%----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_127, type, vPI: !>[X0: $tType]:(((X0 > $o) > $o))).
thf(func_def_128, type, db0: !>[X0: $tType]:(X0)).
thf(func_def_129, type, db2: !>[X0: $tType]:(X0)).
thf(func_def_130, type, db1: !>[X0: $tType]:(X0)).
thf(func_def_131, type, vEQ: !>[X0: $tType]:((X0 > X0 > $o))).
thf(func_def_132, type, vLAM: !>[X0: $tType, X1: $tType]:((X1) > (X0 > X1))).
thf(func_def_135, type, db4: !>[X0: $tType]:(X0)).
thf(func_def_136, type, db3: !>[X0: $tType]:(X0)).
thf(func_def_137, type, sP0: ((term > $o) > $o)).
thf(func_def_138, type, sP1: $o).
thf(func_def_139, type, sP2: $o).
thf(func_def_140, type, sP3: ((subst > term > subst > $o) > $o)).
thf(func_def_141, type, sP4: $o).
thf(func_def_142, type, sP5: $o).
thf(func_def_143, type, sP6: $o).
thf(func_def_144, type, sP7: ((term > $o) > $o)).
thf(func_def_145, type, sP8: $o).
thf(func_def_146, type, sP9: $o).
thf(func_def_147, type, sP10: ((subst > term > subst > $o) > $o)).
thf(func_def_148, type, sP11: $o).
thf(func_def_149, type, sP12: $o).
thf(func_def_150, type, sP13: ((term > $o) > $o)).
thf(func_def_151, type, sP14: $o).
thf(func_def_152, type, sP15: $o).
thf(func_def_153, type, sP16: ((subst > term > subst > $o) > $o)).
thf(func_def_154, type, sP17: $o).
thf(func_def_155, type, sP18: $o).
thf(func_def_156, type, sP19: $o).
thf(func_def_157, type, sP20: ((subst > term > subst > $o) > $o)).
thf(func_def_158, type, sP21: $o).
thf(func_def_159, type, sP22: ((subst > term > subst > $o) > $o)).
thf(func_def_160, type, sP23: ((term > $o) > $o)).
thf(func_def_161, type, sP24: ((term > $o) > $o)).
thf(func_def_162, type, sP25: $o).
thf(func_def_163, type, sP26: $o).
thf(func_def_164, type, sP27: ((subst > term > subst > $o) > $o)).
thf(func_def_165, type, sP28: ((subst > term > subst > $o) > $o)).
thf(func_def_166, type, sP29: ((term > $o) > $o)).
thf(func_def_167, type, sP30: $o).
thf(func_def_168, type, sP31: $o).
thf(func_def_169, type, sP32: $o).
thf(func_def_170, type, sP33: ((term > $o) > $o)).
thf(func_def_171, type, sP34: ((subst > term > subst > $o) > $o)).
thf(func_def_172, type, sP35: $o).
thf(func_def_173, type, sP36: $o).
thf(func_def_174, type, sP37: $o).
thf(func_def_175, type, sP38: ((term > $o) > $o)).
thf(func_def_176, type, sP39: ((term > $o) > $o)).
thf(func_def_177, type, sP40: $o).
thf(func_def_178, type, sP41: $o).
thf(func_def_179, type, sP42: $o).
thf(func_def_180, type, sP43: ((term > $o) > $o)).
thf(func_def_181, type, sP44: ((subst > term > subst > $o) > $o)).
thf(func_def_182, type, sP45: ((subst > term > subst > $o) > $o)).
thf(func_def_183, type, sP46: ((subst > term > subst > $o) > $o)).
thf(func_def_184, type, sP47: ((subst > term > subst > $o) > $o)).
thf(func_def_185, type, sP48: $o).
thf(func_def_186, type, sP49: $o).
thf(func_def_187, type, sP50: $o).
thf(func_def_188, type, sP51: $o).
thf(func_def_189, type, sP52: ((subst > term > subst > $o) > $o)).
thf(func_def_190, type, sP53: $o).
thf(func_def_191, type, sP54: ((subst > term > subst > $o) > $o)).
thf(func_def_192, type, sP55: ((subst > term > subst > $o) > $o)).
thf(func_def_193, type, sP56: $o).
thf(func_def_194, type, sP57: ((subst > term > subst > $o) > $o)).
thf(func_def_195, type, sP58: $o).
thf(func_def_196, type, sK59: (term > $o)).
thf(func_def_197, type, sK60: (subst > term > subst > $o)).
thf(func_def_198, type, sK61: term).
thf(func_def_199, type, sK62: (term > (term > $o) > term)).
thf(func_def_200, type, sK63: ((subst > term > subst > $o) > term)).
thf(func_def_201, type, sK64: ((subst > term > subst > $o) > subst)).
thf(func_def_202, type, sK65: ((subst > term > subst > $o) > subst)).
thf(func_def_203, type, sK66: ((subst > term > subst > $o) > subst)).
thf(func_def_204, type, sK67: ((subst > term > subst > $o) > subst)).
thf(func_def_205, type, sK68: ((subst > term > subst > $o) > subst)).
thf(func_def_206, type, sK69: ((subst > term > subst > $o) > subst)).
thf(func_def_207, type, sK70: ((subst > term > subst > $o) > term)).
thf(func_def_208, type, sK71: ((subst > term > subst > $o) > (term > $o) > term)).
thf(func_def_209, type, sK72: ((subst > term > term) > term)).
thf(func_def_210, type, sK73: ((subst > term > term) > subst)).
thf(func_def_211, type, sK74: ((subst > term > term) > subst)).
thf(func_def_212, type, sK75: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_213, type, sK76: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_214, type, sK77: term).
thf(func_def_215, type, sK78: (subst > term > subst > $o)).
thf(func_def_216, type, sK79: ((subst > term > subst > $o) > subst)).
thf(func_def_217, type, sK80: ((subst > term > subst > $o) > subst)).
thf(func_def_218, type, sK81: ((subst > term > subst > $o) > term)).
thf(func_def_219, type, sK82: ((subst > term > subst > $o) > subst)).
thf(func_def_220, type, sK83: ((subst > term > subst > $o) > term)).
thf(func_def_221, type, sK84: ((subst > term > subst > $o) > subst)).
thf(func_def_222, type, sK85: ((subst > term > subst > $o) > subst)).
thf(func_def_223, type, sK86: ((subst > term > subst > $o) > subst)).
thf(func_def_224, type, sK87: ((subst > term > subst > $o) > term)).
thf(func_def_225, type, sK88: ((subst > term > subst > $o) > term)).
thf(func_def_226, type, sK89: ((subst > term > subst > $o) > term)).
thf(func_def_227, type, sK90: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_228, type, sK91: ((subst > term > term) > subst)).
thf(func_def_229, type, sK92: ((subst > term > term) > term)).
thf(func_def_230, type, sK93: ((subst > term > term) > subst)).
thf(func_def_231, type, sK94: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_232, type, sK95: (subst > term > subst > $o)).
thf(func_def_233, type, sK96: (term > $o)).
thf(func_def_234, type, sK97: term).
thf(func_def_235, type, sK98: term).
thf(func_def_236, type, sK99: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_237, type, sK100: ((subst > term > subst > $o) > subst)).
thf(func_def_238, type, sK101: ((subst > term > subst > $o) > subst)).
thf(func_def_239, type, sK102: ((subst > term > subst > $o) > subst)).
thf(func_def_240, type, sK103: ((subst > term > subst > $o) > term)).
thf(func_def_241, type, sK104: ((subst > term > subst > $o) > subst)).
thf(func_def_242, type, sK105: ((subst > term > subst > $o) > subst)).
thf(func_def_243, type, sK106: ((subst > term > subst > $o) > subst)).
thf(func_def_244, type, sK107: ((subst > term > subst > $o) > term)).
thf(func_def_245, type, sK108: ((subst > term > subst > $o) > term)).
thf(func_def_246, type, sK109: ((subst > term > subst > $o) > term)).
thf(func_def_247, type, sK110: ((subst > term > subst > $o) > subst)).
thf(func_def_248, type, sK111: ((subst > term > subst > $o) > subst)).
thf(func_def_249, type, sK112: ((subst > term > subst > $o) > term)).
thf(func_def_250, type, sK113: ((subst > term > subst > $o) > subst)).
thf(func_def_251, type, sK114: ((subst > term > subst > $o) > subst)).
thf(func_def_252, type, sK115: ((subst > term > subst > $o) > subst)).
thf(func_def_253, type, sK116: ((subst > term > subst > $o) > subst)).
thf(func_def_254, type, sK117: ((subst > term > subst > $o) > term)).
thf(func_def_255, type, sK118: (subst > term > subst > $o)).
thf(func_def_256, type, sK119: term).
thf(func_def_257, type, sK120: term).
thf(func_def_258, type, sK121: ((term > $o) > term)).
thf(func_def_259, type, sK122: ((term > $o) > term)).
thf(func_def_260, type, sK123: term).
thf(func_def_261, type, sK124: (term > $o)).
thf(func_def_262, type, sK125: (term > term)).
thf(func_def_263, type, sK126: (subst > term > subst > $o)).
thf(func_def_264, type, sK127: term).
thf(func_def_265, type, sK128: ((subst > term > subst > $o) > subst)).
thf(func_def_266, type, sK129: ((subst > term > subst > $o) > subst)).
thf(func_def_267, type, sK130: ((subst > term > subst > $o) > term)).
thf(func_def_268, type, sK131: ((subst > term > subst > $o) > subst)).
thf(func_def_269, type, sK132: ((subst > term > subst > $o) > term)).
thf(func_def_270, type, sK133: ((subst > term > subst > $o) > subst)).
thf(func_def_271, type, sK134: ((subst > term > subst > $o) > term)).
thf(func_def_272, type, sK135: ((subst > term > subst > $o) > subst)).
thf(func_def_273, type, sK136: ((subst > term > subst > $o) > subst)).
thf(func_def_274, type, sK137: ((subst > term > subst > $o) > term > term)).
thf(func_def_275, type, sK138: ((subst > term > subst > $o) > subst)).
thf(func_def_276, type, sK139: ((subst > term > subst > $o) > subst)).
thf(func_def_277, type, sK140: ((subst > term > subst > $o) > subst)).
thf(func_def_278, type, sK141: ((subst > term > subst > $o) > term)).
thf(func_def_279, type, sK142: ((subst > term > subst > $o) > subst)).
thf(func_def_280, type, sK143: ((subst > term > subst > $o) > subst)).
thf(func_def_281, type, sK144: ((subst > term > subst > $o) > subst)).
thf(func_def_282, type, sK145: ((subst > term > subst > $o) > term)).
thf(func_def_283, type, sK146: term).
thf(func_def_284, type, sK147: (subst > term > subst > $o)).
thf(func_def_285, type, sK148: ((subst > term > subst > $o) > term)).
thf(func_def_286, type, sK149: ((subst > term > subst > $o) > term)).
thf(func_def_287, type, sK150: ((subst > term > subst > $o) > term)).
thf(func_def_288, type, sK151: ((subst > term > subst > $o) > term)).
thf(func_def_289, type, sK152: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_290, type, sK153: ((subst > term > term) > term)).
thf(func_def_291, type, sK154: ((subst > term > term) > subst)).
thf(func_def_292, type, sK155: ((subst > term > term) > subst)).
thf(func_def_293, type, sK156: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_294, type, sK157: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_295, type, sK158: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_296, type, sK159: ((subst > term > term) > subst)).
thf(func_def_297, type, sK160: ((subst > term > term) > subst)).
thf(func_def_298, type, sK161: ((subst > term > term) > term)).
thf(func_def_299, type, sK162: ((term > $o) > term)).
thf(func_def_300, type, sK163: ((term > $o) > term)).
thf(func_def_301, type, sK164: term).
thf(func_def_302, type, sK165: (subst > term > subst > $o)).
thf(func_def_303, type, sK166: ((subst > term > term) > term > subst)).
thf(func_def_304, type, sK167: ((subst > term > term) > term > term)).
thf(func_def_305, type, sK168: ((subst > term > term) > term > subst)).
thf(func_def_306, type, sK169: ((subst > term > term) > term > subst)).
thf(func_def_307, type, sK170: ((subst > term > term) > term > term)).
thf(func_def_308, type, sK171: ((subst > term > term) > term > subst)).
thf(func_def_309, type, sK172: ((subst > term > term) > term)).
thf(func_def_310, type, sK173: ((subst > term > subst > $o) > term > term)).
thf(func_def_311, type, sK174: ((subst > term > subst > $o) > term)).
thf(func_def_312, type, sK175: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_313, type, sK176: ((term > $o) > term)).
thf(func_def_314, type, sK177: ((term > $o) > term)).
thf(func_def_315, type, sK178: (term > $o)).
thf(func_def_316, type, sK179: term).
thf(func_def_317, type, sK180: subst).
thf(func_def_318, type, sK181: term).
thf(func_def_319, type, sK182: subst).
thf(func_def_320, type, sK183: term).
thf(func_def_321, type, sK184: subst).
thf(func_def_322, type, sK185: (term > $o)).
thf(func_def_323, type, sK186: term).
thf(func_def_324, type, sK187: subst).
thf(func_def_325, type, sK188: (term > term)).
thf(func_def_326, type, sK189: ((term > $o) > term)).
thf(func_def_327, type, sK190: (subst > (term > $o) > term)).
thf(func_def_328, type, sK191: ((term > $o) > term)).
thf(func_def_329, type, sK192: ((term > $o) > term)).
thf(func_def_330, type, sK193: ((term > $o) > term)).
thf(func_def_331, type, sK194: ((term > $o) > term)).
thf(func_def_332, type, sK195: term).
thf(func_def_333, type, sK196: (subst > term > subst > $o)).
thf(func_def_334, type, sK197: ((subst > term > term) > term)).
thf(func_def_335, type, sK198: ((subst > term > term) > subst)).
thf(func_def_336, type, sK199: ((subst > term > term) > term)).
thf(func_def_337, type, sK200: ((subst > term > term) > subst)).
thf(func_def_338, type, sK201: ((subst > term > subst > $o) > term > $o)).
thf(func_def_339, type, sK202: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_340, type, sK203: term).
thf(func_def_341, type, sK204: (subst > term > subst > $o)).
thf(func_def_342, type, sK205: ((term > $o) > term)).
thf(func_def_343, type, sK206: ((term > $o) > term)).
thf(func_def_344, type, sK207: ((term > $o) > term)).
thf(func_def_345, type, sK208: (term > $o)).
thf(func_def_346, type, sK209: term).
thf(func_def_347, type, sK210: (term > term)).
thf(func_def_348, type, sK211: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_349, type, sK212: ((subst > term > subst > $o) > (term > $o) > term)).
thf(func_def_350, type, sK213: (term > (term > $o) > term)).
thf(func_def_351, type, sK214: (term > $o)).
thf(func_def_352, type, sK215: (subst > term > subst > $o)).
thf(func_def_353, type, sK216: term).
thf(func_def_354, type, sK217: ((subst > term > term) > term)).
thf(func_def_355, type, sK218: ((subst > term > term) > subst)).
thf(func_def_356, type, sK219: ((subst > term > term) > subst)).
thf(func_def_357, type, sK220: ((subst > term > term) > term)).
thf(func_def_358, type, sK221: ((subst > term > subst > $o) > term)).
thf(func_def_359, type, sK222: ((subst > term > subst > $o) > term)).
thf(func_def_360, type, sK223: ((term > $o) > term)).
thf(func_def_361, type, sK224: ((term > $o) > term)).
thf(func_def_362, type, sK225: ((term > $o) > term)).
thf(func_def_363, type, sK226: ((term > $o) > term)).
thf(func_def_364, type, sK227: ((subst > term > subst > $o) > term > $o)).
thf(func_def_365, type, sK228: (term > $o)).
thf(func_def_366, type, sK229: term).
thf(func_def_367, type, sK230: (term > term)).
thf(func_def_368, type, sK231: term).
thf(func_def_369, type, sK232: (subst > term > subst > $o)).
thf(func_def_370, type, sK233: ((term > $o) > term)).
thf(func_def_371, type, sK234: term).
thf(func_def_372, type, sK235: (subst > term > subst > $o)).
thf(func_def_373, type, sK236: ((subst > term > subst > $o) > subst)).
thf(func_def_374, type, sK237: ((subst > term > subst > $o) > subst)).
thf(func_def_375, type, sK238: ((subst > term > subst > $o) > subst)).
thf(func_def_376, type, sK239: ((subst > term > subst > $o) > term)).
thf(func_def_377, type, sK240: ((subst > term > subst > $o) > term)).
thf(func_def_378, type, sK241: ((subst > term > subst > $o) > subst)).
thf(func_def_379, type, sK242: ((subst > term > subst > $o) > term)).
thf(func_def_380, type, sK243: ((subst > term > subst > $o) > subst)).
thf(func_def_381, type, sK244: ((subst > term > subst > $o) > subst)).
thf(func_def_382, type, sK245: ((subst > term > subst > $o) > (term > $o) > term)).
thf(func_def_383, type, sK246: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_384, type, sK247: (term > (term > $o) > term)).
thf(func_def_385, type, sK248: (term > $o)).
thf(func_def_386, type, sK249: (subst > term > subst > $o)).
thf(func_def_387, type, sK250: term).
thf(func_def_388, type, sK251: ((subst > term > term) > term)).
thf(func_def_389, type, sK252: ((subst > term > term) > term)).
thf(func_def_390, type, sK253: ((subst > term > term) > subst)).
thf(func_def_391, type, sK254: ((subst > term > term) > subst)).
thf(func_def_392, type, sK255: ((term > $o) > term)).
thf(func_def_393, type, sK256: ((term > $o) > term)).
thf(func_def_394, type, sK257: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_395, type, sK258: ((subst > term > term) > subst)).
thf(func_def_396, type, sK259: ((subst > term > term) > term)).
thf(func_def_397, type, sK260: ((subst > term > term) > subst)).
thf(func_def_398, type, sK261: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_399, type, sK262: ((subst > term > subst > $o) > term)).
thf(func_def_400, type, sK263: ((subst > term > subst > $o) > term)).
thf(func_def_401, type, sK264: ((term > $o) > term)).
thf(func_def_402, type, sK265: ((term > $o) > subst > term)).
thf(func_def_403, type, sK266: term).
thf(func_def_404, type, sK267: subst).
thf(func_def_405, type, sK268: term).
thf(func_def_406, type, sK269: (term > $o)).
thf(func_def_407, type, sK270: (term > term)).
thf(func_def_408, type, sK271: (term > $o)).
thf(func_def_409, type, sK272: term).
thf(func_def_410, type, sK273: (term > term)).
thf(func_def_411, type, sK274: ((term > $o) > term)).
thf(func_def_412, type, sK275: ((term > $o) > term)).
thf(func_def_413, type, sK276: ((term > $o) > term)).
thf(func_def_414, type, sK277: ((term > $o) > term)).
thf(func_def_415, type, sK278: ((term > $o) > term)).
thf(func_def_416, type, sK279: ((term > $o) > term)).
thf(func_def_417, type, sK280: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_418, type, sK281: ((subst > term > term) > term)).
thf(func_def_419, type, sK282: ((subst > term > term) > subst)).
thf(func_def_420, type, sK283: ((subst > term > term) > subst)).
thf(func_def_421, type, sK284: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_422, type, sK285: (subst > term > subst > $o)).
thf(func_def_423, type, sK286: term).
thf(func_def_424, type, sK287: ((subst > term > subst > $o) > subst)).
thf(func_def_425, type, sK288: ((subst > term > subst > $o) > subst)).
thf(func_def_426, type, sK289: ((subst > term > subst > $o) > subst)).
thf(func_def_427, type, sK290: ((subst > term > subst > $o) > term)).
thf(func_def_428, type, sK291: ((subst > term > subst > $o) > term)).
thf(func_def_429, type, sK292: ((subst > term > subst > $o) > subst)).
thf(func_def_430, type, sK293: ((subst > term > subst > $o) > subst)).
thf(func_def_431, type, sK294: ((subst > term > subst > $o) > subst)).
thf(func_def_432, type, sK295: (term > (subst > term > subst > $o) > term)).
thf(func_def_433, type, sK296: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_434, type, sK297: ((subst > term > term) > term)).
thf(func_def_435, type, sK298: ((subst > term > term) > subst)).
thf(func_def_436, type, sK299: ((subst > term > term) > subst)).
thf(func_def_437, type, sK300: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_438, type, sK301: term).
thf(func_def_439, type, sK302: (subst > term > subst > $o)).
thf(func_def_440, type, sK303: ((subst > term > term) > term)).
thf(func_def_441, type, sK304: ((subst > term > term) > term)).
thf(func_def_442, type, sK305: ((subst > term > term) > subst)).
thf(func_def_443, type, sK306: ((subst > term > term) > subst)).
thf(func_def_444, type, sK307: ((subst > term > subst > $o) > term > term)).
thf(func_def_445, type, sK308: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_446, type, sK309: term).
thf(func_def_447, type, sK310: (term > (subst > term > subst > $o) > term)).
thf(func_def_448, type, sK311: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_449, type, sK312: (subst > term > subst > $o)).
thf(func_def_450, type, sK313: term).
thf(func_def_451, type, sK314: ((subst > term > term) > term)).
thf(func_def_452, type, sK315: ((subst > term > term) > term)).
thf(func_def_453, type, sK316: ((subst > term > term) > subst)).
thf(func_def_454, type, sK317: ((subst > term > term) > subst)).
thf(func_def_455, type, sK318: term).
thf(func_def_456, type, sK319: (subst > term > subst > $o)).
thf(func_def_457, type, sK320: (term > $o)).
thf(func_def_458, type, sK321: ((subst > term > subst > $o) > subst)).
thf(func_def_459, type, sK322: ((subst > term > subst > $o) > term)).
thf(func_def_460, type, sK323: ((subst > term > subst > $o) > subst)).
thf(func_def_461, type, sK324: ((subst > term > subst > $o) > subst)).
thf(func_def_462, type, sK325: ((subst > term > subst > $o) > term)).
thf(func_def_463, type, sK326: ((subst > term > subst > $o) > subst)).
thf(func_def_464, type, sK327: ((subst > term > subst > $o) > subst)).
thf(func_def_465, type, sK328: ((subst > term > subst > $o) > subst)).
thf(func_def_466, type, sK329: ((term > $o) > term > term)).
thf(func_def_467, type, sK330: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_468, type, sK331: term).
thf(func_def_469, type, sK332: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_470, type, sK333: ((subst > term > term) > subst)).
thf(func_def_471, type, sK334: ((subst > term > term) > term)).
thf(func_def_472, type, sK335: ((subst > term > term) > subst)).
thf(func_def_473, type, sK336: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_474, type, sK337: (subst > term > subst > $o)).
thf(func_def_475, type, sK338: term).
thf(func_def_476, type, sK339: ((subst > term > term) > term)).
thf(func_def_477, type, sK340: ((subst > term > term) > subst)).
thf(func_def_478, type, sK341: ((subst > term > term) > term)).
thf(func_def_479, type, sK342: ((subst > term > term) > subst)).
thf(func_def_480, type, sK343: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_481, type, sK344: (term > (subst > term > subst > $o) > term)).
thf(func_def_482, type, sK345: (subst > term > subst > $o)).
thf(func_def_483, type, sK346: term).
thf(func_def_484, type, sK347: ((subst > term > term) > term)).
thf(func_def_485, type, sK348: ((subst > term > term) > subst)).
thf(func_def_486, type, sK349: ((subst > term > term) > subst)).
thf(func_def_487, type, sK350: ((subst > term > term) > term)).
thf(func_def_488, type, sK351: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_489, type, sK352: (term > (subst > term > subst > $o) > term)).
thf(func_def_490, type, sK353: ((term > $o) > term)).
thf(func_def_491, type, sK354: ((term > $o) > term)).
thf(func_def_492, type, sK355: (subst > term > subst > $o)).
thf(func_def_493, type, sK356: term).
thf(func_def_494, type, sK357: (term > $o)).
thf(func_def_495, type, sK358: ((subst > term > term) > subst)).
thf(func_def_496, type, sK359: ((subst > term > term) > term)).
thf(func_def_497, type, sK360: ((subst > term > term) > subst)).
thf(func_def_498, type, sK361: ((subst > term > term) > term)).
thf(func_def_499, type, sK362: (term > subst > term > term)).
thf(func_def_500, type, sK363: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_501, type, sK364: ((term > $o) > term > term)).
thf(func_def_502, type, sK365: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_503, type, sK366: term).
thf(func_def_504, type, sK367: ((subst > term > term) > term)).
thf(func_def_505, type, sK368: ((subst > term > term) > subst)).
thf(func_def_506, type, sK369: term).
thf(func_def_507, type, sK370: term).
thf(func_def_508, type, sK371: term).
thf(func_def_509, type, sK372: (subst > term > subst > $o)).
thf(func_def_510, type, sK373: ((subst > term > subst > $o) > subst)).
thf(func_def_511, type, sK374: ((subst > term > subst > $o) > subst)).
thf(func_def_512, type, sK375: ((subst > term > subst > $o) > term)).
thf(func_def_513, type, sK376: ((subst > term > subst > $o) > subst)).
thf(func_def_514, type, sK377: ((subst > term > subst > $o) > subst)).
thf(func_def_515, type, sK378: ((subst > term > subst > $o) > subst)).
thf(func_def_516, type, sK379: ((subst > term > subst > $o) > subst)).
thf(func_def_517, type, sK380: ((subst > term > subst > $o) > term)).
thf(func_def_518, type, sK381: ((subst > term > subst > $o) > term)).
thf(func_def_519, type, sK382: ((subst > term > subst > $o) > term)).
thf(func_def_520, type, sK383: (subst > term > term)).
thf(func_def_521, type, sK384: subst).
thf(func_def_522, type, sK385: (subst > term > term)).
thf(func_def_523, type, sK386: term).
thf(func_def_524, type, sK387: subst).
thf(func_def_525, type, sK388: term).
thf(func_def_526, type, sK389: term).
thf(func_def_527, type, sK390: term).
thf(func_def_528, type, sK391: ((subst > term > term) > subst)).
thf(func_def_529, type, sK392: ((subst > term > term) > subst)).
thf(func_def_530, type, sK393: ((subst > term > term) > term)).
thf(func_def_531, type, sK394: ((subst > term > term) > subst)).
thf(func_def_532, type, sK395: ((subst > term > term) > term)).
thf(func_def_533, type, sK396: ((subst > term > term) > subst)).
thf(func_def_534, type, sK397: subst).
thf(func_def_535, type, sK398: term).
thf(func_def_536, type, sK399: subst).
thf(func_def_537, type, sK400: term).
thf(func_def_538, type, sK401: (term > $o)).
thf(func_def_539, type, sK402: (term > term)).
thf(func_def_540, type, sK403: ((term > $o) > subst > term)).
thf(func_def_541, type, sK404: ((term > $o) > term)).
thf(func_def_542, type, sK405: ((term > $o) > term)).
thf(func_def_543, type, sK406: ((term > $o) > term)).
thf(func_def_544, type, sK407: (term > $o)).
thf(func_def_545, type, sK408: (subst > term > subst > $o)).
thf(func_def_546, type, sK409: term).
thf(func_def_547, type, sK410: ((subst > term > term) > term)).
thf(func_def_548, type, sK411: ((subst > term > term) > subst)).
thf(func_def_549, type, sK412: ((subst > term > term) > subst)).
thf(func_def_550, type, sK413: ((subst > term > term) > term)).
thf(func_def_551, type, sK414: ((subst > term > subst > $o) > (term > $o) > term)).
thf(func_def_552, type, sK415: (term > (term > $o) > term)).
thf(func_def_553, type, sK416: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_554, type, sK417: (subst > term > subst > $o)).
thf(func_def_555, type, sK418: term).
thf(func_def_556, type, sK419: (term > (subst > term > subst > $o) > term)).
thf(func_def_557, type, sK420: ((subst > term > subst > $o) > term)).
thf(func_def_558, type, sK421: ((subst > term > subst > $o) > subst)).
thf(func_def_559, type, sK422: ((subst > term > subst > $o) > subst)).
thf(func_def_560, type, sK423: ((subst > term > subst > $o) > subst)).
thf(func_def_561, type, sK424: ((subst > term > subst > $o) > term)).
thf(func_def_562, type, sK425: ((subst > term > subst > $o) > subst)).
thf(func_def_563, type, sK426: ((subst > term > subst > $o) > subst)).
thf(func_def_564, type, sK427: ((subst > term > subst > $o) > subst)).
thf(func_def_565, type, sK428: term).
thf(func_def_566, type, sK429: (term > (subst > term > subst > $o) > term)).
thf(func_def_567, type, sK430: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_568, type, sK431: (subst > term > subst > $o)).
thf(func_def_569, type, sK432: term).
thf(func_def_570, type, sK433: ((subst > term > term) > term)).
thf(func_def_571, type, sK434: ((subst > term > term) > subst)).
thf(func_def_572, type, sK435: ((subst > term > term) > subst)).
thf(func_def_573, type, sK436: ((subst > term > term) > term)).
thf(func_def_574, type, sK437: ((subst > term > term) > subst)).
thf(func_def_575, type, sK438: ((subst > term > term) > subst)).
thf(func_def_576, type, sK439: ((subst > term > term) > term)).
thf(func_def_577, type, sK440: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_578, type, sK441: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_579, type, sK442: term).
thf(func_def_580, type, sK443: (subst > term > subst > $o)).
thf(func_def_581, type, sK444: ((subst > term > term) > term)).
thf(func_def_582, type, sK445: ((subst > term > term) > subst)).
thf(func_def_583, type, sK446: ((subst > term > term) > term)).
thf(func_def_584, type, sK447: ((subst > term > term) > subst)).
thf(func_def_585, type, sK448: ((subst > term > subst > $o) > term > term)).
thf(func_def_586, type, sK449: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_587, type, sK450: (term > $o)).
thf(func_def_588, type, sK451: term).
thf(func_def_589, type, sK452: (term > term)).
thf(func_def_590, type, sK453: ((term > $o) > term)).
thf(func_def_591, type, sK454: ((term > $o) > term)).
thf(func_def_592, type, sK455: ((term > $o) > term)).
thf(func_def_593, type, sK456: ((term > $o) > term)).
thf(func_def_594, type, sK457: (term > $o)).
thf(func_def_595, type, sK458: term).
thf(func_def_596, type, sK459: (term > $o)).
thf(func_def_597, type, sK460: subst).
thf(func_def_598, type, sK461: (subst > (term > $o) > term > (term > $o) > term)).
thf(func_def_599, type, sK462: term).
thf(func_def_600, type, sK463: term).
thf(func_def_601, type, sK464: term).
thf(func_def_602, type, sK465: ((subst > term > term) > subst)).
thf(func_def_603, type, sK466: ((subst > term > term) > term)).
thf(func_def_604, type, sK467: (term > subst > term > term)).
thf(func_def_605, type, sK468: (term > subst > term > term)).
thf(func_def_606, type, sK469: term).
thf(func_def_607, type, sK470: subst).
thf(func_def_608, type, sK471: (subst > (term > $o) > term > term > $o)).
thf(func_def_609, type, sK472: term).
thf(func_def_610, type, sK473: (term > $o)).
thf(func_def_611, type, sK474: subst).
thf(func_def_612, type, sK475: ((term > $o) > term)).
thf(func_def_613, type, sK476: ((subst > term > subst > $o) > term > $o)).
thf(func_def_614, type, sK477: (subst > term > subst > $o)).
thf(func_def_615, type, sK478: ((term > $o) > term)).
thf(func_def_616, type, sK479: (subst > term > subst > $o)).
thf(func_def_617, type, sK480: ((term > $o) > term)).
thf(func_def_618, type, sK481: ((subst > term > subst > $o) > term > $o)).
thf(func_def_619, type, sK482: (term > d_term)).
thf(func_def_620, type, sK483: ((subst > term > subst > $o) > term)).
thf(func_def_621, type, sK484: ((subst > term > subst > $o) > term)).
thf(func_def_622, type, sK485: term).
thf(func_def_623, type, sK486: (subst > term > subst > $o)).
thf(func_def_624, type, sK487: subst).
thf(func_def_625, type, sK488: subst).
thf(func_def_626, type, sK489: subst).
thf(func_def_627, type, sK490: subst).
thf(func_def_628, type, sK491: subst).
thf(func_def_629, type, sK492: term).
thf(func_def_630, type, sK493: subst).
thf(func_def_631, type, sK494: (subst > d_subst)).
thf(func_def_632, type, sK495: term).
thf(func_def_633, type, sK496: term).
thf(func_def_634, type, sK497: term).
thf(func_def_635, type, sK498: term).
thf(func_def_636, type, sK499: (term > $o)).
thf(func_def_637, type, sK500: term).
thf(func_def_638, type, sK501: (term > $o)).
thf(func_def_639, type, sK502: term).
thf(func_def_640, type, sK503: subst).
thf(func_def_641, type, sK504: subst).
thf(func_def_642, type, sK505: term).
thf(func_def_643, type, sK506: subst).
thf(func_def_644, type, sK507: term).
thf(func_def_645, type, sK508: term).
thf(func_def_646, type, sK509: term).
thf(func_def_647, type, sK510: ((subst > term > term) > subst)).
thf(func_def_648, type, sK511: ((subst > term > term) > term)).
thf(func_def_649, type, sK512: ((subst > term > term) > subst)).
thf(func_def_650, type, sK513: ((subst > term > term) > subst)).
thf(func_def_651, type, sK514: ((subst > term > term) > subst)).
thf(func_def_652, type, sK515: ((subst > term > term) > term)).
thf(func_def_653, type, sK516: (subst > term > term)).
thf(func_def_654, type, sK517: subst).
thf(func_def_655, type, sK518: term).
thf(func_def_656, type, sK519: (subst > term > term)).
thf(func_def_657, type, sK520: term).
thf(func_def_658, type, sK521: (term > subst > term > term)).
thf(func_def_659, type, sK522: term).
thf(func_def_660, type, sK523: (term > $o)).
thf(func_def_661, type, sK524: term).
thf(func_def_662, type, sK525: subst).
thf(func_def_663, type, sK526: (term > $o)).
thf(func_def_664, type, sK527: ((term > $o) > subst > term > (term > $o) > term)).
thf(func_def_665, type, sK528: term).
thf(func_def_666, type, sK529: subst).
thf(func_def_667, type, sK530: (subst > term > (term > $o) > term > $o)).
thf(func_def_668, type, sK531: (term > $o)).
thf(func_def_669, type, sK532: term).
thf(func_def_670, type, sK533: subst).
thf(func_def_671, type, sK534: ((term > $o) > term)).
thf(func_def_672, type, sK535: subst).
thf(func_def_673, type, sK536: subst).
thf(func_def_674, type, sK537: term).
thf(func_def_675, type, sK538: (term > $o)).
thf(func_def_676, type, sK539: ((term > $o) > term)).
thf(func_def_677, type, sK540: ((term > $o) > term)).
thf(func_def_678, type, sK541: term).
thf(func_def_679, type, sK542: term).
thf(func_def_680, type, sK543: term).
thf(func_def_681, type, sK544: term).
thf(func_def_682, type, sK545: (term > $o)).
thf(func_def_683, type, sK546: term).
thf(func_def_684, type, sK547: (term > $o)).
thf(func_def_685, type, sK548: subst).
thf(func_def_686, type, sK549: term).
thf(func_def_687, type, sK550: term).
thf(func_def_688, type, sK551: term).
thf(func_def_689, type, sK552: term).
thf(func_def_690, type, sK553: term).
thf(func_def_691, type, sK554: term).
thf(func_def_692, type, sK555: term).
thf(func_def_693, type, sK556: term).
thf(func_def_694, type, sK557: term).
thf(func_def_695, type, sK558: term).
thf(func_def_696, type, sK559: term).
thf(func_def_697, type, sK560: term).
thf(func_def_698, type, sK561: term).
thf(func_def_699, type, sK562: subst).
thf(func_def_700, type, sK563: term).
thf(func_def_701, type, sK564: ((subst > term > term) > term)).
thf(func_def_702, type, sK565: ((subst > term > term) > subst)).
thf(func_def_703, type, sK566: ((subst > term > term) > subst)).
thf(func_def_704, type, sK567: (subst > term > term)).
thf(func_def_705, type, sK568: term).
thf(func_def_706, type, sK569: term).
thf(func_def_707, type, sK570: term).
thf(func_def_708, type, sK571: term).
thf(func_def_709, type, sK572: term).
thf(func_def_710, type, sK573: term).
thf(func_def_711, type, sK574: term).
thf(func_def_712, type, sK575: term).
thf(func_def_713, type, sK576: term).
thf(func_def_714, type, sK577: term).
thf(func_def_715, type, sK578: term).
thf(func_def_716, type, sK579: term).
thf(func_def_717, type, sK580: term).
thf(func_def_718, type, sK581: term).
thf(func_def_719, type, sK582: term).
thf(func_def_720, type, sK583: (term > subst > term > term)).
thf(func_def_721, type, sK584: term).
thf(func_def_722, type, sK585: ((subst > term > term) > term)).
thf(func_def_723, type, sK586: ((subst > term > term) > subst)).
thf(func_def_724, type, sK587: term).
thf(func_def_725, type, sK588: term).
thf(func_def_726, type, sK589: term).
thf(func_def_727, type, sK590: subst).
thf(func_def_728, type, vNOT: ($o > $o)).
thf(func_def_729, type, sK591: subst).
thf(func_def_730, type, sK592: subst).
thf(func_def_731, type, sK593: term).
thf(f1,conjecture,(
  (! [X0 : term,X2 : subst,X1 : subst] : (((sub @ (sub @ X0 @ X1) @ X2)) = ((sub @ X0 @ (comp @ X1 @ X2)))) = axclos)),
  file('/export/starexec/sandbox/benchmark/theBenchmark.p',axclos)).
thf(f2,negated_conjecture,(
  ~ (! [X0 : term,X2 : subst,X1 : subst] : (((sub @ (sub @ X0 @ X1) @ X2)) = ((sub @ X0 @ (comp @ X1 @ X2)))) = axclos)),
  inference(negated_conjecture,[status(cth)],[f1])).
thf(f3,axiom,(
  (((ap @ (d2term @ d_one) @ (d2term @ d_one))) = ((d2term @ d_one))) & hoasinduction_lem1 & pushprop_lem0_gthm & hoasapnotvar_lthm & (! [X33 : term,X10 : (subst > term > subst > $o)] : (~ ! [X22 : subst,X34 : subst,X8 : term,X9 : subst] : ((X10 @ (comp @ X9 @ X34) @ (sub @ X8 @ X34) @ X22) | ~(X10 @ X9 @ X8 @ (comp @ X34 @ X22))) | ~ ! [X21 : (subst > term > term)] : ((X10 @ id @ (lam @ (X21 @ sh @ one)) @ id) | ~ ! [X22 : subst,X8 : term,X9 : subst] : (((sub @ (X21 @ X9 @ X8) @ X22)) = ((X21 @ (comp @ X9 @ X22) @ (sub @ X8 @ X22)))) | ~ ! [X8 : term] : (~(X10 @ id @ X8 @ id) | (X10 @ id @ (X21 @ id @ X8) @ id))) | ~ ! [X22 : subst,X34 : subst,X8 : term,X9 : subst] : (~(X10 @ (comp @ X9 @ X34) @ (sub @ X8 @ X34) @ X22) | (X10 @ X9 @ X8 @ (comp @ X34 @ X22))) | (X10 @ id @ X33 @ id) | ~ ! [X8 : term] : ((X10 @ id @ X8 @ id) | ~(var @ (sub @ X8 @ id))) | ~ ! [X8 : term,X13 : term] : ((X10 @ id @ (ap @ (sub @ X8 @ id) @ X13) @ id) | ~(X10 @ id @ X8 @ id) | ~(X10 @ id @ X13 @ id))) <=> hoasinduction) & ! [X1 : d_term] : (X1 = d_one) & axvarshift & ! [X7 : d_subst,X6 : d_subst] : ((((d2subst @ X6)) = ((d2subst @ X7))) => (X6 = X7)) & (laminj <=> ! [X8 : term,X13 : term] : ((X8 = X13) | (((lam @ X8)) != ((lam @ X13))))) & hoasinduction_lem1_lthm & ~(hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) & hoasinduction_lem2_gthm & hoasinduction_lem2v2_gthm & (id = ((d2subst @ d_id))) & (substmonoid_lthm <=> (! [X9 : subst] : (X9 = ((comp @ id @ X9))) => (! [X9 : subst] : (X9 = ((comp @ X9 @ id))) => (! [X9 : subst] : (X9 = ((comp @ id @ X9))) & ! [X9 : subst] : (X9 = ((comp @ X9 @ id))))))) & ((^[X8 : term, X9 : subst, X10 : (term > $o), X11 : (term > $o)] : (! [X12 : term] : (((X11 @ X12)) = ((X10 @ (sub @ X12 @ (push @ X8 @ X9))))))) = pushprop_p_and_p_prime) & ((^[X10 : (subst > term > subst > $o), X11 : (term > $o)] : (! [X12 : term] : (((X11 @ X12)) = ((X10 @ id @ X12 @ id))))) = hoasinduction_p_and_p_prime) & pushprop_lem2v2_gthm & (! [X8 : term] : (X8 = ((sub @ X8 @ id))) <=> axvarid) & ((! [X8 : term] : (X8 = ((sub @ X8 @ id))) => (! [X15 : term,X14 : term,X13 : term,X8 : term] : ((((ap @ X8 @ X14)) != ((ap @ X13 @ X15))) | (X8 = X13)) => ! [X13 : term,X14 : term,X15 : term,X8 : term] : ((X8 = X13) | (((ap @ (sub @ X8 @ id) @ X14)) != ((ap @ (sub @ X13 @ id) @ X15)))))) <=> hoasapinj1_lthm) & (hoasinduction_lem0_lthm <=> ! [X10 : (subst > term > subst > $o)] : ~ ! [X11 : (term > $o)] : ~ ! [X12 : term] : (((X11 @ X12)) = ((X10 @ id @ X12 @ id)))) & (hoasinduction_lem3aaa <=> ! [X24 : term,X10 : (subst > term > subst > $o)] : (~ ! [X13 : term] : ((X10 @ id @ (sub @ X24 @ (push @ X13 @ id)) @ id) | ~(X10 @ id @ X13 @ id)) | ~ ! [X25 : term,X21 : (subst > term > term)] : (~ ! [X27 : term,X28 : subst,X26 : subst] : (((sub @ (X21 @ X26 @ X27) @ X28)) = ((sub @ (sub @ X25 @ (push @ X27 @ X26)) @ X28))) | ~ ! [X8 : term] : ((X10 @ id @ (X21 @ id @ X8) @ id) | ~(X10 @ id @ X8 @ id)) | ~ ! [X29 : subst,X31 : subst,X30 : term] : (((X21 @ (comp @ X29 @ X31) @ (sub @ X30 @ X31))) = ((sub @ X25 @ (push @ (sub @ X30 @ X31) @ (comp @ X29 @ X31))))) | (X10 @ id @ (lam @ (X21 @ sh @ one)) @ id)) | (X10 @ id @ (lam @ (sub @ X24 @ (push @ one @ sh))) @ id))) & (! [X13 : term] : ~ ! [X21 : (subst > term > term)] : ~ ! [X8 : term,X9 : subst] : (((sub @ X13 @ (push @ X8 @ X9))) = ((X21 @ X9 @ X8))) <=> hoasinduction_lem3v2_f) & hoasinduction_lem3_gthm & pushprop_lem2v2_lthm & ((! [X10 : (subst > term > subst > $o)] : ~ ! [X11 : (term > $o)] : ~ ! [X12 : term] : (((X11 @ X12)) = ((X10 @ id @ X12 @ id))) => (! [X19 : term,X10 : (term > $o)] : ((X10 @ X19) | ~ ! [X8 : term] : ((X10 @ (lam @ X8)) | ~ ! [X13 : term] : ((X10 @ (sub @ X8 @ (push @ X13 @ id))) | ~(X10 @ X13))) | ~ ! [X13 : term,X8 : term] : (~(X10 @ X13) | ~(X10 @ X8) | (X10 @ (ap @ X8 @ X13))) | ~ ! [X8 : term] : (~(var @ X8) | (X10 @ X8))) => (! [X8 : term] : (X8 = ((sub @ X8 @ id))) => (! [X11 : (term > $o),X20 : term,X10 : (subst > term > subst > $o)] : (~ ! [X21 : (subst > term > term)] : (~ ! [X9 : subst,X22 : subst,X8 : term] : (((sub @ (X21 @ X9 @ X8) @ X22)) = ((X21 @ (comp @ X9 @ X22) @ (sub @ X8 @ X22)))) | (X10 @ id @ (lam @ (X21 @ sh @ one)) @ id) | ~ ! [X8 : term] : (~(X10 @ id @ X8 @ id) | (X10 @ id @ (X21 @ id @ X8) @ id))) | ~ ! [X13 : term] : ((X11 @ (sub @ X20 @ (push @ X13 @ id))) | ~(X11 @ X13)) | (X11 @ (lam @ X20)) | ~ ! [X12 : term] : (((X11 @ X12)) = ((X10 @ id @ X12 @ id)))) => ! [X33 : term,X10 : (subst > term > subst > $o)] : (~ ! [X8 : term] : ((X10 @ id @ X8 @ id) | ~(var @ (sub @ X8 @ id))) | (X10 @ id @ X33 @ id) | ~ ! [X21 : (subst > term > term)] : (~ ! [X9 : subst,X8 : term,X22 : subst] : (((sub @ (X21 @ X9 @ X8) @ X22)) = ((X21 @ (comp @ X9 @ X22) @ (sub @ X8 @ X22)))) | (X10 @ id @ (lam @ (X21 @ sh @ one)) @ id) | ~ ! [X8 : term] : (~(X10 @ id @ X8 @ id) | (X10 @ id @ (X21 @ id @ X8) @ id))) | ~ ! [X13 : term,X8 : term] : (~(X10 @ id @ X8 @ id) | (X10 @ id @ (ap @ (sub @ X8 @ id) @ X13) @ id) | ~(X10 @ id @ X13 @ id)) | ~ ! [X9 : subst,X8 : term,X22 : subst,X34 : subst] : ((X10 @ (comp @ X9 @ X34) @ (sub @ X8 @ X34) @ X22) | ~(X10 @ X9 @ X8 @ (comp @ X34 @ X22))) | ~ ! [X34 : subst,X8 : term,X22 : subst,X9 : subst] : ((X10 @ X9 @ X8 @ (comp @ X34 @ X22)) | ~(X10 @ (comp @ X9 @ X34) @ (sub @ X8 @ X34) @ X22))))))) <=> hoasinduction_lthm_3) & (sh = ((d2subst @ d_id))) & (hoaslam = (^[X47 : subst, X48 : (subst > term > term)] : ((d2term @ d_one)))) & (pushprop_lem0_lthm <=> ! [X8 : term,X10 : (term > $o),X9 : subst] : ~ ! [X11 : (term > $o)] : ~ ! [X12 : term] : (((X11 @ X12)) = ((X10 @ (sub @ X12 @ (push @ X8 @ X9)))))) & (hoasinduction_lem2 <=> ! [X10 : (subst > term > subst > $o),X45 : term,X46 : term] : (~ ! [X9 : subst,X22 : subst,X34 : subst,X8 : term] : (~(X10 @ (comp @ X9 @ X34) @ (sub @ X8 @ X34) @ X22) | (X10 @ X9 @ X8 @ (comp @ X34 @ X22))) | ~(X10 @ id @ X45 @ id) | (X10 @ id @ (ap @ X45 @ X46) @ id) | ~ ! [X8 : term,X13 : term] : ((X10 @ id @ (ap @ (sub @ X8 @ id) @ X13) @ id) | ~(X10 @ id @ X8 @ id) | ~(X10 @ id @ X13 @ id)) | ~ ! [X22 : subst,X8 : term,X9 : subst,X34 : subst] : (~(X10 @ X9 @ X8 @ (comp @ X34 @ X22)) | (X10 @ (comp @ X9 @ X34) @ (sub @ X8 @ X34) @ X22)) | ~(X10 @ id @ X46 @ id))) & (! [X18 : subst,X17 : term,X10 : (term > $o)] : (~ ! [X13 : term] : (~(var @ X13) | (X10 @ (sub @ X13 @ X18))) | ~ ! [X8 : term] : ((X10 @ (lam @ X8)) | ~ ! [X13 : term] : ((X10 @ (sub @ X8 @ (push @ X13 @ id))) | ~(X10 @ X13))) | (X10 @ (sub @ X17 @ X18)) | ~ ! [X8 : term,X13 : term] : (~(X10 @ X13) | (X10 @ (ap @ X8 @ X13)) | ~(X10 @ X8))) <=> induction2lem) & pushprop & (! [X13 : term] : ~ ! [X21 : (subst > term > term)] : ~ ! [X8 : term,X9 : subst] : (((sub @ X13 @ (push @ X8 @ X9))) = ((X21 @ X9 @ X8))) <=> hoasinduction_lem3v2_f_lthm) & ~ulamvar1 & (hoasinduction_lem3v2a_lthm <=> (! [X13 : term] : ~ ! [X21 : (subst > term > term)] : ~ ! [X8 : term,X9 : subst] : (((sub @ X13 @ (push @ X8 @ X9))) = ((X21 @ X9 @ X8))) => (! [X8 : term] : (X8 = ((sub @ X8 @ id))) => ! [X11 : (term > $o),X20 : term,X10 : (subst > term > subst > $o)] : ((X11 @ (lam @ X20)) | ~ ! [X21 : (subst > term > term)] : (~ ! [X22 : subst,X9 : subst,X8 : term] : (((sub @ (X21 @ X9 @ X8) @ X22)) = ((X21 @ (comp @ X9 @ X22) @ (sub @ X8 @ X22)))) | (X10 @ id @ (lam @ (X21 @ sh @ one)) @ id) | ~ ! [X8 : term] : (~(X10 @ id @ X8 @ id) | (X10 @ id @ (X21 @ id @ X8) @ id))) | ~ ! [X12 : term] : (((X11 @ X12)) = ((X10 @ id @ X12 @ id))) | ~ ! [X13 : term] : (~(X11 @ X13) | (X11 @ (sub @ X20 @ (push @ X13 @ id)))))))) & hoasinduction_lem1v2 & (hoasinduction_lthm <=> (! [X19 : term,X10 : (term > $o)] : (~ ! [X8 : term] : (~(var @ X8) | (X10 @ X8)) | ~ ! [X8 : term] : ((X10 @ (lam @ X8)) | ~ ! [X13 : term] : (~(X10 @ X13) | (X10 @ (sub @ X8 @ (push @ X13 @ id))))) | (X10 @ X19) | ~ ! [X13 : term,X8 : term] : (~(X10 @ X13) | ~(X10 @ X8) | (X10 @ (ap @ X8 @ X13)))) => (! [X46 : term,X10 : (subst > term > subst > $o),X45 : term] : (~ ! [X34 : subst,X8 : term,X9 : subst,X22 : subst] : ((X10 @ (comp @ X9 @ X34) @ (sub @ X8 @ X34) @ X22) | ~(X10 @ X9 @ X8 @ (comp @ X34 @ X22))) | ~(X10 @ id @ X46 @ id) | ~(X10 @ id @ X45 @ id) | (X10 @ id @ (ap @ X45 @ X46) @ id) | ~ ! [X34 : subst,X9 : subst,X8 : term,X22 : subst] : (~(X10 @ (comp @ X9 @ X34) @ (sub @ X8 @ X34) @ X22) | (X10 @ X9 @ X8 @ (comp @ X34 @ X22))) | ~ ! [X13 : term,X8 : term] : (~(X10 @ id @ X13 @ id) | ~(X10 @ id @ X8 @ id) | (X10 @ id @ (ap @ (sub @ X8 @ id) @ X13) @ id))) => (! [X10 : (subst > term > subst > $o),X39 : term] : (~ ! [X8 : term,X9 : subst,X34 : subst,X22 : subst] : ((X10 @ X9 @ X8 @ (comp @ X34 @ X22)) | ~(X10 @ (comp @ X9 @ X34) @ (sub @ X8 @ X34) @ X22)) | ~ ! [X21 : (subst > term > term)] : (~ ! [X9 : subst,X22 : subst,X8 : term] : (((sub @ (X21 @ X9 @ X8) @ X22)) = ((X21 @ (comp @ X9 @ X22) @ (sub @ X8 @ X22)))) | (X10 @ id @ (lam @ (X21 @ sh @ one)) @ id) | ~ ! [X8 : term] : (~(X10 @ id @ X8 @ id) | (X10 @ id @ (X21 @ id @ X8) @ id))) | ~ ! [X34 : subst,X8 : term,X22 : subst,X9 : subst] : ((X10 @ (comp @ X9 @ X34) @ (sub @ X8 @ X34) @ X22) | ~(X10 @ X9 @ X8 @ (comp @ X34 @ X22))) | (X10 @ id @ (lam @ X39) @ id) | ~ ! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (sub @ X39 @ (push @ X13 @ id)) @ id))) => ! [X33 : term,X10 : (subst > term > subst > $o)] : (~ ! [X8 : term,X13 : term] : ((X10 @ id @ (ap @ (sub @ X8 @ id) @ X13) @ id) | ~(X10 @ id @ X13 @ id) | ~(X10 @ id @ X8 @ id)) | (X10 @ id @ X33 @ id) | ~ ! [X21 : (subst > term > term)] : (~ ! [X8 : term] : ((X10 @ id @ (X21 @ id @ X8) @ id) | ~(X10 @ id @ X8 @ id)) | ~ ! [X22 : subst,X9 : subst,X8 : term] : (((sub @ (X21 @ X9 @ X8) @ X22)) = ((X21 @ (comp @ X9 @ X22) @ (sub @ X8 @ X22)))) | (X10 @ id @ (lam @ (X21 @ sh @ one)) @ id)) | ~ ! [X8 : term,X34 : subst,X22 : subst,X9 : subst] : (~(X10 @ X9 @ X8 @ (comp @ X34 @ X22)) | (X10 @ (comp @ X9 @ X34) @ (sub @ X8 @ X34) @ X22)) | ~ ! [X8 : term] : ((X10 @ id @ X8 @ id) | ~(var @ (sub @ X8 @ id))) | ~ ! [X22 : subst,X34 : subst,X9 : subst,X8 : term] : ((X10 @ X9 @ X8 @ (comp @ X34 @ X22)) | ~(X10 @ (comp @ X9 @ X34) @ (sub @ X8 @ X34) @ X22))))))) & (pushprop_lem1v2_lthm <=> (! [X9 : subst,X8 : term] : (X8 = ((sub @ one @ (push @ X8 @ X9)))) => ! [X11 : (term > $o),X9 : subst,X10 : (term > $o),X8 : term] : (~(X10 @ X8) | (X11 @ one) | ~ ! [X12 : term] : (((X11 @ X12)) = ((X10 @ (sub @ X12 @ (push @ X8 @ X9)))))))) & (! [X44 : term,X11 : (term > $o),X10 : (subst > term > subst > $o),X43 : term] : (~ ! [X8 : term,X34 : subst,X9 : subst,X22 : subst] : (~(X10 @ X9 @ X8 @ (comp @ X34 @ X22)) | (X10 @ (comp @ X9 @ X34) @ (sub @ X8 @ X34) @ X22)) | ~(X11 @ X44) | ~ ! [X8 : term,X9 : subst,X34 : subst,X22 : subst] : (~(X10 @ (comp @ X9 @ X34) @ (sub @ X8 @ X34) @ X22) | (X10 @ X9 @ X8 @ (comp @ X34 @ X22))) | (X11 @ (ap @ X43 @ X44)) | ~ ! [X12 : term] : (((X11 @ X12)) = ((X10 @ id @ X12 @ id))) | ~(X11 @ X43) | ~ ! [X8 : term,X13 : term] : (~(X10 @ id @ X8 @ id) | (X10 @ id @ (ap @ (sub @ X8 @ id) @ X13) @ id) | ~(X10 @ id @ X13 @ id))) <=> hoasinduction_lem2v2) & pushprop_lem1v2_gthm & (pushprop_lem1 <=> ! [X10 : (term > $o),X8 : term,X34 : (term > $o),X9 : subst,X13 : term] : ((X34 @ (sub @ X8 @ (push @ X13 @ X9))) | ~(X10 @ X8))) & hoasapinj2_gthm & pushprop_lthm_orig & axabs & ! [X2 : d_term,X3 : d_term] : ((((d2term @ X2)) = ((d2term @ X3))) => (X2 = X3)) & induction2_gthm & (((comp @ (d2subst @ d_id) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & ~(var @ (d2term @ d_one)) & (! [X8 : term,X9 : subst] : (X9 = ((comp @ sh @ (push @ X8 @ X9)))) <=> axshiftcons) & hoasinduction_lem3b_gthm & (one = ((d2term @ d_one))) & hoasapnotvar & hoasinduction_lem3a_gthm & hoaslamnotap_gthm & ! [X5 : d_subst] : (X5 = d_id) & hoasinduction_lem2_lthm & (((lam @ (d2term @ d_one))) = ((d2term @ d_one))) & axassoc & (hoasinduction_no_psi_cond_lthm <=> (! [X10 : (subst > term > subst > $o)] : ~ ! [X11 : (term > $o)] : ~ ! [X12 : term] : (((X11 @ X12)) = ((X10 @ id @ X12 @ id))) => (! [X19 : term,X10 : (term > $o)] : ((X10 @ X19) | ~ ! [X8 : term] : ((X10 @ X8) | ~(var @ X8)) | ~ ! [X8 : term,X13 : term] : (~(X10 @ X8) | ~(X10 @ X13) | (X10 @ (ap @ X8 @ X13))) | ~ ! [X8 : term] : (~ ! [X13 : term] : ((X10 @ (sub @ X8 @ (push @ X13 @ id))) | ~(X10 @ X13)) | (X10 @ (lam @ X8)))) => (! [X8 : term] : (X8 = ((sub @ X8 @ id))) => (! [X10 : (subst > term > subst > $o),X20 : term,X11 : (term > $o)] : (~ ! [X21 : (subst > term > term)] : (~ ! [X22 : subst,X9 : subst,X8 : term] : (((sub @ (X21 @ X9 @ X8) @ X22)) = ((X21 @ (comp @ X9 @ X22) @ (sub @ X8 @ X22)))) | ~ ! [X8 : term] : ((X10 @ id @ (X21 @ id @ X8) @ id) | ~(X10 @ id @ X8 @ id)) | (X10 @ id @ (lam @ (X21 @ sh @ one)) @ id)) | ~ ! [X13 : term] : (~(X11 @ X13) | (X11 @ (sub @ X20 @ (push @ X13 @ id)))) | (X11 @ (lam @ X20)) | ~ ! [X12 : term] : (((X11 @ X12)) = ((X10 @ id @ X12 @ id)))) => ! [X23 : term,X10 : (subst > term > subst > $o)] : ((X10 @ id @ X23 @ id) | ~ ! [X21 : (subst > term > term)] : (~ ! [X8 : term] : ((X10 @ id @ (X21 @ id @ X8) @ id) | ~(X10 @ id @ X8 @ id)) | ~ ! [X8 : term,X22 : subst,X9 : subst] : (((sub @ (X21 @ X9 @ X8) @ X22)) = ((X21 @ (comp @ X9 @ X22) @ (sub @ X8 @ X22)))) | (X10 @ id @ (lam @ (X21 @ sh @ one)) @ id)) | ~ ! [X8 : term,X13 : term] : (~(X10 @ id @ X8 @ id) | (X10 @ id @ (ap @ (sub @ X8 @ id) @ X13) @ id) | ~(X10 @ id @ X13 @ id)))))))) & axapp & ((! [X9 : subst,X8 : term] : (X8 = ((sub @ one @ (push @ X8 @ X9)))) => (! [X8 : term,X9 : subst] : (X9 = ((comp @ sh @ (push @ X8 @ X9)))) => (! [X8 : term,X13 : term] : ((((lam @ X8)) != ((lam @ X13))) | (X8 = X13)) => ! [X21 : (subst > term > term),X40 : (subst > term > term),X41 : subst,X42 : term] : ((((X21 @ X41 @ X42)) = ((X40 @ X41 @ X42))) | ~ ! [X9 : subst,X22 : subst,X8 : term] : (((sub @ (X40 @ X9 @ X8) @ X22)) = ((X40 @ (comp @ X9 @ X22) @ (sub @ X8 @ X22)))) | (((lam @ (X21 @ sh @ one))) != ((lam @ (X40 @ sh @ one)))) | ~ ! [X9 : subst,X8 : term,X22 : subst] : (((sub @ (X21 @ X9 @ X8) @ X22)) = ((X21 @ (comp @ X9 @ X22) @ (sub @ X8 @ X22)))))))) <=> hoaslaminj_lthm) & pushprop_gthm & (! [X9 : subst] : (X9 = ((comp @ X9 @ id))) <=> axidr) & pushprop_lem1_gthm & (! [X14 : term,X13 : term,X8 : term,X15 : term] : ((X14 = X15) | (((ap @ (sub @ X8 @ id) @ X14)) != ((ap @ (sub @ X13 @ id) @ X15)))) <=> hoasapinj2) & (substmonoid <=> (! [X9 : subst] : (X9 = ((comp @ id @ X9))) & ! [X9 : subst] : (X9 = ((comp @ X9 @ id))))) & axmap & (((sub @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2term @ d_one))) & (! [X13 : term,X8 : term,X15 : term,X14 : term] : ((X14 = X15) | (((ap @ X8 @ X14)) != ((ap @ X13 @ X15)))) <=> apinj2) & (hoasinduction_lem3aa_lthm <=> ! [X10 : (subst > term > subst > $o),X32 : term] : (~ ! [X21 : (subst > term > term)] : (~ ! [X8 : term] : ((X10 @ id @ (X21 @ id @ X8) @ id) | ~(X10 @ id @ X8 @ id)) | ~ ! [X22 : subst,X9 : subst,X8 : term] : (((sub @ (X21 @ X9 @ X8) @ X22)) = ((X21 @ (comp @ X9 @ X22) @ (sub @ X8 @ X22)))) | (X10 @ id @ (lam @ (X21 @ sh @ one)) @ id)) | ~ ! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (sub @ X32 @ (push @ X13 @ id)) @ id)) | (X10 @ id @ (lam @ (sub @ X32 @ (push @ one @ sh))) @ id))) & hoasinduction_lem1_gthm & hoaslamnotvar & (termmset <=> ! [X8 : term] : (X8 = ((sub @ X8 @ id)))) & pushprop_lem2v2 & ulamvarind & (! [X36 : term,X37 : term,X21 : (subst > term > term)] : (~ ! [X22 : subst,X9 : subst,X8 : term] : (((sub @ (X21 @ X9 @ X8) @ X22)) = ((X21 @ (comp @ X9 @ X22) @ (sub @ X8 @ X22)))) | (((lam @ (X21 @ sh @ one))) != ((ap @ (sub @ X36 @ id) @ X37)))) <=> hoaslamnotap) & hoasapnotvar_gthm & (axscons <=> ! [X9 : subst] : (X9 = ((push @ (sub @ one @ X9) @ (comp @ sh @ X9))))) & lamnotvar & (! [X8 : term,X14 : term,X15 : term,X13 : term] : ((X8 = X13) | (((ap @ (sub @ X8 @ id) @ X14)) != ((ap @ (sub @ X13 @ id) @ X15)))) <=> hoasapinj1) & termmset_gthm & (hoasinduction_lem3v2 <=> ! [X38 : term,X11 : (term > $o),X10 : (subst > term > subst > $o)] : (~ ! [X12 : term] : (((X11 @ X12)) = ((X10 @ id @ X12 @ id))) | ~ ! [X8 : term,X22 : subst,X34 : subst,X9 : subst] : (~(X10 @ X9 @ X8 @ (comp @ X34 @ X22)) | (X10 @ (comp @ X9 @ X34) @ (sub @ X8 @ X34) @ X22)) | ~ ! [X21 : (subst > term > term)] : ((X10 @ id @ (lam @ (X21 @ sh @ one)) @ id) | ~ ! [X9 : subst,X22 : subst,X8 : term] : (((sub @ (X21 @ X9 @ X8) @ X22)) = ((X21 @ (comp @ X9 @ X22) @ (sub @ X8 @ X22)))) | ~ ! [X8 : term] : ((X10 @ id @ (X21 @ id @ X8) @ id) | ~(X10 @ id @ X8 @ id))) | ~ ! [X22 : subst,X34 : subst,X8 : term,X9 : subst] : ((X10 @ X9 @ X8 @ (comp @ X34 @ X22)) | ~(X10 @ (comp @ X9 @ X34) @ (sub @ X8 @ X34) @ X22)) | (X11 @ (lam @ X38)) | ~ ! [X13 : term] : (~(X11 @ X13) | (X11 @ (sub @ X38 @ (push @ X13 @ id)))))) & (! [X40 : (subst > term > term),X21 : (subst > term > term),X42 : term,X41 : subst] : ((((lam @ (X21 @ sh @ one))) != ((lam @ (X40 @ sh @ one)))) | ~ ! [X22 : subst,X8 : term,X9 : subst] : (((sub @ (X40 @ X9 @ X8) @ X22)) = ((X40 @ (comp @ X9 @ X22) @ (sub @ X8 @ X22)))) | ~ ! [X9 : subst,X22 : subst,X8 : term] : (((sub @ (X21 @ X9 @ X8) @ X22)) = ((X21 @ (comp @ X9 @ X22) @ (sub @ X8 @ X22)))) | (((X21 @ X41 @ X42)) = ((X40 @ X41 @ X42)))) <=> hoaslaminj) & substmonoid_gthm & (hoasinduction_lem3v2_lthm <=> (! [X8 : term] : (X8 = ((sub @ X8 @ id))) => ! [X10 : (subst > term > subst > $o),X11 : (term > $o),X38 : term] : (~ ! [X8 : term,X34 : subst,X22 : subst,X9 : subst] : ((X10 @ X9 @ X8 @ (comp @ X34 @ X22)) | ~(X10 @ (comp @ X9 @ X34) @ (sub @ X8 @ X34) @ X22)) | (X11 @ (lam @ X38)) | ~ ! [X12 : term] : (((X11 @ X12)) = ((X10 @ id @ X12 @ id))) | ~ ! [X21 : (subst > term > term)] : (~ ! [X8 : term,X9 : subst,X22 : subst] : (((sub @ (X21 @ X9 @ X8) @ X22)) = ((X21 @ (comp @ X9 @ X22) @ (sub @ X8 @ X22)))) | ~ ! [X8 : term] : (~(X10 @ id @ X8 @ id) | (X10 @ id @ (X21 @ id @ X8) @ id)) | (X10 @ id @ (lam @ (X21 @ sh @ one)) @ id)) | ~ ! [X22 : subst,X34 : subst,X8 : term,X9 : subst] : (~(X10 @ X9 @ X8 @ (comp @ X34 @ X22)) | (X10 @ (comp @ X9 @ X34) @ (sub @ X8 @ X34) @ X22)) | ~ ! [X13 : term] : (~(X11 @ X13) | (X11 @ (sub @ X38 @ (push @ X13 @ id))))))) & (! [X35 : term,X10 : (subst > term > subst > $o)] : (~ ! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (sub @ X35 @ (push @ X13 @ id)) @ id)) | ~ ! [X21 : (subst > term > term)] : (~ ! [X8 : term] : (~(X10 @ id @ X8 @ id) | (X10 @ id @ (X21 @ id @ X8) @ id)) | (X10 @ id @ (lam @ (X21 @ sh @ one)) @ id) | ~ ! [X9 : subst,X22 : subst,X8 : term] : (((sub @ (X21 @ X9 @ X8) @ X22)) = ((X21 @ (comp @ X9 @ X22) @ (sub @ X8 @ X22))))) | (X10 @ id @ (lam @ X35) @ id)) <=> hoasinduction_lem3a) & (! [X10 : (term > $o),X19 : term] : (~ ! [X8 : term] : (~(var @ X8) | (X10 @ X8)) | ~ ! [X8 : term] : ((X10 @ (lam @ X8)) | ~ ! [X13 : term] : (~(X10 @ X13) | (X10 @ (sub @ X8 @ (push @ X13 @ id))))) | ~ ! [X8 : term,X13 : term] : (~(X10 @ X8) | ~(X10 @ X13) | (X10 @ (ap @ X8 @ X13))) | (X10 @ X19)) <=> induction2) & ! [X4 : subst] : ? [X5 : d_subst] : (X4 = ((d2subst @ X5))) & (hoasinduction_lem3v2a <=> ! [X10 : (subst > term > subst > $o),X11 : (term > $o),X20 : term] : (~ ! [X13 : term] : ((X11 @ (sub @ X20 @ (push @ X13 @ id))) | ~(X11 @ X13)) | ~ ! [X12 : term] : (((X11 @ X12)) = ((X10 @ id @ X12 @ id))) | ~ ! [X21 : (subst > term > term)] : ((X10 @ id @ (lam @ (X21 @ sh @ one)) @ id) | ~ ! [X8 : term] : ((X10 @ id @ (X21 @ id @ X8) @ id) | ~(X10 @ id @ X8 @ id)) | ~ ! [X22 : subst,X8 : term,X9 : subst] : (((sub @ (X21 @ X9 @ X8) @ X22)) = ((X21 @ (comp @ X9 @ X22) @ (sub @ X8 @ X22))))) | (X11 @ (lam @ X20)))) & (! [X13 : term,X8 : term,X14 : term,X15 : term] : ((((ap @ X8 @ X14)) != ((ap @ X13 @ X15))) | (X8 = X13)) <=> apinj1) & hoaslamnotap_lthm & (((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) = ((d2term @ d_one))) & (((push @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & (! [X10 : (term > $o),X16 : term] : (~ ! [X8 : term] : (~(var @ X8) | (X10 @ X8)) | ~ ! [X8 : term] : ((X10 @ (lam @ X8)) | ~(X10 @ X8)) | (X10 @ X16) | ~ ! [X8 : term,X13 : term] : ((X10 @ (ap @ X8 @ X13)) | ~(X10 @ X13) | ~(X10 @ X8))) <=> induction) & hoaslaminj_gthm & hoasinduction_lem1v2_gthm & hoaslamnotvar_lthm & (hoasinduction_lem3a_lthm <=> (! [X8 : term] : (X8 = ((sub @ X8 @ id))) => (! [X32 : term,X10 : (subst > term > subst > $o)] : (~ ! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (sub @ X32 @ (push @ X13 @ id)) @ id)) | ~ ! [X21 : (subst > term > term)] : ((X10 @ id @ (lam @ (X21 @ sh @ one)) @ id) | ~ ! [X8 : term] : ((X10 @ id @ (X21 @ id @ X8) @ id) | ~(X10 @ id @ X8 @ id)) | ~ ! [X9 : subst,X22 : subst,X8 : term] : (((sub @ (X21 @ X9 @ X8) @ X22)) = ((X21 @ (comp @ X9 @ X22) @ (sub @ X8 @ X22))))) | (X10 @ id @ (lam @ (sub @ X32 @ (push @ one @ sh))) @ id)) => ! [X10 : (subst > term > subst > $o),X35 : term] : (~ ! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (sub @ X35 @ (push @ X13 @ id)) @ id)) | (X10 @ id @ (lam @ X35) @ id) | ~ ! [X21 : (subst > term > term)] : (~ ! [X8 : term] : (~(X10 @ id @ X8 @ id) | (X10 @ id @ (X21 @ id @ X8) @ id)) | ~ ! [X9 : subst,X8 : term,X22 : subst] : (((sub @ (X21 @ X9 @ X8) @ X22)) = ((X21 @ (comp @ X9 @ X22) @ (sub @ X8 @ X22)))) | (X10 @ id @ (lam @ (X21 @ sh @ one)) @ id)))))) & (axvarcons <=> ! [X8 : term,X9 : subst] : (X8 = ((sub @ one @ (push @ X8 @ X9))))) & (! [X9 : subst] : (X9 = ((comp @ id @ X9))) <=> axidl) & (! [X23 : term,X10 : (subst > term > subst > $o)] : (~ ! [X21 : (subst > term > term)] : (~ ! [X8 : term] : ((X10 @ id @ (X21 @ id @ X8) @ id) | ~(X10 @ id @ X8 @ id)) | (X10 @ id @ (lam @ (X21 @ sh @ one)) @ id) | ~ ! [X9 : subst,X8 : term,X22 : subst] : (((sub @ (X21 @ X9 @ X8) @ X22)) = ((X21 @ (comp @ X9 @ X22) @ (sub @ X8 @ X22))))) | (X10 @ id @ X23 @ id) | ~ ! [X8 : term,X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (ap @ (sub @ X8 @ id) @ X13) @ id) | ~(X10 @ id @ X8 @ id))) <=> hoasinduction_no_psi_cond) & (induction2lem_lthm <=> (! [X8 : term,X9 : subst] : (X8 = ((sub @ one @ (push @ X8 @ X9)))) => (! [X9 : subst,X8 : term] : (X9 = ((comp @ sh @ (push @ X8 @ X9)))) => (! [X9 : subst] : (X9 = ((comp @ X9 @ id))) => (! [X16 : term,X10 : (term > $o)] : (~ ! [X8 : term] : (~(X10 @ X8) | (X10 @ (lam @ X8))) | ~ ! [X8 : term,X13 : term] : ((X10 @ (ap @ X8 @ X13)) | ~(X10 @ X13) | ~(X10 @ X8)) | ~ ! [X8 : term] : (~(var @ X8) | (X10 @ X8)) | (X10 @ X16)) => ! [X10 : (term > $o),X17 : term,X18 : subst] : ((X10 @ (sub @ X17 @ X18)) | ~ ! [X13 : term] : ((X10 @ (sub @ X13 @ X18)) | ~(var @ X13)) | ~ ! [X13 : term,X8 : term] : (~(X10 @ X13) | (X10 @ (ap @ X8 @ X13)) | ~(X10 @ X8)) | ~ ! [X8 : term] : (~ ! [X13 : term] : (~(X10 @ X13) | (X10 @ (sub @ X8 @ (push @ X13 @ id)))) | (X10 @ (lam @ X8))))))))) & (hoasinduction_lem3_lthm <=> (! [X8 : term] : (X8 = ((sub @ X8 @ id))) => (! [X32 : term,X10 : (subst > term > subst > $o)] : (~ ! [X21 : (subst > term > term)] : (~ ! [X8 : term] : ((X10 @ id @ (X21 @ id @ X8) @ id) | ~(X10 @ id @ X8 @ id)) | (X10 @ id @ (lam @ (X21 @ sh @ one)) @ id) | ~ ! [X8 : term,X9 : subst,X22 : subst] : (((sub @ (X21 @ X9 @ X8) @ X22)) = ((X21 @ (comp @ X9 @ X22) @ (sub @ X8 @ X22))))) | (X10 @ id @ (lam @ (sub @ X32 @ (push @ one @ sh))) @ id) | ~ ! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (sub @ X32 @ (push @ X13 @ id)) @ id))) => ! [X39 : term,X10 : (subst > term > subst > $o)] : (~ ! [X8 : term,X9 : subst,X22 : subst,X34 : subst] : (~(X10 @ X9 @ X8 @ (comp @ X34 @ X22)) | (X10 @ (comp @ X9 @ X34) @ (sub @ X8 @ X34) @ X22)) | ~ ! [X21 : (subst > term > term)] : (~ ! [X8 : term] : ((X10 @ id @ (X21 @ id @ X8) @ id) | ~(X10 @ id @ X8 @ id)) | ~ ! [X8 : term,X22 : subst,X9 : subst] : (((sub @ (X21 @ X9 @ X8) @ X22)) = ((X21 @ (comp @ X9 @ X22) @ (sub @ X8 @ X22)))) | (X10 @ id @ (lam @ (X21 @ sh @ one)) @ id)) | ~ ! [X13 : term] : ((X10 @ id @ (sub @ X39 @ (push @ X13 @ id)) @ id) | ~(X10 @ id @ X13 @ id)) | ~ ! [X22 : subst,X8 : term,X9 : subst,X34 : subst] : (~(X10 @ (comp @ X9 @ X34) @ (sub @ X8 @ X34) @ X22) | (X10 @ X9 @ X8 @ (comp @ X34 @ X22))) | (X10 @ id @ (lam @ X39) @ id))))) & hoasinduction_gthm & ((! [X15 : term,X8 : term,X14 : term,X13 : term] : ((((ap @ X8 @ X14)) != ((ap @ X13 @ X15))) | (X14 = X15)) => ! [X8 : term,X15 : term,X13 : term,X14 : term] : ((((ap @ (sub @ X8 @ id) @ X14)) != ((ap @ (sub @ X13 @ id) @ X15))) | (X14 = X15))) <=> hoasapinj2_lthm) & (! [X39 : term,X10 : (subst > term > subst > $o)] : ((X10 @ id @ (lam @ X39) @ id) | ~ ! [X13 : term] : ((X10 @ id @ (sub @ X39 @ (push @ X13 @ id)) @ id) | ~(X10 @ id @ X13 @ id)) | ~ ! [X21 : (subst > term > term)] : ((X10 @ id @ (lam @ (X21 @ sh @ one)) @ id) | ~ ! [X22 : subst,X8 : term,X9 : subst] : (((sub @ (X21 @ X9 @ X8) @ X22)) = ((X21 @ (comp @ X9 @ X22) @ (sub @ X8 @ X22)))) | ~ ! [X8 : term] : (~(X10 @ id @ X8 @ id) | (X10 @ id @ (X21 @ id @ X8) @ id))) | ~ ! [X9 : subst,X22 : subst,X34 : subst,X8 : term] : (~(X10 @ X9 @ X8 @ (comp @ X34 @ X22)) | (X10 @ (comp @ X9 @ X34) @ (sub @ X8 @ X34) @ X22)) | ~ ! [X8 : term,X9 : subst,X22 : subst,X34 : subst] : ((X10 @ X9 @ X8 @ (comp @ X34 @ X22)) | ~(X10 @ (comp @ X9 @ X34) @ (sub @ X8 @ X34) @ X22))) <=> hoasinduction_lem3) & axclos & (induction2_lthm <=> (! [X8 : term] : (X8 = ((sub @ X8 @ id))) => (! [X10 : (term > $o),X17 : term,X18 : subst] : (~ ! [X13 : term] : (~(var @ X13) | (X10 @ (sub @ X13 @ X18))) | (X10 @ (sub @ X17 @ X18)) | ~ ! [X8 : term] : ((X10 @ (lam @ X8)) | ~ ! [X13 : term] : (~(X10 @ X13) | (X10 @ (sub @ X8 @ (push @ X13 @ id))))) | ~ ! [X8 : term,X13 : term] : (~(X10 @ X8) | (X10 @ (ap @ X8 @ X13)) | ~(X10 @ X13))) => ! [X19 : term,X10 : (term > $o)] : (~ ! [X8 : term] : ((X10 @ X8) | ~(var @ X8)) | ~ ! [X8 : term,X13 : term] : ((X10 @ (ap @ X8 @ X13)) | ~(X10 @ X8) | ~(X10 @ X13)) | ~ ! [X8 : term] : ((X10 @ (lam @ X8)) | ~ ! [X13 : term] : ((X10 @ (sub @ X8 @ (push @ X13 @ id))) | ~(X10 @ X13))) | (X10 @ X19))))) & apnotvar & hoasinduction_lem3v2_gthm & hoaslamnotvar_gthm & (hoasinduction_lem3b_lthm <=> ! [X13 : term] : ~ ! [X21 : (subst > term > term)] : (((X21 @ sh @ one)) != ((sub @ X13 @ (push @ one @ sh))))) & pushprop_lthm & (shinj <=> ! [X13 : term,X8 : term] : ((((sub @ X8 @ sh)) != ((sub @ X13 @ sh))) | (X8 = X13))) & (! [X10 : (subst > term > subst > $o)] : ~ ! [X11 : (term > $o)] : ~ ! [X12 : term] : (((X11 @ X12)) = ((X10 @ id @ X12 @ id))) <=> hoasinduction_lem0) & (! [X32 : term,X10 : (subst > term > subst > $o)] : ((X10 @ id @ (lam @ (sub @ X32 @ (push @ one @ sh))) @ id) | ~ ! [X13 : term] : (~(X10 @ id @ X13 @ id) | (X10 @ id @ (sub @ X32 @ (push @ X13 @ id)) @ id)) | ~ ! [X21 : (subst > term > term)] : (~ ! [X8 : term] : ((X10 @ id @ (X21 @ id @ X8) @ id) | ~(X10 @ id @ X8 @ id)) | (X10 @ id @ (lam @ (X21 @ sh @ one)) @ id) | ~ ! [X8 : term,X22 : subst,X9 : subst] : (((sub @ (X21 @ X9 @ X8) @ X22)) = ((X21 @ (comp @ X9 @ X22) @ (sub @ X8 @ X22)))))) <=> hoasinduction_lem3aa) & (hoasinduction_lem3b <=> ! [X13 : term] : ~ ! [X21 : (subst > term > term)] : (((X21 @ sh @ one)) != ((sub @ X13 @ (push @ one @ sh))))) & (termmset_lthm <=> (! [X8 : term] : (X8 = ((sub @ X8 @ id))) => ! [X8 : term] : (X8 = ((sub @ X8 @ id))))) & ulamvarsh & ~lamnotap & (! [X9 : subst,X10 : (term > $o),X8 : term] : ~ ! [X11 : (term > $o)] : ~ ! [X12 : term] : (((X11 @ X12)) = ((X10 @ (sub @ X12 @ (push @ X8 @ X9))))) <=> pushprop_lem0) & induction2lem_gthm & (pushprop_lem1_lthm <=> (! [X8 : term,X9 : subst] : (X8 = ((sub @ one @ (push @ X8 @ X9)))) => (! [X9 : subst,X8 : term] : (X9 = ((comp @ sh @ (push @ X8 @ X9)))) => ! [X13 : term,X10 : (term > $o),X9 : subst,X8 : term,X34 : (term > $o)] : ((X34 @ (sub @ X8 @ (push @ X13 @ X9))) | ~(X10 @ X8))))) & (! [X10 : (term > $o),X11 : (term > $o),X9 : subst,X8 : term] : ((X11 @ one) | ~(X10 @ X8) | ~ ! [X12 : term] : (((X11 @ X12)) = ((X10 @ (sub @ X12 @ (push @ X8 @ X9)))))) <=> pushprop_lem1v2) & pushprop_lem3v2 & hoasapinj1_gthm & ! [X0 : term] : ? [X1 : d_term] : (X0 = ((d2term @ X1))) & pushprop_lem3v2_lthm),
  file('/export/starexec/sandbox/benchmark/theBenchmark.p',alg444_1)).
