%----Output an unreadable proof
% SZS output start Proof for theBenchmark
thf(type_def_5, type, term: $tType).
thf(type_def_6, type, subst: $tType).
thf(type_def_7, type, sTfun: ($tType * $tType) > $tType).
thf(type_def_8, type, d_term: $tType).
thf(type_def_9, type, d_subst: $tType).
thf(func_def_0, type, one: term).
thf(func_def_1, type, ap: (term > term > term)).
thf(func_def_2, type, lam: (term > term)).
thf(func_def_3, type, sub: (term > subst > term)).
thf(func_def_4, type, id: subst).
thf(func_def_5, type, sh: subst).
thf(func_def_6, type, push: (term > subst > subst)).
thf(func_def_7, type, comp: (subst > subst > subst)).
thf(func_def_8, type, var: (term > $o)).
thf(func_def_9, type, pushprop_lem1v2: $o).
thf(func_def_10, type, pushprop_lem1_gthm: $o).
thf(func_def_11, type, axmap: $o).
thf(func_def_12, type, pushprop_lem0_gthm: $o).
thf(func_def_13, type, shinj: $o).
thf(func_def_14, type, hoasinduction_lem1v2: $o).
thf(func_def_15, type, hoasinduction_lem1v2_gthm: $o).
thf(func_def_16, type, hoasap: (subst > term > subst > term > term)).
thf(func_def_17, type, induction2lem: $o).
thf(func_def_18, type, hoasinduction_lem3v2_f: $o).
thf(func_def_19, type, axvarshift: $o).
thf(func_def_20, type, hoasapinj2: $o).
thf(func_def_21, type, hoasapnotvar_gthm: $o).
thf(func_def_22, type, hoasapinj1: $o).
thf(func_def_23, type, ulamvar1: $o).
thf(func_def_24, type, induction2lem_lthm: $o).
thf(func_def_25, type, hoasinduction_lem3v2_gthm: $o).
thf(func_def_26, type, apnotvar: $o).
thf(func_def_27, type, pushprop_lthm_orig: $o).
thf(func_def_28, type, hoasinduction_lem3v2_f_lthm: $o).
thf(func_def_29, type, hoasinduction_lthm: $o).
thf(func_def_30, type, hoasinduction_no_psi_cond_lthm: $o).
thf(func_def_31, type, hoaslaminj: $o).
thf(func_def_32, type, hoasinduction_lem3aaa: $o).
thf(func_def_33, type, induction2lem_gthm: $o).
thf(func_def_34, type, hoasinduction_lem3aa_lthm: $o).
thf(func_def_35, type, hoasinduction_lem3: $o).
thf(func_def_36, type, hoasinduction_lem2: $o).
thf(func_def_37, type, termmset_lthm: $o).
thf(func_def_38, type, hoasinduction_lem1: $o).
thf(func_def_39, type, hoaslamnotap_lthm: $o).
thf(func_def_40, type, pushprop_lem1v2_lthm: $o).
thf(func_def_41, type, hoasapnotvar: $o).
thf(func_def_42, type, hoasinduction_lem0: $o).
thf(func_def_43, type, hoasinduction: $o).
thf(func_def_44, type, hoasinduction_gthm: $o).
thf(func_def_45, type, axapp: $o).
thf(func_def_46, type, hoaslamnotvar_lthm: $o).
thf(func_def_47, type, pushprop_lem3v2_lthm: $o).
thf(func_def_48, type, hoasinduction_lem3b_lthm: $o).
thf(func_def_49, type, ulamvarind: $o).
thf(func_def_50, type, induction: $o).
thf(func_def_51, type, hoasinduction_lem3a_lthm: $o).
thf(func_def_52, type, termmset_gthm: $o).
thf(func_def_53, type, hoasinduction_lem3aa: $o).
thf(func_def_54, type, pushprop_lem1v2_gthm: $o).
thf(func_def_55, type, hoaslamnotap_gthm: $o).
thf(func_def_56, type, hoaslamnotvar_gthm: $o).
thf(func_def_57, type, hoasinduction_lem3b_gthm: $o).
thf(func_def_58, type, pushprop_lem2v2: $o).
thf(func_def_59, type, hoasinduction_lem3a_gthm: $o).
thf(func_def_60, type, axclos: $o).
thf(func_def_61, type, axassoc: $o).
thf(func_def_62, type, hoasinduction_lem2v2: $o).
thf(func_def_63, type, pushprop_lthm: $o).
thf(func_def_64, type, apinj2: $o).
thf(func_def_65, type, apinj1: $o).
thf(func_def_66, type, hoasapinj2_lthm: $o).
thf(func_def_67, type, hoasinduction_lem3v2a: $o).
thf(func_def_68, type, hoasapinj1_lthm: $o).
thf(func_def_69, type, hoaslaminj_lthm: $o).
thf(func_def_70, type, axvarcons: $o).
thf(func_def_71, type, hoaslam: (subst > (subst > term > term) > term)).
thf(func_def_72, type, axscons: $o).
thf(func_def_73, type, hoasinduction_lem2v2_gthm: $o).
thf(func_def_74, type, axidr: $o).
thf(func_def_75, type, pushprop_lem1: $o).
thf(func_def_76, type, laminj: $o).
thf(func_def_77, type, hoasinduction_lem3_lthm: $o).
thf(func_def_78, type, pushprop_lem0: $o).
thf(func_def_79, type, pushprop_gthm: $o).
thf(func_def_80, type, axabs: $o).
thf(func_def_81, type, hoasinduction_lem3v2a_lthm: $o).
thf(func_def_82, type, hoasinduction_lem2_lthm: $o).
thf(func_def_83, type, hoasapinj2_gthm: $o).
thf(func_def_84, type, hoasinduction_p_and_p_prime: ((subst > term > subst > $o) > (term > $o) > $o)).
thf(func_def_85, type, hoasinduction_lem1_lthm: $o).
thf(func_def_86, type, lamnotap: $o).
thf(func_def_87, type, hoasapinj1_gthm: $o).
thf(func_def_88, type, hoaslamnotvar: $o).
thf(func_def_89, type, axidl: $o).
thf(func_def_90, type, hoaslaminj_gthm: $o).
thf(func_def_91, type, induction2_lthm: $o).
thf(func_def_92, type, hoasinduction_lem0_lthm: $o).
thf(func_def_93, type, substmonoid_lthm: $o).
thf(func_def_94, type, pushprop: $o).
thf(func_def_95, type, hoasinduction_lem3_gthm: $o).
thf(func_def_96, type, hoasinduction_lem2_gthm: $o).
thf(func_def_97, type, hoasinduction_lem3b: $o).
thf(func_def_98, type, substmonoid: $o).
thf(func_def_99, type, lamnotvar: $o).
thf(func_def_100, type, hoasinduction_lem3a: $o).
thf(func_def_101, type, hoasinduction_lem1_gthm: $o).
thf(func_def_102, type, hoasinduction_no_psi_cond: $o).
thf(func_def_103, type, induction2_gthm: $o).
thf(func_def_104, type, pushprop_lem2v2_lthm: $o).
thf(func_def_105, type, hoasvar: (subst > term > subst > $o)).
thf(func_def_106, type, hoaslamnotap: $o).
thf(func_def_107, type, substmonoid_gthm: $o).
thf(func_def_108, type, ulamvarsh: $o).
thf(func_def_109, type, induction2: $o).
thf(func_def_110, type, pushprop_lem3v2: $o).
thf(func_def_111, type, pushprop_lem2v2_gthm: $o).
thf(func_def_112, type, pushprop_lem1_lthm: $o).
thf(func_def_113, type, hoasinduction_lem3v2: $o).
thf(func_def_114, type, axshiftcons: $o).
thf(func_def_115, type, termmset: $o).
thf(func_def_116, type, pushprop_lem0_lthm: $o).
thf(func_def_117, type, hoasapnotvar_lthm: $o).
thf(func_def_118, type, hoasinduction_lem3v2_lthm: $o).
thf(func_def_119, type, pushprop_p_and_p_prime: (term > subst > (term > $o) > (term > $o) > $o)).
thf(func_def_120, type, axvarid: $o).
thf(func_def_121, type, hoasinduction_lthm_3: $o).
thf(func_def_122, type, d2term: (d_term > term)).
thf(func_def_123, type, d2subst: (d_subst > subst)).
thf(func_def_124, type, d_one: d_term).
thf(func_def_125, type, d_id: d_subst).
thf(func_def_129, type, vPI: !>[X0: $tType]:(((X0 > $o) > $o))).
thf(func_def_130, type, db1: !>[X0: $tType]:(X0)).
thf(func_def_131, type, db0: !>[X0: $tType]:(X0)).
thf(func_def_132, type, db2: !>[X0: $tType]:(X0)).
thf(func_def_133, type, vEQ: !>[X0: $tType]:((X0 > X0 > $o))).
thf(func_def_134, type, vLAM: !>[X0: $tType, X1: $tType]:((X1) > (X0 > X1))).
thf(func_def_135, type, db4: !>[X0: $tType]:(X0)).
thf(func_def_136, type, db3: !>[X0: $tType]:(X0)).
thf(func_def_137, type, sP0: $o).
thf(func_def_138, type, sP1: $o).
thf(func_def_139, type, sP2: $o).
thf(func_def_140, type, sP3: $o).
thf(func_def_141, type, sP4: ((subst > term > subst > $o) > $o)).
thf(func_def_142, type, sP5: $o).
thf(func_def_143, type, sP6: ((subst > term > subst > $o) > $o)).
thf(func_def_144, type, sP7: ((subst > term > subst > $o) > $o)).
thf(func_def_145, type, sP8: $o).
thf(func_def_146, type, sP9: $o).
thf(func_def_147, type, sP10: $o).
thf(func_def_148, type, sK11: term).
thf(func_def_149, type, sK12: ((subst > term > subst > $o) > term > term)).
thf(func_def_150, type, sK13: ((subst > term > subst > $o) > subst)).
thf(func_def_151, type, sK14: ((subst > term > subst > $o) > subst)).
thf(func_def_152, type, sK15: ((subst > term > subst > $o) > subst)).
thf(func_def_153, type, sK16: ((subst > term > subst > $o) > term)).
thf(func_def_154, type, sK17: ((subst > term > subst > $o) > subst)).
thf(func_def_155, type, sK18: ((subst > term > subst > $o) > subst)).
thf(func_def_156, type, sK19: ((subst > term > subst > $o) > subst)).
thf(func_def_157, type, sK20: ((subst > term > subst > $o) > term)).
thf(func_def_158, type, sK21: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_159, type, sK22: ((term > $o) > term)).
thf(func_def_160, type, sK23: ((term > $o) > term)).
thf(func_def_161, type, sK24: ((term > $o) > term)).
thf(func_def_162, type, sK25: ((term > $o) > term)).
thf(func_def_163, type, sK26: term).
thf(func_def_164, type, sK27: (subst > term > subst > $o)).
thf(func_def_165, type, sK28: ((subst > term > term) > term)).
thf(func_def_166, type, sK29: ((subst > term > term) > subst)).
thf(func_def_167, type, sK30: ((subst > term > term) > term)).
thf(func_def_168, type, sK31: ((subst > term > term) > subst)).
thf(func_def_169, type, sK32: term).
thf(func_def_170, type, sK33: (term > $o)).
thf(func_def_171, type, sK34: (term > term)).
thf(func_def_172, type, sK35: ((subst > term > subst > $o) > term)).
thf(func_def_173, type, sK36: ((subst > term > subst > $o) > term)).
thf(func_def_174, type, sK37: ((subst > term > subst > $o) > subst)).
thf(func_def_175, type, sK38: ((subst > term > subst > $o) > subst)).
thf(func_def_176, type, sK39: ((subst > term > subst > $o) > term)).
thf(func_def_177, type, sK40: ((subst > term > subst > $o) > subst)).
thf(func_def_178, type, sK41: ((subst > term > subst > $o) > term)).
thf(func_def_179, type, sK42: ((subst > term > subst > $o) > subst)).
thf(func_def_180, type, sK43: ((subst > term > subst > $o) > subst)).
thf(func_def_181, type, sK44: ((subst > term > subst > $o) > subst)).
thf(func_def_182, type, sK45: term).
thf(func_def_183, type, sK46: term).
thf(func_def_184, type, sK47: (subst > term > subst > $o)).
thf(func_def_185, type, sK48: (subst > term > subst > $o)).
thf(func_def_186, type, sK49: term).
thf(func_def_187, type, sK50: ((subst > term > subst > $o) > term)).
thf(func_def_188, type, sK51: ((subst > term > subst > $o) > term)).
thf(func_def_189, type, sK52: ((subst > term > subst > $o) > term)).
thf(func_def_190, type, sK53: ((subst > term > subst > $o) > subst)).
thf(func_def_191, type, sK54: ((subst > term > subst > $o) > subst)).
thf(func_def_192, type, sK55: ((subst > term > subst > $o) > term)).
thf(func_def_193, type, sK56: ((subst > term > subst > $o) > subst)).
thf(func_def_194, type, sK57: ((subst > term > subst > $o) > subst)).
thf(func_def_195, type, sK58: ((subst > term > subst > $o) > subst)).
thf(func_def_196, type, sK59: ((subst > term > subst > $o) > subst)).
thf(func_def_197, type, sK60: ((subst > term > subst > $o) > term)).
thf(func_def_198, type, sK61: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_199, type, sK62: ((subst > term > term) > subst)).
thf(func_def_200, type, sK63: ((subst > term > term) > subst)).
thf(func_def_201, type, sK64: ((subst > term > term) > term)).
thf(func_def_202, type, sK65: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_203, type, sK66: ((subst > term > term) > (subst > term > subst > $o) > term)).
thf(func_def_204, type, sK67: ((subst > term > term) > subst)).
thf(func_def_205, type, sK68: ((subst > term > term) > subst)).
thf(func_def_206, type, sK69: ((subst > term > term) > term)).
thf(func_def_207, type, sK70: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_208, type, sK71: term).
thf(func_def_209, type, sK72: (subst > term > subst > $o)).
thf(func_def_210, type, sK73: ((subst > term > term) > term)).
thf(func_def_211, type, sK74: ((subst > term > term) > subst)).
thf(func_def_212, type, sK75: ((subst > term > term) > term)).
thf(func_def_213, type, sK76: ((subst > term > term) > subst)).
thf(func_def_214, type, sK77: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_215, type, sK78: ((subst > term > subst > $o) > term)).
thf(func_def_216, type, sK79: ((subst > term > subst > $o) > subst)).
thf(func_def_217, type, sK80: ((subst > term > subst > $o) > term)).
thf(func_def_218, type, sK81: ((subst > term > subst > $o) > subst)).
thf(func_def_219, type, sK82: ((subst > term > subst > $o) > subst)).
thf(func_def_220, type, sK83: ((subst > term > subst > $o) > subst)).
thf(func_def_221, type, sK84: ((subst > term > subst > $o) > subst)).
thf(func_def_222, type, sK85: ((subst > term > subst > $o) > subst)).
thf(func_def_223, type, sK86: ((subst > term > subst > $o) > term)).
thf(func_def_224, type, sK87: ((subst > term > subst > $o) > term)).
thf(func_def_225, type, sK88: ((subst > term > subst > $o) > term)).
thf(func_def_226, type, sK89: ((term > $o) > term)).
thf(func_def_227, type, sK90: ((term > $o) > term)).
thf(func_def_228, type, sK91: ((term > $o) > term)).
thf(func_def_229, type, sK92: ((term > $o) > term)).
thf(func_def_230, type, sK93: term).
thf(func_def_231, type, sK94: subst).
thf(func_def_232, type, sK95: (term > $o)).
thf(func_def_233, type, sK96: (term > term)).
thf(func_def_234, type, sK97: term).
thf(func_def_235, type, sK98: (term > $o)).
thf(func_def_236, type, sK99: term).
thf(func_def_237, type, sK100: subst).
thf(func_def_238, type, sK101: subst).
thf(func_def_239, type, sK102: term).
thf(func_def_240, type, sK103: subst).
thf(func_def_241, type, sK104: ((term > $o) > term)).
thf(func_def_242, type, sK105: ((term > $o) > subst > term)).
thf(func_def_243, type, sK106: ((term > $o) > term)).
thf(func_def_244, type, sK107: ((term > $o) > term)).
thf(func_def_245, type, sK108: term).
thf(func_def_246, type, sK109: (subst > term > subst > $o)).
thf(func_def_247, type, sK110: ((subst > term > term) > term)).
thf(func_def_248, type, sK111: ((subst > term > term) > term)).
thf(func_def_249, type, sK112: ((subst > term > term) > subst)).
thf(func_def_250, type, sK113: ((subst > term > term) > subst)).
thf(func_def_251, type, sK114: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_252, type, sK115: ((subst > term > subst > $o) > term)).
thf(func_def_253, type, sK116: ((subst > term > subst > $o) > term)).
thf(func_def_254, type, sK117: ((term > $o) > subst > term)).
thf(func_def_255, type, sK118: ((term > $o) > term)).
thf(func_def_256, type, sK119: ((term > $o) > term)).
thf(func_def_257, type, sK120: ((term > $o) > term)).
thf(func_def_258, type, sK121: (term > $o)).
thf(func_def_259, type, sK122: term).
thf(func_def_260, type, sK123: (term > term)).
thf(func_def_261, type, sK124: subst).
thf(func_def_262, type, sK125: (term > $o)).
thf(func_def_263, type, sK126: term).
thf(func_def_264, type, sK127: (term > term)).
thf(func_def_265, type, sK128: ((term > $o) > term)).
thf(func_def_266, type, sK129: ((term > $o) > term)).
thf(func_def_267, type, sK130: ((term > $o) > term)).
thf(func_def_268, type, sK131: ((term > $o) > term)).
thf(func_def_269, type, sK132: term).
thf(func_def_270, type, sK133: (subst > term > subst > $o)).
thf(func_def_271, type, sK134: term).
thf(func_def_272, type, sK135: ((subst > term > term) > term)).
thf(func_def_273, type, sK136: ((subst > term > term) > subst)).
thf(func_def_274, type, sK137: ((subst > term > term) > subst)).
thf(func_def_275, type, sK138: ((subst > term > term) > term)).
thf(func_def_276, type, sK139: (term > (subst > term > subst > $o) > term)).
thf(func_def_277, type, sK140: ((subst > term > subst > $o) > subst)).
thf(func_def_278, type, sK141: ((subst > term > subst > $o) > term)).
thf(func_def_279, type, sK142: ((subst > term > subst > $o) > subst)).
thf(func_def_280, type, sK143: ((subst > term > subst > $o) > subst)).
thf(func_def_281, type, sK144: ((subst > term > subst > $o) > subst)).
thf(func_def_282, type, sK145: ((subst > term > subst > $o) > subst)).
thf(func_def_283, type, sK146: ((subst > term > subst > $o) > term)).
thf(func_def_284, type, sK147: ((subst > term > subst > $o) > subst)).
thf(func_def_285, type, sK148: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_286, type, sK149: (term > (subst > term > subst > $o) > term)).
thf(func_def_287, type, sK150: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_288, type, sK151: (subst > term > subst > $o)).
thf(func_def_289, type, sK152: term).
thf(func_def_290, type, sK153: ((subst > term > term) > subst)).
thf(func_def_291, type, sK154: ((subst > term > term) > term)).
thf(func_def_292, type, sK155: ((subst > term > term) > subst)).
thf(func_def_293, type, sK156: ((subst > term > term) > term)).
thf(func_def_294, type, sK157: term).
thf(func_def_295, type, sK158: subst).
thf(func_def_296, type, sK159: (term > $o)).
thf(func_def_297, type, sK160: (term > term)).
thf(func_def_298, type, sK161: ((term > $o) > term)).
thf(func_def_299, type, sK162: ((term > $o) > term)).
thf(func_def_300, type, sK163: ((term > $o) > term)).
thf(func_def_301, type, sK164: ((term > $o) > subst > term)).
thf(func_def_302, type, sK165: (subst > term > subst > $o)).
thf(func_def_303, type, sK166: (term > $o)).
thf(func_def_304, type, sK167: term).
thf(func_def_305, type, sK168: ((subst > term > term) > term)).
thf(func_def_306, type, sK169: ((subst > term > term) > subst)).
thf(func_def_307, type, sK170: ((subst > term > term) > term)).
thf(func_def_308, type, sK171: ((subst > term > term) > subst)).
thf(func_def_309, type, sK172: (term > subst > term > term)).
thf(func_def_310, type, sK173: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_311, type, sK174: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_312, type, sK175: (term > (term > $o) > term)).
thf(func_def_313, type, sK176: term).
thf(func_def_314, type, sK177: ((subst > term > term) > term)).
thf(func_def_315, type, sK178: ((subst > term > term) > subst)).
thf(func_def_316, type, sK179: term).
thf(func_def_317, type, sK180: term).
thf(func_def_318, type, sK181: term).
thf(func_def_319, type, sK182: term).
thf(func_def_320, type, sK183: term).
thf(func_def_321, type, sK184: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_322, type, sK185: ((subst > term > subst > $o) > term > term)).
thf(func_def_323, type, sK186: term).
thf(func_def_324, type, sK187: (subst > term > subst > $o)).
thf(func_def_325, type, sK188: ((subst > term > term) > term)).
thf(func_def_326, type, sK189: ((subst > term > term) > subst)).
thf(func_def_327, type, sK190: ((subst > term > term) > subst)).
thf(func_def_328, type, sK191: ((subst > term > term) > term)).
thf(func_def_329, type, sK192: subst).
thf(func_def_330, type, sK193: term).
thf(func_def_331, type, sK194: term).
thf(func_def_332, type, sK195: (subst > term > term)).
thf(func_def_333, type, sK196: ((subst > term > term) > subst)).
thf(func_def_334, type, sK197: ((subst > term > term) > subst)).
thf(func_def_335, type, sK198: ((subst > term > term) > term)).
thf(func_def_336, type, sK199: term).
thf(func_def_337, type, sK200: term).
thf(func_def_338, type, sK201: ((subst > term > subst > $o) > subst)).
thf(func_def_339, type, sK202: ((subst > term > subst > $o) > subst)).
thf(func_def_340, type, sK203: ((subst > term > subst > $o) > term)).
thf(func_def_341, type, sK204: ((subst > term > subst > $o) > subst)).
thf(func_def_342, type, sK205: ((subst > term > subst > $o) > subst)).
thf(func_def_343, type, sK206: ((subst > term > subst > $o) > subst)).
thf(func_def_344, type, sK207: ((subst > term > subst > $o) > term)).
thf(func_def_345, type, sK208: ((subst > term > subst > $o) > subst)).
thf(func_def_346, type, sK209: ((subst > term > subst > $o) > term)).
thf(func_def_347, type, sK210: ((subst > term > subst > $o) > term)).
thf(func_def_348, type, sK211: term).
thf(func_def_349, type, sK212: term).
thf(func_def_350, type, sK213: (subst > term > subst > $o)).
thf(func_def_351, type, sK214: term).
thf(func_def_352, type, sK215: term).
thf(func_def_353, type, sK216: term).
thf(func_def_354, type, sK217: term).
thf(func_def_355, type, sK218: subst).
thf(func_def_356, type, sK219: term).
thf(func_def_357, type, sK220: term).
thf(func_def_358, type, sK221: (subst > term > subst > $o)).
thf(func_def_359, type, sK222: ((subst > term > subst > $o) > term)).
thf(func_def_360, type, sK223: ((subst > term > subst > $o) > term)).
thf(func_def_361, type, sK224: ((subst > term > subst > $o) > subst)).
thf(func_def_362, type, sK225: ((subst > term > subst > $o) > subst)).
thf(func_def_363, type, sK226: ((subst > term > subst > $o) > term)).
thf(func_def_364, type, sK227: ((subst > term > subst > $o) > subst)).
thf(func_def_365, type, sK228: ((subst > term > subst > $o) > term)).
thf(func_def_366, type, sK229: ((subst > term > subst > $o) > subst)).
thf(func_def_367, type, sK230: ((subst > term > subst > $o) > subst)).
thf(func_def_368, type, sK231: ((subst > term > subst > $o) > subst)).
thf(func_def_369, type, sK232: ((subst > term > subst > $o) > term)).
thf(func_def_370, type, sK233: subst).
thf(func_def_371, type, sK234: term).
thf(func_def_372, type, sK235: term).
thf(func_def_373, type, sK236: term).
thf(func_def_374, type, sK237: term).
thf(func_def_375, type, sK238: term).
thf(func_def_376, type, sK239: term).
thf(func_def_377, type, sK240: term).
thf(func_def_378, type, sK241: term).
thf(func_def_379, type, sK242: (term > subst > term > term)).
thf(func_def_380, type, sK243: term).
thf(func_def_381, type, sK244: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_382, type, sK245: (term > (subst > term > subst > $o) > term)).
thf(func_def_383, type, sK246: (subst > term > subst > $o)).
thf(func_def_384, type, sK247: term).
thf(func_def_385, type, sK248: ((subst > term > term) > subst)).
thf(func_def_386, type, sK249: ((subst > term > term) > subst)).
thf(func_def_387, type, sK250: ((subst > term > term) > term)).
thf(func_def_388, type, sK251: ((subst > term > term) > term)).
thf(func_def_389, type, sK252: (subst > term > subst > $o)).
thf(func_def_390, type, sK253: term).
thf(func_def_391, type, sK254: ((subst > term > term) > term)).
thf(func_def_392, type, sK255: ((subst > term > term) > subst)).
thf(func_def_393, type, sK256: ((subst > term > term) > subst)).
thf(func_def_394, type, sK257: ((subst > term > term) > term)).
thf(func_def_395, type, sK258: term).
thf(func_def_396, type, sK259: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_397, type, sK260: (term > (subst > term > subst > $o) > term)).
thf(func_def_398, type, sK261: (subst > d_subst)).
thf(func_def_399, type, sK262: (term > $o)).
thf(func_def_400, type, sK263: term).
thf(func_def_401, type, sK264: (subst > term > subst > $o)).
thf(func_def_402, type, sK265: ((subst > term > term) > term)).
thf(func_def_403, type, sK266: ((subst > term > term) > subst)).
thf(func_def_404, type, sK267: ((subst > term > term) > subst)).
thf(func_def_405, type, sK268: ((subst > term > term) > term)).
thf(func_def_406, type, sK269: ((subst > term > subst > $o) > (term > $o) > term)).
thf(func_def_407, type, sK270: (term > (term > $o) > term)).
thf(func_def_408, type, sK271: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_409, type, sK272: (term > d_term)).
thf(func_def_410, type, sK273: (subst > term > subst > $o)).
thf(func_def_411, type, sK274: term).
thf(func_def_412, type, sK275: ((subst > term > term) > subst)).
thf(func_def_413, type, sK276: ((subst > term > term) > subst)).
thf(func_def_414, type, sK277: ((subst > term > term) > term)).
thf(func_def_415, type, sK278: ((subst > term > term) > term)).
thf(func_def_416, type, sK279: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_417, type, sK280: ((subst > term > subst > $o) > term)).
thf(func_def_418, type, sK281: ((subst > term > subst > $o) > term)).
thf(func_def_419, type, sK282: ((subst > term > subst > $o) > term)).
thf(func_def_420, type, sK283: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_421, type, sK284: ((subst > term > subst > $o) > term > term)).
thf(func_def_422, type, sK285: term).
thf(func_def_423, type, sK286: (subst > term > subst > $o)).
thf(func_def_424, type, sK287: ((subst > term > term) > term > subst)).
thf(func_def_425, type, sK288: ((subst > term > term) > term > subst)).
thf(func_def_426, type, sK289: ((subst > term > term) > term > term)).
thf(func_def_427, type, sK290: ((subst > term > term) > term > subst)).
thf(func_def_428, type, sK291: ((subst > term > term) > term > term)).
thf(func_def_429, type, sK292: ((subst > term > term) > term > subst)).
thf(func_def_430, type, sK293: ((subst > term > term) > term)).
thf(func_def_431, type, sK294: subst).
thf(func_def_432, type, sK295: (subst > term > term)).
thf(func_def_433, type, sK296: term).
thf(func_def_434, type, sK297: (subst > term > term)).
thf(func_def_435, type, sK298: ((subst > term > term) > subst)).
thf(func_def_436, type, sK299: ((subst > term > term) > term)).
thf(func_def_437, type, sK300: ((subst > term > term) > subst)).
thf(func_def_438, type, sK301: ((subst > term > term) > subst)).
thf(func_def_439, type, sK302: ((subst > term > term) > subst)).
thf(func_def_440, type, sK303: ((subst > term > term) > term)).
thf(func_def_441, type, sK304: term).
thf(func_def_442, type, sK305: (subst > term > subst > $o)).
thf(func_def_443, type, sK306: ((term > $o) > term)).
thf(func_def_444, type, sK307: ((subst > term > subst > $o) > term > $o)).
thf(func_def_445, type, sK308: (subst > term > term)).
thf(func_def_446, type, sK309: (subst > term > term)).
thf(func_def_447, type, sK310: term).
thf(func_def_448, type, sK311: subst).
thf(func_def_449, type, sK312: subst).
thf(func_def_450, type, sK313: term).
thf(func_def_451, type, sK314: subst).
thf(func_def_452, type, sK315: term).
thf(func_def_453, type, sK316: ((subst > term > term) > term)).
thf(func_def_454, type, sK317: ((subst > term > term) > subst)).
thf(func_def_455, type, sK318: ((subst > term > term) > subst)).
thf(func_def_456, type, sK319: ((subst > term > term) > subst)).
thf(func_def_457, type, sK320: ((subst > term > term) > term)).
thf(func_def_458, type, sK321: ((subst > term > term) > subst)).
thf(func_def_459, type, sK322: term).
thf(func_def_460, type, sK323: term).
thf(func_def_461, type, sK324: ((subst > term > subst > $o) > term > term)).
thf(func_def_462, type, sK325: ((subst > term > subst > $o) > subst)).
thf(func_def_463, type, sK326: ((subst > term > subst > $o) > subst)).
thf(func_def_464, type, sK327: ((subst > term > subst > $o) > subst)).
thf(func_def_465, type, sK328: ((subst > term > subst > $o) > term)).
thf(func_def_466, type, sK329: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_467, type, sK330: ((subst > term > subst > $o) > subst)).
thf(func_def_468, type, sK331: ((subst > term > subst > $o) > term)).
thf(func_def_469, type, sK332: ((subst > term > subst > $o) > subst)).
thf(func_def_470, type, sK333: ((subst > term > subst > $o) > subst)).
thf(func_def_471, type, sK334: term).
thf(func_def_472, type, sK335: (subst > term > subst > $o)).
thf(func_def_473, type, sK336: ((subst > term > term) > subst)).
thf(func_def_474, type, sK337: ((subst > term > term) > term)).
thf(func_def_475, type, sK338: ((subst > term > term) > subst)).
thf(func_def_476, type, sK339: ((subst > term > term) > term)).
thf(func_def_477, type, sK340: term).
thf(func_def_478, type, sK341: term).
thf(func_def_479, type, sK342: subst).
thf(func_def_480, type, sK343: term).
thf(func_def_481, type, sK344: term).
thf(func_def_482, type, sK345: (subst > term > subst > $o)).
thf(func_def_483, type, sK346: ((term > $o) > term)).
thf(func_def_484, type, sK347: term).
thf(func_def_485, type, sK348: (term > $o)).
thf(func_def_486, type, sK349: (subst > term > subst > $o)).
thf(func_def_487, type, sK350: ((subst > term > term) > subst)).
thf(func_def_488, type, sK351: ((subst > term > term) > subst)).
thf(func_def_489, type, sK352: ((subst > term > term) > term)).
thf(func_def_490, type, sK353: ((subst > term > term) > term)).
thf(func_def_491, type, sK354: term).
thf(func_def_492, type, sK355: (term > $o)).
thf(func_def_493, type, sK356: (term > term)).
thf(func_def_494, type, sK357: ((subst > term > subst > $o) > term > $o)).
thf(func_def_495, type, sK358: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_496, type, sK359: ((term > $o) > term > term)).
thf(func_def_497, type, sK360: ((subst > term > subst > $o) > (term > $o) > term)).
thf(func_def_498, type, sK361: ((term > $o) > term)).
thf(func_def_499, type, sK362: ((term > $o) > term)).
thf(func_def_500, type, sK363: ((term > $o) > term)).
thf(func_def_501, type, sK364: ((term > $o) > term)).
thf(func_def_502, type, sK365: (subst > term > subst > $o)).
thf(func_def_503, type, sK366: term).
thf(func_def_504, type, sK367: ((subst > term > term) > term)).
thf(func_def_505, type, sK368: ((subst > term > term) > subst)).
thf(func_def_506, type, sK369: ((subst > term > term) > subst)).
thf(func_def_507, type, sK370: ((subst > term > term) > term)).
thf(func_def_508, type, sK371: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_509, type, sK372: (term > (subst > term > subst > $o) > term)).
thf(func_def_510, type, sK373: term).
thf(func_def_511, type, sK374: term).
thf(func_def_512, type, sK375: term).
thf(func_def_513, type, sK376: ((subst > term > term) > subst)).
thf(func_def_514, type, sK377: ((subst > term > term) > term)).
thf(func_def_515, type, sK378: (term > subst > term > term)).
thf(func_def_516, type, sK379: subst).
thf(func_def_517, type, sK380: subst).
thf(func_def_518, type, sK381: term).
thf(func_def_519, type, sK382: (term > subst > term > term)).
thf(func_def_520, type, sK383: (term > subst > (term > $o) > (term > $o) > term)).
thf(func_def_521, type, sK384: subst).
thf(func_def_522, type, sK385: term).
thf(func_def_523, type, sK386: (term > $o)).
thf(func_def_524, type, sK387: (term > $o)).
thf(func_def_525, type, sK388: subst).
thf(func_def_526, type, sK389: term).
thf(func_def_527, type, sK390: subst).
thf(func_def_528, type, sK391: subst).
thf(func_def_529, type, sK392: subst).
thf(func_def_530, type, sK393: subst).
thf(func_def_531, type, sK394: term).
thf(func_def_532, type, sK395: term).
thf(func_def_533, type, sK396: term).
thf(func_def_534, type, sK397: term).
thf(func_def_535, type, sK398: term).
thf(func_def_536, type, sK399: term).
thf(func_def_537, type, sK400: term).
thf(func_def_538, type, sK401: term).
thf(func_def_539, type, sK402: term).
thf(func_def_540, type, sK403: subst).
thf(func_def_541, type, sK404: (term > $o)).
thf(func_def_542, type, sK405: term).
thf(func_def_543, type, sK406: ((term > $o) > term)).
thf(func_def_544, type, sK407: (term > (term > $o) > subst > term > $o)).
thf(func_def_545, type, sK408: ((subst > term > subst > $o) > (term > $o) > term)).
thf(func_def_546, type, sK409: (term > (term > $o) > term)).
thf(func_def_547, type, sK410: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_548, type, sK411: ((term > $o) > term)).
thf(func_def_549, type, sK412: ((term > $o) > term)).
thf(func_def_550, type, sK413: ((term > $o) > term)).
thf(func_def_551, type, sK414: ((term > $o) > term)).
thf(func_def_552, type, sK415: ((subst > term > subst > $o) > term > $o)).
thf(func_def_553, type, sK416: term).
thf(func_def_554, type, sK417: (term > $o)).
thf(func_def_555, type, sK418: (subst > term > subst > $o)).
thf(func_def_556, type, sK419: term).
thf(func_def_557, type, sK420: ((subst > term > term) > term)).
thf(func_def_558, type, sK421: ((subst > term > term) > term)).
thf(func_def_559, type, sK422: ((subst > term > term) > subst)).
thf(func_def_560, type, sK423: ((subst > term > term) > subst)).
thf(func_def_561, type, sK424: (term > $o)).
thf(func_def_562, type, sK425: term).
thf(func_def_563, type, sK426: (term > term)).
thf(func_def_564, type, sK427: (subst > term > subst > $o)).
thf(func_def_565, type, sK428: ((term > $o) > term)).
thf(func_def_566, type, sK429: (term > $o)).
thf(func_def_567, type, sK430: (term > $o)).
thf(func_def_568, type, sK431: term).
thf(func_def_569, type, sK432: subst).
thf(func_def_570, type, sK433: (subst > term > (term > $o) > (term > $o) > term)).
thf(func_def_571, type, sK434: subst).
thf(func_def_572, type, sK435: term).
thf(func_def_573, type, sK436: subst).
thf(func_def_574, type, sK437: term).
thf(func_def_575, type, sK438: term).
thf(func_def_576, type, sK439: subst).
thf(func_def_577, type, sK440: term).
thf(func_def_578, type, sK441: (term > $o)).
thf(func_def_579, type, sK442: (term > $o)).
thf(func_def_580, type, sK443: ((term > $o) > term)).
thf(func_def_581, type, sK444: ((term > $o) > term)).
thf(func_def_582, type, sK445: ((term > $o) > term)).
thf(func_def_583, type, sK446: ((term > $o) > term)).
thf(func_def_584, type, sK447: (term > $o)).
thf(func_def_585, type, sK448: term).
thf(func_def_586, type, sK449: (term > term)).
thf(func_def_587, type, sK450: (term > $o)).
thf(func_def_588, type, sK451: term).
thf(func_def_589, type, sK452: subst).
thf(func_def_590, type, sK453: term).
thf(func_def_591, type, sK454: (term > $o)).
thf(func_def_592, type, sK455: (subst > term > subst > $o)).
thf(func_def_593, type, sK456: ((term > $o) > term)).
thf(func_def_594, type, sK457: ((subst > term > subst > $o) > term > $o)).
thf(func_def_595, type, sK458: term).
thf(func_def_596, type, sK459: term).
thf(func_def_597, type, sK460: term).
thf(func_def_598, type, sK461: term).
thf(func_def_599, type, sK462: ((term > $o) > term)).
thf(func_def_600, type, sK463: ((term > $o) > term)).
thf(func_def_601, type, sK464: ((term > $o) > term)).
thf(func_def_602, type, sK465: ((term > $o) > term)).
thf(func_def_603, type, sK466: (term > $o)).
thf(func_def_604, type, sK467: term).
thf(func_def_605, type, sK468: (subst > term > subst > $o)).
thf(func_def_606, type, sK469: term).
thf(func_def_607, type, sK470: (term > $o)).
thf(func_def_608, type, sK471: term).
thf(func_def_609, type, sK472: ((subst > term > subst > $o) > subst)).
thf(func_def_610, type, sK473: ((subst > term > subst > $o) > term)).
thf(func_def_611, type, sK474: ((subst > term > subst > $o) > subst)).
thf(func_def_612, type, sK475: ((subst > term > subst > $o) > subst)).
thf(func_def_613, type, sK476: ((subst > term > subst > $o) > term)).
thf(func_def_614, type, sK477: ((subst > term > subst > $o) > term)).
thf(func_def_615, type, sK478: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_616, type, sK479: ((subst > term > subst > $o) > subst)).
thf(func_def_617, type, sK480: ((subst > term > subst > $o) > subst)).
thf(func_def_618, type, sK481: ((subst > term > subst > $o) > subst)).
thf(func_def_619, type, sK482: ((subst > term > subst > $o) > term)).
thf(func_def_620, type, sK483: (term > subst > term > term)).
thf(func_def_621, type, sK484: term).
thf(func_def_622, type, sK485: ((subst > term > term) > term)).
thf(func_def_623, type, sK486: ((subst > term > term) > subst)).
thf(func_def_624, type, sK487: (subst > term > subst > $o)).
thf(func_def_625, type, sK488: term).
thf(func_def_626, type, sK489: (term > $o)).
thf(func_def_627, type, sK490: ((subst > term > term) > subst)).
thf(func_def_628, type, sK491: ((subst > term > term) > subst)).
thf(func_def_629, type, sK492: ((subst > term > term) > term)).
thf(func_def_630, type, sK493: ((subst > term > term) > term)).
thf(func_def_631, type, sK494: term).
thf(func_def_632, type, sK495: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_633, type, sK496: ((subst > term > subst > $o) > subst)).
thf(func_def_634, type, sK497: ((subst > term > subst > $o) > term)).
thf(func_def_635, type, sK498: ((subst > term > subst > $o) > subst)).
thf(func_def_636, type, sK499: ((subst > term > subst > $o) > subst)).
thf(func_def_637, type, sK500: ((subst > term > subst > $o) > term)).
thf(func_def_638, type, sK501: ((subst > term > subst > $o) > subst)).
thf(func_def_639, type, sK502: ((subst > term > subst > $o) > subst)).
thf(func_def_640, type, sK503: ((subst > term > subst > $o) > subst)).
thf(func_def_641, type, sK504: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_642, type, sK505: ((term > $o) > term > term)).
thf(func_def_643, type, sK506: term).
thf(func_def_644, type, sK507: term).
thf(func_def_645, type, sK508: term).
thf(func_def_646, type, sK509: term).
thf(func_def_647, type, sK510: (term > subst > (term > $o) > term > $o)).
thf(func_def_648, type, sK511: (term > $o)).
thf(func_def_649, type, sK512: subst).
thf(func_def_650, type, sK513: term).
thf(func_def_651, type, sK514: ((term > $o) > term)).
thf(func_def_652, type, sK515: ((term > $o) > term > term)).
thf(func_def_653, type, sK516: ((term > $o) > (subst > term > subst > $o) > term)).
thf(func_def_654, type, sK517: ((subst > term > subst > $o) > subst)).
thf(func_def_655, type, sK518: ((subst > term > subst > $o) > subst)).
thf(func_def_656, type, sK519: ((subst > term > subst > $o) > term)).
thf(func_def_657, type, sK520: ((subst > term > subst > $o) > subst)).
thf(func_def_658, type, sK521: ((subst > term > subst > $o) > term)).
thf(func_def_659, type, sK522: ((subst > term > subst > $o) > subst)).
thf(func_def_660, type, sK523: ((subst > term > subst > $o) > subst)).
thf(func_def_661, type, sK524: ((subst > term > subst > $o) > subst)).
thf(func_def_662, type, sK525: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_663, type, sK526: (subst > term > subst > $o)).
thf(func_def_664, type, sK527: term).
thf(func_def_665, type, sK528: (term > $o)).
thf(func_def_666, type, sK529: ((subst > term > term) > term)).
thf(func_def_667, type, sK530: ((subst > term > term) > subst)).
thf(func_def_668, type, sK531: ((subst > term > term) > subst)).
thf(func_def_669, type, sK532: ((subst > term > term) > term)).
thf(func_def_670, type, sK533: term).
thf(func_def_671, type, sK534: subst).
thf(func_def_672, type, sK535: (term > (subst > term > subst > $o) > term)).
thf(func_def_673, type, sK536: ((subst > term > subst > $o) > subst > term > term)).
thf(func_def_674, type, sK537: term).
thf(func_def_675, type, sK538: (subst > term > subst > $o)).
thf(func_def_676, type, sK539: term).
thf(func_def_677, type, sK540: ((subst > term > term) > subst)).
thf(func_def_678, type, sK541: ((subst > term > term) > term)).
thf(func_def_679, type, sK542: ((subst > term > term) > subst)).
thf(func_def_680, type, sK543: ((subst > term > term) > term)).
thf(func_def_681, type, vNOT: ($o > $o)).
thf(f1,conjecture,(
  (! [X0 : term] : ~(var @ (lam @ X0)) = lamnotvar)),
  file('/export/starexec/sandbox2/benchmark/theBenchmark.p',lamnotvar)).
thf(f2,negated_conjecture,(
  ~ (! [X0 : term] : ~(var @ (lam @ X0)) = lamnotvar)),
  inference(negated_conjecture,[status(cth)],[f1])).
thf(f3,axiom,(
  (! [X6 : term] : (X6 = ((sub @ X6 @ id))) <=> axvarid) & axapp & termmset_gthm & (! [X6 : term,X14 : term,X11 : term,X13 : term] : ((X13 = X14) | (((ap @ X6 @ X13)) != ((ap @ X11 @ X14)))) <=> apinj2) & axclos & hoasinduction_lem3b_gthm & hoasinduction_lem2_gthm & (! [X8 : (term > $o),X6 : term,X7 : subst] : ~ ! [X9 : (term > $o)] : ~ ! [X10 : term] : (((X9 @ X10)) = ((X8 @ (sub @ X10 @ (push @ X6 @ X7))))) <=> pushprop_lem0_lthm) & (hoasinduction_lem0_lthm <=> ! [X8 : (subst > term > subst > $o)] : ~ ! [X9 : (term > $o)] : ~ ! [X10 : term] : (((X9 @ X10)) = ((X8 @ id @ X10 @ id)))) & pushprop_lem2v2_gthm & ((! [X11 : term] : ~ ! [X12 : (subst > term > term)] : ~ ! [X6 : term,X7 : subst] : (((sub @ X11 @ (push @ X6 @ X7))) = ((X12 @ X7 @ X6))) => (! [X6 : term] : (X6 = ((sub @ X6 @ id))) => ! [X25 : term,X9 : (term > $o),X8 : (subst > term > subst > $o)] : (~ ! [X10 : term] : (((X9 @ X10)) = ((X8 @ id @ X10 @ id))) | ~ ! [X11 : term] : ((X9 @ (sub @ X25 @ (push @ X11 @ id))) | ~(X9 @ X11)) | (X9 @ (lam @ X25)) | ~ ! [X12 : (subst > term > term)] : ((X8 @ id @ (lam @ (X12 @ sh @ one)) @ id) | ~ ! [X6 : term] : ((X8 @ id @ (X12 @ id @ X6) @ id) | ~(X8 @ id @ X6 @ id)) | ~ ! [X7 : subst,X6 : term,X20 : subst] : (((sub @ (X12 @ X7 @ X6) @ X20)) = ((X12 @ (comp @ X7 @ X20) @ (sub @ X6 @ X20)))))))) <=> hoasinduction_lem3v2a_lthm) & (! [X6 : term,X14 : term,X11 : term,X13 : term] : ((X6 = X11) | (((ap @ (sub @ X6 @ id) @ X13)) != ((ap @ (sub @ X11 @ id) @ X14)))) <=> hoasapinj1) & axmap & induction2_gthm & ((! [X7 : subst,X6 : term] : (X6 = ((sub @ one @ (push @ X6 @ X7)))) => (! [X6 : term,X7 : subst] : (X7 = ((comp @ sh @ (push @ X6 @ X7)))) => (! [X11 : term,X6 : term] : ((((lam @ X6)) != ((lam @ X11))) | (X6 = X11)) => ! [X40 : subst,X12 : (subst > term > term),X39 : (subst > term > term),X41 : term] : (~ ! [X20 : subst,X7 : subst,X6 : term] : (((sub @ (X12 @ X7 @ X6) @ X20)) = ((X12 @ (comp @ X7 @ X20) @ (sub @ X6 @ X20)))) | (((lam @ (X12 @ sh @ one))) != ((lam @ (X39 @ sh @ one)))) | (((X12 @ X40 @ X41)) = ((X39 @ X40 @ X41))) | ~ ! [X7 : subst,X6 : term,X20 : subst] : (((sub @ (X39 @ X7 @ X6) @ X20)) = ((X39 @ (comp @ X7 @ X20) @ (sub @ X6 @ X20)))))))) <=> hoaslaminj_lthm) & hoasinduction_lem1v2_gthm & ! [X3 : d_subst] : (X3 = d_id) & hoaslaminj_gthm & (termmset_lthm <=> (! [X6 : term] : (X6 = ((sub @ X6 @ id))) => ! [X6 : term] : (X6 = ((sub @ X6 @ id))))) & ! [X47 : d_term,X48 : d_term] : ((((d2term @ X47)) = ((d2term @ X48))) => (X47 = X48)) & (hoasinduction_lem3a <=> ! [X35 : term,X8 : (subst > term > subst > $o)] : (~ ! [X12 : (subst > term > term)] : ((X8 @ id @ (lam @ (X12 @ sh @ one)) @ id) | ~ ! [X20 : subst,X7 : subst,X6 : term] : (((sub @ (X12 @ X7 @ X6) @ X20)) = ((X12 @ (comp @ X7 @ X20) @ (sub @ X6 @ X20)))) | ~ ! [X6 : term] : (~(X8 @ id @ X6 @ id) | (X8 @ id @ (X12 @ id @ X6) @ id))) | ~ ! [X11 : term] : (~(X8 @ id @ X11 @ id) | (X8 @ id @ (sub @ X35 @ (push @ X11 @ id)) @ id)) | (X8 @ id @ (lam @ X35) @ id))) & ! [X0 : term] : ? [X1 : d_term] : (X0 = ((d2term @ X1))) & (((comp @ (d2subst @ d_id) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & hoasinduction_gthm & (axvarcons <=> ! [X6 : term,X7 : subst] : (X6 = ((sub @ one @ (push @ X6 @ X7))))) & (! [X8 : (term > $o),X17 : term] : ((X8 @ X17) | ~ ! [X6 : term] : ((X8 @ X6) | ~(var @ X6)) | ~ ! [X6 : term,X11 : term] : (~(X8 @ X6) | ~(X8 @ X11) | (X8 @ (ap @ X6 @ X11))) | ~ ! [X6 : term] : ((X8 @ (lam @ X6)) | ~(X8 @ X6))) <=> induction) & axassoc & hoaslamnotap_gthm & (termmset <=> ! [X6 : term] : (X6 = ((sub @ X6 @ id)))) & pushprop_lthm & (! [X8 : (subst > term > subst > $o),X22 : term] : (~ ! [X11 : term] : (~(X8 @ id @ X11 @ id) | (X8 @ id @ (sub @ X22 @ (push @ X11 @ id)) @ id)) | ~ ! [X7 : subst,X20 : subst,X6 : term,X21 : subst] : ((X8 @ X7 @ X6 @ (comp @ X21 @ X20)) | ~(X8 @ (comp @ X7 @ X21) @ (sub @ X6 @ X21) @ X20)) | (X8 @ id @ (lam @ X22) @ id) | ~ ! [X12 : (subst > term > term)] : (~ ! [X6 : term] : ((X8 @ id @ (X12 @ id @ X6) @ id) | ~(X8 @ id @ X6 @ id)) | (X8 @ id @ (lam @ (X12 @ sh @ one)) @ id) | ~ ! [X6 : term,X20 : subst,X7 : subst] : (((sub @ (X12 @ X7 @ X6) @ X20)) = ((X12 @ (comp @ X7 @ X20) @ (sub @ X6 @ X20))))) | ~ ! [X21 : subst,X7 : subst,X20 : subst,X6 : term] : ((X8 @ (comp @ X7 @ X21) @ (sub @ X6 @ X21) @ X20) | ~(X8 @ X7 @ X6 @ (comp @ X21 @ X20)))) <=> hoasinduction_lem3) & hoasinduction_lem1_lthm & (substmonoid_lthm <=> (! [X7 : subst] : (X7 = ((comp @ id @ X7))) => (! [X7 : subst] : (X7 = ((comp @ X7 @ id))) => (! [X7 : subst] : (X7 = ((comp @ X7 @ id))) & ! [X7 : subst] : (X7 = ((comp @ id @ X7))))))) & (id = ((d2subst @ d_id))) & pushprop_lem3v2 & hoasinduction_lem1 & (((sub @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2term @ d_one))) & (hoaslamnotap <=> ! [X44 : term,X12 : (subst > term > term),X43 : term] : (~ ! [X7 : subst,X6 : term,X20 : subst] : (((sub @ (X12 @ X7 @ X6) @ X20)) = ((X12 @ (comp @ X7 @ X20) @ (sub @ X6 @ X20)))) | (((lam @ (X12 @ sh @ one))) != ((ap @ (sub @ X43 @ id) @ X44))))) & (! [X6 : term,X11 : term] : ((X6 = X11) | (((lam @ X6)) != ((lam @ X11)))) <=> laminj) & (sh = ((d2subst @ d_id))) & (hoasinduction_lem3v2_f <=> ! [X11 : term] : ~ ! [X12 : (subst > term > term)] : ~ ! [X7 : subst,X6 : term] : (((sub @ X11 @ (push @ X6 @ X7))) = ((X12 @ X7 @ X6)))) & hoasinduction_lem3_gthm & ((! [X7 : subst,X6 : term] : (X6 = ((sub @ one @ (push @ X6 @ X7)))) => ! [X9 : (term > $o),X6 : term,X8 : (term > $o),X7 : subst] : (~ ! [X10 : term] : (((X9 @ X10)) = ((X8 @ (sub @ X10 @ (push @ X6 @ X7))))) | ~(X8 @ X6) | (X9 @ one))) <=> pushprop_lem1v2_lthm) & hoaslamnotvar_gthm & induction2lem_gthm & ! [X1 : d_term] : (X1 = d_one) & (one = ((d2term @ d_one))) & hoasapnotvar_gthm & (! [X6 : term,X7 : subst] : (X7 = ((comp @ sh @ (push @ X6 @ X7)))) <=> axshiftcons) & (hoasinduction_lem3aaa <=> ! [X8 : (subst > term > subst > $o),X27 : term] : (~ ! [X11 : term] : (~(X8 @ id @ X11 @ id) | (X8 @ id @ (sub @ X27 @ (push @ X11 @ id)) @ id)) | ~ ! [X12 : (subst > term > term),X28 : term] : ((X8 @ id @ (lam @ (X12 @ sh @ one)) @ id) | ~ ! [X32 : subst,X33 : term,X34 : subst] : (((X12 @ (comp @ X32 @ X34) @ (sub @ X33 @ X34))) = ((sub @ X28 @ (push @ (sub @ X33 @ X34) @ (comp @ X32 @ X34))))) | ~ ! [X29 : subst,X30 : term,X31 : subst] : (((sub @ (X12 @ X29 @ X30) @ X31)) = ((sub @ (sub @ X28 @ (push @ X30 @ X29)) @ X31))) | ~ ! [X6 : term] : ((X8 @ id @ (X12 @ id @ X6) @ id) | ~(X8 @ id @ X6 @ id))) | (X8 @ id @ (lam @ (sub @ X27 @ (push @ one @ sh))) @ id))) & pushprop_lem2v2_lthm & (hoasinduction_lem0 <=> ! [X8 : (subst > term > subst > $o)] : ~ ! [X9 : (term > $o)] : ~ ! [X10 : term] : (((X9 @ X10)) = ((X8 @ id @ X10 @ id)))) & (hoasinduction_lthm <=> (! [X8 : (term > $o),X18 : term] : (~ ! [X6 : term,X11 : term] : (~(X8 @ X6) | (X8 @ (ap @ X6 @ X11)) | ~(X8 @ X11)) | ~ ! [X6 : term] : (~(var @ X6) | (X8 @ X6)) | (X8 @ X18) | ~ ! [X6 : term] : ((X8 @ (lam @ X6)) | ~ ! [X11 : term] : ((X8 @ (sub @ X6 @ (push @ X11 @ id))) | ~(X8 @ X11)))) => (! [X24 : term,X8 : (subst > term > subst > $o),X23 : term] : (~ ! [X21 : subst,X7 : subst,X6 : term,X20 : subst] : ((X8 @ (comp @ X7 @ X21) @ (sub @ X6 @ X21) @ X20) | ~(X8 @ X7 @ X6 @ (comp @ X21 @ X20))) | ~ ! [X6 : term,X11 : term] : ((X8 @ id @ (ap @ (sub @ X6 @ id) @ X11) @ id) | ~(X8 @ id @ X11 @ id) | ~(X8 @ id @ X6 @ id)) | ~(X8 @ id @ X23 @ id) | (X8 @ id @ (ap @ X23 @ X24) @ id) | ~(X8 @ id @ X24 @ id) | ~ ! [X7 : subst,X21 : subst,X6 : term,X20 : subst] : ((X8 @ X7 @ X6 @ (comp @ X21 @ X20)) | ~(X8 @ (comp @ X7 @ X21) @ (sub @ X6 @ X21) @ X20))) => (! [X8 : (subst > term > subst > $o),X22 : term] : (~ ! [X6 : term,X7 : subst,X21 : subst,X20 : subst] : ((X8 @ X7 @ X6 @ (comp @ X21 @ X20)) | ~(X8 @ (comp @ X7 @ X21) @ (sub @ X6 @ X21) @ X20)) | ~ ! [X11 : term] : (~(X8 @ id @ X11 @ id) | (X8 @ id @ (sub @ X22 @ (push @ X11 @ id)) @ id)) | (X8 @ id @ (lam @ X22) @ id) | ~ ! [X20 : subst,X7 : subst,X6 : term,X21 : subst] : (~(X8 @ X7 @ X6 @ (comp @ X21 @ X20)) | (X8 @ (comp @ X7 @ X21) @ (sub @ X6 @ X21) @ X20)) | ~ ! [X12 : (subst > term > term)] : ((X8 @ id @ (lam @ (X12 @ sh @ one)) @ id) | ~ ! [X20 : subst,X6 : term,X7 : subst] : (((sub @ (X12 @ X7 @ X6) @ X20)) = ((X12 @ (comp @ X7 @ X20) @ (sub @ X6 @ X20)))) | ~ ! [X6 : term] : ((X8 @ id @ (X12 @ id @ X6) @ id) | ~(X8 @ id @ X6 @ id)))) => ! [X8 : (subst > term > subst > $o),X19 : term] : (~ ! [X6 : term,X21 : subst,X7 : subst,X20 : subst] : (~(X8 @ X7 @ X6 @ (comp @ X21 @ X20)) | (X8 @ (comp @ X7 @ X21) @ (sub @ X6 @ X21) @ X20)) | ~ ! [X20 : subst,X6 : term,X7 : subst,X21 : subst] : ((X8 @ X7 @ X6 @ (comp @ X21 @ X20)) | ~(X8 @ (comp @ X7 @ X21) @ (sub @ X6 @ X21) @ X20)) | ~ ! [X6 : term,X11 : term] : ((X8 @ id @ (ap @ (sub @ X6 @ id) @ X11) @ id) | ~(X8 @ id @ X6 @ id) | ~(X8 @ id @ X11 @ id)) | ~ ! [X6 : term] : (~(var @ (sub @ X6 @ id)) | (X8 @ id @ X6 @ id)) | ~ ! [X12 : (subst > term > term)] : (~ ! [X7 : subst,X20 : subst,X6 : term] : (((sub @ (X12 @ X7 @ X6) @ X20)) = ((X12 @ (comp @ X7 @ X20) @ (sub @ X6 @ X20)))) | ~ ! [X6 : term] : ((X8 @ id @ (X12 @ id @ X6) @ id) | ~(X8 @ id @ X6 @ id)) | (X8 @ id @ (lam @ (X12 @ sh @ one)) @ id)) | (X8 @ id @ X19 @ id)))))) & hoasapinj1_gthm & (axscons <=> ! [X7 : subst] : (X7 = ((push @ (sub @ one @ X7) @ (comp @ sh @ X7))))) & hoaslamnotvar_lthm & (pushprop_lem1 <=> ! [X21 : (term > $o),X6 : term,X8 : (term > $o),X7 : subst,X11 : term] : (~(X8 @ X6) | (X21 @ (sub @ X6 @ (push @ X11 @ X7))))) & pushprop_lem3v2_lthm & hoasinduction_lem2v2_gthm & ((^[X6 : term, X7 : subst, X8 : (term > $o), X9 : (term > $o)] : (! [X10 : term] : (((X9 @ X10)) = ((X8 @ (sub @ X10 @ (push @ X6 @ X7))))))) = pushprop_p_and_p_prime) & hoasapnotvar & (! [X11 : term] : ~ ! [X12 : (subst > term > term)] : ~ ! [X7 : subst,X6 : term] : (((sub @ X11 @ (push @ X6 @ X7))) = ((X12 @ X7 @ X6))) <=> hoasinduction_lem3v2_f_lthm) & (! [X23 : term,X24 : term,X8 : (subst > term > subst > $o)] : (~(X8 @ id @ X23 @ id) | (X8 @ id @ (ap @ X23 @ X24) @ id) | ~(X8 @ id @ X24 @ id) | ~ ! [X6 : term,X11 : term] : ((X8 @ id @ (ap @ (sub @ X6 @ id) @ X11) @ id) | ~(X8 @ id @ X6 @ id) | ~(X8 @ id @ X11 @ id)) | ~ ! [X6 : term,X20 : subst,X21 : subst,X7 : subst] : ((X8 @ X7 @ X6 @ (comp @ X21 @ X20)) | ~(X8 @ (comp @ X7 @ X21) @ (sub @ X6 @ X21) @ X20)) | ~ ! [X6 : term,X20 : subst,X7 : subst,X21 : subst] : ((X8 @ (comp @ X7 @ X21) @ (sub @ X6 @ X21) @ X20) | ~(X8 @ X7 @ X6 @ (comp @ X21 @ X20)))) <=> hoasinduction_lem2) & ulamvarind & (hoasinduction_no_psi_cond <=> ! [X8 : (subst > term > subst > $o),X26 : term] : (~ ! [X12 : (subst > term > term)] : (~ ! [X6 : term,X20 : subst,X7 : subst] : (((sub @ (X12 @ X7 @ X6) @ X20)) = ((X12 @ (comp @ X7 @ X20) @ (sub @ X6 @ X20)))) | ~ ! [X6 : term] : (~(X8 @ id @ X6 @ id) | (X8 @ id @ (X12 @ id @ X6) @ id)) | (X8 @ id @ (lam @ (X12 @ sh @ one)) @ id)) | (X8 @ id @ X26 @ id) | ~ ! [X6 : term,X11 : term] : (~(X8 @ id @ X6 @ id) | (X8 @ id @ (ap @ (sub @ X6 @ id) @ X11) @ id) | ~(X8 @ id @ X11 @ id)))) & hoasapinj2_gthm & lamnotvar & (! [X16 : subst,X8 : (term > $o),X15 : term] : (~ ! [X11 : term] : ((X8 @ (sub @ X11 @ X16)) | ~(var @ X11)) | ~ ! [X6 : term,X11 : term] : ((X8 @ (ap @ X6 @ X11)) | ~(X8 @ X6) | ~(X8 @ X11)) | (X8 @ (sub @ X15 @ X16)) | ~ ! [X6 : term] : (~ ! [X11 : term] : (~(X8 @ X11) | (X8 @ (sub @ X6 @ (push @ X11 @ id)))) | (X8 @ (lam @ X6)))) <=> induction2lem) & ((! [X7 : subst] : (X7 = ((comp @ id @ X7))) & ! [X7 : subst] : (X7 = ((comp @ X7 @ id)))) <=> substmonoid) & (hoasinduction_lem3b_lthm <=> ! [X11 : term] : ~ ! [X12 : (subst > term > term)] : (((X12 @ sh @ one)) != ((sub @ X11 @ (push @ one @ sh))))) & (! [X9 : (term > $o),X25 : term,X8 : (subst > term > subst > $o)] : (~ ! [X12 : (subst > term > term)] : (~ ! [X7 : subst,X6 : term,X20 : subst] : (((sub @ (X12 @ X7 @ X6) @ X20)) = ((X12 @ (comp @ X7 @ X20) @ (sub @ X6 @ X20)))) | ~ ! [X6 : term] : ((X8 @ id @ (X12 @ id @ X6) @ id) | ~(X8 @ id @ X6 @ id)) | (X8 @ id @ (lam @ (X12 @ sh @ one)) @ id)) | (X9 @ (lam @ X25)) | ~ ! [X10 : term] : (((X9 @ X10)) = ((X8 @ id @ X10 @ id))) | ~ ! [X11 : term] : (~(X9 @ X11) | (X9 @ (sub @ X25 @ (push @ X11 @ id))))) <=> hoasinduction_lem3v2a) & (pushprop_lem0 <=> ! [X7 : subst,X8 : (term > $o),X6 : term] : ~ ! [X9 : (term > $o)] : ~ ! [X10 : term] : (((X9 @ X10)) = ((X8 @ (sub @ X10 @ (push @ X6 @ X7)))))) & (hoasinduction_lem3v2 <=> ! [X42 : term,X8 : (subst > term > subst > $o),X9 : (term > $o)] : (~ ! [X20 : subst,X7 : subst,X6 : term,X21 : subst] : (~(X8 @ X7 @ X6 @ (comp @ X21 @ X20)) | (X8 @ (comp @ X7 @ X21) @ (sub @ X6 @ X21) @ X20)) | ~ ! [X10 : term] : (((X9 @ X10)) = ((X8 @ id @ X10 @ id))) | ~ ! [X20 : subst,X7 : subst,X21 : subst,X6 : term] : (~(X8 @ (comp @ X7 @ X21) @ (sub @ X6 @ X21) @ X20) | (X8 @ X7 @ X6 @ (comp @ X21 @ X20))) | (X9 @ (lam @ X42)) | ~ ! [X12 : (subst > term > term)] : (~ ! [X6 : term] : (~(X8 @ id @ X6 @ id) | (X8 @ id @ (X12 @ id @ X6) @ id)) | (X8 @ id @ (lam @ (X12 @ sh @ one)) @ id) | ~ ! [X20 : subst,X7 : subst,X6 : term] : (((sub @ (X12 @ X7 @ X6) @ X20)) = ((X12 @ (comp @ X7 @ X20) @ (sub @ X6 @ X20))))) | ~ ! [X11 : term] : (~(X9 @ X11) | (X9 @ (sub @ X42 @ (push @ X11 @ id)))))) & (! [X36 : term,X8 : (subst > term > subst > $o)] : (~ ! [X12 : (subst > term > term)] : (~ ! [X20 : subst,X6 : term,X7 : subst] : (((sub @ (X12 @ X7 @ X6) @ X20)) = ((X12 @ (comp @ X7 @ X20) @ (sub @ X6 @ X20)))) | ~ ! [X6 : term] : ((X8 @ id @ (X12 @ id @ X6) @ id) | ~(X8 @ id @ X6 @ id)) | (X8 @ id @ (lam @ (X12 @ sh @ one)) @ id)) | (X8 @ id @ (lam @ (sub @ X36 @ (push @ one @ sh))) @ id) | ~ ! [X11 : term] : (~(X8 @ id @ X11 @ id) | (X8 @ id @ (sub @ X36 @ (push @ X11 @ id)) @ id))) <=> hoasinduction_lem3aa_lthm) & (apinj1 <=> ! [X14 : term,X6 : term,X13 : term,X11 : term] : ((((ap @ X6 @ X13)) != ((ap @ X11 @ X14))) | (X6 = X11))) & (! [X7 : subst] : (X7 = ((comp @ X7 @ id))) <=> axidr) & (hoasinduction_lem3v2_lthm <=> (! [X6 : term] : (X6 = ((sub @ X6 @ id))) => ! [X8 : (subst > term > subst > $o),X42 : term,X9 : (term > $o)] : (~ ! [X10 : term] : (((X9 @ X10)) = ((X8 @ id @ X10 @ id))) | (X9 @ (lam @ X42)) | ~ ! [X12 : (subst > term > term)] : (~ ! [X6 : term] : (~(X8 @ id @ X6 @ id) | (X8 @ id @ (X12 @ id @ X6) @ id)) | ~ ! [X7 : subst,X20 : subst,X6 : term] : (((sub @ (X12 @ X7 @ X6) @ X20)) = ((X12 @ (comp @ X7 @ X20) @ (sub @ X6 @ X20)))) | (X8 @ id @ (lam @ (X12 @ sh @ one)) @ id)) | ~ ! [X6 : term,X20 : subst,X7 : subst,X21 : subst] : ((X8 @ (comp @ X7 @ X21) @ (sub @ X6 @ X21) @ X20) | ~(X8 @ X7 @ X6 @ (comp @ X21 @ X20))) | ~ ! [X11 : term] : (~(X9 @ X11) | (X9 @ (sub @ X42 @ (push @ X11 @ id)))) | ~ ! [X7 : subst,X20 : subst,X6 : term,X21 : subst] : ((X8 @ X7 @ X6 @ (comp @ X21 @ X20)) | ~(X8 @ (comp @ X7 @ X21) @ (sub @ X6 @ X21) @ X20))))) & hoaslamnotvar & (((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) = ((d2term @ d_one))) & (! [X36 : term,X8 : (subst > term > subst > $o)] : (~ ! [X11 : term] : ((X8 @ id @ (sub @ X36 @ (push @ X11 @ id)) @ id) | ~(X8 @ id @ X11 @ id)) | (X8 @ id @ (lam @ (sub @ X36 @ (push @ one @ sh))) @ id) | ~ ! [X12 : (subst > term > term)] : (~ ! [X20 : subst,X6 : term,X7 : subst] : (((sub @ (X12 @ X7 @ X6) @ X20)) = ((X12 @ (comp @ X7 @ X20) @ (sub @ X6 @ X20)))) | ~ ! [X6 : term] : ((X8 @ id @ (X12 @ id @ X6) @ id) | ~(X8 @ id @ X6 @ id)) | (X8 @ id @ (lam @ (X12 @ sh @ one)) @ id))) <=> hoasinduction_lem3aa) & (hoasapinj1_lthm <=> (! [X6 : term] : (X6 = ((sub @ X6 @ id))) => (! [X6 : term,X14 : term,X13 : term,X11 : term] : ((((ap @ X6 @ X13)) != ((ap @ X11 @ X14))) | (X6 = X11)) => ! [X11 : term,X6 : term,X14 : term,X13 : term] : ((X6 = X11) | (((ap @ (sub @ X6 @ id) @ X13)) != ((ap @ (sub @ X11 @ id) @ X14))))))) & pushprop_gthm & (((push @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & ! [X45 : d_subst,X46 : d_subst] : ((((d2subst @ X45)) = ((d2subst @ X46))) => (X45 = X46)) & (hoasinduction_lthm_3 <=> (! [X8 : (subst > term > subst > $o)] : ~ ! [X9 : (term > $o)] : ~ ! [X10 : term] : (((X9 @ X10)) = ((X8 @ id @ X10 @ id))) => (! [X18 : term,X8 : (term > $o)] : ((X8 @ X18) | ~ ! [X11 : term,X6 : term] : (~(X8 @ X6) | ~(X8 @ X11) | (X8 @ (ap @ X6 @ X11))) | ~ ! [X6 : term] : ((X8 @ X6) | ~(var @ X6)) | ~ ! [X6 : term] : (~ ! [X11 : term] : ((X8 @ (sub @ X6 @ (push @ X11 @ id))) | ~(X8 @ X11)) | (X8 @ (lam @ X6)))) => (! [X6 : term] : (X6 = ((sub @ X6 @ id))) => (! [X25 : term,X9 : (term > $o),X8 : (subst > term > subst > $o)] : (~ ! [X11 : term] : (~(X9 @ X11) | (X9 @ (sub @ X25 @ (push @ X11 @ id)))) | ~ ! [X12 : (subst > term > term)] : (~ ! [X6 : term] : ((X8 @ id @ (X12 @ id @ X6) @ id) | ~(X8 @ id @ X6 @ id)) | ~ ! [X7 : subst,X6 : term,X20 : subst] : (((sub @ (X12 @ X7 @ X6) @ X20)) = ((X12 @ (comp @ X7 @ X20) @ (sub @ X6 @ X20)))) | (X8 @ id @ (lam @ (X12 @ sh @ one)) @ id)) | (X9 @ (lam @ X25)) | ~ ! [X10 : term] : (((X9 @ X10)) = ((X8 @ id @ X10 @ id)))) => ! [X8 : (subst > term > subst > $o),X19 : term] : ((X8 @ id @ X19 @ id) | ~ ! [X7 : subst,X6 : term,X21 : subst,X20 : subst] : (~(X8 @ X7 @ X6 @ (comp @ X21 @ X20)) | (X8 @ (comp @ X7 @ X21) @ (sub @ X6 @ X21) @ X20)) | ~ ! [X6 : term,X11 : term] : ((X8 @ id @ (ap @ (sub @ X6 @ id) @ X11) @ id) | ~(X8 @ id @ X6 @ id) | ~(X8 @ id @ X11 @ id)) | ~ ! [X12 : (subst > term > term)] : ((X8 @ id @ (lam @ (X12 @ sh @ one)) @ id) | ~ ! [X6 : term] : ((X8 @ id @ (X12 @ id @ X6) @ id) | ~(X8 @ id @ X6 @ id)) | ~ ! [X6 : term,X7 : subst,X20 : subst] : (((sub @ (X12 @ X7 @ X6) @ X20)) = ((X12 @ (comp @ X7 @ X20) @ (sub @ X6 @ X20))))) | ~ ! [X21 : subst,X7 : subst,X6 : term,X20 : subst] : (~(X8 @ (comp @ X7 @ X21) @ (sub @ X6 @ X21) @ X20) | (X8 @ X7 @ X6 @ (comp @ X21 @ X20))) | ~ ! [X6 : term] : ((X8 @ id @ X6 @ id) | ~(var @ (sub @ X6 @ id))))))))) & axabs & ~lamnotap & ~ulamvar1 & hoasinduction_lem3v2_gthm & ((! [X7 : subst,X6 : term] : (X6 = ((sub @ one @ (push @ X6 @ X7)))) => (! [X7 : subst,X6 : term] : (X7 = ((comp @ sh @ (push @ X6 @ X7)))) => (! [X7 : subst] : (X7 = ((comp @ X7 @ id))) => (! [X17 : term,X8 : (term > $o)] : (~ ! [X6 : term] : (~(var @ X6) | (X8 @ X6)) | ~ ! [X6 : term,X11 : term] : (~(X8 @ X11) | (X8 @ (ap @ X6 @ X11)) | ~(X8 @ X6)) | ~ ! [X6 : term] : (~(X8 @ X6) | (X8 @ (lam @ X6))) | (X8 @ X17)) => ! [X8 : (term > $o),X16 : subst,X15 : term] : (~ ! [X11 : term] : (~(var @ X11) | (X8 @ (sub @ X11 @ X16))) | ~ ! [X6 : term] : ((X8 @ (lam @ X6)) | ~ ! [X11 : term] : ((X8 @ (sub @ X6 @ (push @ X11 @ id))) | ~(X8 @ X11))) | ~ ! [X6 : term,X11 : term] : (~(X8 @ X6) | ~(X8 @ X11) | (X8 @ (ap @ X6 @ X11))) | (X8 @ (sub @ X15 @ X16))))))) <=> induction2lem_lthm) & hoasinduction_lem2_lthm & (! [X41 : term,X39 : (subst > term > term),X12 : (subst > term > term),X40 : subst] : (~ ! [X6 : term,X20 : subst,X7 : subst] : (((sub @ (X39 @ X7 @ X6) @ X20)) = ((X39 @ (comp @ X7 @ X20) @ (sub @ X6 @ X20)))) | (((lam @ (X12 @ sh @ one))) != ((lam @ (X39 @ sh @ one)))) | ~ ! [X7 : subst,X20 : subst,X6 : term] : (((sub @ (X12 @ X7 @ X6) @ X20)) = ((X12 @ (comp @ X7 @ X20) @ (sub @ X6 @ X20)))) | (((X12 @ X40 @ X41)) = ((X39 @ X40 @ X41)))) <=> hoaslaminj) & pushprop_lem2v2 & (pushprop_lem1_lthm <=> (! [X7 : subst,X6 : term] : (X6 = ((sub @ one @ (push @ X6 @ X7)))) => (! [X7 : subst,X6 : term] : (X7 = ((comp @ sh @ (push @ X6 @ X7)))) => ! [X21 : (term > $o),X6 : term,X8 : (term > $o),X11 : term,X7 : subst] : (~(X8 @ X6) | (X21 @ (sub @ X6 @ (push @ X11 @ X7))))))) & (pushprop_lem1v2 <=> ! [X9 : (term > $o),X7 : subst,X6 : term,X8 : (term > $o)] : (~(X8 @ X6) | ~ ! [X10 : term] : (((X9 @ X10)) = ((X8 @ (sub @ X10 @ (push @ X6 @ X7))))) | (X9 @ one))) & (((ap @ (d2term @ d_one) @ (d2term @ d_one))) = ((d2term @ d_one))) & axvarshift & (! [X19 : term,X8 : (subst > term > subst > $o)] : (~ ! [X7 : subst,X6 : term,X21 : subst,X20 : subst] : ((X8 @ (comp @ X7 @ X21) @ (sub @ X6 @ X21) @ X20) | ~(X8 @ X7 @ X6 @ (comp @ X21 @ X20))) | ~ ! [X6 : term,X11 : term] : (~(X8 @ id @ X6 @ id) | ~(X8 @ id @ X11 @ id) | (X8 @ id @ (ap @ (sub @ X6 @ id) @ X11) @ id)) | (X8 @ id @ X19 @ id) | ~ ! [X6 : term] : ((X8 @ id @ X6 @ id) | ~(var @ (sub @ X6 @ id))) | ~ ! [X21 : subst,X6 : term,X7 : subst,X20 : subst] : ((X8 @ X7 @ X6 @ (comp @ X21 @ X20)) | ~(X8 @ (comp @ X7 @ X21) @ (sub @ X6 @ X21) @ X20)) | ~ ! [X12 : (subst > term > term)] : (~ ! [X6 : term] : ((X8 @ id @ (X12 @ id @ X6) @ id) | ~(X8 @ id @ X6 @ id)) | ~ ! [X7 : subst,X20 : subst,X6 : term] : (((sub @ (X12 @ X7 @ X6) @ X20)) = ((X12 @ (comp @ X7 @ X20) @ (sub @ X6 @ X20)))) | (X8 @ id @ (lam @ (X12 @ sh @ one)) @ id))) <=> hoasinduction) & pushprop & ((! [X6 : term] : (X6 = ((sub @ X6 @ id))) => (! [X36 : term,X8 : (subst > term > subst > $o)] : (~ ! [X12 : (subst > term > term)] : ((X8 @ id @ (lam @ (X12 @ sh @ one)) @ id) | ~ ! [X6 : term] : (~(X8 @ id @ X6 @ id) | (X8 @ id @ (X12 @ id @ X6) @ id)) | ~ ! [X20 : subst,X7 : subst,X6 : term] : (((sub @ (X12 @ X7 @ X6) @ X20)) = ((X12 @ (comp @ X7 @ X20) @ (sub @ X6 @ X20))))) | (X8 @ id @ (lam @ (sub @ X36 @ (push @ one @ sh))) @ id) | ~ ! [X11 : term] : ((X8 @ id @ (sub @ X36 @ (push @ X11 @ id)) @ id) | ~(X8 @ id @ X11 @ id))) => ! [X35 : term,X8 : (subst > term > subst > $o)] : (~ ! [X12 : (subst > term > term)] : (~ ! [X7 : subst,X20 : subst,X6 : term] : (((sub @ (X12 @ X7 @ X6) @ X20)) = ((X12 @ (comp @ X7 @ X20) @ (sub @ X6 @ X20)))) | ~ ! [X6 : term] : ((X8 @ id @ (X12 @ id @ X6) @ id) | ~(X8 @ id @ X6 @ id)) | (X8 @ id @ (lam @ (X12 @ sh @ one)) @ id)) | ~ ! [X11 : term] : (~(X8 @ id @ X11 @ id) | (X8 @ id @ (sub @ X35 @ (push @ X11 @ id)) @ id)) | (X8 @ id @ (lam @ X35) @ id)))) <=> hoasinduction_lem3a_lthm) & pushprop_lem1v2_gthm & (shinj <=> ! [X11 : term,X6 : term] : ((((sub @ X6 @ sh)) != ((sub @ X11 @ sh))) | (X6 = X11))) & ~(hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) & ~(var @ (d2term @ d_one)) & substmonoid_gthm & (hoasinduction_lem2v2 <=> ! [X38 : term,X8 : (subst > term > subst > $o),X9 : (term > $o),X37 : term] : (~ ! [X10 : term] : (((X9 @ X10)) = ((X8 @ id @ X10 @ id))) | ~(X9 @ X38) | ~(X9 @ X37) | ~ ! [X6 : term,X11 : term] : (~(X8 @ id @ X6 @ id) | ~(X8 @ id @ X11 @ id) | (X8 @ id @ (ap @ (sub @ X6 @ id) @ X11) @ id)) | ~ ! [X7 : subst,X21 : subst,X6 : term,X20 : subst] : ((X8 @ X7 @ X6 @ (comp @ X21 @ X20)) | ~(X8 @ (comp @ X7 @ X21) @ (sub @ X6 @ X21) @ X20)) | (X9 @ (ap @ X37 @ X38)) | ~ ! [X21 : subst,X7 : subst,X6 : term,X20 : subst] : (~(X8 @ X7 @ X6 @ (comp @ X21 @ X20)) | (X8 @ (comp @ X7 @ X21) @ (sub @ X6 @ X21) @ X20)))) & ((! [X8 : (subst > term > subst > $o)] : ~ ! [X9 : (term > $o)] : ~ ! [X10 : term] : (((X9 @ X10)) = ((X8 @ id @ X10 @ id))) => (! [X18 : term,X8 : (term > $o)] : ((X8 @ X18) | ~ ! [X6 : term,X11 : term] : ((X8 @ (ap @ X6 @ X11)) | ~(X8 @ X11) | ~(X8 @ X6)) | ~ ! [X6 : term] : ((X8 @ (lam @ X6)) | ~ ! [X11 : term] : ((X8 @ (sub @ X6 @ (push @ X11 @ id))) | ~(X8 @ X11))) | ~ ! [X6 : term] : (~(var @ X6) | (X8 @ X6))) => (! [X6 : term] : (X6 = ((sub @ X6 @ id))) => (! [X9 : (term > $o),X8 : (subst > term > subst > $o),X25 : term] : (~ ! [X10 : term] : (((X9 @ X10)) = ((X8 @ id @ X10 @ id))) | ~ ! [X11 : term] : ((X9 @ (sub @ X25 @ (push @ X11 @ id))) | ~(X9 @ X11)) | (X9 @ (lam @ X25)) | ~ ! [X12 : (subst > term > term)] : (~ ! [X6 : term] : ((X8 @ id @ (X12 @ id @ X6) @ id) | ~(X8 @ id @ X6 @ id)) | (X8 @ id @ (lam @ (X12 @ sh @ one)) @ id) | ~ ! [X6 : term,X20 : subst,X7 : subst] : (((sub @ (X12 @ X7 @ X6) @ X20)) = ((X12 @ (comp @ X7 @ X20) @ (sub @ X6 @ X20)))))) => ! [X8 : (subst > term > subst > $o),X26 : term] : ((X8 @ id @ X26 @ id) | ~ ! [X11 : term,X6 : term] : ((X8 @ id @ (ap @ (sub @ X6 @ id) @ X11) @ id) | ~(X8 @ id @ X6 @ id) | ~(X8 @ id @ X11 @ id)) | ~ ! [X12 : (subst > term > term)] : ((X8 @ id @ (lam @ (X12 @ sh @ one)) @ id) | ~ ! [X6 : term,X7 : subst,X20 : subst] : (((sub @ (X12 @ X7 @ X6) @ X20)) = ((X12 @ (comp @ X7 @ X20) @ (sub @ X6 @ X20)))) | ~ ! [X6 : term] : (~(X8 @ id @ X6 @ id) | (X8 @ id @ (X12 @ id @ X6) @ id)))))))) <=> hoasinduction_no_psi_cond_lthm) & (induction2_lthm <=> (! [X6 : term] : (X6 = ((sub @ X6 @ id))) => (! [X16 : subst,X8 : (term > $o),X15 : term] : (~ ! [X6 : term,X11 : term] : (~(X8 @ X6) | (X8 @ (ap @ X6 @ X11)) | ~(X8 @ X11)) | ~ ! [X6 : term] : (~ ! [X11 : term] : (~(X8 @ X11) | (X8 @ (sub @ X6 @ (push @ X11 @ id)))) | (X8 @ (lam @ X6))) | ~ ! [X11 : term] : (~(var @ X11) | (X8 @ (sub @ X11 @ X16))) | (X8 @ (sub @ X15 @ X16))) => ! [X8 : (term > $o),X18 : term] : (~ ! [X6 : term] : (~ ! [X11 : term] : (~(X8 @ X11) | (X8 @ (sub @ X6 @ (push @ X11 @ id)))) | (X8 @ (lam @ X6))) | ~ ! [X11 : term,X6 : term] : (~(X8 @ X6) | (X8 @ (ap @ X6 @ X11)) | ~(X8 @ X11)) | (X8 @ X18) | ~ ! [X6 : term] : (~(var @ X6) | (X8 @ X6)))))) & (! [X8 : (term > $o),X18 : term] : ((X8 @ X18) | ~ ! [X6 : term] : ((X8 @ X6) | ~(var @ X6)) | ~ ! [X11 : term,X6 : term] : ((X8 @ (ap @ X6 @ X11)) | ~(X8 @ X11) | ~(X8 @ X6)) | ~ ! [X6 : term] : (~ ! [X11 : term] : (~(X8 @ X11) | (X8 @ (sub @ X6 @ (push @ X11 @ id)))) | (X8 @ (lam @ X6)))) <=> induction2) & pushprop_lem1_gthm & hoaslamnotap_lthm & hoasinduction_lem1v2 & ((! [X6 : term] : (X6 = ((sub @ X6 @ id))) => (! [X8 : (subst > term > subst > $o),X36 : term] : (~ ! [X11 : term] : ((X8 @ id @ (sub @ X36 @ (push @ X11 @ id)) @ id) | ~(X8 @ id @ X11 @ id)) | ~ ! [X12 : (subst > term > term)] : (~ ! [X6 : term] : ((X8 @ id @ (X12 @ id @ X6) @ id) | ~(X8 @ id @ X6 @ id)) | (X8 @ id @ (lam @ (X12 @ sh @ one)) @ id) | ~ ! [X7 : subst,X20 : subst,X6 : term] : (((sub @ (X12 @ X7 @ X6) @ X20)) = ((X12 @ (comp @ X7 @ X20) @ (sub @ X6 @ X20))))) | (X8 @ id @ (lam @ (sub @ X36 @ (push @ one @ sh))) @ id)) => ! [X22 : term,X8 : (subst > term > subst > $o)] : ((X8 @ id @ (lam @ X22) @ id) | ~ ! [X20 : subst,X7 : subst,X21 : subst,X6 : term] : (~(X8 @ (comp @ X7 @ X21) @ (sub @ X6 @ X21) @ X20) | (X8 @ X7 @ X6 @ (comp @ X21 @ X20))) | ~ ! [X11 : term] : ((X8 @ id @ (sub @ X22 @ (push @ X11 @ id)) @ id) | ~(X8 @ id @ X11 @ id)) | ~ ! [X6 : term,X7 : subst,X20 : subst,X21 : subst] : (~(X8 @ X7 @ X6 @ (comp @ X21 @ X20)) | (X8 @ (comp @ X7 @ X21) @ (sub @ X6 @ X21) @ X20)) | ~ ! [X12 : (subst > term > term)] : (~ ! [X6 : term] : (~(X8 @ id @ X6 @ id) | (X8 @ id @ (X12 @ id @ X6) @ id)) | ~ ! [X20 : subst,X6 : term,X7 : subst] : (((sub @ (X12 @ X7 @ X6) @ X20)) = ((X12 @ (comp @ X7 @ X20) @ (sub @ X6 @ X20)))) | (X8 @ id @ (lam @ (X12 @ sh @ one)) @ id))))) <=> hoasinduction_lem3_lthm) & pushprop_lthm_orig & pushprop_lem0_gthm & ((! [X14 : term,X13 : term,X6 : term,X11 : term] : ((X13 = X14) | (((ap @ X6 @ X13)) != ((ap @ X11 @ X14)))) => ! [X13 : term,X11 : term,X6 : term,X14 : term] : ((((ap @ (sub @ X6 @ id) @ X13)) != ((ap @ (sub @ X11 @ id) @ X14))) | (X13 = X14))) <=> hoasapinj2_lthm) & (hoasapinj2 <=> ! [X11 : term,X14 : term,X13 : term,X6 : term] : ((((ap @ (sub @ X6 @ id) @ X13)) != ((ap @ (sub @ X11 @ id) @ X14))) | (X13 = X14))) & (axidl <=> ! [X7 : subst] : (X7 = ((comp @ id @ X7)))) & (((lam @ (d2term @ d_one))) = ((d2term @ d_one))) & hoasapnotvar_lthm & (hoaslam = (^[X4 : subst, X5 : (subst > term > term)] : ((d2term @ d_one)))) & hoasinduction_lem3a_gthm & ((^[X8 : (subst > term > subst > $o), X9 : (term > $o)] : (! [X10 : term] : (((X9 @ X10)) = ((X8 @ id @ X10 @ id))))) = hoasinduction_p_and_p_prime) & ulamvarsh & apnotvar & hoasinduction_lem1_gthm & ! [X2 : subst] : ? [X3 : d_subst] : (X2 = ((d2subst @ X3))) & (! [X11 : term] : ~ ! [X12 : (subst > term > term)] : (((X12 @ sh @ one)) != ((sub @ X11 @ (push @ one @ sh)))) <=> hoasinduction_lem3b)),
  file('/export/starexec/sandbox2/benchmark/theBenchmark.p',alg444_1)).
thf(f4,plain,(
  ~ (! [X0 : term] : ~(var @ (lam @ X0)) = lamnotvar)),
  inference(rectify,[],[f2])).
thf(f5,plain,(
  ~(! [X0 : term] : ~ (((var @ (lam @ X0))) = $true) <=> (lamnotvar = $true))),
  inference(fool_elimination,[],[f4])).
