%------------------------------------------------------------------------------
% File : Vampire---5.0.1
% Problem : SYO572^7 : TPTP v9.3.1. Released v5.5.0.
% Transfm : none
% Format : tptp:raw
% Command : run_vampire /export/starexec/sandbox/benchmark/theBenchmark.p 300 THM
% Computer : n010.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Sep 30 08:48:26 AM UTC 2026
% Result : Theorem 0.19s 0.28s
% Output : Refutation 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : SYO572^7 : TPTP v9.3.1. Released v5.5.0.
% 0.00/0.05 % Command : run_vampire /export/starexec/sandbox/benchmark/theBenchmark.p 300 THM
% 0.07/0.18 % Computer : n010.cluster.edu
% 0.07/0.18 % Model : x86_64 x86_64
% 0.07/0.18 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.07/0.18 % Memory : 8046.5625MB
% 0.07/0.18 % OS : Linux 6.8.0-71-generic
% 0.07/0.18 % CPULimit : 300
% 0.07/0.18 % WCLimit : 300
% 0.07/0.18 % DateTime : Tue Sep 29 17:15:49 UTC 2026
% 0.07/0.18 % CPUTime :
% 0.07/0.18 Running run_vampire /export/starexec/sandbox/benchmark/theBenchmark.p 300 THM
% 0.07/0.22 Running higher-order theorem proving
% 0.07/0.23 Running: /export/starexec/sandbox/solver/bin/vampire-ho --input_syntax tptp --output_axiom_names on --mode casc -m 16384 --cores 7 -t 300 /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.19/0.28 % (3159316)Detected a higher-order problem, will run a greedy HOL sequence.
% 0.19/0.28 % (3159321)lrs+10_40_drc=off:e2e=on:si=on:uwa=one_side_interpreted:random_seed=2252177557:s2a=on:i=87:rtra=on:ntd=on_2999 on theBenchmark for (2999ds/87Mi)
% 0.19/0.28 % (3159321) found proof, printing to "/export/starexec/sandbox/tmp/vampire-proof-3159316-3159321"...
% 0.19/0.28 % (3159327)WARNING Broken Constraint: if ho_split_queue_ratios(1,8) has been set then ho_split_queue(off) is equal to on
% 0.19/0.28 % (3159327)WARNING Broken Constraint: if sine_to_age_tolerance(5) has been set then sine_to_age(off) is equal to on or sine_to_pred_levels(off) is not equal to off or sine_level_split_queue(off) is equal to on
% 0.19/0.28 % (3159321)...printing done.
% 0.19/0.28 % (3159321)Refutation found. Thanks to Tanya!
% 0.19/0.28 % SZS status Theorem for theBenchmark
% 0.19/0.28 % SZS output start Proof for theBenchmark
% 0.19/0.28 thf(type_def_5, type, mu: $tType).
% 0.19/0.28 thf(type_def_6, type, sTfun: ($tType * $tType) > $tType).
% 0.19/0.28 thf(func_def_0, type, qmltpeq: (mu > mu > $i > $o)).
% 0.19/0.28 thf(func_def_1, type, meq_prop: (($i > $o) > ($i > $o) > $i > $o)).
% 0.19/0.28 thf(func_def_3, type, mnot: (($i > $o) > $i > $o)).
% 0.19/0.28 thf(func_def_4, type, mor: (($i > $o) > ($i > $o) > $i > $o)).
% 0.19/0.28 thf(func_def_5, type, mbox: (($i > $i > $o) > ($i > $o) > $i > $o)).
% 0.19/0.28 thf(func_def_6, type, mforall_prop: ((($i > $o) > $i > $o) > $i > $o)).
% 0.19/0.28 thf(func_def_7, type, mtrue: ($i > $o)).
% 0.19/0.28 thf(func_def_8, type, mfalse: ($i > $o)).
% 0.19/0.28 thf(func_def_9, type, mand: (($i > $o) > ($i > $o) > $i > $o)).
% 0.19/0.28 thf(func_def_10, type, mimplies: (($i > $o) > ($i > $o) > $i > $o)).
% 0.19/0.28 thf(func_def_11, type, mimplied: (($i > $o) > ($i > $o) > $i > $o)).
