%----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: ((subst > term > subst > $o) > $o)).
thf(func_def_139, type, sP1: $o).
thf(func_def_140, type, sP2: ((term > $o) > $o)).
thf(func_def_141, type, sP3: ((subst > term > subst > $o) > $o)).
thf(func_def_142, type, sP4: $o).
thf(func_def_143, type, sP5: ((subst > term > subst > $o) > $o)).
thf(func_def_144, type, sP6: $o).
thf(func_def_145, type, sP7: ((subst > term > subst > $o) > $o)).
thf(func_def_146, type, sP8: $o).
thf(func_def_147, type, sP9: $o).
thf(func_def_148, type, sP10: ((subst > term > subst > $o) > $o)).
thf(func_def_149, type, sP11: ((subst > term > subst > $o) > $o)).
thf(func_def_150, type, sP12: $o).
thf(func_def_151, type, sP13: ((term > $o) > $o)).
thf(func_def_152, type, sP14: $o).
thf(func_def_153, type, sP15: $o).
thf(func_def_154, type, sP16: $o).
thf(func_def_155, type, sP17: ((subst > term > subst > $o) > $o)).
thf(func_def_156, type, sP18: $o).
thf(func_def_157, type, sP19: $o).
thf(func_def_158, type, sP20: ((term > $o) > $o)).
thf(func_def_159, type, sP21: ((term > $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: $o).
thf(func_def_164, type, sP26: $o).
thf(func_def_165, type, sP27: ((subst > term > subst > $o) > $o)).
thf(func_def_166, type, sP28: ((term > $o) > $o)).
thf(func_def_167, type, sP29: ((subst > term > subst > $o) > $o)).
thf(func_def_168, type, sP30: ((subst > term > subst > $o) > $o)).
thf(func_def_169, type, sP31: ((subst > term > subst > $o) > $o)).
thf(func_def_170, type, sP32: $o).
thf(func_def_171, type, sP33: $o).
thf(func_def_172, type, sP34: $o).
thf(func_def_173, type, sP35: $o).
thf(func_def_174, type, sP36: ((term > $o) > $o)).
thf(func_def_175, type, sP37: ((term > $o) > $o)).
thf(func_def_176, type, sP38: $o).
thf(func_def_177, type, sP39: $o).
thf(func_def_178, type, sP40: ((term > $o) > $o)).
thf(func_def_179, type, sP41: $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: ((subst > term > subst > $o) > $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: ((subst > term > subst > $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: ((term > $o) > $o)).
thf(func_def_192, type, sP54: ((subst > term > subst > $o) > $o)).
thf(func_def_193, type, sP55: ((subst > term > subst > $o) > $o)).
thf(func_def_194, type, sP56: $o).
thf(func_def_195, type, sP57: $o).
thf(func_def_196, type, sP58: $o).
thf(func_def_197, type, sK59: ((subst > term > subst > $o) > term > $o)).
thf(func_def_198, type, sK60: ((term > $o) > term)).
thf(func_def_199, type, sK61: ((term > $o) > term)).
thf(func_def_200, type, sK62: term).
thf(func_def_201, type, sK63: (subst > term > subst > $o)).
thf(func_def_202, type, sK64: ((term > $o) > term)).
thf(func_def_203, type, sK65: term).
thf(func_def_204, type, sK66: (term > $o)).
thf(func_def_205, type, sK67: (term > term)).
thf(func_def_206, type, sK68: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_207, type, sK69: ((subst > term > subst > $o) > (term > $o) > term)).
thf(func_def_208, type, sK70: (term > (term > $o) > 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) > subst)).
thf(func_def_221, type, sK83: ((subst > term > subst > $o) > term)).
thf(func_def_222, type, sK84: ((subst > term > subst > $o) > subst)).
thf(func_def_223, type, sK85: ((subst > term > subst > $o) > term)).
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: ((subst > term > subst > $o) > term)).
thf(func_def_228, type, sK90: ((subst > term > subst > $o) > term)).
thf(func_def_229, type, sK91: ((subst > term > term) > (subst > term > subst > $o) > term)).
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)).
thf(func_def_233, type, sK95: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_234, type, sK96: ((term > $o) > term)).
thf(func_def_235, type, sK97: ((term > $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 > subst > $o) > term > $o)).
thf(func_def_239, type, sK101: ((subst > term > subst > $o) > term)).
thf(func_def_240, type, sK102: ((subst > term > subst > $o) > term)).
thf(func_def_241, type, sK103: term).
thf(func_def_242, type, sK104: (subst > term > subst > $o)).
thf(func_def_243, type, sK105: ((term > $o) > term)).
thf(func_def_244, type, sK106: ((term > $o) > term)).
thf(func_def_245, type, sK107: ((term > $o) > term)).
thf(func_def_246, type, sK108: (term > $o)).
thf(func_def_247, type, sK109: term).
thf(func_def_248, type, sK110: (term > term)).
thf(func_def_249, type, sK111: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_250, type, sK112: ((term > $o) > term > term)).
thf(func_def_251, type, sK113: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_252, type, sK114: term).
thf(func_def_253, type, sK115: (subst > term > subst > $o)).
thf(func_def_254, type, sK116: (term > $o)).
thf(func_def_255, type, sK117: ((subst > term > term) > term)).
thf(func_def_256, type, sK118: ((subst > term > term) > subst)).
thf(func_def_257, type, sK119: ((subst > term > term) > subst)).
thf(func_def_258, type, sK120: ((subst > term > term) > term)).
thf(func_def_259, type, sK121: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_260, type, sK122: ((subst > term > term) > term)).
thf(func_def_261, type, sK123: ((subst > term > term) > subst)).
thf(func_def_262, type, sK124: ((subst > term > term) > subst)).
thf(func_def_263, type, sK125: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_264, type, sK126: ((term > $o) > term)).
thf(func_def_265, type, sK127: ((term > $o) > term)).
thf(func_def_266, type, sK128: (subst > term > subst > $o)).
thf(func_def_267, type, sK129: term).
thf(func_def_268, type, sK130: ((subst > term > term) > term)).
thf(func_def_269, type, sK131: ((subst > term > term) > term > subst)).
thf(func_def_270, type, sK132: ((subst > term > term) > term > term)).
thf(func_def_271, type, sK133: ((subst > term > term) > term > subst)).
thf(func_def_272, type, sK134: ((subst > term > term) > term > subst)).
thf(func_def_273, type, sK135: ((subst > term > term) > term > subst)).
thf(func_def_274, type, sK136: ((subst > term > term) > term > term)).
thf(func_def_275, type, sK137: (term > (subst > term > subst > $o) > term)).
thf(func_def_276, type, sK138: ((subst > term > subst > $o) > term)).
thf(func_def_277, type, sK139: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_278, type, sK140: term).
thf(func_def_279, type, sK141: (term > $o)).
thf(func_def_280, type, sK142: (subst > term > subst > $o)).
thf(func_def_281, type, sK143: term).
thf(func_def_282, type, sK144: ((term > $o) > term > term)).
thf(func_def_283, type, sK145: ((subst > term > subst > $o) > subst)).
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) > term)).
thf(func_def_287, type, sK149: ((subst > term > subst > $o) > subst)).
thf(func_def_288, type, sK150: ((subst > term > subst > $o) > term)).
thf(func_def_289, type, sK151: ((subst > term > subst > $o) > subst)).
thf(func_def_290, type, sK152: ((subst > term > subst > $o) > subst)).
thf(func_def_291, type, sK153: ((subst > term > subst > $o) > (term > $o) > term)).
thf(func_def_292, type, sK154: ((subst > term > term) > term)).
thf(func_def_293, type, sK155: ((subst > term > term) > subst)).
thf(func_def_294, type, sK156: ((subst > term > term) > subst)).
thf(func_def_295, type, sK157: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_296, type, sK158: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_297, type, sK159: term).
thf(func_def_298, type, sK160: (subst > term > subst > $o)).
thf(func_def_299, type, sK161: ((subst > term > term) > term)).
thf(func_def_300, type, sK162: ((subst > term > term) > subst)).
thf(func_def_301, type, sK163: ((subst > term > term) > subst)).
thf(func_def_302, type, sK164: ((subst > term > term) > term)).
thf(func_def_303, type, sK165: ((subst > term > subst > $o) > term > term)).
thf(func_def_304, type, sK166: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_305, type, sK167: (subst > term > subst > $o)).
thf(func_def_306, type, sK168: term).
thf(func_def_307, type, sK169: ((subst > term > term) > subst)).
thf(func_def_308, type, sK170: ((subst > term > term) > term)).
thf(func_def_309, type, sK171: ((subst > term > term) > subst)).
thf(func_def_310, type, sK172: ((subst > term > term) > term)).
thf(func_def_311, type, sK173: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_312, type, sK174: (term > (subst > term > subst > $o) > term)).
thf(func_def_313, type, sK175: term).
thf(func_def_314, type, sK176: (term > (subst > term > subst > $o) > term)).
thf(func_def_315, type, sK177: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_316, type, sK178: (subst > term > subst > $o)).
thf(func_def_317, type, sK179: term).
thf(func_def_318, type, sK180: ((subst > term > term) > term)).
thf(func_def_319, type, sK181: ((subst > term > term) > subst)).
thf(func_def_320, type, sK182: ((subst > term > term) > subst)).
thf(func_def_321, type, sK183: ((subst > term > term) > term)).
thf(func_def_322, type, sK184: (term > $o)).
thf(func_def_323, type, sK185: term).
thf(func_def_324, type, sK186: (term > term)).
thf(func_def_325, type, sK187: ((term > $o) > term)).
thf(func_def_326, type, sK188: ((term > $o) > term)).
thf(func_def_327, type, sK189: ((term > $o) > term)).
thf(func_def_328, type, sK190: ((term > $o) > term)).
thf(func_def_329, type, sK191: ((term > $o) > term)).
thf(func_def_330, type, sK192: ((term > $o) > term)).
thf(func_def_331, type, sK193: term).
thf(func_def_332, type, sK194: subst).
thf(func_def_333, type, sK195: subst).
thf(func_def_334, type, sK196: term).
thf(func_def_335, type, sK197: (term > $o)).
thf(func_def_336, type, sK198: subst).
thf(func_def_337, type, sK199: term).
thf(func_def_338, type, sK200: (term > $o)).
thf(func_def_339, type, sK201: subst).
thf(func_def_340, type, sK202: term).
thf(func_def_341, type, sK203: (term > term)).
thf(func_def_342, type, sK204: ((term > $o) > term)).
thf(func_def_343, type, sK205: (subst > (term > $o) > term)).
thf(func_def_344, type, sK206: ((term > $o) > term)).
thf(func_def_345, type, sK207: ((term > $o) > term)).
thf(func_def_346, type, sK208: ((term > $o) > term)).
thf(func_def_347, type, sK209: ((term > $o) > term)).
thf(func_def_348, type, sK210: ((subst > term > subst > $o) > subst)).
thf(func_def_349, type, sK211: ((subst > term > subst > $o) > subst)).
thf(func_def_350, type, sK212: ((subst > term > subst > $o) > subst)).
thf(func_def_351, type, sK213: ((subst > term > subst > $o) > term)).
thf(func_def_352, type, sK214: ((subst > term > subst > $o) > subst)).
thf(func_def_353, type, sK215: ((subst > term > subst > $o) > term)).
thf(func_def_354, type, sK216: ((subst > term > subst > $o) > subst)).
thf(func_def_355, type, sK217: ((subst > term > subst > $o) > subst)).
thf(func_def_356, type, sK218: term).
thf(func_def_357, type, sK219: term).
thf(func_def_358, type, sK220: (subst > term > subst > $o)).
thf(func_def_359, type, sK221: ((term > $o) > term)).
thf(func_def_360, type, sK222: ((term > $o) > term)).
thf(func_def_361, type, sK223: term).
thf(func_def_362, type, sK224: (term > $o)).
thf(func_def_363, type, sK225: (term > term)).
thf(func_def_364, type, sK226: (subst > term > subst > $o)).
thf(func_def_365, type, sK227: term).
thf(func_def_366, type, sK228: ((subst > term > subst > $o) > term)).
thf(func_def_367, type, sK229: ((subst > term > subst > $o) > term)).
thf(func_def_368, type, sK230: ((subst > term > subst > $o) > subst)).
thf(func_def_369, type, sK231: ((subst > term > subst > $o) > subst)).
thf(func_def_370, type, sK232: ((subst > term > subst > $o) > subst)).
thf(func_def_371, type, sK233: ((subst > term > subst > $o) > term)).
thf(func_def_372, type, sK234: ((subst > term > subst > $o) > subst)).
thf(func_def_373, type, sK235: ((subst > term > subst > $o) > subst)).
thf(func_def_374, type, sK236: ((subst > term > subst > $o) > subst)).
thf(func_def_375, type, sK237: ((subst > term > subst > $o) > term > term)).
thf(func_def_376, type, sK238: ((subst > term > subst > $o) > subst)).
thf(func_def_377, type, sK239: ((subst > term > subst > $o) > subst)).
thf(func_def_378, type, sK240: ((subst > term > subst > $o) > subst)).
thf(func_def_379, type, sK241: ((subst > term > subst > $o) > term)).
thf(func_def_380, type, sK242: ((subst > term > subst > $o) > subst)).
thf(func_def_381, type, sK243: ((subst > term > subst > $o) > subst)).
thf(func_def_382, type, sK244: ((subst > term > subst > $o) > term)).
thf(func_def_383, type, sK245: ((subst > term > subst > $o) > subst)).
thf(func_def_384, type, sK246: term).
thf(func_def_385, type, sK247: (subst > term > subst > $o)).
thf(func_def_386, type, sK248: ((subst > term > subst > $o) > term)).
thf(func_def_387, type, sK249: ((subst > term > subst > $o) > term)).
thf(func_def_388, type, sK250: ((subst > term > subst > $o) > term)).
thf(func_def_389, type, sK251: ((subst > term > subst > $o) > term)).
thf(func_def_390, type, sK252: ((subst > term > term) > subst)).
thf(func_def_391, type, sK253: ((subst > term > term) > subst)).
thf(func_def_392, type, sK254: ((subst > term > term) > term)).
thf(func_def_393, type, sK255: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_394, type, sK256: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_395, type, sK257: ((term > $o) > term)).
thf(func_def_396, type, sK258: ((term > $o) > term)).
thf(func_def_397, type, sK259: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_398, type, sK260: ((subst > term > term) > subst)).
thf(func_def_399, type, sK261: ((subst > term > term) > term)).
thf(func_def_400, type, sK262: ((subst > term > term) > subst)).
thf(func_def_401, type, sK263: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_402, type, sK264: (subst > term > subst > $o)).
thf(func_def_403, type, sK265: term).
thf(func_def_404, type, sK266: ((subst > term > term) > term)).
thf(func_def_405, type, sK267: ((subst > term > term) > subst)).
thf(func_def_406, type, sK268: ((subst > term > term) > term)).
thf(func_def_407, type, sK269: ((subst > term > term) > subst)).
thf(func_def_408, type, sK270: (term > (subst > term > subst > $o) > term)).
thf(func_def_409, type, sK271: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_410, type, sK272: term).
thf(func_def_411, type, sK273: term).
thf(func_def_412, type, sK274: (term > $o)).
thf(func_def_413, type, sK275: (subst > term > subst > $o)).
thf(func_def_414, type, sK276: ((subst > term > subst > $o) > subst)).
thf(func_def_415, type, sK277: ((subst > term > subst > $o) > subst)).
thf(func_def_416, type, sK278: ((subst > term > subst > $o) > term)).
thf(func_def_417, type, sK279: ((subst > term > subst > $o) > subst)).
thf(func_def_418, type, sK280: ((subst > term > subst > $o) > (term > $o) > term)).
thf(func_def_419, type, sK281: ((subst > term > subst > $o) > subst)).
thf(func_def_420, type, sK282: ((subst > term > subst > $o) > term)).
thf(func_def_421, type, sK283: ((subst > term > subst > $o) > subst)).
thf(func_def_422, type, sK284: ((subst > term > subst > $o) > subst)).
thf(func_def_423, type, sK285: ((subst > term > subst > $o) > term)).
thf(func_def_424, type, sK286: ((subst > term > subst > $o) > term)).
thf(func_def_425, type, sK287: ((term > $o) > subst > term)).
thf(func_def_426, type, sK288: ((term > $o) > term)).
thf(func_def_427, type, sK289: subst).
thf(func_def_428, type, sK290: term).
thf(func_def_429, type, sK291: (term > $o)).
thf(func_def_430, type, sK292: (term > term)).
thf(func_def_431, type, sK293: term).
thf(func_def_432, type, sK294: term).
thf(func_def_433, type, sK295: (term > $o)).
thf(func_def_434, type, sK296: (term > term)).
thf(func_def_435, type, sK297: ((term > $o) > term)).
thf(func_def_436, type, sK298: ((term > $o) > term)).
thf(func_def_437, type, sK299: ((term > $o) > term)).
thf(func_def_438, type, sK300: ((term > $o) > term)).
thf(func_def_439, type, sK301: ((term > $o) > term)).
thf(func_def_440, type, sK302: ((term > $o) > term)).
thf(func_def_441, type, sK303: term).
thf(func_def_442, type, sK304: subst).
thf(func_def_443, type, sK305: (subst > term > term)).
thf(func_def_444, type, sK306: (subst > term > term)).
thf(func_def_445, type, sK307: subst).
thf(func_def_446, type, sK308: term).
thf(func_def_447, type, sK309: term).
thf(func_def_448, type, sK310: term).
thf(func_def_449, type, sK311: subst).
thf(func_def_450, type, sK312: term).
thf(func_def_451, type, sK313: ((subst > term > term) > subst)).
thf(func_def_452, type, sK314: ((subst > term > term) > term)).
thf(func_def_453, type, sK315: ((subst > term > term) > subst)).
thf(func_def_454, type, sK316: ((subst > term > term) > term)).
thf(func_def_455, type, sK317: ((subst > term > term) > subst)).
thf(func_def_456, type, sK318: ((subst > term > term) > subst)).
thf(func_def_457, type, sK319: (subst > term > subst > $o)).
thf(func_def_458, type, sK320: term).
thf(func_def_459, type, sK321: 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) > subst)).
thf(func_def_465, type, sK327: ((subst > term > subst > $o) > subst)).
thf(func_def_466, type, sK328: ((subst > term > subst > $o) > subst)).
thf(func_def_467, type, sK329: ((subst > term > subst > $o) > term)).
thf(func_def_468, type, sK330: ((subst > term > subst > $o) > term)).
thf(func_def_469, type, sK331: ((subst > term > subst > $o) > term)).
thf(func_def_470, type, sK332: (term > subst > term > term)).
thf(func_def_471, type, sK333: (subst > term > subst > $o)).
thf(func_def_472, type, sK334: term).
thf(func_def_473, type, sK335: (term > $o)).
thf(func_def_474, type, sK336: ((subst > term > term) > subst)).
thf(func_def_475, type, sK337: ((subst > term > term) > subst)).
thf(func_def_476, type, sK338: ((subst > term > term) > term)).
thf(func_def_477, type, sK339: ((subst > term > term) > term)).
thf(func_def_478, type, sK340: term).
thf(func_def_479, type, sK341: term).
thf(func_def_480, type, sK342: ((subst > term > term) > subst)).
thf(func_def_481, type, sK343: ((subst > term > term) > term)).
thf(func_def_482, type, sK344: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_483, type, sK345: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_484, type, sK346: ((term > $o) > term > term)).
thf(func_def_485, type, sK347: term).
thf(func_def_486, type, sK348: (subst > term > subst > $o)).
thf(func_def_487, type, sK349: ((subst > term > term) > subst)).
thf(func_def_488, type, sK350: ((subst > term > term) > subst)).
thf(func_def_489, type, sK351: ((subst > term > term) > term)).
thf(func_def_490, type, sK352: ((subst > term > term) > term)).
thf(func_def_491, type, sK353: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_492, type, sK354: ((subst > term > subst > $o) > term > term)).
thf(func_def_493, type, sK355: subst).
thf(func_def_494, type, sK356: term).
thf(func_def_495, type, sK357: (term > $o)).
thf(func_def_496, type, sK358: (term > term)).
thf(func_def_497, type, sK359: ((term > $o) > term)).
thf(func_def_498, type, sK360: ((term > $o) > subst > term)).
thf(func_def_499, type, sK361: ((term > $o) > term)).
thf(func_def_500, type, sK362: ((term > $o) > term)).
thf(func_def_501, type, sK363: (subst > term > subst > $o)).
thf(func_def_502, type, sK364: term).
thf(func_def_503, type, sK365: ((subst > term > subst > $o) > subst)).
thf(func_def_504, type, sK366: ((subst > term > subst > $o) > subst)).
thf(func_def_505, type, sK367: ((subst > term > subst > $o) > subst)).
thf(func_def_506, type, sK368: ((subst > term > subst > $o) > term)).
thf(func_def_507, type, sK369: ((subst > term > subst > $o) > term)).
thf(func_def_508, type, sK370: ((subst > term > subst > $o) > subst)).
thf(func_def_509, type, sK371: ((subst > term > subst > $o) > subst)).
thf(func_def_510, type, sK372: ((subst > term > subst > $o) > subst)).
thf(func_def_511, type, sK373: ((subst > term > subst > $o) > term)).
thf(func_def_512, type, sK374: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_513, type, sK375: ((subst > term > term) > subst)).
thf(func_def_514, type, sK376: ((subst > term > term) > term)).
thf(func_def_515, type, sK377: ((subst > term > term) > subst)).
thf(func_def_516, type, sK378: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_517, type, sK379: ((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)).
thf(func_def_520, type, sK382: term).
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) > term)).
thf(func_def_526, type, sK388: ((subst > term > subst > $o) > subst)).
thf(func_def_527, type, sK389: ((subst > term > subst > $o) > subst)).
thf(func_def_528, type, sK390: ((subst > term > subst > $o) > subst)).
thf(func_def_529, type, sK391: (term > (subst > term > subst > $o) > term)).
thf(func_def_530, type, sK392: term).
thf(func_def_531, type, sK393: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_532, type, sK394: (term > (subst > term > subst > $o) > 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 > term) > term)).
thf(func_def_536, type, sK398: ((subst > term > term) > subst)).
thf(func_def_537, type, sK399: ((subst > term > term) > subst)).
thf(func_def_538, type, sK400: ((subst > term > term) > term)).
thf(func_def_539, type, sK401: ((subst > term > term) > term)).
thf(func_def_540, type, sK402: ((subst > term > term) > subst)).
thf(func_def_541, type, sK403: ((subst > term > term) > subst)).
thf(func_def_542, type, sK404: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_543, type, sK405: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_544, type, sK406: (subst > term > subst > $o)).
thf(func_def_545, type, sK407: (term > $o)).
thf(func_def_546, type, sK408: term).
thf(func_def_547, type, sK409: ((subst > term > term) > subst)).
thf(func_def_548, type, sK410: ((subst > term > term) > term)).
thf(func_def_549, type, sK411: ((subst > term > term) > subst)).
thf(func_def_550, type, sK412: ((subst > term > term) > term)).
thf(func_def_551, type, sK413: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_552, type, sK414: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_553, type, sK415: (term > (term > $o) > term)).
thf(func_def_554, type, sK416: ((subst > term > subst > $o) > term)).
thf(func_def_555, type, sK417: ((subst > term > subst > $o) > term)).
thf(func_def_556, type, sK418: (subst > term > subst > $o)).
thf(func_def_557, type, sK419: term).
thf(func_def_558, type, sK420: ((subst > term > subst > $o) > subst)).
thf(func_def_559, type, sK421: ((subst > term > subst > $o) > term)).
thf(func_def_560, type, sK422: ((subst > term > subst > $o) > subst)).
thf(func_def_561, type, sK423: ((subst > term > subst > $o) > subst)).
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: (term > (subst > term > subst > $o) > term)).
thf(func_def_567, type, sK429: ((subst > term > term) > subst)).
thf(func_def_568, type, sK430: ((subst > term > term) > term)).
thf(func_def_569, type, sK431: ((subst > term > term) > subst)).
thf(func_def_570, type, sK432: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_571, type, sK433: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_572, type, sK434: ((term > $o) > term)).
thf(func_def_573, type, sK435: ((term > $o) > term)).
thf(func_def_574, type, sK436: (subst > term > subst > $o)).
thf(func_def_575, type, sK437: (term > $o)).
thf(func_def_576, type, sK438: term).
thf(func_def_577, type, sK439: (term > (term > $o) > term)).
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: ((term > $o) > (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) > term)).
thf(func_def_585, type, sK447: ((subst > term > subst > $o) > subst)).
thf(func_def_586, type, sK448: ((subst > term > subst > $o) > subst)).
thf(func_def_587, type, sK449: ((subst > term > term) > (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 > subst > $o) > subst > term > term)).
thf(func_def_592, type, sK454: term).
thf(func_def_593, type, sK455: subst).
thf(func_def_594, type, sK456: term).
thf(func_def_595, type, sK457: (term > $o)).
thf(func_def_596, type, sK458: (term > $o)).
thf(func_def_597, type, sK459: (term > subst > term > term)).
thf(func_def_598, type, sK460: term).
thf(func_def_599, type, sK461: ((subst > term > term) > subst)).
thf(func_def_600, type, sK462: ((subst > term > term) > term)).
thf(func_def_601, type, sK463: term).
thf(func_def_602, type, sK464: term).
thf(func_def_603, type, sK465: term).
thf(func_def_604, type, sK466: subst).
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: subst).
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: (subst > term > term)).
thf(func_def_615, type, sK477: subst).
thf(func_def_616, type, sK478: (subst > term > term)).
thf(func_def_617, type, sK479: term).
thf(func_def_618, type, sK480: ((subst > term > term) > term)).
thf(func_def_619, type, sK481: ((subst > term > term) > subst)).
thf(func_def_620, type, sK482: ((subst > term > term) > subst)).
thf(func_def_621, type, sK483: ((subst > term > term) > term)).
thf(func_def_622, type, sK484: ((subst > term > term) > subst)).
thf(func_def_623, type, sK485: ((subst > term > term) > subst)).
thf(func_def_624, type, sK486: (term > subst > term > term)).
thf(func_def_625, type, sK487: term).
thf(func_def_626, type, sK488: ((subst > term > term) > term)).
thf(func_def_627, type, sK489: ((subst > term > term) > subst)).
thf(func_def_628, type, sK490: term).
thf(func_def_629, type, sK491: subst).
thf(func_def_630, type, sK492: subst).
thf(func_def_631, type, sK493: subst).
thf(func_def_632, type, sK494: subst).
thf(func_def_633, type, sK495: term).
thf(func_def_634, type, sK496: subst).
thf(func_def_635, type, sK497: term).
thf(func_def_636, type, sK498: term).
thf(func_def_637, type, sK499: term).
thf(func_def_638, type, sK500: term).
thf(func_def_639, type, sK501: term).
thf(func_def_640, type, sK502: term).
thf(func_def_641, type, sK503: term).
thf(func_def_642, type, sK504: term).
thf(func_def_643, type, sK505: term).
thf(func_def_644, type, sK506: term).
thf(func_def_645, type, sK507: term).
thf(func_def_646, type, sK508: subst).
thf(func_def_647, type, sK509: (term > $o)).
thf(func_def_648, type, sK510: ((term > $o) > term)).
thf(func_def_649, type, sK511: ((term > $o) > subst > term > term > $o)).
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: (subst > d_subst)).
thf(func_def_655, type, sK517: subst).
thf(func_def_656, type, sK518: term).
thf(func_def_657, type, sK519: (term > (term > $o) > subst > (term > $o) > term)).
thf(func_def_658, type, sK520: (term > $o)).
thf(func_def_659, type, sK521: subst).
thf(func_def_660, type, sK522: (term > $o)).
thf(func_def_661, type, sK523: term).
thf(func_def_662, type, sK524: (term > subst > term > term)).
thf(func_def_663, type, sK525: term).
thf(func_def_664, type, sK526: (subst > term > subst > $o)).
thf(func_def_665, type, sK527: ((term > $o) > term)).
thf(func_def_666, type, sK528: ((subst > term > subst > $o) > term > $o)).
thf(func_def_667, type, sK529: subst).
thf(func_def_668, type, sK530: subst).
thf(func_def_669, type, sK531: term).
thf(func_def_670, type, sK532: (term > subst > term > term)).
thf(func_def_671, type, sK533: (term > subst > (term > $o) > term > $o)).
thf(func_def_672, type, sK534: (term > $o)).
thf(func_def_673, type, sK535: subst).
thf(func_def_674, type, sK536: term).
thf(func_def_675, type, sK537: ((term > $o) > term)).
thf(func_def_676, type, sK538: ((subst > term > term) > subst)).
thf(func_def_677, type, sK539: ((subst > term > term) > term)).
thf(func_def_678, type, sK540: ((subst > term > term) > subst)).
thf(func_def_679, type, sK541: (subst > term > term)).
thf(func_def_680, type, sK542: term).
thf(func_def_681, type, sK543: term).
thf(func_def_682, type, sK544: (subst > term > subst > $o)).
thf(func_def_683, type, sK545: ((term > $o) > term)).
thf(func_def_684, type, sK546: ((subst > term > subst > $o) > term > $o)).
thf(func_def_685, type, sK547: term).
thf(func_def_686, type, sK548: term).
thf(func_def_687, type, sK549: term).
thf(func_def_688, type, sK550: term).
thf(func_def_689, type, sK551: term).
thf(func_def_690, type, sK552: term).
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: term).
thf(func_def_695, type, sK557: term).
thf(func_def_696, type, sK558: term).
thf(func_def_697, type, sK559: term).
thf(func_def_698, type, sK560: (term > d_term)).
thf(func_def_699, type, sK561: subst).
thf(func_def_700, type, sK562: term).
thf(func_def_701, type, sK563: (subst > term > subst > $o)).
thf(func_def_702, type, sK564: ((subst > term > term) > term)).
thf(func_def_703, type, sK565: ((subst > term > term) > subst)).
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 > term > subst > $o) > subst > term > term)).
thf(func_def_707, type, sK569: (term > $o)).
thf(func_def_708, type, sK570: subst).
thf(func_def_709, type, sK571: (term > $o)).
thf(func_def_710, type, sK572: term).
thf(func_def_711, type, sK573: (term > (term > $o) > subst > (term > $o) > term)).
thf(func_def_712, type, sK574: term).
thf(func_def_713, type, sK575: term).
thf(func_def_714, type, sK576: term).
thf(func_def_715, type, sK577: term).
thf(func_def_716, type, sK578: (term > $o)).
thf(func_def_717, type, sK579: term).
thf(func_def_718, type, sK580: ((term > $o) > term)).
thf(func_def_719, type, sK581: ((term > $o) > term)).
thf(func_def_720, type, sK582: term).
thf(func_def_721, type, sK583: subst).
thf(func_def_722, type, sK584: term).
thf(func_def_723, type, sK585: subst).
thf(func_def_724, type, sK586: subst).
thf(func_def_725, type, sK587: (term > $o)).
thf(func_def_726, type, sK588: term).
thf(func_def_727, type, sK589: term).
thf(func_def_728, type, sK590: (term > $o)).
thf(func_def_729, type, sF591: $o).
thf(func_def_730, type, sK592: ((subst > term > subst > $o) > subst)).
thf(func_def_731, type, sK593: ((subst > term > subst > $o) > term)).
thf(func_def_732, type, sK594: ((subst > term > subst > $o) > term)).
thf(func_def_733, type, sK595: ((subst > term > subst > $o) > term)).
thf(func_def_734, type, sK596: ((subst > term > subst > $o) > subst)).
thf(func_def_735, type, sK597: ((subst > term > subst > $o) > subst)).
thf(func_def_736, type, sK598: ((subst > term > subst > $o) > subst)).
thf(func_def_737, type, sK599: ((subst > term > subst > $o) > subst)).
thf(func_def_738, type, sK600: ((subst > term > subst > $o) > subst)).
thf(func_def_739, type, sK601: ((subst > term > subst > $o) > term)).
thf(func_def_740, type, sK602: ((subst > term > subst > $o) > subst)).
thf(func_def_741, type, sK603: ((subst > term > subst > $o) > subst)).
thf(func_def_742, type, sK604: ((subst > term > subst > $o) > subst)).
thf(func_def_743, type, sK605: ((subst > term > subst > $o) > term)).
thf(func_def_744, type, sK606: ((subst > term > subst > $o) > subst)).
thf(func_def_745, type, sK607: ((subst > term > subst > $o) > subst)).
thf(func_def_746, type, sK608: ((subst > term > subst > $o) > term)).
thf(func_def_747, type, sK609: (subst > term > subst > $o)).
thf(func_def_748, type, sK610: term).
thf(func_def_749, type, sK611: term).
thf(f1,axiom,(
  (! [X2 : term,X11 : term,X12 : term,X9 : term] : ((X2 = X9) | (((ap @ X2 @ X11)) != ((ap @ X9 @ X12)))) <=> apinj1) & hoasinduction_lem2_lthm & (hoasinduction_lem3v2a <=> ! [X5 : (term > $o),X15 : term,X4 : (subst > term > subst > $o)] : (~ ! [X10 : (subst > term > term)] : ((X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X2 : term] : (~(X4 @ id @ X2 @ id) | (X4 @ id @ (X10 @ id @ X2) @ id)) | ~ ! [X16 : subst,X2 : term,X3 : subst] : (((sub @ (X10 @ X3 @ X2) @ X16)) = ((X10 @ (comp @ X3 @ X16) @ (sub @ X2 @ X16))))) | (X5 @ (lam @ X15)) | ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id))) | ~ ! [X9 : term] : ((X5 @ (sub @ X15 @ (push @ X9 @ id))) | ~(X5 @ X9)))) & (axvarid <=> ! [X2 : term] : (X2 = ((sub @ X2 @ id)))) & hoasapnotvar_gthm & (((ap @ (d2term @ d_one) @ (d2term @ d_one))) = ((d2term @ d_one))) & (substmonoid_lthm <=> (! [X3 : subst] : (X3 = ((comp @ id @ X3))) => (! [X3 : subst] : (X3 = ((comp @ X3 @ id))) => (! [X3 : subst] : (X3 = ((comp @ X3 @ id))) & ! [X3 : subst] : (X3 = ((comp @ id @ X3))))))) & (hoasapinj1 <=> ! [X12 : term,X11 : term,X2 : term,X9 : term] : ((((ap @ (sub @ X2 @ id) @ X11)) != ((ap @ (sub @ X9 @ id) @ X12))) | (X2 = X9))) & substmonoid_gthm & (! [X38 : term,X4 : (subst > term > subst > $o)] : ((X4 @ id @ (lam @ X38) @ id) | ~ ! [X3 : subst,X2 : term,X16 : subst,X31 : subst] : ((X4 @ X3 @ X2 @ (comp @ X31 @ X16)) | ~(X4 @ (comp @ X3 @ X31) @ (sub @ X2 @ X31) @ X16)) | ~ ! [X10 : (subst > term > term)] : (~ ! [X3 : subst,X2 : term,X16 : subst] : (((sub @ (X10 @ X3 @ X2) @ X16)) = ((X10 @ (comp @ X3 @ X16) @ (sub @ X2 @ X16)))) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X2 : term] : (~(X4 @ id @ X2 @ id) | (X4 @ id @ (X10 @ id @ X2) @ id))) | ~ ! [X2 : term,X31 : subst,X16 : subst,X3 : subst] : ((X4 @ (comp @ X3 @ X31) @ (sub @ X2 @ X31) @ X16) | ~(X4 @ X3 @ X2 @ (comp @ X31 @ X16))) | ~ ! [X9 : term] : (~(X4 @ id @ X9 @ id) | (X4 @ id @ (sub @ X38 @ (push @ X9 @ id)) @ id))) <=> hoasinduction_lem3) & pushprop_gthm & (pushprop_p_and_p_prime = (^[X2 : term, X3 : subst, X4 : (term > $o), X5 : (term > $o)] : (! [X6 : term] : (((X5 @ X6)) = ((X4 @ (sub @ X6 @ (push @ X2 @ X3)))))))) & ! [X43 : d_subst,X44 : d_subst] : ((((d2subst @ X43)) = ((d2subst @ X44))) => (X43 = X44)) & (hoaslaminj <=> ! [X18 : (subst > term > term),X10 : (subst > term > term),X19 : subst,X20 : term] : (~ ! [X2 : term,X3 : subst,X16 : subst] : (((sub @ (X10 @ X3 @ X2) @ X16)) = ((X10 @ (comp @ X3 @ X16) @ (sub @ X2 @ X16)))) | (((X10 @ X19 @ X20)) = ((X18 @ X19 @ X20))) | (((lam @ (X10 @ sh @ one))) != ((lam @ (X18 @ sh @ one)))) | ~ ! [X2 : term,X3 : subst,X16 : subst] : (((sub @ (X18 @ X3 @ X2) @ X16)) = ((X18 @ (comp @ X3 @ X16) @ (sub @ X2 @ X16)))))) & ! [X48 : term] : ? [X47 : d_term] : (X48 = ((d2term @ X47))) & hoaslamnotvar_gthm & axvarshift & (! [X29 : term,X30 : term,X4 : (subst > term > subst > $o)] : (~(X4 @ id @ X29 @ id) | ~ ! [X3 : subst,X2 : term,X31 : subst,X16 : subst] : (~(X4 @ X3 @ X2 @ (comp @ X31 @ X16)) | (X4 @ (comp @ X3 @ X31) @ (sub @ X2 @ X31) @ X16)) | ~ ! [X9 : term,X2 : term] : ((X4 @ id @ (ap @ (sub @ X2 @ id) @ X9) @ id) | ~(X4 @ id @ X9 @ id) | ~(X4 @ id @ X2 @ id)) | (X4 @ id @ (ap @ X29 @ X30) @ id) | ~ ! [X2 : term,X31 : subst,X3 : subst,X16 : subst] : (~(X4 @ (comp @ X3 @ X31) @ (sub @ X2 @ X31) @ X16) | (X4 @ X3 @ X2 @ (comp @ X31 @ X16))) | ~(X4 @ id @ X30 @ id)) <=> hoasinduction_lem2) & (sh = ((d2subst @ d_id))) & pushprop_lem0_gthm & (hoasinduction_no_psi_cond <=> ! [X17 : term,X4 : (subst > term > subst > $o)] : (~ ! [X10 : (subst > term > term)] : (~ ! [X3 : subst,X2 : term,X16 : subst] : (((sub @ (X10 @ X3 @ X2) @ X16)) = ((X10 @ (comp @ X3 @ X16) @ (sub @ X2 @ X16)))) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X2 : term] : (~(X4 @ id @ X2 @ id) | (X4 @ id @ (X10 @ id @ X2) @ id))) | ~ ! [X2 : term,X9 : term] : ((X4 @ id @ (ap @ (sub @ X2 @ id) @ X9) @ id) | ~(X4 @ id @ X9 @ id) | ~(X4 @ id @ X2 @ id)) | (X4 @ id @ X17 @ id))) & hoasinduction_lem1v2_gthm & hoasapnotvar_lthm & ((! [X14 : term,X4 : (term > $o)] : ((X4 @ X14) | ~ ! [X2 : term] : ((X4 @ (lam @ X2)) | ~ ! [X9 : term] : (~(X4 @ X9) | (X4 @ (sub @ X2 @ (push @ X9 @ id))))) | ~ ! [X2 : term,X9 : term] : (~(X4 @ X2) | (X4 @ (ap @ X2 @ X9)) | ~(X4 @ X9)) | ~ ! [X2 : term] : ((X4 @ X2) | ~(var @ X2))) => (! [X30 : term,X29 : term,X4 : (subst > term > subst > $o)] : (~(X4 @ id @ X30 @ id) | ~(X4 @ id @ X29 @ id) | (X4 @ id @ (ap @ X29 @ X30) @ id) | ~ ! [X3 : subst,X31 : subst,X16 : subst,X2 : term] : (~(X4 @ (comp @ X3 @ X31) @ (sub @ X2 @ X31) @ X16) | (X4 @ X3 @ X2 @ (comp @ X31 @ X16))) | ~ ! [X2 : term,X9 : term] : ((X4 @ id @ (ap @ (sub @ X2 @ id) @ X9) @ id) | ~(X4 @ id @ X2 @ id) | ~(X4 @ id @ X9 @ id)) | ~ ! [X31 : subst,X2 : term,X3 : subst,X16 : subst] : (~(X4 @ X3 @ X2 @ (comp @ X31 @ X16)) | (X4 @ (comp @ X3 @ X31) @ (sub @ X2 @ X31) @ X16))) => (! [X38 : term,X4 : (subst > term > subst > $o)] : (~ ! [X2 : term,X3 : subst,X31 : subst,X16 : subst] : (~(X4 @ (comp @ X3 @ X31) @ (sub @ X2 @ X31) @ X16) | (X4 @ X3 @ X2 @ (comp @ X31 @ X16))) | ~ ! [X2 : term,X3 : subst,X31 : subst,X16 : subst] : ((X4 @ (comp @ X3 @ X31) @ (sub @ X2 @ X31) @ X16) | ~(X4 @ X3 @ X2 @ (comp @ X31 @ X16))) | (X4 @ id @ (lam @ X38) @ id) | ~ ! [X9 : term] : ((X4 @ id @ (sub @ X38 @ (push @ X9 @ id)) @ id) | ~(X4 @ id @ X9 @ id)) | ~ ! [X10 : (subst > term > term)] : (~ ! [X2 : term] : (~(X4 @ id @ X2 @ id) | (X4 @ id @ (X10 @ id @ X2) @ id)) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X2 : term,X16 : subst,X3 : subst] : (((sub @ (X10 @ X3 @ X2) @ X16)) = ((X10 @ (comp @ X3 @ X16) @ (sub @ X2 @ X16)))))) => ! [X4 : (subst > term > subst > $o),X32 : term] : (~ ! [X2 : term,X9 : term] : (~(X4 @ id @ X9 @ id) | (X4 @ id @ (ap @ (sub @ X2 @ id) @ X9) @ id) | ~(X4 @ id @ X2 @ id)) | ~ ! [X10 : (subst > term > term)] : ((X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X2 : term,X16 : subst,X3 : subst] : (((sub @ (X10 @ X3 @ X2) @ X16)) = ((X10 @ (comp @ X3 @ X16) @ (sub @ X2 @ X16)))) | ~ ! [X2 : term] : ((X4 @ id @ (X10 @ id @ X2) @ id) | ~(X4 @ id @ X2 @ id))) | (X4 @ id @ X32 @ id) | ~ ! [X31 : subst,X3 : subst,X16 : subst,X2 : term] : (~(X4 @ X3 @ X2 @ (comp @ X31 @ X16)) | (X4 @ (comp @ X3 @ X31) @ (sub @ X2 @ X31) @ X16)) | ~ ! [X2 : term] : (~(var @ (sub @ X2 @ id)) | (X4 @ id @ X2 @ id)) | ~ ! [X3 : subst,X16 : subst,X31 : subst,X2 : term] : (~(X4 @ (comp @ X3 @ X31) @ (sub @ X2 @ X31) @ X16) | (X4 @ X3 @ X2 @ (comp @ X31 @ X16))))))) <=> hoasinduction_lthm) & induction2_gthm & (hoasinduction_lem3aa <=> ! [X33 : term,X4 : (subst > term > subst > $o)] : ((X4 @ id @ (lam @ (sub @ X33 @ (push @ one @ sh))) @ id) | ~ ! [X9 : term] : ((X4 @ id @ (sub @ X33 @ (push @ X9 @ id)) @ id) | ~(X4 @ id @ X9 @ id)) | ~ ! [X10 : (subst > term > term)] : (~ ! [X2 : term] : (~(X4 @ id @ X2 @ id) | (X4 @ id @ (X10 @ id @ X2) @ id)) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X16 : subst,X2 : term,X3 : subst] : (((sub @ (X10 @ X3 @ X2) @ X16)) = ((X10 @ (comp @ X3 @ X16) @ (sub @ X2 @ X16))))))) & (! [X37 : term,X36 : term,X10 : (subst > term > term)] : (~ ! [X16 : subst,X2 : term,X3 : subst] : (((sub @ (X10 @ X3 @ X2) @ X16)) = ((X10 @ (comp @ X3 @ X16) @ (sub @ X2 @ X16)))) | (((lam @ (X10 @ sh @ one))) != ((ap @ (sub @ X36 @ id) @ X37)))) <=> hoaslamnotap) & (! [X2 : term,X4 : (term > $o),X3 : subst] : ~ ! [X5 : (term > $o)] : ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ (sub @ X6 @ (push @ X2 @ X3))))) <=> pushprop_lem0_lthm) & (hoasinduction_lem3v2 <=> ! [X4 : (subst > term > subst > $o),X35 : term,X5 : (term > $o)] : ((X5 @ (lam @ X35)) | ~ ! [X3 : subst,X2 : term,X16 : subst,X31 : subst] : ((X4 @ X3 @ X2 @ (comp @ X31 @ X16)) | ~(X4 @ (comp @ X3 @ X31) @ (sub @ X2 @ X31) @ X16)) | ~ ! [X10 : (subst > term > term)] : ((X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X2 : term] : (~(X4 @ id @ X2 @ id) | (X4 @ id @ (X10 @ id @ X2) @ id)) | ~ ! [X3 : subst,X16 : subst,X2 : term] : (((sub @ (X10 @ X3 @ X2) @ X16)) = ((X10 @ (comp @ X3 @ X16) @ (sub @ X2 @ X16))))) | ~ ! [X31 : subst,X3 : subst,X16 : subst,X2 : term] : (~(X4 @ X3 @ X2 @ (comp @ X31 @ X16)) | (X4 @ (comp @ X3 @ X31) @ (sub @ X2 @ X31) @ X16)) | ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id))) | ~ ! [X9 : term] : (~(X5 @ X9) | (X5 @ (sub @ X35 @ (push @ X9 @ id)))))) & (pushprop_lem0 <=> ! [X3 : subst,X4 : (term > $o),X2 : term] : ~ ! [X5 : (term > $o)] : ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ (sub @ X6 @ (push @ X2 @ X3)))))) & hoasapinj1_gthm & (induction2_lthm <=> (! [X2 : term] : (X2 = ((sub @ X2 @ id))) => (! [X8 : subst,X7 : term,X4 : (term > $o)] : ((X4 @ (sub @ X7 @ X8)) | ~ ! [X2 : term,X9 : term] : (~(X4 @ X2) | ~(X4 @ X9) | (X4 @ (ap @ X2 @ X9))) | ~ ! [X2 : term] : ((X4 @ (lam @ X2)) | ~ ! [X9 : term] : ((X4 @ (sub @ X2 @ (push @ X9 @ id))) | ~(X4 @ X9))) | ~ ! [X9 : term] : (~(var @ X9) | (X4 @ (sub @ X9 @ X8)))) => ! [X14 : term,X4 : (term > $o)] : (~ ! [X9 : term,X2 : term] : (~(X4 @ X9) | ~(X4 @ X2) | (X4 @ (ap @ X2 @ X9))) | ~ ! [X2 : term] : (~(var @ X2) | (X4 @ X2)) | ~ ! [X2 : term] : (~ ! [X9 : term] : (~(X4 @ X9) | (X4 @ (sub @ X2 @ (push @ X9 @ id)))) | (X4 @ (lam @ X2))) | (X4 @ X14))))) & (! [X32 : term,X4 : (subst > term > subst > $o)] : (~ ! [X2 : term] : ((X4 @ id @ X2 @ id) | ~(var @ (sub @ X2 @ id))) | ~ ! [X9 : term,X2 : term] : ((X4 @ id @ (ap @ (sub @ X2 @ id) @ X9) @ id) | ~(X4 @ id @ X2 @ id) | ~(X4 @ id @ X9 @ id)) | (X4 @ id @ X32 @ id) | ~ ! [X10 : (subst > term > term)] : (~ ! [X2 : term] : (~(X4 @ id @ X2 @ id) | (X4 @ id @ (X10 @ id @ X2) @ id)) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X2 : term,X16 : subst,X3 : subst] : (((sub @ (X10 @ X3 @ X2) @ X16)) = ((X10 @ (comp @ X3 @ X16) @ (sub @ X2 @ X16))))) | ~ ! [X31 : subst,X3 : subst,X16 : subst,X2 : term] : (~(X4 @ X3 @ X2 @ (comp @ X31 @ X16)) | (X4 @ (comp @ X3 @ X31) @ (sub @ X2 @ X31) @ X16)) | ~ ! [X31 : subst,X3 : subst,X2 : term,X16 : subst] : (~(X4 @ (comp @ X3 @ X31) @ (sub @ X2 @ X31) @ X16) | (X4 @ X3 @ X2 @ (comp @ X31 @ X16)))) <=> hoasinduction) & (axvarcons <=> ! [X3 : subst,X2 : term] : (X2 = ((sub @ one @ (push @ X2 @ X3))))) & (! [X9 : term] : ~ ! [X10 : (subst > term > term)] : ~ ! [X3 : subst,X2 : term] : (((sub @ X9 @ (push @ X2 @ X3))) = ((X10 @ X3 @ X2))) <=> hoasinduction_lem3v2_f_lthm) & (hoasinduction_lem0 <=> ! [X4 : (subst > term > subst > $o)] : ~ ! [X5 : (term > $o)] : ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id)))) & (hoasinduction_lem3v2_f <=> ! [X9 : term] : ~ ! [X10 : (subst > term > term)] : ~ ! [X2 : term,X3 : subst] : (((sub @ X9 @ (push @ X2 @ X3))) = ((X10 @ X3 @ X2)))) & ((! [X3 : subst,X2 : term] : (X2 = ((sub @ one @ (push @ X2 @ X3)))) => (! [X3 : subst,X2 : term] : (X3 = ((comp @ sh @ (push @ X2 @ X3)))) => ! [X4 : (term > $o),X31 : (term > $o),X2 : term,X9 : term,X3 : subst] : (~(X4 @ X2) | (X31 @ (sub @ X2 @ (push @ X9 @ X3)))))) <=> pushprop_lem1_lthm) & ((! [X9 : term,X12 : term,X11 : term,X2 : term] : ((X11 = X12) | (((ap @ X2 @ X11)) != ((ap @ X9 @ X12)))) => ! [X11 : term,X2 : term,X9 : term,X12 : term] : ((((ap @ (sub @ X2 @ id) @ X11)) != ((ap @ (sub @ X9 @ id) @ X12))) | (X11 = X12))) <=> hoasapinj2_lthm) & ulamvarind & apnotvar & ((! [X2 : term] : (X2 = ((sub @ X2 @ id))) => ! [X35 : term,X4 : (subst > term > subst > $o),X5 : (term > $o)] : (~ ! [X10 : (subst > term > term)] : ((X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X2 : term] : ((X4 @ id @ (X10 @ id @ X2) @ id) | ~(X4 @ id @ X2 @ id)) | ~ ! [X3 : subst,X16 : subst,X2 : term] : (((sub @ (X10 @ X3 @ X2) @ X16)) = ((X10 @ (comp @ X3 @ X16) @ (sub @ X2 @ X16))))) | ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id))) | ~ ! [X2 : term,X31 : subst,X16 : subst,X3 : subst] : ((X4 @ X3 @ X2 @ (comp @ X31 @ X16)) | ~(X4 @ (comp @ X3 @ X31) @ (sub @ X2 @ X31) @ X16)) | (X5 @ (lam @ X35)) | ~ ! [X9 : term] : (~(X5 @ X9) | (X5 @ (sub @ X35 @ (push @ X9 @ id)))) | ~ ! [X2 : term,X31 : subst,X3 : subst,X16 : subst] : (~(X4 @ X3 @ X2 @ (comp @ X31 @ X16)) | (X4 @ (comp @ X3 @ X31) @ (sub @ X2 @ X31) @ X16)))) <=> hoasinduction_lem3v2_lthm) & (pushprop_lem1v2_lthm <=> (! [X3 : subst,X2 : term] : (X2 = ((sub @ one @ (push @ X2 @ X3)))) => ! [X3 : subst,X4 : (term > $o),X5 : (term > $o),X2 : term] : (~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ (sub @ X6 @ (push @ X2 @ X3))))) | ~(X4 @ X2) | (X5 @ one)))) & hoasinduction_lem3a_gthm & (! [X9 : term,X2 : term] : ((X2 = X9) | (((lam @ X2)) != ((lam @ X9)))) <=> laminj) & hoaslamnotap_gthm & (one = ((d2term @ d_one))) & (apinj2 <=> ! [X9 : term,X12 : term,X2 : term,X11 : term] : ((X11 = X12) | (((ap @ X2 @ X11)) != ((ap @ X9 @ X12))))) & lamnotvar & (pushprop_lem1v2 <=> ! [X4 : (term > $o),X3 : subst,X5 : (term > $o),X2 : term] : (~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ (sub @ X6 @ (push @ X2 @ X3))))) | (X5 @ one) | ~(X4 @ X2))) & ~(var @ (d2term @ d_one)) & hoaslamnotvar_lthm & pushprop_lem3v2 & (! [X4 : (term > $o),X8 : subst,X7 : term] : ((X4 @ (sub @ X7 @ X8)) | ~ ! [X2 : term] : (~ ! [X9 : term] : (~(X4 @ X9) | (X4 @ (sub @ X2 @ (push @ X9 @ id)))) | (X4 @ (lam @ X2))) | ~ ! [X2 : term,X9 : term] : (~(X4 @ X9) | ~(X4 @ X2) | (X4 @ (ap @ X2 @ X9))) | ~ ! [X9 : term] : ((X4 @ (sub @ X9 @ X8)) | ~(var @ X9))) <=> induction2lem) & hoasinduction_lem2_gthm & ! [X47 : d_term] : (X47 = d_one) & pushprop_lem2v2 & hoasinduction_lem3v2_gthm & ((! [X9 : term] : ~ ! [X10 : (subst > term > term)] : ~ ! [X3 : subst,X2 : term] : (((sub @ X9 @ (push @ X2 @ X3))) = ((X10 @ X3 @ X2))) => (! [X2 : term] : (X2 = ((sub @ X2 @ id))) => ! [X4 : (subst > term > subst > $o),X5 : (term > $o),X15 : term] : (~ ! [X9 : term] : ((X5 @ (sub @ X15 @ (push @ X9 @ id))) | ~(X5 @ X9)) | (X5 @ (lam @ X15)) | ~ ! [X10 : (subst > term > term)] : (~ ! [X2 : term] : (~(X4 @ id @ X2 @ id) | (X4 @ id @ (X10 @ id @ X2) @ id)) | ~ ! [X16 : subst,X3 : subst,X2 : term] : (((sub @ (X10 @ X3 @ X2) @ X16)) = ((X10 @ (comp @ X3 @ X16) @ (sub @ X2 @ X16)))) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id)) | ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id)))))) <=> hoasinduction_lem3v2a_lthm) & ! [X0 : subst] : ? [X1 : d_subst] : (X0 = ((d2subst @ X1))) & hoaslamnotvar & hoasinduction_lem3b_gthm & termmset_gthm & ((^[X41 : subst, X42 : (subst > term > term)] : ((d2term @ d_one))) = hoaslam) & (shinj <=> ! [X9 : term,X2 : term] : ((((sub @ X2 @ sh)) != ((sub @ X9 @ sh))) | (X2 = X9))) & (hoasinduction_lem3b_lthm <=> ! [X9 : term] : ~ ! [X10 : (subst > term > term)] : (((X10 @ sh @ one)) != ((sub @ X9 @ (push @ one @ sh))))) & ! [X45 : d_term,X46 : d_term] : ((((d2term @ X45)) = ((d2term @ X46))) => (X45 = X46)) & axabs & hoasinduction_lem2v2_gthm & (axidr <=> ! [X3 : subst] : (X3 = ((comp @ X3 @ id)))) & pushprop_lem1_gthm & hoasinduction_lem3_gthm & (hoasinduction_lem3aa_lthm <=> ! [X33 : term,X4 : (subst > term > subst > $o)] : (~ ! [X10 : (subst > term > term)] : ((X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X2 : term] : (~(X4 @ id @ X2 @ id) | (X4 @ id @ (X10 @ id @ X2) @ id)) | ~ ! [X16 : subst,X2 : term,X3 : subst] : (((sub @ (X10 @ X3 @ X2) @ X16)) = ((X10 @ (comp @ X3 @ X16) @ (sub @ X2 @ X16))))) | (X4 @ id @ (lam @ (sub @ X33 @ (push @ one @ sh))) @ id) | ~ ! [X9 : term] : ((X4 @ id @ (sub @ X33 @ (push @ X9 @ id)) @ id) | ~(X4 @ id @ X9 @ id)))) & (hoasinduction_no_psi_cond_lthm <=> (! [X4 : (subst > term > subst > $o)] : ~ ! [X5 : (term > $o)] : ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id))) => (! [X4 : (term > $o),X14 : term] : (~ ! [X9 : term,X2 : term] : (~(X4 @ X9) | (X4 @ (ap @ X2 @ X9)) | ~(X4 @ X2)) | (X4 @ X14) | ~ ! [X2 : term] : ((X4 @ (lam @ X2)) | ~ ! [X9 : term] : ((X4 @ (sub @ X2 @ (push @ X9 @ id))) | ~(X4 @ X9))) | ~ ! [X2 : term] : (~(var @ X2) | (X4 @ X2))) => (! [X2 : term] : (X2 = ((sub @ X2 @ id))) => (! [X4 : (subst > term > subst > $o),X15 : term,X5 : (term > $o)] : (~ ! [X10 : (subst > term > term)] : ((X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X3 : subst,X16 : subst,X2 : term] : (((sub @ (X10 @ X3 @ X2) @ X16)) = ((X10 @ (comp @ X3 @ X16) @ (sub @ X2 @ X16)))) | ~ ! [X2 : term] : ((X4 @ id @ (X10 @ id @ X2) @ id) | ~(X4 @ id @ X2 @ id))) | ~ ! [X9 : term] : (~(X5 @ X9) | (X5 @ (sub @ X15 @ (push @ X9 @ id)))) | ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id))) | (X5 @ (lam @ X15))) => ! [X17 : term,X4 : (subst > term > subst > $o)] : (~ ! [X10 : (subst > term > term)] : (~ ! [X2 : term] : (~(X4 @ id @ X2 @ id) | (X4 @ id @ (X10 @ id @ X2) @ id)) | ~ ! [X2 : term,X16 : subst,X3 : subst] : (((sub @ (X10 @ X3 @ X2) @ X16)) = ((X10 @ (comp @ X3 @ X16) @ (sub @ X2 @ X16)))) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id)) | (X4 @ id @ X17 @ id) | ~ ! [X2 : term,X9 : term] : ((X4 @ id @ (ap @ (sub @ X2 @ id) @ X9) @ id) | ~(X4 @ id @ X9 @ id) | ~(X4 @ id @ X2 @ id)))))))) & (! [X9 : term] : ~ ! [X10 : (subst > term > term)] : (((X10 @ sh @ one)) != ((sub @ X9 @ (push @ one @ sh)))) <=> hoasinduction_lem3b) & pushprop_lthm & axapp & (induction2 <=> ! [X4 : (term > $o),X14 : term] : (~ ! [X2 : term] : ((X4 @ X2) | ~(var @ X2)) | (X4 @ X14) | ~ ! [X2 : term] : (~ ! [X9 : term] : (~(X4 @ X9) | (X4 @ (sub @ X2 @ (push @ X9 @ id)))) | (X4 @ (lam @ X2))) | ~ ! [X9 : term,X2 : term] : (~(X4 @ X9) | (X4 @ (ap @ X2 @ X9)) | ~(X4 @ X2)))) & (hoasinduction_lem3a_lthm <=> (! [X2 : term] : (X2 = ((sub @ X2 @ id))) => (! [X33 : term,X4 : (subst > term > subst > $o)] : (~ ! [X9 : term] : ((X4 @ id @ (sub @ X33 @ (push @ X9 @ id)) @ id) | ~(X4 @ id @ X9 @ id)) | ~ ! [X10 : (subst > term > term)] : (~ ! [X2 : term] : (~(X4 @ id @ X2 @ id) | (X4 @ id @ (X10 @ id @ X2) @ id)) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X3 : subst,X16 : subst,X2 : term] : (((sub @ (X10 @ X3 @ X2) @ X16)) = ((X10 @ (comp @ X3 @ X16) @ (sub @ X2 @ X16))))) | (X4 @ id @ (lam @ (sub @ X33 @ (push @ one @ sh))) @ id)) => ! [X34 : term,X4 : (subst > term > subst > $o)] : (~ ! [X10 : (subst > term > term)] : ((X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X16 : subst,X3 : subst,X2 : term] : (((sub @ (X10 @ X3 @ X2) @ X16)) = ((X10 @ (comp @ X3 @ X16) @ (sub @ X2 @ X16)))) | ~ ! [X2 : term] : ((X4 @ id @ (X10 @ id @ X2) @ id) | ~(X4 @ id @ X2 @ id))) | (X4 @ id @ (lam @ X34) @ id) | ~ ! [X9 : term] : (~(X4 @ id @ X9 @ id) | (X4 @ id @ (sub @ X34 @ (push @ X9 @ id)) @ id)))))) & hoasinduction_lem1_lthm & ((^[X4 : (subst > term > subst > $o), X5 : (term > $o)] : (! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id))))) = hoasinduction_p_and_p_prime) & ! [X1 : d_subst] : (X1 = d_id) & (induction2lem_lthm <=> (! [X3 : subst,X2 : term] : (X2 = ((sub @ one @ (push @ X2 @ X3)))) => (! [X2 : term,X3 : subst] : (X3 = ((comp @ sh @ (push @ X2 @ X3)))) => (! [X3 : subst] : (X3 = ((comp @ X3 @ id))) => (! [X4 : (term > $o),X13 : term] : (~ ! [X9 : term,X2 : term] : (~(X4 @ X9) | (X4 @ (ap @ X2 @ X9)) | ~(X4 @ X2)) | (X4 @ X13) | ~ ! [X2 : term] : ((X4 @ X2) | ~(var @ X2)) | ~ ! [X2 : term] : ((X4 @ (lam @ X2)) | ~(X4 @ X2))) => ! [X4 : (term > $o),X7 : term,X8 : subst] : (~ ! [X2 : term] : ((X4 @ (lam @ X2)) | ~ ! [X9 : term] : (~(X4 @ X9) | (X4 @ (sub @ X2 @ (push @ X9 @ id))))) | ~ ! [X9 : term] : (~(var @ X9) | (X4 @ (sub @ X9 @ X8))) | ~ ! [X2 : term,X9 : term] : ((X4 @ (ap @ X2 @ X9)) | ~(X4 @ X2) | ~(X4 @ X9)) | (X4 @ (sub @ X7 @ X8)))))))) & (! [X31 : (term > $o),X4 : (term > $o),X9 : term,X2 : term,X3 : subst] : (~(X4 @ X2) | (X31 @ (sub @ X2 @ (push @ X9 @ X3)))) <=> pushprop_lem1) & hoasinduction_lem1_gthm & induction2lem_gthm & ((! [X4 : (subst > term > subst > $o)] : ~ ! [X5 : (term > $o)] : ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id))) => (! [X4 : (term > $o),X14 : term] : (~ ! [X2 : term] : ((X4 @ (lam @ X2)) | ~ ! [X9 : term] : ((X4 @ (sub @ X2 @ (push @ X9 @ id))) | ~(X4 @ X9))) | (X4 @ X14) | ~ ! [X2 : term,X9 : term] : (~(X4 @ X9) | ~(X4 @ X2) | (X4 @ (ap @ X2 @ X9))) | ~ ! [X2 : term] : (~(var @ X2) | (X4 @ X2))) => (! [X2 : term] : (X2 = ((sub @ X2 @ id))) => (! [X4 : (subst > term > subst > $o),X15 : term,X5 : (term > $o)] : ((X5 @ (lam @ X15)) | ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id))) | ~ ! [X9 : term] : (~(X5 @ X9) | (X5 @ (sub @ X15 @ (push @ X9 @ id)))) | ~ ! [X10 : (subst > term > term)] : (~ ! [X2 : term] : ((X4 @ id @ (X10 @ id @ X2) @ id) | ~(X4 @ id @ X2 @ id)) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X3 : subst,X2 : term,X16 : subst] : (((sub @ (X10 @ X3 @ X2) @ X16)) = ((X10 @ (comp @ X3 @ X16) @ (sub @ X2 @ X16)))))) => ! [X4 : (subst > term > subst > $o),X32 : term] : (~ ! [X9 : term,X2 : term] : (~(X4 @ id @ X2 @ id) | (X4 @ id @ (ap @ (sub @ X2 @ id) @ X9) @ id) | ~(X4 @ id @ X9 @ id)) | (X4 @ id @ X32 @ id) | ~ ! [X2 : term,X16 : subst,X3 : subst,X31 : subst] : ((X4 @ X3 @ X2 @ (comp @ X31 @ X16)) | ~(X4 @ (comp @ X3 @ X31) @ (sub @ X2 @ X31) @ X16)) | ~ ! [X2 : term] : (~(var @ (sub @ X2 @ id)) | (X4 @ id @ X2 @ id)) | ~ ! [X2 : term,X3 : subst,X16 : subst,X31 : subst] : (~(X4 @ X3 @ X2 @ (comp @ X31 @ X16)) | (X4 @ (comp @ X3 @ X31) @ (sub @ X2 @ X31) @ X16)) | ~ ! [X10 : (subst > term > term)] : ((X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X16 : subst,X2 : term,X3 : subst] : (((sub @ (X10 @ X3 @ X2) @ X16)) = ((X10 @ (comp @ X3 @ X16) @ (sub @ X2 @ X16)))) | ~ ! [X2 : term] : ((X4 @ id @ (X10 @ id @ X2) @ id) | ~(X4 @ id @ X2 @ id)))))))) <=> hoasinduction_lthm_3) & (hoaslaminj_lthm <=> (! [X2 : term,X3 : subst] : (X2 = ((sub @ one @ (push @ X2 @ X3)))) => (! [X3 : subst,X2 : term] : (X3 = ((comp @ sh @ (push @ X2 @ X3)))) => (! [X9 : term,X2 : term] : ((X2 = X9) | (((lam @ X2)) != ((lam @ X9)))) => ! [X20 : term,X18 : (subst > term > term),X10 : (subst > term > term),X19 : subst] : (~ ! [X2 : term,X3 : subst,X16 : subst] : (((sub @ (X10 @ X3 @ X2) @ X16)) = ((X10 @ (comp @ X3 @ X16) @ (sub @ X2 @ X16)))) | ~ ! [X2 : term,X3 : subst,X16 : subst] : (((sub @ (X18 @ X3 @ X2) @ X16)) = ((X18 @ (comp @ X3 @ X16) @ (sub @ X2 @ X16)))) | (((lam @ (X10 @ sh @ one))) != ((lam @ (X18 @ sh @ one)))) | (((X10 @ X19 @ X20)) = ((X18 @ X19 @ X20)))))))) & (! [X3 : subst] : (X3 = ((push @ (sub @ one @ X3) @ (comp @ sh @ X3)))) <=> axscons) & (! [X3 : subst] : (X3 = ((comp @ id @ X3))) <=> axidl) & hoasinduction_lem1v2 & (hoasinduction_lem0_lthm <=> ! [X4 : (subst > term > subst > $o)] : ~ ! [X5 : (term > $o)] : ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id)))) & hoasinduction_gthm & pushprop_lem1v2_gthm & (hoasapinj2 <=> ! [X2 : term,X12 : term,X11 : term,X9 : term] : ((X11 = X12) | (((ap @ (sub @ X2 @ id) @ X11)) != ((ap @ (sub @ X9 @ id) @ X12))))) & hoasapinj2_gthm & pushprop_lem2v2_gthm & axassoc & (hoasinduction_lem2v2 <=> ! [X40 : term,X39 : term,X4 : (subst > term > subst > $o),X5 : (term > $o)] : (~ ! [X2 : term,X9 : term] : (~(X4 @ id @ X2 @ id) | (X4 @ id @ (ap @ (sub @ X2 @ id) @ X9) @ id) | ~(X4 @ id @ X9 @ id)) | ~ ! [X6 : term] : (((X5 @ X6)) = ((X4 @ id @ X6 @ id))) | (X5 @ (ap @ X39 @ X40)) | ~ ! [X31 : subst,X16 : subst,X3 : subst,X2 : term] : ((X4 @ X3 @ X2 @ (comp @ X31 @ X16)) | ~(X4 @ (comp @ X3 @ X31) @ (sub @ X2 @ X31) @ X16)) | ~ ! [X2 : term,X31 : subst,X16 : subst,X3 : subst] : (~(X4 @ X3 @ X2 @ (comp @ X31 @ X16)) | (X4 @ (comp @ X3 @ X31) @ (sub @ X2 @ X31) @ X16)) | ~(X5 @ X40) | ~(X5 @ X39))) & axclos & (! [X2 : term] : (X2 = ((sub @ X2 @ id))) <=> termmset) & (((lam @ (d2term @ d_one))) = ((d2term @ d_one))) & ~(hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) & ~lamnotap & pushprop_lem3v2_lthm & ((! [X2 : term] : (X2 = ((sub @ X2 @ id))) => ! [X2 : term] : (X2 = ((sub @ X2 @ id)))) <=> termmset_lthm) & (id = ((d2subst @ d_id))) & (hoasinduction_lem3aaa <=> ! [X21 : term,X4 : (subst > term > subst > $o)] : (~ ! [X9 : term] : ((X4 @ id @ (sub @ X21 @ (push @ X9 @ id)) @ id) | ~(X4 @ id @ X9 @ id)) | ~ ! [X10 : (subst > term > term),X22 : term] : (~ ! [X28 : subst,X26 : subst,X27 : term] : (((sub @ (X10 @ X26 @ X27) @ X28)) = ((sub @ (sub @ X22 @ (push @ X27 @ X26)) @ X28))) | ~ ! [X2 : term] : (~(X4 @ id @ X2 @ id) | (X4 @ id @ (X10 @ id @ X2) @ id)) | ~ ! [X23 : subst,X24 : term,X25 : subst] : (((X10 @ (comp @ X23 @ X25) @ (sub @ X24 @ X25))) = ((sub @ X22 @ (push @ (sub @ X24 @ X25) @ (comp @ X23 @ X25))))) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id)) | (X4 @ id @ (lam @ (sub @ X21 @ (push @ one @ sh))) @ id))) & pushprop_lthm_orig & (! [X2 : term,X3 : subst] : (X3 = ((comp @ sh @ (push @ X2 @ X3)))) <=> axshiftcons) & (((sub @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2term @ d_one))) & (hoasinduction_lem3_lthm <=> (! [X2 : term] : (X2 = ((sub @ X2 @ id))) => (! [X33 : term,X4 : (subst > term > subst > $o)] : (~ ! [X10 : (subst > term > term)] : (~ ! [X3 : subst,X16 : subst,X2 : term] : (((sub @ (X10 @ X3 @ X2) @ X16)) = ((X10 @ (comp @ X3 @ X16) @ (sub @ X2 @ X16)))) | ~ ! [X2 : term] : (~(X4 @ id @ X2 @ id) | (X4 @ id @ (X10 @ id @ X2) @ id)) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id)) | ~ ! [X9 : term] : (~(X4 @ id @ X9 @ id) | (X4 @ id @ (sub @ X33 @ (push @ X9 @ id)) @ id)) | (X4 @ id @ (lam @ (sub @ X33 @ (push @ one @ sh))) @ id)) => ! [X38 : term,X4 : (subst > term > subst > $o)] : (~ ! [X2 : term,X16 : subst,X31 : subst,X3 : subst] : ((X4 @ X3 @ X2 @ (comp @ X31 @ X16)) | ~(X4 @ (comp @ X3 @ X31) @ (sub @ X2 @ X31) @ X16)) | ~ ! [X10 : (subst > term > term)] : (~ ! [X2 : term] : (~(X4 @ id @ X2 @ id) | (X4 @ id @ (X10 @ id @ X2) @ id)) | ~ ! [X2 : term,X16 : subst,X3 : subst] : (((sub @ (X10 @ X3 @ X2) @ X16)) = ((X10 @ (comp @ X3 @ X16) @ (sub @ X2 @ X16)))) | (X4 @ id @ (lam @ (X10 @ sh @ one)) @ id)) | (X4 @ id @ (lam @ X38) @ id) | ~ ! [X9 : term] : (~(X4 @ id @ X9 @ id) | (X4 @ id @ (sub @ X38 @ (push @ X9 @ id)) @ id)) | ~ ! [X2 : term,X16 : subst,X31 : subst,X3 : subst] : ((X4 @ (comp @ X3 @ X31) @ (sub @ X2 @ X31) @ X16) | ~(X4 @ X3 @ X2 @ (comp @ X31 @ X16))))))) & hoasapnotvar & (((comp @ (d2subst @ d_id) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & hoaslamnotap_lthm & hoasinduction_lem1 & (((push @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & pushprop_lem2v2_lthm & (! [X13 : term,X4 : (term > $o)] : (~ ! [X2 : term,X9 : term] : ((X4 @ (ap @ X2 @ X9)) | ~(X4 @ X9) | ~(X4 @ X2)) | (X4 @ X13) | ~ ! [X2 : term] : ((X4 @ (lam @ X2)) | ~(X4 @ X2)) | ~ ! [X2 : term] : (~(var @ X2) | (X4 @ X2))) <=> induction) & (((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) = ((d2term @ d_one))) & ((! [X3 : subst] : (X3 = ((comp @ X3 @ id))) & ! [X3 : subst] : (X3 = ((comp @ id @ X3)))) <=> substmonoid) & ~ulamvar1 & ulamvarsh & axmap & ((! [X2 : term] : (X2 = ((sub @ X2 @ id))) => (! [X2 : term,X9 : term,X12 : term,X11 : term] : ((X2 = X9) | (((ap @ X2 @ X11)) != ((ap @ X9 @ X12)))) => ! [X9 : term,X11 : term,X2 : term,X12 : term] : ((((ap @ (sub @ X2 @ id) @ X11)) != ((ap @ (sub @ X9 @ id) @ X12))) | (X2 = X9)))) <=> hoasapinj1_lthm) & hoaslaminj_gthm & (! [X34 : term,X4 : (subst > term > subst > $o)] : (~ ! [X9 : term] : ((X4 @ id @ (sub @ X34 @ (push @ X9 @ id)) @ id) | ~(X4 @ id @ X9 @ id)) | ~ ! [X10 : (subst > term > term)] : ((X4 @ id @ (lam @ (X10 @ sh @ one)) @ id) | ~ ! [X2 : term,X16 : subst,X3 : subst] : (((sub @ (X10 @ X3 @ X2) @ X16)) = ((X10 @ (comp @ X3 @ X16) @ (sub @ X2 @ X16)))) | ~ ! [X2 : term] : (~(X4 @ id @ X2 @ id) | (X4 @ id @ (X10 @ id @ X2) @ id))) | (X4 @ id @ (lam @ X34) @ id)) <=> hoasinduction_lem3a) & pushprop),
  file('/export/starexec/sandbox2/benchmark/theBenchmark.p',alg444_1)).