thf(f6,plain,(
  (! [X0 : term] : (((sub @ X0 @ id)) = X0) <=> axvarid) & axapp & termmset_gthm & (! [X1 : term,X2 : term,X3 : term,X4 : term] : ((X2 = X4) | (((ap @ X3 @ X2)) != ((ap @ X1 @ X4)))) <=> apinj2) & axclos & hoasinduction_lem3b_gthm & hoasinduction_lem2_gthm & (! [X5 : (term > $o),X6 : term,X7 : subst] : ~ ! [X8 : (term > $o)] : ~ ! [X9 : term] : (((X5 @ (sub @ X9 @ (push @ X6 @ X7)))) = ((X8 @ X9))) <=> pushprop_lem0_lthm) & (hoasinduction_lem0_lthm <=> ! [X10 : (subst > term > subst > $o)] : ~ ! [X11 : (term > $o)] : ~ ! [X12 : term] : (((X11 @ X12)) = ((X10 @ id @ X12 @ id)))) & pushprop_lem2v2_gthm & ((! [X13 : term] : ~ ! [X14 : (subst > term > term)] : ~ ! [X15 : term,X16 : subst] : (((sub @ X13 @ (push @ X15 @ X16))) = ((X14 @ X16 @ X15))) => (! [X17 : term] : (((sub @ X17 @ id)) = X17) => ! [X18 : term,X19 : (term > $o),X20 : (subst > term > subst > $o)] : (~ ! [X21 : term] : (((X19 @ X21)) = ((X20 @ id @ X21 @ id))) | ~ ! [X22 : term] : ((X19 @ (sub @ X18 @ (push @ X22 @ id))) | ~(X19 @ X22)) | (X19 @ (lam @ X18)) | ~ ! [X23 : (subst > term > term)] : ((X20 @ id @ (lam @ (X23 @ sh @ one)) @ id) | ~ ! [X24 : term] : ((X20 @ id @ (X23 @ id @ X24) @ id) | ~(X20 @ id @ X24 @ id)) | ~ ! [X25 : subst,X26 : term,X27 : subst] : (((sub @ (X23 @ X25 @ X26) @ X27)) = ((X23 @ (comp @ X25 @ X27) @ (sub @ X26 @ X27)))))))) <=> hoasinduction_lem3v2a_lthm) & (! [X28 : term,X29 : term,X30 : term,X31 : term] : ((X28 = X30) | (((ap @ (sub @ X30 @ id) @ X29)) != ((ap @ (sub @ X28 @ id) @ X31)))) <=> hoasapinj1) & axmap & induction2_gthm & ((! [X32 : subst,X33 : term] : (((sub @ one @ (push @ X33 @ X32))) = X33) => (! [X34 : term,X35 : subst] : (((comp @ sh @ (push @ X34 @ X35))) = X35) => (! [X36 : term,X37 : term] : ((((lam @ X37)) != ((lam @ X36))) | (X36 = X37)) => ! [X38 : subst,X39 : (subst > term > term),X40 : (subst > term > term),X41 : term] : (~ ! [X42 : subst,X43 : subst,X44 : term] : (((sub @ (X39 @ X43 @ X44) @ X42)) = ((X39 @ (comp @ X43 @ X42) @ (sub @ X44 @ X42)))) | (((lam @ (X39 @ sh @ one))) != ((lam @ (X40 @ sh @ one)))) | (((X40 @ X38 @ X41)) = ((X39 @ X38 @ X41))) | ~ ! [X45 : subst,X46 : term,X47 : subst] : (((sub @ (X40 @ X45 @ X46) @ X47)) = ((X40 @ (comp @ X45 @ X47) @ (sub @ X46 @ X47)))))))) <=> hoaslaminj_lthm) & hoasinduction_lem1v2_gthm & ! [X48 : d_subst] : (d_id = X48) & hoaslaminj_gthm & (termmset_lthm <=> (! [X49 : term] : (((sub @ X49 @ id)) = X49) => ! [X50 : term] : (((sub @ X50 @ id)) = X50))) & ! [X51 : d_term,X52 : d_term] : ((((d2term @ X51)) = ((d2term @ X52))) => (X51 = X52)) & (hoasinduction_lem3a <=> ! [X53 : term,X54 : (subst > term > subst > $o)] : (~ ! [X55 : (subst > term > term)] : ((X54 @ id @ (lam @ (X55 @ sh @ one)) @ id) | ~ ! [X56 : subst,X57 : subst,X58 : term] : (((X55 @ (comp @ X57 @ X56) @ (sub @ X58 @ X56))) = ((sub @ (X55 @ X57 @ X58) @ X56))) | ~ ! [X59 : term] : (~(X54 @ id @ X59 @ id) | (X54 @ id @ (X55 @ id @ X59) @ id))) | ~ ! [X60 : term] : (~(X54 @ id @ X60 @ id) | (X54 @ id @ (sub @ X53 @ (push @ X60 @ id)) @ id)) | (X54 @ id @ (lam @ X53) @ id))) & ! [X61 : term] : ? [X62 : d_term] : (((d2term @ X62)) = X61) & (((comp @ (d2subst @ d_id) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & hoasinduction_gthm & (axvarcons <=> ! [X63 : term,X64 : subst] : (((sub @ one @ (push @ X63 @ X64))) = X63)) & (! [X65 : (term > $o),X66 : term] : ((X65 @ X66) | ~ ! [X67 : term] : ((X65 @ X67) | ~(var @ X67)) | ~ ! [X68 : term,X69 : term] : (~(X65 @ X68) | ~(X65 @ X69) | (X65 @ (ap @ X68 @ X69))) | ~ ! [X70 : term] : ((X65 @ (lam @ X70)) | ~(X65 @ X70))) <=> induction) & axassoc & hoaslamnotap_gthm & (termmset <=> ! [X71 : term] : (((sub @ X71 @ id)) = X71)) & pushprop_lthm & (! [X72 : (subst > term > subst > $o),X73 : term] : (~ ! [X74 : term] : (~(X72 @ id @ X74 @ id) | (X72 @ id @ (sub @ X73 @ (push @ X74 @ id)) @ id)) | ~ ! [X75 : subst,X76 : subst,X77 : term,X78 : subst] : ((X72 @ X75 @ X77 @ (comp @ X78 @ X76)) | ~(X72 @ (comp @ X75 @ X78) @ (sub @ X77 @ X78) @ X76)) | (X72 @ id @ (lam @ X73) @ id) | ~ ! [X79 : (subst > term > term)] : (~ ! [X80 : term] : ((X72 @ id @ (X79 @ id @ X80) @ id) | ~(X72 @ id @ X80 @ id)) | (X72 @ id @ (lam @ (X79 @ sh @ one)) @ id) | ~ ! [X81 : term,X82 : subst,X83 : subst] : (((X79 @ (comp @ X83 @ X82) @ (sub @ X81 @ X82))) = ((sub @ (X79 @ X83 @ X81) @ X82)))) | ~ ! [X84 : subst,X85 : subst,X86 : subst,X87 : term] : ((X72 @ (comp @ X85 @ X84) @ (sub @ X87 @ X84) @ X86) | ~(X72 @ X85 @ X87 @ (comp @ X84 @ X86)))) <=> hoasinduction_lem3) & hoasinduction_lem1_lthm & (substmonoid_lthm <=> (! [X88 : subst] : (((comp @ id @ X88)) = X88) => (! [X89 : subst] : (((comp @ X89 @ id)) = X89) => (! [X90 : subst] : (((comp @ X90 @ id)) = X90) & ! [X91 : subst] : (((comp @ id @ X91)) = X91))))) & (id = ((d2subst @ d_id))) & pushprop_lem3v2 & hoasinduction_lem1 & (((sub @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2term @ d_one))) & (hoaslamnotap <=> ! [X92 : term,X93 : (subst > term > term),X94 : term] : (~ ! [X95 : subst,X96 : term,X97 : subst] : (((X93 @ (comp @ X95 @ X97) @ (sub @ X96 @ X97))) = ((sub @ (X93 @ X95 @ X96) @ X97))) | (((lam @ (X93 @ sh @ one))) != ((ap @ (sub @ X94 @ id) @ X92))))) & (! [X98 : term,X99 : term] : ((X98 = X99) | (((lam @ X98)) != ((lam @ X99)))) <=> laminj) & (sh = ((d2subst @ d_id))) & (hoasinduction_lem3v2_f <=> ! [X100 : term] : ~ ! [X101 : (subst > term > term)] : ~ ! [X102 : subst,X103 : term] : (((X101 @ X102 @ X103)) = ((sub @ X100 @ (push @ X103 @ X102))))) & hoasinduction_lem3_gthm & ((! [X104 : subst,X105 : term] : (((sub @ one @ (push @ X105 @ X104))) = X105) => ! [X106 : (term > $o),X107 : term,X108 : (term > $o),X109 : subst] : (~ ! [X110 : term] : (((X106 @ X110)) = ((X108 @ (sub @ X110 @ (push @ X107 @ X109))))) | ~(X108 @ X107) | (X106 @ one))) <=> pushprop_lem1v2_lthm) & hoaslamnotvar_gthm & induction2lem_gthm & ! [X111 : d_term] : (d_one = X111) & (one = ((d2term @ d_one))) & hoasapnotvar_gthm & (! [X112 : term,X113 : subst] : (((comp @ sh @ (push @ X112 @ X113))) = X113) <=> axshiftcons) & (hoasinduction_lem3aaa <=> ! [X114 : (subst > term > subst > $o),X115 : term] : (~ ! [X116 : term] : (~(X114 @ id @ X116 @ id) | (X114 @ id @ (sub @ X115 @ (push @ X116 @ id)) @ id)) | ~ ! [X117 : (subst > term > term),X118 : term] : ((X114 @ id @ (lam @ (X117 @ sh @ one)) @ id) | ~ ! [X119 : subst,X120 : term,X121 : subst] : (((X117 @ (comp @ X119 @ X121) @ (sub @ X120 @ X121))) = ((sub @ X118 @ (push @ (sub @ X120 @ X121) @ (comp @ X119 @ X121))))) | ~ ! [X122 : subst,X123 : term,X124 : subst] : (((sub @ (sub @ X118 @ (push @ X123 @ X122)) @ X124)) = ((sub @ (X117 @ X122 @ X123) @ X124))) | ~ ! [X125 : term] : ((X114 @ id @ (X117 @ id @ X125) @ id) | ~(X114 @ id @ X125 @ id))) | (X114 @ id @ (lam @ (sub @ X115 @ (push @ one @ sh))) @ id))) & pushprop_lem2v2_lthm & (hoasinduction_lem0 <=> ! [X126 : (subst > term > subst > $o)] : ~ ! [X127 : (term > $o)] : ~ ! [X128 : term] : (((X127 @ X128)) = ((X126 @ id @ X128 @ id)))) & (hoasinduction_lthm <=> (! [X129 : (term > $o),X130 : term] : (~ ! [X131 : term,X132 : term] : (~(X129 @ X131) | (X129 @ (ap @ X131 @ X132)) | ~(X129 @ X132)) | ~ ! [X133 : term] : (~(var @ X133) | (X129 @ X133)) | (X129 @ X130) | ~ ! [X134 : term] : ((X129 @ (lam @ X134)) | ~ ! [X135 : term] : ((X129 @ (sub @ X134 @ (push @ X135 @ id))) | ~(X129 @ X135)))) => (! [X136 : term,X137 : (subst > term > subst > $o),X138 : term] : (~ ! [X139 : subst,X140 : subst,X141 : term,X142 : subst] : ((X137 @ (comp @ X140 @ X139) @ (sub @ X141 @ X139) @ X142) | ~(X137 @ X140 @ X141 @ (comp @ X139 @ X142))) | ~ ! [X143 : term,X144 : term] : ((X137 @ id @ (ap @ (sub @ X143 @ id) @ X144) @ id) | ~(X137 @ id @ X144 @ id) | ~(X137 @ id @ X143 @ id)) | ~(X137 @ id @ X138 @ id) | (X137 @ id @ (ap @ X138 @ X136) @ id) | ~(X137 @ id @ X136 @ id) | ~ ! [X145 : subst,X146 : subst,X147 : term,X148 : subst] : ((X137 @ X145 @ X147 @ (comp @ X146 @ X148)) | ~(X137 @ (comp @ X145 @ X146) @ (sub @ X147 @ X146) @ X148))) => (! [X149 : (subst > term > subst > $o),X150 : term] : (~ ! [X151 : term,X152 : subst,X153 : subst,X154 : subst] : ((X149 @ X152 @ X151 @ (comp @ X153 @ X154)) | ~(X149 @ (comp @ X152 @ X153) @ (sub @ X151 @ X153) @ X154)) | ~ ! [X155 : term] : (~(X149 @ id @ X155 @ id) | (X149 @ id @ (sub @ X150 @ (push @ X155 @ id)) @ id)) | (X149 @ id @ (lam @ X150) @ id) | ~ ! [X156 : subst,X157 : subst,X158 : term,X159 : subst] : (~(X149 @ X157 @ X158 @ (comp @ X159 @ X156)) | (X149 @ (comp @ X157 @ X159) @ (sub @ X158 @ X159) @ X156)) | ~ ! [X160 : (subst > term > term)] : ((X149 @ id @ (lam @ (X160 @ sh @ one)) @ id) | ~ ! [X161 : subst,X162 : term,X163 : subst] : (((sub @ (X160 @ X163 @ X162) @ X161)) = ((X160 @ (comp @ X163 @ X161) @ (sub @ X162 @ X161)))) | ~ ! [X164 : term] : ((X149 @ id @ (X160 @ id @ X164) @ id) | ~(X149 @ id @ X164 @ id)))) => ! [X165 : (subst > term > subst > $o),X166 : term] : (~ ! [X167 : term,X168 : subst,X169 : subst,X170 : subst] : (~(X165 @ X169 @ X167 @ (comp @ X168 @ X170)) | (X165 @ (comp @ X169 @ X168) @ (sub @ X167 @ X168) @ X170)) | ~ ! [X171 : subst,X172 : term,X173 : subst,X174 : subst] : ((X165 @ X173 @ X172 @ (comp @ X174 @ X171)) | ~(X165 @ (comp @ X173 @ X174) @ (sub @ X172 @ X174) @ X171)) | ~ ! [X175 : term,X176 : term] : ((X165 @ id @ (ap @ (sub @ X175 @ id) @ X176) @ id) | ~(X165 @ id @ X175 @ id) | ~(X165 @ id @ X176 @ id)) | ~ ! [X177 : term] : (~(var @ (sub @ X177 @ id)) | (X165 @ id @ X177 @ id)) | ~ ! [X178 : (subst > term > term)] : (~ ! [X179 : subst,X180 : subst,X181 : term] : (((sub @ (X178 @ X179 @ X181) @ X180)) = ((X178 @ (comp @ X179 @ X180) @ (sub @ X181 @ X180)))) | ~ ! [X182 : term] : ((X165 @ id @ (X178 @ id @ X182) @ id) | ~(X165 @ id @ X182 @ id)) | (X165 @ id @ (lam @ (X178 @ sh @ one)) @ id)) | (X165 @ id @ X166 @ id)))))) & hoasapinj1_gthm & (axscons <=> ! [X183 : subst] : (((push @ (sub @ one @ X183) @ (comp @ sh @ X183))) = X183)) & hoaslamnotvar_lthm & (pushprop_lem1 <=> ! [X184 : (term > $o),X185 : term,X186 : (term > $o),X187 : subst,X188 : term] : (~(X186 @ X185) | (X184 @ (sub @ X185 @ (push @ X188 @ X187))))) & pushprop_lem3v2_lthm & hoasinduction_lem2v2_gthm & ((^[X189 : term, X190 : subst, X191 : (term > $o), X192 : (term > $o)] : (! [X193 : term] : (((X192 @ X193)) = ((X191 @ (sub @ X193 @ (push @ X189 @ X190))))))) = pushprop_p_and_p_prime) & hoasapnotvar & (! [X194 : term] : ~ ! [X195 : (subst > term > term)] : ~ ! [X196 : subst,X197 : term] : (((X195 @ X196 @ X197)) = ((sub @ X194 @ (push @ X197 @ X196)))) <=> hoasinduction_lem3v2_f_lthm) & (! [X198 : term,X199 : term,X200 : (subst > term > subst > $o)] : (~(X200 @ id @ X198 @ id) | (X200 @ id @ (ap @ X198 @ X199) @ id) | ~(X200 @ id @ X199 @ id) | ~ ! [X201 : term,X202 : term] : ((X200 @ id @ (ap @ (sub @ X201 @ id) @ X202) @ id) | ~(X200 @ id @ X201 @ id) | ~(X200 @ id @ X202 @ id)) | ~ ! [X203 : term,X204 : subst,X205 : subst,X206 : subst] : ((X200 @ X206 @ X203 @ (comp @ X205 @ X204)) | ~(X200 @ (comp @ X206 @ X205) @ (sub @ X203 @ X205) @ X204)) | ~ ! [X207 : term,X208 : subst,X209 : subst,X210 : subst] : ((X200 @ (comp @ X209 @ X210) @ (sub @ X207 @ X210) @ X208) | ~(X200 @ X209 @ X207 @ (comp @ X210 @ X208)))) <=> hoasinduction_lem2) & ulamvarind & (hoasinduction_no_psi_cond <=> ! [X211 : (subst > term > subst > $o),X212 : term] : (~ ! [X213 : (subst > term > term)] : (~ ! [X214 : term,X215 : subst,X216 : subst] : (((X213 @ (comp @ X216 @ X215) @ (sub @ X214 @ X215))) = ((sub @ (X213 @ X216 @ X214) @ X215))) | ~ ! [X217 : term] : (~(X211 @ id @ X217 @ id) | (X211 @ id @ (X213 @ id @ X217) @ id)) | (X211 @ id @ (lam @ (X213 @ sh @ one)) @ id)) | (X211 @ id @ X212 @ id) | ~ ! [X218 : term,X219 : term] : (~(X211 @ id @ X218 @ id) | (X211 @ id @ (ap @ (sub @ X218 @ id) @ X219) @ id) | ~(X211 @ id @ X219 @ id)))) & hoasapinj2_gthm & lamnotvar & (! [X220 : subst,X221 : (term > $o),X222 : term] : (~ ! [X223 : term] : ((X221 @ (sub @ X223 @ X220)) | ~(var @ X223)) | ~ ! [X224 : term,X225 : term] : ((X221 @ (ap @ X224 @ X225)) | ~(X221 @ X224) | ~(X221 @ X225)) | (X221 @ (sub @ X222 @ X220)) | ~ ! [X226 : term] : (~ ! [X227 : term] : (~(X221 @ X227) | (X221 @ (sub @ X226 @ (push @ X227 @ id)))) | (X221 @ (lam @ X226)))) <=> induction2lem) & ((! [X228 : subst] : (((comp @ id @ X228)) = X228) & ! [X229 : subst] : (((comp @ X229 @ id)) = X229)) <=> substmonoid) & (hoasinduction_lem3b_lthm <=> ! [X230 : term] : ~ ! [X231 : (subst > term > term)] : (((X231 @ sh @ one)) != ((sub @ X230 @ (push @ one @ sh))))) & (! [X232 : (term > $o),X233 : term,X234 : (subst > term > subst > $o)] : (~ ! [X235 : (subst > term > term)] : (~ ! [X236 : subst,X237 : term,X238 : subst] : (((sub @ (X235 @ X236 @ X237) @ X238)) = ((X235 @ (comp @ X236 @ X238) @ (sub @ X237 @ X238)))) | ~ ! [X239 : term] : ((X234 @ id @ (X235 @ id @ X239) @ id) | ~(X234 @ id @ X239 @ id)) | (X234 @ id @ (lam @ (X235 @ sh @ one)) @ id)) | (X232 @ (lam @ X233)) | ~ ! [X240 : term] : (((X232 @ X240)) = ((X234 @ id @ X240 @ id))) | ~ ! [X241 : term] : (~(X232 @ X241) | (X232 @ (sub @ X233 @ (push @ X241 @ id))))) <=> hoasinduction_lem3v2a) & (pushprop_lem0 <=> ! [X242 : subst,X243 : (term > $o),X244 : term] : ~ ! [X245 : (term > $o)] : ~ ! [X246 : term] : (((X243 @ (sub @ X246 @ (push @ X244 @ X242)))) = ((X245 @ X246)))) & (hoasinduction_lem3v2 <=> ! [X247 : term,X248 : (subst > term > subst > $o),X249 : (term > $o)] : (~ ! [X250 : subst,X251 : subst,X252 : term,X253 : subst] : (~(X248 @ X251 @ X252 @ (comp @ X253 @ X250)) | (X248 @ (comp @ X251 @ X253) @ (sub @ X252 @ X253) @ X250)) | ~ ! [X254 : term] : (((X248 @ id @ X254 @ id)) = ((X249 @ X254))) | ~ ! [X255 : subst,X256 : subst,X257 : subst,X258 : term] : (~(X248 @ (comp @ X256 @ X257) @ (sub @ X258 @ X257) @ X255) | (X248 @ X256 @ X258 @ (comp @ X257 @ X255))) | (X249 @ (lam @ X247)) | ~ ! [X259 : (subst > term > term)] : (~ ! [X260 : term] : (~(X248 @ id @ X260 @ id) | (X248 @ id @ (X259 @ id @ X260) @ id)) | (X248 @ id @ (lam @ (X259 @ sh @ one)) @ id) | ~ ! [X261 : subst,X262 : subst,X263 : term] : (((sub @ (X259 @ X262 @ X263) @ X261)) = ((X259 @ (comp @ X262 @ X261) @ (sub @ X263 @ X261))))) | ~ ! [X264 : term] : (~(X249 @ X264) | (X249 @ (sub @ X247 @ (push @ X264 @ id)))))) & (! [X265 : term,X266 : (subst > term > subst > $o)] : (~ ! [X267 : (subst > term > term)] : (~ ! [X268 : subst,X269 : term,X270 : subst] : (((X267 @ (comp @ X270 @ X268) @ (sub @ X269 @ X268))) = ((sub @ (X267 @ X270 @ X269) @ X268))) | ~ ! [X271 : term] : ((X266 @ id @ (X267 @ id @ X271) @ id) | ~(X266 @ id @ X271 @ id)) | (X266 @ id @ (lam @ (X267 @ sh @ one)) @ id)) | (X266 @ id @ (lam @ (sub @ X265 @ (push @ one @ sh))) @ id) | ~ ! [X272 : term] : (~(X266 @ id @ X272 @ id) | (X266 @ id @ (sub @ X265 @ (push @ X272 @ id)) @ id))) <=> hoasinduction_lem3aa_lthm) & (apinj1 <=> ! [X273 : term,X274 : term,X275 : term,X276 : term] : ((((ap @ X274 @ X275)) != ((ap @ X276 @ X273))) | (X274 = X276))) & (! [X277 : subst] : (((comp @ X277 @ id)) = X277) <=> axidr) & (hoasinduction_lem3v2_lthm <=> (! [X278 : term] : (((sub @ X278 @ id)) = X278) => ! [X279 : (subst > term > subst > $o),X280 : term,X281 : (term > $o)] : (~ ! [X282 : term] : (((X279 @ id @ X282 @ id)) = ((X281 @ X282))) | (X281 @ (lam @ X280)) | ~ ! [X283 : (subst > term > term)] : (~ ! [X284 : term] : (~(X279 @ id @ X284 @ id) | (X279 @ id @ (X283 @ id @ X284) @ id)) | ~ ! [X285 : subst,X286 : subst,X287 : term] : (((sub @ (X283 @ X285 @ X287) @ X286)) = ((X283 @ (comp @ X285 @ X286) @ (sub @ X287 @ X286)))) | (X279 @ id @ (lam @ (X283 @ sh @ one)) @ id)) | ~ ! [X288 : term,X289 : subst,X290 : subst,X291 : subst] : ((X279 @ (comp @ X290 @ X291) @ (sub @ X288 @ X291) @ X289) | ~(X279 @ X290 @ X288 @ (comp @ X291 @ X289))) | ~ ! [X292 : term] : (~(X281 @ X292) | (X281 @ (sub @ X280 @ (push @ X292 @ id)))) | ~ ! [X293 : subst,X294 : subst,X295 : term,X296 : subst] : ((X279 @ X293 @ X295 @ (comp @ X296 @ X294)) | ~(X279 @ (comp @ X293 @ X296) @ (sub @ X295 @ X296) @ X294))))) & hoaslamnotvar & (((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one))) = ((d2term @ d_one))) & (! [X297 : term,X298 : (subst > term > subst > $o)] : (~ ! [X299 : term] : ((X298 @ id @ (sub @ X297 @ (push @ X299 @ id)) @ id) | ~(X298 @ id @ X299 @ id)) | (X298 @ id @ (lam @ (sub @ X297 @ (push @ one @ sh))) @ id) | ~ ! [X300 : (subst > term > term)] : (~ ! [X301 : subst,X302 : term,X303 : subst] : (((sub @ (X300 @ X303 @ X302) @ X301)) = ((X300 @ (comp @ X303 @ X301) @ (sub @ X302 @ X301)))) | ~ ! [X304 : term] : ((X298 @ id @ (X300 @ id @ X304) @ id) | ~(X298 @ id @ X304 @ id)) | (X298 @ id @ (lam @ (X300 @ sh @ one)) @ id))) <=> hoasinduction_lem3aa) & (hoasapinj1_lthm <=> (! [X305 : term] : (((sub @ X305 @ id)) = X305) => (! [X306 : term,X307 : term,X308 : term,X309 : term] : ((((ap @ X306 @ X308)) != ((ap @ X309 @ X307))) | (X306 = X309)) => ! [X310 : term,X311 : term,X312 : term,X313 : term] : ((X310 = X311) | (((ap @ (sub @ X310 @ id) @ X312)) != ((ap @ (sub @ X311 @ id) @ X313))))))) & pushprop_gthm & (((push @ (d2term @ d_one) @ (d2subst @ d_id))) = ((d2subst @ d_id))) & ! [X314 : d_subst,X315 : d_subst] : ((((d2subst @ X314)) = ((d2subst @ X315))) => (X314 = X315)) & (hoasinduction_lthm_3 <=> (! [X316 : (subst > term > subst > $o)] : ~ ! [X317 : (term > $o)] : ~ ! [X318 : term] : (((X317 @ X318)) = ((X316 @ id @ X318 @ id))) => (! [X319 : term,X320 : (term > $o)] : ((X320 @ X319) | ~ ! [X321 : term,X322 : term] : (~(X320 @ X322) | ~(X320 @ X321) | (X320 @ (ap @ X322 @ X321))) | ~ ! [X323 : term] : ((X320 @ X323) | ~(var @ X323)) | ~ ! [X324 : term] : (~ ! [X325 : term] : ((X320 @ (sub @ X324 @ (push @ X325 @ id))) | ~(X320 @ X325)) | (X320 @ (lam @ X324)))) => (! [X326 : term] : (((sub @ X326 @ id)) = X326) => (! [X327 : term,X328 : (term > $o),X329 : (subst > term > subst > $o)] : (~ ! [X330 : term] : (~(X328 @ X330) | (X328 @ (sub @ X327 @ (push @ X330 @ id)))) | ~ ! [X331 : (subst > term > term)] : (~ ! [X332 : term] : ((X329 @ id @ (X331 @ id @ X332) @ id) | ~(X329 @ id @ X332 @ id)) | ~ ! [X333 : subst,X334 : term,X335 : subst] : (((sub @ (X331 @ X333 @ X334) @ X335)) = ((X331 @ (comp @ X333 @ X335) @ (sub @ X334 @ X335)))) | (X329 @ id @ (lam @ (X331 @ sh @ one)) @ id)) | (X328 @ (lam @ X327)) | ~ ! [X336 : term] : (((X329 @ id @ X336 @ id)) = ((X328 @ X336)))) => ! [X337 : (subst > term > subst > $o),X338 : term] : ((X337 @ id @ X338 @ id) | ~ ! [X339 : subst,X340 : term,X341 : subst,X342 : subst] : (~(X337 @ X339 @ X340 @ (comp @ X341 @ X342)) | (X337 @ (comp @ X339 @ X341) @ (sub @ X340 @ X341) @ X342)) | ~ ! [X343 : term,X344 : term] : ((X337 @ id @ (ap @ (sub @ X343 @ id) @ X344) @ id) | ~(X337 @ id @ X343 @ id) | ~(X337 @ id @ X344 @ id)) | ~ ! [X345 : (subst > term > term)] : ((X337 @ id @ (lam @ (X345 @ sh @ one)) @ id) | ~ ! [X346 : term] : ((X337 @ id @ (X345 @ id @ X346) @ id) | ~(X337 @ id @ X346 @ id)) | ~ ! [X347 : term,X348 : subst,X349 : subst] : (((sub @ (X345 @ X348 @ X347) @ X349)) = ((X345 @ (comp @ X348 @ X349) @ (sub @ X347 @ X349))))) | ~ ! [X350 : subst,X351 : subst,X352 : term,X353 : subst] : (~(X337 @ (comp @ X351 @ X350) @ (sub @ X352 @ X350) @ X353) | (X337 @ X351 @ X352 @ (comp @ X350 @ X353))) | ~ ! [X354 : term] : ((X337 @ id @ X354 @ id) | ~(var @ (sub @ X354 @ id))))))))) & axabs & ~lamnotap & ~ulamvar1 & hoasinduction_lem3v2_gthm & ((! [X355 : subst,X356 : term] : (((sub @ one @ (push @ X356 @ X355))) = X356) => (! [X357 : subst,X358 : term] : (((comp @ sh @ (push @ X358 @ X357))) = X357) => (! [X359 : subst] : (((comp @ X359 @ id)) = X359) => (! [X360 : term,X361 : (term > $o)] : (~ ! [X362 : term] : (~(var @ X362) | (X361 @ X362)) | ~ ! [X363 : term,X364 : term] : (~(X361 @ X364) | (X361 @ (ap @ X363 @ X364)) | ~(X361 @ X363)) | ~ ! [X365 : term] : (~(X361 @ X365) | (X361 @ (lam @ X365))) | (X361 @ X360)) => ! [X366 : (term > $o),X367 : subst,X368 : term] : (~ ! [X369 : term] : (~(var @ X369) | (X366 @ (sub @ X369 @ X367))) | ~ ! [X370 : term] : ((X366 @ (lam @ X370)) | ~ ! [X371 : term] : ((X366 @ (sub @ X370 @ (push @ X371 @ id))) | ~(X366 @ X371))) | ~ ! [X372 : term,X373 : term] : (~(X366 @ X372) | ~(X366 @ X373) | (X366 @ (ap @ X372 @ X373))) | (X366 @ (sub @ X368 @ X367))))))) <=> induction2lem_lthm) & hoasinduction_lem2_lthm & (! [X374 : term,X375 : (subst > term > term),X376 : (subst > term > term),X377 : subst] : (~ ! [X378 : term,X379 : subst,X380 : subst] : (((sub @ (X375 @ X380 @ X378) @ X379)) = ((X375 @ (comp @ X380 @ X379) @ (sub @ X378 @ X379)))) | (((lam @ (X375 @ sh @ one))) != ((lam @ (X376 @ sh @ one)))) | ~ ! [X381 : subst,X382 : subst,X383 : term] : (((X376 @ (comp @ X381 @ X382) @ (sub @ X383 @ X382))) = ((sub @ (X376 @ X381 @ X383) @ X382))) | (((X375 @ X377 @ X374)) = ((X376 @ X377 @ X374)))) <=> hoaslaminj) & pushprop_lem2v2 & (pushprop_lem1_lthm <=> (! [X384 : subst,X385 : term] : (((sub @ one @ (push @ X385 @ X384))) = X385) => (! [X386 : subst,X387 : term] : (((comp @ sh @ (push @ X387 @ X386))) = X386) => ! [X388 : (term > $o),X389 : term,X390 : (term > $o),X391 : term,X392 : subst] : (~(X390 @ X389) | (X388 @ (sub @ X389 @ (push @ X391 @ X392))))))) & (pushprop_lem1v2 <=> ! [X393 : (term > $o),X394 : subst,X395 : term,X396 : (term > $o)] : (~(X396 @ X395) | ~ ! [X397 : term] : (((X393 @ X397)) = ((X396 @ (sub @ X397 @ (push @ X395 @ X394))))) | (X393 @ one))) & (((ap @ (d2term @ d_one) @ (d2term @ d_one))) = ((d2term @ d_one))) & axvarshift & (! [X398 : term,X399 : (subst > term > subst > $o)] : (~ ! [X400 : subst,X401 : term,X402 : subst,X403 : subst] : ((X399 @ (comp @ X400 @ X402) @ (sub @ X401 @ X402) @ X403) | ~(X399 @ X400 @ X401 @ (comp @ X402 @ X403))) | ~ ! [X404 : term,X405 : term] : (~(X399 @ id @ X404 @ id) | ~(X399 @ id @ X405 @ id) | (X399 @ id @ (ap @ (sub @ X404 @ id) @ X405) @ id)) | (X399 @ id @ X398 @ id) | ~ ! [X406 : term] : ((X399 @ id @ X406 @ id) | ~(var @ (sub @ X406 @ id))) | ~ ! [X407 : subst,X408 : term,X409 : subst,X410 : subst] : ((X399 @ X409 @ X408 @ (comp @ X407 @ X410)) | ~(X399 @ (comp @ X409 @ X407) @ (sub @ X408 @ X407) @ X410)) | ~ ! [X411 : (subst > term > term)] : (~ ! [X412 : term] : ((X399 @ id @ (X411 @ id @ X412) @ id) | ~(X399 @ id @ X412 @ id)) | ~ ! [X413 : subst,X414 : subst,X415 : term] : (((X411 @ (comp @ X413 @ X414) @ (sub @ X415 @ X414))) = ((sub @ (X411 @ X413 @ X415) @ X414))) | (X399 @ id @ (lam @ (X411 @ sh @ one)) @ id))) <=> hoasinduction) & pushprop & ((! [X416 : term] : (((sub @ X416 @ id)) = X416) => (! [X417 : term,X418 : (subst > term > subst > $o)] : (~ ! [X419 : (subst > term > term)] : ((X418 @ id @ (lam @ (X419 @ sh @ one)) @ id) | ~ ! [X420 : term] : (~(X418 @ id @ X420 @ id) | (X418 @ id @ (X419 @ id @ X420) @ id)) | ~ ! [X421 : subst,X422 : subst,X423 : term] : (((sub @ (X419 @ X422 @ X423) @ X421)) = ((X419 @ (comp @ X422 @ X421) @ (sub @ X423 @ X421))))) | (X418 @ id @ (lam @ (sub @ X417 @ (push @ one @ sh))) @ id) | ~ ! [X424 : term] : ((X418 @ id @ (sub @ X417 @ (push @ X424 @ id)) @ id) | ~(X418 @ id @ X424 @ id))) => ! [X425 : term,X426 : (subst > term > subst > $o)] : (~ ! [X427 : (subst > term > term)] : (~ ! [X428 : subst,X429 : subst,X430 : term] : (((sub @ (X427 @ X428 @ X430) @ X429)) = ((X427 @ (comp @ X428 @ X429) @ (sub @ X430 @ X429)))) | ~ ! [X431 : term] : ((X426 @ id @ (X427 @ id @ X431) @ id) | ~(X426 @ id @ X431 @ id)) | (X426 @ id @ (lam @ (X427 @ sh @ one)) @ id)) | ~ ! [X432 : term] : (~(X426 @ id @ X432 @ id) | (X426 @ id @ (sub @ X425 @ (push @ X432 @ id)) @ id)) | (X426 @ id @ (lam @ X425) @ id)))) <=> hoasinduction_lem3a_lthm) & pushprop_lem1v2_gthm & (shinj <=> ! [X433 : term,X434 : term] : ((((sub @ X433 @ sh)) != ((sub @ X434 @ sh))) | (X433 = X434))) & ~(hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id)) & ~(var @ (d2term @ d_one)) & substmonoid_gthm & (hoasinduction_lem2v2 <=> ! [X435 : term,X436 : (subst > term > subst > $o),X437 : (term > $o),X438 : term] : (~ ! [X439 : term] : (((X436 @ id @ X439 @ id)) = ((X437 @ X439))) | ~(X437 @ X435) | ~(X437 @ X438) | ~ ! [X440 : term,X441 : term] : (~(X436 @ id @ X440 @ id) | ~(X436 @ id @ X441 @ id) | (X436 @ id @ (ap @ (sub @ X440 @ id) @ X441) @ id)) | ~ ! [X442 : subst,X443 : subst,X444 : term,X445 : subst] : ((X436 @ X442 @ X444 @ (comp @ X443 @ X445)) | ~(X436 @ (comp @ X442 @ X443) @ (sub @ X444 @ X443) @ X445)) | (X437 @ (ap @ X438 @ X435)) | ~ ! [X446 : subst,X447 : subst,X448 : term,X449 : subst] : (~(X436 @ X447 @ X448 @ (comp @ X446 @ X449)) | (X436 @ (comp @ X447 @ X446) @ (sub @ X448 @ X446) @ X449)))) & ((! [X450 : (subst > term > subst > $o)] : ~ ! [X451 : (term > $o)] : ~ ! [X452 : term] : (((X450 @ id @ X452 @ id)) = ((X451 @ X452))) => (! [X453 : term,X454 : (term > $o)] : ((X454 @ X453) | ~ ! [X455 : term,X456 : term] : ((X454 @ (ap @ X455 @ X456)) | ~(X454 @ X456) | ~(X454 @ X455)) | ~ ! [X457 : term] : ((X454 @ (lam @ X457)) | ~ ! [X458 : term] : ((X454 @ (sub @ X457 @ (push @ X458 @ id))) | ~(X454 @ X458))) | ~ ! [X459 : term] : (~(var @ X459) | (X454 @ X459))) => (! [X460 : term] : (((sub @ X460 @ id)) = X460) => (! [X461 : (term > $o),X462 : (subst > term > subst > $o),X463 : term] : (~ ! [X464 : term] : (((X461 @ X464)) = ((X462 @ id @ X464 @ id))) | ~ ! [X465 : term] : ((X461 @ (sub @ X463 @ (push @ X465 @ id))) | ~(X461 @ X465)) | (X461 @ (lam @ X463)) | ~ ! [X466 : (subst > term > term)] : (~ ! [X467 : term] : ((X462 @ id @ (X466 @ id @ X467) @ id) | ~(X462 @ id @ X467 @ id)) | (X462 @ id @ (lam @ (X466 @ sh @ one)) @ id) | ~ ! [X468 : term,X469 : subst,X470 : subst] : (((X466 @ (comp @ X470 @ X469) @ (sub @ X468 @ X469))) = ((sub @ (X466 @ X470 @ X468) @ X469))))) => ! [X471 : (subst > term > subst > $o),X472 : term] : ((X471 @ id @ X472 @ id) | ~ ! [X473 : term,X474 : term] : ((X471 @ id @ (ap @ (sub @ X474 @ id) @ X473) @ id) | ~(X471 @ id @ X474 @ id) | ~(X471 @ id @ X473 @ id)) | ~ ! [X475 : (subst > term > term)] : ((X471 @ id @ (lam @ (X475 @ sh @ one)) @ id) | ~ ! [X476 : term,X477 : subst,X478 : subst] : (((X475 @ (comp @ X477 @ X478) @ (sub @ X476 @ X478))) = ((sub @ (X475 @ X477 @ X476) @ X478))) | ~ ! [X479 : term] : (~(X471 @ id @ X479 @ id) | (X471 @ id @ (X475 @ id @ X479) @ id)))))))) <=> hoasinduction_no_psi_cond_lthm) & (induction2_lthm <=> (! [X480 : term] : (((sub @ X480 @ id)) = X480) => (! [X481 : subst,X482 : (term > $o),X483 : term] : (~ ! [X484 : term,X485 : term] : (~(X482 @ X484) | (X482 @ (ap @ X484 @ X485)) | ~(X482 @ X485)) | ~ ! [X486 : term] : (~ ! [X487 : term] : (~(X482 @ X487) | (X482 @ (sub @ X486 @ (push @ X487 @ id)))) | (X482 @ (lam @ X486))) | ~ ! [X488 : term] : (~(var @ X488) | (X482 @ (sub @ X488 @ X481))) | (X482 @ (sub @ X483 @ X481))) => ! [X489 : (term > $o),X490 : term] : (~ ! [X491 : term] : (~ ! [X492 : term] : (~(X489 @ X492) | (X489 @ (sub @ X491 @ (push @ X492 @ id)))) | (X489 @ (lam @ X491))) | ~ ! [X493 : term,X494 : term] : (~(X489 @ X494) | (X489 @ (ap @ X494 @ X493)) | ~(X489 @ X493)) | (X489 @ X490) | ~ ! [X495 : term] : (~(var @ X495) | (X489 @ X495)))))) & (! [X496 : (term > $o),X497 : term] : ((X496 @ X497) | ~ ! [X498 : term] : ((X496 @ X498) | ~(var @ X498)) | ~ ! [X499 : term,X500 : term] : ((X496 @ (ap @ X500 @ X499)) | ~(X496 @ X499) | ~(X496 @ X500)) | ~ ! [X501 : term] : (~ ! [X502 : term] : (~(X496 @ X502) | (X496 @ (sub @ X501 @ (push @ X502 @ id)))) | (X496 @ (lam @ X501)))) <=> induction2) & pushprop_lem1_gthm & hoaslamnotap_lthm & hoasinduction_lem1v2 & ((! [X503 : term] : (((sub @ X503 @ id)) = X503) => (! [X504 : (subst > term > subst > $o),X505 : term] : (~ ! [X506 : term] : ((X504 @ id @ (sub @ X505 @ (push @ X506 @ id)) @ id) | ~(X504 @ id @ X506 @ id)) | ~ ! [X507 : (subst > term > term)] : (~ ! [X508 : term] : ((X504 @ id @ (X507 @ id @ X508) @ id) | ~(X504 @ id @ X508 @ id)) | (X504 @ id @ (lam @ (X507 @ sh @ one)) @ id) | ~ ! [X509 : subst,X510 : subst,X511 : term] : (((X507 @ (comp @ X509 @ X510) @ (sub @ X511 @ X510))) = ((sub @ (X507 @ X509 @ X511) @ X510)))) | (X504 @ id @ (lam @ (sub @ X505 @ (push @ one @ sh))) @ id)) => ! [X512 : term,X513 : (subst > term > subst > $o)] : ((X513 @ id @ (lam @ X512) @ id) | ~ ! [X514 : subst,X515 : subst,X516 : subst,X517 : term] : (~(X513 @ (comp @ X515 @ X516) @ (sub @ X517 @ X516) @ X514) | (X513 @ X515 @ X517 @ (comp @ X516 @ X514))) | ~ ! [X518 : term] : ((X513 @ id @ (sub @ X512 @ (push @ X518 @ id)) @ id) | ~(X513 @ id @ X518 @ id)) | ~ ! [X519 : term,X520 : subst,X521 : subst,X522 : subst] : (~(X513 @ X520 @ X519 @ (comp @ X522 @ X521)) | (X513 @ (comp @ X520 @ X522) @ (sub @ X519 @ X522) @ X521)) | ~ ! [X523 : (subst > term > term)] : (~ ! [X524 : term] : (~(X513 @ id @ X524 @ id) | (X513 @ id @ (X523 @ id @ X524) @ id)) | ~ ! [X525 : subst,X526 : term,X527 : subst] : (((sub @ (X523 @ X527 @ X526) @ X525)) = ((X523 @ (comp @ X527 @ X525) @ (sub @ X526 @ X525)))) | (X513 @ id @ (lam @ (X523 @ sh @ one)) @ id))))) <=> hoasinduction_lem3_lthm) & pushprop_lthm_orig & pushprop_lem0_gthm & ((! [X528 : term,X529 : term,X530 : term,X531 : term] : ((X528 = X529) | (((ap @ X530 @ X529)) != ((ap @ X531 @ X528)))) => ! [X532 : term,X533 : term,X534 : term,X535 : term] : ((((ap @ (sub @ X533 @ id) @ X535)) != ((ap @ (sub @ X534 @ id) @ X532))) | (X532 = X535))) <=> hoasapinj2_lthm) & (hoasapinj2 <=> ! [X536 : term,X537 : term,X538 : term,X539 : term] : ((((ap @ (sub @ X536 @ id) @ X537)) != ((ap @ (sub @ X539 @ id) @ X538))) | (X537 = X538))) & (axidl <=> ! [X540 : subst] : (((comp @ id @ X540)) = X540)) & (((lam @ (d2term @ d_one))) = ((d2term @ d_one))) & hoasapnotvar_lthm & (hoaslam = (^[X541 : subst, X542 : (subst > term > term)] : ((d2term @ d_one)))) & hoasinduction_lem3a_gthm & ((^[X543 : (subst > term > subst > $o), X544 : (term > $o)] : (! [X545 : term] : (((X544 @ X545)) = ((X543 @ id @ X545 @ id))))) = hoasinduction_p_and_p_prime) & ulamvarsh & apnotvar & hoasinduction_lem1_gthm & ! [X546 : subst] : ? [X547 : d_subst] : (((d2subst @ X547)) = X546) & (! [X548 : term] : ~ ! [X549 : (subst > term > term)] : (((sub @ X548 @ (push @ one @ sh))) != ((X549 @ sh @ one))) <=> hoasinduction_lem3b)),
  inference(rectify,[],[f3])).
thf(f7,plain,(
  (! [X0 : term] : (((sub @ X0 @ id)) = X0) <=> (axvarid = $true)) & (axapp = $true) & (termmset_gthm = $true) & (! [X3 : term,X2 : term,X1 : term,X4 : term] : ((((ap @ X3 @ X2)) != ((ap @ X1 @ X4))) | (X2 = X4)) <=> (apinj2 = $true)) & (axclos = $true) & (hoasinduction_lem3b_gthm = $true) & (hoasinduction_lem2_gthm = $true) & (! [X5 : (term > $o),X7 : subst,X6 : term] : ~ ! [X8 : (term > $o)] : ~ ! [X9 : term] : (((X5 @ (sub @ X9 @ (push @ X6 @ X7)))) = ((X8 @ X9))) <=> (pushprop_lem0_lthm = $true)) & ((hoasinduction_lem0_lthm = $true) <=> ! [X10 : (subst > term > subst > $o)] : ~ ! [X11 : (term > $o)] : ~ ! [X12 : term] : (((X11 @ X12)) = ((X10 @ id @ X12 @ id)))) & (pushprop_lem2v2_gthm = $true) & ((! [X13 : term] : ~ ! [X14 : (subst > term > term)] : ~ ! [X16 : subst,X15 : term] : (((sub @ X13 @ (push @ X15 @ X16))) = ((X14 @ X16 @ X15))) => (! [X17 : term] : (((sub @ X17 @ id)) = X17) => ! [X18 : term,X19 : (term > $o),X20 : (subst > term > subst > $o)] : (~ ! [X21 : term] : (((X19 @ X21)) = ((X20 @ id @ X21 @ id))) | ~ ! [X22 : term] : (($true = ((X19 @ (sub @ X18 @ (push @ X22 @ id))))) | ~ (((X19 @ X22)) = $true)) | ($true = ((X19 @ (lam @ X18)))) | ~ ! [X23 : (subst > term > term)] : (($true = ((X20 @ id @ (lam @ (X23 @ sh @ one)) @ id))) | ~ ! [X24 : term] : ((((X20 @ id @ (X23 @ id @ X24) @ id)) = $true) | ~ ($true = ((X20 @ id @ X24 @ id)))) | ~ ! [X25 : subst,X26 : term,X27 : subst] : (((sub @ (X23 @ X25 @ X26) @ X27)) = ((X23 @ (comp @ X25 @ X27) @ (sub @ X26 @ X27)))))))) <=> (hoasinduction_lem3v2a_lthm = $true)) & (! [X30 : term,X31 : term,X29 : term,X28 : term] : ((((ap @ (sub @ X30 @ id) @ X29)) != ((ap @ (sub @ X28 @ id) @ X31))) | (X28 = X30)) <=> (hoasapinj1 = $true)) & (axmap = $true) & (induction2_gthm = $true) & ((hoaslaminj_lthm = $true) <=> (! [X33 : term,X32 : subst] : (((sub @ one @ (push @ X33 @ X32))) = X33) => (! [X34 : term,X35 : subst] : (((comp @ sh @ (push @ X34 @ X35))) = X35) => (! [X36 : term,X37 : term] : ((((lam @ X37)) != ((lam @ X36))) | (X36 = X37)) => ! [X41 : term,X40 : (subst > term > term),X38 : subst,X39 : (subst > term > term)] : ((((X40 @ X38 @ X41)) = ((X39 @ X38 @ X41))) | ~ ! [X45 : subst,X46 : term,X47 : subst] : (((sub @ (X40 @ X45 @ X46) @ X47)) = ((X40 @ (comp @ X45 @ X47) @ (sub @ X46 @ X47)))) | (((lam @ (X39 @ sh @ one))) != ((lam @ (X40 @ sh @ one)))) | ~ ! [X43 : subst,X44 : term,X42 : subst] : (((sub @ (X39 @ X43 @ X44) @ X42)) = ((X39 @ (comp @ X43 @ X42) @ (sub @ X44 @ X42))))))))) & (hoasinduction_lem1v2_gthm = $true) & ! [X48 : d_subst] : (d_id = X48) & (hoaslaminj_gthm = $true) & ((termmset_lthm = $true) <=> (! [X49 : term] : (((sub @ X49 @ id)) = X49) => ! [X50 : term] : (((sub @ X50 @ id)) = X50))) & ! [X51 : d_term,X52 : d_term] : ((((d2term @ X51)) = ((d2term @ X52))) => (X51 = X52)) & ((hoasinduction_lem3a = $true) <=> ! [X53 : term,X54 : (subst > term > subst > $o)] : (~ ! [X55 : (subst > term > term)] : (($true = ((X54 @ id @ (lam @ (X55 @ sh @ one)) @ id))) | ~ ! [X57 : subst,X56 : subst,X58 : term] : (((X55 @ (comp @ X57 @ X56) @ (sub @ X58 @ X56))) = ((sub @ (X55 @ X57 @ X58) @ X56))) | ~ ! [X59 : term] : (~ ($true = ((X54 @ id @ X59 @ id))) | ($true = ((X54 @ id @ (X55 @ id @ X59) @ id))))) | ~ ! [X60 : term] : (~ ($true = ((X54 @ id @ X60 @ id))) | ($true = ((X54 @ id @ (sub @ X53 @ (push @ X60 @ id)) @ id)))) | ($true = ((X54 @ id @ (lam @ X53) @ id))))) & ! [X61 : term] : ? [X62 : d_term] : (((d2term @ X62)) = X61) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (hoasinduction_gthm = $true) & ((axvarcons = $true) <=> ! [X64 : subst,X63 : term] : (((sub @ one @ (push @ X63 @ X64))) = X63)) & (! [X65 : (term > $o),X66 : term] : (($true = ((X65 @ X66))) | ~ ! [X67 : term] : (($true = ((X65 @ X67))) | ~ ($true = ((var @ X67)))) | ~ ! [X68 : term,X69 : term] : (~ ($true = ((X65 @ X68))) | ~ (((X65 @ X69)) = $true) | ($true = ((X65 @ (ap @ X68 @ X69))))) | ~ ! [X70 : term] : ((((X65 @ (lam @ X70))) = $true) | ~ ($true = ((X65 @ X70))))) <=> (induction = $true)) & (axassoc = $true) & (hoaslamnotap_gthm = $true) & (! [X71 : term] : (((sub @ X71 @ id)) = X71) <=> (termmset = $true)) & (pushprop_lthm = $true) & (! [X72 : (subst > term > subst > $o),X73 : term] : (~ ! [X74 : term] : (~ ($true = ((X72 @ id @ X74 @ id))) | ($true = ((X72 @ id @ (sub @ X73 @ (push @ X74 @ id)) @ id)))) | ~ ! [X75 : subst,X76 : subst,X77 : term,X78 : subst] : ((((X72 @ X75 @ X77 @ (comp @ X78 @ X76))) = $true) | ~ ($true = ((X72 @ (comp @ X75 @ X78) @ (sub @ X77 @ X78) @ X76)))) | ($true = ((X72 @ id @ (lam @ X73) @ id))) | ~ ! [X79 : (subst > term > term)] : (~ ! [X80 : term] : (($true = ((X72 @ id @ (X79 @ id @ X80) @ id))) | ~ (((X72 @ id @ X80 @ id)) = $true)) | (((X72 @ id @ (lam @ (X79 @ sh @ one)) @ id)) = $true) | ~ ! [X83 : subst,X81 : term,X82 : subst] : (((X79 @ (comp @ X83 @ X82) @ (sub @ X81 @ X82))) = ((sub @ (X79 @ X83 @ X81) @ X82)))) | ~ ! [X84 : subst,X85 : subst,X86 : subst,X87 : term] : (($true = ((X72 @ (comp @ X85 @ X84) @ (sub @ X87 @ X84) @ X86))) | ~ ($true = ((X72 @ X85 @ X87 @ (comp @ X84 @ X86)))))) <=> (hoasinduction_lem3 = $true)) & (hoasinduction_lem1_lthm = $true) & ((substmonoid_lthm = $true) <=> (! [X88 : subst] : (((comp @ id @ X88)) = X88) => (! [X89 : subst] : (((comp @ X89 @ id)) = X89) => (! [X91 : subst] : (((comp @ id @ X91)) = X91) & ! [X90 : subst] : (((comp @ X90 @ id)) = X90))))) & (id = ((d2subst @ d_id))) & (pushprop_lem3v2 = $true) & (hoasinduction_lem1 = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoaslamnotap = $true) <=> ! [X93 : (subst > term > term),X94 : term,X92 : term] : ((((lam @ (X93 @ sh @ one))) != ((ap @ (sub @ X94 @ id) @ X92))) | ~ ! [X95 : subst,X96 : term,X97 : subst] : (((X93 @ (comp @ X95 @ X97) @ (sub @ X96 @ X97))) = ((sub @ (X93 @ X95 @ X96) @ X97))))) & (! [X99 : term,X98 : term] : ((((lam @ X98)) != ((lam @ X99))) | (X98 = X99)) <=> (laminj = $true)) & (((d2subst @ d_id)) = sh) & ((hoasinduction_lem3v2_f = $true) <=> ! [X100 : term] : ~ ! [X101 : (subst > term > term)] : ~ ! [X102 : subst,X103 : term] : (((X101 @ X102 @ X103)) = ((sub @ X100 @ (push @ X103 @ X102))))) & (hoasinduction_lem3_gthm = $true) & ((! [X104 : subst,X105 : term] : (((sub @ one @ (push @ X105 @ X104))) = X105) => ! [X106 : (term > $o),X107 : term,X108 : (term > $o),X109 : subst] : (~ ! [X110 : term] : (((X106 @ X110)) = ((X108 @ (sub @ X110 @ (push @ X107 @ X109))))) | ~ ($true = ((X108 @ X107))) | ($true = ((X106 @ one))))) <=> (pushprop_lem1v2_lthm = $true)) & (hoaslamnotvar_gthm = $true) & (induction2lem_gthm = $true) & ! [X111 : d_term] : (d_one = X111) & (one = ((d2term @ d_one))) & (hoasapnotvar_gthm = $true) & ((axshiftcons = $true) <=> ! [X112 : term,X113 : subst] : (((comp @ sh @ (push @ X112 @ X113))) = X113)) & ((hoasinduction_lem3aaa = $true) <=> ! [X114 : (subst > term > subst > $o),X115 : term] : (~ ! [X116 : term] : (~ ($true = ((X114 @ id @ X116 @ id))) | ($true = ((X114 @ id @ (sub @ X115 @ (push @ X116 @ id)) @ id)))) | ~ ! [X117 : (subst > term > term),X118 : term] : ((((X114 @ id @ (lam @ (X117 @ sh @ one)) @ id)) = $true) | ~ ! [X121 : subst,X119 : subst,X120 : term] : (((X117 @ (comp @ X119 @ X121) @ (sub @ X120 @ X121))) = ((sub @ X118 @ (push @ (sub @ X120 @ X121) @ (comp @ X119 @ X121))))) | ~ ! [X124 : subst,X123 : term,X122 : subst] : (((sub @ (sub @ X118 @ (push @ X123 @ X122)) @ X124)) = ((sub @ (X117 @ X122 @ X123) @ X124))) | ~ ! [X125 : term] : (($true = ((X114 @ id @ (X117 @ id @ X125) @ id))) | ~ ($true = ((X114 @ id @ X125 @ id))))) | ($true = ((X114 @ id @ (lam @ (sub @ X115 @ (push @ one @ sh))) @ id))))) & (pushprop_lem2v2_lthm = $true) & (! [X126 : (subst > term > subst > $o)] : ~ ! [X127 : (term > $o)] : ~ ! [X128 : term] : (((X127 @ X128)) = ((X126 @ id @ X128 @ id))) <=> (hoasinduction_lem0 = $true)) & ((hoasinduction_lthm = $true) <=> (! [X129 : (term > $o),X130 : term] : (~ ! [X131 : term,X132 : term] : (~ ($true = ((X129 @ X131))) | ($true = ((X129 @ (ap @ X131 @ X132)))) | ~ (((X129 @ X132)) = $true)) | ~ ! [X133 : term] : (~ ($true = ((var @ X133))) | ($true = ((X129 @ X133)))) | ($true = ((X129 @ X130))) | ~ ! [X134 : term] : ((((X129 @ (lam @ X134))) = $true) | ~ ! [X135 : term] : (($true = ((X129 @ (sub @ X134 @ (push @ X135 @ id))))) | ~ ($true = ((X129 @ X135)))))) => (! [X136 : term,X137 : (subst > term > subst > $o),X138 : term] : (~ ! [X139 : subst,X140 : subst,X141 : term,X142 : subst] : ((((X137 @ (comp @ X140 @ X139) @ (sub @ X141 @ X139) @ X142)) = $true) | ~ ($true = ((X137 @ X140 @ X141 @ (comp @ X139 @ X142))))) | ~ ! [X143 : term,X144 : term] : ((((X137 @ id @ (ap @ (sub @ X143 @ id) @ X144) @ id)) = $true) | ~ (((X137 @ id @ X144 @ id)) = $true) | ~ ($true = ((X137 @ id @ X143 @ id)))) | ~ (((X137 @ id @ X138 @ id)) = $true) | (((X137 @ id @ (ap @ X138 @ X136) @ id)) = $true) | ~ ($true = ((X137 @ id @ X136 @ id))) | ~ ! [X145 : subst,X146 : subst,X147 : term,X148 : subst] : (($true = ((X137 @ X145 @ X147 @ (comp @ X146 @ X148)))) | ~ (((X137 @ (comp @ X145 @ X146) @ (sub @ X147 @ X146) @ X148)) = $true))) => (! [X149 : (subst > term > subst > $o),X150 : term] : (~ ! [X151 : term,X152 : subst,X153 : subst,X154 : subst] : ((((X149 @ X152 @ X151 @ (comp @ X153 @ X154))) = $true) | ~ ($true = ((X149 @ (comp @ X152 @ X153) @ (sub @ X151 @ X153) @ X154)))) | ~ ! [X155 : term] : (~ (((X149 @ id @ X155 @ id)) = $true) | (((X149 @ id @ (sub @ X150 @ (push @ X155 @ id)) @ id)) = $true)) | ($true = ((X149 @ id @ (lam @ X150) @ id))) | ~ ! [X156 : subst,X157 : subst,X158 : term,X159 : subst] : (~ ($true = ((X149 @ X157 @ X158 @ (comp @ X159 @ X156)))) | ($true = ((X149 @ (comp @ X157 @ X159) @ (sub @ X158 @ X159) @ X156)))) | ~ ! [X160 : (subst > term > term)] : (($true = ((X149 @ id @ (lam @ (X160 @ sh @ one)) @ id))) | ~ ! [X161 : subst,X163 : subst,X162 : term] : (((sub @ (X160 @ X163 @ X162) @ X161)) = ((X160 @ (comp @ X163 @ X161) @ (sub @ X162 @ X161)))) | ~ ! [X164 : term] : (($true = ((X149 @ id @ (X160 @ id @ X164) @ id))) | ~ ($true = ((X149 @ id @ X164 @ id)))))) => ! [X165 : (subst > term > subst > $o),X166 : term] : (~ ! [X167 : term,X168 : subst,X169 : subst,X170 : subst] : (~ ($true = ((X165 @ X169 @ X167 @ (comp @ X168 @ X170)))) | ($true = ((X165 @ (comp @ X169 @ X168) @ (sub @ X167 @ X168) @ X170)))) | ~ ! [X171 : subst,X172 : term,X173 : subst,X174 : subst] : (($true = ((X165 @ X173 @ X172 @ (comp @ X174 @ X171)))) | ~ ($true = ((X165 @ (comp @ X173 @ X174) @ (sub @ X172 @ X174) @ X171)))) | ~ ! [X175 : term,X176 : term] : (($true = ((X165 @ id @ (ap @ (sub @ X175 @ id) @ X176) @ id))) | ~ ($true = ((X165 @ id @ X175 @ id))) | ~ ($true = ((X165 @ id @ X176 @ id)))) | ~ ! [X177 : term] : (~ (((var @ (sub @ X177 @ id))) = $true) | ($true = ((X165 @ id @ X177 @ id)))) | ~ ! [X178 : (subst > term > term)] : (~ ! [X179 : subst,X180 : subst,X181 : term] : (((sub @ (X178 @ X179 @ X181) @ X180)) = ((X178 @ (comp @ X179 @ X180) @ (sub @ X181 @ X180)))) | ~ ! [X182 : term] : (($true = ((X165 @ id @ (X178 @ id @ X182) @ id))) | ~ (((X165 @ id @ X182 @ id)) = $true)) | ($true = ((X165 @ id @ (lam @ (X178 @ sh @ one)) @ id)))) | ($true = ((X165 @ id @ X166 @ id)))))))) & (hoasapinj1_gthm = $true) & (! [X183 : subst] : (((push @ (sub @ one @ X183) @ (comp @ sh @ X183))) = X183) <=> (axscons = $true)) & (hoaslamnotvar_lthm = $true) & ((pushprop_lem1 = $true) <=> ! [X184 : (term > $o),X185 : term,X186 : (term > $o),X187 : subst,X188 : term] : (~ ($true = ((X186 @ X185))) | (((X184 @ (sub @ X185 @ (push @ X188 @ X187)))) = $true))) & (pushprop_lem3v2_lthm = $true) & (hoasinduction_lem2v2_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (hoasapnotvar = $true) & (! [X194 : term] : ~ ! [X195 : (subst > term > term)] : ~ ! [X196 : subst,X197 : term] : (((X195 @ X196 @ X197)) = ((sub @ X194 @ (push @ X197 @ X196)))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & (! [X198 : term,X199 : term,X200 : (subst > term > subst > $o)] : (~ ($true = ((X200 @ id @ X198 @ id))) | ($true = ((X200 @ id @ (ap @ X198 @ X199) @ id))) | ~ (((X200 @ id @ X199 @ id)) = $true) | ~ ! [X201 : term,X202 : term] : (($true = ((X200 @ id @ (ap @ (sub @ X201 @ id) @ X202) @ id))) | ~ ($true = ((X200 @ id @ X201 @ id))) | ~ ($true = ((X200 @ id @ X202 @ id)))) | ~ ! [X203 : term,X204 : subst,X205 : subst,X206 : subst] : (($true = ((X200 @ X206 @ X203 @ (comp @ X205 @ X204)))) | ~ ($true = ((X200 @ (comp @ X206 @ X205) @ (sub @ X203 @ X205) @ X204)))) | ~ ! [X207 : term,X208 : subst,X209 : subst,X210 : subst] : (($true = ((X200 @ (comp @ X209 @ X210) @ (sub @ X207 @ X210) @ X208))) | ~ ($true = ((X200 @ X209 @ X207 @ (comp @ X210 @ X208)))))) <=> (hoasinduction_lem2 = $true)) & (ulamvarind = $true) & ((hoasinduction_no_psi_cond = $true) <=> ! [X211 : (subst > term > subst > $o),X212 : term] : (~ ! [X213 : (subst > term > term)] : (~ ! [X214 : term,X215 : subst,X216 : subst] : (((X213 @ (comp @ X216 @ X215) @ (sub @ X214 @ X215))) = ((sub @ (X213 @ X216 @ X214) @ X215))) | ~ ! [X217 : term] : (~ (((X211 @ id @ X217 @ id)) = $true) | ($true = ((X211 @ id @ (X213 @ id @ X217) @ id)))) | ($true = ((X211 @ id @ (lam @ (X213 @ sh @ one)) @ id)))) | ($true = ((X211 @ id @ X212 @ id))) | ~ ! [X218 : term,X219 : term] : (~ ($true = ((X211 @ id @ X218 @ id))) | ($true = ((X211 @ id @ (ap @ (sub @ X218 @ id) @ X219) @ id))) | ~ ($true = ((X211 @ id @ X219 @ id)))))) & (hoasapinj2_gthm = $true) & (lamnotvar = $true) & (! [X220 : subst,X221 : (term > $o),X222 : term] : (~ ! [X223 : term] : (($true = ((X221 @ (sub @ X223 @ X220)))) | ~ ($true = ((var @ X223)))) | ~ ! [X224 : term,X225 : term] : (($true = ((X221 @ (ap @ X224 @ X225)))) | ~ ($true = ((X221 @ X224))) | ~ ($true = ((X221 @ X225)))) | ($true = ((X221 @ (sub @ X222 @ X220)))) | ~ ! [X226 : term] : (~ ! [X227 : term] : (~ ($true = ((X221 @ X227))) | (((X221 @ (sub @ X226 @ (push @ X227 @ id)))) = $true)) | ($true = ((X221 @ (lam @ X226)))))) <=> (induction2lem = $true)) & ((! [X228 : subst] : (((comp @ id @ X228)) = X228) & ! [X229 : subst] : (((comp @ X229 @ id)) = X229)) <=> (substmonoid = $true)) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X230 : term] : ~ ! [X231 : (subst > term > term)] : (((X231 @ sh @ one)) != ((sub @ X230 @ (push @ one @ sh))))) & (! [X232 : (term > $o),X233 : term,X234 : (subst > term > subst > $o)] : (~ ! [X235 : (subst > term > term)] : (~ ! [X236 : subst,X237 : term,X238 : subst] : (((sub @ (X235 @ X236 @ X237) @ X238)) = ((X235 @ (comp @ X236 @ X238) @ (sub @ X237 @ X238)))) | ~ ! [X239 : term] : (($true = ((X234 @ id @ (X235 @ id @ X239) @ id))) | ~ ($true = ((X234 @ id @ X239 @ id)))) | ($true = ((X234 @ id @ (lam @ (X235 @ sh @ one)) @ id)))) | ($true = ((X232 @ (lam @ X233)))) | ~ ! [X240 : term] : (((X232 @ X240)) = ((X234 @ id @ X240 @ id))) | ~ ! [X241 : term] : (~ ($true = ((X232 @ X241))) | ($true = ((X232 @ (sub @ X233 @ (push @ X241 @ id))))))) <=> (hoasinduction_lem3v2a = $true)) & ((pushprop_lem0 = $true) <=> ! [X242 : subst,X243 : (term > $o),X244 : term] : ~ ! [X245 : (term > $o)] : ~ ! [X246 : term] : (((X243 @ (sub @ X246 @ (push @ X244 @ X242)))) = ((X245 @ X246)))) & ((hoasinduction_lem3v2 = $true) <=> ! [X247 : term,X248 : (subst > term > subst > $o),X249 : (term > $o)] : (~ ! [X250 : subst,X251 : subst,X252 : term,X253 : subst] : (~ ($true = ((X248 @ X251 @ X252 @ (comp @ X253 @ X250)))) | ($true = ((X248 @ (comp @ X251 @ X253) @ (sub @ X252 @ X253) @ X250)))) | ~ ! [X254 : term] : (((X248 @ id @ X254 @ id)) = ((X249 @ X254))) | ~ ! [X255 : subst,X256 : subst,X257 : subst,X258 : term] : (~ (((X248 @ (comp @ X256 @ X257) @ (sub @ X258 @ X257) @ X255)) = $true) | ($true = ((X248 @ X256 @ X258 @ (comp @ X257 @ X255))))) | ($true = ((X249 @ (lam @ X247)))) | ~ ! [X259 : (subst > term > term)] : (~ ! [X260 : term] : (~ (((X248 @ id @ X260 @ id)) = $true) | ($true = ((X248 @ id @ (X259 @ id @ X260) @ id)))) | ($true = ((X248 @ id @ (lam @ (X259 @ sh @ one)) @ id))) | ~ ! [X261 : subst,X262 : subst,X263 : term] : (((sub @ (X259 @ X262 @ X263) @ X261)) = ((X259 @ (comp @ X262 @ X261) @ (sub @ X263 @ X261))))) | ~ ! [X264 : term] : (~ ($true = ((X249 @ X264))) | ($true = ((X249 @ (sub @ X247 @ (push @ X264 @ id)))))))) & (! [X265 : term,X266 : (subst > term > subst > $o)] : (~ ! [X267 : (subst > term > term)] : (~ ! [X268 : subst,X269 : term,X270 : subst] : (((X267 @ (comp @ X270 @ X268) @ (sub @ X269 @ X268))) = ((sub @ (X267 @ X270 @ X269) @ X268))) | ~ ! [X271 : term] : ((((X266 @ id @ (X267 @ id @ X271) @ id)) = $true) | ~ ($true = ((X266 @ id @ X271 @ id)))) | ($true = ((X266 @ id @ (lam @ (X267 @ sh @ one)) @ id)))) | ($true = ((X266 @ id @ (lam @ (sub @ X265 @ (push @ one @ sh))) @ id))) | ~ ! [X272 : term] : (~ ($true = ((X266 @ id @ X272 @ id))) | (((X266 @ id @ (sub @ X265 @ (push @ X272 @ id)) @ id)) = $true))) <=> (hoasinduction_lem3aa_lthm = $true)) & ((apinj1 = $true) <=> ! [X273 : term,X274 : term,X275 : term,X276 : term] : ((((ap @ X274 @ X275)) != ((ap @ X276 @ X273))) | (X274 = X276))) & (! [X277 : subst] : (((comp @ X277 @ id)) = X277) <=> (axidr = $true)) & ((hoasinduction_lem3v2_lthm = $true) <=> (! [X278 : term] : (((sub @ X278 @ id)) = X278) => ! [X279 : (subst > term > subst > $o),X280 : term,X281 : (term > $o)] : (~ ! [X282 : term] : (((X279 @ id @ X282 @ id)) = ((X281 @ X282))) | (((X281 @ (lam @ X280))) = $true) | ~ ! [X283 : (subst > term > term)] : (~ ! [X284 : term] : (~ ($true = ((X279 @ id @ X284 @ id))) | ($true = ((X279 @ id @ (X283 @ id @ X284) @ id)))) | ~ ! [X285 : subst,X286 : subst,X287 : term] : (((sub @ (X283 @ X285 @ X287) @ X286)) = ((X283 @ (comp @ X285 @ X286) @ (sub @ X287 @ X286)))) | ($true = ((X279 @ id @ (lam @ (X283 @ sh @ one)) @ id)))) | ~ ! [X288 : term,X289 : subst,X290 : subst,X291 : subst] : ((((X279 @ (comp @ X290 @ X291) @ (sub @ X288 @ X291) @ X289)) = $true) | ~ ($true = ((X279 @ X290 @ X288 @ (comp @ X291 @ X289))))) | ~ ! [X292 : term] : (~ ($true = ((X281 @ X292))) | ($true = ((X281 @ (sub @ X280 @ (push @ X292 @ id)))))) | ~ ! [X293 : subst,X294 : subst,X295 : term,X296 : subst] : ((((X279 @ X293 @ X295 @ (comp @ X296 @ X294))) = $true) | ~ ($true = ((X279 @ (comp @ X293 @ X296) @ (sub @ X295 @ X296) @ X294))))))) & (hoaslamnotvar = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (! [X297 : term,X298 : (subst > term > subst > $o)] : (~ ! [X299 : term] : (($true = ((X298 @ id @ (sub @ X297 @ (push @ X299 @ id)) @ id))) | ~ ($true = ((X298 @ id @ X299 @ id)))) | ($true = ((X298 @ id @ (lam @ (sub @ X297 @ (push @ one @ sh))) @ id))) | ~ ! [X300 : (subst > term > term)] : (~ ! [X301 : subst,X302 : term,X303 : subst] : (((sub @ (X300 @ X303 @ X302) @ X301)) = ((X300 @ (comp @ X303 @ X301) @ (sub @ X302 @ X301)))) | ~ ! [X304 : term] : (($true = ((X298 @ id @ (X300 @ id @ X304) @ id))) | ~ (((X298 @ id @ X304 @ id)) = $true)) | ($true = ((X298 @ id @ (lam @ (X300 @ sh @ one)) @ id))))) <=> (hoasinduction_lem3aa = $true)) & ((hoasapinj1_lthm = $true) <=> (! [X305 : term] : (((sub @ X305 @ id)) = X305) => (! [X306 : term,X307 : term,X308 : term,X309 : term] : ((((ap @ X306 @ X308)) != ((ap @ X309 @ X307))) | (X306 = X309)) => ! [X310 : term,X311 : term,X312 : term,X313 : term] : ((X310 = X311) | (((ap @ (sub @ X310 @ id) @ X312)) != ((ap @ (sub @ X311 @ id) @ X313))))))) & (pushprop_gthm = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ! [X314 : d_subst,X315 : d_subst] : ((((d2subst @ X314)) = ((d2subst @ X315))) => (X314 = X315)) & ((hoasinduction_lthm_3 = $true) <=> (! [X316 : (subst > term > subst > $o)] : ~ ! [X317 : (term > $o)] : ~ ! [X318 : term] : (((X317 @ X318)) = ((X316 @ id @ X318 @ id))) => (! [X319 : term,X320 : (term > $o)] : ((((X320 @ X319)) = $true) | ~ ! [X321 : term,X322 : term] : (~ (((X320 @ X322)) = $true) | ~ ($true = ((X320 @ X321))) | ($true = ((X320 @ (ap @ X322 @ X321))))) | ~ ! [X323 : term] : (($true = ((X320 @ X323))) | ~ ($true = ((var @ X323)))) | ~ ! [X324 : term] : (~ ! [X325 : term] : ((((X320 @ (sub @ X324 @ (push @ X325 @ id)))) = $true) | ~ (((X320 @ X325)) = $true)) | ($true = ((X320 @ (lam @ X324)))))) => (! [X326 : term] : (((sub @ X326 @ id)) = X326) => (! [X327 : term,X328 : (term > $o),X329 : (subst > term > subst > $o)] : (~ ! [X330 : term] : (~ ($true = ((X328 @ X330))) | (((X328 @ (sub @ X327 @ (push @ X330 @ id)))) = $true)) | ~ ! [X331 : (subst > term > term)] : (~ ! [X332 : term] : (($true = ((X329 @ id @ (X331 @ id @ X332) @ id))) | ~ (((X329 @ id @ X332 @ id)) = $true)) | ~ ! [X333 : subst,X334 : term,X335 : subst] : (((sub @ (X331 @ X333 @ X334) @ X335)) = ((X331 @ (comp @ X333 @ X335) @ (sub @ X334 @ X335)))) | (((X329 @ id @ (lam @ (X331 @ sh @ one)) @ id)) = $true)) | ($true = ((X328 @ (lam @ X327)))) | ~ ! [X336 : term] : (((X329 @ id @ X336 @ id)) = ((X328 @ X336)))) => ! [X337 : (subst > term > subst > $o),X338 : term] : (($true = ((X337 @ id @ X338 @ id))) | ~ ! [X339 : subst,X340 : term,X341 : subst,X342 : subst] : (~ ($true = ((X337 @ X339 @ X340 @ (comp @ X341 @ X342)))) | ($true = ((X337 @ (comp @ X339 @ X341) @ (sub @ X340 @ X341) @ X342)))) | ~ ! [X343 : term,X344 : term] : ((((X337 @ id @ (ap @ (sub @ X343 @ id) @ X344) @ id)) = $true) | ~ ($true = ((X337 @ id @ X343 @ id))) | ~ ($true = ((X337 @ id @ X344 @ id)))) | ~ ! [X345 : (subst > term > term)] : ((((X337 @ id @ (lam @ (X345 @ sh @ one)) @ id)) = $true) | ~ ! [X346 : term] : (($true = ((X337 @ id @ (X345 @ id @ X346) @ id))) | ~ (((X337 @ id @ X346 @ id)) = $true)) | ~ ! [X347 : term,X348 : subst,X349 : subst] : (((sub @ (X345 @ X348 @ X347) @ X349)) = ((X345 @ (comp @ X348 @ X349) @ (sub @ X347 @ X349))))) | ~ ! [X350 : subst,X351 : subst,X352 : term,X353 : subst] : (~ ($true = ((X337 @ (comp @ X351 @ X350) @ (sub @ X352 @ X350) @ X353))) | (((X337 @ X351 @ X352 @ (comp @ X350 @ X353))) = $true)) | ~ ! [X354 : term] : (($true = ((X337 @ id @ X354 @ id))) | ~ ($true = ((var @ (sub @ X354 @ id))))))))))) & (axabs = $true) & ~ (lamnotap = $true) & ~ (ulamvar1 = $true) & (hoasinduction_lem3v2_gthm = $true) & ((! [X355 : subst,X356 : term] : (((sub @ one @ (push @ X356 @ X355))) = X356) => (! [X357 : subst,X358 : term] : (((comp @ sh @ (push @ X358 @ X357))) = X357) => (! [X359 : subst] : (((comp @ X359 @ id)) = X359) => (! [X360 : term,X361 : (term > $o)] : (~ ! [X362 : term] : (~ ($true = ((var @ X362))) | ($true = ((X361 @ X362)))) | ~ ! [X363 : term,X364 : term] : (~ (((X361 @ X364)) = $true) | ($true = ((X361 @ (ap @ X363 @ X364)))) | ~ ($true = ((X361 @ X363)))) | ~ ! [X365 : term] : (~ ($true = ((X361 @ X365))) | ($true = ((X361 @ (lam @ X365))))) | (((X361 @ X360)) = $true)) => ! [X366 : (term > $o),X367 : subst,X368 : term] : (~ ! [X369 : term] : (~ ($true = ((var @ X369))) | ($true = ((X366 @ (sub @ X369 @ X367))))) | ~ ! [X370 : term] : (($true = ((X366 @ (lam @ X370)))) | ~ ! [X371 : term] : (($true = ((X366 @ (sub @ X370 @ (push @ X371 @ id))))) | ~ ($true = ((X366 @ X371))))) | ~ ! [X372 : term,X373 : term] : (~ (((X366 @ X372)) = $true) | ~ ($true = ((X366 @ X373))) | ($true = ((X366 @ (ap @ X372 @ X373))))) | ($true = ((X366 @ (sub @ X368 @ X367))))))))) <=> (induction2lem_lthm = $true)) & (hoasinduction_lem2_lthm = $true) & (! [X374 : term,X375 : (subst > term > term),X376 : (subst > term > term),X377 : subst] : (~ ! [X378 : term,X379 : subst,X380 : subst] : (((sub @ (X375 @ X380 @ X378) @ X379)) = ((X375 @ (comp @ X380 @ X379) @ (sub @ X378 @ X379)))) | (((lam @ (X375 @ sh @ one))) != ((lam @ (X376 @ sh @ one)))) | ~ ! [X381 : subst,X382 : subst,X383 : term] : (((X376 @ (comp @ X381 @ X382) @ (sub @ X383 @ X382))) = ((sub @ (X376 @ X381 @ X383) @ X382))) | (((X375 @ X377 @ X374)) = ((X376 @ X377 @ X374)))) <=> (hoaslaminj = $true)) & (pushprop_lem2v2 = $true) & ((pushprop_lem1_lthm = $true) <=> (! [X384 : subst,X385 : term] : (((sub @ one @ (push @ X385 @ X384))) = X385) => (! [X386 : subst,X387 : term] : (((comp @ sh @ (push @ X387 @ X386))) = X386) => ! [X388 : (term > $o),X389 : term,X390 : (term > $o),X391 : term,X392 : subst] : (~ (((X390 @ X389)) = $true) | (((X388 @ (sub @ X389 @ (push @ X391 @ X392)))) = $true))))) & ((pushprop_lem1v2 = $true) <=> ! [X393 : (term > $o),X394 : subst,X395 : term,X396 : (term > $o)] : (~ ($true = ((X396 @ X395))) | ~ ! [X397 : term] : (((X393 @ X397)) = ((X396 @ (sub @ X397 @ (push @ X395 @ X394))))) | (((X393 @ one)) = $true))) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (axvarshift = $true) & (! [X398 : term,X399 : (subst > term > subst > $o)] : (~ ! [X400 : subst,X401 : term,X402 : subst,X403 : subst] : (($true = ((X399 @ (comp @ X400 @ X402) @ (sub @ X401 @ X402) @ X403))) | ~ ($true = ((X399 @ X400 @ X401 @ (comp @ X402 @ X403))))) | ~ ! [X404 : term,X405 : term] : (~ ($true = ((X399 @ id @ X404 @ id))) | ~ (((X399 @ id @ X405 @ id)) = $true) | ($true = ((X399 @ id @ (ap @ (sub @ X404 @ id) @ X405) @ id)))) | (((X399 @ id @ X398 @ id)) = $true) | ~ ! [X406 : term] : ((((X399 @ id @ X406 @ id)) = $true) | ~ ($true = ((var @ (sub @ X406 @ id))))) | ~ ! [X407 : subst,X408 : term,X409 : subst,X410 : subst] : (($true = ((X399 @ X409 @ X408 @ (comp @ X407 @ X410)))) | ~ (((X399 @ (comp @ X409 @ X407) @ (sub @ X408 @ X407) @ X410)) = $true)) | ~ ! [X411 : (subst > term > term)] : (~ ! [X412 : term] : (($true = ((X399 @ id @ (X411 @ id @ X412) @ id))) | ~ (((X399 @ id @ X412 @ id)) = $true)) | ~ ! [X413 : subst,X414 : subst,X415 : term] : (((X411 @ (comp @ X413 @ X414) @ (sub @ X415 @ X414))) = ((sub @ (X411 @ X413 @ X415) @ X414))) | ($true = ((X399 @ id @ (lam @ (X411 @ sh @ one)) @ id))))) <=> (hoasinduction = $true)) & (pushprop = $true) & ((! [X416 : term] : (((sub @ X416 @ id)) = X416) => (! [X417 : term,X418 : (subst > term > subst > $o)] : (~ ! [X419 : (subst > term > term)] : ((((X418 @ id @ (lam @ (X419 @ sh @ one)) @ id)) = $true) | ~ ! [X420 : term] : (~ (((X418 @ id @ X420 @ id)) = $true) | ($true = ((X418 @ id @ (X419 @ id @ X420) @ id)))) | ~ ! [X421 : subst,X422 : subst,X423 : term] : (((sub @ (X419 @ X422 @ X423) @ X421)) = ((X419 @ (comp @ X422 @ X421) @ (sub @ X423 @ X421))))) | ($true = ((X418 @ id @ (lam @ (sub @ X417 @ (push @ one @ sh))) @ id))) | ~ ! [X424 : term] : (($true = ((X418 @ id @ (sub @ X417 @ (push @ X424 @ id)) @ id))) | ~ (((X418 @ id @ X424 @ id)) = $true))) => ! [X425 : term,X426 : (subst > term > subst > $o)] : (~ ! [X427 : (subst > term > term)] : (~ ! [X428 : subst,X429 : subst,X430 : term] : (((sub @ (X427 @ X428 @ X430) @ X429)) = ((X427 @ (comp @ X428 @ X429) @ (sub @ X430 @ X429)))) | ~ ! [X431 : term] : (($true = ((X426 @ id @ (X427 @ id @ X431) @ id))) | ~ ($true = ((X426 @ id @ X431 @ id)))) | ($true = ((X426 @ id @ (lam @ (X427 @ sh @ one)) @ id)))) | ~ ! [X432 : term] : (~ ($true = ((X426 @ id @ X432 @ id))) | ($true = ((X426 @ id @ (sub @ X425 @ (push @ X432 @ id)) @ id)))) | ($true = ((X426 @ id @ (lam @ X425) @ id)))))) <=> (hoasinduction_lem3a_lthm = $true)) & (pushprop_lem1v2_gthm = $true) & ((shinj = $true) <=> ! [X433 : term,X434 : term] : ((((sub @ X433 @ sh)) != ((sub @ X434 @ sh))) | (X433 = X434))) & ~ (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) = $true) & ~ (((var @ (d2term @ d_one))) = $true) & (substmonoid_gthm = $true) & ((hoasinduction_lem2v2 = $true) <=> ! [X435 : term,X436 : (subst > term > subst > $o),X437 : (term > $o),X438 : term] : (~ ! [X439 : term] : (((X436 @ id @ X439 @ id)) = ((X437 @ X439))) | ~ (((X437 @ X435)) = $true) | ~ ($true = ((X437 @ X438))) | ~ ! [X440 : term,X441 : term] : (~ ($true = ((X436 @ id @ X440 @ id))) | ~ ($true = ((X436 @ id @ X441 @ id))) | (((X436 @ id @ (ap @ (sub @ X440 @ id) @ X441) @ id)) = $true)) | ~ ! [X442 : subst,X443 : subst,X444 : term,X445 : subst] : ((((X436 @ X442 @ X444 @ (comp @ X443 @ X445))) = $true) | ~ (((X436 @ (comp @ X442 @ X443) @ (sub @ X444 @ X443) @ X445)) = $true)) | ($true = ((X437 @ (ap @ X438 @ X435)))) | ~ ! [X446 : subst,X447 : subst,X448 : term,X449 : subst] : (~ (((X436 @ X447 @ X448 @ (comp @ X446 @ X449))) = $true) | ($true = ((X436 @ (comp @ X447 @ X446) @ (sub @ X448 @ X446) @ X449)))))) & ((! [X450 : (subst > term > subst > $o)] : ~ ! [X451 : (term > $o)] : ~ ! [X452 : term] : (((X450 @ id @ X452 @ id)) = ((X451 @ X452))) => (! [X453 : term,X454 : (term > $o)] : ((((X454 @ X453)) = $true) | ~ ! [X455 : term,X456 : term] : ((((X454 @ (ap @ X455 @ X456))) = $true) | ~ (((X454 @ X456)) = $true) | ~ ($true = ((X454 @ X455)))) | ~ ! [X457 : term] : (($true = ((X454 @ (lam @ X457)))) | ~ ! [X458 : term] : (($true = ((X454 @ (sub @ X457 @ (push @ X458 @ id))))) | ~ (((X454 @ X458)) = $true))) | ~ ! [X459 : term] : (~ (((var @ X459)) = $true) | ($true = ((X454 @ X459))))) => (! [X460 : term] : (((sub @ X460 @ id)) = X460) => (! [X461 : (term > $o),X462 : (subst > term > subst > $o),X463 : term] : (~ ! [X464 : term] : (((X461 @ X464)) = ((X462 @ id @ X464 @ id))) | ~ ! [X465 : term] : (($true = ((X461 @ (sub @ X463 @ (push @ X465 @ id))))) | ~ (((X461 @ X465)) = $true)) | ($true = ((X461 @ (lam @ X463)))) | ~ ! [X466 : (subst > term > term)] : (~ ! [X467 : term] : (($true = ((X462 @ id @ (X466 @ id @ X467) @ id))) | ~ ($true = ((X462 @ id @ X467 @ id)))) | ($true = ((X462 @ id @ (lam @ (X466 @ sh @ one)) @ id))) | ~ ! [X468 : term,X469 : subst,X470 : subst] : (((X466 @ (comp @ X470 @ X469) @ (sub @ X468 @ X469))) = ((sub @ (X466 @ X470 @ X468) @ X469))))) => ! [X471 : (subst > term > subst > $o),X472 : term] : ((((X471 @ id @ X472 @ id)) = $true) | ~ ! [X473 : term,X474 : term] : (($true = ((X471 @ id @ (ap @ (sub @ X474 @ id) @ X473) @ id))) | ~ ($true = ((X471 @ id @ X474 @ id))) | ~ ($true = ((X471 @ id @ X473 @ id)))) | ~ ! [X475 : (subst > term > term)] : (($true = ((X471 @ id @ (lam @ (X475 @ sh @ one)) @ id))) | ~ ! [X476 : term,X477 : subst,X478 : subst] : (((X475 @ (comp @ X477 @ X478) @ (sub @ X476 @ X478))) = ((sub @ (X475 @ X477 @ X476) @ X478))) | ~ ! [X479 : term] : (~ ($true = ((X471 @ id @ X479 @ id))) | (((X471 @ id @ (X475 @ id @ X479) @ id)) = $true)))))))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & ((induction2_lthm = $true) <=> (! [X480 : term] : (((sub @ X480 @ id)) = X480) => (! [X481 : subst,X482 : (term > $o),X483 : term] : (~ ! [X484 : term,X485 : term] : (~ ($true = ((X482 @ X484))) | (((X482 @ (ap @ X484 @ X485))) = $true) | ~ ($true = ((X482 @ X485)))) | ~ ! [X486 : term] : (~ ! [X487 : term] : (~ (((X482 @ X487)) = $true) | ($true = ((X482 @ (sub @ X486 @ (push @ X487 @ id)))))) | (((X482 @ (lam @ X486))) = $true)) | ~ ! [X488 : term] : (~ ($true = ((var @ X488))) | (((X482 @ (sub @ X488 @ X481))) = $true)) | ($true = ((X482 @ (sub @ X483 @ X481))))) => ! [X489 : (term > $o),X490 : term] : (~ ! [X491 : term] : (~ ! [X492 : term] : (~ ($true = ((X489 @ X492))) | ($true = ((X489 @ (sub @ X491 @ (push @ X492 @ id)))))) | ($true = ((X489 @ (lam @ X491))))) | ~ ! [X493 : term,X494 : term] : (~ ($true = ((X489 @ X494))) | ($true = ((X489 @ (ap @ X494 @ X493)))) | ~ (((X489 @ X493)) = $true)) | ($true = ((X489 @ X490))) | ~ ! [X495 : term] : (~ ($true = ((var @ X495))) | ($true = ((X489 @ X495)))))))) & (! [X496 : (term > $o),X497 : term] : (($true = ((X496 @ X497))) | ~ ! [X498 : term] : (($true = ((X496 @ X498))) | ~ ($true = ((var @ X498)))) | ~ ! [X499 : term,X500 : term] : ((((X496 @ (ap @ X500 @ X499))) = $true) | ~ ($true = ((X496 @ X499))) | ~ ($true = ((X496 @ X500)))) | ~ ! [X501 : term] : (~ ! [X502 : term] : (~ ($true = ((X496 @ X502))) | (((X496 @ (sub @ X501 @ (push @ X502 @ id)))) = $true)) | ($true = ((X496 @ (lam @ X501)))))) <=> (induction2 = $true)) & (pushprop_lem1_gthm = $true) & (hoaslamnotap_lthm = $true) & (hoasinduction_lem1v2 = $true) & ((! [X503 : term] : (((sub @ X503 @ id)) = X503) => (! [X504 : (subst > term > subst > $o),X505 : term] : (~ ! [X506 : term] : (($true = ((X504 @ id @ (sub @ X505 @ (push @ X506 @ id)) @ id))) | ~ ($true = ((X504 @ id @ X506 @ id)))) | ~ ! [X507 : (subst > term > term)] : (~ ! [X508 : term] : (($true = ((X504 @ id @ (X507 @ id @ X508) @ id))) | ~ (((X504 @ id @ X508 @ id)) = $true)) | (((X504 @ id @ (lam @ (X507 @ sh @ one)) @ id)) = $true) | ~ ! [X509 : subst,X510 : subst,X511 : term] : (((X507 @ (comp @ X509 @ X510) @ (sub @ X511 @ X510))) = ((sub @ (X507 @ X509 @ X511) @ X510)))) | ($true = ((X504 @ id @ (lam @ (sub @ X505 @ (push @ one @ sh))) @ id)))) => ! [X512 : term,X513 : (subst > term > subst > $o)] : ((((X513 @ id @ (lam @ X512) @ id)) = $true) | ~ ! [X514 : subst,X515 : subst,X516 : subst,X517 : term] : (~ (((X513 @ (comp @ X515 @ X516) @ (sub @ X517 @ X516) @ X514)) = $true) | ($true = ((X513 @ X515 @ X517 @ (comp @ X516 @ X514))))) | ~ ! [X518 : term] : (($true = ((X513 @ id @ (sub @ X512 @ (push @ X518 @ id)) @ id))) | ~ (((X513 @ id @ X518 @ id)) = $true)) | ~ ! [X519 : term,X520 : subst,X521 : subst,X522 : subst] : (~ ($true = ((X513 @ X520 @ X519 @ (comp @ X522 @ X521)))) | ($true = ((X513 @ (comp @ X520 @ X522) @ (sub @ X519 @ X522) @ X521)))) | ~ ! [X523 : (subst > term > term)] : (~ ! [X524 : term] : (~ ($true = ((X513 @ id @ X524 @ id))) | (((X513 @ id @ (X523 @ id @ X524) @ id)) = $true)) | ~ ! [X525 : subst,X526 : term,X527 : subst] : (((sub @ (X523 @ X527 @ X526) @ X525)) = ((X523 @ (comp @ X527 @ X525) @ (sub @ X526 @ X525)))) | (((X513 @ id @ (lam @ (X523 @ sh @ one)) @ id)) = $true))))) <=> (hoasinduction_lem3_lthm = $true)) & (pushprop_lthm_orig = $true) & (pushprop_lem0_gthm = $true) & ((! [X528 : term,X529 : term,X530 : term,X531 : term] : ((X528 = X529) | (((ap @ X530 @ X529)) != ((ap @ X531 @ X528)))) => ! [X532 : term,X533 : term,X534 : term,X535 : term] : ((((ap @ (sub @ X533 @ id) @ X535)) != ((ap @ (sub @ X534 @ id) @ X532))) | (X532 = X535))) <=> (hoasapinj2_lthm = $true)) & ((hoasapinj2 = $true) <=> ! [X536 : term,X537 : term,X538 : term,X539 : term] : ((((ap @ (sub @ X536 @ id) @ X537)) != ((ap @ (sub @ X539 @ id) @ X538))) | (X537 = X538))) & ((axidl = $true) <=> ! [X540 : subst] : (((comp @ id @ X540)) = X540)) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (hoasapnotvar_lthm = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (hoasinduction_lem3a_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (ulamvarsh = $true) & (apnotvar = $true) & (hoasinduction_lem1_gthm = $true) & ! [X546 : subst] : ? [X547 : d_subst] : (((d2subst @ X547)) = X546) & (! [X548 : term] : ~ ! [X549 : (subst > term > term)] : (((sub @ X548 @ (push @ one @ sh))) != ((X549 @ sh @ one))) <=> (hoasinduction_lem3b = $true))),
  inference(fool_elimination,[],[f6])).