thf(f4,plain,(
  ~ (! [X0 : term,X1 : subst,X2 : subst] : (((sub @ (sub @ X0 @ X2) @ X1)) = ((sub @ X0 @ (comp @ X2 @ X1)))) = axclos)),
  inference(rectify,[],[f2])).
thf(f5,plain,(
  ~ (axclos = ((!! @ subst @ (^[Y0 : subst]: (!! @ subst @ (^[Y1 : subst]: (!! @ term @ (^[Y2 : term]: ((sub @ (sub @ Y2 @ Y0) @ Y1) = (sub @ Y2 @ (comp @ Y0 @ Y1)))))))))))),
  inference(fool_elimination,[],[f4])).
thf(f6,plain,(
  (((ap @ (d2term @ d_one) @ (d2term @ d_one))) = ((d2term @ d_one))) & hoasinduction_lem1 & pushprop_lem0_gthm & hoasapnotvar_lthm & (! [X0 : term,X1 : (subst > term > subst > $o)] : (~ ! [X2 : subst,X3 : subst,X4 : term,X5 : subst] : ((X1 @ (comp @ X5 @ X3) @ (sub @ X4 @ X3) @ X2) | ~(X1 @ X5 @ X4 @ (comp @ X3 @ X2))) | ~ ! [X6 : (subst > term > term)] : ((X1 @ id @ (lam @ (X6 @ sh @ one)) @ id) | ~ ! [X7 : subst,X8 : term,X9 : subst] : (((sub @ (X6 @ X9 @ X8) @ X7)) = ((X6 @ (comp @ X9 @ X7) @ (sub @ X8 @ X7)))) | ~ ! [X10 : term] : (~(X1 @ id @ X10 @ id) | (X1 @ id @ (X6 @ id @ X10) @ id))) | ~ ! [X11 : subst,X12 : subst,X13 : term,X14 : subst] : (~(X1 @ (comp @ X14 @ X12) @ (sub @ X13 @ X12) @ X11) | (X1 @ X14 @ X13 @ (comp @ X12 @ X11))) | (X1 @ id @ X0 @ id) | ~ ! [X15 : term] : ((X1 @ id @ X15 @ id) | ~(var @ (sub @ X15 @ id))) | ~ ! [X16 : term,X17 : term] : ((X1 @ id @ (ap @ (sub @ X16 @ id) @ X17) @ id) | ~(X1 @ id @ X16 @ id) | ~(X1 @ id @ X17 @ id))) <=> hoasinduction) & ! [X18 : d_term] : (d_one = X18) & axvarshift & ! [X19 : d_subst,X20 : d_subst] : ((((d2subst @ X19)) = ((d2subst @ X20))) => (X19 = X20)) & (laminj <=> ! [X21 : term,X22 : term] : ((X21 = X22) | (((lam @ X22)) != ((lam @ X21))))) & hoasinduction_lem1_lthm & ~(hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) & hoasinduction_lem2_gthm & hoasinduction_lem2v2_gthm & (id = ((d2subst @ d_id))) & (substmonoid_lthm <=> (! [X23 : subst] : (((comp @ id @ X23)) = X23) => (! [X24 : subst] : (((comp @ X24 @ id)) = X24) => (! [X25 : subst] : (((comp @ id @ X25)) = X25) & ! [X26 : subst] : (((comp @ X26 @ id)) = X26))))) & ((^[X27 : term, X28 : subst, X29 : (term > $o), X30 : (term > $o)] : (! [X31 : term] : (((X30 @ X31)) = ((X29 @ (sub @ X31 @ (push @ X27 @ X28))))))) = pushprop_p_and_p_prime) & ((^[X32 : (subst > term > subst > $o), X33 : (term > $o)] : (! [X34 : term] : (((X32 @ id @ X34 @ id)) = ((X33 @ X34))))) = hoasinduction_p_and_p_prime) & pushprop_lem2v2_gthm & (! [X35 : term] : (((sub @ X35 @ id)) = X35) <=> axvarid) & ((! [X36 : term] : (((sub @ X36 @ id)) = X36) => (! [X37 : term,X38 : term,X39 : term,X40 : term] : ((((ap @ X40 @ X38)) != ((ap @ X39 @ X37))) | (X39 = X40)) => ! [X41 : term,X42 : term,X43 : term,X44 : term] : ((X41 = X44) | (((ap @ (sub @ X44 @ id) @ X42)) != ((ap @ (sub @ X41 @ id) @ X43)))))) <=> hoasapinj1_lthm) & (hoasinduction_lem0_lthm <=> ! [X45 : (subst > term > subst > $o)] : ~ ! [X46 : (term > $o)] : ~ ! [X47 : term] : (((X45 @ id @ X47 @ id)) = ((X46 @ X47)))) & (hoasinduction_lem3aaa <=> ! [X48 : term,X49 : (subst > term > subst > $o)] : (~ ! [X50 : term] : ((X49 @ id @ (sub @ X48 @ (push @ X50 @ id)) @ id) | ~(X49 @ id @ X50 @ id)) | ~ ! [X51 : term,X52 : (subst > term > term)] : (~ ! [X53 : term,X54 : subst,X55 : subst] : (((sub @ (X52 @ X55 @ X53) @ X54)) = ((sub @ (sub @ X51 @ (push @ X53 @ X55)) @ X54))) | ~ ! [X56 : term] : ((X49 @ id @ (X52 @ id @ X56) @ id) | ~(X49 @ id @ X56 @ id)) | ~ ! [X57 : subst,X58 : subst,X59 : term] : (((X52 @ (comp @ X57 @ X58) @ (sub @ X59 @ X58))) = ((sub @ X51 @ (push @ (sub @ X59 @ X58) @ (comp @ X57 @ X58))))) | (X49 @ id @ (lam @ (X52 @ sh @ one)) @ id)) | (X49 @ id @ (lam @ (sub @ X48 @ (push @ one @ sh))) @ id))) & (! [X60 : term] : ~ ! [X61 : (subst > term > term)] : ~ ! [X62 : term,X63 : subst] : (((X61 @ X63 @ X62)) = ((sub @ X60 @ (push @ X62 @ X63)))) <=> hoasinduction_lem3v2_f) & hoasinduction_lem3_gthm & pushprop_lem2v2_lthm & ((! [X64 : (subst > term > subst > $o)] : ~ ! [X65 : (term > $o)] : ~ ! [X66 : term] : (((X64 @ id @ X66 @ id)) = ((X65 @ X66))) => (! [X67 : term,X68 : (term > $o)] : ((X68 @ X67) | ~ ! [X69 : term] : ((X68 @ (lam @ X69)) | ~ ! [X70 : term] : ((X68 @ (sub @ X69 @ (push @ X70 @ id))) | ~(X68 @ X70))) | ~ ! [X71 : term,X72 : term] : (~(X68 @ X71) | ~(X68 @ X72) | (X68 @ (ap @ X72 @ X71))) | ~ ! [X73 : term] : (~(var @ X73) | (X68 @ X73))) => (! [X74 : term] : (((sub @ X74 @ id)) = X74) => (! [X75 : (term > $o),X76 : term,X77 : (subst > term > subst > $o)] : (~ ! [X78 : (subst > term > term)] : (~ ! [X79 : subst,X80 : subst,X81 : term] : (((sub @ (X78 @ X79 @ X81) @ X80)) = ((X78 @ (comp @ X79 @ X80) @ (sub @ X81 @ X80)))) | (X77 @ id @ (lam @ (X78 @ sh @ one)) @ id) | ~ ! [X82 : term] : (~(X77 @ id @ X82 @ id) | (X77 @ id @ (X78 @ id @ X82) @ id))) | ~ ! [X83 : term] : ((X75 @ (sub @ X76 @ (push @ X83 @ id))) | ~(X75 @ X83)) | (X75 @ (lam @ X76)) | ~ ! [X84 : term] : (((X77 @ id @ X84 @ id)) = ((X75 @ X84)))) => ! [X85 : term,X86 : (subst > term > subst > $o)] : (~ ! [X87 : term] : ((X86 @ id @ X87 @ id) | ~(var @ (sub @ X87 @ id))) | (X86 @ id @ X85 @ id) | ~ ! [X88 : (subst > term > term)] : (~ ! [X89 : subst,X90 : term,X91 : subst] : (((X88 @ (comp @ X89 @ X91) @ (sub @ X90 @ X91))) = ((sub @ (X88 @ X89 @ X90) @ X91))) | (X86 @ id @ (lam @ (X88 @ sh @ one)) @ id) | ~ ! [X92 : term] : (~(X86 @ id @ X92 @ id) | (X86 @ id @ (X88 @ id @ X92) @ id))) | ~ ! [X93 : term,X94 : term] : (~(X86 @ id @ X94 @ id) | (X86 @ id @ (ap @ (sub @ X94 @ id) @ X93) @ id) | ~(X86 @ id @ X93 @ id)) | ~ ! [X95 : subst,X96 : term,X97 : subst,X98 : subst] : ((X86 @ (comp @ X95 @ X98) @ (sub @ X96 @ X98) @ X97) | ~(X86 @ X95 @ X96 @ (comp @ X98 @ X97))) | ~ ! [X99 : subst,X100 : term,X101 : subst,X102 : subst] : ((X86 @ X102 @ X100 @ (comp @ X99 @ X101)) | ~(X86 @ (comp @ X102 @ X99) @ (sub @ X100 @ X99) @ X101))))))) <=> hoasinduction_lthm_3) & (sh = ((d2subst @ d_id))) & (hoaslam = (^[X103 : subst, X104 : (subst > term > term)] : ((d2term @ d_one)))) & (pushprop_lem0_lthm <=> ! [X105 : term,X106 : (term > $o),X107 : subst] : ~ ! [X108 : (term > $o)] : ~ ! [X109 : term] : (((X106 @ (sub @ X109 @ (push @ X105 @ X107)))) = ((X108 @ X109)))) & (hoasinduction_lem2 <=> ! [X110 : (subst > term > subst > $o),X111 : term,X112 : term] : (~ ! [X113 : subst,X114 : subst,X115 : subst,X116 : term] : (~(X110 @ (comp @ X113 @ X115) @ (sub @ X116 @ X115) @ X114) | (X110 @ X113 @ X116 @ (comp @ X115 @ X114))) | ~(X110 @ id @ X111 @ id) | (X110 @ id @ (ap @ X111 @ X112) @ id) | ~ ! [X117 : term,X118 : term] : ((X110 @ id @ (ap @ (sub @ X117 @ id) @ X118) @ id) | ~(X110 @ id @ X117 @ id) | ~(X110 @ id @ X118 @ id)) | ~ ! [X119 : subst,X120 : term,X121 : subst,X122 : subst] : (~(X110 @ X121 @ X120 @ (comp @ X122 @ X119)) | (X110 @ (comp @ X121 @ X122) @ (sub @ X120 @ X122) @ X119)) | ~(X110 @ id @ X112 @ id))) & (! [X123 : subst,X124 : term,X125 : (term > $o)] : (~ ! [X126 : term] : (~(var @ X126) | (X125 @ (sub @ X126 @ X123))) | ~ ! [X127 : term] : ((X125 @ (lam @ X127)) | ~ ! [X128 : term] : ((X125 @ (sub @ X127 @ (push @ X128 @ id))) | ~(X125 @ X128))) | (X125 @ (sub @ X124 @ X123)) | ~ ! [X129 : term,X130 : term] : (~(X125 @ X130) | (X125 @ (ap @ X129 @ X130)) | ~(X125 @ X129))) <=> induction2lem) & pushprop & (! [X131 : term] : ~ ! [X132 : (subst > term > term)] : ~ ! [X133 : term,X134 : subst] : (((X132 @ X134 @ X133)) = ((sub @ X131 @ (push @ X133 @ X134)))) <=> hoasinduction_lem3v2_f_lthm) & ~ulamvar1 & (hoasinduction_lem3v2a_lthm <=> (! [X135 : term] : ~ ! [X136 : (subst > term > term)] : ~ ! [X137 : term,X138 : subst] : (((X136 @ X138 @ X137)) = ((sub @ X135 @ (push @ X137 @ X138)))) => (! [X139 : term] : (((sub @ X139 @ id)) = X139) => ! [X140 : (term > $o),X141 : term,X142 : (subst > term > subst > $o)] : ((X140 @ (lam @ X141)) | ~ ! [X143 : (subst > term > term)] : (~ ! [X144 : subst,X145 : subst,X146 : term] : (((sub @ (X143 @ X145 @ X146) @ X144)) = ((X143 @ (comp @ X145 @ X144) @ (sub @ X146 @ X144)))) | (X142 @ id @ (lam @ (X143 @ sh @ one)) @ id) | ~ ! [X147 : term] : (~(X142 @ id @ X147 @ id) | (X142 @ id @ (X143 @ id @ X147) @ id))) | ~ ! [X148 : term] : (((X140 @ X148)) = ((X142 @ id @ X148 @ id))) | ~ ! [X149 : term] : (~(X140 @ X149) | (X140 @ (sub @ X141 @ (push @ X149 @ id)))))))) & hoasinduction_lem1v2 & (hoasinduction_lthm <=> (! [X150 : term,X151 : (term > $o)] : (~ ! [X152 : term] : (~(var @ X152) | (X151 @ X152)) | ~ ! [X153 : term] : ((X151 @ (lam @ X153)) | ~ ! [X154 : term] : (~(X151 @ X154) | (X151 @ (sub @ X153 @ (push @ X154 @ id))))) | (X151 @ X150) | ~ ! [X155 : term,X156 : term] : (~(X151 @ X155) | ~(X151 @ X156) | (X151 @ (ap @ X156 @ X155)))) => (! [X157 : term,X158 : (subst > term > subst > $o),X159 : term] : (~ ! [X160 : subst,X161 : term,X162 : subst,X163 : subst] : ((X158 @ (comp @ X162 @ X160) @ (sub @ X161 @ X160) @ X163) | ~(X158 @ X162 @ X161 @ (comp @ X160 @ X163))) | ~(X158 @ id @ X157 @ id) | ~(X158 @ id @ X159 @ id) | (X158 @ id @ (ap @ X159 @ X157) @ id) | ~ ! [X164 : subst,X165 : subst,X166 : term,X167 : subst] : (~(X158 @ (comp @ X165 @ X164) @ (sub @ X166 @ X164) @ X167) | (X158 @ X165 @ X166 @ (comp @ X164 @ X167))) | ~ ! [X168 : term,X169 : term] : (~(X158 @ id @ X168 @ id) | ~(X158 @ id @ X169 @ id) | (X158 @ id @ (ap @ (sub @ X169 @ id) @ X168) @ id))) => (! [X170 : (subst > term > subst > $o),X171 : term] : (~ ! [X172 : term,X173 : subst,X174 : subst,X175 : subst] : ((X170 @ X173 @ X172 @ (comp @ X174 @ X175)) | ~(X170 @ (comp @ X173 @ X174) @ (sub @ X172 @ X174) @ X175)) | ~ ! [X176 : (subst > term > term)] : (~ ! [X177 : subst,X178 : subst,X179 : term] : (((sub @ (X176 @ X177 @ X179) @ X178)) = ((X176 @ (comp @ X177 @ X178) @ (sub @ X179 @ X178)))) | (X170 @ id @ (lam @ (X176 @ sh @ one)) @ id) | ~ ! [X180 : term] : (~(X170 @ id @ X180 @ id) | (X170 @ id @ (X176 @ id @ X180) @ id))) | ~ ! [X181 : subst,X182 : term,X183 : subst,X184 : subst] : ((X170 @ (comp @ X184 @ X181) @ (sub @ X182 @ X181) @ X183) | ~(X170 @ X184 @ X182 @ (comp @ X181 @ X183))) | (X170 @ id @ (lam @ X171) @ id) | ~ ! [X185 : term] : (~(X170 @ id @ X185 @ id) | (X170 @ id @ (sub @ X171 @ (push @ X185 @ id)) @ id))) => ! [X186 : term,X187 : (subst > term > subst > $o)] : (~ ! [X188 : term,X189 : term] : ((X187 @ id @ (ap @ (sub @ X188 @ id) @ X189) @ id) | ~(X187 @ id @ X189 @ id) | ~(X187 @ id @ X188 @ id)) | (X187 @ id @ X186 @ id) | ~ ! [X190 : (subst > term > term)] : (~ ! [X191 : term] : ((X187 @ id @ (X190 @ id @ X191) @ id) | ~(X187 @ id @ X191 @ id)) | ~ ! [X192 : subst,X193 : subst,X194 : term] : (((X190 @ (comp @ X193 @ X192) @ (sub @ X194 @ X192))) = ((sub @ (X190 @ X193 @ X194) @ X192))) | (X187 @ id @ (lam @ (X190 @ sh @ one)) @ id)) | ~ ! [X195 : term,X196 : subst,X197 : subst,X198 : subst] : (~(X187 @ X198 @ X195 @ (comp @ X196 @ X197)) | (X187 @ (comp @ X198 @ X196) @ (sub @ X195 @ X196) @ X197)) | ~ ! [X199 : term] : ((X187 @ id @ X199 @ id) | ~(var @ (sub @ X199 @ id))) | ~ ! [X200 : subst,X201 : subst,X202 : subst,X203 : term] : ((X187 @ X202 @ X203 @ (comp @ X201 @ X200)) | ~(X187 @ (comp @ X202 @ X201) @ (sub @ X203 @ X201) @ X200))))))) & (pushprop_lem1v2_lthm <=> (! [X204 : subst,X205 : term] : (((sub @ one @ (push @ X205 @ X204))) = X205) => ! [X206 : (term > $o),X207 : subst,X208 : (term > $o),X209 : term] : (~(X208 @ X209) | (X206 @ one) | ~ ! [X210 : term] : (((X208 @ (sub @ X210 @ (push @ X209 @ X207)))) = ((X206 @ X210)))))) & (! [X211 : term,X212 : (term > $o),X213 : (subst > term > subst > $o),X214 : term] : (~ ! [X215 : term,X216 : subst,X217 : subst,X218 : subst] : (~(X213 @ X217 @ X215 @ (comp @ X216 @ X218)) | (X213 @ (comp @ X217 @ X216) @ (sub @ X215 @ X216) @ X218)) | ~(X212 @ X211) | ~ ! [X219 : term,X220 : subst,X221 : subst,X222 : subst] : (~(X213 @ (comp @ X220 @ X221) @ (sub @ X219 @ X221) @ X222) | (X213 @ X220 @ X219 @ (comp @ X221 @ X222))) | (X212 @ (ap @ X214 @ X211)) | ~ ! [X223 : term] : (((X212 @ X223)) = ((X213 @ id @ X223 @ id))) | ~(X212 @ X214) | ~ ! [X224 : term,X225 : term] : (~(X213 @ id @ X224 @ id) | (X213 @ id @ (ap @ (sub @ X224 @ id) @ X225) @ id) | ~(X213 @ id @ X225 @ id))) <=> hoasinduction_lem2v2) & pushprop_lem1v2_gthm & (pushprop_lem1 <=> ! [X226 : (term > $o),X227 : term,X228 : (term > $o),X229 : subst,X230 : term] : ((X228 @ (sub @ X227 @ (push @ X230 @ X229))) | ~(X226 @ X227))) & hoasapinj2_gthm & pushprop_lthm_orig & axabs & ! [X231 : d_term,X232 : d_term] : ((((d2term @ X232)) = ((d2term @ X231))) => (X231 = X232)) & induction2_gthm & (((comp @ (d2subst @ d_id) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & ~(var @ (d2term @ d_one)) & (! [X233 : term,X234 : subst] : (((comp @ sh @ (push @ X233 @ X234))) = X234) <=> axshiftcons) & hoasinduction_lem3b_gthm & (one = ((d2term @ d_one))) & hoasapnotvar & hoasinduction_lem3a_gthm & hoaslamnotap_gthm & ! [X235 : d_subst] : (d_id = X235) & hoasinduction_lem2_lthm & (((lam @ (d2term @ d_one))) = ((d2term @ d_one))) & axassoc & (hoasinduction_no_psi_cond_lthm <=> (! [X236 : (subst > term > subst > $o)] : ~ ! [X237 : (term > $o)] : ~ ! [X238 : term] : (((X237 @ X238)) = ((X236 @ id @ X238 @ id))) => (! [X239 : term,X240 : (term > $o)] : ((X240 @ X239) | ~ ! [X241 : term] : ((X240 @ X241) | ~(var @ X241)) | ~ ! [X242 : term,X243 : term] : (~(X240 @ X242) | ~(X240 @ X243) | (X240 @ (ap @ X242 @ X243))) | ~ ! [X244 : term] : (~ ! [X245 : term] : ((X240 @ (sub @ X244 @ (push @ X245 @ id))) | ~(X240 @ X245)) | (X240 @ (lam @ X244)))) => (! [X246 : term] : (((sub @ X246 @ id)) = X246) => (! [X247 : (subst > term > subst > $o),X248 : term,X249 : (term > $o)] : (~ ! [X250 : (subst > term > term)] : (~ ! [X251 : subst,X252 : subst,X253 : term] : (((sub @ (X250 @ X252 @ X253) @ X251)) = ((X250 @ (comp @ X252 @ X251) @ (sub @ X253 @ X251)))) | ~ ! [X254 : term] : ((X247 @ id @ (X250 @ id @ X254) @ id) | ~(X247 @ id @ X254 @ id)) | (X247 @ id @ (lam @ (X250 @ sh @ one)) @ id)) | ~ ! [X255 : term] : (~(X249 @ X255) | (X249 @ (sub @ X248 @ (push @ X255 @ id)))) | (X249 @ (lam @ X248)) | ~ ! [X256 : term] : (((X247 @ id @ X256 @ id)) = ((X249 @ X256)))) => ! [X257 : term,X258 : (subst > term > subst > $o)] : ((X258 @ id @ X257 @ id) | ~ ! [X259 : (subst > term > term)] : (~ ! [X260 : term] : ((X258 @ id @ (X259 @ id @ X260) @ id) | ~(X258 @ id @ X260 @ id)) | ~ ! [X261 : term,X262 : subst,X263 : subst] : (((sub @ (X259 @ X263 @ X261) @ X262)) = ((X259 @ (comp @ X263 @ X262) @ (sub @ X261 @ X262)))) | (X258 @ id @ (lam @ (X259 @ sh @ one)) @ id)) | ~ ! [X264 : term,X265 : term] : (~(X258 @ id @ X264 @ id) | (X258 @ id @ (ap @ (sub @ X264 @ id) @ X265) @ id) | ~(X258 @ id @ X265 @ id)))))))) & axapp & ((! [X266 : subst,X267 : term] : (((sub @ one @ (push @ X267 @ X266))) = X267) => (! [X268 : term,X269 : subst] : (((comp @ sh @ (push @ X268 @ X269))) = X269) => (! [X270 : term,X271 : term] : ((((lam @ X270)) != ((lam @ X271))) | (X270 = X271)) => ! [X272 : (subst > term > term),X273 : (subst > term > term),X274 : subst,X275 : term] : ((((X273 @ X274 @ X275)) = ((X272 @ X274 @ X275))) | ~ ! [X276 : subst,X277 : subst,X278 : term] : (((X273 @ (comp @ X276 @ X277) @ (sub @ X278 @ X277))) = ((sub @ (X273 @ X276 @ X278) @ X277))) | (((lam @ (X272 @ sh @ one))) != ((lam @ (X273 @ sh @ one)))) | ~ ! [X279 : subst,X280 : term,X281 : subst] : (((X272 @ (comp @ X279 @ X281) @ (sub @ X280 @ X281))) = ((sub @ (X272 @ X279 @ X280) @ X281))))))) <=> hoaslaminj_lthm) & pushprop_gthm & (! [X282 : subst] : (((comp @ X282 @ id)) = X282) <=> axidr) & pushprop_lem1_gthm & (! [X283 : term,X284 : term,X285 : term,X286 : term] : ((X283 = X286) | (((ap @ (sub @ X285 @ id) @ X283)) != ((ap @ (sub @ X284 @ id) @ X286)))) <=> hoasapinj2) & (substmonoid <=> (! [X287 : subst] : (((comp @ id @ X287)) = X287) & ! [X288 : subst] : (((comp @ X288 @ id)) = X288))) & axmap & (((sub @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2term @ d_one))) & (! [X289 : term,X290 : term,X291 : term,X292 : term] : ((X291 = X292) | (((ap @ X290 @ X292)) != ((ap @ X289 @ X291)))) <=> apinj2) & (hoasinduction_lem3aa_lthm <=> ! [X293 : (subst > term > subst > $o),X294 : term] : (~ ! [X295 : (subst > term > term)] : (~ ! [X296 : term] : ((X293 @ id @ (X295 @ id @ X296) @ id) | ~(X293 @ id @ X296 @ id)) | ~ ! [X297 : subst,X298 : subst,X299 : term] : (((sub @ (X295 @ X298 @ X299) @ X297)) = ((X295 @ (comp @ X298 @ X297) @ (sub @ X299 @ X297)))) | (X293 @ id @ (lam @ (X295 @ sh @ one)) @ id)) | ~ ! [X300 : term] : (~(X293 @ id @ X300 @ id) | (X293 @ id @ (sub @ X294 @ (push @ X300 @ id)) @ id)) | (X293 @ id @ (lam @ (sub @ X294 @ (push @ one @ sh))) @ id))) & hoasinduction_lem1_gthm & hoaslamnotvar & (termmset <=> ! [X301 : term] : (((sub @ X301 @ id)) = X301)) & pushprop_lem2v2 & ulamvarind & (! [X302 : term,X303 : term,X304 : (subst > term > term)] : (~ ! [X305 : subst,X306 : subst,X307 : term] : (((X304 @ (comp @ X306 @ X305) @ (sub @ X307 @ X305))) = ((sub @ (X304 @ X306 @ X307) @ X305))) | (((lam @ (X304 @ sh @ one))) != ((ap @ (sub @ X302 @ id) @ X303)))) <=> hoaslamnotap) & hoasapnotvar_gthm & (axscons <=> ! [X308 : subst] : (((push @ (sub @ one @ X308) @ (comp @ sh @ X308))) = X308)) & lamnotvar & (! [X309 : term,X310 : term,X311 : term,X312 : term] : ((X309 = X312) | (((ap @ (sub @ X309 @ id) @ X310)) != ((ap @ (sub @ X312 @ id) @ X311)))) <=> hoasapinj1) & termmset_gthm & (hoasinduction_lem3v2 <=> ! [X313 : term,X314 : (term > $o),X315 : (subst > term > subst > $o)] : (~ ! [X316 : term] : (((X314 @ X316)) = ((X315 @ id @ X316 @ id))) | ~ ! [X317 : term,X318 : subst,X319 : subst,X320 : subst] : (~(X315 @ X320 @ X317 @ (comp @ X319 @ X318)) | (X315 @ (comp @ X320 @ X319) @ (sub @ X317 @ X319) @ X318)) | ~ ! [X321 : (subst > term > term)] : ((X315 @ id @ (lam @ (X321 @ sh @ one)) @ id) | ~ ! [X322 : subst,X323 : subst,X324 : term] : (((X321 @ (comp @ X322 @ X323) @ (sub @ X324 @ X323))) = ((sub @ (X321 @ X322 @ X324) @ X323))) | ~ ! [X325 : term] : ((X315 @ id @ (X321 @ id @ X325) @ id) | ~(X315 @ id @ X325 @ id))) | ~ ! [X326 : subst,X327 : subst,X328 : term,X329 : subst] : ((X315 @ X329 @ X328 @ (comp @ X327 @ X326)) | ~(X315 @ (comp @ X329 @ X327) @ (sub @ X328 @ X327) @ X326)) | (X314 @ (lam @ X313)) | ~ ! [X330 : term] : (~(X314 @ X330) | (X314 @ (sub @ X313 @ (push @ X330 @ id)))))) & (! [X331 : (subst > term > term),X332 : (subst > term > term),X333 : term,X334 : subst] : ((((lam @ (X332 @ sh @ one))) != ((lam @ (X331 @ sh @ one)))) | ~ ! [X335 : subst,X336 : term,X337 : subst] : (((sub @ (X331 @ X337 @ X336) @ X335)) = ((X331 @ (comp @ X337 @ X335) @ (sub @ X336 @ X335)))) | ~ ! [X338 : subst,X339 : subst,X340 : term] : (((sub @ (X332 @ X338 @ X340) @ X339)) = ((X332 @ (comp @ X338 @ X339) @ (sub @ X340 @ X339)))) | (((X331 @ X334 @ X333)) = ((X332 @ X334 @ X333)))) <=> hoaslaminj) & substmonoid_gthm & (hoasinduction_lem3v2_lthm <=> (! [X341 : term] : (((sub @ X341 @ id)) = X341) => ! [X342 : (subst > term > subst > $o),X343 : (term > $o),X344 : term] : (~ ! [X345 : term,X346 : subst,X347 : subst,X348 : subst] : ((X342 @ X348 @ X345 @ (comp @ X346 @ X347)) | ~(X342 @ (comp @ X348 @ X346) @ (sub @ X345 @ X346) @ X347)) | (X343 @ (lam @ X344)) | ~ ! [X349 : term] : (((X343 @ X349)) = ((X342 @ id @ X349 @ id))) | ~ ! [X350 : (subst > term > term)] : (~ ! [X351 : term,X352 : subst,X353 : subst] : (((X350 @ (comp @ X352 @ X353) @ (sub @ X351 @ X353))) = ((sub @ (X350 @ X352 @ X351) @ X353))) | ~ ! [X354 : term] : (~(X342 @ id @ X354 @ id) | (X342 @ id @ (X350 @ id @ X354) @ id)) | (X342 @ id @ (lam @ (X350 @ sh @ one)) @ id)) | ~ ! [X355 : subst,X356 : subst,X357 : term,X358 : subst] : (~(X342 @ X358 @ X357 @ (comp @ X356 @ X355)) | (X342 @ (comp @ X358 @ X356) @ (sub @ X357 @ X356) @ X355)) | ~ ! [X359 : term] : (~(X343 @ X359) | (X343 @ (sub @ X344 @ (push @ X359 @ id))))))) & (! [X360 : term,X361 : (subst > term > subst > $o)] : (~ ! [X362 : term] : (~(X361 @ id @ X362 @ id) | (X361 @ id @ (sub @ X360 @ (push @ X362 @ id)) @ id)) | ~ ! [X363 : (subst > term > term)] : (~ ! [X364 : term] : (~(X361 @ id @ X364 @ id) | (X361 @ id @ (X363 @ id @ X364) @ id)) | (X361 @ id @ (lam @ (X363 @ sh @ one)) @ id) | ~ ! [X365 : subst,X366 : subst,X367 : term] : (((X363 @ (comp @ X365 @ X366) @ (sub @ X367 @ X366))) = ((sub @ (X363 @ X365 @ X367) @ X366)))) | (X361 @ id @ (lam @ X360) @ id)) <=> hoasinduction_lem3a) & (! [X368 : (term > $o),X369 : term] : (~ ! [X370 : term] : (~(var @ X370) | (X368 @ X370)) | ~ ! [X371 : term] : ((X368 @ (lam @ X371)) | ~ ! [X372 : term] : (~(X368 @ X372) | (X368 @ (sub @ X371 @ (push @ X372 @ id))))) | ~ ! [X373 : term,X374 : term] : (~(X368 @ X373) | ~(X368 @ X374) | (X368 @ (ap @ X373 @ X374))) | (X368 @ X369)) <=> induction2) & ! [X375 : subst] : ? [X376 : d_subst] : (((d2subst @ X376)) = X375) & (hoasinduction_lem3v2a <=> ! [X377 : (subst > term > subst > $o),X378 : (term > $o),X379 : term] : (~ ! [X380 : term] : ((X378 @ (sub @ X379 @ (push @ X380 @ id))) | ~(X378 @ X380)) | ~ ! [X381 : term] : (((X378 @ X381)) = ((X377 @ id @ X381 @ id))) | ~ ! [X382 : (subst > term > term)] : ((X377 @ id @ (lam @ (X382 @ sh @ one)) @ id) | ~ ! [X383 : term] : ((X377 @ id @ (X382 @ id @ X383) @ id) | ~(X377 @ id @ X383 @ id)) | ~ ! [X384 : subst,X385 : term,X386 : subst] : (((sub @ (X382 @ X386 @ X385) @ X384)) = ((X382 @ (comp @ X386 @ X384) @ (sub @ X385 @ X384))))) | (X378 @ (lam @ X379)))) & (! [X387 : term,X388 : term,X389 : term,X390 : term] : ((((ap @ X387 @ X390)) != ((ap @ X388 @ X389))) | (X387 = X388)) <=> apinj1) & hoaslamnotap_lthm & (((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) = ((d2term @ d_one))) & (((push @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & (! [X391 : (term > $o),X392 : term] : (~ ! [X393 : term] : (~(var @ X393) | (X391 @ X393)) | ~ ! [X394 : term] : ((X391 @ (lam @ X394)) | ~(X391 @ X394)) | (X391 @ X392) | ~ ! [X395 : term,X396 : term] : ((X391 @ (ap @ X395 @ X396)) | ~(X391 @ X396) | ~(X391 @ X395))) <=> induction) & hoaslaminj_gthm & hoasinduction_lem1v2_gthm & hoaslamnotvar_lthm & (hoasinduction_lem3a_lthm <=> (! [X397 : term] : (((sub @ X397 @ id)) = X397) => (! [X398 : term,X399 : (subst > term > subst > $o)] : (~ ! [X400 : term] : (~(X399 @ id @ X400 @ id) | (X399 @ id @ (sub @ X398 @ (push @ X400 @ id)) @ id)) | ~ ! [X401 : (subst > term > term)] : ((X399 @ id @ (lam @ (X401 @ sh @ one)) @ id) | ~ ! [X402 : term] : ((X399 @ id @ (X401 @ id @ X402) @ id) | ~(X399 @ id @ X402 @ id)) | ~ ! [X403 : subst,X404 : subst,X405 : term] : (((X401 @ (comp @ X403 @ X404) @ (sub @ X405 @ X404))) = ((sub @ (X401 @ X403 @ X405) @ X404)))) | (X399 @ id @ (lam @ (sub @ X398 @ (push @ one @ sh))) @ id)) => ! [X406 : (subst > term > subst > $o),X407 : term] : (~ ! [X408 : term] : (~(X406 @ id @ X408 @ id) | (X406 @ id @ (sub @ X407 @ (push @ X408 @ id)) @ id)) | (X406 @ id @ (lam @ X407) @ id) | ~ ! [X409 : (subst > term > term)] : (~ ! [X410 : term] : (~(X406 @ id @ X410 @ id) | (X406 @ id @ (X409 @ id @ X410) @ id)) | ~ ! [X411 : subst,X412 : term,X413 : subst] : (((X409 @ (comp @ X411 @ X413) @ (sub @ X412 @ X413))) = ((sub @ (X409 @ X411 @ X412) @ X413))) | (X406 @ id @ (lam @ (X409 @ sh @ one)) @ id)))))) & (axvarcons <=> ! [X414 : term,X415 : subst] : (((sub @ one @ (push @ X414 @ X415))) = X414)) & (! [X416 : subst] : (((comp @ id @ X416)) = X416) <=> axidl) & (! [X417 : term,X418 : (subst > term > subst > $o)] : (~ ! [X419 : (subst > term > term)] : (~ ! [X420 : term] : ((X418 @ id @ (X419 @ id @ X420) @ id) | ~(X418 @ id @ X420 @ id)) | (X418 @ id @ (lam @ (X419 @ sh @ one)) @ id) | ~ ! [X421 : subst,X422 : term,X423 : subst] : (((X419 @ (comp @ X421 @ X423) @ (sub @ X422 @ X423))) = ((sub @ (X419 @ X421 @ X422) @ X423)))) | (X418 @ id @ X417 @ id) | ~ ! [X424 : term,X425 : term] : (~(X418 @ id @ X425 @ id) | (X418 @ id @ (ap @ (sub @ X424 @ id) @ X425) @ id) | ~(X418 @ id @ X424 @ id))) <=> hoasinduction_no_psi_cond) & (induction2lem_lthm <=> (! [X426 : term,X427 : subst] : (((sub @ one @ (push @ X426 @ X427))) = X426) => (! [X428 : subst,X429 : term] : (((comp @ sh @ (push @ X429 @ X428))) = X428) => (! [X430 : subst] : (((comp @ X430 @ id)) = X430) => (! [X431 : term,X432 : (term > $o)] : (~ ! [X433 : term] : (~(X432 @ X433) | (X432 @ (lam @ X433))) | ~ ! [X434 : term,X435 : term] : ((X432 @ (ap @ X434 @ X435)) | ~(X432 @ X435) | ~(X432 @ X434)) | ~ ! [X436 : term] : (~(var @ X436) | (X432 @ X436)) | (X432 @ X431)) => ! [X437 : (term > $o),X438 : term,X439 : subst] : ((X437 @ (sub @ X438 @ X439)) | ~ ! [X440 : term] : ((X437 @ (sub @ X440 @ X439)) | ~(var @ X440)) | ~ ! [X441 : term,X442 : term] : (~(X437 @ X441) | (X437 @ (ap @ X442 @ X441)) | ~(X437 @ X442)) | ~ ! [X443 : term] : (~ ! [X444 : term] : (~(X437 @ X444) | (X437 @ (sub @ X443 @ (push @ X444 @ id)))) | (X437 @ (lam @ X443))))))))) & (hoasinduction_lem3_lthm <=> (! [X445 : term] : (((sub @ X445 @ id)) = X445) => (! [X446 : term,X447 : (subst > term > subst > $o)] : (~ ! [X448 : (subst > term > term)] : (~ ! [X449 : term] : ((X447 @ id @ (X448 @ id @ X449) @ id) | ~(X447 @ id @ X449 @ id)) | (X447 @ id @ (lam @ (X448 @ sh @ one)) @ id) | ~ ! [X450 : term,X451 : subst,X452 : subst] : (((sub @ (X448 @ X451 @ X450) @ X452)) = ((X448 @ (comp @ X451 @ X452) @ (sub @ X450 @ X452))))) | (X447 @ id @ (lam @ (sub @ X446 @ (push @ one @ sh))) @ id) | ~ ! [X453 : term] : (~(X447 @ id @ X453 @ id) | (X447 @ id @ (sub @ X446 @ (push @ X453 @ id)) @ id))) => ! [X454 : term,X455 : (subst > term > subst > $o)] : (~ ! [X456 : term,X457 : subst,X458 : subst,X459 : subst] : (~(X455 @ X457 @ X456 @ (comp @ X459 @ X458)) | (X455 @ (comp @ X457 @ X459) @ (sub @ X456 @ X459) @ X458)) | ~ ! [X460 : (subst > term > term)] : (~ ! [X461 : term] : ((X455 @ id @ (X460 @ id @ X461) @ id) | ~(X455 @ id @ X461 @ id)) | ~ ! [X462 : term,X463 : subst,X464 : subst] : (((sub @ (X460 @ X464 @ X462) @ X463)) = ((X460 @ (comp @ X464 @ X463) @ (sub @ X462 @ X463)))) | (X455 @ id @ (lam @ (X460 @ sh @ one)) @ id)) | ~ ! [X465 : term] : ((X455 @ id @ (sub @ X454 @ (push @ X465 @ id)) @ id) | ~(X455 @ id @ X465 @ id)) | ~ ! [X466 : subst,X467 : term,X468 : subst,X469 : subst] : (~(X455 @ (comp @ X468 @ X469) @ (sub @ X467 @ X469) @ X466) | (X455 @ X468 @ X467 @ (comp @ X469 @ X466))) | (X455 @ id @ (lam @ X454) @ id))))) & hoasinduction_gthm & ((! [X470 : term,X471 : term,X472 : term,X473 : term] : ((((ap @ X471 @ X472)) != ((ap @ X473 @ X470))) | (X470 = X472)) => ! [X474 : term,X475 : term,X476 : term,X477 : term] : ((((ap @ (sub @ X476 @ id) @ X475)) != ((ap @ (sub @ X474 @ id) @ X477))) | (X475 = X477))) <=> hoasapinj2_lthm) & (! [X478 : term,X479 : (subst > term > subst > $o)] : ((X479 @ id @ (lam @ X478) @ id) | ~ ! [X480 : term] : ((X479 @ id @ (sub @ X478 @ (push @ X480 @ id)) @ id) | ~(X479 @ id @ X480 @ id)) | ~ ! [X481 : (subst > term > term)] : ((X479 @ id @ (lam @ (X481 @ sh @ one)) @ id) | ~ ! [X482 : subst,X483 : term,X484 : subst] : (((X481 @ (comp @ X484 @ X482) @ (sub @ X483 @ X482))) = ((sub @ (X481 @ X484 @ X483) @ X482))) | ~ ! [X485 : term] : (~(X479 @ id @ X485 @ id) | (X479 @ id @ (X481 @ id @ X485) @ id))) | ~ ! [X486 : subst,X487 : subst,X488 : subst,X489 : term] : (~(X479 @ X486 @ X489 @ (comp @ X488 @ X487)) | (X479 @ (comp @ X486 @ X488) @ (sub @ X489 @ X488) @ X487)) | ~ ! [X490 : term,X491 : subst,X492 : subst,X493 : subst] : ((X479 @ X491 @ X490 @ (comp @ X493 @ X492)) | ~(X479 @ (comp @ X491 @ X493) @ (sub @ X490 @ X493) @ X492))) <=> hoasinduction_lem3) & axclos & (induction2_lthm <=> (! [X494 : term] : (((sub @ X494 @ id)) = X494) => (! [X495 : (term > $o),X496 : term,X497 : subst] : (~ ! [X498 : term] : (~(var @ X498) | (X495 @ (sub @ X498 @ X497))) | (X495 @ (sub @ X496 @ X497)) | ~ ! [X499 : term] : ((X495 @ (lam @ X499)) | ~ ! [X500 : term] : (~(X495 @ X500) | (X495 @ (sub @ X499 @ (push @ X500 @ id))))) | ~ ! [X501 : term,X502 : term] : (~(X495 @ X501) | (X495 @ (ap @ X501 @ X502)) | ~(X495 @ X502))) => ! [X503 : term,X504 : (term > $o)] : (~ ! [X505 : term] : ((X504 @ X505) | ~(var @ X505)) | ~ ! [X506 : term,X507 : term] : ((X504 @ (ap @ X506 @ X507)) | ~(X504 @ X506) | ~(X504 @ X507)) | ~ ! [X508 : term] : ((X504 @ (lam @ X508)) | ~ ! [X509 : term] : ((X504 @ (sub @ X508 @ (push @ X509 @ id))) | ~(X504 @ X509))) | (X504 @ X503))))) & apnotvar & hoasinduction_lem3v2_gthm & hoaslamnotvar_gthm & (hoasinduction_lem3b_lthm <=> ! [X510 : term] : ~ ! [X511 : (subst > term > term)] : (((sub @ X510 @ (push @ one @ sh))) != ((X511 @ sh @ one)))) & pushprop_lthm & (shinj <=> ! [X512 : term,X513 : term] : ((((sub @ X512 @ sh)) != ((sub @ X513 @ sh))) | (X512 = X513))) & (! [X514 : (subst > term > subst > $o)] : ~ ! [X515 : (term > $o)] : ~ ! [X516 : term] : (((X514 @ id @ X516 @ id)) = ((X515 @ X516))) <=> hoasinduction_lem0) & (! [X517 : term,X518 : (subst > term > subst > $o)] : ((X518 @ id @ (lam @ (sub @ X517 @ (push @ one @ sh))) @ id) | ~ ! [X519 : term] : (~(X518 @ id @ X519 @ id) | (X518 @ id @ (sub @ X517 @ (push @ X519 @ id)) @ id)) | ~ ! [X520 : (subst > term > term)] : (~ ! [X521 : term] : ((X518 @ id @ (X520 @ id @ X521) @ id) | ~(X518 @ id @ X521 @ id)) | (X518 @ id @ (lam @ (X520 @ sh @ one)) @ id) | ~ ! [X522 : term,X523 : subst,X524 : subst] : (((X520 @ (comp @ X524 @ X523) @ (sub @ X522 @ X523))) = ((sub @ (X520 @ X524 @ X522) @ X523))))) <=> hoasinduction_lem3aa) & (hoasinduction_lem3b <=> ! [X525 : term] : ~ ! [X526 : (subst > term > term)] : (((sub @ X525 @ (push @ one @ sh))) != ((X526 @ sh @ one)))) & (termmset_lthm <=> (! [X527 : term] : (((sub @ X527 @ id)) = X527) => ! [X528 : term] : (((sub @ X528 @ id)) = X528))) & ulamvarsh & ~lamnotap & (! [X529 : subst,X530 : (term > $o),X531 : term] : ~ ! [X532 : (term > $o)] : ~ ! [X533 : term] : (((X532 @ X533)) = ((X530 @ (sub @ X533 @ (push @ X531 @ X529))))) <=> pushprop_lem0) & induction2lem_gthm & (pushprop_lem1_lthm <=> (! [X534 : term,X535 : subst] : (((sub @ one @ (push @ X534 @ X535))) = X534) => (! [X536 : subst,X537 : term] : (((comp @ sh @ (push @ X537 @ X536))) = X536) => ! [X538 : term,X539 : (term > $o),X540 : subst,X541 : term,X542 : (term > $o)] : ((X542 @ (sub @ X541 @ (push @ X538 @ X540))) | ~(X539 @ X541))))) & (! [X543 : (term > $o),X544 : (term > $o),X545 : subst,X546 : term] : ((X544 @ one) | ~(X543 @ X546) | ~ ! [X547 : term] : (((X543 @ (sub @ X547 @ (push @ X546 @ X545)))) = ((X544 @ X547)))) <=> pushprop_lem1v2) & pushprop_lem3v2 & hoasapinj1_gthm & ! [X548 : term] : ? [X549 : d_term] : (((d2term @ X549)) = X548) & pushprop_lem3v2_lthm),
  inference(rectify,[],[f3])).