% 0.19/0.28 thf(func_def_12, type, mequiv: (($i > $o) > ($i > $o) > $i > $o)).
% 0.19/0.28 thf(func_def_13, type, mxor: (($i > $o) > ($i > $o) > $i > $o)).
% 0.19/0.28 thf(func_def_14, type, mdia: (($i > $i > $o) > ($i > $o) > $i > $o)).
% 0.19/0.28 thf(func_def_15, type, exists_in_world: (mu > $i > $o)).
% 0.19/0.28 thf(func_def_16, type, mforall_ind: ((mu > $i > $o) > $i > $o)).
% 0.19/0.28 thf(func_def_17, type, mexists_ind: ((mu > $i > $o) > $i > $o)).
% 0.19/0.28 thf(func_def_18, type, mexists_prop: ((($i > $o) > $i > $o) > $i > $o)).
% 0.19/0.28 thf(func_def_19, type, mreflexive: (($i > $i > $o) > $o)).
% 0.19/0.28 thf(func_def_20, type, msymmetric: (($i > $i > $o) > $o)).
% 0.19/0.28 thf(func_def_21, type, mserial: (($i > $i > $o) > $o)).
% 0.19/0.28 thf(func_def_22, type, mtransitive: (($i > $i > $o) > $o)).
% 0.19/0.28 thf(func_def_23, type, meuclidean: (($i > $i > $o) > $o)).
% 0.19/0.28 thf(func_def_24, type, mpartially_functional: (($i > $i > $o) > $o)).
% 0.19/0.28 thf(func_def_25, type, mfunctional: (($i > $i > $o) > $o)).
% 0.19/0.28 thf(func_def_26, type, mweakly_dense: (($i > $i > $o) > $o)).
% 0.19/0.28 thf(func_def_27, type, mweakly_connected: (($i > $i > $o) > $o)).
% 0.19/0.28 thf(func_def_28, type, mweakly_directed: (($i > $i > $o) > $o)).
% 0.19/0.28 thf(func_def_29, type, mvalid: (($i > $o) > $o)).
% 0.19/0.28 thf(func_def_30, type, msatisfiable: (($i > $o) > $o)).
% 0.19/0.28 thf(func_def_31, type, mcountersatisfiable: (($i > $o) > $o)).
% 0.19/0.28 thf(func_def_32, type, minvalid: (($i > $o) > $o)).
% 0.19/0.28 thf(func_def_33, type, rel_s4: ($i > $i > $o)).
% 0.19/0.28 thf(func_def_34, type, mbox_s4: (($i > $o) > $i > $o)).
% 0.19/0.28 thf(func_def_35, type, mdia_s4: (($i > $o) > $i > $o)).
% 0.19/0.28 thf(func_def_36, type, f: (mu > $i > $o)).
% 0.19/0.28 thf(func_def_39, type, vPI: !>[X0: $tType]:(((X0 > $o) > $o))).
% 0.19/0.28 thf(func_def_40, type, vIMP: ($o > $o > $o)).
% 0.19/0.28 thf(func_def_41, type, vAND: ($o > $o > $o)).
% 0.19/0.28 thf(func_def_42, type, db3: !>[X0: $tType]:(X0)).
% 0.19/0.28 thf(func_def_43, type, db2: !>[X0: $tType]:(X0)).
% 0.19/0.28 thf(func_def_44, type, db0: !>[X0: $tType]:(X0)).
% 0.19/0.28 thf(func_def_45, type, db1: !>[X0: $tType]:(X0)).
% 0.19/0.28 thf(func_def_46, type, vOR: ($o > $o > $o)).
% 0.19/0.28 thf(func_def_47, type, vEQ: !>[X0: $tType]:((X0 > X0 > $o))).
% 0.19/0.28 thf(func_def_48, type, vLAM: !>[X0: $tType, X1: $tType]:((X1) > (X0 > X1))).
% 0.19/0.28 thf(func_def_49, type, vSIGMA: !>[X0: $tType]:(((X0 > $o) > $o))).
% 0.19/0.28 thf(func_def_50, type, db4: !>[X0: $tType]:(X0)).