thf(f8,plain,(
  ~(! [X0 : term] : (((var @ (lam @ X0))) != $true) <=> (lamnotvar = $true))),
  inference(flattening,[],[f5])).
thf(f9,plain,(
  (! [X0 : term] : (((sub @ X0 @ id)) = X0) <=> (axvarid = $true)) & (axapp = $true) & (termmset_gthm = $true) & (! [X3 : term,X2 : term,X1 : term,X4 : term] : ((((ap @ X3 @ X2)) != ((ap @ X1 @ X4))) | (X2 = X4)) <=> (apinj2 = $true)) & (axclos = $true) & (hoasinduction_lem3b_gthm = $true) & (hoasinduction_lem2_gthm = $true) & (! [X5 : (term > $o),X7 : subst,X6 : term] : ~ ! [X8 : (term > $o)] : ~ ! [X9 : term] : (((X5 @ (sub @ X9 @ (push @ X6 @ X7)))) = ((X8 @ X9))) <=> (pushprop_lem0_lthm = $true)) & ((hoasinduction_lem0_lthm = $true) <=> ! [X10 : (subst > term > subst > $o)] : ~ ! [X11 : (term > $o)] : ~ ! [X12 : term] : (((X11 @ X12)) = ((X10 @ id @ X12 @ id)))) & (pushprop_lem2v2_gthm = $true) & ((! [X13 : term] : ~ ! [X14 : (subst > term > term)] : ~ ! [X16 : subst,X15 : term] : (((sub @ X13 @ (push @ X15 @ X16))) = ((X14 @ X16 @ X15))) => (! [X17 : term] : (((sub @ X17 @ id)) = X17) => ! [X18 : term,X19 : (term > $o),X20 : (subst > term > subst > $o)] : (~ ! [X21 : term] : (((X19 @ X21)) = ((X20 @ id @ X21 @ id))) | ~ ! [X22 : term] : (($true = ((X19 @ (sub @ X18 @ (push @ X22 @ id))))) | ~ (((X19 @ X22)) = $true)) | ($true = ((X19 @ (lam @ X18)))) | ~ ! [X23 : (subst > term > term)] : (($true = ((X20 @ id @ (lam @ (X23 @ sh @ one)) @ id))) | ~ ! [X24 : term] : ((((X20 @ id @ (X23 @ id @ X24) @ id)) = $true) | ~ ($true = ((X20 @ id @ X24 @ id)))) | ~ ! [X25 : subst,X26 : term,X27 : subst] : (((sub @ (X23 @ X25 @ X26) @ X27)) = ((X23 @ (comp @ X25 @ X27) @ (sub @ X26 @ X27)))))))) <=> (hoasinduction_lem3v2a_lthm = $true)) & (! [X30 : term,X31 : term,X29 : term,X28 : term] : ((((ap @ (sub @ X30 @ id) @ X29)) != ((ap @ (sub @ X28 @ id) @ X31))) | (X28 = X30)) <=> (hoasapinj1 = $true)) & (axmap = $true) & (induction2_gthm = $true) & ((hoaslaminj_lthm = $true) <=> (! [X33 : term,X32 : subst] : (((sub @ one @ (push @ X33 @ X32))) = X33) => (! [X34 : term,X35 : subst] : (((comp @ sh @ (push @ X34 @ X35))) = X35) => (! [X36 : term,X37 : term] : ((((lam @ X37)) != ((lam @ X36))) | (X36 = X37)) => ! [X41 : term,X40 : (subst > term > term),X38 : subst,X39 : (subst > term > term)] : ((((X40 @ X38 @ X41)) = ((X39 @ X38 @ X41))) | ~ ! [X45 : subst,X46 : term,X47 : subst] : (((sub @ (X40 @ X45 @ X46) @ X47)) = ((X40 @ (comp @ X45 @ X47) @ (sub @ X46 @ X47)))) | (((lam @ (X39 @ sh @ one))) != ((lam @ (X40 @ sh @ one)))) | ~ ! [X43 : subst,X44 : term,X42 : subst] : (((sub @ (X39 @ X43 @ X44) @ X42)) = ((X39 @ (comp @ X43 @ X42) @ (sub @ X44 @ X42))))))))) & (hoasinduction_lem1v2_gthm = $true) & ! [X48 : d_subst] : (d_id = X48) & (hoaslaminj_gthm = $true) & ((termmset_lthm = $true) <=> (! [X49 : term] : (((sub @ X49 @ id)) = X49) => ! [X50 : term] : (((sub @ X50 @ id)) = X50))) & ! [X51 : d_term,X52 : d_term] : ((((d2term @ X51)) = ((d2term @ X52))) => (X51 = X52)) & ((hoasinduction_lem3a = $true) <=> ! [X53 : term,X54 : (subst > term > subst > $o)] : (~ ! [X55 : (subst > term > term)] : (($true = ((X54 @ id @ (lam @ (X55 @ sh @ one)) @ id))) | ~ ! [X57 : subst,X56 : subst,X58 : term] : (((X55 @ (comp @ X57 @ X56) @ (sub @ X58 @ X56))) = ((sub @ (X55 @ X57 @ X58) @ X56))) | ~ ! [X59 : term] : (~ ($true = ((X54 @ id @ X59 @ id))) | ($true = ((X54 @ id @ (X55 @ id @ X59) @ id))))) | ~ ! [X60 : term] : (~ ($true = ((X54 @ id @ X60 @ id))) | ($true = ((X54 @ id @ (sub @ X53 @ (push @ X60 @ id)) @ id)))) | ($true = ((X54 @ id @ (lam @ X53) @ id))))) & ! [X61 : term] : ? [X62 : d_term] : (((d2term @ X62)) = X61) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (hoasinduction_gthm = $true) & ((axvarcons = $true) <=> ! [X64 : subst,X63 : term] : (((sub @ one @ (push @ X63 @ X64))) = X63)) & (! [X65 : (term > $o),X66 : term] : (($true = ((X65 @ X66))) | ~ ! [X67 : term] : (($true = ((X65 @ X67))) | ~ ($true = ((var @ X67)))) | ~ ! [X68 : term,X69 : term] : (~ ($true = ((X65 @ X68))) | ~ (((X65 @ X69)) = $true) | ($true = ((X65 @ (ap @ X68 @ X69))))) | ~ ! [X70 : term] : ((((X65 @ (lam @ X70))) = $true) | ~ ($true = ((X65 @ X70))))) <=> (induction = $true)) & (axassoc = $true) & (hoaslamnotap_gthm = $true) & (! [X71 : term] : (((sub @ X71 @ id)) = X71) <=> (termmset = $true)) & (pushprop_lthm = $true) & (! [X72 : (subst > term > subst > $o),X73 : term] : (~ ! [X74 : term] : (~ ($true = ((X72 @ id @ X74 @ id))) | ($true = ((X72 @ id @ (sub @ X73 @ (push @ X74 @ id)) @ id)))) | ~ ! [X75 : subst,X76 : subst,X77 : term,X78 : subst] : ((((X72 @ X75 @ X77 @ (comp @ X78 @ X76))) = $true) | ~ ($true = ((X72 @ (comp @ X75 @ X78) @ (sub @ X77 @ X78) @ X76)))) | ($true = ((X72 @ id @ (lam @ X73) @ id))) | ~ ! [X79 : (subst > term > term)] : (~ ! [X80 : term] : (($true = ((X72 @ id @ (X79 @ id @ X80) @ id))) | ~ (((X72 @ id @ X80 @ id)) = $true)) | (((X72 @ id @ (lam @ (X79 @ sh @ one)) @ id)) = $true) | ~ ! [X83 : subst,X81 : term,X82 : subst] : (((X79 @ (comp @ X83 @ X82) @ (sub @ X81 @ X82))) = ((sub @ (X79 @ X83 @ X81) @ X82)))) | ~ ! [X84 : subst,X85 : subst,X86 : subst,X87 : term] : (($true = ((X72 @ (comp @ X85 @ X84) @ (sub @ X87 @ X84) @ X86))) | ~ ($true = ((X72 @ X85 @ X87 @ (comp @ X84 @ X86)))))) <=> (hoasinduction_lem3 = $true)) & (hoasinduction_lem1_lthm = $true) & ((substmonoid_lthm = $true) <=> (! [X88 : subst] : (((comp @ id @ X88)) = X88) => (! [X89 : subst] : (((comp @ X89 @ id)) = X89) => (! [X91 : subst] : (((comp @ id @ X91)) = X91) & ! [X90 : subst] : (((comp @ X90 @ id)) = X90))))) & (id = ((d2subst @ d_id))) & (pushprop_lem3v2 = $true) & (hoasinduction_lem1 = $true) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((hoaslamnotap = $true) <=> ! [X93 : (subst > term > term),X94 : term,X92 : term] : ((((lam @ (X93 @ sh @ one))) != ((ap @ (sub @ X94 @ id) @ X92))) | ~ ! [X95 : subst,X96 : term,X97 : subst] : (((X93 @ (comp @ X95 @ X97) @ (sub @ X96 @ X97))) = ((sub @ (X93 @ X95 @ X96) @ X97))))) & (! [X99 : term,X98 : term] : ((((lam @ X98)) != ((lam @ X99))) | (X98 = X99)) <=> (laminj = $true)) & (((d2subst @ d_id)) = sh) & ((hoasinduction_lem3v2_f = $true) <=> ! [X100 : term] : ~ ! [X101 : (subst > term > term)] : ~ ! [X102 : subst,X103 : term] : (((X101 @ X102 @ X103)) = ((sub @ X100 @ (push @ X103 @ X102))))) & (hoasinduction_lem3_gthm = $true) & ((! [X104 : subst,X105 : term] : (((sub @ one @ (push @ X105 @ X104))) = X105) => ! [X106 : (term > $o),X107 : term,X108 : (term > $o),X109 : subst] : (~ ! [X110 : term] : (((X106 @ X110)) = ((X108 @ (sub @ X110 @ (push @ X107 @ X109))))) | ~ ($true = ((X108 @ X107))) | ($true = ((X106 @ one))))) <=> (pushprop_lem1v2_lthm = $true)) & (hoaslamnotvar_gthm = $true) & (induction2lem_gthm = $true) & ! [X111 : d_term] : (d_one = X111) & (one = ((d2term @ d_one))) & (hoasapnotvar_gthm = $true) & ((axshiftcons = $true) <=> ! [X112 : term,X113 : subst] : (((comp @ sh @ (push @ X112 @ X113))) = X113)) & ((hoasinduction_lem3aaa = $true) <=> ! [X114 : (subst > term > subst > $o),X115 : term] : (~ ! [X116 : term] : (~ ($true = ((X114 @ id @ X116 @ id))) | ($true = ((X114 @ id @ (sub @ X115 @ (push @ X116 @ id)) @ id)))) | ~ ! [X117 : (subst > term > term),X118 : term] : ((((X114 @ id @ (lam @ (X117 @ sh @ one)) @ id)) = $true) | ~ ! [X121 : subst,X119 : subst,X120 : term] : (((X117 @ (comp @ X119 @ X121) @ (sub @ X120 @ X121))) = ((sub @ X118 @ (push @ (sub @ X120 @ X121) @ (comp @ X119 @ X121))))) | ~ ! [X124 : subst,X123 : term,X122 : subst] : (((sub @ (sub @ X118 @ (push @ X123 @ X122)) @ X124)) = ((sub @ (X117 @ X122 @ X123) @ X124))) | ~ ! [X125 : term] : (($true = ((X114 @ id @ (X117 @ id @ X125) @ id))) | ~ ($true = ((X114 @ id @ X125 @ id))))) | ($true = ((X114 @ id @ (lam @ (sub @ X115 @ (push @ one @ sh))) @ id))))) & (pushprop_lem2v2_lthm = $true) & (! [X126 : (subst > term > subst > $o)] : ~ ! [X127 : (term > $o)] : ~ ! [X128 : term] : (((X127 @ X128)) = ((X126 @ id @ X128 @ id))) <=> (hoasinduction_lem0 = $true)) & ((hoasinduction_lthm = $true) <=> (! [X129 : (term > $o),X130 : term] : (~ ! [X131 : term,X132 : term] : (~ ($true = ((X129 @ X131))) | ($true = ((X129 @ (ap @ X131 @ X132)))) | ~ (((X129 @ X132)) = $true)) | ~ ! [X133 : term] : (~ ($true = ((var @ X133))) | ($true = ((X129 @ X133)))) | ($true = ((X129 @ X130))) | ~ ! [X134 : term] : ((((X129 @ (lam @ X134))) = $true) | ~ ! [X135 : term] : (($true = ((X129 @ (sub @ X134 @ (push @ X135 @ id))))) | ~ ($true = ((X129 @ X135)))))) => (! [X136 : term,X137 : (subst > term > subst > $o),X138 : term] : (~ ! [X139 : subst,X140 : subst,X141 : term,X142 : subst] : ((((X137 @ (comp @ X140 @ X139) @ (sub @ X141 @ X139) @ X142)) = $true) | ~ ($true = ((X137 @ X140 @ X141 @ (comp @ X139 @ X142))))) | ~ ! [X143 : term,X144 : term] : ((((X137 @ id @ (ap @ (sub @ X143 @ id) @ X144) @ id)) = $true) | ~ (((X137 @ id @ X144 @ id)) = $true) | ~ ($true = ((X137 @ id @ X143 @ id)))) | ~ (((X137 @ id @ X138 @ id)) = $true) | (((X137 @ id @ (ap @ X138 @ X136) @ id)) = $true) | ~ ($true = ((X137 @ id @ X136 @ id))) | ~ ! [X145 : subst,X146 : subst,X147 : term,X148 : subst] : (($true = ((X137 @ X145 @ X147 @ (comp @ X146 @ X148)))) | ~ (((X137 @ (comp @ X145 @ X146) @ (sub @ X147 @ X146) @ X148)) = $true))) => (! [X149 : (subst > term > subst > $o),X150 : term] : (~ ! [X151 : term,X152 : subst,X153 : subst,X154 : subst] : ((((X149 @ X152 @ X151 @ (comp @ X153 @ X154))) = $true) | ~ ($true = ((X149 @ (comp @ X152 @ X153) @ (sub @ X151 @ X153) @ X154)))) | ~ ! [X155 : term] : (~ (((X149 @ id @ X155 @ id)) = $true) | (((X149 @ id @ (sub @ X150 @ (push @ X155 @ id)) @ id)) = $true)) | ($true = ((X149 @ id @ (lam @ X150) @ id))) | ~ ! [X156 : subst,X157 : subst,X158 : term,X159 : subst] : (~ ($true = ((X149 @ X157 @ X158 @ (comp @ X159 @ X156)))) | ($true = ((X149 @ (comp @ X157 @ X159) @ (sub @ X158 @ X159) @ X156)))) | ~ ! [X160 : (subst > term > term)] : (($true = ((X149 @ id @ (lam @ (X160 @ sh @ one)) @ id))) | ~ ! [X161 : subst,X163 : subst,X162 : term] : (((sub @ (X160 @ X163 @ X162) @ X161)) = ((X160 @ (comp @ X163 @ X161) @ (sub @ X162 @ X161)))) | ~ ! [X164 : term] : (($true = ((X149 @ id @ (X160 @ id @ X164) @ id))) | ~ ($true = ((X149 @ id @ X164 @ id)))))) => ! [X165 : (subst > term > subst > $o),X166 : term] : (~ ! [X167 : term,X168 : subst,X169 : subst,X170 : subst] : (~ ($true = ((X165 @ X169 @ X167 @ (comp @ X168 @ X170)))) | ($true = ((X165 @ (comp @ X169 @ X168) @ (sub @ X167 @ X168) @ X170)))) | ~ ! [X171 : subst,X172 : term,X173 : subst,X174 : subst] : (($true = ((X165 @ X173 @ X172 @ (comp @ X174 @ X171)))) | ~ ($true = ((X165 @ (comp @ X173 @ X174) @ (sub @ X172 @ X174) @ X171)))) | ~ ! [X175 : term,X176 : term] : (($true = ((X165 @ id @ (ap @ (sub @ X175 @ id) @ X176) @ id))) | ~ ($true = ((X165 @ id @ X175 @ id))) | ~ ($true = ((X165 @ id @ X176 @ id)))) | ~ ! [X177 : term] : (~ (((var @ (sub @ X177 @ id))) = $true) | ($true = ((X165 @ id @ X177 @ id)))) | ~ ! [X178 : (subst > term > term)] : (~ ! [X179 : subst,X180 : subst,X181 : term] : (((sub @ (X178 @ X179 @ X181) @ X180)) = ((X178 @ (comp @ X179 @ X180) @ (sub @ X181 @ X180)))) | ~ ! [X182 : term] : (($true = ((X165 @ id @ (X178 @ id @ X182) @ id))) | ~ (((X165 @ id @ X182 @ id)) = $true)) | ($true = ((X165 @ id @ (lam @ (X178 @ sh @ one)) @ id)))) | ($true = ((X165 @ id @ X166 @ id)))))))) & (hoasapinj1_gthm = $true) & (! [X183 : subst] : (((push @ (sub @ one @ X183) @ (comp @ sh @ X183))) = X183) <=> (axscons = $true)) & (hoaslamnotvar_lthm = $true) & ((pushprop_lem1 = $true) <=> ! [X184 : (term > $o),X185 : term,X186 : (term > $o),X187 : subst,X188 : term] : (~ ($true = ((X186 @ X185))) | (((X184 @ (sub @ X185 @ (push @ X188 @ X187)))) = $true))) & (pushprop_lem3v2_lthm = $true) & (hoasinduction_lem2v2_gthm = $true) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (hoasapnotvar = $true) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X189 : term] : ~ ! [X190 : (subst > term > term)] : ~ ! [X192 : term,X191 : subst] : (((X190 @ X191 @ X192)) = ((sub @ X189 @ (push @ X192 @ X191))))) & (! [X193 : term,X194 : term,X195 : (subst > term > subst > $o)] : (~ ($true = ((X195 @ id @ X193 @ id))) | (((X195 @ id @ (ap @ X193 @ X194) @ id)) = $true) | ~ ($true = ((X195 @ id @ X194 @ id))) | ~ ! [X196 : term,X197 : term] : (($true = ((X195 @ id @ (ap @ (sub @ X196 @ id) @ X197) @ id))) | ~ (((X195 @ id @ X196 @ id)) = $true) | ~ ($true = ((X195 @ id @ X197 @ id)))) | ~ ! [X198 : term,X199 : subst,X200 : subst,X201 : subst] : (($true = ((X195 @ X201 @ X198 @ (comp @ X200 @ X199)))) | ~ ($true = ((X195 @ (comp @ X201 @ X200) @ (sub @ X198 @ X200) @ X199)))) | ~ ! [X202 : term,X203 : subst,X204 : subst,X205 : subst] : (($true = ((X195 @ (comp @ X204 @ X205) @ (sub @ X202 @ X205) @ X203))) | ~ ($true = ((X195 @ X204 @ X202 @ (comp @ X205 @ X203)))))) <=> (hoasinduction_lem2 = $true)) & (ulamvarind = $true) & ((hoasinduction_no_psi_cond = $true) <=> ! [X206 : (subst > term > subst > $o),X207 : term] : (~ ! [X208 : (subst > term > term)] : (~ ! [X211 : subst,X210 : subst,X209 : term] : (((sub @ (X208 @ X211 @ X209) @ X210)) = ((X208 @ (comp @ X211 @ X210) @ (sub @ X209 @ X210)))) | ~ ! [X212 : term] : (~ ($true = ((X206 @ id @ X212 @ id))) | ($true = ((X206 @ id @ (X208 @ id @ X212) @ id)))) | ($true = ((X206 @ id @ (lam @ (X208 @ sh @ one)) @ id)))) | ($true = ((X206 @ id @ X207 @ id))) | ~ ! [X213 : term,X214 : term] : (~ ($true = ((X206 @ id @ X213 @ id))) | ($true = ((X206 @ id @ (ap @ (sub @ X213 @ id) @ X214) @ id))) | ~ (((X206 @ id @ X214 @ id)) = $true)))) & (hoasapinj2_gthm = $true) & (lamnotvar = $true) & (! [X215 : subst,X216 : (term > $o),X217 : term] : (~ ! [X218 : term] : (($true = ((X216 @ (sub @ X218 @ X215)))) | ~ ($true = ((var @ X218)))) | ~ ! [X219 : term,X220 : term] : ((((X216 @ (ap @ X219 @ X220))) = $true) | ~ ($true = ((X216 @ X219))) | ~ ($true = ((X216 @ X220)))) | (((X216 @ (sub @ X217 @ X215))) = $true) | ~ ! [X221 : term] : (~ ! [X222 : term] : (~ ($true = ((X216 @ X222))) | ($true = ((X216 @ (sub @ X221 @ (push @ X222 @ id)))))) | ($true = ((X216 @ (lam @ X221)))))) <=> (induction2lem = $true)) & ((substmonoid = $true) <=> (! [X224 : subst] : (((comp @ X224 @ id)) = X224) & ! [X223 : subst] : (((comp @ id @ X223)) = X223))) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X225 : term] : ~ ! [X226 : (subst > term > term)] : (((sub @ X225 @ (push @ one @ sh))) != ((X226 @ sh @ one)))) & (! [X227 : (term > $o),X228 : term,X229 : (subst > term > subst > $o)] : (~ ! [X230 : (subst > term > term)] : (~ ! [X232 : term,X231 : subst,X233 : subst] : (((X230 @ (comp @ X231 @ X233) @ (sub @ X232 @ X233))) = ((sub @ (X230 @ X231 @ X232) @ X233))) | ~ ! [X234 : term] : ((((X229 @ id @ (X230 @ id @ X234) @ id)) = $true) | ~ ($true = ((X229 @ id @ X234 @ id)))) | (((X229 @ id @ (lam @ (X230 @ sh @ one)) @ id)) = $true)) | ($true = ((X227 @ (lam @ X228)))) | ~ ! [X235 : term] : (((X229 @ id @ X235 @ id)) = ((X227 @ X235))) | ~ ! [X236 : term] : (~ ($true = ((X227 @ X236))) | ($true = ((X227 @ (sub @ X228 @ (push @ X236 @ id))))))) <=> (hoasinduction_lem3v2a = $true)) & (! [X238 : (term > $o),X239 : term,X237 : subst] : ~ ! [X240 : (term > $o)] : ~ ! [X241 : term] : (((X238 @ (sub @ X241 @ (push @ X239 @ X237)))) = ((X240 @ X241))) <=> (pushprop_lem0 = $true)) & ((hoasinduction_lem3v2 = $true) <=> ! [X242 : term,X243 : (subst > term > subst > $o),X244 : (term > $o)] : (~ ! [X245 : subst,X246 : subst,X247 : term,X248 : subst] : (~ ($true = ((X243 @ X246 @ X247 @ (comp @ X248 @ X245)))) | ($true = ((X243 @ (comp @ X246 @ X248) @ (sub @ X247 @ X248) @ X245)))) | ~ ! [X249 : term] : (((X244 @ X249)) = ((X243 @ id @ X249 @ id))) | ~ ! [X250 : subst,X251 : subst,X252 : subst,X253 : term] : (~ ($true = ((X243 @ (comp @ X251 @ X252) @ (sub @ X253 @ X252) @ X250))) | (((X243 @ X251 @ X253 @ (comp @ X252 @ X250))) = $true)) | ($true = ((X244 @ (lam @ X242)))) | ~ ! [X254 : (subst > term > term)] : (~ ! [X255 : term] : (~ ($true = ((X243 @ id @ X255 @ id))) | ($true = ((X243 @ id @ (X254 @ id @ X255) @ id)))) | ($true = ((X243 @ id @ (lam @ (X254 @ sh @ one)) @ id))) | ~ ! [X256 : subst,X257 : subst,X258 : term] : (((sub @ (X254 @ X257 @ X258) @ X256)) = ((X254 @ (comp @ X257 @ X256) @ (sub @ X258 @ X256))))) | ~ ! [X259 : term] : (~ ($true = ((X244 @ X259))) | (((X244 @ (sub @ X242 @ (push @ X259 @ id)))) = $true)))) & (! [X260 : term,X261 : (subst > term > subst > $o)] : (~ ! [X262 : (subst > term > term)] : (~ ! [X264 : term,X265 : subst,X263 : subst] : (((X262 @ (comp @ X265 @ X263) @ (sub @ X264 @ X263))) = ((sub @ (X262 @ X265 @ X264) @ X263))) | ~ ! [X266 : term] : ((((X261 @ id @ (X262 @ id @ X266) @ id)) = $true) | ~ ($true = ((X261 @ id @ X266 @ id)))) | ($true = ((X261 @ id @ (lam @ (X262 @ sh @ one)) @ id)))) | ($true = ((X261 @ id @ (lam @ (sub @ X260 @ (push @ one @ sh))) @ id))) | ~ ! [X267 : term] : (~ (((X261 @ id @ X267 @ id)) = $true) | (((X261 @ id @ (sub @ X260 @ (push @ X267 @ id)) @ id)) = $true))) <=> (hoasinduction_lem3aa_lthm = $true)) & (! [X270 : term,X271 : term,X269 : term,X268 : term] : ((((ap @ X269 @ X270)) != ((ap @ X271 @ X268))) | (X269 = X271)) <=> (apinj1 = $true)) & (! [X272 : subst] : (((comp @ X272 @ id)) = X272) <=> (axidr = $true)) & ((hoasinduction_lem3v2_lthm = $true) <=> (! [X273 : term] : (((sub @ X273 @ id)) = X273) => ! [X274 : (subst > term > subst > $o),X275 : term,X276 : (term > $o)] : (~ ! [X277 : term] : (((X276 @ X277)) = ((X274 @ id @ X277 @ id))) | ($true = ((X276 @ (lam @ X275)))) | ~ ! [X278 : (subst > term > term)] : (~ ! [X279 : term] : (~ (((X274 @ id @ X279 @ id)) = $true) | ($true = ((X274 @ id @ (X278 @ id @ X279) @ id)))) | ~ ! [X281 : subst,X280 : subst,X282 : term] : (((sub @ (X278 @ X280 @ X282) @ X281)) = ((X278 @ (comp @ X280 @ X281) @ (sub @ X282 @ X281)))) | ($true = ((X274 @ id @ (lam @ (X278 @ sh @ one)) @ id)))) | ~ ! [X283 : term,X284 : subst,X285 : subst,X286 : subst] : ((((X274 @ (comp @ X285 @ X286) @ (sub @ X283 @ X286) @ X284)) = $true) | ~ (((X274 @ X285 @ X283 @ (comp @ X286 @ X284))) = $true)) | ~ ! [X287 : term] : (~ ($true = ((X276 @ X287))) | ($true = ((X276 @ (sub @ X275 @ (push @ X287 @ id)))))) | ~ ! [X288 : subst,X289 : subst,X290 : term,X291 : subst] : (($true = ((X274 @ X288 @ X290 @ (comp @ X291 @ X289)))) | ~ ($true = ((X274 @ (comp @ X288 @ X291) @ (sub @ X290 @ X291) @ X289))))))) & (hoaslamnotvar = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (! [X292 : term,X293 : (subst > term > subst > $o)] : (~ ! [X294 : term] : (($true = ((X293 @ id @ (sub @ X292 @ (push @ X294 @ id)) @ id))) | ~ (((X293 @ id @ X294 @ id)) = $true)) | ($true = ((X293 @ id @ (lam @ (sub @ X292 @ (push @ one @ sh))) @ id))) | ~ ! [X295 : (subst > term > term)] : (~ ! [X296 : subst,X297 : term,X298 : subst] : (((sub @ (X295 @ X298 @ X297) @ X296)) = ((X295 @ (comp @ X298 @ X296) @ (sub @ X297 @ X296)))) | ~ ! [X299 : term] : (($true = ((X293 @ id @ (X295 @ id @ X299) @ id))) | ~ (((X293 @ id @ X299 @ id)) = $true)) | ($true = ((X293 @ id @ (lam @ (X295 @ sh @ one)) @ id))))) <=> (hoasinduction_lem3aa = $true)) & ((! [X300 : term] : (((sub @ X300 @ id)) = X300) => (! [X303 : term,X304 : term,X302 : term,X301 : term] : ((X301 = X304) | (((ap @ X301 @ X303)) != ((ap @ X304 @ X302)))) => ! [X307 : term,X306 : term,X308 : term,X305 : term] : ((((ap @ (sub @ X305 @ id) @ X307)) != ((ap @ (sub @ X306 @ id) @ X308))) | (X305 = X306)))) <=> (hoasapinj1_lthm = $true)) & (pushprop_gthm = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ! [X309 : d_subst,X310 : d_subst] : ((((d2subst @ X310)) = ((d2subst @ X309))) => (X309 = X310)) & ((hoasinduction_lthm_3 = $true) <=> (! [X311 : (subst > term > subst > $o)] : ~ ! [X312 : (term > $o)] : ~ ! [X313 : term] : (((X312 @ X313)) = ((X311 @ id @ X313 @ id))) => (! [X314 : term,X315 : (term > $o)] : (($true = ((X315 @ X314))) | ~ ! [X316 : term,X317 : term] : (~ ($true = ((X315 @ X317))) | ~ ($true = ((X315 @ X316))) | (((X315 @ (ap @ X317 @ X316))) = $true)) | ~ ! [X318 : term] : ((((X315 @ X318)) = $true) | ~ (((var @ X318)) = $true)) | ~ ! [X319 : term] : (~ ! [X320 : term] : ((((X315 @ (sub @ X319 @ (push @ X320 @ id)))) = $true) | ~ ($true = ((X315 @ X320)))) | ($true = ((X315 @ (lam @ X319)))))) => (! [X321 : term] : (((sub @ X321 @ id)) = X321) => (! [X322 : term,X323 : (term > $o),X324 : (subst > term > subst > $o)] : (~ ! [X325 : term] : (~ (((X323 @ X325)) = $true) | ($true = ((X323 @ (sub @ X322 @ (push @ X325 @ id)))))) | ~ ! [X326 : (subst > term > term)] : (~ ! [X327 : term] : (($true = ((X324 @ id @ (X326 @ id @ X327) @ id))) | ~ (((X324 @ id @ X327 @ id)) = $true)) | ~ ! [X329 : term,X330 : subst,X328 : subst] : (((X326 @ (comp @ X328 @ X330) @ (sub @ X329 @ X330))) = ((sub @ (X326 @ X328 @ X329) @ X330))) | ($true = ((X324 @ id @ (lam @ (X326 @ sh @ one)) @ id)))) | ($true = ((X323 @ (lam @ X322)))) | ~ ! [X331 : term] : (((X324 @ id @ X331 @ id)) = ((X323 @ X331)))) => ! [X332 : (subst > term > subst > $o),X333 : term] : (($true = ((X332 @ id @ X333 @ id))) | ~ ! [X334 : subst,X335 : term,X336 : subst,X337 : subst] : (~ ($true = ((X332 @ X334 @ X335 @ (comp @ X336 @ X337)))) | ($true = ((X332 @ (comp @ X334 @ X336) @ (sub @ X335 @ X336) @ X337)))) | ~ ! [X338 : term,X339 : term] : (($true = ((X332 @ id @ (ap @ (sub @ X338 @ id) @ X339) @ id))) | ~ (((X332 @ id @ X338 @ id)) = $true) | ~ ($true = ((X332 @ id @ X339 @ id)))) | ~ ! [X340 : (subst > term > term)] : (($true = ((X332 @ id @ (lam @ (X340 @ sh @ one)) @ id))) | ~ ! [X341 : term] : (($true = ((X332 @ id @ (X340 @ id @ X341) @ id))) | ~ ($true = ((X332 @ id @ X341 @ id)))) | ~ ! [X344 : subst,X342 : term,X343 : subst] : (((X340 @ (comp @ X343 @ X344) @ (sub @ X342 @ X344))) = ((sub @ (X340 @ X343 @ X342) @ X344)))) | ~ ! [X345 : subst,X346 : subst,X347 : term,X348 : subst] : (~ (((X332 @ (comp @ X346 @ X345) @ (sub @ X347 @ X345) @ X348)) = $true) | ($true = ((X332 @ X346 @ X347 @ (comp @ X345 @ X348))))) | ~ ! [X349 : term] : (($true = ((X332 @ id @ X349 @ id))) | ~ ($true = ((var @ (sub @ X349 @ id))))))))))) & (axabs = $true) & ~ (lamnotap = $true) & ~ (ulamvar1 = $true) & (hoasinduction_lem3v2_gthm = $true) & ((! [X351 : term,X350 : subst] : (((sub @ one @ (push @ X351 @ X350))) = X351) => (! [X352 : subst,X353 : term] : (((comp @ sh @ (push @ X353 @ X352))) = X352) => (! [X354 : subst] : (((comp @ X354 @ id)) = X354) => (! [X355 : term,X356 : (term > $o)] : (~ ! [X357 : term] : (~ ($true = ((var @ X357))) | ($true = ((X356 @ X357)))) | ~ ! [X358 : term,X359 : term] : (~ ($true = ((X356 @ X359))) | (((X356 @ (ap @ X358 @ X359))) = $true) | ~ ($true = ((X356 @ X358)))) | ~ ! [X360 : term] : (~ (((X356 @ X360)) = $true) | ($true = ((X356 @ (lam @ X360))))) | (((X356 @ X355)) = $true)) => ! [X361 : (term > $o),X362 : subst,X363 : term] : (~ ! [X364 : term] : (~ (((var @ X364)) = $true) | (((X361 @ (sub @ X364 @ X362))) = $true)) | ~ ! [X365 : term] : (($true = ((X361 @ (lam @ X365)))) | ~ ! [X366 : term] : (($true = ((X361 @ (sub @ X365 @ (push @ X366 @ id))))) | ~ ($true = ((X361 @ X366))))) | ~ ! [X367 : term,X368 : term] : (~ ($true = ((X361 @ X367))) | ~ ($true = ((X361 @ X368))) | ($true = ((X361 @ (ap @ X367 @ X368))))) | ($true = ((X361 @ (sub @ X363 @ X362))))))))) <=> (induction2lem_lthm = $true)) & (hoasinduction_lem2_lthm = $true) & (! [X372 : subst,X369 : term,X370 : (subst > term > term),X371 : (subst > term > term)] : ((((lam @ (X370 @ sh @ one))) != ((lam @ (X371 @ sh @ one)))) | ~ ! [X377 : subst,X378 : term,X376 : subst] : (((sub @ (X371 @ X376 @ X378) @ X377)) = ((X371 @ (comp @ X376 @ X377) @ (sub @ X378 @ X377)))) | ~ ! [X375 : subst,X374 : subst,X373 : term] : (((sub @ (X370 @ X375 @ X373) @ X374)) = ((X370 @ (comp @ X375 @ X374) @ (sub @ X373 @ X374)))) | (((X371 @ X372 @ X369)) = ((X370 @ X372 @ X369)))) <=> (hoaslaminj = $true)) & (pushprop_lem2v2 = $true) & ((pushprop_lem1_lthm = $true) <=> (! [X379 : subst,X380 : term] : (((sub @ one @ (push @ X380 @ X379))) = X380) => (! [X382 : term,X381 : subst] : (((comp @ sh @ (push @ X382 @ X381))) = X381) => ! [X383 : (term > $o),X384 : term,X385 : (term > $o),X386 : term,X387 : subst] : (~ ($true = ((X385 @ X384))) | ($true = ((X383 @ (sub @ X384 @ (push @ X386 @ X387))))))))) & ((pushprop_lem1v2 = $true) <=> ! [X388 : (term > $o),X389 : subst,X390 : term,X391 : (term > $o)] : (~ ($true = ((X391 @ X390))) | ~ ! [X392 : term] : (((X391 @ (sub @ X392 @ (push @ X390 @ X389)))) = ((X388 @ X392))) | ($true = ((X388 @ one))))) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (axvarshift = $true) & (! [X393 : term,X394 : (subst > term > subst > $o)] : (~ ! [X395 : subst,X396 : term,X397 : subst,X398 : subst] : (($true = ((X394 @ (comp @ X395 @ X397) @ (sub @ X396 @ X397) @ X398))) | ~ ($true = ((X394 @ X395 @ X396 @ (comp @ X397 @ X398))))) | ~ ! [X399 : term,X400 : term] : (~ ($true = ((X394 @ id @ X399 @ id))) | ~ (((X394 @ id @ X400 @ id)) = $true) | (((X394 @ id @ (ap @ (sub @ X399 @ id) @ X400) @ id)) = $true)) | ($true = ((X394 @ id @ X393 @ id))) | ~ ! [X401 : term] : (($true = ((X394 @ id @ X401 @ id))) | ~ (((var @ (sub @ X401 @ id))) = $true)) | ~ ! [X402 : subst,X403 : term,X404 : subst,X405 : subst] : (($true = ((X394 @ X404 @ X403 @ (comp @ X402 @ X405)))) | ~ ($true = ((X394 @ (comp @ X404 @ X402) @ (sub @ X403 @ X402) @ X405)))) | ~ ! [X406 : (subst > term > term)] : (~ ! [X407 : term] : (($true = ((X394 @ id @ (X406 @ id @ X407) @ id))) | ~ (((X394 @ id @ X407 @ id)) = $true)) | ~ ! [X408 : subst,X409 : subst,X410 : term] : (((X406 @ (comp @ X408 @ X409) @ (sub @ X410 @ X409))) = ((sub @ (X406 @ X408 @ X410) @ X409))) | (((X394 @ id @ (lam @ (X406 @ sh @ one)) @ id)) = $true))) <=> (hoasinduction = $true)) & (pushprop = $true) & ((! [X411 : term] : (((sub @ X411 @ id)) = X411) => (! [X412 : term,X413 : (subst > term > subst > $o)] : (~ ! [X414 : (subst > term > term)] : (($true = ((X413 @ id @ (lam @ (X414 @ sh @ one)) @ id))) | ~ ! [X415 : term] : (~ (((X413 @ id @ X415 @ id)) = $true) | (((X413 @ id @ (X414 @ id @ X415) @ id)) = $true)) | ~ ! [X417 : subst,X416 : subst,X418 : term] : (((X414 @ (comp @ X417 @ X416) @ (sub @ X418 @ X416))) = ((sub @ (X414 @ X417 @ X418) @ X416)))) | (((X413 @ id @ (lam @ (sub @ X412 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X419 : term] : (($true = ((X413 @ id @ (sub @ X412 @ (push @ X419 @ id)) @ id))) | ~ ($true = ((X413 @ id @ X419 @ id))))) => ! [X420 : term,X421 : (subst > term > subst > $o)] : (~ ! [X422 : (subst > term > term)] : (~ ! [X423 : subst,X424 : subst,X425 : term] : (((sub @ (X422 @ X423 @ X425) @ X424)) = ((X422 @ (comp @ X423 @ X424) @ (sub @ X425 @ X424)))) | ~ ! [X426 : term] : (($true = ((X421 @ id @ (X422 @ id @ X426) @ id))) | ~ ($true = ((X421 @ id @ X426 @ id)))) | (((X421 @ id @ (lam @ (X422 @ sh @ one)) @ id)) = $true)) | ~ ! [X427 : term] : (~ (((X421 @ id @ X427 @ id)) = $true) | (((X421 @ id @ (sub @ X420 @ (push @ X427 @ id)) @ id)) = $true)) | ($true = ((X421 @ id @ (lam @ X420) @ id)))))) <=> (hoasinduction_lem3a_lthm = $true)) & (pushprop_lem1v2_gthm = $true) & (! [X429 : term,X428 : term] : ((((sub @ X428 @ sh)) != ((sub @ X429 @ sh))) | (X428 = X429)) <=> (shinj = $true)) & ~ (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) = $true) & ~ (((var @ (d2term @ d_one))) = $true) & (substmonoid_gthm = $true) & ((hoasinduction_lem2v2 = $true) <=> ! [X430 : term,X431 : (subst > term > subst > $o),X432 : (term > $o),X433 : term] : (~ ! [X434 : term] : (((X432 @ X434)) = ((X431 @ id @ X434 @ id))) | ~ (((X432 @ X430)) = $true) | ~ ($true = ((X432 @ X433))) | ~ ! [X435 : term,X436 : term] : (~ ($true = ((X431 @ id @ X435 @ id))) | ~ ($true = ((X431 @ id @ X436 @ id))) | ($true = ((X431 @ id @ (ap @ (sub @ X435 @ id) @ X436) @ id)))) | ~ ! [X437 : subst,X438 : subst,X439 : term,X440 : subst] : ((((X431 @ X437 @ X439 @ (comp @ X438 @ X440))) = $true) | ~ ($true = ((X431 @ (comp @ X437 @ X438) @ (sub @ X439 @ X438) @ X440)))) | ($true = ((X432 @ (ap @ X433 @ X430)))) | ~ ! [X441 : subst,X442 : subst,X443 : term,X444 : subst] : (~ ($true = ((X431 @ X442 @ X443 @ (comp @ X441 @ X444)))) | ($true = ((X431 @ (comp @ X442 @ X441) @ (sub @ X443 @ X441) @ X444)))))) & ((! [X445 : (subst > term > subst > $o)] : ~ ! [X446 : (term > $o)] : ~ ! [X447 : term] : (((X445 @ id @ X447 @ id)) = ((X446 @ X447))) => (! [X448 : term,X449 : (term > $o)] : (($true = ((X449 @ X448))) | ~ ! [X450 : term,X451 : term] : (($true = ((X449 @ (ap @ X450 @ X451)))) | ~ (((X449 @ X451)) = $true) | ~ ($true = ((X449 @ X450)))) | ~ ! [X452 : term] : (($true = ((X449 @ (lam @ X452)))) | ~ ! [X453 : term] : (($true = ((X449 @ (sub @ X452 @ (push @ X453 @ id))))) | ~ (((X449 @ X453)) = $true))) | ~ ! [X454 : term] : (~ (((var @ X454)) = $true) | (((X449 @ X454)) = $true))) => (! [X455 : term] : (((sub @ X455 @ id)) = X455) => (! [X456 : (term > $o),X457 : (subst > term > subst > $o),X458 : term] : (~ ! [X459 : term] : (((X457 @ id @ X459 @ id)) = ((X456 @ X459))) | ~ ! [X460 : term] : ((((X456 @ (sub @ X458 @ (push @ X460 @ id)))) = $true) | ~ (((X456 @ X460)) = $true)) | (((X456 @ (lam @ X458))) = $true) | ~ ! [X461 : (subst > term > term)] : (~ ! [X462 : term] : ((((X457 @ id @ (X461 @ id @ X462) @ id)) = $true) | ~ ($true = ((X457 @ id @ X462 @ id)))) | ($true = ((X457 @ id @ (lam @ (X461 @ sh @ one)) @ id))) | ~ ! [X463 : term,X464 : subst,X465 : subst] : (((sub @ (X461 @ X465 @ X463) @ X464)) = ((X461 @ (comp @ X465 @ X464) @ (sub @ X463 @ X464)))))) => ! [X466 : (subst > term > subst > $o),X467 : term] : ((((X466 @ id @ X467 @ id)) = $true) | ~ ! [X468 : term,X469 : term] : (($true = ((X466 @ id @ (ap @ (sub @ X469 @ id) @ X468) @ id))) | ~ ($true = ((X466 @ id @ X469 @ id))) | ~ ($true = ((X466 @ id @ X468 @ id)))) | ~ ! [X470 : (subst > term > term)] : (($true = ((X466 @ id @ (lam @ (X470 @ sh @ one)) @ id))) | ~ ! [X471 : term,X473 : subst,X472 : subst] : (((X470 @ (comp @ X472 @ X473) @ (sub @ X471 @ X473))) = ((sub @ (X470 @ X472 @ X471) @ X473))) | ~ ! [X474 : term] : (~ ($true = ((X466 @ id @ X474 @ id))) | ($true = ((X466 @ id @ (X470 @ id @ X474) @ id)))))))))) <=> (hoasinduction_no_psi_cond_lthm = $true)) & ((induction2_lthm = $true) <=> (! [X475 : term] : (((sub @ X475 @ id)) = X475) => (! [X476 : subst,X477 : (term > $o),X478 : term] : (~ ! [X479 : term,X480 : term] : (~ (((X477 @ X479)) = $true) | ($true = ((X477 @ (ap @ X479 @ X480)))) | ~ ($true = ((X477 @ X480)))) | ~ ! [X481 : term] : (~ ! [X482 : term] : (~ (((X477 @ X482)) = $true) | (((X477 @ (sub @ X481 @ (push @ X482 @ id)))) = $true)) | ($true = ((X477 @ (lam @ X481))))) | ~ ! [X483 : term] : (~ ($true = ((var @ X483))) | ($true = ((X477 @ (sub @ X483 @ X476))))) | ($true = ((X477 @ (sub @ X478 @ X476))))) => ! [X484 : (term > $o),X485 : term] : (~ ! [X486 : term] : (~ ! [X487 : term] : (~ ($true = ((X484 @ X487))) | ($true = ((X484 @ (sub @ X486 @ (push @ X487 @ id)))))) | (((X484 @ (lam @ X486))) = $true)) | ~ ! [X488 : term,X489 : term] : (~ ($true = ((X484 @ X489))) | ($true = ((X484 @ (ap @ X489 @ X488)))) | ~ ($true = ((X484 @ X488)))) | ($true = ((X484 @ X485))) | ~ ! [X490 : term] : (~ ($true = ((var @ X490))) | ($true = ((X484 @ X490)))))))) & (! [X491 : (term > $o),X492 : term] : (($true = ((X491 @ X492))) | ~ ! [X493 : term] : ((((X491 @ X493)) = $true) | ~ ($true = ((var @ X493)))) | ~ ! [X494 : term,X495 : term] : (($true = ((X491 @ (ap @ X495 @ X494)))) | ~ ($true = ((X491 @ X494))) | ~ ($true = ((X491 @ X495)))) | ~ ! [X496 : term] : (~ ! [X497 : term] : (~ ($true = ((X491 @ X497))) | (((X491 @ (sub @ X496 @ (push @ X497 @ id)))) = $true)) | ($true = ((X491 @ (lam @ X496)))))) <=> (induction2 = $true)) & (pushprop_lem1_gthm = $true) & (hoaslamnotap_lthm = $true) & (hoasinduction_lem1v2 = $true) & ((! [X498 : term] : (((sub @ X498 @ id)) = X498) => (! [X499 : (subst > term > subst > $o),X500 : term] : (~ ! [X501 : term] : (($true = ((X499 @ id @ (sub @ X500 @ (push @ X501 @ id)) @ id))) | ~ ($true = ((X499 @ id @ X501 @ id)))) | ~ ! [X502 : (subst > term > term)] : (~ ! [X503 : term] : (($true = ((X499 @ id @ (X502 @ id @ X503) @ id))) | ~ ($true = ((X499 @ id @ X503 @ id)))) | ($true = ((X499 @ id @ (lam @ (X502 @ sh @ one)) @ id))) | ~ ! [X505 : subst,X506 : term,X504 : subst] : (((sub @ (X502 @ X504 @ X506) @ X505)) = ((X502 @ (comp @ X504 @ X505) @ (sub @ X506 @ X505))))) | ($true = ((X499 @ id @ (lam @ (sub @ X500 @ (push @ one @ sh))) @ id)))) => ! [X507 : term,X508 : (subst > term > subst > $o)] : (($true = ((X508 @ id @ (lam @ X507) @ id))) | ~ ! [X509 : subst,X510 : subst,X511 : subst,X512 : term] : (~ ($true = ((X508 @ (comp @ X510 @ X511) @ (sub @ X512 @ X511) @ X509))) | ($true = ((X508 @ X510 @ X512 @ (comp @ X511 @ X509))))) | ~ ! [X513 : term] : ((((X508 @ id @ (sub @ X507 @ (push @ X513 @ id)) @ id)) = $true) | ~ ($true = ((X508 @ id @ X513 @ id)))) | ~ ! [X514 : term,X515 : subst,X516 : subst,X517 : subst] : (~ ($true = ((X508 @ X515 @ X514 @ (comp @ X517 @ X516)))) | ($true = ((X508 @ (comp @ X515 @ X517) @ (sub @ X514 @ X517) @ X516)))) | ~ ! [X518 : (subst > term > term)] : (~ ! [X519 : term] : (~ (((X508 @ id @ X519 @ id)) = $true) | (((X508 @ id @ (X518 @ id @ X519) @ id)) = $true)) | ~ ! [X522 : subst,X520 : subst,X521 : term] : (((X518 @ (comp @ X522 @ X520) @ (sub @ X521 @ X520))) = ((sub @ (X518 @ X522 @ X521) @ X520))) | (((X508 @ id @ (lam @ (X518 @ sh @ one)) @ id)) = $true))))) <=> (hoasinduction_lem3_lthm = $true)) & (pushprop_lthm_orig = $true) & (pushprop_lem0_gthm = $true) & ((hoasapinj2_lthm = $true) <=> (! [X523 : term,X524 : term,X525 : term,X526 : term] : ((((ap @ X526 @ X523)) != ((ap @ X525 @ X524))) | (X523 = X524)) => ! [X529 : term,X528 : term,X527 : term,X530 : term] : ((X527 = X530) | (((ap @ (sub @ X529 @ id) @ X527)) != ((ap @ (sub @ X528 @ id) @ X530)))))) & (! [X533 : term,X532 : term,X531 : term,X534 : term] : ((X532 = X533) | (((ap @ (sub @ X534 @ id) @ X533)) != ((ap @ (sub @ X531 @ id) @ X532)))) <=> (hoasapinj2 = $true)) & (! [X535 : subst] : (((comp @ id @ X535)) = X535) <=> (axidl = $true)) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (hoasapnotvar_lthm = $true) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (hoasinduction_lem3a_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (ulamvarsh = $true) & (apnotvar = $true) & (hoasinduction_lem1_gthm = $true) & ! [X536 : subst] : ? [X537 : d_subst] : (((d2subst @ X537)) = X536) & ((hoasinduction_lem3b = $true) <=> ! [X538 : term] : ~ ! [X539 : (subst > term > term)] : (((X539 @ sh @ one)) != ((sub @ X538 @ (push @ one @ sh)))))),
  inference(rectify,[],[f7])).
thf(f10,plain,(
  (! [X394 : (subst > term > subst > $o),X393 : term] : (~ ! [X399 : term,X400 : term] : ((((X394 @ id @ (ap @ (sub @ X399 @ id) @ X400) @ id)) = $true) | ($true != ((X394 @ id @ X399 @ id))) | (((X394 @ id @ X400 @ id)) != $true)) | ~ ! [X406 : (subst > term > term)] : ((((X394 @ id @ (lam @ (X406 @ sh @ one)) @ id)) = $true) | ~ ! [X408 : subst,X409 : subst,X410 : term] : (((X406 @ (comp @ X408 @ X409) @ (sub @ X410 @ X409))) = ((sub @ (X406 @ X408 @ X410) @ X409))) | ~ ! [X407 : term] : (($true = ((X394 @ id @ (X406 @ id @ X407) @ id))) | (((X394 @ id @ X407 @ id)) != $true))) | ($true = ((X394 @ id @ X393 @ id))) | ~ ! [X404 : subst,X402 : subst,X403 : term,X405 : subst] : (($true = ((X394 @ X404 @ X403 @ (comp @ X402 @ X405)))) | ($true != ((X394 @ (comp @ X404 @ X402) @ (sub @ X403 @ X402) @ X405)))) | ~ ! [X401 : term] : (($true = ((X394 @ id @ X401 @ id))) | (((var @ (sub @ X401 @ id))) != $true)) | ~ ! [X395 : subst,X398 : subst,X396 : term,X397 : subst] : (($true != ((X394 @ X395 @ X396 @ (comp @ X397 @ X398)))) | ($true = ((X394 @ (comp @ X395 @ X397) @ (sub @ X396 @ X397) @ X398))))) <=> (hoasinduction = $true)) & ((hoasinduction_lem3 = $true) <=> ! [X72 : (subst > term > subst > $o),X73 : term] : (~ ! [X74 : term] : (($true = ((X72 @ id @ (sub @ X73 @ (push @ X74 @ id)) @ id))) | ($true != ((X72 @ id @ X74 @ id)))) | ~ ! [X87 : term,X84 : subst,X85 : subst,X86 : subst] : (($true != ((X72 @ X85 @ X87 @ (comp @ X84 @ X86)))) | ($true = ((X72 @ (comp @ X85 @ X84) @ (sub @ X87 @ X84) @ X86)))) | ~ ! [X79 : (subst > term > term)] : (~ ! [X80 : term] : ((((X72 @ id @ X80 @ id)) != $true) | ($true = ((X72 @ id @ (X79 @ id @ X80) @ id)))) | ~ ! [X83 : subst,X81 : term,X82 : subst] : (((X79 @ (comp @ X83 @ X82) @ (sub @ X81 @ X82))) = ((sub @ (X79 @ X83 @ X81) @ X82))) | (((X72 @ id @ (lam @ (X79 @ sh @ one)) @ id)) = $true)) | ($true = ((X72 @ id @ (lam @ X73) @ id))) | ~ ! [X76 : subst,X77 : term,X75 : subst,X78 : subst] : (($true != ((X72 @ (comp @ X75 @ X78) @ (sub @ X77 @ X78) @ X76))) | (((X72 @ X75 @ X77 @ (comp @ X78 @ X76))) = $true)))) & (axapp = $true) & ((hoasapinj2_lthm = $true) <=> (! [X523 : term,X524 : term,X525 : term,X526 : term] : ((((ap @ X526 @ X523)) != ((ap @ X525 @ X524))) | (X523 = X524)) => ! [X529 : term,X528 : term,X527 : term,X530 : term] : ((X527 = X530) | (((ap @ (sub @ X529 @ id) @ X527)) != ((ap @ (sub @ X528 @ id) @ X530)))))) & (! [X272 : subst] : (((comp @ X272 @ id)) = X272) <=> (axidr = $true)) & ((hoasinduction_lem2 = $true) <=> ! [X195 : (subst > term > subst > $o),X193 : term,X194 : term] : (($true != ((X195 @ id @ X193 @ id))) | ~ ! [X201 : subst,X199 : subst,X198 : term,X200 : subst] : (($true != ((X195 @ (comp @ X201 @ X200) @ (sub @ X198 @ X200) @ X199))) | ($true = ((X195 @ X201 @ X198 @ (comp @ X200 @ X199))))) | ~ ! [X196 : term,X197 : term] : (($true != ((X195 @ id @ X197 @ id))) | ($true = ((X195 @ id @ (ap @ (sub @ X196 @ id) @ X197) @ id))) | (((X195 @ id @ X196 @ id)) != $true)) | ($true != ((X195 @ id @ X194 @ id))) | (((X195 @ id @ (ap @ X193 @ X194) @ id)) = $true) | ~ ! [X205 : subst,X202 : term,X203 : subst,X204 : subst] : (($true = ((X195 @ (comp @ X204 @ X205) @ (sub @ X202 @ X205) @ X203))) | ($true != ((X195 @ X204 @ X202 @ (comp @ X205 @ X203))))))) & (! [X242 : term,X244 : (term > $o),X243 : (subst > term > subst > $o)] : (~ ! [X259 : term] : ((((X244 @ (sub @ X242 @ (push @ X259 @ id)))) = $true) | ($true != ((X244 @ X259)))) | ($true = ((X244 @ (lam @ X242)))) | ~ ! [X249 : term] : (((X244 @ X249)) = ((X243 @ id @ X249 @ id))) | ~ ! [X254 : (subst > term > term)] : (~ ! [X256 : subst,X257 : subst,X258 : term] : (((sub @ (X254 @ X257 @ X258) @ X256)) = ((X254 @ (comp @ X257 @ X256) @ (sub @ X258 @ X256)))) | ($true = ((X243 @ id @ (lam @ (X254 @ sh @ one)) @ id))) | ~ ! [X255 : term] : (($true != ((X243 @ id @ X255 @ id))) | ($true = ((X243 @ id @ (X254 @ id @ X255) @ id))))) | ~ ! [X250 : subst,X251 : subst,X252 : subst,X253 : term] : (($true != ((X243 @ (comp @ X251 @ X252) @ (sub @ X253 @ X252) @ X250))) | (((X243 @ X251 @ X253 @ (comp @ X252 @ X250))) = $true)) | ~ ! [X245 : subst,X247 : term,X246 : subst,X248 : subst] : (($true = ((X243 @ (comp @ X246 @ X248) @ (sub @ X247 @ X248) @ X245))) | ($true != ((X243 @ X246 @ X247 @ (comp @ X248 @ X245)))))) <=> (hoasinduction_lem3v2 = $true)) & (pushprop_lem2v2_gthm = $true) & ((substmonoid_lthm = $true) <=> (! [X88 : subst] : (((comp @ id @ X88)) = X88) => (! [X89 : subst] : (((comp @ X89 @ id)) = X89) => (! [X91 : subst] : (((comp @ id @ X91)) = X91) & ! [X90 : subst] : (((comp @ X90 @ id)) = X90))))) & (((var @ (d2term @ d_one))) != $true) & ((hoasinduction_lem3aaa = $true) <=> ! [X114 : (subst > term > subst > $o),X115 : term] : (~ ! [X116 : term] : (($true = ((X114 @ id @ (sub @ X115 @ (push @ X116 @ id)) @ id))) | ($true != ((X114 @ id @ X116 @ id)))) | ~ ! [X117 : (subst > term > term),X118 : term] : ((((X114 @ id @ (lam @ (X117 @ sh @ one)) @ id)) = $true) | ~ ! [X125 : term] : (($true = ((X114 @ id @ (X117 @ id @ X125) @ id))) | ($true != ((X114 @ id @ X125 @ id)))) | ~ ! [X121 : subst,X119 : subst,X120 : term] : (((X117 @ (comp @ X119 @ X121) @ (sub @ X120 @ X121))) = ((sub @ X118 @ (push @ (sub @ X120 @ X121) @ (comp @ X119 @ X121))))) | ~ ! [X124 : subst,X123 : term,X122 : subst] : (((sub @ (sub @ X118 @ (push @ X123 @ X122)) @ X124)) = ((sub @ (X117 @ X122 @ X123) @ X124)))) | ($true = ((X114 @ id @ (lam @ (sub @ X115 @ (push @ one @ sh))) @ id))))) & ((hoasinduction_lem0_lthm = $true) <=> ! [X10 : (subst > term > subst > $o)] : ~ ! [X11 : (term > $o)] : ~ ! [X12 : term] : (((X11 @ X12)) = ((X10 @ id @ X12 @ id)))) & ((! [X300 : term] : (((sub @ X300 @ id)) = X300) => (! [X303 : term,X304 : term,X302 : term,X301 : term] : ((X301 = X304) | (((ap @ X301 @ X303)) != ((ap @ X304 @ X302)))) => ! [X307 : term,X306 : term,X308 : term,X305 : term] : ((((ap @ (sub @ X305 @ id) @ X307)) != ((ap @ (sub @ X306 @ id) @ X308))) | (X305 = X306)))) <=> (hoasapinj1_lthm = $true)) & (termmset_gthm = $true) & ((hoasinduction_lem3v2_lthm = $true) <=> (! [X273 : term] : (((sub @ X273 @ id)) = X273) => ! [X275 : term,X274 : (subst > term > subst > $o),X276 : (term > $o)] : (~ ! [X278 : (subst > term > term)] : (($true = ((X274 @ id @ (lam @ (X278 @ sh @ one)) @ id))) | ~ ! [X281 : subst,X280 : subst,X282 : term] : (((sub @ (X278 @ X280 @ X282) @ X281)) = ((X278 @ (comp @ X280 @ X281) @ (sub @ X282 @ X281)))) | ~ ! [X279 : term] : ((((X274 @ id @ X279 @ id)) != $true) | ($true = ((X274 @ id @ (X278 @ id @ X279) @ id))))) | ~ ! [X283 : term,X284 : subst,X286 : subst,X285 : subst] : ((((X274 @ (comp @ X285 @ X286) @ (sub @ X283 @ X286) @ X284)) = $true) | (((X274 @ X285 @ X283 @ (comp @ X286 @ X284))) != $true)) | ~ ! [X277 : term] : (((X276 @ X277)) = ((X274 @ id @ X277 @ id))) | ($true = ((X276 @ (lam @ X275)))) | ~ ! [X287 : term] : (($true != ((X276 @ X287))) | ($true = ((X276 @ (sub @ X275 @ (push @ X287 @ id)))))) | ~ ! [X289 : subst,X290 : term,X291 : subst,X288 : subst] : (($true = ((X274 @ X288 @ X290 @ (comp @ X291 @ X289)))) | ($true != ((X274 @ (comp @ X288 @ X291) @ (sub @ X290 @ X291) @ X289))))))) & (! [X270 : term,X271 : term,X269 : term,X268 : term] : ((((ap @ X269 @ X270)) != ((ap @ X271 @ X268))) | (X269 = X271)) <=> (apinj1 = $true)) & (hoasapinj2_gthm = $true) & ((hoasinduction_lem3b = $true) <=> ! [X538 : term] : ~ ! [X539 : (subst > term > term)] : (((X539 @ sh @ one)) != ((sub @ X538 @ (push @ one @ sh))))) & (! [X429 : term,X428 : term] : ((((sub @ X428 @ sh)) != ((sub @ X429 @ sh))) | (X428 = X429)) <=> (shinj = $true)) & (one = ((d2term @ d_one))) & (hoasinduction_lem1v2_gthm = $true) & (! [X533 : term,X532 : term,X531 : term,X534 : term] : ((X532 = X533) | (((ap @ (sub @ X534 @ id) @ X533)) != ((ap @ (sub @ X531 @ id) @ X532)))) <=> (hoasapinj2 = $true)) & (hoasinduction_lem1v2 = $true) & (hoasapinj1_gthm = $true) & (hoasinduction_lem2_gthm = $true) & ((induction = $true) <=> ! [X65 : (term > $o),X66 : term] : (~ ! [X70 : term] : (($true != ((X65 @ X70))) | (((X65 @ (lam @ X70))) = $true)) | ~ ! [X68 : term,X69 : term] : (($true != ((X65 @ X68))) | (((X65 @ X69)) != $true) | ($true = ((X65 @ (ap @ X68 @ X69))))) | ~ ! [X67 : term] : (($true = ((X65 @ X67))) | ($true != ((var @ X67)))) | ($true = ((X65 @ X66))))) & (hoasinduction_lem2v2_gthm = $true) & (pushprop_lem1_gthm = $true) & ! [X61 : term] : ? [X62 : d_term] : (((d2term @ X62)) = X61) & (hoasinduction_lem2_lthm = $true) & ((! [X498 : term] : (((sub @ X498 @ id)) = X498) => (! [X499 : (subst > term > subst > $o),X500 : term] : (($true = ((X499 @ id @ (lam @ (sub @ X500 @ (push @ one @ sh))) @ id))) | ~ ! [X502 : (subst > term > term)] : (($true = ((X499 @ id @ (lam @ (X502 @ sh @ one)) @ id))) | ~ ! [X503 : term] : (($true != ((X499 @ id @ X503 @ id))) | ($true = ((X499 @ id @ (X502 @ id @ X503) @ id)))) | ~ ! [X505 : subst,X506 : term,X504 : subst] : (((sub @ (X502 @ X504 @ X506) @ X505)) = ((X502 @ (comp @ X504 @ X505) @ (sub @ X506 @ X505))))) | ~ ! [X501 : term] : (($true != ((X499 @ id @ X501 @ id))) | ($true = ((X499 @ id @ (sub @ X500 @ (push @ X501 @ id)) @ id))))) => ! [X508 : (subst > term > subst > $o),X507 : term] : (~ ! [X512 : term,X511 : subst,X510 : subst,X509 : subst] : (($true = ((X508 @ X510 @ X512 @ (comp @ X511 @ X509)))) | ($true != ((X508 @ (comp @ X510 @ X511) @ (sub @ X512 @ X511) @ X509)))) | ~ ! [X518 : (subst > term > term)] : (~ ! [X522 : subst,X520 : subst,X521 : term] : (((X518 @ (comp @ X522 @ X520) @ (sub @ X521 @ X520))) = ((sub @ (X518 @ X522 @ X521) @ X520))) | ~ ! [X519 : term] : ((((X508 @ id @ X519 @ id)) != $true) | (((X508 @ id @ (X518 @ id @ X519) @ id)) = $true)) | (((X508 @ id @ (lam @ (X518 @ sh @ one)) @ id)) = $true)) | ~ ! [X517 : subst,X516 : subst,X515 : subst,X514 : term] : (($true != ((X508 @ X515 @ X514 @ (comp @ X517 @ X516)))) | ($true = ((X508 @ (comp @ X515 @ X517) @ (sub @ X514 @ X517) @ X516)))) | ($true = ((X508 @ id @ (lam @ X507) @ id))) | ~ ! [X513 : term] : ((((X508 @ id @ (sub @ X507 @ (push @ X513 @ id)) @ id)) = $true) | ($true != ((X508 @ id @ X513 @ id))))))) <=> (hoasinduction_lem3_lthm = $true)) & ((! [X104 : subst,X105 : term] : (((sub @ one @ (push @ X105 @ X104))) = X105) => ! [X108 : (term > $o),X107 : term,X109 : subst,X106 : (term > $o)] : (($true = ((X106 @ one))) | ~ ! [X110 : term] : (((X106 @ X110)) = ((X108 @ (sub @ X110 @ (push @ X107 @ X109))))) | ($true != ((X108 @ X107))))) <=> (pushprop_lem1v2_lthm = $true)) & (hoaslaminj_gthm = $true) & (hoasinduction_lem1_gthm = $true) & (hoasinduction_lem3a_gthm = $true) & ((hoasinduction_lem3a = $true) <=> ! [X54 : (subst > term > subst > $o),X53 : term] : (($true = ((X54 @ id @ (lam @ X53) @ id))) | ~ ! [X60 : term] : (($true = ((X54 @ id @ (sub @ X53 @ (push @ X60 @ id)) @ id))) | ($true != ((X54 @ id @ X60 @ id)))) | ~ ! [X55 : (subst > term > term)] : (~ ! [X57 : subst,X56 : subst,X58 : term] : (((X55 @ (comp @ X57 @ X56) @ (sub @ X58 @ X56))) = ((sub @ (X55 @ X57 @ X58) @ X56))) | ($true = ((X54 @ id @ (lam @ (X55 @ sh @ one)) @ id))) | ~ ! [X59 : term] : (($true = ((X54 @ id @ (X55 @ id @ X59) @ id))) | ($true != ((X54 @ id @ X59 @ id))))))) & ((hoaslamnotap = $true) <=> ! [X93 : (subst > term > term),X94 : term,X92 : term] : ((((lam @ (X93 @ sh @ one))) != ((ap @ (sub @ X94 @ id) @ X92))) | ~ ! [X95 : subst,X96 : term,X97 : subst] : (((X93 @ (comp @ X95 @ X97) @ (sub @ X96 @ X97))) = ((sub @ (X93 @ X95 @ X96) @ X97))))) & ! [X111 : d_term] : (d_one = X111) & (! [X260 : term,X261 : (subst > term > subst > $o)] : (($true = ((X261 @ id @ (lam @ (sub @ X260 @ (push @ one @ sh))) @ id))) | ~ ! [X267 : term] : ((((X261 @ id @ (sub @ X260 @ (push @ X267 @ id)) @ id)) = $true) | (((X261 @ id @ X267 @ id)) != $true)) | ~ ! [X262 : (subst > term > term)] : (($true = ((X261 @ id @ (lam @ (X262 @ sh @ one)) @ id))) | ~ ! [X266 : term] : ((((X261 @ id @ (X262 @ id @ X266) @ id)) = $true) | ($true != ((X261 @ id @ X266 @ id)))) | ~ ! [X264 : term,X265 : subst,X263 : subst] : (((X262 @ (comp @ X265 @ X263) @ (sub @ X264 @ X263))) = ((sub @ (X262 @ X265 @ X264) @ X263))))) <=> (hoasinduction_lem3aa_lthm = $true)) & ((pushprop_lem1v2 = $true) <=> ! [X389 : subst,X390 : term,X391 : (term > $o),X388 : (term > $o)] : (($true != ((X391 @ X390))) | ~ ! [X392 : term] : (((X391 @ (sub @ X392 @ (push @ X390 @ X389)))) = ((X388 @ X392))) | ($true = ((X388 @ one))))) & (axvarshift = $true) & (! [X238 : (term > $o),X239 : term,X237 : subst] : ~ ! [X240 : (term > $o)] : ~ ! [X241 : term] : (((X238 @ (sub @ X241 @ (push @ X239 @ X237)))) = ((X240 @ X241))) <=> (pushprop_lem0 = $true)) & ((substmonoid = $true) <=> (! [X224 : subst] : (((comp @ X224 @ id)) = X224) & ! [X223 : subst] : (((comp @ id @ X223)) = X223))) & ((! [X411 : term] : (((sub @ X411 @ id)) = X411) => (! [X412 : term,X413 : (subst > term > subst > $o)] : ((((X413 @ id @ (lam @ (sub @ X412 @ (push @ one @ sh))) @ id)) = $true) | ~ ! [X419 : term] : (($true != ((X413 @ id @ X419 @ id))) | ($true = ((X413 @ id @ (sub @ X412 @ (push @ X419 @ id)) @ id)))) | ~ ! [X414 : (subst > term > term)] : (($true = ((X413 @ id @ (lam @ (X414 @ sh @ one)) @ id))) | ~ ! [X417 : subst,X416 : subst,X418 : term] : (((X414 @ (comp @ X417 @ X416) @ (sub @ X418 @ X416))) = ((sub @ (X414 @ X417 @ X418) @ X416))) | ~ ! [X415 : term] : ((((X413 @ id @ X415 @ id)) != $true) | (((X413 @ id @ (X414 @ id @ X415) @ id)) = $true)))) => ! [X420 : term,X421 : (subst > term > subst > $o)] : (~ ! [X427 : term] : ((((X421 @ id @ X427 @ id)) != $true) | (((X421 @ id @ (sub @ X420 @ (push @ X427 @ id)) @ id)) = $true)) | ($true = ((X421 @ id @ (lam @ X420) @ id))) | ~ ! [X422 : (subst > term > term)] : (~ ! [X423 : subst,X424 : subst,X425 : term] : (((sub @ (X422 @ X423 @ X425) @ X424)) = ((X422 @ (comp @ X423 @ X424) @ (sub @ X425 @ X424)))) | (((X421 @ id @ (lam @ (X422 @ sh @ one)) @ id)) = $true) | ~ ! [X426 : term] : (($true = ((X421 @ id @ (X422 @ id @ X426) @ id))) | ($true != ((X421 @ id @ X426 @ id)))))))) <=> (hoasinduction_lem3a_lthm = $true)) & ((hoasinduction_lem3v2_f = $true) <=> ! [X100 : term] : ~ ! [X101 : (subst > term > term)] : ~ ! [X102 : subst,X103 : term] : (((X101 @ X102 @ X103)) = ((sub @ X100 @ (push @ X103 @ X102))))) & (! [X535 : subst] : (((comp @ id @ X535)) = X535) <=> (axidl = $true)) & (lamnotap != $true) & (hoaslamnotvar_lthm = $true) & (id = ((d2subst @ d_id))) & (! [X217 : term,X216 : (term > $o),X215 : subst] : (~ ! [X221 : term] : (~ ! [X222 : term] : (($true = ((X216 @ (sub @ X221 @ (push @ X222 @ id))))) | ($true != ((X216 @ X222)))) | ($true = ((X216 @ (lam @ X221))))) | ~ ! [X220 : term,X219 : term] : (($true != ((X216 @ X220))) | (((X216 @ (ap @ X219 @ X220))) = $true) | ($true != ((X216 @ X219)))) | (((X216 @ (sub @ X217 @ X215))) = $true) | ~ ! [X218 : term] : (($true != ((var @ X218))) | ($true = ((X216 @ (sub @ X218 @ X215)))))) <=> (induction2lem = $true)) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (hoasinduction_gthm = $true) & (axabs = $true) & ((hoaslaminj_lthm = $true) <=> (! [X33 : term,X32 : subst] : (((sub @ one @ (push @ X33 @ X32))) = X33) => (! [X34 : term,X35 : subst] : (((comp @ sh @ (push @ X34 @ X35))) = X35) => (! [X36 : term,X37 : term] : ((((lam @ X37)) != ((lam @ X36))) | (X36 = X37)) => ! [X41 : term,X40 : (subst > term > term),X38 : subst,X39 : (subst > term > term)] : ((((X40 @ X38 @ X41)) = ((X39 @ X38 @ X41))) | ~ ! [X45 : subst,X46 : term,X47 : subst] : (((sub @ (X40 @ X45 @ X46) @ X47)) = ((X40 @ (comp @ X45 @ X47) @ (sub @ X46 @ X47)))) | (((lam @ (X39 @ sh @ one))) != ((lam @ (X40 @ sh @ one)))) | ~ ! [X43 : subst,X44 : term,X42 : subst] : (((sub @ (X39 @ X43 @ X44) @ X42)) = ((X39 @ (comp @ X43 @ X42) @ (sub @ X44 @ X42))))))))) & (hoaslamnotvar_gthm = $true) & (! [X206 : (subst > term > subst > $o),X207 : term] : (~ ! [X208 : (subst > term > term)] : (~ ! [X211 : subst,X210 : subst,X209 : term] : (((sub @ (X208 @ X211 @ X209) @ X210)) = ((X208 @ (comp @ X211 @ X210) @ (sub @ X209 @ X210)))) | ~ ! [X212 : term] : (($true = ((X206 @ id @ (X208 @ id @ X212) @ id))) | ($true != ((X206 @ id @ X212 @ id)))) | ($true = ((X206 @ id @ (lam @ (X208 @ sh @ one)) @ id)))) | ~ ! [X214 : term,X213 : term] : (($true = ((X206 @ id @ (ap @ (sub @ X213 @ id) @ X214) @ id))) | ($true != ((X206 @ id @ X213 @ id))) | (((X206 @ id @ X214 @ id)) != $true)) | ($true = ((X206 @ id @ X207 @ id)))) <=> (hoasinduction_no_psi_cond = $true)) & (! [X0 : term] : (((sub @ X0 @ id)) = X0) <=> (axvarid = $true)) & ((pushprop_lem1 = $true) <=> ! [X186 : (term > $o),X185 : term,X187 : subst,X188 : term,X184 : (term > $o)] : ((((X184 @ (sub @ X185 @ (push @ X188 @ X187)))) = $true) | ($true != ((X186 @ X185))))) & ((hoasinduction_lthm_3 = $true) <=> (! [X311 : (subst > term > subst > $o)] : ~ ! [X312 : (term > $o)] : ~ ! [X313 : term] : (((X312 @ X313)) = ((X311 @ id @ X313 @ id))) => (! [X314 : term,X315 : (term > $o)] : (~ ! [X318 : term] : ((((X315 @ X318)) = $true) | (((var @ X318)) != $true)) | ~ ! [X319 : term] : (~ ! [X320 : term] : ((((X315 @ (sub @ X319 @ (push @ X320 @ id)))) = $true) | ($true != ((X315 @ X320)))) | ($true = ((X315 @ (lam @ X319))))) | ~ ! [X317 : term,X316 : term] : (($true != ((X315 @ X317))) | (((X315 @ (ap @ X317 @ X316))) = $true) | ($true != ((X315 @ X316)))) | ($true = ((X315 @ X314)))) => (! [X321 : term] : (((sub @ X321 @ id)) = X321) => (! [X324 : (subst > term > subst > $o),X322 : term,X323 : (term > $o)] : (~ ! [X325 : term] : ((((X323 @ X325)) != $true) | ($true = ((X323 @ (sub @ X322 @ (push @ X325 @ id)))))) | ~ ! [X326 : (subst > term > term)] : (~ ! [X327 : term] : ((((X324 @ id @ X327 @ id)) != $true) | ($true = ((X324 @ id @ (X326 @ id @ X327) @ id)))) | ~ ! [X329 : term,X330 : subst,X328 : subst] : (((X326 @ (comp @ X328 @ X330) @ (sub @ X329 @ X330))) = ((sub @ (X326 @ X328 @ X329) @ X330))) | ($true = ((X324 @ id @ (lam @ (X326 @ sh @ one)) @ id)))) | ~ ! [X331 : term] : (((X324 @ id @ X331 @ id)) = ((X323 @ X331))) | ($true = ((X323 @ (lam @ X322))))) => ! [X333 : term,X332 : (subst > term > subst > $o)] : (($true = ((X332 @ id @ X333 @ id))) | ~ ! [X349 : term] : (($true = ((X332 @ id @ X349 @ id))) | ($true != ((var @ (sub @ X349 @ id))))) | ~ ! [X348 : subst,X347 : term,X345 : subst,X346 : subst] : ((((X332 @ (comp @ X346 @ X345) @ (sub @ X347 @ X345) @ X348)) != $true) | ($true = ((X332 @ X346 @ X347 @ (comp @ X345 @ X348))))) | ~ ! [X339 : term,X338 : term] : (($true != ((X332 @ id @ X339 @ id))) | (((X332 @ id @ X338 @ id)) != $true) | ($true = ((X332 @ id @ (ap @ (sub @ X338 @ id) @ X339) @ id)))) | ~ ! [X340 : (subst > term > term)] : (~ ! [X344 : subst,X342 : term,X343 : subst] : (((X340 @ (comp @ X343 @ X344) @ (sub @ X342 @ X344))) = ((sub @ (X340 @ X343 @ X342) @ X344))) | ($true = ((X332 @ id @ (lam @ (X340 @ sh @ one)) @ id))) | ~ ! [X341 : term] : (($true != ((X332 @ id @ X341 @ id))) | ($true = ((X332 @ id @ (X340 @ id @ X341) @ id))))) | ~ ! [X335 : term,X336 : subst,X334 : subst,X337 : subst] : (($true = ((X332 @ (comp @ X334 @ X336) @ (sub @ X335 @ X336) @ X337))) | ($true != ((X332 @ X334 @ X335 @ (comp @ X336 @ X337))))))))))) & ! [X309 : d_subst,X310 : d_subst] : ((((d2subst @ X310)) = ((d2subst @ X309))) => (X309 = X310)) & (! [X71 : term] : (((sub @ X71 @ id)) = X71) <=> (termmset = $true)) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (! [X445 : (subst > term > subst > $o)] : ~ ! [X446 : (term > $o)] : ~ ! [X447 : term] : (((X445 @ id @ X447 @ id)) = ((X446 @ X447))) => (! [X449 : (term > $o),X448 : term] : (~ ! [X451 : term,X450 : term] : ((((X449 @ X451)) != $true) | ($true = ((X449 @ (ap @ X450 @ X451)))) | ($true != ((X449 @ X450)))) | ($true = ((X449 @ X448))) | ~ ! [X454 : term] : ((((var @ X454)) != $true) | (((X449 @ X454)) = $true)) | ~ ! [X452 : term] : (~ ! [X453 : term] : (($true = ((X449 @ (sub @ X452 @ (push @ X453 @ id))))) | (((X449 @ X453)) != $true)) | ($true = ((X449 @ (lam @ X452)))))) => (! [X455 : term] : (((sub @ X455 @ id)) = X455) => (! [X456 : (term > $o),X458 : term,X457 : (subst > term > subst > $o)] : (~ ! [X461 : (subst > term > term)] : (($true = ((X457 @ id @ (lam @ (X461 @ sh @ one)) @ id))) | ~ ! [X463 : term,X464 : subst,X465 : subst] : (((sub @ (X461 @ X465 @ X463) @ X464)) = ((X461 @ (comp @ X465 @ X464) @ (sub @ X463 @ X464)))) | ~ ! [X462 : term] : ((((X457 @ id @ (X461 @ id @ X462) @ id)) = $true) | ($true != ((X457 @ id @ X462 @ id))))) | ~ ! [X460 : term] : ((((X456 @ (sub @ X458 @ (push @ X460 @ id)))) = $true) | (((X456 @ X460)) != $true)) | ~ ! [X459 : term] : (((X457 @ id @ X459 @ id)) = ((X456 @ X459))) | (((X456 @ (lam @ X458))) = $true)) => ! [X466 : (subst > term > subst > $o),X467 : term] : (~ ! [X470 : (subst > term > term)] : (~ ! [X471 : term,X473 : subst,X472 : subst] : (((X470 @ (comp @ X472 @ X473) @ (sub @ X471 @ X473))) = ((sub @ (X470 @ X472 @ X471) @ X473))) | ($true = ((X466 @ id @ (lam @ (X470 @ sh @ one)) @ id))) | ~ ! [X474 : term] : (($true != ((X466 @ id @ X474 @ id))) | ($true = ((X466 @ id @ (X470 @ id @ X474) @ id))))) | ~ ! [X469 : term,X468 : term] : (($true = ((X466 @ id @ (ap @ (sub @ X469 @ id) @ X468) @ id))) | ($true != ((X466 @ id @ X469 @ id))) | ($true != ((X466 @ id @ X468 @ id)))) | (((X466 @ id @ X467 @ id)) = $true))))))) & ! [X48 : d_subst] : (d_id = X48) & (induction2_gthm = $true) & (! [X30 : term,X31 : term,X29 : term,X28 : term] : ((((ap @ (sub @ X30 @ id) @ X29)) != ((ap @ (sub @ X28 @ id) @ X31))) | (X28 = X30)) <=> (hoasapinj1 = $true)) & (ulamvarind = $true) & (pushprop_lem2v2 = $true) & (pushprop_lem3v2_lthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (apnotvar = $true) & ((induction2_lthm = $true) <=> (! [X475 : term] : (((sub @ X475 @ id)) = X475) => (! [X476 : subst,X478 : term,X477 : (term > $o)] : (~ ! [X479 : term,X480 : term] : (($true != ((X477 @ X480))) | ($true = ((X477 @ (ap @ X479 @ X480)))) | (((X477 @ X479)) != $true)) | ~ ! [X483 : term] : (($true = ((X477 @ (sub @ X483 @ X476)))) | ($true != ((var @ X483)))) | ~ ! [X481 : term] : (~ ! [X482 : term] : ((((X477 @ X482)) != $true) | (((X477 @ (sub @ X481 @ (push @ X482 @ id)))) = $true)) | ($true = ((X477 @ (lam @ X481))))) | ($true = ((X477 @ (sub @ X478 @ X476))))) => ! [X484 : (term > $o),X485 : term] : (~ ! [X490 : term] : (($true != ((var @ X490))) | ($true = ((X484 @ X490)))) | ($true = ((X484 @ X485))) | ~ ! [X486 : term] : ((((X484 @ (lam @ X486))) = $true) | ~ ! [X487 : term] : (($true != ((X484 @ X487))) | ($true = ((X484 @ (sub @ X486 @ (push @ X487 @ id))))))) | ~ ! [X489 : term,X488 : term] : (($true != ((X484 @ X489))) | ($true != ((X484 @ X488))) | ($true = ((X484 @ (ap @ X489 @ X488))))))))) & (ulamvarsh = $true) & ((hoasinduction_lem2v2 = $true) <=> ! [X431 : (subst > term > subst > $o),X432 : (term > $o),X430 : term,X433 : term] : (($true != ((X432 @ X433))) | ~ ! [X436 : term,X435 : term] : (($true = ((X431 @ id @ (ap @ (sub @ X435 @ id) @ X436) @ id))) | ($true != ((X431 @ id @ X435 @ id))) | ($true != ((X431 @ id @ X436 @ id)))) | ~ ! [X439 : term,X437 : subst,X438 : subst,X440 : subst] : ((((X431 @ X437 @ X439 @ (comp @ X438 @ X440))) = $true) | ($true != ((X431 @ (comp @ X437 @ X438) @ (sub @ X439 @ X438) @ X440)))) | ~ ! [X434 : term] : (((X432 @ X434)) = ((X431 @ id @ X434 @ id))) | (((X432 @ X430)) != $true) | ~ ! [X444 : subst,X441 : subst,X443 : term,X442 : subst] : (($true = ((X431 @ (comp @ X442 @ X441) @ (sub @ X443 @ X441) @ X444))) | ($true != ((X431 @ X442 @ X443 @ (comp @ X441 @ X444))))) | ($true = ((X432 @ (ap @ X433 @ X430)))))) & (hoasinduction_lem3v2_gthm = $true) & ((axvarcons = $true) <=> ! [X64 : subst,X63 : term] : (((sub @ one @ (push @ X63 @ X64))) = X63)) & (! [X372 : subst,X369 : term,X370 : (subst > term > term),X371 : (subst > term > term)] : ((((lam @ (X370 @ sh @ one))) != ((lam @ (X371 @ sh @ one)))) | ~ ! [X377 : subst,X378 : term,X376 : subst] : (((sub @ (X371 @ X376 @ X378) @ X377)) = ((X371 @ (comp @ X376 @ X377) @ (sub @ X378 @ X377)))) | ~ ! [X375 : subst,X374 : subst,X373 : term] : (((sub @ (X370 @ X375 @ X373) @ X374)) = ((X370 @ (comp @ X375 @ X374) @ (sub @ X373 @ X374)))) | (((X371 @ X372 @ X369)) = ((X370 @ X372 @ X369)))) <=> (hoaslaminj = $true)) & (hoaslamnotap_gthm = $true) & ((induction2lem_lthm = $true) <=> (! [X351 : term,X350 : subst] : (((sub @ one @ (push @ X351 @ X350))) = X351) => (! [X352 : subst,X353 : term] : (((comp @ sh @ (push @ X353 @ X352))) = X352) => (! [X354 : subst] : (((comp @ X354 @ id)) = X354) => (! [X356 : (term > $o),X355 : term] : (~ ! [X358 : term,X359 : term] : ((((X356 @ (ap @ X358 @ X359))) = $true) | ($true != ((X356 @ X359))) | ($true != ((X356 @ X358)))) | (((X356 @ X355)) = $true) | ~ ! [X357 : term] : (($true = ((X356 @ X357))) | ($true != ((var @ X357)))) | ~ ! [X360 : term] : ((((X356 @ X360)) != $true) | ($true = ((X356 @ (lam @ X360)))))) => ! [X363 : term,X362 : subst,X361 : (term > $o)] : (~ ! [X364 : term] : ((((var @ X364)) != $true) | (((X361 @ (sub @ X364 @ X362))) = $true)) | ($true = ((X361 @ (sub @ X363 @ X362)))) | ~ ! [X365 : term] : (($true = ((X361 @ (lam @ X365)))) | ~ ! [X366 : term] : (($true != ((X361 @ X366))) | ($true = ((X361 @ (sub @ X365 @ (push @ X366 @ id))))))) | ~ ! [X367 : term,X368 : term] : (($true != ((X361 @ X368))) | ($true != ((X361 @ X367))) | ($true = ((X361 @ (ap @ X367 @ X368))))))))))) & (! [X99 : term,X98 : term] : ((((lam @ X98)) != ((lam @ X99))) | (X98 = X99)) <=> (laminj = $true)) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X225 : term] : ~ ! [X226 : (subst > term > term)] : (((sub @ X225 @ (push @ one @ sh))) != ((X226 @ sh @ one)))) & ((hoasinduction_lem3v2_f_lthm = $true) <=> ! [X189 : term] : ~ ! [X190 : (subst > term > term)] : ~ ! [X192 : term,X191 : subst] : (((X190 @ X191 @ X192)) = ((sub @ X189 @ (push @ X192 @ X191))))) & ((hoasinduction_lem3v2a = $true) <=> ! [X227 : (term > $o),X228 : term,X229 : (subst > term > subst > $o)] : (~ ! [X236 : term] : (($true = ((X227 @ (sub @ X228 @ (push @ X236 @ id))))) | ($true != ((X227 @ X236)))) | ($true = ((X227 @ (lam @ X228)))) | ~ ! [X230 : (subst > term > term)] : ((((X229 @ id @ (lam @ (X230 @ sh @ one)) @ id)) = $true) | ~ ! [X234 : term] : ((((X229 @ id @ (X230 @ id @ X234) @ id)) = $true) | ($true != ((X229 @ id @ X234 @ id)))) | ~ ! [X232 : term,X231 : subst,X233 : subst] : (((X230 @ (comp @ X231 @ X233) @ (sub @ X232 @ X233))) = ((sub @ (X230 @ X231 @ X232) @ X233)))) | ~ ! [X235 : term] : (((X229 @ id @ X235 @ id)) = ((X227 @ X235))))) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_lem3_gthm = $true) & (hoasinduction_lem3b_gthm = $true) & (pushprop_lem3v2 = $true) & ((hoasinduction_lthm = $true) <=> (! [X129 : (term > $o),X130 : term] : (~ ! [X134 : term] : ((((X129 @ (lam @ X134))) = $true) | ~ ! [X135 : term] : (($true = ((X129 @ (sub @ X134 @ (push @ X135 @ id))))) | ($true != ((X129 @ X135))))) | ($true = ((X129 @ X130))) | ~ ! [X133 : term] : (($true = ((X129 @ X133))) | ($true != ((var @ X133)))) | ~ ! [X131 : term,X132 : term] : (($true != ((X129 @ X131))) | (((X129 @ X132)) != $true) | ($true = ((X129 @ (ap @ X131 @ X132)))))) => (! [X137 : (subst > term > subst > $o),X136 : term,X138 : term] : (($true != ((X137 @ id @ X136 @ id))) | ~ ! [X141 : term,X139 : subst,X140 : subst,X142 : subst] : ((((X137 @ (comp @ X140 @ X139) @ (sub @ X141 @ X139) @ X142)) = $true) | ($true != ((X137 @ X140 @ X141 @ (comp @ X139 @ X142))))) | ~ ! [X145 : subst,X148 : subst,X147 : term,X146 : subst] : (($true = ((X137 @ X145 @ X147 @ (comp @ X146 @ X148)))) | (((X137 @ (comp @ X145 @ X146) @ (sub @ X147 @ X146) @ X148)) != $true)) | ~ ! [X143 : term,X144 : term] : (($true != ((X137 @ id @ X143 @ id))) | (((X137 @ id @ (ap @ (sub @ X143 @ id) @ X144) @ id)) = $true) | (((X137 @ id @ X144 @ id)) != $true)) | (((X137 @ id @ (ap @ X138 @ X136) @ id)) = $true) | (((X137 @ id @ X138 @ id)) != $true)) => (! [X150 : term,X149 : (subst > term > subst > $o)] : (~ ! [X157 : subst,X159 : subst,X156 : subst,X158 : term] : (($true != ((X149 @ X157 @ X158 @ (comp @ X159 @ X156)))) | ($true = ((X149 @ (comp @ X157 @ X159) @ (sub @ X158 @ X159) @ X156)))) | ~ ! [X152 : subst,X153 : subst,X154 : subst,X151 : term] : ((((X149 @ X152 @ X151 @ (comp @ X153 @ X154))) = $true) | ($true != ((X149 @ (comp @ X152 @ X153) @ (sub @ X151 @ X153) @ X154)))) | ~ ! [X155 : term] : ((((X149 @ id @ X155 @ id)) != $true) | (((X149 @ id @ (sub @ X150 @ (push @ X155 @ id)) @ id)) = $true)) | ~ ! [X160 : (subst > term > term)] : (~ ! [X164 : term] : (($true != ((X149 @ id @ X164 @ id))) | ($true = ((X149 @ id @ (X160 @ id @ X164) @ id)))) | ~ ! [X161 : subst,X163 : subst,X162 : term] : (((sub @ (X160 @ X163 @ X162) @ X161)) = ((X160 @ (comp @ X163 @ X161) @ (sub @ X162 @ X161)))) | ($true = ((X149 @ id @ (lam @ (X160 @ sh @ one)) @ id)))) | ($true = ((X149 @ id @ (lam @ X150) @ id)))) => ! [X166 : term,X165 : (subst > term > subst > $o)] : (~ ! [X167 : term,X168 : subst,X170 : subst,X169 : subst] : (($true != ((X165 @ X169 @ X167 @ (comp @ X168 @ X170)))) | ($true = ((X165 @ (comp @ X169 @ X168) @ (sub @ X167 @ X168) @ X170)))) | ~ ! [X172 : term,X173 : subst,X171 : subst,X174 : subst] : (($true != ((X165 @ (comp @ X173 @ X174) @ (sub @ X172 @ X174) @ X171))) | ($true = ((X165 @ X173 @ X172 @ (comp @ X174 @ X171))))) | ~ ! [X178 : (subst > term > term)] : (~ ! [X179 : subst,X180 : subst,X181 : term] : (((sub @ (X178 @ X179 @ X181) @ X180)) = ((X178 @ (comp @ X179 @ X180) @ (sub @ X181 @ X180)))) | ($true = ((X165 @ id @ (lam @ (X178 @ sh @ one)) @ id))) | ~ ! [X182 : term] : (($true = ((X165 @ id @ (X178 @ id @ X182) @ id))) | (((X165 @ id @ X182 @ id)) != $true))) | ($true = ((X165 @ id @ X166 @ id))) | ~ ! [X177 : term] : ((((var @ (sub @ X177 @ id))) != $true) | ($true = ((X165 @ id @ X177 @ id)))) | ~ ! [X176 : term,X175 : term] : (($true != ((X165 @ id @ X175 @ id))) | ($true = ((X165 @ id @ (ap @ (sub @ X175 @ id) @ X176) @ id))) | ($true != ((X165 @ id @ X176 @ id))))))))) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((termmset_lthm = $true) <=> (! [X49 : term] : (((sub @ X49 @ id)) = X49) => ! [X50 : term] : (((sub @ X50 @ id)) = X50))) & (! [X293 : (subst > term > subst > $o),X292 : term] : (~ ! [X294 : term] : (($true = ((X293 @ id @ (sub @ X292 @ (push @ X294 @ id)) @ id))) | (((X293 @ id @ X294 @ id)) != $true)) | ~ ! [X295 : (subst > term > term)] : (~ ! [X299 : term] : ((((X293 @ id @ X299 @ id)) != $true) | ($true = ((X293 @ id @ (X295 @ id @ X299) @ id)))) | ($true = ((X293 @ id @ (lam @ (X295 @ sh @ one)) @ id))) | ~ ! [X296 : subst,X297 : term,X298 : subst] : (((sub @ (X295 @ X298 @ X297) @ X296)) = ((X295 @ (comp @ X298 @ X296) @ (sub @ X297 @ X296))))) | ($true = ((X293 @ id @ (lam @ (sub @ X292 @ (push @ one @ sh))) @ id)))) <=> (hoasinduction_lem3aa = $true)) & (substmonoid_gthm = $true) & (hoasapnotvar_gthm = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (! [X5 : (term > $o),X7 : subst,X6 : term] : ~ ! [X8 : (term > $o)] : ~ ! [X9 : term] : (((X5 @ (sub @ X9 @ (push @ X6 @ X7)))) = ((X8 @ X9))) <=> (pushprop_lem0_lthm = $true)) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ! [X51 : d_term,X52 : d_term] : ((((d2term @ X51)) = ((d2term @ X52))) => (X51 = X52)) & (! [X492 : term,X491 : (term > $o)] : (~ ! [X493 : term] : (($true != ((var @ X493))) | (((X491 @ X493)) = $true)) | ($true = ((X491 @ X492))) | ~ ! [X494 : term,X495 : term] : (($true = ((X491 @ (ap @ X495 @ X494)))) | ($true != ((X491 @ X494))) | ($true != ((X491 @ X495)))) | ~ ! [X496 : term] : (~ ! [X497 : term] : (($true != ((X491 @ X497))) | (((X491 @ (sub @ X496 @ (push @ X497 @ id)))) = $true)) | ($true = ((X491 @ (lam @ X496)))))) <=> (induction2 = $true)) & (! [X183 : subst] : (((push @ (sub @ one @ X183) @ (comp @ sh @ X183))) = X183) <=> (axscons = $true)) & (((d2subst @ d_id)) = sh) & (! [X3 : term,X2 : term,X1 : term,X4 : term] : ((((ap @ X3 @ X2)) != ((ap @ X1 @ X4))) | (X2 = X4)) <=> (apinj2 = $true)) & (axmap = $true) & (axclos = $true) & (pushprop_lem0_gthm = $true) & (pushprop_lthm_orig = $true) & ! [X536 : subst] : ? [X537 : d_subst] : (((d2subst @ X537)) = X536) & (pushprop_gthm = $true) & (hoaslamnotap_lthm = $true) & (pushprop_lem2v2_lthm = $true) & (! [X126 : (subst > term > subst > $o)] : ~ ! [X127 : (term > $o)] : ~ ! [X128 : term] : (((X127 @ X128)) = ((X126 @ id @ X128 @ id))) <=> (hoasinduction_lem0 = $true)) & ((! [X13 : term] : ~ ! [X14 : (subst > term > term)] : ~ ! [X16 : subst,X15 : term] : (((sub @ X13 @ (push @ X15 @ X16))) = ((X14 @ X16 @ X15))) => (! [X17 : term] : (((sub @ X17 @ id)) = X17) => ! [X18 : term,X19 : (term > $o),X20 : (subst > term > subst > $o)] : (($true = ((X19 @ (lam @ X18)))) | ~ ! [X23 : (subst > term > term)] : (~ ! [X25 : subst,X26 : term,X27 : subst] : (((sub @ (X23 @ X25 @ X26) @ X27)) = ((X23 @ (comp @ X25 @ X27) @ (sub @ X26 @ X27)))) | ~ ! [X24 : term] : ((((X20 @ id @ (X23 @ id @ X24) @ id)) = $true) | ($true != ((X20 @ id @ X24 @ id)))) | ($true = ((X20 @ id @ (lam @ (X23 @ sh @ one)) @ id)))) | ~ ! [X21 : term] : (((X19 @ X21)) = ((X20 @ id @ X21 @ id))) | ~ ! [X22 : term] : (($true = ((X19 @ (sub @ X18 @ (push @ X22 @ id))))) | (((X19 @ X22)) != $true))))) <=> (hoasinduction_lem3v2a_lthm = $true)) & (hoasapnotvar = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (hoasinduction_lem1_lthm = $true) & (pushprop_lthm = $true) & (hoaslamnotvar = $true) & (axassoc = $true) & ((axshiftcons = $true) <=> ! [X112 : term,X113 : subst] : (((comp @ sh @ (push @ X112 @ X113))) = X113)) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (hoasapnotvar_lthm = $true) & (ulamvar1 != $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ((pushprop_lem1_lthm = $true) <=> (! [X379 : subst,X380 : term] : (((sub @ one @ (push @ X380 @ X379))) = X380) => (! [X382 : term,X381 : subst] : (((comp @ sh @ (push @ X382 @ X381))) = X381) => ! [X384 : term,X387 : subst,X386 : term,X385 : (term > $o),X383 : (term > $o)] : (($true = ((X383 @ (sub @ X384 @ (push @ X386 @ X387))))) | ($true != ((X385 @ X384))))))) & (hoasinduction_lem1 = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (lamnotvar = $true) & (induction2lem_gthm = $true) & (pushprop = $true)),
  inference(flattening,[],[f9])).
