%----Output an unreadable proof
% SZS output start Proof for theBenchmark
thf(type_def_5, type, sTfun: ($tType * $tType) > $tType).
thf(func_def_0, type, cP: $o).
thf(func_def_4, type, vOR: ($o > $o > $o)).
thf(func_def_5, type, vNOT: ($o > $o)).
thf(func_def_6, type, vLAM: !>[X0: $tType, X1: $tType]:((X1) > (X0 > X1))).
thf(func_def_7, type, vAND: ($o > $o > $o)).
thf(func_def_8, type, sK0: ((($i > $o) > ($i > $o) > $o) > $i > $o)).
thf(func_def_9, type, sK1: ((($i > $o) > ($i > $o) > $o) > $i > $o)).
thf(func_def_10, type, sK2: ((($i > $o) > ($i > $o) > $o) > $i > $o)).
thf(func_def_11, type, sK3: ((($i > $o) > ($i > $o) > $o) > $i > $o)).
thf(func_def_12, type, db0: !>[X0: $tType]:(X0)).
thf(func_def_13, type, db1: !>[X0: $tType]:(X0)).
thf(func_def_14, type, vEQ: !>[X0: $tType]:((X0 > X0 > $o))).
thf(f1,conjecture,(
  ? [X0 : (($i > $o) > ($i > $o) > $o)] : (! [X1 : ($i > $o)] : (X0 @ X1 @ X1) & ~(X0 @ (^[X1 : $i] : (~cP | cP)) @ (^[X1 : $i] : (~cP & cP))) & ! [X2 : ($i > $o),X1 : ($i > $o),X3 : ($i > $o)] : (((X0 @ X1 @ X2) & (X0 @ X2 @ X3)) => (X0 @ X1 @ X3)))),
  file('/export/starexec/sandbox/benchmark/theBenchmark.p',cTHM120J_pme)).
thf(f2,negated_conjecture,(
  ~ ? [X0 : (($i > $o) > ($i > $o) > $o)] : (! [X1 : ($i > $o)] : (X0 @ X1 @ X1) & ~(X0 @ (^[X1 : $i] : (~cP | cP)) @ (^[X1 : $i] : (~cP & cP))) & ! [X2 : ($i > $o),X1 : ($i > $o),X3 : ($i > $o)] : (((X0 @ X1 @ X2) & (X0 @ X2 @ X3)) => (X0 @ X1 @ X3)))),
  inference(negated_conjecture,[status(cth)],[f1])).
thf(f3,plain,(
  ~ ? [X0 : (($i > $o) > ($i > $o) > $o)] : (! [X1 : ($i > $o)] : (X0 @ X1 @ X1) & ~(X0 @ (^[X2 : $i] : (~cP | cP)) @ (^[X3 : $i] : (~cP & cP))) & ! [X4 : ($i > $o),X5 : ($i > $o),X6 : ($i > $o)] : (((X0 @ X5 @ X4) & (X0 @ X4 @ X6)) => (X0 @ X5 @ X6)))),
  inference(rectify,[],[f2])).
thf(f4,plain,(
  ~ ? [X0 : (($i > $o) > ($i > $o) > $o)] : (! [X1 : ($i > $o)] : (((X0 @ X1 @ X1)) = $true) & ~ ($true = ((X0 @ (^[Y0 : $i]: (cP | (~ cP))) @ (^[Y0 : $i]: (cP & (~ cP)))))) & ! [X4 : ($i > $o),X5 : ($i > $o),X6 : ($i > $o)] : (((((X0 @ X5 @ X4)) = $true) & (((X0 @ X4 @ X6)) = $true)) => ($true = ((X0 @ X5 @ X6)))))),
  inference(fool_elimination,[],[f3])).
thf(f5,plain,(
  ~ ? [X0 : (($i > $o) > ($i > $o) > $o)] : (! [X1 : ($i > $o)] : (((X0 @ X1 @ X1)) = $true) & ~ ($true = ((X0 @ (^[Y0 : $i]: (cP | (~ cP))) @ (^[Y0 : $i]: (cP & (~ cP)))))) & ! [X3 : ($i > $o),X4 : ($i > $o),X2 : ($i > $o)] : (((((X0 @ X3 @ X2)) = $true) & ($true = ((X0 @ X2 @ X4)))) => (((X0 @ X3 @ X4)) = $true)))),
  inference(rectify,[],[f4])).
