%----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) > $o))).
thf(func_def_128, type, vIMP: ($o > $o > $o)).
thf(func_def_129, type, db0: !>[X0: $tType]:(X0)).
thf(func_def_130, type, db2: !>[X0: $tType]:(X0)).
thf(func_def_131, type, db1: !>[X0: $tType]:(X0)).
thf(func_def_132, type, db3: !>[X0: $tType]:(X0)).
thf(func_def_133, type, vLAM: !>[X0: $tType, X1: $tType]:((X1) > (X0 > X1))).
thf(func_def_136, type, db4: !>[X0: $tType]:(X0)).
thf(func_def_137, type, vEQ: !>[X0: $tType]:((X0 > X0 > $o))).
thf(func_def_138, type, sP0: $o).
thf(func_def_139, type, sP1: ((subst > term > subst > $o) > $o)).
thf(func_def_140, type, sP2: ((subst > term > subst > $o) > $o)).
thf(func_def_141, type, sP3: ((subst > term > subst > $o) > $o)).
thf(func_def_142, type, sP4: $o).
thf(func_def_143, type, sP5: ((subst > term > subst > $o) > $o)).
thf(func_def_144, type, sP6: ((term > $o) > $o)).
thf(func_def_145, type, sP7: ((term > $o) > $o)).
thf(func_def_146, type, sP8: ((subst > term > subst > $o) > $o)).
thf(func_def_147, type, sP9: ((subst > term > subst > $o) > $o)).
thf(func_def_148, type, sP10: $o).
thf(func_def_149, type, sP11: $o).
thf(func_def_150, type, sP12: $o).
thf(func_def_151, type, sP13: ((term > $o) > $o)).
thf(func_def_152, type, sP14: ((term > $o) > $o)).
thf(func_def_153, type, sP15: $o).
thf(func_def_154, type, sP16: ((subst > term > subst > $o) > $o)).
thf(func_def_155, type, sP17: $o).
thf(func_def_156, type, sP18: ((subst > term > subst > $o) > $o)).
thf(func_def_157, type, sP19: $o).
thf(func_def_158, type, sP20: ((term > $o) > $o)).
thf(func_def_159, type, sP21: ((term > $o) > $o)).
thf(func_def_160, type, sP22: $o).
thf(func_def_161, type, sP23: ((subst > term > subst > $o) > $o)).
thf(func_def_162, type, sP24: $o).
thf(func_def_163, type, sP25: ((subst > term > subst > $o) > $o)).
thf(func_def_164, type, sP26: (term > (term > $o) > $o)).
thf(func_def_165, type, sP27: ((subst > term > subst > $o) > $o)).
thf(func_def_166, type, sP28: $o).
thf(func_def_167, type, sP29: ((subst > term > subst > $o) > $o)).
thf(func_def_168, type, sP30: ((subst > term > subst > $o) > $o)).
thf(func_def_169, type, sP31: $o).
thf(func_def_170, type, sP32: ((subst > term > subst > $o) > $o)).
thf(func_def_171, type, sP33: $o).
thf(func_def_172, type, sP34: ((subst > term > subst > $o) > $o)).
thf(func_def_173, type, sP35: $o).
thf(func_def_174, type, sP36: $o).
thf(func_def_175, type, sP37: $o).
thf(func_def_176, type, sP38: ((subst > term > subst > $o) > $o)).
thf(func_def_177, type, sP39: ((subst > term > subst > $o) > $o)).
thf(func_def_178, type, sP40: ((subst > term > subst > $o) > $o)).
thf(func_def_179, type, sP41: ((subst > term > subst > $o) > $o)).
thf(func_def_180, type, sP42: ((term > $o) > $o)).
thf(func_def_181, type, sP43: ((term > $o) > $o)).
thf(func_def_182, type, sP44: ((term > $o) > $o)).
thf(func_def_183, type, sP45: ((subst > term > subst > $o) > $o)).
thf(func_def_184, type, sP46: $o).
thf(func_def_185, type, sP47: $o).
thf(func_def_186, type, sP48: $o).
thf(func_def_187, type, sP49: ((subst > term > subst > $o) > $o)).
thf(func_def_188, type, sP50: ((subst > term > subst > $o) > $o)).
thf(func_def_189, type, sP51: ((subst > term > subst > $o) > $o)).
thf(func_def_190, type, sP52: ((subst > term > subst > $o) > $o)).
thf(func_def_191, type, sP53: $o).
thf(func_def_192, type, sP54: $o).
thf(func_def_193, type, sP55: ((term > $o) > $o)).
thf(func_def_194, type, sP56: ((term > $o) > $o)).
thf(func_def_195, type, sP57: ((term > $o) > $o)).
thf(func_def_196, type, sP58: ((term > $o) > $o)).
thf(func_def_197, type, sP59: $o).
thf(func_def_198, type, sP60: $o).
thf(func_def_199, type, sP61: ((subst > term > subst > $o) > $o)).
thf(func_def_200, type, sP62: ((subst > term > subst > $o) > $o)).
thf(func_def_201, type, sP63: ((subst > term > subst > $o) > $o)).
thf(func_def_202, type, sP64: ((subst > term > subst > $o) > $o)).
thf(func_def_203, type, sP65: $o).
thf(func_def_204, type, sP66: ((subst > term > subst > $o) > $o)).
thf(func_def_205, type, sP67: ((subst > term > subst > $o) > $o)).
thf(func_def_206, type, sP68: ((subst > term > subst > $o) > $o)).
thf(func_def_207, type, sP69: ((subst > term > subst > $o) > $o)).
thf(func_def_208, type, sP70: $o).
thf(func_def_209, type, sP71: $o).
thf(func_def_210, type, sP72: ((term > $o) > $o)).
thf(func_def_211, type, sP73: ((term > $o) > $o)).
thf(func_def_212, type, sP74: $o).
thf(func_def_213, type, sP75: $o).
thf(func_def_214, type, sP76: $o).
thf(func_def_215, type, sP77: ((term > $o) > $o)).
thf(func_def_216, type, sP78: ((term > $o) > $o)).
thf(func_def_217, type, sP79: ((subst > term > subst > $o) > $o)).
thf(func_def_218, type, sP80: ((subst > term > subst > $o) > $o)).
thf(func_def_219, type, sP81: ((subst > term > subst > $o) > $o)).
thf(func_def_220, type, sP82: ((subst > term > subst > $o) > $o)).
thf(func_def_221, type, sP83: ((subst > term > subst > $o) > $o)).
thf(func_def_222, type, sP84: (term > (subst > term > subst > $o) > $o)).
thf(func_def_223, type, sP85: ((subst > term > subst > $o) > $o)).
thf(func_def_224, type, sP86: ((subst > term > subst > $o) > $o)).
thf(func_def_225, type, sP87: ((subst > term > subst > $o) > $o)).
thf(func_def_226, type, sP88: $o).
thf(func_def_227, type, sP89: $o).
thf(func_def_228, type, sP90: $o).
thf(func_def_229, type, sP91: $o).
thf(func_def_230, type, sP92: ((term > $o) > $o)).
thf(func_def_231, type, sP93: ((term > $o) > $o)).
thf(func_def_232, type, sP94: ((term > $o) > $o)).
thf(func_def_233, type, sP95: ((term > $o) > $o)).
thf(func_def_234, type, sP96: $o).
thf(func_def_235, type, sP97: $o).
thf(func_def_236, type, sP98: ((subst > term > subst > $o) > $o)).
thf(func_def_237, type, sP99: $o).
thf(func_def_238, type, sP100: ((subst > term > subst > $o) > $o)).
thf(func_def_239, type, sP101: ((subst > term > subst > $o) > $o)).
thf(func_def_240, type, sP102: $o).
thf(func_def_241, type, sP103: $o).
thf(func_def_242, type, sP104: $o).
thf(func_def_243, type, sP105: ((subst > term > subst > $o) > $o)).
thf(func_def_244, type, sP106: (term > (subst > term > subst > $o) > $o)).
thf(func_def_245, type, sP107: ((subst > term > subst > $o) > $o)).
thf(func_def_246, type, sP108: $o).
thf(func_def_247, type, sK109: (subst > term > subst > $o)).
thf(func_def_248, type, sK110: term).
thf(func_def_249, type, sK111: ((subst > term > subst > $o) > subst)).
thf(func_def_250, type, sK112: ((subst > term > subst > $o) > subst)).
thf(func_def_251, type, sK113: ((subst > term > subst > $o) > term)).
thf(func_def_252, type, sK114: ((subst > term > subst > $o) > subst)).
thf(func_def_253, type, sK115: ((subst > term > subst > $o) > term)).
thf(func_def_254, type, sK116: ((subst > term > subst > $o) > subst)).
thf(func_def_255, type, sK117: ((subst > term > subst > $o) > subst)).
thf(func_def_256, type, sK118: ((subst > term > subst > $o) > subst)).
thf(func_def_257, type, sK119: (term > (subst > term > subst > $o) > term)).
thf(func_def_258, type, sK120: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_259, type, sK121: ((subst > term > term) > subst)).
thf(func_def_260, type, sK122: ((subst > term > term) > term)).
thf(func_def_261, type, sK123: ((subst > term > term) > subst)).
thf(func_def_262, type, sK124: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_263, type, sK125: term).
thf(func_def_264, type, sK126: term).
thf(func_def_265, type, sK127: term).
thf(func_def_266, type, sK128: term).
thf(func_def_267, type, sK129: term).
thf(func_def_268, type, sK130: term).
thf(func_def_269, type, sK131: term).
thf(func_def_270, type, sK132: term).
thf(func_def_271, type, sK133: term).
thf(func_def_272, type, sK134: term).
thf(func_def_273, type, sK135: (subst > term > subst > $o)).
thf(func_def_274, type, sK136: term).
thf(func_def_275, type, sK137: (term > $o)).
thf(func_def_276, type, sK138: ((subst > term > subst > $o) > subst)).
thf(func_def_277, type, sK139: ((subst > term > subst > $o) > term)).
thf(func_def_278, type, sK140: ((subst > term > subst > $o) > subst)).
thf(func_def_279, type, sK141: ((subst > term > subst > $o) > subst)).
thf(func_def_280, type, sK142: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_281, type, sK143: ((subst > term > subst > $o) > term)).
thf(func_def_282, type, sK144: ((subst > term > subst > $o) > subst)).
thf(func_def_283, type, sK145: ((subst > term > subst > $o) > subst)).
thf(func_def_284, type, sK146: ((subst > term > subst > $o) > subst)).
thf(func_def_285, type, sK147: ((subst > term > subst > $o) > term)).
thf(func_def_286, type, sK148: ((subst > term > subst > $o) > term)).
thf(func_def_287, type, sK149: (subst > term > subst > $o)).
thf(func_def_288, type, sK150: term).
thf(func_def_289, type, sK151: (term > (subst > term > subst > $o) > term)).
thf(func_def_290, type, sK152: ((subst > term > term) > subst)).
thf(func_def_291, type, sK153: ((subst > term > term) > term)).
thf(func_def_292, type, sK154: ((subst > term > term) > subst)).
thf(func_def_293, type, sK155: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_294, type, sK156: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_295, type, sK157: ((term > $o) > subst > term)).
thf(func_def_296, type, sK158: subst).
thf(func_def_297, type, sK159: (term > $o)).
thf(func_def_298, type, sK160: term).
thf(func_def_299, type, sK161: (term > $o)).
thf(func_def_300, type, sK162: term).
thf(func_def_301, type, sK163: ((term > $o) > term)).
thf(func_def_302, type, sK164: ((term > $o) > term)).
thf(func_def_303, type, sK165: (term > (term > $o) > term)).
thf(func_def_304, type, sK166: ((term > $o) > term)).
thf(func_def_305, type, sK167: ((term > $o) > term)).
thf(func_def_306, type, sK168: ((term > $o) > term)).
thf(func_def_307, type, sK169: ((term > $o) > term)).
thf(func_def_308, type, sK170: (term > (term > $o) > term)).
thf(func_def_309, type, sK171: ((term > $o) > term)).
thf(func_def_310, type, sK172: ((subst > term > subst > $o) > subst)).
thf(func_def_311, type, sK173: ((subst > term > subst > $o) > subst)).
thf(func_def_312, type, sK174: ((subst > term > subst > $o) > term)).
thf(func_def_313, type, sK175: ((subst > term > subst > $o) > subst)).
thf(func_def_314, type, sK176: (subst > term > subst > $o)).
thf(func_def_315, type, sK177: term).
thf(func_def_316, type, sK178: term).
thf(func_def_317, type, sK179: ((subst > term > subst > $o) > subst)).
thf(func_def_318, type, sK180: ((subst > term > subst > $o) > subst)).
thf(func_def_319, type, sK181: ((subst > term > subst > $o) > subst)).
thf(func_def_320, type, sK182: ((subst > term > subst > $o) > term)).
thf(func_def_321, type, sK183: (subst > term > subst > $o)).
thf(func_def_322, type, sK184: term).
thf(func_def_323, type, sK185: ((term > $o) > term)).
thf(func_def_324, type, sK186: (term > $o)).
thf(func_def_325, type, sK187: term).
thf(func_def_326, type, sK188: (subst > term > subst > $o)).
thf(func_def_327, type, sK189: term).
thf(func_def_328, type, sK190: ((subst > term > subst > $o) > term)).
thf(func_def_329, type, sK191: ((subst > term > subst > $o) > subst)).
thf(func_def_330, type, sK192: ((subst > term > subst > $o) > subst)).
thf(func_def_331, type, sK193: ((subst > term > subst > $o) > subst)).
thf(func_def_332, type, sK194: ((subst > term > subst > $o) > subst)).
thf(func_def_333, type, sK195: ((subst > term > subst > $o) > subst)).
thf(func_def_334, type, sK196: ((subst > term > subst > $o) > term)).
thf(func_def_335, type, sK197: ((subst > term > subst > $o) > subst)).
thf(func_def_336, type, sK198: ((subst > term > subst > $o) > term)).
thf(func_def_337, type, sK199: ((subst > term > subst > $o) > term)).
thf(func_def_338, type, sK200: ((subst > term > subst > $o) > subst)).
thf(func_def_339, type, sK201: ((subst > term > subst > $o) > subst)).
thf(func_def_340, type, sK202: ((subst > term > subst > $o) > term)).
thf(func_def_341, type, sK203: ((subst > term > subst > $o) > subst)).
thf(func_def_342, type, sK204: (term > (subst > term > subst > $o) > term)).
thf(func_def_343, type, sK205: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_344, type, sK206: ((subst > term > term) > subst)).
thf(func_def_345, type, sK207: ((subst > term > term) > term)).
thf(func_def_346, type, sK208: ((subst > term > term) > subst)).
thf(func_def_347, type, sK209: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_348, type, sK210: ((subst > term > subst > $o) > subst)).
thf(func_def_349, type, sK211: ((subst > term > subst > $o) > subst)).
thf(func_def_350, type, sK212: ((subst > term > subst > $o) > term)).
thf(func_def_351, type, sK213: ((subst > term > subst > $o) > subst)).
thf(func_def_352, type, sK214: ((subst > term > subst > $o) > term)).
thf(func_def_353, type, sK215: ((subst > term > subst > $o) > term)).
thf(func_def_354, type, sK216: ((subst > term > subst > $o) > term)).
thf(func_def_355, type, sK217: ((subst > term > term) > term)).
thf(func_def_356, type, sK218: ((subst > term > term) > subst)).
thf(func_def_357, type, sK219: ((subst > term > term) > subst)).
thf(func_def_358, type, sK220: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_359, type, sK221: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_360, type, sK222: ((term > $o) > term)).
thf(func_def_361, type, sK223: (term > (term > $o) > term)).
thf(func_def_362, type, sK224: ((term > $o) > term)).
thf(func_def_363, type, sK225: ((term > $o) > term)).
thf(func_def_364, type, sK226: term).
thf(func_def_365, type, sK227: term).
thf(func_def_366, type, sK228: term).
thf(func_def_367, type, sK229: term).
thf(func_def_368, type, sK230: term).
thf(func_def_369, type, sK231: term).
thf(func_def_370, type, sK232: term).
thf(func_def_371, type, sK233: term).
thf(func_def_372, type, sK234: term).
thf(func_def_373, type, sK235: (term > $o)).
thf(func_def_374, type, sK236: ((term > $o) > term)).
thf(func_def_375, type, sK237: ((term > $o) > term)).
thf(func_def_376, type, sK238: ((term > $o) > term)).
thf(func_def_377, type, sK239: (term > (term > $o) > term)).
thf(func_def_378, type, sK240: ((term > $o) > term)).
thf(func_def_379, type, sK241: ((subst > term > subst > $o) > term > term)).
thf(func_def_380, type, sK242: term).
thf(func_def_381, type, sK243: (subst > term > subst > $o)).
thf(func_def_382, type, sK244: term).
thf(func_def_383, type, sK245: (subst > term > subst > $o)).
thf(func_def_384, type, sK246: ((subst > term > subst > $o) > term > term)).
thf(func_def_385, type, sK247: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_386, type, sK248: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_387, type, sK249: ((subst > term > term) > subst)).
thf(func_def_388, type, sK250: ((subst > term > term) > term)).
thf(func_def_389, type, sK251: ((subst > term > term) > subst)).
thf(func_def_390, type, sK252: ((subst > term > subst > $o) > subst)).
thf(func_def_391, type, sK253: ((subst > term > subst > $o) > term)).
thf(func_def_392, type, sK254: ((subst > term > subst > $o) > subst)).
thf(func_def_393, type, sK255: ((subst > term > subst > $o) > subst)).
thf(func_def_394, type, sK256: ((subst > term > subst > $o) > subst)).
thf(func_def_395, type, sK257: ((subst > term > subst > $o) > subst)).
thf(func_def_396, type, sK258: ((subst > term > subst > $o) > subst)).
thf(func_def_397, type, sK259: ((subst > term > subst > $o) > term)).
thf(func_def_398, type, sK260: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_399, type, sK261: ((subst > term > term) > term)).
thf(func_def_400, type, sK262: ((subst > term > term) > subst)).
thf(func_def_401, type, sK263: ((subst > term > term) > subst)).
thf(func_def_402, type, sK264: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_403, type, sK265: term).
thf(func_def_404, type, sK266: (subst > term > subst > $o)).
thf(func_def_405, type, sK267: ((subst > term > subst > $o) > term)).
thf(func_def_406, type, sK268: ((subst > term > subst > $o) > subst)).
thf(func_def_407, type, sK269: ((subst > term > subst > $o) > subst)).
thf(func_def_408, type, sK270: ((subst > term > subst > $o) > subst)).
thf(func_def_409, type, sK271: ((subst > term > subst > $o) > term)).
thf(func_def_410, type, sK272: ((subst > term > subst > $o) > subst)).
thf(func_def_411, type, sK273: ((subst > term > subst > $o) > term)).
thf(func_def_412, type, sK274: ((subst > term > subst > $o) > subst)).
thf(func_def_413, type, sK275: ((subst > term > subst > $o) > subst)).
thf(func_def_414, type, sK276: ((subst > term > subst > $o) > term)).
thf(func_def_415, type, sK277: ((subst > term > subst > $o) > term)).
thf(func_def_416, type, sK278: ((subst > term > term) > subst)).
thf(func_def_417, type, sK279: ((subst > term > term) > term)).
thf(func_def_418, type, sK280: ((subst > term > term) > subst)).
thf(func_def_419, type, sK281: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_420, type, sK282: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_421, type, sK283: ((term > $o) > term)).
thf(func_def_422, type, sK284: (term > $o)).
thf(func_def_423, type, sK285: term).
thf(func_def_424, type, sK286: (term > $o)).
thf(func_def_425, type, sK287: subst).
thf(func_def_426, type, sK288: term).
thf(func_def_427, type, sK289: (subst > (term > $o) > term)).
thf(func_def_428, type, sK290: ((term > $o) > term)).
thf(func_def_429, type, sK291: ((term > $o) > term)).
thf(func_def_430, type, sK292: ((term > $o) > term)).
thf(func_def_431, type, sK293: ((term > $o) > term)).
thf(func_def_432, type, sK294: ((term > $o) > term)).
thf(func_def_433, type, sK295: (term > (term > $o) > term)).
thf(func_def_434, type, sK296: ((term > $o) > term)).
thf(func_def_435, type, sK297: ((subst > term > subst > $o) > term > term)).
thf(func_def_436, type, sK298: term).
thf(func_def_437, type, sK299: (subst > term > subst > $o)).
thf(func_def_438, type, sK300: term).
thf(func_def_439, type, sK301: (subst > term > subst > $o)).
thf(func_def_440, type, sK302: ((subst > term > subst > $o) > term > term)).
thf(func_def_441, type, sK303: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_442, type, sK304: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_443, type, sK305: ((subst > term > term) > term)).
thf(func_def_444, type, sK306: ((subst > term > term) > subst)).
thf(func_def_445, type, sK307: ((subst > term > term) > subst)).
thf(func_def_446, type, sK308: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_447, type, sK309: ((subst > term > term) > term)).
thf(func_def_448, type, sK310: ((subst > term > term) > subst)).
thf(func_def_449, type, sK311: ((subst > term > term) > subst)).
thf(func_def_450, type, sK312: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_451, type, sK313: ((subst > term > subst > $o) > term)).
thf(func_def_452, type, sK314: ((subst > term > subst > $o) > term)).
thf(func_def_453, type, sK315: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_454, type, sK316: ((subst > term > term) > term)).
thf(func_def_455, type, sK317: ((subst > term > term) > subst)).
thf(func_def_456, type, sK318: ((subst > term > term) > subst)).
thf(func_def_457, type, sK319: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_458, type, sK320: (term > $o)).
thf(func_def_459, type, sK321: term).
thf(func_def_460, type, sK322: ((subst > term > subst > $o) > (term > $o) > term)).
thf(func_def_461, type, sK323: ((term > $o) > term > term)).
thf(func_def_462, type, sK324: term).
thf(func_def_463, type, sK325: (term > $o)).
thf(func_def_464, type, sK326: (subst > term > subst > $o)).
thf(func_def_465, type, sK327: term).
thf(func_def_466, type, sK328: (subst > term > subst > $o)).
thf(func_def_467, type, sK329: ((subst > term > subst > $o) > subst)).
thf(func_def_468, type, sK330: ((subst > term > subst > $o) > subst)).
thf(func_def_469, type, sK331: ((subst > term > subst > $o) > term)).
thf(func_def_470, type, sK332: ((subst > term > subst > $o) > subst)).
thf(func_def_471, type, sK333: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_472, type, sK334: ((subst > term > term) > subst)).
thf(func_def_473, type, sK335: ((subst > term > term) > subst)).
thf(func_def_474, type, sK336: ((subst > term > term) > term)).
thf(func_def_475, type, sK337: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_476, type, sK338: ((term > $o) > term)).
thf(func_def_477, type, sK339: (term > (term > $o) > term)).
thf(func_def_478, type, sK340: ((term > $o) > term)).
thf(func_def_479, type, sK341: ((term > $o) > term)).
thf(func_def_480, type, sK342: ((term > $o) > term)).
thf(func_def_481, type, sK343: ((subst > term > subst > $o) > term)).
thf(func_def_482, type, sK344: ((subst > term > subst > $o) > subst)).
thf(func_def_483, type, sK345: ((subst > term > subst > $o) > subst)).
thf(func_def_484, type, sK346: ((subst > term > subst > $o) > subst)).
thf(func_def_485, type, sK347: ((subst > term > subst > $o) > term)).
thf(func_def_486, type, sK348: ((subst > term > subst > $o) > term)).
thf(func_def_487, type, sK349: ((subst > term > subst > $o) > term)).
thf(func_def_488, type, sK350: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_489, type, sK351: ((subst > term > term) > subst)).
thf(func_def_490, type, sK352: ((subst > term > term) > subst)).
thf(func_def_491, type, sK353: ((subst > term > term) > term)).
thf(func_def_492, type, sK354: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_493, type, sK355: subst).
thf(func_def_494, type, sK356: term).
thf(func_def_495, type, sK357: term).
thf(func_def_496, type, sK358: term).
thf(func_def_497, type, sK359: subst).
thf(func_def_498, type, sK360: term).
thf(func_def_499, type, sK361: term).
thf(func_def_500, type, sK362: (subst > term > term)).
thf(func_def_501, type, sK363: subst).
thf(func_def_502, type, sK364: (subst > term > term)).
thf(func_def_503, type, sK365: ((subst > term > term) > subst)).
thf(func_def_504, type, sK366: ((subst > term > term) > term)).
thf(func_def_505, type, sK367: ((subst > term > term) > subst)).
thf(func_def_506, type, sK368: ((subst > term > term) > subst)).
thf(func_def_507, type, sK369: ((subst > term > term) > subst)).
thf(func_def_508, type, sK370: ((subst > term > term) > term)).
thf(func_def_509, type, sK371: (subst > term > subst > $o)).
thf(func_def_510, type, sK372: term).
thf(func_def_511, type, sK373: (term > (subst > term > subst > $o) > term)).
thf(func_def_512, type, sK374: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_513, type, sK375: ((subst > term > term) > subst)).
thf(func_def_514, type, sK376: ((subst > term > term) > term)).
thf(func_def_515, type, sK377: ((subst > term > term) > subst)).
thf(func_def_516, type, sK378: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_517, type, sK379: (term > $o)).
thf(func_def_518, type, sK380: term).
thf(func_def_519, type, sK381: (subst > term > subst > $o)).
thf(func_def_520, type, sK382: ((subst > term > subst > $o) > (term > $o) > term)).
thf(func_def_521, type, sK383: (term > (term > $o) > term)).
thf(func_def_522, type, sK384: ((subst > term > term) > subst)).
thf(func_def_523, type, sK385: ((subst > term > term) > term)).
thf(func_def_524, type, sK386: ((subst > term > term) > subst)).
thf(func_def_525, type, sK387: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_526, type, sK388: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_527, type, sK389: term).
thf(func_def_528, type, sK390: term).
thf(func_def_529, type, sK391: (subst > term > subst > $o)).
thf(func_def_530, type, sK392: ((subst > term > subst > $o) > subst)).
thf(func_def_531, type, sK393: ((subst > term > subst > $o) > subst)).
thf(func_def_532, type, sK394: ((subst > term > subst > $o) > subst)).
thf(func_def_533, type, sK395: ((subst > term > subst > $o) > term)).
thf(func_def_534, type, sK396: ((subst > term > subst > $o) > subst)).
thf(func_def_535, type, sK397: ((subst > term > subst > $o) > term)).
thf(func_def_536, type, sK398: ((subst > term > subst > $o) > subst)).
thf(func_def_537, type, sK399: ((subst > term > subst > $o) > subst)).
thf(func_def_538, type, sK400: ((subst > term > subst > $o) > term)).
thf(func_def_539, type, sK401: ((subst > term > subst > $o) > term)).
thf(func_def_540, type, sK402: (subst > term > subst > $o)).
thf(func_def_541, type, sK403: (term > $o)).
thf(func_def_542, type, sK404: term).
thf(func_def_543, type, sK405: ((subst > term > subst > $o) > subst)).
thf(func_def_544, type, sK406: ((subst > term > subst > $o) > subst)).
thf(func_def_545, type, sK407: ((subst > term > subst > $o) > subst)).
thf(func_def_546, type, sK408: ((subst > term > subst > $o) > term)).
thf(func_def_547, type, sK409: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_548, type, sK410: ((subst > term > subst > $o) > term)).
thf(func_def_549, type, sK411: ((subst > term > subst > $o) > subst)).
thf(func_def_550, type, sK412: ((subst > term > subst > $o) > subst)).
thf(func_def_551, type, sK413: ((subst > term > subst > $o) > subst)).
thf(func_def_552, type, sK414: (term > (term > $o) > term)).
thf(func_def_553, type, sK415: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_554, type, sK416: ((subst > term > term) > subst)).
thf(func_def_555, type, sK417: ((subst > term > term) > term)).
thf(func_def_556, type, sK418: ((subst > term > term) > subst)).
thf(func_def_557, type, sK419: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_558, type, sK420: (subst > term > subst > $o)).
thf(func_def_559, type, sK421: term).
thf(func_def_560, type, sK422: (term > (subst > term > subst > $o) > term)).
thf(func_def_561, type, sK423: ((subst > term > term) > term)).
thf(func_def_562, type, sK424: ((subst > term > term) > subst)).
thf(func_def_563, type, sK425: ((subst > term > term) > subst)).
thf(func_def_564, type, sK426: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_565, type, sK427: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_566, type, sK428: term).
thf(func_def_567, type, sK429: subst).
thf(func_def_568, type, sK430: (term > $o)).
thf(func_def_569, type, sK431: ((term > $o) > subst > term)).
thf(func_def_570, type, sK432: ((term > $o) > term)).
thf(func_def_571, type, sK433: ((term > $o) > term)).
thf(func_def_572, type, sK434: (term > (term > $o) > term)).
thf(func_def_573, type, sK435: ((term > $o) > term)).
thf(func_def_574, type, sK436: term).
thf(func_def_575, type, sK437: (subst > term > subst > $o)).
thf(func_def_576, type, sK438: (term > $o)).
thf(func_def_577, type, sK439: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_578, type, sK440: ((term > $o) > term > term)).
thf(func_def_579, type, sK441: ((subst > term > term) > subst)).
thf(func_def_580, type, sK442: ((subst > term > term) > subst)).
thf(func_def_581, type, sK443: ((subst > term > term) > term)).
thf(func_def_582, type, sK444: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_583, type, sK445: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_584, type, sK446: (subst > term > subst > $o)).
thf(func_def_585, type, sK447: term).
thf(func_def_586, type, sK448: (term > (subst > term > subst > $o) > term)).
thf(func_def_587, type, sK449: ((subst > term > term) > term > subst)).
thf(func_def_588, type, sK450: ((subst > term > term) > term > subst)).
thf(func_def_589, type, sK451: ((subst > term > term) > term > term)).
thf(func_def_590, type, sK452: ((subst > term > term) > term > subst)).
thf(func_def_591, type, sK453: ((subst > term > term) > term > term)).
thf(func_def_592, type, sK454: ((subst > term > term) > term > subst)).
thf(func_def_593, type, sK455: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_594, type, sK456: ((subst > term > subst > $o) > term)).
thf(func_def_595, type, sK457: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_596, type, sK458: (term > $o)).
thf(func_def_597, type, sK459: term).
thf(func_def_598, type, sK460: ((term > $o) > term)).
thf(func_def_599, type, sK461: ((term > $o) > term)).
thf(func_def_600, type, sK462: ((term > $o) > term)).
thf(func_def_601, type, sK463: ((term > $o) > term)).
thf(func_def_602, type, sK464: ((term > $o) > term)).
thf(func_def_603, type, sK465: term).
thf(func_def_604, type, sK466: (term > $o)).
thf(func_def_605, type, sK467: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_606, type, sK468: ((term > $o) > term > term)).
thf(func_def_607, type, sK469: term).
thf(func_def_608, type, sK470: (subst > term > subst > $o)).
thf(func_def_609, type, sK471: (term > $o)).
thf(func_def_610, type, sK472: (subst > term > subst > $o)).
thf(func_def_611, type, sK473: term).
thf(func_def_612, type, sK474: ((subst > term > subst > $o) > term)).
thf(func_def_613, type, sK475: ((subst > term > subst > $o) > term)).
thf(func_def_614, type, sK476: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_615, type, sK477: ((subst > term > term) > subst)).
thf(func_def_616, type, sK478: ((subst > term > term) > subst)).
thf(func_def_617, type, sK479: ((subst > term > term) > term)).
thf(func_def_618, type, sK480: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_619, type, sK481: ((term > $o) > term)).
thf(func_def_620, type, sK482: (term > (term > $o) > term)).
thf(func_def_621, type, sK483: ((term > $o) > term)).
thf(func_def_622, type, sK484: ((term > $o) > term)).
thf(func_def_623, type, sK485: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_624, type, sK486: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_625, type, sK487: ((subst > term > term) > subst)).
thf(func_def_626, type, sK488: ((subst > term > term) > term)).
thf(func_def_627, type, sK489: ((subst > term > term) > subst)).
thf(func_def_628, type, sK490: term).
thf(func_def_629, type, sK491: (subst > term > subst > $o)).
thf(func_def_630, type, sK492: (term > $o)).
thf(func_def_631, type, sK493: ((term > $o) > term > term)).
thf(func_def_632, type, sK494: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_633, type, sK495: ((subst > term > subst > $o) > subst)).
thf(func_def_634, type, sK496: ((subst > term > subst > $o) > subst)).
thf(func_def_635, type, sK497: ((subst > term > subst > $o) > term)).
thf(func_def_636, type, sK498: ((subst > term > subst > $o) > subst)).
thf(func_def_637, type, sK499: ((subst > term > subst > $o) > subst)).
thf(func_def_638, type, sK500: ((subst > term > subst > $o) > term)).
thf(func_def_639, type, sK501: ((subst > term > subst > $o) > subst)).
thf(func_def_640, type, sK502: ((subst > term > subst > $o) > subst)).
thf(func_def_641, type, sK503: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_642, type, sK504: ((subst > term > term) > subst)).
thf(func_def_643, type, sK505: ((subst > term > term) > subst)).
thf(func_def_644, type, sK506: ((subst > term > term) > term)).
thf(func_def_645, type, sK507: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_646, type, sK508: subst).
thf(func_def_647, type, sK509: subst).
thf(func_def_648, type, sK510: subst).
thf(func_def_649, type, sK511: subst).
thf(func_def_650, type, sK512: (subst > term > term)).
thf(func_def_651, type, sK513: term).
thf(func_def_652, type, sK514: subst).
thf(func_def_653, type, sK515: (subst > term > term)).
thf(func_def_654, type, sK516: ((subst > term > term) > subst)).
thf(func_def_655, type, sK517: ((subst > term > term) > term)).
thf(func_def_656, type, sK518: ((subst > term > term) > subst)).
thf(func_def_657, type, sK519: ((subst > term > term) > term)).
thf(func_def_658, type, sK520: ((subst > term > term) > subst)).
thf(func_def_659, type, sK521: ((subst > term > term) > subst)).
thf(func_def_660, type, sK522: subst).
thf(func_def_661, type, sK523: term).
thf(func_def_662, type, sK524: (term > $o)).
thf(func_def_663, type, sK525: term).
thf(func_def_664, type, sK526: subst).
thf(func_def_665, type, sK527: ((term > $o) > term)).
thf(func_def_666, type, sK528: (subst > term > (term > $o) > term > $o)).
thf(func_def_667, type, sK529: term).
thf(func_def_668, type, sK530: ((subst > term > term) > subst)).
thf(func_def_669, type, sK531: ((subst > term > term) > term)).
thf(func_def_670, type, sK532: (term > subst > term > term)).
thf(func_def_671, type, sK533: ((subst > term > term) > term)).
thf(func_def_672, type, sK534: ((subst > term > term) > subst)).
thf(func_def_673, type, sK535: ((subst > term > term) > subst)).
thf(func_def_674, type, sK536: term).
thf(func_def_675, type, sK537: (subst > term > term)).
thf(func_def_676, type, sK538: term).
thf(func_def_677, type, sK539: term).
thf(func_def_678, type, sK540: subst).
thf(func_def_679, type, sK541: term).
thf(func_def_680, type, sK542: term).
thf(func_def_681, type, sK543: subst).
thf(func_def_682, type, sK544: (term > $o)).
thf(func_def_683, type, sK545: (term > $o)).
thf(func_def_684, type, sK546: subst).
thf(func_def_685, type, sK547: term).
thf(func_def_686, type, sK548: subst).
thf(func_def_687, type, sK549: term).
thf(func_def_688, type, sK550: term).
thf(func_def_689, type, sK551: term).
thf(func_def_690, type, sK552: subst).
thf(func_def_691, type, sK553: (term > subst > (term > $o) > (term > $o) > term)).
thf(func_def_692, type, sK554: (term > $o)).
thf(func_def_693, type, sK555: (term > $o)).
thf(func_def_694, type, sK556: subst).
thf(func_def_695, type, sK557: term).
thf(func_def_696, type, sK558: (term > $o)).
thf(func_def_697, type, sK559: subst).
thf(func_def_698, type, sK560: term).
thf(func_def_699, type, sK561: (term > $o)).
thf(func_def_700, type, sK562: ((term > $o) > term > subst > (term > $o) > term)).
thf(func_def_701, type, sK563: term).
thf(func_def_702, type, sK564: term).
thf(func_def_703, type, sK565: term).
thf(func_def_704, type, sK566: term).
thf(func_def_705, type, sK567: term).
thf(func_def_706, type, sK568: subst).
thf(func_def_707, type, sK569: term).
thf(func_def_708, type, sK570: subst).
thf(func_def_709, type, sK571: subst).
thf(func_def_710, type, sK572: term).
thf(func_def_711, type, sK573: term).
thf(func_def_712, type, sK574: term).
thf(func_def_713, type, sK575: term).
thf(func_def_714, type, sK576: (term > subst > term > term)).
thf(func_def_715, type, sK577: term).
thf(func_def_716, type, sK578: term).
thf(func_def_717, type, sK579: (subst > term > subst > $o)).
thf(func_def_718, type, sK580: (term > d_term)).
thf(func_def_719, type, sK581: term).
thf(func_def_720, type, sK582: (subst > term > subst > $o)).
thf(func_def_721, type, sK583: ((term > $o) > term)).
thf(func_def_722, type, sK584: ((subst > term > subst > $o) > term > $o)).
thf(func_def_723, type, sK585: (term > subst > term > term)).
thf(func_def_724, type, sK586: term).
thf(func_def_725, type, sK587: ((subst > term > term) > term)).
thf(func_def_726, type, sK588: ((subst > term > term) > subst)).
thf(func_def_727, type, sK589: term).
thf(func_def_728, type, sK590: term).
thf(func_def_729, type, sK591: term).
thf(func_def_730, type, sK592: term).
thf(func_def_731, type, sK593: term).
thf(func_def_732, type, sK594: subst).
thf(func_def_733, type, sK595: term).
thf(func_def_734, type, sK596: (term > $o)).
thf(func_def_735, type, sK597: term).
thf(func_def_736, type, sK598: (term > $o)).
thf(func_def_737, type, sK599: ((subst > term > subst > $o) > term > $o)).
thf(func_def_738, type, sK600: (subst > term > subst > $o)).
thf(func_def_739, type, sK601: ((term > $o) > term)).
thf(func_def_740, type, sK602: term).
thf(func_def_741, type, sK603: term).
thf(func_def_742, type, sK604: term).
thf(func_def_743, type, sK605: term).
thf(func_def_744, type, sK606: term).
thf(func_def_745, type, sK607: subst).
thf(func_def_746, type, sK608: term).
thf(func_def_747, type, sK609: ((subst > term > term) > subst)).
thf(func_def_748, type, sK610: ((subst > term > term) > term)).
thf(func_def_749, type, sK611: (term > subst > term > term)).
thf(func_def_750, type, sK612: subst).
thf(func_def_751, type, sK613: term).
thf(func_def_752, type, sK614: subst).
thf(func_def_753, type, sK615: term).
thf(func_def_754, type, sK616: term).
thf(func_def_755, type, sK617: term).
thf(func_def_756, type, sK618: term).
thf(func_def_757, type, sK619: ((subst > term > subst > $o) > term > $o)).
thf(func_def_758, type, sK620: (subst > term > subst > $o)).
thf(func_def_759, type, sK621: ((term > $o) > term)).
thf(func_def_760, type, sK622: subst).
thf(func_def_761, type, sK623: subst).
thf(func_def_762, type, sK624: (subst > d_subst)).
thf(func_def_763, type, sK625: (term > subst > term > term)).
thf(func_def_764, type, sK626: term).
thf(func_def_765, type, sK627: ((subst > term > subst > $o) > term > $o)).
thf(func_def_766, type, sK628: (subst > term > subst > $o)).
thf(func_def_767, type, sK629: ((term > $o) > term)).
thf(func_def_768, type, sK630: term).
thf(func_def_769, type, sK631: term).
thf(func_def_770, type, sK632: term).
thf(func_def_771, type, sK633: term).
thf(func_def_772, type, sK634: term).
thf(func_def_773, type, sK635: term).
thf(func_def_774, type, sK636: subst).
thf(func_def_775, type, sK637: term).
thf(func_def_776, type, sK638: (term > $o)).
thf(func_def_777, type, sK639: ((term > $o) > term)).
thf(func_def_778, type, sK640: ((term > $o) > term > subst > term > $o)).
thf(func_def_779, type, sK641: (term > $o)).
thf(func_def_780, type, db6: !>[X0: $tType]:(X0)).
thf(func_def_781, type, db5: !>[X0: $tType]:(X0)).
thf(func_def_782, type, sK642: term).
thf(func_def_783, type, sK643: subst).
thf(func_def_784, type, sK644: (term > $o)).
thf(func_def_785, type, sK645: (subst > (term > $o) > term > (term > $o) > term)).
thf(f1,axiom,(
  (! [X10 : subst] : (X10 = ((comp @ X10 @ id))) <=> axidr) & hoasapinj1_gthm & (termmset_lthm <=> (! [X9 : term] : (X9 = ((sub @ X9 @ id))) => ! [X9 : term] : (X9 = ((sub @ X9 @ id))))) & termmset_gthm & hoasinduction_lem1 & ((! [X10 : subst] : (X10 = ((comp @ X10 @ id))) & ! [X10 : subst] : (X10 = ((comp @ id @ X10)))) <=> substmonoid) & pushprop_lthm & hoaslamnotvar_lthm & ! [X46 : d_subst] : (X46 = d_id) & (((comp @ (d2subst @ d_id) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & (! [X23 : term,X7 : (subst > term > subst > $o)] : (~ ! [X14 : term,X9 : term] : (~(X7 @ id @ X14 @ id) | ~(X7 @ id @ X9 @ id) | (X7 @ id @ (ap @ (sub @ X9 @ id) @ X14) @ id)) | ~ ! [X20 : subst,X10 : subst,X24 : subst,X9 : term] : ((X7 @ (comp @ X10 @ X24) @ (sub @ X9 @ X24) @ X20) | ~(X7 @ X10 @ X9 @ (comp @ X24 @ X20))) | ~ ! [X10 : subst,X9 : term,X24 : subst,X20 : subst] : (~(X7 @ (comp @ X10 @ X24) @ (sub @ X9 @ X24) @ X20) | (X7 @ X10 @ X9 @ (comp @ X24 @ X20))) | (X7 @ id @ X23 @ id) | ~ ! [X9 : term] : (~(var @ (sub @ X9 @ id)) | (X7 @ id @ X9 @ id)) | ~ ! [X17 : (subst > term > term)] : ((X7 @ id @ (lam @ (X17 @ sh @ one)) @ id) | ~ ! [X20 : subst,X9 : term,X10 : subst] : (((sub @ (X17 @ X10 @ X9) @ X20)) = ((X17 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20)))) | ~ ! [X9 : term] : ((X7 @ id @ (X17 @ id @ X9) @ id) | ~(X7 @ id @ X9 @ id)))) <=> hoasinduction) & ((! [X9 : term] : (X9 = ((sub @ X9 @ id))) => (! [X13 : subst,X12 : term,X7 : (term > $o)] : (~ ! [X9 : term] : (~ ! [X14 : term] : (~(X7 @ X14) | (X7 @ (sub @ X9 @ (push @ X14 @ id)))) | (X7 @ (lam @ X9))) | (X7 @ (sub @ X12 @ X13)) | ~ ! [X9 : term,X14 : term] : (~(X7 @ X9) | (X7 @ (ap @ X9 @ X14)) | ~(X7 @ X14)) | ~ ! [X14 : term] : (~(var @ X14) | (X7 @ (sub @ X14 @ X13)))) => ! [X7 : (term > $o),X18 : term] : (~ ! [X9 : term] : ((X7 @ (lam @ X9)) | ~ ! [X14 : term] : (~(X7 @ X14) | (X7 @ (sub @ X9 @ (push @ X14 @ id))))) | (X7 @ X18) | ~ ! [X14 : term,X9 : term] : (~(X7 @ X14) | ~(X7 @ X9) | (X7 @ (ap @ X9 @ X14))) | ~ ! [X9 : term] : ((X7 @ X9) | ~(var @ X9))))) <=> induction2_lthm) & (! [X10 : subst] : (X10 = ((comp @ id @ X10))) <=> axidl) & axclos & pushprop_lem2v2 & pushprop & axmap & ((! [X10 : subst,X9 : term] : (X9 = ((sub @ one @ (push @ X9 @ X10)))) => (! [X10 : subst,X9 : term] : (X10 = ((comp @ sh @ (push @ X9 @ X10)))) => (! [X10 : subst] : (X10 = ((comp @ X10 @ id))) => (! [X7 : (term > $o),X35 : term] : (~ ! [X9 : term] : (~(var @ X9) | (X7 @ X9)) | ~ ! [X9 : term,X14 : term] : (~(X7 @ X9) | ~(X7 @ X14) | (X7 @ (ap @ X9 @ X14))) | ~ ! [X9 : term] : ((X7 @ (lam @ X9)) | ~(X7 @ X9)) | (X7 @ X35)) => ! [X7 : (term > $o),X13 : subst,X12 : term] : (~ ! [X9 : term,X14 : term] : (~(X7 @ X9) | (X7 @ (ap @ X9 @ X14)) | ~(X7 @ X14)) | (X7 @ (sub @ X12 @ X13)) | ~ ! [X9 : term] : ((X7 @ (lam @ X9)) | ~ ! [X14 : term] : ((X7 @ (sub @ X9 @ (push @ X14 @ id))) | ~(X7 @ X14))) | ~ ! [X14 : term] : ((X7 @ (sub @ X14 @ X13)) | ~(var @ X14))))))) <=> induction2lem_lthm) & (! [X9 : term,X14 : term,X16 : term,X15 : term] : ((X9 = X14) | (((ap @ (sub @ X9 @ id) @ X15)) != ((ap @ (sub @ X14 @ id) @ X16)))) <=> hoasapinj1) & hoaslaminj_gthm & (hoasinduction_lem3 <=> ! [X31 : term,X7 : (subst > term > subst > $o)] : (~ ! [X10 : subst,X9 : term,X20 : subst,X24 : subst] : ((X7 @ (comp @ X10 @ X24) @ (sub @ X9 @ X24) @ X20) | ~(X7 @ X10 @ X9 @ (comp @ X24 @ X20))) | (X7 @ id @ (lam @ X31) @ id) | ~ ! [X14 : term] : ((X7 @ id @ (sub @ X31 @ (push @ X14 @ id)) @ id) | ~(X7 @ id @ X14 @ id)) | ~ ! [X17 : (subst > term > term)] : (~ ! [X20 : subst,X10 : subst,X9 : term] : (((sub @ (X17 @ X10 @ X9) @ X20)) = ((X17 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20)))) | (X7 @ id @ (lam @ (X17 @ sh @ one)) @ id) | ~ ! [X9 : term] : (~(X7 @ id @ X9 @ id) | (X7 @ id @ (X17 @ id @ X9) @ id))) | ~ ! [X9 : term,X10 : subst,X20 : subst,X24 : subst] : ((X7 @ X10 @ X9 @ (comp @ X24 @ X20)) | ~(X7 @ (comp @ X10 @ X24) @ (sub @ X9 @ X24) @ X20)))) & (! [X15 : term,X9 : term,X14 : term,X16 : term] : ((X15 = X16) | (((ap @ X9 @ X15)) != ((ap @ X14 @ X16)))) <=> apinj2) & ((! [X9 : term] : (X9 = ((sub @ X9 @ id))) => (! [X7 : (subst > term > subst > $o),X22 : term] : ((X7 @ id @ (lam @ (sub @ X22 @ (push @ one @ sh))) @ id) | ~ ! [X17 : (subst > term > term)] : (~ ! [X10 : subst,X20 : subst,X9 : term] : (((sub @ (X17 @ X10 @ X9) @ X20)) = ((X17 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20)))) | ~ ! [X9 : term] : ((X7 @ id @ (X17 @ id @ X9) @ id) | ~(X7 @ id @ X9 @ id)) | (X7 @ id @ (lam @ (X17 @ sh @ one)) @ id)) | ~ ! [X14 : term] : ((X7 @ id @ (sub @ X22 @ (push @ X14 @ id)) @ id) | ~(X7 @ id @ X14 @ id))) => ! [X7 : (subst > term > subst > $o),X27 : term] : (~ ! [X14 : term] : ((X7 @ id @ (sub @ X27 @ (push @ X14 @ id)) @ id) | ~(X7 @ id @ X14 @ id)) | (X7 @ id @ (lam @ X27) @ id) | ~ ! [X17 : (subst > term > term)] : (~ ! [X9 : term] : ((X7 @ id @ (X17 @ id @ X9) @ id) | ~(X7 @ id @ X9 @ id)) | ~ ! [X20 : subst,X9 : term,X10 : subst] : (((sub @ (X17 @ X10 @ X9) @ X20)) = ((X17 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20)))) | (X7 @ id @ (lam @ (X17 @ sh @ one)) @ id))))) <=> hoasinduction_lem3a_lthm) & ~lamnotap & (! [X10 : subst,X7 : (term > $o),X9 : term] : ~ ! [X8 : (term > $o)] : ~ ! [X11 : term] : (((X8 @ X11)) = ((X7 @ (sub @ X11 @ (push @ X9 @ X10))))) <=> pushprop_lem0) & hoasapinj2_gthm & (hoasinduction_p_and_p_prime = (^[X7 : (subst > term > subst > $o), X8 : (term > $o)] : (! [X11 : term] : (((X8 @ X11)) = ((X7 @ id @ X11 @ id)))))) & pushprop_lem2v2_gthm & (pushprop_lem0_lthm <=> ! [X7 : (term > $o),X10 : subst,X9 : term] : ~ ! [X8 : (term > $o)] : ~ ! [X11 : term] : (((X8 @ X11)) = ((X7 @ (sub @ X11 @ (push @ X9 @ X10)))))) & (pushprop_lem1v2 <=> ! [X7 : (term > $o),X8 : (term > $o),X9 : term,X10 : subst] : (~(X7 @ X9) | (X8 @ one) | ~ ! [X11 : term] : (((X8 @ X11)) = ((X7 @ (sub @ X11 @ (push @ X9 @ X10))))))) & pushprop_lthm_orig & (axvarid <=> ! [X9 : term] : (X9 = ((sub @ X9 @ id)))) & (! [X37 : term,X36 : term,X7 : (subst > term > subst > $o)] : (~ ! [X9 : term,X14 : term] : ((X7 @ id @ (ap @ (sub @ X9 @ id) @ X14) @ id) | ~(X7 @ id @ X9 @ id) | ~(X7 @ id @ X14 @ id)) | ~(X7 @ id @ X36 @ id) | ~(X7 @ id @ X37 @ id) | ~ ! [X9 : term,X10 : subst,X20 : subst,X24 : subst] : ((X7 @ X10 @ X9 @ (comp @ X24 @ X20)) | ~(X7 @ (comp @ X10 @ X24) @ (sub @ X9 @ X24) @ X20)) | ~ ! [X10 : subst,X9 : term,X24 : subst,X20 : subst] : ((X7 @ (comp @ X10 @ X24) @ (sub @ X9 @ X24) @ X20) | ~(X7 @ X10 @ X9 @ (comp @ X24 @ X20))) | (X7 @ id @ (ap @ X36 @ X37) @ id)) <=> hoasinduction_lem2) & pushprop_gthm & axassoc & (((push @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & (one = ((d2term @ d_one))) & hoasinduction_lem3_gthm & (hoaslaminj_lthm <=> (! [X10 : subst,X9 : term] : (X9 = ((sub @ one @ (push @ X9 @ X10)))) => (! [X9 : term,X10 : subst] : (X10 = ((comp @ sh @ (push @ X9 @ X10)))) => (! [X9 : term,X14 : term] : ((X9 = X14) | (((lam @ X9)) != ((lam @ X14)))) => ! [X34 : term,X17 : (subst > term > term),X33 : subst,X32 : (subst > term > term)] : (~ ! [X20 : subst,X9 : term,X10 : subst] : (((sub @ (X32 @ X10 @ X9) @ X20)) = ((X32 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20)))) | (((lam @ (X17 @ sh @ one))) != ((lam @ (X32 @ sh @ one)))) | (((X17 @ X33 @ X34)) = ((X32 @ X33 @ X34))) | ~ ! [X9 : term,X20 : subst,X10 : subst] : (((sub @ (X17 @ X10 @ X9) @ X20)) = ((X17 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20))))))))) & (id = ((d2subst @ d_id))) & ulamvarsh & ! [X47 : subst] : ? [X46 : d_subst] : (X47 = ((d2subst @ X46))) & hoaslamnotvar & pushprop_lem1v2_gthm & hoasapnotvar_lthm & hoasinduction_gthm & (hoasinduction_lthm_3 <=> (! [X7 : (subst > term > subst > $o)] : ~ ! [X8 : (term > $o)] : ~ ! [X11 : term] : (((X8 @ X11)) = ((X7 @ id @ X11 @ id))) => (! [X7 : (term > $o),X18 : term] : ((X7 @ X18) | ~ ! [X14 : term,X9 : term] : (~(X7 @ X9) | ~(X7 @ X14) | (X7 @ (ap @ X9 @ X14))) | ~ ! [X9 : term] : ((X7 @ (lam @ X9)) | ~ ! [X14 : term] : (~(X7 @ X14) | (X7 @ (sub @ X9 @ (push @ X14 @ id))))) | ~ ! [X9 : term] : ((X7 @ X9) | ~(var @ X9))) => (! [X9 : term] : (X9 = ((sub @ X9 @ id))) => (! [X7 : (subst > term > subst > $o),X8 : (term > $o),X21 : term] : ((X8 @ (lam @ X21)) | ~ ! [X11 : term] : (((X8 @ X11)) = ((X7 @ id @ X11 @ id))) | ~ ! [X14 : term] : ((X8 @ (sub @ X21 @ (push @ X14 @ id))) | ~(X8 @ X14)) | ~ ! [X17 : (subst > term > term)] : (~ ! [X9 : term] : ((X7 @ id @ (X17 @ id @ X9) @ id) | ~(X7 @ id @ X9 @ id)) | ~ ! [X10 : subst,X20 : subst,X9 : term] : (((sub @ (X17 @ X10 @ X9) @ X20)) = ((X17 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20)))) | (X7 @ id @ (lam @ (X17 @ sh @ one)) @ id))) => ! [X23 : term,X7 : (subst > term > subst > $o)] : (~ ! [X17 : (subst > term > term)] : (~ ! [X9 : term] : ((X7 @ id @ (X17 @ id @ X9) @ id) | ~(X7 @ id @ X9 @ id)) | ~ ! [X10 : subst,X20 : subst,X9 : term] : (((sub @ (X17 @ X10 @ X9) @ X20)) = ((X17 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20)))) | (X7 @ id @ (lam @ (X17 @ sh @ one)) @ id)) | (X7 @ id @ X23 @ id) | ~ ! [X10 : subst,X24 : subst,X20 : subst,X9 : term] : (~(X7 @ (comp @ X10 @ X24) @ (sub @ X9 @ X24) @ X20) | (X7 @ X10 @ X9 @ (comp @ X24 @ X20))) | ~ ! [X9 : term] : ((X7 @ id @ X9 @ id) | ~(var @ (sub @ X9 @ id))) | ~ ! [X9 : term,X24 : subst,X20 : subst,X10 : subst] : (~(X7 @ X10 @ X9 @ (comp @ X24 @ X20)) | (X7 @ (comp @ X10 @ X24) @ (sub @ X9 @ X24) @ X20)) | ~ ! [X14 : term,X9 : term] : (~(X7 @ id @ X9 @ id) | (X7 @ id @ (ap @ (sub @ X9 @ id) @ X14) @ id) | ~(X7 @ id @ X14 @ id)))))))) & hoasinduction_lem1v2 & (hoasinduction_lem2v2 <=> ! [X26 : term,X25 : term,X7 : (subst > term > subst > $o),X8 : (term > $o)] : (~ ! [X11 : term] : (((X8 @ X11)) = ((X7 @ id @ X11 @ id))) | (X8 @ (ap @ X25 @ X26)) | ~(X8 @ X26) | ~ ! [X20 : subst,X24 : subst,X10 : subst,X9 : term] : (~(X7 @ (comp @ X10 @ X24) @ (sub @ X9 @ X24) @ X20) | (X7 @ X10 @ X9 @ (comp @ X24 @ X20))) | ~ ! [X24 : subst,X20 : subst,X9 : term,X10 : subst] : (~(X7 @ X10 @ X9 @ (comp @ X24 @ X20)) | (X7 @ (comp @ X10 @ X24) @ (sub @ X9 @ X24) @ X20)) | ~ ! [X9 : term,X14 : term] : (~(X7 @ id @ X9 @ id) | (X7 @ id @ (ap @ (sub @ X9 @ id) @ X14) @ id) | ~(X7 @ id @ X14 @ id)) | ~(X8 @ X25))) & (! [X7 : (term > $o),X13 : subst,X12 : term] : (~ ! [X9 : term] : (~ ! [X14 : term] : ((X7 @ (sub @ X9 @ (push @ X14 @ id))) | ~(X7 @ X14)) | (X7 @ (lam @ X9))) | ~ ! [X9 : term,X14 : term] : (~(X7 @ X9) | (X7 @ (ap @ X9 @ X14)) | ~(X7 @ X14)) | (X7 @ (sub @ X12 @ X13)) | ~ ! [X14 : term] : (~(var @ X14) | (X7 @ (sub @ X14 @ X13)))) <=> induction2lem) & ! [X1 : d_term,X2 : d_term] : ((((d2term @ X1)) = ((d2term @ X2))) => (X1 = X2)) & hoasinduction_lem2_gthm & (pushprop_lem1 <=> ! [X10 : subst,X9 : term,X7 : (term > $o),X24 : (term > $o),X14 : term] : (~(X7 @ X9) | (X24 @ (sub @ X9 @ (push @ X14 @ X10))))) & ((! [X9 : term] : (X9 = ((sub @ X9 @ id))) => ! [X30 : term,X7 : (subst > term > subst > $o),X8 : (term > $o)] : (~ ! [X20 : subst,X24 : subst,X9 : term,X10 : subst] : ((X7 @ X10 @ X9 @ (comp @ X24 @ X20)) | ~(X7 @ (comp @ X10 @ X24) @ (sub @ X9 @ X24) @ X20)) | ~ ! [X10 : subst,X9 : term,X20 : subst,X24 : subst] : ((X7 @ (comp @ X10 @ X24) @ (sub @ X9 @ X24) @ X20) | ~(X7 @ X10 @ X9 @ (comp @ X24 @ X20))) | ~ ! [X14 : term] : (~(X8 @ X14) | (X8 @ (sub @ X30 @ (push @ X14 @ id)))) | ~ ! [X11 : term] : (((X8 @ X11)) = ((X7 @ id @ X11 @ id))) | (X8 @ (lam @ X30)) | ~ ! [X17 : (subst > term > term)] : (~ ! [X9 : term,X20 : subst,X10 : subst] : (((sub @ (X17 @ X10 @ X9) @ X20)) = ((X17 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20)))) | ~ ! [X9 : term] : ((X7 @ id @ (X17 @ id @ X9) @ id) | ~(X7 @ id @ X9 @ id)) | (X7 @ id @ (lam @ (X17 @ sh @ one)) @ id)))) <=> hoasinduction_lem3v2_lthm) & ((! [X9 : term] : (X9 = ((sub @ X9 @ id))) => (! [X14 : term,X9 : term,X16 : term,X15 : term] : ((X9 = X14) | (((ap @ X9 @ X15)) != ((ap @ X14 @ X16)))) => ! [X15 : term,X14 : term,X16 : term,X9 : term] : ((((ap @ (sub @ X9 @ id) @ X15)) != ((ap @ (sub @ X14 @ id) @ X16))) | (X9 = X14)))) <=> hoasapinj1_lthm) & (hoasinduction_lthm <=> (! [X18 : term,X7 : (term > $o)] : (~ ! [X9 : term] : ((X7 @ X9) | ~(var @ X9)) | (X7 @ X18) | ~ ! [X9 : term] : ((X7 @ (lam @ X9)) | ~ ! [X14 : term] : (~(X7 @ X14) | (X7 @ (sub @ X9 @ (push @ X14 @ id))))) | ~ ! [X9 : term,X14 : term] : (~(X7 @ X14) | (X7 @ (ap @ X9 @ X14)) | ~(X7 @ X9))) => (! [X7 : (subst > term > subst > $o),X36 : term,X37 : term] : (~ ! [X24 : subst,X10 : subst,X20 : subst,X9 : term] : (~(X7 @ (comp @ X10 @ X24) @ (sub @ X9 @ X24) @ X20) | (X7 @ X10 @ X9 @ (comp @ X24 @ X20))) | ~ ! [X10 : subst,X9 : term,X20 : subst,X24 : subst] : ((X7 @ (comp @ X10 @ X24) @ (sub @ X9 @ X24) @ X20) | ~(X7 @ X10 @ X9 @ (comp @ X24 @ X20))) | ~(X7 @ id @ X37 @ id) | ~ ! [X14 : term,X9 : term] : (~(X7 @ id @ X9 @ id) | (X7 @ id @ (ap @ (sub @ X9 @ id) @ X14) @ id) | ~(X7 @ id @ X14 @ id)) | (X7 @ id @ (ap @ X36 @ X37) @ id) | ~(X7 @ id @ X36 @ id)) => (! [X31 : term,X7 : (subst > term > subst > $o)] : (~ ! [X9 : term,X24 : subst,X10 : subst,X20 : subst] : ((X7 @ X10 @ X9 @ (comp @ X24 @ X20)) | ~(X7 @ (comp @ X10 @ X24) @ (sub @ X9 @ X24) @ X20)) | (X7 @ id @ (lam @ X31) @ id) | ~ ! [X20 : subst,X10 : subst,X24 : subst,X9 : term] : (~(X7 @ X10 @ X9 @ (comp @ X24 @ X20)) | (X7 @ (comp @ X10 @ X24) @ (sub @ X9 @ X24) @ X20)) | ~ ! [X14 : term] : ((X7 @ id @ (sub @ X31 @ (push @ X14 @ id)) @ id) | ~(X7 @ id @ X14 @ id)) | ~ ! [X17 : (subst > term > term)] : (~ ! [X9 : term] : ((X7 @ id @ (X17 @ id @ X9) @ id) | ~(X7 @ id @ X9 @ id)) | (X7 @ id @ (lam @ (X17 @ sh @ one)) @ id) | ~ ! [X10 : subst,X20 : subst,X9 : term] : (((sub @ (X17 @ X10 @ X9) @ X20)) = ((X17 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20)))))) => ! [X7 : (subst > term > subst > $o),X23 : term] : ((X7 @ id @ X23 @ id) | ~ ! [X17 : (subst > term > term)] : ((X7 @ id @ (lam @ (X17 @ sh @ one)) @ id) | ~ ! [X9 : term] : (~(X7 @ id @ X9 @ id) | (X7 @ id @ (X17 @ id @ X9) @ id)) | ~ ! [X9 : term,X10 : subst,X20 : subst] : (((sub @ (X17 @ X10 @ X9) @ X20)) = ((X17 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20))))) | ~ ! [X10 : subst,X24 : subst,X9 : term,X20 : subst] : (~(X7 @ X10 @ X9 @ (comp @ X24 @ X20)) | (X7 @ (comp @ X10 @ X24) @ (sub @ X9 @ X24) @ X20)) | ~ ! [X10 : subst,X20 : subst,X9 : term,X24 : subst] : (~(X7 @ (comp @ X10 @ X24) @ (sub @ X9 @ X24) @ X20) | (X7 @ X10 @ X9 @ (comp @ X24 @ X20))) | ~ ! [X9 : term] : (~(var @ (sub @ X9 @ id)) | (X7 @ id @ X9 @ id)) | ~ ! [X9 : term,X14 : term] : (~(X7 @ id @ X14 @ id) | (X7 @ id @ (ap @ (sub @ X9 @ id) @ X14) @ id) | ~(X7 @ id @ X9 @ id))))))) & ~(hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) & hoasapnotvar & ((! [X14 : term] : ~ ! [X17 : (subst > term > term)] : ~ ! [X10 : subst,X9 : term] : (((sub @ X14 @ (push @ X9 @ X10))) = ((X17 @ X10 @ X9))) => (! [X9 : term] : (X9 = ((sub @ X9 @ id))) => ! [X21 : term,X8 : (term > $o),X7 : (subst > term > subst > $o)] : (~ ! [X17 : (subst > term > term)] : (~ ! [X9 : term,X10 : subst,X20 : subst] : (((sub @ (X17 @ X10 @ X9) @ X20)) = ((X17 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20)))) | (X7 @ id @ (lam @ (X17 @ sh @ one)) @ id) | ~ ! [X9 : term] : ((X7 @ id @ (X17 @ id @ X9) @ id) | ~(X7 @ id @ X9 @ id))) | ~ ! [X11 : term] : (((X8 @ X11)) = ((X7 @ id @ X11 @ id))) | (X8 @ (lam @ X21)) | ~ ! [X14 : term] : ((X8 @ (sub @ X21 @ (push @ X14 @ id))) | ~(X8 @ X14))))) <=> hoasinduction_lem3v2a_lthm) & ((! [X7 : (subst > term > subst > $o)] : ~ ! [X8 : (term > $o)] : ~ ! [X11 : term] : (((X8 @ X11)) = ((X7 @ id @ X11 @ id))) => (! [X7 : (term > $o),X18 : term] : (~ ! [X9 : term] : ((X7 @ (lam @ X9)) | ~ ! [X14 : term] : ((X7 @ (sub @ X9 @ (push @ X14 @ id))) | ~(X7 @ X14))) | ~ ! [X9 : term,X14 : term] : (~(X7 @ X9) | ~(X7 @ X14) | (X7 @ (ap @ X9 @ X14))) | ~ ! [X9 : term] : ((X7 @ X9) | ~(var @ X9)) | (X7 @ X18)) => (! [X9 : term] : (X9 = ((sub @ X9 @ id))) => (! [X8 : (term > $o),X7 : (subst > term > subst > $o),X21 : term] : (~ ! [X14 : term] : ((X8 @ (sub @ X21 @ (push @ X14 @ id))) | ~(X8 @ X14)) | ~ ! [X11 : term] : (((X8 @ X11)) = ((X7 @ id @ X11 @ id))) | ~ ! [X17 : (subst > term > term)] : (~ ! [X20 : subst,X10 : subst,X9 : term] : (((sub @ (X17 @ X10 @ X9) @ X20)) = ((X17 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20)))) | ~ ! [X9 : term] : (~(X7 @ id @ X9 @ id) | (X7 @ id @ (X17 @ id @ X9) @ id)) | (X7 @ id @ (lam @ (X17 @ sh @ one)) @ id)) | (X8 @ (lam @ X21))) => ! [X19 : term,X7 : (subst > term > subst > $o)] : (~ ! [X9 : term,X14 : term] : (~(X7 @ id @ X14 @ id) | (X7 @ id @ (ap @ (sub @ X9 @ id) @ X14) @ id) | ~(X7 @ id @ X9 @ id)) | ~ ! [X17 : (subst > term > term)] : (~ ! [X9 : term] : ((X7 @ id @ (X17 @ id @ X9) @ id) | ~(X7 @ id @ X9 @ id)) | (X7 @ id @ (lam @ (X17 @ sh @ one)) @ id) | ~ ! [X9 : term,X10 : subst,X20 : subst] : (((sub @ (X17 @ X10 @ X9) @ X20)) = ((X17 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20))))) | (X7 @ id @ X19 @ id)))))) <=> hoasinduction_no_psi_cond_lthm) & (hoasinduction_lem0_lthm <=> ! [X7 : (subst > term > subst > $o)] : ~ ! [X8 : (term > $o)] : ~ ! [X11 : term] : (((X8 @ X11)) = ((X7 @ id @ X11 @ id)))) & (((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) = ((d2term @ d_one))) & hoaslamnotvar_gthm & apnotvar & (! [X10 : subst,X9 : term] : (X9 = ((sub @ one @ (push @ X9 @ X10)))) <=> axvarcons) & (hoasinduction_lem3v2a <=> ! [X7 : (subst > term > subst > $o),X8 : (term > $o),X21 : term] : (~ ! [X17 : (subst > term > term)] : ((X7 @ id @ (lam @ (X17 @ sh @ one)) @ id) | ~ ! [X9 : term] : (~(X7 @ id @ X9 @ id) | (X7 @ id @ (X17 @ id @ X9) @ id)) | ~ ! [X9 : term,X20 : subst,X10 : subst] : (((sub @ (X17 @ X10 @ X9) @ X20)) = ((X17 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20))))) | (X8 @ (lam @ X21)) | ~ ! [X11 : term] : (((X8 @ X11)) = ((X7 @ id @ X11 @ id))) | ~ ! [X14 : term] : (~(X8 @ X14) | (X8 @ (sub @ X21 @ (push @ X14 @ id)))))) & (! [X7 : (term > $o),X18 : term] : (~ ! [X9 : term] : (~ ! [X14 : term] : (~(X7 @ X14) | (X7 @ (sub @ X9 @ (push @ X14 @ id)))) | (X7 @ (lam @ X9))) | ~ ! [X9 : term] : (~(var @ X9) | (X7 @ X9)) | (X7 @ X18) | ~ ! [X14 : term,X9 : term] : ((X7 @ (ap @ X9 @ X14)) | ~(X7 @ X14) | ~(X7 @ X9))) <=> induction2) & pushprop_lem0_gthm & hoaslamnotap_gthm & (hoasinduction_lem3aa_lthm <=> ! [X7 : (subst > term > subst > $o),X22 : term] : (~ ! [X17 : (subst > term > term)] : ((X7 @ id @ (lam @ (X17 @ sh @ one)) @ id) | ~ ! [X20 : subst,X10 : subst,X9 : term] : (((sub @ (X17 @ X10 @ X9) @ X20)) = ((X17 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20)))) | ~ ! [X9 : term] : ((X7 @ id @ (X17 @ id @ X9) @ id) | ~(X7 @ id @ X9 @ id))) | ~ ! [X14 : term] : ((X7 @ id @ (sub @ X22 @ (push @ X14 @ id)) @ id) | ~(X7 @ id @ X14 @ id)) | (X7 @ id @ (lam @ (sub @ X22 @ (push @ one @ sh))) @ id))) & (pushprop_p_and_p_prime = (^[X9 : term, X10 : subst, X7 : (term > $o), X8 : (term > $o)] : (! [X11 : term] : (((X8 @ X11)) = ((X7 @ (sub @ X11 @ (push @ X9 @ X10)))))))) & (! [X14 : term] : ~ ! [X17 : (subst > term > term)] : ~ ! [X10 : subst,X9 : term] : (((sub @ X14 @ (push @ X9 @ X10))) = ((X17 @ X10 @ X9))) <=> hoasinduction_lem3v2_f) & ((! [X9 : term] : (X9 = ((sub @ X9 @ id))) => (! [X22 : term,X7 : (subst > term > subst > $o)] : (~ ! [X17 : (subst > term > term)] : (~ ! [X9 : term] : (~(X7 @ id @ X9 @ id) | (X7 @ id @ (X17 @ id @ X9) @ id)) | (X7 @ id @ (lam @ (X17 @ sh @ one)) @ id) | ~ ! [X10 : subst,X9 : term,X20 : subst] : (((sub @ (X17 @ X10 @ X9) @ X20)) = ((X17 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20))))) | ~ ! [X14 : term] : ((X7 @ id @ (sub @ X22 @ (push @ X14 @ id)) @ id) | ~(X7 @ id @ X14 @ id)) | (X7 @ id @ (lam @ (sub @ X22 @ (push @ one @ sh))) @ id)) => ! [X7 : (subst > term > subst > $o),X31 : term] : ((X7 @ id @ (lam @ X31) @ id) | ~ ! [X24 : subst,X20 : subst,X9 : term,X10 : subst] : ((X7 @ (comp @ X10 @ X24) @ (sub @ X9 @ X24) @ X20) | ~(X7 @ X10 @ X9 @ (comp @ X24 @ X20))) | ~ ! [X14 : term] : (~(X7 @ id @ X14 @ id) | (X7 @ id @ (sub @ X31 @ (push @ X14 @ id)) @ id)) | ~ ! [X20 : subst,X24 : subst,X9 : term,X10 : subst] : ((X7 @ X10 @ X9 @ (comp @ X24 @ X20)) | ~(X7 @ (comp @ X10 @ X24) @ (sub @ X9 @ X24) @ X20)) | ~ ! [X17 : (subst > term > term)] : (~ ! [X10 : subst,X9 : term,X20 : subst] : (((sub @ (X17 @ X10 @ X9) @ X20)) = ((X17 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20)))) | ~ ! [X9 : term] : ((X7 @ id @ (X17 @ id @ X9) @ id) | ~(X7 @ id @ X9 @ id)) | (X7 @ id @ (lam @ (X17 @ sh @ one)) @ id))))) <=> hoasinduction_lem3_lthm) & induction2lem_gthm & ulamvarind & (! [X10 : subst] : (X10 = ((push @ (sub @ one @ X10) @ (comp @ sh @ X10)))) <=> axscons) & axapp & pushprop_lem2v2_lthm & ~ulamvar1 & hoasinduction_lem3b_gthm & hoasinduction_lem3a_gthm & pushprop_lem3v2_lthm & (substmonoid_lthm <=> (! [X10 : subst] : (X10 = ((comp @ id @ X10))) => (! [X10 : subst] : (X10 = ((comp @ X10 @ id))) => (! [X10 : subst] : (X10 = ((comp @ X10 @ id))) & ! [X10 : subst] : (X10 = ((comp @ id @ X10))))))) & ! [X3 : d_subst,X4 : d_subst] : ((((d2subst @ X3)) = ((d2subst @ X4))) => (X3 = X4)) & pushprop_lem3v2 & (hoaslamnotap <=> ! [X17 : (subst > term > term),X29 : term,X28 : term] : ((((lam @ (X17 @ sh @ one))) != ((ap @ (sub @ X28 @ id) @ X29))) | ~ ! [X20 : subst,X10 : subst,X9 : term] : (((sub @ (X17 @ X10 @ X9) @ X20)) = ((X17 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20)))))) & axvarshift & substmonoid_gthm & (! [X7 : (subst > term > subst > $o)] : ~ ! [X8 : (term > $o)] : ~ ! [X11 : term] : (((X8 @ X11)) = ((X7 @ id @ X11 @ id))) <=> hoasinduction_lem0) & (hoasinduction_lem3v2_f_lthm <=> ! [X14 : term] : ~ ! [X17 : (subst > term > term)] : ~ ! [X10 : subst,X9 : term] : (((sub @ X14 @ (push @ X9 @ X10))) = ((X17 @ X10 @ X9)))) & hoasinduction_lem1_gthm & ! [X48 : term] : ? [X0 : d_term] : (X48 = ((d2term @ X0))) & (hoasinduction_lem3v2 <=> ! [X30 : term,X8 : (term > $o),X7 : (subst > term > subst > $o)] : (~ ! [X14 : term] : ((X8 @ (sub @ X30 @ (push @ X14 @ id))) | ~(X8 @ X14)) | ~ ! [X20 : subst,X10 : subst,X24 : subst,X9 : term] : ((X7 @ X10 @ X9 @ (comp @ X24 @ X20)) | ~(X7 @ (comp @ X10 @ X24) @ (sub @ X9 @ X24) @ X20)) | ~ ! [X17 : (subst > term > term)] : ((X7 @ id @ (lam @ (X17 @ sh @ one)) @ id) | ~ ! [X9 : term] : (~(X7 @ id @ X9 @ id) | (X7 @ id @ (X17 @ id @ X9) @ id)) | ~ ! [X20 : subst,X9 : term,X10 : subst] : (((sub @ (X17 @ X10 @ X9) @ X20)) = ((X17 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20))))) | ~ ! [X11 : term] : (((X8 @ X11)) = ((X7 @ id @ X11 @ id))) | ~ ! [X20 : subst,X10 : subst,X24 : subst,X9 : term] : (~(X7 @ X10 @ X9 @ (comp @ X24 @ X20)) | (X7 @ (comp @ X10 @ X24) @ (sub @ X9 @ X24) @ X20)) | (X8 @ (lam @ X30)))) & induction2_gthm & hoasapnotvar_gthm & pushprop_lem1_gthm & hoasinduction_lem1_lthm & (axshiftcons <=> ! [X10 : subst,X9 : term] : (X10 = ((comp @ sh @ (push @ X9 @ X10))))) & (! [X14 : term,X9 : term] : ((X9 = X14) | (((lam @ X9)) != ((lam @ X14)))) <=> laminj) & (hoasapinj2_lthm <=> (! [X14 : term,X15 : term,X16 : term,X9 : term] : ((((ap @ X9 @ X15)) != ((ap @ X14 @ X16))) | (X15 = X16)) => ! [X15 : term,X9 : term,X16 : term,X14 : term] : ((X15 = X16) | (((ap @ (sub @ X9 @ id) @ X15)) != ((ap @ (sub @ X14 @ id) @ X16)))))) & (apinj1 <=> ! [X9 : term,X15 : term,X16 : term,X14 : term] : ((X9 = X14) | (((ap @ X9 @ X15)) != ((ap @ X14 @ X16))))) & (! [X33 : subst,X17 : (subst > term > term),X32 : (subst > term > term),X34 : term] : ((((lam @ (X17 @ sh @ one))) != ((lam @ (X32 @ sh @ one)))) | ~ ! [X10 : subst,X9 : term,X20 : subst] : (((sub @ (X17 @ X10 @ X9) @ X20)) = ((X17 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20)))) | ~ ! [X9 : term,X20 : subst,X10 : subst] : (((sub @ (X32 @ X10 @ X9) @ X20)) = ((X32 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20)))) | (((X17 @ X33 @ X34)) = ((X32 @ X33 @ X34)))) <=> hoaslaminj) & (((sub @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2term @ d_one))) & (shinj <=> ! [X9 : term,X14 : term] : ((X9 = X14) | (((sub @ X9 @ sh)) != ((sub @ X14 @ sh))))) & hoaslamnotap_lthm & (hoasapinj2 <=> ! [X14 : term,X15 : term,X16 : term,X9 : term] : ((((ap @ (sub @ X9 @ id) @ X15)) != ((ap @ (sub @ X14 @ id) @ X16))) | (X15 = X16))) & ~(var @ (d2term @ d_one)) & (! [X9 : term] : (X9 = ((sub @ X9 @ id))) <=> termmset) & (((lam @ (d2term @ d_one))) = ((d2term @ d_one))) & axabs & ! [X0 : d_term] : (X0 = d_one) & lamnotvar & hoasinduction_lem3v2_gthm & (! [X14 : term] : ~ ! [X17 : (subst > term > term)] : (((X17 @ sh @ one)) != ((sub @ X14 @ (push @ one @ sh)))) <=> hoasinduction_lem3b) & (pushprop_lem1v2_lthm <=> (! [X9 : term,X10 : subst] : (X9 = ((sub @ one @ (push @ X9 @ X10)))) => ! [X9 : term,X7 : (term > $o),X10 : subst,X8 : (term > $o)] : (~ ! [X11 : term] : (((X8 @ X11)) = ((X7 @ (sub @ X11 @ (push @ X9 @ X10))))) | ~(X7 @ X9) | (X8 @ one)))) & (! [X14 : term] : ~ ! [X17 : (subst > term > term)] : (((X17 @ sh @ one)) != ((sub @ X14 @ (push @ one @ sh)))) <=> hoasinduction_lem3b_lthm) & (! [X7 : (subst > term > subst > $o),X38 : term] : ((X7 @ id @ (lam @ (sub @ X38 @ (push @ one @ sh))) @ id) | ~ ! [X17 : (subst > term > term),X39 : term] : (~ ! [X40 : subst,X42 : subst,X41 : term] : (((X17 @ (comp @ X40 @ X42) @ (sub @ X41 @ X42))) = ((sub @ X39 @ (push @ (sub @ X41 @ X42) @ (comp @ X40 @ X42))))) | ~ ! [X9 : term] : ((X7 @ id @ (X17 @ id @ X9) @ id) | ~(X7 @ id @ X9 @ id)) | (X7 @ id @ (lam @ (X17 @ sh @ one)) @ id) | ~ ! [X44 : term,X45 : subst,X43 : subst] : (((sub @ (X17 @ X43 @ X44) @ X45)) = ((sub @ (sub @ X39 @ (push @ X44 @ X43)) @ X45)))) | ~ ! [X14 : term] : ((X7 @ id @ (sub @ X38 @ (push @ X14 @ id)) @ id) | ~(X7 @ id @ X14 @ id))) <=> hoasinduction_lem3aaa) & (induction <=> ! [X35 : term,X7 : (term > $o)] : (~ ! [X9 : term] : (~(X7 @ X9) | (X7 @ (lam @ X9))) | ~ ! [X9 : term] : (~(var @ X9) | (X7 @ X9)) | (X7 @ X35) | ~ ! [X14 : term,X9 : term] : (~(X7 @ X14) | (X7 @ (ap @ X9 @ X14)) | ~(X7 @ X9)))) & (pushprop_lem1_lthm <=> (! [X9 : term,X10 : subst] : (X9 = ((sub @ one @ (push @ X9 @ X10)))) => (! [X10 : subst,X9 : term] : (X10 = ((comp @ sh @ (push @ X9 @ X10)))) => ! [X10 : subst,X9 : term,X24 : (term > $o),X7 : (term > $o),X14 : term] : (~(X7 @ X9) | (X24 @ (sub @ X9 @ (push @ X14 @ X10))))))) & (! [X7 : (subst > term > subst > $o),X27 : term] : (~ ! [X14 : term] : ((X7 @ id @ (sub @ X27 @ (push @ X14 @ id)) @ id) | ~(X7 @ id @ X14 @ id)) | (X7 @ id @ (lam @ X27) @ id) | ~ ! [X17 : (subst > term > term)] : ((X7 @ id @ (lam @ (X17 @ sh @ one)) @ id) | ~ ! [X9 : term] : (~(X7 @ id @ X9 @ id) | (X7 @ id @ (X17 @ id @ X9) @ id)) | ~ ! [X9 : term,X10 : subst,X20 : subst] : (((sub @ (X17 @ X10 @ X9) @ X20)) = ((X17 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20)))))) <=> hoasinduction_lem3a) & hoasinduction_lem2v2_gthm & hoasinduction_lem2_lthm & (hoaslam = (^[X5 : subst, X6 : (subst > term > term)] : ((d2term @ d_one)))) & (! [X22 : term,X7 : (subst > term > subst > $o)] : ((X7 @ id @ (lam @ (sub @ X22 @ (push @ one @ sh))) @ id) | ~ ! [X17 : (subst > term > term)] : (~ ! [X20 : subst,X10 : subst,X9 : term] : (((sub @ (X17 @ X10 @ X9) @ X20)) = ((X17 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20)))) | (X7 @ id @ (lam @ (X17 @ sh @ one)) @ id) | ~ ! [X9 : term] : (~(X7 @ id @ X9 @ id) | (X7 @ id @ (X17 @ id @ X9) @ id))) | ~ ! [X14 : term] : (~(X7 @ id @ X14 @ id) | (X7 @ id @ (sub @ X22 @ (push @ X14 @ id)) @ id))) <=> hoasinduction_lem3aa) & (hoasinduction_no_psi_cond <=> ! [X19 : term,X7 : (subst > term > subst > $o)] : ((X7 @ id @ X19 @ id) | ~ ! [X17 : (subst > term > term)] : ((X7 @ id @ (lam @ (X17 @ sh @ one)) @ id) | ~ ! [X9 : term] : (~(X7 @ id @ X9 @ id) | (X7 @ id @ (X17 @ id @ X9) @ id)) | ~ ! [X10 : subst,X9 : term,X20 : subst] : (((sub @ (X17 @ X10 @ X9) @ X20)) = ((X17 @ (comp @ X10 @ X20) @ (sub @ X9 @ X20))))) | ~ ! [X14 : term,X9 : term] : (~(X7 @ id @ X14 @ id) | ~(X7 @ id @ X9 @ id) | (X7 @ id @ (ap @ (sub @ X9 @ id) @ X14) @ id)))) & (sh = ((d2subst @ d_id))) & (((ap @ (d2term @ d_one) @ (d2term @ d_one))) = ((d2term @ d_one))) & hoasinduction_lem1v2_gthm),
  file('/export/starexec/sandbox2/benchmark/theBenchmark.p',alg444_1)).
