%----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, db2: !>[X0: $tType]:(X0)).
thf(func_def_131, type, db0: !>[X0: $tType]:(X0)).
thf(func_def_132, type, db4: !>[X0: $tType]:(X0)).
thf(func_def_133, type, db3: !>[X0: $tType]:(X0)).
thf(func_def_134, type, db1: !>[X0: $tType]:(X0)).
thf(func_def_135, type, vEQ: !>[X0: $tType]:((X0 > X0 > $o))).
thf(func_def_136, type, vLAM: !>[X0: $tType, X1: $tType]:((X1) > (X0 > X1))).
thf(func_def_137, type, vIMP: ($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: ((subst > term > subst > $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: ((term > $o) > $o)).
thf(func_def_144, type, sP6: $o).
thf(func_def_145, type, sP7: $o).
thf(func_def_146, type, sP8: $o).
thf(func_def_147, type, sP9: ((subst > term > subst > $o) > $o)).
thf(func_def_148, type, sP10: $o).
thf(func_def_149, type, sP11: ((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: ((subst > term > subst > $o) > $o)).
thf(func_def_153, type, sP15: ((subst > term > subst > $o) > $o)).
thf(func_def_154, type, sP16: ((subst > term > subst > $o) > $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: $o).
thf(func_def_159, type, sP21: $o).
thf(func_def_160, type, sP22: $o).
thf(func_def_161, type, sP23: $o).
thf(func_def_162, type, sP24: ((term > $o) > $o)).
thf(func_def_163, type, sP25: ((subst > term > subst > $o) > $o)).
thf(func_def_164, type, sP26: ((subst > term > subst > $o) > $o)).
thf(func_def_165, type, sP27: $o).
thf(func_def_166, type, sP28: $o).
thf(func_def_167, type, sP29: $o).
thf(func_def_168, type, sP30: $o).
thf(func_def_169, type, sP31: ((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: ((term > $o) > $o)).
thf(func_def_174, type, sP36: ((term > $o) > $o)).
thf(func_def_175, type, sP37: $o).
thf(func_def_176, type, sP38: $o).
thf(func_def_177, type, sP39: ((term > $o) > $o)).
thf(func_def_178, type, sP40: ((term > $o) > $o)).
thf(func_def_179, type, sP41: $o).
thf(func_def_180, type, sP42: $o).
thf(func_def_181, type, sP43: ((subst > term > subst > $o) > $o)).
thf(func_def_182, type, sP44: ((term > $o) > $o)).
thf(func_def_183, type, sP45: $o).
thf(func_def_184, type, sP46: $o).
thf(func_def_185, type, sP47: ((subst > term > subst > $o) > $o)).
thf(func_def_186, type, sP48: $o).
thf(func_def_187, type, sP49: $o).
thf(func_def_188, type, sP50: ((subst > term > subst > $o) > $o)).
thf(func_def_189, type, sP51: $o).
thf(func_def_190, type, sP52: ((term > $o) > $o)).
thf(func_def_191, type, sP53: ((subst > term > subst > $o) > $o)).
thf(func_def_192, type, sP54: $o).
thf(func_def_193, type, sP55: $o).
thf(func_def_194, type, sP56: $o).
thf(func_def_195, type, sP57: ((term > $o) > $o)).
thf(func_def_196, type, sP58: $o).
thf(func_def_197, type, sK59: (subst > term > subst > $o)).
thf(func_def_198, type, sK60: (term > $o)).
thf(func_def_199, type, sK61: term).
thf(func_def_200, type, sK62: ((subst > term > term) > subst)).
thf(func_def_201, type, sK63: ((subst > term > term) > subst)).
thf(func_def_202, type, sK64: ((subst > term > term) > term)).
thf(func_def_203, type, sK65: ((subst > term > term) > term)).
thf(func_def_204, type, sK66: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_205, type, sK67: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_206, type, sK68: (term > (term > $o) > term)).
thf(func_def_207, type, sK69: ((term > $o) > term)).
thf(func_def_208, type, sK70: ((term > $o) > term)).
thf(func_def_209, type, sK71: term).
thf(func_def_210, type, sK72: (subst > term > subst > $o)).
thf(func_def_211, type, sK73: ((subst > term > subst > $o) > term > $o)).
thf(func_def_212, type, sK74: ((subst > term > subst > $o) > term)).
thf(func_def_213, type, sK75: ((subst > term > subst > $o) > term)).
thf(func_def_214, type, sK76: term).
thf(func_def_215, type, sK77: (subst > term > subst > $o)).
thf(func_def_216, type, sK78: ((term > $o) > term)).
thf(func_def_217, type, sK79: ((term > $o) > term)).
thf(func_def_218, type, sK80: ((term > $o) > term)).
thf(func_def_219, type, sK81: (term > $o)).
thf(func_def_220, type, sK82: term).
thf(func_def_221, type, sK83: (term > term)).
thf(func_def_222, type, sK84: (term > (term > $o) > term)).
thf(func_def_223, type, sK85: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_224, type, sK86: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_225, type, sK87: (subst > term > subst > $o)).
thf(func_def_226, type, sK88: (term > $o)).
thf(func_def_227, type, sK89: term).
thf(func_def_228, type, sK90: ((subst > term > term) > subst)).
thf(func_def_229, type, sK91: ((subst > term > term) > 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 > term > subst > $o) > term)).
thf(func_def_233, type, sK95: ((subst > term > term) > subst)).
thf(func_def_234, type, sK96: ((subst > term > term) > term)).
thf(func_def_235, type, sK97: ((subst > term > term) > subst)).
thf(func_def_236, type, sK98: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_237, type, sK99: ((term > $o) > term)).
thf(func_def_238, type, sK100: ((term > $o) > term)).
thf(func_def_239, type, sK101: (subst > term > subst > $o)).
thf(func_def_240, type, sK102: term).
thf(func_def_241, type, sK103: ((subst > term > subst > $o) > subst)).
thf(func_def_242, type, sK104: ((subst > term > subst > $o) > term)).
thf(func_def_243, type, sK105: ((subst > term > subst > $o) > subst)).
thf(func_def_244, type, sK106: ((subst > term > subst > $o) > subst)).
thf(func_def_245, type, sK107: ((subst > term > subst > $o) > subst)).
thf(func_def_246, type, sK108: ((subst > term > subst > $o) > subst)).
thf(func_def_247, type, sK109: ((subst > term > subst > $o) > subst)).
thf(func_def_248, type, sK110: ((subst > term > subst > $o) > term)).
thf(func_def_249, type, sK111: (term > (subst > term > subst > $o) > term)).
thf(func_def_250, type, sK112: ((subst > term > term) > term)).
thf(func_def_251, type, sK113: ((subst > term > term) > subst)).
thf(func_def_252, type, sK114: ((subst > term > term) > subst)).
thf(func_def_253, type, sK115: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_254, type, sK116: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_255, type, sK117: (subst > term > subst > $o)).
thf(func_def_256, type, sK118: term).
thf(func_def_257, type, sK119: ((subst > term > subst > $o) > subst)).
thf(func_def_258, type, sK120: ((subst > term > subst > $o) > subst)).
thf(func_def_259, type, sK121: ((subst > term > subst > $o) > term)).
thf(func_def_260, type, sK122: ((subst > term > subst > $o) > subst)).
thf(func_def_261, type, sK123: (term > (subst > term > subst > $o) > term)).
thf(func_def_262, type, sK124: ((subst > term > subst > $o) > subst)).
thf(func_def_263, type, sK125: ((subst > term > subst > $o) > term)).
thf(func_def_264, type, sK126: ((subst > term > subst > $o) > subst)).
thf(func_def_265, type, sK127: ((subst > term > subst > $o) > subst)).
thf(func_def_266, type, sK128: term).
thf(func_def_267, type, sK129: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_268, type, sK130: (term > (subst > term > subst > $o) > term)).
thf(func_def_269, type, sK131: (subst > term > subst > $o)).
thf(func_def_270, type, sK132: term).
thf(func_def_271, type, sK133: ((subst > term > term) > subst)).
thf(func_def_272, type, sK134: ((subst > term > term) > term)).
thf(func_def_273, type, sK135: ((subst > term > term) > subst)).
thf(func_def_274, type, sK136: ((subst > term > term) > term)).
thf(func_def_275, type, sK137: ((subst > term > term) > subst)).
thf(func_def_276, type, sK138: ((subst > term > term) > subst)).
thf(func_def_277, type, sK139: ((subst > term > term) > term)).
thf(func_def_278, type, sK140: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_279, type, sK141: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_280, type, sK142: term).
thf(func_def_281, type, sK143: (subst > term > subst > $o)).
thf(func_def_282, type, sK144: ((subst > term > term) > subst)).
thf(func_def_283, type, sK145: ((subst > term > term) > term)).
thf(func_def_284, type, sK146: ((subst > term > term) > subst)).
thf(func_def_285, type, sK147: ((subst > term > term) > term)).
thf(func_def_286, type, sK148: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_287, type, sK149: ((subst > term > subst > $o) > term > term)).
thf(func_def_288, type, sK150: (term > $o)).
thf(func_def_289, type, sK151: term).
thf(func_def_290, type, sK152: (term > term)).
thf(func_def_291, type, sK153: ((term > $o) > term)).
thf(func_def_292, type, sK154: ((term > $o) > term)).
thf(func_def_293, type, sK155: ((term > $o) > term)).
thf(func_def_294, type, sK156: ((term > $o) > term)).
thf(func_def_295, type, sK157: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_296, type, sK158: ((subst > term > term) > subst)).
thf(func_def_297, type, sK159: ((subst > term > term) > subst)).
thf(func_def_298, type, sK160: ((subst > term > term) > term)).
thf(func_def_299, type, sK161: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_300, type, sK162: ((term > $o) > term)).
thf(func_def_301, type, sK163: ((term > $o) > term)).
thf(func_def_302, type, sK164: (term > $o)).
thf(func_def_303, type, sK165: term).
thf(func_def_304, type, sK166: subst).
thf(func_def_305, type, sK167: term).
thf(func_def_306, type, sK168: subst).
thf(func_def_307, type, sK169: subst).
thf(func_def_308, type, sK170: term).
thf(func_def_309, type, sK171: term).
thf(func_def_310, type, sK172: subst).
thf(func_def_311, type, sK173: (term > $o)).
thf(func_def_312, type, sK174: (term > term)).
thf(func_def_313, type, sK175: ((term > $o) > subst > term)).
thf(func_def_314, type, sK176: ((term > $o) > term)).
thf(func_def_315, type, sK177: ((term > $o) > term)).
thf(func_def_316, type, sK178: ((term > $o) > term)).
thf(func_def_317, type, sK179: ((term > $o) > term)).
thf(func_def_318, type, sK180: ((term > $o) > term)).
thf(func_def_319, type, sK181: ((term > $o) > term)).
thf(func_def_320, type, sK182: (subst > (term > $o) > term)).
thf(func_def_321, type, sK183: (term > $o)).
thf(func_def_322, type, sK184: subst).
thf(func_def_323, type, sK185: term).
thf(func_def_324, type, sK186: (term > term)).
thf(func_def_325, type, sK187: term).
thf(func_def_326, type, sK188: term).
thf(func_def_327, type, sK189: (term > $o)).
thf(func_def_328, type, sK190: (term > term)).
thf(func_def_329, type, sK191: ((term > $o) > term)).
thf(func_def_330, type, sK192: ((term > $o) > term)).
thf(func_def_331, type, sK193: ((term > $o) > term)).
thf(func_def_332, type, sK194: ((term > $o) > term)).
thf(func_def_333, type, sK195: ((term > $o) > term)).
thf(func_def_334, type, sK196: ((term > $o) > term)).
thf(func_def_335, type, sK197: term).
thf(func_def_336, type, sK198: (subst > term > subst > $o)).
thf(func_def_337, type, sK199: (term > (subst > term > term) > subst)).
thf(func_def_338, type, sK200: (term > (subst > term > term) > term)).
thf(func_def_339, type, sK201: (term > (subst > term > term) > subst)).
thf(func_def_340, type, sK202: ((subst > term > term) > term)).
thf(func_def_341, type, sK203: (term > (subst > term > term) > term)).
thf(func_def_342, type, sK204: (term > (subst > term > term) > subst)).
thf(func_def_343, type, sK205: (term > (subst > term > term) > subst)).
thf(func_def_344, type, sK206: ((subst > term > subst > $o) > term > term)).
thf(func_def_345, type, sK207: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_346, type, sK208: ((subst > term > subst > $o) > term)).
thf(func_def_347, type, sK209: subst).
thf(func_def_348, type, sK210: (subst > term > term)).
thf(func_def_349, type, sK211: (subst > term > term)).
thf(func_def_350, type, sK212: term).
thf(func_def_351, type, sK213: ((subst > term > term) > subst)).
thf(func_def_352, type, sK214: ((subst > term > term) > term)).
thf(func_def_353, type, sK215: ((subst > term > term) > subst)).
thf(func_def_354, type, sK216: ((subst > term > term) > term)).
thf(func_def_355, type, sK217: ((subst > term > term) > subst)).
thf(func_def_356, type, sK218: ((subst > term > term) > subst)).
thf(func_def_357, type, sK219: term).
thf(func_def_358, type, sK220: subst).
thf(func_def_359, type, sK221: term).
thf(func_def_360, type, sK222: term).
thf(func_def_361, type, sK223: term).
thf(func_def_362, type, sK224: subst).
thf(func_def_363, type, sK225: term).
thf(func_def_364, type, sK226: term).
thf(func_def_365, type, sK227: (subst > term > subst > $o)).
thf(func_def_366, type, sK228: ((subst > term > subst > $o) > term)).
thf(func_def_367, type, sK229: ((subst > term > subst > $o) > subst)).
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) > subst)).
thf(func_def_372, type, sK234: ((subst > term > subst > $o) > term)).
thf(func_def_373, type, sK235: ((subst > term > subst > $o) > subst)).
thf(func_def_374, type, sK236: ((subst > term > subst > $o) > term)).
thf(func_def_375, type, sK237: ((subst > term > subst > $o) > term)).
thf(func_def_376, type, sK238: (subst > term > subst > $o)).
thf(func_def_377, type, sK239: term).
thf(func_def_378, type, sK240: ((subst > term > term) > term)).
thf(func_def_379, type, sK241: ((subst > term > term) > term)).
thf(func_def_380, type, sK242: ((subst > term > term) > subst)).
thf(func_def_381, type, sK243: ((subst > term > term) > subst)).
thf(func_def_382, type, sK244: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_383, type, sK245: (term > (subst > term > subst > $o) > term)).
thf(func_def_384, type, sK246: ((subst > term > subst > $o) > term > $o)).
thf(func_def_385, type, sK247: ((term > $o) > term)).
thf(func_def_386, type, sK248: ((term > $o) > term)).
thf(func_def_387, type, sK249: (subst > term > subst > $o)).
thf(func_def_388, type, sK250: ((term > $o) > term)).
thf(func_def_389, type, sK251: term).
thf(func_def_390, type, sK252: (term > $o)).
thf(func_def_391, type, sK253: term).
thf(func_def_392, type, sK254: (term > term)).
thf(func_def_393, type, sK255: term).
thf(func_def_394, type, sK256: (subst > term > subst > $o)).
thf(func_def_395, type, sK257: ((subst > term > subst > $o) > subst)).
thf(func_def_396, type, sK258: ((subst > term > subst > $o) > term)).
thf(func_def_397, type, sK259: ((subst > term > subst > $o) > subst)).
thf(func_def_398, type, sK260: ((subst > term > subst > $o) > subst)).
thf(func_def_399, type, sK261: ((subst > term > subst > $o) > term)).
thf(func_def_400, type, sK262: ((subst > term > subst > $o) > term)).
thf(func_def_401, type, sK263: ((subst > term > subst > $o) > subst)).
thf(func_def_402, type, sK264: ((subst > term > subst > $o) > subst)).
thf(func_def_403, type, sK265: ((subst > term > subst > $o) > subst)).
thf(func_def_404, type, sK266: ((term > $o) > term > term)).
thf(func_def_405, type, sK267: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_406, type, sK268: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_407, type, sK269: term).
thf(func_def_408, type, sK270: (subst > term > subst > $o)).
thf(func_def_409, type, sK271: (term > $o)).
thf(func_def_410, type, sK272: ((subst > term > term) > term)).
thf(func_def_411, type, sK273: ((subst > term > term) > subst)).
thf(func_def_412, type, sK274: ((subst > term > term) > subst)).
thf(func_def_413, type, sK275: ((subst > term > term) > term)).
thf(func_def_414, type, sK276: ((subst > term > term) > subst)).
thf(func_def_415, type, sK277: ((subst > term > term) > term)).
thf(func_def_416, type, sK278: ((subst > term > term) > subst)).
thf(func_def_417, type, sK279: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_418, type, sK280: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_419, type, sK281: ((subst > term > subst > $o) > term)).
thf(func_def_420, type, sK282: ((subst > term > subst > $o) > term)).
thf(func_def_421, type, sK283: ((term > $o) > term)).
thf(func_def_422, type, sK284: ((term > $o) > term)).
thf(func_def_423, type, sK285: term).
thf(func_def_424, type, sK286: (subst > term > subst > $o)).
thf(func_def_425, type, sK287: ((subst > term > term) > subst)).
thf(func_def_426, type, sK288: ((subst > term > term) > term)).
thf(func_def_427, type, sK289: ((subst > term > term) > subst)).
thf(func_def_428, type, sK290: ((subst > term > term) > term)).
thf(func_def_429, type, sK291: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_430, type, sK292: ((subst > term > subst > $o) > term > term)).
thf(func_def_431, type, sK293: (term > $o)).
thf(func_def_432, type, sK294: term).
thf(func_def_433, type, sK295: (subst > term > subst > $o)).
thf(func_def_434, type, sK296: ((subst > term > term) > term)).
thf(func_def_435, type, sK297: ((subst > term > term) > subst)).
thf(func_def_436, type, sK298: ((subst > term > term) > term)).
thf(func_def_437, type, sK299: ((subst > term > term) > subst)).
thf(func_def_438, type, sK300: (term > subst > term > term)).
thf(func_def_439, type, sK301: term).
thf(func_def_440, type, sK302: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_441, type, sK303: (term > (term > $o) > term)).
thf(func_def_442, type, sK304: ((subst > term > subst > $o) > (term > $o) > term)).
thf(func_def_443, type, sK305: term).
thf(func_def_444, type, sK306: ((subst > term > term) > subst)).
thf(func_def_445, type, sK307: ((subst > term > term) > term)).
thf(func_def_446, type, sK308: ((subst > term > subst > $o) > subst)).
thf(func_def_447, type, sK309: ((subst > term > subst > $o) > subst)).
thf(func_def_448, type, sK310: ((subst > term > subst > $o) > subst)).
thf(func_def_449, type, sK311: ((subst > term > subst > $o) > term)).
thf(func_def_450, type, sK312: ((subst > term > subst > $o) > subst)).
thf(func_def_451, type, sK313: ((subst > term > subst > $o) > term)).
thf(func_def_452, type, sK314: ((subst > term > subst > $o) > subst)).
thf(func_def_453, type, sK315: ((subst > term > subst > $o) > subst)).
thf(func_def_454, type, sK316: term).
thf(func_def_455, type, sK317: (subst > term > subst > $o)).
thf(func_def_456, type, sK318: term).
thf(func_def_457, type, sK319: ((term > $o) > term)).
thf(func_def_458, type, sK320: ((term > $o) > term)).
thf(func_def_459, type, sK321: term).
thf(func_def_460, type, sK322: (term > $o)).
thf(func_def_461, type, sK323: (term > term)).
thf(func_def_462, type, sK324: term).
thf(func_def_463, type, sK325: (subst > term > subst > $o)).
thf(func_def_464, type, sK326: ((subst > term > subst > $o) > term)).
thf(func_def_465, type, sK327: ((subst > term > subst > $o) > term)).
thf(func_def_466, type, sK328: ((subst > term > subst > $o) > subst)).
thf(func_def_467, type, sK329: ((subst > term > subst > $o) > subst)).
thf(func_def_468, type, sK330: ((subst > term > subst > $o) > subst)).
thf(func_def_469, type, sK331: ((subst > term > subst > $o) > subst)).
thf(func_def_470, type, sK332: ((subst > term > subst > $o) > term)).
thf(func_def_471, type, sK333: ((subst > term > subst > $o) > subst)).
thf(func_def_472, type, sK334: ((subst > term > subst > $o) > subst)).
thf(func_def_473, type, sK335: ((subst > term > subst > $o) > subst)).
thf(func_def_474, type, sK336: ((subst > term > subst > $o) > subst)).
thf(func_def_475, type, sK337: ((subst > term > subst > $o) > term)).
thf(func_def_476, type, sK338: ((subst > term > subst > $o) > subst)).
thf(func_def_477, type, sK339: ((subst > term > subst > $o) > subst)).
thf(func_def_478, type, sK340: ((subst > term > subst > $o) > subst)).
thf(func_def_479, type, sK341: ((subst > term > subst > $o) > term)).
thf(func_def_480, type, sK342: ((subst > term > subst > $o) > subst)).
thf(func_def_481, type, sK343: ((subst > term > subst > $o) > term > term)).
thf(func_def_482, type, sK344: term).
thf(func_def_483, type, sK345: (subst > term > subst > $o)).
thf(func_def_484, type, sK346: ((subst > term > subst > $o) > term)).
thf(func_def_485, type, sK347: ((subst > term > subst > $o) > term)).
thf(func_def_486, type, sK348: ((subst > term > term) > (subst > term > subst > $o) > term)).
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 > subst > $o) > subst > term > term)).
thf(func_def_491, type, sK353: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_492, type, sK354: ((subst > term > term) > subst)).
thf(func_def_493, type, sK355: ((subst > term > term) > term)).
thf(func_def_494, type, sK356: ((subst > term > term) > subst)).
thf(func_def_495, type, sK357: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_496, type, sK358: ((subst > term > subst > $o) > term)).
thf(func_def_497, type, sK359: ((subst > term > subst > $o) > term)).
thf(func_def_498, type, sK360: ((term > $o) > term)).
thf(func_def_499, type, sK361: ((term > $o) > term)).
thf(func_def_500, type, sK362: (subst > term > subst > $o)).
thf(func_def_501, type, sK363: term).
thf(func_def_502, type, sK364: term).
thf(func_def_503, type, sK365: (term > $o)).
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) > subst)).
thf(func_def_507, type, sK369: ((subst > term > subst > $o) > term)).
thf(func_def_508, type, sK370: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_509, type, sK371: ((subst > term > subst > $o) > subst)).
thf(func_def_510, type, sK372: ((subst > term > subst > $o) > term)).
thf(func_def_511, type, sK373: ((subst > term > subst > $o) > subst)).
thf(func_def_512, type, sK374: ((subst > term > subst > $o) > subst)).
thf(func_def_513, type, sK375: ((subst > term > subst > $o) > term)).
thf(func_def_514, type, sK376: ((subst > term > subst > $o) > term)).
thf(func_def_515, type, sK377: term).
thf(func_def_516, type, sK378: (term > $o)).
thf(func_def_517, type, sK379: (subst > term > subst > $o)).
thf(func_def_518, type, sK380: ((subst > term > subst > $o) > subst)).
thf(func_def_519, type, sK381: ((subst > term > subst > $o) > subst)).
thf(func_def_520, type, sK382: ((subst > term > subst > $o) > term)).
thf(func_def_521, type, sK383: ((subst > term > subst > $o) > subst)).
thf(func_def_522, type, sK384: ((term > $o) > term > term)).
thf(func_def_523, type, sK385: ((subst > term > subst > $o) > (term > $o) > term)).
thf(func_def_524, type, sK386: ((subst > term > subst > $o) > term)).
thf(func_def_525, type, sK387: ((subst > term > subst > $o) > subst)).
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 > term) > (subst > term > subst > $o) > term)).
thf(func_def_529, type, sK391: ((subst > term > term) > subst)).
thf(func_def_530, type, sK392: ((subst > term > term) > term)).
thf(func_def_531, type, sK393: ((subst > term > term) > subst)).
thf(func_def_532, type, sK394: ((subst > term > subst > $o) > subst > term > 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: (term > (subst > term > subst > $o) > term)).
thf(func_def_540, type, sK402: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_541, type, sK403: term).
thf(func_def_542, type, sK404: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_543, type, sK405: (term > (subst > term > subst > $o) > term)).
thf(func_def_544, type, sK406: (subst > term > subst > $o)).
thf(func_def_545, type, sK407: term).
thf(func_def_546, type, sK408: ((subst > term > term) > term)).
thf(func_def_547, type, sK409: ((subst > term > term) > subst)).
thf(func_def_548, type, sK410: ((subst > term > term) > subst)).
thf(func_def_549, type, sK411: ((subst > term > term) > term)).
thf(func_def_550, type, sK412: subst).
thf(func_def_551, type, sK413: term).
thf(func_def_552, type, sK414: (term > $o)).
thf(func_def_553, type, sK415: (term > term)).
thf(func_def_554, type, sK416: ((term > $o) > term)).
thf(func_def_555, type, sK417: ((term > $o) > subst > term)).
thf(func_def_556, type, sK418: ((term > $o) > term)).
thf(func_def_557, type, sK419: ((term > $o) > term)).
thf(func_def_558, type, sK420: term).
thf(func_def_559, type, sK421: (subst > term > subst > $o)).
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) > term)).
thf(func_def_563, type, sK425: ((subst > term > subst > $o) > subst)).
thf(func_def_564, type, sK426: ((subst > term > subst > $o) > term)).
thf(func_def_565, type, sK427: ((subst > term > subst > $o) > term)).
thf(func_def_566, type, sK428: ((subst > term > subst > $o) > subst)).
thf(func_def_567, type, sK429: ((subst > term > subst > $o) > subst)).
thf(func_def_568, type, sK430: ((subst > term > subst > $o) > subst)).
thf(func_def_569, type, sK431: ((subst > term > subst > $o) > term)).
thf(func_def_570, type, sK432: ((subst > term > subst > $o) > term)).
thf(func_def_571, type, sK433: ((subst > term > term) > term)).
thf(func_def_572, type, sK434: ((subst > term > term) > subst)).
thf(func_def_573, type, sK435: ((subst > term > term) > subst)).
thf(func_def_574, type, sK436: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_575, type, sK437: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_576, type, sK438: term).
thf(func_def_577, type, sK439: (subst > term > subst > $o)).
thf(func_def_578, type, sK440: (term > $o)).
thf(func_def_579, type, sK441: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_580, type, sK442: ((term > $o) > term > term)).
thf(func_def_581, type, sK443: ((subst > term > subst > $o) > subst)).
thf(func_def_582, type, sK444: ((subst > term > subst > $o) > term)).
thf(func_def_583, type, sK445: ((subst > term > subst > $o) > subst)).
thf(func_def_584, type, sK446: ((subst > term > subst > $o) > subst)).
thf(func_def_585, type, sK447: ((subst > term > subst > $o) > subst)).
thf(func_def_586, type, sK448: ((subst > term > subst > $o) > term)).
thf(func_def_587, type, sK449: ((subst > term > subst > $o) > subst)).
thf(func_def_588, type, sK450: ((subst > term > subst > $o) > subst)).
thf(func_def_589, type, sK451: term).
thf(func_def_590, type, sK452: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_591, type, sK453: ((subst > term > term) > subst)).
thf(func_def_592, type, sK454: ((subst > term > term) > term)).
thf(func_def_593, type, sK455: ((subst > term > term) > subst)).
thf(func_def_594, type, sK456: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_595, type, sK457: subst).
thf(func_def_596, type, sK458: term).
thf(func_def_597, type, sK459: (term > $o)).
thf(func_def_598, type, sK460: ((term > $o) > term)).
thf(func_def_599, type, sK461: ((term > $o) > term > subst > term > $o)).
thf(func_def_600, type, sK462: subst).
thf(func_def_601, type, sK463: term).
thf(func_def_602, type, sK464: (term > $o)).
thf(func_def_603, type, sK465: term).
thf(func_def_604, type, sK466: ((term > $o) > term)).
thf(func_def_605, type, sK467: ((term > $o) > term)).
thf(func_def_606, type, sK468: term).
thf(func_def_607, type, sK469: subst).
thf(func_def_608, type, sK470: (term > $o)).
thf(func_def_609, type, sK471: (term > $o)).
thf(func_def_610, type, sK472: ((term > $o) > (term > $o) > subst > term > term)).
thf(func_def_611, type, sK473: subst).
thf(func_def_612, type, sK474: term).
thf(func_def_613, type, sK475: term).
thf(func_def_614, type, sK476: subst).
thf(func_def_615, type, sK477: subst).
thf(func_def_616, type, sK478: ((subst > term > term) > term)).
thf(func_def_617, type, sK479: ((subst > term > term) > subst)).
thf(func_def_618, type, sK480: ((subst > term > term) > subst)).
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)).
thf(func_def_623, type, sK485: term).
thf(func_def_624, type, sK486: (subst > term > term)).
thf(func_def_625, type, sK487: subst).
thf(func_def_626, type, sK488: (subst > term > subst > $o)).
thf(func_def_627, type, sK489: term).
thf(func_def_628, type, sK490: ((subst > term > subst > $o) > term)).
thf(func_def_629, type, sK491: ((subst > term > subst > $o) > term)).
thf(func_def_630, type, sK492: subst).
thf(func_def_631, type, sK493: subst).
thf(func_def_632, type, sK494: term).
thf(func_def_633, type, sK495: term).
thf(func_def_634, type, sK496: term).
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: term).
thf(func_def_647, type, sK509: ((subst > term > subst > $o) > term > $o)).
thf(func_def_648, type, sK510: (subst > term > subst > $o)).
thf(func_def_649, type, sK511: ((term > $o) > term)).
thf(func_def_650, type, sK512: term).
thf(func_def_651, type, sK513: (term > $o)).
thf(func_def_652, type, sK514: subst).
thf(func_def_653, type, sK515: term).
thf(func_def_654, type, sK516: (term > $o)).
thf(func_def_655, type, sK517: term).
thf(func_def_656, type, sK518: subst).
thf(func_def_657, type, sK519: subst).
thf(func_def_658, type, sK520: term).
thf(func_def_659, type, sK521: ((term > $o) > subst > (term > $o) > term > term)).
thf(func_def_660, type, sK522: term).
thf(func_def_661, type, sK523: (term > $o)).
thf(func_def_662, type, sK524: subst).
thf(func_def_663, type, sK525: (term > $o)).
thf(func_def_664, type, sK526: term).
thf(func_def_665, type, sK527: term).
thf(func_def_666, type, sK528: (term > subst > term > term)).
thf(func_def_667, type, sK529: term).
thf(func_def_668, type, sK530: (term > subst > term > term)).
thf(func_def_669, type, sK531: term).
thf(func_def_670, type, sK532: ((subst > term > term) > subst)).
thf(func_def_671, type, sK533: ((subst > term > term) > term)).
thf(func_def_672, type, sK534: subst).
thf(func_def_673, type, sK535: subst).
thf(func_def_674, type, sK536: subst).
thf(func_def_675, type, sK537: subst).
thf(func_def_676, type, sK538: (subst > term > subst > $o)).
thf(func_def_677, type, sK539: ((term > $o) > term)).
thf(func_def_678, type, sK540: ((subst > term > subst > $o) > term > $o)).
thf(func_def_679, type, sK541: term).
thf(func_def_680, type, sK542: term).
thf(func_def_681, type, sK543: term).
thf(func_def_682, type, sK544: term).
thf(func_def_683, type, sK545: term).
thf(func_def_684, type, sK546: term).
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: subst).
thf(func_def_692, type, sK554: term).
thf(func_def_693, type, sK555: (term > d_term)).
thf(func_def_694, type, sK556: (term > subst > term > term)).
thf(func_def_695, type, sK557: term).
thf(func_def_696, type, sK558: ((subst > term > term) > term)).
thf(func_def_697, type, sK559: ((subst > term > term) > subst)).
thf(func_def_698, type, sK560: term).
thf(func_def_699, type, sK561: (term > $o)).
thf(func_def_700, type, sK562: term).
thf(func_def_701, type, sK563: (term > $o)).
thf(func_def_702, type, sK564: subst).
thf(func_def_703, type, sK565: subst).
thf(func_def_704, type, sK566: subst).
thf(func_def_705, type, sK567: term).
thf(func_def_706, type, sK568: (term > $o)).
thf(func_def_707, type, sK569: ((term > $o) > term)).
thf(func_def_708, type, sK570: ((term > $o) > term > subst > term > $o)).
thf(func_def_709, type, sK571: term).
thf(func_def_710, type, sK572: (term > subst > term > term)).
thf(func_def_711, type, sK573: term).
thf(func_def_712, type, sK574: term).
thf(func_def_713, type, sK575: term).
thf(func_def_714, type, sK576: term).
thf(func_def_715, type, sK577: (subst > d_subst)).
thf(func_def_716, type, sK578: term).
thf(func_def_717, type, sK579: term).
thf(func_def_718, type, sK580: term).
thf(func_def_719, type, sK581: term).
thf(func_def_720, type, sK582: (subst > term > term)).
thf(func_def_721, type, sK583: term).
thf(func_def_722, type, sK584: term).
thf(func_def_723, type, sK585: ((subst > term > term) > term)).
thf(func_def_724, type, sK586: ((subst > term > term) > subst)).
thf(func_def_725, type, sK587: ((subst > term > term) > subst)).
thf(func_def_726, type, sK588: term).
thf(func_def_727, type, sK589: term).
thf(func_def_728, type, sK590: term).
thf(func_def_729, type, vNOT: ($o > $o)).
thf(func_def_730, type, sK591: subst).
thf(func_def_731, type, sK592: (term > $o)).
thf(func_def_732, type, sK593: (term > $o)).
thf(func_def_733, type, sK594: term).
thf(func_def_734, type, sK595: term).
thf(f1,axiom,(
  pushprop_lem1_gthm & induction2_gthm & (((lam @ (d2term @ d_one))) = ((d2term @ d_one))) & (induction2 <=> ! [X21 : term,X3 : (term > $o)] : (~ ! [X6 : term] : ((X3 @ (lam @ X6)) | ~ ! [X8 : term] : (~(X3 @ X8) | (X3 @ (sub @ X6 @ (push @ X8 @ id))))) | (X3 @ X21) | ~ ! [X8 : term,X6 : term] : (~(X3 @ X8) | (X3 @ (ap @ X6 @ X8)) | ~(X3 @ X6)) | ~ ! [X6 : term] : ((X3 @ X6) | ~(var @ X6)))) & ~lamnotap & hoaslamnotvar_lthm & pushprop_lem2v2_gthm & ((! [X11 : term,X8 : term,X10 : term,X6 : term] : ((X10 = X11) | (((ap @ X6 @ X10)) != ((ap @ X8 @ X11)))) => ! [X10 : term,X11 : term,X8 : term,X6 : term] : ((X10 = X11) | (((ap @ (sub @ X6 @ id) @ X10)) != ((ap @ (sub @ X8 @ id) @ X11))))) <=> hoasapinj2_lthm) & ! [X45 : subst] : ? [X44 : d_subst] : (X45 = ((d2subst @ X44))) & hoasapnotvar_lthm & axmap & axclos & hoasinduction_lem1_lthm & ~ulamvar1 & (! [X8 : term,X11 : term,X10 : term,X6 : term] : ((X6 = X8) | (((ap @ (sub @ X6 @ id) @ X10)) != ((ap @ (sub @ X8 @ id) @ X11)))) <=> hoasapinj1) & hoasapnotvar_gthm & (hoasinduction <=> ! [X3 : (subst > term > subst > $o),X19 : term] : (~ ! [X17 : subst,X18 : subst,X7 : subst,X6 : term] : ((X3 @ (comp @ X7 @ X18) @ (sub @ X6 @ X18) @ X17) | ~(X3 @ X7 @ X6 @ (comp @ X18 @ X17))) | ~ ! [X9 : (subst > term > term)] : ((X3 @ id @ (lam @ (X9 @ sh @ one)) @ id) | ~ ! [X6 : term] : (~(X3 @ id @ X6 @ id) | (X3 @ id @ (X9 @ id @ X6) @ id)) | ~ ! [X7 : subst,X6 : term,X17 : subst] : (((sub @ (X9 @ X7 @ X6) @ X17)) = ((X9 @ (comp @ X7 @ X17) @ (sub @ X6 @ X17))))) | ~ ! [X8 : term,X6 : term] : ((X3 @ id @ (ap @ (sub @ X6 @ id) @ X8) @ id) | ~(X3 @ id @ X6 @ id) | ~(X3 @ id @ X8 @ id)) | (X3 @ id @ X19 @ id) | ~ ! [X6 : term,X18 : subst,X7 : subst,X17 : subst] : (~(X3 @ (comp @ X7 @ X18) @ (sub @ X6 @ X18) @ X17) | (X3 @ X7 @ X6 @ (comp @ X18 @ X17))) | ~ ! [X6 : term] : (~(var @ (sub @ X6 @ id)) | (X3 @ id @ X6 @ id)))) & (sh = ((d2subst @ d_id))) & hoaslamnotvar & pushprop_lem3v2 & (! [X3 : (subst > term > subst > $o),X23 : term] : (~ ! [X6 : term,X8 : term] : ((X3 @ id @ (ap @ (sub @ X6 @ id) @ X8) @ id) | ~(X3 @ id @ X6 @ id) | ~(X3 @ id @ X8 @ id)) | (X3 @ id @ X23 @ id) | ~ ! [X9 : (subst > term > term)] : (~ ! [X17 : subst,X6 : term,X7 : subst] : (((sub @ (X9 @ X7 @ X6) @ X17)) = ((X9 @ (comp @ X7 @ X17) @ (sub @ X6 @ X17)))) | (X3 @ id @ (lam @ (X9 @ sh @ one)) @ id) | ~ ! [X6 : term] : ((X3 @ id @ (X9 @ id @ X6) @ id) | ~(X3 @ id @ X6 @ id)))) <=> hoasinduction_no_psi_cond) & (! [X40 : term,X9 : (subst > term > term),X39 : term] : ((((lam @ (X9 @ sh @ one))) != ((ap @ (sub @ X39 @ id) @ X40))) | ~ ! [X7 : subst,X17 : subst,X6 : term] : (((sub @ (X9 @ X7 @ X6) @ X17)) = ((X9 @ (comp @ X7 @ X17) @ (sub @ X6 @ X17))))) <=> hoaslamnotap) & pushprop_lthm_orig & (hoasinduction_lem3aa_lthm <=> ! [X32 : term,X3 : (subst > term > subst > $o)] : (~ ! [X9 : (subst > term > term)] : (~ ! [X6 : term] : ((X3 @ id @ (X9 @ id @ X6) @ id) | ~(X3 @ id @ X6 @ id)) | ~ ! [X6 : term,X17 : subst,X7 : subst] : (((sub @ (X9 @ X7 @ X6) @ X17)) = ((X9 @ (comp @ X7 @ X17) @ (sub @ X6 @ X17)))) | (X3 @ id @ (lam @ (X9 @ sh @ one)) @ id)) | ~ ! [X8 : term] : (~(X3 @ id @ X8 @ id) | (X3 @ id @ (sub @ X32 @ (push @ X8 @ id)) @ id)) | (X3 @ id @ (lam @ (sub @ X32 @ (push @ one @ sh))) @ id))) & pushprop_lem0_gthm & (hoaslaminj_lthm <=> (! [X6 : term,X7 : subst] : (X6 = ((sub @ one @ (push @ X6 @ X7)))) => (! [X6 : term,X7 : subst] : (X7 = ((comp @ sh @ (push @ X6 @ X7)))) => (! [X6 : term,X8 : term] : ((X6 = X8) | (((lam @ X6)) != ((lam @ X8)))) => ! [X36 : subst,X35 : (subst > term > term),X9 : (subst > term > term),X37 : term] : (~ ! [X7 : subst,X6 : term,X17 : subst] : (((sub @ (X35 @ X7 @ X6) @ X17)) = ((X35 @ (comp @ X7 @ X17) @ (sub @ X6 @ X17)))) | (((X9 @ X36 @ X37)) = ((X35 @ X36 @ X37))) | ~ ! [X17 : subst,X7 : subst,X6 : term] : (((sub @ (X9 @ X7 @ X6) @ X17)) = ((X9 @ (comp @ X7 @ X17) @ (sub @ X6 @ X17)))) | (((lam @ (X9 @ sh @ one))) != ((lam @ (X35 @ sh @ one))))))))) & ! [X44 : d_subst] : (X44 = d_id) & (((push @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & ~(hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) & (pushprop_lem1 <=> ! [X7 : subst,X3 : (term > $o),X6 : term,X18 : (term > $o),X8 : term] : ((X18 @ (sub @ X6 @ (push @ X8 @ X7))) | ~(X3 @ X6))) & hoaslaminj_gthm & (hoasapinj1_lthm <=> (! [X6 : term] : (X6 = ((sub @ X6 @ id))) => (! [X10 : term,X11 : term,X8 : term,X6 : term] : ((X6 = X8) | (((ap @ X6 @ X10)) != ((ap @ X8 @ X11)))) => ! [X8 : term,X10 : term,X11 : term,X6 : term] : ((((ap @ (sub @ X6 @ id) @ X10)) != ((ap @ (sub @ X8 @ id) @ X11))) | (X6 = X8))))) & hoasinduction_lem3b_gthm & ! [X0 : d_term] : (X0 = d_one) & ((! [X6 : term] : (X6 = ((sub @ X6 @ id))) => ! [X6 : term] : (X6 = ((sub @ X6 @ id)))) <=> termmset_lthm) & hoasinduction_lem2_gthm & (! [X7 : subst,X6 : term] : (X6 = ((sub @ one @ (push @ X6 @ X7)))) <=> axvarcons) & ((^[X1 : subst, X2 : (subst > term > term)] : ((d2term @ d_one))) = hoaslam) & (((sub @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2term @ d_one))) & axassoc & hoaslamnotap_lthm & (pushprop_lem1v2 <=> ! [X6 : term,X3 : (term > $o),X7 : subst,X4 : (term > $o)] : (~(X3 @ X6) | ~ ! [X5 : term] : (((X4 @ X5)) = ((X3 @ (sub @ X5 @ (push @ X6 @ X7))))) | (X4 @ one))) & (! [X6 : term] : (X6 = ((sub @ X6 @ id))) <=> termmset) & (((comp @ (d2subst @ d_id) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & hoasinduction_lem1 & hoasapnotvar & ((! [X6 : term,X7 : subst] : (X6 = ((sub @ one @ (push @ X6 @ X7)))) => ! [X3 : (term > $o),X7 : subst,X6 : term,X4 : (term > $o)] : ((X4 @ one) | ~ ! [X5 : term] : (((X4 @ X5)) = ((X3 @ (sub @ X5 @ (push @ X6 @ X7))))) | ~(X3 @ X6))) <=> pushprop_lem1v2_lthm) & (! [X11 : term,X8 : term,X10 : term,X6 : term] : ((((ap @ X6 @ X10)) != ((ap @ X8 @ X11))) | (X6 = X8)) <=> apinj1) & (! [X37 : term,X35 : (subst > term > term),X9 : (subst > term > term),X36 : subst] : (~ ! [X7 : subst,X6 : term,X17 : subst] : (((sub @ (X9 @ X7 @ X6) @ X17)) = ((X9 @ (comp @ X7 @ X17) @ (sub @ X6 @ X17)))) | (((lam @ (X9 @ sh @ one))) != ((lam @ (X35 @ sh @ one)))) | (((X9 @ X36 @ X37)) = ((X35 @ X36 @ X37))) | ~ ! [X17 : subst,X7 : subst,X6 : term] : (((sub @ (X35 @ X7 @ X6) @ X17)) = ((X35 @ (comp @ X7 @ X17) @ (sub @ X6 @ X17))))) <=> hoaslaminj) & (hoasinduction_lem3aa <=> ! [X32 : term,X3 : (subst > term > subst > $o)] : ((X3 @ id @ (lam @ (sub @ X32 @ (push @ one @ sh))) @ id) | ~ ! [X8 : term] : ((X3 @ id @ (sub @ X32 @ (push @ X8 @ id)) @ id) | ~(X3 @ id @ X8 @ id)) | ~ ! [X9 : (subst > term > term)] : ((X3 @ id @ (lam @ (X9 @ sh @ one)) @ id) | ~ ! [X17 : subst,X6 : term,X7 : subst] : (((sub @ (X9 @ X7 @ X6) @ X17)) = ((X9 @ (comp @ X7 @ X17) @ (sub @ X6 @ X17)))) | ~ ! [X6 : term] : (~(X3 @ id @ X6 @ id) | (X3 @ id @ (X9 @ id @ X6) @ id))))) & (induction <=> ! [X3 : (term > $o),X12 : term] : (~ ! [X6 : term] : ((X3 @ (lam @ X6)) | ~(X3 @ X6)) | ~ ! [X6 : term] : (~(var @ X6) | (X3 @ X6)) | (X3 @ X12) | ~ ! [X6 : term,X8 : term] : (~(X3 @ X6) | (X3 @ (ap @ X6 @ X8)) | ~(X3 @ X8)))) & (axscons <=> ! [X7 : subst] : (X7 = ((push @ (sub @ one @ X7) @ (comp @ sh @ X7))))) & axapp & induction2lem_gthm & (hoasinduction_p_and_p_prime = (^[X3 : (subst > term > subst > $o), X4 : (term > $o)] : (! [X5 : term] : (((X4 @ X5)) = ((X3 @ id @ X5 @ id)))))) & pushprop_lthm & hoasinduction_lem3a_gthm & pushprop_lem1v2_gthm & hoasinduction_lem2v2_gthm & (! [X10 : term,X6 : term,X8 : term,X11 : term] : ((((ap @ X6 @ X10)) != ((ap @ X8 @ X11))) | (X10 = X11)) <=> apinj2) & (id = ((d2subst @ d_id))) & hoasinduction_lem3_gthm & pushprop_lem3v2_lthm & (hoasinduction_lthm <=> (! [X3 : (term > $o),X21 : term] : ((X3 @ X21) | ~ ! [X8 : term,X6 : term] : ((X3 @ (ap @ X6 @ X8)) | ~(X3 @ X6) | ~(X3 @ X8)) | ~ ! [X6 : term] : ((X3 @ X6) | ~(var @ X6)) | ~ ! [X6 : term] : (~ ! [X8 : term] : (~(X3 @ X8) | (X3 @ (sub @ X6 @ (push @ X8 @ id)))) | (X3 @ (lam @ X6)))) => (! [X3 : (subst > term > subst > $o),X15 : term,X16 : term] : ((X3 @ id @ (ap @ X15 @ X16) @ id) | ~ ! [X8 : term,X6 : term] : (~(X3 @ id @ X6 @ id) | ~(X3 @ id @ X8 @ id) | (X3 @ id @ (ap @ (sub @ X6 @ id) @ X8) @ id)) | ~ ! [X7 : subst,X17 : subst,X18 : subst,X6 : term] : (~(X3 @ X7 @ X6 @ (comp @ X18 @ X17)) | (X3 @ (comp @ X7 @ X18) @ (sub @ X6 @ X18) @ X17)) | ~ ! [X7 : subst,X18 : subst,X17 : subst,X6 : term] : ((X3 @ X7 @ X6 @ (comp @ X18 @ X17)) | ~(X3 @ (comp @ X7 @ X18) @ (sub @ X6 @ X18) @ X17)) | ~(X3 @ id @ X16 @ id) | ~(X3 @ id @ X15 @ id)) => (! [X3 : (subst > term > subst > $o),X20 : term] : (~ ! [X9 : (subst > term > term)] : (~ ! [X6 : term,X7 : subst,X17 : subst] : (((sub @ (X9 @ X7 @ X6) @ X17)) = ((X9 @ (comp @ X7 @ X17) @ (sub @ X6 @ X17)))) | ~ ! [X6 : term] : (~(X3 @ id @ X6 @ id) | (X3 @ id @ (X9 @ id @ X6) @ id)) | (X3 @ id @ (lam @ (X9 @ sh @ one)) @ id)) | ~ ! [X7 : subst,X18 : subst,X6 : term,X17 : subst] : ((X3 @ (comp @ X7 @ X18) @ (sub @ X6 @ X18) @ X17) | ~(X3 @ X7 @ X6 @ (comp @ X18 @ X17))) | ~ ! [X8 : term] : ((X3 @ id @ (sub @ X20 @ (push @ X8 @ id)) @ id) | ~(X3 @ id @ X8 @ id)) | ~ ! [X7 : subst,X18 : subst,X17 : subst,X6 : term] : ((X3 @ X7 @ X6 @ (comp @ X18 @ X17)) | ~(X3 @ (comp @ X7 @ X18) @ (sub @ X6 @ X18) @ X17)) | (X3 @ id @ (lam @ X20) @ id)) => ! [X3 : (subst > term > subst > $o),X19 : term] : (~ ! [X18 : subst,X6 : term,X7 : subst,X17 : subst] : ((X3 @ X7 @ X6 @ (comp @ X18 @ X17)) | ~(X3 @ (comp @ X7 @ X18) @ (sub @ X6 @ X18) @ X17)) | ~ ! [X18 : subst,X6 : term,X7 : subst,X17 : subst] : ((X3 @ (comp @ X7 @ X18) @ (sub @ X6 @ X18) @ X17) | ~(X3 @ X7 @ X6 @ (comp @ X18 @ X17))) | (X3 @ id @ X19 @ id) | ~ ! [X6 : term] : (~(var @ (sub @ X6 @ id)) | (X3 @ id @ X6 @ id)) | ~ ! [X6 : term,X8 : term] : ((X3 @ id @ (ap @ (sub @ X6 @ id) @ X8) @ id) | ~(X3 @ id @ X8 @ id) | ~(X3 @ id @ X6 @ id)) | ~ ! [X9 : (subst > term > term)] : (~ ! [X6 : term] : ((X3 @ id @ (X9 @ id @ X6) @ id) | ~(X3 @ id @ X6 @ id)) | ~ ! [X17 : subst,X7 : subst,X6 : term] : (((sub @ (X9 @ X7 @ X6) @ X17)) = ((X9 @ (comp @ X7 @ X17) @ (sub @ X6 @ X17)))) | (X3 @ id @ (lam @ (X9 @ sh @ one)) @ id))))))) & termmset_gthm & (pushprop_lem1_lthm <=> (! [X6 : term,X7 : subst] : (X6 = ((sub @ one @ (push @ X6 @ X7)))) => (! [X6 : term,X7 : subst] : (X7 = ((comp @ sh @ (push @ X6 @ X7)))) => ! [X6 : term,X3 : (term > $o),X8 : term,X18 : (term > $o),X7 : subst] : ((X18 @ (sub @ X6 @ (push @ X8 @ X7))) | ~(X3 @ X6))))) & substmonoid_gthm & (axidl <=> ! [X7 : subst] : (X7 = ((comp @ id @ X7)))) & apnotvar & hoaslamnotvar_gthm & (axvarid <=> ! [X6 : term] : (X6 = ((sub @ X6 @ id)))) & (((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) = ((d2term @ d_one))) & (! [X6 : term,X11 : term,X8 : term,X10 : term] : ((((ap @ (sub @ X6 @ id) @ X10)) != ((ap @ (sub @ X8 @ id) @ X11))) | (X10 = X11)) <=> hoasapinj2) & (pushprop_p_and_p_prime = (^[X6 : term, X7 : subst, X3 : (term > $o), X4 : (term > $o)] : (! [X5 : term] : (((X4 @ X5)) = ((X3 @ (sub @ X5 @ (push @ X6 @ X7)))))))) & ulamvarsh & (hoasinduction_lem3b <=> ! [X8 : term] : ~ ! [X9 : (subst > term > term)] : (((X9 @ sh @ one)) != ((sub @ X8 @ (push @ one @ sh))))) & hoasapinj1_gthm & (((ap @ (d2term @ d_one) @ (d2term @ d_one))) = ((d2term @ d_one))) & pushprop_gthm & (! [X38 : term,X3 : (subst > term > subst > $o),X4 : (term > $o)] : (~ ! [X8 : term] : (~(X4 @ X8) | (X4 @ (sub @ X38 @ (push @ X8 @ id)))) | ~ ! [X18 : subst,X17 : subst,X7 : subst,X6 : term] : (~(X3 @ (comp @ X7 @ X18) @ (sub @ X6 @ X18) @ X17) | (X3 @ X7 @ X6 @ (comp @ X18 @ X17))) | ~ ! [X5 : term] : (((X4 @ X5)) = ((X3 @ id @ X5 @ id))) | ~ ! [X9 : (subst > term > term)] : ((X3 @ id @ (lam @ (X9 @ sh @ one)) @ id) | ~ ! [X6 : term] : ((X3 @ id @ (X9 @ id @ X6) @ id) | ~(X3 @ id @ X6 @ id)) | ~ ! [X6 : term,X17 : subst,X7 : subst] : (((sub @ (X9 @ X7 @ X6) @ X17)) = ((X9 @ (comp @ X7 @ X17) @ (sub @ X6 @ X17))))) | ~ ! [X17 : subst,X6 : term,X7 : subst,X18 : subst] : ((X3 @ (comp @ X7 @ X18) @ (sub @ X6 @ X18) @ X17) | ~(X3 @ X7 @ X6 @ (comp @ X18 @ X17))) | (X4 @ (lam @ X38))) <=> hoasinduction_lem3v2) & (hoasinduction_lem2 <=> ! [X3 : (subst > term > subst > $o),X15 : term,X16 : term] : (~ ! [X8 : term,X6 : term] : (~(X3 @ id @ X6 @ id) | (X3 @ id @ (ap @ (sub @ X6 @ id) @ X8) @ id) | ~(X3 @ id @ X8 @ id)) | ~(X3 @ id @ X15 @ id) | ~ ! [X7 : subst,X6 : term,X17 : subst,X18 : subst] : ((X3 @ X7 @ X6 @ (comp @ X18 @ X17)) | ~(X3 @ (comp @ X7 @ X18) @ (sub @ X6 @ X18) @ X17)) | ~(X3 @ id @ X16 @ id) | ~ ! [X18 : subst,X6 : term,X17 : subst,X7 : subst] : (~(X3 @ X7 @ X6 @ (comp @ X18 @ X17)) | (X3 @ (comp @ X7 @ X18) @ (sub @ X6 @ X18) @ X17)) | (X3 @ id @ (ap @ X15 @ X16) @ id))) & (pushprop_lem0_lthm <=> ! [X3 : (term > $o),X6 : term,X7 : subst] : ~ ! [X4 : (term > $o)] : ~ ! [X5 : term] : (((X4 @ X5)) = ((X3 @ (sub @ X5 @ (push @ X6 @ X7)))))) & (substmonoid_lthm <=> (! [X7 : subst] : (X7 = ((comp @ id @ X7))) => (! [X7 : subst] : (X7 = ((comp @ X7 @ id))) => (! [X7 : subst] : (X7 = ((comp @ id @ X7))) & ! [X7 : subst] : (X7 = ((comp @ X7 @ id))))))) & (! [X3 : (subst > term > subst > $o)] : ~ ! [X4 : (term > $o)] : ~ ! [X5 : term] : (((X4 @ X5)) = ((X3 @ id @ X5 @ id))) <=> hoasinduction_lem0) & ((! [X6 : term] : (X6 = ((sub @ X6 @ id))) => (! [X3 : (subst > term > subst > $o),X32 : term] : (~ ! [X9 : (subst > term > term)] : ((X3 @ id @ (lam @ (X9 @ sh @ one)) @ id) | ~ ! [X6 : term] : (~(X3 @ id @ X6 @ id) | (X3 @ id @ (X9 @ id @ X6) @ id)) | ~ ! [X7 : subst,X17 : subst,X6 : term] : (((sub @ (X9 @ X7 @ X6) @ X17)) = ((X9 @ (comp @ X7 @ X17) @ (sub @ X6 @ X17))))) | ~ ! [X8 : term] : (~(X3 @ id @ X8 @ id) | (X3 @ id @ (sub @ X32 @ (push @ X8 @ id)) @ id)) | (X3 @ id @ (lam @ (sub @ X32 @ (push @ one @ sh))) @ id)) => ! [X3 : (subst > term > subst > $o),X20 : term] : (~ ! [X8 : term] : ((X3 @ id @ (sub @ X20 @ (push @ X8 @ id)) @ id) | ~(X3 @ id @ X8 @ id)) | (X3 @ id @ (lam @ X20) @ id) | ~ ! [X18 : subst,X6 : term,X17 : subst,X7 : subst] : (~(X3 @ (comp @ X7 @ X18) @ (sub @ X6 @ X18) @ X17) | (X3 @ X7 @ X6 @ (comp @ X18 @ X17))) | ~ ! [X9 : (subst > term > term)] : (~ ! [X17 : subst,X7 : subst,X6 : term] : (((sub @ (X9 @ X7 @ X6) @ X17)) = ((X9 @ (comp @ X7 @ X17) @ (sub @ X6 @ X17)))) | ~ ! [X6 : term] : ((X3 @ id @ (X9 @ id @ X6) @ id) | ~(X3 @ id @ X6 @ id)) | (X3 @ id @ (lam @ (X9 @ sh @ one)) @ id)) | ~ ! [X17 : subst,X6 : term,X18 : subst,X7 : subst] : ((X3 @ (comp @ X7 @ X18) @ (sub @ X6 @ X18) @ X17) | ~(X3 @ X7 @ X6 @ (comp @ X18 @ X17)))))) <=> hoasinduction_lem3_lthm) & (hoasinduction_lthm_3 <=> (! [X3 : (subst > term > subst > $o)] : ~ ! [X4 : (term > $o)] : ~ ! [X5 : term] : (((X4 @ X5)) = ((X3 @ id @ X5 @ id))) => (! [X21 : term,X3 : (term > $o)] : (~ ! [X6 : term] : (~(var @ X6) | (X3 @ X6)) | ~ ! [X6 : term] : ((X3 @ (lam @ X6)) | ~ ! [X8 : term] : (~(X3 @ X8) | (X3 @ (sub @ X6 @ (push @ X8 @ id))))) | ~ ! [X8 : term,X6 : term] : (~(X3 @ X6) | (X3 @ (ap @ X6 @ X8)) | ~(X3 @ X8)) | (X3 @ X21)) => (! [X6 : term] : (X6 = ((sub @ X6 @ id))) => (! [X22 : term,X3 : (subst > term > subst > $o),X4 : (term > $o)] : ((X4 @ (lam @ X22)) | ~ ! [X9 : (subst > term > term)] : (~ ! [X6 : term,X7 : subst,X17 : subst] : (((sub @ (X9 @ X7 @ X6) @ X17)) = ((X9 @ (comp @ X7 @ X17) @ (sub @ X6 @ X17)))) | (X3 @ id @ (lam @ (X9 @ sh @ one)) @ id) | ~ ! [X6 : term] : ((X3 @ id @ (X9 @ id @ X6) @ id) | ~(X3 @ id @ X6 @ id))) | ~ ! [X8 : term] : ((X4 @ (sub @ X22 @ (push @ X8 @ id))) | ~(X4 @ X8)) | ~ ! [X5 : term] : (((X4 @ X5)) = ((X3 @ id @ X5 @ id)))) => ! [X3 : (subst > term > subst > $o),X19 : term] : (~ ! [X17 : subst,X18 : subst,X7 : subst,X6 : term] : ((X3 @ (comp @ X7 @ X18) @ (sub @ X6 @ X18) @ X17) | ~(X3 @ X7 @ X6 @ (comp @ X18 @ X17))) | ~ ! [X9 : (subst > term > term)] : ((X3 @ id @ (lam @ (X9 @ sh @ one)) @ id) | ~ ! [X6 : term] : (~(X3 @ id @ X6 @ id) | (X3 @ id @ (X9 @ id @ X6) @ id)) | ~ ! [X17 : subst,X6 : term,X7 : subst] : (((sub @ (X9 @ X7 @ X6) @ X17)) = ((X9 @ (comp @ X7 @ X17) @ (sub @ X6 @ X17))))) | (X3 @ id @ X19 @ id) | ~ ! [X6 : term,X17 : subst,X7 : subst,X18 : subst] : ((X3 @ X7 @ X6 @ (comp @ X18 @ X17)) | ~(X3 @ (comp @ X7 @ X18) @ (sub @ X6 @ X18) @ X17)) | ~ ! [X6 : term] : (~(var @ (sub @ X6 @ id)) | (X3 @ id @ X6 @ id)) | ~ ! [X8 : term,X6 : term] : ((X3 @ id @ (ap @ (sub @ X6 @ id) @ X8) @ id) | ~(X3 @ id @ X8 @ id) | ~(X3 @ id @ X6 @ id)))))))) & ulamvarind & (substmonoid <=> (! [X7 : subst] : (X7 = ((comp @ X7 @ id))) & ! [X7 : subst] : (X7 = ((comp @ id @ X7))))) & axabs & hoaslamnotap_gthm & ! [X46 : d_term,X47 : d_term] : ((((d2term @ X46)) = ((d2term @ X47))) => (X46 = X47)) & (! [X7 : subst,X6 : term] : (X7 = ((comp @ sh @ (push @ X6 @ X7)))) <=> axshiftcons) & ((! [X8 : term] : ~ ! [X9 : (subst > term > term)] : ~ ! [X6 : term,X7 : subst] : (((sub @ X8 @ (push @ X6 @ X7))) = ((X9 @ X7 @ X6))) => (! [X6 : term] : (X6 = ((sub @ X6 @ id))) => ! [X22 : term,X3 : (subst > term > subst > $o),X4 : (term > $o)] : (~ ! [X8 : term] : (~(X4 @ X8) | (X4 @ (sub @ X22 @ (push @ X8 @ id)))) | ~ ! [X9 : (subst > term > term)] : (~ ! [X6 : term] : ((X3 @ id @ (X9 @ id @ X6) @ id) | ~(X3 @ id @ X6 @ id)) | ~ ! [X7 : subst,X6 : term,X17 : subst] : (((sub @ (X9 @ X7 @ X6) @ X17)) = ((X9 @ (comp @ X7 @ X17) @ (sub @ X6 @ X17)))) | (X3 @ id @ (lam @ (X9 @ sh @ one)) @ id)) | (X4 @ (lam @ X22)) | ~ ! [X5 : term] : (((X4 @ X5)) = ((X3 @ id @ X5 @ id)))))) <=> hoasinduction_lem3v2a_lthm) & ((! [X6 : term] : (X6 = ((sub @ X6 @ id))) => ! [X4 : (term > $o),X38 : term,X3 : (subst > term > subst > $o)] : (~ ! [X5 : term] : (((X4 @ X5)) = ((X3 @ id @ X5 @ id))) | ~ ! [X7 : subst,X17 : subst,X6 : term,X18 : subst] : (~(X3 @ (comp @ X7 @ X18) @ (sub @ X6 @ X18) @ X17) | (X3 @ X7 @ X6 @ (comp @ X18 @ X17))) | ~ ! [X17 : subst,X18 : subst,X6 : term,X7 : subst] : ((X3 @ (comp @ X7 @ X18) @ (sub @ X6 @ X18) @ X17) | ~(X3 @ X7 @ X6 @ (comp @ X18 @ X17))) | (X4 @ (lam @ X38)) | ~ ! [X9 : (subst > term > term)] : (~ ! [X6 : term] : (~(X3 @ id @ X6 @ id) | (X3 @ id @ (X9 @ id @ X6) @ id)) | ~ ! [X7 : subst,X17 : subst,X6 : term] : (((sub @ (X9 @ X7 @ X6) @ X17)) = ((X9 @ (comp @ X7 @ X17) @ (sub @ X6 @ X17)))) | (X3 @ id @ (lam @ (X9 @ sh @ one)) @ id)) | ~ ! [X8 : term] : ((X4 @ (sub @ X38 @ (push @ X8 @ id))) | ~(X4 @ X8)))) <=> hoasinduction_lem3v2_lthm) & hoasinduction_lem3v2_gthm & (hoasinduction_lem3a_lthm <=> (! [X6 : term] : (X6 = ((sub @ X6 @ id))) => (! [X3 : (subst > term > subst > $o),X32 : term] : ((X3 @ id @ (lam @ (sub @ X32 @ (push @ one @ sh))) @ id) | ~ ! [X8 : term] : ((X3 @ id @ (sub @ X32 @ (push @ X8 @ id)) @ id) | ~(X3 @ id @ X8 @ id)) | ~ ! [X9 : (subst > term > term)] : ((X3 @ id @ (lam @ (X9 @ sh @ one)) @ id) | ~ ! [X17 : subst,X7 : subst,X6 : term] : (((sub @ (X9 @ X7 @ X6) @ X17)) = ((X9 @ (comp @ X7 @ X17) @ (sub @ X6 @ X17)))) | ~ ! [X6 : term] : ((X3 @ id @ (X9 @ id @ X6) @ id) | ~(X3 @ id @ X6 @ id)))) => ! [X3 : (subst > term > subst > $o),X41 : term] : (~ ! [X8 : term] : ((X3 @ id @ (sub @ X41 @ (push @ X8 @ id)) @ id) | ~(X3 @ id @ X8 @ id)) | (X3 @ id @ (lam @ X41) @ id) | ~ ! [X9 : (subst > term > term)] : (~ ! [X6 : term] : ((X3 @ id @ (X9 @ id @ X6) @ id) | ~(X3 @ id @ X6 @ id)) | ~ ! [X17 : subst,X6 : term,X7 : subst] : (((sub @ (X9 @ X7 @ X6) @ X17)) = ((X9 @ (comp @ X7 @ X17) @ (sub @ X6 @ X17)))) | (X3 @ id @ (lam @ (X9 @ sh @ one)) @ id)))))) & axvarshift & hoasinduction_lem1_gthm & (induction2_lthm <=> (! [X6 : term] : (X6 = ((sub @ X6 @ id))) => (! [X13 : term,X3 : (term > $o),X14 : subst] : (~ ! [X6 : term,X8 : term] : (~(X3 @ X8) | ~(X3 @ X6) | (X3 @ (ap @ X6 @ X8))) | (X3 @ (sub @ X13 @ X14)) | ~ ! [X8 : term] : (~(var @ X8) | (X3 @ (sub @ X8 @ X14))) | ~ ! [X6 : term] : ((X3 @ (lam @ X6)) | ~ ! [X8 : term] : ((X3 @ (sub @ X6 @ (push @ X8 @ id))) | ~(X3 @ X8)))) => ! [X3 : (term > $o),X21 : term] : (~ ! [X6 : term] : (~(var @ X6) | (X3 @ X6)) | ~ ! [X6 : term,X8 : term] : (~(X3 @ X6) | ~(X3 @ X8) | (X3 @ (ap @ X6 @ X8))) | (X3 @ X21) | ~ ! [X6 : term] : (~ ! [X8 : term] : ((X3 @ (sub @ X6 @ (push @ X8 @ id))) | ~(X3 @ X8)) | (X3 @ (lam @ X6))))))) & (! [X8 : term,X6 : term] : ((((lam @ X6)) != ((lam @ X8))) | (X6 = X8)) <=> laminj) & (! [X13 : term,X3 : (term > $o),X14 : subst] : (~ ! [X8 : term] : (~(var @ X8) | (X3 @ (sub @ X8 @ X14))) | ~ ! [X6 : term] : ((X3 @ (lam @ X6)) | ~ ! [X8 : term] : (~(X3 @ X8) | (X3 @ (sub @ X6 @ (push @ X8 @ id))))) | (X3 @ (sub @ X13 @ X14)) | ~ ! [X8 : term,X6 : term] : ((X3 @ (ap @ X6 @ X8)) | ~(X3 @ X6) | ~(X3 @ X8))) <=> induction2lem) & hoasapinj2_gthm & (! [X3 : (subst > term > subst > $o),X41 : term] : (~ ! [X9 : (subst > term > term)] : (~ ! [X7 : subst,X6 : term,X17 : subst] : (((sub @ (X9 @ X7 @ X6) @ X17)) = ((X9 @ (comp @ X7 @ X17) @ (sub @ X6 @ X17)))) | ~ ! [X6 : term] : ((X3 @ id @ (X9 @ id @ X6) @ id) | ~(X3 @ id @ X6 @ id)) | (X3 @ id @ (lam @ (X9 @ sh @ one)) @ id)) | ~ ! [X8 : term] : ((X3 @ id @ (sub @ X41 @ (push @ X8 @ id)) @ id) | ~(X3 @ id @ X8 @ id)) | (X3 @ id @ (lam @ X41) @ id)) <=> hoasinduction_lem3a) & hoasinduction_lem2_lthm & ~(var @ (d2term @ d_one)) & pushprop_lem2v2_lthm & pushprop_lem2v2 & ! [X48 : term] : ? [X0 : d_term] : (X48 = ((d2term @ X0))) & (pushprop_lem0 <=> ! [X7 : subst,X6 : term,X3 : (term > $o)] : ~ ! [X4 : (term > $o)] : ~ ! [X5 : term] : (((X4 @ X5)) = ((X3 @ (sub @ X5 @ (push @ X6 @ X7)))))) & (! [X8 : term] : ~ ! [X9 : (subst > term > term)] : (((X9 @ sh @ one)) != ((sub @ X8 @ (push @ one @ sh)))) <=> hoasinduction_lem3b_lthm) & (! [X6 : term,X8 : term] : ((X6 = X8) | (((sub @ X6 @ sh)) != ((sub @ X8 @ sh)))) <=> shinj) & (hoasinduction_lem0_lthm <=> ! [X3 : (subst > term > subst > $o)] : ~ ! [X4 : (term > $o)] : ~ ! [X5 : term] : (((X4 @ X5)) = ((X3 @ id @ X5 @ id)))) & (one = ((d2term @ d_one))) & (hoasinduction_lem3 <=> ! [X20 : term,X3 : (subst > term > subst > $o)] : ((X3 @ id @ (lam @ X20) @ id) | ~ ! [X7 : subst,X18 : subst,X17 : subst,X6 : term] : ((X3 @ X7 @ X6 @ (comp @ X18 @ X17)) | ~(X3 @ (comp @ X7 @ X18) @ (sub @ X6 @ X18) @ X17)) | ~ ! [X7 : subst,X6 : term,X17 : subst,X18 : subst] : ((X3 @ (comp @ X7 @ X18) @ (sub @ X6 @ X18) @ X17) | ~(X3 @ X7 @ X6 @ (comp @ X18 @ X17))) | ~ ! [X9 : (subst > term > term)] : ((X3 @ id @ (lam @ (X9 @ sh @ one)) @ id) | ~ ! [X6 : term,X17 : subst,X7 : subst] : (((sub @ (X9 @ X7 @ X6) @ X17)) = ((X9 @ (comp @ X7 @ X17) @ (sub @ X6 @ X17)))) | ~ ! [X6 : term] : (~(X3 @ id @ X6 @ id) | (X3 @ id @ (X9 @ id @ X6) @ id))) | ~ ! [X8 : term] : ((X3 @ id @ (sub @ X20 @ (push @ X8 @ id)) @ id) | ~(X3 @ id @ X8 @ id)))) & (! [X3 : (subst > term > subst > $o),X33 : term,X4 : (term > $o),X34 : term] : (~ ! [X18 : subst,X17 : subst,X6 : term,X7 : subst] : (~(X3 @ X7 @ X6 @ (comp @ X18 @ X17)) | (X3 @ (comp @ X7 @ X18) @ (sub @ X6 @ X18) @ X17)) | ~(X4 @ X33) | ~(X4 @ X34) | ~ ! [X5 : term] : (((X4 @ X5)) = ((X3 @ id @ X5 @ id))) | ~ ! [X8 : term,X6 : term] : (~(X3 @ id @ X6 @ id) | ~(X3 @ id @ X8 @ id) | (X3 @ id @ (ap @ (sub @ X6 @ id) @ X8) @ id)) | ~ ! [X17 : subst,X18 : subst,X6 : term,X7 : subst] : (~(X3 @ (comp @ X7 @ X18) @ (sub @ X6 @ X18) @ X17) | (X3 @ X7 @ X6 @ (comp @ X18 @ X17))) | (X4 @ (ap @ X33 @ X34))) <=> hoasinduction_lem2v2) & (! [X3 : (subst > term > subst > $o),X22 : term,X4 : (term > $o)] : (~ ! [X8 : term] : (~(X4 @ X8) | (X4 @ (sub @ X22 @ (push @ X8 @ id)))) | (X4 @ (lam @ X22)) | ~ ! [X9 : (subst > term > term)] : (~ ! [X17 : subst,X7 : subst,X6 : term] : (((sub @ (X9 @ X7 @ X6) @ X17)) = ((X9 @ (comp @ X7 @ X17) @ (sub @ X6 @ X17)))) | ~ ! [X6 : term] : ((X3 @ id @ (X9 @ id @ X6) @ id) | ~(X3 @ id @ X6 @ id)) | (X3 @ id @ (lam @ (X9 @ sh @ one)) @ id)) | ~ ! [X5 : term] : (((X4 @ X5)) = ((X3 @ id @ X5 @ id)))) <=> hoasinduction_lem3v2a) & hoasinduction_gthm & pushprop & hoasinduction_lem1v2_gthm & (hoasinduction_no_psi_cond_lthm <=> (! [X3 : (subst > term > subst > $o)] : ~ ! [X4 : (term > $o)] : ~ ! [X5 : term] : (((X4 @ X5)) = ((X3 @ id @ X5 @ id))) => (! [X3 : (term > $o),X21 : term] : (~ ! [X6 : term] : ((X3 @ (lam @ X6)) | ~ ! [X8 : term] : (~(X3 @ X8) | (X3 @ (sub @ X6 @ (push @ X8 @ id))))) | ~ ! [X8 : term,X6 : term] : ((X3 @ (ap @ X6 @ X8)) | ~(X3 @ X8) | ~(X3 @ X6)) | ~ ! [X6 : term] : (~(var @ X6) | (X3 @ X6)) | (X3 @ X21)) => (! [X6 : term] : (X6 = ((sub @ X6 @ id))) => (! [X3 : (subst > term > subst > $o),X4 : (term > $o),X22 : term] : (~ ! [X8 : term] : ((X4 @ (sub @ X22 @ (push @ X8 @ id))) | ~(X4 @ X8)) | ~ ! [X5 : term] : (((X4 @ X5)) = ((X3 @ id @ X5 @ id))) | ~ ! [X9 : (subst > term > term)] : (~ ! [X17 : subst,X7 : subst,X6 : term] : (((sub @ (X9 @ X7 @ X6) @ X17)) = ((X9 @ (comp @ X7 @ X17) @ (sub @ X6 @ X17)))) | (X3 @ id @ (lam @ (X9 @ sh @ one)) @ id) | ~ ! [X6 : term] : (~(X3 @ id @ X6 @ id) | (X3 @ id @ (X9 @ id @ X6) @ id))) | (X4 @ (lam @ X22))) => ! [X23 : term,X3 : (subst > term > subst > $o)] : (~ ! [X6 : term,X8 : term] : (~(X3 @ id @ X6 @ id) | ~(X3 @ id @ X8 @ id) | (X3 @ id @ (ap @ (sub @ X6 @ id) @ X8) @ id)) | (X3 @ id @ X23 @ id) | ~ ! [X9 : (subst > term > term)] : (~ ! [X6 : term] : (~(X3 @ id @ X6 @ id) | (X3 @ id @ (X9 @ id @ X6) @ id)) | (X3 @ id @ (lam @ (X9 @ sh @ one)) @ id) | ~ ! [X17 : subst,X6 : term,X7 : subst] : (((sub @ (X9 @ X7 @ X6) @ X17)) = ((X9 @ (comp @ X7 @ X17) @ (sub @ X6 @ X17))))))))))) & hoasinduction_lem1v2 & (hoasinduction_lem3v2_f_lthm <=> ! [X8 : term] : ~ ! [X9 : (subst > term > term)] : ~ ! [X6 : term,X7 : subst] : (((sub @ X8 @ (push @ X6 @ X7))) = ((X9 @ X7 @ X6)))) & (hoasinduction_lem3aaa <=> ! [X3 : (subst > term > subst > $o),X24 : term] : (~ ! [X9 : (subst > term > term),X25 : term] : (~ ! [X31 : subst,X29 : subst,X30 : term] : (((sub @ (X9 @ X29 @ X30) @ X31)) = ((sub @ (sub @ X25 @ (push @ X30 @ X29)) @ X31))) | ~ ! [X26 : subst,X28 : subst,X27 : term] : (((X9 @ (comp @ X26 @ X28) @ (sub @ X27 @ X28))) = ((sub @ X25 @ (push @ (sub @ X27 @ X28) @ (comp @ X26 @ X28))))) | (X3 @ id @ (lam @ (X9 @ sh @ one)) @ id) | ~ ! [X6 : term] : ((X3 @ id @ (X9 @ id @ X6) @ id) | ~(X3 @ id @ X6 @ id))) | (X3 @ id @ (lam @ (sub @ X24 @ (push @ one @ sh))) @ id) | ~ ! [X8 : term] : (~(X3 @ id @ X8 @ id) | (X3 @ id @ (sub @ X24 @ (push @ X8 @ id)) @ id)))) & ! [X42 : d_subst,X43 : d_subst] : ((((d2subst @ X42)) = ((d2subst @ X43))) => (X42 = X43)) & (hoasinduction_lem3v2_f <=> ! [X8 : term] : ~ ! [X9 : (subst > term > term)] : ~ ! [X7 : subst,X6 : term] : (((sub @ X8 @ (push @ X6 @ X7))) = ((X9 @ X7 @ X6)))) & (axidr <=> ! [X7 : subst] : (X7 = ((comp @ X7 @ id)))) & (induction2lem_lthm <=> (! [X7 : subst,X6 : term] : (X6 = ((sub @ one @ (push @ X6 @ X7)))) => (! [X7 : subst,X6 : term] : (X7 = ((comp @ sh @ (push @ X6 @ X7)))) => (! [X7 : subst] : (X7 = ((comp @ X7 @ id))) => (! [X12 : term,X3 : (term > $o)] : (~ ! [X6 : term] : (~(X3 @ X6) | (X3 @ (lam @ X6))) | (X3 @ X12) | ~ ! [X6 : term] : ((X3 @ X6) | ~(var @ X6)) | ~ ! [X8 : term,X6 : term] : (~(X3 @ X8) | (X3 @ (ap @ X6 @ X8)) | ~(X3 @ X6))) => ! [X14 : subst,X13 : term,X3 : (term > $o)] : ((X3 @ (sub @ X13 @ X14)) | ~ ! [X6 : term] : ((X3 @ (lam @ X6)) | ~ ! [X8 : term] : (~(X3 @ X8) | (X3 @ (sub @ X6 @ (push @ X8 @ id))))) | ~ ! [X6 : term,X8 : term] : (~(X3 @ X8) | (X3 @ (ap @ X6 @ X8)) | ~(X3 @ X6)) | ~ ! [X8 : term] : ((X3 @ (sub @ X8 @ X14)) | ~(var @ X8)))))))) & lamnotvar),
  file('/export/starexec/sandbox2/benchmark/theBenchmark.p',alg444_1)).