thf(f7,plain,(
  (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (hoasinduction_lem1 = $true) & (pushprop_lem0_gthm = $true) & (hoasapnotvar_lthm = $true) & (! [X0 : term,X1 : (subst > term > subst > $o)] : (~ ! [X2 : subst,X3 : subst,X4 : term,X5 : subst] : ((((X1 @ (comp @ X5 @ X3) @ (sub @ X4 @ X3) @ X2)) = $true) | ~ (((X1 @ X5 @ X4 @ (comp @ X3 @ X2))) = $true)) | ~ ! [X6 : (subst > term > term)] : ((((X1 @ id @ (lam @ (X6 @ sh @ one)) @ id)) = $true) | ~ ! [X9 : subst,X8 : term,X7 : subst] : (((sub @ (X6 @ X9 @ X8) @ X7)) = ((X6 @ (comp @ X9 @ X7) @ (sub @ X8 @ X7)))) | ~ ! [X10 : term] : (~ (((X1 @ id @ X10 @ id)) = $true) | (((X1 @ id @ (X6 @ id @ X10) @ id)) = $true))) | ~ ! [X11 : subst,X12 : subst,X13 : term,X14 : subst] : (~ ($true = ((X1 @ (comp @ X14 @ X12) @ (sub @ X13 @ X12) @ X11))) | ($true = ((X1 @ X14 @ X13 @ (comp @ X12 @ X11))))) | (((X1 @ id @ X0 @ id)) = $true) | ~ ! [X15 : term] : ((((X1 @ id @ X15 @ id)) = $true) | ~ ($true = ((var @ (sub @ X15 @ id))))) | ~ ! [X16 : term,X17 : term] : ((((X1 @ id @ (ap @ (sub @ X16 @ id) @ X17) @ id)) = $true) | ~ (((X1 @ id @ X16 @ id)) = $true) | ~ ($true = ((X1 @ id @ X17 @ id))))) <=> (hoasinduction = $true)) & ! [X18 : d_term] : (d_one = X18) & (axvarshift = $true) & ! [X19 : d_subst,X20 : d_subst] : ((((d2subst @ X19)) = ((d2subst @ X20))) => (X19 = X20)) & ((laminj = $true) <=> ! [X21 : term,X22 : term] : ((((lam @ X22)) != ((lam @ X21))) | (X21 = X22))) & (hoasinduction_lem1_lthm = $true) & ~ (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) = $true) & (hoasinduction_lem2_gthm = $true) & (hoasinduction_lem2v2_gthm = $true) & (((d2subst @ d_id)) = id) & ((substmonoid_lthm = $true) <=> (! [X23 : subst] : (((comp @ id @ X23)) = X23) => (! [X24 : subst] : (((comp @ X24 @ id)) = X24) => (! [X26 : subst] : (((comp @ X26 @ id)) = X26) & ! [X25 : subst] : (((comp @ id @ X25)) = X25))))) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (pushprop_lem2v2_gthm = $true) & (! [X35 : term] : (((sub @ X35 @ id)) = X35) <=> (axvarid = $true)) & ((! [X36 : term] : (((sub @ X36 @ id)) = X36) => (! [X37 : term,X38 : term,X39 : term,X40 : term] : ((((ap @ X40 @ X38)) != ((ap @ X39 @ X37))) | (X39 = X40)) => ! [X41 : term,X42 : term,X43 : term,X44 : term] : ((X41 = X44) | (((ap @ (sub @ X44 @ id) @ X42)) != ((ap @ (sub @ X41 @ id) @ X43)))))) <=> (hoasapinj1_lthm = $true)) & ((hoasinduction_lem0_lthm = $true) <=> ! [X45 : (subst > term > subst > $o)] : ~ ! [X46 : (term > $o)] : ~ ! [X47 : term] : (((X45 @ id @ X47 @ id)) = ((X46 @ X47)))) & ((hoasinduction_lem3aaa = $true) <=> ! [X48 : term,X49 : (subst > term > subst > $o)] : (~ ! [X50 : term] : ((((X49 @ id @ (sub @ X48 @ (push @ X50 @ id)) @ id)) = $true) | ~ (((X49 @ id @ X50 @ id)) = $true)) | ~ ! [X51 : term,X52 : (subst > term > term)] : (~ ! [X53 : term,X54 : subst,X55 : subst] : (((sub @ (X52 @ X55 @ X53) @ X54)) = ((sub @ (sub @ X51 @ (push @ X53 @ X55)) @ X54))) | ~ ! [X56 : term] : ((((X49 @ id @ (X52 @ id @ X56) @ id)) = $true) | ~ ($true = ((X49 @ id @ X56 @ id)))) | ~ ! [X57 : subst,X58 : subst,X59 : term] : (((X52 @ (comp @ X57 @ X58) @ (sub @ X59 @ X58))) = ((sub @ X51 @ (push @ (sub @ X59 @ X58) @ (comp @ X57 @ X58))))) | (((X49 @ id @ (lam @ (X52 @ sh @ one)) @ id)) = $true)) | (((X49 @ id @ (lam @ (sub @ X48 @ (push @ one @ sh))) @ id)) = $true))) & (! [X60 : term] : ~ ! [X61 : (subst > term > term)] : ~ ! [X62 : term,X63 : subst] : (((X61 @ X63 @ X62)) = ((sub @ X60 @ (push @ X62 @ X63)))) <=> (hoasinduction_lem3v2_f = $true)) & (hoasinduction_lem3_gthm = $true) & (pushprop_lem2v2_lthm = $true) & ((! [X64 : (subst > term > subst > $o)] : ~ ! [X65 : (term > $o)] : ~ ! [X66 : term] : (((X64 @ id @ X66 @ id)) = ((X65 @ X66))) => (! [X67 : term,X68 : (term > $o)] : ((((X68 @ X67)) = $true) | ~ ! [X69 : term] : ((((X68 @ (lam @ X69))) = $true) | ~ ! [X70 : term] : ((((X68 @ (sub @ X69 @ (push @ X70 @ id)))) = $true) | ~ (((X68 @ X70)) = $true))) | ~ ! [X71 : term,X72 : term] : (~ (((X68 @ X71)) = $true) | ~ (((X68 @ X72)) = $true) | (((X68 @ (ap @ X72 @ X71))) = $true)) | ~ ! [X73 : term] : (~ (((var @ X73)) = $true) | (((X68 @ X73)) = $true))) => (! [X74 : term] : (((sub @ X74 @ id)) = X74) => (! [X75 : (term > $o),X76 : term,X77 : (subst > term > subst > $o)] : (~ ! [X78 : (subst > term > term)] : (~ ! [X79 : subst,X80 : subst,X81 : term] : (((sub @ (X78 @ X79 @ X81) @ X80)) = ((X78 @ (comp @ X79 @ X80) @ (sub @ X81 @ X80)))) | (((X77 @ id @ (lam @ (X78 @ sh @ one)) @ id)) = $true) | ~ ! [X82 : term] : (~ ($true = ((X77 @ id @ X82 @ id))) | ($true = ((X77 @ id @ (X78 @ id @ X82) @ id))))) | ~ ! [X83 : term] : ((((X75 @ (sub @ X76 @ (push @ X83 @ id)))) = $true) | ~ (((X75 @ X83)) = $true)) | (((X75 @ (lam @ X76))) = $true) | ~ ! [X84 : term] : (((X77 @ id @ X84 @ id)) = ((X75 @ X84)))) => ! [X85 : term,X86 : (subst > term > subst > $o)] : (~ ! [X87 : term] : ((((X86 @ id @ X87 @ id)) = $true) | ~ (((var @ (sub @ X87 @ id))) = $true)) | (((X86 @ id @ X85 @ id)) = $true) | ~ ! [X88 : (subst > term > term)] : (~ ! [X89 : subst,X90 : term,X91 : subst] : (((X88 @ (comp @ X89 @ X91) @ (sub @ X90 @ X91))) = ((sub @ (X88 @ X89 @ X90) @ X91))) | ($true = ((X86 @ id @ (lam @ (X88 @ sh @ one)) @ id))) | ~ ! [X92 : term] : (~ ($true = ((X86 @ id @ X92 @ id))) | (((X86 @ id @ (X88 @ id @ X92) @ id)) = $true))) | ~ ! [X93 : term,X94 : term] : (~ ($true = ((X86 @ id @ X94 @ id))) | ($true = ((X86 @ id @ (ap @ (sub @ X94 @ id) @ X93) @ id))) | ~ (((X86 @ id @ X93 @ id)) = $true)) | ~ ! [X95 : subst,X96 : term,X97 : subst,X98 : subst] : ((((X86 @ (comp @ X95 @ X98) @ (sub @ X96 @ X98) @ X97)) = $true) | ~ (((X86 @ X95 @ X96 @ (comp @ X98 @ X97))) = $true)) | ~ ! [X99 : subst,X100 : term,X101 : subst,X102 : subst] : ((((X86 @ X102 @ X100 @ (comp @ X99 @ X101))) = $true) | ~ (((X86 @ (comp @ X102 @ X99) @ (sub @ X100 @ X99) @ X101)) = $true))))))) <=> (hoasinduction_lthm_3 = $true)) & (((d2subst @ d_id)) = sh) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((pushprop_lem0_lthm = $true) <=> ! [X105 : term,X106 : (term > $o),X107 : subst] : ~ ! [X108 : (term > $o)] : ~ ! [X109 : term] : (((X106 @ (sub @ X109 @ (push @ X105 @ X107)))) = ((X108 @ X109)))) & ((hoasinduction_lem2 = $true) <=> ! [X110 : (subst > term > subst > $o),X111 : term,X112 : term] : (~ ! [X113 : subst,X114 : subst,X115 : subst,X116 : term] : (~ (((X110 @ (comp @ X113 @ X115) @ (sub @ X116 @ X115) @ X114)) = $true) | (((X110 @ X113 @ X116 @ (comp @ X115 @ X114))) = $true)) | ~ (((X110 @ id @ X111 @ id)) = $true) | ($true = ((X110 @ id @ (ap @ X111 @ X112) @ id))) | ~ ! [X117 : term,X118 : term] : ((((X110 @ id @ (ap @ (sub @ X117 @ id) @ X118) @ id)) = $true) | ~ (((X110 @ id @ X117 @ id)) = $true) | ~ ($true = ((X110 @ id @ X118 @ id)))) | ~ ! [X119 : subst,X120 : term,X121 : subst,X122 : subst] : (~ (((X110 @ X121 @ X120 @ (comp @ X122 @ X119))) = $true) | (((X110 @ (comp @ X121 @ X122) @ (sub @ X120 @ X122) @ X119)) = $true)) | ~ (((X110 @ id @ X112 @ id)) = $true))) & (! [X123 : subst,X124 : term,X125 : (term > $o)] : (~ ! [X126 : term] : (~ (((var @ X126)) = $true) | (((X125 @ (sub @ X126 @ X123))) = $true)) | ~ ! [X127 : term] : (($true = ((X125 @ (lam @ X127)))) | ~ ! [X128 : term] : ((((X125 @ (sub @ X127 @ (push @ X128 @ id)))) = $true) | ~ (((X125 @ X128)) = $true))) | (((X125 @ (sub @ X124 @ X123))) = $true) | ~ ! [X129 : term,X130 : term] : (~ ($true = ((X125 @ X130))) | ($true = ((X125 @ (ap @ X129 @ X130)))) | ~ (((X125 @ X129)) = $true))) <=> (induction2lem = $true)) & (pushprop = $true) & (! [X131 : term] : ~ ! [X132 : (subst > term > term)] : ~ ! [X133 : term,X134 : subst] : (((X132 @ X134 @ X133)) = ((sub @ X131 @ (push @ X133 @ X134)))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & ~ (ulamvar1 = $true) & ((hoasinduction_lem3v2a_lthm = $true) <=> (! [X135 : term] : ~ ! [X136 : (subst > term > term)] : ~ ! [X137 : term,X138 : subst] : (((X136 @ X138 @ X137)) = ((sub @ X135 @ (push @ X137 @ X138)))) => (! [X139 : term] : (((sub @ X139 @ id)) = X139) => ! [X140 : (term > $o),X141 : term,X142 : (subst > term > subst > $o)] : ((((X140 @ (lam @ X141))) = $true) | ~ ! [X143 : (subst > term > term)] : (~ ! [X144 : subst,X145 : subst,X146 : term] : (((sub @ (X143 @ X145 @ X146) @ X144)) = ((X143 @ (comp @ X145 @ X144) @ (sub @ X146 @ X144)))) | ($true = ((X142 @ id @ (lam @ (X143 @ sh @ one)) @ id))) | ~ ! [X147 : term] : (~ (((X142 @ id @ X147 @ id)) = $true) | (((X142 @ id @ (X143 @ id @ X147) @ id)) = $true))) | ~ ! [X148 : term] : (((X140 @ X148)) = ((X142 @ id @ X148 @ id))) | ~ ! [X149 : term] : (~ (((X140 @ X149)) = $true) | (((X140 @ (sub @ X141 @ (push @ X149 @ id)))) = $true)))))) & (hoasinduction_lem1v2 = $true) & ((hoasinduction_lthm = $true) <=> (! [X150 : term,X151 : (term > $o)] : (~ ! [X152 : term] : (~ (((var @ X152)) = $true) | (((X151 @ X152)) = $true)) | ~ ! [X153 : term] : ((((X151 @ (lam @ X153))) = $true) | ~ ! [X154 : term] : (~ (((X151 @ X154)) = $true) | ($true = ((X151 @ (sub @ X153 @ (push @ X154 @ id))))))) | (((X151 @ X150)) = $true) | ~ ! [X155 : term,X156 : term] : (~ ($true = ((X151 @ X155))) | ~ (((X151 @ X156)) = $true) | (((X151 @ (ap @ X156 @ X155))) = $true))) => (! [X157 : term,X158 : (subst > term > subst > $o),X159 : term] : (~ ! [X160 : subst,X161 : term,X162 : subst,X163 : subst] : ((((X158 @ (comp @ X162 @ X160) @ (sub @ X161 @ X160) @ X163)) = $true) | ~ (((X158 @ X162 @ X161 @ (comp @ X160 @ X163))) = $true)) | ~ (((X158 @ id @ X157 @ id)) = $true) | ~ (((X158 @ id @ X159 @ id)) = $true) | ($true = ((X158 @ id @ (ap @ X159 @ X157) @ id))) | ~ ! [X164 : subst,X165 : subst,X166 : term,X167 : subst] : (~ (((X158 @ (comp @ X165 @ X164) @ (sub @ X166 @ X164) @ X167)) = $true) | (((X158 @ X165 @ X166 @ (comp @ X164 @ X167))) = $true)) | ~ ! [X168 : term,X169 : term] : (~ (((X158 @ id @ X168 @ id)) = $true) | ~ (((X158 @ id @ X169 @ id)) = $true) | (((X158 @ id @ (ap @ (sub @ X169 @ id) @ X168) @ id)) = $true))) => (! [X170 : (subst > term > subst > $o),X171 : term] : (~ ! [X172 : term,X173 : subst,X174 : subst,X175 : subst] : ((((X170 @ X173 @ X172 @ (comp @ X174 @ X175))) = $true) | ~ (((X170 @ (comp @ X173 @ X174) @ (sub @ X172 @ X174) @ X175)) = $true)) | ~ ! [X176 : (subst > term > term)] : (~ ! [X177 : subst,X178 : subst,X179 : term] : (((sub @ (X176 @ X177 @ X179) @ X178)) = ((X176 @ (comp @ X177 @ X178) @ (sub @ X179 @ X178)))) | ($true = ((X170 @ id @ (lam @ (X176 @ sh @ one)) @ id))) | ~ ! [X180 : term] : (~ ($true = ((X170 @ id @ X180 @ id))) | (((X170 @ id @ (X176 @ id @ X180) @ id)) = $true))) | ~ ! [X181 : subst,X182 : term,X183 : subst,X184 : subst] : ((((X170 @ (comp @ X184 @ X181) @ (sub @ X182 @ X181) @ X183)) = $true) | ~ (((X170 @ X184 @ X182 @ (comp @ X181 @ X183))) = $true)) | (((X170 @ id @ (lam @ X171) @ id)) = $true) | ~ ! [X185 : term] : (~ (((X170 @ id @ X185 @ id)) = $true) | ($true = ((X170 @ id @ (sub @ X171 @ (push @ X185 @ id)) @ id))))) => ! [X186 : term,X187 : (subst > term > subst > $o)] : (~ ! [X188 : term,X189 : term] : ((((X187 @ id @ (ap @ (sub @ X188 @ id) @ X189) @ id)) = $true) | ~ (((X187 @ id @ X189 @ id)) = $true) | ~ (((X187 @ id @ X188 @ id)) = $true)) | (((X187 @ id @ X186 @ id)) = $true) | ~ ! [X190 : (subst > term > term)] : (~ ! [X191 : term] : ((((X187 @ id @ (X190 @ id @ X191) @ id)) = $true) | ~ (((X187 @ id @ X191 @ id)) = $true)) | ~ ! [X192 : subst,X193 : subst,X194 : term] : (((X190 @ (comp @ X193 @ X192) @ (sub @ X194 @ X192))) = ((sub @ (X190 @ X193 @ X194) @ X192))) | (((X187 @ id @ (lam @ (X190 @ sh @ one)) @ id)) = $true)) | ~ ! [X195 : term,X196 : subst,X197 : subst,X198 : subst] : (~ (((X187 @ X198 @ X195 @ (comp @ X196 @ X197))) = $true) | (((X187 @ (comp @ X198 @ X196) @ (sub @ X195 @ X196) @ X197)) = $true)) | ~ ! [X199 : term] : ((((X187 @ id @ X199 @ id)) = $true) | ~ (((var @ (sub @ X199 @ id))) = $true)) | ~ ! [X200 : subst,X201 : subst,X202 : subst,X203 : term] : ((((X187 @ X202 @ X203 @ (comp @ X201 @ X200))) = $true) | ~ (((X187 @ (comp @ X202 @ X201) @ (sub @ X203 @ X201) @ X200)) = $true))))))) & ((pushprop_lem1v2_lthm = $true) <=> (! [X204 : subst,X205 : term] : (((sub @ one @ (push @ X205 @ X204))) = X205) => ! [X206 : (term > $o),X207 : subst,X208 : (term > $o),X209 : term] : (~ (((X208 @ X209)) = $true) | (((X206 @ one)) = $true) | ~ ! [X210 : term] : (((X208 @ (sub @ X210 @ (push @ X209 @ X207)))) = ((X206 @ X210)))))) & (! [X211 : term,X212 : (term > $o),X213 : (subst > term > subst > $o),X214 : term] : (~ ! [X215 : term,X216 : subst,X217 : subst,X218 : subst] : (~ ($true = ((X213 @ X217 @ X215 @ (comp @ X216 @ X218)))) | (((X213 @ (comp @ X217 @ X216) @ (sub @ X215 @ X216) @ X218)) = $true)) | ~ ($true = ((X212 @ X211))) | ~ ! [X219 : term,X220 : subst,X221 : subst,X222 : subst] : (~ (((X213 @ (comp @ X220 @ X221) @ (sub @ X219 @ X221) @ X222)) = $true) | (((X213 @ X220 @ X219 @ (comp @ X221 @ X222))) = $true)) | (((X212 @ (ap @ X214 @ X211))) = $true) | ~ ! [X223 : term] : (((X212 @ X223)) = ((X213 @ id @ X223 @ id))) | ~ (((X212 @ X214)) = $true) | ~ ! [X224 : term,X225 : term] : (~ (((X213 @ id @ X224 @ id)) = $true) | (((X213 @ id @ (ap @ (sub @ X224 @ id) @ X225) @ id)) = $true) | ~ (((X213 @ id @ X225 @ id)) = $true))) <=> (hoasinduction_lem2v2 = $true)) & (pushprop_lem1v2_gthm = $true) & ((pushprop_lem1 = $true) <=> ! [X226 : (term > $o),X227 : term,X228 : (term > $o),X229 : subst,X230 : term] : ((((X228 @ (sub @ X227 @ (push @ X230 @ X229)))) = $true) | ~ (((X226 @ X227)) = $true))) & (hoasapinj2_gthm = $true) & (pushprop_lthm_orig = $true) & (axabs = $true) & ! [X231 : d_term,X232 : d_term] : ((((d2term @ X232)) = ((d2term @ X231))) => (X231 = X232)) & (induction2_gthm = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ~ (((var @ (d2term @ d_one))) = $true) & (! [X233 : term,X234 : subst] : (((comp @ sh @ (push @ X233 @ X234))) = X234) <=> (axshiftcons = $true)) & (hoasinduction_lem3b_gthm = $true) & (((d2term @ d_one)) = one) & (hoasapnotvar = $true) & (hoasinduction_lem3a_gthm = $true) & (hoaslamnotap_gthm = $true) & ! [X235 : d_subst] : (d_id = X235) & (hoasinduction_lem2_lthm = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (axassoc = $true) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (! [X236 : (subst > term > subst > $o)] : ~ ! [X237 : (term > $o)] : ~ ! [X238 : term] : (((X237 @ X238)) = ((X236 @ id @ X238 @ id))) => (! [X239 : term,X240 : (term > $o)] : (($true = ((X240 @ X239))) | ~ ! [X241 : term] : ((((X240 @ X241)) = $true) | ~ (((var @ X241)) = $true)) | ~ ! [X242 : term,X243 : term] : (~ ($true = ((X240 @ X242))) | ~ (((X240 @ X243)) = $true) | (((X240 @ (ap @ X242 @ X243))) = $true)) | ~ ! [X244 : term] : (~ ! [X245 : term] : ((((X240 @ (sub @ X244 @ (push @ X245 @ id)))) = $true) | ~ (((X240 @ X245)) = $true)) | (((X240 @ (lam @ X244))) = $true))) => (! [X246 : term] : (((sub @ X246 @ id)) = X246) => (! [X247 : (subst > term > subst > $o),X248 : term,X249 : (term > $o)] : (~ ! [X250 : (subst > term > term)] : (~ ! [X251 : subst,X252 : subst,X253 : term] : (((sub @ (X250 @ X252 @ X253) @ X251)) = ((X250 @ (comp @ X252 @ X251) @ (sub @ X253 @ X251)))) | ~ ! [X254 : term] : ((((X247 @ id @ (X250 @ id @ X254) @ id)) = $true) | ~ (((X247 @ id @ X254 @ id)) = $true)) | (((X247 @ id @ (lam @ (X250 @ sh @ one)) @ id)) = $true)) | ~ ! [X255 : term] : (~ (((X249 @ X255)) = $true) | (((X249 @ (sub @ X248 @ (push @ X255 @ id)))) = $true)) | (((X249 @ (lam @ X248))) = $true) | ~ ! [X256 : term] : (((X247 @ id @ X256 @ id)) = ((X249 @ X256)))) => ! [X257 : term,X258 : (subst > term > subst > $o)] : (($true = ((X258 @ id @ X257 @ id))) | ~ ! [X259 : (subst > term > term)] : (~ ! [X260 : term] : ((((X258 @ id @ (X259 @ id @ X260) @ id)) = $true) | ~ (((X258 @ id @ X260 @ id)) = $true)) | ~ ! [X261 : term,X262 : subst,X263 : subst] : (((sub @ (X259 @ X263 @ X261) @ X262)) = ((X259 @ (comp @ X263 @ X262) @ (sub @ X261 @ X262)))) | (((X258 @ id @ (lam @ (X259 @ sh @ one)) @ id)) = $true)) | ~ ! [X264 : term,X265 : term] : (~ (((X258 @ id @ X264 @ id)) = $true) | (((X258 @ id @ (ap @ (sub @ X264 @ id) @ X265) @ id)) = $true) | ~ (((X258 @ id @ X265 @ id)) = $true)))))))) & (axapp = $true) & ((! [X266 : subst,X267 : term] : (((sub @ one @ (push @ X267 @ X266))) = X267) => (! [X268 : term,X269 : subst] : (((comp @ sh @ (push @ X268 @ X269))) = X269) => (! [X270 : term,X271 : term] : ((((lam @ X270)) != ((lam @ X271))) | (X270 = X271)) => ! [X272 : (subst > term > term),X273 : (subst > term > term),X274 : subst,X275 : term] : ((((X273 @ X274 @ X275)) = ((X272 @ X274 @ X275))) | ~ ! [X276 : subst,X277 : subst,X278 : term] : (((X273 @ (comp @ X276 @ X277) @ (sub @ X278 @ X277))) = ((sub @ (X273 @ X276 @ X278) @ X277))) | (((lam @ (X272 @ sh @ one))) != ((lam @ (X273 @ sh @ one)))) | ~ ! [X279 : subst,X280 : term,X281 : subst] : (((X272 @ (comp @ X279 @ X281) @ (sub @ X280 @ X281))) = ((sub @ (X272 @ X279 @ X280) @ X281))))))) <=> (hoaslaminj_lthm = $true)) & (pushprop_gthm = $true) & (! [X282 : subst] : (((comp @ X282 @ id)) = X282) <=> (axidr = $true)) & (pushprop_lem1_gthm = $true) & (! [X283 : term,X284 : term,X285 : term,X286 : term] : ((X283 = X286) | (((ap @ (sub @ X285 @ id) @ X283)) != ((ap @ (sub @ X284 @ id) @ X286)))) <=> (hoasapinj2 = $true)) & ((substmonoid = $true) <=> (! [X287 : subst] : (((comp @ id @ X287)) = X287) & ! [X288 : subst] : (((comp @ X288 @ id)) = X288))) & (axmap = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (! [X289 : term,X290 : term,X291 : term,X292 : term] : ((X291 = X292) | (((ap @ X290 @ X292)) != ((ap @ X289 @ X291)))) <=> (apinj2 = $true)) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X293 : (subst > term > subst > $o),X294 : term] : (~ ! [X295 : (subst > term > term)] : (~ ! [X296 : term] : ((((X293 @ id @ (X295 @ id @ X296) @ id)) = $true) | ~ (((X293 @ id @ X296 @ id)) = $true)) | ~ ! [X297 : subst,X298 : subst,X299 : term] : (((sub @ (X295 @ X298 @ X299) @ X297)) = ((X295 @ (comp @ X298 @ X297) @ (sub @ X299 @ X297)))) | ($true = ((X293 @ id @ (lam @ (X295 @ sh @ one)) @ id)))) | ~ ! [X300 : term] : (~ (((X293 @ id @ X300 @ id)) = $true) | (((X293 @ id @ (sub @ X294 @ (push @ X300 @ id)) @ id)) = $true)) | (((X293 @ id @ (lam @ (sub @ X294 @ (push @ one @ sh))) @ id)) = $true))) & (hoasinduction_lem1_gthm = $true) & (hoaslamnotvar = $true) & ((termmset = $true) <=> ! [X301 : term] : (((sub @ X301 @ id)) = X301)) & (pushprop_lem2v2 = $true) & (ulamvarind = $true) & (! [X302 : term,X303 : term,X304 : (subst > term > term)] : (~ ! [X305 : subst,X306 : subst,X307 : term] : (((X304 @ (comp @ X306 @ X305) @ (sub @ X307 @ X305))) = ((sub @ (X304 @ X306 @ X307) @ X305))) | (((lam @ (X304 @ sh @ one))) != ((ap @ (sub @ X302 @ id) @ X303)))) <=> (hoaslamnotap = $true)) & (hoasapnotvar_gthm = $true) & ((axscons = $true) <=> ! [X308 : subst] : (((push @ (sub @ one @ X308) @ (comp @ sh @ X308))) = X308)) & (lamnotvar = $true) & (! [X309 : term,X310 : term,X311 : term,X312 : term] : ((X309 = X312) | (((ap @ (sub @ X309 @ id) @ X310)) != ((ap @ (sub @ X312 @ id) @ X311)))) <=> (hoasapinj1 = $true)) & (termmset_gthm = $true) & ((hoasinduction_lem3v2 = $true) <=> ! [X313 : term,X314 : (term > $o),X315 : (subst > term > subst > $o)] : (~ ! [X316 : term] : (((X314 @ X316)) = ((X315 @ id @ X316 @ id))) | ~ ! [X317 : term,X318 : subst,X319 : subst,X320 : subst] : (~ ($true = ((X315 @ X320 @ X317 @ (comp @ X319 @ X318)))) | ($true = ((X315 @ (comp @ X320 @ X319) @ (sub @ X317 @ X319) @ X318)))) | ~ ! [X321 : (subst > term > term)] : (($true = ((X315 @ id @ (lam @ (X321 @ sh @ one)) @ id))) | ~ ! [X322 : subst,X323 : subst,X324 : term] : (((X321 @ (comp @ X322 @ X323) @ (sub @ X324 @ X323))) = ((sub @ (X321 @ X322 @ X324) @ X323))) | ~ ! [X325 : term] : ((((X315 @ id @ (X321 @ id @ X325) @ id)) = $true) | ~ ($true = ((X315 @ id @ X325 @ id))))) | ~ ! [X326 : subst,X327 : subst,X328 : term,X329 : subst] : ((((X315 @ X329 @ X328 @ (comp @ X327 @ X326))) = $true) | ~ (((X315 @ (comp @ X329 @ X327) @ (sub @ X328 @ X327) @ X326)) = $true)) | (((X314 @ (lam @ X313))) = $true) | ~ ! [X330 : term] : (~ (((X314 @ X330)) = $true) | (((X314 @ (sub @ X313 @ (push @ X330 @ id)))) = $true)))) & (! [X331 : (subst > term > term),X332 : (subst > term > term),X333 : term,X334 : subst] : ((((lam @ (X332 @ sh @ one))) != ((lam @ (X331 @ sh @ one)))) | ~ ! [X335 : subst,X336 : term,X337 : subst] : (((sub @ (X331 @ X337 @ X336) @ X335)) = ((X331 @ (comp @ X337 @ X335) @ (sub @ X336 @ X335)))) | ~ ! [X338 : subst,X339 : subst,X340 : term] : (((sub @ (X332 @ X338 @ X340) @ X339)) = ((X332 @ (comp @ X338 @ X339) @ (sub @ X340 @ X339)))) | (((X331 @ X334 @ X333)) = ((X332 @ X334 @ X333)))) <=> (hoaslaminj = $true)) & (substmonoid_gthm = $true) & ((hoasinduction_lem3v2_lthm = $true) <=> (! [X341 : term] : (((sub @ X341 @ id)) = X341) => ! [X342 : (subst > term > subst > $o),X343 : (term > $o),X344 : term] : (~ ! [X345 : term,X346 : subst,X347 : subst,X348 : subst] : (($true = ((X342 @ X348 @ X345 @ (comp @ X346 @ X347)))) | ~ ($true = ((X342 @ (comp @ X348 @ X346) @ (sub @ X345 @ X346) @ X347)))) | (((X343 @ (lam @ X344))) = $true) | ~ ! [X349 : term] : (((X343 @ X349)) = ((X342 @ id @ X349 @ id))) | ~ ! [X350 : (subst > term > term)] : (~ ! [X351 : term,X352 : subst,X353 : subst] : (((X350 @ (comp @ X352 @ X353) @ (sub @ X351 @ X353))) = ((sub @ (X350 @ X352 @ X351) @ X353))) | ~ ! [X354 : term] : (~ (((X342 @ id @ X354 @ id)) = $true) | (((X342 @ id @ (X350 @ id @ X354) @ id)) = $true)) | (((X342 @ id @ (lam @ (X350 @ sh @ one)) @ id)) = $true)) | ~ ! [X355 : subst,X356 : subst,X357 : term,X358 : subst] : (~ (((X342 @ X358 @ X357 @ (comp @ X356 @ X355))) = $true) | (((X342 @ (comp @ X358 @ X356) @ (sub @ X357 @ X356) @ X355)) = $true)) | ~ ! [X359 : term] : (~ (((X343 @ X359)) = $true) | (((X343 @ (sub @ X344 @ (push @ X359 @ id)))) = $true))))) & (! [X360 : term,X361 : (subst > term > subst > $o)] : (~ ! [X362 : term] : (~ ($true = ((X361 @ id @ X362 @ id))) | (((X361 @ id @ (sub @ X360 @ (push @ X362 @ id)) @ id)) = $true)) | ~ ! [X363 : (subst > term > term)] : (~ ! [X364 : term] : (~ (((X361 @ id @ X364 @ id)) = $true) | (((X361 @ id @ (X363 @ id @ X364) @ id)) = $true)) | (((X361 @ id @ (lam @ (X363 @ sh @ one)) @ id)) = $true) | ~ ! [X365 : subst,X366 : subst,X367 : term] : (((X363 @ (comp @ X365 @ X366) @ (sub @ X367 @ X366))) = ((sub @ (X363 @ X365 @ X367) @ X366)))) | ($true = ((X361 @ id @ (lam @ X360) @ id)))) <=> (hoasinduction_lem3a = $true)) & (! [X368 : (term > $o),X369 : term] : (~ ! [X370 : term] : (~ (((var @ X370)) = $true) | (((X368 @ X370)) = $true)) | ~ ! [X371 : term] : ((((X368 @ (lam @ X371))) = $true) | ~ ! [X372 : term] : (~ (((X368 @ X372)) = $true) | ($true = ((X368 @ (sub @ X371 @ (push @ X372 @ id))))))) | ~ ! [X373 : term,X374 : term] : (~ (((X368 @ X373)) = $true) | ~ (((X368 @ X374)) = $true) | (((X368 @ (ap @ X373 @ X374))) = $true)) | ($true = ((X368 @ X369)))) <=> (induction2 = $true)) & ! [X375 : subst] : ? [X376 : d_subst] : (((d2subst @ X376)) = X375) & ((hoasinduction_lem3v2a = $true) <=> ! [X377 : (subst > term > subst > $o),X378 : (term > $o),X379 : term] : (~ ! [X380 : term] : (($true = ((X378 @ (sub @ X379 @ (push @ X380 @ id))))) | ~ (((X378 @ X380)) = $true)) | ~ ! [X381 : term] : (((X378 @ X381)) = ((X377 @ id @ X381 @ id))) | ~ ! [X382 : (subst > term > term)] : ((((X377 @ id @ (lam @ (X382 @ sh @ one)) @ id)) = $true) | ~ ! [X383 : term] : ((((X377 @ id @ (X382 @ id @ X383) @ id)) = $true) | ~ (((X377 @ id @ X383 @ id)) = $true)) | ~ ! [X384 : subst,X385 : term,X386 : subst] : (((sub @ (X382 @ X386 @ X385) @ X384)) = ((X382 @ (comp @ X386 @ X384) @ (sub @ X385 @ X384))))) | (((X378 @ (lam @ X379))) = $true))) & (! [X387 : term,X388 : term,X389 : term,X390 : term] : ((((ap @ X387 @ X390)) != ((ap @ X388 @ X389))) | (X387 = X388)) <=> (apinj1 = $true)) & (hoaslamnotap_lthm = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (! [X391 : (term > $o),X392 : term] : (~ ! [X393 : term] : (~ ($true = ((var @ X393))) | (((X391 @ X393)) = $true)) | ~ ! [X394 : term] : ((((X391 @ (lam @ X394))) = $true) | ~ (((X391 @ X394)) = $true)) | (((X391 @ X392)) = $true) | ~ ! [X395 : term,X396 : term] : ((((X391 @ (ap @ X395 @ X396))) = $true) | ~ (((X391 @ X396)) = $true) | ~ (((X391 @ X395)) = $true))) <=> (induction = $true)) & (hoaslaminj_gthm = $true) & (hoasinduction_lem1v2_gthm = $true) & (hoaslamnotvar_lthm = $true) & ((hoasinduction_lem3a_lthm = $true) <=> (! [X397 : term] : (((sub @ X397 @ id)) = X397) => (! [X398 : term,X399 : (subst > term > subst > $o)] : (~ ! [X400 : term] : (~ ($true = ((X399 @ id @ X400 @ id))) | (((X399 @ id @ (sub @ X398 @ (push @ X400 @ id)) @ id)) = $true)) | ~ ! [X401 : (subst > term > term)] : (($true = ((X399 @ id @ (lam @ (X401 @ sh @ one)) @ id))) | ~ ! [X402 : term] : ((((X399 @ id @ (X401 @ id @ X402) @ id)) = $true) | ~ ($true = ((X399 @ id @ X402 @ id)))) | ~ ! [X403 : subst,X404 : subst,X405 : term] : (((X401 @ (comp @ X403 @ X404) @ (sub @ X405 @ X404))) = ((sub @ (X401 @ X403 @ X405) @ X404)))) | ($true = ((X399 @ id @ (lam @ (sub @ X398 @ (push @ one @ sh))) @ id)))) => ! [X406 : (subst > term > subst > $o),X407 : term] : (~ ! [X408 : term] : (~ (((X406 @ id @ X408 @ id)) = $true) | (((X406 @ id @ (sub @ X407 @ (push @ X408 @ id)) @ id)) = $true)) | ($true = ((X406 @ id @ (lam @ X407) @ id))) | ~ ! [X409 : (subst > term > term)] : (~ ! [X410 : term] : (~ (((X406 @ id @ X410 @ id)) = $true) | ($true = ((X406 @ id @ (X409 @ id @ X410) @ id)))) | ~ ! [X411 : subst,X412 : term,X413 : subst] : (((X409 @ (comp @ X411 @ X413) @ (sub @ X412 @ X413))) = ((sub @ (X409 @ X411 @ X412) @ X413))) | (((X406 @ id @ (lam @ (X409 @ sh @ one)) @ id)) = $true)))))) & ((axvarcons = $true) <=> ! [X414 : term,X415 : subst] : (((sub @ one @ (push @ X414 @ X415))) = X414)) & (! [X416 : subst] : (((comp @ id @ X416)) = X416) <=> (axidl = $true)) & (! [X417 : term,X418 : (subst > term > subst > $o)] : (~ ! [X419 : (subst > term > term)] : (~ ! [X420 : term] : ((((X418 @ id @ (X419 @ id @ X420) @ id)) = $true) | ~ (((X418 @ id @ X420 @ id)) = $true)) | (((X418 @ id @ (lam @ (X419 @ sh @ one)) @ id)) = $true) | ~ ! [X421 : subst,X422 : term,X423 : subst] : (((X419 @ (comp @ X421 @ X423) @ (sub @ X422 @ X423))) = ((sub @ (X419 @ X421 @ X422) @ X423)))) | ($true = ((X418 @ id @ X417 @ id))) | ~ ! [X424 : term,X425 : term] : (~ (((X418 @ id @ X425 @ id)) = $true) | ($true = ((X418 @ id @ (ap @ (sub @ X424 @ id) @ X425) @ id))) | ~ (((X418 @ id @ X424 @ id)) = $true))) <=> (hoasinduction_no_psi_cond = $true)) & ((induction2lem_lthm = $true) <=> (! [X426 : term,X427 : subst] : (((sub @ one @ (push @ X426 @ X427))) = X426) => (! [X428 : subst,X429 : term] : (((comp @ sh @ (push @ X429 @ X428))) = X428) => (! [X430 : subst] : (((comp @ X430 @ id)) = X430) => (! [X431 : term,X432 : (term > $o)] : (~ ! [X433 : term] : (~ (((X432 @ X433)) = $true) | ($true = ((X432 @ (lam @ X433))))) | ~ ! [X434 : term,X435 : term] : ((((X432 @ (ap @ X434 @ X435))) = $true) | ~ ($true = ((X432 @ X435))) | ~ (((X432 @ X434)) = $true)) | ~ ! [X436 : term] : (~ (((var @ X436)) = $true) | (((X432 @ X436)) = $true)) | (((X432 @ X431)) = $true)) => ! [X437 : (term > $o),X438 : term,X439 : subst] : ((((X437 @ (sub @ X438 @ X439))) = $true) | ~ ! [X440 : term] : ((((X437 @ (sub @ X440 @ X439))) = $true) | ~ ($true = ((var @ X440)))) | ~ ! [X441 : term,X442 : term] : (~ (((X437 @ X441)) = $true) | (((X437 @ (ap @ X442 @ X441))) = $true) | ~ (((X437 @ X442)) = $true)) | ~ ! [X443 : term] : (~ ! [X444 : term] : (~ (((X437 @ X444)) = $true) | (((X437 @ (sub @ X443 @ (push @ X444 @ id)))) = $true)) | ($true = ((X437 @ (lam @ X443))))))))))) & ((hoasinduction_lem3_lthm = $true) <=> (! [X445 : term] : (((sub @ X445 @ id)) = X445) => (! [X446 : term,X447 : (subst > term > subst > $o)] : (~ ! [X448 : (subst > term > term)] : (~ ! [X449 : term] : ((((X447 @ id @ (X448 @ id @ X449) @ id)) = $true) | ~ (((X447 @ id @ X449 @ id)) = $true)) | (((X447 @ id @ (lam @ (X448 @ sh @ one)) @ id)) = $true) | ~ ! [X450 : term,X451 : subst,X452 : subst] : (((sub @ (X448 @ X451 @ X450) @ X452)) = ((X448 @ (comp @ X451 @ X452) @ (sub @ X450 @ X452))))) | (((X447 @ id @ (lam @ (sub @ X446 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X453 : term] : (~ (((X447 @ id @ X453 @ id)) = $true) | (((X447 @ id @ (sub @ X446 @ (push @ X453 @ id)) @ id)) = $true))) => ! [X454 : term,X455 : (subst > term > subst > $o)] : (~ ! [X456 : term,X457 : subst,X458 : subst,X459 : subst] : (~ ($true = ((X455 @ X457 @ X456 @ (comp @ X459 @ X458)))) | (((X455 @ (comp @ X457 @ X459) @ (sub @ X456 @ X459) @ X458)) = $true)) | ~ ! [X460 : (subst > term > term)] : (~ ! [X461 : term] : (($true = ((X455 @ id @ (X460 @ id @ X461) @ id))) | ~ (((X455 @ id @ X461 @ id)) = $true)) | ~ ! [X462 : term,X463 : subst,X464 : subst] : (((sub @ (X460 @ X464 @ X462) @ X463)) = ((X460 @ (comp @ X464 @ X463) @ (sub @ X462 @ X463)))) | (((X455 @ id @ (lam @ (X460 @ sh @ one)) @ id)) = $true)) | ~ ! [X465 : term] : ((((X455 @ id @ (sub @ X454 @ (push @ X465 @ id)) @ id)) = $true) | ~ (((X455 @ id @ X465 @ id)) = $true)) | ~ ! [X466 : subst,X467 : term,X468 : subst,X469 : subst] : (~ (((X455 @ (comp @ X468 @ X469) @ (sub @ X467 @ X469) @ X466)) = $true) | (((X455 @ X468 @ X467 @ (comp @ X469 @ X466))) = $true)) | (((X455 @ id @ (lam @ X454) @ id)) = $true))))) & (hoasinduction_gthm = $true) & ((! [X470 : term,X471 : term,X472 : term,X473 : term] : ((((ap @ X471 @ X472)) != ((ap @ X473 @ X470))) | (X470 = X472)) => ! [X474 : term,X475 : term,X476 : term,X477 : term] : ((((ap @ (sub @ X476 @ id) @ X475)) != ((ap @ (sub @ X474 @ id) @ X477))) | (X475 = X477))) <=> (hoasapinj2_lthm = $true)) & (! [X478 : term,X479 : (subst > term > subst > $o)] : ((((X479 @ id @ (lam @ X478) @ id)) = $true) | ~ ! [X480 : term] : ((((X479 @ id @ (sub @ X478 @ (push @ X480 @ id)) @ id)) = $true) | ~ (((X479 @ id @ X480 @ id)) = $true)) | ~ ! [X481 : (subst > term > term)] : ((((X479 @ id @ (lam @ (X481 @ sh @ one)) @ id)) = $true) | ~ ! [X482 : subst,X483 : term,X484 : subst] : (((X481 @ (comp @ X484 @ X482) @ (sub @ X483 @ X482))) = ((sub @ (X481 @ X484 @ X483) @ X482))) | ~ ! [X485 : term] : (~ (((X479 @ id @ X485 @ id)) = $true) | (((X479 @ id @ (X481 @ id @ X485) @ id)) = $true))) | ~ ! [X486 : subst,X487 : subst,X488 : subst,X489 : term] : (~ (((X479 @ X486 @ X489 @ (comp @ X488 @ X487))) = $true) | (((X479 @ (comp @ X486 @ X488) @ (sub @ X489 @ X488) @ X487)) = $true)) | ~ ! [X490 : term,X491 : subst,X492 : subst,X493 : subst] : ((((X479 @ X491 @ X490 @ (comp @ X493 @ X492))) = $true) | ~ ($true = ((X479 @ (comp @ X491 @ X493) @ (sub @ X490 @ X493) @ X492))))) <=> (hoasinduction_lem3 = $true)) & (axclos = $true) & ((induction2_lthm = $true) <=> (! [X494 : term] : (((sub @ X494 @ id)) = X494) => (! [X495 : (term > $o),X496 : term,X497 : subst] : (~ ! [X498 : term] : (~ (((var @ X498)) = $true) | (((X495 @ (sub @ X498 @ X497))) = $true)) | (((X495 @ (sub @ X496 @ X497))) = $true) | ~ ! [X499 : term] : ((((X495 @ (lam @ X499))) = $true) | ~ ! [X500 : term] : (~ (((X495 @ X500)) = $true) | (((X495 @ (sub @ X499 @ (push @ X500 @ id)))) = $true))) | ~ ! [X501 : term,X502 : term] : (~ (((X495 @ X501)) = $true) | (((X495 @ (ap @ X501 @ X502))) = $true) | ~ (((X495 @ X502)) = $true))) => ! [X503 : term,X504 : (term > $o)] : (~ ! [X505 : term] : ((((X504 @ X505)) = $true) | ~ ($true = ((var @ X505)))) | ~ ! [X506 : term,X507 : term] : ((((X504 @ (ap @ X506 @ X507))) = $true) | ~ ($true = ((X504 @ X506))) | ~ (((X504 @ X507)) = $true)) | ~ ! [X508 : term] : ((((X504 @ (lam @ X508))) = $true) | ~ ! [X509 : term] : ((((X504 @ (sub @ X508 @ (push @ X509 @ id)))) = $true) | ~ (((X504 @ X509)) = $true))) | (((X504 @ X503)) = $true))))) & (apnotvar = $true) & (hoasinduction_lem3v2_gthm = $true) & (hoaslamnotvar_gthm = $true) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X510 : term] : ~ ! [X511 : (subst > term > term)] : (((sub @ X510 @ (push @ one @ sh))) != ((X511 @ sh @ one)))) & (pushprop_lthm = $true) & ((shinj = $true) <=> ! [X512 : term,X513 : term] : ((((sub @ X512 @ sh)) != ((sub @ X513 @ sh))) | (X512 = X513))) & (! [X514 : (subst > term > subst > $o)] : ~ ! [X515 : (term > $o)] : ~ ! [X516 : term] : (((X514 @ id @ X516 @ id)) = ((X515 @ X516))) <=> (hoasinduction_lem0 = $true)) & (! [X517 : term,X518 : (subst > term > subst > $o)] : (($true = ((X518 @ id @ (lam @ (sub @ X517 @ (push @ one @ sh))) @ id))) | ~ ! [X519 : term] : (~ (((X518 @ id @ X519 @ id)) = $true) | (((X518 @ id @ (sub @ X517 @ (push @ X519 @ id)) @ id)) = $true)) | ~ ! [X520 : (subst > term > term)] : (~ ! [X521 : term] : ((((X518 @ id @ (X520 @ id @ X521) @ id)) = $true) | ~ (((X518 @ id @ X521 @ id)) = $true)) | ($true = ((X518 @ id @ (lam @ (X520 @ sh @ one)) @ id))) | ~ ! [X522 : term,X523 : subst,X524 : subst] : (((X520 @ (comp @ X524 @ X523) @ (sub @ X522 @ X523))) = ((sub @ (X520 @ X524 @ X522) @ X523))))) <=> (hoasinduction_lem3aa = $true)) & ((hoasinduction_lem3b = $true) <=> ! [X525 : term] : ~ ! [X526 : (subst > term > term)] : (((sub @ X525 @ (push @ one @ sh))) != ((X526 @ sh @ one)))) & ((termmset_lthm = $true) <=> (! [X527 : term] : (((sub @ X527 @ id)) = X527) => ! [X528 : term] : (((sub @ X528 @ id)) = X528))) & (ulamvarsh = $true) & ~ (lamnotap = $true) & (! [X529 : subst,X530 : (term > $o),X531 : term] : ~ ! [X532 : (term > $o)] : ~ ! [X533 : term] : (((X532 @ X533)) = ((X530 @ (sub @ X533 @ (push @ X531 @ X529))))) <=> (pushprop_lem0 = $true)) & (induction2lem_gthm = $true) & ((pushprop_lem1_lthm = $true) <=> (! [X534 : term,X535 : subst] : (((sub @ one @ (push @ X534 @ X535))) = X534) => (! [X536 : subst,X537 : term] : (((comp @ sh @ (push @ X537 @ X536))) = X536) => ! [X538 : term,X539 : (term > $o),X540 : subst,X541 : term,X542 : (term > $o)] : ((((X542 @ (sub @ X541 @ (push @ X538 @ X540)))) = $true) | ~ (((X539 @ X541)) = $true))))) & (! [X543 : (term > $o),X544 : (term > $o),X545 : subst,X546 : term] : (($true = ((X544 @ one))) | ~ (((X543 @ X546)) = $true) | ~ ! [X547 : term] : (((X543 @ (sub @ X547 @ (push @ X546 @ X545)))) = ((X544 @ X547)))) <=> (pushprop_lem1v2 = $true)) & (pushprop_lem3v2 = $true) & (hoasapinj1_gthm = $true) & ! [X548 : term] : ? [X549 : d_term] : (((d2term @ X549)) = X548) & (pushprop_lem3v2_lthm = $true)),
  inference(fool_elimination,[],[f6])).
thf(f8,plain,(
  (axclos != ((!! @ subst @ (^[Y0 : subst]: (!! @ subst @ (^[Y1 : subst]: (!! @ term @ (^[Y2 : term]: ((sub @ (sub @ Y2 @ Y0) @ Y1) = (sub @ Y2 @ (comp @ Y0 @ Y1)))))))))))),
  inference(flattening,[],[f5])).
thf(f9,plain,(
  (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (hoasinduction_lem1 = $true) & (pushprop_lem0_gthm = $true) & (hoasapnotvar_lthm = $true) & (! [X0 : term,X1 : (subst > term > subst > $o)] : (~ ! [X2 : subst,X3 : subst,X4 : term,X5 : subst] : ((((X1 @ (comp @ X5 @ X3) @ (sub @ X4 @ X3) @ X2)) = $true) | ~ (((X1 @ X5 @ X4 @ (comp @ X3 @ X2))) = $true)) | ~ ! [X6 : (subst > term > term)] : ((((X1 @ id @ (lam @ (X6 @ sh @ one)) @ id)) = $true) | ~ ! [X9 : subst,X8 : term,X7 : subst] : (((sub @ (X6 @ X9 @ X8) @ X7)) = ((X6 @ (comp @ X9 @ X7) @ (sub @ X8 @ X7)))) | ~ ! [X10 : term] : (~ (((X1 @ id @ X10 @ id)) = $true) | (((X1 @ id @ (X6 @ id @ X10) @ id)) = $true))) | ~ ! [X11 : subst,X12 : subst,X13 : term,X14 : subst] : (~ ($true = ((X1 @ (comp @ X14 @ X12) @ (sub @ X13 @ X12) @ X11))) | ($true = ((X1 @ X14 @ X13 @ (comp @ X12 @ X11))))) | (((X1 @ id @ X0 @ id)) = $true) | ~ ! [X15 : term] : ((((X1 @ id @ X15 @ id)) = $true) | ~ ($true = ((var @ (sub @ X15 @ id))))) | ~ ! [X16 : term,X17 : term] : ((((X1 @ id @ (ap @ (sub @ X16 @ id) @ X17) @ id)) = $true) | ~ (((X1 @ id @ X16 @ id)) = $true) | ~ ($true = ((X1 @ id @ X17 @ id))))) <=> (hoasinduction = $true)) & ! [X18 : d_term] : (d_one = X18) & (axvarshift = $true) & ! [X19 : d_subst,X20 : d_subst] : ((((d2subst @ X19)) = ((d2subst @ X20))) => (X19 = X20)) & ((laminj = $true) <=> ! [X21 : term,X22 : term] : ((((lam @ X22)) != ((lam @ X21))) | (X21 = X22))) & (hoasinduction_lem1_lthm = $true) & ~ (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) = $true) & (hoasinduction_lem2_gthm = $true) & (hoasinduction_lem2v2_gthm = $true) & (((d2subst @ d_id)) = id) & ((substmonoid_lthm = $true) <=> (! [X23 : subst] : (((comp @ id @ X23)) = X23) => (! [X24 : subst] : (((comp @ X24 @ id)) = X24) => (! [X26 : subst] : (((comp @ X26 @ id)) = X26) & ! [X25 : subst] : (((comp @ id @ X25)) = X25))))) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (pushprop_lem2v2_gthm = $true) & ((axvarid = $true) <=> ! [X27 : term] : (((sub @ X27 @ id)) = X27)) & ((! [X28 : term] : (((sub @ X28 @ id)) = X28) => (! [X30 : term,X32 : term,X31 : term,X29 : term] : ((((ap @ X31 @ X29)) != ((ap @ X32 @ X30))) | (X31 = X32)) => ! [X36 : term,X33 : term,X34 : term,X35 : term] : ((X33 = X36) | (((ap @ (sub @ X33 @ id) @ X35)) != ((ap @ (sub @ X36 @ id) @ X34)))))) <=> (hoasapinj1_lthm = $true)) & (! [X37 : (subst > term > subst > $o)] : ~ ! [X38 : (term > $o)] : ~ ! [X39 : term] : (((X37 @ id @ X39 @ id)) = ((X38 @ X39))) <=> (hoasinduction_lem0_lthm = $true)) & ((hoasinduction_lem3aaa = $true) <=> ! [X40 : term,X41 : (subst > term > subst > $o)] : (~ ! [X42 : term] : ((((X41 @ id @ (sub @ X40 @ (push @ X42 @ id)) @ id)) = $true) | ~ (((X41 @ id @ X42 @ id)) = $true)) | ~ ! [X43 : term,X44 : (subst > term > term)] : (~ ! [X47 : subst,X45 : term,X46 : subst] : (((sub @ (X44 @ X47 @ X45) @ X46)) = ((sub @ (sub @ X43 @ (push @ X45 @ X47)) @ X46))) | ~ ! [X48 : term] : ((((X41 @ id @ (X44 @ id @ X48) @ id)) = $true) | ~ (((X41 @ id @ X48 @ id)) = $true)) | ~ ! [X49 : subst,X51 : term,X50 : subst] : (((sub @ X43 @ (push @ (sub @ X51 @ X50) @ (comp @ X49 @ X50)))) = ((X44 @ (comp @ X49 @ X50) @ (sub @ X51 @ X50)))) | (((X41 @ id @ (lam @ (X44 @ sh @ one)) @ id)) = $true)) | (((X41 @ id @ (lam @ (sub @ X40 @ (push @ one @ sh))) @ id)) = $true))) & (! [X52 : term] : ~ ! [X53 : (subst > term > term)] : ~ ! [X55 : subst,X54 : term] : (((sub @ X52 @ (push @ X54 @ X55))) = ((X53 @ X55 @ X54))) <=> (hoasinduction_lem3v2_f = $true)) & (hoasinduction_lem3_gthm = $true) & (pushprop_lem2v2_lthm = $true) & ((! [X56 : (subst > term > subst > $o)] : ~ ! [X57 : (term > $o)] : ~ ! [X58 : term] : (((X56 @ id @ X58 @ id)) = ((X57 @ X58))) => (! [X59 : term,X60 : (term > $o)] : ((((X60 @ X59)) = $true) | ~ ! [X61 : term] : (($true = ((X60 @ (lam @ X61)))) | ~ ! [X62 : term] : ((((X60 @ (sub @ X61 @ (push @ X62 @ id)))) = $true) | ~ (((X60 @ X62)) = $true))) | ~ ! [X63 : term,X64 : term] : (~ (((X60 @ X63)) = $true) | ~ (((X60 @ X64)) = $true) | (((X60 @ (ap @ X64 @ X63))) = $true)) | ~ ! [X65 : term] : (~ (((var @ X65)) = $true) | (((X60 @ X65)) = $true))) => (! [X66 : term] : (((sub @ X66 @ id)) = X66) => (! [X67 : (term > $o),X68 : term,X69 : (subst > term > subst > $o)] : (~ ! [X70 : (subst > term > term)] : (~ ! [X72 : subst,X73 : term,X71 : subst] : (((X70 @ (comp @ X71 @ X72) @ (sub @ X73 @ X72))) = ((sub @ (X70 @ X71 @ X73) @ X72))) | (((X69 @ id @ (lam @ (X70 @ sh @ one)) @ id)) = $true) | ~ ! [X74 : term] : (~ ($true = ((X69 @ id @ X74 @ id))) | (((X69 @ id @ (X70 @ id @ X74) @ id)) = $true))) | ~ ! [X75 : term] : ((((X67 @ (sub @ X68 @ (push @ X75 @ id)))) = $true) | ~ (((X67 @ X75)) = $true)) | ($true = ((X67 @ (lam @ X68)))) | ~ ! [X76 : term] : (((X69 @ id @ X76 @ id)) = ((X67 @ X76)))) => ! [X77 : term,X78 : (subst > term > subst > $o)] : (~ ! [X79 : term] : ((((X78 @ id @ X79 @ id)) = $true) | ~ (((var @ (sub @ X79 @ id))) = $true)) | (((X78 @ id @ X77 @ id)) = $true) | ~ ! [X80 : (subst > term > term)] : (~ ! [X81 : subst,X82 : term,X83 : subst] : (((sub @ (X80 @ X81 @ X82) @ X83)) = ((X80 @ (comp @ X81 @ X83) @ (sub @ X82 @ X83)))) | ($true = ((X78 @ id @ (lam @ (X80 @ sh @ one)) @ id))) | ~ ! [X84 : term] : (~ (((X78 @ id @ X84 @ id)) = $true) | (((X78 @ id @ (X80 @ id @ X84) @ id)) = $true))) | ~ ! [X85 : term,X86 : term] : (~ (((X78 @ id @ X86 @ id)) = $true) | (((X78 @ id @ (ap @ (sub @ X86 @ id) @ X85) @ id)) = $true) | ~ (((X78 @ id @ X85 @ id)) = $true)) | ~ ! [X87 : subst,X88 : term,X89 : subst,X90 : subst] : ((((X78 @ (comp @ X87 @ X90) @ (sub @ X88 @ X90) @ X89)) = $true) | ~ ($true = ((X78 @ X87 @ X88 @ (comp @ X90 @ X89))))) | ~ ! [X91 : subst,X92 : term,X93 : subst,X94 : subst] : (($true = ((X78 @ X94 @ X92 @ (comp @ X91 @ X93)))) | ~ (((X78 @ (comp @ X94 @ X91) @ (sub @ X92 @ X91) @ X93)) = $true))))))) <=> (hoasinduction_lthm_3 = $true)) & (((d2subst @ d_id)) = sh) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((pushprop_lem0_lthm = $true) <=> ! [X97 : subst,X96 : (term > $o),X95 : term] : ~ ! [X98 : (term > $o)] : ~ ! [X99 : term] : (((X96 @ (sub @ X99 @ (push @ X95 @ X97)))) = ((X98 @ X99)))) & ((hoasinduction_lem2 = $true) <=> ! [X100 : (subst > term > subst > $o),X101 : term,X102 : term] : (~ ! [X103 : subst,X104 : subst,X105 : subst,X106 : term] : (~ ($true = ((X100 @ (comp @ X103 @ X105) @ (sub @ X106 @ X105) @ X104))) | (((X100 @ X103 @ X106 @ (comp @ X105 @ X104))) = $true)) | ~ (((X100 @ id @ X101 @ id)) = $true) | (((X100 @ id @ (ap @ X101 @ X102) @ id)) = $true) | ~ ! [X107 : term,X108 : term] : ((((X100 @ id @ (ap @ (sub @ X107 @ id) @ X108) @ id)) = $true) | ~ ($true = ((X100 @ id @ X107 @ id))) | ~ ($true = ((X100 @ id @ X108 @ id)))) | ~ ! [X109 : subst,X110 : term,X111 : subst,X112 : subst] : (~ (((X100 @ X111 @ X110 @ (comp @ X112 @ X109))) = $true) | (((X100 @ (comp @ X111 @ X112) @ (sub @ X110 @ X112) @ X109)) = $true)) | ~ (((X100 @ id @ X102 @ id)) = $true))) & (! [X113 : subst,X114 : term,X115 : (term > $o)] : (~ ! [X116 : term] : (~ (((var @ X116)) = $true) | (((X115 @ (sub @ X116 @ X113))) = $true)) | ~ ! [X117 : term] : (($true = ((X115 @ (lam @ X117)))) | ~ ! [X118 : term] : (($true = ((X115 @ (sub @ X117 @ (push @ X118 @ id))))) | ~ (((X115 @ X118)) = $true))) | (((X115 @ (sub @ X114 @ X113))) = $true) | ~ ! [X119 : term,X120 : term] : (~ ($true = ((X115 @ X120))) | (((X115 @ (ap @ X119 @ X120))) = $true) | ~ (((X115 @ X119)) = $true))) <=> (induction2lem = $true)) & (pushprop = $true) & (! [X121 : term] : ~ ! [X122 : (subst > term > term)] : ~ ! [X123 : term,X124 : subst] : (((X122 @ X124 @ X123)) = ((sub @ X121 @ (push @ X123 @ X124)))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & ~ (ulamvar1 = $true) & ((hoasinduction_lem3v2a_lthm = $true) <=> (! [X125 : term] : ~ ! [X126 : (subst > term > term)] : ~ ! [X127 : term,X128 : subst] : (((sub @ X125 @ (push @ X127 @ X128))) = ((X126 @ X128 @ X127))) => (! [X129 : term] : (((sub @ X129 @ id)) = X129) => ! [X130 : (term > $o),X131 : term,X132 : (subst > term > subst > $o)] : (($true = ((X130 @ (lam @ X131)))) | ~ ! [X133 : (subst > term > term)] : (~ ! [X134 : subst,X136 : term,X135 : subst] : (((sub @ (X133 @ X135 @ X136) @ X134)) = ((X133 @ (comp @ X135 @ X134) @ (sub @ X136 @ X134)))) | (((X132 @ id @ (lam @ (X133 @ sh @ one)) @ id)) = $true) | ~ ! [X137 : term] : (~ (((X132 @ id @ X137 @ id)) = $true) | (((X132 @ id @ (X133 @ id @ X137) @ id)) = $true))) | ~ ! [X138 : term] : (((X132 @ id @ X138 @ id)) = ((X130 @ X138))) | ~ ! [X139 : term] : (~ (((X130 @ X139)) = $true) | (((X130 @ (sub @ X131 @ (push @ X139 @ id)))) = $true)))))) & (hoasinduction_lem1v2 = $true) & ((hoasinduction_lthm = $true) <=> (! [X140 : term,X141 : (term > $o)] : (~ ! [X142 : term] : (~ (((var @ X142)) = $true) | ($true = ((X141 @ X142)))) | ~ ! [X143 : term] : ((((X141 @ (lam @ X143))) = $true) | ~ ! [X144 : term] : (~ (((X141 @ X144)) = $true) | ($true = ((X141 @ (sub @ X143 @ (push @ X144 @ id))))))) | (((X141 @ X140)) = $true) | ~ ! [X145 : term,X146 : term] : (~ (((X141 @ X145)) = $true) | ~ (((X141 @ X146)) = $true) | ($true = ((X141 @ (ap @ X146 @ X145)))))) => (! [X147 : term,X148 : (subst > term > subst > $o),X149 : term] : (~ ! [X150 : subst,X151 : term,X152 : subst,X153 : subst] : (($true = ((X148 @ (comp @ X152 @ X150) @ (sub @ X151 @ X150) @ X153))) | ~ (((X148 @ X152 @ X151 @ (comp @ X150 @ X153))) = $true)) | ~ (((X148 @ id @ X147 @ id)) = $true) | ~ (((X148 @ id @ X149 @ id)) = $true) | (((X148 @ id @ (ap @ X149 @ X147) @ id)) = $true) | ~ ! [X154 : subst,X155 : subst,X156 : term,X157 : subst] : (~ (((X148 @ (comp @ X155 @ X154) @ (sub @ X156 @ X154) @ X157)) = $true) | (((X148 @ X155 @ X156 @ (comp @ X154 @ X157))) = $true)) | ~ ! [X158 : term,X159 : term] : (~ (((X148 @ id @ X158 @ id)) = $true) | ~ (((X148 @ id @ X159 @ id)) = $true) | (((X148 @ id @ (ap @ (sub @ X159 @ id) @ X158) @ id)) = $true))) => (! [X160 : (subst > term > subst > $o),X161 : term] : (~ ! [X162 : term,X163 : subst,X164 : subst,X165 : subst] : ((((X160 @ X163 @ X162 @ (comp @ X164 @ X165))) = $true) | ~ (((X160 @ (comp @ X163 @ X164) @ (sub @ X162 @ X164) @ X165)) = $true)) | ~ ! [X166 : (subst > term > term)] : (~ ! [X168 : subst,X169 : term,X167 : subst] : (((X166 @ (comp @ X167 @ X168) @ (sub @ X169 @ X168))) = ((sub @ (X166 @ X167 @ X169) @ X168))) | ($true = ((X160 @ id @ (lam @ (X166 @ sh @ one)) @ id))) | ~ ! [X170 : term] : (~ (((X160 @ id @ X170 @ id)) = $true) | (((X160 @ id @ (X166 @ id @ X170) @ id)) = $true))) | ~ ! [X171 : subst,X172 : term,X173 : subst,X174 : subst] : (($true = ((X160 @ (comp @ X174 @ X171) @ (sub @ X172 @ X171) @ X173))) | ~ (((X160 @ X174 @ X172 @ (comp @ X171 @ X173))) = $true)) | (((X160 @ id @ (lam @ X161) @ id)) = $true) | ~ ! [X175 : term] : (~ (((X160 @ id @ X175 @ id)) = $true) | (((X160 @ id @ (sub @ X161 @ (push @ X175 @ id)) @ id)) = $true))) => ! [X176 : term,X177 : (subst > term > subst > $o)] : (~ ! [X178 : term,X179 : term] : ((((X177 @ id @ (ap @ (sub @ X178 @ id) @ X179) @ id)) = $true) | ~ (((X177 @ id @ X179 @ id)) = $true) | ~ (((X177 @ id @ X178 @ id)) = $true)) | (((X177 @ id @ X176 @ id)) = $true) | ~ ! [X180 : (subst > term > term)] : (~ ! [X181 : term] : (($true = ((X177 @ id @ (X180 @ id @ X181) @ id))) | ~ ($true = ((X177 @ id @ X181 @ id)))) | ~ ! [X184 : term,X183 : subst,X182 : subst] : (((sub @ (X180 @ X183 @ X184) @ X182)) = ((X180 @ (comp @ X183 @ X182) @ (sub @ X184 @ X182)))) | (((X177 @ id @ (lam @ (X180 @ sh @ one)) @ id)) = $true)) | ~ ! [X185 : term,X186 : subst,X187 : subst,X188 : subst] : (~ (((X177 @ X188 @ X185 @ (comp @ X186 @ X187))) = $true) | (((X177 @ (comp @ X188 @ X186) @ (sub @ X185 @ X186) @ X187)) = $true)) | ~ ! [X189 : term] : ((((X177 @ id @ X189 @ id)) = $true) | ~ (((var @ (sub @ X189 @ id))) = $true)) | ~ ! [X190 : subst,X191 : subst,X192 : subst,X193 : term] : ((((X177 @ X192 @ X193 @ (comp @ X191 @ X190))) = $true) | ~ (((X177 @ (comp @ X192 @ X191) @ (sub @ X193 @ X191) @ X190)) = $true))))))) & ((pushprop_lem1v2_lthm = $true) <=> (! [X194 : subst,X195 : term] : (((sub @ one @ (push @ X195 @ X194))) = X195) => ! [X196 : (term > $o),X197 : subst,X198 : (term > $o),X199 : term] : (~ (((X198 @ X199)) = $true) | ($true = ((X196 @ one))) | ~ ! [X200 : term] : (((X196 @ X200)) = ((X198 @ (sub @ X200 @ (push @ X199 @ X197)))))))) & (! [X201 : term,X202 : (term > $o),X203 : (subst > term > subst > $o),X204 : term] : (~ ! [X205 : term,X206 : subst,X207 : subst,X208 : subst] : (~ (((X203 @ X207 @ X205 @ (comp @ X206 @ X208))) = $true) | ($true = ((X203 @ (comp @ X207 @ X206) @ (sub @ X205 @ X206) @ X208)))) | ~ (((X202 @ X201)) = $true) | ~ ! [X209 : term,X210 : subst,X211 : subst,X212 : subst] : (~ (((X203 @ (comp @ X210 @ X211) @ (sub @ X209 @ X211) @ X212)) = $true) | (((X203 @ X210 @ X209 @ (comp @ X211 @ X212))) = $true)) | (((X202 @ (ap @ X204 @ X201))) = $true) | ~ ! [X213 : term] : (((X203 @ id @ X213 @ id)) = ((X202 @ X213))) | ~ ($true = ((X202 @ X204))) | ~ ! [X214 : term,X215 : term] : (~ (((X203 @ id @ X214 @ id)) = $true) | (((X203 @ id @ (ap @ (sub @ X214 @ id) @ X215) @ id)) = $true) | ~ (((X203 @ id @ X215 @ id)) = $true))) <=> (hoasinduction_lem2v2 = $true)) & (pushprop_lem1v2_gthm = $true) & ((pushprop_lem1 = $true) <=> ! [X216 : (term > $o),X217 : term,X218 : (term > $o),X219 : subst,X220 : term] : ((((X218 @ (sub @ X217 @ (push @ X220 @ X219)))) = $true) | ~ (((X216 @ X217)) = $true))) & (hoasapinj2_gthm = $true) & (pushprop_lthm_orig = $true) & (axabs = $true) & ! [X221 : d_term,X222 : d_term] : ((((d2term @ X222)) = ((d2term @ X221))) => (X221 = X222)) & (induction2_gthm = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ~ (((var @ (d2term @ d_one))) = $true) & (! [X223 : term,X224 : subst] : (((comp @ sh @ (push @ X223 @ X224))) = X224) <=> (axshiftcons = $true)) & (hoasinduction_lem3b_gthm = $true) & (((d2term @ d_one)) = one) & (hoasapnotvar = $true) & (hoasinduction_lem3a_gthm = $true) & (hoaslamnotap_gthm = $true) & ! [X225 : d_subst] : (d_id = X225) & (hoasinduction_lem2_lthm = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (axassoc = $true) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (! [X226 : (subst > term > subst > $o)] : ~ ! [X227 : (term > $o)] : ~ ! [X228 : term] : (((X227 @ X228)) = ((X226 @ id @ X228 @ id))) => (! [X229 : term,X230 : (term > $o)] : (($true = ((X230 @ X229))) | ~ ! [X231 : term] : (($true = ((X230 @ X231))) | ~ ($true = ((var @ X231)))) | ~ ! [X232 : term,X233 : term] : (~ ($true = ((X230 @ X232))) | ~ (((X230 @ X233)) = $true) | (((X230 @ (ap @ X232 @ X233))) = $true)) | ~ ! [X234 : term] : (~ ! [X235 : term] : ((((X230 @ (sub @ X234 @ (push @ X235 @ id)))) = $true) | ~ (((X230 @ X235)) = $true)) | (((X230 @ (lam @ X234))) = $true))) => (! [X236 : term] : (((sub @ X236 @ id)) = X236) => (! [X237 : (subst > term > subst > $o),X238 : term,X239 : (term > $o)] : (~ ! [X240 : (subst > term > term)] : (~ ! [X241 : subst,X243 : term,X242 : subst] : (((X240 @ (comp @ X242 @ X241) @ (sub @ X243 @ X241))) = ((sub @ (X240 @ X242 @ X243) @ X241))) | ~ ! [X244 : term] : ((((X237 @ id @ (X240 @ id @ X244) @ id)) = $true) | ~ ($true = ((X237 @ id @ X244 @ id)))) | ($true = ((X237 @ id @ (lam @ (X240 @ sh @ one)) @ id)))) | ~ ! [X245 : term] : (~ (((X239 @ X245)) = $true) | ($true = ((X239 @ (sub @ X238 @ (push @ X245 @ id)))))) | (((X239 @ (lam @ X238))) = $true) | ~ ! [X246 : term] : (((X237 @ id @ X246 @ id)) = ((X239 @ X246)))) => ! [X247 : term,X248 : (subst > term > subst > $o)] : ((((X248 @ id @ X247 @ id)) = $true) | ~ ! [X249 : (subst > term > term)] : (~ ! [X250 : term] : (($true = ((X248 @ id @ (X249 @ id @ X250) @ id))) | ~ (((X248 @ id @ X250 @ id)) = $true)) | ~ ! [X253 : subst,X251 : term,X252 : subst] : (((X249 @ (comp @ X253 @ X252) @ (sub @ X251 @ X252))) = ((sub @ (X249 @ X253 @ X251) @ X252))) | (((X248 @ id @ (lam @ (X249 @ sh @ one)) @ id)) = $true)) | ~ ! [X254 : term,X255 : term] : (~ (((X248 @ id @ X254 @ id)) = $true) | (((X248 @ id @ (ap @ (sub @ X254 @ id) @ X255) @ id)) = $true) | ~ (((X248 @ id @ X255 @ id)) = $true)))))))) & (axapp = $true) & ((! [X256 : subst,X257 : term] : (((sub @ one @ (push @ X257 @ X256))) = X257) => (! [X259 : subst,X258 : term] : (((comp @ sh @ (push @ X258 @ X259))) = X259) => (! [X261 : term,X260 : term] : ((X260 = X261) | (((lam @ X260)) != ((lam @ X261)))) => ! [X265 : term,X262 : (subst > term > term),X263 : (subst > term > term),X264 : subst] : (~ ! [X270 : term,X271 : subst,X269 : subst] : (((X262 @ (comp @ X269 @ X271) @ (sub @ X270 @ X271))) = ((sub @ (X262 @ X269 @ X270) @ X271))) | ~ ! [X266 : subst,X267 : subst,X268 : term] : (((X263 @ (comp @ X266 @ X267) @ (sub @ X268 @ X267))) = ((sub @ (X263 @ X266 @ X268) @ X267))) | (((X262 @ X264 @ X265)) = ((X263 @ X264 @ X265))) | (((lam @ (X263 @ sh @ one))) != ((lam @ (X262 @ sh @ one)))))))) <=> (hoaslaminj_lthm = $true)) & (pushprop_gthm = $true) & ((axidr = $true) <=> ! [X272 : subst] : (((comp @ X272 @ id)) = X272)) & (pushprop_lem1_gthm = $true) & ((hoasapinj2 = $true) <=> ! [X274 : term,X273 : term,X276 : term,X275 : term] : ((((ap @ (sub @ X274 @ id) @ X276)) != ((ap @ (sub @ X275 @ id) @ X273))) | (X273 = X276))) & ((! [X278 : subst] : (((comp @ X278 @ id)) = X278) & ! [X277 : subst] : (((comp @ id @ X277)) = X277)) <=> (substmonoid = $true)) & (axmap = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (! [X279 : term,X282 : term,X281 : term,X280 : term] : ((((ap @ X280 @ X282)) != ((ap @ X279 @ X281))) | (X281 = X282)) <=> (apinj2 = $true)) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X283 : (subst > term > subst > $o),X284 : term] : (~ ! [X285 : (subst > term > term)] : (~ ! [X286 : term] : ((((X283 @ id @ (X285 @ id @ X286) @ id)) = $true) | ~ (((X283 @ id @ X286 @ id)) = $true)) | ~ ! [X288 : subst,X289 : term,X287 : subst] : (((X285 @ (comp @ X288 @ X287) @ (sub @ X289 @ X287))) = ((sub @ (X285 @ X288 @ X289) @ X287))) | (((X283 @ id @ (lam @ (X285 @ sh @ one)) @ id)) = $true)) | ~ ! [X290 : term] : (~ ($true = ((X283 @ id @ X290 @ id))) | ($true = ((X283 @ id @ (sub @ X284 @ (push @ X290 @ id)) @ id)))) | (((X283 @ id @ (lam @ (sub @ X284 @ (push @ one @ sh))) @ id)) = $true))) & (hoasinduction_lem1_gthm = $true) & (hoaslamnotvar = $true) & ((termmset = $true) <=> ! [X291 : term] : (((sub @ X291 @ id)) = X291)) & (pushprop_lem2v2 = $true) & (ulamvarind = $true) & ((hoaslamnotap = $true) <=> ! [X293 : term,X294 : (subst > term > term),X292 : term] : ((((lam @ (X294 @ sh @ one))) != ((ap @ (sub @ X292 @ id) @ X293))) | ~ ! [X296 : subst,X297 : term,X295 : subst] : (((sub @ (X294 @ X296 @ X297) @ X295)) = ((X294 @ (comp @ X296 @ X295) @ (sub @ X297 @ X295)))))) & (hoasapnotvar_gthm = $true) & (! [X298 : subst] : (((push @ (sub @ one @ X298) @ (comp @ sh @ X298))) = X298) <=> (axscons = $true)) & (lamnotvar = $true) & ((hoasapinj1 = $true) <=> ! [X302 : term,X299 : term,X301 : term,X300 : term] : ((X299 = X302) | (((ap @ (sub @ X302 @ id) @ X301)) != ((ap @ (sub @ X299 @ id) @ X300))))) & (termmset_gthm = $true) & ((hoasinduction_lem3v2 = $true) <=> ! [X303 : term,X304 : (term > $o),X305 : (subst > term > subst > $o)] : (~ ! [X306 : term] : (((X304 @ X306)) = ((X305 @ id @ X306 @ id))) | ~ ! [X307 : term,X308 : subst,X309 : subst,X310 : subst] : (~ (((X305 @ X310 @ X307 @ (comp @ X309 @ X308))) = $true) | ($true = ((X305 @ (comp @ X310 @ X309) @ (sub @ X307 @ X309) @ X308)))) | ~ ! [X311 : (subst > term > term)] : ((((X305 @ id @ (lam @ (X311 @ sh @ one)) @ id)) = $true) | ~ ! [X314 : term,X312 : subst,X313 : subst] : (((sub @ (X311 @ X312 @ X314) @ X313)) = ((X311 @ (comp @ X312 @ X313) @ (sub @ X314 @ X313)))) | ~ ! [X315 : term] : ((((X305 @ id @ (X311 @ id @ X315) @ id)) = $true) | ~ (((X305 @ id @ X315 @ id)) = $true))) | ~ ! [X316 : subst,X317 : subst,X318 : term,X319 : subst] : (($true = ((X305 @ X319 @ X318 @ (comp @ X317 @ X316)))) | ~ (((X305 @ (comp @ X319 @ X317) @ (sub @ X318 @ X317) @ X316)) = $true)) | (((X304 @ (lam @ X303))) = $true) | ~ ! [X320 : term] : (~ (((X304 @ X320)) = $true) | ($true = ((X304 @ (sub @ X303 @ (push @ X320 @ id)))))))) & ((hoaslaminj = $true) <=> ! [X321 : (subst > term > term),X323 : term,X322 : (subst > term > term),X324 : subst] : (~ ! [X328 : subst,X330 : term,X329 : subst] : (((sub @ (X322 @ X328 @ X330) @ X329)) = ((X322 @ (comp @ X328 @ X329) @ (sub @ X330 @ X329)))) | (((lam @ (X321 @ sh @ one))) != ((lam @ (X322 @ sh @ one)))) | ~ ! [X326 : term,X327 : subst,X325 : subst] : (((sub @ (X321 @ X327 @ X326) @ X325)) = ((X321 @ (comp @ X327 @ X325) @ (sub @ X326 @ X325)))) | (((X322 @ X324 @ X323)) = ((X321 @ X324 @ X323))))) & (substmonoid_gthm = $true) & ((hoasinduction_lem3v2_lthm = $true) <=> (! [X331 : term] : (((sub @ X331 @ id)) = X331) => ! [X332 : (subst > term > subst > $o),X333 : (term > $o),X334 : term] : (~ ! [X335 : term,X336 : subst,X337 : subst,X338 : subst] : ((((X332 @ X338 @ X335 @ (comp @ X336 @ X337))) = $true) | ~ ($true = ((X332 @ (comp @ X338 @ X336) @ (sub @ X335 @ X336) @ X337)))) | (((X333 @ (lam @ X334))) = $true) | ~ ! [X339 : term] : (((X333 @ X339)) = ((X332 @ id @ X339 @ id))) | ~ ! [X340 : (subst > term > term)] : (~ ! [X343 : subst,X341 : term,X342 : subst] : (((X340 @ (comp @ X342 @ X343) @ (sub @ X341 @ X343))) = ((sub @ (X340 @ X342 @ X341) @ X343))) | ~ ! [X344 : term] : (~ ($true = ((X332 @ id @ X344 @ id))) | (((X332 @ id @ (X340 @ id @ X344) @ id)) = $true)) | (((X332 @ id @ (lam @ (X340 @ sh @ one)) @ id)) = $true)) | ~ ! [X345 : subst,X346 : subst,X347 : term,X348 : subst] : (~ ($true = ((X332 @ X348 @ X347 @ (comp @ X346 @ X345)))) | ($true = ((X332 @ (comp @ X348 @ X346) @ (sub @ X347 @ X346) @ X345)))) | ~ ! [X349 : term] : (~ (((X333 @ X349)) = $true) | (((X333 @ (sub @ X334 @ (push @ X349 @ id)))) = $true))))) & (! [X350 : term,X351 : (subst > term > subst > $o)] : (~ ! [X352 : term] : (~ (((X351 @ id @ X352 @ id)) = $true) | ($true = ((X351 @ id @ (sub @ X350 @ (push @ X352 @ id)) @ id)))) | ~ ! [X353 : (subst > term > term)] : (~ ! [X354 : term] : (~ ($true = ((X351 @ id @ X354 @ id))) | (((X351 @ id @ (X353 @ id @ X354) @ id)) = $true)) | (((X351 @ id @ (lam @ (X353 @ sh @ one)) @ id)) = $true) | ~ ! [X355 : subst,X357 : term,X356 : subst] : (((X353 @ (comp @ X355 @ X356) @ (sub @ X357 @ X356))) = ((sub @ (X353 @ X355 @ X357) @ X356)))) | (((X351 @ id @ (lam @ X350) @ id)) = $true)) <=> (hoasinduction_lem3a = $true)) & (! [X358 : (term > $o),X359 : term] : (~ ! [X360 : term] : (~ (((var @ X360)) = $true) | (((X358 @ X360)) = $true)) | ~ ! [X361 : term] : ((((X358 @ (lam @ X361))) = $true) | ~ ! [X362 : term] : (~ (((X358 @ X362)) = $true) | (((X358 @ (sub @ X361 @ (push @ X362 @ id)))) = $true))) | ~ ! [X363 : term,X364 : term] : (~ ($true = ((X358 @ X363))) | ~ (((X358 @ X364)) = $true) | (((X358 @ (ap @ X363 @ X364))) = $true)) | ($true = ((X358 @ X359)))) <=> (induction2 = $true)) & ! [X365 : subst] : ? [X366 : d_subst] : (((d2subst @ X366)) = X365) & ((hoasinduction_lem3v2a = $true) <=> ! [X367 : (subst > term > subst > $o),X368 : (term > $o),X369 : term] : (~ ! [X370 : term] : (($true = ((X368 @ (sub @ X369 @ (push @ X370 @ id))))) | ~ (((X368 @ X370)) = $true)) | ~ ! [X371 : term] : (((X367 @ id @ X371 @ id)) = ((X368 @ X371))) | ~ ! [X372 : (subst > term > term)] : ((((X367 @ id @ (lam @ (X372 @ sh @ one)) @ id)) = $true) | ~ ! [X373 : term] : ((((X367 @ id @ (X372 @ id @ X373) @ id)) = $true) | ~ (((X367 @ id @ X373 @ id)) = $true)) | ~ ! [X375 : term,X376 : subst,X374 : subst] : (((sub @ (X372 @ X376 @ X375) @ X374)) = ((X372 @ (comp @ X376 @ X374) @ (sub @ X375 @ X374))))) | (((X368 @ (lam @ X369))) = $true))) & (! [X379 : term,X377 : term,X380 : term,X378 : term] : ((X377 = X378) | (((ap @ X378 @ X379)) != ((ap @ X377 @ X380)))) <=> (apinj1 = $true)) & (hoaslamnotap_lthm = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (! [X381 : (term > $o),X382 : term] : (~ ! [X383 : term] : (~ (((var @ X383)) = $true) | (((X381 @ X383)) = $true)) | ~ ! [X384 : term] : (($true = ((X381 @ (lam @ X384)))) | ~ ($true = ((X381 @ X384)))) | ($true = ((X381 @ X382))) | ~ ! [X385 : term,X386 : term] : (($true = ((X381 @ (ap @ X385 @ X386)))) | ~ ($true = ((X381 @ X386))) | ~ (((X381 @ X385)) = $true))) <=> (induction = $true)) & (hoaslaminj_gthm = $true) & (hoasinduction_lem1v2_gthm = $true) & (hoaslamnotvar_lthm = $true) & ((hoasinduction_lem3a_lthm = $true) <=> (! [X387 : term] : (((sub @ X387 @ id)) = X387) => (! [X388 : term,X389 : (subst > term > subst > $o)] : (~ ! [X390 : term] : (~ ($true = ((X389 @ id @ X390 @ id))) | (((X389 @ id @ (sub @ X388 @ (push @ X390 @ id)) @ id)) = $true)) | ~ ! [X391 : (subst > term > term)] : ((((X389 @ id @ (lam @ (X391 @ sh @ one)) @ id)) = $true) | ~ ! [X392 : term] : ((((X389 @ id @ (X391 @ id @ X392) @ id)) = $true) | ~ (((X389 @ id @ X392 @ id)) = $true)) | ~ ! [X395 : term,X393 : subst,X394 : subst] : (((sub @ (X391 @ X393 @ X395) @ X394)) = ((X391 @ (comp @ X393 @ X394) @ (sub @ X395 @ X394))))) | (((X389 @ id @ (lam @ (sub @ X388 @ (push @ one @ sh))) @ id)) = $true)) => ! [X396 : (subst > term > subst > $o),X397 : term] : (~ ! [X398 : term] : (~ (((X396 @ id @ X398 @ id)) = $true) | (((X396 @ id @ (sub @ X397 @ (push @ X398 @ id)) @ id)) = $true)) | (((X396 @ id @ (lam @ X397) @ id)) = $true) | ~ ! [X399 : (subst > term > term)] : (~ ! [X400 : term] : (~ (((X396 @ id @ X400 @ id)) = $true) | (((X396 @ id @ (X399 @ id @ X400) @ id)) = $true)) | ~ ! [X402 : term,X403 : subst,X401 : subst] : (((X399 @ (comp @ X401 @ X403) @ (sub @ X402 @ X403))) = ((sub @ (X399 @ X401 @ X402) @ X403))) | ($true = ((X396 @ id @ (lam @ (X399 @ sh @ one)) @ id)))))))) & ((axvarcons = $true) <=> ! [X404 : term,X405 : subst] : (((sub @ one @ (push @ X404 @ X405))) = X404)) & ((axidl = $true) <=> ! [X406 : subst] : (((comp @ id @ X406)) = X406)) & (! [X407 : term,X408 : (subst > term > subst > $o)] : (~ ! [X409 : (subst > term > term)] : (~ ! [X410 : term] : ((((X408 @ id @ (X409 @ id @ X410) @ id)) = $true) | ~ (((X408 @ id @ X410 @ id)) = $true)) | ($true = ((X408 @ id @ (lam @ (X409 @ sh @ one)) @ id))) | ~ ! [X412 : term,X413 : subst,X411 : subst] : (((X409 @ (comp @ X411 @ X413) @ (sub @ X412 @ X413))) = ((sub @ (X409 @ X411 @ X412) @ X413)))) | (((X408 @ id @ X407 @ id)) = $true) | ~ ! [X414 : term,X415 : term] : (~ ($true = ((X408 @ id @ X415 @ id))) | (((X408 @ id @ (ap @ (sub @ X414 @ id) @ X415) @ id)) = $true) | ~ (((X408 @ id @ X414 @ id)) = $true))) <=> (hoasinduction_no_psi_cond = $true)) & ((induction2lem_lthm = $true) <=> (! [X416 : term,X417 : subst] : (((sub @ one @ (push @ X416 @ X417))) = X416) => (! [X419 : term,X418 : subst] : (((comp @ sh @ (push @ X419 @ X418))) = X418) => (! [X420 : subst] : (((comp @ X420 @ id)) = X420) => (! [X421 : term,X422 : (term > $o)] : (~ ! [X423 : term] : (~ (((X422 @ X423)) = $true) | (((X422 @ (lam @ X423))) = $true)) | ~ ! [X424 : term,X425 : term] : ((((X422 @ (ap @ X424 @ X425))) = $true) | ~ (((X422 @ X425)) = $true) | ~ ($true = ((X422 @ X424)))) | ~ ! [X426 : term] : (~ ($true = ((var @ X426))) | (((X422 @ X426)) = $true)) | ($true = ((X422 @ X421)))) => ! [X427 : (term > $o),X428 : term,X429 : subst] : ((((X427 @ (sub @ X428 @ X429))) = $true) | ~ ! [X430 : term] : ((((X427 @ (sub @ X430 @ X429))) = $true) | ~ (((var @ X430)) = $true)) | ~ ! [X431 : term,X432 : term] : (~ (((X427 @ X431)) = $true) | ($true = ((X427 @ (ap @ X432 @ X431)))) | ~ (((X427 @ X432)) = $true)) | ~ ! [X433 : term] : (~ ! [X434 : term] : (~ (((X427 @ X434)) = $true) | (((X427 @ (sub @ X433 @ (push @ X434 @ id)))) = $true)) | (((X427 @ (lam @ X433))) = $true)))))))) & ((hoasinduction_lem3_lthm = $true) <=> (! [X435 : term] : (((sub @ X435 @ id)) = X435) => (! [X436 : term,X437 : (subst > term > subst > $o)] : (~ ! [X438 : (subst > term > term)] : (~ ! [X439 : term] : ((((X437 @ id @ (X438 @ id @ X439) @ id)) = $true) | ~ ($true = ((X437 @ id @ X439 @ id)))) | (((X437 @ id @ (lam @ (X438 @ sh @ one)) @ id)) = $true) | ~ ! [X441 : subst,X442 : subst,X440 : term] : (((X438 @ (comp @ X441 @ X442) @ (sub @ X440 @ X442))) = ((sub @ (X438 @ X441 @ X440) @ X442)))) | ($true = ((X437 @ id @ (lam @ (sub @ X436 @ (push @ one @ sh))) @ id))) | ~ ! [X443 : term] : (~ (((X437 @ id @ X443 @ id)) = $true) | (((X437 @ id @ (sub @ X436 @ (push @ X443 @ id)) @ id)) = $true))) => ! [X444 : term,X445 : (subst > term > subst > $o)] : (~ ! [X446 : term,X447 : subst,X448 : subst,X449 : subst] : (~ (((X445 @ X447 @ X446 @ (comp @ X449 @ X448))) = $true) | ($true = ((X445 @ (comp @ X447 @ X449) @ (sub @ X446 @ X449) @ X448)))) | ~ ! [X450 : (subst > term > term)] : (~ ! [X451 : term] : ((((X445 @ id @ (X450 @ id @ X451) @ id)) = $true) | ~ ($true = ((X445 @ id @ X451 @ id)))) | ~ ! [X453 : subst,X454 : subst,X452 : term] : (((sub @ (X450 @ X454 @ X452) @ X453)) = ((X450 @ (comp @ X454 @ X453) @ (sub @ X452 @ X453)))) | (((X445 @ id @ (lam @ (X450 @ sh @ one)) @ id)) = $true)) | ~ ! [X455 : term] : ((((X445 @ id @ (sub @ X444 @ (push @ X455 @ id)) @ id)) = $true) | ~ (((X445 @ id @ X455 @ id)) = $true)) | ~ ! [X456 : subst,X457 : term,X458 : subst,X459 : subst] : (~ (((X445 @ (comp @ X458 @ X459) @ (sub @ X457 @ X459) @ X456)) = $true) | (((X445 @ X458 @ X457 @ (comp @ X459 @ X456))) = $true)) | (((X445 @ id @ (lam @ X444) @ id)) = $true))))) & (hoasinduction_gthm = $true) & ((hoasapinj2_lthm = $true) <=> (! [X463 : term,X462 : term,X460 : term,X461 : term] : ((((ap @ X463 @ X460)) != ((ap @ X461 @ X462))) | (X460 = X462)) => ! [X466 : term,X464 : term,X465 : term,X467 : term] : ((((ap @ (sub @ X464 @ id) @ X467)) != ((ap @ (sub @ X466 @ id) @ X465))) | (X465 = X467)))) & (! [X468 : term,X469 : (subst > term > subst > $o)] : ((((X469 @ id @ (lam @ X468) @ id)) = $true) | ~ ! [X470 : term] : (($true = ((X469 @ id @ (sub @ X468 @ (push @ X470 @ id)) @ id))) | ~ ($true = ((X469 @ id @ X470 @ id)))) | ~ ! [X471 : (subst > term > term)] : ((((X469 @ id @ (lam @ (X471 @ sh @ one)) @ id)) = $true) | ~ ! [X473 : term,X472 : subst,X474 : subst] : (((X471 @ (comp @ X474 @ X472) @ (sub @ X473 @ X472))) = ((sub @ (X471 @ X474 @ X473) @ X472))) | ~ ! [X475 : term] : (~ (((X469 @ id @ X475 @ id)) = $true) | (((X469 @ id @ (X471 @ id @ X475) @ id)) = $true))) | ~ ! [X476 : subst,X477 : subst,X478 : subst,X479 : term] : (~ (((X469 @ X476 @ X479 @ (comp @ X478 @ X477))) = $true) | (((X469 @ (comp @ X476 @ X478) @ (sub @ X479 @ X478) @ X477)) = $true)) | ~ ! [X480 : term,X481 : subst,X482 : subst,X483 : subst] : ((((X469 @ X481 @ X480 @ (comp @ X483 @ X482))) = $true) | ~ ($true = ((X469 @ (comp @ X481 @ X483) @ (sub @ X480 @ X483) @ X482))))) <=> (hoasinduction_lem3 = $true)) & (axclos = $true) & ((induction2_lthm = $true) <=> (! [X484 : term] : (((sub @ X484 @ id)) = X484) => (! [X485 : (term > $o),X486 : term,X487 : subst] : (~ ! [X488 : term] : (~ (((var @ X488)) = $true) | (((X485 @ (sub @ X488 @ X487))) = $true)) | (((X485 @ (sub @ X486 @ X487))) = $true) | ~ ! [X489 : term] : (($true = ((X485 @ (lam @ X489)))) | ~ ! [X490 : term] : (~ ($true = ((X485 @ X490))) | (((X485 @ (sub @ X489 @ (push @ X490 @ id)))) = $true))) | ~ ! [X491 : term,X492 : term] : (~ (((X485 @ X491)) = $true) | (((X485 @ (ap @ X491 @ X492))) = $true) | ~ (((X485 @ X492)) = $true))) => ! [X493 : term,X494 : (term > $o)] : (~ ! [X495 : term] : ((((X494 @ X495)) = $true) | ~ (((var @ X495)) = $true)) | ~ ! [X496 : term,X497 : term] : ((((X494 @ (ap @ X496 @ X497))) = $true) | ~ (((X494 @ X496)) = $true) | ~ (((X494 @ X497)) = $true)) | ~ ! [X498 : term] : ((((X494 @ (lam @ X498))) = $true) | ~ ! [X499 : term] : ((((X494 @ (sub @ X498 @ (push @ X499 @ id)))) = $true) | ~ (((X494 @ X499)) = $true))) | ($true = ((X494 @ X493))))))) & (apnotvar = $true) & (hoasinduction_lem3v2_gthm = $true) & (hoaslamnotvar_gthm = $true) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X500 : term] : ~ ! [X501 : (subst > term > term)] : (((sub @ X500 @ (push @ one @ sh))) != ((X501 @ sh @ one)))) & (pushprop_lthm = $true) & ((shinj = $true) <=> ! [X502 : term,X503 : term] : ((X502 = X503) | (((sub @ X502 @ sh)) != ((sub @ X503 @ sh))))) & ((hoasinduction_lem0 = $true) <=> ! [X504 : (subst > term > subst > $o)] : ~ ! [X505 : (term > $o)] : ~ ! [X506 : term] : (((X504 @ id @ X506 @ id)) = ((X505 @ X506)))) & (! [X507 : term,X508 : (subst > term > subst > $o)] : (($true = ((X508 @ id @ (lam @ (sub @ X507 @ (push @ one @ sh))) @ id))) | ~ ! [X509 : term] : (~ ($true = ((X508 @ id @ X509 @ id))) | (((X508 @ id @ (sub @ X507 @ (push @ X509 @ id)) @ id)) = $true)) | ~ ! [X510 : (subst > term > term)] : (~ ! [X511 : term] : ((((X508 @ id @ (X510 @ id @ X511) @ id)) = $true) | ~ (((X508 @ id @ X511 @ id)) = $true)) | (((X508 @ id @ (lam @ (X510 @ sh @ one)) @ id)) = $true) | ~ ! [X512 : term,X514 : subst,X513 : subst] : (((X510 @ (comp @ X514 @ X513) @ (sub @ X512 @ X513))) = ((sub @ (X510 @ X514 @ X512) @ X513))))) <=> (hoasinduction_lem3aa = $true)) & (! [X515 : term] : ~ ! [X516 : (subst > term > term)] : (((X516 @ sh @ one)) != ((sub @ X515 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b = $true)) & ((termmset_lthm = $true) <=> (! [X517 : term] : (((sub @ X517 @ id)) = X517) => ! [X518 : term] : (((sub @ X518 @ id)) = X518))) & (ulamvarsh = $true) & ~ (lamnotap = $true) & ((pushprop_lem0 = $true) <=> ! [X520 : (term > $o),X519 : subst,X521 : term] : ~ ! [X522 : (term > $o)] : ~ ! [X523 : term] : (((X520 @ (sub @ X523 @ (push @ X521 @ X519)))) = ((X522 @ X523)))) & (induction2lem_gthm = $true) & ((pushprop_lem1_lthm = $true) <=> (! [X525 : subst,X524 : term] : (((sub @ one @ (push @ X524 @ X525))) = X524) => (! [X527 : term,X526 : subst] : (((comp @ sh @ (push @ X527 @ X526))) = X526) => ! [X528 : term,X529 : (term > $o),X530 : subst,X531 : term,X532 : (term > $o)] : ((((X532 @ (sub @ X531 @ (push @ X528 @ X530)))) = $true) | ~ ($true = ((X529 @ X531))))))) & (! [X533 : (term > $o),X534 : (term > $o),X535 : subst,X536 : term] : ((((X534 @ one)) = $true) | ~ (((X533 @ X536)) = $true) | ~ ! [X537 : term] : (((X533 @ (sub @ X537 @ (push @ X536 @ X535)))) = ((X534 @ X537)))) <=> (pushprop_lem1v2 = $true)) & (pushprop_lem3v2 = $true) & (hoasapinj1_gthm = $true) & ! [X538 : term] : ? [X539 : d_term] : (((d2term @ X539)) = X538) & (pushprop_lem3v2_lthm = $true)),
  inference(rectify,[],[f7])).
thf(f10,plain,(
  (hoasinduction_lem1_gthm = $true) & (((d2term @ d_one)) = one) & ! [X538 : term] : ? [X539 : d_term] : (((d2term @ X539)) = X538) & (! [X535 : subst,X533 : (term > $o),X536 : term,X534 : (term > $o)] : ((((X534 @ one)) = $true) | ~ ! [X537 : term] : (((X533 @ (sub @ X537 @ (push @ X536 @ X535)))) = ((X534 @ X537))) | (((X533 @ X536)) != $true)) <=> (pushprop_lem1v2 = $true)) & ((hoasinduction_lem3a = $true) <=> ! [X350 : term,X351 : (subst > term > subst > $o)] : ((((X351 @ id @ (lam @ X350) @ id)) = $true) | ~ ! [X352 : term] : (($true = ((X351 @ id @ (sub @ X350 @ (push @ X352 @ id)) @ id))) | (((X351 @ id @ X352 @ id)) != $true)) | ~ ! [X353 : (subst > term > term)] : (~ ! [X354 : term] : (($true != ((X351 @ id @ X354 @ id))) | (((X351 @ id @ (X353 @ id @ X354) @ id)) = $true)) | (((X351 @ id @ (lam @ (X353 @ sh @ one)) @ id)) = $true) | ~ ! [X355 : subst,X357 : term,X356 : subst] : (((X353 @ (comp @ X355 @ X356) @ (sub @ X357 @ X356))) = ((sub @ (X353 @ X355 @ X357) @ X356)))))) & (pushprop_lem2v2_gthm = $true) & (pushprop_lthm_orig = $true) & ((pushprop_lem0_lthm = $true) <=> ! [X97 : subst,X96 : (term > $o),X95 : term] : ~ ! [X98 : (term > $o)] : ~ ! [X99 : term] : (((X96 @ (sub @ X99 @ (push @ X95 @ X97)))) = ((X98 @ X99)))) & (axclos = $true) & (! [X102 : term,X101 : term,X100 : (subst > term > subst > $o)] : ((((X100 @ id @ X102 @ id)) != $true) | ~ ! [X110 : term,X112 : subst,X109 : subst,X111 : subst] : ((((X100 @ X111 @ X110 @ (comp @ X112 @ X109))) != $true) | (((X100 @ (comp @ X111 @ X112) @ (sub @ X110 @ X112) @ X109)) = $true)) | ~ ! [X108 : term,X107 : term] : (($true != ((X100 @ id @ X107 @ id))) | (((X100 @ id @ (ap @ (sub @ X107 @ id) @ X108) @ id)) = $true) | ($true != ((X100 @ id @ X108 @ id)))) | (((X100 @ id @ X101 @ id)) != $true) | ~ ! [X104 : subst,X103 : subst,X105 : subst,X106 : term] : (($true != ((X100 @ (comp @ X103 @ X105) @ (sub @ X106 @ X105) @ X104))) | (((X100 @ X103 @ X106 @ (comp @ X105 @ X104))) = $true)) | (((X100 @ id @ (ap @ X101 @ X102) @ id)) = $true)) <=> (hoasinduction_lem2 = $true)) & (axabs = $true) & ((substmonoid_lthm = $true) <=> (! [X23 : subst] : (((comp @ id @ X23)) = X23) => (! [X24 : subst] : (((comp @ X24 @ id)) = X24) => (! [X26 : subst] : (((comp @ X26 @ id)) = X26) & ! [X25 : subst] : (((comp @ id @ X25)) = X25))))) & (hoasapnotvar = $true) & (pushprop_lem0_gthm = $true) & (hoasinduction_lem3b_gthm = $true) & ((hoasinduction_lem3 = $true) <=> ! [X469 : (subst > term > subst > $o),X468 : term] : (~ ! [X482 : subst,X483 : subst,X480 : term,X481 : subst] : (($true != ((X469 @ (comp @ X481 @ X483) @ (sub @ X480 @ X483) @ X482))) | (((X469 @ X481 @ X480 @ (comp @ X483 @ X482))) = $true)) | ~ ! [X470 : term] : (($true = ((X469 @ id @ (sub @ X468 @ (push @ X470 @ id)) @ id))) | ($true != ((X469 @ id @ X470 @ id)))) | ~ ! [X471 : (subst > term > term)] : (~ ! [X475 : term] : ((((X469 @ id @ X475 @ id)) != $true) | (((X469 @ id @ (X471 @ id @ X475) @ id)) = $true)) | ~ ! [X473 : term,X472 : subst,X474 : subst] : (((X471 @ (comp @ X474 @ X472) @ (sub @ X473 @ X472))) = ((sub @ (X471 @ X474 @ X473) @ X472))) | (((X469 @ id @ (lam @ (X471 @ sh @ one)) @ id)) = $true)) | (((X469 @ id @ (lam @ X468) @ id)) = $true) | ~ ! [X477 : subst,X479 : term,X476 : subst,X478 : subst] : ((((X469 @ (comp @ X476 @ X478) @ (sub @ X479 @ X478) @ X477)) = $true) | (((X469 @ X476 @ X479 @ (comp @ X478 @ X477))) != $true)))) & (pushprop_lem2v2 = $true) & ((induction2lem_lthm = $true) <=> (! [X416 : term,X417 : subst] : (((sub @ one @ (push @ X416 @ X417))) = X416) => (! [X419 : term,X418 : subst] : (((comp @ sh @ (push @ X419 @ X418))) = X418) => (! [X420 : subst] : (((comp @ X420 @ id)) = X420) => (! [X422 : (term > $o),X421 : term] : (~ ! [X423 : term] : ((((X422 @ (lam @ X423))) = $true) | (((X422 @ X423)) != $true)) | ~ ! [X426 : term] : (($true != ((var @ X426))) | (((X422 @ X426)) = $true)) | ($true = ((X422 @ X421))) | ~ ! [X425 : term,X424 : term] : (($true != ((X422 @ X424))) | (((X422 @ (ap @ X424 @ X425))) = $true) | (((X422 @ X425)) != $true))) => ! [X429 : subst,X428 : term,X427 : (term > $o)] : ((((X427 @ (sub @ X428 @ X429))) = $true) | ~ ! [X432 : term,X431 : term] : ((((X427 @ X432)) != $true) | (((X427 @ X431)) != $true) | ($true = ((X427 @ (ap @ X432 @ X431))))) | ~ ! [X430 : term] : ((((X427 @ (sub @ X430 @ X429))) = $true) | (((var @ X430)) != $true)) | ~ ! [X433 : term] : ((((X427 @ (lam @ X433))) = $true) | ~ ! [X434 : term] : ((((X427 @ (sub @ X433 @ (push @ X434 @ id)))) = $true) | (((X427 @ X434)) != $true))))))))) & (hoasinduction_lem3v2_gthm = $true) & (induction2_gthm = $true) & (pushprop_lthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (((d2subst @ d_id)) = id) & (hoasapnotvar_gthm = $true) & (hoasapnotvar_lthm = $true) & (hoasinduction_lem2v2_gthm = $true) & ((! [X56 : (subst > term > subst > $o)] : ~ ! [X57 : (term > $o)] : ~ ! [X58 : term] : (((X56 @ id @ X58 @ id)) = ((X57 @ X58))) => (! [X59 : term,X60 : (term > $o)] : (~ ! [X63 : term,X64 : term] : ((((X60 @ X63)) != $true) | (((X60 @ (ap @ X64 @ X63))) = $true) | (((X60 @ X64)) != $true)) | (((X60 @ X59)) = $true) | ~ ! [X65 : term] : ((((X60 @ X65)) = $true) | (((var @ X65)) != $true)) | ~ ! [X61 : term] : (~ ! [X62 : term] : ((((X60 @ X62)) != $true) | (((X60 @ (sub @ X61 @ (push @ X62 @ id)))) = $true)) | ($true = ((X60 @ (lam @ X61)))))) => (! [X66 : term] : (((sub @ X66 @ id)) = X66) => (! [X68 : term,X69 : (subst > term > subst > $o),X67 : (term > $o)] : (~ ! [X75 : term] : ((((X67 @ (sub @ X68 @ (push @ X75 @ id)))) = $true) | (((X67 @ X75)) != $true)) | ~ ! [X70 : (subst > term > term)] : ((((X69 @ id @ (lam @ (X70 @ sh @ one)) @ id)) = $true) | ~ ! [X72 : subst,X73 : term,X71 : subst] : (((X70 @ (comp @ X71 @ X72) @ (sub @ X73 @ X72))) = ((sub @ (X70 @ X71 @ X73) @ X72))) | ~ ! [X74 : term] : ((((X69 @ id @ (X70 @ id @ X74) @ id)) = $true) | ($true != ((X69 @ id @ X74 @ id))))) | ($true = ((X67 @ (lam @ X68)))) | ~ ! [X76 : term] : (((X69 @ id @ X76 @ id)) = ((X67 @ X76)))) => ! [X78 : (subst > term > subst > $o),X77 : term] : (~ ! [X90 : subst,X89 : subst,X87 : subst,X88 : term] : ((((X78 @ (comp @ X87 @ X90) @ (sub @ X88 @ X90) @ X89)) = $true) | ($true != ((X78 @ X87 @ X88 @ (comp @ X90 @ X89))))) | (((X78 @ id @ X77 @ id)) = $true) | ~ ! [X91 : subst,X92 : term,X93 : subst,X94 : subst] : (($true = ((X78 @ X94 @ X92 @ (comp @ X91 @ X93)))) | (((X78 @ (comp @ X94 @ X91) @ (sub @ X92 @ X91) @ X93)) != $true)) | ~ ! [X85 : term,X86 : term] : ((((X78 @ id @ X86 @ id)) != $true) | (((X78 @ id @ X85 @ id)) != $true) | (((X78 @ id @ (ap @ (sub @ X86 @ id) @ X85) @ id)) = $true)) | ~ ! [X80 : (subst > term > term)] : (~ ! [X84 : term] : ((((X78 @ id @ X84 @ id)) != $true) | (((X78 @ id @ (X80 @ id @ X84) @ id)) = $true)) | ~ ! [X81 : subst,X82 : term,X83 : subst] : (((sub @ (X80 @ X81 @ X82) @ X83)) = ((X80 @ (comp @ X81 @ X83) @ (sub @ X82 @ X83)))) | ($true = ((X78 @ id @ (lam @ (X80 @ sh @ one)) @ id)))) | ~ ! [X79 : term] : ((((var @ (sub @ X79 @ id))) != $true) | (((X78 @ id @ X79 @ id)) = $true))))))) <=> (hoasinduction_lthm_3 = $true)) & (pushprop = $true) & ((axidl = $true) <=> ! [X406 : subst] : (((comp @ id @ X406)) = X406)) & ((! [X387 : term] : (((sub @ X387 @ id)) = X387) => (! [X388 : term,X389 : (subst > term > subst > $o)] : (~ ! [X391 : (subst > term > term)] : ((((X389 @ id @ (lam @ (X391 @ sh @ one)) @ id)) = $true) | ~ ! [X392 : term] : ((((X389 @ id @ (X391 @ id @ X392) @ id)) = $true) | (((X389 @ id @ X392 @ id)) != $true)) | ~ ! [X395 : term,X393 : subst,X394 : subst] : (((sub @ (X391 @ X393 @ X395) @ X394)) = ((X391 @ (comp @ X393 @ X394) @ (sub @ X395 @ X394))))) | (((X389 @ id @ (lam @ (sub @ X388 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X390 : term] : (($true != ((X389 @ id @ X390 @ id))) | (((X389 @ id @ (sub @ X388 @ (push @ X390 @ id)) @ id)) = $true))) => ! [X397 : term,X396 : (subst > term > subst > $o)] : ((((X396 @ id @ (lam @ X397) @ id)) = $true) | ~ ! [X398 : term] : ((((X396 @ id @ (sub @ X397 @ (push @ X398 @ id)) @ id)) = $true) | (((X396 @ id @ X398 @ id)) != $true)) | ~ ! [X399 : (subst > term > term)] : (~ ! [X400 : term] : ((((X396 @ id @ (X399 @ id @ X400) @ id)) = $true) | (((X396 @ id @ X400 @ id)) != $true)) | ~ ! [X402 : term,X403 : subst,X401 : subst] : (((X399 @ (comp @ X401 @ X403) @ (sub @ X402 @ X403))) = ((sub @ (X399 @ X401 @ X402) @ X403))) | ($true = ((X396 @ id @ (lam @ (X399 @ sh @ one)) @ id))))))) <=> (hoasinduction_lem3a_lthm = $true)) & (! [X52 : term] : ~ ! [X53 : (subst > term > term)] : ~ ! [X55 : subst,X54 : term] : (((sub @ X52 @ (push @ X54 @ X55))) = ((X53 @ X55 @ X54))) <=> (hoasinduction_lem3v2_f = $true)) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (! [X508 : (subst > term > subst > $o),X507 : term] : (~ ! [X509 : term] : (($true != ((X508 @ id @ X509 @ id))) | (((X508 @ id @ (sub @ X507 @ (push @ X509 @ id)) @ id)) = $true)) | ~ ! [X510 : (subst > term > term)] : (~ ! [X512 : term,X514 : subst,X513 : subst] : (((X510 @ (comp @ X514 @ X513) @ (sub @ X512 @ X513))) = ((sub @ (X510 @ X514 @ X512) @ X513))) | (((X508 @ id @ (lam @ (X510 @ sh @ one)) @ id)) = $true) | ~ ! [X511 : term] : ((((X508 @ id @ X511 @ id)) != $true) | (((X508 @ id @ (X510 @ id @ X511) @ id)) = $true))) | ($true = ((X508 @ id @ (lam @ (sub @ X507 @ (push @ one @ sh))) @ id)))) <=> (hoasinduction_lem3aa = $true)) & (hoasinduction_gthm = $true) & ((laminj = $true) <=> ! [X21 : term,X22 : term] : ((((lam @ X22)) != ((lam @ X21))) | (X21 = X22))) & (hoasinduction_lem1 = $true) & (hoaslaminj_gthm = $true) & (apnotvar = $true) & ((hoasinduction_lem3v2a_lthm = $true) <=> (! [X125 : term] : ~ ! [X126 : (subst > term > term)] : ~ ! [X127 : term,X128 : subst] : (((sub @ X125 @ (push @ X127 @ X128))) = ((X126 @ X128 @ X127))) => (! [X129 : term] : (((sub @ X129 @ id)) = X129) => ! [X132 : (subst > term > subst > $o),X131 : term,X130 : (term > $o)] : (~ ! [X138 : term] : (((X132 @ id @ X138 @ id)) = ((X130 @ X138))) | ($true = ((X130 @ (lam @ X131)))) | ~ ! [X139 : term] : ((((X130 @ X139)) != $true) | (((X130 @ (sub @ X131 @ (push @ X139 @ id)))) = $true)) | ~ ! [X133 : (subst > term > term)] : ((((X132 @ id @ (lam @ (X133 @ sh @ one)) @ id)) = $true) | ~ ! [X134 : subst,X136 : term,X135 : subst] : (((sub @ (X133 @ X135 @ X136) @ X134)) = ((X133 @ (comp @ X135 @ X134) @ (sub @ X136 @ X134)))) | ~ ! [X137 : term] : ((((X132 @ id @ (X133 @ id @ X137) @ id)) = $true) | (((X132 @ id @ X137 @ id)) != $true))))))) & (ulamvar1 != $true) & (((var @ (d2term @ d_one))) != $true) & (hoasinduction_lem1v2_gthm = $true) & (axassoc = $true) & (hoasapinj1_gthm = $true) & ((pushprop_lem1 = $true) <=> ! [X216 : (term > $o),X217 : term,X218 : (term > $o),X219 : subst,X220 : term] : ((((X216 @ X217)) != $true) | (((X218 @ (sub @ X217 @ (push @ X220 @ X219)))) = $true))) & (pushprop_lem3v2_lthm = $true) & ((! [X331 : term] : (((sub @ X331 @ id)) = X331) => ! [X334 : term,X332 : (subst > term > subst > $o),X333 : (term > $o)] : (~ ! [X339 : term] : (((X333 @ X339)) = ((X332 @ id @ X339 @ id))) | ~ ! [X349 : term] : ((((X333 @ X349)) != $true) | (((X333 @ (sub @ X334 @ (push @ X349 @ id)))) = $true)) | ~ ! [X340 : (subst > term > term)] : (~ ! [X344 : term] : (($true != ((X332 @ id @ X344 @ id))) | (((X332 @ id @ (X340 @ id @ X344) @ id)) = $true)) | ~ ! [X343 : subst,X341 : term,X342 : subst] : (((X340 @ (comp @ X342 @ X343) @ (sub @ X341 @ X343))) = ((sub @ (X340 @ X342 @ X341) @ X343))) | (((X332 @ id @ (lam @ (X340 @ sh @ one)) @ id)) = $true)) | ~ ! [X345 : subst,X347 : term,X348 : subst,X346 : subst] : (($true = ((X332 @ (comp @ X348 @ X346) @ (sub @ X347 @ X346) @ X345))) | ($true != ((X332 @ X348 @ X347 @ (comp @ X346 @ X345))))) | ~ ! [X337 : subst,X336 : subst,X335 : term,X338 : subst] : (($true != ((X332 @ (comp @ X338 @ X336) @ (sub @ X335 @ X336) @ X337))) | (((X332 @ X338 @ X335 @ (comp @ X336 @ X337))) = $true)) | (((X333 @ (lam @ X334))) = $true))) <=> (hoasinduction_lem3v2_lthm = $true)) & ! [X19 : d_subst,X20 : d_subst] : ((((d2subst @ X19)) = ((d2subst @ X20))) => (X19 = X20)) & (hoaslamnotvar_lthm = $true) & ((hoaslaminj = $true) <=> ! [X321 : (subst > term > term),X323 : term,X322 : (subst > term > term),X324 : subst] : (~ ! [X328 : subst,X330 : term,X329 : subst] : (((sub @ (X322 @ X328 @ X330) @ X329)) = ((X322 @ (comp @ X328 @ X329) @ (sub @ X330 @ X329)))) | (((lam @ (X321 @ sh @ one))) != ((lam @ (X322 @ sh @ one)))) | ~ ! [X326 : term,X327 : subst,X325 : subst] : (((sub @ (X321 @ X327 @ X326) @ X325)) = ((X321 @ (comp @ X327 @ X325) @ (sub @ X326 @ X325)))) | (((X322 @ X324 @ X323)) = ((X321 @ X324 @ X323))))) & (lamnotvar = $true) & (ulamvarsh = $true) & (ulamvarind = $true) & ((hoasinduction_lem2v2 = $true) <=> ! [X202 : (term > $o),X203 : (subst > term > subst > $o),X201 : term,X204 : term] : ((((X202 @ (ap @ X204 @ X201))) = $true) | ~ ! [X213 : term] : (((X203 @ id @ X213 @ id)) = ((X202 @ X213))) | ~ ! [X206 : subst,X208 : subst,X207 : subst,X205 : term] : ((((X203 @ X207 @ X205 @ (comp @ X206 @ X208))) != $true) | ($true = ((X203 @ (comp @ X207 @ X206) @ (sub @ X205 @ X206) @ X208)))) | (((X202 @ X201)) != $true) | ~ ! [X214 : term,X215 : term] : ((((X203 @ id @ X214 @ id)) != $true) | (((X203 @ id @ (ap @ (sub @ X214 @ id) @ X215) @ id)) = $true) | (((X203 @ id @ X215 @ id)) != $true)) | ~ ! [X211 : subst,X209 : term,X210 : subst,X212 : subst] : ((((X203 @ X210 @ X209 @ (comp @ X211 @ X212))) = $true) | (((X203 @ (comp @ X210 @ X211) @ (sub @ X209 @ X211) @ X212)) != $true)) | ($true != ((X202 @ X204))))) & ((! [X278 : subst] : (((comp @ X278 @ id)) = X278) & ! [X277 : subst] : (((comp @ id @ X277)) = X277)) <=> (substmonoid = $true)) & (hoasinduction_lem3a_gthm = $true) & (substmonoid_gthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (axmap = $true) & ((hoasinduction_lem3aaa = $true) <=> ! [X40 : term,X41 : (subst > term > subst > $o)] : (~ ! [X43 : term,X44 : (subst > term > term)] : (~ ! [X48 : term] : ((((X41 @ id @ X48 @ id)) != $true) | (((X41 @ id @ (X44 @ id @ X48) @ id)) = $true)) | ~ ! [X49 : subst,X51 : term,X50 : subst] : (((sub @ X43 @ (push @ (sub @ X51 @ X50) @ (comp @ X49 @ X50)))) = ((X44 @ (comp @ X49 @ X50) @ (sub @ X51 @ X50)))) | ~ ! [X47 : subst,X45 : term,X46 : subst] : (((sub @ (X44 @ X47 @ X45) @ X46)) = ((sub @ (sub @ X43 @ (push @ X45 @ X47)) @ X46))) | (((X41 @ id @ (lam @ (X44 @ sh @ one)) @ id)) = $true)) | ~ ! [X42 : term] : ((((X41 @ id @ X42 @ id)) != $true) | (((X41 @ id @ (sub @ X40 @ (push @ X42 @ id)) @ id)) = $true)) | (((X41 @ id @ (lam @ (sub @ X40 @ (push @ one @ sh))) @ id)) = $true))) & ((shinj = $true) <=> ! [X502 : term,X503 : term] : ((X502 = X503) | (((sub @ X502 @ sh)) != ((sub @ X503 @ sh))))) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ((axidr = $true) <=> ! [X272 : subst] : (((comp @ X272 @ id)) = X272)) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X500 : term] : ~ ! [X501 : (subst > term > term)] : (((sub @ X500 @ (push @ one @ sh))) != ((X501 @ sh @ one)))) & ((pushprop_lem0 = $true) <=> ! [X520 : (term > $o),X519 : subst,X521 : term] : ~ ! [X522 : (term > $o)] : ~ ! [X523 : term] : (((X520 @ (sub @ X523 @ (push @ X521 @ X519)))) = ((X522 @ X523)))) & (hoasinduction_lem3_gthm = $true) & (induction2lem_gthm = $true) & ! [X221 : d_term,X222 : d_term] : ((((d2term @ X222)) = ((d2term @ X221))) => (X221 = X222)) & ! [X365 : subst] : ? [X366 : d_subst] : (((d2subst @ X366)) = X365) & (hoaslamnotap_lthm = $true) & (hoaslamnotvar_gthm = $true) & ((! [X484 : term] : (((sub @ X484 @ id)) = X484) => (! [X486 : term,X485 : (term > $o),X487 : subst] : ((((X485 @ (sub @ X486 @ X487))) = $true) | ~ ! [X488 : term] : ((((var @ X488)) != $true) | (((X485 @ (sub @ X488 @ X487))) = $true)) | ~ ! [X491 : term,X492 : term] : ((((X485 @ (ap @ X491 @ X492))) = $true) | (((X485 @ X491)) != $true) | (((X485 @ X492)) != $true)) | ~ ! [X489 : term] : (~ ! [X490 : term] : (($true != ((X485 @ X490))) | (((X485 @ (sub @ X489 @ (push @ X490 @ id)))) = $true)) | ($true = ((X485 @ (lam @ X489)))))) => ! [X493 : term,X494 : (term > $o)] : (($true = ((X494 @ X493))) | ~ ! [X497 : term,X496 : term] : ((((X494 @ (ap @ X496 @ X497))) = $true) | (((X494 @ X496)) != $true) | (((X494 @ X497)) != $true)) | ~ ! [X498 : term] : (~ ! [X499 : term] : ((((X494 @ X499)) != $true) | (((X494 @ (sub @ X498 @ (push @ X499 @ id)))) = $true)) | (((X494 @ (lam @ X498))) = $true)) | ~ ! [X495 : term] : ((((X494 @ X495)) = $true) | (((var @ X495)) != $true))))) <=> (induction2_lthm = $true)) & (! [X382 : term,X381 : (term > $o)] : (($true = ((X381 @ X382))) | ~ ! [X385 : term,X386 : term] : (($true = ((X381 @ (ap @ X385 @ X386)))) | ($true != ((X381 @ X386))) | (((X381 @ X385)) != $true)) | ~ ! [X383 : term] : ((((X381 @ X383)) = $true) | (((var @ X383)) != $true)) | ~ ! [X384 : term] : (($true != ((X381 @ X384))) | ($true = ((X381 @ (lam @ X384)))))) <=> (induction = $true)) & ((pushprop_lem1v2_lthm = $true) <=> (! [X194 : subst,X195 : term] : (((sub @ one @ (push @ X195 @ X194))) = X195) => ! [X197 : subst,X199 : term,X198 : (term > $o),X196 : (term > $o)] : (($true = ((X196 @ one))) | (((X198 @ X199)) != $true) | ~ ! [X200 : term] : (((X196 @ X200)) = ((X198 @ (sub @ X200 @ (push @ X199 @ X197)))))))) & ((hoasinduction_lem3v2 = $true) <=> ! [X304 : (term > $o),X303 : term,X305 : (subst > term > subst > $o)] : (~ ! [X318 : term,X319 : subst,X316 : subst,X317 : subst] : (($true = ((X305 @ X319 @ X318 @ (comp @ X317 @ X316)))) | (((X305 @ (comp @ X319 @ X317) @ (sub @ X318 @ X317) @ X316)) != $true)) | ~ ! [X320 : term] : ((((X304 @ X320)) != $true) | ($true = ((X304 @ (sub @ X303 @ (push @ X320 @ id)))))) | ~ ! [X311 : (subst > term > term)] : (~ ! [X314 : term,X312 : subst,X313 : subst] : (((sub @ (X311 @ X312 @ X314) @ X313)) = ((X311 @ (comp @ X312 @ X313) @ (sub @ X314 @ X313)))) | (((X305 @ id @ (lam @ (X311 @ sh @ one)) @ id)) = $true) | ~ ! [X315 : term] : ((((X305 @ id @ (X311 @ id @ X315) @ id)) = $true) | (((X305 @ id @ X315 @ id)) != $true))) | ~ ! [X310 : subst,X309 : subst,X307 : term,X308 : subst] : ((((X305 @ X310 @ X307 @ (comp @ X309 @ X308))) != $true) | ($true = ((X305 @ (comp @ X310 @ X309) @ (sub @ X307 @ X309) @ X308)))) | ~ ! [X306 : term] : (((X304 @ X306)) = ((X305 @ id @ X306 @ id))) | (((X304 @ (lam @ X303))) = $true))) & ((! [X28 : term] : (((sub @ X28 @ id)) = X28) => (! [X30 : term,X32 : term,X31 : term,X29 : term] : ((((ap @ X31 @ X29)) != ((ap @ X32 @ X30))) | (X31 = X32)) => ! [X36 : term,X33 : term,X34 : term,X35 : term] : ((X33 = X36) | (((ap @ (sub @ X33 @ id) @ X35)) != ((ap @ (sub @ X36 @ id) @ X34)))))) <=> (hoasapinj1_lthm = $true)) & (((d2subst @ d_id)) = sh) & (pushprop_lem2v2_lthm = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ((! [X525 : subst,X524 : term] : (((sub @ one @ (push @ X524 @ X525))) = X524) => (! [X527 : term,X526 : subst] : (((comp @ sh @ (push @ X527 @ X526))) = X526) => ! [X532 : (term > $o),X531 : term,X529 : (term > $o),X528 : term,X530 : subst] : ((((X532 @ (sub @ X531 @ (push @ X528 @ X530)))) = $true) | ($true != ((X529 @ X531)))))) <=> (pushprop_lem1_lthm = $true)) & (! [X367 : (subst > term > subst > $o),X368 : (term > $o),X369 : term] : ((((X368 @ (lam @ X369))) = $true) | ~ ! [X371 : term] : (((X367 @ id @ X371 @ id)) = ((X368 @ X371))) | ~ ! [X372 : (subst > term > term)] : (~ ! [X375 : term,X376 : subst,X374 : subst] : (((sub @ (X372 @ X376 @ X375) @ X374)) = ((X372 @ (comp @ X376 @ X374) @ (sub @ X375 @ X374)))) | ~ ! [X373 : term] : ((((X367 @ id @ (X372 @ id @ X373) @ id)) = $true) | (((X367 @ id @ X373 @ id)) != $true)) | (((X367 @ id @ (lam @ (X372 @ sh @ one)) @ id)) = $true)) | ~ ! [X370 : term] : ((((X368 @ X370)) != $true) | ($true = ((X368 @ (sub @ X369 @ (push @ X370 @ id))))))) <=> (hoasinduction_lem3v2a = $true)) & (hoasinduction_lem2_gthm = $true) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_lem2_lthm = $true) & (! [X223 : term,X224 : subst] : (((comp @ sh @ (push @ X223 @ X224))) = X224) <=> (axshiftcons = $true)) & ((hoasapinj1 = $true) <=> ! [X302 : term,X299 : term,X301 : term,X300 : term] : ((X299 = X302) | (((ap @ (sub @ X302 @ id) @ X301)) != ((ap @ (sub @ X299 @ id) @ X300))))) & (pushprop_gthm = $true) & ((hoasapinj2 = $true) <=> ! [X274 : term,X273 : term,X276 : term,X275 : term] : ((((ap @ (sub @ X274 @ id) @ X276)) != ((ap @ (sub @ X275 @ id) @ X273))) | (X273 = X276))) & ((termmset = $true) <=> ! [X291 : term] : (((sub @ X291 @ id)) = X291)) & (hoaslamnotvar = $true) & (! [X1 : (subst > term > subst > $o),X0 : term] : (~ ! [X16 : term,X17 : term] : (($true != ((X1 @ id @ X17 @ id))) | (((X1 @ id @ (ap @ (sub @ X16 @ id) @ X17) @ id)) = $true) | (((X1 @ id @ X16 @ id)) != $true)) | (((X1 @ id @ X0 @ id)) = $true) | ~ ! [X12 : subst,X13 : term,X11 : subst,X14 : subst] : (($true != ((X1 @ (comp @ X14 @ X12) @ (sub @ X13 @ X12) @ X11))) | ($true = ((X1 @ X14 @ X13 @ (comp @ X12 @ X11))))) | ~ ! [X15 : term] : (($true != ((var @ (sub @ X15 @ id)))) | (((X1 @ id @ X15 @ id)) = $true)) | ~ ! [X6 : (subst > term > term)] : (~ ! [X10 : term] : ((((X1 @ id @ (X6 @ id @ X10) @ id)) = $true) | (((X1 @ id @ X10 @ id)) != $true)) | ~ ! [X9 : subst,X8 : term,X7 : subst] : (((sub @ (X6 @ X9 @ X8) @ X7)) = ((X6 @ (comp @ X9 @ X7) @ (sub @ X8 @ X7)))) | (((X1 @ id @ (lam @ (X6 @ sh @ one)) @ id)) = $true)) | ~ ! [X2 : subst,X4 : term,X3 : subst,X5 : subst] : ((((X1 @ X5 @ X4 @ (comp @ X3 @ X2))) != $true) | (((X1 @ (comp @ X5 @ X3) @ (sub @ X4 @ X3) @ X2)) = $true))) <=> (hoasinduction = $true)) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (hoasapinj2_gthm = $true) & (! [X298 : subst] : (((push @ (sub @ one @ X298) @ (comp @ sh @ X298))) = X298) <=> (axscons = $true)) & ((termmset_lthm = $true) <=> (! [X517 : term] : (((sub @ X517 @ id)) = X517) => ! [X518 : term] : (((sub @ X518 @ id)) = X518))) & ((! [X140 : term,X141 : (term > $o)] : ((((X141 @ X140)) = $true) | ~ ! [X146 : term,X145 : term] : ((((X141 @ X145)) != $true) | ($true = ((X141 @ (ap @ X146 @ X145)))) | (((X141 @ X146)) != $true)) | ~ ! [X142 : term] : (($true = ((X141 @ X142))) | (((var @ X142)) != $true)) | ~ ! [X143 : term] : (~ ! [X144 : term] : (($true = ((X141 @ (sub @ X143 @ (push @ X144 @ id))))) | (((X141 @ X144)) != $true)) | (((X141 @ (lam @ X143))) = $true))) => (! [X147 : term,X149 : term,X148 : (subst > term > subst > $o)] : ((((X148 @ id @ X149 @ id)) != $true) | ~ ! [X158 : term,X159 : term] : ((((X148 @ id @ X158 @ id)) != $true) | (((X148 @ id @ X159 @ id)) != $true) | (((X148 @ id @ (ap @ (sub @ X159 @ id) @ X158) @ id)) = $true)) | ~ ! [X155 : subst,X154 : subst,X157 : subst,X156 : term] : ((((X148 @ (comp @ X155 @ X154) @ (sub @ X156 @ X154) @ X157)) != $true) | (((X148 @ X155 @ X156 @ (comp @ X154 @ X157))) = $true)) | ~ ! [X152 : subst,X153 : subst,X151 : term,X150 : subst] : (($true = ((X148 @ (comp @ X152 @ X150) @ (sub @ X151 @ X150) @ X153))) | (((X148 @ X152 @ X151 @ (comp @ X150 @ X153))) != $true)) | (((X148 @ id @ X147 @ id)) != $true) | (((X148 @ id @ (ap @ X149 @ X147) @ id)) = $true)) => (! [X161 : term,X160 : (subst > term > subst > $o)] : (~ ! [X166 : (subst > term > term)] : (($true = ((X160 @ id @ (lam @ (X166 @ sh @ one)) @ id))) | ~ ! [X170 : term] : ((((X160 @ id @ (X166 @ id @ X170) @ id)) = $true) | (((X160 @ id @ X170 @ id)) != $true)) | ~ ! [X168 : subst,X169 : term,X167 : subst] : (((X166 @ (comp @ X167 @ X168) @ (sub @ X169 @ X168))) = ((sub @ (X166 @ X167 @ X169) @ X168)))) | ~ ! [X165 : subst,X163 : subst,X164 : subst,X162 : term] : ((((X160 @ X163 @ X162 @ (comp @ X164 @ X165))) = $true) | (((X160 @ (comp @ X163 @ X164) @ (sub @ X162 @ X164) @ X165)) != $true)) | ~ ! [X175 : term] : ((((X160 @ id @ X175 @ id)) != $true) | (((X160 @ id @ (sub @ X161 @ (push @ X175 @ id)) @ id)) = $true)) | (((X160 @ id @ (lam @ X161) @ id)) = $true) | ~ ! [X174 : subst,X171 : subst,X173 : subst,X172 : term] : ((((X160 @ X174 @ X172 @ (comp @ X171 @ X173))) != $true) | ($true = ((X160 @ (comp @ X174 @ X171) @ (sub @ X172 @ X171) @ X173))))) => ! [X177 : (subst > term > subst > $o),X176 : term] : (~ ! [X178 : term,X179 : term] : ((((X177 @ id @ (ap @ (sub @ X178 @ id) @ X179) @ id)) = $true) | (((X177 @ id @ X178 @ id)) != $true) | (((X177 @ id @ X179 @ id)) != $true)) | ~ ! [X180 : (subst > term > term)] : ((((X177 @ id @ (lam @ (X180 @ sh @ one)) @ id)) = $true) | ~ ! [X184 : term,X183 : subst,X182 : subst] : (((sub @ (X180 @ X183 @ X184) @ X182)) = ((X180 @ (comp @ X183 @ X182) @ (sub @ X184 @ X182)))) | ~ ! [X181 : term] : (($true != ((X177 @ id @ X181 @ id))) | ($true = ((X177 @ id @ (X180 @ id @ X181) @ id))))) | ~ ! [X189 : term] : ((((var @ (sub @ X189 @ id))) != $true) | (((X177 @ id @ X189 @ id)) = $true)) | ~ ! [X191 : subst,X190 : subst,X193 : term,X192 : subst] : ((((X177 @ X192 @ X193 @ (comp @ X191 @ X190))) = $true) | (((X177 @ (comp @ X192 @ X191) @ (sub @ X193 @ X191) @ X190)) != $true)) | (((X177 @ id @ X176 @ id)) = $true) | ~ ! [X188 : subst,X187 : subst,X186 : subst,X185 : term] : ((((X177 @ (comp @ X188 @ X186) @ (sub @ X185 @ X186) @ X187)) = $true) | (((X177 @ X188 @ X185 @ (comp @ X186 @ X187))) != $true)))))) <=> (hoasinduction_lthm = $true)) & (pushprop_lem3v2 = $true) & ((hoasinduction_lem0 = $true) <=> ! [X504 : (subst > term > subst > $o)] : ~ ! [X505 : (term > $o)] : ~ ! [X506 : term] : (((X504 @ id @ X506 @ id)) = ((X505 @ X506)))) & (! [X379 : term,X377 : term,X380 : term,X378 : term] : ((X377 = X378) | (((ap @ X378 @ X379)) != ((ap @ X377 @ X380)))) <=> (apinj1 = $true)) & (! [X284 : term,X283 : (subst > term > subst > $o)] : ((((X283 @ id @ (lam @ (sub @ X284 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X285 : (subst > term > term)] : ((((X283 @ id @ (lam @ (X285 @ sh @ one)) @ id)) = $true) | ~ ! [X288 : subst,X289 : term,X287 : subst] : (((X285 @ (comp @ X288 @ X287) @ (sub @ X289 @ X287))) = ((sub @ (X285 @ X288 @ X289) @ X287))) | ~ ! [X286 : term] : ((((X283 @ id @ (X285 @ id @ X286) @ id)) = $true) | (((X283 @ id @ X286 @ id)) != $true))) | ~ ! [X290 : term] : (($true = ((X283 @ id @ (sub @ X284 @ (push @ X290 @ id)) @ id))) | ($true != ((X283 @ id @ X290 @ id))))) <=> (hoasinduction_lem3aa_lthm = $true)) & ((! [X256 : subst,X257 : term] : (((sub @ one @ (push @ X257 @ X256))) = X257) => (! [X259 : subst,X258 : term] : (((comp @ sh @ (push @ X258 @ X259))) = X259) => (! [X261 : term,X260 : term] : ((X260 = X261) | (((lam @ X260)) != ((lam @ X261)))) => ! [X265 : term,X262 : (subst > term > term),X263 : (subst > term > term),X264 : subst] : (~ ! [X270 : term,X271 : subst,X269 : subst] : (((X262 @ (comp @ X269 @ X271) @ (sub @ X270 @ X271))) = ((sub @ (X262 @ X269 @ X270) @ X271))) | ~ ! [X266 : subst,X267 : subst,X268 : term] : (((X263 @ (comp @ X266 @ X267) @ (sub @ X268 @ X267))) = ((sub @ (X263 @ X266 @ X268) @ X267))) | (((X262 @ X264 @ X265)) = ((X263 @ X264 @ X265))) | (((lam @ (X263 @ sh @ one))) != ((lam @ (X262 @ sh @ one)))))))) <=> (hoaslaminj_lthm = $true)) & (! [X515 : term] : ~ ! [X516 : (subst > term > term)] : (((X516 @ sh @ one)) != ((sub @ X515 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b = $true)) & ! [X18 : d_term] : (d_one = X18) & ((! [X226 : (subst > term > subst > $o)] : ~ ! [X227 : (term > $o)] : ~ ! [X228 : term] : (((X227 @ X228)) = ((X226 @ id @ X228 @ id))) => (! [X230 : (term > $o),X229 : term] : (($true = ((X230 @ X229))) | ~ ! [X231 : term] : (($true != ((var @ X231))) | ($true = ((X230 @ X231)))) | ~ ! [X234 : term] : (~ ! [X235 : term] : ((((X230 @ (sub @ X234 @ (push @ X235 @ id)))) = $true) | (((X230 @ X235)) != $true)) | (((X230 @ (lam @ X234))) = $true)) | ~ ! [X232 : term,X233 : term] : ((((X230 @ X233)) != $true) | (((X230 @ (ap @ X232 @ X233))) = $true) | ($true != ((X230 @ X232))))) => (! [X236 : term] : (((sub @ X236 @ id)) = X236) => (! [X237 : (subst > term > subst > $o),X238 : term,X239 : (term > $o)] : (~ ! [X245 : term] : (($true = ((X239 @ (sub @ X238 @ (push @ X245 @ id))))) | (((X239 @ X245)) != $true)) | ~ ! [X240 : (subst > term > term)] : (~ ! [X241 : subst,X243 : term,X242 : subst] : (((X240 @ (comp @ X242 @ X241) @ (sub @ X243 @ X241))) = ((sub @ (X240 @ X242 @ X243) @ X241))) | ($true = ((X237 @ id @ (lam @ (X240 @ sh @ one)) @ id))) | ~ ! [X244 : term] : (($true != ((X237 @ id @ X244 @ id))) | (((X237 @ id @ (X240 @ id @ X244) @ id)) = $true))) | (((X239 @ (lam @ X238))) = $true) | ~ ! [X246 : term] : (((X237 @ id @ X246 @ id)) = ((X239 @ X246)))) => ! [X248 : (subst > term > subst > $o),X247 : term] : ((((X248 @ id @ X247 @ id)) = $true) | ~ ! [X254 : term,X255 : term] : ((((X248 @ id @ (ap @ (sub @ X254 @ id) @ X255) @ id)) = $true) | (((X248 @ id @ X255 @ id)) != $true) | (((X248 @ id @ X254 @ id)) != $true)) | ~ ! [X249 : (subst > term > term)] : (~ ! [X250 : term] : ((((X248 @ id @ X250 @ id)) != $true) | ($true = ((X248 @ id @ (X249 @ id @ X250) @ id)))) | ~ ! [X253 : subst,X251 : term,X252 : subst] : (((X249 @ (comp @ X253 @ X252) @ (sub @ X251 @ X252))) = ((sub @ (X249 @ X253 @ X251) @ X252))) | (((X248 @ id @ (lam @ (X249 @ sh @ one)) @ id)) = $true))))))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (pushprop_lem1_gthm = $true) & (lamnotap != $true) & ((axvarcons = $true) <=> ! [X404 : term,X405 : subst] : (((sub @ one @ (push @ X404 @ X405))) = X404)) & ((hoaslamnotap = $true) <=> ! [X293 : term,X294 : (subst > term > term),X292 : term] : ((((lam @ (X294 @ sh @ one))) != ((ap @ (sub @ X292 @ id) @ X293))) | ~ ! [X296 : subst,X297 : term,X295 : subst] : (((sub @ (X294 @ X296 @ X297) @ X295)) = ((X294 @ (comp @ X296 @ X295) @ (sub @ X297 @ X295)))))) & (hoasinduction_lem1v2 = $true) & (axapp = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (termmset_gthm = $true) & ! [X225 : d_subst] : (d_id = X225) & ((hoasinduction_no_psi_cond = $true) <=> ! [X407 : term,X408 : (subst > term > subst > $o)] : (~ ! [X409 : (subst > term > term)] : (~ ! [X412 : term,X413 : subst,X411 : subst] : (((X409 @ (comp @ X411 @ X413) @ (sub @ X412 @ X413))) = ((sub @ (X409 @ X411 @ X412) @ X413))) | ($true = ((X408 @ id @ (lam @ (X409 @ sh @ one)) @ id))) | ~ ! [X410 : term] : ((((X408 @ id @ (X409 @ id @ X410) @ id)) = $true) | (((X408 @ id @ X410 @ id)) != $true))) | (((X408 @ id @ X407 @ id)) = $true) | ~ ! [X415 : term,X414 : term] : (($true != ((X408 @ id @ X415 @ id))) | (((X408 @ id @ (ap @ (sub @ X414 @ id) @ X415) @ id)) = $true) | (((X408 @ id @ X414 @ id)) != $true)))) & ((induction2lem = $true) <=> ! [X115 : (term > $o),X113 : subst,X114 : term] : (~ ! [X116 : term] : ((((X115 @ (sub @ X116 @ X113))) = $true) | (((var @ X116)) != $true)) | ~ ! [X120 : term,X119 : term] : ((((X115 @ X119)) != $true) | ($true != ((X115 @ X120))) | (((X115 @ (ap @ X119 @ X120))) = $true)) | ~ ! [X117 : term] : (($true = ((X115 @ (lam @ X117)))) | ~ ! [X118 : term] : (($true = ((X115 @ (sub @ X117 @ (push @ X118 @ id))))) | (((X115 @ X118)) != $true))) | (((X115 @ (sub @ X114 @ X113))) = $true))) & (! [X358 : (term > $o),X359 : term] : (~ ! [X360 : term] : ((((var @ X360)) != $true) | (((X358 @ X360)) = $true)) | ($true = ((X358 @ X359))) | ~ ! [X361 : term] : (~ ! [X362 : term] : ((((X358 @ X362)) != $true) | (((X358 @ (sub @ X361 @ (push @ X362 @ id)))) = $true)) | (((X358 @ (lam @ X361))) = $true)) | ~ ! [X363 : term,X364 : term] : (($true != ((X358 @ X363))) | (((X358 @ X364)) != $true) | (((X358 @ (ap @ X363 @ X364))) = $true))) <=> (induction2 = $true)) & (hoaslamnotap_gthm = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (axvarshift = $true) & (! [X37 : (subst > term > subst > $o)] : ~ ! [X38 : (term > $o)] : ~ ! [X39 : term] : (((X37 @ id @ X39 @ id)) = ((X38 @ X39))) <=> (hoasinduction_lem0_lthm = $true)) & ((! [X435 : term] : (((sub @ X435 @ id)) = X435) => (! [X436 : term,X437 : (subst > term > subst > $o)] : (~ ! [X443 : term] : ((((X437 @ id @ X443 @ id)) != $true) | (((X437 @ id @ (sub @ X436 @ (push @ X443 @ id)) @ id)) = $true)) | ~ ! [X438 : (subst > term > term)] : ((((X437 @ id @ (lam @ (X438 @ sh @ one)) @ id)) = $true) | ~ ! [X441 : subst,X442 : subst,X440 : term] : (((X438 @ (comp @ X441 @ X442) @ (sub @ X440 @ X442))) = ((sub @ (X438 @ X441 @ X440) @ X442))) | ~ ! [X439 : term] : (($true != ((X437 @ id @ X439 @ id))) | (((X437 @ id @ (X438 @ id @ X439) @ id)) = $true))) | ($true = ((X437 @ id @ (lam @ (sub @ X436 @ (push @ one @ sh))) @ id)))) => ! [X445 : (subst > term > subst > $o),X444 : term] : (~ ! [X455 : term] : ((((X445 @ id @ X455 @ id)) != $true) | (((X445 @ id @ (sub @ X444 @ (push @ X455 @ id)) @ id)) = $true)) | ~ ! [X456 : subst,X459 : subst,X458 : subst,X457 : term] : ((((X445 @ X458 @ X457 @ (comp @ X459 @ X456))) = $true) | (((X445 @ (comp @ X458 @ X459) @ (sub @ X457 @ X459) @ X456)) != $true)) | ~ ! [X450 : (subst > term > term)] : ((((X445 @ id @ (lam @ (X450 @ sh @ one)) @ id)) = $true) | ~ ! [X451 : term] : (($true != ((X445 @ id @ X451 @ id))) | (((X445 @ id @ (X450 @ id @ X451) @ id)) = $true)) | ~ ! [X453 : subst,X454 : subst,X452 : term] : (((sub @ (X450 @ X454 @ X452) @ X453)) = ((X450 @ (comp @ X454 @ X453) @ (sub @ X452 @ X453))))) | (((X445 @ id @ (lam @ X444) @ id)) = $true) | ~ ! [X449 : subst,X447 : subst,X448 : subst,X446 : term] : (($true = ((X445 @ (comp @ X447 @ X449) @ (sub @ X446 @ X449) @ X448))) | (((X445 @ X447 @ X446 @ (comp @ X449 @ X448))) != $true))))) <=> (hoasinduction_lem3_lthm = $true)) & (hoasinduction_lem1_lthm = $true) & (! [X121 : term] : ~ ! [X122 : (subst > term > term)] : ~ ! [X123 : term,X124 : subst] : (((X122 @ X124 @ X123)) = ((sub @ X121 @ (push @ X123 @ X124)))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & (! [X279 : term,X282 : term,X281 : term,X280 : term] : ((((ap @ X280 @ X282)) != ((ap @ X279 @ X281))) | (X281 = X282)) <=> (apinj2 = $true)) & ((axvarid = $true) <=> ! [X27 : term] : (((sub @ X27 @ id)) = X27)) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoasapinj2_lthm = $true) <=> (! [X463 : term,X462 : term,X460 : term,X461 : term] : ((((ap @ X463 @ X460)) != ((ap @ X461 @ X462))) | (X460 = X462)) => ! [X466 : term,X464 : term,X465 : term,X467 : term] : ((((ap @ (sub @ X464 @ id) @ X467)) != ((ap @ (sub @ X466 @ id) @ X465))) | (X465 = X467))))),
  inference(flattening,[],[f9])).