thf(f2,conjecture,(
  (pushprop_lem1v2 = ! [X0 : (term > $o),X3 : subst,X2 : term,X1 : (term > $o)] : ((X0 @ X2) => ((pushprop_p_and_p_prime @ X2 @ X3 @ X0 @ X1) => (X1 @ one))))),
  file('/export/starexec/sandbox2/benchmark/theBenchmark.p',pushprop_lem1v2)).
thf(f3,negated_conjecture,(
  ~ (pushprop_lem1v2 = ! [X0 : (term > $o),X3 : subst,X2 : term,X1 : (term > $o)] : ((X0 @ X2) => ((pushprop_p_and_p_prime @ X2 @ X3 @ X0 @ X1) => (X1 @ one))))),
  inference(negated_conjecture,[status(cth)],[f2])).
thf(f4,plain,(
  ~ (pushprop_lem1v2 = ! [X0 : (term > $o),X1 : subst,X2 : term,X3 : (term > $o)] : ((X0 @ X2) => ((pushprop_p_and_p_prime @ X2 @ X1 @ X0 @ X3) => (X3 @ one))))),
  inference(rectify,[],[f3])).
thf(f5,plain,(
  ~ (pushprop_lem1v2 = ((!! @ (term > $o) @ (^[Y0 : term > $o]: (!! @ term @ (^[Y1 : term]: (!! @ subst @ (^[Y2 : subst]: (!! @ (term > $o) @ (^[Y3 : term > $o]: ((Y3 @ Y1) => ((pushprop_p_and_p_prime @ Y1 @ Y2 @ Y3 @ Y0) => (Y0 @ one)))))))))))))),
  inference(fool_elimination,[],[f4])).
thf(f6,plain,(
  (! [X0 : subst] : (((comp @ X0 @ id)) = X0) <=> axidr) & hoasapinj1_gthm & (termmset_lthm <=> (! [X1 : term] : (((sub @ X1 @ id)) = X1) => ! [X2 : term] : (((sub @ X2 @ id)) = X2))) & termmset_gthm & hoasinduction_lem1 & ((! [X3 : subst] : (((comp @ X3 @ id)) = X3) & ! [X4 : subst] : (((comp @ id @ X4)) = X4)) <=> substmonoid) & pushprop_lthm & hoaslamnotvar_lthm & ! [X5 : d_subst] : (d_id = X5) & (((comp @ (d2subst @ d_id) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & (! [X6 : term,X7 : (subst > term > subst > $o)] : (~ ! [X8 : term,X9 : term] : (~(X7 @ id @ X8 @ id) | ~(X7 @ id @ X9 @ id) | (X7 @ id @ (ap @ (sub @ X9 @ id) @ X8) @ id)) | ~ ! [X10 : subst,X11 : subst,X12 : subst,X13 : term] : ((X7 @ (comp @ X11 @ X12) @ (sub @ X13 @ X12) @ X10) | ~(X7 @ X11 @ X13 @ (comp @ X12 @ X10))) | ~ ! [X14 : subst,X15 : term,X16 : subst,X17 : subst] : (~(X7 @ (comp @ X14 @ X16) @ (sub @ X15 @ X16) @ X17) | (X7 @ X14 @ X15 @ (comp @ X16 @ X17))) | (X7 @ id @ X6 @ id) | ~ ! [X18 : term] : (~(var @ (sub @ X18 @ id)) | (X7 @ id @ X18 @ id)) | ~ ! [X19 : (subst > term > term)] : ((X7 @ id @ (lam @ (X19 @ sh @ one)) @ id) | ~ ! [X20 : subst,X21 : term,X22 : subst] : (((sub @ (X19 @ X22 @ X21) @ X20)) = ((X19 @ (comp @ X22 @ X20) @ (sub @ X21 @ X20)))) | ~ ! [X23 : term] : ((X7 @ id @ (X19 @ id @ X23) @ id) | ~(X7 @ id @ X23 @ id)))) <=> hoasinduction) & ((! [X24 : term] : (((sub @ X24 @ id)) = X24) => (! [X25 : subst,X26 : term,X27 : (term > $o)] : (~ ! [X28 : term] : (~ ! [X29 : term] : (~(X27 @ X29) | (X27 @ (sub @ X28 @ (push @ X29 @ id)))) | (X27 @ (lam @ X28))) | (X27 @ (sub @ X26 @ X25)) | ~ ! [X30 : term,X31 : term] : (~(X27 @ X30) | (X27 @ (ap @ X30 @ X31)) | ~(X27 @ X31)) | ~ ! [X32 : term] : (~(var @ X32) | (X27 @ (sub @ X32 @ X25)))) => ! [X33 : (term > $o),X34 : term] : (~ ! [X35 : term] : ((X33 @ (lam @ X35)) | ~ ! [X36 : term] : (~(X33 @ X36) | (X33 @ (sub @ X35 @ (push @ X36 @ id))))) | (X33 @ X34) | ~ ! [X37 : term,X38 : term] : (~(X33 @ X37) | ~(X33 @ X38) | (X33 @ (ap @ X38 @ X37))) | ~ ! [X39 : term] : ((X33 @ X39) | ~(var @ X39))))) <=> induction2_lthm) & (! [X40 : subst] : (((comp @ id @ X40)) = X40) <=> axidl) & axclos & pushprop_lem2v2 & pushprop & axmap & ((! [X41 : subst,X42 : term] : (((sub @ one @ (push @ X42 @ X41))) = X42) => (! [X43 : subst,X44 : term] : (((comp @ sh @ (push @ X44 @ X43))) = X43) => (! [X45 : subst] : (((comp @ X45 @ id)) = X45) => (! [X46 : (term > $o),X47 : term] : (~ ! [X48 : term] : (~(var @ X48) | (X46 @ X48)) | ~ ! [X49 : term,X50 : term] : (~(X46 @ X49) | ~(X46 @ X50) | (X46 @ (ap @ X49 @ X50))) | ~ ! [X51 : term] : ((X46 @ (lam @ X51)) | ~(X46 @ X51)) | (X46 @ X47)) => ! [X52 : (term > $o),X53 : subst,X54 : term] : (~ ! [X55 : term,X56 : term] : (~(X52 @ X55) | (X52 @ (ap @ X55 @ X56)) | ~(X52 @ X56)) | (X52 @ (sub @ X54 @ X53)) | ~ ! [X57 : term] : ((X52 @ (lam @ X57)) | ~ ! [X58 : term] : ((X52 @ (sub @ X57 @ (push @ X58 @ id))) | ~(X52 @ X58))) | ~ ! [X59 : term] : ((X52 @ (sub @ X59 @ X53)) | ~(var @ X59))))))) <=> induction2lem_lthm) & (! [X60 : term,X61 : term,X62 : term,X63 : term] : ((X60 = X61) | (((ap @ (sub @ X61 @ id) @ X62)) != ((ap @ (sub @ X60 @ id) @ X63)))) <=> hoasapinj1) & hoaslaminj_gthm & (hoasinduction_lem3 <=> ! [X64 : term,X65 : (subst > term > subst > $o)] : (~ ! [X66 : subst,X67 : term,X68 : subst,X69 : subst] : ((X65 @ (comp @ X66 @ X69) @ (sub @ X67 @ X69) @ X68) | ~(X65 @ X66 @ X67 @ (comp @ X69 @ X68))) | (X65 @ id @ (lam @ X64) @ id) | ~ ! [X70 : term] : ((X65 @ id @ (sub @ X64 @ (push @ X70 @ id)) @ id) | ~(X65 @ id @ X70 @ id)) | ~ ! [X71 : (subst > term > term)] : (~ ! [X72 : subst,X73 : subst,X74 : term] : (((X71 @ (comp @ X73 @ X72) @ (sub @ X74 @ X72))) = ((sub @ (X71 @ X73 @ X74) @ X72))) | (X65 @ id @ (lam @ (X71 @ sh @ one)) @ id) | ~ ! [X75 : term] : (~(X65 @ id @ X75 @ id) | (X65 @ id @ (X71 @ id @ X75) @ id))) | ~ ! [X76 : term,X77 : subst,X78 : subst,X79 : subst] : ((X65 @ X77 @ X76 @ (comp @ X79 @ X78)) | ~(X65 @ (comp @ X77 @ X79) @ (sub @ X76 @ X79) @ X78)))) & (! [X80 : term,X81 : term,X82 : term,X83 : term] : ((X80 = X83) | (((ap @ X82 @ X83)) != ((ap @ X81 @ X80)))) <=> apinj2) & ((! [X84 : term] : (((sub @ X84 @ id)) = X84) => (! [X85 : (subst > term > subst > $o),X86 : term] : ((X85 @ id @ (lam @ (sub @ X86 @ (push @ one @ sh))) @ id) | ~ ! [X87 : (subst > term > term)] : (~ ! [X88 : subst,X89 : subst,X90 : term] : (((X87 @ (comp @ X88 @ X89) @ (sub @ X90 @ X89))) = ((sub @ (X87 @ X88 @ X90) @ X89))) | ~ ! [X91 : term] : ((X85 @ id @ (X87 @ id @ X91) @ id) | ~(X85 @ id @ X91 @ id)) | (X85 @ id @ (lam @ (X87 @ sh @ one)) @ id)) | ~ ! [X92 : term] : ((X85 @ id @ (sub @ X86 @ (push @ X92 @ id)) @ id) | ~(X85 @ id @ X92 @ id))) => ! [X93 : (subst > term > subst > $o),X94 : term] : (~ ! [X95 : term] : ((X93 @ id @ (sub @ X94 @ (push @ X95 @ id)) @ id) | ~(X93 @ id @ X95 @ id)) | (X93 @ id @ (lam @ X94) @ id) | ~ ! [X96 : (subst > term > term)] : (~ ! [X97 : term] : ((X93 @ id @ (X96 @ id @ X97) @ id) | ~(X93 @ id @ X97 @ id)) | ~ ! [X98 : subst,X99 : term,X100 : subst] : (((sub @ (X96 @ X100 @ X99) @ X98)) = ((X96 @ (comp @ X100 @ X98) @ (sub @ X99 @ X98)))) | (X93 @ id @ (lam @ (X96 @ sh @ one)) @ id))))) <=> hoasinduction_lem3a_lthm) & ~lamnotap & (! [X101 : subst,X102 : (term > $o),X103 : term] : ~ ! [X104 : (term > $o)] : ~ ! [X105 : term] : (((X104 @ X105)) = ((X102 @ (sub @ X105 @ (push @ X103 @ X101))))) <=> pushprop_lem0) & hoasapinj2_gthm & (hoasinduction_p_and_p_prime = (^[X106 : (subst > term > subst > $o), X107 : (term > $o)] : (! [X108 : term] : (((X107 @ X108)) = ((X106 @ id @ X108 @ id)))))) & pushprop_lem2v2_gthm & (pushprop_lem0_lthm <=> ! [X109 : (term > $o),X110 : subst,X111 : term] : ~ ! [X112 : (term > $o)] : ~ ! [X113 : term] : (((X112 @ X113)) = ((X109 @ (sub @ X113 @ (push @ X111 @ X110)))))) & (pushprop_lem1v2 <=> ! [X114 : (term > $o),X115 : (term > $o),X116 : term,X117 : subst] : (~(X114 @ X116) | (X115 @ one) | ~ ! [X118 : term] : (((X115 @ X118)) = ((X114 @ (sub @ X118 @ (push @ X116 @ X117))))))) & pushprop_lthm_orig & (axvarid <=> ! [X119 : term] : (((sub @ X119 @ id)) = X119)) & (! [X120 : term,X121 : term,X122 : (subst > term > subst > $o)] : (~ ! [X123 : term,X124 : term] : ((X122 @ id @ (ap @ (sub @ X123 @ id) @ X124) @ id) | ~(X122 @ id @ X123 @ id) | ~(X122 @ id @ X124 @ id)) | ~(X122 @ id @ X121 @ id) | ~(X122 @ id @ X120 @ id) | ~ ! [X125 : term,X126 : subst,X127 : subst,X128 : subst] : ((X122 @ X126 @ X125 @ (comp @ X128 @ X127)) | ~(X122 @ (comp @ X126 @ X128) @ (sub @ X125 @ X128) @ X127)) | ~ ! [X129 : subst,X130 : term,X131 : subst,X132 : subst] : ((X122 @ (comp @ X129 @ X131) @ (sub @ X130 @ X131) @ X132) | ~(X122 @ X129 @ X130 @ (comp @ X131 @ X132))) | (X122 @ id @ (ap @ X121 @ X120) @ id)) <=> hoasinduction_lem2) & pushprop_gthm & axassoc & (((push @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & (one = ((d2term @ d_one))) & hoasinduction_lem3_gthm & (hoaslaminj_lthm <=> (! [X133 : subst,X134 : term] : (((sub @ one @ (push @ X134 @ X133))) = X134) => (! [X135 : term,X136 : subst] : (((comp @ sh @ (push @ X135 @ X136))) = X136) => (! [X137 : term,X138 : term] : ((X137 = X138) | (((lam @ X138)) != ((lam @ X137)))) => ! [X139 : term,X140 : (subst > term > term),X141 : subst,X142 : (subst > term > term)] : (~ ! [X143 : subst,X144 : term,X145 : subst] : (((X142 @ (comp @ X145 @ X143) @ (sub @ X144 @ X143))) = ((sub @ (X142 @ X145 @ X144) @ X143))) | (((lam @ (X140 @ sh @ one))) != ((lam @ (X142 @ sh @ one)))) | (((X142 @ X141 @ X139)) = ((X140 @ X141 @ X139))) | ~ ! [X146 : term,X147 : subst,X148 : subst] : (((sub @ (X140 @ X148 @ X146) @ X147)) = ((X140 @ (comp @ X148 @ X147) @ (sub @ X146 @ X147))))))))) & (id = ((d2subst @ d_id))) & ulamvarsh & ! [X149 : subst] : ? [X150 : d_subst] : (((d2subst @ X150)) = X149) & hoaslamnotvar & pushprop_lem1v2_gthm & hoasapnotvar_lthm & hoasinduction_gthm & (hoasinduction_lthm_3 <=> (! [X151 : (subst > term > subst > $o)] : ~ ! [X152 : (term > $o)] : ~ ! [X153 : term] : (((X151 @ id @ X153 @ id)) = ((X152 @ X153))) => (! [X154 : (term > $o),X155 : term] : ((X154 @ X155) | ~ ! [X156 : term,X157 : term] : (~(X154 @ X157) | ~(X154 @ X156) | (X154 @ (ap @ X157 @ X156))) | ~ ! [X158 : term] : ((X154 @ (lam @ X158)) | ~ ! [X159 : term] : (~(X154 @ X159) | (X154 @ (sub @ X158 @ (push @ X159 @ id))))) | ~ ! [X160 : term] : ((X154 @ X160) | ~(var @ X160))) => (! [X161 : term] : (((sub @ X161 @ id)) = X161) => (! [X162 : (subst > term > subst > $o),X163 : (term > $o),X164 : term] : ((X163 @ (lam @ X164)) | ~ ! [X165 : term] : (((X163 @ X165)) = ((X162 @ id @ X165 @ id))) | ~ ! [X166 : term] : ((X163 @ (sub @ X164 @ (push @ X166 @ id))) | ~(X163 @ X166)) | ~ ! [X167 : (subst > term > term)] : (~ ! [X168 : term] : ((X162 @ id @ (X167 @ id @ X168) @ id) | ~(X162 @ id @ X168 @ id)) | ~ ! [X169 : subst,X170 : subst,X171 : term] : (((sub @ (X167 @ X169 @ X171) @ X170)) = ((X167 @ (comp @ X169 @ X170) @ (sub @ X171 @ X170)))) | (X162 @ id @ (lam @ (X167 @ sh @ one)) @ id))) => ! [X172 : term,X173 : (subst > term > subst > $o)] : (~ ! [X174 : (subst > term > term)] : (~ ! [X175 : term] : ((X173 @ id @ (X174 @ id @ X175) @ id) | ~(X173 @ id @ X175 @ id)) | ~ ! [X176 : subst,X177 : subst,X178 : term] : (((X174 @ (comp @ X176 @ X177) @ (sub @ X178 @ X177))) = ((sub @ (X174 @ X176 @ X178) @ X177))) | (X173 @ id @ (lam @ (X174 @ sh @ one)) @ id)) | (X173 @ id @ X172 @ id) | ~ ! [X179 : subst,X180 : subst,X181 : subst,X182 : term] : (~(X173 @ (comp @ X179 @ X180) @ (sub @ X182 @ X180) @ X181) | (X173 @ X179 @ X182 @ (comp @ X180 @ X181))) | ~ ! [X183 : term] : ((X173 @ id @ X183 @ id) | ~(var @ (sub @ X183 @ id))) | ~ ! [X184 : term,X185 : subst,X186 : subst,X187 : subst] : (~(X173 @ X187 @ X184 @ (comp @ X185 @ X186)) | (X173 @ (comp @ X187 @ X185) @ (sub @ X184 @ X185) @ X186)) | ~ ! [X188 : term,X189 : term] : (~(X173 @ id @ X189 @ id) | (X173 @ id @ (ap @ (sub @ X189 @ id) @ X188) @ id) | ~(X173 @ id @ X188 @ id)))))))) & hoasinduction_lem1v2 & (hoasinduction_lem2v2 <=> ! [X190 : term,X191 : term,X192 : (subst > term > subst > $o),X193 : (term > $o)] : (~ ! [X194 : term] : (((X192 @ id @ X194 @ id)) = ((X193 @ X194))) | (X193 @ (ap @ X191 @ X190)) | ~(X193 @ X190) | ~ ! [X195 : subst,X196 : subst,X197 : subst,X198 : term] : (~(X192 @ (comp @ X197 @ X196) @ (sub @ X198 @ X196) @ X195) | (X192 @ X197 @ X198 @ (comp @ X196 @ X195))) | ~ ! [X199 : subst,X200 : subst,X201 : term,X202 : subst] : (~(X192 @ X202 @ X201 @ (comp @ X199 @ X200)) | (X192 @ (comp @ X202 @ X199) @ (sub @ X201 @ X199) @ X200)) | ~ ! [X203 : term,X204 : term] : (~(X192 @ id @ X203 @ id) | (X192 @ id @ (ap @ (sub @ X203 @ id) @ X204) @ id) | ~(X192 @ id @ X204 @ id)) | ~(X193 @ X191))) & (! [X205 : (term > $o),X206 : subst,X207 : term] : (~ ! [X208 : term] : (~ ! [X209 : term] : ((X205 @ (sub @ X208 @ (push @ X209 @ id))) | ~(X205 @ X209)) | (X205 @ (lam @ X208))) | ~ ! [X210 : term,X211 : term] : (~(X205 @ X210) | (X205 @ (ap @ X210 @ X211)) | ~(X205 @ X211)) | (X205 @ (sub @ X207 @ X206)) | ~ ! [X212 : term] : (~(var @ X212) | (X205 @ (sub @ X212 @ X206)))) <=> induction2lem) & ! [X213 : d_term,X214 : d_term] : ((((d2term @ X214)) = ((d2term @ X213))) => (X213 = X214)) & hoasinduction_lem2_gthm & (pushprop_lem1 <=> ! [X215 : subst,X216 : term,X217 : (term > $o),X218 : (term > $o),X219 : term] : (~(X217 @ X216) | (X218 @ (sub @ X216 @ (push @ X219 @ X215))))) & ((! [X220 : term] : (((sub @ X220 @ id)) = X220) => ! [X221 : term,X222 : (subst > term > subst > $o),X223 : (term > $o)] : (~ ! [X224 : subst,X225 : subst,X226 : term,X227 : subst] : ((X222 @ X227 @ X226 @ (comp @ X225 @ X224)) | ~(X222 @ (comp @ X227 @ X225) @ (sub @ X226 @ X225) @ X224)) | ~ ! [X228 : subst,X229 : term,X230 : subst,X231 : subst] : ((X222 @ (comp @ X228 @ X231) @ (sub @ X229 @ X231) @ X230) | ~(X222 @ X228 @ X229 @ (comp @ X231 @ X230))) | ~ ! [X232 : term] : (~(X223 @ X232) | (X223 @ (sub @ X221 @ (push @ X232 @ id)))) | ~ ! [X233 : term] : (((X222 @ id @ X233 @ id)) = ((X223 @ X233))) | (X223 @ (lam @ X221)) | ~ ! [X234 : (subst > term > term)] : (~ ! [X235 : term,X236 : subst,X237 : subst] : (((X234 @ (comp @ X237 @ X236) @ (sub @ X235 @ X236))) = ((sub @ (X234 @ X237 @ X235) @ X236))) | ~ ! [X238 : term] : ((X222 @ id @ (X234 @ id @ X238) @ id) | ~(X222 @ id @ X238 @ id)) | (X222 @ id @ (lam @ (X234 @ sh @ one)) @ id)))) <=> hoasinduction_lem3v2_lthm) & ((! [X239 : term] : (((sub @ X239 @ id)) = X239) => (! [X240 : term,X241 : term,X242 : term,X243 : term] : ((X240 = X241) | (((ap @ X241 @ X243)) != ((ap @ X240 @ X242)))) => ! [X244 : term,X245 : term,X246 : term,X247 : term] : ((((ap @ (sub @ X245 @ id) @ X246)) != ((ap @ (sub @ X247 @ id) @ X244))) | (X245 = X247)))) <=> hoasapinj1_lthm) & (hoasinduction_lthm <=> (! [X248 : term,X249 : (term > $o)] : (~ ! [X250 : term] : ((X249 @ X250) | ~(var @ X250)) | (X249 @ X248) | ~ ! [X251 : term] : ((X249 @ (lam @ X251)) | ~ ! [X252 : term] : (~(X249 @ X252) | (X249 @ (sub @ X251 @ (push @ X252 @ id))))) | ~ ! [X253 : term,X254 : term] : (~(X249 @ X254) | (X249 @ (ap @ X253 @ X254)) | ~(X249 @ X253))) => (! [X255 : (subst > term > subst > $o),X256 : term,X257 : term] : (~ ! [X258 : subst,X259 : subst,X260 : subst,X261 : term] : (~(X255 @ (comp @ X259 @ X258) @ (sub @ X261 @ X258) @ X260) | (X255 @ X259 @ X261 @ (comp @ X258 @ X260))) | ~ ! [X262 : subst,X263 : term,X264 : subst,X265 : subst] : ((X255 @ (comp @ X262 @ X265) @ (sub @ X263 @ X265) @ X264) | ~(X255 @ X262 @ X263 @ (comp @ X265 @ X264))) | ~(X255 @ id @ X257 @ id) | ~ ! [X266 : term,X267 : term] : (~(X255 @ id @ X267 @ id) | (X255 @ id @ (ap @ (sub @ X267 @ id) @ X266) @ id) | ~(X255 @ id @ X266 @ id)) | (X255 @ id @ (ap @ X256 @ X257) @ id) | ~(X255 @ id @ X256 @ id)) => (! [X268 : term,X269 : (subst > term > subst > $o)] : (~ ! [X270 : term,X271 : subst,X272 : subst,X273 : subst] : ((X269 @ X272 @ X270 @ (comp @ X271 @ X273)) | ~(X269 @ (comp @ X272 @ X271) @ (sub @ X270 @ X271) @ X273)) | (X269 @ id @ (lam @ X268) @ id) | ~ ! [X274 : subst,X275 : subst,X276 : subst,X277 : term] : (~(X269 @ X275 @ X277 @ (comp @ X276 @ X274)) | (X269 @ (comp @ X275 @ X276) @ (sub @ X277 @ X276) @ X274)) | ~ ! [X278 : term] : ((X269 @ id @ (sub @ X268 @ (push @ X278 @ id)) @ id) | ~(X269 @ id @ X278 @ id)) | ~ ! [X279 : (subst > term > term)] : (~ ! [X280 : term] : ((X269 @ id @ (X279 @ id @ X280) @ id) | ~(X269 @ id @ X280 @ id)) | (X269 @ id @ (lam @ (X279 @ sh @ one)) @ id) | ~ ! [X281 : subst,X282 : subst,X283 : term] : (((X279 @ (comp @ X281 @ X282) @ (sub @ X283 @ X282))) = ((sub @ (X279 @ X281 @ X283) @ X282))))) => ! [X284 : (subst > term > subst > $o),X285 : term] : ((X284 @ id @ X285 @ id) | ~ ! [X286 : (subst > term > term)] : ((X284 @ id @ (lam @ (X286 @ sh @ one)) @ id) | ~ ! [X287 : term] : (~(X284 @ id @ X287 @ id) | (X284 @ id @ (X286 @ id @ X287) @ id)) | ~ ! [X288 : term,X289 : subst,X290 : subst] : (((X286 @ (comp @ X289 @ X290) @ (sub @ X288 @ X290))) = ((sub @ (X286 @ X289 @ X288) @ X290)))) | ~ ! [X291 : subst,X292 : subst,X293 : term,X294 : subst] : (~(X284 @ X291 @ X293 @ (comp @ X292 @ X294)) | (X284 @ (comp @ X291 @ X292) @ (sub @ X293 @ X292) @ X294)) | ~ ! [X295 : subst,X296 : subst,X297 : term,X298 : subst] : (~(X284 @ (comp @ X295 @ X298) @ (sub @ X297 @ X298) @ X296) | (X284 @ X295 @ X297 @ (comp @ X298 @ X296))) | ~ ! [X299 : term] : (~(var @ (sub @ X299 @ id)) | (X284 @ id @ X299 @ id)) | ~ ! [X300 : term,X301 : term] : (~(X284 @ id @ X301 @ id) | (X284 @ id @ (ap @ (sub @ X300 @ id) @ X301) @ id) | ~(X284 @ id @ X300 @ id))))))) & ~(hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) & hoasapnotvar & ((! [X302 : term] : ~ ! [X303 : (subst > term > term)] : ~ ! [X304 : subst,X305 : term] : (((sub @ X302 @ (push @ X305 @ X304))) = ((X303 @ X304 @ X305))) => (! [X306 : term] : (((sub @ X306 @ id)) = X306) => ! [X307 : term,X308 : (term > $o),X309 : (subst > term > subst > $o)] : (~ ! [X310 : (subst > term > term)] : (~ ! [X311 : term,X312 : subst,X313 : subst] : (((X310 @ (comp @ X312 @ X313) @ (sub @ X311 @ X313))) = ((sub @ (X310 @ X312 @ X311) @ X313))) | (X309 @ id @ (lam @ (X310 @ sh @ one)) @ id) | ~ ! [X314 : term] : ((X309 @ id @ (X310 @ id @ X314) @ id) | ~(X309 @ id @ X314 @ id))) | ~ ! [X315 : term] : (((X308 @ X315)) = ((X309 @ id @ X315 @ id))) | (X308 @ (lam @ X307)) | ~ ! [X316 : term] : ((X308 @ (sub @ X307 @ (push @ X316 @ id))) | ~(X308 @ X316))))) <=> hoasinduction_lem3v2a_lthm) & ((! [X317 : (subst > term > subst > $o)] : ~ ! [X318 : (term > $o)] : ~ ! [X319 : term] : (((X317 @ id @ X319 @ id)) = ((X318 @ X319))) => (! [X320 : (term > $o),X321 : term] : (~ ! [X322 : term] : ((X320 @ (lam @ X322)) | ~ ! [X323 : term] : ((X320 @ (sub @ X322 @ (push @ X323 @ id))) | ~(X320 @ X323))) | ~ ! [X324 : term,X325 : term] : (~(X320 @ X324) | ~(X320 @ X325) | (X320 @ (ap @ X324 @ X325))) | ~ ! [X326 : term] : ((X320 @ X326) | ~(var @ X326)) | (X320 @ X321)) => (! [X327 : term] : (((sub @ X327 @ id)) = X327) => (! [X328 : (term > $o),X329 : (subst > term > subst > $o),X330 : term] : (~ ! [X331 : term] : ((X328 @ (sub @ X330 @ (push @ X331 @ id))) | ~(X328 @ X331)) | ~ ! [X332 : term] : (((X329 @ id @ X332 @ id)) = ((X328 @ X332))) | ~ ! [X333 : (subst > term > term)] : (~ ! [X334 : subst,X335 : subst,X336 : term] : (((X333 @ (comp @ X335 @ X334) @ (sub @ X336 @ X334))) = ((sub @ (X333 @ X335 @ X336) @ X334))) | ~ ! [X337 : term] : (~(X329 @ id @ X337 @ id) | (X329 @ id @ (X333 @ id @ X337) @ id)) | (X329 @ id @ (lam @ (X333 @ sh @ one)) @ id)) | (X328 @ (lam @ X330))) => ! [X338 : term,X339 : (subst > term > subst > $o)] : (~ ! [X340 : term,X341 : term] : (~(X339 @ id @ X341 @ id) | (X339 @ id @ (ap @ (sub @ X340 @ id) @ X341) @ id) | ~(X339 @ id @ X340 @ id)) | ~ ! [X342 : (subst > term > term)] : (~ ! [X343 : term] : ((X339 @ id @ (X342 @ id @ X343) @ id) | ~(X339 @ id @ X343 @ id)) | (X339 @ id @ (lam @ (X342 @ sh @ one)) @ id) | ~ ! [X344 : term,X345 : subst,X346 : subst] : (((X342 @ (comp @ X345 @ X346) @ (sub @ X344 @ X346))) = ((sub @ (X342 @ X345 @ X344) @ X346)))) | (X339 @ id @ X338 @ id)))))) <=> hoasinduction_no_psi_cond_lthm) & (hoasinduction_lem0_lthm <=> ! [X347 : (subst > term > subst > $o)] : ~ ! [X348 : (term > $o)] : ~ ! [X349 : term] : (((X347 @ id @ X349 @ id)) = ((X348 @ X349)))) & (((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) = ((d2term @ d_one))) & hoaslamnotvar_gthm & apnotvar & (! [X350 : subst,X351 : term] : (((sub @ one @ (push @ X351 @ X350))) = X351) <=> axvarcons) & (hoasinduction_lem3v2a <=> ! [X352 : (subst > term > subst > $o),X353 : (term > $o),X354 : term] : (~ ! [X355 : (subst > term > term)] : ((X352 @ id @ (lam @ (X355 @ sh @ one)) @ id) | ~ ! [X356 : term] : (~(X352 @ id @ X356 @ id) | (X352 @ id @ (X355 @ id @ X356) @ id)) | ~ ! [X357 : term,X358 : subst,X359 : subst] : (((sub @ (X355 @ X359 @ X357) @ X358)) = ((X355 @ (comp @ X359 @ X358) @ (sub @ X357 @ X358))))) | (X353 @ (lam @ X354)) | ~ ! [X360 : term] : (((X353 @ X360)) = ((X352 @ id @ X360 @ id))) | ~ ! [X361 : term] : (~(X353 @ X361) | (X353 @ (sub @ X354 @ (push @ X361 @ id)))))) & (! [X362 : (term > $o),X363 : term] : (~ ! [X364 : term] : (~ ! [X365 : term] : (~(X362 @ X365) | (X362 @ (sub @ X364 @ (push @ X365 @ id)))) | (X362 @ (lam @ X364))) | ~ ! [X366 : term] : (~(var @ X366) | (X362 @ X366)) | (X362 @ X363) | ~ ! [X367 : term,X368 : term] : ((X362 @ (ap @ X368 @ X367)) | ~(X362 @ X367) | ~(X362 @ X368))) <=> induction2) & pushprop_lem0_gthm & hoaslamnotap_gthm & (hoasinduction_lem3aa_lthm <=> ! [X369 : (subst > term > subst > $o),X370 : term] : (~ ! [X371 : (subst > term > term)] : ((X369 @ id @ (lam @ (X371 @ sh @ one)) @ id) | ~ ! [X372 : subst,X373 : subst,X374 : term] : (((sub @ (X371 @ X373 @ X374) @ X372)) = ((X371 @ (comp @ X373 @ X372) @ (sub @ X374 @ X372)))) | ~ ! [X375 : term] : ((X369 @ id @ (X371 @ id @ X375) @ id) | ~(X369 @ id @ X375 @ id))) | ~ ! [X376 : term] : ((X369 @ id @ (sub @ X370 @ (push @ X376 @ id)) @ id) | ~(X369 @ id @ X376 @ id)) | (X369 @ id @ (lam @ (sub @ X370 @ (push @ one @ sh))) @ id))) & (pushprop_p_and_p_prime = (^[X377 : term, X378 : subst, X379 : (term > $o), X380 : (term > $o)] : (! [X381 : term] : (((X379 @ (sub @ X381 @ (push @ X377 @ X378)))) = ((X380 @ X381)))))) & (! [X382 : term] : ~ ! [X383 : (subst > term > term)] : ~ ! [X384 : subst,X385 : term] : (((X383 @ X384 @ X385)) = ((sub @ X382 @ (push @ X385 @ X384)))) <=> hoasinduction_lem3v2_f) & ((! [X386 : term] : (((sub @ X386 @ id)) = X386) => (! [X387 : term,X388 : (subst > term > subst > $o)] : (~ ! [X389 : (subst > term > term)] : (~ ! [X390 : term] : (~(X388 @ id @ X390 @ id) | (X388 @ id @ (X389 @ id @ X390) @ id)) | (X388 @ id @ (lam @ (X389 @ sh @ one)) @ id) | ~ ! [X391 : subst,X392 : term,X393 : subst] : (((sub @ (X389 @ X391 @ X392) @ X393)) = ((X389 @ (comp @ X391 @ X393) @ (sub @ X392 @ X393))))) | ~ ! [X394 : term] : ((X388 @ id @ (sub @ X387 @ (push @ X394 @ id)) @ id) | ~(X388 @ id @ X394 @ id)) | (X388 @ id @ (lam @ (sub @ X387 @ (push @ one @ sh))) @ id)) => ! [X395 : (subst > term > subst > $o),X396 : term] : ((X395 @ id @ (lam @ X396) @ id) | ~ ! [X397 : subst,X398 : subst,X399 : term,X400 : subst] : ((X395 @ (comp @ X400 @ X397) @ (sub @ X399 @ X397) @ X398) | ~(X395 @ X400 @ X399 @ (comp @ X397 @ X398))) | ~ ! [X401 : term] : (~(X395 @ id @ X401 @ id) | (X395 @ id @ (sub @ X396 @ (push @ X401 @ id)) @ id)) | ~ ! [X402 : subst,X403 : subst,X404 : term,X405 : subst] : ((X395 @ X405 @ X404 @ (comp @ X403 @ X402)) | ~(X395 @ (comp @ X405 @ X403) @ (sub @ X404 @ X403) @ X402)) | ~ ! [X406 : (subst > term > term)] : (~ ! [X407 : subst,X408 : term,X409 : subst] : (((X406 @ (comp @ X407 @ X409) @ (sub @ X408 @ X409))) = ((sub @ (X406 @ X407 @ X408) @ X409))) | ~ ! [X410 : term] : ((X395 @ id @ (X406 @ id @ X410) @ id) | ~(X395 @ id @ X410 @ id)) | (X395 @ id @ (lam @ (X406 @ sh @ one)) @ id))))) <=> hoasinduction_lem3_lthm) & induction2lem_gthm & ulamvarind & (! [X411 : subst] : (((push @ (sub @ one @ X411) @ (comp @ sh @ X411))) = X411) <=> axscons) & axapp & pushprop_lem2v2_lthm & ~ulamvar1 & hoasinduction_lem3b_gthm & hoasinduction_lem3a_gthm & pushprop_lem3v2_lthm & (substmonoid_lthm <=> (! [X412 : subst] : (((comp @ id @ X412)) = X412) => (! [X413 : subst] : (((comp @ X413 @ id)) = X413) => (! [X414 : subst] : (((comp @ X414 @ id)) = X414) & ! [X415 : subst] : (((comp @ id @ X415)) = X415))))) & ! [X416 : d_subst,X417 : d_subst] : ((((d2subst @ X417)) = ((d2subst @ X416))) => (X416 = X417)) & pushprop_lem3v2 & (hoaslamnotap <=> ! [X418 : (subst > term > term),X419 : term,X420 : term] : ((((ap @ (sub @ X420 @ id) @ X419)) != ((lam @ (X418 @ sh @ one)))) | ~ ! [X421 : subst,X422 : subst,X423 : term] : (((sub @ (X418 @ X422 @ X423) @ X421)) = ((X418 @ (comp @ X422 @ X421) @ (sub @ X423 @ X421)))))) & axvarshift & substmonoid_gthm & (! [X424 : (subst > term > subst > $o)] : ~ ! [X425 : (term > $o)] : ~ ! [X426 : term] : (((X424 @ id @ X426 @ id)) = ((X425 @ X426))) <=> hoasinduction_lem0) & (hoasinduction_lem3v2_f_lthm <=> ! [X427 : term] : ~ ! [X428 : (subst > term > term)] : ~ ! [X429 : subst,X430 : term] : (((sub @ X427 @ (push @ X430 @ X429))) = ((X428 @ X429 @ X430)))) & hoasinduction_lem1_gthm & ! [X431 : term] : ? [X432 : d_term] : (((d2term @ X432)) = X431) & (hoasinduction_lem3v2 <=> ! [X433 : term,X434 : (term > $o),X435 : (subst > term > subst > $o)] : (~ ! [X436 : term] : ((X434 @ (sub @ X433 @ (push @ X436 @ id))) | ~(X434 @ X436)) | ~ ! [X437 : subst,X438 : subst,X439 : subst,X440 : term] : ((X435 @ X438 @ X440 @ (comp @ X439 @ X437)) | ~(X435 @ (comp @ X438 @ X439) @ (sub @ X440 @ X439) @ X437)) | ~ ! [X441 : (subst > term > term)] : ((X435 @ id @ (lam @ (X441 @ sh @ one)) @ id) | ~ ! [X442 : term] : (~(X435 @ id @ X442 @ id) | (X435 @ id @ (X441 @ id @ X442) @ id)) | ~ ! [X443 : subst,X444 : term,X445 : subst] : (((X441 @ (comp @ X445 @ X443) @ (sub @ X444 @ X443))) = ((sub @ (X441 @ X445 @ X444) @ X443)))) | ~ ! [X446 : term] : (((X435 @ id @ X446 @ id)) = ((X434 @ X446))) | ~ ! [X447 : subst,X448 : subst,X449 : subst,X450 : term] : (~(X435 @ X448 @ X450 @ (comp @ X449 @ X447)) | (X435 @ (comp @ X448 @ X449) @ (sub @ X450 @ X449) @ X447)) | (X434 @ (lam @ X433)))) & induction2_gthm & hoasapnotvar_gthm & pushprop_lem1_gthm & hoasinduction_lem1_lthm & (axshiftcons <=> ! [X451 : subst,X452 : term] : (((comp @ sh @ (push @ X452 @ X451))) = X451)) & (! [X453 : term,X454 : term] : ((X453 = X454) | (((lam @ X453)) != ((lam @ X454)))) <=> laminj) & (hoasapinj2_lthm <=> (! [X455 : term,X456 : term,X457 : term,X458 : term] : ((((ap @ X458 @ X456)) != ((ap @ X455 @ X457))) | (X456 = X457)) => ! [X459 : term,X460 : term,X461 : term,X462 : term] : ((X459 = X461) | (((ap @ (sub @ X462 @ id) @ X461)) != ((ap @ (sub @ X460 @ id) @ X459)))))) & (apinj1 <=> ! [X463 : term,X464 : term,X465 : term,X466 : term] : ((X463 = X466) | (((ap @ X466 @ X465)) != ((ap @ X463 @ X464))))) & (! [X467 : subst,X468 : (subst > term > term),X469 : (subst > term > term),X470 : term] : ((((lam @ (X469 @ sh @ one))) != ((lam @ (X468 @ sh @ one)))) | ~ ! [X471 : subst,X472 : term,X473 : subst] : (((X468 @ (comp @ X471 @ X473) @ (sub @ X472 @ X473))) = ((sub @ (X468 @ X471 @ X472) @ X473))) | ~ ! [X474 : term,X475 : subst,X476 : subst] : (((X469 @ (comp @ X476 @ X475) @ (sub @ X474 @ X475))) = ((sub @ (X469 @ X476 @ X474) @ X475))) | (((X468 @ X467 @ X470)) = ((X469 @ X467 @ X470)))) <=> hoaslaminj) & (((sub @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2term @ d_one))) & (shinj <=> ! [X477 : term,X478 : term] : ((X477 = X478) | (((sub @ X477 @ sh)) != ((sub @ X478 @ sh))))) & hoaslamnotap_lthm & (hoasapinj2 <=> ! [X479 : term,X480 : term,X481 : term,X482 : term] : ((((ap @ (sub @ X479 @ id) @ X481)) != ((ap @ (sub @ X482 @ id) @ X480))) | (X480 = X481))) & ~(var @ (d2term @ d_one)) & (! [X483 : term] : (((sub @ X483 @ id)) = X483) <=> termmset) & (((lam @ (d2term @ d_one))) = ((d2term @ d_one))) & axabs & ! [X484 : d_term] : (d_one = X484) & lamnotvar & hoasinduction_lem3v2_gthm & (! [X485 : term] : ~ ! [X486 : (subst > term > term)] : (((X486 @ sh @ one)) != ((sub @ X485 @ (push @ one @ sh)))) <=> hoasinduction_lem3b) & (pushprop_lem1v2_lthm <=> (! [X487 : term,X488 : subst] : (((sub @ one @ (push @ X487 @ X488))) = X487) => ! [X489 : term,X490 : (term > $o),X491 : subst,X492 : (term > $o)] : (~ ! [X493 : term] : (((X492 @ X493)) = ((X490 @ (sub @ X493 @ (push @ X489 @ X491))))) | ~(X490 @ X489) | (X492 @ one)))) & (! [X494 : term] : ~ ! [X495 : (subst > term > term)] : (((X495 @ sh @ one)) != ((sub @ X494 @ (push @ one @ sh)))) <=> hoasinduction_lem3b_lthm) & (! [X496 : (subst > term > subst > $o),X497 : term] : ((X496 @ id @ (lam @ (sub @ X497 @ (push @ one @ sh))) @ id) | ~ ! [X498 : (subst > term > term),X499 : term] : (~ ! [X500 : subst,X501 : subst,X502 : term] : (((X498 @ (comp @ X500 @ X501) @ (sub @ X502 @ X501))) = ((sub @ X499 @ (push @ (sub @ X502 @ X501) @ (comp @ X500 @ X501))))) | ~ ! [X503 : term] : ((X496 @ id @ (X498 @ id @ X503) @ id) | ~(X496 @ id @ X503 @ id)) | (X496 @ id @ (lam @ (X498 @ sh @ one)) @ id) | ~ ! [X504 : term,X505 : subst,X506 : subst] : (((sub @ (sub @ X499 @ (push @ X504 @ X506)) @ X505)) = ((sub @ (X498 @ X506 @ X504) @ X505)))) | ~ ! [X507 : term] : ((X496 @ id @ (sub @ X497 @ (push @ X507 @ id)) @ id) | ~(X496 @ id @ X507 @ id))) <=> hoasinduction_lem3aaa) & (induction <=> ! [X508 : term,X509 : (term > $o)] : (~ ! [X510 : term] : (~(X509 @ X510) | (X509 @ (lam @ X510))) | ~ ! [X511 : term] : (~(var @ X511) | (X509 @ X511)) | (X509 @ X508) | ~ ! [X512 : term,X513 : term] : (~(X509 @ X512) | (X509 @ (ap @ X513 @ X512)) | ~(X509 @ X513)))) & (pushprop_lem1_lthm <=> (! [X514 : term,X515 : subst] : (((sub @ one @ (push @ X514 @ X515))) = X514) => (! [X516 : subst,X517 : term] : (((comp @ sh @ (push @ X517 @ X516))) = X516) => ! [X518 : subst,X519 : term,X520 : (term > $o),X521 : (term > $o),X522 : term] : (~(X521 @ X519) | (X520 @ (sub @ X519 @ (push @ X522 @ X518))))))) & (! [X523 : (subst > term > subst > $o),X524 : term] : (~ ! [X525 : term] : ((X523 @ id @ (sub @ X524 @ (push @ X525 @ id)) @ id) | ~(X523 @ id @ X525 @ id)) | (X523 @ id @ (lam @ X524) @ id) | ~ ! [X526 : (subst > term > term)] : ((X523 @ id @ (lam @ (X526 @ sh @ one)) @ id) | ~ ! [X527 : term] : (~(X523 @ id @ X527 @ id) | (X523 @ id @ (X526 @ id @ X527) @ id)) | ~ ! [X528 : term,X529 : subst,X530 : subst] : (((X526 @ (comp @ X529 @ X530) @ (sub @ X528 @ X530))) = ((sub @ (X526 @ X529 @ X528) @ X530))))) <=> hoasinduction_lem3a) & hoasinduction_lem2v2_gthm & hoasinduction_lem2_lthm & (hoaslam = (^[X531 : subst, X532 : (subst > term > term)] : ((d2term @ d_one)))) & (! [X533 : term,X534 : (subst > term > subst > $o)] : ((X534 @ id @ (lam @ (sub @ X533 @ (push @ one @ sh))) @ id) | ~ ! [X535 : (subst > term > term)] : (~ ! [X536 : subst,X537 : subst,X538 : term] : (((X535 @ (comp @ X537 @ X536) @ (sub @ X538 @ X536))) = ((sub @ (X535 @ X537 @ X538) @ X536))) | (X534 @ id @ (lam @ (X535 @ sh @ one)) @ id) | ~ ! [X539 : term] : (~(X534 @ id @ X539 @ id) | (X534 @ id @ (X535 @ id @ X539) @ id))) | ~ ! [X540 : term] : (~(X534 @ id @ X540 @ id) | (X534 @ id @ (sub @ X533 @ (push @ X540 @ id)) @ id))) <=> hoasinduction_lem3aa) & (hoasinduction_no_psi_cond <=> ! [X541 : term,X542 : (subst > term > subst > $o)] : ((X542 @ id @ X541 @ id) | ~ ! [X543 : (subst > term > term)] : ((X542 @ id @ (lam @ (X543 @ sh @ one)) @ id) | ~ ! [X544 : term] : (~(X542 @ id @ X544 @ id) | (X542 @ id @ (X543 @ id @ X544) @ id)) | ~ ! [X545 : subst,X546 : term,X547 : subst] : (((X543 @ (comp @ X545 @ X547) @ (sub @ X546 @ X547))) = ((sub @ (X543 @ X545 @ X546) @ X547)))) | ~ ! [X548 : term,X549 : term] : (~(X542 @ id @ X548 @ id) | ~(X542 @ id @ X549 @ id) | (X542 @ id @ (ap @ (sub @ X549 @ id) @ X548) @ id)))) & (sh = ((d2subst @ d_id))) & (((ap @ (d2term @ d_one) @ (d2term @ d_one))) = ((d2term @ d_one))) & hoasinduction_lem1v2_gthm),
  inference(rectify,[],[f1])).