thf(f2,conjecture,(
  (! [X2 : term,X0 : (term > $o),X1 : (term > $o),X3 : subst] : ((pushprop_p_and_p_prime @ X2 @ X3 @ X0 @ X1) => (! [X4 : term] : ((var @ X4) => (X1 @ X4)) => ! [X4 : term] : ((var @ X4) => (X0 @ (sub @ X4 @ (push @ X2 @ X3)))))) = pushprop_lem3v2)),
  file('/export/starexec/sandbox2/benchmark/theBenchmark.p',pushprop_lem3v2)).
thf(f3,negated_conjecture,(
  ~ (! [X2 : term,X0 : (term > $o),X1 : (term > $o),X3 : subst] : ((pushprop_p_and_p_prime @ X2 @ X3 @ X0 @ X1) => (! [X4 : term] : ((var @ X4) => (X1 @ X4)) => ! [X4 : term] : ((var @ X4) => (X0 @ (sub @ X4 @ (push @ X2 @ X3)))))) = pushprop_lem3v2)),
  inference(negated_conjecture,[status(cth)],[f2])).
thf(f4,plain,(
  pushprop_lem1_gthm & induction2_gthm & (((lam @ (d2term @ d_one))) = ((d2term @ d_one))) & (induction2 <=> ! [X0 : term,X1 : (term > $o)] : (~ ! [X2 : term] : ((X1 @ (lam @ X2)) | ~ ! [X3 : term] : (~(X1 @ X3) | (X1 @ (sub @ X2 @ (push @ X3 @ id))))) | (X1 @ X0) | ~ ! [X4 : term,X5 : term] : (~(X1 @ X4) | (X1 @ (ap @ X5 @ X4)) | ~(X1 @ X5)) | ~ ! [X6 : term] : ((X1 @ X6) | ~(var @ X6)))) & ~lamnotap & hoaslamnotvar_lthm & pushprop_lem2v2_gthm & ((! [X7 : term,X8 : term,X9 : term,X10 : term] : ((X7 = X9) | (((ap @ X8 @ X7)) != ((ap @ X10 @ X9)))) => ! [X11 : term,X12 : term,X13 : term,X14 : term] : ((X11 = X12) | (((ap @ (sub @ X13 @ id) @ X12)) != ((ap @ (sub @ X14 @ id) @ X11))))) <=> hoasapinj2_lthm) & ! [X15 : subst] : ? [X16 : d_subst] : (((d2subst @ X16)) = X15) & hoasapnotvar_lthm & axmap & axclos & hoasinduction_lem1_lthm & ~ulamvar1 & (! [X17 : term,X18 : term,X19 : term,X20 : term] : ((X17 = X20) | (((ap @ (sub @ X20 @ id) @ X19)) != ((ap @ (sub @ X17 @ id) @ X18)))) <=> hoasapinj1) & hoasapnotvar_gthm & (hoasinduction <=> ! [X21 : (subst > term > subst > $o),X22 : term] : (~ ! [X23 : subst,X24 : subst,X25 : subst,X26 : term] : ((X21 @ (comp @ X25 @ X24) @ (sub @ X26 @ X24) @ X23) | ~(X21 @ X25 @ X26 @ (comp @ X24 @ X23))) | ~ ! [X27 : (subst > term > term)] : ((X21 @ id @ (lam @ (X27 @ sh @ one)) @ id) | ~ ! [X28 : term] : (~(X21 @ id @ X28 @ id) | (X21 @ id @ (X27 @ id @ X28) @ id)) | ~ ! [X29 : subst,X30 : term,X31 : subst] : (((X27 @ (comp @ X29 @ X31) @ (sub @ X30 @ X31))) = ((sub @ (X27 @ X29 @ X30) @ X31)))) | ~ ! [X32 : term,X33 : term] : ((X21 @ id @ (ap @ (sub @ X33 @ id) @ X32) @ id) | ~(X21 @ id @ X33 @ id) | ~(X21 @ id @ X32 @ id)) | (X21 @ id @ X22 @ id) | ~ ! [X34 : term,X35 : subst,X36 : subst,X37 : subst] : (~(X21 @ (comp @ X36 @ X35) @ (sub @ X34 @ X35) @ X37) | (X21 @ X36 @ X34 @ (comp @ X35 @ X37))) | ~ ! [X38 : term] : (~(var @ (sub @ X38 @ id)) | (X21 @ id @ X38 @ id)))) & (sh = ((d2subst @ d_id))) & hoaslamnotvar & pushprop_lem3v2 & (! [X39 : (subst > term > subst > $o),X40 : term] : (~ ! [X41 : term,X42 : term] : ((X39 @ id @ (ap @ (sub @ X41 @ id) @ X42) @ id) | ~(X39 @ id @ X41 @ id) | ~(X39 @ id @ X42 @ id)) | (X39 @ id @ X40 @ id) | ~ ! [X43 : (subst > term > term)] : (~ ! [X44 : subst,X45 : term,X46 : subst] : (((sub @ (X43 @ X46 @ X45) @ X44)) = ((X43 @ (comp @ X46 @ X44) @ (sub @ X45 @ X44)))) | (X39 @ id @ (lam @ (X43 @ sh @ one)) @ id) | ~ ! [X47 : term] : ((X39 @ id @ (X43 @ id @ X47) @ id) | ~(X39 @ id @ X47 @ id)))) <=> hoasinduction_no_psi_cond) & (! [X48 : term,X49 : (subst > term > term),X50 : term] : ((((ap @ (sub @ X50 @ id) @ X48)) != ((lam @ (X49 @ sh @ one)))) | ~ ! [X51 : subst,X52 : subst,X53 : term] : (((X49 @ (comp @ X51 @ X52) @ (sub @ X53 @ X52))) = ((sub @ (X49 @ X51 @ X53) @ X52)))) <=> hoaslamnotap) & pushprop_lthm_orig & (hoasinduction_lem3aa_lthm <=> ! [X54 : term,X55 : (subst > term > subst > $o)] : (~ ! [X56 : (subst > term > term)] : (~ ! [X57 : term] : ((X55 @ id @ (X56 @ id @ X57) @ id) | ~(X55 @ id @ X57 @ id)) | ~ ! [X58 : term,X59 : subst,X60 : subst] : (((X56 @ (comp @ X60 @ X59) @ (sub @ X58 @ X59))) = ((sub @ (X56 @ X60 @ X58) @ X59))) | (X55 @ id @ (lam @ (X56 @ sh @ one)) @ id)) | ~ ! [X61 : term] : (~(X55 @ id @ X61 @ id) | (X55 @ id @ (sub @ X54 @ (push @ X61 @ id)) @ id)) | (X55 @ id @ (lam @ (sub @ X54 @ (push @ one @ sh))) @ id))) & pushprop_lem0_gthm & (hoaslaminj_lthm <=> (! [X62 : term,X63 : subst] : (((sub @ one @ (push @ X62 @ X63))) = X62) => (! [X64 : term,X65 : subst] : (((comp @ sh @ (push @ X64 @ X65))) = X65) => (! [X66 : term,X67 : term] : ((X66 = X67) | (((lam @ X66)) != ((lam @ X67)))) => ! [X68 : subst,X69 : (subst > term > term),X70 : (subst > term > term),X71 : term] : (~ ! [X72 : subst,X73 : term,X74 : subst] : (((sub @ (X69 @ X72 @ X73) @ X74)) = ((X69 @ (comp @ X72 @ X74) @ (sub @ X73 @ X74)))) | (((X69 @ X68 @ X71)) = ((X70 @ X68 @ X71))) | ~ ! [X75 : subst,X76 : subst,X77 : term] : (((sub @ (X70 @ X76 @ X77) @ X75)) = ((X70 @ (comp @ X76 @ X75) @ (sub @ X77 @ X75)))) | (((lam @ (X70 @ sh @ one))) != ((lam @ (X69 @ sh @ one))))))))) & ! [X78 : d_subst] : (d_id = X78) & (((push @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & ~(hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) & (pushprop_lem1 <=> ! [X79 : subst,X80 : (term > $o),X81 : term,X82 : (term > $o),X83 : term] : ((X82 @ (sub @ X81 @ (push @ X83 @ X79))) | ~(X80 @ X81))) & hoaslaminj_gthm & (hoasapinj1_lthm <=> (! [X84 : term] : (((sub @ X84 @ id)) = X84) => (! [X85 : term,X86 : term,X87 : term,X88 : term] : ((X87 = X88) | (((ap @ X88 @ X85)) != ((ap @ X87 @ X86)))) => ! [X89 : term,X90 : term,X91 : term,X92 : term] : ((((ap @ (sub @ X89 @ id) @ X91)) != ((ap @ (sub @ X92 @ id) @ X90))) | (X89 = X92))))) & hoasinduction_lem3b_gthm & ! [X93 : d_term] : (d_one = X93) & ((! [X94 : term] : (((sub @ X94 @ id)) = X94) => ! [X95 : term] : (((sub @ X95 @ id)) = X95)) <=> termmset_lthm) & hoasinduction_lem2_gthm & (! [X96 : subst,X97 : term] : (((sub @ one @ (push @ X97 @ X96))) = X97) <=> axvarcons) & ((^[X98 : subst, X99 : (subst > term > term)] : ((d2term @ d_one))) = hoaslam) & (((sub @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2term @ d_one))) & axassoc & hoaslamnotap_lthm & (pushprop_lem1v2 <=> ! [X100 : term,X101 : (term > $o),X102 : subst,X103 : (term > $o)] : (~(X101 @ X100) | ~ ! [X104 : term] : (((X103 @ X104)) = ((X101 @ (sub @ X104 @ (push @ X100 @ X102))))) | (X103 @ one))) & (! [X105 : term] : (((sub @ X105 @ id)) = X105) <=> termmset) & (((comp @ (d2subst @ d_id) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & hoasinduction_lem1 & hoasapnotvar & ((! [X106 : term,X107 : subst] : (((sub @ one @ (push @ X106 @ X107))) = X106) => ! [X108 : (term > $o),X109 : subst,X110 : term,X111 : (term > $o)] : ((X111 @ one) | ~ ! [X112 : term] : (((X108 @ (sub @ X112 @ (push @ X110 @ X109)))) = ((X111 @ X112))) | ~(X108 @ X110))) <=> pushprop_lem1v2_lthm) & (! [X113 : term,X114 : term,X115 : term,X116 : term] : ((((ap @ X116 @ X115)) != ((ap @ X114 @ X113))) | (X114 = X116)) <=> apinj1) & (! [X117 : term,X118 : (subst > term > term),X119 : (subst > term > term),X120 : subst] : (~ ! [X121 : subst,X122 : term,X123 : subst] : (((X119 @ (comp @ X121 @ X123) @ (sub @ X122 @ X123))) = ((sub @ (X119 @ X121 @ X122) @ X123))) | (((lam @ (X119 @ sh @ one))) != ((lam @ (X118 @ sh @ one)))) | (((X118 @ X120 @ X117)) = ((X119 @ X120 @ X117))) | ~ ! [X124 : subst,X125 : subst,X126 : term] : (((sub @ (X118 @ X125 @ X126) @ X124)) = ((X118 @ (comp @ X125 @ X124) @ (sub @ X126 @ X124))))) <=> hoaslaminj) & (hoasinduction_lem3aa <=> ! [X127 : term,X128 : (subst > term > subst > $o)] : ((X128 @ id @ (lam @ (sub @ X127 @ (push @ one @ sh))) @ id) | ~ ! [X129 : term] : ((X128 @ id @ (sub @ X127 @ (push @ X129 @ id)) @ id) | ~(X128 @ id @ X129 @ id)) | ~ ! [X130 : (subst > term > term)] : ((X128 @ id @ (lam @ (X130 @ sh @ one)) @ id) | ~ ! [X131 : subst,X132 : term,X133 : subst] : (((X130 @ (comp @ X133 @ X131) @ (sub @ X132 @ X131))) = ((sub @ (X130 @ X133 @ X132) @ X131))) | ~ ! [X134 : term] : (~(X128 @ id @ X134 @ id) | (X128 @ id @ (X130 @ id @ X134) @ id))))) & (induction <=> ! [X135 : (term > $o),X136 : term] : (~ ! [X137 : term] : ((X135 @ (lam @ X137)) | ~(X135 @ X137)) | ~ ! [X138 : term] : (~(var @ X138) | (X135 @ X138)) | (X135 @ X136) | ~ ! [X139 : term,X140 : term] : (~(X135 @ X139) | (X135 @ (ap @ X139 @ X140)) | ~(X135 @ X140)))) & (axscons <=> ! [X141 : subst] : (((push @ (sub @ one @ X141) @ (comp @ sh @ X141))) = X141)) & axapp & induction2lem_gthm & (hoasinduction_p_and_p_prime = (^[X142 : (subst > term > subst > $o), X143 : (term > $o)] : (! [X144 : term] : (((X142 @ id @ X144 @ id)) = ((X143 @ X144)))))) & pushprop_lthm & hoasinduction_lem3a_gthm & pushprop_lem1v2_gthm & hoasinduction_lem2v2_gthm & (! [X145 : term,X146 : term,X147 : term,X148 : term] : ((((ap @ X146 @ X145)) != ((ap @ X147 @ X148))) | (X145 = X148)) <=> apinj2) & (id = ((d2subst @ d_id))) & hoasinduction_lem3_gthm & pushprop_lem3v2_lthm & (hoasinduction_lthm <=> (! [X149 : (term > $o),X150 : term] : ((X149 @ X150) | ~ ! [X151 : term,X152 : term] : ((X149 @ (ap @ X152 @ X151)) | ~(X149 @ X152) | ~(X149 @ X151)) | ~ ! [X153 : term] : ((X149 @ X153) | ~(var @ X153)) | ~ ! [X154 : term] : (~ ! [X155 : term] : (~(X149 @ X155) | (X149 @ (sub @ X154 @ (push @ X155 @ id)))) | (X149 @ (lam @ X154)))) => (! [X156 : (subst > term > subst > $o),X157 : term,X158 : term] : ((X156 @ id @ (ap @ X157 @ X158) @ id) | ~ ! [X159 : term,X160 : term] : (~(X156 @ id @ X160 @ id) | ~(X156 @ id @ X159 @ id) | (X156 @ id @ (ap @ (sub @ X160 @ id) @ X159) @ id)) | ~ ! [X161 : subst,X162 : subst,X163 : subst,X164 : term] : (~(X156 @ X161 @ X164 @ (comp @ X163 @ X162)) | (X156 @ (comp @ X161 @ X163) @ (sub @ X164 @ X163) @ X162)) | ~ ! [X165 : subst,X166 : subst,X167 : subst,X168 : term] : ((X156 @ X165 @ X168 @ (comp @ X166 @ X167)) | ~(X156 @ (comp @ X165 @ X166) @ (sub @ X168 @ X166) @ X167)) | ~(X156 @ id @ X158 @ id) | ~(X156 @ id @ X157 @ id)) => (! [X169 : (subst > term > subst > $o),X170 : term] : (~ ! [X171 : (subst > term > term)] : (~ ! [X172 : term,X173 : subst,X174 : subst] : (((sub @ (X171 @ X173 @ X172) @ X174)) = ((X171 @ (comp @ X173 @ X174) @ (sub @ X172 @ X174)))) | ~ ! [X175 : term] : (~(X169 @ id @ X175 @ id) | (X169 @ id @ (X171 @ id @ X175) @ id)) | (X169 @ id @ (lam @ (X171 @ sh @ one)) @ id)) | ~ ! [X176 : subst,X177 : subst,X178 : term,X179 : subst] : ((X169 @ (comp @ X176 @ X177) @ (sub @ X178 @ X177) @ X179) | ~(X169 @ X176 @ X178 @ (comp @ X177 @ X179))) | ~ ! [X180 : term] : ((X169 @ id @ (sub @ X170 @ (push @ X180 @ id)) @ id) | ~(X169 @ id @ X180 @ id)) | ~ ! [X181 : subst,X182 : subst,X183 : subst,X184 : term] : ((X169 @ X181 @ X184 @ (comp @ X182 @ X183)) | ~(X169 @ (comp @ X181 @ X182) @ (sub @ X184 @ X182) @ X183)) | (X169 @ id @ (lam @ X170) @ id)) => ! [X185 : (subst > term > subst > $o),X186 : term] : (~ ! [X187 : subst,X188 : term,X189 : subst,X190 : subst] : ((X185 @ X189 @ X188 @ (comp @ X187 @ X190)) | ~(X185 @ (comp @ X189 @ X187) @ (sub @ X188 @ X187) @ X190)) | ~ ! [X191 : subst,X192 : term,X193 : subst,X194 : subst] : ((X185 @ (comp @ X193 @ X191) @ (sub @ X192 @ X191) @ X194) | ~(X185 @ X193 @ X192 @ (comp @ X191 @ X194))) | (X185 @ id @ X186 @ id) | ~ ! [X195 : term] : (~(var @ (sub @ X195 @ id)) | (X185 @ id @ X195 @ id)) | ~ ! [X196 : term,X197 : term] : ((X185 @ id @ (ap @ (sub @ X196 @ id) @ X197) @ id) | ~(X185 @ id @ X197 @ id) | ~(X185 @ id @ X196 @ id)) | ~ ! [X198 : (subst > term > term)] : (~ ! [X199 : term] : ((X185 @ id @ (X198 @ id @ X199) @ id) | ~(X185 @ id @ X199 @ id)) | ~ ! [X200 : subst,X201 : subst,X202 : term] : (((X198 @ (comp @ X201 @ X200) @ (sub @ X202 @ X200))) = ((sub @ (X198 @ X201 @ X202) @ X200))) | (X185 @ id @ (lam @ (X198 @ sh @ one)) @ id))))))) & termmset_gthm & (pushprop_lem1_lthm <=> (! [X203 : term,X204 : subst] : (((sub @ one @ (push @ X203 @ X204))) = X203) => (! [X205 : term,X206 : subst] : (((comp @ sh @ (push @ X205 @ X206))) = X206) => ! [X207 : term,X208 : (term > $o),X209 : term,X210 : (term > $o),X211 : subst] : ((X210 @ (sub @ X207 @ (push @ X209 @ X211))) | ~(X208 @ X207))))) & substmonoid_gthm & (axidl <=> ! [X212 : subst] : (((comp @ id @ X212)) = X212)) & apnotvar & hoaslamnotvar_gthm & (axvarid <=> ! [X213 : term] : (((sub @ X213 @ id)) = X213)) & (((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) = ((d2term @ d_one))) & (! [X214 : term,X215 : term,X216 : term,X217 : term] : ((((ap @ (sub @ X216 @ id) @ X215)) != ((ap @ (sub @ X214 @ id) @ X217))) | (X215 = X217)) <=> hoasapinj2) & (pushprop_p_and_p_prime = (^[X218 : term, X219 : subst, X220 : (term > $o), X221 : (term > $o)] : (! [X222 : term] : (((X220 @ (sub @ X222 @ (push @ X218 @ X219)))) = ((X221 @ X222)))))) & ulamvarsh & (hoasinduction_lem3b <=> ! [X223 : term] : ~ ! [X224 : (subst > term > term)] : (((X224 @ sh @ one)) != ((sub @ X223 @ (push @ one @ sh))))) & hoasapinj1_gthm & (((ap @ (d2term @ d_one) @ (d2term @ d_one))) = ((d2term @ d_one))) & pushprop_gthm & (! [X225 : term,X226 : (subst > term > subst > $o),X227 : (term > $o)] : (~ ! [X228 : term] : (~(X227 @ X228) | (X227 @ (sub @ X225 @ (push @ X228 @ id)))) | ~ ! [X229 : subst,X230 : subst,X231 : subst,X232 : term] : (~(X226 @ (comp @ X231 @ X229) @ (sub @ X232 @ X229) @ X230) | (X226 @ X231 @ X232 @ (comp @ X229 @ X230))) | ~ ! [X233 : term] : (((X226 @ id @ X233 @ id)) = ((X227 @ X233))) | ~ ! [X234 : (subst > term > term)] : ((X226 @ id @ (lam @ (X234 @ sh @ one)) @ id) | ~ ! [X235 : term] : ((X226 @ id @ (X234 @ id @ X235) @ id) | ~(X226 @ id @ X235 @ id)) | ~ ! [X236 : term,X237 : subst,X238 : subst] : (((sub @ (X234 @ X238 @ X236) @ X237)) = ((X234 @ (comp @ X238 @ X237) @ (sub @ X236 @ X237))))) | ~ ! [X239 : subst,X240 : term,X241 : subst,X242 : subst] : ((X226 @ (comp @ X241 @ X242) @ (sub @ X240 @ X242) @ X239) | ~(X226 @ X241 @ X240 @ (comp @ X242 @ X239))) | (X227 @ (lam @ X225))) <=> hoasinduction_lem3v2) & (hoasinduction_lem2 <=> ! [X243 : (subst > term > subst > $o),X244 : term,X245 : term] : (~ ! [X246 : term,X247 : term] : (~(X243 @ id @ X247 @ id) | (X243 @ id @ (ap @ (sub @ X247 @ id) @ X246) @ id) | ~(X243 @ id @ X246 @ id)) | ~(X243 @ id @ X244 @ id) | ~ ! [X248 : subst,X249 : term,X250 : subst,X251 : subst] : ((X243 @ X248 @ X249 @ (comp @ X251 @ X250)) | ~(X243 @ (comp @ X248 @ X251) @ (sub @ X249 @ X251) @ X250)) | ~(X243 @ id @ X245 @ id) | ~ ! [X252 : subst,X253 : term,X254 : subst,X255 : subst] : (~(X243 @ X255 @ X253 @ (comp @ X252 @ X254)) | (X243 @ (comp @ X255 @ X252) @ (sub @ X253 @ X252) @ X254)) | (X243 @ id @ (ap @ X244 @ X245) @ id))) & (pushprop_lem0_lthm <=> ! [X256 : (term > $o),X257 : term,X258 : subst] : ~ ! [X259 : (term > $o)] : ~ ! [X260 : term] : (((X259 @ X260)) = ((X256 @ (sub @ X260 @ (push @ X257 @ X258)))))) & (substmonoid_lthm <=> (! [X261 : subst] : (((comp @ id @ X261)) = X261) => (! [X262 : subst] : (((comp @ X262 @ id)) = X262) => (! [X263 : subst] : (((comp @ id @ X263)) = X263) & ! [X264 : subst] : (((comp @ X264 @ id)) = X264))))) & (! [X265 : (subst > term > subst > $o)] : ~ ! [X266 : (term > $o)] : ~ ! [X267 : term] : (((X266 @ X267)) = ((X265 @ id @ X267 @ id))) <=> hoasinduction_lem0) & ((! [X268 : term] : (((sub @ X268 @ id)) = X268) => (! [X269 : (subst > term > subst > $o),X270 : term] : (~ ! [X271 : (subst > term > term)] : ((X269 @ id @ (lam @ (X271 @ sh @ one)) @ id) | ~ ! [X272 : term] : (~(X269 @ id @ X272 @ id) | (X269 @ id @ (X271 @ id @ X272) @ id)) | ~ ! [X273 : subst,X274 : subst,X275 : term] : (((X271 @ (comp @ X273 @ X274) @ (sub @ X275 @ X274))) = ((sub @ (X271 @ X273 @ X275) @ X274)))) | ~ ! [X276 : term] : (~(X269 @ id @ X276 @ id) | (X269 @ id @ (sub @ X270 @ (push @ X276 @ id)) @ id)) | (X269 @ id @ (lam @ (sub @ X270 @ (push @ one @ sh))) @ id)) => ! [X277 : (subst > term > subst > $o),X278 : term] : (~ ! [X279 : term] : ((X277 @ id @ (sub @ X278 @ (push @ X279 @ id)) @ id) | ~(X277 @ id @ X279 @ id)) | (X277 @ id @ (lam @ X278) @ id) | ~ ! [X280 : subst,X281 : term,X282 : subst,X283 : subst] : (~(X277 @ (comp @ X283 @ X280) @ (sub @ X281 @ X280) @ X282) | (X277 @ X283 @ X281 @ (comp @ X280 @ X282))) | ~ ! [X284 : (subst > term > term)] : (~ ! [X285 : subst,X286 : subst,X287 : term] : (((sub @ (X284 @ X286 @ X287) @ X285)) = ((X284 @ (comp @ X286 @ X285) @ (sub @ X287 @ X285)))) | ~ ! [X288 : term] : ((X277 @ id @ (X284 @ id @ X288) @ id) | ~(X277 @ id @ X288 @ id)) | (X277 @ id @ (lam @ (X284 @ sh @ one)) @ id)) | ~ ! [X289 : subst,X290 : term,X291 : subst,X292 : subst] : ((X277 @ (comp @ X292 @ X291) @ (sub @ X290 @ X291) @ X289) | ~(X277 @ X292 @ X290 @ (comp @ X291 @ X289)))))) <=> hoasinduction_lem3_lthm) & (hoasinduction_lthm_3 <=> (! [X293 : (subst > term > subst > $o)] : ~ ! [X294 : (term > $o)] : ~ ! [X295 : term] : (((X294 @ X295)) = ((X293 @ id @ X295 @ id))) => (! [X296 : term,X297 : (term > $o)] : (~ ! [X298 : term] : (~(var @ X298) | (X297 @ X298)) | ~ ! [X299 : term] : ((X297 @ (lam @ X299)) | ~ ! [X300 : term] : (~(X297 @ X300) | (X297 @ (sub @ X299 @ (push @ X300 @ id))))) | ~ ! [X301 : term,X302 : term] : (~(X297 @ X302) | (X297 @ (ap @ X302 @ X301)) | ~(X297 @ X301)) | (X297 @ X296)) => (! [X303 : term] : (((sub @ X303 @ id)) = X303) => (! [X304 : term,X305 : (subst > term > subst > $o),X306 : (term > $o)] : ((X306 @ (lam @ X304)) | ~ ! [X307 : (subst > term > term)] : (~ ! [X308 : term,X309 : subst,X310 : subst] : (((X307 @ (comp @ X309 @ X310) @ (sub @ X308 @ X310))) = ((sub @ (X307 @ X309 @ X308) @ X310))) | (X305 @ id @ (lam @ (X307 @ sh @ one)) @ id) | ~ ! [X311 : term] : ((X305 @ id @ (X307 @ id @ X311) @ id) | ~(X305 @ id @ X311 @ id))) | ~ ! [X312 : term] : ((X306 @ (sub @ X304 @ (push @ X312 @ id))) | ~(X306 @ X312)) | ~ ! [X313 : term] : (((X305 @ id @ X313 @ id)) = ((X306 @ X313)))) => ! [X314 : (subst > term > subst > $o),X315 : term] : (~ ! [X316 : subst,X317 : subst,X318 : subst,X319 : term] : ((X314 @ (comp @ X318 @ X317) @ (sub @ X319 @ X317) @ X316) | ~(X314 @ X318 @ X319 @ (comp @ X317 @ X316))) | ~ ! [X320 : (subst > term > term)] : ((X314 @ id @ (lam @ (X320 @ sh @ one)) @ id) | ~ ! [X321 : term] : (~(X314 @ id @ X321 @ id) | (X314 @ id @ (X320 @ id @ X321) @ id)) | ~ ! [X322 : subst,X323 : term,X324 : subst] : (((sub @ (X320 @ X324 @ X323) @ X322)) = ((X320 @ (comp @ X324 @ X322) @ (sub @ X323 @ X322))))) | (X314 @ id @ X315 @ id) | ~ ! [X325 : term,X326 : subst,X327 : subst,X328 : subst] : ((X314 @ X327 @ X325 @ (comp @ X328 @ X326)) | ~(X314 @ (comp @ X327 @ X328) @ (sub @ X325 @ X328) @ X326)) | ~ ! [X329 : term] : (~(var @ (sub @ X329 @ id)) | (X314 @ id @ X329 @ id)) | ~ ! [X330 : term,X331 : term] : ((X314 @ id @ (ap @ (sub @ X331 @ id) @ X330) @ id) | ~(X314 @ id @ X330 @ id) | ~(X314 @ id @ X331 @ id)))))))) & ulamvarind & (substmonoid <=> (! [X332 : subst] : (((comp @ X332 @ id)) = X332) & ! [X333 : subst] : (((comp @ id @ X333)) = X333))) & axabs & hoaslamnotap_gthm & ! [X334 : d_term,X335 : d_term] : ((((d2term @ X334)) = ((d2term @ X335))) => (X334 = X335)) & (! [X336 : subst,X337 : term] : (((comp @ sh @ (push @ X337 @ X336))) = X336) <=> axshiftcons) & ((! [X338 : term] : ~ ! [X339 : (subst > term > term)] : ~ ! [X340 : term,X341 : subst] : (((sub @ X338 @ (push @ X340 @ X341))) = ((X339 @ X341 @ X340))) => (! [X342 : term] : (((sub @ X342 @ id)) = X342) => ! [X343 : term,X344 : (subst > term > subst > $o),X345 : (term > $o)] : (~ ! [X346 : term] : (~(X345 @ X346) | (X345 @ (sub @ X343 @ (push @ X346 @ id)))) | ~ ! [X347 : (subst > term > term)] : (~ ! [X348 : term] : ((X344 @ id @ (X347 @ id @ X348) @ id) | ~(X344 @ id @ X348 @ id)) | ~ ! [X349 : subst,X350 : term,X351 : subst] : (((X347 @ (comp @ X349 @ X351) @ (sub @ X350 @ X351))) = ((sub @ (X347 @ X349 @ X350) @ X351))) | (X344 @ id @ (lam @ (X347 @ sh @ one)) @ id)) | (X345 @ (lam @ X343)) | ~ ! [X352 : term] : (((X345 @ X352)) = ((X344 @ id @ X352 @ id)))))) <=> hoasinduction_lem3v2a_lthm) & ((! [X353 : term] : (((sub @ X353 @ id)) = X353) => ! [X354 : (term > $o),X355 : term,X356 : (subst > term > subst > $o)] : (~ ! [X357 : term] : (((X354 @ X357)) = ((X356 @ id @ X357 @ id))) | ~ ! [X358 : subst,X359 : subst,X360 : term,X361 : subst] : (~(X356 @ (comp @ X358 @ X361) @ (sub @ X360 @ X361) @ X359) | (X356 @ X358 @ X360 @ (comp @ X361 @ X359))) | ~ ! [X362 : subst,X363 : subst,X364 : term,X365 : subst] : ((X356 @ (comp @ X365 @ X363) @ (sub @ X364 @ X363) @ X362) | ~(X356 @ X365 @ X364 @ (comp @ X363 @ X362))) | (X354 @ (lam @ X355)) | ~ ! [X366 : (subst > term > term)] : (~ ! [X367 : term] : (~(X356 @ id @ X367 @ id) | (X356 @ id @ (X366 @ id @ X367) @ id)) | ~ ! [X368 : subst,X369 : subst,X370 : term] : (((sub @ (X366 @ X368 @ X370) @ X369)) = ((X366 @ (comp @ X368 @ X369) @ (sub @ X370 @ X369)))) | (X356 @ id @ (lam @ (X366 @ sh @ one)) @ id)) | ~ ! [X371 : term] : ((X354 @ (sub @ X355 @ (push @ X371 @ id))) | ~(X354 @ X371)))) <=> hoasinduction_lem3v2_lthm) & hoasinduction_lem3v2_gthm & (hoasinduction_lem3a_lthm <=> (! [X372 : term] : (((sub @ X372 @ id)) = X372) => (! [X373 : (subst > term > subst > $o),X374 : term] : ((X373 @ id @ (lam @ (sub @ X374 @ (push @ one @ sh))) @ id) | ~ ! [X375 : term] : ((X373 @ id @ (sub @ X374 @ (push @ X375 @ id)) @ id) | ~(X373 @ id @ X375 @ id)) | ~ ! [X376 : (subst > term > term)] : ((X373 @ id @ (lam @ (X376 @ sh @ one)) @ id) | ~ ! [X377 : subst,X378 : subst,X379 : term] : (((X376 @ (comp @ X378 @ X377) @ (sub @ X379 @ X377))) = ((sub @ (X376 @ X378 @ X379) @ X377))) | ~ ! [X380 : term] : ((X373 @ id @ (X376 @ id @ X380) @ id) | ~(X373 @ id @ X380 @ id)))) => ! [X381 : (subst > term > subst > $o),X382 : term] : (~ ! [X383 : term] : ((X381 @ id @ (sub @ X382 @ (push @ X383 @ id)) @ id) | ~(X381 @ id @ X383 @ id)) | (X381 @ id @ (lam @ X382) @ id) | ~ ! [X384 : (subst > term > term)] : (~ ! [X385 : term] : ((X381 @ id @ (X384 @ id @ X385) @ id) | ~(X381 @ id @ X385 @ id)) | ~ ! [X386 : subst,X387 : term,X388 : subst] : (((X384 @ (comp @ X388 @ X386) @ (sub @ X387 @ X386))) = ((sub @ (X384 @ X388 @ X387) @ X386))) | (X381 @ id @ (lam @ (X384 @ sh @ one)) @ id)))))) & axvarshift & hoasinduction_lem1_gthm & (induction2_lthm <=> (! [X389 : term] : (((sub @ X389 @ id)) = X389) => (! [X390 : term,X391 : (term > $o),X392 : subst] : (~ ! [X393 : term,X394 : term] : (~(X391 @ X394) | ~(X391 @ X393) | (X391 @ (ap @ X393 @ X394))) | (X391 @ (sub @ X390 @ X392)) | ~ ! [X395 : term] : (~(var @ X395) | (X391 @ (sub @ X395 @ X392))) | ~ ! [X396 : term] : ((X391 @ (lam @ X396)) | ~ ! [X397 : term] : ((X391 @ (sub @ X396 @ (push @ X397 @ id))) | ~(X391 @ X397)))) => ! [X398 : (term > $o),X399 : term] : (~ ! [X400 : term] : (~(var @ X400) | (X398 @ X400)) | ~ ! [X401 : term,X402 : term] : (~(X398 @ X401) | ~(X398 @ X402) | (X398 @ (ap @ X401 @ X402))) | (X398 @ X399) | ~ ! [X403 : term] : (~ ! [X404 : term] : ((X398 @ (sub @ X403 @ (push @ X404 @ id))) | ~(X398 @ X404)) | (X398 @ (lam @ X403))))))) & (! [X405 : term,X406 : term] : ((((lam @ X406)) != ((lam @ X405))) | (X405 = X406)) <=> laminj) & (! [X407 : term,X408 : (term > $o),X409 : subst] : (~ ! [X410 : term] : (~(var @ X410) | (X408 @ (sub @ X410 @ X409))) | ~ ! [X411 : term] : ((X408 @ (lam @ X411)) | ~ ! [X412 : term] : (~(X408 @ X412) | (X408 @ (sub @ X411 @ (push @ X412 @ id))))) | (X408 @ (sub @ X407 @ X409)) | ~ ! [X413 : term,X414 : term] : ((X408 @ (ap @ X414 @ X413)) | ~(X408 @ X414) | ~(X408 @ X413))) <=> induction2lem) & hoasapinj2_gthm & (! [X415 : (subst > term > subst > $o),X416 : term] : (~ ! [X417 : (subst > term > term)] : (~ ! [X418 : subst,X419 : term,X420 : subst] : (((X417 @ (comp @ X418 @ X420) @ (sub @ X419 @ X420))) = ((sub @ (X417 @ X418 @ X419) @ X420))) | ~ ! [X421 : term] : ((X415 @ id @ (X417 @ id @ X421) @ id) | ~(X415 @ id @ X421 @ id)) | (X415 @ id @ (lam @ (X417 @ sh @ one)) @ id)) | ~ ! [X422 : term] : ((X415 @ id @ (sub @ X416 @ (push @ X422 @ id)) @ id) | ~(X415 @ id @ X422 @ id)) | (X415 @ id @ (lam @ X416) @ id)) <=> hoasinduction_lem3a) & hoasinduction_lem2_lthm & ~(var @ (d2term @ d_one)) & pushprop_lem2v2_lthm & pushprop_lem2v2 & ! [X423 : term] : ? [X424 : d_term] : (((d2term @ X424)) = X423) & (pushprop_lem0 <=> ! [X425 : subst,X426 : term,X427 : (term > $o)] : ~ ! [X428 : (term > $o)] : ~ ! [X429 : term] : (((X428 @ X429)) = ((X427 @ (sub @ X429 @ (push @ X426 @ X425)))))) & (! [X430 : term] : ~ ! [X431 : (subst > term > term)] : (((sub @ X430 @ (push @ one @ sh))) != ((X431 @ sh @ one))) <=> hoasinduction_lem3b_lthm) & (! [X432 : term,X433 : term] : ((X432 = X433) | (((sub @ X433 @ sh)) != ((sub @ X432 @ sh)))) <=> shinj) & (hoasinduction_lem0_lthm <=> ! [X434 : (subst > term > subst > $o)] : ~ ! [X435 : (term > $o)] : ~ ! [X436 : term] : (((X434 @ id @ X436 @ id)) = ((X435 @ X436)))) & (one = ((d2term @ d_one))) & (hoasinduction_lem3 <=> ! [X437 : term,X438 : (subst > term > subst > $o)] : ((X438 @ id @ (lam @ X437) @ id) | ~ ! [X439 : subst,X440 : subst,X441 : subst,X442 : term] : ((X438 @ X439 @ X442 @ (comp @ X440 @ X441)) | ~(X438 @ (comp @ X439 @ X440) @ (sub @ X442 @ X440) @ X441)) | ~ ! [X443 : subst,X444 : term,X445 : subst,X446 : subst] : ((X438 @ (comp @ X443 @ X446) @ (sub @ X444 @ X446) @ X445) | ~(X438 @ X443 @ X444 @ (comp @ X446 @ X445))) | ~ ! [X447 : (subst > term > term)] : ((X438 @ id @ (lam @ (X447 @ sh @ one)) @ id) | ~ ! [X448 : term,X449 : subst,X450 : subst] : (((X447 @ (comp @ X450 @ X449) @ (sub @ X448 @ X449))) = ((sub @ (X447 @ X450 @ X448) @ X449))) | ~ ! [X451 : term] : (~(X438 @ id @ X451 @ id) | (X438 @ id @ (X447 @ id @ X451) @ id))) | ~ ! [X452 : term] : ((X438 @ id @ (sub @ X437 @ (push @ X452 @ id)) @ id) | ~(X438 @ id @ X452 @ id)))) & (! [X453 : (subst > term > subst > $o),X454 : term,X455 : (term > $o),X456 : term] : (~ ! [X457 : subst,X458 : subst,X459 : term,X460 : subst] : (~(X453 @ X460 @ X459 @ (comp @ X457 @ X458)) | (X453 @ (comp @ X460 @ X457) @ (sub @ X459 @ X457) @ X458)) | ~(X455 @ X454) | ~(X455 @ X456) | ~ ! [X461 : term] : (((X455 @ X461)) = ((X453 @ id @ X461 @ id))) | ~ ! [X462 : term,X463 : term] : (~(X453 @ id @ X463 @ id) | ~(X453 @ id @ X462 @ id) | (X453 @ id @ (ap @ (sub @ X463 @ id) @ X462) @ id)) | ~ ! [X464 : subst,X465 : subst,X466 : term,X467 : subst] : (~(X453 @ (comp @ X467 @ X465) @ (sub @ X466 @ X465) @ X464) | (X453 @ X467 @ X466 @ (comp @ X465 @ X464))) | (X455 @ (ap @ X454 @ X456))) <=> hoasinduction_lem2v2) & (! [X468 : (subst > term > subst > $o),X469 : term,X470 : (term > $o)] : (~ ! [X471 : term] : (~(X470 @ X471) | (X470 @ (sub @ X469 @ (push @ X471 @ id)))) | (X470 @ (lam @ X469)) | ~ ! [X472 : (subst > term > term)] : (~ ! [X473 : subst,X474 : subst,X475 : term] : (((sub @ (X472 @ X474 @ X475) @ X473)) = ((X472 @ (comp @ X474 @ X473) @ (sub @ X475 @ X473)))) | ~ ! [X476 : term] : ((X468 @ id @ (X472 @ id @ X476) @ id) | ~(X468 @ id @ X476 @ id)) | (X468 @ id @ (lam @ (X472 @ sh @ one)) @ id)) | ~ ! [X477 : term] : (((X470 @ X477)) = ((X468 @ id @ X477 @ id)))) <=> hoasinduction_lem3v2a) & hoasinduction_gthm & pushprop & hoasinduction_lem1v2_gthm & (hoasinduction_no_psi_cond_lthm <=> (! [X478 : (subst > term > subst > $o)] : ~ ! [X479 : (term > $o)] : ~ ! [X480 : term] : (((X479 @ X480)) = ((X478 @ id @ X480 @ id))) => (! [X481 : (term > $o),X482 : term] : (~ ! [X483 : term] : ((X481 @ (lam @ X483)) | ~ ! [X484 : term] : (~(X481 @ X484) | (X481 @ (sub @ X483 @ (push @ X484 @ id))))) | ~ ! [X485 : term,X486 : term] : ((X481 @ (ap @ X486 @ X485)) | ~(X481 @ X485) | ~(X481 @ X486)) | ~ ! [X487 : term] : (~(var @ X487) | (X481 @ X487)) | (X481 @ X482)) => (! [X488 : term] : (((sub @ X488 @ id)) = X488) => (! [X489 : (subst > term > subst > $o),X490 : (term > $o),X491 : term] : (~ ! [X492 : term] : ((X490 @ (sub @ X491 @ (push @ X492 @ id))) | ~(X490 @ X492)) | ~ ! [X493 : term] : (((X489 @ id @ X493 @ id)) = ((X490 @ X493))) | ~ ! [X494 : (subst > term > term)] : (~ ! [X495 : subst,X496 : subst,X497 : term] : (((sub @ (X494 @ X496 @ X497) @ X495)) = ((X494 @ (comp @ X496 @ X495) @ (sub @ X497 @ X495)))) | (X489 @ id @ (lam @ (X494 @ sh @ one)) @ id) | ~ ! [X498 : term] : (~(X489 @ id @ X498 @ id) | (X489 @ id @ (X494 @ id @ X498) @ id))) | (X490 @ (lam @ X491))) => ! [X499 : term,X500 : (subst > term > subst > $o)] : (~ ! [X501 : term,X502 : term] : (~(X500 @ id @ X501 @ id) | ~(X500 @ id @ X502 @ id) | (X500 @ id @ (ap @ (sub @ X501 @ id) @ X502) @ id)) | (X500 @ id @ X499 @ id) | ~ ! [X503 : (subst > term > term)] : (~ ! [X504 : term] : (~(X500 @ id @ X504 @ id) | (X500 @ id @ (X503 @ id @ X504) @ id)) | (X500 @ id @ (lam @ (X503 @ sh @ one)) @ id) | ~ ! [X505 : subst,X506 : term,X507 : subst] : (((X503 @ (comp @ X507 @ X505) @ (sub @ X506 @ X505))) = ((sub @ (X503 @ X507 @ X506) @ X505)))))))))) & hoasinduction_lem1v2 & (hoasinduction_lem3v2_f_lthm <=> ! [X508 : term] : ~ ! [X509 : (subst > term > term)] : ~ ! [X510 : term,X511 : subst] : (((X509 @ X511 @ X510)) = ((sub @ X508 @ (push @ X510 @ X511))))) & (hoasinduction_lem3aaa <=> ! [X512 : (subst > term > subst > $o),X513 : term] : (~ ! [X514 : (subst > term > term),X515 : term] : (~ ! [X516 : subst,X517 : subst,X518 : term] : (((sub @ (sub @ X515 @ (push @ X518 @ X517)) @ X516)) = ((sub @ (X514 @ X517 @ X518) @ X516))) | ~ ! [X519 : subst,X520 : subst,X521 : term] : (((sub @ X515 @ (push @ (sub @ X521 @ X520) @ (comp @ X519 @ X520)))) = ((X514 @ (comp @ X519 @ X520) @ (sub @ X521 @ X520)))) | (X512 @ id @ (lam @ (X514 @ sh @ one)) @ id) | ~ ! [X522 : term] : ((X512 @ id @ (X514 @ id @ X522) @ id) | ~(X512 @ id @ X522 @ id))) | (X512 @ id @ (lam @ (sub @ X513 @ (push @ one @ sh))) @ id) | ~ ! [X523 : term] : (~(X512 @ id @ X523 @ id) | (X512 @ id @ (sub @ X513 @ (push @ X523 @ id)) @ id)))) & ! [X524 : d_subst,X525 : d_subst] : ((((d2subst @ X524)) = ((d2subst @ X525))) => (X524 = X525)) & (hoasinduction_lem3v2_f <=> ! [X526 : term] : ~ ! [X527 : (subst > term > term)] : ~ ! [X528 : subst,X529 : term] : (((sub @ X526 @ (push @ X529 @ X528))) = ((X527 @ X528 @ X529)))) & (axidr <=> ! [X530 : subst] : (((comp @ X530 @ id)) = X530)) & (induction2lem_lthm <=> (! [X531 : subst,X532 : term] : (((sub @ one @ (push @ X532 @ X531))) = X532) => (! [X533 : subst,X534 : term] : (((comp @ sh @ (push @ X534 @ X533))) = X533) => (! [X535 : subst] : (((comp @ X535 @ id)) = X535) => (! [X536 : term,X537 : (term > $o)] : (~ ! [X538 : term] : (~(X537 @ X538) | (X537 @ (lam @ X538))) | (X537 @ X536) | ~ ! [X539 : term] : ((X537 @ X539) | ~(var @ X539)) | ~ ! [X540 : term,X541 : term] : (~(X537 @ X540) | (X537 @ (ap @ X541 @ X540)) | ~(X537 @ X541))) => ! [X542 : subst,X543 : term,X544 : (term > $o)] : ((X544 @ (sub @ X543 @ X542)) | ~ ! [X545 : term] : ((X544 @ (lam @ X545)) | ~ ! [X546 : term] : (~(X544 @ X546) | (X544 @ (sub @ X545 @ (push @ X546 @ id))))) | ~ ! [X547 : term,X548 : term] : (~(X544 @ X548) | (X544 @ (ap @ X547 @ X548)) | ~(X544 @ X547)) | ~ ! [X549 : term] : ((X544 @ (sub @ X549 @ X542)) | ~(var @ X549)))))))) & lamnotvar),
  inference(rectify,[],[f1])).