thf(f2,conjecture,(
  (hoasinduction_lem2 = ! [X0 : (subst > term > subst > $o)] : (! [X1 : term,X4 : subst,X5 : subst,X3 : subst] : ((X0 @ X3 @ X1 @ (comp @ X5 @ X4)) => (X0 @ (comp @ X3 @ X5) @ (sub @ X1 @ X5) @ X4)) => (! [X4 : subst,X3 : subst,X5 : subst,X1 : term] : ((X0 @ (comp @ X3 @ X5) @ (sub @ X1 @ X5) @ X4) => (X0 @ X3 @ X1 @ (comp @ X5 @ X4))) => (! [X1 : term,X2 : term] : ((X0 @ id @ X1 @ id) => ((X0 @ id @ X2 @ id) => (X0 @ id @ (hoasap @ id @ X1 @ id @ X2) @ id))) => ! [X2 : term,X1 : term] : ((X0 @ id @ X1 @ id) => ((X0 @ id @ X2 @ id) => (X0 @ id @ (ap @ X1 @ X2) @ id)))))))),
  file('/export/starexec/sandbox2/benchmark/theBenchmark.p',hoasinduction_lem2)).
thf(f3,negated_conjecture,(
  ~ (hoasinduction_lem2 = ! [X0 : (subst > term > subst > $o)] : (! [X1 : term,X4 : subst,X5 : subst,X3 : subst] : ((X0 @ X3 @ X1 @ (comp @ X5 @ X4)) => (X0 @ (comp @ X3 @ X5) @ (sub @ X1 @ X5) @ X4)) => (! [X4 : subst,X3 : subst,X5 : subst,X1 : term] : ((X0 @ (comp @ X3 @ X5) @ (sub @ X1 @ X5) @ X4) => (X0 @ X3 @ X1 @ (comp @ X5 @ X4))) => (! [X1 : term,X2 : term] : ((X0 @ id @ X1 @ id) => ((X0 @ id @ X2 @ id) => (X0 @ id @ (hoasap @ id @ X1 @ id @ X2) @ id))) => ! [X2 : term,X1 : term] : ((X0 @ id @ X1 @ id) => ((X0 @ id @ X2 @ id) => (X0 @ id @ (ap @ X1 @ X2) @ id)))))))),
  inference(negated_conjecture,[status(cth)],[f2])).
thf(f4,plain,(
  (! [X0 : term,X1 : term,X2 : term,X3 : term] : ((X0 = X3) | (((ap @ X3 @ X2)) != ((ap @ X0 @ X1)))) <=> apinj1) & hoasinduction_lem2_lthm & (hoasinduction_lem3v2a <=> ! [X4 : (term > $o),X5 : term,X6 : (subst > term > subst > $o)] : (~ ! [X7 : (subst > term > term)] : ((X6 @ id @ (lam @ (X7 @ sh @ one)) @ id) | ~ ! [X8 : term] : (~(X6 @ id @ X8 @ id) | (X6 @ id @ (X7 @ id @ X8) @ id)) | ~ ! [X9 : subst,X10 : term,X11 : subst] : (((sub @ (X7 @ X11 @ X10) @ X9)) = ((X7 @ (comp @ X11 @ X9) @ (sub @ X10 @ X9))))) | (X4 @ (lam @ X5)) | ~ ! [X12 : term] : (((X6 @ id @ X12 @ id)) = ((X4 @ X12))) | ~ ! [X13 : term] : ((X4 @ (sub @ X5 @ (push @ X13 @ id))) | ~(X4 @ X13)))) & (axvarid <=> ! [X14 : term] : (((sub @ X14 @ id)) = X14)) & hoasapnotvar_gthm & (((ap @ (d2term @ d_one) @ (d2term @ d_one))) = ((d2term @ d_one))) & (substmonoid_lthm <=> (! [X15 : subst] : (((comp @ id @ X15)) = X15) => (! [X16 : subst] : (((comp @ X16 @ id)) = X16) => (! [X17 : subst] : (((comp @ X17 @ id)) = X17) & ! [X18 : subst] : (((comp @ id @ X18)) = X18))))) & (hoasapinj1 <=> ! [X19 : term,X20 : term,X21 : term,X22 : term] : ((((ap @ (sub @ X22 @ id) @ X19)) != ((ap @ (sub @ X21 @ id) @ X20))) | (X21 = X22))) & substmonoid_gthm & (! [X23 : term,X24 : (subst > term > subst > $o)] : ((X24 @ id @ (lam @ X23) @ id) | ~ ! [X25 : subst,X26 : term,X27 : subst,X28 : subst] : ((X24 @ X25 @ X26 @ (comp @ X28 @ X27)) | ~(X24 @ (comp @ X25 @ X28) @ (sub @ X26 @ X28) @ X27)) | ~ ! [X29 : (subst > term > term)] : (~ ! [X30 : subst,X31 : term,X32 : subst] : (((sub @ (X29 @ X30 @ X31) @ X32)) = ((X29 @ (comp @ X30 @ X32) @ (sub @ X31 @ X32)))) | (X24 @ id @ (lam @ (X29 @ sh @ one)) @ id) | ~ ! [X33 : term] : (~(X24 @ id @ X33 @ id) | (X24 @ id @ (X29 @ id @ X33) @ id))) | ~ ! [X34 : term,X35 : subst,X36 : subst,X37 : subst] : ((X24 @ (comp @ X37 @ X35) @ (sub @ X34 @ X35) @ X36) | ~(X24 @ X37 @ X34 @ (comp @ X35 @ X36))) | ~ ! [X38 : term] : (~(X24 @ id @ X38 @ id) | (X24 @ id @ (sub @ X23 @ (push @ X38 @ id)) @ id))) <=> hoasinduction_lem3) & pushprop_gthm & (pushprop_p_and_p_prime = (^[X39 : term, X40 : subst, X41 : (term > $o), X42 : (term > $o)] : (! [X43 : term] : (((X41 @ (sub @ X43 @ (push @ X39 @ X40)))) = ((X42 @ X43)))))) & ! [X44 : d_subst,X45 : d_subst] : ((((d2subst @ X44)) = ((d2subst @ X45))) => (X44 = X45)) & (hoaslaminj <=> ! [X46 : (subst > term > term),X47 : (subst > term > term),X48 : subst,X49 : term] : (~ ! [X50 : term,X51 : subst,X52 : subst] : (((sub @ (X47 @ X51 @ X50) @ X52)) = ((X47 @ (comp @ X51 @ X52) @ (sub @ X50 @ X52)))) | (((X47 @ X48 @ X49)) = ((X46 @ X48 @ X49))) | (((lam @ (X47 @ sh @ one))) != ((lam @ (X46 @ sh @ one)))) | ~ ! [X53 : term,X54 : subst,X55 : subst] : (((X46 @ (comp @ X54 @ X55) @ (sub @ X53 @ X55))) = ((sub @ (X46 @ X54 @ X53) @ X55))))) & ! [X56 : term] : ? [X57 : d_term] : (((d2term @ X57)) = X56) & hoaslamnotvar_gthm & axvarshift & (! [X58 : term,X59 : term,X60 : (subst > term > subst > $o)] : (~(X60 @ id @ X58 @ id) | ~ ! [X61 : subst,X62 : term,X63 : subst,X64 : subst] : (~(X60 @ X61 @ X62 @ (comp @ X63 @ X64)) | (X60 @ (comp @ X61 @ X63) @ (sub @ X62 @ X63) @ X64)) | ~ ! [X65 : term,X66 : term] : ((X60 @ id @ (ap @ (sub @ X66 @ id) @ X65) @ id) | ~(X60 @ id @ X65 @ id) | ~(X60 @ id @ X66 @ id)) | (X60 @ id @ (ap @ X58 @ X59) @ id) | ~ ! [X67 : term,X68 : subst,X69 : subst,X70 : subst] : (~(X60 @ (comp @ X69 @ X68) @ (sub @ X67 @ X68) @ X70) | (X60 @ X69 @ X67 @ (comp @ X68 @ X70))) | ~(X60 @ id @ X59 @ id)) <=> hoasinduction_lem2) & (sh = ((d2subst @ d_id))) & pushprop_lem0_gthm & (hoasinduction_no_psi_cond <=> ! [X71 : term,X72 : (subst > term > subst > $o)] : (~ ! [X73 : (subst > term > term)] : (~ ! [X74 : subst,X75 : term,X76 : subst] : (((sub @ (X73 @ X74 @ X75) @ X76)) = ((X73 @ (comp @ X74 @ X76) @ (sub @ X75 @ X76)))) | (X72 @ id @ (lam @ (X73 @ sh @ one)) @ id) | ~ ! [X77 : term] : (~(X72 @ id @ X77 @ id) | (X72 @ id @ (X73 @ id @ X77) @ id))) | ~ ! [X78 : term,X79 : term] : ((X72 @ id @ (ap @ (sub @ X78 @ id) @ X79) @ id) | ~(X72 @ id @ X79 @ id) | ~(X72 @ id @ X78 @ id)) | (X72 @ id @ X71 @ id))) & hoasinduction_lem1v2_gthm & hoasapnotvar_lthm & ((! [X80 : term,X81 : (term > $o)] : ((X81 @ X80) | ~ ! [X82 : term] : ((X81 @ (lam @ X82)) | ~ ! [X83 : term] : (~(X81 @ X83) | (X81 @ (sub @ X82 @ (push @ X83 @ id))))) | ~ ! [X84 : term,X85 : term] : (~(X81 @ X84) | (X81 @ (ap @ X84 @ X85)) | ~(X81 @ X85)) | ~ ! [X86 : term] : ((X81 @ X86) | ~(var @ X86))) => (! [X87 : term,X88 : term,X89 : (subst > term > subst > $o)] : (~(X89 @ id @ X87 @ id) | ~(X89 @ id @ X88 @ id) | (X89 @ id @ (ap @ X88 @ X87) @ id) | ~ ! [X90 : subst,X91 : subst,X92 : subst,X93 : term] : (~(X89 @ (comp @ X90 @ X91) @ (sub @ X93 @ X91) @ X92) | (X89 @ X90 @ X93 @ (comp @ X91 @ X92))) | ~ ! [X94 : term,X95 : term] : ((X89 @ id @ (ap @ (sub @ X94 @ id) @ X95) @ id) | ~(X89 @ id @ X94 @ id) | ~(X89 @ id @ X95 @ id)) | ~ ! [X96 : subst,X97 : term,X98 : subst,X99 : subst] : (~(X89 @ X98 @ X97 @ (comp @ X96 @ X99)) | (X89 @ (comp @ X98 @ X96) @ (sub @ X97 @ X96) @ X99))) => (! [X100 : term,X101 : (subst > term > subst > $o)] : (~ ! [X102 : term,X103 : subst,X104 : subst,X105 : subst] : (~(X101 @ (comp @ X103 @ X104) @ (sub @ X102 @ X104) @ X105) | (X101 @ X103 @ X102 @ (comp @ X104 @ X105))) | ~ ! [X106 : term,X107 : subst,X108 : subst,X109 : subst] : ((X101 @ (comp @ X107 @ X108) @ (sub @ X106 @ X108) @ X109) | ~(X101 @ X107 @ X106 @ (comp @ X108 @ X109))) | (X101 @ id @ (lam @ X100) @ id) | ~ ! [X110 : term] : ((X101 @ id @ (sub @ X100 @ (push @ X110 @ id)) @ id) | ~(X101 @ id @ X110 @ id)) | ~ ! [X111 : (subst > term > term)] : (~ ! [X112 : term] : (~(X101 @ id @ X112 @ id) | (X101 @ id @ (X111 @ id @ X112) @ id)) | (X101 @ id @ (lam @ (X111 @ sh @ one)) @ id) | ~ ! [X113 : term,X114 : subst,X115 : subst] : (((sub @ (X111 @ X115 @ X113) @ X114)) = ((X111 @ (comp @ X115 @ X114) @ (sub @ X113 @ X114)))))) => ! [X116 : (subst > term > subst > $o),X117 : term] : (~ ! [X118 : term,X119 : term] : (~(X116 @ id @ X119 @ id) | (X116 @ id @ (ap @ (sub @ X118 @ id) @ X119) @ id) | ~(X116 @ id @ X118 @ id)) | ~ ! [X120 : (subst > term > term)] : ((X116 @ id @ (lam @ (X120 @ sh @ one)) @ id) | ~ ! [X121 : term,X122 : subst,X123 : subst] : (((sub @ (X120 @ X123 @ X121) @ X122)) = ((X120 @ (comp @ X123 @ X122) @ (sub @ X121 @ X122)))) | ~ ! [X124 : term] : ((X116 @ id @ (X120 @ id @ X124) @ id) | ~(X116 @ id @ X124 @ id))) | (X116 @ id @ X117 @ id) | ~ ! [X125 : subst,X126 : subst,X127 : subst,X128 : term] : (~(X116 @ X126 @ X128 @ (comp @ X125 @ X127)) | (X116 @ (comp @ X126 @ X125) @ (sub @ X128 @ X125) @ X127)) | ~ ! [X129 : term] : (~(var @ (sub @ X129 @ id)) | (X116 @ id @ X129 @ id)) | ~ ! [X130 : subst,X131 : subst,X132 : subst,X133 : term] : (~(X116 @ (comp @ X130 @ X132) @ (sub @ X133 @ X132) @ X131) | (X116 @ X130 @ X133 @ (comp @ X132 @ X131))))))) <=> hoasinduction_lthm) & induction2_gthm & (hoasinduction_lem3aa <=> ! [X134 : term,X135 : (subst > term > subst > $o)] : ((X135 @ id @ (lam @ (sub @ X134 @ (push @ one @ sh))) @ id) | ~ ! [X136 : term] : ((X135 @ id @ (sub @ X134 @ (push @ X136 @ id)) @ id) | ~(X135 @ id @ X136 @ id)) | ~ ! [X137 : (subst > term > term)] : (~ ! [X138 : term] : (~(X135 @ id @ X138 @ id) | (X135 @ id @ (X137 @ id @ X138) @ id)) | (X135 @ id @ (lam @ (X137 @ sh @ one)) @ id) | ~ ! [X139 : subst,X140 : term,X141 : subst] : (((X137 @ (comp @ X141 @ X139) @ (sub @ X140 @ X139))) = ((sub @ (X137 @ X141 @ X140) @ X139)))))) & (! [X142 : term,X143 : term,X144 : (subst > term > term)] : (~ ! [X145 : subst,X146 : term,X147 : subst] : (((sub @ (X144 @ X147 @ X146) @ X145)) = ((X144 @ (comp @ X147 @ X145) @ (sub @ X146 @ X145)))) | (((ap @ (sub @ X143 @ id) @ X142)) != ((lam @ (X144 @ sh @ one))))) <=> hoaslamnotap) & (! [X148 : term,X149 : (term > $o),X150 : subst] : ~ ! [X151 : (term > $o)] : ~ ! [X152 : term] : (((X151 @ X152)) = ((X149 @ (sub @ X152 @ (push @ X148 @ X150))))) <=> pushprop_lem0_lthm) & (hoasinduction_lem3v2 <=> ! [X153 : (subst > term > subst > $o),X154 : term,X155 : (term > $o)] : ((X155 @ (lam @ X154)) | ~ ! [X156 : subst,X157 : term,X158 : subst,X159 : subst] : ((X153 @ X156 @ X157 @ (comp @ X159 @ X158)) | ~(X153 @ (comp @ X156 @ X159) @ (sub @ X157 @ X159) @ X158)) | ~ ! [X160 : (subst > term > term)] : ((X153 @ id @ (lam @ (X160 @ sh @ one)) @ id) | ~ ! [X161 : term] : (~(X153 @ id @ X161 @ id) | (X153 @ id @ (X160 @ id @ X161) @ id)) | ~ ! [X162 : subst,X163 : subst,X164 : term] : (((X160 @ (comp @ X162 @ X163) @ (sub @ X164 @ X163))) = ((sub @ (X160 @ X162 @ X164) @ X163)))) | ~ ! [X165 : subst,X166 : subst,X167 : subst,X168 : term] : (~(X153 @ X166 @ X168 @ (comp @ X165 @ X167)) | (X153 @ (comp @ X166 @ X165) @ (sub @ X168 @ X165) @ X167)) | ~ ! [X169 : term] : (((X155 @ X169)) = ((X153 @ id @ X169 @ id))) | ~ ! [X170 : term] : (~(X155 @ X170) | (X155 @ (sub @ X154 @ (push @ X170 @ id)))))) & (pushprop_lem0 <=> ! [X171 : subst,X172 : (term > $o),X173 : term] : ~ ! [X174 : (term > $o)] : ~ ! [X175 : term] : (((X172 @ (sub @ X175 @ (push @ X173 @ X171)))) = ((X174 @ X175)))) & hoasapinj1_gthm & (induction2_lthm <=> (! [X176 : term] : (((sub @ X176 @ id)) = X176) => (! [X177 : subst,X178 : term,X179 : (term > $o)] : ((X179 @ (sub @ X178 @ X177)) | ~ ! [X180 : term,X181 : term] : (~(X179 @ X180) | ~(X179 @ X181) | (X179 @ (ap @ X180 @ X181))) | ~ ! [X182 : term] : ((X179 @ (lam @ X182)) | ~ ! [X183 : term] : ((X179 @ (sub @ X182 @ (push @ X183 @ id))) | ~(X179 @ X183))) | ~ ! [X184 : term] : (~(var @ X184) | (X179 @ (sub @ X184 @ X177)))) => ! [X185 : term,X186 : (term > $o)] : (~ ! [X187 : term,X188 : term] : (~(X186 @ X187) | ~(X186 @ X188) | (X186 @ (ap @ X188 @ X187))) | ~ ! [X189 : term] : (~(var @ X189) | (X186 @ X189)) | ~ ! [X190 : term] : (~ ! [X191 : term] : (~(X186 @ X191) | (X186 @ (sub @ X190 @ (push @ X191 @ id)))) | (X186 @ (lam @ X190))) | (X186 @ X185))))) & (! [X192 : term,X193 : (subst > term > subst > $o)] : (~ ! [X194 : term] : ((X193 @ id @ X194 @ id) | ~(var @ (sub @ X194 @ id))) | ~ ! [X195 : term,X196 : term] : ((X193 @ id @ (ap @ (sub @ X196 @ id) @ X195) @ id) | ~(X193 @ id @ X196 @ id) | ~(X193 @ id @ X195 @ id)) | (X193 @ id @ X192 @ id) | ~ ! [X197 : (subst > term > term)] : (~ ! [X198 : term] : (~(X193 @ id @ X198 @ id) | (X193 @ id @ (X197 @ id @ X198) @ id)) | (X193 @ id @ (lam @ (X197 @ sh @ one)) @ id) | ~ ! [X199 : term,X200 : subst,X201 : subst] : (((X197 @ (comp @ X201 @ X200) @ (sub @ X199 @ X200))) = ((sub @ (X197 @ X201 @ X199) @ X200)))) | ~ ! [X202 : subst,X203 : subst,X204 : subst,X205 : term] : (~(X193 @ X203 @ X205 @ (comp @ X202 @ X204)) | (X193 @ (comp @ X203 @ X202) @ (sub @ X205 @ X202) @ X204)) | ~ ! [X206 : subst,X207 : subst,X208 : term,X209 : subst] : (~(X193 @ (comp @ X207 @ X206) @ (sub @ X208 @ X206) @ X209) | (X193 @ X207 @ X208 @ (comp @ X206 @ X209)))) <=> hoasinduction) & (axvarcons <=> ! [X210 : subst,X211 : term] : (((sub @ one @ (push @ X211 @ X210))) = X211)) & (! [X212 : term] : ~ ! [X213 : (subst > term > term)] : ~ ! [X214 : subst,X215 : term] : (((sub @ X212 @ (push @ X215 @ X214))) = ((X213 @ X214 @ X215))) <=> hoasinduction_lem3v2_f_lthm) & (hoasinduction_lem0 <=> ! [X216 : (subst > term > subst > $o)] : ~ ! [X217 : (term > $o)] : ~ ! [X218 : term] : (((X217 @ X218)) = ((X216 @ id @ X218 @ id)))) & (hoasinduction_lem3v2_f <=> ! [X219 : term] : ~ ! [X220 : (subst > term > term)] : ~ ! [X221 : term,X222 : subst] : (((X220 @ X222 @ X221)) = ((sub @ X219 @ (push @ X221 @ X222))))) & ((! [X223 : subst,X224 : term] : (((sub @ one @ (push @ X224 @ X223))) = X224) => (! [X225 : subst,X226 : term] : (((comp @ sh @ (push @ X226 @ X225))) = X225) => ! [X227 : (term > $o),X228 : (term > $o),X229 : term,X230 : term,X231 : subst] : (~(X227 @ X229) | (X228 @ (sub @ X229 @ (push @ X230 @ X231)))))) <=> pushprop_lem1_lthm) & ((! [X232 : term,X233 : term,X234 : term,X235 : term] : ((X233 = X234) | (((ap @ X235 @ X234)) != ((ap @ X232 @ X233)))) => ! [X236 : term,X237 : term,X238 : term,X239 : term] : ((((ap @ (sub @ X237 @ id) @ X236)) != ((ap @ (sub @ X238 @ id) @ X239))) | (X236 = X239))) <=> hoasapinj2_lthm) & ulamvarind & apnotvar & ((! [X240 : term] : (((sub @ X240 @ id)) = X240) => ! [X241 : term,X242 : (subst > term > subst > $o),X243 : (term > $o)] : (~ ! [X244 : (subst > term > term)] : ((X242 @ id @ (lam @ (X244 @ sh @ one)) @ id) | ~ ! [X245 : term] : ((X242 @ id @ (X244 @ id @ X245) @ id) | ~(X242 @ id @ X245 @ id)) | ~ ! [X246 : subst,X247 : subst,X248 : term] : (((X244 @ (comp @ X246 @ X247) @ (sub @ X248 @ X247))) = ((sub @ (X244 @ X246 @ X248) @ X247)))) | ~ ! [X249 : term] : (((X243 @ X249)) = ((X242 @ id @ X249 @ id))) | ~ ! [X250 : term,X251 : subst,X252 : subst,X253 : subst] : ((X242 @ X253 @ X250 @ (comp @ X251 @ X252)) | ~(X242 @ (comp @ X253 @ X251) @ (sub @ X250 @ X251) @ X252)) | (X243 @ (lam @ X241)) | ~ ! [X254 : term] : (~(X243 @ X254) | (X243 @ (sub @ X241 @ (push @ X254 @ id)))) | ~ ! [X255 : term,X256 : subst,X257 : subst,X258 : subst] : (~(X242 @ X257 @ X255 @ (comp @ X256 @ X258)) | (X242 @ (comp @ X257 @ X256) @ (sub @ X255 @ X256) @ X258)))) <=> hoasinduction_lem3v2_lthm) & (pushprop_lem1v2_lthm <=> (! [X259 : subst,X260 : term] : (((sub @ one @ (push @ X260 @ X259))) = X260) => ! [X261 : subst,X262 : (term > $o),X263 : (term > $o),X264 : term] : (~ ! [X265 : term] : (((X262 @ (sub @ X265 @ (push @ X264 @ X261)))) = ((X263 @ X265))) | ~(X262 @ X264) | (X263 @ one)))) & hoasinduction_lem3a_gthm & (! [X266 : term,X267 : term] : ((X266 = X267) | (((lam @ X266)) != ((lam @ X267)))) <=> laminj) & hoaslamnotap_gthm & (one = ((d2term @ d_one))) & (apinj2 <=> ! [X268 : term,X269 : term,X270 : term,X271 : term] : ((X269 = X271) | (((ap @ X270 @ X271)) != ((ap @ X268 @ X269))))) & lamnotvar & (pushprop_lem1v2 <=> ! [X272 : (term > $o),X273 : subst,X274 : (term > $o),X275 : term] : (~ ! [X276 : term] : (((X272 @ (sub @ X276 @ (push @ X275 @ X273)))) = ((X274 @ X276))) | (X274 @ one) | ~(X272 @ X275))) & ~(var @ (d2term @ d_one)) & hoaslamnotvar_lthm & pushprop_lem3v2 & (! [X277 : (term > $o),X278 : subst,X279 : term] : ((X277 @ (sub @ X279 @ X278)) | ~ ! [X280 : term] : (~ ! [X281 : term] : (~(X277 @ X281) | (X277 @ (sub @ X280 @ (push @ X281 @ id)))) | (X277 @ (lam @ X280))) | ~ ! [X282 : term,X283 : term] : (~(X277 @ X283) | ~(X277 @ X282) | (X277 @ (ap @ X282 @ X283))) | ~ ! [X284 : term] : ((X277 @ (sub @ X284 @ X278)) | ~(var @ X284))) <=> induction2lem) & hoasinduction_lem2_gthm & ! [X285 : d_term] : (d_one = X285) & pushprop_lem2v2 & hoasinduction_lem3v2_gthm & ((! [X286 : term] : ~ ! [X287 : (subst > term > term)] : ~ ! [X288 : subst,X289 : term] : (((X287 @ X288 @ X289)) = ((sub @ X286 @ (push @ X289 @ X288)))) => (! [X290 : term] : (((sub @ X290 @ id)) = X290) => ! [X291 : (subst > term > subst > $o),X292 : (term > $o),X293 : term] : (~ ! [X294 : term] : ((X292 @ (sub @ X293 @ (push @ X294 @ id))) | ~(X292 @ X294)) | (X292 @ (lam @ X293)) | ~ ! [X295 : (subst > term > term)] : (~ ! [X296 : term] : (~(X291 @ id @ X296 @ id) | (X291 @ id @ (X295 @ id @ X296) @ id)) | ~ ! [X297 : subst,X298 : subst,X299 : term] : (((X295 @ (comp @ X298 @ X297) @ (sub @ X299 @ X297))) = ((sub @ (X295 @ X298 @ X299) @ X297))) | (X291 @ id @ (lam @ (X295 @ sh @ one)) @ id)) | ~ ! [X300 : term] : (((X291 @ id @ X300 @ id)) = ((X292 @ X300)))))) <=> hoasinduction_lem3v2a_lthm) & ! [X301 : subst] : ? [X302 : d_subst] : (((d2subst @ X302)) = X301) & hoaslamnotvar & hoasinduction_lem3b_gthm & termmset_gthm & ((^[X303 : subst, X304 : (subst > term > term)] : ((d2term @ d_one))) = hoaslam) & (shinj <=> ! [X305 : term,X306 : term] : ((((sub @ X305 @ sh)) != ((sub @ X306 @ sh))) | (X305 = X306))) & (hoasinduction_lem3b_lthm <=> ! [X307 : term] : ~ ! [X308 : (subst > term > term)] : (((X308 @ sh @ one)) != ((sub @ X307 @ (push @ one @ sh))))) & ! [X309 : d_term,X310 : d_term] : ((((d2term @ X309)) = ((d2term @ X310))) => (X309 = X310)) & axabs & hoasinduction_lem2v2_gthm & (axidr <=> ! [X311 : subst] : (((comp @ X311 @ id)) = X311)) & pushprop_lem1_gthm & hoasinduction_lem3_gthm & (hoasinduction_lem3aa_lthm <=> ! [X312 : term,X313 : (subst > term > subst > $o)] : (~ ! [X314 : (subst > term > term)] : ((X313 @ id @ (lam @ (X314 @ sh @ one)) @ id) | ~ ! [X315 : term] : (~(X313 @ id @ X315 @ id) | (X313 @ id @ (X314 @ id @ X315) @ id)) | ~ ! [X316 : subst,X317 : term,X318 : subst] : (((X314 @ (comp @ X318 @ X316) @ (sub @ X317 @ X316))) = ((sub @ (X314 @ X318 @ X317) @ X316)))) | (X313 @ id @ (lam @ (sub @ X312 @ (push @ one @ sh))) @ id) | ~ ! [X319 : term] : ((X313 @ id @ (sub @ X312 @ (push @ X319 @ id)) @ id) | ~(X313 @ id @ X319 @ id)))) & (hoasinduction_no_psi_cond_lthm <=> (! [X320 : (subst > term > subst > $o)] : ~ ! [X321 : (term > $o)] : ~ ! [X322 : term] : (((X321 @ X322)) = ((X320 @ id @ X322 @ id))) => (! [X323 : (term > $o),X324 : term] : (~ ! [X325 : term,X326 : term] : (~(X323 @ X325) | (X323 @ (ap @ X326 @ X325)) | ~(X323 @ X326)) | (X323 @ X324) | ~ ! [X327 : term] : ((X323 @ (lam @ X327)) | ~ ! [X328 : term] : ((X323 @ (sub @ X327 @ (push @ X328 @ id))) | ~(X323 @ X328))) | ~ ! [X329 : term] : (~(var @ X329) | (X323 @ X329))) => (! [X330 : term] : (((sub @ X330 @ id)) = X330) => (! [X331 : (subst > term > subst > $o),X332 : term,X333 : (term > $o)] : (~ ! [X334 : (subst > term > term)] : ((X331 @ id @ (lam @ (X334 @ sh @ one)) @ id) | ~ ! [X335 : subst,X336 : subst,X337 : term] : (((X334 @ (comp @ X335 @ X336) @ (sub @ X337 @ X336))) = ((sub @ (X334 @ X335 @ X337) @ X336))) | ~ ! [X338 : term] : ((X331 @ id @ (X334 @ id @ X338) @ id) | ~(X331 @ id @ X338 @ id))) | ~ ! [X339 : term] : (~(X333 @ X339) | (X333 @ (sub @ X332 @ (push @ X339 @ id)))) | ~ ! [X340 : term] : (((X331 @ id @ X340 @ id)) = ((X333 @ X340))) | (X333 @ (lam @ X332))) => ! [X341 : term,X342 : (subst > term > subst > $o)] : (~ ! [X343 : (subst > term > term)] : (~ ! [X344 : term] : (~(X342 @ id @ X344 @ id) | (X342 @ id @ (X343 @ id @ X344) @ id)) | ~ ! [X345 : term,X346 : subst,X347 : subst] : (((sub @ (X343 @ X347 @ X345) @ X346)) = ((X343 @ (comp @ X347 @ X346) @ (sub @ X345 @ X346)))) | (X342 @ id @ (lam @ (X343 @ sh @ one)) @ id)) | (X342 @ id @ X341 @ id) | ~ ! [X348 : term,X349 : term] : ((X342 @ id @ (ap @ (sub @ X348 @ id) @ X349) @ id) | ~(X342 @ id @ X349 @ id) | ~(X342 @ id @ X348 @ id)))))))) & (! [X350 : term] : ~ ! [X351 : (subst > term > term)] : (((X351 @ sh @ one)) != ((sub @ X350 @ (push @ one @ sh)))) <=> hoasinduction_lem3b) & pushprop_lthm & axapp & (induction2 <=> ! [X352 : (term > $o),X353 : term] : (~ ! [X354 : term] : ((X352 @ X354) | ~(var @ X354)) | (X352 @ X353) | ~ ! [X355 : term] : (~ ! [X356 : term] : (~(X352 @ X356) | (X352 @ (sub @ X355 @ (push @ X356 @ id)))) | (X352 @ (lam @ X355))) | ~ ! [X357 : term,X358 : term] : (~(X352 @ X357) | (X352 @ (ap @ X358 @ X357)) | ~(X352 @ X358)))) & (hoasinduction_lem3a_lthm <=> (! [X359 : term] : (((sub @ X359 @ id)) = X359) => (! [X360 : term,X361 : (subst > term > subst > $o)] : (~ ! [X362 : term] : ((X361 @ id @ (sub @ X360 @ (push @ X362 @ id)) @ id) | ~(X361 @ id @ X362 @ id)) | ~ ! [X363 : (subst > term > term)] : (~ ! [X364 : term] : (~(X361 @ id @ X364 @ id) | (X361 @ id @ (X363 @ id @ X364) @ id)) | (X361 @ id @ (lam @ (X363 @ sh @ one)) @ id) | ~ ! [X365 : subst,X366 : subst,X367 : term] : (((sub @ (X363 @ X365 @ X367) @ X366)) = ((X363 @ (comp @ X365 @ X366) @ (sub @ X367 @ X366))))) | (X361 @ id @ (lam @ (sub @ X360 @ (push @ one @ sh))) @ id)) => ! [X368 : term,X369 : (subst > term > subst > $o)] : (~ ! [X370 : (subst > term > term)] : ((X369 @ id @ (lam @ (X370 @ sh @ one)) @ id) | ~ ! [X371 : subst,X372 : subst,X373 : term] : (((sub @ (X370 @ X372 @ X373) @ X371)) = ((X370 @ (comp @ X372 @ X371) @ (sub @ X373 @ X371)))) | ~ ! [X374 : term] : ((X369 @ id @ (X370 @ id @ X374) @ id) | ~(X369 @ id @ X374 @ id))) | (X369 @ id @ (lam @ X368) @ id) | ~ ! [X375 : term] : (~(X369 @ id @ X375 @ id) | (X369 @ id @ (sub @ X368 @ (push @ X375 @ id)) @ id)))))) & hoasinduction_lem1_lthm & ((^[X376 : (subst > term > subst > $o), X377 : (term > $o)] : (! [X378 : term] : (((X377 @ X378)) = ((X376 @ id @ X378 @ id))))) = hoasinduction_p_and_p_prime) & ! [X379 : d_subst] : (d_id = X379) & (induction2lem_lthm <=> (! [X380 : subst,X381 : term] : (((sub @ one @ (push @ X381 @ X380))) = X381) => (! [X382 : term,X383 : subst] : (((comp @ sh @ (push @ X382 @ X383))) = X383) => (! [X384 : subst] : (((comp @ X384 @ id)) = X384) => (! [X385 : (term > $o),X386 : term] : (~ ! [X387 : term,X388 : term] : (~(X385 @ X387) | (X385 @ (ap @ X388 @ X387)) | ~(X385 @ X388)) | (X385 @ X386) | ~ ! [X389 : term] : ((X385 @ X389) | ~(var @ X389)) | ~ ! [X390 : term] : ((X385 @ (lam @ X390)) | ~(X385 @ X390))) => ! [X391 : (term > $o),X392 : term,X393 : subst] : (~ ! [X394 : term] : ((X391 @ (lam @ X394)) | ~ ! [X395 : term] : (~(X391 @ X395) | (X391 @ (sub @ X394 @ (push @ X395 @ id))))) | ~ ! [X396 : term] : (~(var @ X396) | (X391 @ (sub @ X396 @ X393))) | ~ ! [X397 : term,X398 : term] : ((X391 @ (ap @ X397 @ X398)) | ~(X391 @ X397) | ~(X391 @ X398)) | (X391 @ (sub @ X392 @ X393)))))))) & (! [X399 : (term > $o),X400 : (term > $o),X401 : term,X402 : term,X403 : subst] : (~(X400 @ X402) | (X399 @ (sub @ X402 @ (push @ X401 @ X403)))) <=> pushprop_lem1) & hoasinduction_lem1_gthm & induction2lem_gthm & ((! [X404 : (subst > term > subst > $o)] : ~ ! [X405 : (term > $o)] : ~ ! [X406 : term] : (((X404 @ id @ X406 @ id)) = ((X405 @ X406))) => (! [X407 : (term > $o),X408 : term] : (~ ! [X409 : term] : ((X407 @ (lam @ X409)) | ~ ! [X410 : term] : ((X407 @ (sub @ X409 @ (push @ X410 @ id))) | ~(X407 @ X410))) | (X407 @ X408) | ~ ! [X411 : term,X412 : term] : (~(X407 @ X412) | ~(X407 @ X411) | (X407 @ (ap @ X411 @ X412))) | ~ ! [X413 : term] : (~(var @ X413) | (X407 @ X413))) => (! [X414 : term] : (((sub @ X414 @ id)) = X414) => (! [X415 : (subst > term > subst > $o),X416 : term,X417 : (term > $o)] : ((X417 @ (lam @ X416)) | ~ ! [X418 : term] : (((X417 @ X418)) = ((X415 @ id @ X418 @ id))) | ~ ! [X419 : term] : (~(X417 @ X419) | (X417 @ (sub @ X416 @ (push @ X419 @ id)))) | ~ ! [X420 : (subst > term > term)] : (~ ! [X421 : term] : ((X415 @ id @ (X420 @ id @ X421) @ id) | ~(X415 @ id @ X421 @ id)) | (X415 @ id @ (lam @ (X420 @ sh @ one)) @ id) | ~ ! [X422 : subst,X423 : term,X424 : subst] : (((X420 @ (comp @ X422 @ X424) @ (sub @ X423 @ X424))) = ((sub @ (X420 @ X422 @ X423) @ X424))))) => ! [X425 : (subst > term > subst > $o),X426 : term] : (~ ! [X427 : term,X428 : term] : (~(X425 @ id @ X428 @ id) | (X425 @ id @ (ap @ (sub @ X428 @ id) @ X427) @ id) | ~(X425 @ id @ X427 @ id)) | (X425 @ id @ X426 @ id) | ~ ! [X429 : term,X430 : subst,X431 : subst,X432 : subst] : ((X425 @ X431 @ X429 @ (comp @ X432 @ X430)) | ~(X425 @ (comp @ X431 @ X432) @ (sub @ X429 @ X432) @ X430)) | ~ ! [X433 : term] : (~(var @ (sub @ X433 @ id)) | (X425 @ id @ X433 @ id)) | ~ ! [X434 : term,X435 : subst,X436 : subst,X437 : subst] : (~(X425 @ X435 @ X434 @ (comp @ X437 @ X436)) | (X425 @ (comp @ X435 @ X437) @ (sub @ X434 @ X437) @ X436)) | ~ ! [X438 : (subst > term > term)] : ((X425 @ id @ (lam @ (X438 @ sh @ one)) @ id) | ~ ! [X439 : subst,X440 : term,X441 : subst] : (((X438 @ (comp @ X441 @ X439) @ (sub @ X440 @ X439))) = ((sub @ (X438 @ X441 @ X440) @ X439))) | ~ ! [X442 : term] : ((X425 @ id @ (X438 @ id @ X442) @ id) | ~(X425 @ id @ X442 @ id)))))))) <=> hoasinduction_lthm_3) & (hoaslaminj_lthm <=> (! [X443 : term,X444 : subst] : (((sub @ one @ (push @ X443 @ X444))) = X443) => (! [X445 : subst,X446 : term] : (((comp @ sh @ (push @ X446 @ X445))) = X445) => (! [X447 : term,X448 : term] : ((X447 = X448) | (((lam @ X448)) != ((lam @ X447)))) => ! [X449 : term,X450 : (subst > term > term),X451 : (subst > term > term),X452 : subst] : (~ ! [X453 : term,X454 : subst,X455 : subst] : (((sub @ (X451 @ X454 @ X453) @ X455)) = ((X451 @ (comp @ X454 @ X455) @ (sub @ X453 @ X455)))) | ~ ! [X456 : term,X457 : subst,X458 : subst] : (((X450 @ (comp @ X457 @ X458) @ (sub @ X456 @ X458))) = ((sub @ (X450 @ X457 @ X456) @ X458))) | (((lam @ (X451 @ sh @ one))) != ((lam @ (X450 @ sh @ one)))) | (((X451 @ X452 @ X449)) = ((X450 @ X452 @ X449)))))))) & (! [X459 : subst] : (((push @ (sub @ one @ X459) @ (comp @ sh @ X459))) = X459) <=> axscons) & (! [X460 : subst] : (((comp @ id @ X460)) = X460) <=> axidl) & hoasinduction_lem1v2 & (hoasinduction_lem0_lthm <=> ! [X461 : (subst > term > subst > $o)] : ~ ! [X462 : (term > $o)] : ~ ! [X463 : term] : (((X462 @ X463)) = ((X461 @ id @ X463 @ id)))) & hoasinduction_gthm & pushprop_lem1v2_gthm & (hoasapinj2 <=> ! [X464 : term,X465 : term,X466 : term,X467 : term] : ((X465 = X466) | (((ap @ (sub @ X464 @ id) @ X466)) != ((ap @ (sub @ X467 @ id) @ X465))))) & hoasapinj2_gthm & pushprop_lem2v2_gthm & axassoc & (hoasinduction_lem2v2 <=> ! [X468 : term,X469 : term,X470 : (subst > term > subst > $o),X471 : (term > $o)] : (~ ! [X472 : term,X473 : term] : (~(X470 @ id @ X472 @ id) | (X470 @ id @ (ap @ (sub @ X472 @ id) @ X473) @ id) | ~(X470 @ id @ X473 @ id)) | ~ ! [X474 : term] : (((X471 @ X474)) = ((X470 @ id @ X474 @ id))) | (X471 @ (ap @ X469 @ X468)) | ~ ! [X475 : subst,X476 : subst,X477 : subst,X478 : term] : ((X470 @ X477 @ X478 @ (comp @ X475 @ X476)) | ~(X470 @ (comp @ X477 @ X475) @ (sub @ X478 @ X475) @ X476)) | ~ ! [X479 : term,X480 : subst,X481 : subst,X482 : subst] : (~(X470 @ X482 @ X479 @ (comp @ X480 @ X481)) | (X470 @ (comp @ X482 @ X480) @ (sub @ X479 @ X480) @ X481)) | ~(X471 @ X468) | ~(X471 @ X469))) & axclos & (! [X483 : term] : (((sub @ X483 @ id)) = X483) <=> termmset) & (((lam @ (d2term @ d_one))) = ((d2term @ d_one))) & ~(hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) & ~lamnotap & pushprop_lem3v2_lthm & ((! [X484 : term] : (((sub @ X484 @ id)) = X484) => ! [X485 : term] : (((sub @ X485 @ id)) = X485)) <=> termmset_lthm) & (id = ((d2subst @ d_id))) & (hoasinduction_lem3aaa <=> ! [X486 : term,X487 : (subst > term > subst > $o)] : (~ ! [X488 : term] : ((X487 @ id @ (sub @ X486 @ (push @ X488 @ id)) @ id) | ~(X487 @ id @ X488 @ id)) | ~ ! [X489 : (subst > term > term),X490 : term] : (~ ! [X491 : subst,X492 : subst,X493 : term] : (((sub @ (X489 @ X492 @ X493) @ X491)) = ((sub @ (sub @ X490 @ (push @ X493 @ X492)) @ X491))) | ~ ! [X494 : term] : (~(X487 @ id @ X494 @ id) | (X487 @ id @ (X489 @ id @ X494) @ id)) | ~ ! [X495 : subst,X496 : term,X497 : subst] : (((sub @ X490 @ (push @ (sub @ X496 @ X497) @ (comp @ X495 @ X497)))) = ((X489 @ (comp @ X495 @ X497) @ (sub @ X496 @ X497)))) | (X487 @ id @ (lam @ (X489 @ sh @ one)) @ id)) | (X487 @ id @ (lam @ (sub @ X486 @ (push @ one @ sh))) @ id))) & pushprop_lthm_orig & (! [X498 : term,X499 : subst] : (((comp @ sh @ (push @ X498 @ X499))) = X499) <=> axshiftcons) & (((sub @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2term @ d_one))) & (hoasinduction_lem3_lthm <=> (! [X500 : term] : (((sub @ X500 @ id)) = X500) => (! [X501 : term,X502 : (subst > term > subst > $o)] : (~ ! [X503 : (subst > term > term)] : (~ ! [X504 : subst,X505 : subst,X506 : term] : (((X503 @ (comp @ X504 @ X505) @ (sub @ X506 @ X505))) = ((sub @ (X503 @ X504 @ X506) @ X505))) | ~ ! [X507 : term] : (~(X502 @ id @ X507 @ id) | (X502 @ id @ (X503 @ id @ X507) @ id)) | (X502 @ id @ (lam @ (X503 @ sh @ one)) @ id)) | ~ ! [X508 : term] : (~(X502 @ id @ X508 @ id) | (X502 @ id @ (sub @ X501 @ (push @ X508 @ id)) @ id)) | (X502 @ id @ (lam @ (sub @ X501 @ (push @ one @ sh))) @ id)) => ! [X509 : term,X510 : (subst > term > subst > $o)] : (~ ! [X511 : term,X512 : subst,X513 : subst,X514 : subst] : ((X510 @ X514 @ X511 @ (comp @ X513 @ X512)) | ~(X510 @ (comp @ X514 @ X513) @ (sub @ X511 @ X513) @ X512)) | ~ ! [X515 : (subst > term > term)] : (~ ! [X516 : term] : (~(X510 @ id @ X516 @ id) | (X510 @ id @ (X515 @ id @ X516) @ id)) | ~ ! [X517 : term,X518 : subst,X519 : subst] : (((X515 @ (comp @ X519 @ X518) @ (sub @ X517 @ X518))) = ((sub @ (X515 @ X519 @ X517) @ X518))) | (X510 @ id @ (lam @ (X515 @ sh @ one)) @ id)) | (X510 @ id @ (lam @ X509) @ id) | ~ ! [X520 : term] : (~(X510 @ id @ X520 @ id) | (X510 @ id @ (sub @ X509 @ (push @ X520 @ id)) @ id)) | ~ ! [X521 : term,X522 : subst,X523 : subst,X524 : subst] : ((X510 @ (comp @ X524 @ X523) @ (sub @ X521 @ X523) @ X522) | ~(X510 @ X524 @ X521 @ (comp @ X523 @ X522))))))) & hoasapnotvar & (((comp @ (d2subst @ d_id) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & hoaslamnotap_lthm & hoasinduction_lem1 & (((push @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & pushprop_lem2v2_lthm & (! [X525 : term,X526 : (term > $o)] : (~ ! [X527 : term,X528 : term] : ((X526 @ (ap @ X527 @ X528)) | ~(X526 @ X528) | ~(X526 @ X527)) | (X526 @ X525) | ~ ! [X529 : term] : ((X526 @ (lam @ X529)) | ~(X526 @ X529)) | ~ ! [X530 : term] : (~(var @ X530) | (X526 @ X530))) <=> induction) & (((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) = ((d2term @ d_one))) & ((! [X531 : subst] : (((comp @ X531 @ id)) = X531) & ! [X532 : subst] : (((comp @ id @ X532)) = X532)) <=> substmonoid) & ~ulamvar1 & ulamvarsh & axmap & ((! [X533 : term] : (((sub @ X533 @ id)) = X533) => (! [X534 : term,X535 : term,X536 : term,X537 : term] : ((X534 = X535) | (((ap @ X534 @ X537)) != ((ap @ X535 @ X536)))) => ! [X538 : term,X539 : term,X540 : term,X541 : term] : ((((ap @ (sub @ X538 @ id) @ X541)) != ((ap @ (sub @ X540 @ id) @ X539))) | (X538 = X540)))) <=> hoasapinj1_lthm) & hoaslaminj_gthm & (! [X542 : term,X543 : (subst > term > subst > $o)] : (~ ! [X544 : term] : ((X543 @ id @ (sub @ X542 @ (push @ X544 @ id)) @ id) | ~(X543 @ id @ X544 @ id)) | ~ ! [X545 : (subst > term > term)] : ((X543 @ id @ (lam @ (X545 @ sh @ one)) @ id) | ~ ! [X546 : term,X547 : subst,X548 : subst] : (((X545 @ (comp @ X548 @ X547) @ (sub @ X546 @ X547))) = ((sub @ (X545 @ X548 @ X546) @ X547))) | ~ ! [X549 : term] : (~(X543 @ id @ X549 @ id) | (X543 @ id @ (X545 @ id @ X549) @ id))) | (X543 @ id @ (lam @ X542) @ id)) <=> hoasinduction_lem3a) & pushprop),
  inference(rectify,[],[f1])).