thf(f7,plain,(
  (! [X0 : subst] : (((comp @ X0 @ id)) = X0) <=> (axidr = $true)) & (hoasapinj1_gthm = $true) & ((! [X1 : term] : (((sub @ X1 @ id)) = X1) => ! [X2 : term] : (((sub @ X2 @ id)) = X2)) <=> (termmset_lthm = $true)) & (termmset_gthm = $true) & (hoasinduction_lem1 = $true) & ((! [X4 : subst] : (((comp @ id @ X4)) = X4) & ! [X3 : subst] : (((comp @ X3 @ id)) = X3)) <=> (substmonoid = $true)) & (pushprop_lthm = $true) & (hoaslamnotvar_lthm = $true) & ! [X5 : d_subst] : (d_id = X5) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (! [X6 : term,X7 : (subst > term > subst > $o)] : (~ ! [X8 : term,X9 : term] : (~ ($true = ((X7 @ id @ X8 @ id))) | ~ (((X7 @ id @ X9 @ id)) = $true) | ($true = ((X7 @ id @ (ap @ (sub @ X9 @ id) @ X8) @ id)))) | ~ ! [X10 : subst,X11 : subst,X12 : subst,X13 : term] : (($true = ((X7 @ (comp @ X11 @ X12) @ (sub @ X13 @ X12) @ X10))) | ~ ($true = ((X7 @ X11 @ X13 @ (comp @ X12 @ X10))))) | ~ ! [X14 : subst,X15 : term,X16 : subst,X17 : subst] : (~ ($true = ((X7 @ (comp @ X14 @ X16) @ (sub @ X15 @ X16) @ X17))) | ($true = ((X7 @ X14 @ X15 @ (comp @ X16 @ X17))))) | ($true = ((X7 @ id @ X6 @ id))) | ~ ! [X18 : term] : (~ ($true = ((var @ (sub @ X18 @ id)))) | ($true = ((X7 @ id @ X18 @ id)))) | ~ ! [X19 : (subst > term > term)] : (($true = ((X7 @ id @ (lam @ (X19 @ sh @ one)) @ id))) | ~ ! [X22 : subst,X21 : term,X20 : subst] : (((sub @ (X19 @ X22 @ X21) @ X20)) = ((X19 @ (comp @ X22 @ X20) @ (sub @ X21 @ X20)))) | ~ ! [X23 : term] : (($true = ((X7 @ id @ (X19 @ id @ X23) @ id))) | ~ (((X7 @ id @ X23 @ id)) = $true)))) <=> (hoasinduction = $true)) & ((! [X24 : term] : (((sub @ X24 @ id)) = X24) => (! [X25 : subst,X26 : term,X27 : (term > $o)] : (~ ! [X28 : term] : (~ ! [X29 : term] : (~ ($true = ((X27 @ X29))) | ($true = ((X27 @ (sub @ X28 @ (push @ X29 @ id)))))) | ($true = ((X27 @ (lam @ X28))))) | ($true = ((X27 @ (sub @ X26 @ X25)))) | ~ ! [X30 : term,X31 : term] : (~ ($true = ((X27 @ X30))) | ($true = ((X27 @ (ap @ X30 @ X31)))) | ~ ($true = ((X27 @ X31)))) | ~ ! [X32 : term] : (~ ($true = ((var @ X32))) | ($true = ((X27 @ (sub @ X32 @ X25)))))) => ! [X33 : (term > $o),X34 : term] : (~ ! [X35 : term] : (($true = ((X33 @ (lam @ X35)))) | ~ ! [X36 : term] : (~ ($true = ((X33 @ X36))) | ($true = ((X33 @ (sub @ X35 @ (push @ X36 @ id))))))) | ($true = ((X33 @ X34))) | ~ ! [X37 : term,X38 : term] : (~ ($true = ((X33 @ X37))) | ~ ($true = ((X33 @ X38))) | ($true = ((X33 @ (ap @ X38 @ X37))))) | ~ ! [X39 : term] : (($true = ((X33 @ X39))) | ~ ($true = ((var @ X39))))))) <=> (induction2_lthm = $true)) & (! [X40 : subst] : (((comp @ id @ X40)) = X40) <=> (axidl = $true)) & (axclos = $true) & (pushprop_lem2v2 = $true) & (pushprop = $true) & (axmap = $true) & ((! [X41 : subst,X42 : term] : (((sub @ one @ (push @ X42 @ X41))) = X42) => (! [X44 : term,X43 : subst] : (((comp @ sh @ (push @ X44 @ X43))) = X43) => (! [X45 : subst] : (((comp @ X45 @ id)) = X45) => (! [X46 : (term > $o),X47 : term] : (~ ! [X48 : term] : (~ ($true = ((var @ X48))) | ($true = ((X46 @ X48)))) | ~ ! [X49 : term,X50 : term] : (~ ($true = ((X46 @ X49))) | ~ ($true = ((X46 @ X50))) | ($true = ((X46 @ (ap @ X49 @ X50))))) | ~ ! [X51 : term] : (($true = ((X46 @ (lam @ X51)))) | ~ ($true = ((X46 @ X51)))) | ($true = ((X46 @ X47)))) => ! [X52 : (term > $o),X53 : subst,X54 : term] : (~ ! [X55 : term,X56 : term] : (~ (((X52 @ X55)) = $true) | ($true = ((X52 @ (ap @ X55 @ X56)))) | ~ ($true = ((X52 @ X56)))) | ($true = ((X52 @ (sub @ X54 @ X53)))) | ~ ! [X57 : term] : (($true = ((X52 @ (lam @ X57)))) | ~ ! [X58 : term] : (($true = ((X52 @ (sub @ X57 @ (push @ X58 @ id))))) | ~ ($true = ((X52 @ X58))))) | ~ ! [X59 : term] : (($true = ((X52 @ (sub @ X59 @ X53)))) | ~ ($true = ((var @ X59))))))))) <=> (induction2lem_lthm = $true)) & (! [X63 : term,X60 : term,X61 : term,X62 : term] : ((X60 = X61) | (((ap @ (sub @ X61 @ id) @ X62)) != ((ap @ (sub @ X60 @ id) @ X63)))) <=> (hoasapinj1 = $true)) & (hoaslaminj_gthm = $true) & ((hoasinduction_lem3 = $true) <=> ! [X64 : term,X65 : (subst > term > subst > $o)] : (~ ! [X66 : subst,X67 : term,X68 : subst,X69 : subst] : (($true = ((X65 @ (comp @ X66 @ X69) @ (sub @ X67 @ X69) @ X68))) | ~ ($true = ((X65 @ X66 @ X67 @ (comp @ X69 @ X68))))) | ($true = ((X65 @ id @ (lam @ X64) @ id))) | ~ ! [X70 : term] : (($true = ((X65 @ id @ (sub @ X64 @ (push @ X70 @ id)) @ id))) | ~ ($true = ((X65 @ id @ X70 @ id)))) | ~ ! [X71 : (subst > term > term)] : (~ ! [X73 : subst,X74 : term,X72 : subst] : (((X71 @ (comp @ X73 @ X72) @ (sub @ X74 @ X72))) = ((sub @ (X71 @ X73 @ X74) @ X72))) | ($true = ((X65 @ id @ (lam @ (X71 @ sh @ one)) @ id))) | ~ ! [X75 : term] : (~ ($true = ((X65 @ id @ X75 @ id))) | ($true = ((X65 @ id @ (X71 @ id @ X75) @ id))))) | ~ ! [X76 : term,X77 : subst,X78 : subst,X79 : subst] : (($true = ((X65 @ X77 @ X76 @ (comp @ X79 @ X78)))) | ~ ($true = ((X65 @ (comp @ X77 @ X79) @ (sub @ X76 @ X79) @ X78)))))) & (! [X80 : term,X83 : term,X82 : term,X81 : term] : ((X80 = X83) | (((ap @ X82 @ X83)) != ((ap @ X81 @ X80)))) <=> (apinj2 = $true)) & ((! [X84 : term] : (((sub @ X84 @ id)) = X84) => (! [X85 : (subst > term > subst > $o),X86 : term] : (($true = ((X85 @ id @ (lam @ (sub @ X86 @ (push @ one @ sh))) @ id))) | ~ ! [X87 : (subst > term > term)] : (~ ! [X90 : term,X89 : subst,X88 : subst] : (((X87 @ (comp @ X88 @ X89) @ (sub @ X90 @ X89))) = ((sub @ (X87 @ X88 @ X90) @ X89))) | ~ ! [X91 : term] : (($true = ((X85 @ id @ (X87 @ id @ X91) @ id))) | ~ ($true = ((X85 @ id @ X91 @ id)))) | ($true = ((X85 @ id @ (lam @ (X87 @ sh @ one)) @ id)))) | ~ ! [X92 : term] : (($true = ((X85 @ id @ (sub @ X86 @ (push @ X92 @ id)) @ id))) | ~ ($true = ((X85 @ id @ X92 @ id))))) => ! [X93 : (subst > term > subst > $o),X94 : term] : (~ ! [X95 : term] : (($true = ((X93 @ id @ (sub @ X94 @ (push @ X95 @ id)) @ id))) | ~ ($true = ((X93 @ id @ X95 @ id)))) | ($true = ((X93 @ id @ (lam @ X94) @ id))) | ~ ! [X96 : (subst > term > term)] : (~ ! [X97 : term] : (($true = ((X93 @ id @ (X96 @ id @ X97) @ id))) | ~ ($true = ((X93 @ id @ X97 @ id)))) | ~ ! [X99 : term,X100 : subst,X98 : subst] : (((sub @ (X96 @ X100 @ X99) @ X98)) = ((X96 @ (comp @ X100 @ X98) @ (sub @ X99 @ X98)))) | ($true = ((X93 @ id @ (lam @ (X96 @ sh @ one)) @ id))))))) <=> (hoasinduction_lem3a_lthm = $true)) & ~ (lamnotap = $true) & ((pushprop_lem0 = $true) <=> ! [X103 : term,X101 : subst,X102 : (term > $o)] : ~ ! [X104 : (term > $o)] : ~ ! [X105 : term] : (((X104 @ X105)) = ((X102 @ (sub @ X105 @ (push @ X103 @ X101)))))) & (hoasapinj2_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (pushprop_lem2v2_gthm = $true) & ((pushprop_lem0_lthm = $true) <=> ! [X109 : (term > $o),X110 : subst,X111 : term] : ~ ! [X112 : (term > $o)] : ~ ! [X113 : term] : (((X112 @ X113)) = ((X109 @ (sub @ X113 @ (push @ X111 @ X110)))))) & ((pushprop_lem1v2 = $true) <=> ! [X114 : (term > $o),X115 : (term > $o),X116 : term,X117 : subst] : (~ ($true = ((X114 @ X116))) | ($true = ((X115 @ one))) | ~ ! [X118 : term] : (((X115 @ X118)) = ((X114 @ (sub @ X118 @ (push @ X116 @ X117))))))) & (pushprop_lthm_orig = $true) & ((axvarid = $true) <=> ! [X119 : term] : (((sub @ X119 @ id)) = X119)) & (! [X120 : term,X121 : term,X122 : (subst > term > subst > $o)] : (~ ! [X123 : term,X124 : term] : (($true = ((X122 @ id @ (ap @ (sub @ X123 @ id) @ X124) @ id))) | ~ ($true = ((X122 @ id @ X123 @ id))) | ~ ($true = ((X122 @ id @ X124 @ id)))) | ~ (((X122 @ id @ X121 @ id)) = $true) | ~ ($true = ((X122 @ id @ X120 @ id))) | ~ ! [X125 : term,X126 : subst,X127 : subst,X128 : subst] : (($true = ((X122 @ X126 @ X125 @ (comp @ X128 @ X127)))) | ~ ($true = ((X122 @ (comp @ X126 @ X128) @ (sub @ X125 @ X128) @ X127)))) | ~ ! [X129 : subst,X130 : term,X131 : subst,X132 : subst] : (($true = ((X122 @ (comp @ X129 @ X131) @ (sub @ X130 @ X131) @ X132))) | ~ ($true = ((X122 @ X129 @ X130 @ (comp @ X131 @ X132))))) | ($true = ((X122 @ id @ (ap @ X121 @ X120) @ id)))) <=> (hoasinduction_lem2 = $true)) & (pushprop_gthm = $true) & (axassoc = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (one = ((d2term @ d_one))) & (hoasinduction_lem3_gthm = $true) & ((hoaslaminj_lthm = $true) <=> (! [X133 : subst,X134 : term] : (((sub @ one @ (push @ X134 @ X133))) = X134) => (! [X135 : term,X136 : subst] : (((comp @ sh @ (push @ X135 @ X136))) = X136) => (! [X137 : term,X138 : term] : ((X137 = X138) | (((lam @ X138)) != ((lam @ X137)))) => ! [X139 : term,X140 : (subst > term > term),X141 : subst,X142 : (subst > term > term)] : (~ ! [X143 : subst,X144 : term,X145 : subst] : (((X142 @ (comp @ X145 @ X143) @ (sub @ X144 @ X143))) = ((sub @ (X142 @ X145 @ X144) @ X143))) | (((lam @ (X140 @ sh @ one))) != ((lam @ (X142 @ sh @ one)))) | (((X142 @ X141 @ X139)) = ((X140 @ X141 @ X139))) | ~ ! [X146 : term,X147 : subst,X148 : subst] : (((sub @ (X140 @ X148 @ X146) @ X147)) = ((X140 @ (comp @ X148 @ X147) @ (sub @ X146 @ X147))))))))) & (((d2subst @ d_id)) = id) & (ulamvarsh = $true) & ! [X149 : subst] : ? [X150 : d_subst] : (((d2subst @ X150)) = X149) & (hoaslamnotvar = $true) & (pushprop_lem1v2_gthm = $true) & (hoasapnotvar_lthm = $true) & (hoasinduction_gthm = $true) & ((hoasinduction_lthm_3 = $true) <=> (! [X151 : (subst > term > subst > $o)] : ~ ! [X152 : (term > $o)] : ~ ! [X153 : term] : (((X151 @ id @ X153 @ id)) = ((X152 @ X153))) => (! [X154 : (term > $o),X155 : term] : (($true = ((X154 @ X155))) | ~ ! [X156 : term,X157 : term] : (~ ($true = ((X154 @ X157))) | ~ ($true = ((X154 @ X156))) | ($true = ((X154 @ (ap @ X157 @ X156))))) | ~ ! [X158 : term] : (($true = ((X154 @ (lam @ X158)))) | ~ ! [X159 : term] : (~ ($true = ((X154 @ X159))) | ($true = ((X154 @ (sub @ X158 @ (push @ X159 @ id))))))) | ~ ! [X160 : term] : (($true = ((X154 @ X160))) | ~ ($true = ((var @ X160))))) => (! [X161 : term] : (((sub @ X161 @ id)) = X161) => (! [X162 : (subst > term > subst > $o),X163 : (term > $o),X164 : term] : (($true = ((X163 @ (lam @ X164)))) | ~ ! [X165 : term] : (((X163 @ X165)) = ((X162 @ id @ X165 @ id))) | ~ ! [X166 : term] : (($true = ((X163 @ (sub @ X164 @ (push @ X166 @ id))))) | ~ ($true = ((X163 @ X166)))) | ~ ! [X167 : (subst > term > term)] : (~ ! [X168 : term] : (($true = ((X162 @ id @ (X167 @ id @ X168) @ id))) | ~ ($true = ((X162 @ id @ X168 @ id)))) | ~ ! [X169 : subst,X170 : subst,X171 : term] : (((sub @ (X167 @ X169 @ X171) @ X170)) = ((X167 @ (comp @ X169 @ X170) @ (sub @ X171 @ X170)))) | ($true = ((X162 @ id @ (lam @ (X167 @ sh @ one)) @ id))))) => ! [X172 : term,X173 : (subst > term > subst > $o)] : (~ ! [X174 : (subst > term > term)] : (~ ! [X175 : term] : (($true = ((X173 @ id @ (X174 @ id @ X175) @ id))) | ~ ($true = ((X173 @ id @ X175 @ id)))) | ~ ! [X176 : subst,X177 : subst,X178 : term] : (((X174 @ (comp @ X176 @ X177) @ (sub @ X178 @ X177))) = ((sub @ (X174 @ X176 @ X178) @ X177))) | ($true = ((X173 @ id @ (lam @ (X174 @ sh @ one)) @ id)))) | ($true = ((X173 @ id @ X172 @ id))) | ~ ! [X179 : subst,X180 : subst,X181 : subst,X182 : term] : (~ ($true = ((X173 @ (comp @ X179 @ X180) @ (sub @ X182 @ X180) @ X181))) | ($true = ((X173 @ X179 @ X182 @ (comp @ X180 @ X181))))) | ~ ! [X183 : term] : (($true = ((X173 @ id @ X183 @ id))) | ~ ($true = ((var @ (sub @ X183 @ id))))) | ~ ! [X184 : term,X185 : subst,X186 : subst,X187 : subst] : (~ ($true = ((X173 @ X187 @ X184 @ (comp @ X185 @ X186)))) | ($true = ((X173 @ (comp @ X187 @ X185) @ (sub @ X184 @ X185) @ X186)))) | ~ ! [X188 : term,X189 : term] : (~ ($true = ((X173 @ id @ X189 @ id))) | ($true = ((X173 @ id @ (ap @ (sub @ X189 @ id) @ X188) @ id))) | ~ ($true = ((X173 @ id @ X188 @ id)))))))))) & (hoasinduction_lem1v2 = $true) & ((hoasinduction_lem2v2 = $true) <=> ! [X190 : term,X191 : term,X192 : (subst > term > subst > $o),X193 : (term > $o)] : (~ ! [X194 : term] : (((X192 @ id @ X194 @ id)) = ((X193 @ X194))) | ($true = ((X193 @ (ap @ X191 @ X190)))) | ~ ($true = ((X193 @ X190))) | ~ ! [X195 : subst,X196 : subst,X197 : subst,X198 : term] : (~ ($true = ((X192 @ (comp @ X197 @ X196) @ (sub @ X198 @ X196) @ X195))) | ($true = ((X192 @ X197 @ X198 @ (comp @ X196 @ X195))))) | ~ ! [X199 : subst,X200 : subst,X201 : term,X202 : subst] : (~ ($true = ((X192 @ X202 @ X201 @ (comp @ X199 @ X200)))) | ($true = ((X192 @ (comp @ X202 @ X199) @ (sub @ X201 @ X199) @ X200)))) | ~ ! [X203 : term,X204 : term] : (~ ($true = ((X192 @ id @ X203 @ id))) | ($true = ((X192 @ id @ (ap @ (sub @ X203 @ id) @ X204) @ id))) | ~ ($true = ((X192 @ id @ X204 @ id)))) | ~ ($true = ((X193 @ X191))))) & (! [X205 : (term > $o),X206 : subst,X207 : term] : (~ ! [X208 : term] : (~ ! [X209 : term] : (($true = ((X205 @ (sub @ X208 @ (push @ X209 @ id))))) | ~ ($true = ((X205 @ X209)))) | ($true = ((X205 @ (lam @ X208))))) | ~ ! [X210 : term,X211 : term] : (~ ($true = ((X205 @ X210))) | ($true = ((X205 @ (ap @ X210 @ X211)))) | ~ ($true = ((X205 @ X211)))) | (((X205 @ (sub @ X207 @ X206))) = $true) | ~ ! [X212 : term] : (~ ($true = ((var @ X212))) | ($true = ((X205 @ (sub @ X212 @ X206)))))) <=> (induction2lem = $true)) & ! [X213 : d_term,X214 : d_term] : ((((d2term @ X214)) = ((d2term @ X213))) => (X213 = X214)) & (hoasinduction_lem2_gthm = $true) & ((pushprop_lem1 = $true) <=> ! [X215 : subst,X216 : term,X217 : (term > $o),X218 : (term > $o),X219 : term] : (~ ($true = ((X217 @ X216))) | ($true = ((X218 @ (sub @ X216 @ (push @ X219 @ X215))))))) & ((! [X220 : term] : (((sub @ X220 @ id)) = X220) => ! [X221 : term,X222 : (subst > term > subst > $o),X223 : (term > $o)] : (~ ! [X224 : subst,X225 : subst,X226 : term,X227 : subst] : (($true = ((X222 @ X227 @ X226 @ (comp @ X225 @ X224)))) | ~ ($true = ((X222 @ (comp @ X227 @ X225) @ (sub @ X226 @ X225) @ X224)))) | ~ ! [X228 : subst,X229 : term,X230 : subst,X231 : subst] : (($true = ((X222 @ (comp @ X228 @ X231) @ (sub @ X229 @ X231) @ X230))) | ~ ($true = ((X222 @ X228 @ X229 @ (comp @ X231 @ X230))))) | ~ ! [X232 : term] : (~ ($true = ((X223 @ X232))) | ($true = ((X223 @ (sub @ X221 @ (push @ X232 @ id)))))) | ~ ! [X233 : term] : (((X222 @ id @ X233 @ id)) = ((X223 @ X233))) | ($true = ((X223 @ (lam @ X221)))) | ~ ! [X234 : (subst > term > term)] : (~ ! [X235 : term,X236 : subst,X237 : subst] : (((X234 @ (comp @ X237 @ X236) @ (sub @ X235 @ X236))) = ((sub @ (X234 @ X237 @ X235) @ X236))) | ~ ! [X238 : term] : (($true = ((X222 @ id @ (X234 @ id @ X238) @ id))) | ~ ($true = ((X222 @ id @ X238 @ id)))) | ($true = ((X222 @ id @ (lam @ (X234 @ sh @ one)) @ id)))))) <=> (hoasinduction_lem3v2_lthm = $true)) & ((! [X239 : term] : (((sub @ X239 @ id)) = X239) => (! [X240 : term,X241 : term,X242 : term,X243 : term] : ((X240 = X241) | (((ap @ X241 @ X243)) != ((ap @ X240 @ X242)))) => ! [X244 : term,X245 : term,X246 : term,X247 : term] : ((((ap @ (sub @ X245 @ id) @ X246)) != ((ap @ (sub @ X247 @ id) @ X244))) | (X245 = X247)))) <=> (hoasapinj1_lthm = $true)) & ((hoasinduction_lthm = $true) <=> (! [X248 : term,X249 : (term > $o)] : (~ ! [X250 : term] : (($true = ((X249 @ X250))) | ~ ($true = ((var @ X250)))) | ($true = ((X249 @ X248))) | ~ ! [X251 : term] : (($true = ((X249 @ (lam @ X251)))) | ~ ! [X252 : term] : (~ ($true = ((X249 @ X252))) | ($true = ((X249 @ (sub @ X251 @ (push @ X252 @ id))))))) | ~ ! [X253 : term,X254 : term] : (~ ($true = ((X249 @ X254))) | ($true = ((X249 @ (ap @ X253 @ X254)))) | ~ ($true = ((X249 @ X253))))) => (! [X255 : (subst > term > subst > $o),X256 : term,X257 : term] : (~ ! [X258 : subst,X259 : subst,X260 : subst,X261 : term] : (~ ($true = ((X255 @ (comp @ X259 @ X258) @ (sub @ X261 @ X258) @ X260))) | ($true = ((X255 @ X259 @ X261 @ (comp @ X258 @ X260))))) | ~ ! [X262 : subst,X263 : term,X264 : subst,X265 : subst] : (($true = ((X255 @ (comp @ X262 @ X265) @ (sub @ X263 @ X265) @ X264))) | ~ ($true = ((X255 @ X262 @ X263 @ (comp @ X265 @ X264))))) | ~ ($true = ((X255 @ id @ X257 @ id))) | ~ ! [X266 : term,X267 : term] : (~ ($true = ((X255 @ id @ X267 @ id))) | ($true = ((X255 @ id @ (ap @ (sub @ X267 @ id) @ X266) @ id))) | ~ ($true = ((X255 @ id @ X266 @ id)))) | ($true = ((X255 @ id @ (ap @ X256 @ X257) @ id))) | ~ ($true = ((X255 @ id @ X256 @ id)))) => (! [X268 : term,X269 : (subst > term > subst > $o)] : (~ ! [X270 : term,X271 : subst,X272 : subst,X273 : subst] : (($true = ((X269 @ X272 @ X270 @ (comp @ X271 @ X273)))) | ~ ($true = ((X269 @ (comp @ X272 @ X271) @ (sub @ X270 @ X271) @ X273)))) | ($true = ((X269 @ id @ (lam @ X268) @ id))) | ~ ! [X274 : subst,X275 : subst,X276 : subst,X277 : term] : (~ ($true = ((X269 @ X275 @ X277 @ (comp @ X276 @ X274)))) | ($true = ((X269 @ (comp @ X275 @ X276) @ (sub @ X277 @ X276) @ X274)))) | ~ ! [X278 : term] : (($true = ((X269 @ id @ (sub @ X268 @ (push @ X278 @ id)) @ id))) | ~ ($true = ((X269 @ id @ X278 @ id)))) | ~ ! [X279 : (subst > term > term)] : (~ ! [X280 : term] : (($true = ((X269 @ id @ (X279 @ id @ X280) @ id))) | ~ ($true = ((X269 @ id @ X280 @ id)))) | ($true = ((X269 @ id @ (lam @ (X279 @ sh @ one)) @ id))) | ~ ! [X281 : subst,X282 : subst,X283 : term] : (((X279 @ (comp @ X281 @ X282) @ (sub @ X283 @ X282))) = ((sub @ (X279 @ X281 @ X283) @ X282))))) => ! [X284 : (subst > term > subst > $o),X285 : term] : (($true = ((X284 @ id @ X285 @ id))) | ~ ! [X286 : (subst > term > term)] : (($true = ((X284 @ id @ (lam @ (X286 @ sh @ one)) @ id))) | ~ ! [X287 : term] : (~ ($true = ((X284 @ id @ X287 @ id))) | ($true = ((X284 @ id @ (X286 @ id @ X287) @ id)))) | ~ ! [X288 : term,X289 : subst,X290 : subst] : (((X286 @ (comp @ X289 @ X290) @ (sub @ X288 @ X290))) = ((sub @ (X286 @ X289 @ X288) @ X290)))) | ~ ! [X291 : subst,X292 : subst,X293 : term,X294 : subst] : (~ ($true = ((X284 @ X291 @ X293 @ (comp @ X292 @ X294)))) | ($true = ((X284 @ (comp @ X291 @ X292) @ (sub @ X293 @ X292) @ X294)))) | ~ ! [X295 : subst,X296 : subst,X297 : term,X298 : subst] : (~ ($true = ((X284 @ (comp @ X295 @ X298) @ (sub @ X297 @ X298) @ X296))) | ($true = ((X284 @ X295 @ X297 @ (comp @ X298 @ X296))))) | ~ ! [X299 : term] : (~ ($true = ((var @ (sub @ X299 @ id)))) | ($true = ((X284 @ id @ X299 @ id)))) | ~ ! [X300 : term,X301 : term] : (~ ($true = ((X284 @ id @ X301 @ id))) | ($true = ((X284 @ id @ (ap @ (sub @ X300 @ id) @ X301) @ id))) | ~ ($true = ((X284 @ id @ X300 @ id))))))))) & ~ (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) = $true) & (hoasapnotvar = $true) & ((! [X302 : term] : ~ ! [X303 : (subst > term > term)] : ~ ! [X304 : subst,X305 : term] : (((sub @ X302 @ (push @ X305 @ X304))) = ((X303 @ X304 @ X305))) => (! [X306 : term] : (((sub @ X306 @ id)) = X306) => ! [X307 : term,X308 : (term > $o),X309 : (subst > term > subst > $o)] : (~ ! [X310 : (subst > term > term)] : (~ ! [X311 : term,X312 : subst,X313 : subst] : (((X310 @ (comp @ X312 @ X313) @ (sub @ X311 @ X313))) = ((sub @ (X310 @ X312 @ X311) @ X313))) | ($true = ((X309 @ id @ (lam @ (X310 @ sh @ one)) @ id))) | ~ ! [X314 : term] : (($true = ((X309 @ id @ (X310 @ id @ X314) @ id))) | ~ ($true = ((X309 @ id @ X314 @ id))))) | ~ ! [X315 : term] : (((X308 @ X315)) = ((X309 @ id @ X315 @ id))) | ($true = ((X308 @ (lam @ X307)))) | ~ ! [X316 : term] : (($true = ((X308 @ (sub @ X307 @ (push @ X316 @ id))))) | ~ ($true = ((X308 @ X316))))))) <=> (hoasinduction_lem3v2a_lthm = $true)) & ((! [X317 : (subst > term > subst > $o)] : ~ ! [X318 : (term > $o)] : ~ ! [X319 : term] : (((X317 @ id @ X319 @ id)) = ((X318 @ X319))) => (! [X320 : (term > $o),X321 : term] : (~ ! [X322 : term] : (($true = ((X320 @ (lam @ X322)))) | ~ ! [X323 : term] : (($true = ((X320 @ (sub @ X322 @ (push @ X323 @ id))))) | ~ ($true = ((X320 @ X323))))) | ~ ! [X324 : term,X325 : term] : (~ ($true = ((X320 @ X324))) | ~ ($true = ((X320 @ X325))) | ($true = ((X320 @ (ap @ X324 @ X325))))) | ~ ! [X326 : term] : (($true = ((X320 @ X326))) | ~ ($true = ((var @ X326)))) | ($true = ((X320 @ X321)))) => (! [X327 : term] : (((sub @ X327 @ id)) = X327) => (! [X328 : (term > $o),X329 : (subst > term > subst > $o),X330 : term] : (~ ! [X331 : term] : (($true = ((X328 @ (sub @ X330 @ (push @ X331 @ id))))) | ~ ($true = ((X328 @ X331)))) | ~ ! [X332 : term] : (((X329 @ id @ X332 @ id)) = ((X328 @ X332))) | ~ ! [X333 : (subst > term > term)] : (~ ! [X334 : subst,X335 : subst,X336 : term] : (((X333 @ (comp @ X335 @ X334) @ (sub @ X336 @ X334))) = ((sub @ (X333 @ X335 @ X336) @ X334))) | ~ ! [X337 : term] : (~ ($true = ((X329 @ id @ X337 @ id))) | ($true = ((X329 @ id @ (X333 @ id @ X337) @ id)))) | ($true = ((X329 @ id @ (lam @ (X333 @ sh @ one)) @ id)))) | ($true = ((X328 @ (lam @ X330))))) => ! [X338 : term,X339 : (subst > term > subst > $o)] : (~ ! [X340 : term,X341 : term] : (~ ($true = ((X339 @ id @ X341 @ id))) | ($true = ((X339 @ id @ (ap @ (sub @ X340 @ id) @ X341) @ id))) | ~ ($true = ((X339 @ id @ X340 @ id)))) | ~ ! [X342 : (subst > term > term)] : (~ ! [X343 : term] : (($true = ((X339 @ id @ (X342 @ id @ X343) @ id))) | ~ ($true = ((X339 @ id @ X343 @ id)))) | ($true = ((X339 @ id @ (lam @ (X342 @ sh @ one)) @ id))) | ~ ! [X344 : term,X345 : subst,X346 : subst] : (((X342 @ (comp @ X345 @ X346) @ (sub @ X344 @ X346))) = ((sub @ (X342 @ X345 @ X344) @ X346)))) | ($true = ((X339 @ id @ X338 @ id)))))))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & ((hoasinduction_lem0_lthm = $true) <=> ! [X347 : (subst > term > subst > $o)] : ~ ! [X348 : (term > $o)] : ~ ! [X349 : term] : (((X347 @ id @ X349 @ id)) = ((X348 @ X349)))) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (hoaslamnotvar_gthm = $true) & (apnotvar = $true) & (! [X350 : subst,X351 : term] : (((sub @ one @ (push @ X351 @ X350))) = X351) <=> (axvarcons = $true)) & ((hoasinduction_lem3v2a = $true) <=> ! [X352 : (subst > term > subst > $o),X353 : (term > $o),X354 : term] : (~ ! [X355 : (subst > term > term)] : (($true = ((X352 @ id @ (lam @ (X355 @ sh @ one)) @ id))) | ~ ! [X356 : term] : (~ ($true = ((X352 @ id @ X356 @ id))) | ($true = ((X352 @ id @ (X355 @ id @ X356) @ id)))) | ~ ! [X357 : term,X358 : subst,X359 : subst] : (((sub @ (X355 @ X359 @ X357) @ X358)) = ((X355 @ (comp @ X359 @ X358) @ (sub @ X357 @ X358))))) | ($true = ((X353 @ (lam @ X354)))) | ~ ! [X360 : term] : (((X353 @ X360)) = ((X352 @ id @ X360 @ id))) | ~ ! [X361 : term] : (~ ($true = ((X353 @ X361))) | ($true = ((X353 @ (sub @ X354 @ (push @ X361 @ id)))))))) & (! [X362 : (term > $o),X363 : term] : (~ ! [X364 : term] : (~ ! [X365 : term] : (~ ($true = ((X362 @ X365))) | ($true = ((X362 @ (sub @ X364 @ (push @ X365 @ id)))))) | ($true = ((X362 @ (lam @ X364))))) | ~ ! [X366 : term] : (~ ($true = ((var @ X366))) | ($true = ((X362 @ X366)))) | ($true = ((X362 @ X363))) | ~ ! [X367 : term,X368 : term] : (($true = ((X362 @ (ap @ X368 @ X367)))) | ~ ($true = ((X362 @ X367))) | ~ ($true = ((X362 @ X368))))) <=> (induction2 = $true)) & (pushprop_lem0_gthm = $true) & (hoaslamnotap_gthm = $true) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X369 : (subst > term > subst > $o),X370 : term] : (~ ! [X371 : (subst > term > term)] : (($true = ((X369 @ id @ (lam @ (X371 @ sh @ one)) @ id))) | ~ ! [X372 : subst,X373 : subst,X374 : term] : (((sub @ (X371 @ X373 @ X374) @ X372)) = ((X371 @ (comp @ X373 @ X372) @ (sub @ X374 @ X372)))) | ~ ! [X375 : term] : (($true = ((X369 @ id @ (X371 @ id @ X375) @ id))) | ~ ($true = ((X369 @ id @ X375 @ id))))) | ~ ! [X376 : term] : (($true = ((X369 @ id @ (sub @ X370 @ (push @ X376 @ id)) @ id))) | ~ ($true = ((X369 @ id @ X376 @ id)))) | ($true = ((X369 @ id @ (lam @ (sub @ X370 @ (push @ one @ sh))) @ id))))) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & (! [X382 : term] : ~ ! [X383 : (subst > term > term)] : ~ ! [X384 : subst,X385 : term] : (((X383 @ X384 @ X385)) = ((sub @ X382 @ (push @ X385 @ X384)))) <=> (hoasinduction_lem3v2_f = $true)) & ((! [X386 : term] : (((sub @ X386 @ id)) = X386) => (! [X387 : term,X388 : (subst > term > subst > $o)] : (~ ! [X389 : (subst > term > term)] : (~ ! [X390 : term] : (~ ($true = ((X388 @ id @ X390 @ id))) | ($true = ((X388 @ id @ (X389 @ id @ X390) @ id)))) | ($true = ((X388 @ id @ (lam @ (X389 @ sh @ one)) @ id))) | ~ ! [X391 : subst,X392 : term,X393 : subst] : (((sub @ (X389 @ X391 @ X392) @ X393)) = ((X389 @ (comp @ X391 @ X393) @ (sub @ X392 @ X393))))) | ~ ! [X394 : term] : (($true = ((X388 @ id @ (sub @ X387 @ (push @ X394 @ id)) @ id))) | ~ ($true = ((X388 @ id @ X394 @ id)))) | ($true = ((X388 @ id @ (lam @ (sub @ X387 @ (push @ one @ sh))) @ id)))) => ! [X395 : (subst > term > subst > $o),X396 : term] : (($true = ((X395 @ id @ (lam @ X396) @ id))) | ~ ! [X397 : subst,X398 : subst,X399 : term,X400 : subst] : (($true = ((X395 @ (comp @ X400 @ X397) @ (sub @ X399 @ X397) @ X398))) | ~ ($true = ((X395 @ X400 @ X399 @ (comp @ X397 @ X398))))) | ~ ! [X401 : term] : (~ ($true = ((X395 @ id @ X401 @ id))) | ($true = ((X395 @ id @ (sub @ X396 @ (push @ X401 @ id)) @ id)))) | ~ ! [X402 : subst,X403 : subst,X404 : term,X405 : subst] : (($true = ((X395 @ X405 @ X404 @ (comp @ X403 @ X402)))) | ~ ($true = ((X395 @ (comp @ X405 @ X403) @ (sub @ X404 @ X403) @ X402)))) | ~ ! [X406 : (subst > term > term)] : (~ ! [X407 : subst,X408 : term,X409 : subst] : (((X406 @ (comp @ X407 @ X409) @ (sub @ X408 @ X409))) = ((sub @ (X406 @ X407 @ X408) @ X409))) | ~ ! [X410 : term] : (($true = ((X395 @ id @ (X406 @ id @ X410) @ id))) | ~ ($true = ((X395 @ id @ X410 @ id)))) | ($true = ((X395 @ id @ (lam @ (X406 @ sh @ one)) @ id))))))) <=> (hoasinduction_lem3_lthm = $true)) & (induction2lem_gthm = $true) & (ulamvarind = $true) & (! [X411 : subst] : (((push @ (sub @ one @ X411) @ (comp @ sh @ X411))) = X411) <=> (axscons = $true)) & (axapp = $true) & (pushprop_lem2v2_lthm = $true) & ~ (ulamvar1 = $true) & (hoasinduction_lem3b_gthm = $true) & (hoasinduction_lem3a_gthm = $true) & (pushprop_lem3v2_lthm = $true) & ((substmonoid_lthm = $true) <=> (! [X412 : subst] : (((comp @ id @ X412)) = X412) => (! [X413 : subst] : (((comp @ X413 @ id)) = X413) => (! [X414 : subst] : (((comp @ X414 @ id)) = X414) & ! [X415 : subst] : (((comp @ id @ X415)) = X415))))) & ! [X416 : d_subst,X417 : d_subst] : ((((d2subst @ X417)) = ((d2subst @ X416))) => (X416 = X417)) & (pushprop_lem3v2 = $true) & ((hoaslamnotap = $true) <=> ! [X418 : (subst > term > term),X419 : term,X420 : term] : ((((ap @ (sub @ X420 @ id) @ X419)) != ((lam @ (X418 @ sh @ one)))) | ~ ! [X421 : subst,X422 : subst,X423 : term] : (((sub @ (X418 @ X422 @ X423) @ X421)) = ((X418 @ (comp @ X422 @ X421) @ (sub @ X423 @ X421)))))) & (axvarshift = $true) & (substmonoid_gthm = $true) & (! [X424 : (subst > term > subst > $o)] : ~ ! [X425 : (term > $o)] : ~ ! [X426 : term] : (((X424 @ id @ X426 @ id)) = ((X425 @ X426))) <=> (hoasinduction_lem0 = $true)) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X427 : term] : ~ ! [X428 : (subst > term > term)] : ~ ! [X429 : subst,X430 : term] : (((sub @ X427 @ (push @ X430 @ X429))) = ((X428 @ X429 @ X430)))) & (hoasinduction_lem1_gthm = $true) & ! [X431 : term] : ? [X432 : d_term] : (((d2term @ X432)) = X431) & ((hoasinduction_lem3v2 = $true) <=> ! [X433 : term,X434 : (term > $o),X435 : (subst > term > subst > $o)] : (~ ! [X436 : term] : (($true = ((X434 @ (sub @ X433 @ (push @ X436 @ id))))) | ~ ($true = ((X434 @ X436)))) | ~ ! [X437 : subst,X438 : subst,X439 : subst,X440 : term] : (($true = ((X435 @ X438 @ X440 @ (comp @ X439 @ X437)))) | ~ ($true = ((X435 @ (comp @ X438 @ X439) @ (sub @ X440 @ X439) @ X437)))) | ~ ! [X441 : (subst > term > term)] : (($true = ((X435 @ id @ (lam @ (X441 @ sh @ one)) @ id))) | ~ ! [X442 : term] : (~ ($true = ((X435 @ id @ X442 @ id))) | ($true = ((X435 @ id @ (X441 @ id @ X442) @ id)))) | ~ ! [X443 : subst,X444 : term,X445 : subst] : (((X441 @ (comp @ X445 @ X443) @ (sub @ X444 @ X443))) = ((sub @ (X441 @ X445 @ X444) @ X443)))) | ~ ! [X446 : term] : (((X435 @ id @ X446 @ id)) = ((X434 @ X446))) | ~ ! [X447 : subst,X448 : subst,X449 : subst,X450 : term] : (~ ($true = ((X435 @ X448 @ X450 @ (comp @ X449 @ X447)))) | ($true = ((X435 @ (comp @ X448 @ X449) @ (sub @ X450 @ X449) @ X447)))) | ($true = ((X434 @ (lam @ X433)))))) & (induction2_gthm = $true) & (hoasapnotvar_gthm = $true) & (pushprop_lem1_gthm = $true) & (hoasinduction_lem1_lthm = $true) & ((axshiftcons = $true) <=> ! [X451 : subst,X452 : term] : (((comp @ sh @ (push @ X452 @ X451))) = X451)) & (! [X453 : term,X454 : term] : ((X453 = X454) | (((lam @ X453)) != ((lam @ X454)))) <=> (laminj = $true)) & ((hoasapinj2_lthm = $true) <=> (! [X455 : term,X456 : term,X457 : term,X458 : term] : ((((ap @ X458 @ X456)) != ((ap @ X455 @ X457))) | (X456 = X457)) => ! [X459 : term,X460 : term,X461 : term,X462 : term] : ((X459 = X461) | (((ap @ (sub @ X462 @ id) @ X461)) != ((ap @ (sub @ X460 @ id) @ X459)))))) & ((apinj1 = $true) <=> ! [X463 : term,X464 : term,X465 : term,X466 : term] : ((X463 = X466) | (((ap @ X466 @ X465)) != ((ap @ X463 @ X464))))) & (! [X467 : subst,X468 : (subst > term > term),X469 : (subst > term > term),X470 : term] : ((((lam @ (X469 @ sh @ one))) != ((lam @ (X468 @ sh @ one)))) | ~ ! [X471 : subst,X472 : term,X473 : subst] : (((X468 @ (comp @ X471 @ X473) @ (sub @ X472 @ X473))) = ((sub @ (X468 @ X471 @ X472) @ X473))) | ~ ! [X474 : term,X475 : subst,X476 : subst] : (((X469 @ (comp @ X476 @ X475) @ (sub @ X474 @ X475))) = ((sub @ (X469 @ X476 @ X474) @ X475))) | (((X468 @ X467 @ X470)) = ((X469 @ X467 @ X470)))) <=> (hoaslaminj = $true)) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((shinj = $true) <=> ! [X477 : term,X478 : term] : ((X477 = X478) | (((sub @ X477 @ sh)) != ((sub @ X478 @ sh))))) & (hoaslamnotap_lthm = $true) & ((hoasapinj2 = $true) <=> ! [X479 : term,X480 : term,X481 : term,X482 : term] : ((((ap @ (sub @ X479 @ id) @ X481)) != ((ap @ (sub @ X482 @ id) @ X480))) | (X480 = X481))) & ~ (((var @ (d2term @ d_one))) = $true) & (! [X483 : term] : (((sub @ X483 @ id)) = X483) <=> (termmset = $true)) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (axabs = $true) & ! [X484 : d_term] : (d_one = X484) & (lamnotvar = $true) & (hoasinduction_lem3v2_gthm = $true) & (! [X485 : term] : ~ ! [X486 : (subst > term > term)] : (((X486 @ sh @ one)) != ((sub @ X485 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b = $true)) & ((pushprop_lem1v2_lthm = $true) <=> (! [X487 : term,X488 : subst] : (((sub @ one @ (push @ X487 @ X488))) = X487) => ! [X489 : term,X490 : (term > $o),X491 : subst,X492 : (term > $o)] : (~ ! [X493 : term] : (((X492 @ X493)) = ((X490 @ (sub @ X493 @ (push @ X489 @ X491))))) | ~ ($true = ((X490 @ X489))) | (((X492 @ one)) = $true)))) & (! [X494 : term] : ~ ! [X495 : (subst > term > term)] : (((X495 @ sh @ one)) != ((sub @ X494 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b_lthm = $true)) & (! [X496 : (subst > term > subst > $o),X497 : term] : ((((X496 @ id @ (lam @ (sub @ X497 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X498 : (subst > term > term),X499 : term] : (~ ! [X500 : subst,X501 : subst,X502 : term] : (((X498 @ (comp @ X500 @ X501) @ (sub @ X502 @ X501))) = ((sub @ X499 @ (push @ (sub @ X502 @ X501) @ (comp @ X500 @ X501))))) | ~ ! [X503 : term] : (($true = ((X496 @ id @ (X498 @ id @ X503) @ id))) | ~ ($true = ((X496 @ id @ X503 @ id)))) | ($true = ((X496 @ id @ (lam @ (X498 @ sh @ one)) @ id))) | ~ ! [X504 : term,X505 : subst,X506 : subst] : (((sub @ (sub @ X499 @ (push @ X504 @ X506)) @ X505)) = ((sub @ (X498 @ X506 @ X504) @ X505)))) | ~ ! [X507 : term] : (($true = ((X496 @ id @ (sub @ X497 @ (push @ X507 @ id)) @ id))) | ~ ($true = ((X496 @ id @ X507 @ id))))) <=> (hoasinduction_lem3aaa = $true)) & ((induction = $true) <=> ! [X508 : term,X509 : (term > $o)] : (~ ! [X510 : term] : (~ ($true = ((X509 @ X510))) | ($true = ((X509 @ (lam @ X510))))) | ~ ! [X511 : term] : (~ ($true = ((var @ X511))) | ($true = ((X509 @ X511)))) | ($true = ((X509 @ X508))) | ~ ! [X512 : term,X513 : term] : (~ ($true = ((X509 @ X512))) | ($true = ((X509 @ (ap @ X513 @ X512)))) | ~ ($true = ((X509 @ X513)))))) & ((pushprop_lem1_lthm = $true) <=> (! [X514 : term,X515 : subst] : (((sub @ one @ (push @ X514 @ X515))) = X514) => (! [X516 : subst,X517 : term] : (((comp @ sh @ (push @ X517 @ X516))) = X516) => ! [X518 : subst,X519 : term,X520 : (term > $o),X521 : (term > $o),X522 : term] : (~ ($true = ((X521 @ X519))) | ($true = ((X520 @ (sub @ X519 @ (push @ X522 @ X518))))))))) & (! [X523 : (subst > term > subst > $o),X524 : term] : (~ ! [X525 : term] : (($true = ((X523 @ id @ (sub @ X524 @ (push @ X525 @ id)) @ id))) | ~ ($true = ((X523 @ id @ X525 @ id)))) | ($true = ((X523 @ id @ (lam @ X524) @ id))) | ~ ! [X526 : (subst > term > term)] : (($true = ((X523 @ id @ (lam @ (X526 @ sh @ one)) @ id))) | ~ ! [X527 : term] : (~ ($true = ((X523 @ id @ X527 @ id))) | ($true = ((X523 @ id @ (X526 @ id @ X527) @ id)))) | ~ ! [X528 : term,X529 : subst,X530 : subst] : (((X526 @ (comp @ X529 @ X530) @ (sub @ X528 @ X530))) = ((sub @ (X526 @ X529 @ X528) @ X530))))) <=> (hoasinduction_lem3a = $true)) & (hoasinduction_lem2v2_gthm = $true) & (hoasinduction_lem2_lthm = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (! [X533 : term,X534 : (subst > term > subst > $o)] : (($true = ((X534 @ id @ (lam @ (sub @ X533 @ (push @ one @ sh))) @ id))) | ~ ! [X535 : (subst > term > term)] : (~ ! [X536 : subst,X537 : subst,X538 : term] : (((X535 @ (comp @ X537 @ X536) @ (sub @ X538 @ X536))) = ((sub @ (X535 @ X537 @ X538) @ X536))) | ($true = ((X534 @ id @ (lam @ (X535 @ sh @ one)) @ id))) | ~ ! [X539 : term] : (~ ($true = ((X534 @ id @ X539 @ id))) | ($true = ((X534 @ id @ (X535 @ id @ X539) @ id))))) | ~ ! [X540 : term] : (~ ($true = ((X534 @ id @ X540 @ id))) | ($true = ((X534 @ id @ (sub @ X533 @ (push @ X540 @ id)) @ id))))) <=> (hoasinduction_lem3aa = $true)) & ((hoasinduction_no_psi_cond = $true) <=> ! [X541 : term,X542 : (subst > term > subst > $o)] : (($true = ((X542 @ id @ X541 @ id))) | ~ ! [X543 : (subst > term > term)] : (($true = ((X542 @ id @ (lam @ (X543 @ sh @ one)) @ id))) | ~ ! [X544 : term] : (~ (((X542 @ id @ X544 @ id)) = $true) | ($true = ((X542 @ id @ (X543 @ id @ X544) @ id)))) | ~ ! [X545 : subst,X546 : term,X547 : subst] : (((X543 @ (comp @ X545 @ X547) @ (sub @ X546 @ X547))) = ((sub @ (X543 @ X545 @ X546) @ X547)))) | ~ ! [X548 : term,X549 : term] : (~ ($true = ((X542 @ id @ X548 @ id))) | ~ ($true = ((X542 @ id @ X549 @ id))) | ($true = ((X542 @ id @ (ap @ (sub @ X549 @ id) @ X548) @ id)))))) & (sh = ((d2subst @ d_id))) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (hoasinduction_lem1v2_gthm = $true)),
  inference(fool_elimination,[],[f6])).
thf(f8,plain,(
  (pushprop_lem1v2 != ((!! @ (term > $o) @ (^[Y0 : term > $o]: (!! @ term @ (^[Y1 : term]: (!! @ subst @ (^[Y2 : subst]: (!! @ (term > $o) @ (^[Y3 : term > $o]: ((Y3 @ Y1) => ((pushprop_p_and_p_prime @ Y1 @ Y2 @ Y3 @ Y0) => (Y0 @ one)))))))))))))),
  inference(flattening,[],[f5])).
thf(f9,plain,(
  (! [X0 : subst] : (((comp @ X0 @ id)) = X0) <=> (axidr = $true)) & (hoasapinj1_gthm = $true) & ((! [X1 : term] : (((sub @ X1 @ id)) = X1) => ! [X2 : term] : (((sub @ X2 @ id)) = X2)) <=> (termmset_lthm = $true)) & (termmset_gthm = $true) & (hoasinduction_lem1 = $true) & ((! [X4 : subst] : (((comp @ id @ X4)) = X4) & ! [X3 : subst] : (((comp @ X3 @ id)) = X3)) <=> (substmonoid = $true)) & (pushprop_lthm = $true) & (hoaslamnotvar_lthm = $true) & ! [X5 : d_subst] : (d_id = X5) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (! [X6 : term,X7 : (subst > term > subst > $o)] : (~ ! [X8 : term,X9 : term] : (~ ($true = ((X7 @ id @ X8 @ id))) | ~ (((X7 @ id @ X9 @ id)) = $true) | ($true = ((X7 @ id @ (ap @ (sub @ X9 @ id) @ X8) @ id)))) | ~ ! [X10 : subst,X11 : subst,X12 : subst,X13 : term] : (($true = ((X7 @ (comp @ X11 @ X12) @ (sub @ X13 @ X12) @ X10))) | ~ ($true = ((X7 @ X11 @ X13 @ (comp @ X12 @ X10))))) | ~ ! [X14 : subst,X15 : term,X16 : subst,X17 : subst] : (~ ($true = ((X7 @ (comp @ X14 @ X16) @ (sub @ X15 @ X16) @ X17))) | ($true = ((X7 @ X14 @ X15 @ (comp @ X16 @ X17))))) | ($true = ((X7 @ id @ X6 @ id))) | ~ ! [X18 : term] : (~ ($true = ((var @ (sub @ X18 @ id)))) | ($true = ((X7 @ id @ X18 @ id)))) | ~ ! [X19 : (subst > term > term)] : (($true = ((X7 @ id @ (lam @ (X19 @ sh @ one)) @ id))) | ~ ! [X22 : subst,X21 : term,X20 : subst] : (((sub @ (X19 @ X22 @ X21) @ X20)) = ((X19 @ (comp @ X22 @ X20) @ (sub @ X21 @ X20)))) | ~ ! [X23 : term] : (($true = ((X7 @ id @ (X19 @ id @ X23) @ id))) | ~ (((X7 @ id @ X23 @ id)) = $true)))) <=> (hoasinduction = $true)) & ((! [X24 : term] : (((sub @ X24 @ id)) = X24) => (! [X25 : subst,X26 : term,X27 : (term > $o)] : (~ ! [X28 : term] : (~ ! [X29 : term] : (~ ($true = ((X27 @ X29))) | ($true = ((X27 @ (sub @ X28 @ (push @ X29 @ id)))))) | ($true = ((X27 @ (lam @ X28))))) | ($true = ((X27 @ (sub @ X26 @ X25)))) | ~ ! [X30 : term,X31 : term] : (~ ($true = ((X27 @ X30))) | ($true = ((X27 @ (ap @ X30 @ X31)))) | ~ ($true = ((X27 @ X31)))) | ~ ! [X32 : term] : (~ ($true = ((var @ X32))) | ($true = ((X27 @ (sub @ X32 @ X25)))))) => ! [X33 : (term > $o),X34 : term] : (~ ! [X35 : term] : (($true = ((X33 @ (lam @ X35)))) | ~ ! [X36 : term] : (~ ($true = ((X33 @ X36))) | ($true = ((X33 @ (sub @ X35 @ (push @ X36 @ id))))))) | ($true = ((X33 @ X34))) | ~ ! [X37 : term,X38 : term] : (~ ($true = ((X33 @ X37))) | ~ ($true = ((X33 @ X38))) | ($true = ((X33 @ (ap @ X38 @ X37))))) | ~ ! [X39 : term] : (($true = ((X33 @ X39))) | ~ ($true = ((var @ X39))))))) <=> (induction2_lthm = $true)) & (! [X40 : subst] : (((comp @ id @ X40)) = X40) <=> (axidl = $true)) & (axclos = $true) & (pushprop_lem2v2 = $true) & (pushprop = $true) & (axmap = $true) & ((! [X41 : subst,X42 : term] : (((sub @ one @ (push @ X42 @ X41))) = X42) => (! [X44 : term,X43 : subst] : (((comp @ sh @ (push @ X44 @ X43))) = X43) => (! [X45 : subst] : (((comp @ X45 @ id)) = X45) => (! [X46 : (term > $o),X47 : term] : (~ ! [X48 : term] : (~ ($true = ((var @ X48))) | ($true = ((X46 @ X48)))) | ~ ! [X49 : term,X50 : term] : (~ ($true = ((X46 @ X49))) | ~ ($true = ((X46 @ X50))) | ($true = ((X46 @ (ap @ X49 @ X50))))) | ~ ! [X51 : term] : (($true = ((X46 @ (lam @ X51)))) | ~ ($true = ((X46 @ X51)))) | ($true = ((X46 @ X47)))) => ! [X52 : (term > $o),X53 : subst,X54 : term] : (~ ! [X55 : term,X56 : term] : (~ (((X52 @ X55)) = $true) | ($true = ((X52 @ (ap @ X55 @ X56)))) | ~ ($true = ((X52 @ X56)))) | ($true = ((X52 @ (sub @ X54 @ X53)))) | ~ ! [X57 : term] : (($true = ((X52 @ (lam @ X57)))) | ~ ! [X58 : term] : (($true = ((X52 @ (sub @ X57 @ (push @ X58 @ id))))) | ~ ($true = ((X52 @ X58))))) | ~ ! [X59 : term] : (($true = ((X52 @ (sub @ X59 @ X53)))) | ~ ($true = ((var @ X59))))))))) <=> (induction2lem_lthm = $true)) & (! [X63 : term,X60 : term,X61 : term,X62 : term] : ((X60 = X61) | (((ap @ (sub @ X61 @ id) @ X62)) != ((ap @ (sub @ X60 @ id) @ X63)))) <=> (hoasapinj1 = $true)) & (hoaslaminj_gthm = $true) & ((hoasinduction_lem3 = $true) <=> ! [X64 : term,X65 : (subst > term > subst > $o)] : (~ ! [X66 : subst,X67 : term,X68 : subst,X69 : subst] : (($true = ((X65 @ (comp @ X66 @ X69) @ (sub @ X67 @ X69) @ X68))) | ~ ($true = ((X65 @ X66 @ X67 @ (comp @ X69 @ X68))))) | ($true = ((X65 @ id @ (lam @ X64) @ id))) | ~ ! [X70 : term] : (($true = ((X65 @ id @ (sub @ X64 @ (push @ X70 @ id)) @ id))) | ~ ($true = ((X65 @ id @ X70 @ id)))) | ~ ! [X71 : (subst > term > term)] : (~ ! [X73 : subst,X74 : term,X72 : subst] : (((X71 @ (comp @ X73 @ X72) @ (sub @ X74 @ X72))) = ((sub @ (X71 @ X73 @ X74) @ X72))) | ($true = ((X65 @ id @ (lam @ (X71 @ sh @ one)) @ id))) | ~ ! [X75 : term] : (~ ($true = ((X65 @ id @ X75 @ id))) | ($true = ((X65 @ id @ (X71 @ id @ X75) @ id))))) | ~ ! [X76 : term,X77 : subst,X78 : subst,X79 : subst] : (($true = ((X65 @ X77 @ X76 @ (comp @ X79 @ X78)))) | ~ ($true = ((X65 @ (comp @ X77 @ X79) @ (sub @ X76 @ X79) @ X78)))))) & (! [X80 : term,X83 : term,X82 : term,X81 : term] : ((X80 = X83) | (((ap @ X82 @ X83)) != ((ap @ X81 @ X80)))) <=> (apinj2 = $true)) & ((! [X84 : term] : (((sub @ X84 @ id)) = X84) => (! [X85 : (subst > term > subst > $o),X86 : term] : (($true = ((X85 @ id @ (lam @ (sub @ X86 @ (push @ one @ sh))) @ id))) | ~ ! [X87 : (subst > term > term)] : (~ ! [X90 : term,X89 : subst,X88 : subst] : (((X87 @ (comp @ X88 @ X89) @ (sub @ X90 @ X89))) = ((sub @ (X87 @ X88 @ X90) @ X89))) | ~ ! [X91 : term] : (($true = ((X85 @ id @ (X87 @ id @ X91) @ id))) | ~ ($true = ((X85 @ id @ X91 @ id)))) | ($true = ((X85 @ id @ (lam @ (X87 @ sh @ one)) @ id)))) | ~ ! [X92 : term] : (($true = ((X85 @ id @ (sub @ X86 @ (push @ X92 @ id)) @ id))) | ~ ($true = ((X85 @ id @ X92 @ id))))) => ! [X93 : (subst > term > subst > $o),X94 : term] : (~ ! [X95 : term] : (($true = ((X93 @ id @ (sub @ X94 @ (push @ X95 @ id)) @ id))) | ~ ($true = ((X93 @ id @ X95 @ id)))) | ($true = ((X93 @ id @ (lam @ X94) @ id))) | ~ ! [X96 : (subst > term > term)] : (~ ! [X97 : term] : (($true = ((X93 @ id @ (X96 @ id @ X97) @ id))) | ~ ($true = ((X93 @ id @ X97 @ id)))) | ~ ! [X99 : term,X100 : subst,X98 : subst] : (((sub @ (X96 @ X100 @ X99) @ X98)) = ((X96 @ (comp @ X100 @ X98) @ (sub @ X99 @ X98)))) | ($true = ((X93 @ id @ (lam @ (X96 @ sh @ one)) @ id))))))) <=> (hoasinduction_lem3a_lthm = $true)) & ~ (lamnotap = $true) & ((pushprop_lem0 = $true) <=> ! [X103 : term,X101 : subst,X102 : (term > $o)] : ~ ! [X104 : (term > $o)] : ~ ! [X105 : term] : (((X104 @ X105)) = ((X102 @ (sub @ X105 @ (push @ X103 @ X101)))))) & (hoasapinj2_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (pushprop_lem2v2_gthm = $true) & (! [X107 : subst,X108 : term,X106 : (term > $o)] : ~ ! [X109 : (term > $o)] : ~ ! [X110 : term] : (((X106 @ (sub @ X110 @ (push @ X108 @ X107)))) = ((X109 @ X110))) <=> (pushprop_lem0_lthm = $true)) & ((pushprop_lem1v2 = $true) <=> ! [X111 : (term > $o),X112 : (term > $o),X113 : term,X114 : subst] : (~ ($true = ((X111 @ X113))) | ($true = ((X112 @ one))) | ~ ! [X115 : term] : (((X112 @ X115)) = ((X111 @ (sub @ X115 @ (push @ X113 @ X114))))))) & (pushprop_lthm_orig = $true) & (! [X116 : term] : (((sub @ X116 @ id)) = X116) <=> (axvarid = $true)) & (! [X117 : term,X118 : term,X119 : (subst > term > subst > $o)] : (~ ! [X120 : term,X121 : term] : (($true = ((X119 @ id @ (ap @ (sub @ X120 @ id) @ X121) @ id))) | ~ ($true = ((X119 @ id @ X120 @ id))) | ~ ($true = ((X119 @ id @ X121 @ id)))) | ~ ($true = ((X119 @ id @ X118 @ id))) | ~ ($true = ((X119 @ id @ X117 @ id))) | ~ ! [X122 : term,X123 : subst,X124 : subst,X125 : subst] : (($true = ((X119 @ X123 @ X122 @ (comp @ X125 @ X124)))) | ~ ($true = ((X119 @ (comp @ X123 @ X125) @ (sub @ X122 @ X125) @ X124)))) | ~ ! [X126 : subst,X127 : term,X128 : subst,X129 : subst] : (($true = ((X119 @ (comp @ X126 @ X128) @ (sub @ X127 @ X128) @ X129))) | ~ ($true = ((X119 @ X126 @ X127 @ (comp @ X128 @ X129))))) | ($true = ((X119 @ id @ (ap @ X118 @ X117) @ id)))) <=> (hoasinduction_lem2 = $true)) & (pushprop_gthm = $true) & (axassoc = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (one = ((d2term @ d_one))) & (hoasinduction_lem3_gthm = $true) & ((! [X130 : subst,X131 : term] : (((sub @ one @ (push @ X131 @ X130))) = X131) => (! [X133 : subst,X132 : term] : (((comp @ sh @ (push @ X132 @ X133))) = X133) => (! [X134 : term,X135 : term] : ((((lam @ X134)) != ((lam @ X135))) | (X134 = X135)) => ! [X136 : term,X138 : subst,X139 : (subst > term > term),X137 : (subst > term > term)] : ((((lam @ (X139 @ sh @ one))) != ((lam @ (X137 @ sh @ one)))) | (((X137 @ X138 @ X136)) = ((X139 @ X138 @ X136))) | ~ ! [X143 : term,X145 : subst,X144 : subst] : (((X137 @ (comp @ X145 @ X144) @ (sub @ X143 @ X144))) = ((sub @ (X137 @ X145 @ X143) @ X144))) | ~ ! [X142 : subst,X141 : term,X140 : subst] : (((X139 @ (comp @ X142 @ X140) @ (sub @ X141 @ X140))) = ((sub @ (X139 @ X142 @ X141) @ X140))))))) <=> (hoaslaminj_lthm = $true)) & (((d2subst @ d_id)) = id) & (ulamvarsh = $true) & ! [X146 : subst] : ? [X147 : d_subst] : (((d2subst @ X147)) = X146) & (hoaslamnotvar = $true) & (pushprop_lem1v2_gthm = $true) & (hoasapnotvar_lthm = $true) & (hoasinduction_gthm = $true) & ((hoasinduction_lthm_3 = $true) <=> (! [X148 : (subst > term > subst > $o)] : ~ ! [X149 : (term > $o)] : ~ ! [X150 : term] : (((X148 @ id @ X150 @ id)) = ((X149 @ X150))) => (! [X151 : (term > $o),X152 : term] : (($true = ((X151 @ X152))) | ~ ! [X153 : term,X154 : term] : (~ ($true = ((X151 @ X154))) | ~ ($true = ((X151 @ X153))) | ($true = ((X151 @ (ap @ X154 @ X153))))) | ~ ! [X155 : term] : (($true = ((X151 @ (lam @ X155)))) | ~ ! [X156 : term] : (~ ($true = ((X151 @ X156))) | ($true = ((X151 @ (sub @ X155 @ (push @ X156 @ id))))))) | ~ ! [X157 : term] : (($true = ((X151 @ X157))) | ~ ($true = ((var @ X157))))) => (! [X158 : term] : (((sub @ X158 @ id)) = X158) => (! [X159 : (subst > term > subst > $o),X160 : (term > $o),X161 : term] : (($true = ((X160 @ (lam @ X161)))) | ~ ! [X162 : term] : (((X160 @ X162)) = ((X159 @ id @ X162 @ id))) | ~ ! [X163 : term] : (($true = ((X160 @ (sub @ X161 @ (push @ X163 @ id))))) | ~ ($true = ((X160 @ X163)))) | ~ ! [X164 : (subst > term > term)] : (~ ! [X165 : term] : (($true = ((X159 @ id @ (X164 @ id @ X165) @ id))) | ~ ($true = ((X159 @ id @ X165 @ id)))) | ~ ! [X166 : subst,X168 : term,X167 : subst] : (((X164 @ (comp @ X166 @ X167) @ (sub @ X168 @ X167))) = ((sub @ (X164 @ X166 @ X168) @ X167))) | ($true = ((X159 @ id @ (lam @ (X164 @ sh @ one)) @ id))))) => ! [X169 : term,X170 : (subst > term > subst > $o)] : (~ ! [X171 : (subst > term > term)] : (~ ! [X172 : term] : (($true = ((X170 @ id @ (X171 @ id @ X172) @ id))) | ~ (((X170 @ id @ X172 @ id)) = $true)) | ~ ! [X173 : subst,X174 : subst,X175 : term] : (((sub @ (X171 @ X173 @ X175) @ X174)) = ((X171 @ (comp @ X173 @ X174) @ (sub @ X175 @ X174)))) | ($true = ((X170 @ id @ (lam @ (X171 @ sh @ one)) @ id)))) | ($true = ((X170 @ id @ X169 @ id))) | ~ ! [X176 : subst,X177 : subst,X178 : subst,X179 : term] : (~ ($true = ((X170 @ (comp @ X176 @ X177) @ (sub @ X179 @ X177) @ X178))) | ($true = ((X170 @ X176 @ X179 @ (comp @ X177 @ X178))))) | ~ ! [X180 : term] : (($true = ((X170 @ id @ X180 @ id))) | ~ ($true = ((var @ (sub @ X180 @ id))))) | ~ ! [X181 : term,X182 : subst,X183 : subst,X184 : subst] : (~ ($true = ((X170 @ X184 @ X181 @ (comp @ X182 @ X183)))) | ($true = ((X170 @ (comp @ X184 @ X182) @ (sub @ X181 @ X182) @ X183)))) | ~ ! [X185 : term,X186 : term] : (~ ($true = ((X170 @ id @ X186 @ id))) | ($true = ((X170 @ id @ (ap @ (sub @ X186 @ id) @ X185) @ id))) | ~ ($true = ((X170 @ id @ X185 @ id)))))))))) & (hoasinduction_lem1v2 = $true) & ((hoasinduction_lem2v2 = $true) <=> ! [X187 : term,X188 : term,X189 : (subst > term > subst > $o),X190 : (term > $o)] : (~ ! [X191 : term] : (((X190 @ X191)) = ((X189 @ id @ X191 @ id))) | ($true = ((X190 @ (ap @ X188 @ X187)))) | ~ ($true = ((X190 @ X187))) | ~ ! [X192 : subst,X193 : subst,X194 : subst,X195 : term] : (~ ($true = ((X189 @ (comp @ X194 @ X193) @ (sub @ X195 @ X193) @ X192))) | (((X189 @ X194 @ X195 @ (comp @ X193 @ X192))) = $true)) | ~ ! [X196 : subst,X197 : subst,X198 : term,X199 : subst] : (~ ($true = ((X189 @ X199 @ X198 @ (comp @ X196 @ X197)))) | ($true = ((X189 @ (comp @ X199 @ X196) @ (sub @ X198 @ X196) @ X197)))) | ~ ! [X200 : term,X201 : term] : (~ ($true = ((X189 @ id @ X200 @ id))) | ($true = ((X189 @ id @ (ap @ (sub @ X200 @ id) @ X201) @ id))) | ~ ($true = ((X189 @ id @ X201 @ id)))) | ~ ($true = ((X190 @ X188))))) & (! [X202 : (term > $o),X203 : subst,X204 : term] : (~ ! [X205 : term] : (~ ! [X206 : term] : (($true = ((X202 @ (sub @ X205 @ (push @ X206 @ id))))) | ~ ($true = ((X202 @ X206)))) | ($true = ((X202 @ (lam @ X205))))) | ~ ! [X207 : term,X208 : term] : (~ ($true = ((X202 @ X207))) | ($true = ((X202 @ (ap @ X207 @ X208)))) | ~ ($true = ((X202 @ X208)))) | ($true = ((X202 @ (sub @ X204 @ X203)))) | ~ ! [X209 : term] : (~ ($true = ((var @ X209))) | ($true = ((X202 @ (sub @ X209 @ X203)))))) <=> (induction2lem = $true)) & ! [X210 : d_term,X211 : d_term] : ((((d2term @ X210)) = ((d2term @ X211))) => (X210 = X211)) & (hoasinduction_lem2_gthm = $true) & ((pushprop_lem1 = $true) <=> ! [X212 : subst,X213 : term,X214 : (term > $o),X215 : (term > $o),X216 : term] : (~ ($true = ((X214 @ X213))) | ($true = ((X215 @ (sub @ X213 @ (push @ X216 @ X212))))))) & ((! [X217 : term] : (((sub @ X217 @ id)) = X217) => ! [X218 : term,X219 : (subst > term > subst > $o),X220 : (term > $o)] : (~ ! [X221 : subst,X222 : subst,X223 : term,X224 : subst] : (($true = ((X219 @ X224 @ X223 @ (comp @ X222 @ X221)))) | ~ ($true = ((X219 @ (comp @ X224 @ X222) @ (sub @ X223 @ X222) @ X221)))) | ~ ! [X225 : subst,X226 : term,X227 : subst,X228 : subst] : (($true = ((X219 @ (comp @ X225 @ X228) @ (sub @ X226 @ X228) @ X227))) | ~ ($true = ((X219 @ X225 @ X226 @ (comp @ X228 @ X227))))) | ~ ! [X229 : term] : (~ ($true = ((X220 @ X229))) | ($true = ((X220 @ (sub @ X218 @ (push @ X229 @ id)))))) | ~ ! [X230 : term] : (((X219 @ id @ X230 @ id)) = ((X220 @ X230))) | ($true = ((X220 @ (lam @ X218)))) | ~ ! [X231 : (subst > term > term)] : (~ ! [X234 : subst,X233 : subst,X232 : term] : (((sub @ (X231 @ X234 @ X232) @ X233)) = ((X231 @ (comp @ X234 @ X233) @ (sub @ X232 @ X233)))) | ~ ! [X235 : term] : (($true = ((X219 @ id @ (X231 @ id @ X235) @ id))) | ~ ($true = ((X219 @ id @ X235 @ id)))) | ($true = ((X219 @ id @ (lam @ (X231 @ sh @ one)) @ id)))))) <=> (hoasinduction_lem3v2_lthm = $true)) & ((! [X236 : term] : (((sub @ X236 @ id)) = X236) => (! [X238 : term,X239 : term,X237 : term,X240 : term] : ((((ap @ X237 @ X239)) != ((ap @ X238 @ X240))) | (X237 = X238)) => ! [X242 : term,X244 : term,X241 : term,X243 : term] : ((X242 = X244) | (((ap @ (sub @ X244 @ id) @ X241)) != ((ap @ (sub @ X242 @ id) @ X243)))))) <=> (hoasapinj1_lthm = $true)) & ((hoasinduction_lthm = $true) <=> (! [X245 : term,X246 : (term > $o)] : (~ ! [X247 : term] : (($true = ((X246 @ X247))) | ~ ($true = ((var @ X247)))) | ($true = ((X246 @ X245))) | ~ ! [X248 : term] : (($true = ((X246 @ (lam @ X248)))) | ~ ! [X249 : term] : (~ ($true = ((X246 @ X249))) | ($true = ((X246 @ (sub @ X248 @ (push @ X249 @ id))))))) | ~ ! [X250 : term,X251 : term] : (~ ($true = ((X246 @ X251))) | ($true = ((X246 @ (ap @ X250 @ X251)))) | ~ ($true = ((X246 @ X250))))) => (! [X252 : (subst > term > subst > $o),X253 : term,X254 : term] : (~ ! [X255 : subst,X256 : subst,X257 : subst,X258 : term] : (~ ($true = ((X252 @ (comp @ X256 @ X255) @ (sub @ X258 @ X255) @ X257))) | ($true = ((X252 @ X256 @ X258 @ (comp @ X255 @ X257))))) | ~ ! [X259 : subst,X260 : term,X261 : subst,X262 : subst] : (($true = ((X252 @ (comp @ X259 @ X262) @ (sub @ X260 @ X262) @ X261))) | ~ ($true = ((X252 @ X259 @ X260 @ (comp @ X262 @ X261))))) | ~ ($true = ((X252 @ id @ X254 @ id))) | ~ ! [X263 : term,X264 : term] : (~ ($true = ((X252 @ id @ X264 @ id))) | ($true = ((X252 @ id @ (ap @ (sub @ X264 @ id) @ X263) @ id))) | ~ ($true = ((X252 @ id @ X263 @ id)))) | ($true = ((X252 @ id @ (ap @ X253 @ X254) @ id))) | ~ ($true = ((X252 @ id @ X253 @ id)))) => (! [X265 : term,X266 : (subst > term > subst > $o)] : (~ ! [X267 : term,X268 : subst,X269 : subst,X270 : subst] : (($true = ((X266 @ X269 @ X267 @ (comp @ X268 @ X270)))) | ~ ($true = ((X266 @ (comp @ X269 @ X268) @ (sub @ X267 @ X268) @ X270)))) | ($true = ((X266 @ id @ (lam @ X265) @ id))) | ~ ! [X271 : subst,X272 : subst,X273 : subst,X274 : term] : (~ ($true = ((X266 @ X272 @ X274 @ (comp @ X273 @ X271)))) | ($true = ((X266 @ (comp @ X272 @ X273) @ (sub @ X274 @ X273) @ X271)))) | ~ ! [X275 : term] : (($true = ((X266 @ id @ (sub @ X265 @ (push @ X275 @ id)) @ id))) | ~ ($true = ((X266 @ id @ X275 @ id)))) | ~ ! [X276 : (subst > term > term)] : (~ ! [X277 : term] : (($true = ((X266 @ id @ (X276 @ id @ X277) @ id))) | ~ ($true = ((X266 @ id @ X277 @ id)))) | ($true = ((X266 @ id @ (lam @ (X276 @ sh @ one)) @ id))) | ~ ! [X278 : subst,X279 : subst,X280 : term] : (((sub @ (X276 @ X278 @ X280) @ X279)) = ((X276 @ (comp @ X278 @ X279) @ (sub @ X280 @ X279)))))) => ! [X281 : (subst > term > subst > $o),X282 : term] : (($true = ((X281 @ id @ X282 @ id))) | ~ ! [X283 : (subst > term > term)] : (($true = ((X281 @ id @ (lam @ (X283 @ sh @ one)) @ id))) | ~ ! [X284 : term] : (~ ($true = ((X281 @ id @ X284 @ id))) | ($true = ((X281 @ id @ (X283 @ id @ X284) @ id)))) | ~ ! [X285 : term,X286 : subst,X287 : subst] : (((sub @ (X283 @ X286 @ X285) @ X287)) = ((X283 @ (comp @ X286 @ X287) @ (sub @ X285 @ X287))))) | ~ ! [X288 : subst,X289 : subst,X290 : term,X291 : subst] : (~ ($true = ((X281 @ X288 @ X290 @ (comp @ X289 @ X291)))) | ($true = ((X281 @ (comp @ X288 @ X289) @ (sub @ X290 @ X289) @ X291)))) | ~ ! [X292 : subst,X293 : subst,X294 : term,X295 : subst] : (~ ($true = ((X281 @ (comp @ X292 @ X295) @ (sub @ X294 @ X295) @ X293))) | ($true = ((X281 @ X292 @ X294 @ (comp @ X295 @ X293))))) | ~ ! [X296 : term] : (~ ($true = ((var @ (sub @ X296 @ id)))) | ($true = ((X281 @ id @ X296 @ id)))) | ~ ! [X297 : term,X298 : term] : (~ ($true = ((X281 @ id @ X298 @ id))) | ($true = ((X281 @ id @ (ap @ (sub @ X297 @ id) @ X298) @ id))) | ~ ($true = ((X281 @ id @ X297 @ id))))))))) & ~ (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) = $true) & (hoasapnotvar = $true) & ((! [X299 : term] : ~ ! [X300 : (subst > term > term)] : ~ ! [X302 : term,X301 : subst] : (((X300 @ X301 @ X302)) = ((sub @ X299 @ (push @ X302 @ X301)))) => (! [X303 : term] : (((sub @ X303 @ id)) = X303) => ! [X304 : term,X305 : (term > $o),X306 : (subst > term > subst > $o)] : (~ ! [X307 : (subst > term > term)] : (~ ! [X309 : subst,X308 : term,X310 : subst] : (((X307 @ (comp @ X309 @ X310) @ (sub @ X308 @ X310))) = ((sub @ (X307 @ X309 @ X308) @ X310))) | ($true = ((X306 @ id @ (lam @ (X307 @ sh @ one)) @ id))) | ~ ! [X311 : term] : (($true = ((X306 @ id @ (X307 @ id @ X311) @ id))) | ~ ($true = ((X306 @ id @ X311 @ id))))) | ~ ! [X312 : term] : (((X305 @ X312)) = ((X306 @ id @ X312 @ id))) | ($true = ((X305 @ (lam @ X304)))) | ~ ! [X313 : term] : (($true = ((X305 @ (sub @ X304 @ (push @ X313 @ id))))) | ~ ($true = ((X305 @ X313))))))) <=> (hoasinduction_lem3v2a_lthm = $true)) & ((! [X314 : (subst > term > subst > $o)] : ~ ! [X315 : (term > $o)] : ~ ! [X316 : term] : (((X314 @ id @ X316 @ id)) = ((X315 @ X316))) => (! [X317 : (term > $o),X318 : term] : (~ ! [X319 : term] : (($true = ((X317 @ (lam @ X319)))) | ~ ! [X320 : term] : (($true = ((X317 @ (sub @ X319 @ (push @ X320 @ id))))) | ~ ($true = ((X317 @ X320))))) | ~ ! [X321 : term,X322 : term] : (~ ($true = ((X317 @ X321))) | ~ ($true = ((X317 @ X322))) | ($true = ((X317 @ (ap @ X321 @ X322))))) | ~ ! [X323 : term] : (($true = ((X317 @ X323))) | ~ ($true = ((var @ X323)))) | ($true = ((X317 @ X318)))) => (! [X324 : term] : (((sub @ X324 @ id)) = X324) => (! [X325 : (term > $o),X326 : (subst > term > subst > $o),X327 : term] : (~ ! [X328 : term] : (($true = ((X325 @ (sub @ X327 @ (push @ X328 @ id))))) | ~ ($true = ((X325 @ X328)))) | ~ ! [X329 : term] : (((X325 @ X329)) = ((X326 @ id @ X329 @ id))) | ~ ! [X330 : (subst > term > term)] : (~ ! [X332 : subst,X331 : subst,X333 : term] : (((X330 @ (comp @ X332 @ X331) @ (sub @ X333 @ X331))) = ((sub @ (X330 @ X332 @ X333) @ X331))) | ~ ! [X334 : term] : (~ ($true = ((X326 @ id @ X334 @ id))) | ($true = ((X326 @ id @ (X330 @ id @ X334) @ id)))) | ($true = ((X326 @ id @ (lam @ (X330 @ sh @ one)) @ id)))) | ($true = ((X325 @ (lam @ X327))))) => ! [X335 : term,X336 : (subst > term > subst > $o)] : (~ ! [X337 : term,X338 : term] : (~ ($true = ((X336 @ id @ X338 @ id))) | ($true = ((X336 @ id @ (ap @ (sub @ X337 @ id) @ X338) @ id))) | ~ (((X336 @ id @ X337 @ id)) = $true)) | ~ ! [X339 : (subst > term > term)] : (~ ! [X340 : term] : (($true = ((X336 @ id @ (X339 @ id @ X340) @ id))) | ~ ($true = ((X336 @ id @ X340 @ id)))) | ($true = ((X336 @ id @ (lam @ (X339 @ sh @ one)) @ id))) | ~ ! [X343 : subst,X342 : subst,X341 : term] : (((sub @ (X339 @ X342 @ X341) @ X343)) = ((X339 @ (comp @ X342 @ X343) @ (sub @ X341 @ X343))))) | ($true = ((X336 @ id @ X335 @ id)))))))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (! [X344 : (subst > term > subst > $o)] : ~ ! [X345 : (term > $o)] : ~ ! [X346 : term] : (((X345 @ X346)) = ((X344 @ id @ X346 @ id))) <=> (hoasinduction_lem0_lthm = $true)) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (hoaslamnotvar_gthm = $true) & (apnotvar = $true) & (! [X348 : term,X347 : subst] : (((sub @ one @ (push @ X348 @ X347))) = X348) <=> (axvarcons = $true)) & ((hoasinduction_lem3v2a = $true) <=> ! [X349 : (subst > term > subst > $o),X350 : (term > $o),X351 : term] : (~ ! [X352 : (subst > term > term)] : (($true = ((X349 @ id @ (lam @ (X352 @ sh @ one)) @ id))) | ~ ! [X353 : term] : (~ ($true = ((X349 @ id @ X353 @ id))) | ($true = ((X349 @ id @ (X352 @ id @ X353) @ id)))) | ~ ! [X356 : subst,X355 : subst,X354 : term] : (((X352 @ (comp @ X356 @ X355) @ (sub @ X354 @ X355))) = ((sub @ (X352 @ X356 @ X354) @ X355)))) | ($true = ((X350 @ (lam @ X351)))) | ~ ! [X357 : term] : (((X350 @ X357)) = ((X349 @ id @ X357 @ id))) | ~ ! [X358 : term] : (~ ($true = ((X350 @ X358))) | ($true = ((X350 @ (sub @ X351 @ (push @ X358 @ id)))))))) & (! [X359 : (term > $o),X360 : term] : (~ ! [X361 : term] : (~ ! [X362 : term] : (~ ($true = ((X359 @ X362))) | ($true = ((X359 @ (sub @ X361 @ (push @ X362 @ id)))))) | ($true = ((X359 @ (lam @ X361))))) | ~ ! [X363 : term] : (~ ($true = ((var @ X363))) | ($true = ((X359 @ X363)))) | ($true = ((X359 @ X360))) | ~ ! [X364 : term,X365 : term] : (($true = ((X359 @ (ap @ X365 @ X364)))) | ~ ($true = ((X359 @ X364))) | ~ ($true = ((X359 @ X365))))) <=> (induction2 = $true)) & (pushprop_lem0_gthm = $true) & (hoaslamnotap_gthm = $true) & ((hoasinduction_lem3aa_lthm = $true) <=> ! [X366 : (subst > term > subst > $o),X367 : term] : (~ ! [X368 : (subst > term > term)] : (($true = ((X366 @ id @ (lam @ (X368 @ sh @ one)) @ id))) | ~ ! [X370 : subst,X371 : term,X369 : subst] : (((X368 @ (comp @ X370 @ X369) @ (sub @ X371 @ X369))) = ((sub @ (X368 @ X370 @ X371) @ X369))) | ~ ! [X372 : term] : (($true = ((X366 @ id @ (X368 @ id @ X372) @ id))) | ~ ($true = ((X366 @ id @ X372 @ id))))) | ~ ! [X373 : term] : (($true = ((X366 @ id @ (sub @ X367 @ (push @ X373 @ id)) @ id))) | ~ ($true = ((X366 @ id @ X373 @ id)))) | ($true = ((X366 @ id @ (lam @ (sub @ X367 @ (push @ one @ sh))) @ id))))) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & (! [X374 : term] : ~ ! [X375 : (subst > term > term)] : ~ ! [X376 : subst,X377 : term] : (((sub @ X374 @ (push @ X377 @ X376))) = ((X375 @ X376 @ X377))) <=> (hoasinduction_lem3v2_f = $true)) & ((! [X378 : term] : (((sub @ X378 @ id)) = X378) => (! [X379 : term,X380 : (subst > term > subst > $o)] : (~ ! [X381 : (subst > term > term)] : (~ ! [X382 : term] : (~ ($true = ((X380 @ id @ X382 @ id))) | ($true = ((X380 @ id @ (X381 @ id @ X382) @ id)))) | ($true = ((X380 @ id @ (lam @ (X381 @ sh @ one)) @ id))) | ~ ! [X383 : subst,X384 : term,X385 : subst] : (((X381 @ (comp @ X383 @ X385) @ (sub @ X384 @ X385))) = ((sub @ (X381 @ X383 @ X384) @ X385)))) | ~ ! [X386 : term] : (($true = ((X380 @ id @ (sub @ X379 @ (push @ X386 @ id)) @ id))) | ~ ($true = ((X380 @ id @ X386 @ id)))) | ($true = ((X380 @ id @ (lam @ (sub @ X379 @ (push @ one @ sh))) @ id)))) => ! [X387 : (subst > term > subst > $o),X388 : term] : (($true = ((X387 @ id @ (lam @ X388) @ id))) | ~ ! [X389 : subst,X390 : subst,X391 : term,X392 : subst] : (($true = ((X387 @ (comp @ X392 @ X389) @ (sub @ X391 @ X389) @ X390))) | ~ ($true = ((X387 @ X392 @ X391 @ (comp @ X389 @ X390))))) | ~ ! [X393 : term] : (~ ($true = ((X387 @ id @ X393 @ id))) | ($true = ((X387 @ id @ (sub @ X388 @ (push @ X393 @ id)) @ id)))) | ~ ! [X394 : subst,X395 : subst,X396 : term,X397 : subst] : (($true = ((X387 @ X397 @ X396 @ (comp @ X395 @ X394)))) | ~ ($true = ((X387 @ (comp @ X397 @ X395) @ (sub @ X396 @ X395) @ X394)))) | ~ ! [X398 : (subst > term > term)] : (~ ! [X400 : term,X401 : subst,X399 : subst] : (((X398 @ (comp @ X399 @ X401) @ (sub @ X400 @ X401))) = ((sub @ (X398 @ X399 @ X400) @ X401))) | ~ ! [X402 : term] : (($true = ((X387 @ id @ (X398 @ id @ X402) @ id))) | ~ ($true = ((X387 @ id @ X402 @ id)))) | ($true = ((X387 @ id @ (lam @ (X398 @ sh @ one)) @ id))))))) <=> (hoasinduction_lem3_lthm = $true)) & (induction2lem_gthm = $true) & (ulamvarind = $true) & ((axscons = $true) <=> ! [X403 : subst] : (((push @ (sub @ one @ X403) @ (comp @ sh @ X403))) = X403)) & (axapp = $true) & (pushprop_lem2v2_lthm = $true) & ~ (ulamvar1 = $true) & (hoasinduction_lem3b_gthm = $true) & (hoasinduction_lem3a_gthm = $true) & (pushprop_lem3v2_lthm = $true) & ((! [X404 : subst] : (((comp @ id @ X404)) = X404) => (! [X405 : subst] : (((comp @ X405 @ id)) = X405) => (! [X407 : subst] : (((comp @ id @ X407)) = X407) & ! [X406 : subst] : (((comp @ X406 @ id)) = X406)))) <=> (substmonoid_lthm = $true)) & ! [X409 : d_subst,X408 : d_subst] : ((((d2subst @ X409)) = ((d2subst @ X408))) => (X408 = X409)) & (pushprop_lem3v2 = $true) & (! [X410 : (subst > term > term),X412 : term,X411 : term] : (~ ! [X414 : subst,X413 : subst,X415 : term] : (((X410 @ (comp @ X414 @ X413) @ (sub @ X415 @ X413))) = ((sub @ (X410 @ X414 @ X415) @ X413))) | (((ap @ (sub @ X412 @ id) @ X411)) != ((lam @ (X410 @ sh @ one))))) <=> (hoaslamnotap = $true)) & (axvarshift = $true) & (substmonoid_gthm = $true) & (! [X416 : (subst > term > subst > $o)] : ~ ! [X417 : (term > $o)] : ~ ! [X418 : term] : (((X416 @ id @ X418 @ id)) = ((X417 @ X418))) <=> (hoasinduction_lem0 = $true)) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X419 : term] : ~ ! [X420 : (subst > term > term)] : ~ ! [X421 : subst,X422 : term] : (((X420 @ X421 @ X422)) = ((sub @ X419 @ (push @ X422 @ X421))))) & (hoasinduction_lem1_gthm = $true) & ! [X423 : term] : ? [X424 : d_term] : (((d2term @ X424)) = X423) & ((hoasinduction_lem3v2 = $true) <=> ! [X425 : term,X426 : (term > $o),X427 : (subst > term > subst > $o)] : (~ ! [X428 : term] : (($true = ((X426 @ (sub @ X425 @ (push @ X428 @ id))))) | ~ ($true = ((X426 @ X428)))) | ~ ! [X429 : subst,X430 : subst,X431 : subst,X432 : term] : (($true = ((X427 @ X430 @ X432 @ (comp @ X431 @ X429)))) | ~ ($true = ((X427 @ (comp @ X430 @ X431) @ (sub @ X432 @ X431) @ X429)))) | ~ ! [X433 : (subst > term > term)] : (($true = ((X427 @ id @ (lam @ (X433 @ sh @ one)) @ id))) | ~ ! [X434 : term] : (~ ($true = ((X427 @ id @ X434 @ id))) | ($true = ((X427 @ id @ (X433 @ id @ X434) @ id)))) | ~ ! [X437 : subst,X436 : term,X435 : subst] : (((X433 @ (comp @ X437 @ X435) @ (sub @ X436 @ X435))) = ((sub @ (X433 @ X437 @ X436) @ X435)))) | ~ ! [X438 : term] : (((X426 @ X438)) = ((X427 @ id @ X438 @ id))) | ~ ! [X439 : subst,X440 : subst,X441 : subst,X442 : term] : (~ ($true = ((X427 @ X440 @ X442 @ (comp @ X441 @ X439)))) | ($true = ((X427 @ (comp @ X440 @ X441) @ (sub @ X442 @ X441) @ X439)))) | ($true = ((X426 @ (lam @ X425)))))) & (induction2_gthm = $true) & (hoasapnotvar_gthm = $true) & (pushprop_lem1_gthm = $true) & (hoasinduction_lem1_lthm = $true) & ((axshiftcons = $true) <=> ! [X443 : subst,X444 : term] : (((comp @ sh @ (push @ X444 @ X443))) = X443)) & (! [X446 : term,X445 : term] : ((((lam @ X446)) != ((lam @ X445))) | (X445 = X446)) <=> (laminj = $true)) & ((! [X447 : term,X450 : term,X448 : term,X449 : term] : ((((ap @ X447 @ X449)) != ((ap @ X450 @ X448))) | (X448 = X449)) => ! [X453 : term,X454 : term,X451 : term,X452 : term] : ((((ap @ (sub @ X452 @ id) @ X451)) != ((ap @ (sub @ X454 @ id) @ X453))) | (X451 = X453))) <=> (hoasapinj2_lthm = $true)) & (! [X456 : term,X455 : term,X457 : term,X458 : term] : ((X455 = X458) | (((ap @ X458 @ X457)) != ((ap @ X455 @ X456)))) <=> (apinj1 = $true)) & (! [X461 : (subst > term > term),X460 : (subst > term > term),X459 : subst,X462 : term] : ((((lam @ (X461 @ sh @ one))) != ((lam @ (X460 @ sh @ one)))) | ~ ! [X463 : subst,X464 : term,X465 : subst] : (((X460 @ (comp @ X463 @ X465) @ (sub @ X464 @ X465))) = ((sub @ (X460 @ X463 @ X464) @ X465))) | ~ ! [X468 : subst,X466 : term,X467 : subst] : (((X461 @ (comp @ X468 @ X467) @ (sub @ X466 @ X467))) = ((sub @ (X461 @ X468 @ X466) @ X467))) | (((X460 @ X459 @ X462)) = ((X461 @ X459 @ X462)))) <=> (hoaslaminj = $true)) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((shinj = $true) <=> ! [X470 : term,X469 : term] : ((X469 = X470) | (((sub @ X470 @ sh)) != ((sub @ X469 @ sh))))) & (hoaslamnotap_lthm = $true) & ((hoasapinj2 = $true) <=> ! [X473 : term,X472 : term,X471 : term,X474 : term] : ((X472 = X473) | (((ap @ (sub @ X474 @ id) @ X472)) != ((ap @ (sub @ X471 @ id) @ X473))))) & ~ (((var @ (d2term @ d_one))) = $true) & (! [X475 : term] : (((sub @ X475 @ id)) = X475) <=> (termmset = $true)) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (axabs = $true) & ! [X476 : d_term] : (d_one = X476) & (lamnotvar = $true) & (hoasinduction_lem3v2_gthm = $true) & ((hoasinduction_lem3b = $true) <=> ! [X477 : term] : ~ ! [X478 : (subst > term > term)] : (((X478 @ sh @ one)) != ((sub @ X477 @ (push @ one @ sh))))) & ((pushprop_lem1v2_lthm = $true) <=> (! [X479 : term,X480 : subst] : (((sub @ one @ (push @ X479 @ X480))) = X479) => ! [X481 : term,X482 : (term > $o),X483 : subst,X484 : (term > $o)] : (~ ! [X485 : term] : (((X484 @ X485)) = ((X482 @ (sub @ X485 @ (push @ X481 @ X483))))) | ~ ($true = ((X482 @ X481))) | ($true = ((X484 @ one)))))) & (! [X486 : term] : ~ ! [X487 : (subst > term > term)] : (((sub @ X486 @ (push @ one @ sh))) != ((X487 @ sh @ one))) <=> (hoasinduction_lem3b_lthm = $true)) & (! [X488 : (subst > term > subst > $o),X489 : term] : ((((X488 @ id @ (lam @ (sub @ X489 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X490 : (subst > term > term),X491 : term] : (~ ! [X492 : subst,X494 : term,X493 : subst] : (((X490 @ (comp @ X492 @ X493) @ (sub @ X494 @ X493))) = ((sub @ X491 @ (push @ (sub @ X494 @ X493) @ (comp @ X492 @ X493))))) | ~ ! [X495 : term] : (($true = ((X488 @ id @ (X490 @ id @ X495) @ id))) | ~ ($true = ((X488 @ id @ X495 @ id)))) | ($true = ((X488 @ id @ (lam @ (X490 @ sh @ one)) @ id))) | ~ ! [X497 : subst,X498 : subst,X496 : term] : (((sub @ (X490 @ X498 @ X496) @ X497)) = ((sub @ (sub @ X491 @ (push @ X496 @ X498)) @ X497)))) | ~ ! [X499 : term] : (($true = ((X488 @ id @ (sub @ X489 @ (push @ X499 @ id)) @ id))) | ~ ($true = ((X488 @ id @ X499 @ id))))) <=> (hoasinduction_lem3aaa = $true)) & ((induction = $true) <=> ! [X500 : term,X501 : (term > $o)] : (~ ! [X502 : term] : (~ ($true = ((X501 @ X502))) | ($true = ((X501 @ (lam @ X502))))) | ~ ! [X503 : term] : (~ ($true = ((var @ X503))) | ($true = ((X501 @ X503)))) | ($true = ((X501 @ X500))) | ~ ! [X504 : term,X505 : term] : (~ ($true = ((X501 @ X504))) | ($true = ((X501 @ (ap @ X505 @ X504)))) | ~ ($true = ((X501 @ X505)))))) & ((pushprop_lem1_lthm = $true) <=> (! [X507 : subst,X506 : term] : (((sub @ one @ (push @ X506 @ X507))) = X506) => (! [X508 : subst,X509 : term] : (((comp @ sh @ (push @ X509 @ X508))) = X508) => ! [X510 : subst,X511 : term,X512 : (term > $o),X513 : (term > $o),X514 : term] : (~ ($true = ((X513 @ X511))) | ($true = ((X512 @ (sub @ X511 @ (push @ X514 @ X510))))))))) & (! [X515 : (subst > term > subst > $o),X516 : term] : (~ ! [X517 : term] : (($true = ((X515 @ id @ (sub @ X516 @ (push @ X517 @ id)) @ id))) | ~ ($true = ((X515 @ id @ X517 @ id)))) | ($true = ((X515 @ id @ (lam @ X516) @ id))) | ~ ! [X518 : (subst > term > term)] : (($true = ((X515 @ id @ (lam @ (X518 @ sh @ one)) @ id))) | ~ ! [X519 : term] : (~ ($true = ((X515 @ id @ X519 @ id))) | ($true = ((X515 @ id @ (X518 @ id @ X519) @ id)))) | ~ ! [X521 : subst,X520 : term,X522 : subst] : (((X518 @ (comp @ X521 @ X522) @ (sub @ X520 @ X522))) = ((sub @ (X518 @ X521 @ X520) @ X522))))) <=> (hoasinduction_lem3a = $true)) & (hoasinduction_lem2v2_gthm = $true) & (hoasinduction_lem2_lthm = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (! [X523 : term,X524 : (subst > term > subst > $o)] : (($true = ((X524 @ id @ (lam @ (sub @ X523 @ (push @ one @ sh))) @ id))) | ~ ! [X525 : (subst > term > term)] : (~ ! [X528 : term,X527 : subst,X526 : subst] : (((sub @ (X525 @ X527 @ X528) @ X526)) = ((X525 @ (comp @ X527 @ X526) @ (sub @ X528 @ X526)))) | ($true = ((X524 @ id @ (lam @ (X525 @ sh @ one)) @ id))) | ~ ! [X529 : term] : (~ ($true = ((X524 @ id @ X529 @ id))) | ($true = ((X524 @ id @ (X525 @ id @ X529) @ id))))) | ~ ! [X530 : term] : (~ ($true = ((X524 @ id @ X530 @ id))) | ($true = ((X524 @ id @ (sub @ X523 @ (push @ X530 @ id)) @ id))))) <=> (hoasinduction_lem3aa = $true)) & ((hoasinduction_no_psi_cond = $true) <=> ! [X531 : term,X532 : (subst > term > subst > $o)] : (($true = ((X532 @ id @ X531 @ id))) | ~ ! [X533 : (subst > term > term)] : (($true = ((X532 @ id @ (lam @ (X533 @ sh @ one)) @ id))) | ~ ! [X534 : term] : (~ ($true = ((X532 @ id @ X534 @ id))) | ($true = ((X532 @ id @ (X533 @ id @ X534) @ id)))) | ~ ! [X536 : term,X537 : subst,X535 : subst] : (((X533 @ (comp @ X535 @ X537) @ (sub @ X536 @ X537))) = ((sub @ (X533 @ X535 @ X536) @ X537)))) | ~ ! [X538 : term,X539 : term] : (~ ($true = ((X532 @ id @ X538 @ id))) | ~ ($true = ((X532 @ id @ X539 @ id))) | ($true = ((X532 @ id @ (ap @ (sub @ X539 @ id) @ X538) @ id)))))) & (sh = ((d2subst @ d_id))) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (hoasinduction_lem1v2_gthm = $true)),
  inference(rectify,[],[f7])).