thf(f5,plain,(
  (pushprop_lem1_gthm = $true) & (induction2_gthm = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((induction2 = $true) <=> ! [X0 : term,X1 : (term > $o)] : (~ ! [X2 : term] : ((((X1 @ (lam @ X2))) = $true) | ~ ! [X3 : term] : (~ (((X1 @ X3)) = $true) | (((X1 @ (sub @ X2 @ (push @ X3 @ id)))) = $true))) | (((X1 @ X0)) = $true) | ~ ! [X4 : term,X5 : term] : (~ (((X1 @ X4)) = $true) | (((X1 @ (ap @ X5 @ X4))) = $true) | ~ (((X1 @ X5)) = $true)) | ~ ! [X6 : term] : ((((X1 @ X6)) = $true) | ~ (((var @ X6)) = $true)))) & ~ (lamnotap = $true) & (hoaslamnotvar_lthm = $true) & (pushprop_lem2v2_gthm = $true) & ((! [X7 : term,X10 : term,X8 : term,X9 : term] : ((X7 = X9) | (((ap @ X8 @ X7)) != ((ap @ X10 @ X9)))) => ! [X14 : term,X11 : term,X13 : term,X12 : term] : ((((ap @ (sub @ X13 @ id) @ X12)) != ((ap @ (sub @ X14 @ id) @ X11))) | (X11 = X12))) <=> (hoasapinj2_lthm = $true)) & ! [X15 : subst] : ? [X16 : d_subst] : (((d2subst @ X16)) = X15) & (hoasapnotvar_lthm = $true) & (axmap = $true) & (axclos = $true) & (hoasinduction_lem1_lthm = $true) & ~ (ulamvar1 = $true) & ((hoasapinj1 = $true) <=> ! [X18 : term,X19 : term,X17 : term,X20 : term] : ((((ap @ (sub @ X20 @ id) @ X19)) != ((ap @ (sub @ X17 @ id) @ X18))) | (X17 = X20))) & (hoasapnotvar_gthm = $true) & ((hoasinduction = $true) <=> ! [X21 : (subst > term > subst > $o),X22 : term] : (~ ! [X23 : subst,X24 : subst,X25 : subst,X26 : term] : ((((X21 @ (comp @ X25 @ X24) @ (sub @ X26 @ X24) @ X23)) = $true) | ~ (((X21 @ X25 @ X26 @ (comp @ X24 @ X23))) = $true)) | ~ ! [X27 : (subst > term > term)] : ((((X21 @ id @ (lam @ (X27 @ sh @ one)) @ id)) = $true) | ~ ! [X28 : term] : (~ (((X21 @ id @ X28 @ id)) = $true) | (((X21 @ id @ (X27 @ id @ X28) @ id)) = $true)) | ~ ! [X30 : term,X29 : subst,X31 : subst] : (((X27 @ (comp @ X29 @ X31) @ (sub @ X30 @ X31))) = ((sub @ (X27 @ X29 @ X30) @ X31)))) | ~ ! [X32 : term,X33 : term] : ((((X21 @ id @ (ap @ (sub @ X33 @ id) @ X32) @ id)) = $true) | ~ ($true = ((X21 @ id @ X33 @ id))) | ~ (((X21 @ id @ X32 @ id)) = $true)) | (((X21 @ id @ X22 @ id)) = $true) | ~ ! [X34 : term,X35 : subst,X36 : subst,X37 : subst] : (~ (((X21 @ (comp @ X36 @ X35) @ (sub @ X34 @ X35) @ X37)) = $true) | (((X21 @ X36 @ X34 @ (comp @ X35 @ X37))) = $true)) | ~ ! [X38 : term] : (~ (((var @ (sub @ X38 @ id))) = $true) | (((X21 @ id @ X38 @ id)) = $true)))) & (((d2subst @ d_id)) = sh) & (hoaslamnotvar = $true) & (pushprop_lem3v2 = $true) & (! [X39 : (subst > term > subst > $o),X40 : term] : (~ ! [X41 : term,X42 : term] : ((((X39 @ id @ (ap @ (sub @ X41 @ id) @ X42) @ id)) = $true) | ~ (((X39 @ id @ X41 @ id)) = $true) | ~ (((X39 @ id @ X42 @ id)) = $true)) | ($true = ((X39 @ id @ X40 @ id))) | ~ ! [X43 : (subst > term > term)] : (~ ! [X46 : subst,X44 : subst,X45 : term] : (((sub @ (X43 @ X46 @ X45) @ X44)) = ((X43 @ (comp @ X46 @ X44) @ (sub @ X45 @ X44)))) | (((X39 @ id @ (lam @ (X43 @ sh @ one)) @ id)) = $true) | ~ ! [X47 : term] : ((((X39 @ id @ (X43 @ id @ X47) @ id)) = $true) | ~ (((X39 @ id @ X47 @ id)) = $true)))) <=> (hoasinduction_no_psi_cond = $true)) & (! [X49 : (subst > term > term),X48 : term,X50 : term] : (~ ! [X51 : subst,X53 : term,X52 : subst] : (((X49 @ (comp @ X51 @ X52) @ (sub @ X53 @ X52))) = ((sub @ (X49 @ X51 @ X53) @ X52))) | (((ap @ (sub @ X50 @ id) @ X48)) != ((lam @ (X49 @ sh @ one))))) <=> (hoaslamnotap = $true)) & (pushprop_lthm_orig = $true) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X54 : term,X55 : (subst > term > subst > $o)] : (~ ! [X56 : (subst > term > term)] : (~ ! [X57 : term] : ((((X55 @ id @ (X56 @ id @ X57) @ id)) = $true) | ~ (((X55 @ id @ X57 @ id)) = $true)) | ~ ! [X58 : term,X59 : subst,X60 : subst] : (((X56 @ (comp @ X60 @ X59) @ (sub @ X58 @ X59))) = ((sub @ (X56 @ X60 @ X58) @ X59))) | (((X55 @ id @ (lam @ (X56 @ sh @ one)) @ id)) = $true)) | ~ ! [X61 : term] : (~ ($true = ((X55 @ id @ X61 @ id))) | (((X55 @ id @ (sub @ X54 @ (push @ X61 @ id)) @ id)) = $true)) | (((X55 @ id @ (lam @ (sub @ X54 @ (push @ one @ sh))) @ id)) = $true))) & (pushprop_lem0_gthm = $true) & ((hoaslaminj_lthm = $true) <=> (! [X62 : term,X63 : subst] : (((sub @ one @ (push @ X62 @ X63))) = X62) => (! [X64 : term,X65 : subst] : (((comp @ sh @ (push @ X64 @ X65))) = X65) => (! [X66 : term,X67 : term] : ((X66 = X67) | (((lam @ X66)) != ((lam @ X67)))) => ! [X71 : term,X68 : subst,X70 : (subst > term > term),X69 : (subst > term > term)] : (~ ! [X77 : term,X75 : subst,X76 : subst] : (((sub @ (X70 @ X76 @ X77) @ X75)) = ((X70 @ (comp @ X76 @ X75) @ (sub @ X77 @ X75)))) | ~ ! [X73 : term,X72 : subst,X74 : subst] : (((sub @ (X69 @ X72 @ X73) @ X74)) = ((X69 @ (comp @ X72 @ X74) @ (sub @ X73 @ X74)))) | (((lam @ (X70 @ sh @ one))) != ((lam @ (X69 @ sh @ one)))) | (((X69 @ X68 @ X71)) = ((X70 @ X68 @ X71)))))))) & ! [X78 : d_subst] : (d_id = X78) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ~ (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) = $true) & ((pushprop_lem1 = $true) <=> ! [X79 : subst,X80 : (term > $o),X81 : term,X82 : (term > $o),X83 : term] : ((((X82 @ (sub @ X81 @ (push @ X83 @ X79)))) = $true) | ~ (((X80 @ X81)) = $true))) & (hoaslaminj_gthm = $true) & ((hoasapinj1_lthm = $true) <=> (! [X84 : term] : (((sub @ X84 @ id)) = X84) => (! [X87 : term,X85 : term,X86 : term,X88 : term] : ((((ap @ X88 @ X85)) != ((ap @ X87 @ X86))) | (X87 = X88)) => ! [X92 : term,X89 : term,X90 : term,X91 : term] : ((X89 = X92) | (((ap @ (sub @ X89 @ id) @ X91)) != ((ap @ (sub @ X92 @ id) @ X90))))))) & (hoasinduction_lem3b_gthm = $true) & ! [X93 : d_term] : (d_one = X93) & ((termmset_lthm = $true) <=> (! [X94 : term] : (((sub @ X94 @ id)) = X94) => ! [X95 : term] : (((sub @ X95 @ id)) = X95))) & (hoasinduction_lem2_gthm = $true) & (! [X96 : subst,X97 : term] : (((sub @ one @ (push @ X97 @ X96))) = X97) <=> (axvarcons = $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (axassoc = $true) & (hoaslamnotap_lthm = $true) & ((pushprop_lem1v2 = $true) <=> ! [X100 : term,X101 : (term > $o),X102 : subst,X103 : (term > $o)] : (~ (((X101 @ X100)) = $true) | ~ ! [X104 : term] : (((X103 @ X104)) = ((X101 @ (sub @ X104 @ (push @ X100 @ X102))))) | (((X103 @ one)) = $true))) & (! [X105 : term] : (((sub @ X105 @ id)) = X105) <=> (termmset = $true)) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (hoasinduction_lem1 = $true) & (hoasapnotvar = $true) & ((! [X106 : term,X107 : subst] : (((sub @ one @ (push @ X106 @ X107))) = X106) => ! [X108 : (term > $o),X109 : subst,X110 : term,X111 : (term > $o)] : ((((X111 @ one)) = $true) | ~ ! [X112 : term] : (((X108 @ (sub @ X112 @ (push @ X110 @ X109)))) = ((X111 @ X112))) | ~ (((X108 @ X110)) = $true))) <=> (pushprop_lem1v2_lthm = $true)) & (! [X113 : term,X114 : term,X115 : term,X116 : term] : ((((ap @ X116 @ X115)) != ((ap @ X114 @ X113))) | (X114 = X116)) <=> (apinj1 = $true)) & (! [X117 : term,X118 : (subst > term > term),X119 : (subst > term > term),X120 : subst] : (~ ! [X121 : subst,X122 : term,X123 : subst] : (((X119 @ (comp @ X121 @ X123) @ (sub @ X122 @ X123))) = ((sub @ (X119 @ X121 @ X122) @ X123))) | (((lam @ (X119 @ sh @ one))) != ((lam @ (X118 @ sh @ one)))) | (((X118 @ X120 @ X117)) = ((X119 @ X120 @ X117))) | ~ ! [X124 : subst,X125 : subst,X126 : term] : (((sub @ (X118 @ X125 @ X126) @ X124)) = ((X118 @ (comp @ X125 @ X124) @ (sub @ X126 @ X124))))) <=> (hoaslaminj = $true)) & ((hoasinduction_lem3aa = $true) <=> ! [X127 : term,X128 : (subst > term > subst > $o)] : (($true = ((X128 @ id @ (lam @ (sub @ X127 @ (push @ one @ sh))) @ id))) | ~ ! [X129 : term] : (($true = ((X128 @ id @ (sub @ X127 @ (push @ X129 @ id)) @ id))) | ~ (((X128 @ id @ X129 @ id)) = $true)) | ~ ! [X130 : (subst > term > term)] : ((((X128 @ id @ (lam @ (X130 @ sh @ one)) @ id)) = $true) | ~ ! [X131 : subst,X132 : term,X133 : subst] : (((X130 @ (comp @ X133 @ X131) @ (sub @ X132 @ X131))) = ((sub @ (X130 @ X133 @ X132) @ X131))) | ~ ! [X134 : term] : (~ (((X128 @ id @ X134 @ id)) = $true) | (((X128 @ id @ (X130 @ id @ X134) @ id)) = $true))))) & ((induction = $true) <=> ! [X135 : (term > $o),X136 : term] : (~ ! [X137 : term] : ((((X135 @ (lam @ X137))) = $true) | ~ (((X135 @ X137)) = $true)) | ~ ! [X138 : term] : (~ (((var @ X138)) = $true) | (((X135 @ X138)) = $true)) | (((X135 @ X136)) = $true) | ~ ! [X139 : term,X140 : term] : (~ ($true = ((X135 @ X139))) | (((X135 @ (ap @ X139 @ X140))) = $true) | ~ (((X135 @ X140)) = $true)))) & ((axscons = $true) <=> ! [X141 : subst] : (((push @ (sub @ one @ X141) @ (comp @ sh @ X141))) = X141)) & (axapp = $true) & (induction2lem_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (pushprop_lthm = $true) & (hoasinduction_lem3a_gthm = $true) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_lem2v2_gthm = $true) & (! [X145 : term,X146 : term,X147 : term,X148 : term] : ((((ap @ X146 @ X145)) != ((ap @ X147 @ X148))) | (X145 = X148)) <=> (apinj2 = $true)) & (((d2subst @ d_id)) = id) & (hoasinduction_lem3_gthm = $true) & (pushprop_lem3v2_lthm = $true) & ((hoasinduction_lthm = $true) <=> (! [X149 : (term > $o),X150 : term] : ((((X149 @ X150)) = $true) | ~ ! [X151 : term,X152 : term] : ((((X149 @ (ap @ X152 @ X151))) = $true) | ~ (((X149 @ X152)) = $true) | ~ (((X149 @ X151)) = $true)) | ~ ! [X153 : term] : ((((X149 @ X153)) = $true) | ~ (((var @ X153)) = $true)) | ~ ! [X154 : term] : (~ ! [X155 : term] : (~ (((X149 @ X155)) = $true) | (((X149 @ (sub @ X154 @ (push @ X155 @ id)))) = $true)) | (((X149 @ (lam @ X154))) = $true))) => (! [X156 : (subst > term > subst > $o),X157 : term,X158 : term] : ((((X156 @ id @ (ap @ X157 @ X158) @ id)) = $true) | ~ ! [X159 : term,X160 : term] : (~ (((X156 @ id @ X160 @ id)) = $true) | ~ (((X156 @ id @ X159 @ id)) = $true) | (((X156 @ id @ (ap @ (sub @ X160 @ id) @ X159) @ id)) = $true)) | ~ ! [X161 : subst,X162 : subst,X163 : subst,X164 : term] : (~ (((X156 @ X161 @ X164 @ (comp @ X163 @ X162))) = $true) | (((X156 @ (comp @ X161 @ X163) @ (sub @ X164 @ X163) @ X162)) = $true)) | ~ ! [X165 : subst,X166 : subst,X167 : subst,X168 : term] : ((((X156 @ X165 @ X168 @ (comp @ X166 @ X167))) = $true) | ~ (((X156 @ (comp @ X165 @ X166) @ (sub @ X168 @ X166) @ X167)) = $true)) | ~ (((X156 @ id @ X158 @ id)) = $true) | ~ (((X156 @ id @ X157 @ id)) = $true)) => (! [X169 : (subst > term > subst > $o),X170 : term] : (~ ! [X171 : (subst > term > term)] : (~ ! [X172 : term,X173 : subst,X174 : subst] : (((sub @ (X171 @ X173 @ X172) @ X174)) = ((X171 @ (comp @ X173 @ X174) @ (sub @ X172 @ X174)))) | ~ ! [X175 : term] : (~ (((X169 @ id @ X175 @ id)) = $true) | (((X169 @ id @ (X171 @ id @ X175) @ id)) = $true)) | (((X169 @ id @ (lam @ (X171 @ sh @ one)) @ id)) = $true)) | ~ ! [X176 : subst,X177 : subst,X178 : term,X179 : subst] : ((((X169 @ (comp @ X176 @ X177) @ (sub @ X178 @ X177) @ X179)) = $true) | ~ (((X169 @ X176 @ X178 @ (comp @ X177 @ X179))) = $true)) | ~ ! [X180 : term] : (($true = ((X169 @ id @ (sub @ X170 @ (push @ X180 @ id)) @ id))) | ~ (((X169 @ id @ X180 @ id)) = $true)) | ~ ! [X181 : subst,X182 : subst,X183 : subst,X184 : term] : ((((X169 @ X181 @ X184 @ (comp @ X182 @ X183))) = $true) | ~ (((X169 @ (comp @ X181 @ X182) @ (sub @ X184 @ X182) @ X183)) = $true)) | (((X169 @ id @ (lam @ X170) @ id)) = $true)) => ! [X185 : (subst > term > subst > $o),X186 : term] : (~ ! [X187 : subst,X188 : term,X189 : subst,X190 : subst] : ((((X185 @ X189 @ X188 @ (comp @ X187 @ X190))) = $true) | ~ (((X185 @ (comp @ X189 @ X187) @ (sub @ X188 @ X187) @ X190)) = $true)) | ~ ! [X191 : subst,X192 : term,X193 : subst,X194 : subst] : (($true = ((X185 @ (comp @ X193 @ X191) @ (sub @ X192 @ X191) @ X194))) | ~ (((X185 @ X193 @ X192 @ (comp @ X191 @ X194))) = $true)) | (((X185 @ id @ X186 @ id)) = $true) | ~ ! [X195 : term] : (~ (((var @ (sub @ X195 @ id))) = $true) | (((X185 @ id @ X195 @ id)) = $true)) | ~ ! [X196 : term,X197 : term] : ((((X185 @ id @ (ap @ (sub @ X196 @ id) @ X197) @ id)) = $true) | ~ (((X185 @ id @ X197 @ id)) = $true) | ~ (((X185 @ id @ X196 @ id)) = $true)) | ~ ! [X198 : (subst > term > term)] : (~ ! [X199 : term] : ((((X185 @ id @ (X198 @ id @ X199) @ id)) = $true) | ~ (((X185 @ id @ X199 @ id)) = $true)) | ~ ! [X200 : subst,X201 : subst,X202 : term] : (((X198 @ (comp @ X201 @ X200) @ (sub @ X202 @ X200))) = ((sub @ (X198 @ X201 @ X202) @ X200))) | (((X185 @ id @ (lam @ (X198 @ sh @ one)) @ id)) = $true))))))) & (termmset_gthm = $true) & ((pushprop_lem1_lthm = $true) <=> (! [X203 : term,X204 : subst] : (((sub @ one @ (push @ X203 @ X204))) = X203) => (! [X205 : term,X206 : subst] : (((comp @ sh @ (push @ X205 @ X206))) = X206) => ! [X207 : term,X208 : (term > $o),X209 : term,X210 : (term > $o),X211 : subst] : (($true = ((X210 @ (sub @ X207 @ (push @ X209 @ X211))))) | ~ (((X208 @ X207)) = $true))))) & (substmonoid_gthm = $true) & ((axidl = $true) <=> ! [X212 : subst] : (((comp @ id @ X212)) = X212)) & (apnotvar = $true) & (hoaslamnotvar_gthm = $true) & ((axvarid = $true) <=> ! [X213 : term] : (((sub @ X213 @ id)) = X213)) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (! [X214 : term,X215 : term,X216 : term,X217 : term] : ((((ap @ (sub @ X216 @ id) @ X215)) != ((ap @ (sub @ X214 @ id) @ X217))) | (X215 = X217)) <=> (hoasapinj2 = $true)) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & (ulamvarsh = $true) & ((hoasinduction_lem3b = $true) <=> ! [X223 : term] : ~ ! [X224 : (subst > term > term)] : (((X224 @ sh @ one)) != ((sub @ X223 @ (push @ one @ sh))))) & (hoasapinj1_gthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (pushprop_gthm = $true) & (! [X225 : term,X226 : (subst > term > subst > $o),X227 : (term > $o)] : (~ ! [X228 : term] : (~ (((X227 @ X228)) = $true) | (((X227 @ (sub @ X225 @ (push @ X228 @ id)))) = $true)) | ~ ! [X229 : subst,X230 : subst,X231 : subst,X232 : term] : (~ ($true = ((X226 @ (comp @ X231 @ X229) @ (sub @ X232 @ X229) @ X230))) | (((X226 @ X231 @ X232 @ (comp @ X229 @ X230))) = $true)) | ~ ! [X233 : term] : (((X226 @ id @ X233 @ id)) = ((X227 @ X233))) | ~ ! [X234 : (subst > term > term)] : ((((X226 @ id @ (lam @ (X234 @ sh @ one)) @ id)) = $true) | ~ ! [X235 : term] : ((((X226 @ id @ (X234 @ id @ X235) @ id)) = $true) | ~ (((X226 @ id @ X235 @ id)) = $true)) | ~ ! [X236 : term,X237 : subst,X238 : subst] : (((sub @ (X234 @ X238 @ X236) @ X237)) = ((X234 @ (comp @ X238 @ X237) @ (sub @ X236 @ X237))))) | ~ ! [X239 : subst,X240 : term,X241 : subst,X242 : subst] : ((((X226 @ (comp @ X241 @ X242) @ (sub @ X240 @ X242) @ X239)) = $true) | ~ (((X226 @ X241 @ X240 @ (comp @ X242 @ X239))) = $true)) | (((X227 @ (lam @ X225))) = $true)) <=> (hoasinduction_lem3v2 = $true)) & ((hoasinduction_lem2 = $true) <=> ! [X243 : (subst > term > subst > $o),X244 : term,X245 : term] : (~ ! [X246 : term,X247 : term] : (~ (((X243 @ id @ X247 @ id)) = $true) | (((X243 @ id @ (ap @ (sub @ X247 @ id) @ X246) @ id)) = $true) | ~ (((X243 @ id @ X246 @ id)) = $true)) | ~ (((X243 @ id @ X244 @ id)) = $true) | ~ ! [X248 : subst,X249 : term,X250 : subst,X251 : subst] : ((((X243 @ X248 @ X249 @ (comp @ X251 @ X250))) = $true) | ~ (((X243 @ (comp @ X248 @ X251) @ (sub @ X249 @ X251) @ X250)) = $true)) | ~ (((X243 @ id @ X245 @ id)) = $true) | ~ ! [X252 : subst,X253 : term,X254 : subst,X255 : subst] : (~ (((X243 @ X255 @ X253 @ (comp @ X252 @ X254))) = $true) | (((X243 @ (comp @ X255 @ X252) @ (sub @ X253 @ X252) @ X254)) = $true)) | (((X243 @ id @ (ap @ X244 @ X245) @ id)) = $true))) & ((pushprop_lem0_lthm = $true) <=> ! [X256 : (term > $o),X257 : term,X258 : subst] : ~ ! [X259 : (term > $o)] : ~ ! [X260 : term] : (((X259 @ X260)) = ((X256 @ (sub @ X260 @ (push @ X257 @ X258)))))) & ((substmonoid_lthm = $true) <=> (! [X261 : subst] : (((comp @ id @ X261)) = X261) => (! [X262 : subst] : (((comp @ X262 @ id)) = X262) => (! [X263 : subst] : (((comp @ id @ X263)) = X263) & ! [X264 : subst] : (((comp @ X264 @ id)) = X264))))) & (! [X265 : (subst > term > subst > $o)] : ~ ! [X266 : (term > $o)] : ~ ! [X267 : term] : (((X266 @ X267)) = ((X265 @ id @ X267 @ id))) <=> (hoasinduction_lem0 = $true)) & ((! [X268 : term] : (((sub @ X268 @ id)) = X268) => (! [X269 : (subst > term > subst > $o),X270 : term] : (~ ! [X271 : (subst > term > term)] : ((((X269 @ id @ (lam @ (X271 @ sh @ one)) @ id)) = $true) | ~ ! [X272 : term] : (~ (((X269 @ id @ X272 @ id)) = $true) | (((X269 @ id @ (X271 @ id @ X272) @ id)) = $true)) | ~ ! [X273 : subst,X274 : subst,X275 : term] : (((X271 @ (comp @ X273 @ X274) @ (sub @ X275 @ X274))) = ((sub @ (X271 @ X273 @ X275) @ X274)))) | ~ ! [X276 : term] : (~ (((X269 @ id @ X276 @ id)) = $true) | (((X269 @ id @ (sub @ X270 @ (push @ X276 @ id)) @ id)) = $true)) | (((X269 @ id @ (lam @ (sub @ X270 @ (push @ one @ sh))) @ id)) = $true)) => ! [X277 : (subst > term > subst > $o),X278 : term] : (~ ! [X279 : term] : ((((X277 @ id @ (sub @ X278 @ (push @ X279 @ id)) @ id)) = $true) | ~ (((X277 @ id @ X279 @ id)) = $true)) | (((X277 @ id @ (lam @ X278) @ id)) = $true) | ~ ! [X280 : subst,X281 : term,X282 : subst,X283 : subst] : (~ (((X277 @ (comp @ X283 @ X280) @ (sub @ X281 @ X280) @ X282)) = $true) | (((X277 @ X283 @ X281 @ (comp @ X280 @ X282))) = $true)) | ~ ! [X284 : (subst > term > term)] : (~ ! [X285 : subst,X286 : subst,X287 : term] : (((sub @ (X284 @ X286 @ X287) @ X285)) = ((X284 @ (comp @ X286 @ X285) @ (sub @ X287 @ X285)))) | ~ ! [X288 : term] : ((((X277 @ id @ (X284 @ id @ X288) @ id)) = $true) | ~ (((X277 @ id @ X288 @ id)) = $true)) | (((X277 @ id @ (lam @ (X284 @ sh @ one)) @ id)) = $true)) | ~ ! [X289 : subst,X290 : term,X291 : subst,X292 : subst] : ((((X277 @ (comp @ X292 @ X291) @ (sub @ X290 @ X291) @ X289)) = $true) | ~ (((X277 @ X292 @ X290 @ (comp @ X291 @ X289))) = $true))))) <=> (hoasinduction_lem3_lthm = $true)) & ((hoasinduction_lthm_3 = $true) <=> (! [X293 : (subst > term > subst > $o)] : ~ ! [X294 : (term > $o)] : ~ ! [X295 : term] : (((X294 @ X295)) = ((X293 @ id @ X295 @ id))) => (! [X296 : term,X297 : (term > $o)] : (~ ! [X298 : term] : (~ (((var @ X298)) = $true) | (((X297 @ X298)) = $true)) | ~ ! [X299 : term] : ((((X297 @ (lam @ X299))) = $true) | ~ ! [X300 : term] : (~ ($true = ((X297 @ X300))) | (((X297 @ (sub @ X299 @ (push @ X300 @ id)))) = $true))) | ~ ! [X301 : term,X302 : term] : (~ (((X297 @ X302)) = $true) | (((X297 @ (ap @ X302 @ X301))) = $true) | ~ (((X297 @ X301)) = $true)) | (((X297 @ X296)) = $true)) => (! [X303 : term] : (((sub @ X303 @ id)) = X303) => (! [X304 : term,X305 : (subst > term > subst > $o),X306 : (term > $o)] : ((((X306 @ (lam @ X304))) = $true) | ~ ! [X307 : (subst > term > term)] : (~ ! [X308 : term,X309 : subst,X310 : subst] : (((X307 @ (comp @ X309 @ X310) @ (sub @ X308 @ X310))) = ((sub @ (X307 @ X309 @ X308) @ X310))) | (((X305 @ id @ (lam @ (X307 @ sh @ one)) @ id)) = $true) | ~ ! [X311 : term] : ((((X305 @ id @ (X307 @ id @ X311) @ id)) = $true) | ~ (((X305 @ id @ X311 @ id)) = $true))) | ~ ! [X312 : term] : ((((X306 @ (sub @ X304 @ (push @ X312 @ id)))) = $true) | ~ (((X306 @ X312)) = $true)) | ~ ! [X313 : term] : (((X305 @ id @ X313 @ id)) = ((X306 @ X313)))) => ! [X314 : (subst > term > subst > $o),X315 : term] : (~ ! [X316 : subst,X317 : subst,X318 : subst,X319 : term] : ((((X314 @ (comp @ X318 @ X317) @ (sub @ X319 @ X317) @ X316)) = $true) | ~ (((X314 @ X318 @ X319 @ (comp @ X317 @ X316))) = $true)) | ~ ! [X320 : (subst > term > term)] : ((((X314 @ id @ (lam @ (X320 @ sh @ one)) @ id)) = $true) | ~ ! [X321 : term] : (~ (((X314 @ id @ X321 @ id)) = $true) | (((X314 @ id @ (X320 @ id @ X321) @ id)) = $true)) | ~ ! [X322 : subst,X323 : term,X324 : subst] : (((sub @ (X320 @ X324 @ X323) @ X322)) = ((X320 @ (comp @ X324 @ X322) @ (sub @ X323 @ X322))))) | ($true = ((X314 @ id @ X315 @ id))) | ~ ! [X325 : term,X326 : subst,X327 : subst,X328 : subst] : ((((X314 @ X327 @ X325 @ (comp @ X328 @ X326))) = $true) | ~ ($true = ((X314 @ (comp @ X327 @ X328) @ (sub @ X325 @ X328) @ X326)))) | ~ ! [X329 : term] : (~ (((var @ (sub @ X329 @ id))) = $true) | ($true = ((X314 @ id @ X329 @ id)))) | ~ ! [X330 : term,X331 : term] : ((((X314 @ id @ (ap @ (sub @ X331 @ id) @ X330) @ id)) = $true) | ~ (((X314 @ id @ X330 @ id)) = $true) | ~ (((X314 @ id @ X331 @ id)) = $true)))))))) & (ulamvarind = $true) & ((substmonoid = $true) <=> (! [X332 : subst] : (((comp @ X332 @ id)) = X332) & ! [X333 : subst] : (((comp @ id @ X333)) = X333))) & (axabs = $true) & (hoaslamnotap_gthm = $true) & ! [X334 : d_term,X335 : d_term] : ((((d2term @ X334)) = ((d2term @ X335))) => (X334 = X335)) & (! [X336 : subst,X337 : term] : (((comp @ sh @ (push @ X337 @ X336))) = X336) <=> (axshiftcons = $true)) & ((! [X338 : term] : ~ ! [X339 : (subst > term > term)] : ~ ! [X340 : term,X341 : subst] : (((sub @ X338 @ (push @ X340 @ X341))) = ((X339 @ X341 @ X340))) => (! [X342 : term] : (((sub @ X342 @ id)) = X342) => ! [X343 : term,X344 : (subst > term > subst > $o),X345 : (term > $o)] : (~ ! [X346 : term] : (~ (((X345 @ X346)) = $true) | (((X345 @ (sub @ X343 @ (push @ X346 @ id)))) = $true)) | ~ ! [X347 : (subst > term > term)] : (~ ! [X348 : term] : ((((X344 @ id @ (X347 @ id @ X348) @ id)) = $true) | ~ ($true = ((X344 @ id @ X348 @ id)))) | ~ ! [X349 : subst,X350 : term,X351 : subst] : (((X347 @ (comp @ X349 @ X351) @ (sub @ X350 @ X351))) = ((sub @ (X347 @ X349 @ X350) @ X351))) | (((X344 @ id @ (lam @ (X347 @ sh @ one)) @ id)) = $true)) | (((X345 @ (lam @ X343))) = $true) | ~ ! [X352 : term] : (((X345 @ X352)) = ((X344 @ id @ X352 @ id)))))) <=> (hoasinduction_lem3v2a_lthm = $true)) & ((! [X353 : term] : (((sub @ X353 @ id)) = X353) => ! [X354 : (term > $o),X355 : term,X356 : (subst > term > subst > $o)] : (~ ! [X357 : term] : (((X354 @ X357)) = ((X356 @ id @ X357 @ id))) | ~ ! [X358 : subst,X359 : subst,X360 : term,X361 : subst] : (~ (((X356 @ (comp @ X358 @ X361) @ (sub @ X360 @ X361) @ X359)) = $true) | (((X356 @ X358 @ X360 @ (comp @ X361 @ X359))) = $true)) | ~ ! [X362 : subst,X363 : subst,X364 : term,X365 : subst] : ((((X356 @ (comp @ X365 @ X363) @ (sub @ X364 @ X363) @ X362)) = $true) | ~ (((X356 @ X365 @ X364 @ (comp @ X363 @ X362))) = $true)) | (((X354 @ (lam @ X355))) = $true) | ~ ! [X366 : (subst > term > term)] : (~ ! [X367 : term] : (~ (((X356 @ id @ X367 @ id)) = $true) | (((X356 @ id @ (X366 @ id @ X367) @ id)) = $true)) | ~ ! [X368 : subst,X369 : subst,X370 : term] : (((sub @ (X366 @ X368 @ X370) @ X369)) = ((X366 @ (comp @ X368 @ X369) @ (sub @ X370 @ X369)))) | (((X356 @ id @ (lam @ (X366 @ sh @ one)) @ id)) = $true)) | ~ ! [X371 : term] : ((((X354 @ (sub @ X355 @ (push @ X371 @ id)))) = $true) | ~ (((X354 @ X371)) = $true)))) <=> (hoasinduction_lem3v2_lthm = $true)) & (hoasinduction_lem3v2_gthm = $true) & ((hoasinduction_lem3a_lthm = $true) <=> (! [X372 : term] : (((sub @ X372 @ id)) = X372) => (! [X373 : (subst > term > subst > $o),X374 : term] : ((((X373 @ id @ (lam @ (sub @ X374 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X375 : term] : ((((X373 @ id @ (sub @ X374 @ (push @ X375 @ id)) @ id)) = $true) | ~ (((X373 @ id @ X375 @ id)) = $true)) | ~ ! [X376 : (subst > term > term)] : ((((X373 @ id @ (lam @ (X376 @ sh @ one)) @ id)) = $true) | ~ ! [X377 : subst,X378 : subst,X379 : term] : (((X376 @ (comp @ X378 @ X377) @ (sub @ X379 @ X377))) = ((sub @ (X376 @ X378 @ X379) @ X377))) | ~ ! [X380 : term] : ((((X373 @ id @ (X376 @ id @ X380) @ id)) = $true) | ~ (((X373 @ id @ X380 @ id)) = $true)))) => ! [X381 : (subst > term > subst > $o),X382 : term] : (~ ! [X383 : term] : ((((X381 @ id @ (sub @ X382 @ (push @ X383 @ id)) @ id)) = $true) | ~ (((X381 @ id @ X383 @ id)) = $true)) | (((X381 @ id @ (lam @ X382) @ id)) = $true) | ~ ! [X384 : (subst > term > term)] : (~ ! [X385 : term] : ((((X381 @ id @ (X384 @ id @ X385) @ id)) = $true) | ~ ($true = ((X381 @ id @ X385 @ id)))) | ~ ! [X386 : subst,X387 : term,X388 : subst] : (((X384 @ (comp @ X388 @ X386) @ (sub @ X387 @ X386))) = ((sub @ (X384 @ X388 @ X387) @ X386))) | (((X381 @ id @ (lam @ (X384 @ sh @ one)) @ id)) = $true)))))) & (axvarshift = $true) & (hoasinduction_lem1_gthm = $true) & ((induction2_lthm = $true) <=> (! [X389 : term] : (((sub @ X389 @ id)) = X389) => (! [X390 : term,X391 : (term > $o),X392 : subst] : (~ ! [X393 : term,X394 : term] : (~ (((X391 @ X394)) = $true) | ~ (((X391 @ X393)) = $true) | ($true = ((X391 @ (ap @ X393 @ X394))))) | (((X391 @ (sub @ X390 @ X392))) = $true) | ~ ! [X395 : term] : (~ (((var @ X395)) = $true) | (((X391 @ (sub @ X395 @ X392))) = $true)) | ~ ! [X396 : term] : ((((X391 @ (lam @ X396))) = $true) | ~ ! [X397 : term] : ((((X391 @ (sub @ X396 @ (push @ X397 @ id)))) = $true) | ~ (((X391 @ X397)) = $true)))) => ! [X398 : (term > $o),X399 : term] : (~ ! [X400 : term] : (~ (((var @ X400)) = $true) | (((X398 @ X400)) = $true)) | ~ ! [X401 : term,X402 : term] : (~ (((X398 @ X401)) = $true) | ~ (((X398 @ X402)) = $true) | (((X398 @ (ap @ X401 @ X402))) = $true)) | (((X398 @ X399)) = $true) | ~ ! [X403 : term] : (~ ! [X404 : term] : ((((X398 @ (sub @ X403 @ (push @ X404 @ id)))) = $true) | ~ (((X398 @ X404)) = $true)) | (((X398 @ (lam @ X403))) = $true)))))) & (! [X405 : term,X406 : term] : ((((lam @ X406)) != ((lam @ X405))) | (X405 = X406)) <=> (laminj = $true)) & (! [X407 : term,X408 : (term > $o),X409 : subst] : (~ ! [X410 : term] : (~ (((var @ X410)) = $true) | (((X408 @ (sub @ X410 @ X409))) = $true)) | ~ ! [X411 : term] : ((((X408 @ (lam @ X411))) = $true) | ~ ! [X412 : term] : (~ (((X408 @ X412)) = $true) | (((X408 @ (sub @ X411 @ (push @ X412 @ id)))) = $true))) | (((X408 @ (sub @ X407 @ X409))) = $true) | ~ ! [X413 : term,X414 : term] : ((((X408 @ (ap @ X414 @ X413))) = $true) | ~ (((X408 @ X414)) = $true) | ~ ($true = ((X408 @ X413))))) <=> (induction2lem = $true)) & (hoasapinj2_gthm = $true) & (! [X415 : (subst > term > subst > $o),X416 : term] : (~ ! [X417 : (subst > term > term)] : (~ ! [X418 : subst,X419 : term,X420 : subst] : (((X417 @ (comp @ X418 @ X420) @ (sub @ X419 @ X420))) = ((sub @ (X417 @ X418 @ X419) @ X420))) | ~ ! [X421 : term] : ((((X415 @ id @ (X417 @ id @ X421) @ id)) = $true) | ~ (((X415 @ id @ X421 @ id)) = $true)) | (((X415 @ id @ (lam @ (X417 @ sh @ one)) @ id)) = $true)) | ~ ! [X422 : term] : ((((X415 @ id @ (sub @ X416 @ (push @ X422 @ id)) @ id)) = $true) | ~ (((X415 @ id @ X422 @ id)) = $true)) | (((X415 @ id @ (lam @ X416) @ id)) = $true)) <=> (hoasinduction_lem3a = $true)) & (hoasinduction_lem2_lthm = $true) & ~ (((var @ (d2term @ d_one))) = $true) & (pushprop_lem2v2_lthm = $true) & (pushprop_lem2v2 = $true) & ! [X423 : term] : ? [X424 : d_term] : (((d2term @ X424)) = X423) & ((pushprop_lem0 = $true) <=> ! [X425 : subst,X426 : term,X427 : (term > $o)] : ~ ! [X428 : (term > $o)] : ~ ! [X429 : term] : (((X428 @ X429)) = ((X427 @ (sub @ X429 @ (push @ X426 @ X425)))))) & (! [X430 : term] : ~ ! [X431 : (subst > term > term)] : (((sub @ X430 @ (push @ one @ sh))) != ((X431 @ sh @ one))) <=> (hoasinduction_lem3b_lthm = $true)) & (! [X432 : term,X433 : term] : ((X432 = X433) | (((sub @ X433 @ sh)) != ((sub @ X432 @ sh)))) <=> (shinj = $true)) & ((hoasinduction_lem0_lthm = $true) <=> ! [X434 : (subst > term > subst > $o)] : ~ ! [X435 : (term > $o)] : ~ ! [X436 : term] : (((X434 @ id @ X436 @ id)) = ((X435 @ X436)))) & (((d2term @ d_one)) = one) & ((hoasinduction_lem3 = $true) <=> ! [X437 : term,X438 : (subst > term > subst > $o)] : ((((X438 @ id @ (lam @ X437) @ id)) = $true) | ~ ! [X439 : subst,X440 : subst,X441 : subst,X442 : term] : ((((X438 @ X439 @ X442 @ (comp @ X440 @ X441))) = $true) | ~ (((X438 @ (comp @ X439 @ X440) @ (sub @ X442 @ X440) @ X441)) = $true)) | ~ ! [X443 : subst,X444 : term,X445 : subst,X446 : subst] : ((((X438 @ (comp @ X443 @ X446) @ (sub @ X444 @ X446) @ X445)) = $true) | ~ (((X438 @ X443 @ X444 @ (comp @ X446 @ X445))) = $true)) | ~ ! [X447 : (subst > term > term)] : ((((X438 @ id @ (lam @ (X447 @ sh @ one)) @ id)) = $true) | ~ ! [X448 : term,X449 : subst,X450 : subst] : (((X447 @ (comp @ X450 @ X449) @ (sub @ X448 @ X449))) = ((sub @ (X447 @ X450 @ X448) @ X449))) | ~ ! [X451 : term] : (~ (((X438 @ id @ X451 @ id)) = $true) | (((X438 @ id @ (X447 @ id @ X451) @ id)) = $true))) | ~ ! [X452 : term] : ((((X438 @ id @ (sub @ X437 @ (push @ X452 @ id)) @ id)) = $true) | ~ (((X438 @ id @ X452 @ id)) = $true)))) & (! [X453 : (subst > term > subst > $o),X454 : term,X455 : (term > $o),X456 : term] : (~ ! [X457 : subst,X458 : subst,X459 : term,X460 : subst] : (~ (((X453 @ X460 @ X459 @ (comp @ X457 @ X458))) = $true) | (((X453 @ (comp @ X460 @ X457) @ (sub @ X459 @ X457) @ X458)) = $true)) | ~ (((X455 @ X454)) = $true) | ~ (((X455 @ X456)) = $true) | ~ ! [X461 : term] : (((X455 @ X461)) = ((X453 @ id @ X461 @ id))) | ~ ! [X462 : term,X463 : term] : (~ (((X453 @ id @ X463 @ id)) = $true) | ~ (((X453 @ id @ X462 @ id)) = $true) | (((X453 @ id @ (ap @ (sub @ X463 @ id) @ X462) @ id)) = $true)) | ~ ! [X464 : subst,X465 : subst,X466 : term,X467 : subst] : (~ (((X453 @ (comp @ X467 @ X465) @ (sub @ X466 @ X465) @ X464)) = $true) | (((X453 @ X467 @ X466 @ (comp @ X465 @ X464))) = $true)) | (((X455 @ (ap @ X454 @ X456))) = $true)) <=> (hoasinduction_lem2v2 = $true)) & (! [X468 : (subst > term > subst > $o),X469 : term,X470 : (term > $o)] : (~ ! [X471 : term] : (~ (((X470 @ X471)) = $true) | (((X470 @ (sub @ X469 @ (push @ X471 @ id)))) = $true)) | (((X470 @ (lam @ X469))) = $true) | ~ ! [X472 : (subst > term > term)] : (~ ! [X473 : subst,X474 : subst,X475 : term] : (((sub @ (X472 @ X474 @ X475) @ X473)) = ((X472 @ (comp @ X474 @ X473) @ (sub @ X475 @ X473)))) | ~ ! [X476 : term] : ((((X468 @ id @ (X472 @ id @ X476) @ id)) = $true) | ~ (((X468 @ id @ X476 @ id)) = $true)) | (((X468 @ id @ (lam @ (X472 @ sh @ one)) @ id)) = $true)) | ~ ! [X477 : term] : (((X470 @ X477)) = ((X468 @ id @ X477 @ id)))) <=> (hoasinduction_lem3v2a = $true)) & (hoasinduction_gthm = $true) & (pushprop = $true) & (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (! [X478 : (subst > term > subst > $o)] : ~ ! [X479 : (term > $o)] : ~ ! [X480 : term] : (((X479 @ X480)) = ((X478 @ id @ X480 @ id))) => (! [X481 : (term > $o),X482 : term] : (~ ! [X483 : term] : ((((X481 @ (lam @ X483))) = $true) | ~ ! [X484 : term] : (~ (((X481 @ X484)) = $true) | (((X481 @ (sub @ X483 @ (push @ X484 @ id)))) = $true))) | ~ ! [X485 : term,X486 : term] : ((((X481 @ (ap @ X486 @ X485))) = $true) | ~ (((X481 @ X485)) = $true) | ~ (((X481 @ X486)) = $true)) | ~ ! [X487 : term] : (~ (((var @ X487)) = $true) | (((X481 @ X487)) = $true)) | (((X481 @ X482)) = $true)) => (! [X488 : term] : (((sub @ X488 @ id)) = X488) => (! [X489 : (subst > term > subst > $o),X490 : (term > $o),X491 : term] : (~ ! [X492 : term] : ((((X490 @ (sub @ X491 @ (push @ X492 @ id)))) = $true) | ~ (((X490 @ X492)) = $true)) | ~ ! [X493 : term] : (((X489 @ id @ X493 @ id)) = ((X490 @ X493))) | ~ ! [X494 : (subst > term > term)] : (~ ! [X495 : subst,X496 : subst,X497 : term] : (((sub @ (X494 @ X496 @ X497) @ X495)) = ((X494 @ (comp @ X496 @ X495) @ (sub @ X497 @ X495)))) | (((X489 @ id @ (lam @ (X494 @ sh @ one)) @ id)) = $true) | ~ ! [X498 : term] : (~ (((X489 @ id @ X498 @ id)) = $true) | (((X489 @ id @ (X494 @ id @ X498) @ id)) = $true))) | (((X490 @ (lam @ X491))) = $true)) => ! [X499 : term,X500 : (subst > term > subst > $o)] : (~ ! [X501 : term,X502 : term] : (~ (((X500 @ id @ X501 @ id)) = $true) | ~ (((X500 @ id @ X502 @ id)) = $true) | (((X500 @ id @ (ap @ (sub @ X501 @ id) @ X502) @ id)) = $true)) | ($true = ((X500 @ id @ X499 @ id))) | ~ ! [X503 : (subst > term > term)] : (~ ! [X504 : term] : (~ (((X500 @ id @ X504 @ id)) = $true) | (((X500 @ id @ (X503 @ id @ X504) @ id)) = $true)) | (((X500 @ id @ (lam @ (X503 @ sh @ one)) @ id)) = $true) | ~ ! [X505 : subst,X506 : term,X507 : subst] : (((X503 @ (comp @ X507 @ X505) @ (sub @ X506 @ X505))) = ((sub @ (X503 @ X507 @ X506) @ X505)))))))))) & (hoasinduction_lem1v2 = $true) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X508 : term] : ~ ! [X509 : (subst > term > term)] : ~ ! [X510 : term,X511 : subst] : (((X509 @ X511 @ X510)) = ((sub @ X508 @ (push @ X510 @ X511))))) & ((hoasinduction_lem3aaa = $true) <=> ! [X512 : (subst > term > subst > $o),X513 : term] : (~ ! [X514 : (subst > term > term),X515 : term] : (~ ! [X516 : subst,X517 : subst,X518 : term] : (((sub @ (sub @ X515 @ (push @ X518 @ X517)) @ X516)) = ((sub @ (X514 @ X517 @ X518) @ X516))) | ~ ! [X519 : subst,X520 : subst,X521 : term] : (((sub @ X515 @ (push @ (sub @ X521 @ X520) @ (comp @ X519 @ X520)))) = ((X514 @ (comp @ X519 @ X520) @ (sub @ X521 @ X520)))) | (((X512 @ id @ (lam @ (X514 @ sh @ one)) @ id)) = $true) | ~ ! [X522 : term] : ((((X512 @ id @ (X514 @ id @ X522) @ id)) = $true) | ~ (((X512 @ id @ X522 @ id)) = $true))) | (((X512 @ id @ (lam @ (sub @ X513 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X523 : term] : (~ (((X512 @ id @ X523 @ id)) = $true) | (((X512 @ id @ (sub @ X513 @ (push @ X523 @ id)) @ id)) = $true)))) & ! [X524 : d_subst,X525 : d_subst] : ((((d2subst @ X524)) = ((d2subst @ X525))) => (X524 = X525)) & ((hoasinduction_lem3v2_f = $true) <=> ! [X526 : term] : ~ ! [X527 : (subst > term > term)] : ~ ! [X528 : subst,X529 : term] : (((sub @ X526 @ (push @ X529 @ X528))) = ((X527 @ X528 @ X529)))) & ((axidr = $true) <=> ! [X530 : subst] : (((comp @ X530 @ id)) = X530)) & ((induction2lem_lthm = $true) <=> (! [X531 : subst,X532 : term] : (((sub @ one @ (push @ X532 @ X531))) = X532) => (! [X533 : subst,X534 : term] : (((comp @ sh @ (push @ X534 @ X533))) = X533) => (! [X535 : subst] : (((comp @ X535 @ id)) = X535) => (! [X536 : term,X537 : (term > $o)] : (~ ! [X538 : term] : (~ (((X537 @ X538)) = $true) | (((X537 @ (lam @ X538))) = $true)) | (((X537 @ X536)) = $true) | ~ ! [X539 : term] : ((((X537 @ X539)) = $true) | ~ (((var @ X539)) = $true)) | ~ ! [X540 : term,X541 : term] : (~ (((X537 @ X540)) = $true) | (((X537 @ (ap @ X541 @ X540))) = $true) | ~ (((X537 @ X541)) = $true))) => ! [X542 : subst,X543 : term,X544 : (term > $o)] : ((((X544 @ (sub @ X543 @ X542))) = $true) | ~ ! [X545 : term] : ((((X544 @ (lam @ X545))) = $true) | ~ ! [X546 : term] : (~ (((X544 @ X546)) = $true) | (((X544 @ (sub @ X545 @ (push @ X546 @ id)))) = $true))) | ~ ! [X547 : term,X548 : term] : (~ (((X544 @ X548)) = $true) | (((X544 @ (ap @ X547 @ X548))) = $true) | ~ (((X544 @ X547)) = $true)) | ~ ! [X549 : term] : ((((X544 @ (sub @ X549 @ X542))) = $true) | ~ (((var @ X549)) = $true)))))))) & (lamnotvar = $true)),
  inference(fool_elimination,[],[f4])).
thf(f6,plain,(
  ~ (! [X0 : term,X1 : (term > $o),X2 : (term > $o),X3 : subst] : ((pushprop_p_and_p_prime @ X0 @ X3 @ X1 @ X2) => (! [X4 : term] : ((var @ X4) => (X2 @ X4)) => ! [X5 : term] : ((var @ X5) => (X1 @ (sub @ X5 @ (push @ X0 @ X3)))))) = pushprop_lem3v2)),
  inference(rectify,[],[f3])).
thf(f7,plain,(
  ~ (pushprop_lem3v2 = ((!! @ subst @ (^[Y0 : subst]: (!! @ (term > $o) @ (^[Y1 : term > $o]: (!! @ (term > $o) @ (^[Y2 : term > $o]: (!! @ term @ (^[Y3 : term]: ((pushprop_p_and_p_prime @ Y3 @ Y0 @ Y2 @ Y1) => ((!! @ term @ (^[Y4 : term]: ((var @ Y4) => (Y1 @ Y4)))) => (!! @ term @ (^[Y4 : term]: ((var @ Y4) => (Y2 @ (sub @ Y4 @ (push @ Y3 @ Y0))))))))))))))))))),
  inference(fool_elimination,[],[f6])).
thf(f8,plain,(
  (pushprop_lem3v2 != ((!! @ subst @ (^[Y0 : subst]: (!! @ (term > $o) @ (^[Y1 : term > $o]: (!! @ (term > $o) @ (^[Y2 : term > $o]: (!! @ term @ (^[Y3 : term]: ((pushprop_p_and_p_prime @ Y3 @ Y0 @ Y2 @ Y1) => ((!! @ term @ (^[Y4 : term]: ((var @ Y4) => (Y1 @ Y4)))) => (!! @ term @ (^[Y4 : term]: ((var @ Y4) => (Y2 @ (sub @ Y4 @ (push @ Y3 @ Y0))))))))))))))))))),
  inference(flattening,[],[f7])).
thf(f9,plain,(
  (pushprop_lem1_gthm = $true) & (induction2_gthm = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((induction2 = $true) <=> ! [X0 : term,X1 : (term > $o)] : (~ ! [X2 : term] : ((((X1 @ (lam @ X2))) = $true) | ~ ! [X3 : term] : (~ (((X1 @ X3)) = $true) | (((X1 @ (sub @ X2 @ (push @ X3 @ id)))) = $true))) | (((X1 @ X0)) = $true) | ~ ! [X4 : term,X5 : term] : (~ (((X1 @ X4)) = $true) | (((X1 @ (ap @ X5 @ X4))) = $true) | ~ (((X1 @ X5)) = $true)) | ~ ! [X6 : term] : ((((X1 @ X6)) = $true) | ~ (((var @ X6)) = $true)))) & ~ (lamnotap = $true) & (hoaslamnotvar_lthm = $true) & (pushprop_lem2v2_gthm = $true) & ((! [X7 : term,X10 : term,X8 : term,X9 : term] : ((X7 = X9) | (((ap @ X8 @ X7)) != ((ap @ X10 @ X9)))) => ! [X14 : term,X11 : term,X13 : term,X12 : term] : ((((ap @ (sub @ X13 @ id) @ X12)) != ((ap @ (sub @ X14 @ id) @ X11))) | (X11 = X12))) <=> (hoasapinj2_lthm = $true)) & ! [X15 : subst] : ? [X16 : d_subst] : (((d2subst @ X16)) = X15) & (hoasapnotvar_lthm = $true) & (axmap = $true) & (axclos = $true) & (hoasinduction_lem1_lthm = $true) & ~ (ulamvar1 = $true) & ((hoasapinj1 = $true) <=> ! [X18 : term,X19 : term,X17 : term,X20 : term] : ((((ap @ (sub @ X20 @ id) @ X19)) != ((ap @ (sub @ X17 @ id) @ X18))) | (X17 = X20))) & (hoasapnotvar_gthm = $true) & ((hoasinduction = $true) <=> ! [X21 : (subst > term > subst > $o),X22 : term] : (~ ! [X23 : subst,X24 : subst,X25 : subst,X26 : term] : ((((X21 @ (comp @ X25 @ X24) @ (sub @ X26 @ X24) @ X23)) = $true) | ~ (((X21 @ X25 @ X26 @ (comp @ X24 @ X23))) = $true)) | ~ ! [X27 : (subst > term > term)] : ((((X21 @ id @ (lam @ (X27 @ sh @ one)) @ id)) = $true) | ~ ! [X28 : term] : (~ (((X21 @ id @ X28 @ id)) = $true) | (((X21 @ id @ (X27 @ id @ X28) @ id)) = $true)) | ~ ! [X30 : term,X29 : subst,X31 : subst] : (((X27 @ (comp @ X29 @ X31) @ (sub @ X30 @ X31))) = ((sub @ (X27 @ X29 @ X30) @ X31)))) | ~ ! [X32 : term,X33 : term] : ((((X21 @ id @ (ap @ (sub @ X33 @ id) @ X32) @ id)) = $true) | ~ ($true = ((X21 @ id @ X33 @ id))) | ~ (((X21 @ id @ X32 @ id)) = $true)) | (((X21 @ id @ X22 @ id)) = $true) | ~ ! [X34 : term,X35 : subst,X36 : subst,X37 : subst] : (~ (((X21 @ (comp @ X36 @ X35) @ (sub @ X34 @ X35) @ X37)) = $true) | (((X21 @ X36 @ X34 @ (comp @ X35 @ X37))) = $true)) | ~ ! [X38 : term] : (~ (((var @ (sub @ X38 @ id))) = $true) | (((X21 @ id @ X38 @ id)) = $true)))) & (((d2subst @ d_id)) = sh) & (hoaslamnotvar = $true) & (pushprop_lem3v2 = $true) & (! [X39 : (subst > term > subst > $o),X40 : term] : (~ ! [X41 : term,X42 : term] : ((((X39 @ id @ (ap @ (sub @ X41 @ id) @ X42) @ id)) = $true) | ~ (((X39 @ id @ X41 @ id)) = $true) | ~ (((X39 @ id @ X42 @ id)) = $true)) | ($true = ((X39 @ id @ X40 @ id))) | ~ ! [X43 : (subst > term > term)] : (~ ! [X46 : subst,X44 : subst,X45 : term] : (((sub @ (X43 @ X46 @ X45) @ X44)) = ((X43 @ (comp @ X46 @ X44) @ (sub @ X45 @ X44)))) | (((X39 @ id @ (lam @ (X43 @ sh @ one)) @ id)) = $true) | ~ ! [X47 : term] : ((((X39 @ id @ (X43 @ id @ X47) @ id)) = $true) | ~ (((X39 @ id @ X47 @ id)) = $true)))) <=> (hoasinduction_no_psi_cond = $true)) & (! [X49 : (subst > term > term),X48 : term,X50 : term] : (~ ! [X51 : subst,X53 : term,X52 : subst] : (((X49 @ (comp @ X51 @ X52) @ (sub @ X53 @ X52))) = ((sub @ (X49 @ X51 @ X53) @ X52))) | (((ap @ (sub @ X50 @ id) @ X48)) != ((lam @ (X49 @ sh @ one))))) <=> (hoaslamnotap = $true)) & (pushprop_lthm_orig = $true) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X54 : term,X55 : (subst > term > subst > $o)] : (~ ! [X56 : (subst > term > term)] : (~ ! [X57 : term] : ((((X55 @ id @ (X56 @ id @ X57) @ id)) = $true) | ~ (((X55 @ id @ X57 @ id)) = $true)) | ~ ! [X58 : term,X59 : subst,X60 : subst] : (((X56 @ (comp @ X60 @ X59) @ (sub @ X58 @ X59))) = ((sub @ (X56 @ X60 @ X58) @ X59))) | (((X55 @ id @ (lam @ (X56 @ sh @ one)) @ id)) = $true)) | ~ ! [X61 : term] : (~ ($true = ((X55 @ id @ X61 @ id))) | (((X55 @ id @ (sub @ X54 @ (push @ X61 @ id)) @ id)) = $true)) | (((X55 @ id @ (lam @ (sub @ X54 @ (push @ one @ sh))) @ id)) = $true))) & (pushprop_lem0_gthm = $true) & ((hoaslaminj_lthm = $true) <=> (! [X62 : term,X63 : subst] : (((sub @ one @ (push @ X62 @ X63))) = X62) => (! [X64 : term,X65 : subst] : (((comp @ sh @ (push @ X64 @ X65))) = X65) => (! [X66 : term,X67 : term] : ((X66 = X67) | (((lam @ X66)) != ((lam @ X67)))) => ! [X71 : term,X68 : subst,X70 : (subst > term > term),X69 : (subst > term > term)] : (~ ! [X77 : term,X75 : subst,X76 : subst] : (((sub @ (X70 @ X76 @ X77) @ X75)) = ((X70 @ (comp @ X76 @ X75) @ (sub @ X77 @ X75)))) | ~ ! [X73 : term,X72 : subst,X74 : subst] : (((sub @ (X69 @ X72 @ X73) @ X74)) = ((X69 @ (comp @ X72 @ X74) @ (sub @ X73 @ X74)))) | (((lam @ (X70 @ sh @ one))) != ((lam @ (X69 @ sh @ one)))) | (((X69 @ X68 @ X71)) = ((X70 @ X68 @ X71)))))))) & ! [X78 : d_subst] : (d_id = X78) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ~ (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) = $true) & ((pushprop_lem1 = $true) <=> ! [X79 : subst,X80 : (term > $o),X81 : term,X82 : (term > $o),X83 : term] : ((((X82 @ (sub @ X81 @ (push @ X83 @ X79)))) = $true) | ~ (((X80 @ X81)) = $true))) & (hoaslaminj_gthm = $true) & ((hoasapinj1_lthm = $true) <=> (! [X84 : term] : (((sub @ X84 @ id)) = X84) => (! [X87 : term,X85 : term,X86 : term,X88 : term] : ((((ap @ X88 @ X85)) != ((ap @ X87 @ X86))) | (X87 = X88)) => ! [X92 : term,X89 : term,X90 : term,X91 : term] : ((X89 = X92) | (((ap @ (sub @ X89 @ id) @ X91)) != ((ap @ (sub @ X92 @ id) @ X90))))))) & (hoasinduction_lem3b_gthm = $true) & ! [X93 : d_term] : (d_one = X93) & ((termmset_lthm = $true) <=> (! [X94 : term] : (((sub @ X94 @ id)) = X94) => ! [X95 : term] : (((sub @ X95 @ id)) = X95))) & (hoasinduction_lem2_gthm = $true) & (! [X96 : subst,X97 : term] : (((sub @ one @ (push @ X97 @ X96))) = X97) <=> (axvarcons = $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (axassoc = $true) & (hoaslamnotap_lthm = $true) & ((pushprop_lem1v2 = $true) <=> ! [X98 : term,X99 : (term > $o),X100 : subst,X101 : (term > $o)] : (~ (((X99 @ X98)) = $true) | ~ ! [X102 : term] : (((X101 @ X102)) = ((X99 @ (sub @ X102 @ (push @ X98 @ X100))))) | (((X101 @ one)) = $true))) & ((termmset = $true) <=> ! [X103 : term] : (((sub @ X103 @ id)) = X103)) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (hoasinduction_lem1 = $true) & (hoasapnotvar = $true) & ((! [X104 : term,X105 : subst] : (((sub @ one @ (push @ X104 @ X105))) = X104) => ! [X106 : (term > $o),X107 : subst,X108 : term,X109 : (term > $o)] : ((((X109 @ one)) = $true) | ~ ! [X110 : term] : (((X109 @ X110)) = ((X106 @ (sub @ X110 @ (push @ X108 @ X107))))) | ~ (((X106 @ X108)) = $true))) <=> (pushprop_lem1v2_lthm = $true)) & ((apinj1 = $true) <=> ! [X111 : term,X113 : term,X114 : term,X112 : term] : ((X112 = X114) | (((ap @ X112 @ X111)) != ((ap @ X114 @ X113))))) & ((hoaslaminj = $true) <=> ! [X117 : (subst > term > term),X115 : term,X118 : subst,X116 : (subst > term > term)] : ((((X117 @ X118 @ X115)) = ((X116 @ X118 @ X115))) | (((lam @ (X116 @ sh @ one))) != ((lam @ (X117 @ sh @ one)))) | ~ ! [X119 : subst,X120 : term,X121 : subst] : (((X117 @ (comp @ X119 @ X121) @ (sub @ X120 @ X121))) = ((sub @ (X117 @ X119 @ X120) @ X121))) | ~ ! [X122 : subst,X123 : subst,X124 : term] : (((sub @ (X116 @ X123 @ X124) @ X122)) = ((X116 @ (comp @ X123 @ X122) @ (sub @ X124 @ X122)))))) & ((hoasinduction_lem3aa = $true) <=> ! [X125 : term,X126 : (subst > term > subst > $o)] : ((((X126 @ id @ (lam @ (sub @ X125 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X127 : term] : ((((X126 @ id @ (sub @ X125 @ (push @ X127 @ id)) @ id)) = $true) | ~ (((X126 @ id @ X127 @ id)) = $true)) | ~ ! [X128 : (subst > term > term)] : ((((X126 @ id @ (lam @ (X128 @ sh @ one)) @ id)) = $true) | ~ ! [X131 : subst,X130 : term,X129 : subst] : (((sub @ (X128 @ X131 @ X130) @ X129)) = ((X128 @ (comp @ X131 @ X129) @ (sub @ X130 @ X129)))) | ~ ! [X132 : term] : (~ (((X126 @ id @ X132 @ id)) = $true) | (((X126 @ id @ (X128 @ id @ X132) @ id)) = $true))))) & ((induction = $true) <=> ! [X133 : (term > $o),X134 : term] : (~ ! [X135 : term] : ((((X133 @ (lam @ X135))) = $true) | ~ (((X133 @ X135)) = $true)) | ~ ! [X136 : term] : (~ (((var @ X136)) = $true) | (((X133 @ X136)) = $true)) | (((X133 @ X134)) = $true) | ~ ! [X137 : term,X138 : term] : (~ (((X133 @ X137)) = $true) | (((X133 @ (ap @ X137 @ X138))) = $true) | ~ (((X133 @ X138)) = $true)))) & (! [X139 : subst] : (((push @ (sub @ one @ X139) @ (comp @ sh @ X139))) = X139) <=> (axscons = $true)) & (axapp = $true) & (induction2lem_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (pushprop_lthm = $true) & (hoasinduction_lem3a_gthm = $true) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_lem2v2_gthm = $true) & ((apinj2 = $true) <=> ! [X140 : term,X142 : term,X141 : term,X143 : term] : ((X140 = X143) | (((ap @ X141 @ X140)) != ((ap @ X142 @ X143))))) & (((d2subst @ d_id)) = id) & (hoasinduction_lem3_gthm = $true) & (pushprop_lem3v2_lthm = $true) & ((hoasinduction_lthm = $true) <=> (! [X144 : (term > $o),X145 : term] : ((((X144 @ X145)) = $true) | ~ ! [X146 : term,X147 : term] : ((((X144 @ (ap @ X147 @ X146))) = $true) | ~ (((X144 @ X147)) = $true) | ~ (((X144 @ X146)) = $true)) | ~ ! [X148 : term] : (($true = ((X144 @ X148))) | ~ (((var @ X148)) = $true)) | ~ ! [X149 : term] : (~ ! [X150 : term] : (~ ($true = ((X144 @ X150))) | ($true = ((X144 @ (sub @ X149 @ (push @ X150 @ id)))))) | (((X144 @ (lam @ X149))) = $true))) => (! [X151 : (subst > term > subst > $o),X152 : term,X153 : term] : ((((X151 @ id @ (ap @ X152 @ X153) @ id)) = $true) | ~ ! [X154 : term,X155 : term] : (~ (((X151 @ id @ X155 @ id)) = $true) | ~ (((X151 @ id @ X154 @ id)) = $true) | (((X151 @ id @ (ap @ (sub @ X155 @ id) @ X154) @ id)) = $true)) | ~ ! [X156 : subst,X157 : subst,X158 : subst,X159 : term] : (~ (((X151 @ X156 @ X159 @ (comp @ X158 @ X157))) = $true) | (((X151 @ (comp @ X156 @ X158) @ (sub @ X159 @ X158) @ X157)) = $true)) | ~ ! [X160 : subst,X161 : subst,X162 : subst,X163 : term] : ((((X151 @ X160 @ X163 @ (comp @ X161 @ X162))) = $true) | ~ (((X151 @ (comp @ X160 @ X161) @ (sub @ X163 @ X161) @ X162)) = $true)) | ~ (((X151 @ id @ X153 @ id)) = $true) | ~ (((X151 @ id @ X152 @ id)) = $true)) => (! [X164 : (subst > term > subst > $o),X165 : term] : (~ ! [X166 : (subst > term > term)] : (~ ! [X168 : subst,X169 : subst,X167 : term] : (((sub @ (X166 @ X168 @ X167) @ X169)) = ((X166 @ (comp @ X168 @ X169) @ (sub @ X167 @ X169)))) | ~ ! [X170 : term] : (~ (((X164 @ id @ X170 @ id)) = $true) | (((X164 @ id @ (X166 @ id @ X170) @ id)) = $true)) | (((X164 @ id @ (lam @ (X166 @ sh @ one)) @ id)) = $true)) | ~ ! [X171 : subst,X172 : subst,X173 : term,X174 : subst] : ((((X164 @ (comp @ X171 @ X172) @ (sub @ X173 @ X172) @ X174)) = $true) | ~ (((X164 @ X171 @ X173 @ (comp @ X172 @ X174))) = $true)) | ~ ! [X175 : term] : ((((X164 @ id @ (sub @ X165 @ (push @ X175 @ id)) @ id)) = $true) | ~ (((X164 @ id @ X175 @ id)) = $true)) | ~ ! [X176 : subst,X177 : subst,X178 : subst,X179 : term] : ((((X164 @ X176 @ X179 @ (comp @ X177 @ X178))) = $true) | ~ (((X164 @ (comp @ X176 @ X177) @ (sub @ X179 @ X177) @ X178)) = $true)) | (((X164 @ id @ (lam @ X165) @ id)) = $true)) => ! [X180 : (subst > term > subst > $o),X181 : term] : (~ ! [X182 : subst,X183 : term,X184 : subst,X185 : subst] : ((((X180 @ X184 @ X183 @ (comp @ X182 @ X185))) = $true) | ~ (((X180 @ (comp @ X184 @ X182) @ (sub @ X183 @ X182) @ X185)) = $true)) | ~ ! [X186 : subst,X187 : term,X188 : subst,X189 : subst] : ((((X180 @ (comp @ X188 @ X186) @ (sub @ X187 @ X186) @ X189)) = $true) | ~ (((X180 @ X188 @ X187 @ (comp @ X186 @ X189))) = $true)) | (((X180 @ id @ X181 @ id)) = $true) | ~ ! [X190 : term] : (~ (((var @ (sub @ X190 @ id))) = $true) | (((X180 @ id @ X190 @ id)) = $true)) | ~ ! [X191 : term,X192 : term] : ((((X180 @ id @ (ap @ (sub @ X191 @ id) @ X192) @ id)) = $true) | ~ (((X180 @ id @ X192 @ id)) = $true) | ~ (((X180 @ id @ X191 @ id)) = $true)) | ~ ! [X193 : (subst > term > term)] : (~ ! [X194 : term] : ((((X180 @ id @ (X193 @ id @ X194) @ id)) = $true) | ~ (((X180 @ id @ X194 @ id)) = $true)) | ~ ! [X195 : subst,X196 : subst,X197 : term] : (((sub @ (X193 @ X196 @ X197) @ X195)) = ((X193 @ (comp @ X196 @ X195) @ (sub @ X197 @ X195)))) | (((X180 @ id @ (lam @ (X193 @ sh @ one)) @ id)) = $true))))))) & (termmset_gthm = $true) & ((pushprop_lem1_lthm = $true) <=> (! [X198 : term,X199 : subst] : (((sub @ one @ (push @ X198 @ X199))) = X198) => (! [X201 : subst,X200 : term] : (((comp @ sh @ (push @ X200 @ X201))) = X201) => ! [X202 : term,X203 : (term > $o),X204 : term,X205 : (term > $o),X206 : subst] : ((((X205 @ (sub @ X202 @ (push @ X204 @ X206)))) = $true) | ~ (((X203 @ X202)) = $true))))) & (substmonoid_gthm = $true) & ((axidl = $true) <=> ! [X207 : subst] : (((comp @ id @ X207)) = X207)) & (apnotvar = $true) & (hoaslamnotvar_gthm = $true) & (! [X208 : term] : (((sub @ X208 @ id)) = X208) <=> (axvarid = $true)) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (! [X211 : term,X212 : term,X209 : term,X210 : term] : ((X210 = X212) | (((ap @ (sub @ X209 @ id) @ X212)) != ((ap @ (sub @ X211 @ id) @ X210)))) <=> (hoasapinj2 = $true)) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & (ulamvarsh = $true) & ((hoasinduction_lem3b = $true) <=> ! [X213 : term] : ~ ! [X214 : (subst > term > term)] : (((X214 @ sh @ one)) != ((sub @ X213 @ (push @ one @ sh))))) & (hoasapinj1_gthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (pushprop_gthm = $true) & (! [X215 : term,X216 : (subst > term > subst > $o),X217 : (term > $o)] : (~ ! [X218 : term] : (~ (((X217 @ X218)) = $true) | ($true = ((X217 @ (sub @ X215 @ (push @ X218 @ id)))))) | ~ ! [X219 : subst,X220 : subst,X221 : subst,X222 : term] : (~ (((X216 @ (comp @ X221 @ X219) @ (sub @ X222 @ X219) @ X220)) = $true) | (((X216 @ X221 @ X222 @ (comp @ X219 @ X220))) = $true)) | ~ ! [X223 : term] : (((X216 @ id @ X223 @ id)) = ((X217 @ X223))) | ~ ! [X224 : (subst > term > term)] : ((((X216 @ id @ (lam @ (X224 @ sh @ one)) @ id)) = $true) | ~ ! [X225 : term] : ((((X216 @ id @ (X224 @ id @ X225) @ id)) = $true) | ~ (((X216 @ id @ X225 @ id)) = $true)) | ~ ! [X227 : subst,X226 : term,X228 : subst] : (((X224 @ (comp @ X228 @ X227) @ (sub @ X226 @ X227))) = ((sub @ (X224 @ X228 @ X226) @ X227)))) | ~ ! [X229 : subst,X230 : term,X231 : subst,X232 : subst] : ((((X216 @ (comp @ X231 @ X232) @ (sub @ X230 @ X232) @ X229)) = $true) | ~ (((X216 @ X231 @ X230 @ (comp @ X232 @ X229))) = $true)) | (((X217 @ (lam @ X215))) = $true)) <=> (hoasinduction_lem3v2 = $true)) & ((hoasinduction_lem2 = $true) <=> ! [X233 : (subst > term > subst > $o),X234 : term,X235 : term] : (~ ! [X236 : term,X237 : term] : (~ (((X233 @ id @ X237 @ id)) = $true) | (((X233 @ id @ (ap @ (sub @ X237 @ id) @ X236) @ id)) = $true) | ~ (((X233 @ id @ X236 @ id)) = $true)) | ~ (((X233 @ id @ X234 @ id)) = $true) | ~ ! [X238 : subst,X239 : term,X240 : subst,X241 : subst] : ((((X233 @ X238 @ X239 @ (comp @ X241 @ X240))) = $true) | ~ (((X233 @ (comp @ X238 @ X241) @ (sub @ X239 @ X241) @ X240)) = $true)) | ~ (((X233 @ id @ X235 @ id)) = $true) | ~ ! [X242 : subst,X243 : term,X244 : subst,X245 : subst] : (~ (((X233 @ X245 @ X243 @ (comp @ X242 @ X244))) = $true) | (((X233 @ (comp @ X245 @ X242) @ (sub @ X243 @ X242) @ X244)) = $true)) | (((X233 @ id @ (ap @ X234 @ X235) @ id)) = $true))) & (! [X246 : (term > $o),X247 : term,X248 : subst] : ~ ! [X249 : (term > $o)] : ~ ! [X250 : term] : (((X246 @ (sub @ X250 @ (push @ X247 @ X248)))) = ((X249 @ X250))) <=> (pushprop_lem0_lthm = $true)) & ((! [X251 : subst] : (((comp @ id @ X251)) = X251) => (! [X252 : subst] : (((comp @ X252 @ id)) = X252) => (! [X254 : subst] : (((comp @ X254 @ id)) = X254) & ! [X253 : subst] : (((comp @ id @ X253)) = X253)))) <=> (substmonoid_lthm = $true)) & ((hoasinduction_lem0 = $true) <=> ! [X255 : (subst > term > subst > $o)] : ~ ! [X256 : (term > $o)] : ~ ! [X257 : term] : (((X256 @ X257)) = ((X255 @ id @ X257 @ id)))) & ((! [X258 : term] : (((sub @ X258 @ id)) = X258) => (! [X259 : (subst > term > subst > $o),X260 : term] : (~ ! [X261 : (subst > term > term)] : ((((X259 @ id @ (lam @ (X261 @ sh @ one)) @ id)) = $true) | ~ ! [X262 : term] : (~ (((X259 @ id @ X262 @ id)) = $true) | (((X259 @ id @ (X261 @ id @ X262) @ id)) = $true)) | ~ ! [X265 : term,X264 : subst,X263 : subst] : (((X261 @ (comp @ X263 @ X264) @ (sub @ X265 @ X264))) = ((sub @ (X261 @ X263 @ X265) @ X264)))) | ~ ! [X266 : term] : (~ (((X259 @ id @ X266 @ id)) = $true) | (((X259 @ id @ (sub @ X260 @ (push @ X266 @ id)) @ id)) = $true)) | (((X259 @ id @ (lam @ (sub @ X260 @ (push @ one @ sh))) @ id)) = $true)) => ! [X267 : (subst > term > subst > $o),X268 : term] : (~ ! [X269 : term] : ((((X267 @ id @ (sub @ X268 @ (push @ X269 @ id)) @ id)) = $true) | ~ (((X267 @ id @ X269 @ id)) = $true)) | (((X267 @ id @ (lam @ X268) @ id)) = $true) | ~ ! [X270 : subst,X271 : term,X272 : subst,X273 : subst] : (~ (((X267 @ (comp @ X273 @ X270) @ (sub @ X271 @ X270) @ X272)) = $true) | (((X267 @ X273 @ X271 @ (comp @ X270 @ X272))) = $true)) | ~ ! [X274 : (subst > term > term)] : (~ ! [X276 : subst,X275 : subst,X277 : term] : (((X274 @ (comp @ X276 @ X275) @ (sub @ X277 @ X275))) = ((sub @ (X274 @ X276 @ X277) @ X275))) | ~ ! [X278 : term] : ((((X267 @ id @ (X274 @ id @ X278) @ id)) = $true) | ~ (((X267 @ id @ X278 @ id)) = $true)) | (((X267 @ id @ (lam @ (X274 @ sh @ one)) @ id)) = $true)) | ~ ! [X279 : subst,X280 : term,X281 : subst,X282 : subst] : ((((X267 @ (comp @ X282 @ X281) @ (sub @ X280 @ X281) @ X279)) = $true) | ~ (((X267 @ X282 @ X280 @ (comp @ X281 @ X279))) = $true))))) <=> (hoasinduction_lem3_lthm = $true)) & ((hoasinduction_lthm_3 = $true) <=> (! [X283 : (subst > term > subst > $o)] : ~ ! [X284 : (term > $o)] : ~ ! [X285 : term] : (((X283 @ id @ X285 @ id)) = ((X284 @ X285))) => (! [X286 : term,X287 : (term > $o)] : (~ ! [X288 : term] : (~ (((var @ X288)) = $true) | (((X287 @ X288)) = $true)) | ~ ! [X289 : term] : ((((X287 @ (lam @ X289))) = $true) | ~ ! [X290 : term] : (~ (((X287 @ X290)) = $true) | (((X287 @ (sub @ X289 @ (push @ X290 @ id)))) = $true))) | ~ ! [X291 : term,X292 : term] : (~ (((X287 @ X292)) = $true) | (((X287 @ (ap @ X292 @ X291))) = $true) | ~ (((X287 @ X291)) = $true)) | (((X287 @ X286)) = $true)) => (! [X293 : term] : (((sub @ X293 @ id)) = X293) => (! [X294 : term,X295 : (subst > term > subst > $o),X296 : (term > $o)] : ((((X296 @ (lam @ X294))) = $true) | ~ ! [X297 : (subst > term > term)] : (~ ! [X300 : subst,X298 : term,X299 : subst] : (((X297 @ (comp @ X299 @ X300) @ (sub @ X298 @ X300))) = ((sub @ (X297 @ X299 @ X298) @ X300))) | (((X295 @ id @ (lam @ (X297 @ sh @ one)) @ id)) = $true) | ~ ! [X301 : term] : ((((X295 @ id @ (X297 @ id @ X301) @ id)) = $true) | ~ (((X295 @ id @ X301 @ id)) = $true))) | ~ ! [X302 : term] : ((((X296 @ (sub @ X294 @ (push @ X302 @ id)))) = $true) | ~ (((X296 @ X302)) = $true)) | ~ ! [X303 : term] : (((X296 @ X303)) = ((X295 @ id @ X303 @ id)))) => ! [X304 : (subst > term > subst > $o),X305 : term] : (~ ! [X306 : subst,X307 : subst,X308 : subst,X309 : term] : ((((X304 @ (comp @ X308 @ X307) @ (sub @ X309 @ X307) @ X306)) = $true) | ~ (((X304 @ X308 @ X309 @ (comp @ X307 @ X306))) = $true)) | ~ ! [X310 : (subst > term > term)] : ((((X304 @ id @ (lam @ (X310 @ sh @ one)) @ id)) = $true) | ~ ! [X311 : term] : (~ (((X304 @ id @ X311 @ id)) = $true) | (((X304 @ id @ (X310 @ id @ X311) @ id)) = $true)) | ~ ! [X312 : subst,X313 : term,X314 : subst] : (((sub @ (X310 @ X314 @ X313) @ X312)) = ((X310 @ (comp @ X314 @ X312) @ (sub @ X313 @ X312))))) | (((X304 @ id @ X305 @ id)) = $true) | ~ ! [X315 : term,X316 : subst,X317 : subst,X318 : subst] : ((((X304 @ X317 @ X315 @ (comp @ X318 @ X316))) = $true) | ~ (((X304 @ (comp @ X317 @ X318) @ (sub @ X315 @ X318) @ X316)) = $true)) | ~ ! [X319 : term] : (~ (((var @ (sub @ X319 @ id))) = $true) | (((X304 @ id @ X319 @ id)) = $true)) | ~ ! [X320 : term,X321 : term] : ((((X304 @ id @ (ap @ (sub @ X321 @ id) @ X320) @ id)) = $true) | ~ (((X304 @ id @ X320 @ id)) = $true) | ~ (((X304 @ id @ X321 @ id)) = $true)))))))) & (ulamvarind = $true) & ((! [X323 : subst] : (((comp @ id @ X323)) = X323) & ! [X322 : subst] : (((comp @ X322 @ id)) = X322)) <=> (substmonoid = $true)) & (axabs = $true) & (hoaslamnotap_gthm = $true) & ! [X324 : d_term,X325 : d_term] : ((((d2term @ X325)) = ((d2term @ X324))) => (X324 = X325)) & ((axshiftcons = $true) <=> ! [X326 : subst,X327 : term] : (((comp @ sh @ (push @ X327 @ X326))) = X326)) & ((! [X328 : term] : ~ ! [X329 : (subst > term > term)] : ~ ! [X330 : term,X331 : subst] : (((sub @ X328 @ (push @ X330 @ X331))) = ((X329 @ X331 @ X330))) => (! [X332 : term] : (((sub @ X332 @ id)) = X332) => ! [X333 : term,X334 : (subst > term > subst > $o),X335 : (term > $o)] : (~ ! [X336 : term] : (~ (((X335 @ X336)) = $true) | (((X335 @ (sub @ X333 @ (push @ X336 @ id)))) = $true)) | ~ ! [X337 : (subst > term > term)] : (~ ! [X338 : term] : ((((X334 @ id @ (X337 @ id @ X338) @ id)) = $true) | ~ (((X334 @ id @ X338 @ id)) = $true)) | ~ ! [X339 : subst,X340 : term,X341 : subst] : (((sub @ (X337 @ X339 @ X340) @ X341)) = ((X337 @ (comp @ X339 @ X341) @ (sub @ X340 @ X341)))) | (((X334 @ id @ (lam @ (X337 @ sh @ one)) @ id)) = $true)) | (((X335 @ (lam @ X333))) = $true) | ~ ! [X342 : term] : (((X335 @ X342)) = ((X334 @ id @ X342 @ id)))))) <=> (hoasinduction_lem3v2a_lthm = $true)) & ((! [X343 : term] : (((sub @ X343 @ id)) = X343) => ! [X344 : (term > $o),X345 : term,X346 : (subst > term > subst > $o)] : (~ ! [X347 : term] : (((X344 @ X347)) = ((X346 @ id @ X347 @ id))) | ~ ! [X348 : subst,X349 : subst,X350 : term,X351 : subst] : (~ (((X346 @ (comp @ X348 @ X351) @ (sub @ X350 @ X351) @ X349)) = $true) | (((X346 @ X348 @ X350 @ (comp @ X351 @ X349))) = $true)) | ~ ! [X352 : subst,X353 : subst,X354 : term,X355 : subst] : ((((X346 @ (comp @ X355 @ X353) @ (sub @ X354 @ X353) @ X352)) = $true) | ~ (((X346 @ X355 @ X354 @ (comp @ X353 @ X352))) = $true)) | (((X344 @ (lam @ X345))) = $true) | ~ ! [X356 : (subst > term > term)] : (~ ! [X357 : term] : (~ (((X346 @ id @ X357 @ id)) = $true) | (((X346 @ id @ (X356 @ id @ X357) @ id)) = $true)) | ~ ! [X358 : subst,X360 : term,X359 : subst] : (((X356 @ (comp @ X358 @ X359) @ (sub @ X360 @ X359))) = ((sub @ (X356 @ X358 @ X360) @ X359))) | (((X346 @ id @ (lam @ (X356 @ sh @ one)) @ id)) = $true)) | ~ ! [X361 : term] : ((((X344 @ (sub @ X345 @ (push @ X361 @ id)))) = $true) | ~ (((X344 @ X361)) = $true)))) <=> (hoasinduction_lem3v2_lthm = $true)) & (hoasinduction_lem3v2_gthm = $true) & ((hoasinduction_lem3a_lthm = $true) <=> (! [X362 : term] : (((sub @ X362 @ id)) = X362) => (! [X363 : (subst > term > subst > $o),X364 : term] : ((((X363 @ id @ (lam @ (sub @ X364 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X365 : term] : ((((X363 @ id @ (sub @ X364 @ (push @ X365 @ id)) @ id)) = $true) | ~ (((X363 @ id @ X365 @ id)) = $true)) | ~ ! [X366 : (subst > term > term)] : ((((X363 @ id @ (lam @ (X366 @ sh @ one)) @ id)) = $true) | ~ ! [X369 : term,X367 : subst,X368 : subst] : (((X366 @ (comp @ X368 @ X367) @ (sub @ X369 @ X367))) = ((sub @ (X366 @ X368 @ X369) @ X367))) | ~ ! [X370 : term] : ((((X363 @ id @ (X366 @ id @ X370) @ id)) = $true) | ~ (((X363 @ id @ X370 @ id)) = $true)))) => ! [X371 : (subst > term > subst > $o),X372 : term] : (~ ! [X373 : term] : ((((X371 @ id @ (sub @ X372 @ (push @ X373 @ id)) @ id)) = $true) | ~ (((X371 @ id @ X373 @ id)) = $true)) | (((X371 @ id @ (lam @ X372) @ id)) = $true) | ~ ! [X374 : (subst > term > term)] : (~ ! [X375 : term] : ((((X371 @ id @ (X374 @ id @ X375) @ id)) = $true) | ~ (((X371 @ id @ X375 @ id)) = $true)) | ~ ! [X377 : term,X376 : subst,X378 : subst] : (((X374 @ (comp @ X378 @ X376) @ (sub @ X377 @ X376))) = ((sub @ (X374 @ X378 @ X377) @ X376))) | (((X371 @ id @ (lam @ (X374 @ sh @ one)) @ id)) = $true)))))) & (axvarshift = $true) & (hoasinduction_lem1_gthm = $true) & ((induction2_lthm = $true) <=> (! [X379 : term] : (((sub @ X379 @ id)) = X379) => (! [X380 : term,X381 : (term > $o),X382 : subst] : (~ ! [X383 : term,X384 : term] : (~ (((X381 @ X384)) = $true) | ~ (((X381 @ X383)) = $true) | (((X381 @ (ap @ X383 @ X384))) = $true)) | (((X381 @ (sub @ X380 @ X382))) = $true) | ~ ! [X385 : term] : (~ (((var @ X385)) = $true) | (((X381 @ (sub @ X385 @ X382))) = $true)) | ~ ! [X386 : term] : ((((X381 @ (lam @ X386))) = $true) | ~ ! [X387 : term] : ((((X381 @ (sub @ X386 @ (push @ X387 @ id)))) = $true) | ~ (((X381 @ X387)) = $true)))) => ! [X388 : (term > $o),X389 : term] : (~ ! [X390 : term] : (~ (((var @ X390)) = $true) | (((X388 @ X390)) = $true)) | ~ ! [X391 : term,X392 : term] : (~ (((X388 @ X391)) = $true) | ~ (((X388 @ X392)) = $true) | (((X388 @ (ap @ X391 @ X392))) = $true)) | (((X388 @ X389)) = $true) | ~ ! [X393 : term] : (~ ! [X394 : term] : ((((X388 @ (sub @ X393 @ (push @ X394 @ id)))) = $true) | ~ (((X388 @ X394)) = $true)) | (((X388 @ (lam @ X393))) = $true)))))) & ((laminj = $true) <=> ! [X395 : term,X396 : term] : ((X395 = X396) | (((lam @ X395)) != ((lam @ X396))))) & (! [X397 : term,X398 : (term > $o),X399 : subst] : (~ ! [X400 : term] : (~ (((var @ X400)) = $true) | (((X398 @ (sub @ X400 @ X399))) = $true)) | ~ ! [X401 : term] : ((((X398 @ (lam @ X401))) = $true) | ~ ! [X402 : term] : (~ (((X398 @ X402)) = $true) | (((X398 @ (sub @ X401 @ (push @ X402 @ id)))) = $true))) | (((X398 @ (sub @ X397 @ X399))) = $true) | ~ ! [X403 : term,X404 : term] : ((((X398 @ (ap @ X404 @ X403))) = $true) | ~ (((X398 @ X404)) = $true) | ~ (((X398 @ X403)) = $true))) <=> (induction2lem = $true)) & (hoasapinj2_gthm = $true) & (! [X405 : (subst > term > subst > $o),X406 : term] : (~ ! [X407 : (subst > term > term)] : (~ ! [X408 : subst,X409 : term,X410 : subst] : (((sub @ (X407 @ X408 @ X409) @ X410)) = ((X407 @ (comp @ X408 @ X410) @ (sub @ X409 @ X410)))) | ~ ! [X411 : term] : ((((X405 @ id @ (X407 @ id @ X411) @ id)) = $true) | ~ (((X405 @ id @ X411 @ id)) = $true)) | (((X405 @ id @ (lam @ (X407 @ sh @ one)) @ id)) = $true)) | ~ ! [X412 : term] : ((((X405 @ id @ (sub @ X406 @ (push @ X412 @ id)) @ id)) = $true) | ~ (((X405 @ id @ X412 @ id)) = $true)) | (((X405 @ id @ (lam @ X406) @ id)) = $true)) <=> (hoasinduction_lem3a = $true)) & (hoasinduction_lem2_lthm = $true) & ~ (((var @ (d2term @ d_one))) = $true) & (pushprop_lem2v2_lthm = $true) & (pushprop_lem2v2 = $true) & ! [X413 : term] : ? [X414 : d_term] : (((d2term @ X414)) = X413) & (! [X417 : (term > $o),X415 : subst,X416 : term] : ~ ! [X418 : (term > $o)] : ~ ! [X419 : term] : (((X417 @ (sub @ X419 @ (push @ X416 @ X415)))) = ((X418 @ X419))) <=> (pushprop_lem0 = $true)) & (! [X420 : term] : ~ ! [X421 : (subst > term > term)] : (((sub @ X420 @ (push @ one @ sh))) != ((X421 @ sh @ one))) <=> (hoasinduction_lem3b_lthm = $true)) & (! [X423 : term,X422 : term] : ((((sub @ X422 @ sh)) != ((sub @ X423 @ sh))) | (X422 = X423)) <=> (shinj = $true)) & ((hoasinduction_lem0_lthm = $true) <=> ! [X424 : (subst > term > subst > $o)] : ~ ! [X425 : (term > $o)] : ~ ! [X426 : term] : (((X425 @ X426)) = ((X424 @ id @ X426 @ id)))) & (((d2term @ d_one)) = one) & ((hoasinduction_lem3 = $true) <=> ! [X427 : term,X428 : (subst > term > subst > $o)] : ((((X428 @ id @ (lam @ X427) @ id)) = $true) | ~ ! [X429 : subst,X430 : subst,X431 : subst,X432 : term] : ((((X428 @ X429 @ X432 @ (comp @ X430 @ X431))) = $true) | ~ (((X428 @ (comp @ X429 @ X430) @ (sub @ X432 @ X430) @ X431)) = $true)) | ~ ! [X433 : subst,X434 : term,X435 : subst,X436 : subst] : ((((X428 @ (comp @ X433 @ X436) @ (sub @ X434 @ X436) @ X435)) = $true) | ~ (((X428 @ X433 @ X434 @ (comp @ X436 @ X435))) = $true)) | ~ ! [X437 : (subst > term > term)] : ((((X428 @ id @ (lam @ (X437 @ sh @ one)) @ id)) = $true) | ~ ! [X438 : term,X439 : subst,X440 : subst] : (((X437 @ (comp @ X440 @ X439) @ (sub @ X438 @ X439))) = ((sub @ (X437 @ X440 @ X438) @ X439))) | ~ ! [X441 : term] : (~ (((X428 @ id @ X441 @ id)) = $true) | (((X428 @ id @ (X437 @ id @ X441) @ id)) = $true))) | ~ ! [X442 : term] : ((((X428 @ id @ (sub @ X427 @ (push @ X442 @ id)) @ id)) = $true) | ~ (((X428 @ id @ X442 @ id)) = $true)))) & (! [X443 : (subst > term > subst > $o),X444 : term,X445 : (term > $o),X446 : term] : (~ ! [X447 : subst,X448 : subst,X449 : term,X450 : subst] : (~ (((X443 @ X450 @ X449 @ (comp @ X447 @ X448))) = $true) | (((X443 @ (comp @ X450 @ X447) @ (sub @ X449 @ X447) @ X448)) = $true)) | ~ (((X445 @ X444)) = $true) | ~ (((X445 @ X446)) = $true) | ~ ! [X451 : term] : (((X443 @ id @ X451 @ id)) = ((X445 @ X451))) | ~ ! [X452 : term,X453 : term] : (~ (((X443 @ id @ X453 @ id)) = $true) | ~ (((X443 @ id @ X452 @ id)) = $true) | (((X443 @ id @ (ap @ (sub @ X453 @ id) @ X452) @ id)) = $true)) | ~ ! [X454 : subst,X455 : subst,X456 : term,X457 : subst] : (~ (((X443 @ (comp @ X457 @ X455) @ (sub @ X456 @ X455) @ X454)) = $true) | (((X443 @ X457 @ X456 @ (comp @ X455 @ X454))) = $true)) | (((X445 @ (ap @ X444 @ X446))) = $true)) <=> (hoasinduction_lem2v2 = $true)) & (! [X458 : (subst > term > subst > $o),X459 : term,X460 : (term > $o)] : (~ ! [X461 : term] : (~ (((X460 @ X461)) = $true) | (((X460 @ (sub @ X459 @ (push @ X461 @ id)))) = $true)) | (((X460 @ (lam @ X459))) = $true) | ~ ! [X462 : (subst > term > term)] : (~ ! [X463 : subst,X464 : subst,X465 : term] : (((X462 @ (comp @ X464 @ X463) @ (sub @ X465 @ X463))) = ((sub @ (X462 @ X464 @ X465) @ X463))) | ~ ! [X466 : term] : ((((X458 @ id @ (X462 @ id @ X466) @ id)) = $true) | ~ (((X458 @ id @ X466 @ id)) = $true)) | (((X458 @ id @ (lam @ (X462 @ sh @ one)) @ id)) = $true)) | ~ ! [X467 : term] : (((X458 @ id @ X467 @ id)) = ((X460 @ X467)))) <=> (hoasinduction_lem3v2a = $true)) & (hoasinduction_gthm = $true) & (pushprop = $true) & (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (! [X468 : (subst > term > subst > $o)] : ~ ! [X469 : (term > $o)] : ~ ! [X470 : term] : (((X468 @ id @ X470 @ id)) = ((X469 @ X470))) => (! [X471 : (term > $o),X472 : term] : (~ ! [X473 : term] : ((((X471 @ (lam @ X473))) = $true) | ~ ! [X474 : term] : (~ (((X471 @ X474)) = $true) | (((X471 @ (sub @ X473 @ (push @ X474 @ id)))) = $true))) | ~ ! [X475 : term,X476 : term] : (($true = ((X471 @ (ap @ X476 @ X475)))) | ~ ($true = ((X471 @ X475))) | ~ (((X471 @ X476)) = $true)) | ~ ! [X477 : term] : (~ (((var @ X477)) = $true) | ($true = ((X471 @ X477)))) | (((X471 @ X472)) = $true)) => (! [X478 : term] : (((sub @ X478 @ id)) = X478) => (! [X479 : (subst > term > subst > $o),X480 : (term > $o),X481 : term] : (~ ! [X482 : term] : ((((X480 @ (sub @ X481 @ (push @ X482 @ id)))) = $true) | ~ (((X480 @ X482)) = $true)) | ~ ! [X483 : term] : (((X480 @ X483)) = ((X479 @ id @ X483 @ id))) | ~ ! [X484 : (subst > term > term)] : (~ ! [X486 : subst,X487 : term,X485 : subst] : (((sub @ (X484 @ X486 @ X487) @ X485)) = ((X484 @ (comp @ X486 @ X485) @ (sub @ X487 @ X485)))) | ($true = ((X479 @ id @ (lam @ (X484 @ sh @ one)) @ id))) | ~ ! [X488 : term] : (~ (((X479 @ id @ X488 @ id)) = $true) | (((X479 @ id @ (X484 @ id @ X488) @ id)) = $true))) | (((X480 @ (lam @ X481))) = $true)) => ! [X489 : term,X490 : (subst > term > subst > $o)] : (~ ! [X491 : term,X492 : term] : (~ (((X490 @ id @ X491 @ id)) = $true) | ~ (((X490 @ id @ X492 @ id)) = $true) | (((X490 @ id @ (ap @ (sub @ X491 @ id) @ X492) @ id)) = $true)) | (((X490 @ id @ X489 @ id)) = $true) | ~ ! [X493 : (subst > term > term)] : (~ ! [X494 : term] : (~ ($true = ((X490 @ id @ X494 @ id))) | (((X490 @ id @ (X493 @ id @ X494) @ id)) = $true)) | (((X490 @ id @ (lam @ (X493 @ sh @ one)) @ id)) = $true) | ~ ! [X495 : subst,X496 : term,X497 : subst] : (((sub @ (X493 @ X497 @ X496) @ X495)) = ((X493 @ (comp @ X497 @ X495) @ (sub @ X496 @ X495))))))))))) & (hoasinduction_lem1v2 = $true) & (! [X498 : term] : ~ ! [X499 : (subst > term > term)] : ~ ! [X501 : subst,X500 : term] : (((X499 @ X501 @ X500)) = ((sub @ X498 @ (push @ X500 @ X501)))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & ((hoasinduction_lem3aaa = $true) <=> ! [X502 : (subst > term > subst > $o),X503 : term] : (~ ! [X504 : (subst > term > term),X505 : term] : (~ ! [X506 : subst,X508 : term,X507 : subst] : (((sub @ (sub @ X505 @ (push @ X508 @ X507)) @ X506)) = ((sub @ (X504 @ X507 @ X508) @ X506))) | ~ ! [X511 : term,X509 : subst,X510 : subst] : (((X504 @ (comp @ X509 @ X510) @ (sub @ X511 @ X510))) = ((sub @ X505 @ (push @ (sub @ X511 @ X510) @ (comp @ X509 @ X510))))) | (((X502 @ id @ (lam @ (X504 @ sh @ one)) @ id)) = $true) | ~ ! [X512 : term] : ((((X502 @ id @ (X504 @ id @ X512) @ id)) = $true) | ~ ($true = ((X502 @ id @ X512 @ id))))) | (((X502 @ id @ (lam @ (sub @ X503 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X513 : term] : (~ (((X502 @ id @ X513 @ id)) = $true) | (((X502 @ id @ (sub @ X503 @ (push @ X513 @ id)) @ id)) = $true)))) & ! [X514 : d_subst,X515 : d_subst] : ((((d2subst @ X514)) = ((d2subst @ X515))) => (X514 = X515)) & (! [X516 : term] : ~ ! [X517 : (subst > term > term)] : ~ ! [X519 : term,X518 : subst] : (((X517 @ X518 @ X519)) = ((sub @ X516 @ (push @ X519 @ X518)))) <=> (hoasinduction_lem3v2_f = $true)) & (! [X520 : subst] : (((comp @ X520 @ id)) = X520) <=> (axidr = $true)) & ((induction2lem_lthm = $true) <=> (! [X521 : subst,X522 : term] : (((sub @ one @ (push @ X522 @ X521))) = X522) => (! [X524 : term,X523 : subst] : (((comp @ sh @ (push @ X524 @ X523))) = X523) => (! [X525 : subst] : (((comp @ X525 @ id)) = X525) => (! [X526 : term,X527 : (term > $o)] : (~ ! [X528 : term] : (~ (((X527 @ X528)) = $true) | (((X527 @ (lam @ X528))) = $true)) | (((X527 @ X526)) = $true) | ~ ! [X529 : term] : ((((X527 @ X529)) = $true) | ~ (((var @ X529)) = $true)) | ~ ! [X530 : term,X531 : term] : (~ (((X527 @ X530)) = $true) | (((X527 @ (ap @ X531 @ X530))) = $true) | ~ (((X527 @ X531)) = $true))) => ! [X532 : subst,X533 : term,X534 : (term > $o)] : ((((X534 @ (sub @ X533 @ X532))) = $true) | ~ ! [X535 : term] : ((((X534 @ (lam @ X535))) = $true) | ~ ! [X536 : term] : (~ (((X534 @ X536)) = $true) | (((X534 @ (sub @ X535 @ (push @ X536 @ id)))) = $true))) | ~ ! [X537 : term,X538 : term] : (~ (((X534 @ X538)) = $true) | (((X534 @ (ap @ X537 @ X538))) = $true) | ~ (((X534 @ X537)) = $true)) | ~ ! [X539 : term] : ((((X534 @ (sub @ X539 @ X532))) = $true) | ~ (((var @ X539)) = $true)))))))) & (lamnotvar = $true)),
  inference(rectify,[],[f5])).
thf(f10,plain,(
  ((apinj2 = $true) <=> ! [X140 : term,X142 : term,X141 : term,X143 : term] : ((X140 = X143) | (((ap @ X141 @ X140)) != ((ap @ X142 @ X143))))) & ((induction2lem_lthm = $true) <=> (! [X521 : subst,X522 : term] : (((sub @ one @ (push @ X522 @ X521))) = X522) => (! [X524 : term,X523 : subst] : (((comp @ sh @ (push @ X524 @ X523))) = X523) => (! [X525 : subst] : (((comp @ X525 @ id)) = X525) => (! [X526 : term,X527 : (term > $o)] : ((((X527 @ X526)) = $true) | ~ ! [X528 : term] : ((((X527 @ (lam @ X528))) = $true) | (((X527 @ X528)) != $true)) | ~ ! [X529 : term] : ((((var @ X529)) != $true) | (((X527 @ X529)) = $true)) | ~ ! [X531 : term,X530 : term] : ((((X527 @ X530)) != $true) | (((X527 @ X531)) != $true) | (((X527 @ (ap @ X531 @ X530))) = $true))) => ! [X532 : subst,X533 : term,X534 : (term > $o)] : (~ ! [X539 : term] : ((((X534 @ (sub @ X539 @ X532))) = $true) | (((var @ X539)) != $true)) | ~ ! [X537 : term,X538 : term] : ((((X534 @ (ap @ X537 @ X538))) = $true) | (((X534 @ X538)) != $true) | (((X534 @ X537)) != $true)) | (((X534 @ (sub @ X533 @ X532))) = $true) | ~ ! [X535 : term] : ((((X534 @ (lam @ X535))) = $true) | ~ ! [X536 : term] : ((((X534 @ X536)) != $true) | (((X534 @ (sub @ X535 @ (push @ X536 @ id)))) = $true))))))))) & ((! [X251 : subst] : (((comp @ id @ X251)) = X251) => (! [X252 : subst] : (((comp @ X252 @ id)) = X252) => (! [X254 : subst] : (((comp @ X254 @ id)) = X254) & ! [X253 : subst] : (((comp @ id @ X253)) = X253)))) <=> (substmonoid_lthm = $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (hoasinduction_lem1v2 = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoasapnotvar = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (apnotvar = $true) & (hoasapnotvar_gthm = $true) & ((laminj = $true) <=> ! [X395 : term,X396 : term] : ((X395 = X396) | (((lam @ X395)) != ((lam @ X396))))) & ! [X78 : d_subst] : (d_id = X78) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (! [X468 : (subst > term > subst > $o)] : ~ ! [X469 : (term > $o)] : ~ ! [X470 : term] : (((X468 @ id @ X470 @ id)) = ((X469 @ X470))) => (! [X471 : (term > $o),X472 : term] : (~ ! [X477 : term] : (($true = ((X471 @ X477))) | (((var @ X477)) != $true)) | ~ ! [X475 : term,X476 : term] : ((((X471 @ X476)) != $true) | ($true = ((X471 @ (ap @ X476 @ X475)))) | ($true != ((X471 @ X475)))) | ~ ! [X473 : term] : (~ ! [X474 : term] : ((((X471 @ (sub @ X473 @ (push @ X474 @ id)))) = $true) | (((X471 @ X474)) != $true)) | (((X471 @ (lam @ X473))) = $true)) | (((X471 @ X472)) = $true)) => (! [X478 : term] : (((sub @ X478 @ id)) = X478) => (! [X480 : (term > $o),X479 : (subst > term > subst > $o),X481 : term] : ((((X480 @ (lam @ X481))) = $true) | ~ ! [X483 : term] : (((X480 @ X483)) = ((X479 @ id @ X483 @ id))) | ~ ! [X484 : (subst > term > term)] : (($true = ((X479 @ id @ (lam @ (X484 @ sh @ one)) @ id))) | ~ ! [X486 : subst,X487 : term,X485 : subst] : (((sub @ (X484 @ X486 @ X487) @ X485)) = ((X484 @ (comp @ X486 @ X485) @ (sub @ X487 @ X485)))) | ~ ! [X488 : term] : ((((X479 @ id @ (X484 @ id @ X488) @ id)) = $true) | (((X479 @ id @ X488 @ id)) != $true))) | ~ ! [X482 : term] : ((((X480 @ (sub @ X481 @ (push @ X482 @ id)))) = $true) | (((X480 @ X482)) != $true))) => ! [X490 : (subst > term > subst > $o),X489 : term] : (~ ! [X493 : (subst > term > term)] : ((((X490 @ id @ (lam @ (X493 @ sh @ one)) @ id)) = $true) | ~ ! [X495 : subst,X496 : term,X497 : subst] : (((sub @ (X493 @ X497 @ X496) @ X495)) = ((X493 @ (comp @ X497 @ X495) @ (sub @ X496 @ X495)))) | ~ ! [X494 : term] : ((((X490 @ id @ (X493 @ id @ X494) @ id)) = $true) | ($true != ((X490 @ id @ X494 @ id))))) | ~ ! [X492 : term,X491 : term] : ((((X490 @ id @ X491 @ id)) != $true) | (((X490 @ id @ (ap @ (sub @ X491 @ id) @ X492) @ id)) = $true) | (((X490 @ id @ X492 @ id)) != $true)) | (((X490 @ id @ X489 @ id)) = $true))))))) & (induction2_gthm = $true) & ((! [X323 : subst] : (((comp @ id @ X323)) = X323) & ! [X322 : subst] : (((comp @ X322 @ id)) = X322)) <=> (substmonoid = $true)) & (hoasinduction_lem3_gthm = $true) & ((hoaslaminj = $true) <=> ! [X117 : (subst > term > term),X115 : term,X118 : subst,X116 : (subst > term > term)] : ((((X117 @ X118 @ X115)) = ((X116 @ X118 @ X115))) | (((lam @ (X116 @ sh @ one))) != ((lam @ (X117 @ sh @ one)))) | ~ ! [X119 : subst,X120 : term,X121 : subst] : (((X117 @ (comp @ X119 @ X121) @ (sub @ X120 @ X121))) = ((sub @ (X117 @ X119 @ X120) @ X121))) | ~ ! [X122 : subst,X123 : subst,X124 : term] : (((sub @ (X116 @ X123 @ X124) @ X122)) = ((X116 @ (comp @ X123 @ X122) @ (sub @ X124 @ X122)))))) & ((apinj1 = $true) <=> ! [X111 : term,X113 : term,X114 : term,X112 : term] : ((X112 = X114) | (((ap @ X112 @ X111)) != ((ap @ X114 @ X113))))) & ((! [X258 : term] : (((sub @ X258 @ id)) = X258) => (! [X260 : term,X259 : (subst > term > subst > $o)] : (~ ! [X266 : term] : ((((X259 @ id @ X266 @ id)) != $true) | (((X259 @ id @ (sub @ X260 @ (push @ X266 @ id)) @ id)) = $true)) | ~ ! [X261 : (subst > term > term)] : (~ ! [X265 : term,X264 : subst,X263 : subst] : (((X261 @ (comp @ X263 @ X264) @ (sub @ X265 @ X264))) = ((sub @ (X261 @ X263 @ X265) @ X264))) | ~ ! [X262 : term] : ((((X259 @ id @ (X261 @ id @ X262) @ id)) = $true) | (((X259 @ id @ X262 @ id)) != $true)) | (((X259 @ id @ (lam @ (X261 @ sh @ one)) @ id)) = $true)) | (((X259 @ id @ (lam @ (sub @ X260 @ (push @ one @ sh))) @ id)) = $true)) => ! [X268 : term,X267 : (subst > term > subst > $o)] : (~ ! [X280 : term,X279 : subst,X282 : subst,X281 : subst] : ((((X267 @ X282 @ X280 @ (comp @ X281 @ X279))) != $true) | (((X267 @ (comp @ X282 @ X281) @ (sub @ X280 @ X281) @ X279)) = $true)) | (((X267 @ id @ (lam @ X268) @ id)) = $true) | ~ ! [X269 : term] : ((((X267 @ id @ (sub @ X268 @ (push @ X269 @ id)) @ id)) = $true) | (((X267 @ id @ X269 @ id)) != $true)) | ~ ! [X272 : subst,X270 : subst,X271 : term,X273 : subst] : ((((X267 @ X273 @ X271 @ (comp @ X270 @ X272))) = $true) | (((X267 @ (comp @ X273 @ X270) @ (sub @ X271 @ X270) @ X272)) != $true)) | ~ ! [X274 : (subst > term > term)] : (~ ! [X276 : subst,X275 : subst,X277 : term] : (((X274 @ (comp @ X276 @ X275) @ (sub @ X277 @ X275))) = ((sub @ (X274 @ X276 @ X277) @ X275))) | (((X267 @ id @ (lam @ (X274 @ sh @ one)) @ id)) = $true) | ~ ! [X278 : term] : ((((X267 @ id @ X278 @ id)) != $true) | (((X267 @ id @ (X274 @ id @ X278) @ id)) = $true)))))) <=> (hoasinduction_lem3_lthm = $true)) & (hoasapinj2_gthm = $true) & (hoasinduction_lem1 = $true) & (pushprop_lem2v2_gthm = $true) & ((pushprop_lem1 = $true) <=> ! [X81 : term,X82 : (term > $o),X83 : term,X80 : (term > $o),X79 : subst] : ((((X80 @ X81)) != $true) | (((X82 @ (sub @ X81 @ (push @ X83 @ X79)))) = $true))) & (hoasinduction_gthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((pushprop_lem1v2_lthm = $true) <=> (! [X104 : term,X105 : subst] : (((sub @ one @ (push @ X104 @ X105))) = X104) => ! [X106 : (term > $o),X108 : term,X109 : (term > $o),X107 : subst] : (~ ! [X110 : term] : (((X109 @ X110)) = ((X106 @ (sub @ X110 @ (push @ X108 @ X107))))) | (((X106 @ X108)) != $true) | (((X109 @ one)) = $true)))) & (ulamvarind = $true) & ! [X514 : d_subst,X515 : d_subst] : ((((d2subst @ X514)) = ((d2subst @ X515))) => (X514 = X515)) & (hoasinduction_lem2v2_gthm = $true) & (hoasinduction_lem1_gthm = $true) & (hoasinduction_lem3a_gthm = $true) & (hoaslamnotvar_gthm = $true) & (! [X246 : (term > $o),X247 : term,X248 : subst] : ~ ! [X249 : (term > $o)] : ~ ! [X250 : term] : (((X246 @ (sub @ X250 @ (push @ X247 @ X248)))) = ((X249 @ X250))) <=> (pushprop_lem0_lthm = $true)) & (! [X98 : term,X101 : (term > $o),X100 : subst,X99 : (term > $o)] : ((((X101 @ one)) = $true) | (((X99 @ X98)) != $true) | ~ ! [X102 : term] : (((X101 @ X102)) = ((X99 @ (sub @ X102 @ (push @ X98 @ X100)))))) <=> (pushprop_lem1v2 = $true)) & (substmonoid_gthm = $true) & (hoasapinj1_gthm = $true) & ((! [X362 : term] : (((sub @ X362 @ id)) = X362) => (! [X364 : term,X363 : (subst > term > subst > $o)] : ((((X363 @ id @ (lam @ (sub @ X364 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X366 : (subst > term > term)] : ((((X363 @ id @ (lam @ (X366 @ sh @ one)) @ id)) = $true) | ~ ! [X370 : term] : ((((X363 @ id @ X370 @ id)) != $true) | (((X363 @ id @ (X366 @ id @ X370) @ id)) = $true)) | ~ ! [X369 : term,X367 : subst,X368 : subst] : (((X366 @ (comp @ X368 @ X367) @ (sub @ X369 @ X367))) = ((sub @ (X366 @ X368 @ X369) @ X367)))) | ~ ! [X365 : term] : ((((X363 @ id @ X365 @ id)) != $true) | (((X363 @ id @ (sub @ X364 @ (push @ X365 @ id)) @ id)) = $true))) => ! [X371 : (subst > term > subst > $o),X372 : term] : (~ ! [X374 : (subst > term > term)] : (~ ! [X377 : term,X376 : subst,X378 : subst] : (((X374 @ (comp @ X378 @ X376) @ (sub @ X377 @ X376))) = ((sub @ (X374 @ X378 @ X377) @ X376))) | (((X371 @ id @ (lam @ (X374 @ sh @ one)) @ id)) = $true) | ~ ! [X375 : term] : ((((X371 @ id @ (X374 @ id @ X375) @ id)) = $true) | (((X371 @ id @ X375 @ id)) != $true))) | (((X371 @ id @ (lam @ X372) @ id)) = $true) | ~ ! [X373 : term] : ((((X371 @ id @ (sub @ X372 @ (push @ X373 @ id)) @ id)) = $true) | (((X371 @ id @ X373 @ id)) != $true))))) <=> (hoasinduction_lem3a_lthm = $true)) & (hoasinduction_lem2_gthm = $true) & (pushprop = $true) & (pushprop_lem3v2 = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & ! [X15 : subst] : ? [X16 : d_subst] : (((d2subst @ X16)) = X15) & (! [X406 : term,X405 : (subst > term > subst > $o)] : (~ ! [X407 : (subst > term > term)] : (~ ! [X408 : subst,X409 : term,X410 : subst] : (((sub @ (X407 @ X408 @ X409) @ X410)) = ((X407 @ (comp @ X408 @ X410) @ (sub @ X409 @ X410)))) | (((X405 @ id @ (lam @ (X407 @ sh @ one)) @ id)) = $true) | ~ ! [X411 : term] : ((((X405 @ id @ X411 @ id)) != $true) | (((X405 @ id @ (X407 @ id @ X411) @ id)) = $true))) | ~ ! [X412 : term] : ((((X405 @ id @ (sub @ X406 @ (push @ X412 @ id)) @ id)) = $true) | (((X405 @ id @ X412 @ id)) != $true)) | (((X405 @ id @ (lam @ X406) @ id)) = $true)) <=> (hoasinduction_lem3a = $true)) & (! [X417 : (term > $o),X415 : subst,X416 : term] : ~ ! [X418 : (term > $o)] : ~ ! [X419 : term] : (((X417 @ (sub @ X419 @ (push @ X416 @ X415)))) = ((X418 @ X419))) <=> (pushprop_lem0 = $true)) & (! [X458 : (subst > term > subst > $o),X459 : term,X460 : (term > $o)] : (~ ! [X462 : (subst > term > term)] : (~ ! [X463 : subst,X464 : subst,X465 : term] : (((X462 @ (comp @ X464 @ X463) @ (sub @ X465 @ X463))) = ((sub @ (X462 @ X464 @ X465) @ X463))) | (((X458 @ id @ (lam @ (X462 @ sh @ one)) @ id)) = $true) | ~ ! [X466 : term] : ((((X458 @ id @ X466 @ id)) != $true) | (((X458 @ id @ (X462 @ id @ X466) @ id)) = $true))) | (((X460 @ (lam @ X459))) = $true) | ~ ! [X467 : term] : (((X458 @ id @ X467 @ id)) = ((X460 @ X467))) | ~ ! [X461 : term] : ((((X460 @ (sub @ X459 @ (push @ X461 @ id)))) = $true) | (((X460 @ X461)) != $true))) <=> (hoasinduction_lem3v2a = $true)) & (axassoc = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (((var @ (d2term @ d_one))) != $true) & ((induction2_lthm = $true) <=> (! [X379 : term] : (((sub @ X379 @ id)) = X379) => (! [X381 : (term > $o),X382 : subst,X380 : term] : (~ ! [X384 : term,X383 : term] : ((((X381 @ X383)) != $true) | (((X381 @ X384)) != $true) | (((X381 @ (ap @ X383 @ X384))) = $true)) | (((X381 @ (sub @ X380 @ X382))) = $true) | ~ ! [X385 : term] : ((((X381 @ (sub @ X385 @ X382))) = $true) | (((var @ X385)) != $true)) | ~ ! [X386 : term] : ((((X381 @ (lam @ X386))) = $true) | ~ ! [X387 : term] : ((((X381 @ (sub @ X386 @ (push @ X387 @ id)))) = $true) | (((X381 @ X387)) != $true)))) => ! [X389 : term,X388 : (term > $o)] : (~ ! [X390 : term] : ((((X388 @ X390)) = $true) | (((var @ X390)) != $true)) | ~ ! [X393 : term] : ((((X388 @ (lam @ X393))) = $true) | ~ ! [X394 : term] : ((((X388 @ X394)) != $true) | (((X388 @ (sub @ X393 @ (push @ X394 @ id)))) = $true))) | ~ ! [X391 : term,X392 : term] : ((((X388 @ X391)) != $true) | (((X388 @ X392)) != $true) | (((X388 @ (ap @ X391 @ X392))) = $true)) | (((X388 @ X389)) = $true))))) & (pushprop_lem1_gthm = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (lamnotvar = $true) & ((axshiftcons = $true) <=> ! [X326 : subst,X327 : term] : (((comp @ sh @ (push @ X327 @ X326))) = X326)) & (! [X502 : (subst > term > subst > $o),X503 : term] : (~ ! [X513 : term] : ((((X502 @ id @ X513 @ id)) != $true) | (((X502 @ id @ (sub @ X503 @ (push @ X513 @ id)) @ id)) = $true)) | (((X502 @ id @ (lam @ (sub @ X503 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X505 : term,X504 : (subst > term > term)] : (~ ! [X506 : subst,X508 : term,X507 : subst] : (((sub @ (sub @ X505 @ (push @ X508 @ X507)) @ X506)) = ((sub @ (X504 @ X507 @ X508) @ X506))) | (((X502 @ id @ (lam @ (X504 @ sh @ one)) @ id)) = $true) | ~ ! [X511 : term,X509 : subst,X510 : subst] : (((X504 @ (comp @ X509 @ X510) @ (sub @ X511 @ X510))) = ((sub @ X505 @ (push @ (sub @ X511 @ X510) @ (comp @ X509 @ X510))))) | ~ ! [X512 : term] : ((((X502 @ id @ (X504 @ id @ X512) @ id)) = $true) | ($true != ((X502 @ id @ X512 @ id)))))) <=> (hoasinduction_lem3aaa = $true)) & (hoasinduction_lem2_lthm = $true) & (! [X520 : subst] : (((comp @ X520 @ id)) = X520) <=> (axidr = $true)) & (pushprop_gthm = $true) & ((termmset_lthm = $true) <=> (! [X94 : term] : (((sub @ X94 @ id)) = X94) => ! [X95 : term] : (((sub @ X95 @ id)) = X95))) & ((hoasinduction_lem2 = $true) <=> ! [X235 : term,X234 : term,X233 : (subst > term > subst > $o)] : ((((X233 @ id @ X235 @ id)) != $true) | ~ ! [X244 : subst,X243 : term,X245 : subst,X242 : subst] : ((((X233 @ X245 @ X243 @ (comp @ X242 @ X244))) != $true) | (((X233 @ (comp @ X245 @ X242) @ (sub @ X243 @ X242) @ X244)) = $true)) | (((X233 @ id @ X234 @ id)) != $true) | ~ ! [X238 : subst,X240 : subst,X241 : subst,X239 : term] : ((((X233 @ (comp @ X238 @ X241) @ (sub @ X239 @ X241) @ X240)) != $true) | (((X233 @ X238 @ X239 @ (comp @ X241 @ X240))) = $true)) | ~ ! [X236 : term,X237 : term] : ((((X233 @ id @ X236 @ id)) != $true) | (((X233 @ id @ X237 @ id)) != $true) | (((X233 @ id @ (ap @ (sub @ X237 @ id) @ X236) @ id)) = $true)) | (((X233 @ id @ (ap @ X234 @ X235) @ id)) = $true))) & (hoaslamnotvar = $true) & (hoaslamnotap_lthm = $true) & (! [X423 : term,X422 : term] : ((((sub @ X422 @ sh)) != ((sub @ X423 @ sh))) | (X422 = X423)) <=> (shinj = $true)) & (! [X96 : subst,X97 : term] : (((sub @ one @ (push @ X97 @ X96))) = X97) <=> (axvarcons = $true)) & (! [X516 : term] : ~ ! [X517 : (subst > term > term)] : ~ ! [X519 : term,X518 : subst] : (((X517 @ X518 @ X519)) = ((sub @ X516 @ (push @ X519 @ X518)))) <=> (hoasinduction_lem3v2_f = $true)) & (! [X133 : (term > $o),X134 : term] : (~ ! [X138 : term,X137 : term] : ((((X133 @ X137)) != $true) | (((X133 @ (ap @ X137 @ X138))) = $true) | (((X133 @ X138)) != $true)) | ~ ! [X136 : term] : ((((X133 @ X136)) = $true) | (((var @ X136)) != $true)) | ~ ! [X135 : term] : ((((X133 @ X135)) != $true) | (((X133 @ (lam @ X135))) = $true)) | (((X133 @ X134)) = $true)) <=> (induction = $true)) & (pushprop_lem2v2 = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (! [X427 : term,X428 : (subst > term > subst > $o)] : (~ ! [X436 : subst,X433 : subst,X434 : term,X435 : subst] : ((((X428 @ X433 @ X434 @ (comp @ X436 @ X435))) != $true) | (((X428 @ (comp @ X433 @ X436) @ (sub @ X434 @ X436) @ X435)) = $true)) | ~ ! [X437 : (subst > term > term)] : (~ ! [X441 : term] : ((((X428 @ id @ X441 @ id)) != $true) | (((X428 @ id @ (X437 @ id @ X441) @ id)) = $true)) | (((X428 @ id @ (lam @ (X437 @ sh @ one)) @ id)) = $true) | ~ ! [X438 : term,X439 : subst,X440 : subst] : (((X437 @ (comp @ X440 @ X439) @ (sub @ X438 @ X439))) = ((sub @ (X437 @ X440 @ X438) @ X439)))) | ~ ! [X430 : subst,X432 : term,X431 : subst,X429 : subst] : ((((X428 @ X429 @ X432 @ (comp @ X430 @ X431))) = $true) | (((X428 @ (comp @ X429 @ X430) @ (sub @ X432 @ X430) @ X431)) != $true)) | ~ ! [X442 : term] : ((((X428 @ id @ X442 @ id)) != $true) | (((X428 @ id @ (sub @ X427 @ (push @ X442 @ id)) @ id)) = $true)) | (((X428 @ id @ (lam @ X427) @ id)) = $true)) <=> (hoasinduction_lem3 = $true)) & (! [X49 : (subst > term > term),X48 : term,X50 : term] : (~ ! [X51 : subst,X53 : term,X52 : subst] : (((X49 @ (comp @ X51 @ X52) @ (sub @ X53 @ X52))) = ((sub @ (X49 @ X51 @ X53) @ X52))) | (((ap @ (sub @ X50 @ id) @ X48)) != ((lam @ (X49 @ sh @ one))))) <=> (hoaslamnotap = $true)) & (pushprop_lthm_orig = $true) & ! [X93 : d_term] : (d_one = X93) & (! [X21 : (subst > term > subst > $o),X22 : term] : (~ ! [X36 : subst,X34 : term,X37 : subst,X35 : subst] : ((((X21 @ X36 @ X34 @ (comp @ X35 @ X37))) = $true) | (((X21 @ (comp @ X36 @ X35) @ (sub @ X34 @ X35) @ X37)) != $true)) | ~ ! [X26 : term,X24 : subst,X23 : subst,X25 : subst] : ((((X21 @ X25 @ X26 @ (comp @ X24 @ X23))) != $true) | (((X21 @ (comp @ X25 @ X24) @ (sub @ X26 @ X24) @ X23)) = $true)) | ~ ! [X33 : term,X32 : term] : ((((X21 @ id @ X32 @ id)) != $true) | (((X21 @ id @ (ap @ (sub @ X33 @ id) @ X32) @ id)) = $true) | ($true != ((X21 @ id @ X33 @ id)))) | (((X21 @ id @ X22 @ id)) = $true) | ~ ! [X38 : term] : ((((var @ (sub @ X38 @ id))) != $true) | (((X21 @ id @ X38 @ id)) = $true)) | ~ ! [X27 : (subst > term > term)] : (~ ! [X30 : term,X29 : subst,X31 : subst] : (((X27 @ (comp @ X29 @ X31) @ (sub @ X30 @ X31))) = ((sub @ (X27 @ X29 @ X30) @ X31))) | ~ ! [X28 : term] : ((((X21 @ id @ (X27 @ id @ X28) @ id)) = $true) | (((X21 @ id @ X28 @ id)) != $true)) | (((X21 @ id @ (lam @ (X27 @ sh @ one)) @ id)) = $true))) <=> (hoasinduction = $true)) & (((d2subst @ d_id)) = sh) & (! [X498 : term] : ~ ! [X499 : (subst > term > term)] : ~ ! [X501 : subst,X500 : term] : (((X499 @ X501 @ X500)) = ((sub @ X498 @ (push @ X500 @ X501)))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & ! [X324 : d_term,X325 : d_term] : ((((d2term @ X325)) = ((d2term @ X324))) => (X324 = X325)) & (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)))))))))))) & ((hoasapinj1_lthm = $true) <=> (! [X84 : term] : (((sub @ X84 @ id)) = X84) => (! [X87 : term,X85 : term,X86 : term,X88 : term] : ((((ap @ X88 @ X85)) != ((ap @ X87 @ X86))) | (X87 = X88)) => ! [X92 : term,X89 : term,X90 : term,X91 : term] : ((X89 = X92) | (((ap @ (sub @ X89 @ id) @ X91)) != ((ap @ (sub @ X92 @ id) @ X90))))))) & (! [X208 : term] : (((sub @ X208 @ id)) = X208) <=> (axvarid = $true)) & ((! [X198 : term,X199 : subst] : (((sub @ one @ (push @ X198 @ X199))) = X198) => (! [X201 : subst,X200 : term] : (((comp @ sh @ (push @ X200 @ X201))) = X201) => ! [X202 : term,X205 : (term > $o),X206 : subst,X204 : term,X203 : (term > $o)] : ((((X203 @ X202)) != $true) | (((X205 @ (sub @ X202 @ (push @ X204 @ X206)))) = $true)))) <=> (pushprop_lem1_lthm = $true)) & (induction2lem_gthm = $true) & (pushprop_lem1v2_gthm = $true) & (axabs = $true) & (ulamvarsh = $true) & ((! [X283 : (subst > term > subst > $o)] : ~ ! [X284 : (term > $o)] : ~ ! [X285 : term] : (((X283 @ id @ X285 @ id)) = ((X284 @ X285))) => (! [X287 : (term > $o),X286 : term] : (~ ! [X289 : term] : (~ ! [X290 : term] : ((((X287 @ X290)) != $true) | (((X287 @ (sub @ X289 @ (push @ X290 @ id)))) = $true)) | (((X287 @ (lam @ X289))) = $true)) | ~ ! [X292 : term,X291 : term] : ((((X287 @ X291)) != $true) | (((X287 @ (ap @ X292 @ X291))) = $true) | (((X287 @ X292)) != $true)) | (((X287 @ X286)) = $true) | ~ ! [X288 : term] : ((((X287 @ X288)) = $true) | (((var @ X288)) != $true))) => (! [X293 : term] : (((sub @ X293 @ id)) = X293) => (! [X295 : (subst > term > subst > $o),X294 : term,X296 : (term > $o)] : (~ ! [X303 : term] : (((X296 @ X303)) = ((X295 @ id @ X303 @ id))) | (((X296 @ (lam @ X294))) = $true) | ~ ! [X297 : (subst > term > term)] : ((((X295 @ id @ (lam @ (X297 @ sh @ one)) @ id)) = $true) | ~ ! [X301 : term] : ((((X295 @ id @ (X297 @ id @ X301) @ id)) = $true) | (((X295 @ id @ X301 @ id)) != $true)) | ~ ! [X300 : subst,X298 : term,X299 : subst] : (((X297 @ (comp @ X299 @ X300) @ (sub @ X298 @ X300))) = ((sub @ (X297 @ X299 @ X298) @ X300)))) | ~ ! [X302 : term] : ((((X296 @ (sub @ X294 @ (push @ X302 @ id)))) = $true) | (((X296 @ X302)) != $true))) => ! [X305 : term,X304 : (subst > term > subst > $o)] : (~ ! [X310 : (subst > term > term)] : (~ ! [X311 : term] : ((((X304 @ id @ X311 @ id)) != $true) | (((X304 @ id @ (X310 @ id @ X311) @ id)) = $true)) | (((X304 @ id @ (lam @ (X310 @ sh @ one)) @ id)) = $true) | ~ ! [X312 : subst,X313 : term,X314 : subst] : (((sub @ (X310 @ X314 @ X313) @ X312)) = ((X310 @ (comp @ X314 @ X312) @ (sub @ X313 @ X312))))) | ~ ! [X320 : term,X321 : term] : ((((X304 @ id @ (ap @ (sub @ X321 @ id) @ X320) @ id)) = $true) | (((X304 @ id @ X321 @ id)) != $true) | (((X304 @ id @ X320 @ id)) != $true)) | ~ ! [X319 : term] : ((((X304 @ id @ X319 @ id)) = $true) | (((var @ (sub @ X319 @ id))) != $true)) | (((X304 @ id @ X305 @ id)) = $true) | ~ ! [X307 : subst,X309 : term,X308 : subst,X306 : subst] : ((((X304 @ X308 @ X309 @ (comp @ X307 @ X306))) != $true) | (((X304 @ (comp @ X308 @ X307) @ (sub @ X309 @ X307) @ X306)) = $true)) | ~ ! [X317 : subst,X316 : subst,X315 : term,X318 : subst] : ((((X304 @ (comp @ X317 @ X318) @ (sub @ X315 @ X318) @ X316)) != $true) | (((X304 @ X317 @ X315 @ (comp @ X318 @ X316))) = $true))))))) <=> (hoasinduction_lthm_3 = $true)) & ((termmset = $true) <=> ! [X103 : term] : (((sub @ X103 @ id)) = X103)) & (pushprop_lem0_gthm = $true) & ((hoasinduction_lthm = $true) <=> (! [X145 : term,X144 : (term > $o)] : (~ ! [X149 : term] : (~ ! [X150 : term] : (($true != ((X144 @ X150))) | ($true = ((X144 @ (sub @ X149 @ (push @ X150 @ id)))))) | (((X144 @ (lam @ X149))) = $true)) | (((X144 @ X145)) = $true) | ~ ! [X146 : term,X147 : term] : ((((X144 @ (ap @ X147 @ X146))) = $true) | (((X144 @ X147)) != $true) | (((X144 @ X146)) != $true)) | ~ ! [X148 : term] : ((((var @ X148)) != $true) | ($true = ((X144 @ X148))))) => (! [X152 : term,X151 : (subst > term > subst > $o),X153 : term] : ((((X151 @ id @ X153 @ id)) != $true) | ~ ! [X155 : term,X154 : term] : ((((X151 @ id @ (ap @ (sub @ X155 @ id) @ X154) @ id)) = $true) | (((X151 @ id @ X155 @ id)) != $true) | (((X151 @ id @ X154 @ id)) != $true)) | ~ ! [X162 : subst,X163 : term,X160 : subst,X161 : subst] : ((((X151 @ (comp @ X160 @ X161) @ (sub @ X163 @ X161) @ X162)) != $true) | (((X151 @ X160 @ X163 @ (comp @ X161 @ X162))) = $true)) | (((X151 @ id @ (ap @ X152 @ X153) @ id)) = $true) | ~ ! [X158 : subst,X157 : subst,X156 : subst,X159 : term] : ((((X151 @ X156 @ X159 @ (comp @ X158 @ X157))) != $true) | (((X151 @ (comp @ X156 @ X158) @ (sub @ X159 @ X158) @ X157)) = $true)) | (((X151 @ id @ X152 @ id)) != $true)) => (! [X164 : (subst > term > subst > $o),X165 : term] : (~ ! [X172 : subst,X171 : subst,X173 : term,X174 : subst] : ((((X164 @ (comp @ X171 @ X172) @ (sub @ X173 @ X172) @ X174)) = $true) | (((X164 @ X171 @ X173 @ (comp @ X172 @ X174))) != $true)) | ~ ! [X166 : (subst > term > term)] : ((((X164 @ id @ (lam @ (X166 @ sh @ one)) @ id)) = $true) | ~ ! [X170 : term] : ((((X164 @ id @ X170 @ id)) != $true) | (((X164 @ id @ (X166 @ id @ X170) @ id)) = $true)) | ~ ! [X168 : subst,X169 : subst,X167 : term] : (((sub @ (X166 @ X168 @ X167) @ X169)) = ((X166 @ (comp @ X168 @ X169) @ (sub @ X167 @ X169))))) | ~ ! [X177 : subst,X178 : subst,X179 : term,X176 : subst] : ((((X164 @ X176 @ X179 @ (comp @ X177 @ X178))) = $true) | (((X164 @ (comp @ X176 @ X177) @ (sub @ X179 @ X177) @ X178)) != $true)) | ~ ! [X175 : term] : ((((X164 @ id @ X175 @ id)) != $true) | (((X164 @ id @ (sub @ X165 @ (push @ X175 @ id)) @ id)) = $true)) | (((X164 @ id @ (lam @ X165) @ id)) = $true)) => ! [X180 : (subst > term > subst > $o),X181 : term] : (~ ! [X192 : term,X191 : term] : ((((X180 @ id @ X191 @ id)) != $true) | (((X180 @ id @ X192 @ id)) != $true) | (((X180 @ id @ (ap @ (sub @ X191 @ id) @ X192) @ id)) = $true)) | ~ ! [X190 : term] : ((((var @ (sub @ X190 @ id))) != $true) | (((X180 @ id @ X190 @ id)) = $true)) | (((X180 @ id @ X181 @ id)) = $true) | ~ ! [X189 : subst,X186 : subst,X187 : term,X188 : subst] : ((((X180 @ (comp @ X188 @ X186) @ (sub @ X187 @ X186) @ X189)) = $true) | (((X180 @ X188 @ X187 @ (comp @ X186 @ X189))) != $true)) | ~ ! [X193 : (subst > term > term)] : ((((X180 @ id @ (lam @ (X193 @ sh @ one)) @ id)) = $true) | ~ ! [X195 : subst,X196 : subst,X197 : term] : (((sub @ (X193 @ X196 @ X197) @ X195)) = ((X193 @ (comp @ X196 @ X195) @ (sub @ X197 @ X195)))) | ~ ! [X194 : term] : ((((X180 @ id @ (X193 @ id @ X194) @ id)) = $true) | (((X180 @ id @ X194 @ id)) != $true))) | ~ ! [X184 : subst,X183 : term,X182 : subst,X185 : subst] : ((((X180 @ X184 @ X183 @ (comp @ X182 @ X185))) = $true) | (((X180 @ (comp @ X184 @ X182) @ (sub @ X183 @ X182) @ X185)) != $true))))))) & (! [X215 : term,X217 : (term > $o),X216 : (subst > term > subst > $o)] : (~ ! [X218 : term] : (($true = ((X217 @ (sub @ X215 @ (push @ X218 @ id))))) | (((X217 @ X218)) != $true)) | ~ ! [X224 : (subst > term > term)] : (~ ! [X225 : term] : ((((X216 @ id @ (X224 @ id @ X225) @ id)) = $true) | (((X216 @ id @ X225 @ id)) != $true)) | (((X216 @ id @ (lam @ (X224 @ sh @ one)) @ id)) = $true) | ~ ! [X227 : subst,X226 : term,X228 : subst] : (((X224 @ (comp @ X228 @ X227) @ (sub @ X226 @ X227))) = ((sub @ (X224 @ X228 @ X226) @ X227)))) | ~ ! [X219 : subst,X221 : subst,X222 : term,X220 : subst] : ((((X216 @ (comp @ X221 @ X219) @ (sub @ X222 @ X219) @ X220)) != $true) | (((X216 @ X221 @ X222 @ (comp @ X219 @ X220))) = $true)) | ~ ! [X223 : term] : (((X216 @ id @ X223 @ id)) = ((X217 @ X223))) | ~ ! [X231 : subst,X232 : subst,X229 : subst,X230 : term] : ((((X216 @ (comp @ X231 @ X232) @ (sub @ X230 @ X232) @ X229)) = $true) | (((X216 @ X231 @ X230 @ (comp @ X232 @ X229))) != $true)) | (((X217 @ (lam @ X215))) = $true)) <=> (hoasinduction_lem3v2 = $true)) & ! [X413 : term] : ? [X414 : d_term] : (((d2term @ X414)) = X413) & (hoasinduction_lem3v2_gthm = $true) & (! [X55 : (subst > term > subst > $o),X54 : term] : (~ ! [X56 : (subst > term > term)] : (~ ! [X58 : term,X59 : subst,X60 : subst] : (((X56 @ (comp @ X60 @ X59) @ (sub @ X58 @ X59))) = ((sub @ (X56 @ X60 @ X58) @ X59))) | (((X55 @ id @ (lam @ (X56 @ sh @ one)) @ id)) = $true) | ~ ! [X57 : term] : ((((X55 @ id @ X57 @ id)) != $true) | (((X55 @ id @ (X56 @ id @ X57) @ id)) = $true))) | (((X55 @ id @ (lam @ (sub @ X54 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X61 : term] : (($true != ((X55 @ id @ X61 @ id))) | (((X55 @ id @ (sub @ X54 @ (push @ X61 @ id)) @ id)) = $true))) <=> (hoasinduction_lem3aa_lthm = $true)) & (pushprop_lem3v2_lthm = $true) & ((hoasinduction_lem0 = $true) <=> ! [X255 : (subst > term > subst > $o)] : ~ ! [X256 : (term > $o)] : ~ ! [X257 : term] : (((X256 @ X257)) = ((X255 @ id @ X257 @ id)))) & (((d2term @ d_one)) = one) & (! [X420 : term] : ~ ! [X421 : (subst > term > term)] : (((sub @ X420 @ (push @ one @ sh))) != ((X421 @ sh @ one))) <=> (hoasinduction_lem3b_lthm = $true)) & (axclos = $true) & (hoaslaminj_gthm = $true) & (hoasinduction_lem1_lthm = $true) & ((induction2 = $true) <=> ! [X1 : (term > $o),X0 : term] : (~ ! [X6 : term] : ((((var @ X6)) != $true) | (((X1 @ X6)) = $true)) | ~ ! [X4 : term,X5 : term] : ((((X1 @ X4)) != $true) | (((X1 @ (ap @ X5 @ X4))) = $true) | (((X1 @ X5)) != $true)) | (((X1 @ X0)) = $true) | ~ ! [X2 : term] : ((((X1 @ (lam @ X2))) = $true) | ~ ! [X3 : term] : ((((X1 @ (sub @ X2 @ (push @ X3 @ id)))) = $true) | (((X1 @ X3)) != $true))))) & (axvarshift = $true) & ((! [X7 : term,X10 : term,X8 : term,X9 : term] : ((X7 = X9) | (((ap @ X8 @ X7)) != ((ap @ X10 @ X9)))) => ! [X14 : term,X11 : term,X13 : term,X12 : term] : ((((ap @ (sub @ X13 @ id) @ X12)) != ((ap @ (sub @ X14 @ id) @ X11))) | (X11 = X12))) <=> (hoasapinj2_lthm = $true)) & ((hoasapinj1 = $true) <=> ! [X18 : term,X19 : term,X17 : term,X20 : term] : ((((ap @ (sub @ X20 @ id) @ X19)) != ((ap @ (sub @ X17 @ id) @ X18))) | (X17 = X20))) & (hoaslamnotap_gthm = $true) & ((axidl = $true) <=> ! [X207 : subst] : (((comp @ id @ X207)) = X207)) & ((hoaslaminj_lthm = $true) <=> (! [X62 : term,X63 : subst] : (((sub @ one @ (push @ X62 @ X63))) = X62) => (! [X64 : term,X65 : subst] : (((comp @ sh @ (push @ X64 @ X65))) = X65) => (! [X66 : term,X67 : term] : ((X66 = X67) | (((lam @ X66)) != ((lam @ X67)))) => ! [X71 : term,X68 : subst,X70 : (subst > term > term),X69 : (subst > term > term)] : (~ ! [X77 : term,X75 : subst,X76 : subst] : (((sub @ (X70 @ X76 @ X77) @ X75)) = ((X70 @ (comp @ X76 @ X75) @ (sub @ X77 @ X75)))) | ~ ! [X73 : term,X72 : subst,X74 : subst] : (((sub @ (X69 @ X72 @ X73) @ X74)) = ((X69 @ (comp @ X72 @ X74) @ (sub @ X73 @ X74)))) | (((lam @ (X70 @ sh @ one))) != ((lam @ (X69 @ sh @ one)))) | (((X69 @ X68 @ X71)) = ((X70 @ X68 @ X71)))))))) & (pushprop_lem2v2_lthm = $true) & ((hoasinduction_lem0_lthm = $true) <=> ! [X424 : (subst > term > subst > $o)] : ~ ! [X425 : (term > $o)] : ~ ! [X426 : term] : (((X425 @ X426)) = ((X424 @ id @ X426 @ id)))) & ((hoasinduction_lem3v2_lthm = $true) <=> (! [X343 : term] : (((sub @ X343 @ id)) = X343) => ! [X345 : term,X346 : (subst > term > subst > $o),X344 : (term > $o)] : (~ ! [X354 : term,X352 : subst,X353 : subst,X355 : subst] : ((((X346 @ (comp @ X355 @ X353) @ (sub @ X354 @ X353) @ X352)) = $true) | (((X346 @ X355 @ X354 @ (comp @ X353 @ X352))) != $true)) | ~ ! [X348 : subst,X351 : subst,X350 : term,X349 : subst] : ((((X346 @ (comp @ X348 @ X351) @ (sub @ X350 @ X351) @ X349)) != $true) | (((X346 @ X348 @ X350 @ (comp @ X351 @ X349))) = $true)) | ~ ! [X356 : (subst > term > term)] : ((((X346 @ id @ (lam @ (X356 @ sh @ one)) @ id)) = $true) | ~ ! [X358 : subst,X360 : term,X359 : subst] : (((X356 @ (comp @ X358 @ X359) @ (sub @ X360 @ X359))) = ((sub @ (X356 @ X358 @ X360) @ X359))) | ~ ! [X357 : term] : ((((X346 @ id @ (X356 @ id @ X357) @ id)) = $true) | (((X346 @ id @ X357 @ id)) != $true))) | ~ ! [X347 : term] : (((X344 @ X347)) = ((X346 @ id @ X347 @ id))) | (((X344 @ (lam @ X345))) = $true) | ~ ! [X361 : term] : ((((X344 @ X361)) != $true) | (((X344 @ (sub @ X345 @ (push @ X361 @ id)))) = $true))))) & (! [X39 : (subst > term > subst > $o),X40 : term] : (~ ! [X43 : (subst > term > term)] : (~ ! [X46 : subst,X44 : subst,X45 : term] : (((sub @ (X43 @ X46 @ X45) @ X44)) = ((X43 @ (comp @ X46 @ X44) @ (sub @ X45 @ X44)))) | ~ ! [X47 : term] : ((((X39 @ id @ (X43 @ id @ X47) @ id)) = $true) | (((X39 @ id @ X47 @ id)) != $true)) | (((X39 @ id @ (lam @ (X43 @ sh @ one)) @ id)) = $true)) | ~ ! [X42 : term,X41 : term] : ((((X39 @ id @ (ap @ (sub @ X41 @ id) @ X42) @ id)) = $true) | (((X39 @ id @ X42 @ id)) != $true) | (((X39 @ id @ X41 @ id)) != $true)) | ($true = ((X39 @ id @ X40 @ id)))) <=> (hoasinduction_no_psi_cond = $true)) & (pushprop_lthm = $true) & (lamnotap != $true) & (! [X125 : term,X126 : (subst > term > subst > $o)] : (~ ! [X127 : term] : ((((X126 @ id @ X127 @ id)) != $true) | (((X126 @ id @ (sub @ X125 @ (push @ X127 @ id)) @ id)) = $true)) | ~ ! [X128 : (subst > term > term)] : (~ ! [X131 : subst,X130 : term,X129 : subst] : (((sub @ (X128 @ X131 @ X130) @ X129)) = ((X128 @ (comp @ X131 @ X129) @ (sub @ X130 @ X129)))) | ~ ! [X132 : term] : ((((X126 @ id @ X132 @ id)) != $true) | (((X126 @ id @ (X128 @ id @ X132) @ id)) = $true)) | (((X126 @ id @ (lam @ (X128 @ sh @ one)) @ id)) = $true)) | (((X126 @ id @ (lam @ (sub @ X125 @ (push @ one @ sh))) @ id)) = $true)) <=> (hoasinduction_lem3aa = $true)) & ((hoasinduction_lem3b = $true) <=> ! [X213 : term] : ~ ! [X214 : (subst > term > term)] : (((X214 @ sh @ one)) != ((sub @ X213 @ (push @ one @ sh))))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (hoasinduction_lem1v2_gthm = $true) & ((! [X328 : term] : ~ ! [X329 : (subst > term > term)] : ~ ! [X330 : term,X331 : subst] : (((sub @ X328 @ (push @ X330 @ X331))) = ((X329 @ X331 @ X330))) => (! [X332 : term] : (((sub @ X332 @ id)) = X332) => ! [X335 : (term > $o),X333 : term,X334 : (subst > term > subst > $o)] : (~ ! [X342 : term] : (((X335 @ X342)) = ((X334 @ id @ X342 @ id))) | ~ ! [X336 : term] : ((((X335 @ X336)) != $true) | (((X335 @ (sub @ X333 @ (push @ X336 @ id)))) = $true)) | (((X335 @ (lam @ X333))) = $true) | ~ ! [X337 : (subst > term > term)] : (~ ! [X338 : term] : ((((X334 @ id @ X338 @ id)) != $true) | (((X334 @ id @ (X337 @ id @ X338) @ id)) = $true)) | ~ ! [X339 : subst,X340 : term,X341 : subst] : (((sub @ (X337 @ X339 @ X340) @ X341)) = ((X337 @ (comp @ X339 @ X341) @ (sub @ X340 @ X341)))) | (((X334 @ id @ (lam @ (X337 @ sh @ one)) @ id)) = $true))))) <=> (hoasinduction_lem3v2a_lthm = $true)) & (hoasinduction_lem3b_gthm = $true) & (ulamvar1 != $true) & (termmset_gthm = $true) & (! [X211 : term,X212 : term,X209 : term,X210 : term] : ((X210 = X212) | (((ap @ (sub @ X209 @ id) @ X212)) != ((ap @ (sub @ X211 @ id) @ X210)))) <=> (hoasapinj2 = $true)) & (! [X139 : subst] : (((push @ (sub @ one @ X139) @ (comp @ sh @ X139))) = X139) <=> (axscons = $true)) & (((d2subst @ d_id)) = id) & (hoasapnotvar_lthm = $true) & (hoaslamnotvar_lthm = $true) & ((hoasinduction_lem2v2 = $true) <=> ! [X443 : (subst > term > subst > $o),X444 : term,X446 : term,X445 : (term > $o)] : (~ ! [X456 : term,X454 : subst,X457 : subst,X455 : subst] : ((((X443 @ (comp @ X457 @ X455) @ (sub @ X456 @ X455) @ X454)) != $true) | (((X443 @ X457 @ X456 @ (comp @ X455 @ X454))) = $true)) | (((X445 @ X444)) != $true) | ~ ! [X452 : term,X453 : term] : ((((X443 @ id @ X453 @ id)) != $true) | (((X443 @ id @ X452 @ id)) != $true) | (((X443 @ id @ (ap @ (sub @ X453 @ id) @ X452) @ id)) = $true)) | (((X445 @ X446)) != $true) | ~ ! [X447 : subst,X450 : subst,X449 : term,X448 : subst] : ((((X443 @ X450 @ X449 @ (comp @ X447 @ X448))) != $true) | (((X443 @ (comp @ X450 @ X447) @ (sub @ X449 @ X447) @ X448)) = $true)) | ~ ! [X451 : term] : (((X443 @ id @ X451 @ id)) = ((X445 @ X451))) | (((X445 @ (ap @ X444 @ X446))) = $true))) & ((induction2lem = $true) <=> ! [X397 : term,X399 : subst,X398 : (term > $o)] : (~ ! [X401 : term] : (~ ! [X402 : term] : ((((X398 @ (sub @ X401 @ (push @ X402 @ id)))) = $true) | (((X398 @ X402)) != $true)) | (((X398 @ (lam @ X401))) = $true)) | (((X398 @ (sub @ X397 @ X399))) = $true) | ~ ! [X400 : term] : ((((X398 @ (sub @ X400 @ X399))) = $true) | (((var @ X400)) != $true)) | ~ ! [X404 : term,X403 : term] : ((((X398 @ X404)) != $true) | (((X398 @ (ap @ X404 @ X403))) = $true) | (((X398 @ X403)) != $true)))) & (axapp = $true) & (axmap = $true)),
  inference(flattening,[],[f9])).