thf(f6,plain,(
  ~ ? [X0 : (($i > $o) > ($i > $o) > $o)] : (! [X3 : ($i > $o),X4 : ($i > $o),X2 : ($i > $o)] : (((((X0 @ X3 @ X2)) = $true) & ($true = ((X0 @ X2 @ X4)))) => (((X0 @ X3 @ X4)) = $true)) & ($true != ((X0 @ (^[Y0 : $i]: (cP | (~ cP))) @ (^[Y0 : $i]: (cP & (~ cP)))))) & ! [X1 : ($i > $o)] : (((X0 @ X1 @ X1)) = $true))),
  inference(flattening,[],[f5])).
thf(f7,plain,(
  ! [X0 : (($i > $o) > ($i > $o) > $o)] : (? [X3 : ($i > $o),X4 : ($i > $o),X2 : ($i > $o)] : ((((X0 @ X3 @ X4)) != $true) & ((((X0 @ X3 @ X2)) = $true) & ($true = ((X0 @ X2 @ X4))))) | ($true = ((X0 @ (^[Y0 : $i]: (cP | (~ cP))) @ (^[Y0 : $i]: (cP & (~ cP)))))) | ? [X1 : ($i > $o)] : (((X0 @ X1 @ X1)) != $true))),
  inference(ennf_transformation,[],[f6])).
thf(f8,plain,(
  ! [X0 : (($i > $o) > ($i > $o) > $o)] : (? [X4 : ($i > $o),X2 : ($i > $o),X3 : ($i > $o)] : ((((X0 @ X3 @ X4)) != $true) & ($true = ((X0 @ X2 @ X4))) & (((X0 @ X3 @ X2)) = $true)) | ? [X1 : ($i > $o)] : (((X0 @ X1 @ X1)) != $true) | ($true = ((X0 @ (^[Y0 : $i]: (cP | (~ cP))) @ (^[Y0 : $i]: (cP & (~ cP)))))))),
  inference(flattening,[],[f7])).
thf(f9,plain,(
  ! [X0 : (($i > $o) > ($i > $o) > $o)] : (? [X1 : ($i > $o),X2 : ($i > $o),X3 : ($i > $o)] : (($true != ((X0 @ X3 @ X1))) & ($true = ((X0 @ X2 @ X1))) & (((X0 @ X3 @ X2)) = $true)) | ? [X4 : ($i > $o)] : ($true != ((X0 @ X4 @ X4))) | ($true = ((X0 @ (^[Y0 : $i]: (cP | (~ cP))) @ (^[Y0 : $i]: (cP & (~ cP)))))))),
  inference(rectify,[],[f8])).
thf(f10,plain,(
  ! [X0 : (($i > $o) > ($i > $o) > $o)] : (((((X0 @ (sK2 @ X0) @ (sK0 @ X0))) != $true) & (((X0 @ (sK1 @ X0) @ (sK0 @ X0))) = $true) & (((X0 @ (sK2 @ X0) @ (sK1 @ X0))) = $true)) | ($true != ((X0 @ (sK3 @ X0) @ (sK3 @ X0)))) | ($true = ((X0 @ (^[Y0 : $i]: (cP | (~ cP))) @ (^[Y0 : $i]: (cP & (~ cP)))))))),
  inference(skolemize,[status(esa),new_symbols(skolem,[vAPP]),skolemize(X4,$thf(sK3 @ X0)),skolemize(X4,$thf(sK3 @ X0)),skolemize(X4,$thf(sK3 @ X0)),skolemize(X4,$thf(sK3 @ X0))],[f9])).
thf(f11,plain,(
  ( ! [X0 : (($i > $o) > ($i > $o) > $o)] : (($true != ((X0 @ (sK3 @ X0) @ (sK3 @ X0)))) | ($true = ((X0 @ (^[Y0 : $i]: (cP | (~ cP))) @ (^[Y0 : $i]: (cP & (~ cP)))))) | (((X0 @ (sK2 @ X0) @ (sK1 @ X0))) = $true)) )),
  inference(cnf_transformation,[],[f10])).
thf(f12,plain,(
  ( ! [X0 : (($i > $o) > ($i > $o) > $o)] : (($true = ((X0 @ (^[Y0 : $i]: (cP | (~ cP))) @ (^[Y0 : $i]: (cP & (~ cP)))))) | ($true != ((X0 @ (sK3 @ X0) @ (sK3 @ X0)))) | (((X0 @ (sK1 @ X0) @ (sK0 @ X0))) = $true)) )),
  inference(cnf_transformation,[],[f10])).