thf(f11,plain,(
  (hoasinduction_lem1_gthm = $true) & (((d2term @ d_one)) = one) & ! [X538 : term] : ? [X539 : d_term] : (((d2term @ X539)) = X538) & ((pushprop_lem1v2 = $true) <=> ! [X534 : (term > $o),X536 : term,X533 : (term > $o),X535 : subst] : (? [X537 : term] : (((X533 @ (sub @ X537 @ (push @ X536 @ X535)))) != ((X534 @ X537))) | (((X534 @ one)) = $true) | (((X533 @ X536)) != $true))) & ((hoasinduction_lem3a = $true) <=> ! [X351 : (subst > term > subst > $o),X350 : term] : ((((X351 @ id @ (lam @ X350) @ id)) = $true) | ? [X353 : (subst > term > term)] : (! [X354 : term] : (($true != ((X351 @ id @ X354 @ id))) | (((X351 @ id @ (X353 @ id @ X354) @ id)) = $true)) & (((X351 @ id @ (lam @ (X353 @ sh @ one)) @ id)) != $true) & ! [X355 : subst,X357 : term,X356 : subst] : (((X353 @ (comp @ X355 @ X356) @ (sub @ X357 @ X356))) = ((sub @ (X353 @ X355 @ X357) @ X356)))) | ? [X352 : term] : (($true != ((X351 @ id @ (sub @ X350 @ (push @ X352 @ id)) @ id))) & (((X351 @ id @ X352 @ id)) = $true)))) & (pushprop_lem2v2_gthm = $true) & (pushprop_lthm_orig = $true) & (! [X96 : (term > $o),X95 : term,X97 : subst] : ? [X98 : (term > $o)] : ! [X99 : term] : (((X96 @ (sub @ X99 @ (push @ X95 @ X97)))) = ((X98 @ X99))) <=> (pushprop_lem0_lthm = $true)) & (axclos = $true) & ((hoasinduction_lem2 = $true) <=> ! [X101 : term,X102 : term,X100 : (subst > term > subst > $o)] : ((((X100 @ id @ X101 @ id)) != $true) | (((X100 @ id @ X102 @ id)) != $true) | ? [X105 : subst,X104 : subst,X106 : term,X103 : subst] : (($true = ((X100 @ (comp @ X103 @ X105) @ (sub @ X106 @ X105) @ X104))) & (((X100 @ X103 @ X106 @ (comp @ X105 @ X104))) != $true)) | (((X100 @ id @ (ap @ X101 @ X102) @ id)) = $true) | ? [X112 : subst,X111 : subst,X109 : subst,X110 : term] : ((((X100 @ X111 @ X110 @ (comp @ X112 @ X109))) = $true) & (((X100 @ (comp @ X111 @ X112) @ (sub @ X110 @ X112) @ X109)) != $true)) | ? [X108 : term,X107 : term] : ((((X100 @ id @ (ap @ (sub @ X107 @ id) @ X108) @ id)) != $true) & ($true = ((X100 @ id @ X107 @ id))) & ($true = ((X100 @ id @ X108 @ id)))))) & (axabs = $true) & ((substmonoid_lthm = $true) <=> (((! [X26 : subst] : (((comp @ X26 @ id)) = X26) & ! [X25 : subst] : (((comp @ id @ X25)) = X25)) | ? [X24 : subst] : (((comp @ X24 @ id)) != X24)) | ? [X23 : subst] : (((comp @ id @ X23)) != X23))) & (hoasapnotvar = $true) & (pushprop_lem0_gthm = $true) & (hoasinduction_lem3b_gthm = $true) & ((hoasinduction_lem3 = $true) <=> ! [X469 : (subst > term > subst > $o),X468 : term] : (? [X477 : subst,X476 : subst,X478 : subst,X479 : term] : ((((X469 @ X476 @ X479 @ (comp @ X478 @ X477))) = $true) & (((X469 @ (comp @ X476 @ X478) @ (sub @ X479 @ X478) @ X477)) != $true)) | ? [X480 : term,X483 : subst,X482 : subst,X481 : subst] : ((((X469 @ X481 @ X480 @ (comp @ X483 @ X482))) != $true) & ($true = ((X469 @ (comp @ X481 @ X483) @ (sub @ X480 @ X483) @ X482)))) | ? [X470 : term] : (($true = ((X469 @ id @ X470 @ id))) & ($true != ((X469 @ id @ (sub @ X468 @ (push @ X470 @ id)) @ id)))) | (((X469 @ id @ (lam @ X468) @ id)) = $true) | ? [X471 : (subst > term > term)] : (! [X475 : term] : ((((X469 @ id @ X475 @ id)) != $true) | (((X469 @ id @ (X471 @ id @ X475) @ id)) = $true)) & (((X469 @ id @ (lam @ (X471 @ sh @ one)) @ id)) != $true) & ! [X473 : term,X472 : subst,X474 : subst] : (((X471 @ (comp @ X474 @ X472) @ (sub @ X473 @ X472))) = ((sub @ (X471 @ X474 @ X473) @ X472)))))) & (pushprop_lem2v2 = $true) & ((induction2lem_lthm = $true) <=> ((((! [X427 : (term > $o),X428 : term,X429 : subst] : (? [X433 : term] : (! [X434 : term] : ((((X427 @ (sub @ X433 @ (push @ X434 @ id)))) = $true) | (((X427 @ X434)) != $true)) & (((X427 @ (lam @ X433))) != $true)) | ? [X430 : term] : ((((X427 @ (sub @ X430 @ X429))) != $true) & (((var @ X430)) = $true)) | ? [X432 : term,X431 : term] : ((((X427 @ X431)) = $true) & ($true != ((X427 @ (ap @ X432 @ X431)))) & (((X427 @ X432)) = $true)) | (((X427 @ (sub @ X428 @ X429))) = $true)) | ? [X422 : (term > $o),X421 : term] : (($true != ((X422 @ X421))) & ! [X426 : term] : (($true != ((var @ X426))) | (((X422 @ X426)) = $true)) & ! [X423 : term] : ((((X422 @ (lam @ X423))) = $true) | (((X422 @ X423)) != $true)) & ! [X425 : term,X424 : term] : (($true != ((X422 @ X424))) | (((X422 @ (ap @ X424 @ X425))) = $true) | (((X422 @ X425)) != $true)))) | ? [X420 : subst] : (((comp @ X420 @ id)) != X420)) | ? [X418 : subst,X419 : term] : (((comp @ sh @ (push @ X419 @ X418))) != X418)) | ? [X417 : subst,X416 : term] : (((sub @ one @ (push @ X416 @ X417))) != X416))) & (hoasinduction_lem3v2_gthm = $true) & (induction2_gthm = $true) & (pushprop_lthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (((d2subst @ d_id)) = id) & (hoasapnotvar_gthm = $true) & (hoasapnotvar_lthm = $true) & (hoasinduction_lem2v2_gthm = $true) & (((((! [X77 : term,X78 : (subst > term > subst > $o)] : (? [X93 : subst,X94 : subst,X91 : subst,X92 : term] : (($true != ((X78 @ X94 @ X92 @ (comp @ X91 @ X93)))) & (((X78 @ (comp @ X94 @ X91) @ (sub @ X92 @ X91) @ X93)) = $true)) | (((X78 @ id @ X77 @ id)) = $true) | ? [X80 : (subst > term > term)] : (! [X84 : term] : ((((X78 @ id @ X84 @ id)) != $true) | (((X78 @ id @ (X80 @ id @ X84) @ id)) = $true)) & ! [X81 : subst,X82 : term,X83 : subst] : (((sub @ (X80 @ X81 @ X82) @ X83)) = ((X80 @ (comp @ X81 @ X83) @ (sub @ X82 @ X83)))) & ($true != ((X78 @ id @ (lam @ (X80 @ sh @ one)) @ id)))) | ? [X79 : term] : ((((var @ (sub @ X79 @ id))) = $true) & (((X78 @ id @ X79 @ id)) != $true)) | ? [X90 : subst,X88 : term,X89 : subst,X87 : subst] : ((((X78 @ (comp @ X87 @ X90) @ (sub @ X88 @ X90) @ X89)) != $true) & ($true = ((X78 @ X87 @ X88 @ (comp @ X90 @ X89))))) | ? [X86 : term,X85 : term] : ((((X78 @ id @ X85 @ id)) = $true) & (((X78 @ id @ (ap @ (sub @ X86 @ id) @ X85) @ id)) != $true) & (((X78 @ id @ X86 @ id)) = $true))) | ? [X67 : (term > $o),X69 : (subst > term > subst > $o),X68 : term] : (! [X76 : term] : (((X69 @ id @ X76 @ id)) = ((X67 @ X76))) & ! [X70 : (subst > term > term)] : ((((X69 @ id @ (lam @ (X70 @ sh @ one)) @ id)) = $true) | ? [X74 : term] : ((((X69 @ id @ (X70 @ id @ X74) @ id)) != $true) & ($true = ((X69 @ id @ X74 @ id)))) | ? [X73 : term,X72 : subst,X71 : subst] : (((X70 @ (comp @ X71 @ X72) @ (sub @ X73 @ X72))) != ((sub @ (X70 @ X71 @ X73) @ X72)))) & ! [X75 : term] : ((((X67 @ (sub @ X68 @ (push @ X75 @ id)))) = $true) | (((X67 @ X75)) != $true)) & ($true != ((X67 @ (lam @ X68)))))) | ? [X66 : term] : (((sub @ X66 @ id)) != X66)) | ? [X60 : (term > $o),X59 : term] : ((((X60 @ X59)) != $true) & ! [X65 : term] : ((((X60 @ X65)) = $true) | (((var @ X65)) != $true)) & ! [X63 : term,X64 : term] : ((((X60 @ X63)) != $true) | (((X60 @ (ap @ X64 @ X63))) = $true) | (((X60 @ X64)) != $true)) & ! [X61 : term] : (? [X62 : term] : ((((X60 @ X62)) = $true) & (((X60 @ (sub @ X61 @ (push @ X62 @ id)))) != $true)) | ($true = ((X60 @ (lam @ X61))))))) | ? [X56 : (subst > term > subst > $o)] : ! [X57 : (term > $o)] : ? [X58 : term] : (((X56 @ id @ X58 @ id)) != ((X57 @ X58)))) <=> (hoasinduction_lthm_3 = $true)) & (pushprop = $true) & ((axidl = $true) <=> ! [X406 : subst] : (((comp @ id @ X406)) = X406)) & (((! [X397 : term,X396 : (subst > term > subst > $o)] : (? [X398 : term] : ((((X396 @ id @ (sub @ X397 @ (push @ X398 @ id)) @ id)) != $true) & (((X396 @ id @ X398 @ id)) = $true)) | (((X396 @ id @ (lam @ X397) @ id)) = $true) | ? [X399 : (subst > term > term)] : (! [X400 : term] : ((((X396 @ id @ (X399 @ id @ X400) @ id)) = $true) | (((X396 @ id @ X400 @ id)) != $true)) & ! [X402 : term,X403 : subst,X401 : subst] : (((X399 @ (comp @ X401 @ X403) @ (sub @ X402 @ X403))) = ((sub @ (X399 @ X401 @ X402) @ X403))) & ($true != ((X396 @ id @ (lam @ (X399 @ sh @ one)) @ id))))) | ? [X389 : (subst > term > subst > $o),X388 : term] : (! [X390 : term] : (($true != ((X389 @ id @ X390 @ id))) | (((X389 @ id @ (sub @ X388 @ (push @ X390 @ id)) @ id)) = $true)) & ! [X391 : (subst > term > term)] : (? [X392 : term] : ((((X389 @ id @ (X391 @ id @ X392) @ id)) != $true) & (((X389 @ id @ X392 @ id)) = $true)) | ? [X395 : term,X393 : subst,X394 : subst] : (((sub @ (X391 @ X393 @ X395) @ X394)) != ((X391 @ (comp @ X393 @ X394) @ (sub @ X395 @ X394)))) | (((X389 @ id @ (lam @ (X391 @ sh @ one)) @ id)) = $true)) & (((X389 @ id @ (lam @ (sub @ X388 @ (push @ one @ sh))) @ id)) != $true))) | ? [X387 : term] : (((sub @ X387 @ id)) != X387)) <=> (hoasinduction_lem3a_lthm = $true)) & ((hoasinduction_lem3v2_f = $true) <=> ! [X52 : term] : ? [X53 : (subst > term > term)] : ! [X55 : subst,X54 : term] : (((sub @ X52 @ (push @ X54 @ X55))) = ((X53 @ X55 @ X54)))) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (! [X508 : (subst > term > subst > $o),X507 : term] : (? [X510 : (subst > term > term)] : ((((X508 @ id @ (lam @ (X510 @ sh @ one)) @ id)) != $true) & ! [X512 : term,X514 : subst,X513 : subst] : (((X510 @ (comp @ X514 @ X513) @ (sub @ X512 @ X513))) = ((sub @ (X510 @ X514 @ X512) @ X513))) & ! [X511 : term] : ((((X508 @ id @ X511 @ id)) != $true) | (((X508 @ id @ (X510 @ id @ X511) @ id)) = $true))) | ($true = ((X508 @ id @ (lam @ (sub @ X507 @ (push @ one @ sh))) @ id))) | ? [X509 : term] : (($true = ((X508 @ id @ X509 @ id))) & (((X508 @ id @ (sub @ X507 @ (push @ X509 @ id)) @ id)) != $true))) <=> (hoasinduction_lem3aa = $true)) & (hoasinduction_gthm = $true) & ((laminj = $true) <=> ! [X21 : term,X22 : term] : ((((lam @ X22)) != ((lam @ X21))) | (X21 = X22))) & (hoasinduction_lem1 = $true) & (hoaslaminj_gthm = $true) & (apnotvar = $true) & ((hoasinduction_lem3v2a_lthm = $true) <=> ((! [X132 : (subst > term > subst > $o),X131 : term,X130 : (term > $o)] : (? [X133 : (subst > term > term)] : (! [X134 : subst,X136 : term,X135 : subst] : (((sub @ (X133 @ X135 @ X136) @ X134)) = ((X133 @ (comp @ X135 @ X134) @ (sub @ X136 @ X134)))) & (((X132 @ id @ (lam @ (X133 @ sh @ one)) @ id)) != $true) & ! [X137 : term] : ((((X132 @ id @ (X133 @ id @ X137) @ id)) = $true) | (((X132 @ id @ X137 @ id)) != $true))) | ($true = ((X130 @ (lam @ X131)))) | ? [X139 : term] : ((((X130 @ X139)) = $true) & (((X130 @ (sub @ X131 @ (push @ X139 @ id)))) != $true)) | ? [X138 : term] : (((X132 @ id @ X138 @ id)) != ((X130 @ X138)))) | ? [X129 : term] : (((sub @ X129 @ id)) != X129)) | ? [X125 : term] : ! [X126 : (subst > term > term)] : ? [X127 : term,X128 : subst] : (((sub @ X125 @ (push @ X127 @ X128))) != ((X126 @ X128 @ X127))))) & (ulamvar1 != $true) & (((var @ (d2term @ d_one))) != $true) & (hoasinduction_lem1v2_gthm = $true) & (axassoc = $true) & (hoasapinj1_gthm = $true) & ((pushprop_lem1 = $true) <=> ! [X216 : (term > $o),X217 : term,X218 : (term > $o),X219 : subst,X220 : term] : ((((X216 @ X217)) != $true) | (((X218 @ (sub @ X217 @ (push @ X220 @ X219)))) = $true))) & (pushprop_lem3v2_lthm = $true) & ((! [X334 : term,X332 : (subst > term > subst > $o),X333 : (term > $o)] : (? [X346 : subst,X347 : term,X348 : subst,X345 : subst] : (($true != ((X332 @ (comp @ X348 @ X346) @ (sub @ X347 @ X346) @ X345))) & ($true = ((X332 @ X348 @ X347 @ (comp @ X346 @ X345))))) | (((X333 @ (lam @ X334))) = $true) | ? [X335 : term,X338 : subst,X336 : subst,X337 : subst] : (($true = ((X332 @ (comp @ X338 @ X336) @ (sub @ X335 @ X336) @ X337))) & (((X332 @ X338 @ X335 @ (comp @ X336 @ X337))) != $true)) | ? [X349 : term] : ((((X333 @ (sub @ X334 @ (push @ X349 @ id)))) != $true) & (((X333 @ X349)) = $true)) | ? [X339 : term] : (((X333 @ X339)) != ((X332 @ id @ X339 @ id))) | ? [X340 : (subst > term > term)] : (! [X344 : term] : (($true != ((X332 @ id @ X344 @ id))) | (((X332 @ id @ (X340 @ id @ X344) @ id)) = $true)) & ! [X343 : subst,X341 : term,X342 : subst] : (((X340 @ (comp @ X342 @ X343) @ (sub @ X341 @ X343))) = ((sub @ (X340 @ X342 @ X341) @ X343))) & (((X332 @ id @ (lam @ (X340 @ sh @ one)) @ id)) != $true))) | ? [X331 : term] : (((sub @ X331 @ id)) != X331)) <=> (hoasinduction_lem3v2_lthm = $true)) & ! [X20 : d_subst,X19 : d_subst] : ((((d2subst @ X19)) != ((d2subst @ X20))) | (X19 = X20)) & (hoaslamnotvar_lthm = $true) & (! [X321 : (subst > term > term),X324 : subst,X323 : term,X322 : (subst > term > term)] : ((((lam @ (X321 @ sh @ one))) != ((lam @ (X322 @ sh @ one)))) | ? [X325 : subst,X326 : term,X327 : subst] : (((sub @ (X321 @ X327 @ X326) @ X325)) != ((X321 @ (comp @ X327 @ X325) @ (sub @ X326 @ X325)))) | ? [X328 : subst,X329 : subst,X330 : term] : (((sub @ (X322 @ X328 @ X330) @ X329)) != ((X322 @ (comp @ X328 @ X329) @ (sub @ X330 @ X329)))) | (((X322 @ X324 @ X323)) = ((X321 @ X324 @ X323)))) <=> (hoaslaminj = $true)) & (lamnotvar = $true) & (ulamvarsh = $true) & (ulamvarind = $true) & ((hoasinduction_lem2v2 = $true) <=> ! [X203 : (subst > term > subst > $o),X202 : (term > $o),X204 : term,X201 : term] : (? [X215 : term,X214 : term] : ((((X203 @ id @ (ap @ (sub @ X214 @ id) @ X215) @ id)) != $true) & (((X203 @ id @ X215 @ id)) = $true) & (((X203 @ id @ X214 @ id)) = $true)) | (((X202 @ (ap @ X204 @ X201))) = $true) | ? [X213 : term] : (((X203 @ id @ X213 @ id)) != ((X202 @ X213))) | (((X202 @ X201)) != $true) | ? [X212 : subst,X210 : subst,X211 : subst,X209 : term] : ((((X203 @ (comp @ X210 @ X211) @ (sub @ X209 @ X211) @ X212)) = $true) & (((X203 @ X210 @ X209 @ (comp @ X211 @ X212))) != $true)) | ? [X207 : subst,X208 : subst,X206 : subst,X205 : term] : (($true != ((X203 @ (comp @ X207 @ X206) @ (sub @ X205 @ X206) @ X208))) & (((X203 @ X207 @ X205 @ (comp @ X206 @ X208))) = $true)) | ($true != ((X202 @ X204))))) & ((! [X278 : subst] : (((comp @ X278 @ id)) = X278) & ! [X277 : subst] : (((comp @ id @ X277)) = X277)) <=> (substmonoid = $true)) & (hoasinduction_lem3a_gthm = $true) & (substmonoid_gthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (axmap = $true) & (! [X40 : term,X41 : (subst > term > subst > $o)] : (? [X42 : term] : ((((X41 @ id @ X42 @ id)) = $true) & (((X41 @ id @ (sub @ X40 @ (push @ X42 @ id)) @ id)) != $true)) | (((X41 @ id @ (lam @ (sub @ X40 @ (push @ one @ sh))) @ id)) = $true) | ? [X43 : term,X44 : (subst > term > term)] : (! [X47 : subst,X45 : term,X46 : subst] : (((sub @ (X44 @ X47 @ X45) @ X46)) = ((sub @ (sub @ X43 @ (push @ X45 @ X47)) @ X46))) & ! [X49 : subst,X51 : term,X50 : subst] : (((sub @ X43 @ (push @ (sub @ X51 @ X50) @ (comp @ X49 @ X50)))) = ((X44 @ (comp @ X49 @ X50) @ (sub @ X51 @ X50)))) & ! [X48 : term] : ((((X41 @ id @ X48 @ id)) != $true) | (((X41 @ id @ (X44 @ id @ X48) @ id)) = $true)) & (((X41 @ id @ (lam @ (X44 @ sh @ one)) @ id)) != $true))) <=> (hoasinduction_lem3aaa = $true)) & ((shinj = $true) <=> ! [X502 : term,X503 : term] : ((X502 = X503) | (((sub @ X502 @ sh)) != ((sub @ X503 @ sh))))) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ((axidr = $true) <=> ! [X272 : subst] : (((comp @ X272 @ id)) = X272)) & (! [X500 : term] : ? [X501 : (subst > term > term)] : (((sub @ X500 @ (push @ one @ sh))) = ((X501 @ sh @ one))) <=> (hoasinduction_lem3b_lthm = $true)) & (! [X521 : term,X520 : (term > $o),X519 : subst] : ? [X522 : (term > $o)] : ! [X523 : term] : (((X520 @ (sub @ X523 @ (push @ X521 @ X519)))) = ((X522 @ X523))) <=> (pushprop_lem0 = $true)) & (hoasinduction_lem3_gthm = $true) & (induction2lem_gthm = $true) & ! [X221 : d_term,X222 : d_term] : ((X221 = X222) | (((d2term @ X222)) != ((d2term @ X221)))) & ! [X365 : subst] : ? [X366 : d_subst] : (((d2subst @ X366)) = X365) & (hoaslamnotap_lthm = $true) & (hoaslamnotvar_gthm = $true) & (((! [X494 : (term > $o),X493 : term] : (? [X495 : term] : ((((var @ X495)) = $true) & (((X494 @ X495)) != $true)) | ? [X498 : term] : (! [X499 : term] : ((((X494 @ X499)) != $true) | (((X494 @ (sub @ X498 @ (push @ X499 @ id)))) = $true)) & (((X494 @ (lam @ X498))) != $true)) | ? [X497 : term,X496 : term] : ((((X494 @ X497)) = $true) & (((X494 @ (ap @ X496 @ X497))) != $true) & (((X494 @ X496)) = $true)) | ($true = ((X494 @ X493)))) | ? [X487 : subst,X486 : term,X485 : (term > $o)] : (! [X491 : term,X492 : term] : ((((X485 @ (ap @ X491 @ X492))) = $true) | (((X485 @ X491)) != $true) | (((X485 @ X492)) != $true)) & ! [X489 : term] : (? [X490 : term] : (($true = ((X485 @ X490))) & (((X485 @ (sub @ X489 @ (push @ X490 @ id)))) != $true)) | ($true = ((X485 @ (lam @ X489))))) & ! [X488 : term] : ((((var @ X488)) != $true) | (((X485 @ (sub @ X488 @ X487))) = $true)) & (((X485 @ (sub @ X486 @ X487))) != $true))) | ? [X484 : term] : (((sub @ X484 @ id)) != X484)) <=> (induction2_lthm = $true)) & ((induction = $true) <=> ! [X382 : term,X381 : (term > $o)] : (? [X386 : term,X385 : term] : (($true != ((X381 @ (ap @ X385 @ X386)))) & ($true = ((X381 @ X386))) & (((X381 @ X385)) = $true)) | ($true = ((X381 @ X382))) | ? [X384 : term] : (($true = ((X381 @ X384))) & ($true != ((X381 @ (lam @ X384))))) | ? [X383 : term] : ((((var @ X383)) = $true) & (((X381 @ X383)) != $true)))) & ((pushprop_lem1v2_lthm = $true) <=> (! [X196 : (term > $o),X199 : term,X197 : subst,X198 : (term > $o)] : ((((X198 @ X199)) != $true) | ($true = ((X196 @ one))) | ? [X200 : term] : (((X196 @ X200)) != ((X198 @ (sub @ X200 @ (push @ X199 @ X197)))))) | ? [X195 : term,X194 : subst] : (((sub @ one @ (push @ X195 @ X194))) != X195))) & ((hoasinduction_lem3v2 = $true) <=> ! [X304 : (term > $o),X305 : (subst > term > subst > $o),X303 : term] : (? [X320 : term] : (($true != ((X304 @ (sub @ X303 @ (push @ X320 @ id))))) & (((X304 @ X320)) = $true)) | ? [X318 : term,X319 : subst,X317 : subst,X316 : subst] : ((((X305 @ (comp @ X319 @ X317) @ (sub @ X318 @ X317) @ X316)) = $true) & ($true != ((X305 @ X319 @ X318 @ (comp @ X317 @ X316))))) | (((X304 @ (lam @ X303))) = $true) | ? [X310 : subst,X308 : subst,X309 : subst,X307 : term] : (($true != ((X305 @ (comp @ X310 @ X309) @ (sub @ X307 @ X309) @ X308))) & (((X305 @ X310 @ X307 @ (comp @ X309 @ X308))) = $true)) | ? [X306 : term] : (((X304 @ X306)) != ((X305 @ id @ X306 @ id))) | ? [X311 : (subst > term > term)] : (! [X314 : term,X312 : subst,X313 : subst] : (((sub @ (X311 @ X312 @ X314) @ X313)) = ((X311 @ (comp @ X312 @ X313) @ (sub @ X314 @ X313)))) & (((X305 @ id @ (lam @ (X311 @ sh @ one)) @ id)) != $true) & ! [X315 : term] : ((((X305 @ id @ (X311 @ id @ X315) @ id)) = $true) | (((X305 @ id @ X315 @ id)) != $true))))) & (((! [X36 : term,X33 : term,X34 : term,X35 : term] : ((X33 = X36) | (((ap @ (sub @ X33 @ id) @ X35)) != ((ap @ (sub @ X36 @ id) @ X34)))) | ? [X31 : term,X32 : term,X30 : term,X29 : term] : ((((ap @ X31 @ X29)) = ((ap @ X32 @ X30))) & (X31 != X32))) | ? [X28 : term] : (((sub @ X28 @ id)) != X28)) <=> (hoasapinj1_lthm = $true)) & (((d2subst @ d_id)) = sh) & (pushprop_lem2v2_lthm = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (((! [X532 : (term > $o),X531 : term,X529 : (term > $o),X528 : term,X530 : subst] : ((((X532 @ (sub @ X531 @ (push @ X528 @ X530)))) = $true) | ($true != ((X529 @ X531)))) | ? [X526 : subst,X527 : term] : (((comp @ sh @ (push @ X527 @ X526))) != X526)) | ? [X525 : subst,X524 : term] : (((sub @ one @ (push @ X524 @ X525))) != X524)) <=> (pushprop_lem1_lthm = $true)) & ((hoasinduction_lem3v2a = $true) <=> ! [X368 : (term > $o),X367 : (subst > term > subst > $o),X369 : term] : ((((X368 @ (lam @ X369))) = $true) | ? [X371 : term] : (((X367 @ id @ X371 @ id)) != ((X368 @ X371))) | ? [X370 : term] : (($true != ((X368 @ (sub @ X369 @ (push @ X370 @ id))))) & (((X368 @ X370)) = $true)) | ? [X372 : (subst > term > term)] : (! [X375 : term,X376 : subst,X374 : subst] : (((sub @ (X372 @ X376 @ X375) @ X374)) = ((X372 @ (comp @ X376 @ X374) @ (sub @ X375 @ X374)))) & ! [X373 : term] : ((((X367 @ id @ (X372 @ id @ X373) @ id)) = $true) | (((X367 @ id @ X373 @ id)) != $true)) & (((X367 @ id @ (lam @ (X372 @ sh @ one)) @ id)) != $true)))) & (hoasinduction_lem2_gthm = $true) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_lem2_lthm = $true) & (! [X223 : term,X224 : subst] : (((comp @ sh @ (push @ X223 @ X224))) = X224) <=> (axshiftcons = $true)) & ((hoasapinj1 = $true) <=> ! [X302 : term,X299 : term,X301 : term,X300 : term] : ((X299 = X302) | (((ap @ (sub @ X302 @ id) @ X301)) != ((ap @ (sub @ X299 @ id) @ X300))))) & (pushprop_gthm = $true) & ((hoasapinj2 = $true) <=> ! [X274 : term,X273 : term,X276 : term,X275 : term] : ((((ap @ (sub @ X274 @ id) @ X276)) != ((ap @ (sub @ X275 @ id) @ X273))) | (X273 = X276))) & ((termmset = $true) <=> ! [X291 : term] : (((sub @ X291 @ id)) = X291)) & (hoaslamnotvar = $true) & ((hoasinduction = $true) <=> ! [X0 : term,X1 : (subst > term > subst > $o)] : (? [X2 : subst,X3 : subst,X4 : term,X5 : subst] : ((((X1 @ X5 @ X4 @ (comp @ X3 @ X2))) = $true) & (((X1 @ (comp @ X5 @ X3) @ (sub @ X4 @ X3) @ X2)) != $true)) | ? [X13 : term,X12 : subst,X11 : subst,X14 : subst] : (($true != ((X1 @ X14 @ X13 @ (comp @ X12 @ X11)))) & ($true = ((X1 @ (comp @ X14 @ X12) @ (sub @ X13 @ X12) @ X11)))) | (((X1 @ id @ X0 @ id)) = $true) | ? [X17 : term,X16 : term] : ((((X1 @ id @ X16 @ id)) = $true) & (((X1 @ id @ (ap @ (sub @ X16 @ id) @ X17) @ id)) != $true) & ($true = ((X1 @ id @ X17 @ id)))) | ? [X15 : term] : (($true = ((var @ (sub @ X15 @ id)))) & (((X1 @ id @ X15 @ id)) != $true)) | ? [X6 : (subst > term > term)] : (! [X10 : term] : ((((X1 @ id @ (X6 @ id @ X10) @ id)) = $true) | (((X1 @ id @ X10 @ id)) != $true)) & (((X1 @ id @ (lam @ (X6 @ sh @ one)) @ id)) != $true) & ! [X9 : subst,X8 : term,X7 : subst] : (((sub @ (X6 @ X9 @ X8) @ X7)) = ((X6 @ (comp @ X9 @ X7) @ (sub @ X8 @ X7))))))) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (hoasapinj2_gthm = $true) & (! [X298 : subst] : (((push @ (sub @ one @ X298) @ (comp @ sh @ X298))) = X298) <=> (axscons = $true)) & ((! [X518 : term] : (((sub @ X518 @ id)) = X518) | ? [X517 : term] : (((sub @ X517 @ id)) != X517)) <=> (termmset_lthm = $true)) & ((((! [X177 : (subst > term > subst > $o),X176 : term] : (? [X187 : subst,X186 : subst,X185 : term,X188 : subst] : ((((X177 @ X188 @ X185 @ (comp @ X186 @ X187))) = $true) & (((X177 @ (comp @ X188 @ X186) @ (sub @ X185 @ X186) @ X187)) != $true)) | ? [X179 : term,X178 : term] : ((((X177 @ id @ X179 @ id)) = $true) & (((X177 @ id @ (ap @ (sub @ X178 @ id) @ X179) @ id)) != $true) & (((X177 @ id @ X178 @ id)) = $true)) | (((X177 @ id @ X176 @ id)) = $true) | ? [X189 : term] : ((((var @ (sub @ X189 @ id))) = $true) & (((X177 @ id @ X189 @ id)) != $true)) | ? [X180 : (subst > term > term)] : (! [X181 : term] : (($true != ((X177 @ id @ X181 @ id))) | ($true = ((X177 @ id @ (X180 @ id @ X181) @ id)))) & (((X177 @ id @ (lam @ (X180 @ sh @ one)) @ id)) != $true) & ! [X184 : term,X183 : subst,X182 : subst] : (((sub @ (X180 @ X183 @ X184) @ X182)) = ((X180 @ (comp @ X183 @ X182) @ (sub @ X184 @ X182))))) | ? [X191 : subst,X193 : term,X192 : subst,X190 : subst] : ((((X177 @ X192 @ X193 @ (comp @ X191 @ X190))) != $true) & (((X177 @ (comp @ X192 @ X191) @ (sub @ X193 @ X191) @ X190)) = $true))) | ? [X161 : term,X160 : (subst > term > subst > $o)] : (! [X175 : term] : ((((X160 @ id @ X175 @ id)) != $true) | (((X160 @ id @ (sub @ X161 @ (push @ X175 @ id)) @ id)) = $true)) & ! [X165 : subst,X163 : subst,X164 : subst,X162 : term] : ((((X160 @ X163 @ X162 @ (comp @ X164 @ X165))) = $true) | (((X160 @ (comp @ X163 @ X164) @ (sub @ X162 @ X164) @ X165)) != $true)) & ! [X174 : subst,X171 : subst,X173 : subst,X172 : term] : ((((X160 @ X174 @ X172 @ (comp @ X171 @ X173))) != $true) | ($true = ((X160 @ (comp @ X174 @ X171) @ (sub @ X172 @ X171) @ X173)))) & (((X160 @ id @ (lam @ X161) @ id)) != $true) & ! [X166 : (subst > term > term)] : (? [X170 : term] : ((((X160 @ id @ X170 @ id)) = $true) & (((X160 @ id @ (X166 @ id @ X170) @ id)) != $true)) | ? [X167 : subst,X168 : subst,X169 : term] : (((X166 @ (comp @ X167 @ X168) @ (sub @ X169 @ X168))) != ((sub @ (X166 @ X167 @ X169) @ X168))) | ($true = ((X160 @ id @ (lam @ (X166 @ sh @ one)) @ id)))))) | ? [X148 : (subst > term > subst > $o),X147 : term,X149 : term] : ((((X148 @ id @ (ap @ X149 @ X147) @ id)) != $true) & ! [X152 : subst,X153 : subst,X151 : term,X150 : subst] : (($true = ((X148 @ (comp @ X152 @ X150) @ (sub @ X151 @ X150) @ X153))) | (((X148 @ X152 @ X151 @ (comp @ X150 @ X153))) != $true)) & ! [X155 : subst,X154 : subst,X157 : subst,X156 : term] : ((((X148 @ (comp @ X155 @ X154) @ (sub @ X156 @ X154) @ X157)) != $true) | (((X148 @ X155 @ X156 @ (comp @ X154 @ X157))) = $true)) & ! [X158 : term,X159 : term] : ((((X148 @ id @ X158 @ id)) != $true) | (((X148 @ id @ X159 @ id)) != $true) | (((X148 @ id @ (ap @ (sub @ X159 @ id) @ X158) @ id)) = $true)) & (((X148 @ id @ X149 @ id)) = $true) & (((X148 @ id @ X147 @ id)) = $true))) | ? [X140 : term,X141 : (term > $o)] : (! [X143 : term] : (? [X144 : term] : (($true != ((X141 @ (sub @ X143 @ (push @ X144 @ id))))) & (((X141 @ X144)) = $true)) | (((X141 @ (lam @ X143))) = $true)) & (((X141 @ X140)) != $true) & ! [X146 : term,X145 : term] : ((((X141 @ X145)) != $true) | ($true = ((X141 @ (ap @ X146 @ X145)))) | (((X141 @ X146)) != $true)) & ! [X142 : term] : (($true = ((X141 @ X142))) | (((var @ X142)) != $true)))) <=> (hoasinduction_lthm = $true)) & (pushprop_lem3v2 = $true) & ((hoasinduction_lem0 = $true) <=> ! [X504 : (subst > term > subst > $o)] : ? [X505 : (term > $o)] : ! [X506 : term] : (((X504 @ id @ X506 @ id)) = ((X505 @ X506)))) & (! [X379 : term,X377 : term,X380 : term,X378 : term] : ((X377 = X378) | (((ap @ X378 @ X379)) != ((ap @ X377 @ X380)))) <=> (apinj1 = $true)) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X284 : term,X283 : (subst > term > subst > $o)] : ((((X283 @ id @ (lam @ (sub @ X284 @ (push @ one @ sh))) @ id)) = $true) | ? [X290 : term] : (($true != ((X283 @ id @ (sub @ X284 @ (push @ X290 @ id)) @ id))) & ($true = ((X283 @ id @ X290 @ id)))) | ? [X285 : (subst > term > term)] : (! [X286 : term] : ((((X283 @ id @ (X285 @ id @ X286) @ id)) = $true) | (((X283 @ id @ X286 @ id)) != $true)) & ! [X288 : subst,X289 : term,X287 : subst] : (((X285 @ (comp @ X288 @ X287) @ (sub @ X289 @ X287))) = ((sub @ (X285 @ X288 @ X289) @ X287))) & (((X283 @ id @ (lam @ (X285 @ sh @ one)) @ id)) != $true)))) & ((((! [X263 : (subst > term > term),X264 : subst,X262 : (subst > term > term),X265 : term] : (? [X266 : subst,X267 : subst,X268 : term] : (((X263 @ (comp @ X266 @ X267) @ (sub @ X268 @ X267))) != ((sub @ (X263 @ X266 @ X268) @ X267))) | (((X262 @ X264 @ X265)) = ((X263 @ X264 @ X265))) | (((lam @ (X263 @ sh @ one))) != ((lam @ (X262 @ sh @ one)))) | ? [X269 : subst,X270 : term,X271 : subst] : (((X262 @ (comp @ X269 @ X271) @ (sub @ X270 @ X271))) != ((sub @ (X262 @ X269 @ X270) @ X271)))) | ? [X261 : term,X260 : term] : ((X260 != X261) & (((lam @ X260)) = ((lam @ X261))))) | ? [X259 : subst,X258 : term] : (((comp @ sh @ (push @ X258 @ X259))) != X259)) | ? [X256 : subst,X257 : term] : (((sub @ one @ (push @ X257 @ X256))) != X257)) <=> (hoaslaminj_lthm = $true)) & (! [X515 : term] : ? [X516 : (subst > term > term)] : (((X516 @ sh @ one)) = ((sub @ X515 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b = $true)) & ! [X18 : d_term] : (d_one = X18) & (((((! [X247 : term,X248 : (subst > term > subst > $o)] : (? [X249 : (subst > term > term)] : (! [X250 : term] : ((((X248 @ id @ X250 @ id)) != $true) | ($true = ((X248 @ id @ (X249 @ id @ X250) @ id)))) & (((X248 @ id @ (lam @ (X249 @ sh @ one)) @ id)) != $true) & ! [X253 : subst,X251 : term,X252 : subst] : (((X249 @ (comp @ X253 @ X252) @ (sub @ X251 @ X252))) = ((sub @ (X249 @ X253 @ X251) @ X252)))) | (((X248 @ id @ X247 @ id)) = $true) | ? [X254 : term,X255 : term] : ((((X248 @ id @ X255 @ id)) = $true) & (((X248 @ id @ X254 @ id)) = $true) & (((X248 @ id @ (ap @ (sub @ X254 @ id) @ X255) @ id)) != $true))) | ? [X239 : (term > $o),X237 : (subst > term > subst > $o),X238 : term] : (! [X240 : (subst > term > term)] : (? [X244 : term] : (($true = ((X237 @ id @ X244 @ id))) & (((X237 @ id @ (X240 @ id @ X244) @ id)) != $true)) | ($true = ((X237 @ id @ (lam @ (X240 @ sh @ one)) @ id))) | ? [X241 : subst,X242 : subst,X243 : term] : (((X240 @ (comp @ X242 @ X241) @ (sub @ X243 @ X241))) != ((sub @ (X240 @ X242 @ X243) @ X241)))) & (((X239 @ (lam @ X238))) != $true) & ! [X246 : term] : (((X237 @ id @ X246 @ id)) = ((X239 @ X246))) & ! [X245 : term] : (($true = ((X239 @ (sub @ X238 @ (push @ X245 @ id))))) | (((X239 @ X245)) != $true)))) | ? [X236 : term] : (((sub @ X236 @ id)) != X236)) | ? [X230 : (term > $o),X229 : term] : (! [X231 : term] : (($true != ((var @ X231))) | ($true = ((X230 @ X231)))) & ! [X234 : term] : (? [X235 : term] : ((((X230 @ X235)) = $true) & (((X230 @ (sub @ X234 @ (push @ X235 @ id)))) != $true)) | (((X230 @ (lam @ X234))) = $true)) & ! [X232 : term,X233 : term] : ((((X230 @ X233)) != $true) | (((X230 @ (ap @ X232 @ X233))) = $true) | ($true != ((X230 @ X232)))) & ($true != ((X230 @ X229))))) | ? [X226 : (subst > term > subst > $o)] : ! [X227 : (term > $o)] : ? [X228 : term] : (((X227 @ X228)) != ((X226 @ id @ X228 @ id)))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (pushprop_lem1_gthm = $true) & (lamnotap != $true) & ((axvarcons = $true) <=> ! [X404 : term,X405 : subst] : (((sub @ one @ (push @ X404 @ X405))) = X404)) & (! [X294 : (subst > term > term),X293 : term,X292 : term] : (? [X297 : term,X295 : subst,X296 : subst] : (((sub @ (X294 @ X296 @ X297) @ X295)) != ((X294 @ (comp @ X296 @ X295) @ (sub @ X297 @ X295)))) | (((lam @ (X294 @ sh @ one))) != ((ap @ (sub @ X292 @ id) @ X293)))) <=> (hoaslamnotap = $true)) & (hoasinduction_lem1v2 = $true) & (axapp = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2)))))))) & (termmset_gthm = $true) & ! [X225 : d_subst] : (d_id = X225) & (! [X407 : term,X408 : (subst > term > subst > $o)] : (? [X415 : term,X414 : term] : (($true = ((X408 @ id @ X415 @ id))) & (((X408 @ id @ (ap @ (sub @ X414 @ id) @ X415) @ id)) != $true) & (((X408 @ id @ X414 @ id)) = $true)) | ? [X409 : (subst > term > term)] : (! [X410 : term] : ((((X408 @ id @ (X409 @ id @ X410) @ id)) = $true) | (((X408 @ id @ X410 @ id)) != $true)) & ($true != ((X408 @ id @ (lam @ (X409 @ sh @ one)) @ id))) & ! [X412 : term,X413 : subst,X411 : subst] : (((X409 @ (comp @ X411 @ X413) @ (sub @ X412 @ X413))) = ((sub @ (X409 @ X411 @ X412) @ X413)))) | (((X408 @ id @ X407 @ id)) = $true)) <=> (hoasinduction_no_psi_cond = $true)) & ((induction2lem = $true) <=> ! [X113 : subst,X114 : term,X115 : (term > $o)] : (? [X116 : term] : ((((X115 @ (sub @ X116 @ X113))) != $true) & (((var @ X116)) = $true)) | ? [X117 : term] : (! [X118 : term] : (($true = ((X115 @ (sub @ X117 @ (push @ X118 @ id))))) | (((X115 @ X118)) != $true)) & ($true != ((X115 @ (lam @ X117))))) | ? [X120 : term,X119 : term] : ((((X115 @ (ap @ X119 @ X120))) != $true) & (((X115 @ X119)) = $true) & ($true = ((X115 @ X120)))) | (((X115 @ (sub @ X114 @ X113))) = $true))) & (! [X358 : (term > $o),X359 : term] : (($true = ((X358 @ X359))) | ? [X360 : term] : ((((X358 @ X360)) != $true) & (((var @ X360)) = $true)) | ? [X364 : term,X363 : term] : ((((X358 @ (ap @ X363 @ X364))) != $true) & (((X358 @ X364)) = $true) & ($true = ((X358 @ X363)))) | ? [X361 : term] : ((((X358 @ (lam @ X361))) != $true) & ! [X362 : term] : ((((X358 @ X362)) != $true) | (((X358 @ (sub @ X361 @ (push @ X362 @ id)))) = $true)))) <=> (induction2 = $true)) & (hoaslamnotap_gthm = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (axvarshift = $true) & (! [X37 : (subst > term > subst > $o)] : ? [X38 : (term > $o)] : ! [X39 : term] : (((X37 @ id @ X39 @ id)) = ((X38 @ X39))) <=> (hoasinduction_lem0_lthm = $true)) & (((! [X445 : (subst > term > subst > $o),X444 : term] : (? [X455 : term] : ((((X445 @ id @ (sub @ X444 @ (push @ X455 @ id)) @ id)) != $true) & (((X445 @ id @ X455 @ id)) = $true)) | ? [X446 : term,X447 : subst,X449 : subst,X448 : subst] : (($true != ((X445 @ (comp @ X447 @ X449) @ (sub @ X446 @ X449) @ X448))) & (((X445 @ X447 @ X446 @ (comp @ X449 @ X448))) = $true)) | ? [X450 : (subst > term > term)] : (! [X453 : subst,X454 : subst,X452 : term] : (((sub @ (X450 @ X454 @ X452) @ X453)) = ((X450 @ (comp @ X454 @ X453) @ (sub @ X452 @ X453)))) & ! [X451 : term] : (($true != ((X445 @ id @ X451 @ id))) | (((X445 @ id @ (X450 @ id @ X451) @ id)) = $true)) & (((X445 @ id @ (lam @ (X450 @ sh @ one)) @ id)) != $true)) | (((X445 @ id @ (lam @ X444) @ id)) = $true) | ? [X457 : term,X459 : subst,X456 : subst,X458 : subst] : ((((X445 @ X458 @ X457 @ (comp @ X459 @ X456))) != $true) & (((X445 @ (comp @ X458 @ X459) @ (sub @ X457 @ X459) @ X456)) = $true))) | ? [X437 : (subst > term > subst > $o),X436 : term] : (! [X443 : term] : ((((X437 @ id @ X443 @ id)) != $true) | (((X437 @ id @ (sub @ X436 @ (push @ X443 @ id)) @ id)) = $true)) & ($true != ((X437 @ id @ (lam @ (sub @ X436 @ (push @ one @ sh))) @ id))) & ! [X438 : (subst > term > term)] : (? [X439 : term] : ((((X437 @ id @ (X438 @ id @ X439) @ id)) != $true) & ($true = ((X437 @ id @ X439 @ id)))) | (((X437 @ id @ (lam @ (X438 @ sh @ one)) @ id)) = $true) | ? [X442 : subst,X441 : subst,X440 : term] : (((X438 @ (comp @ X441 @ X442) @ (sub @ X440 @ X442))) != ((sub @ (X438 @ X441 @ X440) @ X442)))))) | ? [X435 : term] : (((sub @ X435 @ id)) != X435)) <=> (hoasinduction_lem3_lthm = $true)) & (hoasinduction_lem1_lthm = $true) & (! [X121 : term] : ? [X122 : (subst > term > term)] : ! [X123 : term,X124 : subst] : (((X122 @ X124 @ X123)) = ((sub @ X121 @ (push @ X123 @ X124)))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & (! [X279 : term,X282 : term,X281 : term,X280 : term] : ((((ap @ X280 @ X282)) != ((ap @ X279 @ X281))) | (X281 = X282)) <=> (apinj2 = $true)) & ((axvarid = $true) <=> ! [X27 : term] : (((sub @ X27 @ id)) = X27)) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((? [X462 : term,X463 : term,X460 : term,X461 : term] : ((((ap @ X463 @ X460)) = ((ap @ X461 @ X462))) & (X460 != X462)) | ! [X466 : term,X464 : term,X465 : term,X467 : term] : ((((ap @ (sub @ X464 @ id) @ X467)) != ((ap @ (sub @ X466 @ id) @ X465))) | (X465 = X467))) <=> (hoasapinj2_lthm = $true))),
  inference(ennf_transformation,[],[f10])).