thf(f10,plain,(
  (induction2lem_gthm = $true) & (! [X107 : subst,X108 : term,X106 : (term > $o)] : ~ ! [X109 : (term > $o)] : ~ ! [X110 : term] : (((X106 @ (sub @ X110 @ (push @ X108 @ X107)))) = ((X109 @ X110))) <=> (pushprop_lem0_lthm = $true)) & (pushprop_lem3v2_lthm = $true) & ((shinj = $true) <=> ! [X470 : term,X469 : term] : ((X469 = X470) | (((sub @ X470 @ sh)) != ((sub @ X469 @ sh))))) & (axassoc = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & ((hoasinduction = $true) <=> ! [X7 : (subst > term > subst > $o),X6 : term] : (($true = ((X7 @ id @ X6 @ id))) | ~ ! [X8 : term,X9 : term] : (($true != ((X7 @ id @ X8 @ id))) | (((X7 @ id @ X9 @ id)) != $true) | ($true = ((X7 @ id @ (ap @ (sub @ X9 @ id) @ X8) @ id)))) | ~ ! [X18 : term] : (($true != ((var @ (sub @ X18 @ id)))) | ($true = ((X7 @ id @ X18 @ id)))) | ~ ! [X14 : subst,X16 : subst,X15 : term,X17 : subst] : (($true != ((X7 @ (comp @ X14 @ X16) @ (sub @ X15 @ X16) @ X17))) | ($true = ((X7 @ X14 @ X15 @ (comp @ X16 @ X17))))) | ~ ! [X10 : subst,X13 : term,X11 : subst,X12 : subst] : (($true = ((X7 @ (comp @ X11 @ X12) @ (sub @ X13 @ X12) @ X10))) | ($true != ((X7 @ X11 @ X13 @ (comp @ X12 @ X10))))) | ~ ! [X19 : (subst > term > term)] : (~ ! [X23 : term] : ((((X7 @ id @ X23 @ id)) != $true) | ($true = ((X7 @ id @ (X19 @ id @ X23) @ id)))) | ~ ! [X22 : subst,X21 : term,X20 : subst] : (((sub @ (X19 @ X22 @ X21) @ X20)) = ((X19 @ (comp @ X22 @ X20) @ (sub @ X21 @ X20)))) | ($true = ((X7 @ id @ (lam @ (X19 @ sh @ one)) @ id)))))) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ((! [X447 : term,X450 : term,X448 : term,X449 : term] : ((((ap @ X447 @ X449)) != ((ap @ X450 @ X448))) | (X448 = X449)) => ! [X453 : term,X454 : term,X451 : term,X452 : term] : ((((ap @ (sub @ X452 @ id) @ X451)) != ((ap @ (sub @ X454 @ id) @ X453))) | (X451 = X453))) <=> (hoasapinj2_lthm = $true)) & (! [X501 : (term > $o),X500 : term] : (~ ! [X505 : term,X504 : term] : (($true != ((X501 @ X505))) | ($true != ((X501 @ X504))) | ($true = ((X501 @ (ap @ X505 @ X504))))) | ($true = ((X501 @ X500))) | ~ ! [X502 : term] : (($true != ((X501 @ X502))) | ($true = ((X501 @ (lam @ X502))))) | ~ ! [X503 : term] : (($true = ((X501 @ X503))) | ($true != ((var @ X503))))) <=> (induction = $true)) & (! [X486 : term] : ~ ! [X487 : (subst > term > term)] : (((sub @ X486 @ (push @ one @ sh))) != ((X487 @ sh @ one))) <=> (hoasinduction_lem3b_lthm = $true)) & ! [X409 : d_subst,X408 : d_subst] : ((((d2subst @ X409)) = ((d2subst @ X408))) => (X408 = X409)) & ((! [X236 : term] : (((sub @ X236 @ id)) = X236) => (! [X238 : term,X239 : term,X237 : term,X240 : term] : ((((ap @ X237 @ X239)) != ((ap @ X238 @ X240))) | (X237 = X238)) => ! [X242 : term,X244 : term,X241 : term,X243 : term] : ((X242 = X244) | (((ap @ (sub @ X244 @ id) @ X241)) != ((ap @ (sub @ X242 @ id) @ X243)))))) <=> (hoasapinj1_lthm = $true)) & ((! [X4 : subst] : (((comp @ id @ X4)) = X4) & ! [X3 : subst] : (((comp @ X3 @ id)) = X3)) <=> (substmonoid = $true)) & ((! [X148 : (subst > term > subst > $o)] : ~ ! [X149 : (term > $o)] : ~ ! [X150 : term] : (((X148 @ id @ X150 @ id)) = ((X149 @ X150))) => (! [X152 : term,X151 : (term > $o)] : (~ ! [X157 : term] : (($true != ((var @ X157))) | ($true = ((X151 @ X157)))) | ~ ! [X155 : term] : (~ ! [X156 : term] : (($true != ((X151 @ X156))) | ($true = ((X151 @ (sub @ X155 @ (push @ X156 @ id)))))) | ($true = ((X151 @ (lam @ X155))))) | ~ ! [X153 : term,X154 : term] : (($true = ((X151 @ (ap @ X154 @ X153)))) | ($true != ((X151 @ X154))) | ($true != ((X151 @ X153)))) | ($true = ((X151 @ X152)))) => (! [X158 : term] : (((sub @ X158 @ id)) = X158) => (! [X159 : (subst > term > subst > $o),X160 : (term > $o),X161 : term] : (~ ! [X164 : (subst > term > term)] : (($true = ((X159 @ id @ (lam @ (X164 @ sh @ one)) @ id))) | ~ ! [X165 : term] : (($true != ((X159 @ id @ X165 @ id))) | ($true = ((X159 @ id @ (X164 @ id @ X165) @ id)))) | ~ ! [X166 : subst,X168 : term,X167 : subst] : (((X164 @ (comp @ X166 @ X167) @ (sub @ X168 @ X167))) = ((sub @ (X164 @ X166 @ X168) @ X167)))) | ~ ! [X163 : term] : (($true != ((X160 @ X163))) | ($true = ((X160 @ (sub @ X161 @ (push @ X163 @ id)))))) | ($true = ((X160 @ (lam @ X161)))) | ~ ! [X162 : term] : (((X160 @ X162)) = ((X159 @ id @ X162 @ id)))) => ! [X169 : term,X170 : (subst > term > subst > $o)] : (~ ! [X171 : (subst > term > term)] : (~ ! [X172 : term] : ((((X170 @ id @ X172 @ id)) != $true) | ($true = ((X170 @ id @ (X171 @ id @ X172) @ id)))) | ~ ! [X173 : subst,X174 : subst,X175 : term] : (((sub @ (X171 @ X173 @ X175) @ X174)) = ((X171 @ (comp @ X173 @ X174) @ (sub @ X175 @ X174)))) | ($true = ((X170 @ id @ (lam @ (X171 @ sh @ one)) @ id)))) | ~ ! [X180 : term] : (($true = ((X170 @ id @ X180 @ id))) | ($true != ((var @ (sub @ X180 @ id))))) | ~ ! [X186 : term,X185 : term] : (($true = ((X170 @ id @ (ap @ (sub @ X186 @ id) @ X185) @ id))) | ($true != ((X170 @ id @ X186 @ id))) | ($true != ((X170 @ id @ X185 @ id)))) | ($true = ((X170 @ id @ X169 @ id))) | ~ ! [X183 : subst,X181 : term,X182 : subst,X184 : subst] : (($true = ((X170 @ (comp @ X184 @ X182) @ (sub @ X181 @ X182) @ X183))) | ($true != ((X170 @ X184 @ X181 @ (comp @ X182 @ X183))))) | ~ ! [X178 : subst,X177 : subst,X179 : term,X176 : subst] : (($true != ((X170 @ (comp @ X176 @ X177) @ (sub @ X179 @ X177) @ X178))) | ($true = ((X170 @ X176 @ X179 @ (comp @ X177 @ X178)))))))))) <=> (hoasinduction_lthm_3 = $true)) & (axmap = $true) & ((! [X130 : subst,X131 : term] : (((sub @ one @ (push @ X131 @ X130))) = X131) => (! [X133 : subst,X132 : term] : (((comp @ sh @ (push @ X132 @ X133))) = X133) => (! [X134 : term,X135 : term] : ((((lam @ X134)) != ((lam @ X135))) | (X134 = X135)) => ! [X136 : term,X138 : subst,X139 : (subst > term > term),X137 : (subst > term > term)] : ((((lam @ (X139 @ sh @ one))) != ((lam @ (X137 @ sh @ one)))) | (((X137 @ X138 @ X136)) = ((X139 @ X138 @ X136))) | ~ ! [X143 : term,X145 : subst,X144 : subst] : (((X137 @ (comp @ X145 @ X144) @ (sub @ X143 @ X144))) = ((sub @ (X137 @ X145 @ X143) @ X144))) | ~ ! [X142 : subst,X141 : term,X140 : subst] : (((X139 @ (comp @ X142 @ X140) @ (sub @ X141 @ X140))) = ((sub @ (X139 @ X142 @ X141) @ X140))))))) <=> (hoaslaminj_lthm = $true)) & (hoasapnotvar_lthm = $true) & (axclos = $true) & ((pushprop_lem1 = $true) <=> ! [X212 : subst,X216 : term,X215 : (term > $o),X213 : term,X214 : (term > $o)] : (($true = ((X215 @ (sub @ X213 @ (push @ X216 @ X212))))) | ($true != ((X214 @ X213))))) & (! [X80 : term,X83 : term,X82 : term,X81 : term] : ((X80 = X83) | (((ap @ X82 @ X83)) != ((ap @ X81 @ X80)))) <=> (apinj2 = $true)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoasinduction_lem3v2_lthm = $true) <=> (! [X217 : term] : (((sub @ X217 @ id)) = X217) => ! [X219 : (subst > term > subst > $o),X220 : (term > $o),X218 : term] : (($true = ((X220 @ (lam @ X218)))) | ~ ! [X231 : (subst > term > term)] : (~ ! [X234 : subst,X233 : subst,X232 : term] : (((sub @ (X231 @ X234 @ X232) @ X233)) = ((X231 @ (comp @ X234 @ X233) @ (sub @ X232 @ X233)))) | ($true = ((X219 @ id @ (lam @ (X231 @ sh @ one)) @ id))) | ~ ! [X235 : term] : (($true != ((X219 @ id @ X235 @ id))) | ($true = ((X219 @ id @ (X231 @ id @ X235) @ id))))) | ~ ! [X228 : subst,X226 : term,X225 : subst,X227 : subst] : (($true != ((X219 @ X225 @ X226 @ (comp @ X228 @ X227)))) | ($true = ((X219 @ (comp @ X225 @ X228) @ (sub @ X226 @ X228) @ X227)))) | ~ ! [X230 : term] : (((X219 @ id @ X230 @ id)) = ((X220 @ X230))) | ~ ! [X229 : term] : (($true != ((X220 @ X229))) | ($true = ((X220 @ (sub @ X218 @ (push @ X229 @ id)))))) | ~ ! [X222 : subst,X221 : subst,X223 : term,X224 : subst] : (($true = ((X219 @ X224 @ X223 @ (comp @ X222 @ X221)))) | ($true != ((X219 @ (comp @ X224 @ X222) @ (sub @ X223 @ X222) @ X221))))))) & (apnotvar = $true) & ((hoasinduction_lem3a_lthm = $true) <=> (! [X84 : term] : (((sub @ X84 @ id)) = X84) => (! [X85 : (subst > term > subst > $o),X86 : term] : (~ ! [X92 : term] : (($true = ((X85 @ id @ (sub @ X86 @ (push @ X92 @ id)) @ id))) | ($true != ((X85 @ id @ X92 @ id)))) | ~ ! [X87 : (subst > term > term)] : (($true = ((X85 @ id @ (lam @ (X87 @ sh @ one)) @ id))) | ~ ! [X90 : term,X89 : subst,X88 : subst] : (((X87 @ (comp @ X88 @ X89) @ (sub @ X90 @ X89))) = ((sub @ (X87 @ X88 @ X90) @ X89))) | ~ ! [X91 : term] : (($true != ((X85 @ id @ X91 @ id))) | ($true = ((X85 @ id @ (X87 @ id @ X91) @ id))))) | ($true = ((X85 @ id @ (lam @ (sub @ X86 @ (push @ one @ sh))) @ id)))) => ! [X93 : (subst > term > subst > $o),X94 : term] : (~ ! [X95 : term] : (($true = ((X93 @ id @ (sub @ X94 @ (push @ X95 @ id)) @ id))) | ($true != ((X93 @ id @ X95 @ id)))) | ~ ! [X96 : (subst > term > term)] : (($true = ((X93 @ id @ (lam @ (X96 @ sh @ one)) @ id))) | ~ ! [X97 : term] : (($true != ((X93 @ id @ X97 @ id))) | ($true = ((X93 @ id @ (X96 @ id @ X97) @ id)))) | ~ ! [X99 : term,X100 : subst,X98 : subst] : (((sub @ (X96 @ X100 @ X99) @ X98)) = ((X96 @ (comp @ X100 @ X98) @ (sub @ X99 @ X98))))) | ($true = ((X93 @ id @ (lam @ X94) @ id))))))) & (one = ((d2term @ d_one))) & (! [X475 : term] : (((sub @ X475 @ id)) = X475) <=> (termmset = $true)) & ((! [X41 : subst,X42 : term] : (((sub @ one @ (push @ X42 @ X41))) = X42) => (! [X44 : term,X43 : subst] : (((comp @ sh @ (push @ X44 @ X43))) = X43) => (! [X45 : subst] : (((comp @ X45 @ id)) = X45) => (! [X46 : (term > $o),X47 : term] : (~ ! [X51 : term] : (($true != ((X46 @ X51))) | ($true = ((X46 @ (lam @ X51))))) | ($true = ((X46 @ X47))) | ~ ! [X48 : term] : (($true = ((X46 @ X48))) | ($true != ((var @ X48)))) | ~ ! [X50 : term,X49 : term] : (($true != ((X46 @ X49))) | ($true != ((X46 @ X50))) | ($true = ((X46 @ (ap @ X49 @ X50)))))) => ! [X52 : (term > $o),X53 : subst,X54 : term] : (($true = ((X52 @ (sub @ X54 @ X53)))) | ~ ! [X59 : term] : (($true = ((X52 @ (sub @ X59 @ X53)))) | ($true != ((var @ X59)))) | ~ ! [X57 : term] : (($true = ((X52 @ (lam @ X57)))) | ~ ! [X58 : term] : (($true != ((X52 @ X58))) | ($true = ((X52 @ (sub @ X57 @ (push @ X58 @ id))))))) | ~ ! [X56 : term,X55 : term] : ((((X52 @ X55)) != $true) | ($true = ((X52 @ (ap @ X55 @ X56)))) | ($true != ((X52 @ X56))))))))) <=> (induction2lem_lthm = $true)) & (! [X0 : subst] : (((comp @ X0 @ id)) = X0) <=> (axidr = $true)) & ((pushprop_lem0 = $true) <=> ! [X103 : term,X101 : subst,X102 : (term > $o)] : ~ ! [X104 : (term > $o)] : ~ ! [X105 : term] : (((X104 @ X105)) = ((X102 @ (sub @ X105 @ (push @ X103 @ X101)))))) & (hoaslamnotvar_gthm = $true) & ((hoasapinj2 = $true) <=> ! [X473 : term,X472 : term,X471 : term,X474 : term] : ((X472 = X473) | (((ap @ (sub @ X474 @ id) @ X472)) != ((ap @ (sub @ X471 @ id) @ X473))))) & (hoasapinj1_gthm = $true) & (axabs = $true) & (! [X374 : term] : ~ ! [X375 : (subst > term > term)] : ~ ! [X376 : subst,X377 : term] : (((sub @ X374 @ (push @ X377 @ X376))) = ((X375 @ X376 @ X377))) <=> (hoasinduction_lem3v2_f = $true)) & (! [X117 : term,X118 : term,X119 : (subst > term > subst > $o)] : (($true != ((X119 @ id @ X117 @ id))) | ~ ! [X121 : term,X120 : term] : (($true != ((X119 @ id @ X121 @ id))) | ($true != ((X119 @ id @ X120 @ id))) | ($true = ((X119 @ id @ (ap @ (sub @ X120 @ id) @ X121) @ id)))) | ~ ! [X125 : subst,X123 : subst,X124 : subst,X122 : term] : (($true = ((X119 @ X123 @ X122 @ (comp @ X125 @ X124)))) | ($true != ((X119 @ (comp @ X123 @ X125) @ (sub @ X122 @ X125) @ X124)))) | ($true = ((X119 @ id @ (ap @ X118 @ X117) @ id))) | ~ ! [X128 : subst,X127 : term,X126 : subst,X129 : subst] : (($true = ((X119 @ (comp @ X126 @ X128) @ (sub @ X127 @ X128) @ X129))) | ($true != ((X119 @ X126 @ X127 @ (comp @ X128 @ X129))))) | ($true != ((X119 @ id @ X118 @ id)))) <=> (hoasinduction_lem2 = $true)) & (hoasinduction_lem1 = $true) & (axapp = $true) & ((hoasinduction_lthm = $true) <=> (! [X245 : term,X246 : (term > $o)] : (~ ! [X250 : term,X251 : term] : (($true = ((X246 @ (ap @ X250 @ X251)))) | ($true != ((X246 @ X250))) | ($true != ((X246 @ X251)))) | ~ ! [X248 : term] : (($true = ((X246 @ (lam @ X248)))) | ~ ! [X249 : term] : (($true = ((X246 @ (sub @ X248 @ (push @ X249 @ id))))) | ($true != ((X246 @ X249))))) | ($true = ((X246 @ X245))) | ~ ! [X247 : term] : (($true != ((var @ X247))) | ($true = ((X246 @ X247))))) => (! [X252 : (subst > term > subst > $o),X254 : term,X253 : term] : (~ ! [X259 : subst,X261 : subst,X260 : term,X262 : subst] : (($true != ((X252 @ X259 @ X260 @ (comp @ X262 @ X261)))) | ($true = ((X252 @ (comp @ X259 @ X262) @ (sub @ X260 @ X262) @ X261)))) | ($true = ((X252 @ id @ (ap @ X253 @ X254) @ id))) | ~ ! [X263 : term,X264 : term] : (($true != ((X252 @ id @ X264 @ id))) | ($true = ((X252 @ id @ (ap @ (sub @ X264 @ id) @ X263) @ id))) | ($true != ((X252 @ id @ X263 @ id)))) | ($true != ((X252 @ id @ X253 @ id))) | ($true != ((X252 @ id @ X254 @ id))) | ~ ! [X256 : subst,X255 : subst,X258 : term,X257 : subst] : (($true = ((X252 @ X256 @ X258 @ (comp @ X255 @ X257)))) | ($true != ((X252 @ (comp @ X256 @ X255) @ (sub @ X258 @ X255) @ X257))))) => (! [X265 : term,X266 : (subst > term > subst > $o)] : (~ ! [X276 : (subst > term > term)] : (~ ! [X277 : term] : (($true != ((X266 @ id @ X277 @ id))) | ($true = ((X266 @ id @ (X276 @ id @ X277) @ id)))) | ~ ! [X278 : subst,X279 : subst,X280 : term] : (((sub @ (X276 @ X278 @ X280) @ X279)) = ((X276 @ (comp @ X278 @ X279) @ (sub @ X280 @ X279)))) | ($true = ((X266 @ id @ (lam @ (X276 @ sh @ one)) @ id)))) | ~ ! [X275 : term] : (($true != ((X266 @ id @ X275 @ id))) | ($true = ((X266 @ id @ (sub @ X265 @ (push @ X275 @ id)) @ id)))) | ($true = ((X266 @ id @ (lam @ X265) @ id))) | ~ ! [X271 : subst,X272 : subst,X274 : term,X273 : subst] : (($true = ((X266 @ (comp @ X272 @ X273) @ (sub @ X274 @ X273) @ X271))) | ($true != ((X266 @ X272 @ X274 @ (comp @ X273 @ X271))))) | ~ ! [X270 : subst,X269 : subst,X268 : subst,X267 : term] : (($true = ((X266 @ X269 @ X267 @ (comp @ X268 @ X270)))) | ($true != ((X266 @ (comp @ X269 @ X268) @ (sub @ X267 @ X268) @ X270))))) => ! [X281 : (subst > term > subst > $o),X282 : term] : (~ ! [X298 : term,X297 : term] : (($true != ((X281 @ id @ X298 @ id))) | ($true = ((X281 @ id @ (ap @ (sub @ X297 @ id) @ X298) @ id))) | ($true != ((X281 @ id @ X297 @ id)))) | ~ ! [X293 : subst,X292 : subst,X294 : term,X295 : subst] : (($true = ((X281 @ X292 @ X294 @ (comp @ X295 @ X293)))) | ($true != ((X281 @ (comp @ X292 @ X295) @ (sub @ X294 @ X295) @ X293)))) | ~ ! [X296 : term] : (($true = ((X281 @ id @ X296 @ id))) | ($true != ((var @ (sub @ X296 @ id))))) | ($true = ((X281 @ id @ X282 @ id))) | ~ ! [X288 : subst,X291 : subst,X289 : subst,X290 : term] : (($true = ((X281 @ (comp @ X288 @ X289) @ (sub @ X290 @ X289) @ X291))) | ($true != ((X281 @ X288 @ X290 @ (comp @ X289 @ X291))))) | ~ ! [X283 : (subst > term > term)] : (~ ! [X285 : term,X286 : subst,X287 : subst] : (((sub @ (X283 @ X286 @ X285) @ X287)) = ((X283 @ (comp @ X286 @ X287) @ (sub @ X285 @ X287)))) | ~ ! [X284 : term] : (($true != ((X281 @ id @ X284 @ id))) | ($true = ((X281 @ id @ (X283 @ id @ X284) @ id)))) | ($true = ((X281 @ id @ (lam @ (X283 @ sh @ one)) @ id))))))))) & ((! [X24 : term] : (((sub @ X24 @ id)) = X24) => (! [X26 : term,X25 : subst,X27 : (term > $o)] : (~ ! [X28 : term] : (~ ! [X29 : term] : (($true != ((X27 @ X29))) | ($true = ((X27 @ (sub @ X28 @ (push @ X29 @ id)))))) | ($true = ((X27 @ (lam @ X28))))) | ~ ! [X30 : term,X31 : term] : (($true != ((X27 @ X31))) | ($true = ((X27 @ (ap @ X30 @ X31)))) | ($true != ((X27 @ X30)))) | ~ ! [X32 : term] : (($true != ((var @ X32))) | ($true = ((X27 @ (sub @ X32 @ X25))))) | ($true = ((X27 @ (sub @ X26 @ X25))))) => ! [X33 : (term > $o),X34 : term] : (($true = ((X33 @ X34))) | ~ ! [X35 : term] : (($true = ((X33 @ (lam @ X35)))) | ~ ! [X36 : term] : (($true != ((X33 @ X36))) | ($true = ((X33 @ (sub @ X35 @ (push @ X36 @ id))))))) | ~ ! [X38 : term,X37 : term] : (($true != ((X33 @ X37))) | ($true = ((X33 @ (ap @ X38 @ X37)))) | ($true != ((X33 @ X38)))) | ~ ! [X39 : term] : (($true != ((var @ X39))) | ($true = ((X33 @ X39))))))) <=> (induction2_lthm = $true)) & (hoaslaminj_gthm = $true) & (hoasinduction_lem1v2 = $true) & (hoasinduction_lem1_gthm = $true) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X419 : term] : ~ ! [X420 : (subst > term > term)] : ~ ! [X421 : subst,X422 : term] : (((X420 @ X421 @ X422)) = ((sub @ X419 @ (push @ X422 @ X421))))) & (hoaslamnotap_lthm = $true) & (! [X446 : term,X445 : term] : ((((lam @ X446)) != ((lam @ X445))) | (X445 = X446)) <=> (laminj = $true)) & (lamnotvar = $true) & (termmset_gthm = $true) & (pushprop_lthm = $true) & (ulamvarsh = $true) & (hoasapinj2_gthm = $true) & (hoasinduction_lem3v2_gthm = $true) & (pushprop_gthm = $true) & (! [X344 : (subst > term > subst > $o)] : ~ ! [X345 : (term > $o)] : ~ ! [X346 : term] : (((X345 @ X346)) = ((X344 @ id @ X346 @ id))) <=> (hoasinduction_lem0_lthm = $true)) & (pushprop_lthm_orig = $true) & ((pushprop_lem1_lthm = $true) <=> (! [X507 : subst,X506 : term] : (((sub @ one @ (push @ X506 @ X507))) = X506) => (! [X508 : subst,X509 : term] : (((comp @ sh @ (push @ X509 @ X508))) = X508) => ! [X511 : term,X514 : term,X510 : subst,X512 : (term > $o),X513 : (term > $o)] : (($true = ((X512 @ (sub @ X511 @ (push @ X514 @ X510))))) | ($true != ((X513 @ X511))))))) & (hoasapnotvar_gthm = $true) & (! [X188 : term,X190 : (term > $o),X187 : term,X189 : (subst > term > subst > $o)] : (~ ! [X191 : term] : (((X190 @ X191)) = ((X189 @ id @ X191 @ id))) | ~ ! [X198 : term,X196 : subst,X199 : subst,X197 : subst] : (($true = ((X189 @ (comp @ X199 @ X196) @ (sub @ X198 @ X196) @ X197))) | ($true != ((X189 @ X199 @ X198 @ (comp @ X196 @ X197))))) | ($true = ((X190 @ (ap @ X188 @ X187)))) | ~ ! [X195 : term,X193 : subst,X192 : subst,X194 : subst] : ((((X189 @ X194 @ X195 @ (comp @ X193 @ X192))) = $true) | ($true != ((X189 @ (comp @ X194 @ X193) @ (sub @ X195 @ X193) @ X192)))) | ~ ! [X200 : term,X201 : term] : (($true != ((X189 @ id @ X200 @ id))) | ($true = ((X189 @ id @ (ap @ (sub @ X200 @ id) @ X201) @ id))) | ($true != ((X189 @ id @ X201 @ id)))) | ($true != ((X190 @ X187))) | ($true != ((X190 @ X188)))) <=> (hoasinduction_lem2v2 = $true)) & (pushprop_lem2v2_lthm = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (lamnotap != $true) & (induction2_gthm = $true) & (ulamvar1 != $true) & (! [X113 : term,X114 : subst,X112 : (term > $o),X111 : (term > $o)] : (($true = ((X112 @ one))) | ($true != ((X111 @ X113))) | ~ ! [X115 : term] : (((X112 @ X115)) = ((X111 @ (sub @ X115 @ (push @ X113 @ X114)))))) <=> (pushprop_lem1v2 = $true)) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (ulamvarind = $true) & (hoasapnotvar = $true) & (hoasinduction_lem1_lthm = $true) & (! [X426 : (term > $o),X427 : (subst > term > subst > $o),X425 : term] : (~ ! [X433 : (subst > term > term)] : (($true = ((X427 @ id @ (lam @ (X433 @ sh @ one)) @ id))) | ~ ! [X434 : term] : (($true != ((X427 @ id @ X434 @ id))) | ($true = ((X427 @ id @ (X433 @ id @ X434) @ id)))) | ~ ! [X437 : subst,X436 : term,X435 : subst] : (((X433 @ (comp @ X437 @ X435) @ (sub @ X436 @ X435))) = ((sub @ (X433 @ X437 @ X436) @ X435)))) | ~ ! [X432 : term,X429 : subst,X430 : subst,X431 : subst] : (($true = ((X427 @ X430 @ X432 @ (comp @ X431 @ X429)))) | ($true != ((X427 @ (comp @ X430 @ X431) @ (sub @ X432 @ X431) @ X429)))) | ($true = ((X426 @ (lam @ X425)))) | ~ ! [X428 : term] : (($true != ((X426 @ X428))) | ($true = ((X426 @ (sub @ X425 @ (push @ X428 @ id)))))) | ~ ! [X438 : term] : (((X426 @ X438)) = ((X427 @ id @ X438 @ id))) | ~ ! [X441 : subst,X439 : subst,X440 : subst,X442 : term] : (($true != ((X427 @ X440 @ X442 @ (comp @ X441 @ X439)))) | ($true = ((X427 @ (comp @ X440 @ X441) @ (sub @ X442 @ X441) @ X439))))) <=> (hoasinduction_lem3v2 = $true)) & (pushprop_lem1v2_gthm = $true) & (axvarshift = $true) & (pushprop_lem0_gthm = $true) & (pushprop = $true) & (! [X360 : term,X359 : (term > $o)] : (~ ! [X363 : term] : (($true = ((X359 @ X363))) | ($true != ((var @ X363)))) | ($true = ((X359 @ X360))) | ~ ! [X361 : term] : (($true = ((X359 @ (lam @ X361)))) | ~ ! [X362 : term] : (($true != ((X359 @ X362))) | ($true = ((X359 @ (sub @ X361 @ (push @ X362 @ id))))))) | ~ ! [X364 : term,X365 : term] : (($true != ((X359 @ X364))) | ($true = ((X359 @ (ap @ X365 @ X364)))) | ($true != ((X359 @ X365))))) <=> (induction2 = $true)) & ((hoasinduction_lem3a = $true) <=> ! [X516 : term,X515 : (subst > term > subst > $o)] : (~ ! [X517 : term] : (($true != ((X515 @ id @ X517 @ id))) | ($true = ((X515 @ id @ (sub @ X516 @ (push @ X517 @ id)) @ id)))) | ~ ! [X518 : (subst > term > term)] : (~ ! [X521 : subst,X520 : term,X522 : subst] : (((X518 @ (comp @ X521 @ X522) @ (sub @ X520 @ X522))) = ((sub @ (X518 @ X521 @ X520) @ X522))) | ($true = ((X515 @ id @ (lam @ (X518 @ sh @ one)) @ id))) | ~ ! [X519 : term] : (($true != ((X515 @ id @ X519 @ id))) | ($true = ((X515 @ id @ (X518 @ id @ X519) @ id))))) | ($true = ((X515 @ id @ (lam @ X516) @ id))))) & ((! [X378 : term] : (((sub @ X378 @ id)) = X378) => (! [X380 : (subst > term > subst > $o),X379 : term] : (~ ! [X386 : term] : (($true = ((X380 @ id @ (sub @ X379 @ (push @ X386 @ id)) @ id))) | ($true != ((X380 @ id @ X386 @ id)))) | ~ ! [X381 : (subst > term > term)] : (~ ! [X383 : subst,X384 : term,X385 : subst] : (((X381 @ (comp @ X383 @ X385) @ (sub @ X384 @ X385))) = ((sub @ (X381 @ X383 @ X384) @ X385))) | ~ ! [X382 : term] : (($true = ((X380 @ id @ (X381 @ id @ X382) @ id))) | ($true != ((X380 @ id @ X382 @ id)))) | ($true = ((X380 @ id @ (lam @ (X381 @ sh @ one)) @ id)))) | ($true = ((X380 @ id @ (lam @ (sub @ X379 @ (push @ one @ sh))) @ id)))) => ! [X388 : term,X387 : (subst > term > subst > $o)] : (~ ! [X397 : subst,X394 : subst,X395 : subst,X396 : term] : (($true != ((X387 @ (comp @ X397 @ X395) @ (sub @ X396 @ X395) @ X394))) | ($true = ((X387 @ X397 @ X396 @ (comp @ X395 @ X394))))) | ~ ! [X393 : term] : (($true = ((X387 @ id @ (sub @ X388 @ (push @ X393 @ id)) @ id))) | ($true != ((X387 @ id @ X393 @ id)))) | ~ ! [X398 : (subst > term > term)] : (($true = ((X387 @ id @ (lam @ (X398 @ sh @ one)) @ id))) | ~ ! [X402 : term] : (($true = ((X387 @ id @ (X398 @ id @ X402) @ id))) | ($true != ((X387 @ id @ X402 @ id)))) | ~ ! [X400 : term,X401 : subst,X399 : subst] : (((X398 @ (comp @ X399 @ X401) @ (sub @ X400 @ X401))) = ((sub @ (X398 @ X399 @ X400) @ X401)))) | ($true = ((X387 @ id @ (lam @ X388) @ id))) | ~ ! [X389 : subst,X392 : subst,X391 : term,X390 : subst] : (($true != ((X387 @ X392 @ X391 @ (comp @ X389 @ X390)))) | ($true = ((X387 @ (comp @ X392 @ X389) @ (sub @ X391 @ X389) @ X390))))))) <=> (hoasinduction_lem3_lthm = $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (! [X456 : term,X455 : term,X457 : term,X458 : term] : ((X455 = X458) | (((ap @ X458 @ X457)) != ((ap @ X455 @ X456)))) <=> (apinj1 = $true)) & ((hoasinduction_lem3b = $true) <=> ! [X477 : term] : ~ ! [X478 : (subst > term > term)] : (((X478 @ sh @ one)) != ((sub @ X477 @ (push @ one @ sh))))) & ! [X5 : d_subst] : (d_id = X5) & (! [X203 : subst,X202 : (term > $o),X204 : term] : (($true = ((X202 @ (sub @ X204 @ X203)))) | ~ ! [X209 : term] : (($true != ((var @ X209))) | ($true = ((X202 @ (sub @ X209 @ X203))))) | ~ ! [X205 : term] : (~ ! [X206 : term] : (($true != ((X202 @ X206))) | ($true = ((X202 @ (sub @ X205 @ (push @ X206 @ id)))))) | ($true = ((X202 @ (lam @ X205))))) | ~ ! [X207 : term,X208 : term] : (($true != ((X202 @ X208))) | ($true = ((X202 @ (ap @ X207 @ X208)))) | ($true != ((X202 @ X207))))) <=> (induction2lem = $true)) & (hoasinduction_lem3b_gthm = $true) & (pushprop_lem3v2 = $true) & (hoasinduction_lem3_gthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (! [X348 : term,X347 : subst] : (((sub @ one @ (push @ X348 @ X347))) = X348) <=> (axvarcons = $true)) & (substmonoid_gthm = $true) & ((! [X1 : term] : (((sub @ X1 @ id)) = X1) => ! [X2 : term] : (((sub @ X2 @ id)) = X2)) <=> (termmset_lthm = $true)) & (! [X416 : (subst > term > subst > $o)] : ~ ! [X417 : (term > $o)] : ~ ! [X418 : term] : (((X416 @ id @ X418 @ id)) = ((X417 @ X418))) <=> (hoasinduction_lem0 = $true)) & ((axshiftcons = $true) <=> ! [X443 : subst,X444 : term] : (((comp @ sh @ (push @ X444 @ X443))) = X443)) & ((pushprop_lem1v2_lthm = $true) <=> (! [X479 : term,X480 : subst] : (((sub @ one @ (push @ X479 @ X480))) = X479) => ! [X483 : subst,X482 : (term > $o),X481 : term,X484 : (term > $o)] : (~ ! [X485 : term] : (((X484 @ X485)) = ((X482 @ (sub @ X485 @ (push @ X481 @ X483))))) | ($true != ((X482 @ X481))) | ($true = ((X484 @ one)))))) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (! [X410 : (subst > term > term),X412 : term,X411 : term] : (~ ! [X414 : subst,X413 : subst,X415 : term] : (((X410 @ (comp @ X414 @ X413) @ (sub @ X415 @ X413))) = ((sub @ (X410 @ X414 @ X415) @ X413))) | (((ap @ (sub @ X412 @ id) @ X411)) != ((lam @ (X410 @ sh @ one))))) <=> (hoaslamnotap = $true)) & (hoaslamnotvar_lthm = $true) & ((hoasinduction_no_psi_cond = $true) <=> ! [X532 : (subst > term > subst > $o),X531 : term] : (($true = ((X532 @ id @ X531 @ id))) | ~ ! [X538 : term,X539 : term] : (($true != ((X532 @ id @ X538 @ id))) | ($true != ((X532 @ id @ X539 @ id))) | ($true = ((X532 @ id @ (ap @ (sub @ X539 @ id) @ X538) @ id)))) | ~ ! [X533 : (subst > term > term)] : (($true = ((X532 @ id @ (lam @ (X533 @ sh @ one)) @ id))) | ~ ! [X534 : term] : (($true = ((X532 @ id @ (X533 @ id @ X534) @ id))) | ($true != ((X532 @ id @ X534 @ id)))) | ~ ! [X536 : term,X537 : subst,X535 : subst] : (((X533 @ (comp @ X535 @ X537) @ (sub @ X536 @ X537))) = ((sub @ (X533 @ X535 @ X536) @ X537)))))) & (pushprop_lem2v2_gthm = $true) & ((! [X314 : (subst > term > subst > $o)] : ~ ! [X315 : (term > $o)] : ~ ! [X316 : term] : (((X314 @ id @ X316 @ id)) = ((X315 @ X316))) => (! [X317 : (term > $o),X318 : term] : (~ ! [X323 : term] : (($true != ((var @ X323))) | ($true = ((X317 @ X323)))) | ($true = ((X317 @ X318))) | ~ ! [X319 : term] : (($true = ((X317 @ (lam @ X319)))) | ~ ! [X320 : term] : (($true != ((X317 @ X320))) | ($true = ((X317 @ (sub @ X319 @ (push @ X320 @ id))))))) | ~ ! [X322 : term,X321 : term] : (($true = ((X317 @ (ap @ X321 @ X322)))) | ($true != ((X317 @ X322))) | ($true != ((X317 @ X321))))) => (! [X324 : term] : (((sub @ X324 @ id)) = X324) => (! [X327 : term,X325 : (term > $o),X326 : (subst > term > subst > $o)] : (~ ! [X329 : term] : (((X325 @ X329)) = ((X326 @ id @ X329 @ id))) | ~ ! [X328 : term] : (($true != ((X325 @ X328))) | ($true = ((X325 @ (sub @ X327 @ (push @ X328 @ id)))))) | ($true = ((X325 @ (lam @ X327)))) | ~ ! [X330 : (subst > term > term)] : (~ ! [X332 : subst,X331 : subst,X333 : term] : (((X330 @ (comp @ X332 @ X331) @ (sub @ X333 @ X331))) = ((sub @ (X330 @ X332 @ X333) @ X331))) | ~ ! [X334 : term] : (($true = ((X326 @ id @ (X330 @ id @ X334) @ id))) | ($true != ((X326 @ id @ X334 @ id)))) | ($true = ((X326 @ id @ (lam @ (X330 @ sh @ one)) @ id))))) => ! [X335 : term,X336 : (subst > term > subst > $o)] : (~ ! [X339 : (subst > term > term)] : (~ ! [X343 : subst,X342 : subst,X341 : term] : (((sub @ (X339 @ X342 @ X341) @ X343)) = ((X339 @ (comp @ X342 @ X343) @ (sub @ X341 @ X343)))) | ~ ! [X340 : term] : (($true != ((X336 @ id @ X340 @ id))) | ($true = ((X336 @ id @ (X339 @ id @ X340) @ id)))) | ($true = ((X336 @ id @ (lam @ (X339 @ sh @ one)) @ id)))) | ($true = ((X336 @ id @ X335 @ id))) | ~ ! [X337 : term,X338 : term] : (($true = ((X336 @ id @ (ap @ (sub @ X337 @ id) @ X338) @ id))) | (((X336 @ id @ X337 @ id)) != $true) | ($true != ((X336 @ id @ X338 @ id))))))))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (((var @ (d2term @ d_one))) != $true) & ((hoasinduction_lem3v2a = $true) <=> ! [X349 : (subst > term > subst > $o),X351 : term,X350 : (term > $o)] : (~ ! [X352 : (subst > term > term)] : (~ ! [X353 : term] : (($true != ((X349 @ id @ X353 @ id))) | ($true = ((X349 @ id @ (X352 @ id @ X353) @ id)))) | ($true = ((X349 @ id @ (lam @ (X352 @ sh @ one)) @ id))) | ~ ! [X356 : subst,X355 : subst,X354 : term] : (((X352 @ (comp @ X356 @ X355) @ (sub @ X354 @ X355))) = ((sub @ (X352 @ X356 @ X354) @ X355)))) | ($true = ((X350 @ (lam @ X351)))) | ~ ! [X358 : term] : (($true != ((X350 @ X358))) | ($true = ((X350 @ (sub @ X351 @ (push @ X358 @ id)))))) | ~ ! [X357 : term] : (((X350 @ X357)) = ((X349 @ id @ X357 @ id))))) & ((hoasinduction_lem3aaa = $true) <=> ! [X489 : term,X488 : (subst > term > subst > $o)] : ((((X488 @ id @ (lam @ (sub @ X489 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X499 : term] : (($true = ((X488 @ id @ (sub @ X489 @ (push @ X499 @ id)) @ id))) | ($true != ((X488 @ id @ X499 @ id)))) | ~ ! [X491 : term,X490 : (subst > term > term)] : (~ ! [X497 : subst,X498 : subst,X496 : term] : (((sub @ (X490 @ X498 @ X496) @ X497)) = ((sub @ (sub @ X491 @ (push @ X496 @ X498)) @ X497))) | ~ ! [X492 : subst,X494 : term,X493 : subst] : (((X490 @ (comp @ X492 @ X493) @ (sub @ X494 @ X493))) = ((sub @ X491 @ (push @ (sub @ X494 @ X493) @ (comp @ X492 @ X493))))) | ($true = ((X488 @ id @ (lam @ (X490 @ sh @ one)) @ id))) | ~ ! [X495 : term] : (($true = ((X488 @ id @ (X490 @ id @ X495) @ id))) | ($true != ((X488 @ id @ X495 @ id))))))) & (! [X63 : term,X60 : term,X61 : term,X62 : term] : ((X60 = X61) | (((ap @ (sub @ X61 @ id) @ X62)) != ((ap @ (sub @ X60 @ id) @ X63)))) <=> (hoasapinj1 = $true)) & ! [X210 : d_term,X211 : d_term] : ((((d2term @ X210)) = ((d2term @ X211))) => (X210 = X211)) & (hoasinduction_lem2_lthm = $true) & (! [X366 : (subst > term > subst > $o),X367 : term] : (($true = ((X366 @ id @ (lam @ (sub @ X367 @ (push @ one @ sh))) @ id))) | ~ ! [X368 : (subst > term > term)] : (~ ! [X372 : term] : (($true != ((X366 @ id @ X372 @ id))) | ($true = ((X366 @ id @ (X368 @ id @ X372) @ id)))) | ($true = ((X366 @ id @ (lam @ (X368 @ sh @ one)) @ id))) | ~ ! [X370 : subst,X371 : term,X369 : subst] : (((X368 @ (comp @ X370 @ X369) @ (sub @ X371 @ X369))) = ((sub @ (X368 @ X370 @ X371) @ X369)))) | ~ ! [X373 : term] : (($true = ((X366 @ id @ (sub @ X367 @ (push @ X373 @ id)) @ id))) | ($true != ((X366 @ id @ X373 @ id))))) <=> (hoasinduction_lem3aa_lthm = $true)) & (hoasinduction_lem1v2_gthm = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (! [X40 : subst] : (((comp @ id @ X40)) = X40) <=> (axidl = $true)) & (hoasinduction_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (sh = ((d2subst @ d_id))) & ! [X146 : subst] : ? [X147 : d_subst] : (((d2subst @ X147)) = X146) & ((axscons = $true) <=> ! [X403 : subst] : (((push @ (sub @ one @ X403) @ (comp @ sh @ X403))) = X403)) & ! [X476 : d_term] : (d_one = X476) & (hoasinduction_lem2_gthm = $true) & (! [X116 : term] : (((sub @ X116 @ id)) = X116) <=> (axvarid = $true)) & (hoasinduction_lem2v2_gthm = $true) & ((! [X404 : subst] : (((comp @ id @ X404)) = X404) => (! [X405 : subst] : (((comp @ X405 @ id)) = X405) => (! [X407 : subst] : (((comp @ id @ X407)) = X407) & ! [X406 : subst] : (((comp @ X406 @ id)) = X406)))) <=> (substmonoid_lthm = $true)) & ((! [X299 : term] : ~ ! [X300 : (subst > term > term)] : ~ ! [X302 : term,X301 : subst] : (((X300 @ X301 @ X302)) = ((sub @ X299 @ (push @ X302 @ X301)))) => (! [X303 : term] : (((sub @ X303 @ id)) = X303) => ! [X304 : term,X306 : (subst > term > subst > $o),X305 : (term > $o)] : (~ ! [X312 : term] : (((X305 @ X312)) = ((X306 @ id @ X312 @ id))) | ~ ! [X313 : term] : (($true != ((X305 @ X313))) | ($true = ((X305 @ (sub @ X304 @ (push @ X313 @ id)))))) | ~ ! [X307 : (subst > term > term)] : (($true = ((X306 @ id @ (lam @ (X307 @ sh @ one)) @ id))) | ~ ! [X311 : term] : (($true = ((X306 @ id @ (X307 @ id @ X311) @ id))) | ($true != ((X306 @ id @ X311 @ id)))) | ~ ! [X309 : subst,X308 : term,X310 : subst] : (((X307 @ (comp @ X309 @ X310) @ (sub @ X308 @ X310))) = ((sub @ (X307 @ X309 @ X308) @ X310)))) | ($true = ((X305 @ (lam @ X304))))))) <=> (hoasinduction_lem3v2a_lthm = $true)) & ((hoasinduction_lem3 = $true) <=> ! [X64 : term,X65 : (subst > term > subst > $o)] : (~ ! [X67 : term,X68 : subst,X69 : subst,X66 : subst] : (($true = ((X65 @ (comp @ X66 @ X69) @ (sub @ X67 @ X69) @ X68))) | ($true != ((X65 @ X66 @ X67 @ (comp @ X69 @ X68))))) | ~ ! [X70 : term] : (($true = ((X65 @ id @ (sub @ X64 @ (push @ X70 @ id)) @ id))) | ($true != ((X65 @ id @ X70 @ id)))) | ~ ! [X71 : (subst > term > term)] : (~ ! [X73 : subst,X74 : term,X72 : subst] : (((X71 @ (comp @ X73 @ X72) @ (sub @ X74 @ X72))) = ((sub @ (X71 @ X73 @ X74) @ X72))) | ~ ! [X75 : term] : (($true != ((X65 @ id @ X75 @ id))) | ($true = ((X65 @ id @ (X71 @ id @ X75) @ id)))) | ($true = ((X65 @ id @ (lam @ (X71 @ sh @ one)) @ id)))) | ($true = ((X65 @ id @ (lam @ X64) @ id))) | ~ ! [X78 : subst,X77 : subst,X79 : subst,X76 : term] : (($true = ((X65 @ X77 @ X76 @ (comp @ X79 @ X78)))) | ($true != ((X65 @ (comp @ X77 @ X79) @ (sub @ X76 @ X79) @ X78)))))) & (! [X523 : term,X524 : (subst > term > subst > $o)] : (~ ! [X530 : term] : (($true = ((X524 @ id @ (sub @ X523 @ (push @ X530 @ id)) @ id))) | ($true != ((X524 @ id @ X530 @ id)))) | ~ ! [X525 : (subst > term > term)] : (($true = ((X524 @ id @ (lam @ (X525 @ sh @ one)) @ id))) | ~ ! [X529 : term] : (($true != ((X524 @ id @ X529 @ id))) | ($true = ((X524 @ id @ (X525 @ id @ X529) @ id)))) | ~ ! [X528 : term,X527 : subst,X526 : subst] : (((sub @ (X525 @ X527 @ X528) @ X526)) = ((X525 @ (comp @ X527 @ X526) @ (sub @ X528 @ X526))))) | ($true = ((X524 @ id @ (lam @ (sub @ X523 @ (push @ one @ sh))) @ id)))) <=> (hoasinduction_lem3aa = $true)) & (((d2subst @ d_id)) = id) & (! [X461 : (subst > term > term),X460 : (subst > term > term),X459 : subst,X462 : term] : ((((lam @ (X461 @ sh @ one))) != ((lam @ (X460 @ sh @ one)))) | ~ ! [X463 : subst,X464 : term,X465 : subst] : (((X460 @ (comp @ X463 @ X465) @ (sub @ X464 @ X465))) = ((sub @ (X460 @ X463 @ X464) @ X465))) | ~ ! [X468 : subst,X466 : term,X467 : subst] : (((X461 @ (comp @ X468 @ X467) @ (sub @ X466 @ X467))) = ((sub @ (X461 @ X468 @ X466) @ X467))) | (((X460 @ X459 @ X462)) = ((X461 @ X459 @ X462)))) <=> (hoaslaminj = $true)) & (hoaslamnotap_gthm = $true) & (pushprop_lem1_gthm = $true) & ! [X423 : term] : ? [X424 : d_term] : (((d2term @ X424)) = X423) & (pushprop_lem2v2 = $true) & (hoaslamnotvar = $true) & (hoasinduction_lem3a_gthm = $true)),
  inference(flattening,[],[f9])).
