%----Output an unreadable proof
% SZS output start Proof for theBenchmark
thf(type_def_5, type, sTfun: ($tType * $tType) > $tType).
thf(func_def_1, type, cS: ($i > $i)).
thf(func_def_2, type, cIND: $o).
thf(func_def_6, type, sK0: (($i > $o) > $i)).
thf(func_def_7, type, sK1: ($i > $o)).
thf(func_def_10, type, vNOT: ($o > $o)).
thf(func_def_11, type, vEQ: !>[X0: $tType]:((X0 > X0 > $o))).
thf(func_def_12, type, db0: !>[X0: $tType]:(X0)).
thf(func_def_13, type, vLAM: !>[X0: $tType, X1: $tType]:((X1) > (X0 > X1))).
thf(f1,axiom,(
  (! [X0 : ($i > $o)] : ((! [X1 : $i] : ((X0 @ X1) => (X0 @ (cS @ X1))) & (X0 @ c0)) => ! [X1 : $i] : (X0 @ X1)) = cIND)),
  file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cIND_def)).
thf(f2,conjecture,(
  cIND => ! [X0 : $i] : ((X0 = c0) | ? [X1 : $i] : (X0 = ((cS @ X1))))),
  file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cTHM578)).
thf(f3,negated_conjecture,(
  ~(cIND => ! [X0 : $i] : ((X0 = c0) | ? [X1 : $i] : (X0 = ((cS @ X1)))))),
  inference(negated_conjecture,[status(cth)],[f2])).
thf(f4,plain,(
  ~(cIND => ! [X0 : $i] : ((X0 = c0) | ? [X1 : $i] : (X0 = ((cS @ X1)))))),
  inference(rectify,[],[f3])).
thf(f5,plain,(
  ~((cIND = $true) => ! [X0 : $i] : (? [X1 : $i] : (((cS @ X1)) = X0) | (c0 = X0)))),
  inference(fool_elimination,[],[f4])).
thf(f6,plain,(
  (! [X0 : ($i > $o)] : ((! [X1 : $i] : ((X0 @ X1) => (X0 @ (cS @ X1))) & (X0 @ c0)) => ! [X2 : $i] : (X0 @ X2)) = cIND)),
  inference(rectify,[],[f1])).
thf(f7,plain,(
  ! [X0 : ($i > $o)] : (((((X0 @ c0)) = $true) & ! [X1 : $i] : ((((X0 @ X1)) = $true) => (((X0 @ (cS @ X1))) = $true))) => ! [X2 : $i] : ($true = ((X0 @ X2)))) <=> (cIND = $true)),
  inference(fool_elimination,[],[f6])).
thf(f8,plain,(
  ! [X0 : ($i > $o)] : (! [X2 : $i] : ($true = ((X0 @ X2))) | ((((X0 @ c0)) != $true) | ? [X1 : $i] : ((((X0 @ X1)) = $true) & (((X0 @ (cS @ X1))) != $true)))) <=> (cIND = $true)),
  inference(ennf_transformation,[],[f7])).
thf(f9,plain,(
  ! [X0 : ($i > $o)] : (! [X2 : $i] : ($true = ((X0 @ X2))) | (((X0 @ c0)) != $true) | ? [X1 : $i] : ((((X0 @ X1)) = $true) & (((X0 @ (cS @ X1))) != $true))) <=> (cIND = $true)),
  inference(flattening,[],[f8])).
thf(f10,plain,(
  ? [X0 : $i] : ((c0 != X0) & ! [X1 : $i] : (((cS @ X1)) != X0)) & (cIND = $true)),
  inference(ennf_transformation,[],[f5])).
thf(f11,plain,(
  (! [X0 : ($i > $o)] : (! [X2 : $i] : ($true = ((X0 @ X2))) | (((X0 @ c0)) != $true) | ? [X1 : $i] : ((((X0 @ X1)) = $true) & (((X0 @ (cS @ X1))) != $true))) | (cIND != $true)) & ((cIND = $true) | ? [X0 : ($i > $o)] : (? [X2 : $i] : ($true != ((X0 @ X2))) & (((X0 @ c0)) = $true) & ! [X1 : $i] : ((((X0 @ X1)) != $true) | (((X0 @ (cS @ X1))) = $true))))),
  inference(nnf_transformation,[],[f9])).