thf(f12,plain,(
  ((hoasinduction_lem3v2 = $true) <=> ! [X304 : (term > $o),X305 : (subst > term > subst > $o),X303 : term] : (? [X320 : term] : (($true != ((X304 @ (sub @ X303 @ (push @ X320 @ id))))) & (((X304 @ X320)) = $true)) | ? [X318 : term,X319 : subst,X317 : subst,X316 : subst] : ((((X305 @ (comp @ X319 @ X317) @ (sub @ X318 @ X317) @ X316)) = $true) & ($true != ((X305 @ X319 @ X318 @ (comp @ X317 @ X316))))) | (((X304 @ (lam @ X303))) = $true) | ? [X310 : subst,X308 : subst,X309 : subst,X307 : term] : (($true != ((X305 @ (comp @ X310 @ X309) @ (sub @ X307 @ X309) @ X308))) & (((X305 @ X310 @ X307 @ (comp @ X309 @ X308))) = $true)) | ? [X306 : term] : (((X304 @ X306)) != ((X305 @ id @ X306 @ id))) | ? [X311 : (subst > term > term)] : (! [X314 : term,X312 : subst,X313 : subst] : (((sub @ (X311 @ X312 @ X314) @ X313)) = ((X311 @ (comp @ X312 @ X313) @ (sub @ X314 @ X313)))) & (((X305 @ id @ (lam @ (X311 @ sh @ one)) @ id)) != $true) & ! [X315 : term] : ((((X305 @ id @ (X311 @ id @ X315) @ id)) = $true) | (((X305 @ id @ X315 @ id)) != $true))))) & ((hoasinduction = $true) <=> ! [X0 : term,X1 : (subst > term > subst > $o)] : (? [X2 : subst,X3 : subst,X4 : term,X5 : subst] : ((((X1 @ X5 @ X4 @ (comp @ X3 @ X2))) = $true) & (((X1 @ (comp @ X5 @ X3) @ (sub @ X4 @ X3) @ X2)) != $true)) | ? [X13 : term,X12 : subst,X11 : subst,X14 : subst] : (($true != ((X1 @ X14 @ X13 @ (comp @ X12 @ X11)))) & ($true = ((X1 @ (comp @ X14 @ X12) @ (sub @ X13 @ X12) @ X11)))) | (((X1 @ id @ X0 @ id)) = $true) | ? [X17 : term,X16 : term] : ((((X1 @ id @ X16 @ id)) = $true) & (((X1 @ id @ (ap @ (sub @ X16 @ id) @ X17) @ id)) != $true) & ($true = ((X1 @ id @ X17 @ id)))) | ? [X15 : term] : (($true = ((var @ (sub @ X15 @ id)))) & (((X1 @ id @ X15 @ id)) != $true)) | ? [X6 : (subst > term > term)] : (! [X10 : term] : ((((X1 @ id @ (X6 @ id @ X10) @ id)) = $true) | (((X1 @ id @ X10 @ id)) != $true)) & (((X1 @ id @ (lam @ (X6 @ sh @ one)) @ id)) != $true) & ! [X9 : subst,X8 : term,X7 : subst] : (((sub @ (X6 @ X9 @ X8) @ X7)) = ((X6 @ (comp @ X9 @ X7) @ (sub @ X8 @ X7))))))) & (axabs = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((pushprop_lem1v2 = $true) <=> ! [X534 : (term > $o),X536 : term,X533 : (term > $o),X535 : subst] : (? [X537 : term] : (((X533 @ (sub @ X537 @ (push @ X536 @ X535)))) != ((X534 @ X537))) | (((X534 @ one)) = $true) | (((X533 @ X536)) != $true))) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ((shinj = $true) <=> ! [X502 : term,X503 : term] : ((X502 = X503) | (((sub @ X502 @ sh)) != ((sub @ X503 @ sh))))) & ((hoasinduction_lem2v2 = $true) <=> ! [X203 : (subst > term > subst > $o),X202 : (term > $o),X204 : term,X201 : term] : (? [X215 : term,X214 : term] : ((((X203 @ id @ (ap @ (sub @ X214 @ id) @ X215) @ id)) != $true) & (((X203 @ id @ X215 @ id)) = $true) & (((X203 @ id @ X214 @ id)) = $true)) | (((X202 @ (ap @ X204 @ X201))) = $true) | ? [X213 : term] : (((X203 @ id @ X213 @ id)) != ((X202 @ X213))) | (((X202 @ X201)) != $true) | ? [X212 : subst,X210 : subst,X211 : subst,X209 : term] : ((((X203 @ (comp @ X210 @ X211) @ (sub @ X209 @ X211) @ X212)) = $true) & (((X203 @ X210 @ X209 @ (comp @ X211 @ X212))) != $true)) | ? [X207 : subst,X208 : subst,X206 : subst,X205 : term] : (($true != ((X203 @ (comp @ X207 @ X206) @ (sub @ X205 @ X206) @ X208))) & (((X203 @ X207 @ X205 @ (comp @ X206 @ X208))) = $true)) | ($true != ((X202 @ X204))))) & ! [X221 : d_term,X222 : d_term] : ((X221 = X222) | (((d2term @ X222)) != ((d2term @ X221)))) & ((hoasinduction_lem3v2_f = $true) <=> ! [X52 : term] : ? [X53 : (subst > term > term)] : ! [X55 : subst,X54 : term] : (((sub @ X52 @ (push @ X54 @ X55))) = ((X53 @ X55 @ X54)))) & ! [X18 : d_term] : (d_one = X18) & (axapp = $true) & (ulamvarind = $true) & (hoasinduction_lem2v2_gthm = $true) & ((hoasinduction_lthm = $true) <=> (? [X148 : (subst > term > subst > $o),X147 : term,X149 : term] : ((((X148 @ id @ (ap @ X149 @ X147) @ id)) != $true) & ! [X152 : subst,X153 : subst,X151 : term,X150 : subst] : (($true = ((X148 @ (comp @ X152 @ X150) @ (sub @ X151 @ X150) @ X153))) | (((X148 @ X152 @ X151 @ (comp @ X150 @ X153))) != $true)) & ! [X155 : subst,X154 : subst,X157 : subst,X156 : term] : ((((X148 @ (comp @ X155 @ X154) @ (sub @ X156 @ X154) @ X157)) != $true) | (((X148 @ X155 @ X156 @ (comp @ X154 @ X157))) = $true)) & ! [X158 : term,X159 : term] : ((((X148 @ id @ X158 @ id)) != $true) | (((X148 @ id @ X159 @ id)) != $true) | (((X148 @ id @ (ap @ (sub @ X159 @ id) @ X158) @ id)) = $true)) & (((X148 @ id @ X149 @ id)) = $true) & (((X148 @ id @ X147 @ id)) = $true)) | ! [X177 : (subst > term > subst > $o),X176 : term] : (? [X187 : subst,X186 : subst,X185 : term,X188 : subst] : ((((X177 @ X188 @ X185 @ (comp @ X186 @ X187))) = $true) & (((X177 @ (comp @ X188 @ X186) @ (sub @ X185 @ X186) @ X187)) != $true)) | ? [X179 : term,X178 : term] : ((((X177 @ id @ X179 @ id)) = $true) & (((X177 @ id @ (ap @ (sub @ X178 @ id) @ X179) @ id)) != $true) & (((X177 @ id @ X178 @ id)) = $true)) | (((X177 @ id @ X176 @ id)) = $true) | ? [X189 : term] : ((((var @ (sub @ X189 @ id))) = $true) & (((X177 @ id @ X189 @ id)) != $true)) | ? [X180 : (subst > term > term)] : (! [X181 : term] : (($true != ((X177 @ id @ X181 @ id))) | ($true = ((X177 @ id @ (X180 @ id @ X181) @ id)))) & (((X177 @ id @ (lam @ (X180 @ sh @ one)) @ id)) != $true) & ! [X184 : term,X183 : subst,X182 : subst] : (((sub @ (X180 @ X183 @ X184) @ X182)) = ((X180 @ (comp @ X183 @ X182) @ (sub @ X184 @ X182))))) | ? [X191 : subst,X193 : term,X192 : subst,X190 : subst] : ((((X177 @ X192 @ X193 @ (comp @ X191 @ X190))) != $true) & (((X177 @ (comp @ X192 @ X191) @ (sub @ X193 @ X191) @ X190)) = $true))) | ? [X161 : term,X160 : (subst > term > subst > $o)] : (! [X175 : term] : ((((X160 @ id @ X175 @ id)) != $true) | (((X160 @ id @ (sub @ X161 @ (push @ X175 @ id)) @ id)) = $true)) & ! [X165 : subst,X163 : subst,X164 : subst,X162 : term] : ((((X160 @ X163 @ X162 @ (comp @ X164 @ X165))) = $true) | (((X160 @ (comp @ X163 @ X164) @ (sub @ X162 @ X164) @ X165)) != $true)) & ! [X174 : subst,X171 : subst,X173 : subst,X172 : term] : ((((X160 @ X174 @ X172 @ (comp @ X171 @ X173))) != $true) | ($true = ((X160 @ (comp @ X174 @ X171) @ (sub @ X172 @ X171) @ X173)))) & (((X160 @ id @ (lam @ X161) @ id)) != $true) & ! [X166 : (subst > term > term)] : (? [X170 : term] : ((((X160 @ id @ X170 @ id)) = $true) & (((X160 @ id @ (X166 @ id @ X170) @ id)) != $true)) | ? [X167 : subst,X168 : subst,X169 : term] : (((X166 @ (comp @ X167 @ X168) @ (sub @ X169 @ X168))) != ((sub @ (X166 @ X167 @ X169) @ X168))) | ($true = ((X160 @ id @ (lam @ (X166 @ sh @ one)) @ id))))) | ? [X140 : term,X141 : (term > $o)] : (! [X143 : term] : (? [X144 : term] : (($true != ((X141 @ (sub @ X143 @ (push @ X144 @ id))))) & (((X141 @ X144)) = $true)) | (((X141 @ (lam @ X143))) = $true)) & (((X141 @ X140)) != $true) & ! [X146 : term,X145 : term] : ((((X141 @ X145)) != $true) | ($true = ((X141 @ (ap @ X146 @ X145)))) | (((X141 @ X146)) != $true)) & ! [X142 : term] : (($true = ((X141 @ X142))) | (((var @ X142)) != $true))))) & (! [X500 : term] : ? [X501 : (subst > term > term)] : (((sub @ X500 @ (push @ one @ sh))) = ((X501 @ sh @ one))) <=> (hoasinduction_lem3b_lthm = $true)) & (! [X40 : term,X41 : (subst > term > subst > $o)] : (? [X42 : term] : ((((X41 @ id @ X42 @ id)) = $true) & (((X41 @ id @ (sub @ X40 @ (push @ X42 @ id)) @ id)) != $true)) | (((X41 @ id @ (lam @ (sub @ X40 @ (push @ one @ sh))) @ id)) = $true) | ? [X43 : term,X44 : (subst > term > term)] : (! [X47 : subst,X45 : term,X46 : subst] : (((sub @ (X44 @ X47 @ X45) @ X46)) = ((sub @ (sub @ X43 @ (push @ X45 @ X47)) @ X46))) & ! [X49 : subst,X51 : term,X50 : subst] : (((sub @ X43 @ (push @ (sub @ X51 @ X50) @ (comp @ X49 @ X50)))) = ((X44 @ (comp @ X49 @ X50) @ (sub @ X51 @ X50)))) & ! [X48 : term] : ((((X41 @ id @ X48 @ id)) != $true) | (((X41 @ id @ (X44 @ id @ X48) @ id)) = $true)) & (((X41 @ id @ (lam @ (X44 @ sh @ one)) @ id)) != $true))) <=> (hoasinduction_lem3aaa = $true)) & ((axidr = $true) <=> ! [X272 : subst] : (((comp @ X272 @ id)) = X272)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (induction2_gthm = $true) & (hoaslamnotvar_gthm = $true) & (pushprop = $true) & (! [X521 : term,X520 : (term > $o),X519 : subst] : ? [X522 : (term > $o)] : ! [X523 : term] : (((X520 @ (sub @ X523 @ (push @ X521 @ X519)))) = ((X522 @ X523))) <=> (pushprop_lem0 = $true)) & (apnotvar = $true) & (hoasinduction_lem1v2 = $true) & (! [X37 : (subst > term > subst > $o)] : ? [X38 : (term > $o)] : ! [X39 : term] : (((X37 @ id @ X39 @ id)) = ((X38 @ X39))) <=> (hoasinduction_lem0_lthm = $true)) & ((hoasinduction_lem0 = $true) <=> ! [X504 : (subst > term > subst > $o)] : ? [X505 : (term > $o)] : ! [X506 : term] : (((X504 @ id @ X506 @ id)) = ((X505 @ X506)))) & ((! [X427 : (term > $o),X428 : term,X429 : subst] : (? [X433 : term] : (! [X434 : term] : ((((X427 @ (sub @ X433 @ (push @ X434 @ id)))) = $true) | (((X427 @ X434)) != $true)) & (((X427 @ (lam @ X433))) != $true)) | ? [X430 : term] : ((((X427 @ (sub @ X430 @ X429))) != $true) & (((var @ X430)) = $true)) | ? [X432 : term,X431 : term] : ((((X427 @ X431)) = $true) & ($true != ((X427 @ (ap @ X432 @ X431)))) & (((X427 @ X432)) = $true)) | (((X427 @ (sub @ X428 @ X429))) = $true)) | ? [X422 : (term > $o),X421 : term] : (($true != ((X422 @ X421))) & ! [X426 : term] : (($true != ((var @ X426))) | (((X422 @ X426)) = $true)) & ! [X423 : term] : ((((X422 @ (lam @ X423))) = $true) | (((X422 @ X423)) != $true)) & ! [X425 : term,X424 : term] : (($true != ((X422 @ X424))) | (((X422 @ (ap @ X424 @ X425))) = $true) | (((X422 @ X425)) != $true))) | ? [X417 : subst,X416 : term] : (((sub @ one @ (push @ X416 @ X417))) != X416) | ? [X418 : subst,X419 : term] : (((comp @ sh @ (push @ X419 @ X418))) != X418) | ? [X420 : subst] : (((comp @ X420 @ id)) != X420)) <=> (induction2lem_lthm = $true)) & (pushprop_lem0_gthm = $true) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (! [X247 : term,X248 : (subst > term > subst > $o)] : (? [X249 : (subst > term > term)] : (! [X250 : term] : ((((X248 @ id @ X250 @ id)) != $true) | ($true = ((X248 @ id @ (X249 @ id @ X250) @ id)))) & (((X248 @ id @ (lam @ (X249 @ sh @ one)) @ id)) != $true) & ! [X253 : subst,X251 : term,X252 : subst] : (((X249 @ (comp @ X253 @ X252) @ (sub @ X251 @ X252))) = ((sub @ (X249 @ X253 @ X251) @ X252)))) | (((X248 @ id @ X247 @ id)) = $true) | ? [X254 : term,X255 : term] : ((((X248 @ id @ X255 @ id)) = $true) & (((X248 @ id @ X254 @ id)) = $true) & (((X248 @ id @ (ap @ (sub @ X254 @ id) @ X255) @ id)) != $true))) | ? [X236 : term] : (((sub @ X236 @ id)) != X236) | ? [X239 : (term > $o),X237 : (subst > term > subst > $o),X238 : term] : (! [X240 : (subst > term > term)] : (? [X244 : term] : (($true = ((X237 @ id @ X244 @ id))) & (((X237 @ id @ (X240 @ id @ X244) @ id)) != $true)) | ($true = ((X237 @ id @ (lam @ (X240 @ sh @ one)) @ id))) | ? [X241 : subst,X242 : subst,X243 : term] : (((X240 @ (comp @ X242 @ X241) @ (sub @ X243 @ X241))) != ((sub @ (X240 @ X242 @ X243) @ X241)))) & (((X239 @ (lam @ X238))) != $true) & ! [X246 : term] : (((X237 @ id @ X246 @ id)) = ((X239 @ X246))) & ! [X245 : term] : (($true = ((X239 @ (sub @ X238 @ (push @ X245 @ id))))) | (((X239 @ X245)) != $true))) | ? [X230 : (term > $o),X229 : term] : (! [X231 : term] : (($true != ((var @ X231))) | ($true = ((X230 @ X231)))) & ! [X234 : term] : (? [X235 : term] : ((((X230 @ X235)) = $true) & (((X230 @ (sub @ X234 @ (push @ X235 @ id)))) != $true)) | (((X230 @ (lam @ X234))) = $true)) & ! [X232 : term,X233 : term] : ((((X230 @ X233)) != $true) | (((X230 @ (ap @ X232 @ X233))) = $true) | ($true != ((X230 @ X232)))) & ($true != ((X230 @ X229)))) | ? [X226 : (subst > term > subst > $o)] : ! [X227 : (term > $o)] : ? [X228 : term] : (((X227 @ X228)) != ((X226 @ id @ X228 @ id))))) & (hoaslamnotap_gthm = $true) & ((? [X60 : (term > $o),X59 : term] : ((((X60 @ X59)) != $true) & ! [X65 : term] : ((((X60 @ X65)) = $true) | (((var @ X65)) != $true)) & ! [X63 : term,X64 : term] : ((((X60 @ X63)) != $true) | (((X60 @ (ap @ X64 @ X63))) = $true) | (((X60 @ X64)) != $true)) & ! [X61 : term] : (? [X62 : term] : ((((X60 @ X62)) = $true) & (((X60 @ (sub @ X61 @ (push @ X62 @ id)))) != $true)) | ($true = ((X60 @ (lam @ X61)))))) | ? [X66 : term] : (((sub @ X66 @ id)) != X66) | ! [X77 : term,X78 : (subst > term > subst > $o)] : (? [X93 : subst,X94 : subst,X91 : subst,X92 : term] : (($true != ((X78 @ X94 @ X92 @ (comp @ X91 @ X93)))) & (((X78 @ (comp @ X94 @ X91) @ (sub @ X92 @ X91) @ X93)) = $true)) | (((X78 @ id @ X77 @ id)) = $true) | ? [X80 : (subst > term > term)] : (! [X84 : term] : ((((X78 @ id @ X84 @ id)) != $true) | (((X78 @ id @ (X80 @ id @ X84) @ id)) = $true)) & ! [X81 : subst,X82 : term,X83 : subst] : (((sub @ (X80 @ X81 @ X82) @ X83)) = ((X80 @ (comp @ X81 @ X83) @ (sub @ X82 @ X83)))) & ($true != ((X78 @ id @ (lam @ (X80 @ sh @ one)) @ id)))) | ? [X79 : term] : ((((var @ (sub @ X79 @ id))) = $true) & (((X78 @ id @ X79 @ id)) != $true)) | ? [X90 : subst,X88 : term,X89 : subst,X87 : subst] : ((((X78 @ (comp @ X87 @ X90) @ (sub @ X88 @ X90) @ X89)) != $true) & ($true = ((X78 @ X87 @ X88 @ (comp @ X90 @ X89))))) | ? [X86 : term,X85 : term] : ((((X78 @ id @ X85 @ id)) = $true) & (((X78 @ id @ (ap @ (sub @ X86 @ id) @ X85) @ id)) != $true) & (((X78 @ id @ X86 @ id)) = $true))) | ? [X56 : (subst > term > subst > $o)] : ! [X57 : (term > $o)] : ? [X58 : term] : (((X56 @ id @ X58 @ id)) != ((X57 @ X58))) | ? [X67 : (term > $o),X69 : (subst > term > subst > $o),X68 : term] : (! [X76 : term] : (((X69 @ id @ X76 @ id)) = ((X67 @ X76))) & ! [X70 : (subst > term > term)] : ((((X69 @ id @ (lam @ (X70 @ sh @ one)) @ id)) = $true) | ? [X74 : term] : ((((X69 @ id @ (X70 @ id @ X74) @ id)) != $true) & ($true = ((X69 @ id @ X74 @ id)))) | ? [X73 : term,X72 : subst,X71 : subst] : (((X70 @ (comp @ X71 @ X72) @ (sub @ X73 @ X72))) != ((sub @ (X70 @ X71 @ X73) @ X72)))) & ! [X75 : term] : ((((X67 @ (sub @ X68 @ (push @ X75 @ id)))) = $true) | (((X67 @ X75)) != $true)) & ($true != ((X67 @ (lam @ X68)))))) <=> (hoasinduction_lthm_3 = $true)) & (pushprop_gthm = $true) & (hoasinduction_lem3b_gthm = $true) & (substmonoid_gthm = $true) & ! [X538 : term] : ? [X539 : d_term] : (((d2term @ X539)) = X538) & ((? [X484 : term] : (((sub @ X484 @ id)) != X484) | ? [X487 : subst,X486 : term,X485 : (term > $o)] : (! [X491 : term,X492 : term] : ((((X485 @ (ap @ X491 @ X492))) = $true) | (((X485 @ X491)) != $true) | (((X485 @ X492)) != $true)) & ! [X489 : term] : (? [X490 : term] : (($true = ((X485 @ X490))) & (((X485 @ (sub @ X489 @ (push @ X490 @ id)))) != $true)) | ($true = ((X485 @ (lam @ X489))))) & ! [X488 : term] : ((((var @ X488)) != $true) | (((X485 @ (sub @ X488 @ X487))) = $true)) & (((X485 @ (sub @ X486 @ X487))) != $true)) | ! [X494 : (term > $o),X493 : term] : (? [X495 : term] : ((((var @ X495)) = $true) & (((X494 @ X495)) != $true)) | ? [X498 : term] : (! [X499 : term] : ((((X494 @ X499)) != $true) | (((X494 @ (sub @ X498 @ (push @ X499 @ id)))) = $true)) & (((X494 @ (lam @ X498))) != $true)) | ? [X497 : term,X496 : term] : ((((X494 @ X497)) = $true) & (((X494 @ (ap @ X496 @ X497))) != $true) & (((X494 @ X496)) = $true)) | ($true = ((X494 @ X493))))) <=> (induction2_lthm = $true)) & (! [X407 : term,X408 : (subst > term > subst > $o)] : (? [X415 : term,X414 : term] : (($true = ((X408 @ id @ X415 @ id))) & (((X408 @ id @ (ap @ (sub @ X414 @ id) @ X415) @ id)) != $true) & (((X408 @ id @ X414 @ id)) = $true)) | ? [X409 : (subst > term > term)] : (! [X410 : term] : ((((X408 @ id @ (X409 @ id @ X410) @ id)) = $true) | (((X408 @ id @ X410 @ id)) != $true)) & ($true != ((X408 @ id @ (lam @ (X409 @ sh @ one)) @ id))) & ! [X412 : term,X413 : subst,X411 : subst] : (((X409 @ (comp @ X411 @ X413) @ (sub @ X412 @ X413))) = ((sub @ (X409 @ X411 @ X412) @ X413)))) | (((X408 @ id @ X407 @ id)) = $true)) <=> (hoasinduction_no_psi_cond = $true)) & ((axidl = $true) <=> ! [X406 : subst] : (((comp @ id @ X406)) = X406)) & (hoasapnotvar_gthm = $true) & (((! [X26 : subst] : (((comp @ X26 @ id)) = X26) & ! [X25 : subst] : (((comp @ id @ X25)) = X25)) | ? [X23 : subst] : (((comp @ id @ X23)) != X23) | ? [X24 : subst] : (((comp @ X24 @ id)) != X24)) <=> (substmonoid_lthm = $true)) & ((hoasinduction_lem3 = $true) <=> ! [X469 : (subst > term > subst > $o),X468 : term] : (? [X477 : subst,X476 : subst,X478 : subst,X479 : term] : ((((X469 @ X476 @ X479 @ (comp @ X478 @ X477))) = $true) & (((X469 @ (comp @ X476 @ X478) @ (sub @ X479 @ X478) @ X477)) != $true)) | ? [X480 : term,X483 : subst,X482 : subst,X481 : subst] : ((((X469 @ X481 @ X480 @ (comp @ X483 @ X482))) != $true) & ($true = ((X469 @ (comp @ X481 @ X483) @ (sub @ X480 @ X483) @ X482)))) | ? [X470 : term] : (($true = ((X469 @ id @ X470 @ id))) & ($true != ((X469 @ id @ (sub @ X468 @ (push @ X470 @ id)) @ id)))) | (((X469 @ id @ (lam @ X468) @ id)) = $true) | ? [X471 : (subst > term > term)] : (! [X475 : term] : ((((X469 @ id @ X475 @ id)) != $true) | (((X469 @ id @ (X471 @ id @ X475) @ id)) = $true)) & (((X469 @ id @ (lam @ (X471 @ sh @ one)) @ id)) != $true) & ! [X473 : term,X472 : subst,X474 : subst] : (((X471 @ (comp @ X474 @ X472) @ (sub @ X473 @ X472))) = ((sub @ (X471 @ X474 @ X473) @ X472)))))) & ((axvarcons = $true) <=> ! [X404 : term,X405 : subst] : (((sub @ one @ (push @ X404 @ X405))) = X404)) & ((hoasinduction_lem3a_lthm = $true) <=> (! [X397 : term,X396 : (subst > term > subst > $o)] : (? [X398 : term] : ((((X396 @ id @ (sub @ X397 @ (push @ X398 @ id)) @ id)) != $true) & (((X396 @ id @ X398 @ id)) = $true)) | (((X396 @ id @ (lam @ X397) @ id)) = $true) | ? [X399 : (subst > term > term)] : (! [X400 : term] : ((((X396 @ id @ (X399 @ id @ X400) @ id)) = $true) | (((X396 @ id @ X400 @ id)) != $true)) & ! [X402 : term,X403 : subst,X401 : subst] : (((X399 @ (comp @ X401 @ X403) @ (sub @ X402 @ X403))) = ((sub @ (X399 @ X401 @ X402) @ X403))) & ($true != ((X396 @ id @ (lam @ (X399 @ sh @ one)) @ id))))) | ? [X387 : term] : (((sub @ X387 @ id)) != X387) | ? [X389 : (subst > term > subst > $o),X388 : term] : (! [X390 : term] : (($true != ((X389 @ id @ X390 @ id))) | (((X389 @ id @ (sub @ X388 @ (push @ X390 @ id)) @ id)) = $true)) & ! [X391 : (subst > term > term)] : (? [X392 : term] : ((((X389 @ id @ (X391 @ id @ X392) @ id)) != $true) & (((X389 @ id @ X392 @ id)) = $true)) | ? [X395 : term,X393 : subst,X394 : subst] : (((sub @ (X391 @ X393 @ X395) @ X394)) != ((X391 @ (comp @ X393 @ X394) @ (sub @ X395 @ X394)))) | (((X389 @ id @ (lam @ (X391 @ sh @ one)) @ id)) = $true)) & (((X389 @ id @ (lam @ (sub @ X388 @ (push @ one @ sh))) @ id)) != $true)))) & (hoaslamnotap_lthm = $true) & ((! [X334 : term,X332 : (subst > term > subst > $o),X333 : (term > $o)] : (? [X346 : subst,X347 : term,X348 : subst,X345 : subst] : (($true != ((X332 @ (comp @ X348 @ X346) @ (sub @ X347 @ X346) @ X345))) & ($true = ((X332 @ X348 @ X347 @ (comp @ X346 @ X345))))) | (((X333 @ (lam @ X334))) = $true) | ? [X335 : term,X338 : subst,X336 : subst,X337 : subst] : (($true = ((X332 @ (comp @ X338 @ X336) @ (sub @ X335 @ X336) @ X337))) & (((X332 @ X338 @ X335 @ (comp @ X336 @ X337))) != $true)) | ? [X349 : term] : ((((X333 @ (sub @ X334 @ (push @ X349 @ id)))) != $true) & (((X333 @ X349)) = $true)) | ? [X339 : term] : (((X333 @ X339)) != ((X332 @ id @ X339 @ id))) | ? [X340 : (subst > term > term)] : (! [X344 : term] : (($true != ((X332 @ id @ X344 @ id))) | (((X332 @ id @ (X340 @ id @ X344) @ id)) = $true)) & ! [X343 : subst,X341 : term,X342 : subst] : (((X340 @ (comp @ X342 @ X343) @ (sub @ X341 @ X343))) = ((sub @ (X340 @ X342 @ X341) @ X343))) & (((X332 @ id @ (lam @ (X340 @ sh @ one)) @ id)) != $true))) | ? [X331 : term] : (((sub @ X331 @ id)) != X331)) <=> (hoasinduction_lem3v2_lthm = $true)) & (axclos = $true) & ! [X365 : subst] : ? [X366 : d_subst] : (((d2subst @ X366)) = X365) & (hoaslamnotvar = $true) & (hoasinduction_lem2_lthm = $true) & ! [X225 : d_subst] : (d_id = X225) & ((hoasapinj2 = $true) <=> ! [X274 : term,X273 : term,X276 : term,X275 : term] : ((((ap @ (sub @ X274 @ id) @ X276)) != ((ap @ (sub @ X275 @ id) @ X273))) | (X273 = X276))) & (((var @ (d2term @ d_one))) != $true) & (hoasinduction_lem1_gthm = $true) & ((pushprop_lem1_lthm = $true) <=> (? [X525 : subst,X524 : term] : (((sub @ one @ (push @ X524 @ X525))) != X524) | ! [X532 : (term > $o),X531 : term,X529 : (term > $o),X528 : term,X530 : subst] : ((((X532 @ (sub @ X531 @ (push @ X528 @ X530)))) = $true) | ($true != ((X529 @ X531)))) | ? [X526 : subst,X527 : term] : (((comp @ sh @ (push @ X527 @ X526))) != X526))) & ((laminj = $true) <=> ! [X21 : term,X22 : term] : ((((lam @ X22)) != ((lam @ X21))) | (X21 = X22))) & (lamnotap != $true) & (! [X321 : (subst > term > term),X324 : subst,X323 : term,X322 : (subst > term > term)] : ((((lam @ (X321 @ sh @ one))) != ((lam @ (X322 @ sh @ one)))) | ? [X325 : subst,X326 : term,X327 : subst] : (((sub @ (X321 @ X327 @ X326) @ X325)) != ((X321 @ (comp @ X327 @ X325) @ (sub @ X326 @ X325)))) | ? [X328 : subst,X329 : subst,X330 : term] : (((sub @ (X322 @ X328 @ X330) @ X329)) != ((X322 @ (comp @ X328 @ X329) @ (sub @ X330 @ X329)))) | (((X322 @ X324 @ X323)) = ((X321 @ X324 @ X323)))) <=> (hoaslaminj = $true)) & ((hoasinduction_lem3a = $true) <=> ! [X351 : (subst > term > subst > $o),X350 : term] : ((((X351 @ id @ (lam @ X350) @ id)) = $true) | ? [X353 : (subst > term > term)] : (! [X354 : term] : (($true != ((X351 @ id @ X354 @ id))) | (((X351 @ id @ (X353 @ id @ X354) @ id)) = $true)) & (((X351 @ id @ (lam @ (X353 @ sh @ one)) @ id)) != $true) & ! [X355 : subst,X357 : term,X356 : subst] : (((X353 @ (comp @ X355 @ X356) @ (sub @ X357 @ X356))) = ((sub @ (X353 @ X355 @ X357) @ X356)))) | ? [X352 : term] : (($true != ((X351 @ id @ (sub @ X350 @ (push @ X352 @ id)) @ id))) & (((X351 @ id @ X352 @ id)) = $true)))) & ((axvarid = $true) <=> ! [X27 : term] : (((sub @ X27 @ id)) = X27)) & (! [X508 : (subst > term > subst > $o),X507 : term] : (? [X510 : (subst > term > term)] : ((((X508 @ id @ (lam @ (X510 @ sh @ one)) @ id)) != $true) & ! [X512 : term,X514 : subst,X513 : subst] : (((X510 @ (comp @ X514 @ X513) @ (sub @ X512 @ X513))) = ((sub @ (X510 @ X514 @ X512) @ X513))) & ! [X511 : term] : ((((X508 @ id @ X511 @ id)) != $true) | (((X508 @ id @ (X510 @ id @ X511) @ id)) = $true))) | ($true = ((X508 @ id @ (lam @ (sub @ X507 @ (push @ one @ sh))) @ id))) | ? [X509 : term] : (($true = ((X508 @ id @ X509 @ id))) & (((X508 @ id @ (sub @ X507 @ (push @ X509 @ id)) @ id)) != $true))) <=> (hoasinduction_lem3aa = $true)) & (! [X515 : term] : ? [X516 : (subst > term > term)] : (((X516 @ sh @ one)) = ((sub @ X515 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b = $true)) & (pushprop_lem1_gthm = $true) & (axvarshift = $true) & (((d2term @ d_one)) = one) & (pushprop_lem3v2_lthm = $true) & (hoasinduction_lem1v2_gthm = $true) & (hoasinduction_lem2_gthm = $true) & ((pushprop_lem1v2_lthm = $true) <=> (! [X196 : (term > $o),X199 : term,X197 : subst,X198 : (term > $o)] : ((((X198 @ X199)) != $true) | ($true = ((X196 @ one))) | ? [X200 : term] : (((X196 @ X200)) != ((X198 @ (sub @ X200 @ (push @ X199 @ X197)))))) | ? [X195 : term,X194 : subst] : (((sub @ one @ (push @ X195 @ X194))) != X195))) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (! [X96 : (term > $o),X95 : term,X97 : subst] : ? [X98 : (term > $o)] : ! [X99 : term] : (((X96 @ (sub @ X99 @ (push @ X95 @ X97)))) = ((X98 @ X99))) <=> (pushprop_lem0_lthm = $true)) & ((! [X278 : subst] : (((comp @ X278 @ id)) = X278) & ! [X277 : subst] : (((comp @ id @ X277)) = X277)) <=> (substmonoid = $true)) & (hoasinduction_lem1_lthm = $true) & (termmset_gthm = $true) & (pushprop_lem2v2_lthm = $true) & (pushprop_lem2v2 = $true) & ((induction = $true) <=> ! [X382 : term,X381 : (term > $o)] : (? [X386 : term,X385 : term] : (($true != ((X381 @ (ap @ X385 @ X386)))) & ($true = ((X381 @ X386))) & (((X381 @ X385)) = $true)) | ($true = ((X381 @ X382))) | ? [X384 : term] : (($true = ((X381 @ X384))) & ($true != ((X381 @ (lam @ X384))))) | ? [X383 : term] : ((((var @ X383)) = $true) & (((X381 @ X383)) != $true)))) & (hoaslaminj_gthm = $true) & ((hoasapinj1 = $true) <=> ! [X302 : term,X299 : term,X301 : term,X300 : term] : ((X299 = X302) | (((ap @ (sub @ X302 @ id) @ X301)) != ((ap @ (sub @ X299 @ id) @ X300))))) & ((hoasinduction_lem3v2a_lthm = $true) <=> (! [X132 : (subst > term > subst > $o),X131 : term,X130 : (term > $o)] : (? [X133 : (subst > term > term)] : (! [X134 : subst,X136 : term,X135 : subst] : (((sub @ (X133 @ X135 @ X136) @ X134)) = ((X133 @ (comp @ X135 @ X134) @ (sub @ X136 @ X134)))) & (((X132 @ id @ (lam @ (X133 @ sh @ one)) @ id)) != $true) & ! [X137 : term] : ((((X132 @ id @ (X133 @ id @ X137) @ id)) = $true) | (((X132 @ id @ X137 @ id)) != $true))) | ($true = ((X130 @ (lam @ X131)))) | ? [X139 : term] : ((((X130 @ X139)) = $true) & (((X130 @ (sub @ X131 @ (push @ X139 @ id)))) != $true)) | ? [X138 : term] : (((X132 @ id @ X138 @ id)) != ((X130 @ X138)))) | ? [X125 : term] : ! [X126 : (subst > term > term)] : ? [X127 : term,X128 : subst] : (((sub @ X125 @ (push @ X127 @ X128))) != ((X126 @ X128 @ X127))) | ? [X129 : term] : (((sub @ X129 @ id)) != X129))) & ((pushprop_lem1 = $true) <=> ! [X216 : (term > $o),X217 : term,X218 : (term > $o),X219 : subst,X220 : term] : ((((X216 @ X217)) != $true) | (((X218 @ (sub @ X217 @ (push @ X220 @ X219)))) = $true))) & (! [X379 : term,X377 : term,X380 : term,X378 : term] : ((X377 = X378) | (((ap @ X378 @ X379)) != ((ap @ X377 @ X380)))) <=> (apinj1 = $true)) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (pushprop_lem3v2 = $true) & ! [X20 : d_subst,X19 : d_subst] : ((((d2subst @ X19)) != ((d2subst @ X20))) | (X19 = X20)) & (ulamvar1 != $true) & (pushprop_lem1v2_gthm = $true) & (hoasapnotvar_lthm = $true) & (hoasinduction_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & ((? [X462 : term,X463 : term,X460 : term,X461 : term] : ((((ap @ X463 @ X460)) = ((ap @ X461 @ X462))) & (X460 != X462)) | ! [X466 : term,X464 : term,X465 : term,X467 : term] : ((((ap @ (sub @ X464 @ id) @ X467)) != ((ap @ (sub @ X466 @ id) @ X465))) | (X465 = X467))) <=> (hoasapinj2_lthm = $true)) & ((hoasinduction_lem2 = $true) <=> ! [X101 : term,X102 : term,X100 : (subst > term > subst > $o)] : ((((X100 @ id @ X101 @ id)) != $true) | (((X100 @ id @ X102 @ id)) != $true) | ? [X105 : subst,X104 : subst,X106 : term,X103 : subst] : (($true = ((X100 @ (comp @ X103 @ X105) @ (sub @ X106 @ X105) @ X104))) & (((X100 @ X103 @ X106 @ (comp @ X105 @ X104))) != $true)) | (((X100 @ id @ (ap @ X101 @ X102) @ id)) = $true) | ? [X112 : subst,X111 : subst,X109 : subst,X110 : term] : ((((X100 @ X111 @ X110 @ (comp @ X112 @ X109))) = $true) & (((X100 @ (comp @ X111 @ X112) @ (sub @ X110 @ X112) @ X109)) != $true)) | ? [X108 : term,X107 : term] : ((((X100 @ id @ (ap @ (sub @ X107 @ id) @ X108) @ id)) != $true) & ($true = ((X100 @ id @ X107 @ id))) & ($true = ((X100 @ id @ X108 @ id)))))) & ((? [X259 : subst,X258 : term] : (((comp @ sh @ (push @ X258 @ X259))) != X259) | ? [X261 : term,X260 : term] : ((X260 != X261) & (((lam @ X260)) = ((lam @ X261)))) | ! [X263 : (subst > term > term),X264 : subst,X262 : (subst > term > term),X265 : term] : (? [X266 : subst,X267 : subst,X268 : term] : (((X263 @ (comp @ X266 @ X267) @ (sub @ X268 @ X267))) != ((sub @ (X263 @ X266 @ X268) @ X267))) | (((X262 @ X264 @ X265)) = ((X263 @ X264 @ X265))) | (((lam @ (X263 @ sh @ one))) != ((lam @ (X262 @ sh @ one)))) | ? [X269 : subst,X270 : term,X271 : subst] : (((X262 @ (comp @ X269 @ X271) @ (sub @ X270 @ X271))) != ((sub @ (X262 @ X269 @ X270) @ X271)))) | ? [X256 : subst,X257 : term] : (((sub @ one @ (push @ X257 @ X256))) != X257)) <=> (hoaslaminj_lthm = $true)) & (hoasapnotvar = $true) & (ulamvarsh = $true) & (hoasinduction_lem3v2_gthm = $true) & ((induction2lem = $true) <=> ! [X113 : subst,X114 : term,X115 : (term > $o)] : (? [X116 : term] : ((((X115 @ (sub @ X116 @ X113))) != $true) & (((var @ X116)) = $true)) | ? [X117 : term] : (! [X118 : term] : (($true = ((X115 @ (sub @ X117 @ (push @ X118 @ id))))) | (((X115 @ X118)) != $true)) & ($true != ((X115 @ (lam @ X117))))) | ? [X120 : term,X119 : term] : ((((X115 @ (ap @ X119 @ X120))) != $true) & (((X115 @ X119)) = $true) & ($true = ((X115 @ X120)))) | (((X115 @ (sub @ X114 @ X113))) = $true))) & (! [X298 : subst] : (((push @ (sub @ one @ X298) @ (comp @ sh @ X298))) = X298) <=> (axscons = $true)) & (pushprop_lem2v2_gthm = $true) & (hoaslamnotvar_lthm = $true) & (lamnotvar = $true) & ((termmset = $true) <=> ! [X291 : term] : (((sub @ X291 @ id)) = X291)) & ((hoasinduction_lem3v2a = $true) <=> ! [X368 : (term > $o),X367 : (subst > term > subst > $o),X369 : term] : ((((X368 @ (lam @ X369))) = $true) | ? [X371 : term] : (((X367 @ id @ X371 @ id)) != ((X368 @ X371))) | ? [X370 : term] : (($true != ((X368 @ (sub @ X369 @ (push @ X370 @ id))))) & (((X368 @ X370)) = $true)) | ? [X372 : (subst > term > term)] : (! [X375 : term,X376 : subst,X374 : subst] : (((sub @ (X372 @ X376 @ X375) @ X374)) = ((X372 @ (comp @ X376 @ X374) @ (sub @ X375 @ X374)))) & ! [X373 : term] : ((((X367 @ id @ (X372 @ id @ X373) @ id)) = $true) | (((X367 @ id @ X373 @ id)) != $true)) & (((X367 @ id @ (lam @ (X372 @ sh @ one)) @ id)) != $true)))) & ((hoasinduction_lem3_lthm = $true) <=> (? [X437 : (subst > term > subst > $o),X436 : term] : (! [X443 : term] : ((((X437 @ id @ X443 @ id)) != $true) | (((X437 @ id @ (sub @ X436 @ (push @ X443 @ id)) @ id)) = $true)) & ($true != ((X437 @ id @ (lam @ (sub @ X436 @ (push @ one @ sh))) @ id))) & ! [X438 : (subst > term > term)] : (? [X439 : term] : ((((X437 @ id @ (X438 @ id @ X439) @ id)) != $true) & ($true = ((X437 @ id @ X439 @ id)))) | (((X437 @ id @ (lam @ (X438 @ sh @ one)) @ id)) = $true) | ? [X442 : subst,X441 : subst,X440 : term] : (((X438 @ (comp @ X441 @ X442) @ (sub @ X440 @ X442))) != ((sub @ (X438 @ X441 @ X440) @ X442))))) | ! [X445 : (subst > term > subst > $o),X444 : term] : (? [X455 : term] : ((((X445 @ id @ (sub @ X444 @ (push @ X455 @ id)) @ id)) != $true) & (((X445 @ id @ X455 @ id)) = $true)) | ? [X446 : term,X447 : subst,X449 : subst,X448 : subst] : (($true != ((X445 @ (comp @ X447 @ X449) @ (sub @ X446 @ X449) @ X448))) & (((X445 @ X447 @ X446 @ (comp @ X449 @ X448))) = $true)) | ? [X450 : (subst > term > term)] : (! [X453 : subst,X454 : subst,X452 : term] : (((sub @ (X450 @ X454 @ X452) @ X453)) = ((X450 @ (comp @ X454 @ X453) @ (sub @ X452 @ X453)))) & ! [X451 : term] : (($true != ((X445 @ id @ X451 @ id))) | (((X445 @ id @ (X450 @ id @ X451) @ id)) = $true)) & (((X445 @ id @ (lam @ (X450 @ sh @ one)) @ id)) != $true)) | (((X445 @ id @ (lam @ X444) @ id)) = $true) | ? [X457 : term,X459 : subst,X456 : subst,X458 : subst] : ((((X445 @ X458 @ X457 @ (comp @ X459 @ X456))) != $true) & (((X445 @ (comp @ X458 @ X459) @ (sub @ X457 @ X459) @ X456)) = $true))) | ? [X435 : term] : (((sub @ X435 @ id)) != X435))) & (pushprop_lthm = $true) & (((d2subst @ d_id)) = sh) & (! [X294 : (subst > term > term),X293 : term,X292 : term] : (? [X297 : term,X295 : subst,X296 : subst] : (((sub @ (X294 @ X296 @ X297) @ X295)) != ((X294 @ (comp @ X296 @ X295) @ (sub @ X297 @ X295)))) | (((lam @ (X294 @ sh @ one))) != ((ap @ (sub @ X292 @ id) @ X293)))) <=> (hoaslamnotap = $true)) & (((d2subst @ d_id)) = id) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X284 : term,X283 : (subst > term > subst > $o)] : ((((X283 @ id @ (lam @ (sub @ X284 @ (push @ one @ sh))) @ id)) = $true) | ? [X290 : term] : (($true != ((X283 @ id @ (sub @ X284 @ (push @ X290 @ id)) @ id))) & ($true = ((X283 @ id @ X290 @ id)))) | ? [X285 : (subst > term > term)] : (! [X286 : term] : ((((X283 @ id @ (X285 @ id @ X286) @ id)) = $true) | (((X283 @ id @ X286 @ id)) != $true)) & ! [X288 : subst,X289 : term,X287 : subst] : (((X285 @ (comp @ X288 @ X287) @ (sub @ X289 @ X287))) = ((sub @ (X285 @ X288 @ X289) @ X287))) & (((X283 @ id @ (lam @ (X285 @ sh @ one)) @ id)) != $true)))) & (! [X279 : term,X282 : term,X281 : term,X280 : term] : ((((ap @ X280 @ X282)) != ((ap @ X279 @ X281))) | (X281 = X282)) <=> (apinj2 = $true)) & ((hoasapinj1_lthm = $true) <=> (? [X31 : term,X32 : term,X30 : term,X29 : term] : ((((ap @ X31 @ X29)) = ((ap @ X32 @ X30))) & (X31 != X32)) | ! [X36 : term,X33 : term,X34 : term,X35 : term] : ((X33 = X36) | (((ap @ (sub @ X33 @ id) @ X35)) != ((ap @ (sub @ X36 @ id) @ X34)))) | ? [X28 : term] : (((sub @ X28 @ id)) != X28))) & (! [X121 : term] : ? [X122 : (subst > term > term)] : ! [X123 : term,X124 : subst] : (((X122 @ X124 @ X123)) = ((sub @ X121 @ (push @ X123 @ X124)))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & (hoasinduction_lem3a_gthm = $true) & (hoasinduction_lem1 = $true) & (axmap = $true) & ((! [X518 : term] : (((sub @ X518 @ id)) = X518) | ? [X517 : term] : (((sub @ X517 @ id)) != X517)) <=> (termmset_lthm = $true)) & (! [X358 : (term > $o),X359 : term] : (($true = ((X358 @ X359))) | ? [X360 : term] : ((((X358 @ X360)) != $true) & (((var @ X360)) = $true)) | ? [X364 : term,X363 : term] : ((((X358 @ (ap @ X363 @ X364))) != $true) & (((X358 @ X364)) = $true) & ($true = ((X358 @ X363)))) | ? [X361 : term] : ((((X358 @ (lam @ X361))) != $true) & ! [X362 : term] : ((((X358 @ X362)) != $true) | (((X358 @ (sub @ X361 @ (push @ X362 @ id)))) = $true)))) <=> (induction2 = $true)) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (induction2lem_gthm = $true) & (hoasapinj2_gthm = $true) & (hoasinduction_lem3_gthm = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (hoasapinj1_gthm = $true) & (pushprop_lthm_orig = $true) & (axassoc = $true) & (! [X223 : term,X224 : subst] : (((comp @ sh @ (push @ X223 @ X224))) = X224) <=> (axshiftcons = $true)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2))))))))),
  inference(flattening,[],[f11])).
thf(f13,definition,(
  ! [X358 : (term > $o)] : ((((sP0 @ X358)) = $true) <=> ? [X364 : term,X363 : term] : ((((X358 @ (ap @ X363 @ X364))) != $true) & (((X358 @ X364)) = $true) & ($true = ((X358 @ X363)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f14,definition,(
  (sP1 = $true) <=> ! [X358 : (term > $o),X359 : term] : (($true = ((X358 @ X359))) | ? [X360 : term] : ((((X358 @ X360)) != $true) & (((var @ X360)) = $true)) | (((sP0 @ X358)) = $true) | ? [X361 : term] : ((((X358 @ (lam @ X361))) != $true) & ! [X362 : term] : ((((X358 @ X362)) != $true) | (((X358 @ (sub @ X361 @ (push @ X362 @ id)))) = $true))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f15,definition,(
  (sP2 = $true) <=> ! [X284 : term,X283 : (subst > term > subst > $o)] : ((((X283 @ id @ (lam @ (sub @ X284 @ (push @ one @ sh))) @ id)) = $true) | ? [X290 : term] : (($true != ((X283 @ id @ (sub @ X284 @ (push @ X290 @ id)) @ id))) & ($true = ((X283 @ id @ X290 @ id)))) | ? [X285 : (subst > term > term)] : (! [X286 : term] : ((((X283 @ id @ (X285 @ id @ X286) @ id)) = $true) | (((X283 @ id @ X286 @ id)) != $true)) & ! [X288 : subst,X289 : term,X287 : subst] : (((X285 @ (comp @ X288 @ X287) @ (sub @ X289 @ X287))) = ((sub @ (X285 @ X288 @ X289) @ X287))) & (((X283 @ id @ (lam @ (X285 @ sh @ one)) @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f16,definition,(
  ! [X445 : (subst > term > subst > $o)] : (($true = ((sP3 @ X445))) <=> ? [X450 : (subst > term > term)] : (! [X453 : subst,X454 : subst,X452 : term] : (((sub @ (X450 @ X454 @ X452) @ X453)) = ((X450 @ (comp @ X454 @ X453) @ (sub @ X452 @ X453)))) & ! [X451 : term] : (($true != ((X445 @ id @ X451 @ id))) | (((X445 @ id @ (X450 @ id @ X451) @ id)) = $true)) & (((X445 @ id @ (lam @ (X450 @ sh @ one)) @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f17,definition,(
  ($true = sP4) <=> ? [X437 : (subst > term > subst > $o),X436 : term] : (! [X443 : term] : ((((X437 @ id @ X443 @ id)) != $true) | (((X437 @ id @ (sub @ X436 @ (push @ X443 @ id)) @ id)) = $true)) & ($true != ((X437 @ id @ (lam @ (sub @ X436 @ (push @ one @ sh))) @ id))) & ! [X438 : (subst > term > term)] : (? [X439 : term] : ((((X437 @ id @ (X438 @ id @ X439) @ id)) != $true) & ($true = ((X437 @ id @ X439 @ id)))) | (((X437 @ id @ (lam @ (X438 @ sh @ one)) @ id)) = $true) | ? [X442 : subst,X441 : subst,X440 : term] : (((X438 @ (comp @ X441 @ X442) @ (sub @ X440 @ X442))) != ((sub @ (X438 @ X441 @ X440) @ X442)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f18,definition,(
  (sP5 = $true) <=> (($true = sP4) | ! [X445 : (subst > term > subst > $o),X444 : term] : (? [X455 : term] : ((((X445 @ id @ (sub @ X444 @ (push @ X455 @ id)) @ id)) != $true) & (((X445 @ id @ X455 @ id)) = $true)) | ? [X446 : term,X447 : subst,X449 : subst,X448 : subst] : (($true != ((X445 @ (comp @ X447 @ X449) @ (sub @ X446 @ X449) @ X448))) & (((X445 @ X447 @ X446 @ (comp @ X449 @ X448))) = $true)) | ($true = ((sP3 @ X445))) | (((X445 @ id @ (lam @ X444) @ id)) = $true) | ? [X457 : term,X459 : subst,X456 : subst,X458 : subst] : ((((X445 @ X458 @ X457 @ (comp @ X459 @ X456))) != $true) & (((X445 @ (comp @ X458 @ X459) @ (sub @ X457 @ X459) @ X456)) = $true))) | ? [X435 : term] : (((sub @ X435 @ id)) != X435))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f19,definition,(
  (sP6 = $true) <=> ! [X368 : (term > $o),X367 : (subst > term > subst > $o),X369 : term] : ((((X368 @ (lam @ X369))) = $true) | ? [X371 : term] : (((X367 @ id @ X371 @ id)) != ((X368 @ X371))) | ? [X370 : term] : (($true != ((X368 @ (sub @ X369 @ (push @ X370 @ id))))) & (((X368 @ X370)) = $true)) | ? [X372 : (subst > term > term)] : (! [X375 : term,X376 : subst,X374 : subst] : (((sub @ (X372 @ X376 @ X375) @ X374)) = ((X372 @ (comp @ X376 @ X374) @ (sub @ X375 @ X374)))) & ! [X373 : term] : ((((X367 @ id @ (X372 @ id @ X373) @ id)) = $true) | (((X367 @ id @ X373 @ id)) != $true)) & (((X367 @ id @ (lam @ (X372 @ sh @ one)) @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f20,definition,(
  ! [X115 : (term > $o)] : ((((sP7 @ X115)) = $true) <=> ? [X120 : term,X119 : term] : ((((X115 @ (ap @ X119 @ X120))) != $true) & (((X115 @ X119)) = $true) & ($true = ((X115 @ X120)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f21,definition,(
  (sP8 = $true) <=> ! [X113 : subst,X114 : term,X115 : (term > $o)] : (? [X116 : term] : ((((X115 @ (sub @ X116 @ X113))) != $true) & (((var @ X116)) = $true)) | ? [X117 : term] : (! [X118 : term] : (($true = ((X115 @ (sub @ X117 @ (push @ X118 @ id))))) | (((X115 @ X118)) != $true)) & ($true != ((X115 @ (lam @ X117))))) | (((sP7 @ X115)) = $true) | (((X115 @ (sub @ X114 @ X113))) = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f22,definition,(
  (sP9 = $true) <=> (? [X259 : subst,X258 : term] : (((comp @ sh @ (push @ X258 @ X259))) != X259) | ? [X261 : term,X260 : term] : ((X260 != X261) & (((lam @ X260)) = ((lam @ X261)))) | ! [X263 : (subst > term > term),X264 : subst,X262 : (subst > term > term),X265 : term] : (? [X266 : subst,X267 : subst,X268 : term] : (((X263 @ (comp @ X266 @ X267) @ (sub @ X268 @ X267))) != ((sub @ (X263 @ X266 @ X268) @ X267))) | (((X262 @ X264 @ X265)) = ((X263 @ X264 @ X265))) | (((lam @ (X263 @ sh @ one))) != ((lam @ (X262 @ sh @ one)))) | ? [X269 : subst,X270 : term,X271 : subst] : (((X262 @ (comp @ X269 @ X271) @ (sub @ X270 @ X271))) != ((sub @ (X262 @ X269 @ X270) @ X271)))) | ? [X256 : subst,X257 : term] : (((sub @ one @ (push @ X257 @ X256))) != X257))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f23,definition,(
  ! [X100 : (subst > term > subst > $o)] : ((((sP10 @ X100)) = $true) <=> ? [X108 : term,X107 : term] : ((((X100 @ id @ (ap @ (sub @ X107 @ id) @ X108) @ id)) != $true) & ($true = ((X100 @ id @ X107 @ id))) & ($true = ((X100 @ id @ X108 @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f24,definition,(
  (sP11 = $true) <=> ! [X101 : term,X102 : term,X100 : (subst > term > subst > $o)] : ((((X100 @ id @ X101 @ id)) != $true) | (((X100 @ id @ X102 @ id)) != $true) | ? [X105 : subst,X104 : subst,X106 : term,X103 : subst] : (($true = ((X100 @ (comp @ X103 @ X105) @ (sub @ X106 @ X105) @ X104))) & (((X100 @ X103 @ X106 @ (comp @ X105 @ X104))) != $true)) | (((X100 @ id @ (ap @ X101 @ X102) @ id)) = $true) | ? [X112 : subst,X111 : subst,X109 : subst,X110 : term] : ((((X100 @ X111 @ X110 @ (comp @ X112 @ X109))) = $true) & (((X100 @ (comp @ X111 @ X112) @ (sub @ X110 @ X112) @ X109)) != $true)) | (((sP10 @ X100)) = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f25,definition,(
  (sP12 = $true) <=> (! [X132 : (subst > term > subst > $o),X131 : term,X130 : (term > $o)] : (? [X133 : (subst > term > term)] : (! [X134 : subst,X136 : term,X135 : subst] : (((sub @ (X133 @ X135 @ X136) @ X134)) = ((X133 @ (comp @ X135 @ X134) @ (sub @ X136 @ X134)))) & (((X132 @ id @ (lam @ (X133 @ sh @ one)) @ id)) != $true) & ! [X137 : term] : ((((X132 @ id @ (X133 @ id @ X137) @ id)) = $true) | (((X132 @ id @ X137 @ id)) != $true))) | ($true = ((X130 @ (lam @ X131)))) | ? [X139 : term] : ((((X130 @ X139)) = $true) & (((X130 @ (sub @ X131 @ (push @ X139 @ id)))) != $true)) | ? [X138 : term] : (((X132 @ id @ X138 @ id)) != ((X130 @ X138)))) | ? [X125 : term] : ! [X126 : (subst > term > term)] : ? [X127 : term,X128 : subst] : (((sub @ X125 @ (push @ X127 @ X128))) != ((X126 @ X128 @ X127))) | ? [X129 : term] : (((sub @ X129 @ id)) != X129))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f26,definition,(
  ! [X381 : (term > $o)] : (($true = ((sP13 @ X381))) <=> ? [X386 : term,X385 : term] : (($true != ((X381 @ (ap @ X385 @ X386)))) & ($true = ((X381 @ X386))) & (((X381 @ X385)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f27,definition,(
  (sP14 = $true) <=> ! [X508 : (subst > term > subst > $o),X507 : term] : (? [X510 : (subst > term > term)] : ((((X508 @ id @ (lam @ (X510 @ sh @ one)) @ id)) != $true) & ! [X512 : term,X514 : subst,X513 : subst] : (((X510 @ (comp @ X514 @ X513) @ (sub @ X512 @ X513))) = ((sub @ (X510 @ X514 @ X512) @ X513))) & ! [X511 : term] : ((((X508 @ id @ X511 @ id)) != $true) | (((X508 @ id @ (X510 @ id @ X511) @ id)) = $true))) | ($true = ((X508 @ id @ (lam @ (sub @ X507 @ (push @ one @ sh))) @ id))) | ? [X509 : term] : (($true = ((X508 @ id @ X509 @ id))) & (((X508 @ id @ (sub @ X507 @ (push @ X509 @ id)) @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f28,definition,(
  (sP15 = $true) <=> ! [X351 : (subst > term > subst > $o),X350 : term] : ((((X351 @ id @ (lam @ X350) @ id)) = $true) | ? [X353 : (subst > term > term)] : (! [X354 : term] : (($true != ((X351 @ id @ X354 @ id))) | (((X351 @ id @ (X353 @ id @ X354) @ id)) = $true)) & (((X351 @ id @ (lam @ (X353 @ sh @ one)) @ id)) != $true) & ! [X355 : subst,X357 : term,X356 : subst] : (((X353 @ (comp @ X355 @ X356) @ (sub @ X357 @ X356))) = ((sub @ (X353 @ X355 @ X357) @ X356)))) | ? [X352 : term] : (($true != ((X351 @ id @ (sub @ X350 @ (push @ X352 @ id)) @ id))) & (((X351 @ id @ X352 @ id)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f29,definition,(
  ! [X332 : (subst > term > subst > $o)] : ((((sP16 @ X332)) = $true) <=> ? [X340 : (subst > term > term)] : (! [X344 : term] : (($true != ((X332 @ id @ X344 @ id))) | (((X332 @ id @ (X340 @ id @ X344) @ id)) = $true)) & ! [X343 : subst,X341 : term,X342 : subst] : (((X340 @ (comp @ X342 @ X343) @ (sub @ X341 @ X343))) = ((sub @ (X340 @ X342 @ X341) @ X343))) & (((X332 @ id @ (lam @ (X340 @ sh @ one)) @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f30,definition,(
  (sP17 = $true) <=> (! [X334 : term,X332 : (subst > term > subst > $o),X333 : (term > $o)] : (? [X346 : subst,X347 : term,X348 : subst,X345 : subst] : (($true != ((X332 @ (comp @ X348 @ X346) @ (sub @ X347 @ X346) @ X345))) & ($true = ((X332 @ X348 @ X347 @ (comp @ X346 @ X345))))) | (((X333 @ (lam @ X334))) = $true) | ? [X335 : term,X338 : subst,X336 : subst,X337 : subst] : (($true = ((X332 @ (comp @ X338 @ X336) @ (sub @ X335 @ X336) @ X337))) & (((X332 @ X338 @ X335 @ (comp @ X336 @ X337))) != $true)) | ? [X349 : term] : ((((X333 @ (sub @ X334 @ (push @ X349 @ id)))) != $true) & (((X333 @ X349)) = $true)) | ? [X339 : term] : (((X333 @ X339)) != ((X332 @ id @ X339 @ id))) | (((sP16 @ X332)) = $true)) | ? [X331 : term] : (((sub @ X331 @ id)) != X331))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f31,definition,(
  ($true = sP18) <=> ? [X389 : (subst > term > subst > $o),X388 : term] : (! [X390 : term] : (($true != ((X389 @ id @ X390 @ id))) | (((X389 @ id @ (sub @ X388 @ (push @ X390 @ id)) @ id)) = $true)) & ! [X391 : (subst > term > term)] : (? [X392 : term] : ((((X389 @ id @ (X391 @ id @ X392) @ id)) != $true) & (((X389 @ id @ X392 @ id)) = $true)) | ? [X395 : term,X393 : subst,X394 : subst] : (((sub @ (X391 @ X393 @ X395) @ X394)) != ((X391 @ (comp @ X393 @ X394) @ (sub @ X395 @ X394)))) | (((X389 @ id @ (lam @ (X391 @ sh @ one)) @ id)) = $true)) & (((X389 @ id @ (lam @ (sub @ X388 @ (push @ one @ sh))) @ id)) != $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f32,definition,(
  ($true = sP19) <=> (! [X397 : term,X396 : (subst > term > subst > $o)] : (? [X398 : term] : ((((X396 @ id @ (sub @ X397 @ (push @ X398 @ id)) @ id)) != $true) & (((X396 @ id @ X398 @ id)) = $true)) | (((X396 @ id @ (lam @ X397) @ id)) = $true) | ? [X399 : (subst > term > term)] : (! [X400 : term] : ((((X396 @ id @ (X399 @ id @ X400) @ id)) = $true) | (((X396 @ id @ X400 @ id)) != $true)) & ! [X402 : term,X403 : subst,X401 : subst] : (((X399 @ (comp @ X401 @ X403) @ (sub @ X402 @ X403))) = ((sub @ (X399 @ X401 @ X402) @ X403))) & ($true != ((X396 @ id @ (lam @ (X399 @ sh @ one)) @ id))))) | ? [X387 : term] : (((sub @ X387 @ id)) != X387) | ($true = sP18))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f33,definition,(
  ! [X469 : (subst > term > subst > $o)] : ((((sP20 @ X469)) = $true) <=> ? [X471 : (subst > term > term)] : (! [X475 : term] : ((((X469 @ id @ X475 @ id)) != $true) | (((X469 @ id @ (X471 @ id @ X475) @ id)) = $true)) & (((X469 @ id @ (lam @ (X471 @ sh @ one)) @ id)) != $true) & ! [X473 : term,X472 : subst,X474 : subst] : (((X471 @ (comp @ X474 @ X472) @ (sub @ X473 @ X472))) = ((sub @ (X471 @ X474 @ X473) @ X472)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f34,definition,(
  ($true = sP21) <=> ! [X469 : (subst > term > subst > $o),X468 : term] : (? [X477 : subst,X476 : subst,X478 : subst,X479 : term] : ((((X469 @ X476 @ X479 @ (comp @ X478 @ X477))) = $true) & (((X469 @ (comp @ X476 @ X478) @ (sub @ X479 @ X478) @ X477)) != $true)) | ? [X480 : term,X483 : subst,X482 : subst,X481 : subst] : ((((X469 @ X481 @ X480 @ (comp @ X483 @ X482))) != $true) & ($true = ((X469 @ (comp @ X481 @ X483) @ (sub @ X480 @ X483) @ X482)))) | ? [X470 : term] : (($true = ((X469 @ id @ X470 @ id))) & ($true != ((X469 @ id @ (sub @ X468 @ (push @ X470 @ id)) @ id)))) | (((X469 @ id @ (lam @ X468) @ id)) = $true) | (((sP20 @ X469)) = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f35,definition,(
  ! [X408 : (subst > term > subst > $o)] : ((((sP22 @ X408)) = $true) <=> ? [X409 : (subst > term > term)] : (! [X410 : term] : ((((X408 @ id @ (X409 @ id @ X410) @ id)) = $true) | (((X408 @ id @ X410 @ id)) != $true)) & ($true != ((X408 @ id @ (lam @ (X409 @ sh @ one)) @ id))) & ! [X412 : term,X413 : subst,X411 : subst] : (((X409 @ (comp @ X411 @ X413) @ (sub @ X412 @ X413))) = ((sub @ (X409 @ X411 @ X412) @ X413)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f36,definition,(
  ! [X494 : (term > $o)] : ((((sP23 @ X494)) = $true) <=> ? [X497 : term,X496 : term] : ((((X494 @ X497)) = $true) & (((X494 @ (ap @ X496 @ X497))) != $true) & (((X494 @ X496)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f37,definition,(
  ! [X485 : (term > $o)] : (($true = ((sP24 @ X485))) <=> ! [X491 : term,X492 : term] : ((((X485 @ (ap @ X491 @ X492))) = $true) | (((X485 @ X491)) != $true) | (((X485 @ X492)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f38,definition,(
  ($true = sP25) <=> ! [X494 : (term > $o),X493 : term] : (? [X495 : term] : ((((var @ X495)) = $true) & (((X494 @ X495)) != $true)) | ? [X498 : term] : (! [X499 : term] : ((((X494 @ X499)) != $true) | (((X494 @ (sub @ X498 @ (push @ X499 @ id)))) = $true)) & (((X494 @ (lam @ X498))) != $true)) | (((sP23 @ X494)) = $true) | ($true = ((X494 @ X493))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f39,definition,(
  (sP26 = $true) <=> (? [X484 : term] : (((sub @ X484 @ id)) != X484) | ? [X487 : subst,X486 : term,X485 : (term > $o)] : (($true = ((sP24 @ X485))) & ! [X489 : term] : (? [X490 : term] : (($true = ((X485 @ X490))) & (((X485 @ (sub @ X489 @ (push @ X490 @ id)))) != $true)) | ($true = ((X485 @ (lam @ X489))))) & ! [X488 : term] : ((((var @ X488)) != $true) | (((X485 @ (sub @ X488 @ X487))) = $true)) & (((X485 @ (sub @ X486 @ X487))) != $true)) | ($true = sP25))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f40,definition,(
  ! [X78 : (subst > term > subst > $o)] : (($true = ((sP27 @ X78))) <=> ? [X86 : term,X85 : term] : ((((X78 @ id @ X85 @ id)) = $true) & (((X78 @ id @ (ap @ (sub @ X86 @ id) @ X85) @ id)) != $true) & (((X78 @ id @ X86 @ id)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f41,definition,(
  ! [X78 : (subst > term > subst > $o)] : ((((sP28 @ X78)) = $true) <=> ? [X80 : (subst > term > term)] : (! [X84 : term] : ((((X78 @ id @ X84 @ id)) != $true) | (((X78 @ id @ (X80 @ id @ X84) @ id)) = $true)) & ! [X81 : subst,X82 : term,X83 : subst] : (((sub @ (X80 @ X81 @ X82) @ X83)) = ((X80 @ (comp @ X81 @ X83) @ (sub @ X82 @ X83)))) & ($true != ((X78 @ id @ (lam @ (X80 @ sh @ one)) @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f42,definition,(
  ! [X60 : (term > $o)] : ((((sP29 @ X60)) = $true) <=> ! [X63 : term,X64 : term] : ((((X60 @ X63)) != $true) | (((X60 @ (ap @ X64 @ X63))) = $true) | (((X60 @ X64)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f43,definition,(
  ($true = sP30) <=> ? [X67 : (term > $o),X69 : (subst > term > subst > $o),X68 : term] : (! [X76 : term] : (((X69 @ id @ X76 @ id)) = ((X67 @ X76))) & ! [X70 : (subst > term > term)] : ((((X69 @ id @ (lam @ (X70 @ sh @ one)) @ id)) = $true) | ? [X74 : term] : ((((X69 @ id @ (X70 @ id @ X74) @ id)) != $true) & ($true = ((X69 @ id @ X74 @ id)))) | ? [X73 : term,X72 : subst,X71 : subst] : (((X70 @ (comp @ X71 @ X72) @ (sub @ X73 @ X72))) != ((sub @ (X70 @ X71 @ X73) @ X72)))) & ! [X75 : term] : ((((X67 @ (sub @ X68 @ (push @ X75 @ id)))) = $true) | (((X67 @ X75)) != $true)) & ($true != ((X67 @ (lam @ X68)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f44,definition,(
  ($true = sP31) <=> ! [X77 : term,X78 : (subst > term > subst > $o)] : (? [X93 : subst,X94 : subst,X91 : subst,X92 : term] : (($true != ((X78 @ X94 @ X92 @ (comp @ X91 @ X93)))) & (((X78 @ (comp @ X94 @ X91) @ (sub @ X92 @ X91) @ X93)) = $true)) | (((X78 @ id @ X77 @ id)) = $true) | (((sP28 @ X78)) = $true) | ? [X79 : term] : ((((var @ (sub @ X79 @ id))) = $true) & (((X78 @ id @ X79 @ id)) != $true)) | ? [X90 : subst,X88 : term,X89 : subst,X87 : subst] : ((((X78 @ (comp @ X87 @ X90) @ (sub @ X88 @ X90) @ X89)) != $true) & ($true = ((X78 @ X87 @ X88 @ (comp @ X90 @ X89))))) | ($true = ((sP27 @ X78))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f45,definition,(
  (sP32 = $true) <=> (? [X60 : (term > $o),X59 : term] : ((((X60 @ X59)) != $true) & ! [X65 : term] : ((((X60 @ X65)) = $true) | (((var @ X65)) != $true)) & (((sP29 @ X60)) = $true) & ! [X61 : term] : (? [X62 : term] : ((((X60 @ X62)) = $true) & (((X60 @ (sub @ X61 @ (push @ X62 @ id)))) != $true)) | ($true = ((X60 @ (lam @ X61)))))) | ? [X66 : term] : (((sub @ X66 @ id)) != X66) | ($true = sP31) | ? [X56 : (subst > term > subst > $o)] : ! [X57 : (term > $o)] : ? [X58 : term] : (((X56 @ id @ X58 @ id)) != ((X57 @ X58))) | ($true = sP30))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f46,definition,(
  ! [X230 : (term > $o)] : ((((sP33 @ X230)) = $true) <=> ! [X232 : term,X233 : term] : ((((X230 @ X233)) != $true) | (((X230 @ (ap @ X232 @ X233))) = $true) | ($true != ((X230 @ X232)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f47,definition,(
  ! [X248 : (subst > term > subst > $o)] : ((((sP34 @ X248)) = $true) <=> ? [X254 : term,X255 : term] : ((((X248 @ id @ X255 @ id)) = $true) & (((X248 @ id @ X254 @ id)) = $true) & (((X248 @ id @ (ap @ (sub @ X254 @ id) @ X255) @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f48,definition,(
  (sP35 = $true) <=> ? [X239 : (term > $o),X237 : (subst > term > subst > $o),X238 : term] : (! [X240 : (subst > term > term)] : (? [X244 : term] : (($true = ((X237 @ id @ X244 @ id))) & (((X237 @ id @ (X240 @ id @ X244) @ id)) != $true)) | ($true = ((X237 @ id @ (lam @ (X240 @ sh @ one)) @ id))) | ? [X241 : subst,X242 : subst,X243 : term] : (((X240 @ (comp @ X242 @ X241) @ (sub @ X243 @ X241))) != ((sub @ (X240 @ X242 @ X243) @ X241)))) & (((X239 @ (lam @ X238))) != $true) & ! [X246 : term] : (((X237 @ id @ X246 @ id)) = ((X239 @ X246))) & ! [X245 : term] : (($true = ((X239 @ (sub @ X238 @ (push @ X245 @ id))))) | (((X239 @ X245)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f49,definition,(
  (sP36 = $true) <=> ? [X230 : (term > $o),X229 : term] : (! [X231 : term] : (($true != ((var @ X231))) | ($true = ((X230 @ X231)))) & ! [X234 : term] : (? [X235 : term] : ((((X230 @ X235)) = $true) & (((X230 @ (sub @ X234 @ (push @ X235 @ id)))) != $true)) | (((X230 @ (lam @ X234))) = $true)) & (((sP33 @ X230)) = $true) & ($true != ((X230 @ X229))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f50,definition,(
  ($true = sP37) <=> (! [X247 : term,X248 : (subst > term > subst > $o)] : (? [X249 : (subst > term > term)] : (! [X250 : term] : ((((X248 @ id @ X250 @ id)) != $true) | ($true = ((X248 @ id @ (X249 @ id @ X250) @ id)))) & (((X248 @ id @ (lam @ (X249 @ sh @ one)) @ id)) != $true) & ! [X253 : subst,X251 : term,X252 : subst] : (((X249 @ (comp @ X253 @ X252) @ (sub @ X251 @ X252))) = ((sub @ (X249 @ X253 @ X251) @ X252)))) | (((X248 @ id @ X247 @ id)) = $true) | (((sP34 @ X248)) = $true)) | ? [X236 : term] : (((sub @ X236 @ id)) != X236) | (sP35 = $true) | (sP36 = $true) | ? [X226 : (subst > term > subst > $o)] : ! [X227 : (term > $o)] : ? [X228 : term] : (((X227 @ X228)) != ((X226 @ id @ X228 @ id))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f51,definition,(
  ! [X422 : (term > $o)] : (($true = ((sP38 @ X422))) <=> ! [X425 : term,X424 : term] : (($true != ((X422 @ X424))) | (((X422 @ (ap @ X424 @ X425))) = $true) | (((X422 @ X425)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f52,definition,(
  ! [X427 : (term > $o)] : ((((sP39 @ X427)) = $true) <=> ? [X432 : term,X431 : term] : ((((X427 @ X431)) = $true) & ($true != ((X427 @ (ap @ X432 @ X431)))) & (((X427 @ X432)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f53,definition,(
  (sP40 = $true) <=> ! [X427 : (term > $o),X428 : term,X429 : subst] : (? [X433 : term] : (! [X434 : term] : ((((X427 @ (sub @ X433 @ (push @ X434 @ id)))) = $true) | (((X427 @ X434)) != $true)) & (((X427 @ (lam @ X433))) != $true)) | ? [X430 : term] : ((((X427 @ (sub @ X430 @ X429))) != $true) & (((var @ X430)) = $true)) | (((sP39 @ X427)) = $true) | (((X427 @ (sub @ X428 @ X429))) = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f54,definition,(
