ((((X375 @ id @ X381 @ id)) = $true) & ($true != ((X375 @ id @ (X377 @ id @ X381) @ id)))) | (((X375 @ id @ (lam @ (X377 @ sh @ one)) @ id)) = $true))))) & (pushprop_lem3v2_lthm = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (pushprop_lem2v2_lthm = $true) & ((axvarid = $true) <=> ! [X53 : term] : (((sub @ X53 @ id)) = X53)) & (! [X0 : term,X3 : term,X1 : term,X2 : term] : ((X0 = X2) | (((ap @ (sub @ X2 @ id) @ X1)) != ((ap @ (sub @ X0 @ id) @ X3)))) <=> (hoasapinj1 = $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((termmset = $true) <=> ! [X290 : term] : (((sub @ X290 @ id)) = X290)) & ((hoasapinj2 = $true) <=> ! [X415 : term,X413 : term,X414 : term,X416 : term] : ((X413 = X414) | (((ap @ (sub @ X415 @ id) @ X413)) != ((ap @ (sub @ X416 @ id) @ X414))))) & (axabs = $true) & (pushprop = $true) & (one = ((d2term @ d_one))) & ((hoaslaminj = $true) <=> ! [X306 : subst,X304 : (subst > term > term),X307 : term,X305 : (subst > term > term)] : (? [X310 : subst,X308 : term,X309 : subst] : (((X304 @ (comp @ X310 @ X309) @ (sub @ X308 @ X309))) != ((sub @ (X304 @ X310 @ X308) @ X309))) | (((lam @ (X305 @ sh @ one))) != ((lam @ (X304 @ sh @ one)))) | (((X305 @ X306 @ X307)) = ((X304 @ X306 @ X307))) | ? [X312 : subst,X311 : term,X313 : subst] : (((sub @ (X305 @ X313 @ X311) @ X312)) != ((X305 @ (comp @ X313 @ X312) @ (sub @ X311 @ X312)))))) & (pushprop_lthm = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (hoasinduction_lem3v2_gthm = $true) & (hoasapnotvar = $true) & (! [X314 : (term > $o),X316 : term,X315 : subst] : ? [X317 : (term > $o)] : ! [X318 : term] : (((X314 @ (sub @ X318 @ (push @ X316 @ X315)))) = ((X317 @ X318))) <=> (pushprop_lem0_lthm = $true)) & ((hoaslaminj_lthm = $true) <=> (? [X39 : subst,X40 : term] : (((comp @ sh @ (push @ X40 @ X39))) != X39) | ! [X45 : subst,X46 : (subst > term > term),X43 : term,X44 : (subst > term > term)] : ((((lam @ (X44 @ sh @ one))) != ((lam @ (X46 @ sh @ one)))) | ? [X51 : term,X50 : subst,X52 : subst] : (((X46 @ (comp @ X52 @ X50) @ (sub @ X51 @ X50))) != ((sub @ (X46 @ X52 @ X51) @ X50))) | (((X44 @ X45 @ X43)) = ((X46 @ X45 @ X43))) | ? [X47 : term,X49 : subst,X48 : subst] : (((X44 @ (comp @ X48 @ X49) @ (sub @ X47 @ X49))) != ((sub @ (X44 @ X48 @ X47) @ X49)))) | ? [X38 : subst,X37 : term] : (((sub @ one @ (push @ X37 @ X38))) != X37) | ? [X42 : term,X41 : term] : ((X41 != X42) & (((lam @ X41)) = ((lam @ X42)))))) & (pushprop_lem3v2 = $true) & ((hoasinduction_lem3v2a = $true) <=> ! [X169 : (term > $o),X168 : term,X170 : (subst > term > subst > $o)] : (? [X171 : term] : (($true = ((X169 @ X171))) & ($true != ((X169 @ (sub @ X168 @ (push @ X171 @ id)))))) | ? [X172 : (subst > term > term)] : ((((X170 @ id @ (lam @ (X172 @ sh @ one)) @ id)) != $true) & ! [X173 : subst,X174 : term,X175 : subst] : (((sub @ (X172 @ X173 @ X174) @ X175)) = ((X172 @ (comp @ X173 @ X175) @ (sub @ X174 @ X175)))) & ! [X176 : term] : ((((X170 @ id @ (X172 @ id @ X176) @ id)) = $true) | (((X170 @ id @ X176 @ id)) != $true))) | (((X169 @ (lam @ X168))) = $true) | ? [X177 : term] : (((X169 @ X177)) != ((X170 @ id @ X177 @ id))))) & ((! [X179 : term] : (((sub @ X179 @ id)) = X179) | ? [X178 : term] : (((sub @ X178 @ id)) != X178)) <=> (termmset_lthm = $true)) & (induction2lem_gthm = $true) & (axapp = $true) & ((? [X267 : term,X268 : (term > $o)] : (! [X270 : term,X269 : term] : ((((X268 @ X269)) != $true) | (((X268 @ (ap @ X269 @ X270))) = $true) | (((X268 @ X270)) != $true)) & ! [X272 : term] : ((((X268 @ (lam @ X272))) = $true) | (((X268 @ X272)) != $true)) & ! [X271 : term] : (($true != ((var @ X271))) | (((X268 @ X271)) = $true)) & (((X268 @ X267)) != $true)) | ? [X265 : term,X264 : subst] : (((comp @ sh @ (push @ X265 @ X264))) != X264) | ? [X262 : subst,X263 : term] : (((sub @ one @ (push @ X263 @ X262))) != X263) | ! [X274 : term,X273 : (term > $o),X275 : subst] : (($true = ((X273 @ (sub @ X274 @ X275)))) | ? [X276 : term] : ((((var @ X276)) = $true) & (((X273 @ (sub @ X276 @ X275))) != $true)) | ? [X280 : term,X279 : term] : ((((X273 @ X279)) = $true) & (((X273 @ X280)) = $true) & ($true != ((X273 @ (ap @ X279 @ X280))))) | ? [X277 : term] : ((((X273 @ (lam @ X277))) != $true) & ! [X278 : term] : ((((X273 @ (sub @ X277 @ (push @ X278 @ id)))) = $true) | ($true != ((X273 @ X278)))))) | ? [X266 : subst] : (((comp @ X266 @ id)) != X266)) <=> (induction2lem_lthm = $true)) & (hoasinduction_lem1 = $true) & (pushprop_lem0_gthm = $true) & (hoasinduction_lem3_gthm = $true) & (hoaslamnotap_lthm = $true) & (! [X181 : (subst > term > subst > $o),X180 : term] : ((((X181 @ id @ X180 @ id)) = $true) | ? [X188 : term,X187 : term] : (($true = ((X181 @ id @ X188 @ id))) & (((X181 @ id @ (ap @ (sub @ X187 @ id) @ X188) @ id)) != $true) & ($true = ((X181 @ id @ X187 @ id)))) | ? [X182 : (subst > term > term)] : (! [X183 : term] : ((((X181 @ id @ (X182 @ id @ X183) @ id)) = $true) | (((X181 @ id @ X183 @ id)) != $true)) & ($true != ((X181 @ id @ (lam @ (X182 @ sh @ one)) @ id))) & ! [X186 : term,X185 : subst,X184 : subst] : (((X182 @ (comp @ X185 @ X184) @ (sub @ X186 @ X184))) = ((sub @ (X182 @ X185 @ X186) @ X184))))) <=> (hoasinduction_no_psi_cond = $true)) & (((var @ (d2term @ d_one))) != $true) & ((axscons = $true) <=> ! [X454 : subst] : (((push @ (sub @ one @ X454) @ (comp @ sh @ X454))) = X454)) & (! [X358 : term,X361 : term,X359 : term,X360 : term] : ((X359 = X360) | (((ap @ X359 @ X361)) != ((ap @ X360 @ X358)))) <=> (apinj1 = $true)) & (pushprop_lthm_orig = $true) & (pushprop_lem1_gthm = $true) & (hoasapinj2_gthm = $true) & ((hoasinduction_lem3_lthm = $true) <=> (? [X427 : term,X426 : (subst > term > subst > $o)] : (! [X433 : term] : (($true != ((X426 @ id @ X433 @ id))) | (((X426 @ id @ (sub @ X427 @ (push @ X433 @ id)) @ id)) = $true)) & ($true != ((X426 @ id @ (lam @ (sub @ X427 @ (push @ one @ sh))) @ id))) & ! [X428 : (subst > term > term)] : (? [X431 : term,X432 : subst,X430 : subst] : (((sub @ (X428 @ X432 @ X431) @ X430)) != ((X428 @ (comp @ X432 @ X430) @ (sub @ X431 @ X430)))) | ($true = ((X426 @ id @ (lam @ (X428 @ sh @ one)) @ id))) | ? [X429 : term] : (($true = ((X426 @ id @ X429 @ id))) & (((X426 @ id @ (X428 @ id @ X429) @ id)) != $true)))) | ! [X435 : (subst > term > subst > $o),X434 : term] : ((((X435 @ id @ (lam @ X434) @ id)) = $true) | ? [X447 : subst,X446 : subst,X448 : term,X449 : subst] : ((((X435 @ X446 @ X448 @ (comp @ X449 @ X447))) != $true) & (((X435 @ (comp @ X446 @ X449) @ (sub @ X448 @ X449) @ X447)) = $true)) | ? [X438 : subst,X436 : term,X439 : subst,X437 : subst] : (($true != ((X435 @ (comp @ X438 @ X439) @ (sub @ X436 @ X439) @ X437))) & ($true = ((X435 @ X438 @ X436 @ (comp @ X439 @ X437))))) | ? [X440 : term] : ((((X435 @ id @ X440 @ id)) = $true) & ($true != ((X435 @ id @ (sub @ X434 @ (push @ X440 @ id)) @ id)))) | ? [X441 : (subst > term > term)] : (! [X443 : term,X444 : subst,X442 : subst] : (((X441 @ (comp @ X442 @ X444) @ (sub @ X443 @ X444))) = ((sub @ (X441 @ X442 @ X443) @ X444))) & ($true != ((X435 @ id @ (lam @ (X441 @ sh @ one)) @ id))) & ! [X445 : term] : ((((X435 @ id @ X445 @ id)) != $true) | (((X435 @ id @ (X441 @ id @ X445) @ id)) = $true)))) | ? [X425 : term] : (((sub @ X425 @ id)) != X425))) & ((! [X19 : term,X17 : term,X18 : term,X16 : term] : ((X16 = X19) | (((ap @ (sub @ X17 @ id) @ X19)) != ((ap @ (sub @ X18 @ id) @ X16)))) | ? [X13 : term,X12 : term,X15 : term,X14 : term] : ((((ap @ X13 @ X14)) = ((ap @ X15 @ X12))) & (X12 != X14))) <=> (hoasapinj2_lthm = $true)) & ((hoasinduction_lem2v2 = $true) <=> ! [X148 : term,X147 : term,X146 : (term > $o),X145 : (subst > term > subst > $o)] : ((((X146 @ X147)) != $true) | ? [X157 : subst,X159 : subst,X158 : subst,X156 : term] : ((((X145 @ X157 @ X156 @ (comp @ X158 @ X159))) = $true) & (((X145 @ (comp @ X157 @ X158) @ (sub @ X156 @ X158) @ X159)) != $true)) | (((X146 @ (ap @ X147 @ X148))) = $true) | ? [X149 : term] : (((X145 @ id @ X149 @ id)) != ((X146 @ X149))) | (((X146 @ X148)) != $true) | ? [X154 : term,X155 : term] : ((((X145 @ id @ X155 @ id)) = $true) & ($true != ((X145 @ id @ (ap @ (sub @ X154 @ id) @ X155) @ id))) & (((X145 @ id @ X154 @ id)) = $true)) | ? [X152 : subst,X150 : term,X153 : subst,X151 : subst] : ((((X145 @ X153 @ X150 @ (comp @ X151 @ X152))) != $true) & ($true = ((X145 @ (comp @ X153 @ X151) @ (sub @ X150 @ X151) @ X152)))))) & (hoasinduction_lem3b_gthm = $true) & (! [X409 : term,X408 : subst] : (((comp @ sh @ (push @ X409 @ X408))) = X408) <=> (axshiftcons = $true)) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (! [X521 : term,X520 : term] : ((((lam @ X521)) != ((lam @ X520))) | (X520 = X521)) <=> (laminj = $true)) & ((pushprop_lem1 = $true) <=> ! [X163 : (term > $o),X162 : term,X165 : term,X161 : (term > $o),X164 : subst] : ((((X161 @ X162)) != $true) | (((X163 @ (sub @ X162 @ (push @ X165 @ X164)))) = $true))) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X85 : term] : ? [X86 : (subst > term > term)] : (((sub @ X85 @ (push @ one @ sh))) = ((X86 @ sh @ one)))) & (! [X222 : subst] : (((comp @ id @ X222)) = X222) <=> (axidl = $true)) & ((? [X393 : term,X392 : subst] : (((sub @ one @ (push @ X393 @ X392))) != X393) | ! [X395 : (term > $o),X397 : term,X396 : subst,X394 : (term > $o)] : (? [X398 : term] : (((X395 @ (sub @ X398 @ (push @ X397 @ X396)))) != ((X394 @ X398))) | ($true != ((X395 @ X397))) | ($true = ((X394 @ one))))) <=> (pushprop_lem1v2_lthm = $true)) & (hoasinduction_lem2_lthm = $true) & (! [X142 : (subst > term > subst > $o)] : ? [X143 : (term > $o)] : ! [X144 : term] : (((X142 @ id @ X144 @ id)) = ((X143 @ X144))) <=> (hoasinduction_lem0_lthm = $true)) & (! [X320 : term,X319 : (subst > term > subst > $o)] : (? [X330 : term] : ((((X319 @ id @ X330 @ id)) = $true) & (((X319 @ id @ (sub @ X320 @ (push @ X330 @ id)) @ id)) != $true)) | (((X319 @ id @ (lam @ X320) @ id)) = $true) | ? [X321 : term,X324 : subst,X322 : subst,X323 : subst] : ((((X319 @ X324 @ X321 @ (comp @ X322 @ X323))) = $true) & ($true != ((X319 @ (comp @ X324 @ X322) @ (sub @ X321 @ X322) @ X323)))) | ? [X333 : subst,X331 : subst,X332 : subst,X334 : term] : ((((X319 @ X332 @ X334 @ (comp @ X333 @ X331))) != $true) & (((X319 @ (comp @ X332 @ X333) @ (sub @ X334 @ X333) @ X331)) = $true)) | ? [X325 : (subst > term > term)] : ((((X319 @ id @ (lam @ (X325 @ sh @ one)) @ id)) != $true) & ! [X328 : term,X326 : subst,X327 : subst] : (((sub @ (X325 @ X326 @ X328) @ X327)) = ((X325 @ (comp @ X326 @ X327) @ (sub @ X328 @ X327)))) & ! [X329 : term] : (($true = ((X319 @ id @ (X325 @ id @ X329) @ id))) | (((X319 @ id @ X329 @ id)) != $true)))) <=> (hoasinduction_lem3 = $true)) & (hoaslamnotvar = $true) & ! [X450 : d_term,X451 : d_term] : ((((d2term @ X450)) != ((d2term @ X451))) | (X450 = X451)) & (termmset_gthm = $true) & (! [X117 : term,X115 : (term > $o),X116 : subst] : (? [X118 : term] : (($true = ((var @ X118))) & (((X115 @ (sub @ X118 @ X116))) != $true)) | (((X115 @ (sub @ X117 @ X116))) = $true) | ? [X122 : term,X121 : term] : ((((X115 @ (ap @ X122 @ X121))) != $true) & (((X115 @ X122)) = $true) & (((X115 @ X121)) = $true)) | ? [X119 : term] : (($true != ((X115 @ (lam @ X119)))) & ! [X120 : term] : (($true != ((X115 @ X120))) | (((X115 @ (sub @ X119 @ (push @ X120 @ id)))) = $true)))) <=> (induction2lem = $true)) & ((? [X294 : subst,X293 : term] : (((comp @ sh @ (push @ X293 @ X294))) != X294) | ! [X298 : term,X295 : subst,X297 : term,X299 : (term > $o),X296 : (term > $o)] : ((((X299 @ X297)) != $true) | (((X296 @ (sub @ X297 @ (push @ X298 @ X295)))) = $true)) | ? [X292 : term,X291 : subst] : (((sub @ one @ (push @ X292 @ X291))) != X292)) <=> (pushprop_lem1_lthm = $true)) & (! [X107 : term,X106 : (term > $o)] : (? [X112 : term] : ((((X106 @ X112)) != $true) & (((var @ X112)) = $true)) | ($true = ((X106 @ X107))) | ? [X110 : term,X111 : term] : ((((X106 @ X110)) = $true) & (((X106 @ (ap @ X110 @ X111))) != $true) & (((X106 @ X111)) = $true)) | ? [X108 : term] : (($true != ((X106 @ (lam @ X108)))) & ! [X109 : term] : ((((X106 @ (sub @ X108 @ (push @ X109 @ id)))) = $true) | (((X106 @ X109)) != $true)))) <=> (induction2 = $true)) & ((! [X113 : subst] : (((comp @ X113 @ id)) = X113) & ! [X114 : subst] : (((comp @ id @ X114)) = X114)) <=> (substmonoid = $true)) & (axvarshift = $true) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_lem1_lthm = $true) & ! [X166 : subst] : ? [X167 : d_subst] : (((d2subst @ X167)) = X166) & ((hoasinduction_lem3aa = $true) <=> ! [X457 : (subst > term > subst > $o),X458 : term] : (? [X459 : (subst > term > term)] : (! [X463 : subst,X461 : term,X462 : subst] : (((X459 @ (comp @ X463 @ X462) @ (sub @ X461 @ X462))) = ((sub @ (X459 @ X463 @ X461) @ X462))) & (((X457 @ id @ (lam @ (X459 @ sh @ one)) @ id)) != $true) & ! [X460 : term] : (($true != ((X457 @ id @ X460 @ id))) | (((X457 @ id @ (X459 @ id @ X460) @ id)) = $true))) | (((X457 @ id @ (lam @ (sub @ X458 @ (push @ one @ sh))) @ id)) = $true) | ? [X464 : term] : ((((X457 @ id @ X464 @ id)) = $true) & (((X457 @ id @ (sub @ X458 @ (push @ X464 @ id)) @ id)) != $true)))) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem1v2_gthm = $true) & ((shinj = $true) <=> ! [X455 : term,X456 : term] : ((((sub @ X455 @ sh)) != ((sub @ X456 @ sh))) | (X455 = X456))) & ! [X74 : d_subst] : (d_id = X74) & (pushprop_lem2v2 = $true) & (apnotvar = $true) & ((? [X300 : subst] : (((comp @ id @ X300)) != X300) | ? [X301 : subst] : (((comp @ X301 @ id)) != X301) | (! [X302 : subst] : (((comp @ X302 @ id)) = X302) & ! [X303 : subst] : (((comp @ id @ X303)) = X303))) <=> (substmonoid_lthm = $true)) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & ((pushprop_lem0 = $true) <=> ! [X22 : (term > $o),X21 : subst,X20 : term] : ? [X23 : (term > $o)] : ! [X24 : term] : (((X23 @ X24)) = ((X22 @ (sub @ X24 @ (push @ X20 @ X21)))))) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (! [X123 : term] : ? [X124 : (subst > term > term)] : ! [X125 : subst,X126 : term] : (((sub @ X123 @ (push @ X126 @ X125))) = ((X124 @ X125 @ X126))) <=> (hoasinduction_lem3v2_f = $true)) & (hoasinduction_lem3a_gthm = $true) & ((hoasinduction_lem3a = $true) <=> ! [X351 : term,X350 : (subst > term > subst > $o)] : (? [X357 : term] : ((((X350 @ id @ X357 @ id)) = $true) & (((X350 @ id @ (sub @ X351 @ (push @ X357 @ id)) @ id)) != $true)) | (((X350 @ id @ (lam @ X351) @ id)) = $true) | ? [X352 : (subst > term > term)] : (! [X355 : subst,X354 : subst,X353 : term] : (((X352 @ (comp @ X355 @ X354) @ (sub @ X353 @ X354))) = ((sub @ (X352 @ X355 @ X353) @ X354))) & (((X350 @ id @ (lam @ (X352 @ sh @ one)) @ id)) != $true) & ! [X356 : term] : ((((X350 @ id @ X356 @ id)) != $true) | (((X350 @ id @ (X352 @ id @ X356) @ id)) = $true))))) & ((? [X87 : term] : (((sub @ X87 @ id)) != X87) | ! [X88 : (subst > term > subst > $o),X90 : (term > $o),X89 : term] : (? [X100 : term] : (((X90 @ X100)) != ((X88 @ id @ X100 @ id))) | ? [X91 : term,X94 : subst,X93 : subst,X92 : subst] : (($true != ((X88 @ (comp @ X93 @ X92) @ (sub @ X91 @ X92) @ X94))) & (((X88 @ X93 @ X91 @ (comp @ X92 @ X94))) = $true)) | ? [X103 : subst,X104 : term,X101 : subst,X102 : subst] : ((((X88 @ (comp @ X102 @ X101) @ (sub @ X104 @ X101) @ X103)) = $true) & (((X88 @ X102 @ X104 @ (comp @ X101 @ X103))) != $true)) | ? [X105 : term] : ((((X90 @ X105)) = $true) & (((X90 @ (sub @ X89 @ (push @ X105 @ id)))) != $true)) | ? [X95 : (subst > term > term)] : (! [X96 : term] : (($true = ((X88 @ id @ (X95 @ id @ X96) @ id))) | (((X88 @ id @ X96 @ id)) != $true)) & (((X88 @ id @ (lam @ (X95 @ sh @ one)) @ id)) != $true) & ! [X98 : term,X99 : subst,X97 : subst] : (((sub @ (X95 @ X97 @ X98) @ X99)) = ((X95 @ (comp @ X97 @ X99) @ (sub @ X98 @ X99))))) | (((X90 @ (lam @ X89))) = $true))) <=> (hoasinduction_lem3v2_lthm = $true)) & (! [X337 : term,X335 : (subst > term > subst > $o),X336 : term] : ((((X335 @ id @ (ap @ X337 @ X336) @ id)) = $true) | ($true != ((X335 @ id @ X337 @ id))) | ? [X345 : subst,X343 : subst,X342 : subst,X344 : term] : (($true = ((X335 @ (comp @ X343 @ X345) @ (sub @ X344 @ X345) @ X342))) & ($true != ((X335 @ X343 @ X344 @ (comp @ X345 @ X342))))) | ? [X340 : subst,X341 : term,X339 : subst,X338 : subst] : ((((X335 @ (comp @ X339 @ X338) @ (sub @ X341 @ X338) @ X340)) != $true) & ($true = ((X335 @ X339 @ X341 @ (comp @ X338 @ X340))))) | ? [X346 : term,X347 : term] : ((((X335 @ id @ X347 @ id)) = $true) & (((X335 @ id @ (ap @ (sub @ X346 @ id) @ X347) @ id)) != $true) & ($true = ((X335 @ id @ X346 @ id)))) | ($true != ((X335 @ id @ X336 @ id)))) <=> (hoasinduction_lem2 = $true)) & (ulamvarsh = $true) & (hoasapnotvar_lthm = $true) & ((hoasinduction_lem3aaa = $true) <=> ! [X26 : (subst > term > subst > $o),X25 : term] : (? [X28 : term,X29 : (subst > term > term)] : (! [X30 : term,X31 : subst,X32 : subst] : (((sub @ (sub @ X28 @ (push @ X30 @ X31)) @ X32)) = ((sub @ (X29 @ X31 @ X30) @ X32))) & ! [X36 : term] : ((((X26 @ id @ (X29 @ id @ X36) @ id)) = $true) | (((X26 @ id @ X36 @ id)) != $true)) & ! [X33 : subst,X35 : term,X34 : subst] : (((X29 @ (comp @ X33 @ X34) @ (sub @ X35 @ X34))) = ((sub @ X28 @ (push @ (sub @ X35 @ X34) @ (comp @ X33 @ X34))))) & (((X26 @ id @ (lam @ (X29 @ sh @ one)) @ id)) != $true)) | ? [X27 : term] : ((((X26 @ id @ X27 @ id)) = $true) & ($true != ((X26 @ id @ (sub @ X25 @ (push @ X27 @ id)) @ id)))) | (((X26 @ id @ (lam @ (sub @ X25 @ (push @ one @ sh))) @ id)) = $true))) & ((axidr = $true) <=> ! [X519 : subst] : (((comp @ X519 @ id)) = X519)) & (hoaslamnotap_gthm = $true) & ((hoaslamnotap = $true) <=> ! [X419 : term,X417 : (subst > term > term),X418 : term] : (? [X421 : subst,X422 : term,X420 : subst] : (((X417 @ (comp @ X420 @ X421) @ (sub @ X422 @ X421))) != ((sub @ (X417 @ X420 @ X422) @ X421))) | (((lam @ (X417 @ sh @ one))) != ((ap @ (sub @ X419 @ id) @ X418))))) & (id = ((d2subst @ d_id))) & ! [X348 : term] : ? [X349 : d_term] : (((d2term @ X349)) = X348) & (axmap = $true) & (! [X523 : term,X522 : (term > $o),X524 : (subst > term > subst > $o)] : (? [X533 : term] : ((((X522 @ (sub @ X523 @ (push @ X533 @ id)))) != $true) & ($true = ((X522 @ X533)))) | ? [X539 : term] : (((X524 @ id @ X539 @ id)) != ((X522 @ X539))) | ? [X530 : subst,X529 : subst,X532 : subst,X531 : term] : ((((X524 @ X532 @ X531 @ (comp @ X530 @ X529))) != $true) & (((X524 @ (comp @ X532 @ X530) @ (sub @ X531 @ X530) @ X529)) = $true)) | ? [X534 : (subst > term > term)] : (! [X538 : subst,X537 : term,X536 : subst] : (((X534 @ (comp @ X536 @ X538) @ (sub @ X537 @ X538))) = ((sub @ (X534 @ X536 @ X537) @ X538))) & ! [X535 : term] : ((((X524 @ id @ (X534 @ id @ X535) @ id)) = $true) | ($true != ((X524 @ id @ X535 @ id)))) & (((X524 @ id @ (lam @ (X534 @ sh @ one)) @ id)) != $true)) | ? [X528 : subst,X526 : subst,X525 : subst,X527 : term] : ((((X524 @ (comp @ X525 @ X526) @ (sub @ X527 @ X526) @ X528)) != $true) & ($true = ((X524 @ X525 @ X527 @ (comp @ X526 @ X528))))) | (((X522 @ (lam @ X523))) = $true)) <=> (hoasinduction_lem3v2 = $true)) & (hoasinduction_lem2v2_gthm = $true) & (! [X423 : subst,X424 : term] : (((sub @ one @ (push @ X424 @ X423))) = X424) <=> (axvarcons = $true)) & ((hoasinduction = $true) <=> ! [X57 : (subst > term > subst > $o),X56 : term] : (? [X69 : term,X68 : term] : ((((X57 @ id @ (ap @ (sub @ X69 @ id) @ X68) @ id)) != $true) & ($true = ((X57 @ id @ X68 @ id))) & (((X57 @ id @ X69 @ id)) = $true)) | ? [X60 : subst,X59 : term,X61 : subst,X62 : subst] : ((((X57 @ (comp @ X60 @ X61) @ (sub @ X59 @ X61) @ X62)) != $true) & (((X57 @ X60 @ X59 @ (comp @ X61 @ X62))) = $true)) | ? [X72 : subst,X71 : term,X73 : subst,X70 : subst] : (($true != ((X57 @ X70 @ X71 @ (comp @ X73 @ X72)))) & (((X57 @ (comp @ X70 @ X73) @ (sub @ X71 @ X73) @ X72)) = $true)) | ? [X58 : term] : ((((var @ (sub @ X58 @ id))) = $true) & (((X57 @ id @ X58 @ id)) != $true)) | (((X57 @ id @ X56 @ id)) = $true) | ? [X63 : (subst > term > term)] : (($true != ((X57 @ id @ (lam @ (X63 @ sh @ one)) @ id))) & ! [X64 : term,X66 : subst,X65 : subst] : (((X63 @ (comp @ X65 @ X66) @ (sub @ X64 @ X66))) = ((sub @ (X63 @ X65 @ X64) @ X66))) & ! [X67 : term] : ((((X57 @ id @ X67 @ id)) != $true) | (((X57 @ id @ (X63 @ id @ X67) @ id)) = $true))))) & (axclos = $true) & (hoaslamnotvar_gthm = $true) & (lamnotap != $true) & ((hoasinduction_lem0 = $true) <=> ! [X410 : (subst > term > subst > $o)] : ? [X411 : (term > $o)] : ! [X412 : term] : (((X410 @ id @ X412 @ id)) = ((X411 @ X412)))) & ((hoasinduction_lem3b = $true) <=> ! [X54 : term] : ? [X55 : (subst > term > term)] : (((X55 @ sh @ one)) = ((sub @ X54 @ (push @ one @ sh))))) & (hoaslaminj_gthm = $true) & ((induction = $true) <=> ! [X75 : term,X76 : (term > $o)] : ((((X76 @ X75)) = $true) | ? [X78 : term] : (($true = ((X76 @ X78))) & (((X76 @ (lam @ X78))) != $true)) | ? [X79 : term,X80 : term] : (($true = ((X76 @ X80))) & (((X76 @ X79)) = $true) & (((X76 @ (ap @ X79 @ X80))) != $true)) | ? [X77 : term] : ((((var @ X77)) = $true) & (((X76 @ X77)) != $true)))) & (lamnotvar = $true) & (hoasinduction_lem1_gthm = $true) & ((apinj2 = $true) <=> ! [X405 : term,X404 : term,X406 : term,X407 : term] : ((((ap @ X404 @ X407)) != ((ap @ X405 @ X406))) | (X406 = X407))) & (sh = ((d2subst @ d_id))) & (ulamvarind = $true) & (substmonoid_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & ! [X160 : d_term] : (d_one = X160) & (hoaslamnotvar_lthm = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))))),
  inference(flattening,[],[f11])).