thf(f5,plain,(
  (! [X1 : term,X0 : term,X3 : term,X2 : term] : ((X0 = X3) | (((ap @ X3 @ X2)) != ((ap @ X0 @ X1)))) <=> (apinj1 = $true)) & (hoasinduction_lem2_lthm = $true) & ((hoasinduction_lem3v2a = $true) <=> ! [X4 : (term > $o),X5 : term,X6 : (subst > term > subst > $o)] : (~ ! [X7 : (subst > term > term)] : (($true = ((X6 @ id @ (lam @ (X7 @ sh @ one)) @ id))) | ~ ! [X8 : term] : (~ ($true = ((X6 @ id @ X8 @ id))) | (((X6 @ id @ (X7 @ id @ X8) @ id)) = $true)) | ~ ! [X11 : subst,X10 : term,X9 : subst] : (((sub @ (X7 @ X11 @ X10) @ X9)) = ((X7 @ (comp @ X11 @ X9) @ (sub @ X10 @ X9))))) | ($true = ((X4 @ (lam @ X5)))) | ~ ! [X12 : term] : (((X6 @ id @ X12 @ id)) = ((X4 @ X12))) | ~ ! [X13 : term] : (($true = ((X4 @ (sub @ X5 @ (push @ X13 @ id))))) | ~ (((X4 @ X13)) = $true)))) & ((axvarid = $true) <=> ! [X14 : term] : (((sub @ X14 @ id)) = X14)) & (hoasapnotvar_gthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((substmonoid_lthm = $true) <=> (! [X15 : subst] : (((comp @ id @ X15)) = X15) => (! [X16 : subst] : (((comp @ X16 @ id)) = X16) => (! [X17 : subst] : (((comp @ X17 @ id)) = X17) & ! [X18 : subst] : (((comp @ id @ X18)) = X18))))) & ((hoasapinj1 = $true) <=> ! [X21 : term,X19 : term,X20 : term,X22 : term] : ((X21 = X22) | (((ap @ (sub @ X22 @ id) @ X19)) != ((ap @ (sub @ X21 @ id) @ X20))))) & (substmonoid_gthm = $true) & (! [X23 : term,X24 : (subst > term > subst > $o)] : (($true = ((X24 @ id @ (lam @ X23) @ id))) | ~ ! [X25 : subst,X26 : term,X27 : subst,X28 : subst] : (($true = ((X24 @ X25 @ X26 @ (comp @ X28 @ X27)))) | ~ (((X24 @ (comp @ X25 @ X28) @ (sub @ X26 @ X28) @ X27)) = $true)) | ~ ! [X29 : (subst > term > term)] : (~ ! [X32 : subst,X31 : term,X30 : subst] : (((sub @ (X29 @ X30 @ X31) @ X32)) = ((X29 @ (comp @ X30 @ X32) @ (sub @ X31 @ X32)))) | (((X24 @ id @ (lam @ (X29 @ sh @ one)) @ id)) = $true) | ~ ! [X33 : term] : (~ (((X24 @ id @ X33 @ id)) = $true) | (((X24 @ id @ (X29 @ id @ X33) @ id)) = $true))) | ~ ! [X34 : term,X35 : subst,X36 : subst,X37 : subst] : ((((X24 @ (comp @ X37 @ X35) @ (sub @ X34 @ X35) @ X36)) = $true) | ~ ($true = ((X24 @ X37 @ X34 @ (comp @ X35 @ X36))))) | ~ ! [X38 : term] : (~ (((X24 @ id @ X38 @ id)) = $true) | ($true = ((X24 @ id @ (sub @ X23 @ (push @ X38 @ id)) @ id))))) <=> (hoasinduction_lem3 = $true)) & (pushprop_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)))))))))))) & ! [X44 : d_subst,X45 : d_subst] : ((((d2subst @ X44)) = ((d2subst @ X45))) => (X44 = X45)) & ((hoaslaminj = $true) <=> ! [X46 : (subst > term > term),X47 : (subst > term > term),X48 : subst,X49 : term] : (~ ! [X50 : term,X51 : subst,X52 : subst] : (((sub @ (X47 @ X51 @ X50) @ X52)) = ((X47 @ (comp @ X51 @ X52) @ (sub @ X50 @ X52)))) | (((X47 @ X48 @ X49)) = ((X46 @ X48 @ X49))) | (((lam @ (X47 @ sh @ one))) != ((lam @ (X46 @ sh @ one)))) | ~ ! [X53 : term,X54 : subst,X55 : subst] : (((X46 @ (comp @ X54 @ X55) @ (sub @ X53 @ X55))) = ((sub @ (X46 @ X54 @ X53) @ X55))))) & ! [X56 : term] : ? [X57 : d_term] : (((d2term @ X57)) = X56) & (hoaslamnotvar_gthm = $true) & (axvarshift = $true) & (! [X58 : term,X59 : term,X60 : (subst > term > subst > $o)] : (~ ($true = ((X60 @ id @ X58 @ id))) | ~ ! [X61 : subst,X62 : term,X63 : subst,X64 : subst] : (~ ($true = ((X60 @ X61 @ X62 @ (comp @ X63 @ X64)))) | ($true = ((X60 @ (comp @ X61 @ X63) @ (sub @ X62 @ X63) @ X64)))) | ~ ! [X65 : term,X66 : term] : ((((X60 @ id @ (ap @ (sub @ X66 @ id) @ X65) @ id)) = $true) | ~ ($true = ((X60 @ id @ X65 @ id))) | ~ (((X60 @ id @ X66 @ id)) = $true)) | ($true = ((X60 @ id @ (ap @ X58 @ X59) @ id))) | ~ ! [X67 : term,X68 : subst,X69 : subst,X70 : subst] : (~ ($true = ((X60 @ (comp @ X69 @ X68) @ (sub @ X67 @ X68) @ X70))) | ($true = ((X60 @ X69 @ X67 @ (comp @ X68 @ X70))))) | ~ (((X60 @ id @ X59 @ id)) = $true)) <=> (hoasinduction_lem2 = $true)) & (sh = ((d2subst @ d_id))) & (pushprop_lem0_gthm = $true) & ((hoasinduction_no_psi_cond = $true) <=> ! [X71 : term,X72 : (subst > term > subst > $o)] : (~ ! [X73 : (subst > term > term)] : (~ ! [X74 : subst,X75 : term,X76 : subst] : (((sub @ (X73 @ X74 @ X75) @ X76)) = ((X73 @ (comp @ X74 @ X76) @ (sub @ X75 @ X76)))) | ($true = ((X72 @ id @ (lam @ (X73 @ sh @ one)) @ id))) | ~ ! [X77 : term] : (~ ($true = ((X72 @ id @ X77 @ id))) | (((X72 @ id @ (X73 @ id @ X77) @ id)) = $true))) | ~ ! [X78 : term,X79 : term] : ((((X72 @ id @ (ap @ (sub @ X78 @ id) @ X79) @ id)) = $true) | ~ ($true = ((X72 @ id @ X79 @ id))) | ~ (((X72 @ id @ X78 @ id)) = $true)) | ($true = ((X72 @ id @ X71 @ id))))) & (hoasinduction_lem1v2_gthm = $true) & (hoasapnotvar_lthm = $true) & ((! [X80 : term,X81 : (term > $o)] : ((((X81 @ X80)) = $true) | ~ ! [X82 : term] : ((((X81 @ (lam @ X82))) = $true) | ~ ! [X83 : term] : (~ (((X81 @ X83)) = $true) | ($true = ((X81 @ (sub @ X82 @ (push @ X83 @ id))))))) | ~ ! [X84 : term,X85 : term] : (~ (((X81 @ X84)) = $true) | ($true = ((X81 @ (ap @ X84 @ X85)))) | ~ ($true = ((X81 @ X85)))) | ~ ! [X86 : term] : ((((X81 @ X86)) = $true) | ~ (((var @ X86)) = $true))) => (! [X87 : term,X88 : term,X89 : (subst > term > subst > $o)] : (~ ($true = ((X89 @ id @ X87 @ id))) | ~ (((X89 @ id @ X88 @ id)) = $true) | ($true = ((X89 @ id @ (ap @ X88 @ X87) @ id))) | ~ ! [X90 : subst,X91 : subst,X92 : subst,X93 : term] : (~ (((X89 @ (comp @ X90 @ X91) @ (sub @ X93 @ X91) @ X92)) = $true) | (((X89 @ X90 @ X93 @ (comp @ X91 @ X92))) = $true)) | ~ ! [X94 : term,X95 : term] : (($true = ((X89 @ id @ (ap @ (sub @ X94 @ id) @ X95) @ id))) | ~ ($true = ((X89 @ id @ X94 @ id))) | ~ ($true = ((X89 @ id @ X95 @ id)))) | ~ ! [X96 : subst,X97 : term,X98 : subst,X99 : subst] : (~ ($true = ((X89 @ X98 @ X97 @ (comp @ X96 @ X99)))) | (((X89 @ (comp @ X98 @ X96) @ (sub @ X97 @ X96) @ X99)) = $true))) => (! [X100 : term,X101 : (subst > term > subst > $o)] : (~ ! [X102 : term,X103 : subst,X104 : subst,X105 : subst] : (~ ($true = ((X101 @ (comp @ X103 @ X104) @ (sub @ X102 @ X104) @ X105))) | (((X101 @ X103 @ X102 @ (comp @ X104 @ X105))) = $true)) | ~ ! [X106 : term,X107 : subst,X108 : subst,X109 : subst] : (($true = ((X101 @ (comp @ X107 @ X108) @ (sub @ X106 @ X108) @ X109))) | ~ ($true = ((X101 @ X107 @ X106 @ (comp @ X108 @ X109))))) | ($true = ((X101 @ id @ (lam @ X100) @ id))) | ~ ! [X110 : term] : ((((X101 @ id @ (sub @ X100 @ (push @ X110 @ id)) @ id)) = $true) | ~ (((X101 @ id @ X110 @ id)) = $true)) | ~ ! [X111 : (subst > term > term)] : (~ ! [X112 : term] : (~ ($true = ((X101 @ id @ X112 @ id))) | ($true = ((X101 @ id @ (X111 @ id @ X112) @ id)))) | ($true = ((X101 @ id @ (lam @ (X111 @ sh @ one)) @ id))) | ~ ! [X113 : term,X114 : subst,X115 : subst] : (((sub @ (X111 @ X115 @ X113) @ X114)) = ((X111 @ (comp @ X115 @ X114) @ (sub @ X113 @ X114)))))) => ! [X116 : (subst > term > subst > $o),X117 : term] : (~ ! [X118 : term,X119 : term] : (~ ($true = ((X116 @ id @ X119 @ id))) | (((X116 @ id @ (ap @ (sub @ X118 @ id) @ X119) @ id)) = $true) | ~ (((X116 @ id @ X118 @ id)) = $true)) | ~ ! [X120 : (subst > term > term)] : ((((X116 @ id @ (lam @ (X120 @ sh @ one)) @ id)) = $true) | ~ ! [X121 : term,X122 : subst,X123 : subst] : (((sub @ (X120 @ X123 @ X121) @ X122)) = ((X120 @ (comp @ X123 @ X122) @ (sub @ X121 @ X122)))) | ~ ! [X124 : term] : (($true = ((X116 @ id @ (X120 @ id @ X124) @ id))) | ~ (((X116 @ id @ X124 @ id)) = $true))) | (((X116 @ id @ X117 @ id)) = $true) | ~ ! [X125 : subst,X126 : subst,X127 : subst,X128 : term] : (~ ($true = ((X116 @ X126 @ X128 @ (comp @ X125 @ X127)))) | (((X116 @ (comp @ X126 @ X125) @ (sub @ X128 @ X125) @ X127)) = $true)) | ~ ! [X129 : term] : (~ (((var @ (sub @ X129 @ id))) = $true) | (((X116 @ id @ X129 @ id)) = $true)) | ~ ! [X130 : subst,X131 : subst,X132 : subst,X133 : term] : (~ (((X116 @ (comp @ X130 @ X132) @ (sub @ X133 @ X132) @ X131)) = $true) | (((X116 @ X130 @ X133 @ (comp @ X132 @ X131))) = $true)))))) <=> (hoasinduction_lthm = $true)) & (induction2_gthm = $true) & ((hoasinduction_lem3aa = $true) <=> ! [X134 : term,X135 : (subst > term > subst > $o)] : ((((X135 @ id @ (lam @ (sub @ X134 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X136 : term] : (($true = ((X135 @ id @ (sub @ X134 @ (push @ X136 @ id)) @ id))) | ~ (((X135 @ id @ X136 @ id)) = $true)) | ~ ! [X137 : (subst > term > term)] : (~ ! [X138 : term] : (~ (((X135 @ id @ X138 @ id)) = $true) | ($true = ((X135 @ id @ (X137 @ id @ X138) @ id)))) | ($true = ((X135 @ id @ (lam @ (X137 @ sh @ one)) @ id))) | ~ ! [X139 : subst,X140 : term,X141 : subst] : (((X137 @ (comp @ X141 @ X139) @ (sub @ X140 @ X139))) = ((sub @ (X137 @ X141 @ X140) @ X139)))))) & (! [X142 : term,X143 : term,X144 : (subst > term > term)] : (~ ! [X145 : subst,X146 : term,X147 : subst] : (((sub @ (X144 @ X147 @ X146) @ X145)) = ((X144 @ (comp @ X147 @ X145) @ (sub @ X146 @ X145)))) | (((ap @ (sub @ X143 @ id) @ X142)) != ((lam @ (X144 @ sh @ one))))) <=> (hoaslamnotap = $true)) & (! [X148 : term,X149 : (term > $o),X150 : subst] : ~ ! [X151 : (term > $o)] : ~ ! [X152 : term] : (((X151 @ X152)) = ((X149 @ (sub @ X152 @ (push @ X148 @ X150))))) <=> (pushprop_lem0_lthm = $true)) & ((hoasinduction_lem3v2 = $true) <=> ! [X153 : (subst > term > subst > $o),X154 : term,X155 : (term > $o)] : (($true = ((X155 @ (lam @ X154)))) | ~ ! [X156 : subst,X157 : term,X158 : subst,X159 : subst] : ((((X153 @ X156 @ X157 @ (comp @ X159 @ X158))) = $true) | ~ ($true = ((X153 @ (comp @ X156 @ X159) @ (sub @ X157 @ X159) @ X158)))) | ~ ! [X160 : (subst > term > term)] : (($true = ((X153 @ id @ (lam @ (X160 @ sh @ one)) @ id))) | ~ ! [X161 : term] : (~ (((X153 @ id @ X161 @ id)) = $true) | (((X153 @ id @ (X160 @ id @ X161) @ id)) = $true)) | ~ ! [X162 : subst,X163 : subst,X164 : term] : (((X160 @ (comp @ X162 @ X163) @ (sub @ X164 @ X163))) = ((sub @ (X160 @ X162 @ X164) @ X163)))) | ~ ! [X165 : subst,X166 : subst,X167 : subst,X168 : term] : (~ ($true = ((X153 @ X166 @ X168 @ (comp @ X165 @ X167)))) | ($true = ((X153 @ (comp @ X166 @ X165) @ (sub @ X168 @ X165) @ X167)))) | ~ ! [X169 : term] : (((X155 @ X169)) = ((X153 @ id @ X169 @ id))) | ~ ! [X170 : term] : (~ (((X155 @ X170)) = $true) | (((X155 @ (sub @ X154 @ (push @ X170 @ id)))) = $true)))) & ((pushprop_lem0 = $true) <=> ! [X171 : subst,X172 : (term > $o),X173 : term] : ~ ! [X174 : (term > $o)] : ~ ! [X175 : term] : (((X172 @ (sub @ X175 @ (push @ X173 @ X171)))) = ((X174 @ X175)))) & (hoasapinj1_gthm = $true) & ((induction2_lthm = $true) <=> (! [X176 : term] : (((sub @ X176 @ id)) = X176) => (! [X177 : subst,X178 : term,X179 : (term > $o)] : (($true = ((X179 @ (sub @ X178 @ X177)))) | ~ ! [X180 : term,X181 : term] : (~ (((X179 @ X180)) = $true) | ~ (((X179 @ X181)) = $true) | (((X179 @ (ap @ X180 @ X181))) = $true)) | ~ ! [X182 : term] : ((((X179 @ (lam @ X182))) = $true) | ~ ! [X183 : term] : (($true = ((X179 @ (sub @ X182 @ (push @ X183 @ id))))) | ~ ($true = ((X179 @ X183))))) | ~ ! [X184 : term] : (~ (((var @ X184)) = $true) | ($true = ((X179 @ (sub @ X184 @ X177)))))) => ! [X185 : term,X186 : (term > $o)] : (~ ! [X187 : term,X188 : term] : (~ (((X186 @ X187)) = $true) | ~ ($true = ((X186 @ X188))) | (((X186 @ (ap @ X188 @ X187))) = $true)) | ~ ! [X189 : term] : (~ ($true = ((var @ X189))) | ($true = ((X186 @ X189)))) | ~ ! [X190 : term] : (~ ! [X191 : term] : (~ (((X186 @ X191)) = $true) | ($true = ((X186 @ (sub @ X190 @ (push @ X191 @ id)))))) | (((X186 @ (lam @ X190))) = $true)) | ($true = ((X186 @ X185))))))) & (! [X192 : term,X193 : (subst > term > subst > $o)] : (~ ! [X194 : term] : ((((X193 @ id @ X194 @ id)) = $true) | ~ ($true = ((var @ (sub @ X194 @ id))))) | ~ ! [X195 : term,X196 : term] : ((((X193 @ id @ (ap @ (sub @ X196 @ id) @ X195) @ id)) = $true) | ~ (((X193 @ id @ X196 @ id)) = $true) | ~ (((X193 @ id @ X195 @ id)) = $true)) | (((X193 @ id @ X192 @ id)) = $true) | ~ ! [X197 : (subst > term > term)] : (~ ! [X198 : term] : (~ (((X193 @ id @ X198 @ id)) = $true) | (((X193 @ id @ (X197 @ id @ X198) @ id)) = $true)) | (((X193 @ id @ (lam @ (X197 @ sh @ one)) @ id)) = $true) | ~ ! [X199 : term,X200 : subst,X201 : subst] : (((X197 @ (comp @ X201 @ X200) @ (sub @ X199 @ X200))) = ((sub @ (X197 @ X201 @ X199) @ X200)))) | ~ ! [X202 : subst,X203 : subst,X204 : subst,X205 : term] : (~ ($true = ((X193 @ X203 @ X205 @ (comp @ X202 @ X204)))) | ($true = ((X193 @ (comp @ X203 @ X202) @ (sub @ X205 @ X202) @ X204)))) | ~ ! [X206 : subst,X207 : subst,X208 : term,X209 : subst] : (~ (((X193 @ (comp @ X207 @ X206) @ (sub @ X208 @ X206) @ X209)) = $true) | ($true = ((X193 @ X207 @ X208 @ (comp @ X206 @ X209)))))) <=> (hoasinduction = $true)) & ((axvarcons = $true) <=> ! [X210 : subst,X211 : term] : (((sub @ one @ (push @ X211 @ X210))) = X211)) & (! [X212 : term] : ~ ! [X213 : (subst > term > term)] : ~ ! [X214 : subst,X215 : term] : (((sub @ X212 @ (push @ X215 @ X214))) = ((X213 @ X214 @ X215))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & ((hoasinduction_lem0 = $true) <=> ! [X216 : (subst > term > subst > $o)] : ~ ! [X217 : (term > $o)] : ~ ! [X218 : term] : (((X217 @ X218)) = ((X216 @ id @ X218 @ id)))) & ((hoasinduction_lem3v2_f = $true) <=> ! [X219 : term] : ~ ! [X220 : (subst > term > term)] : ~ ! [X221 : term,X222 : subst] : (((X220 @ X222 @ X221)) = ((sub @ X219 @ (push @ X221 @ X222))))) & ((! [X223 : subst,X224 : term] : (((sub @ one @ (push @ X224 @ X223))) = X224) => (! [X225 : subst,X226 : term] : (((comp @ sh @ (push @ X226 @ X225))) = X225) => ! [X227 : (term > $o),X228 : (term > $o),X229 : term,X230 : term,X231 : subst] : (~ (((X227 @ X229)) = $true) | ($true = ((X228 @ (sub @ X229 @ (push @ X230 @ X231)))))))) <=> (pushprop_lem1_lthm = $true)) & ((! [X232 : term,X233 : term,X234 : term,X235 : term] : ((X233 = X234) | (((ap @ X235 @ X234)) != ((ap @ X232 @ X233)))) => ! [X236 : term,X237 : term,X238 : term,X239 : term] : ((((ap @ (sub @ X237 @ id) @ X236)) != ((ap @ (sub @ X238 @ id) @ X239))) | (X236 = X239))) <=> (hoasapinj2_lthm = $true)) & (ulamvarind = $true) & (apnotvar = $true) & ((! [X240 : term] : (((sub @ X240 @ id)) = X240) => ! [X241 : term,X242 : (subst > term > subst > $o),X243 : (term > $o)] : (~ ! [X244 : (subst > term > term)] : (($true = ((X242 @ id @ (lam @ (X244 @ sh @ one)) @ id))) | ~ ! [X245 : term] : (($true = ((X242 @ id @ (X244 @ id @ X245) @ id))) | ~ ($true = ((X242 @ id @ X245 @ id)))) | ~ ! [X246 : subst,X247 : subst,X248 : term] : (((X244 @ (comp @ X246 @ X247) @ (sub @ X248 @ X247))) = ((sub @ (X244 @ X246 @ X248) @ X247)))) | ~ ! [X249 : term] : (((X243 @ X249)) = ((X242 @ id @ X249 @ id))) | ~ ! [X250 : term,X251 : subst,X252 : subst,X253 : subst] : ((((X242 @ X253 @ X250 @ (comp @ X251 @ X252))) = $true) | ~ (((X242 @ (comp @ X253 @ X251) @ (sub @ X250 @ X251) @ X252)) = $true)) | (((X243 @ (lam @ X241))) = $true) | ~ ! [X254 : term] : (~ ($true = ((X243 @ X254))) | (((X243 @ (sub @ X241 @ (push @ X254 @ id)))) = $true)) | ~ ! [X255 : term,X256 : subst,X257 : subst,X258 : subst] : (~ ($true = ((X242 @ X257 @ X255 @ (comp @ X256 @ X258)))) | ($true = ((X242 @ (comp @ X257 @ X256) @ (sub @ X255 @ X256) @ X258)))))) <=> (hoasinduction_lem3v2_lthm = $true)) & ((pushprop_lem1v2_lthm = $true) <=> (! [X259 : subst,X260 : term] : (((sub @ one @ (push @ X260 @ X259))) = X260) => ! [X261 : subst,X262 : (term > $o),X263 : (term > $o),X264 : term] : (~ ! [X265 : term] : (((X262 @ (sub @ X265 @ (push @ X264 @ X261)))) = ((X263 @ X265))) | ~ (((X262 @ X264)) = $true) | ($true = ((X263 @ one)))))) & (hoasinduction_lem3a_gthm = $true) & (! [X266 : term,X267 : term] : ((X266 = X267) | (((lam @ X266)) != ((lam @ X267)))) <=> (laminj = $true)) & (hoaslamnotap_gthm = $true) & (one = ((d2term @ d_one))) & ((apinj2 = $true) <=> ! [X268 : term,X269 : term,X270 : term,X271 : term] : ((X269 = X271) | (((ap @ X270 @ X271)) != ((ap @ X268 @ X269))))) & (lamnotvar = $true) & ((pushprop_lem1v2 = $true) <=> ! [X272 : (term > $o),X273 : subst,X274 : (term > $o),X275 : term] : (~ ! [X276 : term] : (((X272 @ (sub @ X276 @ (push @ X275 @ X273)))) = ((X274 @ X276))) | (((X274 @ one)) = $true) | ~ ($true = ((X272 @ X275))))) & ~ (((var @ (d2term @ d_one))) = $true) & (hoaslamnotvar_lthm = $true) & (pushprop_lem3v2 = $true) & (! [X277 : (term > $o),X278 : subst,X279 : term] : (($true = ((X277 @ (sub @ X279 @ X278)))) | ~ ! [X280 : term] : (~ ! [X281 : term] : (~ (((X277 @ X281)) = $true) | ($true = ((X277 @ (sub @ X280 @ (push @ X281 @ id)))))) | (((X277 @ (lam @ X280))) = $true)) | ~ ! [X282 : term,X283 : term] : (~ (((X277 @ X283)) = $true) | ~ ($true = ((X277 @ X282))) | ($true = ((X277 @ (ap @ X282 @ X283))))) | ~ ! [X284 : term] : ((((X277 @ (sub @ X284 @ X278))) = $true) | ~ ($true = ((var @ X284))))) <=> (induction2lem = $true)) & (hoasinduction_lem2_gthm = $true) & ! [X285 : d_term] : (d_one = X285) & (pushprop_lem2v2 = $true) & (hoasinduction_lem3v2_gthm = $true) & ((! [X286 : term] : ~ ! [X287 : (subst > term > term)] : ~ ! [X288 : subst,X289 : term] : (((X287 @ X288 @ X289)) = ((sub @ X286 @ (push @ X289 @ X288)))) => (! [X290 : term] : (((sub @ X290 @ id)) = X290) => ! [X291 : (subst > term > subst > $o),X292 : (term > $o),X293 : term] : (~ ! [X294 : term] : ((((X292 @ (sub @ X293 @ (push @ X294 @ id)))) = $true) | ~ (((X292 @ X294)) = $true)) | (((X292 @ (lam @ X293))) = $true) | ~ ! [X295 : (subst > term > term)] : (~ ! [X296 : term] : (~ ($true = ((X291 @ id @ X296 @ id))) | ($true = ((X291 @ id @ (X295 @ id @ X296) @ id)))) | ~ ! [X297 : subst,X298 : subst,X299 : term] : (((X295 @ (comp @ X298 @ X297) @ (sub @ X299 @ X297))) = ((sub @ (X295 @ X298 @ X299) @ X297))) | ($true = ((X291 @ id @ (lam @ (X295 @ sh @ one)) @ id)))) | ~ ! [X300 : term] : (((X291 @ id @ X300 @ id)) = ((X292 @ X300)))))) <=> (hoasinduction_lem3v2a_lthm = $true)) & ! [X301 : subst] : ? [X302 : d_subst] : (((d2subst @ X302)) = X301) & (hoaslamnotvar = $true) & (hoasinduction_lem3b_gthm = $true) & (termmset_gthm = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((shinj = $true) <=> ! [X305 : term,X306 : term] : ((((sub @ X305 @ sh)) != ((sub @ X306 @ sh))) | (X305 = X306))) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X307 : term] : ~ ! [X308 : (subst > term > term)] : (((X308 @ sh @ one)) != ((sub @ X307 @ (push @ one @ sh))))) & ! [X309 : d_term,X310 : d_term] : ((((d2term @ X309)) = ((d2term @ X310))) => (X309 = X310)) & (axabs = $true) & (hoasinduction_lem2v2_gthm = $true) & ((axidr = $true) <=> ! [X311 : subst] : (((comp @ X311 @ id)) = X311)) & (pushprop_lem1_gthm = $true) & (hoasinduction_lem3_gthm = $true) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X312 : term,X313 : (subst > term > subst > $o)] : (~ ! [X314 : (subst > term > term)] : ((((X313 @ id @ (lam @ (X314 @ sh @ one)) @ id)) = $true) | ~ ! [X315 : term] : (~ ($true = ((X313 @ id @ X315 @ id))) | (((X313 @ id @ (X314 @ id @ X315) @ id)) = $true)) | ~ ! [X316 : subst,X317 : term,X318 : subst] : (((X314 @ (comp @ X318 @ X316) @ (sub @ X317 @ X316))) = ((sub @ (X314 @ X318 @ X317) @ X316)))) | ($true = ((X313 @ id @ (lam @ (sub @ X312 @ (push @ one @ sh))) @ id))) | ~ ! [X319 : term] : (($true = ((X313 @ id @ (sub @ X312 @ (push @ X319 @ id)) @ id))) | ~ (((X313 @ id @ X319 @ id)) = $true)))) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (! [X320 : (subst > term > subst > $o)] : ~ ! [X321 : (term > $o)] : ~ ! [X322 : term] : (((X321 @ X322)) = ((X320 @ id @ X322 @ id))) => (! [X323 : (term > $o),X324 : term] : (~ ! [X325 : term,X326 : term] : (~ ($true = ((X323 @ X325))) | (((X323 @ (ap @ X326 @ X325))) = $true) | ~ ($true = ((X323 @ X326)))) | (((X323 @ X324)) = $true) | ~ ! [X327 : term] : (($true = ((X323 @ (lam @ X327)))) | ~ ! [X328 : term] : (($true = ((X323 @ (sub @ X327 @ (push @ X328 @ id))))) | ~ (((X323 @ X328)) = $true))) | ~ ! [X329 : term] : (~ (((var @ X329)) = $true) | (((X323 @ X329)) = $true))) => (! [X330 : term] : (((sub @ X330 @ id)) = X330) => (! [X331 : (subst > term > subst > $o),X332 : term,X333 : (term > $o)] : (~ ! [X334 : (subst > term > term)] : ((((X331 @ id @ (lam @ (X334 @ sh @ one)) @ id)) = $true) | ~ ! [X335 : subst,X336 : subst,X337 : term] : (((X334 @ (comp @ X335 @ X336) @ (sub @ X337 @ X336))) = ((sub @ (X334 @ X335 @ X337) @ X336))) | ~ ! [X338 : term] : ((((X331 @ id @ (X334 @ id @ X338) @ id)) = $true) | ~ ($true = ((X331 @ id @ X338 @ id))))) | ~ ! [X339 : term] : (~ (((X333 @ X339)) = $true) | (((X333 @ (sub @ X332 @ (push @ X339 @ id)))) = $true)) | ~ ! [X340 : term] : (((X331 @ id @ X340 @ id)) = ((X333 @ X340))) | (((X333 @ (lam @ X332))) = $true)) => ! [X341 : term,X342 : (subst > term > subst > $o)] : (~ ! [X343 : (subst > term > term)] : (~ ! [X344 : term] : (~ (((X342 @ id @ X344 @ id)) = $true) | (((X342 @ id @ (X343 @ id @ X344) @ id)) = $true)) | ~ ! [X345 : term,X346 : subst,X347 : subst] : (((sub @ (X343 @ X347 @ X345) @ X346)) = ((X343 @ (comp @ X347 @ X346) @ (sub @ X345 @ X346)))) | ($true = ((X342 @ id @ (lam @ (X343 @ sh @ one)) @ id)))) | (((X342 @ id @ X341 @ id)) = $true) | ~ ! [X348 : term,X349 : term] : ((((X342 @ id @ (ap @ (sub @ X348 @ id) @ X349) @ id)) = $true) | ~ (((X342 @ id @ X349 @ id)) = $true) | ~ ($true = ((X342 @ id @ X348 @ id)))))))))) & (! [X350 : term] : ~ ! [X351 : (subst > term > term)] : (((X351 @ sh @ one)) != ((sub @ X350 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b = $true)) & (pushprop_lthm = $true) & (axapp = $true) & ((induction2 = $true) <=> ! [X352 : (term > $o),X353 : term] : (~ ! [X354 : term] : (($true = ((X352 @ X354))) | ~ ($true = ((var @ X354)))) | ($true = ((X352 @ X353))) | ~ ! [X355 : term] : (~ ! [X356 : term] : (~ ($true = ((X352 @ X356))) | (((X352 @ (sub @ X355 @ (push @ X356 @ id)))) = $true)) | ($true = ((X352 @ (lam @ X355))))) | ~ ! [X357 : term,X358 : term] : (~ ($true = ((X352 @ X357))) | (((X352 @ (ap @ X358 @ X357))) = $true) | ~ ($true = ((X352 @ X358)))))) & ((hoasinduction_lem3a_lthm = $true) <=> (! [X359 : term] : (((sub @ X359 @ id)) = X359) => (! [X360 : term,X361 : (subst > term > subst > $o)] : (~ ! [X362 : term] : (($true = ((X361 @ id @ (sub @ X360 @ (push @ X362 @ id)) @ id))) | ~ ($true = ((X361 @ id @ X362 @ id)))) | ~ ! [X363 : (subst > term > term)] : (~ ! [X364 : term] : (~ (((X361 @ id @ X364 @ id)) = $true) | (((X361 @ id @ (X363 @ id @ X364) @ id)) = $true)) | (((X361 @ id @ (lam @ (X363 @ sh @ one)) @ id)) = $true) | ~ ! [X365 : subst,X366 : subst,X367 : term] : (((sub @ (X363 @ X365 @ X367) @ X366)) = ((X363 @ (comp @ X365 @ X366) @ (sub @ X367 @ X366))))) | ($true = ((X361 @ id @ (lam @ (sub @ X360 @ (push @ one @ sh))) @ id)))) => ! [X368 : term,X369 : (subst > term > subst > $o)] : (~ ! [X370 : (subst > term > term)] : ((((X369 @ id @ (lam @ (X370 @ sh @ one)) @ id)) = $true) | ~ ! [X371 : subst,X372 : subst,X373 : term] : (((sub @ (X370 @ X372 @ X373) @ X371)) = ((X370 @ (comp @ X372 @ X371) @ (sub @ X373 @ X371)))) | ~ ! [X374 : term] : ((((X369 @ id @ (X370 @ id @ X374) @ id)) = $true) | ~ ($true = ((X369 @ id @ X374 @ id))))) | (((X369 @ id @ (lam @ X368) @ id)) = $true) | ~ ! [X375 : term] : (~ (((X369 @ id @ X375 @ id)) = $true) | ($true = ((X369 @ id @ (sub @ X368 @ (push @ X375 @ id)) @ id)))))))) & (hoasinduction_lem1_lthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & ! [X379 : d_subst] : (d_id = X379) & ((induction2lem_lthm = $true) <=> (! [X380 : subst,X381 : term] : (((sub @ one @ (push @ X381 @ X380))) = X381) => (! [X382 : term,X383 : subst] : (((comp @ sh @ (push @ X382 @ X383))) = X383) => (! [X384 : subst] : (((comp @ X384 @ id)) = X384) => (! [X385 : (term > $o),X386 : term] : (~ ! [X387 : term,X388 : term] : (~ (((X385 @ X387)) = $true) | (((X385 @ (ap @ X388 @ X387))) = $true) | ~ ($true = ((X385 @ X388)))) | ($true = ((X385 @ X386))) | ~ ! [X389 : term] : (($true = ((X385 @ X389))) | ~ (((var @ X389)) = $true)) | ~ ! [X390 : term] : ((((X385 @ (lam @ X390))) = $true) | ~ (((X385 @ X390)) = $true))) => ! [X391 : (term > $o),X392 : term,X393 : subst] : (~ ! [X394 : term] : ((((X391 @ (lam @ X394))) = $true) | ~ ! [X395 : term] : (~ (((X391 @ X395)) = $true) | (((X391 @ (sub @ X394 @ (push @ X395 @ id)))) = $true))) | ~ ! [X396 : term] : (~ (((var @ X396)) = $true) | ($true = ((X391 @ (sub @ X396 @ X393))))) | ~ ! [X397 : term,X398 : term] : ((((X391 @ (ap @ X397 @ X398))) = $true) | ~ ($true = ((X391 @ X397))) | ~ (((X391 @ X398)) = $true)) | ($true = ((X391 @ (sub @ X392 @ X393)))))))))) & (! [X399 : (term > $o),X400 : (term > $o),X401 : term,X402 : term,X403 : subst] : (~ ($true = ((X400 @ X402))) | ($true = ((X399 @ (sub @ X402 @ (push @ X401 @ X403)))))) <=> (pushprop_lem1 = $true)) & (hoasinduction_lem1_gthm = $true) & (induction2lem_gthm = $true) & ((! [X404 : (subst > term > subst > $o)] : ~ ! [X405 : (term > $o)] : ~ ! [X406 : term] : (((X404 @ id @ X406 @ id)) = ((X405 @ X406))) => (! [X407 : (term > $o),X408 : term] : (~ ! [X409 : term] : ((((X407 @ (lam @ X409))) = $true) | ~ ! [X410 : term] : (($true = ((X407 @ (sub @ X409 @ (push @ X410 @ id))))) | ~ (((X407 @ X410)) = $true))) | ($true = ((X407 @ X408))) | ~ ! [X411 : term,X412 : term] : (~ (((X407 @ X412)) = $true) | ~ (((X407 @ X411)) = $true) | (((X407 @ (ap @ X411 @ X412))) = $true)) | ~ ! [X413 : term] : (~ ($true = ((var @ X413))) | (((X407 @ X413)) = $true))) => (! [X414 : term] : (((sub @ X414 @ id)) = X414) => (! [X415 : (subst > term > subst > $o),X416 : term,X417 : (term > $o)] : (($true = ((X417 @ (lam @ X416)))) | ~ ! [X418 : term] : (((X417 @ X418)) = ((X415 @ id @ X418 @ id))) | ~ ! [X419 : term] : (~ ($true = ((X417 @ X419))) | (((X417 @ (sub @ X416 @ (push @ X419 @ id)))) = $true)) | ~ ! [X420 : (subst > term > term)] : (~ ! [X421 : term] : ((((X415 @ id @ (X420 @ id @ X421) @ id)) = $true) | ~ ($true = ((X415 @ id @ X421 @ id)))) | (((X415 @ id @ (lam @ (X420 @ sh @ one)) @ id)) = $true) | ~ ! [X422 : subst,X423 : term,X424 : subst] : (((X420 @ (comp @ X422 @ X424) @ (sub @ X423 @ X424))) = ((sub @ (X420 @ X422 @ X423) @ X424))))) => ! [X425 : (subst > term > subst > $o),X426 : term] : (~ ! [X427 : term,X428 : term] : (~ (((X425 @ id @ X428 @ id)) = $true) | (((X425 @ id @ (ap @ (sub @ X428 @ id) @ X427) @ id)) = $true) | ~ (((X425 @ id @ X427 @ id)) = $true)) | (((X425 @ id @ X426 @ id)) = $true) | ~ ! [X429 : term,X430 : subst,X431 : subst,X432 : subst] : (($true = ((X425 @ X431 @ X429 @ (comp @ X432 @ X430)))) | ~ (((X425 @ (comp @ X431 @ X432) @ (sub @ X429 @ X432) @ X430)) = $true)) | ~ ! [X433 : term] : (~ (((var @ (sub @ X433 @ id))) = $true) | ($true = ((X425 @ id @ X433 @ id)))) | ~ ! [X434 : term,X435 : subst,X436 : subst,X437 : subst] : (~ (((X425 @ X435 @ X434 @ (comp @ X437 @ X436))) = $true) | (((X425 @ (comp @ X435 @ X437) @ (sub @ X434 @ X437) @ X436)) = $true)) | ~ ! [X438 : (subst > term > term)] : (($true = ((X425 @ id @ (lam @ (X438 @ sh @ one)) @ id))) | ~ ! [X439 : subst,X440 : term,X441 : subst] : (((X438 @ (comp @ X441 @ X439) @ (sub @ X440 @ X439))) = ((sub @ (X438 @ X441 @ X440) @ X439))) | ~ ! [X442 : term] : (($true = ((X425 @ id @ (X438 @ id @ X442) @ id))) | ~ (((X425 @ id @ X442 @ id)) = $true)))))))) <=> (hoasinduction_lthm_3 = $true)) & ((hoaslaminj_lthm = $true) <=> (! [X443 : term,X444 : subst] : (((sub @ one @ (push @ X443 @ X444))) = X443) => (! [X445 : subst,X446 : term] : (((comp @ sh @ (push @ X446 @ X445))) = X445) => (! [X447 : term,X448 : term] : ((X447 = X448) | (((lam @ X448)) != ((lam @ X447)))) => ! [X449 : term,X450 : (subst > term > term),X451 : (subst > term > term),X452 : subst] : (~ ! [X453 : term,X454 : subst,X455 : subst] : (((sub @ (X451 @ X454 @ X453) @ X455)) = ((X451 @ (comp @ X454 @ X455) @ (sub @ X453 @ X455)))) | ~ ! [X456 : term,X457 : subst,X458 : subst] : (((X450 @ (comp @ X457 @ X458) @ (sub @ X456 @ X458))) = ((sub @ (X450 @ X457 @ X456) @ X458))) | (((lam @ (X451 @ sh @ one))) != ((lam @ (X450 @ sh @ one)))) | (((X451 @ X452 @ X449)) = ((X450 @ X452 @ X449)))))))) & (! [X459 : subst] : (((push @ (sub @ one @ X459) @ (comp @ sh @ X459))) = X459) <=> (axscons = $true)) & (! [X460 : subst] : (((comp @ id @ X460)) = X460) <=> (axidl = $true)) & (hoasinduction_lem1v2 = $true) & ((hoasinduction_lem0_lthm = $true) <=> ! [X461 : (subst > term > subst > $o)] : ~ ! [X462 : (term > $o)] : ~ ! [X463 : term] : (((X462 @ X463)) = ((X461 @ id @ X463 @ id)))) & (hoasinduction_gthm = $true) & (pushprop_lem1v2_gthm = $true) & ((hoasapinj2 = $true) <=> ! [X464 : term,X465 : term,X466 : term,X467 : term] : ((X465 = X466) | (((ap @ (sub @ X464 @ id) @ X466)) != ((ap @ (sub @ X467 @ id) @ X465))))) & (hoasapinj2_gthm = $true) & (pushprop_lem2v2_gthm = $true) & (axassoc = $true) & ((hoasinduction_lem2v2 = $true) <=> ! [X468 : term,X469 : term,X470 : (subst > term > subst > $o),X471 : (term > $o)] : (~ ! [X472 : term,X473 : term] : (~ ($true = ((X470 @ id @ X472 @ id))) | (((X470 @ id @ (ap @ (sub @ X472 @ id) @ X473) @ id)) = $true) | ~ ($true = ((X470 @ id @ X473 @ id)))) | ~ ! [X474 : term] : (((X471 @ X474)) = ((X470 @ id @ X474 @ id))) | ($true = ((X471 @ (ap @ X469 @ X468)))) | ~ ! [X475 : subst,X476 : subst,X477 : subst,X478 : term] : ((((X470 @ X477 @ X478 @ (comp @ X475 @ X476))) = $true) | ~ (((X470 @ (comp @ X477 @ X475) @ (sub @ X478 @ X475) @ X476)) = $true)) | ~ ! [X479 : term,X480 : subst,X481 : subst,X482 : subst] : (~ (((X470 @ X482 @ X479 @ (comp @ X480 @ X481))) = $true) | (((X470 @ (comp @ X482 @ X480) @ (sub @ X479 @ X480) @ X481)) = $true)) | ~ (((X471 @ X468)) = $true) | ~ (((X471 @ X469)) = $true))) & (axclos = $true) & (! [X483 : term] : (((sub @ X483 @ id)) = X483) <=> (termmset = $true)) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ~ (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) = $true) & ~ (lamnotap = $true) & (pushprop_lem3v2_lthm = $true) & ((! [X484 : term] : (((sub @ X484 @ id)) = X484) => ! [X485 : term] : (((sub @ X485 @ id)) = X485)) <=> (termmset_lthm = $true)) & (((d2subst @ d_id)) = id) & ((hoasinduction_lem3aaa = $true) <=> ! [X486 : term,X487 : (subst > term > subst > $o)] : (~ ! [X488 : term] : ((((X487 @ id @ (sub @ X486 @ (push @ X488 @ id)) @ id)) = $true) | ~ (((X487 @ id @ X488 @ id)) = $true)) | ~ ! [X489 : (subst > term > term),X490 : term] : (~ ! [X491 : subst,X492 : subst,X493 : term] : (((sub @ (X489 @ X492 @ X493) @ X491)) = ((sub @ (sub @ X490 @ (push @ X493 @ X492)) @ X491))) | ~ ! [X494 : term] : (~ ($true = ((X487 @ id @ X494 @ id))) | (((X487 @ id @ (X489 @ id @ X494) @ id)) = $true)) | ~ ! [X495 : subst,X496 : term,X497 : subst] : (((sub @ X490 @ (push @ (sub @ X496 @ X497) @ (comp @ X495 @ X497)))) = ((X489 @ (comp @ X495 @ X497) @ (sub @ X496 @ X497)))) | (((X487 @ id @ (lam @ (X489 @ sh @ one)) @ id)) = $true)) | ($true = ((X487 @ id @ (lam @ (sub @ X486 @ (push @ one @ sh))) @ id))))) & (pushprop_lthm_orig = $true) & (! [X498 : term,X499 : subst] : (((comp @ sh @ (push @ X498 @ X499))) = X499) <=> (axshiftcons = $true)) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoasinduction_lem3_lthm = $true) <=> (! [X500 : term] : (((sub @ X500 @ id)) = X500) => (! [X501 : term,X502 : (subst > term > subst > $o)] : (~ ! [X503 : (subst > term > term)] : (~ ! [X504 : subst,X505 : subst,X506 : term] : (((X503 @ (comp @ X504 @ X505) @ (sub @ X506 @ X505))) = ((sub @ (X503 @ X504 @ X506) @ X505))) | ~ ! [X507 : term] : (~ ($true = ((X502 @ id @ X507 @ id))) | ($true = ((X502 @ id @ (X503 @ id @ X507) @ id)))) | ($true = ((X502 @ id @ (lam @ (X503 @ sh @ one)) @ id)))) | ~ ! [X508 : term] : (~ ($true = ((X502 @ id @ X508 @ id))) | ($true = ((X502 @ id @ (sub @ X501 @ (push @ X508 @ id)) @ id)))) | ($true = ((X502 @ id @ (lam @ (sub @ X501 @ (push @ one @ sh))) @ id)))) => ! [X509 : term,X510 : (subst > term > subst > $o)] : (~ ! [X511 : term,X512 : subst,X513 : subst,X514 : subst] : ((((X510 @ X514 @ X511 @ (comp @ X513 @ X512))) = $true) | ~ ($true = ((X510 @ (comp @ X514 @ X513) @ (sub @ X511 @ X513) @ X512)))) | ~ ! [X515 : (subst > term > term)] : (~ ! [X516 : term] : (~ ($true = ((X510 @ id @ X516 @ id))) | ($true = ((X510 @ id @ (X515 @ id @ X516) @ id)))) | ~ ! [X517 : term,X518 : subst,X519 : subst] : (((X515 @ (comp @ X519 @ X518) @ (sub @ X517 @ X518))) = ((sub @ (X515 @ X519 @ X517) @ X518))) | ($true = ((X510 @ id @ (lam @ (X515 @ sh @ one)) @ id)))) | (((X510 @ id @ (lam @ X509) @ id)) = $true) | ~ ! [X520 : term] : (~ ($true = ((X510 @ id @ X520 @ id))) | (((X510 @ id @ (sub @ X509 @ (push @ X520 @ id)) @ id)) = $true)) | ~ ! [X521 : term,X522 : subst,X523 : subst,X524 : subst] : ((((X510 @ (comp @ X524 @ X523) @ (sub @ X521 @ X523) @ X522)) = $true) | ~ ($true = ((X510 @ X524 @ X521 @ (comp @ X523 @ X522))))))))) & (hoasapnotvar = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (hoaslamnotap_lthm = $true) & (hoasinduction_lem1 = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (pushprop_lem2v2_lthm = $true) & (! [X525 : term,X526 : (term > $o)] : (~ ! [X527 : term,X528 : term] : ((((X526 @ (ap @ X527 @ X528))) = $true) | ~ (((X526 @ X528)) = $true) | ~ (((X526 @ X527)) = $true)) | (((X526 @ X525)) = $true) | ~ ! [X529 : term] : ((((X526 @ (lam @ X529))) = $true) | ~ (((X526 @ X529)) = $true)) | ~ ! [X530 : term] : (~ ($true = ((var @ X530))) | (((X526 @ X530)) = $true))) <=> (induction = $true)) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ((! [X531 : subst] : (((comp @ X531 @ id)) = X531) & ! [X532 : subst] : (((comp @ id @ X532)) = X532)) <=> (substmonoid = $true)) & ~ (ulamvar1 = $true) & (ulamvarsh = $true) & (axmap = $true) & ((! [X533 : term] : (((sub @ X533 @ id)) = X533) => (! [X534 : term,X535 : term,X536 : term,X537 : term] : ((X534 = X535) | (((ap @ X534 @ X537)) != ((ap @ X535 @ X536)))) => ! [X538 : term,X539 : term,X540 : term,X541 : term] : ((((ap @ (sub @ X538 @ id) @ X541)) != ((ap @ (sub @ X540 @ id) @ X539))) | (X538 = X540)))) <=> (hoasapinj1_lthm = $true)) & (hoaslaminj_gthm = $true) & (! [X542 : term,X543 : (subst > term > subst > $o)] : (~ ! [X544 : term] : (($true = ((X543 @ id @ (sub @ X542 @ (push @ X544 @ id)) @ id))) | ~ ($true = ((X543 @ id @ X544 @ id)))) | ~ ! [X545 : (subst > term > term)] : (($true = ((X543 @ id @ (lam @ (X545 @ sh @ one)) @ id))) | ~ ! [X546 : term,X547 : subst,X548 : subst] : (((X545 @ (comp @ X548 @ X547) @ (sub @ X546 @ X547))) = ((sub @ (X545 @ X548 @ X546) @ X547))) | ~ ! [X549 : term] : (~ (((X543 @ id @ X549 @ id)) = $true) | (((X543 @ id @ (X545 @ id @ X549) @ id)) = $true))) | (((X543 @ id @ (lam @ X542) @ id)) = $true)) <=> (hoasinduction_lem3a = $true)) & (pushprop = $true)),
  inference(fool_elimination,[],[f4])).
thf(f6,plain,(
  ~ (hoasinduction_lem2 = ! [X0 : (subst > term > subst > $o)] : (! [X1 : term,X2 : subst,X3 : subst,X4 : subst] : ((X0 @ X4 @ X1 @ (comp @ X3 @ X2)) => (X0 @ (comp @ X4 @ X3) @ (sub @ X1 @ X3) @ X2)) => (! [X5 : subst,X6 : subst,X7 : subst,X8 : term] : ((X0 @ (comp @ X6 @ X7) @ (sub @ X8 @ X7) @ X5) => (X0 @ X6 @ X8 @ (comp @ X7 @ X5))) => (! [X9 : term,X10 : term] : ((X0 @ id @ X9 @ id) => ((X0 @ id @ X10 @ id) => (X0 @ id @ (hoasap @ id @ X9 @ id @ X10) @ id))) => ! [X11 : term,X12 : term] : ((X0 @ id @ X12 @ id) => ((X0 @ id @ X11 @ id) => (X0 @ id @ (ap @ X12 @ X11) @ id)))))))),
  inference(rectify,[],[f3])).
thf(f7,plain,(
  ~ (hoasinduction_lem2 = ((!! @ (subst > term > subst > $o) @ (^[Y0 : subst > term > subst > $o]: ((!! @ subst @ (^[Y1 : subst]: (!! @ subst @ (^[Y2 : subst]: (!! @ subst @ (^[Y3 : subst]: (!! @ term @ (^[Y4 : term]: ((Y0 @ Y1 @ Y4 @ (comp @ Y2 @ Y3)) => (Y0 @ (comp @ Y1 @ Y2) @ (sub @ Y4 @ Y2) @ Y3)))))))))) => ((!! @ term @ (^[Y1 : term]: (!! @ subst @ (^[Y2 : subst]: (!! @ subst @ (^[Y3 : subst]: (!! @ subst @ (^[Y4 : subst]: ((Y0 @ (comp @ Y3 @ Y2) @ (sub @ Y1 @ Y2) @ Y4) => (Y0 @ Y3 @ Y1 @ (comp @ Y2 @ Y4))))))))))) => ((!! @ term @ (^[Y1 : term]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) => ((Y0 @ id @ Y1 @ id) => (Y0 @ id @ (hoasap @ id @ Y2 @ id @ Y1) @ id))))))) => (!! @ term @ (^[Y1 : term]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y1 @ id) => ((Y0 @ id @ Y2 @ id) => (Y0 @ id @ (ap @ Y1 @ Y2) @ id))))))))))))))),
  inference(fool_elimination,[],[f6])).
thf(f8,plain,(
  (! [X1 : term,X0 : term,X3 : term,X2 : term] : ((X0 = X3) | (((ap @ X3 @ X2)) != ((ap @ X0 @ X1)))) <=> (apinj1 = $true)) & (hoasinduction_lem2_lthm = $true) & ((hoasinduction_lem3v2a = $true) <=> ! [X4 : (term > $o),X5 : term,X6 : (subst > term > subst > $o)] : (~ ! [X7 : (subst > term > term)] : (($true = ((X6 @ id @ (lam @ (X7 @ sh @ one)) @ id))) | ~ ! [X8 : term] : (~ ($true = ((X6 @ id @ X8 @ id))) | (((X6 @ id @ (X7 @ id @ X8) @ id)) = $true)) | ~ ! [X11 : subst,X10 : term,X9 : subst] : (((sub @ (X7 @ X11 @ X10) @ X9)) = ((X7 @ (comp @ X11 @ X9) @ (sub @ X10 @ X9))))) | ($true = ((X4 @ (lam @ X5)))) | ~ ! [X12 : term] : (((X6 @ id @ X12 @ id)) = ((X4 @ X12))) | ~ ! [X13 : term] : (($true = ((X4 @ (sub @ X5 @ (push @ X13 @ id))))) | ~ (((X4 @ X13)) = $true)))) & ((axvarid = $true) <=> ! [X14 : term] : (((sub @ X14 @ id)) = X14)) & (hoasapnotvar_gthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((substmonoid_lthm = $true) <=> (! [X15 : subst] : (((comp @ id @ X15)) = X15) => (! [X16 : subst] : (((comp @ X16 @ id)) = X16) => (! [X17 : subst] : (((comp @ X17 @ id)) = X17) & ! [X18 : subst] : (((comp @ id @ X18)) = X18))))) & ((hoasapinj1 = $true) <=> ! [X21 : term,X19 : term,X20 : term,X22 : term] : ((X21 = X22) | (((ap @ (sub @ X22 @ id) @ X19)) != ((ap @ (sub @ X21 @ id) @ X20))))) & (substmonoid_gthm = $true) & (! [X23 : term,X24 : (subst > term > subst > $o)] : (($true = ((X24 @ id @ (lam @ X23) @ id))) | ~ ! [X25 : subst,X26 : term,X27 : subst,X28 : subst] : (($true = ((X24 @ X25 @ X26 @ (comp @ X28 @ X27)))) | ~ (((X24 @ (comp @ X25 @ X28) @ (sub @ X26 @ X28) @ X27)) = $true)) | ~ ! [X29 : (subst > term > term)] : (~ ! [X32 : subst,X31 : term,X30 : subst] : (((sub @ (X29 @ X30 @ X31) @ X32)) = ((X29 @ (comp @ X30 @ X32) @ (sub @ X31 @ X32)))) | (((X24 @ id @ (lam @ (X29 @ sh @ one)) @ id)) = $true) | ~ ! [X33 : term] : (~ (((X24 @ id @ X33 @ id)) = $true) | (((X24 @ id @ (X29 @ id @ X33) @ id)) = $true))) | ~ ! [X34 : term,X35 : subst,X36 : subst,X37 : subst] : ((((X24 @ (comp @ X37 @ X35) @ (sub @ X34 @ X35) @ X36)) = $true) | ~ ($true = ((X24 @ X37 @ X34 @ (comp @ X35 @ X36))))) | ~ ! [X38 : term] : (~ (((X24 @ id @ X38 @ id)) = $true) | ($true = ((X24 @ id @ (sub @ X23 @ (push @ X38 @ id)) @ id))))) <=> (hoasinduction_lem3 = $true)) & (pushprop_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)))))))))))) & ! [X40 : d_subst,X39 : d_subst] : ((((d2subst @ X40)) = ((d2subst @ X39))) => (X39 = X40)) & ((hoaslaminj = $true) <=> ! [X42 : (subst > term > term),X41 : (subst > term > term),X43 : subst,X44 : term] : (~ ! [X49 : subst,X50 : subst,X48 : term] : (((sub @ (X41 @ X49 @ X48) @ X50)) = ((X41 @ (comp @ X49 @ X50) @ (sub @ X48 @ X50)))) | ~ ! [X46 : subst,X45 : term,X47 : subst] : (((sub @ (X42 @ X46 @ X45) @ X47)) = ((X42 @ (comp @ X46 @ X47) @ (sub @ X45 @ X47)))) | (((lam @ (X41 @ sh @ one))) != ((lam @ (X42 @ sh @ one)))) | (((X41 @ X43 @ X44)) = ((X42 @ X43 @ X44))))) & ! [X51 : term] : ? [X52 : d_term] : (((d2term @ X52)) = X51) & (hoaslamnotvar_gthm = $true) & (axvarshift = $true) & (! [X53 : term,X54 : term,X55 : (subst > term > subst > $o)] : (~ ($true = ((X55 @ id @ X53 @ id))) | ~ ! [X56 : subst,X57 : term,X58 : subst,X59 : subst] : (~ ($true = ((X55 @ X56 @ X57 @ (comp @ X58 @ X59)))) | (((X55 @ (comp @ X56 @ X58) @ (sub @ X57 @ X58) @ X59)) = $true)) | ~ ! [X60 : term,X61 : term] : (($true = ((X55 @ id @ (ap @ (sub @ X61 @ id) @ X60) @ id))) | ~ ($true = ((X55 @ id @ X60 @ id))) | ~ ($true = ((X55 @ id @ X61 @ id)))) | ($true = ((X55 @ id @ (ap @ X53 @ X54) @ id))) | ~ ! [X62 : term,X63 : subst,X64 : subst,X65 : subst] : (~ (((X55 @ (comp @ X64 @ X63) @ (sub @ X62 @ X63) @ X65)) = $true) | ($true = ((X55 @ X64 @ X62 @ (comp @ X63 @ X65))))) | ~ ($true = ((X55 @ id @ X54 @ id)))) <=> (hoasinduction_lem2 = $true)) & (sh = ((d2subst @ d_id))) & (pushprop_lem0_gthm = $true) & ((hoasinduction_no_psi_cond = $true) <=> ! [X66 : term,X67 : (subst > term > subst > $o)] : (~ ! [X68 : (subst > term > term)] : (~ ! [X69 : subst,X71 : subst,X70 : term] : (((X68 @ (comp @ X69 @ X71) @ (sub @ X70 @ X71))) = ((sub @ (X68 @ X69 @ X70) @ X71))) | ($true = ((X67 @ id @ (lam @ (X68 @ sh @ one)) @ id))) | ~ ! [X72 : term] : (~ (((X67 @ id @ X72 @ id)) = $true) | (((X67 @ id @ (X68 @ id @ X72) @ id)) = $true))) | ~ ! [X73 : term,X74 : term] : ((((X67 @ id @ (ap @ (sub @ X73 @ id) @ X74) @ id)) = $true) | ~ (((X67 @ id @ X74 @ id)) = $true) | ~ (((X67 @ id @ X73 @ id)) = $true)) | ($true = ((X67 @ id @ X66 @ id))))) & (hoasinduction_lem1v2_gthm = $true) & (hoasapnotvar_lthm = $true) & ((! [X75 : term,X76 : (term > $o)] : ((((X76 @ X75)) = $true) | ~ ! [X77 : term] : ((((X76 @ (lam @ X77))) = $true) | ~ ! [X78 : term] : (~ (((X76 @ X78)) = $true) | ($true = ((X76 @ (sub @ X77 @ (push @ X78 @ id))))))) | ~ ! [X79 : term,X80 : term] : (~ ($true = ((X76 @ X79))) | (((X76 @ (ap @ X79 @ X80))) = $true) | ~ (((X76 @ X80)) = $true)) | ~ ! [X81 : term] : ((((X76 @ X81)) = $true) | ~ ($true = ((var @ X81))))) => (! [X82 : term,X83 : term,X84 : (subst > term > subst > $o)] : (~ (((X84 @ id @ X82 @ id)) = $true) | ~ (((X84 @ id @ X83 @ id)) = $true) | (((X84 @ id @ (ap @ X83 @ X82) @ id)) = $true) | ~ ! [X85 : subst,X86 : subst,X87 : subst,X88 : term] : (~ ($true = ((X84 @ (comp @ X85 @ X86) @ (sub @ X88 @ X86) @ X87))) | (((X84 @ X85 @ X88 @ (comp @ X86 @ X87))) = $true)) | ~ ! [X89 : term,X90 : term] : ((((X84 @ id @ (ap @ (sub @ X89 @ id) @ X90) @ id)) = $true) | ~ ($true = ((X84 @ id @ X89 @ id))) | ~ ($true = ((X84 @ id @ X90 @ id)))) | ~ ! [X91 : subst,X92 : term,X93 : subst,X94 : subst] : (~ (((X84 @ X93 @ X92 @ (comp @ X91 @ X94))) = $true) | (((X84 @ (comp @ X93 @ X91) @ (sub @ X92 @ X91) @ X94)) = $true))) => (! [X95 : term,X96 : (subst > term > subst > $o)] : (~ ! [X97 : term,X98 : subst,X99 : subst,X100 : subst] : (~ (((X96 @ (comp @ X98 @ X99) @ (sub @ X97 @ X99) @ X100)) = $true) | (((X96 @ X98 @ X97 @ (comp @ X99 @ X100))) = $true)) | ~ ! [X101 : term,X102 : subst,X103 : subst,X104 : subst] : ((((X96 @ (comp @ X102 @ X103) @ (sub @ X101 @ X103) @ X104)) = $true) | ~ ($true = ((X96 @ X102 @ X101 @ (comp @ X103 @ X104))))) | (((X96 @ id @ (lam @ X95) @ id)) = $true) | ~ ! [X105 : term] : ((((X96 @ id @ (sub @ X95 @ (push @ X105 @ id)) @ id)) = $true) | ~ ($true = ((X96 @ id @ X105 @ id)))) | ~ ! [X106 : (subst > term > term)] : (~ ! [X107 : term] : (~ ($true = ((X96 @ id @ X107 @ id))) | (((X96 @ id @ (X106 @ id @ X107) @ id)) = $true)) | (((X96 @ id @ (lam @ (X106 @ sh @ one)) @ id)) = $true) | ~ ! [X109 : subst,X108 : term,X110 : subst] : (((X106 @ (comp @ X110 @ X109) @ (sub @ X108 @ X109))) = ((sub @ (X106 @ X110 @ X108) @ X109))))) => ! [X111 : (subst > term > subst > $o),X112 : term] : (~ ! [X113 : term,X114 : term] : (~ (((X111 @ id @ X114 @ id)) = $true) | ($true = ((X111 @ id @ (ap @ (sub @ X113 @ id) @ X114) @ id))) | ~ ($true = ((X111 @ id @ X113 @ id)))) | ~ ! [X115 : (subst > term > term)] : (($true = ((X111 @ id @ (lam @ (X115 @ sh @ one)) @ id))) | ~ ! [X117 : subst,X118 : subst,X116 : term] : (((sub @ (X115 @ X118 @ X116) @ X117)) = ((X115 @ (comp @ X118 @ X117) @ (sub @ X116 @ X117)))) | ~ ! [X119 : term] : ((((X111 @ id @ (X115 @ id @ X119) @ id)) = $true) | ~ (((X111 @ id @ X119 @ id)) = $true))) | (((X111 @ id @ X112 @ id)) = $true) | ~ ! [X120 : subst,X121 : subst,X122 : subst,X123 : term] : (~ ($true = ((X111 @ X121 @ X123 @ (comp @ X120 @ X122)))) | ($true = ((X111 @ (comp @ X121 @ X120) @ (sub @ X123 @ X120) @ X122)))) | ~ ! [X124 : term] : (~ (((var @ (sub @ X124 @ id))) = $true) | (((X111 @ id @ X124 @ id)) = $true)) | ~ ! [X125 : subst,X126 : subst,X127 : subst,X128 : term] : (~ (((X111 @ (comp @ X125 @ X127) @ (sub @ X128 @ X127) @ X126)) = $true) | (((X111 @ X125 @ X128 @ (comp @ X127 @ X126))) = $true)))))) <=> (hoasinduction_lthm = $true)) & (induction2_gthm = $true) & ((hoasinduction_lem3aa = $true) <=> ! [X129 : term,X130 : (subst > term > subst > $o)] : (($true = ((X130 @ id @ (lam @ (sub @ X129 @ (push @ one @ sh))) @ id))) | ~ ! [X131 : term] : ((((X130 @ id @ (sub @ X129 @ (push @ X131 @ id)) @ id)) = $true) | ~ (((X130 @ id @ X131 @ id)) = $true)) | ~ ! [X132 : (subst > term > term)] : (~ ! [X133 : term] : (~ ($true = ((X130 @ id @ X133 @ id))) | ($true = ((X130 @ id @ (X132 @ id @ X133) @ id)))) | ($true = ((X130 @ id @ (lam @ (X132 @ sh @ one)) @ id))) | ~ ! [X136 : subst,X135 : term,X134 : subst] : (((sub @ (X132 @ X136 @ X135) @ X134)) = ((X132 @ (comp @ X136 @ X134) @ (sub @ X135 @ X134))))))) & (! [X139 : (subst > term > term),X137 : term,X138 : term] : (~ ! [X140 : subst,X142 : subst,X141 : term] : (((sub @ (X139 @ X142 @ X141) @ X140)) = ((X139 @ (comp @ X142 @ X140) @ (sub @ X141 @ X140)))) | (((ap @ (sub @ X138 @ id) @ X137)) != ((lam @ (X139 @ sh @ one))))) <=> (hoaslamnotap = $true)) & (! [X145 : subst,X143 : term,X144 : (term > $o)] : ~ ! [X146 : (term > $o)] : ~ ! [X147 : term] : (((X144 @ (sub @ X147 @ (push @ X143 @ X145)))) = ((X146 @ X147))) <=> (pushprop_lem0_lthm = $true)) & ((hoasinduction_lem3v2 = $true) <=> ! [X148 : (subst > term > subst > $o),X149 : term,X150 : (term > $o)] : ((((X150 @ (lam @ X149))) = $true) | ~ ! [X151 : subst,X152 : term,X153 : subst,X154 : subst] : ((((X148 @ X151 @ X152 @ (comp @ X154 @ X153))) = $true) | ~ (((X148 @ (comp @ X151 @ X154) @ (sub @ X152 @ X154) @ X153)) = $true)) | ~ ! [X155 : (subst > term > term)] : ((((X148 @ id @ (lam @ (X155 @ sh @ one)) @ id)) = $true) | ~ ! [X156 : term] : (~ ($true = ((X148 @ id @ X156 @ id))) | ($true = ((X148 @ id @ (X155 @ id @ X156) @ id)))) | ~ ! [X159 : term,X158 : subst,X157 : subst] : (((X155 @ (comp @ X157 @ X158) @ (sub @ X159 @ X158))) = ((sub @ (X155 @ X157 @ X159) @ X158)))) | ~ ! [X160 : subst,X161 : subst,X162 : subst,X163 : term] : (~ ($true = ((X148 @ X161 @ X163 @ (comp @ X160 @ X162)))) | (((X148 @ (comp @ X161 @ X160) @ (sub @ X163 @ X160) @ X162)) = $true)) | ~ ! [X164 : term] : (((X148 @ id @ X164 @ id)) = ((X150 @ X164))) | ~ ! [X165 : term] : (~ (((X150 @ X165)) = $true) | (((X150 @ (sub @ X149 @ (push @ X165 @ id)))) = $true)))) & (! [X167 : (term > $o),X168 : term,X166 : subst] : ~ ! [X169 : (term > $o)] : ~ ! [X170 : term] : (((X169 @ X170)) = ((X167 @ (sub @ X170 @ (push @ X168 @ X166))))) <=> (pushprop_lem0 = $true)) & (hoasapinj1_gthm = $true) & ((induction2_lthm = $true) <=> (! [X171 : term] : (((sub @ X171 @ id)) = X171) => (! [X172 : subst,X173 : term,X174 : (term > $o)] : ((((X174 @ (sub @ X173 @ X172))) = $true) | ~ ! [X175 : term,X176 : term] : (~ ($true = ((X174 @ X175))) | ~ (((X174 @ X176)) = $true) | ($true = ((X174 @ (ap @ X175 @ X176))))) | ~ ! [X177 : term] : (($true = ((X174 @ (lam @ X177)))) | ~ ! [X178 : term] : (($true = ((X174 @ (sub @ X177 @ (push @ X178 @ id))))) | ~ (((X174 @ X178)) = $true))) | ~ ! [X179 : term] : (~ (((var @ X179)) = $true) | ($true = ((X174 @ (sub @ X179 @ X172)))))) => ! [X180 : term,X181 : (term > $o)] : (~ ! [X182 : term,X183 : term] : (~ (((X181 @ X182)) = $true) | ~ (((X181 @ X183)) = $true) | (((X181 @ (ap @ X183 @ X182))) = $true)) | ~ ! [X184 : term] : (~ (((var @ X184)) = $true) | ($true = ((X181 @ X184)))) | ~ ! [X185 : term] : (~ ! [X186 : term] : (~ (((X181 @ X186)) = $true) | ($true = ((X181 @ (sub @ X185 @ (push @ X186 @ id)))))) | ($true = ((X181 @ (lam @ X185))))) | ($true = ((X181 @ X180))))))) & (! [X187 : term,X188 : (subst > term > subst > $o)] : (~ ! [X189 : term] : (($true = ((X188 @ id @ X189 @ id))) | ~ (((var @ (sub @ X189 @ id))) = $true)) | ~ ! [X190 : term,X191 : term] : ((((X188 @ id @ (ap @ (sub @ X191 @ id) @ X190) @ id)) = $true) | ~ ($true = ((X188 @ id @ X191 @ id))) | ~ (((X188 @ id @ X190 @ id)) = $true)) | (((X188 @ id @ X187 @ id)) = $true) | ~ ! [X192 : (subst > term > term)] : (~ ! [X193 : term] : (~ (((X188 @ id @ X193 @ id)) = $true) | ($true = ((X188 @ id @ (X192 @ id @ X193) @ id)))) | (((X188 @ id @ (lam @ (X192 @ sh @ one)) @ id)) = $true) | ~ ! [X196 : subst,X194 : term,X195 : subst] : (((X192 @ (comp @ X196 @ X195) @ (sub @ X194 @ X195))) = ((sub @ (X192 @ X196 @ X194) @ X195)))) | ~ ! [X197 : subst,X198 : subst,X199 : subst,X200 : term] : (~ (((X188 @ X198 @ X200 @ (comp @ X197 @ X199))) = $true) | ($true = ((X188 @ (comp @ X198 @ X197) @ (sub @ X200 @ X197) @ X199)))) | ~ ! [X201 : subst,X202 : subst,X203 : term,X204 : subst] : (~ (((X188 @ (comp @ X202 @ X201) @ (sub @ X203 @ X201) @ X204)) = $true) | ($true = ((X188 @ X202 @ X203 @ (comp @ X201 @ X204)))))) <=> (hoasinduction = $true)) & ((axvarcons = $true) <=> ! [X206 : term,X205 : subst] : (((sub @ one @ (push @ X206 @ X205))) = X206)) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X207 : term] : ~ ! [X208 : (subst > term > term)] : ~ ! [X210 : term,X209 : subst] : (((sub @ X207 @ (push @ X210 @ X209))) = ((X208 @ X209 @ X210)))) & ((hoasinduction_lem0 = $true) <=> ! [X211 : (subst > term > subst > $o)] : ~ ! [X212 : (term > $o)] : ~ ! [X213 : term] : (((X212 @ X213)) = ((X211 @ id @ X213 @ id)))) & ((hoasinduction_lem3v2_f = $true) <=> ! [X214 : term] : ~ ! [X215 : (subst > term > term)] : ~ ! [X217 : subst,X216 : term] : (((X215 @ X217 @ X216)) = ((sub @ X214 @ (push @ X216 @ X217))))) & ((! [X218 : subst,X219 : term] : (((sub @ one @ (push @ X219 @ X218))) = X219) => (! [X220 : subst,X221 : term] : (((comp @ sh @ (push @ X221 @ X220))) = X220) => ! [X222 : (term > $o),X223 : (term > $o),X224 : term,X225 : term,X226 : subst] : (~ ($true = ((X222 @ X224))) | ($true = ((X223 @ (sub @ X224 @ (push @ X225 @ X226)))))))) <=> (pushprop_lem1_lthm = $true)) & ((hoasapinj2_lthm = $true) <=> (! [X227 : term,X228 : term,X229 : term,X230 : term] : ((((ap @ X227 @ X228)) != ((ap @ X230 @ X229))) | (X228 = X229)) => ! [X234 : term,X231 : term,X233 : term,X232 : term] : ((((ap @ (sub @ X233 @ id) @ X234)) != ((ap @ (sub @ X232 @ id) @ X231))) | (X231 = X234)))) & (ulamvarind = $true) & (apnotvar = $true) & ((! [X235 : term] : (((sub @ X235 @ id)) = X235) => ! [X236 : term,X237 : (subst > term > subst > $o),X238 : (term > $o)] : (~ ! [X239 : (subst > term > term)] : (($true = ((X237 @ id @ (lam @ (X239 @ sh @ one)) @ id))) | ~ ! [X240 : term] : ((((X237 @ id @ (X239 @ id @ X240) @ id)) = $true) | ~ ($true = ((X237 @ id @ X240 @ id)))) | ~ ! [X243 : term,X241 : subst,X242 : subst] : (((sub @ (X239 @ X241 @ X243) @ X242)) = ((X239 @ (comp @ X241 @ X242) @ (sub @ X243 @ X242))))) | ~ ! [X244 : term] : (((X238 @ X244)) = ((X237 @ id @ X244 @ id))) | ~ ! [X245 : term,X246 : subst,X247 : subst,X248 : subst] : (($true = ((X237 @ X248 @ X245 @ (comp @ X246 @ X247)))) | ~ (((X237 @ (comp @ X248 @ X246) @ (sub @ X245 @ X246) @ X247)) = $true)) | (((X238 @ (lam @ X236))) = $true) | ~ ! [X249 : term] : (~ (((X238 @ X249)) = $true) | (((X238 @ (sub @ X236 @ (push @ X249 @ id)))) = $true)) | ~ ! [X250 : term,X251 : subst,X252 : subst,X253 : subst] : (~ (((X237 @ X252 @ X250 @ (comp @ X251 @ X253))) = $true) | (((X237 @ (comp @ X252 @ X251) @ (sub @ X250 @ X251) @ X253)) = $true)))) <=> (hoasinduction_lem3v2_lthm = $true)) & ((pushprop_lem1v2_lthm = $true) <=> (! [X255 : term,X254 : subst] : (((sub @ one @ (push @ X255 @ X254))) = X255) => ! [X256 : subst,X257 : (term > $o),X258 : (term > $o),X259 : term] : (~ ! [X260 : term] : (((X257 @ (sub @ X260 @ (push @ X259 @ X256)))) = ((X258 @ X260))) | ~ ($true = ((X257 @ X259))) | ($true = ((X258 @ one)))))) & (hoasinduction_lem3a_gthm = $true) & ((laminj = $true) <=> ! [X262 : term,X261 : term] : ((X261 = X262) | (((lam @ X262)) != ((lam @ X261))))) & (hoaslamnotap_gthm = $true) & (one = ((d2term @ d_one))) & ((apinj2 = $true) <=> ! [X263 : term,X264 : term,X266 : term,X265 : term] : ((((ap @ X265 @ X266)) != ((ap @ X263 @ X264))) | (X264 = X266))) & (lamnotvar = $true) & ((pushprop_lem1v2 = $true) <=> ! [X267 : (term > $o),X268 : subst,X269 : (term > $o),X270 : term] : (~ ! [X271 : term] : (((X267 @ (sub @ X271 @ (push @ X270 @ X268)))) = ((X269 @ X271))) | (((X269 @ one)) = $true) | ~ (((X267 @ X270)) = $true))) & ~ (((var @ (d2term @ d_one))) = $true) & (hoaslamnotvar_lthm = $true) & (pushprop_lem3v2 = $true) & (! [X272 : (term > $o),X273 : subst,X274 : term] : ((((X272 @ (sub @ X274 @ X273))) = $true) | ~ ! [X275 : term] : (~ ! [X276 : term] : (~ (((X272 @ X276)) = $true) | ($true = ((X272 @ (sub @ X275 @ (push @ X276 @ id)))))) | ($true = ((X272 @ (lam @ X275))))) | ~ ! [X277 : term,X278 : term] : (~ (((X272 @ X278)) = $true) | ~ ($true = ((X272 @ X277))) | (((X272 @ (ap @ X277 @ X278))) = $true)) | ~ ! [X279 : term] : ((((X272 @ (sub @ X279 @ X273))) = $true) | ~ ($true = ((var @ X279))))) <=> (induction2lem = $true)) & (hoasinduction_lem2_gthm = $true) & ! [X280 : d_term] : (d_one = X280) & (pushprop_lem2v2 = $true) & (hoasinduction_lem3v2_gthm = $true) & ((! [X281 : term] : ~ ! [X282 : (subst > term > term)] : ~ ! [X284 : term,X283 : subst] : (((X282 @ X283 @ X284)) = ((sub @ X281 @ (push @ X284 @ X283)))) => (! [X285 : term] : (((sub @ X285 @ id)) = X285) => ! [X286 : (subst > term > subst > $o),X287 : (term > $o),X288 : term] : (~ ! [X289 : term] : (($true = ((X287 @ (sub @ X288 @ (push @ X289 @ id))))) | ~ (((X287 @ X289)) = $true)) | (((X287 @ (lam @ X288))) = $true) | ~ ! [X290 : (subst > term > term)] : (~ ! [X291 : term] : (~ (((X286 @ id @ X291 @ id)) = $true) | (((X286 @ id @ (X290 @ id @ X291) @ id)) = $true)) | ~ ! [X293 : subst,X292 : subst,X294 : term] : (((X290 @ (comp @ X293 @ X292) @ (sub @ X294 @ X292))) = ((sub @ (X290 @ X293 @ X294) @ X292))) | ($true = ((X286 @ id @ (lam @ (X290 @ sh @ one)) @ id)))) | ~ ! [X295 : term] : (((X286 @ id @ X295 @ id)) = ((X287 @ X295)))))) <=> (hoasinduction_lem3v2a_lthm = $true)) & ! [X296 : subst] : ? [X297 : d_subst] : (((d2subst @ X297)) = X296) & (hoaslamnotvar = $true) & (hoasinduction_lem3b_gthm = $true) & (termmset_gthm = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (! [X298 : term,X299 : term] : ((((sub @ X298 @ sh)) != ((sub @ X299 @ sh))) | (X298 = X299)) <=> (shinj = $true)) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X300 : term] : ~ ! [X301 : (subst > term > term)] : (((sub @ X300 @ (push @ one @ sh))) != ((X301 @ sh @ one)))) & ! [X302 : d_term,X303 : d_term] : ((((d2term @ X302)) = ((d2term @ X303))) => (X302 = X303)) & (axabs = $true) & (hoasinduction_lem2v2_gthm = $true) & (! [X304 : subst] : (((comp @ X304 @ id)) = X304) <=> (axidr = $true)) & (pushprop_lem1_gthm = $true) & (hoasinduction_lem3_gthm = $true) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X305 : term,X306 : (subst > term > subst > $o)] : (~ ! [X307 : (subst > term > term)] : (($true = ((X306 @ id @ (lam @ (X307 @ sh @ one)) @ id))) | ~ ! [X308 : term] : (~ (((X306 @ id @ X308 @ id)) = $true) | ($true = ((X306 @ id @ (X307 @ id @ X308) @ id)))) | ~ ! [X309 : subst,X311 : subst,X310 : term] : (((X307 @ (comp @ X311 @ X309) @ (sub @ X310 @ X309))) = ((sub @ (X307 @ X311 @ X310) @ X309)))) | ($true = ((X306 @ id @ (lam @ (sub @ X305 @ (push @ one @ sh))) @ id))) | ~ ! [X312 : term] : (($true = ((X306 @ id @ (sub @ X305 @ (push @ X312 @ id)) @ id))) | ~ (((X306 @ id @ X312 @ id)) = $true)))) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (! [X313 : (subst > term > subst > $o)] : ~ ! [X314 : (term > $o)] : ~ ! [X315 : term] : (((X314 @ X315)) = ((X313 @ id @ X315 @ id))) => (! [X316 : (term > $o),X317 : term] : (~ ! [X318 : term,X319 : term] : (~ (((X316 @ X318)) = $true) | ($true = ((X316 @ (ap @ X319 @ X318)))) | ~ (((X316 @ X319)) = $true)) | (((X316 @ X317)) = $true) | ~ ! [X320 : term] : ((((X316 @ (lam @ X320))) = $true) | ~ ! [X321 : term] : (($true = ((X316 @ (sub @ X320 @ (push @ X321 @ id))))) | ~ ($true = ((X316 @ X321))))) | ~ ! [X322 : term] : (~ ($true = ((var @ X322))) | (((X316 @ X322)) = $true))) => (! [X323 : term] : (((sub @ X323 @ id)) = X323) => (! [X324 : (subst > term > subst > $o),X325 : term,X326 : (term > $o)] : (~ ! [X327 : (subst > term > term)] : (($true = ((X324 @ id @ (lam @ (X327 @ sh @ one)) @ id))) | ~ ! [X330 : term,X329 : subst,X328 : subst] : (((sub @ (X327 @ X328 @ X330) @ X329)) = ((X327 @ (comp @ X328 @ X329) @ (sub @ X330 @ X329)))) | ~ ! [X331 : term] : (($true = ((X324 @ id @ (X327 @ id @ X331) @ id))) | ~ ($true = ((X324 @ id @ X331 @ id))))) | ~ ! [X332 : term] : (~ (((X326 @ X332)) = $true) | (((X326 @ (sub @ X325 @ (push @ X332 @ id)))) = $true)) | ~ ! [X333 : term] : (((X326 @ X333)) = ((X324 @ id @ X333 @ id))) | (((X326 @ (lam @ X325))) = $true)) => ! [X334 : term,X335 : (subst > term > subst > $o)] : (~ ! [X336 : (subst > term > term)] : (~ ! [X337 : term] : (~ (((X335 @ id @ X337 @ id)) = $true) | ($true = ((X335 @ id @ (X336 @ id @ X337) @ id)))) | ~ ! [X338 : term,X340 : subst,X339 : subst] : (((sub @ (X336 @ X340 @ X338) @ X339)) = ((X336 @ (comp @ X340 @ X339) @ (sub @ X338 @ X339)))) | (((X335 @ id @ (lam @ (X336 @ sh @ one)) @ id)) = $true)) | ($true = ((X335 @ id @ X334 @ id))) | ~ ! [X341 : term,X342 : term] : (($true = ((X335 @ id @ (ap @ (sub @ X341 @ id) @ X342) @ id))) | ~ (((X335 @ id @ X342 @ id)) = $true) | ~ ($true = ((X335 @ id @ X341 @ id)))))))))) & (! [X343 : term] : ~ ! [X344 : (subst > term > term)] : (((sub @ X343 @ (push @ one @ sh))) != ((X344 @ sh @ one))) <=> (hoasinduction_lem3b = $true)) & (pushprop_lthm = $true) & (axapp = $true) & ((induction2 = $true) <=> ! [X345 : (term > $o),X346 : term] : (~ ! [X347 : term] : ((((X345 @ X347)) = $true) | ~ (((var @ X347)) = $true)) | ($true = ((X345 @ X346))) | ~ ! [X348 : term] : (~ ! [X349 : term] : (~ ($true = ((X345 @ X349))) | ($true = ((X345 @ (sub @ X348 @ (push @ X349 @ id)))))) | ($true = ((X345 @ (lam @ X348))))) | ~ ! [X350 : term,X351 : term] : (~ ($true = ((X345 @ X350))) | (((X345 @ (ap @ X351 @ X350))) = $true) | ~ ($true = ((X345 @ X351)))))) & ((hoasinduction_lem3a_lthm = $true) <=> (! [X352 : term] : (((sub @ X352 @ id)) = X352) => (! [X353 : term,X354 : (subst > term > subst > $o)] : (~ ! [X355 : term] : (($true = ((X354 @ id @ (sub @ X353 @ (push @ X355 @ id)) @ id))) | ~ (((X354 @ id @ X355 @ id)) = $true)) | ~ ! [X356 : (subst > term > term)] : (~ ! [X357 : term] : (~ ($true = ((X354 @ id @ X357 @ id))) | (((X354 @ id @ (X356 @ id @ X357) @ id)) = $true)) | ($true = ((X354 @ id @ (lam @ (X356 @ sh @ one)) @ id))) | ~ ! [X358 : subst,X359 : subst,X360 : term] : (((sub @ (X356 @ X358 @ X360) @ X359)) = ((X356 @ (comp @ X358 @ X359) @ (sub @ X360 @ X359))))) | (((X354 @ id @ (lam @ (sub @ X353 @ (push @ one @ sh))) @ id)) = $true)) => ! [X361 : term,X362 : (subst > term > subst > $o)] : (~ ! [X363 : (subst > term > term)] : (($true = ((X362 @ id @ (lam @ (X363 @ sh @ one)) @ id))) | ~ ! [X365 : subst,X366 : term,X364 : subst] : (((sub @ (X363 @ X365 @ X366) @ X364)) = ((X363 @ (comp @ X365 @ X364) @ (sub @ X366 @ X364)))) | ~ ! [X367 : term] : ((((X362 @ id @ (X363 @ id @ X367) @ id)) = $true) | ~ (((X362 @ id @ X367 @ id)) = $true))) | ($true = ((X362 @ id @ (lam @ X361) @ id))) | ~ ! [X368 : term] : (~ (((X362 @ id @ X368 @ id)) = $true) | ($true = ((X362 @ id @ (sub @ X361 @ (push @ X368 @ id)) @ id)))))))) & (hoasinduction_lem1_lthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & ! [X369 : d_subst] : (d_id = X369) & ((induction2lem_lthm = $true) <=> (! [X370 : subst,X371 : term] : (((sub @ one @ (push @ X371 @ X370))) = X371) => (! [X372 : term,X373 : subst] : (((comp @ sh @ (push @ X372 @ X373))) = X373) => (! [X374 : subst] : (((comp @ X374 @ id)) = X374) => (! [X375 : (term > $o),X376 : term] : (~ ! [X377 : term,X378 : term] : (~ ($true = ((X375 @ X377))) | (((X375 @ (ap @ X378 @ X377))) = $true) | ~ (((X375 @ X378)) = $true)) | (((X375 @ X376)) = $true) | ~ ! [X379 : term] : ((((X375 @ X379)) = $true) | ~ ($true = ((var @ X379)))) | ~ ! [X380 : term] : (($true = ((X375 @ (lam @ X380)))) | ~ ($true = ((X375 @ X380))))) => ! [X381 : (term > $o),X382 : term,X383 : subst] : (~ ! [X384 : term] : (($true = ((X381 @ (lam @ X384)))) | ~ ! [X385 : term] : (~ (((X381 @ X385)) = $true) | (((X381 @ (sub @ X384 @ (push @ X385 @ id)))) = $true))) | ~ ! [X386 : term] : (~ (((var @ X386)) = $true) | (((X381 @ (sub @ X386 @ X383))) = $true)) | ~ ! [X387 : term,X388 : term] : ((((X381 @ (ap @ X387 @ X388))) = $true) | ~ (((X381 @ X387)) = $true) | ~ (((X381 @ X388)) = $true)) | (((X381 @ (sub @ X382 @ X383))) = $true))))))) & (! [X389 : (term > $o),X390 : (term > $o),X391 : term,X392 : term,X393 : subst] : (~ ($true = ((X390 @ X392))) | ($true = ((X389 @ (sub @ X392 @ (push @ X391 @ X393)))))) <=> (pushprop_lem1 = $true)) & (hoasinduction_lem1_gthm = $true) & (induction2lem_gthm = $true) & ((! [X394 : (subst > term > subst > $o)] : ~ ! [X395 : (term > $o)] : ~ ! [X396 : term] : (((X395 @ X396)) = ((X394 @ id @ X396 @ id))) => (! [X397 : (term > $o),X398 : term] : (~ ! [X399 : term] : ((((X397 @ (lam @ X399))) = $true) | ~ ! [X400 : term] : (($true = ((X397 @ (sub @ X399 @ (push @ X400 @ id))))) | ~ (((X397 @ X400)) = $true))) | (((X397 @ X398)) = $true) | ~ ! [X401 : term,X402 : term] : (~ ($true = ((X397 @ X402))) | ~ ($true = ((X397 @ X401))) | ($true = ((X397 @ (ap @ X401 @ X402))))) | ~ ! [X403 : term] : (~ ($true = ((var @ X403))) | ($true = ((X397 @ X403))))) => (! [X404 : term] : (((sub @ X404 @ id)) = X404) => (! [X405 : (subst > term > subst > $o),X406 : term,X407 : (term > $o)] : ((((X407 @ (lam @ X406))) = $true) | ~ ! [X408 : term] : (((X405 @ id @ X408 @ id)) = ((X407 @ X408))) | ~ ! [X409 : term] : (~ (((X407 @ X409)) = $true) | (((X407 @ (sub @ X406 @ (push @ X409 @ id)))) = $true)) | ~ ! [X410 : (subst > term > term)] : (~ ! [X411 : term] : (($true = ((X405 @ id @ (X410 @ id @ X411) @ id))) | ~ ($true = ((X405 @ id @ X411 @ id)))) | (((X405 @ id @ (lam @ (X410 @ sh @ one)) @ id)) = $true) | ~ ! [X413 : term,X414 : subst,X412 : subst] : (((X410 @ (comp @ X412 @ X414) @ (sub @ X413 @ X414))) = ((sub @ (X410 @ X412 @ X413) @ X414))))) => ! [X415 : (subst > term > subst > $o),X416 : term] : (~ ! [X417 : term,X418 : term] : (~ (((X415 @ id @ X418 @ id)) = $true) | ($true = ((X415 @ id @ (ap @ (sub @ X418 @ id) @ X417) @ id))) | ~ ($true = ((X415 @ id @ X417 @ id)))) | ($true = ((X415 @ id @ X416 @ id))) | ~ ! [X419 : term,X420 : subst,X421 : subst,X422 : subst] : (($true = ((X415 @ X421 @ X419 @ (comp @ X422 @ X420)))) | ~ ($true = ((X415 @ (comp @ X421 @ X422) @ (sub @ X419 @ X422) @ X420)))) | ~ ! [X423 : term] : (~ (((var @ (sub @ X423 @ id))) = $true) | (((X415 @ id @ X423 @ id)) = $true)) | ~ ! [X424 : term,X425 : subst,X426 : subst,X427 : subst] : (~ (((X415 @ X425 @ X424 @ (comp @ X427 @ X426))) = $true) | (((X415 @ (comp @ X425 @ X427) @ (sub @ X424 @ X427) @ X426)) = $true)) | ~ ! [X428 : (subst > term > term)] : (($true = ((X415 @ id @ (lam @ (X428 @ sh @ one)) @ id))) | ~ ! [X429 : subst,X430 : term,X431 : subst] : (((X428 @ (comp @ X431 @ X429) @ (sub @ X430 @ X429))) = ((sub @ (X428 @ X431 @ X430) @ X429))) | ~ ! [X432 : term] : (($true = ((X415 @ id @ (X428 @ id @ X432) @ id))) | ~ (((X415 @ id @ X432 @ id)) = $true)))))))) <=> (hoasinduction_lthm_3 = $true)) & ((! [X434 : subst,X433 : term] : (((sub @ one @ (push @ X433 @ X434))) = X433) => (! [X436 : term,X435 : subst] : (((comp @ sh @ (push @ X436 @ X435))) = X435) => (! [X437 : term,X438 : term] : ((X437 = X438) | (((lam @ X438)) != ((lam @ X437)))) => ! [X442 : subst,X441 : (subst > term > term),X440 : (subst > term > term),X439 : term] : ((((lam @ (X441 @ sh @ one))) != ((lam @ (X440 @ sh @ one)))) | (((X440 @ X442 @ X439)) = ((X441 @ X442 @ X439))) | ~ ! [X448 : subst,X447 : subst,X446 : term] : (((sub @ (X440 @ X447 @ X446) @ X448)) = ((X440 @ (comp @ X447 @ X448) @ (sub @ X446 @ X448)))) | ~ ! [X444 : subst,X445 : subst,X443 : term] : (((sub @ (X441 @ X444 @ X443) @ X445)) = ((X441 @ (comp @ X444 @ X445) @ (sub @ X443 @ X445)))))))) <=> (hoaslaminj_lthm = $true)) & ((axscons = $true) <=> ! [X449 : subst] : (((push @ (sub @ one @ X449) @ (comp @ sh @ X449))) = X449)) & ((axidl = $true) <=> ! [X450 : subst] : (((comp @ id @ X450)) = X450)) & (hoasinduction_lem1v2 = $true) & ((hoasinduction_lem0_lthm = $true) <=> ! [X451 : (subst > term > subst > $o)] : ~ ! [X452 : (term > $o)] : ~ ! [X453 : term] : (((X451 @ id @ X453 @ id)) = ((X452 @ X453)))) & (hoasinduction_gthm = $true) & (pushprop_lem1v2_gthm = $true) & ((hoasapinj2 = $true) <=> ! [X455 : term,X454 : term,X456 : term,X457 : term] : ((((ap @ (sub @ X457 @ id) @ X455)) != ((ap @ (sub @ X454 @ id) @ X456))) | (X455 = X456))) & (hoasapinj2_gthm = $true) & (pushprop_lem2v2_gthm = $true) & (axassoc = $true) & ((hoasinduction_lem2v2 = $true) <=> ! [X458 : term,X459 : term,X460 : (subst > term > subst > $o),X461 : (term > $o)] : (~ ! [X462 : term,X463 : term] : (~ (((X460 @ id @ X462 @ id)) = $true) | ($true = ((X460 @ id @ (ap @ (sub @ X462 @ id) @ X463) @ id))) | ~ ($true = ((X460 @ id @ X463 @ id)))) | ~ ! [X464 : term] : (((X460 @ id @ X464 @ id)) = ((X461 @ X464))) | (((X461 @ (ap @ X459 @ X458))) = $true) | ~ ! [X465 : subst,X466 : subst,X467 : subst,X468 : term] : ((((X460 @ X467 @ X468 @ (comp @ X465 @ X466))) = $true) | ~ (((X460 @ (comp @ X467 @ X465) @ (sub @ X468 @ X465) @ X466)) = $true)) | ~ ! [X469 : term,X470 : subst,X471 : subst,X472 : subst] : (~ ($true = ((X460 @ X472 @ X469 @ (comp @ X470 @ X471)))) | (((X460 @ (comp @ X472 @ X470) @ (sub @ X469 @ X470) @ X471)) = $true)) | ~ ($true = ((X461 @ X458))) | ~ ($true = ((X461 @ X459))))) & (axclos = $true) & (! [X473 : term] : (((sub @ X473 @ id)) = X473) <=> (termmset = $true)) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ~ (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) = $true) & ~ (lamnotap = $true) & (pushprop_lem3v2_lthm = $true) & ((! [X474 : term] : (((sub @ X474 @ id)) = X474) => ! [X475 : term] : (((sub @ X475 @ id)) = X475)) <=> (termmset_lthm = $true)) & (((d2subst @ d_id)) = id) & ((hoasinduction_lem3aaa = $true) <=> ! [X476 : term,X477 : (subst > term > subst > $o)] : (~ ! [X478 : term] : ((((X477 @ id @ (sub @ X476 @ (push @ X478 @ id)) @ id)) = $true) | ~ (((X477 @ id @ X478 @ id)) = $true)) | ~ ! [X479 : (subst > term > term),X480 : term] : (~ ! [X482 : subst,X483 : term,X481 : subst] : (((sub @ (sub @ X480 @ (push @ X483 @ X482)) @ X481)) = ((sub @ (X479 @ X482 @ X483) @ X481))) | ~ ! [X484 : term] : (~ ($true = ((X477 @ id @ X484 @ id))) | ($true = ((X477 @ id @ (X479 @ id @ X484) @ id)))) | ~ ! [X487 : subst,X485 : subst,X486 : term] : (((X479 @ (comp @ X485 @ X487) @ (sub @ X486 @ X487))) = ((sub @ X480 @ (push @ (sub @ X486 @ X487) @ (comp @ X485 @ X487))))) | ($true = ((X477 @ id @ (lam @ (X479 @ sh @ one)) @ id)))) | ($true = ((X477 @ id @ (lam @ (sub @ X476 @ (push @ one @ sh))) @ id))))) & (pushprop_lthm_orig = $true) & ((axshiftcons = $true) <=> ! [X489 : subst,X488 : term] : (((comp @ sh @ (push @ X488 @ X489))) = X489)) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoasinduction_lem3_lthm = $true) <=> (! [X490 : term] : (((sub @ X490 @ id)) = X490) => (! [X491 : term,X492 : (subst > term > subst > $o)] : (~ ! [X493 : (subst > term > term)] : (~ ! [X496 : term,X494 : subst,X495 : subst] : (((sub @ (X493 @ X494 @ X496) @ X495)) = ((X493 @ (comp @ X494 @ X495) @ (sub @ X496 @ X495)))) | ~ ! [X497 : term] : (~ (((X492 @ id @ X497 @ id)) = $true) | ($true = ((X492 @ id @ (X493 @ id @ X497) @ id)))) | (((X492 @ id @ (lam @ (X493 @ sh @ one)) @ id)) = $true)) | ~ ! [X498 : term] : (~ ($true = ((X492 @ id @ X498 @ id))) | (((X492 @ id @ (sub @ X491 @ (push @ X498 @ id)) @ id)) = $true)) | (((X492 @ id @ (lam @ (sub @ X491 @ (push @ one @ sh))) @ id)) = $true)) => ! [X499 : term,X500 : (subst > term > subst > $o)] : (~ ! [X501 : term,X502 : subst,X503 : subst,X504 : subst] : ((((X500 @ X504 @ X501 @ (comp @ X503 @ X502))) = $true) | ~ (((X500 @ (comp @ X504 @ X503) @ (sub @ X501 @ X503) @ X502)) = $true)) | ~ ! [X505 : (subst > term > term)] : (~ ! [X506 : term] : (~ (((X500 @ id @ X506 @ id)) = $true) | (((X500 @ id @ (X505 @ id @ X506) @ id)) = $true)) | ~ ! [X507 : term,X509 : subst,X508 : subst] : (((X505 @ (comp @ X509 @ X508) @ (sub @ X507 @ X508))) = ((sub @ (X505 @ X509 @ X507) @ X508))) | (((X500 @ id @ (lam @ (X505 @ sh @ one)) @ id)) = $true)) | ($true = ((X500 @ id @ (lam @ X499) @ id))) | ~ ! [X510 : term] : (~ (((X500 @ id @ X510 @ id)) = $true) | (((X500 @ id @ (sub @ X499 @ (push @ X510 @ id)) @ id)) = $true)) | ~ ! [X511 : term,X512 : subst,X513 : subst,X514 : subst] : ((((X500 @ (comp @ X514 @ X513) @ (sub @ X511 @ X513) @ X512)) = $true) | ~ (((X500 @ X514 @ X511 @ (comp @ X513 @ X512))) = $true)))))) & (hoasapnotvar = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (hoaslamnotap_lthm = $true) & (hoasinduction_lem1 = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (pushprop_lem2v2_lthm = $true) & (! [X515 : term,X516 : (term > $o)] : (~ ! [X517 : term,X518 : term] : ((((X516 @ (ap @ X517 @ X518))) = $true) | ~ (((X516 @ X518)) = $true) | ~ ($true = ((X516 @ X517)))) | (((X516 @ X515)) = $true) | ~ ! [X519 : term] : ((((X516 @ (lam @ X519))) = $true) | ~ ($true = ((X516 @ X519)))) | ~ ! [X520 : term] : (~ ($true = ((var @ X520))) | ($true = ((X516 @ X520))))) <=> (induction = $true)) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ((! [X521 : subst] : (((comp @ X521 @ id)) = X521) & ! [X522 : subst] : (((comp @ id @ X522)) = X522)) <=> (substmonoid = $true)) & ~ (ulamvar1 = $true) & (ulamvarsh = $true) & (axmap = $true) & ((hoasapinj1_lthm = $true) <=> (! [X523 : term] : (((sub @ X523 @ id)) = X523) => (! [X526 : term,X525 : term,X524 : term,X527 : term] : ((X524 = X525) | (((ap @ X525 @ X526)) != ((ap @ X524 @ X527)))) => ! [X531 : term,X528 : term,X530 : term,X529 : term] : ((X528 = X530) | (((ap @ (sub @ X530 @ id) @ X529)) != ((ap @ (sub @ X528 @ id) @ X531))))))) & (hoaslaminj_gthm = $true) & (! [X532 : term,X533 : (subst > term > subst > $o)] : (~ ! [X534 : term] : ((((X533 @ id @ (sub @ X532 @ (push @ X534 @ id)) @ id)) = $true) | ~ ($true = ((X533 @ id @ X534 @ id)))) | ~ ! [X535 : (subst > term > term)] : ((((X533 @ id @ (lam @ (X535 @ sh @ one)) @ id)) = $true) | ~ ! [X537 : subst,X538 : subst,X536 : term] : (((sub @ (X535 @ X538 @ X536) @ X537)) = ((X535 @ (comp @ X538 @ X537) @ (sub @ X536 @ X537)))) | ~ ! [X539 : term] : (~ ($true = ((X533 @ id @ X539 @ id))) | ($true = ((X533 @ id @ (X535 @ id @ X539) @ id))))) | (((X533 @ id @ (lam @ X532) @ id)) = $true)) <=> (hoasinduction_lem3a = $true)) & (pushprop = $true)),
  inference(rectify,[],[f5])).