thf(f11,plain,(
  (induction2lem_gthm = $true) & ((pushprop_lem0_lthm = $true) <=> ! [X107 : subst,X108 : term,X106 : (term > $o)] : ? [X109 : (term > $o)] : ! [X110 : term] : (((X106 @ (sub @ X110 @ (push @ X108 @ X107)))) = ((X109 @ X110)))) & (pushprop_lem3v2_lthm = $true) & ((shinj = $true) <=> ! [X470 : term,X469 : term] : ((X469 = X470) | (((sub @ X470 @ sh)) != ((sub @ X469 @ sh))))) & (axassoc = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & ((hoasinduction = $true) <=> ! [X6 : term,X7 : (subst > term > subst > $o)] : (? [X13 : term,X12 : subst,X11 : subst,X10 : subst] : (($true = ((X7 @ X11 @ X13 @ (comp @ X12 @ X10)))) & ($true != ((X7 @ (comp @ X11 @ X12) @ (sub @ X13 @ X12) @ X10)))) | ? [X19 : (subst > term > term)] : (($true != ((X7 @ id @ (lam @ (X19 @ sh @ one)) @ id))) & ! [X22 : subst,X21 : term,X20 : subst] : (((sub @ (X19 @ X22 @ X21) @ X20)) = ((X19 @ (comp @ X22 @ X20) @ (sub @ X21 @ X20)))) & ! [X23 : term] : ((((X7 @ id @ X23 @ id)) != $true) | ($true = ((X7 @ id @ (X19 @ id @ X23) @ id))))) | ? [X18 : term] : (($true != ((X7 @ id @ X18 @ id))) & ($true = ((var @ (sub @ X18 @ id))))) | ? [X16 : subst,X15 : term,X17 : subst,X14 : subst] : (($true != ((X7 @ X14 @ X15 @ (comp @ X16 @ X17)))) & ($true = ((X7 @ (comp @ X14 @ X16) @ (sub @ X15 @ X16) @ X17)))) | ? [X8 : term,X9 : term] : (($true = ((X7 @ id @ X8 @ id))) & (((X7 @ id @ X9 @ id)) = $true) & ($true != ((X7 @ id @ (ap @ (sub @ X9 @ id) @ X8) @ id)))) | ($true = ((X7 @ id @ X6 @ id))))) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ((? [X448 : term,X449 : term,X450 : term,X447 : term] : ((((ap @ X447 @ X449)) = ((ap @ X450 @ X448))) & (X448 != X449)) | ! [X453 : term,X454 : term,X451 : term,X452 : term] : ((((ap @ (sub @ X452 @ id) @ X451)) != ((ap @ (sub @ X454 @ id) @ X453))) | (X451 = X453))) <=> (hoasapinj2_lthm = $true)) & ((induction = $true) <=> ! [X501 : (term > $o),X500 : term] : (? [X504 : term,X505 : term] : (($true != ((X501 @ (ap @ X505 @ X504)))) & ($true = ((X501 @ X504))) & ($true = ((X501 @ X505)))) | ? [X503 : term] : (($true = ((var @ X503))) & ($true != ((X501 @ X503)))) | ? [X502 : term] : (($true = ((X501 @ X502))) & ($true != ((X501 @ (lam @ X502))))) | ($true = ((X501 @ X500))))) & (! [X486 : term] : ? [X487 : (subst > term > term)] : (((sub @ X486 @ (push @ one @ sh))) = ((X487 @ sh @ one))) <=> (hoasinduction_lem3b_lthm = $true)) & ! [X409 : d_subst,X408 : d_subst] : ((((d2subst @ X409)) != ((d2subst @ X408))) | (X408 = X409)) & (((! [X242 : term,X244 : term,X241 : term,X243 : term] : ((X242 = X244) | (((ap @ (sub @ X244 @ id) @ X241)) != ((ap @ (sub @ X242 @ id) @ X243)))) | ? [X240 : term,X239 : term,X238 : term,X237 : term] : ((X237 != X238) & (((ap @ X237 @ X239)) = ((ap @ X238 @ X240))))) | ? [X236 : term] : (((sub @ X236 @ id)) != X236)) <=> (hoasapinj1_lthm = $true)) & ((! [X4 : subst] : (((comp @ id @ X4)) = X4) & ! [X3 : subst] : (((comp @ X3 @ id)) = X3)) <=> (substmonoid = $true)) & (((((! [X169 : term,X170 : (subst > term > subst > $o)] : (($true = ((X170 @ id @ X169 @ id))) | ? [X185 : term,X186 : term] : (($true = ((X170 @ id @ X185 @ id))) & ($true = ((X170 @ id @ X186 @ id))) & ($true != ((X170 @ id @ (ap @ (sub @ X186 @ id) @ X185) @ id)))) | ? [X183 : subst,X184 : subst,X181 : term,X182 : subst] : (($true != ((X170 @ (comp @ X184 @ X182) @ (sub @ X181 @ X182) @ X183))) & ($true = ((X170 @ X184 @ X181 @ (comp @ X182 @ X183))))) | ? [X180 : term] : (($true != ((X170 @ id @ X180 @ id))) & ($true = ((var @ (sub @ X180 @ id))))) | ? [X178 : subst,X177 : subst,X176 : subst,X179 : term] : (($true != ((X170 @ X176 @ X179 @ (comp @ X177 @ X178)))) & ($true = ((X170 @ (comp @ X176 @ X177) @ (sub @ X179 @ X177) @ X178)))) | ? [X171 : (subst > term > term)] : (! [X172 : term] : ((((X170 @ id @ X172 @ id)) != $true) | ($true = ((X170 @ id @ (X171 @ id @ X172) @ id)))) & ($true != ((X170 @ id @ (lam @ (X171 @ sh @ one)) @ id))) & ! [X173 : subst,X174 : subst,X175 : term] : (((sub @ (X171 @ X173 @ X175) @ X174)) = ((X171 @ (comp @ X173 @ X174) @ (sub @ X175 @ X174)))))) | ? [X161 : term,X160 : (term > $o),X159 : (subst > term > subst > $o)] : (! [X164 : (subst > term > term)] : (? [X166 : subst,X167 : subst,X168 : term] : (((X164 @ (comp @ X166 @ X167) @ (sub @ X168 @ X167))) != ((sub @ (X164 @ X166 @ X168) @ X167))) | ($true = ((X159 @ id @ (lam @ (X164 @ sh @ one)) @ id))) | ? [X165 : term] : (($true = ((X159 @ id @ X165 @ id))) & ($true != ((X159 @ id @ (X164 @ id @ X165) @ id))))) & ($true != ((X160 @ (lam @ X161)))) & ! [X162 : term] : (((X160 @ X162)) = ((X159 @ id @ X162 @ id))) & ! [X163 : term] : (($true != ((X160 @ X163))) | ($true = ((X160 @ (sub @ X161 @ (push @ X163 @ id)))))))) | ? [X158 : term] : (((sub @ X158 @ id)) != X158)) | ? [X151 : (term > $o),X152 : term] : (! [X153 : term,X154 : term] : (($true = ((X151 @ (ap @ X154 @ X153)))) | ($true != ((X151 @ X154))) | ($true != ((X151 @ X153)))) & ! [X155 : term] : (? [X156 : term] : (($true != ((X151 @ (sub @ X155 @ (push @ X156 @ id))))) & ($true = ((X151 @ X156)))) | ($true = ((X151 @ (lam @ X155))))) & ! [X157 : term] : (($true != ((var @ X157))) | ($true = ((X151 @ X157)))) & ($true != ((X151 @ X152))))) | ? [X148 : (subst > term > subst > $o)] : ! [X149 : (term > $o)] : ? [X150 : term] : (((X148 @ id @ X150 @ id)) != ((X149 @ X150)))) <=> (hoasinduction_lthm_3 = $true)) & (axmap = $true) & ((((! [X136 : term,X139 : (subst > term > term),X138 : subst,X137 : (subst > term > term)] : ((((lam @ (X139 @ sh @ one))) != ((lam @ (X137 @ sh @ one)))) | ? [X144 : subst,X143 : term,X145 : subst] : (((X137 @ (comp @ X145 @ X144) @ (sub @ X143 @ X144))) != ((sub @ (X137 @ X145 @ X143) @ X144))) | ? [X142 : subst,X140 : subst,X141 : term] : (((X139 @ (comp @ X142 @ X140) @ (sub @ X141 @ X140))) != ((sub @ (X139 @ X142 @ X141) @ X140))) | (((X137 @ X138 @ X136)) = ((X139 @ X138 @ X136)))) | ? [X134 : term,X135 : term] : ((((lam @ X134)) = ((lam @ X135))) & (X134 != X135))) | ? [X133 : subst,X132 : term] : (((comp @ sh @ (push @ X132 @ X133))) != X133)) | ? [X130 : subst,X131 : term] : (((sub @ one @ (push @ X131 @ X130))) != X131)) <=> (hoaslaminj_lthm = $true)) & (hoasapnotvar_lthm = $true) & (axclos = $true) & ((pushprop_lem1 = $true) <=> ! [X212 : subst,X216 : term,X215 : (term > $o),X213 : term,X214 : (term > $o)] : (($true = ((X215 @ (sub @ X213 @ (push @ X216 @ X212))))) | ($true != ((X214 @ X213))))) & (! [X80 : term,X83 : term,X82 : term,X81 : term] : ((X80 = X83) | (((ap @ X82 @ X83)) != ((ap @ X81 @ X80)))) <=> (apinj2 = $true)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoasinduction_lem3v2_lthm = $true) <=> (! [X218 : term,X219 : (subst > term > subst > $o),X220 : (term > $o)] : (($true = ((X220 @ (lam @ X218)))) | ? [X229 : term] : (($true != ((X220 @ (sub @ X218 @ (push @ X229 @ id))))) & ($true = ((X220 @ X229)))) | ? [X230 : term] : (((X219 @ id @ X230 @ id)) != ((X220 @ X230))) | ? [X221 : subst,X222 : subst,X223 : term,X224 : subst] : (($true != ((X219 @ X224 @ X223 @ (comp @ X222 @ X221)))) & ($true = ((X219 @ (comp @ X224 @ X222) @ (sub @ X223 @ X222) @ X221)))) | ? [X231 : (subst > term > term)] : (! [X235 : term] : (($true != ((X219 @ id @ X235 @ id))) | ($true = ((X219 @ id @ (X231 @ id @ X235) @ id)))) & ($true != ((X219 @ id @ (lam @ (X231 @ sh @ one)) @ id))) & ! [X234 : subst,X233 : subst,X232 : term] : (((sub @ (X231 @ X234 @ X232) @ X233)) = ((X231 @ (comp @ X234 @ X233) @ (sub @ X232 @ X233))))) | ? [X225 : subst,X226 : term,X227 : subst,X228 : subst] : (($true != ((X219 @ (comp @ X225 @ X228) @ (sub @ X226 @ X228) @ X227))) & ($true = ((X219 @ X225 @ X226 @ (comp @ X228 @ X227)))))) | ? [X217 : term] : (((sub @ X217 @ id)) != X217))) & (apnotvar = $true) & ((hoasinduction_lem3a_lthm = $true) <=> ((! [X94 : term,X93 : (subst > term > subst > $o)] : (($true = ((X93 @ id @ (lam @ X94) @ id))) | ? [X95 : term] : (($true != ((X93 @ id @ (sub @ X94 @ (push @ X95 @ id)) @ id))) & ($true = ((X93 @ id @ X95 @ id)))) | ? [X96 : (subst > term > term)] : (($true != ((X93 @ id @ (lam @ (X96 @ sh @ one)) @ id))) & ! [X97 : term] : (($true != ((X93 @ id @ X97 @ id))) | ($true = ((X93 @ id @ (X96 @ id @ X97) @ id)))) & ! [X99 : term,X100 : subst,X98 : subst] : (((sub @ (X96 @ X100 @ X99) @ X98)) = ((X96 @ (comp @ X100 @ X98) @ (sub @ X99 @ X98)))))) | ? [X86 : term,X85 : (subst > term > subst > $o)] : (! [X92 : term] : (($true = ((X85 @ id @ (sub @ X86 @ (push @ X92 @ id)) @ id))) | ($true != ((X85 @ id @ X92 @ id)))) & ! [X87 : (subst > term > term)] : (? [X91 : term] : (($true != ((X85 @ id @ (X87 @ id @ X91) @ id))) & ($true = ((X85 @ id @ X91 @ id)))) | ? [X90 : term,X88 : subst,X89 : subst] : (((X87 @ (comp @ X88 @ X89) @ (sub @ X90 @ X89))) != ((sub @ (X87 @ X88 @ X90) @ X89))) | ($true = ((X85 @ id @ (lam @ (X87 @ sh @ one)) @ id)))) & ($true != ((X85 @ id @ (lam @ (sub @ X86 @ (push @ one @ sh))) @ id))))) | ? [X84 : term] : (((sub @ X84 @ id)) != X84))) & (one = ((d2term @ d_one))) & (! [X475 : term] : (((sub @ X475 @ id)) = X475) <=> (termmset = $true)) & (((((! [X52 : (term > $o),X53 : subst,X54 : term] : (? [X56 : term,X55 : term] : (($true != ((X52 @ (ap @ X55 @ X56)))) & ($true = ((X52 @ X56))) & (((X52 @ X55)) = $true)) | ? [X57 : term] : (! [X58 : term] : (($true != ((X52 @ X58))) | ($true = ((X52 @ (sub @ X57 @ (push @ X58 @ id)))))) & ($true != ((X52 @ (lam @ X57))))) | ? [X59 : term] : (($true != ((X52 @ (sub @ X59 @ X53)))) & ($true = ((var @ X59)))) | ($true = ((X52 @ (sub @ X54 @ X53))))) | ? [X46 : (term > $o),X47 : term] : (! [X51 : term] : (($true != ((X46 @ X51))) | ($true = ((X46 @ (lam @ X51))))) & ! [X48 : term] : (($true = ((X46 @ X48))) | ($true != ((var @ X48)))) & ! [X50 : term,X49 : term] : (($true != ((X46 @ X49))) | ($true != ((X46 @ X50))) | ($true = ((X46 @ (ap @ X49 @ X50))))) & ($true != ((X46 @ X47))))) | ? [X45 : subst] : (((comp @ X45 @ id)) != X45)) | ? [X44 : term,X43 : subst] : (((comp @ sh @ (push @ X44 @ X43))) != X43)) | ? [X41 : subst,X42 : term] : (((sub @ one @ (push @ X42 @ X41))) != X42)) <=> (induction2lem_lthm = $true)) & (! [X0 : subst] : (((comp @ X0 @ id)) = X0) <=> (axidr = $true)) & ((pushprop_lem0 = $true) <=> ! [X102 : (term > $o),X103 : term,X101 : subst] : ? [X104 : (term > $o)] : ! [X105 : term] : (((X104 @ X105)) = ((X102 @ (sub @ X105 @ (push @ X103 @ X101)))))) & (hoaslamnotvar_gthm = $true) & ((hoasapinj2 = $true) <=> ! [X473 : term,X472 : term,X471 : term,X474 : term] : ((X472 = X473) | (((ap @ (sub @ X474 @ id) @ X472)) != ((ap @ (sub @ X471 @ id) @ X473))))) & (hoasapinj1_gthm = $true) & (axabs = $true) & ((hoasinduction_lem3v2_f = $true) <=> ! [X374 : term] : ? [X375 : (subst > term > term)] : ! [X376 : subst,X377 : term] : (((sub @ X374 @ (push @ X377 @ X376))) = ((X375 @ X376 @ X377)))) & (! [X117 : term,X118 : term,X119 : (subst > term > subst > $o)] : (($true != ((X119 @ id @ X117 @ id))) | ? [X124 : subst,X125 : subst,X123 : subst,X122 : term] : (($true = ((X119 @ (comp @ X123 @ X125) @ (sub @ X122 @ X125) @ X124))) & ($true != ((X119 @ X123 @ X122 @ (comp @ X125 @ X124))))) | ? [X128 : subst,X127 : term,X126 : subst,X129 : subst] : (($true != ((X119 @ (comp @ X126 @ X128) @ (sub @ X127 @ X128) @ X129))) & ($true = ((X119 @ X126 @ X127 @ (comp @ X128 @ X129))))) | ($true != ((X119 @ id @ X118 @ id))) | ($true = ((X119 @ id @ (ap @ X118 @ X117) @ id))) | ? [X120 : term,X121 : term] : (($true = ((X119 @ id @ X120 @ id))) & ($true != ((X119 @ id @ (ap @ (sub @ X120 @ id) @ X121) @ id))) & ($true = ((X119 @ id @ X121 @ id))))) <=> (hoasinduction_lem2 = $true)) & (hoasinduction_lem1 = $true) & (axapp = $true) & ((hoasinduction_lthm = $true) <=> (((! [X281 : (subst > term > subst > $o),X282 : term] : (($true = ((X281 @ id @ X282 @ id))) | ? [X297 : term,X298 : term] : (($true != ((X281 @ id @ (ap @ (sub @ X297 @ id) @ X298) @ id))) & ($true = ((X281 @ id @ X297 @ id))) & ($true = ((X281 @ id @ X298 @ id)))) | ? [X290 : term,X289 : subst,X288 : subst,X291 : subst] : (($true != ((X281 @ (comp @ X288 @ X289) @ (sub @ X290 @ X289) @ X291))) & ($true = ((X281 @ X288 @ X290 @ (comp @ X289 @ X291))))) | ? [X293 : subst,X295 : subst,X294 : term,X292 : subst] : (($true = ((X281 @ (comp @ X292 @ X295) @ (sub @ X294 @ X295) @ X293))) & ($true != ((X281 @ X292 @ X294 @ (comp @ X295 @ X293))))) | ? [X283 : (subst > term > term)] : (! [X285 : term,X286 : subst,X287 : subst] : (((sub @ (X283 @ X286 @ X285) @ X287)) = ((X283 @ (comp @ X286 @ X287) @ (sub @ X285 @ X287)))) & ($true != ((X281 @ id @ (lam @ (X283 @ sh @ one)) @ id))) & ! [X284 : term] : (($true != ((X281 @ id @ X284 @ id))) | ($true = ((X281 @ id @ (X283 @ id @ X284) @ id))))) | ? [X296 : term] : (($true != ((X281 @ id @ X296 @ id))) & ($true = ((var @ (sub @ X296 @ id)))))) | ? [X266 : (subst > term > subst > $o),X265 : term] : (! [X270 : subst,X269 : subst,X268 : subst,X267 : term] : (($true = ((X266 @ X269 @ X267 @ (comp @ X268 @ X270)))) | ($true != ((X266 @ (comp @ X269 @ X268) @ (sub @ X267 @ X268) @ X270)))) & ! [X271 : subst,X272 : subst,X274 : term,X273 : subst] : (($true = ((X266 @ (comp @ X272 @ X273) @ (sub @ X274 @ X273) @ X271))) | ($true != ((X266 @ X272 @ X274 @ (comp @ X273 @ X271))))) & ! [X275 : term] : (($true != ((X266 @ id @ X275 @ id))) | ($true = ((X266 @ id @ (sub @ X265 @ (push @ X275 @ id)) @ id)))) & ($true != ((X266 @ id @ (lam @ X265) @ id))) & ! [X276 : (subst > term > term)] : (? [X278 : subst,X280 : term,X279 : subst] : (((sub @ (X276 @ X278 @ X280) @ X279)) != ((X276 @ (comp @ X278 @ X279) @ (sub @ X280 @ X279)))) | ($true = ((X266 @ id @ (lam @ (X276 @ sh @ one)) @ id))) | ? [X277 : term] : (($true = ((X266 @ id @ X277 @ id))) & ($true != ((X266 @ id @ (X276 @ id @ X277) @ id))))))) | ? [X252 : (subst > term > subst > $o),X254 : term,X253 : term] : (($true != ((X252 @ id @ (ap @ X253 @ X254) @ id))) & ! [X259 : subst,X261 : subst,X260 : term,X262 : subst] : (($true != ((X252 @ X259 @ X260 @ (comp @ X262 @ X261)))) | ($true = ((X252 @ (comp @ X259 @ X262) @ (sub @ X260 @ X262) @ X261)))) & ! [X263 : term,X264 : term] : (($true != ((X252 @ id @ X264 @ id))) | ($true = ((X252 @ id @ (ap @ (sub @ X264 @ id) @ X263) @ id))) | ($true != ((X252 @ id @ X263 @ id)))) & ! [X256 : subst,X255 : subst,X258 : term,X257 : subst] : (($true = ((X252 @ X256 @ X258 @ (comp @ X255 @ X257)))) | ($true != ((X252 @ (comp @ X256 @ X255) @ (sub @ X258 @ X255) @ X257)))) & ($true = ((X252 @ id @ X253 @ id))) & ($true = ((X252 @ id @ X254 @ id))))) | ? [X246 : (term > $o),X245 : term] : (! [X247 : term] : (($true != ((var @ X247))) | ($true = ((X246 @ X247)))) & ! [X248 : term] : (($true = ((X246 @ (lam @ X248)))) | ? [X249 : term] : (($true != ((X246 @ (sub @ X248 @ (push @ X249 @ id))))) & ($true = ((X246 @ X249))))) & ! [X250 : term,X251 : term] : (($true = ((X246 @ (ap @ X250 @ X251)))) | ($true != ((X246 @ X250))) | ($true != ((X246 @ X251)))) & ($true != ((X246 @ X245)))))) & (((! [X33 : (term > $o),X34 : term] : (? [X39 : term] : (($true = ((var @ X39))) & ($true != ((X33 @ X39)))) | ? [X38 : term,X37 : term] : (($true = ((X33 @ X37))) & ($true = ((X33 @ X38))) & ($true != ((X33 @ (ap @ X38 @ X37))))) | ($true = ((X33 @ X34))) | ? [X35 : term] : (($true != ((X33 @ (lam @ X35)))) & ! [X36 : term] : (($true != ((X33 @ X36))) | ($true = ((X33 @ (sub @ X35 @ (push @ X36 @ id)))))))) | ? [X25 : subst,X27 : (term > $o),X26 : term] : (($true != ((X27 @ (sub @ X26 @ X25)))) & ! [X32 : term] : (($true != ((var @ X32))) | ($true = ((X27 @ (sub @ X32 @ X25))))) & ! [X28 : term] : (($true = ((X27 @ (lam @ X28)))) | ? [X29 : term] : (($true != ((X27 @ (sub @ X28 @ (push @ X29 @ id))))) & ($true = ((X27 @ X29))))) & ! [X30 : term,X31 : term] : (($true != ((X27 @ X31))) | ($true = ((X27 @ (ap @ X30 @ X31)))) | ($true != ((X27 @ X30)))))) | ? [X24 : term] : (((sub @ X24 @ id)) != X24)) <=> (induction2_lthm = $true)) & (hoaslaminj_gthm = $true) & (hoasinduction_lem1v2 = $true) & (hoasinduction_lem1_gthm = $true) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X419 : term] : ? [X420 : (subst > term > term)] : ! [X421 : subst,X422 : term] : (((X420 @ X421 @ X422)) = ((sub @ X419 @ (push @ X422 @ X421))))) & (hoaslamnotap_lthm = $true) & (! [X446 : term,X445 : term] : ((((lam @ X446)) != ((lam @ X445))) | (X445 = X446)) <=> (laminj = $true)) & (lamnotvar = $true) & (termmset_gthm = $true) & (pushprop_lthm = $true) & (ulamvarsh = $true) & (hoasapinj2_gthm = $true) & (hoasinduction_lem3v2_gthm = $true) & (pushprop_gthm = $true) & (! [X344 : (subst > term > subst > $o)] : ? [X345 : (term > $o)] : ! [X346 : term] : (((X345 @ X346)) = ((X344 @ id @ X346 @ id))) <=> (hoasinduction_lem0_lthm = $true)) & (pushprop_lthm_orig = $true) & ((pushprop_lem1_lthm = $true) <=> ((! [X511 : term,X514 : term,X510 : subst,X512 : (term > $o),X513 : (term > $o)] : (($true = ((X512 @ (sub @ X511 @ (push @ X514 @ X510))))) | ($true != ((X513 @ X511)))) | ? [X508 : subst,X509 : term] : (((comp @ sh @ (push @ X509 @ X508))) != X508)) | ? [X507 : subst,X506 : term] : (((sub @ one @ (push @ X506 @ X507))) != X506))) & (hoasapnotvar_gthm = $true) & (! [X188 : term,X189 : (subst > term > subst > $o),X187 : term,X190 : (term > $o)] : (($true != ((X190 @ X187))) | ($true != ((X190 @ X188))) | ? [X194 : subst,X195 : term,X193 : subst,X192 : subst] : (($true = ((X189 @ (comp @ X194 @ X193) @ (sub @ X195 @ X193) @ X192))) & (((X189 @ X194 @ X195 @ (comp @ X193 @ X192))) != $true)) | ? [X191 : term] : (((X190 @ X191)) != ((X189 @ id @ X191 @ id))) | ? [X198 : term,X199 : subst,X197 : subst,X196 : subst] : (($true = ((X189 @ X199 @ X198 @ (comp @ X196 @ X197)))) & ($true != ((X189 @ (comp @ X199 @ X196) @ (sub @ X198 @ X196) @ X197)))) | ? [X200 : term,X201 : term] : (($true = ((X189 @ id @ X201 @ id))) & ($true = ((X189 @ id @ X200 @ id))) & ($true != ((X189 @ id @ (ap @ (sub @ X200 @ id) @ X201) @ id)))) | ($true = ((X190 @ (ap @ X188 @ X187))))) <=> (hoasinduction_lem2v2 = $true)) & (pushprop_lem2v2_lthm = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (lamnotap != $true) & (induction2_gthm = $true) & (ulamvar1 != $true) & ((pushprop_lem1v2 = $true) <=> ! [X111 : (term > $o),X114 : subst,X113 : term,X112 : (term > $o)] : (($true = ((X112 @ one))) | ($true != ((X111 @ X113))) | ? [X115 : term] : (((X112 @ X115)) != ((X111 @ (sub @ X115 @ (push @ X113 @ X114))))))) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (ulamvarind = $true) & (hoasapnotvar = $true) & (hoasinduction_lem1_lthm = $true) & ((hoasinduction_lem3v2 = $true) <=> ! [X427 : (subst > term > subst > $o),X426 : (term > $o),X425 : term] : (? [X433 : (subst > term > term)] : (($true != ((X427 @ id @ (lam @ (X433 @ sh @ one)) @ id))) & ! [X434 : term] : (($true != ((X427 @ id @ X434 @ id))) | ($true = ((X427 @ id @ (X433 @ id @ X434) @ id)))) & ! [X437 : subst,X436 : term,X435 : subst] : (((X433 @ (comp @ X437 @ X435) @ (sub @ X436 @ X435))) = ((sub @ (X433 @ X437 @ X436) @ X435)))) | ? [X441 : subst,X440 : subst,X439 : subst,X442 : term] : (($true != ((X427 @ (comp @ X440 @ X441) @ (sub @ X442 @ X441) @ X439))) & ($true = ((X427 @ X440 @ X442 @ (comp @ X441 @ X439))))) | ? [X432 : term,X429 : subst,X431 : subst,X430 : subst] : (($true = ((X427 @ (comp @ X430 @ X431) @ (sub @ X432 @ X431) @ X429))) & ($true != ((X427 @ X430 @ X432 @ (comp @ X431 @ X429))))) | ? [X438 : term] : (((X426 @ X438)) != ((X427 @ id @ X438 @ id))) | ? [X428 : term] : (($true = ((X426 @ X428))) & ($true != ((X426 @ (sub @ X425 @ (push @ X428 @ id)))))) | ($true = ((X426 @ (lam @ X425)))))) & (pushprop_lem1v2_gthm = $true) & (axvarshift = $true) & (pushprop_lem0_gthm = $true) & (pushprop = $true) & ((induction2 = $true) <=> ! [X360 : term,X359 : (term > $o)] : (? [X363 : term] : (($true != ((X359 @ X363))) & ($true = ((var @ X363)))) | ? [X361 : term] : (($true != ((X359 @ (lam @ X361)))) & ! [X362 : term] : (($true != ((X359 @ X362))) | ($true = ((X359 @ (sub @ X361 @ (push @ X362 @ id))))))) | ($true = ((X359 @ X360))) | ? [X365 : term,X364 : term] : (($true = ((X359 @ X365))) & ($true != ((X359 @ (ap @ X365 @ X364)))) & ($true = ((X359 @ X364)))))) & ((hoasinduction_lem3a = $true) <=> ! [X515 : (subst > term > subst > $o),X516 : term] : (($true = ((X515 @ id @ (lam @ X516) @ id))) | ? [X518 : (subst > term > term)] : (! [X521 : subst,X520 : term,X522 : subst] : (((X518 @ (comp @ X521 @ X522) @ (sub @ X520 @ X522))) = ((sub @ (X518 @ X521 @ X520) @ X522))) & ! [X519 : term] : (($true != ((X515 @ id @ X519 @ id))) | ($true = ((X515 @ id @ (X518 @ id @ X519) @ id)))) & ($true != ((X515 @ id @ (lam @ (X518 @ sh @ one)) @ id)))) | ? [X517 : term] : (($true = ((X515 @ id @ X517 @ id))) & ($true != ((X515 @ id @ (sub @ X516 @ (push @ X517 @ id)) @ id)))))) & (((! [X388 : term,X387 : (subst > term > subst > $o)] : (? [X398 : (subst > term > term)] : (($true != ((X387 @ id @ (lam @ (X398 @ sh @ one)) @ id))) & ! [X402 : term] : (($true = ((X387 @ id @ (X398 @ id @ X402) @ id))) | ($true != ((X387 @ id @ X402 @ id)))) & ! [X400 : term,X401 : subst,X399 : subst] : (((X398 @ (comp @ X399 @ X401) @ (sub @ X400 @ X401))) = ((sub @ (X398 @ X399 @ X400) @ X401)))) | ? [X393 : term] : (($true != ((X387 @ id @ (sub @ X388 @ (push @ X393 @ id)) @ id))) & ($true = ((X387 @ id @ X393 @ id)))) | ? [X389 : subst,X391 : term,X390 : subst,X392 : subst] : (($true != ((X387 @ (comp @ X392 @ X389) @ (sub @ X391 @ X389) @ X390))) & ($true = ((X387 @ X392 @ X391 @ (comp @ X389 @ X390))))) | ($true = ((X387 @ id @ (lam @ X388) @ id))) | ? [X395 : subst,X394 : subst,X397 : subst,X396 : term] : (($true = ((X387 @ (comp @ X397 @ X395) @ (sub @ X396 @ X395) @ X394))) & ($true != ((X387 @ X397 @ X396 @ (comp @ X395 @ X394)))))) | ? [X379 : term,X380 : (subst > term > subst > $o)] : (! [X381 : (subst > term > term)] : (? [X382 : term] : (($true = ((X380 @ id @ X382 @ id))) & ($true != ((X380 @ id @ (X381 @ id @ X382) @ id)))) | ($true = ((X380 @ id @ (lam @ (X381 @ sh @ one)) @ id))) | ? [X383 : subst,X384 : term,X385 : subst] : (((X381 @ (comp @ X383 @ X385) @ (sub @ X384 @ X385))) != ((sub @ (X381 @ X383 @ X384) @ X385)))) & ($true != ((X380 @ id @ (lam @ (sub @ X379 @ (push @ one @ sh))) @ id))) & ! [X386 : term] : (($true = ((X380 @ id @ (sub @ X379 @ (push @ X386 @ id)) @ id))) | ($true != ((X380 @ id @ X386 @ id)))))) | ? [X378 : term] : (((sub @ X378 @ id)) != X378)) <=> (hoasinduction_lem3_lthm = $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (! [X456 : term,X455 : term,X457 : term,X458 : term] : ((X455 = X458) | (((ap @ X458 @ X457)) != ((ap @ X455 @ X456)))) <=> (apinj1 = $true)) & (! [X477 : term] : ? [X478 : (subst > term > term)] : (((X478 @ sh @ one)) = ((sub @ X477 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b = $true)) & ! [X5 : d_subst] : (d_id = X5) & (! [X204 : term,X203 : subst,X202 : (term > $o)] : (? [X209 : term] : (($true != ((X202 @ (sub @ X209 @ X203)))) & ($true = ((var @ X209)))) | ? [X205 : term] : (($true != ((X202 @ (lam @ X205)))) & ! [X206 : term] : (($true != ((X202 @ X206))) | ($true = ((X202 @ (sub @ X205 @ (push @ X206 @ id))))))) | ? [X207 : term,X208 : term] : (($true = ((X202 @ X208))) & ($true = ((X202 @ X207))) & ($true != ((X202 @ (ap @ X207 @ X208))))) | ($true = ((X202 @ (sub @ X204 @ X203))))) <=> (induction2lem = $true)) & (hoasinduction_lem3b_gthm = $true) & (pushprop_lem3v2 = $true) & (hoasinduction_lem3_gthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (! [X348 : term,X347 : subst] : (((sub @ one @ (push @ X348 @ X347))) = X348) <=> (axvarcons = $true)) & (substmonoid_gthm = $true) & ((termmset_lthm = $true) <=> (? [X1 : term] : (((sub @ X1 @ id)) != X1) | ! [X2 : term] : (((sub @ X2 @ id)) = X2))) & (! [X416 : (subst > term > subst > $o)] : ? [X417 : (term > $o)] : ! [X418 : term] : (((X416 @ id @ X418 @ id)) = ((X417 @ X418))) <=> (hoasinduction_lem0 = $true)) & ((axshiftcons = $true) <=> ! [X443 : subst,X444 : term] : (((comp @ sh @ (push @ X444 @ X443))) = X443)) & ((? [X479 : term,X480 : subst] : (((sub @ one @ (push @ X479 @ X480))) != X479) | ! [X484 : (term > $o),X482 : (term > $o),X483 : subst,X481 : term] : (($true != ((X482 @ X481))) | ? [X485 : term] : (((X484 @ X485)) != ((X482 @ (sub @ X485 @ (push @ X481 @ X483))))) | ($true = ((X484 @ one))))) <=> (pushprop_lem1v2_lthm = $true)) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (! [X411 : term,X410 : (subst > term > term),X412 : term] : (? [X415 : term,X414 : subst,X413 : subst] : (((X410 @ (comp @ X414 @ X413) @ (sub @ X415 @ X413))) != ((sub @ (X410 @ X414 @ X415) @ X413))) | (((ap @ (sub @ X412 @ id) @ X411)) != ((lam @ (X410 @ sh @ one))))) <=> (hoaslamnotap = $true)) & (hoaslamnotvar_lthm = $true) & (! [X531 : term,X532 : (subst > term > subst > $o)] : (($true = ((X532 @ id @ X531 @ id))) | ? [X533 : (subst > term > term)] : (($true != ((X532 @ id @ (lam @ (X533 @ sh @ one)) @ id))) & ! [X534 : term] : (($true = ((X532 @ id @ (X533 @ id @ X534) @ id))) | ($true != ((X532 @ id @ X534 @ id)))) & ! [X536 : term,X537 : subst,X535 : subst] : (((X533 @ (comp @ X535 @ X537) @ (sub @ X536 @ X537))) = ((sub @ (X533 @ X535 @ X536) @ X537)))) | ? [X538 : term,X539 : term] : (($true != ((X532 @ id @ (ap @ (sub @ X539 @ id) @ X538) @ id))) & ($true = ((X532 @ id @ X539 @ id))) & ($true = ((X532 @ id @ X538 @ id))))) <=> (hoasinduction_no_psi_cond = $true)) & (pushprop_lem2v2_gthm = $true) & (((((! [X336 : (subst > term > subst > $o),X335 : term] : (? [X339 : (subst > term > term)] : (($true != ((X336 @ id @ (lam @ (X339 @ sh @ one)) @ id))) & ! [X340 : term] : (($true != ((X336 @ id @ X340 @ id))) | ($true = ((X336 @ id @ (X339 @ id @ X340) @ id)))) & ! [X343 : subst,X342 : subst,X341 : term] : (((sub @ (X339 @ X342 @ X341) @ X343)) = ((X339 @ (comp @ X342 @ X343) @ (sub @ X341 @ X343))))) | ? [X337 : term,X338 : term] : ((((X336 @ id @ X337 @ id)) = $true) & ($true != ((X336 @ id @ (ap @ (sub @ X337 @ id) @ X338) @ id))) & ($true = ((X336 @ id @ X338 @ id)))) | ($true = ((X336 @ id @ X335 @ id)))) | ? [X327 : term,X326 : (subst > term > subst > $o),X325 : (term > $o)] : (! [X329 : term] : (((X325 @ X329)) = ((X326 @ id @ X329 @ id))) & ($true != ((X325 @ (lam @ X327)))) & ! [X330 : (subst > term > term)] : (($true = ((X326 @ id @ (lam @ (X330 @ sh @ one)) @ id))) | ? [X334 : term] : (($true != ((X326 @ id @ (X330 @ id @ X334) @ id))) & ($true = ((X326 @ id @ X334 @ id)))) | ? [X332 : subst,X333 : term,X331 : subst] : (((X330 @ (comp @ X332 @ X331) @ (sub @ X333 @ X331))) != ((sub @ (X330 @ X332 @ X333) @ X331)))) & ! [X328 : term] : (($true != ((X325 @ X328))) | ($true = ((X325 @ (sub @ X327 @ (push @ X328 @ id)))))))) | ? [X324 : term] : (((sub @ X324 @ id)) != X324)) | ? [X318 : term,X317 : (term > $o)] : (! [X322 : term,X321 : term] : (($true = ((X317 @ (ap @ X321 @ X322)))) | ($true != ((X317 @ X322))) | ($true != ((X317 @ X321)))) & ($true != ((X317 @ X318))) & ! [X323 : term] : (($true != ((var @ X323))) | ($true = ((X317 @ X323)))) & ! [X319 : term] : (($true = ((X317 @ (lam @ X319)))) | ? [X320 : term] : (($true != ((X317 @ (sub @ X319 @ (push @ X320 @ id))))) & ($true = ((X317 @ X320))))))) | ? [X314 : (subst > term > subst > $o)] : ! [X315 : (term > $o)] : ? [X316 : term] : (((X314 @ id @ X316 @ id)) != ((X315 @ X316)))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & (((var @ (d2term @ d_one))) != $true) & (! [X351 : term,X349 : (subst > term > subst > $o),X350 : (term > $o)] : (? [X352 : (subst > term > term)] : (! [X356 : subst,X355 : subst,X354 : term] : (((X352 @ (comp @ X356 @ X355) @ (sub @ X354 @ X355))) = ((sub @ (X352 @ X356 @ X354) @ X355))) & ! [X353 : term] : (($true != ((X349 @ id @ X353 @ id))) | ($true = ((X349 @ id @ (X352 @ id @ X353) @ id)))) & ($true != ((X349 @ id @ (lam @ (X352 @ sh @ one)) @ id)))) | ($true = ((X350 @ (lam @ X351)))) | ? [X357 : term] : (((X350 @ X357)) != ((X349 @ id @ X357 @ id))) | ? [X358 : term] : (($true = ((X350 @ X358))) & ($true != ((X350 @ (sub @ X351 @ (push @ X358 @ id))))))) <=> (hoasinduction_lem3v2a = $true)) & (! [X488 : (subst > term > subst > $o),X489 : term] : ((((X488 @ id @ (lam @ (sub @ X489 @ (push @ one @ sh))) @ id)) = $true) | ? [X491 : term,X490 : (subst > term > term)] : (($true != ((X488 @ id @ (lam @ (X490 @ sh @ one)) @ id))) & ! [X497 : subst,X498 : subst,X496 : term] : (((sub @ (X490 @ X498 @ X496) @ X497)) = ((sub @ (sub @ X491 @ (push @ X496 @ X498)) @ X497))) & ! [X492 : subst,X494 : term,X493 : subst] : (((X490 @ (comp @ X492 @ X493) @ (sub @ X494 @ X493))) = ((sub @ X491 @ (push @ (sub @ X494 @ X493) @ (comp @ X492 @ X493))))) & ! [X495 : term] : (($true = ((X488 @ id @ (X490 @ id @ X495) @ id))) | ($true != ((X488 @ id @ X495 @ id))))) | ? [X499 : term] : (($true != ((X488 @ id @ (sub @ X489 @ (push @ X499 @ id)) @ id))) & ($true = ((X488 @ id @ X499 @ id))))) <=> (hoasinduction_lem3aaa = $true)) & (! [X63 : term,X60 : term,X61 : term,X62 : term] : ((X60 = X61) | (((ap @ (sub @ X61 @ id) @ X62)) != ((ap @ (sub @ X60 @ id) @ X63)))) <=> (hoasapinj1 = $true)) & ! [X210 : d_term,X211 : d_term] : ((((d2term @ X210)) != ((d2term @ X211))) | (X210 = X211)) & (hoasinduction_lem2_lthm = $true) & (! [X366 : (subst > term > subst > $o),X367 : term] : (? [X373 : term] : (($true = ((X366 @ id @ X373 @ id))) & ($true != ((X366 @ id @ (sub @ X367 @ (push @ X373 @ id)) @ id)))) | ? [X368 : (subst > term > term)] : (($true != ((X366 @ id @ (lam @ (X368 @ sh @ one)) @ id))) & ! [X372 : term] : (($true != ((X366 @ id @ X372 @ id))) | ($true = ((X366 @ id @ (X368 @ id @ X372) @ id)))) & ! [X370 : subst,X371 : term,X369 : subst] : (((X368 @ (comp @ X370 @ X369) @ (sub @ X371 @ X369))) = ((sub @ (X368 @ X370 @ X371) @ X369)))) | ($true = ((X366 @ id @ (lam @ (sub @ X367 @ (push @ one @ sh))) @ id)))) <=> (hoasinduction_lem3aa_lthm = $true)) & (hoasinduction_lem1v2_gthm = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (! [X40 : subst] : (((comp @ id @ X40)) = X40) <=> (axidl = $true)) & (hoasinduction_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (sh = ((d2subst @ d_id))) & ! [X146 : subst] : ? [X147 : d_subst] : (((d2subst @ X147)) = X146) & ((axscons = $true) <=> ! [X403 : subst] : (((push @ (sub @ one @ X403) @ (comp @ sh @ X403))) = X403)) & ! [X476 : d_term] : (d_one = X476) & (hoasinduction_lem2_gthm = $true) & (! [X116 : term] : (((sub @ X116 @ id)) = X116) <=> (axvarid = $true)) & (hoasinduction_lem2v2_gthm = $true) & ((((! [X407 : subst] : (((comp @ id @ X407)) = X407) & ! [X406 : subst] : (((comp @ X406 @ id)) = X406)) | ? [X405 : subst] : (((comp @ X405 @ id)) != X405)) | ? [X404 : subst] : (((comp @ id @ X404)) != X404)) <=> (substmonoid_lthm = $true)) & (((! [X305 : (term > $o),X304 : term,X306 : (subst > term > subst > $o)] : (($true = ((X305 @ (lam @ X304)))) | ? [X307 : (subst > term > term)] : (! [X309 : subst,X308 : term,X310 : subst] : (((X307 @ (comp @ X309 @ X310) @ (sub @ X308 @ X310))) = ((sub @ (X307 @ X309 @ X308) @ X310))) & ! [X311 : term] : (($true = ((X306 @ id @ (X307 @ id @ X311) @ id))) | ($true != ((X306 @ id @ X311 @ id)))) & ($true != ((X306 @ id @ (lam @ (X307 @ sh @ one)) @ id)))) | ? [X312 : term] : (((X305 @ X312)) != ((X306 @ id @ X312 @ id))) | ? [X313 : term] : (($true = ((X305 @ X313))) & ($true != ((X305 @ (sub @ X304 @ (push @ X313 @ id))))))) | ? [X303 : term] : (((sub @ X303 @ id)) != X303)) | ? [X299 : term] : ! [X300 : (subst > term > term)] : ? [X302 : term,X301 : subst] : (((X300 @ X301 @ X302)) != ((sub @ X299 @ (push @ X302 @ X301))))) <=> (hoasinduction_lem3v2a_lthm = $true)) & (! [X65 : (subst > term > subst > $o),X64 : term] : (? [X68 : subst,X69 : subst,X67 : term,X66 : subst] : (($true = ((X65 @ X66 @ X67 @ (comp @ X69 @ X68)))) & ($true != ((X65 @ (comp @ X66 @ X69) @ (sub @ X67 @ X69) @ X68)))) | ($true = ((X65 @ id @ (lam @ X64) @ id))) | ? [X71 : (subst > term > term)] : (! [X75 : term] : (($true != ((X65 @ id @ X75 @ id))) | ($true = ((X65 @ id @ (X71 @ id @ X75) @ id)))) & ! [X73 : subst,X74 : term,X72 : subst] : (((X71 @ (comp @ X73 @ X72) @ (sub @ X74 @ X72))) = ((sub @ (X71 @ X73 @ X74) @ X72))) & ($true != ((X65 @ id @ (lam @ (X71 @ sh @ one)) @ id)))) | ? [X76 : term,X79 : subst,X78 : subst,X77 : subst] : (($true = ((X65 @ (comp @ X77 @ X79) @ (sub @ X76 @ X79) @ X78))) & ($true != ((X65 @ X77 @ X76 @ (comp @ X79 @ X78))))) | ? [X70 : term] : (($true = ((X65 @ id @ X70 @ id))) & ($true != ((X65 @ id @ (sub @ X64 @ (push @ X70 @ id)) @ id))))) <=> (hoasinduction_lem3 = $true)) & ((hoasinduction_lem3aa = $true) <=> ! [X524 : (subst > term > subst > $o),X523 : term] : (($true = ((X524 @ id @ (lam @ (sub @ X523 @ (push @ one @ sh))) @ id))) | ? [X525 : (subst > term > term)] : (($true != ((X524 @ id @ (lam @ (X525 @ sh @ one)) @ id))) & ! [X528 : term,X527 : subst,X526 : subst] : (((sub @ (X525 @ X527 @ X528) @ X526)) = ((X525 @ (comp @ X527 @ X526) @ (sub @ X528 @ X526)))) & ! [X529 : term] : (($true != ((X524 @ id @ X529 @ id))) | ($true = ((X524 @ id @ (X525 @ id @ X529) @ id))))) | ? [X530 : term] : (($true != ((X524 @ id @ (sub @ X523 @ (push @ X530 @ id)) @ id))) & ($true = ((X524 @ id @ X530 @ id)))))) & (((d2subst @ d_id)) = id) & ((hoaslaminj = $true) <=> ! [X461 : (subst > term > term),X462 : term,X459 : subst,X460 : (subst > term > term)] : ((((lam @ (X461 @ sh @ one))) != ((lam @ (X460 @ sh @ one)))) | ? [X468 : subst,X466 : term,X467 : subst] : (((X461 @ (comp @ X468 @ X467) @ (sub @ X466 @ X467))) != ((sub @ (X461 @ X468 @ X466) @ X467))) | ? [X464 : term,X465 : subst,X463 : subst] : (((X460 @ (comp @ X463 @ X465) @ (sub @ X464 @ X465))) != ((sub @ (X460 @ X463 @ X464) @ X465))) | (((X460 @ X459 @ X462)) = ((X461 @ X459 @ X462))))) & (hoaslamnotap_gthm = $true) & (pushprop_lem1_gthm = $true) & ! [X423 : term] : ? [X424 : d_term] : (((d2term @ X424)) = X423) & (pushprop_lem2v2 = $true) & (hoaslamnotvar = $true) & (hoasinduction_lem3a_gthm = $true)),
  inference(ennf_transformation,[],[f10])).
thf(f12,plain,(
  ((hoaslaminj = $true) <=> ! [X461 : (subst > term > term),X462 : term,X459 : subst,X460 : (subst > term > term)] : ((((lam @ (X461 @ sh @ one))) != ((lam @ (X460 @ sh @ one)))) | ? [X468 : subst,X466 : term,X467 : subst] : (((X461 @ (comp @ X468 @ X467) @ (sub @ X466 @ X467))) != ((sub @ (X461 @ X468 @ X466) @ X467))) | ? [X464 : term,X465 : subst,X463 : subst] : (((X460 @ (comp @ X463 @ X465) @ (sub @ X464 @ X465))) != ((sub @ (X460 @ X463 @ X464) @ X465))) | (((X460 @ X459 @ X462)) = ((X461 @ X459 @ X462))))) & ((axshiftcons = $true) <=> ! [X443 : subst,X444 : term] : (((comp @ sh @ (push @ X444 @ X443))) = X443)) & ((pushprop_lem0 = $true) <=> ! [X102 : (term > $o),X103 : term,X101 : subst] : ? [X104 : (term > $o)] : ! [X105 : term] : (((X104 @ X105)) = ((X102 @ (sub @ X105 @ (push @ X103 @ X101)))))) & ! [X5 : d_subst] : (d_id = X5) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X419 : term] : ? [X420 : (subst > term > term)] : ! [X421 : subst,X422 : term] : (((X420 @ X421 @ X422)) = ((sub @ X419 @ (push @ X422 @ X421))))) & (axclos = $true) & (hoasinduction_lem1_lthm = $true) & (termmset_gthm = $true) & (! [X65 : (subst > term > subst > $o),X64 : term] : (? [X68 : subst,X69 : subst,X67 : term,X66 : subst] : (($true = ((X65 @ X66 @ X67 @ (comp @ X69 @ X68)))) & ($true != ((X65 @ (comp @ X66 @ X69) @ (sub @ X67 @ X69) @ X68)))) | ($true = ((X65 @ id @ (lam @ X64) @ id))) | ? [X71 : (subst > term > term)] : (! [X75 : term] : (($true != ((X65 @ id @ X75 @ id))) | ($true = ((X65 @ id @ (X71 @ id @ X75) @ id)))) & ! [X73 : subst,X74 : term,X72 : subst] : (((X71 @ (comp @ X73 @ X72) @ (sub @ X74 @ X72))) = ((sub @ (X71 @ X73 @ X74) @ X72))) & ($true != ((X65 @ id @ (lam @ (X71 @ sh @ one)) @ id)))) | ? [X76 : term,X79 : subst,X78 : subst,X77 : subst] : (($true = ((X65 @ (comp @ X77 @ X79) @ (sub @ X76 @ X79) @ X78))) & ($true != ((X65 @ X77 @ X76 @ (comp @ X79 @ X78))))) | ? [X70 : term] : (($true = ((X65 @ id @ X70 @ id))) & ($true != ((X65 @ id @ (sub @ X64 @ (push @ X70 @ id)) @ id))))) <=> (hoasinduction_lem3 = $true)) & ((! [X242 : term,X244 : term,X241 : term,X243 : term] : ((X242 = X244) | (((ap @ (sub @ X244 @ id) @ X241)) != ((ap @ (sub @ X242 @ id) @ X243)))) | ? [X236 : term] : (((sub @ X236 @ id)) != X236) | ? [X240 : term,X239 : term,X238 : term,X237 : term] : ((X237 != X238) & (((ap @ X237 @ X239)) = ((ap @ X238 @ X240))))) <=> (hoasapinj1_lthm = $true)) & (axvarshift = $true) & (pushprop_gthm = $true) & (! [X411 : term,X410 : (subst > term > term),X412 : term] : (? [X415 : term,X414 : subst,X413 : subst] : (((X410 @ (comp @ X414 @ X413) @ (sub @ X415 @ X413))) != ((sub @ (X410 @ X414 @ X415) @ X413))) | (((ap @ (sub @ X412 @ id) @ X411)) != ((lam @ (X410 @ sh @ one))))) <=> (hoaslamnotap = $true)) & (! [X188 : term,X189 : (subst > term > subst > $o),X187 : term,X190 : (term > $o)] : (($true != ((X190 @ X187))) | ($true != ((X190 @ X188))) | ? [X194 : subst,X195 : term,X193 : subst,X192 : subst] : (($true = ((X189 @ (comp @ X194 @ X193) @ (sub @ X195 @ X193) @ X192))) & (((X189 @ X194 @ X195 @ (comp @ X193 @ X192))) != $true)) | ? [X191 : term] : (((X190 @ X191)) != ((X189 @ id @ X191 @ id))) | ? [X198 : term,X199 : subst,X197 : subst,X196 : subst] : (($true = ((X189 @ X199 @ X198 @ (comp @ X196 @ X197)))) & ($true != ((X189 @ (comp @ X199 @ X196) @ (sub @ X198 @ X196) @ X197)))) | ? [X200 : term,X201 : term] : (($true = ((X189 @ id @ X201 @ id))) & ($true = ((X189 @ id @ X200 @ id))) & ($true != ((X189 @ id @ (ap @ (sub @ X200 @ id) @ X201) @ id)))) | ($true = ((X190 @ (ap @ X188 @ X187))))) <=> (hoasinduction_lem2v2 = $true)) & (pushprop_lem0_gthm = $true) & ! [X409 : d_subst,X408 : d_subst] : ((((d2subst @ X409)) != ((d2subst @ X408))) | (X408 = X409)) & (hoasinduction_lem1_gthm = $true) & (hoasinduction_lem1v2 = $true) & (substmonoid_gthm = $true) & (pushprop_lem3v2_lthm = $true) & (pushprop_lem2v2_lthm = $true) & (((d2subst @ d_id)) = id) & (((var @ (d2term @ d_one))) != $true) & (axassoc = $true) & (lamnotap != $true) & ((hoasinduction_lem3a = $true) <=> ! [X515 : (subst > term > subst > $o),X516 : term] : (($true = ((X515 @ id @ (lam @ X516) @ id))) | ? [X518 : (subst > term > term)] : (! [X521 : subst,X520 : term,X522 : subst] : (((X518 @ (comp @ X521 @ X522) @ (sub @ X520 @ X522))) = ((sub @ (X518 @ X521 @ X520) @ X522))) & ! [X519 : term] : (($true != ((X515 @ id @ X519 @ id))) | ($true = ((X515 @ id @ (X518 @ id @ X519) @ id)))) & ($true != ((X515 @ id @ (lam @ (X518 @ sh @ one)) @ id)))) | ? [X517 : term] : (($true = ((X515 @ id @ X517 @ id))) & ($true != ((X515 @ id @ (sub @ X516 @ (push @ X517 @ id)) @ id)))))) & ((induction2_lthm = $true) <=> (? [X25 : subst,X27 : (term > $o),X26 : term] : (($true != ((X27 @ (sub @ X26 @ X25)))) & ! [X32 : term] : (($true != ((var @ X32))) | ($true = ((X27 @ (sub @ X32 @ X25))))) & ! [X28 : term] : (($true = ((X27 @ (lam @ X28)))) | ? [X29 : term] : (($true != ((X27 @ (sub @ X28 @ (push @ X29 @ id))))) & ($true = ((X27 @ X29))))) & ! [X30 : term,X31 : term] : (($true != ((X27 @ X31))) | ($true = ((X27 @ (ap @ X30 @ X31)))) | ($true != ((X27 @ X30))))) | ! [X33 : (term > $o),X34 : term] : (? [X39 : term] : (($true = ((var @ X39))) & ($true != ((X33 @ X39)))) | ? [X38 : term,X37 : term] : (($true = ((X33 @ X37))) & ($true = ((X33 @ X38))) & ($true != ((X33 @ (ap @ X38 @ X37))))) | ($true = ((X33 @ X34))) | ? [X35 : term] : (($true != ((X33 @ (lam @ X35)))) & ! [X36 : term] : (($true != ((X33 @ X36))) | ($true = ((X33 @ (sub @ X35 @ (push @ X36 @ id)))))))) | ? [X24 : term] : (((sub @ X24 @ id)) != X24))) & ((? [X252 : (subst > term > subst > $o),X254 : term,X253 : term] : (($true != ((X252 @ id @ (ap @ X253 @ X254) @ id))) & ! [X259 : subst,X261 : subst,X260 : term,X262 : subst] : (($true != ((X252 @ X259 @ X260 @ (comp @ X262 @ X261)))) | ($true = ((X252 @ (comp @ X259 @ X262) @ (sub @ X260 @ X262) @ X261)))) & ! [X263 : term,X264 : term] : (($true != ((X252 @ id @ X264 @ id))) | ($true = ((X252 @ id @ (ap @ (sub @ X264 @ id) @ X263) @ id))) | ($true != ((X252 @ id @ X263 @ id)))) & ! [X256 : subst,X255 : subst,X258 : term,X257 : subst] : (($true = ((X252 @ X256 @ X258 @ (comp @ X255 @ X257)))) | ($true != ((X252 @ (comp @ X256 @ X255) @ (sub @ X258 @ X255) @ X257)))) & ($true = ((X252 @ id @ X253 @ id))) & ($true = ((X252 @ id @ X254 @ id)))) | ? [X266 : (subst > term > subst > $o),X265 : term] : (! [X270 : subst,X269 : subst,X268 : subst,X267 : term] : (($true = ((X266 @ X269 @ X267 @ (comp @ X268 @ X270)))) | ($true != ((X266 @ (comp @ X269 @ X268) @ (sub @ X267 @ X268) @ X270)))) & ! [X271 : subst,X272 : subst,X274 : term,X273 : subst] : (($true = ((X266 @ (comp @ X272 @ X273) @ (sub @ X274 @ X273) @ X271))) | ($true != ((X266 @ X272 @ X274 @ (comp @ X273 @ X271))))) & ! [X275 : term] : (($true != ((X266 @ id @ X275 @ id))) | ($true = ((X266 @ id @ (sub @ X265 @ (push @ X275 @ id)) @ id)))) & ($true != ((X266 @ id @ (lam @ X265) @ id))) & ! [X276 : (subst > term > term)] : (? [X278 : subst,X280 : term,X279 : subst] : (((sub @ (X276 @ X278 @ X280) @ X279)) != ((X276 @ (comp @ X278 @ X279) @ (sub @ X280 @ X279)))) | ($true = ((X266 @ id @ (lam @ (X276 @ sh @ one)) @ id))) | ? [X277 : term] : (($true = ((X266 @ id @ X277 @ id))) & ($true != ((X266 @ id @ (X276 @ id @ X277) @ id)))))) | ! [X281 : (subst > term > subst > $o),X282 : term] : (($true = ((X281 @ id @ X282 @ id))) | ? [X297 : term,X298 : term] : (($true != ((X281 @ id @ (ap @ (sub @ X297 @ id) @ X298) @ id))) & ($true = ((X281 @ id @ X297 @ id))) & ($true = ((X281 @ id @ X298 @ id)))) | ? [X290 : term,X289 : subst,X288 : subst,X291 : subst] : (($true != ((X281 @ (comp @ X288 @ X289) @ (sub @ X290 @ X289) @ X291))) & ($true = ((X281 @ X288 @ X290 @ (comp @ X289 @ X291))))) | ? [X293 : subst,X295 : subst,X294 : term,X292 : subst] : (($true = ((X281 @ (comp @ X292 @ X295) @ (sub @ X294 @ X295) @ X293))) & ($true != ((X281 @ X292 @ X294 @ (comp @ X295 @ X293))))) | ? [X283 : (subst > term > term)] : (! [X285 : term,X286 : subst,X287 : subst] : (((sub @ (X283 @ X286 @ X285) @ X287)) = ((X283 @ (comp @ X286 @ X287) @ (sub @ X285 @ X287)))) & ($true != ((X281 @ id @ (lam @ (X283 @ sh @ one)) @ id))) & ! [X284 : term] : (($true != ((X281 @ id @ X284 @ id))) | ($true = ((X281 @ id @ (X283 @ id @ X284) @ id))))) | ? [X296 : term] : (($true != ((X281 @ id @ X296 @ id))) & ($true = ((var @ (sub @ X296 @ id)))))) | ? [X246 : (term > $o),X245 : term] : (! [X247 : term] : (($true != ((var @ X247))) | ($true = ((X246 @ X247)))) & ! [X248 : term] : (($true = ((X246 @ (lam @ X248)))) | ? [X249 : term] : (($true != ((X246 @ (sub @ X248 @ (push @ X249 @ id))))) & ($true = ((X246 @ X249))))) & ! [X250 : term,X251 : term] : (($true = ((X246 @ (ap @ X250 @ X251)))) | ($true != ((X246 @ X250))) | ($true != ((X246 @ X251)))) & ($true != ((X246 @ X245))))) <=> (hoasinduction_lthm = $true)) & (hoasinduction_lem2_gthm = $true) & (hoasinduction_lem2v2_gthm = $true) & (pushprop_lem3v2 = $true) & (pushprop_lthm = $true) & (! [X40 : subst] : (((comp @ id @ X40)) = X40) <=> (axidl = $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((pushprop_lem1_lthm = $true) <=> (? [X508 : subst,X509 : term] : (((comp @ sh @ (push @ X509 @ X508))) != X508) | ! [X511 : term,X514 : term,X510 : subst,X512 : (term > $o),X513 : (term > $o)] : (($true = ((X512 @ (sub @ X511 @ (push @ X514 @ X510))))) | ($true != ((X513 @ X511)))) | ? [X507 : subst,X506 : term] : (((sub @ one @ (push @ X506 @ X507))) != X506))) & (pushprop_lem1v2_gthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & ((? [X448 : term,X449 : term,X450 : term,X447 : term] : ((((ap @ X447 @ X449)) = ((ap @ X450 @ X448))) & (X448 != X449)) | ! [X453 : term,X454 : term,X451 : term,X452 : term] : ((((ap @ (sub @ X452 @ id) @ X451)) != ((ap @ (sub @ X454 @ id) @ X453))) | (X451 = X453))) <=> (hoasapinj2_lthm = $true)) & (! [X475 : term] : (((sub @ X475 @ id)) = X475) <=> (termmset = $true)) & (pushprop_lthm_orig = $true) & ((? [X479 : term,X480 : subst] : (((sub @ one @ (push @ X479 @ X480))) != X479) | ! [X484 : (term > $o),X482 : (term > $o),X483 : subst,X481 : term] : (($true != ((X482 @ X481))) | ? [X485 : term] : (((X484 @ X485)) != ((X482 @ (sub @ X485 @ (push @ X481 @ X483))))) | ($true = ((X484 @ one))))) <=> (pushprop_lem1v2_lthm = $true)) & (ulamvar1 != $true) & ((induction2 = $true) <=> ! [X360 : term,X359 : (term > $o)] : (? [X363 : term] : (($true != ((X359 @ X363))) & ($true = ((var @ X363)))) | ? [X361 : term] : (($true != ((X359 @ (lam @ X361)))) & ! [X362 : term] : (($true != ((X359 @ X362))) | ($true = ((X359 @ (sub @ X361 @ (push @ X362 @ id))))))) | ($true = ((X359 @ X360))) | ? [X365 : term,X364 : term] : (($true = ((X359 @ X365))) & ($true != ((X359 @ (ap @ X365 @ X364)))) & ($true = ((X359 @ X364)))))) & ! [X210 : d_term,X211 : d_term] : ((((d2term @ X210)) != ((d2term @ X211))) | (X210 = X211)) & ((pushprop_lem1v2 = $true) <=> ! [X111 : (term > $o),X114 : subst,X113 : term,X112 : (term > $o)] : (($true = ((X112 @ one))) | ($true != ((X111 @ X113))) | ? [X115 : term] : (((X112 @ X115)) != ((X111 @ (sub @ X115 @ (push @ X113 @ X114))))))) & (hoasinduction_lem3b_gthm = $true) & ((hoasinduction_lem3_lthm = $true) <=> (? [X379 : term,X380 : (subst > term > subst > $o)] : (! [X381 : (subst > term > term)] : (? [X382 : term] : (($true = ((X380 @ id @ X382 @ id))) & ($true != ((X380 @ id @ (X381 @ id @ X382) @ id)))) | ($true = ((X380 @ id @ (lam @ (X381 @ sh @ one)) @ id))) | ? [X383 : subst,X384 : term,X385 : subst] : (((X381 @ (comp @ X383 @ X385) @ (sub @ X384 @ X385))) != ((sub @ (X381 @ X383 @ X384) @ X385)))) & ($true != ((X380 @ id @ (lam @ (sub @ X379 @ (push @ one @ sh))) @ id))) & ! [X386 : term] : (($true = ((X380 @ id @ (sub @ X379 @ (push @ X386 @ id)) @ id))) | ($true != ((X380 @ id @ X386 @ id))))) | ? [X378 : term] : (((sub @ X378 @ id)) != X378) | ! [X388 : term,X387 : (subst > term > subst > $o)] : (? [X398 : (subst > term > term)] : (($true != ((X387 @ id @ (lam @ (X398 @ sh @ one)) @ id))) & ! [X402 : term] : (($true = ((X387 @ id @ (X398 @ id @ X402) @ id))) | ($true != ((X387 @ id @ X402 @ id)))) & ! [X400 : term,X401 : subst,X399 : subst] : (((X398 @ (comp @ X399 @ X401) @ (sub @ X400 @ X401))) = ((sub @ (X398 @ X399 @ X400) @ X401)))) | ? [X393 : term] : (($true != ((X387 @ id @ (sub @ X388 @ (push @ X393 @ id)) @ id))) & ($true = ((X387 @ id @ X393 @ id)))) | ? [X389 : subst,X391 : term,X390 : subst,X392 : subst] : (($true != ((X387 @ (comp @ X392 @ X389) @ (sub @ X391 @ X389) @ X390))) & ($true = ((X387 @ X392 @ X391 @ (comp @ X389 @ X390))))) | ($true = ((X387 @ id @ (lam @ X388) @ id))) | ? [X395 : subst,X394 : subst,X397 : subst,X396 : term] : (($true = ((X387 @ (comp @ X397 @ X395) @ (sub @ X396 @ X395) @ X394))) & ($true != ((X387 @ X397 @ X396 @ (comp @ X395 @ X394)))))))) & (! [X456 : term,X455 : term,X457 : term,X458 : term] : ((X455 = X458) | (((ap @ X458 @ X457)) != ((ap @ X455 @ X456)))) <=> (apinj1 = $true)) & ((hoasinduction = $true) <=> ! [X6 : term,X7 : (subst > term > subst > $o)] : (? [X13 : term,X12 : subst,X11 : subst,X10 : subst] : (($true = ((X7 @ X11 @ X13 @ (comp @ X12 @ X10)))) & ($true != ((X7 @ (comp @ X11 @ X12) @ (sub @ X13 @ X12) @ X10)))) | ? [X19 : (subst > term > term)] : (($true != ((X7 @ id @ (lam @ (X19 @ sh @ one)) @ id))) & ! [X22 : subst,X21 : term,X20 : subst] : (((sub @ (X19 @ X22 @ X21) @ X20)) = ((X19 @ (comp @ X22 @ X20) @ (sub @ X21 @ X20)))) & ! [X23 : term] : ((((X7 @ id @ X23 @ id)) != $true) | ($true = ((X7 @ id @ (X19 @ id @ X23) @ id))))) | ? [X18 : term] : (($true != ((X7 @ id @ X18 @ id))) & ($true = ((var @ (sub @ X18 @ id))))) | ? [X16 : subst,X15 : term,X17 : subst,X14 : subst] : (($true != ((X7 @ X14 @ X15 @ (comp @ X16 @ X17)))) & ($true = ((X7 @ (comp @ X14 @ X16) @ (sub @ X15 @ X16) @ X17)))) | ? [X8 : term,X9 : term] : (($true = ((X7 @ id @ X8 @ id))) & (((X7 @ id @ X9 @ id)) = $true) & ($true != ((X7 @ id @ (ap @ (sub @ X9 @ id) @ X8) @ id)))) | ($true = ((X7 @ id @ X6 @ id))))) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((? [X45 : subst] : (((comp @ X45 @ id)) != X45) | ? [X44 : term,X43 : subst] : (((comp @ sh @ (push @ X44 @ X43))) != X43) | ? [X41 : subst,X42 : term] : (((sub @ one @ (push @ X42 @ X41))) != X42) | ? [X46 : (term > $o),X47 : term] : (! [X51 : term] : (($true != ((X46 @ X51))) | ($true = ((X46 @ (lam @ X51))))) & ! [X48 : term] : (($true = ((X46 @ X48))) | ($true != ((var @ X48)))) & ! [X50 : term,X49 : term] : (($true != ((X46 @ X49))) | ($true != ((X46 @ X50))) | ($true = ((X46 @ (ap @ X49 @ X50))))) & ($true != ((X46 @ X47)))) | ! [X52 : (term > $o),X53 : subst,X54 : term] : (? [X56 : term,X55 : term] : (($true != ((X52 @ (ap @ X55 @ X56)))) & ($true = ((X52 @ X56))) & (((X52 @ X55)) = $true)) | ? [X57 : term] : (! [X58 : term] : (($true != ((X52 @ X58))) | ($true = ((X52 @ (sub @ X57 @ (push @ X58 @ id)))))) & ($true != ((X52 @ (lam @ X57))))) | ? [X59 : term] : (($true != ((X52 @ (sub @ X59 @ X53)))) & ($true = ((var @ X59)))) | ($true = ((X52 @ (sub @ X54 @ X53)))))) <=> (induction2lem_lthm = $true)) & (pushprop_lem1_gthm = $true) & (one = ((d2term @ d_one))) & (hoasinduction_gthm = $true) & (hoaslamnotvar = $true) & (hoasinduction_lem2_lthm = $true) & ((shinj = $true) <=> ! [X470 : term,X469 : term] : ((X469 = X470) | (((sub @ X470 @ sh)) != ((sub @ X469 @ sh))))) & (hoasinduction_lem3v2_gthm = $true) & ((? [X84 : term] : (((sub @ X84 @ id)) != X84) | ? [X86 : term,X85 : (subst > term > subst > $o)] : (! [X92 : term] : (($true = ((X85 @ id @ (sub @ X86 @ (push @ X92 @ id)) @ id))) | ($true != ((X85 @ id @ X92 @ id)))) & ! [X87 : (subst > term > term)] : (? [X91 : term] : (($true != ((X85 @ id @ (X87 @ id @ X91) @ id))) & ($true = ((X85 @ id @ X91 @ id)))) | ? [X90 : term,X88 : subst,X89 : subst] : (((X87 @ (comp @ X88 @ X89) @ (sub @ X90 @ X89))) != ((sub @ (X87 @ X88 @ X90) @ X89))) | ($true = ((X85 @ id @ (lam @ (X87 @ sh @ one)) @ id)))) & ($true != ((X85 @ id @ (lam @ (sub @ X86 @ (push @ one @ sh))) @ id)))) | ! [X94 : term,X93 : (subst > term > subst > $o)] : (($true = ((X93 @ id @ (lam @ X94) @ id))) | ? [X95 : term] : (($true != ((X93 @ id @ (sub @ X94 @ (push @ X95 @ id)) @ id))) & ($true = ((X93 @ id @ X95 @ id)))) | ? [X96 : (subst > term > term)] : (($true != ((X93 @ id @ (lam @ (X96 @ sh @ one)) @ id))) & ! [X97 : term] : (($true != ((X93 @ id @ X97 @ id))) | ($true = ((X93 @ id @ (X96 @ id @ X97) @ id)))) & ! [X99 : term,X100 : subst,X98 : subst] : (((sub @ (X96 @ X100 @ X99) @ X98)) = ((X96 @ (comp @ X100 @ X98) @ (sub @ X99 @ X98))))))) <=> (hoasinduction_lem3a_lthm = $true)) & (! [X477 : term] : ? [X478 : (subst > term > term)] : (((X478 @ sh @ one)) = ((sub @ X477 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b = $true)) & (hoasapnotvar_lthm = $true) & (axmap = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (! [X531 : term,X532 : (subst > term > subst > $o)] : (($true = ((X532 @ id @ X531 @ id))) | ? [X533 : (subst > term > term)] : (($true != ((X532 @ id @ (lam @ (X533 @ sh @ one)) @ id))) & ! [X534 : term] : (($true = ((X532 @ id @ (X533 @ id @ X534) @ id))) | ($true != ((X532 @ id @ X534 @ id)))) & ! [X536 : term,X537 : subst,X535 : subst] : (((X533 @ (comp @ X535 @ X537) @ (sub @ X536 @ X537))) = ((sub @ (X533 @ X535 @ X536) @ X537)))) | ? [X538 : term,X539 : term] : (($true != ((X532 @ id @ (ap @ (sub @ X539 @ id) @ X538) @ id))) & ($true = ((X532 @ id @ X539 @ id))) & ($true = ((X532 @ id @ X538 @ id))))) <=> (hoasinduction_no_psi_cond = $true)) & ! [X423 : term] : ? [X424 : d_term] : (((d2term @ X424)) = X423) & (hoasinduction_lem1 = $true) & ((? [X161 : term,X160 : (term > $o),X159 : (subst > term > subst > $o)] : (! [X164 : (subst > term > term)] : (? [X166 : subst,X167 : subst,X168 : term] : (((X164 @ (comp @ X166 @ X167) @ (sub @ X168 @ X167))) != ((sub @ (X164 @ X166 @ X168) @ X167))) | ($true = ((X159 @ id @ (lam @ (X164 @ sh @ one)) @ id))) | ? [X165 : term] : (($true = ((X159 @ id @ X165 @ id))) & ($true != ((X159 @ id @ (X164 @ id @ X165) @ id))))) & ($true != ((X160 @ (lam @ X161)))) & ! [X162 : term] : (((X160 @ X162)) = ((X159 @ id @ X162 @ id))) & ! [X163 : term] : (($true != ((X160 @ X163))) | ($true = ((X160 @ (sub @ X161 @ (push @ X163 @ id))))))) | ? [X151 : (term > $o),X152 : term] : (! [X153 : term,X154 : term] : (($true = ((X151 @ (ap @ X154 @ X153)))) | ($true != ((X151 @ X154))) | ($true != ((X151 @ X153)))) & ! [X155 : term] : (? [X156 : term] : (($true != ((X151 @ (sub @ X155 @ (push @ X156 @ id))))) & ($true = ((X151 @ X156)))) | ($true = ((X151 @ (lam @ X155))))) & ! [X157 : term] : (($true != ((var @ X157))) | ($true = ((X151 @ X157)))) & ($true != ((X151 @ X152)))) | ! [X169 : term,X170 : (subst > term > subst > $o)] : (($true = ((X170 @ id @ X169 @ id))) | ? [X185 : term,X186 : term] : (($true = ((X170 @ id @ X185 @ id))) & ($true = ((X170 @ id @ X186 @ id))) & ($true != ((X170 @ id @ (ap @ (sub @ X186 @ id) @ X185) @ id)))) | ? [X183 : subst,X184 : subst,X181 : term,X182 : subst] : (($true != ((X170 @ (comp @ X184 @ X182) @ (sub @ X181 @ X182) @ X183))) & ($true = ((X170 @ X184 @ X181 @ (comp @ X182 @ X183))))) | ? [X180 : term] : (($true != ((X170 @ id @ X180 @ id))) & ($true = ((var @ (sub @ X180 @ id))))) | ? [X178 : subst,X177 : subst,X176 : subst,X179 : term] : (($true != ((X170 @ X176 @ X179 @ (comp @ X177 @ X178)))) & ($true = ((X170 @ (comp @ X176 @ X177) @ (sub @ X179 @ X177) @ X178)))) | ? [X171 : (subst > term > term)] : (! [X172 : term] : ((((X170 @ id @ X172 @ id)) != $true) | ($true = ((X170 @ id @ (X171 @ id @ X172) @ id)))) & ($true != ((X170 @ id @ (lam @ (X171 @ sh @ one)) @ id))) & ! [X173 : subst,X174 : subst,X175 : term] : (((sub @ (X171 @ X173 @ X175) @ X174)) = ((X171 @ (comp @ X173 @ X174) @ (sub @ X175 @ X174)))))) | ? [X158 : term] : (((sub @ X158 @ id)) != X158) | ? [X148 : (subst > term > subst > $o)] : ! [X149 : (term > $o)] : ? [X150 : term] : (((X148 @ id @ X150 @ id)) != ((X149 @ X150)))) <=> (hoasinduction_lthm_3 = $true)) & (hoaslamnotap_lthm = $true) & (induction2_gthm = $true) & ((hoaslaminj_lthm = $true) <=> (? [X130 : subst,X131 : term] : (((sub @ one @ (push @ X131 @ X130))) != X131) | ? [X134 : term,X135 : term] : ((((lam @ X134)) = ((lam @ X135))) & (X134 != X135)) | ? [X133 : subst,X132 : term] : (((comp @ sh @ (push @ X132 @ X133))) != X133) | ! [X136 : term,X139 : (subst > term > term),X138 : subst,X137 : (subst > term > term)] : ((((lam @ (X139 @ sh @ one))) != ((lam @ (X137 @ sh @ one)))) | ? [X144 : subst,X143 : term,X145 : subst] : (((X137 @ (comp @ X145 @ X144) @ (sub @ X143 @ X144))) != ((sub @ (X137 @ X145 @ X143) @ X144))) | ? [X142 : subst,X140 : subst,X141 : term] : (((X139 @ (comp @ X142 @ X140) @ (sub @ X141 @ X140))) != ((sub @ (X139 @ X142 @ X141) @ X140))) | (((X137 @ X138 @ X136)) = ((X139 @ X138 @ X136)))))) & (! [X366 : (subst > term > subst > $o),X367 : term] : (? [X373 : term] : (($true = ((X366 @ id @ X373 @ id))) & ($true != ((X366 @ id @ (sub @ X367 @ (push @ X373 @ id)) @ id)))) | ? [X368 : (subst > term > term)] : (($true != ((X366 @ id @ (lam @ (X368 @ sh @ one)) @ id))) & ! [X372 : term] : (($true != ((X366 @ id @ X372 @ id))) | ($true = ((X366 @ id @ (X368 @ id @ X372) @ id)))) & ! [X370 : subst,X371 : term,X369 : subst] : (((X368 @ (comp @ X370 @ X369) @ (sub @ X371 @ X369))) = ((sub @ (X368 @ X370 @ X371) @ X369)))) | ($true = ((X366 @ id @ (lam @ (sub @ X367 @ (push @ one @ sh))) @ id)))) <=> (hoasinduction_lem3aa_lthm = $true)) & ((hoasinduction_lem3v2a_lthm = $true) <=> (? [X299 : term] : ! [X300 : (subst > term > term)] : ? [X302 : term,X301 : subst] : (((X300 @ X301 @ X302)) != ((sub @ X299 @ (push @ X302 @ X301)))) | ! [X305 : (term > $o),X304 : term,X306 : (subst > term > subst > $o)] : (($true = ((X305 @ (lam @ X304)))) | ? [X307 : (subst > term > term)] : (! [X309 : subst,X308 : term,X310 : subst] : (((X307 @ (comp @ X309 @ X310) @ (sub @ X308 @ X310))) = ((sub @ (X307 @ X309 @ X308) @ X310))) & ! [X311 : term] : (($true = ((X306 @ id @ (X307 @ id @ X311) @ id))) | ($true != ((X306 @ id @ X311 @ id)))) & ($true != ((X306 @ id @ (lam @ (X307 @ sh @ one)) @ id)))) | ? [X312 : term] : (((X305 @ X312)) != ((X306 @ id @ X312 @ id))) | ? [X313 : term] : (($true = ((X305 @ X313))) & ($true != ((X305 @ (sub @ X304 @ (push @ X313 @ id))))))) | ? [X303 : term] : (((sub @ X303 @ id)) != X303))) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (ulamvarind = $true) & ! [X476 : d_term] : (d_one = X476) & (hoasapinj1_gthm = $true) & (! [X80 : term,X83 : term,X82 : term,X81 : term] : ((X80 = X83) | (((ap @ X82 @ X83)) != ((ap @ X81 @ X80)))) <=> (apinj2 = $true)) & (hoaslaminj_gthm = $true) & (! [X117 : term,X118 : term,X119 : (subst > term > subst > $o)] : (($true != ((X119 @ id @ X117 @ id))) | ? [X124 : subst,X125 : subst,X123 : subst,X122 : term] : (($true = ((X119 @ (comp @ X123 @ X125) @ (sub @ X122 @ X125) @ X124))) & ($true != ((X119 @ X123 @ X122 @ (comp @ X125 @ X124))))) | ? [X128 : subst,X127 : term,X126 : subst,X129 : subst] : (($true != ((X119 @ (comp @ X126 @ X128) @ (sub @ X127 @ X128) @ X129))) & ($true = ((X119 @ X126 @ X127 @ (comp @ X128 @ X129))))) | ($true != ((X119 @ id @ X118 @ id))) | ($true = ((X119 @ id @ (ap @ X118 @ X117) @ id))) | ? [X120 : term,X121 : term] : (($true = ((X119 @ id @ X120 @ id))) & ($true != ((X119 @ id @ (ap @ (sub @ X120 @ id) @ X121) @ id))) & ($true = ((X119 @ id @ X121 @ id))))) <=> (hoasinduction_lem2 = $true)) & ((hoasinduction_lem3v2 = $true) <=> ! [X427 : (subst > term > subst > $o),X426 : (term > $o),X425 : term] : (? [X433 : (subst > term > term)] : (($true != ((X427 @ id @ (lam @ (X433 @ sh @ one)) @ id))) & ! [X434 : term] : (($true != ((X427 @ id @ X434 @ id))) | ($true = ((X427 @ id @ (X433 @ id @ X434) @ id)))) & ! [X437 : subst,X436 : term,X435 : subst] : (((X433 @ (comp @ X437 @ X435) @ (sub @ X436 @ X435))) = ((sub @ (X433 @ X437 @ X436) @ X435)))) | ? [X441 : subst,X440 : subst,X439 : subst,X442 : term] : (($true != ((X427 @ (comp @ X440 @ X441) @ (sub @ X442 @ X441) @ X439))) & ($true = ((X427 @ X440 @ X442 @ (comp @ X441 @ X439))))) | ? [X432 : term,X429 : subst,X431 : subst,X430 : subst] : (($true = ((X427 @ (comp @ X430 @ X431) @ (sub @ X432 @ X431) @ X429))) & ($true != ((X427 @ X430 @ X432 @ (comp @ X431 @ X429))))) | ? [X438 : term] : (((X426 @ X438)) != ((X427 @ id @ X438 @ id))) | ? [X428 : term] : (($true = ((X426 @ X428))) & ($true != ((X426 @ (sub @ X425 @ (push @ X428 @ id)))))) | ($true = ((X426 @ (lam @ X425)))))) & ((pushprop_lem1 = $true) <=> ! [X212 : subst,X216 : term,X215 : (term > $o),X213 : term,X214 : (term > $o)] : (($true = ((X215 @ (sub @ X213 @ (push @ X216 @ X212))))) | ($true != ((X214 @ X213))))) & (axabs = $true) & (! [X416 : (subst > term > subst > $o)] : ? [X417 : (term > $o)] : ! [X418 : term] : (((X416 @ id @ X418 @ id)) = ((X417 @ X418))) <=> (hoasinduction_lem0 = $true)) & (! [X446 : term,X445 : term] : ((((lam @ X446)) != ((lam @ X445))) | (X445 = X446)) <=> (laminj = $true)) & (hoasapnotvar = $true) & (pushprop_lem2v2_gthm = $true) & ((termmset_lthm = $true) <=> (? [X1 : term] : (((sub @ X1 @ id)) != X1) | ! [X2 : term] : (((sub @ X2 @ id)) = X2))) & ((hoasinduction_lem3aa = $true) <=> ! [X524 : (subst > term > subst > $o),X523 : term] : (($true = ((X524 @ id @ (lam @ (sub @ X523 @ (push @ one @ sh))) @ id))) | ? [X525 : (subst > term > term)] : (($true != ((X524 @ id @ (lam @ (X525 @ sh @ one)) @ id))) & ! [X528 : term,X527 : subst,X526 : subst] : (((sub @ (X525 @ X527 @ X528) @ X526)) = ((X525 @ (comp @ X527 @ X526) @ (sub @ X528 @ X526)))) & ! [X529 : term] : (($true != ((X524 @ id @ X529 @ id))) | ($true = ((X524 @ id @ (X525 @ id @ X529) @ id))))) | ? [X530 : term] : (($true != ((X524 @ id @ (sub @ X523 @ (push @ X530 @ id)) @ id))) & ($true = ((X524 @ id @ X530 @ id)))))) & (pushprop = $true) & (! [X204 : term,X203 : subst,X202 : (term > $o)] : (? [X209 : term] : (($true != ((X202 @ (sub @ X209 @ X203)))) & ($true = ((var @ X209)))) | ? [X205 : term] : (($true != ((X202 @ (lam @ X205)))) & ! [X206 : term] : (($true != ((X202 @ X206))) | ($true = ((X202 @ (sub @ X205 @ (push @ X206 @ id))))))) | ? [X207 : term,X208 : term] : (($true = ((X202 @ X208))) & ($true = ((X202 @ X207))) & ($true != ((X202 @ (ap @ X207 @ X208))))) | ($true = ((X202 @ (sub @ X204 @ X203))))) <=> (induction2lem = $true)) & (hoaslamnotvar_gthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (! [X348 : term,X347 : subst] : (((sub @ one @ (push @ X348 @ X347))) = X348) <=> (axvarcons = $true)) & ((hoasinduction_lem3v2_f = $true) <=> ! [X374 : term] : ? [X375 : (subst > term > term)] : ! [X376 : subst,X377 : term] : (((sub @ X374 @ (push @ X377 @ X376))) = ((X375 @ X376 @ X377)))) & ((axscons = $true) <=> ! [X403 : subst] : (((push @ (sub @ one @ X403) @ (comp @ sh @ X403))) = X403)) & (ulamvarsh = $true) & (! [X351 : term,X349 : (subst > term > subst > $o),X350 : (term > $o)] : (? [X352 : (subst > term > term)] : (! [X356 : subst,X355 : subst,X354 : term] : (((X352 @ (comp @ X356 @ X355) @ (sub @ X354 @ X355))) = ((sub @ (X352 @ X356 @ X354) @ X355))) & ! [X353 : term] : (($true != ((X349 @ id @ X353 @ id))) | ($true = ((X349 @ id @ (X352 @ id @ X353) @ id)))) & ($true != ((X349 @ id @ (lam @ (X352 @ sh @ one)) @ id)))) | ($true = ((X350 @ (lam @ X351)))) | ? [X357 : term] : (((X350 @ X357)) != ((X349 @ id @ X357 @ id))) | ? [X358 : term] : (($true = ((X350 @ X358))) & ($true != ((X350 @ (sub @ X351 @ (push @ X358 @ id))))))) <=> (hoasinduction_lem3v2a = $true)) & (hoasapinj2_gthm = $true) & (hoasinduction_lem3a_gthm = $true) & (sh = ((d2subst @ d_id))) & (axapp = $true) & (! [X116 : term] : (((sub @ X116 @ id)) = X116) <=> (axvarid = $true)) & (! [X0 : subst] : (((comp @ X0 @ id)) = X0) <=> (axidr = $true)) & (! [X63 : term,X60 : term,X61 : term,X62 : term] : ((X60 = X61) | (((ap @ (sub @ X61 @ id) @ X62)) != ((ap @ (sub @ X60 @ id) @ X63)))) <=> (hoasapinj1 = $true)) & (apnotvar = $true) & (! [X344 : (subst > term > subst > $o)] : ? [X345 : (term > $o)] : ! [X346 : term] : (((X345 @ X346)) = ((X344 @ id @ X346 @ id))) <=> (hoasinduction_lem0_lthm = $true)) & (! [X488 : (subst > term > subst > $o),X489 : term] : ((((X488 @ id @ (lam @ (sub @ X489 @ (push @ one @ sh))) @ id)) = $true) | ? [X491 : term,X490 : (subst > term > term)] : (($true != ((X488 @ id @ (lam @ (X490 @ sh @ one)) @ id))) & ! [X497 : subst,X498 : subst,X496 : term] : (((sub @ (X490 @ X498 @ X496) @ X497)) = ((sub @ (sub @ X491 @ (push @ X496 @ X498)) @ X497))) & ! [X492 : subst,X494 : term,X493 : subst] : (((X490 @ (comp @ X492 @ X493) @ (sub @ X494 @ X493))) = ((sub @ X491 @ (push @ (sub @ X494 @ X493) @ (comp @ X492 @ X493))))) & ! [X495 : term] : (($true = ((X488 @ id @ (X490 @ id @ X495) @ id))) | ($true != ((X488 @ id @ X495 @ id))))) | ? [X499 : term] : (($true != ((X488 @ id @ (sub @ X489 @ (push @ X499 @ id)) @ id))) & ($true = ((X488 @ id @ X499 @ id))))) <=> (hoasinduction_lem3aaa = $true)) & ((! [X4 : subst] : (((comp @ id @ X4)) = X4) & ! [X3 : subst] : (((comp @ X3 @ id)) = X3)) <=> (substmonoid = $true)) & (hoaslamnotap_gthm = $true) & (lamnotvar = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & (hoasinduction_lem3_gthm = $true) & (induction2lem_gthm = $true) & (hoaslamnotvar_lthm = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ! [X146 : subst] : ? [X147 : d_subst] : (((d2subst @ X147)) = X146) & (! [X486 : term] : ? [X487 : (subst > term > term)] : (((sub @ X486 @ (push @ one @ sh))) = ((X487 @ sh @ one))) <=> (hoasinduction_lem3b_lthm = $true)) & ((induction = $true) <=> ! [X501 : (term > $o),X500 : term] : (? [X504 : term,X505 : term] : (($true != ((X501 @ (ap @ X505 @ X504)))) & ($true = ((X501 @ X504))) & ($true = ((X501 @ X505)))) | ? [X503 : term] : (($true = ((var @ X503))) & ($true != ((X501 @ X503)))) | ? [X502 : term] : (($true = ((X501 @ X502))) & ($true != ((X501 @ (lam @ X502))))) | ($true = ((X501 @ X500))))) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (? [X314 : (subst > term > subst > $o)] : ! [X315 : (term > $o)] : ? [X316 : term] : (((X314 @ id @ X316 @ id)) != ((X315 @ X316))) | ! [X336 : (subst > term > subst > $o),X335 : term] : (? [X339 : (subst > term > term)] : (($true != ((X336 @ id @ (lam @ (X339 @ sh @ one)) @ id))) & ! [X340 : term] : (($true != ((X336 @ id @ X340 @ id))) | ($true = ((X336 @ id @ (X339 @ id @ X340) @ id)))) & ! [X343 : subst,X342 : subst,X341 : term] : (((sub @ (X339 @ X342 @ X341) @ X343)) = ((X339 @ (comp @ X342 @ X343) @ (sub @ X341 @ X343))))) | ? [X337 : term,X338 : term] : ((((X336 @ id @ X337 @ id)) = $true) & ($true != ((X336 @ id @ (ap @ (sub @ X337 @ id) @ X338) @ id))) & ($true = ((X336 @ id @ X338 @ id)))) | ($true = ((X336 @ id @ X335 @ id)))) | ? [X324 : term] : (((sub @ X324 @ id)) != X324) | ? [X318 : term,X317 : (term > $o)] : (! [X322 : term,X321 : term] : (($true = ((X317 @ (ap @ X321 @ X322)))) | ($true != ((X317 @ X322))) | ($true != ((X317 @ X321)))) & ($true != ((X317 @ X318))) & ! [X323 : term] : (($true != ((var @ X323))) | ($true = ((X317 @ X323)))) & ! [X319 : term] : (($true = ((X317 @ (lam @ X319)))) | ? [X320 : term] : (($true != ((X317 @ (sub @ X319 @ (push @ X320 @ id))))) & ($true = ((X317 @ X320)))))) | ? [X327 : term,X326 : (subst > term > subst > $o),X325 : (term > $o)] : (! [X329 : term] : (((X325 @ X329)) = ((X326 @ id @ X329 @ id))) & ($true != ((X325 @ (lam @ X327)))) & ! [X330 : (subst > term > term)] : (($true = ((X326 @ id @ (lam @ (X330 @ sh @ one)) @ id))) | ? [X334 : term] : (($true != ((X326 @ id @ (X330 @ id @ X334) @ id))) & ($true = ((X326 @ id @ X334 @ id)))) | ? [X332 : subst,X333 : term,X331 : subst] : (((X330 @ (comp @ X332 @ X331) @ (sub @ X333 @ X331))) != ((sub @ (X330 @ X332 @ X333) @ X331)))) & ! [X328 : term] : (($true != ((X325 @ X328))) | ($true = ((X325 @ (sub @ X327 @ (push @ X328 @ id))))))))) & ((hoasinduction_lem3v2_lthm = $true) <=> (! [X218 : term,X219 : (subst > term > subst > $o),X220 : (term > $o)] : (($true = ((X220 @ (lam @ X218)))) | ? [X229 : term] : (($true != ((X220 @ (sub @ X218 @ (push @ X229 @ id))))) & ($true = ((X220 @ X229)))) | ? [X230 : term] : (((X219 @ id @ X230 @ id)) != ((X220 @ X230))) | ? [X221 : subst,X222 : subst,X223 : term,X224 : subst] : (($true != ((X219 @ X224 @ X223 @ (comp @ X222 @ X221)))) & ($true = ((X219 @ (comp @ X224 @ X222) @ (sub @ X223 @ X222) @ X221)))) | ? [X231 : (subst > term > term)] : (! [X235 : term] : (($true != ((X219 @ id @ X235 @ id))) | ($true = ((X219 @ id @ (X231 @ id @ X235) @ id)))) & ($true != ((X219 @ id @ (lam @ (X231 @ sh @ one)) @ id))) & ! [X234 : subst,X233 : subst,X232 : term] : (((sub @ (X231 @ X234 @ X232) @ X233)) = ((X231 @ (comp @ X234 @ X233) @ (sub @ X232 @ X233))))) | ? [X225 : subst,X226 : term,X227 : subst,X228 : subst] : (($true != ((X219 @ (comp @ X225 @ X228) @ (sub @ X226 @ X228) @ X227))) & ($true = ((X219 @ X225 @ X226 @ (comp @ X228 @ X227)))))) | ? [X217 : term] : (((sub @ X217 @ id)) != X217))) & ((hoasapinj2 = $true) <=> ! [X473 : term,X472 : term,X471 : term,X474 : term] : ((X472 = X473) | (((ap @ (sub @ X474 @ id) @ X472)) != ((ap @ (sub @ X471 @ id) @ X473))))) & ((? [X405 : subst] : (((comp @ X405 @ id)) != X405) | (! [X407 : subst] : (((comp @ id @ X407)) = X407) & ! [X406 : subst] : (((comp @ X406 @ id)) = X406)) | ? [X404 : subst] : (((comp @ id @ X404)) != X404)) <=> (substmonoid_lthm = $true)) & ((pushprop_lem0_lthm = $true) <=> ! [X107 : subst,X108 : term,X106 : (term > $o)] : ? [X109 : (term > $o)] : ! [X110 : term] : (((X106 @ (sub @ X110 @ (push @ X108 @ X107)))) = ((X109 @ X110)))) & (pushprop_lem2v2 = $true)),
  inference(flattening,[],[f11])).