thf(f13,definition,(
  ! [X76 : (term > $o)] : ((((sP0 @ X76)) = $true) <=> ? [X79 : term,X80 : term] : (($true = ((X76 @ X80))) & (((X76 @ X79)) = $true) & (((X76 @ (ap @ X79 @ X80))) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f14,definition,(
  ! [X57 : (subst > term > subst > $o)] : (($true = ((sP1 @ X57))) <=> ? [X63 : (subst > term > term)] : (($true != ((X57 @ id @ (lam @ (X63 @ sh @ one)) @ id))) & ! [X64 : term,X66 : subst,X65 : subst] : (((X63 @ (comp @ X65 @ X66) @ (sub @ X64 @ X66))) = ((sub @ (X63 @ X65 @ X64) @ X66))) & ! [X67 : term] : ((((X57 @ id @ X67 @ id)) != $true) | (((X57 @ id @ (X63 @ id @ X67) @ id)) = $true))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f15,definition,(
  ! [X57 : (subst > term > subst > $o)] : ((((sP2 @ X57)) = $true) <=> ? [X69 : term,X68 : term] : ((((X57 @ id @ (ap @ (sub @ X69 @ id) @ X68) @ id)) != $true) & ($true = ((X57 @ id @ X68 @ id))) & (((X57 @ id @ X69 @ id)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f16,definition,(
  (sP3 = $true) <=> ! [X57 : (subst > term > subst > $o),X56 : term] : ((((sP2 @ X57)) = $true) | ? [X60 : subst,X59 : term,X61 : subst,X62 : subst] : ((((X57 @ (comp @ X60 @ X61) @ (sub @ X59 @ X61) @ X62)) != $true) & (((X57 @ X60 @ X59 @ (comp @ X61 @ X62))) = $true)) | ? [X72 : subst,X71 : term,X73 : subst,X70 : subst] : (($true != ((X57 @ X70 @ X71 @ (comp @ X73 @ X72)))) & (((X57 @ (comp @ X70 @ X73) @ (sub @ X71 @ X73) @ X72)) = $true)) | ? [X58 : term] : ((((var @ (sub @ X58 @ id))) = $true) & (((X57 @ id @ X58 @ id)) != $true)) | (((X57 @ id @ X56 @ id)) = $true) | ($true = ((sP1 @ X57))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f17,definition,(
  ! [X524 : (subst > term > subst > $o)] : ((((sP4 @ X524)) = $true) <=> ? [X534 : (subst > term > term)] : (! [X538 : subst,X537 : term,X536 : subst] : (((X534 @ (comp @ X536 @ X538) @ (sub @ X537 @ X538))) = ((sub @ (X534 @ X536 @ X537) @ X538))) & ! [X535 : term] : ((((X524 @ id @ (X534 @ id @ X535) @ id)) = $true) | ($true != ((X524 @ id @ X535 @ id)))) & (((X524 @ id @ (lam @ (X534 @ sh @ one)) @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f18,definition,(
  (sP5 = $true) <=> ! [X523 : term,X522 : (term > $o),X524 : (subst > term > subst > $o)] : (? [X533 : term] : ((((X522 @ (sub @ X523 @ (push @ X533 @ id)))) != $true) & ($true = ((X522 @ X533)))) | ? [X539 : term] : (((X524 @ id @ X539 @ id)) != ((X522 @ X539))) | ? [X530 : subst,X529 : subst,X532 : subst,X531 : term] : ((((X524 @ X532 @ X531 @ (comp @ X530 @ X529))) != $true) & (((X524 @ (comp @ X532 @ X530) @ (sub @ X531 @ X530) @ X529)) = $true)) | (((sP4 @ X524)) = $true) | ? [X528 : subst,X526 : subst,X525 : subst,X527 : term] : ((((X524 @ (comp @ X525 @ X526) @ (sub @ X527 @ X526) @ X528)) != $true) & ($true = ((X524 @ X525 @ X527 @ (comp @ X526 @ X528))))) | (((X522 @ (lam @ X523))) = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f19,definition,(
  ($true = sP6) <=> ! [X26 : (subst > term > subst > $o),X25 : term] : (? [X28 : term,X29 : (subst > term > term)] : (! [X30 : term,X31 : subst,X32 : subst] : (((sub @ (sub @ X28 @ (push @ X30 @ X31)) @ X32)) = ((sub @ (X29 @ X31 @ X30) @ X32))) & ! [X36 : term] : ((((X26 @ id @ (X29 @ id @ X36) @ id)) = $true) | (((X26 @ id @ X36 @ id)) != $true)) & ! [X33 : subst,X35 : term,X34 : subst] : (((X29 @ (comp @ X33 @ X34) @ (sub @ X35 @ X34))) = ((sub @ X28 @ (push @ (sub @ X35 @ X34) @ (comp @ X33 @ X34))))) & (((X26 @ id @ (lam @ (X29 @ sh @ one)) @ id)) != $true)) | ? [X27 : term] : ((((X26 @ id @ X27 @ id)) = $true) & ($true != ((X26 @ id @ (sub @ X25 @ (push @ X27 @ id)) @ id)))) | (((X26 @ id @ (lam @ (sub @ X25 @ (push @ one @ sh))) @ id)) = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f20,definition,(
  ! [X335 : (subst > term > subst > $o)] : ((((sP7 @ X335)) = $true) <=> ? [X346 : term,X347 : term] : ((((X335 @ id @ X347 @ id)) = $true) & (((X335 @ id @ (ap @ (sub @ X346 @ id) @ X347) @ id)) != $true) & ($true = ((X335 @ id @ X346 @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f21,definition,(
  (sP8 = $true) <=> ! [X337 : term,X335 : (subst > term > subst > $o),X336 : term] : ((((X335 @ id @ (ap @ X337 @ X336) @ id)) = $true) | ($true != ((X335 @ id @ X337 @ id))) | ? [X345 : subst,X343 : subst,X342 : subst,X344 : term] : (($true = ((X335 @ (comp @ X343 @ X345) @ (sub @ X344 @ X345) @ X342))) & ($true != ((X335 @ X343 @ X344 @ (comp @ X345 @ X342))))) | ? [X340 : subst,X341 : term,X339 : subst,X338 : subst] : ((((X335 @ (comp @ X339 @ X338) @ (sub @ X341 @ X338) @ X340)) != $true) & ($true = ((X335 @ X339 @ X341 @ (comp @ X338 @ X340))))) | (((sP7 @ X335)) = $true) | ($true != ((X335 @ id @ X336 @ id))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f22,definition,(
  ! [X88 : (subst > term > subst > $o)] : (($true = ((sP9 @ X88))) <=> ? [X95 : (subst > term > term)] : (! [X96 : term] : (($true = ((X88 @ id @ (X95 @ id @ X96) @ id))) | (((X88 @ id @ X96 @ id)) != $true)) & (((X88 @ id @ (lam @ (X95 @ sh @ one)) @ id)) != $true) & ! [X98 : term,X99 : subst,X97 : subst] : (((sub @ (X95 @ X97 @ X98) @ X99)) = ((X95 @ (comp @ X97 @ X99) @ (sub @ X98 @ X99))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f23,definition,(
  ($true = sP10) <=> (? [X87 : term] : (((sub @ X87 @ id)) != X87) | ! [X88 : (subst > term > subst > $o),X90 : (term > $o),X89 : term] : (? [X100 : term] : (((X90 @ X100)) != ((X88 @ id @ X100 @ id))) | ? [X91 : term,X94 : subst,X93 : subst,X92 : subst] : (($true != ((X88 @ (comp @ X93 @ X92) @ (sub @ X91 @ X92) @ X94))) & (((X88 @ X93 @ X91 @ (comp @ X92 @ X94))) = $true)) | ? [X103 : subst,X104 : term,X101 : subst,X102 : subst] : ((((X88 @ (comp @ X102 @ X101) @ (sub @ X104 @ X101) @ X103)) = $true) & (((X88 @ X102 @ X104 @ (comp @ X101 @ X103))) != $true)) | ? [X105 : term] : ((((X90 @ X105)) = $true) & (((X90 @ (sub @ X89 @ (push @ X105 @ id)))) != $true)) | ($true = ((sP9 @ X88))) | (((X90 @ (lam @ X89))) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f24,definition,(
  ($true = sP11) <=> ! [X351 : term,X350 : (subst > term > subst > $o)] : (? [X357 : term] : ((((X350 @ id @ X357 @ id)) = $true) & (((X350 @ id @ (sub @ X351 @ (push @ X357 @ id)) @ id)) != $true)) | (((X350 @ id @ (lam @ X351) @ id)) = $true) | ? [X352 : (subst > term > term)] : (! [X355 : subst,X354 : subst,X353 : term] : (((X352 @ (comp @ X355 @ X354) @ (sub @ X353 @ X354))) = ((sub @ (X352 @ X355 @ X353) @ X354))) & (((X350 @ id @ (lam @ (X352 @ sh @ one)) @ id)) != $true) & ! [X356 : term] : ((((X350 @ id @ X356 @ id)) != $true) | (((X350 @ id @ (X352 @ id @ X356) @ id)) = $true))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f25,definition,(
  (sP12 = $true) <=> ! [X457 : (subst > term > subst > $o),X458 : term] : (? [X459 : (subst > term > term)] : (! [X463 : subst,X461 : term,X462 : subst] : (((X459 @ (comp @ X463 @ X462) @ (sub @ X461 @ X462))) = ((sub @ (X459 @ X463 @ X461) @ X462))) & (((X457 @ id @ (lam @ (X459 @ sh @ one)) @ id)) != $true) & ! [X460 : term] : (($true != ((X457 @ id @ X460 @ id))) | (((X457 @ id @ (X459 @ id @ X460) @ id)) = $true))) | (((X457 @ id @ (lam @ (sub @ X458 @ (push @ one @ sh))) @ id)) = $true) | ? [X464 : term] : ((((X457 @ id @ X464 @ id)) = $true) & (((X457 @ id @ (sub @ X458 @ (push @ X464 @ id)) @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f26,definition,(
  ! [X106 : (term > $o)] : (($true = ((sP13 @ X106))) <=> ? [X110 : term,X111 : term] : ((((X106 @ X110)) = $true) & (((X106 @ (ap @ X110 @ X111))) != $true) & (((X106 @ X111)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f27,definition,(
  (sP14 = $true) <=> ! [X107 : term,X106 : (term > $o)] : (? [X112 : term] : ((((X106 @ X112)) != $true) & (((var @ X112)) = $true)) | ($true = ((X106 @ X107))) | ($true = ((sP13 @ X106))) | ? [X108 : term] : (($true != ((X106 @ (lam @ X108)))) & ! [X109 : term] : ((((X106 @ (sub @ X108 @ (push @ X109 @ id)))) = $true) | (((X106 @ X109)) != $true))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f28,definition,(
  ! [X115 : (term > $o)] : (($true = ((sP15 @ X115))) <=> ? [X122 : term,X121 : term] : ((((X115 @ (ap @ X122 @ X121))) != $true) & (((X115 @ X122)) = $true) & (((X115 @ X121)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f29,definition,(
  (sP16 = $true) <=> ! [X117 : term,X115 : (term > $o),X116 : subst] : (? [X118 : term] : (($true = ((var @ X118))) & (((X115 @ (sub @ X118 @ X116))) != $true)) | (((X115 @ (sub @ X117 @ X116))) = $true) | ($true = ((sP15 @ X115))) | ? [X119 : term] : (($true != ((X115 @ (lam @ X119)))) & ! [X120 : term] : (($true != ((X115 @ X120))) | (((X115 @ (sub @ X119 @ (push @ X120 @ id)))) = $true))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f30,definition,(
  ! [X319 : (subst > term > subst > $o)] : ((((sP17 @ X319)) = $true) <=> ? [X325 : (subst > term > term)] : ((((X319 @ id @ (lam @ (X325 @ sh @ one)) @ id)) != $true) & ! [X328 : term,X326 : subst,X327 : subst] : (((sub @ (X325 @ X326 @ X328) @ X327)) = ((X325 @ (comp @ X326 @ X327) @ (sub @ X328 @ X327)))) & ! [X329 : term] : (($true = ((X319 @ id @ (X325 @ id @ X329) @ id))) | (((X319 @ id @ X329 @ id)) != $true))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f31,definition,(
  ($true = sP18) <=> ! [X320 : term,X319 : (subst > term > subst > $o)] : (? [X330 : term] : ((((X319 @ id @ X330 @ id)) = $true) & (((X319 @ id @ (sub @ X320 @ (push @ X330 @ id)) @ id)) != $true)) | (((X319 @ id @ (lam @ X320) @ id)) = $true) | ? [X321 : term,X324 : subst,X322 : subst,X323 : subst] : ((((X319 @ X324 @ X321 @ (comp @ X322 @ X323))) = $true) & ($true != ((X319 @ (comp @ X324 @ X322) @ (sub @ X321 @ X322) @ X323)))) | ? [X333 : subst,X331 : subst,X332 : subst,X334 : term] : ((((X319 @ X332 @ X334 @ (comp @ X333 @ X331))) != $true) & (((X319 @ (comp @ X332 @ X333) @ (sub @ X334 @ X333) @ X331)) = $true)) | (((sP17 @ X319)) = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f32,definition,(
  ! [X145 : (subst > term > subst > $o)] : (($true = ((sP19 @ X145))) <=> ? [X154 : term,X155 : term] : ((((X145 @ id @ X155 @ id)) = $true) & ($true != ((X145 @ id @ (ap @ (sub @ X154 @ id) @ X155) @ id))) & (((X145 @ id @ X154 @ id)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f33,definition,(
  ($true = sP20) <=> ! [X148 : term,X147 : term,X146 : (term > $o),X145 : (subst > term > subst > $o)] : ((((X146 @ X147)) != $true) | ? [X157 : subst,X159 : subst,X158 : subst,X156 : term] : ((((X145 @ X157 @ X156 @ (comp @ X158 @ X159))) = $true) & (((X145 @ (comp @ X157 @ X158) @ (sub @ X156 @ X158) @ X159)) != $true)) | (((X146 @ (ap @ X147 @ X148))) = $true) | ? [X149 : term] : (((X145 @ id @ X149 @ id)) != ((X146 @ X149))) | (((X146 @ X148)) != $true) | ($true = ((sP19 @ X145))) | ? [X152 : subst,X150 : term,X153 : subst,X151 : subst] : ((((X145 @ X153 @ X150 @ (comp @ X151 @ X152))) != $true) & ($true = ((X145 @ (comp @ X153 @ X151) @ (sub @ X150 @ X151) @ X152)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f34,definition,(
  ! [X435 : (subst > term > subst > $o)] : ((((sP21 @ X435)) = $true) <=> ? [X441 : (subst > term > term)] : (! [X443 : term,X444 : subst,X442 : subst] : (((X441 @ (comp @ X442 @ X444) @ (sub @ X443 @ X444))) = ((sub @ (X441 @ X442 @ X443) @ X444))) & ($true != ((X435 @ id @ (lam @ (X441 @ sh @ one)) @ id))) & ! [X445 : term] : ((((X435 @ id @ X445 @ id)) != $true) | (((X435 @ id @ (X441 @ id @ X445) @ id)) = $true))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f35,definition,(
  (sP22 = $true) <=> ? [X427 : term,X426 : (subst > term > subst > $o)] : (! [X433 : term] : (($true != ((X426 @ id @ X433 @ id))) | (((X426 @ id @ (sub @ X427 @ (push @ X433 @ id)) @ id)) = $true)) & ($true != ((X426 @ id @ (lam @ (sub @ X427 @ (push @ one @ sh))) @ id))) & ! [X428 : (subst > term > term)] : (? [X431 : term,X432 : subst,X430 : subst] : (((sub @ (X428 @ X432 @ X431) @ X430)) != ((X428 @ (comp @ X432 @ X430) @ (sub @ X431 @ X430)))) | ($true = ((X426 @ id @ (lam @ (X428 @ sh @ one)) @ id))) | ? [X429 : term] : (($true = ((X426 @ id @ X429 @ id))) & (((X426 @ id @ (X428 @ id @ X429) @ id)) != $true))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f36,definition,(
  ($true = sP23) <=> ((sP22 = $true) | ! [X435 : (subst > term > subst > $o),X434 : term] : ((((X435 @ id @ (lam @ X434) @ id)) = $true) | ? [X447 : subst,X446 : subst,X448 : term,X449 : subst] : ((((X435 @ X446 @ X448 @ (comp @ X449 @ X447))) != $true) & (((X435 @ (comp @ X446 @ X449) @ (sub @ X448 @ X449) @ X447)) = $true)) | ? [X438 : subst,X436 : term,X439 : subst,X437 : subst] : (($true != ((X435 @ (comp @ X438 @ X439) @ (sub @ X436 @ X439) @ X437))) & ($true = ((X435 @ X438 @ X436 @ (comp @ X439 @ X437))))) | ? [X440 : term] : ((((X435 @ id @ X440 @ id)) = $true) & ($true != ((X435 @ id @ (sub @ X434 @ (push @ X440 @ id)) @ id)))) | (((sP21 @ X435)) = $true)) | ? [X425 : term] : (((sub @ X425 @ id)) != X425))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f37,definition,(
  ! [X181 : (subst > term > subst > $o)] : ((((sP24 @ X181)) = $true) <=> ? [X182 : (subst > term > term)] : (! [X183 : term] : ((((X181 @ id @ (X182 @ id @ X183) @ id)) = $true) | (((X181 @ id @ X183 @ id)) != $true)) & ($true != ((X181 @ id @ (lam @ (X182 @ sh @ one)) @ id))) & ! [X186 : term,X185 : subst,X184 : subst] : (((X182 @ (comp @ X185 @ X184) @ (sub @ X186 @ X184))) = ((sub @ (X182 @ X185 @ X186) @ X184)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f38,definition,(
  ! [X273 : (term > $o)] : ((((sP25 @ X273)) = $true) <=> ? [X280 : term,X279 : term] : ((((X273 @ X279)) = $true) & (((X273 @ X280)) = $true) & ($true != ((X273 @ (ap @ X279 @ X280))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f39,definition,(
  ! [X268 : (term > $o)] : ((((sP26 @ X268)) = $true) <=> ! [X270 : term,X269 : term] : ((((X268 @ X269)) != $true) | (((X268 @ (ap @ X269 @ X270))) = $true) | (((X268 @ X270)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f40,definition,(
  ($true = sP27) <=> ! [X274 : term,X273 : (term > $o),X275 : subst] : (($true = ((X273 @ (sub @ X274 @ X275)))) | ? [X276 : term] : ((((var @ X276)) = $true) & (((X273 @ (sub @ X276 @ X275))) != $true)) | (((sP25 @ X273)) = $true) | ? [X277 : term] : ((((X273 @ (lam @ X277))) != $true) & ! [X278 : term] : ((((X273 @ (sub @ X277 @ (push @ X278 @ id)))) = $true) | ($true != ((X273 @ X278))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f41,definition,(
  ($true = sP28) <=> (? [X267 : term,X268 : (term > $o)] : ((((sP26 @ X268)) = $true) & ! [X272 : term] : ((((X268 @ (lam @ X272))) = $true) | (((X268 @ X272)) != $true)) & ! [X271 : term] : (($true != ((var @ X271))) | (((X268 @ X271)) = $true)) & (((X268 @ X267)) != $true)) | ? [X265 : term,X264 : subst] : (((comp @ sh @ (push @ X265 @ X264))) != X264) | ? [X262 : subst,X263 : term] : (((sub @ one @ (push @ X263 @ X262))) != X263) | ($true = sP27) | ? [X266 : subst] : (((comp @ X266 @ id)) != X266))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f42,definition,(
  ($true = sP29) <=> ! [X169 : (term > $o),X168 : term,X170 : (subst > term > subst > $o)] : (? [X171 : term] : (($true = ((X169 @ X171))) & ($true != ((X169 @ (sub @ X168 @ (push @ X171 @ id)))))) | ? [X172 : (subst > term > term)] : ((((X170 @ id @ (lam @ (X172 @ sh @ one)) @ id)) != $true) & ! [X173 : subst,X174 : term,X175 : subst] : (((sub @ (X172 @ X173 @ X174) @ X175)) = ((X172 @ (comp @ X173 @ X175) @ (sub @ X174 @ X175)))) & ! [X176 : term] : ((((X170 @ id @ (X172 @ id @ X176) @ id)) = $true) | (((X170 @ id @ X176 @ id)) != $true))) | (((X169 @ (lam @ X168))) = $true) | ? [X177 : term] : (((X169 @ X177)) != ((X170 @ id @ X177 @ id))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f43,definition,(
  ($true = sP30) <=> (? [X39 : subst,X40 : term] : (((comp @ sh @ (push @ X40 @ X39))) != X39) | ! [X45 : subst,X46 : (subst > term > term),X43 : term,X44 : (subst > term > term)] : ((((lam @ (X44 @ sh @ one))) != ((lam @ (X46 @ sh @ one)))) | ? [X51 : term,X50 : subst,X52 : subst] : (((X46 @ (comp @ X52 @ X50) @ (sub @ X51 @ X50))) != ((sub @ (X46 @ X52 @ X51) @ X50))) | (((X44 @ X45 @ X43)) = ((X46 @ X45 @ X43))) | ? [X47 : term,X49 : subst,X48 : subst] : (((X44 @ (comp @ X48 @ X49) @ (sub @ X47 @ X49))) != ((sub @ (X44 @ X48 @ X47) @ X49)))) | ? [X38 : subst,X37 : term] : (((sub @ one @ (push @ X37 @ X38))) != X37) | ? [X42 : term,X41 : term] : ((X41 != X42) & (((lam @ X41)) = ((lam @ X42)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f44,definition,(
  ! [X365 : (term > $o)] : ((((sP31 @ X365)) = $true) <=> ! [X369 : term,X368 : term] : ((((X365 @ X368)) != $true) | (((X365 @ X369)) != $true) | (((X365 @ (ap @ X369 @ X368))) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f45,definition,(
  ! [X384 : (subst > term > subst > $o)] : ((((sP32 @ X384)) = $true) <=> ? [X387 : (subst > term > term)] : (! [X391 : term] : (($true != ((X384 @ id @ X391 @ id))) | (((X384 @ id @ (X387 @ id @ X391) @ id)) = $true)) & (((X384 @ id @ (lam @ (X387 @ sh @ one)) @ id)) != $true) & ! [X389 : subst,X390 : term,X388 : subst] : (((X387 @ (comp @ X389 @ X388) @ (sub @ X390 @ X388))) = ((sub @ (X387 @ X389 @ X390) @ X388)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f46,definition,(
  (sP33 = $true) <=> ? [X374 : (term > $o),X373 : term,X375 : (subst > term > subst > $o)] : (($true != ((X374 @ (lam @ X373)))) & ! [X376 : term] : (($true = ((X374 @ (sub @ X373 @ (push @ X376 @ id))))) | ($true != ((X374 @ X376)))) & ! [X382 : term] : (((X375 @ id @ X382 @ id)) = ((X374 @ X382))) & ! [X377 : (subst > term > term)] : (? [X380 : term,X379 : subst,X378 : subst] : (((sub @ (X377 @ X379 @ X380) @ X378)) != ((X377 @ (comp @ X379 @ X378) @ (sub @ X380 @ X378)))) | ? [X381 : term] : ((((X375 @ id @ X381 @ id)) = $true) & ($true != ((X375 @ id @ (X377 @ id @ X381) @ id)))) | (((X375 @ id @ (lam @ (X377 @ sh @ one)) @ id)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f47,definition,(
  (sP34 = $true) <=> ? [X365 : (term > $o),X366 : term] : (! [X367 : term] : (($true != ((var @ X367))) | (((X365 @ X367)) = $true)) & ! [X370 : term] : (? [X371 : term] : ((((X365 @ (sub @ X370 @ (push @ X371 @ id)))) != $true) & (((X365 @ X371)) = $true)) | (((X365 @ (lam @ X370))) = $true)) & (((X365 @ X366)) != $true) & (((sP31 @ X365)) = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f48,definition,(
  (sP35 = $true) <=> (? [X372 : term] : (((sub @ X372 @ id)) != X372) | ! [X384 : (subst > term > subst > $o),X383 : term] : (($true = ((X384 @ id @ X383 @ id))) | ? [X385 : term,X386 : term] : ((((X384 @ id @ (ap @ (sub @ X385 @ id) @ X386) @ id)) != $true) & ($true = ((X384 @ id @ X385 @ id))) & (((X384 @ id @ X386 @ id)) = $true)) | (((sP32 @ X384)) = $true)) | ? [X362 : (subst > term > subst > $o)] : ! [X363 : (term > $o)] : ? [X364 : term] : (((X362 @ id @ X364 @ id)) != ((X363 @ X364))) | (sP34 = $true) | (sP33 = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f49,definition,(
  (sP36 = $true) <=> (? [X127 : term] : ! [X128 : (subst > term > term)] : ? [X129 : subst,X130 : term] : (((X128 @ X129 @ X130)) != ((sub @ X127 @ (push @ X130 @ X129)))) | ? [X131 : term] : (((sub @ X131 @ id)) != X131) | ! [X132 : (subst > term > subst > $o),X134 : term,X133 : (term > $o)] : (? [X136 : (subst > term > term)] : ((((X132 @ id @ (lam @ (X136 @ sh @ one)) @ id)) != $true) & ! [X139 : subst,X137 : term,X138 : subst] : (((X136 @ (comp @ X139 @ X138) @ (sub @ X137 @ X138))) = ((sub @ (X136 @ X139 @ X137) @ X138))) & ! [X140 : term] : ((((X132 @ id @ (X136 @ id @ X140) @ id)) = $true) | ($true != ((X132 @ id @ X140 @ id))))) | ? [X135 : term] : (((X133 @ X135)) != ((X132 @ id @ X135 @ id))) | (((X133 @ (lam @ X134))) = $true) | ? [X141 : term] : ((((X133 @ X141)) = $true) & ($true != ((X133 @ (sub @ X134 @ (push @ X141 @ id))))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f50,definition,(
  ! [X486 : (subst > term > subst > $o)] : ((((sP37 @ X486)) = $true) <=> ! [X491 : (subst > term > term)] : (? [X493 : subst,X492 : subst,X494 : term] : (((sub @ (X491 @ X492 @ X494) @ X493)) != ((X491 @ (comp @ X492 @ X493) @ (sub @ X494 @ X493)))) | (((X486 @ id @ (lam @ (X491 @ sh @ one)) @ id)) = $true) | ? [X495 : term] : ((((X486 @ id @ X495 @ id)) = $true) & (((X486 @ id @ (X491 @ id @ X495) @ id)) != $true))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f51,definition,(
  ! [X474 : (subst > term > subst > $o)] : ((((sP38 @ X474)) = $true) <=> ! [X484 : term,X483 : term] : ((((X474 @ id @ X483 @ id)) != $true) | (((X474 @ id @ X484 @ id)) != $true) | (((X474 @ id @ (ap @ (sub @ X483 @ id) @ X484) @ id)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f52,definition,(