thf(f9,plain,(
  (! [X1 : term,X0 : term,X3 : term,X2 : term] : ((X0 = X3) | (((ap @ X3 @ X2)) != ((ap @ X0 @ X1)))) <=> (apinj1 = $true)) & (((d2subst @ d_id)) = id) & ((hoasinduction_lem0 = $true) <=> ! [X211 : (subst > term > subst > $o)] : ~ ! [X212 : (term > $o)] : ~ ! [X213 : term] : (((X212 @ X213)) = ((X211 @ id @ X213 @ id)))) & (pushprop_gthm = $true) & (hoasapinj2_gthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoasinduction_lem1v2 = $true) & (axassoc = $true) & ((hoasinduction_lthm_3 = $true) <=> (! [X394 : (subst > term > subst > $o)] : ~ ! [X395 : (term > $o)] : ~ ! [X396 : term] : (((X395 @ X396)) = ((X394 @ id @ X396 @ id))) => (! [X398 : term,X397 : (term > $o)] : (~ ! [X399 : term] : (~ ! [X400 : term] : (($true = ((X397 @ (sub @ X399 @ (push @ X400 @ id))))) | (((X397 @ X400)) != $true)) | (((X397 @ (lam @ X399))) = $true)) | (((X397 @ X398)) = $true) | ~ ! [X401 : term,X402 : term] : (($true = ((X397 @ (ap @ X401 @ X402)))) | ($true != ((X397 @ X401))) | ($true != ((X397 @ X402)))) | ~ ! [X403 : term] : (($true != ((var @ X403))) | ($true = ((X397 @ X403))))) => (! [X404 : term] : (((sub @ X404 @ id)) = X404) => (! [X406 : term,X407 : (term > $o),X405 : (subst > term > subst > $o)] : (~ ! [X408 : term] : (((X405 @ id @ X408 @ id)) = ((X407 @ X408))) | ~ ! [X409 : term] : ((((X407 @ (sub @ X406 @ (push @ X409 @ id)))) = $true) | (((X407 @ X409)) != $true)) | (((X407 @ (lam @ X406))) = $true) | ~ ! [X410 : (subst > term > term)] : (~ ! [X413 : term,X414 : subst,X412 : subst] : (((X410 @ (comp @ X412 @ X414) @ (sub @ X413 @ X414))) = ((sub @ (X410 @ X412 @ X413) @ X414))) | (((X405 @ id @ (lam @ (X410 @ sh @ one)) @ id)) = $true) | ~ ! [X411 : term] : (($true != ((X405 @ id @ X411 @ id))) | ($true = ((X405 @ id @ (X410 @ id @ X411) @ id)))))) => ! [X415 : (subst > term > subst > $o),X416 : term] : (~ ! [X422 : subst,X421 : subst,X419 : term,X420 : subst] : (($true = ((X415 @ X421 @ X419 @ (comp @ X422 @ X420)))) | ($true != ((X415 @ (comp @ X421 @ X422) @ (sub @ X419 @ X422) @ X420)))) | ~ ! [X423 : term] : ((((var @ (sub @ X423 @ id))) != $true) | (((X415 @ id @ X423 @ id)) = $true)) | ~ ! [X425 : subst,X427 : subst,X424 : term,X426 : subst] : ((((X415 @ (comp @ X425 @ X427) @ (sub @ X424 @ X427) @ X426)) = $true) | (((X415 @ X425 @ X424 @ (comp @ X427 @ X426))) != $true)) | ($true = ((X415 @ id @ X416 @ id))) | ~ ! [X428 : (subst > term > term)] : (~ ! [X432 : term] : ((((X415 @ id @ X432 @ id)) != $true) | ($true = ((X415 @ id @ (X428 @ id @ X432) @ id)))) | ($true = ((X415 @ id @ (lam @ (X428 @ sh @ one)) @ id))) | ~ ! [X429 : subst,X430 : term,X431 : subst] : (((X428 @ (comp @ X431 @ X429) @ (sub @ X430 @ X429))) = ((sub @ (X428 @ X431 @ X430) @ X429)))) | ~ ! [X417 : term,X418 : term] : ((((X415 @ id @ X418 @ id)) != $true) | ($true != ((X415 @ id @ X417 @ id))) | ($true = ((X415 @ id @ (ap @ (sub @ X418 @ id) @ X417) @ id)))))))))) & (! [X149 : term,X148 : (subst > term > subst > $o),X150 : (term > $o)] : (~ ! [X153 : subst,X152 : term,X151 : subst,X154 : subst] : ((((X148 @ (comp @ X151 @ X154) @ (sub @ X152 @ X154) @ X153)) != $true) | (((X148 @ X151 @ X152 @ (comp @ X154 @ X153))) = $true)) | (((X150 @ (lam @ X149))) = $true) | ~ ! [X164 : term] : (((X148 @ id @ X164 @ id)) = ((X150 @ X164))) | ~ ! [X161 : subst,X160 : subst,X162 : subst,X163 : term] : ((((X148 @ (comp @ X161 @ X160) @ (sub @ X163 @ X160) @ X162)) = $true) | ($true != ((X148 @ X161 @ X163 @ (comp @ X160 @ X162))))) | ~ ! [X155 : (subst > term > term)] : ((((X148 @ id @ (lam @ (X155 @ sh @ one)) @ id)) = $true) | ~ ! [X156 : term] : (($true != ((X148 @ id @ X156 @ id))) | ($true = ((X148 @ id @ (X155 @ id @ X156) @ id)))) | ~ ! [X159 : term,X158 : subst,X157 : subst] : (((X155 @ (comp @ X157 @ X158) @ (sub @ X159 @ X158))) = ((sub @ (X155 @ X157 @ X159) @ X158)))) | ~ ! [X165 : term] : ((((X150 @ (sub @ X149 @ (push @ X165 @ id)))) = $true) | (((X150 @ X165)) != $true))) <=> (hoasinduction_lem3v2 = $true)) & (! [X458 : term,X461 : (term > $o),X460 : (subst > term > subst > $o),X459 : term] : (~ ! [X464 : term] : (((X460 @ id @ X464 @ id)) = ((X461 @ X464))) | ($true != ((X461 @ X459))) | (((X461 @ (ap @ X459 @ X458))) = $true) | ~ ! [X465 : subst,X466 : subst,X467 : subst,X468 : term] : ((((X460 @ X467 @ X468 @ (comp @ X465 @ X466))) = $true) | (((X460 @ (comp @ X467 @ X465) @ (sub @ X468 @ X465) @ X466)) != $true)) | ~ ! [X469 : term,X471 : subst,X470 : subst,X472 : subst] : (($true != ((X460 @ X472 @ X469 @ (comp @ X470 @ X471)))) | (((X460 @ (comp @ X472 @ X470) @ (sub @ X469 @ X470) @ X471)) = $true)) | ($true != ((X461 @ X458))) | ~ ! [X463 : term,X462 : term] : (($true = ((X460 @ id @ (ap @ (sub @ X462 @ id) @ X463) @ id))) | (((X460 @ id @ X462 @ id)) != $true) | ($true != ((X460 @ id @ X463 @ id))))) <=> (hoasinduction_lem2v2 = $true)) & (hoasinduction_lem3a_gthm = $true) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X207 : term] : ~ ! [X208 : (subst > term > term)] : ~ ! [X210 : term,X209 : subst] : (((sub @ X207 @ (push @ X210 @ X209))) = ((X208 @ X209 @ X210)))) & (pushprop_lem2v2_lthm = $true) & ((hoaslaminj = $true) <=> ! [X42 : (subst > term > term),X41 : (subst > term > term),X43 : subst,X44 : term] : (~ ! [X49 : subst,X50 : subst,X48 : term] : (((sub @ (X41 @ X49 @ X48) @ X50)) = ((X41 @ (comp @ X49 @ X50) @ (sub @ X48 @ X50)))) | ~ ! [X46 : subst,X45 : term,X47 : subst] : (((sub @ (X42 @ X46 @ X45) @ X47)) = ((X42 @ (comp @ X46 @ X47) @ (sub @ X45 @ X47)))) | (((lam @ (X41 @ sh @ one))) != ((lam @ (X42 @ sh @ one)))) | (((X41 @ X43 @ X44)) = ((X42 @ X43 @ X44))))) & (hoasinduction_lem2v2_gthm = $true) & (! [X532 : term,X533 : (subst > term > subst > $o)] : (~ ! [X534 : term] : ((((X533 @ id @ (sub @ X532 @ (push @ X534 @ id)) @ id)) = $true) | ($true != ((X533 @ id @ X534 @ id)))) | ~ ! [X535 : (subst > term > term)] : ((((X533 @ id @ (lam @ (X535 @ sh @ one)) @ id)) = $true) | ~ ! [X537 : subst,X538 : subst,X536 : term] : (((sub @ (X535 @ X538 @ X536) @ X537)) = ((X535 @ (comp @ X538 @ X537) @ (sub @ X536 @ X537)))) | ~ ! [X539 : term] : (($true = ((X533 @ id @ (X535 @ id @ X539) @ id))) | ($true != ((X533 @ id @ X539 @ id))))) | (((X533 @ id @ (lam @ X532) @ id)) = $true)) <=> (hoasinduction_lem3a = $true)) & (pushprop_lem0_gthm = $true) & (((var @ (d2term @ d_one))) != $true) & ((! [X171 : term] : (((sub @ X171 @ id)) = X171) => (! [X172 : subst,X174 : (term > $o),X173 : term] : (~ ! [X177 : term] : (($true = ((X174 @ (lam @ X177)))) | ~ ! [X178 : term] : ((((X174 @ X178)) != $true) | ($true = ((X174 @ (sub @ X177 @ (push @ X178 @ id))))))) | ~ ! [X179 : term] : (($true = ((X174 @ (sub @ X179 @ X172)))) | (((var @ X179)) != $true)) | (((X174 @ (sub @ X173 @ X172))) = $true) | ~ ! [X176 : term,X175 : term] : (($true != ((X174 @ X175))) | ($true = ((X174 @ (ap @ X175 @ X176)))) | (((X174 @ X176)) != $true))) => ! [X180 : term,X181 : (term > $o)] : (($true = ((X181 @ X180))) | ~ ! [X184 : term] : (($true = ((X181 @ X184))) | (((var @ X184)) != $true)) | ~ ! [X185 : term] : (($true = ((X181 @ (lam @ X185)))) | ~ ! [X186 : term] : (($true = ((X181 @ (sub @ X185 @ (push @ X186 @ id))))) | (((X181 @ X186)) != $true))) | ~ ! [X183 : term,X182 : term] : ((((X181 @ X183)) != $true) | (((X181 @ (ap @ X183 @ X182))) = $true) | (((X181 @ X182)) != $true))))) <=> (induction2_lthm = $true)) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (pushprop_lem1_gthm = $true) & (axvarshift = $true) & (apnotvar = $true) & (pushprop_lem3v2_lthm = $true) & ((! [X474 : term] : (((sub @ X474 @ id)) = X474) => ! [X475 : term] : (((sub @ X475 @ id)) = X475)) <=> (termmset_lthm = $true)) & ((! [X521 : subst] : (((comp @ X521 @ id)) = X521) & ! [X522 : subst] : (((comp @ id @ X522)) = X522)) <=> (substmonoid = $true)) & (! [X129 : term,X130 : (subst > term > subst > $o)] : (~ ! [X131 : term] : ((((X130 @ id @ X131 @ id)) != $true) | (((X130 @ id @ (sub @ X129 @ (push @ X131 @ id)) @ id)) = $true)) | ~ ! [X132 : (subst > term > term)] : (~ ! [X133 : term] : (($true = ((X130 @ id @ (X132 @ id @ X133) @ id))) | ($true != ((X130 @ id @ X133 @ id)))) | ~ ! [X136 : subst,X135 : term,X134 : subst] : (((sub @ (X132 @ X136 @ X135) @ X134)) = ((X132 @ (comp @ X136 @ X134) @ (sub @ X135 @ X134)))) | ($true = ((X130 @ id @ (lam @ (X132 @ sh @ one)) @ id)))) | ($true = ((X130 @ id @ (lam @ (sub @ X129 @ (push @ one @ sh))) @ id)))) <=> (hoasinduction_lem3aa = $true)) & (hoaslamnotvar_gthm = $true) & (! [X267 : (term > $o),X269 : (term > $o),X268 : subst,X270 : term] : ((((X267 @ X270)) != $true) | ~ ! [X271 : term] : (((X267 @ (sub @ X271 @ (push @ X270 @ X268)))) = ((X269 @ X271))) | (((X269 @ one)) = $true)) <=> (pushprop_lem1v2 = $true)) & (hoaslamnotap_gthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (hoasinduction_lem1v2_gthm = $true) & (! [X145 : subst,X143 : term,X144 : (term > $o)] : ~ ! [X146 : (term > $o)] : ~ ! [X147 : term] : (((X144 @ (sub @ X147 @ (push @ X143 @ X145)))) = ((X146 @ X147))) <=> (pushprop_lem0_lthm = $true)) & (ulamvarsh = $true) & (! [X392 : term,X393 : subst,X391 : term,X389 : (term > $o),X390 : (term > $o)] : (($true = ((X389 @ (sub @ X392 @ (push @ X391 @ X393))))) | ($true != ((X390 @ X392)))) <=> (pushprop_lem1 = $true)) & ((! [X352 : term] : (((sub @ X352 @ id)) = X352) => (! [X354 : (subst > term > subst > $o),X353 : term] : (~ ! [X355 : term] : (($true = ((X354 @ id @ (sub @ X353 @ (push @ X355 @ id)) @ id))) | (((X354 @ id @ X355 @ id)) != $true)) | ~ ! [X356 : (subst > term > term)] : (~ ! [X358 : subst,X359 : subst,X360 : term] : (((sub @ (X356 @ X358 @ X360) @ X359)) = ((X356 @ (comp @ X358 @ X359) @ (sub @ X360 @ X359)))) | ~ ! [X357 : term] : ((((X354 @ id @ (X356 @ id @ X357) @ id)) = $true) | ($true != ((X354 @ id @ X357 @ id)))) | ($true = ((X354 @ id @ (lam @ (X356 @ sh @ one)) @ id)))) | (((X354 @ id @ (lam @ (sub @ X353 @ (push @ one @ sh))) @ id)) = $true)) => ! [X362 : (subst > term > subst > $o),X361 : term] : (($true = ((X362 @ id @ (lam @ X361) @ id))) | ~ ! [X363 : (subst > term > term)] : (~ ! [X367 : term] : ((((X362 @ id @ (X363 @ id @ X367) @ id)) = $true) | (((X362 @ id @ X367 @ id)) != $true)) | ~ ! [X365 : subst,X366 : term,X364 : subst] : (((sub @ (X363 @ X365 @ X366) @ X364)) = ((X363 @ (comp @ X365 @ X364) @ (sub @ X366 @ X364)))) | ($true = ((X362 @ id @ (lam @ (X363 @ sh @ one)) @ id)))) | ~ ! [X368 : term] : (($true = ((X362 @ id @ (sub @ X361 @ (push @ X368 @ id)) @ id))) | (((X362 @ id @ X368 @ id)) != $true))))) <=> (hoasinduction_lem3a_lthm = $true)) & (hoaslamnotvar = $true) & (pushprop_lthm_orig = $true) & ((induction = $true) <=> ! [X515 : term,X516 : (term > $o)] : (~ ! [X519 : term] : ((((X516 @ (lam @ X519))) = $true) | ($true != ((X516 @ X519)))) | (((X516 @ X515)) = $true) | ~ ! [X517 : term,X518 : term] : ((((X516 @ X518)) != $true) | (((X516 @ (ap @ X517 @ X518))) = $true) | ($true != ((X516 @ X517)))) | ~ ! [X520 : term] : (($true = ((X516 @ X520))) | ($true != ((var @ X520)))))) & (axapp = $true) & ((hoasinduction_lem3v2a = $true) <=> ! [X5 : term,X4 : (term > $o),X6 : (subst > term > subst > $o)] : (~ ! [X12 : term] : (((X6 @ id @ X12 @ id)) = ((X4 @ X12))) | ~ ! [X7 : (subst > term > term)] : (~ ! [X8 : term] : ((((X6 @ id @ (X7 @ id @ X8) @ id)) = $true) | ($true != ((X6 @ id @ X8 @ id)))) | ~ ! [X11 : subst,X10 : term,X9 : subst] : (((sub @ (X7 @ X11 @ X10) @ X9)) = ((X7 @ (comp @ X11 @ X9) @ (sub @ X10 @ X9)))) | ($true = ((X6 @ id @ (lam @ (X7 @ sh @ one)) @ id)))) | ~ ! [X13 : term] : ((((X4 @ X13)) != $true) | ($true = ((X4 @ (sub @ X5 @ (push @ X13 @ id)))))) | ($true = ((X4 @ (lam @ X5)))))) & (pushprop = $true) & ((axidl = $true) <=> ! [X450 : subst] : (((comp @ id @ X450)) = X450)) & ((hoasinduction_lthm = $true) <=> (! [X76 : (term > $o),X75 : term] : (~ ! [X80 : term,X79 : term] : ((((X76 @ X80)) != $true) | (((X76 @ (ap @ X79 @ X80))) = $true) | ($true != ((X76 @ X79)))) | ~ ! [X81 : term] : (($true != ((var @ X81))) | (((X76 @ X81)) = $true)) | ~ ! [X77 : term] : (~ ! [X78 : term] : ((((X76 @ X78)) != $true) | ($true = ((X76 @ (sub @ X77 @ (push @ X78 @ id)))))) | (((X76 @ (lam @ X77))) = $true)) | (((X76 @ X75)) = $true)) => (! [X83 : term,X84 : (subst > term > subst > $o),X82 : term] : (~ ! [X94 : subst,X92 : term,X91 : subst,X93 : subst] : ((((X84 @ (comp @ X93 @ X91) @ (sub @ X92 @ X91) @ X94)) = $true) | (((X84 @ X93 @ X92 @ (comp @ X91 @ X94))) != $true)) | (((X84 @ id @ X83 @ id)) != $true) | (((X84 @ id @ (ap @ X83 @ X82) @ id)) = $true) | ~ ! [X85 : subst,X86 : subst,X87 : subst,X88 : term] : ((((X84 @ X85 @ X88 @ (comp @ X86 @ X87))) = $true) | ($true != ((X84 @ (comp @ X85 @ X86) @ (sub @ X88 @ X86) @ X87)))) | ~ ! [X89 : term,X90 : term] : ((((X84 @ id @ (ap @ (sub @ X89 @ id) @ X90) @ id)) = $true) | ($true != ((X84 @ id @ X89 @ id))) | ($true != ((X84 @ id @ X90 @ id)))) | (((X84 @ id @ X82 @ id)) != $true)) => (! [X95 : term,X96 : (subst > term > subst > $o)] : (~ ! [X106 : (subst > term > term)] : ((((X96 @ id @ (lam @ (X106 @ sh @ one)) @ id)) = $true) | ~ ! [X107 : term] : (($true != ((X96 @ id @ X107 @ id))) | (((X96 @ id @ (X106 @ id @ X107) @ id)) = $true)) | ~ ! [X109 : subst,X108 : term,X110 : subst] : (((X106 @ (comp @ X110 @ X109) @ (sub @ X108 @ X109))) = ((sub @ (X106 @ X110 @ X108) @ X109)))) | ~ ! [X105 : term] : ((((X96 @ id @ (sub @ X95 @ (push @ X105 @ id)) @ id)) = $true) | ($true != ((X96 @ id @ X105 @ id)))) | ~ ! [X104 : subst,X102 : subst,X101 : term,X103 : subst] : ((((X96 @ (comp @ X102 @ X103) @ (sub @ X101 @ X103) @ X104)) = $true) | ($true != ((X96 @ X102 @ X101 @ (comp @ X103 @ X104))))) | (((X96 @ id @ (lam @ X95) @ id)) = $true) | ~ ! [X98 : subst,X99 : subst,X100 : subst,X97 : term] : ((((X96 @ (comp @ X98 @ X99) @ (sub @ X97 @ X99) @ X100)) != $true) | (((X96 @ X98 @ X97 @ (comp @ X99 @ X100))) = $true))) => ! [X112 : term,X111 : (subst > term > subst > $o)] : (~ ! [X115 : (subst > term > term)] : (~ ! [X119 : term] : ((((X111 @ id @ X119 @ id)) != $true) | (((X111 @ id @ (X115 @ id @ X119) @ id)) = $true)) | ($true = ((X111 @ id @ (lam @ (X115 @ sh @ one)) @ id))) | ~ ! [X117 : subst,X118 : subst,X116 : term] : (((sub @ (X115 @ X118 @ X116) @ X117)) = ((X115 @ (comp @ X118 @ X117) @ (sub @ X116 @ X117))))) | ~ ! [X124 : term] : ((((X111 @ id @ X124 @ id)) = $true) | (((var @ (sub @ X124 @ id))) != $true)) | (((X111 @ id @ X112 @ id)) = $true) | ~ ! [X114 : term,X113 : term] : ((((X111 @ id @ X114 @ id)) != $true) | ($true = ((X111 @ id @ (ap @ (sub @ X113 @ id) @ X114) @ id))) | ($true != ((X111 @ id @ X113 @ id)))) | ~ ! [X120 : subst,X123 : term,X121 : subst,X122 : subst] : (($true != ((X111 @ X121 @ X123 @ (comp @ X120 @ X122)))) | ($true = ((X111 @ (comp @ X121 @ X120) @ (sub @ X123 @ X120) @ X122)))) | ~ ! [X126 : subst,X128 : term,X125 : subst,X127 : subst] : ((((X111 @ X125 @ X128 @ (comp @ X127 @ X126))) = $true) | (((X111 @ (comp @ X125 @ X127) @ (sub @ X128 @ X127) @ X126)) != $true))))))) & (induction2lem_gthm = $true) & (axabs = $true) & (pushprop_lem3v2 = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ! [X51 : term] : ? [X52 : d_term] : (((d2term @ X52)) = X51) & (hoaslamnotvar_lthm = $true) & (pushprop_lem2v2 = $true) & ((hoasinduction_no_psi_cond = $true) <=> ! [X66 : term,X67 : (subst > term > subst > $o)] : (~ ! [X68 : (subst > term > term)] : (($true = ((X67 @ id @ (lam @ (X68 @ sh @ one)) @ id))) | ~ ! [X69 : subst,X71 : subst,X70 : term] : (((X68 @ (comp @ X69 @ X71) @ (sub @ X70 @ X71))) = ((sub @ (X68 @ X69 @ X70) @ X71))) | ~ ! [X72 : term] : ((((X67 @ id @ X72 @ id)) != $true) | (((X67 @ id @ (X68 @ id @ X72) @ id)) = $true))) | ($true = ((X67 @ id @ X66 @ id))) | ~ ! [X73 : term,X74 : term] : ((((X67 @ id @ (ap @ (sub @ X73 @ id) @ X74) @ id)) = $true) | (((X67 @ id @ X73 @ id)) != $true) | (((X67 @ id @ X74 @ id)) != $true)))) & ((axscons = $true) <=> ! [X449 : subst] : (((push @ (sub @ one @ X449) @ (comp @ sh @ X449))) = X449)) & (hoasinduction_gthm = $true) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X306 : (subst > term > subst > $o),X305 : term] : (~ ! [X307 : (subst > term > term)] : (~ ! [X308 : term] : (($true = ((X306 @ id @ (X307 @ id @ X308) @ id))) | (((X306 @ id @ X308 @ id)) != $true)) | ~ ! [X309 : subst,X311 : subst,X310 : term] : (((X307 @ (comp @ X311 @ X309) @ (sub @ X310 @ X309))) = ((sub @ (X307 @ X311 @ X310) @ X309))) | ($true = ((X306 @ id @ (lam @ (X307 @ sh @ one)) @ id)))) | ($true = ((X306 @ id @ (lam @ (sub @ X305 @ (push @ one @ sh))) @ id))) | ~ ! [X312 : term] : (($true = ((X306 @ id @ (sub @ X305 @ (push @ X312 @ id)) @ id))) | (((X306 @ id @ X312 @ id)) != $true)))) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((hoasinduction_lem0_lthm = $true) <=> ! [X451 : (subst > term > subst > $o)] : ~ ! [X452 : (term > $o)] : ~ ! [X453 : term] : (((X451 @ id @ X453 @ id)) = ((X452 @ X453)))) & ! [X280 : d_term] : (d_one = X280) & (hoasinduction_lem2_gthm = $true) & (hoasapnotvar_lthm = $true) & ((hoasinduction_lem3v2_f = $true) <=> ! [X214 : term] : ~ ! [X215 : (subst > term > term)] : ~ ! [X217 : subst,X216 : term] : (((X215 @ X217 @ X216)) = ((sub @ X214 @ (push @ X216 @ X217))))) & (lamnotap != $true) & ((axvarid = $true) <=> ! [X14 : term] : (((sub @ X14 @ id)) = X14)) & (! [X298 : term,X299 : term] : ((((sub @ X298 @ sh)) != ((sub @ X299 @ sh))) | (X298 = X299)) <=> (shinj = $true)) & (hoasinduction_lem1_gthm = $true) & ((! [X255 : term,X254 : subst] : (((sub @ one @ (push @ X255 @ X254))) = X255) => ! [X257 : (term > $o),X259 : term,X258 : (term > $o),X256 : subst] : (~ ! [X260 : term] : (((X257 @ (sub @ X260 @ (push @ X259 @ X256)))) = ((X258 @ X260))) | ($true != ((X257 @ X259))) | ($true = ((X258 @ one))))) <=> (pushprop_lem1v2_lthm = $true)) & ((hoasinduction_lem3 = $true) <=> ! [X24 : (subst > term > subst > $o),X23 : term] : (~ ! [X27 : subst,X28 : subst,X25 : subst,X26 : term] : (($true = ((X24 @ X25 @ X26 @ (comp @ X28 @ X27)))) | (((X24 @ (comp @ X25 @ X28) @ (sub @ X26 @ X28) @ X27)) != $true)) | ~ ! [X38 : term] : (($true = ((X24 @ id @ (sub @ X23 @ (push @ X38 @ id)) @ id))) | (((X24 @ id @ X38 @ id)) != $true)) | ($true = ((X24 @ id @ (lam @ X23) @ id))) | ~ ! [X36 : subst,X35 : subst,X34 : term,X37 : subst] : (($true != ((X24 @ X37 @ X34 @ (comp @ X35 @ X36)))) | (((X24 @ (comp @ X37 @ X35) @ (sub @ X34 @ X35) @ X36)) = $true)) | ~ ! [X29 : (subst > term > term)] : (~ ! [X32 : subst,X31 : term,X30 : subst] : (((sub @ (X29 @ X30 @ X31) @ X32)) = ((X29 @ (comp @ X30 @ X32) @ (sub @ X31 @ X32)))) | ~ ! [X33 : term] : ((((X24 @ id @ X33 @ id)) != $true) | (((X24 @ id @ (X29 @ id @ X33) @ id)) = $true)) | (((X24 @ id @ (lam @ (X29 @ sh @ one)) @ id)) = $true)))) & ((axshiftcons = $true) <=> ! [X489 : subst,X488 : term] : (((comp @ sh @ (push @ X488 @ X489))) = X489)) & (hoaslaminj_gthm = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ((laminj = $true) <=> ! [X262 : term,X261 : term] : ((X261 = X262) | (((lam @ X262)) != ((lam @ X261))))) & ((pushprop_lem1_lthm = $true) <=> (! [X218 : subst,X219 : term] : (((sub @ one @ (push @ X219 @ X218))) = X219) => (! [X220 : subst,X221 : term] : (((comp @ sh @ (push @ X221 @ X220))) = X220) => ! [X226 : subst,X223 : (term > $o),X224 : term,X225 : term,X222 : (term > $o)] : (($true != ((X222 @ X224))) | ($true = ((X223 @ (sub @ X224 @ (push @ X225 @ X226))))))))) & (! [X343 : term] : ~ ! [X344 : (subst > term > term)] : (((sub @ X343 @ (push @ one @ sh))) != ((X344 @ sh @ one))) <=> (hoasinduction_lem3b = $true)) & (! [X167 : (term > $o),X168 : term,X166 : subst] : ~ ! [X169 : (term > $o)] : ~ ! [X170 : term] : (((X169 @ X170)) = ((X167 @ (sub @ X170 @ (push @ X168 @ X166))))) <=> (pushprop_lem0 = $true)) & ! [X302 : d_term,X303 : d_term] : ((((d2term @ X302)) = ((d2term @ X303))) => (X302 = X303)) & ! [X296 : subst] : ? [X297 : d_subst] : (((d2subst @ X297)) = X296) & ((! [X370 : subst,X371 : term] : (((sub @ one @ (push @ X371 @ X370))) = X371) => (! [X372 : term,X373 : subst] : (((comp @ sh @ (push @ X372 @ X373))) = X373) => (! [X374 : subst] : (((comp @ X374 @ id)) = X374) => (! [X375 : (term > $o),X376 : term] : (~ ! [X379 : term] : (($true != ((var @ X379))) | (((X375 @ X379)) = $true)) | ~ ! [X378 : term,X377 : term] : ((((X375 @ (ap @ X378 @ X377))) = $true) | ($true != ((X375 @ X377))) | (((X375 @ X378)) != $true)) | ~ ! [X380 : term] : (($true = ((X375 @ (lam @ X380)))) | ($true != ((X375 @ X380)))) | (((X375 @ X376)) = $true)) => ! [X383 : subst,X382 : term,X381 : (term > $o)] : (~ ! [X387 : term,X388 : term] : ((((X381 @ X388)) != $true) | (((X381 @ X387)) != $true) | (((X381 @ (ap @ X387 @ X388))) = $true)) | ~ ! [X386 : term] : ((((var @ X386)) != $true) | (((X381 @ (sub @ X386 @ X383))) = $true)) | (((X381 @ (sub @ X382 @ X383))) = $true) | ~ ! [X384 : term] : (($true = ((X381 @ (lam @ X384)))) | ~ ! [X385 : term] : ((((X381 @ (sub @ X384 @ (push @ X385 @ id)))) = $true) | (((X381 @ X385)) != $true)))))))) <=> (induction2lem_lthm = $true)) & ((hoasapinj1 = $true) <=> ! [X21 : term,X19 : term,X20 : term,X22 : term] : ((X21 = X22) | (((ap @ (sub @ X22 @ id) @ X19)) != ((ap @ (sub @ X21 @ id) @ X20))))) & ((! [X313 : (subst > term > subst > $o)] : ~ ! [X314 : (term > $o)] : ~ ! [X315 : term] : (((X314 @ X315)) = ((X313 @ id @ X315 @ id))) => (! [X317 : term,X316 : (term > $o)] : (~ ! [X318 : term,X319 : term] : (($true = ((X316 @ (ap @ X319 @ X318)))) | (((X316 @ X319)) != $true) | (((X316 @ X318)) != $true)) | ~ ! [X320 : term] : (~ ! [X321 : term] : (($true = ((X316 @ (sub @ X320 @ (push @ X321 @ id))))) | ($true != ((X316 @ X321)))) | (((X316 @ (lam @ X320))) = $true)) | (((X316 @ X317)) = $true) | ~ ! [X322 : term] : ((((X316 @ X322)) = $true) | ($true != ((var @ X322))))) => (! [X323 : term] : (((sub @ X323 @ id)) = X323) => (! [X324 : (subst > term > subst > $o),X325 : term,X326 : (term > $o)] : (~ ! [X327 : (subst > term > term)] : (~ ! [X330 : term,X329 : subst,X328 : subst] : (((sub @ (X327 @ X328 @ X330) @ X329)) = ((X327 @ (comp @ X328 @ X329) @ (sub @ X330 @ X329)))) | ~ ! [X331 : term] : (($true = ((X324 @ id @ (X327 @ id @ X331) @ id))) | ($true != ((X324 @ id @ X331 @ id)))) | ($true = ((X324 @ id @ (lam @ (X327 @ sh @ one)) @ id)))) | (((X326 @ (lam @ X325))) = $true) | ~ ! [X333 : term] : (((X326 @ X333)) = ((X324 @ id @ X333 @ id))) | ~ ! [X332 : term] : ((((X326 @ X332)) != $true) | (((X326 @ (sub @ X325 @ (push @ X332 @ id)))) = $true))) => ! [X334 : term,X335 : (subst > term > subst > $o)] : (($true = ((X335 @ id @ X334 @ id))) | ~ ! [X336 : (subst > term > term)] : (~ ! [X337 : term] : ((((X335 @ id @ X337 @ id)) != $true) | ($true = ((X335 @ id @ (X336 @ id @ X337) @ id)))) | ~ ! [X338 : term,X340 : subst,X339 : subst] : (((sub @ (X336 @ X340 @ X338) @ X339)) = ((X336 @ (comp @ X340 @ X339) @ (sub @ X338 @ X339)))) | (((X335 @ id @ (lam @ (X336 @ sh @ one)) @ id)) = $true)) | ~ ! [X342 : term,X341 : term] : ((((X335 @ id @ X342 @ id)) != $true) | ($true != ((X335 @ id @ X341 @ id))) | ($true = ((X335 @ id @ (ap @ (sub @ X341 @ id) @ X342) @ id))))))))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (pushprop_lthm = $true) & (ulamvar1 != $true) & (pushprop_lem1v2_gthm = $true) & (substmonoid_gthm = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (! [X304 : subst] : (((comp @ X304 @ id)) = X304) <=> (axidr = $true)) & ((hoasinduction_lem3v2_lthm = $true) <=> (! [X235 : term] : (((sub @ X235 @ id)) = X235) => ! [X237 : (subst > term > subst > $o),X236 : term,X238 : (term > $o)] : (~ ! [X244 : term] : (((X238 @ X244)) = ((X237 @ id @ X244 @ id))) | ~ ! [X251 : subst,X252 : subst,X250 : term,X253 : subst] : ((((X237 @ (comp @ X252 @ X251) @ (sub @ X250 @ X251) @ X253)) = $true) | (((X237 @ X252 @ X250 @ (comp @ X251 @ X253))) != $true)) | ~ ! [X248 : subst,X246 : subst,X245 : term,X247 : subst] : ((((X237 @ (comp @ X248 @ X246) @ (sub @ X245 @ X246) @ X247)) != $true) | ($true = ((X237 @ X248 @ X245 @ (comp @ X246 @ X247))))) | ~ ! [X239 : (subst > term > term)] : (~ ! [X243 : term,X241 : subst,X242 : subst] : (((sub @ (X239 @ X241 @ X243) @ X242)) = ((X239 @ (comp @ X241 @ X242) @ (sub @ X243 @ X242)))) | ~ ! [X240 : term] : (($true != ((X237 @ id @ X240 @ id))) | (((X237 @ id @ (X239 @ id @ X240) @ id)) = $true)) | ($true = ((X237 @ id @ (lam @ (X239 @ sh @ one)) @ id)))) | (((X238 @ (lam @ X236))) = $true) | ~ ! [X249 : term] : ((((X238 @ X249)) != $true) | (((X238 @ (sub @ X236 @ (push @ X249 @ id)))) = $true))))) & ! [X40 : d_subst,X39 : d_subst] : ((((d2subst @ X40)) = ((d2subst @ X39))) => (X39 = X40)) & (! [X139 : (subst > term > term),X137 : term,X138 : term] : (~ ! [X140 : subst,X142 : subst,X141 : term] : (((sub @ (X139 @ X142 @ X141) @ X140)) = ((X139 @ (comp @ X142 @ X140) @ (sub @ X141 @ X140)))) | (((ap @ (sub @ X138 @ id) @ X137)) != ((lam @ (X139 @ sh @ one))))) <=> (hoaslamnotap = $true)) & (one = ((d2term @ d_one))) & (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)))))))))))) & ((hoasapinj2 = $true) <=> ! [X455 : term,X454 : term,X456 : term,X457 : term] : ((((ap @ (sub @ X457 @ id) @ X455)) != ((ap @ (sub @ X454 @ id) @ X456))) | (X455 = X456))) & ((axvarcons = $true) <=> ! [X206 : term,X205 : subst] : (((sub @ one @ (push @ X206 @ X205))) = X206)) & (hoasinduction_lem1 = $true) & ((induction2 = $true) <=> ! [X345 : (term > $o),X346 : term] : (~ ! [X350 : term,X351 : term] : (($true != ((X345 @ X350))) | ($true != ((X345 @ X351))) | (((X345 @ (ap @ X351 @ X350))) = $true)) | ~ ! [X347 : term] : ((((X345 @ X347)) = $true) | (((var @ X347)) != $true)) | ~ ! [X348 : term] : (~ ! [X349 : term] : (($true = ((X345 @ (sub @ X348 @ (push @ X349 @ id))))) | ($true != ((X345 @ X349)))) | ($true = ((X345 @ (lam @ X348))))) | ($true = ((X345 @ X346))))) & ! [X369 : d_subst] : (d_id = X369) & (pushprop_lem2v2_gthm = $true) & (! [X476 : term,X477 : (subst > term > subst > $o)] : (~ ! [X478 : term] : ((((X477 @ id @ (sub @ X476 @ (push @ X478 @ id)) @ id)) = $true) | (((X477 @ id @ X478 @ id)) != $true)) | ($true = ((X477 @ id @ (lam @ (sub @ X476 @ (push @ one @ sh))) @ id))) | ~ ! [X479 : (subst > term > term),X480 : term] : (~ ! [X487 : subst,X485 : subst,X486 : term] : (((X479 @ (comp @ X485 @ X487) @ (sub @ X486 @ X487))) = ((sub @ X480 @ (push @ (sub @ X486 @ X487) @ (comp @ X485 @ X487))))) | ~ ! [X484 : term] : (($true = ((X477 @ id @ (X479 @ id @ X484) @ id))) | ($true != ((X477 @ id @ X484 @ id)))) | ($true = ((X477 @ id @ (lam @ (X479 @ sh @ one)) @ id))) | ~ ! [X482 : subst,X483 : term,X481 : subst] : (((sub @ (sub @ X480 @ (push @ X483 @ X482)) @ X481)) = ((sub @ (X479 @ X482 @ X483) @ X481))))) <=> (hoasinduction_lem3aaa = $true)) & (hoasinduction_lem3b_gthm = $true) & ((apinj2 = $true) <=> ! [X263 : term,X264 : term,X266 : term,X265 : term] : ((((ap @ X265 @ X266)) != ((ap @ X263 @ X264))) | (X264 = X266))) & (ulamvarind = $true) & (lamnotvar = $true) & (hoasinduction_lem3_gthm = $true) & (hoaslamnotap_lthm = $true) & ((hoasapinj1_lthm = $true) <=> (! [X523 : term] : (((sub @ X523 @ id)) = X523) => (! [X526 : term,X525 : term,X524 : term,X527 : term] : ((X524 = X525) | (((ap @ X525 @ X526)) != ((ap @ X524 @ X527)))) => ! [X531 : term,X528 : term,X530 : term,X529 : term] : ((X528 = X530) | (((ap @ (sub @ X530 @ id) @ X529)) != ((ap @ (sub @ X528 @ id) @ X531))))))) & ((hoasapinj2_lthm = $true) <=> (! [X227 : term,X228 : term,X229 : term,X230 : term] : ((((ap @ X227 @ X228)) != ((ap @ X230 @ X229))) | (X228 = X229)) => ! [X234 : term,X231 : term,X233 : term,X232 : term] : ((((ap @ (sub @ X233 @ id) @ X234)) != ((ap @ (sub @ X232 @ id) @ X231))) | (X231 = X234)))) & ((hoasinduction_lem3_lthm = $true) <=> (! [X490 : term] : (((sub @ X490 @ id)) = X490) => (! [X491 : term,X492 : (subst > term > subst > $o)] : ((((X492 @ id @ (lam @ (sub @ X491 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X498 : term] : (($true != ((X492 @ id @ X498 @ id))) | (((X492 @ id @ (sub @ X491 @ (push @ X498 @ id)) @ id)) = $true)) | ~ ! [X493 : (subst > term > term)] : (~ ! [X497 : term] : ((((X492 @ id @ X497 @ id)) != $true) | ($true = ((X492 @ id @ (X493 @ id @ X497) @ id)))) | (((X492 @ id @ (lam @ (X493 @ sh @ one)) @ id)) = $true) | ~ ! [X496 : term,X494 : subst,X495 : subst] : (((sub @ (X493 @ X494 @ X496) @ X495)) = ((X493 @ (comp @ X494 @ X495) @ (sub @ X496 @ X495)))))) => ! [X499 : term,X500 : (subst > term > subst > $o)] : (($true = ((X500 @ id @ (lam @ X499) @ id))) | ~ ! [X502 : subst,X501 : term,X504 : subst,X503 : subst] : ((((X500 @ X504 @ X501 @ (comp @ X503 @ X502))) = $true) | (((X500 @ (comp @ X504 @ X503) @ (sub @ X501 @ X503) @ X502)) != $true)) | ~ ! [X505 : (subst > term > term)] : (~ ! [X506 : term] : ((((X500 @ id @ (X505 @ id @ X506) @ id)) = $true) | (((X500 @ id @ X506 @ id)) != $true)) | (((X500 @ id @ (lam @ (X505 @ sh @ one)) @ id)) = $true) | ~ ! [X507 : term,X509 : subst,X508 : subst] : (((X505 @ (comp @ X509 @ X508) @ (sub @ X507 @ X508))) = ((sub @ (X505 @ X509 @ X507) @ X508)))) | ~ ! [X512 : subst,X514 : subst,X511 : term,X513 : subst] : ((((X500 @ X514 @ X511 @ (comp @ X513 @ X512))) != $true) | (((X500 @ (comp @ X514 @ X513) @ (sub @ X511 @ X513) @ X512)) = $true)) | ~ ! [X510 : term] : ((((X500 @ id @ X510 @ id)) != $true) | (((X500 @ id @ (sub @ X499 @ (push @ X510 @ id)) @ id)) = $true)))))) & ((hoasinduction = $true) <=> ! [X187 : term,X188 : (subst > term > subst > $o)] : ((((X188 @ id @ X187 @ id)) = $true) | ~ ! [X191 : term,X190 : term] : ((((X188 @ id @ (ap @ (sub @ X191 @ id) @ X190) @ id)) = $true) | ($true != ((X188 @ id @ X191 @ id))) | (((X188 @ id @ X190 @ id)) != $true)) | ~ ! [X202 : subst,X201 : subst,X203 : term,X204 : subst] : ((((X188 @ (comp @ X202 @ X201) @ (sub @ X203 @ X201) @ X204)) != $true) | ($true = ((X188 @ X202 @ X203 @ (comp @ X201 @ X204))))) | ~ ! [X192 : (subst > term > term)] : ((((X188 @ id @ (lam @ (X192 @ sh @ one)) @ id)) = $true) | ~ ! [X196 : subst,X194 : term,X195 : subst] : (((X192 @ (comp @ X196 @ X195) @ (sub @ X194 @ X195))) = ((sub @ (X192 @ X196 @ X194) @ X195))) | ~ ! [X193 : term] : (($true = ((X188 @ id @ (X192 @ id @ X193) @ id))) | (((X188 @ id @ X193 @ id)) != $true))) | ~ ! [X199 : subst,X198 : subst,X197 : subst,X200 : term] : (($true = ((X188 @ (comp @ X198 @ X197) @ (sub @ X200 @ X197) @ X199))) | (((X188 @ X198 @ X200 @ (comp @ X197 @ X199))) != $true)) | ~ ! [X189 : term] : (($true = ((X188 @ id @ X189 @ id))) | (((var @ (sub @ X189 @ id))) != $true)))) & (! [X473 : term] : (((sub @ X473 @ id)) = X473) <=> (termmset = $true)) & (hoasapinj1_gthm = $true) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X300 : term] : ~ ! [X301 : (subst > term > term)] : (((sub @ X300 @ (push @ one @ sh))) != ((X301 @ sh @ one)))) & ((! [X434 : subst,X433 : term] : (((sub @ one @ (push @ X433 @ X434))) = X433) => (! [X436 : term,X435 : subst] : (((comp @ sh @ (push @ X436 @ X435))) = X435) => (! [X437 : term,X438 : term] : ((X437 = X438) | (((lam @ X438)) != ((lam @ X437)))) => ! [X442 : subst,X441 : (subst > term > term),X440 : (subst > term > term),X439 : term] : ((((lam @ (X441 @ sh @ one))) != ((lam @ (X440 @ sh @ one)))) | (((X440 @ X442 @ X439)) = ((X441 @ X442 @ X439))) | ~ ! [X448 : subst,X447 : subst,X446 : term] : (((sub @ (X440 @ X447 @ X446) @ X448)) = ((X440 @ (comp @ X447 @ X448) @ (sub @ X446 @ X448)))) | ~ ! [X444 : subst,X445 : subst,X443 : term] : (((sub @ (X441 @ X444 @ X443) @ X445)) = ((X441 @ (comp @ X444 @ X445) @ (sub @ X443 @ X445)))))))) <=> (hoaslaminj_lthm = $true)) & (! [X53 : term,X55 : (subst > term > subst > $o),X54 : term] : (($true != ((X55 @ id @ X53 @ id))) | ($true != ((X55 @ id @ X54 @ id))) | ~ ! [X61 : term,X60 : term] : (($true != ((X55 @ id @ X61 @ id))) | ($true = ((X55 @ id @ (ap @ (sub @ X61 @ id) @ X60) @ id))) | ($true != ((X55 @ id @ X60 @ id)))) | ~ ! [X62 : term,X64 : subst,X65 : subst,X63 : subst] : (($true = ((X55 @ X64 @ X62 @ (comp @ X63 @ X65)))) | (((X55 @ (comp @ X64 @ X63) @ (sub @ X62 @ X63) @ X65)) != $true)) | ~ ! [X57 : term,X56 : subst,X59 : subst,X58 : subst] : ((((X55 @ (comp @ X56 @ X58) @ (sub @ X57 @ X58) @ X59)) = $true) | ($true != ((X55 @ X56 @ X57 @ (comp @ X58 @ X59))))) | ($true = ((X55 @ id @ (ap @ X53 @ X54) @ id)))) <=> (hoasinduction_lem2 = $true)) & ((induction2lem = $true) <=> ! [X274 : term,X272 : (term > $o),X273 : subst] : ((((X272 @ (sub @ X274 @ X273))) = $true) | ~ ! [X279 : term] : (($true != ((var @ X279))) | (((X272 @ (sub @ X279 @ X273))) = $true)) | ~ ! [X275 : term] : (~ ! [X276 : term] : (($true = ((X272 @ (sub @ X275 @ (push @ X276 @ id))))) | (((X272 @ X276)) != $true)) | ($true = ((X272 @ (lam @ X275))))) | ~ ! [X277 : term,X278 : term] : ((((X272 @ (ap @ X277 @ X278))) = $true) | (((X272 @ X278)) != $true) | ($true != ((X272 @ X277)))))) & (axclos = $true) & (hoasinduction_lem3v2_gthm = $true) & ((substmonoid_lthm = $true) <=> (! [X15 : subst] : (((comp @ id @ X15)) = X15) => (! [X16 : subst] : (((comp @ X16 @ id)) = X16) => (! [X17 : subst] : (((comp @ X17 @ id)) = X17) & ! [X18 : subst] : (((comp @ id @ X18)) = X18))))) & (induction2_gthm = $true) & (axmap = $true) & (hoasapnotvar = $true) & (sh = ((d2subst @ d_id))) & ((hoasinduction_lem3v2a_lthm = $true) <=> (! [X281 : term] : ~ ! [X282 : (subst > term > term)] : ~ ! [X284 : term,X283 : subst] : (((X282 @ X283 @ X284)) = ((sub @ X281 @ (push @ X284 @ X283)))) => (! [X285 : term] : (((sub @ X285 @ id)) = X285) => ! [X287 : (term > $o),X288 : term,X286 : (subst > term > subst > $o)] : (~ ! [X289 : term] : ((((X287 @ X289)) != $true) | ($true = ((X287 @ (sub @ X288 @ (push @ X289 @ id)))))) | ~ ! [X290 : (subst > term > term)] : (~ ! [X293 : subst,X292 : subst,X294 : term] : (((X290 @ (comp @ X293 @ X292) @ (sub @ X294 @ X292))) = ((sub @ (X290 @ X293 @ X294) @ X292))) | ~ ! [X291 : term] : ((((X286 @ id @ (X290 @ id @ X291) @ id)) = $true) | (((X286 @ id @ X291 @ id)) != $true)) | ($true = ((X286 @ id @ (lam @ (X290 @ sh @ one)) @ id)))) | (((X287 @ (lam @ X288))) = $true) | ~ ! [X295 : term] : (((X286 @ id @ X295 @ id)) = ((X287 @ X295))))))) & (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) & (hoasinduction_lem1_lthm = $true) & (termmset_gthm = $true) & (hoasapnotvar_gthm = $true)),
  inference(flattening,[],[f8])).
thf(f10,plain,(
  (hoasinduction_lem2 != ((!! @ (subst > term > subst > $o) @ (^[Y0 : subst > term > subst > $o]: ((!! @ subst @ (^[Y1 : subst]: (!! @ subst @ (^[Y2 : subst]: (!! @ subst @ (^[Y3 : subst]: (!! @ term @ (^[Y4 : term]: ((Y0 @ Y1 @ Y4 @ (comp @ Y2 @ Y3)) => (Y0 @ (comp @ Y1 @ Y2) @ (sub @ Y4 @ Y2) @ Y3)))))))))) => ((!! @ term @ (^[Y1 : term]: (!! @ subst @ (^[Y2 : subst]: (!! @ subst @ (^[Y3 : subst]: (!! @ subst @ (^[Y4 : subst]: ((Y0 @ (comp @ Y3 @ Y2) @ (sub @ Y1 @ Y2) @ Y4) => (Y0 @ Y3 @ Y1 @ (comp @ Y2 @ Y4))))))))))) => ((!! @ term @ (^[Y1 : term]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) => ((Y0 @ id @ Y1 @ id) => (Y0 @ id @ (hoasap @ id @ Y2 @ id @ Y1) @ id))))))) => (!! @ term @ (^[Y1 : term]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y1 @ id) => ((Y0 @ id @ Y2 @ id) => (Y0 @ id @ (ap @ Y1 @ Y2) @ id))))))))))))))),
  inference(flattening,[],[f7])).