thf(f13,definition,(
  ($true = sP0) <=> (? [X405 : subst] : (((comp @ X405 @ id)) != X405) | (! [X407 : subst] : (((comp @ id @ X407)) = X407) & ! [X406 : subst] : (((comp @ X406 @ id)) = X406)) | ? [X404 : subst] : (((comp @ id @ X404)) != X404))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f14,definition,(
  ! [X219 : (subst > term > subst > $o)] : (($true = ((sP1 @ X219))) <=> ? [X231 : (subst > term > term)] : (! [X235 : term] : (($true != ((X219 @ id @ X235 @ id))) | ($true = ((X219 @ id @ (X231 @ id @ X235) @ id)))) & ($true != ((X219 @ id @ (lam @ (X231 @ sh @ one)) @ id))) & ! [X234 : subst,X233 : subst,X232 : term] : (((sub @ (X231 @ X234 @ X232) @ X233)) = ((X231 @ (comp @ X234 @ X233) @ (sub @ X232 @ X233))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f15,definition,(
  ! [X219 : (subst > term > subst > $o)] : (($true = ((sP2 @ X219))) <=> ? [X225 : subst,X226 : term,X227 : subst,X228 : subst] : (($true != ((X219 @ (comp @ X225 @ X228) @ (sub @ X226 @ X228) @ X227))) & ($true = ((X219 @ X225 @ X226 @ (comp @ X228 @ X227))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f16,definition,(
  ! [X219 : (subst > term > subst > $o)] : (($true = ((sP3 @ X219))) <=> ? [X221 : subst,X222 : subst,X223 : term,X224 : subst] : (($true != ((X219 @ X224 @ X223 @ (comp @ X222 @ X221)))) & ($true = ((X219 @ (comp @ X224 @ X222) @ (sub @ X223 @ X222) @ X221)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f17,definition,(
  ($true = sP4) <=> ! [X218 : term,X219 : (subst > term > subst > $o),X220 : (term > $o)] : (($true = ((X220 @ (lam @ X218)))) | ? [X229 : term] : (($true != ((X220 @ (sub @ X218 @ (push @ X229 @ id))))) & ($true = ((X220 @ X229)))) | ? [X230 : term] : (((X219 @ id @ X230 @ id)) != ((X220 @ X230))) | ($true = ((sP3 @ X219))) | ($true = ((sP1 @ X219))) | ($true = ((sP2 @ X219))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f18,definition,(
  ! [X326 : (subst > term > subst > $o)] : (($true = ((sP5 @ X326))) <=> ! [X330 : (subst > term > term)] : (($true = ((X326 @ id @ (lam @ (X330 @ sh @ one)) @ id))) | ? [X334 : term] : (($true != ((X326 @ id @ (X330 @ id @ X334) @ id))) & ($true = ((X326 @ id @ X334 @ id)))) | ? [X332 : subst,X333 : term,X331 : subst] : (((X330 @ (comp @ X332 @ X331) @ (sub @ X333 @ X331))) != ((sub @ (X330 @ X332 @ X333) @ X331)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f19,definition,(
  ! [X317 : (term > $o)] : (($true = ((sP6 @ X317))) <=> ! [X322 : term,X321 : term] : (($true = ((X317 @ (ap @ X321 @ X322)))) | ($true != ((X317 @ X322))) | ($true != ((X317 @ X321)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f20,definition,(
  ! [X317 : (term > $o)] : (($true = ((sP7 @ X317))) <=> ! [X319 : term] : (($true = ((X317 @ (lam @ X319)))) | ? [X320 : term] : (($true != ((X317 @ (sub @ X319 @ (push @ X320 @ id))))) & ($true = ((X317 @ X320))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f21,definition,(
  ! [X336 : (subst > term > subst > $o)] : (($true = ((sP8 @ X336))) <=> ? [X339 : (subst > term > term)] : (($true != ((X336 @ id @ (lam @ (X339 @ sh @ one)) @ id))) & ! [X340 : term] : (($true != ((X336 @ id @ X340 @ id))) | ($true = ((X336 @ id @ (X339 @ id @ X340) @ id)))) & ! [X343 : subst,X342 : subst,X341 : term] : (((sub @ (X339 @ X342 @ X341) @ X343)) = ((X339 @ (comp @ X342 @ X343) @ (sub @ X341 @ X343))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f22,definition,(
  ! [X336 : (subst > term > subst > $o)] : (($true = ((sP9 @ X336))) <=> ? [X337 : term,X338 : term] : ((((X336 @ id @ X337 @ id)) = $true) & ($true != ((X336 @ id @ (ap @ (sub @ X337 @ id) @ X338) @ id))) & ($true = ((X336 @ id @ X338 @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f23,definition,(
  ($true = sP10) <=> ! [X336 : (subst > term > subst > $o),X335 : term] : (($true = ((sP8 @ X336))) | ($true = ((sP9 @ X336))) | ($true = ((X336 @ id @ X335 @ id))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f24,definition,(
  ($true = sP11) <=> ? [X327 : term,X326 : (subst > term > subst > $o),X325 : (term > $o)] : (! [X329 : term] : (((X325 @ X329)) = ((X326 @ id @ X329 @ id))) & ($true != ((X325 @ (lam @ X327)))) & ($true = ((sP5 @ X326))) & ! [X328 : term] : (($true != ((X325 @ X328))) | ($true = ((X325 @ (sub @ X327 @ (push @ X328 @ id)))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f25,definition,(
  ($true = sP12) <=> ? [X318 : term,X317 : (term > $o)] : (($true = ((sP6 @ X317))) & ($true != ((X317 @ X318))) & ! [X323 : term] : (($true != ((var @ X323))) | ($true = ((X317 @ X323)))) & ($true = ((sP7 @ X317))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f26,definition,(
  ! [X501 : (term > $o)] : ((((sP13 @ X501)) = $true) <=> ? [X504 : term,X505 : term] : (($true != ((X501 @ (ap @ X505 @ X504)))) & ($true = ((X501 @ X504))) & ($true = ((X501 @ X505)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f27,definition,(
  ! [X501 : (term > $o)] : (($true = ((sP14 @ X501))) <=> ? [X502 : term] : (($true = ((X501 @ X502))) & ($true != ((X501 @ (lam @ X502))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f28,definition,(
  ($true = sP15) <=> ! [X501 : (term > $o),X500 : term] : ((((sP13 @ X501)) = $true) | ? [X503 : term] : (($true = ((var @ X503))) & ($true != ((X501 @ X503)))) | ($true = ((sP14 @ X501))) | ($true = ((X501 @ X500))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f29,definition,(
  ! [X488 : (subst > term > subst > $o)] : (($true = ((sP16 @ X488))) <=> ? [X491 : term,X490 : (subst > term > term)] : (($true != ((X488 @ id @ (lam @ (X490 @ sh @ one)) @ id))) & ! [X497 : subst,X498 : subst,X496 : term] : (((sub @ (X490 @ X498 @ X496) @ X497)) = ((sub @ (sub @ X491 @ (push @ X496 @ X498)) @ X497))) & ! [X492 : subst,X494 : term,X493 : subst] : (((X490 @ (comp @ X492 @ X493) @ (sub @ X494 @ X493))) = ((sub @ X491 @ (push @ (sub @ X494 @ X493) @ (comp @ X492 @ X493))))) & ! [X495 : term] : (($true = ((X488 @ id @ (X490 @ id @ X495) @ id))) | ($true != ((X488 @ id @ X495 @ id))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f30,definition,(
  ($true = sP17) <=> ! [X488 : (subst > term > subst > $o),X489 : term] : ((((X488 @ id @ (lam @ (sub @ X489 @ (push @ one @ sh))) @ id)) = $true) | ($true = ((sP16 @ X488))) | ? [X499 : term] : (($true != ((X488 @ id @ (sub @ X489 @ (push @ X499 @ id)) @ id))) & ($true = ((X488 @ id @ X499 @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f31,definition,(
  ! [X349 : (subst > term > subst > $o)] : (($true = ((sP18 @ X349))) <=> ? [X352 : (subst > term > term)] : (! [X356 : subst,X355 : subst,X354 : term] : (((X352 @ (comp @ X356 @ X355) @ (sub @ X354 @ X355))) = ((sub @ (X352 @ X356 @ X354) @ X355))) & ! [X353 : term] : (($true != ((X349 @ id @ X353 @ id))) | ($true = ((X349 @ id @ (X352 @ id @ X353) @ id)))) & ($true != ((X349 @ id @ (lam @ (X352 @ sh @ one)) @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f32,definition,(
  ($true = sP19) <=> ! [X351 : term,X349 : (subst > term > subst > $o),X350 : (term > $o)] : (($true = ((sP18 @ X349))) | ($true = ((X350 @ (lam @ X351)))) | ? [X357 : term] : (((X350 @ X357)) != ((X349 @ id @ X357 @ id))) | ? [X358 : term] : (($true = ((X350 @ X358))) & ($true != ((X350 @ (sub @ X351 @ (push @ X358 @ id)))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f33,definition,(
  ! [X202 : (term > $o)] : (($true = ((sP20 @ X202))) <=> ? [X205 : term] : (($true != ((X202 @ (lam @ X205)))) & ! [X206 : term] : (($true != ((X202 @ X206))) | ($true = ((X202 @ (sub @ X205 @ (push @ X206 @ id))))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f34,definition,(
  ! [X202 : (term > $o)] : (($true = ((sP21 @ X202))) <=> ? [X207 : term,X208 : term] : (($true = ((X202 @ X208))) & ($true = ((X202 @ X207))) & ($true != ((X202 @ (ap @ X207 @ X208))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f35,definition,(
  ($true = sP22) <=> ! [X204 : term,X203 : subst,X202 : (term > $o)] : (? [X209 : term] : (($true != ((X202 @ (sub @ X209 @ X203)))) & ($true = ((var @ X209)))) | ($true = ((sP20 @ X202))) | ($true = ((sP21 @ X202))) | ($true = ((X202 @ (sub @ X204 @ X203)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f36,definition,(
  ! [X524 : (subst > term > subst > $o)] : (($true = ((sP23 @ X524))) <=> ? [X525 : (subst > term > term)] : (($true != ((X524 @ id @ (lam @ (X525 @ sh @ one)) @ id))) & ! [X528 : term,X527 : subst,X526 : subst] : (((sub @ (X525 @ X527 @ X528) @ X526)) = ((X525 @ (comp @ X527 @ X526) @ (sub @ X528 @ X526)))) & ! [X529 : term] : (($true != ((X524 @ id @ X529 @ id))) | ($true = ((X524 @ id @ (X525 @ id @ X529) @ id))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f37,definition,(
  ($true = sP24) <=> ! [X524 : (subst > term > subst > $o),X523 : term] : (($true = ((X524 @ id @ (lam @ (sub @ X523 @ (push @ one @ sh))) @ id))) | ($true = ((sP23 @ X524))) | ? [X530 : term] : (($true != ((X524 @ id @ (sub @ X523 @ (push @ X530 @ id)) @ id))) & ($true = ((X524 @ id @ X530 @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f38,definition,(
  ! [X427 : (subst > term > subst > $o)] : (($true = ((sP25 @ X427))) <=> ? [X433 : (subst > term > term)] : (($true != ((X427 @ id @ (lam @ (X433 @ sh @ one)) @ id))) & ! [X434 : term] : (($true != ((X427 @ id @ X434 @ id))) | ($true = ((X427 @ id @ (X433 @ id @ X434) @ id)))) & ! [X437 : subst,X436 : term,X435 : subst] : (((X433 @ (comp @ X437 @ X435) @ (sub @ X436 @ X435))) = ((sub @ (X433 @ X437 @ X436) @ X435)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f39,definition,(
  ! [X426 : (term > $o),X425 : term] : (($true = ((sP26 @ X425 @ X426))) <=> ? [X428 : term] : (($true = ((X426 @ X428))) & ($true != ((X426 @ (sub @ X425 @ (push @ X428 @ id)))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f40,definition,(
  ! [X427 : (subst > term > subst > $o)] : (($true = ((sP27 @ X427))) <=> ? [X432 : term,X429 : subst,X431 : subst,X430 : subst] : (($true = ((X427 @ (comp @ X430 @ X431) @ (sub @ X432 @ X431) @ X429))) & ($true != ((X427 @ X430 @ X432 @ (comp @ X431 @ X429))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f41,definition,(
  ($true = sP28) <=> ! [X427 : (subst > term > subst > $o),X426 : (term > $o),X425 : term] : (($true = ((sP25 @ X427))) | ? [X441 : subst,X440 : subst,X439 : subst,X442 : term] : (($true != ((X427 @ (comp @ X440 @ X441) @ (sub @ X442 @ X441) @ X439))) & ($true = ((X427 @ X440 @ X442 @ (comp @ X441 @ X439))))) | ($true = ((sP27 @ X427))) | ? [X438 : term] : (((X426 @ X438)) != ((X427 @ id @ X438 @ id))) | ($true = ((sP26 @ X425 @ X426))) | ($true = ((X426 @ (lam @ X425)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f42,definition,(
  ! [X119 : (subst > term > subst > $o)] : (($true = ((sP29 @ X119))) <=> ? [X120 : term,X121 : term] : (($true = ((X119 @ id @ X120 @ id))) & ($true != ((X119 @ id @ (ap @ (sub @ X120 @ id) @ X121) @ id))) & ($true = ((X119 @ id @ X121 @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f43,definition,(
  ! [X119 : (subst > term > subst > $o)] : (($true = ((sP30 @ X119))) <=> ? [X128 : subst,X127 : term,X126 : subst,X129 : subst] : (($true != ((X119 @ (comp @ X126 @ X128) @ (sub @ X127 @ X128) @ X129))) & ($true = ((X119 @ X126 @ X127 @ (comp @ X128 @ X129))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f44,definition,(
  ($true = sP31) <=> ! [X117 : term,X118 : term,X119 : (subst > term > subst > $o)] : (($true != ((X119 @ id @ X117 @ id))) | ? [X124 : subst,X125 : subst,X123 : subst,X122 : term] : (($true = ((X119 @ (comp @ X123 @ X125) @ (sub @ X122 @ X125) @ X124))) & ($true != ((X119 @ X123 @ X122 @ (comp @ X125 @ X124))))) | ($true = ((sP30 @ X119))) | ($true != ((X119 @ id @ X118 @ id))) | ($true = ((X119 @ id @ (ap @ X118 @ X117) @ id))) | ($true = ((sP29 @ X119))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f45,definition,(
  ! [X306 : (subst > term > subst > $o)] : (($true = ((sP32 @ X306))) <=> ? [X307 : (subst > term > term)] : (! [X309 : subst,X308 : term,X310 : subst] : (((X307 @ (comp @ X309 @ X310) @ (sub @ X308 @ X310))) = ((sub @ (X307 @ X309 @ X308) @ X310))) & ! [X311 : term] : (($true = ((X306 @ id @ (X307 @ id @ X311) @ id))) | ($true != ((X306 @ id @ X311 @ id)))) & ($true != ((X306 @ id @ (lam @ (X307 @ sh @ one)) @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f46,definition,(
  ($true = sP33) <=> ! [X305 : (term > $o),X304 : term,X306 : (subst > term > subst > $o)] : (($true = ((X305 @ (lam @ X304)))) | ($true = ((sP32 @ X306))) | ? [X312 : term] : (((X305 @ X312)) != ((X306 @ id @ X312 @ id))) | ? [X313 : term] : (($true = ((X305 @ X313))) & ($true != ((X305 @ (sub @ X304 @ (push @ X313 @ id)))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f47,definition,(
  ! [X366 : (subst > term > subst > $o)] : (($true = ((sP34 @ X366))) <=> ? [X368 : (subst > term > term)] : (($true != ((X366 @ id @ (lam @ (X368 @ sh @ one)) @ id))) & ! [X372 : term] : (($true != ((X366 @ id @ X372 @ id))) | ($true = ((X366 @ id @ (X368 @ id @ X372) @ id)))) & ! [X370 : subst,X371 : term,X369 : subst] : (((X368 @ (comp @ X370 @ X369) @ (sub @ X371 @ X369))) = ((sub @ (X368 @ X370 @ X371) @ X369)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f48,definition,(
  ($true = sP35) <=> ! [X366 : (subst > term > subst > $o),X367 : term] : (? [X373 : term] : (($true = ((X366 @ id @ X373 @ id))) & ($true != ((X366 @ id @ (sub @ X367 @ (push @ X373 @ id)) @ id)))) | ($true = ((sP34 @ X366))) | ($true = ((X366 @ id @ (lam @ (sub @ X367 @ (push @ one @ sh))) @ id))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f49,definition,(
  ($true = sP36) <=> ! [X136 : term,X139 : (subst > term > term),X138 : subst,X137 : (subst > term > term)] : ((((lam @ (X139 @ sh @ one))) != ((lam @ (X137 @ sh @ one)))) | ? [X144 : subst,X143 : term,X145 : subst] : (((X137 @ (comp @ X145 @ X144) @ (sub @ X143 @ X144))) != ((sub @ (X137 @ X145 @ X143) @ X144))) | ? [X142 : subst,X140 : subst,X141 : term] : (((X139 @ (comp @ X142 @ X140) @ (sub @ X141 @ X140))) != ((sub @ (X139 @ X142 @ X141) @ X140))) | (((X137 @ X138 @ X136)) = ((X139 @ X138 @ X136))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f50,definition,(
  ($true = sP37) <=> (? [X130 : subst,X131 : term] : (((sub @ one @ (push @ X131 @ X130))) != X131) | ? [X134 : term,X135 : term] : ((((lam @ X134)) = ((lam @ X135))) & (X134 != X135)) | ? [X133 : subst,X132 : term] : (((comp @ sh @ (push @ X132 @ X133))) != X133) | ($true = sP36))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f51,definition,(
  ! [X170 : (subst > term > subst > $o)] : (($true = ((sP38 @ X170))) <=> ? [X171 : (subst > term > term)] : (! [X172 : term] : ((((X170 @ id @ X172 @ id)) != $true) | ($true = ((X170 @ id @ (X171 @ id @ X172) @ id)))) & ($true != ((X170 @ id @ (lam @ (X171 @ sh @ one)) @ id))) & ! [X173 : subst,X174 : subst,X175 : term] : (((sub @ (X171 @ X173 @ X175) @ X174)) = ((X171 @ (comp @ X173 @ X174) @ (sub @ X175 @ X174))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f52,definition,(
  ! [X170 : (subst > term > subst > $o)] : (($true = ((sP39 @ X170))) <=> ? [X185 : term,X186 : term] : (($true = ((X170 @ id @ X185 @ id))) & ($true = ((X170 @ id @ X186 @ id))) & ($true != ((X170 @ id @ (ap @ (sub @ X186 @ id) @ X185) @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f53,definition,(
  ! [X170 : (subst > term > subst > $o)] : (($true = ((sP40 @ X170))) <=> ? [X178 : subst,X177 : subst,X176 : subst,X179 : term] : (($true != ((X170 @ X176 @ X179 @ (comp @ X177 @ X178)))) & ($true = ((X170 @ (comp @ X176 @ X177) @ (sub @ X179 @ X177) @ X178)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f54,definition,(
  ! [X170 : (subst > term > subst > $o)] : (($true = ((sP41 @ X170))) <=> ? [X180 : term] : (($true != ((X170 @ id @ X180 @ id))) & ($true = ((var @ (sub @ X180 @ id))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f55,definition,(
  ! [X151 : (term > $o)] : (($true = ((sP42 @ X151))) <=> ! [X153 : term,X154 : term] : (($true = ((X151 @ (ap @ X154 @ X153)))) | ($true != ((X151 @ X154))) | ($true != ((X151 @ X153)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f56,definition,(
  ! [X151 : (term > $o)] : (($true = ((sP43 @ X151))) <=> ! [X157 : term] : (($true != ((var @ X157))) | ($true = ((X151 @ X157)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f57,definition,(
  ! [X151 : (term > $o)] : (($true = ((sP44 @ X151))) <=> ! [X155 : term] : (? [X156 : term] : (($true != ((X151 @ (sub @ X155 @ (push @ X156 @ id))))) & ($true = ((X151 @ X156)))) | ($true = ((X151 @ (lam @ X155))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f58,definition,(
  ! [X159 : (subst > term > subst > $o)] : (($true = ((sP45 @ X159))) <=> ! [X164 : (subst > term > term)] : (? [X166 : subst,X167 : subst,X168 : term] : (((X164 @ (comp @ X166 @ X167) @ (sub @ X168 @ X167))) != ((sub @ (X164 @ X166 @ X168) @ X167))) | ($true = ((X159 @ id @ (lam @ (X164 @ sh @ one)) @ id))) | ? [X165 : term] : (($true = ((X159 @ id @ X165 @ id))) & ($true != ((X159 @ id @ (X164 @ id @ X165) @ id))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f59,definition,(
  ($true = sP46) <=> ! [X169 : term,X170 : (subst > term > subst > $o)] : (($true = ((X170 @ id @ X169 @ id))) | ($true = ((sP39 @ X170))) | ? [X183 : subst,X184 : subst,X181 : term,X182 : subst] : (($true != ((X170 @ (comp @ X184 @ X182) @ (sub @ X181 @ X182) @ X183))) & ($true = ((X170 @ X184 @ X181 @ (comp @ X182 @ X183))))) | ($true = ((sP41 @ X170))) | ($true = ((sP40 @ X170))) | ($true = ((sP38 @ X170))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f60,definition,(
  ($true = sP47) <=> ? [X161 : term,X160 : (term > $o),X159 : (subst > term > subst > $o)] : (($true = ((sP45 @ X159))) & ($true != ((X160 @ (lam @ X161)))) & ! [X162 : term] : (((X160 @ X162)) = ((X159 @ id @ X162 @ id))) & ! [X163 : term] : (($true != ((X160 @ X163))) | ($true = ((X160 @ (sub @ X161 @ (push @ X163 @ id)))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f61,definition,(
  ($true = sP48) <=> ? [X151 : (term > $o),X152 : term] : (($true = ((sP42 @ X151))) & ($true = ((sP44 @ X151))) & ($true = ((sP43 @ X151))) & ($true != ((X151 @ X152))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f62,definition,(
  ! [X532 : (subst > term > subst > $o)] : (($true = ((sP49 @ X532))) <=> ? [X533 : (subst > term > term)] : (($true != ((X532 @ id @ (lam @ (X533 @ sh @ one)) @ id))) & ! [X534 : term] : (($true = ((X532 @ id @ (X533 @ id @ X534) @ id))) | ($true != ((X532 @ id @ X534 @ id)))) & ! [X536 : term,X537 : subst,X535 : subst] : (((X533 @ (comp @ X535 @ X537) @ (sub @ X536 @ X537))) = ((sub @ (X533 @ X535 @ X536) @ X537)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f63,definition,(
  ! [X532 : (subst > term > subst > $o)] : (($true = ((sP50 @ X532))) <=> ? [X538 : term,X539 : term] : (($true != ((X532 @ id @ (ap @ (sub @ X539 @ id) @ X538) @ id))) & ($true = ((X532 @ id @ X539 @ id))) & ($true = ((X532 @ id @ X538 @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f64,definition,(
  ! [X93 : (subst > term > subst > $o)] : (($true = ((sP51 @ X93))) <=> ? [X96 : (subst > term > term)] : (($true != ((X93 @ id @ (lam @ (X96 @ sh @ one)) @ id))) & ! [X97 : term] : (($true != ((X93 @ id @ X97 @ id))) | ($true = ((X93 @ id @ (X96 @ id @ X97) @ id)))) & ! [X99 : term,X100 : subst,X98 : subst] : (((sub @ (X96 @ X100 @ X99) @ X98)) = ((X96 @ (comp @ X100 @ X98) @ (sub @ X99 @ X98))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f65,definition,(
  ! [X85 : (subst > term > subst > $o)] : (($true = ((sP52 @ X85))) <=> ! [X87 : (subst > term > term)] : (? [X91 : term] : (($true != ((X85 @ id @ (X87 @ id @ X91) @ id))) & ($true = ((X85 @ id @ X91 @ id)))) | ? [X90 : term,X88 : subst,X89 : subst] : (((X87 @ (comp @ X88 @ X89) @ (sub @ X90 @ X89))) != ((sub @ (X87 @ X88 @ X90) @ X89))) | ($true = ((X85 @ id @ (lam @ (X87 @ sh @ one)) @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f66,definition,(
  ($true = sP53) <=> ! [X94 : term,X93 : (subst > term > subst > $o)] : (($true = ((X93 @ id @ (lam @ X94) @ id))) | ? [X95 : term] : (($true != ((X93 @ id @ (sub @ X94 @ (push @ X95 @ id)) @ id))) & ($true = ((X93 @ id @ X95 @ id)))) | ($true = ((sP51 @ X93))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f67,definition,(
  ($true = sP54) <=> ? [X86 : term,X85 : (subst > term > subst > $o)] : (! [X92 : term] : (($true = ((X85 @ id @ (sub @ X86 @ (push @ X92 @ id)) @ id))) | ($true != ((X85 @ id @ X92 @ id)))) & ($true = ((sP52 @ X85))) & ($true != ((X85 @ id @ (lam @ (sub @ X86 @ (push @ one @ sh))) @ id))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f68,definition,(
  ! [X52 : (term > $o)] : (($true = ((sP55 @ X52))) <=> ? [X57 : term] : (! [X58 : term] : (($true != ((X52 @ X58))) | ($true = ((X52 @ (sub @ X57 @ (push @ X58 @ id)))))) & ($true != ((X52 @ (lam @ X57))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f69,definition,(
  ! [X52 : (term > $o)] : (($true = ((sP56 @ X52))) <=> ? [X56 : term,X55 : term] : (($true != ((X52 @ (ap @ X55 @ X56)))) & ($true = ((X52 @ X56))) & (((X52 @ X55)) = $true)))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f70,definition,(
  ! [X46 : (term > $o)] : (($true = ((sP57 @ X46))) <=> ! [X50 : term,X49 : term] : (($true != ((X46 @ X49))) | ($true != ((X46 @ X50))) | ($true = ((X46 @ (ap @ X49 @ X50))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f71,definition,(
  ! [X46 : (term > $o)] : (($true = ((sP58 @ X46))) <=> ! [X48 : term] : (($true = ((X46 @ X48))) | ($true != ((var @ X48)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f72,definition,(
  ($true = sP59) <=> ! [X52 : (term > $o),X53 : subst,X54 : term] : (($true = ((sP56 @ X52))) | ($true = ((sP55 @ X52))) | ? [X59 : term] : (($true != ((X52 @ (sub @ X59 @ X53)))) & ($true = ((var @ X59)))) | ($true = ((X52 @ (sub @ X54 @ X53)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f73,definition,(
  ($true = sP60) <=> ? [X46 : (term > $o),X47 : term] : (! [X51 : term] : (($true != ((X46 @ X51))) | ($true = ((X46 @ (lam @ X51))))) & ($true = ((sP58 @ X46))) & ($true = ((sP57 @ X46))) & ($true != ((X46 @ X47))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f74,definition,(
  ! [X7 : (subst > term > subst > $o)] : (($true = ((sP61 @ X7))) <=> ? [X19 : (subst > term > term)] : (($true != ((X7 @ id @ (lam @ (X19 @ sh @ one)) @ id))) & ! [X22 : subst,X21 : term,X20 : subst] : (((sub @ (X19 @ X22 @ X21) @ X20)) = ((X19 @ (comp @ X22 @ X20) @ (sub @ X21 @ X20)))) & ! [X23 : term] : ((((X7 @ id @ X23 @ id)) != $true) | ($true = ((X7 @ id @ (X19 @ id @ X23) @ id))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f75,definition,(
  ! [X7 : (subst > term > subst > $o)] : (($true = ((sP62 @ X7))) <=> ? [X8 : term,X9 : term] : (($true = ((X7 @ id @ X8 @ id))) & (((X7 @ id @ X9 @ id)) = $true) & ($true != ((X7 @ id @ (ap @ (sub @ X9 @ id) @ X8) @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f76,definition,(
  ! [X7 : (subst > term > subst > $o)] : (($true = ((sP63 @ X7))) <=> ? [X16 : subst,X15 : term,X17 : subst,X14 : subst] : (($true != ((X7 @ X14 @ X15 @ (comp @ X16 @ X17)))) & ($true = ((X7 @ (comp @ X14 @ X16) @ (sub @ X15 @ X16) @ X17)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f77,definition,(
  ! [X7 : (subst > term > subst > $o)] : (($true = ((sP64 @ X7))) <=> ? [X18 : term] : (($true != ((X7 @ id @ X18 @ id))) & ($true = ((var @ (sub @ X18 @ id))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f78,definition,(
  ($true = sP65) <=> ! [X6 : term,X7 : (subst > term > subst > $o)] : (? [X13 : term,X12 : subst,X11 : subst,X10 : subst] : (($true = ((X7 @ X11 @ X13 @ (comp @ X12 @ X10)))) & ($true != ((X7 @ (comp @ X11 @ X12) @ (sub @ X13 @ X12) @ X10)))) | ($true = ((sP61 @ X7))) | ($true = ((sP64 @ X7))) | ($true = ((sP63 @ X7))) | ($true = ((sP62 @ X7))) | ($true = ((X7 @ id @ X6 @ id))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f79,definition,(
  ! [X387 : (subst > term > subst > $o)] : (($true = ((sP66 @ X387))) <=> ? [X398 : (subst > term > term)] : (($true != ((X387 @ id @ (lam @ (X398 @ sh @ one)) @ id))) & ! [X402 : term] : (($true = ((X387 @ id @ (X398 @ id @ X402) @ id))) | ($true != ((X387 @ id @ X402 @ id)))) & ! [X400 : term,X401 : subst,X399 : subst] : (((X398 @ (comp @ X399 @ X401) @ (sub @ X400 @ X401))) = ((sub @ (X398 @ X399 @ X400) @ X401)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f80,definition,(
  ! [X387 : (subst > term > subst > $o)] : (($true = ((sP67 @ X387))) <=> ? [X395 : subst,X394 : subst,X397 : subst,X396 : term] : (($true = ((X387 @ (comp @ X397 @ X395) @ (sub @ X396 @ X395) @ X394))) & ($true != ((X387 @ X397 @ X396 @ (comp @ X395 @ X394))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f81,definition,(
  ! [X387 : (subst > term > subst > $o)] : (($true = ((sP68 @ X387))) <=> ? [X389 : subst,X391 : term,X390 : subst,X392 : subst] : (($true != ((X387 @ (comp @ X392 @ X389) @ (sub @ X391 @ X389) @ X390))) & ($true = ((X387 @ X392 @ X391 @ (comp @ X389 @ X390))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f82,definition,(
  ! [X380 : (subst > term > subst > $o)] : (($true = ((sP69 @ X380))) <=> ! [X381 : (subst > term > term)] : (? [X382 : term] : (($true = ((X380 @ id @ X382 @ id))) & ($true != ((X380 @ id @ (X381 @ id @ X382) @ id)))) | ($true = ((X380 @ id @ (lam @ (X381 @ sh @ one)) @ id))) | ? [X383 : subst,X384 : term,X385 : subst] : (((X381 @ (comp @ X383 @ X385) @ (sub @ X384 @ X385))) != ((sub @ (X381 @ X383 @ X384) @ X385)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f83,definition,(
  ($true = sP70) <=> ! [X388 : term,X387 : (subst > term > subst > $o)] : (($true = ((sP66 @ X387))) | ? [X393 : term] : (($true != ((X387 @ id @ (sub @ X388 @ (push @ X393 @ id)) @ id))) & ($true = ((X387 @ id @ X393 @ id)))) | ($true = ((sP68 @ X387))) | ($true = ((X387 @ id @ (lam @ X388) @ id))) | ($true = ((sP67 @ X387))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f84,definition,(
  ($true = sP71) <=> ? [X379 : term,X380 : (subst > term > subst > $o)] : (($true = ((sP69 @ X380))) & ($true != ((X380 @ id @ (lam @ (sub @ X379 @ (push @ one @ sh))) @ id))) & ! [X386 : term] : (($true = ((X380 @ id @ (sub @ X379 @ (push @ X386 @ id)) @ id))) | ($true != ((X380 @ id @ X386 @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f85,definition,(
  ! [X359 : (term > $o)] : (($true = ((sP72 @ X359))) <=> ? [X361 : term] : (($true != ((X359 @ (lam @ X361)))) & ! [X362 : term] : (($true != ((X359 @ X362))) | ($true = ((X359 @ (sub @ X361 @ (push @ X362 @ id))))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f86,definition,(
  ! [X359 : (term > $o)] : (($true = ((sP73 @ X359))) <=> ? [X365 : term,X364 : term] : (($true = ((X359 @ X365))) & ($true != ((X359 @ (ap @ X365 @ X364)))) & ($true = ((X359 @ X364)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f87,definition,(
  ($true = sP74) <=> ! [X360 : term,X359 : (term > $o)] : (? [X363 : term] : (($true != ((X359 @ X363))) & ($true = ((var @ X363)))) | ($true = ((sP72 @ X359))) | ($true = ((X359 @ X360))) | ($true = ((sP73 @ X359))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f88,definition,(
  ($true = sP75) <=> ! [X453 : term,X454 : term,X451 : term,X452 : term] : ((((ap @ (sub @ X452 @ id) @ X451)) != ((ap @ (sub @ X454 @ id) @ X453))) | (X451 = X453))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f89,definition,(
  ($true = sP76) <=> (? [X448 : term,X449 : term,X450 : term,X447 : term] : ((((ap @ X447 @ X449)) = ((ap @ X450 @ X448))) & (X448 != X449)) | ($true = sP75))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f90,definition,(
  ! [X246 : (term > $o)] : (($true = ((sP77 @ X246))) <=> ! [X250 : term,X251 : term] : (($true = ((X246 @ (ap @ X250 @ X251)))) | ($true != ((X246 @ X250))) | ($true != ((X246 @ X251)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f91,definition,(
  ! [X246 : (term > $o)] : (($true = ((sP78 @ X246))) <=> ! [X248 : term] : (($true = ((X246 @ (lam @ X248)))) | ? [X249 : term] : (($true != ((X246 @ (sub @ X248 @ (push @ X249 @ id))))) & ($true = ((X246 @ X249))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f92,definition,(
  ! [X281 : (subst > term > subst > $o)] : (($true = ((sP79 @ X281))) <=> ? [X283 : (subst > term > term)] : (! [X285 : term,X286 : subst,X287 : subst] : (((sub @ (X283 @ X286 @ X285) @ X287)) = ((X283 @ (comp @ X286 @ X287) @ (sub @ X285 @ X287)))) & ($true != ((X281 @ id @ (lam @ (X283 @ sh @ one)) @ id))) & ! [X284 : term] : (($true != ((X281 @ id @ X284 @ id))) | ($true = ((X281 @ id @ (X283 @ id @ X284) @ id))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f93,definition,(
  ! [X281 : (subst > term > subst > $o)] : (($true = ((sP80 @ X281))) <=> ? [X297 : term,X298 : term] : (($true != ((X281 @ id @ (ap @ (sub @ X297 @ id) @ X298) @ id))) & ($true = ((X281 @ id @ X297 @ id))) & ($true = ((X281 @ id @ X298 @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f94,definition,(
  ! [X281 : (subst > term > subst > $o)] : (($true = ((sP81 @ X281))) <=> ? [X296 : term] : (($true != ((X281 @ id @ X296 @ id))) & ($true = ((var @ (sub @ X296 @ id))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f95,definition,(
  ! [X281 : (subst > term > subst > $o)] : (($true = ((sP82 @ X281))) <=> ? [X293 : subst,X295 : subst,X294 : term,X292 : subst] : (($true = ((X281 @ (comp @ X292 @ X295) @ (sub @ X294 @ X295) @ X293))) & ($true != ((X281 @ X292 @ X294 @ (comp @ X295 @ X293))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f96,definition,(
  ! [X266 : (subst > term > subst > $o)] : (($true = ((sP83 @ X266))) <=> ! [X276 : (subst > term > term)] : (? [X278 : subst,X280 : term,X279 : subst] : (((sub @ (X276 @ X278 @ X280) @ X279)) != ((X276 @ (comp @ X278 @ X279) @ (sub @ X280 @ X279)))) | ($true = ((X266 @ id @ (lam @ (X276 @ sh @ one)) @ id))) | ? [X277 : term] : (($true = ((X266 @ id @ X277 @ id))) & ($true != ((X266 @ id @ (X276 @ id @ X277) @ id))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f97,definition,(
  ! [X266 : (subst > term > subst > $o),X265 : term] : (($true = ((sP84 @ X265 @ X266))) <=> ! [X275 : term] : (($true != ((X266 @ id @ X275 @ id))) | ($true = ((X266 @ id @ (sub @ X265 @ (push @ X275 @ id)) @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f98,definition,(
  ! [X266 : (subst > term > subst > $o)] : (($true = ((sP85 @ X266))) <=> ! [X271 : subst,X272 : subst,X274 : term,X273 : subst] : (($true = ((X266 @ (comp @ X272 @ X273) @ (sub @ X274 @ X273) @ X271))) | ($true != ((X266 @ X272 @ X274 @ (comp @ X273 @ X271))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f99,definition,(
  ! [X252 : (subst > term > subst > $o)] : (($true = ((sP86 @ X252))) <=> ! [X263 : term,X264 : term] : (($true != ((X252 @ id @ X264 @ id))) | ($true = ((X252 @ id @ (ap @ (sub @ X264 @ id) @ X263) @ id))) | ($true != ((X252 @ id @ X263 @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f100,definition,(
  ! [X252 : (subst > term > subst > $o)] : (($true = ((sP87 @ X252))) <=> ! [X256 : subst,X255 : subst,X258 : term,X257 : subst] : (($true = ((X252 @ X256 @ X258 @ (comp @ X255 @ X257)))) | ($true != ((X252 @ (comp @ X256 @ X255) @ (sub @ X258 @ X255) @ X257)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f101,definition,(
  ($true = sP88) <=> ! [X281 : (subst > term > subst > $o),X282 : term] : (($true = ((X281 @ id @ X282 @ id))) | ($true = ((sP80 @ X281))) | ? [X290 : term,X289 : subst,X288 : subst,X291 : subst] : (($true != ((X281 @ (comp @ X288 @ X289) @ (sub @ X290 @ X289) @ X291))) & ($true = ((X281 @ X288 @ X290 @ (comp @ X289 @ X291))))) | ($true = ((sP82 @ X281))) | ($true = ((sP79 @ X281))) | ($true = ((sP81 @ X281))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f102,definition,(
  ($true = sP89) <=> ? [X246 : (term > $o),X245 : term] : (! [X247 : term] : (($true != ((var @ X247))) | ($true = ((X246 @ X247)))) & ($true = ((sP78 @ X246))) & ($true = ((sP77 @ X246))) & ($true != ((X246 @ X245))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f103,definition,(
  ($true = sP90) <=> ? [X266 : (subst > term > subst > $o),X265 : term] : (! [X270 : subst,X269 : subst,X268 : subst,X267 : term] : (($true = ((X266 @ X269 @ X267 @ (comp @ X268 @ X270)))) | ($true != ((X266 @ (comp @ X269 @ X268) @ (sub @ X267 @ X268) @ X270)))) & ($true = ((sP85 @ X266))) & ($true = ((sP84 @ X265 @ X266))) & ($true != ((X266 @ id @ (lam @ X265) @ id))) & ($true = ((sP83 @ X266))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f104,definition,(
  ($true = sP91) <=> ? [X252 : (subst > term > subst > $o),X254 : term,X253 : term] : (($true != ((X252 @ id @ (ap @ X253 @ X254) @ id))) & ! [X259 : subst,X261 : subst,X260 : term,X262 : subst] : (($true != ((X252 @ X259 @ X260 @ (comp @ X262 @ X261)))) | ($true = ((X252 @ (comp @ X259 @ X262) @ (sub @ X260 @ X262) @ X261)))) & ($true = ((sP86 @ X252))) & ($true = ((sP87 @ X252))) & ($true = ((X252 @ id @ X253 @ id))) & ($true = ((X252 @ id @ X254 @ id))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f105,definition,(
  ! [X33 : (term > $o)] : (($true = ((sP92 @ X33))) <=> ? [X35 : term] : (($true != ((X33 @ (lam @ X35)))) & ! [X36 : term] : (($true != ((X33 @ X36))) | ($true = ((X33 @ (sub @ X35 @ (push @ X36 @ id))))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f106,definition,(
  ! [X33 : (term > $o)] : (($true = ((sP93 @ X33))) <=> ? [X38 : term,X37 : term] : (($true = ((X33 @ X37))) & ($true = ((X33 @ X38))) & ($true != ((X33 @ (ap @ X38 @ X37))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f107,definition,(
  ! [X27 : (term > $o)] : (($true = ((sP94 @ X27))) <=> ! [X30 : term,X31 : term] : (($true != ((X27 @ X31))) | ($true = ((X27 @ (ap @ X30 @ X31)))) | ($true != ((X27 @ X30)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f108,definition,(
  ! [X27 : (term > $o)] : (($true = ((sP95 @ X27))) <=> ! [X28 : term] : (($true = ((X27 @ (lam @ X28)))) | ? [X29 : term] : (($true != ((X27 @ (sub @ X28 @ (push @ X29 @ id))))) & ($true = ((X27 @ X29))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f109,definition,(
  ($true = sP96) <=> ! [X33 : (term > $o),X34 : term] : (? [X39 : term] : (($true = ((var @ X39))) & ($true != ((X33 @ X39)))) | ($true = ((sP93 @ X33))) | ($true = ((X33 @ X34))) | ($true = ((sP92 @ X33))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f110,definition,(
  ($true = sP97) <=> ? [X25 : subst,X27 : (term > $o),X26 : term] : (($true != ((X27 @ (sub @ X26 @ X25)))) & ! [X32 : term] : (($true != ((var @ X32))) | ($true = ((X27 @ (sub @ X32 @ X25))))) & ($true = ((sP95 @ X27))) & ($true = ((sP94 @ X27))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f111,definition,(
  ! [X515 : (subst > term > subst > $o)] : (($true = ((sP98 @ X515))) <=> ? [X518 : (subst > term > term)] : (! [X521 : subst,X520 : term,X522 : subst] : (((X518 @ (comp @ X521 @ X522) @ (sub @ X520 @ X522))) = ((sub @ (X518 @ X521 @ X520) @ X522))) & ! [X519 : term] : (($true != ((X515 @ id @ X519 @ id))) | ($true = ((X515 @ id @ (X518 @ id @ X519) @ id)))) & ($true != ((X515 @ id @ (lam @ (X518 @ sh @ one)) @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f112,definition,(
  ($true = sP99) <=> ! [X515 : (subst > term > subst > $o),X516 : term] : (($true = ((X515 @ id @ (lam @ X516) @ id))) | ($true = ((sP98 @ X515))) | ? [X517 : term] : (($true = ((X515 @ id @ X517 @ id))) & ($true != ((X515 @ id @ (sub @ X516 @ (push @ X517 @ id)) @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f113,definition,(
  ! [X189 : (subst > term > subst > $o)] : (($true = ((sP100 @ X189))) <=> ? [X200 : term,X201 : term] : (($true = ((X189 @ id @ X201 @ id))) & ($true = ((X189 @ id @ X200 @ id))) & ($true != ((X189 @ id @ (ap @ (sub @ X200 @ id) @ X201) @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f114,definition,(
  ! [X189 : (subst > term > subst > $o)] : (($true = ((sP101 @ X189))) <=> ? [X198 : term,X199 : subst,X197 : subst,X196 : subst] : (($true = ((X189 @ X199 @ X198 @ (comp @ X196 @ X197)))) & ($true != ((X189 @ (comp @ X199 @ X196) @ (sub @ X198 @ X196) @ X197)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f115,definition,(
  ($true = sP102) <=> ! [X188 : term,X189 : (subst > term > subst > $o),X187 : term,X190 : (term > $o)] : (($true != ((X190 @ X187))) | ($true != ((X190 @ X188))) | ? [X194 : subst,X195 : term,X193 : subst,X192 : subst] : (($true = ((X189 @ (comp @ X194 @ X193) @ (sub @ X195 @ X193) @ X192))) & (((X189 @ X194 @ X195 @ (comp @ X193 @ X192))) != $true)) | ? [X191 : term] : (((X190 @ X191)) != ((X189 @ id @ X191 @ id))) | ($true = ((sP101 @ X189))) | ($true = ((sP100 @ X189))) | ($true = ((X190 @ (ap @ X188 @ X187)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f116,definition,(
  ($true = sP103) <=> ! [X242 : term,X244 : term,X241 : term,X243 : term] : ((X242 = X244) | (((ap @ (sub @ X244 @ id) @ X241)) != ((ap @ (sub @ X242 @ id) @ X243))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f117,definition,(
  ($true = sP104) <=> (($true = sP103) | ? [X236 : term] : (((sub @ X236 @ id)) != X236) | ? [X240 : term,X239 : term,X238 : term,X237 : term] : ((X237 != X238) & (((ap @ X237 @ X239)) = ((ap @ X238 @ X240)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f118,definition,(
  ! [X65 : (subst > term > subst > $o)] : (($true = ((sP105 @ X65))) <=> ? [X71 : (subst > term > term)] : (! [X75 : term] : (($true != ((X65 @ id @ X75 @ id))) | ($true = ((X65 @ id @ (X71 @ id @ X75) @ id)))) & ! [X73 : subst,X74 : term,X72 : subst] : (((X71 @ (comp @ X73 @ X72) @ (sub @ X74 @ X72))) = ((sub @ (X71 @ X73 @ X74) @ X72))) & ($true != ((X65 @ id @ (lam @ (X71 @ sh @ one)) @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f119,definition,(
  ! [X65 : (subst > term > subst > $o),X64 : term] : (($true = ((sP106 @ X64 @ X65))) <=> ? [X70 : term] : (($true = ((X65 @ id @ X70 @ id))) & ($true != ((X65 @ id @ (sub @ X64 @ (push @ X70 @ id)) @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f120,definition,(
  ! [X65 : (subst > term > subst > $o)] : (($true = ((sP107 @ X65))) <=> ? [X76 : term,X79 : subst,X78 : subst,X77 : subst] : (($true = ((X65 @ (comp @ X77 @ X79) @ (sub @ X76 @ X79) @ X78))) & ($true != ((X65 @ X77 @ X76 @ (comp @ X79 @ X78))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f121,definition,(
  ($true = sP108) <=> ! [X65 : (subst > term > subst > $o),X64 : term] : (? [X68 : subst,X69 : subst,X67 : term,X66 : subst] : (($true = ((X65 @ X66 @ X67 @ (comp @ X69 @ X68)))) & ($true != ((X65 @ (comp @ X66 @ X69) @ (sub @ X67 @ X69) @ X68)))) | ($true = ((X65 @ id @ (lam @ X64) @ id))) | ($true = ((sP105 @ X65))) | ($true = ((sP107 @ X65))) | ($true = ((sP106 @ X64 @ X65))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f122,plain,(
  ((hoaslaminj = $true) <=> ! [X461 : (subst > term > term),X462 : term,X459 : subst,X460 : (subst > term > term)] : ((((lam @ (X461 @ sh @ one))) != ((lam @ (X460 @ sh @ one)))) | ? [X468 : subst,X466 : term,X467 : subst] : (((X461 @ (comp @ X468 @ X467) @ (sub @ X466 @ X467))) != ((sub @ (X461 @ X468 @ X466) @ X467))) | ? [X464 : term,X465 : subst,X463 : subst] : (((X460 @ (comp @ X463 @ X465) @ (sub @ X464 @ X465))) != ((sub @ (X460 @ X463 @ X464) @ X465))) | (((X460 @ X459 @ X462)) = ((X461 @ X459 @ X462))))) & ((axshiftcons = $true) <=> ! [X443 : subst,X444 : term] : (((comp @ sh @ (push @ X444 @ X443))) = X443)) & ((pushprop_lem0 = $true) <=> ! [X102 : (term > $o),X103 : term,X101 : subst] : ? [X104 : (term > $o)] : ! [X105 : term] : (((X104 @ X105)) = ((X102 @ (sub @ X105 @ (push @ X103 @ X101)))))) & ! [X5 : d_subst] : (d_id = X5) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X419 : term] : ? [X420 : (subst > term > term)] : ! [X421 : subst,X422 : term] : (((X420 @ X421 @ X422)) = ((sub @ X419 @ (push @ X422 @ X421))))) & (axclos = $true) & (hoasinduction_lem1_lthm = $true) & (termmset_gthm = $true) & (($true = sP108) <=> (hoasinduction_lem3 = $true)) & (($true = sP104) <=> (hoasapinj1_lthm = $true)) & (axvarshift = $true) & (pushprop_gthm = $true) & (! [X411 : term,X410 : (subst > term > term),X412 : term] : (? [X415 : term,X414 : subst,X413 : subst] : (((X410 @ (comp @ X414 @ X413) @ (sub @ X415 @ X413))) != ((sub @ (X410 @ X414 @ X415) @ X413))) | (((ap @ (sub @ X412 @ id) @ X411)) != ((lam @ (X410 @ sh @ one))))) <=> (hoaslamnotap = $true)) & (($true = sP102) <=> (hoasinduction_lem2v2 = $true)) & (pushprop_lem0_gthm = $true) & ! [X409 : d_subst,X408 : d_subst] : ((((d2subst @ X409)) != ((d2subst @ X408))) | (X408 = X409)) & (hoasinduction_lem1_gthm = $true) & (hoasinduction_lem1v2 = $true) & (substmonoid_gthm = $true) & (pushprop_lem3v2_lthm = $true) & (pushprop_lem2v2_lthm = $true) & (((d2subst @ d_id)) = id) & (((var @ (d2term @ d_one))) != $true) & (axassoc = $true) & (lamnotap != $true) & ((hoasinduction_lem3a = $true) <=> ($true = sP99)) & ((induction2_lthm = $true) <=> (($true = sP97) | ($true = sP96) | ? [X24 : term] : (((sub @ X24 @ id)) != X24))) & ((($true = sP91) | ($true = sP90) | ($true = sP88) | ($true = sP89)) <=> (hoasinduction_lthm = $true)) & (hoasinduction_lem2_gthm = $true) & (hoasinduction_lem2v2_gthm = $true) & (pushprop_lem3v2 = $true) & (pushprop_lthm = $true) & (! [X40 : subst] : (((comp @ id @ X40)) = X40) <=> (axidl = $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((pushprop_lem1_lthm = $true) <=> (? [X508 : subst,X509 : term] : (((comp @ sh @ (push @ X509 @ X508))) != X508) | ! [X511 : term,X514 : term,X510 : subst,X512 : (term > $o),X513 : (term > $o)] : (($true = ((X512 @ (sub @ X511 @ (push @ X514 @ X510))))) | ($true != ((X513 @ X511)))) | ? [X507 : subst,X506 : term] : (((sub @ one @ (push @ X506 @ X507))) != X506))) & (pushprop_lem1v2_gthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (($true = sP76) <=> (hoasapinj2_lthm = $true)) & (! [X475 : term] : (((sub @ X475 @ id)) = X475) <=> (termmset = $true)) & (pushprop_lthm_orig = $true) & ((? [X479 : term,X480 : subst] : (((sub @ one @ (push @ X479 @ X480))) != X479) | ! [X484 : (term > $o),X482 : (term > $o),X483 : subst,X481 : term] : (($true != ((X482 @ X481))) | ? [X485 : term] : (((X484 @ X485)) != ((X482 @ (sub @ X485 @ (push @ X481 @ X483))))) | ($true = ((X484 @ one))))) <=> (pushprop_lem1v2_lthm = $true)) & (ulamvar1 != $true) & ((induction2 = $true) <=> ($true = sP74)) & ! [X210 : d_term,X211 : d_term] : ((((d2term @ X210)) != ((d2term @ X211))) | (X210 = X211)) & ((pushprop_lem1v2 = $true) <=> ! [X111 : (term > $o),X114 : subst,X113 : term,X112 : (term > $o)] : (($true = ((X112 @ one))) | ($true != ((X111 @ X113))) | ? [X115 : term] : (((X112 @ X115)) != ((X111 @ (sub @ X115 @ (push @ X113 @ X114))))))) & (hoasinduction_lem3b_gthm = $true) & ((hoasinduction_lem3_lthm = $true) <=> (($true = sP71) | ? [X378 : term] : (((sub @ X378 @ id)) != X378) | ($true = sP70))) & (! [X456 : term,X455 : term,X457 : term,X458 : term] : ((X455 = X458) | (((ap @ X458 @ X457)) != ((ap @ X455 @ X456)))) <=> (apinj1 = $true)) & ((hoasinduction = $true) <=> ($true = sP65)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((? [X45 : subst] : (((comp @ X45 @ id)) != X45) | ? [X44 : term,X43 : subst] : (((comp @ sh @ (push @ X44 @ X43))) != X43) | ? [X41 : subst,X42 : term] : (((sub @ one @ (push @ X42 @ X41))) != X42) | ($true = sP60) | ($true = sP59)) <=> (induction2lem_lthm = $true)) & (pushprop_lem1_gthm = $true) & (one = ((d2term @ d_one))) & (hoasinduction_gthm = $true) & (hoaslamnotvar = $true) & (hoasinduction_lem2_lthm = $true) & ((shinj = $true) <=> ! [X470 : term,X469 : term] : ((X469 = X470) | (((sub @ X470 @ sh)) != ((sub @ X469 @ sh))))) & (hoasinduction_lem3v2_gthm = $true) & ((? [X84 : term] : (((sub @ X84 @ id)) != X84) | ($true = sP54) | ($true = sP53)) <=> (hoasinduction_lem3a_lthm = $true)) & (! [X477 : term] : ? [X478 : (subst > term > term)] : (((X478 @ sh @ one)) = ((sub @ X477 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b = $true)) & (hoasapnotvar_lthm = $true) & (axmap = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (! [X531 : term,X532 : (subst > term > subst > $o)] : (($true = ((X532 @ id @ X531 @ id))) | ($true = ((sP49 @ X532))) | ($true = ((sP50 @ X532)))) <=> (hoasinduction_no_psi_cond = $true)) & ! [X423 : term] : ? [X424 : d_term] : (((d2term @ X424)) = X423) & (hoasinduction_lem1 = $true) & ((($true = sP47) | ($true = sP48) | ($true = sP46) | ? [X158 : term] : (((sub @ X158 @ id)) != X158) | ? [X148 : (subst > term > subst > $o)] : ! [X149 : (term > $o)] : ? [X150 : term] : (((X148 @ id @ X150 @ id)) != ((X149 @ X150)))) <=> (hoasinduction_lthm_3 = $true)) & (hoaslamnotap_lthm = $true) & (induction2_gthm = $true) & ((hoaslaminj_lthm = $true) <=> ($true = sP37)) & (($true = sP35) <=> (hoasinduction_lem3aa_lthm = $true)) & ((hoasinduction_lem3v2a_lthm = $true) <=> (? [X299 : term] : ! [X300 : (subst > term > term)] : ? [X302 : term,X301 : subst] : (((X300 @ X301 @ X302)) != ((sub @ X299 @ (push @ X302 @ X301)))) | ($true = sP33) | ? [X303 : term] : (((sub @ X303 @ id)) != X303))) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (ulamvarind = $true) & ! [X476 : d_term] : (d_one = X476) & (hoasapinj1_gthm = $true) & (! [X80 : term,X83 : term,X82 : term,X81 : term] : ((X80 = X83) | (((ap @ X82 @ X83)) != ((ap @ X81 @ X80)))) <=> (apinj2 = $true)) & (hoaslaminj_gthm = $true) & (($true = sP31) <=> (hoasinduction_lem2 = $true)) & ((hoasinduction_lem3v2 = $true) <=> ($true = sP28)) & ((pushprop_lem1 = $true) <=> ! [X212 : subst,X216 : term,X215 : (term > $o),X213 : term,X214 : (term > $o)] : (($true = ((X215 @ (sub @ X213 @ (push @ X216 @ X212))))) | ($true != ((X214 @ X213))))) & (axabs = $true) & (! [X416 : (subst > term > subst > $o)] : ? [X417 : (term > $o)] : ! [X418 : term] : (((X416 @ id @ X418 @ id)) = ((X417 @ X418))) <=> (hoasinduction_lem0 = $true)) & (! [X446 : term,X445 : term] : ((((lam @ X446)) != ((lam @ X445))) | (X445 = X446)) <=> (laminj = $true)) & (hoasapnotvar = $true) & (pushprop_lem2v2_gthm = $true) & ((termmset_lthm = $true) <=> (? [X1 : term] : (((sub @ X1 @ id)) != X1) | ! [X2 : term] : (((sub @ X2 @ id)) = X2))) & ((hoasinduction_lem3aa = $true) <=> ($true = sP24)) & (pushprop = $true) & (($true = sP22) <=> (induction2lem = $true)) & (hoaslamnotvar_gthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (! [X348 : term,X347 : subst] : (((sub @ one @ (push @ X348 @ X347))) = X348) <=> (axvarcons = $true)) & ((hoasinduction_lem3v2_f = $true) <=> ! [X374 : term] : ? [X375 : (subst > term > term)] : ! [X376 : subst,X377 : term] : (((sub @ X374 @ (push @ X377 @ X376))) = ((X375 @ X376 @ X377)))) & ((axscons = $true) <=> ! [X403 : subst] : (((push @ (sub @ one @ X403) @ (comp @ sh @ X403))) = X403)) & (ulamvarsh = $true) & (($true = sP19) <=> (hoasinduction_lem3v2a = $true)) & (hoasapinj2_gthm = $true) & (hoasinduction_lem3a_gthm = $true) & (sh = ((d2subst @ d_id))) & (axapp = $true) & (! [X116 : term] : (((sub @ X116 @ id)) = X116) <=> (axvarid = $true)) & (! [X0 : subst] : (((comp @ X0 @ id)) = X0) <=> (axidr = $true)) & (! [X63 : term,X60 : term,X61 : term,X62 : term] : ((X60 = X61) | (((ap @ (sub @ X61 @ id) @ X62)) != ((ap @ (sub @ X60 @ id) @ X63)))) <=> (hoasapinj1 = $true)) & (apnotvar = $true) & (! [X344 : (subst > term > subst > $o)] : ? [X345 : (term > $o)] : ! [X346 : term] : (((X345 @ X346)) = ((X344 @ id @ X346 @ id))) <=> (hoasinduction_lem0_lthm = $true)) & (($true = sP17) <=> (hoasinduction_lem3aaa = $true)) & ((! [X4 : subst] : (((comp @ id @ X4)) = X4) & ! [X3 : subst] : (((comp @ X3 @ id)) = X3)) <=> (substmonoid = $true)) & (hoaslamnotap_gthm = $true) & (lamnotvar = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & (hoasinduction_lem3_gthm = $true) & (induction2lem_gthm = $true) & (hoaslamnotvar_lthm = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ! [X146 : subst] : ? [X147 : d_subst] : (((d2subst @ X147)) = X146) & (! [X486 : term] : ? [X487 : (subst > term > term)] : (((sub @ X486 @ (push @ one @ sh))) = ((X487 @ sh @ one))) <=> (hoasinduction_lem3b_lthm = $true)) & ((induction = $true) <=> ($true = sP15)) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (? [X314 : (subst > term > subst > $o)] : ! [X315 : (term > $o)] : ? [X316 : term] : (((X314 @ id @ X316 @ id)) != ((X315 @ X316))) | ($true = sP10) | ? [X324 : term] : (((sub @ X324 @ id)) != X324) | ($true = sP12) | ($true = sP11))) & ((hoasinduction_lem3v2_lthm = $true) <=> (($true = sP4) | ? [X217 : term] : (((sub @ X217 @ id)) != X217))) & ((hoasapinj2 = $true) <=> ! [X473 : term,X472 : term,X471 : term,X474 : term] : ((X472 = X473) | (((ap @ (sub @ X474 @ id) @ X472)) != ((ap @ (sub @ X471 @ id) @ X473))))) & (($true = sP0) <=> (substmonoid_lthm = $true)) & ((pushprop_lem0_lthm = $true) <=> ! [X107 : subst,X108 : term,X106 : (term > $o)] : ? [X109 : (term > $o)] : ! [X110 : term] : (((X106 @ (sub @ X110 @ (push @ X108 @ X107)))) = ((X109 @ X110)))) & (pushprop_lem2v2 = $true)),
  inference(definition_folding,[],[f12,f121,f120,f119,f118,f117,f116,f115,f114,f113,f112,f111,f110,f109,f108,f107,f106,f105,f104,f103,f102,f101,f100,f99,f98,f97,f96,f95,f94,f93,f92,f91,f90,f89,f88,f87,f86,f85,f84,f83,f82,f81,f80,f79,f78,f77,f76,f75,f74,f73,f72,f71,f70,f69,f68,f67,f66,f65,f64,f63,f62,f61,f60,f59,f58,f57,f56,f55,f54,f53,f52,f51,f50,f49,f48,f47,f46,f45,f44,f43,f42,f41,f40,f39,f38,f37,f36,f35,f34,f33,f32,f31,f30,f29,f28,f27,f26,f25,f24,f23,f22,f21,f20,f19,f18,f17,f16,f15,f14,f13])).