  ! [X466 : (term > $o)] : ((((sP39 @ X466)) = $true) <=> ! [X469 : term,X468 : term] : (($true = ((X466 @ (ap @ X468 @ X469)))) | (((X466 @ X468)) != $true) | ($true != ((X466 @ X469)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f53,definition,(
  ! [X501 : (subst > term > subst > $o)] : (($true = ((sP40 @ X501))) <=> ? [X506 : (subst > term > term)] : (! [X509 : subst,X510 : subst,X508 : term] : (((X506 @ (comp @ X510 @ X509) @ (sub @ X508 @ X509))) = ((sub @ (X506 @ X510 @ X508) @ X509))) & (((X501 @ id @ (lam @ (X506 @ sh @ one)) @ id)) != $true) & ! [X507 : term] : ((((X501 @ id @ (X506 @ id @ X507) @ id)) = $true) | ($true != ((X501 @ id @ X507 @ id))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f54,definition,(
  ! [X501 : (subst > term > subst > $o)] : ((((sP41 @ X501)) = $true) <=> ? [X504 : term,X503 : term] : ((((X501 @ id @ X504 @ id)) = $true) & (((X501 @ id @ X503 @ id)) = $true) & (((X501 @ id @ (ap @ (sub @ X504 @ id) @ X503) @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f55,definition,(
  (sP42 = $true) <=> ? [X485 : term,X486 : (subst > term > subst > $o)] : ((((X486 @ id @ (lam @ X485) @ id)) != $true) & ! [X499 : subst,X496 : term,X498 : subst,X497 : subst] : (($true = ((X486 @ (comp @ X499 @ X497) @ (sub @ X496 @ X497) @ X498))) | (((X486 @ X499 @ X496 @ (comp @ X497 @ X498))) != $true)) & ! [X500 : term] : ((((X486 @ id @ X500 @ id)) != $true) | (((X486 @ id @ (sub @ X485 @ (push @ X500 @ id)) @ id)) = $true)) & ! [X488 : term,X490 : subst,X487 : subst,X489 : subst] : (($true = ((X486 @ X489 @ X488 @ (comp @ X490 @ X487)))) | ($true != ((X486 @ (comp @ X489 @ X490) @ (sub @ X488 @ X490) @ X487)))) & (((sP37 @ X486)) = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f56,definition,(
  ($true = sP43) <=> ! [X501 : (subst > term > subst > $o),X502 : term] : (? [X514 : subst,X513 : subst,X512 : term,X511 : subst] : ((((X501 @ (comp @ X514 @ X513) @ (sub @ X512 @ X513) @ X511)) = $true) & (((X501 @ X514 @ X512 @ (comp @ X513 @ X511))) != $true)) | (((X501 @ id @ X502 @ id)) = $true) | ? [X505 : term] : ((((X501 @ id @ X505 @ id)) != $true) & ($true = ((var @ (sub @ X505 @ id))))) | (((sP41 @ X501)) = $true) | ($true = ((sP40 @ X501))) | ? [X517 : subst,X516 : subst,X515 : term,X518 : subst] : ((((X501 @ (comp @ X517 @ X516) @ (sub @ X515 @ X516) @ X518)) != $true) & (((X501 @ X517 @ X515 @ (comp @ X516 @ X518))) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f57,definition,(
  (sP44 = $true) <=> ? [X472 : term,X474 : (subst > term > subst > $o),X473 : term] : (! [X477 : subst,X475 : term,X476 : subst,X478 : subst] : ((((X474 @ (comp @ X476 @ X477) @ (sub @ X475 @ X477) @ X478)) = $true) | ($true != ((X474 @ X476 @ X475 @ (comp @ X477 @ X478))))) & (((X474 @ id @ (ap @ X472 @ X473) @ id)) != $true) & (((X474 @ id @ X473 @ id)) = $true) & ! [X480 : subst,X482 : term,X479 : subst,X481 : subst] : ((((X474 @ X480 @ X482 @ (comp @ X481 @ X479))) = $true) | ($true != ((X474 @ (comp @ X480 @ X481) @ (sub @ X482 @ X481) @ X479)))) & (((X474 @ id @ X472 @ id)) = $true) & (((sP38 @ X474)) = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f58,definition,(
  (sP45 = $true) <=> (($true = sP43) | ? [X465 : term,X466 : (term > $o)] : ((((sP39 @ X466)) = $true) & ! [X467 : term] : (($true = ((X466 @ X467))) | (((var @ X467)) != $true)) & ! [X470 : term] : ((((X466 @ (lam @ X470))) = $true) | ? [X471 : term] : ((((X466 @ (sub @ X470 @ (push @ X471 @ id)))) != $true) & (((X466 @ X471)) = $true))) & ($true != ((X466 @ X465)))) | (sP44 = $true) | (sP42 = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f59,definition,(
  (sP46 = $true) <=> ? [X191 : (subst > term > subst > $o),X190 : term] : (! [X192 : (subst > term > term)] : (? [X195 : subst,X196 : term,X194 : subst] : (((X192 @ (comp @ X195 @ X194) @ (sub @ X196 @ X194))) != ((sub @ (X192 @ X195 @ X196) @ X194))) | ? [X193 : term] : ((((X191 @ id @ (X192 @ id @ X193) @ id)) != $true) & (((X191 @ id @ X193 @ id)) = $true)) | ($true = ((X191 @ id @ (lam @ (X192 @ sh @ one)) @ id)))) & (((X191 @ id @ (lam @ (sub @ X190 @ (push @ one @ sh))) @ id)) != $true) & ! [X197 : term] : ((((X191 @ id @ (sub @ X190 @ (push @ X197 @ id)) @ id)) = $true) | ($true != ((X191 @ id @ X197 @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f60,definition,(
  (sP47 = $true) <=> ((sP46 = $true) | ! [X199 : term,X198 : (subst > term > subst > $o)] : (($true = ((X198 @ id @ (lam @ X199) @ id))) | ? [X201 : (subst > term > term)] : (($true != ((X198 @ id @ (lam @ (X201 @ sh @ one)) @ id))) & ! [X202 : subst,X203 : term,X204 : subst] : (((sub @ (X201 @ X204 @ X203) @ X202)) = ((X201 @ (comp @ X204 @ X202) @ (sub @ X203 @ X202)))) & ! [X205 : term] : ((((X198 @ id @ (X201 @ id @ X205) @ id)) = $true) | (((X198 @ id @ X205 @ id)) != $true))) | ? [X200 : term] : ((((X198 @ id @ (sub @ X199 @ (push @ X200 @ id)) @ id)) != $true) & (((X198 @ id @ X200 @ id)) = $true))) | ? [X189 : term] : (((sub @ X189 @ id)) != X189))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f61,definition,(
  ! [X215 : (term > $o)] : ((((sP48 @ X215)) = $true) <=> ? [X217 : term,X218 : term] : ((((X215 @ X217)) = $true) & ($true = ((X215 @ X218))) & (((X215 @ (ap @ X218 @ X217))) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f62,definition,(
  ! [X209 : (term > $o)] : ((((sP49 @ X209)) = $true) <=> ! [X213 : term,X212 : term] : ((((X209 @ X212)) != $true) | (((X209 @ X213)) != $true) | (((X209 @ (ap @ X212 @ X213))) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f63,definition,(
  (sP50 = $true) <=> ! [X216 : term,X215 : (term > $o)] : ((((X215 @ X216)) = $true) | ? [X219 : term] : ((((var @ X219)) = $true) & (((X215 @ X219)) != $true)) | (((sP48 @ X215)) = $true) | ? [X220 : term] : ((((X215 @ (lam @ X220))) != $true) & ! [X221 : term] : ((((X215 @ X221)) != $true) | (((X215 @ (sub @ X220 @ (push @ X221 @ id)))) = $true))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f64,definition,(
  ($true = sP51) <=> (? [X208 : term,X207 : subst,X209 : (term > $o)] : ((((X209 @ (sub @ X208 @ X207))) != $true) & ! [X214 : term] : ((((X209 @ (sub @ X214 @ X207))) = $true) | (((var @ X214)) != $true)) & (((sP49 @ X209)) = $true) & ! [X210 : term] : (? [X211 : term] : ((((X209 @ X211)) = $true) & ($true != ((X209 @ (sub @ X210 @ (push @ X211 @ id)))))) | (((X209 @ (lam @ X210))) = $true))) | ? [X206 : term] : (((sub @ X206 @ id)) != X206) | (sP50 = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f65,definition,(
  (sP52 = $true) <=> ! [X4 : (subst > term > subst > $o),X5 : term] : (($true = ((X4 @ id @ (lam @ (sub @ X5 @ (push @ one @ sh))) @ id))) | ? [X11 : term] : ((((X4 @ id @ X11 @ id)) = $true) & (((X4 @ id @ (sub @ X5 @ (push @ X11 @ id)) @ id)) != $true)) | ? [X6 : (subst > term > term)] : (! [X10 : term,X8 : subst,X9 : subst] : (((sub @ (X6 @ X9 @ X10) @ X8)) = ((X6 @ (comp @ X9 @ X8) @ (sub @ X10 @ X8)))) & ! [X7 : term] : (($true = ((X4 @ id @ (X6 @ id @ X7) @ id))) | (((X4 @ id @ X7 @ id)) != $true)) & (((X4 @ id @ (lam @ (X6 @ sh @ one)) @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f66,definition,(
  ! [X244 : (subst > term > subst > $o)] : ((((sP53 @ X244)) = $true) <=> ? [X255 : term,X254 : term] : ((((X244 @ id @ (ap @ (sub @ X254 @ id) @ X255) @ id)) != $true) & (((X244 @ id @ X255 @ id)) = $true) & (((X244 @ id @ X254 @ id)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f67,definition,(
  ! [X244 : (subst > term > subst > $o)] : ((((sP54 @ X244)) = $true) <=> ? [X257 : (subst > term > term)] : (! [X258 : subst,X260 : subst,X259 : term] : (((sub @ (X257 @ X260 @ X259) @ X258)) = ((X257 @ (comp @ X260 @ X258) @ (sub @ X259 @ X258)))) & ! [X261 : term] : ((((X244 @ id @ (X257 @ id @ X261) @ id)) = $true) | (((X244 @ id @ X261 @ id)) != $true)) & (((X244 @ id @ (lam @ (X257 @ sh @ one)) @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f68,definition,(
  ! [X226 : (term > $o)] : ((((sP55 @ X226)) = $true) <=> ! [X230 : term,X229 : term] : ((((X226 @ (ap @ X230 @ X229))) = $true) | (((X226 @ X229)) != $true) | (((X226 @ X230)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f69,definition,(
  (sP56 = $true) <=> ! [X244 : (subst > term > subst > $o),X245 : term] : ((((sP54 @ X244)) = $true) | ? [X250 : subst,X253 : subst,X252 : term,X251 : subst] : (($true != ((X244 @ X253 @ X252 @ (comp @ X251 @ X250)))) & ($true = ((X244 @ (comp @ X253 @ X251) @ (sub @ X252 @ X251) @ X250)))) | ? [X247 : term,X249 : subst,X248 : subst,X246 : subst] : ((((X244 @ X246 @ X247 @ (comp @ X249 @ X248))) = $true) & (((X244 @ (comp @ X246 @ X249) @ (sub @ X247 @ X249) @ X248)) != $true)) | (((sP53 @ X244)) = $true) | ? [X256 : term] : ((((X244 @ id @ X256 @ id)) != $true) & (((var @ (sub @ X256 @ id))) = $true)) | ($true = ((X244 @ id @ X245 @ id))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f70,definition,(
  (sP57 = $true) <=> ? [X234 : (term > $o),X236 : (subst > term > subst > $o),X235 : term] : ((((X234 @ (lam @ X235))) != $true) & ! [X237 : term] : (((X234 @ X237)) = ((X236 @ id @ X237 @ id))) & ! [X243 : term] : ((((X234 @ X243)) != $true) | ($true = ((X234 @ (sub @ X235 @ (push @ X243 @ id)))))) & ! [X238 : (subst > term > term)] : (? [X239 : term] : ((((X236 @ id @ (X238 @ id @ X239) @ id)) != $true) & (((X236 @ id @ X239 @ id)) = $true)) | ? [X242 : term,X241 : subst,X240 : subst] : (((sub @ (X238 @ X241 @ X242) @ X240)) != ((X238 @ (comp @ X241 @ X240) @ (sub @ X242 @ X240)))) | ($true = ((X236 @ id @ (lam @ (X238 @ sh @ one)) @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f71,definition,(
  (sP58 = $true) <=> (? [X233 : term] : (((sub @ X233 @ id)) != X233) | ? [X226 : (term > $o),X227 : term] : ((((sP55 @ X226)) = $true) & ! [X228 : term] : ((((var @ X228)) != $true) | (((X226 @ X228)) = $true)) & ! [X231 : term] : ((((X226 @ (lam @ X231))) = $true) | ? [X232 : term] : ((((X226 @ (sub @ X231 @ (push @ X232 @ id)))) != $true) & (((X226 @ X232)) = $true))) & (((X226 @ X227)) != $true)) | (sP57 = $true) | ? [X223 : (subst > term > subst > $o)] : ! [X224 : (term > $o)] : ? [X225 : term] : (((X224 @ X225)) != ((X223 @ id @ X225 @ id))) | (sP56 = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f72,plain,(
  ((sP58 = $true) <=> (hoasinduction_lthm_3 = $true)) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (induction2_gthm = $true) & ((hoasinduction_lem3aa_lthm = $true) <=> (sP52 = $true)) & (($true = sP51) <=> (induction2_lthm = $true)) & (hoasinduction_lem2_gthm = $true) & (pushprop_lem2v2_gthm = $true) & ! [X453 : d_subst,X452 : d_subst] : ((X452 = X453) | (((d2subst @ X453)) != ((d2subst @ X452)))) & ((sP47 = $true) <=> (hoasinduction_lem3a_lthm = $true)) & ((hoasinduction_lthm = $true) <=> (sP45 = $true)) & (axassoc = $true) & (hoasinduction_gthm = $true) & (! [X399 : subst,X400 : (term > $o),X402 : term,X401 : (term > $o)] : (? [X403 : term] : (((X400 @ X403)) != ((X401 @ (sub @ X403 @ (push @ X402 @ X399))))) | ($true = ((X400 @ one))) | (((X401 @ X402)) != $true)) <=> (pushprop_lem1v2 = $true)) & (hoasapinj1_gthm = $true) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X81 : term] : ? [X82 : (subst > term > term)] : ! [X83 : subst,X84 : term] : (((sub @ X81 @ (push @ X84 @ X83))) = ((X82 @ X83 @ X84)))) & (pushprop_gthm = $true) & ((sP36 = $true) <=> (hoasinduction_lem3v2a_lthm = $true)) & ((? [X282 : term,X283 : term,X284 : term,X285 : term] : ((X283 != X284) & (((ap @ X283 @ X285)) = ((ap @ X284 @ X282)))) | ? [X281 : term] : (((sub @ X281 @ id)) != X281) | ! [X286 : term,X289 : term,X288 : term,X287 : term] : ((X286 = X288) | (((ap @ (sub @ X286 @ id) @ X287)) != ((ap @ (sub @ X288 @ id) @ X289))))) <=> (hoasapinj1_lthm = $true)) & (ulamvar1 != $true) & (hoasinduction_lem1v2 = $true) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (sP35 = $true)) & (pushprop_lem3v2_lthm = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (pushprop_lem2v2_lthm = $true) & ((axvarid = $true) <=> ! [X53 : term] : (((sub @ X53 @ id)) = X53)) & (! [X0 : term,X3 : term,X1 : term,X2 : term] : ((X0 = X2) | (((ap @ (sub @ X2 @ id) @ X1)) != ((ap @ (sub @ X0 @ id) @ X3)))) <=> (hoasapinj1 = $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((termmset = $true) <=> ! [X290 : term] : (((sub @ X290 @ id)) = X290)) & ((hoasapinj2 = $true) <=> ! [X415 : term,X413 : term,X414 : term,X416 : term] : ((X413 = X414) | (((ap @ (sub @ X415 @ id) @ X413)) != ((ap @ (sub @ X416 @ id) @ X414))))) & (axabs = $true) & (pushprop = $true) & (one = ((d2term @ d_one))) & ((hoaslaminj = $true) <=> ! [X306 : subst,X304 : (subst > term > term),X307 : term,X305 : (subst > term > term)] : (? [X310 : subst,X308 : term,X309 : subst] : (((X304 @ (comp @ X310 @ X309) @ (sub @ X308 @ X309))) != ((sub @ (X304 @ X310 @ X308) @ X309))) | (((lam @ (X305 @ sh @ one))) != ((lam @ (X304 @ sh @ one)))) | (((X305 @ X306 @ X307)) = ((X304 @ X306 @ X307))) | ? [X312 : subst,X311 : term,X313 : subst] : (((sub @ (X305 @ X313 @ X311) @ X312)) != ((X305 @ (comp @ X313 @ X312) @ (sub @ X311 @ X312)))))) & (pushprop_lthm = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (hoasinduction_lem3v2_gthm = $true) & (hoasapnotvar = $true) & (! [X314 : (term > $o),X316 : term,X315 : subst] : ? [X317 : (term > $o)] : ! [X318 : term] : (((X314 @ (sub @ X318 @ (push @ X316 @ X315)))) = ((X317 @ X318))) <=> (pushprop_lem0_lthm = $true)) & ((hoaslaminj_lthm = $true) <=> ($true = sP30)) & (pushprop_lem3v2 = $true) & ((hoasinduction_lem3v2a = $true) <=> ($true = sP29)) & ((! [X179 : term] : (((sub @ X179 @ id)) = X179) | ? [X178 : term] : (((sub @ X178 @ id)) != X178)) <=> (termmset_lthm = $true)) & (induction2lem_gthm = $true) & (axapp = $true) & (($true = sP28) <=> (induction2lem_lthm = $true)) & (hoasinduction_lem1 = $true) & (pushprop_lem0_gthm = $true) & (hoasinduction_lem3_gthm = $true) & (hoaslamnotap_lthm = $true) & (! [X181 : (subst > term > subst > $o),X180 : term] : ((((X181 @ id @ X180 @ id)) = $true) | ? [X188 : term,X187 : term] : (($true = ((X181 @ id @ X188 @ id))) & (((X181 @ id @ (ap @ (sub @ X187 @ id) @ X188) @ id)) != $true) & ($true = ((X181 @ id @ X187 @ id)))) | (((sP24 @ X181)) = $true)) <=> (hoasinduction_no_psi_cond = $true)) & (((var @ (d2term @ d_one))) != $true) & ((axscons = $true) <=> ! [X454 : subst] : (((push @ (sub @ one @ X454) @ (comp @ sh @ X454))) = X454)) & (! [X358 : term,X361 : term,X359 : term,X360 : term] : ((X359 = X360) | (((ap @ X359 @ X361)) != ((ap @ X360 @ X358)))) <=> (apinj1 = $true)) & (pushprop_lthm_orig = $true) & (pushprop_lem1_gthm = $true) & (hoasapinj2_gthm = $true) & ((hoasinduction_lem3_lthm = $true) <=> ($true = sP23)) & ((! [X19 : term,X17 : term,X18 : term,X16 : term] : ((X16 = X19) | (((ap @ (sub @ X17 @ id) @ X19)) != ((ap @ (sub @ X18 @ id) @ X16)))) | ? [X13 : term,X12 : term,X15 : term,X14 : term] : ((((ap @ X13 @ X14)) = ((ap @ X15 @ X12))) & (X12 != X14))) <=> (hoasapinj2_lthm = $true)) & ((hoasinduction_lem2v2 = $true) <=> ($true = sP20)) & (hoasinduction_lem3b_gthm = $true) & (! [X409 : term,X408 : subst] : (((comp @ sh @ (push @ X409 @ X408))) = X408) <=> (axshiftcons = $true)) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (! [X521 : term,X520 : term] : ((((lam @ X521)) != ((lam @ X520))) | (X520 = X521)) <=> (laminj = $true)) & ((pushprop_lem1 = $true) <=> ! [X163 : (term > $o),X162 : term,X165 : term,X161 : (term > $o),X164 : subst] : ((((X161 @ X162)) != $true) | (((X163 @ (sub @ X162 @ (push @ X165 @ X164)))) = $true))) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X85 : term] : ? [X86 : (subst > term > term)] : (((sub @ X85 @ (push @ one @ sh))) = ((X86 @ sh @ one)))) & (! [X222 : subst] : (((comp @ id @ X222)) = X222) <=> (axidl = $true)) & ((? [X393 : term,X392 : subst] : (((sub @ one @ (push @ X393 @ X392))) != X393) | ! [X395 : (term > $o),X397 : term,X396 : subst,X394 : (term > $o)] : (? [X398 : term] : (((X395 @ (sub @ X398 @ (push @ X397 @ X396)))) != ((X394 @ X398))) | ($true != ((X395 @ X397))) | ($true = ((X394 @ one))))) <=> (pushprop_lem1v2_lthm = $true)) & (hoasinduction_lem2_lthm = $true) & (! [X142 : (subst > term > subst > $o)] : ? [X143 : (term > $o)] : ! [X144 : term] : (((X142 @ id @ X144 @ id)) = ((X143 @ X144))) <=> (hoasinduction_lem0_lthm = $true)) & (($true = sP18) <=> (hoasinduction_lem3 = $true)) & (hoaslamnotvar = $true) & ! [X450 : d_term,X451 : d_term] : ((((d2term @ X450)) != ((d2term @ X451))) | (X450 = X451)) & (termmset_gthm = $true) & ((sP16 = $true) <=> (induction2lem = $true)) & ((? [X294 : subst,X293 : term] : (((comp @ sh @ (push @ X293 @ X294))) != X294) | ! [X298 : term,X295 : subst,X297 : term,X299 : (term > $o),X296 : (term > $o)] : ((((X299 @ X297)) != $true) | (((X296 @ (sub @ X297 @ (push @ X298 @ X295)))) = $true)) | ? [X292 : term,X291 : subst] : (((sub @ one @ (push @ X292 @ X291))) != X292)) <=> (pushprop_lem1_lthm = $true)) & ((sP14 = $true) <=> (induction2 = $true)) & ((! [X113 : subst] : (((comp @ X113 @ id)) = X113) & ! [X114 : subst] : (((comp @ id @ X114)) = X114)) <=> (substmonoid = $true)) & (axvarshift = $true) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_lem1_lthm = $true) & ! [X166 : subst] : ? [X167 : d_subst] : (((d2subst @ X167)) = X166) & ((hoasinduction_lem3aa = $true) <=> (sP12 = $true)) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem1v2_gthm = $true) & ((shinj = $true) <=> ! [X455 : term,X456 : term] : ((((sub @ X455 @ sh)) != ((sub @ X456 @ sh))) | (X455 = X456))) & ! [X74 : d_subst] : (d_id = X74) & (pushprop_lem2v2 = $true) & (apnotvar = $true) & ((? [X300 : subst] : (((comp @ id @ X300)) != X300) | ? [X301 : subst] : (((comp @ X301 @ id)) != X301) | (! [X302 : subst] : (((comp @ X302 @ id)) = X302) & ! [X303 : subst] : (((comp @ id @ X303)) = X303))) <=> (substmonoid_lthm = $true)) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & ((pushprop_lem0 = $true) <=> ! [X22 : (term > $o),X21 : subst,X20 : term] : ? [X23 : (term > $o)] : ! [X24 : term] : (((X23 @ X24)) = ((X22 @ (sub @ X24 @ (push @ X20 @ X21)))))) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (! [X123 : term] : ? [X124 : (subst > term > term)] : ! [X125 : subst,X126 : term] : (((sub @ X123 @ (push @ X126 @ X125))) = ((X124 @ X125 @ X126))) <=> (hoasinduction_lem3v2_f = $true)) & (hoasinduction_lem3a_gthm = $true) & ((hoasinduction_lem3a = $true) <=> ($true = sP11)) & (($true = sP10) <=> (hoasinduction_lem3v2_lthm = $true)) & ((sP8 = $true) <=> (hoasinduction_lem2 = $true)) & (ulamvarsh = $true) & (hoasapnotvar_lthm = $true) & ((hoasinduction_lem3aaa = $true) <=> ($true = sP6)) & ((axidr = $true) <=> ! [X519 : subst] : (((comp @ X519 @ id)) = X519)) & (hoaslamnotap_gthm = $true) & ((hoaslamnotap = $true) <=> ! [X419 : term,X417 : (subst > term > term),X418 : term] : (? [X421 : subst,X422 : term,X420 : subst] : (((X417 @ (comp @ X420 @ X421) @ (sub @ X422 @ X421))) != ((sub @ (X417 @ X420 @ X422) @ X421))) | (((lam @ (X417 @ sh @ one))) != ((ap @ (sub @ X419 @ id) @ X418))))) & (id = ((d2subst @ d_id))) & ! [X348 : term] : ? [X349 : d_term] : (((d2term @ X349)) = X348) & (axmap = $true) & ((sP5 = $true) <=> (hoasinduction_lem3v2 = $true)) & (hoasinduction_lem2v2_gthm = $true) & (! [X423 : subst,X424 : term] : (((sub @ one @ (push @ X424 @ X423))) = X424) <=> (axvarcons = $true)) & ((hoasinduction = $true) <=> (sP3 = $true)) & (axclos = $true) & (hoaslamnotvar_gthm = $true) & (lamnotap != $true) & ((hoasinduction_lem0 = $true) <=> ! [X410 : (subst > term > subst > $o)] : ? [X411 : (term > $o)] : ! [X412 : term] : (((X410 @ id @ X412 @ id)) = ((X411 @ X412)))) & ((hoasinduction_lem3b = $true) <=> ! [X54 : term] : ? [X55 : (subst > term > term)] : (((X55 @ sh @ one)) = ((sub @ X54 @ (push @ one @ sh))))) & (hoaslaminj_gthm = $true) & ((induction = $true) <=> ! [X75 : term,X76 : (term > $o)] : ((((X76 @ X75)) = $true) | ? [X78 : term] : (($true = ((X76 @ X78))) & (((X76 @ (lam @ X78))) != $true)) | (((sP0 @ X76)) = $true) | ? [X77 : term] : ((((var @ X77)) = $true) & (((X76 @ X77)) != $true)))) & (lamnotvar = $true) & (hoasinduction_lem1_gthm = $true) & ((apinj2 = $true) <=> ! [X405 : term,X404 : term,X406 : term,X407 : term] : ((((ap @ X404 @ X407)) != ((ap @ X405 @ X406))) | (X406 = X407))) & (sh = ((d2subst @ d_id))) & (ulamvarind = $true) & (substmonoid_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & ! [X160 : d_term] : (d_one = X160) & (hoaslamnotvar_lthm = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))))),
  inference(definition_folding,[],[f12,f71,f70,f69,f68,f67,f66,f65,f64,f63,f62,f61,f60,f59,f58,f57,f56,f55,f54,f53,f52,f51,f50,f49,f48,f47,f46,f45,f44,f43,f42,f41,f40,f39,f38,f37,f36,f35,f34,f33,f32,f31,f30,f29,f28,f27,f26,f25,f24,f23,f22,f21,f20,f19,f18,f17,f16,f15,f14,f13])).