thf(f11,plain,(
  ((hoasinduction = $true) <=> ! [X393 : term,X394 : (subst > term > subst > $o)] : (? [X400 : term,X399 : term] : ((((X394 @ id @ (ap @ (sub @ X399 @ id) @ X400) @ id)) != $true) & ($true = ((X394 @ id @ X399 @ id))) & (((X394 @ id @ X400 @ id)) = $true)) | ? [X397 : subst,X395 : subst,X396 : term,X398 : subst] : (($true != ((X394 @ (comp @ X395 @ X397) @ (sub @ X396 @ X397) @ X398))) & ($true = ((X394 @ X395 @ X396 @ (comp @ X397 @ X398))))) | ? [X401 : term] : ((((var @ (sub @ X401 @ id))) = $true) & ($true != ((X394 @ id @ X401 @ id)))) | ($true = ((X394 @ id @ X393 @ id))) | ? [X406 : (subst > term > term)] : ((((X394 @ id @ (lam @ (X406 @ sh @ one)) @ id)) != $true) & ! [X407 : term] : (($true = ((X394 @ id @ (X406 @ id @ X407) @ id))) | (((X394 @ id @ X407 @ id)) != $true)) & ! [X408 : subst,X409 : subst,X410 : term] : (((X406 @ (comp @ X408 @ X409) @ (sub @ X410 @ X409))) = ((sub @ (X406 @ X408 @ X410) @ X409)))) | ? [X402 : subst,X404 : subst,X405 : subst,X403 : term] : (($true = ((X394 @ (comp @ X404 @ X402) @ (sub @ X403 @ X402) @ X405))) & ($true != ((X394 @ X404 @ X403 @ (comp @ X402 @ X405))))))) & (! [X73 : term,X72 : (subst > term > subst > $o)] : (? [X74 : term] : (($true != ((X72 @ id @ (sub @ X73 @ (push @ X74 @ id)) @ id))) & ($true = ((X72 @ id @ X74 @ id)))) | ? [X85 : subst,X86 : subst,X84 : subst,X87 : term] : (($true != ((X72 @ (comp @ X85 @ X84) @ (sub @ X87 @ X84) @ X86))) & ($true = ((X72 @ X85 @ X87 @ (comp @ X84 @ X86))))) | ? [X79 : (subst > term > term)] : (! [X83 : subst,X81 : term,X82 : subst] : (((X79 @ (comp @ X83 @ X82) @ (sub @ X81 @ X82))) = ((sub @ (X79 @ X83 @ X81) @ X82))) & ! [X80 : term] : ((((X72 @ id @ X80 @ id)) != $true) | ($true = ((X72 @ id @ (X79 @ id @ X80) @ id)))) & (((X72 @ id @ (lam @ (X79 @ sh @ one)) @ id)) != $true)) | ($true = ((X72 @ id @ (lam @ X73) @ id))) | ? [X76 : subst,X77 : term,X78 : subst,X75 : subst] : (($true = ((X72 @ (comp @ X75 @ X78) @ (sub @ X77 @ X78) @ X76))) & (((X72 @ X75 @ X77 @ (comp @ X78 @ X76))) != $true))) <=> (hoasinduction_lem3 = $true)) & (axapp = $true) & ((hoasapinj2_lthm = $true) <=> (? [X524 : term,X525 : term,X523 : term,X526 : term] : ((X523 != X524) & (((ap @ X526 @ X523)) = ((ap @ X525 @ X524)))) | ! [X529 : term,X528 : term,X527 : term,X530 : term] : ((X527 = X530) | (((ap @ (sub @ X529 @ id) @ X527)) != ((ap @ (sub @ X528 @ id) @ X530)))))) & (! [X272 : subst] : (((comp @ X272 @ id)) = X272) <=> (axidr = $true)) & (! [X193 : term,X194 : term,X195 : (subst > term > subst > $o)] : (($true != ((X195 @ id @ X194 @ id))) | ($true != ((X195 @ id @ X193 @ id))) | (((X195 @ id @ (ap @ X193 @ X194) @ id)) = $true) | ? [X199 : subst,X200 : subst,X198 : term,X201 : subst] : (($true = ((X195 @ (comp @ X201 @ X200) @ (sub @ X198 @ X200) @ X199))) & ($true != ((X195 @ X201 @ X198 @ (comp @ X200 @ X199))))) | ? [X204 : subst,X203 : subst,X202 : term,X205 : subst] : (($true != ((X195 @ (comp @ X204 @ X205) @ (sub @ X202 @ X205) @ X203))) & ($true = ((X195 @ X204 @ X202 @ (comp @ X205 @ X203))))) | ? [X197 : term,X196 : term] : (($true != ((X195 @ id @ (ap @ (sub @ X196 @ id) @ X197) @ id))) & (((X195 @ id @ X196 @ id)) = $true) & ($true = ((X195 @ id @ X197 @ id))))) <=> (hoasinduction_lem2 = $true)) & (! [X243 : (subst > term > subst > $o),X242 : term,X244 : (term > $o)] : (? [X259 : term] : (($true = ((X244 @ X259))) & (((X244 @ (sub @ X242 @ (push @ X259 @ id)))) != $true)) | ? [X249 : term] : (((X244 @ X249)) != ((X243 @ id @ X249 @ id))) | ? [X248 : subst,X246 : subst,X247 : term,X245 : subst] : (($true = ((X243 @ X246 @ X247 @ (comp @ X248 @ X245)))) & ($true != ((X243 @ (comp @ X246 @ X248) @ (sub @ X247 @ X248) @ X245)))) | ? [X253 : term,X251 : subst,X252 : subst,X250 : subst] : (($true = ((X243 @ (comp @ X251 @ X252) @ (sub @ X253 @ X252) @ X250))) & (((X243 @ X251 @ X253 @ (comp @ X252 @ X250))) != $true)) | ($true = ((X244 @ (lam @ X242)))) | ? [X254 : (subst > term > term)] : (! [X255 : term] : (($true != ((X243 @ id @ X255 @ id))) | ($true = ((X243 @ id @ (X254 @ id @ X255) @ id)))) & ! [X256 : subst,X257 : subst,X258 : term] : (((sub @ (X254 @ X257 @ X258) @ X256)) = ((X254 @ (comp @ X257 @ X256) @ (sub @ X258 @ X256)))) & ($true != ((X243 @ id @ (lam @ (X254 @ sh @ one)) @ id))))) <=> (hoasinduction_lem3v2 = $true)) & (pushprop_lem2v2_gthm = $true) & ((substmonoid_lthm = $true) <=> (((! [X91 : subst] : (((comp @ id @ X91)) = X91) & ! [X90 : subst] : (((comp @ X90 @ id)) = X90)) | ? [X89 : subst] : (((comp @ X89 @ id)) != X89)) | ? [X88 : subst] : (((comp @ id @ X88)) != X88))) & (((var @ (d2term @ d_one))) != $true) & (! [X115 : term,X114 : (subst > term > subst > $o)] : (($true = ((X114 @ id @ (lam @ (sub @ X115 @ (push @ one @ sh))) @ id))) | ? [X118 : term,X117 : (subst > term > term)] : (! [X121 : subst,X119 : subst,X120 : term] : (((X117 @ (comp @ X119 @ X121) @ (sub @ X120 @ X121))) = ((sub @ X118 @ (push @ (sub @ X120 @ X121) @ (comp @ X119 @ X121))))) & ! [X124 : subst,X123 : term,X122 : subst] : (((sub @ (sub @ X118 @ (push @ X123 @ X122)) @ X124)) = ((sub @ (X117 @ X122 @ X123) @ X124))) & (((X114 @ id @ (lam @ (X117 @ sh @ one)) @ id)) != $true) & ! [X125 : term] : (($true = ((X114 @ id @ (X117 @ id @ X125) @ id))) | ($true != ((X114 @ id @ X125 @ id))))) | ? [X116 : term] : (($true != ((X114 @ id @ (sub @ X115 @ (push @ X116 @ id)) @ id))) & ($true = ((X114 @ id @ X116 @ id))))) <=> (hoasinduction_lem3aaa = $true)) & ((hoasinduction_lem0_lthm = $true) <=> ! [X10 : (subst > term > subst > $o)] : ? [X11 : (term > $o)] : ! [X12 : term] : (((X11 @ X12)) = ((X10 @ id @ X12 @ id)))) & (((! [X307 : term,X306 : term,X308 : term,X305 : term] : ((((ap @ (sub @ X305 @ id) @ X307)) != ((ap @ (sub @ X306 @ id) @ X308))) | (X305 = X306)) | ? [X304 : term,X302 : term,X303 : term,X301 : term] : ((((ap @ X301 @ X303)) = ((ap @ X304 @ X302))) & (X301 != X304))) | ? [X300 : term] : (((sub @ X300 @ id)) != X300)) <=> (hoasapinj1_lthm = $true)) & (termmset_gthm = $true) & ((hoasinduction_lem3v2_lthm = $true) <=> (? [X273 : term] : (((sub @ X273 @ id)) != X273) | ! [X274 : (subst > term > subst > $o),X275 : term,X276 : (term > $o)] : (? [X277 : term] : (((X276 @ X277)) != ((X274 @ id @ X277 @ id))) | ($true = ((X276 @ (lam @ X275)))) | ? [X289 : subst,X290 : term,X288 : subst,X291 : subst] : (($true = ((X274 @ (comp @ X288 @ X291) @ (sub @ X290 @ X291) @ X289))) & ($true != ((X274 @ X288 @ X290 @ (comp @ X291 @ X289))))) | ? [X283 : term,X285 : subst,X284 : subst,X286 : subst] : ((((X274 @ (comp @ X285 @ X286) @ (sub @ X283 @ X286) @ X284)) != $true) & (((X274 @ X285 @ X283 @ (comp @ X286 @ X284))) = $true)) | ? [X278 : (subst > term > term)] : (! [X281 : subst,X280 : subst,X282 : term] : (((sub @ (X278 @ X280 @ X282) @ X281)) = ((X278 @ (comp @ X280 @ X281) @ (sub @ X282 @ X281)))) & ! [X279 : term] : ((((X274 @ id @ X279 @ id)) != $true) | ($true = ((X274 @ id @ (X278 @ id @ X279) @ id)))) & ($true != ((X274 @ id @ (lam @ (X278 @ sh @ one)) @ id)))) | ? [X287 : term] : (($true != ((X276 @ (sub @ X275 @ (push @ X287 @ id))))) & ($true = ((X276 @ X287))))))) & (! [X270 : term,X271 : term,X269 : term,X268 : term] : ((((ap @ X269 @ X270)) != ((ap @ X271 @ X268))) | (X269 = X271)) <=> (apinj1 = $true)) & (hoasapinj2_gthm = $true) & (! [X538 : term] : ? [X539 : (subst > term > term)] : (((X539 @ sh @ one)) = ((sub @ X538 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b = $true)) & (! [X429 : term,X428 : term] : ((((sub @ X428 @ sh)) != ((sub @ X429 @ sh))) | (X428 = X429)) <=> (shinj = $true)) & (one = ((d2term @ d_one))) & (hoasinduction_lem1v2_gthm = $true) & (! [X533 : term,X532 : term,X531 : term,X534 : term] : ((X532 = X533) | (((ap @ (sub @ X534 @ id) @ X533)) != ((ap @ (sub @ X531 @ id) @ X532)))) <=> (hoasapinj2 = $true)) & (hoasinduction_lem1v2 = $true) & (hoasapinj1_gthm = $true) & (hoasinduction_lem2_gthm = $true) & (! [X65 : (term > $o),X66 : term] : (? [X70 : term] : ((((X65 @ (lam @ X70))) != $true) & ($true = ((X65 @ X70)))) | ($true = ((X65 @ X66))) | ? [X69 : term,X68 : term] : (($true != ((X65 @ (ap @ X68 @ X69)))) & ($true = ((X65 @ X68))) & (((X65 @ X69)) = $true)) | ? [X67 : term] : (($true = ((var @ X67))) & ($true != ((X65 @ X67))))) <=> (induction = $true)) & (hoasinduction_lem2v2_gthm = $true) & (pushprop_lem1_gthm = $true) & ! [X61 : term] : ? [X62 : d_term] : (((d2term @ X62)) = X61) & (hoasinduction_lem2_lthm = $true) & (((! [X508 : (subst > term > subst > $o),X507 : term] : (($true = ((X508 @ id @ (lam @ X507) @ id))) | ? [X513 : term] : (($true = ((X508 @ id @ X513 @ id))) & (((X508 @ id @ (sub @ X507 @ (push @ X513 @ id)) @ id)) != $true)) | ? [X516 : subst,X514 : term,X517 : subst,X515 : subst] : (($true != ((X508 @ (comp @ X515 @ X517) @ (sub @ X514 @ X517) @ X516))) & ($true = ((X508 @ X515 @ X514 @ (comp @ X517 @ X516))))) | ? [X509 : subst,X511 : subst,X512 : term,X510 : subst] : (($true != ((X508 @ X510 @ X512 @ (comp @ X511 @ X509)))) & ($true = ((X508 @ (comp @ X510 @ X511) @ (sub @ X512 @ X511) @ X509)))) | ? [X518 : (subst > term > term)] : (! [X519 : term] : ((((X508 @ id @ X519 @ id)) != $true) | (((X508 @ id @ (X518 @ id @ X519) @ id)) = $true)) & (((X508 @ id @ (lam @ (X518 @ sh @ one)) @ id)) != $true) & ! [X522 : subst,X520 : subst,X521 : term] : (((X518 @ (comp @ X522 @ X520) @ (sub @ X521 @ X520))) = ((sub @ (X518 @ X522 @ X521) @ X520))))) | ? [X499 : (subst > term > subst > $o),X500 : term] : (! [X501 : term] : (($true != ((X499 @ id @ X501 @ id))) | ($true = ((X499 @ id @ (sub @ X500 @ (push @ X501 @ id)) @ id)))) & ($true != ((X499 @ id @ (lam @ (sub @ X500 @ (push @ one @ sh))) @ id))) & ! [X502 : (subst > term > term)] : (? [X505 : subst,X506 : term,X504 : subst] : (((sub @ (X502 @ X504 @ X506) @ X505)) != ((X502 @ (comp @ X504 @ X505) @ (sub @ X506 @ X505)))) | ? [X503 : term] : (($true != ((X499 @ id @ (X502 @ id @ X503) @ id))) & ($true = ((X499 @ id @ X503 @ id)))) | ($true = ((X499 @ id @ (lam @ (X502 @ sh @ one)) @ id)))))) | ? [X498 : term] : (((sub @ X498 @ id)) != X498)) <=> (hoasinduction_lem3_lthm = $true)) & ((! [X108 : (term > $o),X106 : (term > $o),X109 : subst,X107 : term] : (? [X110 : term] : (((X106 @ X110)) != ((X108 @ (sub @ X110 @ (push @ X107 @ X109))))) | ($true != ((X108 @ X107))) | ($true = ((X106 @ one)))) | ? [X104 : subst,X105 : term] : (((sub @ one @ (push @ X105 @ X104))) != X105)) <=> (pushprop_lem1v2_lthm = $true)) & (hoaslaminj_gthm = $true) & (hoasinduction_lem1_gthm = $true) & (hoasinduction_lem3a_gthm = $true) & ((hoasinduction_lem3a = $true) <=> ! [X54 : (subst > term > subst > $o),X53 : term] : (? [X55 : (subst > term > term)] : (! [X59 : term] : (($true = ((X54 @ id @ (X55 @ id @ X59) @ id))) | ($true != ((X54 @ id @ X59 @ id)))) & ($true != ((X54 @ id @ (lam @ (X55 @ sh @ one)) @ id))) & ! [X57 : subst,X56 : subst,X58 : term] : (((X55 @ (comp @ X57 @ X56) @ (sub @ X58 @ X56))) = ((sub @ (X55 @ X57 @ X58) @ X56)))) | ? [X60 : term] : (($true = ((X54 @ id @ X60 @ id))) & ($true != ((X54 @ id @ (sub @ X53 @ (push @ X60 @ id)) @ id)))) | ($true = ((X54 @ id @ (lam @ X53) @ id))))) & ((hoaslamnotap = $true) <=> ! [X94 : term,X92 : term,X93 : (subst > term > term)] : (? [X95 : subst,X97 : subst,X96 : term] : (((X93 @ (comp @ X95 @ X97) @ (sub @ X96 @ X97))) != ((sub @ (X93 @ X95 @ X96) @ X97))) | (((lam @ (X93 @ sh @ one))) != ((ap @ (sub @ X94 @ id) @ X92))))) & ! [X111 : d_term] : (d_one = X111) & (! [X260 : term,X261 : (subst > term > subst > $o)] : (? [X262 : (subst > term > term)] : (($true != ((X261 @ id @ (lam @ (X262 @ sh @ one)) @ id))) & ! [X264 : term,X265 : subst,X263 : subst] : (((X262 @ (comp @ X265 @ X263) @ (sub @ X264 @ X263))) = ((sub @ (X262 @ X265 @ X264) @ X263))) & ! [X266 : term] : ((((X261 @ id @ (X262 @ id @ X266) @ id)) = $true) | ($true != ((X261 @ id @ X266 @ id))))) | ? [X267 : term] : ((((X261 @ id @ (sub @ X260 @ (push @ X267 @ id)) @ id)) != $true) & (((X261 @ id @ X267 @ id)) = $true)) | ($true = ((X261 @ id @ (lam @ (sub @ X260 @ (push @ one @ sh))) @ id)))) <=> (hoasinduction_lem3aa_lthm = $true)) & ((pushprop_lem1v2 = $true) <=> ! [X388 : (term > $o),X391 : (term > $o),X390 : term,X389 : subst] : (? [X392 : term] : (((X391 @ (sub @ X392 @ (push @ X390 @ X389)))) != ((X388 @ X392))) | ($true != ((X391 @ X390))) | ($true = ((X388 @ one))))) & (axvarshift = $true) & ((pushprop_lem0 = $true) <=> ! [X237 : subst,X238 : (term > $o),X239 : term] : ? [X240 : (term > $o)] : ! [X241 : term] : (((X238 @ (sub @ X241 @ (push @ X239 @ X237)))) = ((X240 @ X241)))) & ((substmonoid = $true) <=> (! [X224 : subst] : (((comp @ X224 @ id)) = X224) & ! [X223 : subst] : (((comp @ id @ X223)) = X223))) & (((! [X421 : (subst > term > subst > $o),X420 : term] : (? [X422 : (subst > term > term)] : (! [X423 : subst,X424 : subst,X425 : term] : (((sub @ (X422 @ X423 @ X425) @ X424)) = ((X422 @ (comp @ X423 @ X424) @ (sub @ X425 @ X424)))) & ! [X426 : term] : (($true = ((X421 @ id @ (X422 @ id @ X426) @ id))) | ($true != ((X421 @ id @ X426 @ id)))) & (((X421 @ id @ (lam @ (X422 @ sh @ one)) @ id)) != $true)) | ? [X427 : term] : ((((X421 @ id @ (sub @ X420 @ (push @ X427 @ id)) @ id)) != $true) & (((X421 @ id @ X427 @ id)) = $true)) | ($true = ((X421 @ id @ (lam @ X420) @ id)))) | ? [X413 : (subst > term > subst > $o),X412 : term] : (! [X414 : (subst > term > term)] : (? [X418 : term,X417 : subst,X416 : subst] : (((X414 @ (comp @ X417 @ X416) @ (sub @ X418 @ X416))) != ((sub @ (X414 @ X417 @ X418) @ X416))) | ? [X415 : term] : ((((X413 @ id @ X415 @ id)) = $true) & (((X413 @ id @ (X414 @ id @ X415) @ id)) != $true)) | ($true = ((X413 @ id @ (lam @ (X414 @ sh @ one)) @ id)))) & ! [X419 : term] : (($true != ((X413 @ id @ X419 @ id))) | ($true = ((X413 @ id @ (sub @ X412 @ (push @ X419 @ id)) @ id)))) & (((X413 @ id @ (lam @ (sub @ X412 @ (push @ one @ sh))) @ id)) != $true))) | ? [X411 : term] : (((sub @ X411 @ id)) != X411)) <=> (hoasinduction_lem3a_lthm = $true)) & ((hoasinduction_lem3v2_f = $true) <=> ! [X100 : term] : ? [X101 : (subst > term > term)] : ! [X102 : subst,X103 : term] : (((X101 @ X102 @ X103)) = ((sub @ X100 @ (push @ X103 @ X102))))) & (! [X535 : subst] : (((comp @ id @ X535)) = X535) <=> (axidl = $true)) & (lamnotap != $true) & (hoaslamnotvar_lthm = $true) & (id = ((d2subst @ d_id))) & ((induction2lem = $true) <=> ! [X217 : term,X215 : subst,X216 : (term > $o)] : (? [X221 : term] : (($true != ((X216 @ (lam @ X221)))) & ! [X222 : term] : (($true = ((X216 @ (sub @ X221 @ (push @ X222 @ id))))) | ($true != ((X216 @ X222))))) | ? [X219 : term,X220 : term] : (($true = ((X216 @ X220))) & (((X216 @ (ap @ X219 @ X220))) != $true) & ($true = ((X216 @ X219)))) | ? [X218 : term] : (($true != ((X216 @ (sub @ X218 @ X215)))) & ($true = ((var @ X218)))) | (((X216 @ (sub @ X217 @ X215))) = $true))) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (hoasinduction_gthm = $true) & (axabs = $true) & ((hoaslaminj_lthm = $true) <=> (((! [X39 : (subst > term > term),X40 : (subst > term > term),X41 : term,X38 : subst] : (? [X46 : term,X47 : subst,X45 : subst] : (((sub @ (X40 @ X45 @ X46) @ X47)) != ((X40 @ (comp @ X45 @ X47) @ (sub @ X46 @ X47)))) | (((lam @ (X39 @ sh @ one))) != ((lam @ (X40 @ sh @ one)))) | ? [X42 : subst,X44 : term,X43 : subst] : (((sub @ (X39 @ X43 @ X44) @ X42)) != ((X39 @ (comp @ X43 @ X42) @ (sub @ X44 @ X42)))) | (((X40 @ X38 @ X41)) = ((X39 @ X38 @ X41)))) | ? [X36 : term,X37 : term] : ((X36 != X37) & (((lam @ X37)) = ((lam @ X36))))) | ? [X35 : subst,X34 : term] : (((comp @ sh @ (push @ X34 @ X35))) != X35)) | ? [X32 : subst,X33 : term] : (((sub @ one @ (push @ X33 @ X32))) != X33))) & (hoaslamnotvar_gthm = $true) & ((hoasinduction_no_psi_cond = $true) <=> ! [X206 : (subst > term > subst > $o),X207 : term] : (($true = ((X206 @ id @ X207 @ id))) | ? [X208 : (subst > term > term)] : (($true != ((X206 @ id @ (lam @ (X208 @ sh @ one)) @ id))) & ! [X211 : subst,X210 : subst,X209 : term] : (((sub @ (X208 @ X211 @ X209) @ X210)) = ((X208 @ (comp @ X211 @ X210) @ (sub @ X209 @ X210)))) & ! [X212 : term] : (($true = ((X206 @ id @ (X208 @ id @ X212) @ id))) | ($true != ((X206 @ id @ X212 @ id))))) | ? [X214 : term,X213 : term] : ((((X206 @ id @ X214 @ id)) = $true) & ($true = ((X206 @ id @ X213 @ id))) & ($true != ((X206 @ id @ (ap @ (sub @ X213 @ id) @ X214) @ id)))))) & (! [X0 : term] : (((sub @ X0 @ id)) = X0) <=> (axvarid = $true)) & ((pushprop_lem1 = $true) <=> ! [X186 : (term > $o),X185 : term,X187 : subst,X188 : term,X184 : (term > $o)] : ((((X184 @ (sub @ X185 @ (push @ X188 @ X187)))) = $true) | ($true != ((X186 @ X185))))) & ((hoasinduction_lthm_3 = $true) <=> ((((! [X333 : term,X332 : (subst > term > subst > $o)] : (? [X340 : (subst > term > term)] : (! [X341 : term] : (($true != ((X332 @ id @ X341 @ id))) | ($true = ((X332 @ id @ (X340 @ id @ X341) @ id)))) & ($true != ((X332 @ id @ (lam @ (X340 @ sh @ one)) @ id))) & ! [X344 : subst,X342 : term,X343 : subst] : (((X340 @ (comp @ X343 @ X344) @ (sub @ X342 @ X344))) = ((sub @ (X340 @ X343 @ X342) @ X344)))) | ($true = ((X332 @ id @ X333 @ id))) | ? [X349 : term] : (($true != ((X332 @ id @ X349 @ id))) & ($true = ((var @ (sub @ X349 @ id))))) | ? [X338 : term,X339 : term] : (($true = ((X332 @ id @ X339 @ id))) & (((X332 @ id @ X338 @ id)) = $true) & ($true != ((X332 @ id @ (ap @ (sub @ X338 @ id) @ X339) @ id)))) | ? [X337 : subst,X335 : term,X334 : subst,X336 : subst] : (($true = ((X332 @ X334 @ X335 @ (comp @ X336 @ X337)))) & ($true != ((X332 @ (comp @ X334 @ X336) @ (sub @ X335 @ X336) @ X337)))) | ? [X345 : subst,X346 : subst,X348 : subst,X347 : term] : ((((X332 @ (comp @ X346 @ X345) @ (sub @ X347 @ X345) @ X348)) = $true) & ($true != ((X332 @ X346 @ X347 @ (comp @ X345 @ X348)))))) | ? [X322 : term,X323 : (term > $o),X324 : (subst > term > subst > $o)] : (($true != ((X323 @ (lam @ X322)))) & ! [X326 : (subst > term > term)] : (($true = ((X324 @ id @ (lam @ (X326 @ sh @ one)) @ id))) | ? [X330 : subst,X328 : subst,X329 : term] : (((X326 @ (comp @ X328 @ X330) @ (sub @ X329 @ X330))) != ((sub @ (X326 @ X328 @ X329) @ X330))) | ? [X327 : term] : ((((X324 @ id @ X327 @ id)) = $true) & ($true != ((X324 @ id @ (X326 @ id @ X327) @ id))))) & ! [X325 : term] : ((((X323 @ X325)) != $true) | ($true = ((X323 @ (sub @ X322 @ (push @ X325 @ id)))))) & ! [X331 : term] : (((X324 @ id @ X331 @ id)) = ((X323 @ X331))))) | ? [X321 : term] : (((sub @ X321 @ id)) != X321)) | ? [X314 : term,X315 : (term > $o)] : (! [X318 : term] : ((((X315 @ X318)) = $true) | (((var @ X318)) != $true)) & ($true != ((X315 @ X314))) & ! [X319 : term] : (? [X320 : term] : (($true = ((X315 @ X320))) & (((X315 @ (sub @ X319 @ (push @ X320 @ id)))) != $true)) | ($true = ((X315 @ (lam @ X319))))) & ! [X317 : term,X316 : term] : (($true != ((X315 @ X317))) | (((X315 @ (ap @ X317 @ X316))) = $true) | ($true != ((X315 @ X316)))))) | ? [X311 : (subst > term > subst > $o)] : ! [X312 : (term > $o)] : ? [X313 : term] : (((X312 @ X313)) != ((X311 @ id @ X313 @ id))))) & ! [X309 : d_subst,X310 : d_subst] : ((((d2subst @ X310)) != ((d2subst @ X309))) | (X309 = X310)) & (! [X71 : term] : (((sub @ X71 @ id)) = X71) <=> (termmset = $true)) & ((hoasinduction_no_psi_cond_lthm = $true) <=> ((((! [X467 : term,X466 : (subst > term > subst > $o)] : (? [X470 : (subst > term > term)] : (! [X474 : term] : (($true != ((X466 @ id @ X474 @ id))) | ($true = ((X466 @ id @ (X470 @ id @ X474) @ id)))) & ($true != ((X466 @ id @ (lam @ (X470 @ sh @ one)) @ id))) & ! [X471 : term,X473 : subst,X472 : subst] : (((X470 @ (comp @ X472 @ X473) @ (sub @ X471 @ X473))) = ((sub @ (X470 @ X472 @ X471) @ X473)))) | ? [X468 : term,X469 : term] : (($true = ((X466 @ id @ X469 @ id))) & ($true != ((X466 @ id @ (ap @ (sub @ X469 @ id) @ X468) @ id))) & ($true = ((X466 @ id @ X468 @ id)))) | (((X466 @ id @ X467 @ id)) = $true)) | ? [X456 : (term > $o),X457 : (subst > term > subst > $o),X458 : term] : (! [X459 : term] : (((X457 @ id @ X459 @ id)) = ((X456 @ X459))) & (((X456 @ (lam @ X458))) != $true) & ! [X460 : term] : ((((X456 @ (sub @ X458 @ (push @ X460 @ id)))) = $true) | (((X456 @ X460)) != $true)) & ! [X461 : (subst > term > term)] : (($true = ((X457 @ id @ (lam @ (X461 @ sh @ one)) @ id))) | ? [X462 : term] : (($true = ((X457 @ id @ X462 @ id))) & (((X457 @ id @ (X461 @ id @ X462) @ id)) != $true)) | ? [X463 : term,X465 : subst,X464 : subst] : (((sub @ (X461 @ X465 @ X463) @ X464)) != ((X461 @ (comp @ X465 @ X464) @ (sub @ X463 @ X464))))))) | ? [X455 : term] : (((sub @ X455 @ id)) != X455)) | ? [X449 : (term > $o),X448 : term] : (! [X451 : term,X450 : term] : ((((X449 @ X451)) != $true) | ($true = ((X449 @ (ap @ X450 @ X451)))) | ($true != ((X449 @ X450)))) & ($true != ((X449 @ X448))) & ! [X454 : term] : ((((var @ X454)) != $true) | (((X449 @ X454)) = $true)) & ! [X452 : term] : (? [X453 : term] : ((((X449 @ X453)) = $true) & ($true != ((X449 @ (sub @ X452 @ (push @ X453 @ id)))))) | ($true = ((X449 @ (lam @ X452))))))) | ? [X445 : (subst > term > subst > $o)] : ! [X446 : (term > $o)] : ? [X447 : term] : (((X445 @ id @ X447 @ id)) != ((X446 @ X447))))) & ! [X48 : d_subst] : (d_id = X48) & (induction2_gthm = $true) & (! [X30 : term,X31 : term,X29 : term,X28 : term] : ((((ap @ (sub @ X30 @ id) @ X29)) != ((ap @ (sub @ X28 @ id) @ X31))) | (X28 = X30)) <=> (hoasapinj1 = $true)) & (ulamvarind = $true) & (pushprop_lem2v2 = $true) & (pushprop_lem3v2_lthm = $true) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (apnotvar = $true) & ((induction2_lthm = $true) <=> ((! [X484 : (term > $o),X485 : term] : (? [X486 : term] : (! [X487 : term] : (($true != ((X484 @ X487))) | ($true = ((X484 @ (sub @ X486 @ (push @ X487 @ id)))))) & (((X484 @ (lam @ X486))) != $true)) | ? [X490 : term] : (($true = ((var @ X490))) & ($true != ((X484 @ X490)))) | ? [X489 : term,X488 : term] : (($true != ((X484 @ (ap @ X489 @ X488)))) & ($true = ((X484 @ X488))) & ($true = ((X484 @ X489)))) | ($true = ((X484 @ X485)))) | ? [X476 : subst,X477 : (term > $o),X478 : term] : (! [X483 : term] : (($true = ((X477 @ (sub @ X483 @ X476)))) | ($true != ((var @ X483)))) & ! [X479 : term,X480 : term] : (($true != ((X477 @ X480))) | ($true = ((X477 @ (ap @ X479 @ X480)))) | (((X477 @ X479)) != $true)) & ! [X481 : term] : (($true = ((X477 @ (lam @ X481)))) | ? [X482 : term] : ((((X477 @ X482)) = $true) & (((X477 @ (sub @ X481 @ (push @ X482 @ id)))) != $true))) & ($true != ((X477 @ (sub @ X478 @ X476)))))) | ? [X475 : term] : (((sub @ X475 @ id)) != X475))) & (ulamvarsh = $true) & ((hoasinduction_lem2v2 = $true) <=> ! [X431 : (subst > term > subst > $o),X430 : term,X432 : (term > $o),X433 : term] : (? [X438 : subst,X439 : term,X440 : subst,X437 : subst] : ((((X431 @ X437 @ X439 @ (comp @ X438 @ X440))) != $true) & ($true = ((X431 @ (comp @ X437 @ X438) @ (sub @ X439 @ X438) @ X440)))) | ($true != ((X432 @ X433))) | (((X432 @ X430)) != $true) | ? [X435 : term,X436 : term] : (($true = ((X431 @ id @ X435 @ id))) & ($true != ((X431 @ id @ (ap @ (sub @ X435 @ id) @ X436) @ id))) & ($true = ((X431 @ id @ X436 @ id)))) | ($true = ((X432 @ (ap @ X433 @ X430)))) | ? [X434 : term] : (((X432 @ X434)) != ((X431 @ id @ X434 @ id))) | ? [X444 : subst,X442 : subst,X441 : subst,X443 : term] : (($true != ((X431 @ (comp @ X442 @ X441) @ (sub @ X443 @ X441) @ X444))) & ($true = ((X431 @ X442 @ X443 @ (comp @ X441 @ X444))))))) & (hoasinduction_lem3v2_gthm = $true) & ((axvarcons = $true) <=> ! [X64 : subst,X63 : term] : (((sub @ one @ (push @ X63 @ X64))) = X63)) & ((hoaslaminj = $true) <=> ! [X372 : subst,X370 : (subst > term > term),X369 : term,X371 : (subst > term > term)] : ((((lam @ (X370 @ sh @ one))) != ((lam @ (X371 @ sh @ one)))) | ? [X376 : subst,X378 : term,X377 : subst] : (((sub @ (X371 @ X376 @ X378) @ X377)) != ((X371 @ (comp @ X376 @ X377) @ (sub @ X378 @ X377)))) | (((X371 @ X372 @ X369)) = ((X370 @ X372 @ X369))) | ? [X375 : subst,X374 : subst,X373 : term] : (((sub @ (X370 @ X375 @ X373) @ X374)) != ((X370 @ (comp @ X375 @ X374) @ (sub @ X373 @ X374)))))) & (hoaslamnotap_gthm = $true) & ((induction2lem_lthm = $true) <=> ((((! [X363 : term,X362 : subst,X361 : (term > $o)] : (? [X365 : term] : (($true != ((X361 @ (lam @ X365)))) & ! [X366 : term] : (($true != ((X361 @ X366))) | ($true = ((X361 @ (sub @ X365 @ (push @ X366 @ id))))))) | ($true = ((X361 @ (sub @ X363 @ X362)))) | ? [X364 : term] : ((((X361 @ (sub @ X364 @ X362))) != $true) & (((var @ X364)) = $true)) | ? [X368 : term,X367 : term] : (($true = ((X361 @ X368))) & ($true = ((X361 @ X367))) & ($true != ((X361 @ (ap @ X367 @ X368)))))) | ? [X355 : term,X356 : (term > $o)] : (! [X360 : term] : ((((X356 @ X360)) != $true) | ($true = ((X356 @ (lam @ X360))))) & ! [X358 : term,X359 : term] : ((((X356 @ (ap @ X358 @ X359))) = $true) | ($true != ((X356 @ X359))) | ($true != ((X356 @ X358)))) & ! [X357 : term] : (($true = ((X356 @ X357))) | ($true != ((var @ X357)))) & (((X356 @ X355)) != $true))) | ? [X354 : subst] : (((comp @ X354 @ id)) != X354)) | ? [X353 : term,X352 : subst] : (((comp @ sh @ (push @ X353 @ X352))) != X352)) | ? [X350 : subst,X351 : term] : (((sub @ one @ (push @ X351 @ X350))) != X351))) & (! [X99 : term,X98 : term] : ((((lam @ X98)) != ((lam @ X99))) | (X98 = X99)) <=> (laminj = $true)) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X225 : term] : ? [X226 : (subst > term > term)] : (((sub @ X225 @ (push @ one @ sh))) = ((X226 @ sh @ one)))) & (! [X189 : term] : ? [X190 : (subst > term > term)] : ! [X192 : term,X191 : subst] : (((X190 @ X191 @ X192)) = ((sub @ X189 @ (push @ X192 @ X191)))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & ((hoasinduction_lem3v2a = $true) <=> ! [X227 : (term > $o),X228 : term,X229 : (subst > term > subst > $o)] : (? [X235 : term] : (((X229 @ id @ X235 @ id)) != ((X227 @ X235))) | ($true = ((X227 @ (lam @ X228)))) | ? [X236 : term] : (($true != ((X227 @ (sub @ X228 @ (push @ X236 @ id))))) & ($true = ((X227 @ X236)))) | ? [X230 : (subst > term > term)] : (! [X232 : term,X231 : subst,X233 : subst] : (((X230 @ (comp @ X231 @ X233) @ (sub @ X232 @ X233))) = ((sub @ (X230 @ X231 @ X232) @ X233))) & ! [X234 : term] : ((((X229 @ id @ (X230 @ id @ X234) @ id)) = $true) | ($true != ((X229 @ id @ X234 @ id)))) & (((X229 @ id @ (lam @ (X230 @ sh @ one)) @ id)) != $true)))) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (pushprop_lem1v2_gthm = $true) & (hoasinduction_lem3_gthm = $true) & (hoasinduction_lem3b_gthm = $true) & (pushprop_lem3v2 = $true) & ((hoasinduction_lthm = $true) <=> (((! [X165 : (subst > term > subst > $o),X166 : term] : (? [X175 : term,X176 : term] : (($true = ((X165 @ id @ X175 @ id))) & ($true = ((X165 @ id @ X176 @ id))) & ($true != ((X165 @ id @ (ap @ (sub @ X175 @ id) @ X176) @ id)))) | ? [X178 : (subst > term > term)] : (! [X182 : term] : (($true = ((X165 @ id @ (X178 @ id @ X182) @ id))) | (((X165 @ id @ X182 @ id)) != $true)) & ! [X179 : subst,X180 : subst,X181 : term] : (((sub @ (X178 @ X179 @ X181) @ X180)) = ((X178 @ (comp @ X179 @ X180) @ (sub @ X181 @ X180)))) & ($true != ((X165 @ id @ (lam @ (X178 @ sh @ one)) @ id)))) | ? [X177 : term] : (($true != ((X165 @ id @ X177 @ id))) & (((var @ (sub @ X177 @ id))) = $true)) | ($true = ((X165 @ id @ X166 @ id))) | ? [X171 : subst,X174 : subst,X172 : term,X173 : subst] : (($true != ((X165 @ X173 @ X172 @ (comp @ X174 @ X171)))) & ($true = ((X165 @ (comp @ X173 @ X174) @ (sub @ X172 @ X174) @ X171)))) | ? [X169 : subst,X170 : subst,X168 : subst,X167 : term] : (($true != ((X165 @ (comp @ X169 @ X168) @ (sub @ X167 @ X168) @ X170))) & ($true = ((X165 @ X169 @ X167 @ (comp @ X168 @ X170)))))) | ? [X150 : term,X149 : (subst > term > subst > $o)] : (! [X155 : term] : ((((X149 @ id @ X155 @ id)) != $true) | (((X149 @ id @ (sub @ X150 @ (push @ X155 @ id)) @ id)) = $true)) & ($true != ((X149 @ id @ (lam @ X150) @ id))) & ! [X157 : subst,X159 : subst,X156 : subst,X158 : term] : (($true != ((X149 @ X157 @ X158 @ (comp @ X159 @ X156)))) | ($true = ((X149 @ (comp @ X157 @ X159) @ (sub @ X158 @ X159) @ X156)))) & ! [X152 : subst,X153 : subst,X154 : subst,X151 : term] : ((((X149 @ X152 @ X151 @ (comp @ X153 @ X154))) = $true) | ($true != ((X149 @ (comp @ X152 @ X153) @ (sub @ X151 @ X153) @ X154)))) & ! [X160 : (subst > term > term)] : (($true = ((X149 @ id @ (lam @ (X160 @ sh @ one)) @ id))) | ? [X164 : term] : (($true = ((X149 @ id @ X164 @ id))) & ($true != ((X149 @ id @ (X160 @ id @ X164) @ id)))) | ? [X161 : subst,X162 : term,X163 : subst] : (((sub @ (X160 @ X163 @ X162) @ X161)) != ((X160 @ (comp @ X163 @ X161) @ (sub @ X162 @ X161))))))) | ? [X138 : term,X136 : term,X137 : (subst > term > subst > $o)] : (! [X143 : term,X144 : term] : (($true != ((X137 @ id @ X143 @ id))) | (((X137 @ id @ (ap @ (sub @ X143 @ id) @ X144) @ id)) = $true) | (((X137 @ id @ X144 @ id)) != $true)) & (((X137 @ id @ (ap @ X138 @ X136) @ id)) != $true) & (((X137 @ id @ X138 @ id)) = $true) & ! [X145 : subst,X148 : subst,X147 : term,X146 : subst] : (($true = ((X137 @ X145 @ X147 @ (comp @ X146 @ X148)))) | (((X137 @ (comp @ X145 @ X146) @ (sub @ X147 @ X146) @ X148)) != $true)) & ! [X141 : term,X139 : subst,X140 : subst,X142 : subst] : ((((X137 @ (comp @ X140 @ X139) @ (sub @ X141 @ X139) @ X142)) = $true) | ($true != ((X137 @ X140 @ X141 @ (comp @ X139 @ X142))))) & ($true = ((X137 @ id @ X136 @ id))))) | ? [X130 : term,X129 : (term > $o)] : (! [X131 : term,X132 : term] : (($true != ((X129 @ X131))) | (((X129 @ X132)) != $true) | ($true = ((X129 @ (ap @ X131 @ X132))))) & ($true != ((X129 @ X130))) & ! [X133 : term] : (($true = ((X129 @ X133))) | ($true != ((var @ X133)))) & ! [X134 : term] : ((((X129 @ (lam @ X134))) = $true) | ? [X135 : term] : (($true = ((X129 @ X135))) & ($true != ((X129 @ (sub @ X134 @ (push @ X135 @ id)))))))))) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ((? [X49 : term] : (((sub @ X49 @ id)) != X49) | ! [X50 : term] : (((sub @ X50 @ id)) = X50)) <=> (termmset_lthm = $true)) & (! [X293 : (subst > term > subst > $o),X292 : term] : (($true = ((X293 @ id @ (lam @ (sub @ X292 @ (push @ one @ sh))) @ id))) | ? [X294 : term] : ((((X293 @ id @ X294 @ id)) = $true) & ($true != ((X293 @ id @ (sub @ X292 @ (push @ X294 @ id)) @ id)))) | ? [X295 : (subst > term > term)] : (($true != ((X293 @ id @ (lam @ (X295 @ sh @ one)) @ id))) & ! [X296 : subst,X297 : term,X298 : subst] : (((sub @ (X295 @ X298 @ X297) @ X296)) = ((X295 @ (comp @ X298 @ X296) @ (sub @ X297 @ X296)))) & ! [X299 : term] : ((((X293 @ id @ X299 @ id)) != $true) | ($true = ((X293 @ id @ (X295 @ id @ X299) @ id)))))) <=> (hoasinduction_lem3aa = $true)) & (substmonoid_gthm = $true) & (hoasapnotvar_gthm = $true) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (! [X5 : (term > $o),X7 : subst,X6 : term] : ? [X8 : (term > $o)] : ! [X9 : term] : (((X5 @ (sub @ X9 @ (push @ X6 @ X7)))) = ((X8 @ X9))) <=> (pushprop_lem0_lthm = $true)) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & ! [X51 : d_term,X52 : d_term] : ((((d2term @ X51)) != ((d2term @ X52))) | (X51 = X52)) & (! [X491 : (term > $o),X492 : term] : (? [X494 : term,X495 : term] : (($true = ((X491 @ X494))) & ($true = ((X491 @ X495))) & ($true != ((X491 @ (ap @ X495 @ X494))))) | ? [X496 : term] : (($true != ((X491 @ (lam @ X496)))) & ! [X497 : term] : (($true != ((X491 @ X497))) | (((X491 @ (sub @ X496 @ (push @ X497 @ id)))) = $true))) | ($true = ((X491 @ X492))) | ? [X493 : term] : (($true = ((var @ X493))) & (((X491 @ X493)) != $true))) <=> (induction2 = $true)) & (! [X183 : subst] : (((push @ (sub @ one @ X183) @ (comp @ sh @ X183))) = X183) <=> (axscons = $true)) & (((d2subst @ d_id)) = sh) & (! [X3 : term,X2 : term,X1 : term,X4 : term] : ((((ap @ X3 @ X2)) != ((ap @ X1 @ X4))) | (X2 = X4)) <=> (apinj2 = $true)) & (axmap = $true) & (axclos = $true) & (pushprop_lem0_gthm = $true) & (pushprop_lthm_orig = $true) & ! [X536 : subst] : ? [X537 : d_subst] : (((d2subst @ X537)) = X536) & (pushprop_gthm = $true) & (hoaslamnotap_lthm = $true) & (pushprop_lem2v2_lthm = $true) & ((hoasinduction_lem0 = $true) <=> ! [X126 : (subst > term > subst > $o)] : ? [X127 : (term > $o)] : ! [X128 : term] : (((X127 @ X128)) = ((X126 @ id @ X128 @ id)))) & (((! [X20 : (subst > term > subst > $o),X19 : (term > $o),X18 : term] : (($true = ((X19 @ (lam @ X18)))) | ? [X21 : term] : (((X19 @ X21)) != ((X20 @ id @ X21 @ id))) | ? [X23 : (subst > term > term)] : (($true != ((X20 @ id @ (lam @ (X23 @ sh @ one)) @ id))) & ! [X24 : term] : ((((X20 @ id @ (X23 @ id @ X24) @ id)) = $true) | ($true != ((X20 @ id @ X24 @ id)))) & ! [X25 : subst,X26 : term,X27 : subst] : (((sub @ (X23 @ X25 @ X26) @ X27)) = ((X23 @ (comp @ X25 @ X27) @ (sub @ X26 @ X27))))) | ? [X22 : term] : (($true != ((X19 @ (sub @ X18 @ (push @ X22 @ id))))) & (((X19 @ X22)) = $true))) | ? [X17 : term] : (((sub @ X17 @ id)) != X17)) | ? [X13 : term] : ! [X14 : (subst > term > term)] : ? [X15 : term,X16 : subst] : (((sub @ X13 @ (push @ X15 @ X16))) != ((X14 @ X16 @ X15)))) <=> (hoasinduction_lem3v2a_lthm = $true)) & (hoasapnotvar = $true) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (hoasinduction_lem1_lthm = $true) & (pushprop_lthm = $true) & (hoaslamnotvar = $true) & (axassoc = $true) & ((axshiftcons = $true) <=> ! [X112 : term,X113 : subst] : (((comp @ sh @ (push @ X112 @ X113))) = X113)) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (hoasapnotvar_lthm = $true) & (ulamvar1 != $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & ((pushprop_lem1_lthm = $true) <=> ((! [X384 : term,X387 : subst,X386 : term,X385 : (term > $o),X383 : (term > $o)] : (($true = ((X383 @ (sub @ X384 @ (push @ X386 @ X387))))) | ($true != ((X385 @ X384)))) | ? [X381 : subst,X382 : term] : (((comp @ sh @ (push @ X382 @ X381))) != X381)) | ? [X379 : subst,X380 : term] : (((sub @ one @ (push @ X380 @ X379))) != X380))) & (hoasinduction_lem1 = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (lamnotvar = $true) & (induction2lem_gthm = $true) & (pushprop = $true)),
  inference(ennf_transformation,[],[f10])).
thf(f12,plain,(
  (hoaslamnotap_lthm = $true) & (axabs = $true) & (! [X293 : (subst > term > subst > $o),X292 : term] : (($true = ((X293 @ id @ (lam @ (sub @ X292 @ (push @ one @ sh))) @ id))) | ? [X294 : term] : ((((X293 @ id @ X294 @ id)) = $true) & ($true != ((X293 @ id @ (sub @ X292 @ (push @ X294 @ id)) @ id)))) | ? [X295 : (subst > term > term)] : (($true != ((X293 @ id @ (lam @ (X295 @ sh @ one)) @ id))) & ! [X296 : subst,X297 : term,X298 : subst] : (((sub @ (X295 @ X298 @ X297) @ X296)) = ((X295 @ (comp @ X298 @ X296) @ (sub @ X297 @ X296)))) & ! [X299 : term] : ((((X293 @ id @ X299 @ id)) != $true) | ($true = ((X293 @ id @ (X295 @ id @ X299) @ id)))))) <=> (hoasinduction_lem3aa = $true)) & (pushprop_lem2v2_gthm = $true) & ((induction2lem = $true) <=> ! [X217 : term,X215 : subst,X216 : (term > $o)] : (? [X221 : term] : (($true != ((X216 @ (lam @ X221)))) & ! [X222 : term] : (($true = ((X216 @ (sub @ X221 @ (push @ X222 @ id))))) | ($true != ((X216 @ X222))))) | ? [X219 : term,X220 : term] : (($true = ((X216 @ X220))) & (((X216 @ (ap @ X219 @ X220))) != $true) & ($true = ((X216 @ X219)))) | ? [X218 : term] : (($true != ((X216 @ (sub @ X218 @ X215)))) & ($true = ((var @ X218)))) | (((X216 @ (sub @ X217 @ X215))) = $true))) & (lamnotap != $true) & ((hoasinduction_lem3v2a_lthm = $true) <=> (! [X20 : (subst > term > subst > $o),X19 : (term > $o),X18 : term] : (($true = ((X19 @ (lam @ X18)))) | ? [X21 : term] : (((X19 @ X21)) != ((X20 @ id @ X21 @ id))) | ? [X23 : (subst > term > term)] : (($true != ((X20 @ id @ (lam @ (X23 @ sh @ one)) @ id))) & ! [X24 : term] : ((((X20 @ id @ (X23 @ id @ X24) @ id)) = $true) | ($true != ((X20 @ id @ X24 @ id)))) & ! [X25 : subst,X26 : term,X27 : subst] : (((sub @ (X23 @ X25 @ X26) @ X27)) = ((X23 @ (comp @ X25 @ X27) @ (sub @ X26 @ X27))))) | ? [X22 : term] : (($true != ((X19 @ (sub @ X18 @ (push @ X22 @ id))))) & (((X19 @ X22)) = $true))) | ? [X13 : term] : ! [X14 : (subst > term > term)] : ? [X15 : term,X16 : subst] : (((sub @ X13 @ (push @ X15 @ X16))) != ((X14 @ X16 @ X15))) | ? [X17 : term] : (((sub @ X17 @ id)) != X17))) & (! [X533 : term,X532 : term,X531 : term,X534 : term] : ((X532 = X533) | (((ap @ (sub @ X534 @ id) @ X533)) != ((ap @ (sub @ X531 @ id) @ X532)))) <=> (hoasapinj2 = $true)) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem1_gthm = $true) & (! [X260 : term,X261 : (subst > term > subst > $o)] : (? [X262 : (subst > term > term)] : (($true != ((X261 @ id @ (lam @ (X262 @ sh @ one)) @ id))) & ! [X264 : term,X265 : subst,X263 : subst] : (((X262 @ (comp @ X265 @ X263) @ (sub @ X264 @ X263))) = ((sub @ (X262 @ X265 @ X264) @ X263))) & ! [X266 : term] : ((((X261 @ id @ (X262 @ id @ X266) @ id)) = $true) | ($true != ((X261 @ id @ X266 @ id))))) | ? [X267 : term] : ((((X261 @ id @ (sub @ X260 @ (push @ X267 @ id)) @ id)) != $true) & (((X261 @ id @ X267 @ id)) = $true)) | ($true = ((X261 @ id @ (lam @ (sub @ X260 @ (push @ one @ sh))) @ id)))) <=> (hoasinduction_lem3aa_lthm = $true)) & (! [X535 : subst] : (((comp @ id @ X535)) = X535) <=> (axidl = $true)) & ((hoaslamnotap = $true) <=> ! [X94 : term,X92 : term,X93 : (subst > term > term)] : (? [X95 : subst,X97 : subst,X96 : term] : (((X93 @ (comp @ X95 @ X97) @ (sub @ X96 @ X97))) != ((sub @ (X93 @ X95 @ X96) @ X97))) | (((lam @ (X93 @ sh @ one))) != ((ap @ (sub @ X94 @ id) @ X92))))) & (! [X429 : term,X428 : term] : ((((sub @ X428 @ sh)) != ((sub @ X429 @ sh))) | (X428 = X429)) <=> (shinj = $true)) & (! [X193 : term,X194 : term,X195 : (subst > term > subst > $o)] : (($true != ((X195 @ id @ X194 @ id))) | ($true != ((X195 @ id @ X193 @ id))) | (((X195 @ id @ (ap @ X193 @ X194) @ id)) = $true) | ? [X199 : subst,X200 : subst,X198 : term,X201 : subst] : (($true = ((X195 @ (comp @ X201 @ X200) @ (sub @ X198 @ X200) @ X199))) & ($true != ((X195 @ X201 @ X198 @ (comp @ X200 @ X199))))) | ? [X204 : subst,X203 : subst,X202 : term,X205 : subst] : (($true != ((X195 @ (comp @ X204 @ X205) @ (sub @ X202 @ X205) @ X203))) & ($true = ((X195 @ X204 @ X202 @ (comp @ X205 @ X203))))) | ? [X197 : term,X196 : term] : (($true != ((X195 @ id @ (ap @ (sub @ X196 @ id) @ X197) @ id))) & (((X195 @ id @ X196 @ id)) = $true) & ($true = ((X195 @ id @ X197 @ id))))) <=> (hoasinduction_lem2 = $true)) & (pushprop_lem3v2 = $true) & ! [X309 : d_subst,X310 : d_subst] : ((((d2subst @ X310)) != ((d2subst @ X309))) | (X309 = X310)) & (! [X30 : term,X31 : term,X29 : term,X28 : term] : ((((ap @ (sub @ X30 @ id) @ X29)) != ((ap @ (sub @ X28 @ id) @ X31))) | (X28 = X30)) <=> (hoasapinj1 = $true)) & ((axvarcons = $true) <=> ! [X64 : subst,X63 : term] : (((sub @ one @ (push @ X63 @ X64))) = X63)) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((hoasinduction_lthm = $true) <=> (? [X150 : term,X149 : (subst > term > subst > $o)] : (! [X155 : term] : ((((X149 @ id @ X155 @ id)) != $true) | (((X149 @ id @ (sub @ X150 @ (push @ X155 @ id)) @ id)) = $true)) & ($true != ((X149 @ id @ (lam @ X150) @ id))) & ! [X157 : subst,X159 : subst,X156 : subst,X158 : term] : (($true != ((X149 @ X157 @ X158 @ (comp @ X159 @ X156)))) | ($true = ((X149 @ (comp @ X157 @ X159) @ (sub @ X158 @ X159) @ X156)))) & ! [X152 : subst,X153 : subst,X154 : subst,X151 : term] : ((((X149 @ X152 @ X151 @ (comp @ X153 @ X154))) = $true) | ($true != ((X149 @ (comp @ X152 @ X153) @ (sub @ X151 @ X153) @ X154)))) & ! [X160 : (subst > term > term)] : (($true = ((X149 @ id @ (lam @ (X160 @ sh @ one)) @ id))) | ? [X164 : term] : (($true = ((X149 @ id @ X164 @ id))) & ($true != ((X149 @ id @ (X160 @ id @ X164) @ id)))) | ? [X161 : subst,X162 : term,X163 : subst] : (((sub @ (X160 @ X163 @ X162) @ X161)) != ((X160 @ (comp @ X163 @ X161) @ (sub @ X162 @ X161)))))) | ? [X130 : term,X129 : (term > $o)] : (! [X131 : term,X132 : term] : (($true != ((X129 @ X131))) | (((X129 @ X132)) != $true) | ($true = ((X129 @ (ap @ X131 @ X132))))) & ($true != ((X129 @ X130))) & ! [X133 : term] : (($true = ((X129 @ X133))) | ($true != ((var @ X133)))) & ! [X134 : term] : ((((X129 @ (lam @ X134))) = $true) | ? [X135 : term] : (($true = ((X129 @ X135))) & ($true != ((X129 @ (sub @ X134 @ (push @ X135 @ id)))))))) | ! [X165 : (subst > term > subst > $o),X166 : term] : (? [X175 : term,X176 : term] : (($true = ((X165 @ id @ X175 @ id))) & ($true = ((X165 @ id @ X176 @ id))) & ($true != ((X165 @ id @ (ap @ (sub @ X175 @ id) @ X176) @ id)))) | ? [X178 : (subst > term > term)] : (! [X182 : term] : (($true = ((X165 @ id @ (X178 @ id @ X182) @ id))) | (((X165 @ id @ X182 @ id)) != $true)) & ! [X179 : subst,X180 : subst,X181 : term] : (((sub @ (X178 @ X179 @ X181) @ X180)) = ((X178 @ (comp @ X179 @ X180) @ (sub @ X181 @ X180)))) & ($true != ((X165 @ id @ (lam @ (X178 @ sh @ one)) @ id)))) | ? [X177 : term] : (($true != ((X165 @ id @ X177 @ id))) & (((var @ (sub @ X177 @ id))) = $true)) | ($true = ((X165 @ id @ X166 @ id))) | ? [X171 : subst,X174 : subst,X172 : term,X173 : subst] : (($true != ((X165 @ X173 @ X172 @ (comp @ X174 @ X171)))) & ($true = ((X165 @ (comp @ X173 @ X174) @ (sub @ X172 @ X174) @ X171)))) | ? [X169 : subst,X170 : subst,X168 : subst,X167 : term] : (($true != ((X165 @ (comp @ X169 @ X168) @ (sub @ X167 @ X168) @ X170))) & ($true = ((X165 @ X169 @ X167 @ (comp @ X168 @ X170)))))) | ? [X138 : term,X136 : term,X137 : (subst > term > subst > $o)] : (! [X143 : term,X144 : term] : (($true != ((X137 @ id @ X143 @ id))) | (((X137 @ id @ (ap @ (sub @ X143 @ id) @ X144) @ id)) = $true) | (((X137 @ id @ X144 @ id)) != $true)) & (((X137 @ id @ (ap @ X138 @ X136) @ id)) != $true) & (((X137 @ id @ X138 @ id)) = $true) & ! [X145 : subst,X148 : subst,X147 : term,X146 : subst] : (($true = ((X137 @ X145 @ X147 @ (comp @ X146 @ X148)))) | (((X137 @ (comp @ X145 @ X146) @ (sub @ X147 @ X146) @ X148)) != $true)) & ! [X141 : term,X139 : subst,X140 : subst,X142 : subst] : ((((X137 @ (comp @ X140 @ X139) @ (sub @ X141 @ X139) @ X142)) = $true) | ($true != ((X137 @ X140 @ X141 @ (comp @ X139 @ X142))))) & ($true = ((X137 @ id @ X136 @ id)))))) & ! [X51 : d_term,X52 : d_term] : ((((d2term @ X51)) != ((d2term @ X52))) | (X51 = X52)) & (hoaslamnotvar_gthm = $true) & (hoaslamnotvar = $true) & ((hoasinduction = $true) <=> ! [X393 : term,X394 : (subst > term > subst > $o)] : (? [X400 : term,X399 : term] : ((((X394 @ id @ (ap @ (sub @ X399 @ id) @ X400) @ id)) != $true) & ($true = ((X394 @ id @ X399 @ id))) & (((X394 @ id @ X400 @ id)) = $true)) | ? [X397 : subst,X395 : subst,X396 : term,X398 : subst] : (($true != ((X394 @ (comp @ X395 @ X397) @ (sub @ X396 @ X397) @ X398))) & ($true = ((X394 @ X395 @ X396 @ (comp @ X397 @ X398))))) | ? [X401 : term] : ((((var @ (sub @ X401 @ id))) = $true) & ($true != ((X394 @ id @ X401 @ id)))) | ($true = ((X394 @ id @ X393 @ id))) | ? [X406 : (subst > term > term)] : ((((X394 @ id @ (lam @ (X406 @ sh @ one)) @ id)) != $true) & ! [X407 : term] : (($true = ((X394 @ id @ (X406 @ id @ X407) @ id))) | (((X394 @ id @ X407 @ id)) != $true)) & ! [X408 : subst,X409 : subst,X410 : term] : (((X406 @ (comp @ X408 @ X409) @ (sub @ X410 @ X409))) = ((sub @ (X406 @ X408 @ X410) @ X409)))) | ? [X402 : subst,X404 : subst,X405 : subst,X403 : term] : (($true = ((X394 @ (comp @ X404 @ X402) @ (sub @ X403 @ X402) @ X405))) & ($true != ((X394 @ X404 @ X403 @ (comp @ X402 @ X405))))))) & (pushprop_lem3v2_lthm = $true) & (! [X183 : subst] : (((push @ (sub @ one @ X183) @ (comp @ sh @ X183))) = X183) <=> (axscons = $true)) & (hoasinduction_lem3b_gthm = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ! [X111 : d_term] : (d_one = X111) & ((hoasapinj2_lthm = $true) <=> (? [X524 : term,X525 : term,X523 : term,X526 : term] : ((X523 != X524) & (((ap @ X526 @ X523)) = ((ap @ X525 @ X524)))) | ! [X529 : term,X528 : term,X527 : term,X530 : term] : ((X527 = X530) | (((ap @ (sub @ X529 @ id) @ X527)) != ((ap @ (sub @ X528 @ id) @ X530)))))) & (axvarshift = $true) & (! [X538 : term] : ? [X539 : (subst > term > term)] : (((X539 @ sh @ one)) = ((sub @ X538 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b = $true)) & ((hoasinduction_lem3a_lthm = $true) <=> (? [X413 : (subst > term > subst > $o),X412 : term] : (! [X414 : (subst > term > term)] : (? [X418 : term,X417 : subst,X416 : subst] : (((X414 @ (comp @ X417 @ X416) @ (sub @ X418 @ X416))) != ((sub @ (X414 @ X417 @ X418) @ X416))) | ? [X415 : term] : ((((X413 @ id @ X415 @ id)) = $true) & (((X413 @ id @ (X414 @ id @ X415) @ id)) != $true)) | ($true = ((X413 @ id @ (lam @ (X414 @ sh @ one)) @ id)))) & ! [X419 : term] : (($true != ((X413 @ id @ X419 @ id))) | ($true = ((X413 @ id @ (sub @ X412 @ (push @ X419 @ id)) @ id)))) & (((X413 @ id @ (lam @ (sub @ X412 @ (push @ one @ sh))) @ id)) != $true)) | ? [X411 : term] : (((sub @ X411 @ id)) != X411) | ! [X421 : (subst > term > subst > $o),X420 : term] : (? [X422 : (subst > term > term)] : (! [X423 : subst,X424 : subst,X425 : term] : (((sub @ (X422 @ X423 @ X425) @ X424)) = ((X422 @ (comp @ X423 @ X424) @ (sub @ X425 @ X424)))) & ! [X426 : term] : (($true = ((X421 @ id @ (X422 @ id @ X426) @ id))) | ($true != ((X421 @ id @ X426 @ id)))) & (((X421 @ id @ (lam @ (X422 @ sh @ one)) @ id)) != $true)) | ? [X427 : term] : ((((X421 @ id @ (sub @ X420 @ (push @ X427 @ id)) @ id)) != $true) & (((X421 @ id @ X427 @ id)) = $true)) | ($true = ((X421 @ id @ (lam @ X420) @ id)))))) & (induction2lem_gthm = $true) & ! [X48 : d_subst] : (d_id = X48) & (pushprop_lem2v2_lthm = $true) & (ulamvar1 != $true) & ! [X536 : subst] : ? [X537 : d_subst] : (((d2subst @ X537)) = X536) & (pushprop_lthm = $true) & (hoasinduction_lem1v2 = $true) & (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_lem3v2a = $true) <=> ! [X227 : (term > $o),X228 : term,X229 : (subst > term > subst > $o)] : (? [X235 : term] : (((X229 @ id @ X235 @ id)) != ((X227 @ X235))) | ($true = ((X227 @ (lam @ X228)))) | ? [X236 : term] : (($true != ((X227 @ (sub @ X228 @ (push @ X236 @ id))))) & ($true = ((X227 @ X236)))) | ? [X230 : (subst > term > term)] : (! [X232 : term,X231 : subst,X233 : subst] : (((X230 @ (comp @ X231 @ X233) @ (sub @ X232 @ X233))) = ((sub @ (X230 @ X231 @ X232) @ X233))) & ! [X234 : term] : ((((X229 @ id @ (X230 @ id @ X234) @ id)) = $true) | ($true != ((X229 @ id @ X234 @ id)))) & (((X229 @ id @ (lam @ (X230 @ sh @ one)) @ id)) != $true)))) & (pushprop_lthm_orig = $true) & (pushprop_lem1v2_gthm = $true) & (pushprop_lem2v2 = $true) & ! [X61 : term] : ? [X62 : d_term] : (((d2term @ X62)) = X61) & ((hoasinduction_no_psi_cond = $true) <=> ! [X206 : (subst > term > subst > $o),X207 : term] : (($true = ((X206 @ id @ X207 @ id))) | ? [X208 : (subst > term > term)] : (($true != ((X206 @ id @ (lam @ (X208 @ sh @ one)) @ id))) & ! [X211 : subst,X210 : subst,X209 : term] : (((sub @ (X208 @ X211 @ X209) @ X210)) = ((X208 @ (comp @ X211 @ X210) @ (sub @ X209 @ X210)))) & ! [X212 : term] : (($true = ((X206 @ id @ (X208 @ id @ X212) @ id))) | ($true != ((X206 @ id @ X212 @ id))))) | ? [X214 : term,X213 : term] : ((((X206 @ id @ X214 @ id)) = $true) & ($true = ((X206 @ id @ X213 @ id))) & ($true != ((X206 @ id @ (ap @ (sub @ X213 @ id) @ X214) @ id)))))) & (! [X115 : term,X114 : (subst > term > subst > $o)] : (($true = ((X114 @ id @ (lam @ (sub @ X115 @ (push @ one @ sh))) @ id))) | ? [X118 : term,X117 : (subst > term > term)] : (! [X121 : subst,X119 : subst,X120 : term] : (((X117 @ (comp @ X119 @ X121) @ (sub @ X120 @ X121))) = ((sub @ X118 @ (push @ (sub @ X120 @ X121) @ (comp @ X119 @ X121))))) & ! [X124 : subst,X123 : term,X122 : subst] : (((sub @ (sub @ X118 @ (push @ X123 @ X122)) @ X124)) = ((sub @ (X117 @ X122 @ X123) @ X124))) & (((X114 @ id @ (lam @ (X117 @ sh @ one)) @ id)) != $true) & ! [X125 : term] : (($true = ((X114 @ id @ (X117 @ id @ X125) @ id))) | ($true != ((X114 @ id @ X125 @ id))))) | ? [X116 : term] : (($true != ((X114 @ id @ (sub @ X115 @ (push @ X116 @ id)) @ id))) & ($true = ((X114 @ id @ X116 @ id))))) <=> (hoasinduction_lem3aaa = $true)) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (pushprop_lem0_gthm = $true) & (apnotvar = $true) & (hoasapnotvar = $true) & (hoasinduction_lem1_lthm = $true) & ((hoaslaminj = $true) <=> ! [X372 : subst,X370 : (subst > term > term),X369 : term,X371 : (subst > term > term)] : ((((lam @ (X370 @ sh @ one))) != ((lam @ (X371 @ sh @ one)))) | ? [X376 : subst,X378 : term,X377 : subst] : (((sub @ (X371 @ X376 @ X378) @ X377)) != ((X371 @ (comp @ X376 @ X377) @ (sub @ X378 @ X377)))) | (((X371 @ X372 @ X369)) = ((X370 @ X372 @ X369))) | ? [X375 : subst,X374 : subst,X373 : term] : (((sub @ (X370 @ X375 @ X373) @ X374)) != ((X370 @ (comp @ X375 @ X374) @ (sub @ X373 @ X374)))))) & (induction2_gthm = $true) & (! [X71 : term] : (((sub @ X71 @ id)) = X71) <=> (termmset = $true)) & ((hoasinduction_lem0_lthm = $true) <=> ! [X10 : (subst > term > subst > $o)] : ? [X11 : (term > $o)] : ! [X12 : term] : (((X11 @ X12)) = ((X10 @ id @ X12 @ id)))) & (hoaslamnotap_gthm = $true) & (((d2subst @ d_id)) = sh) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (hoasinduction_lem2v2_gthm = $true) & ((hoaslaminj_lthm = $true) <=> (? [X35 : subst,X34 : term] : (((comp @ sh @ (push @ X34 @ X35))) != X35) | ? [X32 : subst,X33 : term] : (((sub @ one @ (push @ X33 @ X32))) != X33) | ! [X39 : (subst > term > term),X40 : (subst > term > term),X41 : term,X38 : subst] : (? [X46 : term,X47 : subst,X45 : subst] : (((sub @ (X40 @ X45 @ X46) @ X47)) != ((X40 @ (comp @ X45 @ X47) @ (sub @ X46 @ X47)))) | (((lam @ (X39 @ sh @ one))) != ((lam @ (X40 @ sh @ one)))) | ? [X42 : subst,X44 : term,X43 : subst] : (((sub @ (X39 @ X43 @ X44) @ X42)) != ((X39 @ (comp @ X43 @ X42) @ (sub @ X44 @ X42)))) | (((X40 @ X38 @ X41)) = ((X39 @ X38 @ X41)))) | ? [X36 : term,X37 : term] : ((X36 != X37) & (((lam @ X37)) = ((lam @ X36)))))) & (! [X73 : term,X72 : (subst > term > subst > $o)] : (? [X74 : term] : (($true != ((X72 @ id @ (sub @ X73 @ (push @ X74 @ id)) @ id))) & ($true = ((X72 @ id @ X74 @ id)))) | ? [X85 : subst,X86 : subst,X84 : subst,X87 : term] : (($true != ((X72 @ (comp @ X85 @ X84) @ (sub @ X87 @ X84) @ X86))) & ($true = ((X72 @ X85 @ X87 @ (comp @ X84 @ X86))))) | ? [X79 : (subst > term > term)] : (! [X83 : subst,X81 : term,X82 : subst] : (((X79 @ (comp @ X83 @ X82) @ (sub @ X81 @ X82))) = ((sub @ (X79 @ X83 @ X81) @ X82))) & ! [X80 : term] : ((((X72 @ id @ X80 @ id)) != $true) | ($true = ((X72 @ id @ (X79 @ id @ X80) @ id)))) & (((X72 @ id @ (lam @ (X79 @ sh @ one)) @ id)) != $true)) | ($true = ((X72 @ id @ (lam @ X73) @ id))) | ? [X76 : subst,X77 : term,X78 : subst,X75 : subst] : (($true = ((X72 @ (comp @ X75 @ X78) @ (sub @ X77 @ X78) @ X76))) & (((X72 @ X75 @ X77 @ (comp @ X78 @ X76))) != $true))) <=> (hoasinduction_lem3 = $true)) & (termmset_gthm = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (hoasinduction_lem2_lthm = $true) & (! [X99 : term,X98 : term] : ((((lam @ X98)) != ((lam @ X99))) | (X98 = X99)) <=> (laminj = $true)) & (((var @ (d2term @ d_one))) != $true) & (! [X272 : subst] : (((comp @ X272 @ id)) = X272) <=> (axidr = $true)) & (! [X0 : term] : (((sub @ X0 @ id)) = X0) <=> (axvarid = $true)) & ((? [X321 : term] : (((sub @ X321 @ id)) != X321) | ! [X333 : term,X332 : (subst > term > subst > $o)] : (? [X340 : (subst > term > term)] : (! [X341 : term] : (($true != ((X332 @ id @ X341 @ id))) | ($true = ((X332 @ id @ (X340 @ id @ X341) @ id)))) & ($true != ((X332 @ id @ (lam @ (X340 @ sh @ one)) @ id))) & ! [X344 : subst,X342 : term,X343 : subst] : (((X340 @ (comp @ X343 @ X344) @ (sub @ X342 @ X344))) = ((sub @ (X340 @ X343 @ X342) @ X344)))) | ($true = ((X332 @ id @ X333 @ id))) | ? [X349 : term] : (($true != ((X332 @ id @ X349 @ id))) & ($true = ((var @ (sub @ X349 @ id))))) | ? [X338 : term,X339 : term] : (($true = ((X332 @ id @ X339 @ id))) & (((X332 @ id @ X338 @ id)) = $true) & ($true != ((X332 @ id @ (ap @ (sub @ X338 @ id) @ X339) @ id)))) | ? [X337 : subst,X335 : term,X334 : subst,X336 : subst] : (($true = ((X332 @ X334 @ X335 @ (comp @ X336 @ X337)))) & ($true != ((X332 @ (comp @ X334 @ X336) @ (sub @ X335 @ X336) @ X337)))) | ? [X345 : subst,X346 : subst,X348 : subst,X347 : term] : ((((X332 @ (comp @ X346 @ X345) @ (sub @ X347 @ X345) @ X348)) = $true) & ($true != ((X332 @ X346 @ X347 @ (comp @ X345 @ X348)))))) | ? [X311 : (subst > term > subst > $o)] : ! [X312 : (term > $o)] : ? [X313 : term] : (((X312 @ X313)) != ((X311 @ id @ X313 @ id))) | ? [X322 : term,X323 : (term > $o),X324 : (subst > term > subst > $o)] : (($true != ((X323 @ (lam @ X322)))) & ! [X326 : (subst > term > term)] : (($true = ((X324 @ id @ (lam @ (X326 @ sh @ one)) @ id))) | ? [X330 : subst,X328 : subst,X329 : term] : (((X326 @ (comp @ X328 @ X330) @ (sub @ X329 @ X330))) != ((sub @ (X326 @ X328 @ X329) @ X330))) | ? [X327 : term] : ((((X324 @ id @ X327 @ id)) = $true) & ($true != ((X324 @ id @ (X326 @ id @ X327) @ id))))) & ! [X325 : term] : ((((X323 @ X325)) != $true) | ($true = ((X323 @ (sub @ X322 @ (push @ X325 @ id)))))) & ! [X331 : term] : (((X324 @ id @ X331 @ id)) = ((X323 @ X331)))) | ? [X314 : term,X315 : (term > $o)] : (! [X318 : term] : ((((X315 @ X318)) = $true) | (((var @ X318)) != $true)) & ($true != ((X315 @ X314))) & ! [X319 : term] : (? [X320 : term] : (($true = ((X315 @ X320))) & (((X315 @ (sub @ X319 @ (push @ X320 @ id)))) != $true)) | ($true = ((X315 @ (lam @ X319))))) & ! [X317 : term,X316 : term] : (($true != ((X315 @ X317))) | (((X315 @ (ap @ X317 @ X316))) = $true) | ($true != ((X315 @ X316)))))) <=> (hoasinduction_lthm_3 = $true)) & (substmonoid_gthm = $true) & (pushprop = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (axclos = $true) & ((hoasinduction_lem3a = $true) <=> ! [X54 : (subst > term > subst > $o),X53 : term] : (? [X55 : (subst > term > term)] : (! [X59 : term] : (($true = ((X54 @ id @ (X55 @ id @ X59) @ id))) | ($true != ((X54 @ id @ X59 @ id)))) & ($true != ((X54 @ id @ (lam @ (X55 @ sh @ one)) @ id))) & ! [X57 : subst,X56 : subst,X58 : term] : (((X55 @ (comp @ X57 @ X56) @ (sub @ X58 @ X56))) = ((sub @ (X55 @ X57 @ X58) @ X56)))) | ? [X60 : term] : (($true = ((X54 @ id @ X60 @ id))) & ($true != ((X54 @ id @ (sub @ X53 @ (push @ X60 @ id)) @ id)))) | ($true = ((X54 @ id @ (lam @ X53) @ id))))) & ((? [X49 : term] : (((sub @ X49 @ id)) != X49) | ! [X50 : term] : (((sub @ X50 @ id)) = X50)) <=> (termmset_lthm = $true)) & (ulamvarsh = $true) & ((hoasinduction_lem3v2_f = $true) <=> ! [X100 : term] : ? [X101 : (subst > term > term)] : ! [X102 : subst,X103 : term] : (((X101 @ X102 @ X103)) = ((sub @ X100 @ (push @ X103 @ X102))))) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((substmonoid = $true) <=> (! [X224 : subst] : (((comp @ X224 @ id)) = X224) & ! [X223 : subst] : (((comp @ id @ X223)) = X223))) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X225 : term] : ? [X226 : (subst > term > term)] : (((sub @ X225 @ (push @ one @ sh))) = ((X226 @ sh @ one)))) & ((! [X108 : (term > $o),X106 : (term > $o),X109 : subst,X107 : term] : (? [X110 : term] : (((X106 @ X110)) != ((X108 @ (sub @ X110 @ (push @ X107 @ X109))))) | ($true != ((X108 @ X107))) | ($true = ((X106 @ one)))) | ? [X104 : subst,X105 : term] : (((sub @ one @ (push @ X105 @ X104))) != X105)) <=> (pushprop_lem1v2_lthm = $true)) & (axassoc = $true) & ((? [X88 : subst] : (((comp @ id @ X88)) != X88) | (! [X91 : subst] : (((comp @ id @ X91)) = X91) & ! [X90 : subst] : (((comp @ X90 @ id)) = X90)) | ? [X89 : subst] : (((comp @ X89 @ id)) != X89)) <=> (substmonoid_lthm = $true)) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((? [X304 : term,X302 : term,X303 : term,X301 : term] : ((((ap @ X301 @ X303)) = ((ap @ X304 @ X302))) & (X301 != X304)) | ? [X300 : term] : (((sub @ X300 @ id)) != X300) | ! [X307 : term,X306 : term,X308 : term,X305 : term] : ((((ap @ (sub @ X305 @ id) @ X307)) != ((ap @ (sub @ X306 @ id) @ X308))) | (X305 = X306))) <=> (hoasapinj1_lthm = $true)) & (hoasapinj1_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (hoasapnotvar_lthm = $true) & (one = ((d2term @ d_one))) & (axapp = $true) & (hoasinduction_lem1 = $true) & ((pushprop_lem0 = $true) <=> ! [X237 : subst,X238 : (term > $o),X239 : term] : ? [X240 : (term > $o)] : ! [X241 : term] : (((X238 @ (sub @ X241 @ (push @ X239 @ X237)))) = ((X240 @ X241)))) & ((induction2lem_lthm = $true) <=> (? [X355 : term,X356 : (term > $o)] : (! [X360 : term] : ((((X356 @ X360)) != $true) | ($true = ((X356 @ (lam @ X360))))) & ! [X358 : term,X359 : term] : ((((X356 @ (ap @ X358 @ X359))) = $true) | ($true != ((X356 @ X359))) | ($true != ((X356 @ X358)))) & ! [X357 : term] : (($true = ((X356 @ X357))) | ($true != ((var @ X357)))) & (((X356 @ X355)) != $true)) | ? [X353 : term,X352 : subst] : (((comp @ sh @ (push @ X353 @ X352))) != X352) | ? [X350 : subst,X351 : term] : (((sub @ one @ (push @ X351 @ X350))) != X351) | ? [X354 : subst] : (((comp @ X354 @ id)) != X354) | ! [X363 : term,X362 : subst,X361 : (term > $o)] : (? [X365 : term] : (($true != ((X361 @ (lam @ X365)))) & ! [X366 : term] : (($true != ((X361 @ X366))) | ($true = ((X361 @ (sub @ X365 @ (push @ X366 @ id))))))) | ($true = ((X361 @ (sub @ X363 @ X362)))) | ? [X364 : term] : ((((X361 @ (sub @ X364 @ X362))) != $true) & (((var @ X364)) = $true)) | ? [X368 : term,X367 : term] : (($true = ((X361 @ X368))) & ($true = ((X361 @ X367))) & ($true != ((X361 @ (ap @ X367 @ X368)))))))) & (ulamvarind = $true) & (axmap = $true) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (? [X455 : term] : (((sub @ X455 @ id)) != X455) | ? [X456 : (term > $o),X457 : (subst > term > subst > $o),X458 : term] : (! [X459 : term] : (((X457 @ id @ X459 @ id)) = ((X456 @ X459))) & (((X456 @ (lam @ X458))) != $true) & ! [X460 : term] : ((((X456 @ (sub @ X458 @ (push @ X460 @ id)))) = $true) | (((X456 @ X460)) != $true)) & ! [X461 : (subst > term > term)] : (($true = ((X457 @ id @ (lam @ (X461 @ sh @ one)) @ id))) | ? [X462 : term] : (($true = ((X457 @ id @ X462 @ id))) & (((X457 @ id @ (X461 @ id @ X462) @ id)) != $true)) | ? [X463 : term,X465 : subst,X464 : subst] : (((sub @ (X461 @ X465 @ X463) @ X464)) != ((X461 @ (comp @ X465 @ X464) @ (sub @ X463 @ X464)))))) | ? [X449 : (term > $o),X448 : term] : (! [X451 : term,X450 : term] : ((((X449 @ X451)) != $true) | ($true = ((X449 @ (ap @ X450 @ X451)))) | ($true != ((X449 @ X450)))) & ($true != ((X449 @ X448))) & ! [X454 : term] : ((((var @ X454)) != $true) | (((X449 @ X454)) = $true)) & ! [X452 : term] : (? [X453 : term] : ((((X449 @ X453)) = $true) & ($true != ((X449 @ (sub @ X452 @ (push @ X453 @ id)))))) | ($true = ((X449 @ (lam @ X452)))))) | ? [X445 : (subst > term > subst > $o)] : ! [X446 : (term > $o)] : ? [X447 : term] : (((X445 @ id @ X447 @ id)) != ((X446 @ X447))) | ! [X467 : term,X466 : (subst > term > subst > $o)] : (? [X470 : (subst > term > term)] : (! [X474 : term] : (($true != ((X466 @ id @ X474 @ id))) | ($true = ((X466 @ id @ (X470 @ id @ X474) @ id)))) & ($true != ((X466 @ id @ (lam @ (X470 @ sh @ one)) @ id))) & ! [X471 : term,X473 : subst,X472 : subst] : (((X470 @ (comp @ X472 @ X473) @ (sub @ X471 @ X473))) = ((sub @ (X470 @ X472 @ X471) @ X473)))) | ? [X468 : term,X469 : term] : (($true = ((X466 @ id @ X469 @ id))) & ($true != ((X466 @ id @ (ap @ (sub @ X469 @ id) @ X468) @ id))) & ($true = ((X466 @ id @ X468 @ id)))) | (((X466 @ id @ X467 @ id)) = $true)))) & ((pushprop_lem1v2 = $true) <=> ! [X388 : (term > $o),X391 : (term > $o),X390 : term,X389 : subst] : (? [X392 : term] : (((X391 @ (sub @ X392 @ (push @ X390 @ X389)))) != ((X388 @ X392))) | ($true != ((X391 @ X390))) | ($true = ((X388 @ one))))) & (hoasinduction_gthm = $true) & ((! [X384 : term,X387 : subst,X386 : term,X385 : (term > $o),X383 : (term > $o)] : (($true = ((X383 @ (sub @ X384 @ (push @ X386 @ X387))))) | ($true != ((X385 @ X384)))) | ? [X379 : subst,X380 : term] : (((sub @ one @ (push @ X380 @ X379))) != X380) | ? [X381 : subst,X382 : term] : (((comp @ sh @ (push @ X382 @ X381))) != X381)) <=> (pushprop_lem1_lthm = $true)) & (! [X491 : (term > $o),X492 : term] : (? [X494 : term,X495 : term] : (($true = ((X491 @ X494))) & ($true = ((X491 @ X495))) & ($true != ((X491 @ (ap @ X495 @ X494))))) | ? [X496 : term] : (($true != ((X491 @ (lam @ X496)))) & ! [X497 : term] : (($true != ((X491 @ X497))) | (((X491 @ (sub @ X496 @ (push @ X497 @ id)))) = $true))) | ($true = ((X491 @ X492))) | ? [X493 : term] : (($true = ((var @ X493))) & (((X491 @ X493)) != $true))) <=> (induction2 = $true)) & ((pushprop_lem1 = $true) <=> ! [X186 : (term > $o),X185 : term,X187 : subst,X188 : term,X184 : (term > $o)] : ((((X184 @ (sub @ X185 @ (push @ X188 @ X187)))) = $true) | ($true != ((X186 @ X185))))) & ((induction2_lthm = $true) <=> (? [X476 : subst,X477 : (term > $o),X478 : term] : (! [X483 : term] : (($true = ((X477 @ (sub @ X483 @ X476)))) | ($true != ((var @ X483)))) & ! [X479 : term,X480 : term] : (($true != ((X477 @ X480))) | ($true = ((X477 @ (ap @ X479 @ X480)))) | (((X477 @ X479)) != $true)) & ! [X481 : term] : (($true = ((X477 @ (lam @ X481)))) | ? [X482 : term] : ((((X477 @ X482)) = $true) & (((X477 @ (sub @ X481 @ (push @ X482 @ id)))) != $true))) & ($true != ((X477 @ (sub @ X478 @ X476))))) | ! [X484 : (term > $o),X485 : term] : (? [X486 : term] : (! [X487 : term] : (($true != ((X484 @ X487))) | ($true = ((X484 @ (sub @ X486 @ (push @ X487 @ id)))))) & (((X484 @ (lam @ X486))) != $true)) | ? [X490 : term] : (($true = ((var @ X490))) & ($true != ((X484 @ X490)))) | ? [X489 : term,X488 : term] : (($true != ((X484 @ (ap @ X489 @ X488)))) & ($true = ((X484 @ X488))) & ($true = ((X484 @ X489)))) | ($true = ((X484 @ X485)))) | ? [X475 : term] : (((sub @ X475 @ id)) != X475))) & ((hoasinduction_lem0 = $true) <=> ! [X126 : (subst > term > subst > $o)] : ? [X127 : (term > $o)] : ! [X128 : term] : (((X127 @ X128)) = ((X126 @ id @ X128 @ id)))) & (lamnotvar = $true) & (! [X3 : term,X2 : term,X1 : term,X4 : term] : ((((ap @ X3 @ X2)) != ((ap @ X1 @ X4))) | (X2 = X4)) <=> (apinj2 = $true)) & (! [X65 : (term > $o),X66 : term] : (? [X70 : term] : ((((X65 @ (lam @ X70))) != $true) & ($true = ((X65 @ X70)))) | ($true = ((X65 @ X66))) | ? [X69 : term,X68 : term] : (($true != ((X65 @ (ap @ X68 @ X69)))) & ($true = ((X65 @ X68))) & (((X65 @ X69)) = $true)) | ? [X67 : term] : (($true = ((var @ X67))) & ($true != ((X65 @ X67))))) <=> (induction = $true)) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoaslamnotvar_lthm = $true) & ((hoasinduction_lem2v2 = $true) <=> ! [X431 : (subst > term > subst > $o),X430 : term,X432 : (term > $o),X433 : term] : (? [X438 : subst,X439 : term,X440 : subst,X437 : subst] : ((((X431 @ X437 @ X439 @ (comp @ X438 @ X440))) != $true) & ($true = ((X431 @ (comp @ X437 @ X438) @ (sub @ X439 @ X438) @ X440)))) | ($true != ((X432 @ X433))) | (((X432 @ X430)) != $true) | ? [X435 : term,X436 : term] : (($true = ((X431 @ id @ X435 @ id))) & ($true != ((X431 @ id @ (ap @ (sub @ X435 @ id) @ X436) @ id))) & ($true = ((X431 @ id @ X436 @ id)))) | ($true = ((X432 @ (ap @ X433 @ X430)))) | ? [X434 : term] : (((X432 @ X434)) != ((X431 @ id @ X434 @ id))) | ? [X444 : subst,X442 : subst,X441 : subst,X443 : term] : (($true != ((X431 @ (comp @ X442 @ X441) @ (sub @ X443 @ X441) @ X444))) & ($true = ((X431 @ X442 @ X443 @ (comp @ X441 @ X444))))))) & (hoasinduction_lem3a_gthm = $true) & (pushprop_lem1_gthm = $true) & (hoasapinj2_gthm = $true) & (id = ((d2subst @ d_id))) & (pushprop_gthm = $true) & (! [X189 : term] : ? [X190 : (subst > term > term)] : ! [X192 : term,X191 : subst] : (((X190 @ X191 @ X192)) = ((sub @ X189 @ (push @ X192 @ X191)))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & ((hoasinduction_lem3v2_lthm = $true) <=> (? [X273 : term] : (((sub @ X273 @ id)) != X273) | ! [X274 : (subst > term > subst > $o),X275 : term,X276 : (term > $o)] : (? [X277 : term] : (((X276 @ X277)) != ((X274 @ id @ X277 @ id))) | ($true = ((X276 @ (lam @ X275)))) | ? [X289 : subst,X290 : term,X288 : subst,X291 : subst] : (($true = ((X274 @ (comp @ X288 @ X291) @ (sub @ X290 @ X291) @ X289))) & ($true != ((X274 @ X288 @ X290 @ (comp @ X291 @ X289))))) | ? [X283 : term,X285 : subst,X284 : subst,X286 : subst] : ((((X274 @ (comp @ X285 @ X286) @ (sub @ X283 @ X286) @ X284)) != $true) & (((X274 @ X285 @ X283 @ (comp @ X286 @ X284))) = $true)) | ? [X278 : (subst > term > term)] : (! [X281 : subst,X280 : subst,X282 : term] : (((sub @ (X278 @ X280 @ X282) @ X281)) = ((X278 @ (comp @ X280 @ X281) @ (sub @ X282 @ X281)))) & ! [X279 : term] : ((((X274 @ id @ X279 @ id)) != $true) | ($true = ((X274 @ id @ (X278 @ id @ X279) @ id)))) & ($true != ((X274 @ id @ (lam @ (X278 @ sh @ one)) @ id)))) | ? [X287 : term] : (($true != ((X276 @ (sub @ X275 @ (push @ X287 @ id))))) & ($true = ((X276 @ X287))))))) & (! [X270 : term,X271 : term,X269 : term,X268 : term] : ((((ap @ X269 @ X270)) != ((ap @ X271 @ X268))) | (X269 = X271)) <=> (apinj1 = $true)) & (! [X5 : (term > $o),X7 : subst,X6 : term] : ? [X8 : (term > $o)] : ! [X9 : term] : (((X5 @ (sub @ X9 @ (push @ X6 @ X7)))) = ((X8 @ X9))) <=> (pushprop_lem0_lthm = $true)) & (hoasinduction_lem3v2_gthm = $true) & (hoasinduction_lem3_gthm = $true) & (hoaslaminj_gthm = $true) & (hoasinduction_lem2_gthm = $true) & (! [X243 : (subst > term > subst > $o),X242 : term,X244 : (term > $o)] : (? [X259 : term] : (($true = ((X244 @ X259))) & (((X244 @ (sub @ X242 @ (push @ X259 @ id)))) != $true)) | ? [X249 : term] : (((X244 @ X249)) != ((X243 @ id @ X249 @ id))) | ? [X248 : subst,X246 : subst,X247 : term,X245 : subst] : (($true = ((X243 @ X246 @ X247 @ (comp @ X248 @ X245)))) & ($true != ((X243 @ (comp @ X246 @ X248) @ (sub @ X247 @ X248) @ X245)))) | ? [X253 : term,X251 : subst,X252 : subst,X250 : subst] : (($true = ((X243 @ (comp @ X251 @ X252) @ (sub @ X253 @ X252) @ X250))) & (((X243 @ X251 @ X253 @ (comp @ X252 @ X250))) != $true)) | ($true = ((X244 @ (lam @ X242)))) | ? [X254 : (subst > term > term)] : (! [X255 : term] : (($true != ((X243 @ id @ X255 @ id))) | ($true = ((X243 @ id @ (X254 @ id @ X255) @ id)))) & ! [X256 : subst,X257 : subst,X258 : term] : (((sub @ (X254 @ X257 @ X258) @ X256)) = ((X254 @ (comp @ X257 @ X256) @ (sub @ X258 @ X256)))) & ($true != ((X243 @ id @ (lam @ (X254 @ sh @ one)) @ id))))) <=> (hoasinduction_lem3v2 = $true)) & ((axshiftcons = $true) <=> ! [X112 : term,X113 : subst] : (((comp @ sh @ (push @ X112 @ X113))) = X113)) & ((hoasinduction_lem3_lthm = $true) <=> (! [X508 : (subst > term > subst > $o),X507 : term] : (($true = ((X508 @ id @ (lam @ X507) @ id))) | ? [X513 : term] : (($true = ((X508 @ id @ X513 @ id))) & (((X508 @ id @ (sub @ X507 @ (push @ X513 @ id)) @ id)) != $true)) | ? [X516 : subst,X514 : term,X517 : subst,X515 : subst] : (($true != ((X508 @ (comp @ X515 @ X517) @ (sub @ X514 @ X517) @ X516))) & ($true = ((X508 @ X515 @ X514 @ (comp @ X517 @ X516))))) | ? [X509 : subst,X511 : subst,X512 : term,X510 : subst] : (($true != ((X508 @ X510 @ X512 @ (comp @ X511 @ X509)))) & ($true = ((X508 @ (comp @ X510 @ X511) @ (sub @ X512 @ X511) @ X509)))) | ? [X518 : (subst > term > term)] : (! [X519 : term] : ((((X508 @ id @ X519 @ id)) != $true) | (((X508 @ id @ (X518 @ id @ X519) @ id)) = $true)) & (((X508 @ id @ (lam @ (X518 @ sh @ one)) @ id)) != $true) & ! [X522 : subst,X520 : subst,X521 : term] : (((X518 @ (comp @ X522 @ X520) @ (sub @ X521 @ X520))) = ((sub @ (X518 @ X522 @ X521) @ X520))))) | ? [X498 : term] : (((sub @ X498 @ id)) != X498) | ? [X499 : (subst > term > subst > $o),X500 : term] : (! [X501 : term] : (($true != ((X499 @ id @ X501 @ id))) | ($true = ((X499 @ id @ (sub @ X500 @ (push @ X501 @ id)) @ id)))) & ($true != ((X499 @ id @ (lam @ (sub @ X500 @ (push @ one @ sh))) @ id))) & ! [X502 : (subst > term > term)] : (? [X505 : subst,X506 : term,X504 : subst] : (((sub @ (X502 @ X504 @ X506) @ X505)) != ((X502 @ (comp @ X504 @ X505) @ (sub @ X506 @ X505)))) | ? [X503 : term] : (($true != ((X499 @ id @ (X502 @ id @ X503) @ id))) & ($true = ((X499 @ id @ X503 @ id)))) | ($true = ((X499 @ id @ (lam @ (X502 @ sh @ one)) @ id)))))))),
  inference(flattening,[],[f11])).