thf(f454,plain,(
  (((hoaslaminj = $true) | ? [X461 : (subst > term > term),X462 : term,X459 : subst,X460 : (subst > term > term)] : ((((lam @ (X461 @ sh @ one))) = ((lam @ (X460 @ sh @ one)))) & ! [X468 : subst,X466 : term,X467 : subst] : (((X461 @ (comp @ X468 @ X467) @ (sub @ X466 @ X467))) = ((sub @ (X461 @ X468 @ X466) @ X467))) & ! [X464 : term,X465 : subst,X463 : subst] : (((X460 @ (comp @ X463 @ X465) @ (sub @ X464 @ X465))) = ((sub @ (X460 @ X463 @ X464) @ X465))) & (((X460 @ X459 @ X462)) != ((X461 @ X459 @ X462))))) & (! [X461 : (subst > term > term),X462 : term,X459 : subst,X460 : (subst > term > term)] : ((((lam @ (X461 @ sh @ one))) != ((lam @ (X460 @ sh @ one)))) | ? [X468 : subst,X466 : term,X467 : subst] : (((X461 @ (comp @ X468 @ X467) @ (sub @ X466 @ X467))) != ((sub @ (X461 @ X468 @ X466) @ X467))) | ? [X464 : term,X465 : subst,X463 : subst] : (((X460 @ (comp @ X463 @ X465) @ (sub @ X464 @ X465))) != ((sub @ (X460 @ X463 @ X464) @ X465))) | (((X460 @ X459 @ X462)) = ((X461 @ X459 @ X462)))) | (hoaslaminj != $true))) & (((axshiftcons = $true) | ? [X443 : subst,X444 : term] : (((comp @ sh @ (push @ X444 @ X443))) != X443)) & (! [X443 : subst,X444 : term] : (((comp @ sh @ (push @ X444 @ X443))) = X443) | (axshiftcons != $true))) & (((pushprop_lem0 = $true) | ? [X102 : (term > $o),X103 : term,X101 : subst] : ! [X104 : (term > $o)] : ? [X105 : term] : (((X104 @ X105)) != ((X102 @ (sub @ X105 @ (push @ X103 @ X101)))))) & (! [X102 : (term > $o),X103 : term,X101 : subst] : ? [X104 : (term > $o)] : ! [X105 : term] : (((X104 @ X105)) = ((X102 @ (sub @ X105 @ (push @ X103 @ X101))))) | (pushprop_lem0 != $true))) & ! [X5 : d_subst] : (d_id = X5) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem1v2_gthm = $true) & (((hoasinduction_lem3v2_f_lthm = $true) | ? [X419 : term] : ! [X420 : (subst > term > term)] : ? [X421 : subst,X422 : term] : (((X420 @ X421 @ X422)) != ((sub @ X419 @ (push @ X422 @ X421))))) & (! [X419 : term] : ? [X420 : (subst > term > term)] : ! [X421 : subst,X422 : term] : (((X420 @ X421 @ X422)) = ((sub @ X419 @ (push @ X422 @ X421)))) | (hoasinduction_lem3v2_f_lthm != $true))) & (axclos = $true) & (hoasinduction_lem1_lthm = $true) & (termmset_gthm = $true) & ((($true = sP108) | (hoasinduction_lem3 != $true)) & ((hoasinduction_lem3 = $true) | ($true != sP108))) & ((($true = sP104) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | ($true != sP104))) & (axvarshift = $true) & (pushprop_gthm = $true) & ((! [X411 : term,X410 : (subst > term > term),X412 : term] : (? [X415 : term,X414 : subst,X413 : subst] : (((X410 @ (comp @ X414 @ X413) @ (sub @ X415 @ X413))) != ((sub @ (X410 @ X414 @ X415) @ X413))) | (((ap @ (sub @ X412 @ id) @ X411)) != ((lam @ (X410 @ sh @ one))))) | (hoaslamnotap != $true)) & ((hoaslamnotap = $true) | ? [X411 : term,X410 : (subst > term > term),X412 : term] : (! [X415 : term,X414 : subst,X413 : subst] : (((X410 @ (comp @ X414 @ X413) @ (sub @ X415 @ X413))) = ((sub @ (X410 @ X414 @ X415) @ X413))) & (((ap @ (sub @ X412 @ id) @ X411)) = ((lam @ (X410 @ sh @ one))))))) & ((($true = sP102) | (hoasinduction_lem2v2 != $true)) & ((hoasinduction_lem2v2 = $true) | ($true != sP102))) & (pushprop_lem0_gthm = $true) & ! [X409 : d_subst,X408 : d_subst] : ((((d2subst @ X409)) != ((d2subst @ X408))) | (X408 = X409)) & (hoasinduction_lem1_gthm = $true) & (hoasinduction_lem1v2 = $true) & (substmonoid_gthm = $true) & (pushprop_lem3v2_lthm = $true) & (pushprop_lem2v2_lthm = $true) & (((d2subst @ d_id)) = id) & (((var @ (d2term @ d_one))) != $true) & (axassoc = $true) & (lamnotap != $true) & (((hoasinduction_lem3a = $true) | ($true != sP99)) & (($true = sP99) | (hoasinduction_lem3a != $true))) & (((induction2_lthm = $true) | (($true != sP97) & ($true != sP96) & ! [X24 : term] : (((sub @ X24 @ id)) = X24))) & ((($true = sP97) | ($true = sP96) | ? [X24 : term] : (((sub @ X24 @ id)) != X24)) | (induction2_lthm != $true))) & (((($true = sP91) | ($true = sP90) | ($true = sP88) | ($true = sP89)) | (hoasinduction_lthm != $true)) & ((hoasinduction_lthm = $true) | (($true != sP91) & ($true != sP90) & ($true != sP88) & ($true != sP89)))) & (hoasinduction_lem2_gthm = $true) & (hoasinduction_lem2v2_gthm = $true) & (pushprop_lem3v2 = $true) & (pushprop_lthm = $true) & ((! [X40 : subst] : (((comp @ id @ X40)) = X40) | (axidl != $true)) & ((axidl = $true) | ? [X40 : subst] : (((comp @ id @ X40)) != X40))) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (((pushprop_lem1_lthm = $true) | (! [X508 : subst,X509 : term] : (((comp @ sh @ (push @ X509 @ X508))) = X508) & ? [X511 : term,X514 : term,X510 : subst,X512 : (term > $o),X513 : (term > $o)] : (($true != ((X512 @ (sub @ X511 @ (push @ X514 @ X510))))) & ($true = ((X513 @ X511)))) & ! [X507 : subst,X506 : term] : (((sub @ one @ (push @ X506 @ X507))) = X506))) & ((? [X508 : subst,X509 : term] : (((comp @ sh @ (push @ X509 @ X508))) != X508) | ! [X511 : term,X514 : term,X510 : subst,X512 : (term > $o),X513 : (term > $o)] : (($true = ((X512 @ (sub @ X511 @ (push @ X514 @ X510))))) | ($true != ((X513 @ X511)))) | ? [X507 : subst,X506 : term] : (((sub @ one @ (push @ X506 @ X507))) != X506)) | (pushprop_lem1_lthm != $true))) & (pushprop_lem1v2_gthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & ((($true = sP76) | (hoasapinj2_lthm != $true)) & ((hoasapinj2_lthm = $true) | ($true != sP76))) & ((! [X475 : term] : (((sub @ X475 @ id)) = X475) | (termmset != $true)) & ((termmset = $true) | ? [X475 : term] : (((sub @ X475 @ id)) != X475))) & (pushprop_lthm_orig = $true) & (((? [X479 : term,X480 : subst] : (((sub @ one @ (push @ X479 @ X480))) != X479) | ! [X484 : (term > $o),X482 : (term > $o),X483 : subst,X481 : term] : (($true != ((X482 @ X481))) | ? [X485 : term] : (((X484 @ X485)) != ((X482 @ (sub @ X485 @ (push @ X481 @ X483))))) | ($true = ((X484 @ one))))) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (! [X479 : term,X480 : subst] : (((sub @ one @ (push @ X479 @ X480))) = X479) & ? [X484 : (term > $o),X482 : (term > $o),X483 : subst,X481 : term] : (($true = ((X482 @ X481))) & ! [X485 : term] : (((X484 @ X485)) = ((X482 @ (sub @ X485 @ (push @ X481 @ X483))))) & ($true != ((X484 @ one))))))) & (ulamvar1 != $true) & (((induction2 = $true) | ($true != sP74)) & (($true = sP74) | (induction2 != $true))) & ! [X210 : d_term,X211 : d_term] : ((((d2term @ X210)) != ((d2term @ X211))) | (X210 = X211)) & (((pushprop_lem1v2 = $true) | ? [X111 : (term > $o),X114 : subst,X113 : term,X112 : (term > $o)] : (($true != ((X112 @ one))) & ($true = ((X111 @ X113))) & ! [X115 : term] : (((X112 @ X115)) = ((X111 @ (sub @ X115 @ (push @ X113 @ X114))))))) & (! [X111 : (term > $o),X114 : subst,X113 : term,X112 : (term > $o)] : (($true = ((X112 @ one))) | ($true != ((X111 @ X113))) | ? [X115 : term] : (((X112 @ X115)) != ((X111 @ (sub @ X115 @ (push @ X113 @ X114)))))) | (pushprop_lem1v2 != $true))) & (hoasinduction_lem3b_gthm = $true) & (((hoasinduction_lem3_lthm = $true) | (($true != sP71) & ! [X378 : term] : (((sub @ X378 @ id)) = X378) & ($true != sP70))) & ((($true = sP71) | ? [X378 : term] : (((sub @ X378 @ id)) != X378) | ($true = sP70)) | (hoasinduction_lem3_lthm != $true))) & ((! [X456 : term,X455 : term,X457 : term,X458 : term] : ((X455 = X458) | (((ap @ X458 @ X457)) != ((ap @ X455 @ X456)))) | (apinj1 != $true)) & ((apinj1 = $true) | ? [X456 : term,X455 : term,X457 : term,X458 : term] : ((X455 != X458) & (((ap @ X458 @ X457)) = ((ap @ X455 @ X456)))))) & (((hoasinduction = $true) | ($true != sP65)) & (($true = sP65) | (hoasinduction != $true))) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (((? [X45 : subst] : (((comp @ X45 @ id)) != X45) | ? [X44 : term,X43 : subst] : (((comp @ sh @ (push @ X44 @ X43))) != X43) | ? [X41 : subst,X42 : term] : (((sub @ one @ (push @ X42 @ X41))) != X42) | ($true = sP60) | ($true = sP59)) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | (! [X45 : subst] : (((comp @ X45 @ id)) = X45) & ! [X44 : term,X43 : subst] : (((comp @ sh @ (push @ X44 @ X43))) = X43) & ! [X41 : subst,X42 : term] : (((sub @ one @ (push @ X42 @ X41))) = X42) & ($true != sP60) & ($true != sP59)))) & (pushprop_lem1_gthm = $true) & (one = ((d2term @ d_one))) & (hoasinduction_gthm = $true) & (hoaslamnotvar = $true) & (hoasinduction_lem2_lthm = $true) & (((shinj = $true) | ? [X470 : term,X469 : term] : ((X469 != X470) & (((sub @ X470 @ sh)) = ((sub @ X469 @ sh))))) & (! [X470 : term,X469 : term] : ((X469 = X470) | (((sub @ X470 @ sh)) != ((sub @ X469 @ sh)))) | (shinj != $true))) & (hoasinduction_lem3v2_gthm = $true) & (((? [X84 : term] : (((sub @ X84 @ id)) != X84) | ($true = sP54) | ($true = sP53)) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | (! [X84 : term] : (((sub @ X84 @ id)) = X84) & ($true != sP54) & ($true != sP53)))) & ((! [X477 : term] : ? [X478 : (subst > term > term)] : (((X478 @ sh @ one)) = ((sub @ X477 @ (push @ one @ sh)))) | (hoasinduction_lem3b != $true)) & ((hoasinduction_lem3b = $true) | ? [X477 : term] : ! [X478 : (subst > term > term)] : (((X478 @ sh @ one)) != ((sub @ X477 @ (push @ one @ sh)))))) & (hoasapnotvar_lthm = $true) & (axmap = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ((! [X531 : term,X532 : (subst > term > subst > $o)] : (($true = ((X532 @ id @ X531 @ id))) | ($true = ((sP49 @ X532))) | ($true = ((sP50 @ X532)))) | (hoasinduction_no_psi_cond != $true)) & ((hoasinduction_no_psi_cond = $true) | ? [X531 : term,X532 : (subst > term > subst > $o)] : (($true != ((X532 @ id @ X531 @ id))) & ($true != ((sP49 @ X532))) & ($true != ((sP50 @ X532)))))) & ! [X423 : term] : ? [X424 : d_term] : (((d2term @ X424)) = X423) & (hoasinduction_lem1 = $true) & (((($true = sP47) | ($true = sP48) | ($true = sP46) | ? [X158 : term] : (((sub @ X158 @ id)) != X158) | ? [X148 : (subst > term > subst > $o)] : ! [X149 : (term > $o)] : ? [X150 : term] : (((X148 @ id @ X150 @ id)) != ((X149 @ X150)))) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | (($true != sP47) & ($true != sP48) & ($true != sP46) & ! [X158 : term] : (((sub @ X158 @ id)) = X158) & ! [X148 : (subst > term > subst > $o)] : ? [X149 : (term > $o)] : ! [X150 : term] : (((X148 @ id @ X150 @ id)) = ((X149 @ X150)))))) & (hoaslamnotap_lthm = $true) & (induction2_gthm = $true) & (((hoaslaminj_lthm = $true) | ($true != sP37)) & (($true = sP37) | (hoaslaminj_lthm != $true))) & ((($true = sP35) | (hoasinduction_lem3aa_lthm != $true)) & ((hoasinduction_lem3aa_lthm = $true) | ($true != sP35))) & (((hoasinduction_lem3v2a_lthm = $true) | (! [X299 : term] : ? [X300 : (subst > term > term)] : ! [X302 : term,X301 : subst] : (((X300 @ X301 @ X302)) = ((sub @ X299 @ (push @ X302 @ X301)))) & ($true != sP33) & ! [X303 : term] : (((sub @ X303 @ id)) = X303))) & ((? [X299 : term] : ! [X300 : (subst > term > term)] : ? [X302 : term,X301 : subst] : (((X300 @ X301 @ X302)) != ((sub @ X299 @ (push @ X302 @ X301)))) | ($true = sP33) | ? [X303 : term] : (((sub @ X303 @ id)) != X303)) | (hoasinduction_lem3v2a_lthm != $true))) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (ulamvarind = $true) & ! [X476 : d_term] : (d_one = X476) & (hoasapinj1_gthm = $true) & ((! [X80 : term,X83 : term,X82 : term,X81 : term] : ((X80 = X83) | (((ap @ X82 @ X83)) != ((ap @ X81 @ X80)))) | (apinj2 != $true)) & ((apinj2 = $true) | ? [X80 : term,X83 : term,X82 : term,X81 : term] : ((X80 != X83) & (((ap @ X82 @ X83)) = ((ap @ X81 @ X80)))))) & (hoaslaminj_gthm = $true) & ((($true = sP31) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | ($true != sP31))) & (((hoasinduction_lem3v2 = $true) | ($true != sP28)) & (($true = sP28) | (hoasinduction_lem3v2 != $true))) & (((pushprop_lem1 = $true) | ? [X212 : subst,X216 : term,X215 : (term > $o),X213 : term,X214 : (term > $o)] : (($true != ((X215 @ (sub @ X213 @ (push @ X216 @ X212))))) & ($true = ((X214 @ X213))))) & (! [X212 : subst,X216 : term,X215 : (term > $o),X213 : term,X214 : (term > $o)] : (($true = ((X215 @ (sub @ X213 @ (push @ X216 @ X212))))) | ($true != ((X214 @ X213)))) | (pushprop_lem1 != $true))) & (axabs = $true) & ((! [X416 : (subst > term > subst > $o)] : ? [X417 : (term > $o)] : ! [X418 : term] : (((X416 @ id @ X418 @ id)) = ((X417 @ X418))) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem0 = $true) | ? [X416 : (subst > term > subst > $o)] : ! [X417 : (term > $o)] : ? [X418 : term] : (((X416 @ id @ X418 @ id)) != ((X417 @ X418))))) & ((! [X446 : term,X445 : term] : ((((lam @ X446)) != ((lam @ X445))) | (X445 = X446)) | (laminj != $true)) & ((laminj = $true) | ? [X446 : term,X445 : term] : ((((lam @ X446)) = ((lam @ X445))) & (X445 != X446)))) & (hoasapnotvar = $true) & (pushprop_lem2v2_gthm = $true) & (((termmset_lthm = $true) | (! [X1 : term] : (((sub @ X1 @ id)) = X1) & ? [X2 : term] : (((sub @ X2 @ id)) != X2))) & ((? [X1 : term] : (((sub @ X1 @ id)) != X1) | ! [X2 : term] : (((sub @ X2 @ id)) = X2)) | (termmset_lthm != $true))) & (((hoasinduction_lem3aa = $true) | ($true != sP24)) & (($true = sP24) | (hoasinduction_lem3aa != $true))) & (pushprop = $true) & ((($true = sP22) | (induction2lem != $true)) & ((induction2lem = $true) | ($true != sP22))) & (hoaslamnotvar_gthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & ((! [X348 : term,X347 : subst] : (((sub @ one @ (push @ X348 @ X347))) = X348) | (axvarcons != $true)) & ((axvarcons = $true) | ? [X348 : term,X347 : subst] : (((sub @ one @ (push @ X348 @ X347))) != X348))) & (((hoasinduction_lem3v2_f = $true) | ? [X374 : term] : ! [X375 : (subst > term > term)] : ? [X376 : subst,X377 : term] : (((sub @ X374 @ (push @ X377 @ X376))) != ((X375 @ X376 @ X377)))) & (! [X374 : term] : ? [X375 : (subst > term > term)] : ! [X376 : subst,X377 : term] : (((sub @ X374 @ (push @ X377 @ X376))) = ((X375 @ X376 @ X377))) | (hoasinduction_lem3v2_f != $true))) & (((axscons = $true) | ? [X403 : subst] : (((push @ (sub @ one @ X403) @ (comp @ sh @ X403))) != X403)) & (! [X403 : subst] : (((push @ (sub @ one @ X403) @ (comp @ sh @ X403))) = X403) | (axscons != $true))) & (ulamvarsh = $true) & ((($true = sP19) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | ($true != sP19))) & (hoasapinj2_gthm = $true) & (hoasinduction_lem3a_gthm = $true) & (sh = ((d2subst @ d_id))) & (axapp = $true) & ((! [X116 : term] : (((sub @ X116 @ id)) = X116) | (axvarid != $true)) & ((axvarid = $true) | ? [X116 : term] : (((sub @ X116 @ id)) != X116))) & ((! [X0 : subst] : (((comp @ X0 @ id)) = X0) | (axidr != $true)) & ((axidr = $true) | ? [X0 : subst] : (((comp @ X0 @ id)) != X0))) & ((! [X63 : term,X60 : term,X61 : term,X62 : term] : ((X60 = X61) | (((ap @ (sub @ X61 @ id) @ X62)) != ((ap @ (sub @ X60 @ id) @ X63)))) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | ? [X63 : term,X60 : term,X61 : term,X62 : term] : ((X60 != X61) & (((ap @ (sub @ X61 @ id) @ X62)) = ((ap @ (sub @ X60 @ id) @ X63)))))) & (apnotvar = $true) & ((! [X344 : (subst > term > subst > $o)] : ? [X345 : (term > $o)] : ! [X346 : term] : (((X345 @ X346)) = ((X344 @ id @ X346 @ id))) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ? [X344 : (subst > term > subst > $o)] : ! [X345 : (term > $o)] : ? [X346 : term] : (((X345 @ X346)) != ((X344 @ id @ X346 @ id))))) & ((($true = sP17) | (hoasinduction_lem3aaa != $true)) & ((hoasinduction_lem3aaa = $true) | ($true != sP17))) & (((! [X4 : subst] : (((comp @ id @ X4)) = X4) & ! [X3 : subst] : (((comp @ X3 @ id)) = X3)) | (substmonoid != $true)) & ((substmonoid = $true) | (? [X4 : subst] : (((comp @ id @ X4)) != X4) | ? [X3 : subst] : (((comp @ X3 @ id)) != X3)))) & (hoaslamnotap_gthm = $true) & (lamnotvar = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & (hoasinduction_lem3_gthm = $true) & (induction2lem_gthm = $true) & (hoaslamnotvar_lthm = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ! [X146 : subst] : ? [X147 : d_subst] : (((d2subst @ X147)) = X146) & ((! [X486 : term] : ? [X487 : (subst > term > term)] : (((sub @ X486 @ (push @ one @ sh))) = ((X487 @ sh @ one))) | (hoasinduction_lem3b_lthm != $true)) & ((hoasinduction_lem3b_lthm = $true) | ? [X486 : term] : ! [X487 : (subst > term > term)] : (((sub @ X486 @ (push @ one @ sh))) != ((X487 @ sh @ one))))) & (((induction = $true) | ($true != sP15)) & (($true = sP15) | (induction != $true))) & (((hoasinduction_no_psi_cond_lthm = $true) | (! [X314 : (subst > term > subst > $o)] : ? [X315 : (term > $o)] : ! [X316 : term] : (((X314 @ id @ X316 @ id)) = ((X315 @ X316))) & ($true != sP10) & ! [X324 : term] : (((sub @ X324 @ id)) = X324) & ($true != sP12) & ($true != sP11))) & ((? [X314 : (subst > term > subst > $o)] : ! [X315 : (term > $o)] : ? [X316 : term] : (((X314 @ id @ X316 @ id)) != ((X315 @ X316))) | ($true = sP10) | ? [X324 : term] : (((sub @ X324 @ id)) != X324) | ($true = sP12) | ($true = sP11)) | (hoasinduction_no_psi_cond_lthm != $true))) & (((hoasinduction_lem3v2_lthm = $true) | (($true != sP4) & ! [X217 : term] : (((sub @ X217 @ id)) = X217))) & ((($true = sP4) | ? [X217 : term] : (((sub @ X217 @ id)) != X217)) | (hoasinduction_lem3v2_lthm != $true))) & (((hoasapinj2 = $true) | ? [X473 : term,X472 : term,X471 : term,X474 : term] : ((X472 != X473) & (((ap @ (sub @ X474 @ id) @ X472)) = ((ap @ (sub @ X471 @ id) @ X473))))) & (! [X473 : term,X472 : term,X471 : term,X474 : term] : ((X472 = X473) | (((ap @ (sub @ X474 @ id) @ X472)) != ((ap @ (sub @ X471 @ id) @ X473)))) | (hoasapinj2 != $true))) & ((($true = sP0) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | ($true != sP0))) & (((pushprop_lem0_lthm = $true) | ? [X107 : subst,X108 : term,X106 : (term > $o)] : ! [X109 : (term > $o)] : ? [X110 : term] : (((X106 @ (sub @ X110 @ (push @ X108 @ X107)))) != ((X109 @ X110)))) & (! [X107 : subst,X108 : term,X106 : (term > $o)] : ? [X109 : (term > $o)] : ! [X110 : term] : (((X106 @ (sub @ X110 @ (push @ X108 @ X107)))) = ((X109 @ X110))) | (pushprop_lem0_lthm != $true))) & (pushprop_lem2v2 = $true)),
  inference(nnf_transformation,[],[f122])).
thf(f455,plain,(
  ((hoaslaminj = $true) | ? [X461 : (subst > term > term),X462 : term,X459 : subst,X460 : (subst > term > term)] : ((((lam @ (X461 @ sh @ one))) = ((lam @ (X460 @ sh @ one)))) & ! [X468 : subst,X466 : term,X467 : subst] : (((X461 @ (comp @ X468 @ X467) @ (sub @ X466 @ X467))) = ((sub @ (X461 @ X468 @ X466) @ X467))) & ! [X464 : term,X465 : subst,X463 : subst] : (((X460 @ (comp @ X463 @ X465) @ (sub @ X464 @ X465))) = ((sub @ (X460 @ X463 @ X464) @ X465))) & (((X460 @ X459 @ X462)) != ((X461 @ X459 @ X462))))) & (! [X461 : (subst > term > term),X462 : term,X459 : subst,X460 : (subst > term > term)] : ((((lam @ (X461 @ sh @ one))) != ((lam @ (X460 @ sh @ one)))) | ? [X468 : subst,X466 : term,X467 : subst] : (((X461 @ (comp @ X468 @ X467) @ (sub @ X466 @ X467))) != ((sub @ (X461 @ X468 @ X466) @ X467))) | ? [X464 : term,X465 : subst,X463 : subst] : (((X460 @ (comp @ X463 @ X465) @ (sub @ X464 @ X465))) != ((sub @ (X460 @ X463 @ X464) @ X465))) | (((X460 @ X459 @ X462)) = ((X461 @ X459 @ X462)))) | (hoaslaminj != $true)) & ((axshiftcons = $true) | ? [X443 : subst,X444 : term] : (((comp @ sh @ (push @ X444 @ X443))) != X443)) & (! [X443 : subst,X444 : term] : (((comp @ sh @ (push @ X444 @ X443))) = X443) | (axshiftcons != $true)) & ((pushprop_lem0 = $true) | ? [X102 : (term > $o),X103 : term,X101 : subst] : ! [X104 : (term > $o)] : ? [X105 : term] : (((X104 @ X105)) != ((X102 @ (sub @ X105 @ (push @ X103 @ X101)))))) & (! [X102 : (term > $o),X103 : term,X101 : subst] : ? [X104 : (term > $o)] : ! [X105 : term] : (((X104 @ X105)) = ((X102 @ (sub @ X105 @ (push @ X103 @ X101))))) | (pushprop_lem0 != $true)) & ! [X5 : d_subst] : (d_id = X5) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X419 : term] : ! [X420 : (subst > term > term)] : ? [X421 : subst,X422 : term] : (((X420 @ X421 @ X422)) != ((sub @ X419 @ (push @ X422 @ X421))))) & (! [X419 : term] : ? [X420 : (subst > term > term)] : ! [X421 : subst,X422 : term] : (((X420 @ X421 @ X422)) = ((sub @ X419 @ (push @ X422 @ X421)))) | (hoasinduction_lem3v2_f_lthm != $true)) & (axclos = $true) & (hoasinduction_lem1_lthm = $true) & (termmset_gthm = $true) & (($true = sP108) | (hoasinduction_lem3 != $true)) & ((hoasinduction_lem3 = $true) | ($true != sP108)) & (($true = sP104) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | ($true != sP104)) & (axvarshift = $true) & (pushprop_gthm = $true) & (! [X411 : term,X410 : (subst > term > term),X412 : term] : (? [X415 : term,X414 : subst,X413 : subst] : (((X410 @ (comp @ X414 @ X413) @ (sub @ X415 @ X413))) != ((sub @ (X410 @ X414 @ X415) @ X413))) | (((ap @ (sub @ X412 @ id) @ X411)) != ((lam @ (X410 @ sh @ one))))) | (hoaslamnotap != $true)) & ((hoaslamnotap = $true) | ? [X411 : term,X410 : (subst > term > term),X412 : term] : (! [X415 : term,X414 : subst,X413 : subst] : (((X410 @ (comp @ X414 @ X413) @ (sub @ X415 @ X413))) = ((sub @ (X410 @ X414 @ X415) @ X413))) & (((ap @ (sub @ X412 @ id) @ X411)) = ((lam @ (X410 @ sh @ one)))))) & (($true = sP102) | (hoasinduction_lem2v2 != $true)) & ((hoasinduction_lem2v2 = $true) | ($true != sP102)) & (pushprop_lem0_gthm = $true) & ! [X409 : d_subst,X408 : d_subst] : ((((d2subst @ X409)) != ((d2subst @ X408))) | (X408 = X409)) & (hoasinduction_lem1_gthm = $true) & (hoasinduction_lem1v2 = $true) & (substmonoid_gthm = $true) & (pushprop_lem3v2_lthm = $true) & (pushprop_lem2v2_lthm = $true) & (((d2subst @ d_id)) = id) & (((var @ (d2term @ d_one))) != $true) & (axassoc = $true) & (lamnotap != $true) & ((hoasinduction_lem3a = $true) | ($true != sP99)) & (($true = sP99) | (hoasinduction_lem3a != $true)) & ((induction2_lthm = $true) | (($true != sP97) & ($true != sP96) & ! [X24 : term] : (((sub @ X24 @ id)) = X24))) & (($true = sP97) | ($true = sP96) | ? [X24 : term] : (((sub @ X24 @ id)) != X24) | (induction2_lthm != $true)) & (($true = sP91) | ($true = sP90) | ($true = sP88) | ($true = sP89) | (hoasinduction_lthm != $true)) & ((hoasinduction_lthm = $true) | (($true != sP91) & ($true != sP90) & ($true != sP88) & ($true != sP89))) & (hoasinduction_lem2_gthm = $true) & (hoasinduction_lem2v2_gthm = $true) & (pushprop_lem3v2 = $true) & (pushprop_lthm = $true) & (! [X40 : subst] : (((comp @ id @ X40)) = X40) | (axidl != $true)) & ((axidl = $true) | ? [X40 : subst] : (((comp @ id @ X40)) != X40)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((pushprop_lem1_lthm = $true) | (! [X508 : subst,X509 : term] : (((comp @ sh @ (push @ X509 @ X508))) = X508) & ? [X511 : term,X514 : term,X510 : subst,X512 : (term > $o),X513 : (term > $o)] : (($true != ((X512 @ (sub @ X511 @ (push @ X514 @ X510))))) & ($true = ((X513 @ X511)))) & ! [X507 : subst,X506 : term] : (((sub @ one @ (push @ X506 @ X507))) = X506))) & (? [X508 : subst,X509 : term] : (((comp @ sh @ (push @ X509 @ X508))) != X508) | ! [X511 : term,X514 : term,X510 : subst,X512 : (term > $o),X513 : (term > $o)] : (($true = ((X512 @ (sub @ X511 @ (push @ X514 @ X510))))) | ($true != ((X513 @ X511)))) | ? [X507 : subst,X506 : term] : (((sub @ one @ (push @ X506 @ X507))) != X506) | (pushprop_lem1_lthm != $true)) & (pushprop_lem1v2_gthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (($true = sP76) | (hoasapinj2_lthm != $true)) & ((hoasapinj2_lthm = $true) | ($true != sP76)) & (! [X475 : term] : (((sub @ X475 @ id)) = X475) | (termmset != $true)) & ((termmset = $true) | ? [X475 : term] : (((sub @ X475 @ id)) != X475)) & (pushprop_lthm_orig = $true) & (? [X479 : term,X480 : subst] : (((sub @ one @ (push @ X479 @ X480))) != X479) | ! [X484 : (term > $o),X482 : (term > $o),X483 : subst,X481 : term] : (($true != ((X482 @ X481))) | ? [X485 : term] : (((X484 @ X485)) != ((X482 @ (sub @ X485 @ (push @ X481 @ X483))))) | ($true = ((X484 @ one)))) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (! [X479 : term,X480 : subst] : (((sub @ one @ (push @ X479 @ X480))) = X479) & ? [X484 : (term > $o),X482 : (term > $o),X483 : subst,X481 : term] : (($true = ((X482 @ X481))) & ! [X485 : term] : (((X484 @ X485)) = ((X482 @ (sub @ X485 @ (push @ X481 @ X483))))) & ($true != ((X484 @ one)))))) & (ulamvar1 != $true) & ((induction2 = $true) | ($true != sP74)) & (($true = sP74) | (induction2 != $true)) & ! [X210 : d_term,X211 : d_term] : ((((d2term @ X210)) != ((d2term @ X211))) | (X210 = X211)) & ((pushprop_lem1v2 = $true) | ? [X111 : (term > $o),X114 : subst,X113 : term,X112 : (term > $o)] : (($true != ((X112 @ one))) & ($true = ((X111 @ X113))) & ! [X115 : term] : (((X112 @ X115)) = ((X111 @ (sub @ X115 @ (push @ X113 @ X114))))))) & (! [X111 : (term > $o),X114 : subst,X113 : term,X112 : (term > $o)] : (($true = ((X112 @ one))) | ($true != ((X111 @ X113))) | ? [X115 : term] : (((X112 @ X115)) != ((X111 @ (sub @ X115 @ (push @ X113 @ X114)))))) | (pushprop_lem1v2 != $true)) & (hoasinduction_lem3b_gthm = $true) & ((hoasinduction_lem3_lthm = $true) | (($true != sP71) & ! [X378 : term] : (((sub @ X378 @ id)) = X378) & ($true != sP70))) & (($true = sP71) | ? [X378 : term] : (((sub @ X378 @ id)) != X378) | ($true = sP70) | (hoasinduction_lem3_lthm != $true)) & (! [X456 : term,X455 : term,X457 : term,X458 : term] : ((X455 = X458) | (((ap @ X458 @ X457)) != ((ap @ X455 @ X456)))) | (apinj1 != $true)) & ((apinj1 = $true) | ? [X456 : term,X455 : term,X457 : term,X458 : term] : ((X455 != X458) & (((ap @ X458 @ X457)) = ((ap @ X455 @ X456))))) & ((hoasinduction = $true) | ($true != sP65)) & (($true = sP65) | (hoasinduction != $true)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (? [X45 : subst] : (((comp @ X45 @ id)) != X45) | ? [X44 : term,X43 : subst] : (((comp @ sh @ (push @ X44 @ X43))) != X43) | ? [X41 : subst,X42 : term] : (((sub @ one @ (push @ X42 @ X41))) != X42) | ($true = sP60) | ($true = sP59) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | (! [X45 : subst] : (((comp @ X45 @ id)) = X45) & ! [X44 : term,X43 : subst] : (((comp @ sh @ (push @ X44 @ X43))) = X43) & ! [X41 : subst,X42 : term] : (((sub @ one @ (push @ X42 @ X41))) = X42) & ($true != sP60) & ($true != sP59))) & (pushprop_lem1_gthm = $true) & (one = ((d2term @ d_one))) & (hoasinduction_gthm = $true) & (hoaslamnotvar = $true) & (hoasinduction_lem2_lthm = $true) & ((shinj = $true) | ? [X470 : term,X469 : term] : ((X469 != X470) & (((sub @ X470 @ sh)) = ((sub @ X469 @ sh))))) & (! [X470 : term,X469 : term] : ((X469 = X470) | (((sub @ X470 @ sh)) != ((sub @ X469 @ sh)))) | (shinj != $true)) & (hoasinduction_lem3v2_gthm = $true) & (? [X84 : term] : (((sub @ X84 @ id)) != X84) | ($true = sP54) | ($true = sP53) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | (! [X84 : term] : (((sub @ X84 @ id)) = X84) & ($true != sP54) & ($true != sP53))) & (! [X477 : term] : ? [X478 : (subst > term > term)] : (((X478 @ sh @ one)) = ((sub @ X477 @ (push @ one @ sh)))) | (hoasinduction_lem3b != $true)) & ((hoasinduction_lem3b = $true) | ? [X477 : term] : ! [X478 : (subst > term > term)] : (((X478 @ sh @ one)) != ((sub @ X477 @ (push @ one @ sh))))) & (hoasapnotvar_lthm = $true) & (axmap = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (! [X531 : term,X532 : (subst > term > subst > $o)] : (($true = ((X532 @ id @ X531 @ id))) | ($true = ((sP49 @ X532))) | ($true = ((sP50 @ X532)))) | (hoasinduction_no_psi_cond != $true)) & ((hoasinduction_no_psi_cond = $true) | ? [X531 : term,X532 : (subst > term > subst > $o)] : (($true != ((X532 @ id @ X531 @ id))) & ($true != ((sP49 @ X532))) & ($true != ((sP50 @ X532))))) & ! [X423 : term] : ? [X424 : d_term] : (((d2term @ X424)) = X423) & (hoasinduction_lem1 = $true) & (($true = sP47) | ($true = sP48) | ($true = sP46) | ? [X158 : term] : (((sub @ X158 @ id)) != X158) | ? [X148 : (subst > term > subst > $o)] : ! [X149 : (term > $o)] : ? [X150 : term] : (((X148 @ id @ X150 @ id)) != ((X149 @ X150))) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | (($true != sP47) & ($true != sP48) & ($true != sP46) & ! [X158 : term] : (((sub @ X158 @ id)) = X158) & ! [X148 : (subst > term > subst > $o)] : ? [X149 : (term > $o)] : ! [X150 : term] : (((X148 @ id @ X150 @ id)) = ((X149 @ X150))))) & (hoaslamnotap_lthm = $true) & (induction2_gthm = $true) & ((hoaslaminj_lthm = $true) | ($true != sP37)) & (($true = sP37) | (hoaslaminj_lthm != $true)) & (($true = sP35) | (hoasinduction_lem3aa_lthm != $true)) & ((hoasinduction_lem3aa_lthm = $true) | ($true != sP35)) & ((hoasinduction_lem3v2a_lthm = $true) | (! [X299 : term] : ? [X300 : (subst > term > term)] : ! [X302 : term,X301 : subst] : (((X300 @ X301 @ X302)) = ((sub @ X299 @ (push @ X302 @ X301)))) & ($true != sP33) & ! [X303 : term] : (((sub @ X303 @ id)) = X303))) & (? [X299 : term] : ! [X300 : (subst > term > term)] : ? [X302 : term,X301 : subst] : (((X300 @ X301 @ X302)) != ((sub @ X299 @ (push @ X302 @ X301)))) | ($true = sP33) | ? [X303 : term] : (((sub @ X303 @ id)) != X303) | (hoasinduction_lem3v2a_lthm != $true)) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (ulamvarind = $true) & ! [X476 : d_term] : (d_one = X476) & (hoasapinj1_gthm = $true) & (! [X80 : term,X83 : term,X82 : term,X81 : term] : ((X80 = X83) | (((ap @ X82 @ X83)) != ((ap @ X81 @ X80)))) | (apinj2 != $true)) & ((apinj2 = $true) | ? [X80 : term,X83 : term,X82 : term,X81 : term] : ((X80 != X83) & (((ap @ X82 @ X83)) = ((ap @ X81 @ X80))))) & (hoaslaminj_gthm = $true) & (($true = sP31) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | ($true != sP31)) & ((hoasinduction_lem3v2 = $true) | ($true != sP28)) & (($true = sP28) | (hoasinduction_lem3v2 != $true)) & ((pushprop_lem1 = $true) | ? [X212 : subst,X216 : term,X215 : (term > $o),X213 : term,X214 : (term > $o)] : (($true != ((X215 @ (sub @ X213 @ (push @ X216 @ X212))))) & ($true = ((X214 @ X213))))) & (! [X212 : subst,X216 : term,X215 : (term > $o),X213 : term,X214 : (term > $o)] : (($true = ((X215 @ (sub @ X213 @ (push @ X216 @ X212))))) | ($true != ((X214 @ X213)))) | (pushprop_lem1 != $true)) & (axabs = $true) & (! [X416 : (subst > term > subst > $o)] : ? [X417 : (term > $o)] : ! [X418 : term] : (((X416 @ id @ X418 @ id)) = ((X417 @ X418))) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem0 = $true) | ? [X416 : (subst > term > subst > $o)] : ! [X417 : (term > $o)] : ? [X418 : term] : (((X416 @ id @ X418 @ id)) != ((X417 @ X418)))) & (! [X446 : term,X445 : term] : ((((lam @ X446)) != ((lam @ X445))) | (X445 = X446)) | (laminj != $true)) & ((laminj = $true) | ? [X446 : term,X445 : term] : ((((lam @ X446)) = ((lam @ X445))) & (X445 != X446))) & (hoasapnotvar = $true) & (pushprop_lem2v2_gthm = $true) & ((termmset_lthm = $true) | (! [X1 : term] : (((sub @ X1 @ id)) = X1) & ? [X2 : term] : (((sub @ X2 @ id)) != X2))) & (? [X1 : term] : (((sub @ X1 @ id)) != X1) | ! [X2 : term] : (((sub @ X2 @ id)) = X2) | (termmset_lthm != $true)) & ((hoasinduction_lem3aa = $true) | ($true != sP24)) & (($true = sP24) | (hoasinduction_lem3aa != $true)) & (pushprop = $true) & (($true = sP22) | (induction2lem != $true)) & ((induction2lem = $true) | ($true != sP22)) & (hoaslamnotvar_gthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (! [X348 : term,X347 : subst] : (((sub @ one @ (push @ X348 @ X347))) = X348) | (axvarcons != $true)) & ((axvarcons = $true) | ? [X348 : term,X347 : subst] : (((sub @ one @ (push @ X348 @ X347))) != X348)) & ((hoasinduction_lem3v2_f = $true) | ? [X374 : term] : ! [X375 : (subst > term > term)] : ? [X376 : subst,X377 : term] : (((sub @ X374 @ (push @ X377 @ X376))) != ((X375 @ X376 @ X377)))) & (! [X374 : term] : ? [X375 : (subst > term > term)] : ! [X376 : subst,X377 : term] : (((sub @ X374 @ (push @ X377 @ X376))) = ((X375 @ X376 @ X377))) | (hoasinduction_lem3v2_f != $true)) & ((axscons = $true) | ? [X403 : subst] : (((push @ (sub @ one @ X403) @ (comp @ sh @ X403))) != X403)) & (! [X403 : subst] : (((push @ (sub @ one @ X403) @ (comp @ sh @ X403))) = X403) | (axscons != $true)) & (ulamvarsh = $true) & (($true = sP19) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | ($true != sP19)) & (hoasapinj2_gthm = $true) & (hoasinduction_lem3a_gthm = $true) & (sh = ((d2subst @ d_id))) & (axapp = $true) & (! [X116 : term] : (((sub @ X116 @ id)) = X116) | (axvarid != $true)) & ((axvarid = $true) | ? [X116 : term] : (((sub @ X116 @ id)) != X116)) & (! [X0 : subst] : (((comp @ X0 @ id)) = X0) | (axidr != $true)) & ((axidr = $true) | ? [X0 : subst] : (((comp @ X0 @ id)) != X0)) & (! [X63 : term,X60 : term,X61 : term,X62 : term] : ((X60 = X61) | (((ap @ (sub @ X61 @ id) @ X62)) != ((ap @ (sub @ X60 @ id) @ X63)))) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | ? [X63 : term,X60 : term,X61 : term,X62 : term] : ((X60 != X61) & (((ap @ (sub @ X61 @ id) @ X62)) = ((ap @ (sub @ X60 @ id) @ X63))))) & (apnotvar = $true) & (! [X344 : (subst > term > subst > $o)] : ? [X345 : (term > $o)] : ! [X346 : term] : (((X345 @ X346)) = ((X344 @ id @ X346 @ id))) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ? [X344 : (subst > term > subst > $o)] : ! [X345 : (term > $o)] : ? [X346 : term] : (((X345 @ X346)) != ((X344 @ id @ X346 @ id)))) & (($true = sP17) | (hoasinduction_lem3aaa != $true)) & ((hoasinduction_lem3aaa = $true) | ($true != sP17)) & ((! [X4 : subst] : (((comp @ id @ X4)) = X4) & ! [X3 : subst] : (((comp @ X3 @ id)) = X3)) | (substmonoid != $true)) & ((substmonoid = $true) | ? [X4 : subst] : (((comp @ id @ X4)) != X4) | ? [X3 : subst] : (((comp @ X3 @ id)) != X3)) & (hoaslamnotap_gthm = $true) & (lamnotvar = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & (hoasinduction_lem3_gthm = $true) & (induction2lem_gthm = $true) & (hoaslamnotvar_lthm = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ! [X146 : subst] : ? [X147 : d_subst] : (((d2subst @ X147)) = X146) & (! [X486 : term] : ? [X487 : (subst > term > term)] : (((sub @ X486 @ (push @ one @ sh))) = ((X487 @ sh @ one))) | (hoasinduction_lem3b_lthm != $true)) & ((hoasinduction_lem3b_lthm = $true) | ? [X486 : term] : ! [X487 : (subst > term > term)] : (((sub @ X486 @ (push @ one @ sh))) != ((X487 @ sh @ one)))) & ((induction = $true) | ($true != sP15)) & (($true = sP15) | (induction != $true)) & ((hoasinduction_no_psi_cond_lthm = $true) | (! [X314 : (subst > term > subst > $o)] : ? [X315 : (term > $o)] : ! [X316 : term] : (((X314 @ id @ X316 @ id)) = ((X315 @ X316))) & ($true != sP10) & ! [X324 : term] : (((sub @ X324 @ id)) = X324) & ($true != sP12) & ($true != sP11))) & (? [X314 : (subst > term > subst > $o)] : ! [X315 : (term > $o)] : ? [X316 : term] : (((X314 @ id @ X316 @ id)) != ((X315 @ X316))) | ($true = sP10) | ? [X324 : term] : (((sub @ X324 @ id)) != X324) | ($true = sP12) | ($true = sP11) | (hoasinduction_no_psi_cond_lthm != $true)) & ((hoasinduction_lem3v2_lthm = $true) | (($true != sP4) & ! [X217 : term] : (((sub @ X217 @ id)) = X217))) & (($true = sP4) | ? [X217 : term] : (((sub @ X217 @ id)) != X217) | (hoasinduction_lem3v2_lthm != $true)) & ((hoasapinj2 = $true) | ? [X473 : term,X472 : term,X471 : term,X474 : term] : ((X472 != X473) & (((ap @ (sub @ X474 @ id) @ X472)) = ((ap @ (sub @ X471 @ id) @ X473))))) & (! [X473 : term,X472 : term,X471 : term,X474 : term] : ((X472 = X473) | (((ap @ (sub @ X474 @ id) @ X472)) != ((ap @ (sub @ X471 @ id) @ X473)))) | (hoasapinj2 != $true)) & (($true = sP0) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | ($true != sP0)) & ((pushprop_lem0_lthm = $true) | ? [X107 : subst,X108 : term,X106 : (term > $o)] : ! [X109 : (term > $o)] : ? [X110 : term] : (((X106 @ (sub @ X110 @ (push @ X108 @ X107)))) != ((X109 @ X110)))) & (! [X107 : subst,X108 : term,X106 : (term > $o)] : ? [X109 : (term > $o)] : ! [X110 : term] : (((X106 @ (sub @ X110 @ (push @ X108 @ X107)))) = ((X109 @ X110))) | (pushprop_lem0_lthm != $true)) & (pushprop_lem2v2 = $true)),
  inference(flattening,[],[f454])).
thf(f456,plain,(
  ((hoaslaminj = $true) | ? [X0 : (subst > term > term),X1 : term,X2 : subst,X3 : (subst > term > term)] : ((((lam @ (X0 @ sh @ one))) = ((lam @ (X3 @ sh @ one)))) & ! [X4 : subst,X5 : term,X6 : subst] : (((sub @ (X0 @ X4 @ X5) @ X6)) = ((X0 @ (comp @ X4 @ X6) @ (sub @ X5 @ X6)))) & ! [X7 : term,X8 : subst,X9 : subst] : (((X3 @ (comp @ X9 @ X8) @ (sub @ X7 @ X8))) = ((sub @ (X3 @ X9 @ X7) @ X8))) & (((X3 @ X2 @ X1)) != ((X0 @ X2 @ X1))))) & (! [X10 : (subst > term > term),X11 : term,X12 : subst,X13 : (subst > term > term)] : ((((lam @ (X13 @ sh @ one))) != ((lam @ (X10 @ sh @ one)))) | ? [X14 : subst,X15 : term,X16 : subst] : (((sub @ (X10 @ X14 @ X15) @ X16)) != ((X10 @ (comp @ X14 @ X16) @ (sub @ X15 @ X16)))) | ? [X17 : term,X18 : subst,X19 : subst] : (((X13 @ (comp @ X19 @ X18) @ (sub @ X17 @ X18))) != ((sub @ (X13 @ X19 @ X17) @ X18))) | (((X10 @ X12 @ X11)) = ((X13 @ X12 @ X11)))) | (hoaslaminj != $true)) & ((axshiftcons = $true) | ? [X20 : subst,X21 : term] : (((comp @ sh @ (push @ X21 @ X20))) != X20)) & (! [X22 : subst,X23 : term] : (((comp @ sh @ (push @ X23 @ X22))) = X22) | (axshiftcons != $true)) & ((pushprop_lem0 = $true) | ? [X24 : (term > $o),X25 : term,X26 : subst] : ! [X27 : (term > $o)] : ? [X28 : term] : (((X24 @ (sub @ X28 @ (push @ X25 @ X26)))) != ((X27 @ X28)))) & (! [X29 : (term > $o),X30 : term,X31 : subst] : ? [X32 : (term > $o)] : ! [X33 : term] : (((X32 @ X33)) = ((X29 @ (sub @ X33 @ (push @ X30 @ X31))))) | (pushprop_lem0 != $true)) & ! [X34 : d_subst] : (d_id = X34) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X35 : term] : ! [X36 : (subst > term > term)] : ? [X37 : subst,X38 : term] : (((sub @ X35 @ (push @ X38 @ X37))) != ((X36 @ X37 @ X38)))) & (! [X39 : term] : ? [X40 : (subst > term > term)] : ! [X41 : subst,X42 : term] : (((sub @ X39 @ (push @ X42 @ X41))) = ((X40 @ X41 @ X42))) | (hoasinduction_lem3v2_f_lthm != $true)) & (axclos = $true) & (hoasinduction_lem1_lthm = $true) & (termmset_gthm = $true) & (($true = sP108) | (hoasinduction_lem3 != $true)) & ((hoasinduction_lem3 = $true) | ($true != sP108)) & (($true = sP104) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | ($true != sP104)) & (axvarshift = $true) & (pushprop_gthm = $true) & (! [X43 : term,X44 : (subst > term > term),X45 : term] : (? [X46 : term,X47 : subst,X48 : subst] : (((X44 @ (comp @ X47 @ X48) @ (sub @ X46 @ X48))) != ((sub @ (X44 @ X47 @ X46) @ X48))) | (((ap @ (sub @ X45 @ id) @ X43)) != ((lam @ (X44 @ sh @ one))))) | (hoaslamnotap != $true)) & ((hoaslamnotap = $true) | ? [X49 : term,X50 : (subst > term > term),X51 : term] : (! [X52 : term,X53 : subst,X54 : subst] : (((X50 @ (comp @ X53 @ X54) @ (sub @ X52 @ X54))) = ((sub @ (X50 @ X53 @ X52) @ X54))) & (((ap @ (sub @ X51 @ id) @ X49)) = ((lam @ (X50 @ sh @ one)))))) & (($true = sP102) | (hoasinduction_lem2v2 != $true)) & ((hoasinduction_lem2v2 = $true) | ($true != sP102)) & (pushprop_lem0_gthm = $true) & ! [X55 : d_subst,X56 : d_subst] : ((((d2subst @ X55)) != ((d2subst @ X56))) | (X55 = X56)) & (hoasinduction_lem1_gthm = $true) & (hoasinduction_lem1v2 = $true) & (substmonoid_gthm = $true) & (pushprop_lem3v2_lthm = $true) & (pushprop_lem2v2_lthm = $true) & (((d2subst @ d_id)) = id) & (((var @ (d2term @ d_one))) != $true) & (axassoc = $true) & (lamnotap != $true) & ((hoasinduction_lem3a = $true) | ($true != sP99)) & (($true = sP99) | (hoasinduction_lem3a != $true)) & ((induction2_lthm = $true) | (($true != sP97) & ($true != sP96) & ! [X57 : term] : (((sub @ X57 @ id)) = X57))) & (($true = sP97) | ($true = sP96) | ? [X58 : term] : (((sub @ X58 @ id)) != X58) | (induction2_lthm != $true)) & (($true = sP91) | ($true = sP90) | ($true = sP88) | ($true = sP89) | (hoasinduction_lthm != $true)) & ((hoasinduction_lthm = $true) | (($true != sP91) & ($true != sP90) & ($true != sP88) & ($true != sP89))) & (hoasinduction_lem2_gthm = $true) & (hoasinduction_lem2v2_gthm = $true) & (pushprop_lem3v2 = $true) & (pushprop_lthm = $true) & (! [X59 : subst] : (((comp @ id @ X59)) = X59) | (axidl != $true)) & ((axidl = $true) | ? [X60 : subst] : (((comp @ id @ X60)) != X60)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((pushprop_lem1_lthm = $true) | (! [X61 : subst,X62 : term] : (((comp @ sh @ (push @ X62 @ X61))) = X61) & ? [X63 : term,X64 : term,X65 : subst,X66 : (term > $o),X67 : (term > $o)] : (($true != ((X66 @ (sub @ X63 @ (push @ X64 @ X65))))) & ($true = ((X67 @ X63)))) & ! [X68 : subst,X69 : term] : (((sub @ one @ (push @ X69 @ X68))) = X69))) & (? [X70 : subst,X71 : term] : (((comp @ sh @ (push @ X71 @ X70))) != X70) | ! [X72 : term,X73 : term,X74 : subst,X75 : (term > $o),X76 : (term > $o)] : (($true = ((X75 @ (sub @ X72 @ (push @ X73 @ X74))))) | (((X76 @ X72)) != $true)) | ? [X77 : subst,X78 : term] : (((sub @ one @ (push @ X78 @ X77))) != X78) | (pushprop_lem1_lthm != $true)) & (pushprop_lem1v2_gthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (($true = sP76) | (hoasapinj2_lthm != $true)) & ((hoasapinj2_lthm = $true) | ($true != sP76)) & (! [X79 : term] : (((sub @ X79 @ id)) = X79) | (termmset != $true)) & ((termmset = $true) | ? [X80 : term] : (((sub @ X80 @ id)) != X80)) & (pushprop_lthm_orig = $true) & (? [X81 : term,X82 : subst] : (((sub @ one @ (push @ X81 @ X82))) != X81) | ! [X83 : (term > $o),X84 : (term > $o),X85 : subst,X86 : term] : (($true != ((X84 @ X86))) | ? [X87 : term] : (((X83 @ X87)) != ((X84 @ (sub @ X87 @ (push @ X86 @ X85))))) | ($true = ((X83 @ one)))) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (! [X88 : term,X89 : subst] : (((sub @ one @ (push @ X88 @ X89))) = X88) & ? [X90 : (term > $o),X91 : (term > $o),X92 : subst,X93 : term] : (($true = ((X91 @ X93))) & ! [X94 : term] : (((X90 @ X94)) = ((X91 @ (sub @ X94 @ (push @ X93 @ X92))))) & ($true != ((X90 @ one)))))) & (ulamvar1 != $true) & ((induction2 = $true) | ($true != sP74)) & (($true = sP74) | (induction2 != $true)) & ! [X95 : d_term,X96 : d_term] : ((((d2term @ X95)) != ((d2term @ X96))) | (X95 = X96)) & ((pushprop_lem1v2 = $true) | ? [X97 : (term > $o),X98 : subst,X99 : term,X100 : (term > $o)] : (($true != ((X100 @ one))) & ($true = ((X97 @ X99))) & ! [X101 : term] : (((X97 @ (sub @ X101 @ (push @ X99 @ X98)))) = ((X100 @ X101))))) & (! [X102 : (term > $o),X103 : subst,X104 : term,X105 : (term > $o)] : (($true = ((X105 @ one))) | ($true != ((X102 @ X104))) | ? [X106 : term] : (((X102 @ (sub @ X106 @ (push @ X104 @ X103)))) != ((X105 @ X106)))) | (pushprop_lem1v2 != $true)) & (hoasinduction_lem3b_gthm = $true) & ((hoasinduction_lem3_lthm = $true) | (($true != sP71) & ! [X107 : term] : (((sub @ X107 @ id)) = X107) & ($true != sP70))) & (($true = sP71) | ? [X108 : term] : (((sub @ X108 @ id)) != X108) | ($true = sP70) | (hoasinduction_lem3_lthm != $true)) & (! [X109 : term,X110 : term,X111 : term,X112 : term] : ((X110 = X112) | (((ap @ X112 @ X111)) != ((ap @ X110 @ X109)))) | (apinj1 != $true)) & ((apinj1 = $true) | ? [X113 : term,X114 : term,X115 : term,X116 : term] : ((X114 != X116) & (((ap @ X116 @ X115)) = ((ap @ X114 @ X113))))) & ((hoasinduction = $true) | ($true != sP65)) & (($true = sP65) | (hoasinduction != $true)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (? [X117 : subst] : (((comp @ X117 @ id)) != X117) | ? [X118 : term,X119 : subst] : (((comp @ sh @ (push @ X118 @ X119))) != X119) | ? [X120 : subst,X121 : term] : (((sub @ one @ (push @ X121 @ X120))) != X121) | ($true = sP60) | ($true = sP59) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | (! [X122 : subst] : (((comp @ X122 @ id)) = X122) & ! [X123 : term,X124 : subst] : (((comp @ sh @ (push @ X123 @ X124))) = X124) & ! [X125 : subst,X126 : term] : (((sub @ one @ (push @ X126 @ X125))) = X126) & ($true != sP60) & ($true != sP59))) & (pushprop_lem1_gthm = $true) & (one = ((d2term @ d_one))) & (hoasinduction_gthm = $true) & (hoaslamnotvar = $true) & (hoasinduction_lem2_lthm = $true) & ((shinj = $true) | ? [X127 : term,X128 : term] : ((X127 != X128) & (((sub @ X127 @ sh)) = ((sub @ X128 @ sh))))) & (! [X129 : term,X130 : term] : ((X129 = X130) | (((sub @ X130 @ sh)) != ((sub @ X129 @ sh)))) | (shinj != $true)) & (hoasinduction_lem3v2_gthm = $true) & (? [X131 : term] : (((sub @ X131 @ id)) != X131) | ($true = sP54) | ($true = sP53) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | (! [X132 : term] : (((sub @ X132 @ id)) = X132) & ($true != sP54) & ($true != sP53))) & (! [X133 : term] : ? [X134 : (subst > term > term)] : (((sub @ X133 @ (push @ one @ sh))) = ((X134 @ sh @ one))) | (hoasinduction_lem3b != $true)) & ((hoasinduction_lem3b = $true) | ? [X135 : term] : ! [X136 : (subst > term > term)] : (((sub @ X135 @ (push @ one @ sh))) != ((X136 @ sh @ one)))) & (hoasapnotvar_lthm = $true) & (axmap = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (! [X137 : term,X138 : (subst > term > subst > $o)] : (($true = ((X138 @ id @ X137 @ id))) | ($true = ((sP49 @ X138))) | ($true = ((sP50 @ X138)))) | (hoasinduction_no_psi_cond != $true)) & ((hoasinduction_no_psi_cond = $true) | ? [X139 : term,X140 : (subst > term > subst > $o)] : (($true != ((X140 @ id @ X139 @ id))) & ($true != ((sP49 @ X140))) & ($true != ((sP50 @ X140))))) & ! [X141 : term] : ? [X142 : d_term] : (((d2term @ X142)) = X141) & (hoasinduction_lem1 = $true) & (($true = sP47) | ($true = sP48) | ($true = sP46) | ? [X143 : term] : (((sub @ X143 @ id)) != X143) | ? [X144 : (subst > term > subst > $o)] : ! [X145 : (term > $o)] : ? [X146 : term] : (((X145 @ X146)) != ((X144 @ id @ X146 @ id))) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | (($true != sP47) & ($true != sP48) & ($true != sP46) & ! [X147 : term] : (((sub @ X147 @ id)) = X147) & ! [X148 : (subst > term > subst > $o)] : ? [X149 : (term > $o)] : ! [X150 : term] : (((X148 @ id @ X150 @ id)) = ((X149 @ X150))))) & (hoaslamnotap_lthm = $true) & (induction2_gthm = $true) & ((hoaslaminj_lthm = $true) | ($true != sP37)) & (($true = sP37) | (hoaslaminj_lthm != $true)) & (($true = sP35) | (hoasinduction_lem3aa_lthm != $true)) & ((hoasinduction_lem3aa_lthm = $true) | ($true != sP35)) & ((hoasinduction_lem3v2a_lthm = $true) | (! [X151 : term] : ? [X152 : (subst > term > term)] : ! [X153 : term,X154 : subst] : (((sub @ X151 @ (push @ X153 @ X154))) = ((X152 @ X154 @ X153))) & ($true != sP33) & ! [X155 : term] : (((sub @ X155 @ id)) = X155))) & (? [X156 : term] : ! [X157 : (subst > term > term)] : ? [X158 : term,X159 : subst] : (((X157 @ X159 @ X158)) != ((sub @ X156 @ (push @ X158 @ X159)))) | ($true = sP33) | ? [X160 : term] : (((sub @ X160 @ id)) != X160) | (hoasinduction_lem3v2a_lthm != $true)) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (ulamvarind = $true) & ! [X161 : d_term] : (d_one = X161) & (hoasapinj1_gthm = $true) & (! [X162 : term,X163 : term,X164 : term,X165 : term] : ((X162 = X163) | (((ap @ X164 @ X163)) != ((ap @ X165 @ X162)))) | (apinj2 != $true)) & ((apinj2 = $true) | ? [X166 : term,X167 : term,X168 : term,X169 : term] : ((X166 != X167) & (((ap @ X168 @ X167)) = ((ap @ X169 @ X166))))) & (hoaslaminj_gthm = $true) & (($true = sP31) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | ($true != sP31)) & ((hoasinduction_lem3v2 = $true) | ($true != sP28)) & (($true = sP28) | (hoasinduction_lem3v2 != $true)) & ((pushprop_lem1 = $true) | ? [X170 : subst,X171 : term,X172 : (term > $o),X173 : term,X174 : (term > $o)] : (($true != ((X172 @ (sub @ X173 @ (push @ X171 @ X170))))) & ($true = ((X174 @ X173))))) & (! [X175 : subst,X176 : term,X177 : (term > $o),X178 : term,X179 : (term > $o)] : (($true = ((X177 @ (sub @ X178 @ (push @ X176 @ X175))))) | ($true != ((X179 @ X178)))) | (pushprop_lem1 != $true)) & (axabs = $true) & (! [X180 : (subst > term > subst > $o)] : ? [X181 : (term > $o)] : ! [X182 : term] : (((X181 @ X182)) = ((X180 @ id @ X182 @ id))) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem0 = $true) | ? [X183 : (subst > term > subst > $o)] : ! [X184 : (term > $o)] : ? [X185 : term] : (((X183 @ id @ X185 @ id)) != ((X184 @ X185)))) & (! [X186 : term,X187 : term] : ((((lam @ X186)) != ((lam @ X187))) | (X186 = X187)) | (laminj != $true)) & ((laminj = $true) | ? [X188 : term,X189 : term] : ((((lam @ X189)) = ((lam @ X188))) & (X188 != X189))) & (hoasapnotvar = $true) & (pushprop_lem2v2_gthm = $true) & ((termmset_lthm = $true) | (! [X190 : term] : (((sub @ X190 @ id)) = X190) & ? [X191 : term] : (((sub @ X191 @ id)) != X191))) & (? [X192 : term] : (((sub @ X192 @ id)) != X192) | ! [X193 : term] : (((sub @ X193 @ id)) = X193) | (termmset_lthm != $true)) & ((hoasinduction_lem3aa = $true) | ($true != sP24)) & (($true = sP24) | (hoasinduction_lem3aa != $true)) & (pushprop = $true) & (($true = sP22) | (induction2lem != $true)) & ((induction2lem = $true) | ($true != sP22)) & (hoaslamnotvar_gthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (! [X194 : term,X195 : subst] : (((sub @ one @ (push @ X194 @ X195))) = X194) | (axvarcons != $true)) & ((axvarcons = $true) | ? [X196 : term,X197 : subst] : (((sub @ one @ (push @ X196 @ X197))) != X196)) & ((hoasinduction_lem3v2_f = $true) | ? [X198 : term] : ! [X199 : (subst > term > term)] : ? [X200 : subst,X201 : term] : (((sub @ X198 @ (push @ X201 @ X200))) != ((X199 @ X200 @ X201)))) & (! [X202 : term] : ? [X203 : (subst > term > term)] : ! [X204 : subst,X205 : term] : (((sub @ X202 @ (push @ X205 @ X204))) = ((X203 @ X204 @ X205))) | (hoasinduction_lem3v2_f != $true)) & ((axscons = $true) | ? [X206 : subst] : (((push @ (sub @ one @ X206) @ (comp @ sh @ X206))) != X206)) & (! [X207 : subst] : (((push @ (sub @ one @ X207) @ (comp @ sh @ X207))) = X207) | (axscons != $true)) & (ulamvarsh = $true) & (($true = sP19) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | ($true != sP19)) & (hoasapinj2_gthm = $true) & (hoasinduction_lem3a_gthm = $true) & (sh = ((d2subst @ d_id))) & (axapp = $true) & (! [X208 : term] : (((sub @ X208 @ id)) = X208) | (axvarid != $true)) & ((axvarid = $true) | ? [X209 : term] : (((sub @ X209 @ id)) != X209)) & (! [X210 : subst] : (((comp @ X210 @ id)) = X210) | (axidr != $true)) & ((axidr = $true) | ? [X211 : subst] : (((comp @ X211 @ id)) != X211)) & (! [X212 : term,X213 : term,X214 : term,X215 : term] : ((X213 = X214) | (((ap @ (sub @ X213 @ id) @ X212)) != ((ap @ (sub @ X214 @ id) @ X215)))) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | ? [X216 : term,X217 : term,X218 : term,X219 : term] : ((X217 != X218) & (((ap @ (sub @ X217 @ id) @ X216)) = ((ap @ (sub @ X218 @ id) @ X219))))) & (apnotvar = $true) & (! [X220 : (subst > term > subst > $o)] : ? [X221 : (term > $o)] : ! [X222 : term] : (((X220 @ id @ X222 @ id)) = ((X221 @ X222))) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ? [X223 : (subst > term > subst > $o)] : ! [X224 : (term > $o)] : ? [X225 : term] : (((X224 @ X225)) != ((X223 @ id @ X225 @ id)))) & (($true = sP17) | (hoasinduction_lem3aaa != $true)) & ((hoasinduction_lem3aaa = $true) | ($true != sP17)) & ((! [X226 : subst] : (((comp @ id @ X226)) = X226) & ! [X227 : subst] : (((comp @ X227 @ id)) = X227)) | (substmonoid != $true)) & ((substmonoid = $true) | ? [X228 : subst] : (((comp @ id @ X228)) != X228) | ? [X229 : subst] : (((comp @ X229 @ id)) != X229)) & (hoaslamnotap_gthm = $true) & (lamnotvar = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & (hoasinduction_lem3_gthm = $true) & (induction2lem_gthm = $true) & (hoaslamnotvar_lthm = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ! [X230 : subst] : ? [X231 : d_subst] : (((d2subst @ X231)) = X230) & (! [X232 : term] : ? [X233 : (subst > term > term)] : (((sub @ X232 @ (push @ one @ sh))) = ((X233 @ sh @ one))) | (hoasinduction_lem3b_lthm != $true)) & ((hoasinduction_lem3b_lthm = $true) | ? [X234 : term] : ! [X235 : (subst > term > term)] : (((sub @ X234 @ (push @ one @ sh))) != ((X235 @ sh @ one)))) & ((induction = $true) | ($true != sP15)) & (($true = sP15) | (induction != $true)) & ((hoasinduction_no_psi_cond_lthm = $true) | (! [X236 : (subst > term > subst > $o)] : ? [X237 : (term > $o)] : ! [X238 : term] : (((X237 @ X238)) = ((X236 @ id @ X238 @ id))) & ($true != sP10) & ! [X239 : term] : (((sub @ X239 @ id)) = X239) & ($true != sP12) & ($true != sP11))) & (? [X240 : (subst > term > subst > $o)] : ! [X241 : (term > $o)] : ? [X242 : term] : (((X240 @ id @ X242 @ id)) != ((X241 @ X242))) | ($true = sP10) | ? [X243 : term] : (((sub @ X243 @ id)) != X243) | ($true = sP12) | ($true = sP11) | (hoasinduction_no_psi_cond_lthm != $true)) & ((hoasinduction_lem3v2_lthm = $true) | (($true != sP4) & ! [X244 : term] : (((sub @ X244 @ id)) = X244))) & (($true = sP4) | ? [X245 : term] : (((sub @ X245 @ id)) != X245) | (hoasinduction_lem3v2_lthm != $true)) & ((hoasapinj2 = $true) | ? [X246 : term,X247 : term,X248 : term,X249 : term] : ((X246 != X247) & (((ap @ (sub @ X249 @ id) @ X247)) = ((ap @ (sub @ X248 @ id) @ X246))))) & (! [X250 : term,X251 : term,X252 : term,X253 : term] : ((X250 = X251) | (((ap @ (sub @ X253 @ id) @ X251)) != ((ap @ (sub @ X252 @ id) @ X250)))) | (hoasapinj2 != $true)) & (($true = sP0) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | ($true != sP0)) & ((pushprop_lem0_lthm = $true) | ? [X254 : subst,X255 : term,X256 : (term > $o)] : ! [X257 : (term > $o)] : ? [X258 : term] : (((X256 @ (sub @ X258 @ (push @ X255 @ X254)))) != ((X257 @ X258)))) & (! [X259 : subst,X260 : term,X261 : (term > $o)] : ? [X262 : (term > $o)] : ! [X263 : term] : (((X262 @ X263)) = ((X261 @ (sub @ X263 @ (push @ X260 @ X259))))) | (pushprop_lem0_lthm != $true)) & (pushprop_lem2v2 = $true)),
  inference(rectify,[],[f455])).