thf(f11,plain,(
  (! [X1 : term,X0 : term,X3 : term,X2 : term] : ((X0 = X3) | (((ap @ X3 @ X2)) != ((ap @ X0 @ X1)))) <=> (apinj1 = $true)) & (((d2subst @ d_id)) = id) & ((hoasinduction_lem0 = $true) <=> ! [X211 : (subst > term > subst > $o)] : ? [X212 : (term > $o)] : ! [X213 : term] : (((X212 @ X213)) = ((X211 @ id @ X213 @ id)))) & (pushprop_gthm = $true) & (hoasapinj2_gthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoasinduction_lem1v2 = $true) & (axassoc = $true) & ((hoasinduction_lthm_3 = $true) <=> ((((! [X415 : (subst > term > subst > $o),X416 : term] : (? [X420 : subst,X421 : subst,X422 : subst,X419 : term] : (($true != ((X415 @ X421 @ X419 @ (comp @ X422 @ X420)))) & ($true = ((X415 @ (comp @ X421 @ X422) @ (sub @ X419 @ X422) @ X420)))) | ? [X417 : term,X418 : term] : (($true != ((X415 @ id @ (ap @ (sub @ X418 @ id) @ X417) @ id))) & ($true = ((X415 @ id @ X417 @ id))) & (((X415 @ id @ X418 @ id)) = $true)) | ? [X428 : (subst > term > term)] : (! [X432 : term] : ((((X415 @ id @ X432 @ id)) != $true) | ($true = ((X415 @ id @ (X428 @ id @ X432) @ id)))) & ($true != ((X415 @ id @ (lam @ (X428 @ sh @ one)) @ id))) & ! [X429 : subst,X430 : term,X431 : subst] : (((X428 @ (comp @ X431 @ X429) @ (sub @ X430 @ X429))) = ((sub @ (X428 @ X431 @ X430) @ X429)))) | ($true = ((X415 @ id @ X416 @ id))) | ? [X426 : subst,X424 : term,X425 : subst,X427 : subst] : ((((X415 @ (comp @ X425 @ X427) @ (sub @ X424 @ X427) @ X426)) != $true) & (((X415 @ X425 @ X424 @ (comp @ X427 @ X426))) = $true)) | ? [X423 : term] : ((((X415 @ id @ X423 @ id)) != $true) & (((var @ (sub @ X423 @ id))) = $true))) | ? [X407 : (term > $o),X405 : (subst > term > subst > $o),X406 : term] : ((((X407 @ (lam @ X406))) != $true) & ! [X410 : (subst > term > term)] : (? [X411 : term] : (($true != ((X405 @ id @ (X410 @ id @ X411) @ id))) & ($true = ((X405 @ id @ X411 @ id)))) | ? [X413 : term,X412 : subst,X414 : subst] : (((X410 @ (comp @ X412 @ X414) @ (sub @ X413 @ X414))) != ((sub @ (X410 @ X412 @ X413) @ X414))) | (((X405 @ id @ (lam @ (X410 @ sh @ one)) @ id)) = $true)) & ! [X408 : term] : (((X405 @ id @ X408 @ id)) = ((X407 @ X408))) & ! [X409 : term] : ((((X407 @ (sub @ X406 @ (push @ X409 @ id)))) = $true) | (((X407 @ X409)) != $true)))) | ? [X404 : term] : (((sub @ X404 @ id)) != X404)) | ? [X398 : term,X397 : (term > $o)] : ((((X397 @ X398)) != $true) & ! [X399 : term] : ((((X397 @ (lam @ X399))) = $true) | ? [X400 : term] : (($true != ((X397 @ (sub @ X399 @ (push @ X400 @ id))))) & (((X397 @ X400)) = $true))) & ! [X401 : term,X402 : term] : (($true = ((X397 @ (ap @ X401 @ X402)))) | ($true != ((X397 @ X401))) | ($true != ((X397 @ X402)))) & ! [X403 : term] : (($true != ((var @ X403))) | ($true = ((X397 @ X403)))))) | ? [X394 : (subst > term > subst > $o)] : ! [X395 : (term > $o)] : ? [X396 : term] : (((X395 @ X396)) != ((X394 @ id @ X396 @ id))))) & ((hoasinduction_lem3v2 = $true) <=> ! [X148 : (subst > term > subst > $o),X150 : (term > $o),X149 : term] : (? [X155 : (subst > term > term)] : (! [X156 : term] : (($true != ((X148 @ id @ X156 @ id))) | ($true = ((X148 @ id @ (X155 @ id @ X156) @ id)))) & (((X148 @ id @ (lam @ (X155 @ sh @ one)) @ id)) != $true) & ! [X159 : term,X158 : subst,X157 : subst] : (((X155 @ (comp @ X157 @ X158) @ (sub @ X159 @ X158))) = ((sub @ (X155 @ X157 @ X159) @ X158)))) | ? [X165 : term] : ((((X150 @ (sub @ X149 @ (push @ X165 @ id)))) != $true) & (((X150 @ X165)) = $true)) | ? [X152 : term,X151 : subst,X153 : subst,X154 : subst] : ((((X148 @ X151 @ X152 @ (comp @ X154 @ X153))) != $true) & (((X148 @ (comp @ X151 @ X154) @ (sub @ X152 @ X154) @ X153)) = $true)) | ? [X164 : term] : (((X148 @ id @ X164 @ id)) != ((X150 @ X164))) | (((X150 @ (lam @ X149))) = $true) | ? [X161 : subst,X163 : term,X160 : subst,X162 : subst] : ((((X148 @ (comp @ X161 @ X160) @ (sub @ X163 @ X160) @ X162)) != $true) & ($true = ((X148 @ X161 @ X163 @ (comp @ X160 @ X162))))))) & ((hoasinduction_lem2v2 = $true) <=> ! [X459 : term,X458 : term,X461 : (term > $o),X460 : (subst > term > subst > $o)] : (? [X471 : subst,X470 : subst,X469 : term,X472 : subst] : ((((X460 @ (comp @ X472 @ X470) @ (sub @ X469 @ X470) @ X471)) != $true) & ($true = ((X460 @ X472 @ X469 @ (comp @ X470 @ X471))))) | ? [X464 : term] : (((X460 @ id @ X464 @ id)) != ((X461 @ X464))) | ? [X465 : subst,X468 : term,X467 : subst,X466 : subst] : ((((X460 @ X467 @ X468 @ (comp @ X465 @ X466))) != $true) & (((X460 @ (comp @ X467 @ X465) @ (sub @ X468 @ X465) @ X466)) = $true)) | (((X461 @ (ap @ X459 @ X458))) = $true) | ? [X463 : term,X462 : term] : (($true = ((X460 @ id @ X463 @ id))) & ($true != ((X460 @ id @ (ap @ (sub @ X462 @ id) @ X463) @ id))) & (((X460 @ id @ X462 @ id)) = $true)) | ($true != ((X461 @ X458))) | ($true != ((X461 @ X459))))) & (hoasinduction_lem3a_gthm = $true) & (! [X207 : term] : ? [X208 : (subst > term > term)] : ! [X210 : term,X209 : subst] : (((sub @ X207 @ (push @ X210 @ X209))) = ((X208 @ X209 @ X210))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & (pushprop_lem2v2_lthm = $true) & ((hoaslaminj = $true) <=> ! [X42 : (subst > term > term),X43 : subst,X41 : (subst > term > term),X44 : term] : ((((lam @ (X41 @ sh @ one))) != ((lam @ (X42 @ sh @ one)))) | ? [X45 : term,X47 : subst,X46 : subst] : (((sub @ (X42 @ X46 @ X45) @ X47)) != ((X42 @ (comp @ X46 @ X47) @ (sub @ X45 @ X47)))) | ? [X48 : term,X49 : subst,X50 : subst] : (((sub @ (X41 @ X49 @ X48) @ X50)) != ((X41 @ (comp @ X49 @ X50) @ (sub @ X48 @ X50)))) | (((X41 @ X43 @ X44)) = ((X42 @ X43 @ X44))))) & (hoasinduction_lem2v2_gthm = $true) & (! [X532 : term,X533 : (subst > term > subst > $o)] : (? [X534 : term] : ((((X533 @ id @ (sub @ X532 @ (push @ X534 @ id)) @ id)) != $true) & ($true = ((X533 @ id @ X534 @ id)))) | (((X533 @ id @ (lam @ X532) @ id)) = $true) | ? [X535 : (subst > term > term)] : (! [X539 : term] : (($true = ((X533 @ id @ (X535 @ id @ X539) @ id))) | ($true != ((X533 @ id @ X539 @ id)))) & ! [X537 : subst,X538 : subst,X536 : term] : (((sub @ (X535 @ X538 @ X536) @ X537)) = ((X535 @ (comp @ X538 @ X537) @ (sub @ X536 @ X537)))) & (((X533 @ id @ (lam @ (X535 @ sh @ one)) @ id)) != $true))) <=> (hoasinduction_lem3a = $true)) & (pushprop_lem0_gthm = $true) & (((var @ (d2term @ d_one))) != $true) & (((! [X180 : term,X181 : (term > $o)] : (? [X184 : term] : ((((var @ X184)) = $true) & ($true != ((X181 @ X184)))) | ($true = ((X181 @ X180))) | ? [X185 : term] : (! [X186 : term] : (($true = ((X181 @ (sub @ X185 @ (push @ X186 @ id))))) | (((X181 @ X186)) != $true)) & ($true != ((X181 @ (lam @ X185))))) | ? [X183 : term,X182 : term] : ((((X181 @ X182)) = $true) & (((X181 @ X183)) = $true) & (((X181 @ (ap @ X183 @ X182))) != $true))) | ? [X172 : subst,X173 : term,X174 : (term > $o)] : ((((X174 @ (sub @ X173 @ X172))) != $true) & ! [X176 : term,X175 : term] : (($true != ((X174 @ X175))) | ($true = ((X174 @ (ap @ X175 @ X176)))) | (((X174 @ X176)) != $true)) & ! [X179 : term] : (($true = ((X174 @ (sub @ X179 @ X172)))) | (((var @ X179)) != $true)) & ! [X177 : term] : (? [X178 : term] : ((((X174 @ X178)) = $true) & ($true != ((X174 @ (sub @ X177 @ (push @ X178 @ id)))))) | ($true = ((X174 @ (lam @ X177))))))) | ? [X171 : term] : (((sub @ X171 @ id)) != X171)) <=> (induction2_lthm = $true)) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (pushprop_lem1_gthm = $true) & (axvarshift = $true) & (apnotvar = $true) & (pushprop_lem3v2_lthm = $true) & ((termmset_lthm = $true) <=> (! [X475 : term] : (((sub @ X475 @ id)) = X475) | ? [X474 : term] : (((sub @ X474 @ id)) != X474))) & ((! [X521 : subst] : (((comp @ X521 @ id)) = X521) & ! [X522 : subst] : (((comp @ id @ X522)) = X522)) <=> (substmonoid = $true)) & ((hoasinduction_lem3aa = $true) <=> ! [X130 : (subst > term > subst > $o),X129 : term] : (($true = ((X130 @ id @ (lam @ (sub @ X129 @ (push @ one @ sh))) @ id))) | ? [X131 : term] : ((((X130 @ id @ (sub @ X129 @ (push @ X131 @ id)) @ id)) != $true) & (((X130 @ id @ X131 @ id)) = $true)) | ? [X132 : (subst > term > term)] : (! [X133 : term] : (($true = ((X130 @ id @ (X132 @ id @ X133) @ id))) | ($true != ((X130 @ id @ X133 @ id)))) & ($true != ((X130 @ id @ (lam @ (X132 @ sh @ one)) @ id))) & ! [X136 : subst,X135 : term,X134 : subst] : (((sub @ (X132 @ X136 @ X135) @ X134)) = ((X132 @ (comp @ X136 @ X134) @ (sub @ X135 @ X134))))))) & (hoaslamnotvar_gthm = $true) & ((pushprop_lem1v2 = $true) <=> ! [X267 : (term > $o),X268 : subst,X269 : (term > $o),X270 : term] : ((((X269 @ one)) = $true) | (((X267 @ X270)) != $true) | ? [X271 : term] : (((X267 @ (sub @ X271 @ (push @ X270 @ X268)))) != ((X269 @ X271))))) & (hoaslamnotap_gthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (hoasinduction_lem1v2_gthm = $true) & (! [X144 : (term > $o),X145 : subst,X143 : term] : ? [X146 : (term > $o)] : ! [X147 : term] : (((X144 @ (sub @ X147 @ (push @ X143 @ X145)))) = ((X146 @ X147))) <=> (pushprop_lem0_lthm = $true)) & (ulamvarsh = $true) & (! [X392 : term,X393 : subst,X391 : term,X389 : (term > $o),X390 : (term > $o)] : (($true = ((X389 @ (sub @ X392 @ (push @ X391 @ X393))))) | ($true != ((X390 @ X392)))) <=> (pushprop_lem1 = $true)) & (((! [X362 : (subst > term > subst > $o),X361 : term] : (? [X363 : (subst > term > term)] : (! [X365 : subst,X366 : term,X364 : subst] : (((sub @ (X363 @ X365 @ X366) @ X364)) = ((X363 @ (comp @ X365 @ X364) @ (sub @ X366 @ X364)))) & ! [X367 : term] : ((((X362 @ id @ (X363 @ id @ X367) @ id)) = $true) | (((X362 @ id @ X367 @ id)) != $true)) & ($true != ((X362 @ id @ (lam @ (X363 @ sh @ one)) @ id)))) | ? [X368 : term] : ((((X362 @ id @ X368 @ id)) = $true) & ($true != ((X362 @ id @ (sub @ X361 @ (push @ X368 @ id)) @ id)))) | ($true = ((X362 @ id @ (lam @ X361) @ id)))) | ? [X354 : (subst > term > subst > $o),X353 : term] : ((((X354 @ id @ (lam @ (sub @ X353 @ (push @ one @ sh))) @ id)) != $true) & ! [X355 : term] : (($true = ((X354 @ id @ (sub @ X353 @ (push @ X355 @ id)) @ id))) | (((X354 @ id @ X355 @ id)) != $true)) & ! [X356 : (subst > term > term)] : (($true = ((X354 @ id @ (lam @ (X356 @ sh @ one)) @ id))) | ? [X357 : term] : (($true = ((X354 @ id @ X357 @ id))) & (((X354 @ id @ (X356 @ id @ X357) @ id)) != $true)) | ? [X359 : subst,X358 : subst,X360 : term] : (((sub @ (X356 @ X358 @ X360) @ X359)) != ((X356 @ (comp @ X358 @ X359) @ (sub @ X360 @ X359))))))) | ? [X352 : term] : (((sub @ X352 @ id)) != X352)) <=> (hoasinduction_lem3a_lthm = $true)) & (hoaslamnotvar = $true) & (pushprop_lthm_orig = $true) & ((induction = $true) <=> ! [X516 : (term > $o),X515 : term] : (? [X520 : term] : (($true != ((X516 @ X520))) & ($true = ((var @ X520)))) | (((X516 @ X515)) = $true) | ? [X519 : term] : (($true = ((X516 @ X519))) & (((X516 @ (lam @ X519))) != $true)) | ? [X517 : term,X518 : term] : ((((X516 @ (ap @ X517 @ X518))) != $true) & ($true = ((X516 @ X517))) & (((X516 @ X518)) = $true)))) & (axapp = $true) & (! [X6 : (subst > term > subst > $o),X4 : (term > $o),X5 : term] : (($true = ((X4 @ (lam @ X5)))) | ? [X7 : (subst > term > term)] : (! [X11 : subst,X10 : term,X9 : subst] : (((sub @ (X7 @ X11 @ X10) @ X9)) = ((X7 @ (comp @ X11 @ X9) @ (sub @ X10 @ X9)))) & ($true != ((X6 @ id @ (lam @ (X7 @ sh @ one)) @ id))) & ! [X8 : term] : ((((X6 @ id @ (X7 @ id @ X8) @ id)) = $true) | ($true != ((X6 @ id @ X8 @ id))))) | ? [X12 : term] : (((X6 @ id @ X12 @ id)) != ((X4 @ X12))) | ? [X13 : term] : ((((X4 @ X13)) = $true) & ($true != ((X4 @ (sub @ X5 @ (push @ X13 @ id))))))) <=> (hoasinduction_lem3v2a = $true)) & (pushprop = $true) & ((axidl = $true) <=> ! [X450 : subst] : (((comp @ id @ X450)) = X450)) & ((hoasinduction_lthm = $true) <=> (((! [X111 : (subst > term > subst > $o),X112 : term] : (? [X114 : term,X113 : term] : ((((X111 @ id @ X114 @ id)) = $true) & ($true != ((X111 @ id @ (ap @ (sub @ X113 @ id) @ X114) @ id))) & ($true = ((X111 @ id @ X113 @ id)))) | (((X111 @ id @ X112 @ id)) = $true) | ? [X124 : term] : ((((var @ (sub @ X124 @ id))) = $true) & (((X111 @ id @ X124 @ id)) != $true)) | ? [X115 : (subst > term > term)] : (! [X117 : subst,X118 : subst,X116 : term] : (((sub @ (X115 @ X118 @ X116) @ X117)) = ((X115 @ (comp @ X118 @ X117) @ (sub @ X116 @ X117)))) & ! [X119 : term] : ((((X111 @ id @ X119 @ id)) != $true) | (((X111 @ id @ (X115 @ id @ X119) @ id)) = $true)) & ($true != ((X111 @ id @ (lam @ (X115 @ sh @ one)) @ id)))) | ? [X128 : term,X126 : subst,X125 : subst,X127 : subst] : ((((X111 @ X125 @ X128 @ (comp @ X127 @ X126))) != $true) & (((X111 @ (comp @ X125 @ X127) @ (sub @ X128 @ X127) @ X126)) = $true)) | ? [X123 : term,X121 : subst,X122 : subst,X120 : subst] : (($true = ((X111 @ X121 @ X123 @ (comp @ X120 @ X122)))) & ($true != ((X111 @ (comp @ X121 @ X120) @ (sub @ X123 @ X120) @ X122))))) | ? [X95 : term,X96 : (subst > term > subst > $o)] : (! [X105 : term] : ((((X96 @ id @ (sub @ X95 @ (push @ X105 @ id)) @ id)) = $true) | ($true != ((X96 @ id @ X105 @ id)))) & (((X96 @ id @ (lam @ X95) @ id)) != $true) & ! [X98 : subst,X99 : subst,X100 : subst,X97 : term] : ((((X96 @ (comp @ X98 @ X99) @ (sub @ X97 @ X99) @ X100)) != $true) | (((X96 @ X98 @ X97 @ (comp @ X99 @ X100))) = $true)) & ! [X106 : (subst > term > term)] : ((((X96 @ id @ (lam @ (X106 @ sh @ one)) @ id)) = $true) | ? [X110 : subst,X108 : term,X109 : subst] : (((X106 @ (comp @ X110 @ X109) @ (sub @ X108 @ X109))) != ((sub @ (X106 @ X110 @ X108) @ X109))) | ? [X107 : term] : ((((X96 @ id @ (X106 @ id @ X107) @ id)) != $true) & ($true = ((X96 @ id @ X107 @ id))))) & ! [X104 : subst,X102 : subst,X101 : term,X103 : subst] : ((((X96 @ (comp @ X102 @ X103) @ (sub @ X101 @ X103) @ X104)) = $true) | ($true != ((X96 @ X102 @ X101 @ (comp @ X103 @ X104))))))) | ? [X83 : term,X82 : term,X84 : (subst > term > subst > $o)] : (! [X85 : subst,X86 : subst,X87 : subst,X88 : term] : ((((X84 @ X85 @ X88 @ (comp @ X86 @ X87))) = $true) | ($true != ((X84 @ (comp @ X85 @ X86) @ (sub @ X88 @ X86) @ X87)))) & (((X84 @ id @ X82 @ id)) = $true) & (((X84 @ id @ X83 @ id)) = $true) & ! [X94 : subst,X92 : term,X91 : subst,X93 : subst] : ((((X84 @ (comp @ X93 @ X91) @ (sub @ X92 @ X91) @ X94)) = $true) | (((X84 @ X93 @ X92 @ (comp @ X91 @ X94))) != $true)) & (((X84 @ id @ (ap @ X83 @ X82) @ id)) != $true) & ! [X89 : term,X90 : term] : ((((X84 @ id @ (ap @ (sub @ X89 @ id) @ X90) @ id)) = $true) | ($true != ((X84 @ id @ X89 @ id))) | ($true != ((X84 @ id @ X90 @ id)))))) | ? [X75 : term,X76 : (term > $o)] : (! [X77 : term] : ((((X76 @ (lam @ X77))) = $true) | ? [X78 : term] : ((((X76 @ X78)) = $true) & ($true != ((X76 @ (sub @ X77 @ (push @ X78 @ id))))))) & (((X76 @ X75)) != $true) & ! [X81 : term] : (($true != ((var @ X81))) | (((X76 @ X81)) = $true)) & ! [X80 : term,X79 : term] : ((((X76 @ X80)) != $true) | (((X76 @ (ap @ X79 @ X80))) = $true) | ($true != ((X76 @ X79))))))) & (induction2lem_gthm = $true) & (axabs = $true) & (pushprop_lem3v2 = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ! [X51 : term] : ? [X52 : d_term] : (((d2term @ X52)) = X51) & (hoaslamnotvar_lthm = $true) & (pushprop_lem2v2 = $true) & ((hoasinduction_no_psi_cond = $true) <=> ! [X66 : term,X67 : (subst > term > subst > $o)] : (? [X68 : (subst > term > term)] : (! [X72 : term] : ((((X67 @ id @ X72 @ id)) != $true) | (((X67 @ id @ (X68 @ id @ X72) @ id)) = $true)) & ($true != ((X67 @ id @ (lam @ (X68 @ sh @ one)) @ id))) & ! [X69 : subst,X71 : subst,X70 : term] : (((X68 @ (comp @ X69 @ X71) @ (sub @ X70 @ X71))) = ((sub @ (X68 @ X69 @ X70) @ X71)))) | ? [X74 : term,X73 : term] : ((((X67 @ id @ (ap @ (sub @ X73 @ id) @ X74) @ id)) != $true) & (((X67 @ id @ X74 @ id)) = $true) & (((X67 @ id @ X73 @ id)) = $true)) | ($true = ((X67 @ id @ X66 @ id))))) & ((axscons = $true) <=> ! [X449 : subst] : (((push @ (sub @ one @ X449) @ (comp @ sh @ X449))) = X449)) & (hoasinduction_gthm = $true) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X305 : term,X306 : (subst > term > subst > $o)] : (? [X307 : (subst > term > term)] : (($true != ((X306 @ id @ (lam @ (X307 @ sh @ one)) @ id))) & ! [X309 : subst,X311 : subst,X310 : term] : (((X307 @ (comp @ X311 @ X309) @ (sub @ X310 @ X309))) = ((sub @ (X307 @ X311 @ X310) @ X309))) & ! [X308 : term] : (($true = ((X306 @ id @ (X307 @ id @ X308) @ id))) | (((X306 @ id @ X308 @ id)) != $true))) | ($true = ((X306 @ id @ (lam @ (sub @ X305 @ (push @ one @ sh))) @ id))) | ? [X312 : term] : (($true != ((X306 @ id @ (sub @ X305 @ (push @ X312 @ id)) @ id))) & (((X306 @ id @ X312 @ id)) = $true)))) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((hoasinduction_lem0_lthm = $true) <=> ! [X451 : (subst > term > subst > $o)] : ? [X452 : (term > $o)] : ! [X453 : term] : (((X451 @ id @ X453 @ id)) = ((X452 @ X453)))) & ! [X280 : d_term] : (d_one = X280) & (hoasinduction_lem2_gthm = $true) & (hoasapnotvar_lthm = $true) & (! [X214 : term] : ? [X215 : (subst > term > term)] : ! [X217 : subst,X216 : term] : (((X215 @ X217 @ X216)) = ((sub @ X214 @ (push @ X216 @ X217)))) <=> (hoasinduction_lem3v2_f = $true)) & (lamnotap != $true) & ((axvarid = $true) <=> ! [X14 : term] : (((sub @ X14 @ id)) = X14)) & (! [X298 : term,X299 : term] : ((((sub @ X298 @ sh)) != ((sub @ X299 @ sh))) | (X298 = X299)) <=> (shinj = $true)) & (hoasinduction_lem1_gthm = $true) & ((? [X254 : subst,X255 : term] : (((sub @ one @ (push @ X255 @ X254))) != X255) | ! [X257 : (term > $o),X256 : subst,X258 : (term > $o),X259 : term] : (? [X260 : term] : (((X257 @ (sub @ X260 @ (push @ X259 @ X256)))) != ((X258 @ X260))) | ($true != ((X257 @ X259))) | ($true = ((X258 @ one))))) <=> (pushprop_lem1v2_lthm = $true)) & ((hoasinduction_lem3 = $true) <=> ! [X24 : (subst > term > subst > $o),X23 : term] : (? [X27 : subst,X26 : term,X28 : subst,X25 : subst] : (($true != ((X24 @ X25 @ X26 @ (comp @ X28 @ X27)))) & (((X24 @ (comp @ X25 @ X28) @ (sub @ X26 @ X28) @ X27)) = $true)) | ? [X29 : (subst > term > term)] : (! [X32 : subst,X31 : term,X30 : subst] : (((sub @ (X29 @ X30 @ X31) @ X32)) = ((X29 @ (comp @ X30 @ X32) @ (sub @ X31 @ X32)))) & (((X24 @ id @ (lam @ (X29 @ sh @ one)) @ id)) != $true) & ! [X33 : term] : ((((X24 @ id @ X33 @ id)) != $true) | (((X24 @ id @ (X29 @ id @ X33) @ id)) = $true))) | ? [X36 : subst,X37 : subst,X35 : subst,X34 : term] : ((((X24 @ (comp @ X37 @ X35) @ (sub @ X34 @ X35) @ X36)) != $true) & ($true = ((X24 @ X37 @ X34 @ (comp @ X35 @ X36))))) | ($true = ((X24 @ id @ (lam @ X23) @ id))) | ? [X38 : term] : (($true != ((X24 @ id @ (sub @ X23 @ (push @ X38 @ id)) @ id))) & (((X24 @ id @ X38 @ id)) = $true)))) & ((axshiftcons = $true) <=> ! [X489 : subst,X488 : term] : (((comp @ sh @ (push @ X488 @ X489))) = X489)) & (hoaslaminj_gthm = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ((laminj = $true) <=> ! [X262 : term,X261 : term] : ((X261 = X262) | (((lam @ X262)) != ((lam @ X261))))) & ((pushprop_lem1_lthm = $true) <=> ((! [X226 : subst,X223 : (term > $o),X224 : term,X225 : term,X222 : (term > $o)] : (($true != ((X222 @ X224))) | ($true = ((X223 @ (sub @ X224 @ (push @ X225 @ X226)))))) | ? [X221 : term,X220 : subst] : (((comp @ sh @ (push @ X221 @ X220))) != X220)) | ? [X219 : term,X218 : subst] : (((sub @ one @ (push @ X219 @ X218))) != X219))) & ((hoasinduction_lem3b = $true) <=> ! [X343 : term] : ? [X344 : (subst > term > term)] : (((sub @ X343 @ (push @ one @ sh))) = ((X344 @ sh @ one)))) & ((pushprop_lem0 = $true) <=> ! [X168 : term,X166 : subst,X167 : (term > $o)] : ? [X169 : (term > $o)] : ! [X170 : term] : (((X169 @ X170)) = ((X167 @ (sub @ X170 @ (push @ X168 @ X166)))))) & ! [X302 : d_term,X303 : d_term] : ((((d2term @ X302)) != ((d2term @ X303))) | (X302 = X303)) & ! [X296 : subst] : ? [X297 : d_subst] : (((d2subst @ X297)) = X296) & (((((! [X381 : (term > $o),X383 : subst,X382 : term] : (? [X384 : term] : (! [X385 : term] : ((((X381 @ (sub @ X384 @ (push @ X385 @ id)))) = $true) | (((X381 @ X385)) != $true)) & ($true != ((X381 @ (lam @ X384))))) | ? [X387 : term,X388 : term] : ((((X381 @ (ap @ X387 @ X388))) != $true) & (((X381 @ X388)) = $true) & (((X381 @ X387)) = $true)) | (((X381 @ (sub @ X382 @ X383))) = $true) | ? [X386 : term] : ((((var @ X386)) = $true) & (((X381 @ (sub @ X386 @ X383))) != $true))) | ? [X376 : term,X375 : (term > $o)] : ((((X375 @ X376)) != $true) & ! [X380 : term] : (($true = ((X375 @ (lam @ X380)))) | ($true != ((X375 @ X380)))) & ! [X379 : term] : (($true != ((var @ X379))) | (((X375 @ X379)) = $true)) & ! [X378 : term,X377 : term] : ((((X375 @ (ap @ X378 @ X377))) = $true) | ($true != ((X375 @ X377))) | (((X375 @ X378)) != $true)))) | ? [X374 : subst] : (((comp @ X374 @ id)) != X374)) | ? [X373 : subst,X372 : term] : (((comp @ sh @ (push @ X372 @ X373))) != X373)) | ? [X371 : term,X370 : subst] : (((sub @ one @ (push @ X371 @ X370))) != X371)) <=> (induction2lem_lthm = $true)) & ((hoasapinj1 = $true) <=> ! [X21 : term,X19 : term,X20 : term,X22 : term] : ((X21 = X22) | (((ap @ (sub @ X22 @ id) @ X19)) != ((ap @ (sub @ X21 @ id) @ X20))))) & (((((! [X335 : (subst > term > subst > $o),X334 : term] : (($true = ((X335 @ id @ X334 @ id))) | ? [X341 : term,X342 : term] : (($true != ((X335 @ id @ (ap @ (sub @ X341 @ id) @ X342) @ id))) & (((X335 @ id @ X342 @ id)) = $true) & ($true = ((X335 @ id @ X341 @ id)))) | ? [X336 : (subst > term > term)] : (! [X337 : term] : ((((X335 @ id @ X337 @ id)) != $true) | ($true = ((X335 @ id @ (X336 @ id @ X337) @ id)))) & ! [X338 : term,X340 : subst,X339 : subst] : (((sub @ (X336 @ X340 @ X338) @ X339)) = ((X336 @ (comp @ X340 @ X339) @ (sub @ X338 @ X339)))) & (((X335 @ id @ (lam @ (X336 @ sh @ one)) @ id)) != $true))) | ? [X325 : term,X324 : (subst > term > subst > $o),X326 : (term > $o)] : ((((X326 @ (lam @ X325))) != $true) & ! [X327 : (subst > term > term)] : (? [X331 : term] : (($true = ((X324 @ id @ X331 @ id))) & ($true != ((X324 @ id @ (X327 @ id @ X331) @ id)))) | ? [X329 : subst,X328 : subst,X330 : term] : (((sub @ (X327 @ X328 @ X330) @ X329)) != ((X327 @ (comp @ X328 @ X329) @ (sub @ X330 @ X329)))) | ($true = ((X324 @ id @ (lam @ (X327 @ sh @ one)) @ id)))) & ! [X332 : term] : ((((X326 @ X332)) != $true) | (((X326 @ (sub @ X325 @ (push @ X332 @ id)))) = $true)) & ! [X333 : term] : (((X326 @ X333)) = ((X324 @ id @ X333 @ id))))) | ? [X323 : term] : (((sub @ X323 @ id)) != X323)) | ? [X316 : (term > $o),X317 : term] : (! [X322 : term] : ((((X316 @ X322)) = $true) | ($true != ((var @ X322)))) & (((X316 @ X317)) != $true) & ! [X318 : term,X319 : term] : (($true = ((X316 @ (ap @ X319 @ X318)))) | (((X316 @ X319)) != $true) | (((X316 @ X318)) != $true)) & ! [X320 : term] : (? [X321 : term] : (($true != ((X316 @ (sub @ X320 @ (push @ X321 @ id))))) & ($true = ((X316 @ X321)))) | (((X316 @ (lam @ X320))) = $true)))) | ? [X313 : (subst > term > subst > $o)] : ! [X314 : (term > $o)] : ? [X315 : term] : (((X314 @ X315)) != ((X313 @ id @ X315 @ id)))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (pushprop_lthm = $true) & (ulamvar1 != $true) & (pushprop_lem1v2_gthm = $true) & (substmonoid_gthm = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (! [X304 : subst] : (((comp @ X304 @ id)) = X304) <=> (axidr = $true)) & ((hoasinduction_lem3v2_lthm = $true) <=> (? [X235 : term] : (((sub @ X235 @ id)) != X235) | ! [X236 : term,X238 : (term > $o),X237 : (subst > term > subst > $o)] : (? [X249 : term] : ((((X238 @ (sub @ X236 @ (push @ X249 @ id)))) != $true) & (((X238 @ X249)) = $true)) | ? [X246 : subst,X248 : subst,X247 : subst,X245 : term] : (($true != ((X237 @ X248 @ X245 @ (comp @ X246 @ X247)))) & (((X237 @ (comp @ X248 @ X246) @ (sub @ X245 @ X246) @ X247)) = $true)) | (((X238 @ (lam @ X236))) = $true) | ? [X252 : subst,X250 : term,X251 : subst,X253 : subst] : ((((X237 @ (comp @ X252 @ X251) @ (sub @ X250 @ X251) @ X253)) != $true) & (((X237 @ X252 @ X250 @ (comp @ X251 @ X253))) = $true)) | ? [X239 : (subst > term > term)] : (! [X243 : term,X241 : subst,X242 : subst] : (((sub @ (X239 @ X241 @ X243) @ X242)) = ((X239 @ (comp @ X241 @ X242) @ (sub @ X243 @ X242)))) & ! [X240 : term] : (($true != ((X237 @ id @ X240 @ id))) | (((X237 @ id @ (X239 @ id @ X240) @ id)) = $true)) & ($true != ((X237 @ id @ (lam @ (X239 @ sh @ one)) @ id)))) | ? [X244 : term] : (((X238 @ X244)) != ((X237 @ id @ X244 @ id)))))) & ! [X40 : d_subst,X39 : d_subst] : ((X39 = X40) | (((d2subst @ X40)) != ((d2subst @ X39)))) & (! [X139 : (subst > term > term),X138 : term,X137 : term] : (? [X140 : subst,X141 : term,X142 : subst] : (((sub @ (X139 @ X142 @ X141) @ X140)) != ((X139 @ (comp @ X142 @ X140) @ (sub @ X141 @ X140)))) | (((ap @ (sub @ X138 @ id) @ X137)) != ((lam @ (X139 @ sh @ one))))) <=> (hoaslamnotap = $true)) & (one = ((d2term @ d_one))) & (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)))))))))))) & ((hoasapinj2 = $true) <=> ! [X455 : term,X454 : term,X456 : term,X457 : term] : ((((ap @ (sub @ X457 @ id) @ X455)) != ((ap @ (sub @ X454 @ id) @ X456))) | (X455 = X456))) & ((axvarcons = $true) <=> ! [X206 : term,X205 : subst] : (((sub @ one @ (push @ X206 @ X205))) = X206)) & (hoasinduction_lem1 = $true) & ((induction2 = $true) <=> ! [X345 : (term > $o),X346 : term] : (? [X347 : term] : ((((var @ X347)) = $true) & (((X345 @ X347)) != $true)) | ($true = ((X345 @ X346))) | ? [X348 : term] : (! [X349 : term] : (($true = ((X345 @ (sub @ X348 @ (push @ X349 @ id))))) | ($true != ((X345 @ X349)))) & ($true != ((X345 @ (lam @ X348))))) | ? [X350 : term,X351 : term] : (($true = ((X345 @ X350))) & ($true = ((X345 @ X351))) & (((X345 @ (ap @ X351 @ X350))) != $true)))) & ! [X369 : d_subst] : (d_id = X369) & (pushprop_lem2v2_gthm = $true) & ((hoasinduction_lem3aaa = $true) <=> ! [X477 : (subst > term > subst > $o),X476 : term] : (($true = ((X477 @ id @ (lam @ (sub @ X476 @ (push @ one @ sh))) @ id))) | ? [X478 : term] : ((((X477 @ id @ X478 @ id)) = $true) & (((X477 @ id @ (sub @ X476 @ (push @ X478 @ id)) @ id)) != $true)) | ? [X480 : term,X479 : (subst > term > term)] : (! [X484 : term] : (($true = ((X477 @ id @ (X479 @ id @ X484) @ id))) | ($true != ((X477 @ id @ X484 @ id)))) & ! [X482 : subst,X483 : term,X481 : subst] : (((sub @ (sub @ X480 @ (push @ X483 @ X482)) @ X481)) = ((sub @ (X479 @ X482 @ X483) @ X481))) & ($true != ((X477 @ id @ (lam @ (X479 @ sh @ one)) @ id))) & ! [X487 : subst,X485 : subst,X486 : term] : (((X479 @ (comp @ X485 @ X487) @ (sub @ X486 @ X487))) = ((sub @ X480 @ (push @ (sub @ X486 @ X487) @ (comp @ X485 @ X487)))))))) & (hoasinduction_lem3b_gthm = $true) & ((apinj2 = $true) <=> ! [X263 : term,X264 : term,X266 : term,X265 : term] : ((((ap @ X265 @ X266)) != ((ap @ X263 @ X264))) | (X264 = X266))) & (ulamvarind = $true) & (lamnotvar = $true) & (hoasinduction_lem3_gthm = $true) & (hoaslamnotap_lthm = $true) & ((hoasapinj1_lthm = $true) <=> ((! [X531 : term,X528 : term,X530 : term,X529 : term] : ((X528 = X530) | (((ap @ (sub @ X530 @ id) @ X529)) != ((ap @ (sub @ X528 @ id) @ X531)))) | ? [X524 : term,X526 : term,X525 : term,X527 : term] : ((X524 != X525) & (((ap @ X525 @ X526)) = ((ap @ X524 @ X527))))) | ? [X523 : term] : (((sub @ X523 @ id)) != X523))) & ((? [X228 : term,X230 : term,X229 : term,X227 : term] : ((X228 != X229) & (((ap @ X227 @ X228)) = ((ap @ X230 @ X229)))) | ! [X234 : term,X231 : term,X233 : term,X232 : term] : ((((ap @ (sub @ X233 @ id) @ X234)) != ((ap @ (sub @ X232 @ id) @ X231))) | (X231 = X234))) <=> (hoasapinj2_lthm = $true)) & ((hoasinduction_lem3_lthm = $true) <=> ((! [X500 : (subst > term > subst > $o),X499 : term] : (? [X502 : subst,X503 : subst,X501 : term,X504 : subst] : ((((X500 @ X504 @ X501 @ (comp @ X503 @ X502))) != $true) & (((X500 @ (comp @ X504 @ X503) @ (sub @ X501 @ X503) @ X502)) = $true)) | ? [X511 : term,X514 : subst,X512 : subst,X513 : subst] : ((((X500 @ X514 @ X511 @ (comp @ X513 @ X512))) = $true) & (((X500 @ (comp @ X514 @ X513) @ (sub @ X511 @ X513) @ X512)) != $true)) | ? [X505 : (subst > term > term)] : (! [X507 : term,X509 : subst,X508 : subst] : (((X505 @ (comp @ X509 @ X508) @ (sub @ X507 @ X508))) = ((sub @ (X505 @ X509 @ X507) @ X508))) & ! [X506 : term] : ((((X500 @ id @ (X505 @ id @ X506) @ id)) = $true) | (((X500 @ id @ X506 @ id)) != $true)) & (((X500 @ id @ (lam @ (X505 @ sh @ one)) @ id)) != $true)) | ($true = ((X500 @ id @ (lam @ X499) @ id))) | ? [X510 : term] : ((((X500 @ id @ X510 @ id)) = $true) & (((X500 @ id @ (sub @ X499 @ (push @ X510 @ id)) @ id)) != $true))) | ? [X492 : (subst > term > subst > $o),X491 : term] : ((((X492 @ id @ (lam @ (sub @ X491 @ (push @ one @ sh))) @ id)) != $true) & ! [X493 : (subst > term > term)] : ((((X492 @ id @ (lam @ (X493 @ sh @ one)) @ id)) = $true) | ? [X497 : term] : ((((X492 @ id @ X497 @ id)) = $true) & ($true != ((X492 @ id @ (X493 @ id @ X497) @ id)))) | ? [X494 : subst,X495 : subst,X496 : term] : (((sub @ (X493 @ X494 @ X496) @ X495)) != ((X493 @ (comp @ X494 @ X495) @ (sub @ X496 @ X495))))) & ! [X498 : term] : (($true != ((X492 @ id @ X498 @ id))) | (((X492 @ id @ (sub @ X491 @ (push @ X498 @ id)) @ id)) = $true)))) | ? [X490 : term] : (((sub @ X490 @ id)) != X490))) & ((hoasinduction = $true) <=> ! [X188 : (subst > term > subst > $o),X187 : term] : (? [X202 : subst,X204 : subst,X201 : subst,X203 : term] : ((((X188 @ (comp @ X202 @ X201) @ (sub @ X203 @ X201) @ X204)) = $true) & ($true != ((X188 @ X202 @ X203 @ (comp @ X201 @ X204))))) | ? [X200 : term,X199 : subst,X197 : subst,X198 : subst] : ((((X188 @ X198 @ X200 @ (comp @ X197 @ X199))) = $true) & ($true != ((X188 @ (comp @ X198 @ X197) @ (sub @ X200 @ X197) @ X199)))) | ? [X192 : (subst > term > term)] : (! [X193 : term] : (($true = ((X188 @ id @ (X192 @ id @ X193) @ id))) | (((X188 @ id @ X193 @ id)) != $true)) & (((X188 @ id @ (lam @ (X192 @ sh @ one)) @ id)) != $true) & ! [X196 : subst,X194 : term,X195 : subst] : (((X192 @ (comp @ X196 @ X195) @ (sub @ X194 @ X195))) = ((sub @ (X192 @ X196 @ X194) @ X195)))) | (((X188 @ id @ X187 @ id)) = $true) | ? [X189 : term] : (($true != ((X188 @ id @ X189 @ id))) & (((var @ (sub @ X189 @ id))) = $true)) | ? [X191 : term,X190 : term] : ((((X188 @ id @ (ap @ (sub @ X191 @ id) @ X190) @ id)) != $true) & (((X188 @ id @ X190 @ id)) = $true) & ($true = ((X188 @ id @ X191 @ id)))))) & (! [X473 : term] : (((sub @ X473 @ id)) = X473) <=> (termmset = $true)) & (hoasapinj1_gthm = $true) & (! [X300 : term] : ? [X301 : (subst > term > term)] : (((sub @ X300 @ (push @ one @ sh))) = ((X301 @ sh @ one))) <=> (hoasinduction_lem3b_lthm = $true)) & ((((! [X439 : term,X442 : subst,X440 : (subst > term > term),X441 : (subst > term > term)] : (? [X445 : subst,X443 : term,X444 : subst] : (((sub @ (X441 @ X444 @ X443) @ X445)) != ((X441 @ (comp @ X444 @ X445) @ (sub @ X443 @ X445)))) | ? [X446 : term,X447 : subst,X448 : subst] : (((sub @ (X440 @ X447 @ X446) @ X448)) != ((X440 @ (comp @ X447 @ X448) @ (sub @ X446 @ X448)))) | (((lam @ (X441 @ sh @ one))) != ((lam @ (X440 @ sh @ one)))) | (((X440 @ X442 @ X439)) = ((X441 @ X442 @ X439)))) | ? [X437 : term,X438 : term] : ((((lam @ X438)) = ((lam @ X437))) & (X437 != X438))) | ? [X435 : subst,X436 : term] : (((comp @ sh @ (push @ X436 @ X435))) != X435)) | ? [X434 : subst,X433 : term] : (((sub @ one @ (push @ X433 @ X434))) != X433)) <=> (hoaslaminj_lthm = $true)) & ((hoasinduction_lem2 = $true) <=> ! [X55 : (subst > term > subst > $o),X53 : term,X54 : term] : (? [X64 : subst,X62 : term,X65 : subst,X63 : subst] : ((((X55 @ (comp @ X64 @ X63) @ (sub @ X62 @ X63) @ X65)) = $true) & ($true != ((X55 @ X64 @ X62 @ (comp @ X63 @ X65))))) | ? [X60 : term,X61 : term] : (($true = ((X55 @ id @ X60 @ id))) & ($true = ((X55 @ id @ X61 @ id))) & ($true != ((X55 @ id @ (ap @ (sub @ X61 @ id) @ X60) @ id)))) | ($true != ((X55 @ id @ X54 @ id))) | ? [X56 : subst,X59 : subst,X58 : subst,X57 : term] : ((((X55 @ (comp @ X56 @ X58) @ (sub @ X57 @ X58) @ X59)) != $true) & ($true = ((X55 @ X56 @ X57 @ (comp @ X58 @ X59))))) | ($true = ((X55 @ id @ (ap @ X53 @ X54) @ id))) | ($true != ((X55 @ id @ X53 @ id))))) & ((induction2lem = $true) <=> ! [X273 : subst,X274 : term,X272 : (term > $o)] : ((((X272 @ (sub @ X274 @ X273))) = $true) | ? [X277 : term,X278 : term] : ((((X272 @ X278)) = $true) & (((X272 @ (ap @ X277 @ X278))) != $true) & ($true = ((X272 @ X277)))) | ? [X275 : term] : (($true != ((X272 @ (lam @ X275)))) & ! [X276 : term] : (($true = ((X272 @ (sub @ X275 @ (push @ X276 @ id))))) | (((X272 @ X276)) != $true))) | ? [X279 : term] : (($true = ((var @ X279))) & (((X272 @ (sub @ X279 @ X273))) != $true)))) & (axclos = $true) & (hoasinduction_lem3v2_gthm = $true) & ((substmonoid_lthm = $true) <=> (((! [X17 : subst] : (((comp @ X17 @ id)) = X17) & ! [X18 : subst] : (((comp @ id @ X18)) = X18)) | ? [X16 : subst] : (((comp @ X16 @ id)) != X16)) | ? [X15 : subst] : (((comp @ id @ X15)) != X15))) & (induction2_gthm = $true) & (axmap = $true) & (hoasapnotvar = $true) & (sh = ((d2subst @ d_id))) & ((hoasinduction_lem3v2a_lthm = $true) <=> ((! [X286 : (subst > term > subst > $o),X288 : term,X287 : (term > $o)] : (? [X295 : term] : (((X286 @ id @ X295 @ id)) != ((X287 @ X295))) | ? [X290 : (subst > term > term)] : (! [X293 : subst,X292 : subst,X294 : term] : (((X290 @ (comp @ X293 @ X292) @ (sub @ X294 @ X292))) = ((sub @ (X290 @ X293 @ X294) @ X292))) & ! [X291 : term] : ((((X286 @ id @ (X290 @ id @ X291) @ id)) = $true) | (((X286 @ id @ X291 @ id)) != $true)) & ($true != ((X286 @ id @ (lam @ (X290 @ sh @ one)) @ id)))) | ? [X289 : term] : (($true != ((X287 @ (sub @ X288 @ (push @ X289 @ id))))) & (((X287 @ X289)) = $true)) | (((X287 @ (lam @ X288))) = $true)) | ? [X285 : term] : (((sub @ X285 @ id)) != X285)) | ? [X281 : term] : ! [X282 : (subst > term > term)] : ? [X283 : subst,X284 : term] : (((X282 @ X283 @ X284)) != ((sub @ X281 @ (push @ X284 @ X283)))))) & (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) & (hoasinduction_lem1_lthm = $true) & (termmset_gthm = $true) & (hoasapnotvar_gthm = $true)),
  inference(ennf_transformation,[],[f9])).
thf(f12,plain,(
  (! [X392 : term,X393 : subst,X391 : term,X389 : (term > $o),X390 : (term > $o)] : (($true = ((X389 @ (sub @ X392 @ (push @ X391 @ X393))))) | ($true != ((X390 @ X392)))) <=> (pushprop_lem1 = $true)) & (hoasinduction_lem3a_gthm = $true) & (! [X214 : term] : ? [X215 : (subst > term > term)] : ! [X217 : subst,X216 : term] : (((X215 @ X217 @ X216)) = ((sub @ X214 @ (push @ X216 @ X217)))) <=> (hoasinduction_lem3v2_f = $true)) & ((? [X407 : (term > $o),X405 : (subst > term > subst > $o),X406 : term] : ((((X407 @ (lam @ X406))) != $true) & ! [X410 : (subst > term > term)] : (? [X411 : term] : (($true != ((X405 @ id @ (X410 @ id @ X411) @ id))) & ($true = ((X405 @ id @ X411 @ id)))) | ? [X413 : term,X412 : subst,X414 : subst] : (((X410 @ (comp @ X412 @ X414) @ (sub @ X413 @ X414))) != ((sub @ (X410 @ X412 @ X413) @ X414))) | (((X405 @ id @ (lam @ (X410 @ sh @ one)) @ id)) = $true)) & ! [X408 : term] : (((X405 @ id @ X408 @ id)) = ((X407 @ X408))) & ! [X409 : term] : ((((X407 @ (sub @ X406 @ (push @ X409 @ id)))) = $true) | (((X407 @ X409)) != $true))) | ! [X415 : (subst > term > subst > $o),X416 : term] : (? [X420 : subst,X421 : subst,X422 : subst,X419 : term] : (($true != ((X415 @ X421 @ X419 @ (comp @ X422 @ X420)))) & ($true = ((X415 @ (comp @ X421 @ X422) @ (sub @ X419 @ X422) @ X420)))) | ? [X417 : term,X418 : term] : (($true != ((X415 @ id @ (ap @ (sub @ X418 @ id) @ X417) @ id))) & ($true = ((X415 @ id @ X417 @ id))) & (((X415 @ id @ X418 @ id)) = $true)) | ? [X428 : (subst > term > term)] : (! [X432 : term] : ((((X415 @ id @ X432 @ id)) != $true) | ($true = ((X415 @ id @ (X428 @ id @ X432) @ id)))) & ($true != ((X415 @ id @ (lam @ (X428 @ sh @ one)) @ id))) & ! [X429 : subst,X430 : term,X431 : subst] : (((X428 @ (comp @ X431 @ X429) @ (sub @ X430 @ X429))) = ((sub @ (X428 @ X431 @ X430) @ X429)))) | ($true = ((X415 @ id @ X416 @ id))) | ? [X426 : subst,X424 : term,X425 : subst,X427 : subst] : ((((X415 @ (comp @ X425 @ X427) @ (sub @ X424 @ X427) @ X426)) != $true) & (((X415 @ X425 @ X424 @ (comp @ X427 @ X426))) = $true)) | ? [X423 : term] : ((((X415 @ id @ X423 @ id)) != $true) & (((var @ (sub @ X423 @ id))) = $true))) | ? [X404 : term] : (((sub @ X404 @ id)) != X404) | ? [X394 : (subst > term > subst > $o)] : ! [X395 : (term > $o)] : ? [X396 : term] : (((X395 @ X396)) != ((X394 @ id @ X396 @ id))) | ? [X398 : term,X397 : (term > $o)] : ((((X397 @ X398)) != $true) & ! [X399 : term] : ((((X397 @ (lam @ X399))) = $true) | ? [X400 : term] : (($true != ((X397 @ (sub @ X399 @ (push @ X400 @ id))))) & (((X397 @ X400)) = $true))) & ! [X401 : term,X402 : term] : (($true = ((X397 @ (ap @ X401 @ X402)))) | ($true != ((X397 @ X401))) | ($true != ((X397 @ X402)))) & ! [X403 : term] : (($true != ((var @ X403))) | ($true = ((X397 @ X403)))))) <=> (hoasinduction_lthm_3 = $true)) & ((termmset_lthm = $true) <=> (! [X475 : term] : (((sub @ X475 @ id)) = X475) | ? [X474 : term] : (((sub @ X474 @ id)) != X474))) & (hoasinduction_lem3_gthm = $true) & (hoasinduction_lem1_lthm = $true) & (hoasapinj1_gthm = $true) & ((axvarcons = $true) <=> ! [X206 : term,X205 : subst] : (((sub @ one @ (push @ X206 @ X205))) = X206)) & (substmonoid_gthm = $true) & ((apinj2 = $true) <=> ! [X263 : term,X264 : term,X266 : term,X265 : term] : ((((ap @ X265 @ X266)) != ((ap @ X263 @ X264))) | (X264 = X266))) & ((axidl = $true) <=> ! [X450 : subst] : (((comp @ id @ X450)) = X450)) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (! [X335 : (subst > term > subst > $o),X334 : term] : (($true = ((X335 @ id @ X334 @ id))) | ? [X341 : term,X342 : term] : (($true != ((X335 @ id @ (ap @ (sub @ X341 @ id) @ X342) @ id))) & (((X335 @ id @ X342 @ id)) = $true) & ($true = ((X335 @ id @ X341 @ id)))) | ? [X336 : (subst > term > term)] : (! [X337 : term] : ((((X335 @ id @ X337 @ id)) != $true) | ($true = ((X335 @ id @ (X336 @ id @ X337) @ id)))) & ! [X338 : term,X340 : subst,X339 : subst] : (((sub @ (X336 @ X340 @ X338) @ X339)) = ((X336 @ (comp @ X340 @ X339) @ (sub @ X338 @ X339)))) & (((X335 @ id @ (lam @ (X336 @ sh @ one)) @ id)) != $true))) | ? [X316 : (term > $o),X317 : term] : (! [X322 : term] : ((((X316 @ X322)) = $true) | ($true != ((var @ X322)))) & (((X316 @ X317)) != $true) & ! [X318 : term,X319 : term] : (($true = ((X316 @ (ap @ X319 @ X318)))) | (((X316 @ X319)) != $true) | (((X316 @ X318)) != $true)) & ! [X320 : term] : (? [X321 : term] : (($true != ((X316 @ (sub @ X320 @ (push @ X321 @ id))))) & ($true = ((X316 @ X321)))) | (((X316 @ (lam @ X320))) = $true))) | ? [X323 : term] : (((sub @ X323 @ id)) != X323) | ? [X313 : (subst > term > subst > $o)] : ! [X314 : (term > $o)] : ? [X315 : term] : (((X314 @ X315)) != ((X313 @ id @ X315 @ id))) | ? [X325 : term,X324 : (subst > term > subst > $o),X326 : (term > $o)] : ((((X326 @ (lam @ X325))) != $true) & ! [X327 : (subst > term > term)] : (? [X331 : term] : (($true = ((X324 @ id @ X331 @ id))) & ($true != ((X324 @ id @ (X327 @ id @ X331) @ id)))) | ? [X329 : subst,X328 : subst,X330 : term] : (((sub @ (X327 @ X328 @ X330) @ X329)) != ((X327 @ (comp @ X328 @ X329) @ (sub @ X330 @ X329)))) | ($true = ((X324 @ id @ (lam @ (X327 @ sh @ one)) @ id)))) & ! [X332 : term] : ((((X326 @ X332)) != $true) | (((X326 @ (sub @ X325 @ (push @ X332 @ id)))) = $true)) & ! [X333 : term] : (((X326 @ X333)) = ((X324 @ id @ X333 @ id)))))) & (hoaslamnotap_lthm = $true) & (hoasinduction_lem1 = $true) & (pushprop = $true) & (sh = ((d2subst @ d_id))) & (pushprop_lem1_gthm = $true) & (lamnotvar = $true) & (! [X1 : term,X0 : term,X3 : term,X2 : term] : ((X0 = X3) | (((ap @ X3 @ X2)) != ((ap @ X0 @ X1)))) <=> (apinj1 = $true)) & (axassoc = $true) & (apnotvar = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (hoasinduction_lem2v2_gthm = $true) & (induction2_gthm = $true) & ((hoaslaminj = $true) <=> ! [X42 : (subst > term > term),X43 : subst,X41 : (subst > term > term),X44 : term] : ((((lam @ (X41 @ sh @ one))) != ((lam @ (X42 @ sh @ one)))) | ? [X45 : term,X47 : subst,X46 : subst] : (((sub @ (X42 @ X46 @ X45) @ X47)) != ((X42 @ (comp @ X46 @ X47) @ (sub @ X45 @ X47)))) | ? [X48 : term,X49 : subst,X50 : subst] : (((sub @ (X41 @ X49 @ X48) @ X50)) != ((X41 @ (comp @ X49 @ X50) @ (sub @ X48 @ X50)))) | (((X41 @ X43 @ X44)) = ((X42 @ X43 @ X44))))) & (axclos = $true) & ((hoasinduction_lem3aaa = $true) <=> ! [X477 : (subst > term > subst > $o),X476 : term] : (($true = ((X477 @ id @ (lam @ (sub @ X476 @ (push @ one @ sh))) @ id))) | ? [X478 : term] : ((((X477 @ id @ X478 @ id)) = $true) & (((X477 @ id @ (sub @ X476 @ (push @ X478 @ id)) @ id)) != $true)) | ? [X480 : term,X479 : (subst > term > term)] : (! [X484 : term] : (($true = ((X477 @ id @ (X479 @ id @ X484) @ id))) | ($true != ((X477 @ id @ X484 @ id)))) & ! [X482 : subst,X483 : term,X481 : subst] : (((sub @ (sub @ X480 @ (push @ X483 @ X482)) @ X481)) = ((sub @ (X479 @ X482 @ X483) @ X481))) & ($true != ((X477 @ id @ (lam @ (X479 @ sh @ one)) @ id))) & ! [X487 : subst,X485 : subst,X486 : term] : (((X479 @ (comp @ X485 @ X487) @ (sub @ X486 @ X487))) = ((sub @ X480 @ (push @ (sub @ X486 @ X487) @ (comp @ X485 @ X487)))))))) & (hoaslamnotvar_lthm = $true) & ((hoasinduction_lem3v2_lthm = $true) <=> (? [X235 : term] : (((sub @ X235 @ id)) != X235) | ! [X236 : term,X238 : (term > $o),X237 : (subst > term > subst > $o)] : (? [X249 : term] : ((((X238 @ (sub @ X236 @ (push @ X249 @ id)))) != $true) & (((X238 @ X249)) = $true)) | ? [X246 : subst,X248 : subst,X247 : subst,X245 : term] : (($true != ((X237 @ X248 @ X245 @ (comp @ X246 @ X247)))) & (((X237 @ (comp @ X248 @ X246) @ (sub @ X245 @ X246) @ X247)) = $true)) | (((X238 @ (lam @ X236))) = $true) | ? [X252 : subst,X250 : term,X251 : subst,X253 : subst] : ((((X237 @ (comp @ X252 @ X251) @ (sub @ X250 @ X251) @ X253)) != $true) & (((X237 @ X252 @ X250 @ (comp @ X251 @ X253))) = $true)) | ? [X239 : (subst > term > term)] : (! [X243 : term,X241 : subst,X242 : subst] : (((sub @ (X239 @ X241 @ X243) @ X242)) = ((X239 @ (comp @ X241 @ X242) @ (sub @ X243 @ X242)))) & ! [X240 : term] : (($true != ((X237 @ id @ X240 @ id))) | (((X237 @ id @ (X239 @ id @ X240) @ id)) = $true)) & ($true != ((X237 @ id @ (lam @ (X239 @ sh @ one)) @ id)))) | ? [X244 : term] : (((X238 @ X244)) != ((X237 @ id @ X244 @ id)))))) & (! [X207 : term] : ? [X208 : (subst > term > term)] : ! [X210 : term,X209 : subst] : (((sub @ X207 @ (push @ X210 @ X209))) = ((X208 @ X209 @ X210))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & (! [X473 : term] : (((sub @ X473 @ id)) = X473) <=> (termmset = $true)) & ((? [X15 : subst] : (((comp @ id @ X15)) != X15) | (! [X17 : subst] : (((comp @ X17 @ id)) = X17) & ! [X18 : subst] : (((comp @ id @ X18)) = X18)) | ? [X16 : subst] : (((comp @ X16 @ id)) != X16)) <=> (substmonoid_lthm = $true)) & (hoaslamnotap_gthm = $true) & (hoaslamnotvar = $true) & (axmap = $true) & ((axvarid = $true) <=> ! [X14 : term] : (((sub @ X14 @ id)) = X14)) & (! [X532 : term,X533 : (subst > term > subst > $o)] : (? [X534 : term] : ((((X533 @ id @ (sub @ X532 @ (push @ X534 @ id)) @ id)) != $true) & ($true = ((X533 @ id @ X534 @ id)))) | (((X533 @ id @ (lam @ X532) @ id)) = $true) | ? [X535 : (subst > term > term)] : (! [X539 : term] : (($true = ((X533 @ id @ (X535 @ id @ X539) @ id))) | ($true != ((X533 @ id @ X539 @ id)))) & ! [X537 : subst,X538 : subst,X536 : term] : (((sub @ (X535 @ X538 @ X536) @ X537)) = ((X535 @ (comp @ X538 @ X537) @ (sub @ X536 @ X537)))) & (((X533 @ id @ (lam @ (X535 @ sh @ one)) @ id)) != $true))) <=> (hoasinduction_lem3a = $true)) & (termmset_gthm = $true) & (axapp = $true) & ((axshiftcons = $true) <=> ! [X489 : subst,X488 : term] : (((comp @ sh @ (push @ X488 @ X489))) = X489)) & ((! [X531 : term,X528 : term,X530 : term,X529 : term] : ((X528 = X530) | (((ap @ (sub @ X530 @ id) @ X529)) != ((ap @ (sub @ X528 @ id) @ X531)))) | ? [X524 : term,X526 : term,X525 : term,X527 : term] : ((X524 != X525) & (((ap @ X525 @ X526)) = ((ap @ X524 @ X527)))) | ? [X523 : term] : (((sub @ X523 @ id)) != X523)) <=> (hoasapinj1_lthm = $true)) & ((pushprop_lem0 = $true) <=> ! [X168 : term,X166 : subst,X167 : (term > $o)] : ? [X169 : (term > $o)] : ! [X170 : term] : (((X169 @ X170)) = ((X167 @ (sub @ X170 @ (push @ X168 @ X166)))))) & ((hoasinduction_lem3a_lthm = $true) <=> (! [X362 : (subst > term > subst > $o),X361 : term] : (? [X363 : (subst > term > term)] : (! [X365 : subst,X366 : term,X364 : subst] : (((sub @ (X363 @ X365 @ X366) @ X364)) = ((X363 @ (comp @ X365 @ X364) @ (sub @ X366 @ X364)))) & ! [X367 : term] : ((((X362 @ id @ (X363 @ id @ X367) @ id)) = $true) | (((X362 @ id @ X367 @ id)) != $true)) & ($true != ((X362 @ id @ (lam @ (X363 @ sh @ one)) @ id)))) | ? [X368 : term] : ((((X362 @ id @ X368 @ id)) = $true) & ($true != ((X362 @ id @ (sub @ X361 @ (push @ X368 @ id)) @ id)))) | ($true = ((X362 @ id @ (lam @ X361) @ id)))) | ? [X352 : term] : (((sub @ X352 @ id)) != X352) | ? [X354 : (subst > term > subst > $o),X353 : term] : ((((X354 @ id @ (lam @ (sub @ X353 @ (push @ one @ sh))) @ id)) != $true) & ! [X355 : term] : (($true = ((X354 @ id @ (sub @ X353 @ (push @ X355 @ id)) @ id))) | (((X354 @ id @ X355 @ id)) != $true)) & ! [X356 : (subst > term > term)] : (($true = ((X354 @ id @ (lam @ (X356 @ sh @ one)) @ id))) | ? [X357 : term] : (($true = ((X354 @ id @ X357 @ id))) & (((X354 @ id @ (X356 @ id @ X357) @ id)) != $true)) | ? [X359 : subst,X358 : subst,X360 : term] : (((sub @ (X356 @ X358 @ X360) @ X359)) != ((X356 @ (comp @ X358 @ X359) @ (sub @ X360 @ X359)))))))) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoasapinj2_gthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (one = ((d2term @ d_one))) & ((hoasapinj1 = $true) <=> ! [X21 : term,X19 : term,X20 : term,X22 : term] : ((X21 = X22) | (((ap @ (sub @ X22 @ id) @ X19)) != ((ap @ (sub @ X21 @ id) @ X20))))) & ((induction2 = $true) <=> ! [X345 : (term > $o),X346 : term] : (? [X347 : term] : ((((var @ X347)) = $true) & (((X345 @ X347)) != $true)) | ($true = ((X345 @ X346))) | ? [X348 : term] : (! [X349 : term] : (($true = ((X345 @ (sub @ X348 @ (push @ X349 @ id))))) | ($true != ((X345 @ X349)))) & ($true != ((X345 @ (lam @ X348))))) | ? [X350 : term,X351 : term] : (($true = ((X345 @ X350))) & ($true = ((X345 @ X351))) & (((X345 @ (ap @ X351 @ X350))) != $true)))) & ! [X296 : subst] : ? [X297 : d_subst] : (((d2subst @ X297)) = X296) & (hoasinduction_lem2_gthm = $true) & ((? [X254 : subst,X255 : term] : (((sub @ one @ (push @ X255 @ X254))) != X255) | ! [X257 : (term > $o),X256 : subst,X258 : (term > $o),X259 : term] : (? [X260 : term] : (((X257 @ (sub @ X260 @ (push @ X259 @ X256)))) != ((X258 @ X260))) | ($true != ((X257 @ X259))) | ($true = ((X258 @ one))))) <=> (pushprop_lem1v2_lthm = $true)) & (! [X300 : term] : ? [X301 : (subst > term > term)] : (((sub @ X300 @ (push @ one @ sh))) = ((X301 @ sh @ one))) <=> (hoasinduction_lem3b_lthm = $true)) & (pushprop_lem3v2 = $true) & ((hoasinduction_lem0_lthm = $true) <=> ! [X451 : (subst > term > subst > $o)] : ? [X452 : (term > $o)] : ! [X453 : term] : (((X451 @ id @ X453 @ id)) = ((X452 @ X453)))) & ((! [X521 : subst] : (((comp @ X521 @ id)) = X521) & ! [X522 : subst] : (((comp @ id @ X522)) = X522)) <=> (substmonoid = $true)) & (((d2subst @ d_id)) = id) & ((hoasinduction_lem3b = $true) <=> ! [X343 : term] : ? [X344 : (subst > term > term)] : (((sub @ X343 @ (push @ one @ sh))) = ((X344 @ sh @ one)))) & (! [X144 : (term > $o),X145 : subst,X143 : term] : ? [X146 : (term > $o)] : ! [X147 : term] : (((X144 @ (sub @ X147 @ (push @ X143 @ X145)))) = ((X146 @ X147))) <=> (pushprop_lem0_lthm = $true)) & (pushprop_lem2v2 = $true) & ! [X302 : d_term,X303 : d_term] : ((((d2term @ X302)) != ((d2term @ X303))) | (X302 = X303)) & (hoasinduction_lem1v2 = $true) & (! [X139 : (subst > term > term),X138 : term,X137 : term] : (? [X140 : subst,X141 : term,X142 : subst] : (((sub @ (X139 @ X142 @ X141) @ X140)) != ((X139 @ (comp @ X142 @ X140) @ (sub @ X141 @ X140)))) | (((ap @ (sub @ X138 @ id) @ X137)) != ((lam @ (X139 @ sh @ one))))) <=> (hoaslamnotap = $true)) & ((? [X371 : term,X370 : subst] : (((sub @ one @ (push @ X371 @ X370))) != X371) | ? [X374 : subst] : (((comp @ X374 @ id)) != X374) | ? [X376 : term,X375 : (term > $o)] : ((((X375 @ X376)) != $true) & ! [X380 : term] : (($true = ((X375 @ (lam @ X380)))) | ($true != ((X375 @ X380)))) & ! [X379 : term] : (($true != ((var @ X379))) | (((X375 @ X379)) = $true)) & ! [X378 : term,X377 : term] : ((((X375 @ (ap @ X378 @ X377))) = $true) | ($true != ((X375 @ X377))) | (((X375 @ X378)) != $true))) | ! [X381 : (term > $o),X383 : subst,X382 : term] : (? [X384 : term] : (! [X385 : term] : ((((X381 @ (sub @ X384 @ (push @ X385 @ id)))) = $true) | (((X381 @ X385)) != $true)) & ($true != ((X381 @ (lam @ X384))))) | ? [X387 : term,X388 : term] : ((((X381 @ (ap @ X387 @ X388))) != $true) & (((X381 @ X388)) = $true) & (((X381 @ X387)) = $true)) | (((X381 @ (sub @ X382 @ X383))) = $true) | ? [X386 : term] : ((((var @ X386)) = $true) & (((X381 @ (sub @ X386 @ X383))) != $true))) | ? [X373 : subst,X372 : term] : (((comp @ sh @ (push @ X372 @ X373))) != X373)) <=> (induction2lem_lthm = $true)) & (ulamvarsh = $true) & ((hoasinduction_lem0 = $true) <=> ! [X211 : (subst > term > subst > $o)] : ? [X212 : (term > $o)] : ! [X213 : term] : (((X212 @ X213)) = ((X211 @ id @ X213 @ id)))) & ((hoasinduction_lthm = $true) <=> (? [X83 : term,X82 : term,X84 : (subst > term > subst > $o)] : (! [X85 : subst,X86 : subst,X87 : subst,X88 : term] : ((((X84 @ X85 @ X88 @ (comp @ X86 @ X87))) = $true) | ($true != ((X84 @ (comp @ X85 @ X86) @ (sub @ X88 @ X86) @ X87)))) & (((X84 @ id @ X82 @ id)) = $true) & (((X84 @ id @ X83 @ id)) = $true) & ! [X94 : subst,X92 : term,X91 : subst,X93 : subst] : ((((X84 @ (comp @ X93 @ X91) @ (sub @ X92 @ X91) @ X94)) = $true) | (((X84 @ X93 @ X92 @ (comp @ X91 @ X94))) != $true)) & (((X84 @ id @ (ap @ X83 @ X82) @ id)) != $true) & ! [X89 : term,X90 : term] : ((((X84 @ id @ (ap @ (sub @ X89 @ id) @ X90) @ id)) = $true) | ($true != ((X84 @ id @ X89 @ id))) | ($true != ((X84 @ id @ X90 @ id))))) | ! [X111 : (subst > term > subst > $o),X112 : term] : (? [X114 : term,X113 : term] : ((((X111 @ id @ X114 @ id)) = $true) & ($true != ((X111 @ id @ (ap @ (sub @ X113 @ id) @ X114) @ id))) & ($true = ((X111 @ id @ X113 @ id)))) | (((X111 @ id @ X112 @ id)) = $true) | ? [X124 : term] : ((((var @ (sub @ X124 @ id))) = $true) & (((X111 @ id @ X124 @ id)) != $true)) | ? [X115 : (subst > term > term)] : (! [X117 : subst,X118 : subst,X116 : term] : (((sub @ (X115 @ X118 @ X116) @ X117)) = ((X115 @ (comp @ X118 @ X117) @ (sub @ X116 @ X117)))) & ! [X119 : term] : ((((X111 @ id @ X119 @ id)) != $true) | (((X111 @ id @ (X115 @ id @ X119) @ id)) = $true)) & ($true != ((X111 @ id @ (lam @ (X115 @ sh @ one)) @ id)))) | ? [X128 : term,X126 : subst,X125 : subst,X127 : subst] : ((((X111 @ X125 @ X128 @ (comp @ X127 @ X126))) != $true) & (((X111 @ (comp @ X125 @ X127) @ (sub @ X128 @ X127) @ X126)) = $true)) | ? [X123 : term,X121 : subst,X122 : subst,X120 : subst] : (($true = ((X111 @ X121 @ X123 @ (comp @ X120 @ X122)))) & ($true != ((X111 @ (comp @ X121 @ X120) @ (sub @ X123 @ X120) @ X122))))) | ? [X75 : term,X76 : (term > $o)] : (! [X77 : term] : ((((X76 @ (lam @ X77))) = $true) | ? [X78 : term] : ((((X76 @ X78)) = $true) & ($true != ((X76 @ (sub @ X77 @ (push @ X78 @ id))))))) & (((X76 @ X75)) != $true) & ! [X81 : term] : (($true != ((var @ X81))) | (((X76 @ X81)) = $true)) & ! [X80 : term,X79 : term] : ((((X76 @ X80)) != $true) | (((X76 @ (ap @ X79 @ X80))) = $true) | ($true != ((X76 @ X79))))) | ? [X95 : term,X96 : (subst > term > subst > $o)] : (! [X105 : term] : ((((X96 @ id @ (sub @ X95 @ (push @ X105 @ id)) @ id)) = $true) | ($true != ((X96 @ id @ X105 @ id)))) & (((X96 @ id @ (lam @ X95) @ id)) != $true) & ! [X98 : subst,X99 : subst,X100 : subst,X97 : term] : ((((X96 @ (comp @ X98 @ X99) @ (sub @ X97 @ X99) @ X100)) != $true) | (((X96 @ X98 @ X97 @ (comp @ X99 @ X100))) = $true)) & ! [X106 : (subst > term > term)] : ((((X96 @ id @ (lam @ (X106 @ sh @ one)) @ id)) = $true) | ? [X110 : subst,X108 : term,X109 : subst] : (((X106 @ (comp @ X110 @ X109) @ (sub @ X108 @ X109))) != ((sub @ (X106 @ X110 @ X108) @ X109))) | ? [X107 : term] : ((((X96 @ id @ (X106 @ id @ X107) @ id)) != $true) & ($true = ((X96 @ id @ X107 @ id))))) & ! [X104 : subst,X102 : subst,X101 : term,X103 : subst] : ((((X96 @ (comp @ X102 @ X103) @ (sub @ X101 @ X103) @ X104)) = $true) | ($true != ((X96 @ X102 @ X101 @ (comp @ X103 @ X104)))))))) & ((? [X228 : term,X230 : term,X229 : term,X227 : term] : ((X228 != X229) & (((ap @ X227 @ X228)) = ((ap @ X230 @ X229)))) | ! [X234 : term,X231 : term,X233 : term,X232 : term] : ((((ap @ (sub @ X233 @ id) @ X234)) != ((ap @ (sub @ X232 @ id) @ X231))) | (X231 = X234))) <=> (hoasapinj2_lthm = $true)) & (ulamvarind = $true) & (pushprop_lem3v2_lthm = $true) & (! [X304 : subst] : (((comp @ X304 @ id)) = X304) <=> (axidr = $true)) & (hoasinduction_lem2_lthm = $true) & ((hoasinduction_lem3aa = $true) <=> ! [X130 : (subst > term > subst > $o),X129 : term] : (($true = ((X130 @ id @ (lam @ (sub @ X129 @ (push @ one @ sh))) @ id))) | ? [X131 : term] : ((((X130 @ id @ (sub @ X129 @ (push @ X131 @ id)) @ id)) != $true) & (((X130 @ id @ X131 @ id)) = $true)) | ? [X132 : (subst > term > term)] : (! [X133 : term] : (($true = ((X130 @ id @ (X132 @ id @ X133) @ id))) | ($true != ((X130 @ id @ X133 @ id)))) & ($true != ((X130 @ id @ (lam @ (X132 @ sh @ one)) @ id))) & ! [X136 : subst,X135 : term,X134 : subst] : (((sub @ (X132 @ X136 @ X135) @ X134)) = ((X132 @ (comp @ X136 @ X134) @ (sub @ X135 @ X134))))))) & (axabs = $true) & (hoasapnotvar = $true) & (hoasinduction_lem1_gthm = $true) & ((hoasapinj2 = $true) <=> ! [X455 : term,X454 : term,X456 : term,X457 : term] : ((((ap @ (sub @ X457 @ id) @ X455)) != ((ap @ (sub @ X454 @ id) @ X456))) | (X455 = X456))) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((hoasinduction_lem2v2 = $true) <=> ! [X459 : term,X458 : term,X461 : (term > $o),X460 : (subst > term > subst > $o)] : (? [X471 : subst,X470 : subst,X469 : term,X472 : subst] : ((((X460 @ (comp @ X472 @ X470) @ (sub @ X469 @ X470) @ X471)) != $true) & ($true = ((X460 @ X472 @ X469 @ (comp @ X470 @ X471))))) | ? [X464 : term] : (((X460 @ id @ X464 @ id)) != ((X461 @ X464))) | ? [X465 : subst,X468 : term,X467 : subst,X466 : subst] : ((((X460 @ X467 @ X468 @ (comp @ X465 @ X466))) != $true) & (((X460 @ (comp @ X467 @ X465) @ (sub @ X468 @ X465) @ X466)) = $true)) | (((X461 @ (ap @ X459 @ X458))) = $true) | ? [X463 : term,X462 : term] : (($true = ((X460 @ id @ X463 @ id))) & ($true != ((X460 @ id @ (ap @ (sub @ X462 @ id) @ X463) @ id))) & (((X460 @ id @ X462 @ id)) = $true)) | ($true != ((X461 @ X458))) | ($true != ((X461 @ X459))))) & (lamnotap != $true) & (pushprop_lem2v2_lthm = $true) & (pushprop_lem0_gthm = $true) & ! [X51 : term] : ? [X52 : d_term] : (((d2term @ X52)) = X51) & ((induction2_lthm = $true) <=> (! [X180 : term,X181 : (term > $o)] : (? [X184 : term] : ((((var @ X184)) = $true) & ($true != ((X181 @ X184)))) | ($true = ((X181 @ X180))) | ? [X185 : term] : (! [X186 : term] : (($true = ((X181 @ (sub @ X185 @ (push @ X186 @ id))))) | (((X181 @ X186)) != $true)) & ($true != ((X181 @ (lam @ X185))))) | ? [X183 : term,X182 : term] : ((((X181 @ X182)) = $true) & (((X181 @ X183)) = $true) & (((X181 @ (ap @ X183 @ X182))) != $true))) | ? [X172 : subst,X173 : term,X174 : (term > $o)] : ((((X174 @ (sub @ X173 @ X172))) != $true) & ! [X176 : term,X175 : term] : (($true != ((X174 @ X175))) | ($true = ((X174 @ (ap @ X175 @ X176)))) | (((X174 @ X176)) != $true)) & ! [X179 : term] : (($true = ((X174 @ (sub @ X179 @ X172)))) | (((var @ X179)) != $true)) & ! [X177 : term] : (? [X178 : term] : ((((X174 @ X178)) = $true) & ($true != ((X174 @ (sub @ X177 @ (push @ X178 @ id)))))) | ($true = ((X174 @ (lam @ X177)))))) | ? [X171 : term] : (((sub @ X171 @ id)) != X171))) & (hoaslamnotvar_gthm = $true) & (induction2lem_gthm = $true) & ((? [X435 : subst,X436 : term] : (((comp @ sh @ (push @ X436 @ X435))) != X435) | ? [X437 : term,X438 : term] : ((((lam @ X438)) = ((lam @ X437))) & (X437 != X438)) | ? [X434 : subst,X433 : term] : (((sub @ one @ (push @ X433 @ X434))) != X433) | ! [X439 : term,X442 : subst,X440 : (subst > term > term),X441 : (subst > term > term)] : (? [X445 : subst,X443 : term,X444 : subst] : (((sub @ (X441 @ X444 @ X443) @ X445)) != ((X441 @ (comp @ X444 @ X445) @ (sub @ X443 @ X445)))) | ? [X446 : term,X447 : subst,X448 : subst] : (((sub @ (X440 @ X447 @ X446) @ X448)) != ((X440 @ (comp @ X447 @ X448) @ (sub @ X446 @ X448)))) | (((lam @ (X441 @ sh @ one))) != ((lam @ (X440 @ sh @ one)))) | (((X440 @ X442 @ X439)) = ((X441 @ X442 @ X439))))) <=> (hoaslaminj_lthm = $true)) & (pushprop_lthm = $true) & ((hoasinduction_lem2 = $true) <=> ! [X55 : (subst > term > subst > $o),X53 : term,X54 : term] : (? [X64 : subst,X62 : term,X65 : subst,X63 : subst] : ((((X55 @ (comp @ X64 @ X63) @ (sub @ X62 @ X63) @ X65)) = $true) & ($true != ((X55 @ X64 @ X62 @ (comp @ X63 @ X65))))) | ? [X60 : term,X61 : term] : (($true = ((X55 @ id @ X60 @ id))) & ($true = ((X55 @ id @ X61 @ id))) & ($true != ((X55 @ id @ (ap @ (sub @ X61 @ id) @ X60) @ id)))) | ($true != ((X55 @ id @ X54 @ id))) | ? [X56 : subst,X59 : subst,X58 : subst,X57 : term] : ((((X55 @ (comp @ X56 @ X58) @ (sub @ X57 @ X58) @ X59)) != $true) & ($true = ((X55 @ X56 @ X57 @ (comp @ X58 @ X59))))) | ($true = ((X55 @ id @ (ap @ X53 @ X54) @ id))) | ($true != ((X55 @ id @ X53 @ id))))) & (pushprop_lem1v2_gthm = $true) & ((hoasinduction_lem3v2a_lthm = $true) <=> (? [X285 : term] : (((sub @ X285 @ id)) != X285) | ? [X281 : term] : ! [X282 : (subst > term > term)] : ? [X283 : subst,X284 : term] : (((X282 @ X283 @ X284)) != ((sub @ X281 @ (push @ X284 @ X283)))) | ! [X286 : (subst > term > subst > $o),X288 : term,X287 : (term > $o)] : (? [X295 : term] : (((X286 @ id @ X295 @ id)) != ((X287 @ X295))) | ? [X290 : (subst > term > term)] : (! [X293 : subst,X292 : subst,X294 : term] : (((X290 @ (comp @ X293 @ X292) @ (sub @ X294 @ X292))) = ((sub @ (X290 @ X293 @ X294) @ X292))) & ! [X291 : term] : ((((X286 @ id @ (X290 @ id @ X291) @ id)) = $true) | (((X286 @ id @ X291 @ id)) != $true)) & ($true != ((X286 @ id @ (lam @ (X290 @ sh @ one)) @ id)))) | ? [X289 : term] : (($true != ((X287 @ (sub @ X288 @ (push @ X289 @ id))))) & (((X287 @ X289)) = $true)) | (((X287 @ (lam @ X288))) = $true)))) & (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X305 : term,X306 : (subst > term > subst > $o)] : (? [X307 : (subst > term > term)] : (($true != ((X306 @ id @ (lam @ (X307 @ sh @ one)) @ id))) & ! [X309 : subst,X311 : subst,X310 : term] : (((X307 @ (comp @ X311 @ X309) @ (sub @ X310 @ X309))) = ((sub @ (X307 @ X311 @ X310) @ X309))) & ! [X308 : term] : (($true = ((X306 @ id @ (X307 @ id @ X308) @ id))) | (((X306 @ id @ X308 @ id)) != $true))) | ($true = ((X306 @ id @ (lam @ (sub @ X305 @ (push @ one @ sh))) @ id))) | ? [X312 : term] : (($true != ((X306 @ id @ (sub @ X305 @ (push @ X312 @ id)) @ id))) & (((X306 @ id @ X312 @ id)) = $true)))) & ((induction2lem = $true) <=> ! [X273 : subst,X274 : term,X272 : (term > $o)] : ((((X272 @ (sub @ X274 @ X273))) = $true) | ? [X277 : term,X278 : term] : ((((X272 @ X278)) = $true) & (((X272 @ (ap @ X277 @ X278))) != $true) & ($true = ((X272 @ X277)))) | ? [X275 : term] : (($true != ((X272 @ (lam @ X275)))) & ! [X276 : term] : (($true = ((X272 @ (sub @ X275 @ (push @ X276 @ id))))) | (((X272 @ X276)) != $true))) | ? [X279 : term] : (($true = ((var @ X279))) & (((X272 @ (sub @ X279 @ X273))) != $true)))) & (hoasinduction_lem3b_gthm = $true) & (pushprop_gthm = $true) & ((axscons = $true) <=> ! [X449 : subst] : (((push @ (sub @ one @ X449) @ (comp @ sh @ X449))) = X449)) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((hoasinduction = $true) <=> ! [X188 : (subst > term > subst > $o),X187 : term] : (? [X202 : subst,X204 : subst,X201 : subst,X203 : term] : ((((X188 @ (comp @ X202 @ X201) @ (sub @ X203 @ X201) @ X204)) = $true) & ($true != ((X188 @ X202 @ X203 @ (comp @ X201 @ X204))))) | ? [X200 : term,X199 : subst,X197 : subst,X198 : subst] : ((((X188 @ X198 @ X200 @ (comp @ X197 @ X199))) = $true) & ($true != ((X188 @ (comp @ X198 @ X197) @ (sub @ X200 @ X197) @ X199)))) | ? [X192 : (subst > term > term)] : (! [X193 : term] : (($true = ((X188 @ id @ (X192 @ id @ X193) @ id))) | (((X188 @ id @ X193 @ id)) != $true)) & (((X188 @ id @ (lam @ (X192 @ sh @ one)) @ id)) != $true) & ! [X196 : subst,X194 : term,X195 : subst] : (((X192 @ (comp @ X196 @ X195) @ (sub @ X194 @ X195))) = ((sub @ (X192 @ X196 @ X194) @ X195)))) | (((X188 @ id @ X187 @ id)) = $true) | ? [X189 : term] : (($true != ((X188 @ id @ X189 @ id))) & (((var @ (sub @ X189 @ id))) = $true)) | ? [X191 : term,X190 : term] : ((((X188 @ id @ (ap @ (sub @ X191 @ id) @ X190) @ id)) != $true) & (((X188 @ id @ X190 @ id)) = $true) & ($true = ((X188 @ id @ X191 @ id)))))) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (axvarshift = $true) & (hoasinduction_lem3v2_gthm = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoaslaminj_gthm = $true) & (pushprop_lem2v2_gthm = $true) & ((! [X500 : (subst > term > subst > $o),X499 : term] : (? [X502 : subst,X503 : subst,X501 : term,X504 : subst] : ((((X500 @ X504 @ X501 @ (comp @ X503 @ X502))) != $true) & (((X500 @ (comp @ X504 @ X503) @ (sub @ X501 @ X503) @ X502)) = $true)) | ? [X511 : term,X514 : subst,X512 : subst,X513 : subst] : ((((X500 @ X514 @ X511 @ (comp @ X513 @ X512))) = $true) & (((X500 @ (comp @ X514 @ X513) @ (sub @ X511 @ X513) @ X512)) != $true)) | ? [X505 : (subst > term > term)] : (! [X507 : term,X509 : subst,X508 : subst] : (((X505 @ (comp @ X509 @ X508) @ (sub @ X507 @ X508))) = ((sub @ (X505 @ X509 @ X507) @ X508))) & ! [X506 : term] : ((((X500 @ id @ (X505 @ id @ X506) @ id)) = $true) | (((X500 @ id @ X506 @ id)) != $true)) & (((X500 @ id @ (lam @ (X505 @ sh @ one)) @ id)) != $true)) | ($true = ((X500 @ id @ (lam @ X499) @ id))) | ? [X510 : term] : ((((X500 @ id @ X510 @ id)) = $true) & (((X500 @ id @ (sub @ X499 @ (push @ X510 @ id)) @ id)) != $true))) | ? [X492 : (subst > term > subst > $o),X491 : term] : ((((X492 @ id @ (lam @ (sub @ X491 @ (push @ one @ sh))) @ id)) != $true) & ! [X493 : (subst > term > term)] : ((((X492 @ id @ (lam @ (X493 @ sh @ one)) @ id)) = $true) | ? [X497 : term] : ((((X492 @ id @ X497 @ id)) = $true) & ($true != ((X492 @ id @ (X493 @ id @ X497) @ id)))) | ? [X494 : subst,X495 : subst,X496 : term] : (((sub @ (X493 @ X494 @ X496) @ X495)) != ((X493 @ (comp @ X494 @ X495) @ (sub @ X496 @ X495))))) & ! [X498 : term] : (($true != ((X492 @ id @ X498 @ id))) | (((X492 @ id @ (sub @ X491 @ (push @ X498 @ id)) @ id)) = $true))) | ? [X490 : term] : (((sub @ X490 @ id)) != X490)) <=> (hoasinduction_lem3_lthm = $true)) & (! [X6 : (subst > term > subst > $o),X4 : (term > $o),X5 : term] : (($true = ((X4 @ (lam @ X5)))) | ? [X7 : (subst > term > term)] : (! [X11 : subst,X10 : term,X9 : subst] : (((sub @ (X7 @ X11 @ X10) @ X9)) = ((X7 @ (comp @ X11 @ X9) @ (sub @ X10 @ X9)))) & ($true != ((X6 @ id @ (lam @ (X7 @ sh @ one)) @ id))) & ! [X8 : term] : ((((X6 @ id @ (X7 @ id @ X8) @ id)) = $true) | ($true != ((X6 @ id @ X8 @ id))))) | ? [X12 : term] : (((X6 @ id @ X12 @ id)) != ((X4 @ X12))) | ? [X13 : term] : ((((X4 @ X13)) = $true) & ($true != ((X4 @ (sub @ X5 @ (push @ X13 @ id))))))) <=> (hoasinduction_lem3v2a = $true)) & (((var @ (d2term @ d_one))) != $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)))))))))))) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ((hoasinduction_no_psi_cond = $true) <=> ! [X66 : term,X67 : (subst > term > subst > $o)] : (? [X68 : (subst > term > term)] : (! [X72 : term] : ((((X67 @ id @ X72 @ id)) != $true) | (((X67 @ id @ (X68 @ id @ X72) @ id)) = $true)) & ($true != ((X67 @ id @ (lam @ (X68 @ sh @ one)) @ id))) & ! [X69 : subst,X71 : subst,X70 : term] : (((X68 @ (comp @ X69 @ X71) @ (sub @ X70 @ X71))) = ((sub @ (X68 @ X69 @ X70) @ X71)))) | ? [X74 : term,X73 : term] : ((((X67 @ id @ (ap @ (sub @ X73 @ id) @ X74) @ id)) != $true) & (((X67 @ id @ X74 @ id)) = $true) & (((X67 @ id @ X73 @ id)) = $true)) | ($true = ((X67 @ id @ X66 @ id))))) & (ulamvar1 != $true) & ((pushprop_lem1v2 = $true) <=> ! [X267 : (term > $o),X268 : subst,X269 : (term > $o),X270 : term] : ((((X269 @ one)) = $true) | (((X267 @ X270)) != $true) | ? [X271 : term] : (((X267 @ (sub @ X271 @ (push @ X270 @ X268)))) != ((X269 @ X271))))) & ! [X280 : d_term] : (d_one = X280) & ! [X369 : d_subst] : (d_id = X369) & ((hoasinduction_lem3 = $true) <=> ! [X24 : (subst > term > subst > $o),X23 : term] : (? [X27 : subst,X26 : term,X28 : subst,X25 : subst] : (($true != ((X24 @ X25 @ X26 @ (comp @ X28 @ X27)))) & (((X24 @ (comp @ X25 @ X28) @ (sub @ X26 @ X28) @ X27)) = $true)) | ? [X29 : (subst > term > term)] : (! [X32 : subst,X31 : term,X30 : subst] : (((sub @ (X29 @ X30 @ X31) @ X32)) = ((X29 @ (comp @ X30 @ X32) @ (sub @ X31 @ X32)))) & (((X24 @ id @ (lam @ (X29 @ sh @ one)) @ id)) != $true) & ! [X33 : term] : ((((X24 @ id @ X33 @ id)) != $true) | (((X24 @ id @ (X29 @ id @ X33) @ id)) = $true))) | ? [X36 : subst,X37 : subst,X35 : subst,X34 : term] : ((((X24 @ (comp @ X37 @ X35) @ (sub @ X34 @ X35) @ X36)) != $true) & ($true = ((X24 @ X37 @ X34 @ (comp @ X35 @ X36))))) | ($true = ((X24 @ id @ (lam @ X23) @ id))) | ? [X38 : term] : (($true != ((X24 @ id @ (sub @ X23 @ (push @ X38 @ id)) @ id))) & (((X24 @ id @ X38 @ id)) = $true)))) & (hoasapnotvar_gthm = $true) & ((laminj = $true) <=> ! [X262 : term,X261 : term] : ((X261 = X262) | (((lam @ X262)) != ((lam @ X261))))) & (hoasinduction_gthm = $true) & (hoasapnotvar_lthm = $true) & (! [X298 : term,X299 : term] : ((((sub @ X298 @ sh)) != ((sub @ X299 @ sh))) | (X298 = X299)) <=> (shinj = $true)) & ((induction = $true) <=> ! [X516 : (term > $o),X515 : term] : (? [X520 : term] : (($true != ((X516 @ X520))) & ($true = ((var @ X520)))) | (((X516 @ X515)) = $true) | ? [X519 : term] : (($true = ((X516 @ X519))) & (((X516 @ (lam @ X519))) != $true)) | ? [X517 : term,X518 : term] : ((((X516 @ (ap @ X517 @ X518))) != $true) & ($true = ((X516 @ X517))) & (((X516 @ X518)) = $true)))) & ! [X40 : d_subst,X39 : d_subst] : ((X39 = X40) | (((d2subst @ X40)) != ((d2subst @ X39)))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & ((hoasinduction_lem3v2 = $true) <=> ! [X148 : (subst > term > subst > $o),X150 : (term > $o),X149 : term] : (? [X155 : (subst > term > term)] : (! [X156 : term] : (($true != ((X148 @ id @ X156 @ id))) | ($true = ((X148 @ id @ (X155 @ id @ X156) @ id)))) & (((X148 @ id @ (lam @ (X155 @ sh @ one)) @ id)) != $true) & ! [X159 : term,X158 : subst,X157 : subst] : (((X155 @ (comp @ X157 @ X158) @ (sub @ X159 @ X158))) = ((sub @ (X155 @ X157 @ X159) @ X158)))) | ? [X165 : term] : ((((X150 @ (sub @ X149 @ (push @ X165 @ id)))) != $true) & (((X150 @ X165)) = $true)) | ? [X152 : term,X151 : subst,X153 : subst,X154 : subst] : ((((X148 @ X151 @ X152 @ (comp @ X154 @ X153))) != $true) & (((X148 @ (comp @ X151 @ X154) @ (sub @ X152 @ X154) @ X153)) = $true)) | ? [X164 : term] : (((X148 @ id @ X164 @ id)) != ((X150 @ X164))) | (((X150 @ (lam @ X149))) = $true) | ? [X161 : subst,X163 : term,X160 : subst,X162 : subst] : ((((X148 @ (comp @ X161 @ X160) @ (sub @ X163 @ X160) @ X162)) != $true) & ($true = ((X148 @ X161 @ X163 @ (comp @ X160 @ X162))))))) & ((! [X226 : subst,X223 : (term > $o),X224 : term,X225 : term,X222 : (term > $o)] : (($true != ((X222 @ X224))) | ($true = ((X223 @ (sub @ X224 @ (push @ X225 @ X226)))))) | ? [X219 : term,X218 : subst] : (((sub @ one @ (push @ X219 @ X218))) != X219) | ? [X221 : term,X220 : subst] : (((comp @ sh @ (push @ X221 @ X220))) != X220)) <=> (pushprop_lem1_lthm = $true)) & (pushprop_lthm_orig = $true)),
  inference(flattening,[],[f11])).
thf(f13,definition,(
  ! [X148 : (subst > term > subst > $o)] : ((((sP0 @ X148)) = $true) <=> ? [X155 : (subst > term > term)] : (! [X156 : term] : (($true != ((X148 @ id @ X156 @ id))) | ($true = ((X148 @ id @ (X155 @ id @ X156) @ id)))) & (((X148 @ id @ (lam @ (X155 @ sh @ one)) @ id)) != $true) & ! [X159 : term,X158 : subst,X157 : subst] : (((X155 @ (comp @ X157 @ X158) @ (sub @ X159 @ X158))) = ((sub @ (X155 @ X157 @ X159) @ X158)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f14,definition,(
  (sP1 = $true) <=> ! [X148 : (subst > term > subst > $o),X150 : (term > $o),X149 : term] : ((((sP0 @ X148)) = $true) | ? [X165 : term] : ((((X150 @ (sub @ X149 @ (push @ X165 @ id)))) != $true) & (((X150 @ X165)) = $true)) | ? [X152 : term,X151 : subst,X153 : subst,X154 : subst] : ((((X148 @ X151 @ X152 @ (comp @ X154 @ X153))) != $true) & (((X148 @ (comp @ X151 @ X154) @ (sub @ X152 @ X154) @ X153)) = $true)) | ? [X164 : term] : (((X148 @ id @ X164 @ id)) != ((X150 @ X164))) | (((X150 @ (lam @ X149))) = $true) | ? [X161 : subst,X163 : term,X160 : subst,X162 : subst] : ((((X148 @ (comp @ X161 @ X160) @ (sub @ X163 @ X160) @ X162)) != $true) & ($true = ((X148 @ X161 @ X163 @ (comp @ X160 @ X162))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f15,definition,(
  ! [X516 : (term > $o)] : ((((sP2 @ X516)) = $true) <=> ? [X517 : term,X518 : term] : ((((X516 @ (ap @ X517 @ X518))) != $true) & ($true = ((X516 @ X517))) & (((X516 @ X518)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f16,definition,(
  ! [X24 : (subst > term > subst > $o)] : ((((sP3 @ X24)) = $true) <=> ? [X29 : (subst > term > term)] : (! [X32 : subst,X31 : term,X30 : subst] : (((sub @ (X29 @ X30 @ X31) @ X32)) = ((X29 @ (comp @ X30 @ X32) @ (sub @ X31 @ X32)))) & (((X24 @ id @ (lam @ (X29 @ sh @ one)) @ id)) != $true) & ! [X33 : term] : ((((X24 @ id @ X33 @ id)) != $true) | (((X24 @ id @ (X29 @ id @ X33) @ id)) = $true))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f17,definition,(
  ($true = sP4) <=> ! [X24 : (subst > term > subst > $o),X23 : term] : (? [X27 : subst,X26 : term,X28 : subst,X25 : subst] : (($true != ((X24 @ X25 @ X26 @ (comp @ X28 @ X27)))) & (((X24 @ (comp @ X25 @ X28) @ (sub @ X26 @ X28) @ X27)) = $true)) | (((sP3 @ X24)) = $true) | ? [X36 : subst,X37 : subst,X35 : subst,X34 : term] : ((((X24 @ (comp @ X37 @ X35) @ (sub @ X34 @ X35) @ X36)) != $true) & ($true = ((X24 @ X37 @ X34 @ (comp @ X35 @ X36))))) | ($true = ((X24 @ id @ (lam @ X23) @ id))) | ? [X38 : term] : (($true != ((X24 @ id @ (sub @ X23 @ (push @ X38 @ id)) @ id))) & (((X24 @ id @ X38 @ id)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f18,definition,(
  ! [X67 : (subst > term > subst > $o)] : ((((sP5 @ X67)) = $true) <=> ? [X74 : term,X73 : term] : ((((X67 @ id @ (ap @ (sub @ X73 @ id) @ X74) @ id)) != $true) & (((X67 @ id @ X74 @ id)) = $true) & (((X67 @ id @ X73 @ id)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f19,definition,(
  (sP6 = $true) <=> ! [X6 : (subst > term > subst > $o),X4 : (term > $o),X5 : term] : (($true = ((X4 @ (lam @ X5)))) | ? [X7 : (subst > term > term)] : (! [X11 : subst,X10 : term,X9 : subst] : (((sub @ (X7 @ X11 @ X10) @ X9)) = ((X7 @ (comp @ X11 @ X9) @ (sub @ X10 @ X9)))) & ($true != ((X6 @ id @ (lam @ (X7 @ sh @ one)) @ id))) & ! [X8 : term] : ((((X6 @ id @ (X7 @ id @ X8) @ id)) = $true) | ($true != ((X6 @ id @ X8 @ id))))) | ? [X12 : term] : (((X6 @ id @ X12 @ id)) != ((X4 @ X12))) | ? [X13 : term] : ((((X4 @ X13)) = $true) & ($true != ((X4 @ (sub @ X5 @ (push @ X13 @ id)))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f20,definition,(
  ! [X500 : (subst > term > subst > $o)] : ((((sP7 @ X500)) = $true) <=> ? [X505 : (subst > term > term)] : (! [X507 : term,X509 : subst,X508 : subst] : (((X505 @ (comp @ X509 @ X508) @ (sub @ X507 @ X508))) = ((sub @ (X505 @ X509 @ X507) @ X508))) & ! [X506 : term] : ((((X500 @ id @ (X505 @ id @ X506) @ id)) = $true) | (((X500 @ id @ X506 @ id)) != $true)) & (((X500 @ id @ (lam @ (X505 @ sh @ one)) @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f21,definition,(
  (sP8 = $true) <=> ? [X492 : (subst > term > subst > $o),X491 : term] : ((((X492 @ id @ (lam @ (sub @ X491 @ (push @ one @ sh))) @ id)) != $true) & ! [X493 : (subst > term > term)] : ((((X492 @ id @ (lam @ (X493 @ sh @ one)) @ id)) = $true) | ? [X497 : term] : ((((X492 @ id @ X497 @ id)) = $true) & ($true != ((X492 @ id @ (X493 @ id @ X497) @ id)))) | ? [X494 : subst,X495 : subst,X496 : term] : (((sub @ (X493 @ X494 @ X496) @ X495)) != ((X493 @ (comp @ X494 @ X495) @ (sub @ X496 @ X495))))) & ! [X498 : term] : (($true != ((X492 @ id @ X498 @ id))) | (((X492 @ id @ (sub @ X491 @ (push @ X498 @ id)) @ id)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f22,definition,(
  ($true = sP9) <=> (! [X500 : (subst > term > subst > $o),X499 : term] : (? [X502 : subst,X503 : subst,X501 : term,X504 : subst] : ((((X500 @ X504 @ X501 @ (comp @ X503 @ X502))) != $true) & (((X500 @ (comp @ X504 @ X503) @ (sub @ X501 @ X503) @ X502)) = $true)) | ? [X511 : term,X514 : subst,X512 : subst,X513 : subst] : ((((X500 @ X514 @ X511 @ (comp @ X513 @ X512))) = $true) & (((X500 @ (comp @ X514 @ X513) @ (sub @ X511 @ X513) @ X512)) != $true)) | (((sP7 @ X500)) = $true) | ($true = ((X500 @ id @ (lam @ X499) @ id))) | ? [X510 : term] : ((((X500 @ id @ X510 @ id)) = $true) & (((X500 @ id @ (sub @ X499 @ (push @ X510 @ id)) @ id)) != $true))) | (sP8 = $true) | ? [X490 : term] : (((sub @ X490 @ id)) != X490))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f23,definition,(
  ! [X188 : (subst > term > subst > $o)] : ((((sP10 @ X188)) = $true) <=> ? [X191 : term,X190 : term] : ((((X188 @ id @ (ap @ (sub @ X191 @ id) @ X190) @ id)) != $true) & (((X188 @ id @ X190 @ id)) = $true) & ($true = ((X188 @ id @ X191 @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f24,definition,(
  ! [X188 : (subst > term > subst > $o)] : ((((sP11 @ X188)) = $true) <=> ? [X192 : (subst > term > term)] : (! [X193 : term] : (($true = ((X188 @ id @ (X192 @ id @ X193) @ id))) | (((X188 @ id @ X193 @ id)) != $true)) & (((X188 @ id @ (lam @ (X192 @ sh @ one)) @ id)) != $true) & ! [X196 : subst,X194 : term,X195 : subst] : (((X192 @ (comp @ X196 @ X195) @ (sub @ X194 @ X195))) = ((sub @ (X192 @ X196 @ X194) @ X195)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f25,definition,(
  (sP12 = $true) <=> ! [X188 : (subst > term > subst > $o),X187 : term] : (? [X202 : subst,X204 : subst,X201 : subst,X203 : term] : ((((X188 @ (comp @ X202 @ X201) @ (sub @ X203 @ X201) @ X204)) = $true) & ($true != ((X188 @ X202 @ X203 @ (comp @ X201 @ X204))))) | ? [X200 : term,X199 : subst,X197 : subst,X198 : subst] : ((((X188 @ X198 @ X200 @ (comp @ X197 @ X199))) = $true) & ($true != ((X188 @ (comp @ X198 @ X197) @ (sub @ X200 @ X197) @ X199)))) | (((sP11 @ X188)) = $true) | (((X188 @ id @ X187 @ id)) = $true) | ? [X189 : term] : (($true != ((X188 @ id @ X189 @ id))) & (((var @ (sub @ X189 @ id))) = $true)) | (((sP10 @ X188)) = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f26,definition,(
  ! [X272 : (term > $o)] : (($true = ((sP13 @ X272))) <=> ? [X277 : term,X278 : term] : ((((X272 @ X278)) = $true) & (((X272 @ (ap @ X277 @ X278))) != $true) & ($true = ((X272 @ X277)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f27,definition,(
  ($true = sP14) <=> ! [X273 : subst,X274 : term,X272 : (term > $o)] : ((((X272 @ (sub @ X274 @ X273))) = $true) | ($true = ((sP13 @ X272))) | ? [X275 : term] : (($true != ((X272 @ (lam @ X275)))) & ! [X276 : term] : (($true = ((X272 @ (sub @ X275 @ (push @ X276 @ id))))) | (((X272 @ X276)) != $true))) | ? [X279 : term] : (($true = ((var @ X279))) & (((X272 @ (sub @ X279 @ X273))) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f28,definition,(
  (sP15 = $true) <=> ! [X305 : term,X306 : (subst > term > subst > $o)] : (? [X307 : (subst > term > term)] : (($true != ((X306 @ id @ (lam @ (X307 @ sh @ one)) @ id))) & ! [X309 : subst,X311 : subst,X310 : term] : (((X307 @ (comp @ X311 @ X309) @ (sub @ X310 @ X309))) = ((sub @ (X307 @ X311 @ X310) @ X309))) & ! [X308 : term] : (($true = ((X306 @ id @ (X307 @ id @ X308) @ id))) | (((X306 @ id @ X308 @ id)) != $true))) | ($true = ((X306 @ id @ (lam @ (sub @ X305 @ (push @ one @ sh))) @ id))) | ? [X312 : term] : (($true != ((X306 @ id @ (sub @ X305 @ (push @ X312 @ id)) @ id))) & (((X306 @ id @ X312 @ id)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f29,definition,(
  ($true = sP16) <=> (? [X285 : term] : (((sub @ X285 @ id)) != X285) | ? [X281 : term] : ! [X282 : (subst > term > term)] : ? [X283 : subst,X284 : term] : (((X282 @ X283 @ X284)) != ((sub @ X281 @ (push @ X284 @ X283)))) | ! [X286 : (subst > term > subst > $o),X288 : term,X287 : (term > $o)] : (? [X295 : term] : (((X286 @ id @ X295 @ id)) != ((X287 @ X295))) | ? [X290 : (subst > term > term)] : (! [X293 : subst,X292 : subst,X294 : term] : (((X290 @ (comp @ X293 @ X292) @ (sub @ X294 @ X292))) = ((sub @ (X290 @ X293 @ X294) @ X292))) & ! [X291 : term] : ((((X286 @ id @ (X290 @ id @ X291) @ id)) = $true) | (((X286 @ id @ X291 @ id)) != $true)) & ($true != ((X286 @ id @ (lam @ (X290 @ sh @ one)) @ id)))) | ? [X289 : term] : (($true != ((X287 @ (sub @ X288 @ (push @ X289 @ id))))) & (((X287 @ X289)) = $true)) | (((X287 @ (lam @ X288))) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f30,definition,(
  ! [X55 : (subst > term > subst > $o)] : (($true = ((sP17 @ X55))) <=> ? [X60 : term,X61 : term] : (($true = ((X55 @ id @ X60 @ id))) & ($true = ((X55 @ id @ X61 @ id))) & ($true != ((X55 @ id @ (ap @ (sub @ X61 @ id) @ X60) @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f31,definition,(
  ($true = sP18) <=> ! [X55 : (subst > term > subst > $o),X53 : term,X54 : term] : (? [X64 : subst,X62 : term,X65 : subst,X63 : subst] : ((((X55 @ (comp @ X64 @ X63) @ (sub @ X62 @ X63) @ X65)) = $true) & ($true != ((X55 @ X64 @ X62 @ (comp @ X63 @ X65))))) | ($true = ((sP17 @ X55))) | ($true != ((X55 @ id @ X54 @ id))) | ? [X56 : subst,X59 : subst,X58 : subst,X57 : term] : ((((X55 @ (comp @ X56 @ X58) @ (sub @ X57 @ X58) @ X59)) != $true) & ($true = ((X55 @ X56 @ X57 @ (comp @ X58 @ X59))))) | ($true = ((X55 @ id @ (ap @ X53 @ X54) @ id))) | ($true != ((X55 @ id @ X53 @ id))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f32,definition,(
  ($true = sP19) <=> (? [X435 : subst,X436 : term] : (((comp @ sh @ (push @ X436 @ X435))) != X435) | ? [X437 : term,X438 : term] : ((((lam @ X438)) = ((lam @ X437))) & (X437 != X438)) | ? [X434 : subst,X433 : term] : (((sub @ one @ (push @ X433 @ X434))) != X433) | ! [X439 : term,X442 : subst,X440 : (subst > term > term),X441 : (subst > term > term)] : (? [X445 : subst,X443 : term,X444 : subst] : (((sub @ (X441 @ X444 @ X443) @ X445)) != ((X441 @ (comp @ X444 @ X445) @ (sub @ X443 @ X445)))) | ? [X446 : term,X447 : subst,X448 : subst] : (((sub @ (X440 @ X447 @ X446) @ X448)) != ((X440 @ (comp @ X447 @ X448) @ (sub @ X446 @ X448)))) | (((lam @ (X441 @ sh @ one))) != ((lam @ (X440 @ sh @ one)))) | (((X440 @ X442 @ X439)) = ((X441 @ X442 @ X439)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f33,definition,(
  ! [X174 : (term > $o)] : (($true = ((sP20 @ X174))) <=> ! [X176 : term,X175 : term] : (($true != ((X174 @ X175))) | ($true = ((X174 @ (ap @ X175 @ X176)))) | (((X174 @ X176)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f34,definition,(
  ! [X181 : (term > $o)] : (($true = ((sP21 @ X181))) <=> ? [X183 : term,X182 : term] : ((((X181 @ X182)) = $true) & (((X181 @ X183)) = $true) & (((X181 @ (ap @ X183 @ X182))) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f35,definition,(
  (sP22 = $true) <=> ! [X180 : term,X181 : (term > $o)] : (? [X184 : term] : ((((var @ X184)) = $true) & ($true != ((X181 @ X184)))) | ($true = ((X181 @ X180))) | ? [X185 : term] : (! [X186 : term] : (($true = ((X181 @ (sub @ X185 @ (push @ X186 @ id))))) | (((X181 @ X186)) != $true)) & ($true != ((X181 @ (lam @ X185))))) | ($true = ((sP21 @ X181))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f36,definition,(
  (sP23 = $true) <=> ((sP22 = $true) | ? [X172 : subst,X173 : term,X174 : (term > $o)] : ((((X174 @ (sub @ X173 @ X172))) != $true) & ($true = ((sP20 @ X174))) & ! [X179 : term] : (($true = ((X174 @ (sub @ X179 @ X172)))) | (((var @ X179)) != $true)) & ! [X177 : term] : (? [X178 : term] : ((((X174 @ X178)) = $true) & ($true != ((X174 @ (sub @ X177 @ (push @ X178 @ id)))))) | ($true = ((X174 @ (lam @ X177)))))) | ? [X171 : term] : (((sub @ X171 @ id)) != X171))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f37,definition,(
  ! [X460 : (subst > term > subst > $o)] : ((((sP24 @ X460)) = $true) <=> ? [X463 : term,X462 : term] : (($true = ((X460 @ id @ X463 @ id))) & ($true != ((X460 @ id @ (ap @ (sub @ X462 @ id) @ X463) @ id))) & (((X460 @ id @ X462 @ id)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f38,definition,(
  (sP25 = $true) <=> ! [X459 : term,X458 : term,X461 : (term > $o),X460 : (subst > term > subst > $o)] : (? [X471 : subst,X470 : subst,X469 : term,X472 : subst] : ((((X460 @ (comp @ X472 @ X470) @ (sub @ X469 @ X470) @ X471)) != $true) & ($true = ((X460 @ X472 @ X469 @ (comp @ X470 @ X471))))) | ? [X464 : term] : (((X460 @ id @ X464 @ id)) != ((X461 @ X464))) | ? [X465 : subst,X468 : term,X467 : subst,X466 : subst] : ((((X460 @ X467 @ X468 @ (comp @ X465 @ X466))) != $true) & (((X460 @ (comp @ X467 @ X465) @ (sub @ X468 @ X465) @ X466)) = $true)) | (((X461 @ (ap @ X459 @ X458))) = $true) | (((sP24 @ X460)) = $true) | ($true != ((X461 @ X458))) | ($true != ((X461 @ X459))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f39,definition,(
  (sP26 = $true) <=> ! [X130 : (subst > term > subst > $o),X129 : term] : (($true = ((X130 @ id @ (lam @ (sub @ X129 @ (push @ one @ sh))) @ id))) | ? [X131 : term] : ((((X130 @ id @ (sub @ X129 @ (push @ X131 @ id)) @ id)) != $true) & (((X130 @ id @ X131 @ id)) = $true)) | ? [X132 : (subst > term > term)] : (! [X133 : term] : (($true = ((X130 @ id @ (X132 @ id @ X133) @ id))) | ($true != ((X130 @ id @ X133 @ id)))) & ($true != ((X130 @ id @ (lam @ (X132 @ sh @ one)) @ id))) & ! [X136 : subst,X135 : term,X134 : subst] : (((sub @ (X132 @ X136 @ X135) @ X134)) = ((X132 @ (comp @ X136 @ X134) @ (sub @ X135 @ X134))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f40,definition,(
  ! [X96 : (subst > term > subst > $o)] : (($true = ((sP27 @ X96))) <=> ! [X106 : (subst > term > term)] : ((((X96 @ id @ (lam @ (X106 @ sh @ one)) @ id)) = $true) | ? [X110 : subst,X108 : term,X109 : subst] : (((X106 @ (comp @ X110 @ X109) @ (sub @ X108 @ X109))) != ((sub @ (X106 @ X110 @ X108) @ X109))) | ? [X107 : term] : ((((X96 @ id @ (X106 @ id @ X107) @ id)) != $true) & ($true = ((X96 @ id @ X107 @ id))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f41,definition,(
  ! [X76 : (term > $o)] : (($true = ((sP28 @ X76))) <=> ! [X80 : term,X79 : term] : ((((X76 @ X80)) != $true) | (((X76 @ (ap @ X79 @ X80))) = $true) | ($true != ((X76 @ X79)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f42,definition,(
  ! [X111 : (subst > term > subst > $o)] : ((((sP29 @ X111)) = $true) <=> ? [X115 : (subst > term > term)] : (! [X117 : subst,X118 : subst,X116 : term] : (((sub @ (X115 @ X118 @ X116) @ X117)) = ((X115 @ (comp @ X118 @ X117) @ (sub @ X116 @ X117)))) & ! [X119 : term] : ((((X111 @ id @ X119 @ id)) != $true) | (((X111 @ id @ (X115 @ id @ X119) @ id)) = $true)) & ($true != ((X111 @ id @ (lam @ (X115 @ sh @ one)) @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f43,definition,(
  ! [X111 : (subst > term > subst > $o)] : ((((sP30 @ X111)) = $true) <=> ? [X114 : term,X113 : term] : ((((X111 @ id @ X114 @ id)) = $true) & ($true != ((X111 @ id @ (ap @ (sub @ X113 @ id) @ X114) @ id))) & ($true = ((X111 @ id @ X113 @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f44,definition,(
  ! [X84 : (subst > term > subst > $o)] : (($true = ((sP31 @ X84))) <=> ! [X89 : term,X90 : term] : ((((X84 @ id @ (ap @ (sub @ X89 @ id) @ X90) @ id)) = $true) | ($true != ((X84 @ id @ X89 @ id))) | ($true != ((X84 @ id @ X90 @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f45,definition,(
  ($true = sP32) <=> ? [X95 : term,X96 : (subst > term > subst > $o)] : (! [X105 : term] : ((((X96 @ id @ (sub @ X95 @ (push @ X105 @ id)) @ id)) = $true) | ($true != ((X96 @ id @ X105 @ id)))) & (((X96 @ id @ (lam @ X95) @ id)) != $true) & ! [X98 : subst,X99 : subst,X100 : subst,X97 : term] : ((((X96 @ (comp @ X98 @ X99) @ (sub @ X97 @ X99) @ X100)) != $true) | (((X96 @ X98 @ X97 @ (comp @ X99 @ X100))) = $true)) & ($true = ((sP27 @ X96))) & ! [X104 : subst,X102 : subst,X101 : term,X103 : subst] : ((((X96 @ (comp @ X102 @ X103) @ (sub @ X101 @ X103) @ X104)) = $true) | ($true != ((X96 @ X102 @ X101 @ (comp @ X103 @ X104))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f46,definition,(
  (sP33 = $true) <=> ! [X111 : (subst > term > subst > $o),X112 : term] : ((((sP30 @ X111)) = $true) | (((X111 @ id @ X112 @ id)) = $true) | ? [X124 : term] : ((((var @ (sub @ X124 @ id))) = $true) & (((X111 @ id @ X124 @ id)) != $true)) | (((sP29 @ X111)) = $true) | ? [X128 : term,X126 : subst,X125 : subst,X127 : subst] : ((((X111 @ X125 @ X128 @ (comp @ X127 @ X126))) != $true) & (((X111 @ (comp @ X125 @ X127) @ (sub @ X128 @ X127) @ X126)) = $true)) | ? [X123 : term,X121 : subst,X122 : subst,X120 : subst] : (($true = ((X111 @ X121 @ X123 @ (comp @ X120 @ X122)))) & ($true != ((X111 @ (comp @ X121 @ X120) @ (sub @ X123 @ X120) @ X122)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f47,definition,(
  ($true = sP34) <=> ? [X75 : term,X76 : (term > $o)] : (! [X77 : term] : ((((X76 @ (lam @ X77))) = $true) | ? [X78 : term] : ((((X76 @ X78)) = $true) & ($true != ((X76 @ (sub @ X77 @ (push @ X78 @ id))))))) & (((X76 @ X75)) != $true) & ! [X81 : term] : (($true != ((var @ X81))) | (((X76 @ X81)) = $true)) & ($true = ((sP28 @ X76))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f48,definition,(
  ($true = sP35) <=> (? [X83 : term,X82 : term,X84 : (subst > term > subst > $o)] : (! [X85 : subst,X86 : subst,X87 : subst,X88 : term] : ((((X84 @ X85 @ X88 @ (comp @ X86 @ X87))) = $true) | ($true != ((X84 @ (comp @ X85 @ X86) @ (sub @ X88 @ X86) @ X87)))) & (((X84 @ id @ X82 @ id)) = $true) & (((X84 @ id @ X83 @ id)) = $true) & ! [X94 : subst,X92 : term,X91 : subst,X93 : subst] : ((((X84 @ (comp @ X93 @ X91) @ (sub @ X92 @ X91) @ X94)) = $true) | (((X84 @ X93 @ X92 @ (comp @ X91 @ X94))) != $true)) & (((X84 @ id @ (ap @ X83 @ X82) @ id)) != $true) & ($true = ((sP31 @ X84)))) | (sP33 = $true) | ($true = sP34) | ($true = sP32))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f49,definition,(
  ! [X381 : (term > $o)] : (($true = ((sP36 @ X381))) <=> ? [X387 : term,X388 : term] : ((((X381 @ (ap @ X387 @ X388))) != $true) & (((X381 @ X388)) = $true) & (((X381 @ X387)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f50,definition,(
  ! [X375 : (term > $o)] : (($true = ((sP37 @ X375))) <=> ! [X378 : term,X377 : term] : ((((X375 @ (ap @ X378 @ X377))) = $true) | ($true != ((X375 @ X377))) | (((X375 @ X378)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f51,definition,(
  ($true = sP38) <=> ! [X381 : (term > $o),X383 : subst,X382 : term] : (? [X384 : term] : (! [X385 : term] : ((((X381 @ (sub @ X384 @ (push @ X385 @ id)))) = $true) | (((X381 @ X385)) != $true)) & ($true != ((X381 @ (lam @ X384))))) | ($true = ((sP36 @ X381))) | (((X381 @ (sub @ X382 @ X383))) = $true) | ? [X386 : term] : ((((var @ X386)) = $true) & (((X381 @ (sub @ X386 @ X383))) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f52,definition,(
  (sP39 = $true) <=> (? [X371 : term,X370 : subst] : (((sub @ one @ (push @ X371 @ X370))) != X371) | ? [X374 : subst] : (((comp @ X374 @ id)) != X374) | ? [X376 : term,X375 : (term > $o)] : ((((X375 @ X376)) != $true) & ! [X380 : term] : (($true = ((X375 @ (lam @ X380)))) | ($true != ((X375 @ X380)))) & ! [X379 : term] : (($true != ((var @ X379))) | (((X375 @ X379)) = $true)) & ($true = ((sP37 @ X375)))) | ($true = sP38) | ? [X373 : subst,X372 : term] : (((comp @ sh @ (push @ X372 @ X373))) != X373))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f53,definition,(
  ! [X345 : (term > $o)] : (($true = ((sP40 @ X345))) <=> ? [X350 : term,X351 : term] : (($true = ((X345 @ X350))) & ($true = ((X345 @ X351))) & (((X345 @ (ap @ X351 @ X350))) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f54,definition,(
  (sP41 = $true) <=> ! [X345 : (term > $o),X346 : term] : (? [X347 : term] : ((((var @ X347)) = $true) & (((X345 @ X347)) != $true)) | ($true = ((X345 @ X346))) | ? [X348 : term] : (! [X349 : term] : (($true = ((X345 @ (sub @ X348 @ (push @ X349 @ id))))) | ($true != ((X345 @ X349)))) & ($true != ((X345 @ (lam @ X348))))) | ($true = ((sP40 @ X345))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f55,definition,(
  (sP42 = $true) <=> ? [X354 : (subst > term > subst > $o),X353 : term] : ((((X354 @ id @ (lam @ (sub @ X353 @ (push @ one @ sh))) @ id)) != $true) & ! [X355 : term] : (($true = ((X354 @ id @ (sub @ X353 @ (push @ X355 @ id)) @ id))) | (((X354 @ id @ X355 @ id)) != $true)) & ! [X356 : (subst > term > term)] : (($true = ((X354 @ id @ (lam @ (X356 @ sh @ one)) @ id))) | ? [X357 : term] : (($true = ((X354 @ id @ X357 @ id))) & (((X354 @ id @ (X356 @ id @ X357) @ id)) != $true)) | ? [X359 : subst,X358 : subst,X360 : term] : (((sub @ (X356 @ X358 @ X360) @ X359)) != ((X356 @ (comp @ X358 @ X359) @ (sub @ X360 @ X359))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f56,definition,(
  ($true = sP43) <=> (! [X362 : (subst > term > subst > $o),X361 : term] : (? [X363 : (subst > term > term)] : (! [X365 : subst,X366 : term,X364 : subst] : (((sub @ (X363 @ X365 @ X366) @ X364)) = ((X363 @ (comp @ X365 @ X364) @ (sub @ X366 @ X364)))) & ! [X367 : term] : ((((X362 @ id @ (X363 @ id @ X367) @ id)) = $true) | (((X362 @ id @ X367 @ id)) != $true)) & ($true != ((X362 @ id @ (lam @ (X363 @ sh @ one)) @ id)))) | ? [X368 : term] : ((((X362 @ id @ X368 @ id)) = $true) & ($true != ((X362 @ id @ (sub @ X361 @ (push @ X368 @ id)) @ id)))) | ($true = ((X362 @ id @ (lam @ X361) @ id)))) | ? [X352 : term] : (((sub @ X352 @ id)) != X352) | (sP42 = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f57,definition,(
  (sP44 = $true) <=> ! [X532 : term,X533 : (subst > term > subst > $o)] : (? [X534 : term] : ((((X533 @ id @ (sub @ X532 @ (push @ X534 @ id)) @ id)) != $true) & ($true = ((X533 @ id @ X534 @ id)))) | (((X533 @ id @ (lam @ X532) @ id)) = $true) | ? [X535 : (subst > term > term)] : (! [X539 : term] : (($true = ((X533 @ id @ (X535 @ id @ X539) @ id))) | ($true != ((X533 @ id @ X539 @ id)))) & ! [X537 : subst,X538 : subst,X536 : term] : (((sub @ (X535 @ X538 @ X536) @ X537)) = ((X535 @ (comp @ X538 @ X537) @ (sub @ X536 @ X537)))) & (((X533 @ id @ (lam @ (X535 @ sh @ one)) @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f58,definition,(
  ! [X237 : (subst > term > subst > $o)] : (($true = ((sP45 @ X237))) <=> ? [X239 : (subst > term > term)] : (! [X243 : term,X241 : subst,X242 : subst] : (((sub @ (X239 @ X241 @ X243) @ X242)) = ((X239 @ (comp @ X241 @ X242) @ (sub @ X243 @ X242)))) & ! [X240 : term] : (($true != ((X237 @ id @ X240 @ id))) | (((X237 @ id @ (X239 @ id @ X240) @ id)) = $true)) & ($true != ((X237 @ id @ (lam @ (X239 @ sh @ one)) @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f59,definition,(
  ($true = sP46) <=> (? [X235 : term] : (((sub @ X235 @ id)) != X235) | ! [X236 : term,X238 : (term > $o),X237 : (subst > term > subst > $o)] : (? [X249 : term] : ((((X238 @ (sub @ X236 @ (push @ X249 @ id)))) != $true) & (((X238 @ X249)) = $true)) | ? [X246 : subst,X248 : subst,X247 : subst,X245 : term] : (($true != ((X237 @ X248 @ X245 @ (comp @ X246 @ X247)))) & (((X237 @ (comp @ X248 @ X246) @ (sub @ X245 @ X246) @ X247)) = $true)) | (((X238 @ (lam @ X236))) = $true) | ? [X252 : subst,X250 : term,X251 : subst,X253 : subst] : ((((X237 @ (comp @ X252 @ X251) @ (sub @ X250 @ X251) @ X253)) != $true) & (((X237 @ X252 @ X250 @ (comp @ X251 @ X253))) = $true)) | ($true = ((sP45 @ X237))) | ? [X244 : term] : (((X238 @ X244)) != ((X237 @ id @ X244 @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f60,definition,(
  (sP47 = $true) <=> ! [X477 : (subst > term > subst > $o),X476 : term] : (($true = ((X477 @ id @ (lam @ (sub @ X476 @ (push @ one @ sh))) @ id))) | ? [X478 : term] : ((((X477 @ id @ X478 @ id)) = $true) & (((X477 @ id @ (sub @ X476 @ (push @ X478 @ id)) @ id)) != $true)) | ? [X480 : term,X479 : (subst > term > term)] : (! [X484 : term] : (($true = ((X477 @ id @ (X479 @ id @ X484) @ id))) | ($true != ((X477 @ id @ X484 @ id)))) & ! [X482 : subst,X483 : term,X481 : subst] : (((sub @ (sub @ X480 @ (push @ X483 @ X482)) @ X481)) = ((sub @ (X479 @ X482 @ X483) @ X481))) & ($true != ((X477 @ id @ (lam @ (X479 @ sh @ one)) @ id))) & ! [X487 : subst,X485 : subst,X486 : term] : (((X479 @ (comp @ X485 @ X487) @ (sub @ X486 @ X487))) = ((sub @ X480 @ (push @ (sub @ X486 @ X487) @ (comp @ X485 @ X487)))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f61,definition,(
  ! [X316 : (term > $o)] : ((((sP48 @ X316)) = $true) <=> ! [X318 : term,X319 : term] : (($true = ((X316 @ (ap @ X319 @ X318)))) | (((X316 @ X319)) != $true) | (((X316 @ X318)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f62,definition,(
  ! [X335 : (subst > term > subst > $o)] : (($true = ((sP49 @ X335))) <=> ? [X336 : (subst > term > term)] : (! [X337 : term] : ((((X335 @ id @ X337 @ id)) != $true) | ($true = ((X335 @ id @ (X336 @ id @ X337) @ id)))) & ! [X338 : term,X340 : subst,X339 : subst] : (((sub @ (X336 @ X340 @ X338) @ X339)) = ((X336 @ (comp @ X340 @ X339) @ (sub @ X338 @ X339)))) & (((X335 @ id @ (lam @ (X336 @ sh @ one)) @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f63,definition,(
  (sP50 = $true) <=> ? [X325 : term,X324 : (subst > term > subst > $o),X326 : (term > $o)] : ((((X326 @ (lam @ X325))) != $true) & ! [X327 : (subst > term > term)] : (? [X331 : term] : (($true = ((X324 @ id @ X331 @ id))) & ($true != ((X324 @ id @ (X327 @ id @ X331) @ id)))) | ? [X329 : subst,X328 : subst,X330 : term] : (((sub @ (X327 @ X328 @ X330) @ X329)) != ((X327 @ (comp @ X328 @ X329) @ (sub @ X330 @ X329)))) | ($true = ((X324 @ id @ (lam @ (X327 @ sh @ one)) @ id)))) & ! [X332 : term] : ((((X326 @ X332)) != $true) | (((X326 @ (sub @ X325 @ (push @ X332 @ id)))) = $true)) & ! [X333 : term] : (((X326 @ X333)) = ((X324 @ id @ X333 @ id))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f64,definition,(
  (sP51 = $true) <=> ? [X316 : (term > $o),X317 : term] : (! [X322 : term] : ((((X316 @ X322)) = $true) | ($true != ((var @ X322)))) & (((X316 @ X317)) != $true) & (((sP48 @ X316)) = $true) & ! [X320 : term] : (? [X321 : term] : (($true != ((X316 @ (sub @ X320 @ (push @ X321 @ id))))) & ($true = ((X316 @ X321)))) | (((X316 @ (lam @ X320))) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f65,definition,(
  ($true = sP52) <=> (! [X335 : (subst > term > subst > $o),X334 : term] : (($true = ((X335 @ id @ X334 @ id))) | ? [X341 : term,X342 : term] : (($true != ((X335 @ id @ (ap @ (sub @ X341 @ id) @ X342) @ id))) & (((X335 @ id @ X342 @ id)) = $true) & ($true = ((X335 @ id @ X341 @ id)))) | ($true = ((sP49 @ X335)))) | (sP51 = $true) | ? [X323 : term] : (((sub @ X323 @ id)) != X323) | ? [X313 : (subst > term > subst > $o)] : ! [X314 : (term > $o)] : ? [X315 : term] : (((X314 @ X315)) != ((X313 @ id @ X315 @ id))) | (sP50 = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f66,definition,(
  ! [X397 : (term > $o)] : ((((sP53 @ X397)) = $true) <=> ! [X401 : term,X402 : term] : (($true = ((X397 @ (ap @ X401 @ X402)))) | ($true != ((X397 @ X401))) | ($true != ((X397 @ X402)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f67,definition,(
  ! [X415 : (subst > term > subst > $o)] : ((((sP54 @ X415)) = $true) <=> ? [X428 : (subst > term > term)] : (! [X432 : term] : ((((X415 @ id @ X432 @ id)) != $true) | ($true = ((X415 @ id @ (X428 @ id @ X432) @ id)))) & ($true != ((X415 @ id @ (lam @ (X428 @ sh @ one)) @ id))) & ! [X429 : subst,X430 : term,X431 : subst] : (((X428 @ (comp @ X431 @ X429) @ (sub @ X430 @ X429))) = ((sub @ (X428 @ X431 @ X430) @ X429)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f68,definition,(
  ! [X415 : (subst > term > subst > $o)] : (($true = ((sP55 @ X415))) <=> ? [X417 : term,X418 : term] : (($true != ((X415 @ id @ (ap @ (sub @ X418 @ id) @ X417) @ id))) & ($true = ((X415 @ id @ X417 @ id))) & (((X415 @ id @ X418 @ id)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f69,definition,(
  ($true = sP56) <=> ! [X415 : (subst > term > subst > $o),X416 : term] : (? [X420 : subst,X421 : subst,X422 : subst,X419 : term] : (($true != ((X415 @ X421 @ X419 @ (comp @ X422 @ X420)))) & ($true = ((X415 @ (comp @ X421 @ X422) @ (sub @ X419 @ X422) @ X420)))) | ($true = ((sP55 @ X415))) | (((sP54 @ X415)) = $true) | ($true = ((X415 @ id @ X416 @ id))) | ? [X426 : subst,X424 : term,X425 : subst,X427 : subst] : ((((X415 @ (comp @ X425 @ X427) @ (sub @ X424 @ X427) @ X426)) != $true) & (((X415 @ X425 @ X424 @ (comp @ X427 @ X426))) = $true)) | ? [X423 : term] : ((((X415 @ id @ X423 @ id)) != $true) & (((var @ (sub @ X423 @ id))) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f70,definition,(
  (sP57 = $true) <=> ? [X407 : (term > $o),X405 : (subst > term > subst > $o),X406 : term] : ((((X407 @ (lam @ X406))) != $true) & ! [X410 : (subst > term > term)] : (? [X411 : term] : (($true != ((X405 @ id @ (X410 @ id @ X411) @ id))) & ($true = ((X405 @ id @ X411 @ id)))) | ? [X413 : term,X412 : subst,X414 : subst] : (((X410 @ (comp @ X412 @ X414) @ (sub @ X413 @ X414))) != ((sub @ (X410 @ X412 @ X413) @ X414))) | (((X405 @ id @ (lam @ (X410 @ sh @ one)) @ id)) = $true)) & ! [X408 : term] : (((X405 @ id @ X408 @ id)) = ((X407 @ X408))) & ! [X409 : term] : ((((X407 @ (sub @ X406 @ (push @ X409 @ id)))) = $true) | (((X407 @ X409)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f71,definition,(
  (sP58 = $true) <=> ((sP57 = $true) | ($true = sP56) | ? [X404 : term] : (((sub @ X404 @ id)) != X404) | ? [X394 : (subst > term > subst > $o)] : ! [X395 : (term > $o)] : ? [X396 : term] : (((X395 @ X396)) != ((X394 @ id @ X396 @ id))) | ? [X398 : term,X397 : (term > $o)] : ((((X397 @ X398)) != $true) & ! [X399 : term] : ((((X397 @ (lam @ X399))) = $true) | ? [X400 : term] : (($true != ((X397 @ (sub @ X399 @ (push @ X400 @ id))))) & (((X397 @ X400)) = $true))) & (((sP53 @ X397)) = $true) & ! [X403 : term] : (($true != ((var @ X403))) | ($true = ((X397 @ X403))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f72,plain,(
  (! [X392 : term,X393 : subst,X391 : term,X389 : (term > $o),X390 : (term > $o)] : (($true = ((X389 @ (sub @ X392 @ (push @ X391 @ X393))))) | ($true != ((X390 @ X392)))) <=> (pushprop_lem1 = $true)) & (hoasinduction_lem3a_gthm = $true) & (! [X214 : term] : ? [X215 : (subst > term > term)] : ! [X217 : subst,X216 : term] : (((X215 @ X217 @ X216)) = ((sub @ X214 @ (push @ X216 @ X217)))) <=> (hoasinduction_lem3v2_f = $true)) & ((sP58 = $true) <=> (hoasinduction_lthm_3 = $true)) & ((termmset_lthm = $true) <=> (! [X475 : term] : (((sub @ X475 @ id)) = X475) | ? [X474 : term] : (((sub @ X474 @ id)) != X474))) & (hoasinduction_lem3_gthm = $true) & (hoasinduction_lem1_lthm = $true) & (hoasapinj1_gthm = $true) & ((axvarcons = $true) <=> ! [X206 : term,X205 : subst] : (((sub @ one @ (push @ X206 @ X205))) = X206)) & (substmonoid_gthm = $true) & ((apinj2 = $true) <=> ! [X263 : term,X264 : term,X266 : term,X265 : term] : ((((ap @ X265 @ X266)) != ((ap @ X263 @ X264))) | (X264 = X266))) & ((axidl = $true) <=> ! [X450 : subst] : (((comp @ id @ X450)) = X450)) & ((hoasinduction_no_psi_cond_lthm = $true) <=> ($true = sP52)) & (hoaslamnotap_lthm = $true) & (hoasinduction_lem1 = $true) & (pushprop = $true) & (sh = ((d2subst @ d_id))) & (pushprop_lem1_gthm = $true) & (lamnotvar = $true) & (! [X1 : term,X0 : term,X3 : term,X2 : term] : ((X0 = X3) | (((ap @ X3 @ X2)) != ((ap @ X0 @ X1)))) <=> (apinj1 = $true)) & (axassoc = $true) & (apnotvar = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (hoasinduction_lem2v2_gthm = $true) & (induction2_gthm = $true) & ((hoaslaminj = $true) <=> ! [X42 : (subst > term > term),X43 : subst,X41 : (subst > term > term),X44 : term] : ((((lam @ (X41 @ sh @ one))) != ((lam @ (X42 @ sh @ one)))) | ? [X45 : term,X47 : subst,X46 : subst] : (((sub @ (X42 @ X46 @ X45) @ X47)) != ((X42 @ (comp @ X46 @ X47) @ (sub @ X45 @ X47)))) | ? [X48 : term,X49 : subst,X50 : subst] : (((sub @ (X41 @ X49 @ X48) @ X50)) != ((X41 @ (comp @ X49 @ X50) @ (sub @ X48 @ X50)))) | (((X41 @ X43 @ X44)) = ((X42 @ X43 @ X44))))) & (axclos = $true) & ((hoasinduction_lem3aaa = $true) <=> (sP47 = $true)) & (hoaslamnotvar_lthm = $true) & ((hoasinduction_lem3v2_lthm = $true) <=> ($true = sP46)) & (! [X207 : term] : ? [X208 : (subst > term > term)] : ! [X210 : term,X209 : subst] : (((sub @ X207 @ (push @ X210 @ X209))) = ((X208 @ X209 @ X210))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & (! [X473 : term] : (((sub @ X473 @ id)) = X473) <=> (termmset = $true)) & ((? [X15 : subst] : (((comp @ id @ X15)) != X15) | (! [X17 : subst] : (((comp @ X17 @ id)) = X17) & ! [X18 : subst] : (((comp @ id @ X18)) = X18)) | ? [X16 : subst] : (((comp @ X16 @ id)) != X16)) <=> (substmonoid_lthm = $true)) & (hoaslamnotap_gthm = $true) & (hoaslamnotvar = $true) & (axmap = $true) & ((axvarid = $true) <=> ! [X14 : term] : (((sub @ X14 @ id)) = X14)) & ((sP44 = $true) <=> (hoasinduction_lem3a = $true)) & (termmset_gthm = $true) & (axapp = $true) & ((axshiftcons = $true) <=> ! [X489 : subst,X488 : term] : (((comp @ sh @ (push @ X488 @ X489))) = X489)) & ((! [X531 : term,X528 : term,X530 : term,X529 : term] : ((X528 = X530) | (((ap @ (sub @ X530 @ id) @ X529)) != ((ap @ (sub @ X528 @ id) @ X531)))) | ? [X524 : term,X526 : term,X525 : term,X527 : term] : ((X524 != X525) & (((ap @ X525 @ X526)) = ((ap @ X524 @ X527)))) | ? [X523 : term] : (((sub @ X523 @ id)) != X523)) <=> (hoasapinj1_lthm = $true)) & ((pushprop_lem0 = $true) <=> ! [X168 : term,X166 : subst,X167 : (term > $o)] : ? [X169 : (term > $o)] : ! [X170 : term] : (((X169 @ X170)) = ((X167 @ (sub @ X170 @ (push @ X168 @ X166)))))) & ((hoasinduction_lem3a_lthm = $true) <=> ($true = sP43)) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoasapinj2_gthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (one = ((d2term @ d_one))) & ((hoasapinj1 = $true) <=> ! [X21 : term,X19 : term,X20 : term,X22 : term] : ((X21 = X22) | (((ap @ (sub @ X22 @ id) @ X19)) != ((ap @ (sub @ X21 @ id) @ X20))))) & ((induction2 = $true) <=> (sP41 = $true)) & ! [X296 : subst] : ? [X297 : d_subst] : (((d2subst @ X297)) = X296) & (hoasinduction_lem2_gthm = $true) & ((? [X254 : subst,X255 : term] : (((sub @ one @ (push @ X255 @ X254))) != X255) | ! [X257 : (term > $o),X256 : subst,X258 : (term > $o),X259 : term] : (? [X260 : term] : (((X257 @ (sub @ X260 @ (push @ X259 @ X256)))) != ((X258 @ X260))) | ($true != ((X257 @ X259))) | ($true = ((X258 @ one))))) <=> (pushprop_lem1v2_lthm = $true)) & (! [X300 : term] : ? [X301 : (subst > term > term)] : (((sub @ X300 @ (push @ one @ sh))) = ((X301 @ sh @ one))) <=> (hoasinduction_lem3b_lthm = $true)) & (pushprop_lem3v2 = $true) & ((hoasinduction_lem0_lthm = $true) <=> ! [X451 : (subst > term > subst > $o)] : ? [X452 : (term > $o)] : ! [X453 : term] : (((X451 @ id @ X453 @ id)) = ((X452 @ X453)))) & ((! [X521 : subst] : (((comp @ X521 @ id)) = X521) & ! [X522 : subst] : (((comp @ id @ X522)) = X522)) <=> (substmonoid = $true)) & (((d2subst @ d_id)) = id) & ((hoasinduction_lem3b = $true) <=> ! [X343 : term] : ? [X344 : (subst > term > term)] : (((sub @ X343 @ (push @ one @ sh))) = ((X344 @ sh @ one)))) & (! [X144 : (term > $o),X145 : subst,X143 : term] : ? [X146 : (term > $o)] : ! [X147 : term] : (((X144 @ (sub @ X147 @ (push @ X143 @ X145)))) = ((X146 @ X147))) <=> (pushprop_lem0_lthm = $true)) & (pushprop_lem2v2 = $true) & ! [X302 : d_term,X303 : d_term] : ((((d2term @ X302)) != ((d2term @ X303))) | (X302 = X303)) & (hoasinduction_lem1v2 = $true) & (! [X139 : (subst > term > term),X138 : term,X137 : term] : (? [X140 : subst,X141 : term,X142 : subst] : (((sub @ (X139 @ X142 @ X141) @ X140)) != ((X139 @ (comp @ X142 @ X140) @ (sub @ X141 @ X140)))) | (((ap @ (sub @ X138 @ id) @ X137)) != ((lam @ (X139 @ sh @ one))))) <=> (hoaslamnotap = $true)) & ((sP39 = $true) <=> (induction2lem_lthm = $true)) & (ulamvarsh = $true) & ((hoasinduction_lem0 = $true) <=> ! [X211 : (subst > term > subst > $o)] : ? [X212 : (term > $o)] : ! [X213 : term] : (((X212 @ X213)) = ((X211 @ id @ X213 @ id)))) & ((hoasinduction_lthm = $true) <=> ($true = sP35)) & ((? [X228 : term,X230 : term,X229 : term,X227 : term] : ((X228 != X229) & (((ap @ X227 @ X228)) = ((ap @ X230 @ X229)))) | ! [X234 : term,X231 : term,X233 : term,X232 : term] : ((((ap @ (sub @ X233 @ id) @ X234)) != ((ap @ (sub @ X232 @ id) @ X231))) | (X231 = X234))) <=> (hoasapinj2_lthm = $true)) & (ulamvarind = $true) & (pushprop_lem3v2_lthm = $true) & (! [X304 : subst] : (((comp @ X304 @ id)) = X304) <=> (axidr = $true)) & (hoasinduction_lem2_lthm = $true) & ((hoasinduction_lem3aa = $true) <=> (sP26 = $true)) & (axabs = $true) & (hoasapnotvar = $true) & (hoasinduction_lem1_gthm = $true) & ((hoasapinj2 = $true) <=> ! [X455 : term,X454 : term,X456 : term,X457 : term] : ((((ap @ (sub @ X457 @ id) @ X455)) != ((ap @ (sub @ X454 @ id) @ X456))) | (X455 = X456))) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((hoasinduction_lem2v2 = $true) <=> (sP25 = $true)) & (lamnotap != $true) & (pushprop_lem2v2_lthm = $true) & (pushprop_lem0_gthm = $true) & ! [X51 : term] : ? [X52 : d_term] : (((d2term @ X52)) = X51) & ((induction2_lthm = $true) <=> (sP23 = $true)) & (hoaslamnotvar_gthm = $true) & (induction2lem_gthm = $true) & (($true = sP19) <=> (hoaslaminj_lthm = $true)) & (pushprop_lthm = $true) & ((hoasinduction_lem2 = $true) <=> ($true = sP18)) & (pushprop_lem1v2_gthm = $true) & ((hoasinduction_lem3v2a_lthm = $true) <=> ($true = sP16)) & (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_lem3aa_lthm = $true) <=> (sP15 = $true)) & ((induction2lem = $true) <=> ($true = sP14)) & (hoasinduction_lem3b_gthm = $true) & (pushprop_gthm = $true) & ((axscons = $true) <=> ! [X449 : subst] : (((push @ (sub @ one @ X449) @ (comp @ sh @ X449))) = X449)) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((hoasinduction = $true) <=> (sP12 = $true)) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (axvarshift = $true) & (hoasinduction_lem3v2_gthm = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoaslaminj_gthm = $true) & (pushprop_lem2v2_gthm = $true) & (($true = sP9) <=> (hoasinduction_lem3_lthm = $true)) & ((sP6 = $true) <=> (hoasinduction_lem3v2a = $true)) & (((var @ (d2term @ d_one))) != $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)))))))))))) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ((hoasinduction_no_psi_cond = $true) <=> ! [X66 : term,X67 : (subst > term > subst > $o)] : (? [X68 : (subst > term > term)] : (! [X72 : term] : ((((X67 @ id @ X72 @ id)) != $true) | (((X67 @ id @ (X68 @ id @ X72) @ id)) = $true)) & ($true != ((X67 @ id @ (lam @ (X68 @ sh @ one)) @ id))) & ! [X69 : subst,X71 : subst,X70 : term] : (((X68 @ (comp @ X69 @ X71) @ (sub @ X70 @ X71))) = ((sub @ (X68 @ X69 @ X70) @ X71)))) | (((sP5 @ X67)) = $true) | ($true = ((X67 @ id @ X66 @ id))))) & (ulamvar1 != $true) & ((pushprop_lem1v2 = $true) <=> ! [X267 : (term > $o),X268 : subst,X269 : (term > $o),X270 : term] : ((((X269 @ one)) = $true) | (((X267 @ X270)) != $true) | ? [X271 : term] : (((X267 @ (sub @ X271 @ (push @ X270 @ X268)))) != ((X269 @ X271))))) & ! [X280 : d_term] : (d_one = X280) & ! [X369 : d_subst] : (d_id = X369) & ((hoasinduction_lem3 = $true) <=> ($true = sP4)) & (hoasapnotvar_gthm = $true) & ((laminj = $true) <=> ! [X262 : term,X261 : term] : ((X261 = X262) | (((lam @ X262)) != ((lam @ X261))))) & (hoasinduction_gthm = $true) & (hoasapnotvar_lthm = $true) & (! [X298 : term,X299 : term] : ((((sub @ X298 @ sh)) != ((sub @ X299 @ sh))) | (X298 = X299)) <=> (shinj = $true)) & ((induction = $true) <=> ! [X516 : (term > $o),X515 : term] : (? [X520 : term] : (($true != ((X516 @ X520))) & ($true = ((var @ X520)))) | (((X516 @ X515)) = $true) | ? [X519 : term] : (($true = ((X516 @ X519))) & (((X516 @ (lam @ X519))) != $true)) | (((sP2 @ X516)) = $true))) & ! [X40 : d_subst,X39 : d_subst] : ((X39 = X40) | (((d2subst @ X40)) != ((d2subst @ X39)))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & ((hoasinduction_lem3v2 = $true) <=> (sP1 = $true)) & ((! [X226 : subst,X223 : (term > $o),X224 : term,X225 : term,X222 : (term > $o)] : (($true != ((X222 @ X224))) | ($true = ((X223 @ (sub @ X224 @ (push @ X225 @ X226)))))) | ? [X219 : term,X218 : subst] : (((sub @ one @ (push @ X219 @ X218))) != X219) | ? [X221 : term,X220 : subst] : (((comp @ sh @ (push @ X221 @ X220))) != X220)) <=> (pushprop_lem1_lthm = $true)) & (pushprop_lthm_orig = $true)),
  inference(definition_folding,[],[f12,f71,f70,f69,f68,f67,f66,f65,f64,f63,f62,f61,f60,f59,f58,f57,f56,f55,f54,f53,f52,f51,f50,f49,f48,f47,f46,f45,f44,f43,f42,f41,f40,f39,f38,f37,f36,f35,f34,f33,f32,f31,f30,f29,f28,f27,f26,f25,f24,f23,f22,f21,f20,f19,f18,f17,f16,f15,f14,f13])).