thf(f13,plain,(
  ! [X0 : term] : (((var @ (lam @ X0))) != $true) <~> (lamnotvar = $true)),
  inference(ennf_transformation,[],[f8])).
thf(f14,definition,(
  ($true = sP0) <=> ! [X508 : (subst > term > subst > $o),X507 : term] : (($true = ((X508 @ id @ (lam @ X507) @ id))) | ? [X513 : term] : (($true = ((X508 @ id @ X513 @ id))) & (((X508 @ id @ (sub @ X507 @ (push @ X513 @ id)) @ id)) != $true)) | ? [X516 : subst,X514 : term,X517 : subst,X515 : subst] : (($true != ((X508 @ (comp @ X515 @ X517) @ (sub @ X514 @ X517) @ X516))) & ($true = ((X508 @ X515 @ X514 @ (comp @ X517 @ X516))))) | ? [X509 : subst,X511 : subst,X512 : term,X510 : subst] : (($true != ((X508 @ X510 @ X512 @ (comp @ X511 @ X509)))) & ($true = ((X508 @ (comp @ X510 @ X511) @ (sub @ X512 @ X511) @ X509)))) | ? [X518 : (subst > term > term)] : (! [X519 : term] : ((((X508 @ id @ X519 @ id)) != $true) | (((X508 @ id @ (X518 @ id @ X519) @ id)) = $true)) & (((X508 @ id @ (lam @ (X518 @ sh @ one)) @ id)) != $true) & ! [X522 : subst,X520 : subst,X521 : term] : (((X518 @ (comp @ X522 @ X520) @ (sub @ X521 @ X520))) = ((sub @ (X518 @ X522 @ X521) @ X520)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f15,definition,(
  ($true = sP1) <=> (? [X476 : subst,X477 : (term > $o),X478 : term] : (! [X483 : term] : (($true = ((X477 @ (sub @ X483 @ X476)))) | ($true != ((var @ X483)))) & ! [X479 : term,X480 : term] : (($true != ((X477 @ X480))) | ($true = ((X477 @ (ap @ X479 @ X480)))) | (((X477 @ X479)) != $true)) & ! [X481 : term] : (($true = ((X477 @ (lam @ X481)))) | ? [X482 : term] : ((((X477 @ X482)) = $true) & (((X477 @ (sub @ X481 @ (push @ X482 @ id)))) != $true))) & ($true != ((X477 @ (sub @ X478 @ X476))))) | ! [X484 : (term > $o),X485 : term] : (? [X486 : term] : (! [X487 : term] : (($true != ((X484 @ X487))) | ($true = ((X484 @ (sub @ X486 @ (push @ X487 @ id)))))) & (((X484 @ (lam @ X486))) != $true)) | ? [X490 : term] : (($true = ((var @ X490))) & ($true != ((X484 @ X490)))) | ? [X489 : term,X488 : term] : (($true != ((X484 @ (ap @ X489 @ X488)))) & ($true = ((X484 @ X488))) & ($true = ((X484 @ X489)))) | ($true = ((X484 @ X485)))) | ? [X475 : term] : (((sub @ X475 @ id)) != X475))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f16,definition,(
  ($true = sP2) <=> ! [X467 : term,X466 : (subst > term > subst > $o)] : (? [X470 : (subst > term > term)] : (! [X474 : term] : (($true != ((X466 @ id @ X474 @ id))) | ($true = ((X466 @ id @ (X470 @ id @ X474) @ id)))) & ($true != ((X466 @ id @ (lam @ (X470 @ sh @ one)) @ id))) & ! [X471 : term,X473 : subst,X472 : subst] : (((X470 @ (comp @ X472 @ X473) @ (sub @ X471 @ X473))) = ((sub @ (X470 @ X472 @ X471) @ X473)))) | ? [X468 : term,X469 : term] : (($true = ((X466 @ id @ X469 @ id))) & ($true != ((X466 @ id @ (ap @ (sub @ X469 @ id) @ X468) @ id))) & ($true = ((X466 @ id @ X468 @ id)))) | (((X466 @ id @ X467 @ id)) = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f17,definition,(
  ($true = sP3) <=> (? [X355 : term,X356 : (term > $o)] : (! [X360 : term] : ((((X356 @ X360)) != $true) | ($true = ((X356 @ (lam @ X360))))) & ! [X358 : term,X359 : term] : ((((X356 @ (ap @ X358 @ X359))) = $true) | ($true != ((X356 @ X359))) | ($true != ((X356 @ X358)))) & ! [X357 : term] : (($true = ((X356 @ X357))) | ($true != ((var @ X357)))) & (((X356 @ X355)) != $true)) | ? [X353 : term,X352 : subst] : (((comp @ sh @ (push @ X353 @ X352))) != X352) | ? [X350 : subst,X351 : term] : (((sub @ one @ (push @ X351 @ X350))) != X351) | ? [X354 : subst] : (((comp @ X354 @ id)) != X354) | ! [X363 : term,X362 : subst,X361 : (term > $o)] : (? [X365 : term] : (($true != ((X361 @ (lam @ X365)))) & ! [X366 : term] : (($true != ((X361 @ X366))) | ($true = ((X361 @ (sub @ X365 @ (push @ X366 @ id))))))) | ($true = ((X361 @ (sub @ X363 @ X362)))) | ? [X364 : term] : ((((X361 @ (sub @ X364 @ X362))) != $true) & (((var @ X364)) = $true)) | ? [X368 : term,X367 : term] : (($true = ((X361 @ X368))) & ($true = ((X361 @ X367))) & ($true != ((X361 @ (ap @ X367 @ X368)))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f18,definition,(
  ! [X332 : (subst > term > subst > $o)] : ((((sP4 @ X332)) = $true) <=> ? [X338 : term,X339 : term] : (($true = ((X332 @ id @ X339 @ id))) & (((X332 @ id @ X338 @ id)) = $true) & ($true != ((X332 @ id @ (ap @ (sub @ X338 @ id) @ X339) @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f19,definition,(
  ($true = sP5) <=> ! [X333 : term,X332 : (subst > term > subst > $o)] : (? [X340 : (subst > term > term)] : (! [X341 : term] : (($true != ((X332 @ id @ X341 @ id))) | ($true = ((X332 @ id @ (X340 @ id @ X341) @ id)))) & ($true != ((X332 @ id @ (lam @ (X340 @ sh @ one)) @ id))) & ! [X344 : subst,X342 : term,X343 : subst] : (((X340 @ (comp @ X343 @ X344) @ (sub @ X342 @ X344))) = ((sub @ (X340 @ X343 @ X342) @ X344)))) | ($true = ((X332 @ id @ X333 @ id))) | ? [X349 : term] : (($true != ((X332 @ id @ X349 @ id))) & ($true = ((var @ (sub @ X349 @ id))))) | (((sP4 @ X332)) = $true) | ? [X337 : subst,X335 : term,X334 : subst,X336 : subst] : (($true = ((X332 @ X334 @ X335 @ (comp @ X336 @ X337)))) & ($true != ((X332 @ (comp @ X334 @ X336) @ (sub @ X335 @ X336) @ X337)))) | ? [X345 : subst,X346 : subst,X348 : subst,X347 : term] : ((((X332 @ (comp @ X346 @ X345) @ (sub @ X347 @ X345) @ X348)) = $true) & ($true != ((X332 @ X346 @ X347 @ (comp @ X345 @ X348))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f20,definition,(
  ! [X394 : (subst > term > subst > $o)] : (($true = ((sP6 @ X394))) <=> ? [X406 : (subst > term > term)] : ((((X394 @ id @ (lam @ (X406 @ sh @ one)) @ id)) != $true) & ! [X407 : term] : (($true = ((X394 @ id @ (X406 @ id @ X407) @ id))) | (((X394 @ id @ X407 @ id)) != $true)) & ! [X408 : subst,X409 : subst,X410 : term] : (((X406 @ (comp @ X408 @ X409) @ (sub @ X410 @ X409))) = ((sub @ (X406 @ X408 @ X410) @ X409)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f21,definition,(
  ! [X165 : (subst > term > subst > $o)] : (($true = ((sP7 @ X165))) <=> ? [X178 : (subst > term > term)] : (! [X182 : term] : (($true = ((X165 @ id @ (X178 @ id @ X182) @ id))) | (((X165 @ id @ X182 @ id)) != $true)) & ! [X179 : subst,X180 : subst,X181 : term] : (((sub @ (X178 @ X179 @ X181) @ X180)) = ((X178 @ (comp @ X179 @ X180) @ (sub @ X181 @ X180)))) & ($true != ((X165 @ id @ (lam @ (X178 @ sh @ one)) @ id)))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f22,definition,(
  ($true = sP8) <=> ! [X165 : (subst > term > subst > $o),X166 : term] : (? [X175 : term,X176 : term] : (($true = ((X165 @ id @ X175 @ id))) & ($true = ((X165 @ id @ X176 @ id))) & ($true != ((X165 @ id @ (ap @ (sub @ X175 @ id) @ X176) @ id)))) | ($true = ((sP7 @ X165))) | ? [X177 : term] : (($true != ((X165 @ id @ X177 @ id))) & (((var @ (sub @ X177 @ id))) = $true)) | ($true = ((X165 @ id @ X166 @ id))) | ? [X171 : subst,X174 : subst,X172 : term,X173 : subst] : (($true != ((X165 @ X173 @ X172 @ (comp @ X174 @ X171)))) & ($true = ((X165 @ (comp @ X173 @ X174) @ (sub @ X172 @ X174) @ X171)))) | ? [X169 : subst,X170 : subst,X168 : subst,X167 : term] : (($true != ((X165 @ (comp @ X169 @ X168) @ (sub @ X167 @ X168) @ X170))) & ($true = ((X165 @ X169 @ X167 @ (comp @ X168 @ X170))))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f23,definition,(
  (sP9 = $true) <=> ? [X138 : term,X136 : term,X137 : (subst > term > subst > $o)] : (! [X143 : term,X144 : term] : (($true != ((X137 @ id @ X143 @ id))) | (((X137 @ id @ (ap @ (sub @ X143 @ id) @ X144) @ id)) = $true) | (((X137 @ id @ X144 @ id)) != $true)) & (((X137 @ id @ (ap @ X138 @ X136) @ id)) != $true) & (((X137 @ id @ X138 @ id)) = $true) & ! [X145 : subst,X148 : subst,X147 : term,X146 : subst] : (($true = ((X137 @ X145 @ X147 @ (comp @ X146 @ X148)))) | (((X137 @ (comp @ X145 @ X146) @ (sub @ X147 @ X146) @ X148)) != $true)) & ! [X141 : term,X139 : subst,X140 : subst,X142 : subst] : ((((X137 @ (comp @ X140 @ X139) @ (sub @ X141 @ X139) @ X142)) = $true) | ($true != ((X137 @ X140 @ X141 @ (comp @ X139 @ X142))))) & ($true = ((X137 @ id @ X136 @ id))))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f24,definition,(
  ($true = sP10) <=> (? [X150 : term,X149 : (subst > term > subst > $o)] : (! [X155 : term] : ((((X149 @ id @ X155 @ id)) != $true) | (((X149 @ id @ (sub @ X150 @ (push @ X155 @ id)) @ id)) = $true)) & ($true != ((X149 @ id @ (lam @ X150) @ id))) & ! [X157 : subst,X159 : subst,X156 : subst,X158 : term] : (($true != ((X149 @ X157 @ X158 @ (comp @ X159 @ X156)))) | ($true = ((X149 @ (comp @ X157 @ X159) @ (sub @ X158 @ X159) @ X156)))) & ! [X152 : subst,X153 : subst,X154 : subst,X151 : term] : ((((X149 @ X152 @ X151 @ (comp @ X153 @ X154))) = $true) | ($true != ((X149 @ (comp @ X152 @ X153) @ (sub @ X151 @ X153) @ X154)))) & ! [X160 : (subst > term > term)] : (($true = ((X149 @ id @ (lam @ (X160 @ sh @ one)) @ id))) | ? [X164 : term] : (($true = ((X149 @ id @ X164 @ id))) & ($true != ((X149 @ id @ (X160 @ id @ X164) @ id)))) | ? [X161 : subst,X162 : term,X163 : subst] : (((sub @ (X160 @ X163 @ X162) @ X161)) != ((X160 @ (comp @ X163 @ X161) @ (sub @ X162 @ X161)))))) | ? [X130 : term,X129 : (term > $o)] : (! [X131 : term,X132 : term] : (($true != ((X129 @ X131))) | (((X129 @ X132)) != $true) | ($true = ((X129 @ (ap @ X131 @ X132))))) & ($true != ((X129 @ X130))) & ! [X133 : term] : (($true = ((X129 @ X133))) | ($true != ((var @ X133)))) & ! [X134 : term] : ((((X129 @ (lam @ X134))) = $true) | ? [X135 : term] : (($true = ((X129 @ X135))) & ($true != ((X129 @ (sub @ X134 @ (push @ X135 @ id)))))))) | ($true = sP8) | (sP9 = $true))),
  introduced(definition,[new_symbols(definition,[=])],[predicate_definition_introduction])).
thf(f25,plain,(
  (hoaslamnotap_lthm = $true) & (axabs = $true) & (! [X293 : (subst > term > subst > $o),X292 : term] : (($true = ((X293 @ id @ (lam @ (sub @ X292 @ (push @ one @ sh))) @ id))) | ? [X294 : term] : ((((X293 @ id @ X294 @ id)) = $true) & ($true != ((X293 @ id @ (sub @ X292 @ (push @ X294 @ id)) @ id)))) | ? [X295 : (subst > term > term)] : (($true != ((X293 @ id @ (lam @ (X295 @ sh @ one)) @ id))) & ! [X296 : subst,X297 : term,X298 : subst] : (((sub @ (X295 @ X298 @ X297) @ X296)) = ((X295 @ (comp @ X298 @ X296) @ (sub @ X297 @ X296)))) & ! [X299 : term] : ((((X293 @ id @ X299 @ id)) != $true) | ($true = ((X293 @ id @ (X295 @ id @ X299) @ id)))))) <=> (hoasinduction_lem3aa = $true)) & (pushprop_lem2v2_gthm = $true) & ((induction2lem = $true) <=> ! [X217 : term,X215 : subst,X216 : (term > $o)] : (? [X221 : term] : (($true != ((X216 @ (lam @ X221)))) & ! [X222 : term] : (($true = ((X216 @ (sub @ X221 @ (push @ X222 @ id))))) | ($true != ((X216 @ X222))))) | ? [X219 : term,X220 : term] : (($true = ((X216 @ X220))) & (((X216 @ (ap @ X219 @ X220))) != $true) & ($true = ((X216 @ X219)))) | ? [X218 : term] : (($true != ((X216 @ (sub @ X218 @ X215)))) & ($true = ((var @ X218)))) | (((X216 @ (sub @ X217 @ X215))) = $true))) & (lamnotap != $true) & ((hoasinduction_lem3v2a_lthm = $true) <=> (! [X20 : (subst > term > subst > $o),X19 : (term > $o),X18 : term] : (($true = ((X19 @ (lam @ X18)))) | ? [X21 : term] : (((X19 @ X21)) != ((X20 @ id @ X21 @ id))) | ? [X23 : (subst > term > term)] : (($true != ((X20 @ id @ (lam @ (X23 @ sh @ one)) @ id))) & ! [X24 : term] : ((((X20 @ id @ (X23 @ id @ X24) @ id)) = $true) | ($true != ((X20 @ id @ X24 @ id)))) & ! [X25 : subst,X26 : term,X27 : subst] : (((sub @ (X23 @ X25 @ X26) @ X27)) = ((X23 @ (comp @ X25 @ X27) @ (sub @ X26 @ X27))))) | ? [X22 : term] : (($true != ((X19 @ (sub @ X18 @ (push @ X22 @ id))))) & (((X19 @ X22)) = $true))) | ? [X13 : term] : ! [X14 : (subst > term > term)] : ? [X15 : term,X16 : subst] : (((sub @ X13 @ (push @ X15 @ X16))) != ((X14 @ X16 @ X15))) | ? [X17 : term] : (((sub @ X17 @ id)) != X17))) & (! [X533 : term,X532 : term,X531 : term,X534 : term] : ((X532 = X533) | (((ap @ (sub @ X534 @ id) @ X533)) != ((ap @ (sub @ X531 @ id) @ X532)))) <=> (hoasapinj2 = $true)) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem1_gthm = $true) & (! [X260 : term,X261 : (subst > term > subst > $o)] : (? [X262 : (subst > term > term)] : (($true != ((X261 @ id @ (lam @ (X262 @ sh @ one)) @ id))) & ! [X264 : term,X265 : subst,X263 : subst] : (((X262 @ (comp @ X265 @ X263) @ (sub @ X264 @ X263))) = ((sub @ (X262 @ X265 @ X264) @ X263))) & ! [X266 : term] : ((((X261 @ id @ (X262 @ id @ X266) @ id)) = $true) | ($true != ((X261 @ id @ X266 @ id))))) | ? [X267 : term] : ((((X261 @ id @ (sub @ X260 @ (push @ X267 @ id)) @ id)) != $true) & (((X261 @ id @ X267 @ id)) = $true)) | ($true = ((X261 @ id @ (lam @ (sub @ X260 @ (push @ one @ sh))) @ id)))) <=> (hoasinduction_lem3aa_lthm = $true)) & (! [X535 : subst] : (((comp @ id @ X535)) = X535) <=> (axidl = $true)) & ((hoaslamnotap = $true) <=> ! [X94 : term,X92 : term,X93 : (subst > term > term)] : (? [X95 : subst,X97 : subst,X96 : term] : (((X93 @ (comp @ X95 @ X97) @ (sub @ X96 @ X97))) != ((sub @ (X93 @ X95 @ X96) @ X97))) | (((lam @ (X93 @ sh @ one))) != ((ap @ (sub @ X94 @ id) @ X92))))) & (! [X429 : term,X428 : term] : ((((sub @ X428 @ sh)) != ((sub @ X429 @ sh))) | (X428 = X429)) <=> (shinj = $true)) & (! [X193 : term,X194 : term,X195 : (subst > term > subst > $o)] : (($true != ((X195 @ id @ X194 @ id))) | ($true != ((X195 @ id @ X193 @ id))) | (((X195 @ id @ (ap @ X193 @ X194) @ id)) = $true) | ? [X199 : subst,X200 : subst,X198 : term,X201 : subst] : (($true = ((X195 @ (comp @ X201 @ X200) @ (sub @ X198 @ X200) @ X199))) & ($true != ((X195 @ X201 @ X198 @ (comp @ X200 @ X199))))) | ? [X204 : subst,X203 : subst,X202 : term,X205 : subst] : (($true != ((X195 @ (comp @ X204 @ X205) @ (sub @ X202 @ X205) @ X203))) & ($true = ((X195 @ X204 @ X202 @ (comp @ X205 @ X203))))) | ? [X197 : term,X196 : term] : (($true != ((X195 @ id @ (ap @ (sub @ X196 @ id) @ X197) @ id))) & (((X195 @ id @ X196 @ id)) = $true) & ($true = ((X195 @ id @ X197 @ id))))) <=> (hoasinduction_lem2 = $true)) & (pushprop_lem3v2 = $true) & ! [X309 : d_subst,X310 : d_subst] : ((((d2subst @ X310)) != ((d2subst @ X309))) | (X309 = X310)) & (! [X30 : term,X31 : term,X29 : term,X28 : term] : ((((ap @ (sub @ X30 @ id) @ X29)) != ((ap @ (sub @ X28 @ id) @ X31))) | (X28 = X30)) <=> (hoasapinj1 = $true)) & ((axvarcons = $true) <=> ! [X64 : subst,X63 : term] : (((sub @ one @ (push @ X63 @ X64))) = X63)) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((hoasinduction_lthm = $true) <=> ($true = sP10)) & ! [X51 : d_term,X52 : d_term] : ((((d2term @ X51)) != ((d2term @ X52))) | (X51 = X52)) & (hoaslamnotvar_gthm = $true) & (hoaslamnotvar = $true) & ((hoasinduction = $true) <=> ! [X393 : term,X394 : (subst > term > subst > $o)] : (? [X400 : term,X399 : term] : ((((X394 @ id @ (ap @ (sub @ X399 @ id) @ X400) @ id)) != $true) & ($true = ((X394 @ id @ X399 @ id))) & (((X394 @ id @ X400 @ id)) = $true)) | ? [X397 : subst,X395 : subst,X396 : term,X398 : subst] : (($true != ((X394 @ (comp @ X395 @ X397) @ (sub @ X396 @ X397) @ X398))) & ($true = ((X394 @ X395 @ X396 @ (comp @ X397 @ X398))))) | ? [X401 : term] : ((((var @ (sub @ X401 @ id))) = $true) & ($true != ((X394 @ id @ X401 @ id)))) | ($true = ((X394 @ id @ X393 @ id))) | ($true = ((sP6 @ X394))) | ? [X402 : subst,X404 : subst,X405 : subst,X403 : term] : (($true = ((X394 @ (comp @ X404 @ X402) @ (sub @ X403 @ X402) @ X405))) & ($true != ((X394 @ X404 @ X403 @ (comp @ X402 @ X405))))))) & (pushprop_lem3v2_lthm = $true) & (! [X183 : subst] : (((push @ (sub @ one @ X183) @ (comp @ sh @ X183))) = X183) <=> (axscons = $true)) & (hoasinduction_lem3b_gthm = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ! [X111 : d_term] : (d_one = X111) & ((hoasapinj2_lthm = $true) <=> (? [X524 : term,X525 : term,X523 : term,X526 : term] : ((X523 != X524) & (((ap @ X526 @ X523)) = ((ap @ X525 @ X524)))) | ! [X529 : term,X528 : term,X527 : term,X530 : term] : ((X527 = X530) | (((ap @ (sub @ X529 @ id) @ X527)) != ((ap @ (sub @ X528 @ id) @ X530)))))) & (axvarshift = $true) & (! [X538 : term] : ? [X539 : (subst > term > term)] : (((X539 @ sh @ one)) = ((sub @ X538 @ (push @ one @ sh)))) <=> (hoasinduction_lem3b = $true)) & ((hoasinduction_lem3a_lthm = $true) <=> (? [X413 : (subst > term > subst > $o),X412 : term] : (! [X414 : (subst > term > term)] : (? [X418 : term,X417 : subst,X416 : subst] : (((X414 @ (comp @ X417 @ X416) @ (sub @ X418 @ X416))) != ((sub @ (X414 @ X417 @ X418) @ X416))) | ? [X415 : term] : ((((X413 @ id @ X415 @ id)) = $true) & (((X413 @ id @ (X414 @ id @ X415) @ id)) != $true)) | ($true = ((X413 @ id @ (lam @ (X414 @ sh @ one)) @ id)))) & ! [X419 : term] : (($true != ((X413 @ id @ X419 @ id))) | ($true = ((X413 @ id @ (sub @ X412 @ (push @ X419 @ id)) @ id)))) & (((X413 @ id @ (lam @ (sub @ X412 @ (push @ one @ sh))) @ id)) != $true)) | ? [X411 : term] : (((sub @ X411 @ id)) != X411) | ! [X421 : (subst > term > subst > $o),X420 : term] : (? [X422 : (subst > term > term)] : (! [X423 : subst,X424 : subst,X425 : term] : (((sub @ (X422 @ X423 @ X425) @ X424)) = ((X422 @ (comp @ X423 @ X424) @ (sub @ X425 @ X424)))) & ! [X426 : term] : (($true = ((X421 @ id @ (X422 @ id @ X426) @ id))) | ($true != ((X421 @ id @ X426 @ id)))) & (((X421 @ id @ (lam @ (X422 @ sh @ one)) @ id)) != $true)) | ? [X427 : term] : ((((X421 @ id @ (sub @ X420 @ (push @ X427 @ id)) @ id)) != $true) & (((X421 @ id @ X427 @ id)) = $true)) | ($true = ((X421 @ id @ (lam @ X420) @ id)))))) & (induction2lem_gthm = $true) & ! [X48 : d_subst] : (d_id = X48) & (pushprop_lem2v2_lthm = $true) & (ulamvar1 != $true) & ! [X536 : subst] : ? [X537 : d_subst] : (((d2subst @ X537)) = X536) & (pushprop_lthm = $true) & (hoasinduction_lem1v2 = $true) & (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_lem3v2a = $true) <=> ! [X227 : (term > $o),X228 : term,X229 : (subst > term > subst > $o)] : (? [X235 : term] : (((X229 @ id @ X235 @ id)) != ((X227 @ X235))) | ($true = ((X227 @ (lam @ X228)))) | ? [X236 : term] : (($true != ((X227 @ (sub @ X228 @ (push @ X236 @ id))))) & ($true = ((X227 @ X236)))) | ? [X230 : (subst > term > term)] : (! [X232 : term,X231 : subst,X233 : subst] : (((X230 @ (comp @ X231 @ X233) @ (sub @ X232 @ X233))) = ((sub @ (X230 @ X231 @ X232) @ X233))) & ! [X234 : term] : ((((X229 @ id @ (X230 @ id @ X234) @ id)) = $true) | ($true != ((X229 @ id @ X234 @ id)))) & (((X229 @ id @ (lam @ (X230 @ sh @ one)) @ id)) != $true)))) & (pushprop_lthm_orig = $true) & (pushprop_lem1v2_gthm = $true) & (pushprop_lem2v2 = $true) & ! [X61 : term] : ? [X62 : d_term] : (((d2term @ X62)) = X61) & ((hoasinduction_no_psi_cond = $true) <=> ! [X206 : (subst > term > subst > $o),X207 : term] : (($true = ((X206 @ id @ X207 @ id))) | ? [X208 : (subst > term > term)] : (($true != ((X206 @ id @ (lam @ (X208 @ sh @ one)) @ id))) & ! [X211 : subst,X210 : subst,X209 : term] : (((sub @ (X208 @ X211 @ X209) @ X210)) = ((X208 @ (comp @ X211 @ X210) @ (sub @ X209 @ X210)))) & ! [X212 : term] : (($true = ((X206 @ id @ (X208 @ id @ X212) @ id))) | ($true != ((X206 @ id @ X212 @ id))))) | ? [X214 : term,X213 : term] : ((((X206 @ id @ X214 @ id)) = $true) & ($true = ((X206 @ id @ X213 @ id))) & ($true != ((X206 @ id @ (ap @ (sub @ X213 @ id) @ X214) @ id)))))) & (! [X115 : term,X114 : (subst > term > subst > $o)] : (($true = ((X114 @ id @ (lam @ (sub @ X115 @ (push @ one @ sh))) @ id))) | ? [X118 : term,X117 : (subst > term > term)] : (! [X121 : subst,X119 : subst,X120 : term] : (((X117 @ (comp @ X119 @ X121) @ (sub @ X120 @ X121))) = ((sub @ X118 @ (push @ (sub @ X120 @ X121) @ (comp @ X119 @ X121))))) & ! [X124 : subst,X123 : term,X122 : subst] : (((sub @ (sub @ X118 @ (push @ X123 @ X122)) @ X124)) = ((sub @ (X117 @ X122 @ X123) @ X124))) & (((X114 @ id @ (lam @ (X117 @ sh @ one)) @ id)) != $true) & ! [X125 : term] : (($true = ((X114 @ id @ (X117 @ id @ X125) @ id))) | ($true != ((X114 @ id @ X125 @ id))))) | ? [X116 : term] : (($true != ((X114 @ id @ (sub @ X115 @ (push @ X116 @ id)) @ id))) & ($true = ((X114 @ id @ X116 @ id))))) <=> (hoasinduction_lem3aaa = $true)) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (pushprop_lem0_gthm = $true) & (apnotvar = $true) & (hoasapnotvar = $true) & (hoasinduction_lem1_lthm = $true) & ((hoaslaminj = $true) <=> ! [X372 : subst,X370 : (subst > term > term),X369 : term,X371 : (subst > term > term)] : ((((lam @ (X370 @ sh @ one))) != ((lam @ (X371 @ sh @ one)))) | ? [X376 : subst,X378 : term,X377 : subst] : (((sub @ (X371 @ X376 @ X378) @ X377)) != ((X371 @ (comp @ X376 @ X377) @ (sub @ X378 @ X377)))) | (((X371 @ X372 @ X369)) = ((X370 @ X372 @ X369))) | ? [X375 : subst,X374 : subst,X373 : term] : (((sub @ (X370 @ X375 @ X373) @ X374)) != ((X370 @ (comp @ X375 @ X374) @ (sub @ X373 @ X374)))))) & (induction2_gthm = $true) & (! [X71 : term] : (((sub @ X71 @ id)) = X71) <=> (termmset = $true)) & ((hoasinduction_lem0_lthm = $true) <=> ! [X10 : (subst > term > subst > $o)] : ? [X11 : (term > $o)] : ! [X12 : term] : (((X11 @ X12)) = ((X10 @ id @ X12 @ id)))) & (hoaslamnotap_gthm = $true) & (((d2subst @ d_id)) = sh) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (hoasinduction_lem2v2_gthm = $true) & ((hoaslaminj_lthm = $true) <=> (? [X35 : subst,X34 : term] : (((comp @ sh @ (push @ X34 @ X35))) != X35) | ? [X32 : subst,X33 : term] : (((sub @ one @ (push @ X33 @ X32))) != X33) | ! [X39 : (subst > term > term),X40 : (subst > term > term),X41 : term,X38 : subst] : (? [X46 : term,X47 : subst,X45 : subst] : (((sub @ (X40 @ X45 @ X46) @ X47)) != ((X40 @ (comp @ X45 @ X47) @ (sub @ X46 @ X47)))) | (((lam @ (X39 @ sh @ one))) != ((lam @ (X40 @ sh @ one)))) | ? [X42 : subst,X44 : term,X43 : subst] : (((sub @ (X39 @ X43 @ X44) @ X42)) != ((X39 @ (comp @ X43 @ X42) @ (sub @ X44 @ X42)))) | (((X40 @ X38 @ X41)) = ((X39 @ X38 @ X41)))) | ? [X36 : term,X37 : term] : ((X36 != X37) & (((lam @ X37)) = ((lam @ X36)))))) & (! [X73 : term,X72 : (subst > term > subst > $o)] : (? [X74 : term] : (($true != ((X72 @ id @ (sub @ X73 @ (push @ X74 @ id)) @ id))) & ($true = ((X72 @ id @ X74 @ id)))) | ? [X85 : subst,X86 : subst,X84 : subst,X87 : term] : (($true != ((X72 @ (comp @ X85 @ X84) @ (sub @ X87 @ X84) @ X86))) & ($true = ((X72 @ X85 @ X87 @ (comp @ X84 @ X86))))) | ? [X79 : (subst > term > term)] : (! [X83 : subst,X81 : term,X82 : subst] : (((X79 @ (comp @ X83 @ X82) @ (sub @ X81 @ X82))) = ((sub @ (X79 @ X83 @ X81) @ X82))) & ! [X80 : term] : ((((X72 @ id @ X80 @ id)) != $true) | ($true = ((X72 @ id @ (X79 @ id @ X80) @ id)))) & (((X72 @ id @ (lam @ (X79 @ sh @ one)) @ id)) != $true)) | ($true = ((X72 @ id @ (lam @ X73) @ id))) | ? [X76 : subst,X77 : term,X78 : subst,X75 : subst] : (($true = ((X72 @ (comp @ X75 @ X78) @ (sub @ X77 @ X78) @ X76))) & (((X72 @ X75 @ X77 @ (comp @ X78 @ X76))) != $true))) <=> (hoasinduction_lem3 = $true)) & (termmset_gthm = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (hoasinduction_lem2_lthm = $true) & (! [X99 : term,X98 : term] : ((((lam @ X98)) != ((lam @ X99))) | (X98 = X99)) <=> (laminj = $true)) & (((var @ (d2term @ d_one))) != $true) & (! [X272 : subst] : (((comp @ X272 @ id)) = X272) <=> (axidr = $true)) & (! [X0 : term] : (((sub @ X0 @ id)) = X0) <=> (axvarid = $true)) & ((? [X321 : term] : (((sub @ X321 @ id)) != X321) | ($true = sP5) | ? [X311 : (subst > term > subst > $o)] : ! [X312 : (term > $o)] : ? [X313 : term] : (((X312 @ X313)) != ((X311 @ id @ X313 @ id))) | ? [X322 : term,X323 : (term > $o),X324 : (subst > term > subst > $o)] : (($true != ((X323 @ (lam @ X322)))) & ! [X326 : (subst > term > term)] : (($true = ((X324 @ id @ (lam @ (X326 @ sh @ one)) @ id))) | ? [X330 : subst,X328 : subst,X329 : term] : (((X326 @ (comp @ X328 @ X330) @ (sub @ X329 @ X330))) != ((sub @ (X326 @ X328 @ X329) @ X330))) | ? [X327 : term] : ((((X324 @ id @ X327 @ id)) = $true) & ($true != ((X324 @ id @ (X326 @ id @ X327) @ id))))) & ! [X325 : term] : ((((X323 @ X325)) != $true) | ($true = ((X323 @ (sub @ X322 @ (push @ X325 @ id)))))) & ! [X331 : term] : (((X324 @ id @ X331 @ id)) = ((X323 @ X331)))) | ? [X314 : term,X315 : (term > $o)] : (! [X318 : term] : ((((X315 @ X318)) = $true) | (((var @ X318)) != $true)) & ($true != ((X315 @ X314))) & ! [X319 : term] : (? [X320 : term] : (($true = ((X315 @ X320))) & (((X315 @ (sub @ X319 @ (push @ X320 @ id)))) != $true)) | ($true = ((X315 @ (lam @ X319))))) & ! [X317 : term,X316 : term] : (($true != ((X315 @ X317))) | (((X315 @ (ap @ X317 @ X316))) = $true) | ($true != ((X315 @ X316)))))) <=> (hoasinduction_lthm_3 = $true)) & (substmonoid_gthm = $true) & (pushprop = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (axclos = $true) & ((hoasinduction_lem3a = $true) <=> ! [X54 : (subst > term > subst > $o),X53 : term] : (? [X55 : (subst > term > term)] : (! [X59 : term] : (($true = ((X54 @ id @ (X55 @ id @ X59) @ id))) | ($true != ((X54 @ id @ X59 @ id)))) & ($true != ((X54 @ id @ (lam @ (X55 @ sh @ one)) @ id))) & ! [X57 : subst,X56 : subst,X58 : term] : (((X55 @ (comp @ X57 @ X56) @ (sub @ X58 @ X56))) = ((sub @ (X55 @ X57 @ X58) @ X56)))) | ? [X60 : term] : (($true = ((X54 @ id @ X60 @ id))) & ($true != ((X54 @ id @ (sub @ X53 @ (push @ X60 @ id)) @ id)))) | ($true = ((X54 @ id @ (lam @ X53) @ id))))) & ((? [X49 : term] : (((sub @ X49 @ id)) != X49) | ! [X50 : term] : (((sub @ X50 @ id)) = X50)) <=> (termmset_lthm = $true)) & (ulamvarsh = $true) & ((hoasinduction_lem3v2_f = $true) <=> ! [X100 : term] : ? [X101 : (subst > term > term)] : ! [X102 : subst,X103 : term] : (((X101 @ X102 @ X103)) = ((sub @ X100 @ (push @ X103 @ X102))))) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((substmonoid = $true) <=> (! [X224 : subst] : (((comp @ X224 @ id)) = X224) & ! [X223 : subst] : (((comp @ id @ X223)) = X223))) & ((hoasinduction_lem3b_lthm = $true) <=> ! [X225 : term] : ? [X226 : (subst > term > term)] : (((sub @ X225 @ (push @ one @ sh))) = ((X226 @ sh @ one)))) & ((! [X108 : (term > $o),X106 : (term > $o),X109 : subst,X107 : term] : (? [X110 : term] : (((X106 @ X110)) != ((X108 @ (sub @ X110 @ (push @ X107 @ X109))))) | ($true != ((X108 @ X107))) | ($true = ((X106 @ one)))) | ? [X104 : subst,X105 : term] : (((sub @ one @ (push @ X105 @ X104))) != X105)) <=> (pushprop_lem1v2_lthm = $true)) & (axassoc = $true) & ((? [X88 : subst] : (((comp @ id @ X88)) != X88) | (! [X91 : subst] : (((comp @ id @ X91)) = X91) & ! [X90 : subst] : (((comp @ X90 @ id)) = X90)) | ? [X89 : subst] : (((comp @ X89 @ id)) != X89)) <=> (substmonoid_lthm = $true)) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & ((? [X304 : term,X302 : term,X303 : term,X301 : term] : ((((ap @ X301 @ X303)) = ((ap @ X304 @ X302))) & (X301 != X304)) | ? [X300 : term] : (((sub @ X300 @ id)) != X300) | ! [X307 : term,X306 : term,X308 : term,X305 : term] : ((((ap @ (sub @ X305 @ id) @ X307)) != ((ap @ (sub @ X306 @ id) @ X308))) | (X305 = X306))) <=> (hoasapinj1_lthm = $true)) & (hoasapinj1_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (hoasapnotvar_lthm = $true) & (one = ((d2term @ d_one))) & (axapp = $true) & (hoasinduction_lem1 = $true) & ((pushprop_lem0 = $true) <=> ! [X237 : subst,X238 : (term > $o),X239 : term] : ? [X240 : (term > $o)] : ! [X241 : term] : (((X238 @ (sub @ X241 @ (push @ X239 @ X237)))) = ((X240 @ X241)))) & ((induction2lem_lthm = $true) <=> ($true = sP3)) & (ulamvarind = $true) & (axmap = $true) & ((hoasinduction_no_psi_cond_lthm = $true) <=> (? [X455 : term] : (((sub @ X455 @ id)) != X455) | ? [X456 : (term > $o),X457 : (subst > term > subst > $o),X458 : term] : (! [X459 : term] : (((X457 @ id @ X459 @ id)) = ((X456 @ X459))) & (((X456 @ (lam @ X458))) != $true) & ! [X460 : term] : ((((X456 @ (sub @ X458 @ (push @ X460 @ id)))) = $true) | (((X456 @ X460)) != $true)) & ! [X461 : (subst > term > term)] : (($true = ((X457 @ id @ (lam @ (X461 @ sh @ one)) @ id))) | ? [X462 : term] : (($true = ((X457 @ id @ X462 @ id))) & (((X457 @ id @ (X461 @ id @ X462) @ id)) != $true)) | ? [X463 : term,X465 : subst,X464 : subst] : (((sub @ (X461 @ X465 @ X463) @ X464)) != ((X461 @ (comp @ X465 @ X464) @ (sub @ X463 @ X464)))))) | ? [X449 : (term > $o),X448 : term] : (! [X451 : term,X450 : term] : ((((X449 @ X451)) != $true) | ($true = ((X449 @ (ap @ X450 @ X451)))) | ($true != ((X449 @ X450)))) & ($true != ((X449 @ X448))) & ! [X454 : term] : ((((var @ X454)) != $true) | (((X449 @ X454)) = $true)) & ! [X452 : term] : (? [X453 : term] : ((((X449 @ X453)) = $true) & ($true != ((X449 @ (sub @ X452 @ (push @ X453 @ id)))))) | ($true = ((X449 @ (lam @ X452)))))) | ? [X445 : (subst > term > subst > $o)] : ! [X446 : (term > $o)] : ? [X447 : term] : (((X445 @ id @ X447 @ id)) != ((X446 @ X447))) | ($true = sP2))) & ((pushprop_lem1v2 = $true) <=> ! [X388 : (term > $o),X391 : (term > $o),X390 : term,X389 : subst] : (? [X392 : term] : (((X391 @ (sub @ X392 @ (push @ X390 @ X389)))) != ((X388 @ X392))) | ($true != ((X391 @ X390))) | ($true = ((X388 @ one))))) & (hoasinduction_gthm = $true) & ((! [X384 : term,X387 : subst,X386 : term,X385 : (term > $o),X383 : (term > $o)] : (($true = ((X383 @ (sub @ X384 @ (push @ X386 @ X387))))) | ($true != ((X385 @ X384)))) | ? [X379 : subst,X380 : term] : (((sub @ one @ (push @ X380 @ X379))) != X380) | ? [X381 : subst,X382 : term] : (((comp @ sh @ (push @ X382 @ X381))) != X381)) <=> (pushprop_lem1_lthm = $true)) & (! [X491 : (term > $o),X492 : term] : (? [X494 : term,X495 : term] : (($true = ((X491 @ X494))) & ($true = ((X491 @ X495))) & ($true != ((X491 @ (ap @ X495 @ X494))))) | ? [X496 : term] : (($true != ((X491 @ (lam @ X496)))) & ! [X497 : term] : (($true != ((X491 @ X497))) | (((X491 @ (sub @ X496 @ (push @ X497 @ id)))) = $true))) | ($true = ((X491 @ X492))) | ? [X493 : term] : (($true = ((var @ X493))) & (((X491 @ X493)) != $true))) <=> (induction2 = $true)) & ((pushprop_lem1 = $true) <=> ! [X186 : (term > $o),X185 : term,X187 : subst,X188 : term,X184 : (term > $o)] : ((((X184 @ (sub @ X185 @ (push @ X188 @ X187)))) = $true) | ($true != ((X186 @ X185))))) & ((induction2_lthm = $true) <=> ($true = sP1)) & ((hoasinduction_lem0 = $true) <=> ! [X126 : (subst > term > subst > $o)] : ? [X127 : (term > $o)] : ! [X128 : term] : (((X127 @ X128)) = ((X126 @ id @ X128 @ id)))) & (lamnotvar = $true) & (! [X3 : term,X2 : term,X1 : term,X4 : term] : ((((ap @ X3 @ X2)) != ((ap @ X1 @ X4))) | (X2 = X4)) <=> (apinj2 = $true)) & (! [X65 : (term > $o),X66 : term] : (? [X70 : term] : ((((X65 @ (lam @ X70))) != $true) & ($true = ((X65 @ X70)))) | ($true = ((X65 @ X66))) | ? [X69 : term,X68 : term] : (($true != ((X65 @ (ap @ X68 @ X69)))) & ($true = ((X65 @ X68))) & (((X65 @ X69)) = $true)) | ? [X67 : term] : (($true = ((var @ X67))) & ($true != ((X65 @ X67))))) <=> (induction = $true)) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoaslamnotvar_lthm = $true) & ((hoasinduction_lem2v2 = $true) <=> ! [X431 : (subst > term > subst > $o),X430 : term,X432 : (term > $o),X433 : term] : (? [X438 : subst,X439 : term,X440 : subst,X437 : subst] : ((((X431 @ X437 @ X439 @ (comp @ X438 @ X440))) != $true) & ($true = ((X431 @ (comp @ X437 @ X438) @ (sub @ X439 @ X438) @ X440)))) | ($true != ((X432 @ X433))) | (((X432 @ X430)) != $true) | ? [X435 : term,X436 : term] : (($true = ((X431 @ id @ X435 @ id))) & ($true != ((X431 @ id @ (ap @ (sub @ X435 @ id) @ X436) @ id))) & ($true = ((X431 @ id @ X436 @ id)))) | ($true = ((X432 @ (ap @ X433 @ X430)))) | ? [X434 : term] : (((X432 @ X434)) != ((X431 @ id @ X434 @ id))) | ? [X444 : subst,X442 : subst,X441 : subst,X443 : term] : (($true != ((X431 @ (comp @ X442 @ X441) @ (sub @ X443 @ X441) @ X444))) & ($true = ((X431 @ X442 @ X443 @ (comp @ X441 @ X444))))))) & (hoasinduction_lem3a_gthm = $true) & (pushprop_lem1_gthm = $true) & (hoasapinj2_gthm = $true) & (id = ((d2subst @ d_id))) & (pushprop_gthm = $true) & (! [X189 : term] : ? [X190 : (subst > term > term)] : ! [X192 : term,X191 : subst] : (((X190 @ X191 @ X192)) = ((sub @ X189 @ (push @ X192 @ X191)))) <=> (hoasinduction_lem3v2_f_lthm = $true)) & ((hoasinduction_lem3v2_lthm = $true) <=> (? [X273 : term] : (((sub @ X273 @ id)) != X273) | ! [X274 : (subst > term > subst > $o),X275 : term,X276 : (term > $o)] : (? [X277 : term] : (((X276 @ X277)) != ((X274 @ id @ X277 @ id))) | ($true = ((X276 @ (lam @ X275)))) | ? [X289 : subst,X290 : term,X288 : subst,X291 : subst] : (($true = ((X274 @ (comp @ X288 @ X291) @ (sub @ X290 @ X291) @ X289))) & ($true != ((X274 @ X288 @ X290 @ (comp @ X291 @ X289))))) | ? [X283 : term,X285 : subst,X284 : subst,X286 : subst] : ((((X274 @ (comp @ X285 @ X286) @ (sub @ X283 @ X286) @ X284)) != $true) & (((X274 @ X285 @ X283 @ (comp @ X286 @ X284))) = $true)) | ? [X278 : (subst > term > term)] : (! [X281 : subst,X280 : subst,X282 : term] : (((sub @ (X278 @ X280 @ X282) @ X281)) = ((X278 @ (comp @ X280 @ X281) @ (sub @ X282 @ X281)))) & ! [X279 : term] : ((((X274 @ id @ X279 @ id)) != $true) | ($true = ((X274 @ id @ (X278 @ id @ X279) @ id)))) & ($true != ((X274 @ id @ (lam @ (X278 @ sh @ one)) @ id)))) | ? [X287 : term] : (($true != ((X276 @ (sub @ X275 @ (push @ X287 @ id))))) & ($true = ((X276 @ X287))))))) & (! [X270 : term,X271 : term,X269 : term,X268 : term] : ((((ap @ X269 @ X270)) != ((ap @ X271 @ X268))) | (X269 = X271)) <=> (apinj1 = $true)) & (! [X5 : (term > $o),X7 : subst,X6 : term] : ? [X8 : (term > $o)] : ! [X9 : term] : (((X5 @ (sub @ X9 @ (push @ X6 @ X7)))) = ((X8 @ X9))) <=> (pushprop_lem0_lthm = $true)) & (hoasinduction_lem3v2_gthm = $true) & (hoasinduction_lem3_gthm = $true) & (hoaslaminj_gthm = $true) & (hoasinduction_lem2_gthm = $true) & (! [X243 : (subst > term > subst > $o),X242 : term,X244 : (term > $o)] : (? [X259 : term] : (($true = ((X244 @ X259))) & (((X244 @ (sub @ X242 @ (push @ X259 @ id)))) != $true)) | ? [X249 : term] : (((X244 @ X249)) != ((X243 @ id @ X249 @ id))) | ? [X248 : subst,X246 : subst,X247 : term,X245 : subst] : (($true = ((X243 @ X246 @ X247 @ (comp @ X248 @ X245)))) & ($true != ((X243 @ (comp @ X246 @ X248) @ (sub @ X247 @ X248) @ X245)))) | ? [X253 : term,X251 : subst,X252 : subst,X250 : subst] : (($true = ((X243 @ (comp @ X251 @ X252) @ (sub @ X253 @ X252) @ X250))) & (((X243 @ X251 @ X253 @ (comp @ X252 @ X250))) != $true)) | ($true = ((X244 @ (lam @ X242)))) | ? [X254 : (subst > term > term)] : (! [X255 : term] : (($true != ((X243 @ id @ X255 @ id))) | ($true = ((X243 @ id @ (X254 @ id @ X255) @ id)))) & ! [X256 : subst,X257 : subst,X258 : term] : (((sub @ (X254 @ X257 @ X258) @ X256)) = ((X254 @ (comp @ X257 @ X256) @ (sub @ X258 @ X256)))) & ($true != ((X243 @ id @ (lam @ (X254 @ sh @ one)) @ id))))) <=> (hoasinduction_lem3v2 = $true)) & ((axshiftcons = $true) <=> ! [X112 : term,X113 : subst] : (((comp @ sh @ (push @ X112 @ X113))) = X113)) & ((hoasinduction_lem3_lthm = $true) <=> (($true = sP0) | ? [X498 : term] : (((sub @ X498 @ id)) != X498) | ? [X499 : (subst > term > subst > $o),X500 : term] : (! [X501 : term] : (($true != ((X499 @ id @ X501 @ id))) | ($true = ((X499 @ id @ (sub @ X500 @ (push @ X501 @ id)) @ id)))) & ($true != ((X499 @ id @ (lam @ (sub @ X500 @ (push @ one @ sh))) @ id))) & ! [X502 : (subst > term > term)] : (? [X505 : subst,X506 : term,X504 : subst] : (((sub @ (X502 @ X504 @ X506) @ X505)) != ((X502 @ (comp @ X504 @ X505) @ (sub @ X506 @ X505)))) | ? [X503 : term] : (($true != ((X499 @ id @ (X502 @ id @ X503) @ id))) & ($true = ((X499 @ id @ X503 @ id)))) | ($true = ((X499 @ id @ (lam @ (X502 @ sh @ one)) @ id)))))))),
  inference(definition_folding,[],[f12,f24,f23,f22,f21,f20,f19,f18,f17,f16,f15,f14])).
thf(f26,plain,(
  ((lamnotvar != $true) | ? [X0 : term] : (((var @ (lam @ X0))) = $true)) & ((lamnotvar = $true) | ! [X0 : term] : (((var @ (lam @ X0))) != $true))),
  inference(nnf_transformation,[],[f13])).
thf(f27,plain,(
  ((lamnotvar != $true) | ? [X0 : term] : (((var @ (lam @ X0))) = $true)) & ((lamnotvar = $true) | ! [X1 : term] : ($true != ((var @ (lam @ X1)))))),
  inference(rectify,[],[f26])).
thf(f28,plain,(
  ((lamnotvar != $true) | ($true = ((var @ (lam @ sK11))))) & ((lamnotvar = $true) | ! [X1 : term] : ($true != ((var @ (lam @ X1)))))),
  inference(skolemize,[status(esa),new_symbols(skolem,[sK11]),skolemize(X0,$thf($thf(sK11)))],[f27])).