thf(f13,plain,(
  ( ! [X0 : (($i > $o) > ($i > $o) > $o)] : (($true != ((X0 @ (sK3 @ X0) @ (sK3 @ X0)))) | (((X0 @ (sK2 @ X0) @ (sK0 @ X0))) != $true) | ($true = ((X0 @ (^[Y0 : $i]: (cP | (~ cP))) @ (^[Y0 : $i]: (cP & (~ cP))))))) )),
  inference(cnf_transformation,[],[f10])).
thf(f16,plain,(
  ( ! [X0 : (($i > $o) > ($i > $o) > $o)] : (($true = ((X0 @ (^[Y0 : $i]: (cP | (~ cP))) @ (^[Y0 : $i]: ($false))))) | (((X0 @ (sK1 @ X0) @ (sK0 @ X0))) = $true) | ($true != ((X0 @ (sK3 @ X0) @ (sK3 @ X0))))) )),
  inference(boolean_simplification,[],[f12])).
thf(f17,plain,(
  ( ! [X0 : (($i > $o) > ($i > $o) > $o)] : (($true != ((X0 @ (sK3 @ X0) @ (sK3 @ X0)))) | (((X0 @ (sK1 @ X0) @ (sK0 @ X0))) = $true) | ($true = ((X0 @ (^[Y0 : $i]: ($true)) @ (^[Y0 : $i]: ($false)))))) )),
  inference(boolean_simplification,[],[f16])).
thf(f18,plain,(
  ( ! [X0 : (($i > $o) > ($i > $o) > $o)] : (($true = ((X0 @ (^[Y0 : $i]: (cP | (~ cP))) @ (^[Y0 : $i]: ($false))))) | (((X0 @ (sK2 @ X0) @ (sK0 @ X0))) != $true) | ($true != ((X0 @ (sK3 @ X0) @ (sK3 @ X0))))) )),
  inference(boolean_simplification,[],[f13])).
thf(f19,plain,(
  ( ! [X0 : (($i > $o) > ($i > $o) > $o)] : (($true != ((X0 @ (sK3 @ X0) @ (sK3 @ X0)))) | ($true = ((X0 @ (^[Y0 : $i]: ($true)) @ (^[Y0 : $i]: ($false))))) | (((X0 @ (sK2 @ X0) @ (sK0 @ X0))) != $true)) )),
  inference(boolean_simplification,[],[f18])).
thf(f20,plain,(
  ( ! [X0 : (($i > $o) > ($i > $o) > $o)] : (($true = ((X0 @ (^[Y0 : $i]: (cP | (~ cP))) @ (^[Y0 : $i]: ($false))))) | ($true != ((X0 @ (sK3 @ X0) @ (sK3 @ X0)))) | (((X0 @ (sK2 @ X0) @ (sK1 @ X0))) = $true)) )),
  inference(boolean_simplification,[],[f11])).
thf(f21,plain,(
  ( ! [X0 : (($i > $o) > ($i > $o) > $o)] : (($true != ((X0 @ (sK3 @ X0) @ (sK3 @ X0)))) | (((X0 @ (sK2 @ X0) @ (sK1 @ X0))) = $true) | ($true = ((X0 @ (^[Y0 : $i]: ($true)) @ (^[Y0 : $i]: ($false)))))) )),
  inference(boolean_simplification,[],[f20])).
thf(f22,plain,(
  ($true = (((^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: (Y0 = Y1)))) @ (sK1 @ (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: (Y0 = Y1))))) @ (sK0 @ (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: (Y0 = Y1)))))))) | ((((^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: (Y0 = Y1)))) @ (sK3 @ (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: (Y0 = Y1))))) @ (sK3 @ (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: (Y0 = Y1))))))) != $true) | ((((^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: (Y0 = Y1)))) @ (^[Y0 : $i]: ($true)) @ (^[Y0 : $i]: ($false)))) = $true)),
  inference(primitive_instantiation,[],[f17])).
thf(f36,plain,(
  ((((^[Y0 : $i]: ($true)) = (^[Y0 : $i]: ($false)))) = $true) | ($true != (((sK3 @ =) = (sK3 @ =)))) | ((((sK1 @ =) = (sK0 @ =))) = $true)),
  inference(beta-eta_normalization,[],[f22])).
thf(f37,plain,(
  ((((sK1 @ =) = (sK0 @ =))) = $true) | ((^[Y0 : $i]: ($false)) = (^[Y0 : $i]: ($true))) | ($true != (((sK3 @ =) = (sK3 @ =))))),
  inference(equality_proxy_clausification,[],[f36])).