thf(f220,plain,(
  ((sP12 = $true) | ? [X457 : (subst > term > subst > $o),X458 : term] : (! [X459 : (subst > term > term)] : (? [X463 : subst,X461 : term,X462 : subst] : (((X459 @ (comp @ X463 @ X462) @ (sub @ X461 @ X462))) != ((sub @ (X459 @ X463 @ X461) @ X462))) | (((X457 @ id @ (lam @ (X459 @ sh @ one)) @ id)) = $true) | ? [X460 : term] : (($true = ((X457 @ id @ X460 @ id))) & (((X457 @ id @ (X459 @ id @ X460) @ id)) != $true))) & (((X457 @ id @ (lam @ (sub @ X458 @ (push @ one @ sh))) @ id)) != $true) & ! [X464 : term] : ((((X457 @ id @ X464 @ id)) != $true) | (((X457 @ id @ (sub @ X458 @ (push @ X464 @ id)) @ id)) = $true)))) & (! [X457 : (subst > term > subst > $o),X458 : term] : (? [X459 : (subst > term > term)] : (! [X463 : subst,X461 : term,X462 : subst] : (((X459 @ (comp @ X463 @ X462) @ (sub @ X461 @ X462))) = ((sub @ (X459 @ X463 @ X461) @ X462))) & (((X457 @ id @ (lam @ (X459 @ sh @ one)) @ id)) != $true) & ! [X460 : term] : (($true != ((X457 @ id @ X460 @ id))) | (((X457 @ id @ (X459 @ id @ X460) @ id)) = $true))) | (((X457 @ id @ (lam @ (sub @ X458 @ (push @ one @ sh))) @ id)) = $true) | ? [X464 : term] : ((((X457 @ id @ X464 @ id)) = $true) & (((X457 @ id @ (sub @ X458 @ (push @ X464 @ id)) @ id)) != $true))) | (sP12 != $true))),
  inference(nnf_transformation,[],[f25])).
thf(f221,plain,(
  ((sP12 = $true) | ? [X0 : (subst > term > subst > $o),X1 : term] : (! [X2 : (subst > term > term)] : (? [X3 : subst,X4 : term,X5 : subst] : (((sub @ (X2 @ X3 @ X4) @ X5)) != ((X2 @ (comp @ X3 @ X5) @ (sub @ X4 @ X5)))) | (((X0 @ id @ (lam @ (X2 @ sh @ one)) @ id)) = $true) | ? [X6 : term] : ((((X0 @ id @ X6 @ id)) = $true) & (((X0 @ id @ (X2 @ id @ X6) @ id)) != $true))) & (((X0 @ id @ (lam @ (sub @ X1 @ (push @ one @ sh))) @ id)) != $true) & ! [X7 : term] : ((((X0 @ id @ X7 @ id)) != $true) | (((X0 @ id @ (sub @ X1 @ (push @ X7 @ id)) @ id)) = $true)))) & (! [X8 : (subst > term > subst > $o),X9 : term] : (? [X10 : (subst > term > term)] : (! [X11 : subst,X12 : term,X13 : subst] : (((X10 @ (comp @ X11 @ X13) @ (sub @ X12 @ X13))) = ((sub @ (X10 @ X11 @ X12) @ X13))) & (((X8 @ id @ (lam @ (X10 @ sh @ one)) @ id)) != $true) & ! [X14 : term] : (($true != ((X8 @ id @ X14 @ id))) | (((X8 @ id @ (X10 @ id @ X14) @ id)) = $true))) | (((X8 @ id @ (lam @ (sub @ X9 @ (push @ one @ sh))) @ id)) = $true) | ? [X15 : term] : ((((X8 @ id @ X15 @ id)) = $true) & (((X8 @ id @ (sub @ X9 @ (push @ X15 @ id)) @ id)) != $true))) | (sP12 != $true))),
  inference(rectify,[],[f220])).
thf(f222,plain,(
  ((sP12 = $true) | (! [X2 : (subst > term > term)] : ((((X2 @ (comp @ (sK361 @ X2) @ (sK363 @ X2)) @ (sub @ (sK362 @ X2) @ (sK363 @ X2)))) != ((sub @ (X2 @ (sK361 @ X2) @ (sK362 @ X2)) @ (sK363 @ X2)))) | (((sK359 @ id @ (lam @ (X2 @ sh @ one)) @ id)) = $true) | ((((sK359 @ id @ (sK364 @ X2) @ id)) = $true) & (((sK359 @ id @ (X2 @ id @ (sK364 @ X2)) @ id)) != $true))) & (((sK359 @ id @ (lam @ (sub @ sK360 @ (push @ one @ sh))) @ id)) != $true) & ! [X7 : term] : ((((sK359 @ id @ X7 @ id)) != $true) | (((sK359 @ id @ (sub @ sK360 @ (push @ X7 @ id)) @ id)) = $true)))) & (! [X8 : (subst > term > subst > $o),X9 : term] : ((! [X11 : subst,X12 : term,X13 : subst] : (((sK365 @ X8 @ (comp @ X11 @ X13) @ (sub @ X12 @ X13))) = ((sub @ (sK365 @ X8 @ X11 @ X12) @ X13))) & (((X8 @ id @ (lam @ (sK365 @ X8 @ sh @ one)) @ id)) != $true) & ! [X14 : term] : (($true != ((X8 @ id @ X14 @ id))) | (((X8 @ id @ (sK365 @ X8 @ id @ X14) @ id)) = $true))) | (((X8 @ id @ (lam @ (sub @ X9 @ (push @ one @ sh))) @ id)) = $true) | (($true = ((X8 @ id @ (sK366 @ X9 @ X8) @ id))) & (((X8 @ id @ (sub @ X9 @ (push @ (sK366 @ X9 @ X8) @ id)) @ id)) != $true))) | (sP12 != $true))),
  inference(skolemize,[status(esa),new_symbols(skolem,[sK359,sK360,vAPP]),skolemize(X0,$thf(sK359)),skolemize(X1,$thf(sK360)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16))],[f221])).
thf(f260,plain,(
  (((sP58 = $true) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | (sP58 != $true))) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (induction2_gthm = $true) & (((hoasinduction_lem3aa_lthm = $true) | (sP52 != $true)) & ((sP52 = $true) | (hoasinduction_lem3aa_lthm != $true))) & ((($true = sP51) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | ($true != sP51))) & (hoasinduction_lem2_gthm = $true) & (pushprop_lem2v2_gthm = $true) & ! [X453 : d_subst,X452 : d_subst] : ((X452 = X453) | (((d2subst @ X453)) != ((d2subst @ X452)))) & (((sP47 = $true) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | (sP47 != $true))) & (((hoasinduction_lthm = $true) | (sP45 != $true)) & ((sP45 = $true) | (hoasinduction_lthm != $true))) & (axassoc = $true) & (hoasinduction_gthm = $true) & ((! [X399 : subst,X400 : (term > $o),X402 : term,X401 : (term > $o)] : (? [X403 : term] : (((X400 @ X403)) != ((X401 @ (sub @ X403 @ (push @ X402 @ X399))))) | ($true = ((X400 @ one))) | (((X401 @ X402)) != $true)) | (pushprop_lem1v2 != $true)) & ((pushprop_lem1v2 = $true) | ? [X399 : subst,X400 : (term > $o),X402 : term,X401 : (term > $o)] : (! [X403 : term] : (((X400 @ X403)) = ((X401 @ (sub @ X403 @ (push @ X402 @ X399))))) & ($true != ((X400 @ one))) & (((X401 @ X402)) = $true)))) & (hoasapinj1_gthm = $true) & (((hoasinduction_lem3v2_f_lthm = $true) | ? [X81 : term] : ! [X82 : (subst > term > term)] : ? [X83 : subst,X84 : term] : (((sub @ X81 @ (push @ X84 @ X83))) != ((X82 @ X83 @ X84)))) & (! [X81 : term] : ? [X82 : (subst > term > term)] : ! [X83 : subst,X84 : term] : (((sub @ X81 @ (push @ X84 @ X83))) = ((X82 @ X83 @ X84))) | (hoasinduction_lem3v2_f_lthm != $true))) & (pushprop_gthm = $true) & (((sP36 = $true) | (hoasinduction_lem3v2a_lthm != $true)) & ((hoasinduction_lem3v2a_lthm = $true) | (sP36 != $true))) & (((? [X282 : term,X283 : term,X284 : term,X285 : term] : ((X283 != X284) & (((ap @ X283 @ X285)) = ((ap @ X284 @ X282)))) | ? [X281 : term] : (((sub @ X281 @ id)) != X281) | ! [X286 : term,X289 : term,X288 : term,X287 : term] : ((X286 = X288) | (((ap @ (sub @ X286 @ id) @ X287)) != ((ap @ (sub @ X288 @ id) @ X289))))) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | (! [X282 : term,X283 : term,X284 : term,X285 : term] : ((X283 = X284) | (((ap @ X283 @ X285)) != ((ap @ X284 @ X282)))) & ! [X281 : term] : (((sub @ X281 @ id)) = X281) & ? [X286 : term,X289 : term,X288 : term,X287 : term] : ((X286 != X288) & (((ap @ (sub @ X286 @ id) @ X287)) = ((ap @ (sub @ X288 @ id) @ X289))))))) & (ulamvar1 != $true) & (hoasinduction_lem1v2 = $true) & (((hoasinduction_no_psi_cond_lthm = $true) | (sP35 != $true)) & ((sP35 = $true) | (hoasinduction_no_psi_cond_lthm != $true))) & (pushprop_lem3v2_lthm = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (pushprop_lem2v2_lthm = $true) & (((axvarid = $true) | ? [X53 : term] : (((sub @ X53 @ id)) != X53)) & (! [X53 : term] : (((sub @ X53 @ id)) = X53) | (axvarid != $true))) & ((! [X0 : term,X3 : term,X1 : term,X2 : term] : ((X0 = X2) | (((ap @ (sub @ X2 @ id) @ X1)) != ((ap @ (sub @ X0 @ id) @ X3)))) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | ? [X0 : term,X3 : term,X1 : term,X2 : term] : ((X0 != X2) & (((ap @ (sub @ X2 @ id) @ X1)) = ((ap @ (sub @ X0 @ id) @ X3)))))) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (((termmset = $true) | ? [X290 : term] : (((sub @ X290 @ id)) != X290)) & (! [X290 : term] : (((sub @ X290 @ id)) = X290) | (termmset != $true))) & (((hoasapinj2 = $true) | ? [X415 : term,X413 : term,X414 : term,X416 : term] : ((X413 != X414) & (((ap @ (sub @ X415 @ id) @ X413)) = ((ap @ (sub @ X416 @ id) @ X414))))) & (! [X415 : term,X413 : term,X414 : term,X416 : term] : ((X413 = X414) | (((ap @ (sub @ X415 @ id) @ X413)) != ((ap @ (sub @ X416 @ id) @ X414)))) | (hoasapinj2 != $true))) & (axabs = $true) & (pushprop = $true) & (one = ((d2term @ d_one))) & (((hoaslaminj = $true) | ? [X306 : subst,X304 : (subst > term > term),X307 : term,X305 : (subst > term > term)] : (! [X310 : subst,X308 : term,X309 : subst] : (((X304 @ (comp @ X310 @ X309) @ (sub @ X308 @ X309))) = ((sub @ (X304 @ X310 @ X308) @ X309))) & (((lam @ (X305 @ sh @ one))) = ((lam @ (X304 @ sh @ one)))) & (((X305 @ X306 @ X307)) != ((X304 @ X306 @ X307))) & ! [X312 : subst,X311 : term,X313 : subst] : (((sub @ (X305 @ X313 @ X311) @ X312)) = ((X305 @ (comp @ X313 @ X312) @ (sub @ X311 @ X312)))))) & (! [X306 : subst,X304 : (subst > term > term),X307 : term,X305 : (subst > term > term)] : (? [X310 : subst,X308 : term,X309 : subst] : (((X304 @ (comp @ X310 @ X309) @ (sub @ X308 @ X309))) != ((sub @ (X304 @ X310 @ X308) @ X309))) | (((lam @ (X305 @ sh @ one))) != ((lam @ (X304 @ sh @ one)))) | (((X305 @ X306 @ X307)) = ((X304 @ X306 @ X307))) | ? [X312 : subst,X311 : term,X313 : subst] : (((sub @ (X305 @ X313 @ X311) @ X312)) != ((X305 @ (comp @ X313 @ X312) @ (sub @ X311 @ X312))))) | (hoaslaminj != $true))) & (pushprop_lthm = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (hoasinduction_lem3v2_gthm = $true) & (hoasapnotvar = $true) & ((! [X314 : (term > $o),X316 : term,X315 : subst] : ? [X317 : (term > $o)] : ! [X318 : term] : (((X314 @ (sub @ X318 @ (push @ X316 @ X315)))) = ((X317 @ X318))) | (pushprop_lem0_lthm != $true)) & ((pushprop_lem0_lthm = $true) | ? [X314 : (term > $o),X316 : term,X315 : subst] : ! [X317 : (term > $o)] : ? [X318 : term] : (((X314 @ (sub @ X318 @ (push @ X316 @ X315)))) != ((X317 @ X318))))) & (((hoaslaminj_lthm = $true) | ($true != sP30)) & (($true = sP30) | (hoaslaminj_lthm != $true))) & (pushprop_lem3v2 = $true) & (((hoasinduction_lem3v2a = $true) | ($true != sP29)) & (($true = sP29) | (hoasinduction_lem3v2a != $true))) & (((! [X179 : term] : (((sub @ X179 @ id)) = X179) | ? [X178 : term] : (((sub @ X178 @ id)) != X178)) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | (? [X179 : term] : (((sub @ X179 @ id)) != X179) & ! [X178 : term] : (((sub @ X178 @ id)) = X178)))) & (induction2lem_gthm = $true) & (axapp = $true) & ((($true = sP28) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | ($true != sP28))) & (hoasinduction_lem1 = $true) & (pushprop_lem0_gthm = $true) & (hoasinduction_lem3_gthm = $true) & (hoaslamnotap_lthm = $true) & ((! [X181 : (subst > term > subst > $o),X180 : term] : ((((X181 @ id @ X180 @ id)) = $true) | ? [X188 : term,X187 : term] : (($true = ((X181 @ id @ X188 @ id))) & (((X181 @ id @ (ap @ (sub @ X187 @ id) @ X188) @ id)) != $true) & ($true = ((X181 @ id @ X187 @ id)))) | (((sP24 @ X181)) = $true)) | (hoasinduction_no_psi_cond != $true)) & ((hoasinduction_no_psi_cond = $true) | ? [X181 : (subst > term > subst > $o),X180 : term] : ((((X181 @ id @ X180 @ id)) != $true) & ! [X188 : term,X187 : term] : (($true != ((X181 @ id @ X188 @ id))) | (((X181 @ id @ (ap @ (sub @ X187 @ id) @ X188) @ id)) = $true) | ($true != ((X181 @ id @ X187 @ id)))) & (((sP24 @ X181)) != $true)))) & (((var @ (d2term @ d_one))) != $true) & (((axscons = $true) | ? [X454 : subst] : (((push @ (sub @ one @ X454) @ (comp @ sh @ X454))) != X454)) & (! [X454 : subst] : (((push @ (sub @ one @ X454) @ (comp @ sh @ X454))) = X454) | (axscons != $true))) & ((! [X358 : term,X361 : term,X359 : term,X360 : term] : ((X359 = X360) | (((ap @ X359 @ X361)) != ((ap @ X360 @ X358)))) | (apinj1 != $true)) & ((apinj1 = $true) | ? [X358 : term,X361 : term,X359 : term,X360 : term] : ((X359 != X360) & (((ap @ X359 @ X361)) = ((ap @ X360 @ X358)))))) & (pushprop_lthm_orig = $true) & (pushprop_lem1_gthm = $true) & (hoasapinj2_gthm = $true) & (((hoasinduction_lem3_lthm = $true) | ($true != sP23)) & (($true = sP23) | (hoasinduction_lem3_lthm != $true))) & (((! [X19 : term,X17 : term,X18 : term,X16 : term] : ((X16 = X19) | (((ap @ (sub @ X17 @ id) @ X19)) != ((ap @ (sub @ X18 @ id) @ X16)))) | ? [X13 : term,X12 : term,X15 : term,X14 : term] : ((((ap @ X13 @ X14)) = ((ap @ X15 @ X12))) & (X12 != X14))) | (hoasapinj2_lthm != $true)) & ((hoasapinj2_lthm = $true) | (? [X19 : term,X17 : term,X18 : term,X16 : term] : ((X16 != X19) & (((ap @ (sub @ X17 @ id) @ X19)) = ((ap @ (sub @ X18 @ id) @ X16)))) & ! [X13 : term,X12 : term,X15 : term,X14 : term] : ((((ap @ X13 @ X14)) != ((ap @ X15 @ X12))) | (X12 = X14))))) & (((hoasinduction_lem2v2 = $true) | ($true != sP20)) & (($true = sP20) | (hoasinduction_lem2v2 != $true))) & (hoasinduction_lem3b_gthm = $true) & ((! [X409 : term,X408 : subst] : (((comp @ sh @ (push @ X409 @ X408))) = X408) | (axshiftcons != $true)) & ((axshiftcons = $true) | ? [X409 : term,X408 : subst] : (((comp @ sh @ (push @ X409 @ X408))) != X408))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & ((! [X521 : term,X520 : term] : ((((lam @ X521)) != ((lam @ X520))) | (X520 = X521)) | (laminj != $true)) & ((laminj = $true) | ? [X521 : term,X520 : term] : ((((lam @ X521)) = ((lam @ X520))) & (X520 != X521)))) & (((pushprop_lem1 = $true) | ? [X163 : (term > $o),X162 : term,X165 : term,X161 : (term > $o),X164 : subst] : ((((X161 @ X162)) = $true) & (((X163 @ (sub @ X162 @ (push @ X165 @ X164)))) != $true))) & (! [X163 : (term > $o),X162 : term,X165 : term,X161 : (term > $o),X164 : subst] : ((((X161 @ X162)) != $true) | (((X163 @ (sub @ X162 @ (push @ X165 @ X164)))) = $true)) | (pushprop_lem1 != $true))) & (((hoasinduction_lem3b_lthm = $true) | ? [X85 : term] : ! [X86 : (subst > term > term)] : (((sub @ X85 @ (push @ one @ sh))) != ((X86 @ sh @ one)))) & (! [X85 : term] : ? [X86 : (subst > term > term)] : (((sub @ X85 @ (push @ one @ sh))) = ((X86 @ sh @ one))) | (hoasinduction_lem3b_lthm != $true))) & ((! [X222 : subst] : (((comp @ id @ X222)) = X222) | (axidl != $true)) & ((axidl = $true) | ? [X222 : subst] : (((comp @ id @ X222)) != X222))) & (((? [X393 : term,X392 : subst] : (((sub @ one @ (push @ X393 @ X392))) != X393) | ! [X395 : (term > $o),X397 : term,X396 : subst,X394 : (term > $o)] : (? [X398 : term] : (((X395 @ (sub @ X398 @ (push @ X397 @ X396)))) != ((X394 @ X398))) | ($true != ((X395 @ X397))) | ($true = ((X394 @ one))))) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (! [X393 : term,X392 : subst] : (((sub @ one @ (push @ X393 @ X392))) = X393) & ? [X395 : (term > $o),X397 : term,X396 : subst,X394 : (term > $o)] : (! [X398 : term] : (((X395 @ (sub @ X398 @ (push @ X397 @ X396)))) = ((X394 @ X398))) & ($true = ((X395 @ X397))) & ($true != ((X394 @ one))))))) & (hoasinduction_lem2_lthm = $true) & ((! [X142 : (subst > term > subst > $o)] : ? [X143 : (term > $o)] : ! [X144 : term] : (((X142 @ id @ X144 @ id)) = ((X143 @ X144))) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ? [X142 : (subst > term > subst > $o)] : ! [X143 : (term > $o)] : ? [X144 : term] : (((X142 @ id @ X144 @ id)) != ((X143 @ X144))))) & ((($true = sP18) | (hoasinduction_lem3 != $true)) & ((hoasinduction_lem3 = $true) | ($true != sP18))) & (hoaslamnotvar = $true) & ! [X450 : d_term,X451 : d_term] : ((((d2term @ X450)) != ((d2term @ X451))) | (X450 = X451)) & (termmset_gthm = $true) & (((sP16 = $true) | (induction2lem != $true)) & ((induction2lem = $true) | (sP16 != $true))) & (((? [X294 : subst,X293 : term] : (((comp @ sh @ (push @ X293 @ X294))) != X294) | ! [X298 : term,X295 : subst,X297 : term,X299 : (term > $o),X296 : (term > $o)] : ((((X299 @ X297)) != $true) | (((X296 @ (sub @ X297 @ (push @ X298 @ X295)))) = $true)) | ? [X292 : term,X291 : subst] : (((sub @ one @ (push @ X292 @ X291))) != X292)) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (! [X294 : subst,X293 : term] : (((comp @ sh @ (push @ X293 @ X294))) = X294) & ? [X298 : term,X295 : subst,X297 : term,X299 : (term > $o),X296 : (term > $o)] : ((((X299 @ X297)) = $true) & (((X296 @ (sub @ X297 @ (push @ X298 @ X295)))) != $true)) & ! [X292 : term,X291 : subst] : (((sub @ one @ (push @ X292 @ X291))) = X292)))) & (((sP14 = $true) | (induction2 != $true)) & ((induction2 = $true) | (sP14 != $true))) & (((! [X113 : subst] : (((comp @ X113 @ id)) = X113) & ! [X114 : subst] : (((comp @ id @ X114)) = X114)) | (substmonoid != $true)) & ((substmonoid = $true) | (? [X113 : subst] : (((comp @ X113 @ id)) != X113) | ? [X114 : subst] : (((comp @ id @ X114)) != X114)))) & (axvarshift = $true) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_lem1_lthm = $true) & ! [X166 : subst] : ? [X167 : d_subst] : (((d2subst @ X167)) = X166) & (((hoasinduction_lem3aa = $true) | (sP12 != $true)) & ((sP12 = $true) | (hoasinduction_lem3aa != $true))) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem1v2_gthm = $true) & (((shinj = $true) | ? [X455 : term,X456 : term] : ((((sub @ X455 @ sh)) = ((sub @ X456 @ sh))) & (X455 != X456))) & (! [X455 : term,X456 : term] : ((((sub @ X455 @ sh)) != ((sub @ X456 @ sh))) | (X455 = X456)) | (shinj != $true))) & ! [X74 : d_subst] : (d_id = X74) & (pushprop_lem2v2 = $true) & (apnotvar = $true) & (((? [X300 : subst] : (((comp @ id @ X300)) != X300) | ? [X301 : subst] : (((comp @ X301 @ id)) != X301) | (! [X302 : subst] : (((comp @ X302 @ id)) = X302) & ! [X303 : subst] : (((comp @ id @ X303)) = X303))) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | (! [X300 : subst] : (((comp @ id @ X300)) = X300) & ! [X301 : subst] : (((comp @ X301 @ id)) = X301) & (? [X302 : subst] : (((comp @ X302 @ id)) != X302) | ? [X303 : subst] : (((comp @ id @ X303)) != X303))))) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (((pushprop_lem0 = $true) | ? [X22 : (term > $o),X21 : subst,X20 : term] : ! [X23 : (term > $o)] : ? [X24 : term] : (((X23 @ X24)) != ((X22 @ (sub @ X24 @ (push @ X20 @ X21)))))) & (! [X22 : (term > $o),X21 : subst,X20 : term] : ? [X23 : (term > $o)] : ! [X24 : term] : (((X23 @ X24)) = ((X22 @ (sub @ X24 @ (push @ X20 @ X21))))) | (pushprop_lem0 != $true))) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((! [X123 : term] : ? [X124 : (subst > term > term)] : ! [X125 : subst,X126 : term] : (((sub @ X123 @ (push @ X126 @ X125))) = ((X124 @ X125 @ X126))) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ? [X123 : term] : ! [X124 : (subst > term > term)] : ? [X125 : subst,X126 : term] : (((sub @ X123 @ (push @ X126 @ X125))) != ((X124 @ X125 @ X126))))) & (hoasinduction_lem3a_gthm = $true) & (((hoasinduction_lem3a = $true) | ($true != sP11)) & (($true = sP11) | (hoasinduction_lem3a != $true))) & ((($true = sP10) | (hoasinduction_lem3v2_lthm != $true)) & ((hoasinduction_lem3v2_lthm = $true) | ($true != sP10))) & (((sP8 = $true) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | (sP8 != $true))) & (ulamvarsh = $true) & (hoasapnotvar_lthm = $true) & (((hoasinduction_lem3aaa = $true) | ($true != sP6)) & (($true = sP6) | (hoasinduction_lem3aaa != $true))) & (((axidr = $true) | ? [X519 : subst] : (((comp @ X519 @ id)) != X519)) & (! [X519 : subst] : (((comp @ X519 @ id)) = X519) | (axidr != $true))) & (hoaslamnotap_gthm = $true) & (((hoaslamnotap = $true) | ? [X419 : term,X417 : (subst > term > term),X418 : term] : (! [X421 : subst,X422 : term,X420 : subst] : (((X417 @ (comp @ X420 @ X421) @ (sub @ X422 @ X421))) = ((sub @ (X417 @ X420 @ X422) @ X421))) & (((lam @ (X417 @ sh @ one))) = ((ap @ (sub @ X419 @ id) @ X418))))) & (! [X419 : term,X417 : (subst > term > term),X418 : term] : (? [X421 : subst,X422 : term,X420 : subst] : (((X417 @ (comp @ X420 @ X421) @ (sub @ X422 @ X421))) != ((sub @ (X417 @ X420 @ X422) @ X421))) | (((lam @ (X417 @ sh @ one))) != ((ap @ (sub @ X419 @ id) @ X418)))) | (hoaslamnotap != $true))) & (id = ((d2subst @ d_id))) & ! [X348 : term] : ? [X349 : d_term] : (((d2term @ X349)) = X348) & (axmap = $true) & (((sP5 = $true) | (hoasinduction_lem3v2 != $true)) & ((hoasinduction_lem3v2 = $true) | (sP5 != $true))) & (hoasinduction_lem2v2_gthm = $true) & ((! [X423 : subst,X424 : term] : (((sub @ one @ (push @ X424 @ X423))) = X424) | (axvarcons != $true)) & ((axvarcons = $true) | ? [X423 : subst,X424 : term] : (((sub @ one @ (push @ X424 @ X423))) != X424))) & (((hoasinduction = $true) | (sP3 != $true)) & ((sP3 = $true) | (hoasinduction != $true))) & (axclos = $true) & (hoaslamnotvar_gthm = $true) & (lamnotap != $true) & (((hoasinduction_lem0 = $true) | ? [X410 : (subst > term > subst > $o)] : ! [X411 : (term > $o)] : ? [X412 : term] : (((X410 @ id @ X412 @ id)) != ((X411 @ X412)))) & (! [X410 : (subst > term > subst > $o)] : ? [X411 : (term > $o)] : ! [X412 : term] : (((X410 @ id @ X412 @ id)) = ((X411 @ X412))) | (hoasinduction_lem0 != $true))) & (((hoasinduction_lem3b = $true) | ? [X54 : term] : ! [X55 : (subst > term > term)] : (((X55 @ sh @ one)) != ((sub @ X54 @ (push @ one @ sh))))) & (! [X54 : term] : ? [X55 : (subst > term > term)] : (((X55 @ sh @ one)) = ((sub @ X54 @ (push @ one @ sh)))) | (hoasinduction_lem3b != $true))) & (hoaslaminj_gthm = $true) & (((induction = $true) | ? [X75 : term,X76 : (term > $o)] : ((((X76 @ X75)) != $true) & ! [X78 : term] : (($true != ((X76 @ X78))) | (((X76 @ (lam @ X78))) = $true)) & (((sP0 @ X76)) != $true) & ! [X77 : term] : ((((var @ X77)) != $true) | (((X76 @ X77)) = $true)))) & (! [X75 : term,X76 : (term > $o)] : ((((X76 @ X75)) = $true) | ? [X78 : term] : (($true = ((X76 @ X78))) & (((X76 @ (lam @ X78))) != $true)) | (((sP0 @ X76)) = $true) | ? [X77 : term] : ((((var @ X77)) = $true) & (((X76 @ X77)) != $true))) | (induction != $true))) & (lamnotvar = $true) & (hoasinduction_lem1_gthm = $true) & (((apinj2 = $true) | ? [X405 : term,X404 : term,X406 : term,X407 : term] : ((((ap @ X404 @ X407)) = ((ap @ X405 @ X406))) & (X406 != X407))) & (! [X405 : term,X404 : term,X406 : term,X407 : term] : ((((ap @ X404 @ X407)) != ((ap @ X405 @ X406))) | (X406 = X407)) | (apinj2 != $true))) & (sh = ((d2subst @ d_id))) & (ulamvarind = $true) & (substmonoid_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & ! [X160 : d_term] : (d_one = X160) & (hoaslamnotvar_lthm = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))))),
  inference(nnf_transformation,[],[f72])).