thf(f65,plain,(
  (hoaslamnotap_lthm = $true) & (axabs = $true) & ((! [X293 : (subst > term > subst > $o),X292 : term] : (($true = ((X293 @ id @ (lam @ (sub @ X292 @ (push @ one @ sh))) @ id))) | ? [X294 : term] : ((((X293 @ id @ X294 @ id)) = $true) & ($true != ((X293 @ id @ (sub @ X292 @ (push @ X294 @ id)) @ id)))) | ? [X295 : (subst > term > term)] : (($true != ((X293 @ id @ (lam @ (X295 @ sh @ one)) @ id))) & ! [X296 : subst,X297 : term,X298 : subst] : (((sub @ (X295 @ X298 @ X297) @ X296)) = ((X295 @ (comp @ X298 @ X296) @ (sub @ X297 @ X296)))) & ! [X299 : term] : ((((X293 @ id @ X299 @ id)) != $true) | ($true = ((X293 @ id @ (X295 @ id @ X299) @ id)))))) | (hoasinduction_lem3aa != $true)) & ((hoasinduction_lem3aa = $true) | ? [X293 : (subst > term > subst > $o),X292 : term] : (($true != ((X293 @ id @ (lam @ (sub @ X292 @ (push @ one @ sh))) @ id))) & ! [X294 : term] : ((((X293 @ id @ X294 @ id)) != $true) | ($true = ((X293 @ id @ (sub @ X292 @ (push @ X294 @ id)) @ id)))) & ! [X295 : (subst > term > term)] : (($true = ((X293 @ id @ (lam @ (X295 @ sh @ one)) @ id))) | ? [X296 : subst,X297 : term,X298 : subst] : (((sub @ (X295 @ X298 @ X297) @ X296)) != ((X295 @ (comp @ X298 @ X296) @ (sub @ X297 @ X296)))) | ? [X299 : term] : ((((X293 @ id @ X299 @ id)) = $true) & ($true != ((X293 @ id @ (X295 @ id @ X299) @ id)))))))) & (pushprop_lem2v2_gthm = $true) & (((induction2lem = $true) | ? [X217 : term,X215 : subst,X216 : (term > $o)] : (! [X221 : term] : (($true = ((X216 @ (lam @ X221)))) | ? [X222 : term] : (($true != ((X216 @ (sub @ X221 @ (push @ X222 @ id))))) & ($true = ((X216 @ X222))))) & ! [X219 : term,X220 : term] : (($true != ((X216 @ X220))) | (((X216 @ (ap @ X219 @ X220))) = $true) | ($true != ((X216 @ X219)))) & ! [X218 : term] : (($true = ((X216 @ (sub @ X218 @ X215)))) | ($true != ((var @ X218)))) & (((X216 @ (sub @ X217 @ X215))) != $true))) & (! [X217 : term,X215 : subst,X216 : (term > $o)] : (? [X221 : term] : (($true != ((X216 @ (lam @ X221)))) & ! [X222 : term] : (($true = ((X216 @ (sub @ X221 @ (push @ X222 @ id))))) | ($true != ((X216 @ X222))))) | ? [X219 : term,X220 : term] : (($true = ((X216 @ X220))) & (((X216 @ (ap @ X219 @ X220))) != $true) & ($true = ((X216 @ X219)))) | ? [X218 : term] : (($true != ((X216 @ (sub @ X218 @ X215)))) & ($true = ((var @ X218)))) | (((X216 @ (sub @ X217 @ X215))) = $true)) | (induction2lem != $true))) & (lamnotap != $true) & (((hoasinduction_lem3v2a_lthm = $true) | (? [X20 : (subst > term > subst > $o),X19 : (term > $o),X18 : term] : (($true != ((X19 @ (lam @ X18)))) & ! [X21 : term] : (((X19 @ X21)) = ((X20 @ id @ X21 @ id))) & ! [X23 : (subst > term > term)] : (($true = ((X20 @ id @ (lam @ (X23 @ sh @ one)) @ id))) | ? [X24 : term] : ((((X20 @ id @ (X23 @ id @ X24) @ id)) != $true) & ($true = ((X20 @ id @ X24 @ id)))) | ? [X25 : subst,X26 : term,X27 : subst] : (((sub @ (X23 @ X25 @ X26) @ X27)) != ((X23 @ (comp @ X25 @ X27) @ (sub @ X26 @ X27))))) & ! [X22 : term] : (($true = ((X19 @ (sub @ X18 @ (push @ X22 @ id))))) | (((X19 @ X22)) != $true))) & ! [X13 : term] : ? [X14 : (subst > term > term)] : ! [X15 : term,X16 : subst] : (((sub @ X13 @ (push @ X15 @ X16))) = ((X14 @ X16 @ X15))) & ! [X17 : term] : (((sub @ X17 @ id)) = X17))) & ((! [X20 : (subst > term > subst > $o),X19 : (term > $o),X18 : term] : (($true = ((X19 @ (lam @ X18)))) | ? [X21 : term] : (((X19 @ X21)) != ((X20 @ id @ X21 @ id))) | ? [X23 : (subst > term > term)] : (($true != ((X20 @ id @ (lam @ (X23 @ sh @ one)) @ id))) & ! [X24 : term] : ((((X20 @ id @ (X23 @ id @ X24) @ id)) = $true) | ($true != ((X20 @ id @ X24 @ id)))) & ! [X25 : subst,X26 : term,X27 : subst] : (((sub @ (X23 @ X25 @ X26) @ X27)) = ((X23 @ (comp @ X25 @ X27) @ (sub @ X26 @ X27))))) | ? [X22 : term] : (($true != ((X19 @ (sub @ X18 @ (push @ X22 @ id))))) & (((X19 @ X22)) = $true))) | ? [X13 : term] : ! [X14 : (subst > term > term)] : ? [X15 : term,X16 : subst] : (((sub @ X13 @ (push @ X15 @ X16))) != ((X14 @ X16 @ X15))) | ? [X17 : term] : (((sub @ X17 @ id)) != X17)) | (hoasinduction_lem3v2a_lthm != $true))) & ((! [X533 : term,X532 : term,X531 : term,X534 : term] : ((X532 = X533) | (((ap @ (sub @ X534 @ id) @ X533)) != ((ap @ (sub @ X531 @ id) @ X532)))) | (hoasapinj2 != $true)) & ((hoasapinj2 = $true) | ? [X533 : term,X532 : term,X531 : term,X534 : term] : ((X532 != X533) & (((ap @ (sub @ X534 @ id) @ X533)) = ((ap @ (sub @ X531 @ id) @ X532)))))) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem1_gthm = $true) & ((! [X260 : term,X261 : (subst > term > subst > $o)] : (? [X262 : (subst > term > term)] : (($true != ((X261 @ id @ (lam @ (X262 @ sh @ one)) @ id))) & ! [X264 : term,X265 : subst,X263 : subst] : (((X262 @ (comp @ X265 @ X263) @ (sub @ X264 @ X263))) = ((sub @ (X262 @ X265 @ X264) @ X263))) & ! [X266 : term] : ((((X261 @ id @ (X262 @ id @ X266) @ id)) = $true) | ($true != ((X261 @ id @ X266 @ id))))) | ? [X267 : term] : ((((X261 @ id @ (sub @ X260 @ (push @ X267 @ id)) @ id)) != $true) & (((X261 @ id @ X267 @ id)) = $true)) | ($true = ((X261 @ id @ (lam @ (sub @ X260 @ (push @ one @ sh))) @ id)))) | (hoasinduction_lem3aa_lthm != $true)) & ((hoasinduction_lem3aa_lthm = $true) | ? [X260 : term,X261 : (subst > term > subst > $o)] : (! [X262 : (subst > term > term)] : (($true = ((X261 @ id @ (lam @ (X262 @ sh @ one)) @ id))) | ? [X264 : term,X265 : subst,X263 : subst] : (((X262 @ (comp @ X265 @ X263) @ (sub @ X264 @ X263))) != ((sub @ (X262 @ X265 @ X264) @ X263))) | ? [X266 : term] : ((((X261 @ id @ (X262 @ id @ X266) @ id)) != $true) & ($true = ((X261 @ id @ X266 @ id))))) & ! [X267 : term] : ((((X261 @ id @ (sub @ X260 @ (push @ X267 @ id)) @ id)) = $true) | (((X261 @ id @ X267 @ id)) != $true)) & ($true != ((X261 @ id @ (lam @ (sub @ X260 @ (push @ one @ sh))) @ id)))))) & ((! [X535 : subst] : (((comp @ id @ X535)) = X535) | (axidl != $true)) & ((axidl = $true) | ? [X535 : subst] : (((comp @ id @ X535)) != X535))) & (((hoaslamnotap = $true) | ? [X94 : term,X92 : term,X93 : (subst > term > term)] : (! [X95 : subst,X97 : subst,X96 : term] : (((X93 @ (comp @ X95 @ X97) @ (sub @ X96 @ X97))) = ((sub @ (X93 @ X95 @ X96) @ X97))) & (((lam @ (X93 @ sh @ one))) = ((ap @ (sub @ X94 @ id) @ X92))))) & (! [X94 : term,X92 : term,X93 : (subst > term > term)] : (? [X95 : subst,X97 : subst,X96 : term] : (((X93 @ (comp @ X95 @ X97) @ (sub @ X96 @ X97))) != ((sub @ (X93 @ X95 @ X96) @ X97))) | (((lam @ (X93 @ sh @ one))) != ((ap @ (sub @ X94 @ id) @ X92)))) | (hoaslamnotap != $true))) & ((! [X429 : term,X428 : term] : ((((sub @ X428 @ sh)) != ((sub @ X429 @ sh))) | (X428 = X429)) | (shinj != $true)) & ((shinj = $true) | ? [X429 : term,X428 : term] : ((((sub @ X428 @ sh)) = ((sub @ X429 @ sh))) & (X428 != X429)))) & ((! [X193 : term,X194 : term,X195 : (subst > term > subst > $o)] : (($true != ((X195 @ id @ X194 @ id))) | ($true != ((X195 @ id @ X193 @ id))) | (((X195 @ id @ (ap @ X193 @ X194) @ id)) = $true) | ? [X199 : subst,X200 : subst,X198 : term,X201 : subst] : (($true = ((X195 @ (comp @ X201 @ X200) @ (sub @ X198 @ X200) @ X199))) & ($true != ((X195 @ X201 @ X198 @ (comp @ X200 @ X199))))) | ? [X204 : subst,X203 : subst,X202 : term,X205 : subst] : (($true != ((X195 @ (comp @ X204 @ X205) @ (sub @ X202 @ X205) @ X203))) & ($true = ((X195 @ X204 @ X202 @ (comp @ X205 @ X203))))) | ? [X197 : term,X196 : term] : (($true != ((X195 @ id @ (ap @ (sub @ X196 @ id) @ X197) @ id))) & (((X195 @ id @ X196 @ id)) = $true) & ($true = ((X195 @ id @ X197 @ id))))) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | ? [X193 : term,X194 : term,X195 : (subst > term > subst > $o)] : (($true = ((X195 @ id @ X194 @ id))) & ($true = ((X195 @ id @ X193 @ id))) & (((X195 @ id @ (ap @ X193 @ X194) @ id)) != $true) & ! [X199 : subst,X200 : subst,X198 : term,X201 : subst] : (($true != ((X195 @ (comp @ X201 @ X200) @ (sub @ X198 @ X200) @ X199))) | ($true = ((X195 @ X201 @ X198 @ (comp @ X200 @ X199))))) & ! [X204 : subst,X203 : subst,X202 : term,X205 : subst] : (($true = ((X195 @ (comp @ X204 @ X205) @ (sub @ X202 @ X205) @ X203))) | ($true != ((X195 @ X204 @ X202 @ (comp @ X205 @ X203))))) & ! [X197 : term,X196 : term] : (($true = ((X195 @ id @ (ap @ (sub @ X196 @ id) @ X197) @ id))) | (((X195 @ id @ X196 @ id)) != $true) | ($true != ((X195 @ id @ X197 @ id))))))) & (pushprop_lem3v2 = $true) & ! [X309 : d_subst,X310 : d_subst] : ((((d2subst @ X310)) != ((d2subst @ X309))) | (X309 = X310)) & ((! [X30 : term,X31 : term,X29 : term,X28 : term] : ((((ap @ (sub @ X30 @ id) @ X29)) != ((ap @ (sub @ X28 @ id) @ X31))) | (X28 = X30)) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | ? [X30 : term,X31 : term,X29 : term,X28 : term] : ((((ap @ (sub @ X30 @ id) @ X29)) = ((ap @ (sub @ X28 @ id) @ X31))) & (X28 != X30)))) & (((axvarcons = $true) | ? [X64 : subst,X63 : term] : (((sub @ one @ (push @ X63 @ X64))) != X63)) & (! [X64 : subst,X63 : term] : (((sub @ one @ (push @ X63 @ X64))) = X63) | (axvarcons != $true))) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & (((hoasinduction_lthm = $true) | ($true != sP10)) & (($true = sP10) | (hoasinduction_lthm != $true))) & ! [X51 : d_term,X52 : d_term] : ((((d2term @ X51)) != ((d2term @ X52))) | (X51 = X52)) & (hoaslamnotvar_gthm = $true) & (hoaslamnotvar = $true) & (((hoasinduction = $true) | ? [X393 : term,X394 : (subst > term > subst > $o)] : (! [X400 : term,X399 : term] : ((((X394 @ id @ (ap @ (sub @ X399 @ id) @ X400) @ id)) = $true) | ($true != ((X394 @ id @ X399 @ id))) | (((X394 @ id @ X400 @ id)) != $true)) & ! [X397 : subst,X395 : subst,X396 : term,X398 : subst] : (($true = ((X394 @ (comp @ X395 @ X397) @ (sub @ X396 @ X397) @ X398))) | ($true != ((X394 @ X395 @ X396 @ (comp @ X397 @ X398))))) & ! [X401 : term] : ((((var @ (sub @ X401 @ id))) != $true) | ($true = ((X394 @ id @ X401 @ id)))) & ($true != ((X394 @ id @ X393 @ id))) & ($true != ((sP6 @ X394))) & ! [X402 : subst,X404 : subst,X405 : subst,X403 : term] : (($true != ((X394 @ (comp @ X404 @ X402) @ (sub @ X403 @ X402) @ X405))) | ($true = ((X394 @ X404 @ X403 @ (comp @ X402 @ X405))))))) & (! [X393 : term,X394 : (subst > term > subst > $o)] : (? [X400 : term,X399 : term] : ((((X394 @ id @ (ap @ (sub @ X399 @ id) @ X400) @ id)) != $true) & ($true = ((X394 @ id @ X399 @ id))) & (((X394 @ id @ X400 @ id)) = $true)) | ? [X397 : subst,X395 : subst,X396 : term,X398 : subst] : (($true != ((X394 @ (comp @ X395 @ X397) @ (sub @ X396 @ X397) @ X398))) & ($true = ((X394 @ X395 @ X396 @ (comp @ X397 @ X398))))) | ? [X401 : term] : ((((var @ (sub @ X401 @ id))) = $true) & ($true != ((X394 @ id @ X401 @ id)))) | ($true = ((X394 @ id @ X393 @ id))) | ($true = ((sP6 @ X394))) | ? [X402 : subst,X404 : subst,X405 : subst,X403 : term] : (($true = ((X394 @ (comp @ X404 @ X402) @ (sub @ X403 @ X402) @ X405))) & ($true != ((X394 @ X404 @ X403 @ (comp @ X402 @ X405)))))) | (hoasinduction != $true))) & (pushprop_lem3v2_lthm = $true) & ((! [X183 : subst] : (((push @ (sub @ one @ X183) @ (comp @ sh @ X183))) = X183) | (axscons != $true)) & ((axscons = $true) | ? [X183 : subst] : (((push @ (sub @ one @ X183) @ (comp @ sh @ X183))) != X183))) & (hoasinduction_lem3b_gthm = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ! [X111 : d_term] : (d_one = X111) & (((hoasapinj2_lthm = $true) | (! [X524 : term,X525 : term,X523 : term,X526 : term] : ((X523 = X524) | (((ap @ X526 @ X523)) != ((ap @ X525 @ X524)))) & ? [X529 : term,X528 : term,X527 : term,X530 : term] : ((X527 != X530) & (((ap @ (sub @ X529 @ id) @ X527)) = ((ap @ (sub @ X528 @ id) @ X530)))))) & ((? [X524 : term,X525 : term,X523 : term,X526 : term] : ((X523 != X524) & (((ap @ X526 @ X523)) = ((ap @ X525 @ X524)))) | ! [X529 : term,X528 : term,X527 : term,X530 : term] : ((X527 = X530) | (((ap @ (sub @ X529 @ id) @ X527)) != ((ap @ (sub @ X528 @ id) @ X530))))) | (hoasapinj2_lthm != $true))) & (axvarshift = $true) & ((! [X538 : term] : ? [X539 : (subst > term > term)] : (((X539 @ sh @ one)) = ((sub @ X538 @ (push @ one @ sh)))) | (hoasinduction_lem3b != $true)) & ((hoasinduction_lem3b = $true) | ? [X538 : term] : ! [X539 : (subst > term > term)] : (((X539 @ sh @ one)) != ((sub @ X538 @ (push @ one @ sh)))))) & (((hoasinduction_lem3a_lthm = $true) | (! [X413 : (subst > term > subst > $o),X412 : term] : (? [X414 : (subst > term > term)] : (! [X418 : term,X417 : subst,X416 : subst] : (((X414 @ (comp @ X417 @ X416) @ (sub @ X418 @ X416))) = ((sub @ (X414 @ X417 @ X418) @ X416))) & ! [X415 : term] : ((((X413 @ id @ X415 @ id)) != $true) | (((X413 @ id @ (X414 @ id @ X415) @ id)) = $true)) & ($true != ((X413 @ id @ (lam @ (X414 @ sh @ one)) @ id)))) | ? [X419 : term] : (($true = ((X413 @ id @ X419 @ id))) & ($true != ((X413 @ id @ (sub @ X412 @ (push @ X419 @ id)) @ id)))) | (((X413 @ id @ (lam @ (sub @ X412 @ (push @ one @ sh))) @ id)) = $true)) & ! [X411 : term] : (((sub @ X411 @ id)) = X411) & ? [X421 : (subst > term > subst > $o),X420 : term] : (! [X422 : (subst > term > term)] : (? [X423 : subst,X424 : subst,X425 : term] : (((sub @ (X422 @ X423 @ X425) @ X424)) != ((X422 @ (comp @ X423 @ X424) @ (sub @ X425 @ X424)))) | ? [X426 : term] : (($true != ((X421 @ id @ (X422 @ id @ X426) @ id))) & ($true = ((X421 @ id @ X426 @ id)))) | (((X421 @ id @ (lam @ (X422 @ sh @ one)) @ id)) = $true)) & ! [X427 : term] : ((((X421 @ id @ (sub @ X420 @ (push @ X427 @ id)) @ id)) = $true) | (((X421 @ id @ X427 @ id)) != $true)) & ($true != ((X421 @ id @ (lam @ X420) @ id)))))) & ((? [X413 : (subst > term > subst > $o),X412 : term] : (! [X414 : (subst > term > term)] : (? [X418 : term,X417 : subst,X416 : subst] : (((X414 @ (comp @ X417 @ X416) @ (sub @ X418 @ X416))) != ((sub @ (X414 @ X417 @ X418) @ X416))) | ? [X415 : term] : ((((X413 @ id @ X415 @ id)) = $true) & (((X413 @ id @ (X414 @ id @ X415) @ id)) != $true)) | ($true = ((X413 @ id @ (lam @ (X414 @ sh @ one)) @ id)))) & ! [X419 : term] : (($true != ((X413 @ id @ X419 @ id))) | ($true = ((X413 @ id @ (sub @ X412 @ (push @ X419 @ id)) @ id)))) & (((X413 @ id @ (lam @ (sub @ X412 @ (push @ one @ sh))) @ id)) != $true)) | ? [X411 : term] : (((sub @ X411 @ id)) != X411) | ! [X421 : (subst > term > subst > $o),X420 : term] : (? [X422 : (subst > term > term)] : (! [X423 : subst,X424 : subst,X425 : term] : (((sub @ (X422 @ X423 @ X425) @ X424)) = ((X422 @ (comp @ X423 @ X424) @ (sub @ X425 @ X424)))) & ! [X426 : term] : (($true = ((X421 @ id @ (X422 @ id @ X426) @ id))) | ($true != ((X421 @ id @ X426 @ id)))) & (((X421 @ id @ (lam @ (X422 @ sh @ one)) @ id)) != $true)) | ? [X427 : term] : ((((X421 @ id @ (sub @ X420 @ (push @ X427 @ id)) @ id)) != $true) & (((X421 @ id @ X427 @ id)) = $true)) | ($true = ((X421 @ id @ (lam @ X420) @ id))))) | (hoasinduction_lem3a_lthm != $true))) & (induction2lem_gthm = $true) & ! [X48 : d_subst] : (d_id = X48) & (pushprop_lem2v2_lthm = $true) & (ulamvar1 != $true) & ! [X536 : subst] : ? [X537 : d_subst] : (((d2subst @ X537)) = X536) & (pushprop_lthm = $true) & (hoasinduction_lem1v2 = $true) & (hoasinduction_lem1v2_gthm = $true) & (((hoasinduction_lem3v2a = $true) | ? [X227 : (term > $o),X228 : term,X229 : (subst > term > subst > $o)] : (! [X235 : term] : (((X229 @ id @ X235 @ id)) = ((X227 @ X235))) & ($true != ((X227 @ (lam @ X228)))) & ! [X236 : term] : (($true = ((X227 @ (sub @ X228 @ (push @ X236 @ id))))) | ($true != ((X227 @ X236)))) & ! [X230 : (subst > term > term)] : (? [X232 : term,X231 : subst,X233 : subst] : (((X230 @ (comp @ X231 @ X233) @ (sub @ X232 @ X233))) != ((sub @ (X230 @ X231 @ X232) @ X233))) | ? [X234 : term] : ((((X229 @ id @ (X230 @ id @ X234) @ id)) != $true) & ($true = ((X229 @ id @ X234 @ id)))) | (((X229 @ id @ (lam @ (X230 @ sh @ one)) @ id)) = $true)))) & (! [X227 : (term > $o),X228 : term,X229 : (subst > term > subst > $o)] : (? [X235 : term] : (((X229 @ id @ X235 @ id)) != ((X227 @ X235))) | ($true = ((X227 @ (lam @ X228)))) | ? [X236 : term] : (($true != ((X227 @ (sub @ X228 @ (push @ X236 @ id))))) & ($true = ((X227 @ X236)))) | ? [X230 : (subst > term > term)] : (! [X232 : term,X231 : subst,X233 : subst] : (((X230 @ (comp @ X231 @ X233) @ (sub @ X232 @ X233))) = ((sub @ (X230 @ X231 @ X232) @ X233))) & ! [X234 : term] : ((((X229 @ id @ (X230 @ id @ X234) @ id)) = $true) | ($true != ((X229 @ id @ X234 @ id)))) & (((X229 @ id @ (lam @ (X230 @ sh @ one)) @ id)) != $true))) | (hoasinduction_lem3v2a != $true))) & (pushprop_lthm_orig = $true) & (pushprop_lem1v2_gthm = $true) & (pushprop_lem2v2 = $true) & ! [X61 : term] : ? [X62 : d_term] : (((d2term @ X62)) = X61) & (((hoasinduction_no_psi_cond = $true) | ? [X206 : (subst > term > subst > $o),X207 : term] : (($true != ((X206 @ id @ X207 @ id))) & ! [X208 : (subst > term > term)] : (($true = ((X206 @ id @ (lam @ (X208 @ sh @ one)) @ id))) | ? [X211 : subst,X210 : subst,X209 : term] : (((sub @ (X208 @ X211 @ X209) @ X210)) != ((X208 @ (comp @ X211 @ X210) @ (sub @ X209 @ X210)))) | ? [X212 : term] : (($true != ((X206 @ id @ (X208 @ id @ X212) @ id))) & ($true = ((X206 @ id @ X212 @ id))))) & ! [X214 : term,X213 : term] : ((((X206 @ id @ X214 @ id)) != $true) | ($true != ((X206 @ id @ X213 @ id))) | ($true = ((X206 @ id @ (ap @ (sub @ X213 @ id) @ X214) @ id)))))) & (! [X206 : (subst > term > subst > $o),X207 : term] : (($true = ((X206 @ id @ X207 @ id))) | ? [X208 : (subst > term > term)] : (($true != ((X206 @ id @ (lam @ (X208 @ sh @ one)) @ id))) & ! [X211 : subst,X210 : subst,X209 : term] : (((sub @ (X208 @ X211 @ X209) @ X210)) = ((X208 @ (comp @ X211 @ X210) @ (sub @ X209 @ X210)))) & ! [X212 : term] : (($true = ((X206 @ id @ (X208 @ id @ X212) @ id))) | ($true != ((X206 @ id @ X212 @ id))))) | ? [X214 : term,X213 : term] : ((((X206 @ id @ X214 @ id)) = $true) & ($true = ((X206 @ id @ X213 @ id))) & ($true != ((X206 @ id @ (ap @ (sub @ X213 @ id) @ X214) @ id))))) | (hoasinduction_no_psi_cond != $true))) & ((! [X115 : term,X114 : (subst > term > subst > $o)] : (($true = ((X114 @ id @ (lam @ (sub @ X115 @ (push @ one @ sh))) @ id))) | ? [X118 : term,X117 : (subst > term > term)] : (! [X121 : subst,X119 : subst,X120 : term] : (((X117 @ (comp @ X119 @ X121) @ (sub @ X120 @ X121))) = ((sub @ X118 @ (push @ (sub @ X120 @ X121) @ (comp @ X119 @ X121))))) & ! [X124 : subst,X123 : term,X122 : subst] : (((sub @ (sub @ X118 @ (push @ X123 @ X122)) @ X124)) = ((sub @ (X117 @ X122 @ X123) @ X124))) & (((X114 @ id @ (lam @ (X117 @ sh @ one)) @ id)) != $true) & ! [X125 : term] : (($true = ((X114 @ id @ (X117 @ id @ X125) @ id))) | ($true != ((X114 @ id @ X125 @ id))))) | ? [X116 : term] : (($true != ((X114 @ id @ (sub @ X115 @ (push @ X116 @ id)) @ id))) & ($true = ((X114 @ id @ X116 @ id))))) | (hoasinduction_lem3aaa != $true)) & ((hoasinduction_lem3aaa = $true) | ? [X115 : term,X114 : (subst > term > subst > $o)] : (($true != ((X114 @ id @ (lam @ (sub @ X115 @ (push @ one @ sh))) @ id))) & ! [X118 : term,X117 : (subst > term > term)] : (? [X121 : subst,X119 : subst,X120 : term] : (((X117 @ (comp @ X119 @ X121) @ (sub @ X120 @ X121))) != ((sub @ X118 @ (push @ (sub @ X120 @ X121) @ (comp @ X119 @ X121))))) | ? [X124 : subst,X123 : term,X122 : subst] : (((sub @ (sub @ X118 @ (push @ X123 @ X122)) @ X124)) != ((sub @ (X117 @ X122 @ X123) @ X124))) | (((X114 @ id @ (lam @ (X117 @ sh @ one)) @ id)) = $true) | ? [X125 : term] : (($true != ((X114 @ id @ (X117 @ id @ X125) @ id))) & ($true = ((X114 @ id @ X125 @ id))))) & ! [X116 : term] : (($true = ((X114 @ id @ (sub @ X115 @ (push @ X116 @ id)) @ id))) | ($true != ((X114 @ id @ X116 @ id))))))) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (pushprop_lem0_gthm = $true) & (apnotvar = $true) & (hoasapnotvar = $true) & (hoasinduction_lem1_lthm = $true) & (((hoaslaminj = $true) | ? [X372 : subst,X370 : (subst > term > term),X369 : term,X371 : (subst > term > term)] : ((((lam @ (X370 @ sh @ one))) = ((lam @ (X371 @ sh @ one)))) & ! [X376 : subst,X378 : term,X377 : subst] : (((sub @ (X371 @ X376 @ X378) @ X377)) = ((X371 @ (comp @ X376 @ X377) @ (sub @ X378 @ X377)))) & (((X371 @ X372 @ X369)) != ((X370 @ X372 @ X369))) & ! [X375 : subst,X374 : subst,X373 : term] : (((sub @ (X370 @ X375 @ X373) @ X374)) = ((X370 @ (comp @ X375 @ X374) @ (sub @ X373 @ X374)))))) & (! [X372 : subst,X370 : (subst > term > term),X369 : term,X371 : (subst > term > term)] : ((((lam @ (X370 @ sh @ one))) != ((lam @ (X371 @ sh @ one)))) | ? [X376 : subst,X378 : term,X377 : subst] : (((sub @ (X371 @ X376 @ X378) @ X377)) != ((X371 @ (comp @ X376 @ X377) @ (sub @ X378 @ X377)))) | (((X371 @ X372 @ X369)) = ((X370 @ X372 @ X369))) | ? [X375 : subst,X374 : subst,X373 : term] : (((sub @ (X370 @ X375 @ X373) @ X374)) != ((X370 @ (comp @ X375 @ X374) @ (sub @ X373 @ X374))))) | (hoaslaminj != $true))) & (induction2_gthm = $true) & ((! [X71 : term] : (((sub @ X71 @ id)) = X71) | (termmset != $true)) & ((termmset = $true) | ? [X71 : term] : (((sub @ X71 @ id)) != X71))) & (((hoasinduction_lem0_lthm = $true) | ? [X10 : (subst > term > subst > $o)] : ! [X11 : (term > $o)] : ? [X12 : term] : (((X11 @ X12)) != ((X10 @ id @ X12 @ id)))) & (! [X10 : (subst > term > subst > $o)] : ? [X11 : (term > $o)] : ! [X12 : term] : (((X11 @ X12)) = ((X10 @ id @ X12 @ id))) | (hoasinduction_lem0_lthm != $true))) & (hoaslamnotap_gthm = $true) & (((d2subst @ d_id)) = sh) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (hoasinduction_lem2v2_gthm = $true) & (((hoaslaminj_lthm = $true) | (! [X35 : subst,X34 : term] : (((comp @ sh @ (push @ X34 @ X35))) = X35) & ! [X32 : subst,X33 : term] : (((sub @ one @ (push @ X33 @ X32))) = X33) & ? [X39 : (subst > term > term),X40 : (subst > term > term),X41 : term,X38 : subst] : (! [X46 : term,X47 : subst,X45 : subst] : (((sub @ (X40 @ X45 @ X46) @ X47)) = ((X40 @ (comp @ X45 @ X47) @ (sub @ X46 @ X47)))) & (((lam @ (X39 @ sh @ one))) = ((lam @ (X40 @ sh @ one)))) & ! [X42 : subst,X44 : term,X43 : subst] : (((sub @ (X39 @ X43 @ X44) @ X42)) = ((X39 @ (comp @ X43 @ X42) @ (sub @ X44 @ X42)))) & (((X40 @ X38 @ X41)) != ((X39 @ X38 @ X41)))) & ! [X36 : term,X37 : term] : ((X36 = X37) | (((lam @ X37)) != ((lam @ X36)))))) & ((? [X35 : subst,X34 : term] : (((comp @ sh @ (push @ X34 @ X35))) != X35) | ? [X32 : subst,X33 : term] : (((sub @ one @ (push @ X33 @ X32))) != X33) | ! [X39 : (subst > term > term),X40 : (subst > term > term),X41 : term,X38 : subst] : (? [X46 : term,X47 : subst,X45 : subst] : (((sub @ (X40 @ X45 @ X46) @ X47)) != ((X40 @ (comp @ X45 @ X47) @ (sub @ X46 @ X47)))) | (((lam @ (X39 @ sh @ one))) != ((lam @ (X40 @ sh @ one)))) | ? [X42 : subst,X44 : term,X43 : subst] : (((sub @ (X39 @ X43 @ X44) @ X42)) != ((X39 @ (comp @ X43 @ X42) @ (sub @ X44 @ X42)))) | (((X40 @ X38 @ X41)) = ((X39 @ X38 @ X41)))) | ? [X36 : term,X37 : term] : ((X36 != X37) & (((lam @ X37)) = ((lam @ X36))))) | (hoaslaminj_lthm != $true))) & ((! [X73 : term,X72 : (subst > term > subst > $o)] : (? [X74 : term] : (($true != ((X72 @ id @ (sub @ X73 @ (push @ X74 @ id)) @ id))) & ($true = ((X72 @ id @ X74 @ id)))) | ? [X85 : subst,X86 : subst,X84 : subst,X87 : term] : (($true != ((X72 @ (comp @ X85 @ X84) @ (sub @ X87 @ X84) @ X86))) & ($true = ((X72 @ X85 @ X87 @ (comp @ X84 @ X86))))) | ? [X79 : (subst > term > term)] : (! [X83 : subst,X81 : term,X82 : subst] : (((X79 @ (comp @ X83 @ X82) @ (sub @ X81 @ X82))) = ((sub @ (X79 @ X83 @ X81) @ X82))) & ! [X80 : term] : ((((X72 @ id @ X80 @ id)) != $true) | ($true = ((X72 @ id @ (X79 @ id @ X80) @ id)))) & (((X72 @ id @ (lam @ (X79 @ sh @ one)) @ id)) != $true)) | ($true = ((X72 @ id @ (lam @ X73) @ id))) | ? [X76 : subst,X77 : term,X78 : subst,X75 : subst] : (($true = ((X72 @ (comp @ X75 @ X78) @ (sub @ X77 @ X78) @ X76))) & (((X72 @ X75 @ X77 @ (comp @ X78 @ X76))) != $true))) | (hoasinduction_lem3 != $true)) & ((hoasinduction_lem3 = $true) | ? [X73 : term,X72 : (subst > term > subst > $o)] : (! [X74 : term] : (($true = ((X72 @ id @ (sub @ X73 @ (push @ X74 @ id)) @ id))) | ($true != ((X72 @ id @ X74 @ id)))) & ! [X85 : subst,X86 : subst,X84 : subst,X87 : term] : (($true = ((X72 @ (comp @ X85 @ X84) @ (sub @ X87 @ X84) @ X86))) | ($true != ((X72 @ X85 @ X87 @ (comp @ X84 @ X86))))) & ! [X79 : (subst > term > term)] : (? [X83 : subst,X81 : term,X82 : subst] : (((X79 @ (comp @ X83 @ X82) @ (sub @ X81 @ X82))) != ((sub @ (X79 @ X83 @ X81) @ X82))) | ? [X80 : term] : ((((X72 @ id @ X80 @ id)) = $true) & ($true != ((X72 @ id @ (X79 @ id @ X80) @ id)))) | (((X72 @ id @ (lam @ (X79 @ sh @ one)) @ id)) = $true)) & ($true != ((X72 @ id @ (lam @ X73) @ id))) & ! [X76 : subst,X77 : term,X78 : subst,X75 : subst] : (($true != ((X72 @ (comp @ X75 @ X78) @ (sub @ X77 @ X78) @ X76))) | (((X72 @ X75 @ X77 @ (comp @ X78 @ X76))) = $true))))) & (termmset_gthm = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (hoasinduction_lem2_lthm = $true) & ((! [X99 : term,X98 : term] : ((((lam @ X98)) != ((lam @ X99))) | (X98 = X99)) | (laminj != $true)) & ((laminj = $true) | ? [X99 : term,X98 : term] : ((((lam @ X98)) = ((lam @ X99))) & (X98 != X99)))) & (((var @ (d2term @ d_one))) != $true) & ((! [X272 : subst] : (((comp @ X272 @ id)) = X272) | (axidr != $true)) & ((axidr = $true) | ? [X272 : subst] : (((comp @ X272 @ id)) != X272))) & ((! [X0 : term] : (((sub @ X0 @ id)) = X0) | (axvarid != $true)) & ((axvarid = $true) | ? [X0 : term] : (((sub @ X0 @ id)) != X0))) & (((? [X321 : term] : (((sub @ X321 @ id)) != X321) | ($true = sP5) | ? [X311 : (subst > term > subst > $o)] : ! [X312 : (term > $o)] : ? [X313 : term] : (((X312 @ X313)) != ((X311 @ id @ X313 @ id))) | ? [X322 : term,X323 : (term > $o),X324 : (subst > term > subst > $o)] : (($true != ((X323 @ (lam @ X322)))) & ! [X326 : (subst > term > term)] : (($true = ((X324 @ id @ (lam @ (X326 @ sh @ one)) @ id))) | ? [X330 : subst,X328 : subst,X329 : term] : (((X326 @ (comp @ X328 @ X330) @ (sub @ X329 @ X330))) != ((sub @ (X326 @ X328 @ X329) @ X330))) | ? [X327 : term] : ((((X324 @ id @ X327 @ id)) = $true) & ($true != ((X324 @ id @ (X326 @ id @ X327) @ id))))) & ! [X325 : term] : ((((X323 @ X325)) != $true) | ($true = ((X323 @ (sub @ X322 @ (push @ X325 @ id)))))) & ! [X331 : term] : (((X324 @ id @ X331 @ id)) = ((X323 @ X331)))) | ? [X314 : term,X315 : (term > $o)] : (! [X318 : term] : ((((X315 @ X318)) = $true) | (((var @ X318)) != $true)) & ($true != ((X315 @ X314))) & ! [X319 : term] : (? [X320 : term] : (($true = ((X315 @ X320))) & (((X315 @ (sub @ X319 @ (push @ X320 @ id)))) != $true)) | ($true = ((X315 @ (lam @ X319))))) & ! [X317 : term,X316 : term] : (($true != ((X315 @ X317))) | (((X315 @ (ap @ X317 @ X316))) = $true) | ($true != ((X315 @ X316)))))) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | (! [X321 : term] : (((sub @ X321 @ id)) = X321) & ($true != sP5) & ! [X311 : (subst > term > subst > $o)] : ? [X312 : (term > $o)] : ! [X313 : term] : (((X312 @ X313)) = ((X311 @ id @ X313 @ id))) & ! [X322 : term,X323 : (term > $o),X324 : (subst > term > subst > $o)] : (($true = ((X323 @ (lam @ X322)))) | ? [X326 : (subst > term > term)] : (($true != ((X324 @ id @ (lam @ (X326 @ sh @ one)) @ id))) & ! [X330 : subst,X328 : subst,X329 : term] : (((X326 @ (comp @ X328 @ X330) @ (sub @ X329 @ X330))) = ((sub @ (X326 @ X328 @ X329) @ X330))) & ! [X327 : term] : ((((X324 @ id @ X327 @ id)) != $true) | ($true = ((X324 @ id @ (X326 @ id @ X327) @ id))))) | ? [X325 : term] : ((((X323 @ X325)) = $true) & ($true != ((X323 @ (sub @ X322 @ (push @ X325 @ id)))))) | ? [X331 : term] : (((X324 @ id @ X331 @ id)) != ((X323 @ X331)))) & ! [X314 : term,X315 : (term > $o)] : (? [X318 : term] : ((((X315 @ X318)) != $true) & (((var @ X318)) = $true)) | ($true = ((X315 @ X314))) | ? [X319 : term] : (! [X320 : term] : (($true != ((X315 @ X320))) | (((X315 @ (sub @ X319 @ (push @ X320 @ id)))) = $true)) & ($true != ((X315 @ (lam @ X319))))) | ? [X317 : term,X316 : term] : (($true = ((X315 @ X317))) & (((X315 @ (ap @ X317 @ X316))) != $true) & ($true = ((X315 @ X316)))))))) & (substmonoid_gthm = $true) & (pushprop = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (axclos = $true) & (((hoasinduction_lem3a = $true) | ? [X54 : (subst > term > subst > $o),X53 : term] : (! [X55 : (subst > term > term)] : (? [X59 : term] : (($true != ((X54 @ id @ (X55 @ id @ X59) @ id))) & ($true = ((X54 @ id @ X59 @ id)))) | ($true = ((X54 @ id @ (lam @ (X55 @ sh @ one)) @ id))) | ? [X57 : subst,X56 : subst,X58 : term] : (((X55 @ (comp @ X57 @ X56) @ (sub @ X58 @ X56))) != ((sub @ (X55 @ X57 @ X58) @ X56)))) & ! [X60 : term] : (($true != ((X54 @ id @ X60 @ id))) | ($true = ((X54 @ id @ (sub @ X53 @ (push @ X60 @ id)) @ id)))) & ($true != ((X54 @ id @ (lam @ X53) @ id))))) & (! [X54 : (subst > term > subst > $o),X53 : term] : (? [X55 : (subst > term > term)] : (! [X59 : term] : (($true = ((X54 @ id @ (X55 @ id @ X59) @ id))) | ($true != ((X54 @ id @ X59 @ id)))) & ($true != ((X54 @ id @ (lam @ (X55 @ sh @ one)) @ id))) & ! [X57 : subst,X56 : subst,X58 : term] : (((X55 @ (comp @ X57 @ X56) @ (sub @ X58 @ X56))) = ((sub @ (X55 @ X57 @ X58) @ X56)))) | ? [X60 : term] : (($true = ((X54 @ id @ X60 @ id))) & ($true != ((X54 @ id @ (sub @ X53 @ (push @ X60 @ id)) @ id)))) | ($true = ((X54 @ id @ (lam @ X53) @ id)))) | (hoasinduction_lem3a != $true))) & (((? [X49 : term] : (((sub @ X49 @ id)) != X49) | ! [X50 : term] : (((sub @ X50 @ id)) = X50)) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | (! [X49 : term] : (((sub @ X49 @ id)) = X49) & ? [X50 : term] : (((sub @ X50 @ id)) != X50)))) & (ulamvarsh = $true) & (((hoasinduction_lem3v2_f = $true) | ? [X100 : term] : ! [X101 : (subst > term > term)] : ? [X102 : subst,X103 : term] : (((X101 @ X102 @ X103)) != ((sub @ X100 @ (push @ X103 @ X102))))) & (! [X100 : term] : ? [X101 : (subst > term > term)] : ! [X102 : subst,X103 : term] : (((X101 @ X102 @ X103)) = ((sub @ X100 @ (push @ X103 @ X102)))) | (hoasinduction_lem3v2_f != $true))) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & (((substmonoid = $true) | (? [X224 : subst] : (((comp @ X224 @ id)) != X224) | ? [X223 : subst] : (((comp @ id @ X223)) != X223))) & ((! [X224 : subst] : (((comp @ X224 @ id)) = X224) & ! [X223 : subst] : (((comp @ id @ X223)) = X223)) | (substmonoid != $true))) & (((hoasinduction_lem3b_lthm = $true) | ? [X225 : term] : ! [X226 : (subst > term > term)] : (((sub @ X225 @ (push @ one @ sh))) != ((X226 @ sh @ one)))) & (! [X225 : term] : ? [X226 : (subst > term > term)] : (((sub @ X225 @ (push @ one @ sh))) = ((X226 @ sh @ one))) | (hoasinduction_lem3b_lthm != $true))) & (((! [X108 : (term > $o),X106 : (term > $o),X109 : subst,X107 : term] : (? [X110 : term] : (((X106 @ X110)) != ((X108 @ (sub @ X110 @ (push @ X107 @ X109))))) | ($true != ((X108 @ X107))) | ($true = ((X106 @ one)))) | ? [X104 : subst,X105 : term] : (((sub @ one @ (push @ X105 @ X104))) != X105)) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (? [X108 : (term > $o),X106 : (term > $o),X109 : subst,X107 : term] : (! [X110 : term] : (((X106 @ X110)) = ((X108 @ (sub @ X110 @ (push @ X107 @ X109))))) & ($true = ((X108 @ X107))) & ($true != ((X106 @ one)))) & ! [X104 : subst,X105 : term] : (((sub @ one @ (push @ X105 @ X104))) = X105)))) & (axassoc = $true) & (((? [X88 : subst] : (((comp @ id @ X88)) != X88) | (! [X91 : subst] : (((comp @ id @ X91)) = X91) & ! [X90 : subst] : (((comp @ X90 @ id)) = X90)) | ? [X89 : subst] : (((comp @ X89 @ id)) != X89)) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | (! [X88 : subst] : (((comp @ id @ X88)) = X88) & (? [X91 : subst] : (((comp @ id @ X91)) != X91) | ? [X90 : subst] : (((comp @ X90 @ id)) != X90)) & ! [X89 : subst] : (((comp @ X89 @ id)) = X89)))) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (((? [X304 : term,X302 : term,X303 : term,X301 : term] : ((((ap @ X301 @ X303)) = ((ap @ X304 @ X302))) & (X301 != X304)) | ? [X300 : term] : (((sub @ X300 @ id)) != X300) | ! [X307 : term,X306 : term,X308 : term,X305 : term] : ((((ap @ (sub @ X305 @ id) @ X307)) != ((ap @ (sub @ X306 @ id) @ X308))) | (X305 = X306))) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | (! [X304 : term,X302 : term,X303 : term,X301 : term] : ((((ap @ X301 @ X303)) != ((ap @ X304 @ X302))) | (X301 = X304)) & ! [X300 : term] : (((sub @ X300 @ id)) = X300) & ? [X307 : term,X306 : term,X308 : term,X305 : term] : ((((ap @ (sub @ X305 @ id) @ X307)) = ((ap @ (sub @ X306 @ id) @ X308))) & (X305 != X306))))) & (hoasapinj1_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (hoasapnotvar_lthm = $true) & (one = ((d2term @ d_one))) & (axapp = $true) & (hoasinduction_lem1 = $true) & (((pushprop_lem0 = $true) | ? [X237 : subst,X238 : (term > $o),X239 : term] : ! [X240 : (term > $o)] : ? [X241 : term] : (((X238 @ (sub @ X241 @ (push @ X239 @ X237)))) != ((X240 @ X241)))) & (! [X237 : subst,X238 : (term > $o),X239 : term] : ? [X240 : (term > $o)] : ! [X241 : term] : (((X238 @ (sub @ X241 @ (push @ X239 @ X237)))) = ((X240 @ X241))) | (pushprop_lem0 != $true))) & (((induction2lem_lthm = $true) | ($true != sP3)) & (($true = sP3) | (induction2lem_lthm != $true))) & (ulamvarind = $true) & (axmap = $true) & (((hoasinduction_no_psi_cond_lthm = $true) | (! [X455 : term] : (((sub @ X455 @ id)) = X455) & ! [X456 : (term > $o),X457 : (subst > term > subst > $o),X458 : term] : (? [X459 : term] : (((X457 @ id @ X459 @ id)) != ((X456 @ X459))) | (((X456 @ (lam @ X458))) = $true) | ? [X460 : term] : ((((X456 @ (sub @ X458 @ (push @ X460 @ id)))) != $true) & (((X456 @ X460)) = $true)) | ? [X461 : (subst > term > term)] : (($true != ((X457 @ id @ (lam @ (X461 @ sh @ one)) @ id))) & ! [X462 : term] : (($true != ((X457 @ id @ X462 @ id))) | (((X457 @ id @ (X461 @ id @ X462) @ id)) = $true)) & ! [X463 : term,X465 : subst,X464 : subst] : (((sub @ (X461 @ X465 @ X463) @ X464)) = ((X461 @ (comp @ X465 @ X464) @ (sub @ X463 @ X464)))))) & ! [X449 : (term > $o),X448 : term] : (? [X451 : term,X450 : term] : ((((X449 @ X451)) = $true) & ($true != ((X449 @ (ap @ X450 @ X451)))) & ($true = ((X449 @ X450)))) | ($true = ((X449 @ X448))) | ? [X454 : term] : ((((var @ X454)) = $true) & (((X449 @ X454)) != $true)) | ? [X452 : term] : (! [X453 : term] : ((((X449 @ X453)) != $true) | ($true = ((X449 @ (sub @ X452 @ (push @ X453 @ id)))))) & ($true != ((X449 @ (lam @ X452)))))) & ! [X445 : (subst > term > subst > $o)] : ? [X446 : (term > $o)] : ! [X447 : term] : (((X445 @ id @ X447 @ id)) = ((X446 @ X447))) & ($true != sP2))) & ((? [X455 : term] : (((sub @ X455 @ id)) != X455) | ? [X456 : (term > $o),X457 : (subst > term > subst > $o),X458 : term] : (! [X459 : term] : (((X457 @ id @ X459 @ id)) = ((X456 @ X459))) & (((X456 @ (lam @ X458))) != $true) & ! [X460 : term] : ((((X456 @ (sub @ X458 @ (push @ X460 @ id)))) = $true) | (((X456 @ X460)) != $true)) & ! [X461 : (subst > term > term)] : (($true = ((X457 @ id @ (lam @ (X461 @ sh @ one)) @ id))) | ? [X462 : term] : (($true = ((X457 @ id @ X462 @ id))) & (((X457 @ id @ (X461 @ id @ X462) @ id)) != $true)) | ? [X463 : term,X465 : subst,X464 : subst] : (((sub @ (X461 @ X465 @ X463) @ X464)) != ((X461 @ (comp @ X465 @ X464) @ (sub @ X463 @ X464)))))) | ? [X449 : (term > $o),X448 : term] : (! [X451 : term,X450 : term] : ((((X449 @ X451)) != $true) | ($true = ((X449 @ (ap @ X450 @ X451)))) | ($true != ((X449 @ X450)))) & ($true != ((X449 @ X448))) & ! [X454 : term] : ((((var @ X454)) != $true) | (((X449 @ X454)) = $true)) & ! [X452 : term] : (? [X453 : term] : ((((X449 @ X453)) = $true) & ($true != ((X449 @ (sub @ X452 @ (push @ X453 @ id)))))) | ($true = ((X449 @ (lam @ X452)))))) | ? [X445 : (subst > term > subst > $o)] : ! [X446 : (term > $o)] : ? [X447 : term] : (((X445 @ id @ X447 @ id)) != ((X446 @ X447))) | ($true = sP2)) | (hoasinduction_no_psi_cond_lthm != $true))) & (((pushprop_lem1v2 = $true) | ? [X388 : (term > $o),X391 : (term > $o),X390 : term,X389 : subst] : (! [X392 : term] : (((X391 @ (sub @ X392 @ (push @ X390 @ X389)))) = ((X388 @ X392))) & ($true = ((X391 @ X390))) & ($true != ((X388 @ one))))) & (! [X388 : (term > $o),X391 : (term > $o),X390 : term,X389 : subst] : (? [X392 : term] : (((X391 @ (sub @ X392 @ (push @ X390 @ X389)))) != ((X388 @ X392))) | ($true != ((X391 @ X390))) | ($true = ((X388 @ one)))) | (pushprop_lem1v2 != $true))) & (hoasinduction_gthm = $true) & (((! [X384 : term,X387 : subst,X386 : term,X385 : (term > $o),X383 : (term > $o)] : (($true = ((X383 @ (sub @ X384 @ (push @ X386 @ X387))))) | ($true != ((X385 @ X384)))) | ? [X379 : subst,X380 : term] : (((sub @ one @ (push @ X380 @ X379))) != X380) | ? [X381 : subst,X382 : term] : (((comp @ sh @ (push @ X382 @ X381))) != X381)) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (? [X384 : term,X387 : subst,X386 : term,X385 : (term > $o),X383 : (term > $o)] : (($true != ((X383 @ (sub @ X384 @ (push @ X386 @ X387))))) & ($true = ((X385 @ X384)))) & ! [X379 : subst,X380 : term] : (((sub @ one @ (push @ X380 @ X379))) = X380) & ! [X381 : subst,X382 : term] : (((comp @ sh @ (push @ X382 @ X381))) = X381)))) & ((! [X491 : (term > $o),X492 : term] : (? [X494 : term,X495 : term] : (($true = ((X491 @ X494))) & ($true = ((X491 @ X495))) & ($true != ((X491 @ (ap @ X495 @ X494))))) | ? [X496 : term] : (($true != ((X491 @ (lam @ X496)))) & ! [X497 : term] : (($true != ((X491 @ X497))) | (((X491 @ (sub @ X496 @ (push @ X497 @ id)))) = $true))) | ($true = ((X491 @ X492))) | ? [X493 : term] : (($true = ((var @ X493))) & (((X491 @ X493)) != $true))) | (induction2 != $true)) & ((induction2 = $true) | ? [X491 : (term > $o),X492 : term] : (! [X494 : term,X495 : term] : (($true != ((X491 @ X494))) | ($true != ((X491 @ X495))) | ($true = ((X491 @ (ap @ X495 @ X494))))) & ! [X496 : term] : (($true = ((X491 @ (lam @ X496)))) | ? [X497 : term] : (($true = ((X491 @ X497))) & (((X491 @ (sub @ X496 @ (push @ X497 @ id)))) != $true))) & ($true != ((X491 @ X492))) & ! [X493 : term] : (($true != ((var @ X493))) | (((X491 @ X493)) = $true))))) & (((pushprop_lem1 = $true) | ? [X186 : (term > $o),X185 : term,X187 : subst,X188 : term,X184 : (term > $o)] : ((((X184 @ (sub @ X185 @ (push @ X188 @ X187)))) != $true) & ($true = ((X186 @ X185))))) & (! [X186 : (term > $o),X185 : term,X187 : subst,X188 : term,X184 : (term > $o)] : ((((X184 @ (sub @ X185 @ (push @ X188 @ X187)))) = $true) | ($true != ((X186 @ X185)))) | (pushprop_lem1 != $true))) & (((induction2_lthm = $true) | ($true != sP1)) & (($true = sP1) | (induction2_lthm != $true))) & (((hoasinduction_lem0 = $true) | ? [X126 : (subst > term > subst > $o)] : ! [X127 : (term > $o)] : ? [X128 : term] : (((X127 @ X128)) != ((X126 @ id @ X128 @ id)))) & (! [X126 : (subst > term > subst > $o)] : ? [X127 : (term > $o)] : ! [X128 : term] : (((X127 @ X128)) = ((X126 @ id @ X128 @ id))) | (hoasinduction_lem0 != $true))) & (lamnotvar = $true) & ((! [X3 : term,X2 : term,X1 : term,X4 : term] : ((((ap @ X3 @ X2)) != ((ap @ X1 @ X4))) | (X2 = X4)) | (apinj2 != $true)) & ((apinj2 = $true) | ? [X3 : term,X2 : term,X1 : term,X4 : term] : ((((ap @ X3 @ X2)) = ((ap @ X1 @ X4))) & (X2 != X4)))) & ((! [X65 : (term > $o),X66 : term] : (? [X70 : term] : ((((X65 @ (lam @ X70))) != $true) & ($true = ((X65 @ X70)))) | ($true = ((X65 @ X66))) | ? [X69 : term,X68 : term] : (($true != ((X65 @ (ap @ X68 @ X69)))) & ($true = ((X65 @ X68))) & (((X65 @ X69)) = $true)) | ? [X67 : term] : (($true = ((var @ X67))) & ($true != ((X65 @ X67))))) | (induction != $true)) & ((induction = $true) | ? [X65 : (term > $o),X66 : term] : (! [X70 : term] : ((((X65 @ (lam @ X70))) = $true) | ($true != ((X65 @ X70)))) & ($true != ((X65 @ X66))) & ! [X69 : term,X68 : term] : (($true = ((X65 @ (ap @ X68 @ X69)))) | ($true != ((X65 @ X68))) | (((X65 @ X69)) != $true)) & ! [X67 : term] : (($true != ((var @ X67))) | ($true = ((X65 @ X67))))))) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoaslamnotvar_lthm = $true) & (((hoasinduction_lem2v2 = $true) | ? [X431 : (subst > term > subst > $o),X430 : term,X432 : (term > $o),X433 : term] : (! [X438 : subst,X439 : term,X440 : subst,X437 : subst] : ((((X431 @ X437 @ X439 @ (comp @ X438 @ X440))) = $true) | ($true != ((X431 @ (comp @ X437 @ X438) @ (sub @ X439 @ X438) @ X440)))) & ($true = ((X432 @ X433))) & (((X432 @ X430)) = $true) & ! [X435 : term,X436 : term] : (($true != ((X431 @ id @ X435 @ id))) | ($true = ((X431 @ id @ (ap @ (sub @ X435 @ id) @ X436) @ id))) | ($true != ((X431 @ id @ X436 @ id)))) & ($true != ((X432 @ (ap @ X433 @ X430)))) & ! [X434 : term] : (((X432 @ X434)) = ((X431 @ id @ X434 @ id))) & ! [X444 : subst,X442 : subst,X441 : subst,X443 : term] : (($true = ((X431 @ (comp @ X442 @ X441) @ (sub @ X443 @ X441) @ X444))) | ($true != ((X431 @ X442 @ X443 @ (comp @ X441 @ X444))))))) & (! [X431 : (subst > term > subst > $o),X430 : term,X432 : (term > $o),X433 : term] : (? [X438 : subst,X439 : term,X440 : subst,X437 : subst] : ((((X431 @ X437 @ X439 @ (comp @ X438 @ X440))) != $true) & ($true = ((X431 @ (comp @ X437 @ X438) @ (sub @ X439 @ X438) @ X440)))) | ($true != ((X432 @ X433))) | (((X432 @ X430)) != $true) | ? [X435 : term,X436 : term] : (($true = ((X431 @ id @ X435 @ id))) & ($true != ((X431 @ id @ (ap @ (sub @ X435 @ id) @ X436) @ id))) & ($true = ((X431 @ id @ X436 @ id)))) | ($true = ((X432 @ (ap @ X433 @ X430)))) | ? [X434 : term] : (((X432 @ X434)) != ((X431 @ id @ X434 @ id))) | ? [X444 : subst,X442 : subst,X441 : subst,X443 : term] : (($true != ((X431 @ (comp @ X442 @ X441) @ (sub @ X443 @ X441) @ X444))) & ($true = ((X431 @ X442 @ X443 @ (comp @ X441 @ X444)))))) | (hoasinduction_lem2v2 != $true))) & (hoasinduction_lem3a_gthm = $true) & (pushprop_lem1_gthm = $true) & (hoasapinj2_gthm = $true) & (id = ((d2subst @ d_id))) & (pushprop_gthm = $true) & ((! [X189 : term] : ? [X190 : (subst > term > term)] : ! [X192 : term,X191 : subst] : (((X190 @ X191 @ X192)) = ((sub @ X189 @ (push @ X192 @ X191)))) | (hoasinduction_lem3v2_f_lthm != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X189 : term] : ! [X190 : (subst > term > term)] : ? [X192 : term,X191 : subst] : (((X190 @ X191 @ X192)) != ((sub @ X189 @ (push @ X192 @ X191)))))) & (((hoasinduction_lem3v2_lthm = $true) | (! [X273 : term] : (((sub @ X273 @ id)) = X273) & ? [X274 : (subst > term > subst > $o),X275 : term,X276 : (term > $o)] : (! [X277 : term] : (((X276 @ X277)) = ((X274 @ id @ X277 @ id))) & ($true != ((X276 @ (lam @ X275)))) & ! [X289 : subst,X290 : term,X288 : subst,X291 : subst] : (($true != ((X274 @ (comp @ X288 @ X291) @ (sub @ X290 @ X291) @ X289))) | ($true = ((X274 @ X288 @ X290 @ (comp @ X291 @ X289))))) & ! [X283 : term,X285 : subst,X284 : subst,X286 : subst] : ((((X274 @ (comp @ X285 @ X286) @ (sub @ X283 @ X286) @ X284)) = $true) | (((X274 @ X285 @ X283 @ (comp @ X286 @ X284))) != $true)) & ! [X278 : (subst > term > term)] : (? [X281 : subst,X280 : subst,X282 : term] : (((sub @ (X278 @ X280 @ X282) @ X281)) != ((X278 @ (comp @ X280 @ X281) @ (sub @ X282 @ X281)))) | ? [X279 : term] : ((((X274 @ id @ X279 @ id)) = $true) & ($true != ((X274 @ id @ (X278 @ id @ X279) @ id)))) | ($true = ((X274 @ id @ (lam @ (X278 @ sh @ one)) @ id)))) & ! [X287 : term] : (($true = ((X276 @ (sub @ X275 @ (push @ X287 @ id))))) | ($true != ((X276 @ X287))))))) & ((? [X273 : term] : (((sub @ X273 @ id)) != X273) | ! [X274 : (subst > term > subst > $o),X275 : term,X276 : (term > $o)] : (? [X277 : term] : (((X276 @ X277)) != ((X274 @ id @ X277 @ id))) | ($true = ((X276 @ (lam @ X275)))) | ? [X289 : subst,X290 : term,X288 : subst,X291 : subst] : (($true = ((X274 @ (comp @ X288 @ X291) @ (sub @ X290 @ X291) @ X289))) & ($true != ((X274 @ X288 @ X290 @ (comp @ X291 @ X289))))) | ? [X283 : term,X285 : subst,X284 : subst,X286 : subst] : ((((X274 @ (comp @ X285 @ X286) @ (sub @ X283 @ X286) @ X284)) != $true) & (((X274 @ X285 @ X283 @ (comp @ X286 @ X284))) = $true)) | ? [X278 : (subst > term > term)] : (! [X281 : subst,X280 : subst,X282 : term] : (((sub @ (X278 @ X280 @ X282) @ X281)) = ((X278 @ (comp @ X280 @ X281) @ (sub @ X282 @ X281)))) & ! [X279 : term] : ((((X274 @ id @ X279 @ id)) != $true) | ($true = ((X274 @ id @ (X278 @ id @ X279) @ id)))) & ($true != ((X274 @ id @ (lam @ (X278 @ sh @ one)) @ id)))) | ? [X287 : term] : (($true != ((X276 @ (sub @ X275 @ (push @ X287 @ id))))) & ($true = ((X276 @ X287)))))) | (hoasinduction_lem3v2_lthm != $true))) & ((! [X270 : term,X271 : term,X269 : term,X268 : term] : ((((ap @ X269 @ X270)) != ((ap @ X271 @ X268))) | (X269 = X271)) | (apinj1 != $true)) & ((apinj1 = $true) | ? [X270 : term,X271 : term,X269 : term,X268 : term] : ((((ap @ X269 @ X270)) = ((ap @ X271 @ X268))) & (X269 != X271)))) & ((! [X5 : (term > $o),X7 : subst,X6 : term] : ? [X8 : (term > $o)] : ! [X9 : term] : (((X5 @ (sub @ X9 @ (push @ X6 @ X7)))) = ((X8 @ X9))) | (pushprop_lem0_lthm != $true)) & ((pushprop_lem0_lthm = $true) | ? [X5 : (term > $o),X7 : subst,X6 : term] : ! [X8 : (term > $o)] : ? [X9 : term] : (((X5 @ (sub @ X9 @ (push @ X6 @ X7)))) != ((X8 @ X9))))) & (hoasinduction_lem3v2_gthm = $true) & (hoasinduction_lem3_gthm = $true) & (hoaslaminj_gthm = $true) & (hoasinduction_lem2_gthm = $true) & ((! [X243 : (subst > term > subst > $o),X242 : term,X244 : (term > $o)] : (? [X259 : term] : (($true = ((X244 @ X259))) & (((X244 @ (sub @ X242 @ (push @ X259 @ id)))) != $true)) | ? [X249 : term] : (((X244 @ X249)) != ((X243 @ id @ X249 @ id))) | ? [X248 : subst,X246 : subst,X247 : term,X245 : subst] : (($true = ((X243 @ X246 @ X247 @ (comp @ X248 @ X245)))) & ($true != ((X243 @ (comp @ X246 @ X248) @ (sub @ X247 @ X248) @ X245)))) | ? [X253 : term,X251 : subst,X252 : subst,X250 : subst] : (($true = ((X243 @ (comp @ X251 @ X252) @ (sub @ X253 @ X252) @ X250))) & (((X243 @ X251 @ X253 @ (comp @ X252 @ X250))) != $true)) | ($true = ((X244 @ (lam @ X242)))) | ? [X254 : (subst > term > term)] : (! [X255 : term] : (($true != ((X243 @ id @ X255 @ id))) | ($true = ((X243 @ id @ (X254 @ id @ X255) @ id)))) & ! [X256 : subst,X257 : subst,X258 : term] : (((sub @ (X254 @ X257 @ X258) @ X256)) = ((X254 @ (comp @ X257 @ X256) @ (sub @ X258 @ X256)))) & ($true != ((X243 @ id @ (lam @ (X254 @ sh @ one)) @ id))))) | (hoasinduction_lem3v2 != $true)) & ((hoasinduction_lem3v2 = $true) | ? [X243 : (subst > term > subst > $o),X242 : term,X244 : (term > $o)] : (! [X259 : term] : (($true != ((X244 @ X259))) | (((X244 @ (sub @ X242 @ (push @ X259 @ id)))) = $true)) & ! [X249 : term] : (((X244 @ X249)) = ((X243 @ id @ X249 @ id))) & ! [X248 : subst,X246 : subst,X247 : term,X245 : subst] : (($true != ((X243 @ X246 @ X247 @ (comp @ X248 @ X245)))) | ($true = ((X243 @ (comp @ X246 @ X248) @ (sub @ X247 @ X248) @ X245)))) & ! [X253 : term,X251 : subst,X252 : subst,X250 : subst] : (($true != ((X243 @ (comp @ X251 @ X252) @ (sub @ X253 @ X252) @ X250))) | (((X243 @ X251 @ X253 @ (comp @ X252 @ X250))) = $true)) & ($true != ((X244 @ (lam @ X242)))) & ! [X254 : (subst > term > term)] : (? [X255 : term] : (($true = ((X243 @ id @ X255 @ id))) & ($true != ((X243 @ id @ (X254 @ id @ X255) @ id)))) | ? [X256 : subst,X257 : subst,X258 : term] : (((sub @ (X254 @ X257 @ X258) @ X256)) != ((X254 @ (comp @ X257 @ X256) @ (sub @ X258 @ X256)))) | ($true = ((X243 @ id @ (lam @ (X254 @ sh @ one)) @ id))))))) & (((axshiftcons = $true) | ? [X112 : term,X113 : subst] : (((comp @ sh @ (push @ X112 @ X113))) != X113)) & (! [X112 : term,X113 : subst] : (((comp @ sh @ (push @ X112 @ X113))) = X113) | (axshiftcons != $true))) & (((hoasinduction_lem3_lthm = $true) | (($true != sP0) & ! [X498 : term] : (((sub @ X498 @ id)) = X498) & ! [X499 : (subst > term > subst > $o),X500 : term] : (? [X501 : term] : (($true = ((X499 @ id @ X501 @ id))) & ($true != ((X499 @ id @ (sub @ X500 @ (push @ X501 @ id)) @ id)))) | ($true = ((X499 @ id @ (lam @ (sub @ X500 @ (push @ one @ sh))) @ id))) | ? [X502 : (subst > term > term)] : (! [X505 : subst,X506 : term,X504 : subst] : (((sub @ (X502 @ X504 @ X506) @ X505)) = ((X502 @ (comp @ X504 @ X505) @ (sub @ X506 @ X505)))) & ! [X503 : term] : (($true = ((X499 @ id @ (X502 @ id @ X503) @ id))) | ($true != ((X499 @ id @ X503 @ id)))) & ($true != ((X499 @ id @ (lam @ (X502 @ sh @ one)) @ id))))))) & ((($true = sP0) | ? [X498 : term] : (((sub @ X498 @ id)) != X498) | ? [X499 : (subst > term > subst > $o),X500 : term] : (! [X501 : term] : (($true != ((X499 @ id @ X501 @ id))) | ($true = ((X499 @ id @ (sub @ X500 @ (push @ X501 @ id)) @ id)))) & ($true != ((X499 @ id @ (lam @ (sub @ X500 @ (push @ one @ sh))) @ id))) & ! [X502 : (subst > term > term)] : (? [X505 : subst,X506 : term,X504 : subst] : (((sub @ (X502 @ X504 @ X506) @ X505)) != ((X502 @ (comp @ X504 @ X505) @ (sub @ X506 @ X505)))) | ? [X503 : term] : (($true != ((X499 @ id @ (X502 @ id @ X503) @ id))) & ($true = ((X499 @ id @ X503 @ id)))) | ($true = ((X499 @ id @ (lam @ (X502 @ sh @ one)) @ id)))))) | (hoasinduction_lem3_lthm != $true)))),
  inference(nnf_transformation,[],[f25])).
thf(f66,plain,(
  (hoaslamnotap_lthm = $true) & (axabs = $true) & (! [X293 : (subst > term > subst > $o),X292 : term] : (($true = ((X293 @ id @ (lam @ (sub @ X292 @ (push @ one @ sh))) @ id))) | ? [X294 : term] : ((((X293 @ id @ X294 @ id)) = $true) & ($true != ((X293 @ id @ (sub @ X292 @ (push @ X294 @ id)) @ id)))) | ? [X295 : (subst > term > term)] : (($true != ((X293 @ id @ (lam @ (X295 @ sh @ one)) @ id))) & ! [X296 : subst,X297 : term,X298 : subst] : (((sub @ (X295 @ X298 @ X297) @ X296)) = ((X295 @ (comp @ X298 @ X296) @ (sub @ X297 @ X296)))) & ! [X299 : term] : ((((X293 @ id @ X299 @ id)) != $true) | ($true = ((X293 @ id @ (X295 @ id @ X299) @ id)))))) | (hoasinduction_lem3aa != $true)) & ((hoasinduction_lem3aa = $true) | ? [X293 : (subst > term > subst > $o),X292 : term] : (($true != ((X293 @ id @ (lam @ (sub @ X292 @ (push @ one @ sh))) @ id))) & ! [X294 : term] : ((((X293 @ id @ X294 @ id)) != $true) | ($true = ((X293 @ id @ (sub @ X292 @ (push @ X294 @ id)) @ id)))) & ! [X295 : (subst > term > term)] : (($true = ((X293 @ id @ (lam @ (X295 @ sh @ one)) @ id))) | ? [X296 : subst,X297 : term,X298 : subst] : (((sub @ (X295 @ X298 @ X297) @ X296)) != ((X295 @ (comp @ X298 @ X296) @ (sub @ X297 @ X296)))) | ? [X299 : term] : ((((X293 @ id @ X299 @ id)) = $true) & ($true != ((X293 @ id @ (X295 @ id @ X299) @ id))))))) & (pushprop_lem2v2_gthm = $true) & ((induction2lem = $true) | ? [X217 : term,X215 : subst,X216 : (term > $o)] : (! [X221 : term] : (($true = ((X216 @ (lam @ X221)))) | ? [X222 : term] : (($true != ((X216 @ (sub @ X221 @ (push @ X222 @ id))))) & ($true = ((X216 @ X222))))) & ! [X219 : term,X220 : term] : (($true != ((X216 @ X220))) | (((X216 @ (ap @ X219 @ X220))) = $true) | ($true != ((X216 @ X219)))) & ! [X218 : term] : (($true = ((X216 @ (sub @ X218 @ X215)))) | ($true != ((var @ X218)))) & (((X216 @ (sub @ X217 @ X215))) != $true))) & (! [X217 : term,X215 : subst,X216 : (term > $o)] : (? [X221 : term] : (($true != ((X216 @ (lam @ X221)))) & ! [X222 : term] : (($true = ((X216 @ (sub @ X221 @ (push @ X222 @ id))))) | ($true != ((X216 @ X222))))) | ? [X219 : term,X220 : term] : (($true = ((X216 @ X220))) & (((X216 @ (ap @ X219 @ X220))) != $true) & ($true = ((X216 @ X219)))) | ? [X218 : term] : (($true != ((X216 @ (sub @ X218 @ X215)))) & ($true = ((var @ X218)))) | (((X216 @ (sub @ X217 @ X215))) = $true)) | (induction2lem != $true)) & (lamnotap != $true) & ((hoasinduction_lem3v2a_lthm = $true) | (? [X20 : (subst > term > subst > $o),X19 : (term > $o),X18 : term] : (($true != ((X19 @ (lam @ X18)))) & ! [X21 : term] : (((X19 @ X21)) = ((X20 @ id @ X21 @ id))) & ! [X23 : (subst > term > term)] : (($true = ((X20 @ id @ (lam @ (X23 @ sh @ one)) @ id))) | ? [X24 : term] : ((((X20 @ id @ (X23 @ id @ X24) @ id)) != $true) & ($true = ((X20 @ id @ X24 @ id)))) | ? [X25 : subst,X26 : term,X27 : subst] : (((sub @ (X23 @ X25 @ X26) @ X27)) != ((X23 @ (comp @ X25 @ X27) @ (sub @ X26 @ X27))))) & ! [X22 : term] : (($true = ((X19 @ (sub @ X18 @ (push @ X22 @ id))))) | (((X19 @ X22)) != $true))) & ! [X13 : term] : ? [X14 : (subst > term > term)] : ! [X15 : term,X16 : subst] : (((sub @ X13 @ (push @ X15 @ X16))) = ((X14 @ X16 @ X15))) & ! [X17 : term] : (((sub @ X17 @ id)) = X17))) & (! [X20 : (subst > term > subst > $o),X19 : (term > $o),X18 : term] : (($true = ((X19 @ (lam @ X18)))) | ? [X21 : term] : (((X19 @ X21)) != ((X20 @ id @ X21 @ id))) | ? [X23 : (subst > term > term)] : (($true != ((X20 @ id @ (lam @ (X23 @ sh @ one)) @ id))) & ! [X24 : term] : ((((X20 @ id @ (X23 @ id @ X24) @ id)) = $true) | ($true != ((X20 @ id @ X24 @ id)))) & ! [X25 : subst,X26 : term,X27 : subst] : (((sub @ (X23 @ X25 @ X26) @ X27)) = ((X23 @ (comp @ X25 @ X27) @ (sub @ X26 @ X27))))) | ? [X22 : term] : (($true != ((X19 @ (sub @ X18 @ (push @ X22 @ id))))) & (((X19 @ X22)) = $true))) | ? [X13 : term] : ! [X14 : (subst > term > term)] : ? [X15 : term,X16 : subst] : (((sub @ X13 @ (push @ X15 @ X16))) != ((X14 @ X16 @ X15))) | ? [X17 : term] : (((sub @ X17 @ id)) != X17) | (hoasinduction_lem3v2a_lthm != $true)) & (! [X533 : term,X532 : term,X531 : term,X534 : term] : ((X532 = X533) | (((ap @ (sub @ X534 @ id) @ X533)) != ((ap @ (sub @ X531 @ id) @ X532)))) | (hoasapinj2 != $true)) & ((hoasapinj2 = $true) | ? [X533 : term,X532 : term,X531 : term,X534 : term] : ((X532 != X533) & (((ap @ (sub @ X534 @ id) @ X533)) = ((ap @ (sub @ X531 @ id) @ X532))))) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem1_gthm = $true) & (! [X260 : term,X261 : (subst > term > subst > $o)] : (? [X262 : (subst > term > term)] : (($true != ((X261 @ id @ (lam @ (X262 @ sh @ one)) @ id))) & ! [X264 : term,X265 : subst,X263 : subst] : (((X262 @ (comp @ X265 @ X263) @ (sub @ X264 @ X263))) = ((sub @ (X262 @ X265 @ X264) @ X263))) & ! [X266 : term] : ((((X261 @ id @ (X262 @ id @ X266) @ id)) = $true) | ($true != ((X261 @ id @ X266 @ id))))) | ? [X267 : term] : ((((X261 @ id @ (sub @ X260 @ (push @ X267 @ id)) @ id)) != $true) & (((X261 @ id @ X267 @ id)) = $true)) | ($true = ((X261 @ id @ (lam @ (sub @ X260 @ (push @ one @ sh))) @ id)))) | (hoasinduction_lem3aa_lthm != $true)) & ((hoasinduction_lem3aa_lthm = $true) | ? [X260 : term,X261 : (subst > term > subst > $o)] : (! [X262 : (subst > term > term)] : (($true = ((X261 @ id @ (lam @ (X262 @ sh @ one)) @ id))) | ? [X264 : term,X265 : subst,X263 : subst] : (((X262 @ (comp @ X265 @ X263) @ (sub @ X264 @ X263))) != ((sub @ (X262 @ X265 @ X264) @ X263))) | ? [X266 : term] : ((((X261 @ id @ (X262 @ id @ X266) @ id)) != $true) & ($true = ((X261 @ id @ X266 @ id))))) & ! [X267 : term] : ((((X261 @ id @ (sub @ X260 @ (push @ X267 @ id)) @ id)) = $true) | (((X261 @ id @ X267 @ id)) != $true)) & ($true != ((X261 @ id @ (lam @ (sub @ X260 @ (push @ one @ sh))) @ id))))) & (! [X535 : subst] : (((comp @ id @ X535)) = X535) | (axidl != $true)) & ((axidl = $true) | ? [X535 : subst] : (((comp @ id @ X535)) != X535)) & ((hoaslamnotap = $true) | ? [X94 : term,X92 : term,X93 : (subst > term > term)] : (! [X95 : subst,X97 : subst,X96 : term] : (((X93 @ (comp @ X95 @ X97) @ (sub @ X96 @ X97))) = ((sub @ (X93 @ X95 @ X96) @ X97))) & (((lam @ (X93 @ sh @ one))) = ((ap @ (sub @ X94 @ id) @ X92))))) & (! [X94 : term,X92 : term,X93 : (subst > term > term)] : (? [X95 : subst,X97 : subst,X96 : term] : (((X93 @ (comp @ X95 @ X97) @ (sub @ X96 @ X97))) != ((sub @ (X93 @ X95 @ X96) @ X97))) | (((lam @ (X93 @ sh @ one))) != ((ap @ (sub @ X94 @ id) @ X92)))) | (hoaslamnotap != $true)) & (! [X429 : term,X428 : term] : ((((sub @ X428 @ sh)) != ((sub @ X429 @ sh))) | (X428 = X429)) | (shinj != $true)) & ((shinj = $true) | ? [X429 : term,X428 : term] : ((((sub @ X428 @ sh)) = ((sub @ X429 @ sh))) & (X428 != X429))) & (! [X193 : term,X194 : term,X195 : (subst > term > subst > $o)] : (($true != ((X195 @ id @ X194 @ id))) | ($true != ((X195 @ id @ X193 @ id))) | (((X195 @ id @ (ap @ X193 @ X194) @ id)) = $true) | ? [X199 : subst,X200 : subst,X198 : term,X201 : subst] : (($true = ((X195 @ (comp @ X201 @ X200) @ (sub @ X198 @ X200) @ X199))) & ($true != ((X195 @ X201 @ X198 @ (comp @ X200 @ X199))))) | ? [X204 : subst,X203 : subst,X202 : term,X205 : subst] : (($true != ((X195 @ (comp @ X204 @ X205) @ (sub @ X202 @ X205) @ X203))) & ($true = ((X195 @ X204 @ X202 @ (comp @ X205 @ X203))))) | ? [X197 : term,X196 : term] : (($true != ((X195 @ id @ (ap @ (sub @ X196 @ id) @ X197) @ id))) & (((X195 @ id @ X196 @ id)) = $true) & ($true = ((X195 @ id @ X197 @ id))))) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | ? [X193 : term,X194 : term,X195 : (subst > term > subst > $o)] : (($true = ((X195 @ id @ X194 @ id))) & ($true = ((X195 @ id @ X193 @ id))) & (((X195 @ id @ (ap @ X193 @ X194) @ id)) != $true) & ! [X199 : subst,X200 : subst,X198 : term,X201 : subst] : (($true != ((X195 @ (comp @ X201 @ X200) @ (sub @ X198 @ X200) @ X199))) | ($true = ((X195 @ X201 @ X198 @ (comp @ X200 @ X199))))) & ! [X204 : subst,X203 : subst,X202 : term,X205 : subst] : (($true = ((X195 @ (comp @ X204 @ X205) @ (sub @ X202 @ X205) @ X203))) | ($true != ((X195 @ X204 @ X202 @ (comp @ X205 @ X203))))) & ! [X197 : term,X196 : term] : (($true = ((X195 @ id @ (ap @ (sub @ X196 @ id) @ X197) @ id))) | (((X195 @ id @ X196 @ id)) != $true) | ($true != ((X195 @ id @ X197 @ id)))))) & (pushprop_lem3v2 = $true) & ! [X309 : d_subst,X310 : d_subst] : ((((d2subst @ X310)) != ((d2subst @ X309))) | (X309 = X310)) & (! [X30 : term,X31 : term,X29 : term,X28 : term] : ((((ap @ (sub @ X30 @ id) @ X29)) != ((ap @ (sub @ X28 @ id) @ X31))) | (X28 = X30)) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | ? [X30 : term,X31 : term,X29 : term,X28 : term] : ((((ap @ (sub @ X30 @ id) @ X29)) = ((ap @ (sub @ X28 @ id) @ X31))) & (X28 != X30))) & ((axvarcons = $true) | ? [X64 : subst,X63 : term] : (((sub @ one @ (push @ X63 @ X64))) != X63)) & (! [X64 : subst,X63 : term] : (((sub @ one @ (push @ X63 @ X64))) = X63) | (axvarcons != $true)) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((hoasinduction_lthm = $true) | ($true != sP10)) & (($true = sP10) | (hoasinduction_lthm != $true)) & ! [X51 : d_term,X52 : d_term] : ((((d2term @ X51)) != ((d2term @ X52))) | (X51 = X52)) & (hoaslamnotvar_gthm = $true) & (hoaslamnotvar = $true) & ((hoasinduction = $true) | ? [X393 : term,X394 : (subst > term > subst > $o)] : (! [X400 : term,X399 : term] : ((((X394 @ id @ (ap @ (sub @ X399 @ id) @ X400) @ id)) = $true) | ($true != ((X394 @ id @ X399 @ id))) | (((X394 @ id @ X400 @ id)) != $true)) & ! [X397 : subst,X395 : subst,X396 : term,X398 : subst] : (($true = ((X394 @ (comp @ X395 @ X397) @ (sub @ X396 @ X397) @ X398))) | ($true != ((X394 @ X395 @ X396 @ (comp @ X397 @ X398))))) & ! [X401 : term] : ((((var @ (sub @ X401 @ id))) != $true) | ($true = ((X394 @ id @ X401 @ id)))) & ($true != ((X394 @ id @ X393 @ id))) & ($true != ((sP6 @ X394))) & ! [X402 : subst,X404 : subst,X405 : subst,X403 : term] : (($true != ((X394 @ (comp @ X404 @ X402) @ (sub @ X403 @ X402) @ X405))) | ($true = ((X394 @ X404 @ X403 @ (comp @ X402 @ X405))))))) & (! [X393 : term,X394 : (subst > term > subst > $o)] : (? [X400 : term,X399 : term] : ((((X394 @ id @ (ap @ (sub @ X399 @ id) @ X400) @ id)) != $true) & ($true = ((X394 @ id @ X399 @ id))) & (((X394 @ id @ X400 @ id)) = $true)) | ? [X397 : subst,X395 : subst,X396 : term,X398 : subst] : (($true != ((X394 @ (comp @ X395 @ X397) @ (sub @ X396 @ X397) @ X398))) & ($true = ((X394 @ X395 @ X396 @ (comp @ X397 @ X398))))) | ? [X401 : term] : ((((var @ (sub @ X401 @ id))) = $true) & ($true != ((X394 @ id @ X401 @ id)))) | ($true = ((X394 @ id @ X393 @ id))) | ($true = ((sP6 @ X394))) | ? [X402 : subst,X404 : subst,X405 : subst,X403 : term] : (($true = ((X394 @ (comp @ X404 @ X402) @ (sub @ X403 @ X402) @ X405))) & ($true != ((X394 @ X404 @ X403 @ (comp @ X402 @ X405)))))) | (hoasinduction != $true)) & (pushprop_lem3v2_lthm = $true) & (! [X183 : subst] : (((push @ (sub @ one @ X183) @ (comp @ sh @ X183))) = X183) | (axscons != $true)) & ((axscons = $true) | ? [X183 : subst] : (((push @ (sub @ one @ X183) @ (comp @ sh @ X183))) != X183)) & (hoasinduction_lem3b_gthm = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ! [X111 : d_term] : (d_one = X111) & ((hoasapinj2_lthm = $true) | (! [X524 : term,X525 : term,X523 : term,X526 : term] : ((X523 = X524) | (((ap @ X526 @ X523)) != ((ap @ X525 @ X524)))) & ? [X529 : term,X528 : term,X527 : term,X530 : term] : ((X527 != X530) & (((ap @ (sub @ X529 @ id) @ X527)) = ((ap @ (sub @ X528 @ id) @ X530)))))) & (? [X524 : term,X525 : term,X523 : term,X526 : term] : ((X523 != X524) & (((ap @ X526 @ X523)) = ((ap @ X525 @ X524)))) | ! [X529 : term,X528 : term,X527 : term,X530 : term] : ((X527 = X530) | (((ap @ (sub @ X529 @ id) @ X527)) != ((ap @ (sub @ X528 @ id) @ X530)))) | (hoasapinj2_lthm != $true)) & (axvarshift = $true) & (! [X538 : term] : ? [X539 : (subst > term > term)] : (((X539 @ sh @ one)) = ((sub @ X538 @ (push @ one @ sh)))) | (hoasinduction_lem3b != $true)) & ((hoasinduction_lem3b = $true) | ? [X538 : term] : ! [X539 : (subst > term > term)] : (((X539 @ sh @ one)) != ((sub @ X538 @ (push @ one @ sh))))) & ((hoasinduction_lem3a_lthm = $true) | (! [X413 : (subst > term > subst > $o),X412 : term] : (? [X414 : (subst > term > term)] : (! [X418 : term,X417 : subst,X416 : subst] : (((X414 @ (comp @ X417 @ X416) @ (sub @ X418 @ X416))) = ((sub @ (X414 @ X417 @ X418) @ X416))) & ! [X415 : term] : ((((X413 @ id @ X415 @ id)) != $true) | (((X413 @ id @ (X414 @ id @ X415) @ id)) = $true)) & ($true != ((X413 @ id @ (lam @ (X414 @ sh @ one)) @ id)))) | ? [X419 : term] : (($true = ((X413 @ id @ X419 @ id))) & ($true != ((X413 @ id @ (sub @ X412 @ (push @ X419 @ id)) @ id)))) | (((X413 @ id @ (lam @ (sub @ X412 @ (push @ one @ sh))) @ id)) = $true)) & ! [X411 : term] : (((sub @ X411 @ id)) = X411) & ? [X421 : (subst > term > subst > $o),X420 : term] : (! [X422 : (subst > term > term)] : (? [X423 : subst,X424 : subst,X425 : term] : (((sub @ (X422 @ X423 @ X425) @ X424)) != ((X422 @ (comp @ X423 @ X424) @ (sub @ X425 @ X424)))) | ? [X426 : term] : (($true != ((X421 @ id @ (X422 @ id @ X426) @ id))) & ($true = ((X421 @ id @ X426 @ id)))) | (((X421 @ id @ (lam @ (X422 @ sh @ one)) @ id)) = $true)) & ! [X427 : term] : ((((X421 @ id @ (sub @ X420 @ (push @ X427 @ id)) @ id)) = $true) | (((X421 @ id @ X427 @ id)) != $true)) & ($true != ((X421 @ id @ (lam @ X420) @ id)))))) & (? [X413 : (subst > term > subst > $o),X412 : term] : (! [X414 : (subst > term > term)] : (? [X418 : term,X417 : subst,X416 : subst] : (((X414 @ (comp @ X417 @ X416) @ (sub @ X418 @ X416))) != ((sub @ (X414 @ X417 @ X418) @ X416))) | ? [X415 : term] : ((((X413 @ id @ X415 @ id)) = $true) & (((X413 @ id @ (X414 @ id @ X415) @ id)) != $true)) | ($true = ((X413 @ id @ (lam @ (X414 @ sh @ one)) @ id)))) & ! [X419 : term] : (($true != ((X413 @ id @ X419 @ id))) | ($true = ((X413 @ id @ (sub @ X412 @ (push @ X419 @ id)) @ id)))) & (((X413 @ id @ (lam @ (sub @ X412 @ (push @ one @ sh))) @ id)) != $true)) | ? [X411 : term] : (((sub @ X411 @ id)) != X411) | ! [X421 : (subst > term > subst > $o),X420 : term] : (? [X422 : (subst > term > term)] : (! [X423 : subst,X424 : subst,X425 : term] : (((sub @ (X422 @ X423 @ X425) @ X424)) = ((X422 @ (comp @ X423 @ X424) @ (sub @ X425 @ X424)))) & ! [X426 : term] : (($true = ((X421 @ id @ (X422 @ id @ X426) @ id))) | ($true != ((X421 @ id @ X426 @ id)))) & (((X421 @ id @ (lam @ (X422 @ sh @ one)) @ id)) != $true)) | ? [X427 : term] : ((((X421 @ id @ (sub @ X420 @ (push @ X427 @ id)) @ id)) != $true) & (((X421 @ id @ X427 @ id)) = $true)) | ($true = ((X421 @ id @ (lam @ X420) @ id)))) | (hoasinduction_lem3a_lthm != $true)) & (induction2lem_gthm = $true) & ! [X48 : d_subst] : (d_id = X48) & (pushprop_lem2v2_lthm = $true) & (ulamvar1 != $true) & ! [X536 : subst] : ? [X537 : d_subst] : (((d2subst @ X537)) = X536) & (pushprop_lthm = $true) & (hoasinduction_lem1v2 = $true) & (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_lem3v2a = $true) | ? [X227 : (term > $o),X228 : term,X229 : (subst > term > subst > $o)] : (! [X235 : term] : (((X229 @ id @ X235 @ id)) = ((X227 @ X235))) & ($true != ((X227 @ (lam @ X228)))) & ! [X236 : term] : (($true = ((X227 @ (sub @ X228 @ (push @ X236 @ id))))) | ($true != ((X227 @ X236)))) & ! [X230 : (subst > term > term)] : (? [X232 : term,X231 : subst,X233 : subst] : (((X230 @ (comp @ X231 @ X233) @ (sub @ X232 @ X233))) != ((sub @ (X230 @ X231 @ X232) @ X233))) | ? [X234 : term] : ((((X229 @ id @ (X230 @ id @ X234) @ id)) != $true) & ($true = ((X229 @ id @ X234 @ id)))) | (((X229 @ id @ (lam @ (X230 @ sh @ one)) @ id)) = $true)))) & (! [X227 : (term > $o),X228 : term,X229 : (subst > term > subst > $o)] : (? [X235 : term] : (((X229 @ id @ X235 @ id)) != ((X227 @ X235))) | ($true = ((X227 @ (lam @ X228)))) | ? [X236 : term] : (($true != ((X227 @ (sub @ X228 @ (push @ X236 @ id))))) & ($true = ((X227 @ X236)))) | ? [X230 : (subst > term > term)] : (! [X232 : term,X231 : subst,X233 : subst] : (((X230 @ (comp @ X231 @ X233) @ (sub @ X232 @ X233))) = ((sub @ (X230 @ X231 @ X232) @ X233))) & ! [X234 : term] : ((((X229 @ id @ (X230 @ id @ X234) @ id)) = $true) | ($true != ((X229 @ id @ X234 @ id)))) & (((X229 @ id @ (lam @ (X230 @ sh @ one)) @ id)) != $true))) | (hoasinduction_lem3v2a != $true)) & (pushprop_lthm_orig = $true) & (pushprop_lem1v2_gthm = $true) & (pushprop_lem2v2 = $true) & ! [X61 : term] : ? [X62 : d_term] : (((d2term @ X62)) = X61) & ((hoasinduction_no_psi_cond = $true) | ? [X206 : (subst > term > subst > $o),X207 : term] : (($true != ((X206 @ id @ X207 @ id))) & ! [X208 : (subst > term > term)] : (($true = ((X206 @ id @ (lam @ (X208 @ sh @ one)) @ id))) | ? [X211 : subst,X210 : subst,X209 : term] : (((sub @ (X208 @ X211 @ X209) @ X210)) != ((X208 @ (comp @ X211 @ X210) @ (sub @ X209 @ X210)))) | ? [X212 : term] : (($true != ((X206 @ id @ (X208 @ id @ X212) @ id))) & ($true = ((X206 @ id @ X212 @ id))))) & ! [X214 : term,X213 : term] : ((((X206 @ id @ X214 @ id)) != $true) | ($true != ((X206 @ id @ X213 @ id))) | ($true = ((X206 @ id @ (ap @ (sub @ X213 @ id) @ X214) @ id)))))) & (! [X206 : (subst > term > subst > $o),X207 : term] : (($true = ((X206 @ id @ X207 @ id))) | ? [X208 : (subst > term > term)] : (($true != ((X206 @ id @ (lam @ (X208 @ sh @ one)) @ id))) & ! [X211 : subst,X210 : subst,X209 : term] : (((sub @ (X208 @ X211 @ X209) @ X210)) = ((X208 @ (comp @ X211 @ X210) @ (sub @ X209 @ X210)))) & ! [X212 : term] : (($true = ((X206 @ id @ (X208 @ id @ X212) @ id))) | ($true != ((X206 @ id @ X212 @ id))))) | ? [X214 : term,X213 : term] : ((((X206 @ id @ X214 @ id)) = $true) & ($true = ((X206 @ id @ X213 @ id))) & ($true != ((X206 @ id @ (ap @ (sub @ X213 @ id) @ X214) @ id))))) | (hoasinduction_no_psi_cond != $true)) & (! [X115 : term,X114 : (subst > term > subst > $o)] : (($true = ((X114 @ id @ (lam @ (sub @ X115 @ (push @ one @ sh))) @ id))) | ? [X118 : term,X117 : (subst > term > term)] : (! [X121 : subst,X119 : subst,X120 : term] : (((X117 @ (comp @ X119 @ X121) @ (sub @ X120 @ X121))) = ((sub @ X118 @ (push @ (sub @ X120 @ X121) @ (comp @ X119 @ X121))))) & ! [X124 : subst,X123 : term,X122 : subst] : (((sub @ (sub @ X118 @ (push @ X123 @ X122)) @ X124)) = ((sub @ (X117 @ X122 @ X123) @ X124))) & (((X114 @ id @ (lam @ (X117 @ sh @ one)) @ id)) != $true) & ! [X125 : term] : (($true = ((X114 @ id @ (X117 @ id @ X125) @ id))) | ($true != ((X114 @ id @ X125 @ id))))) | ? [X116 : term] : (($true != ((X114 @ id @ (sub @ X115 @ (push @ X116 @ id)) @ id))) & ($true = ((X114 @ id @ X116 @ id))))) | (hoasinduction_lem3aaa != $true)) & ((hoasinduction_lem3aaa = $true) | ? [X115 : term,X114 : (subst > term > subst > $o)] : (($true != ((X114 @ id @ (lam @ (sub @ X115 @ (push @ one @ sh))) @ id))) & ! [X118 : term,X117 : (subst > term > term)] : (? [X121 : subst,X119 : subst,X120 : term] : (((X117 @ (comp @ X119 @ X121) @ (sub @ X120 @ X121))) != ((sub @ X118 @ (push @ (sub @ X120 @ X121) @ (comp @ X119 @ X121))))) | ? [X124 : subst,X123 : term,X122 : subst] : (((sub @ (sub @ X118 @ (push @ X123 @ X122)) @ X124)) != ((sub @ (X117 @ X122 @ X123) @ X124))) | (((X114 @ id @ (lam @ (X117 @ sh @ one)) @ id)) = $true) | ? [X125 : term] : (($true != ((X114 @ id @ (X117 @ id @ X125) @ id))) & ($true = ((X114 @ id @ X125 @ id))))) & ! [X116 : term] : (($true = ((X114 @ id @ (sub @ X115 @ (push @ X116 @ id)) @ id))) | ($true != ((X114 @ id @ X116 @ id)))))) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (pushprop_lem0_gthm = $true) & (apnotvar = $true) & (hoasapnotvar = $true) & (hoasinduction_lem1_lthm = $true) & ((hoaslaminj = $true) | ? [X372 : subst,X370 : (subst > term > term),X369 : term,X371 : (subst > term > term)] : ((((lam @ (X370 @ sh @ one))) = ((lam @ (X371 @ sh @ one)))) & ! [X376 : subst,X378 : term,X377 : subst] : (((sub @ (X371 @ X376 @ X378) @ X377)) = ((X371 @ (comp @ X376 @ X377) @ (sub @ X378 @ X377)))) & (((X371 @ X372 @ X369)) != ((X370 @ X372 @ X369))) & ! [X375 : subst,X374 : subst,X373 : term] : (((sub @ (X370 @ X375 @ X373) @ X374)) = ((X370 @ (comp @ X375 @ X374) @ (sub @ X373 @ X374)))))) & (! [X372 : subst,X370 : (subst > term > term),X369 : term,X371 : (subst > term > term)] : ((((lam @ (X370 @ sh @ one))) != ((lam @ (X371 @ sh @ one)))) | ? [X376 : subst,X378 : term,X377 : subst] : (((sub @ (X371 @ X376 @ X378) @ X377)) != ((X371 @ (comp @ X376 @ X377) @ (sub @ X378 @ X377)))) | (((X371 @ X372 @ X369)) = ((X370 @ X372 @ X369))) | ? [X375 : subst,X374 : subst,X373 : term] : (((sub @ (X370 @ X375 @ X373) @ X374)) != ((X370 @ (comp @ X375 @ X374) @ (sub @ X373 @ X374))))) | (hoaslaminj != $true)) & (induction2_gthm = $true) & (! [X71 : term] : (((sub @ X71 @ id)) = X71) | (termmset != $true)) & ((termmset = $true) | ? [X71 : term] : (((sub @ X71 @ id)) != X71)) & ((hoasinduction_lem0_lthm = $true) | ? [X10 : (subst > term > subst > $o)] : ! [X11 : (term > $o)] : ? [X12 : term] : (((X11 @ X12)) != ((X10 @ id @ X12 @ id)))) & (! [X10 : (subst > term > subst > $o)] : ? [X11 : (term > $o)] : ! [X12 : term] : (((X11 @ X12)) = ((X10 @ id @ X12 @ id))) | (hoasinduction_lem0_lthm != $true)) & (hoaslamnotap_gthm = $true) & (((d2subst @ d_id)) = sh) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (hoasinduction_lem2v2_gthm = $true) & ((hoaslaminj_lthm = $true) | (! [X35 : subst,X34 : term] : (((comp @ sh @ (push @ X34 @ X35))) = X35) & ! [X32 : subst,X33 : term] : (((sub @ one @ (push @ X33 @ X32))) = X33) & ? [X39 : (subst > term > term),X40 : (subst > term > term),X41 : term,X38 : subst] : (! [X46 : term,X47 : subst,X45 : subst] : (((sub @ (X40 @ X45 @ X46) @ X47)) = ((X40 @ (comp @ X45 @ X47) @ (sub @ X46 @ X47)))) & (((lam @ (X39 @ sh @ one))) = ((lam @ (X40 @ sh @ one)))) & ! [X42 : subst,X44 : term,X43 : subst] : (((sub @ (X39 @ X43 @ X44) @ X42)) = ((X39 @ (comp @ X43 @ X42) @ (sub @ X44 @ X42)))) & (((X40 @ X38 @ X41)) != ((X39 @ X38 @ X41)))) & ! [X36 : term,X37 : term] : ((X36 = X37) | (((lam @ X37)) != ((lam @ X36)))))) & (? [X35 : subst,X34 : term] : (((comp @ sh @ (push @ X34 @ X35))) != X35) | ? [X32 : subst,X33 : term] : (((sub @ one @ (push @ X33 @ X32))) != X33) | ! [X39 : (subst > term > term),X40 : (subst > term > term),X41 : term,X38 : subst] : (? [X46 : term,X47 : subst,X45 : subst] : (((sub @ (X40 @ X45 @ X46) @ X47)) != ((X40 @ (comp @ X45 @ X47) @ (sub @ X46 @ X47)))) | (((lam @ (X39 @ sh @ one))) != ((lam @ (X40 @ sh @ one)))) | ? [X42 : subst,X44 : term,X43 : subst] : (((sub @ (X39 @ X43 @ X44) @ X42)) != ((X39 @ (comp @ X43 @ X42) @ (sub @ X44 @ X42)))) | (((X40 @ X38 @ X41)) = ((X39 @ X38 @ X41)))) | ? [X36 : term,X37 : term] : ((X36 != X37) & (((lam @ X37)) = ((lam @ X36)))) | (hoaslaminj_lthm != $true)) & (! [X73 : term,X72 : (subst > term > subst > $o)] : (? [X74 : term] : (($true != ((X72 @ id @ (sub @ X73 @ (push @ X74 @ id)) @ id))) & ($true = ((X72 @ id @ X74 @ id)))) | ? [X85 : subst,X86 : subst,X84 : subst,X87 : term] : (($true != ((X72 @ (comp @ X85 @ X84) @ (sub @ X87 @ X84) @ X86))) & ($true = ((X72 @ X85 @ X87 @ (comp @ X84 @ X86))))) | ? [X79 : (subst > term > term)] : (! [X83 : subst,X81 : term,X82 : subst] : (((X79 @ (comp @ X83 @ X82) @ (sub @ X81 @ X82))) = ((sub @ (X79 @ X83 @ X81) @ X82))) & ! [X80 : term] : ((((X72 @ id @ X80 @ id)) != $true) | ($true = ((X72 @ id @ (X79 @ id @ X80) @ id)))) & (((X72 @ id @ (lam @ (X79 @ sh @ one)) @ id)) != $true)) | ($true = ((X72 @ id @ (lam @ X73) @ id))) | ? [X76 : subst,X77 : term,X78 : subst,X75 : subst] : (($true = ((X72 @ (comp @ X75 @ X78) @ (sub @ X77 @ X78) @ X76))) & (((X72 @ X75 @ X77 @ (comp @ X78 @ X76))) != $true))) | (hoasinduction_lem3 != $true)) & ((hoasinduction_lem3 = $true) | ? [X73 : term,X72 : (subst > term > subst > $o)] : (! [X74 : term] : (($true = ((X72 @ id @ (sub @ X73 @ (push @ X74 @ id)) @ id))) | ($true != ((X72 @ id @ X74 @ id)))) & ! [X85 : subst,X86 : subst,X84 : subst,X87 : term] : (($true = ((X72 @ (comp @ X85 @ X84) @ (sub @ X87 @ X84) @ X86))) | ($true != ((X72 @ X85 @ X87 @ (comp @ X84 @ X86))))) & ! [X79 : (subst > term > term)] : (? [X83 : subst,X81 : term,X82 : subst] : (((X79 @ (comp @ X83 @ X82) @ (sub @ X81 @ X82))) != ((sub @ (X79 @ X83 @ X81) @ X82))) | ? [X80 : term] : ((((X72 @ id @ X80 @ id)) = $true) & ($true != ((X72 @ id @ (X79 @ id @ X80) @ id)))) | (((X72 @ id @ (lam @ (X79 @ sh @ one)) @ id)) = $true)) & ($true != ((X72 @ id @ (lam @ X73) @ id))) & ! [X76 : subst,X77 : term,X78 : subst,X75 : subst] : (($true != ((X72 @ (comp @ X75 @ X78) @ (sub @ X77 @ X78) @ X76))) | (((X72 @ X75 @ X77 @ (comp @ X78 @ X76))) = $true)))) & (termmset_gthm = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (hoasinduction_lem2_lthm = $true) & (! [X99 : term,X98 : term] : ((((lam @ X98)) != ((lam @ X99))) | (X98 = X99)) | (laminj != $true)) & ((laminj = $true) | ? [X99 : term,X98 : term] : ((((lam @ X98)) = ((lam @ X99))) & (X98 != X99))) & (((var @ (d2term @ d_one))) != $true) & (! [X272 : subst] : (((comp @ X272 @ id)) = X272) | (axidr != $true)) & ((axidr = $true) | ? [X272 : subst] : (((comp @ X272 @ id)) != X272)) & (! [X0 : term] : (((sub @ X0 @ id)) = X0) | (axvarid != $true)) & ((axvarid = $true) | ? [X0 : term] : (((sub @ X0 @ id)) != X0)) & (? [X321 : term] : (((sub @ X321 @ id)) != X321) | ($true = sP5) | ? [X311 : (subst > term > subst > $o)] : ! [X312 : (term > $o)] : ? [X313 : term] : (((X312 @ X313)) != ((X311 @ id @ X313 @ id))) | ? [X322 : term,X323 : (term > $o),X324 : (subst > term > subst > $o)] : (($true != ((X323 @ (lam @ X322)))) & ! [X326 : (subst > term > term)] : (($true = ((X324 @ id @ (lam @ (X326 @ sh @ one)) @ id))) | ? [X330 : subst,X328 : subst,X329 : term] : (((X326 @ (comp @ X328 @ X330) @ (sub @ X329 @ X330))) != ((sub @ (X326 @ X328 @ X329) @ X330))) | ? [X327 : term] : ((((X324 @ id @ X327 @ id)) = $true) & ($true != ((X324 @ id @ (X326 @ id @ X327) @ id))))) & ! [X325 : term] : ((((X323 @ X325)) != $true) | ($true = ((X323 @ (sub @ X322 @ (push @ X325 @ id)))))) & ! [X331 : term] : (((X324 @ id @ X331 @ id)) = ((X323 @ X331)))) | ? [X314 : term,X315 : (term > $o)] : (! [X318 : term] : ((((X315 @ X318)) = $true) | (((var @ X318)) != $true)) & ($true != ((X315 @ X314))) & ! [X319 : term] : (? [X320 : term] : (($true = ((X315 @ X320))) & (((X315 @ (sub @ X319 @ (push @ X320 @ id)))) != $true)) | ($true = ((X315 @ (lam @ X319))))) & ! [X317 : term,X316 : term] : (($true != ((X315 @ X317))) | (((X315 @ (ap @ X317 @ X316))) = $true) | ($true != ((X315 @ X316))))) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | (! [X321 : term] : (((sub @ X321 @ id)) = X321) & ($true != sP5) & ! [X311 : (subst > term > subst > $o)] : ? [X312 : (term > $o)] : ! [X313 : term] : (((X312 @ X313)) = ((X311 @ id @ X313 @ id))) & ! [X322 : term,X323 : (term > $o),X324 : (subst > term > subst > $o)] : (($true = ((X323 @ (lam @ X322)))) | ? [X326 : (subst > term > term)] : (($true != ((X324 @ id @ (lam @ (X326 @ sh @ one)) @ id))) & ! [X330 : subst,X328 : subst,X329 : term] : (((X326 @ (comp @ X328 @ X330) @ (sub @ X329 @ X330))) = ((sub @ (X326 @ X328 @ X329) @ X330))) & ! [X327 : term] : ((((X324 @ id @ X327 @ id)) != $true) | ($true = ((X324 @ id @ (X326 @ id @ X327) @ id))))) | ? [X325 : term] : ((((X323 @ X325)) = $true) & ($true != ((X323 @ (sub @ X322 @ (push @ X325 @ id)))))) | ? [X331 : term] : (((X324 @ id @ X331 @ id)) != ((X323 @ X331)))) & ! [X314 : term,X315 : (term > $o)] : (? [X318 : term] : ((((X315 @ X318)) != $true) & (((var @ X318)) = $true)) | ($true = ((X315 @ X314))) | ? [X319 : term] : (! [X320 : term] : (($true != ((X315 @ X320))) | (((X315 @ (sub @ X319 @ (push @ X320 @ id)))) = $true)) & ($true != ((X315 @ (lam @ X319))))) | ? [X317 : term,X316 : term] : (($true = ((X315 @ X317))) & (((X315 @ (ap @ X317 @ X316))) != $true) & ($true = ((X315 @ X316))))))) & (substmonoid_gthm = $true) & (pushprop = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (axclos = $true) & ((hoasinduction_lem3a = $true) | ? [X54 : (subst > term > subst > $o),X53 : term] : (! [X55 : (subst > term > term)] : (? [X59 : term] : (($true != ((X54 @ id @ (X55 @ id @ X59) @ id))) & ($true = ((X54 @ id @ X59 @ id)))) | ($true = ((X54 @ id @ (lam @ (X55 @ sh @ one)) @ id))) | ? [X57 : subst,X56 : subst,X58 : term] : (((X55 @ (comp @ X57 @ X56) @ (sub @ X58 @ X56))) != ((sub @ (X55 @ X57 @ X58) @ X56)))) & ! [X60 : term] : (($true != ((X54 @ id @ X60 @ id))) | ($true = ((X54 @ id @ (sub @ X53 @ (push @ X60 @ id)) @ id)))) & ($true != ((X54 @ id @ (lam @ X53) @ id))))) & (! [X54 : (subst > term > subst > $o),X53 : term] : (? [X55 : (subst > term > term)] : (! [X59 : term] : (($true = ((X54 @ id @ (X55 @ id @ X59) @ id))) | ($true != ((X54 @ id @ X59 @ id)))) & ($true != ((X54 @ id @ (lam @ (X55 @ sh @ one)) @ id))) & ! [X57 : subst,X56 : subst,X58 : term] : (((X55 @ (comp @ X57 @ X56) @ (sub @ X58 @ X56))) = ((sub @ (X55 @ X57 @ X58) @ X56)))) | ? [X60 : term] : (($true = ((X54 @ id @ X60 @ id))) & ($true != ((X54 @ id @ (sub @ X53 @ (push @ X60 @ id)) @ id)))) | ($true = ((X54 @ id @ (lam @ X53) @ id)))) | (hoasinduction_lem3a != $true)) & (? [X49 : term] : (((sub @ X49 @ id)) != X49) | ! [X50 : term] : (((sub @ X50 @ id)) = X50) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | (! [X49 : term] : (((sub @ X49 @ id)) = X49) & ? [X50 : term] : (((sub @ X50 @ id)) != X50))) & (ulamvarsh = $true) & ((hoasinduction_lem3v2_f = $true) | ? [X100 : term] : ! [X101 : (subst > term > term)] : ? [X102 : subst,X103 : term] : (((X101 @ X102 @ X103)) != ((sub @ X100 @ (push @ X103 @ X102))))) & (! [X100 : term] : ? [X101 : (subst > term > term)] : ! [X102 : subst,X103 : term] : (((X101 @ X102 @ X103)) = ((sub @ X100 @ (push @ X103 @ X102)))) | (hoasinduction_lem3v2_f != $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((substmonoid = $true) | ? [X224 : subst] : (((comp @ X224 @ id)) != X224) | ? [X223 : subst] : (((comp @ id @ X223)) != X223)) & ((! [X224 : subst] : (((comp @ X224 @ id)) = X224) & ! [X223 : subst] : (((comp @ id @ X223)) = X223)) | (substmonoid != $true)) & ((hoasinduction_lem3b_lthm = $true) | ? [X225 : term] : ! [X226 : (subst > term > term)] : (((sub @ X225 @ (push @ one @ sh))) != ((X226 @ sh @ one)))) & (! [X225 : term] : ? [X226 : (subst > term > term)] : (((sub @ X225 @ (push @ one @ sh))) = ((X226 @ sh @ one))) | (hoasinduction_lem3b_lthm != $true)) & (! [X108 : (term > $o),X106 : (term > $o),X109 : subst,X107 : term] : (? [X110 : term] : (((X106 @ X110)) != ((X108 @ (sub @ X110 @ (push @ X107 @ X109))))) | ($true != ((X108 @ X107))) | ($true = ((X106 @ one)))) | ? [X104 : subst,X105 : term] : (((sub @ one @ (push @ X105 @ X104))) != X105) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (? [X108 : (term > $o),X106 : (term > $o),X109 : subst,X107 : term] : (! [X110 : term] : (((X106 @ X110)) = ((X108 @ (sub @ X110 @ (push @ X107 @ X109))))) & ($true = ((X108 @ X107))) & ($true != ((X106 @ one)))) & ! [X104 : subst,X105 : term] : (((sub @ one @ (push @ X105 @ X104))) = X105))) & (axassoc = $true) & (? [X88 : subst] : (((comp @ id @ X88)) != X88) | (! [X91 : subst] : (((comp @ id @ X91)) = X91) & ! [X90 : subst] : (((comp @ X90 @ id)) = X90)) | ? [X89 : subst] : (((comp @ X89 @ id)) != X89) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | (! [X88 : subst] : (((comp @ id @ X88)) = X88) & (? [X91 : subst] : (((comp @ id @ X91)) != X91) | ? [X90 : subst] : (((comp @ X90 @ id)) != X90)) & ! [X89 : subst] : (((comp @ X89 @ id)) = X89))) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (? [X304 : term,X302 : term,X303 : term,X301 : term] : ((((ap @ X301 @ X303)) = ((ap @ X304 @ X302))) & (X301 != X304)) | ? [X300 : term] : (((sub @ X300 @ id)) != X300) | ! [X307 : term,X306 : term,X308 : term,X305 : term] : ((((ap @ (sub @ X305 @ id) @ X307)) != ((ap @ (sub @ X306 @ id) @ X308))) | (X305 = X306)) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | (! [X304 : term,X302 : term,X303 : term,X301 : term] : ((((ap @ X301 @ X303)) != ((ap @ X304 @ X302))) | (X301 = X304)) & ! [X300 : term] : (((sub @ X300 @ id)) = X300) & ? [X307 : term,X306 : term,X308 : term,X305 : term] : ((((ap @ (sub @ X305 @ id) @ X307)) = ((ap @ (sub @ X306 @ id) @ X308))) & (X305 != X306)))) & (hoasapinj1_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (hoasapnotvar_lthm = $true) & (one = ((d2term @ d_one))) & (axapp = $true) & (hoasinduction_lem1 = $true) & ((pushprop_lem0 = $true) | ? [X237 : subst,X238 : (term > $o),X239 : term] : ! [X240 : (term > $o)] : ? [X241 : term] : (((X238 @ (sub @ X241 @ (push @ X239 @ X237)))) != ((X240 @ X241)))) & (! [X237 : subst,X238 : (term > $o),X239 : term] : ? [X240 : (term > $o)] : ! [X241 : term] : (((X238 @ (sub @ X241 @ (push @ X239 @ X237)))) = ((X240 @ X241))) | (pushprop_lem0 != $true)) & ((induction2lem_lthm = $true) | ($true != sP3)) & (($true = sP3) | (induction2lem_lthm != $true)) & (ulamvarind = $true) & (axmap = $true) & ((hoasinduction_no_psi_cond_lthm = $true) | (! [X455 : term] : (((sub @ X455 @ id)) = X455) & ! [X456 : (term > $o),X457 : (subst > term > subst > $o),X458 : term] : (? [X459 : term] : (((X457 @ id @ X459 @ id)) != ((X456 @ X459))) | (((X456 @ (lam @ X458))) = $true) | ? [X460 : term] : ((((X456 @ (sub @ X458 @ (push @ X460 @ id)))) != $true) & (((X456 @ X460)) = $true)) | ? [X461 : (subst > term > term)] : (($true != ((X457 @ id @ (lam @ (X461 @ sh @ one)) @ id))) & ! [X462 : term] : (($true != ((X457 @ id @ X462 @ id))) | (((X457 @ id @ (X461 @ id @ X462) @ id)) = $true)) & ! [X463 : term,X465 : subst,X464 : subst] : (((sub @ (X461 @ X465 @ X463) @ X464)) = ((X461 @ (comp @ X465 @ X464) @ (sub @ X463 @ X464)))))) & ! [X449 : (term > $o),X448 : term] : (? [X451 : term,X450 : term] : ((((X449 @ X451)) = $true) & ($true != ((X449 @ (ap @ X450 @ X451)))) & ($true = ((X449 @ X450)))) | ($true = ((X449 @ X448))) | ? [X454 : term] : ((((var @ X454)) = $true) & (((X449 @ X454)) != $true)) | ? [X452 : term] : (! [X453 : term] : ((((X449 @ X453)) != $true) | ($true = ((X449 @ (sub @ X452 @ (push @ X453 @ id)))))) & ($true != ((X449 @ (lam @ X452)))))) & ! [X445 : (subst > term > subst > $o)] : ? [X446 : (term > $o)] : ! [X447 : term] : (((X445 @ id @ X447 @ id)) = ((X446 @ X447))) & ($true != sP2))) & (? [X455 : term] : (((sub @ X455 @ id)) != X455) | ? [X456 : (term > $o),X457 : (subst > term > subst > $o),X458 : term] : (! [X459 : term] : (((X457 @ id @ X459 @ id)) = ((X456 @ X459))) & (((X456 @ (lam @ X458))) != $true) & ! [X460 : term] : ((((X456 @ (sub @ X458 @ (push @ X460 @ id)))) = $true) | (((X456 @ X460)) != $true)) & ! [X461 : (subst > term > term)] : (($true = ((X457 @ id @ (lam @ (X461 @ sh @ one)) @ id))) | ? [X462 : term] : (($true = ((X457 @ id @ X462 @ id))) & (((X457 @ id @ (X461 @ id @ X462) @ id)) != $true)) | ? [X463 : term,X465 : subst,X464 : subst] : (((sub @ (X461 @ X465 @ X463) @ X464)) != ((X461 @ (comp @ X465 @ X464) @ (sub @ X463 @ X464)))))) | ? [X449 : (term > $o),X448 : term] : (! [X451 : term,X450 : term] : ((((X449 @ X451)) != $true) | ($true = ((X449 @ (ap @ X450 @ X451)))) | ($true != ((X449 @ X450)))) & ($true != ((X449 @ X448))) & ! [X454 : term] : ((((var @ X454)) != $true) | (((X449 @ X454)) = $true)) & ! [X452 : term] : (? [X453 : term] : ((((X449 @ X453)) = $true) & ($true != ((X449 @ (sub @ X452 @ (push @ X453 @ id)))))) | ($true = ((X449 @ (lam @ X452)))))) | ? [X445 : (subst > term > subst > $o)] : ! [X446 : (term > $o)] : ? [X447 : term] : (((X445 @ id @ X447 @ id)) != ((X446 @ X447))) | ($true = sP2) | (hoasinduction_no_psi_cond_lthm != $true)) & ((pushprop_lem1v2 = $true) | ? [X388 : (term > $o),X391 : (term > $o),X390 : term,X389 : subst] : (! [X392 : term] : (((X391 @ (sub @ X392 @ (push @ X390 @ X389)))) = ((X388 @ X392))) & ($true = ((X391 @ X390))) & ($true != ((X388 @ one))))) & (! [X388 : (term > $o),X391 : (term > $o),X390 : term,X389 : subst] : (? [X392 : term] : (((X391 @ (sub @ X392 @ (push @ X390 @ X389)))) != ((X388 @ X392))) | ($true != ((X391 @ X390))) | ($true = ((X388 @ one)))) | (pushprop_lem1v2 != $true)) & (hoasinduction_gthm = $true) & (! [X384 : term,X387 : subst,X386 : term,X385 : (term > $o),X383 : (term > $o)] : (($true = ((X383 @ (sub @ X384 @ (push @ X386 @ X387))))) | ($true != ((X385 @ X384)))) | ? [X379 : subst,X380 : term] : (((sub @ one @ (push @ X380 @ X379))) != X380) | ? [X381 : subst,X382 : term] : (((comp @ sh @ (push @ X382 @ X381))) != X381) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (? [X384 : term,X387 : subst,X386 : term,X385 : (term > $o),X383 : (term > $o)] : (($true != ((X383 @ (sub @ X384 @ (push @ X386 @ X387))))) & ($true = ((X385 @ X384)))) & ! [X379 : subst,X380 : term] : (((sub @ one @ (push @ X380 @ X379))) = X380) & ! [X381 : subst,X382 : term] : (((comp @ sh @ (push @ X382 @ X381))) = X381))) & (! [X491 : (term > $o),X492 : term] : (? [X494 : term,X495 : term] : (($true = ((X491 @ X494))) & ($true = ((X491 @ X495))) & ($true != ((X491 @ (ap @ X495 @ X494))))) | ? [X496 : term] : (($true != ((X491 @ (lam @ X496)))) & ! [X497 : term] : (($true != ((X491 @ X497))) | (((X491 @ (sub @ X496 @ (push @ X497 @ id)))) = $true))) | ($true = ((X491 @ X492))) | ? [X493 : term] : (($true = ((var @ X493))) & (((X491 @ X493)) != $true))) | (induction2 != $true)) & ((induction2 = $true) | ? [X491 : (term > $o),X492 : term] : (! [X494 : term,X495 : term] : (($true != ((X491 @ X494))) | ($true != ((X491 @ X495))) | ($true = ((X491 @ (ap @ X495 @ X494))))) & ! [X496 : term] : (($true = ((X491 @ (lam @ X496)))) | ? [X497 : term] : (($true = ((X491 @ X497))) & (((X491 @ (sub @ X496 @ (push @ X497 @ id)))) != $true))) & ($true != ((X491 @ X492))) & ! [X493 : term] : (($true != ((var @ X493))) | (((X491 @ X493)) = $true)))) & ((pushprop_lem1 = $true) | ? [X186 : (term > $o),X185 : term,X187 : subst,X188 : term,X184 : (term > $o)] : ((((X184 @ (sub @ X185 @ (push @ X188 @ X187)))) != $true) & ($true = ((X186 @ X185))))) & (! [X186 : (term > $o),X185 : term,X187 : subst,X188 : term,X184 : (term > $o)] : ((((X184 @ (sub @ X185 @ (push @ X188 @ X187)))) = $true) | ($true != ((X186 @ X185)))) | (pushprop_lem1 != $true)) & ((induction2_lthm = $true) | ($true != sP1)) & (($true = sP1) | (induction2_lthm != $true)) & ((hoasinduction_lem0 = $true) | ? [X126 : (subst > term > subst > $o)] : ! [X127 : (term > $o)] : ? [X128 : term] : (((X127 @ X128)) != ((X126 @ id @ X128 @ id)))) & (! [X126 : (subst > term > subst > $o)] : ? [X127 : (term > $o)] : ! [X128 : term] : (((X127 @ X128)) = ((X126 @ id @ X128 @ id))) | (hoasinduction_lem0 != $true)) & (lamnotvar = $true) & (! [X3 : term,X2 : term,X1 : term,X4 : term] : ((((ap @ X3 @ X2)) != ((ap @ X1 @ X4))) | (X2 = X4)) | (apinj2 != $true)) & ((apinj2 = $true) | ? [X3 : term,X2 : term,X1 : term,X4 : term] : ((((ap @ X3 @ X2)) = ((ap @ X1 @ X4))) & (X2 != X4))) & (! [X65 : (term > $o),X66 : term] : (? [X70 : term] : ((((X65 @ (lam @ X70))) != $true) & ($true = ((X65 @ X70)))) | ($true = ((X65 @ X66))) | ? [X69 : term,X68 : term] : (($true != ((X65 @ (ap @ X68 @ X69)))) & ($true = ((X65 @ X68))) & (((X65 @ X69)) = $true)) | ? [X67 : term] : (($true = ((var @ X67))) & ($true != ((X65 @ X67))))) | (induction != $true)) & ((induction = $true) | ? [X65 : (term > $o),X66 : term] : (! [X70 : term] : ((((X65 @ (lam @ X70))) = $true) | ($true != ((X65 @ X70)))) & ($true != ((X65 @ X66))) & ! [X69 : term,X68 : term] : (($true = ((X65 @ (ap @ X68 @ X69)))) | ($true != ((X65 @ X68))) | (((X65 @ X69)) != $true)) & ! [X67 : term] : (($true != ((var @ X67))) | ($true = ((X65 @ X67)))))) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoaslamnotvar_lthm = $true) & ((hoasinduction_lem2v2 = $true) | ? [X431 : (subst > term > subst > $o),X430 : term,X432 : (term > $o),X433 : term] : (! [X438 : subst,X439 : term,X440 : subst,X437 : subst] : ((((X431 @ X437 @ X439 @ (comp @ X438 @ X440))) = $true) | ($true != ((X431 @ (comp @ X437 @ X438) @ (sub @ X439 @ X438) @ X440)))) & ($true = ((X432 @ X433))) & (((X432 @ X430)) = $true) & ! [X435 : term,X436 : term] : (($true != ((X431 @ id @ X435 @ id))) | ($true = ((X431 @ id @ (ap @ (sub @ X435 @ id) @ X436) @ id))) | ($true != ((X431 @ id @ X436 @ id)))) & ($true != ((X432 @ (ap @ X433 @ X430)))) & ! [X434 : term] : (((X432 @ X434)) = ((X431 @ id @ X434 @ id))) & ! [X444 : subst,X442 : subst,X441 : subst,X443 : term] : (($true = ((X431 @ (comp @ X442 @ X441) @ (sub @ X443 @ X441) @ X444))) | ($true != ((X431 @ X442 @ X443 @ (comp @ X441 @ X444))))))) & (! [X431 : (subst > term > subst > $o),X430 : term,X432 : (term > $o),X433 : term] : (? [X438 : subst,X439 : term,X440 : subst,X437 : subst] : ((((X431 @ X437 @ X439 @ (comp @ X438 @ X440))) != $true) & ($true = ((X431 @ (comp @ X437 @ X438) @ (sub @ X439 @ X438) @ X440)))) | ($true != ((X432 @ X433))) | (((X432 @ X430)) != $true) | ? [X435 : term,X436 : term] : (($true = ((X431 @ id @ X435 @ id))) & ($true != ((X431 @ id @ (ap @ (sub @ X435 @ id) @ X436) @ id))) & ($true = ((X431 @ id @ X436 @ id)))) | ($true = ((X432 @ (ap @ X433 @ X430)))) | ? [X434 : term] : (((X432 @ X434)) != ((X431 @ id @ X434 @ id))) | ? [X444 : subst,X442 : subst,X441 : subst,X443 : term] : (($true != ((X431 @ (comp @ X442 @ X441) @ (sub @ X443 @ X441) @ X444))) & ($true = ((X431 @ X442 @ X443 @ (comp @ X441 @ X444)))))) | (hoasinduction_lem2v2 != $true)) & (hoasinduction_lem3a_gthm = $true) & (pushprop_lem1_gthm = $true) & (hoasapinj2_gthm = $true) & (id = ((d2subst @ d_id))) & (pushprop_gthm = $true) & (! [X189 : term] : ? [X190 : (subst > term > term)] : ! [X192 : term,X191 : subst] : (((X190 @ X191 @ X192)) = ((sub @ X189 @ (push @ X192 @ X191)))) | (hoasinduction_lem3v2_f_lthm != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X189 : term] : ! [X190 : (subst > term > term)] : ? [X192 : term,X191 : subst] : (((X190 @ X191 @ X192)) != ((sub @ X189 @ (push @ X192 @ X191))))) & ((hoasinduction_lem3v2_lthm = $true) | (! [X273 : term] : (((sub @ X273 @ id)) = X273) & ? [X274 : (subst > term > subst > $o),X275 : term,X276 : (term > $o)] : (! [X277 : term] : (((X276 @ X277)) = ((X274 @ id @ X277 @ id))) & ($true != ((X276 @ (lam @ X275)))) & ! [X289 : subst,X290 : term,X288 : subst,X291 : subst] : (($true != ((X274 @ (comp @ X288 @ X291) @ (sub @ X290 @ X291) @ X289))) | ($true = ((X274 @ X288 @ X290 @ (comp @ X291 @ X289))))) & ! [X283 : term,X285 : subst,X284 : subst,X286 : subst] : ((((X274 @ (comp @ X285 @ X286) @ (sub @ X283 @ X286) @ X284)) = $true) | (((X274 @ X285 @ X283 @ (comp @ X286 @ X284))) != $true)) & ! [X278 : (subst > term > term)] : (? [X281 : subst,X280 : subst,X282 : term] : (((sub @ (X278 @ X280 @ X282) @ X281)) != ((X278 @ (comp @ X280 @ X281) @ (sub @ X282 @ X281)))) | ? [X279 : term] : ((((X274 @ id @ X279 @ id)) = $true) & ($true != ((X274 @ id @ (X278 @ id @ X279) @ id)))) | ($true = ((X274 @ id @ (lam @ (X278 @ sh @ one)) @ id)))) & ! [X287 : term] : (($true = ((X276 @ (sub @ X275 @ (push @ X287 @ id))))) | ($true != ((X276 @ X287))))))) & (? [X273 : term] : (((sub @ X273 @ id)) != X273) | ! [X274 : (subst > term > subst > $o),X275 : term,X276 : (term > $o)] : (? [X277 : term] : (((X276 @ X277)) != ((X274 @ id @ X277 @ id))) | ($true = ((X276 @ (lam @ X275)))) | ? [X289 : subst,X290 : term,X288 : subst,X291 : subst] : (($true = ((X274 @ (comp @ X288 @ X291) @ (sub @ X290 @ X291) @ X289))) & ($true != ((X274 @ X288 @ X290 @ (comp @ X291 @ X289))))) | ? [X283 : term,X285 : subst,X284 : subst,X286 : subst] : ((((X274 @ (comp @ X285 @ X286) @ (sub @ X283 @ X286) @ X284)) != $true) & (((X274 @ X285 @ X283 @ (comp @ X286 @ X284))) = $true)) | ? [X278 : (subst > term > term)] : (! [X281 : subst,X280 : subst,X282 : term] : (((sub @ (X278 @ X280 @ X282) @ X281)) = ((X278 @ (comp @ X280 @ X281) @ (sub @ X282 @ X281)))) & ! [X279 : term] : ((((X274 @ id @ X279 @ id)) != $true) | ($true = ((X274 @ id @ (X278 @ id @ X279) @ id)))) & ($true != ((X274 @ id @ (lam @ (X278 @ sh @ one)) @ id)))) | ? [X287 : term] : (($true != ((X276 @ (sub @ X275 @ (push @ X287 @ id))))) & ($true = ((X276 @ X287))))) | (hoasinduction_lem3v2_lthm != $true)) & (! [X270 : term,X271 : term,X269 : term,X268 : term] : ((((ap @ X269 @ X270)) != ((ap @ X271 @ X268))) | (X269 = X271)) | (apinj1 != $true)) & ((apinj1 = $true) | ? [X270 : term,X271 : term,X269 : term,X268 : term] : ((((ap @ X269 @ X270)) = ((ap @ X271 @ X268))) & (X269 != X271))) & (! [X5 : (term > $o),X7 : subst,X6 : term] : ? [X8 : (term > $o)] : ! [X9 : term] : (((X5 @ (sub @ X9 @ (push @ X6 @ X7)))) = ((X8 @ X9))) | (pushprop_lem0_lthm != $true)) & ((pushprop_lem0_lthm = $true) | ? [X5 : (term > $o),X7 : subst,X6 : term] : ! [X8 : (term > $o)] : ? [X9 : term] : (((X5 @ (sub @ X9 @ (push @ X6 @ X7)))) != ((X8 @ X9)))) & (hoasinduction_lem3v2_gthm = $true) & (hoasinduction_lem3_gthm = $true) & (hoaslaminj_gthm = $true) & (hoasinduction_lem2_gthm = $true) & (! [X243 : (subst > term > subst > $o),X242 : term,X244 : (term > $o)] : (? [X259 : term] : (($true = ((X244 @ X259))) & (((X244 @ (sub @ X242 @ (push @ X259 @ id)))) != $true)) | ? [X249 : term] : (((X244 @ X249)) != ((X243 @ id @ X249 @ id))) | ? [X248 : subst,X246 : subst,X247 : term,X245 : subst] : (($true = ((X243 @ X246 @ X247 @ (comp @ X248 @ X245)))) & ($true != ((X243 @ (comp @ X246 @ X248) @ (sub @ X247 @ X248) @ X245)))) | ? [X253 : term,X251 : subst,X252 : subst,X250 : subst] : (($true = ((X243 @ (comp @ X251 @ X252) @ (sub @ X253 @ X252) @ X250))) & (((X243 @ X251 @ X253 @ (comp @ X252 @ X250))) != $true)) | ($true = ((X244 @ (lam @ X242)))) | ? [X254 : (subst > term > term)] : (! [X255 : term] : (($true != ((X243 @ id @ X255 @ id))) | ($true = ((X243 @ id @ (X254 @ id @ X255) @ id)))) & ! [X256 : subst,X257 : subst,X258 : term] : (((sub @ (X254 @ X257 @ X258) @ X256)) = ((X254 @ (comp @ X257 @ X256) @ (sub @ X258 @ X256)))) & ($true != ((X243 @ id @ (lam @ (X254 @ sh @ one)) @ id))))) | (hoasinduction_lem3v2 != $true)) & ((hoasinduction_lem3v2 = $true) | ? [X243 : (subst > term > subst > $o),X242 : term,X244 : (term > $o)] : (! [X259 : term] : (($true != ((X244 @ X259))) | (((X244 @ (sub @ X242 @ (push @ X259 @ id)))) = $true)) & ! [X249 : term] : (((X244 @ X249)) = ((X243 @ id @ X249 @ id))) & ! [X248 : subst,X246 : subst,X247 : term,X245 : subst] : (($true != ((X243 @ X246 @ X247 @ (comp @ X248 @ X245)))) | ($true = ((X243 @ (comp @ X246 @ X248) @ (sub @ X247 @ X248) @ X245)))) & ! [X253 : term,X251 : subst,X252 : subst,X250 : subst] : (($true != ((X243 @ (comp @ X251 @ X252) @ (sub @ X253 @ X252) @ X250))) | (((X243 @ X251 @ X253 @ (comp @ X252 @ X250))) = $true)) & ($true != ((X244 @ (lam @ X242)))) & ! [X254 : (subst > term > term)] : (? [X255 : term] : (($true = ((X243 @ id @ X255 @ id))) & ($true != ((X243 @ id @ (X254 @ id @ X255) @ id)))) | ? [X256 : subst,X257 : subst,X258 : term] : (((sub @ (X254 @ X257 @ X258) @ X256)) != ((X254 @ (comp @ X257 @ X256) @ (sub @ X258 @ X256)))) | ($true = ((X243 @ id @ (lam @ (X254 @ sh @ one)) @ id)))))) & ((axshiftcons = $true) | ? [X112 : term,X113 : subst] : (((comp @ sh @ (push @ X112 @ X113))) != X113)) & (! [X112 : term,X113 : subst] : (((comp @ sh @ (push @ X112 @ X113))) = X113) | (axshiftcons != $true)) & ((hoasinduction_lem3_lthm = $true) | (($true != sP0) & ! [X498 : term] : (((sub @ X498 @ id)) = X498) & ! [X499 : (subst > term > subst > $o),X500 : term] : (? [X501 : term] : (($true = ((X499 @ id @ X501 @ id))) & ($true != ((X499 @ id @ (sub @ X500 @ (push @ X501 @ id)) @ id)))) | ($true = ((X499 @ id @ (lam @ (sub @ X500 @ (push @ one @ sh))) @ id))) | ? [X502 : (subst > term > term)] : (! [X505 : subst,X506 : term,X504 : subst] : (((sub @ (X502 @ X504 @ X506) @ X505)) = ((X502 @ (comp @ X504 @ X505) @ (sub @ X506 @ X505)))) & ! [X503 : term] : (($true = ((X499 @ id @ (X502 @ id @ X503) @ id))) | ($true != ((X499 @ id @ X503 @ id)))) & ($true != ((X499 @ id @ (lam @ (X502 @ sh @ one)) @ id))))))) & (($true = sP0) | ? [X498 : term] : (((sub @ X498 @ id)) != X498) | ? [X499 : (subst > term > subst > $o),X500 : term] : (! [X501 : term] : (($true != ((X499 @ id @ X501 @ id))) | ($true = ((X499 @ id @ (sub @ X500 @ (push @ X501 @ id)) @ id)))) & ($true != ((X499 @ id @ (lam @ (sub @ X500 @ (push @ one @ sh))) @ id))) & ! [X502 : (subst > term > term)] : (? [X505 : subst,X506 : term,X504 : subst] : (((sub @ (X502 @ X504 @ X506) @ X505)) != ((X502 @ (comp @ X504 @ X505) @ (sub @ X506 @ X505)))) | ? [X503 : term] : (($true != ((X499 @ id @ (X502 @ id @ X503) @ id))) & ($true = ((X499 @ id @ X503 @ id)))) | ($true = ((X499 @ id @ (lam @ (X502 @ sh @ one)) @ id))))) | (hoasinduction_lem3_lthm != $true))),
  inference(flattening,[],[f65])).