thf(f201,plain,(
  (($true = sP18) | ? [X55 : (subst > term > subst > $o),X53 : term,X54 : term] : (! [X64 : subst,X62 : term,X65 : subst,X63 : subst] : ((((X55 @ (comp @ X64 @ X63) @ (sub @ X62 @ X63) @ X65)) != $true) | ($true = ((X55 @ X64 @ X62 @ (comp @ X63 @ X65))))) & ($true != ((sP17 @ X55))) & ($true = ((X55 @ id @ X54 @ id))) & ! [X56 : subst,X59 : subst,X58 : subst,X57 : term] : ((((X55 @ (comp @ X56 @ X58) @ (sub @ X57 @ X58) @ X59)) = $true) | ($true != ((X55 @ X56 @ X57 @ (comp @ X58 @ X59))))) & ($true != ((X55 @ id @ (ap @ X53 @ X54) @ id))) & ($true = ((X55 @ id @ X53 @ id))))) & (! [X55 : (subst > term > subst > $o),X53 : term,X54 : term] : (? [X64 : subst,X62 : term,X65 : subst,X63 : subst] : ((((X55 @ (comp @ X64 @ X63) @ (sub @ X62 @ X63) @ X65)) = $true) & ($true != ((X55 @ X64 @ X62 @ (comp @ X63 @ X65))))) | ($true = ((sP17 @ X55))) | ($true != ((X55 @ id @ X54 @ id))) | ? [X56 : subst,X59 : subst,X58 : subst,X57 : term] : ((((X55 @ (comp @ X56 @ X58) @ (sub @ X57 @ X58) @ X59)) != $true) & ($true = ((X55 @ X56 @ X57 @ (comp @ X58 @ X59))))) | ($true = ((X55 @ id @ (ap @ X53 @ X54) @ id))) | ($true != ((X55 @ id @ X53 @ id)))) | ($true != sP18))),
  inference(nnf_transformation,[],[f31])).
thf(f202,plain,(
  (($true = sP18) | ? [X0 : (subst > term > subst > $o),X1 : term,X2 : term] : (! [X3 : subst,X4 : term,X5 : subst,X6 : subst] : ((((X0 @ (comp @ X3 @ X6) @ (sub @ X4 @ X6) @ X5)) != $true) | (((X0 @ X3 @ X4 @ (comp @ X6 @ X5))) = $true)) & ($true != ((sP17 @ X0))) & (((X0 @ id @ X2 @ id)) = $true) & ! [X7 : subst,X8 : subst,X9 : subst,X10 : term] : (($true = ((X0 @ (comp @ X7 @ X9) @ (sub @ X10 @ X9) @ X8))) | (((X0 @ X7 @ X10 @ (comp @ X9 @ X8))) != $true)) & (((X0 @ id @ (ap @ X1 @ X2) @ id)) != $true) & (((X0 @ id @ X1 @ id)) = $true))) & (! [X11 : (subst > term > subst > $o),X12 : term,X13 : term] : (? [X14 : subst,X15 : term,X16 : subst,X17 : subst] : (($true = ((X11 @ (comp @ X14 @ X17) @ (sub @ X15 @ X17) @ X16))) & (((X11 @ X14 @ X15 @ (comp @ X17 @ X16))) != $true)) | (((sP17 @ X11)) = $true) | ($true != ((X11 @ id @ X13 @ id))) | ? [X18 : subst,X19 : subst,X20 : subst,X21 : term] : ((((X11 @ (comp @ X18 @ X20) @ (sub @ X21 @ X20) @ X19)) != $true) & ($true = ((X11 @ X18 @ X21 @ (comp @ X20 @ X19))))) | (((X11 @ id @ (ap @ X12 @ X13) @ id)) = $true) | (((X11 @ id @ X12 @ id)) != $true)) | ($true != sP18))),
  inference(rectify,[],[f201])).
thf(f203,plain,(
  (($true = sP18) | (! [X3 : subst,X4 : term,X5 : subst,X6 : subst] : (($true != ((sK319 @ (comp @ X3 @ X6) @ (sub @ X4 @ X6) @ X5))) | ($true = ((sK319 @ X3 @ X4 @ (comp @ X6 @ X5))))) & (((sP17 @ sK319)) != $true) & ($true = ((sK319 @ id @ sK321 @ id))) & ! [X7 : subst,X8 : subst,X9 : subst,X10 : term] : (($true = ((sK319 @ (comp @ X7 @ X9) @ (sub @ X10 @ X9) @ X8))) | ($true != ((sK319 @ X7 @ X10 @ (comp @ X9 @ X8))))) & ($true != ((sK319 @ id @ (ap @ sK320 @ sK321) @ id))) & ($true = ((sK319 @ id @ sK320 @ id))))) & (! [X11 : (subst > term > subst > $o),X12 : term,X13 : term] : (((((X11 @ (comp @ (sK322 @ X11) @ (sK325 @ X11)) @ (sub @ (sK323 @ X11) @ (sK325 @ X11)) @ (sK324 @ X11))) = $true) & ($true != ((X11 @ (sK322 @ X11) @ (sK323 @ X11) @ (comp @ (sK325 @ X11) @ (sK324 @ X11)))))) | (((sP17 @ X11)) = $true) | ($true != ((X11 @ id @ X13 @ id))) | ((((X11 @ (comp @ (sK326 @ X11) @ (sK328 @ X11)) @ (sub @ (sK329 @ X11) @ (sK328 @ X11)) @ (sK327 @ X11))) != $true) & ($true = ((X11 @ (sK326 @ X11) @ (sK329 @ X11) @ (comp @ (sK328 @ X11) @ (sK327 @ X11)))))) | (((X11 @ id @ (ap @ X12 @ X13) @ id)) = $true) | (((X11 @ id @ X12 @ id)) != $true)) | ($true != sP18))),
  inference(skolemize,[status(esa),new_symbols(skolem,[sK319,sK320,sK321,vAPP]),skolemize(X0,$thf(sK319)),skolemize(X1,$thf(sK320)),skolemize(X2,$thf(sK321)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X16,$thf(sK67 @ X15))],[f202])).
thf(f260,plain,(
  ((! [X392 : term,X393 : subst,X391 : term,X389 : (term > $o),X390 : (term > $o)] : (($true = ((X389 @ (sub @ X392 @ (push @ X391 @ X393))))) | ($true != ((X390 @ X392)))) | (pushprop_lem1 != $true)) & ((pushprop_lem1 = $true) | ? [X392 : term,X393 : subst,X391 : term,X389 : (term > $o),X390 : (term > $o)] : (($true != ((X389 @ (sub @ X392 @ (push @ X391 @ X393))))) & ($true = ((X390 @ X392)))))) & (hoasinduction_lem3a_gthm = $true) & ((! [X214 : term] : ? [X215 : (subst > term > term)] : ! [X217 : subst,X216 : term] : (((X215 @ X217 @ X216)) = ((sub @ X214 @ (push @ X216 @ X217)))) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ? [X214 : term] : ! [X215 : (subst > term > term)] : ? [X217 : subst,X216 : term] : (((X215 @ X217 @ X216)) != ((sub @ X214 @ (push @ X216 @ X217)))))) & (((sP58 = $true) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | (sP58 != $true))) & (((termmset_lthm = $true) | (? [X475 : term] : (((sub @ X475 @ id)) != X475) & ! [X474 : term] : (((sub @ X474 @ id)) = X474))) & ((! [X475 : term] : (((sub @ X475 @ id)) = X475) | ? [X474 : term] : (((sub @ X474 @ id)) != X474)) | (termmset_lthm != $true))) & (hoasinduction_lem3_gthm = $true) & (hoasinduction_lem1_lthm = $true) & (hoasapinj1_gthm = $true) & (((axvarcons = $true) | ? [X206 : term,X205 : subst] : (((sub @ one @ (push @ X206 @ X205))) != X206)) & (! [X206 : term,X205 : subst] : (((sub @ one @ (push @ X206 @ X205))) = X206) | (axvarcons != $true))) & (substmonoid_gthm = $true) & (((apinj2 = $true) | ? [X263 : term,X264 : term,X266 : term,X265 : term] : ((((ap @ X265 @ X266)) = ((ap @ X263 @ X264))) & (X264 != X266))) & (! [X263 : term,X264 : term,X266 : term,X265 : term] : ((((ap @ X265 @ X266)) != ((ap @ X263 @ X264))) | (X264 = X266)) | (apinj2 != $true))) & (((axidl = $true) | ? [X450 : subst] : (((comp @ id @ X450)) != X450)) & (! [X450 : subst] : (((comp @ id @ X450)) = X450) | (axidl != $true))) & (((hoasinduction_no_psi_cond_lthm = $true) | ($true != sP52)) & (($true = sP52) | (hoasinduction_no_psi_cond_lthm != $true))) & (hoaslamnotap_lthm = $true) & (hoasinduction_lem1 = $true) & (pushprop = $true) & (sh = ((d2subst @ d_id))) & (pushprop_lem1_gthm = $true) & (lamnotvar = $true) & ((! [X1 : term,X0 : term,X3 : term,X2 : term] : ((X0 = X3) | (((ap @ X3 @ X2)) != ((ap @ X0 @ X1)))) | (apinj1 != $true)) & ((apinj1 = $true) | ? [X1 : term,X0 : term,X3 : term,X2 : term] : ((X0 != X3) & (((ap @ X3 @ X2)) = ((ap @ X0 @ X1)))))) & (axassoc = $true) & (apnotvar = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (hoasinduction_lem2v2_gthm = $true) & (induction2_gthm = $true) & (((hoaslaminj = $true) | ? [X42 : (subst > term > term),X43 : subst,X41 : (subst > term > term),X44 : term] : ((((lam @ (X41 @ sh @ one))) = ((lam @ (X42 @ sh @ one)))) & ! [X45 : term,X47 : subst,X46 : subst] : (((sub @ (X42 @ X46 @ X45) @ X47)) = ((X42 @ (comp @ X46 @ X47) @ (sub @ X45 @ X47)))) & ! [X48 : term,X49 : subst,X50 : subst] : (((sub @ (X41 @ X49 @ X48) @ X50)) = ((X41 @ (comp @ X49 @ X50) @ (sub @ X48 @ X50)))) & (((X41 @ X43 @ X44)) != ((X42 @ X43 @ X44))))) & (! [X42 : (subst > term > term),X43 : subst,X41 : (subst > term > term),X44 : term] : ((((lam @ (X41 @ sh @ one))) != ((lam @ (X42 @ sh @ one)))) | ? [X45 : term,X47 : subst,X46 : subst] : (((sub @ (X42 @ X46 @ X45) @ X47)) != ((X42 @ (comp @ X46 @ X47) @ (sub @ X45 @ X47)))) | ? [X48 : term,X49 : subst,X50 : subst] : (((sub @ (X41 @ X49 @ X48) @ X50)) != ((X41 @ (comp @ X49 @ X50) @ (sub @ X48 @ X50)))) | (((X41 @ X43 @ X44)) = ((X42 @ X43 @ X44)))) | (hoaslaminj != $true))) & (axclos = $true) & (((hoasinduction_lem3aaa = $true) | (sP47 != $true)) & ((sP47 = $true) | (hoasinduction_lem3aaa != $true))) & (hoaslamnotvar_lthm = $true) & (((hoasinduction_lem3v2_lthm = $true) | ($true != sP46)) & (($true = sP46) | (hoasinduction_lem3v2_lthm != $true))) & ((! [X207 : term] : ? [X208 : (subst > term > term)] : ! [X210 : term,X209 : subst] : (((sub @ X207 @ (push @ X210 @ X209))) = ((X208 @ X209 @ X210))) | (hoasinduction_lem3v2_f_lthm != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X207 : term] : ! [X208 : (subst > term > term)] : ? [X210 : term,X209 : subst] : (((sub @ X207 @ (push @ X210 @ X209))) != ((X208 @ X209 @ X210))))) & ((! [X473 : term] : (((sub @ X473 @ id)) = X473) | (termmset != $true)) & ((termmset = $true) | ? [X473 : term] : (((sub @ X473 @ id)) != X473))) & (((? [X15 : subst] : (((comp @ id @ X15)) != X15) | (! [X17 : subst] : (((comp @ X17 @ id)) = X17) & ! [X18 : subst] : (((comp @ id @ X18)) = X18)) | ? [X16 : subst] : (((comp @ X16 @ id)) != X16)) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | (! [X15 : subst] : (((comp @ id @ X15)) = X15) & (? [X17 : subst] : (((comp @ X17 @ id)) != X17) | ? [X18 : subst] : (((comp @ id @ X18)) != X18)) & ! [X16 : subst] : (((comp @ X16 @ id)) = X16)))) & (hoaslamnotap_gthm = $true) & (hoaslamnotvar = $true) & (axmap = $true) & (((axvarid = $true) | ? [X14 : term] : (((sub @ X14 @ id)) != X14)) & (! [X14 : term] : (((sub @ X14 @ id)) = X14) | (axvarid != $true))) & (((sP44 = $true) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | (sP44 != $true))) & (termmset_gthm = $true) & (axapp = $true) & (((axshiftcons = $true) | ? [X489 : subst,X488 : term] : (((comp @ sh @ (push @ X488 @ X489))) != X489)) & (! [X489 : subst,X488 : term] : (((comp @ sh @ (push @ X488 @ X489))) = X489) | (axshiftcons != $true))) & (((! [X531 : term,X528 : term,X530 : term,X529 : term] : ((X528 = X530) | (((ap @ (sub @ X530 @ id) @ X529)) != ((ap @ (sub @ X528 @ id) @ X531)))) | ? [X524 : term,X526 : term,X525 : term,X527 : term] : ((X524 != X525) & (((ap @ X525 @ X526)) = ((ap @ X524 @ X527)))) | ? [X523 : term] : (((sub @ X523 @ id)) != X523)) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | (? [X531 : term,X528 : term,X530 : term,X529 : term] : ((X528 != X530) & (((ap @ (sub @ X530 @ id) @ X529)) = ((ap @ (sub @ X528 @ id) @ X531)))) & ! [X524 : term,X526 : term,X525 : term,X527 : term] : ((X524 = X525) | (((ap @ X525 @ X526)) != ((ap @ X524 @ X527)))) & ! [X523 : term] : (((sub @ X523 @ id)) = X523)))) & (((pushprop_lem0 = $true) | ? [X168 : term,X166 : subst,X167 : (term > $o)] : ! [X169 : (term > $o)] : ? [X170 : term] : (((X169 @ X170)) != ((X167 @ (sub @ X170 @ (push @ X168 @ X166)))))) & (! [X168 : term,X166 : subst,X167 : (term > $o)] : ? [X169 : (term > $o)] : ! [X170 : term] : (((X169 @ X170)) = ((X167 @ (sub @ X170 @ (push @ X168 @ X166))))) | (pushprop_lem0 != $true))) & (((hoasinduction_lem3a_lthm = $true) | ($true != sP43)) & (($true = sP43) | (hoasinduction_lem3a_lthm != $true))) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoasapinj2_gthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (one = ((d2term @ d_one))) & (((hoasapinj1 = $true) | ? [X21 : term,X19 : term,X20 : term,X22 : term] : ((X21 != X22) & (((ap @ (sub @ X22 @ id) @ X19)) = ((ap @ (sub @ X21 @ id) @ X20))))) & (! [X21 : term,X19 : term,X20 : term,X22 : term] : ((X21 = X22) | (((ap @ (sub @ X22 @ id) @ X19)) != ((ap @ (sub @ X21 @ id) @ X20)))) | (hoasapinj1 != $true))) & (((induction2 = $true) | (sP41 != $true)) & ((sP41 = $true) | (induction2 != $true))) & ! [X296 : subst] : ? [X297 : d_subst] : (((d2subst @ X297)) = X296) & (hoasinduction_lem2_gthm = $true) & (((? [X254 : subst,X255 : term] : (((sub @ one @ (push @ X255 @ X254))) != X255) | ! [X257 : (term > $o),X256 : subst,X258 : (term > $o),X259 : term] : (? [X260 : term] : (((X257 @ (sub @ X260 @ (push @ X259 @ X256)))) != ((X258 @ X260))) | ($true != ((X257 @ X259))) | ($true = ((X258 @ one))))) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (! [X254 : subst,X255 : term] : (((sub @ one @ (push @ X255 @ X254))) = X255) & ? [X257 : (term > $o),X256 : subst,X258 : (term > $o),X259 : term] : (! [X260 : term] : (((X257 @ (sub @ X260 @ (push @ X259 @ X256)))) = ((X258 @ X260))) & ($true = ((X257 @ X259))) & ($true != ((X258 @ one))))))) & ((! [X300 : term] : ? [X301 : (subst > term > term)] : (((sub @ X300 @ (push @ one @ sh))) = ((X301 @ sh @ one))) | (hoasinduction_lem3b_lthm != $true)) & ((hoasinduction_lem3b_lthm = $true) | ? [X300 : term] : ! [X301 : (subst > term > term)] : (((sub @ X300 @ (push @ one @ sh))) != ((X301 @ sh @ one))))) & (pushprop_lem3v2 = $true) & (((hoasinduction_lem0_lthm = $true) | ? [X451 : (subst > term > subst > $o)] : ! [X452 : (term > $o)] : ? [X453 : term] : (((X451 @ id @ X453 @ id)) != ((X452 @ X453)))) & (! [X451 : (subst > term > subst > $o)] : ? [X452 : (term > $o)] : ! [X453 : term] : (((X451 @ id @ X453 @ id)) = ((X452 @ X453))) | (hoasinduction_lem0_lthm != $true))) & (((! [X521 : subst] : (((comp @ X521 @ id)) = X521) & ! [X522 : subst] : (((comp @ id @ X522)) = X522)) | (substmonoid != $true)) & ((substmonoid = $true) | (? [X521 : subst] : (((comp @ X521 @ id)) != X521) | ? [X522 : subst] : (((comp @ id @ X522)) != X522)))) & (((d2subst @ d_id)) = id) & (((hoasinduction_lem3b = $true) | ? [X343 : term] : ! [X344 : (subst > term > term)] : (((sub @ X343 @ (push @ one @ sh))) != ((X344 @ sh @ one)))) & (! [X343 : term] : ? [X344 : (subst > term > term)] : (((sub @ X343 @ (push @ one @ sh))) = ((X344 @ sh @ one))) | (hoasinduction_lem3b != $true))) & ((! [X144 : (term > $o),X145 : subst,X143 : term] : ? [X146 : (term > $o)] : ! [X147 : term] : (((X144 @ (sub @ X147 @ (push @ X143 @ X145)))) = ((X146 @ X147))) | (pushprop_lem0_lthm != $true)) & ((pushprop_lem0_lthm = $true) | ? [X144 : (term > $o),X145 : subst,X143 : term] : ! [X146 : (term > $o)] : ? [X147 : term] : (((X144 @ (sub @ X147 @ (push @ X143 @ X145)))) != ((X146 @ X147))))) & (pushprop_lem2v2 = $true) & ! [X302 : d_term,X303 : d_term] : ((((d2term @ X302)) != ((d2term @ X303))) | (X302 = X303)) & (hoasinduction_lem1v2 = $true) & ((! [X139 : (subst > term > term),X138 : term,X137 : term] : (? [X140 : subst,X141 : term,X142 : subst] : (((sub @ (X139 @ X142 @ X141) @ X140)) != ((X139 @ (comp @ X142 @ X140) @ (sub @ X141 @ X140)))) | (((ap @ (sub @ X138 @ id) @ X137)) != ((lam @ (X139 @ sh @ one))))) | (hoaslamnotap != $true)) & ((hoaslamnotap = $true) | ? [X139 : (subst > term > term),X138 : term,X137 : term] : (! [X140 : subst,X141 : term,X142 : subst] : (((sub @ (X139 @ X142 @ X141) @ X140)) = ((X139 @ (comp @ X142 @ X140) @ (sub @ X141 @ X140)))) & (((ap @ (sub @ X138 @ id) @ X137)) = ((lam @ (X139 @ sh @ one))))))) & (((sP39 = $true) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | (sP39 != $true))) & (ulamvarsh = $true) & (((hoasinduction_lem0 = $true) | ? [X211 : (subst > term > subst > $o)] : ! [X212 : (term > $o)] : ? [X213 : term] : (((X212 @ X213)) != ((X211 @ id @ X213 @ id)))) & (! [X211 : (subst > term > subst > $o)] : ? [X212 : (term > $o)] : ! [X213 : term] : (((X212 @ X213)) = ((X211 @ id @ X213 @ id))) | (hoasinduction_lem0 != $true))) & (((hoasinduction_lthm = $true) | ($true != sP35)) & (($true = sP35) | (hoasinduction_lthm != $true))) & (((? [X228 : term,X230 : term,X229 : term,X227 : term] : ((X228 != X229) & (((ap @ X227 @ X228)) = ((ap @ X230 @ X229)))) | ! [X234 : term,X231 : term,X233 : term,X232 : term] : ((((ap @ (sub @ X233 @ id) @ X234)) != ((ap @ (sub @ X232 @ id) @ X231))) | (X231 = X234))) | (hoasapinj2_lthm != $true)) & ((hoasapinj2_lthm = $true) | (! [X228 : term,X230 : term,X229 : term,X227 : term] : ((X228 = X229) | (((ap @ X227 @ X228)) != ((ap @ X230 @ X229)))) & ? [X234 : term,X231 : term,X233 : term,X232 : term] : ((((ap @ (sub @ X233 @ id) @ X234)) = ((ap @ (sub @ X232 @ id) @ X231))) & (X231 != X234))))) & (ulamvarind = $true) & (pushprop_lem3v2_lthm = $true) & ((! [X304 : subst] : (((comp @ X304 @ id)) = X304) | (axidr != $true)) & ((axidr = $true) | ? [X304 : subst] : (((comp @ X304 @ id)) != X304))) & (hoasinduction_lem2_lthm = $true) & (((hoasinduction_lem3aa = $true) | (sP26 != $true)) & ((sP26 = $true) | (hoasinduction_lem3aa != $true))) & (axabs = $true) & (hoasapnotvar = $true) & (hoasinduction_lem1_gthm = $true) & (((hoasapinj2 = $true) | ? [X455 : term,X454 : term,X456 : term,X457 : term] : ((((ap @ (sub @ X457 @ id) @ X455)) = ((ap @ (sub @ X454 @ id) @ X456))) & (X455 != X456))) & (! [X455 : term,X454 : term,X456 : term,X457 : term] : ((((ap @ (sub @ X457 @ id) @ X455)) != ((ap @ (sub @ X454 @ id) @ X456))) | (X455 = X456)) | (hoasapinj2 != $true))) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (((hoasinduction_lem2v2 = $true) | (sP25 != $true)) & ((sP25 = $true) | (hoasinduction_lem2v2 != $true))) & (lamnotap != $true) & (pushprop_lem2v2_lthm = $true) & (pushprop_lem0_gthm = $true) & ! [X51 : term] : ? [X52 : d_term] : (((d2term @ X52)) = X51) & (((induction2_lthm = $true) | (sP23 != $true)) & ((sP23 = $true) | (induction2_lthm != $true))) & (hoaslamnotvar_gthm = $true) & (induction2lem_gthm = $true) & ((($true = sP19) | (hoaslaminj_lthm != $true)) & ((hoaslaminj_lthm = $true) | ($true != sP19))) & (pushprop_lthm = $true) & (((hoasinduction_lem2 = $true) | ($true != sP18)) & (($true = sP18) | (hoasinduction_lem2 != $true))) & (pushprop_lem1v2_gthm = $true) & (((hoasinduction_lem3v2a_lthm = $true) | ($true != sP16)) & (($true = sP16) | (hoasinduction_lem3v2a_lthm != $true))) & (hoasinduction_lem1v2_gthm = $true) & (((hoasinduction_lem3aa_lthm = $true) | (sP15 != $true)) & ((sP15 = $true) | (hoasinduction_lem3aa_lthm != $true))) & (((induction2lem = $true) | ($true != sP14)) & (($true = sP14) | (induction2lem != $true))) & (hoasinduction_lem3b_gthm = $true) & (pushprop_gthm = $true) & (((axscons = $true) | ? [X449 : subst] : (((push @ (sub @ one @ X449) @ (comp @ sh @ X449))) != X449)) & (! [X449 : subst] : (((push @ (sub @ one @ X449) @ (comp @ sh @ X449))) = X449) | (axscons != $true))) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (((hoasinduction = $true) | (sP12 != $true)) & ((sP12 = $true) | (hoasinduction != $true))) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (axvarshift = $true) & (hoasinduction_lem3v2_gthm = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoaslaminj_gthm = $true) & (pushprop_lem2v2_gthm = $true) & ((($true = sP9) | (hoasinduction_lem3_lthm != $true)) & ((hoasinduction_lem3_lthm = $true) | ($true != sP9))) & (((sP6 = $true) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | (sP6 != $true))) & (((var @ (d2term @ d_one))) != $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)))))))))))) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (((hoasinduction_no_psi_cond = $true) | ? [X66 : term,X67 : (subst > term > subst > $o)] : (! [X68 : (subst > term > term)] : (? [X72 : term] : ((((X67 @ id @ X72 @ id)) = $true) & (((X67 @ id @ (X68 @ id @ X72) @ id)) != $true)) | ($true = ((X67 @ id @ (lam @ (X68 @ sh @ one)) @ id))) | ? [X69 : subst,X71 : subst,X70 : term] : (((X68 @ (comp @ X69 @ X71) @ (sub @ X70 @ X71))) != ((sub @ (X68 @ X69 @ X70) @ X71)))) & (((sP5 @ X67)) != $true) & ($true != ((X67 @ id @ X66 @ id))))) & (! [X66 : term,X67 : (subst > term > subst > $o)] : (? [X68 : (subst > term > term)] : (! [X72 : term] : ((((X67 @ id @ X72 @ id)) != $true) | (((X67 @ id @ (X68 @ id @ X72) @ id)) = $true)) & ($true != ((X67 @ id @ (lam @ (X68 @ sh @ one)) @ id))) & ! [X69 : subst,X71 : subst,X70 : term] : (((X68 @ (comp @ X69 @ X71) @ (sub @ X70 @ X71))) = ((sub @ (X68 @ X69 @ X70) @ X71)))) | (((sP5 @ X67)) = $true) | ($true = ((X67 @ id @ X66 @ id)))) | (hoasinduction_no_psi_cond != $true))) & (ulamvar1 != $true) & (((pushprop_lem1v2 = $true) | ? [X267 : (term > $o),X268 : subst,X269 : (term > $o),X270 : term] : ((((X269 @ one)) != $true) & (((X267 @ X270)) = $true) & ! [X271 : term] : (((X267 @ (sub @ X271 @ (push @ X270 @ X268)))) = ((X269 @ X271))))) & (! [X267 : (term > $o),X268 : subst,X269 : (term > $o),X270 : term] : ((((X269 @ one)) = $true) | (((X267 @ X270)) != $true) | ? [X271 : term] : (((X267 @ (sub @ X271 @ (push @ X270 @ X268)))) != ((X269 @ X271)))) | (pushprop_lem1v2 != $true))) & ! [X280 : d_term] : (d_one = X280) & ! [X369 : d_subst] : (d_id = X369) & (((hoasinduction_lem3 = $true) | ($true != sP4)) & (($true = sP4) | (hoasinduction_lem3 != $true))) & (hoasapnotvar_gthm = $true) & (((laminj = $true) | ? [X262 : term,X261 : term] : ((X261 != X262) & (((lam @ X262)) = ((lam @ X261))))) & (! [X262 : term,X261 : term] : ((X261 = X262) | (((lam @ X262)) != ((lam @ X261)))) | (laminj != $true))) & (hoasinduction_gthm = $true) & (hoasapnotvar_lthm = $true) & ((! [X298 : term,X299 : term] : ((((sub @ X298 @ sh)) != ((sub @ X299 @ sh))) | (X298 = X299)) | (shinj != $true)) & ((shinj = $true) | ? [X298 : term,X299 : term] : ((((sub @ X298 @ sh)) = ((sub @ X299 @ sh))) & (X298 != X299)))) & (((induction = $true) | ? [X516 : (term > $o),X515 : term] : (! [X520 : term] : (($true = ((X516 @ X520))) | ($true != ((var @ X520)))) & (((X516 @ X515)) != $true) & ! [X519 : term] : (($true != ((X516 @ X519))) | (((X516 @ (lam @ X519))) = $true)) & (((sP2 @ X516)) != $true))) & (! [X516 : (term > $o),X515 : term] : (? [X520 : term] : (($true != ((X516 @ X520))) & ($true = ((var @ X520)))) | (((X516 @ X515)) = $true) | ? [X519 : term] : (($true = ((X516 @ X519))) & (((X516 @ (lam @ X519))) != $true)) | (((sP2 @ X516)) = $true)) | (induction != $true))) & ! [X40 : d_subst,X39 : d_subst] : ((X39 = X40) | (((d2subst @ X40)) != ((d2subst @ X39)))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (((hoasinduction_lem3v2 = $true) | (sP1 != $true)) & ((sP1 = $true) | (hoasinduction_lem3v2 != $true))) & (((! [X226 : subst,X223 : (term > $o),X224 : term,X225 : term,X222 : (term > $o)] : (($true != ((X222 @ X224))) | ($true = ((X223 @ (sub @ X224 @ (push @ X225 @ X226)))))) | ? [X219 : term,X218 : subst] : (((sub @ one @ (push @ X219 @ X218))) != X219) | ? [X221 : term,X220 : subst] : (((comp @ sh @ (push @ X221 @ X220))) != X220)) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (? [X226 : subst,X223 : (term > $o),X224 : term,X225 : term,X222 : (term > $o)] : (($true = ((X222 @ X224))) & ($true != ((X223 @ (sub @ X224 @ (push @ X225 @ X226)))))) & ! [X219 : term,X218 : subst] : (((sub @ one @ (push @ X219 @ X218))) = X219) & ! [X221 : term,X220 : subst] : (((comp @ sh @ (push @ X221 @ X220))) = X220)))) & (pushprop_lthm_orig = $true)),
  inference(nnf_transformation,[],[f72])).
thf(f261,plain,(
  (! [X392 : term,X393 : subst,X391 : term,X389 : (term > $o),X390 : (term > $o)] : (($true = ((X389 @ (sub @ X392 @ (push @ X391 @ X393))))) | ($true != ((X390 @ X392)))) | (pushprop_lem1 != $true)) & ((pushprop_lem1 = $true) | ? [X392 : term,X393 : subst,X391 : term,X389 : (term > $o),X390 : (term > $o)] : (($true != ((X389 @ (sub @ X392 @ (push @ X391 @ X393))))) & ($true = ((X390 @ X392))))) & (hoasinduction_lem3a_gthm = $true) & (! [X214 : term] : ? [X215 : (subst > term > term)] : ! [X217 : subst,X216 : term] : (((X215 @ X217 @ X216)) = ((sub @ X214 @ (push @ X216 @ X217)))) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ? [X214 : term] : ! [X215 : (subst > term > term)] : ? [X217 : subst,X216 : term] : (((X215 @ X217 @ X216)) != ((sub @ X214 @ (push @ X216 @ X217))))) & ((sP58 = $true) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | (sP58 != $true)) & ((termmset_lthm = $true) | (? [X475 : term] : (((sub @ X475 @ id)) != X475) & ! [X474 : term] : (((sub @ X474 @ id)) = X474))) & (! [X475 : term] : (((sub @ X475 @ id)) = X475) | ? [X474 : term] : (((sub @ X474 @ id)) != X474) | (termmset_lthm != $true)) & (hoasinduction_lem3_gthm = $true) & (hoasinduction_lem1_lthm = $true) & (hoasapinj1_gthm = $true) & ((axvarcons = $true) | ? [X206 : term,X205 : subst] : (((sub @ one @ (push @ X206 @ X205))) != X206)) & (! [X206 : term,X205 : subst] : (((sub @ one @ (push @ X206 @ X205))) = X206) | (axvarcons != $true)) & (substmonoid_gthm = $true) & ((apinj2 = $true) | ? [X263 : term,X264 : term,X266 : term,X265 : term] : ((((ap @ X265 @ X266)) = ((ap @ X263 @ X264))) & (X264 != X266))) & (! [X263 : term,X264 : term,X266 : term,X265 : term] : ((((ap @ X265 @ X266)) != ((ap @ X263 @ X264))) | (X264 = X266)) | (apinj2 != $true)) & ((axidl = $true) | ? [X450 : subst] : (((comp @ id @ X450)) != X450)) & (! [X450 : subst] : (((comp @ id @ X450)) = X450) | (axidl != $true)) & ((hoasinduction_no_psi_cond_lthm = $true) | ($true != sP52)) & (($true = sP52) | (hoasinduction_no_psi_cond_lthm != $true)) & (hoaslamnotap_lthm = $true) & (hoasinduction_lem1 = $true) & (pushprop = $true) & (sh = ((d2subst @ d_id))) & (pushprop_lem1_gthm = $true) & (lamnotvar = $true) & (! [X1 : term,X0 : term,X3 : term,X2 : term] : ((X0 = X3) | (((ap @ X3 @ X2)) != ((ap @ X0 @ X1)))) | (apinj1 != $true)) & ((apinj1 = $true) | ? [X1 : term,X0 : term,X3 : term,X2 : term] : ((X0 != X3) & (((ap @ X3 @ X2)) = ((ap @ X0 @ X1))))) & (axassoc = $true) & (apnotvar = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (hoasinduction_lem2v2_gthm = $true) & (induction2_gthm = $true) & ((hoaslaminj = $true) | ? [X42 : (subst > term > term),X43 : subst,X41 : (subst > term > term),X44 : term] : ((((lam @ (X41 @ sh @ one))) = ((lam @ (X42 @ sh @ one)))) & ! [X45 : term,X47 : subst,X46 : subst] : (((sub @ (X42 @ X46 @ X45) @ X47)) = ((X42 @ (comp @ X46 @ X47) @ (sub @ X45 @ X47)))) & ! [X48 : term,X49 : subst,X50 : subst] : (((sub @ (X41 @ X49 @ X48) @ X50)) = ((X41 @ (comp @ X49 @ X50) @ (sub @ X48 @ X50)))) & (((X41 @ X43 @ X44)) != ((X42 @ X43 @ X44))))) & (! [X42 : (subst > term > term),X43 : subst,X41 : (subst > term > term),X44 : term] : ((((lam @ (X41 @ sh @ one))) != ((lam @ (X42 @ sh @ one)))) | ? [X45 : term,X47 : subst,X46 : subst] : (((sub @ (X42 @ X46 @ X45) @ X47)) != ((X42 @ (comp @ X46 @ X47) @ (sub @ X45 @ X47)))) | ? [X48 : term,X49 : subst,X50 : subst] : (((sub @ (X41 @ X49 @ X48) @ X50)) != ((X41 @ (comp @ X49 @ X50) @ (sub @ X48 @ X50)))) | (((X41 @ X43 @ X44)) = ((X42 @ X43 @ X44)))) | (hoaslaminj != $true)) & (axclos = $true) & ((hoasinduction_lem3aaa = $true) | (sP47 != $true)) & ((sP47 = $true) | (hoasinduction_lem3aaa != $true)) & (hoaslamnotvar_lthm = $true) & ((hoasinduction_lem3v2_lthm = $true) | ($true != sP46)) & (($true = sP46) | (hoasinduction_lem3v2_lthm != $true)) & (! [X207 : term] : ? [X208 : (subst > term > term)] : ! [X210 : term,X209 : subst] : (((sub @ X207 @ (push @ X210 @ X209))) = ((X208 @ X209 @ X210))) | (hoasinduction_lem3v2_f_lthm != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X207 : term] : ! [X208 : (subst > term > term)] : ? [X210 : term,X209 : subst] : (((sub @ X207 @ (push @ X210 @ X209))) != ((X208 @ X209 @ X210)))) & (! [X473 : term] : (((sub @ X473 @ id)) = X473) | (termmset != $true)) & ((termmset = $true) | ? [X473 : term] : (((sub @ X473 @ id)) != X473)) & (? [X15 : subst] : (((comp @ id @ X15)) != X15) | (! [X17 : subst] : (((comp @ X17 @ id)) = X17) & ! [X18 : subst] : (((comp @ id @ X18)) = X18)) | ? [X16 : subst] : (((comp @ X16 @ id)) != X16) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | (! [X15 : subst] : (((comp @ id @ X15)) = X15) & (? [X17 : subst] : (((comp @ X17 @ id)) != X17) | ? [X18 : subst] : (((comp @ id @ X18)) != X18)) & ! [X16 : subst] : (((comp @ X16 @ id)) = X16))) & (hoaslamnotap_gthm = $true) & (hoaslamnotvar = $true) & (axmap = $true) & ((axvarid = $true) | ? [X14 : term] : (((sub @ X14 @ id)) != X14)) & (! [X14 : term] : (((sub @ X14 @ id)) = X14) | (axvarid != $true)) & ((sP44 = $true) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | (sP44 != $true)) & (termmset_gthm = $true) & (axapp = $true) & ((axshiftcons = $true) | ? [X489 : subst,X488 : term] : (((comp @ sh @ (push @ X488 @ X489))) != X489)) & (! [X489 : subst,X488 : term] : (((comp @ sh @ (push @ X488 @ X489))) = X489) | (axshiftcons != $true)) & (! [X531 : term,X528 : term,X530 : term,X529 : term] : ((X528 = X530) | (((ap @ (sub @ X530 @ id) @ X529)) != ((ap @ (sub @ X528 @ id) @ X531)))) | ? [X524 : term,X526 : term,X525 : term,X527 : term] : ((X524 != X525) & (((ap @ X525 @ X526)) = ((ap @ X524 @ X527)))) | ? [X523 : term] : (((sub @ X523 @ id)) != X523) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | (? [X531 : term,X528 : term,X530 : term,X529 : term] : ((X528 != X530) & (((ap @ (sub @ X530 @ id) @ X529)) = ((ap @ (sub @ X528 @ id) @ X531)))) & ! [X524 : term,X526 : term,X525 : term,X527 : term] : ((X524 = X525) | (((ap @ X525 @ X526)) != ((ap @ X524 @ X527)))) & ! [X523 : term] : (((sub @ X523 @ id)) = X523))) & ((pushprop_lem0 = $true) | ? [X168 : term,X166 : subst,X167 : (term > $o)] : ! [X169 : (term > $o)] : ? [X170 : term] : (((X169 @ X170)) != ((X167 @ (sub @ X170 @ (push @ X168 @ X166)))))) & (! [X168 : term,X166 : subst,X167 : (term > $o)] : ? [X169 : (term > $o)] : ! [X170 : term] : (((X169 @ X170)) = ((X167 @ (sub @ X170 @ (push @ X168 @ X166))))) | (pushprop_lem0 != $true)) & ((hoasinduction_lem3a_lthm = $true) | ($true != sP43)) & (($true = sP43) | (hoasinduction_lem3a_lthm != $true)) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoasapinj2_gthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (one = ((d2term @ d_one))) & ((hoasapinj1 = $true) | ? [X21 : term,X19 : term,X20 : term,X22 : term] : ((X21 != X22) & (((ap @ (sub @ X22 @ id) @ X19)) = ((ap @ (sub @ X21 @ id) @ X20))))) & (! [X21 : term,X19 : term,X20 : term,X22 : term] : ((X21 = X22) | (((ap @ (sub @ X22 @ id) @ X19)) != ((ap @ (sub @ X21 @ id) @ X20)))) | (hoasapinj1 != $true)) & ((induction2 = $true) | (sP41 != $true)) & ((sP41 = $true) | (induction2 != $true)) & ! [X296 : subst] : ? [X297 : d_subst] : (((d2subst @ X297)) = X296) & (hoasinduction_lem2_gthm = $true) & (? [X254 : subst,X255 : term] : (((sub @ one @ (push @ X255 @ X254))) != X255) | ! [X257 : (term > $o),X256 : subst,X258 : (term > $o),X259 : term] : (? [X260 : term] : (((X257 @ (sub @ X260 @ (push @ X259 @ X256)))) != ((X258 @ X260))) | ($true != ((X257 @ X259))) | ($true = ((X258 @ one)))) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (! [X254 : subst,X255 : term] : (((sub @ one @ (push @ X255 @ X254))) = X255) & ? [X257 : (term > $o),X256 : subst,X258 : (term > $o),X259 : term] : (! [X260 : term] : (((X257 @ (sub @ X260 @ (push @ X259 @ X256)))) = ((X258 @ X260))) & ($true = ((X257 @ X259))) & ($true != ((X258 @ one)))))) & (! [X300 : term] : ? [X301 : (subst > term > term)] : (((sub @ X300 @ (push @ one @ sh))) = ((X301 @ sh @ one))) | (hoasinduction_lem3b_lthm != $true)) & ((hoasinduction_lem3b_lthm = $true) | ? [X300 : term] : ! [X301 : (subst > term > term)] : (((sub @ X300 @ (push @ one @ sh))) != ((X301 @ sh @ one)))) & (pushprop_lem3v2 = $true) & ((hoasinduction_lem0_lthm = $true) | ? [X451 : (subst > term > subst > $o)] : ! [X452 : (term > $o)] : ? [X453 : term] : (((X451 @ id @ X453 @ id)) != ((X452 @ X453)))) & (! [X451 : (subst > term > subst > $o)] : ? [X452 : (term > $o)] : ! [X453 : term] : (((X451 @ id @ X453 @ id)) = ((X452 @ X453))) | (hoasinduction_lem0_lthm != $true)) & ((! [X521 : subst] : (((comp @ X521 @ id)) = X521) & ! [X522 : subst] : (((comp @ id @ X522)) = X522)) | (substmonoid != $true)) & ((substmonoid = $true) | ? [X521 : subst] : (((comp @ X521 @ id)) != X521) | ? [X522 : subst] : (((comp @ id @ X522)) != X522)) & (((d2subst @ d_id)) = id) & ((hoasinduction_lem3b = $true) | ? [X343 : term] : ! [X344 : (subst > term > term)] : (((sub @ X343 @ (push @ one @ sh))) != ((X344 @ sh @ one)))) & (! [X343 : term] : ? [X344 : (subst > term > term)] : (((sub @ X343 @ (push @ one @ sh))) = ((X344 @ sh @ one))) | (hoasinduction_lem3b != $true)) & (! [X144 : (term > $o),X145 : subst,X143 : term] : ? [X146 : (term > $o)] : ! [X147 : term] : (((X144 @ (sub @ X147 @ (push @ X143 @ X145)))) = ((X146 @ X147))) | (pushprop_lem0_lthm != $true)) & ((pushprop_lem0_lthm = $true) | ? [X144 : (term > $o),X145 : subst,X143 : term] : ! [X146 : (term > $o)] : ? [X147 : term] : (((X144 @ (sub @ X147 @ (push @ X143 @ X145)))) != ((X146 @ X147)))) & (pushprop_lem2v2 = $true) & ! [X302 : d_term,X303 : d_term] : ((((d2term @ X302)) != ((d2term @ X303))) | (X302 = X303)) & (hoasinduction_lem1v2 = $true) & (! [X139 : (subst > term > term),X138 : term,X137 : term] : (? [X140 : subst,X141 : term,X142 : subst] : (((sub @ (X139 @ X142 @ X141) @ X140)) != ((X139 @ (comp @ X142 @ X140) @ (sub @ X141 @ X140)))) | (((ap @ (sub @ X138 @ id) @ X137)) != ((lam @ (X139 @ sh @ one))))) | (hoaslamnotap != $true)) & ((hoaslamnotap = $true) | ? [X139 : (subst > term > term),X138 : term,X137 : term] : (! [X140 : subst,X141 : term,X142 : subst] : (((sub @ (X139 @ X142 @ X141) @ X140)) = ((X139 @ (comp @ X142 @ X140) @ (sub @ X141 @ X140)))) & (((ap @ (sub @ X138 @ id) @ X137)) = ((lam @ (X139 @ sh @ one)))))) & ((sP39 = $true) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | (sP39 != $true)) & (ulamvarsh = $true) & ((hoasinduction_lem0 = $true) | ? [X211 : (subst > term > subst > $o)] : ! [X212 : (term > $o)] : ? [X213 : term] : (((X212 @ X213)) != ((X211 @ id @ X213 @ id)))) & (! [X211 : (subst > term > subst > $o)] : ? [X212 : (term > $o)] : ! [X213 : term] : (((X212 @ X213)) = ((X211 @ id @ X213 @ id))) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lthm = $true) | ($true != sP35)) & (($true = sP35) | (hoasinduction_lthm != $true)) & (? [X228 : term,X230 : term,X229 : term,X227 : term] : ((X228 != X229) & (((ap @ X227 @ X228)) = ((ap @ X230 @ X229)))) | ! [X234 : term,X231 : term,X233 : term,X232 : term] : ((((ap @ (sub @ X233 @ id) @ X234)) != ((ap @ (sub @ X232 @ id) @ X231))) | (X231 = X234)) | (hoasapinj2_lthm != $true)) & ((hoasapinj2_lthm = $true) | (! [X228 : term,X230 : term,X229 : term,X227 : term] : ((X228 = X229) | (((ap @ X227 @ X228)) != ((ap @ X230 @ X229)))) & ? [X234 : term,X231 : term,X233 : term,X232 : term] : ((((ap @ (sub @ X233 @ id) @ X234)) = ((ap @ (sub @ X232 @ id) @ X231))) & (X231 != X234)))) & (ulamvarind = $true) & (pushprop_lem3v2_lthm = $true) & (! [X304 : subst] : (((comp @ X304 @ id)) = X304) | (axidr != $true)) & ((axidr = $true) | ? [X304 : subst] : (((comp @ X304 @ id)) != X304)) & (hoasinduction_lem2_lthm = $true) & ((hoasinduction_lem3aa = $true) | (sP26 != $true)) & ((sP26 = $true) | (hoasinduction_lem3aa != $true)) & (axabs = $true) & (hoasapnotvar = $true) & (hoasinduction_lem1_gthm = $true) & ((hoasapinj2 = $true) | ? [X455 : term,X454 : term,X456 : term,X457 : term] : ((((ap @ (sub @ X457 @ id) @ X455)) = ((ap @ (sub @ X454 @ id) @ X456))) & (X455 != X456))) & (! [X455 : term,X454 : term,X456 : term,X457 : term] : ((((ap @ (sub @ X457 @ id) @ X455)) != ((ap @ (sub @ X454 @ id) @ X456))) | (X455 = X456)) | (hoasapinj2 != $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((hoasinduction_lem2v2 = $true) | (sP25 != $true)) & ((sP25 = $true) | (hoasinduction_lem2v2 != $true)) & (lamnotap != $true) & (pushprop_lem2v2_lthm = $true) & (pushprop_lem0_gthm = $true) & ! [X51 : term] : ? [X52 : d_term] : (((d2term @ X52)) = X51) & ((induction2_lthm = $true) | (sP23 != $true)) & ((sP23 = $true) | (induction2_lthm != $true)) & (hoaslamnotvar_gthm = $true) & (induction2lem_gthm = $true) & (($true = sP19) | (hoaslaminj_lthm != $true)) & ((hoaslaminj_lthm = $true) | ($true != sP19)) & (pushprop_lthm = $true) & ((hoasinduction_lem2 = $true) | ($true != sP18)) & (($true = sP18) | (hoasinduction_lem2 != $true)) & (pushprop_lem1v2_gthm = $true) & ((hoasinduction_lem3v2a_lthm = $true) | ($true != sP16)) & (($true = sP16) | (hoasinduction_lem3v2a_lthm != $true)) & (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_lem3aa_lthm = $true) | (sP15 != $true)) & ((sP15 = $true) | (hoasinduction_lem3aa_lthm != $true)) & ((induction2lem = $true) | ($true != sP14)) & (($true = sP14) | (induction2lem != $true)) & (hoasinduction_lem3b_gthm = $true) & (pushprop_gthm = $true) & ((axscons = $true) | ? [X449 : subst] : (((push @ (sub @ one @ X449) @ (comp @ sh @ X449))) != X449)) & (! [X449 : subst] : (((push @ (sub @ one @ X449) @ (comp @ sh @ X449))) = X449) | (axscons != $true)) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((hoasinduction = $true) | (sP12 != $true)) & ((sP12 = $true) | (hoasinduction != $true)) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (axvarshift = $true) & (hoasinduction_lem3v2_gthm = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoaslaminj_gthm = $true) & (pushprop_lem2v2_gthm = $true) & (($true = sP9) | (hoasinduction_lem3_lthm != $true)) & ((hoasinduction_lem3_lthm = $true) | ($true != sP9)) & ((sP6 = $true) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | (sP6 != $true)) & (((var @ (d2term @ d_one))) != $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)))))))))))) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ((hoasinduction_no_psi_cond = $true) | ? [X66 : term,X67 : (subst > term > subst > $o)] : (! [X68 : (subst > term > term)] : (? [X72 : term] : ((((X67 @ id @ X72 @ id)) = $true) & (((X67 @ id @ (X68 @ id @ X72) @ id)) != $true)) | ($true = ((X67 @ id @ (lam @ (X68 @ sh @ one)) @ id))) | ? [X69 : subst,X71 : subst,X70 : term] : (((X68 @ (comp @ X69 @ X71) @ (sub @ X70 @ X71))) != ((sub @ (X68 @ X69 @ X70) @ X71)))) & (((sP5 @ X67)) != $true) & ($true != ((X67 @ id @ X66 @ id))))) & (! [X66 : term,X67 : (subst > term > subst > $o)] : (? [X68 : (subst > term > term)] : (! [X72 : term] : ((((X67 @ id @ X72 @ id)) != $true) | (((X67 @ id @ (X68 @ id @ X72) @ id)) = $true)) & ($true != ((X67 @ id @ (lam @ (X68 @ sh @ one)) @ id))) & ! [X69 : subst,X71 : subst,X70 : term] : (((X68 @ (comp @ X69 @ X71) @ (sub @ X70 @ X71))) = ((sub @ (X68 @ X69 @ X70) @ X71)))) | (((sP5 @ X67)) = $true) | ($true = ((X67 @ id @ X66 @ id)))) | (hoasinduction_no_psi_cond != $true)) & (ulamvar1 != $true) & ((pushprop_lem1v2 = $true) | ? [X267 : (term > $o),X268 : subst,X269 : (term > $o),X270 : term] : ((((X269 @ one)) != $true) & (((X267 @ X270)) = $true) & ! [X271 : term] : (((X267 @ (sub @ X271 @ (push @ X270 @ X268)))) = ((X269 @ X271))))) & (! [X267 : (term > $o),X268 : subst,X269 : (term > $o),X270 : term] : ((((X269 @ one)) = $true) | (((X267 @ X270)) != $true) | ? [X271 : term] : (((X267 @ (sub @ X271 @ (push @ X270 @ X268)))) != ((X269 @ X271)))) | (pushprop_lem1v2 != $true)) & ! [X280 : d_term] : (d_one = X280) & ! [X369 : d_subst] : (d_id = X369) & ((hoasinduction_lem3 = $true) | ($true != sP4)) & (($true = sP4) | (hoasinduction_lem3 != $true)) & (hoasapnotvar_gthm = $true) & ((laminj = $true) | ? [X262 : term,X261 : term] : ((X261 != X262) & (((lam @ X262)) = ((lam @ X261))))) & (! [X262 : term,X261 : term] : ((X261 = X262) | (((lam @ X262)) != ((lam @ X261)))) | (laminj != $true)) & (hoasinduction_gthm = $true) & (hoasapnotvar_lthm = $true) & (! [X298 : term,X299 : term] : ((((sub @ X298 @ sh)) != ((sub @ X299 @ sh))) | (X298 = X299)) | (shinj != $true)) & ((shinj = $true) | ? [X298 : term,X299 : term] : ((((sub @ X298 @ sh)) = ((sub @ X299 @ sh))) & (X298 != X299))) & ((induction = $true) | ? [X516 : (term > $o),X515 : term] : (! [X520 : term] : (($true = ((X516 @ X520))) | ($true != ((var @ X520)))) & (((X516 @ X515)) != $true) & ! [X519 : term] : (($true != ((X516 @ X519))) | (((X516 @ (lam @ X519))) = $true)) & (((sP2 @ X516)) != $true))) & (! [X516 : (term > $o),X515 : term] : (? [X520 : term] : (($true != ((X516 @ X520))) & ($true = ((var @ X520)))) | (((X516 @ X515)) = $true) | ? [X519 : term] : (($true = ((X516 @ X519))) & (((X516 @ (lam @ X519))) != $true)) | (((sP2 @ X516)) = $true)) | (induction != $true)) & ! [X40 : d_subst,X39 : d_subst] : ((X39 = X40) | (((d2subst @ X40)) != ((d2subst @ X39)))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & ((hoasinduction_lem3v2 = $true) | (sP1 != $true)) & ((sP1 = $true) | (hoasinduction_lem3v2 != $true)) & (! [X226 : subst,X223 : (term > $o),X224 : term,X225 : term,X222 : (term > $o)] : (($true != ((X222 @ X224))) | ($true = ((X223 @ (sub @ X224 @ (push @ X225 @ X226)))))) | ? [X219 : term,X218 : subst] : (((sub @ one @ (push @ X219 @ X218))) != X219) | ? [X221 : term,X220 : subst] : (((comp @ sh @ (push @ X221 @ X220))) != X220) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (? [X226 : subst,X223 : (term > $o),X224 : term,X225 : term,X222 : (term > $o)] : (($true = ((X222 @ X224))) & ($true != ((X223 @ (sub @ X224 @ (push @ X225 @ X226)))))) & ! [X219 : term,X218 : subst] : (((sub @ one @ (push @ X219 @ X218))) = X219) & ! [X221 : term,X220 : subst] : (((comp @ sh @ (push @ X221 @ X220))) = X220))) & (pushprop_lthm_orig = $true)),
  inference(flattening,[],[f260])).