thf(f11,plain,(
  ((apinj2 = $true) <=> ! [X140 : term,X142 : term,X141 : term,X143 : term] : ((X140 = X143) | (((ap @ X141 @ X140)) != ((ap @ X142 @ X143))))) & ((induction2lem_lthm = $true) <=> ((((! [X533 : term,X532 : subst,X534 : (term > $o)] : (? [X537 : term,X538 : term] : ((((X534 @ X538)) = $true) & (((X534 @ X537)) = $true) & (((X534 @ (ap @ X537 @ X538))) != $true)) | (((X534 @ (sub @ X533 @ X532))) = $true) | ? [X539 : term] : ((((X534 @ (sub @ X539 @ X532))) != $true) & (((var @ X539)) = $true)) | ? [X535 : term] : (! [X536 : term] : ((((X534 @ X536)) != $true) | (((X534 @ (sub @ X535 @ (push @ X536 @ id)))) = $true)) & (((X534 @ (lam @ X535))) != $true))) | ? [X527 : (term > $o),X526 : term] : ((((X527 @ X526)) != $true) & ! [X531 : term,X530 : term] : ((((X527 @ X530)) != $true) | (((X527 @ X531)) != $true) | (((X527 @ (ap @ X531 @ X530))) = $true)) & ! [X528 : term] : ((((X527 @ (lam @ X528))) = $true) | (((X527 @ X528)) != $true)) & ! [X529 : term] : ((((var @ X529)) != $true) | (((X527 @ X529)) = $true)))) | ? [X525 : subst] : (((comp @ X525 @ id)) != X525)) | ? [X523 : subst,X524 : term] : (((comp @ sh @ (push @ X524 @ X523))) != X523)) | ? [X522 : term,X521 : subst] : (((sub @ one @ (push @ X522 @ X521))) != X522))) & ((((! [X254 : subst] : (((comp @ X254 @ id)) = X254) & ! [X253 : subst] : (((comp @ id @ X253)) = X253)) | ? [X252 : subst] : (((comp @ X252 @ id)) != X252)) | ? [X251 : subst] : (((comp @ id @ X251)) != X251)) <=> (substmonoid_lthm = $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (hoasinduction_lem1v2 = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoasapnotvar = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (apnotvar = $true) & (hoasapnotvar_gthm = $true) & ((laminj = $true) <=> ! [X395 : term,X396 : term] : ((X395 = X396) | (((lam @ X395)) != ((lam @ X396))))) & ! [X78 : d_subst] : (d_id = X78) & ((hoasinduction_no_psi_cond_lthm = $true) <=> ((((! [X489 : term,X490 : (subst > term > subst > $o)] : ((((X490 @ id @ X489 @ id)) = $true) | ? [X492 : term,X491 : term] : ((((X490 @ id @ (ap @ (sub @ X491 @ id) @ X492) @ id)) != $true) & (((X490 @ id @ X492 @ id)) = $true) & (((X490 @ id @ X491 @ id)) = $true)) | ? [X493 : (subst > term > term)] : (! [X494 : term] : ((((X490 @ id @ (X493 @ id @ X494) @ id)) = $true) | ($true != ((X490 @ id @ X494 @ id)))) & (((X490 @ id @ (lam @ (X493 @ sh @ one)) @ id)) != $true) & ! [X495 : subst,X496 : term,X497 : subst] : (((sub @ (X493 @ X497 @ X496) @ X495)) = ((X493 @ (comp @ X497 @ X495) @ (sub @ X496 @ X495)))))) | ? [X479 : (subst > term > subst > $o),X480 : (term > $o),X481 : term] : (! [X482 : term] : ((((X480 @ (sub @ X481 @ (push @ X482 @ id)))) = $true) | (((X480 @ X482)) != $true)) & ! [X483 : term] : (((X480 @ X483)) = ((X479 @ id @ X483 @ id))) & ! [X484 : (subst > term > term)] : (? [X485 : subst,X487 : term,X486 : subst] : (((sub @ (X484 @ X486 @ X487) @ X485)) != ((X484 @ (comp @ X486 @ X485) @ (sub @ X487 @ X485)))) | ? [X488 : term] : ((((X479 @ id @ (X484 @ id @ X488) @ id)) != $true) & (((X479 @ id @ X488 @ id)) = $true)) | ($true = ((X479 @ id @ (lam @ (X484 @ sh @ one)) @ id)))) & (((X480 @ (lam @ X481))) != $true))) | ? [X478 : term] : (((sub @ X478 @ id)) != X478)) | ? [X471 : (term > $o),X472 : term] : (! [X473 : term] : (? [X474 : term] : ((((X471 @ X474)) = $true) & (((X471 @ (sub @ X473 @ (push @ X474 @ id)))) != $true)) | (((X471 @ (lam @ X473))) = $true)) & ! [X475 : term,X476 : term] : ((((X471 @ X476)) != $true) | ($true = ((X471 @ (ap @ X476 @ X475)))) | ($true != ((X471 @ X475)))) & (((X471 @ X472)) != $true) & ! [X477 : term] : (($true = ((X471 @ X477))) | (((var @ X477)) != $true)))) | ? [X468 : (subst > term > subst > $o)] : ! [X469 : (term > $o)] : ? [X470 : term] : (((X468 @ id @ X470 @ id)) != ((X469 @ X470))))) & (induction2_gthm = $true) & ((! [X323 : subst] : (((comp @ id @ X323)) = X323) & ! [X322 : subst] : (((comp @ X322 @ id)) = X322)) <=> (substmonoid = $true)) & (hoasinduction_lem3_gthm = $true) & (! [X116 : (subst > term > term),X115 : term,X117 : (subst > term > term),X118 : subst] : (? [X124 : term,X122 : subst,X123 : subst] : (((sub @ (X116 @ X123 @ X124) @ X122)) != ((X116 @ (comp @ X123 @ X122) @ (sub @ X124 @ X122)))) | ? [X121 : subst,X119 : subst,X120 : term] : (((X117 @ (comp @ X119 @ X121) @ (sub @ X120 @ X121))) != ((sub @ (X117 @ X119 @ X120) @ X121))) | (((lam @ (X116 @ sh @ one))) != ((lam @ (X117 @ sh @ one)))) | (((X117 @ X118 @ X115)) = ((X116 @ X118 @ X115)))) <=> (hoaslaminj = $true)) & ((apinj1 = $true) <=> ! [X111 : term,X113 : term,X114 : term,X112 : term] : ((X112 = X114) | (((ap @ X112 @ X111)) != ((ap @ X114 @ X113))))) & (((! [X267 : (subst > term > subst > $o),X268 : term] : ((((X267 @ id @ (lam @ X268) @ id)) = $true) | ? [X273 : subst,X270 : subst,X271 : term,X272 : subst] : ((((X267 @ X273 @ X271 @ (comp @ X270 @ X272))) != $true) & (((X267 @ (comp @ X273 @ X270) @ (sub @ X271 @ X270) @ X272)) = $true)) | ? [X274 : (subst > term > term)] : ((((X267 @ id @ (lam @ (X274 @ sh @ one)) @ id)) != $true) & ! [X276 : subst,X275 : subst,X277 : term] : (((X274 @ (comp @ X276 @ X275) @ (sub @ X277 @ X275))) = ((sub @ (X274 @ X276 @ X277) @ X275))) & ! [X278 : term] : ((((X267 @ id @ X278 @ id)) != $true) | (((X267 @ id @ (X274 @ id @ X278) @ id)) = $true))) | ? [X269 : term] : ((((X267 @ id @ X269 @ id)) = $true) & (((X267 @ id @ (sub @ X268 @ (push @ X269 @ id)) @ id)) != $true)) | ? [X279 : subst,X280 : term,X281 : subst,X282 : subst] : ((((X267 @ X282 @ X280 @ (comp @ X281 @ X279))) = $true) & (((X267 @ (comp @ X282 @ X281) @ (sub @ X280 @ X281) @ X279)) != $true))) | ? [X259 : (subst > term > subst > $o),X260 : term] : (! [X261 : (subst > term > term)] : (? [X264 : subst,X265 : term,X263 : subst] : (((X261 @ (comp @ X263 @ X264) @ (sub @ X265 @ X264))) != ((sub @ (X261 @ X263 @ X265) @ X264))) | (((X259 @ id @ (lam @ (X261 @ sh @ one)) @ id)) = $true) | ? [X262 : term] : ((((X259 @ id @ X262 @ id)) = $true) & (((X259 @ id @ (X261 @ id @ X262) @ id)) != $true))) & ! [X266 : term] : ((((X259 @ id @ X266 @ id)) != $true) | (((X259 @ id @ (sub @ X260 @ (push @ X266 @ id)) @ id)) = $true)) & (((X259 @ id @ (lam @ (sub @ X260 @ (push @ one @ sh))) @ id)) != $true))) | ? [X258 : term] : (((sub @ X258 @ id)) != X258)) <=> (hoasinduction_lem3_lthm = $true)) & (hoasapinj2_gthm = $true) & (hoasinduction_lem1 = $true) & (pushprop_lem2v2_gthm = $true) & ((pushprop_lem1 = $true) <=> ! [X81 : term,X82 : (term > $o),X83 : term,X80 : (term > $o),X79 : subst] : ((((X80 @ X81)) != $true) | (((X82 @ (sub @ X81 @ (push @ X83 @ X79)))) = $true))) & (hoasinduction_gthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((pushprop_lem1v2_lthm = $true) <=> (! [X108 : term,X107 : subst,X106 : (term > $o),X109 : (term > $o)] : (? [X110 : term] : (((X109 @ X110)) != ((X106 @ (sub @ X110 @ (push @ X108 @ X107))))) | (((X106 @ X108)) != $true) | (((X109 @ one)) = $true)) | ? [X105 : subst,X104 : term] : (((sub @ one @ (push @ X104 @ X105))) != X104))) & (ulamvarind = $true) & ! [X515 : d_subst,X514 : d_subst] : ((((d2subst @ X514)) != ((d2subst @ X515))) | (X514 = X515)) & (hoasinduction_lem2v2_gthm = $true) & (hoasinduction_lem1_gthm = $true) & (hoasinduction_lem3a_gthm = $true) & (hoaslamnotvar_gthm = $true) & ((pushprop_lem0_lthm = $true) <=> ! [X248 : subst,X247 : term,X246 : (term > $o)] : ? [X249 : (term > $o)] : ! [X250 : term] : (((X246 @ (sub @ X250 @ (push @ X247 @ X248)))) = ((X249 @ X250)))) & (! [X98 : term,X101 : (term > $o),X100 : subst,X99 : (term > $o)] : ((((X101 @ one)) = $true) | (((X99 @ X98)) != $true) | ? [X102 : term] : (((X101 @ X102)) != ((X99 @ (sub @ X102 @ (push @ X98 @ X100)))))) <=> (pushprop_lem1v2 = $true)) & (substmonoid_gthm = $true) & (hoasapinj1_gthm = $true) & (((! [X371 : (subst > term > subst > $o),X372 : term] : ((((X371 @ id @ (lam @ X372) @ id)) = $true) | ? [X373 : term] : ((((X371 @ id @ (sub @ X372 @ (push @ X373 @ id)) @ id)) != $true) & (((X371 @ id @ X373 @ id)) = $true)) | ? [X374 : (subst > term > term)] : ((((X371 @ id @ (lam @ (X374 @ sh @ one)) @ id)) != $true) & ! [X377 : term,X376 : subst,X378 : subst] : (((X374 @ (comp @ X378 @ X376) @ (sub @ X377 @ X376))) = ((sub @ (X374 @ X378 @ X377) @ X376))) & ! [X375 : term] : ((((X371 @ id @ (X374 @ id @ X375) @ id)) = $true) | (((X371 @ id @ X375 @ id)) != $true)))) | ? [X363 : (subst > term > subst > $o),X364 : term] : ((((X363 @ id @ (lam @ (sub @ X364 @ (push @ one @ sh))) @ id)) != $true) & ! [X366 : (subst > term > term)] : (? [X370 : term] : ((((X363 @ id @ (X366 @ id @ X370) @ id)) != $true) & (((X363 @ id @ X370 @ id)) = $true)) | ? [X368 : subst,X367 : subst,X369 : term] : (((X366 @ (comp @ X368 @ X367) @ (sub @ X369 @ X367))) != ((sub @ (X366 @ X368 @ X369) @ X367))) | (((X363 @ id @ (lam @ (X366 @ sh @ one)) @ id)) = $true)) & ! [X365 : term] : ((((X363 @ id @ X365 @ id)) != $true) | (((X363 @ id @ (sub @ X364 @ (push @ X365 @ id)) @ id)) = $true)))) | ? [X362 : term] : (((sub @ X362 @ id)) != X362)) <=> (hoasinduction_lem3a_lthm = $true)) & (hoasinduction_lem2_gthm = $true) & (pushprop = $true) & (pushprop_lem3v2 = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & ! [X15 : subst] : ? [X16 : d_subst] : (((d2subst @ X16)) = X15) & (! [X406 : term,X405 : (subst > term > subst > $o)] : (? [X407 : (subst > term > term)] : ((((X405 @ id @ (lam @ (X407 @ sh @ one)) @ id)) != $true) & ! [X408 : subst,X409 : term,X410 : subst] : (((sub @ (X407 @ X408 @ X409) @ X410)) = ((X407 @ (comp @ X408 @ X410) @ (sub @ X409 @ X410)))) & ! [X411 : term] : ((((X405 @ id @ X411 @ id)) != $true) | (((X405 @ id @ (X407 @ id @ X411) @ id)) = $true))) | (((X405 @ id @ (lam @ X406) @ id)) = $true) | ? [X412 : term] : ((((X405 @ id @ X412 @ id)) = $true) & (((X405 @ id @ (sub @ X406 @ (push @ X412 @ id)) @ id)) != $true))) <=> (hoasinduction_lem3a = $true)) & ((pushprop_lem0 = $true) <=> ! [X415 : subst,X416 : term,X417 : (term > $o)] : ? [X418 : (term > $o)] : ! [X419 : term] : (((X417 @ (sub @ X419 @ (push @ X416 @ X415)))) = ((X418 @ X419)))) & (! [X458 : (subst > term > subst > $o),X460 : (term > $o),X459 : term] : (? [X467 : term] : (((X458 @ id @ X467 @ id)) != ((X460 @ X467))) | ? [X462 : (subst > term > term)] : ((((X458 @ id @ (lam @ (X462 @ sh @ one)) @ id)) != $true) & ! [X463 : subst,X464 : subst,X465 : term] : (((X462 @ (comp @ X464 @ X463) @ (sub @ X465 @ X463))) = ((sub @ (X462 @ X464 @ X465) @ X463))) & ! [X466 : term] : ((((X458 @ id @ X466 @ id)) != $true) | (((X458 @ id @ (X462 @ id @ X466) @ id)) = $true))) | (((X460 @ (lam @ X459))) = $true) | ? [X461 : term] : ((((X460 @ (sub @ X459 @ (push @ X461 @ id)))) != $true) & (((X460 @ X461)) = $true))) <=> (hoasinduction_lem3v2a = $true)) & (axassoc = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (((var @ (d2term @ d_one))) != $true) & ((induction2_lthm = $true) <=> ((! [X389 : term,X388 : (term > $o)] : (? [X391 : term,X392 : term] : ((((X388 @ X391)) = $true) & (((X388 @ (ap @ X391 @ X392))) != $true) & (((X388 @ X392)) = $true)) | (((X388 @ X389)) = $true) | ? [X393 : term] : ((((X388 @ (lam @ X393))) != $true) & ! [X394 : term] : ((((X388 @ X394)) != $true) | (((X388 @ (sub @ X393 @ (push @ X394 @ id)))) = $true))) | ? [X390 : term] : ((((X388 @ X390)) != $true) & (((var @ X390)) = $true))) | ? [X381 : (term > $o),X382 : subst,X380 : term] : ((((X381 @ (sub @ X380 @ X382))) != $true) & ! [X384 : term,X383 : term] : ((((X381 @ X383)) != $true) | (((X381 @ X384)) != $true) | (((X381 @ (ap @ X383 @ X384))) = $true)) & ! [X386 : term] : ((((X381 @ (lam @ X386))) = $true) | ? [X387 : term] : ((((X381 @ X387)) = $true) & (((X381 @ (sub @ X386 @ (push @ X387 @ id)))) != $true))) & ! [X385 : term] : ((((X381 @ (sub @ X385 @ X382))) = $true) | (((var @ X385)) != $true)))) | ? [X379 : term] : (((sub @ X379 @ id)) != X379))) & (pushprop_lem1_gthm = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (lamnotvar = $true) & ((axshiftcons = $true) <=> ! [X326 : subst,X327 : term] : (((comp @ sh @ (push @ X327 @ X326))) = X326)) & (! [X503 : term,X502 : (subst > term > subst > $o)] : (? [X513 : term] : ((((X502 @ id @ (sub @ X503 @ (push @ X513 @ id)) @ id)) != $true) & (((X502 @ id @ X513 @ id)) = $true)) | (((X502 @ id @ (lam @ (sub @ X503 @ (push @ one @ sh))) @ id)) = $true) | ? [X504 : (subst > term > term),X505 : term] : (! [X506 : subst,X508 : term,X507 : subst] : (((sub @ (sub @ X505 @ (push @ X508 @ X507)) @ X506)) = ((sub @ (X504 @ X507 @ X508) @ X506))) & (((X502 @ id @ (lam @ (X504 @ sh @ one)) @ id)) != $true) & ! [X512 : term] : ((((X502 @ id @ (X504 @ id @ X512) @ id)) = $true) | ($true != ((X502 @ id @ X512 @ id)))) & ! [X511 : term,X509 : subst,X510 : subst] : (((X504 @ (comp @ X509 @ X510) @ (sub @ X511 @ X510))) = ((sub @ X505 @ (push @ (sub @ X511 @ X510) @ (comp @ X509 @ X510))))))) <=> (hoasinduction_lem3aaa = $true)) & (hoasinduction_lem2_lthm = $true) & (! [X520 : subst] : (((comp @ X520 @ id)) = X520) <=> (axidr = $true)) & (pushprop_gthm = $true) & ((? [X94 : term] : (((sub @ X94 @ id)) != X94) | ! [X95 : term] : (((sub @ X95 @ id)) = X95)) <=> (termmset_lthm = $true)) & (! [X234 : term,X235 : term,X233 : (subst > term > subst > $o)] : (? [X239 : term,X241 : subst,X238 : subst,X240 : subst] : ((((X233 @ X238 @ X239 @ (comp @ X241 @ X240))) != $true) & (((X233 @ (comp @ X238 @ X241) @ (sub @ X239 @ X241) @ X240)) = $true)) | (((X233 @ id @ (ap @ X234 @ X235) @ id)) = $true) | ? [X245 : subst,X244 : subst,X243 : term,X242 : subst] : ((((X233 @ (comp @ X245 @ X242) @ (sub @ X243 @ X242) @ X244)) != $true) & (((X233 @ X245 @ X243 @ (comp @ X242 @ X244))) = $true)) | (((X233 @ id @ X235 @ id)) != $true) | ? [X237 : term,X236 : term] : ((((X233 @ id @ X236 @ id)) = $true) & (((X233 @ id @ X237 @ id)) = $true) & (((X233 @ id @ (ap @ (sub @ X237 @ id) @ X236) @ id)) != $true)) | (((X233 @ id @ X234 @ id)) != $true)) <=> (hoasinduction_lem2 = $true)) & (hoaslamnotvar = $true) & (hoaslamnotap_lthm = $true) & (! [X423 : term,X422 : term] : ((((sub @ X422 @ sh)) != ((sub @ X423 @ sh))) | (X422 = X423)) <=> (shinj = $true)) & (! [X96 : subst,X97 : term] : (((sub @ one @ (push @ X97 @ X96))) = X97) <=> (axvarcons = $true)) & (! [X516 : term] : ? [X517 : (subst > term > term)] : ! [X519 : term,X518 : subst] : (((X517 @ X518 @ X519)) = ((sub @ X516 @ (push @ X519 @ X518)))) <=> (hoasinduction_lem3v2_f = $true)) & ((induction = $true) <=> ! [X133 : (term > $o),X134 : term] : (? [X135 : term] : ((((X133 @ (lam @ X135))) != $true) & (((X133 @ X135)) = $true)) | (((X133 @ X134)) = $true) | ? [X137 : term,X138 : term] : ((((X133 @ (ap @ X137 @ X138))) != $true) & (((X133 @ X137)) = $true) & (((X133 @ X138)) = $true)) | ? [X136 : term] : ((((var @ X136)) = $true) & (((X133 @ X136)) != $true)))) & (pushprop_lem2v2 = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ((hoasinduction_lem3 = $true) <=> ! [X428 : (subst > term > subst > $o),X427 : term] : (? [X436 : subst,X434 : term,X435 : subst,X433 : subst] : ((((X428 @ (comp @ X433 @ X436) @ (sub @ X434 @ X436) @ X435)) != $true) & (((X428 @ X433 @ X434 @ (comp @ X436 @ X435))) = $true)) | ? [X430 : subst,X431 : subst,X429 : subst,X432 : term] : ((((X428 @ X429 @ X432 @ (comp @ X430 @ X431))) != $true) & (((X428 @ (comp @ X429 @ X430) @ (sub @ X432 @ X430) @ X431)) = $true)) | ? [X437 : (subst > term > term)] : (! [X438 : term,X439 : subst,X440 : subst] : (((X437 @ (comp @ X440 @ X439) @ (sub @ X438 @ X439))) = ((sub @ (X437 @ X440 @ X438) @ X439))) & ! [X441 : term] : ((((X428 @ id @ X441 @ id)) != $true) | (((X428 @ id @ (X437 @ id @ X441) @ id)) = $true)) & (((X428 @ id @ (lam @ (X437 @ sh @ one)) @ id)) != $true)) | (((X428 @ id @ (lam @ X427) @ id)) = $true) | ? [X442 : term] : ((((X428 @ id @ X442 @ id)) = $true) & (((X428 @ id @ (sub @ X427 @ (push @ X442 @ id)) @ id)) != $true)))) & ((hoaslamnotap = $true) <=> ! [X49 : (subst > term > term),X48 : term,X50 : term] : ((((ap @ (sub @ X50 @ id) @ X48)) != ((lam @ (X49 @ sh @ one)))) | ? [X53 : term,X51 : subst,X52 : subst] : (((X49 @ (comp @ X51 @ X52) @ (sub @ X53 @ X52))) != ((sub @ (X49 @ X51 @ X53) @ X52))))) & (pushprop_lthm_orig = $true) & ! [X93 : d_term] : (d_one = X93) & (! [X22 : term,X21 : (subst > term > subst > $o)] : (? [X33 : term,X32 : term] : ((((X21 @ id @ (ap @ (sub @ X33 @ id) @ X32) @ id)) != $true) & ($true = ((X21 @ id @ X33 @ id))) & (((X21 @ id @ X32 @ id)) = $true)) | (((X21 @ id @ X22 @ id)) = $true) | ? [X27 : (subst > term > term)] : ((((X21 @ id @ (lam @ (X27 @ sh @ one)) @ id)) != $true) & ! [X30 : term,X29 : subst,X31 : subst] : (((X27 @ (comp @ X29 @ X31) @ (sub @ X30 @ X31))) = ((sub @ (X27 @ X29 @ X30) @ X31))) & ! [X28 : term] : ((((X21 @ id @ (X27 @ id @ X28) @ id)) = $true) | (((X21 @ id @ X28 @ id)) != $true))) | ? [X23 : subst,X24 : subst,X26 : term,X25 : subst] : ((((X21 @ X25 @ X26 @ (comp @ X24 @ X23))) = $true) & (((X21 @ (comp @ X25 @ X24) @ (sub @ X26 @ X24) @ X23)) != $true)) | ? [X38 : term] : ((((var @ (sub @ X38 @ id))) = $true) & (((X21 @ id @ X38 @ id)) != $true)) | ? [X34 : term,X36 : subst,X37 : subst,X35 : subst] : ((((X21 @ (comp @ X36 @ X35) @ (sub @ X34 @ X35) @ X37)) = $true) & (((X21 @ X36 @ X34 @ (comp @ X35 @ X37))) != $true))) <=> (hoasinduction = $true)) & (((d2subst @ d_id)) = sh) & (! [X498 : term] : ? [X499 : (subst > term > term)] : ! [X501 : subst,X500 : term] : (((X499 @ X501 @ X500)) = ((sub @ X498 @ (push @ X500 @ X501)))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & ! [X325 : d_term,X324 : d_term] : ((X324 = X325) | (((d2term @ X325)) != ((d2term @ X324)))) & (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)))))))))))) & ((hoasapinj1_lthm = $true) <=> ((! [X92 : term,X89 : term,X90 : term,X91 : term] : ((X89 = X92) | (((ap @ (sub @ X89 @ id) @ X91)) != ((ap @ (sub @ X92 @ id) @ X90)))) | ? [X86 : term,X85 : term,X87 : term,X88 : term] : ((((ap @ X88 @ X85)) = ((ap @ X87 @ X86))) & (X87 != X88))) | ? [X84 : term] : (((sub @ X84 @ id)) != X84))) & (! [X208 : term] : (((sub @ X208 @ id)) = X208) <=> (axvarid = $true)) & (((! [X202 : term,X205 : (term > $o),X206 : subst,X204 : term,X203 : (term > $o)] : ((((X203 @ X202)) != $true) | (((X205 @ (sub @ X202 @ (push @ X204 @ X206)))) = $true)) | ? [X200 : term,X201 : subst] : (((comp @ sh @ (push @ X200 @ X201))) != X201)) | ? [X199 : subst,X198 : term] : (((sub @ one @ (push @ X198 @ X199))) != X198)) <=> (pushprop_lem1_lthm = $true)) & (induction2lem_gthm = $true) & (pushprop_lem1v2_gthm = $true) & (axabs = $true) & (ulamvarsh = $true) & (((((! [X305 : term,X304 : (subst > term > subst > $o)] : (? [X310 : (subst > term > term)] : ((((X304 @ id @ (lam @ (X310 @ sh @ one)) @ id)) != $true) & ! [X312 : subst,X313 : term,X314 : subst] : (((sub @ (X310 @ X314 @ X313) @ X312)) = ((X310 @ (comp @ X314 @ X312) @ (sub @ X313 @ X312)))) & ! [X311 : term] : ((((X304 @ id @ X311 @ id)) != $true) | (((X304 @ id @ (X310 @ id @ X311) @ id)) = $true))) | (((X304 @ id @ X305 @ id)) = $true) | ? [X320 : term,X321 : term] : ((((X304 @ id @ X320 @ id)) = $true) & (((X304 @ id @ X321 @ id)) = $true) & (((X304 @ id @ (ap @ (sub @ X321 @ id) @ X320) @ id)) != $true)) | ? [X308 : subst,X309 : term,X306 : subst,X307 : subst] : ((((X304 @ X308 @ X309 @ (comp @ X307 @ X306))) = $true) & (((X304 @ (comp @ X308 @ X307) @ (sub @ X309 @ X307) @ X306)) != $true)) | ? [X319 : term] : ((((X304 @ id @ X319 @ id)) != $true) & (((var @ (sub @ X319 @ id))) = $true)) | ? [X315 : term,X317 : subst,X318 : subst,X316 : subst] : ((((X304 @ X317 @ X315 @ (comp @ X318 @ X316))) != $true) & (((X304 @ (comp @ X317 @ X318) @ (sub @ X315 @ X318) @ X316)) = $true))) | ? [X294 : term,X295 : (subst > term > subst > $o),X296 : (term > $o)] : (! [X302 : term] : ((((X296 @ (sub @ X294 @ (push @ X302 @ id)))) = $true) | (((X296 @ X302)) != $true)) & (((X296 @ (lam @ X294))) != $true) & ! [X297 : (subst > term > term)] : (? [X301 : term] : ((((X295 @ id @ X301 @ id)) = $true) & (((X295 @ id @ (X297 @ id @ X301) @ id)) != $true)) | ? [X300 : subst,X299 : subst,X298 : term] : (((X297 @ (comp @ X299 @ X300) @ (sub @ X298 @ X300))) != ((sub @ (X297 @ X299 @ X298) @ X300))) | (((X295 @ id @ (lam @ (X297 @ sh @ one)) @ id)) = $true)) & ! [X303 : term] : (((X296 @ X303)) = ((X295 @ id @ X303 @ id))))) | ? [X293 : term] : (((sub @ X293 @ id)) != X293)) | ? [X287 : (term > $o),X286 : term] : ((((X287 @ X286)) != $true) & ! [X289 : term] : (? [X290 : term] : ((((X287 @ (sub @ X289 @ (push @ X290 @ id)))) != $true) & (((X287 @ X290)) = $true)) | (((X287 @ (lam @ X289))) = $true)) & ! [X292 : term,X291 : term] : ((((X287 @ X291)) != $true) | (((X287 @ (ap @ X292 @ X291))) = $true) | (((X287 @ X292)) != $true)) & ! [X288 : term] : ((((X287 @ X288)) = $true) | (((var @ X288)) != $true)))) | ? [X283 : (subst > term > subst > $o)] : ! [X284 : (term > $o)] : ? [X285 : term] : (((X283 @ id @ X285 @ id)) != ((X284 @ X285)))) <=> (hoasinduction_lthm_3 = $true)) & ((termmset = $true) <=> ! [X103 : term] : (((sub @ X103 @ id)) = X103)) & (pushprop_lem0_gthm = $true) & ((hoasinduction_lthm = $true) <=> (((! [X181 : term,X180 : (subst > term > subst > $o)] : ((((X180 @ id @ X181 @ id)) = $true) | ? [X190 : term] : ((((var @ (sub @ X190 @ id))) = $true) & (((X180 @ id @ X190 @ id)) != $true)) | ? [X192 : term,X191 : term] : ((((X180 @ id @ X191 @ id)) = $true) & (((X180 @ id @ (ap @ (sub @ X191 @ id) @ X192) @ id)) != $true) & (((X180 @ id @ X192 @ id)) = $true)) | ? [X187 : term,X189 : subst,X188 : subst,X186 : subst] : ((((X180 @ X188 @ X187 @ (comp @ X186 @ X189))) = $true) & (((X180 @ (comp @ X188 @ X186) @ (sub @ X187 @ X186) @ X189)) != $true)) | ? [X193 : (subst > term > term)] : ((((X180 @ id @ (lam @ (X193 @ sh @ one)) @ id)) != $true) & ! [X194 : term] : ((((X180 @ id @ (X193 @ id @ X194) @ id)) = $true) | (((X180 @ id @ X194 @ id)) != $true)) & ! [X195 : subst,X196 : subst,X197 : term] : (((sub @ (X193 @ X196 @ X197) @ X195)) = ((X193 @ (comp @ X196 @ X195) @ (sub @ X197 @ X195))))) | ? [X184 : subst,X183 : term,X182 : subst,X185 : subst] : ((((X180 @ (comp @ X184 @ X182) @ (sub @ X183 @ X182) @ X185)) = $true) & (((X180 @ X184 @ X183 @ (comp @ X182 @ X185))) != $true))) | ? [X165 : term,X164 : (subst > term > subst > $o)] : ((((X164 @ id @ (lam @ X165) @ id)) != $true) & ! [X177 : subst,X178 : subst,X179 : term,X176 : subst] : ((((X164 @ X176 @ X179 @ (comp @ X177 @ X178))) = $true) | (((X164 @ (comp @ X176 @ X177) @ (sub @ X179 @ X177) @ X178)) != $true)) & ! [X166 : (subst > term > term)] : ((((X164 @ id @ (lam @ (X166 @ sh @ one)) @ id)) = $true) | ? [X168 : subst,X167 : term,X169 : subst] : (((sub @ (X166 @ X168 @ X167) @ X169)) != ((X166 @ (comp @ X168 @ X169) @ (sub @ X167 @ X169)))) | ? [X170 : term] : ((((X164 @ id @ X170 @ id)) = $true) & (((X164 @ id @ (X166 @ id @ X170) @ id)) != $true))) & ! [X172 : subst,X171 : subst,X173 : term,X174 : subst] : ((((X164 @ (comp @ X171 @ X172) @ (sub @ X173 @ X172) @ X174)) = $true) | (((X164 @ X171 @ X173 @ (comp @ X172 @ X174))) != $true)) & ! [X175 : term] : ((((X164 @ id @ X175 @ id)) != $true) | (((X164 @ id @ (sub @ X165 @ (push @ X175 @ id)) @ id)) = $true)))) | ? [X153 : term,X151 : (subst > term > subst > $o),X152 : term] : (! [X155 : term,X154 : term] : ((((X151 @ id @ (ap @ (sub @ X155 @ id) @ X154) @ id)) = $true) | (((X151 @ id @ X155 @ id)) != $true) | (((X151 @ id @ X154 @ id)) != $true)) & (((X151 @ id @ (ap @ X152 @ X153) @ id)) != $true) & (((X151 @ id @ X153 @ id)) = $true) & (((X151 @ id @ X152 @ id)) = $true) & ! [X158 : subst,X157 : subst,X156 : subst,X159 : term] : ((((X151 @ X156 @ X159 @ (comp @ X158 @ X157))) != $true) | (((X151 @ (comp @ X156 @ X158) @ (sub @ X159 @ X158) @ X157)) = $true)) & ! [X162 : subst,X163 : term,X160 : subst,X161 : subst] : ((((X151 @ (comp @ X160 @ X161) @ (sub @ X163 @ X161) @ X162)) != $true) | (((X151 @ X160 @ X163 @ (comp @ X161 @ X162))) = $true)))) | ? [X145 : term,X144 : (term > $o)] : (! [X146 : term,X147 : term] : ((((X144 @ (ap @ X147 @ X146))) = $true) | (((X144 @ X147)) != $true) | (((X144 @ X146)) != $true)) & ! [X148 : term] : ((((var @ X148)) != $true) | ($true = ((X144 @ X148)))) & ! [X149 : term] : ((((X144 @ (lam @ X149))) = $true) | ? [X150 : term] : (($true != ((X144 @ (sub @ X149 @ (push @ X150 @ id))))) & ($true = ((X144 @ X150))))) & (((X144 @ X145)) != $true)))) & ((hoasinduction_lem3v2 = $true) <=> ! [X215 : term,X217 : (term > $o),X216 : (subst > term > subst > $o)] : (? [X232 : subst,X231 : subst,X230 : term,X229 : subst] : ((((X216 @ X231 @ X230 @ (comp @ X232 @ X229))) = $true) & (((X216 @ (comp @ X231 @ X232) @ (sub @ X230 @ X232) @ X229)) != $true)) | (((X217 @ (lam @ X215))) = $true) | ? [X218 : term] : (($true != ((X217 @ (sub @ X215 @ (push @ X218 @ id))))) & (((X217 @ X218)) = $true)) | ? [X223 : term] : (((X216 @ id @ X223 @ id)) != ((X217 @ X223))) | ? [X224 : (subst > term > term)] : ((((X216 @ id @ (lam @ (X224 @ sh @ one)) @ id)) != $true) & ! [X225 : term] : ((((X216 @ id @ (X224 @ id @ X225) @ id)) = $true) | (((X216 @ id @ X225 @ id)) != $true)) & ! [X227 : subst,X226 : term,X228 : subst] : (((X224 @ (comp @ X228 @ X227) @ (sub @ X226 @ X227))) = ((sub @ (X224 @ X228 @ X226) @ X227)))) | ? [X222 : term,X220 : subst,X219 : subst,X221 : subst] : ((((X216 @ (comp @ X221 @ X219) @ (sub @ X222 @ X219) @ X220)) = $true) & (((X216 @ X221 @ X222 @ (comp @ X219 @ X220))) != $true)))) & ! [X413 : term] : ? [X414 : d_term] : (((d2term @ X414)) = X413) & (hoasinduction_lem3v2_gthm = $true) & (! [X55 : (subst > term > subst > $o),X54 : term] : ((((X55 @ id @ (lam @ (sub @ X54 @ (push @ one @ sh))) @ id)) = $true) | ? [X56 : (subst > term > term)] : (! [X57 : term] : ((((X55 @ id @ X57 @ id)) != $true) | (((X55 @ id @ (X56 @ id @ X57) @ id)) = $true)) & (((X55 @ id @ (lam @ (X56 @ sh @ one)) @ id)) != $true) & ! [X58 : term,X59 : subst,X60 : subst] : (((X56 @ (comp @ X60 @ X59) @ (sub @ X58 @ X59))) = ((sub @ (X56 @ X60 @ X58) @ X59)))) | ? [X61 : term] : (($true = ((X55 @ id @ X61 @ id))) & (((X55 @ id @ (sub @ X54 @ (push @ X61 @ id)) @ id)) != $true))) <=> (hoasinduction_lem3aa_lthm = $true)) & (pushprop_lem3v2_lthm = $true) & (! [X255 : (subst > term > subst > $o)] : ? [X256 : (term > $o)] : ! [X257 : term] : (((X256 @ X257)) = ((X255 @ id @ X257 @ id))) <=> (hoasinduction_lem0 = $true)) & (((d2term @ d_one)) = one) & (! [X420 : term] : ? [X421 : (subst > term > term)] : (((sub @ X420 @ (push @ one @ sh))) = ((X421 @ sh @ one))) <=> (hoasinduction_lem3b_lthm = $true)) & (axclos = $true) & (hoaslaminj_gthm = $true) & (hoasinduction_lem1_lthm = $true) & (! [X1 : (term > $o),X0 : term] : ((((X1 @ X0)) = $true) | ? [X2 : term] : (! [X3 : term] : ((((X1 @ (sub @ X2 @ (push @ X3 @ id)))) = $true) | (((X1 @ X3)) != $true)) & (((X1 @ (lam @ X2))) != $true)) | ? [X5 : term,X4 : term] : ((((X1 @ (ap @ X5 @ X4))) != $true) & (((X1 @ X4)) = $true) & (((X1 @ X5)) = $true)) | ? [X6 : term] : ((((X1 @ X6)) != $true) & (((var @ X6)) = $true))) <=> (induction2 = $true)) & (axvarshift = $true) & ((! [X14 : term,X11 : term,X13 : term,X12 : term] : ((((ap @ (sub @ X13 @ id) @ X12)) != ((ap @ (sub @ X14 @ id) @ X11))) | (X11 = X12)) | ? [X10 : term,X9 : term,X8 : term,X7 : term] : ((X7 != X9) & (((ap @ X8 @ X7)) = ((ap @ X10 @ X9))))) <=> (hoasapinj2_lthm = $true)) & ((hoasapinj1 = $true) <=> ! [X18 : term,X19 : term,X17 : term,X20 : term] : ((((ap @ (sub @ X20 @ id) @ X19)) != ((ap @ (sub @ X17 @ id) @ X18))) | (X17 = X20))) & (hoaslamnotap_gthm = $true) & ((axidl = $true) <=> ! [X207 : subst] : (((comp @ id @ X207)) = X207)) & ((hoaslaminj_lthm = $true) <=> (((! [X68 : subst,X70 : (subst > term > term),X69 : (subst > term > term),X71 : term] : (? [X72 : subst,X73 : term,X74 : subst] : (((sub @ (X69 @ X72 @ X73) @ X74)) != ((X69 @ (comp @ X72 @ X74) @ (sub @ X73 @ X74)))) | (((X69 @ X68 @ X71)) = ((X70 @ X68 @ X71))) | ? [X77 : term,X76 : subst,X75 : subst] : (((sub @ (X70 @ X76 @ X77) @ X75)) != ((X70 @ (comp @ X76 @ X75) @ (sub @ X77 @ X75)))) | (((lam @ (X70 @ sh @ one))) != ((lam @ (X69 @ sh @ one))))) | ? [X66 : term,X67 : term] : ((((lam @ X66)) = ((lam @ X67))) & (X66 != X67))) | ? [X64 : term,X65 : subst] : (((comp @ sh @ (push @ X64 @ X65))) != X65)) | ? [X62 : term,X63 : subst] : (((sub @ one @ (push @ X62 @ X63))) != X62))) & (pushprop_lem2v2_lthm = $true) & ((hoasinduction_lem0_lthm = $true) <=> ! [X424 : (subst > term > subst > $o)] : ? [X425 : (term > $o)] : ! [X426 : term] : (((X425 @ X426)) = ((X424 @ id @ X426 @ id)))) & ((! [X345 : term,X346 : (subst > term > subst > $o),X344 : (term > $o)] : (? [X347 : term] : (((X344 @ X347)) != ((X346 @ id @ X347 @ id))) | (((X344 @ (lam @ X345))) = $true) | ? [X361 : term] : ((((X344 @ X361)) = $true) & (((X344 @ (sub @ X345 @ (push @ X361 @ id)))) != $true)) | ? [X348 : subst,X350 : term,X349 : subst,X351 : subst] : ((((X346 @ X348 @ X350 @ (comp @ X351 @ X349))) != $true) & (((X346 @ (comp @ X348 @ X351) @ (sub @ X350 @ X351) @ X349)) = $true)) | ? [X352 : subst,X354 : term,X355 : subst,X353 : subst] : ((((X346 @ (comp @ X355 @ X353) @ (sub @ X354 @ X353) @ X352)) != $true) & (((X346 @ X355 @ X354 @ (comp @ X353 @ X352))) = $true)) | ? [X356 : (subst > term > term)] : (! [X357 : term] : ((((X346 @ id @ (X356 @ id @ X357) @ id)) = $true) | (((X346 @ id @ X357 @ id)) != $true)) & (((X346 @ id @ (lam @ (X356 @ sh @ one)) @ id)) != $true) & ! [X358 : subst,X360 : term,X359 : subst] : (((X356 @ (comp @ X358 @ X359) @ (sub @ X360 @ X359))) = ((sub @ (X356 @ X358 @ X360) @ X359))))) | ? [X343 : term] : (((sub @ X343 @ id)) != X343)) <=> (hoasinduction_lem3v2_lthm = $true)) & ((hoasinduction_no_psi_cond = $true) <=> ! [X39 : (subst > term > subst > $o),X40 : term] : (? [X41 : term,X42 : term] : ((((X39 @ id @ X42 @ id)) = $true) & (((X39 @ id @ (ap @ (sub @ X41 @ id) @ X42) @ id)) != $true) & (((X39 @ id @ X41 @ id)) = $true)) | ($true = ((X39 @ id @ X40 @ id))) | ? [X43 : (subst > term > term)] : (! [X47 : term] : ((((X39 @ id @ (X43 @ id @ X47) @ id)) = $true) | (((X39 @ id @ X47 @ id)) != $true)) & (((X39 @ id @ (lam @ (X43 @ sh @ one)) @ id)) != $true) & ! [X46 : subst,X44 : subst,X45 : term] : (((sub @ (X43 @ X46 @ X45) @ X44)) = ((X43 @ (comp @ X46 @ X44) @ (sub @ X45 @ X44))))))) & (pushprop_lthm = $true) & (lamnotap != $true) & (! [X125 : term,X126 : (subst > term > subst > $o)] : (? [X128 : (subst > term > term)] : (! [X131 : subst,X130 : term,X129 : subst] : (((sub @ (X128 @ X131 @ X130) @ X129)) = ((X128 @ (comp @ X131 @ X129) @ (sub @ X130 @ X129)))) & (((X126 @ id @ (lam @ (X128 @ sh @ one)) @ id)) != $true) & ! [X132 : term] : ((((X126 @ id @ X132 @ id)) != $true) | (((X126 @ id @ (X128 @ id @ X132) @ id)) = $true))) | ? [X127 : term] : ((((X126 @ id @ X127 @ id)) = $true) & (((X126 @ id @ (sub @ X125 @ (push @ X127 @ id)) @ id)) != $true)) | (((X126 @ id @ (lam @ (sub @ X125 @ (push @ one @ sh))) @ id)) = $true)) <=> (hoasinduction_lem3aa = $true)) & ((hoasinduction_lem3b = $true) <=> ! [X213 : term] : ? [X214 : (subst > term > term)] : (((X214 @ sh @ one)) = ((sub @ X213 @ (push @ one @ sh))))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (hoasinduction_lem1v2_gthm = $true) & (((! [X335 : (term > $o),X333 : term,X334 : (subst > term > subst > $o)] : (? [X337 : (subst > term > term)] : (! [X338 : term] : ((((X334 @ id @ X338 @ id)) != $true) | (((X334 @ id @ (X337 @ id @ X338) @ id)) = $true)) & ! [X339 : subst,X340 : term,X341 : subst] : (((sub @ (X337 @ X339 @ X340) @ X341)) = ((X337 @ (comp @ X339 @ X341) @ (sub @ X340 @ X341)))) & (((X334 @ id @ (lam @ (X337 @ sh @ one)) @ id)) != $true)) | ? [X336 : term] : ((((X335 @ (sub @ X333 @ (push @ X336 @ id)))) != $true) & (((X335 @ X336)) = $true)) | (((X335 @ (lam @ X333))) = $true) | ? [X342 : term] : (((X335 @ X342)) != ((X334 @ id @ X342 @ id)))) | ? [X332 : term] : (((sub @ X332 @ id)) != X332)) | ? [X328 : term] : ! [X329 : (subst > term > term)] : ? [X331 : subst,X330 : term] : (((sub @ X328 @ (push @ X330 @ X331))) != ((X329 @ X331 @ X330)))) <=> (hoasinduction_lem3v2a_lthm = $true)) & (hoasinduction_lem3b_gthm = $true) & (ulamvar1 != $true) & (termmset_gthm = $true) & (! [X211 : term,X212 : term,X209 : term,X210 : term] : ((X210 = X212) | (((ap @ (sub @ X209 @ id) @ X212)) != ((ap @ (sub @ X211 @ id) @ X210)))) <=> (hoasapinj2 = $true)) & (! [X139 : subst] : (((push @ (sub @ one @ X139) @ (comp @ sh @ X139))) = X139) <=> (axscons = $true)) & (((d2subst @ d_id)) = id) & (hoasapnotvar_lthm = $true) & (hoaslamnotvar_lthm = $true) & (! [X443 : (subst > term > subst > $o),X444 : term,X446 : term,X445 : (term > $o)] : (? [X455 : subst,X457 : subst,X454 : subst,X456 : term] : ((((X443 @ X457 @ X456 @ (comp @ X455 @ X454))) != $true) & (((X443 @ (comp @ X457 @ X455) @ (sub @ X456 @ X455) @ X454)) = $true)) | (((X445 @ (ap @ X444 @ X446))) = $true) | ? [X453 : term,X452 : term] : ((((X443 @ id @ X453 @ id)) = $true) & (((X443 @ id @ X452 @ id)) = $true) & (((X443 @ id @ (ap @ (sub @ X453 @ id) @ X452) @ id)) != $true)) | (((X445 @ X444)) != $true) | ? [X451 : term] : (((X443 @ id @ X451 @ id)) != ((X445 @ X451))) | ? [X450 : subst,X449 : term,X448 : subst,X447 : subst] : ((((X443 @ X450 @ X449 @ (comp @ X447 @ X448))) = $true) & (((X443 @ (comp @ X450 @ X447) @ (sub @ X449 @ X447) @ X448)) != $true)) | (((X445 @ X446)) != $true)) <=> (hoasinduction_lem2v2 = $true)) & ((induction2lem = $true) <=> ! [X399 : subst,X397 : term,X398 : (term > $o)] : (? [X404 : term,X403 : term] : ((((X398 @ X403)) = $true) & (((X398 @ (ap @ X404 @ X403))) != $true) & (((X398 @ X404)) = $true)) | ? [X401 : term] : (! [X402 : term] : ((((X398 @ (sub @ X401 @ (push @ X402 @ id)))) = $true) | (((X398 @ X402)) != $true)) & (((X398 @ (lam @ X401))) != $true)) | ? [X400 : term] : ((((var @ X400)) = $true) & (((X398 @ (sub @ X400 @ X399))) != $true)) | (((X398 @ (sub @ X397 @ X399))) = $true))) & (axapp = $true) & (axmap = $true)),
  inference(ennf_transformation,[],[f10])).