  ($true = sP41) <=> ((sP40 = $true) | ? [X422 : (term > $o),X421 : term] : (($true != ((X422 @ X421))) & ! [X426 : term] : (($true != ((var @ X426))) | (((X422 @ X426)) = $true)) & ! [X423 : term] : ((((X422 @ (lam @ X423))) = $true) | (((X422 @ X423)) != $true)) & ($true = ((sP38 @ X422)))) | ? [X417 : subst,X416 : term] : (((sub @ one @ (push @ X416 @ X417))) != X416) | ? [X418 : subst,X419 : term] : (((comp @ sh @ (push @ X419 @ X418))) != X418) | ? [X420 : subst] : (((comp @ X420 @ id)) != X420))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f55,definition,(
  (sP42 = $true) <=> ! [X40 : term,X41 : (subst > term > subst > $o)] : (? [X42 : term] : ((((X41 @ id @ X42 @ id)) = $true) & (((X41 @ id @ (sub @ X40 @ (push @ X42 @ id)) @ id)) != $true)) | (((X41 @ id @ (lam @ (sub @ X40 @ (push @ one @ sh))) @ id)) = $true) | ? [X43 : term,X44 : (subst > term > term)] : (! [X47 : subst,X45 : term,X46 : subst] : (((sub @ (X44 @ X47 @ X45) @ X46)) = ((sub @ (sub @ X43 @ (push @ X45 @ X47)) @ X46))) & ! [X49 : subst,X51 : term,X50 : subst] : (((sub @ X43 @ (push @ (sub @ X51 @ X50) @ (comp @ X49 @ X50)))) = ((X44 @ (comp @ X49 @ X50) @ (sub @ X51 @ X50)))) & ! [X48 : term] : ((((X41 @ id @ X48 @ id)) != $true) | (((X41 @ id @ (X44 @ id @ X48) @ id)) = $true)) & (((X41 @ id @ (lam @ (X44 @ sh @ one)) @ id)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f56,definition,(
  ! [X141 : (term > $o)] : ((((sP43 @ X141)) = $true) <=> ! [X146 : term,X145 : term] : ((((X141 @ X145)) != $true) | ($true = ((X141 @ (ap @ X146 @ X145)))) | (((X141 @ X146)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f57,definition,(
  ! [X160 : (subst > term > subst > $o)] : ((((sP44 @ X160)) = $true) <=> ! [X166 : (subst > term > term)] : (? [X170 : term] : ((((X160 @ id @ X170 @ id)) = $true) & (((X160 @ id @ (X166 @ id @ X170) @ id)) != $true)) | ? [X167 : subst,X168 : subst,X169 : term] : (((X166 @ (comp @ X167 @ X168) @ (sub @ X169 @ X168))) != ((sub @ (X166 @ X167 @ X169) @ X168))) | ($true = ((X160 @ id @ (lam @ (X166 @ sh @ one)) @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f58,definition,(
  ! [X177 : (subst > term > subst > $o)] : (($true = ((sP45 @ X177))) <=> ? [X180 : (subst > term > term)] : (! [X181 : term] : (($true != ((X177 @ id @ X181 @ id))) | ($true = ((X177 @ id @ (X180 @ id @ X181) @ id)))) & (((X177 @ id @ (lam @ (X180 @ sh @ one)) @ id)) != $true) & ! [X184 : term,X183 : subst,X182 : subst] : (((sub @ (X180 @ X183 @ X184) @ X182)) = ((X180 @ (comp @ X183 @ X182) @ (sub @ X184 @ X182))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f59,definition,(
  ! [X177 : (subst > term > subst > $o)] : ((((sP46 @ X177)) = $true) <=> ? [X179 : term,X178 : term] : ((((X177 @ id @ X179 @ id)) = $true) & (((X177 @ id @ (ap @ (sub @ X178 @ id) @ X179) @ id)) != $true) & (((X177 @ id @ X178 @ id)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f60,definition,(
  ! [X148 : (subst > term > subst > $o)] : ((((sP47 @ X148)) = $true) <=> ! [X158 : term,X159 : term] : ((((X148 @ id @ X158 @ id)) != $true) | (((X148 @ id @ X159 @ id)) != $true) | (((X148 @ id @ (ap @ (sub @ X159 @ id) @ X158) @ id)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f61,definition,(
  (sP48 = $true) <=> ? [X161 : term,X160 : (subst > term > subst > $o)] : (! [X175 : term] : ((((X160 @ id @ X175 @ id)) != $true) | (((X160 @ id @ (sub @ X161 @ (push @ X175 @ id)) @ id)) = $true)) & ! [X165 : subst,X163 : subst,X164 : subst,X162 : term] : ((((X160 @ X163 @ X162 @ (comp @ X164 @ X165))) = $true) | (((X160 @ (comp @ X163 @ X164) @ (sub @ X162 @ X164) @ X165)) != $true)) & ! [X174 : subst,X171 : subst,X173 : subst,X172 : term] : ((((X160 @ X174 @ X172 @ (comp @ X171 @ X173))) != $true) | ($true = ((X160 @ (comp @ X174 @ X171) @ (sub @ X172 @ X171) @ X173)))) & (((X160 @ id @ (lam @ X161) @ id)) != $true) & (((sP44 @ X160)) = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f62,definition,(
  (sP49 = $true) <=> ! [X177 : (subst > term > subst > $o),X176 : term] : (? [X187 : subst,X186 : subst,X185 : term,X188 : subst] : ((((X177 @ X188 @ X185 @ (comp @ X186 @ X187))) = $true) & (((X177 @ (comp @ X188 @ X186) @ (sub @ X185 @ X186) @ X187)) != $true)) | (((sP46 @ X177)) = $true) | (((X177 @ id @ X176 @ id)) = $true) | ? [X189 : term] : ((((var @ (sub @ X189 @ id))) = $true) & (((X177 @ id @ X189 @ id)) != $true)) | ($true = ((sP45 @ X177))) | ? [X191 : subst,X193 : term,X192 : subst,X190 : subst] : ((((X177 @ X192 @ X193 @ (comp @ X191 @ X190))) != $true) & (((X177 @ (comp @ X192 @ X191) @ (sub @ X193 @ X191) @ X190)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f63,definition,(
  (sP50 = $true) <=> ? [X140 : term,X141 : (term > $o)] : (! [X143 : term] : (? [X144 : term] : (($true != ((X141 @ (sub @ X143 @ (push @ X144 @ id))))) & (((X141 @ X144)) = $true)) | (((X141 @ (lam @ X143))) = $true)) & (((X141 @ X140)) != $true) & (((sP43 @ X141)) = $true) & ! [X142 : term] : (($true = ((X141 @ X142))) | (((var @ X142)) != $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f64,definition,(
  ($true = sP51) <=> (? [X148 : (subst > term > subst > $o),X147 : term,X149 : term] : ((((X148 @ id @ (ap @ X149 @ X147) @ id)) != $true) & ! [X152 : subst,X153 : subst,X151 : term,X150 : subst] : (($true = ((X148 @ (comp @ X152 @ X150) @ (sub @ X151 @ X150) @ X153))) | (((X148 @ X152 @ X151 @ (comp @ X150 @ X153))) != $true)) & ! [X155 : subst,X154 : subst,X157 : subst,X156 : term] : ((((X148 @ (comp @ X155 @ X154) @ (sub @ X156 @ X154) @ X157)) != $true) | (((X148 @ X155 @ X156 @ (comp @ X154 @ X157))) = $true)) & (((sP47 @ X148)) = $true) & (((X148 @ id @ X149 @ id)) = $true) & (((X148 @ id @ X147 @ id)) = $true)) | (sP49 = $true) | (sP48 = $true) | (sP50 = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f65,definition,(
  ! [X203 : (subst > term > subst > $o)] : ((((sP52 @ X203)) = $true) <=> ? [X215 : term,X214 : term] : ((((X203 @ id @ (ap @ (sub @ X214 @ id) @ X215) @ id)) != $true) & (((X203 @ id @ X215 @ id)) = $true) & (((X203 @ id @ X214 @ id)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f66,definition,(
  (sP53 = $true) <=> ! [X203 : (subst > term > subst > $o),X202 : (term > $o),X204 : term,X201 : term] : ((((sP52 @ X203)) = $true) | (((X202 @ (ap @ X204 @ X201))) = $true) | ? [X213 : term] : (((X203 @ id @ X213 @ id)) != ((X202 @ X213))) | (((X202 @ X201)) != $true) | ? [X212 : subst,X210 : subst,X211 : subst,X209 : term] : ((((X203 @ (comp @ X210 @ X211) @ (sub @ X209 @ X211) @ X212)) = $true) & (((X203 @ X210 @ X209 @ (comp @ X211 @ X212))) != $true)) | ? [X207 : subst,X208 : subst,X206 : subst,X205 : term] : (($true != ((X203 @ (comp @ X207 @ X206) @ (sub @ X205 @ X206) @ X208))) & (((X203 @ X207 @ X205 @ (comp @ X206 @ X208))) = $true)) | ($true != ((X202 @ X204))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f67,definition,(
  ! [X1 : (subst > term > subst > $o)] : ((((sP54 @ X1)) = $true) <=> ? [X6 : (subst > term > term)] : (! [X10 : term] : ((((X1 @ id @ (X6 @ id @ X10) @ id)) = $true) | (((X1 @ id @ X10 @ id)) != $true)) & (((X1 @ id @ (lam @ (X6 @ sh @ one)) @ id)) != $true) & ! [X9 : subst,X8 : term,X7 : subst] : (((sub @ (X6 @ X9 @ X8) @ X7)) = ((X6 @ (comp @ X9 @ X7) @ (sub @ X8 @ X7))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f68,definition,(
  ! [X1 : (subst > term > subst > $o)] : ((((sP55 @ X1)) = $true) <=> ? [X17 : term,X16 : term] : ((((X1 @ id @ X16 @ id)) = $true) & (((X1 @ id @ (ap @ (sub @ X16 @ id) @ X17) @ id)) != $true) & ($true = ((X1 @ id @ X17 @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f69,definition,(
  (sP56 = $true) <=> ! [X0 : term,X1 : (subst > term > subst > $o)] : (? [X2 : subst,X3 : subst,X4 : term,X5 : subst] : ((((X1 @ X5 @ X4 @ (comp @ X3 @ X2))) = $true) & (((X1 @ (comp @ X5 @ X3) @ (sub @ X4 @ X3) @ X2)) != $true)) | ? [X13 : term,X12 : subst,X11 : subst,X14 : subst] : (($true != ((X1 @ X14 @ X13 @ (comp @ X12 @ X11)))) & ($true = ((X1 @ (comp @ X14 @ X12) @ (sub @ X13 @ X12) @ X11)))) | (((X1 @ id @ X0 @ id)) = $true) | (((sP55 @ X1)) = $true) | ? [X15 : term] : (($true = ((var @ (sub @ X15 @ id)))) & (((X1 @ id @ X15 @ id)) != $true)) | (((sP54 @ X1)) = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f70,definition,(
  ! [X305 : (subst > term > subst > $o)] : ((((sP57 @ X305)) = $true) <=> ? [X311 : (subst > term > term)] : (! [X314 : term,X312 : subst,X313 : subst] : (((sub @ (X311 @ X312 @ X314) @ X313)) = ((X311 @ (comp @ X312 @ X313) @ (sub @ X314 @ X313)))) & (((X305 @ id @ (lam @ (X311 @ sh @ one)) @ id)) != $true) & ! [X315 : term] : ((((X305 @ id @ (X311 @ id @ X315) @ id)) = $true) | (((X305 @ id @ X315 @ id)) != $true))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f71,definition,(
  (sP58 = $true) <=> ! [X304 : (term > $o),X305 : (subst > term > subst > $o),X303 : term] : (? [X320 : term] : (($true != ((X304 @ (sub @ X303 @ (push @ X320 @ id))))) & (((X304 @ X320)) = $true)) | ? [X318 : term,X319 : subst,X317 : subst,X316 : subst] : ((((X305 @ (comp @ X319 @ X317) @ (sub @ X318 @ X317) @ X316)) = $true) & ($true != ((X305 @ X319 @ X318 @ (comp @ X317 @ X316))))) | (((X304 @ (lam @ X303))) = $true) | ? [X310 : subst,X308 : subst,X309 : subst,X307 : term] : (($true != ((X305 @ (comp @ X310 @ X309) @ (sub @ X307 @ X309) @ X308))) & (((X305 @ X310 @ X307 @ (comp @ X309 @ X308))) = $true)) | ? [X306 : term] : (((X304 @ X306)) != ((X305 @ id @ X306 @ id))) | (((sP57 @ X305)) = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f72,plain,(
  ((hoasinduction_lem3v2 = $true) <=> (sP58 = $true)) & ((hoasinduction = $true) <=> (sP56 = $true)) & (axabs = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((pushprop_lem1v2 = $true) <=> ! [X534 : (term > $o),X536 : term,X533 : (term > $o),X535 : subst] : (? [X537 : term] : (((X533 @ (sub @ X537 @ (push @ X536 @ X535)))) != ((X534 @ X537))) | (((X534 @ one)) = $true) | (((X533 @ X536)) != $true))) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ((shinj = $true) <=> ! [X502 : term,X503 : term] : ((X502 = X503) | (((sub @ X502 @ sh)) != ((sub @ X503 @ sh))))) & ((hoasinduction_lem2v2 = $true) <=> (sP53 = $true)) & ! [X221 : d_term,X222 : d_term] : ((X221 = X222) | (((d2term @ X222)) != ((d2term @ X221)))) & ((hoasinduction_lem3v2_f = $true) <=> ! [X52 : term] : ? [X53 : (subst > term > term)] : ! [X55 : subst,X54 : term] : (((sub @ X52 @ (push @ X54 @ X55))) = ((X53 @ X55 @ X54)))) & ! [X18 : d_term] : (d_one = X18) & (axapp = $true) & (ulamvarind = $true) & (hoasinduction_lem2v2_gthm = $true) & ((hoasinduction_lthm = $true) <=> ($true = sP51)) & (! [X500 : term] : ? [X501 : (subst > term > term)] : (((sub @ X500 @ (push @ one @ sh))) = ((X501 @ sh @ one))) <=> (hoasinduction_lem3b_lthm = $true)) & ((sP42 = $true) <=> (hoasinduction_lem3aaa = $true)) & ((axidr = $true) <=> ! [X272 : subst] : (((comp @ X272 @ id)) = X272)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (induction2_gthm = $true) & (hoaslamnotvar_gthm = $true) & (pushprop = $true) & (! [X521 : term,X520 : (term > $o),X519 : subst] : ? [X522 : (term > $o)] : ! [X523 : term] : (((X520 @ (sub @ X523 @ (push @ X521 @ X519)))) = ((X522 @ X523))) <=> (pushprop_lem0 = $true)) & (apnotvar = $true) & (hoasinduction_lem1v2 = $true) & (! [X37 : (subst > term > subst > $o)] : ? [X38 : (term > $o)] : ! [X39 : term] : (((X37 @ id @ X39 @ id)) = ((X38 @ X39))) <=> (hoasinduction_lem0_lthm = $true)) & ((hoasinduction_lem0 = $true) <=> ! [X504 : (subst > term > subst > $o)] : ? [X505 : (term > $o)] : ! [X506 : term] : (((X504 @ id @ X506 @ id)) = ((X505 @ X506)))) & (($true = sP41) <=> (induction2lem_lthm = $true)) & (pushprop_lem0_gthm = $true) & ((hoasinduction_no_psi_cond_lthm = $true) <=> ($true = sP37)) & (hoaslamnotap_gthm = $true) & ((sP32 = $true) <=> (hoasinduction_lthm_3 = $true)) & (pushprop_gthm = $true) & (hoasinduction_lem3b_gthm = $true) & (substmonoid_gthm = $true) & ! [X538 : term] : ? [X539 : d_term] : (((d2term @ X539)) = X538) & ((sP26 = $true) <=> (induction2_lthm = $true)) & (! [X407 : term,X408 : (subst > term > subst > $o)] : (? [X415 : term,X414 : term] : (($true = ((X408 @ id @ X415 @ id))) & (((X408 @ id @ (ap @ (sub @ X414 @ id) @ X415) @ id)) != $true) & (((X408 @ id @ X414 @ id)) = $true)) | (((sP22 @ X408)) = $true) | (((X408 @ id @ X407 @ id)) = $true)) <=> (hoasinduction_no_psi_cond = $true)) & ((axidl = $true) <=> ! [X406 : subst] : (((comp @ id @ X406)) = X406)) & (hoasapnotvar_gthm = $true) & (((! [X26 : subst] : (((comp @ X26 @ id)) = X26) & ! [X25 : subst] : (((comp @ id @ X25)) = X25)) | ? [X23 : subst] : (((comp @ id @ X23)) != X23) | ? [X24 : subst] : (((comp @ X24 @ id)) != X24)) <=> (substmonoid_lthm = $true)) & ((hoasinduction_lem3 = $true) <=> ($true = sP21)) & ((axvarcons = $true) <=> ! [X404 : term,X405 : subst] : (((sub @ one @ (push @ X404 @ X405))) = X404)) & ((hoasinduction_lem3a_lthm = $true) <=> ($true = sP19)) & (hoaslamnotap_lthm = $true) & ((sP17 = $true) <=> (hoasinduction_lem3v2_lthm = $true)) & (axclos = $true) & ! [X365 : subst] : ? [X366 : d_subst] : (((d2subst @ X366)) = X365) & (hoaslamnotvar = $true) & (hoasinduction_lem2_lthm = $true) & ! [X225 : d_subst] : (d_id = X225) & ((hoasapinj2 = $true) <=> ! [X274 : term,X273 : term,X276 : term,X275 : term] : ((((ap @ (sub @ X274 @ id) @ X276)) != ((ap @ (sub @ X275 @ id) @ X273))) | (X273 = X276))) & (((var @ (d2term @ d_one))) != $true) & (hoasinduction_lem1_gthm = $true) & ((pushprop_lem1_lthm = $true) <=> (? [X525 : subst,X524 : term] : (((sub @ one @ (push @ X524 @ X525))) != X524) | ! [X532 : (term > $o),X531 : term,X529 : (term > $o),X528 : term,X530 : subst] : ((((X532 @ (sub @ X531 @ (push @ X528 @ X530)))) = $true) | ($true != ((X529 @ X531)))) | ? [X526 : subst,X527 : term] : (((comp @ sh @ (push @ X527 @ X526))) != X526))) & ((laminj = $true) <=> ! [X21 : term,X22 : term] : ((((lam @ X22)) != ((lam @ X21))) | (X21 = X22))) & (lamnotap != $true) & (! [X321 : (subst > term > term),X324 : subst,X323 : term,X322 : (subst > term > term)] : ((((lam @ (X321 @ sh @ one))) != ((lam @ (X322 @ sh @ one)))) | ? [X325 : subst,X326 : term,X327 : subst] : (((sub @ (X321 @ X327 @ X326) @ X325)) != ((X321 @ (comp @ X327 @ X325) @ (sub @ X326 @ X325)))) | ? [X328 : subst,X329 : subst,X330 : term] : (((sub @ (X322 @ X328 @ X330) @ X329)) != ((X322 @ (comp @ X328 @ X329) @ (sub @ X330 @ X329)))) | (((X322 @ X324 @ X323)) = ((X321 @ X324 @ X323)))) <=> (hoaslaminj = $true)) & ((hoasinduction_lem3a = $true) <=> (sP15 = $true)) & ((axvarid = $true) <=> ! [X27 : term] : (((sub @ X27 @ id)) = X27)) & ((sP14 = $true) <=> (hoasinduction_lem3aa = $true)) & (! [X515 : term] : ? [X516 : (subst > term > term)] : (((X516 @ sh @ one)) = ((sub @ X515 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b = $true)) & (pushprop_lem1_gthm = $true) & (axvarshift = $true) & (((d2term @ d_one)) = one) & (pushprop_lem3v2_lthm = $true) & (hoasinduction_lem1v2_gthm = $true) & (hoasinduction_lem2_gthm = $true) & ((pushprop_lem1v2_lthm = $true) <=> (! [X196 : (term > $o),X199 : term,X197 : subst,X198 : (term > $o)] : ((((X198 @ X199)) != $true) | ($true = ((X196 @ one))) | ? [X200 : term] : (((X196 @ X200)) != ((X198 @ (sub @ X200 @ (push @ X199 @ X197)))))) | ? [X195 : term,X194 : subst] : (((sub @ one @ (push @ X195 @ X194))) != X195))) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (! [X96 : (term > $o),X95 : term,X97 : subst] : ? [X98 : (term > $o)] : ! [X99 : term] : (((X96 @ (sub @ X99 @ (push @ X95 @ X97)))) = ((X98 @ X99))) <=> (pushprop_lem0_lthm = $true)) & ((! [X278 : subst] : (((comp @ X278 @ id)) = X278) & ! [X277 : subst] : (((comp @ id @ X277)) = X277)) <=> (substmonoid = $true)) & (hoasinduction_lem1_lthm = $true) & (termmset_gthm = $true) & (pushprop_lem2v2_lthm = $true) & (pushprop_lem2v2 = $true) & ((induction = $true) <=> ! [X382 : term,X381 : (term > $o)] : (($true = ((sP13 @ X381))) | ($true = ((X381 @ X382))) | ? [X384 : term] : (($true = ((X381 @ X384))) & ($true != ((X381 @ (lam @ X384))))) | ? [X383 : term] : ((((var @ X383)) = $true) & (((X381 @ X383)) != $true)))) & (hoaslaminj_gthm = $true) & ((hoasapinj1 = $true) <=> ! [X302 : term,X299 : term,X301 : term,X300 : term] : ((X299 = X302) | (((ap @ (sub @ X302 @ id) @ X301)) != ((ap @ (sub @ X299 @ id) @ X300))))) & ((hoasinduction_lem3v2a_lthm = $true) <=> (sP12 = $true)) & ((pushprop_lem1 = $true) <=> ! [X216 : (term > $o),X217 : term,X218 : (term > $o),X219 : subst,X220 : term] : ((((X216 @ X217)) != $true) | (((X218 @ (sub @ X217 @ (push @ X220 @ X219)))) = $true))) & (! [X379 : term,X377 : term,X380 : term,X378 : term] : ((X377 = X378) | (((ap @ X378 @ X379)) != ((ap @ X377 @ X380)))) <=> (apinj1 = $true)) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (pushprop_lem3v2 = $true) & ! [X20 : d_subst,X19 : d_subst] : ((((d2subst @ X19)) != ((d2subst @ X20))) | (X19 = X20)) & (ulamvar1 != $true) & (pushprop_lem1v2_gthm = $true) & (hoasapnotvar_lthm = $true) & (hoasinduction_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & ((? [X462 : term,X463 : term,X460 : term,X461 : term] : ((((ap @ X463 @ X460)) = ((ap @ X461 @ X462))) & (X460 != X462)) | ! [X466 : term,X464 : term,X465 : term,X467 : term] : ((((ap @ (sub @ X464 @ id) @ X467)) != ((ap @ (sub @ X466 @ id) @ X465))) | (X465 = X467))) <=> (hoasapinj2_lthm = $true)) & ((hoasinduction_lem2 = $true) <=> (sP11 = $true)) & ((sP9 = $true) <=> (hoaslaminj_lthm = $true)) & (hoasapnotvar = $true) & (ulamvarsh = $true) & (hoasinduction_lem3v2_gthm = $true) & ((induction2lem = $true) <=> (sP8 = $true)) & (! [X298 : subst] : (((push @ (sub @ one @ X298) @ (comp @ sh @ X298))) = X298) <=> (axscons = $true)) & (pushprop_lem2v2_gthm = $true) & (hoaslamnotvar_lthm = $true) & (lamnotvar = $true) & ((termmset = $true) <=> ! [X291 : term] : (((sub @ X291 @ id)) = X291)) & ((hoasinduction_lem3v2a = $true) <=> (sP6 = $true)) & ((hoasinduction_lem3_lthm = $true) <=> (sP5 = $true)) & (pushprop_lthm = $true) & (((d2subst @ d_id)) = sh) & (! [X294 : (subst > term > term),X293 : term,X292 : term] : (? [X297 : term,X295 : subst,X296 : subst] : (((sub @ (X294 @ X296 @ X297) @ X295)) != ((X294 @ (comp @ X296 @ X295) @ (sub @ X297 @ X295)))) | (((lam @ (X294 @ sh @ one))) != ((ap @ (sub @ X292 @ id) @ X293)))) <=> (hoaslamnotap = $true)) & (((d2subst @ d_id)) = id) & ((hoasinduction_lem3aa_lthm = $true) <=> (sP2 = $true)) & (! [X279 : term,X282 : term,X281 : term,X280 : term] : ((((ap @ X280 @ X282)) != ((ap @ X279 @ X281))) | (X281 = X282)) <=> (apinj2 = $true)) & ((hoasapinj1_lthm = $true) <=> (? [X31 : term,X32 : term,X30 : term,X29 : term] : ((((ap @ X31 @ X29)) = ((ap @ X32 @ X30))) & (X31 != X32)) | ! [X36 : term,X33 : term,X34 : term,X35 : term] : ((X33 = X36) | (((ap @ (sub @ X33 @ id) @ X35)) != ((ap @ (sub @ X36 @ id) @ X34)))) | ? [X28 : term] : (((sub @ X28 @ id)) != X28))) & (! [X121 : term] : ? [X122 : (subst > term > term)] : ! [X123 : term,X124 : subst] : (((X122 @ X124 @ X123)) = ((sub @ X121 @ (push @ X123 @ X124)))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & (hoasinduction_lem3a_gthm = $true) & (hoasinduction_lem1 = $true) & (axmap = $true) & ((! [X518 : term] : (((sub @ X518 @ id)) = X518) | ? [X517 : term] : (((sub @ X517 @ id)) != X517)) <=> (termmset_lthm = $true)) & ((sP1 = $true) <=> (induction2 = $true)) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (induction2lem_gthm = $true) & (hoasapinj2_gthm = $true) & (hoasinduction_lem3_gthm = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (hoasapinj1_gthm = $true) & (pushprop_lthm_orig = $true) & (axassoc = $true) & (! [X223 : term,X224 : subst] : (((comp @ sh @ (push @ X223 @ X224))) = X224) <=> (axshiftcons = $true)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2))))))))),
  inference(definition_folding,[],[f12,f71,f70,f69,f68,f67,f66,f65,f64,f63,f62,f61,f60,f59,f58,f57,f56,f55,f54,f53,f52,f51,f50,f49,f48,f47,f46,f45,f44,f43,f42,f41,f40,f39,f38,f37,f36,f35,f34,f33,f32,f31,f30,f29,f28,f27,f26,f25,f24,f23,f22,f21,f20,f19,f18,f17,f16,f15,f14,f13])).
thf(f260,plain,(
  (((hoasinduction_lem3v2 = $true) | (sP58 != $true)) & ((sP58 = $true) | (hoasinduction_lem3v2 != $true))) & (((hoasinduction = $true) | (sP56 != $true)) & ((sP56 = $true) | (hoasinduction != $true))) & (axabs = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (((pushprop_lem1v2 = $true) | ? [X534 : (term > $o),X536 : term,X533 : (term > $o),X535 : subst] : (! [X537 : term] : (((X533 @ (sub @ X537 @ (push @ X536 @ X535)))) = ((X534 @ X537))) & (((X534 @ one)) != $true) & (((X533 @ X536)) = $true))) & (! [X534 : (term > $o),X536 : term,X533 : (term > $o),X535 : subst] : (? [X537 : term] : (((X533 @ (sub @ X537 @ (push @ X536 @ X535)))) != ((X534 @ X537))) | (((X534 @ one)) = $true) | (((X533 @ X536)) != $true)) | (pushprop_lem1v2 != $true))) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (((shinj = $true) | ? [X502 : term,X503 : term] : ((X502 != X503) & (((sub @ X502 @ sh)) = ((sub @ X503 @ sh))))) & (! [X502 : term,X503 : term] : ((X502 = X503) | (((sub @ X502 @ sh)) != ((sub @ X503 @ sh)))) | (shinj != $true))) & (((hoasinduction_lem2v2 = $true) | (sP53 != $true)) & ((sP53 = $true) | (hoasinduction_lem2v2 != $true))) & ! [X221 : d_term,X222 : d_term] : ((X221 = X222) | (((d2term @ X222)) != ((d2term @ X221)))) & (((hoasinduction_lem3v2_f = $true) | ? [X52 : term] : ! [X53 : (subst > term > term)] : ? [X55 : subst,X54 : term] : (((sub @ X52 @ (push @ X54 @ X55))) != ((X53 @ X55 @ X54)))) & (! [X52 : term] : ? [X53 : (subst > term > term)] : ! [X55 : subst,X54 : term] : (((sub @ X52 @ (push @ X54 @ X55))) = ((X53 @ X55 @ X54))) | (hoasinduction_lem3v2_f != $true))) & ! [X18 : d_term] : (d_one = X18) & (axapp = $true) & (ulamvarind = $true) & (hoasinduction_lem2v2_gthm = $true) & (((hoasinduction_lthm = $true) | ($true != sP51)) & (($true = sP51) | (hoasinduction_lthm != $true))) & ((! [X500 : term] : ? [X501 : (subst > term > term)] : (((sub @ X500 @ (push @ one @ sh))) = ((X501 @ sh @ one))) | (hoasinduction_lem3b_lthm != $true)) & ((hoasinduction_lem3b_lthm = $true) | ? [X500 : term] : ! [X501 : (subst > term > term)] : (((sub @ X500 @ (push @ one @ sh))) != ((X501 @ sh @ one))))) & (((sP42 = $true) | (hoasinduction_lem3aaa != $true)) & ((hoasinduction_lem3aaa = $true) | (sP42 != $true))) & (((axidr = $true) | ? [X272 : subst] : (((comp @ X272 @ id)) != X272)) & (! [X272 : subst] : (((comp @ X272 @ id)) = X272) | (axidr != $true))) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (induction2_gthm = $true) & (hoaslamnotvar_gthm = $true) & (pushprop = $true) & ((! [X521 : term,X520 : (term > $o),X519 : subst] : ? [X522 : (term > $o)] : ! [X523 : term] : (((X520 @ (sub @ X523 @ (push @ X521 @ X519)))) = ((X522 @ X523))) | (pushprop_lem0 != $true)) & ((pushprop_lem0 = $true) | ? [X521 : term,X520 : (term > $o),X519 : subst] : ! [X522 : (term > $o)] : ? [X523 : term] : (((X520 @ (sub @ X523 @ (push @ X521 @ X519)))) != ((X522 @ X523))))) & (apnotvar = $true) & (hoasinduction_lem1v2 = $true) & ((! [X37 : (subst > term > subst > $o)] : ? [X38 : (term > $o)] : ! [X39 : term] : (((X37 @ id @ X39 @ id)) = ((X38 @ X39))) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ? [X37 : (subst > term > subst > $o)] : ! [X38 : (term > $o)] : ? [X39 : term] : (((X37 @ id @ X39 @ id)) != ((X38 @ X39))))) & (((hoasinduction_lem0 = $true) | ? [X504 : (subst > term > subst > $o)] : ! [X505 : (term > $o)] : ? [X506 : term] : (((X504 @ id @ X506 @ id)) != ((X505 @ X506)))) & (! [X504 : (subst > term > subst > $o)] : ? [X505 : (term > $o)] : ! [X506 : term] : (((X504 @ id @ X506 @ id)) = ((X505 @ X506))) | (hoasinduction_lem0 != $true))) & ((($true = sP41) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | ($true != sP41))) & (pushprop_lem0_gthm = $true) & (((hoasinduction_no_psi_cond_lthm = $true) | ($true != sP37)) & (($true = sP37) | (hoasinduction_no_psi_cond_lthm != $true))) & (hoaslamnotap_gthm = $true) & (((sP32 = $true) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | (sP32 != $true))) & (pushprop_gthm = $true) & (hoasinduction_lem3b_gthm = $true) & (substmonoid_gthm = $true) & ! [X538 : term] : ? [X539 : d_term] : (((d2term @ X539)) = X538) & (((sP26 = $true) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | (sP26 != $true))) & ((! [X407 : term,X408 : (subst > term > subst > $o)] : (? [X415 : term,X414 : term] : (($true = ((X408 @ id @ X415 @ id))) & (((X408 @ id @ (ap @ (sub @ X414 @ id) @ X415) @ id)) != $true) & (((X408 @ id @ X414 @ id)) = $true)) | (((sP22 @ X408)) = $true) | (((X408 @ id @ X407 @ id)) = $true)) | (hoasinduction_no_psi_cond != $true)) & ((hoasinduction_no_psi_cond = $true) | ? [X407 : term,X408 : (subst > term > subst > $o)] : (! [X415 : term,X414 : term] : (($true != ((X408 @ id @ X415 @ id))) | (((X408 @ id @ (ap @ (sub @ X414 @ id) @ X415) @ id)) = $true) | (((X408 @ id @ X414 @ id)) != $true)) & (((sP22 @ X408)) != $true) & (((X408 @ id @ X407 @ id)) != $true)))) & (((axidl = $true) | ? [X406 : subst] : (((comp @ id @ X406)) != X406)) & (! [X406 : subst] : (((comp @ id @ X406)) = X406) | (axidl != $true))) & (hoasapnotvar_gthm = $true) & ((((! [X26 : subst] : (((comp @ X26 @ id)) = X26) & ! [X25 : subst] : (((comp @ id @ X25)) = X25)) | ? [X23 : subst] : (((comp @ id @ X23)) != X23) | ? [X24 : subst] : (((comp @ X24 @ id)) != X24)) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | ((? [X26 : subst] : (((comp @ X26 @ id)) != X26) | ? [X25 : subst] : (((comp @ id @ X25)) != X25)) & ! [X23 : subst] : (((comp @ id @ X23)) = X23) & ! [X24 : subst] : (((comp @ X24 @ id)) = X24)))) & (((hoasinduction_lem3 = $true) | ($true != sP21)) & (($true = sP21) | (hoasinduction_lem3 != $true))) & (((axvarcons = $true) | ? [X404 : term,X405 : subst] : (((sub @ one @ (push @ X404 @ X405))) != X404)) & (! [X404 : term,X405 : subst] : (((sub @ one @ (push @ X404 @ X405))) = X404) | (axvarcons != $true))) & (((hoasinduction_lem3a_lthm = $true) | ($true != sP19)) & (($true = sP19) | (hoasinduction_lem3a_lthm != $true))) & (hoaslamnotap_lthm = $true) & (((sP17 = $true) | (hoasinduction_lem3v2_lthm != $true)) & ((hoasinduction_lem3v2_lthm = $true) | (sP17 != $true))) & (axclos = $true) & ! [X365 : subst] : ? [X366 : d_subst] : (((d2subst @ X366)) = X365) & (hoaslamnotvar = $true) & (hoasinduction_lem2_lthm = $true) & ! [X225 : d_subst] : (d_id = X225) & (((hoasapinj2 = $true) | ? [X274 : term,X273 : term,X276 : term,X275 : term] : ((((ap @ (sub @ X274 @ id) @ X276)) = ((ap @ (sub @ X275 @ id) @ X273))) & (X273 != X276))) & (! [X274 : term,X273 : term,X276 : term,X275 : term] : ((((ap @ (sub @ X274 @ id) @ X276)) != ((ap @ (sub @ X275 @ id) @ X273))) | (X273 = X276)) | (hoasapinj2 != $true))) & (((var @ (d2term @ d_one))) != $true) & (hoasinduction_lem1_gthm = $true) & (((pushprop_lem1_lthm = $true) | (! [X525 : subst,X524 : term] : (((sub @ one @ (push @ X524 @ X525))) = X524) & ? [X532 : (term > $o),X531 : term,X529 : (term > $o),X528 : term,X530 : subst] : ((((X532 @ (sub @ X531 @ (push @ X528 @ X530)))) != $true) & ($true = ((X529 @ X531)))) & ! [X526 : subst,X527 : term] : (((comp @ sh @ (push @ X527 @ X526))) = X526))) & ((? [X525 : subst,X524 : term] : (((sub @ one @ (push @ X524 @ X525))) != X524) | ! [X532 : (term > $o),X531 : term,X529 : (term > $o),X528 : term,X530 : subst] : ((((X532 @ (sub @ X531 @ (push @ X528 @ X530)))) = $true) | ($true != ((X529 @ X531)))) | ? [X526 : subst,X527 : term] : (((comp @ sh @ (push @ X527 @ X526))) != X526)) | (pushprop_lem1_lthm != $true))) & (((laminj = $true) | ? [X21 : term,X22 : term] : ((((lam @ X22)) = ((lam @ X21))) & (X21 != X22))) & (! [X21 : term,X22 : term] : ((((lam @ X22)) != ((lam @ X21))) | (X21 = X22)) | (laminj != $true))) & (lamnotap != $true) & ((! [X321 : (subst > term > term),X324 : subst,X323 : term,X322 : (subst > term > term)] : ((((lam @ (X321 @ sh @ one))) != ((lam @ (X322 @ sh @ one)))) | ? [X325 : subst,X326 : term,X327 : subst] : (((sub @ (X321 @ X327 @ X326) @ X325)) != ((X321 @ (comp @ X327 @ X325) @ (sub @ X326 @ X325)))) | ? [X328 : subst,X329 : subst,X330 : term] : (((sub @ (X322 @ X328 @ X330) @ X329)) != ((X322 @ (comp @ X328 @ X329) @ (sub @ X330 @ X329)))) | (((X322 @ X324 @ X323)) = ((X321 @ X324 @ X323)))) | (hoaslaminj != $true)) & ((hoaslaminj = $true) | ? [X321 : (subst > term > term),X324 : subst,X323 : term,X322 : (subst > term > term)] : ((((lam @ (X321 @ sh @ one))) = ((lam @ (X322 @ sh @ one)))) & ! [X325 : subst,X326 : term,X327 : subst] : (((sub @ (X321 @ X327 @ X326) @ X325)) = ((X321 @ (comp @ X327 @ X325) @ (sub @ X326 @ X325)))) & ! [X328 : subst,X329 : subst,X330 : term] : (((sub @ (X322 @ X328 @ X330) @ X329)) = ((X322 @ (comp @ X328 @ X329) @ (sub @ X330 @ X329)))) & (((X322 @ X324 @ X323)) != ((X321 @ X324 @ X323)))))) & (((hoasinduction_lem3a = $true) | (sP15 != $true)) & ((sP15 = $true) | (hoasinduction_lem3a != $true))) & (((axvarid = $true) | ? [X27 : term] : (((sub @ X27 @ id)) != X27)) & (! [X27 : term] : (((sub @ X27 @ id)) = X27) | (axvarid != $true))) & (((sP14 = $true) | (hoasinduction_lem3aa != $true)) & ((hoasinduction_lem3aa = $true) | (sP14 != $true))) & ((! [X515 : term] : ? [X516 : (subst > term > term)] : (((X516 @ sh @ one)) = ((sub @ X515 @ (push @ one @ sh)))) | (hoasinduction_lem3b != $true)) & ((hoasinduction_lem3b = $true) | ? [X515 : term] : ! [X516 : (subst > term > term)] : (((X516 @ sh @ one)) != ((sub @ X515 @ (push @ one @ sh)))))) & (pushprop_lem1_gthm = $true) & (axvarshift = $true) & (((d2term @ d_one)) = one) & (pushprop_lem3v2_lthm = $true) & (hoasinduction_lem1v2_gthm = $true) & (hoasinduction_lem2_gthm = $true) & (((pushprop_lem1v2_lthm = $true) | (? [X196 : (term > $o),X199 : term,X197 : subst,X198 : (term > $o)] : ((((X198 @ X199)) = $true) & ($true != ((X196 @ one))) & ! [X200 : term] : (((X196 @ X200)) = ((X198 @ (sub @ X200 @ (push @ X199 @ X197)))))) & ! [X195 : term,X194 : subst] : (((sub @ one @ (push @ X195 @ X194))) = X195))) & ((! [X196 : (term > $o),X199 : term,X197 : subst,X198 : (term > $o)] : ((((X198 @ X199)) != $true) | ($true = ((X196 @ one))) | ? [X200 : term] : (((X196 @ X200)) != ((X198 @ (sub @ X200 @ (push @ X199 @ X197)))))) | ? [X195 : term,X194 : subst] : (((sub @ one @ (push @ X195 @ X194))) != X195)) | (pushprop_lem1v2_lthm != $true))) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((! [X96 : (term > $o),X95 : term,X97 : subst] : ? [X98 : (term > $o)] : ! [X99 : term] : (((X96 @ (sub @ X99 @ (push @ X95 @ X97)))) = ((X98 @ X99))) | (pushprop_lem0_lthm != $true)) & ((pushprop_lem0_lthm = $true) | ? [X96 : (term > $o),X95 : term,X97 : subst] : ! [X98 : (term > $o)] : ? [X99 : term] : (((X96 @ (sub @ X99 @ (push @ X95 @ X97)))) != ((X98 @ X99))))) & (((! [X278 : subst] : (((comp @ X278 @ id)) = X278) & ! [X277 : subst] : (((comp @ id @ X277)) = X277)) | (substmonoid != $true)) & ((substmonoid = $true) | (? [X278 : subst] : (((comp @ X278 @ id)) != X278) | ? [X277 : subst] : (((comp @ id @ X277)) != X277)))) & (hoasinduction_lem1_lthm = $true) & (termmset_gthm = $true) & (pushprop_lem2v2_lthm = $true) & (pushprop_lem2v2 = $true) & (((induction = $true) | ? [X382 : term,X381 : (term > $o)] : (($true != ((sP13 @ X381))) & ($true != ((X381 @ X382))) & ! [X384 : term] : (($true != ((X381 @ X384))) | ($true = ((X381 @ (lam @ X384))))) & ! [X383 : term] : ((((var @ X383)) != $true) | (((X381 @ X383)) = $true)))) & (! [X382 : term,X381 : (term > $o)] : (($true = ((sP13 @ X381))) | ($true = ((X381 @ X382))) | ? [X384 : term] : (($true = ((X381 @ X384))) & ($true != ((X381 @ (lam @ X384))))) | ? [X383 : term] : ((((var @ X383)) = $true) & (((X381 @ X383)) != $true))) | (induction != $true))) & (hoaslaminj_gthm = $true) & (((hoasapinj1 = $true) | ? [X302 : term,X299 : term,X301 : term,X300 : term] : ((X299 != X302) & (((ap @ (sub @ X302 @ id) @ X301)) = ((ap @ (sub @ X299 @ id) @ X300))))) & (! [X302 : term,X299 : term,X301 : term,X300 : term] : ((X299 = X302) | (((ap @ (sub @ X302 @ id) @ X301)) != ((ap @ (sub @ X299 @ id) @ X300)))) | (hoasapinj1 != $true))) & (((hoasinduction_lem3v2a_lthm = $true) | (sP12 != $true)) & ((sP12 = $true) | (hoasinduction_lem3v2a_lthm != $true))) & (((pushprop_lem1 = $true) | ? [X216 : (term > $o),X217 : term,X218 : (term > $o),X219 : subst,X220 : term] : ((((X216 @ X217)) = $true) & (((X218 @ (sub @ X217 @ (push @ X220 @ X219)))) != $true))) & (! [X216 : (term > $o),X217 : term,X218 : (term > $o),X219 : subst,X220 : term] : ((((X216 @ X217)) != $true) | (((X218 @ (sub @ X217 @ (push @ X220 @ X219)))) = $true)) | (pushprop_lem1 != $true))) & ((! [X379 : term,X377 : term,X380 : term,X378 : term] : ((X377 = X378) | (((ap @ X378 @ X379)) != ((ap @ X377 @ X380)))) | (apinj1 != $true)) & ((apinj1 = $true) | ? [X379 : term,X377 : term,X380 : term,X378 : term] : ((X377 != X378) & (((ap @ X378 @ X379)) = ((ap @ X377 @ X380)))))) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (pushprop_lem3v2 = $true) & ! [X20 : d_subst,X19 : d_subst] : ((((d2subst @ X19)) != ((d2subst @ X20))) | (X19 = X20)) & (ulamvar1 != $true) & (pushprop_lem1v2_gthm = $true) & (hoasapnotvar_lthm = $true) & (hoasinduction_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (((? [X462 : term,X463 : term,X460 : term,X461 : term] : ((((ap @ X463 @ X460)) = ((ap @ X461 @ X462))) & (X460 != X462)) | ! [X466 : term,X464 : term,X465 : term,X467 : term] : ((((ap @ (sub @ X464 @ id) @ X467)) != ((ap @ (sub @ X466 @ id) @ X465))) | (X465 = X467))) | (hoasapinj2_lthm != $true)) & ((hoasapinj2_lthm = $true) | (! [X462 : term,X463 : term,X460 : term,X461 : term] : ((((ap @ X463 @ X460)) != ((ap @ X461 @ X462))) | (X460 = X462)) & ? [X466 : term,X464 : term,X465 : term,X467 : term] : ((((ap @ (sub @ X464 @ id) @ X467)) = ((ap @ (sub @ X466 @ id) @ X465))) & (X465 != X467))))) & (((hoasinduction_lem2 = $true) | (sP11 != $true)) & ((sP11 = $true) | (hoasinduction_lem2 != $true))) & (((sP9 = $true) | (hoaslaminj_lthm != $true)) & ((hoaslaminj_lthm = $true) | (sP9 != $true))) & (hoasapnotvar = $true) & (ulamvarsh = $true) & (hoasinduction_lem3v2_gthm = $true) & (((induction2lem = $true) | (sP8 != $true)) & ((sP8 = $true) | (induction2lem != $true))) & ((! [X298 : subst] : (((push @ (sub @ one @ X298) @ (comp @ sh @ X298))) = X298) | (axscons != $true)) & ((axscons = $true) | ? [X298 : subst] : (((push @ (sub @ one @ X298) @ (comp @ sh @ X298))) != X298))) & (pushprop_lem2v2_gthm = $true) & (hoaslamnotvar_lthm = $true) & (lamnotvar = $true) & (((termmset = $true) | ? [X291 : term] : (((sub @ X291 @ id)) != X291)) & (! [X291 : term] : (((sub @ X291 @ id)) = X291) | (termmset != $true))) & (((hoasinduction_lem3v2a = $true) | (sP6 != $true)) & ((sP6 = $true) | (hoasinduction_lem3v2a != $true))) & (((hoasinduction_lem3_lthm = $true) | (sP5 != $true)) & ((sP5 = $true) | (hoasinduction_lem3_lthm != $true))) & (pushprop_lthm = $true) & (((d2subst @ d_id)) = sh) & ((! [X294 : (subst > term > term),X293 : term,X292 : term] : (? [X297 : term,X295 : subst,X296 : subst] : (((sub @ (X294 @ X296 @ X297) @ X295)) != ((X294 @ (comp @ X296 @ X295) @ (sub @ X297 @ X295)))) | (((lam @ (X294 @ sh @ one))) != ((ap @ (sub @ X292 @ id) @ X293)))) | (hoaslamnotap != $true)) & ((hoaslamnotap = $true) | ? [X294 : (subst > term > term),X293 : term,X292 : term] : (! [X297 : term,X295 : subst,X296 : subst] : (((sub @ (X294 @ X296 @ X297) @ X295)) = ((X294 @ (comp @ X296 @ X295) @ (sub @ X297 @ X295)))) & (((lam @ (X294 @ sh @ one))) = ((ap @ (sub @ X292 @ id) @ X293)))))) & (((d2subst @ d_id)) = id) & (((hoasinduction_lem3aa_lthm = $true) | (sP2 != $true)) & ((sP2 = $true) | (hoasinduction_lem3aa_lthm != $true))) & ((! [X279 : term,X282 : term,X281 : term,X280 : term] : ((((ap @ X280 @ X282)) != ((ap @ X279 @ X281))) | (X281 = X282)) | (apinj2 != $true)) & ((apinj2 = $true) | ? [X279 : term,X282 : term,X281 : term,X280 : term] : ((((ap @ X280 @ X282)) = ((ap @ X279 @ X281))) & (X281 != X282)))) & (((hoasapinj1_lthm = $true) | (! [X31 : term,X32 : term,X30 : term,X29 : term] : ((((ap @ X31 @ X29)) != ((ap @ X32 @ X30))) | (X31 = X32)) & ? [X36 : term,X33 : term,X34 : term,X35 : term] : ((X33 != X36) & (((ap @ (sub @ X33 @ id) @ X35)) = ((ap @ (sub @ X36 @ id) @ X34)))) & ! [X28 : term] : (((sub @ X28 @ id)) = X28))) & ((? [X31 : term,X32 : term,X30 : term,X29 : term] : ((((ap @ X31 @ X29)) = ((ap @ X32 @ X30))) & (X31 != X32)) | ! [X36 : term,X33 : term,X34 : term,X35 : term] : ((X33 = X36) | (((ap @ (sub @ X33 @ id) @ X35)) != ((ap @ (sub @ X36 @ id) @ X34)))) | ? [X28 : term] : (((sub @ X28 @ id)) != X28)) | (hoasapinj1_lthm != $true))) & ((! [X121 : term] : ? [X122 : (subst > term > term)] : ! [X123 : term,X124 : subst] : (((X122 @ X124 @ X123)) = ((sub @ X121 @ (push @ X123 @ X124)))) | (hoasinduction_lem3v2_f_lthm != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X121 : term] : ! [X122 : (subst > term > term)] : ? [X123 : term,X124 : subst] : (((X122 @ X124 @ X123)) != ((sub @ X121 @ (push @ X123 @ X124)))))) & (hoasinduction_lem3a_gthm = $true) & (hoasinduction_lem1 = $true) & (axmap = $true) & (((! [X518 : term] : (((sub @ X518 @ id)) = X518) | ? [X517 : term] : (((sub @ X517 @ id)) != X517)) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | (? [X518 : term] : (((sub @ X518 @ id)) != X518) & ! [X517 : term] : (((sub @ X517 @ id)) = X517)))) & (((sP1 = $true) | (induction2 != $true)) & ((induction2 = $true) | (sP1 != $true))) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (induction2lem_gthm = $true) & (hoasapinj2_gthm = $true) & (hoasinduction_lem3_gthm = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (hoasapinj1_gthm = $true) & (pushprop_lthm_orig = $true) & (axassoc = $true) & ((! [X223 : term,X224 : subst] : (((comp @ sh @ (push @ X223 @ X224))) = X224) | (axshiftcons != $true)) & ((axshiftcons = $true) | ? [X223 : term,X224 : subst] : (((comp @ sh @ (push @ X223 @ X224))) != X224))) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2))))))))),
  inference(nnf_transformation,[],[f72])).
thf(f261,plain,(
  ((hoasinduction_lem3v2 = $true) | (sP58 != $true)) & ((sP58 = $true) | (hoasinduction_lem3v2 != $true)) & ((hoasinduction = $true) | (sP56 != $true)) & ((sP56 = $true) | (hoasinduction != $true)) & (axabs = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((pushprop_lem1v2 = $true) | ? [X534 : (term > $o),X536 : term,X533 : (term > $o),X535 : subst] : (! [X537 : term] : (((X533 @ (sub @ X537 @ (push @ X536 @ X535)))) = ((X534 @ X537))) & (((X534 @ one)) != $true) & (((X533 @ X536)) = $true))) & (! [X534 : (term > $o),X536 : term,X533 : (term > $o),X535 : subst] : (? [X537 : term] : (((X533 @ (sub @ X537 @ (push @ X536 @ X535)))) != ((X534 @ X537))) | (((X534 @ one)) = $true) | (((X533 @ X536)) != $true)) | (pushprop_lem1v2 != $true)) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ((shinj = $true) | ? [X502 : term,X503 : term] : ((X502 != X503) & (((sub @ X502 @ sh)) = ((sub @ X503 @ sh))))) & (! [X502 : term,X503 : term] : ((X502 = X503) | (((sub @ X502 @ sh)) != ((sub @ X503 @ sh)))) | (shinj != $true)) & ((hoasinduction_lem2v2 = $true) | (sP53 != $true)) & ((sP53 = $true) | (hoasinduction_lem2v2 != $true)) & ! [X221 : d_term,X222 : d_term] : ((X221 = X222) | (((d2term @ X222)) != ((d2term @ X221)))) & ((hoasinduction_lem3v2_f = $true) | ? [X52 : term] : ! [X53 : (subst > term > term)] : ? [X55 : subst,X54 : term] : (((sub @ X52 @ (push @ X54 @ X55))) != ((X53 @ X55 @ X54)))) & (! [X52 : term] : ? [X53 : (subst > term > term)] : ! [X55 : subst,X54 : term] : (((sub @ X52 @ (push @ X54 @ X55))) = ((X53 @ X55 @ X54))) | (hoasinduction_lem3v2_f != $true)) & ! [X18 : d_term] : (d_one = X18) & (axapp = $true) & (ulamvarind = $true) & (hoasinduction_lem2v2_gthm = $true) & ((hoasinduction_lthm = $true) | ($true != sP51)) & (($true = sP51) | (hoasinduction_lthm != $true)) & (! [X500 : term] : ? [X501 : (subst > term > term)] : (((sub @ X500 @ (push @ one @ sh))) = ((X501 @ sh @ one))) | (hoasinduction_lem3b_lthm != $true)) & ((hoasinduction_lem3b_lthm = $true) | ? [X500 : term] : ! [X501 : (subst > term > term)] : (((sub @ X500 @ (push @ one @ sh))) != ((X501 @ sh @ one)))) & ((sP42 = $true) | (hoasinduction_lem3aaa != $true)) & ((hoasinduction_lem3aaa = $true) | (sP42 != $true)) & ((axidr = $true) | ? [X272 : subst] : (((comp @ X272 @ id)) != X272)) & (! [X272 : subst] : (((comp @ X272 @ id)) = X272) | (axidr != $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (induction2_gthm = $true) & (hoaslamnotvar_gthm = $true) & (pushprop = $true) & (! [X521 : term,X520 : (term > $o),X519 : subst] : ? [X522 : (term > $o)] : ! [X523 : term] : (((X520 @ (sub @ X523 @ (push @ X521 @ X519)))) = ((X522 @ X523))) | (pushprop_lem0 != $true)) & ((pushprop_lem0 = $true) | ? [X521 : term,X520 : (term > $o),X519 : subst] : ! [X522 : (term > $o)] : ? [X523 : term] : (((X520 @ (sub @ X523 @ (push @ X521 @ X519)))) != ((X522 @ X523)))) & (apnotvar = $true) & (hoasinduction_lem1v2 = $true) & (! [X37 : (subst > term > subst > $o)] : ? [X38 : (term > $o)] : ! [X39 : term] : (((X37 @ id @ X39 @ id)) = ((X38 @ X39))) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ? [X37 : (subst > term > subst > $o)] : ! [X38 : (term > $o)] : ? [X39 : term] : (((X37 @ id @ X39 @ id)) != ((X38 @ X39)))) & ((hoasinduction_lem0 = $true) | ? [X504 : (subst > term > subst > $o)] : ! [X505 : (term > $o)] : ? [X506 : term] : (((X504 @ id @ X506 @ id)) != ((X505 @ X506)))) & (! [X504 : (subst > term > subst > $o)] : ? [X505 : (term > $o)] : ! [X506 : term] : (((X504 @ id @ X506 @ id)) = ((X505 @ X506))) | (hoasinduction_lem0 != $true)) & (($true = sP41) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | ($true != sP41)) & (pushprop_lem0_gthm = $true) & ((hoasinduction_no_psi_cond_lthm = $true) | ($true != sP37)) & (($true = sP37) | (hoasinduction_no_psi_cond_lthm != $true)) & (hoaslamnotap_gthm = $true) & ((sP32 = $true) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | (sP32 != $true)) & (pushprop_gthm = $true) & (hoasinduction_lem3b_gthm = $true) & (substmonoid_gthm = $true) & ! [X538 : term] : ? [X539 : d_term] : (((d2term @ X539)) = X538) & ((sP26 = $true) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | (sP26 != $true)) & (! [X407 : term,X408 : (subst > term > subst > $o)] : (? [X415 : term,X414 : term] : (($true = ((X408 @ id @ X415 @ id))) & (((X408 @ id @ (ap @ (sub @ X414 @ id) @ X415) @ id)) != $true) & (((X408 @ id @ X414 @ id)) = $true)) | (((sP22 @ X408)) = $true) | (((X408 @ id @ X407 @ id)) = $true)) | (hoasinduction_no_psi_cond != $true)) & ((hoasinduction_no_psi_cond = $true) | ? [X407 : term,X408 : (subst > term > subst > $o)] : (! [X415 : term,X414 : term] : (($true != ((X408 @ id @ X415 @ id))) | (((X408 @ id @ (ap @ (sub @ X414 @ id) @ X415) @ id)) = $true) | (((X408 @ id @ X414 @ id)) != $true)) & (((sP22 @ X408)) != $true) & (((X408 @ id @ X407 @ id)) != $true))) & ((axidl = $true) | ? [X406 : subst] : (((comp @ id @ X406)) != X406)) & (! [X406 : subst] : (((comp @ id @ X406)) = X406) | (axidl != $true)) & (hoasapnotvar_gthm = $true) & ((! [X26 : subst] : (((comp @ X26 @ id)) = X26) & ! [X25 : subst] : (((comp @ id @ X25)) = X25)) | ? [X23 : subst] : (((comp @ id @ X23)) != X23) | ? [X24 : subst] : (((comp @ X24 @ id)) != X24) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | ((? [X26 : subst] : (((comp @ X26 @ id)) != X26) | ? [X25 : subst] : (((comp @ id @ X25)) != X25)) & ! [X23 : subst] : (((comp @ id @ X23)) = X23) & ! [X24 : subst] : (((comp @ X24 @ id)) = X24))) & ((hoasinduction_lem3 = $true) | ($true != sP21)) & (($true = sP21) | (hoasinduction_lem3 != $true)) & ((axvarcons = $true) | ? [X404 : term,X405 : subst] : (((sub @ one @ (push @ X404 @ X405))) != X404)) & (! [X404 : term,X405 : subst] : (((sub @ one @ (push @ X404 @ X405))) = X404) | (axvarcons != $true)) & ((hoasinduction_lem3a_lthm = $true) | ($true != sP19)) & (($true = sP19) | (hoasinduction_lem3a_lthm != $true)) & (hoaslamnotap_lthm = $true) & ((sP17 = $true) | (hoasinduction_lem3v2_lthm != $true)) & ((hoasinduction_lem3v2_lthm = $true) | (sP17 != $true)) & (axclos = $true) & ! [X365 : subst] : ? [X366 : d_subst] : (((d2subst @ X366)) = X365) & (hoaslamnotvar = $true) & (hoasinduction_lem2_lthm = $true) & ! [X225 : d_subst] : (d_id = X225) & ((hoasapinj2 = $true) | ? [X274 : term,X273 : term,X276 : term,X275 : term] : ((((ap @ (sub @ X274 @ id) @ X276)) = ((ap @ (sub @ X275 @ id) @ X273))) & (X273 != X276))) & (! [X274 : term,X273 : term,X276 : term,X275 : term] : ((((ap @ (sub @ X274 @ id) @ X276)) != ((ap @ (sub @ X275 @ id) @ X273))) | (X273 = X276)) | (hoasapinj2 != $true)) & (((var @ (d2term @ d_one))) != $true) & (hoasinduction_lem1_gthm = $true) & ((pushprop_lem1_lthm = $true) | (! [X525 : subst,X524 : term] : (((sub @ one @ (push @ X524 @ X525))) = X524) & ? [X532 : (term > $o),X531 : term,X529 : (term > $o),X528 : term,X530 : subst] : ((((X532 @ (sub @ X531 @ (push @ X528 @ X530)))) != $true) & ($true = ((X529 @ X531)))) & ! [X526 : subst,X527 : term] : (((comp @ sh @ (push @ X527 @ X526))) = X526))) & (? [X525 : subst,X524 : term] : (((sub @ one @ (push @ X524 @ X525))) != X524) | ! [X532 : (term > $o),X531 : term,X529 : (term > $o),X528 : term,X530 : subst] : ((((X532 @ (sub @ X531 @ (push @ X528 @ X530)))) = $true) | ($true != ((X529 @ X531)))) | ? [X526 : subst,X527 : term] : (((comp @ sh @ (push @ X527 @ X526))) != X526) | (pushprop_lem1_lthm != $true)) & ((laminj = $true) | ? [X21 : term,X22 : term] : ((((lam @ X22)) = ((lam @ X21))) & (X21 != X22))) & (! [X21 : term,X22 : term] : ((((lam @ X22)) != ((lam @ X21))) | (X21 = X22)) | (laminj != $true)) & (lamnotap != $true) & (! [X321 : (subst > term > term),X324 : subst,X323 : term,X322 : (subst > term > term)] : ((((lam @ (X321 @ sh @ one))) != ((lam @ (X322 @ sh @ one)))) | ? [X325 : subst,X326 : term,X327 : subst] : (((sub @ (X321 @ X327 @ X326) @ X325)) != ((X321 @ (comp @ X327 @ X325) @ (sub @ X326 @ X325)))) | ? [X328 : subst,X329 : subst,X330 : term] : (((sub @ (X322 @ X328 @ X330) @ X329)) != ((X322 @ (comp @ X328 @ X329) @ (sub @ X330 @ X329)))) | (((X322 @ X324 @ X323)) = ((X321 @ X324 @ X323)))) | (hoaslaminj != $true)) & ((hoaslaminj = $true) | ? [X321 : (subst > term > term),X324 : subst,X323 : term,X322 : (subst > term > term)] : ((((lam @ (X321 @ sh @ one))) = ((lam @ (X322 @ sh @ one)))) & ! [X325 : subst,X326 : term,X327 : subst] : (((sub @ (X321 @ X327 @ X326) @ X325)) = ((X321 @ (comp @ X327 @ X325) @ (sub @ X326 @ X325)))) & ! [X328 : subst,X329 : subst,X330 : term] : (((sub @ (X322 @ X328 @ X330) @ X329)) = ((X322 @ (comp @ X328 @ X329) @ (sub @ X330 @ X329)))) & (((X322 @ X324 @ X323)) != ((X321 @ X324 @ X323))))) & ((hoasinduction_lem3a = $true) | (sP15 != $true)) & ((sP15 = $true) | (hoasinduction_lem3a != $true)) & ((axvarid = $true) | ? [X27 : term] : (((sub @ X27 @ id)) != X27)) & (! [X27 : term] : (((sub @ X27 @ id)) = X27) | (axvarid != $true)) & ((sP14 = $true) | (hoasinduction_lem3aa != $true)) & ((hoasinduction_lem3aa = $true) | (sP14 != $true)) & (! [X515 : term] : ? [X516 : (subst > term > term)] : (((X516 @ sh @ one)) = ((sub @ X515 @ (push @ one @ sh)))) | (hoasinduction_lem3b != $true)) & ((hoasinduction_lem3b = $true) | ? [X515 : term] : ! [X516 : (subst > term > term)] : (((X516 @ sh @ one)) != ((sub @ X515 @ (push @ one @ sh))))) & (pushprop_lem1_gthm = $true) & (axvarshift = $true) & (((d2term @ d_one)) = one) & (pushprop_lem3v2_lthm = $true) & (hoasinduction_lem1v2_gthm = $true) & (hoasinduction_lem2_gthm = $true) & ((pushprop_lem1v2_lthm = $true) | (? [X196 : (term > $o),X199 : term,X197 : subst,X198 : (term > $o)] : ((((X198 @ X199)) = $true) & ($true != ((X196 @ one))) & ! [X200 : term] : (((X196 @ X200)) = ((X198 @ (sub @ X200 @ (push @ X199 @ X197)))))) & ! [X195 : term,X194 : subst] : (((sub @ one @ (push @ X195 @ X194))) = X195))) & (! [X196 : (term > $o),X199 : term,X197 : subst,X198 : (term > $o)] : ((((X198 @ X199)) != $true) | ($true = ((X196 @ one))) | ? [X200 : term] : (((X196 @ X200)) != ((X198 @ (sub @ X200 @ (push @ X199 @ X197)))))) | ? [X195 : term,X194 : subst] : (((sub @ one @ (push @ X195 @ X194))) != X195) | (pushprop_lem1v2_lthm != $true)) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (! [X96 : (term > $o),X95 : term,X97 : subst] : ? [X98 : (term > $o)] : ! [X99 : term] : (((X96 @ (sub @ X99 @ (push @ X95 @ X97)))) = ((X98 @ X99))) | (pushprop_lem0_lthm != $true)) & ((pushprop_lem0_lthm = $true) | ? [X96 : (term > $o),X95 : term,X97 : subst] : ! [X98 : (term > $o)] : ? [X99 : term] : (((X96 @ (sub @ X99 @ (push @ X95 @ X97)))) != ((X98 @ X99)))) & ((! [X278 : subst] : (((comp @ X278 @ id)) = X278) & ! [X277 : subst] : (((comp @ id @ X277)) = X277)) | (substmonoid != $true)) & ((substmonoid = $true) | ? [X278 : subst] : (((comp @ X278 @ id)) != X278) | ? [X277 : subst] : (((comp @ id @ X277)) != X277)) & (hoasinduction_lem1_lthm = $true) & (termmset_gthm = $true) & (pushprop_lem2v2_lthm = $true) & (pushprop_lem2v2 = $true) & ((induction = $true) | ? [X382 : term,X381 : (term > $o)] : (($true != ((sP13 @ X381))) & ($true != ((X381 @ X382))) & ! [X384 : term] : (($true != ((X381 @ X384))) | ($true = ((X381 @ (lam @ X384))))) & ! [X383 : term] : ((((var @ X383)) != $true) | (((X381 @ X383)) = $true)))) & (! [X382 : term,X381 : (term > $o)] : (($true = ((sP13 @ X381))) | ($true = ((X381 @ X382))) | ? [X384 : term] : (($true = ((X381 @ X384))) & ($true != ((X381 @ (lam @ X384))))) | ? [X383 : term] : ((((var @ X383)) = $true) & (((X381 @ X383)) != $true))) | (induction != $true)) & (hoaslaminj_gthm = $true) & ((hoasapinj1 = $true) | ? [X302 : term,X299 : term,X301 : term,X300 : term] : ((X299 != X302) & (((ap @ (sub @ X302 @ id) @ X301)) = ((ap @ (sub @ X299 @ id) @ X300))))) & (! [X302 : term,X299 : term,X301 : term,X300 : term] : ((X299 = X302) | (((ap @ (sub @ X302 @ id) @ X301)) != ((ap @ (sub @ X299 @ id) @ X300)))) | (hoasapinj1 != $true)) & ((hoasinduction_lem3v2a_lthm = $true) | (sP12 != $true)) & ((sP12 = $true) | (hoasinduction_lem3v2a_lthm != $true)) & ((pushprop_lem1 = $true) | ? [X216 : (term > $o),X217 : term,X218 : (term > $o),X219 : subst,X220 : term] : ((((X216 @ X217)) = $true) & (((X218 @ (sub @ X217 @ (push @ X220 @ X219)))) != $true))) & (! [X216 : (term > $o),X217 : term,X218 : (term > $o),X219 : subst,X220 : term] : ((((X216 @ X217)) != $true) | (((X218 @ (sub @ X217 @ (push @ X220 @ X219)))) = $true)) | (pushprop_lem1 != $true)) & (! [X379 : term,X377 : term,X380 : term,X378 : term] : ((X377 = X378) | (((ap @ X378 @ X379)) != ((ap @ X377 @ X380)))) | (apinj1 != $true)) & ((apinj1 = $true) | ? [X379 : term,X377 : term,X380 : term,X378 : term] : ((X377 != X378) & (((ap @ X378 @ X379)) = ((ap @ X377 @ X380))))) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (pushprop_lem3v2 = $true) & ! [X20 : d_subst,X19 : d_subst] : ((((d2subst @ X19)) != ((d2subst @ X20))) | (X19 = X20)) & (ulamvar1 != $true) & (pushprop_lem1v2_gthm = $true) & (hoasapnotvar_lthm = $true) & (hoasinduction_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (? [X462 : term,X463 : term,X460 : term,X461 : term] : ((((ap @ X463 @ X460)) = ((ap @ X461 @ X462))) & (X460 != X462)) | ! [X466 : term,X464 : term,X465 : term,X467 : term] : ((((ap @ (sub @ X464 @ id) @ X467)) != ((ap @ (sub @ X466 @ id) @ X465))) | (X465 = X467)) | (hoasapinj2_lthm != $true)) & ((hoasapinj2_lthm = $true) | (! [X462 : term,X463 : term,X460 : term,X461 : term] : ((((ap @ X463 @ X460)) != ((ap @ X461 @ X462))) | (X460 = X462)) & ? [X466 : term,X464 : term,X465 : term,X467 : term] : ((((ap @ (sub @ X464 @ id) @ X467)) = ((ap @ (sub @ X466 @ id) @ X465))) & (X465 != X467)))) & ((hoasinduction_lem2 = $true) | (sP11 != $true)) & ((sP11 = $true) | (hoasinduction_lem2 != $true)) & ((sP9 = $true) | (hoaslaminj_lthm != $true)) & ((hoaslaminj_lthm = $true) | (sP9 != $true)) & (hoasapnotvar = $true) & (ulamvarsh = $true) & (hoasinduction_lem3v2_gthm = $true) & ((induction2lem = $true) | (sP8 != $true)) & ((sP8 = $true) | (induction2lem != $true)) & (! [X298 : subst] : (((push @ (sub @ one @ X298) @ (comp @ sh @ X298))) = X298) | (axscons != $true)) & ((axscons = $true) | ? [X298 : subst] : (((push @ (sub @ one @ X298) @ (comp @ sh @ X298))) != X298)) & (pushprop_lem2v2_gthm = $true) & (hoaslamnotvar_lthm = $true) & (lamnotvar = $true) & ((termmset = $true) | ? [X291 : term] : (((sub @ X291 @ id)) != X291)) & (! [X291 : term] : (((sub @ X291 @ id)) = X291) | (termmset != $true)) & ((hoasinduction_lem3v2a = $true) | (sP6 != $true)) & ((sP6 = $true) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3_lthm = $true) | (sP5 != $true)) & ((sP5 = $true) | (hoasinduction_lem3_lthm != $true)) & (pushprop_lthm = $true) & (((d2subst @ d_id)) = sh) & (! [X294 : (subst > term > term),X293 : term,X292 : term] : (? [X297 : term,X295 : subst,X296 : subst] : (((sub @ (X294 @ X296 @ X297) @ X295)) != ((X294 @ (comp @ X296 @ X295) @ (sub @ X297 @ X295)))) | (((lam @ (X294 @ sh @ one))) != ((ap @ (sub @ X292 @ id) @ X293)))) | (hoaslamnotap != $true)) & ((hoaslamnotap = $true) | ? [X294 : (subst > term > term),X293 : term,X292 : term] : (! [X297 : term,X295 : subst,X296 : subst] : (((sub @ (X294 @ X296 @ X297) @ X295)) = ((X294 @ (comp @ X296 @ X295) @ (sub @ X297 @ X295)))) & (((lam @ (X294 @ sh @ one))) = ((ap @ (sub @ X292 @ id) @ X293))))) & (((d2subst @ d_id)) = id) & ((hoasinduction_lem3aa_lthm = $true) | (sP2 != $true)) & ((sP2 = $true) | (hoasinduction_lem3aa_lthm != $true)) & (! [X279 : term,X282 : term,X281 : term,X280 : term] : ((((ap @ X280 @ X282)) != ((ap @ X279 @ X281))) | (X281 = X282)) | (apinj2 != $true)) & ((apinj2 = $true) | ? [X279 : term,X282 : term,X281 : term,X280 : term] : ((((ap @ X280 @ X282)) = ((ap @ X279 @ X281))) & (X281 != X282))) & ((hoasapinj1_lthm = $true) | (! [X31 : term,X32 : term,X30 : term,X29 : term] : ((((ap @ X31 @ X29)) != ((ap @ X32 @ X30))) | (X31 = X32)) & ? [X36 : term,X33 : term,X34 : term,X35 : term] : ((X33 != X36) & (((ap @ (sub @ X33 @ id) @ X35)) = ((ap @ (sub @ X36 @ id) @ X34)))) & ! [X28 : term] : (((sub @ X28 @ id)) = X28))) & (? [X31 : term,X32 : term,X30 : term,X29 : term] : ((((ap @ X31 @ X29)) = ((ap @ X32 @ X30))) & (X31 != X32)) | ! [X36 : term,X33 : term,X34 : term,X35 : term] : ((X33 = X36) | (((ap @ (sub @ X33 @ id) @ X35)) != ((ap @ (sub @ X36 @ id) @ X34)))) | ? [X28 : term] : (((sub @ X28 @ id)) != X28) | (hoasapinj1_lthm != $true)) & (! [X121 : term] : ? [X122 : (subst > term > term)] : ! [X123 : term,X124 : subst] : (((X122 @ X124 @ X123)) = ((sub @ X121 @ (push @ X123 @ X124)))) | (hoasinduction_lem3v2_f_lthm != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X121 : term] : ! [X122 : (subst > term > term)] : ? [X123 : term,X124 : subst] : (((X122 @ X124 @ X123)) != ((sub @ X121 @ (push @ X123 @ X124))))) & (hoasinduction_lem3a_gthm = $true) & (hoasinduction_lem1 = $true) & (axmap = $true) & (! [X518 : term] : (((sub @ X518 @ id)) = X518) | ? [X517 : term] : (((sub @ X517 @ id)) != X517) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | (? [X518 : term] : (((sub @ X518 @ id)) != X518) & ! [X517 : term] : (((sub @ X517 @ id)) = X517))) & ((sP1 = $true) | (induction2 != $true)) & ((induction2 = $true) | (sP1 != $true)) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (induction2lem_gthm = $true) & (hoasapinj2_gthm = $true) & (hoasinduction_lem3_gthm = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (hoasapinj1_gthm = $true) & (pushprop_lthm_orig = $true) & (axassoc = $true) & (! [X223 : term,X224 : subst] : (((comp @ sh @ (push @ X223 @ X224))) = X224) | (axshiftcons != $true)) & ((axshiftcons = $true) | ? [X223 : term,X224 : subst] : (((comp @ sh @ (push @ X223 @ X224))) != X224)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2))))))))),
  inference(flattening,[],[f260])).