thf(f457,plain,(
  ((hoaslaminj = $true) | ((((lam @ (sK515 @ sh @ one))) = ((lam @ (sK512 @ sh @ one)))) & ! [X4 : subst,X5 : term,X6 : subst] : (((sub @ (sK512 @ X4 @ X5) @ X6)) = ((sK512 @ (comp @ X4 @ X6) @ (sub @ X5 @ X6)))) & ! [X7 : term,X8 : subst,X9 : subst] : (((sub @ (sK515 @ X9 @ X7) @ X8)) = ((sK515 @ (comp @ X9 @ X8) @ (sub @ X7 @ X8)))) & (((sK515 @ sK514 @ sK513)) != ((sK512 @ sK514 @ sK513))))) & (! [X10 : (subst > term > term),X11 : term,X12 : subst,X13 : (subst > term > term)] : ((((lam @ (X13 @ sh @ one))) != ((lam @ (X10 @ sh @ one)))) | (((sub @ (X10 @ (sK516 @ X10) @ (sK517 @ X10)) @ (sK518 @ X10))) != ((X10 @ (comp @ (sK516 @ X10) @ (sK518 @ X10)) @ (sub @ (sK517 @ X10) @ (sK518 @ X10))))) | (((X13 @ (comp @ (sK521 @ X13) @ (sK520 @ X13)) @ (sub @ (sK519 @ X13) @ (sK520 @ X13)))) != ((sub @ (X13 @ (sK521 @ X13) @ (sK519 @ X13)) @ (sK520 @ X13)))) | (((X10 @ X12 @ X11)) = ((X13 @ X12 @ X11)))) | (hoaslaminj != $true)) & ((axshiftcons = $true) | (((comp @ sh @ (push @ sK523 @ sK522))) != sK522)) & (! [X22 : subst,X23 : term] : (((comp @ sh @ (push @ X23 @ X22))) = X22) | (axshiftcons != $true)) & ((pushprop_lem0 = $true) | ! [X27 : (term > $o)] : (((X27 @ (sK527 @ X27))) != ((sK524 @ (sub @ (sK527 @ X27) @ (push @ sK525 @ sK526)))))) & (! [X29 : (term > $o),X30 : term,X31 : subst] : ! [X33 : term] : (((X29 @ (sub @ X33 @ (push @ X30 @ X31)))) = ((sK528 @ X31 @ X30 @ X29 @ X33))) | (pushprop_lem0 != $true)) & ! [X34 : d_subst] : (d_id = X34) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_lem3v2_f_lthm = $true) | ! [X36 : (subst > term > term)] : (((sub @ sK529 @ (push @ (sK531 @ X36) @ (sK530 @ X36)))) != ((X36 @ (sK530 @ X36) @ (sK531 @ X36))))) & (! [X39 : term] : ! [X41 : subst,X42 : term] : (((sub @ X39 @ (push @ X42 @ X41))) = ((sK532 @ X39 @ X41 @ X42))) | (hoasinduction_lem3v2_f_lthm != $true)) & (axclos = $true) & (hoasinduction_lem1_lthm = $true) & (termmset_gthm = $true) & (($true = sP108) | (hoasinduction_lem3 != $true)) & ((hoasinduction_lem3 = $true) | ($true != sP108)) & (($true = sP104) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | ($true != sP104)) & (axvarshift = $true) & (pushprop_gthm = $true) & (! [X43 : term,X44 : (subst > term > term),X45 : term] : ((((X44 @ (comp @ (sK534 @ X44) @ (sK535 @ X44)) @ (sub @ (sK533 @ X44) @ (sK535 @ X44)))) != ((sub @ (X44 @ (sK534 @ X44) @ (sK533 @ X44)) @ (sK535 @ X44)))) | (((ap @ (sub @ X45 @ id) @ X43)) != ((lam @ (X44 @ sh @ one))))) | (hoaslamnotap != $true)) & ((hoaslamnotap = $true) | (! [X52 : term,X53 : subst,X54 : subst] : (((sK537 @ (comp @ X53 @ X54) @ (sub @ X52 @ X54))) = ((sub @ (sK537 @ X53 @ X52) @ X54))) & (((lam @ (sK537 @ sh @ one))) = ((ap @ (sub @ sK538 @ id) @ sK536))))) & (($true = sP102) | (hoasinduction_lem2v2 != $true)) & ((hoasinduction_lem2v2 = $true) | ($true != sP102)) & (pushprop_lem0_gthm = $true) & ! [X55 : d_subst,X56 : d_subst] : ((((d2subst @ X55)) != ((d2subst @ X56))) | (X55 = X56)) & (hoasinduction_lem1_gthm = $true) & (hoasinduction_lem1v2 = $true) & (substmonoid_gthm = $true) & (pushprop_lem3v2_lthm = $true) & (pushprop_lem2v2_lthm = $true) & (((d2subst @ d_id)) = id) & (((var @ (d2term @ d_one))) != $true) & (axassoc = $true) & (lamnotap != $true) & ((hoasinduction_lem3a = $true) | ($true != sP99)) & (($true = sP99) | (hoasinduction_lem3a != $true)) & ((induction2_lthm = $true) | (($true != sP97) & ($true != sP96) & ! [X57 : term] : (((sub @ X57 @ id)) = X57))) & (($true = sP97) | ($true = sP96) | (((sub @ sK539 @ id)) != sK539) | (induction2_lthm != $true)) & (($true = sP91) | ($true = sP90) | ($true = sP88) | ($true = sP89) | (hoasinduction_lthm != $true)) & ((hoasinduction_lthm = $true) | (($true != sP91) & ($true != sP90) & ($true != sP88) & ($true != sP89))) & (hoasinduction_lem2_gthm = $true) & (hoasinduction_lem2v2_gthm = $true) & (pushprop_lem3v2 = $true) & (pushprop_lthm = $true) & (! [X59 : subst] : (((comp @ id @ X59)) = X59) | (axidl != $true)) & ((axidl = $true) | (((comp @ id @ sK540)) != sK540)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((pushprop_lem1_lthm = $true) | (! [X61 : subst,X62 : term] : (((comp @ sh @ (push @ X62 @ X61))) = X61) & (($true != ((sK544 @ (sub @ sK541 @ (push @ sK542 @ sK543))))) & ($true = ((sK545 @ sK541)))) & ! [X68 : subst,X69 : term] : (((sub @ one @ (push @ X69 @ X68))) = X69))) & ((sK546 != ((comp @ sh @ (push @ sK547 @ sK546)))) | ! [X72 : term,X73 : term,X74 : subst,X75 : (term > $o),X76 : (term > $o)] : (($true = ((X75 @ (sub @ X72 @ (push @ X73 @ X74))))) | (((X76 @ X72)) != $true)) | (sK549 != ((sub @ one @ (push @ sK549 @ sK548)))) | (pushprop_lem1_lthm != $true)) & (pushprop_lem1v2_gthm = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (($true = sP76) | (hoasapinj2_lthm != $true)) & ((hoasapinj2_lthm = $true) | ($true != sP76)) & (! [X79 : term] : (((sub @ X79 @ id)) = X79) | (termmset != $true)) & ((termmset = $true) | (sK550 != ((sub @ sK550 @ id)))) & (pushprop_lthm_orig = $true) & ((((sub @ one @ (push @ sK551 @ sK552))) != sK551) | ! [X83 : (term > $o),X84 : (term > $o),X85 : subst,X86 : term] : (($true != ((X84 @ X86))) | (((X84 @ (sub @ (sK553 @ X86 @ X85 @ X84 @ X83) @ (push @ X86 @ X85)))) != ((X83 @ (sK553 @ X86 @ X85 @ X84 @ X83)))) | ($true = ((X83 @ one)))) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (! [X88 : term,X89 : subst] : (((sub @ one @ (push @ X88 @ X89))) = X88) & (($true = ((sK555 @ sK557))) & ! [X94 : term] : (((sK554 @ X94)) = ((sK555 @ (sub @ X94 @ (push @ sK557 @ sK556))))) & ($true != ((sK554 @ one)))))) & (ulamvar1 != $true) & ((induction2 = $true) | ($true != sP74)) & (($true = sP74) | (induction2 != $true)) & ! [X95 : d_term,X96 : d_term] : ((((d2term @ X95)) != ((d2term @ X96))) | (X95 = X96)) & ((pushprop_lem1v2 = $true) | (($true != ((sK561 @ one))) & ($true = ((sK558 @ sK560))) & ! [X101 : term] : (((sK561 @ X101)) = ((sK558 @ (sub @ X101 @ (push @ sK560 @ sK559))))))) & (! [X102 : (term > $o),X103 : subst,X104 : term,X105 : (term > $o)] : (($true = ((X105 @ one))) | ($true != ((X102 @ X104))) | (((X102 @ (sub @ (sK562 @ X105 @ X104 @ X103 @ X102) @ (push @ X104 @ X103)))) != ((X105 @ (sK562 @ X105 @ X104 @ X103 @ X102))))) | (pushprop_lem1v2 != $true)) & (hoasinduction_lem3b_gthm = $true) & ((hoasinduction_lem3_lthm = $true) | (($true != sP71) & ! [X107 : term] : (((sub @ X107 @ id)) = X107) & ($true != sP70))) & (($true = sP71) | (((sub @ sK563 @ id)) != sK563) | ($true = sP70) | (hoasinduction_lem3_lthm != $true)) & (! [X109 : term,X110 : term,X111 : term,X112 : term] : ((X110 = X112) | (((ap @ X112 @ X111)) != ((ap @ X110 @ X109)))) | (apinj1 != $true)) & ((apinj1 = $true) | ((sK567 != sK565) & (((ap @ sK567 @ sK566)) = ((ap @ sK565 @ sK564))))) & ((hoasinduction = $true) | ($true != sP65)) & (($true = sP65) | (hoasinduction != $true)) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((((comp @ sK568 @ id)) != sK568) | (sK570 != ((comp @ sh @ (push @ sK569 @ sK570)))) | (((sub @ one @ (push @ sK572 @ sK571))) != sK572) | ($true = sP60) | ($true = sP59) | (induction2lem_lthm != $true)) & ((induction2lem_lthm = $true) | (! [X122 : subst] : (((comp @ X122 @ id)) = X122) & ! [X123 : term,X124 : subst] : (((comp @ sh @ (push @ X123 @ X124))) = X124) & ! [X125 : subst,X126 : term] : (((sub @ one @ (push @ X126 @ X125))) = X126) & ($true != sP60) & ($true != sP59))) & (pushprop_lem1_gthm = $true) & (one = ((d2term @ d_one))) & (hoasinduction_gthm = $true) & (hoaslamnotvar = $true) & (hoasinduction_lem2_lthm = $true) & ((shinj = $true) | ((sK573 != sK574) & (((sub @ sK574 @ sh)) = ((sub @ sK573 @ sh))))) & (! [X129 : term,X130 : term] : ((X129 = X130) | (((sub @ X130 @ sh)) != ((sub @ X129 @ sh)))) | (shinj != $true)) & (hoasinduction_lem3v2_gthm = $true) & ((((sub @ sK575 @ id)) != sK575) | ($true = sP54) | ($true = sP53) | (hoasinduction_lem3a_lthm != $true)) & ((hoasinduction_lem3a_lthm = $true) | (! [X132 : term] : (((sub @ X132 @ id)) = X132) & ($true != sP54) & ($true != sP53))) & (! [X133 : term] : (((sub @ X133 @ (push @ one @ sh))) = ((sK576 @ X133 @ sh @ one))) | (hoasinduction_lem3b != $true)) & ((hoasinduction_lem3b = $true) | ! [X136 : (subst > term > term)] : (((X136 @ sh @ one)) != ((sub @ sK577 @ (push @ one @ sh))))) & (hoasapnotvar_lthm = $true) & (axmap = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (! [X137 : term,X138 : (subst > term > subst > $o)] : (($true = ((X138 @ id @ X137 @ id))) | ($true = ((sP49 @ X138))) | ($true = ((sP50 @ X138)))) | (hoasinduction_no_psi_cond != $true)) & ((hoasinduction_no_psi_cond = $true) | (($true != ((sK579 @ id @ sK578 @ id))) & ($true != ((sP49 @ sK579))) & ($true != ((sP50 @ sK579))))) & ! [X141 : term] : (((d2term @ (sK580 @ X141))) = X141) & (hoasinduction_lem1 = $true) & (($true = sP47) | ($true = sP48) | ($true = sP46) | (sK581 != ((sub @ sK581 @ id))) | ! [X145 : (term > $o)] : (((X145 @ (sK583 @ X145))) != ((sK582 @ id @ (sK583 @ X145) @ id))) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | (($true != sP47) & ($true != sP48) & ($true != sP46) & ! [X147 : term] : (((sub @ X147 @ id)) = X147) & ! [X148 : (subst > term > subst > $o)] : ! [X150 : term] : (((X148 @ id @ X150 @ id)) = ((sK584 @ X148 @ X150))))) & (hoaslamnotap_lthm = $true) & (induction2_gthm = $true) & ((hoaslaminj_lthm = $true) | ($true != sP37)) & (($true = sP37) | (hoaslaminj_lthm != $true)) & (($true = sP35) | (hoasinduction_lem3aa_lthm != $true)) & ((hoasinduction_lem3aa_lthm = $true) | ($true != sP35)) & ((hoasinduction_lem3v2a_lthm = $true) | (! [X151 : term] : ! [X153 : term,X154 : subst] : (((sub @ X151 @ (push @ X153 @ X154))) = ((sK585 @ X151 @ X154 @ X153))) & ($true != sP33) & ! [X155 : term] : (((sub @ X155 @ id)) = X155))) & (! [X157 : (subst > term > term)] : (((sub @ sK586 @ (push @ (sK587 @ X157) @ (sK588 @ X157)))) != ((X157 @ (sK588 @ X157) @ (sK587 @ X157)))) | ($true = sP33) | (sK589 != ((sub @ sK589 @ id))) | (hoasinduction_lem3v2a_lthm != $true)) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (ulamvarind = $true) & ! [X161 : d_term] : (d_one = X161) & (hoasapinj1_gthm = $true) & (! [X162 : term,X163 : term,X164 : term,X165 : term] : ((X162 = X163) | (((ap @ X164 @ X163)) != ((ap @ X165 @ X162)))) | (apinj2 != $true)) & ((apinj2 = $true) | ((sK591 != sK590) & (((ap @ sK592 @ sK591)) = ((ap @ sK593 @ sK590))))) & (hoaslaminj_gthm = $true) & (($true = sP31) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | ($true != sP31)) & ((hoasinduction_lem3v2 = $true) | ($true != sP28)) & (($true = sP28) | (hoasinduction_lem3v2 != $true)) & ((pushprop_lem1 = $true) | (($true != ((sK596 @ (sub @ sK597 @ (push @ sK595 @ sK594))))) & ($true = ((sK598 @ sK597))))) & (! [X175 : subst,X176 : term,X177 : (term > $o),X178 : term,X179 : (term > $o)] : (($true = ((X177 @ (sub @ X178 @ (push @ X176 @ X175))))) | ($true != ((X179 @ X178)))) | (pushprop_lem1 != $true)) & (axabs = $true) & (! [X180 : (subst > term > subst > $o)] : ! [X182 : term] : (((X180 @ id @ X182 @ id)) = ((sK599 @ X180 @ X182))) | (hoasinduction_lem0 != $true)) & ((hoasinduction_lem0 = $true) | ! [X184 : (term > $o)] : (((X184 @ (sK601 @ X184))) != ((sK600 @ id @ (sK601 @ X184) @ id)))) & (! [X186 : term,X187 : term] : ((((lam @ X186)) != ((lam @ X187))) | (X186 = X187)) | (laminj != $true)) & ((laminj = $true) | ((((lam @ sK603)) = ((lam @ sK602))) & (sK602 != sK603))) & (hoasapnotvar = $true) & (pushprop_lem2v2_gthm = $true) & ((termmset_lthm = $true) | (! [X190 : term] : (((sub @ X190 @ id)) = X190) & (((sub @ sK604 @ id)) != sK604))) & ((sK605 != ((sub @ sK605 @ id))) | ! [X193 : term] : (((sub @ X193 @ id)) = X193) | (termmset_lthm != $true)) & ((hoasinduction_lem3aa = $true) | ($true != sP24)) & (($true = sP24) | (hoasinduction_lem3aa != $true)) & (pushprop = $true) & (($true = sP22) | (induction2lem != $true)) & ((induction2lem = $true) | ($true != sP22)) & (hoaslamnotvar_gthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (! [X194 : term,X195 : subst] : (((sub @ one @ (push @ X194 @ X195))) = X194) | (axvarcons != $true)) & ((axvarcons = $true) | (((sub @ one @ (push @ sK606 @ sK607))) != sK606)) & ((hoasinduction_lem3v2_f = $true) | ! [X199 : (subst > term > term)] : (((sub @ sK608 @ (push @ (sK610 @ X199) @ (sK609 @ X199)))) != ((X199 @ (sK609 @ X199) @ (sK610 @ X199))))) & (! [X202 : term] : ! [X204 : subst,X205 : term] : (((sub @ X202 @ (push @ X205 @ X204))) = ((sK611 @ X202 @ X204 @ X205))) | (hoasinduction_lem3v2_f != $true)) & ((axscons = $true) | (sK612 != ((push @ (sub @ one @ sK612) @ (comp @ sh @ sK612))))) & (! [X207 : subst] : (((push @ (sub @ one @ X207) @ (comp @ sh @ X207))) = X207) | (axscons != $true)) & (ulamvarsh = $true) & (($true = sP19) | (hoasinduction_lem3v2a != $true)) & ((hoasinduction_lem3v2a = $true) | ($true != sP19)) & (hoasapinj2_gthm = $true) & (hoasinduction_lem3a_gthm = $true) & (sh = ((d2subst @ d_id))) & (axapp = $true) & (! [X208 : term] : (((sub @ X208 @ id)) = X208) | (axvarid != $true)) & ((axvarid = $true) | (((sub @ sK613 @ id)) != sK613)) & (! [X210 : subst] : (((comp @ X210 @ id)) = X210) | (axidr != $true)) & ((axidr = $true) | (((comp @ sK614 @ id)) != sK614)) & (! [X212 : term,X213 : term,X214 : term,X215 : term] : ((X213 = X214) | (((ap @ (sub @ X213 @ id) @ X212)) != ((ap @ (sub @ X214 @ id) @ X215)))) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | ((sK616 != sK617) & (((ap @ (sub @ sK616 @ id) @ sK615)) = ((ap @ (sub @ sK617 @ id) @ sK618))))) & (apnotvar = $true) & (! [X220 : (subst > term > subst > $o)] : ! [X222 : term] : (((sK619 @ X220 @ X222)) = ((X220 @ id @ X222 @ id))) | (hoasinduction_lem0_lthm != $true)) & ((hoasinduction_lem0_lthm = $true) | ! [X224 : (term > $o)] : (((X224 @ (sK621 @ X224))) != ((sK620 @ id @ (sK621 @ X224) @ id)))) & (($true = sP17) | (hoasinduction_lem3aaa != $true)) & ((hoasinduction_lem3aaa = $true) | ($true != sP17)) & ((! [X226 : subst] : (((comp @ id @ X226)) = X226) & ! [X227 : subst] : (((comp @ X227 @ id)) = X227)) | (substmonoid != $true)) & ((substmonoid = $true) | (((comp @ id @ sK622)) != sK622) | (((comp @ sK623 @ id)) != sK623)) & (hoaslamnotap_gthm = $true) & (lamnotvar = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4)))))))))))) & (hoasinduction_lem3_gthm = $true) & (induction2lem_gthm = $true) & (hoaslamnotvar_lthm = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & ! [X230 : subst] : (((d2subst @ (sK624 @ X230))) = X230) & (! [X232 : term] : (((sub @ X232 @ (push @ one @ sh))) = ((sK625 @ X232 @ sh @ one))) | (hoasinduction_lem3b_lthm != $true)) & ((hoasinduction_lem3b_lthm = $true) | ! [X235 : (subst > term > term)] : (((X235 @ sh @ one)) != ((sub @ sK626 @ (push @ one @ sh))))) & ((induction = $true) | ($true != sP15)) & (($true = sP15) | (induction != $true)) & ((hoasinduction_no_psi_cond_lthm = $true) | (! [X236 : (subst > term > subst > $o)] : ! [X238 : term] : (((X236 @ id @ X238 @ id)) = ((sK627 @ X236 @ X238))) & ($true != sP10) & ! [X239 : term] : (((sub @ X239 @ id)) = X239) & ($true != sP12) & ($true != sP11))) & (! [X241 : (term > $o)] : (((sK628 @ id @ (sK629 @ X241) @ id)) != ((X241 @ (sK629 @ X241)))) | ($true = sP10) | (sK630 != ((sub @ sK630 @ id))) | ($true = sP12) | ($true = sP11) | (hoasinduction_no_psi_cond_lthm != $true)) & ((hoasinduction_lem3v2_lthm = $true) | (($true != sP4) & ! [X244 : term] : (((sub @ X244 @ id)) = X244))) & (($true = sP4) | (((sub @ sK631 @ id)) != sK631) | (hoasinduction_lem3v2_lthm != $true)) & ((hoasapinj2 = $true) | ((sK633 != sK632) & (((ap @ (sub @ sK635 @ id) @ sK633)) = ((ap @ (sub @ sK634 @ id) @ sK632))))) & (! [X250 : term,X251 : term,X252 : term,X253 : term] : ((X250 = X251) | (((ap @ (sub @ X253 @ id) @ X251)) != ((ap @ (sub @ X252 @ id) @ X250)))) | (hoasapinj2 != $true)) & (($true = sP0) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | ($true != sP0)) & ((pushprop_lem0_lthm = $true) | ! [X257 : (term > $o)] : (((sK638 @ (sub @ (sK639 @ X257) @ (push @ sK637 @ sK636)))) != ((X257 @ (sK639 @ X257))))) & (! [X259 : subst,X260 : term,X261 : (term > $o)] : ! [X263 : term] : (((X261 @ (sub @ X263 @ (push @ X260 @ X259)))) = ((sK640 @ X261 @ X260 @ X259 @ X263))) | (pushprop_lem0_lthm != $true)) & (pushprop_lem2v2 = $true)),
  inference(skolemize,[status(esa),new_symbols(skolem,[sK512,sK513,sK514,sK515,vAPP,sK522,sK523,sK524,sK525,sK526,vAPP,sK529,vAPP,sK536,sK537,sK538,sK539,sK540,sK541,sK542,sK543,sK544,sK545,sK546,sK547,sK548,sK549,sK550,sK551,sK552,vAPP,sK554,sK555,sK556,sK557,sK558,sK559,sK560,sK561,vAPP,sK563,sK564,sK565,sK566,sK567,sK568,sK569,sK570,sK571,sK572,sK573,sK574,sK575,vAPP,sK577,sK578,sK579,vAPP,sK581,sK582,vAPP,sK586,vAPP,sK589,sK590,sK591,sK592,sK593,sK594,sK595,sK596,sK597,sK598,vAPP,sK600,vAPP,sK602,sK603,sK604,sK605,sK606,sK607,sK608,vAPP,sK612,sK613,sK614,sK615,sK616,sK617,sK618,vAPP,sK620,vAPP,sK622,sK623,vAPP,sK626,vAPP,sK628,vAPP,sK630,sK631,sK632,sK633,sK634,sK635,sK636,sK637,sK638,vAPP]),skolemize(X0,$thf(sK512)),skolemize(X1,$thf(sK513)),skolemize(X2,$thf(sK514)),skolemize(X3,$thf(sK515)),skolemize(X11,$thf(sK114 @ X6)),skolemize(X11,$thf(sK114 @ X6)),skolemize(X11,$thf(sK114 @ X6)),skolemize(X11,$thf(sK114 @ X6)),skolemize(X11,$thf(sK114 @ X6)),skolemize(X11,$thf(sK114 @ X6)),skolemize(X20,$thf(sK522)),skolemize(X21,$thf(sK523)),skolemize(X24,$thf(sK524)),skolemize(X25,$thf(sK525)),skolemize(X26,$thf(sK526)),skolemize(X11,$thf(sK114 @ X6)),skolemize(X11,$thf(sK114 @ X6)),skolemize(X35,$thf(sK529)),skolemize(X11,$thf(sK114 @ X6)),skolemize(X11,$thf(sK114 @ X6)),skolemize(X11,$thf(sK114 @ X6)),skolemize(X11,$thf(sK114 @ X6)),skolemize(X11,$thf(sK114 @ X6)),skolemize(X11,$thf(sK114 @ X6)),skolemize(X49,$thf(sK536)),skolemize(X50,$thf(sK537)),skolemize(X51,$thf(sK538)),skolemize(X58,$thf(sK539)),skolemize(X60,$thf(sK540)),skolemize(X63,$thf(sK541)),skolemize(X64,$thf(sK542)),skolemize(X65,$thf(sK543)),skolemize(X66,$thf(sK544)),skolemize(X67,$thf(sK545)),skolemize(X70,$thf(sK546)),skolemize(X71,$thf(sK547)),skolemize(X77,$thf(sK548)),skolemize(X78,$thf(sK549)),skolemize(X80,$thf(sK550)),skolemize(X81,$thf(sK551)),skolemize(X82,$thf(sK552)),skolemize(X11,$thf(sK114 @ X6)),skolemize(X90,$thf(sK554)),skolemize(X91,$thf(sK555)),skolemize(X92,$thf(sK556)),skolemize(X93,$thf(sK557)),skolemize(X97,$thf(sK558)),skolemize(X98,$thf(sK559)),skolemize(X99,$thf(sK560)),skolemize(X100,$thf(sK561)),skolemize(X11,$thf(sK114 @ X6)),skolemize(X108,$thf(sK563)),skolemize(X113,$thf(sK564)),skolemize(X114,$thf(sK565)),skolemize(X115,$thf(sK566)),skolemize(X116,$thf(sK567)),skolemize(X117,$thf(sK568)),skolemize(X118,$thf(sK569)),skolemize(X119,$thf(sK570)),skolemize(X120,$thf(sK571)),skolemize(X121,$thf(sK572)),skolemize(X127,$thf(sK573)),skolemize(X128,$thf(sK574)),skolemize(X131,$thf(sK575)),skolemize(X11,$thf(sK114 @ X6)),skolemize(X135,$thf(sK577)),skolemize(X139,$thf(sK578)),skolemize(X140,$thf(sK579)),skolemize(X11,$thf(sK114 @ X6)),skolemize(X143,$thf(sK581)),skolemize(X144,$thf(sK582)),skolemize(X11,$thf(sK114 @ X6)),skolemize(X11,$thf(sK114 @ X6)),skolemize(X11,$thf(sK114 @ X6)),skolemize(X156,$thf(sK586)),skolemize(X11,$thf(sK114 @ X6)),skolemize(X11,$thf(sK114 @ X6)),skolemize(X160,$thf(sK589)),skolemize(X166,$thf(sK590)),skolemize(X167,$thf(sK591)),skolemize(X168,$thf(sK592)),skolemize(X169,$thf(sK593)),skolemize(X170,$thf(sK594)),skolemize(X171,$thf(sK595)),skolemize(X172,$thf(sK596)),skolemize(X173,$thf(sK597)),skolemize(X174,$thf(sK598)),skolemize(X11,$thf(sK114 @ X6)),skolemize(X183,$thf(sK600)),skolemize(X11,$thf(sK114 @ X6)),skolemize(X188,$thf(sK602)),skolemize(X189,$thf(sK603)),skolemize(X191,$thf(sK604)),skolemize(X192,$thf(sK605)),skolemize(X196,$thf(sK606)),skolemize(X197,$thf(sK607)),skolemize(X198,$thf(sK608)),skolemize(X11,$thf(sK114 @ X6)),skolemize(X11,$thf(sK114 @ X6)),skolemize(X11,$thf(sK114 @ X6)),skolemize(X206,$thf(sK612)),skolemize(X209,$thf(sK613)),skolemize(X211,$thf(sK614)),skolemize(X216,$thf(sK615)),skolemize(X217,$thf(sK616)),skolemize(X218,$thf(sK617)),skolemize(X219,$thf(sK618)),skolemize(X11,$thf(sK114 @ X6)),skolemize(X223,$thf(sK620)),skolemize(X11,$thf(sK114 @ X6)),skolemize(X228,$thf(sK622)),skolemize(X229,$thf(sK623)),skolemize(X11,$thf(sK114 @ X6)),skolemize(X11,$thf(sK114 @ X6)),skolemize(X234,$thf(sK626)),skolemize(X11,$thf(sK114 @ X6)),skolemize(X240,$thf(sK628)),skolemize(X11,$thf(sK114 @ X6)),skolemize(X243,$thf(sK630)),skolemize(X245,$thf(sK631)),skolemize(X246,$thf(sK632)),skolemize(X247,$thf(sK633)),skolemize(X248,$thf(sK634)),skolemize(X249,$thf(sK635)),skolemize(X254,$thf(sK636)),skolemize(X255,$thf(sK637)),skolemize(X256,$thf(sK638)),skolemize(X11,$thf(sK114 @ X6)),skolemize(X11,$thf(sK114 @ X6))],[f456])).
thf(f999,plain,(
  (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1))) = (Y3 @ Y4))))))))))))),
  inference(cnf_transformation,[],[f457])).
thf(f1061,plain,(
  ( ! [X161 : d_term] : ((d_one = X161)) )),
  inference(cnf_transformation,[],[f457])).
thf(f1081,plain,(
  ( ! [X141 : term] : ((((d2term @ (sK580 @ X141))) = X141)) )),
  inference(cnf_transformation,[],[f457])).
thf(f1102,plain,(
  (one = ((d2term @ d_one)))),
  inference(cnf_transformation,[],[f457])).
thf(f1123,plain,(
  ( ! [X101 : term] : ((pushprop_lem1v2 = $true) | (((sK561 @ X101)) = ((sK558 @ (sub @ X101 @ (push @ sK560 @ sK559)))))) )),
  inference(cnf_transformation,[],[f457])).
thf(f1124,plain,(
  (pushprop_lem1v2 = $true) | ($true = ((sK558 @ sK560)))),
  inference(cnf_transformation,[],[f457])).
thf(f1125,plain,(
  (pushprop_lem1v2 = $true) | ($true != ((sK561 @ one)))),
  inference(cnf_transformation,[],[f457])).
thf(f1165,plain,(
  (lamnotap != $true)),
  inference(cnf_transformation,[],[f457])).
thf(f1204,plain,(
  (pushprop_lem1v2 != ((!! @ (term > $o) @ (^[Y0 : term > $o]: (!! @ term @ (^[Y1 : term]: (!! @ subst @ (^[Y2 : subst]: (!! @ (term > $o) @ (^[Y3 : term > $o]: ((Y3 @ Y1) => ((pushprop_p_and_p_prime @ Y1 @ Y2 @ Y3 @ Y0) => (Y0 @ one)))))))))))))),
  inference(cnf_transformation,[],[f8])).
thf(f1206,definition,(
  ( ! [X0 : $o] : (($true = X0) | ($false = X0)) )),
  introduced(theory,[fool_exhaustiveness_axiom])).
thf(f1207,plain,(
  (pushprop_lem1v2 != ((!! @ (term > $o) @ (^[Y0 : term > $o]: (!! @ term @ (^[Y1 : term]: (!! @ subst @ (^[Y2 : subst]: (!! @ (term > $o) @ (^[Y3 : term > $o]: ((Y3 @ Y1) => (((^[Y4 : term]: ((^[Y5 : subst]: ((^[Y6 : term > $o]: ((^[Y7 : term > $o]: (!! @ term @ (^[Y8 : term]: ((Y6 @ (sub @ Y8 @ (push @ Y4 @ Y5))) = (Y7 @ Y8))))))))))) @ Y1 @ Y2 @ Y3 @ Y0) => (Y0 @ one)))))))))))))),
  inference(definition_unfolding,[],[f1204,f999])).
thf(f1228,plain,(
  ( ! [X101 : term] : (($true = ((sK561 @ X101))) | ($false = ((sK558 @ (sub @ X101 @ (push @ sK560 @ sK559))))) | (pushprop_lem1v2 = $true)) )),
  inference(iff_proxy_clausification,[],[f1123])).
thf(f1282,plain,(
  (pushprop_lem1v2 = $true) | ($true = ((!! @ (term > $o) @ (^[Y0 : term > $o]: (!! @ term @ (^[Y1 : term]: (!! @ subst @ (^[Y2 : subst]: (!! @ (term > $o) @ (^[Y3 : term > $o]: ((Y3 @ Y1) => (((^[Y4 : term]: ((^[Y5 : subst]: ((^[Y6 : term > $o]: ((^[Y7 : term > $o]: (!! @ term @ (^[Y8 : term]: ((Y6 @ (sub @ Y8 @ (push @ Y4 @ Y5))) = (Y7 @ Y8))))))))))) @ Y1 @ Y2 @ Y3 @ Y0) => (Y0 @ one)))))))))))))),
  inference(xor_proxy_clausification,[],[f1207])).
thf(f1283,plain,(
  ($false = ((!! @ (term > $o) @ (^[Y0 : term > $o]: (!! @ term @ (^[Y1 : term]: (!! @ subst @ (^[Y2 : subst]: (!! @ (term > $o) @ (^[Y3 : term > $o]: ((Y3 @ Y1) => (((^[Y4 : term]: ((^[Y5 : subst]: ((^[Y6 : term > $o]: ((^[Y7 : term > $o]: (!! @ term @ (^[Y8 : term]: ((Y6 @ (sub @ Y8 @ (push @ Y4 @ Y5))) = (Y7 @ Y8))))))))))) @ Y1 @ Y2 @ Y3 @ Y0) => (Y0 @ one))))))))))))) | (pushprop_lem1v2 = $false)),
  inference(xor_proxy_clausification,[],[f1207])).
thf(f1284,plain,(
  ($false = (((^[Y0 : term > $o]: (!! @ term @ (^[Y1 : term]: (!! @ subst @ (^[Y2 : subst]: (!! @ (term > $o) @ (^[Y3 : term > $o]: ((Y3 @ Y1) => (((^[Y4 : term]: ((^[Y5 : subst]: ((^[Y6 : term > $o]: ((^[Y7 : term > $o]: (!! @ term @ (^[Y8 : term]: ((Y6 @ (sub @ Y8 @ (push @ Y4 @ Y5))) = (Y7 @ Y8))))))))))) @ Y1 @ Y2 @ Y3 @ Y0) => (Y0 @ one)))))))))) @ sK641))) | (pushprop_lem1v2 = $false)),
  inference(sigma_proxy_clausification,[],[f1283])).
thf(f1285,plain,(
  (((!! @ term @ (^[Y0 : term]: (!! @ subst @ (^[Y1 : subst]: (!! @ (term > $o) @ (^[Y2 : term > $o]: ((Y2 @ Y0) => ((!! @ term @ (^[Y3 : term]: ((Y2 @ (sub @ Y3 @ (push @ Y0 @ Y1))) = (sK641 @ Y3)))) => (sK641 @ one)))))))))) = $false) | (pushprop_lem1v2 = $false)),
  inference(beta-eta_normalization,[],[f1284])).
thf(f1286,plain,(
  (pushprop_lem1v2 = $false) | ((((^[Y0 : term]: (!! @ subst @ (^[Y1 : subst]: (!! @ (term > $o) @ (^[Y2 : term > $o]: ((Y2 @ Y0) => ((!! @ term @ (^[Y3 : term]: ((Y2 @ (sub @ Y3 @ (push @ Y0 @ Y1))) = (sK641 @ Y3)))) => (sK641 @ one)))))))) @ sK642)) = $false)),
  inference(sigma_proxy_clausification,[],[f1285])).
thf(f1287,plain,(
  (pushprop_lem1v2 = $false) | (((!! @ subst @ (^[Y0 : subst]: (!! @ (term > $o) @ (^[Y1 : term > $o]: ((Y1 @ sK642) => ((!! @ term @ (^[Y2 : term]: ((Y1 @ (sub @ Y2 @ (push @ sK642 @ Y0))) = (sK641 @ Y2)))) => (sK641 @ one)))))))) = $false)),
  inference(beta-eta_normalization,[],[f1286])).
thf(f1288,plain,(
  (pushprop_lem1v2 = $false) | ($false = (((^[Y0 : subst]: (!! @ (term > $o) @ (^[Y1 : term > $o]: ((Y1 @ sK642) => ((!! @ term @ (^[Y2 : term]: ((Y1 @ (sub @ Y2 @ (push @ sK642 @ Y0))) = (sK641 @ Y2)))) => (sK641 @ one)))))) @ sK643)))),
  inference(sigma_proxy_clausification,[],[f1287])).
thf(f1289,plain,(
  (pushprop_lem1v2 = $false) | (((!! @ (term > $o) @ (^[Y0 : term > $o]: ((Y0 @ sK642) => ((!! @ term @ (^[Y1 : term]: ((Y0 @ (sub @ Y1 @ (push @ sK642 @ sK643))) = (sK641 @ Y1)))) => (sK641 @ one)))))) = $false)),
  inference(beta-eta_normalization,[],[f1288])).
thf(f1290,plain,(
  ($false = (((^[Y0 : term > $o]: ((Y0 @ sK642) => ((!! @ term @ (^[Y1 : term]: ((Y0 @ (sub @ Y1 @ (push @ sK642 @ sK643))) = (sK641 @ Y1)))) => (sK641 @ one)))) @ sK644))) | (pushprop_lem1v2 = $false)),
  inference(sigma_proxy_clausification,[],[f1289])).
thf(f1291,plain,(
  ((((sK644 @ sK642) => ((!! @ term @ (^[Y0 : term]: ((sK644 @ (sub @ Y0 @ (push @ sK642 @ sK643))) = (sK641 @ Y0)))) => (sK641 @ one)))) = $false) | (pushprop_lem1v2 = $false)),
  inference(beta-eta_normalization,[],[f1290])).
thf(f1292,plain,(
  (pushprop_lem1v2 = $false) | ((((!! @ term @ (^[Y0 : term]: ((sK644 @ (sub @ Y0 @ (push @ sK642 @ sK643))) = (sK641 @ Y0)))) => (sK641 @ one))) = $false)),
  inference(imp_proxy_clausification,[],[f1291])).
thf(f1293,plain,(
  ($true = ((sK644 @ sK642))) | (pushprop_lem1v2 = $false)),
  inference(imp_proxy_clausification,[],[f1291])).
thf(f1294,plain,(
  (pushprop_lem1v2 = $false) | (((sK641 @ one)) = $false)),
  inference(imp_proxy_clausification,[],[f1292])).
thf(f1295,plain,(
  ($true = ((!! @ term @ (^[Y0 : term]: ((sK644 @ (sub @ Y0 @ (push @ sK642 @ sK643))) = (sK641 @ Y0)))))) | (pushprop_lem1v2 = $false)),
  inference(imp_proxy_clausification,[],[f1292])).
thf(f1296,plain,(
  ( ! [X1 : term] : ((pushprop_lem1v2 = $false) | ($true = (((^[Y0 : term]: ((sK644 @ (sub @ Y0 @ (push @ sK642 @ sK643))) = (sK641 @ Y0))) @ X1)))) )),
  inference(pi_proxy_clausification,[],[f1295])).
thf(f1297,plain,(
  ( ! [X1 : term] : (($true = (((sK644 @ (sub @ X1 @ (push @ sK642 @ sK643))) = (sK641 @ X1)))) | (pushprop_lem1v2 = $false)) )),
  inference(beta-eta_normalization,[],[f1296])).
thf(f1298,plain,(
  ( ! [X1 : term] : ((((sK641 @ X1)) = ((sK644 @ (sub @ X1 @ (push @ sK642 @ sK643))))) | (pushprop_lem1v2 = $false)) )),
  inference(equality_proxy_clausification,[],[f1297])).
thf(f1299,plain,(
  ( ! [X1 : term] : (($false = ((sK644 @ (sub @ X1 @ (push @ sK642 @ sK643))))) | (pushprop_lem1v2 = $false) | ($true = ((sK641 @ X1)))) )),
  inference(iff_proxy_clausification,[],[f1298])).
thf(f1301,plain,(
  ( ! [X1 : (term > $o)] : (($true = (((^[Y0 : term > $o]: (!! @ term @ (^[Y1 : term]: (!! @ subst @ (^[Y2 : subst]: (!! @ (term > $o) @ (^[Y3 : term > $o]: ((Y3 @ Y1) => (((^[Y4 : term]: ((^[Y5 : subst]: ((^[Y6 : term > $o]: ((^[Y7 : term > $o]: (!! @ term @ (^[Y8 : term]: ((Y6 @ (sub @ Y8 @ (push @ Y4 @ Y5))) = (Y7 @ Y8))))))))))) @ Y1 @ Y2 @ Y3 @ Y0) => (Y0 @ one)))))))))) @ X1))) | (pushprop_lem1v2 = $true)) )),
  inference(pi_proxy_clausification,[],[f1282])).
thf(f1302,plain,(
  ( ! [X1 : (term > $o)] : ((pushprop_lem1v2 = $true) | ($true = ((!! @ term @ (^[Y0 : term]: (!! @ subst @ (^[Y1 : subst]: (!! @ (term > $o) @ (^[Y2 : term > $o]: ((Y2 @ Y0) => ((!! @ term @ (^[Y3 : term]: ((Y2 @ (sub @ Y3 @ (push @ Y0 @ Y1))) = (X1 @ Y3)))) => (X1 @ one)))))))))))) )),
  inference(beta-eta_normalization,[],[f1301])).
thf(f1303,plain,(
  ( ! [X2 : term,X1 : (term > $o)] : (($true = (((^[Y0 : term]: (!! @ subst @ (^[Y1 : subst]: (!! @ (term > $o) @ (^[Y2 : term > $o]: ((Y2 @ Y0) => ((!! @ term @ (^[Y3 : term]: ((Y2 @ (sub @ Y3 @ (push @ Y0 @ Y1))) = (X1 @ Y3)))) => (X1 @ one)))))))) @ X2))) | (pushprop_lem1v2 = $true)) )),
  inference(pi_proxy_clausification,[],[f1302])).
thf(f1304,plain,(
  ( ! [X2 : term,X1 : (term > $o)] : (($true = ((!! @ subst @ (^[Y0 : subst]: (!! @ (term > $o) @ (^[Y1 : term > $o]: ((Y1 @ X2) => ((!! @ term @ (^[Y2 : term]: ((Y1 @ (sub @ Y2 @ (push @ X2 @ Y0))) = (X1 @ Y2)))) => (X1 @ one))))))))) | (pushprop_lem1v2 = $true)) )),
  inference(beta-eta_normalization,[],[f1303])).
thf(f1305,plain,(
  ( ! [X2 : term,X3 : subst,X1 : (term > $o)] : (($true = (((^[Y0 : subst]: (!! @ (term > $o) @ (^[Y1 : term > $o]: ((Y1 @ X2) => ((!! @ term @ (^[Y2 : term]: ((Y1 @ (sub @ Y2 @ (push @ X2 @ Y0))) = (X1 @ Y2)))) => (X1 @ one)))))) @ X3))) | (pushprop_lem1v2 = $true)) )),
  inference(pi_proxy_clausification,[],[f1304])).
thf(f1306,plain,(
  ( ! [X2 : term,X3 : subst,X1 : (term > $o)] : (($true = ((!! @ (term > $o) @ (^[Y0 : term > $o]: ((Y0 @ X2) => ((!! @ term @ (^[Y1 : term]: ((Y0 @ (sub @ Y1 @ (push @ X2 @ X3))) = (X1 @ Y1)))) => (X1 @ one))))))) | (pushprop_lem1v2 = $true)) )),
  inference(beta-eta_normalization,[],[f1305])).
thf(f1307,plain,(
  ( ! [X2 : term,X3 : subst,X1 : (term > $o),X4 : (term > $o)] : ((pushprop_lem1v2 = $true) | ($true = (((^[Y0 : term > $o]: ((Y0 @ X2) => ((!! @ term @ (^[Y1 : term]: ((Y0 @ (sub @ Y1 @ (push @ X2 @ X3))) = (X1 @ Y1)))) => (X1 @ one)))) @ X4)))) )),
  inference(pi_proxy_clausification,[],[f1306])).
thf(f1308,plain,(
  ( ! [X2 : term,X3 : subst,X1 : (term > $o),X4 : (term > $o)] : (($true = (((X4 @ X2) => ((!! @ term @ (^[Y0 : term]: ((X4 @ (sub @ Y0 @ (push @ X2 @ X3))) = (X1 @ Y0)))) => (X1 @ one))))) | (pushprop_lem1v2 = $true)) )),
  inference(beta-eta_normalization,[],[f1307])).
thf(f1309,plain,(
  ( ! [X2 : term,X3 : subst,X1 : (term > $o),X4 : (term > $o)] : (($true = (((!! @ term @ (^[Y0 : term]: ((X4 @ (sub @ Y0 @ (push @ X2 @ X3))) = (X1 @ Y0)))) => (X1 @ one)))) | (((X4 @ X2)) = $false) | (pushprop_lem1v2 = $true)) )),
  inference(imp_proxy_clausification,[],[f1308])).
thf(f1310,plain,(
  ( ! [X2 : term,X3 : subst,X1 : (term > $o),X4 : (term > $o)] : ((pushprop_lem1v2 = $true) | (((X1 @ one)) = $true) | (((X4 @ X2)) = $false) | ($false = ((!! @ term @ (^[Y0 : term]: ((X4 @ (sub @ Y0 @ (push @ X2 @ X3))) = (X1 @ Y0))))))) )),
  inference(imp_proxy_clausification,[],[f1309])).
thf(f1311,plain,(
  ( ! [X2 : term,X3 : subst,X1 : (term > $o),X4 : (term > $o)] : ((pushprop_lem1v2 = $true) | ($false = (((^[Y0 : term]: ((X4 @ (sub @ Y0 @ (push @ X2 @ X3))) = (X1 @ Y0))) @ (sK645 @ X3 @ X1 @ X2 @ X4)))) | (((X1 @ one)) = $true) | (((X4 @ X2)) = $false)) )),
  inference(sigma_proxy_clausification,[],[f1310])).
thf(f1312,plain,(
  ( ! [X2 : term,X3 : subst,X1 : (term > $o),X4 : (term > $o)] : ((((X1 @ one)) = $true) | (pushprop_lem1v2 = $true) | ((((X4 @ (sub @ (sK645 @ X3 @ X1 @ X2 @ X4) @ (push @ X2 @ X3))) = (X1 @ (sK645 @ X3 @ X1 @ X2 @ X4)))) = $false) | (((X4 @ X2)) = $false)) )),
  inference(beta-eta_normalization,[],[f1311])).
thf(f1313,plain,(
  ( ! [X2 : term,X3 : subst,X1 : (term > $o),X4 : (term > $o)] : ((((X1 @ one)) = $true) | (pushprop_lem1v2 = $true) | (((X4 @ X2)) = $false) | (((X4 @ (sub @ (sK645 @ X3 @ X1 @ X2 @ X4) @ (push @ X2 @ X3)))) != ((X1 @ (sK645 @ X3 @ X1 @ X2 @ X4))))) )),
  inference(equality_proxy_clausification,[],[f1312])).
thf(f1314,plain,(
  ( ! [X2 : term,X3 : subst,X1 : (term > $o),X4 : (term > $o)] : ((pushprop_lem1v2 = $true) | (((X4 @ X2)) = $false) | ($true = ((X1 @ (sK645 @ X3 @ X1 @ X2 @ X4)))) | ($true = ((X4 @ (sub @ (sK645 @ X3 @ X1 @ X2 @ X4) @ (push @ X2 @ X3))))) | (((X1 @ one)) = $true)) )),
  inference(xor_proxy_clausification,[],[f1313])).
thf(f2294,definition,(
  spl646_223 <=> (pushprop_lem1v2 = $true)),
  introduced(definition,[new_symbols(definition,[spl646_223])],[avatar_definition])).
thf(f2296,plain,(
  (pushprop_lem1v2 = $true) | ~spl646_223),
  inference(avatar_component_clause,[],[f2294])).
thf(f2299,definition,(
  spl646_224 <=> ! [X101 : term] : (($true = ((sK561 @ X101))) | ($false = ((sK558 @ (sub @ X101 @ (push @ sK560 @ sK559))))))),
  introduced(definition,[new_symbols(definition,[spl646_224])],[avatar_definition])).
thf(f2300,plain,(
  ( ! [X101 : term] : (($true = ((sK561 @ X101))) | ($false = ((sK558 @ (sub @ X101 @ (push @ sK560 @ sK559)))))) ) | ~spl646_224),
  inference(avatar_component_clause,[],[f2299])).
thf(f2301,plain,(
  spl646_223 | spl646_224),
  inference(avatar_split_clause,[],[f1228,f2299,f2294])).
thf(f2463,definition,(
  spl646_259 <=> ($true = ((sK561 @ one)))),
  introduced(definition,[new_symbols(definition,[spl646_259])],[avatar_definition])).
thf(f2465,plain,(
  ($true != ((sK561 @ one))) | spl646_259),
  inference(avatar_component_clause,[],[f2463])).
thf(f2466,plain,(
  ~spl646_259 | spl646_223),
  inference(avatar_split_clause,[],[f1125,f2294,f2463])).
thf(f3082,definition,(
  spl646_393 <=> ($true = ((sK558 @ sK560)))),
  introduced(definition,[new_symbols(definition,[spl646_393])],[avatar_definition])).
thf(f3084,plain,(
  ($true = ((sK558 @ sK560))) | ~spl646_393),
  inference(avatar_component_clause,[],[f3082])).
thf(f3085,plain,(
  spl646_393 | spl646_223),
  inference(avatar_split_clause,[],[f1124,f2294,f3082])).
thf(f3396,definition,(
  spl646_459 <=> ($true = ((sK644 @ sK642)))),
  introduced(definition,[new_symbols(definition,[spl646_459])],[avatar_definition])).
thf(f3398,plain,(
  ($true = ((sK644 @ sK642))) | ~spl646_459),
  inference(avatar_component_clause,[],[f3396])).
thf(f3400,definition,(
  spl646_460 <=> (pushprop_lem1v2 = $false)),
  introduced(definition,[new_symbols(definition,[spl646_460])],[avatar_definition])).
thf(f3402,plain,(
  (pushprop_lem1v2 = $false) | ~spl646_460),
  inference(avatar_component_clause,[],[f3400])).
thf(f3403,plain,(
  spl646_459 | spl646_460),
  inference(avatar_split_clause,[],[f1293,f3400,f3396])).
thf(f3409,definition,(
  spl646_462 <=> ! [X1 : term] : (($false = ((sK644 @ (sub @ X1 @ (push @ sK642 @ sK643))))) | ($true = ((sK641 @ X1))))),
  introduced(definition,[new_symbols(definition,[spl646_462])],[avatar_definition])).
thf(f3410,plain,(
  ( ! [X1 : term] : (($false = ((sK644 @ (sub @ X1 @ (push @ sK642 @ sK643))))) | ($true = ((sK641 @ X1)))) ) | ~spl646_462),
  inference(avatar_component_clause,[],[f3409])).
thf(f3411,plain,(
  spl646_460 | spl646_462),
  inference(avatar_split_clause,[],[f1299,f3409,f3400])).
thf(f3413,definition,(
  spl646_463 <=> (((sK641 @ one)) = $false)),
  introduced(definition,[new_symbols(definition,[spl646_463])],[avatar_definition])).
thf(f3415,plain,(
  (((sK641 @ one)) = $false) | ~spl646_463),
  inference(avatar_component_clause,[],[f3413])).
thf(f3416,plain,(
  spl646_460 | spl646_463),
  inference(avatar_split_clause,[],[f1294,f3413,f3400])).
thf(f3422,definition,(
  spl646_465 <=> ! [X4 : (term > $o),X3 : subst,X2 : term,X1 : (term > $o)] : ((((X4 @ X2)) = $false) | (((X1 @ one)) = $true) | ($true = ((X4 @ (sub @ (sK645 @ X3 @ X1 @ X2 @ X4) @ (push @ X2 @ X3))))) | ($true = ((X1 @ (sK645 @ X3 @ X1 @ X2 @ X4)))))),
  introduced(definition,[new_symbols(definition,[spl646_465])],[avatar_definition])).
thf(f3423,plain,(
  ( ! [X2 : term,X3 : subst,X1 : (term > $o),X4 : (term > $o)] : (($true = ((X4 @ (sub @ (sK645 @ X3 @ X1 @ X2 @ X4) @ (push @ X2 @ X3))))) | (((X1 @ one)) = $true) | ($true = ((X1 @ (sK645 @ X3 @ X1 @ X2 @ X4)))) | (((X4 @ X2)) = $false)) ) | ~spl646_465),
  inference(avatar_component_clause,[],[f3422])).
thf(f3424,plain,(
  spl646_465 | spl646_223),
  inference(avatar_split_clause,[],[f1314,f2294,f3422])).
thf(f3432,plain,(
  (lamnotap = $false) | ($true != $true)),
  inference(constrained_superposition,[],[f1165,f1206])).
thf(f3439,plain,(
  (lamnotap = $false)),
  inference(trivial_inequality_removal,[],[f3432])).
thf(f3442,plain,(
  ($true = $false) | (~spl646_223 | ~spl646_460)),
  inference(forward_demodulation,[],[f2296,f3402])).
thf(f3443,plain,(
  $false | (~spl646_223 | ~spl646_460)),
  inference(trivial_inequality_removal,[],[f3442])).
thf(f3444,plain,(
  ~spl646_223 | ~spl646_460),
  inference(avatar_contradiction_clause,[],[f3443])).
thf(f3455,plain,(
  ( ! [X0 : d_term] : ((one = ((d2term @ X0)))) )),
  inference(constrained_superposition,[],[f1102,f1061])).
thf(f3509,plain,(
  ( ! [X141 : term] : ((one = X141)) )),
  inference(forward_demodulation,[],[f1081,f3455])).
thf(f3519,plain,(
  ( ! [X0 : term] : ((((sK641 @ X0)) = $false)) ) | ~spl646_463),
  inference(constrained_superposition,[],[f3415,f3509])).
thf(f3532,plain,(
  ( ! [X0 : term] : (($true = ((sK641 @ X0))) | ($false = ((sK644 @ (sub @ X0 @ (push @ one @ sK643)))))) ) | ~spl646_462),
  inference(constrained_superposition,[],[f3410,f3509])).
thf(f3534,plain,(
  ($true = ((sK644 @ one))) | ~spl646_459),
  inference(constrained_superposition,[],[f3398,f3509])).
thf(f3536,definition,(
  spl646_466 <=> ($false = ((sK644 @ one)))),
  introduced(definition,[new_symbols(definition,[spl646_466])],[avatar_definition])).
thf(f3538,plain,(
  ($false = ((sK644 @ one))) | ~spl646_466),
  inference(avatar_component_clause,[],[f3536])).
thf(f3540,definition,(
  spl646_467 <=> ! [X0 : term] : ($true = ((sK641 @ X0)))),
  introduced(definition,[new_symbols(definition,[spl646_467])],[avatar_definition])).
thf(f3541,plain,(
  ( ! [X0 : term] : (($true = ((sK641 @ X0)))) ) | ~spl646_467),
  inference(avatar_component_clause,[],[f3540])).
thf(f3544,definition,(
  spl646_468 <=> ! [X0 : term] : (((sK641 @ X0)) = $false)),
  introduced(definition,[new_symbols(definition,[spl646_468])],[avatar_definition])).
thf(f3545,plain,(
  ( ! [X0 : term] : ((((sK641 @ X0)) = $false)) ) | ~spl646_468),
  inference(avatar_component_clause,[],[f3544])).
thf(f3547,definition,(
  spl646_469 <=> ($true = ((sK644 @ one)))),
  introduced(definition,[new_symbols(definition,[spl646_469])],[avatar_definition])).
thf(f3549,plain,(
  ($true = ((sK644 @ one))) | ~spl646_469),
  inference(avatar_component_clause,[],[f3547])).
thf(f3552,plain,(
  ( ! [X0 : term] : (($true = ((sK641 @ X0))) | ($false = ((sK644 @ one)))) ) | ~spl646_462),
  inference(forward_demodulation,[],[f3532,f3509])).
thf(f3554,plain,(
  spl646_468 | ~spl646_463),
  inference(avatar_split_clause,[],[f3519,f3413,f3544])).
thf(f3555,plain,(
  spl646_469 | ~spl646_459),
  inference(avatar_split_clause,[],[f3534,f3396,f3547])).
thf(f3557,plain,(
  spl646_466 | spl646_467 | ~spl646_462),
  inference(avatar_split_clause,[],[f3552,f3409,f3540,f3536])).
thf(f3565,plain,(
  ($true = $false) | (~spl646_466 | ~spl646_469)),
  inference(constrained_superposition,[],[f3549,f3538])).
thf(f3569,plain,(
  $false | (~spl646_466 | ~spl646_469)),
  inference(trivial_inequality_removal,[],[f3565])).
thf(f3570,plain,(
  ~spl646_466 | ~spl646_469),
  inference(avatar_contradiction_clause,[],[f3569])).
thf(f3571,plain,(
  ($true = $false) | (~spl646_467 | ~spl646_468)),
  inference(forward_demodulation,[],[f3541,f3545])).
thf(f3572,plain,(
  $false | (~spl646_467 | ~spl646_468)),
  inference(trivial_inequality_removal,[],[f3571])).
thf(f3573,plain,(
  ~spl646_467 | ~spl646_468),
  inference(avatar_contradiction_clause,[],[f3572])).
thf(f3582,plain,(
  ( ! [X101 : term] : (($true = ((sK561 @ X101))) | (((sK558 @ one)) = $false)) ) | ~spl646_224),
  inference(forward_demodulation,[],[f2300,f3509])).
thf(f3594,plain,(
  ( ! [X2 : term,X3 : subst,X1 : (term > $o),X4 : (term > $o)] : ((((X4 @ X2)) = $false) | (((X1 @ one)) = $true) | ($true = ((X1 @ (sK645 @ X3 @ X1 @ X2 @ X4)))) | ($true = ((X4 @ one)))) ) | ~spl646_465),
  inference(forward_demodulation,[],[f3423,f3509])).
thf(f3614,plain,(
  ($true = ((sK558 @ one))) | ~spl646_393),
  inference(forward_demodulation,[],[f3084,f3509])).
thf(f3676,definition,(
  spl646_474 <=> ! [X101 : term] : ($true = ((sK561 @ X101)))),
  introduced(definition,[new_symbols(definition,[spl646_474])],[avatar_definition])).
thf(f3677,plain,(
  ( ! [X101 : term] : (($true = ((sK561 @ X101)))) ) | ~spl646_474),
  inference(avatar_component_clause,[],[f3676])).
thf(f3679,definition,(
  spl646_475 <=> (((sK558 @ one)) = $false)),
  introduced(definition,[new_symbols(definition,[spl646_475])],[avatar_definition])).
thf(f3681,plain,(
  (((sK558 @ one)) = $false) | ~spl646_475),
  inference(avatar_component_clause,[],[f3679])).
thf(f3682,plain,(
  spl646_474 | spl646_475 | ~spl646_224),
  inference(avatar_split_clause,[],[f3582,f2299,f3679,f3676])).
thf(f3696,definition,(
  spl646_478 <=> ! [X101 : term] : ($false = ((sK561 @ X101)))),
  introduced(definition,[new_symbols(definition,[spl646_478])],[avatar_definition])).
thf(f3697,plain,(
  ( ! [X101 : term] : (($false = ((sK561 @ X101)))) ) | ~spl646_478),
  inference(avatar_component_clause,[],[f3696])).
thf(f3699,definition,(
  spl646_479 <=> ($true = ((sK558 @ one)))),
  introduced(definition,[new_symbols(definition,[spl646_479])],[avatar_definition])).
thf(f3701,plain,(
  ($true = ((sK558 @ one))) | ~spl646_479),
  inference(avatar_component_clause,[],[f3699])).
thf(f3717,plain,(
  ( ! [X2 : term,X1 : (term > $o),X4 : (term > $o)] : ((((X1 @ one)) = $true) | (((X4 @ X2)) = $false) | (((X1 @ one)) = $true) | ($true = ((X4 @ one)))) ) | ~spl646_465),
  inference(forward_demodulation,[],[f3594,f3509])).
thf(f3718,plain,(
  ( ! [X2 : term,X1 : (term > $o),X4 : (term > $o)] : ((((X1 @ one)) = $true) | ($true = ((X4 @ one))) | (((X4 @ X2)) = $false)) ) | ~spl646_465),
  inference(duplicate_literal_removal,[],[f3717])).
thf(f3748,plain,(
  spl646_479 | ~spl646_393),
  inference(avatar_split_clause,[],[f3614,f3082,f3699])).
thf(f3818,definition,(
  spl646_493 <=> ! [X4 : (term > $o),X2 : term] : (($true = ((X4 @ one))) | (((X4 @ X2)) = $false))),
  introduced(definition,[new_symbols(definition,[spl646_493])],[avatar_definition])).
thf(f3819,plain,(
  ( ! [X2 : term,X4 : (term > $o)] : (($true = ((X4 @ one))) | (((X4 @ X2)) = $false)) ) | ~spl646_493),
  inference(avatar_component_clause,[],[f3818])).
thf(f3821,definition,(
  spl646_494 <=> ! [X1 : (term > $o)] : (((X1 @ one)) = $true)),
  introduced(definition,[new_symbols(definition,[spl646_494])],[avatar_definition])).
thf(f3822,plain,(
  ( ! [X1 : (term > $o)] : ((((X1 @ one)) = $true)) ) | ~spl646_494),
  inference(avatar_component_clause,[],[f3821])).
thf(f3823,plain,(
  spl646_493 | spl646_494 | ~spl646_465),
  inference(avatar_split_clause,[],[f3718,f3422,f3821,f3818])).
thf(f4132,plain,(
  ($true = $false) | ~spl646_494),
  inference(constrained_superposition,[],[f3439,f3822])).
thf(f4148,plain,(
  $false | ~spl646_494),
  inference(trivial_inequality_removal,[],[f4132])).
thf(f4149,plain,(
  ~spl646_494),
  inference(avatar_contradiction_clause,[],[f4148])).
thf(f4901,plain,(
  ( ! [X0 : term] : (((((^[Y0 : term]: (sK561 @ ((^[Y1 : term]: (Y1)) @ Y0))) @ X0)) = $false) | ($true != $true)) ) | (spl646_259 | ~spl646_493)),
  inference(constrained_superposition,[],[f2465,f3819])).
thf(f4910,plain,(
  ( ! [X0 : term] : (((((^[Y0 : term]: (sK561 @ ((^[Y1 : term]: (Y1)) @ Y0))) @ X0)) = $false)) ) | (spl646_259 | ~spl646_493)),
  inference(trivial_inequality_removal,[],[f4901])).
thf(f4911,plain,(
  ( ! [X0 : term] : ((((sK561 @ X0)) = $false)) ) | (spl646_259 | ~spl646_493)),
  inference(beta-eta_normalization,[],[f4910])).
thf(f4915,plain,(
  spl646_478 | spl646_259 | ~spl646_493),
  inference(avatar_split_clause,[],[f4911,f3818,f2463,f3696])).
thf(f5225,plain,(
  ($true = $false) | (~spl646_475 | ~spl646_479)),
  inference(forward_demodulation,[],[f3701,f3681])).
thf(f5226,plain,(
  $false | (~spl646_475 | ~spl646_479)),
  inference(trivial_inequality_removal,[],[f5225])).
thf(f5227,plain,(
  ~spl646_475 | ~spl646_479),
  inference(avatar_contradiction_clause,[],[f5226])).
thf(f5228,plain,(
  ($true = $false) | (~spl646_474 | ~spl646_478)),
  inference(forward_demodulation,[],[f3677,f3697])).
thf(f5229,plain,(
  $false | (~spl646_474 | ~spl646_478)),
  inference(trivial_inequality_removal,[],[f5228])).
thf(f5230,plain,(
  ~spl646_474 | ~spl646_478),
  inference(avatar_contradiction_clause,[],[f5229])).
cnf(s166, plain, spl646_223 | spl646_224, inference(sat_conversion,[],[f2301])).
cnf(s206, plain, spl646_223 | ~spl646_259, inference(sat_conversion,[],[f2466])).
cnf(s356, plain, spl646_223 | spl646_393, inference(sat_conversion,[],[f3085])).
cnf(s435, plain, spl646_459 | spl646_460, inference(sat_conversion,[],[f3403])).
cnf(s437, plain, spl646_460 | spl646_462, inference(sat_conversion,[],[f3411])).
cnf(s438, plain, spl646_460 | spl646_463, inference(sat_conversion,[],[f3416])).
cnf(s440, plain, spl646_223 | spl646_465, inference(sat_conversion,[],[f3424])).
cnf(s442, plain, ~spl646_223 | ~spl646_460, inference(sat_conversion,[],[f3444])).
cnf(s447, plain, ~spl646_463 | spl646_468, inference(sat_conversion,[],[f3554])).
cnf(s448, plain, ~spl646_459 | spl646_469, inference(sat_conversion,[],[f3555])).
cnf(s449, plain, ~spl646_462 | spl646_466 | spl646_467, inference(sat_conversion,[],[f3557])).
cnf(s452, plain, ~spl646_466 | ~spl646_469, inference(sat_conversion,[],[f3570])).
cnf(s453, plain, ~spl646_467 | ~spl646_468, inference(sat_conversion,[],[f3573])).
cnf(s457, plain, ~spl646_224 | spl646_474 | spl646_475, inference(sat_conversion,[],[f3682])).
cnf(s474, plain, ~spl646_393 | spl646_479, inference(sat_conversion,[],[f3748])).
cnf(s494, plain, ~spl646_465 | spl646_493 | spl646_494, inference(sat_conversion,[],[f3823])).
cnf(s525, plain, ~spl646_494, inference(sat_conversion,[],[f4149])).
cnf(s526, plain, spl646_259 | spl646_478 | ~spl646_493, inference(sat_conversion,[],[f4915])).
cnf(s542, plain, ~spl646_475 | ~spl646_479, inference(sat_conversion,[],[f5227])).
cnf(s543, plain, ~spl646_474 | ~spl646_478, inference(sat_conversion,[],[f5230])).
cnf(s555, plain, ~spl646_465 | spl646_493, inference(rat,[],[s494,s525])).
cnf(s596, plain, spl646_460, inference(rat,[],[s449,s452,s453,s448,s447,s435,s437,s438])).
cnf(s597, plain, ~spl646_223, inference(rat,[],[s442,s596])).
cnf(s598, plain, spl646_465, inference(rat,[],[s440,s597])).
cnf(s600, plain, spl646_393, inference(rat,[],[s356,s597])).
cnf(s601, plain, ~spl646_259, inference(rat,[],[s206,s597])).
cnf(s602, plain, spl646_224, inference(rat,[],[s166,s597])).
cnf(s604, plain, spl646_493, inference(rat,[],[s555,s598])).
cnf(s605, plain, spl646_479, inference(rat,[],[s474,s600])).
cnf(s606, plain, spl646_478, inference(rat,[],[s526,s604,s601])).
cnf(s607, plain, ~spl646_475, inference(rat,[],[s542,s605])).
cnf(s608, plain, ~spl646_474, inference(rat,[],[s543,s606])).
cnf(s609, plain, $false, inference(rat,[],[s457,s602,s608,s607])).
thf(f5231,plain,(
  $false),
  inference(avatar_sat_refutation,[],[s609])).
% SZS output end Proof for theBenchmark