thf(f38,plain,(
  ($true != (((sK3 @ =) = (sK3 @ =)))) | ((^[Y0 : $i]: ($false)) = (^[Y0 : $i]: ($true))) | (((sK0 @ =)) = ((sK1 @ =)))),
  inference(equality_proxy_clausification,[],[f37])).
thf(f39,plain,(
  (((sK3 @ =)) != ((sK3 @ =))) | ((^[Y0 : $i]: ($false)) = (^[Y0 : $i]: ($true))) | (((sK0 @ =)) = ((sK1 @ =)))),
  inference(equality_proxy_clausification,[],[f38])).
thf(f40,plain,(
  (((sK0 @ =)) = ((sK1 @ =))) | ((^[Y0 : $i]: ($false)) = (^[Y0 : $i]: ($true)))),
  inference(trivial_inequality_removal,[],[f39])).
thf(f51,definition,(
  spl4_1 <=> ((^[Y0 : $i]: ($false)) = (^[Y0 : $i]: ($true)))),
  introduced(definition,[new_symbols(definition,[spl4_1])],[avatar_definition])).
thf(f53,plain,(
  ((^[Y0 : $i]: ($false)) = (^[Y0 : $i]: ($true))) | ~spl4_1),
  inference(avatar_component_clause,[],[f51])).
thf(f55,definition,(
  spl4_2 <=> (((sK0 @ =)) = ((sK1 @ =)))),
  introduced(definition,[new_symbols(definition,[spl4_2])],[avatar_definition])).
thf(f57,plain,(
  (((sK0 @ =)) = ((sK1 @ =))) | ~spl4_2),
  inference(avatar_component_clause,[],[f55])).
thf(f58,plain,(
  spl4_1 | spl4_2),
  inference(avatar_split_clause,[],[f40,f55,f51])).
thf(f59,plain,(
  ((((^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: (Y0 = Y1)))) @ (sK2 @ (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: (Y0 = Y1))))) @ (sK0 @ (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: (Y0 = Y1))))))) != $true) | ((((^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: (Y0 = Y1)))) @ (^[Y0 : $i]: ($true)) @ (^[Y0 : $i]: ($false)))) = $true) | ((((^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: (Y0 = Y1)))) @ (sK3 @ (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: (Y0 = Y1))))) @ (sK3 @ (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: (Y0 = Y1))))))) != $true)),
  inference(primitive_instantiation,[],[f19])).
thf(f71,plain,(
  ((((sK2 @ =) = (sK0 @ =))) != $true) | ((((^[Y0 : $i]: ($true)) = (^[Y0 : $i]: ($false)))) = $true) | ($true != (((sK3 @ =) = (sK3 @ =))))),
  inference(beta-eta_normalization,[],[f59])).
thf(f72,plain,(
  ($true != (((sK3 @ =) = (sK3 @ =)))) | ((((^[Y0 : $i]: ($true)) = (^[Y0 : $i]: ($false)))) = $true) | (((sK2 @ =)) != ((sK0 @ =)))),
  inference(equality_proxy_clausification,[],[f71])).
thf(f73,plain,(
  (((sK3 @ =)) != ((sK3 @ =))) | ((((^[Y0 : $i]: ($true)) = (^[Y0 : $i]: ($false)))) = $true) | (((sK2 @ =)) != ((sK0 @ =)))),
  inference(equality_proxy_clausification,[],[f72])).
thf(f74,plain,(
  (((sK2 @ =)) != ((sK0 @ =))) | (((sK3 @ =)) != ((sK3 @ =))) | ((^[Y0 : $i]: ($false)) = (^[Y0 : $i]: ($true)))),
  inference(equality_proxy_clausification,[],[f73])).
thf(f75,plain,(
  (((sK2 @ =)) != ((sK0 @ =))) | ((^[Y0 : $i]: ($false)) = (^[Y0 : $i]: ($true)))),
  inference(trivial_inequality_removal,[],[f74])).
thf(f87,plain,(
  ((^[Y0 : $i]: ($false)) = (^[Y0 : $i]: ($true))) | (((sK2 @ =)) != ((sK1 @ =))) | ~spl4_2),
  inference(forward_demodulation,[],[f75,f57])).
thf(f89,definition,(
  spl4_3 <=> (((sK2 @ =)) = ((sK1 @ =)))),
  introduced(definition,[new_symbols(definition,[spl4_3])],[avatar_definition])).
thf(f91,plain,(
  (((sK2 @ =)) != ((sK1 @ =))) | spl4_3),
  inference(avatar_component_clause,[],[f89])).