thf(f12,plain,(
  (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)))))))))))) & (axabs = $true) & (! [X458 : (subst > term > subst > $o),X460 : (term > $o),X459 : term] : (? [X467 : term] : (((X458 @ id @ X467 @ id)) != ((X460 @ X467))) | ? [X462 : (subst > term > term)] : ((((X458 @ id @ (lam @ (X462 @ sh @ one)) @ id)) != $true) & ! [X463 : subst,X464 : subst,X465 : term] : (((X462 @ (comp @ X464 @ X463) @ (sub @ X465 @ X463))) = ((sub @ (X462 @ X464 @ X465) @ X463))) & ! [X466 : term] : ((((X458 @ id @ X466 @ id)) != $true) | (((X458 @ id @ (X462 @ id @ X466) @ id)) = $true))) | (((X460 @ (lam @ X459))) = $true) | ? [X461 : term] : ((((X460 @ (sub @ X459 @ (push @ X461 @ id)))) != $true) & (((X460 @ X461)) = $true))) <=> (hoasinduction_lem3v2a = $true)) & ((pushprop_lem0_lthm = $true) <=> ! [X248 : subst,X247 : term,X246 : (term > $o)] : ? [X249 : (term > $o)] : ! [X250 : term] : (((X246 @ (sub @ X250 @ (push @ X247 @ X248)))) = ((X249 @ X250)))) & (pushprop_lthm_orig = $true) & ((axshiftcons = $true) <=> ! [X326 : subst,X327 : term] : (((comp @ sh @ (push @ X327 @ X326))) = X326)) & (pushprop_lem2v2 = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ((induction = $true) <=> ! [X133 : (term > $o),X134 : term] : (? [X135 : term] : ((((X133 @ (lam @ X135))) != $true) & (((X133 @ X135)) = $true)) | (((X133 @ X134)) = $true) | ? [X137 : term,X138 : term] : ((((X133 @ (ap @ X137 @ X138))) != $true) & (((X133 @ X137)) = $true) & (((X133 @ X138)) = $true)) | ? [X136 : term] : ((((var @ X136)) = $true) & (((X133 @ X136)) != $true)))) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (! [X489 : term,X490 : (subst > term > subst > $o)] : ((((X490 @ id @ X489 @ id)) = $true) | ? [X492 : term,X491 : term] : ((((X490 @ id @ (ap @ (sub @ X491 @ id) @ X492) @ id)) != $true) & (((X490 @ id @ X492 @ id)) = $true) & (((X490 @ id @ X491 @ id)) = $true)) | ? [X493 : (subst > term > term)] : (! [X494 : term] : ((((X490 @ id @ (X493 @ id @ X494) @ id)) = $true) | ($true != ((X490 @ id @ X494 @ id)))) & (((X490 @ id @ (lam @ (X493 @ sh @ one)) @ id)) != $true) & ! [X495 : subst,X496 : term,X497 : subst] : (((sub @ (X493 @ X497 @ X496) @ X495)) = ((X493 @ (comp @ X497 @ X495) @ (sub @ X496 @ X495)))))) | ? [X478 : term] : (((sub @ X478 @ id)) != X478) | ? [X468 : (subst > term > subst > $o)] : ! [X469 : (term > $o)] : ? [X470 : term] : (((X468 @ id @ X470 @ id)) != ((X469 @ X470))) | ? [X471 : (term > $o),X472 : term] : (! [X473 : term] : (? [X474 : term] : ((((X471 @ X474)) = $true) & (((X471 @ (sub @ X473 @ (push @ X474 @ id)))) != $true)) | (((X471 @ (lam @ X473))) = $true)) & ! [X475 : term,X476 : term] : ((((X471 @ X476)) != $true) | ($true = ((X471 @ (ap @ X476 @ X475)))) | ($true != ((X471 @ X475)))) & (((X471 @ X472)) != $true) & ! [X477 : term] : (($true = ((X471 @ X477))) | (((var @ X477)) != $true))) | ? [X479 : (subst > term > subst > $o),X480 : (term > $o),X481 : term] : (! [X482 : term] : ((((X480 @ (sub @ X481 @ (push @ X482 @ id)))) = $true) | (((X480 @ X482)) != $true)) & ! [X483 : term] : (((X480 @ X483)) = ((X479 @ id @ X483 @ id))) & ! [X484 : (subst > term > term)] : (? [X485 : subst,X487 : term,X486 : subst] : (((sub @ (X484 @ X486 @ X487) @ X485)) != ((X484 @ (comp @ X486 @ X485) @ (sub @ X487 @ X485)))) | ? [X488 : term] : ((((X479 @ id @ (X484 @ id @ X488) @ id)) != $true) & (((X479 @ id @ X488 @ id)) = $true)) | ($true = ((X479 @ id @ (lam @ (X484 @ sh @ one)) @ id)))) & (((X480 @ (lam @ X481))) != $true)))) & ((pushprop_lem1v2_lthm = $true) <=> (! [X108 : term,X107 : subst,X106 : (term > $o),X109 : (term > $o)] : (? [X110 : term] : (((X109 @ X110)) != ((X106 @ (sub @ X110 @ (push @ X108 @ X107))))) | (((X106 @ X108)) != $true) | (((X109 @ one)) = $true)) | ? [X105 : subst,X104 : term] : (((sub @ one @ (push @ X104 @ X105))) != X104))) & ((hoasinduction_lem3 = $true) <=> ! [X428 : (subst > term > subst > $o),X427 : term] : (? [X436 : subst,X434 : term,X435 : subst,X433 : subst] : ((((X428 @ (comp @ X433 @ X436) @ (sub @ X434 @ X436) @ X435)) != $true) & (((X428 @ X433 @ X434 @ (comp @ X436 @ X435))) = $true)) | ? [X430 : subst,X431 : subst,X429 : subst,X432 : term] : ((((X428 @ X429 @ X432 @ (comp @ X430 @ X431))) != $true) & (((X428 @ (comp @ X429 @ X430) @ (sub @ X432 @ X430) @ X431)) = $true)) | ? [X437 : (subst > term > term)] : (! [X438 : term,X439 : subst,X440 : subst] : (((X437 @ (comp @ X440 @ X439) @ (sub @ X438 @ X439))) = ((sub @ (X437 @ X440 @ X438) @ X439))) & ! [X441 : term] : ((((X428 @ id @ X441 @ id)) != $true) | (((X428 @ id @ (X437 @ id @ X441) @ id)) = $true)) & (((X428 @ id @ (lam @ (X437 @ sh @ one)) @ id)) != $true)) | (((X428 @ id @ (lam @ X427) @ id)) = $true) | ? [X442 : term] : ((((X428 @ id @ X442 @ id)) = $true) & (((X428 @ id @ (sub @ X427 @ (push @ X442 @ id)) @ id)) != $true)))) & (! [X208 : term] : (((sub @ X208 @ id)) = X208) <=> (axvarid = $true)) & (axapp = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (pushprop_lem2v2_gthm = $true) & ((! [X267 : (subst > term > subst > $o),X268 : term] : ((((X267 @ id @ (lam @ X268) @ id)) = $true) | ? [X273 : subst,X270 : subst,X271 : term,X272 : subst] : ((((X267 @ X273 @ X271 @ (comp @ X270 @ X272))) != $true) & (((X267 @ (comp @ X273 @ X270) @ (sub @ X271 @ X270) @ X272)) = $true)) | ? [X274 : (subst > term > term)] : ((((X267 @ id @ (lam @ (X274 @ sh @ one)) @ id)) != $true) & ! [X276 : subst,X275 : subst,X277 : term] : (((X274 @ (comp @ X276 @ X275) @ (sub @ X277 @ X275))) = ((sub @ (X274 @ X276 @ X277) @ X275))) & ! [X278 : term] : ((((X267 @ id @ X278 @ id)) != $true) | (((X267 @ id @ (X274 @ id @ X278) @ id)) = $true))) | ? [X269 : term] : ((((X267 @ id @ X269 @ id)) = $true) & (((X267 @ id @ (sub @ X268 @ (push @ X269 @ id)) @ id)) != $true)) | ? [X279 : subst,X280 : term,X281 : subst,X282 : subst] : ((((X267 @ X282 @ X280 @ (comp @ X281 @ X279))) = $true) & (((X267 @ (comp @ X282 @ X281) @ (sub @ X280 @ X281) @ X279)) != $true))) | ? [X259 : (subst > term > subst > $o),X260 : term] : (! [X261 : (subst > term > term)] : (? [X264 : subst,X265 : term,X263 : subst] : (((X261 @ (comp @ X263 @ X264) @ (sub @ X265 @ X264))) != ((sub @ (X261 @ X263 @ X265) @ X264))) | (((X259 @ id @ (lam @ (X261 @ sh @ one)) @ id)) = $true) | ? [X262 : term] : ((((X259 @ id @ X262 @ id)) = $true) & (((X259 @ id @ (X261 @ id @ X262) @ id)) != $true))) & ! [X266 : term] : ((((X259 @ id @ X266 @ id)) != $true) | (((X259 @ id @ (sub @ X260 @ (push @ X266 @ id)) @ id)) = $true)) & (((X259 @ id @ (lam @ (sub @ X260 @ (push @ one @ sh))) @ id)) != $true)) | ? [X258 : term] : (((sub @ X258 @ id)) != X258)) <=> (hoasinduction_lem3_lthm = $true)) & (axclos = $true) & (hoasinduction_lem3v2_gthm = $true) & (! [X520 : subst] : (((comp @ X520 @ id)) = X520) <=> (axidr = $true)) & ((axidl = $true) <=> ! [X207 : subst] : (((comp @ id @ X207)) = X207)) & (hoasinduction_lem3_gthm = $true) & (pushprop_lem3v2 = $true) & (! [X116 : (subst > term > term),X115 : term,X117 : (subst > term > term),X118 : subst] : (? [X124 : term,X122 : subst,X123 : subst] : (((sub @ (X116 @ X123 @ X124) @ X122)) != ((X116 @ (comp @ X123 @ X122) @ (sub @ X124 @ X122)))) | ? [X121 : subst,X119 : subst,X120 : term] : (((X117 @ (comp @ X119 @ X121) @ (sub @ X120 @ X121))) != ((sub @ (X117 @ X119 @ X120) @ X121))) | (((lam @ (X116 @ sh @ one))) != ((lam @ (X117 @ sh @ one)))) | (((X117 @ X118 @ X115)) = ((X116 @ X118 @ X115)))) <=> (hoaslaminj = $true)) & (hoasinduction_lem3a_gthm = $true) & (hoasapinj2_gthm = $true) & (hoasapnotvar_gthm = $true) & (pushprop_lem3v2_lthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (! [X125 : term,X126 : (subst > term > subst > $o)] : (? [X128 : (subst > term > term)] : (! [X131 : subst,X130 : term,X129 : subst] : (((sub @ (X128 @ X131 @ X130) @ X129)) = ((X128 @ (comp @ X131 @ X129) @ (sub @ X130 @ X129)))) & (((X126 @ id @ (lam @ (X128 @ sh @ one)) @ id)) != $true) & ! [X132 : term] : ((((X126 @ id @ X132 @ id)) != $true) | (((X126 @ id @ (X128 @ id @ X132) @ id)) = $true))) | ? [X127 : term] : ((((X126 @ id @ X127 @ id)) = $true) & (((X126 @ id @ (sub @ X125 @ (push @ X127 @ id)) @ id)) != $true)) | (((X126 @ id @ (lam @ (sub @ X125 @ (push @ one @ sh))) @ id)) = $true)) <=> (hoasinduction_lem3aa = $true)) & (! [X1 : (term > $o),X0 : term] : ((((X1 @ X0)) = $true) | ? [X2 : term] : (! [X3 : term] : ((((X1 @ (sub @ X2 @ (push @ X3 @ id)))) = $true) | (((X1 @ X3)) != $true)) & (((X1 @ (lam @ X2))) != $true)) | ? [X5 : term,X4 : term] : ((((X1 @ (ap @ X5 @ X4))) != $true) & (((X1 @ X4)) = $true) & (((X1 @ X5)) = $true)) | ? [X6 : term] : ((((X1 @ X6)) != $true) & (((var @ X6)) = $true))) <=> (induction2 = $true)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoasinduction_no_psi_cond = $true) <=> ! [X39 : (subst > term > subst > $o),X40 : term] : (? [X41 : term,X42 : term] : ((((X39 @ id @ X42 @ id)) = $true) & (((X39 @ id @ (ap @ (sub @ X41 @ id) @ X42) @ id)) != $true) & (((X39 @ id @ X41 @ id)) = $true)) | ($true = ((X39 @ id @ X40 @ id))) | ? [X43 : (subst > term > term)] : (! [X47 : term] : ((((X39 @ id @ (X43 @ id @ X47) @ id)) = $true) | (((X39 @ id @ X47 @ id)) != $true)) & (((X39 @ id @ (lam @ (X43 @ sh @ one)) @ id)) != $true) & ! [X46 : subst,X44 : subst,X45 : term] : (((sub @ (X43 @ X46 @ X45) @ X44)) = ((X43 @ (comp @ X46 @ X44) @ (sub @ X45 @ X44))))))) & ((! [X323 : subst] : (((comp @ id @ X323)) = X323) & ! [X322 : subst] : (((comp @ X322 @ id)) = X322)) <=> (substmonoid = $true)) & ((! [X533 : term,X532 : subst,X534 : (term > $o)] : (? [X537 : term,X538 : term] : ((((X534 @ X538)) = $true) & (((X534 @ X537)) = $true) & (((X534 @ (ap @ X537 @ X538))) != $true)) | (((X534 @ (sub @ X533 @ X532))) = $true) | ? [X539 : term] : ((((X534 @ (sub @ X539 @ X532))) != $true) & (((var @ X539)) = $true)) | ? [X535 : term] : (! [X536 : term] : ((((X534 @ X536)) != $true) | (((X534 @ (sub @ X535 @ (push @ X536 @ id)))) = $true)) & (((X534 @ (lam @ X535))) != $true))) | ? [X527 : (term > $o),X526 : term] : ((((X527 @ X526)) != $true) & ! [X531 : term,X530 : term] : ((((X527 @ X530)) != $true) | (((X527 @ X531)) != $true) | (((X527 @ (ap @ X531 @ X530))) = $true)) & ! [X528 : term] : ((((X527 @ (lam @ X528))) = $true) | (((X527 @ X528)) != $true)) & ! [X529 : term] : ((((var @ X529)) != $true) | (((X527 @ X529)) = $true))) | ? [X525 : subst] : (((comp @ X525 @ id)) != X525) | ? [X522 : term,X521 : subst] : (((sub @ one @ (push @ X522 @ X521))) != X522) | ? [X523 : subst,X524 : term] : (((comp @ sh @ (push @ X524 @ X523))) != X523)) <=> (induction2lem_lthm = $true)) & (hoasinduction_lem2_gthm = $true) & (pushprop_lem1v2_gthm = $true) & (termmset_gthm = $true) & ((apinj2 = $true) <=> ! [X140 : term,X142 : term,X141 : term,X143 : term] : ((X140 = X143) | (((ap @ X141 @ X140)) != ((ap @ X142 @ X143))))) & ((! [X389 : term,X388 : (term > $o)] : (? [X391 : term,X392 : term] : ((((X388 @ X391)) = $true) & (((X388 @ (ap @ X391 @ X392))) != $true) & (((X388 @ X392)) = $true)) | (((X388 @ X389)) = $true) | ? [X393 : term] : ((((X388 @ (lam @ X393))) != $true) & ! [X394 : term] : ((((X388 @ X394)) != $true) | (((X388 @ (sub @ X393 @ (push @ X394 @ id)))) = $true))) | ? [X390 : term] : ((((X388 @ X390)) != $true) & (((var @ X390)) = $true))) | ? [X381 : (term > $o),X382 : subst,X380 : term] : ((((X381 @ (sub @ X380 @ X382))) != $true) & ! [X384 : term,X383 : term] : ((((X381 @ X383)) != $true) | (((X381 @ X384)) != $true) | (((X381 @ (ap @ X383 @ X384))) = $true)) & ! [X386 : term] : ((((X381 @ (lam @ X386))) = $true) | ? [X387 : term] : ((((X381 @ X387)) = $true) & (((X381 @ (sub @ X386 @ (push @ X387 @ id)))) != $true))) & ! [X385 : term] : ((((X381 @ (sub @ X385 @ X382))) = $true) | (((var @ X385)) != $true))) | ? [X379 : term] : (((sub @ X379 @ id)) != X379)) <=> (induction2_lthm = $true)) & (! [X503 : term,X502 : (subst > term > subst > $o)] : (? [X513 : term] : ((((X502 @ id @ (sub @ X503 @ (push @ X513 @ id)) @ id)) != $true) & (((X502 @ id @ X513 @ id)) = $true)) | (((X502 @ id @ (lam @ (sub @ X503 @ (push @ one @ sh))) @ id)) = $true) | ? [X504 : (subst > term > term),X505 : term] : (! [X506 : subst,X508 : term,X507 : subst] : (((sub @ (sub @ X505 @ (push @ X508 @ X507)) @ X506)) = ((sub @ (X504 @ X507 @ X508) @ X506))) & (((X502 @ id @ (lam @ (X504 @ sh @ one)) @ id)) != $true) & ! [X512 : term] : ((((X502 @ id @ (X504 @ id @ X512) @ id)) = $true) | ($true != ((X502 @ id @ X512 @ id)))) & ! [X511 : term,X509 : subst,X510 : subst] : (((X504 @ (comp @ X509 @ X510) @ (sub @ X511 @ X510))) = ((sub @ X505 @ (push @ (sub @ X511 @ X510) @ (comp @ X509 @ X510))))))) <=> (hoasinduction_lem3aaa = $true)) & ((? [X86 : term,X85 : term,X87 : term,X88 : term] : ((((ap @ X88 @ X85)) = ((ap @ X87 @ X86))) & (X87 != X88)) | ? [X84 : term] : (((sub @ X84 @ id)) != X84) | ! [X92 : term,X89 : term,X90 : term,X91 : term] : ((X89 = X92) | (((ap @ (sub @ X89 @ id) @ X91)) != ((ap @ (sub @ X92 @ id) @ X90))))) <=> (hoasapinj1_lthm = $true)) & (ulamvarind = $true) & ((hoaslaminj_lthm = $true) <=> (! [X68 : subst,X70 : (subst > term > term),X69 : (subst > term > term),X71 : term] : (? [X72 : subst,X73 : term,X74 : subst] : (((sub @ (X69 @ X72 @ X73) @ X74)) != ((X69 @ (comp @ X72 @ X74) @ (sub @ X73 @ X74)))) | (((X69 @ X68 @ X71)) = ((X70 @ X68 @ X71))) | ? [X77 : term,X76 : subst,X75 : subst] : (((sub @ (X70 @ X76 @ X77) @ X75)) != ((X70 @ (comp @ X76 @ X75) @ (sub @ X77 @ X75)))) | (((lam @ (X70 @ sh @ one))) != ((lam @ (X69 @ sh @ one))))) | ? [X62 : term,X63 : subst] : (((sub @ one @ (push @ X62 @ X63))) != X62) | ? [X66 : term,X67 : term] : ((((lam @ X66)) = ((lam @ X67))) & (X66 != X67)) | ? [X64 : term,X65 : subst] : (((comp @ sh @ (push @ X64 @ X65))) != X65))) & (! [X423 : term,X422 : term] : ((((sub @ X422 @ sh)) != ((sub @ X423 @ sh))) | (X422 = X423)) <=> (shinj = $true)) & ! [X325 : d_term,X324 : d_term] : ((X324 = X325) | (((d2term @ X325)) != ((d2term @ X324)))) & (lamnotvar = $true) & (((var @ (d2term @ d_one))) != $true) & (hoaslaminj_gthm = $true) & (! [X234 : term,X235 : term,X233 : (subst > term > subst > $o)] : (? [X239 : term,X241 : subst,X238 : subst,X240 : subst] : ((((X233 @ X238 @ X239 @ (comp @ X241 @ X240))) != $true) & (((X233 @ (comp @ X238 @ X241) @ (sub @ X239 @ X241) @ X240)) = $true)) | (((X233 @ id @ (ap @ X234 @ X235) @ id)) = $true) | ? [X245 : subst,X244 : subst,X243 : term,X242 : subst] : ((((X233 @ (comp @ X245 @ X242) @ (sub @ X243 @ X242) @ X244)) != $true) & (((X233 @ X245 @ X243 @ (comp @ X242 @ X244))) = $true)) | (((X233 @ id @ X235 @ id)) != $true) | ? [X237 : term,X236 : term] : ((((X233 @ id @ X236 @ id)) = $true) & (((X233 @ id @ X237 @ id)) = $true) & (((X233 @ id @ (ap @ (sub @ X237 @ id) @ X236) @ id)) != $true)) | (((X233 @ id @ X234 @ id)) != $true)) <=> (hoasinduction_lem2 = $true)) & (hoasinduction_lem1_lthm = $true) & ! [X515 : d_subst,X514 : d_subst] : ((((d2subst @ X514)) != ((d2subst @ X515))) | (X514 = X515)) & (hoasinduction_lem1 = $true) & (! [X55 : (subst > term > subst > $o),X54 : term] : ((((X55 @ id @ (lam @ (sub @ X54 @ (push @ one @ sh))) @ id)) = $true) | ? [X56 : (subst > term > term)] : (! [X57 : term] : ((((X55 @ id @ X57 @ id)) != $true) | (((X55 @ id @ (X56 @ id @ X57) @ id)) = $true)) & (((X55 @ id @ (lam @ (X56 @ sh @ one)) @ id)) != $true) & ! [X58 : term,X59 : subst,X60 : subst] : (((X56 @ (comp @ X60 @ X59) @ (sub @ X58 @ X59))) = ((sub @ (X56 @ X60 @ X58) @ X59)))) | ? [X61 : term] : (($true = ((X55 @ id @ X61 @ id))) & (((X55 @ id @ (sub @ X54 @ (push @ X61 @ id)) @ id)) != $true))) <=> (hoasinduction_lem3aa_lthm = $true)) & ((hoasinduction_lthm_3 = $true) <=> (? [X283 : (subst > term > subst > $o)] : ! [X284 : (term > $o)] : ? [X285 : term] : (((X283 @ id @ X285 @ id)) != ((X284 @ X285))) | ? [X293 : term] : (((sub @ X293 @ id)) != X293) | ! [X305 : term,X304 : (subst > term > subst > $o)] : (? [X310 : (subst > term > term)] : ((((X304 @ id @ (lam @ (X310 @ sh @ one)) @ id)) != $true) & ! [X312 : subst,X313 : term,X314 : subst] : (((sub @ (X310 @ X314 @ X313) @ X312)) = ((X310 @ (comp @ X314 @ X312) @ (sub @ X313 @ X312)))) & ! [X311 : term] : ((((X304 @ id @ X311 @ id)) != $true) | (((X304 @ id @ (X310 @ id @ X311) @ id)) = $true))) | (((X304 @ id @ X305 @ id)) = $true) | ? [X320 : term,X321 : term] : ((((X304 @ id @ X320 @ id)) = $true) & (((X304 @ id @ X321 @ id)) = $true) & (((X304 @ id @ (ap @ (sub @ X321 @ id) @ X320) @ id)) != $true)) | ? [X308 : subst,X309 : term,X306 : subst,X307 : subst] : ((((X304 @ X308 @ X309 @ (comp @ X307 @ X306))) = $true) & (((X304 @ (comp @ X308 @ X307) @ (sub @ X309 @ X307) @ X306)) != $true)) | ? [X319 : term] : ((((X304 @ id @ X319 @ id)) != $true) & (((var @ (sub @ X319 @ id))) = $true)) | ? [X315 : term,X317 : subst,X318 : subst,X316 : subst] : ((((X304 @ X317 @ X315 @ (comp @ X318 @ X316))) != $true) & (((X304 @ (comp @ X317 @ X318) @ (sub @ X315 @ X318) @ X316)) = $true))) | ? [X294 : term,X295 : (subst > term > subst > $o),X296 : (term > $o)] : (! [X302 : term] : ((((X296 @ (sub @ X294 @ (push @ X302 @ id)))) = $true) | (((X296 @ X302)) != $true)) & (((X296 @ (lam @ X294))) != $true) & ! [X297 : (subst > term > term)] : (? [X301 : term] : ((((X295 @ id @ X301 @ id)) = $true) & (((X295 @ id @ (X297 @ id @ X301) @ id)) != $true)) | ? [X300 : subst,X299 : subst,X298 : term] : (((X297 @ (comp @ X299 @ X300) @ (sub @ X298 @ X300))) != ((sub @ (X297 @ X299 @ X298) @ X300))) | (((X295 @ id @ (lam @ (X297 @ sh @ one)) @ id)) = $true)) & ! [X303 : term] : (((X296 @ X303)) = ((X295 @ id @ X303 @ id)))) | ? [X287 : (term > $o),X286 : term] : ((((X287 @ X286)) != $true) & ! [X289 : term] : (? [X290 : term] : ((((X287 @ (sub @ X289 @ (push @ X290 @ id)))) != $true) & (((X287 @ X290)) = $true)) | (((X287 @ (lam @ X289))) = $true)) & ! [X292 : term,X291 : term] : ((((X287 @ X291)) != $true) | (((X287 @ (ap @ X292 @ X291))) = $true) | (((X287 @ X292)) != $true)) & ! [X288 : term] : ((((X287 @ X288)) = $true) | (((var @ X288)) != $true))))) & (! [X255 : (subst > term > subst > $o)] : ? [X256 : (term > $o)] : ! [X257 : term] : (((X256 @ X257)) = ((X255 @ id @ X257 @ id))) <=> (hoasinduction_lem0 = $true)) & (hoasinduction_lem2v2_gthm = $true) & (hoaslamnotap_gthm = $true) & (hoasinduction_lem3b_gthm = $true) & (axmap = $true) & ((pushprop_lem1_lthm = $true) <=> (? [X200 : term,X201 : subst] : (((comp @ sh @ (push @ X200 @ X201))) != X201) | ? [X199 : subst,X198 : term] : (((sub @ one @ (push @ X198 @ X199))) != X198) | ! [X202 : term,X205 : (term > $o),X206 : subst,X204 : term,X203 : (term > $o)] : ((((X203 @ X202)) != $true) | (((X205 @ (sub @ X202 @ (push @ X204 @ X206)))) = $true)))) & (axassoc = $true) & (axvarshift = $true) & (lamnotap != $true) & (hoaslamnotvar_lthm = $true) & (! [X406 : term,X405 : (subst > term > subst > $o)] : (? [X407 : (subst > term > term)] : ((((X405 @ id @ (lam @ (X407 @ sh @ one)) @ id)) != $true) & ! [X408 : subst,X409 : term,X410 : subst] : (((sub @ (X407 @ X408 @ X409) @ X410)) = ((X407 @ (comp @ X408 @ X410) @ (sub @ X409 @ X410)))) & ! [X411 : term] : ((((X405 @ id @ X411 @ id)) != $true) | (((X405 @ id @ (X407 @ id @ X411) @ id)) = $true))) | (((X405 @ id @ (lam @ X406) @ id)) = $true) | ? [X412 : term] : ((((X405 @ id @ X412 @ id)) = $true) & (((X405 @ id @ (sub @ X406 @ (push @ X412 @ id)) @ id)) != $true))) <=> (hoasinduction_lem3a = $true)) & (ulamvarsh = $true) & (hoasinduction_gthm = $true) & ((? [X332 : term] : (((sub @ X332 @ id)) != X332) | ! [X335 : (term > $o),X333 : term,X334 : (subst > term > subst > $o)] : (? [X337 : (subst > term > term)] : (! [X338 : term] : ((((X334 @ id @ X338 @ id)) != $true) | (((X334 @ id @ (X337 @ id @ X338) @ id)) = $true)) & ! [X339 : subst,X340 : term,X341 : subst] : (((sub @ (X337 @ X339 @ X340) @ X341)) = ((X337 @ (comp @ X339 @ X341) @ (sub @ X340 @ X341)))) & (((X334 @ id @ (lam @ (X337 @ sh @ one)) @ id)) != $true)) | ? [X336 : term] : ((((X335 @ (sub @ X333 @ (push @ X336 @ id)))) != $true) & (((X335 @ X336)) = $true)) | (((X335 @ (lam @ X333))) = $true) | ? [X342 : term] : (((X335 @ X342)) != ((X334 @ id @ X342 @ id)))) | ? [X328 : term] : ! [X329 : (subst > term > term)] : ? [X331 : subst,X330 : term] : (((sub @ X328 @ (push @ X330 @ X331))) != ((X329 @ X331 @ X330)))) <=> (hoasinduction_lem3v2a_lthm = $true)) & ((! [X181 : term,X180 : (subst > term > subst > $o)] : ((((X180 @ id @ X181 @ id)) = $true) | ? [X190 : term] : ((((var @ (sub @ X190 @ id))) = $true) & (((X180 @ id @ X190 @ id)) != $true)) | ? [X192 : term,X191 : term] : ((((X180 @ id @ X191 @ id)) = $true) & (((X180 @ id @ (ap @ (sub @ X191 @ id) @ X192) @ id)) != $true) & (((X180 @ id @ X192 @ id)) = $true)) | ? [X187 : term,X189 : subst,X188 : subst,X186 : subst] : ((((X180 @ X188 @ X187 @ (comp @ X186 @ X189))) = $true) & (((X180 @ (comp @ X188 @ X186) @ (sub @ X187 @ X186) @ X189)) != $true)) | ? [X193 : (subst > term > term)] : ((((X180 @ id @ (lam @ (X193 @ sh @ one)) @ id)) != $true) & ! [X194 : term] : ((((X180 @ id @ (X193 @ id @ X194) @ id)) = $true) | (((X180 @ id @ X194 @ id)) != $true)) & ! [X195 : subst,X196 : subst,X197 : term] : (((sub @ (X193 @ X196 @ X197) @ X195)) = ((X193 @ (comp @ X196 @ X195) @ (sub @ X197 @ X195))))) | ? [X184 : subst,X183 : term,X182 : subst,X185 : subst] : ((((X180 @ (comp @ X184 @ X182) @ (sub @ X183 @ X182) @ X185)) = $true) & (((X180 @ X184 @ X183 @ (comp @ X182 @ X185))) != $true))) | ? [X165 : term,X164 : (subst > term > subst > $o)] : ((((X164 @ id @ (lam @ X165) @ id)) != $true) & ! [X177 : subst,X178 : subst,X179 : term,X176 : subst] : ((((X164 @ X176 @ X179 @ (comp @ X177 @ X178))) = $true) | (((X164 @ (comp @ X176 @ X177) @ (sub @ X179 @ X177) @ X178)) != $true)) & ! [X166 : (subst > term > term)] : ((((X164 @ id @ (lam @ (X166 @ sh @ one)) @ id)) = $true) | ? [X168 : subst,X167 : term,X169 : subst] : (((sub @ (X166 @ X168 @ X167) @ X169)) != ((X166 @ (comp @ X168 @ X169) @ (sub @ X167 @ X169)))) | ? [X170 : term] : ((((X164 @ id @ X170 @ id)) = $true) & (((X164 @ id @ (X166 @ id @ X170) @ id)) != $true))) & ! [X172 : subst,X171 : subst,X173 : term,X174 : subst] : ((((X164 @ (comp @ X171 @ X172) @ (sub @ X173 @ X172) @ X174)) = $true) | (((X164 @ X171 @ X173 @ (comp @ X172 @ X174))) != $true)) & ! [X175 : term] : ((((X164 @ id @ X175 @ id)) != $true) | (((X164 @ id @ (sub @ X165 @ (push @ X175 @ id)) @ id)) = $true))) | ? [X153 : term,X151 : (subst > term > subst > $o),X152 : term] : (! [X155 : term,X154 : term] : ((((X151 @ id @ (ap @ (sub @ X155 @ id) @ X154) @ id)) = $true) | (((X151 @ id @ X155 @ id)) != $true) | (((X151 @ id @ X154 @ id)) != $true)) & (((X151 @ id @ (ap @ X152 @ X153) @ id)) != $true) & (((X151 @ id @ X153 @ id)) = $true) & (((X151 @ id @ X152 @ id)) = $true) & ! [X158 : subst,X157 : subst,X156 : subst,X159 : term] : ((((X151 @ X156 @ X159 @ (comp @ X158 @ X157))) != $true) | (((X151 @ (comp @ X156 @ X158) @ (sub @ X159 @ X158) @ X157)) = $true)) & ! [X162 : subst,X163 : term,X160 : subst,X161 : subst] : ((((X151 @ (comp @ X160 @ X161) @ (sub @ X163 @ X161) @ X162)) != $true) | (((X151 @ X160 @ X163 @ (comp @ X161 @ X162))) = $true))) | ? [X145 : term,X144 : (term > $o)] : (! [X146 : term,X147 : term] : ((((X144 @ (ap @ X147 @ X146))) = $true) | (((X144 @ X147)) != $true) | (((X144 @ X146)) != $true)) & ! [X148 : term] : ((((var @ X148)) != $true) | ($true = ((X144 @ X148)))) & ! [X149 : term] : ((((X144 @ (lam @ X149))) = $true) | ? [X150 : term] : (($true != ((X144 @ (sub @ X149 @ (push @ X150 @ id))))) & ($true = ((X144 @ X150))))) & (((X144 @ X145)) != $true))) <=> (hoasinduction_lthm = $true)) & (((d2subst @ d_id)) = id) & (! [X443 : (subst > term > subst > $o),X444 : term,X446 : term,X445 : (term > $o)] : (? [X455 : subst,X457 : subst,X454 : subst,X456 : term] : ((((X443 @ X457 @ X456 @ (comp @ X455 @ X454))) != $true) & (((X443 @ (comp @ X457 @ X455) @ (sub @ X456 @ X455) @ X454)) = $true)) | (((X445 @ (ap @ X444 @ X446))) = $true) | ? [X453 : term,X452 : term] : ((((X443 @ id @ X453 @ id)) = $true) & (((X443 @ id @ X452 @ id)) = $true) & (((X443 @ id @ (ap @ (sub @ X453 @ id) @ X452) @ id)) != $true)) | (((X445 @ X444)) != $true) | ? [X451 : term] : (((X443 @ id @ X451 @ id)) != ((X445 @ X451))) | ? [X450 : subst,X449 : term,X448 : subst,X447 : subst] : ((((X443 @ X450 @ X449 @ (comp @ X447 @ X448))) = $true) & (((X443 @ (comp @ X450 @ X447) @ (sub @ X449 @ X447) @ X448)) != $true)) | (((X445 @ X446)) != $true)) <=> (hoasinduction_lem2v2 = $true)) & (pushprop_lem2v2_lthm = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (! [X98 : term,X101 : (term > $o),X100 : subst,X99 : (term > $o)] : ((((X101 @ one)) = $true) | (((X99 @ X98)) != $true) | ? [X102 : term] : (((X101 @ X102)) != ((X99 @ (sub @ X102 @ (push @ X98 @ X100)))))) <=> (pushprop_lem1v2 = $true)) & ((? [X94 : term] : (((sub @ X94 @ id)) != X94) | ! [X95 : term] : (((sub @ X95 @ id)) = X95)) <=> (termmset_lthm = $true)) & (pushprop_lem0_gthm = $true) & (hoasapnotvar_lthm = $true) & (! [X420 : term] : ? [X421 : (subst > term > term)] : (((sub @ X420 @ (push @ one @ sh))) = ((X421 @ sh @ one))) <=> (hoasinduction_lem3b_lthm = $true)) & (! [X498 : term] : ? [X499 : (subst > term > term)] : ! [X501 : subst,X500 : term] : (((X499 @ X501 @ X500)) = ((sub @ X498 @ (push @ X500 @ X501)))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & ! [X78 : d_subst] : (d_id = X78) & (hoasinduction_lem2_lthm = $true) & ((hoasinduction_lem3v2 = $true) <=> ! [X215 : term,X217 : (term > $o),X216 : (subst > term > subst > $o)] : (? [X232 : subst,X231 : subst,X230 : term,X229 : subst] : ((((X216 @ X231 @ X230 @ (comp @ X232 @ X229))) = $true) & (((X216 @ (comp @ X231 @ X232) @ (sub @ X230 @ X232) @ X229)) != $true)) | (((X217 @ (lam @ X215))) = $true) | ? [X218 : term] : (($true != ((X217 @ (sub @ X215 @ (push @ X218 @ id))))) & (((X217 @ X218)) = $true)) | ? [X223 : term] : (((X216 @ id @ X223 @ id)) != ((X217 @ X223))) | ? [X224 : (subst > term > term)] : ((((X216 @ id @ (lam @ (X224 @ sh @ one)) @ id)) != $true) & ! [X225 : term] : ((((X216 @ id @ (X224 @ id @ X225) @ id)) = $true) | (((X216 @ id @ X225 @ id)) != $true)) & ! [X227 : subst,X226 : term,X228 : subst] : (((X224 @ (comp @ X228 @ X227) @ (sub @ X226 @ X227))) = ((sub @ (X224 @ X228 @ X226) @ X227)))) | ? [X222 : term,X220 : subst,X219 : subst,X221 : subst] : ((((X216 @ (comp @ X221 @ X219) @ (sub @ X222 @ X219) @ X220)) = $true) & (((X216 @ X221 @ X222 @ (comp @ X219 @ X220))) != $true)))) & (hoasapnotvar = $true) & (apnotvar = $true) & (substmonoid_gthm = $true) & (hoaslamnotvar_gthm = $true) & (hoaslamnotvar = $true) & ((substmonoid_lthm = $true) <=> (? [X251 : subst] : (((comp @ id @ X251)) != X251) | ? [X252 : subst] : (((comp @ X252 @ id)) != X252) | (! [X254 : subst] : (((comp @ X254 @ id)) = X254) & ! [X253 : subst] : (((comp @ id @ X253)) = X253)))) & (induction2lem_gthm = $true) & ((hoasinduction_lem0_lthm = $true) <=> ! [X424 : (subst > term > subst > $o)] : ? [X425 : (term > $o)] : ! [X426 : term] : (((X425 @ X426)) = ((X424 @ id @ X426 @ id)))) & ((! [X371 : (subst > term > subst > $o),X372 : term] : ((((X371 @ id @ (lam @ X372) @ id)) = $true) | ? [X373 : term] : ((((X371 @ id @ (sub @ X372 @ (push @ X373 @ id)) @ id)) != $true) & (((X371 @ id @ X373 @ id)) = $true)) | ? [X374 : (subst > term > term)] : ((((X371 @ id @ (lam @ (X374 @ sh @ one)) @ id)) != $true) & ! [X377 : term,X376 : subst,X378 : subst] : (((X374 @ (comp @ X378 @ X376) @ (sub @ X377 @ X376))) = ((sub @ (X374 @ X378 @ X377) @ X376))) & ! [X375 : term] : ((((X371 @ id @ (X374 @ id @ X375) @ id)) = $true) | (((X371 @ id @ X375 @ id)) != $true)))) | ? [X362 : term] : (((sub @ X362 @ id)) != X362) | ? [X363 : (subst > term > subst > $o),X364 : term] : ((((X363 @ id @ (lam @ (sub @ X364 @ (push @ one @ sh))) @ id)) != $true) & ! [X366 : (subst > term > term)] : (? [X370 : term] : ((((X363 @ id @ (X366 @ id @ X370) @ id)) != $true) & (((X363 @ id @ X370 @ id)) = $true)) | ? [X368 : subst,X367 : subst,X369 : term] : (((X366 @ (comp @ X368 @ X367) @ (sub @ X369 @ X367))) != ((sub @ (X366 @ X368 @ X369) @ X367))) | (((X363 @ id @ (lam @ (X366 @ sh @ one)) @ id)) = $true)) & ! [X365 : term] : ((((X363 @ id @ X365 @ id)) != $true) | (((X363 @ id @ (sub @ X364 @ (push @ X365 @ id)) @ id)) = $true)))) <=> (hoasinduction_lem3a_lthm = $true)) & ((apinj1 = $true) <=> ! [X111 : term,X113 : term,X114 : term,X112 : term] : ((X112 = X114) | (((ap @ X112 @ X111)) != ((ap @ X114 @ X113))))) & (pushprop_lthm = $true) & ((! [X14 : term,X11 : term,X13 : term,X12 : term] : ((((ap @ (sub @ X13 @ id) @ X12)) != ((ap @ (sub @ X14 @ id) @ X11))) | (X11 = X12)) | ? [X10 : term,X9 : term,X8 : term,X7 : term] : ((X7 != X9) & (((ap @ X8 @ X7)) = ((ap @ X10 @ X9))))) <=> (hoasapinj2_lthm = $true)) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (! [X96 : subst,X97 : term] : (((sub @ one @ (push @ X97 @ X96))) = X97) <=> (axvarcons = $true)) & ! [X413 : term] : ? [X414 : d_term] : (((d2term @ X414)) = X413) & (! [X516 : term] : ? [X517 : (subst > term > term)] : ! [X519 : term,X518 : subst] : (((X517 @ X518 @ X519)) = ((sub @ X516 @ (push @ X519 @ X518)))) <=> (hoasinduction_lem3v2_f = $true)) & ((induction2lem = $true) <=> ! [X399 : subst,X397 : term,X398 : (term > $o)] : (? [X404 : term,X403 : term] : ((((X398 @ X403)) = $true) & (((X398 @ (ap @ X404 @ X403))) != $true) & (((X398 @ X404)) = $true)) | ? [X401 : term] : (! [X402 : term] : ((((X398 @ (sub @ X401 @ (push @ X402 @ id)))) = $true) | (((X398 @ X402)) != $true)) & (((X398 @ (lam @ X401))) != $true)) | ? [X400 : term] : ((((var @ X400)) = $true) & (((X398 @ (sub @ X400 @ X399))) != $true)) | (((X398 @ (sub @ X397 @ X399))) = $true))) & ! [X93 : d_term] : (d_one = X93) & ((pushprop_lem1 = $true) <=> ! [X81 : term,X82 : (term > $o),X83 : term,X80 : (term > $o),X79 : subst] : ((((X80 @ X81)) != $true) | (((X82 @ (sub @ X81 @ (push @ X83 @ X79)))) = $true))) & (! [X22 : term,X21 : (subst > term > subst > $o)] : (? [X33 : term,X32 : term] : ((((X21 @ id @ (ap @ (sub @ X33 @ id) @ X32) @ id)) != $true) & ($true = ((X21 @ id @ X33 @ id))) & (((X21 @ id @ X32 @ id)) = $true)) | (((X21 @ id @ X22 @ id)) = $true) | ? [X27 : (subst > term > term)] : ((((X21 @ id @ (lam @ (X27 @ sh @ one)) @ id)) != $true) & ! [X30 : term,X29 : subst,X31 : subst] : (((X27 @ (comp @ X29 @ X31) @ (sub @ X30 @ X31))) = ((sub @ (X27 @ X29 @ X30) @ X31))) & ! [X28 : term] : ((((X21 @ id @ (X27 @ id @ X28) @ id)) = $true) | (((X21 @ id @ X28 @ id)) != $true))) | ? [X23 : subst,X24 : subst,X26 : term,X25 : subst] : ((((X21 @ X25 @ X26 @ (comp @ X24 @ X23))) = $true) & (((X21 @ (comp @ X25 @ X24) @ (sub @ X26 @ X24) @ X23)) != $true)) | ? [X38 : term] : ((((var @ (sub @ X38 @ id))) = $true) & (((X21 @ id @ X38 @ id)) != $true)) | ? [X34 : term,X36 : subst,X37 : subst,X35 : subst] : ((((X21 @ (comp @ X36 @ X35) @ (sub @ X34 @ X35) @ X37)) = $true) & (((X21 @ X36 @ X34 @ (comp @ X35 @ X37))) != $true))) <=> (hoasinduction = $true)) & (ulamvar1 != $true) & (! [X139 : subst] : (((push @ (sub @ one @ X139) @ (comp @ sh @ X139))) = X139) <=> (axscons = $true)) & (pushprop_lem1_gthm = $true) & ((pushprop_lem0 = $true) <=> ! [X415 : subst,X416 : term,X417 : (term > $o)] : ? [X418 : (term > $o)] : ! [X419 : term] : (((X417 @ (sub @ X419 @ (push @ X416 @ X415)))) = ((X418 @ X419)))) & ((hoasinduction_lem3b = $true) <=> ! [X213 : term] : ? [X214 : (subst > term > term)] : (((X214 @ sh @ one)) = ((sub @ X213 @ (push @ one @ sh))))) & (((d2term @ d_one)) = one) & (((d2subst @ d_id)) = sh) & ((! [X345 : term,X346 : (subst > term > subst > $o),X344 : (term > $o)] : (? [X347 : term] : (((X344 @ X347)) != ((X346 @ id @ X347 @ id))) | (((X344 @ (lam @ X345))) = $true) | ? [X361 : term] : ((((X344 @ X361)) = $true) & (((X344 @ (sub @ X345 @ (push @ X361 @ id)))) != $true)) | ? [X348 : subst,X350 : term,X349 : subst,X351 : subst] : ((((X346 @ X348 @ X350 @ (comp @ X351 @ X349))) != $true) & (((X346 @ (comp @ X348 @ X351) @ (sub @ X350 @ X351) @ X349)) = $true)) | ? [X352 : subst,X354 : term,X355 : subst,X353 : subst] : ((((X346 @ (comp @ X355 @ X353) @ (sub @ X354 @ X353) @ X352)) != $true) & (((X346 @ X355 @ X354 @ (comp @ X353 @ X352))) = $true)) | ? [X356 : (subst > term > term)] : (! [X357 : term] : ((((X346 @ id @ (X356 @ id @ X357) @ id)) = $true) | (((X346 @ id @ X357 @ id)) != $true)) & (((X346 @ id @ (lam @ (X356 @ sh @ one)) @ id)) != $true) & ! [X358 : subst,X360 : term,X359 : subst] : (((X356 @ (comp @ X358 @ X359) @ (sub @ X360 @ X359))) = ((sub @ (X356 @ X358 @ X360) @ X359))))) | ? [X343 : term] : (((sub @ X343 @ id)) != X343)) <=> (hoasinduction_lem3v2_lthm = $true)) & (pushprop_gthm = $true) & (hoasinduction_lem1v2 = $true) & (induction2_gthm = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoasapinj1 = $true) <=> ! [X18 : term,X19 : term,X17 : term,X20 : term] : ((((ap @ (sub @ X20 @ id) @ X19)) != ((ap @ (sub @ X17 @ id) @ X18))) | (X17 = X20))) & (hoaslamnotap_lthm = $true) & (hoasapinj1_gthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (hoasinduction_lem1_gthm = $true) & (pushprop = $true) & ! [X15 : subst] : ? [X16 : d_subst] : (((d2subst @ X16)) = X15) & (! [X211 : term,X212 : term,X209 : term,X210 : term] : ((X210 = X212) | (((ap @ (sub @ X209 @ id) @ X212)) != ((ap @ (sub @ X211 @ id) @ X210)))) <=> (hoasapinj2 = $true)) & ((hoaslamnotap = $true) <=> ! [X49 : (subst > term > term),X48 : term,X50 : term] : ((((ap @ (sub @ X50 @ id) @ X48)) != ((lam @ (X49 @ sh @ one)))) | ? [X53 : term,X51 : subst,X52 : subst] : (((X49 @ (comp @ X51 @ X52) @ (sub @ X53 @ X52))) != ((sub @ (X49 @ X51 @ X53) @ X52))))) & ((termmset = $true) <=> ! [X103 : term] : (((sub @ X103 @ id)) = X103)) & ((laminj = $true) <=> ! [X395 : term,X396 : term] : ((X395 = X396) | (((lam @ X395)) != ((lam @ X396))))) & (hoasinduction_lem1v2_gthm = $true)),
  inference(flattening,[],[f11])).
thf(f13,definition,(
  ! [X346 : (subst > term > subst > $o)] : ((((sP0 @ X346)) = $true) <=> ? [X356 : (subst > term > term)] : (! [X357 : term] : ((((X346 @ id @ (X356 @ id @ X357) @ id)) = $true) | (((X346 @ id @ X357 @ id)) != $true)) & (((X346 @ id @ (lam @ (X356 @ sh @ one)) @ id)) != $true) & ! [X358 : subst,X360 : term,X359 : subst] : (((X356 @ (comp @ X358 @ X359) @ (sub @ X360 @ X359))) = ((sub @ (X356 @ X358 @ X360) @ X359)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f14,definition,(
  (sP1 = $true) <=> (! [X345 : term,X346 : (subst > term > subst > $o),X344 : (term > $o)] : (? [X347 : term] : (((X344 @ X347)) != ((X346 @ id @ X347 @ id))) | (((X344 @ (lam @ X345))) = $true) | ? [X361 : term] : ((((X344 @ X361)) = $true) & (((X344 @ (sub @ X345 @ (push @ X361 @ id)))) != $true)) | ? [X348 : subst,X350 : term,X349 : subst,X351 : subst] : ((((X346 @ X348 @ X350 @ (comp @ X351 @ X349))) != $true) & (((X346 @ (comp @ X348 @ X351) @ (sub @ X350 @ X351) @ X349)) = $true)) | ? [X352 : subst,X354 : term,X355 : subst,X353 : subst] : ((((X346 @ (comp @ X355 @ X353) @ (sub @ X354 @ X353) @ X352)) != $true) & (((X346 @ X355 @ X354 @ (comp @ X353 @ X352))) = $true)) | (((sP0 @ X346)) = $true)) | ? [X343 : term] : (((sub @ X343 @ id)) != X343))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f15,definition,(
  ! [X21 : (subst > term > subst > $o)] : ((((sP2 @ X21)) = $true) <=> ? [X27 : (subst > term > term)] : ((((X21 @ id @ (lam @ (X27 @ sh @ one)) @ id)) != $true) & ! [X30 : term,X29 : subst,X31 : subst] : (((X27 @ (comp @ X29 @ X31) @ (sub @ X30 @ X31))) = ((sub @ (X27 @ X29 @ X30) @ X31))) & ! [X28 : term] : ((((X21 @ id @ (X27 @ id @ X28) @ id)) = $true) | (((X21 @ id @ X28 @ id)) != $true))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f16,definition,(
  ! [X21 : (subst > term > subst > $o)] : ((((sP3 @ X21)) = $true) <=> ? [X33 : term,X32 : term] : ((((X21 @ id @ (ap @ (sub @ X33 @ id) @ X32) @ id)) != $true) & ($true = ((X21 @ id @ X33 @ id))) & (((X21 @ id @ X32 @ id)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f17,definition,(
  (sP4 = $true) <=> ! [X22 : term,X21 : (subst > term > subst > $o)] : ((((sP3 @ X21)) = $true) | (((X21 @ id @ X22 @ id)) = $true) | (((sP2 @ X21)) = $true) | ? [X23 : subst,X24 : subst,X26 : term,X25 : subst] : ((((X21 @ X25 @ X26 @ (comp @ X24 @ X23))) = $true) & (((X21 @ (comp @ X25 @ X24) @ (sub @ X26 @ X24) @ X23)) != $true)) | ? [X38 : term] : ((((var @ (sub @ X38 @ id))) = $true) & (((X21 @ id @ X38 @ id)) != $true)) | ? [X34 : term,X36 : subst,X37 : subst,X35 : subst] : ((((X21 @ (comp @ X36 @ X35) @ (sub @ X34 @ X35) @ X37)) = $true) & (((X21 @ X36 @ X34 @ (comp @ X35 @ X37))) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f18,definition,(
  ! [X398 : (term > $o)] : ((((sP5 @ X398)) = $true) <=> ? [X404 : term,X403 : term] : ((((X398 @ X403)) = $true) & (((X398 @ (ap @ X404 @ X403))) != $true) & (((X398 @ X404)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f19,definition,(
  (sP6 = $true) <=> ! [X399 : subst,X397 : term,X398 : (term > $o)] : ((((sP5 @ X398)) = $true) | ? [X401 : term] : (! [X402 : term] : ((((X398 @ (sub @ X401 @ (push @ X402 @ id)))) = $true) | (((X398 @ X402)) != $true)) & (((X398 @ (lam @ X401))) != $true)) | ? [X400 : term] : ((((var @ X400)) = $true) & (((X398 @ (sub @ X400 @ X399))) != $true)) | (((X398 @ (sub @ X397 @ X399))) = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f20,definition,(
  (sP7 = $true) <=> ? [X363 : (subst > term > subst > $o),X364 : term] : ((((X363 @ id @ (lam @ (sub @ X364 @ (push @ one @ sh))) @ id)) != $true) & ! [X366 : (subst > term > term)] : (? [X370 : term] : ((((X363 @ id @ (X366 @ id @ X370) @ id)) != $true) & (((X363 @ id @ X370 @ id)) = $true)) | ? [X368 : subst,X367 : subst,X369 : term] : (((X366 @ (comp @ X368 @ X367) @ (sub @ X369 @ X367))) != ((sub @ (X366 @ X368 @ X369) @ X367))) | (((X363 @ id @ (lam @ (X366 @ sh @ one)) @ id)) = $true)) & ! [X365 : term] : ((((X363 @ id @ X365 @ id)) != $true) | (((X363 @ id @ (sub @ X364 @ (push @ X365 @ id)) @ id)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f21,definition,(
  (sP8 = $true) <=> (! [X371 : (subst > term > subst > $o),X372 : term] : ((((X371 @ id @ (lam @ X372) @ id)) = $true) | ? [X373 : term] : ((((X371 @ id @ (sub @ X372 @ (push @ X373 @ id)) @ id)) != $true) & (((X371 @ id @ X373 @ id)) = $true)) | ? [X374 : (subst > term > term)] : ((((X371 @ id @ (lam @ (X374 @ sh @ one)) @ id)) != $true) & ! [X377 : term,X376 : subst,X378 : subst] : (((X374 @ (comp @ X378 @ X376) @ (sub @ X377 @ X376))) = ((sub @ (X374 @ X378 @ X377) @ X376))) & ! [X375 : term] : ((((X371 @ id @ (X374 @ id @ X375) @ id)) = $true) | (((X371 @ id @ X375 @ id)) != $true)))) | ? [X362 : term] : (((sub @ X362 @ id)) != X362) | (sP7 = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f22,definition,(
  ! [X216 : (subst > term > subst > $o)] : ((((sP9 @ X216)) = $true) <=> ? [X224 : (subst > term > term)] : ((((X216 @ id @ (lam @ (X224 @ sh @ one)) @ id)) != $true) & ! [X225 : term] : ((((X216 @ id @ (X224 @ id @ X225) @ id)) = $true) | (((X216 @ id @ X225 @ id)) != $true)) & ! [X227 : subst,X226 : term,X228 : subst] : (((X224 @ (comp @ X228 @ X227) @ (sub @ X226 @ X227))) = ((sub @ (X224 @ X228 @ X226) @ X227)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f23,definition,(
  (sP10 = $true) <=> ! [X215 : term,X217 : (term > $o),X216 : (subst > term > subst > $o)] : (? [X232 : subst,X231 : subst,X230 : term,X229 : subst] : ((((X216 @ X231 @ X230 @ (comp @ X232 @ X229))) = $true) & (((X216 @ (comp @ X231 @ X232) @ (sub @ X230 @ X232) @ X229)) != $true)) | (((X217 @ (lam @ X215))) = $true) | ? [X218 : term] : (($true != ((X217 @ (sub @ X215 @ (push @ X218 @ id))))) & (((X217 @ X218)) = $true)) | ? [X223 : term] : (((X216 @ id @ X223 @ id)) != ((X217 @ X223))) | (((sP9 @ X216)) = $true) | ? [X222 : term,X220 : subst,X219 : subst,X221 : subst] : ((((X216 @ (comp @ X221 @ X219) @ (sub @ X222 @ X219) @ X220)) = $true) & (((X216 @ X221 @ X222 @ (comp @ X219 @ X220))) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f24,definition,(
  ! [X443 : (subst > term > subst > $o)] : ((((sP11 @ X443)) = $true) <=> ? [X453 : term,X452 : term] : ((((X443 @ id @ X453 @ id)) = $true) & (((X443 @ id @ X452 @ id)) = $true) & (((X443 @ id @ (ap @ (sub @ X453 @ id) @ X452) @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f25,definition,(
  (sP12 = $true) <=> ! [X443 : (subst > term > subst > $o),X444 : term,X446 : term,X445 : (term > $o)] : (? [X455 : subst,X457 : subst,X454 : subst,X456 : term] : ((((X443 @ X457 @ X456 @ (comp @ X455 @ X454))) != $true) & (((X443 @ (comp @ X457 @ X455) @ (sub @ X456 @ X455) @ X454)) = $true)) | (((X445 @ (ap @ X444 @ X446))) = $true) | (((sP11 @ X443)) = $true) | (((X445 @ X444)) != $true) | ? [X451 : term] : (((X443 @ id @ X451 @ id)) != ((X445 @ X451))) | ? [X450 : subst,X449 : term,X448 : subst,X447 : subst] : ((((X443 @ X450 @ X449 @ (comp @ X447 @ X448))) = $true) & (((X443 @ (comp @ X450 @ X447) @ (sub @ X449 @ X447) @ X448)) != $true)) | (((X445 @ X446)) != $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f26,definition,(
  ! [X144 : (term > $o)] : ((((sP13 @ X144)) = $true) <=> ! [X146 : term,X147 : term] : ((((X144 @ (ap @ X147 @ X146))) = $true) | (((X144 @ X147)) != $true) | (((X144 @ X146)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f27,definition,(
  ! [X151 : (subst > term > subst > $o)] : ((((sP14 @ X151)) = $true) <=> ! [X155 : term,X154 : term] : ((((X151 @ id @ (ap @ (sub @ X155 @ id) @ X154) @ id)) = $true) | (((X151 @ id @ X155 @ id)) != $true) | (((X151 @ id @ X154 @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f28,definition,(
  ! [X164 : (subst > term > subst > $o)] : ((((sP15 @ X164)) = $true) <=> ! [X166 : (subst > term > term)] : ((((X164 @ id @ (lam @ (X166 @ sh @ one)) @ id)) = $true) | ? [X168 : subst,X167 : term,X169 : subst] : (((sub @ (X166 @ X168 @ X167) @ X169)) != ((X166 @ (comp @ X168 @ X169) @ (sub @ X167 @ X169)))) | ? [X170 : term] : ((((X164 @ id @ X170 @ id)) = $true) & (((X164 @ id @ (X166 @ id @ X170) @ id)) != $true))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f29,definition,(
  ! [X180 : (subst > term > subst > $o)] : ((((sP16 @ X180)) = $true) <=> ? [X193 : (subst > term > term)] : ((((X180 @ id @ (lam @ (X193 @ sh @ one)) @ id)) != $true) & ! [X194 : term] : ((((X180 @ id @ (X193 @ id @ X194) @ id)) = $true) | (((X180 @ id @ X194 @ id)) != $true)) & ! [X195 : subst,X196 : subst,X197 : term] : (((sub @ (X193 @ X196 @ X197) @ X195)) = ((X193 @ (comp @ X196 @ X195) @ (sub @ X197 @ X195))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f30,definition,(
  ! [X180 : (subst > term > subst > $o)] : ((((sP17 @ X180)) = $true) <=> ? [X192 : term,X191 : term] : ((((X180 @ id @ X191 @ id)) = $true) & (((X180 @ id @ (ap @ (sub @ X191 @ id) @ X192) @ id)) != $true) & (((X180 @ id @ X192 @ id)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f31,definition,(
  (sP18 = $true) <=> ? [X165 : term,X164 : (subst > term > subst > $o)] : ((((X164 @ id @ (lam @ X165) @ id)) != $true) & ! [X177 : subst,X178 : subst,X179 : term,X176 : subst] : ((((X164 @ X176 @ X179 @ (comp @ X177 @ X178))) = $true) | (((X164 @ (comp @ X176 @ X177) @ (sub @ X179 @ X177) @ X178)) != $true)) & (((sP15 @ X164)) = $true) & ! [X172 : subst,X171 : subst,X173 : term,X174 : subst] : ((((X164 @ (comp @ X171 @ X172) @ (sub @ X173 @ X172) @ X174)) = $true) | (((X164 @ X171 @ X173 @ (comp @ X172 @ X174))) != $true)) & ! [X175 : term] : ((((X164 @ id @ X175 @ id)) != $true) | (((X164 @ id @ (sub @ X165 @ (push @ X175 @ id)) @ id)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f32,definition,(
  (sP19 = $true) <=> ! [X181 : term,X180 : (subst > term > subst > $o)] : ((((X180 @ id @ X181 @ id)) = $true) | ? [X190 : term] : ((((var @ (sub @ X190 @ id))) = $true) & (((X180 @ id @ X190 @ id)) != $true)) | (((sP17 @ X180)) = $true) | ? [X187 : term,X189 : subst,X188 : subst,X186 : subst] : ((((X180 @ X188 @ X187 @ (comp @ X186 @ X189))) = $true) & (((X180 @ (comp @ X188 @ X186) @ (sub @ X187 @ X186) @ X189)) != $true)) | (((sP16 @ X180)) = $true) | ? [X184 : subst,X183 : term,X182 : subst,X185 : subst] : ((((X180 @ (comp @ X184 @ X182) @ (sub @ X183 @ X182) @ X185)) = $true) & (((X180 @ X184 @ X183 @ (comp @ X182 @ X185))) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f33,definition,(
  (sP20 = $true) <=> ? [X145 : term,X144 : (term > $o)] : ((((sP13 @ X144)) = $true) & ! [X148 : term] : ((((var @ X148)) != $true) | ($true = ((X144 @ X148)))) & ! [X149 : term] : ((((X144 @ (lam @ X149))) = $true) | ? [X150 : term] : (($true != ((X144 @ (sub @ X149 @ (push @ X150 @ id))))) & ($true = ((X144 @ X150))))) & (((X144 @ X145)) != $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f34,definition,(
  (sP21 = $true) <=> ((sP19 = $true) | (sP18 = $true) | ? [X153 : term,X151 : (subst > term > subst > $o),X152 : term] : ((((sP14 @ X151)) = $true) & (((X151 @ id @ (ap @ X152 @ X153) @ id)) != $true) & (((X151 @ id @ X153 @ id)) = $true) & (((X151 @ id @ X152 @ id)) = $true) & ! [X158 : subst,X157 : subst,X156 : subst,X159 : term] : ((((X151 @ X156 @ X159 @ (comp @ X158 @ X157))) != $true) | (((X151 @ (comp @ X156 @ X158) @ (sub @ X159 @ X158) @ X157)) = $true)) & ! [X162 : subst,X163 : term,X160 : subst,X161 : subst] : ((((X151 @ (comp @ X160 @ X161) @ (sub @ X163 @ X161) @ X162)) != $true) | (((X151 @ X160 @ X163 @ (comp @ X161 @ X162))) = $true))) | (sP20 = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f35,definition,(
  (sP22 = $true) <=> (? [X332 : term] : (((sub @ X332 @ id)) != X332) | ! [X335 : (term > $o),X333 : term,X334 : (subst > term > subst > $o)] : (? [X337 : (subst > term > term)] : (! [X338 : term] : ((((X334 @ id @ X338 @ id)) != $true) | (((X334 @ id @ (X337 @ id @ X338) @ id)) = $true)) & ! [X339 : subst,X340 : term,X341 : subst] : (((sub @ (X337 @ X339 @ X340) @ X341)) = ((X337 @ (comp @ X339 @ X341) @ (sub @ X340 @ X341)))) & (((X334 @ id @ (lam @ (X337 @ sh @ one)) @ id)) != $true)) | ? [X336 : term] : ((((X335 @ (sub @ X333 @ (push @ X336 @ id)))) != $true) & (((X335 @ X336)) = $true)) | (((X335 @ (lam @ X333))) = $true) | ? [X342 : term] : (((X335 @ X342)) != ((X334 @ id @ X342 @ id)))) | ? [X328 : term] : ! [X329 : (subst > term > term)] : ? [X331 : subst,X330 : term] : (((sub @ X328 @ (push @ X330 @ X331))) != ((X329 @ X331 @ X330))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f36,definition,(
  (sP23 = $true) <=> ! [X406 : term,X405 : (subst > term > subst > $o)] : (? [X407 : (subst > term > term)] : ((((X405 @ id @ (lam @ (X407 @ sh @ one)) @ id)) != $true) & ! [X408 : subst,X409 : term,X410 : subst] : (((sub @ (X407 @ X408 @ X409) @ X410)) = ((X407 @ (comp @ X408 @ X410) @ (sub @ X409 @ X410)))) & ! [X411 : term] : ((((X405 @ id @ X411 @ id)) != $true) | (((X405 @ id @ (X407 @ id @ X411) @ id)) = $true))) | (((X405 @ id @ (lam @ X406) @ id)) = $true) | ? [X412 : term] : ((((X405 @ id @ X412 @ id)) = $true) & (((X405 @ id @ (sub @ X406 @ (push @ X412 @ id)) @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f37,definition,(
  ! [X287 : (term > $o)] : ((((sP24 @ X287)) = $true) <=> ! [X292 : term,X291 : term] : ((((X287 @ X291)) != $true) | (((X287 @ (ap @ X292 @ X291))) = $true) | (((X287 @ X292)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f38,definition,(
  ! [X304 : (subst > term > subst > $o)] : ((((sP25 @ X304)) = $true) <=> ? [X320 : term,X321 : term] : ((((X304 @ id @ X320 @ id)) = $true) & (((X304 @ id @ X321 @ id)) = $true) & (((X304 @ id @ (ap @ (sub @ X321 @ id) @ X320) @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f39,definition,(
  ! [X304 : (subst > term > subst > $o)] : ((((sP26 @ X304)) = $true) <=> ? [X310 : (subst > term > term)] : ((((X304 @ id @ (lam @ (X310 @ sh @ one)) @ id)) != $true) & ! [X312 : subst,X313 : term,X314 : subst] : (((sub @ (X310 @ X314 @ X313) @ X312)) = ((X310 @ (comp @ X314 @ X312) @ (sub @ X313 @ X312)))) & ! [X311 : term] : ((((X304 @ id @ X311 @ id)) != $true) | (((X304 @ id @ (X310 @ id @ X311) @ id)) = $true))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f40,definition,(
  (sP27 = $true) <=> ? [X294 : term,X295 : (subst > term > subst > $o),X296 : (term > $o)] : (! [X302 : term] : ((((X296 @ (sub @ X294 @ (push @ X302 @ id)))) = $true) | (((X296 @ X302)) != $true)) & (((X296 @ (lam @ X294))) != $true) & ! [X297 : (subst > term > term)] : (? [X301 : term] : ((((X295 @ id @ X301 @ id)) = $true) & (((X295 @ id @ (X297 @ id @ X301) @ id)) != $true)) | ? [X300 : subst,X299 : subst,X298 : term] : (((X297 @ (comp @ X299 @ X300) @ (sub @ X298 @ X300))) != ((sub @ (X297 @ X299 @ X298) @ X300))) | (((X295 @ id @ (lam @ (X297 @ sh @ one)) @ id)) = $true)) & ! [X303 : term] : (((X296 @ X303)) = ((X295 @ id @ X303 @ id))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f41,definition,(
  (sP28 = $true) <=> ! [X305 : term,X304 : (subst > term > subst > $o)] : ((((sP26 @ X304)) = $true) | (((X304 @ id @ X305 @ id)) = $true) | (((sP25 @ X304)) = $true) | ? [X308 : subst,X309 : term,X306 : subst,X307 : subst] : ((((X304 @ X308 @ X309 @ (comp @ X307 @ X306))) = $true) & (((X304 @ (comp @ X308 @ X307) @ (sub @ X309 @ X307) @ X306)) != $true)) | ? [X319 : term] : ((((X304 @ id @ X319 @ id)) != $true) & (((var @ (sub @ X319 @ id))) = $true)) | ? [X315 : term,X317 : subst,X318 : subst,X316 : subst] : ((((X304 @ X317 @ X315 @ (comp @ X318 @ X316))) != $true) & (((X304 @ (comp @ X317 @ X318) @ (sub @ X315 @ X318) @ X316)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f42,definition,(
  (sP29 = $true) <=> (? [X283 : (subst > term > subst > $o)] : ! [X284 : (term > $o)] : ? [X285 : term] : (((X283 @ id @ X285 @ id)) != ((X284 @ X285))) | ? [X293 : term] : (((sub @ X293 @ id)) != X293) | (sP28 = $true) | (sP27 = $true) | ? [X287 : (term > $o),X286 : term] : ((((X287 @ X286)) != $true) & ! [X289 : term] : (? [X290 : term] : ((((X287 @ (sub @ X289 @ (push @ X290 @ id)))) != $true) & (((X287 @ X290)) = $true)) | (((X287 @ (lam @ X289))) = $true)) & (((sP24 @ X287)) = $true) & ! [X288 : term] : ((((X287 @ X288)) = $true) | (((var @ X288)) != $true))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f43,definition,(
  (sP30 = $true) <=> ! [X55 : (subst > term > subst > $o),X54 : term] : ((((X55 @ id @ (lam @ (sub @ X54 @ (push @ one @ sh))) @ id)) = $true) | ? [X56 : (subst > term > term)] : (! [X57 : term] : ((((X55 @ id @ X57 @ id)) != $true) | (((X55 @ id @ (X56 @ id @ X57) @ id)) = $true)) & (((X55 @ id @ (lam @ (X56 @ sh @ one)) @ id)) != $true) & ! [X58 : term,X59 : subst,X60 : subst] : (((X56 @ (comp @ X60 @ X59) @ (sub @ X58 @ X59))) = ((sub @ (X56 @ X60 @ X58) @ X59)))) | ? [X61 : term] : (($true = ((X55 @ id @ X61 @ id))) & (((X55 @ id @ (sub @ X54 @ (push @ X61 @ id)) @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f44,definition,(
  ! [X233 : (subst > term > subst > $o)] : ((((sP31 @ X233)) = $true) <=> ? [X237 : term,X236 : term] : ((((X233 @ id @ X236 @ id)) = $true) & (((X233 @ id @ X237 @ id)) = $true) & (((X233 @ id @ (ap @ (sub @ X237 @ id) @ X236) @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f45,definition,(
  ($true = sP32) <=> ! [X234 : term,X235 : term,X233 : (subst > term > subst > $o)] : (? [X239 : term,X241 : subst,X238 : subst,X240 : subst] : ((((X233 @ X238 @ X239 @ (comp @ X241 @ X240))) != $true) & (((X233 @ (comp @ X238 @ X241) @ (sub @ X239 @ X241) @ X240)) = $true)) | (((X233 @ id @ (ap @ X234 @ X235) @ id)) = $true) | ? [X245 : subst,X244 : subst,X243 : term,X242 : subst] : ((((X233 @ (comp @ X245 @ X242) @ (sub @ X243 @ X242) @ X244)) != $true) & (((X233 @ X245 @ X243 @ (comp @ X242 @ X244))) = $true)) | (((X233 @ id @ X235 @ id)) != $true) | (((sP31 @ X233)) = $true) | (((X233 @ id @ X234 @ id)) != $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f46,definition,(
  (sP33 = $true) <=> (! [X68 : subst,X70 : (subst > term > term),X69 : (subst > term > term),X71 : term] : (? [X72 : subst,X73 : term,X74 : subst] : (((sub @ (X69 @ X72 @ X73) @ X74)) != ((X69 @ (comp @ X72 @ X74) @ (sub @ X73 @ X74)))) | (((X69 @ X68 @ X71)) = ((X70 @ X68 @ X71))) | ? [X77 : term,X76 : subst,X75 : subst] : (((sub @ (X70 @ X76 @ X77) @ X75)) != ((X70 @ (comp @ X76 @ X75) @ (sub @ X77 @ X75)))) | (((lam @ (X70 @ sh @ one))) != ((lam @ (X69 @ sh @ one))))) | ? [X62 : term,X63 : subst] : (((sub @ one @ (push @ X62 @ X63))) != X62) | ? [X66 : term,X67 : term] : ((((lam @ X66)) = ((lam @ X67))) & (X66 != X67)) | ? [X64 : term,X65 : subst] : (((comp @ sh @ (push @ X64 @ X65))) != X65))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f47,definition,(
  (sP34 = $true) <=> ! [X503 : term,X502 : (subst > term > subst > $o)] : (? [X513 : term] : ((((X502 @ id @ (sub @ X503 @ (push @ X513 @ id)) @ id)) != $true) & (((X502 @ id @ X513 @ id)) = $true)) | (((X502 @ id @ (lam @ (sub @ X503 @ (push @ one @ sh))) @ id)) = $true) | ? [X504 : (subst > term > term),X505 : term] : (! [X506 : subst,X508 : term,X507 : subst] : (((sub @ (sub @ X505 @ (push @ X508 @ X507)) @ X506)) = ((sub @ (X504 @ X507 @ X508) @ X506))) & (((X502 @ id @ (lam @ (X504 @ sh @ one)) @ id)) != $true) & ! [X512 : term] : ((((X502 @ id @ (X504 @ id @ X512) @ id)) = $true) | ($true != ((X502 @ id @ X512 @ id)))) & ! [X511 : term,X509 : subst,X510 : subst] : (((X504 @ (comp @ X509 @ X510) @ (sub @ X511 @ X510))) = ((sub @ X505 @ (push @ (sub @ X511 @ X510) @ (comp @ X509 @ X510)))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f48,definition,(
  ! [X381 : (term > $o)] : ((((sP35 @ X381)) = $true) <=> ! [X384 : term,X383 : term] : ((((X381 @ X383)) != $true) | (((X381 @ X384)) != $true) | (((X381 @ (ap @ X383 @ X384))) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f49,definition,(
  ! [X388 : (term > $o)] : ((((sP36 @ X388)) = $true) <=> ? [X391 : term,X392 : term] : ((((X388 @ X391)) = $true) & (((X388 @ (ap @ X391 @ X392))) != $true) & (((X388 @ X392)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f50,definition,(
  (sP37 = $true) <=> ! [X389 : term,X388 : (term > $o)] : ((((sP36 @ X388)) = $true) | (((X388 @ X389)) = $true) | ? [X393 : term] : ((((X388 @ (lam @ X393))) != $true) & ! [X394 : term] : ((((X388 @ X394)) != $true) | (((X388 @ (sub @ X393 @ (push @ X394 @ id)))) = $true))) | ? [X390 : term] : ((((X388 @ X390)) != $true) & (((var @ X390)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f51,definition,(
  (sP38 = $true) <=> ((sP37 = $true) | ? [X381 : (term > $o),X382 : subst,X380 : term] : ((((X381 @ (sub @ X380 @ X382))) != $true) & (((sP35 @ X381)) = $true) & ! [X386 : term] : ((((X381 @ (lam @ X386))) = $true) | ? [X387 : term] : ((((X381 @ X387)) = $true) & (((X381 @ (sub @ X386 @ (push @ X387 @ id)))) != $true))) & ! [X385 : term] : ((((X381 @ (sub @ X385 @ X382))) = $true) | (((var @ X385)) != $true))) | ? [X379 : term] : (((sub @ X379 @ id)) != X379))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f52,definition,(
  ! [X527 : (term > $o)] : ((((sP39 @ X527)) = $true) <=> ! [X531 : term,X530 : term] : ((((X527 @ X530)) != $true) | (((X527 @ X531)) != $true) | (((X527 @ (ap @ X531 @ X530))) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f53,definition,(
  ! [X534 : (term > $o)] : ((((sP40 @ X534)) = $true) <=> ? [X537 : term,X538 : term] : ((((X534 @ X538)) = $true) & (((X534 @ X537)) = $true) & (((X534 @ (ap @ X537 @ X538))) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f54,definition,(