thf(f67,plain,(
  (hoaslamnotap_lthm = $true) & (axabs = $true) & (! [X0 : (subst > term > subst > $o),X1 : term] : ((((X0 @ id @ (lam @ (sub @ X1 @ (push @ one @ sh))) @ id)) = $true) | ? [X2 : term] : (($true = ((X0 @ id @ X2 @ id))) & ($true != ((X0 @ id @ (sub @ X1 @ (push @ X2 @ id)) @ id)))) | ? [X3 : (subst > term > term)] : (($true != ((X0 @ id @ (lam @ (X3 @ sh @ one)) @ id))) & ! [X4 : subst,X5 : term,X6 : subst] : (((X3 @ (comp @ X6 @ X4) @ (sub @ X5 @ X4))) = ((sub @ (X3 @ X6 @ X5) @ X4))) & ! [X7 : term] : ((((X0 @ id @ X7 @ id)) != $true) | (((X0 @ id @ (X3 @ id @ X7) @ id)) = $true)))) | (hoasinduction_lem3aa != $true)) & ((hoasinduction_lem3aa = $true) | ? [X8 : (subst > term > subst > $o),X9 : term] : (($true != ((X8 @ id @ (lam @ (sub @ X9 @ (push @ one @ sh))) @ id))) & ! [X10 : term] : ((((X8 @ id @ X10 @ id)) != $true) | (((X8 @ id @ (sub @ X9 @ (push @ X10 @ id)) @ id)) = $true)) & ! [X11 : (subst > term > term)] : (($true = ((X8 @ id @ (lam @ (X11 @ sh @ one)) @ id))) | ? [X12 : subst,X13 : term,X14 : subst] : (((X11 @ (comp @ X14 @ X12) @ (sub @ X13 @ X12))) != ((sub @ (X11 @ X14 @ X13) @ X12))) | ? [X15 : term] : (($true = ((X8 @ id @ X15 @ id))) & ($true != ((X8 @ id @ (X11 @ id @ X15) @ id))))))) & (pushprop_lem2v2_gthm = $true) & ((induction2lem = $true) | ? [X16 : term,X17 : subst,X18 : (term > $o)] : (! [X19 : term] : (($true = ((X18 @ (lam @ X19)))) | ? [X20 : term] : (($true != ((X18 @ (sub @ X19 @ (push @ X20 @ id))))) & ($true = ((X18 @ X20))))) & ! [X21 : term,X22 : term] : (($true != ((X18 @ X22))) | (((X18 @ (ap @ X21 @ X22))) = $true) | ($true != ((X18 @ X21)))) & ! [X23 : term] : (($true = ((X18 @ (sub @ X23 @ X17)))) | (((var @ X23)) != $true)) & ($true != ((X18 @ (sub @ X16 @ X17)))))) & (! [X24 : term,X25 : subst,X26 : (term > $o)] : (? [X27 : term] : (($true != ((X26 @ (lam @ X27)))) & ! [X28 : term] : (($true = ((X26 @ (sub @ X27 @ (push @ X28 @ id))))) | ($true != ((X26 @ X28))))) | ? [X29 : term,X30 : term] : ((((X26 @ X30)) = $true) & ($true != ((X26 @ (ap @ X29 @ X30)))) & ($true = ((X26 @ X29)))) | ? [X31 : term] : (($true != ((X26 @ (sub @ X31 @ X25)))) & ($true = ((var @ X31)))) | (((X26 @ (sub @ X24 @ X25))) = $true)) | (induction2lem != $true)) & (lamnotap != $true) & ((hoasinduction_lem3v2a_lthm = $true) | (? [X32 : (subst > term > subst > $o),X33 : (term > $o),X34 : term] : (($true != ((X33 @ (lam @ X34)))) & ! [X35 : term] : (((X33 @ X35)) = ((X32 @ id @ X35 @ id))) & ! [X36 : (subst > term > term)] : ((((X32 @ id @ (lam @ (X36 @ sh @ one)) @ id)) = $true) | ? [X37 : term] : (($true != ((X32 @ id @ (X36 @ id @ X37) @ id))) & (((X32 @ id @ X37 @ id)) = $true)) | ? [X38 : subst,X39 : term,X40 : subst] : (((X36 @ (comp @ X38 @ X40) @ (sub @ X39 @ X40))) != ((sub @ (X36 @ X38 @ X39) @ X40)))) & ! [X41 : term] : ((((X33 @ (sub @ X34 @ (push @ X41 @ id)))) = $true) | (((X33 @ X41)) != $true))) & ! [X42 : term] : ? [X43 : (subst > term > term)] : ! [X44 : term,X45 : subst] : (((X43 @ X45 @ X44)) = ((sub @ X42 @ (push @ X44 @ X45)))) & ! [X46 : term] : (((sub @ X46 @ id)) = X46))) & (! [X47 : (subst > term > subst > $o),X48 : (term > $o),X49 : term] : (($true = ((X48 @ (lam @ X49)))) | ? [X50 : term] : (((X47 @ id @ X50 @ id)) != ((X48 @ X50))) | ? [X51 : (subst > term > term)] : (($true != ((X47 @ id @ (lam @ (X51 @ sh @ one)) @ id))) & ! [X52 : term] : (($true = ((X47 @ id @ (X51 @ id @ X52) @ id))) | ($true != ((X47 @ id @ X52 @ id)))) & ! [X53 : subst,X54 : term,X55 : subst] : (((X51 @ (comp @ X53 @ X55) @ (sub @ X54 @ X55))) = ((sub @ (X51 @ X53 @ X54) @ X55)))) | ? [X56 : term] : ((((X48 @ (sub @ X49 @ (push @ X56 @ id)))) != $true) & ($true = ((X48 @ X56))))) | ? [X57 : term] : ! [X58 : (subst > term > term)] : ? [X59 : term,X60 : subst] : (((sub @ X57 @ (push @ X59 @ X60))) != ((X58 @ X60 @ X59))) | ? [X61 : term] : (((sub @ X61 @ id)) != X61) | (hoasinduction_lem3v2a_lthm != $true)) & (! [X62 : term,X63 : term,X64 : term,X65 : term] : ((X62 = X63) | (((ap @ (sub @ X64 @ id) @ X63)) != ((ap @ (sub @ X65 @ id) @ X62)))) | (hoasapinj2 != $true)) & ((hoasapinj2 = $true) | ? [X66 : term,X67 : term,X68 : term,X69 : term] : ((X66 != X67) & (((ap @ (sub @ X69 @ id) @ X66)) = ((ap @ (sub @ X68 @ id) @ X67))))) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem1_gthm = $true) & (! [X70 : term,X71 : (subst > term > subst > $o)] : (? [X72 : (subst > term > term)] : (($true != ((X71 @ id @ (lam @ (X72 @ sh @ one)) @ id))) & ! [X73 : term,X74 : subst,X75 : subst] : (((X72 @ (comp @ X74 @ X75) @ (sub @ X73 @ X75))) = ((sub @ (X72 @ X74 @ X73) @ X75))) & ! [X76 : term] : ((((X71 @ id @ (X72 @ id @ X76) @ id)) = $true) | ($true != ((X71 @ id @ X76 @ id))))) | ? [X77 : term] : (($true != ((X71 @ id @ (sub @ X70 @ (push @ X77 @ id)) @ id))) & (((X71 @ id @ X77 @ id)) = $true)) | ($true = ((X71 @ id @ (lam @ (sub @ X70 @ (push @ one @ sh))) @ id)))) | (hoasinduction_lem3aa_lthm != $true)) & ((hoasinduction_lem3aa_lthm = $true) | ? [X78 : term,X79 : (subst > term > subst > $o)] : (! [X80 : (subst > term > term)] : ((((X79 @ id @ (lam @ (X80 @ sh @ one)) @ id)) = $true) | ? [X81 : term,X82 : subst,X83 : subst] : (((sub @ (X80 @ X82 @ X81) @ X83)) != ((X80 @ (comp @ X82 @ X83) @ (sub @ X81 @ X83)))) | ? [X84 : term] : (($true != ((X79 @ id @ (X80 @ id @ X84) @ id))) & ($true = ((X79 @ id @ X84 @ id))))) & ! [X85 : term] : (($true = ((X79 @ id @ (sub @ X78 @ (push @ X85 @ id)) @ id))) | ($true != ((X79 @ id @ X85 @ id)))) & ($true != ((X79 @ id @ (lam @ (sub @ X78 @ (push @ one @ sh))) @ id))))) & (! [X86 : subst] : (((comp @ id @ X86)) = X86) | (axidl != $true)) & ((axidl = $true) | ? [X87 : subst] : (((comp @ id @ X87)) != X87)) & ((hoaslamnotap = $true) | ? [X88 : term,X89 : term,X90 : (subst > term > term)] : (! [X91 : subst,X92 : subst,X93 : term] : (((sub @ (X90 @ X91 @ X93) @ X92)) = ((X90 @ (comp @ X91 @ X92) @ (sub @ X93 @ X92)))) & (((lam @ (X90 @ sh @ one))) = ((ap @ (sub @ X88 @ id) @ X89))))) & (! [X94 : term,X95 : term,X96 : (subst > term > term)] : (? [X97 : subst,X98 : subst,X99 : term] : (((sub @ (X96 @ X97 @ X99) @ X98)) != ((X96 @ (comp @ X97 @ X98) @ (sub @ X99 @ X98)))) | (((ap @ (sub @ X94 @ id) @ X95)) != ((lam @ (X96 @ sh @ one))))) | (hoaslamnotap != $true)) & (! [X100 : term,X101 : term] : ((((sub @ X101 @ sh)) != ((sub @ X100 @ sh))) | (X100 = X101)) | (shinj != $true)) & ((shinj = $true) | ? [X102 : term,X103 : term] : ((((sub @ X103 @ sh)) = ((sub @ X102 @ sh))) & (X102 != X103))) & (! [X104 : term,X105 : term,X106 : (subst > term > subst > $o)] : (($true != ((X106 @ id @ X105 @ id))) | (((X106 @ id @ X104 @ id)) != $true) | ($true = ((X106 @ id @ (ap @ X104 @ X105) @ id))) | ? [X107 : subst,X108 : subst,X109 : term,X110 : subst] : (($true = ((X106 @ (comp @ X110 @ X108) @ (sub @ X109 @ X108) @ X107))) & ($true != ((X106 @ X110 @ X109 @ (comp @ X108 @ X107))))) | ? [X111 : subst,X112 : subst,X113 : term,X114 : subst] : ((((X106 @ (comp @ X111 @ X114) @ (sub @ X113 @ X114) @ X112)) != $true) & ($true = ((X106 @ X111 @ X113 @ (comp @ X114 @ X112))))) | ? [X115 : term,X116 : term] : (($true != ((X106 @ id @ (ap @ (sub @ X116 @ id) @ X115) @ id))) & ($true = ((X106 @ id @ X116 @ id))) & ($true = ((X106 @ id @ X115 @ id))))) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | ? [X117 : term,X118 : term,X119 : (subst > term > subst > $o)] : (($true = ((X119 @ id @ X118 @ id))) & ($true = ((X119 @ id @ X117 @ id))) & ($true != ((X119 @ id @ (ap @ X117 @ X118) @ id))) & ! [X120 : subst,X121 : subst,X122 : term,X123 : subst] : (($true != ((X119 @ (comp @ X123 @ X121) @ (sub @ X122 @ X121) @ X120))) | ($true = ((X119 @ X123 @ X122 @ (comp @ X121 @ X120))))) & ! [X124 : subst,X125 : subst,X126 : term,X127 : subst] : (($true = ((X119 @ (comp @ X124 @ X127) @ (sub @ X126 @ X127) @ X125))) | (((X119 @ X124 @ X126 @ (comp @ X127 @ X125))) != $true)) & ! [X128 : term,X129 : term] : ((((X119 @ id @ (ap @ (sub @ X129 @ id) @ X128) @ id)) = $true) | ($true != ((X119 @ id @ X129 @ id))) | ($true != ((X119 @ id @ X128 @ id)))))) & (pushprop_lem3v2 = $true) & ! [X130 : d_subst,X131 : d_subst] : ((((d2subst @ X130)) != ((d2subst @ X131))) | (X130 = X131)) & (! [X132 : term,X133 : term,X134 : term,X135 : term] : ((((ap @ (sub @ X135 @ id) @ X133)) != ((ap @ (sub @ X132 @ id) @ X134))) | (X132 = X135)) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | ? [X136 : term,X137 : term,X138 : term,X139 : term] : ((((ap @ (sub @ X139 @ id) @ X137)) = ((ap @ (sub @ X136 @ id) @ X138))) & (X136 != X139))) & ((axvarcons = $true) | ? [X140 : subst,X141 : term] : (((sub @ one @ (push @ X141 @ X140))) != X141)) & (! [X142 : subst,X143 : term] : (((sub @ one @ (push @ X143 @ X142))) = X143) | (axvarcons != $true)) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((hoasinduction_lthm = $true) | ($true != sP10)) & (($true = sP10) | (hoasinduction_lthm != $true)) & ! [X144 : d_term,X145 : d_term] : ((((d2term @ X144)) != ((d2term @ X145))) | (X144 = X145)) & (hoaslamnotvar_gthm = $true) & (hoaslamnotvar = $true) & ((hoasinduction = $true) | ? [X146 : term,X147 : (subst > term > subst > $o)] : (! [X148 : term,X149 : term] : ((((X147 @ id @ (ap @ (sub @ X149 @ id) @ X148) @ id)) = $true) | ($true != ((X147 @ id @ X149 @ id))) | (((X147 @ id @ X148 @ id)) != $true)) & ! [X150 : subst,X151 : subst,X152 : term,X153 : subst] : (($true = ((X147 @ (comp @ X151 @ X150) @ (sub @ X152 @ X150) @ X153))) | ($true != ((X147 @ X151 @ X152 @ (comp @ X150 @ X153))))) & ! [X154 : term] : (($true != ((var @ (sub @ X154 @ id)))) | ($true = ((X147 @ id @ X154 @ id)))) & ($true != ((X147 @ id @ X146 @ id))) & ($true != ((sP6 @ X147))) & ! [X155 : subst,X156 : subst,X157 : subst,X158 : term] : (($true != ((X147 @ (comp @ X156 @ X155) @ (sub @ X158 @ X155) @ X157))) | ($true = ((X147 @ X156 @ X158 @ (comp @ X155 @ X157))))))) & (! [X159 : term,X160 : (subst > term > subst > $o)] : (? [X161 : term,X162 : term] : (($true != ((X160 @ id @ (ap @ (sub @ X162 @ id) @ X161) @ id))) & ($true = ((X160 @ id @ X162 @ id))) & ($true = ((X160 @ id @ X161 @ id)))) | ? [X163 : subst,X164 : subst,X165 : term,X166 : subst] : ((((X160 @ (comp @ X164 @ X163) @ (sub @ X165 @ X163) @ X166)) != $true) & ($true = ((X160 @ X164 @ X165 @ (comp @ X163 @ X166))))) | ? [X167 : term] : (($true = ((var @ (sub @ X167 @ id)))) & ($true != ((X160 @ id @ X167 @ id)))) | ($true = ((X160 @ id @ X159 @ id))) | ($true = ((sP6 @ X160))) | ? [X168 : subst,X169 : subst,X170 : subst,X171 : term] : (($true = ((X160 @ (comp @ X169 @ X168) @ (sub @ X171 @ X168) @ X170))) & ($true != ((X160 @ X169 @ X171 @ (comp @ X168 @ X170)))))) | (hoasinduction != $true)) & (pushprop_lem3v2_lthm = $true) & (! [X172 : subst] : (((push @ (sub @ one @ X172) @ (comp @ sh @ X172))) = X172) | (axscons != $true)) & ((axscons = $true) | ? [X173 : subst] : (((push @ (sub @ one @ X173) @ (comp @ sh @ X173))) != X173)) & (hoasinduction_lem3b_gthm = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ! [X174 : d_term] : (d_one = X174) & ((hoasapinj2_lthm = $true) | (! [X175 : term,X176 : term,X177 : term,X178 : term] : ((X175 = X177) | (((ap @ X176 @ X175)) != ((ap @ X178 @ X177)))) & ? [X179 : term,X180 : term,X181 : term,X182 : term] : ((X181 != X182) & (((ap @ (sub @ X179 @ id) @ X181)) = ((ap @ (sub @ X180 @ id) @ X182)))))) & (? [X183 : term,X184 : term,X185 : term,X186 : term] : ((X183 != X185) & (((ap @ X186 @ X185)) = ((ap @ X184 @ X183)))) | ! [X187 : term,X188 : term,X189 : term,X190 : term] : ((X189 = X190) | (((ap @ (sub @ X187 @ id) @ X189)) != ((ap @ (sub @ X188 @ id) @ X190)))) | (hoasapinj2_lthm != $true)) & (axvarshift = $true) & (! [X191 : term] : ? [X192 : (subst > term > term)] : (((sub @ X191 @ (push @ one @ sh))) = ((X192 @ sh @ one))) | (hoasinduction_lem3b != $true)) & ((hoasinduction_lem3b = $true) | ? [X193 : term] : ! [X194 : (subst > term > term)] : (((X194 @ sh @ one)) != ((sub @ X193 @ (push @ one @ sh))))) & ((hoasinduction_lem3a_lthm = $true) | (! [X195 : (subst > term > subst > $o),X196 : term] : (? [X197 : (subst > term > term)] : (! [X198 : term,X199 : subst,X200 : subst] : (((sub @ (X197 @ X199 @ X198) @ X200)) = ((X197 @ (comp @ X199 @ X200) @ (sub @ X198 @ X200)))) & ! [X201 : term] : ((((X195 @ id @ X201 @ id)) != $true) | (((X195 @ id @ (X197 @ id @ X201) @ id)) = $true)) & (((X195 @ id @ (lam @ (X197 @ sh @ one)) @ id)) != $true)) | ? [X202 : term] : (($true = ((X195 @ id @ X202 @ id))) & (((X195 @ id @ (sub @ X196 @ (push @ X202 @ id)) @ id)) != $true)) | ($true = ((X195 @ id @ (lam @ (sub @ X196 @ (push @ one @ sh))) @ id)))) & ! [X203 : term] : (((sub @ X203 @ id)) = X203) & ? [X204 : (subst > term > subst > $o),X205 : term] : (! [X206 : (subst > term > term)] : (? [X207 : subst,X208 : subst,X209 : term] : (((sub @ (X206 @ X207 @ X209) @ X208)) != ((X206 @ (comp @ X207 @ X208) @ (sub @ X209 @ X208)))) | ? [X210 : term] : (($true != ((X204 @ id @ (X206 @ id @ X210) @ id))) & ($true = ((X204 @ id @ X210 @ id)))) | (((X204 @ id @ (lam @ (X206 @ sh @ one)) @ id)) = $true)) & ! [X211 : term] : (($true = ((X204 @ id @ (sub @ X205 @ (push @ X211 @ id)) @ id))) | ($true != ((X204 @ id @ X211 @ id)))) & ($true != ((X204 @ id @ (lam @ X205) @ id)))))) & (? [X212 : (subst > term > subst > $o),X213 : term] : (! [X214 : (subst > term > term)] : (? [X215 : term,X216 : subst,X217 : subst] : (((sub @ (X214 @ X216 @ X215) @ X217)) != ((X214 @ (comp @ X216 @ X217) @ (sub @ X215 @ X217)))) | ? [X218 : term] : ((((X212 @ id @ X218 @ id)) = $true) & ($true != ((X212 @ id @ (X214 @ id @ X218) @ id)))) | (((X212 @ id @ (lam @ (X214 @ sh @ one)) @ id)) = $true)) & ! [X219 : term] : (($true != ((X212 @ id @ X219 @ id))) | ($true = ((X212 @ id @ (sub @ X213 @ (push @ X219 @ id)) @ id)))) & (((X212 @ id @ (lam @ (sub @ X213 @ (push @ one @ sh))) @ id)) != $true)) | ? [X220 : term] : (((sub @ X220 @ id)) != X220) | ! [X221 : (subst > term > subst > $o),X222 : term] : (? [X223 : (subst > term > term)] : (! [X224 : subst,X225 : subst,X226 : term] : (((X223 @ (comp @ X224 @ X225) @ (sub @ X226 @ X225))) = ((sub @ (X223 @ X224 @ X226) @ X225))) & ! [X227 : term] : (($true = ((X221 @ id @ (X223 @ id @ X227) @ id))) | ($true != ((X221 @ id @ X227 @ id)))) & ($true != ((X221 @ id @ (lam @ (X223 @ sh @ one)) @ id)))) | ? [X228 : term] : (($true != ((X221 @ id @ (sub @ X222 @ (push @ X228 @ id)) @ id))) & ($true = ((X221 @ id @ X228 @ id)))) | (((X221 @ id @ (lam @ X222) @ id)) = $true)) | (hoasinduction_lem3a_lthm != $true)) & (induction2lem_gthm = $true) & ! [X229 : d_subst] : (d_id = X229) & (pushprop_lem2v2_lthm = $true) & (ulamvar1 != $true) & ! [X230 : subst] : ? [X231 : d_subst] : (((d2subst @ X231)) = X230) & (pushprop_lthm = $true) & (hoasinduction_lem1v2 = $true) & (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_lem3v2a = $true) | ? [X232 : (term > $o),X233 : term,X234 : (subst > term > subst > $o)] : (! [X235 : term] : (((X234 @ id @ X235 @ id)) = ((X232 @ X235))) & ($true != ((X232 @ (lam @ X233)))) & ! [X236 : term] : ((((X232 @ (sub @ X233 @ (push @ X236 @ id)))) = $true) | ($true != ((X232 @ X236)))) & ! [X237 : (subst > term > term)] : (? [X238 : term,X239 : subst,X240 : subst] : (((X237 @ (comp @ X239 @ X240) @ (sub @ X238 @ X240))) != ((sub @ (X237 @ X239 @ X238) @ X240))) | ? [X241 : term] : (($true != ((X234 @ id @ (X237 @ id @ X241) @ id))) & ($true = ((X234 @ id @ X241 @ id)))) | (((X234 @ id @ (lam @ (X237 @ sh @ one)) @ id)) = $true)))) & (! [X242 : (term > $o),X243 : term,X244 : (subst > term > subst > $o)] : (? [X245 : term] : (((X244 @ id @ X245 @ id)) != ((X242 @ X245))) | (((X242 @ (lam @ X243))) = $true) | ? [X246 : term] : (($true != ((X242 @ (sub @ X243 @ (push @ X246 @ id))))) & ($true = ((X242 @ X246)))) | ? [X247 : (subst > term > term)] : (! [X248 : term,X249 : subst,X250 : subst] : (((sub @ (X247 @ X249 @ X248) @ X250)) = ((X247 @ (comp @ X249 @ X250) @ (sub @ X248 @ X250)))) & ! [X251 : term] : ((((X244 @ id @ (X247 @ id @ X251) @ id)) = $true) | ($true != ((X244 @ id @ X251 @ id)))) & ($true != ((X244 @ id @ (lam @ (X247 @ sh @ one)) @ id))))) | (hoasinduction_lem3v2a != $true)) & (pushprop_lthm_orig = $true) & (pushprop_lem1v2_gthm = $true) & (pushprop_lem2v2 = $true) & ! [X252 : term] : ? [X253 : d_term] : (((d2term @ X253)) = X252) & ((hoasinduction_no_psi_cond = $true) | ? [X254 : (subst > term > subst > $o),X255 : term] : ((((X254 @ id @ X255 @ id)) != $true) & ! [X256 : (subst > term > term)] : ((((X254 @ id @ (lam @ (X256 @ sh @ one)) @ id)) = $true) | ? [X257 : subst,X258 : subst,X259 : term] : (((sub @ (X256 @ X257 @ X259) @ X258)) != ((X256 @ (comp @ X257 @ X258) @ (sub @ X259 @ X258)))) | ? [X260 : term] : (($true != ((X254 @ id @ (X256 @ id @ X260) @ id))) & ($true = ((X254 @ id @ X260 @ id))))) & ! [X261 : term,X262 : term] : ((((X254 @ id @ X261 @ id)) != $true) | ($true != ((X254 @ id @ X262 @ id))) | ($true = ((X254 @ id @ (ap @ (sub @ X262 @ id) @ X261) @ id)))))) & (! [X263 : (subst > term > subst > $o),X264 : term] : (($true = ((X263 @ id @ X264 @ id))) | ? [X265 : (subst > term > term)] : (($true != ((X263 @ id @ (lam @ (X265 @ sh @ one)) @ id))) & ! [X266 : subst,X267 : subst,X268 : term] : (((sub @ (X265 @ X266 @ X268) @ X267)) = ((X265 @ (comp @ X266 @ X267) @ (sub @ X268 @ X267)))) & ! [X269 : term] : (($true = ((X263 @ id @ (X265 @ id @ X269) @ id))) | (((X263 @ id @ X269 @ id)) != $true))) | ? [X270 : term,X271 : term] : (($true = ((X263 @ id @ X270 @ id))) & ($true = ((X263 @ id @ X271 @ id))) & ($true != ((X263 @ id @ (ap @ (sub @ X271 @ id) @ X270) @ id))))) | (hoasinduction_no_psi_cond != $true)) & (! [X272 : term,X273 : (subst > term > subst > $o)] : (($true = ((X273 @ id @ (lam @ (sub @ X272 @ (push @ one @ sh))) @ id))) | ? [X274 : term,X275 : (subst > term > term)] : (! [X276 : subst,X277 : subst,X278 : term] : (((sub @ X274 @ (push @ (sub @ X278 @ X276) @ (comp @ X277 @ X276)))) = ((X275 @ (comp @ X277 @ X276) @ (sub @ X278 @ X276)))) & ! [X279 : subst,X280 : term,X281 : subst] : (((sub @ (X275 @ X281 @ X280) @ X279)) = ((sub @ (sub @ X274 @ (push @ X280 @ X281)) @ X279))) & (((X273 @ id @ (lam @ (X275 @ sh @ one)) @ id)) != $true) & ! [X282 : term] : (($true = ((X273 @ id @ (X275 @ id @ X282) @ id))) | (((X273 @ id @ X282 @ id)) != $true))) | ? [X283 : term] : (($true != ((X273 @ id @ (sub @ X272 @ (push @ X283 @ id)) @ id))) & ($true = ((X273 @ id @ X283 @ id))))) | (hoasinduction_lem3aaa != $true)) & ((hoasinduction_lem3aaa = $true) | ? [X284 : term,X285 : (subst > term > subst > $o)] : (($true != ((X285 @ id @ (lam @ (sub @ X284 @ (push @ one @ sh))) @ id))) & ! [X286 : term,X287 : (subst > term > term)] : (? [X288 : subst,X289 : subst,X290 : term] : (((sub @ X286 @ (push @ (sub @ X290 @ X288) @ (comp @ X289 @ X288)))) != ((X287 @ (comp @ X289 @ X288) @ (sub @ X290 @ X288)))) | ? [X291 : subst,X292 : term,X293 : subst] : (((sub @ (X287 @ X293 @ X292) @ X291)) != ((sub @ (sub @ X286 @ (push @ X292 @ X293)) @ X291))) | ($true = ((X285 @ id @ (lam @ (X287 @ sh @ one)) @ id))) | ? [X294 : term] : ((((X285 @ id @ (X287 @ id @ X294) @ id)) != $true) & ($true = ((X285 @ id @ X294 @ id))))) & ! [X295 : term] : (($true = ((X285 @ id @ (sub @ X284 @ (push @ X295 @ id)) @ id))) | (((X285 @ id @ X295 @ id)) != $true)))) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (pushprop_lem0_gthm = $true) & (apnotvar = $true) & (hoasapnotvar = $true) & (hoasinduction_lem1_lthm = $true) & ((hoaslaminj = $true) | ? [X296 : subst,X297 : (subst > term > term),X298 : term,X299 : (subst > term > term)] : ((((lam @ (X297 @ sh @ one))) = ((lam @ (X299 @ sh @ one)))) & ! [X300 : subst,X301 : term,X302 : subst] : (((sub @ (X299 @ X300 @ X301) @ X302)) = ((X299 @ (comp @ X300 @ X302) @ (sub @ X301 @ X302)))) & (((X299 @ X296 @ X298)) != ((X297 @ X296 @ X298))) & ! [X303 : subst,X304 : subst,X305 : term] : (((sub @ (X297 @ X303 @ X305) @ X304)) = ((X297 @ (comp @ X303 @ X304) @ (sub @ X305 @ X304)))))) & (! [X306 : subst,X307 : (subst > term > term),X308 : term,X309 : (subst > term > term)] : ((((lam @ (X309 @ sh @ one))) != ((lam @ (X307 @ sh @ one)))) | ? [X310 : subst,X311 : term,X312 : subst] : (((X309 @ (comp @ X310 @ X312) @ (sub @ X311 @ X312))) != ((sub @ (X309 @ X310 @ X311) @ X312))) | (((X307 @ X306 @ X308)) = ((X309 @ X306 @ X308))) | ? [X313 : subst,X314 : subst,X315 : term] : (((X307 @ (comp @ X313 @ X314) @ (sub @ X315 @ X314))) != ((sub @ (X307 @ X313 @ X315) @ X314)))) | (hoaslaminj != $true)) & (induction2_gthm = $true) & (! [X316 : term] : (((sub @ X316 @ id)) = X316) | (termmset != $true)) & ((termmset = $true) | ? [X317 : term] : (((sub @ X317 @ id)) != X317)) & ((hoasinduction_lem0_lthm = $true) | ? [X318 : (subst > term > subst > $o)] : ! [X319 : (term > $o)] : ? [X320 : term] : (((X318 @ id @ X320 @ id)) != ((X319 @ X320)))) & (! [X321 : (subst > term > subst > $o)] : ? [X322 : (term > $o)] : ! [X323 : term] : (((X321 @ id @ X323 @ id)) = ((X322 @ X323))) | (hoasinduction_lem0_lthm != $true)) & (hoaslamnotap_gthm = $true) & (((d2subst @ d_id)) = sh) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (hoasinduction_lem2v2_gthm = $true) & ((hoaslaminj_lthm = $true) | (! [X324 : subst,X325 : term] : (((comp @ sh @ (push @ X325 @ X324))) = X324) & ! [X326 : subst,X327 : term] : (((sub @ one @ (push @ X327 @ X326))) = X327) & ? [X328 : (subst > term > term),X329 : (subst > term > term),X330 : term,X331 : subst] : (! [X332 : term,X333 : subst,X334 : subst] : (((sub @ (X329 @ X334 @ X332) @ X333)) = ((X329 @ (comp @ X334 @ X333) @ (sub @ X332 @ X333)))) & (((lam @ (X328 @ sh @ one))) = ((lam @ (X329 @ sh @ one)))) & ! [X335 : subst,X336 : term,X337 : subst] : (((sub @ (X328 @ X337 @ X336) @ X335)) = ((X328 @ (comp @ X337 @ X335) @ (sub @ X336 @ X335)))) & (((X329 @ X331 @ X330)) != ((X328 @ X331 @ X330)))) & ! [X338 : term,X339 : term] : ((X338 = X339) | (((lam @ X339)) != ((lam @ X338)))))) & (? [X340 : subst,X341 : term] : (((comp @ sh @ (push @ X341 @ X340))) != X340) | ? [X342 : subst,X343 : term] : (((sub @ one @ (push @ X343 @ X342))) != X343) | ! [X344 : (subst > term > term),X345 : (subst > term > term),X346 : term,X347 : subst] : (? [X348 : term,X349 : subst,X350 : subst] : (((sub @ (X345 @ X350 @ X348) @ X349)) != ((X345 @ (comp @ X350 @ X349) @ (sub @ X348 @ X349)))) | (((lam @ (X344 @ sh @ one))) != ((lam @ (X345 @ sh @ one)))) | ? [X351 : subst,X352 : term,X353 : subst] : (((sub @ (X344 @ X353 @ X352) @ X351)) != ((X344 @ (comp @ X353 @ X351) @ (sub @ X352 @ X351)))) | (((X345 @ X347 @ X346)) = ((X344 @ X347 @ X346)))) | ? [X354 : term,X355 : term] : ((X354 != X355) & (((lam @ X354)) = ((lam @ X355)))) | (hoaslaminj_lthm != $true)) & (! [X356 : term,X357 : (subst > term > subst > $o)] : (? [X358 : term] : (($true != ((X357 @ id @ (sub @ X356 @ (push @ X358 @ id)) @ id))) & ($true = ((X357 @ id @ X358 @ id)))) | ? [X359 : subst,X360 : subst,X361 : subst,X362 : term] : ((((X357 @ (comp @ X359 @ X361) @ (sub @ X362 @ X361) @ X360)) != $true) & ($true = ((X357 @ X359 @ X362 @ (comp @ X361 @ X360))))) | ? [X363 : (subst > term > term)] : (! [X364 : subst,X365 : term,X366 : subst] : (((sub @ (X363 @ X364 @ X365) @ X366)) = ((X363 @ (comp @ X364 @ X366) @ (sub @ X365 @ X366)))) & ! [X367 : term] : (($true != ((X357 @ id @ X367 @ id))) | (((X357 @ id @ (X363 @ id @ X367) @ id)) = $true)) & ($true != ((X357 @ id @ (lam @ (X363 @ sh @ one)) @ id)))) | ($true = ((X357 @ id @ (lam @ X356) @ id))) | ? [X368 : subst,X369 : term,X370 : subst,X371 : subst] : (($true = ((X357 @ (comp @ X371 @ X370) @ (sub @ X369 @ X370) @ X368))) & ($true != ((X357 @ X371 @ X369 @ (comp @ X370 @ X368)))))) | (hoasinduction_lem3 != $true)) & ((hoasinduction_lem3 = $true) | ? [X372 : term,X373 : (subst > term > subst > $o)] : (! [X374 : term] : ((((X373 @ id @ (sub @ X372 @ (push @ X374 @ id)) @ id)) = $true) | ($true != ((X373 @ id @ X374 @ id)))) & ! [X375 : subst,X376 : subst,X377 : subst,X378 : term] : (($true = ((X373 @ (comp @ X375 @ X377) @ (sub @ X378 @ X377) @ X376))) | (((X373 @ X375 @ X378 @ (comp @ X377 @ X376))) != $true)) & ! [X379 : (subst > term > term)] : (? [X380 : subst,X381 : term,X382 : subst] : (((sub @ (X379 @ X380 @ X381) @ X382)) != ((X379 @ (comp @ X380 @ X382) @ (sub @ X381 @ X382)))) | ? [X383 : term] : (($true = ((X373 @ id @ X383 @ id))) & (((X373 @ id @ (X379 @ id @ X383) @ id)) != $true)) | (((X373 @ id @ (lam @ (X379 @ sh @ one)) @ id)) = $true)) & ($true != ((X373 @ id @ (lam @ X372) @ id))) & ! [X384 : subst,X385 : term,X386 : subst,X387 : subst] : (($true != ((X373 @ (comp @ X387 @ X386) @ (sub @ X385 @ X386) @ X384))) | ($true = ((X373 @ X387 @ X385 @ (comp @ X386 @ X384))))))) & (termmset_gthm = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (hoasinduction_lem2_lthm = $true) & (! [X388 : term,X389 : term] : ((((lam @ X388)) != ((lam @ X389))) | (X388 = X389)) | (laminj != $true)) & ((laminj = $true) | ? [X390 : term,X391 : term] : ((((lam @ X391)) = ((lam @ X390))) & (X390 != X391))) & (((var @ (d2term @ d_one))) != $true) & (! [X392 : subst] : (((comp @ X392 @ id)) = X392) | (axidr != $true)) & ((axidr = $true) | ? [X393 : subst] : (((comp @ X393 @ id)) != X393)) & (! [X394 : term] : (((sub @ X394 @ id)) = X394) | (axvarid != $true)) & ((axvarid = $true) | ? [X395 : term] : (((sub @ X395 @ id)) != X395)) & (? [X396 : term] : (((sub @ X396 @ id)) != X396) | ($true = sP5) | ? [X397 : (subst > term > subst > $o)] : ! [X398 : (term > $o)] : ? [X399 : term] : (((X397 @ id @ X399 @ id)) != ((X398 @ X399))) | ? [X400 : term,X401 : (term > $o),X402 : (subst > term > subst > $o)] : (($true != ((X401 @ (lam @ X400)))) & ! [X403 : (subst > term > term)] : ((((X402 @ id @ (lam @ (X403 @ sh @ one)) @ id)) = $true) | ? [X404 : subst,X405 : subst,X406 : term] : (((X403 @ (comp @ X405 @ X404) @ (sub @ X406 @ X404))) != ((sub @ (X403 @ X405 @ X406) @ X404))) | ? [X407 : term] : (($true = ((X402 @ id @ X407 @ id))) & ($true != ((X402 @ id @ (X403 @ id @ X407) @ id))))) & ! [X408 : term] : (($true != ((X401 @ X408))) | ($true = ((X401 @ (sub @ X400 @ (push @ X408 @ id)))))) & ! [X409 : term] : (((X401 @ X409)) = ((X402 @ id @ X409 @ id)))) | ? [X410 : term,X411 : (term > $o)] : (! [X412 : term] : (($true = ((X411 @ X412))) | ($true != ((var @ X412)))) & (((X411 @ X410)) != $true) & ! [X413 : term] : (? [X414 : term] : (($true = ((X411 @ X414))) & (((X411 @ (sub @ X413 @ (push @ X414 @ id)))) != $true)) | ($true = ((X411 @ (lam @ X413))))) & ! [X415 : term,X416 : term] : (($true != ((X411 @ X415))) | ($true = ((X411 @ (ap @ X415 @ X416)))) | ($true != ((X411 @ X416))))) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | (! [X417 : term] : (((sub @ X417 @ id)) = X417) & ($true != sP5) & ! [X418 : (subst > term > subst > $o)] : ? [X419 : (term > $o)] : ! [X420 : term] : (((X418 @ id @ X420 @ id)) = ((X419 @ X420))) & ! [X421 : term,X422 : (term > $o),X423 : (subst > term > subst > $o)] : (($true = ((X422 @ (lam @ X421)))) | ? [X424 : (subst > term > term)] : (($true != ((X423 @ id @ (lam @ (X424 @ sh @ one)) @ id))) & ! [X425 : subst,X426 : subst,X427 : term] : (((X424 @ (comp @ X426 @ X425) @ (sub @ X427 @ X425))) = ((sub @ (X424 @ X426 @ X427) @ X425))) & ! [X428 : term] : ((((X423 @ id @ X428 @ id)) != $true) | ($true = ((X423 @ id @ (X424 @ id @ X428) @ id))))) | ? [X429 : term] : (($true = ((X422 @ X429))) & ($true != ((X422 @ (sub @ X421 @ (push @ X429 @ id)))))) | ? [X430 : term] : (((X423 @ id @ X430 @ id)) != ((X422 @ X430)))) & ! [X431 : term,X432 : (term > $o)] : (? [X433 : term] : (($true != ((X432 @ X433))) & ($true = ((var @ X433)))) | ($true = ((X432 @ X431))) | ? [X434 : term] : (! [X435 : term] : ((((X432 @ X435)) != $true) | (((X432 @ (sub @ X434 @ (push @ X435 @ id)))) = $true)) & ($true != ((X432 @ (lam @ X434))))) | ? [X436 : term,X437 : term] : (($true = ((X432 @ X436))) & (((X432 @ (ap @ X436 @ X437))) != $true) & ($true = ((X432 @ X437))))))) & (substmonoid_gthm = $true) & (pushprop = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (axclos = $true) & ((hoasinduction_lem3a = $true) | ? [X438 : (subst > term > subst > $o),X439 : term] : (! [X440 : (subst > term > term)] : (? [X441 : term] : ((((X438 @ id @ (X440 @ id @ X441) @ id)) != $true) & ($true = ((X438 @ id @ X441 @ id)))) | ($true = ((X438 @ id @ (lam @ (X440 @ sh @ one)) @ id))) | ? [X442 : subst,X443 : subst,X444 : term] : (((X440 @ (comp @ X442 @ X443) @ (sub @ X444 @ X443))) != ((sub @ (X440 @ X442 @ X444) @ X443)))) & ! [X445 : term] : ((((X438 @ id @ X445 @ id)) != $true) | ($true = ((X438 @ id @ (sub @ X439 @ (push @ X445 @ id)) @ id)))) & ($true != ((X438 @ id @ (lam @ X439) @ id))))) & (! [X446 : (subst > term > subst > $o),X447 : term] : (? [X448 : (subst > term > term)] : (! [X449 : term] : (($true = ((X446 @ id @ (X448 @ id @ X449) @ id))) | ($true != ((X446 @ id @ X449 @ id)))) & ($true != ((X446 @ id @ (lam @ (X448 @ sh @ one)) @ id))) & ! [X450 : subst,X451 : subst,X452 : term] : (((sub @ (X448 @ X450 @ X452) @ X451)) = ((X448 @ (comp @ X450 @ X451) @ (sub @ X452 @ X451))))) | ? [X453 : term] : (($true = ((X446 @ id @ X453 @ id))) & ($true != ((X446 @ id @ (sub @ X447 @ (push @ X453 @ id)) @ id)))) | (((X446 @ id @ (lam @ X447) @ id)) = $true)) | (hoasinduction_lem3a != $true)) & (? [X454 : term] : (((sub @ X454 @ id)) != X454) | ! [X455 : term] : (((sub @ X455 @ id)) = X455) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | (! [X456 : term] : (((sub @ X456 @ id)) = X456) & ? [X457 : term] : (((sub @ X457 @ id)) != X457))) & (ulamvarsh = $true) & ((hoasinduction_lem3v2_f = $true) | ? [X458 : term] : ! [X459 : (subst > term > term)] : ? [X460 : subst,X461 : term] : (((sub @ X458 @ (push @ X461 @ X460))) != ((X459 @ X460 @ X461)))) & (! [X462 : term] : ? [X463 : (subst > term > term)] : ! [X464 : subst,X465 : term] : (((sub @ X462 @ (push @ X465 @ X464))) = ((X463 @ X464 @ X465))) | (hoasinduction_lem3v2_f != $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((substmonoid = $true) | ? [X466 : subst] : (((comp @ X466 @ id)) != X466) | ? [X467 : subst] : (((comp @ id @ X467)) != X467)) & ((! [X468 : subst] : (((comp @ X468 @ id)) = X468) & ! [X469 : subst] : (((comp @ id @ X469)) = X469)) | (substmonoid != $true)) & ((hoasinduction_lem3b_lthm = $true) | ? [X470 : term] : ! [X471 : (subst > term > term)] : (((X471 @ sh @ one)) != ((sub @ X470 @ (push @ one @ sh))))) & (! [X472 : term] : ? [X473 : (subst > term > term)] : (((X473 @ sh @ one)) = ((sub @ X472 @ (push @ one @ sh)))) | (hoasinduction_lem3b_lthm != $true)) & (! [X474 : (term > $o),X475 : (term > $o),X476 : subst,X477 : term] : (? [X478 : term] : (((X474 @ (sub @ X478 @ (push @ X477 @ X476)))) != ((X475 @ X478))) | ($true != ((X474 @ X477))) | (((X475 @ one)) = $true)) | ? [X479 : subst,X480 : term] : (((sub @ one @ (push @ X480 @ X479))) != X480) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | (? [X481 : (term > $o),X482 : (term > $o),X483 : subst,X484 : term] : (! [X485 : term] : (((X481 @ (sub @ X485 @ (push @ X484 @ X483)))) = ((X482 @ X485))) & ($true = ((X481 @ X484))) & ($true != ((X482 @ one)))) & ! [X486 : subst,X487 : term] : (((sub @ one @ (push @ X487 @ X486))) = X487))) & (axassoc = $true) & (? [X488 : subst] : (((comp @ id @ X488)) != X488) | (! [X489 : subst] : (((comp @ id @ X489)) = X489) & ! [X490 : subst] : (((comp @ X490 @ id)) = X490)) | ? [X491 : subst] : (((comp @ X491 @ id)) != X491) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | (! [X492 : subst] : (((comp @ id @ X492)) = X492) & (? [X493 : subst] : (((comp @ id @ X493)) != X493) | ? [X494 : subst] : (((comp @ X494 @ id)) != X494)) & ! [X495 : subst] : (((comp @ X495 @ id)) = X495))) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (? [X496 : term,X497 : term,X498 : term,X499 : term] : ((((ap @ X496 @ X497)) = ((ap @ X499 @ X498))) & (X496 != X499)) | ? [X500 : term] : (((sub @ X500 @ id)) != X500) | ! [X501 : term,X502 : term,X503 : term,X504 : term] : ((((ap @ (sub @ X502 @ id) @ X503)) != ((ap @ (sub @ X504 @ id) @ X501))) | (X502 = X504)) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | (! [X505 : term,X506 : term,X507 : term,X508 : term] : ((((ap @ X505 @ X506)) != ((ap @ X508 @ X507))) | (X505 = X508)) & ! [X509 : term] : (((sub @ X509 @ id)) = X509) & ? [X510 : term,X511 : term,X512 : term,X513 : term] : ((((ap @ (sub @ X513 @ id) @ X510)) = ((ap @ (sub @ X511 @ id) @ X512))) & (X511 != X513)))) & (hoasapinj1_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (hoasapnotvar_lthm = $true) & (one = ((d2term @ d_one))) & (axapp = $true) & (hoasinduction_lem1 = $true) & ((pushprop_lem0 = $true) | ? [X514 : subst,X515 : (term > $o),X516 : term] : ! [X517 : (term > $o)] : ? [X518 : term] : (((X515 @ (sub @ X518 @ (push @ X516 @ X514)))) != ((X517 @ X518)))) & (! [X519 : subst,X520 : (term > $o),X521 : term] : ? [X522 : (term > $o)] : ! [X523 : term] : (((X522 @ X523)) = ((X520 @ (sub @ X523 @ (push @ X521 @ X519))))) | (pushprop_lem0 != $true)) & ((induction2lem_lthm = $true) | ($true != sP3)) & (($true = sP3) | (induction2lem_lthm != $true)) & (ulamvarind = $true) & (axmap = $true) & ((hoasinduction_no_psi_cond_lthm = $true) | (! [X524 : term] : (((sub @ X524 @ id)) = X524) & ! [X525 : (term > $o),X526 : (subst > term > subst > $o),X527 : term] : (? [X528 : term] : (((X525 @ X528)) != ((X526 @ id @ X528 @ id))) | ($true = ((X525 @ (lam @ X527)))) | ? [X529 : term] : (($true != ((X525 @ (sub @ X527 @ (push @ X529 @ id))))) & (((X525 @ X529)) = $true)) | ? [X530 : (subst > term > term)] : (($true != ((X526 @ id @ (lam @ (X530 @ sh @ one)) @ id))) & ! [X531 : term] : (($true != ((X526 @ id @ X531 @ id))) | (((X526 @ id @ (X530 @ id @ X531) @ id)) = $true)) & ! [X532 : term,X533 : subst,X534 : subst] : (((sub @ (X530 @ X533 @ X532) @ X534)) = ((X530 @ (comp @ X533 @ X534) @ (sub @ X532 @ X534)))))) & ! [X535 : (term > $o),X536 : term] : (? [X537 : term,X538 : term] : ((((X535 @ X537)) = $true) & (((X535 @ (ap @ X538 @ X537))) != $true) & ($true = ((X535 @ X538)))) | ($true = ((X535 @ X536))) | ? [X539 : term] : (($true = ((var @ X539))) & ($true != ((X535 @ X539)))) | ? [X540 : term] : (! [X541 : term] : (($true != ((X535 @ X541))) | ($true = ((X535 @ (sub @ X540 @ (push @ X541 @ id)))))) & (((X535 @ (lam @ X540))) != $true))) & ! [X542 : (subst > term > subst > $o)] : ? [X543 : (term > $o)] : ! [X544 : term] : (((X542 @ id @ X544 @ id)) = ((X543 @ X544))) & ($true != sP2))) & (? [X545 : term] : (((sub @ X545 @ id)) != X545) | ? [X546 : (term > $o),X547 : (subst > term > subst > $o),X548 : term] : (! [X549 : term] : (((X547 @ id @ X549 @ id)) = ((X546 @ X549))) & ($true != ((X546 @ (lam @ X548)))) & ! [X550 : term] : (($true = ((X546 @ (sub @ X548 @ (push @ X550 @ id))))) | (((X546 @ X550)) != $true)) & ! [X551 : (subst > term > term)] : (($true = ((X547 @ id @ (lam @ (X551 @ sh @ one)) @ id))) | ? [X552 : term] : ((((X547 @ id @ X552 @ id)) = $true) & (((X547 @ id @ (X551 @ id @ X552) @ id)) != $true)) | ? [X553 : term,X554 : subst,X555 : subst] : (((X551 @ (comp @ X554 @ X555) @ (sub @ X553 @ X555))) != ((sub @ (X551 @ X554 @ X553) @ X555))))) | ? [X556 : (term > $o),X557 : term] : (! [X558 : term,X559 : term] : (($true != ((X556 @ X558))) | ($true = ((X556 @ (ap @ X559 @ X558)))) | ($true != ((X556 @ X559)))) & (((X556 @ X557)) != $true) & ! [X560 : term] : (($true != ((var @ X560))) | ($true = ((X556 @ X560)))) & ! [X561 : term] : (? [X562 : term] : ((((X556 @ X562)) = $true) & ($true != ((X556 @ (sub @ X561 @ (push @ X562 @ id)))))) | (((X556 @ (lam @ X561))) = $true))) | ? [X563 : (subst > term > subst > $o)] : ! [X564 : (term > $o)] : ? [X565 : term] : (((X564 @ X565)) != ((X563 @ id @ X565 @ id))) | ($true = sP2) | (hoasinduction_no_psi_cond_lthm != $true)) & ((pushprop_lem1v2 = $true) | ? [X566 : (term > $o),X567 : (term > $o),X568 : term,X569 : subst] : (! [X570 : term] : (((X567 @ (sub @ X570 @ (push @ X568 @ X569)))) = ((X566 @ X570))) & ($true = ((X567 @ X568))) & ($true != ((X566 @ one))))) & (! [X571 : (term > $o),X572 : (term > $o),X573 : term,X574 : subst] : (? [X575 : term] : (((X571 @ X575)) != ((X572 @ (sub @ X575 @ (push @ X573 @ X574))))) | ($true != ((X572 @ X573))) | ($true = ((X571 @ one)))) | (pushprop_lem1v2 != $true)) & (hoasinduction_gthm = $true) & (! [X576 : term,X577 : subst,X578 : term,X579 : (term > $o),X580 : (term > $o)] : (($true = ((X580 @ (sub @ X576 @ (push @ X578 @ X577))))) | ($true != ((X579 @ X576)))) | ? [X581 : subst,X582 : term] : (((sub @ one @ (push @ X582 @ X581))) != X582) | ? [X583 : subst,X584 : term] : (((comp @ sh @ (push @ X584 @ X583))) != X583) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | (? [X585 : term,X586 : subst,X587 : term,X588 : (term > $o),X589 : (term > $o)] : (($true != ((X589 @ (sub @ X585 @ (push @ X587 @ X586))))) & (((X588 @ X585)) = $true)) & ! [X590 : subst,X591 : term] : (((sub @ one @ (push @ X591 @ X590))) = X591) & ! [X592 : subst,X593 : term] : (((comp @ sh @ (push @ X593 @ X592))) = X592))) & (! [X594 : (term > $o),X595 : term] : (? [X596 : term,X597 : term] : ((((X594 @ X596)) = $true) & ($true = ((X594 @ X597))) & ($true != ((X594 @ (ap @ X597 @ X596))))) | ? [X598 : term] : ((((X594 @ (lam @ X598))) != $true) & ! [X599 : term] : (($true != ((X594 @ X599))) | (((X594 @ (sub @ X598 @ (push @ X599 @ id)))) = $true))) | ($true = ((X594 @ X595))) | ? [X600 : term] : ((((var @ X600)) = $true) & ($true != ((X594 @ X600))))) | (induction2 != $true)) & ((induction2 = $true) | ? [X601 : (term > $o),X602 : term] : (! [X603 : term,X604 : term] : (($true != ((X601 @ X603))) | ($true != ((X601 @ X604))) | (((X601 @ (ap @ X604 @ X603))) = $true)) & ! [X605 : term] : (($true = ((X601 @ (lam @ X605)))) | ? [X606 : term] : ((((X601 @ X606)) = $true) & (((X601 @ (sub @ X605 @ (push @ X606 @ id)))) != $true))) & ($true != ((X601 @ X602))) & ! [X607 : term] : (($true != ((var @ X607))) | ($true = ((X601 @ X607)))))) & ((pushprop_lem1 = $true) | ? [X608 : (term > $o),X609 : term,X610 : subst,X611 : term,X612 : (term > $o)] : (($true != ((X612 @ (sub @ X609 @ (push @ X611 @ X610))))) & ($true = ((X608 @ X609))))) & (! [X613 : (term > $o),X614 : term,X615 : subst,X616 : term,X617 : (term > $o)] : (($true = ((X617 @ (sub @ X614 @ (push @ X616 @ X615))))) | ($true != ((X613 @ X614)))) | (pushprop_lem1 != $true)) & ((induction2_lthm = $true) | ($true != sP1)) & (($true = sP1) | (induction2_lthm != $true)) & ((hoasinduction_lem0 = $true) | ? [X618 : (subst > term > subst > $o)] : ! [X619 : (term > $o)] : ? [X620 : term] : (((X619 @ X620)) != ((X618 @ id @ X620 @ id)))) & (! [X621 : (subst > term > subst > $o)] : ? [X622 : (term > $o)] : ! [X623 : term] : (((X622 @ X623)) = ((X621 @ id @ X623 @ id))) | (hoasinduction_lem0 != $true)) & (lamnotvar = $true) & (! [X624 : term,X625 : term,X626 : term,X627 : term] : ((((ap @ X626 @ X627)) != ((ap @ X624 @ X625))) | (X625 = X627)) | (apinj2 != $true)) & ((apinj2 = $true) | ? [X628 : term,X629 : term,X630 : term,X631 : term] : ((((ap @ X630 @ X631)) = ((ap @ X628 @ X629))) & (X629 != X631))) & (! [X632 : (term > $o),X633 : term] : (? [X634 : term] : (($true != ((X632 @ (lam @ X634)))) & ($true = ((X632 @ X634)))) | ($true = ((X632 @ X633))) | ? [X635 : term,X636 : term] : (($true != ((X632 @ (ap @ X636 @ X635)))) & ($true = ((X632 @ X636))) & (((X632 @ X635)) = $true)) | ? [X637 : term] : (($true = ((var @ X637))) & (((X632 @ X637)) != $true))) | (induction != $true)) & ((induction = $true) | ? [X638 : (term > $o),X639 : term] : (! [X640 : term] : (($true = ((X638 @ (lam @ X640)))) | (((X638 @ X640)) != $true)) & ($true != ((X638 @ X639))) & ! [X641 : term,X642 : term] : ((((X638 @ (ap @ X642 @ X641))) = $true) | ($true != ((X638 @ X642))) | ($true != ((X638 @ X641)))) & ! [X643 : term] : (($true != ((var @ X643))) | (((X638 @ X643)) = $true)))) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoaslamnotvar_lthm = $true) & ((hoasinduction_lem2v2 = $true) | ? [X644 : (subst > term > subst > $o),X645 : term,X646 : (term > $o),X647 : term] : (! [X648 : subst,X649 : term,X650 : subst,X651 : subst] : (($true = ((X644 @ X651 @ X649 @ (comp @ X648 @ X650)))) | ($true != ((X644 @ (comp @ X651 @ X648) @ (sub @ X649 @ X648) @ X650)))) & (((X646 @ X647)) = $true) & ($true = ((X646 @ X645))) & ! [X652 : term,X653 : term] : (($true != ((X644 @ id @ X652 @ id))) | ($true = ((X644 @ id @ (ap @ (sub @ X652 @ id) @ X653) @ id))) | ($true != ((X644 @ id @ X653 @ id)))) & ($true != ((X646 @ (ap @ X647 @ X645)))) & ! [X654 : term] : (((X646 @ X654)) = ((X644 @ id @ X654 @ id))) & ! [X655 : subst,X656 : subst,X657 : subst,X658 : term] : (($true = ((X644 @ (comp @ X656 @ X657) @ (sub @ X658 @ X657) @ X655))) | ($true != ((X644 @ X656 @ X658 @ (comp @ X657 @ X655))))))) & (! [X659 : (subst > term > subst > $o),X660 : term,X661 : (term > $o),X662 : term] : (? [X663 : subst,X664 : term,X665 : subst,X666 : subst] : (($true != ((X659 @ X666 @ X664 @ (comp @ X663 @ X665)))) & (((X659 @ (comp @ X666 @ X663) @ (sub @ X664 @ X663) @ X665)) = $true)) | ($true != ((X661 @ X662))) | ($true != ((X661 @ X660))) | ? [X667 : term,X668 : term] : (($true = ((X659 @ id @ X667 @ id))) & ($true != ((X659 @ id @ (ap @ (sub @ X667 @ id) @ X668) @ id))) & ($true = ((X659 @ id @ X668 @ id)))) | ($true = ((X661 @ (ap @ X662 @ X660)))) | ? [X669 : term] : (((X661 @ X669)) != ((X659 @ id @ X669 @ id))) | ? [X670 : subst,X671 : subst,X672 : subst,X673 : term] : (($true != ((X659 @ (comp @ X671 @ X672) @ (sub @ X673 @ X672) @ X670))) & ($true = ((X659 @ X671 @ X673 @ (comp @ X672 @ X670)))))) | (hoasinduction_lem2v2 != $true)) & (hoasinduction_lem3a_gthm = $true) & (pushprop_lem1_gthm = $true) & (hoasapinj2_gthm = $true) & (id = ((d2subst @ d_id))) & (pushprop_gthm = $true) & (! [X674 : term] : ? [X675 : (subst > term > term)] : ! [X676 : term,X677 : subst] : (((X675 @ X677 @ X676)) = ((sub @ X674 @ (push @ X676 @ X677)))) | (hoasinduction_lem3v2_f_lthm != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ? [X678 : term] : ! [X679 : (subst > term > term)] : ? [X680 : term,X681 : subst] : (((sub @ X678 @ (push @ X680 @ X681))) != ((X679 @ X681 @ X680)))) & ((hoasinduction_lem3v2_lthm = $true) | (! [X682 : term] : (((sub @ X682 @ id)) = X682) & ? [X683 : (subst > term > subst > $o),X684 : term,X685 : (term > $o)] : (! [X686 : term] : (((X683 @ id @ X686 @ id)) = ((X685 @ X686))) & ($true != ((X685 @ (lam @ X684)))) & ! [X687 : subst,X688 : term,X689 : subst,X690 : subst] : ((((X683 @ (comp @ X689 @ X690) @ (sub @ X688 @ X690) @ X687)) != $true) | ($true = ((X683 @ X689 @ X688 @ (comp @ X690 @ X687))))) & ! [X691 : term,X692 : subst,X693 : subst,X694 : subst] : (($true = ((X683 @ (comp @ X692 @ X694) @ (sub @ X691 @ X694) @ X693))) | ($true != ((X683 @ X692 @ X691 @ (comp @ X694 @ X693))))) & ! [X695 : (subst > term > term)] : (? [X696 : subst,X697 : subst,X698 : term] : (((X695 @ (comp @ X697 @ X696) @ (sub @ X698 @ X696))) != ((sub @ (X695 @ X697 @ X698) @ X696))) | ? [X699 : term] : (($true = ((X683 @ id @ X699 @ id))) & (((X683 @ id @ (X695 @ id @ X699) @ id)) != $true)) | ($true = ((X683 @ id @ (lam @ (X695 @ sh @ one)) @ id)))) & ! [X700 : term] : (($true = ((X685 @ (sub @ X684 @ (push @ X700 @ id))))) | (((X685 @ X700)) != $true))))) & (? [X701 : term] : (((sub @ X701 @ id)) != X701) | ! [X702 : (subst > term > subst > $o),X703 : term,X704 : (term > $o)] : (? [X705 : term] : (((X704 @ X705)) != ((X702 @ id @ X705 @ id))) | ($true = ((X704 @ (lam @ X703)))) | ? [X706 : subst,X707 : term,X708 : subst,X709 : subst] : ((((X702 @ (comp @ X708 @ X709) @ (sub @ X707 @ X709) @ X706)) = $true) & ($true != ((X702 @ X708 @ X707 @ (comp @ X709 @ X706))))) | ? [X710 : term,X711 : subst,X712 : subst,X713 : subst] : (($true != ((X702 @ (comp @ X711 @ X713) @ (sub @ X710 @ X713) @ X712))) & (((X702 @ X711 @ X710 @ (comp @ X713 @ X712))) = $true)) | ? [X714 : (subst > term > term)] : (! [X715 : subst,X716 : subst,X717 : term] : (((sub @ (X714 @ X716 @ X717) @ X715)) = ((X714 @ (comp @ X716 @ X715) @ (sub @ X717 @ X715)))) & ! [X718 : term] : (($true != ((X702 @ id @ X718 @ id))) | ($true = ((X702 @ id @ (X714 @ id @ X718) @ id)))) & ($true != ((X702 @ id @ (lam @ (X714 @ sh @ one)) @ id)))) | ? [X719 : term] : ((((X704 @ (sub @ X703 @ (push @ X719 @ id)))) != $true) & (((X704 @ X719)) = $true))) | (hoasinduction_lem3v2_lthm != $true)) & (! [X720 : term,X721 : term,X722 : term,X723 : term] : ((((ap @ X721 @ X723)) != ((ap @ X722 @ X720))) | (X721 = X722)) | (apinj1 != $true)) & ((apinj1 = $true) | ? [X724 : term,X725 : term,X726 : term,X727 : term] : ((((ap @ X726 @ X724)) = ((ap @ X725 @ X727))) & (X725 != X726))) & (! [X728 : (term > $o),X729 : subst,X730 : term] : ? [X731 : (term > $o)] : ! [X732 : term] : (((X731 @ X732)) = ((X728 @ (sub @ X732 @ (push @ X730 @ X729))))) | (pushprop_lem0_lthm != $true)) & ((pushprop_lem0_lthm = $true) | ? [X733 : (term > $o),X734 : subst,X735 : term] : ! [X736 : (term > $o)] : ? [X737 : term] : (((X733 @ (sub @ X737 @ (push @ X735 @ X734)))) != ((X736 @ X737)))) & (hoasinduction_lem3v2_gthm = $true) & (hoasinduction_lem3_gthm = $true) & (hoaslaminj_gthm = $true) & (hoasinduction_lem2_gthm = $true) & (! [X738 : (subst > term > subst > $o),X739 : term,X740 : (term > $o)] : (? [X741 : term] : (($true = ((X740 @ X741))) & (((X740 @ (sub @ X739 @ (push @ X741 @ id)))) != $true)) | ? [X742 : term] : (((X738 @ id @ X742 @ id)) != ((X740 @ X742))) | ? [X743 : subst,X744 : subst,X745 : term,X746 : subst] : (($true = ((X738 @ X744 @ X745 @ (comp @ X743 @ X746)))) & ($true != ((X738 @ (comp @ X744 @ X743) @ (sub @ X745 @ X743) @ X746)))) | ? [X747 : term,X748 : subst,X749 : subst,X750 : subst] : (($true = ((X738 @ (comp @ X748 @ X749) @ (sub @ X747 @ X749) @ X750))) & ($true != ((X738 @ X748 @ X747 @ (comp @ X749 @ X750))))) | ($true = ((X740 @ (lam @ X739)))) | ? [X751 : (subst > term > term)] : (! [X752 : term] : ((((X738 @ id @ X752 @ id)) != $true) | ($true = ((X738 @ id @ (X751 @ id @ X752) @ id)))) & ! [X753 : subst,X754 : subst,X755 : term] : (((X751 @ (comp @ X754 @ X753) @ (sub @ X755 @ X753))) = ((sub @ (X751 @ X754 @ X755) @ X753))) & ($true != ((X738 @ id @ (lam @ (X751 @ sh @ one)) @ id))))) | (hoasinduction_lem3v2 != $true)) & ((hoasinduction_lem3v2 = $true) | ? [X756 : (subst > term > subst > $o),X757 : term,X758 : (term > $o)] : (! [X759 : term] : (($true != ((X758 @ X759))) | ($true = ((X758 @ (sub @ X757 @ (push @ X759 @ id)))))) & ! [X760 : term] : (((X756 @ id @ X760 @ id)) = ((X758 @ X760))) & ! [X761 : subst,X762 : subst,X763 : term,X764 : subst] : (($true != ((X756 @ X762 @ X763 @ (comp @ X761 @ X764)))) | (((X756 @ (comp @ X762 @ X761) @ (sub @ X763 @ X761) @ X764)) = $true)) & ! [X765 : term,X766 : subst,X767 : subst,X768 : subst] : ((((X756 @ (comp @ X766 @ X767) @ (sub @ X765 @ X767) @ X768)) != $true) | ($true = ((X756 @ X766 @ X765 @ (comp @ X767 @ X768))))) & ($true != ((X758 @ (lam @ X757)))) & ! [X769 : (subst > term > term)] : (? [X770 : term] : (($true = ((X756 @ id @ X770 @ id))) & ($true != ((X756 @ id @ (X769 @ id @ X770) @ id)))) | ? [X771 : subst,X772 : subst,X773 : term] : (((sub @ (X769 @ X772 @ X773) @ X771)) != ((X769 @ (comp @ X772 @ X771) @ (sub @ X773 @ X771)))) | (((X756 @ id @ (lam @ (X769 @ sh @ one)) @ id)) = $true)))) & ((axshiftcons = $true) | ? [X774 : term,X775 : subst] : (((comp @ sh @ (push @ X774 @ X775))) != X775)) & (! [X776 : term,X777 : subst] : (((comp @ sh @ (push @ X776 @ X777))) = X777) | (axshiftcons != $true)) & ((hoasinduction_lem3_lthm = $true) | (($true != sP0) & ! [X778 : term] : (((sub @ X778 @ id)) = X778) & ! [X779 : (subst > term > subst > $o),X780 : term] : (? [X781 : term] : (($true = ((X779 @ id @ X781 @ id))) & ($true != ((X779 @ id @ (sub @ X780 @ (push @ X781 @ id)) @ id)))) | (((X779 @ id @ (lam @ (sub @ X780 @ (push @ one @ sh))) @ id)) = $true) | ? [X782 : (subst > term > term)] : (! [X783 : subst,X784 : term,X785 : subst] : (((sub @ (X782 @ X785 @ X784) @ X783)) = ((X782 @ (comp @ X785 @ X783) @ (sub @ X784 @ X783)))) & ! [X786 : term] : ((((X779 @ id @ (X782 @ id @ X786) @ id)) = $true) | (((X779 @ id @ X786 @ id)) != $true)) & ($true != ((X779 @ id @ (lam @ (X782 @ sh @ one)) @ id))))))) & (($true = sP0) | ? [X787 : term] : (((sub @ X787 @ id)) != X787) | ? [X788 : (subst > term > subst > $o),X789 : term] : (! [X790 : term] : (($true != ((X788 @ id @ X790 @ id))) | ($true = ((X788 @ id @ (sub @ X789 @ (push @ X790 @ id)) @ id)))) & (((X788 @ id @ (lam @ (sub @ X789 @ (push @ one @ sh))) @ id)) != $true) & ! [X791 : (subst > term > term)] : (? [X792 : subst,X793 : term,X794 : subst] : (((sub @ (X791 @ X794 @ X793) @ X792)) != ((X791 @ (comp @ X794 @ X792) @ (sub @ X793 @ X792)))) | ? [X795 : term] : ((((X788 @ id @ (X791 @ id @ X795) @ id)) != $true) & ($true = ((X788 @ id @ X795 @ id)))) | ($true = ((X788 @ id @ (lam @ (X791 @ sh @ one)) @ id))))) | (hoasinduction_lem3_lthm != $true))),
  inference(rectify,[],[f66])).