thf(f262,plain,(
  (! [X0 : term,X1 : subst,X2 : term,X3 : (term > $o),X4 : (term > $o)] : ((((X3 @ (sub @ X0 @ (push @ X2 @ X1)))) = $true) | ($true != ((X4 @ X0)))) | (pushprop_lem1 != $true)) & ((pushprop_lem1 = $true) | ? [X5 : term,X6 : subst,X7 : term,X8 : (term > $o),X9 : (term > $o)] : ((((X8 @ (sub @ X5 @ (push @ X7 @ X6)))) != $true) & ($true = ((X9 @ X5))))) & (hoasinduction_lem3a_gthm = $true) & (! [X10 : term] : ? [X11 : (subst > term > term)] : ! [X12 : subst,X13 : term] : (((X11 @ X12 @ X13)) = ((sub @ X10 @ (push @ X13 @ X12)))) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ? [X14 : term] : ! [X15 : (subst > term > term)] : ? [X16 : subst,X17 : term] : (((X15 @ X16 @ X17)) != ((sub @ X14 @ (push @ X17 @ X16))))) & ((sP58 = $true) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | (sP58 != $true)) & ((termmset_lthm = $true) | (? [X18 : term] : (((sub @ X18 @ id)) != X18) & ! [X19 : term] : (((sub @ X19 @ id)) = X19))) & (! [X20 : term] : (((sub @ X20 @ id)) = X20) | ? [X21 : term] : (((sub @ X21 @ id)) != X21) | (termmset_lthm != $true)) & (hoasinduction_lem3_gthm = $true) & (hoasinduction_lem1_lthm = $true) & (hoasapinj1_gthm = $true) & ((axvarcons = $true) | ? [X22 : term,X23 : subst] : (((sub @ one @ (push @ X22 @ X23))) != X22)) & (! [X24 : term,X25 : subst] : (((sub @ one @ (push @ X24 @ X25))) = X24) | (axvarcons != $true)) & (substmonoid_gthm = $true) & ((apinj2 = $true) | ? [X26 : term,X27 : term,X28 : term,X29 : term] : ((((ap @ X26 @ X27)) = ((ap @ X29 @ X28))) & (X27 != X28))) & (! [X30 : term,X31 : term,X32 : term,X33 : term] : ((((ap @ X33 @ X32)) != ((ap @ X30 @ X31))) | (X31 = X32)) | (apinj2 != $true)) & ((axidl = $true) | ? [X34 : subst] : (((comp @ id @ X34)) != X34)) & (! [X35 : subst] : (((comp @ id @ X35)) = X35) | (axidl != $true)) & ((hoasinduction_no_psi_cond_lthm = $true) | ($true != sP52)) & (($true = sP52) | (hoasinduction_no_psi_cond_lthm != $true)) & (hoaslamnotap_lthm = $true) & (hoasinduction_lem1 = $true) & (pushprop = $true) & (sh = ((d2subst @ d_id))) & (pushprop_lem1_gthm = $true) & (lamnotvar = $true) & (! [X36 : term,X37 : term,X38 : term,X39 : term] : ((X37 = X38) | (((ap @ X38 @ X39)) != ((ap @ X37 @ X36)))) | (apinj1 != $true)) & ((apinj1 = $true) | ? [X40 : term,X41 : term,X42 : term,X43 : term] : ((X41 != X42) & (((ap @ X42 @ X43)) = ((ap @ X41 @ X40))))) & (axassoc = $true) & (apnotvar = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (hoasinduction_lem2v2_gthm = $true) & (induction2_gthm = $true) & ((hoaslaminj = $true) | ? [X44 : (subst > term > term),X45 : subst,X46 : (subst > term > term),X47 : term] : ((((lam @ (X44 @ sh @ one))) = ((lam @ (X46 @ sh @ one)))) & ! [X48 : term,X49 : subst,X50 : subst] : (((sub @ (X44 @ X50 @ X48) @ X49)) = ((X44 @ (comp @ X50 @ X49) @ (sub @ X48 @ X49)))) & ! [X51 : term,X52 : subst,X53 : subst] : (((X46 @ (comp @ X52 @ X53) @ (sub @ X51 @ X53))) = ((sub @ (X46 @ X52 @ X51) @ X53))) & (((X46 @ X45 @ X47)) != ((X44 @ X45 @ X47))))) & (! [X54 : (subst > term > term),X55 : subst,X56 : (subst > term > term),X57 : term] : ((((lam @ (X54 @ sh @ one))) != ((lam @ (X56 @ sh @ one)))) | ? [X58 : term,X59 : subst,X60 : subst] : (((X54 @ (comp @ X60 @ X59) @ (sub @ X58 @ X59))) != ((sub @ (X54 @ X60 @ X58) @ X59))) | ? [X61 : term,X62 : subst,X63 : subst] : (((X56 @ (comp @ X62 @ X63) @ (sub @ X61 @ X63))) != ((sub @ (X56 @ X62 @ X61) @ X63))) | (((X56 @ X55 @ X57)) = ((X54 @ X55 @ X57)))) | (hoaslaminj != $true)) & (axclos = $true) & ((hoasinduction_lem3aaa = $true) | (sP47 != $true)) & ((sP47 = $true) | (hoasinduction_lem3aaa != $true)) & (hoaslamnotvar_lthm = $true) & ((hoasinduction_lem3v2_lthm = $true) | ($true != sP46)) & (($true = sP46) | (hoasinduction_lem3v2_lthm != $true)) & (! [X64 : term] : ? [X65 : (subst > term > term)] : ! [X66 : term,X67 : subst] : (((sub @ X64 @ (push @ X66 @ X67))) = ((X65 @ X67 @ X66))) | (hoasinduction_lem3v2_f_lthm != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X68 : term] : ! [X69 : (subst > term > term)] : ? [X70 : term,X71 : subst] : (((sub @ X68 @ (push @ X70 @ X71))) != ((X69 @ X71 @ X70)))) & (! [X72 : term] : (((sub @ X72 @ id)) = X72) | (termmset != $true)) & ((termmset = $true) | ? [X73 : term] : (((sub @ X73 @ id)) != X73)) & (? [X74 : subst] : (((comp @ id @ X74)) != X74) | (! [X75 : subst] : (((comp @ X75 @ id)) = X75) & ! [X76 : subst] : (((comp @ id @ X76)) = X76)) | ? [X77 : subst] : (((comp @ X77 @ id)) != X77) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | (! [X78 : subst] : (((comp @ id @ X78)) = X78) & (? [X79 : subst] : (((comp @ X79 @ id)) != X79) | ? [X80 : subst] : (((comp @ id @ X80)) != X80)) & ! [X81 : subst] : (((comp @ X81 @ id)) = X81))) & (hoaslamnotap_gthm = $true) & (hoaslamnotvar = $true) & (axmap = $true) & ((axvarid = $true) | ? [X82 : term] : (((sub @ X82 @ id)) != X82)) & (! [X83 : term] : (((sub @ X83 @ id)) = X83) | (axvarid != $true)) & ((sP44 = $true) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | (sP44 != $true)) & (termmset_gthm = $true) & (axapp = $true) & ((axshiftcons = $true) | ? [X84 : subst,X85 : term] : (((comp @ sh @ (push @ X85 @ X84))) != X84)) & (! [X86 : subst,X87 : term] : (((comp @ sh @ (push @ X87 @ X86))) = X86) | (axshiftcons != $true)) & (! [X88 : term,X89 : term,X90 : term,X91 : term] : ((X89 = X90) | (((ap @ (sub @ X89 @ id) @ X88)) != ((ap @ (sub @ X90 @ id) @ X91)))) | ? [X92 : term,X93 : term,X94 : term,X95 : term] : ((X92 != X94) & (((ap @ X94 @ X93)) = ((ap @ X92 @ X95)))) | ? [X96 : term] : (((sub @ X96 @ id)) != X96) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | (? [X97 : term,X98 : term,X99 : term,X100 : term] : ((X98 != X99) & (((ap @ (sub @ X98 @ id) @ X97)) = ((ap @ (sub @ X99 @ id) @ X100)))) & ! [X101 : term,X102 : term,X103 : term,X104 : term] : ((X101 = X103) | (((ap @ X101 @ X104)) != ((ap @ X103 @ X102)))) & ! [X105 : term] : (((sub @ X105 @ id)) = X105))) & ((pushprop_lem0 = $true) | ? [X106 : term,X107 : subst,X108 : (term > $o)] : ! [X109 : (term > $o)] : ? [X110 : term] : (((X108 @ (sub @ X110 @ (push @ X106 @ X107)))) != ((X109 @ X110)))) & (! [X111 : term,X112 : subst,X113 : (term > $o)] : ? [X114 : (term > $o)] : ! [X115 : term] : (((X113 @ (sub @ X115 @ (push @ X111 @ X112)))) = ((X114 @ X115))) | (pushprop_lem0 != $true)) & ((hoasinduction_lem3a_lthm = $true) | ($true != sP43)) & (($true = sP43) | (hoasinduction_lem3a_lthm != $true)) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoasapinj2_gthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (one = ((d2term @ d_one))) & ((hoasapinj1 = $true) | ? [X116 : term,X117 : term,X118 : term,X119 : term] : ((X116 != X119) & (((ap @ (sub @ X116 @ id) @ X118)) = ((ap @ (sub @ X119 @ id) @ X117))))) & (! [X120 : term,X121 : term,X122 : term,X123 : term] : ((X120 = X123) | (((ap @ (sub @ X120 @ id) @ X122)) != ((ap @ (sub @ X123 @ id) @ X121)))) | (hoasapinj1 != $true)) & ((induction2 = $true) | (sP41 != $true)) & ((sP41 = $true) | (induction2 != $true)) & ! [X124 : subst] : ? [X125 : d_subst] : (((d2subst @ X125)) = X124) & (hoasinduction_lem2_gthm = $true) & (? [X126 : subst,X127 : term] : (((sub @ one @ (push @ X127 @ X126))) != X127) | ! [X128 : (term > $o),X129 : subst,X130 : (term > $o),X131 : term] : (? [X132 : term] : (((X128 @ (sub @ X132 @ (push @ X131 @ X129)))) != ((X130 @ X132))) | ($true != ((X128 @ X131))) | (((X130 @ one)) = $true)) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (! [X133 : subst,X134 : term] : (((sub @ one @ (push @ X134 @ X133))) = X134) & ? [X135 : (term > $o),X136 : subst,X137 : (term > $o),X138 : term] : (! [X139 : term] : (((X135 @ (sub @ X139 @ (push @ X138 @ X136)))) = ((X137 @ X139))) & ($true = ((X135 @ X138))) & ($true != ((X137 @ one)))))) & (! [X140 : term] : ? [X141 : (subst > term > term)] : (((X141 @ sh @ one)) = ((sub @ X140 @ (push @ one @ sh)))) | (hoasinduction_lem3b_lthm != $true)) & ((hoasinduction_lem3b_lthm = $true) | ? [X142 : term] : ! [X143 : (subst > term > term)] : (((sub @ X142 @ (push @ one @ sh))) != ((X143 @ sh @ one)))) & (pushprop_lem3v2 = $true) & ((hoasinduction_lem0_lthm = $true) | ? [X144 : (subst > term > subst > $o)] : ! [X145 : (term > $o)] : ? [X146 : term] : (((X145 @ X146)) != ((X144 @ id @ X146 @ id)))) & (! [X147 : (subst > term > subst > $o)] : ? [X148 : (term > $o)] : ! [X149 : term] : (((X148 @ X149)) = ((X147 @ id @ X149 @ id))) | (hoasinduction_lem0_lthm != $true)) & ((! [X150 : subst] : (((comp @ X150 @ id)) = X150) & ! [X151 : subst] : (((comp @ id @ X151)) = X151)) | (substmonoid != $true)) & ((substmonoid = $true) | ? [X152 : subst] : (((comp @ X152 @ id)) != X152) | ? [X153 : subst] : (((comp @ id @ X153)) != X153)) & (((d2subst @ d_id)) = id) & ((hoasinduction_lem3b = $true) | ? [X154 : term] : ! [X155 : (subst > term > term)] : (((X155 @ sh @ one)) != ((sub @ X154 @ (push @ one @ sh))))) & (! [X156 : term] : ? [X157 : (subst > term > term)] : (((sub @ X156 @ (push @ one @ sh))) = ((X157 @ sh @ one))) | (hoasinduction_lem3b != $true)) & (! [X158 : (term > $o),X159 : subst,X160 : term] : ? [X161 : (term > $o)] : ! [X162 : term] : (((X161 @ X162)) = ((X158 @ (sub @ X162 @ (push @ X160 @ X159))))) | (pushprop_lem0_lthm != $true)) & ((pushprop_lem0_lthm = $true) | ? [X163 : (term > $o),X164 : subst,X165 : term] : ! [X166 : (term > $o)] : ? [X167 : term] : (((X163 @ (sub @ X167 @ (push @ X165 @ X164)))) != ((X166 @ X167)))) & (pushprop_lem2v2 = $true) & ! [X168 : d_term,X169 : d_term] : ((((d2term @ X169)) != ((d2term @ X168))) | (X168 = X169)) & (hoasinduction_lem1v2 = $true) & (! [X170 : (subst > term > term),X171 : term,X172 : term] : (? [X173 : subst,X174 : term,X175 : subst] : (((sub @ (X170 @ X175 @ X174) @ X173)) != ((X170 @ (comp @ X175 @ X173) @ (sub @ X174 @ X173)))) | (((lam @ (X170 @ sh @ one))) != ((ap @ (sub @ X171 @ id) @ X172)))) | (hoaslamnotap != $true)) & ((hoaslamnotap = $true) | ? [X176 : (subst > term > term),X177 : term,X178 : term] : (! [X179 : subst,X180 : term,X181 : subst] : (((X176 @ (comp @ X181 @ X179) @ (sub @ X180 @ X179))) = ((sub @ (X176 @ X181 @ X180) @ X179))) & (((ap @ (sub @ X177 @ id) @ X178)) = ((lam @ (X176 @ sh @ one)))))) & ((sP39 = $true) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | (sP39 != $true)) & (ulamvarsh = $true) & ((hoasinduction_lem0 = $true) | ? [X182 : (subst > term > subst > $o)] : ! [X183 : (term > $o)] : ? [X184 : term] : (((X182 @ id @ X184 @ id)) != ((X183 @ X184)))) & (! [X185 : (subst > term > subst > $o)] : ? [X186 : (term > $o)] : ! [X187 : term] : (((X186 @ X187)) = ((X185 @ id @ X187 @ id))) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lthm = $true) | ($true != sP35)) & (($true = sP35) | (hoasinduction_lthm != $true)) & (? [X188 : term,X189 : term,X190 : term,X191 : term] : ((X188 != X190) & (((ap @ X191 @ X188)) = ((ap @ X189 @ X190)))) | ! [X192 : term,X193 : term,X194 : term,X195 : term] : ((((ap @ (sub @ X195 @ id) @ X193)) != ((ap @ (sub @ X194 @ id) @ X192))) | (X192 = X193)) | (hoasapinj2_lthm != $true)) & ((hoasapinj2_lthm = $true) | (! [X196 : term,X197 : term,X198 : term,X199 : term] : ((X196 = X198) | (((ap @ X199 @ X196)) != ((ap @ X197 @ X198)))) & ? [X200 : term,X201 : term,X202 : term,X203 : term] : ((((ap @ (sub @ X203 @ id) @ X201)) = ((ap @ (sub @ X202 @ id) @ X200))) & (X200 != X201)))) & (ulamvarind = $true) & (pushprop_lem3v2_lthm = $true) & (! [X204 : subst] : (((comp @ X204 @ id)) = X204) | (axidr != $true)) & ((axidr = $true) | ? [X205 : subst] : (((comp @ X205 @ id)) != X205)) & (hoasinduction_lem2_lthm = $true) & ((hoasinduction_lem3aa = $true) | (sP26 != $true)) & ((sP26 = $true) | (hoasinduction_lem3aa != $true)) & (axabs = $true) & (hoasapnotvar = $true) & (hoasinduction_lem1_gthm = $true) & ((hoasapinj2 = $true) | ? [X206 : term,X207 : term,X208 : term,X209 : term] : ((((ap @ (sub @ X207 @ id) @ X208)) = ((ap @ (sub @ X209 @ id) @ X206))) & (X206 != X208))) & (! [X210 : term,X211 : term,X212 : term,X213 : term] : ((((ap @ (sub @ X211 @ id) @ X212)) != ((ap @ (sub @ X213 @ id) @ X210))) | (X210 = X212)) | (hoasapinj2 != $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((hoasinduction_lem2v2 = $true) | (sP25 != $true)) & ((sP25 = $true) | (hoasinduction_lem2v2 != $true)) & (lamnotap != $true) & (pushprop_lem2v2_lthm = $true) & (pushprop_lem0_gthm = $true) & ! [X214 : term] : ? [X215 : d_term] : (((d2term @ X215)) = X214) & ((induction2_lthm = $true) | (sP23 != $true)) & ((sP23 = $true) | (induction2_lthm != $true)) & (hoaslamnotvar_gthm = $true) & (induction2lem_gthm = $true) & (($true = sP19) | (hoaslaminj_lthm != $true)) & ((hoaslaminj_lthm = $true) | ($true != sP19)) & (pushprop_lthm = $true) & ((hoasinduction_lem2 = $true) | ($true != sP18)) & (($true = sP18) | (hoasinduction_lem2 != $true)) & (pushprop_lem1v2_gthm = $true) & ((hoasinduction_lem3v2a_lthm = $true) | ($true != sP16)) & (($true = sP16) | (hoasinduction_lem3v2a_lthm != $true)) & (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_lem3aa_lthm = $true) | (sP15 != $true)) & ((sP15 = $true) | (hoasinduction_lem3aa_lthm != $true)) & ((induction2lem = $true) | ($true != sP14)) & (($true = sP14) | (induction2lem != $true)) & (hoasinduction_lem3b_gthm = $true) & (pushprop_gthm = $true) & ((axscons = $true) | ? [X216 : subst] : (((push @ (sub @ one @ X216) @ (comp @ sh @ X216))) != X216)) & (! [X217 : subst] : (((push @ (sub @ one @ X217) @ (comp @ sh @ X217))) = X217) | (axscons != $true)) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((hoasinduction = $true) | (sP12 != $true)) & ((sP12 = $true) | (hoasinduction != $true)) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (axvarshift = $true) & (hoasinduction_lem3v2_gthm = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoaslaminj_gthm = $true) & (pushprop_lem2v2_gthm = $true) & (($true = sP9) | (hoasinduction_lem3_lthm != $true)) & ((hoasinduction_lem3_lthm = $true) | ($true != sP9)) & ((sP6 = $true) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | (sP6 != $true)) & (((var @ (d2term @ d_one))) != $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)))))))))))) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ((hoasinduction_no_psi_cond = $true) | ? [X218 : term,X219 : (subst > term > subst > $o)] : (! [X220 : (subst > term > term)] : (? [X221 : term] : (($true = ((X219 @ id @ X221 @ id))) & (((X219 @ id @ (X220 @ id @ X221) @ id)) != $true)) | (((X219 @ id @ (lam @ (X220 @ sh @ one)) @ id)) = $true) | ? [X222 : subst,X223 : subst,X224 : term] : (((sub @ (X220 @ X222 @ X224) @ X223)) != ((X220 @ (comp @ X222 @ X223) @ (sub @ X224 @ X223))))) & ($true != ((sP5 @ X219))) & ($true != ((X219 @ id @ X218 @ id))))) & (! [X225 : term,X226 : (subst > term > subst > $o)] : (? [X227 : (subst > term > term)] : (! [X228 : term] : ((((X226 @ id @ X228 @ id)) != $true) | (((X226 @ id @ (X227 @ id @ X228) @ id)) = $true)) & ($true != ((X226 @ id @ (lam @ (X227 @ sh @ one)) @ id))) & ! [X229 : subst,X230 : subst,X231 : term] : (((sub @ (X227 @ X229 @ X231) @ X230)) = ((X227 @ (comp @ X229 @ X230) @ (sub @ X231 @ X230))))) | (((sP5 @ X226)) = $true) | (((X226 @ id @ X225 @ id)) = $true)) | (hoasinduction_no_psi_cond != $true)) & (ulamvar1 != $true) & ((pushprop_lem1v2 = $true) | ? [X232 : (term > $o),X233 : subst,X234 : (term > $o),X235 : term] : ((((X234 @ one)) != $true) & ($true = ((X232 @ X235))) & ! [X236 : term] : (((X234 @ X236)) = ((X232 @ (sub @ X236 @ (push @ X235 @ X233))))))) & (! [X237 : (term > $o),X238 : subst,X239 : (term > $o),X240 : term] : (($true = ((X239 @ one))) | (((X237 @ X240)) != $true) | ? [X241 : term] : (((X237 @ (sub @ X241 @ (push @ X240 @ X238)))) != ((X239 @ X241)))) | (pushprop_lem1v2 != $true)) & ! [X242 : d_term] : (d_one = X242) & ! [X243 : d_subst] : (d_id = X243) & ((hoasinduction_lem3 = $true) | ($true != sP4)) & (($true = sP4) | (hoasinduction_lem3 != $true)) & (hoasapnotvar_gthm = $true) & ((laminj = $true) | ? [X244 : term,X245 : term] : ((X244 != X245) & (((lam @ X244)) = ((lam @ X245))))) & (! [X246 : term,X247 : term] : ((X246 = X247) | (((lam @ X246)) != ((lam @ X247)))) | (laminj != $true)) & (hoasinduction_gthm = $true) & (hoasapnotvar_lthm = $true) & (! [X248 : term,X249 : term] : ((((sub @ X249 @ sh)) != ((sub @ X248 @ sh))) | (X248 = X249)) | (shinj != $true)) & ((shinj = $true) | ? [X250 : term,X251 : term] : ((((sub @ X250 @ sh)) = ((sub @ X251 @ sh))) & (X250 != X251))) & ((induction = $true) | ? [X252 : (term > $o),X253 : term] : (! [X254 : term] : (($true = ((X252 @ X254))) | ($true != ((var @ X254)))) & ($true != ((X252 @ X253))) & ! [X255 : term] : ((((X252 @ X255)) != $true) | ($true = ((X252 @ (lam @ X255))))) & (((sP2 @ X252)) != $true))) & (! [X256 : (term > $o),X257 : term] : (? [X258 : term] : ((((X256 @ X258)) != $true) & ($true = ((var @ X258)))) | ($true = ((X256 @ X257))) | ? [X259 : term] : (($true = ((X256 @ X259))) & (((X256 @ (lam @ X259))) != $true)) | (((sP2 @ X256)) = $true)) | (induction != $true)) & ! [X260 : d_subst,X261 : d_subst] : ((X260 = X261) | (((d2subst @ X261)) != ((d2subst @ X260)))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & ((hoasinduction_lem3v2 = $true) | (sP1 != $true)) & ((sP1 = $true) | (hoasinduction_lem3v2 != $true)) & (! [X262 : subst,X263 : (term > $o),X264 : term,X265 : term,X266 : (term > $o)] : (($true != ((X266 @ X264))) | (((X263 @ (sub @ X264 @ (push @ X265 @ X262)))) = $true)) | ? [X267 : term,X268 : subst] : (((sub @ one @ (push @ X267 @ X268))) != X267) | ? [X269 : term,X270 : subst] : (((comp @ sh @ (push @ X269 @ X270))) != X270) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (? [X271 : subst,X272 : (term > $o),X273 : term,X274 : term,X275 : (term > $o)] : (($true = ((X275 @ X273))) & (((X272 @ (sub @ X273 @ (push @ X274 @ X271)))) != $true)) & ! [X276 : term,X277 : subst] : (((sub @ one @ (push @ X276 @ X277))) = X276) & ! [X278 : term,X279 : subst] : (((comp @ sh @ (push @ X278 @ X279))) = X279))) & (pushprop_lthm_orig = $true)),
  inference(rectify,[],[f261])).
thf(f263,plain,(
  (! [X0 : term,X1 : subst,X2 : term,X3 : (term > $o),X4 : (term > $o)] : ((((X3 @ (sub @ X0 @ (push @ X2 @ X1)))) = $true) | ($true != ((X4 @ X0)))) | (pushprop_lem1 != $true)) & ((pushprop_lem1 = $true) | (($true != ((sK457 @ (sub @ sK454 @ (push @ sK456 @ sK455))))) & ($true = ((sK458 @ sK454))))) & (hoasinduction_lem3a_gthm = $true) & (! [X10 : term] : ! [X12 : subst,X13 : term] : (((sub @ X10 @ (push @ X13 @ X12))) = ((sK459 @ X10 @ X12 @ X13))) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ! [X15 : (subst > term > term)] : (((sub @ sK460 @ (push @ (sK462 @ X15) @ (sK461 @ X15)))) != ((X15 @ (sK461 @ X15) @ (sK462 @ X15))))) & ((sP58 = $true) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | (sP58 != $true)) & ((termmset_lthm = $true) | ((sK463 != ((sub @ sK463 @ id))) & ! [X19 : term] : (((sub @ X19 @ id)) = X19))) & (! [X20 : term] : (((sub @ X20 @ id)) = X20) | (sK464 != ((sub @ sK464 @ id))) | (termmset_lthm != $true)) & (hoasinduction_lem3_gthm = $true) & (hoasinduction_lem1_lthm = $true) & (hoasapinj1_gthm = $true) & ((axvarcons = $true) | (sK465 != ((sub @ one @ (push @ sK465 @ sK466))))) & (! [X24 : term,X25 : subst] : (((sub @ one @ (push @ X24 @ X25))) = X24) | (axvarcons != $true)) & (substmonoid_gthm = $true) & ((apinj2 = $true) | ((((ap @ sK470 @ sK469)) = ((ap @ sK467 @ sK468))) & (sK469 != sK468))) & (! [X30 : term,X31 : term,X32 : term,X33 : term] : ((((ap @ X33 @ X32)) != ((ap @ X30 @ X31))) | (X31 = X32)) | (apinj2 != $true)) & ((axidl = $true) | (((comp @ id @ sK471)) != sK471)) & (! [X35 : subst] : (((comp @ id @ X35)) = X35) | (axidl != $true)) & ((hoasinduction_no_psi_cond_lthm = $true) | ($true != sP52)) & (($true = sP52) | (hoasinduction_no_psi_cond_lthm != $true)) & (hoaslamnotap_lthm = $true) & (hoasinduction_lem1 = $true) & (pushprop = $true) & (sh = ((d2subst @ d_id))) & (pushprop_lem1_gthm = $true) & (lamnotvar = $true) & (! [X36 : term,X37 : term,X38 : term,X39 : term] : ((X37 = X38) | (((ap @ X38 @ X39)) != ((ap @ X37 @ X36)))) | (apinj1 != $true)) & ((apinj1 = $true) | ((sK473 != sK474) & (((ap @ sK474 @ sK475)) = ((ap @ sK473 @ sK472))))) & (axassoc = $true) & (apnotvar = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (hoasinduction_lem2v2_gthm = $true) & (induction2_gthm = $true) & ((hoaslaminj = $true) | ((((lam @ (sK478 @ sh @ one))) = ((lam @ (sK476 @ sh @ one)))) & ! [X48 : term,X49 : subst,X50 : subst] : (((sub @ (sK476 @ X50 @ X48) @ X49)) = ((sK476 @ (comp @ X50 @ X49) @ (sub @ X48 @ X49)))) & ! [X51 : term,X52 : subst,X53 : subst] : (((sub @ (sK478 @ X52 @ X51) @ X53)) = ((sK478 @ (comp @ X52 @ X53) @ (sub @ X51 @ X53)))) & (((sK478 @ sK477 @ sK479)) != ((sK476 @ sK477 @ sK479))))) & (! [X54 : (subst > term > term),X55 : subst,X56 : (subst > term > term),X57 : term] : ((((lam @ (X54 @ sh @ one))) != ((lam @ (X56 @ sh @ one)))) | (((X54 @ (comp @ (sK482 @ X54) @ (sK481 @ X54)) @ (sub @ (sK480 @ X54) @ (sK481 @ X54)))) != ((sub @ (X54 @ (sK482 @ X54) @ (sK480 @ X54)) @ (sK481 @ X54)))) | (((X56 @ (comp @ (sK484 @ X56) @ (sK485 @ X56)) @ (sub @ (sK483 @ X56) @ (sK485 @ X56)))) != ((sub @ (X56 @ (sK484 @ X56) @ (sK483 @ X56)) @ (sK485 @ X56)))) | (((X56 @ X55 @ X57)) = ((X54 @ X55 @ X57)))) | (hoaslaminj != $true)) & (axclos = $true) & ((hoasinduction_lem3aaa = $true) | (sP47 != $true)) & ((sP47 = $true) | (hoasinduction_lem3aaa != $true)) & (hoaslamnotvar_lthm = $true) & ((hoasinduction_lem3v2_lthm = $true) | ($true != sP46)) & (($true = sP46) | (hoasinduction_lem3v2_lthm != $true)) & (! [X64 : term] : ! [X66 : term,X67 : subst] : (((sK486 @ X64 @ X67 @ X66)) = ((sub @ X64 @ (push @ X66 @ X67)))) | (hoasinduction_lem3v2_f_lthm != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ! [X69 : (subst > term > term)] : (((X69 @ (sK489 @ X69) @ (sK488 @ X69))) != ((sub @ sK487 @ (push @ (sK488 @ X69) @ (sK489 @ X69)))))) & (! [X72 : term] : (((sub @ X72 @ id)) = X72) | (termmset != $true)) & ((termmset = $true) | (((sub @ sK490 @ id)) != sK490)) & ((((comp @ id @ sK491)) != sK491) | (! [X75 : subst] : (((comp @ X75 @ id)) = X75) & ! [X76 : subst] : (((comp @ id @ X76)) = X76)) | (sK492 != ((comp @ sK492 @ id))) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | (! [X78 : subst] : (((comp @ id @ X78)) = X78) & ((sK493 != ((comp @ sK493 @ id))) | (sK494 != ((comp @ id @ sK494)))) & ! [X81 : subst] : (((comp @ X81 @ id)) = X81))) & (hoaslamnotap_gthm = $true) & (hoaslamnotvar = $true) & (axmap = $true) & ((axvarid = $true) | (sK495 != ((sub @ sK495 @ id)))) & (! [X83 : term] : (((sub @ X83 @ id)) = X83) | (axvarid != $true)) & ((sP44 = $true) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | (sP44 != $true)) & (termmset_gthm = $true) & (axapp = $true) & ((axshiftcons = $true) | (sK496 != ((comp @ sh @ (push @ sK497 @ sK496))))) & (! [X86 : subst,X87 : term] : (((comp @ sh @ (push @ X87 @ X86))) = X86) | (axshiftcons != $true)) & (! [X88 : term,X89 : term,X90 : term,X91 : term] : ((X89 = X90) | (((ap @ (sub @ X89 @ id) @ X88)) != ((ap @ (sub @ X90 @ id) @ X91)))) | ((sK500 != sK498) & (((ap @ sK500 @ sK499)) = ((ap @ sK498 @ sK501)))) | (((sub @ sK502 @ id)) != sK502) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | (((sK504 != sK505) & (((ap @ (sub @ sK504 @ id) @ sK503)) = ((ap @ (sub @ sK505 @ id) @ sK506)))) & ! [X101 : term,X102 : term,X103 : term,X104 : term] : ((X101 = X103) | (((ap @ X101 @ X104)) != ((ap @ X103 @ X102)))) & ! [X105 : term] : (((sub @ X105 @ id)) = X105))) & ((pushprop_lem0 = $true) | ! [X109 : (term > $o)] : (((sK509 @ (sub @ (sK510 @ X109) @ (push @ sK507 @ sK508)))) != ((X109 @ (sK510 @ X109))))) & (! [X111 : term,X112 : subst,X113 : (term > $o)] : ! [X115 : term] : (((sK511 @ X113 @ X112 @ X111 @ X115)) = ((X113 @ (sub @ X115 @ (push @ X111 @ X112))))) | (pushprop_lem0 != $true)) & ((hoasinduction_lem3a_lthm = $true) | ($true != sP43)) & (($true = sP43) | (hoasinduction_lem3a_lthm != $true)) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoasapinj2_gthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (one = ((d2term @ d_one))) & ((hoasapinj1 = $true) | ((sK515 != sK512) & (((ap @ (sub @ sK512 @ id) @ sK514)) = ((ap @ (sub @ sK515 @ id) @ sK513))))) & (! [X120 : term,X121 : term,X122 : term,X123 : term] : ((X120 = X123) | (((ap @ (sub @ X120 @ id) @ X122)) != ((ap @ (sub @ X123 @ id) @ X121)))) | (hoasapinj1 != $true)) & ((induction2 = $true) | (sP41 != $true)) & ((sP41 = $true) | (induction2 != $true)) & ! [X124 : subst] : (((d2subst @ (sK516 @ X124))) = X124) & (hoasinduction_lem2_gthm = $true) & ((((sub @ one @ (push @ sK518 @ sK517))) != sK518) | ! [X128 : (term > $o),X129 : subst,X130 : (term > $o),X131 : term] : ((((X128 @ (sub @ (sK519 @ X131 @ X130 @ X129 @ X128) @ (push @ X131 @ X129)))) != ((X130 @ (sK519 @ X131 @ X130 @ X129 @ X128)))) | ($true != ((X128 @ X131))) | (((X130 @ one)) = $true)) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (! [X133 : subst,X134 : term] : (((sub @ one @ (push @ X134 @ X133))) = X134) & (! [X139 : term] : (((sK522 @ X139)) = ((sK520 @ (sub @ X139 @ (push @ sK523 @ sK521))))) & ($true = ((sK520 @ sK523))) & ($true != ((sK522 @ one)))))) & (! [X140 : term] : (((sK524 @ X140 @ sh @ one)) = ((sub @ X140 @ (push @ one @ sh)))) | (hoasinduction_lem3b_lthm != $true)) & ((hoasinduction_lem3b_lthm = $true) | ! [X143 : (subst > term > term)] : (((sub @ sK525 @ (push @ one @ sh))) != ((X143 @ sh @ one)))) & (pushprop_lem3v2 = $true) & ((hoasinduction_lem0_lthm = $true) | ! [X145 : (term > $o)] : (((X145 @ (sK527 @ X145))) != ((sK526 @ id @ (sK527 @ X145) @ id)))) & (! [X147 : (subst > term > subst > $o)] : ! [X149 : term] : (((X147 @ id @ X149 @ id)) = ((sK528 @ X147 @ X149))) | (hoasinduction_lem0_lthm != $true)) & ((! [X150 : subst] : (((comp @ X150 @ id)) = X150) & ! [X151 : subst] : (((comp @ id @ X151)) = X151)) | (substmonoid != $true)) & ((substmonoid = $true) | (((comp @ sK529 @ id)) != sK529) | (sK530 != ((comp @ id @ sK530)))) & (((d2subst @ d_id)) = id) & ((hoasinduction_lem3b = $true) | ! [X155 : (subst > term > term)] : (((sub @ sK531 @ (push @ one @ sh))) != ((X155 @ sh @ one)))) & (! [X156 : term] : (((sK532 @ X156 @ sh @ one)) = ((sub @ X156 @ (push @ one @ sh)))) | (hoasinduction_lem3b != $true)) & (! [X158 : (term > $o),X159 : subst,X160 : term] : ! [X162 : term] : (((sK533 @ X160 @ X159 @ X158 @ X162)) = ((X158 @ (sub @ X162 @ (push @ X160 @ X159))))) | (pushprop_lem0_lthm != $true)) & ((pushprop_lem0_lthm = $true) | ! [X166 : (term > $o)] : (((X166 @ (sK537 @ X166))) != ((sK534 @ (sub @ (sK537 @ X166) @ (push @ sK536 @ sK535)))))) & (pushprop_lem2v2 = $true) & ! [X168 : d_term,X169 : d_term] : ((((d2term @ X169)) != ((d2term @ X168))) | (X168 = X169)) & (hoasinduction_lem1v2 = $true) & (! [X170 : (subst > term > term),X171 : term,X172 : term] : ((((sub @ (X170 @ (sK540 @ X170) @ (sK539 @ X170)) @ (sK538 @ X170))) != ((X170 @ (comp @ (sK540 @ X170) @ (sK538 @ X170)) @ (sub @ (sK539 @ X170) @ (sK538 @ X170))))) | (((lam @ (X170 @ sh @ one))) != ((ap @ (sub @ X171 @ id) @ X172)))) | (hoaslamnotap != $true)) & ((hoaslamnotap = $true) | (! [X179 : subst,X180 : term,X181 : subst] : (((sK541 @ (comp @ X181 @ X179) @ (sub @ X180 @ X179))) = ((sub @ (sK541 @ X181 @ X180) @ X179))) & (((lam @ (sK541 @ sh @ one))) = ((ap @ (sub @ sK542 @ id) @ sK543))))) & ((sP39 = $true) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | (sP39 != $true)) & (ulamvarsh = $true) & ((hoasinduction_lem0 = $true) | ! [X183 : (term > $o)] : (((X183 @ (sK545 @ X183))) != ((sK544 @ id @ (sK545 @ X183) @ id)))) & (! [X185 : (subst > term > subst > $o)] : ! [X187 : term] : (((X185 @ id @ X187 @ id)) = ((sK546 @ X185 @ X187))) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lthm = $true) | ($true != sP35)) & (($true = sP35) | (hoasinduction_lthm != $true)) & (((sK549 != sK547) & (((ap @ sK550 @ sK547)) = ((ap @ sK548 @ sK549)))) | ! [X192 : term,X193 : term,X194 : term,X195 : term] : ((((ap @ (sub @ X195 @ id) @ X193)) != ((ap @ (sub @ X194 @ id) @ X192))) | (X192 = X193)) | (hoasapinj2_lthm != $true)) & ((hoasapinj2_lthm = $true) | (! [X196 : term,X197 : term,X198 : term,X199 : term] : ((X196 = X198) | (((ap @ X199 @ X196)) != ((ap @ X197 @ X198)))) & ((((ap @ (sub @ sK554 @ id) @ sK552)) = ((ap @ (sub @ sK553 @ id) @ sK551))) & (sK551 != sK552)))) & (ulamvarind = $true) & (pushprop_lem3v2_lthm = $true) & (! [X204 : subst] : (((comp @ X204 @ id)) = X204) | (axidr != $true)) & ((axidr = $true) | (((comp @ sK555 @ id)) != sK555)) & (hoasinduction_lem2_lthm = $true) & ((hoasinduction_lem3aa = $true) | (sP26 != $true)) & ((sP26 = $true) | (hoasinduction_lem3aa != $true)) & (axabs = $true) & (hoasapnotvar = $true) & (hoasinduction_lem1_gthm = $true) & ((hoasapinj2 = $true) | ((((ap @ (sub @ sK557 @ id) @ sK558)) = ((ap @ (sub @ sK559 @ id) @ sK556))) & (sK556 != sK558))) & (! [X210 : term,X211 : term,X212 : term,X213 : term] : ((((ap @ (sub @ X211 @ id) @ X212)) != ((ap @ (sub @ X213 @ id) @ X210))) | (X210 = X212)) | (hoasapinj2 != $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((hoasinduction_lem2v2 = $true) | (sP25 != $true)) & ((sP25 = $true) | (hoasinduction_lem2v2 != $true)) & (lamnotap != $true) & (pushprop_lem2v2_lthm = $true) & (pushprop_lem0_gthm = $true) & ! [X214 : term] : (((d2term @ (sK560 @ X214))) = X214) & ((induction2_lthm = $true) | (sP23 != $true)) & ((sP23 = $true) | (induction2_lthm != $true)) & (hoaslamnotvar_gthm = $true) & (induction2lem_gthm = $true) & (($true = sP19) | (hoaslaminj_lthm != $true)) & ((hoaslaminj_lthm = $true) | ($true != sP19)) & (pushprop_lthm = $true) & ((hoasinduction_lem2 = $true) | ($true != sP18)) & (($true = sP18) | (hoasinduction_lem2 != $true)) & (pushprop_lem1v2_gthm = $true) & ((hoasinduction_lem3v2a_lthm = $true) | ($true != sP16)) & (($true = sP16) | (hoasinduction_lem3v2a_lthm != $true)) & (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_lem3aa_lthm = $true) | (sP15 != $true)) & ((sP15 = $true) | (hoasinduction_lem3aa_lthm != $true)) & ((induction2lem = $true) | ($true != sP14)) & (($true = sP14) | (induction2lem != $true)) & (hoasinduction_lem3b_gthm = $true) & (pushprop_gthm = $true) & ((axscons = $true) | (((push @ (sub @ one @ sK561) @ (comp @ sh @ sK561))) != sK561)) & (! [X217 : subst] : (((push @ (sub @ one @ X217) @ (comp @ sh @ X217))) = X217) | (axscons != $true)) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((hoasinduction = $true) | (sP12 != $true)) & ((sP12 = $true) | (hoasinduction != $true)) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (axvarshift = $true) & (hoasinduction_lem3v2_gthm = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoaslaminj_gthm = $true) & (pushprop_lem2v2_gthm = $true) & (($true = sP9) | (hoasinduction_lem3_lthm != $true)) & ((hoasinduction_lem3_lthm = $true) | ($true != sP9)) & ((sP6 = $true) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | (sP6 != $true)) & (((var @ (d2term @ d_one))) != $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)))))))))))) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ((hoasinduction_no_psi_cond = $true) | (! [X220 : (subst > term > term)] : ((($true = ((sK563 @ id @ (sK564 @ X220) @ id))) & ($true != ((sK563 @ id @ (X220 @ id @ (sK564 @ X220)) @ id)))) | ($true = ((sK563 @ id @ (lam @ (X220 @ sh @ one)) @ id))) | (((X220 @ (comp @ (sK565 @ X220) @ (sK566 @ X220)) @ (sub @ (sK567 @ X220) @ (sK566 @ X220)))) != ((sub @ (X220 @ (sK565 @ X220) @ (sK567 @ X220)) @ (sK566 @ X220))))) & ($true != ((sP5 @ sK563))) & (((sK563 @ id @ sK562 @ id)) != $true))) & (! [X225 : term,X226 : (subst > term > subst > $o)] : ((! [X228 : term] : ((((X226 @ id @ X228 @ id)) != $true) | (((X226 @ id @ (sK568 @ X226 @ id @ X228) @ id)) = $true)) & ($true != ((X226 @ id @ (lam @ (sK568 @ X226 @ sh @ one)) @ id))) & ! [X229 : subst,X230 : subst,X231 : term] : (((sK568 @ X226 @ (comp @ X229 @ X230) @ (sub @ X231 @ X230))) = ((sub @ (sK568 @ X226 @ X229 @ X231) @ X230)))) | (((sP5 @ X226)) = $true) | (((X226 @ id @ X225 @ id)) = $true)) | (hoasinduction_no_psi_cond != $true)) & (ulamvar1 != $true) & ((pushprop_lem1v2 = $true) | (($true != ((sK571 @ one))) & (((sK569 @ sK572)) = $true) & ! [X236 : term] : (((sK569 @ (sub @ X236 @ (push @ sK572 @ sK570)))) = ((sK571 @ X236))))) & (! [X237 : (term > $o),X238 : subst,X239 : (term > $o),X240 : term] : (($true = ((X239 @ one))) | (((X237 @ X240)) != $true) | (((X237 @ (sub @ (sK573 @ X240 @ X239 @ X238 @ X237) @ (push @ X240 @ X238)))) != ((X239 @ (sK573 @ X240 @ X239 @ X238 @ X237))))) | (pushprop_lem1v2 != $true)) & ! [X242 : d_term] : (d_one = X242) & ! [X243 : d_subst] : (d_id = X243) & ((hoasinduction_lem3 = $true) | ($true != sP4)) & (($true = sP4) | (hoasinduction_lem3 != $true)) & (hoasapnotvar_gthm = $true) & ((laminj = $true) | ((sK574 != sK575) & (((lam @ sK574)) = ((lam @ sK575))))) & (! [X246 : term,X247 : term] : ((X246 = X247) | (((lam @ X246)) != ((lam @ X247)))) | (laminj != $true)) & (hoasinduction_gthm = $true) & (hoasapnotvar_lthm = $true) & (! [X248 : term,X249 : term] : ((((sub @ X249 @ sh)) != ((sub @ X248 @ sh))) | (X248 = X249)) | (shinj != $true)) & ((shinj = $true) | ((((sub @ sK576 @ sh)) = ((sub @ sK577 @ sh))) & (sK577 != sK576))) & ((induction = $true) | (! [X254 : term] : ((((sK578 @ X254)) = $true) | ($true != ((var @ X254)))) & (((sK578 @ sK579)) != $true) & ! [X255 : term] : ((((sK578 @ X255)) != $true) | (((sK578 @ (lam @ X255))) = $true)) & ($true != ((sP2 @ sK578))))) & (! [X256 : (term > $o),X257 : term] : (((((X256 @ (sK580 @ X256))) != $true) & (((var @ (sK580 @ X256))) = $true)) | ($true = ((X256 @ X257))) | (($true = ((X256 @ (sK581 @ X256)))) & ($true != ((X256 @ (lam @ (sK581 @ X256)))))) | (((sP2 @ X256)) = $true)) | (induction != $true)) & ! [X260 : d_subst,X261 : d_subst] : ((X260 = X261) | (((d2subst @ X261)) != ((d2subst @ X260)))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & ((hoasinduction_lem3v2 = $true) | (sP1 != $true)) & ((sP1 = $true) | (hoasinduction_lem3v2 != $true)) & (! [X262 : subst,X263 : (term > $o),X264 : term,X265 : term,X266 : (term > $o)] : (($true != ((X266 @ X264))) | (((X263 @ (sub @ X264 @ (push @ X265 @ X262)))) = $true)) | (sK582 != ((sub @ one @ (push @ sK582 @ sK583)))) | (((comp @ sh @ (push @ sK584 @ sK585))) != sK585) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (((((sK590 @ sK588)) = $true) & ($true != ((sK587 @ (sub @ sK588 @ (push @ sK589 @ sK586)))))) & ! [X276 : term,X277 : subst] : (((sub @ one @ (push @ X276 @ X277))) = X276) & ! [X278 : term,X279 : subst] : (((comp @ sh @ (push @ X278 @ X279))) = X279))) & (pushprop_lthm_orig = $true)),
  inference(skolemize,[status(esa),new_symbols(skolem,[sK454,sK455,sK456,sK457,sK458,vAPP,sK460,vAPP,sK463,sK464,sK465,sK466,sK467,sK468,sK469,sK470,sK471,sK472,sK473,sK474,sK475,sK476,sK477,sK478,sK479,vAPP,sK487,vAPP,sK490,sK491,sK492,sK493,sK494,sK495,sK496,sK497,sK498,sK499,sK500,sK501,sK502,sK503,sK504,sK505,sK506,sK507,sK508,sK509,vAPP,sK512,sK513,sK514,sK515,vAPP,sK517,sK518,vAPP,sK520,sK521,sK522,sK523,vAPP,sK525,sK526,vAPP,sK529,sK530,sK531,vAPP,sK534,sK535,sK536,vAPP,sK541,sK542,sK543,sK544,vAPP,sK547,sK548,sK549,sK550,sK551,sK552,sK553,sK554,sK555,sK556,sK557,sK558,sK559,vAPP,sK561,sK562,sK563,vAPP,sK569,sK570,sK571,sK572,vAPP,sK574,sK575,sK576,sK577,sK578,sK579,vAPP,sK582,sK583,sK584,sK585,sK586,sK587,sK588,sK589,sK590]),skolemize(X5,$thf(sK454)),skolemize(X6,$thf(sK455)),skolemize(X7,$thf(sK456)),skolemize(X8,$thf(sK457)),skolemize(X9,$thf(sK458)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X14,$thf(sK460)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X18,$thf(sK463)),skolemize(X21,$thf(sK464)),skolemize(X22,$thf(sK465)),skolemize(X23,$thf(sK466)),skolemize(X26,$thf(sK467)),skolemize(X27,$thf(sK468)),skolemize(X28,$thf(sK469)),skolemize(X29,$thf(sK470)),skolemize(X34,$thf(sK471)),skolemize(X40,$thf(sK472)),skolemize(X41,$thf(sK473)),skolemize(X42,$thf(sK474)),skolemize(X43,$thf(sK475)),skolemize(X44,$thf(sK476)),skolemize(X45,$thf(sK477)),skolemize(X46,$thf(sK478)),skolemize(X47,$thf(sK479)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X68,$thf(sK487)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X73,$thf(sK490)),skolemize(X74,$thf(sK491)),skolemize(X77,$thf(sK492)),skolemize(X79,$thf(sK493)),skolemize(X80,$thf(sK494)),skolemize(X82,$thf(sK495)),skolemize(X84,$thf(sK496)),skolemize(X85,$thf(sK497)),skolemize(X92,$thf(sK498)),skolemize(X93,$thf(sK499)),skolemize(X94,$thf(sK500)),skolemize(X95,$thf(sK501)),skolemize(X96,$thf(sK502)),skolemize(X97,$thf(sK503)),skolemize(X98,$thf(sK504)),skolemize(X99,$thf(sK505)),skolemize(X100,$thf(sK506)),skolemize(X106,$thf(sK507)),skolemize(X107,$thf(sK508)),skolemize(X108,$thf(sK509)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X116,$thf(sK512)),skolemize(X117,$thf(sK513)),skolemize(X118,$thf(sK514)),skolemize(X119,$thf(sK515)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X126,$thf(sK517)),skolemize(X127,$thf(sK518)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X135,$thf(sK520)),skolemize(X136,$thf(sK521)),skolemize(X137,$thf(sK522)),skolemize(X138,$thf(sK523)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X142,$thf(sK525)),skolemize(X144,$thf(sK526)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X152,$thf(sK529)),skolemize(X153,$thf(sK530)),skolemize(X154,$thf(sK531)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X163,$thf(sK534)),skolemize(X164,$thf(sK535)),skolemize(X165,$thf(sK536)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X176,$thf(sK541)),skolemize(X177,$thf(sK542)),skolemize(X178,$thf(sK543)),skolemize(X182,$thf(sK544)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X188,$thf(sK547)),skolemize(X189,$thf(sK548)),skolemize(X190,$thf(sK549)),skolemize(X191,$thf(sK550)),skolemize(X200,$thf(sK551)),skolemize(X201,$thf(sK552)),skolemize(X202,$thf(sK553)),skolemize(X203,$thf(sK554)),skolemize(X205,$thf(sK555)),skolemize(X206,$thf(sK556)),skolemize(X207,$thf(sK557)),skolemize(X208,$thf(sK558)),skolemize(X209,$thf(sK559)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X216,$thf(sK561)),skolemize(X218,$thf(sK562)),skolemize(X219,$thf(sK563)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X232,$thf(sK569)),skolemize(X233,$thf(sK570)),skolemize(X234,$thf(sK571)),skolemize(X235,$thf(sK572)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X244,$thf(sK574)),skolemize(X245,$thf(sK575)),skolemize(X250,$thf(sK576)),skolemize(X251,$thf(sK577)),skolemize(X252,$thf(sK578)),skolemize(X253,$thf(sK579)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X16,$thf(sK67 @ X15)),skolemize(X267,$thf(sK582)),skolemize(X268,$thf(sK583)),skolemize(X269,$thf(sK584)),skolemize(X270,$thf(sK585)),skolemize(X271,$thf(sK586)),skolemize(X272,$thf(sK587)),skolemize(X273,$thf(sK588)),skolemize(X274,$thf(sK589)),skolemize(X275,$thf(sK590))],[f262])).
thf(f264,plain,(
  (hoasinduction_lem2 != ((!! @ (subst > term > subst > $o) @ (^[Y0 : subst > term > subst > $o]: ((!! @ subst @ (^[Y1 : subst]: (!! @ subst @ (^[Y2 : subst]: (!! @ subst @ (^[Y3 : subst]: (!! @ term @ (^[Y4 : term]: ((Y0 @ Y1 @ Y4 @ (comp @ Y2 @ Y3)) => (Y0 @ (comp @ Y1 @ Y2) @ (sub @ Y4 @ Y2) @ Y3)))))))))) => ((!! @ term @ (^[Y1 : term]: (!! @ subst @ (^[Y2 : subst]: (!! @ subst @ (^[Y3 : subst]: (!! @ subst @ (^[Y4 : subst]: ((Y0 @ (comp @ Y3 @ Y2) @ (sub @ Y1 @ Y2) @ Y4) => (Y0 @ Y3 @ Y1 @ (comp @ Y2 @ Y4))))))))))) => ((!! @ term @ (^[Y1 : term]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) => ((Y0 @ id @ Y1 @ id) => (Y0 @ id @ (hoasap @ id @ Y2 @ id @ Y1) @ id))))))) => (!! @ term @ (^[Y1 : term]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y1 @ id) => ((Y0 @ id @ Y2 @ id) => (Y0 @ id @ (ap @ Y1 @ Y2) @ id))))))))))))))),
  inference(cnf_transformation,[],[f10])).
thf(f599,plain,(
  ($true = sP18) | ($true != ((sK319 @ id @ (ap @ sK320 @ sK321) @ id)))),
  inference(cnf_transformation,[],[f203])).
thf(f600,plain,(
  ( ! [X10 : term,X8 : subst,X9 : subst,X7 : subst] : (($true = ((sK319 @ (comp @ X7 @ X9) @ (sub @ X10 @ X9) @ X8))) | ($true != ((sK319 @ X7 @ X10 @ (comp @ X9 @ X8)))) | ($true = sP18)) )),
  inference(cnf_transformation,[],[f203])).
thf(f601,plain,(
  ($true = ((sK319 @ id @ sK321 @ id))) | ($true = sP18)),
  inference(cnf_transformation,[],[f203])).
thf(f782,plain,(
  ( ! [X243 : d_subst] : ((d_id = X243)) )),
  inference(cnf_transformation,[],[f263])).
thf(f783,plain,(
  ( ! [X242 : d_term] : ((d_one = X242)) )),
  inference(cnf_transformation,[],[f263])).
thf(f825,plain,(
  (hoasinduction_lem2 = $true) | ($true != sP18)),
  inference(cnf_transformation,[],[f263])).
thf(f833,plain,(
  ( ! [X214 : term] : ((((d2term @ (sK560 @ X214))) = X214)) )),
  inference(cnf_transformation,[],[f263])).
thf(f890,plain,(
  ( ! [X124 : subst] : ((((d2subst @ (sK516 @ X124))) = X124)) )),
  inference(cnf_transformation,[],[f263])).
thf(f896,plain,(
  (one = ((d2term @ d_one)))),
  inference(cnf_transformation,[],[f263])).
thf(f951,plain,(
  (sh = ((d2subst @ d_id)))),
  inference(cnf_transformation,[],[f263])).
thf(f981,definition,(
  (sF591 = ((!! @ (subst > term > subst > $o) @ (^[Y0 : subst > term > subst > $o]: ((!! @ subst @ (^[Y1 : subst]: (!! @ subst @ (^[Y2 : subst]: (!! @ subst @ (^[Y3 : subst]: (!! @ term @ (^[Y4 : term]: ((Y0 @ Y1 @ Y4 @ (comp @ Y2 @ Y3)) => (Y0 @ (comp @ Y1 @ Y2) @ (sub @ Y4 @ Y2) @ Y3)))))))))) => ((!! @ term @ (^[Y1 : term]: (!! @ subst @ (^[Y2 : subst]: (!! @ subst @ (^[Y3 : subst]: (!! @ subst @ (^[Y4 : subst]: ((Y0 @ (comp @ Y3 @ Y2) @ (sub @ Y1 @ Y2) @ Y4) => (Y0 @ Y3 @ Y1 @ (comp @ Y2 @ Y4))))))))))) => ((!! @ term @ (^[Y1 : term]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) => ((Y0 @ id @ Y1 @ id) => (Y0 @ id @ (hoasap @ id @ Y2 @ id @ Y1) @ id))))))) => (!! @ term @ (^[Y1 : term]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y1 @ id) => ((Y0 @ id @ Y2 @ id) => (Y0 @ id @ (ap @ Y1 @ Y2) @ id))))))))))))))),
  introduced(definition,[new_symbols(definition,[sF591])],[function_definition])).
thf(f982,plain,(
  (hoasinduction_lem2 != sF591)),
  inference(definition_folding,[],[f264,f981])).
thf(f1025,plain,(
  ($false = ((!! @ (subst > term > subst > $o) @ (^[Y0 : subst > term > subst > $o]: ((!! @ subst @ (^[Y1 : subst]: (!! @ subst @ (^[Y2 : subst]: (!! @ subst @ (^[Y3 : subst]: (!! @ term @ (^[Y4 : term]: ((Y0 @ Y1 @ Y4 @ (comp @ Y2 @ Y3)) => (Y0 @ (comp @ Y1 @ Y2) @ (sub @ Y4 @ Y2) @ Y3)))))))))) => ((!! @ term @ (^[Y1 : term]: (!! @ subst @ (^[Y2 : subst]: (!! @ subst @ (^[Y3 : subst]: (!! @ subst @ (^[Y4 : subst]: ((Y0 @ (comp @ Y3 @ Y2) @ (sub @ Y1 @ Y2) @ Y4) => (Y0 @ Y3 @ Y1 @ (comp @ Y2 @ Y4))))))))))) => ((!! @ term @ (^[Y1 : term]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) => ((Y0 @ id @ Y1 @ id) => (Y0 @ id @ (hoasap @ id @ Y2 @ id @ Y1) @ id))))))) => (!! @ term @ (^[Y1 : term]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y1 @ id) => ((Y0 @ id @ Y2 @ id) => (Y0 @ id @ (ap @ Y1 @ Y2) @ id)))))))))))))) | (sF591 = $true)),
  inference(iff_proxy_clausification,[],[f981])).
thf(f1085,plain,(
  (sF591 = $true) | ($false = (((^[Y0 : subst > term > subst > $o]: ((!! @ subst @ (^[Y1 : subst]: (!! @ subst @ (^[Y2 : subst]: (!! @ subst @ (^[Y3 : subst]: (!! @ term @ (^[Y4 : term]: ((Y0 @ Y1 @ Y4 @ (comp @ Y2 @ Y3)) => (Y0 @ (comp @ Y1 @ Y2) @ (sub @ Y4 @ Y2) @ Y3)))))))))) => ((!! @ term @ (^[Y1 : term]: (!! @ subst @ (^[Y2 : subst]: (!! @ subst @ (^[Y3 : subst]: (!! @ subst @ (^[Y4 : subst]: ((Y0 @ (comp @ Y3 @ Y2) @ (sub @ Y1 @ Y2) @ Y4) => (Y0 @ Y3 @ Y1 @ (comp @ Y2 @ Y4))))))))))) => ((!! @ term @ (^[Y1 : term]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) => ((Y0 @ id @ Y1 @ id) => (Y0 @ id @ (hoasap @ id @ Y2 @ id @ Y1) @ id))))))) => (!! @ term @ (^[Y1 : term]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y1 @ id) => ((Y0 @ id @ Y2 @ id) => (Y0 @ id @ (ap @ Y1 @ Y2) @ id))))))))))) @ sK609)))),
  inference(sigma_proxy_clausification,[],[f1025])).
thf(f1086,plain,(
  (sF591 = $true) | ((((!! @ subst @ (^[Y0 : subst]: (!! @ subst @ (^[Y1 : subst]: (!! @ subst @ (^[Y2 : subst]: (!! @ term @ (^[Y3 : term]: ((sK609 @ Y0 @ Y3 @ (comp @ Y1 @ Y2)) => (sK609 @ (comp @ Y0 @ Y1) @ (sub @ Y3 @ Y1) @ Y2)))))))))) => ((!! @ term @ (^[Y0 : term]: (!! @ subst @ (^[Y1 : subst]: (!! @ subst @ (^[Y2 : subst]: (!! @ subst @ (^[Y3 : subst]: ((sK609 @ (comp @ Y2 @ Y1) @ (sub @ Y0 @ Y1) @ Y3) => (sK609 @ Y2 @ Y0 @ (comp @ Y1 @ Y3))))))))))) => ((!! @ term @ (^[Y0 : term]: (!! @ term @ (^[Y1 : term]: ((sK609 @ id @ Y1 @ id) => ((sK609 @ id @ Y0 @ id) => (sK609 @ id @ (hoasap @ id @ Y1 @ id @ Y0) @ id))))))) => (!! @ term @ (^[Y0 : term]: (!! @ term @ (^[Y1 : term]: ((sK609 @ id @ Y0 @ id) => ((sK609 @ id @ Y1 @ id) => (sK609 @ id @ (ap @ Y0 @ Y1) @ id))))))))))) = $false)),
  inference(beta-eta_normalization,[],[f1085])).
thf(f1087,plain,(
  (sF591 = $true) | ($false = (((!! @ term @ (^[Y0 : term]: (!! @ subst @ (^[Y1 : subst]: (!! @ subst @ (^[Y2 : subst]: (!! @ subst @ (^[Y3 : subst]: ((sK609 @ (comp @ Y2 @ Y1) @ (sub @ Y0 @ Y1) @ Y3) => (sK609 @ Y2 @ Y0 @ (comp @ Y1 @ Y3))))))))))) => ((!! @ term @ (^[Y0 : term]: (!! @ term @ (^[Y1 : term]: ((sK609 @ id @ Y1 @ id) => ((sK609 @ id @ Y0 @ id) => (sK609 @ id @ (hoasap @ id @ Y1 @ id @ Y0) @ id))))))) => (!! @ term @ (^[Y0 : term]: (!! @ term @ (^[Y1 : term]: ((sK609 @ id @ Y0 @ id) => ((sK609 @ id @ Y1 @ id) => (sK609 @ id @ (ap @ Y0 @ Y1) @ id)))))))))))),
  inference(imp_proxy_clausification,[],[f1086])).
thf(f1098,plain,(
  ($false = (((!! @ term @ (^[Y0 : term]: (!! @ term @ (^[Y1 : term]: ((sK609 @ id @ Y1 @ id) => ((sK609 @ id @ Y0 @ id) => (sK609 @ id @ (hoasap @ id @ Y1 @ id @ Y0) @ id))))))) => (!! @ term @ (^[Y0 : term]: (!! @ term @ (^[Y1 : term]: ((sK609 @ id @ Y0 @ id) => ((sK609 @ id @ Y1 @ id) => (sK609 @ id @ (ap @ Y0 @ Y1) @ id)))))))))) | (sF591 = $true)),
  inference(imp_proxy_clausification,[],[f1087])).
thf(f1109,plain,(
  ($false = ((!! @ term @ (^[Y0 : term]: (!! @ term @ (^[Y1 : term]: ((sK609 @ id @ Y0 @ id) => ((sK609 @ id @ Y1 @ id) => (sK609 @ id @ (ap @ Y0 @ Y1) @ id))))))))) | (sF591 = $true)),
  inference(imp_proxy_clausification,[],[f1098])).
thf(f1117,plain,(
  (sF591 = $true) | ($false = (((^[Y0 : term]: (!! @ term @ (^[Y1 : term]: ((sK609 @ id @ Y0 @ id) => ((sK609 @ id @ Y1 @ id) => (sK609 @ id @ (ap @ Y0 @ Y1) @ id)))))) @ sK610)))),
  inference(sigma_proxy_clausification,[],[f1109])).
thf(f1118,plain,(
  (((!! @ term @ (^[Y0 : term]: ((sK609 @ id @ sK610 @ id) => ((sK609 @ id @ Y0 @ id) => (sK609 @ id @ (ap @ sK610 @ Y0) @ id)))))) = $false) | (sF591 = $true)),
  inference(beta-eta_normalization,[],[f1117])).
thf(f1119,plain,(
  ($false = (((^[Y0 : term]: ((sK609 @ id @ sK610 @ id) => ((sK609 @ id @ Y0 @ id) => (sK609 @ id @ (ap @ sK610 @ Y0) @ id)))) @ sK611))) | (sF591 = $true)),
  inference(sigma_proxy_clausification,[],[f1118])).
thf(f1120,plain,(
  (sF591 = $true) | ($false = (((sK609 @ id @ sK610 @ id) => ((sK609 @ id @ sK611 @ id) => (sK609 @ id @ (ap @ sK610 @ sK611) @ id)))))),
  inference(beta-eta_normalization,[],[f1119])).
thf(f1121,plain,(
  (sF591 = $true) | ($false = (((sK609 @ id @ sK611 @ id) => (sK609 @ id @ (ap @ sK610 @ sK611) @ id))))),
  inference(imp_proxy_clausification,[],[f1120])).
thf(f1122,plain,(
  ($true = ((sK609 @ id @ sK610 @ id))) | (sF591 = $true)),
  inference(imp_proxy_clausification,[],[f1120])).
thf(f1123,plain,(
  ($false = ((sK609 @ id @ (ap @ sK610 @ sK611) @ id))) | (sF591 = $true)),
  inference(imp_proxy_clausification,[],[f1121])).
thf(f1132,plain,(
  (sF591 = $false) | (hoasinduction_lem2 = $false)),
  inference(xor_proxy_clausification,[],[f982])).
thf(f1566,definition,(
  spl612_82 <=> ($true = sP18)),
  introduced(definition,[new_symbols(definition,[spl612_82])],[avatar_definition])).
thf(f1868,definition,(
  spl612_152 <=> (hoasinduction_lem2 = $true)),
  introduced(definition,[new_symbols(definition,[spl612_152])],[avatar_definition])).
thf(f1870,plain,(
  (hoasinduction_lem2 = $true) | ~spl612_152),
  inference(avatar_component_clause,[],[f1868])).
thf(f1871,plain,(
  ~spl612_82 | spl612_152),
  inference(avatar_split_clause,[],[f825,f1868,f1566])).
thf(f2127,definition,(
  spl612_212 <=> (sF591 = $false)),
  introduced(definition,[new_symbols(definition,[spl612_212])],[avatar_definition])).
thf(f2129,plain,(
  (sF591 = $false) | ~spl612_212),
  inference(avatar_component_clause,[],[f2127])).
thf(f2179,definition,(
  spl612_225 <=> (sF591 = $true)),
  introduced(definition,[new_symbols(definition,[spl612_225])],[avatar_definition])).
thf(f2181,plain,(
  (sF591 = $true) | ~spl612_225),
  inference(avatar_component_clause,[],[f2179])).
thf(f2192,definition,(
  spl612_228 <=> ($true = ((sK609 @ id @ sK610 @ id)))),
  introduced(definition,[new_symbols(definition,[spl612_228])],[avatar_definition])).
thf(f2194,plain,(
  ($true = ((sK609 @ id @ sK610 @ id))) | ~spl612_228),
  inference(avatar_component_clause,[],[f2192])).
thf(f2195,plain,(
  spl612_228 | spl612_225),
  inference(avatar_split_clause,[],[f1122,f2179,f2192])).
thf(f2202,definition,(
  spl612_230 <=> ($false = ((sK609 @ id @ (ap @ sK610 @ sK611) @ id)))),
  introduced(definition,[new_symbols(definition,[spl612_230])],[avatar_definition])).
thf(f2204,plain,(
  ($false = ((sK609 @ id @ (ap @ sK610 @ sK611) @ id))) | ~spl612_230),
  inference(avatar_component_clause,[],[f2202])).
thf(f2205,plain,(
  spl612_230 | spl612_225),
  inference(avatar_split_clause,[],[f1123,f2179,f2202])).
thf(f2328,definition,(
  spl612_259 <=> (hoasinduction_lem2 = $false)),
  introduced(definition,[new_symbols(definition,[spl612_259])],[avatar_definition])).
thf(f2330,plain,(
  (hoasinduction_lem2 = $false) | ~spl612_259),
  inference(avatar_component_clause,[],[f2328])).
thf(f2331,plain,(
  spl612_212 | spl612_259),
  inference(avatar_split_clause,[],[f1132,f2328,f2127])).
thf(f3031,definition,(
  spl612_423 <=> ($true = ((sK319 @ id @ (ap @ sK320 @ sK321) @ id)))),
  introduced(definition,[new_symbols(definition,[spl612_423])],[avatar_definition])).
thf(f3033,plain,(
  ($true != ((sK319 @ id @ (ap @ sK320 @ sK321) @ id))) | spl612_423),
  inference(avatar_component_clause,[],[f3031])).
thf(f3034,plain,(
  spl612_82 | ~spl612_423),
  inference(avatar_split_clause,[],[f599,f3031,f1566])).
thf(f3438,definition,(
  spl612_515 <=> ($true = ((sK319 @ id @ sK321 @ id)))),
  introduced(definition,[new_symbols(definition,[spl612_515])],[avatar_definition])).
thf(f3440,plain,(
  ($true = ((sK319 @ id @ sK321 @ id))) | ~spl612_515),
  inference(avatar_component_clause,[],[f3438])).
thf(f3441,plain,(
  spl612_515 | spl612_82),
  inference(avatar_split_clause,[],[f601,f1566,f3438])).
thf(f3818,definition,(
  spl612_602 <=> ! [X10 : term,X7 : subst,X9 : subst,X8 : subst] : (($true = ((sK319 @ (comp @ X7 @ X9) @ (sub @ X10 @ X9) @ X8))) | ($true != ((sK319 @ X7 @ X10 @ (comp @ X9 @ X8)))))),
  introduced(definition,[new_symbols(definition,[spl612_602])],[avatar_definition])).
thf(f3819,plain,(
  ( ! [X10 : term,X8 : subst,X9 : subst,X7 : subst] : (($true != ((sK319 @ X7 @ X10 @ (comp @ X9 @ X8)))) | ($true = ((sK319 @ (comp @ X7 @ X9) @ (sub @ X10 @ X9) @ X8)))) ) | ~spl612_602),
  inference(avatar_component_clause,[],[f3818])).
thf(f3820,plain,(
  spl612_82 | spl612_602),
  inference(avatar_split_clause,[],[f600,f3818,f1566])).
thf(f4030,plain,(
  ($false = $true) | (~spl612_212 | ~spl612_225)),
  inference(superposition,[],[f2129,f2181])).
thf(f4034,plain,(
  $false | (~spl612_212 | ~spl612_225)),
  inference(trivial_inequality_removal,[],[f4030])).
thf(f4035,plain,(
  ~spl612_212 | ~spl612_225),
  inference(avatar_contradiction_clause,[],[f4034])).
thf(f4037,plain,(
  ( ! [X0 : d_term] : ((one = ((d2term @ X0)))) )),
  inference(superposition,[],[f896,f783])).
thf(f4040,plain,(
  ( ! [X0 : d_subst] : ((sh = ((d2subst @ X0)))) )),
  inference(superposition,[],[f951,f782])).
thf(f4051,plain,(
  ( ! [X214 : term] : ((one = X214)) )),
  inference(forward_demodulation,[],[f833,f4037])).
thf(f4087,plain,(
  ( ! [X124 : subst] : ((sh = X124)) )),
  inference(forward_demodulation,[],[f890,f4040])).
thf(f4448,plain,(
  ($true = ((sK609 @ sh @ sK610 @ sh))) | ~spl612_228),
  inference(forward_demodulation,[],[f2194,f4087])).
thf(f4449,plain,(
  (((sK609 @ sh @ one @ sh)) = $true) | ~spl612_228),
  inference(forward_demodulation,[],[f4448,f4051])).
thf(f4450,plain,(
  ( ! [X0 : term] : (($true = ((sK609 @ sh @ X0 @ sh)))) ) | ~spl612_228),
  inference(superposition,[],[f4449,f4051])).
thf(f4458,plain,(
  ( ! [X0 : subst,X1 : term] : ((((sK609 @ X0 @ X1 @ X0)) = $true)) ) | ~spl612_228),
  inference(superposition,[],[f4450,f4087])).
thf(f4877,plain,(
  ($false = $true) | (~spl612_228 | ~spl612_230)),
  inference(forward_demodulation,[],[f2204,f4458])).
thf(f4878,plain,(
  $false | (~spl612_228 | ~spl612_230)),
  inference(trivial_inequality_removal,[],[f4877])).
thf(f4879,plain,(
  ~spl612_228 | ~spl612_230),
  inference(avatar_contradiction_clause,[],[f4878])).
thf(f5004,plain,(
  ($false = $true) | (~spl612_152 | ~spl612_259)),
  inference(superposition,[],[f1870,f2330])).
thf(f5007,plain,(
  $false | (~spl612_152 | ~spl612_259)),
  inference(trivial_inequality_removal,[],[f5004])).
thf(f5008,plain,(
  ~spl612_152 | ~spl612_259),
  inference(avatar_contradiction_clause,[],[f5007])).
thf(f5009,plain,(
  (((sK319 @ sh @ sK321 @ sh)) = $true) | ~spl612_515),
  inference(forward_demodulation,[],[f3440,f4087])).
thf(f5011,plain,(
  (((sK319 @ sh @ (ap @ sK320 @ sK321) @ sh)) != $true) | spl612_423),
  inference(forward_demodulation,[],[f3033,f4087])).
thf(f5012,plain,(
  ( ! [X10 : term,X8 : subst,X9 : subst,X7 : subst] : (($true = ((sK319 @ (comp @ X7 @ X9) @ (sub @ X10 @ X9) @ X8))) | (((sK319 @ X7 @ X10 @ sh)) != $true)) ) | ~spl612_602),
  inference(forward_demodulation,[],[f3819,f4087])).
thf(f5015,plain,(
  (((sK319 @ sh @ one @ sh)) != $true) | spl612_423),
  inference(forward_demodulation,[],[f5011,f4051])).
thf(f5016,plain,(
  ( ! [X10 : term,X8 : subst,X9 : subst,X7 : subst] : ((((sK319 @ sh @ (sub @ X10 @ X9) @ X8)) = $true) | (((sK319 @ X7 @ X10 @ sh)) != $true)) ) | ~spl612_602),
  inference(forward_demodulation,[],[f5012,f4087])).
thf(f5018,plain,(
  ( ! [X10 : term,X8 : subst,X7 : subst] : ((((sK319 @ X7 @ X10 @ sh)) != $true) | ($true = ((sK319 @ sh @ one @ X8)))) ) | ~spl612_602),
  inference(forward_demodulation,[],[f5016,f4051])).
thf(f5021,definition,(
  spl612_666 <=> ! [X7 : subst,X10 : term] : (((sK319 @ X7 @ X10 @ sh)) != $true)),
  introduced(definition,[new_symbols(definition,[spl612_666])],[avatar_definition])).
thf(f5022,plain,(
  ( ! [X10 : term,X7 : subst] : ((((sK319 @ X7 @ X10 @ sh)) != $true)) ) | ~spl612_666),
  inference(avatar_component_clause,[],[f5021])).
thf(f5024,definition,(
  spl612_667 <=> ! [X8 : subst] : ($true = ((sK319 @ sh @ one @ X8)))),
  introduced(definition,[new_symbols(definition,[spl612_667])],[avatar_definition])).
thf(f5025,plain,(
  ( ! [X8 : subst] : (($true = ((sK319 @ sh @ one @ X8)))) ) | ~spl612_667),
  inference(avatar_component_clause,[],[f5024])).
thf(f5026,plain,(
  spl612_666 | spl612_667 | ~spl612_602),
  inference(avatar_split_clause,[],[f5018,f3818,f5024,f5021])).
thf(f5057,plain,(
  ($true != $true) | (spl612_423 | ~spl612_667)),
  inference(superposition,[],[f5015,f5025])).
thf(f5058,plain,(
  $false | (spl612_423 | ~spl612_667)),
  inference(trivial_inequality_removal,[],[f5057])).
thf(f5059,plain,(
  spl612_423 | ~spl612_667),
  inference(avatar_contradiction_clause,[],[f5058])).
thf(f5064,plain,(
  ($true != $true) | (~spl612_515 | ~spl612_666)),
  inference(superposition,[],[f5022,f5009])).
thf(f5065,plain,(
  $false | (~spl612_515 | ~spl612_666)),
  inference(trivial_inequality_removal,[],[f5064])).
thf(f5066,plain,(
  ~spl612_515 | ~spl612_666),
  inference(avatar_contradiction_clause,[],[f5065])).
cnf(s104, plain, ~spl612_82 | spl612_152, inference(sat_conversion,[],[f1871])).
cnf(s175, plain, spl612_225 | spl612_228, inference(sat_conversion,[],[f2195])).
cnf(s177, plain, spl612_225 | spl612_230, inference(sat_conversion,[],[f2205])).
cnf(s203, plain, spl612_212 | spl612_259, inference(sat_conversion,[],[f2331])).
cnf(s362, plain, spl612_82 | ~spl612_423, inference(sat_conversion,[],[f3034])).
cnf(s469, plain, spl612_82 | spl612_515, inference(sat_conversion,[],[f3441])).
cnf(s569, plain, spl612_82 | spl612_602, inference(sat_conversion,[],[f3820])).
cnf(s624, plain, ~spl612_212 | ~spl612_225, inference(sat_conversion,[],[f4035])).
cnf(s723, plain, ~spl612_228 | ~spl612_230, inference(sat_conversion,[],[f4879])).
cnf(s725, plain, ~spl612_152 | ~spl612_259, inference(sat_conversion,[],[f5008])).
cnf(s726, plain, ~spl612_602 | spl612_666 | spl612_667, inference(sat_conversion,[],[f5026])).
cnf(s728, plain, spl612_423 | ~spl612_667, inference(sat_conversion,[],[f5059])).
cnf(s729, plain, ~spl612_515 | ~spl612_666, inference(sat_conversion,[],[f5066])).
cnf(s845, plain, spl612_82, inference(rat,[],[s726,s728,s729,s362,s469,s569])).
cnf(s848, plain, spl612_152, inference(rat,[],[s104,s845])).
cnf(s851, plain, ~spl612_259, inference(rat,[],[s725,s848])).
cnf(s852, plain, spl612_212, inference(rat,[],[s203,s851])).
cnf(s853, plain, ~spl612_225, inference(rat,[],[s624,s852])).
cnf(s854, plain, spl612_230, inference(rat,[],[s177,s853])).
cnf(s856, plain, spl612_228, inference(rat,[],[s175,s853])).
cnf(s860, plain, $false, inference(rat,[],[s723,s854,s856])).
thf(f5069,plain,(
  $false),
  inference(avatar_sat_refutation,[],[s860])).
% SZS output end Proof for theBenchmark