thf(f262,plain,(
  ((hoasinduction_lem3v2 = $true) | (sP58 != $true)) & ((sP58 = $true) | (hoasinduction_lem3v2 != $true)) & ((hoasinduction = $true) | (sP56 != $true)) & ((sP56 = $true) | (hoasinduction != $true)) & (axabs = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((pushprop_lem1v2 = $true) | ? [X0 : (term > $o),X1 : term,X2 : (term > $o),X3 : subst] : (! [X4 : term] : (((X0 @ X4)) = ((X2 @ (sub @ X4 @ (push @ X1 @ X3))))) & ($true != ((X0 @ one))) & (((X2 @ X1)) = $true))) & (! [X5 : (term > $o),X6 : term,X7 : (term > $o),X8 : subst] : (? [X9 : term] : (((X7 @ (sub @ X9 @ (push @ X6 @ X8)))) != ((X5 @ X9))) | (((X5 @ one)) = $true) | (((X7 @ X6)) != $true)) | (pushprop_lem1v2 != $true)) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ((shinj = $true) | ? [X10 : term,X11 : term] : ((X10 != X11) & (((sub @ X10 @ sh)) = ((sub @ X11 @ sh))))) & (! [X12 : term,X13 : term] : ((X12 = X13) | (((sub @ X13 @ sh)) != ((sub @ X12 @ sh)))) | (shinj != $true)) & ((hoasinduction_lem2v2 = $true) | (sP53 != $true)) & ((sP53 = $true) | (hoasinduction_lem2v2 != $true)) & ! [X14 : d_term,X15 : d_term] : ((X14 = X15) | (((d2term @ X14)) != ((d2term @ X15)))) & ((hoasinduction_lem3v2_f = $true) | ? [X16 : term] : ! [X17 : (subst > term > term)] : ? [X18 : subst,X19 : term] : (((X17 @ X18 @ X19)) != ((sub @ X16 @ (push @ X19 @ X18))))) & (! [X20 : term] : ? [X21 : (subst > term > term)] : ! [X22 : subst,X23 : term] : (((sub @ X20 @ (push @ X23 @ X22))) = ((X21 @ X22 @ X23))) | (hoasinduction_lem3v2_f != $true)) & ! [X24 : d_term] : (d_one = X24) & (axapp = $true) & (ulamvarind = $true) & (hoasinduction_lem2v2_gthm = $true) & ((hoasinduction_lthm = $true) | ($true != sP51)) & (($true = sP51) | (hoasinduction_lthm != $true)) & (! [X25 : term] : ? [X26 : (subst > term > term)] : (((X26 @ sh @ one)) = ((sub @ X25 @ (push @ one @ sh)))) | (hoasinduction_lem3b_lthm != $true)) & ((hoasinduction_lem3b_lthm = $true) | ? [X27 : term] : ! [X28 : (subst > term > term)] : (((X28 @ sh @ one)) != ((sub @ X27 @ (push @ one @ sh))))) & ((sP42 = $true) | (hoasinduction_lem3aaa != $true)) & ((hoasinduction_lem3aaa = $true) | (sP42 != $true)) & ((axidr = $true) | ? [X29 : subst] : (((comp @ X29 @ id)) != X29)) & (! [X30 : subst] : (((comp @ X30 @ id)) = X30) | (axidr != $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (induction2_gthm = $true) & (hoaslamnotvar_gthm = $true) & (pushprop = $true) & (! [X31 : term,X32 : (term > $o),X33 : subst] : ? [X34 : (term > $o)] : ! [X35 : term] : (((X34 @ X35)) = ((X32 @ (sub @ X35 @ (push @ X31 @ X33))))) | (pushprop_lem0 != $true)) & ((pushprop_lem0 = $true) | ? [X36 : term,X37 : (term > $o),X38 : subst] : ! [X39 : (term > $o)] : ? [X40 : term] : (((X39 @ X40)) != ((X37 @ (sub @ X40 @ (push @ X36 @ X38)))))) & (apnotvar = $true) & (hoasinduction_lem1v2 = $true) & (! [X41 : (subst > term > subst > $o)] : ? [X42 : (term > $o)] : ! [X43 : term] : (((X42 @ X43)) = ((X41 @ id @ X43 @ id))) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ? [X44 : (subst > term > subst > $o)] : ! [X45 : (term > $o)] : ? [X46 : term] : (((X44 @ id @ X46 @ id)) != ((X45 @ X46)))) & ((hoasinduction_lem0 = $true) | ? [X47 : (subst > term > subst > $o)] : ! [X48 : (term > $o)] : ? [X49 : term] : (((X48 @ X49)) != ((X47 @ id @ X49 @ id)))) & (! [X50 : (subst > term > subst > $o)] : ? [X51 : (term > $o)] : ! [X52 : term] : (((X50 @ id @ X52 @ id)) = ((X51 @ X52))) | (hoasinduction_lem0 != $true)) & (($true = sP41) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | ($true != sP41)) & (pushprop_lem0_gthm = $true) & ((hoasinduction_no_psi_cond_lthm = $true) | ($true != sP37)) & (($true = sP37) | (hoasinduction_no_psi_cond_lthm != $true)) & (hoaslamnotap_gthm = $true) & ((sP32 = $true) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | (sP32 != $true)) & (pushprop_gthm = $true) & (hoasinduction_lem3b_gthm = $true) & (substmonoid_gthm = $true) & ! [X53 : term] : ? [X54 : d_term] : (((d2term @ X54)) = X53) & ((sP26 = $true) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | (sP26 != $true)) & (! [X55 : term,X56 : (subst > term > subst > $o)] : (? [X57 : term,X58 : term] : ((((X56 @ id @ X57 @ id)) = $true) & (((X56 @ id @ (ap @ (sub @ X58 @ id) @ X57) @ id)) != $true) & (((X56 @ id @ X58 @ id)) = $true)) | (((sP22 @ X56)) = $true) | ($true = ((X56 @ id @ X55 @ id)))) | (hoasinduction_no_psi_cond != $true)) & ((hoasinduction_no_psi_cond = $true) | ? [X59 : term,X60 : (subst > term > subst > $o)] : (! [X61 : term,X62 : term] : (($true != ((X60 @ id @ X61 @ id))) | (((X60 @ id @ (ap @ (sub @ X62 @ id) @ X61) @ id)) = $true) | (((X60 @ id @ X62 @ id)) != $true)) & (((sP22 @ X60)) != $true) & (((X60 @ id @ X59 @ id)) != $true))) & ((axidl = $true) | ? [X63 : subst] : (((comp @ id @ X63)) != X63)) & (! [X64 : subst] : (((comp @ id @ X64)) = X64) | (axidl != $true)) & (hoasapnotvar_gthm = $true) & ((! [X65 : subst] : (((comp @ X65 @ id)) = X65) & ! [X66 : subst] : (((comp @ id @ X66)) = X66)) | ? [X67 : subst] : (((comp @ id @ X67)) != X67) | ? [X68 : subst] : (((comp @ X68 @ id)) != X68) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | ((? [X69 : subst] : (((comp @ X69 @ id)) != X69) | ? [X70 : subst] : (((comp @ id @ X70)) != X70)) & ! [X71 : subst] : (((comp @ id @ X71)) = X71) & ! [X72 : subst] : (((comp @ X72 @ id)) = X72))) & ((hoasinduction_lem3 = $true) | ($true != sP21)) & (($true = sP21) | (hoasinduction_lem3 != $true)) & ((axvarcons = $true) | ? [X73 : term,X74 : subst] : (((sub @ one @ (push @ X73 @ X74))) != X73)) & (! [X75 : term,X76 : subst] : (((sub @ one @ (push @ X75 @ X76))) = X75) | (axvarcons != $true)) & ((hoasinduction_lem3a_lthm = $true) | ($true != sP19)) & (($true = sP19) | (hoasinduction_lem3a_lthm != $true)) & (hoaslamnotap_lthm = $true) & ((sP17 = $true) | (hoasinduction_lem3v2_lthm != $true)) & ((hoasinduction_lem3v2_lthm = $true) | (sP17 != $true)) & (axclos = $true) & ! [X77 : subst] : ? [X78 : d_subst] : (((d2subst @ X78)) = X77) & (hoaslamnotvar = $true) & (hoasinduction_lem2_lthm = $true) & ! [X79 : d_subst] : (d_id = X79) & ((hoasapinj2 = $true) | ? [X80 : term,X81 : term,X82 : term,X83 : term] : ((((ap @ (sub @ X83 @ id) @ X81)) = ((ap @ (sub @ X80 @ id) @ X82))) & (X81 != X82))) & (! [X84 : term,X85 : term,X86 : term,X87 : term] : ((((ap @ (sub @ X84 @ id) @ X86)) != ((ap @ (sub @ X87 @ id) @ X85))) | (X85 = X86)) | (hoasapinj2 != $true)) & (((var @ (d2term @ d_one))) != $true) & (hoasinduction_lem1_gthm = $true) & ((pushprop_lem1_lthm = $true) | (! [X88 : subst,X89 : term] : (((sub @ one @ (push @ X89 @ X88))) = X89) & ? [X90 : (term > $o),X91 : term,X92 : (term > $o),X93 : term,X94 : subst] : ((((X90 @ (sub @ X91 @ (push @ X93 @ X94)))) != $true) & (((X92 @ X91)) = $true)) & ! [X95 : subst,X96 : term] : (((comp @ sh @ (push @ X96 @ X95))) = X95))) & (? [X97 : subst,X98 : term] : (((sub @ one @ (push @ X98 @ X97))) != X98) | ! [X99 : (term > $o),X100 : term,X101 : (term > $o),X102 : term,X103 : subst] : ((((X99 @ (sub @ X100 @ (push @ X102 @ X103)))) = $true) | (((X101 @ X100)) != $true)) | ? [X104 : subst,X105 : term] : (((comp @ sh @ (push @ X105 @ X104))) != X104) | (pushprop_lem1_lthm != $true)) & ((laminj = $true) | ? [X106 : term,X107 : term] : ((((lam @ X107)) = ((lam @ X106))) & (X106 != X107))) & (! [X108 : term,X109 : term] : ((((lam @ X109)) != ((lam @ X108))) | (X108 = X109)) | (laminj != $true)) & (lamnotap != $true) & (! [X110 : (subst > term > term),X111 : subst,X112 : term,X113 : (subst > term > term)] : ((((lam @ (X113 @ sh @ one))) != ((lam @ (X110 @ sh @ one)))) | ? [X114 : subst,X115 : term,X116 : subst] : (((X110 @ (comp @ X116 @ X114) @ (sub @ X115 @ X114))) != ((sub @ (X110 @ X116 @ X115) @ X114))) | ? [X117 : subst,X118 : subst,X119 : term] : (((X113 @ (comp @ X117 @ X118) @ (sub @ X119 @ X118))) != ((sub @ (X113 @ X117 @ X119) @ X118))) | (((X113 @ X111 @ X112)) = ((X110 @ X111 @ X112)))) | (hoaslaminj != $true)) & ((hoaslaminj = $true) | ? [X120 : (subst > term > term),X121 : subst,X122 : term,X123 : (subst > term > term)] : ((((lam @ (X123 @ sh @ one))) = ((lam @ (X120 @ sh @ one)))) & ! [X124 : subst,X125 : term,X126 : subst] : (((sub @ (X120 @ X126 @ X125) @ X124)) = ((X120 @ (comp @ X126 @ X124) @ (sub @ X125 @ X124)))) & ! [X127 : subst,X128 : subst,X129 : term] : (((X123 @ (comp @ X127 @ X128) @ (sub @ X129 @ X128))) = ((sub @ (X123 @ X127 @ X129) @ X128))) & (((X123 @ X121 @ X122)) != ((X120 @ X121 @ X122))))) & ((hoasinduction_lem3a = $true) | (sP15 != $true)) & ((sP15 = $true) | (hoasinduction_lem3a != $true)) & ((axvarid = $true) | ? [X130 : term] : (((sub @ X130 @ id)) != X130)) & (! [X131 : term] : (((sub @ X131 @ id)) = X131) | (axvarid != $true)) & ((sP14 = $true) | (hoasinduction_lem3aa != $true)) & ((hoasinduction_lem3aa = $true) | (sP14 != $true)) & (! [X132 : term] : ? [X133 : (subst > term > term)] : (((X133 @ sh @ one)) = ((sub @ X132 @ (push @ one @ sh)))) | (hoasinduction_lem3b != $true)) & ((hoasinduction_lem3b = $true) | ? [X134 : term] : ! [X135 : (subst > term > term)] : (((X135 @ sh @ one)) != ((sub @ X134 @ (push @ one @ sh))))) & (pushprop_lem1_gthm = $true) & (axvarshift = $true) & (((d2term @ d_one)) = one) & (pushprop_lem3v2_lthm = $true) & (hoasinduction_lem1v2_gthm = $true) & (hoasinduction_lem2_gthm = $true) & ((pushprop_lem1v2_lthm = $true) | (? [X136 : (term > $o),X137 : term,X138 : subst,X139 : (term > $o)] : ((((X139 @ X137)) = $true) & ($true != ((X136 @ one))) & ! [X140 : term] : (((X136 @ X140)) = ((X139 @ (sub @ X140 @ (push @ X137 @ X138)))))) & ! [X141 : term,X142 : subst] : (((sub @ one @ (push @ X141 @ X142))) = X141))) & (! [X143 : (term > $o),X144 : term,X145 : subst,X146 : (term > $o)] : ((((X146 @ X144)) != $true) | (((X143 @ one)) = $true) | ? [X147 : term] : (((X146 @ (sub @ X147 @ (push @ X144 @ X145)))) != ((X143 @ X147)))) | ? [X148 : term,X149 : subst] : (((sub @ one @ (push @ X148 @ X149))) != X148) | (pushprop_lem1v2_lthm != $true)) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (! [X150 : (term > $o),X151 : term,X152 : subst] : ? [X153 : (term > $o)] : ! [X154 : term] : (((X153 @ X154)) = ((X150 @ (sub @ X154 @ (push @ X151 @ X152))))) | (pushprop_lem0_lthm != $true)) & ((pushprop_lem0_lthm = $true) | ? [X155 : (term > $o),X156 : term,X157 : subst] : ! [X158 : (term > $o)] : ? [X159 : term] : (((X158 @ X159)) != ((X155 @ (sub @ X159 @ (push @ X156 @ X157)))))) & ((! [X160 : subst] : (((comp @ X160 @ id)) = X160) & ! [X161 : subst] : (((comp @ id @ X161)) = X161)) | (substmonoid != $true)) & ((substmonoid = $true) | ? [X162 : subst] : (((comp @ X162 @ id)) != X162) | ? [X163 : subst] : (((comp @ id @ X163)) != X163)) & (hoasinduction_lem1_lthm = $true) & (termmset_gthm = $true) & (pushprop_lem2v2_lthm = $true) & (pushprop_lem2v2 = $true) & ((induction = $true) | ? [X164 : term,X165 : (term > $o)] : ((((sP13 @ X165)) != $true) & (((X165 @ X164)) != $true) & ! [X166 : term] : ((((X165 @ X166)) != $true) | (((X165 @ (lam @ X166))) = $true)) & ! [X167 : term] : ((((var @ X167)) != $true) | (((X165 @ X167)) = $true)))) & (! [X168 : term,X169 : (term > $o)] : (($true = ((sP13 @ X169))) | (((X169 @ X168)) = $true) | ? [X170 : term] : ((((X169 @ X170)) = $true) & (((X169 @ (lam @ X170))) != $true)) | ? [X171 : term] : ((((var @ X171)) = $true) & (((X169 @ X171)) != $true))) | (induction != $true)) & (hoaslaminj_gthm = $true) & ((hoasapinj1 = $true) | ? [X172 : term,X173 : term,X174 : term,X175 : term] : ((X172 != X173) & (((ap @ (sub @ X173 @ id) @ X175)) = ((ap @ (sub @ X172 @ id) @ X174))))) & (! [X176 : term,X177 : term,X178 : term,X179 : term] : ((X176 = X177) | (((ap @ (sub @ X177 @ id) @ X179)) != ((ap @ (sub @ X176 @ id) @ X178)))) | (hoasapinj1 != $true)) & ((hoasinduction_lem3v2a_lthm = $true) | (sP12 != $true)) & ((sP12 = $true) | (hoasinduction_lem3v2a_lthm != $true)) & ((pushprop_lem1 = $true) | ? [X180 : (term > $o),X181 : term,X182 : (term > $o),X183 : subst,X184 : term] : ((((X180 @ X181)) = $true) & (((X182 @ (sub @ X181 @ (push @ X184 @ X183)))) != $true))) & (! [X185 : (term > $o),X186 : term,X187 : (term > $o),X188 : subst,X189 : term] : ((((X185 @ X186)) != $true) | (((X187 @ (sub @ X186 @ (push @ X189 @ X188)))) = $true)) | (pushprop_lem1 != $true)) & (! [X190 : term,X191 : term,X192 : term,X193 : term] : ((X191 = X193) | (((ap @ X193 @ X190)) != ((ap @ X191 @ X192)))) | (apinj1 != $true)) & ((apinj1 = $true) | ? [X194 : term,X195 : term,X196 : term,X197 : term] : ((X195 != X197) & (((ap @ X197 @ X194)) = ((ap @ X195 @ X196))))) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (pushprop_lem3v2 = $true) & ! [X198 : d_subst,X199 : d_subst] : ((((d2subst @ X198)) != ((d2subst @ X199))) | (X198 = X199)) & (ulamvar1 != $true) & (pushprop_lem1v2_gthm = $true) & (hoasapnotvar_lthm = $true) & (hoasinduction_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (? [X200 : term,X201 : term,X202 : term,X203 : term] : ((((ap @ X201 @ X202)) = ((ap @ X203 @ X200))) & (X200 != X202)) | ! [X204 : term,X205 : term,X206 : term,X207 : term] : ((((ap @ (sub @ X205 @ id) @ X207)) != ((ap @ (sub @ X204 @ id) @ X206))) | (X206 = X207)) | (hoasapinj2_lthm != $true)) & ((hoasapinj2_lthm = $true) | (! [X208 : term,X209 : term,X210 : term,X211 : term] : ((((ap @ X211 @ X208)) != ((ap @ X209 @ X210))) | (X208 = X210)) & ? [X212 : term,X213 : term,X214 : term,X215 : term] : ((((ap @ (sub @ X213 @ id) @ X215)) = ((ap @ (sub @ X212 @ id) @ X214))) & (X214 != X215)))) & ((hoasinduction_lem2 = $true) | (sP11 != $true)) & ((sP11 = $true) | (hoasinduction_lem2 != $true)) & ((sP9 = $true) | (hoaslaminj_lthm != $true)) & ((hoaslaminj_lthm = $true) | (sP9 != $true)) & (hoasapnotvar = $true) & (ulamvarsh = $true) & (hoasinduction_lem3v2_gthm = $true) & ((induction2lem = $true) | (sP8 != $true)) & ((sP8 = $true) | (induction2lem != $true)) & (! [X216 : subst] : (((push @ (sub @ one @ X216) @ (comp @ sh @ X216))) = X216) | (axscons != $true)) & ((axscons = $true) | ? [X217 : subst] : (((push @ (sub @ one @ X217) @ (comp @ sh @ X217))) != X217)) & (pushprop_lem2v2_gthm = $true) & (hoaslamnotvar_lthm = $true) & (lamnotvar = $true) & ((termmset = $true) | ? [X218 : term] : (((sub @ X218 @ id)) != X218)) & (! [X219 : term] : (((sub @ X219 @ id)) = X219) | (termmset != $true)) & ((hoasinduction_lem3v2a = $true) | (sP6 != $true)) & ((sP6 = $true) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3_lthm = $true) | (sP5 != $true)) & ((sP5 = $true) | (hoasinduction_lem3_lthm != $true)) & (pushprop_lthm = $true) & (((d2subst @ d_id)) = sh) & (! [X220 : (subst > term > term),X221 : term,X222 : term] : (? [X223 : term,X224 : subst,X225 : subst] : (((sub @ (X220 @ X225 @ X223) @ X224)) != ((X220 @ (comp @ X225 @ X224) @ (sub @ X223 @ X224)))) | (((lam @ (X220 @ sh @ one))) != ((ap @ (sub @ X222 @ id) @ X221)))) | (hoaslamnotap != $true)) & ((hoaslamnotap = $true) | ? [X226 : (subst > term > term),X227 : term,X228 : term] : (! [X229 : term,X230 : subst,X231 : subst] : (((sub @ (X226 @ X231 @ X229) @ X230)) = ((X226 @ (comp @ X231 @ X230) @ (sub @ X229 @ X230)))) & (((lam @ (X226 @ sh @ one))) = ((ap @ (sub @ X228 @ id) @ X227))))) & (((d2subst @ d_id)) = id) & ((hoasinduction_lem3aa_lthm = $true) | (sP2 != $true)) & ((sP2 = $true) | (hoasinduction_lem3aa_lthm != $true)) & (! [X232 : term,X233 : term,X234 : term,X235 : term] : ((((ap @ X235 @ X233)) != ((ap @ X232 @ X234))) | (X233 = X234)) | (apinj2 != $true)) & ((apinj2 = $true) | ? [X236 : term,X237 : term,X238 : term,X239 : term] : ((((ap @ X236 @ X238)) = ((ap @ X239 @ X237))) & (X237 != X238))) & ((hoasapinj1_lthm = $true) | (! [X240 : term,X241 : term,X242 : term,X243 : term] : ((((ap @ X240 @ X243)) != ((ap @ X241 @ X242))) | (X240 = X241)) & ? [X244 : term,X245 : term,X246 : term,X247 : term] : ((X244 != X245) & (((ap @ (sub @ X245 @ id) @ X247)) = ((ap @ (sub @ X244 @ id) @ X246)))) & ! [X248 : term] : (((sub @ X248 @ id)) = X248))) & (? [X249 : term,X250 : term,X251 : term,X252 : term] : ((((ap @ X250 @ X251)) = ((ap @ X249 @ X252))) & (X249 != X250)) | ! [X253 : term,X254 : term,X255 : term,X256 : term] : ((X253 = X254) | (((ap @ (sub @ X254 @ id) @ X256)) != ((ap @ (sub @ X253 @ id) @ X255)))) | ? [X257 : term] : (((sub @ X257 @ id)) != X257) | (hoasapinj1_lthm != $true)) & (! [X258 : term] : ? [X259 : (subst > term > term)] : ! [X260 : term,X261 : subst] : (((sub @ X258 @ (push @ X260 @ X261))) = ((X259 @ X261 @ X260))) | (hoasinduction_lem3v2_f_lthm != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X262 : term] : ! [X263 : (subst > term > term)] : ? [X264 : term,X265 : subst] : (((sub @ X262 @ (push @ X264 @ X265))) != ((X263 @ X265 @ X264)))) & (hoasinduction_lem3a_gthm = $true) & (hoasinduction_lem1 = $true) & (axmap = $true) & (! [X266 : term] : (((sub @ X266 @ id)) = X266) | ? [X267 : term] : (((sub @ X267 @ id)) != X267) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | (? [X268 : term] : (((sub @ X268 @ id)) != X268) & ! [X269 : term] : (((sub @ X269 @ id)) = X269))) & ((sP1 = $true) | (induction2 != $true)) & ((induction2 = $true) | (sP1 != $true)) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (induction2lem_gthm = $true) & (hoasapinj2_gthm = $true) & (hoasinduction_lem3_gthm = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (hoasapinj1_gthm = $true) & (pushprop_lthm_orig = $true) & (axassoc = $true) & (! [X270 : term,X271 : subst] : (((comp @ sh @ (push @ X270 @ X271))) = X271) | (axshiftcons != $true)) & ((axshiftcons = $true) | ? [X272 : term,X273 : subst] : (((comp @ sh @ (push @ X272 @ X273))) != X273)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2))))))))),
  inference(rectify,[],[f261])).
thf(f263,plain,(
  ((hoasinduction_lem3v2 = $true) | (sP58 != $true)) & ((sP58 = $true) | (hoasinduction_lem3v2 != $true)) & ((hoasinduction = $true) | (sP56 != $true)) & ((sP56 = $true) | (hoasinduction != $true)) & (axabs = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((pushprop_lem1v2 = $true) | (! [X4 : term] : (((sK459 @ (sub @ X4 @ (push @ sK458 @ sK460)))) = ((sK457 @ X4))) & (((sK457 @ one)) != $true) & (((sK459 @ sK458)) = $true))) & (! [X5 : (term > $o),X6 : term,X7 : (term > $o),X8 : subst] : ((((X5 @ (sK461 @ X8 @ X7 @ X6 @ X5))) != ((X7 @ (sub @ (sK461 @ X8 @ X7 @ X6 @ X5) @ (push @ X6 @ X8))))) | (((X5 @ one)) = $true) | (((X7 @ X6)) != $true)) | (pushprop_lem1v2 != $true)) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ((shinj = $true) | ((sK463 != sK462) & (((sub @ sK462 @ sh)) = ((sub @ sK463 @ sh))))) & (! [X12 : term,X13 : term] : ((X12 = X13) | (((sub @ X13 @ sh)) != ((sub @ X12 @ sh)))) | (shinj != $true)) & ((hoasinduction_lem2v2 = $true) | (sP53 != $true)) & ((sP53 = $true) | (hoasinduction_lem2v2 != $true)) & ! [X14 : d_term,X15 : d_term] : ((X14 = X15) | (((d2term @ X14)) != ((d2term @ X15)))) & ((hoasinduction_lem3v2_f = $true) | ! [X17 : (subst > term > term)] : (((sub @ sK464 @ (push @ (sK466 @ X17) @ (sK465 @ X17)))) != ((X17 @ (sK465 @ X17) @ (sK466 @ X17))))) & (! [X20 : term] : ! [X22 : subst,X23 : term] : (((sK467 @ X20 @ X22 @ X23)) = ((sub @ X20 @ (push @ X23 @ X22)))) | (hoasinduction_lem3v2_f != $true)) & ! [X24 : d_term] : (d_one = X24) & (axapp = $true) & (ulamvarind = $true) & (hoasinduction_lem2v2_gthm = $true) & ((hoasinduction_lthm = $true) | ($true != sP51)) & (($true = sP51) | (hoasinduction_lthm != $true)) & (! [X25 : term] : (((sK468 @ X25 @ sh @ one)) = ((sub @ X25 @ (push @ one @ sh)))) | (hoasinduction_lem3b_lthm != $true)) & ((hoasinduction_lem3b_lthm = $true) | ! [X28 : (subst > term > term)] : (((X28 @ sh @ one)) != ((sub @ sK469 @ (push @ one @ sh))))) & ((sP42 = $true) | (hoasinduction_lem3aaa != $true)) & ((hoasinduction_lem3aaa = $true) | (sP42 != $true)) & ((axidr = $true) | (sK470 != ((comp @ sK470 @ id)))) & (! [X30 : subst] : (((comp @ X30 @ id)) = X30) | (axidr != $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (induction2_gthm = $true) & (hoaslamnotvar_gthm = $true) & (pushprop = $true) & (! [X31 : term,X32 : (term > $o),X33 : subst] : ! [X35 : term] : (((X32 @ (sub @ X35 @ (push @ X31 @ X33)))) = ((sK471 @ X33 @ X32 @ X31 @ X35))) | (pushprop_lem0 != $true)) & ((pushprop_lem0 = $true) | ! [X39 : (term > $o)] : (((X39 @ (sK475 @ X39))) != ((sK473 @ (sub @ (sK475 @ X39) @ (push @ sK472 @ sK474)))))) & (apnotvar = $true) & (hoasinduction_lem1v2 = $true) & (! [X41 : (subst > term > subst > $o)] : ! [X43 : term] : (((sK476 @ X41 @ X43)) = ((X41 @ id @ X43 @ id))) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ! [X45 : (term > $o)] : (((sK477 @ id @ (sK478 @ X45) @ id)) != ((X45 @ (sK478 @ X45))))) & ((hoasinduction_lem0 = $true) | ! [X48 : (term > $o)] : (((sK479 @ id @ (sK480 @ X48) @ id)) != ((X48 @ (sK480 @ X48))))) & (! [X50 : (subst > term > subst > $o)] : ! [X52 : term] : (((sK481 @ X50 @ X52)) = ((X50 @ id @ X52 @ id))) | (hoasinduction_lem0 != $true)) & (($true = sP41) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | ($true != sP41)) & (pushprop_lem0_gthm = $true) & ((hoasinduction_no_psi_cond_lthm = $true) | ($true != sP37)) & (($true = sP37) | (hoasinduction_no_psi_cond_lthm != $true)) & (hoaslamnotap_gthm = $true) & ((sP32 = $true) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | (sP32 != $true)) & (pushprop_gthm = $true) & (hoasinduction_lem3b_gthm = $true) & (substmonoid_gthm = $true) & ! [X53 : term] : (((d2term @ (sK482 @ X53))) = X53) & ((sP26 = $true) | (induction2_lthm != $true)) & ((induction2_lthm = $true) | (sP26 != $true)) & (! [X55 : term,X56 : (subst > term > subst > $o)] : (((((X56 @ id @ (sK483 @ X56) @ id)) = $true) & ($true != ((X56 @ id @ (ap @ (sub @ (sK484 @ X56) @ id) @ (sK483 @ X56)) @ id))) & ($true = ((X56 @ id @ (sK484 @ X56) @ id)))) | (((sP22 @ X56)) = $true) | ($true = ((X56 @ id @ X55 @ id)))) | (hoasinduction_no_psi_cond != $true)) & ((hoasinduction_no_psi_cond = $true) | (! [X61 : term,X62 : term] : ((((sK486 @ id @ X61 @ id)) != $true) | ($true = ((sK486 @ id @ (ap @ (sub @ X62 @ id) @ X61) @ id))) | (((sK486 @ id @ X62 @ id)) != $true)) & (((sP22 @ sK486)) != $true) & ($true != ((sK486 @ id @ sK485 @ id))))) & ((axidl = $true) | (sK487 != ((comp @ id @ sK487)))) & (! [X64 : subst] : (((comp @ id @ X64)) = X64) | (axidl != $true)) & (hoasapnotvar_gthm = $true) & ((! [X65 : subst] : (((comp @ X65 @ id)) = X65) & ! [X66 : subst] : (((comp @ id @ X66)) = X66)) | (sK488 != ((comp @ id @ sK488))) | (sK489 != ((comp @ sK489 @ id))) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | (((sK490 != ((comp @ sK490 @ id))) | (sK491 != ((comp @ id @ sK491)))) & ! [X71 : subst] : (((comp @ id @ X71)) = X71) & ! [X72 : subst] : (((comp @ X72 @ id)) = X72))) & ((hoasinduction_lem3 = $true) | ($true != sP21)) & (($true = sP21) | (hoasinduction_lem3 != $true)) & ((axvarcons = $true) | (((sub @ one @ (push @ sK492 @ sK493))) != sK492)) & (! [X75 : term,X76 : subst] : (((sub @ one @ (push @ X75 @ X76))) = X75) | (axvarcons != $true)) & ((hoasinduction_lem3a_lthm = $true) | ($true != sP19)) & (($true = sP19) | (hoasinduction_lem3a_lthm != $true)) & (hoaslamnotap_lthm = $true) & ((sP17 = $true) | (hoasinduction_lem3v2_lthm != $true)) & ((hoasinduction_lem3v2_lthm = $true) | (sP17 != $true)) & (axclos = $true) & ! [X77 : subst] : (((d2subst @ (sK494 @ X77))) = X77) & (hoaslamnotvar = $true) & (hoasinduction_lem2_lthm = $true) & ! [X79 : d_subst] : (d_id = X79) & ((hoasapinj2 = $true) | ((((ap @ (sub @ sK495 @ id) @ sK497)) = ((ap @ (sub @ sK498 @ id) @ sK496))) & (sK496 != sK497))) & (! [X84 : term,X85 : term,X86 : term,X87 : term] : ((((ap @ (sub @ X84 @ id) @ X86)) != ((ap @ (sub @ X87 @ id) @ X85))) | (X85 = X86)) | (hoasapinj2 != $true)) & (((var @ (d2term @ d_one))) != $true) & (hoasinduction_lem1_gthm = $true) & ((pushprop_lem1_lthm = $true) | (! [X88 : subst,X89 : term] : (((sub @ one @ (push @ X89 @ X88))) = X89) & (($true != ((sK499 @ (sub @ sK500 @ (push @ sK502 @ sK503))))) & ($true = ((sK501 @ sK500)))) & ! [X95 : subst,X96 : term] : (((comp @ sh @ (push @ X96 @ X95))) = X95))) & ((sK505 != ((sub @ one @ (push @ sK505 @ sK504)))) | ! [X99 : (term > $o),X100 : term,X101 : (term > $o),X102 : term,X103 : subst] : ((((X99 @ (sub @ X100 @ (push @ X102 @ X103)))) = $true) | (((X101 @ X100)) != $true)) | (sK506 != ((comp @ sh @ (push @ sK507 @ sK506)))) | (pushprop_lem1_lthm != $true)) & ((laminj = $true) | ((((lam @ sK509)) = ((lam @ sK508))) & (sK509 != sK508))) & (! [X108 : term,X109 : term] : ((((lam @ X109)) != ((lam @ X108))) | (X108 = X109)) | (laminj != $true)) & (lamnotap != $true) & (! [X110 : (subst > term > term),X111 : subst,X112 : term,X113 : (subst > term > term)] : ((((lam @ (X113 @ sh @ one))) != ((lam @ (X110 @ sh @ one)))) | (((X110 @ (comp @ (sK512 @ X110) @ (sK510 @ X110)) @ (sub @ (sK511 @ X110) @ (sK510 @ X110)))) != ((sub @ (X110 @ (sK512 @ X110) @ (sK511 @ X110)) @ (sK510 @ X110)))) | (((sub @ (X113 @ (sK513 @ X113) @ (sK515 @ X113)) @ (sK514 @ X113))) != ((X113 @ (comp @ (sK513 @ X113) @ (sK514 @ X113)) @ (sub @ (sK515 @ X113) @ (sK514 @ X113))))) | (((X113 @ X111 @ X112)) = ((X110 @ X111 @ X112)))) | (hoaslaminj != $true)) & ((hoaslaminj = $true) | ((((lam @ (sK519 @ sh @ one))) = ((lam @ (sK516 @ sh @ one)))) & ! [X124 : subst,X125 : term,X126 : subst] : (((sub @ (sK516 @ X126 @ X125) @ X124)) = ((sK516 @ (comp @ X126 @ X124) @ (sub @ X125 @ X124)))) & ! [X127 : subst,X128 : subst,X129 : term] : (((sK519 @ (comp @ X127 @ X128) @ (sub @ X129 @ X128))) = ((sub @ (sK519 @ X127 @ X129) @ X128))) & (((sK519 @ sK517 @ sK518)) != ((sK516 @ sK517 @ sK518))))) & ((hoasinduction_lem3a = $true) | (sP15 != $true)) & ((sP15 = $true) | (hoasinduction_lem3a != $true)) & ((axvarid = $true) | (((sub @ sK520 @ id)) != sK520)) & (! [X131 : term] : (((sub @ X131 @ id)) = X131) | (axvarid != $true)) & ((sP14 = $true) | (hoasinduction_lem3aa != $true)) & ((hoasinduction_lem3aa = $true) | (sP14 != $true)) & (! [X132 : term] : (((sK521 @ X132 @ sh @ one)) = ((sub @ X132 @ (push @ one @ sh)))) | (hoasinduction_lem3b != $true)) & ((hoasinduction_lem3b = $true) | ! [X135 : (subst > term > term)] : (((X135 @ sh @ one)) != ((sub @ sK522 @ (push @ one @ sh))))) & (pushprop_lem1_gthm = $true) & (axvarshift = $true) & (((d2term @ d_one)) = one) & (pushprop_lem3v2_lthm = $true) & (hoasinduction_lem1v2_gthm = $true) & (hoasinduction_lem2_gthm = $true) & ((pushprop_lem1v2_lthm = $true) | (((((sK526 @ sK524)) = $true) & (((sK523 @ one)) != $true) & ! [X140 : term] : (((sK526 @ (sub @ X140 @ (push @ sK524 @ sK525)))) = ((sK523 @ X140)))) & ! [X141 : term,X142 : subst] : (((sub @ one @ (push @ X141 @ X142))) = X141))) & (! [X143 : (term > $o),X144 : term,X145 : subst,X146 : (term > $o)] : ((((X146 @ X144)) != $true) | (((X143 @ one)) = $true) | (((X143 @ (sK527 @ X146 @ X145 @ X144 @ X143))) != ((X146 @ (sub @ (sK527 @ X146 @ X145 @ X144 @ X143) @ (push @ X144 @ X145)))))) | (sK528 != ((sub @ one @ (push @ sK528 @ sK529)))) | (pushprop_lem1v2_lthm != $true)) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (! [X150 : (term > $o),X151 : term,X152 : subst] : ! [X154 : term] : (((sK530 @ X152 @ X151 @ X150 @ X154)) = ((X150 @ (sub @ X154 @ (push @ X151 @ X152))))) | (pushprop_lem0_lthm != $true)) & ((pushprop_lem0_lthm = $true) | ! [X158 : (term > $o)] : (((sK531 @ (sub @ (sK534 @ X158) @ (push @ sK532 @ sK533)))) != ((X158 @ (sK534 @ X158))))) & ((! [X160 : subst] : (((comp @ X160 @ id)) = X160) & ! [X161 : subst] : (((comp @ id @ X161)) = X161)) | (substmonoid != $true)) & ((substmonoid = $true) | (((comp @ sK535 @ id)) != sK535) | (((comp @ id @ sK536)) != sK536)) & (hoasinduction_lem1_lthm = $true) & (termmset_gthm = $true) & (pushprop_lem2v2_lthm = $true) & (pushprop_lem2v2 = $true) & ((induction = $true) | ((((sP13 @ sK538)) != $true) & (((sK538 @ sK537)) != $true) & ! [X166 : term] : ((((sK538 @ X166)) != $true) | ($true = ((sK538 @ (lam @ X166))))) & ! [X167 : term] : ((((var @ X167)) != $true) | (((sK538 @ X167)) = $true)))) & (! [X168 : term,X169 : (term > $o)] : (($true = ((sP13 @ X169))) | (((X169 @ X168)) = $true) | ((((X169 @ (sK539 @ X169))) = $true) & ($true != ((X169 @ (lam @ (sK539 @ X169)))))) | ((((var @ (sK540 @ X169))) = $true) & ($true != ((X169 @ (sK540 @ X169)))))) | (induction != $true)) & (hoaslaminj_gthm = $true) & ((hoasapinj1 = $true) | ((sK542 != sK541) & (((ap @ (sub @ sK541 @ id) @ sK543)) = ((ap @ (sub @ sK542 @ id) @ sK544))))) & (! [X176 : term,X177 : term,X178 : term,X179 : term] : ((X176 = X177) | (((ap @ (sub @ X177 @ id) @ X179)) != ((ap @ (sub @ X176 @ id) @ X178)))) | (hoasapinj1 != $true)) & ((hoasinduction_lem3v2a_lthm = $true) | (sP12 != $true)) & ((sP12 = $true) | (hoasinduction_lem3v2a_lthm != $true)) & ((pushprop_lem1 = $true) | ((((sK545 @ sK546)) = $true) & (((sK547 @ (sub @ sK546 @ (push @ sK549 @ sK548)))) != $true))) & (! [X185 : (term > $o),X186 : term,X187 : (term > $o),X188 : subst,X189 : term] : ((((X185 @ X186)) != $true) | (((X187 @ (sub @ X186 @ (push @ X189 @ X188)))) = $true)) | (pushprop_lem1 != $true)) & (! [X190 : term,X191 : term,X192 : term,X193 : term] : ((X191 = X193) | (((ap @ X193 @ X190)) != ((ap @ X191 @ X192)))) | (apinj1 != $true)) & ((apinj1 = $true) | ((sK551 != sK553) & (((ap @ sK553 @ sK550)) = ((ap @ sK551 @ sK552))))) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (pushprop_lem3v2 = $true) & ! [X198 : d_subst,X199 : d_subst] : ((((d2subst @ X198)) != ((d2subst @ X199))) | (X198 = X199)) & (ulamvar1 != $true) & (pushprop_lem1v2_gthm = $true) & (hoasapnotvar_lthm = $true) & (hoasinduction_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (((((ap @ sK557 @ sK554)) = ((ap @ sK555 @ sK556))) & (sK554 != sK556)) | ! [X204 : term,X205 : term,X206 : term,X207 : term] : ((((ap @ (sub @ X205 @ id) @ X207)) != ((ap @ (sub @ X204 @ id) @ X206))) | (X206 = X207)) | (hoasapinj2_lthm != $true)) & ((hoasapinj2_lthm = $true) | (! [X208 : term,X209 : term,X210 : term,X211 : term] : ((((ap @ X211 @ X208)) != ((ap @ X209 @ X210))) | (X208 = X210)) & ((((ap @ (sub @ sK559 @ id) @ sK561)) = ((ap @ (sub @ sK558 @ id) @ sK560))) & (sK560 != sK561)))) & ((hoasinduction_lem2 = $true) | (sP11 != $true)) & ((sP11 = $true) | (hoasinduction_lem2 != $true)) & ((sP9 = $true) | (hoaslaminj_lthm != $true)) & ((hoaslaminj_lthm = $true) | (sP9 != $true)) & (hoasapnotvar = $true) & (ulamvarsh = $true) & (hoasinduction_lem3v2_gthm = $true) & ((induction2lem = $true) | (sP8 != $true)) & ((sP8 = $true) | (induction2lem != $true)) & (! [X216 : subst] : (((push @ (sub @ one @ X216) @ (comp @ sh @ X216))) = X216) | (axscons != $true)) & ((axscons = $true) | (((push @ (sub @ one @ sK562) @ (comp @ sh @ sK562))) != sK562)) & (pushprop_lem2v2_gthm = $true) & (hoaslamnotvar_lthm = $true) & (lamnotvar = $true) & ((termmset = $true) | (((sub @ sK563 @ id)) != sK563)) & (! [X219 : term] : (((sub @ X219 @ id)) = X219) | (termmset != $true)) & ((hoasinduction_lem3v2a = $true) | (sP6 != $true)) & ((sP6 = $true) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3_lthm = $true) | (sP5 != $true)) & ((sP5 = $true) | (hoasinduction_lem3_lthm != $true)) & (pushprop_lthm = $true) & (((d2subst @ d_id)) = sh) & (! [X220 : (subst > term > term),X221 : term,X222 : term] : ((((X220 @ (comp @ (sK566 @ X220) @ (sK565 @ X220)) @ (sub @ (sK564 @ X220) @ (sK565 @ X220)))) != ((sub @ (X220 @ (sK566 @ X220) @ (sK564 @ X220)) @ (sK565 @ X220)))) | (((lam @ (X220 @ sh @ one))) != ((ap @ (sub @ X222 @ id) @ X221)))) | (hoaslamnotap != $true)) & ((hoaslamnotap = $true) | (! [X229 : term,X230 : subst,X231 : subst] : (((sK567 @ (comp @ X231 @ X230) @ (sub @ X229 @ X230))) = ((sub @ (sK567 @ X231 @ X229) @ X230))) & (((ap @ (sub @ sK569 @ id) @ sK568)) = ((lam @ (sK567 @ sh @ one)))))) & (((d2subst @ d_id)) = id) & ((hoasinduction_lem3aa_lthm = $true) | (sP2 != $true)) & ((sP2 = $true) | (hoasinduction_lem3aa_lthm != $true)) & (! [X232 : term,X233 : term,X234 : term,X235 : term] : ((((ap @ X235 @ X233)) != ((ap @ X232 @ X234))) | (X233 = X234)) | (apinj2 != $true)) & ((apinj2 = $true) | ((((ap @ sK573 @ sK571)) = ((ap @ sK570 @ sK572))) & (sK572 != sK571))) & ((hoasapinj1_lthm = $true) | (! [X240 : term,X241 : term,X242 : term,X243 : term] : ((((ap @ X240 @ X243)) != ((ap @ X241 @ X242))) | (X240 = X241)) & ((sK574 != sK575) & (((ap @ (sub @ sK575 @ id) @ sK577)) = ((ap @ (sub @ sK574 @ id) @ sK576)))) & ! [X248 : term] : (((sub @ X248 @ id)) = X248))) & (((((ap @ sK578 @ sK581)) = ((ap @ sK579 @ sK580))) & (sK579 != sK578)) | ! [X253 : term,X254 : term,X255 : term,X256 : term] : ((X253 = X254) | (((ap @ (sub @ X254 @ id) @ X256)) != ((ap @ (sub @ X253 @ id) @ X255)))) | (sK582 != ((sub @ sK582 @ id))) | (hoasapinj1_lthm != $true)) & (! [X258 : term] : ! [X260 : term,X261 : subst] : (((sub @ X258 @ (push @ X260 @ X261))) = ((sK583 @ X258 @ X261 @ X260))) | (hoasinduction_lem3v2_f_lthm != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ! [X263 : (subst > term > term)] : (((X263 @ (sK586 @ X263) @ (sK585 @ X263))) != ((sub @ sK584 @ (push @ (sK585 @ X263) @ (sK586 @ X263)))))) & (hoasinduction_lem3a_gthm = $true) & (hoasinduction_lem1 = $true) & (axmap = $true) & (! [X266 : term] : (((sub @ X266 @ id)) = X266) | (sK587 != ((sub @ sK587 @ id))) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | ((((sub @ sK588 @ id)) != sK588) & ! [X269 : term] : (((sub @ X269 @ id)) = X269))) & ((sP1 = $true) | (induction2 != $true)) & ((induction2 = $true) | (sP1 != $true)) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (induction2lem_gthm = $true) & (hoasapinj2_gthm = $true) & (hoasinduction_lem3_gthm = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (hoasapinj1_gthm = $true) & (pushprop_lthm_orig = $true) & (axassoc = $true) & (! [X270 : term,X271 : subst] : (((comp @ sh @ (push @ X270 @ X271))) = X271) | (axshiftcons != $true)) & ((axshiftcons = $true) | (sK590 != ((comp @ sh @ (push @ sK589 @ sK590))))) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y0 @ id @ Y2 @ id) = (Y1 @ Y2))))))))),
  inference(skolemize,[status(esa),new_symbols(skolem,[sK457,sK458,sK459,sK460,vAPP,sK462,sK463,sK464,vAPP,sK469,sK470,vAPP,sK472,sK473,sK474,vAPP,sK477,vAPP,sK479,vAPP,sK485,sK486,sK487,sK488,sK489,sK490,sK491,sK492,sK493,vAPP,sK495,sK496,sK497,sK498,sK499,sK500,sK501,sK502,sK503,sK504,sK505,sK506,sK507,sK508,sK509,vAPP,sK516,sK517,sK518,sK519,sK520,vAPP,sK522,sK523,sK524,sK525,sK526,vAPP,sK528,sK529,vAPP,sK531,sK532,sK533,vAPP,sK535,sK536,sK537,sK538,vAPP,sK541,sK542,sK543,sK544,sK545,sK546,sK547,sK548,sK549,sK550,sK551,sK552,sK553,sK554,sK555,sK556,sK557,sK558,sK559,sK560,sK561,sK562,sK563,vAPP,sK567,sK568,sK569,sK570,sK571,sK572,sK573,sK574,sK575,sK576,sK577,sK578,sK579,sK580,sK581,sK582,vAPP,sK584,vAPP,sK587,sK588,sK589,sK590]),skolemize(X0,$thf(sK457)),skolemize(X1,$thf(sK458)),skolemize(X2,$thf(sK459)),skolemize(X3,$thf(sK460)),skolemize(X25,$thf(sK71 @ X14 @ X13)),skolemize(X10,$thf(sK462)),skolemize(X11,$thf(sK463)),skolemize(X16,$thf(sK464)),skolemize(X25,$thf(sK71 @ X14 @ X13)),skolemize(X25,$thf(sK71 @ X14 @ X13)),skolemize(X25,$thf(sK71 @ X14 @ X13)),skolemize(X25,$thf(sK71 @ X14 @ X13)),skolemize(X27,$thf(sK469)),skolemize(X29,$thf(sK470)),skolemize(X25,$thf(sK71 @ X14 @ X13)),skolemize(X36,$thf(sK472)),skolemize(X37,$thf(sK473)),skolemize(X38,$thf(sK474)),skolemize(X25,$thf(sK71 @ X14 @ X13)),skolemize(X25,$thf(sK71 @ X14 @ X13)),skolemize(X44,$thf(sK477)),skolemize(X25,$thf(sK71 @ X14 @ X13)),skolemize(X47,$thf(sK479)),skolemize(X25,$thf(sK71 @ X14 @ X13)),skolemize(X25,$thf(sK71 @ X14 @ X13)),skolemize(X25,$thf(sK71 @ X14 @ X13)),skolemize(X25,$thf(sK71 @ X14 @ X13)),skolemize(X25,$thf(sK71 @ X14 @ X13)),skolemize(X59,$thf(sK485)),skolemize(X60,$thf(sK486)),skolemize(X63,$thf(sK487)),skolemize(X67,$thf(sK488)),skolemize(X68,$thf(sK489)),skolemize(X69,$thf(sK490)),skolemize(X70,$thf(sK491)),skolemize(X73,$thf(sK492)),skolemize(X74,$thf(sK493)),skolemize(X25,$thf(sK71 @ X14 @ X13)),skolemize(X80,$thf(sK495)),skolemize(X81,$thf(sK496)),skolemize(X82,$thf(sK497)),skolemize(X83,$thf(sK498)),skolemize(X90,$thf(sK499)),skolemize(X91,$thf(sK500)),skolemize(X92,$thf(sK501)),skolemize(X93,$thf(sK502)),skolemize(X94,$thf(sK503)),skolemize(X97,$thf(sK504)),skolemize(X98,$thf(sK505)),skolemize(X104,$thf(sK506)),skolemize(X105,$thf(sK507)),skolemize(X106,$thf(sK508)),skolemize(X107,$thf(sK509)),skolemize(X25,$thf(sK71 @ X14 @ X13)),skolemize(X25,$thf(sK71 @ X14 @ X13)),skolemize(X25,$thf(sK71 @ X14 @ X13)),skolemize(X25,$thf(sK71 @ X14 @ X13)),skolemize(X25,$thf(sK71 @ X14 @ X13)),skolemize(X25,$thf(sK71 @ X14 @ X13)),skolemize(X120,$thf(sK516)),skolemize(X121,$thf(sK517)),skolemize(X122,$thf(sK518)),skolemize(X123,$thf(sK519)),skolemize(X130,$thf(sK520)),skolemize(X25,$thf(sK71 @ X14 @ X13)),skolemize(X134,$thf(sK522)),skolemize(X136,$thf(sK523)),skolemize(X137,$thf(sK524)),skolemize(X138,$thf(sK525)),skolemize(X139,$thf(sK526)),skolemize(X25,$thf(sK71 @ X14 @ X13)),skolemize(X148,$thf(sK528)),skolemize(X149,$thf(sK529)),skolemize(X25,$thf(sK71 @ X14 @ X13)),skolemize(X155,$thf(sK531)),skolemize(X156,$thf(sK532)),skolemize(X157,$thf(sK533)),skolemize(X25,$thf(sK71 @ X14 @ X13)),skolemize(X162,$thf(sK535)),skolemize(X163,$thf(sK536)),skolemize(X164,$thf(sK537)),skolemize(X165,$thf(sK538)),skolemize(X25,$thf(sK71 @ X14 @ X13)),skolemize(X25,$thf(sK71 @ X14 @ X13)),skolemize(X172,$thf(sK541)),skolemize(X173,$thf(sK542)),skolemize(X174,$thf(sK543)),skolemize(X175,$thf(sK544)),skolemize(X180,$thf(sK545)),skolemize(X181,$thf(sK546)),skolemize(X182,$thf(sK547)),skolemize(X183,$thf(sK548)),skolemize(X184,$thf(sK549)),skolemize(X194,$thf(sK550)),skolemize(X195,$thf(sK551)),skolemize(X196,$thf(sK552)),skolemize(X197,$thf(sK553)),skolemize(X200,$thf(sK554)),skolemize(X201,$thf(sK555)),skolemize(X202,$thf(sK556)),skolemize(X203,$thf(sK557)),skolemize(X212,$thf(sK558)),skolemize(X213,$thf(sK559)),skolemize(X214,$thf(sK560)),skolemize(X215,$thf(sK561)),skolemize(X217,$thf(sK562)),skolemize(X218,$thf(sK563)),skolemize(X25,$thf(sK71 @ X14 @ X13)),skolemize(X25,$thf(sK71 @ X14 @ X13)),skolemize(X25,$thf(sK71 @ X14 @ X13)),skolemize(X226,$thf(sK567)),skolemize(X227,$thf(sK568)),skolemize(X228,$thf(sK569)),skolemize(X236,$thf(sK570)),skolemize(X237,$thf(sK571)),skolemize(X238,$thf(sK572)),skolemize(X239,$thf(sK573)),skolemize(X244,$thf(sK574)),skolemize(X245,$thf(sK575)),skolemize(X246,$thf(sK576)),skolemize(X247,$thf(sK577)),skolemize(X249,$thf(sK578)),skolemize(X250,$thf(sK579)),skolemize(X251,$thf(sK580)),skolemize(X252,$thf(sK581)),skolemize(X257,$thf(sK582)),skolemize(X25,$thf(sK71 @ X14 @ X13)),skolemize(X262,$thf(sK584)),skolemize(X25,$thf(sK71 @ X14 @ X13)),skolemize(X25,$thf(sK71 @ X14 @ X13)),skolemize(X267,$thf(sK587)),skolemize(X268,$thf(sK588)),skolemize(X272,$thf(sK589)),skolemize(X273,$thf(sK590))],[f262])).
thf(f264,plain,(
  (axclos != ((!! @ subst @ (^[Y0 : subst]: (!! @ subst @ (^[Y1 : subst]: (!! @ term @ (^[Y2 : term]: ((sub @ (sub @ Y2 @ Y0) @ Y1) = (sub @ Y2 @ (comp @ Y0 @ Y1)))))))))))),
  inference(cnf_transformation,[],[f8])).
thf(f898,plain,(
  (axclos = $true)),
  inference(cnf_transformation,[],[f263])).
thf(f924,plain,(
  ( ! [X53 : term] : ((((d2term @ (sK482 @ X53))) = X53)) )),
  inference(cnf_transformation,[],[f263])).
thf(f959,plain,(
  ( ! [X24 : d_term] : ((d_one = X24)) )),
  inference(cnf_transformation,[],[f263])).
thf(f981,plain,(
  (((!! @ subst @ (^[Y0 : subst]: (!! @ subst @ (^[Y1 : subst]: (!! @ term @ (^[Y2 : term]: ((sub @ (sub @ Y2 @ Y0) @ Y1) = (sub @ Y2 @ (comp @ Y0 @ Y1)))))))))) != $true)),
  inference(definition_unfolding,[],[f264,f898])).
thf(f1026,plain,(
  ((((^[Y0 : subst]: (!! @ subst @ (^[Y1 : subst]: (!! @ term @ (^[Y2 : term]: ((sub @ (sub @ Y2 @ Y0) @ Y1) = (sub @ Y2 @ (comp @ Y0 @ Y1)))))))) @ sK591)) = $false)),
  inference(sigma_proxy_clausification,[],[f981])).
thf(f1027,plain,(
  (((!! @ subst @ (^[Y0 : subst]: (!! @ term @ (^[Y1 : term]: ((sub @ (sub @ Y1 @ sK591) @ Y0) = (sub @ Y1 @ (comp @ sK591 @ Y0)))))))) = $false)),
  inference(beta-eta_normalization,[],[f1026])).
thf(f1028,plain,(
  ($false = (((^[Y0 : subst]: (!! @ term @ (^[Y1 : term]: ((sub @ (sub @ Y1 @ sK591) @ Y0) = (sub @ Y1 @ (comp @ sK591 @ Y0)))))) @ sK592)))),
  inference(sigma_proxy_clausification,[],[f1027])).
thf(f1029,plain,(
  (((!! @ term @ (^[Y0 : term]: ((sub @ (sub @ Y0 @ sK591) @ sK592) = (sub @ Y0 @ (comp @ sK591 @ sK592)))))) = $false)),
  inference(beta-eta_normalization,[],[f1028])).
thf(f1030,plain,(
  ($false = (((^[Y0 : term]: ((sub @ (sub @ Y0 @ sK591) @ sK592) = (sub @ Y0 @ (comp @ sK591 @ sK592)))) @ sK593)))),
  inference(sigma_proxy_clausification,[],[f1029])).
thf(f1031,plain,(
  ($false = (((sub @ (sub @ sK593 @ sK591) @ sK592) = (sub @ sK593 @ (comp @ sK591 @ sK592)))))),
  inference(beta-eta_normalization,[],[f1030])).
thf(f1032,plain,(
  (((sub @ sK593 @ (comp @ sK591 @ sK592))) != ((sub @ (sub @ sK593 @ sK591) @ sK592)))),
  inference(equality_proxy_clausification,[],[f1031])).
thf(f3979,plain,(
  ( ! [X53 : term] : ((((d2term @ d_one)) = X53)) )),
  inference(forward_demodulation,[],[f924,f959])).
thf(f3990,plain,(
  (((d2term @ d_one)) != ((sub @ sK593 @ (comp @ sK591 @ sK592))))),
  inference(superposition,[],[f1032,f3979])).
thf(f3992,plain,(
  $false),
  inference(forward_subsumption_resolution,[],[f3990,f3979])).
% SZS output end Proof for theBenchmark