thf(f261,plain,(
  ((sP58 = $true) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | (sP58 != $true)) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (induction2_gthm = $true) & ((hoasinduction_lem3aa_lthm = $true) | (sP52 != $true)) & ((sP52 = $true) | (hoasinduction_lem3aa_lthm != $true)) & (($true = sP51) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | ($true != sP51)) & (hoasinduction_lem2_gthm = $true) & (pushprop_lem2v2_gthm = $true) & ! [X453 : d_subst,X452 : d_subst] : ((X452 = X453) | (((d2subst @ X453)) != ((d2subst @ X452)))) & ((sP47 = $true) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | (sP47 != $true)) & ((hoasinduction_lthm = $true) | (sP45 != $true)) & ((sP45 = $true) | (hoasinduction_lthm != $true)) & (axassoc = $true) & (hoasinduction_gthm = $true) & (! [X399 : subst,X400 : (term > $o),X402 : term,X401 : (term > $o)] : (? [X403 : term] : (((X400 @ X403)) != ((X401 @ (sub @ X403 @ (push @ X402 @ X399))))) | ($true = ((X400 @ one))) | (((X401 @ X402)) != $true)) | (pushprop_lem1v2 != $true)) & ((pushprop_lem1v2 = $true) | ? [X399 : subst,X400 : (term > $o),X402 : term,X401 : (term > $o)] : (! [X403 : term] : (((X400 @ X403)) = ((X401 @ (sub @ X403 @ (push @ X402 @ X399))))) & ($true != ((X400 @ one))) & (((X401 @ X402)) = $true))) & (hoasapinj1_gthm = $true) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X81 : term] : ! [X82 : (subst > term > term)] : ? [X83 : subst,X84 : term] : (((sub @ X81 @ (push @ X84 @ X83))) != ((X82 @ X83 @ X84)))) & (! [X81 : term] : ? [X82 : (subst > term > term)] : ! [X83 : subst,X84 : term] : (((sub @ X81 @ (push @ X84 @ X83))) = ((X82 @ X83 @ X84))) | (hoasinduction_lem3v2_f_lthm != $true)) & (pushprop_gthm = $true) & ((sP36 = $true) | (hoasinduction_lem3v2a_lthm != $true)) & ((hoasinduction_lem3v2a_lthm = $true) | (sP36 != $true)) & (? [X282 : term,X283 : term,X284 : term,X285 : term] : ((X283 != X284) & (((ap @ X283 @ X285)) = ((ap @ X284 @ X282)))) | ? [X281 : term] : (((sub @ X281 @ id)) != X281) | ! [X286 : term,X289 : term,X288 : term,X287 : term] : ((X286 = X288) | (((ap @ (sub @ X286 @ id) @ X287)) != ((ap @ (sub @ X288 @ id) @ X289)))) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | (! [X282 : term,X283 : term,X284 : term,X285 : term] : ((X283 = X284) | (((ap @ X283 @ X285)) != ((ap @ X284 @ X282)))) & ! [X281 : term] : (((sub @ X281 @ id)) = X281) & ? [X286 : term,X289 : term,X288 : term,X287 : term] : ((X286 != X288) & (((ap @ (sub @ X286 @ id) @ X287)) = ((ap @ (sub @ X288 @ id) @ X289)))))) & (ulamvar1 != $true) & (hoasinduction_lem1v2 = $true) & ((hoasinduction_no_psi_cond_lthm = $true) | (sP35 != $true)) & ((sP35 = $true) | (hoasinduction_no_psi_cond_lthm != $true)) & (pushprop_lem3v2_lthm = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (pushprop_lem2v2_lthm = $true) & ((axvarid = $true) | ? [X53 : term] : (((sub @ X53 @ id)) != X53)) & (! [X53 : term] : (((sub @ X53 @ id)) = X53) | (axvarid != $true)) & (! [X0 : term,X3 : term,X1 : term,X2 : term] : ((X0 = X2) | (((ap @ (sub @ X2 @ id) @ X1)) != ((ap @ (sub @ X0 @ id) @ X3)))) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | ? [X0 : term,X3 : term,X1 : term,X2 : term] : ((X0 != X2) & (((ap @ (sub @ X2 @ id) @ X1)) = ((ap @ (sub @ X0 @ id) @ X3))))) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((termmset = $true) | ? [X290 : term] : (((sub @ X290 @ id)) != X290)) & (! [X290 : term] : (((sub @ X290 @ id)) = X290) | (termmset != $true)) & ((hoasapinj2 = $true) | ? [X415 : term,X413 : term,X414 : term,X416 : term] : ((X413 != X414) & (((ap @ (sub @ X415 @ id) @ X413)) = ((ap @ (sub @ X416 @ id) @ X414))))) & (! [X415 : term,X413 : term,X414 : term,X416 : term] : ((X413 = X414) | (((ap @ (sub @ X415 @ id) @ X413)) != ((ap @ (sub @ X416 @ id) @ X414)))) | (hoasapinj2 != $true)) & (axabs = $true) & (pushprop = $true) & (one = ((d2term @ d_one))) & ((hoaslaminj = $true) | ? [X306 : subst,X304 : (subst > term > term),X307 : term,X305 : (subst > term > term)] : (! [X310 : subst,X308 : term,X309 : subst] : (((X304 @ (comp @ X310 @ X309) @ (sub @ X308 @ X309))) = ((sub @ (X304 @ X310 @ X308) @ X309))) & (((lam @ (X305 @ sh @ one))) = ((lam @ (X304 @ sh @ one)))) & (((X305 @ X306 @ X307)) != ((X304 @ X306 @ X307))) & ! [X312 : subst,X311 : term,X313 : subst] : (((sub @ (X305 @ X313 @ X311) @ X312)) = ((X305 @ (comp @ X313 @ X312) @ (sub @ X311 @ X312)))))) & (! [X306 : subst,X304 : (subst > term > term),X307 : term,X305 : (subst > term > term)] : (? [X310 : subst,X308 : term,X309 : subst] : (((X304 @ (comp @ X310 @ X309) @ (sub @ X308 @ X309))) != ((sub @ (X304 @ X310 @ X308) @ X309))) | (((lam @ (X305 @ sh @ one))) != ((lam @ (X304 @ sh @ one)))) | (((X305 @ X306 @ X307)) = ((X304 @ X306 @ X307))) | ? [X312 : subst,X311 : term,X313 : subst] : (((sub @ (X305 @ X313 @ X311) @ X312)) != ((X305 @ (comp @ X313 @ X312) @ (sub @ X311 @ X312))))) | (hoaslaminj != $true)) & (pushprop_lthm = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (hoasinduction_lem3v2_gthm = $true) & (hoasapnotvar = $true) & (! [X314 : (term > $o),X316 : term,X315 : subst] : ? [X317 : (term > $o)] : ! [X318 : term] : (((X314 @ (sub @ X318 @ (push @ X316 @ X315)))) = ((X317 @ X318))) | (pushprop_lem0_lthm != $true)) & ((pushprop_lem0_lthm = $true) | ? [X314 : (term > $o),X316 : term,X315 : subst] : ! [X317 : (term > $o)] : ? [X318 : term] : (((X314 @ (sub @ X318 @ (push @ X316 @ X315)))) != ((X317 @ X318)))) & ((hoaslaminj_lthm = $true) | ($true != sP30)) & (($true = sP30) | (hoaslaminj_lthm != $true)) & (pushprop_lem3v2 = $true) & ((hoasinduction_lem3v2a = $true) | ($true != sP29)) & (($true = sP29) | (hoasinduction_lem3v2a != $true)) & (! [X179 : term] : (((sub @ X179 @ id)) = X179) | ? [X178 : term] : (((sub @ X178 @ id)) != X178) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | (? [X179 : term] : (((sub @ X179 @ id)) != X179) & ! [X178 : term] : (((sub @ X178 @ id)) = X178))) & (induction2lem_gthm = $true) & (axapp = $true) & (($true = sP28) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | ($true != sP28)) & (hoasinduction_lem1 = $true) & (pushprop_lem0_gthm = $true) & (hoasinduction_lem3_gthm = $true) & (hoaslamnotap_lthm = $true) & (! [X181 : (subst > term > subst > $o),X180 : term] : ((((X181 @ id @ X180 @ id)) = $true) | ? [X188 : term,X187 : term] : (($true = ((X181 @ id @ X188 @ id))) & (((X181 @ id @ (ap @ (sub @ X187 @ id) @ X188) @ id)) != $true) & ($true = ((X181 @ id @ X187 @ id)))) | (((sP24 @ X181)) = $true)) | (hoasinduction_no_psi_cond != $true)) & ((hoasinduction_no_psi_cond = $true) | ? [X181 : (subst > term > subst > $o),X180 : term] : ((((X181 @ id @ X180 @ id)) != $true) & ! [X188 : term,X187 : term] : (($true != ((X181 @ id @ X188 @ id))) | (((X181 @ id @ (ap @ (sub @ X187 @ id) @ X188) @ id)) = $true) | ($true != ((X181 @ id @ X187 @ id)))) & (((sP24 @ X181)) != $true))) & (((var @ (d2term @ d_one))) != $true) & ((axscons = $true) | ? [X454 : subst] : (((push @ (sub @ one @ X454) @ (comp @ sh @ X454))) != X454)) & (! [X454 : subst] : (((push @ (sub @ one @ X454) @ (comp @ sh @ X454))) = X454) | (axscons != $true)) & (! [X358 : term,X361 : term,X359 : term,X360 : term] : ((X359 = X360) | (((ap @ X359 @ X361)) != ((ap @ X360 @ X358)))) | (apinj1 != $true)) & ((apinj1 = $true) | ? [X358 : term,X361 : term,X359 : term,X360 : term] : ((X359 != X360) & (((ap @ X359 @ X361)) = ((ap @ X360 @ X358))))) & (pushprop_lthm_orig = $true) & (pushprop_lem1_gthm = $true) & (hoasapinj2_gthm = $true) & ((hoasinduction_lem3_lthm = $true) | ($true != sP23)) & (($true = sP23) | (hoasinduction_lem3_lthm != $true)) & (! [X19 : term,X17 : term,X18 : term,X16 : term] : ((X16 = X19) | (((ap @ (sub @ X17 @ id) @ X19)) != ((ap @ (sub @ X18 @ id) @ X16)))) | ? [X13 : term,X12 : term,X15 : term,X14 : term] : ((((ap @ X13 @ X14)) = ((ap @ X15 @ X12))) & (X12 != X14)) | (hoasapinj2_lthm != $true)) & ((hoasapinj2_lthm = $true) | (? [X19 : term,X17 : term,X18 : term,X16 : term] : ((X16 != X19) & (((ap @ (sub @ X17 @ id) @ X19)) = ((ap @ (sub @ X18 @ id) @ X16)))) & ! [X13 : term,X12 : term,X15 : term,X14 : term] : ((((ap @ X13 @ X14)) != ((ap @ X15 @ X12))) | (X12 = X14)))) & ((hoasinduction_lem2v2 = $true) | ($true != sP20)) & (($true = sP20) | (hoasinduction_lem2v2 != $true)) & (hoasinduction_lem3b_gthm = $true) & (! [X409 : term,X408 : subst] : (((comp @ sh @ (push @ X409 @ X408))) = X408) | (axshiftcons != $true)) & ((axshiftcons = $true) | ? [X409 : term,X408 : subst] : (((comp @ sh @ (push @ X409 @ X408))) != X408)) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (! [X521 : term,X520 : term] : ((((lam @ X521)) != ((lam @ X520))) | (X520 = X521)) | (laminj != $true)) & ((laminj = $true) | ? [X521 : term,X520 : term] : ((((lam @ X521)) = ((lam @ X520))) & (X520 != X521))) & ((pushprop_lem1 = $true) | ? [X163 : (term > $o),X162 : term,X165 : term,X161 : (term > $o),X164 : subst] : ((((X161 @ X162)) = $true) & (((X163 @ (sub @ X162 @ (push @ X165 @ X164)))) != $true))) & (! [X163 : (term > $o),X162 : term,X165 : term,X161 : (term > $o),X164 : subst] : ((((X161 @ X162)) != $true) | (((X163 @ (sub @ X162 @ (push @ X165 @ X164)))) = $true)) | (pushprop_lem1 != $true)) & ((hoasinduction_lem3b_lthm = $true) | ? [X85 : term] : ! [X86 : (subst > term > term)] : (((sub @ X85 @ (push @ one @ sh))) != ((X86 @ sh @ one)))) & (! [X85 : term] : ? [X86 : (subst > term > term)] : (((sub @ X85 @ (push @ one @ sh))) = ((X86 @ sh @ one))) | (hoasinduction_lem3b_lthm != $true)) & (! [X222 : subst] : (((comp @ id @ X222)) = X222) | (axidl != $true)) & ((axidl = $true) | ? [X222 : subst] : (((comp @ id @ X222)) != X222)) & (? [X393 : term,X392 : subst] : (((sub @ one @ (push @ X393 @ X392))) != X393) | ! [X395 : (term > $o),X397 : term,X396 : subst,X394 : (term > $o)] : (? [X398 : term] : (((X395 @ (sub @ X398 @ (push @ X397 @ X396)))) != ((X394 @ X398))) | ($true != ((X395 @ X397))) | ($true = ((X394 @ one)))) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (! [X393 : term,X392 : subst] : (((sub @ one @ (push @ X393 @ X392))) = X393) & ? [X395 : (term > $o),X397 : term,X396 : subst,X394 : (term > $o)] : (! [X398 : term] : (((X395 @ (sub @ X398 @ (push @ X397 @ X396)))) = ((X394 @ X398))) & ($true = ((X395 @ X397))) & ($true != ((X394 @ one)))))) & (hoasinduction_lem2_lthm = $true) & (! [X142 : (subst > term > subst > $o)] : ? [X143 : (term > $o)] : ! [X144 : term] : (((X142 @ id @ X144 @ id)) = ((X143 @ X144))) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ? [X142 : (subst > term > subst > $o)] : ! [X143 : (term > $o)] : ? [X144 : term] : (((X142 @ id @ X144 @ id)) != ((X143 @ X144)))) & (($true = sP18) | (hoasinduction_lem3 != $true)) & ((hoasinduction_lem3 = $true) | ($true != sP18)) & (hoaslamnotvar = $true) & ! [X450 : d_term,X451 : d_term] : ((((d2term @ X450)) != ((d2term @ X451))) | (X450 = X451)) & (termmset_gthm = $true) & ((sP16 = $true) | (induction2lem != $true)) & ((induction2lem = $true) | (sP16 != $true)) & (? [X294 : subst,X293 : term] : (((comp @ sh @ (push @ X293 @ X294))) != X294) | ! [X298 : term,X295 : subst,X297 : term,X299 : (term > $o),X296 : (term > $o)] : ((((X299 @ X297)) != $true) | (((X296 @ (sub @ X297 @ (push @ X298 @ X295)))) = $true)) | ? [X292 : term,X291 : subst] : (((sub @ one @ (push @ X292 @ X291))) != X292) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (! [X294 : subst,X293 : term] : (((comp @ sh @ (push @ X293 @ X294))) = X294) & ? [X298 : term,X295 : subst,X297 : term,X299 : (term > $o),X296 : (term > $o)] : ((((X299 @ X297)) = $true) & (((X296 @ (sub @ X297 @ (push @ X298 @ X295)))) != $true)) & ! [X292 : term,X291 : subst] : (((sub @ one @ (push @ X292 @ X291))) = X292))) & ((sP14 = $true) | (induction2 != $true)) & ((induction2 = $true) | (sP14 != $true)) & ((! [X113 : subst] : (((comp @ X113 @ id)) = X113) & ! [X114 : subst] : (((comp @ id @ X114)) = X114)) | (substmonoid != $true)) & ((substmonoid = $true) | ? [X113 : subst] : (((comp @ X113 @ id)) != X113) | ? [X114 : subst] : (((comp @ id @ X114)) != X114)) & (axvarshift = $true) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_lem1_lthm = $true) & ! [X166 : subst] : ? [X167 : d_subst] : (((d2subst @ X167)) = X166) & ((hoasinduction_lem3aa = $true) | (sP12 != $true)) & ((sP12 = $true) | (hoasinduction_lem3aa != $true)) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem1v2_gthm = $true) & ((shinj = $true) | ? [X455 : term,X456 : term] : ((((sub @ X455 @ sh)) = ((sub @ X456 @ sh))) & (X455 != X456))) & (! [X455 : term,X456 : term] : ((((sub @ X455 @ sh)) != ((sub @ X456 @ sh))) | (X455 = X456)) | (shinj != $true)) & ! [X74 : d_subst] : (d_id = X74) & (pushprop_lem2v2 = $true) & (apnotvar = $true) & (? [X300 : subst] : (((comp @ id @ X300)) != X300) | ? [X301 : subst] : (((comp @ X301 @ id)) != X301) | (! [X302 : subst] : (((comp @ X302 @ id)) = X302) & ! [X303 : subst] : (((comp @ id @ X303)) = X303)) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | (! [X300 : subst] : (((comp @ id @ X300)) = X300) & ! [X301 : subst] : (((comp @ X301 @ id)) = X301) & (? [X302 : subst] : (((comp @ X302 @ id)) != X302) | ? [X303 : subst] : (((comp @ id @ X303)) != X303)))) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & ((pushprop_lem0 = $true) | ? [X22 : (term > $o),X21 : subst,X20 : term] : ! [X23 : (term > $o)] : ? [X24 : term] : (((X23 @ X24)) != ((X22 @ (sub @ X24 @ (push @ X20 @ X21)))))) & (! [X22 : (term > $o),X21 : subst,X20 : term] : ? [X23 : (term > $o)] : ! [X24 : term] : (((X23 @ X24)) = ((X22 @ (sub @ X24 @ (push @ X20 @ X21))))) | (pushprop_lem0 != $true)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (! [X123 : term] : ? [X124 : (subst > term > term)] : ! [X125 : subst,X126 : term] : (((sub @ X123 @ (push @ X126 @ X125))) = ((X124 @ X125 @ X126))) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ? [X123 : term] : ! [X124 : (subst > term > term)] : ? [X125 : subst,X126 : term] : (((sub @ X123 @ (push @ X126 @ X125))) != ((X124 @ X125 @ X126)))) & (hoasinduction_lem3a_gthm = $true) & ((hoasinduction_lem3a = $true) | ($true != sP11)) & (($true = sP11) | (hoasinduction_lem3a != $true)) & (($true = sP10) | (hoasinduction_lem3v2_lthm != $true)) & ((hoasinduction_lem3v2_lthm = $true) | ($true != sP10)) & ((sP8 = $true) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | (sP8 != $true)) & (ulamvarsh = $true) & (hoasapnotvar_lthm = $true) & ((hoasinduction_lem3aaa = $true) | ($true != sP6)) & (($true = sP6) | (hoasinduction_lem3aaa != $true)) & ((axidr = $true) | ? [X519 : subst] : (((comp @ X519 @ id)) != X519)) & (! [X519 : subst] : (((comp @ X519 @ id)) = X519) | (axidr != $true)) & (hoaslamnotap_gthm = $true) & ((hoaslamnotap = $true) | ? [X419 : term,X417 : (subst > term > term),X418 : term] : (! [X421 : subst,X422 : term,X420 : subst] : (((X417 @ (comp @ X420 @ X421) @ (sub @ X422 @ X421))) = ((sub @ (X417 @ X420 @ X422) @ X421))) & (((lam @ (X417 @ sh @ one))) = ((ap @ (sub @ X419 @ id) @ X418))))) & (! [X419 : term,X417 : (subst > term > term),X418 : term] : (? [X421 : subst,X422 : term,X420 : subst] : (((X417 @ (comp @ X420 @ X421) @ (sub @ X422 @ X421))) != ((sub @ (X417 @ X420 @ X422) @ X421))) | (((lam @ (X417 @ sh @ one))) != ((ap @ (sub @ X419 @ id) @ X418)))) | (hoaslamnotap != $true)) & (id = ((d2subst @ d_id))) & ! [X348 : term] : ? [X349 : d_term] : (((d2term @ X349)) = X348) & (axmap = $true) & ((sP5 = $true) | (hoasinduction_lem3v2 != $true)) & ((hoasinduction_lem3v2 = $true) | (sP5 != $true)) & (hoasinduction_lem2v2_gthm = $true) & (! [X423 : subst,X424 : term] : (((sub @ one @ (push @ X424 @ X423))) = X424) | (axvarcons != $true)) & ((axvarcons = $true) | ? [X423 : subst,X424 : term] : (((sub @ one @ (push @ X424 @ X423))) != X424)) & ((hoasinduction = $true) | (sP3 != $true)) & ((sP3 = $true) | (hoasinduction != $true)) & (axclos = $true) & (hoaslamnotvar_gthm = $true) & (lamnotap != $true) & ((hoasinduction_lem0 = $true) | ? [X410 : (subst > term > subst > $o)] : ! [X411 : (term > $o)] : ? [X412 : term] : (((X410 @ id @ X412 @ id)) != ((X411 @ X412)))) & (! [X410 : (subst > term > subst > $o)] : ? [X411 : (term > $o)] : ! [X412 : term] : (((X410 @ id @ X412 @ id)) = ((X411 @ X412))) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem3b = $true) | ? [X54 : term] : ! [X55 : (subst > term > term)] : (((X55 @ sh @ one)) != ((sub @ X54 @ (push @ one @ sh))))) & (! [X54 : term] : ? [X55 : (subst > term > term)] : (((X55 @ sh @ one)) = ((sub @ X54 @ (push @ one @ sh)))) | (hoasinduction_lem3b != $true)) & (hoaslaminj_gthm = $true) & ((induction = $true) | ? [X75 : term,X76 : (term > $o)] : ((((X76 @ X75)) != $true) & ! [X78 : term] : (($true != ((X76 @ X78))) | (((X76 @ (lam @ X78))) = $true)) & (((sP0 @ X76)) != $true) & ! [X77 : term] : ((((var @ X77)) != $true) | (((X76 @ X77)) = $true)))) & (! [X75 : term,X76 : (term > $o)] : ((((X76 @ X75)) = $true) | ? [X78 : term] : (($true = ((X76 @ X78))) & (((X76 @ (lam @ X78))) != $true)) | (((sP0 @ X76)) = $true) | ? [X77 : term] : ((((var @ X77)) = $true) & (((X76 @ X77)) != $true))) | (induction != $true)) & (lamnotvar = $true) & (hoasinduction_lem1_gthm = $true) & ((apinj2 = $true) | ? [X405 : term,X404 : term,X406 : term,X407 : term] : ((((ap @ X404 @ X407)) = ((ap @ X405 @ X406))) & (X406 != X407))) & (! [X405 : term,X404 : term,X406 : term,X407 : term] : ((((ap @ X404 @ X407)) != ((ap @ X405 @ X406))) | (X406 = X407)) | (apinj2 != $true)) & (sh = ((d2subst @ d_id))) & (ulamvarind = $true) & (substmonoid_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & ! [X160 : d_term] : (d_one = X160) & (hoaslamnotvar_lthm = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))))),
  inference(flattening,[],[f260])).