thf(f68,plain,(
  (hoaslamnotap_lthm = $true) & (axabs = $true) & (! [X0 : (subst > term > subst > $o),X1 : term] : ((((X0 @ id @ (lam @ (sub @ X1 @ (push @ one @ sh))) @ id)) = $true) | (($true = ((X0 @ id @ (sK149 @ X1 @ X0) @ id))) & ($true != ((X0 @ id @ (sub @ X1 @ (push @ (sK149 @ X1 @ X0) @ id)) @ id)))) | ((((X0 @ id @ (lam @ (sK150 @ X0 @ sh @ one)) @ id)) != $true) & ! [X4 : subst,X5 : term,X6 : subst] : (((sub @ (sK150 @ X0 @ X6 @ X5) @ X4)) = ((sK150 @ X0 @ (comp @ X6 @ X4) @ (sub @ X5 @ X4)))) & ! [X7 : term] : ((((X0 @ id @ X7 @ id)) != $true) | ($true = ((X0 @ id @ (sK150 @ X0 @ id @ X7) @ id)))))) | (hoasinduction_lem3aa != $true)) & ((hoasinduction_lem3aa = $true) | (($true != ((sK151 @ id @ (lam @ (sub @ sK152 @ (push @ one @ sh))) @ id))) & ! [X10 : term] : ((((sK151 @ id @ X10 @ id)) != $true) | ($true = ((sK151 @ id @ (sub @ sK152 @ (push @ X10 @ id)) @ id)))) & ! [X11 : (subst > term > term)] : (($true = ((sK151 @ id @ (lam @ (X11 @ sh @ one)) @ id))) | (((X11 @ (comp @ (sK155 @ X11) @ (sK153 @ X11)) @ (sub @ (sK154 @ X11) @ (sK153 @ X11)))) != ((sub @ (X11 @ (sK155 @ X11) @ (sK154 @ X11)) @ (sK153 @ X11)))) | ((((sK151 @ id @ (sK156 @ X11) @ id)) = $true) & (((sK151 @ id @ (X11 @ id @ (sK156 @ X11)) @ id)) != $true))))) & (pushprop_lem2v2_gthm = $true) & ((induction2lem = $true) | (! [X19 : term] : (($true = ((sK159 @ (lam @ X19)))) | ((((sK159 @ (sub @ X19 @ (push @ (sK160 @ X19) @ id)))) != $true) & ($true = ((sK159 @ (sK160 @ X19)))))) & ! [X21 : term,X22 : term] : (($true != ((sK159 @ X22))) | ($true = ((sK159 @ (ap @ X21 @ X22)))) | ($true != ((sK159 @ X21)))) & ! [X23 : term] : (($true = ((sK159 @ (sub @ X23 @ sK158)))) | (((var @ X23)) != $true)) & ($true != ((sK159 @ (sub @ sK157 @ sK158)))))) & (! [X24 : term,X25 : subst,X26 : (term > $o)] : ((($true != ((X26 @ (lam @ (sK161 @ X26))))) & ! [X28 : term] : (($true = ((X26 @ (sub @ (sK161 @ X26) @ (push @ X28 @ id))))) | ($true != ((X26 @ X28))))) | (($true = ((X26 @ (sK163 @ X26)))) & ($true != ((X26 @ (ap @ (sK162 @ X26) @ (sK163 @ X26))))) & ($true = ((X26 @ (sK162 @ X26))))) | ((((X26 @ (sub @ (sK164 @ X26 @ X25) @ X25))) != $true) & (((var @ (sK164 @ X26 @ X25))) = $true)) | (((X26 @ (sub @ X24 @ X25))) = $true)) | (induction2lem != $true)) & (lamnotap != $true) & ((hoasinduction_lem3v2a_lthm = $true) | (((((sK166 @ (lam @ sK167))) != $true) & ! [X35 : term] : (((sK165 @ id @ X35 @ id)) = ((sK166 @ X35))) & ! [X36 : (subst > term > term)] : ((((sK165 @ id @ (lam @ (X36 @ sh @ one)) @ id)) = $true) | (($true != ((sK165 @ id @ (X36 @ id @ (sK168 @ X36)) @ id))) & ($true = ((sK165 @ id @ (sK168 @ X36) @ id)))) | (((X36 @ (comp @ (sK169 @ X36) @ (sK171 @ X36)) @ (sub @ (sK170 @ X36) @ (sK171 @ X36)))) != ((sub @ (X36 @ (sK169 @ X36) @ (sK170 @ X36)) @ (sK171 @ X36))))) & ! [X41 : term] : ((((sK166 @ (sub @ sK167 @ (push @ X41 @ id)))) = $true) | (((sK166 @ X41)) != $true))) & ! [X42 : term] : ! [X44 : term,X45 : subst] : (((sK172 @ X42 @ X45 @ X44)) = ((sub @ X42 @ (push @ X44 @ X45)))) & ! [X46 : term] : (((sub @ X46 @ id)) = X46))) & (! [X47 : (subst > term > subst > $o),X48 : (term > $o),X49 : term] : (($true = ((X48 @ (lam @ X49)))) | (((X47 @ id @ (sK173 @ X48 @ X47) @ id)) != ((X48 @ (sK173 @ X48 @ X47)))) | (($true != ((X47 @ id @ (lam @ (sK174 @ X47 @ sh @ one)) @ id))) & ! [X52 : term] : (($true = ((X47 @ id @ (sK174 @ X47 @ id @ X52) @ id))) | ($true != ((X47 @ id @ X52 @ id)))) & ! [X53 : subst,X54 : term,X55 : subst] : (((sK174 @ X47 @ (comp @ X53 @ X55) @ (sub @ X54 @ X55))) = ((sub @ (sK174 @ X47 @ X53 @ X54) @ X55)))) | (($true != ((X48 @ (sub @ X49 @ (push @ (sK175 @ X49 @ X48) @ id))))) & (((X48 @ (sK175 @ X49 @ X48))) = $true))) | ! [X58 : (subst > term > term)] : (((sub @ sK176 @ (push @ (sK177 @ X58) @ (sK178 @ X58)))) != ((X58 @ (sK178 @ X58) @ (sK177 @ X58)))) | (((sub @ sK179 @ id)) != sK179) | (hoasinduction_lem3v2a_lthm != $true)) & (! [X62 : term,X63 : term,X64 : term,X65 : term] : ((X62 = X63) | (((ap @ (sub @ X64 @ id) @ X63)) != ((ap @ (sub @ X65 @ id) @ X62)))) | (hoasapinj2 != $true)) & ((hoasapinj2 = $true) | ((sK180 != sK181) & (((ap @ (sub @ sK183 @ id) @ sK180)) = ((ap @ (sub @ sK182 @ id) @ sK181))))) & (hoasapnotvar_gthm = $true) & (hoasinduction_lem1_gthm = $true) & (! [X70 : term,X71 : (subst > term > subst > $o)] : ((($true != ((X71 @ id @ (lam @ (sK184 @ X71 @ sh @ one)) @ id))) & ! [X73 : term,X74 : subst,X75 : subst] : (((sK184 @ X71 @ (comp @ X74 @ X75) @ (sub @ X73 @ X75))) = ((sub @ (sK184 @ X71 @ X74 @ X73) @ X75))) & ! [X76 : term] : (($true = ((X71 @ id @ (sK184 @ X71 @ id @ X76) @ id))) | ($true != ((X71 @ id @ X76 @ id))))) | ((((X71 @ id @ (sub @ X70 @ (push @ (sK185 @ X71 @ X70) @ id)) @ id)) != $true) & (((X71 @ id @ (sK185 @ X71 @ X70) @ id)) = $true)) | ($true = ((X71 @ id @ (lam @ (sub @ X70 @ (push @ one @ sh))) @ id)))) | (hoasinduction_lem3aa_lthm != $true)) & ((hoasinduction_lem3aa_lthm = $true) | (! [X80 : (subst > term > term)] : (($true = ((sK187 @ id @ (lam @ (X80 @ sh @ one)) @ id))) | (((X80 @ (comp @ (sK189 @ X80) @ (sK190 @ X80)) @ (sub @ (sK188 @ X80) @ (sK190 @ X80)))) != ((sub @ (X80 @ (sK189 @ X80) @ (sK188 @ X80)) @ (sK190 @ X80)))) | (($true != ((sK187 @ id @ (X80 @ id @ (sK191 @ X80)) @ id))) & (((sK187 @ id @ (sK191 @ X80) @ id)) = $true))) & ! [X85 : term] : ((((sK187 @ id @ (sub @ sK186 @ (push @ X85 @ id)) @ id)) = $true) | ($true != ((sK187 @ id @ X85 @ id)))) & ($true != ((sK187 @ id @ (lam @ (sub @ sK186 @ (push @ one @ sh))) @ id))))) & (! [X86 : subst] : (((comp @ id @ X86)) = X86) | (axidl != $true)) & ((axidl = $true) | (sK192 != ((comp @ id @ sK192)))) & ((hoaslamnotap = $true) | (! [X91 : subst,X92 : subst,X93 : term] : (((sub @ (sK195 @ X91 @ X93) @ X92)) = ((sK195 @ (comp @ X91 @ X92) @ (sub @ X93 @ X92)))) & (((ap @ (sub @ sK193 @ id) @ sK194)) = ((lam @ (sK195 @ sh @ one)))))) & (! [X94 : term,X95 : term,X96 : (subst > term > term)] : ((((X96 @ (comp @ (sK196 @ X96) @ (sK197 @ X96)) @ (sub @ (sK198 @ X96) @ (sK197 @ X96)))) != ((sub @ (X96 @ (sK196 @ X96) @ (sK198 @ X96)) @ (sK197 @ X96)))) | (((ap @ (sub @ X94 @ id) @ X95)) != ((lam @ (X96 @ sh @ one))))) | (hoaslamnotap != $true)) & (! [X100 : term,X101 : term] : ((((sub @ X101 @ sh)) != ((sub @ X100 @ sh))) | (X100 = X101)) | (shinj != $true)) & ((shinj = $true) | ((((sub @ sK199 @ sh)) = ((sub @ sK200 @ sh))) & (sK200 != sK199))) & (! [X104 : term,X105 : term,X106 : (subst > term > subst > $o)] : (($true != ((X106 @ id @ X105 @ id))) | (((X106 @ id @ X104 @ id)) != $true) | ($true = ((X106 @ id @ (ap @ X104 @ X105) @ id))) | (($true = ((X106 @ (comp @ (sK204 @ X106) @ (sK202 @ X106)) @ (sub @ (sK203 @ X106) @ (sK202 @ X106)) @ (sK201 @ X106)))) & ($true != ((X106 @ (sK204 @ X106) @ (sK203 @ X106) @ (comp @ (sK202 @ X106) @ (sK201 @ X106)))))) | (($true != ((X106 @ (comp @ (sK205 @ X106) @ (sK208 @ X106)) @ (sub @ (sK207 @ X106) @ (sK208 @ X106)) @ (sK206 @ X106)))) & ($true = ((X106 @ (sK205 @ X106) @ (sK207 @ X106) @ (comp @ (sK208 @ X106) @ (sK206 @ X106)))))) | ((((X106 @ id @ (ap @ (sub @ (sK210 @ X106) @ id) @ (sK209 @ X106)) @ id)) != $true) & ($true = ((X106 @ id @ (sK210 @ X106) @ id))) & ($true = ((X106 @ id @ (sK209 @ X106) @ id))))) | (hoasinduction_lem2 != $true)) & ((hoasinduction_lem2 = $true) | ((((sK213 @ id @ sK212 @ id)) = $true) & ($true = ((sK213 @ id @ sK211 @ id))) & ($true != ((sK213 @ id @ (ap @ sK211 @ sK212) @ id))) & ! [X120 : subst,X121 : subst,X122 : term,X123 : subst] : (($true != ((sK213 @ (comp @ X123 @ X121) @ (sub @ X122 @ X121) @ X120))) | ($true = ((sK213 @ X123 @ X122 @ (comp @ X121 @ X120))))) & ! [X124 : subst,X125 : subst,X126 : term,X127 : subst] : (($true = ((sK213 @ (comp @ X124 @ X127) @ (sub @ X126 @ X127) @ X125))) | (((sK213 @ X124 @ X126 @ (comp @ X127 @ X125))) != $true)) & ! [X128 : term,X129 : term] : ((((sK213 @ id @ (ap @ (sub @ X129 @ id) @ X128) @ id)) = $true) | ($true != ((sK213 @ id @ X129 @ id))) | ($true != ((sK213 @ id @ X128 @ id)))))) & (pushprop_lem3v2 = $true) & ! [X130 : d_subst,X131 : d_subst] : ((((d2subst @ X130)) != ((d2subst @ X131))) | (X130 = X131)) & (! [X132 : term,X133 : term,X134 : term,X135 : term] : ((((ap @ (sub @ X135 @ id) @ X133)) != ((ap @ (sub @ X132 @ id) @ X134))) | (X132 = X135)) | (hoasapinj1 != $true)) & ((hoasapinj1 = $true) | ((((ap @ (sub @ sK217 @ id) @ sK215)) = ((ap @ (sub @ sK214 @ id) @ sK216))) & (sK217 != sK214))) & ((axvarcons = $true) | (sK219 != ((sub @ one @ (push @ sK219 @ sK218))))) & (! [X142 : subst,X143 : term] : (((sub @ one @ (push @ X143 @ X142))) = X143) | (axvarcons != $true)) & (((d2term @ d_one)) = ((ap @ (d2term @ d_one) @ (d2term @ d_one)))) & ((hoasinduction_lthm = $true) | ($true != sP10)) & (($true = sP10) | (hoasinduction_lthm != $true)) & ! [X144 : d_term,X145 : d_term] : ((((d2term @ X144)) != ((d2term @ X145))) | (X144 = X145)) & (hoaslamnotvar_gthm = $true) & (hoaslamnotvar = $true) & ((hoasinduction = $true) | (! [X148 : term,X149 : term] : (($true = ((sK221 @ id @ (ap @ (sub @ X149 @ id) @ X148) @ id))) | ($true != ((sK221 @ id @ X149 @ id))) | ($true != ((sK221 @ id @ X148 @ id)))) & ! [X150 : subst,X151 : subst,X152 : term,X153 : subst] : (($true = ((sK221 @ (comp @ X151 @ X150) @ (sub @ X152 @ X150) @ X153))) | (((sK221 @ X151 @ X152 @ (comp @ X150 @ X153))) != $true)) & ! [X154 : term] : (($true != ((var @ (sub @ X154 @ id)))) | ($true = ((sK221 @ id @ X154 @ id)))) & ($true != ((sK221 @ id @ sK220 @ id))) & (((sP6 @ sK221)) != $true) & ! [X155 : subst,X156 : subst,X157 : subst,X158 : term] : (($true != ((sK221 @ (comp @ X156 @ X155) @ (sub @ X158 @ X155) @ X157))) | ($true = ((sK221 @ X156 @ X158 @ (comp @ X155 @ X157))))))) & (! [X159 : term,X160 : (subst > term > subst > $o)] : (((((X160 @ id @ (ap @ (sub @ (sK223 @ X160) @ id) @ (sK222 @ X160)) @ id)) != $true) & ($true = ((X160 @ id @ (sK223 @ X160) @ id))) & ($true = ((X160 @ id @ (sK222 @ X160) @ id)))) | ((((X160 @ (comp @ (sK225 @ X160) @ (sK224 @ X160)) @ (sub @ (sK226 @ X160) @ (sK224 @ X160)) @ (sK227 @ X160))) != $true) & ($true = ((X160 @ (sK225 @ X160) @ (sK226 @ X160) @ (comp @ (sK224 @ X160) @ (sK227 @ X160)))))) | ((((var @ (sub @ (sK228 @ X160) @ id))) = $true) & ($true != ((X160 @ id @ (sK228 @ X160) @ id)))) | ($true = ((X160 @ id @ X159 @ id))) | ($true = ((sP6 @ X160))) | (($true = ((X160 @ (comp @ (sK230 @ X160) @ (sK229 @ X160)) @ (sub @ (sK232 @ X160) @ (sK229 @ X160)) @ (sK231 @ X160)))) & ($true != ((X160 @ (sK230 @ X160) @ (sK232 @ X160) @ (comp @ (sK229 @ X160) @ (sK231 @ X160))))))) | (hoasinduction != $true)) & (pushprop_lem3v2_lthm = $true) & (! [X172 : subst] : (((push @ (sub @ one @ X172) @ (comp @ sh @ X172))) = X172) | (axscons != $true)) & ((axscons = $true) | (((push @ (sub @ one @ sK233) @ (comp @ sh @ sK233))) != sK233)) & (hoasinduction_lem3b_gthm = $true) & (((d2subst @ d_id)) = ((push @ (d2term @ d_one) @ (d2subst @ d_id)))) & ! [X174 : d_term] : (d_one = X174) & ((hoasapinj2_lthm = $true) | (! [X175 : term,X176 : term,X177 : term,X178 : term] : ((X175 = X177) | (((ap @ X176 @ X175)) != ((ap @ X178 @ X177)))) & ((sK236 != sK237) & (((ap @ (sub @ sK234 @ id) @ sK236)) = ((ap @ (sub @ sK235 @ id) @ sK237)))))) & (((sK240 != sK238) & (((ap @ sK239 @ sK238)) = ((ap @ sK241 @ sK240)))) | ! [X187 : term,X188 : term,X189 : term,X190 : term] : ((X189 = X190) | (((ap @ (sub @ X187 @ id) @ X189)) != ((ap @ (sub @ X188 @ id) @ X190)))) | (hoasapinj2_lthm != $true)) & (axvarshift = $true) & (! [X191 : term] : (((sub @ X191 @ (push @ one @ sh))) = ((sK242 @ X191 @ sh @ one))) | (hoasinduction_lem3b != $true)) & ((hoasinduction_lem3b = $true) | ! [X194 : (subst > term > term)] : (((X194 @ sh @ one)) != ((sub @ sK243 @ (push @ one @ sh))))) & ((hoasinduction_lem3a_lthm = $true) | (! [X195 : (subst > term > subst > $o),X196 : term] : ((! [X198 : term,X199 : subst,X200 : subst] : (((sK244 @ X195 @ (comp @ X199 @ X200) @ (sub @ X198 @ X200))) = ((sub @ (sK244 @ X195 @ X199 @ X198) @ X200))) & ! [X201 : term] : ((((X195 @ id @ X201 @ id)) != $true) | (((X195 @ id @ (sK244 @ X195 @ id @ X201) @ id)) = $true)) & ($true != ((X195 @ id @ (lam @ (sK244 @ X195 @ sh @ one)) @ id)))) | ((((X195 @ id @ (sK245 @ X196 @ X195) @ id)) = $true) & (((X195 @ id @ (sub @ X196 @ (push @ (sK245 @ X196 @ X195) @ id)) @ id)) != $true)) | ($true = ((X195 @ id @ (lam @ (sub @ X196 @ (push @ one @ sh))) @ id)))) & ! [X203 : term] : (((sub @ X203 @ id)) = X203) & (! [X206 : (subst > term > term)] : ((((X206 @ (comp @ (sK248 @ X206) @ (sK249 @ X206)) @ (sub @ (sK250 @ X206) @ (sK249 @ X206)))) != ((sub @ (X206 @ (sK248 @ X206) @ (sK250 @ X206)) @ (sK249 @ X206)))) | (($true != ((sK246 @ id @ (X206 @ id @ (sK251 @ X206)) @ id))) & ($true = ((sK246 @ id @ (sK251 @ X206) @ id)))) | (((sK246 @ id @ (lam @ (X206 @ sh @ one)) @ id)) = $true)) & ! [X211 : term] : ((((sK246 @ id @ (sub @ sK247 @ (push @ X211 @ id)) @ id)) = $true) | (((sK246 @ id @ X211 @ id)) != $true)) & ($true != ((sK246 @ id @ (lam @ sK247) @ id)))))) & ((! [X214 : (subst > term > term)] : ((((X214 @ (comp @ (sK255 @ X214) @ (sK256 @ X214)) @ (sub @ (sK254 @ X214) @ (sK256 @ X214)))) != ((sub @ (X214 @ (sK255 @ X214) @ (sK254 @ X214)) @ (sK256 @ X214)))) | (($true = ((sK252 @ id @ (sK257 @ X214) @ id))) & ($true != ((sK252 @ id @ (X214 @ id @ (sK257 @ X214)) @ id)))) | (((sK252 @ id @ (lam @ (X214 @ sh @ one)) @ id)) = $true)) & ! [X219 : term] : (($true != ((sK252 @ id @ X219 @ id))) | ($true = ((sK252 @ id @ (sub @ sK253 @ (push @ X219 @ id)) @ id)))) & ($true != ((sK252 @ id @ (lam @ (sub @ sK253 @ (push @ one @ sh))) @ id)))) | (sK258 != ((sub @ sK258 @ id))) | ! [X221 : (subst > term > subst > $o),X222 : term] : ((! [X224 : subst,X225 : subst,X226 : term] : (((sub @ (sK259 @ X221 @ X224 @ X226) @ X225)) = ((sK259 @ X221 @ (comp @ X224 @ X225) @ (sub @ X226 @ X225)))) & ! [X227 : term] : (($true = ((X221 @ id @ (sK259 @ X221 @ id @ X227) @ id))) | ($true != ((X221 @ id @ X227 @ id)))) & (((X221 @ id @ (lam @ (sK259 @ X221 @ sh @ one)) @ id)) != $true)) | (($true != ((X221 @ id @ (sub @ X222 @ (push @ (sK260 @ X222 @ X221) @ id)) @ id))) & ($true = ((X221 @ id @ (sK260 @ X222 @ X221) @ id)))) | (((X221 @ id @ (lam @ X222) @ id)) = $true)) | (hoasinduction_lem3a_lthm != $true)) & (induction2lem_gthm = $true) & ! [X229 : d_subst] : (d_id = X229) & (pushprop_lem2v2_lthm = $true) & (ulamvar1 != $true) & ! [X230 : subst] : (((d2subst @ (sK261 @ X230))) = X230) & (pushprop_lthm = $true) & (hoasinduction_lem1v2 = $true) & (hoasinduction_lem1v2_gthm = $true) & ((hoasinduction_lem3v2a = $true) | (! [X235 : term] : (((sK262 @ X235)) = ((sK264 @ id @ X235 @ id))) & ($true != ((sK262 @ (lam @ sK263)))) & ! [X236 : term] : ((((sK262 @ (sub @ sK263 @ (push @ X236 @ id)))) = $true) | ($true != ((sK262 @ X236)))) & ! [X237 : (subst > term > term)] : ((((X237 @ (comp @ (sK266 @ X237) @ (sK267 @ X237)) @ (sub @ (sK265 @ X237) @ (sK267 @ X237)))) != ((sub @ (X237 @ (sK266 @ X237) @ (sK265 @ X237)) @ (sK267 @ X237)))) | (($true != ((sK264 @ id @ (X237 @ id @ (sK268 @ X237)) @ id))) & ($true = ((sK264 @ id @ (sK268 @ X237) @ id)))) | ($true = ((sK264 @ id @ (lam @ (X237 @ sh @ one)) @ id)))))) & (! [X242 : (term > $o),X243 : term,X244 : (subst > term > subst > $o)] : ((((X242 @ (sK269 @ X244 @ X242))) != ((X244 @ id @ (sK269 @ X244 @ X242) @ id))) | (((X242 @ (lam @ X243))) = $true) | (($true != ((X242 @ (sub @ X243 @ (push @ (sK270 @ X243 @ X242) @ id))))) & ($true = ((X242 @ (sK270 @ X243 @ X242))))) | (! [X248 : term,X249 : subst,X250 : subst] : (((sub @ (sK271 @ X244 @ X249 @ X248) @ X250)) = ((sK271 @ X244 @ (comp @ X249 @ X250) @ (sub @ X248 @ X250)))) & ! [X251 : term] : ((((X244 @ id @ (sK271 @ X244 @ id @ X251) @ id)) = $true) | ($true != ((X244 @ id @ X251 @ id)))) & ($true != ((X244 @ id @ (lam @ (sK271 @ X244 @ sh @ one)) @ id))))) | (hoasinduction_lem3v2a != $true)) & (pushprop_lthm_orig = $true) & (pushprop_lem1v2_gthm = $true) & (pushprop_lem2v2 = $true) & ! [X252 : term] : (((d2term @ (sK272 @ X252))) = X252) & ((hoasinduction_no_psi_cond = $true) | (($true != ((sK273 @ id @ sK274 @ id))) & ! [X256 : (subst > term > term)] : ((((sK273 @ id @ (lam @ (X256 @ sh @ one)) @ id)) = $true) | (((sub @ (X256 @ (sK275 @ X256) @ (sK277 @ X256)) @ (sK276 @ X256))) != ((X256 @ (comp @ (sK275 @ X256) @ (sK276 @ X256)) @ (sub @ (sK277 @ X256) @ (sK276 @ X256))))) | (($true != ((sK273 @ id @ (X256 @ id @ (sK278 @ X256)) @ id))) & ($true = ((sK273 @ id @ (sK278 @ X256) @ id))))) & ! [X261 : term,X262 : term] : (($true != ((sK273 @ id @ X261 @ id))) | ($true != ((sK273 @ id @ X262 @ id))) | ($true = ((sK273 @ id @ (ap @ (sub @ X262 @ id) @ X261) @ id)))))) & (! [X263 : (subst > term > subst > $o),X264 : term] : (($true = ((X263 @ id @ X264 @ id))) | (($true != ((X263 @ id @ (lam @ (sK279 @ X263 @ sh @ one)) @ id))) & ! [X266 : subst,X267 : subst,X268 : term] : (((sK279 @ X263 @ (comp @ X266 @ X267) @ (sub @ X268 @ X267))) = ((sub @ (sK279 @ X263 @ X266 @ X268) @ X267))) & ! [X269 : term] : (($true = ((X263 @ id @ (sK279 @ X263 @ id @ X269) @ id))) | (((X263 @ id @ X269 @ id)) != $true))) | (($true = ((X263 @ id @ (sK280 @ X263) @ id))) & ($true = ((X263 @ id @ (sK281 @ X263) @ id))) & ($true != ((X263 @ id @ (ap @ (sub @ (sK281 @ X263) @ id) @ (sK280 @ X263)) @ id))))) | (hoasinduction_no_psi_cond != $true)) & (! [X272 : term,X273 : (subst > term > subst > $o)] : (($true = ((X273 @ id @ (lam @ (sub @ X272 @ (push @ one @ sh))) @ id))) | (! [X276 : subst,X277 : subst,X278 : term] : (((sub @ (sK282 @ X273) @ (push @ (sub @ X278 @ X276) @ (comp @ X277 @ X276)))) = ((sK283 @ X273 @ (comp @ X277 @ X276) @ (sub @ X278 @ X276)))) & ! [X279 : subst,X280 : term,X281 : subst] : (((sub @ (sK283 @ X273 @ X281 @ X280) @ X279)) = ((sub @ (sub @ (sK282 @ X273) @ (push @ X280 @ X281)) @ X279))) & (((X273 @ id @ (lam @ (sK283 @ X273 @ sh @ one)) @ id)) != $true) & ! [X282 : term] : (($true = ((X273 @ id @ (sK283 @ X273 @ id @ X282) @ id))) | (((X273 @ id @ X282 @ id)) != $true))) | (($true != ((X273 @ id @ (sub @ X272 @ (push @ (sK284 @ X273 @ X272) @ id)) @ id))) & (((X273 @ id @ (sK284 @ X273 @ X272) @ id)) = $true))) | (hoasinduction_lem3aaa != $true)) & ((hoasinduction_lem3aaa = $true) | ((((sK286 @ id @ (lam @ (sub @ sK285 @ (push @ one @ sh))) @ id)) != $true) & ! [X286 : term,X287 : (subst > term > term)] : ((((X287 @ (comp @ (sK288 @ X287 @ X286) @ (sK287 @ X287 @ X286)) @ (sub @ (sK289 @ X287 @ X286) @ (sK287 @ X287 @ X286)))) != ((sub @ X286 @ (push @ (sub @ (sK289 @ X287 @ X286) @ (sK287 @ X287 @ X286)) @ (comp @ (sK288 @ X287 @ X286) @ (sK287 @ X287 @ X286)))))) | (((sub @ (sub @ X286 @ (push @ (sK291 @ X287 @ X286) @ (sK292 @ X287 @ X286))) @ (sK290 @ X287 @ X286))) != ((sub @ (X287 @ (sK292 @ X287 @ X286) @ (sK291 @ X287 @ X286)) @ (sK290 @ X287 @ X286)))) | ($true = ((sK286 @ id @ (lam @ (X287 @ sh @ one)) @ id))) | ((((sK286 @ id @ (X287 @ id @ (sK293 @ X287)) @ id)) != $true) & ($true = ((sK286 @ id @ (sK293 @ X287) @ id))))) & ! [X295 : term] : ((((sK286 @ id @ (sub @ sK285 @ (push @ X295 @ id)) @ id)) = $true) | (((sK286 @ id @ X295 @ id)) != $true)))) & (pushprop_p_and_p_prime = (^[Y0 : term]: ((^[Y1 : subst]: ((^[Y2 : term > $o]: ((^[Y3 : term > $o]: (!! @ term @ (^[Y4 : term]: ((Y3 @ Y4) = (Y2 @ (sub @ Y4 @ (push @ Y0 @ Y1)))))))))))))) & (pushprop_lem0_gthm = $true) & (apnotvar = $true) & (hoasapnotvar = $true) & (hoasinduction_lem1_lthm = $true) & ((hoaslaminj = $true) | ((((lam @ (sK297 @ sh @ one))) = ((lam @ (sK295 @ sh @ one)))) & ! [X300 : subst,X301 : term,X302 : subst] : (((sK297 @ (comp @ X300 @ X302) @ (sub @ X301 @ X302))) = ((sub @ (sK297 @ X300 @ X301) @ X302))) & (((sK295 @ sK294 @ sK296)) != ((sK297 @ sK294 @ sK296))) & ! [X303 : subst,X304 : subst,X305 : term] : (((sub @ (sK295 @ X303 @ X305) @ X304)) = ((sK295 @ (comp @ X303 @ X304) @ (sub @ X305 @ X304)))))) & (! [X306 : subst,X307 : (subst > term > term),X308 : term,X309 : (subst > term > term)] : ((((lam @ (X309 @ sh @ one))) != ((lam @ (X307 @ sh @ one)))) | (((sub @ (X309 @ (sK298 @ X309) @ (sK299 @ X309)) @ (sK300 @ X309))) != ((X309 @ (comp @ (sK298 @ X309) @ (sK300 @ X309)) @ (sub @ (sK299 @ X309) @ (sK300 @ X309))))) | (((X307 @ X306 @ X308)) = ((X309 @ X306 @ X308))) | (((sub @ (X307 @ (sK301 @ X307) @ (sK303 @ X307)) @ (sK302 @ X307))) != ((X307 @ (comp @ (sK301 @ X307) @ (sK302 @ X307)) @ (sub @ (sK303 @ X307) @ (sK302 @ X307)))))) | (hoaslaminj != $true)) & (induction2_gthm = $true) & (! [X316 : term] : (((sub @ X316 @ id)) = X316) | (termmset != $true)) & ((termmset = $true) | (sK304 != ((sub @ sK304 @ id)))) & ((hoasinduction_lem0_lthm = $true) | ! [X319 : (term > $o)] : (((X319 @ (sK306 @ X319))) != ((sK305 @ id @ (sK306 @ X319) @ id)))) & (! [X321 : (subst > term > subst > $o)] : ! [X323 : term] : (((X321 @ id @ X323 @ id)) = ((sK307 @ X321 @ X323))) | (hoasinduction_lem0_lthm != $true)) & (hoaslamnotap_gthm = $true) & (((d2subst @ d_id)) = sh) & (((d2term @ d_one)) = ((hoasap @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id) @ (d2term @ d_one)))) & (hoasinduction_lem2v2_gthm = $true) & ((hoaslaminj_lthm = $true) | (! [X324 : subst,X325 : term] : (((comp @ sh @ (push @ X325 @ X324))) = X324) & ! [X326 : subst,X327 : term] : (((sub @ one @ (push @ X327 @ X326))) = X327) & (! [X332 : term,X333 : subst,X334 : subst] : (((sK309 @ (comp @ X334 @ X333) @ (sub @ X332 @ X333))) = ((sub @ (sK309 @ X334 @ X332) @ X333))) & (((lam @ (sK308 @ sh @ one))) = ((lam @ (sK309 @ sh @ one)))) & ! [X335 : subst,X336 : term,X337 : subst] : (((sub @ (sK308 @ X337 @ X336) @ X335)) = ((sK308 @ (comp @ X337 @ X335) @ (sub @ X336 @ X335)))) & (((sK309 @ sK311 @ sK310)) != ((sK308 @ sK311 @ sK310)))) & ! [X338 : term,X339 : term] : ((X338 = X339) | (((lam @ X339)) != ((lam @ X338)))))) & ((sK312 != ((comp @ sh @ (push @ sK313 @ sK312)))) | (((sub @ one @ (push @ sK315 @ sK314))) != sK315) | ! [X344 : (subst > term > term),X345 : (subst > term > term),X346 : term,X347 : subst] : ((((sub @ (X345 @ (sK318 @ X345) @ (sK316 @ X345)) @ (sK317 @ X345))) != ((X345 @ (comp @ (sK318 @ X345) @ (sK317 @ X345)) @ (sub @ (sK316 @ X345) @ (sK317 @ X345))))) | (((lam @ (X344 @ sh @ one))) != ((lam @ (X345 @ sh @ one)))) | (((sub @ (X344 @ (sK321 @ X344) @ (sK320 @ X344)) @ (sK319 @ X344))) != ((X344 @ (comp @ (sK321 @ X344) @ (sK319 @ X344)) @ (sub @ (sK320 @ X344) @ (sK319 @ X344))))) | (((X345 @ X347 @ X346)) = ((X344 @ X347 @ X346)))) | ((sK323 != sK322) & (((lam @ sK322)) = ((lam @ sK323)))) | (hoaslaminj_lthm != $true)) & (! [X356 : term,X357 : (subst > term > subst > $o)] : ((($true != ((X357 @ id @ (sub @ X356 @ (push @ (sK324 @ X357 @ X356) @ id)) @ id))) & ($true = ((X357 @ id @ (sK324 @ X357 @ X356) @ id)))) | ((((X357 @ (comp @ (sK325 @ X357) @ (sK327 @ X357)) @ (sub @ (sK328 @ X357) @ (sK327 @ X357)) @ (sK326 @ X357))) != $true) & (((X357 @ (sK325 @ X357) @ (sK328 @ X357) @ (comp @ (sK327 @ X357) @ (sK326 @ X357)))) = $true)) | (! [X364 : subst,X365 : term,X366 : subst] : (((sK329 @ X357 @ (comp @ X364 @ X366) @ (sub @ X365 @ X366))) = ((sub @ (sK329 @ X357 @ X364 @ X365) @ X366))) & ! [X367 : term] : (($true != ((X357 @ id @ X367 @ id))) | ($true = ((X357 @ id @ (sK329 @ X357 @ id @ X367) @ id)))) & (((X357 @ id @ (lam @ (sK329 @ X357 @ sh @ one)) @ id)) != $true)) | ($true = ((X357 @ id @ (lam @ X356) @ id))) | ((((X357 @ (comp @ (sK333 @ X357) @ (sK332 @ X357)) @ (sub @ (sK331 @ X357) @ (sK332 @ X357)) @ (sK330 @ X357))) = $true) & (((X357 @ (sK333 @ X357) @ (sK331 @ X357) @ (comp @ (sK332 @ X357) @ (sK330 @ X357)))) != $true))) | (hoasinduction_lem3 != $true)) & ((hoasinduction_lem3 = $true) | (! [X374 : term] : ((((sK335 @ id @ (sub @ sK334 @ (push @ X374 @ id)) @ id)) = $true) | (((sK335 @ id @ X374 @ id)) != $true)) & ! [X375 : subst,X376 : subst,X377 : subst,X378 : term] : (($true = ((sK335 @ (comp @ X375 @ X377) @ (sub @ X378 @ X377) @ X376))) | ($true != ((sK335 @ X375 @ X378 @ (comp @ X377 @ X376))))) & ! [X379 : (subst > term > term)] : ((((sub @ (X379 @ (sK336 @ X379) @ (sK337 @ X379)) @ (sK338 @ X379))) != ((X379 @ (comp @ (sK336 @ X379) @ (sK338 @ X379)) @ (sub @ (sK337 @ X379) @ (sK338 @ X379))))) | (($true = ((sK335 @ id @ (sK339 @ X379) @ id))) & (((sK335 @ id @ (X379 @ id @ (sK339 @ X379)) @ id)) != $true)) | ($true = ((sK335 @ id @ (lam @ (X379 @ sh @ one)) @ id)))) & ($true != ((sK335 @ id @ (lam @ sK334) @ id))) & ! [X384 : subst,X385 : term,X386 : subst,X387 : subst] : (($true != ((sK335 @ (comp @ X387 @ X386) @ (sub @ X385 @ X386) @ X384))) | (((sK335 @ X387 @ X385 @ (comp @ X386 @ X384))) = $true)))) & (termmset_gthm = $true) & (((d2subst @ d_id)) = ((comp @ (d2subst @ d_id) @ (d2subst @ d_id)))) & (hoasinduction_lem2_lthm = $true) & (! [X388 : term,X389 : term] : ((((lam @ X388)) != ((lam @ X389))) | (X388 = X389)) | (laminj != $true)) & ((laminj = $true) | ((((lam @ sK341)) = ((lam @ sK340))) & (sK340 != sK341))) & (((var @ (d2term @ d_one))) != $true) & (! [X392 : subst] : (((comp @ X392 @ id)) = X392) | (axidr != $true)) & ((axidr = $true) | (((comp @ sK342 @ id)) != sK342)) & (! [X394 : term] : (((sub @ X394 @ id)) = X394) | (axvarid != $true)) & ((axvarid = $true) | (sK343 != ((sub @ sK343 @ id)))) & ((sK344 != ((sub @ sK344 @ id))) | ($true = sP5) | ! [X398 : (term > $o)] : (((X398 @ (sK346 @ X398))) != ((sK345 @ id @ (sK346 @ X398) @ id))) | (($true != ((sK348 @ (lam @ sK347)))) & ! [X403 : (subst > term > term)] : (($true = ((sK349 @ id @ (lam @ (X403 @ sh @ one)) @ id))) | (((X403 @ (comp @ (sK351 @ X403) @ (sK350 @ X403)) @ (sub @ (sK352 @ X403) @ (sK350 @ X403)))) != ((sub @ (X403 @ (sK351 @ X403) @ (sK352 @ X403)) @ (sK350 @ X403)))) | ((((sK349 @ id @ (sK353 @ X403) @ id)) = $true) & ($true != ((sK349 @ id @ (X403 @ id @ (sK353 @ X403)) @ id))))) & ! [X408 : term] : (($true != ((sK348 @ X408))) | ($true = ((sK348 @ (sub @ sK347 @ (push @ X408 @ id)))))) & ! [X409 : term] : (((sK348 @ X409)) = ((sK349 @ id @ X409 @ id)))) | (! [X412 : term] : (($true = ((sK355 @ X412))) | ($true != ((var @ X412)))) & ($true != ((sK355 @ sK354))) & ! [X413 : term] : ((($true = ((sK355 @ (sK356 @ X413)))) & ($true != ((sK355 @ (sub @ X413 @ (push @ (sK356 @ X413) @ id)))))) | ($true = ((sK355 @ (lam @ X413))))) & ! [X415 : term,X416 : term] : (($true != ((sK355 @ X415))) | ($true = ((sK355 @ (ap @ X415 @ X416)))) | ($true != ((sK355 @ X416))))) | (hoasinduction_lthm_3 != $true)) & ((hoasinduction_lthm_3 = $true) | (! [X417 : term] : (((sub @ X417 @ id)) = X417) & ($true != sP5) & ! [X418 : (subst > term > subst > $o)] : ! [X420 : term] : (((X418 @ id @ X420 @ id)) = ((sK357 @ X418 @ X420))) & ! [X421 : term,X422 : (term > $o),X423 : (subst > term > subst > $o)] : (($true = ((X422 @ (lam @ X421)))) | (($true != ((X423 @ id @ (lam @ (sK358 @ X423 @ sh @ one)) @ id))) & ! [X425 : subst,X426 : subst,X427 : term] : (((sK358 @ X423 @ (comp @ X426 @ X425) @ (sub @ X427 @ X425))) = ((sub @ (sK358 @ X423 @ X426 @ X427) @ X425))) & ! [X428 : term] : ((((X423 @ id @ X428 @ id)) != $true) | ($true = ((X423 @ id @ (sK358 @ X423 @ id @ X428) @ id))))) | (($true = ((X422 @ (sK359 @ X422 @ X421)))) & (((X422 @ (sub @ X421 @ (push @ (sK359 @ X422 @ X421) @ id)))) != $true)) | (((X422 @ (sK360 @ X423 @ X422))) != ((X423 @ id @ (sK360 @ X423 @ X422) @ id)))) & ! [X431 : term,X432 : (term > $o)] : (((((X432 @ (sK361 @ X432))) != $true) & ($true = ((var @ (sK361 @ X432))))) | ($true = ((X432 @ X431))) | (! [X435 : term] : ((((X432 @ X435)) != $true) | ($true = ((X432 @ (sub @ (sK362 @ X432) @ (push @ X435 @ id)))))) & ($true != ((X432 @ (lam @ (sK362 @ X432)))))) | (($true = ((X432 @ (sK363 @ X432)))) & ($true != ((X432 @ (ap @ (sK363 @ X432) @ (sK364 @ X432))))) & (((X432 @ (sK364 @ X432))) = $true))))) & (substmonoid_gthm = $true) & (pushprop = $true) & (((hoasvar @ (d2subst @ d_id) @ (d2term @ d_one) @ (d2subst @ d_id))) != $true) & (axclos = $true) & ((hoasinduction_lem3a = $true) | (! [X440 : (subst > term > term)] : ((($true != ((sK365 @ id @ (X440 @ id @ (sK367 @ X440)) @ id))) & ($true = ((sK365 @ id @ (sK367 @ X440) @ id)))) | (((sK365 @ id @ (lam @ (X440 @ sh @ one)) @ id)) = $true) | (((sub @ (X440 @ (sK368 @ X440) @ (sK370 @ X440)) @ (sK369 @ X440))) != ((X440 @ (comp @ (sK368 @ X440) @ (sK369 @ X440)) @ (sub @ (sK370 @ X440) @ (sK369 @ X440)))))) & ! [X445 : term] : (($true != ((sK365 @ id @ X445 @ id))) | ($true = ((sK365 @ id @ (sub @ sK366 @ (push @ X445 @ id)) @ id)))) & ($true != ((sK365 @ id @ (lam @ sK366) @ id))))) & (! [X446 : (subst > term > subst > $o),X447 : term] : ((! [X449 : term] : (($true = ((X446 @ id @ (sK371 @ X446 @ id @ X449) @ id))) | ($true != ((X446 @ id @ X449 @ id)))) & ($true != ((X446 @ id @ (lam @ (sK371 @ X446 @ sh @ one)) @ id))) & ! [X450 : subst,X451 : subst,X452 : term] : (((sub @ (sK371 @ X446 @ X450 @ X452) @ X451)) = ((sK371 @ X446 @ (comp @ X450 @ X451) @ (sub @ X452 @ X451))))) | (($true = ((X446 @ id @ (sK372 @ X447 @ X446) @ id))) & ($true != ((X446 @ id @ (sub @ X447 @ (push @ (sK372 @ X447 @ X446) @ id)) @ id)))) | (((X446 @ id @ (lam @ X447) @ id)) = $true)) | (hoasinduction_lem3a != $true)) & ((sK373 != ((sub @ sK373 @ id))) | ! [X455 : term] : (((sub @ X455 @ id)) = X455) | (termmset_lthm != $true)) & ((termmset_lthm = $true) | (! [X456 : term] : (((sub @ X456 @ id)) = X456) & (((sub @ sK374 @ id)) != sK374))) & (ulamvarsh = $true) & ((hoasinduction_lem3v2_f = $true) | ! [X459 : (subst > term > term)] : (((X459 @ (sK376 @ X459) @ (sK377 @ X459))) != ((sub @ sK375 @ (push @ (sK377 @ X459) @ (sK376 @ X459)))))) & (! [X462 : term] : ! [X464 : subst,X465 : term] : (((sub @ X462 @ (push @ X465 @ X464))) = ((sK378 @ X462 @ X464 @ X465))) | (hoasinduction_lem3v2_f != $true)) & (hoaslam = (^[Y0 : subst]: ((^[Y1 : subst > term > term]: (d2term @ d_one))))) & ((substmonoid = $true) | (((comp @ sK379 @ id)) != sK379) | (sK380 != ((comp @ id @ sK380)))) & ((! [X468 : subst] : (((comp @ X468 @ id)) = X468) & ! [X469 : subst] : (((comp @ id @ X469)) = X469)) | (substmonoid != $true)) & ((hoasinduction_lem3b_lthm = $true) | ! [X471 : (subst > term > term)] : (((X471 @ sh @ one)) != ((sub @ sK381 @ (push @ one @ sh))))) & (! [X472 : term] : (((sK382 @ X472 @ sh @ one)) = ((sub @ X472 @ (push @ one @ sh)))) | (hoasinduction_lem3b_lthm != $true)) & (! [X474 : (term > $o),X475 : (term > $o),X476 : subst,X477 : term] : ((((X474 @ (sub @ (sK383 @ X477 @ X476 @ X475 @ X474) @ (push @ X477 @ X476)))) != ((X475 @ (sK383 @ X477 @ X476 @ X475 @ X474)))) | ($true != ((X474 @ X477))) | (((X475 @ one)) = $true)) | (sK385 != ((sub @ one @ (push @ sK385 @ sK384)))) | (pushprop_lem1v2_lthm != $true)) & ((pushprop_lem1v2_lthm = $true) | ((! [X485 : term] : (((sK386 @ (sub @ X485 @ (push @ sK389 @ sK388)))) = ((sK387 @ X485))) & (((sK386 @ sK389)) = $true) & ($true != ((sK387 @ one)))) & ! [X486 : subst,X487 : term] : (((sub @ one @ (push @ X487 @ X486))) = X487))) & (axassoc = $true) & ((((comp @ id @ sK390)) != sK390) | (! [X489 : subst] : (((comp @ id @ X489)) = X489) & ! [X490 : subst] : (((comp @ X490 @ id)) = X490)) | (sK391 != ((comp @ sK391 @ id))) | (substmonoid_lthm != $true)) & ((substmonoid_lthm = $true) | (! [X492 : subst] : (((comp @ id @ X492)) = X492) & ((sK392 != ((comp @ id @ sK392))) | (sK393 != ((comp @ sK393 @ id)))) & ! [X495 : subst] : (((comp @ X495 @ id)) = X495))) & (((d2term @ d_one)) = ((lam @ (d2term @ d_one)))) & (((((ap @ sK397 @ sK396)) = ((ap @ sK394 @ sK395))) & (sK397 != sK394)) | (sK398 != ((sub @ sK398 @ id))) | ! [X501 : term,X502 : term,X503 : term,X504 : term] : ((((ap @ (sub @ X502 @ id) @ X503)) != ((ap @ (sub @ X504 @ id) @ X501))) | (X502 = X504)) | (hoasapinj1_lthm != $true)) & ((hoasapinj1_lthm = $true) | (! [X505 : term,X506 : term,X507 : term,X508 : term] : ((((ap @ X505 @ X506)) != ((ap @ X508 @ X507))) | (X505 = X508)) & ! [X509 : term] : (((sub @ X509 @ id)) = X509) & ((((ap @ (sub @ sK402 @ id) @ sK399)) = ((ap @ (sub @ sK400 @ id) @ sK401))) & (sK400 != sK402)))) & (hoasapinj1_gthm = $true) & (hoasinduction_p_and_p_prime = (^[Y0 : subst > term > subst > $o]: ((^[Y1 : term > $o]: (!! @ term @ (^[Y2 : term]: ((Y1 @ Y2) = (Y0 @ id @ Y2 @ id)))))))) & (hoasapnotvar_lthm = $true) & (one = ((d2term @ d_one))) & (axapp = $true) & (hoasinduction_lem1 = $true) & ((pushprop_lem0 = $true) | ! [X517 : (term > $o)] : (((sK404 @ (sub @ (sK406 @ X517) @ (push @ sK405 @ sK403)))) != ((X517 @ (sK406 @ X517))))) & (! [X519 : subst,X520 : (term > $o),X521 : term] : ! [X523 : term] : (((sK407 @ X521 @ X520 @ X519 @ X523)) = ((X520 @ (sub @ X523 @ (push @ X521 @ X519))))) | (pushprop_lem0 != $true)) & ((induction2lem_lthm = $true) | ($true != sP3)) & (($true = sP3) | (induction2lem_lthm != $true)) & (ulamvarind = $true) & (axmap = $true) & ((hoasinduction_no_psi_cond_lthm = $true) | (! [X524 : term] : (((sub @ X524 @ id)) = X524) & ! [X525 : (term > $o),X526 : (subst > term > subst > $o),X527 : term] : ((((X526 @ id @ (sK408 @ X526 @ X525) @ id)) != ((X525 @ (sK408 @ X526 @ X525)))) | ($true = ((X525 @ (lam @ X527)))) | ((((X525 @ (sub @ X527 @ (push @ (sK409 @ X527 @ X525) @ id)))) != $true) & ($true = ((X525 @ (sK409 @ X527 @ X525))))) | (($true != ((X526 @ id @ (lam @ (sK410 @ X526 @ sh @ one)) @ id))) & ! [X531 : term] : (($true != ((X526 @ id @ X531 @ id))) | ($true = ((X526 @ id @ (sK410 @ X526 @ id @ X531) @ id)))) & ! [X532 : term,X533 : subst,X534 : subst] : (((sK410 @ X526 @ (comp @ X533 @ X534) @ (sub @ X532 @ X534))) = ((sub @ (sK410 @ X526 @ X533 @ X532) @ X534))))) & ! [X535 : (term > $o),X536 : term] : ((($true = ((X535 @ (sK411 @ X535)))) & (((X535 @ (ap @ (sK412 @ X535) @ (sK411 @ X535)))) != $true) & ($true = ((X535 @ (sK412 @ X535))))) | ($true = ((X535 @ X536))) | (($true = ((var @ (sK413 @ X535)))) & ($true != ((X535 @ (sK413 @ X535))))) | (! [X541 : term] : (($true != ((X535 @ X541))) | (((X535 @ (sub @ (sK414 @ X535) @ (push @ X541 @ id)))) = $true)) & ($true != ((X535 @ (lam @ (sK414 @ X535))))))) & ! [X542 : (subst > term > subst > $o)] : ! [X544 : term] : (((X542 @ id @ X544 @ id)) = ((sK415 @ X542 @ X544))) & ($true != sP2))) & ((sK416 != ((sub @ sK416 @ id))) | (! [X549 : term] : (((sK417 @ X549)) = ((sK418 @ id @ X549 @ id))) & ($true != ((sK417 @ (lam @ sK419)))) & ! [X550 : term] : (($true = ((sK417 @ (sub @ sK419 @ (push @ X550 @ id))))) | (((sK417 @ X550)) != $true)) & ! [X551 : (subst > term > term)] : ((((sK418 @ id @ (lam @ (X551 @ sh @ one)) @ id)) = $true) | ((((sK418 @ id @ (sK420 @ X551) @ id)) = $true) & ($true != ((sK418 @ id @ (X551 @ id @ (sK420 @ X551)) @ id)))) | (((sub @ (X551 @ (sK422 @ X551) @ (sK421 @ X551)) @ (sK423 @ X551))) != ((X551 @ (comp @ (sK422 @ X551) @ (sK423 @ X551)) @ (sub @ (sK421 @ X551) @ (sK423 @ X551))))))) | (! [X558 : term,X559 : term] : ((((sK424 @ X558)) != $true) | ($true = ((sK424 @ (ap @ X559 @ X558)))) | ($true != ((sK424 @ X559)))) & (((sK424 @ sK425)) != $true) & ! [X560 : term] : (($true != ((var @ X560))) | ($true = ((sK424 @ X560)))) & ! [X561 : term] : ((($true = ((sK424 @ (sK426 @ X561)))) & (((sK424 @ (sub @ X561 @ (push @ (sK426 @ X561) @ id)))) != $true)) | (((sK424 @ (lam @ X561))) = $true))) | ! [X564 : (term > $o)] : (((X564 @ (sK428 @ X564))) != ((sK427 @ id @ (sK428 @ X564) @ id))) | ($true = sP2) | (hoasinduction_no_psi_cond_lthm != $true)) & ((pushprop_lem1v2 = $true) | (! [X570 : term] : (((sK429 @ X570)) = ((sK430 @ (sub @ X570 @ (push @ sK431 @ sK432))))) & ($true = ((sK430 @ sK431))) & ($true != ((sK429 @ one))))) & (! [X571 : (term > $o),X572 : (term > $o),X573 : term,X574 : subst] : ((((X571 @ (sK433 @ X574 @ X573 @ X572 @ X571))) != ((X572 @ (sub @ (sK433 @ X574 @ X573 @ X572 @ X571) @ (push @ X573 @ X574))))) | ($true != ((X572 @ X573))) | ($true = ((X571 @ one)))) | (pushprop_lem1v2 != $true)) & (hoasinduction_gthm = $true) & (! [X576 : term,X577 : subst,X578 : term,X579 : (term > $o),X580 : (term > $o)] : (($true = ((X580 @ (sub @ X576 @ (push @ X578 @ X577))))) | ($true != ((X579 @ X576)))) | (((sub @ one @ (push @ sK435 @ sK434))) != sK435) | (sK436 != ((comp @ sh @ (push @ sK437 @ sK436)))) | (pushprop_lem1_lthm != $true)) & ((pushprop_lem1_lthm = $true) | ((($true != ((sK442 @ (sub @ sK438 @ (push @ sK440 @ sK439))))) & (((sK441 @ sK438)) = $true)) & ! [X590 : subst,X591 : term] : (((sub @ one @ (push @ X591 @ X590))) = X591) & ! [X592 : subst,X593 : term] : (((comp @ sh @ (push @ X593 @ X592))) = X592))) & (! [X594 : (term > $o),X595 : term] : ((($true = ((X594 @ (sK443 @ X594)))) & (((X594 @ (sK444 @ X594))) = $true) & (((X594 @ (ap @ (sK444 @ X594) @ (sK443 @ X594)))) != $true)) | (($true != ((X594 @ (lam @ (sK445 @ X594))))) & ! [X599 : term] : (($true != ((X594 @ X599))) | ($true = ((X594 @ (sub @ (sK445 @ X594) @ (push @ X599 @ id))))))) | ($true = ((X594 @ X595))) | (($true = ((var @ (sK446 @ X594)))) & ($true != ((X594 @ (sK446 @ X594)))))) | (induction2 != $true)) & ((induction2 = $true) | (! [X603 : term,X604 : term] : (($true != ((sK447 @ X603))) | (((sK447 @ X604)) != $true) | ($true = ((sK447 @ (ap @ X604 @ X603))))) & ! [X605 : term] : (($true = ((sK447 @ (lam @ X605)))) | ((((sK447 @ (sK449 @ X605))) = $true) & ($true != ((sK447 @ (sub @ X605 @ (push @ (sK449 @ X605) @ id))))))) & (((sK447 @ sK448)) != $true) & ! [X607 : term] : (($true != ((var @ X607))) | ($true = ((sK447 @ X607)))))) & ((pushprop_lem1 = $true) | (($true != ((sK454 @ (sub @ sK451 @ (push @ sK453 @ sK452))))) & ($true = ((sK450 @ sK451))))) & (! [X613 : (term > $o),X614 : term,X615 : subst,X616 : term,X617 : (term > $o)] : (($true = ((X617 @ (sub @ X614 @ (push @ X616 @ X615))))) | ($true != ((X613 @ X614)))) | (pushprop_lem1 != $true)) & ((induction2_lthm = $true) | ($true != sP1)) & (($true = sP1) | (induction2_lthm != $true)) & ((hoasinduction_lem0 = $true) | ! [X619 : (term > $o)] : (((X619 @ (sK456 @ X619))) != ((sK455 @ id @ (sK456 @ X619) @ id)))) & (! [X621 : (subst > term > subst > $o)] : ! [X623 : term] : (((X621 @ id @ X623 @ id)) = ((sK457 @ X621 @ X623))) | (hoasinduction_lem0 != $true)) & (lamnotvar = $true) & (! [X624 : term,X625 : term,X626 : term,X627 : term] : ((((ap @ X626 @ X627)) != ((ap @ X624 @ X625))) | (X625 = X627)) | (apinj2 != $true)) & ((apinj2 = $true) | ((((ap @ sK460 @ sK461)) = ((ap @ sK458 @ sK459))) & (sK461 != sK459))) & (! [X632 : (term > $o),X633 : term] : ((($true != ((X632 @ (lam @ (sK462 @ X632))))) & (((X632 @ (sK462 @ X632))) = $true)) | ($true = ((X632 @ X633))) | (($true != ((X632 @ (ap @ (sK464 @ X632) @ (sK463 @ X632))))) & ($true = ((X632 @ (sK464 @ X632)))) & ($true = ((X632 @ (sK463 @ X632))))) | (($true = ((var @ (sK465 @ X632)))) & ($true != ((X632 @ (sK465 @ X632)))))) | (induction != $true)) & ((induction = $true) | (! [X640 : term] : ((((sK466 @ (lam @ X640))) = $true) | ($true != ((sK466 @ X640)))) & (((sK466 @ sK467)) != $true) & ! [X641 : term,X642 : term] : ((((sK466 @ (ap @ X642 @ X641))) = $true) | ($true != ((sK466 @ X642))) | ($true != ((sK466 @ X641)))) & ! [X643 : term] : (($true != ((var @ X643))) | (((sK466 @ X643)) = $true)))) & (((d2term @ d_one)) = ((sub @ (d2term @ d_one) @ (d2subst @ d_id)))) & (hoaslamnotvar_lthm = $true) & ((hoasinduction_lem2v2 = $true) | (! [X648 : subst,X649 : term,X650 : subst,X651 : subst] : ((((sK468 @ X651 @ X649 @ (comp @ X648 @ X650))) = $true) | ($true != ((sK468 @ (comp @ X651 @ X648) @ (sub @ X649 @ X648) @ X650)))) & ($true = ((sK470 @ sK471))) & ($true = ((sK470 @ sK469))) & ! [X652 : term,X653 : term] : (($true != ((sK468 @ id @ X652 @ id))) | ($true = ((sK468 @ id @ (ap @ (sub @ X652 @ id) @ X653) @ id))) | ($true != ((sK468 @ id @ X653 @ id)))) & ($true != ((sK470 @ (ap @ sK471 @ sK469)))) & ! [X654 : term] : (((sK468 @ id @ X654 @ id)) = ((sK470 @ X654))) & ! [X655 : subst,X656 : subst,X657 : subst,X658 : term] : (($true = ((sK468 @ (comp @ X656 @ X657) @ (sub @ X658 @ X657) @ X655))) | (((sK468 @ X656 @ X658 @ (comp @ X657 @ X655))) != $true)))) & (! [X659 : (subst > term > subst > $o),X660 : term,X661 : (term > $o),X662 : term] : ((($true != ((X659 @ (sK475 @ X659) @ (sK473 @ X659) @ (comp @ (sK472 @ X659) @ (sK474 @ X659))))) & ($true = ((X659 @ (comp @ (sK475 @ X659) @ (sK472 @ X659)) @ (sub @ (sK473 @ X659) @ (sK472 @ X659)) @ (sK474 @ X659))))) | ($true != ((X661 @ X662))) | ($true != ((X661 @ X660))) | ((((X659 @ id @ (sK476 @ X659) @ id)) = $true) & ($true != ((X659 @ id @ (ap @ (sub @ (sK476 @ X659) @ id) @ (sK477 @ X659)) @ id))) & ($true = ((X659 @ id @ (sK477 @ X659) @ id)))) | ($true = ((X661 @ (ap @ X662 @ X660)))) | (((X661 @ (sK478 @ X661 @ X659))) != ((X659 @ id @ (sK478 @ X661 @ X659) @ id))) | (($true != ((X659 @ (comp @ (sK480 @ X659) @ (sK481 @ X659)) @ (sub @ (sK482 @ X659) @ (sK481 @ X659)) @ (sK479 @ X659)))) & ($true = ((X659 @ (sK480 @ X659) @ (sK482 @ X659) @ (comp @ (sK481 @ X659) @ (sK479 @ X659))))))) | (hoasinduction_lem2v2 != $true)) & (hoasinduction_lem3a_gthm = $true) & (pushprop_lem1_gthm = $true) & (hoasapinj2_gthm = $true) & (id = ((d2subst @ d_id))) & (pushprop_gthm = $true) & (! [X674 : term] : ! [X676 : term,X677 : subst] : (((sub @ X674 @ (push @ X676 @ X677))) = ((sK483 @ X674 @ X677 @ X676))) | (hoasinduction_lem3v2_f_lthm != $true)) & ((hoasinduction_lem3v2_f_lthm = $true) | ! [X679 : (subst > term > term)] : (((sub @ sK484 @ (push @ (sK485 @ X679) @ (sK486 @ X679)))) != ((X679 @ (sK486 @ X679) @ (sK485 @ X679))))) & ((hoasinduction_lem3v2_lthm = $true) | (! [X682 : term] : (((sub @ X682 @ id)) = X682) & (! [X686 : term] : (((sK487 @ id @ X686 @ id)) = ((sK489 @ X686))) & (((sK489 @ (lam @ sK488))) != $true) & ! [X687 : subst,X688 : term,X689 : subst,X690 : subst] : (($true != ((sK487 @ (comp @ X689 @ X690) @ (sub @ X688 @ X690) @ X687))) | ($true = ((sK487 @ X689 @ X688 @ (comp @ X690 @ X687))))) & ! [X691 : term,X692 : subst,X693 : subst,X694 : subst] : (($true = ((sK487 @ (comp @ X692 @ X694) @ (sub @ X691 @ X694) @ X693))) | (((sK487 @ X692 @ X691 @ (comp @ X694 @ X693))) != $true)) & ! [X695 : (subst > term > term)] : ((((X695 @ (comp @ (sK491 @ X695) @ (sK490 @ X695)) @ (sub @ (sK492 @ X695) @ (sK490 @ X695)))) != ((sub @ (X695 @ (sK491 @ X695) @ (sK492 @ X695)) @ (sK490 @ X695)))) | ((((sK487 @ id @ (sK493 @ X695) @ id)) = $true) & (((sK487 @ id @ (X695 @ id @ (sK493 @ X695)) @ id)) != $true)) | ($true = ((sK487 @ id @ (lam @ (X695 @ sh @ one)) @ id)))) & ! [X700 : term] : (($true = ((sK489 @ (sub @ sK488 @ (push @ X700 @ id))))) | ($true != ((sK489 @ X700))))))) & ((sK494 != ((sub @ sK494 @ id))) | ! [X702 : (subst > term > subst > $o),X703 : term,X704 : (term > $o)] : ((((X702 @ id @ (sK495 @ X704 @ X702) @ id)) != ((X704 @ (sK495 @ X704 @ X702)))) | ($true = ((X704 @ (lam @ X703)))) | (($true = ((X702 @ (comp @ (sK498 @ X702) @ (sK499 @ X702)) @ (sub @ (sK497 @ X702) @ (sK499 @ X702)) @ (sK496 @ X702)))) & ($true != ((X702 @ (sK498 @ X702) @ (sK497 @ X702) @ (comp @ (sK499 @ X702) @ (sK496 @ X702)))))) | (($true != ((X702 @ (comp @ (sK501 @ X702) @ (sK503 @ X702)) @ (sub @ (sK500 @ X702) @ (sK503 @ X702)) @ (sK502 @ X702)))) & ($true = ((X702 @ (sK501 @ X702) @ (sK500 @ X702) @ (comp @ (sK503 @ X702) @ (sK502 @ X702)))))) | (! [X715 : subst,X716 : subst,X717 : term] : (((sK504 @ X702 @ (comp @ X716 @ X715) @ (sub @ X717 @ X715))) = ((sub @ (sK504 @ X702 @ X716 @ X717) @ X715))) & ! [X718 : term] : (($true != ((X702 @ id @ X718 @ id))) | ($true = ((X702 @ id @ (sK504 @ X702 @ id @ X718) @ id)))) & ($true != ((X702 @ id @ (lam @ (sK504 @ X702 @ sh @ one)) @ id)))) | ((((X704 @ (sub @ X703 @ (push @ (sK505 @ X704 @ X703) @ id)))) != $true) & (((X704 @ (sK505 @ X704 @ X703))) = $true))) | (hoasinduction_lem3v2_lthm != $true)) & (! [X720 : term,X721 : term,X722 : term,X723 : term] : ((((ap @ X721 @ X723)) != ((ap @ X722 @ X720))) | (X721 = X722)) | (apinj1 != $true)) & ((apinj1 = $true) | ((((ap @ sK508 @ sK506)) = ((ap @ sK507 @ sK509))) & (sK508 != sK507))) & (! [X728 : (term > $o),X729 : subst,X730 : term] : ! [X732 : term] : (((sK510 @ X730 @ X729 @ X728 @ X732)) = ((X728 @ (sub @ X732 @ (push @ X730 @ X729))))) | (pushprop_lem0_lthm != $true)) & ((pushprop_lem0_lthm = $true) | ! [X736 : (term > $o)] : (((X736 @ (sK514 @ X736))) != ((sK511 @ (sub @ (sK514 @ X736) @ (push @ sK513 @ sK512)))))) & (hoasinduction_lem3v2_gthm = $true) & (hoasinduction_lem3_gthm = $true) & (hoaslaminj_gthm = $true) & (hoasinduction_lem2_gthm = $true) & (! [X738 : (subst > term > subst > $o),X739 : term,X740 : (term > $o)] : (((((X740 @ (sK515 @ X740 @ X739))) = $true) & ($true != ((X740 @ (sub @ X739 @ (push @ (sK515 @ X740 @ X739) @ id)))))) | (((X740 @ (sK516 @ X740 @ X738))) != ((X738 @ id @ (sK516 @ X740 @ X738) @ id))) | (($true = ((X738 @ (sK518 @ X738) @ (sK519 @ X738) @ (comp @ (sK517 @ X738) @ (sK520 @ X738))))) & ($true != ((X738 @ (comp @ (sK518 @ X738) @ (sK517 @ X738)) @ (sub @ (sK519 @ X738) @ (sK517 @ X738)) @ (sK520 @ X738))))) | ((((X738 @ (comp @ (sK522 @ X738) @ (sK523 @ X738)) @ (sub @ (sK521 @ X738) @ (sK523 @ X738)) @ (sK524 @ X738))) = $true) & ($true != ((X738 @ (sK522 @ X738) @ (sK521 @ X738) @ (comp @ (sK523 @ X738) @ (sK524 @ X738)))))) | ($true = ((X740 @ (lam @ X739)))) | (! [X752 : term] : ((((X738 @ id @ X752 @ id)) != $true) | ($true = ((X738 @ id @ (sK525 @ X738 @ id @ X752) @ id)))) & ! [X753 : subst,X754 : subst,X755 : term] : (((sub @ (sK525 @ X738 @ X754 @ X755) @ X753)) = ((sK525 @ X738 @ (comp @ X754 @ X753) @ (sub @ X755 @ X753)))) & (((X738 @ id @ (lam @ (sK525 @ X738 @ sh @ one)) @ id)) != $true))) | (hoasinduction_lem3v2 != $true)) & ((hoasinduction_lem3v2 = $true) | (! [X759 : term] : (($true != ((sK528 @ X759))) | (((sK528 @ (sub @ sK527 @ (push @ X759 @ id)))) = $true)) & ! [X760 : term] : (((sK528 @ X760)) = ((sK526 @ id @ X760 @ id))) & ! [X761 : subst,X762 : subst,X763 : term,X764 : subst] : ((((sK526 @ X762 @ X763 @ (comp @ X761 @ X764))) != $true) | ($true = ((sK526 @ (comp @ X762 @ X761) @ (sub @ X763 @ X761) @ X764)))) & ! [X765 : term,X766 : subst,X767 : subst,X768 : subst] : (($true != ((sK526 @ (comp @ X766 @ X767) @ (sub @ X765 @ X767) @ X768))) | (((sK526 @ X766 @ X765 @ (comp @ X767 @ X768))) = $true)) & (((sK528 @ (lam @ sK527))) != $true) & ! [X769 : (subst > term > term)] : ((($true = ((sK526 @ id @ (sK529 @ X769) @ id))) & (((sK526 @ id @ (X769 @ id @ (sK529 @ X769)) @ id)) != $true)) | (((sub @ (X769 @ (sK531 @ X769) @ (sK532 @ X769)) @ (sK530 @ X769))) != ((X769 @ (comp @ (sK531 @ X769) @ (sK530 @ X769)) @ (sub @ (sK532 @ X769) @ (sK530 @ X769))))) | (((sK526 @ id @ (lam @ (X769 @ sh @ one)) @ id)) = $true)))) & ((axshiftcons = $true) | (((comp @ sh @ (push @ sK533 @ sK534))) != sK534)) & (! [X776 : term,X777 : subst] : (((comp @ sh @ (push @ X776 @ X777))) = X777) | (axshiftcons != $true)) & ((hoasinduction_lem3_lthm = $true) | (($true != sP0) & ! [X778 : term] : (((sub @ X778 @ id)) = X778) & ! [X779 : (subst > term > subst > $o),X780 : term] : ((($true = ((X779 @ id @ (sK535 @ X780 @ X779) @ id))) & ($true != ((X779 @ id @ (sub @ X780 @ (push @ (sK535 @ X780 @ X779) @ id)) @ id)))) | (((X779 @ id @ (lam @ (sub @ X780 @ (push @ one @ sh))) @ id)) = $true) | (! [X783 : subst,X784 : term,X785 : subst] : (((sub @ (sK536 @ X779 @ X785 @ X784) @ X783)) = ((sK536 @ X779 @ (comp @ X785 @ X783) @ (sub @ X784 @ X783)))) & ! [X786 : term] : ((((X779 @ id @ (sK536 @ X779 @ id @ X786) @ id)) = $true) | (((X779 @ id @ X786 @ id)) != $true)) & ($true != ((X779 @ id @ (lam @ (sK536 @ X779 @ sh @ one)) @ id))))))) & (($true = sP0) | (sK537 != ((sub @ sK537 @ id))) | (! [X790 : term] : (($true != ((sK538 @ id @ X790 @ id))) | ($true = ((sK538 @ id @ (sub @ sK539 @ (push @ X790 @ id)) @ id)))) & ($true != ((sK538 @ id @ (lam @ (sub @ sK539 @ (push @ one @ sh))) @ id))) & ! [X791 : (subst > term > term)] : ((((sub @ (X791 @ (sK542 @ X791) @ (sK541 @ X791)) @ (sK540 @ X791))) != ((X791 @ (comp @ (sK542 @ X791) @ (sK540 @ X791)) @ (sub @ (sK541 @ X791) @ (sK540 @ X791))))) | (($true != ((sK538 @ id @ (X791 @ id @ (sK543 @ X791)) @ id))) & (((sK538 @ id @ (sK543 @ X791) @ id)) = $true)) | ($true = ((sK538 @ id @ (lam @ (X791 @ sh @ one)) @ id))))) | (hoasinduction_lem3_lthm != $true))),
  inference(skolemize,[status(esa),new_symbols(skolem,[vAPP,sK151,sK152,vAPP,sK157,sK158,sK159,vAPP,sK165,sK166,sK167,vAPP,sK176,vAPP,sK179,sK180,sK181,sK182,sK183,vAPP,sK186,sK187,vAPP,sK192,sK193,sK194,sK195,vAPP,sK199,sK200,vAPP,sK211,sK212,sK213,sK214,sK215,sK216,sK217,sK218,sK219,sK220,sK221,vAPP,sK233,sK234,sK235,sK236,sK237,sK238,sK239,sK240,sK241,vAPP,sK243,vAPP,sK246,sK247,vAPP,sK252,sK253,vAPP,sK258,vAPP,sK262,sK263,sK264,vAPP,sK273,sK274,vAPP,sK285,sK286,vAPP,sK294,sK295,sK296,sK297,vAPP,sK304,sK305,vAPP,sK308,sK309,sK310,sK311,sK312,sK313,sK314,sK315,vAPP,sK322,sK323,vAPP,sK334,sK335,vAPP,sK340,sK341,sK342,sK343,sK344,sK345,vAPP,sK347,sK348,sK349,vAPP,sK354,sK355,vAPP,sK365,sK366,vAPP,sK373,sK374,sK375,vAPP,sK379,sK380,sK381,vAPP,sK384,sK385,sK386,sK387,sK388,sK389,sK390,sK391,sK392,sK393,sK394,sK395,sK396,sK397,sK398,sK399,sK400,sK401,sK402,sK403,sK404,sK405,vAPP,sK416,sK417,sK418,sK419,vAPP,sK424,sK425,vAPP,sK427,vAPP,sK429,sK430,sK431,sK432,vAPP,sK434,sK435,sK436,sK437,sK438,sK439,sK440,sK441,sK442,vAPP,sK447,sK448,vAPP,sK450,sK451,sK452,sK453,sK454,sK455,vAPP,sK458,sK459,sK460,sK461,vAPP,sK466,sK467,sK468,sK469,sK470,sK471,vAPP,sK484,vAPP,sK487,sK488,sK489,vAPP,sK494,vAPP,sK506,sK507,sK508,sK509,vAPP,sK511,sK512,sK513,vAPP,sK526,sK527,sK528,vAPP,sK533,sK534,vAPP,sK537,sK538,sK539,vAPP]),skolemize(X45,$thf($thf(sK34 @ X44))),skolemize(X45,$thf(sK34 @ X44)),skolemize(X8,$thf(sK151)),skolemize(X9,$thf(sK152)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X16,$thf(sK157)),skolemize(X17,$thf(sK158)),skolemize(X18,$thf(sK159)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X32,$thf(sK165)),skolemize(X33,$thf(sK166)),skolemize(X34,$thf(sK167)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X57,$thf(sK176)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X61,$thf(sK179)),skolemize(X66,$thf(sK180)),skolemize(X67,$thf(sK181)),skolemize(X68,$thf(sK182)),skolemize(X69,$thf(sK183)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X78,$thf(sK186)),skolemize(X79,$thf(sK187)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X87,$thf(sK192)),skolemize(X88,$thf(sK193)),skolemize(X89,$thf(sK194)),skolemize(X90,$thf(sK195)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X102,$thf(sK199)),skolemize(X103,$thf(sK200)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X117,$thf(sK211)),skolemize(X118,$thf(sK212)),skolemize(X119,$thf(sK213)),skolemize(X136,$thf(sK214)),skolemize(X137,$thf(sK215)),skolemize(X138,$thf(sK216)),skolemize(X139,$thf(sK217)),skolemize(X140,$thf(sK218)),skolemize(X141,$thf(sK219)),skolemize(X146,$thf(sK220)),skolemize(X147,$thf(sK221)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X173,$thf(sK233)),skolemize(X179,$thf(sK234)),skolemize(X180,$thf(sK235)),skolemize(X181,$thf(sK236)),skolemize(X182,$thf(sK237)),skolemize(X183,$thf(sK238)),skolemize(X184,$thf(sK239)),skolemize(X185,$thf(sK240)),skolemize(X186,$thf(sK241)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X193,$thf(sK243)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X204,$thf(sK246)),skolemize(X205,$thf(sK247)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X212,$thf(sK252)),skolemize(X213,$thf(sK253)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X220,$thf(sK258)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X232,$thf(sK262)),skolemize(X233,$thf(sK263)),skolemize(X234,$thf(sK264)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X254,$thf(sK273)),skolemize(X255,$thf(sK274)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X284,$thf(sK285)),skolemize(X285,$thf(sK286)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X296,$thf(sK294)),skolemize(X297,$thf(sK295)),skolemize(X298,$thf(sK296)),skolemize(X299,$thf(sK297)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X317,$thf(sK304)),skolemize(X318,$thf(sK305)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X328,$thf(sK308)),skolemize(X329,$thf(sK309)),skolemize(X330,$thf(sK310)),skolemize(X331,$thf(sK311)),skolemize(X340,$thf(sK312)),skolemize(X341,$thf(sK313)),skolemize(X342,$thf(sK314)),skolemize(X343,$thf(sK315)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X354,$thf(sK322)),skolemize(X355,$thf(sK323)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X372,$thf(sK334)),skolemize(X373,$thf(sK335)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X390,$thf(sK340)),skolemize(X391,$thf(sK341)),skolemize(X393,$thf(sK342)),skolemize(X395,$thf(sK343)),skolemize(X396,$thf(sK344)),skolemize(X397,$thf(sK345)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X400,$thf(sK347)),skolemize(X401,$thf(sK348)),skolemize(X402,$thf(sK349)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X410,$thf(sK354)),skolemize(X411,$thf(sK355)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X438,$thf(sK365)),skolemize(X439,$thf(sK366)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X454,$thf(sK373)),skolemize(X457,$thf(sK374)),skolemize(X458,$thf(sK375)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X466,$thf(sK379)),skolemize(X467,$thf(sK380)),skolemize(X470,$thf(sK381)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X479,$thf(sK384)),skolemize(X480,$thf(sK385)),skolemize(X481,$thf(sK386)),skolemize(X482,$thf(sK387)),skolemize(X483,$thf(sK388)),skolemize(X484,$thf(sK389)),skolemize(X488,$thf(sK390)),skolemize(X491,$thf(sK391)),skolemize(X493,$thf(sK392)),skolemize(X494,$thf(sK393)),skolemize(X496,$thf(sK394)),skolemize(X497,$thf(sK395)),skolemize(X498,$thf(sK396)),skolemize(X499,$thf(sK397)),skolemize(X500,$thf(sK398)),skolemize(X510,$thf(sK399)),skolemize(X511,$thf(sK400)),skolemize(X512,$thf(sK401)),skolemize(X513,$thf(sK402)),skolemize(X514,$thf(sK403)),skolemize(X515,$thf(sK404)),skolemize(X516,$thf(sK405)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X545,$thf(sK416)),skolemize(X546,$thf(sK417)),skolemize(X547,$thf(sK418)),skolemize(X548,$thf(sK419)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X556,$thf(sK424)),skolemize(X557,$thf(sK425)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X563,$thf(sK427)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X566,$thf(sK429)),skolemize(X567,$thf(sK430)),skolemize(X568,$thf(sK431)),skolemize(X569,$thf(sK432)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X581,$thf(sK434)),skolemize(X582,$thf(sK435)),skolemize(X583,$thf(sK436)),skolemize(X584,$thf(sK437)),skolemize(X585,$thf(sK438)),skolemize(X586,$thf(sK439)),skolemize(X587,$thf(sK440)),skolemize(X588,$thf(sK441)),skolemize(X589,$thf(sK442)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X601,$thf(sK447)),skolemize(X602,$thf(sK448)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X608,$thf(sK450)),skolemize(X609,$thf(sK451)),skolemize(X610,$thf(sK452)),skolemize(X611,$thf(sK453)),skolemize(X612,$thf(sK454)),skolemize(X618,$thf(sK455)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X628,$thf(sK458)),skolemize(X629,$thf(sK459)),skolemize(X630,$thf(sK460)),skolemize(X631,$thf(sK461)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X638,$thf(sK466)),skolemize(X639,$thf(sK467)),skolemize(X644,$thf(sK468)),skolemize(X645,$thf(sK469)),skolemize(X646,$thf(sK470)),skolemize(X647,$thf(sK471)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X678,$thf(sK484)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X683,$thf(sK487)),skolemize(X684,$thf(sK488)),skolemize(X685,$thf(sK489)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X701,$thf(sK494)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X724,$thf(sK506)),skolemize(X725,$thf(sK507)),skolemize(X726,$thf(sK508)),skolemize(X727,$thf(sK509)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X733,$thf(sK511)),skolemize(X734,$thf(sK512)),skolemize(X735,$thf(sK513)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X756,$thf(sK526)),skolemize(X757,$thf(sK527)),skolemize(X758,$thf(sK528)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X774,$thf(sK533)),skolemize(X775,$thf(sK534)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X787,$thf(sK537)),skolemize(X788,$thf(sK538)),skolemize(X789,$thf(sK539)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44)),skolemize(X45,$thf(sK34 @ X44))],[f67])).
thf(f70,plain,(
  (lamnotvar != $true) | ($true = ((var @ (lam @ sK11))))),
  inference(cnf_transformation,[],[f28])).
thf(f554,plain,(
  (lamnotvar = $true)),
  inference(cnf_transformation,[],[f68])).
thf(f643,plain,(
  (one = ((d2term @ d_one)))),
  inference(cnf_transformation,[],[f68])).
thf(f741,plain,(
  (((var @ (d2term @ d_one))) != $true)),
  inference(cnf_transformation,[],[f68])).
thf(f831,plain,(
  ( ! [X252 : term] : ((((d2term @ (sK272 @ X252))) = X252)) )),
  inference(cnf_transformation,[],[f68])).
thf(f897,plain,(
  ( ! [X174 : d_term] : ((d_one = X174)) )),
  inference(cnf_transformation,[],[f68])).
thf(f1745,definition,(
  spl544_94 <=> (one = ((d2term @ d_one)))),
  introduced(definition,[new_symbols(definition,[spl544_94])],[avatar_definition])).
thf(f1747,plain,(
  (one = ((d2term @ d_one))) | ~spl544_94),
  inference(avatar_component_clause,[],[f1745])).
thf(f1748,plain,(
  spl544_94),
  inference(avatar_split_clause,[],[f643,f1745])).
thf(f2939,definition,(
  spl544_374 <=> (lamnotvar = $true)),
  introduced(definition,[new_symbols(definition,[spl544_374])],[avatar_definition])).
thf(f2943,definition,(
  spl544_375 <=> ($true = ((var @ (lam @ sK11))))),
  introduced(definition,[new_symbols(definition,[spl544_375])],[avatar_definition])).
thf(f2945,plain,(
  ($true = ((var @ (lam @ sK11)))) | ~spl544_375),
  inference(avatar_component_clause,[],[f2943])).
thf(f2946,plain,(
  ~spl544_374 | spl544_375),
  inference(avatar_split_clause,[],[f70,f2943,f2939])).
thf(f3442,plain,(
  spl544_374),
  inference(avatar_split_clause,[],[f554,f2939])).
thf(f4251,definition,(
  spl544_668 <=> (((var @ (d2term @ d_one))) = $true)),
  introduced(definition,[new_symbols(definition,[spl544_668])],[avatar_definition])).
thf(f4254,plain,(
  ~spl544_668),
  inference(avatar_split_clause,[],[f741,f4251])).
thf(f5356,plain,(
  ( ! [X0 : d_term] : ((one = ((d2term @ X0)))) ) | ~spl544_94),
  inference(superposition,[],[f1747,f897])).
thf(f5506,definition,(
  spl544_925 <=> ($true = ((var @ one)))),
  introduced(definition,[new_symbols(definition,[spl544_925])],[avatar_definition])).
thf(f5523,plain,(
  ( ! [X252 : term] : ((one = X252)) ) | ~spl544_94),
  inference(backward_demodulation,[],[f831,f5356])).
thf(f5528,plain,(
  ($true = ((var @ one))) | (~spl544_94 | ~spl544_375)),
  inference(backward_demodulation,[],[f2945,f5523])).
thf(f5580,plain,(
  spl544_925 | ~spl544_94 | ~spl544_375),
  inference(avatar_split_clause,[],[f5528,f2943,f1745,f5506])).
thf(f5590,definition,(
  (one != ((d2term @ d_one))) | ($true != ((var @ one))) | (((var @ (d2term @ d_one))) = $true)),
  introduced(theory,[theory_tautology_sat_conflict])).
cnf(s50, plain, spl544_94, inference(sat_conversion,[],[f1748])).
cnf(s339, plain, ~spl544_374 | spl544_375, inference(sat_conversion,[],[f2946])).
cnf(s483, plain, spl544_374, inference(sat_conversion,[],[f3442])).
cnf(s718, plain, ~spl544_668, inference(sat_conversion,[],[f4254])).
cnf(s1092, plain, ~spl544_94 | ~spl544_375 | spl544_925, inference(sat_conversion,[],[f5580])).
cnf(s1096, plain, ~spl544_94 | spl544_668 | ~spl544_925, inference(sat_conversion,[],[f5590])).
cnf(s1097, plain, spl544_375, inference(rat,[],[s339,s483])).
cnf(s1100, plain, ~spl544_925, inference(rat,[],[s1096,s718,s50])).
cnf(s1101, plain, $false, inference(rat,[],[s1092,s1097,s1100,s50])).
thf(f5591,plain,(
  $false),
  inference(avatar_sat_refutation,[],[s1101])).
% SZS output end Proof for theBenchmark