thf(f92,plain,(
  spl4_1 | ~spl4_3 | ~spl4_2),
  inference(avatar_split_clause,[],[f87,f55,f89,f51])).
thf(f123,plain,(
  ((((^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: (Y0 = Y1)))) @ (sK3 @ (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: (Y0 = Y1))))) @ (sK3 @ (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: (Y0 = Y1))))))) != $true) | ((((^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: (Y0 = Y1)))) @ (sK2 @ (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: (Y0 = Y1))))) @ (sK1 @ (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: (Y0 = Y1))))))) = $true) | ((((^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: (Y0 = Y1)))) @ (^[Y0 : $i]: ($true)) @ (^[Y0 : $i]: ($false)))) = $true)),
  inference(primitive_instantiation,[],[f21])).
thf(f129,plain,(
  ($true != (((sK3 @ =) = (sK3 @ =)))) | ((((^[Y0 : $i]: ($true)) = (^[Y0 : $i]: ($false)))) = $true) | ((((sK2 @ =) = (sK1 @ =))) = $true)),
  inference(beta-eta_normalization,[],[f123])).
thf(f130,plain,(
  ((((^[Y0 : $i]: ($true)) = (^[Y0 : $i]: ($false)))) = $true) | (((sK3 @ =)) != ((sK3 @ =))) | ((((sK2 @ =) = (sK1 @ =))) = $true)),
  inference(equality_proxy_clausification,[],[f129])).
thf(f131,plain,(
  ((((sK2 @ =) = (sK1 @ =))) = $true) | (((sK3 @ =)) != ((sK3 @ =))) | ((^[Y0 : $i]: ($false)) = (^[Y0 : $i]: ($true)))),
  inference(equality_proxy_clausification,[],[f130])).
thf(f132,plain,(
  ((^[Y0 : $i]: ($false)) = (^[Y0 : $i]: ($true))) | (((sK3 @ =)) != ((sK3 @ =))) | (((sK2 @ =)) = ((sK1 @ =)))),
  inference(equality_proxy_clausification,[],[f131])).
thf(f133,plain,(
  ((^[Y0 : $i]: ($false)) = (^[Y0 : $i]: ($true))) | (((sK2 @ =)) = ((sK1 @ =)))),
  inference(trivial_inequality_removal,[],[f132])).
thf(f151,plain,(
  ((^[Y0 : $i]: ($false)) = (^[Y0 : $i]: ($true))) | spl4_3),
  inference(forward_subsumption_resolution,[],[f133,f91])).
thf(f152,plain,(
  spl4_1 | spl4_3),
  inference(avatar_split_clause,[],[f151,f89,f51])).
thf(f153,plain,(
  ( ! [X1 : $i] : (((((^[Y0 : $i]: ($false)) @ X1)) = (((^[Y0 : $i]: ($true)) @ X1)))) ) | ~spl4_1),
  inference(argument_congruence,[],[f53])).
thf(f154,plain,(
  ( ! [X1 : $i] : (((((^[Y0 : $i]: ($true)) @ X1)) = $false) | ((((^[Y0 : $i]: ($false)) @ X1)) = $true)) ) | ~spl4_1),
  inference(iff_proxy_clausification,[],[f153])).
thf(f157,plain,(
  ($false = $true) | ($false = $true) | ~spl4_1),
  inference(beta-eta_normalization,[],[f154])).
thf(f158,plain,(
  ($false = $true) | ~spl4_1),
  inference(duplicate_literal_removal,[],[f157])).
thf(f159,plain,(
  $false | ~spl4_1),
  inference(trivial_inequality_removal,[],[f158])).
thf(f160,plain,(
  ~spl4_1),
  inference(avatar_contradiction_clause,[],[f159])).
cnf(s1, plain, spl4_1 | spl4_2, inference(sat_conversion,[],[f58])).
cnf(s2, plain, spl4_1 | ~spl4_2 | ~spl4_3, inference(sat_conversion,[],[f92])).
cnf(s6, plain, spl4_1 | spl4_3, inference(sat_conversion,[],[f152])).
cnf(s7, plain, ~spl4_1, inference(sat_conversion,[],[f160])).
cnf(s8, plain, spl4_3, inference(rat,[],[s6,s7])).
cnf(s9, plain, ~spl4_2, inference(rat,[],[s2,s8,s7])).
cnf(s10, plain, $false, inference(rat,[],[s1,s9,s7])).
thf(f161,plain,(
  $false),
  inference(avatar_sat_refutation,[],[s10])).
% SZS output end Proof for theBenchmark