thf(f12,plain,(
  (! [X0 : ($i > $o)] : (! [X1 : $i] : (((X0 @ X1)) = $true) | (((X0 @ c0)) != $true) | ? [X2 : $i] : (($true = ((X0 @ X2))) & ($true != ((X0 @ (cS @ X2)))))) | (cIND != $true)) & ((cIND = $true) | ? [X3 : ($i > $o)] : (? [X4 : $i] : (((X3 @ X4)) != $true) & (((X3 @ c0)) = $true) & ! [X5 : $i] : ((((X3 @ X5)) != $true) | ($true = ((X3 @ (cS @ X5)))))))),
  inference(rectify,[],[f11])).
thf(f13,plain,(
  (! [X0 : ($i > $o)] : (! [X1 : $i] : (((X0 @ X1)) = $true) | (((X0 @ c0)) != $true) | ((((X0 @ (sK0 @ X0))) = $true) & ($true != ((X0 @ (cS @ (sK0 @ X0))))))) | (cIND != $true)) & ((cIND = $true) | ((((sK1 @ sK2)) != $true) & (((sK1 @ c0)) = $true) & ! [X5 : $i] : ((((sK1 @ X5)) != $true) | ($true = ((sK1 @ (cS @ X5)))))))),
  inference(skolemize,[status(esa),new_symbols(skolem,[vAPP,sK1,sK2]),skolemize(X2,$thf(sK0 @ X0)),skolemize(X3,$thf(sK1)),skolemize(X4,$thf(sK2))],[f12])).
thf(f14,plain,(
  ((c0 != sK3) & ! [X1 : $i] : (((cS @ X1)) != sK3)) & (cIND = $true)),
  inference(skolemize,[status(esa),new_symbols(skolem,[sK3]),skolemize(X0,$thf(sK3))],[f10])).
thf(f18,plain,(
  ( ! [X0 : ($i > $o),X1 : $i] : ((((X0 @ X1)) = $true) | (((X0 @ c0)) != $true) | ($true != ((X0 @ (cS @ (sK0 @ X0))))) | (cIND != $true)) )),
  inference(cnf_transformation,[],[f13])).
thf(f20,plain,(
  (cIND = $true)),
  inference(cnf_transformation,[],[f14])).
thf(f21,plain,(
  ( ! [X1 : $i] : ((((cS @ X1)) != sK3)) )),
  inference(cnf_transformation,[],[f14])).
thf(f22,plain,(
  (c0 != sK3)),
  inference(cnf_transformation,[],[f14])).
thf(f26,plain,(
  ( ! [X0 : ($i > $o),X1 : $i] : ((((X0 @ c0)) != $true) | (((X0 @ X1)) = $true) | ($true != ((X0 @ (cS @ (sK0 @ X0))))) | ($true != $true)) )),
  inference(definition_unfolding,[],[f18,f20])).
thf(f30,plain,(
  ( ! [X0 : ($i > $o),X1 : $i] : (($true != ((X0 @ (cS @ (sK0 @ X0))))) | (((X0 @ c0)) != $true) | (((X0 @ X1)) = $true)) )),
  inference(trivial_inequality_removal,[],[f26])).
thf(f67,plain,(
  ( ! [X0 : $i] : (((((^[Y0 : $i]: (~ (Y0 = X0))) @ c0)) != $true) | (((cS @ (sK0 @ (^[Y0 : $i]: (~ (Y0 = X0)))))) = X0)) )),
  inference(leibniz_equality_elimination,[],[f30])).
thf(f75,plain,(
  ( ! [X0 : $i] : ((((cS @ (sK0 @ (^[Y0 : $i]: (~ (Y0 = X0)))))) = X0) | (((~ (c0 = X0))) != $true)) )),
  inference(beta-eta_normalization,[],[f67])).
thf(f76,plain,(
  ( ! [X0 : $i] : ((((cS @ (sK0 @ (^[Y0 : $i]: (~ (Y0 = X0)))))) = X0) | (((c0 = X0)) = $true)) )),
  inference(not_proxy_clausification,[],[f75])).
thf(f77,plain,(
  ( ! [X0 : $i] : ((((cS @ (sK0 @ (^[Y0 : $i]: (~ (Y0 = X0)))))) = X0) | (c0 = X0)) )),
  inference(equality_proxy_clausification,[],[f76])).
thf(f96,plain,(
  ( ! [X0 : $i] : ((sK3 != X0) | (c0 = X0)) )),
  inference(constrained_superposition,[],[f21,f77])).
thf(f106,plain,(
  (sK3 != sK3) | (c0 = sK3)),
  inference(imitation,[],[f96])).
thf(f108,plain,(
  (c0 = sK3)),
  inference(trivial_inequality_removal,[],[f106])).
thf(f109,plain,(
  $false),
  inference(forward_subsumption_resolution,[],[f108,f22])).
% SZS output end Proof for theBenchmark