  (sP41 = $true) <=> ! [X533 : term,X532 : subst,X534 : (term > $o)] : ((((sP40 @ X534)) = $true) | (((X534 @ (sub @ X533 @ X532))) = $true) | ? [X539 : term] : ((((X534 @ (sub @ X539 @ X532))) != $true) & (((var @ X539)) = $true)) | ? [X535 : term] : (! [X536 : term] : ((((X534 @ X536)) != $true) | (((X534 @ (sub @ X535 @ (push @ X536 @ id)))) = $true)) & (((X534 @ (lam @ X535))) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f55,definition,(
  (sP42 = $true) <=> ((sP41 = $true) | ? [X527 : (term > $o),X526 : term] : ((((X527 @ X526)) != $true) & (((sP39 @ X527)) = $true) & ! [X528 : term] : ((((X527 @ (lam @ X528))) = $true) | (((X527 @ X528)) != $true)) & ! [X529 : term] : ((((var @ X529)) != $true) | (((X527 @ X529)) = $true))) | ? [X525 : subst] : (((comp @ X525 @ id)) != X525) | ? [X522 : term,X521 : subst] : (((sub @ one @ (push @ X522 @ X521))) != X522) | ? [X523 : subst,X524 : term] : (((comp @ sh @ (push @ X524 @ X523))) != X523))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f56,definition,(
  ! [X39 : (subst > term > subst > $o)] : ((((sP43 @ X39)) = $true) <=> ? [X43 : (subst > term > term)] : (! [X47 : term] : ((((X39 @ id @ (X43 @ id @ X47) @ id)) = $true) | (((X39 @ id @ X47 @ id)) != $true)) & (((X39 @ id @ (lam @ (X43 @ sh @ one)) @ id)) != $true) & ! [X46 : subst,X44 : subst,X45 : term] : (((sub @ (X43 @ X46 @ X45) @ X44)) = ((X43 @ (comp @ X46 @ X44) @ (sub @ X45 @ X44))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f57,definition,(
  ! [X1 : (term > $o)] : ((((sP44 @ X1)) = $true) <=> ? [X5 : term,X4 : term] : ((((X1 @ (ap @ X5 @ X4))) != $true) & (((X1 @ X4)) = $true) & (((X1 @ X5)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f58,definition,(
  (sP45 = $true) <=> ! [X1 : (term > $o),X0 : term] : ((((X1 @ X0)) = $true) | ? [X2 : term] : (! [X3 : term] : ((((X1 @ (sub @ X2 @ (push @ X3 @ id)))) = $true) | (((X1 @ X3)) != $true)) & (((X1 @ (lam @ X2))) != $true)) | (((sP44 @ X1)) = $true) | ? [X6 : term] : ((((X1 @ X6)) != $true) & (((var @ X6)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f59,definition,(
  (sP46 = $true) <=> ! [X125 : term,X126 : (subst > term > subst > $o)] : (? [X128 : (subst > term > term)] : (! [X131 : subst,X130 : term,X129 : subst] : (((sub @ (X128 @ X131 @ X130) @ X129)) = ((X128 @ (comp @ X131 @ X129) @ (sub @ X130 @ X129)))) & (((X126 @ id @ (lam @ (X128 @ sh @ one)) @ id)) != $true) & ! [X132 : term] : ((((X126 @ id @ X132 @ id)) != $true) | (((X126 @ id @ (X128 @ id @ X132) @ id)) = $true))) | ? [X127 : term] : ((((X126 @ id @ X127 @ id)) = $true) & (((X126 @ id @ (sub @ X125 @ (push @ X127 @ id)) @ id)) != $true)) | (((X126 @ id @ (lam @ (sub @ X125 @ (push @ one @ sh))) @ id)) = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f60,definition,(
  ! [X267 : (subst > term > subst > $o)] : (($true = ((sP47 @ X267))) <=> ? [X274 : (subst > term > term)] : ((((X267 @ id @ (lam @ (X274 @ sh @ one)) @ id)) != $true) & ! [X276 : subst,X275 : subst,X277 : term] : (((X274 @ (comp @ X276 @ X275) @ (sub @ X277 @ X275))) = ((sub @ (X274 @ X276 @ X277) @ X275))) & ! [X278 : term] : ((((X267 @ id @ X278 @ id)) != $true) | (((X267 @ id @ (X274 @ id @ X278) @ id)) = $true))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f61,definition,(
  (sP48 = $true) <=> ? [X259 : (subst > term > subst > $o),X260 : term] : (! [X261 : (subst > term > term)] : (? [X264 : subst,X265 : term,X263 : subst] : (((X261 @ (comp @ X263 @ X264) @ (sub @ X265 @ X264))) != ((sub @ (X261 @ X263 @ X265) @ X264))) | (((X259 @ id @ (lam @ (X261 @ sh @ one)) @ id)) = $true) | ? [X262 : term] : ((((X259 @ id @ X262 @ id)) = $true) & (((X259 @ id @ (X261 @ id @ X262) @ id)) != $true))) & ! [X266 : term] : ((((X259 @ id @ X266 @ id)) != $true) | (((X259 @ id @ (sub @ X260 @ (push @ X266 @ id)) @ id)) = $true)) & (((X259 @ id @ (lam @ (sub @ X260 @ (push @ one @ sh))) @ id)) != $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f62,definition,(
  (sP49 = $true) <=> (! [X267 : (subst > term > subst > $o),X268 : term] : ((((X267 @ id @ (lam @ X268) @ id)) = $true) | ? [X273 : subst,X270 : subst,X271 : term,X272 : subst] : ((((X267 @ X273 @ X271 @ (comp @ X270 @ X272))) != $true) & (((X267 @ (comp @ X273 @ X270) @ (sub @ X271 @ X270) @ X272)) = $true)) | ($true = ((sP47 @ X267))) | ? [X269 : term] : ((((X267 @ id @ X269 @ id)) = $true) & (((X267 @ id @ (sub @ X268 @ (push @ X269 @ id)) @ id)) != $true)) | ? [X279 : subst,X280 : term,X281 : subst,X282 : subst] : ((((X267 @ X282 @ X280 @ (comp @ X281 @ X279))) = $true) & (((X267 @ (comp @ X282 @ X281) @ (sub @ X280 @ X281) @ X279)) != $true))) | (sP48 = $true) | ? [X258 : term] : (((sub @ X258 @ id)) != X258))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f63,definition,(
  ! [X428 : (subst > term > subst > $o)] : ((((sP50 @ X428)) = $true) <=> ? [X437 : (subst > term > term)] : (! [X438 : term,X439 : subst,X440 : subst] : (((X437 @ (comp @ X440 @ X439) @ (sub @ X438 @ X439))) = ((sub @ (X437 @ X440 @ X438) @ X439))) & ! [X441 : term] : ((((X428 @ id @ X441 @ id)) != $true) | (((X428 @ id @ (X437 @ id @ X441) @ id)) = $true)) & (((X428 @ id @ (lam @ (X437 @ sh @ one)) @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f64,definition,(
  (sP51 = $true) <=> ! [X428 : (subst > term > subst > $o),X427 : term] : (? [X436 : subst,X434 : term,X435 : subst,X433 : subst] : ((((X428 @ (comp @ X433 @ X436) @ (sub @ X434 @ X436) @ X435)) != $true) & (((X428 @ X433 @ X434 @ (comp @ X436 @ X435))) = $true)) | ? [X430 : subst,X431 : subst,X429 : subst,X432 : term] : ((((X428 @ X429 @ X432 @ (comp @ X430 @ X431))) != $true) & (((X428 @ (comp @ X429 @ X430) @ (sub @ X432 @ X430) @ X431)) = $true)) | (((sP50 @ X428)) = $true) | (((X428 @ id @ (lam @ X427) @ id)) = $true) | ? [X442 : term] : ((((X428 @ id @ X442 @ id)) = $true) & (((X428 @ id @ (sub @ X427 @ (push @ X442 @ id)) @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f65,definition,(
  ! [X471 : (term > $o)] : ((((sP52 @ X471)) = $true) <=> ! [X475 : term,X476 : term] : ((((X471 @ X476)) != $true) | ($true = ((X471 @ (ap @ X476 @ X475)))) | ($true != ((X471 @ X475)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f66,definition,(
  ! [X490 : (subst > term > subst > $o)] : ((((sP53 @ X490)) = $true) <=> ? [X493 : (subst > term > term)] : (! [X494 : term] : ((((X490 @ id @ (X493 @ id @ X494) @ id)) = $true) | ($true != ((X490 @ id @ X494 @ id)))) & (((X490 @ id @ (lam @ (X493 @ sh @ one)) @ id)) != $true) & ! [X495 : subst,X496 : term,X497 : subst] : (((sub @ (X493 @ X497 @ X496) @ X495)) = ((X493 @ (comp @ X497 @ X495) @ (sub @ X496 @ X495))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f67,definition,(
  (sP54 = $true) <=> ? [X479 : (subst > term > subst > $o),X480 : (term > $o),X481 : term] : (! [X482 : term] : ((((X480 @ (sub @ X481 @ (push @ X482 @ id)))) = $true) | (((X480 @ X482)) != $true)) & ! [X483 : term] : (((X480 @ X483)) = ((X479 @ id @ X483 @ id))) & ! [X484 : (subst > term > term)] : (? [X485 : subst,X487 : term,X486 : subst] : (((sub @ (X484 @ X486 @ X487) @ X485)) != ((X484 @ (comp @ X486 @ X485) @ (sub @ X487 @ X485)))) | ? [X488 : term] : ((((X479 @ id @ (X484 @ id @ X488) @ id)) != $true) & (((X479 @ id @ X488 @ id)) = $true)) | ($true = ((X479 @ id @ (lam @ (X484 @ sh @ one)) @ id)))) & (((X480 @ (lam @ X481))) != $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f68,definition,(
  (sP55 = $true) <=> ? [X471 : (term > $o),X472 : term] : (! [X473 : term] : (? [X474 : term] : ((((X471 @ X474)) = $true) & (((X471 @ (sub @ X473 @ (push @ X474 @ id)))) != $true)) | (((X471 @ (lam @ X473))) = $true)) & (((sP52 @ X471)) = $true) & (((X471 @ X472)) != $true) & ! [X477 : term] : (($true = ((X471 @ X477))) | (((var @ X477)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f69,definition,(
  (sP56 = $true) <=> (! [X489 : term,X490 : (subst > term > subst > $o)] : ((((X490 @ id @ X489 @ id)) = $true) | ? [X492 : term,X491 : term] : ((((X490 @ id @ (ap @ (sub @ X491 @ id) @ X492) @ id)) != $true) & (((X490 @ id @ X492 @ id)) = $true) & (((X490 @ id @ X491 @ id)) = $true)) | (((sP53 @ X490)) = $true)) | ? [X478 : term] : (((sub @ X478 @ id)) != X478) | ? [X468 : (subst > term > subst > $o)] : ! [X469 : (term > $o)] : ? [X470 : term] : (((X468 @ id @ X470 @ id)) != ((X469 @ X470))) | (sP55 = $true) | (sP54 = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f70,definition,(
  ! [X133 : (term > $o)] : ((((sP57 @ X133)) = $true) <=> ? [X137 : term,X138 : term] : ((((X133 @ (ap @ X137 @ X138))) != $true) & (((X133 @ X137)) = $true) & (((X133 @ X138)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f71,definition,(
  ($true = sP58) <=> ! [X458 : (subst > term > subst > $o),X460 : (term > $o),X459 : term] : (? [X467 : term] : (((X458 @ id @ X467 @ id)) != ((X460 @ X467))) | ? [X462 : (subst > term > term)] : ((((X458 @ id @ (lam @ (X462 @ sh @ one)) @ id)) != $true) & ! [X463 : subst,X464 : subst,X465 : term] : (((X462 @ (comp @ X464 @ X463) @ (sub @ X465 @ X463))) = ((sub @ (X462 @ X464 @ X465) @ X463))) & ! [X466 : term] : ((((X458 @ id @ X466 @ id)) != $true) | (((X458 @ id @ (X462 @ id @ X466) @ id)) = $true))) | (((X460 @ (lam @ X459))) = $true) | ? [X461 : term] : ((((X460 @ (sub @ X459 @ (push @ X461 @ id)))) != $true) & (((X460 @ X461)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f72,plain,(
  (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)))))))))))) & (axabs = $true) & (($true = sP58) <=> (hoasinduction_lem3v2a = $true)) & ((pushprop_lem0_lthm = $true) <=> ! [X248 : subst,X247 : term,X246 : (term > $o)] : ? [X249 : (term > $o)] : ! [X250 : term] : (((X246 @ (sub @ X250 @ (push @ X247 @ X248)))) = ((X249 @ X250)))) & (pushprop_lthm_orig = $true) & ((axshiftcons = $true) <=> ! [X326 : subst,X327 : term] : (((comp @ sh @ (push @ X327 @ X326))) = X326)) & (pushprop_lem2v2 = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ((induction = $true) <=> ! [X133 : (term > $o),X134 : term] : (? [X135 : term] : ((((X133 @ (lam @ X135))) != $true) & (((X133 @ X135)) = $true)) | (((X133 @ X134)) = $true) | (((sP57 @ X133)) = $true) | ? [X136 : term] : ((((var @ X136)) = $true) & (((X133 @ X136)) != $true)))) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (sP56 = $true)) & ((pushprop_lem1v2_lthm = $true) <=> (! [X108 : term,X107 : subst,X106 : (term > $o),X109 : (term > $o)] : (? [X110 : term] : (((X109 @ X110)) != ((X106 @ (sub @ X110 @ (push @ X108 @ X107))))) | (((X106 @ X108)) != $true) | (((X109 @ one)) = $true)) | ? [X105 : subst,X104 : term] : (((sub @ one @ (push @ X104 @ X105))) != X104))) & ((hoasinduction_lem3 = $true) <=> (sP51 = $true)) & (! [X208 : term] : (((sub @ X208 @ id)) = X208) <=> (axvarid = $true)) & (axapp = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (pushprop_lem2v2_gthm = $true) & ((sP49 = $true) <=> (hoasinduction_lem3_lthm = $true)) & (axclos = $true) & (hoasinduction_lem3v2_gthm = $true) & (! [X520 : subst] : (((comp @ X520 @ id)) = X520) <=> (axidr = $true)) & ((axidl = $true) <=> ! [X207 : subst] : (((comp @ id @ X207)) = X207)) & (hoasinduction_lem3_gthm = $true) & (pushprop_lem3v2 = $true) & (! [X116 : (subst > term > term),X115 : term,X117 : (subst > term > term),X118 : subst] : (? [X124 : term,X122 : subst,X123 : subst] : (((sub @ (X116 @ X123 @ X124) @ X122)) != ((X116 @ (comp @ X123 @ X122) @ (sub @ X124 @ X122)))) | ? [X121 : subst,X119 : subst,X120 : term] : (((X117 @ (comp @ X119 @ X121) @ (sub @ X120 @ X121))) != ((sub @ (X117 @ X119 @ X120) @ X121))) | (((lam @ (X116 @ sh @ one))) != ((lam @ (X117 @ sh @ one)))) | (((X117 @ X118 @ X115)) = ((X116 @ X118 @ X115)))) <=> (hoaslaminj = $true)) & (hoasinduction_lem3a_gthm = $true) & (hoasapinj2_gthm = $true) & (hoasapnotvar_gthm = $true) & (pushprop_lem3v2_lthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & ((sP46 = $true) <=> (hoasinduction_lem3aa = $true)) & ((sP45 = $true) <=> (induction2 = $true)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoasinduction_no_psi_cond = $true) <=> ! [X39 : (subst > term > subst > $o),X40 : term] : (? [X41 : term,X42 : term] : ((((X39 @ id @ X42 @ id)) = $true) & (((X39 @ id @ (ap @ (sub @ X41 @ id) @ X42) @ id)) != $true) & (((X39 @ id @ X41 @ id)) = $true)) | ($true = ((X39 @ id @ X40 @ id))) | (((sP43 @ X39)) = $true))) & ((! [X323 : subst] : (((comp @ id @ X323)) = X323) & ! [X322 : subst] : (((comp @ X322 @ id)) = X322)) <=> (substmonoid = $true)) & ((sP42 = $true) <=> (induction2lem_lthm = $true)) & (hoasinduction_lem2_gthm = $true) & (pushprop_lem1v2_gthm = $true) & (termmset_gthm = $true) & ((apinj2 = $true) <=> ! [X140 : term,X142 : term,X141 : term,X143 : term] : ((X140 = X143) | (((ap @ X141 @ X140)) != ((ap @ X142 @ X143))))) & ((sP38 = $true) <=> (induction2_lthm = $true)) & ((sP34 = $true) <=> (hoasinduction_lem3aaa = $true)) & ((? [X86 : term,X85 : term,X87 : term,X88 : term] : ((((ap @ X88 @ X85)) = ((ap @ X87 @ X86))) & (X87 != X88)) | ? [X84 : term] : (((sub @ X84 @ id)) != X84) | ! [X92 : term,X89 : term,X90 : term,X91 : term] : ((X89 = X92) | (((ap @ (sub @ X89 @ id) @ X91)) != ((ap @ (sub @ X92 @ id) @ X90))))) <=> (hoasapinj1_lthm = $true)) & (ulamvarind = $true) & ((hoaslaminj_lthm = $true) <=> (sP33 = $true)) & (! [X423 : term,X422 : term] : ((((sub @ X422 @ sh)) != ((sub @ X423 @ sh))) | (X422 = X423)) <=> (shinj = $true)) & ! [X325 : d_term,X324 : d_term] : ((X324 = X325) | (((d2term @ X325)) != ((d2term @ X324)))) & (lamnotvar = $true) & (((var @ (d2term @ d_one))) != $true) & (hoaslaminj_gthm = $true) & (($true = sP32) <=> (hoasinduction_lem2 = $true)) & (hoasinduction_lem1_lthm = $true) & ! [X515 : d_subst,X514 : d_subst] : ((((d2subst @ X514)) != ((d2subst @ X515))) | (X514 = X515)) & (hoasinduction_lem1 = $true) & ((sP30 = $true) <=> (hoasinduction_lem3aa_lthm = $true)) & ((hoasinduction_lthm_3 = $true) <=> (sP29 = $true)) & (! [X255 : (subst > term > subst > $o)] : ? [X256 : (term > $o)] : ! [X257 : term] : (((X256 @ X257)) = ((X255 @ id @ X257 @ id))) <=> (hoasinduction_lem0 = $true)) & (hoasinduction_lem2v2_gthm = $true) & (hoaslamnotap_gthm = $true) & (hoasinduction_lem3b_gthm = $true) & (axmap = $true) & ((pushprop_lem1_lthm = $true) <=> (? [X200 : term,X201 : subst] : (((comp @ sh @ (push @ X200 @ X201))) != X201) | ? [X199 : subst,X198 : term] : (((sub @ one @ (push @ X198 @ X199))) != X198) | ! [X202 : term,X205 : (term > $o),X206 : subst,X204 : term,X203 : (term > $o)] : ((((X203 @ X202)) != $true) | (((X205 @ (sub @ X202 @ (push @ X204 @ X206)))) = $true)))) & (axassoc = $true) & (axvarshift = $true) & (lamnotap != $true) & (hoaslamnotvar_lthm = $true) & ((sP23 = $true) <=> (hoasinduction_lem3a = $true)) & (ulamvarsh = $true) & (hoasinduction_gthm = $true) & ((sP22 = $true) <=> (hoasinduction_lem3v2a_lthm = $true)) & ((sP21 = $true) <=> (hoasinduction_lthm = $true)) & (((d2subst @ d_id)) = id) & ((sP12 = $true) <=> (hoasinduction_lem2v2 = $true)) & (pushprop_lem2v2_lthm = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (! [X98 : term,X101 : (term > $o),X100 : subst,X99 : (term > $o)] : ((((X101 @ one)) = $true) | (((X99 @ X98)) != $true) | ? [X102 : term] : (((X101 @ X102)) != ((X99 @ (sub @ X102 @ (push @ X98 @ X100)))))) <=> (pushprop_lem1v2 = $true)) & ((? [X94 : term] : (((sub @ X94 @ id)) != X94) | ! [X95 : term] : (((sub @ X95 @ id)) = X95)) <=> (termmset_lthm = $true)) & (pushprop_lem0_gthm = $true) & (hoasapnotvar_lthm = $true) & (! [X420 : term] : ? [X421 : (subst > term > term)] : (((sub @ X420 @ (push @ one @ sh))) = ((X421 @ sh @ one))) <=> (hoasinduction_lem3b_lthm = $true)) & (! [X498 : term] : ? [X499 : (subst > term > term)] : ! [X501 : subst,X500 : term] : (((X499 @ X501 @ X500)) = ((sub @ X498 @ (push @ X500 @ X501)))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & ! [X78 : d_subst] : (d_id = X78) & (hoasinduction_lem2_lthm = $true) & ((hoasinduction_lem3v2 = $true) <=> (sP10 = $true)) & (hoasapnotvar = $true) & (apnotvar = $true) & (substmonoid_gthm = $true) & (hoaslamnotvar_gthm = $true) & (hoaslamnotvar = $true) & ((substmonoid_lthm = $true) <=> (? [X251 : subst] : (((comp @ id @ X251)) != X251) | ? [X252 : subst] : (((comp @ X252 @ id)) != X252) | (! [X254 : subst] : (((comp @ X254 @ id)) = X254) & ! [X253 : subst] : (((comp @ id @ X253)) = X253)))) & (induction2lem_gthm = $true) & ((hoasinduction_lem0_lthm = $true) <=> ! [X424 : (subst > term > subst > $o)] : ? [X425 : (term > $o)] : ! [X426 : term] : (((X425 @ X426)) = ((X424 @ id @ X426 @ id)))) & ((sP8 = $true) <=> (hoasinduction_lem3a_lthm = $true)) & ((apinj1 = $true) <=> ! [X111 : term,X113 : term,X114 : term,X112 : term] : ((X112 = X114) | (((ap @ X112 @ X111)) != ((ap @ X114 @ X113))))) & (pushprop_lthm = $true) & ((! [X14 : term,X11 : term,X13 : term,X12 : term] : ((((ap @ (sub @ X13 @ id) @ X12)) != ((ap @ (sub @ X14 @ id) @ X11))) | (X11 = X12)) | ? [X10 : term,X9 : term,X8 : term,X7 : term] : ((X7 != X9) & (((ap @ X8 @ X7)) = ((ap @ X10 @ X9))))) <=> (hoasapinj2_lthm = $true)) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (! [X96 : subst,X97 : term] : (((sub @ one @ (push @ X97 @ X96))) = X97) <=> (axvarcons = $true)) & ! [X413 : term] : ? [X414 : d_term] : (((d2term @ X414)) = X413) & (! [X516 : term] : ? [X517 : (subst > term > term)] : ! [X519 : term,X518 : subst] : (((X517 @ X518 @ X519)) = ((sub @ X516 @ (push @ X519 @ X518)))) <=> (hoasinduction_lem3v2_f = $true)) & ((induction2lem = $true) <=> (sP6 = $true)) & ! [X93 : d_term] : (d_one = X93) & ((pushprop_lem1 = $true) <=> ! [X81 : term,X82 : (term > $o),X83 : term,X80 : (term > $o),X79 : subst] : ((((X80 @ X81)) != $true) | (((X82 @ (sub @ X81 @ (push @ X83 @ X79)))) = $true))) & ((sP4 = $true) <=> (hoasinduction = $true)) & (ulamvar1 != $true) & (! [X139 : subst] : (((push @ (sub @ one @ X139) @ (comp @ sh @ X139))) = X139) <=> (axscons = $true)) & (pushprop_lem1_gthm = $true) & ((pushprop_lem0 = $true) <=> ! [X415 : subst,X416 : term,X417 : (term > $o)] : ? [X418 : (term > $o)] : ! [X419 : term] : (((X417 @ (sub @ X419 @ (push @ X416 @ X415)))) = ((X418 @ X419)))) & ((hoasinduction_lem3b = $true) <=> ! [X213 : term] : ? [X214 : (subst > term > term)] : (((X214 @ sh @ one)) = ((sub @ X213 @ (push @ one @ sh))))) & (((d2term @ d_one)) = one) & (((d2subst @ d_id)) = sh) & ((sP1 = $true) <=> (hoasinduction_lem3v2_lthm = $true)) & (pushprop_gthm = $true) & (hoasinduction_lem1v2 = $true) & (induction2_gthm = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoasapinj1 = $true) <=> ! [X18 : term,X19 : term,X17 : term,X20 : term] : ((((ap @ (sub @ X20 @ id) @ X19)) != ((ap @ (sub @ X17 @ id) @ X18))) | (X17 = X20))) & (hoaslamnotap_lthm = $true) & (hoasapinj1_gthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (hoasinduction_lem1_gthm = $true) & (pushprop = $true) & ! [X15 : subst] : ? [X16 : d_subst] : (((d2subst @ X16)) = X15) & (! [X211 : term,X212 : term,X209 : term,X210 : term] : ((X210 = X212) | (((ap @ (sub @ X209 @ id) @ X212)) != ((ap @ (sub @ X211 @ id) @ X210)))) <=> (hoasapinj2 = $true)) & ((hoaslamnotap = $true) <=> ! [X49 : (subst > term > term),X48 : term,X50 : term] : ((((ap @ (sub @ X50 @ id) @ X48)) != ((lam @ (X49 @ sh @ one)))) | ? [X53 : term,X51 : subst,X52 : subst] : (((X49 @ (comp @ X51 @ X52) @ (sub @ X53 @ X52))) != ((sub @ (X49 @ X51 @ X53) @ X52))))) & ((termmset = $true) <=> ! [X103 : term] : (((sub @ X103 @ id)) = X103)) & ((laminj = $true) <=> ! [X395 : term,X396 : term] : ((X395 = X396) | (((lam @ X395)) != ((lam @ X396))))) & (hoasinduction_lem1v2_gthm = $true)),
  inference(definition_folding,[],[f12,f71,f70,f69,f68,f67,f66,f65,f64,f63,f62,f61,f60,f59,f58,f57,f56,f55,f54,f53,f52,f51,f50,f49,f48,f47,f46,f45,f44,f43,f42,f41,f40,f39,f38,f37,f36,f35,f34,f33,f32,f31,f30,f29,f28,f27,f26,f25,f24,f23,f22,f21,f20,f19,f18,f17,f16,f15,f14,f13])).
thf(f260,plain,(
  (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)))))))))))) & (axabs = $true) & ((($true = sP58) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | ($true != sP58))) & (((pushprop_lem0_lthm = $true) | ? [X248 : subst,X247 : term,X246 : (term > $o)] : ! [X249 : (term > $o)] : ? [X250 : term] : (((X246 @ (sub @ X250 @ (push @ X247 @ X248)))) != ((X249 @ X250)))) & (! [X248 : subst,X247 : term,X246 : (term > $o)] : ? [X249 : (term > $o)] : ! [X250 : term] : (((X246 @ (sub @ X250 @ (push @ X247 @ X248)))) = ((X249 @ X250))) | (pushprop_lem0_lthm != $true))) & (pushprop_lthm_orig = $true) & (((axshiftcons = $true) | ? [X326 : subst,X327 : term] : (((comp @ sh @ (push @ X327 @ X326))) != X326)) & (! [X326 : subst,X327 : term] : (((comp @ sh @ (push @ X327 @ X326))) = X326) | (axshiftcons != $true))) & (pushprop_lem2v2 = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (((induction = $true) | ? [X133 : (term > $o),X134 : term] : (! [X135 : term] : ((((X133 @ (lam @ X135))) = $true) | (((X133 @ X135)) != $true)) & (((X133 @ X134)) != $true) & (((sP57 @ X133)) != $true) & ! [X136 : term] : ((((var @ X136)) != $true) | (((X133 @ X136)) = $true)))) & (! [X133 : (term > $o),X134 : term] : (? [X135 : term] : ((((X133 @ (lam @ X135))) != $true) & (((X133 @ X135)) = $true)) | (((X133 @ X134)) = $true) | (((sP57 @ X133)) = $true) | ? [X136 : term] : ((((var @ X136)) = $true) & (((X133 @ X136)) != $true))) | (induction != $true))) & (((hoasinduction_no_psi_cond_lthm = $true) | (sP56 != $true)) & ((sP56 = $true) | (hoasinduction_no_psi_cond_lthm != $true))) & (((pushprop_lem1v2_lthm = $true) | (? [X108 : term,X107 : subst,X106 : (term > $o),X109 : (term > $o)] : (! [X110 : term] : (((X109 @ X110)) = ((X106 @ (sub @ X110 @ (push @ X108 @ X107))))) & (((X106 @ X108)) = $true) & (((X109 @ one)) != $true)) & ! [X105 : subst,X104 : term] : (((sub @ one @ (push @ X104 @ X105))) = X104))) & ((! [X108 : term,X107 : subst,X106 : (term > $o),X109 : (term > $o)] : (? [X110 : term] : (((X109 @ X110)) != ((X106 @ (sub @ X110 @ (push @ X108 @ X107))))) | (((X106 @ X108)) != $true) | (((X109 @ one)) = $true)) | ? [X105 : subst,X104 : term] : (((sub @ one @ (push @ X104 @ X105))) != X104)) | (pushprop_lem1v2_lthm != $true))) & (((hoasinduction_lem3 = $true) | (sP51 != $true)) & ((sP51 = $true) | (hoasinduction_lem3 != $true))) & ((! [X208 : term] : (((sub @ X208 @ id)) = X208) | (axvarid != $true)) & ((axvarid = $true) | ? [X208 : term] : (((sub @ X208 @ id)) != X208))) & (axapp = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (pushprop_lem2v2_gthm = $true) & (((sP49 = $true) | (hoasinduction_lem3_lthm != $true)) & ((hoasinduction_lem3_lthm = $true) | (sP49 != $true))) & (axclos = $true) & (hoasinduction_lem3v2_gthm = $true) & ((! [X520 : subst] : (((comp @ X520 @ id)) = X520) | (axidr != $true)) & ((axidr = $true) | ? [X520 : subst] : (((comp @ X520 @ id)) != X520))) & (((axidl = $true) | ? [X207 : subst] : (((comp @ id @ X207)) != X207)) & (! [X207 : subst] : (((comp @ id @ X207)) = X207) | (axidl != $true))) & (hoasinduction_lem3_gthm = $true) & (pushprop_lem3v2 = $true) & ((! [X116 : (subst > term > term),X115 : term,X117 : (subst > term > term),X118 : subst] : (? [X124 : term,X122 : subst,X123 : subst] : (((sub @ (X116 @ X123 @ X124) @ X122)) != ((X116 @ (comp @ X123 @ X122) @ (sub @ X124 @ X122)))) | ? [X121 : subst,X119 : subst,X120 : term] : (((X117 @ (comp @ X119 @ X121) @ (sub @ X120 @ X121))) != ((sub @ (X117 @ X119 @ X120) @ X121))) | (((lam @ (X116 @ sh @ one))) != ((lam @ (X117 @ sh @ one)))) | (((X117 @ X118 @ X115)) = ((X116 @ X118 @ X115)))) | (hoaslaminj != $true)) & ((hoaslaminj = $true) | ? [X116 : (subst > term > term),X115 : term,X117 : (subst > term > term),X118 : subst] : (! [X124 : term,X122 : subst,X123 : subst] : (((sub @ (X116 @ X123 @ X124) @ X122)) = ((X116 @ (comp @ X123 @ X122) @ (sub @ X124 @ X122)))) & ! [X121 : subst,X119 : subst,X120 : term] : (((X117 @ (comp @ X119 @ X121) @ (sub @ X120 @ X121))) = ((sub @ (X117 @ X119 @ X120) @ X121))) & (((lam @ (X116 @ sh @ one))) = ((lam @ (X117 @ sh @ one)))) & (((X117 @ X118 @ X115)) != ((X116 @ X118 @ X115)))))) & (hoasinduction_lem3a_gthm = $true) & (hoasapinj2_gthm = $true) & (hoasapnotvar_gthm = $true) & (pushprop_lem3v2_lthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (((sP46 = $true) | (hoasinduction_lem3aa != $true)) & ((hoasinduction_lem3aa = $true) | (sP46 != $true))) & (((sP45 = $true) | (induction2 != $true)) & ((induction2 = $true) | (sP45 != $true))) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (((hoasinduction_no_psi_cond = $true) | ? [X39 : (subst > term > subst > $o),X40 : term] : (! [X41 : term,X42 : term] : ((((X39 @ id @ X42 @ id)) != $true) | (((X39 @ id @ (ap @ (sub @ X41 @ id) @ X42) @ id)) = $true) | (((X39 @ id @ X41 @ id)) != $true)) & ($true != ((X39 @ id @ X40 @ id))) & (((sP43 @ X39)) != $true))) & (! [X39 : (subst > term > subst > $o),X40 : term] : (? [X41 : term,X42 : term] : ((((X39 @ id @ X42 @ id)) = $true) & (((X39 @ id @ (ap @ (sub @ X41 @ id) @ X42) @ id)) != $true) & (((X39 @ id @ X41 @ id)) = $true)) | ($true = ((X39 @ id @ X40 @ id))) | (((sP43 @ X39)) = $true)) | (hoasinduction_no_psi_cond != $true))) & (((! [X323 : subst] : (((comp @ id @ X323)) = X323) & ! [X322 : subst] : (((comp @ X322 @ id)) = X322)) | (substmonoid != $true)) & ((substmonoid = $true) | (? [X323 : subst] : (((comp @ id @ X323)) != X323) | ? [X322 : subst] : (((comp @ X322 @ id)) != X322)))) & (((sP42 = $true) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | (sP42 != $true))) & (hoasinduction_lem2_gthm = $true) & (pushprop_lem1v2_gthm = $true) & (termmset_gthm = $true) & (((apinj2 = $true) | ? [X140 : term,X142 : term,X141 : term,X143 : term] : ((X140 != X143) & (((ap @ X141 @ X140)) = ((ap @ X142 @ X143))))) & (! [X140 : term,X142 : term,X141 : term,X143 : term] : ((X140 = X143) | (((ap @ X141 @ X140)) != ((ap @ X142 @ X143)))) | (apinj2 != $true))) & (((sP38 = $true) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | (sP38 != $true))) & (((sP34 = $true) | (hoasinduction_lem3aaa != $true)) & ((hoasinduction_lem3aaa = $true) | (sP34 != $true))) & (((? [X86 : term,X85 : term,X87 : term,X88 : term] : ((((ap @ X88 @ X85)) = ((ap @ X87 @ X86))) & (X87 != X88)) | ? [X84 : term] : (((sub @ X84 @ id)) != X84) | ! [X92 : term,X89 : term,X90 : term,X91 : term] : ((X89 = X92) | (((ap @ (sub @ X89 @ id) @ X91)) != ((ap @ (sub @ X92 @ id) @ X90))))) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | (! [X86 : term,X85 : term,X87 : term,X88 : term] : ((((ap @ X88 @ X85)) != ((ap @ X87 @ X86))) | (X87 = X88)) & ! [X84 : term] : (((sub @ X84 @ id)) = X84) & ? [X92 : term,X89 : term,X90 : term,X91 : term] : ((X89 != X92) & (((ap @ (sub @ X89 @ id) @ X91)) = ((ap @ (sub @ X92 @ id) @ X90))))))) & (ulamvarind = $true) & (((hoaslaminj_lthm = $true) | (sP33 != $true)) & ((sP33 = $true) | (hoaslaminj_lthm != $true))) & ((! [X423 : term,X422 : term] : ((((sub @ X422 @ sh)) != ((sub @ X423 @ sh))) | (X422 = X423)) | (shinj != $true)) & ((shinj = $true) | ? [X423 : term,X422 : term] : ((((sub @ X422 @ sh)) = ((sub @ X423 @ sh))) & (X422 != X423)))) & ! [X325 : d_term,X324 : d_term] : ((X324 = X325) | (((d2term @ X325)) != ((d2term @ X324)))) & (lamnotvar = $true) & (((var @ (d2term @ d_one))) != $true) & (hoaslaminj_gthm = $true) & ((($true = sP32) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | ($true != sP32))) & (hoasinduction_lem1_lthm = $true) & ! [X515 : d_subst,X514 : d_subst] : ((((d2subst @ X514)) != ((d2subst @ X515))) | (X514 = X515)) & (hoasinduction_lem1 = $true) & (((sP30 = $true) | (hoasinduction_lem3aa_lthm != $true)) & ((hoasinduction_lem3aa_lthm = $true) | (sP30 != $true))) & (((hoasinduction_lthm_3 = $true) | (sP29 != $true)) & ((sP29 = $true) | (hoasinduction_lthm_3 != $true))) & ((! [X255 : (subst > term > subst > $o)] : ? [X256 : (term > $o)] : ! [X257 : term] : (((X256 @ X257)) = ((X255 @ id @ X257 @ id))) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem0 = $true) | ? [X255 : (subst > term > subst > $o)] : ! [X256 : (term > $o)] : ? [X257 : term] : (((X256 @ X257)) != ((X255 @ id @ X257 @ id))))) & (hoasinduction_lem2v2_gthm = $true) & (hoaslamnotap_gthm = $true) & (hoasinduction_lem3b_gthm = $true) & (axmap = $true) & (((pushprop_lem1_lthm = $true) | (! [X200 : term,X201 : subst] : (((comp @ sh @ (push @ X200 @ X201))) = X201) & ! [X199 : subst,X198 : term] : (((sub @ one @ (push @ X198 @ X199))) = X198) & ? [X202 : term,X205 : (term > $o),X206 : subst,X204 : term,X203 : (term > $o)] : ((((X203 @ X202)) = $true) & (((X205 @ (sub @ X202 @ (push @ X204 @ X206)))) != $true)))) & ((? [X200 : term,X201 : subst] : (((comp @ sh @ (push @ X200 @ X201))) != X201) | ? [X199 : subst,X198 : term] : (((sub @ one @ (push @ X198 @ X199))) != X198) | ! [X202 : term,X205 : (term > $o),X206 : subst,X204 : term,X203 : (term > $o)] : ((((X203 @ X202)) != $true) | (((X205 @ (sub @ X202 @ (push @ X204 @ X206)))) = $true))) | (pushprop_lem1_lthm != $true))) & (axassoc = $true) & (axvarshift = $true) & (lamnotap != $true) & (hoaslamnotvar_lthm = $true) & (((sP23 = $true) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | (sP23 != $true))) & (ulamvarsh = $true) & (hoasinduction_gthm = $true) & (((sP22 = $true) | (hoasinduction_lem3v2a_lthm != $true)) & ((hoasinduction_lem3v2a_lthm = $true) | (sP22 != $true))) & (((sP21 = $true) | (hoasinduction_lthm != $true)) & ((hoasinduction_lthm = $true) | (sP21 != $true))) & (((d2subst @ d_id)) = id) & (((sP12 = $true) | (hoasinduction_lem2v2 != $true)) & ((hoasinduction_lem2v2 = $true) | (sP12 != $true))) & (pushprop_lem2v2_lthm = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & ((! [X98 : term,X101 : (term > $o),X100 : subst,X99 : (term > $o)] : ((((X101 @ one)) = $true) | (((X99 @ X98)) != $true) | ? [X102 : term] : (((X101 @ X102)) != ((X99 @ (sub @ X102 @ (push @ X98 @ X100)))))) | (pushprop_lem1v2 != $true)) & ((pushprop_lem1v2 = $true) | ? [X98 : term,X101 : (term > $o),X100 : subst,X99 : (term > $o)] : ((((X101 @ one)) != $true) & (((X99 @ X98)) = $true) & ! [X102 : term] : (((X101 @ X102)) = ((X99 @ (sub @ X102 @ (push @ X98 @ X100)))))))) & (((? [X94 : term] : (((sub @ X94 @ id)) != X94) | ! [X95 : term] : (((sub @ X95 @ id)) = X95)) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | (! [X94 : term] : (((sub @ X94 @ id)) = X94) & ? [X95 : term] : (((sub @ X95 @ id)) != X95)))) & (pushprop_lem0_gthm = $true) & (hoasapnotvar_lthm = $true) & ((! [X420 : term] : ? [X421 : (subst > term > term)] : (((sub @ X420 @ (push @ one @ sh))) = ((X421 @ sh @ one))) | (hoasinduction_lem3b_lthm != $true)) & ((hoasinduction_lem3b_lthm = $true) | ? [X420 : term] : ! [X421 : (subst > term > term)] : (((sub @ X420 @ (push @ one @ sh))) != ((X421 @ sh @ one))))) & ((! [X498 : term] : ? [X499 : (subst > term > term)] : ! [X501 : subst,X500 : term] : (((X499 @ X501 @ X500)) = ((sub @ X498 @ (push @ X500 @ X501)))) | (hoasinduction_lem3v2_f_lthm != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X498 : term] : ! [X499 : (subst > term > term)] : ? [X501 : subst,X500 : term] : (((X499 @ X501 @ X500)) != ((sub @ X498 @ (push @ X500 @ X501)))))) & ! [X78 : d_subst] : (d_id = X78) & (hoasinduction_lem2_lthm = $true) & (((hoasinduction_lem3v2 = $true) | (sP10 != $true)) & ((sP10 = $true) | (hoasinduction_lem3v2 != $true))) & (hoasapnotvar = $true) & (apnotvar = $true) & (substmonoid_gthm = $true) & (hoaslamnotvar_gthm = $true) & (hoaslamnotvar = $true) & (((substmonoid_lthm = $true) | (! [X251 : subst] : (((comp @ id @ X251)) = X251) & ! [X252 : subst] : (((comp @ X252 @ id)) = X252) & (? [X254 : subst] : (((comp @ X254 @ id)) != X254) | ? [X253 : subst] : (((comp @ id @ X253)) != X253)))) & ((? [X251 : subst] : (((comp @ id @ X251)) != X251) | ? [X252 : subst] : (((comp @ X252 @ id)) != X252) | (! [X254 : subst] : (((comp @ X254 @ id)) = X254) & ! [X253 : subst] : (((comp @ id @ X253)) = X253))) | (substmonoid_lthm != $true))) & (induction2lem_gthm = $true) & (((hoasinduction_lem0_lthm = $true) | ? [X424 : (subst > term > subst > $o)] : ! [X425 : (term > $o)] : ? [X426 : term] : (((X425 @ X426)) != ((X424 @ id @ X426 @ id)))) & (! [X424 : (subst > term > subst > $o)] : ? [X425 : (term > $o)] : ! [X426 : term] : (((X425 @ X426)) = ((X424 @ id @ X426 @ id))) | (hoasinduction_lem0_lthm != $true))) & (((sP8 = $true) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | (sP8 != $true))) & (((apinj1 = $true) | ? [X111 : term,X113 : term,X114 : term,X112 : term] : ((X112 != X114) & (((ap @ X112 @ X111)) = ((ap @ X114 @ X113))))) & (! [X111 : term,X113 : term,X114 : term,X112 : term] : ((X112 = X114) | (((ap @ X112 @ X111)) != ((ap @ X114 @ X113)))) | (apinj1 != $true))) & (pushprop_lthm = $true) & (((! [X14 : term,X11 : term,X13 : term,X12 : term] : ((((ap @ (sub @ X13 @ id) @ X12)) != ((ap @ (sub @ X14 @ id) @ X11))) | (X11 = X12)) | ? [X10 : term,X9 : term,X8 : term,X7 : term] : ((X7 != X9) & (((ap @ X8 @ X7)) = ((ap @ X10 @ X9))))) | (hoasapinj2_lthm != $true)) & ((hoasapinj2_lthm = $true) | (? [X14 : term,X11 : term,X13 : term,X12 : term] : ((((ap @ (sub @ X13 @ id) @ X12)) = ((ap @ (sub @ X14 @ id) @ X11))) & (X11 != X12)) & ! [X10 : term,X9 : term,X8 : term,X7 : term] : ((X7 = X9) | (((ap @ X8 @ X7)) != ((ap @ X10 @ X9))))))) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ((! [X96 : subst,X97 : term] : (((sub @ one @ (push @ X97 @ X96))) = X97) | (axvarcons != $true)) & ((axvarcons = $true) | ? [X96 : subst,X97 : term] : (((sub @ one @ (push @ X97 @ X96))) != X97))) & ! [X413 : term] : ? [X414 : d_term] : (((d2term @ X414)) = X413) & ((! [X516 : term] : ? [X517 : (subst > term > term)] : ! [X519 : term,X518 : subst] : (((X517 @ X518 @ X519)) = ((sub @ X516 @ (push @ X519 @ X518)))) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ? [X516 : term] : ! [X517 : (subst > term > term)] : ? [X519 : term,X518 : subst] : (((X517 @ X518 @ X519)) != ((sub @ X516 @ (push @ X519 @ X518)))))) & (((induction2lem = $true) | (sP6 != $true)) & ((sP6 = $true) | (induction2lem != $true))) & ! [X93 : d_term] : (d_one = X93) & (((pushprop_lem1 = $true) | ? [X81 : term,X82 : (term > $o),X83 : term,X80 : (term > $o),X79 : subst] : ((((X80 @ X81)) = $true) & (((X82 @ (sub @ X81 @ (push @ X83 @ X79)))) != $true))) & (! [X81 : term,X82 : (term > $o),X83 : term,X80 : (term > $o),X79 : subst] : ((((X80 @ X81)) != $true) | (((X82 @ (sub @ X81 @ (push @ X83 @ X79)))) = $true)) | (pushprop_lem1 != $true))) & (((sP4 = $true) | (hoasinduction != $true)) & ((hoasinduction = $true) | (sP4 != $true))) & (ulamvar1 != $true) & ((! [X139 : subst] : (((push @ (sub @ one @ X139) @ (comp @ sh @ X139))) = X139) | (axscons != $true)) & ((axscons = $true) | ? [X139 : subst] : (((push @ (sub @ one @ X139) @ (comp @ sh @ X139))) != X139))) & (pushprop_lem1_gthm = $true) & (((pushprop_lem0 = $true) | ? [X415 : subst,X416 : term,X417 : (term > $o)] : ! [X418 : (term > $o)] : ? [X419 : term] : (((X417 @ (sub @ X419 @ (push @ X416 @ X415)))) != ((X418 @ X419)))) & (! [X415 : subst,X416 : term,X417 : (term > $o)] : ? [X418 : (term > $o)] : ! [X419 : term] : (((X417 @ (sub @ X419 @ (push @ X416 @ X415)))) = ((X418 @ X419))) | (pushprop_lem0 != $true))) & (((hoasinduction_lem3b = $true) | ? [X213 : term] : ! [X214 : (subst > term > term)] : (((X214 @ sh @ one)) != ((sub @ X213 @ (push @ one @ sh))))) & (! [X213 : term] : ? [X214 : (subst > term > term)] : (((X214 @ sh @ one)) = ((sub @ X213 @ (push @ one @ sh)))) | (hoasinduction_lem3b != $true))) & (((d2term @ d_one)) = one) & (((d2subst @ d_id)) = sh) & (((sP1 = $true) | (hoasinduction_lem3v2_lthm != $true)) & ((hoasinduction_lem3v2_lthm = $true) | (sP1 != $true))) & (pushprop_gthm = $true) & (hoasinduction_lem1v2 = $true) & (induction2_gthm = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (((hoasapinj1 = $true) | ? [X18 : term,X19 : term,X17 : term,X20 : term] : ((((ap @ (sub @ X20 @ id) @ X19)) = ((ap @ (sub @ X17 @ id) @ X18))) & (X17 != X20))) & (! [X18 : term,X19 : term,X17 : term,X20 : term] : ((((ap @ (sub @ X20 @ id) @ X19)) != ((ap @ (sub @ X17 @ id) @ X18))) | (X17 = X20)) | (hoasapinj1 != $true))) & (hoaslamnotap_lthm = $true) & (hoasapinj1_gthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (hoasinduction_lem1_gthm = $true) & (pushprop = $true) & ! [X15 : subst] : ? [X16 : d_subst] : (((d2subst @ X16)) = X15) & ((! [X211 : term,X212 : term,X209 : term,X210 : term] : ((X210 = X212) | (((ap @ (sub @ X209 @ id) @ X212)) != ((ap @ (sub @ X211 @ id) @ X210)))) | (hoasapinj2 != $true)) & ((hoasapinj2 = $true) | ? [X211 : term,X212 : term,X209 : term,X210 : term] : ((X210 != X212) & (((ap @ (sub @ X209 @ id) @ X212)) = ((ap @ (sub @ X211 @ id) @ X210)))))) & (((hoaslamnotap = $true) | ? [X49 : (subst > term > term),X48 : term,X50 : term] : ((((ap @ (sub @ X50 @ id) @ X48)) = ((lam @ (X49 @ sh @ one)))) & ! [X53 : term,X51 : subst,X52 : subst] : (((X49 @ (comp @ X51 @ X52) @ (sub @ X53 @ X52))) = ((sub @ (X49 @ X51 @ X53) @ X52))))) & (! [X49 : (subst > term > term),X48 : term,X50 : term] : ((((ap @ (sub @ X50 @ id) @ X48)) != ((lam @ (X49 @ sh @ one)))) | ? [X53 : term,X51 : subst,X52 : subst] : (((X49 @ (comp @ X51 @ X52) @ (sub @ X53 @ X52))) != ((sub @ (X49 @ X51 @ X53) @ X52)))) | (hoaslamnotap != $true))) & (((termmset = $true) | ? [X103 : term] : (((sub @ X103 @ id)) != X103)) & (! [X103 : term] : (((sub @ X103 @ id)) = X103) | (termmset != $true))) & (((laminj = $true) | ? [X395 : term,X396 : term] : ((X395 != X396) & (((lam @ X395)) = ((lam @ X396))))) & (! [X395 : term,X396 : term] : ((X395 = X396) | (((lam @ X395)) != ((lam @ X396)))) | (laminj != $true))) & (hoasinduction_lem1v2_gthm = $true)),
  inference(nnf_transformation,[],[f72])).
thf(f261,plain,(
  (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)))))))))))) & (axabs = $true) & (($true = sP58) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | ($true != sP58)) & ((pushprop_lem0_lthm = $true) | ? [X248 : subst,X247 : term,X246 : (term > $o)] : ! [X249 : (term > $o)] : ? [X250 : term] : (((X246 @ (sub @ X250 @ (push @ X247 @ X248)))) != ((X249 @ X250)))) & (! [X248 : subst,X247 : term,X246 : (term > $o)] : ? [X249 : (term > $o)] : ! [X250 : term] : (((X246 @ (sub @ X250 @ (push @ X247 @ X248)))) = ((X249 @ X250))) | (pushprop_lem0_lthm != $true)) & (pushprop_lthm_orig = $true) & ((axshiftcons = $true) | ? [X326 : subst,X327 : term] : (((comp @ sh @ (push @ X327 @ X326))) != X326)) & (! [X326 : subst,X327 : term] : (((comp @ sh @ (push @ X327 @ X326))) = X326) | (axshiftcons != $true)) & (pushprop_lem2v2 = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ((induction = $true) | ? [X133 : (term > $o),X134 : term] : (! [X135 : term] : ((((X133 @ (lam @ X135))) = $true) | (((X133 @ X135)) != $true)) & (((X133 @ X134)) != $true) & (((sP57 @ X133)) != $true) & ! [X136 : term] : ((((var @ X136)) != $true) | (((X133 @ X136)) = $true)))) & (! [X133 : (term > $o),X134 : term] : (? [X135 : term] : ((((X133 @ (lam @ X135))) != $true) & (((X133 @ X135)) = $true)) | (((X133 @ X134)) = $true) | (((sP57 @ X133)) = $true) | ? [X136 : term] : ((((var @ X136)) = $true) & (((X133 @ X136)) != $true))) | (induction != $true)) & ((hoasinduction_no_psi_cond_lthm = $true) | (sP56 != $true)) & ((sP56 = $true) | (hoasinduction_no_psi_cond_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (? [X108 : term,X107 : subst,X106 : (term > $o),X109 : (term > $o)] : (! [X110 : term] : (((X109 @ X110)) = ((X106 @ (sub @ X110 @ (push @ X108 @ X107))))) & (((X106 @ X108)) = $true) & (((X109 @ one)) != $true)) & ! [X105 : subst,X104 : term] : (((sub @ one @ (push @ X104 @ X105))) = X104))) & (! [X108 : term,X107 : subst,X106 : (term > $o),X109 : (term > $o)] : (? [X110 : term] : (((X109 @ X110)) != ((X106 @ (sub @ X110 @ (push @ X108 @ X107))))) | (((X106 @ X108)) != $true) | (((X109 @ one)) = $true)) | ? [X105 : subst,X104 : term] : (((sub @ one @ (push @ X104 @ X105))) != X104) | (pushprop_lem1v2_lthm != $true)) & ((hoasinduction_lem3 = $true) | (sP51 != $true)) & ((sP51 = $true) | (hoasinduction_lem3 != $true)) & (! [X208 : term] : (((sub @ X208 @ id)) = X208) | (axvarid != $true)) & ((axvarid = $true) | ? [X208 : term] : (((sub @ X208 @ id)) != X208)) & (axapp = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (pushprop_lem2v2_gthm = $true) & ((sP49 = $true) | (hoasinduction_lem3_lthm != $true)) & ((hoasinduction_lem3_lthm = $true) | (sP49 != $true)) & (axclos = $true) & (hoasinduction_lem3v2_gthm = $true) & (! [X520 : subst] : (((comp @ X520 @ id)) = X520) | (axidr != $true)) & ((axidr = $true) | ? [X520 : subst] : (((comp @ X520 @ id)) != X520)) & ((axidl = $true) | ? [X207 : subst] : (((comp @ id @ X207)) != X207)) & (! [X207 : subst] : (((comp @ id @ X207)) = X207) | (axidl != $true)) & (hoasinduction_lem3_gthm = $true) & (pushprop_lem3v2 = $true) & (! [X116 : (subst > term > term),X115 : term,X117 : (subst > term > term),X118 : subst] : (? [X124 : term,X122 : subst,X123 : subst] : (((sub @ (X116 @ X123 @ X124) @ X122)) != ((X116 @ (comp @ X123 @ X122) @ (sub @ X124 @ X122)))) | ? [X121 : subst,X119 : subst,X120 : term] : (((X117 @ (comp @ X119 @ X121) @ (sub @ X120 @ X121))) != ((sub @ (X117 @ X119 @ X120) @ X121))) | (((lam @ (X116 @ sh @ one))) != ((lam @ (X117 @ sh @ one)))) | (((X117 @ X118 @ X115)) = ((X116 @ X118 @ X115)))) | (hoaslaminj != $true)) & ((hoaslaminj = $true) | ? [X116 : (subst > term > term),X115 : term,X117 : (subst > term > term),X118 : subst] : (! [X124 : term,X122 : subst,X123 : subst] : (((sub @ (X116 @ X123 @ X124) @ X122)) = ((X116 @ (comp @ X123 @ X122) @ (sub @ X124 @ X122)))) & ! [X121 : subst,X119 : subst,X120 : term] : (((X117 @ (comp @ X119 @ X121) @ (sub @ X120 @ X121))) = ((sub @ (X117 @ X119 @ X120) @ X121))) & (((lam @ (X116 @ sh @ one))) = ((lam @ (X117 @ sh @ one)))) & (((X117 @ X118 @ X115)) != ((X116 @ X118 @ X115))))) & (hoasinduction_lem3a_gthm = $true) & (hoasapinj2_gthm = $true) & (hoasapnotvar_gthm = $true) & (pushprop_lem3v2_lthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & ((sP46 = $true) | (hoasinduction_lem3aa != $true)) & ((hoasinduction_lem3aa = $true) | (sP46 != $true)) & ((sP45 = $true) | (induction2 != $true)) & ((induction2 = $true) | (sP45 != $true)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoasinduction_no_psi_cond = $true) | ? [X39 : (subst > term > subst > $o),X40 : term] : (! [X41 : term,X42 : term] : ((((X39 @ id @ X42 @ id)) != $true) | (((X39 @ id @ (ap @ (sub @ X41 @ id) @ X42) @ id)) = $true) | (((X39 @ id @ X41 @ id)) != $true)) & ($true != ((X39 @ id @ X40 @ id))) & (((sP43 @ X39)) != $true))) & (! [X39 : (subst > term > subst > $o),X40 : term] : (? [X41 : term,X42 : term] : ((((X39 @ id @ X42 @ id)) = $true) & (((X39 @ id @ (ap @ (sub @ X41 @ id) @ X42) @ id)) != $true) & (((X39 @ id @ X41 @ id)) = $true)) | ($true = ((X39 @ id @ X40 @ id))) | (((sP43 @ X39)) = $true)) | (hoasinduction_no_psi_cond != $true)) & ((! [X323 : subst] : (((comp @ id @ X323)) = X323) & ! [X322 : subst] : (((comp @ X322 @ id)) = X322)) | (substmonoid != $true)) & ((substmonoid = $true) | ? [X323 : subst] : (((comp @ id @ X323)) != X323) | ? [X322 : subst] : (((comp @ X322 @ id)) != X322)) & ((sP42 = $true) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | (sP42 != $true)) & (hoasinduction_lem2_gthm = $true) & (pushprop_lem1v2_gthm = $true) & (termmset_gthm = $true) & ((apinj2 = $true) | ? [X140 : term,X142 : term,X141 : term,X143 : term] : ((X140 != X143) & (((ap @ X141 @ X140)) = ((ap @ X142 @ X143))))) & (! [X140 : term,X142 : term,X141 : term,X143 : term] : ((X140 = X143) | (((ap @ X141 @ X140)) != ((ap @ X142 @ X143)))) | (apinj2 != $true)) & ((sP38 = $true) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | (sP38 != $true)) & ((sP34 = $true) | (hoasinduction_lem3aaa != $true)) & ((hoasinduction_lem3aaa = $true) | (sP34 != $true)) & (? [X86 : term,X85 : term,X87 : term,X88 : term] : ((((ap @ X88 @ X85)) = ((ap @ X87 @ X86))) & (X87 != X88)) | ? [X84 : term] : (((sub @ X84 @ id)) != X84) | ! [X92 : term,X89 : term,X90 : term,X91 : term] : ((X89 = X92) | (((ap @ (sub @ X89 @ id) @ X91)) != ((ap @ (sub @ X92 @ id) @ X90)))) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | (! [X86 : term,X85 : term,X87 : term,X88 : term] : ((((ap @ X88 @ X85)) != ((ap @ X87 @ X86))) | (X87 = X88)) & ! [X84 : term] : (((sub @ X84 @ id)) = X84) & ? [X92 : term,X89 : term,X90 : term,X91 : term] : ((X89 != X92) & (((ap @ (sub @ X89 @ id) @ X91)) = ((ap @ (sub @ X92 @ id) @ X90)))))) & (ulamvarind = $true) & ((hoaslaminj_lthm = $true) | (sP33 != $true)) & ((sP33 = $true) | (hoaslaminj_lthm != $true)) & (! [X423 : term,X422 : term] : ((((sub @ X422 @ sh)) != ((sub @ X423 @ sh))) | (X422 = X423)) | (shinj != $true)) & ((shinj = $true) | ? [X423 : term,X422 : term] : ((((sub @ X422 @ sh)) = ((sub @ X423 @ sh))) & (X422 != X423))) & ! [X325 : d_term,X324 : d_term] : ((X324 = X325) | (((d2term @ X325)) != ((d2term @ X324)))) & (lamnotvar = $true) & (((var @ (d2term @ d_one))) != $true) & (hoaslaminj_gthm = $true) & (($true = sP32) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | ($true != sP32)) & (hoasinduction_lem1_lthm = $true) & ! [X515 : d_subst,X514 : d_subst] : ((((d2subst @ X514)) != ((d2subst @ X515))) | (X514 = X515)) & (hoasinduction_lem1 = $true) & ((sP30 = $true) | (hoasinduction_lem3aa_lthm != $true)) & ((hoasinduction_lem3aa_lthm = $true) | (sP30 != $true)) & ((hoasinduction_lthm_3 = $true) | (sP29 != $true)) & ((sP29 = $true) | (hoasinduction_lthm_3 != $true)) & (! [X255 : (subst > term > subst > $o)] : ? [X256 : (term > $o)] : ! [X257 : term] : (((X256 @ X257)) = ((X255 @ id @ X257 @ id))) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem0 = $true) | ? [X255 : (subst > term > subst > $o)] : ! [X256 : (term > $o)] : ? [X257 : term] : (((X256 @ X257)) != ((X255 @ id @ X257 @ id)))) & (hoasinduction_lem2v2_gthm = $true) & (hoaslamnotap_gthm = $true) & (hoasinduction_lem3b_gthm = $true) & (axmap = $true) & ((pushprop_lem1_lthm = $true) | (! [X200 : term,X201 : subst] : (((comp @ sh @ (push @ X200 @ X201))) = X201) & ! [X199 : subst,X198 : term] : (((sub @ one @ (push @ X198 @ X199))) = X198) & ? [X202 : term,X205 : (term > $o),X206 : subst,X204 : term,X203 : (term > $o)] : ((((X203 @ X202)) = $true) & (((X205 @ (sub @ X202 @ (push @ X204 @ X206)))) != $true)))) & (? [X200 : term,X201 : subst] : (((comp @ sh @ (push @ X200 @ X201))) != X201) | ? [X199 : subst,X198 : term] : (((sub @ one @ (push @ X198 @ X199))) != X198) | ! [X202 : term,X205 : (term > $o),X206 : subst,X204 : term,X203 : (term > $o)] : ((((X203 @ X202)) != $true) | (((X205 @ (sub @ X202 @ (push @ X204 @ X206)))) = $true)) | (pushprop_lem1_lthm != $true)) & (axassoc = $true) & (axvarshift = $true) & (lamnotap != $true) & (hoaslamnotvar_lthm = $true) & ((sP23 = $true) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | (sP23 != $true)) & (ulamvarsh = $true) & (hoasinduction_gthm = $true) & ((sP22 = $true) | (hoasinduction_lem3v2a_lthm != $true)) & ((hoasinduction_lem3v2a_lthm = $true) | (sP22 != $true)) & ((sP21 = $true) | (hoasinduction_lthm != $true)) & ((hoasinduction_lthm = $true) | (sP21 != $true)) & (((d2subst @ d_id)) = id) & ((sP12 = $true) | (hoasinduction_lem2v2 != $true)) & ((hoasinduction_lem2v2 = $true) | (sP12 != $true)) & (pushprop_lem2v2_lthm = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (! [X98 : term,X101 : (term > $o),X100 : subst,X99 : (term > $o)] : ((((X101 @ one)) = $true) | (((X99 @ X98)) != $true) | ? [X102 : term] : (((X101 @ X102)) != ((X99 @ (sub @ X102 @ (push @ X98 @ X100)))))) | (pushprop_lem1v2 != $true)) & ((pushprop_lem1v2 = $true) | ? [X98 : term,X101 : (term > $o),X100 : subst,X99 : (term > $o)] : ((((X101 @ one)) != $true) & (((X99 @ X98)) = $true) & ! [X102 : term] : (((X101 @ X102)) = ((X99 @ (sub @ X102 @ (push @ X98 @ X100))))))) & (? [X94 : term] : (((sub @ X94 @ id)) != X94) | ! [X95 : term] : (((sub @ X95 @ id)) = X95) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | (! [X94 : term] : (((sub @ X94 @ id)) = X94) & ? [X95 : term] : (((sub @ X95 @ id)) != X95))) & (pushprop_lem0_gthm = $true) & (hoasapnotvar_lthm = $true) & (! [X420 : term] : ? [X421 : (subst > term > term)] : (((sub @ X420 @ (push @ one @ sh))) = ((X421 @ sh @ one))) | (hoasinduction_lem3b_lthm != $true)) & ((hoasinduction_lem3b_lthm = $true) | ? [X420 : term] : ! [X421 : (subst > term > term)] : (((sub @ X420 @ (push @ one @ sh))) != ((X421 @ sh @ one)))) & (! [X498 : term] : ? [X499 : (subst > term > term)] : ! [X501 : subst,X500 : term] : (((X499 @ X501 @ X500)) = ((sub @ X498 @ (push @ X500 @ X501)))) | (hoasinduction_lem3v2_f_lthm != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X498 : term] : ! [X499 : (subst > term > term)] : ? [X501 : subst,X500 : term] : (((X499 @ X501 @ X500)) != ((sub @ X498 @ (push @ X500 @ X501))))) & ! [X78 : d_subst] : (d_id = X78) & (hoasinduction_lem2_lthm = $true) & ((hoasinduction_lem3v2 = $true) | (sP10 != $true)) & ((sP10 = $true) | (hoasinduction_lem3v2 != $true)) & (hoasapnotvar = $true) & (apnotvar = $true) & (substmonoid_gthm = $true) & (hoaslamnotvar_gthm = $true) & (hoaslamnotvar = $true) & ((substmonoid_lthm = $true) | (! [X251 : subst] : (((comp @ id @ X251)) = X251) & ! [X252 : subst] : (((comp @ X252 @ id)) = X252) & (? [X254 : subst] : (((comp @ X254 @ id)) != X254) | ? [X253 : subst] : (((comp @ id @ X253)) != X253)))) & (? [X251 : subst] : (((comp @ id @ X251)) != X251) | ? [X252 : subst] : (((comp @ X252 @ id)) != X252) | (! [X254 : subst] : (((comp @ X254 @ id)) = X254) & ! [X253 : subst] : (((comp @ id @ X253)) = X253)) | (substmonoid_lthm != $true)) & (induction2lem_gthm = $true) & ((hoasinduction_lem0_lthm = $true) | ? [X424 : (subst > term > subst > $o)] : ! [X425 : (term > $o)] : ? [X426 : term] : (((X425 @ X426)) != ((X424 @ id @ X426 @ id)))) & (! [X424 : (subst > term > subst > $o)] : ? [X425 : (term > $o)] : ! [X426 : term] : (((X425 @ X426)) = ((X424 @ id @ X426 @ id))) | (hoasinduction_lem0_lthm != $true)) & ((sP8 = $true) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | (sP8 != $true)) & ((apinj1 = $true) | ? [X111 : term,X113 : term,X114 : term,X112 : term] : ((X112 != X114) & (((ap @ X112 @ X111)) = ((ap @ X114 @ X113))))) & (! [X111 : term,X113 : term,X114 : term,X112 : term] : ((X112 = X114) | (((ap @ X112 @ X111)) != ((ap @ X114 @ X113)))) | (apinj1 != $true)) & (pushprop_lthm = $true) & (! [X14 : term,X11 : term,X13 : term,X12 : term] : ((((ap @ (sub @ X13 @ id) @ X12)) != ((ap @ (sub @ X14 @ id) @ X11))) | (X11 = X12)) | ? [X10 : term,X9 : term,X8 : term,X7 : term] : ((X7 != X9) & (((ap @ X8 @ X7)) = ((ap @ X10 @ X9)))) | (hoasapinj2_lthm != $true)) & ((hoasapinj2_lthm = $true) | (? [X14 : term,X11 : term,X13 : term,X12 : term] : ((((ap @ (sub @ X13 @ id) @ X12)) = ((ap @ (sub @ X14 @ id) @ X11))) & (X11 != X12)) & ! [X10 : term,X9 : term,X8 : term,X7 : term] : ((X7 = X9) | (((ap @ X8 @ X7)) != ((ap @ X10 @ X9)))))) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (! [X96 : subst,X97 : term] : (((sub @ one @ (push @ X97 @ X96))) = X97) | (axvarcons != $true)) & ((axvarcons = $true) | ? [X96 : subst,X97 : term] : (((sub @ one @ (push @ X97 @ X96))) != X97)) & ! [X413 : term] : ? [X414 : d_term] : (((d2term @ X414)) = X413) & (! [X516 : term] : ? [X517 : (subst > term > term)] : ! [X519 : term,X518 : subst] : (((X517 @ X518 @ X519)) = ((sub @ X516 @ (push @ X519 @ X518)))) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ? [X516 : term] : ! [X517 : (subst > term > term)] : ? [X519 : term,X518 : subst] : (((X517 @ X518 @ X519)) != ((sub @ X516 @ (push @ X519 @ X518))))) & ((induction2lem = $true) | (sP6 != $true)) & ((sP6 = $true) | (induction2lem != $true)) & ! [X93 : d_term] : (d_one = X93) & ((pushprop_lem1 = $true) | ? [X81 : term,X82 : (term > $o),X83 : term,X80 : (term > $o),X79 : subst] : ((((X80 @ X81)) = $true) & (((X82 @ (sub @ X81 @ (push @ X83 @ X79)))) != $true))) & (! [X81 : term,X82 : (term > $o),X83 : term,X80 : (term > $o),X79 : subst] : ((((X80 @ X81)) != $true) | (((X82 @ (sub @ X81 @ (push @ X83 @ X79)))) = $true)) | (pushprop_lem1 != $true)) & ((sP4 = $true) | (hoasinduction != $true)) & ((hoasinduction = $true) | (sP4 != $true)) & (ulamvar1 != $true) & (! [X139 : subst] : (((push @ (sub @ one @ X139) @ (comp @ sh @ X139))) = X139) | (axscons != $true)) & ((axscons = $true) | ? [X139 : subst] : (((push @ (sub @ one @ X139) @ (comp @ sh @ X139))) != X139)) & (pushprop_lem1_gthm = $true) & ((pushprop_lem0 = $true) | ? [X415 : subst,X416 : term,X417 : (term > $o)] : ! [X418 : (term > $o)] : ? [X419 : term] : (((X417 @ (sub @ X419 @ (push @ X416 @ X415)))) != ((X418 @ X419)))) & (! [X415 : subst,X416 : term,X417 : (term > $o)] : ? [X418 : (term > $o)] : ! [X419 : term] : (((X417 @ (sub @ X419 @ (push @ X416 @ X415)))) = ((X418 @ X419))) | (pushprop_lem0 != $true)) & ((hoasinduction_lem3b = $true) | ? [X213 : term] : ! [X214 : (subst > term > term)] : (((X214 @ sh @ one)) != ((sub @ X213 @ (push @ one @ sh))))) & (! [X213 : term] : ? [X214 : (subst > term > term)] : (((X214 @ sh @ one)) = ((sub @ X213 @ (push @ one @ sh)))) | (hoasinduction_lem3b != $true)) & (((d2term @ d_one)) = one) & (((d2subst @ d_id)) = sh) & ((sP1 = $true) | (hoasinduction_lem3v2_lthm != $true)) & ((hoasinduction_lem3v2_lthm = $true) | (sP1 != $true)) & (pushprop_gthm = $true) & (hoasinduction_lem1v2 = $true) & (induction2_gthm = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoasapinj1 = $true) | ? [X18 : term,X19 : term,X17 : term,X20 : term] : ((((ap @ (sub @ X20 @ id) @ X19)) = ((ap @ (sub @ X17 @ id) @ X18))) & (X17 != X20))) & (! [X18 : term,X19 : term,X17 : term,X20 : term] : ((((ap @ (sub @ X20 @ id) @ X19)) != ((ap @ (sub @ X17 @ id) @ X18))) | (X17 = X20)) | (hoasapinj1 != $true)) & (hoaslamnotap_lthm = $true) & (hoasapinj1_gthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (hoasinduction_lem1_gthm = $true) & (pushprop = $true) & ! [X15 : subst] : ? [X16 : d_subst] : (((d2subst @ X16)) = X15) & (! [X211 : term,X212 : term,X209 : term,X210 : term] : ((X210 = X212) | (((ap @ (sub @ X209 @ id) @ X212)) != ((ap @ (sub @ X211 @ id) @ X210)))) | (hoasapinj2 != $true)) & ((hoasapinj2 = $true) | ? [X211 : term,X212 : term,X209 : term,X210 : term] : ((X210 != X212) & (((ap @ (sub @ X209 @ id) @ X212)) = ((ap @ (sub @ X211 @ id) @ X210))))) & ((hoaslamnotap = $true) | ? [X49 : (subst > term > term),X48 : term,X50 : term] : ((((ap @ (sub @ X50 @ id) @ X48)) = ((lam @ (X49 @ sh @ one)))) & ! [X53 : term,X51 : subst,X52 : subst] : (((X49 @ (comp @ X51 @ X52) @ (sub @ X53 @ X52))) = ((sub @ (X49 @ X51 @ X53) @ X52))))) & (! [X49 : (subst > term > term),X48 : term,X50 : term] : ((((ap @ (sub @ X50 @ id) @ X48)) != ((lam @ (X49 @ sh @ one)))) | ? [X53 : term,X51 : subst,X52 : subst] : (((X49 @ (comp @ X51 @ X52) @ (sub @ X53 @ X52))) != ((sub @ (X49 @ X51 @ X53) @ X52)))) | (hoaslamnotap != $true)) & ((termmset = $true) | ? [X103 : term] : (((sub @ X103 @ id)) != X103)) & (! [X103 : term] : (((sub @ X103 @ id)) = X103) | (termmset != $true)) & ((laminj = $true) | ? [X395 : term,X396 : term] : ((X395 != X396) & (((lam @ X395)) = ((lam @ X396))))) & (! [X395 : term,X396 : term] : ((X395 = X396) | (((lam @ X395)) != ((lam @ X396)))) | (laminj != $true)) & (hoasinduction_lem1v2_gthm = $true)),
  inference(flattening,[],[f260])).