thf(f262,plain,(
  ((sP58 = $true) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | (sP58 != $true)) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (induction2_gthm = $true) & ((hoasinduction_lem3aa_lthm = $true) | (sP52 != $true)) & ((sP52 = $true) | (hoasinduction_lem3aa_lthm != $true)) & (($true = sP51) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | ($true != sP51)) & (hoasinduction_lem2_gthm = $true) & (pushprop_lem2v2_gthm = $true) & ! [X0 : d_subst,X1 : d_subst] : ((X0 = X1) | (((d2subst @ X1)) != ((d2subst @ X0)))) & ((sP47 = $true) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | (sP47 != $true)) & ((hoasinduction_lthm = $true) | (sP45 != $true)) & ((sP45 = $true) | (hoasinduction_lthm != $true)) & (axassoc = $true) & (hoasinduction_gthm = $true) & (! [X2 : subst,X3 : (term > $o),X4 : term,X5 : (term > $o)] : (? [X6 : term] : (((X5 @ (sub @ X6 @ (push @ X4 @ X2)))) != ((X3 @ X6))) | ($true = ((X3 @ one))) | ($true != ((X5 @ X4)))) | (pushprop_lem1v2 != $true)) & ((pushprop_lem1v2 = $true) | ? [X7 : subst,X8 : (term > $o),X9 : term,X10 : (term > $o)] : (! [X11 : term] : (((X8 @ X11)) = ((X10 @ (sub @ X11 @ (push @ X9 @ X7))))) & (((X8 @ one)) != $true) & ($true = ((X10 @ X9))))) & (hoasapinj1_gthm = $true) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X12 : term] : ! [X13 : (subst > term > term)] : ? [X14 : subst,X15 : term] : (((sub @ X12 @ (push @ X15 @ X14))) != ((X13 @ X14 @ X15)))) & (! [X16 : term] : ? [X17 : (subst > term > term)] : ! [X18 : subst,X19 : term] : (((sub @ X16 @ (push @ X19 @ X18))) = ((X17 @ X18 @ X19))) | (hoasinduction_lem3v2_f_lthm != $true)) & (pushprop_gthm = $true) & ((sP36 = $true) | (hoasinduction_lem3v2a_lthm != $true)) & ((hoasinduction_lem3v2a_lthm = $true) | (sP36 != $true)) & (? [X20 : term,X21 : term,X22 : term,X23 : term] : ((X21 != X22) & (((ap @ X22 @ X20)) = ((ap @ X21 @ X23)))) | ? [X24 : term] : (((sub @ X24 @ id)) != X24) | ! [X25 : term,X26 : term,X27 : term,X28 : term] : ((X25 = X27) | (((ap @ (sub @ X27 @ id) @ X26)) != ((ap @ (sub @ X25 @ id) @ X28)))) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | (! [X29 : term,X30 : term,X31 : term,X32 : term] : ((X30 = X31) | (((ap @ X30 @ X32)) != ((ap @ X31 @ X29)))) & ! [X33 : term] : (((sub @ X33 @ id)) = X33) & ? [X34 : term,X35 : term,X36 : term,X37 : term] : ((X34 != X36) & (((ap @ (sub @ X36 @ id) @ X35)) = ((ap @ (sub @ X34 @ id) @ X37)))))) & (ulamvar1 != $true) & (hoasinduction_lem1v2 = $true) & ((hoasinduction_no_psi_cond_lthm = $true) | (sP35 != $true)) & ((sP35 = $true) | (hoasinduction_no_psi_cond_lthm != $true)) & (pushprop_lem3v2_lthm = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (pushprop_lem2v2_lthm = $true) & ((axvarid = $true) | ? [X38 : term] : (((sub @ X38 @ id)) != X38)) & (! [X39 : term] : (((sub @ X39 @ id)) = X39) | (axvarid != $true)) & (! [X40 : term,X41 : term,X42 : term,X43 : term] : ((X40 = X43) | (((ap @ (sub @ X40 @ id) @ X41)) != ((ap @ (sub @ X43 @ id) @ X42)))) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | ? [X44 : term,X45 : term,X46 : term,X47 : term] : ((X44 != X47) & (((ap @ (sub @ X47 @ id) @ X46)) = ((ap @ (sub @ X44 @ id) @ X45))))) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((termmset = $true) | ? [X48 : term] : (((sub @ X48 @ id)) != X48)) & (! [X49 : term] : (((sub @ X49 @ id)) = X49) | (termmset != $true)) & ((hoasapinj2 = $true) | ? [X50 : term,X51 : term,X52 : term,X53 : term] : ((X51 != X52) & (((ap @ (sub @ X50 @ id) @ X51)) = ((ap @ (sub @ X53 @ id) @ X52))))) & (! [X54 : term,X55 : term,X56 : term,X57 : term] : ((X55 = X56) | (((ap @ (sub @ X54 @ id) @ X55)) != ((ap @ (sub @ X57 @ id) @ X56)))) | (hoasapinj2 != $true)) & (axabs = $true) & (pushprop = $true) & (one = ((d2term @ d_one))) & ((hoaslaminj = $true) | ? [X58 : subst,X59 : (subst > term > term),X60 : term,X61 : (subst > term > term)] : (! [X62 : subst,X63 : term,X64 : subst] : (((X59 @ (comp @ X62 @ X64) @ (sub @ X63 @ X64))) = ((sub @ (X59 @ X62 @ X63) @ X64))) & (((lam @ (X61 @ sh @ one))) = ((lam @ (X59 @ sh @ one)))) & (((X59 @ X58 @ X60)) != ((X61 @ X58 @ X60))) & ! [X65 : subst,X66 : term,X67 : subst] : (((sub @ (X61 @ X67 @ X66) @ X65)) = ((X61 @ (comp @ X67 @ X65) @ (sub @ X66 @ X65)))))) & (! [X68 : subst,X69 : (subst > term > term),X70 : term,X71 : (subst > term > term)] : (? [X72 : subst,X73 : term,X74 : subst] : (((X69 @ (comp @ X72 @ X74) @ (sub @ X73 @ X74))) != ((sub @ (X69 @ X72 @ X73) @ X74))) | (((lam @ (X69 @ sh @ one))) != ((lam @ (X71 @ sh @ one)))) | (((X69 @ X68 @ X70)) = ((X71 @ X68 @ X70))) | ? [X75 : subst,X76 : term,X77 : subst] : (((X71 @ (comp @ X77 @ X75) @ (sub @ X76 @ X75))) != ((sub @ (X71 @ X77 @ X76) @ X75)))) | (hoaslaminj != $true)) & (pushprop_lthm = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (hoasinduction_lem3v2_gthm = $true) & (hoasapnotvar = $true) & (! [X78 : (term > $o),X79 : term,X80 : subst] : ? [X81 : (term > $o)] : ! [X82 : term] : (((X78 @ (sub @ X82 @ (push @ X79 @ X80)))) = ((X81 @ X82))) | (pushprop_lem0_lthm != $true)) & ((pushprop_lem0_lthm = $true) | ? [X83 : (term > $o),X84 : term,X85 : subst] : ! [X86 : (term > $o)] : ? [X87 : term] : (((X83 @ (sub @ X87 @ (push @ X84 @ X85)))) != ((X86 @ X87)))) & ((hoaslaminj_lthm = $true) | ($true != sP30)) & (($true = sP30) | (hoaslaminj_lthm != $true)) & (pushprop_lem3v2 = $true) & ((hoasinduction_lem3v2a = $true) | ($true != sP29)) & (($true = sP29) | (hoasinduction_lem3v2a != $true)) & (! [X88 : term] : (((sub @ X88 @ id)) = X88) | ? [X89 : term] : (((sub @ X89 @ id)) != X89) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | (? [X90 : term] : (((sub @ X90 @ id)) != X90) & ! [X91 : term] : (((sub @ X91 @ id)) = X91))) & (induction2lem_gthm = $true) & (axapp = $true) & (($true = sP28) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | ($true != sP28)) & (hoasinduction_lem1 = $true) & (pushprop_lem0_gthm = $true) & (hoasinduction_lem3_gthm = $true) & (hoaslamnotap_lthm = $true) & (! [X92 : (subst > term > subst > $o),X93 : term] : (($true = ((X92 @ id @ X93 @ id))) | ? [X94 : term,X95 : term] : (($true = ((X92 @ id @ X94 @ id))) & ($true != ((X92 @ id @ (ap @ (sub @ X95 @ id) @ X94) @ id))) & (((X92 @ id @ X95 @ id)) = $true)) | (((sP24 @ X92)) = $true)) | (hoasinduction_no_psi_cond != $true)) & ((hoasinduction_no_psi_cond = $true) | ? [X96 : (subst > term > subst > $o),X97 : term] : ((((X96 @ id @ X97 @ id)) != $true) & ! [X98 : term,X99 : term] : ((((X96 @ id @ X98 @ id)) != $true) | ($true = ((X96 @ id @ (ap @ (sub @ X99 @ id) @ X98) @ id))) | (((X96 @ id @ X99 @ id)) != $true)) & (((sP24 @ X96)) != $true))) & (((var @ (d2term @ d_one))) != $true) & ((axscons = $true) | ? [X100 : subst] : (((push @ (sub @ one @ X100) @ (comp @ sh @ X100))) != X100)) & (! [X101 : subst] : (((push @ (sub @ one @ X101) @ (comp @ sh @ X101))) = X101) | (axscons != $true)) & (! [X102 : term,X103 : term,X104 : term,X105 : term] : ((X104 = X105) | (((ap @ X104 @ X103)) != ((ap @ X105 @ X102)))) | (apinj1 != $true)) & ((apinj1 = $true) | ? [X106 : term,X107 : term,X108 : term,X109 : term] : ((X108 != X109) & (((ap @ X109 @ X106)) = ((ap @ X108 @ X107))))) & (pushprop_lthm_orig = $true) & (pushprop_lem1_gthm = $true) & (hoasapinj2_gthm = $true) & ((hoasinduction_lem3_lthm = $true) | ($true != sP23)) & (($true = sP23) | (hoasinduction_lem3_lthm != $true)) & (! [X110 : term,X111 : term,X112 : term,X113 : term] : ((X110 = X113) | (((ap @ (sub @ X111 @ id) @ X110)) != ((ap @ (sub @ X112 @ id) @ X113)))) | ? [X114 : term,X115 : term,X116 : term,X117 : term] : ((((ap @ X114 @ X117)) = ((ap @ X116 @ X115))) & (X115 != X117)) | (hoasapinj2_lthm != $true)) & ((hoasapinj2_lthm = $true) | (? [X118 : term,X119 : term,X120 : term,X121 : term] : ((X118 != X121) & (((ap @ (sub @ X119 @ id) @ X118)) = ((ap @ (sub @ X120 @ id) @ X121)))) & ! [X122 : term,X123 : term,X124 : term,X125 : term] : ((((ap @ X124 @ X123)) != ((ap @ X122 @ X125))) | (X123 = X125)))) & ((hoasinduction_lem2v2 = $true) | ($true != sP20)) & (($true = sP20) | (hoasinduction_lem2v2 != $true)) & (hoasinduction_lem3b_gthm = $true) & (! [X126 : term,X127 : subst] : (((comp @ sh @ (push @ X126 @ X127))) = X127) | (axshiftcons != $true)) & ((axshiftcons = $true) | ? [X128 : term,X129 : subst] : (((comp @ sh @ (push @ X128 @ X129))) != X129)) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (! [X130 : term,X131 : term] : ((((lam @ X130)) != ((lam @ X131))) | (X130 = X131)) | (laminj != $true)) & ((laminj = $true) | ? [X132 : term,X133 : term] : ((((lam @ X133)) = ((lam @ X132))) & (X132 != X133))) & ((pushprop_lem1 = $true) | ? [X134 : (term > $o),X135 : term,X136 : term,X137 : (term > $o),X138 : subst] : ((((X137 @ X135)) = $true) & ($true != ((X134 @ (sub @ X135 @ (push @ X136 @ X138))))))) & (! [X139 : (term > $o),X140 : term,X141 : term,X142 : (term > $o),X143 : subst] : ((((X142 @ X140)) != $true) | (((X139 @ (sub @ X140 @ (push @ X141 @ X143)))) = $true)) | (pushprop_lem1 != $true)) & ((hoasinduction_lem3b_lthm = $true) | ? [X144 : term] : ! [X145 : (subst > term > term)] : (((sub @ X144 @ (push @ one @ sh))) != ((X145 @ sh @ one)))) & (! [X146 : term] : ? [X147 : (subst > term > term)] : (((sub @ X146 @ (push @ one @ sh))) = ((X147 @ sh @ one))) | (hoasinduction_lem3b_lthm != $true)) & (! [X148 : subst] : (((comp @ id @ X148)) = X148) | (axidl != $true)) & ((axidl = $true) | ? [X149 : subst] : (((comp @ id @ X149)) != X149)) & (? [X150 : term,X151 : subst] : (((sub @ one @ (push @ X150 @ X151))) != X150) | ! [X152 : (term > $o),X153 : term,X154 : subst,X155 : (term > $o)] : (? [X156 : term] : (((X155 @ X156)) != ((X152 @ (sub @ X156 @ (push @ X153 @ X154))))) | (((X152 @ X153)) != $true) | (((X155 @ one)) = $true)) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (! [X157 : term,X158 : subst] : (((sub @ one @ (push @ X157 @ X158))) = X157) & ? [X159 : (term > $o),X160 : term,X161 : subst,X162 : (term > $o)] : (! [X163 : term] : (((X162 @ X163)) = ((X159 @ (sub @ X163 @ (push @ X160 @ X161))))) & (((X159 @ X160)) = $true) & (((X162 @ one)) != $true)))) & (hoasinduction_lem2_lthm = $true) & (! [X164 : (subst > term > subst > $o)] : ? [X165 : (term > $o)] : ! [X166 : term] : (((X164 @ id @ X166 @ id)) = ((X165 @ X166))) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ? [X167 : (subst > term > subst > $o)] : ! [X168 : (term > $o)] : ? [X169 : term] : (((X167 @ id @ X169 @ id)) != ((X168 @ X169)))) & (($true = sP18) | (hoasinduction_lem3 != $true)) & ((hoasinduction_lem3 = $true) | ($true != sP18)) & (hoaslamnotvar = $true) & ! [X170 : d_term,X171 : d_term] : ((((d2term @ X170)) != ((d2term @ X171))) | (X170 = X171)) & (termmset_gthm = $true) & ((sP16 = $true) | (induction2lem != $true)) & ((induction2lem = $true) | (sP16 != $true)) & (? [X172 : subst,X173 : term] : (((comp @ sh @ (push @ X173 @ X172))) != X172) | ! [X174 : term,X175 : subst,X176 : term,X177 : (term > $o),X178 : (term > $o)] : (($true != ((X177 @ X176))) | (((X178 @ (sub @ X176 @ (push @ X174 @ X175)))) = $true)) | ? [X179 : term,X180 : subst] : (((sub @ one @ (push @ X179 @ X180))) != X179) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (! [X181 : subst,X182 : term] : (((comp @ sh @ (push @ X182 @ X181))) = X181) & ? [X183 : term,X184 : subst,X185 : term,X186 : (term > $o),X187 : (term > $o)] : ((((X186 @ X185)) = $true) & (((X187 @ (sub @ X185 @ (push @ X183 @ X184)))) != $true)) & ! [X188 : term,X189 : subst] : (((sub @ one @ (push @ X188 @ X189))) = X188))) & ((sP14 = $true) | (induction2 != $true)) & ((induction2 = $true) | (sP14 != $true)) & ((! [X190 : subst] : (((comp @ X190 @ id)) = X190) & ! [X191 : subst] : (((comp @ id @ X191)) = X191)) | (substmonoid != $true)) & ((substmonoid = $true) | ? [X192 : subst] : (((comp @ X192 @ id)) != X192) | ? [X193 : subst] : (((comp @ id @ X193)) != X193)) & (axvarshift = $true) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_lem1_lthm = $true) & ! [X194 : subst] : ? [X195 : d_subst] : (((d2subst @ X195)) = X194) & ((hoasinduction_lem3aa = $true) | (sP12 != $true)) & ((sP12 = $true) | (hoasinduction_lem3aa != $true)) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem1v2_gthm = $true) & ((shinj = $true) | ? [X196 : term,X197 : term] : ((((sub @ X196 @ sh)) = ((sub @ X197 @ sh))) & (X196 != X197))) & (! [X198 : term,X199 : term] : ((((sub @ X198 @ sh)) != ((sub @ X199 @ sh))) | (X198 = X199)) | (shinj != $true)) & ! [X200 : d_subst] : (d_id = X200) & (pushprop_lem2v2 = $true) & (apnotvar = $true) & (? [X201 : subst] : (((comp @ id @ X201)) != X201) | ? [X202 : subst] : (((comp @ X202 @ id)) != X202) | (! [X203 : subst] : (((comp @ X203 @ id)) = X203) & ! [X204 : subst] : (((comp @ id @ X204)) = X204)) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | (! [X205 : subst] : (((comp @ id @ X205)) = X205) & ! [X206 : subst] : (((comp @ X206 @ id)) = X206) & (? [X207 : subst] : (((comp @ X207 @ id)) != X207) | ? [X208 : subst] : (((comp @ id @ X208)) != X208)))) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & ((pushprop_lem0 = $true) | ? [X209 : (term > $o),X210 : subst,X211 : term] : ! [X212 : (term > $o)] : ? [X213 : term] : (((X209 @ (sub @ X213 @ (push @ X211 @ X210)))) != ((X212 @ X213)))) & (! [X214 : (term > $o),X215 : subst,X216 : term] : ? [X217 : (term > $o)] : ! [X218 : term] : (((X217 @ X218)) = ((X214 @ (sub @ X218 @ (push @ X216 @ X215))))) | (pushprop_lem0 != $true)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (! [X219 : term] : ? [X220 : (subst > term > term)] : ! [X221 : subst,X222 : term] : (((X220 @ X221 @ X222)) = ((sub @ X219 @ (push @ X222 @ X221)))) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ? [X223 : term] : ! [X224 : (subst > term > term)] : ? [X225 : subst,X226 : term] : (((sub @ X223 @ (push @ X226 @ X225))) != ((X224 @ X225 @ X226)))) & (hoasinduction_lem3a_gthm = $true) & ((hoasinduction_lem3a = $true) | ($true != sP11)) & (($true = sP11) | (hoasinduction_lem3a != $true)) & (($true = sP10) | (hoasinduction_lem3v2_lthm != $true)) & ((hoasinduction_lem3v2_lthm = $true) | ($true != sP10)) & ((sP8 = $true) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | (sP8 != $true)) & (ulamvarsh = $true) & (hoasapnotvar_lthm = $true) & ((hoasinduction_lem3aaa = $true) | ($true != sP6)) & (($true = sP6) | (hoasinduction_lem3aaa != $true)) & ((axidr = $true) | ? [X227 : subst] : (((comp @ X227 @ id)) != X227)) & (! [X228 : subst] : (((comp @ X228 @ id)) = X228) | (axidr != $true)) & (hoaslamnotap_gthm = $true) & ((hoaslamnotap = $true) | ? [X229 : term,X230 : (subst > term > term),X231 : term] : (! [X232 : subst,X233 : term,X234 : subst] : (((sub @ (X230 @ X234 @ X233) @ X232)) = ((X230 @ (comp @ X234 @ X232) @ (sub @ X233 @ X232)))) & (((lam @ (X230 @ sh @ one))) = ((ap @ (sub @ X229 @ id) @ X231))))) & (! [X235 : term,X236 : (subst > term > term),X237 : term] : (? [X238 : subst,X239 : term,X240 : subst] : (((X236 @ (comp @ X240 @ X238) @ (sub @ X239 @ X238))) != ((sub @ (X236 @ X240 @ X239) @ X238))) | (((lam @ (X236 @ sh @ one))) != ((ap @ (sub @ X235 @ id) @ X237)))) | (hoaslamnotap != $true)) & (id = ((d2subst @ d_id))) & ! [X241 : term] : ? [X242 : d_term] : (((d2term @ X242)) = X241) & (axmap = $true) & ((sP5 = $true) | (hoasinduction_lem3v2 != $true)) & ((hoasinduction_lem3v2 = $true) | (sP5 != $true)) & (hoasinduction_lem2v2_gthm = $true) & (! [X243 : subst,X244 : term] : (((sub @ one @ (push @ X244 @ X243))) = X244) | (axvarcons != $true)) & ((axvarcons = $true) | ? [X245 : subst,X246 : term] : (((sub @ one @ (push @ X246 @ X245))) != X246)) & ((hoasinduction = $true) | (sP3 != $true)) & ((sP3 = $true) | (hoasinduction != $true)) & (axclos = $true) & (hoaslamnotvar_gthm = $true) & (lamnotap != $true) & ((hoasinduction_lem0 = $true) | ? [X247 : (subst > term > subst > $o)] : ! [X248 : (term > $o)] : ? [X249 : term] : (((X247 @ id @ X249 @ id)) != ((X248 @ X249)))) & (! [X250 : (subst > term > subst > $o)] : ? [X251 : (term > $o)] : ! [X252 : term] : (((X251 @ X252)) = ((X250 @ id @ X252 @ id))) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem3b = $true) | ? [X253 : term] : ! [X254 : (subst > term > term)] : (((X254 @ sh @ one)) != ((sub @ X253 @ (push @ one @ sh))))) & (! [X255 : term] : ? [X256 : (subst > term > term)] : (((X256 @ sh @ one)) = ((sub @ X255 @ (push @ one @ sh)))) | (hoasinduction_lem3b != $true)) & (hoaslaminj_gthm = $true) & ((induction = $true) | ? [X257 : term,X258 : (term > $o)] : ((((X258 @ X257)) != $true) & ! [X259 : term] : ((((X258 @ X259)) != $true) | (((X258 @ (lam @ X259))) = $true)) & (((sP0 @ X258)) != $true) & ! [X260 : term] : ((((var @ X260)) != $true) | (((X258 @ X260)) = $true)))) & (! [X261 : term,X262 : (term > $o)] : ((((X262 @ X261)) = $true) | ? [X263 : term] : ((((X262 @ X263)) = $true) & ($true != ((X262 @ (lam @ X263))))) | ($true = ((sP0 @ X262))) | ? [X264 : term] : ((((var @ X264)) = $true) & ($true != ((X262 @ X264))))) | (induction != $true)) & (lamnotvar = $true) & (hoasinduction_lem1_gthm = $true) & ((apinj2 = $true) | ? [X265 : term,X266 : term,X267 : term,X268 : term] : ((((ap @ X265 @ X267)) = ((ap @ X266 @ X268))) & (X267 != X268))) & (! [X269 : term,X270 : term,X271 : term,X272 : term] : ((((ap @ X270 @ X272)) != ((ap @ X269 @ X271))) | (X271 = X272)) | (apinj2 != $true)) & (sh = ((d2subst @ d_id))) & (ulamvarind = $true) & (substmonoid_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & ! [X273 : d_term] : (d_one = X273) & (hoaslamnotvar_lthm = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))))),
  inference(rectify,[],[f261])).
thf(f263,plain,(
  ((sP58 = $true) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | (sP58 != $true)) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (induction2_gthm = $true) & ((hoasinduction_lem3aa_lthm = $true) | (sP52 != $true)) & ((sP52 = $true) | (hoasinduction_lem3aa_lthm != $true)) & (($true = sP51) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | ($true != sP51)) & (hoasinduction_lem2_gthm = $true) & (pushprop_lem2v2_gthm = $true) & ! [X0 : d_subst,X1 : d_subst] : ((X0 = X1) | (((d2subst @ X1)) != ((d2subst @ X0)))) & ((sP47 = $true) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | (sP47 != $true)) & ((hoasinduction_lthm = $true) | (sP45 != $true)) & ((sP45 = $true) | (hoasinduction_lthm != $true)) & (axassoc = $true) & (hoasinduction_gthm = $true) & (! [X2 : subst,X3 : (term > $o),X4 : term,X5 : (term > $o)] : ((((X3 @ (sK457 @ X5 @ X4 @ X3 @ X2))) != ((X5 @ (sub @ (sK457 @ X5 @ X4 @ X3 @ X2) @ (push @ X4 @ X2))))) | ($true = ((X3 @ one))) | ($true != ((X5 @ X4)))) | (pushprop_lem1v2 != $true)) & ((pushprop_lem1v2 = $true) | (! [X11 : term] : (((sK459 @ X11)) = ((sK461 @ (sub @ X11 @ (push @ sK460 @ sK458))))) & (((sK459 @ one)) != $true) & (((sK461 @ sK460)) = $true))) & (hoasapinj1_gthm = $true) & ((hoasinduction_lem3v2_f_lthm = $true) | ! [X13 : (subst > term > term)] : (((X13 @ (sK463 @ X13) @ (sK464 @ X13))) != ((sub @ sK462 @ (push @ (sK464 @ X13) @ (sK463 @ X13)))))) & (! [X16 : term] : ! [X18 : subst,X19 : term] : (((sub @ X16 @ (push @ X19 @ X18))) = ((sK465 @ X16 @ X18 @ X19))) | (hoasinduction_lem3v2_f_lthm != $true)) & (pushprop_gthm = $true) & ((sP36 = $true) | (hoasinduction_lem3v2a_lthm != $true)) & ((hoasinduction_lem3v2a_lthm = $true) | (sP36 != $true)) & (((sK467 != sK468) & (((ap @ sK468 @ sK466)) = ((ap @ sK467 @ sK469)))) | (((sub @ sK470 @ id)) != sK470) | ! [X25 : term,X26 : term,X27 : term,X28 : term] : ((X25 = X27) | (((ap @ (sub @ X27 @ id) @ X26)) != ((ap @ (sub @ X25 @ id) @ X28)))) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | (! [X29 : term,X30 : term,X31 : term,X32 : term] : ((X30 = X31) | (((ap @ X30 @ X32)) != ((ap @ X31 @ X29)))) & ! [X33 : term] : (((sub @ X33 @ id)) = X33) & ((sK473 != sK471) & (((ap @ (sub @ sK473 @ id) @ sK472)) = ((ap @ (sub @ sK471 @ id) @ sK474)))))) & (ulamvar1 != $true) & (hoasinduction_lem1v2 = $true) & ((hoasinduction_no_psi_cond_lthm = $true) | (sP35 != $true)) & ((sP35 = $true) | (hoasinduction_no_psi_cond_lthm != $true)) & (pushprop_lem3v2_lthm = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (pushprop_lem2v2_lthm = $true) & ((axvarid = $true) | (sK475 != ((sub @ sK475 @ id)))) & (! [X39 : term] : (((sub @ X39 @ id)) = X39) | (axvarid != $true)) & (! [X40 : term,X41 : term,X42 : term,X43 : term] : ((X40 = X43) | (((ap @ (sub @ X40 @ id) @ X41)) != ((ap @ (sub @ X43 @ id) @ X42)))) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | ((sK479 != sK476) & (((ap @ (sub @ sK476 @ id) @ sK477)) = ((ap @ (sub @ sK479 @ id) @ sK478))))) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((termmset = $true) | (((sub @ sK480 @ id)) != sK480)) & (! [X49 : term] : (((sub @ X49 @ id)) = X49) | (termmset != $true)) & ((hoasapinj2 = $true) | ((sK483 != sK482) & (((ap @ (sub @ sK484 @ id) @ sK483)) = ((ap @ (sub @ sK481 @ id) @ sK482))))) & (! [X54 : term,X55 : term,X56 : term,X57 : term] : ((X55 = X56) | (((ap @ (sub @ X54 @ id) @ X55)) != ((ap @ (sub @ X57 @ id) @ X56)))) | (hoasapinj2 != $true)) & (axabs = $true) & (pushprop = $true) & (one = ((d2term @ d_one))) & ((hoaslaminj = $true) | (! [X62 : subst,X63 : term,X64 : subst] : (((sK486 @ (comp @ X62 @ X64) @ (sub @ X63 @ X64))) = ((sub @ (sK486 @ X62 @ X63) @ X64))) & (((lam @ (sK488 @ sh @ one))) = ((lam @ (sK486 @ sh @ one)))) & (((sK488 @ sK485 @ sK487)) != ((sK486 @ sK485 @ sK487))) & ! [X65 : subst,X66 : term,X67 : subst] : (((sK488 @ (comp @ X67 @ X65) @ (sub @ X66 @ X65))) = ((sub @ (sK488 @ X67 @ X66) @ X65))))) & (! [X68 : subst,X69 : (subst > term > term),X70 : term,X71 : (subst > term > term)] : ((((X69 @ (comp @ (sK489 @ X69) @ (sK491 @ X69)) @ (sub @ (sK490 @ X69) @ (sK491 @ X69)))) != ((sub @ (X69 @ (sK489 @ X69) @ (sK490 @ X69)) @ (sK491 @ X69)))) | (((lam @ (X69 @ sh @ one))) != ((lam @ (X71 @ sh @ one)))) | (((X69 @ X68 @ X70)) = ((X71 @ X68 @ X70))) | (((sub @ (X71 @ (sK494 @ X71) @ (sK493 @ X71)) @ (sK492 @ X71))) != ((X71 @ (comp @ (sK494 @ X71) @ (sK492 @ X71)) @ (sub @ (sK493 @ X71) @ (sK492 @ X71)))))) | (hoaslaminj != $true)) & (pushprop_lthm = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (hoasinduction_lem3v2_gthm = $true) & (hoasapnotvar = $true) & (! [X78 : (term > $o),X79 : term,X80 : subst] : ! [X82 : term] : (((X78 @ (sub @ X82 @ (push @ X79 @ X80)))) = ((sK495 @ X80 @ X79 @ X78 @ X82))) | (pushprop_lem0_lthm != $true)) & ((pushprop_lem0_lthm = $true) | ! [X86 : (term > $o)] : (((X86 @ (sK499 @ X86))) != ((sK496 @ (sub @ (sK499 @ X86) @ (push @ sK497 @ sK498)))))) & ((hoaslaminj_lthm = $true) | ($true != sP30)) & (($true = sP30) | (hoaslaminj_lthm != $true)) & (pushprop_lem3v2 = $true) & ((hoasinduction_lem3v2a = $true) | ($true != sP29)) & (($true = sP29) | (hoasinduction_lem3v2a != $true)) & (! [X88 : term] : (((sub @ X88 @ id)) = X88) | (sK500 != ((sub @ sK500 @ id))) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | ((((sub @ sK501 @ id)) != sK501) & ! [X91 : term] : (((sub @ X91 @ id)) = X91))) & (induction2lem_gthm = $true) & (axapp = $true) & (($true = sP28) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | ($true != sP28)) & (hoasinduction_lem1 = $true) & (pushprop_lem0_gthm = $true) & (hoasinduction_lem3_gthm = $true) & (hoaslamnotap_lthm = $true) & (! [X92 : (subst > term > subst > $o),X93 : term] : (($true = ((X92 @ id @ X93 @ id))) | ((((X92 @ id @ (sK502 @ X92) @ id)) = $true) & ($true != ((X92 @ id @ (ap @ (sub @ (sK503 @ X92) @ id) @ (sK502 @ X92)) @ id))) & (((X92 @ id @ (sK503 @ X92) @ id)) = $true)) | (((sP24 @ X92)) = $true)) | (hoasinduction_no_psi_cond != $true)) & ((hoasinduction_no_psi_cond = $true) | (($true != ((sK504 @ id @ sK505 @ id))) & ! [X98 : term,X99 : term] : ((((sK504 @ id @ X98 @ id)) != $true) | (((sK504 @ id @ (ap @ (sub @ X99 @ id) @ X98) @ id)) = $true) | (((sK504 @ id @ X99 @ id)) != $true)) & ($true != ((sP24 @ sK504))))) & (((var @ (d2term @ d_one))) != $true) & ((axscons = $true) | (sK506 != ((push @ (sub @ one @ sK506) @ (comp @ sh @ sK506))))) & (! [X101 : subst] : (((push @ (sub @ one @ X101) @ (comp @ sh @ X101))) = X101) | (axscons != $true)) & (! [X102 : term,X103 : term,X104 : term,X105 : term] : ((X104 = X105) | (((ap @ X104 @ X103)) != ((ap @ X105 @ X102)))) | (apinj1 != $true)) & ((apinj1 = $true) | ((sK510 != sK509) & (((ap @ sK510 @ sK507)) = ((ap @ sK509 @ sK508))))) & (pushprop_lthm_orig = $true) & (pushprop_lem1_gthm = $true) & (hoasapinj2_gthm = $true) & ((hoasinduction_lem3_lthm = $true) | ($true != sP23)) & (($true = sP23) | (hoasinduction_lem3_lthm != $true)) & (! [X110 : term,X111 : term,X112 : term,X113 : term] : ((X110 = X113) | (((ap @ (sub @ X111 @ id) @ X110)) != ((ap @ (sub @ X112 @ id) @ X113)))) | ((((ap @ sK511 @ sK514)) = ((ap @ sK513 @ sK512))) & (sK512 != sK514)) | (hoasapinj2_lthm != $true)) & ((hoasapinj2_lthm = $true) | (((sK518 != sK515) & (((ap @ (sub @ sK516 @ id) @ sK515)) = ((ap @ (sub @ sK517 @ id) @ sK518)))) & ! [X122 : term,X123 : term,X124 : term,X125 : term] : ((((ap @ X124 @ X123)) != ((ap @ X122 @ X125))) | (X123 = X125)))) & ((hoasinduction_lem2v2 = $true) | ($true != sP20)) & (($true = sP20) | (hoasinduction_lem2v2 != $true)) & (hoasinduction_lem3b_gthm = $true) & (! [X126 : term,X127 : subst] : (((comp @ sh @ (push @ X126 @ X127))) = X127) | (axshiftcons != $true)) & ((axshiftcons = $true) | (((comp @ sh @ (push @ sK519 @ sK520))) != sK520)) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (! [X130 : term,X131 : term] : ((((lam @ X130)) != ((lam @ X131))) | (X130 = X131)) | (laminj != $true)) & ((laminj = $true) | ((((lam @ sK522)) = ((lam @ sK521))) & (sK522 != sK521))) & ((pushprop_lem1 = $true) | (($true = ((sK526 @ sK524))) & ($true != ((sK523 @ (sub @ sK524 @ (push @ sK525 @ sK527))))))) & (! [X139 : (term > $o),X140 : term,X141 : term,X142 : (term > $o),X143 : subst] : ((((X142 @ X140)) != $true) | (((X139 @ (sub @ X140 @ (push @ X141 @ X143)))) = $true)) | (pushprop_lem1 != $true)) & ((hoasinduction_lem3b_lthm = $true) | ! [X145 : (subst > term > term)] : (((sub @ sK528 @ (push @ one @ sh))) != ((X145 @ sh @ one)))) & (! [X146 : term] : (((sub @ X146 @ (push @ one @ sh))) = ((sK529 @ X146 @ sh @ one))) | (hoasinduction_lem3b_lthm != $true)) & (! [X148 : subst] : (((comp @ id @ X148)) = X148) | (axidl != $true)) & ((axidl = $true) | (sK530 != ((comp @ id @ sK530)))) & ((sK531 != ((sub @ one @ (push @ sK531 @ sK532)))) | ! [X152 : (term > $o),X153 : term,X154 : subst,X155 : (term > $o)] : ((((X155 @ (sK533 @ X155 @ X154 @ X153 @ X152))) != ((X152 @ (sub @ (sK533 @ X155 @ X154 @ X153 @ X152) @ (push @ X153 @ X154))))) | (((X152 @ X153)) != $true) | (((X155 @ one)) = $true)) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (! [X157 : term,X158 : subst] : (((sub @ one @ (push @ X157 @ X158))) = X157) & (! [X163 : term] : (((sK537 @ X163)) = ((sK534 @ (sub @ X163 @ (push @ sK535 @ sK536))))) & (((sK534 @ sK535)) = $true) & ($true != ((sK537 @ one)))))) & (hoasinduction_lem2_lthm = $true) & (! [X164 : (subst > term > subst > $o)] : ! [X166 : term] : (((X164 @ id @ X166 @ id)) = ((sK538 @ X164 @ X166))) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ! [X168 : (term > $o)] : (((sK539 @ id @ (sK540 @ X168) @ id)) != ((X168 @ (sK540 @ X168))))) & (($true = sP18) | (hoasinduction_lem3 != $true)) & ((hoasinduction_lem3 = $true) | ($true != sP18)) & (hoaslamnotvar = $true) & ! [X170 : d_term,X171 : d_term] : ((((d2term @ X170)) != ((d2term @ X171))) | (X170 = X171)) & (termmset_gthm = $true) & ((sP16 = $true) | (induction2lem != $true)) & ((induction2lem = $true) | (sP16 != $true)) & ((sK541 != ((comp @ sh @ (push @ sK542 @ sK541)))) | ! [X174 : term,X175 : subst,X176 : term,X177 : (term > $o),X178 : (term > $o)] : (($true != ((X177 @ X176))) | (((X178 @ (sub @ X176 @ (push @ X174 @ X175)))) = $true)) | (sK543 != ((sub @ one @ (push @ sK543 @ sK544)))) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (! [X181 : subst,X182 : term] : (((comp @ sh @ (push @ X182 @ X181))) = X181) & ((((sK548 @ sK547)) = $true) & (((sK549 @ (sub @ sK547 @ (push @ sK545 @ sK546)))) != $true)) & ! [X188 : term,X189 : subst] : (((sub @ one @ (push @ X188 @ X189))) = X188))) & ((sP14 = $true) | (induction2 != $true)) & ((induction2 = $true) | (sP14 != $true)) & ((! [X190 : subst] : (((comp @ X190 @ id)) = X190) & ! [X191 : subst] : (((comp @ id @ X191)) = X191)) | (substmonoid != $true)) & ((substmonoid = $true) | (sK550 != ((comp @ sK550 @ id))) | (((comp @ id @ sK551)) != sK551)) & (axvarshift = $true) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_lem1_lthm = $true) & ! [X194 : subst] : (((d2subst @ (sK552 @ X194))) = X194) & ((hoasinduction_lem3aa = $true) | (sP12 != $true)) & ((sP12 = $true) | (hoasinduction_lem3aa != $true)) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem1v2_gthm = $true) & ((shinj = $true) | ((((sub @ sK554 @ sh)) = ((sub @ sK553 @ sh))) & (sK554 != sK553))) & (! [X198 : term,X199 : term] : ((((sub @ X198 @ sh)) != ((sub @ X199 @ sh))) | (X198 = X199)) | (shinj != $true)) & ! [X200 : d_subst] : (d_id = X200) & (pushprop_lem2v2 = $true) & (apnotvar = $true) & ((((comp @ id @ sK555)) != sK555) | (((comp @ sK556 @ id)) != sK556) | (! [X203 : subst] : (((comp @ X203 @ id)) = X203) & ! [X204 : subst] : (((comp @ id @ X204)) = X204)) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | (! [X205 : subst] : (((comp @ id @ X205)) = X205) & ! [X206 : subst] : (((comp @ X206 @ id)) = X206) & ((((comp @ sK557 @ id)) != sK557) | (((comp @ id @ sK558)) != sK558)))) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & ((pushprop_lem0 = $true) | ! [X212 : (term > $o)] : (((X212 @ (sK562 @ X212))) != ((sK559 @ (sub @ (sK562 @ X212) @ (push @ sK561 @ sK560)))))) & (! [X214 : (term > $o),X215 : subst,X216 : term] : ! [X218 : term] : (((X214 @ (sub @ X218 @ (push @ X216 @ X215)))) = ((sK563 @ X216 @ X215 @ X214 @ X218))) | (pushprop_lem0 != $true)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (! [X219 : term] : ! [X221 : subst,X222 : term] : (((sK564 @ X219 @ X221 @ X222)) = ((sub @ X219 @ (push @ X222 @ X221)))) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ! [X224 : (subst > term > term)] : (((X224 @ (sK566 @ X224) @ (sK567 @ X224))) != ((sub @ sK565 @ (push @ (sK567 @ X224) @ (sK566 @ X224)))))) & (hoasinduction_lem3a_gthm = $true) & ((hoasinduction_lem3a = $true) | ($true != sP11)) & (($true = sP11) | (hoasinduction_lem3a != $true)) & (($true = sP10) | (hoasinduction_lem3v2_lthm != $true)) & ((hoasinduction_lem3v2_lthm = $true) | ($true != sP10)) & ((sP8 = $true) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | (sP8 != $true)) & (ulamvarsh = $true) & (hoasapnotvar_lthm = $true) & ((hoasinduction_lem3aaa = $true) | ($true != sP6)) & (($true = sP6) | (hoasinduction_lem3aaa != $true)) & ((axidr = $true) | (sK568 != ((comp @ sK568 @ id)))) & (! [X228 : subst] : (((comp @ X228 @ id)) = X228) | (axidr != $true)) & (hoaslamnotap_gthm = $true) & ((hoaslamnotap = $true) | (! [X232 : subst,X233 : term,X234 : subst] : (((sK570 @ (comp @ X234 @ X232) @ (sub @ X233 @ X232))) = ((sub @ (sK570 @ X234 @ X233) @ X232))) & (((lam @ (sK570 @ sh @ one))) = ((ap @ (sub @ sK569 @ id) @ sK571))))) & (! [X235 : term,X236 : (subst > term > term),X237 : term] : ((((X236 @ (comp @ (sK574 @ X236) @ (sK572 @ X236)) @ (sub @ (sK573 @ X236) @ (sK572 @ X236)))) != ((sub @ (X236 @ (sK574 @ X236) @ (sK573 @ X236)) @ (sK572 @ X236)))) | (((lam @ (X236 @ sh @ one))) != ((ap @ (sub @ X235 @ id) @ X237)))) | (hoaslamnotap != $true)) & (id = ((d2subst @ d_id))) & ! [X241 : term] : (((d2term @ (sK575 @ X241))) = X241) & (axmap = $true) & ((sP5 = $true) | (hoasinduction_lem3v2 != $true)) & ((hoasinduction_lem3v2 = $true) | (sP5 != $true)) & (hoasinduction_lem2v2_gthm = $true) & (! [X243 : subst,X244 : term] : (((sub @ one @ (push @ X244 @ X243))) = X244) | (axvarcons != $true)) & ((axvarcons = $true) | (sK577 != ((sub @ one @ (push @ sK577 @ sK576))))) & ((hoasinduction = $true) | (sP3 != $true)) & ((sP3 = $true) | (hoasinduction != $true)) & (axclos = $true) & (hoaslamnotvar_gthm = $true) & (lamnotap != $true) & ((hoasinduction_lem0 = $true) | ! [X248 : (term > $o)] : (((X248 @ (sK579 @ X248))) != ((sK578 @ id @ (sK579 @ X248) @ id)))) & (! [X250 : (subst > term > subst > $o)] : ! [X252 : term] : (((X250 @ id @ X252 @ id)) = ((sK580 @ X250 @ X252))) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem3b = $true) | ! [X254 : (subst > term > term)] : (((X254 @ sh @ one)) != ((sub @ sK581 @ (push @ one @ sh))))) & (! [X255 : term] : (((sub @ X255 @ (push @ one @ sh))) = ((sK582 @ X255 @ sh @ one))) | (hoasinduction_lem3b != $true)) & (hoaslaminj_gthm = $true) & ((induction = $true) | (($true != ((sK584 @ sK583))) & ! [X259 : term] : (($true != ((sK584 @ X259))) | ($true = ((sK584 @ (lam @ X259))))) & ($true != ((sP0 @ sK584))) & ! [X260 : term] : ((((var @ X260)) != $true) | (((sK584 @ X260)) = $true)))) & (! [X261 : term,X262 : (term > $o)] : ((((X262 @ X261)) = $true) | ((((X262 @ (sK585 @ X262))) = $true) & ($true != ((X262 @ (lam @ (sK585 @ X262)))))) | ($true = ((sP0 @ X262))) | ((((var @ (sK586 @ X262))) = $true) & ($true != ((X262 @ (sK586 @ X262)))))) | (induction != $true)) & (lamnotvar = $true) & (hoasinduction_lem1_gthm = $true) & ((apinj2 = $true) | ((((ap @ sK587 @ sK589)) = ((ap @ sK588 @ sK590))) & (sK589 != sK590))) & (! [X269 : term,X270 : term,X271 : term,X272 : term] : ((((ap @ X270 @ X272)) != ((ap @ X269 @ X271))) | (X271 = X272)) | (apinj2 != $true)) & (sh = ((d2subst @ d_id))) & (ulamvarind = $true) & (substmonoid_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & ! [X273 : d_term] : (d_one = X273) & (hoaslamnotvar_lthm = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))))),
  inference(skolemize,[status(esa),new_symbols(skolem,[vAPP,sK458,sK459,sK460,sK461,sK462,vAPP,sK466,sK467,sK468,sK469,sK470,sK471,sK472,sK473,sK474,sK475,sK476,sK477,sK478,sK479,sK480,sK481,sK482,sK483,sK484,sK485,sK486,sK487,sK488,vAPP,sK496,sK497,sK498,vAPP,sK500,sK501,vAPP,sK504,sK505,sK506,sK507,sK508,sK509,sK510,sK511,sK512,sK513,sK514,sK515,sK516,sK517,sK518,sK519,sK520,sK521,sK522,sK523,sK524,sK525,sK526,sK527,sK528,vAPP,sK530,sK531,sK532,vAPP,sK534,sK535,sK536,sK537,vAPP,sK539,vAPP,sK541,sK542,sK543,sK544,sK545,sK546,sK547,sK548,sK549,sK550,sK551,vAPP,sK553,sK554,sK555,sK556,sK557,sK558,sK559,sK560,sK561,vAPP,sK565,vAPP,sK568,sK569,sK570,sK571,vAPP,sK576,sK577,sK578,vAPP,sK581,vAPP,sK583,sK584,vAPP,sK587,sK588,sK589,sK590]),skolemize(X17,$thf(sK67 @ X16)),skolemize(X7,$thf(sK458)),skolemize(X8,$thf(sK459)),skolemize(X9,$thf(sK460)),skolemize(X10,$thf(sK461)),skolemize(X12,$thf(sK462)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X20,$thf(sK466)),skolemize(X21,$thf(sK467)),skolemize(X22,$thf(sK468)),skolemize(X23,$thf(sK469)),skolemize(X24,$thf(sK470)),skolemize(X34,$thf(sK471)),skolemize(X35,$thf(sK472)),skolemize(X36,$thf(sK473)),skolemize(X37,$thf(sK474)),skolemize(X38,$thf(sK475)),skolemize(X44,$thf(sK476)),skolemize(X45,$thf(sK477)),skolemize(X46,$thf(sK478)),skolemize(X47,$thf(sK479)),skolemize(X48,$thf(sK480)),skolemize(X50,$thf(sK481)),skolemize(X51,$thf(sK482)),skolemize(X52,$thf(sK483)),skolemize(X53,$thf(sK484)),skolemize(X58,$thf(sK485)),skolemize(X59,$thf(sK486)),skolemize(X60,$thf(sK487)),skolemize(X61,$thf(sK488)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X83,$thf(sK496)),skolemize(X84,$thf(sK497)),skolemize(X85,$thf(sK498)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X89,$thf(sK500)),skolemize(X90,$thf(sK501)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X96,$thf(sK504)),skolemize(X97,$thf(sK505)),skolemize(X100,$thf(sK506)),skolemize(X106,$thf(sK507)),skolemize(X107,$thf(sK508)),skolemize(X108,$thf(sK509)),skolemize(X109,$thf(sK510)),skolemize(X114,$thf(sK511)),skolemize(X115,$thf(sK512)),skolemize(X116,$thf(sK513)),skolemize(X117,$thf(sK514)),skolemize(X118,$thf(sK515)),skolemize(X119,$thf(sK516)),skolemize(X120,$thf(sK517)),skolemize(X121,$thf(sK518)),skolemize(X128,$thf(sK519)),skolemize(X129,$thf(sK520)),skolemize(X132,$thf(sK521)),skolemize(X133,$thf(sK522)),skolemize(X134,$thf(sK523)),skolemize(X135,$thf(sK524)),skolemize(X136,$thf(sK525)),skolemize(X137,$thf(sK526)),skolemize(X138,$thf(sK527)),skolemize(X144,$thf(sK528)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X149,$thf(sK530)),skolemize(X150,$thf(sK531)),skolemize(X151,$thf(sK532)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X159,$thf(sK534)),skolemize(X160,$thf(sK535)),skolemize(X161,$thf(sK536)),skolemize(X162,$thf(sK537)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X167,$thf(sK539)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X172,$thf(sK541)),skolemize(X173,$thf(sK542)),skolemize(X179,$thf(sK543)),skolemize(X180,$thf(sK544)),skolemize(X183,$thf(sK545)),skolemize(X184,$thf(sK546)),skolemize(X185,$thf(sK547)),skolemize(X186,$thf(sK548)),skolemize(X187,$thf(sK549)),skolemize(X192,$thf(sK550)),skolemize(X193,$thf(sK551)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X196,$thf(sK553)),skolemize(X197,$thf(sK554)),skolemize(X201,$thf(sK555)),skolemize(X202,$thf(sK556)),skolemize(X207,$thf(sK557)),skolemize(X208,$thf(sK558)),skolemize(X209,$thf(sK559)),skolemize(X210,$thf(sK560)),skolemize(X211,$thf(sK561)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X223,$thf(sK565)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X227,$thf(sK568)),skolemize(X229,$thf(sK569)),skolemize(X230,$thf(sK570)),skolemize(X231,$thf(sK571)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X245,$thf(sK576)),skolemize(X246,$thf(sK577)),skolemize(X247,$thf(sK578)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X253,$thf(sK581)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X257,$thf(sK583)),skolemize(X258,$thf(sK584)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X17,$thf(sK67 @ X16)),skolemize(X265,$thf(sK587)),skolemize(X266,$thf(sK588)),skolemize(X267,$thf(sK589)),skolemize(X268,$thf(sK590))],[f262])).
thf(f264,plain,(
  (hoasinduction_lem3aa != ((!! @ (subst > term > subst > $o) @ (^[Y0 : subst > term > subst > $o]: ((!! @ (subst > term > term) @ (^[Y1 : subst > term > term]: ((!! @ term @ (^[Y2 : term]: (!! @ subst @ (^[Y3 : subst]: (!! @ subst @ (^[Y4 : subst]: ((Y1 @ (comp @ Y3 @ Y4) @ (sub @ Y2 @ Y4)) = (sub @ (Y1 @ Y3 @ Y2) @ Y4)))))))) => ((!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) => (Y0 @ id @ (Y1 @ id @ Y2) @ id)))) => (Y0 @ id @ (hoaslam @ id @ (^[Y2 : subst]: ((^[Y3 : term]: (Y1 @ Y2 @ Y3))))) @ id))))) => (!! @ term @ (^[Y1 : term]: ((!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) => (Y0 @ id @ (sub @ Y1 @ (push @ Y2 @ id)) @ id)))) => (Y0 @ id @ (lam @ (sub @ Y1 @ (push @ one @ sh))) @ id)))))))))),
  inference(cnf_transformation,[],[f10])).
thf(f649,plain,(
  (((sK359 @ id @ (lam @ (sub @ sK360 @ (push @ one @ sh))) @ id)) != $true) | (sP12 = $true)),
  inference(cnf_transformation,[],[f222])).
thf(f651,plain,(
  ( ! [X2 : (subst > term > term)] : ((((X2 @ (comp @ (sK361 @ X2) @ (sK363 @ X2)) @ (sub @ (sK362 @ X2) @ (sK363 @ X2)))) != ((sub @ (X2 @ (sK361 @ X2) @ (sK362 @ X2)) @ (sK363 @ X2)))) | (((sK359 @ id @ (sK364 @ X2) @ id)) = $true) | (((sK359 @ id @ (lam @ (X2 @ sh @ one)) @ id)) = $true) | (sP12 = $true)) )),
  inference(cnf_transformation,[],[f222])).
thf(f756,plain,(
  ( ! [X273 : d_term] : ((d_one = X273)) )),
  inference(cnf_transformation,[],[f263])).
thf(f760,plain,(
  (sh = ((d2subst @ d_id)))),
  inference(cnf_transformation,[],[f263])).
thf(f761,plain,(
  ( ! [X271 : term,X272 : term,X270 : term,X269 : term] : ((X271 = X272) | (apinj2 != $true) | (((ap @ X270 @ X272)) != ((ap @ X269 @ X271)))) )),
  inference(cnf_transformation,[],[f263])).
thf(f762,plain,(
  (sK589 != sK590) | (apinj2 = $true)),
  inference(cnf_transformation,[],[f263])).
thf(f790,plain,(
  ( ! [X241 : term] : ((((d2term @ (sK575 @ X241))) = X241)) )),
  inference(cnf_transformation,[],[f263])).
thf(f822,plain,(
  ( ! [X200 : d_subst] : ((d_id = X200)) )),
  inference(cnf_transformation,[],[f263])).
thf(f829,plain,(
  (sP12 != $true) | (hoasinduction_lem3aa = $true)),
  inference(cnf_transformation,[],[f263])).
thf(f830,plain,(
  ( ! [X194 : subst] : ((((d2subst @ (sK552 @ X194))) = X194)) )),
  inference(cnf_transformation,[],[f263])).
thf(f932,plain,(
  (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one)))))),
  inference(cnf_transformation,[],[f263])).
thf(f981,plain,(
  (hoasinduction_lem3aa != ((!! @ (subst > term > subst > $o) @ (^[Y0 : subst > term > subst > $o]: ((!! @ (subst > term > term) @ (^[Y1 : subst > term > term]: ((!! @ term @ (^[Y2 : term]: (!! @ subst @ (^[Y3 : subst]: (!! @ subst @ (^[Y4 : subst]: ((Y1 @ (comp @ Y3 @ Y4) @ (sub @ Y2 @ Y4)) = (sub @ (Y1 @ Y3 @ Y2) @ Y4)))))))) => ((!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) => (Y0 @ id @ (Y1 @ id @ Y2) @ id)))) => (Y0 @ id @ ((^[Y2 : subst]: ((^[Y3 : subst > term > term]: (d2term @ d_one)))) @ id @ (^[Y2 : subst]: ((^[Y3 : term]: (Y1 @ Y2 @ Y3))))) @ id))))) => (!! @ term @ (^[Y1 : term]: ((!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) => (Y0 @ id @ (sub @ Y1 @ (push @ Y2 @ id)) @ id)))) => (Y0 @ id @ (lam @ (sub @ Y1 @ (push @ one @ sh))) @ id)))))))))),
  inference(definition_unfolding,[],[f264,f932])).
thf(f982,definition,(
  (sF591 = ((d2term @ d_one)))),
  introduced(definition,[new_symbols(definition,[sF591])],[function_definition])).
thf(f983,plain,(
  (((d2term @ d_one)) = sF591)),
  inference(reorient_equations,[],[f982])).
thf(f984,definition,(
  (sF592 = ((push @ one @ sh)))),
  introduced(definition,[new_symbols(definition,[sF592])],[function_definition])).
thf(f985,plain,(
  (((push @ one @ sh)) = sF592)),
  inference(reorient_equations,[],[f984])).
thf(f986,definition,(
  (sF593 = ((!! @ (subst > term > subst > $o) @ (^[Y0 : subst > term > subst > $o]: ((!! @ (subst > term > term) @ (^[Y1 : subst > term > term]: ((!! @ term @ (^[Y2 : term]: (!! @ subst @ (^[Y3 : subst]: (!! @ subst @ (^[Y4 : subst]: ((Y1 @ (comp @ Y3 @ Y4) @ (sub @ Y2 @ Y4)) = (sub @ (Y1 @ Y3 @ Y2) @ Y4)))))))) => ((!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) => (Y0 @ id @ (Y1 @ id @ Y2) @ id)))) => (Y0 @ id @ ((^[Y2 : subst]: ((^[Y3 : subst > term > term]: (sF591)))) @ id @ (^[Y2 : subst]: ((^[Y3 : term]: (Y1 @ Y2 @ Y3))))) @ id))))) => (!! @ term @ (^[Y1 : term]: ((!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) => (Y0 @ id @ (sub @ Y1 @ (push @ Y2 @ id)) @ id)))) => (Y0 @ id @ (lam @ (sub @ Y1 @ sF592)) @ id)))))))))),
  introduced(definition,[new_symbols(definition,[sF593])],[function_definition])).
thf(f987,plain,(
  (hoasinduction_lem3aa != sF593)),
  inference(definition_folding,[],[f981,f986,f985,f983])).
thf(f1047,plain,(
  (hoasinduction_lem3aa = $false) | (sF593 = $false)),
  inference(xor_proxy_clausification,[],[f987])).
thf(f1098,plain,(
  (((!! @ (subst > term > subst > $o) @ (^[Y0 : subst > term > subst > $o]: ((!! @ (subst > term > term) @ (^[Y1 : subst > term > term]: ((!! @ term @ (^[Y2 : term]: (!! @ subst @ (^[Y3 : subst]: (!! @ subst @ (^[Y4 : subst]: ((Y1 @ (comp @ Y3 @ Y4) @ (sub @ Y2 @ Y4)) = (sub @ (Y1 @ Y3 @ Y2) @ Y4)))))))) => ((!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) => (Y0 @ id @ (Y1 @ id @ Y2) @ id)))) => (Y0 @ id @ ((^[Y2 : subst]: ((^[Y3 : subst > term > term]: (sF591)))) @ id @ (^[Y2 : subst]: ((^[Y3 : term]: (Y1 @ Y2 @ Y3))))) @ id))))) => (!! @ term @ (^[Y1 : term]: ((!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) => (Y0 @ id @ (sub @ Y1 @ (push @ Y2 @ id)) @ id)))) => (Y0 @ id @ (lam @ (sub @ Y1 @ sF592)) @ id)))))))) = $false) | (sF593 = $true)),
  inference(iff_proxy_clausification,[],[f986])).
thf(f1136,plain,(
  ($false = (((^[Y0 : subst > term > subst > $o]: ((!! @ (subst > term > term) @ (^[Y1 : subst > term > term]: ((!! @ term @ (^[Y2 : term]: (!! @ subst @ (^[Y3 : subst]: (!! @ subst @ (^[Y4 : subst]: ((Y1 @ (comp @ Y3 @ Y4) @ (sub @ Y2 @ Y4)) = (sub @ (Y1 @ Y3 @ Y2) @ Y4)))))))) => ((!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) => (Y0 @ id @ (Y1 @ id @ Y2) @ id)))) => (Y0 @ id @ ((^[Y2 : subst]: ((^[Y3 : subst > term > term]: (sF591)))) @ id @ (^[Y2 : subst]: ((^[Y3 : term]: (Y1 @ Y2 @ Y3))))) @ id))))) => (!! @ term @ (^[Y1 : term]: ((!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) => (Y0 @ id @ (sub @ Y1 @ (push @ Y2 @ id)) @ id)))) => (Y0 @ id @ (lam @ (sub @ Y1 @ sF592)) @ id)))))) @ sK597))) | (sF593 = $true)),
  inference(sigma_proxy_clausification,[],[f1098])).
thf(f1137,plain,(
  ((((!! @ (subst > term > term) @ (^[Y0 : subst > term > term]: ((!! @ term @ (^[Y1 : term]: (!! @ subst @ (^[Y2 : subst]: (!! @ subst @ (^[Y3 : subst]: ((Y0 @ (comp @ Y2 @ Y3) @ (sub @ Y1 @ Y3)) = (sub @ (Y0 @ Y2 @ Y1) @ Y3)))))))) => ((!! @ term @ (^[Y1 : term]: ((sK597 @ id @ Y1 @ id) => (sK597 @ id @ (Y0 @ id @ Y1) @ id)))) => (sK597 @ id @ sF591 @ id))))) => (!! @ term @ (^[Y0 : term]: ((!! @ term @ (^[Y1 : term]: ((sK597 @ id @ Y1 @ id) => (sK597 @ id @ (sub @ Y0 @ (push @ Y1 @ id)) @ id)))) => (sK597 @ id @ (lam @ (sub @ Y0 @ sF592)) @ id)))))) = $false) | (sF593 = $true)),
  inference(beta-eta_normalization,[],[f1136])).
thf(f1138,plain,(
  ($false = ((!! @ term @ (^[Y0 : term]: ((!! @ term @ (^[Y1 : term]: ((sK597 @ id @ Y1 @ id) => (sK597 @ id @ (sub @ Y0 @ (push @ Y1 @ id)) @ id)))) => (sK597 @ id @ (lam @ (sub @ Y0 @ sF592)) @ id)))))) | (sF593 = $true)),
  inference(imp_proxy_clausification,[],[f1137])).