thf(f262,plain,(
  (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)))))))))))) & (axabs = $true) & (($true = sP58) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | ($true != sP58)) & ((pushprop_lem0_lthm = $true) | ? [X0 : subst,X1 : term,X2 : (term > $o)] : ! [X3 : (term > $o)] : ? [X4 : term] : (((X3 @ X4)) != ((X2 @ (sub @ X4 @ (push @ X1 @ X0)))))) & (! [X5 : subst,X6 : term,X7 : (term > $o)] : ? [X8 : (term > $o)] : ! [X9 : term] : (((X7 @ (sub @ X9 @ (push @ X6 @ X5)))) = ((X8 @ X9))) | (pushprop_lem0_lthm != $true)) & (pushprop_lthm_orig = $true) & ((axshiftcons = $true) | ? [X10 : subst,X11 : term] : (((comp @ sh @ (push @ X11 @ X10))) != X10)) & (! [X12 : subst,X13 : term] : (((comp @ sh @ (push @ X13 @ X12))) = X12) | (axshiftcons != $true)) & (pushprop_lem2v2 = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ((induction = $true) | ? [X14 : (term > $o),X15 : term] : (! [X16 : term] : ((((X14 @ (lam @ X16))) = $true) | (((X14 @ X16)) != $true)) & (((X14 @ X15)) != $true) & (((sP57 @ X14)) != $true) & ! [X17 : term] : ((((var @ X17)) != $true) | (((X14 @ X17)) = $true)))) & (! [X18 : (term > $o),X19 : term] : (? [X20 : term] : (($true != ((X18 @ (lam @ X20)))) & (((X18 @ X20)) = $true)) | (((X18 @ X19)) = $true) | (((sP57 @ X18)) = $true) | ? [X21 : term] : (($true = ((var @ X21))) & (((X18 @ X21)) != $true))) | (induction != $true)) & ((hoasinduction_no_psi_cond_lthm = $true) | (sP56 != $true)) & ((sP56 = $true) | (hoasinduction_no_psi_cond_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (? [X22 : term,X23 : subst,X24 : (term > $o),X25 : (term > $o)] : (! [X26 : term] : (((X24 @ (sub @ X26 @ (push @ X22 @ X23)))) = ((X25 @ X26))) & (((X24 @ X22)) = $true) & ($true != ((X25 @ one)))) & ! [X27 : subst,X28 : term] : (((sub @ one @ (push @ X28 @ X27))) = X28))) & (! [X29 : term,X30 : subst,X31 : (term > $o),X32 : (term > $o)] : (? [X33 : term] : (((X31 @ (sub @ X33 @ (push @ X29 @ X30)))) != ((X32 @ X33))) | (((X31 @ X29)) != $true) | (((X32 @ one)) = $true)) | ? [X34 : subst,X35 : term] : (((sub @ one @ (push @ X35 @ X34))) != X35) | (pushprop_lem1v2_lthm != $true)) & ((hoasinduction_lem3 = $true) | (sP51 != $true)) & ((sP51 = $true) | (hoasinduction_lem3 != $true)) & (! [X36 : term] : (((sub @ X36 @ id)) = X36) | (axvarid != $true)) & ((axvarid = $true) | ? [X37 : term] : (((sub @ X37 @ id)) != X37)) & (axapp = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (pushprop_lem2v2_gthm = $true) & ((sP49 = $true) | (hoasinduction_lem3_lthm != $true)) & ((hoasinduction_lem3_lthm = $true) | (sP49 != $true)) & (axclos = $true) & (hoasinduction_lem3v2_gthm = $true) & (! [X38 : subst] : (((comp @ X38 @ id)) = X38) | (axidr != $true)) & ((axidr = $true) | ? [X39 : subst] : (((comp @ X39 @ id)) != X39)) & ((axidl = $true) | ? [X40 : subst] : (((comp @ id @ X40)) != X40)) & (! [X41 : subst] : (((comp @ id @ X41)) = X41) | (axidl != $true)) & (hoasinduction_lem3_gthm = $true) & (pushprop_lem3v2 = $true) & (! [X42 : (subst > term > term),X43 : term,X44 : (subst > term > term),X45 : subst] : (? [X46 : term,X47 : subst,X48 : subst] : (((sub @ (X42 @ X48 @ X46) @ X47)) != ((X42 @ (comp @ X48 @ X47) @ (sub @ X46 @ X47)))) | ? [X49 : subst,X50 : subst,X51 : term] : (((sub @ (X44 @ X50 @ X51) @ X49)) != ((X44 @ (comp @ X50 @ X49) @ (sub @ X51 @ X49)))) | (((lam @ (X42 @ sh @ one))) != ((lam @ (X44 @ sh @ one)))) | (((X42 @ X45 @ X43)) = ((X44 @ X45 @ X43)))) | (hoaslaminj != $true)) & ((hoaslaminj = $true) | ? [X52 : (subst > term > term),X53 : term,X54 : (subst > term > term),X55 : subst] : (! [X56 : term,X57 : subst,X58 : subst] : (((X52 @ (comp @ X58 @ X57) @ (sub @ X56 @ X57))) = ((sub @ (X52 @ X58 @ X56) @ X57))) & ! [X59 : subst,X60 : subst,X61 : term] : (((sub @ (X54 @ X60 @ X61) @ X59)) = ((X54 @ (comp @ X60 @ X59) @ (sub @ X61 @ X59)))) & (((lam @ (X54 @ sh @ one))) = ((lam @ (X52 @ sh @ one)))) & (((X54 @ X55 @ X53)) != ((X52 @ X55 @ X53))))) & (hoasinduction_lem3a_gthm = $true) & (hoasapinj2_gthm = $true) & (hoasapnotvar_gthm = $true) & (pushprop_lem3v2_lthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & ((sP46 = $true) | (hoasinduction_lem3aa != $true)) & ((hoasinduction_lem3aa = $true) | (sP46 != $true)) & ((sP45 = $true) | (induction2 != $true)) & ((induction2 = $true) | (sP45 != $true)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoasinduction_no_psi_cond = $true) | ? [X62 : (subst > term > subst > $o),X63 : term] : (! [X64 : term,X65 : term] : (($true != ((X62 @ id @ X65 @ id))) | (((X62 @ id @ (ap @ (sub @ X64 @ id) @ X65) @ id)) = $true) | ($true != ((X62 @ id @ X64 @ id)))) & (((X62 @ id @ X63 @ id)) != $true) & (((sP43 @ X62)) != $true))) & (! [X66 : (subst > term > subst > $o),X67 : term] : (? [X68 : term,X69 : term] : ((((X66 @ id @ X69 @ id)) = $true) & (((X66 @ id @ (ap @ (sub @ X68 @ id) @ X69) @ id)) != $true) & (((X66 @ id @ X68 @ id)) = $true)) | (((X66 @ id @ X67 @ id)) = $true) | (((sP43 @ X66)) = $true)) | (hoasinduction_no_psi_cond != $true)) & ((! [X70 : subst] : (((comp @ id @ X70)) = X70) & ! [X71 : subst] : (((comp @ X71 @ id)) = X71)) | (substmonoid != $true)) & ((substmonoid = $true) | ? [X72 : subst] : (((comp @ id @ X72)) != X72) | ? [X73 : subst] : (((comp @ X73 @ id)) != X73)) & ((sP42 = $true) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | (sP42 != $true)) & (hoasinduction_lem2_gthm = $true) & (pushprop_lem1v2_gthm = $true) & (termmset_gthm = $true) & ((apinj2 = $true) | ? [X74 : term,X75 : term,X76 : term,X77 : term] : ((X74 != X77) & (((ap @ X75 @ X77)) = ((ap @ X76 @ X74))))) & (! [X78 : term,X79 : term,X80 : term,X81 : term] : ((X78 = X81) | (((ap @ X79 @ X81)) != ((ap @ X80 @ X78)))) | (apinj2 != $true)) & ((sP38 = $true) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | (sP38 != $true)) & ((sP34 = $true) | (hoasinduction_lem3aaa != $true)) & ((hoasinduction_lem3aaa = $true) | (sP34 != $true)) & (? [X82 : term,X83 : term,X84 : term,X85 : term] : ((((ap @ X84 @ X82)) = ((ap @ X85 @ X83))) & (X84 != X85)) | ? [X86 : term] : (((sub @ X86 @ id)) != X86) | ! [X87 : term,X88 : term,X89 : term,X90 : term] : ((X87 = X88) | (((ap @ (sub @ X88 @ id) @ X90)) != ((ap @ (sub @ X87 @ id) @ X89)))) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | (! [X91 : term,X92 : term,X93 : term,X94 : term] : ((((ap @ X94 @ X92)) != ((ap @ X93 @ X91))) | (X93 = X94)) & ! [X95 : term] : (((sub @ X95 @ id)) = X95) & ? [X96 : term,X97 : term,X98 : term,X99 : term] : ((X96 != X97) & (((ap @ (sub @ X97 @ id) @ X99)) = ((ap @ (sub @ X96 @ id) @ X98)))))) & (ulamvarind = $true) & ((hoaslaminj_lthm = $true) | (sP33 != $true)) & ((sP33 = $true) | (hoaslaminj_lthm != $true)) & (! [X100 : term,X101 : term] : ((((sub @ X100 @ sh)) != ((sub @ X101 @ sh))) | (X100 = X101)) | (shinj != $true)) & ((shinj = $true) | ? [X102 : term,X103 : term] : ((((sub @ X102 @ sh)) = ((sub @ X103 @ sh))) & (X102 != X103))) & ! [X104 : d_term,X105 : d_term] : ((X104 = X105) | (((d2term @ X104)) != ((d2term @ X105)))) & (lamnotvar = $true) & (((var @ (d2term @ d_one))) != $true) & (hoaslaminj_gthm = $true) & (($true = sP32) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | ($true != sP32)) & (hoasinduction_lem1_lthm = $true) & ! [X106 : d_subst,X107 : d_subst] : ((((d2subst @ X106)) != ((d2subst @ X107))) | (X106 = X107)) & (hoasinduction_lem1 = $true) & ((sP30 = $true) | (hoasinduction_lem3aa_lthm != $true)) & ((hoasinduction_lem3aa_lthm = $true) | (sP30 != $true)) & ((hoasinduction_lthm_3 = $true) | (sP29 != $true)) & ((sP29 = $true) | (hoasinduction_lthm_3 != $true)) & (! [X108 : (subst > term > subst > $o)] : ? [X109 : (term > $o)] : ! [X110 : term] : (((X109 @ X110)) = ((X108 @ id @ X110 @ id))) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem0 = $true) | ? [X111 : (subst > term > subst > $o)] : ! [X112 : (term > $o)] : ? [X113 : term] : (((X112 @ X113)) != ((X111 @ id @ X113 @ id)))) & (hoasinduction_lem2v2_gthm = $true) & (hoaslamnotap_gthm = $true) & (hoasinduction_lem3b_gthm = $true) & (axmap = $true) & ((pushprop_lem1_lthm = $true) | (! [X114 : term,X115 : subst] : (((comp @ sh @ (push @ X114 @ X115))) = X115) & ! [X116 : subst,X117 : term] : (((sub @ one @ (push @ X117 @ X116))) = X117) & ? [X118 : term,X119 : (term > $o),X120 : subst,X121 : term,X122 : (term > $o)] : ((((X122 @ X118)) = $true) & (((X119 @ (sub @ X118 @ (push @ X121 @ X120)))) != $true)))) & (? [X123 : term,X124 : subst] : (((comp @ sh @ (push @ X123 @ X124))) != X124) | ? [X125 : subst,X126 : term] : (((sub @ one @ (push @ X126 @ X125))) != X126) | ! [X127 : term,X128 : (term > $o),X129 : subst,X130 : term,X131 : (term > $o)] : ((((X131 @ X127)) != $true) | (((X128 @ (sub @ X127 @ (push @ X130 @ X129)))) = $true)) | (pushprop_lem1_lthm != $true)) & (axassoc = $true) & (axvarshift = $true) & (lamnotap != $true) & (hoaslamnotvar_lthm = $true) & ((sP23 = $true) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | (sP23 != $true)) & (ulamvarsh = $true) & (hoasinduction_gthm = $true) & ((sP22 = $true) | (hoasinduction_lem3v2a_lthm != $true)) & ((hoasinduction_lem3v2a_lthm = $true) | (sP22 != $true)) & ((sP21 = $true) | (hoasinduction_lthm != $true)) & ((hoasinduction_lthm = $true) | (sP21 != $true)) & (((d2subst @ d_id)) = id) & ((sP12 = $true) | (hoasinduction_lem2v2 != $true)) & ((hoasinduction_lem2v2 = $true) | (sP12 != $true)) & (pushprop_lem2v2_lthm = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (! [X132 : term,X133 : (term > $o),X134 : subst,X135 : (term > $o)] : ((((X133 @ one)) = $true) | (((X135 @ X132)) != $true) | ? [X136 : term] : (((X133 @ X136)) != ((X135 @ (sub @ X136 @ (push @ X132 @ X134)))))) | (pushprop_lem1v2 != $true)) & ((pushprop_lem1v2 = $true) | ? [X137 : term,X138 : (term > $o),X139 : subst,X140 : (term > $o)] : ((((X138 @ one)) != $true) & (((X140 @ X137)) = $true) & ! [X141 : term] : (((X140 @ (sub @ X141 @ (push @ X137 @ X139)))) = ((X138 @ X141))))) & (? [X142 : term] : (((sub @ X142 @ id)) != X142) | ! [X143 : term] : (((sub @ X143 @ id)) = X143) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | (! [X144 : term] : (((sub @ X144 @ id)) = X144) & ? [X145 : term] : (((sub @ X145 @ id)) != X145))) & (pushprop_lem0_gthm = $true) & (hoasapnotvar_lthm = $true) & (! [X146 : term] : ? [X147 : (subst > term > term)] : (((sub @ X146 @ (push @ one @ sh))) = ((X147 @ sh @ one))) | (hoasinduction_lem3b_lthm != $true)) & ((hoasinduction_lem3b_lthm = $true) | ? [X148 : term] : ! [X149 : (subst > term > term)] : (((sub @ X148 @ (push @ one @ sh))) != ((X149 @ sh @ one)))) & (! [X150 : term] : ? [X151 : (subst > term > term)] : ! [X152 : subst,X153 : term] : (((sub @ X150 @ (push @ X153 @ X152))) = ((X151 @ X152 @ X153))) | (hoasinduction_lem3v2_f_lthm != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X154 : term] : ! [X155 : (subst > term > term)] : ? [X156 : subst,X157 : term] : (((sub @ X154 @ (push @ X157 @ X156))) != ((X155 @ X156 @ X157)))) & ! [X158 : d_subst] : (d_id = X158) & (hoasinduction_lem2_lthm = $true) & ((hoasinduction_lem3v2 = $true) | (sP10 != $true)) & ((sP10 = $true) | (hoasinduction_lem3v2 != $true)) & (hoasapnotvar = $true) & (apnotvar = $true) & (substmonoid_gthm = $true) & (hoaslamnotvar_gthm = $true) & (hoaslamnotvar = $true) & ((substmonoid_lthm = $true) | (! [X159 : subst] : (((comp @ id @ X159)) = X159) & ! [X160 : subst] : (((comp @ X160 @ id)) = X160) & (? [X161 : subst] : (((comp @ X161 @ id)) != X161) | ? [X162 : subst] : (((comp @ id @ X162)) != X162)))) & (? [X163 : subst] : (((comp @ id @ X163)) != X163) | ? [X164 : subst] : (((comp @ X164 @ id)) != X164) | (! [X165 : subst] : (((comp @ X165 @ id)) = X165) & ! [X166 : subst] : (((comp @ id @ X166)) = X166)) | (substmonoid_lthm != $true)) & (induction2lem_gthm = $true) & ((hoasinduction_lem0_lthm = $true) | ? [X167 : (subst > term > subst > $o)] : ! [X168 : (term > $o)] : ? [X169 : term] : (((X167 @ id @ X169 @ id)) != ((X168 @ X169)))) & (! [X170 : (subst > term > subst > $o)] : ? [X171 : (term > $o)] : ! [X172 : term] : (((X170 @ id @ X172 @ id)) = ((X171 @ X172))) | (hoasinduction_lem0_lthm != $true)) & ((sP8 = $true) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | (sP8 != $true)) & ((apinj1 = $true) | ? [X173 : term,X174 : term,X175 : term,X176 : term] : ((X175 != X176) & (((ap @ X176 @ X173)) = ((ap @ X175 @ X174))))) & (! [X177 : term,X178 : term,X179 : term,X180 : term] : ((X179 = X180) | (((ap @ X180 @ X177)) != ((ap @ X179 @ X178)))) | (apinj1 != $true)) & (pushprop_lthm = $true) & (! [X181 : term,X182 : term,X183 : term,X184 : term] : ((((ap @ (sub @ X181 @ id) @ X182)) != ((ap @ (sub @ X183 @ id) @ X184))) | (X182 = X184)) | ? [X185 : term,X186 : term,X187 : term,X188 : term] : ((X186 != X188) & (((ap @ X187 @ X188)) = ((ap @ X185 @ X186)))) | (hoasapinj2_lthm != $true)) & ((hoasapinj2_lthm = $true) | (? [X189 : term,X190 : term,X191 : term,X192 : term] : ((((ap @ (sub @ X189 @ id) @ X190)) = ((ap @ (sub @ X191 @ id) @ X192))) & (X190 != X192)) & ! [X193 : term,X194 : term,X195 : term,X196 : term] : ((X194 = X196) | (((ap @ X195 @ X196)) != ((ap @ X193 @ X194)))))) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (! [X197 : subst,X198 : term] : (((sub @ one @ (push @ X198 @ X197))) = X198) | (axvarcons != $true)) & ((axvarcons = $true) | ? [X199 : subst,X200 : term] : (((sub @ one @ (push @ X200 @ X199))) != X200)) & ! [X201 : term] : ? [X202 : d_term] : (((d2term @ X202)) = X201) & (! [X203 : term] : ? [X204 : (subst > term > term)] : ! [X205 : term,X206 : subst] : (((sub @ X203 @ (push @ X205 @ X206))) = ((X204 @ X206 @ X205))) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ? [X207 : term] : ! [X208 : (subst > term > term)] : ? [X209 : term,X210 : subst] : (((X208 @ X210 @ X209)) != ((sub @ X207 @ (push @ X209 @ X210))))) & ((induction2lem = $true) | (sP6 != $true)) & ((sP6 = $true) | (induction2lem != $true)) & ! [X211 : d_term] : (d_one = X211) & ((pushprop_lem1 = $true) | ? [X212 : term,X213 : (term > $o),X214 : term,X215 : (term > $o),X216 : subst] : ((((X215 @ X212)) = $true) & (((X213 @ (sub @ X212 @ (push @ X214 @ X216)))) != $true))) & (! [X217 : term,X218 : (term > $o),X219 : term,X220 : (term > $o),X221 : subst] : ((((X220 @ X217)) != $true) | (((X218 @ (sub @ X217 @ (push @ X219 @ X221)))) = $true)) | (pushprop_lem1 != $true)) & ((sP4 = $true) | (hoasinduction != $true)) & ((hoasinduction = $true) | (sP4 != $true)) & (ulamvar1 != $true) & (! [X222 : subst] : (((push @ (sub @ one @ X222) @ (comp @ sh @ X222))) = X222) | (axscons != $true)) & ((axscons = $true) | ? [X223 : subst] : (((push @ (sub @ one @ X223) @ (comp @ sh @ X223))) != X223)) & (pushprop_lem1_gthm = $true) & ((pushprop_lem0 = $true) | ? [X224 : subst,X225 : term,X226 : (term > $o)] : ! [X227 : (term > $o)] : ? [X228 : term] : (((X226 @ (sub @ X228 @ (push @ X225 @ X224)))) != ((X227 @ X228)))) & (! [X229 : subst,X230 : term,X231 : (term > $o)] : ? [X232 : (term > $o)] : ! [X233 : term] : (((X232 @ X233)) = ((X231 @ (sub @ X233 @ (push @ X230 @ X229))))) | (pushprop_lem0 != $true)) & ((hoasinduction_lem3b = $true) | ? [X234 : term] : ! [X235 : (subst > term > term)] : (((X235 @ sh @ one)) != ((sub @ X234 @ (push @ one @ sh))))) & (! [X236 : term] : ? [X237 : (subst > term > term)] : (((sub @ X236 @ (push @ one @ sh))) = ((X237 @ sh @ one))) | (hoasinduction_lem3b != $true)) & (((d2term @ d_one)) = one) & (((d2subst @ d_id)) = sh) & ((sP1 = $true) | (hoasinduction_lem3v2_lthm != $true)) & ((hoasinduction_lem3v2_lthm = $true) | (sP1 != $true)) & (pushprop_gthm = $true) & (hoasinduction_lem1v2 = $true) & (induction2_gthm = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoasapinj1 = $true) | ? [X238 : term,X239 : term,X240 : term,X241 : term] : ((((ap @ (sub @ X240 @ id) @ X238)) = ((ap @ (sub @ X241 @ id) @ X239))) & (X240 != X241))) & (! [X242 : term,X243 : term,X244 : term,X245 : term] : ((((ap @ (sub @ X245 @ id) @ X243)) != ((ap @ (sub @ X244 @ id) @ X242))) | (X244 = X245)) | (hoasapinj1 != $true)) & (hoaslamnotap_lthm = $true) & (hoasapinj1_gthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (hoasinduction_lem1_gthm = $true) & (pushprop = $true) & ! [X246 : subst] : ? [X247 : d_subst] : (((d2subst @ X247)) = X246) & (! [X248 : term,X249 : term,X250 : term,X251 : term] : ((X249 = X251) | (((ap @ (sub @ X248 @ id) @ X251)) != ((ap @ (sub @ X250 @ id) @ X249)))) | (hoasapinj2 != $true)) & ((hoasapinj2 = $true) | ? [X252 : term,X253 : term,X254 : term,X255 : term] : ((X253 != X255) & (((ap @ (sub @ X254 @ id) @ X253)) = ((ap @ (sub @ X252 @ id) @ X255))))) & ((hoaslamnotap = $true) | ? [X256 : (subst > term > term),X257 : term,X258 : term] : ((((ap @ (sub @ X258 @ id) @ X257)) = ((lam @ (X256 @ sh @ one)))) & ! [X259 : term,X260 : subst,X261 : subst] : (((sub @ (X256 @ X260 @ X259) @ X261)) = ((X256 @ (comp @ X260 @ X261) @ (sub @ X259 @ X261)))))) & (! [X262 : (subst > term > term),X263 : term,X264 : term] : ((((ap @ (sub @ X264 @ id) @ X263)) != ((lam @ (X262 @ sh @ one)))) | ? [X265 : term,X266 : subst,X267 : subst] : (((X262 @ (comp @ X266 @ X267) @ (sub @ X265 @ X267))) != ((sub @ (X262 @ X266 @ X265) @ X267)))) | (hoaslamnotap != $true)) & ((termmset = $true) | ? [X268 : term] : (((sub @ X268 @ id)) != X268)) & (! [X269 : term] : (((sub @ X269 @ id)) = X269) | (termmset != $true)) & ((laminj = $true) | ? [X270 : term,X271 : term] : ((X270 != X271) & (((lam @ X271)) = ((lam @ X270))))) & (! [X272 : term,X273 : term] : ((X272 = X273) | (((lam @ X272)) != ((lam @ X273)))) | (laminj != $true)) & (hoasinduction_lem1v2_gthm = $true)),
  inference(rectify,[],[f261])).
thf(f263,plain,(
  (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)))))))))))) & (axabs = $true) & (($true = sP58) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | ($true != sP58)) & ((pushprop_lem0_lthm = $true) | ! [X3 : (term > $o)] : (((sK459 @ (sub @ (sK460 @ X3) @ (push @ sK458 @ sK457)))) != ((X3 @ (sK460 @ X3))))) & (! [X5 : subst,X6 : term,X7 : (term > $o)] : ! [X9 : term] : (((X7 @ (sub @ X9 @ (push @ X6 @ X5)))) = ((sK461 @ X7 @ X6 @ X5 @ X9))) | (pushprop_lem0_lthm != $true)) & (pushprop_lthm_orig = $true) & ((axshiftcons = $true) | (sK462 != ((comp @ sh @ (push @ sK463 @ sK462))))) & (! [X12 : subst,X13 : term] : (((comp @ sh @ (push @ X13 @ X12))) = X12) | (axshiftcons != $true)) & (pushprop_lem2v2 = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ((induction = $true) | (! [X16 : term] : ((((sK464 @ (lam @ X16))) = $true) | (((sK464 @ X16)) != $true)) & (((sK464 @ sK465)) != $true) & (((sP57 @ sK464)) != $true) & ! [X17 : term] : ((((var @ X17)) != $true) | (((sK464 @ X17)) = $true)))) & (! [X18 : (term > $o),X19 : term] : (((((X18 @ (lam @ (sK466 @ X18)))) != $true) & (((X18 @ (sK466 @ X18))) = $true)) | (((X18 @ X19)) = $true) | (((sP57 @ X18)) = $true) | ((((var @ (sK467 @ X18))) = $true) & (((X18 @ (sK467 @ X18))) != $true))) | (induction != $true)) & ((hoasinduction_no_psi_cond_lthm = $true) | (sP56 != $true)) & ((sP56 = $true) | (hoasinduction_no_psi_cond_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | ((! [X26 : term] : (((sK470 @ (sub @ X26 @ (push @ sK468 @ sK469)))) = ((sK471 @ X26))) & (((sK470 @ sK468)) = $true) & (((sK471 @ one)) != $true)) & ! [X27 : subst,X28 : term] : (((sub @ one @ (push @ X28 @ X27))) = X28))) & (! [X29 : term,X30 : subst,X31 : (term > $o),X32 : (term > $o)] : ((((X32 @ (sK472 @ X32 @ X31 @ X30 @ X29))) != ((X31 @ (sub @ (sK472 @ X32 @ X31 @ X30 @ X29) @ (push @ X29 @ X30))))) | (((X31 @ X29)) != $true) | (((X32 @ one)) = $true)) | (((sub @ one @ (push @ sK474 @ sK473))) != sK474) | (pushprop_lem1v2_lthm != $true)) & ((hoasinduction_lem3 = $true) | (sP51 != $true)) & ((sP51 = $true) | (hoasinduction_lem3 != $true)) & (! [X36 : term] : (((sub @ X36 @ id)) = X36) | (axvarid != $true)) & ((axvarid = $true) | (sK475 != ((sub @ sK475 @ id)))) & (axapp = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (pushprop_lem2v2_gthm = $true) & ((sP49 = $true) | (hoasinduction_lem3_lthm != $true)) & ((hoasinduction_lem3_lthm = $true) | (sP49 != $true)) & (axclos = $true) & (hoasinduction_lem3v2_gthm = $true) & (! [X38 : subst] : (((comp @ X38 @ id)) = X38) | (axidr != $true)) & ((axidr = $true) | (sK476 != ((comp @ sK476 @ id)))) & ((axidl = $true) | (sK477 != ((comp @ id @ sK477)))) & (! [X41 : subst] : (((comp @ id @ X41)) = X41) | (axidl != $true)) & (hoasinduction_lem3_gthm = $true) & (pushprop_lem3v2 = $true) & (! [X42 : (subst > term > term),X43 : term,X44 : (subst > term > term),X45 : subst] : ((((sub @ (X42 @ (sK480 @ X42) @ (sK478 @ X42)) @ (sK479 @ X42))) != ((X42 @ (comp @ (sK480 @ X42) @ (sK479 @ X42)) @ (sub @ (sK478 @ X42) @ (sK479 @ X42))))) | (((sub @ (X44 @ (sK482 @ X44) @ (sK483 @ X44)) @ (sK481 @ X44))) != ((X44 @ (comp @ (sK482 @ X44) @ (sK481 @ X44)) @ (sub @ (sK483 @ X44) @ (sK481 @ X44))))) | (((lam @ (X42 @ sh @ one))) != ((lam @ (X44 @ sh @ one)))) | (((X42 @ X45 @ X43)) = ((X44 @ X45 @ X43)))) | (hoaslaminj != $true)) & ((hoaslaminj = $true) | (! [X56 : term,X57 : subst,X58 : subst] : (((sK484 @ (comp @ X58 @ X57) @ (sub @ X56 @ X57))) = ((sub @ (sK484 @ X58 @ X56) @ X57))) & ! [X59 : subst,X60 : subst,X61 : term] : (((sub @ (sK486 @ X60 @ X61) @ X59)) = ((sK486 @ (comp @ X60 @ X59) @ (sub @ X61 @ X59)))) & (((lam @ (sK486 @ sh @ one))) = ((lam @ (sK484 @ sh @ one)))) & (((sK486 @ sK487 @ sK485)) != ((sK484 @ sK487 @ sK485))))) & (hoasinduction_lem3a_gthm = $true) & (hoasapinj2_gthm = $true) & (hoasapnotvar_gthm = $true) & (pushprop_lem3v2_lthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & ((sP46 = $true) | (hoasinduction_lem3aa != $true)) & ((hoasinduction_lem3aa = $true) | (sP46 != $true)) & ((sP45 = $true) | (induction2 != $true)) & ((induction2 = $true) | (sP45 != $true)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoasinduction_no_psi_cond = $true) | (! [X64 : term,X65 : term] : ((((sK488 @ id @ X65 @ id)) != $true) | (((sK488 @ id @ (ap @ (sub @ X64 @ id) @ X65) @ id)) = $true) | (((sK488 @ id @ X64 @ id)) != $true)) & (((sK488 @ id @ sK489 @ id)) != $true) & (((sP43 @ sK488)) != $true))) & (! [X66 : (subst > term > subst > $o),X67 : term] : (((((X66 @ id @ (sK491 @ X66) @ id)) = $true) & (((X66 @ id @ (ap @ (sub @ (sK490 @ X66) @ id) @ (sK491 @ X66)) @ id)) != $true) & ($true = ((X66 @ id @ (sK490 @ X66) @ id)))) | (((X66 @ id @ X67 @ id)) = $true) | (((sP43 @ X66)) = $true)) | (hoasinduction_no_psi_cond != $true)) & ((! [X70 : subst] : (((comp @ id @ X70)) = X70) & ! [X71 : subst] : (((comp @ X71 @ id)) = X71)) | (substmonoid != $true)) & ((substmonoid = $true) | (sK492 != ((comp @ id @ sK492))) | (sK493 != ((comp @ sK493 @ id)))) & ((sP42 = $true) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | (sP42 != $true)) & (hoasinduction_lem2_gthm = $true) & (pushprop_lem1v2_gthm = $true) & (termmset_gthm = $true) & ((apinj2 = $true) | ((sK497 != sK494) & (((ap @ sK495 @ sK497)) = ((ap @ sK496 @ sK494))))) & (! [X78 : term,X79 : term,X80 : term,X81 : term] : ((X78 = X81) | (((ap @ X79 @ X81)) != ((ap @ X80 @ X78)))) | (apinj2 != $true)) & ((sP38 = $true) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | (sP38 != $true)) & ((sP34 = $true) | (hoasinduction_lem3aaa != $true)) & ((hoasinduction_lem3aaa = $true) | (sP34 != $true)) & (((((ap @ sK500 @ sK498)) = ((ap @ sK501 @ sK499))) & (sK500 != sK501)) | (((sub @ sK502 @ id)) != sK502) | ! [X87 : term,X88 : term,X89 : term,X90 : term] : ((X87 = X88) | (((ap @ (sub @ X88 @ id) @ X90)) != ((ap @ (sub @ X87 @ id) @ X89)))) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | (! [X91 : term,X92 : term,X93 : term,X94 : term] : ((((ap @ X94 @ X92)) != ((ap @ X93 @ X91))) | (X93 = X94)) & ! [X95 : term] : (((sub @ X95 @ id)) = X95) & ((sK504 != sK503) & (((ap @ (sub @ sK503 @ id) @ sK505)) = ((ap @ (sub @ sK504 @ id) @ sK506)))))) & (ulamvarind = $true) & ((hoaslaminj_lthm = $true) | (sP33 != $true)) & ((sP33 = $true) | (hoaslaminj_lthm != $true)) & (! [X100 : term,X101 : term] : ((((sub @ X100 @ sh)) != ((sub @ X101 @ sh))) | (X100 = X101)) | (shinj != $true)) & ((shinj = $true) | ((((sub @ sK508 @ sh)) = ((sub @ sK507 @ sh))) & (sK507 != sK508))) & ! [X104 : d_term,X105 : d_term] : ((X104 = X105) | (((d2term @ X104)) != ((d2term @ X105)))) & (lamnotvar = $true) & (((var @ (d2term @ d_one))) != $true) & (hoaslaminj_gthm = $true) & (($true = sP32) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | ($true != sP32)) & (hoasinduction_lem1_lthm = $true) & ! [X106 : d_subst,X107 : d_subst] : ((((d2subst @ X106)) != ((d2subst @ X107))) | (X106 = X107)) & (hoasinduction_lem1 = $true) & ((sP30 = $true) | (hoasinduction_lem3aa_lthm != $true)) & ((hoasinduction_lem3aa_lthm = $true) | (sP30 != $true)) & ((hoasinduction_lthm_3 = $true) | (sP29 != $true)) & ((sP29 = $true) | (hoasinduction_lthm_3 != $true)) & (! [X108 : (subst > term > subst > $o)] : ! [X110 : term] : (((X108 @ id @ X110 @ id)) = ((sK509 @ X108 @ X110))) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem0 = $true) | ! [X112 : (term > $o)] : (((X112 @ (sK511 @ X112))) != ((sK510 @ id @ (sK511 @ X112) @ id)))) & (hoasinduction_lem2v2_gthm = $true) & (hoaslamnotap_gthm = $true) & (hoasinduction_lem3b_gthm = $true) & (axmap = $true) & ((pushprop_lem1_lthm = $true) | (! [X114 : term,X115 : subst] : (((comp @ sh @ (push @ X114 @ X115))) = X115) & ! [X116 : subst,X117 : term] : (((sub @ one @ (push @ X117 @ X116))) = X117) & ((((sK516 @ sK512)) = $true) & (((sK513 @ (sub @ sK512 @ (push @ sK515 @ sK514)))) != $true)))) & ((sK518 != ((comp @ sh @ (push @ sK517 @ sK518)))) | (sK520 != ((sub @ one @ (push @ sK520 @ sK519)))) | ! [X127 : term,X128 : (term > $o),X129 : subst,X130 : term,X131 : (term > $o)] : ((((X131 @ X127)) != $true) | (((X128 @ (sub @ X127 @ (push @ X130 @ X129)))) = $true)) | (pushprop_lem1_lthm != $true)) & (axassoc = $true) & (axvarshift = $true) & (lamnotap != $true) & (hoaslamnotvar_lthm = $true) & ((sP23 = $true) | (hoasinduction_lem3a != $true)) & ((hoasinduction_lem3a = $true) | (sP23 != $true)) & (ulamvarsh = $true) & (hoasinduction_gthm = $true) & ((sP22 = $true) | (hoasinduction_lem3v2a_lthm != $true)) & ((hoasinduction_lem3v2a_lthm = $true) | (sP22 != $true)) & ((sP21 = $true) | (hoasinduction_lthm != $true)) & ((hoasinduction_lthm = $true) | (sP21 != $true)) & (((d2subst @ d_id)) = id) & ((sP12 = $true) | (hoasinduction_lem2v2 != $true)) & ((hoasinduction_lem2v2 = $true) | (sP12 != $true)) & (pushprop_lem2v2_lthm = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (! [X132 : term,X133 : (term > $o),X134 : subst,X135 : (term > $o)] : ((((X133 @ one)) = $true) | (((X135 @ X132)) != $true) | (((X135 @ (sub @ (sK521 @ X135 @ X134 @ X133 @ X132) @ (push @ X132 @ X134)))) != ((X133 @ (sK521 @ X135 @ X134 @ X133 @ X132))))) | (pushprop_lem1v2 != $true)) & ((pushprop_lem1v2 = $true) | ((((sK523 @ one)) != $true) & (((sK525 @ sK522)) = $true) & ! [X141 : term] : (((sK525 @ (sub @ X141 @ (push @ sK522 @ sK524)))) = ((sK523 @ X141))))) & ((sK526 != ((sub @ sK526 @ id))) | ! [X143 : term] : (((sub @ X143 @ id)) = X143) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | (! [X144 : term] : (((sub @ X144 @ id)) = X144) & (((sub @ sK527 @ id)) != sK527))) & (pushprop_lem0_gthm = $true) & (hoasapnotvar_lthm = $true) & (! [X146 : term] : (((sK528 @ X146 @ sh @ one)) = ((sub @ X146 @ (push @ one @ sh)))) | (hoasinduction_lem3b_lthm != $true)) & ((hoasinduction_lem3b_lthm = $true) | ! [X149 : (subst > term > term)] : (((X149 @ sh @ one)) != ((sub @ sK529 @ (push @ one @ sh))))) & (! [X150 : term] : ! [X152 : subst,X153 : term] : (((sK530 @ X150 @ X152 @ X153)) = ((sub @ X150 @ (push @ X153 @ X152)))) | (hoasinduction_lem3v2_f_lthm != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ! [X155 : (subst > term > term)] : (((X155 @ (sK532 @ X155) @ (sK533 @ X155))) != ((sub @ sK531 @ (push @ (sK533 @ X155) @ (sK532 @ X155)))))) & ! [X158 : d_subst] : (d_id = X158) & (hoasinduction_lem2_lthm = $true) & ((hoasinduction_lem3v2 = $true) | (sP10 != $true)) & ((sP10 = $true) | (hoasinduction_lem3v2 != $true)) & (hoasapnotvar = $true) & (apnotvar = $true) & (substmonoid_gthm = $true) & (hoaslamnotvar_gthm = $true) & (hoaslamnotvar = $true) & ((substmonoid_lthm = $true) | (! [X159 : subst] : (((comp @ id @ X159)) = X159) & ! [X160 : subst] : (((comp @ X160 @ id)) = X160) & ((sK534 != ((comp @ sK534 @ id))) | (sK535 != ((comp @ id @ sK535)))))) & ((((comp @ id @ sK536)) != sK536) | (((comp @ sK537 @ id)) != sK537) | (! [X165 : subst] : (((comp @ X165 @ id)) = X165) & ! [X166 : subst] : (((comp @ id @ X166)) = X166)) | (substmonoid_lthm != $true)) & (induction2lem_gthm = $true) & ((hoasinduction_lem0_lthm = $true) | ! [X168 : (term > $o)] : (((sK538 @ id @ (sK539 @ X168) @ id)) != ((X168 @ (sK539 @ X168))))) & (! [X170 : (subst > term > subst > $o)] : ! [X172 : term] : (((sK540 @ X170 @ X172)) = ((X170 @ id @ X172 @ id))) | (hoasinduction_lem0_lthm != $true)) & ((sP8 = $true) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | (sP8 != $true)) & ((apinj1 = $true) | ((sK543 != sK544) & (((ap @ sK544 @ sK541)) = ((ap @ sK543 @ sK542))))) & (! [X177 : term,X178 : term,X179 : term,X180 : term] : ((X179 = X180) | (((ap @ X180 @ X177)) != ((ap @ X179 @ X178)))) | (apinj1 != $true)) & (pushprop_lthm = $true) & (! [X181 : term,X182 : term,X183 : term,X184 : term] : ((((ap @ (sub @ X181 @ id) @ X182)) != ((ap @ (sub @ X183 @ id) @ X184))) | (X182 = X184)) | ((sK548 != sK546) & (((ap @ sK547 @ sK548)) = ((ap @ sK545 @ sK546)))) | (hoasapinj2_lthm != $true)) & ((hoasapinj2_lthm = $true) | (((((ap @ (sub @ sK549 @ id) @ sK550)) = ((ap @ (sub @ sK551 @ id) @ sK552))) & (sK552 != sK550)) & ! [X193 : term,X194 : term,X195 : term,X196 : term] : ((X194 = X196) | (((ap @ X195 @ X196)) != ((ap @ X193 @ X194)))))) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (! [X197 : subst,X198 : term] : (((sub @ one @ (push @ X198 @ X197))) = X198) | (axvarcons != $true)) & ((axvarcons = $true) | (sK554 != ((sub @ one @ (push @ sK554 @ sK553))))) & ! [X201 : term] : (((d2term @ (sK555 @ X201))) = X201) & (! [X203 : term] : ! [X205 : term,X206 : subst] : (((sub @ X203 @ (push @ X205 @ X206))) = ((sK556 @ X203 @ X206 @ X205))) | (hoasinduction_lem3v2_f != $true)) & ((hoasinduction_lem3v2_f = $true) | ! [X208 : (subst > term > term)] : (((sub @ sK557 @ (push @ (sK558 @ X208) @ (sK559 @ X208)))) != ((X208 @ (sK559 @ X208) @ (sK558 @ X208))))) & ((induction2lem = $true) | (sP6 != $true)) & ((sP6 = $true) | (induction2lem != $true)) & ! [X211 : d_term] : (d_one = X211) & ((pushprop_lem1 = $true) | ((((sK563 @ sK560)) = $true) & (((sK561 @ (sub @ sK560 @ (push @ sK562 @ sK564)))) != $true))) & (! [X217 : term,X218 : (term > $o),X219 : term,X220 : (term > $o),X221 : subst] : ((((X220 @ X217)) != $true) | (((X218 @ (sub @ X217 @ (push @ X219 @ X221)))) = $true)) | (pushprop_lem1 != $true)) & ((sP4 = $true) | (hoasinduction != $true)) & ((hoasinduction = $true) | (sP4 != $true)) & (ulamvar1 != $true) & (! [X222 : subst] : (((push @ (sub @ one @ X222) @ (comp @ sh @ X222))) = X222) | (axscons != $true)) & ((axscons = $true) | (sK565 != ((push @ (sub @ one @ sK565) @ (comp @ sh @ sK565))))) & (pushprop_lem1_gthm = $true) & ((pushprop_lem0 = $true) | ! [X227 : (term > $o)] : (((sK568 @ (sub @ (sK569 @ X227) @ (push @ sK567 @ sK566)))) != ((X227 @ (sK569 @ X227))))) & (! [X229 : subst,X230 : term,X231 : (term > $o)] : ! [X233 : term] : (((sK570 @ X231 @ X230 @ X229 @ X233)) = ((X231 @ (sub @ X233 @ (push @ X230 @ X229))))) | (pushprop_lem0 != $true)) & ((hoasinduction_lem3b = $true) | ! [X235 : (subst > term > term)] : (((X235 @ sh @ one)) != ((sub @ sK571 @ (push @ one @ sh))))) & (! [X236 : term] : (((sK572 @ X236 @ sh @ one)) = ((sub @ X236 @ (push @ one @ sh)))) | (hoasinduction_lem3b != $true)) & (((d2term @ d_one)) = one) & (((d2subst @ d_id)) = sh) & ((sP1 = $true) | (hoasinduction_lem3v2_lthm != $true)) & ((hoasinduction_lem3v2_lthm = $true) | (sP1 != $true)) & (pushprop_gthm = $true) & (hoasinduction_lem1v2 = $true) & (induction2_gthm = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoasapinj1 = $true) | ((((ap @ (sub @ sK575 @ id) @ sK573)) = ((ap @ (sub @ sK576 @ id) @ sK574))) & (sK575 != sK576))) & (! [X242 : term,X243 : term,X244 : term,X245 : term] : ((((ap @ (sub @ X245 @ id) @ X243)) != ((ap @ (sub @ X244 @ id) @ X242))) | (X244 = X245)) | (hoasapinj1 != $true)) & (hoaslamnotap_lthm = $true) & (hoasapinj1_gthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (hoasinduction_lem1_gthm = $true) & (pushprop = $true) & ! [X246 : subst] : (((d2subst @ (sK577 @ X246))) = X246) & (! [X248 : term,X249 : term,X250 : term,X251 : term] : ((X249 = X251) | (((ap @ (sub @ X248 @ id) @ X251)) != ((ap @ (sub @ X250 @ id) @ X249)))) | (hoasapinj2 != $true)) & ((hoasapinj2 = $true) | ((sK579 != sK581) & (((ap @ (sub @ sK580 @ id) @ sK579)) = ((ap @ (sub @ sK578 @ id) @ sK581))))) & ((hoaslamnotap = $true) | ((((ap @ (sub @ sK584 @ id) @ sK583)) = ((lam @ (sK582 @ sh @ one)))) & ! [X259 : term,X260 : subst,X261 : subst] : (((sub @ (sK582 @ X260 @ X259) @ X261)) = ((sK582 @ (comp @ X260 @ X261) @ (sub @ X259 @ X261)))))) & (! [X262 : (subst > term > term),X263 : term,X264 : term] : ((((ap @ (sub @ X264 @ id) @ X263)) != ((lam @ (X262 @ sh @ one)))) | (((sub @ (X262 @ (sK586 @ X262) @ (sK585 @ X262)) @ (sK587 @ X262))) != ((X262 @ (comp @ (sK586 @ X262) @ (sK587 @ X262)) @ (sub @ (sK585 @ X262) @ (sK587 @ X262)))))) | (hoaslamnotap != $true)) & ((termmset = $true) | (sK588 != ((sub @ sK588 @ id)))) & (! [X269 : term] : (((sub @ X269 @ id)) = X269) | (termmset != $true)) & ((laminj = $true) | ((sK590 != sK589) & (((lam @ sK590)) = ((lam @ sK589))))) & (! [X272 : term,X273 : term] : ((X272 = X273) | (((lam @ X272)) != ((lam @ X273)))) | (laminj != $true)) & (hoasinduction_lem1v2_gthm = $true)),
  inference(skolemize,[status(esa),new_symbols(skolem,[sK457,sK458,sK459,vAPP,sK462,sK463,sK464,sK465,vAPP,sK468,sK469,sK470,sK471,vAPP,sK473,sK474,sK475,sK476,sK477,vAPP,sK484,sK485,sK486,sK487,sK488,sK489,vAPP,sK492,sK493,sK494,sK495,sK496,sK497,sK498,sK499,sK500,sK501,sK502,sK503,sK504,sK505,sK506,sK507,sK508,vAPP,sK510,vAPP,sK512,sK513,sK514,sK515,sK516,sK517,sK518,sK519,sK520,vAPP,sK522,sK523,sK524,sK525,sK526,sK527,vAPP,sK529,vAPP,sK531,vAPP,sK534,sK535,sK536,sK537,sK538,vAPP,sK541,sK542,sK543,sK544,sK545,sK546,sK547,sK548,sK549,sK550,sK551,sK552,sK553,sK554,vAPP,sK557,vAPP,sK560,sK561,sK562,sK563,sK564,sK565,sK566,sK567,sK568,vAPP,sK571,vAPP,sK573,sK574,sK575,sK576,vAPP,sK578,sK579,sK580,sK581,sK582,sK583,sK584,vAPP,sK588,sK589,sK590]),skolemize(X0,$thf(sK457)),skolemize(X1,$thf(sK458)),skolemize(X2,$thf(sK459)),skolemize(X19,$thf(sK68 @ X12 @ X11)),skolemize(X19,$thf(sK68 @ X12 @ X11)),skolemize(X10,$thf(sK462)),skolemize(X11,$thf(sK463)),skolemize(X14,$thf(sK464)),skolemize(X15,$thf(sK465)),skolemize(X19,$thf(sK68 @ X12 @ X11)),skolemize(X19,$thf(sK68 @ X12 @ X11)),skolemize(X22,$thf(sK468)),skolemize(X23,$thf(sK469)),skolemize(X24,$thf(sK470)),skolemize(X25,$thf(sK471)),skolemize(X19,$thf(sK68 @ X12 @ X11)),skolemize(X34,$thf(sK473)),skolemize(X35,$thf(sK474)),skolemize(X37,$thf(sK475)),skolemize(X39,$thf(sK476)),skolemize(X40,$thf(sK477)),skolemize(X19,$thf(sK68 @ X12 @ X11)),skolemize(X19,$thf(sK68 @ X12 @ X11)),skolemize(X19,$thf(sK68 @ X12 @ X11)),skolemize(X19,$thf(sK68 @ X12 @ X11)),skolemize(X19,$thf(sK68 @ X12 @ X11)),skolemize(X19,$thf(sK68 @ X12 @ X11)),skolemize(X52,$thf(sK484)),skolemize(X53,$thf(sK485)),skolemize(X54,$thf(sK486)),skolemize(X55,$thf(sK487)),skolemize(X62,$thf(sK488)),skolemize(X63,$thf(sK489)),skolemize(X19,$thf(sK68 @ X12 @ X11)),skolemize(X19,$thf(sK68 @ X12 @ X11)),skolemize(X72,$thf(sK492)),skolemize(X73,$thf(sK493)),skolemize(X74,$thf(sK494)),skolemize(X75,$thf(sK495)),skolemize(X76,$thf(sK496)),skolemize(X77,$thf(sK497)),skolemize(X82,$thf(sK498)),skolemize(X83,$thf(sK499)),skolemize(X84,$thf(sK500)),skolemize(X85,$thf(sK501)),skolemize(X86,$thf(sK502)),skolemize(X96,$thf(sK503)),skolemize(X97,$thf(sK504)),skolemize(X98,$thf(sK505)),skolemize(X99,$thf(sK506)),skolemize(X102,$thf(sK507)),skolemize(X103,$thf(sK508)),skolemize(X19,$thf(sK68 @ X12 @ X11)),skolemize(X111,$thf(sK510)),skolemize(X19,$thf(sK68 @ X12 @ X11)),skolemize(X118,$thf(sK512)),skolemize(X119,$thf(sK513)),skolemize(X120,$thf(sK514)),skolemize(X121,$thf(sK515)),skolemize(X122,$thf(sK516)),skolemize(X123,$thf(sK517)),skolemize(X124,$thf(sK518)),skolemize(X125,$thf(sK519)),skolemize(X126,$thf(sK520)),skolemize(X19,$thf(sK68 @ X12 @ X11)),skolemize(X137,$thf(sK522)),skolemize(X138,$thf(sK523)),skolemize(X139,$thf(sK524)),skolemize(X140,$thf(sK525)),skolemize(X142,$thf(sK526)),skolemize(X145,$thf(sK527)),skolemize(X19,$thf(sK68 @ X12 @ X11)),skolemize(X148,$thf(sK529)),skolemize(X19,$thf(sK68 @ X12 @ X11)),skolemize(X154,$thf(sK531)),skolemize(X19,$thf(sK68 @ X12 @ X11)),skolemize(X19,$thf(sK68 @ X12 @ X11)),skolemize(X161,$thf(sK534)),skolemize(X162,$thf(sK535)),skolemize(X163,$thf(sK536)),skolemize(X164,$thf(sK537)),skolemize(X167,$thf(sK538)),skolemize(X19,$thf(sK68 @ X12 @ X11)),skolemize(X19,$thf(sK68 @ X12 @ X11)),skolemize(X173,$thf(sK541)),skolemize(X174,$thf(sK542)),skolemize(X175,$thf(sK543)),skolemize(X176,$thf(sK544)),skolemize(X185,$thf(sK545)),skolemize(X186,$thf(sK546)),skolemize(X187,$thf(sK547)),skolemize(X188,$thf(sK548)),skolemize(X189,$thf(sK549)),skolemize(X190,$thf(sK550)),skolemize(X191,$thf(sK551)),skolemize(X192,$thf(sK552)),skolemize(X199,$thf(sK553)),skolemize(X200,$thf(sK554)),skolemize(X19,$thf(sK68 @ X12 @ X11)),skolemize(X19,$thf(sK68 @ X12 @ X11)),skolemize(X207,$thf(sK557)),skolemize(X19,$thf(sK68 @ X12 @ X11)),skolemize(X19,$thf(sK68 @ X12 @ X11)),skolemize(X212,$thf(sK560)),skolemize(X213,$thf(sK561)),skolemize(X214,$thf(sK562)),skolemize(X215,$thf(sK563)),skolemize(X216,$thf(sK564)),skolemize(X223,$thf(sK565)),skolemize(X224,$thf(sK566)),skolemize(X225,$thf(sK567)),skolemize(X226,$thf(sK568)),skolemize(X19,$thf(sK68 @ X12 @ X11)),skolemize(X19,$thf(sK68 @ X12 @ X11)),skolemize(X234,$thf(sK571)),skolemize(X19,$thf(sK68 @ X12 @ X11)),skolemize(X238,$thf(sK573)),skolemize(X239,$thf(sK574)),skolemize(X240,$thf(sK575)),skolemize(X241,$thf(sK576)),skolemize(X19,$thf(sK68 @ X12 @ X11)),skolemize(X252,$thf(sK578)),skolemize(X253,$thf(sK579)),skolemize(X254,$thf(sK580)),skolemize(X255,$thf(sK581)),skolemize(X256,$thf(sK582)),skolemize(X257,$thf(sK583)),skolemize(X258,$thf(sK584)),skolemize(X19,$thf(sK68 @ X12 @ X11)),skolemize(X19,$thf(sK68 @ X12 @ X11)),skolemize(X19,$thf(sK68 @ X12 @ X11)),skolemize(X268,$thf(sK588)),skolemize(X270,$thf(sK589)),skolemize(X271,$thf(sK590))],[f262])).
thf(f935,plain,(
  (pushprop_lem3v2 = $true)),
  inference(cnf_transformation,[],[f263])).
thf(f977,plain,(
  (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))))))))))))),
  inference(cnf_transformation,[],[f263])).
thf(f978,plain,(
  (pushprop_lem3v2 != ((!! @ subst @ (^[Y0 : subst]: (!! @ (term > $o) @ (^[Y1 : term > $o]: (!! @ (term > $o) @ (^[Y2 : term > $o]: (!! @ term @ (^[Y3 : term]: ((pushprop_p_and_p_prime @ Y3 @ Y0 @ Y2 @ Y1) => ((!! @ term @ (^[Y4 : term]: ((var @ Y4) => (Y1 @ Y4)))) => (!! @ term @ (^[Y4 : term]: ((var @ Y4) => (Y2 @ (sub @ Y4 @ (push @ Y3 @ Y0))))))))))))))))))),
  inference(cnf_transformation,[],[f8])).
thf(f981,plain,(
  (((!! @ subst @ (^[Y0 : subst]: (!! @ (term > $o) @ (^[Y1 : term > $o]: (!! @ (term > $o) @ (^[Y2 : term > $o]: (!! @ term @ (^[Y3 : term]: (((^[Y4 : term]: ((^[Y5 : subst]: ((^[Y6 : term > $o]: ((^[Y7 : term > $o]: (!! @ term @ (^[Y8 : term]: ((Y6 @ (sub @ Y8 @ (push @ Y4 @ Y5))) = (Y7 @ Y8))))))))))) @ Y3 @ Y0 @ Y2 @ Y1) => ((!! @ term @ (^[Y4 : term]: ((var @ Y4) => (Y1 @ Y4)))) => (!! @ term @ (^[Y4 : term]: ((var @ Y4) => (Y2 @ (sub @ Y4 @ (push @ Y3 @ Y0))))))))))))))))) != $true)),
  inference(definition_unfolding,[],[f978,f935,f977])).
thf(f1018,plain,(
  ($false = (((^[Y0 : subst]: (!! @ (term > $o) @ (^[Y1 : term > $o]: (!! @ (term > $o) @ (^[Y2 : term > $o]: (!! @ term @ (^[Y3 : term]: (((^[Y4 : term]: ((^[Y5 : subst]: ((^[Y6 : term > $o]: ((^[Y7 : term > $o]: (!! @ term @ (^[Y8 : term]: ((Y6 @ (sub @ Y8 @ (push @ Y4 @ Y5))) = (Y7 @ Y8))))))))))) @ Y3 @ Y0 @ Y2 @ Y1) => ((!! @ term @ (^[Y4 : term]: ((var @ Y4) => (Y1 @ Y4)))) => (!! @ term @ (^[Y4 : term]: ((var @ Y4) => (Y2 @ (sub @ Y4 @ (push @ Y3 @ Y0))))))))))))))) @ sK591)))),
  inference(sigma_proxy_clausification,[],[f981])).
thf(f1019,plain,(
  (((!! @ (term > $o) @ (^[Y0 : term > $o]: (!! @ (term > $o) @ (^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((!! @ term @ (^[Y3 : term]: ((Y1 @ (sub @ Y3 @ (push @ Y2 @ sK591))) = (Y0 @ Y3)))) => ((!! @ term @ (^[Y3 : term]: ((var @ Y3) => (Y0 @ Y3)))) => (!! @ term @ (^[Y3 : term]: ((var @ Y3) => (Y1 @ (sub @ Y3 @ (push @ Y2 @ sK591))))))))))))))) = $false)),
  inference(beta-eta_normalization,[],[f1018])).
thf(f1020,plain,(
  ((((^[Y0 : term > $o]: (!! @ (term > $o) @ (^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((!! @ term @ (^[Y3 : term]: ((Y1 @ (sub @ Y3 @ (push @ Y2 @ sK591))) = (Y0 @ Y3)))) => ((!! @ term @ (^[Y3 : term]: ((var @ Y3) => (Y0 @ Y3)))) => (!! @ term @ (^[Y3 : term]: ((var @ Y3) => (Y1 @ (sub @ Y3 @ (push @ Y2 @ sK591))))))))))))) @ sK592)) = $false)),
  inference(sigma_proxy_clausification,[],[f1019])).
thf(f1021,plain,(
  (((!! @ (term > $o) @ (^[Y0 : term > $o]: (!! @ term @ (^[Y1 : term]: ((!! @ term @ (^[Y2 : term]: ((Y0 @ (sub @ Y2 @ (push @ Y1 @ sK591))) = (sK592 @ Y2)))) => ((!! @ term @ (^[Y2 : term]: ((var @ Y2) => (sK592 @ Y2)))) => (!! @ term @ (^[Y2 : term]: ((var @ Y2) => (Y0 @ (sub @ Y2 @ (push @ Y1 @ sK591))))))))))))) = $false)),
  inference(beta-eta_normalization,[],[f1020])).
thf(f1022,plain,(
  ($false = (((^[Y0 : term > $o]: (!! @ term @ (^[Y1 : term]: ((!! @ term @ (^[Y2 : term]: ((Y0 @ (sub @ Y2 @ (push @ Y1 @ sK591))) = (sK592 @ Y2)))) => ((!! @ term @ (^[Y2 : term]: ((var @ Y2) => (sK592 @ Y2)))) => (!! @ term @ (^[Y2 : term]: ((var @ Y2) => (Y0 @ (sub @ Y2 @ (push @ Y1 @ sK591))))))))))) @ sK593)))),
  inference(sigma_proxy_clausification,[],[f1021])).
thf(f1023,plain,(
  ($false = ((!! @ term @ (^[Y0 : term]: ((!! @ term @ (^[Y1 : term]: ((sK593 @ (sub @ Y1 @ (push @ Y0 @ sK591))) = (sK592 @ Y1)))) => ((!! @ term @ (^[Y1 : term]: ((var @ Y1) => (sK592 @ Y1)))) => (!! @ term @ (^[Y1 : term]: ((var @ Y1) => (sK593 @ (sub @ Y1 @ (push @ Y0 @ sK591))))))))))))),
  inference(beta-eta_normalization,[],[f1022])).
thf(f1024,plain,(
  ((((^[Y0 : term]: ((!! @ term @ (^[Y1 : term]: ((sK593 @ (sub @ Y1 @ (push @ Y0 @ sK591))) = (sK592 @ Y1)))) => ((!! @ term @ (^[Y1 : term]: ((var @ Y1) => (sK592 @ Y1)))) => (!! @ term @ (^[Y1 : term]: ((var @ Y1) => (sK593 @ (sub @ Y1 @ (push @ Y0 @ sK591))))))))) @ sK594)) = $false)),
  inference(sigma_proxy_clausification,[],[f1023])).
thf(f1025,plain,(
  ($false = (((!! @ term @ (^[Y0 : term]: ((sK593 @ (sub @ Y0 @ (push @ sK594 @ sK591))) = (sK592 @ Y0)))) => ((!! @ term @ (^[Y0 : term]: ((var @ Y0) => (sK592 @ Y0)))) => (!! @ term @ (^[Y0 : term]: ((var @ Y0) => (sK593 @ (sub @ Y0 @ (push @ sK594 @ sK591))))))))))),
  inference(beta-eta_normalization,[],[f1024])).
thf(f1026,plain,(
  ((((!! @ term @ (^[Y0 : term]: ((var @ Y0) => (sK592 @ Y0)))) => (!! @ term @ (^[Y0 : term]: ((var @ Y0) => (sK593 @ (sub @ Y0 @ (push @ sK594 @ sK591)))))))) = $false)),
  inference(imp_proxy_clausification,[],[f1025])).
thf(f1027,plain,(
  (((!! @ term @ (^[Y0 : term]: ((sK593 @ (sub @ Y0 @ (push @ sK594 @ sK591))) = (sK592 @ Y0))))) = $true)),
  inference(imp_proxy_clausification,[],[f1025])).
thf(f1028,plain,(
  ( ! [X1 : term] : (((((^[Y0 : term]: ((sK593 @ (sub @ Y0 @ (push @ sK594 @ sK591))) = (sK592 @ Y0))) @ X1)) = $true)) )),
  inference(pi_proxy_clausification,[],[f1027])).
thf(f1029,plain,(
  ( ! [X1 : term] : (($true = (((sK593 @ (sub @ X1 @ (push @ sK594 @ sK591))) = (sK592 @ X1))))) )),
  inference(beta-eta_normalization,[],[f1028])).
thf(f1030,plain,(
  ( ! [X1 : term] : ((((sK592 @ X1)) = ((sK593 @ (sub @ X1 @ (push @ sK594 @ sK591)))))) )),
  inference(equality_proxy_clausification,[],[f1029])).
thf(f1032,plain,(
  ( ! [X1 : term] : (($true = ((sK593 @ (sub @ X1 @ (push @ sK594 @ sK591))))) | ($false = ((sK592 @ X1)))) )),
  inference(iff_proxy_clausification,[],[f1030])).
thf(f1033,plain,(
  (((!! @ term @ (^[Y0 : term]: ((var @ Y0) => (sK593 @ (sub @ Y0 @ (push @ sK594 @ sK591))))))) = $false)),
  inference(imp_proxy_clausification,[],[f1026])).
thf(f1034,plain,(
  ($true = ((!! @ term @ (^[Y0 : term]: ((var @ Y0) => (sK592 @ Y0))))))),
  inference(imp_proxy_clausification,[],[f1026])).
thf(f1035,plain,(
  ( ! [X1 : term] : (((((^[Y0 : term]: ((var @ Y0) => (sK592 @ Y0))) @ X1)) = $true)) )),
  inference(pi_proxy_clausification,[],[f1034])).
thf(f1036,plain,(
  ( ! [X1 : term] : (((((var @ X1) => (sK592 @ X1))) = $true)) )),
  inference(beta-eta_normalization,[],[f1035])).
thf(f1037,plain,(
  ( ! [X1 : term] : (($false = ((var @ X1))) | (((sK592 @ X1)) = $true)) )),
  inference(imp_proxy_clausification,[],[f1036])).
thf(f1038,plain,(
  ((((^[Y0 : term]: ((var @ Y0) => (sK593 @ (sub @ Y0 @ (push @ sK594 @ sK591))))) @ sK595)) = $false)),
  inference(sigma_proxy_clausification,[],[f1033])).
thf(f1039,plain,(
  ((((var @ sK595) => (sK593 @ (sub @ sK595 @ (push @ sK594 @ sK591))))) = $false)),
  inference(beta-eta_normalization,[],[f1038])).
thf(f1040,plain,(
  ($false = ((sK593 @ (sub @ sK595 @ (push @ sK594 @ sK591)))))),
  inference(imp_proxy_clausification,[],[f1039])).
thf(f1041,plain,(
  (((var @ sK595)) = $true)),
  inference(imp_proxy_clausification,[],[f1039])).
thf(f3850,plain,(
  (((sK592 @ sK595)) = $true) | ($false = $true)),
  inference(superposition,[],[f1041,f1037])).
thf(f3851,plain,(
  (((sK592 @ sK595)) = $true)),
  inference(trivial_inequality_removal,[],[f3850])).
thf(f3867,plain,(
  ($false = ((sK592 @ sK595))) | ($false = $true)),
  inference(superposition,[],[f1032,f1040])).
thf(f3868,plain,(
  ($false = ((sK592 @ sK595)))),
  inference(trivial_inequality_removal,[],[f3867])).
thf(f3872,plain,(
  ($false = $true)),
  inference(superposition,[],[f3851,f3868])).
thf(f3873,plain,(
  $false),
  inference(trivial_inequality_removal,[],[f3872])).
% SZS output end Proof for theBenchmark