thf(f1139,plain,(
  (sF593 = $true) | (((!! @ (subst > term > term) @ (^[Y0 : subst > term > term]: ((!! @ term @ (^[Y1 : term]: (!! @ subst @ (^[Y2 : subst]: (!! @ subst @ (^[Y3 : subst]: ((Y0 @ (comp @ Y2 @ Y3) @ (sub @ Y1 @ Y3)) = (sub @ (Y0 @ Y2 @ Y1) @ Y3)))))))) => ((!! @ term @ (^[Y1 : term]: ((sK597 @ id @ Y1 @ id) => (sK597 @ id @ (Y0 @ id @ Y1) @ id)))) => (sK597 @ id @ sF591 @ id)))))) = $true)),
  inference(imp_proxy_clausification,[],[f1137])).
thf(f1140,plain,(
  ( ! [X1 : (subst > term > term)] : (((((^[Y0 : subst > term > term]: ((!! @ term @ (^[Y1 : term]: (!! @ subst @ (^[Y2 : subst]: (!! @ subst @ (^[Y3 : subst]: ((Y0 @ (comp @ Y2 @ Y3) @ (sub @ Y1 @ Y3)) = (sub @ (Y0 @ Y2 @ Y1) @ Y3)))))))) => ((!! @ term @ (^[Y1 : term]: ((sK597 @ id @ Y1 @ id) => (sK597 @ id @ (Y0 @ id @ Y1) @ id)))) => (sK597 @ id @ sF591 @ id)))) @ X1)) = $true) | (sF593 = $true)) )),
  inference(pi_proxy_clausification,[],[f1139])).
thf(f1141,plain,(
  ( ! [X1 : (subst > term > term)] : (($true = (((!! @ term @ (^[Y0 : term]: (!! @ subst @ (^[Y1 : subst]: (!! @ subst @ (^[Y2 : subst]: ((X1 @ (comp @ Y1 @ Y2) @ (sub @ Y0 @ Y2)) = (sub @ (X1 @ Y1 @ Y0) @ Y2)))))))) => ((!! @ term @ (^[Y0 : term]: ((sK597 @ id @ Y0 @ id) => (sK597 @ id @ (X1 @ id @ Y0) @ id)))) => (sK597 @ id @ sF591 @ id))))) | (sF593 = $true)) )),
  inference(beta-eta_normalization,[],[f1140])).
thf(f1142,plain,(
  ( ! [X1 : (subst > term > term)] : ((sF593 = $true) | ($true = (((!! @ term @ (^[Y0 : term]: ((sK597 @ id @ Y0 @ id) => (sK597 @ id @ (X1 @ id @ Y0) @ id)))) => (sK597 @ id @ sF591 @ id)))) | (((!! @ term @ (^[Y0 : term]: (!! @ subst @ (^[Y1 : subst]: (!! @ subst @ (^[Y2 : subst]: ((X1 @ (comp @ Y1 @ Y2) @ (sub @ Y0 @ Y2)) = (sub @ (X1 @ Y1 @ Y0) @ Y2))))))))) = $false)) )),
  inference(imp_proxy_clausification,[],[f1141])).
thf(f1143,plain,(
  ( ! [X1 : (subst > term > term)] : ((sF593 = $true) | (((!! @ term @ (^[Y0 : term]: (!! @ subst @ (^[Y1 : subst]: (!! @ subst @ (^[Y2 : subst]: ((X1 @ (comp @ Y1 @ Y2) @ (sub @ Y0 @ Y2)) = (sub @ (X1 @ Y1 @ Y0) @ Y2))))))))) = $false) | (((!! @ term @ (^[Y0 : term]: ((sK597 @ id @ Y0 @ id) => (sK597 @ id @ (X1 @ id @ Y0) @ id))))) = $false) | (((sK597 @ id @ sF591 @ id)) = $true)) )),
  inference(imp_proxy_clausification,[],[f1142])).
thf(f1144,plain,(
  ( ! [X1 : (subst > term > term)] : ((sF593 = $true) | (((!! @ term @ (^[Y0 : term]: ((sK597 @ id @ Y0 @ id) => (sK597 @ id @ (X1 @ id @ Y0) @ id))))) = $false) | (((sK597 @ id @ sF591 @ id)) = $true) | ((((^[Y0 : term]: (!! @ subst @ (^[Y1 : subst]: (!! @ subst @ (^[Y2 : subst]: ((X1 @ (comp @ Y1 @ Y2) @ (sub @ Y0 @ Y2)) = (sub @ (X1 @ Y1 @ Y0) @ Y2))))))) @ (sK598 @ X1))) = $false)) )),
  inference(sigma_proxy_clausification,[],[f1143])).
thf(f1145,plain,(
  ( ! [X1 : (subst > term > term)] : ((((sK597 @ id @ sF591 @ id)) = $true) | ((((^[Y0 : term]: (!! @ subst @ (^[Y1 : subst]: (!! @ subst @ (^[Y2 : subst]: ((X1 @ (comp @ Y1 @ Y2) @ (sub @ Y0 @ Y2)) = (sub @ (X1 @ Y1 @ Y0) @ Y2))))))) @ (sK598 @ X1))) = $false) | (sF593 = $true) | ((((^[Y0 : term]: ((sK597 @ id @ Y0 @ id) => (sK597 @ id @ (X1 @ id @ Y0) @ id))) @ (sK599 @ X1))) = $false)) )),
  inference(sigma_proxy_clausification,[],[f1144])).
thf(f1146,plain,(
  ( ! [X1 : (subst > term > term)] : (($false = (((sK597 @ id @ (sK599 @ X1) @ id) => (sK597 @ id @ (X1 @ id @ (sK599 @ X1)) @ id)))) | ($false = ((!! @ subst @ (^[Y0 : subst]: (!! @ subst @ (^[Y1 : subst]: ((X1 @ (comp @ Y0 @ Y1) @ (sub @ (sK598 @ X1) @ Y1)) = (sub @ (X1 @ Y0 @ (sK598 @ X1)) @ Y1)))))))) | (((sK597 @ id @ sF591 @ id)) = $true) | (sF593 = $true)) )),
  inference(beta-eta_normalization,[],[f1145])).
thf(f1148,plain,(
  ( ! [X1 : (subst > term > term)] : ((((sK597 @ id @ (sK599 @ X1) @ id)) = $true) | (((sK597 @ id @ sF591 @ id)) = $true) | ($false = ((!! @ subst @ (^[Y0 : subst]: (!! @ subst @ (^[Y1 : subst]: ((X1 @ (comp @ Y0 @ Y1) @ (sub @ (sK598 @ X1) @ Y1)) = (sub @ (X1 @ Y0 @ (sK598 @ X1)) @ Y1)))))))) | (sF593 = $true)) )),
  inference(imp_proxy_clausification,[],[f1146])).
thf(f1149,plain,(
  ( ! [X1 : (subst > term > term)] : (((((^[Y0 : subst]: (!! @ subst @ (^[Y1 : subst]: ((X1 @ (comp @ Y0 @ Y1) @ (sub @ (sK598 @ X1) @ Y1)) = (sub @ (X1 @ Y0 @ (sK598 @ X1)) @ Y1))))) @ (sK600 @ X1))) = $false) | (((sK597 @ id @ sF591 @ id)) = $true) | (sF593 = $true) | (((sK597 @ id @ (sK599 @ X1) @ id)) = $true)) )),
  inference(sigma_proxy_clausification,[],[f1148])).
thf(f1150,plain,(
  ( ! [X1 : (subst > term > term)] : ((sF593 = $true) | (((sK597 @ id @ sF591 @ id)) = $true) | (((sK597 @ id @ (sK599 @ X1) @ id)) = $true) | (((!! @ subst @ (^[Y0 : subst]: ((X1 @ (comp @ (sK600 @ X1) @ Y0) @ (sub @ (sK598 @ X1) @ Y0)) = (sub @ (X1 @ (sK600 @ X1) @ (sK598 @ X1)) @ Y0))))) = $false)) )),
  inference(beta-eta_normalization,[],[f1149])).
thf(f1151,plain,(
  ( ! [X1 : (subst > term > term)] : ((((sK597 @ id @ (sK599 @ X1) @ id)) = $true) | (sF593 = $true) | ((((^[Y0 : subst]: ((X1 @ (comp @ (sK600 @ X1) @ Y0) @ (sub @ (sK598 @ X1) @ Y0)) = (sub @ (X1 @ (sK600 @ X1) @ (sK598 @ X1)) @ Y0))) @ (sK601 @ X1))) = $false) | (((sK597 @ id @ sF591 @ id)) = $true)) )),
  inference(sigma_proxy_clausification,[],[f1150])).
thf(f1152,plain,(
  ( ! [X1 : (subst > term > term)] : ((((sK597 @ id @ sF591 @ id)) = $true) | (sF593 = $true) | ((((X1 @ (comp @ (sK600 @ X1) @ (sK601 @ X1)) @ (sub @ (sK598 @ X1) @ (sK601 @ X1))) = (sub @ (X1 @ (sK600 @ X1) @ (sK598 @ X1)) @ (sK601 @ X1)))) = $false) | (((sK597 @ id @ (sK599 @ X1) @ id)) = $true)) )),
  inference(beta-eta_normalization,[],[f1151])).
thf(f1153,plain,(
  ( ! [X1 : (subst > term > term)] : ((((sK597 @ id @ (sK599 @ X1) @ id)) = $true) | (sF593 = $true) | (((sub @ (X1 @ (sK600 @ X1) @ (sK598 @ X1)) @ (sK601 @ X1))) != ((X1 @ (comp @ (sK600 @ X1) @ (sK601 @ X1)) @ (sub @ (sK598 @ X1) @ (sK601 @ X1))))) | (((sK597 @ id @ sF591 @ id)) = $true)) )),
  inference(equality_proxy_clausification,[],[f1152])).
thf(f1159,plain,(
  ($false = (((^[Y0 : term]: ((!! @ term @ (^[Y1 : term]: ((sK597 @ id @ Y1 @ id) => (sK597 @ id @ (sub @ Y0 @ (push @ Y1 @ id)) @ id)))) => (sK597 @ id @ (lam @ (sub @ Y0 @ sF592)) @ id))) @ sK604))) | (sF593 = $true)),
  inference(sigma_proxy_clausification,[],[f1138])).
thf(f1160,plain,(
  (sF593 = $true) | ((((!! @ term @ (^[Y0 : term]: ((sK597 @ id @ Y0 @ id) => (sK597 @ id @ (sub @ sK604 @ (push @ Y0 @ id)) @ id)))) => (sK597 @ id @ (lam @ (sub @ sK604 @ sF592)) @ id))) = $false)),
  inference(beta-eta_normalization,[],[f1159])).
thf(f1161,plain,(
  (sF593 = $true) | (((sK597 @ id @ (lam @ (sub @ sK604 @ sF592)) @ id)) = $false)),
  inference(imp_proxy_clausification,[],[f1160])).
thf(f1849,definition,(
  spl605_155 <=> (apinj2 = $true)),
  introduced(definition,[new_symbols(definition,[spl605_155])],[avatar_definition])).
thf(f2111,definition,(
  spl605_218 <=> (sP12 = $true)),
  introduced(definition,[new_symbols(definition,[spl605_218])],[avatar_definition])).
thf(f2309,definition,(
  spl605_265 <=> (hoasinduction_lem3aa = $false)),
  introduced(definition,[new_symbols(definition,[spl605_265])],[avatar_definition])).
thf(f2311,plain,(
  (hoasinduction_lem3aa = $false) | ~spl605_265),
  inference(avatar_component_clause,[],[f2309])).
thf(f2313,definition,(
  spl605_266 <=> (sF593 = $false)),
  introduced(definition,[new_symbols(definition,[spl605_266])],[avatar_definition])).
thf(f2315,plain,(
  (sF593 = $false) | ~spl605_266),
  inference(avatar_component_clause,[],[f2313])).
thf(f2316,plain,(
  spl605_265 | spl605_266),
  inference(avatar_split_clause,[],[f1047,f2313,f2309])).
thf(f2318,definition,(
  spl605_267 <=> (sF593 = $true)),
  introduced(definition,[new_symbols(definition,[spl605_267])],[avatar_definition])).
thf(f2320,plain,(
  (sF593 = $true) | ~spl605_267),
  inference(avatar_component_clause,[],[f2318])).
thf(f2322,definition,(
  spl605_268 <=> (hoasinduction_lem3aa = $true)),
  introduced(definition,[new_symbols(definition,[spl605_268])],[avatar_definition])).
thf(f2324,plain,(
  (hoasinduction_lem3aa = $true) | ~spl605_268),
  inference(avatar_component_clause,[],[f2322])).
thf(f2372,definition,(
  spl605_280 <=> ! [X125 : term,X124 : term,X123 : term,X122 : term] : ((((ap @ X124 @ X123)) != ((ap @ X122 @ X125))) | (X123 = X125))),
  introduced(definition,[new_symbols(definition,[spl605_280])],[avatar_definition])).
thf(f2373,plain,(
  ( ! [X124 : term,X125 : term,X122 : term,X123 : term] : ((X123 = X125) | (((ap @ X124 @ X123)) != ((ap @ X122 @ X125)))) ) | ~spl605_280),
  inference(avatar_component_clause,[],[f2372])).
thf(f2380,definition,(
  spl605_282 <=> (sK589 = sK590)),
  introduced(definition,[new_symbols(definition,[spl605_282])],[avatar_definition])).
thf(f2382,plain,(
  (sK589 != sK590) | spl605_282),
  inference(avatar_component_clause,[],[f2380])).
thf(f2383,plain,(
  ~spl605_282 | spl605_155),
  inference(avatar_split_clause,[],[f762,f1849,f2380])).
thf(f2793,definition,(
  spl605_374 <=> (((sK359 @ id @ (lam @ (sub @ sK360 @ (push @ one @ sh))) @ id)) = $true)),
  introduced(definition,[new_symbols(definition,[spl605_374])],[avatar_definition])).
thf(f2795,plain,(
  (((sK359 @ id @ (lam @ (sub @ sK360 @ (push @ one @ sh))) @ id)) != $true) | spl605_374),
  inference(avatar_component_clause,[],[f2793])).
thf(f2796,plain,(
  spl605_218 | ~spl605_374),
  inference(avatar_split_clause,[],[f649,f2793,f2111])).
thf(f3308,definition,(
  spl605_491 <=> ! [X1 : (subst > term > term)] : ((((sK597 @ id @ (sK599 @ X1) @ id)) = $true) | (((sub @ (X1 @ (sK600 @ X1) @ (sK598 @ X1)) @ (sK601 @ X1))) != ((X1 @ (comp @ (sK600 @ X1) @ (sK601 @ X1)) @ (sub @ (sK598 @ X1) @ (sK601 @ X1))))))),
  introduced(definition,[new_symbols(definition,[spl605_491])],[avatar_definition])).
thf(f3309,plain,(
  ( ! [X1 : (subst > term > term)] : ((((sub @ (X1 @ (sK600 @ X1) @ (sK598 @ X1)) @ (sK601 @ X1))) != ((X1 @ (comp @ (sK600 @ X1) @ (sK601 @ X1)) @ (sub @ (sK598 @ X1) @ (sK601 @ X1))))) | (((sK597 @ id @ (sK599 @ X1) @ id)) = $true)) ) | ~spl605_491),
  inference(avatar_component_clause,[],[f3308])).
thf(f3311,definition,(
  spl605_492 <=> (((sK597 @ id @ sF591 @ id)) = $true)),
  introduced(definition,[new_symbols(definition,[spl605_492])],[avatar_definition])).
thf(f3312,plain,(
  (((sK597 @ id @ sF591 @ id)) != $true) | spl605_492),
  inference(avatar_component_clause,[],[f3311])).
thf(f3313,plain,(
  (((sK597 @ id @ sF591 @ id)) = $true) | ~spl605_492),
  inference(avatar_component_clause,[],[f3311])).
thf(f3314,plain,(
  spl605_267 | spl605_491 | spl605_492),
  inference(avatar_split_clause,[],[f1153,f3311,f3308,f2318])).
thf(f3324,definition,(
  spl605_495 <=> (((sK597 @ id @ (lam @ (sub @ sK604 @ sF592)) @ id)) = $false)),
  introduced(definition,[new_symbols(definition,[spl605_495])],[avatar_definition])).
thf(f3326,plain,(
  (((sK597 @ id @ (lam @ (sub @ sK604 @ sF592)) @ id)) = $false) | ~spl605_495),
  inference(avatar_component_clause,[],[f3324])).
thf(f3327,plain,(
  spl605_495 | spl605_267),
  inference(avatar_split_clause,[],[f1161,f2318,f3324])).
thf(f3484,definition,(
  spl605_531 <=> ! [X2 : (subst > term > term)] : ((((X2 @ (comp @ (sK361 @ X2) @ (sK363 @ X2)) @ (sub @ (sK362 @ X2) @ (sK363 @ X2)))) != ((sub @ (X2 @ (sK361 @ X2) @ (sK362 @ X2)) @ (sK363 @ X2)))) | (((sK359 @ id @ (lam @ (X2 @ sh @ one)) @ id)) = $true) | (((sK359 @ id @ (sK364 @ X2) @ id)) = $true))),
  introduced(definition,[new_symbols(definition,[spl605_531])],[avatar_definition])).
thf(f3485,plain,(
  ( ! [X2 : (subst > term > term)] : ((((sK359 @ id @ (lam @ (X2 @ sh @ one)) @ id)) = $true) | (((X2 @ (comp @ (sK361 @ X2) @ (sK363 @ X2)) @ (sub @ (sK362 @ X2) @ (sK363 @ X2)))) != ((sub @ (X2 @ (sK361 @ X2) @ (sK362 @ X2)) @ (sK363 @ X2)))) | (((sK359 @ id @ (sK364 @ X2) @ id)) = $true)) ) | ~spl605_531),
  inference(avatar_component_clause,[],[f3484])).
thf(f3486,plain,(
  spl605_531 | spl605_218),
  inference(avatar_split_clause,[],[f651,f2111,f3484])).
thf(f3725,plain,(
  spl605_280 | ~spl605_155),
  inference(avatar_split_clause,[],[f761,f1849,f2372])).
thf(f3954,plain,(
  spl605_268 | ~spl605_218),
  inference(avatar_split_clause,[],[f829,f2111,f2322])).
thf(f3961,plain,(
  ($false = $true) | (~spl605_266 | ~spl605_267)),
  inference(forward_demodulation,[],[f2320,f2315])).
thf(f3962,plain,(
  $false | (~spl605_266 | ~spl605_267)),
  inference(trivial_inequality_removal,[],[f3961])).
thf(f3963,plain,(
  ~spl605_266 | ~spl605_267),
  inference(avatar_contradiction_clause,[],[f3962])).
thf(f3965,plain,(
  ($false = $true) | (~spl605_265 | ~spl605_268)),
  inference(forward_demodulation,[],[f2324,f2311])).
thf(f3966,plain,(
  $false | (~spl605_265 | ~spl605_268)),
  inference(trivial_inequality_removal,[],[f3965])).
thf(f3967,plain,(
  ~spl605_265 | ~spl605_268),
  inference(avatar_contradiction_clause,[],[f3966])).
thf(f3971,plain,(
  ( ! [X0 : d_subst] : ((sh = ((d2subst @ X0)))) )),
  inference(superposition,[],[f760,f822])).
thf(f3982,plain,(
  ( ! [X0 : d_term] : ((sF591 = ((d2term @ X0)))) )),
  inference(superposition,[],[f983,f756])).
thf(f3992,plain,(
  ( ! [X241 : term] : ((sF591 = X241)) )),
  inference(forward_demodulation,[],[f790,f3982])).
thf(f3995,plain,(
  (sF591 != sK589) | spl605_282),
  inference(superposition,[],[f2382,f3992])).
thf(f3996,plain,(
  $false | spl605_282),
  inference(forward_subsumption_resolution,[],[f3995,f3992])).
thf(f3997,plain,(
  spl605_282),
  inference(avatar_contradiction_clause,[],[f3996])).
thf(f3998,plain,(
  ( ! [X124 : term,X125 : term,X123 : term] : ((X123 = X125) | (sF591 != ((ap @ X124 @ X123)))) ) | ~spl605_280),
  inference(forward_demodulation,[],[f2373,f3992])).
thf(f4000,plain,(
  ( ! [X125 : term,X123 : term] : ((X123 = X125)) ) | ~spl605_280),
  inference(forward_subsumption_resolution,[],[f3998,f3992])).
thf(f4027,plain,(
  ( ! [X194 : subst] : ((sh = X194)) )),
  inference(forward_demodulation,[],[f830,f3971])).
thf(f4364,plain,(
  (((sK597 @ sh @ sF591 @ sh)) = $true) | ~spl605_492),
  inference(forward_demodulation,[],[f3313,f4027])).
thf(f4365,plain,(
  ( ! [X0 : term] : ((((sK597 @ sh @ X0 @ sh)) = $true)) ) | ~spl605_492),
  inference(superposition,[],[f4364,f3992])).
thf(f4371,plain,(
  ( ! [X0 : subst,X1 : term] : (($true = ((sK597 @ X0 @ X1 @ X0)))) ) | ~spl605_492),
  inference(superposition,[],[f4365,f4027])).
thf(f5177,plain,(
  ($false = $true) | (~spl605_492 | ~spl605_495)),
  inference(forward_demodulation,[],[f3326,f4371])).
thf(f5178,plain,(
  $false | (~spl605_492 | ~spl605_495)),
  inference(trivial_inequality_removal,[],[f5177])).
thf(f5179,plain,(
  ~spl605_492 | ~spl605_495),
  inference(avatar_contradiction_clause,[],[f5178])).
thf(f5182,plain,(
  ( ! [X1 : (subst > term > term)] : ((((sK597 @ id @ (sK599 @ X1) @ id)) = $true)) ) | (~spl605_280 | ~spl605_491)),
  inference(forward_subsumption_resolution,[],[f3309,f4000])).
thf(f5183,plain,(
  (((sK597 @ sh @ sF591 @ sh)) != $true) | spl605_492),
  inference(forward_demodulation,[],[f3312,f4027])).
thf(f5186,plain,(
  ( ! [X1 : (subst > term > term)] : ((((sK597 @ sh @ (sK599 @ X1) @ sh)) = $true)) ) | (~spl605_280 | ~spl605_491)),
  inference(forward_demodulation,[],[f5182,f4027])).
thf(f5188,plain,(
  (((sK597 @ sh @ sF591 @ sh)) = $true) | (~spl605_280 | ~spl605_491)),
  inference(forward_demodulation,[],[f5186,f3992])).
thf(f5230,plain,(
  $false | (~spl605_280 | ~spl605_491 | spl605_492)),
  inference(forward_subsumption_resolution,[],[f5188,f5183])).
thf(f5231,plain,(
  ~spl605_280 | ~spl605_491 | spl605_492),
  inference(avatar_contradiction_clause,[],[f5230])).
thf(f5233,plain,(
  (((sK359 @ sh @ (lam @ (sub @ sK360 @ (push @ one @ sh))) @ sh)) != $true) | spl605_374),
  inference(forward_demodulation,[],[f2795,f4027])).
thf(f5235,plain,(
  ( ! [X2 : (subst > term > term)] : ((((sK359 @ id @ (sK364 @ X2) @ id)) = $true) | (((sK359 @ id @ (lam @ (X2 @ sh @ one)) @ id)) = $true)) ) | (~spl605_280 | ~spl605_531)),
  inference(forward_subsumption_resolution,[],[f3485,f4000])).
thf(f5237,plain,(
  (((sK359 @ sh @ sF591 @ sh)) != $true) | spl605_374),
  inference(forward_demodulation,[],[f5233,f3992])).
thf(f5239,plain,(
  ( ! [X2 : (subst > term > term)] : ((((sK359 @ id @ (lam @ (X2 @ sh @ one)) @ id)) = $true) | (((sK359 @ sh @ (sK364 @ X2) @ sh)) = $true)) ) | (~spl605_280 | ~spl605_531)),
  inference(forward_demodulation,[],[f5235,f4027])).
thf(f5242,plain,(
  ( ! [X2 : (subst > term > term)] : ((((sK359 @ sh @ (lam @ (X2 @ sh @ one)) @ sh)) = $true) | (((sK359 @ sh @ (sK364 @ X2) @ sh)) = $true)) ) | (~spl605_280 | ~spl605_531)),
  inference(forward_demodulation,[],[f5239,f4027])).
thf(f5245,definition,(
  spl605_661 <=> (((sK359 @ sh @ sF591 @ sh)) = $true)),
  introduced(definition,[new_symbols(definition,[spl605_661])],[avatar_definition])).
thf(f5252,plain,(
  ( ! [X2 : (subst > term > term)] : ((((sK359 @ sh @ (sK364 @ X2) @ sh)) = $true) | (((sK359 @ sh @ sF591 @ sh)) = $true)) ) | (~spl605_280 | ~spl605_531)),
  inference(forward_demodulation,[],[f5242,f3992])).
thf(f5254,plain,(
  (((sK359 @ sh @ sF591 @ sh)) = $true) | (((sK359 @ sh @ sF591 @ sh)) = $true) | (~spl605_280 | ~spl605_531)),
  inference(forward_demodulation,[],[f5252,f3992])).
thf(f5255,plain,(
  (((sK359 @ sh @ sF591 @ sh)) = $true) | (~spl605_280 | ~spl605_531)),
  inference(duplicate_literal_removal,[],[f5254])).
thf(f5256,plain,(
  spl605_661 | ~spl605_280 | ~spl605_531),
  inference(avatar_split_clause,[],[f5255,f3484,f2372,f5245])).
thf(f5259,plain,(
  ~spl605_661 | spl605_374),
  inference(avatar_split_clause,[],[f5237,f2793,f5245])).
cnf(s204, plain, spl605_265 | spl605_266, inference(sat_conversion,[],[f2316])).
cnf(s215, plain, spl605_155 | ~spl605_282, inference(sat_conversion,[],[f2383])).
cnf(s314, plain, spl605_218 | ~spl605_374, inference(sat_conversion,[],[f2796])).
cnf(s442, plain, spl605_267 | spl605_491 | spl605_492, inference(sat_conversion,[],[f3314])).
cnf(s445, plain, spl605_267 | spl605_495, inference(sat_conversion,[],[f3327])).
cnf(s488, plain, spl605_218 | spl605_531, inference(sat_conversion,[],[f3486])).
cnf(s549, plain, ~spl605_155 | spl605_280, inference(sat_conversion,[],[f3725])).
cnf(s611, plain, ~spl605_218 | spl605_268, inference(sat_conversion,[],[f3954])).
cnf(s614, plain, ~spl605_266 | ~spl605_267, inference(sat_conversion,[],[f3963])).
cnf(s615, plain, ~spl605_265 | ~spl605_268, inference(sat_conversion,[],[f3967])).
cnf(s616, plain, spl605_282, inference(sat_conversion,[],[f3997])).
cnf(s685, plain, ~spl605_492 | ~spl605_495, inference(sat_conversion,[],[f5179])).
cnf(s686, plain, ~spl605_280 | ~spl605_491 | spl605_492, inference(sat_conversion,[],[f5231])).
cnf(s688, plain, ~spl605_280 | ~spl605_531 | spl605_661, inference(sat_conversion,[],[f5256])).
cnf(s689, plain, spl605_374 | ~spl605_661, inference(sat_conversion,[],[f5259])).
cnf(s700, plain, spl605_155, inference(rat,[],[s215,s616])).
cnf(s701, plain, spl605_280, inference(rat,[],[s549,s700])).
cnf(s765, plain, spl605_218, inference(rat,[],[s689,s688,s314,s488,s701])).
cnf(s766, plain, spl605_268, inference(rat,[],[s611,s765])).
cnf(s773, plain, ~spl605_265, inference(rat,[],[s615,s766])).
cnf(s774, plain, spl605_266, inference(rat,[],[s204,s773])).
cnf(s775, plain, ~spl605_267, inference(rat,[],[s614,s774])).
cnf(s776, plain, spl605_495, inference(rat,[],[s445,s775])).
cnf(s778, plain, ~spl605_492, inference(rat,[],[s685,s776])).
cnf(s780, plain, ~spl605_491, inference(rat,[],[s686,s701,s778])).
cnf(s781, plain, $false, inference(rat,[],[s442,s775,s778,s780])).
thf(f5260,plain,(
  $false),
  inference(avatar_sat_refutation,[],[s781])).
% SZS output end Proof for theBenchmark