% 0.19/0.28 thf(func_def_51, type, vNOT: ($o > $o)).
% 0.19/0.28 thf(func_def_52, type, sK0: ($i > mu)).
% 0.19/0.28 thf(f2,axiom,(
% 0.19/0.28 ((^[X0 : ($i > $o), X1 : $i] : (~(X0 @ X1))) = mnot)),
% 0.19/0.28 file('/export/starexec/sandbox/benchmark/Axioms/LCL015^0.ax',mnot)).
% 0.19/0.28 thf(f3,axiom,(
% 0.19/0.28 ((^[X0 : ($i > $o), X1 : ($i > $o), X2 : $i] : ((X0 @ X2) | (X1 @ X2))) = mor)),
% 0.19/0.28 file('/export/starexec/sandbox/benchmark/Axioms/LCL015^0.ax',mor)).
% 0.19/0.28 thf(f9,axiom,(
% 0.19/0.28 ((^[X0 : ($i > $o), X1 : ($i > $o)] : ((mor @ (mnot @ X0) @ X1))) = mimplies)),
% 0.19/0.28 file('/export/starexec/sandbox/benchmark/Axioms/LCL015^0.ax',mimplies)).
% 0.19/0.28 thf(f14,axiom,(
% 0.19/0.28 ! [X0 : $i] : ? [X1 : mu] : (exists_in_world @ X1 @ X0)),
% 0.19/0.28 file('/export/starexec/sandbox/benchmark/Axioms/LCL015^0.ax',nonempty_ax)).
% 0.19/0.28 thf(f15,axiom,(
% 0.19/0.28 (mforall_ind = (^[X0 : (mu > $i > $o), X1 : $i] : (! [X2 : mu] : ((exists_in_world @ X2 @ X1) => (X0 @ X2 @ X1)))))),
% 0.19/0.28 file('/export/starexec/sandbox/benchmark/Axioms/LCL015^0.ax',mforall_ind)).
% 0.19/0.28 thf(f16,axiom,(
% 0.19/0.28 (mexists_ind = (^[X0 : (mu > $i > $o)] : ((mnot @ (mforall_ind @ (^[X1 : mu] : ((mnot @ (X0 @ X1)))))))))),
% 0.19/0.28 file('/export/starexec/sandbox/benchmark/Axioms/LCL015^0.ax',mexists_ind)).
% 0.19/0.28 thf(f28,axiom,(
% 0.19/0.28 ((^[X0 : ($i > $o)] : (! [X1 : $i] : (X0 @ X1))) = mvalid)),
% 0.19/0.28 file('/export/starexec/sandbox/benchmark/Axioms/LCL015^0.ax',mvalid)).
% 0.19/0.28 thf(f32,axiom,(
% 0.19/0.28 (mbox_s4 = (^[X0 : ($i > $o), X1 : $i] : (! [X2 : $i] : (~(rel_s4 @ X1 @ X2) | (X0 @ X2)))))),
% 0.19/0.28 file('/export/starexec/sandbox/benchmark/Axioms/LCL013^5.ax',mbox_s4)).
% 0.19/0.28 thf(f33,axiom,(
% 0.19/0.28 ((^[X0 : ($i > $o)] : ((mnot @ (mbox_s4 @ (mnot @ X0))))) = mdia_s4)),
% 0.19/0.28 file('/export/starexec/sandbox/benchmark/Axioms/LCL013^5.ax',mdia_s4)).
% 0.19/0.28 thf(f36,axiom,(
% 0.19/0.28 ! [X1 : $i,X2 : $i,X0 : mu] : (((rel_s4 @ X1 @ X2) & (exists_in_world @ X0 @ X1)) => (exists_in_world @ X0 @ X2))),
% 0.19/0.28 file('/export/starexec/sandbox/benchmark/Axioms/LCL015^1.ax',cumulative_ax)).
% 0.19/0.28 thf(f37,conjecture,(
% 0.19/0.28 (mvalid @ (mimplies @ (mdia_s4 @ (mforall_ind @ (^[X0 : mu] : ((f @ X0))))) @ (mexists_ind @ (^[X0 : mu] : ((mdia_s4 @ (f @ X0)))))))),
% 0.19/0.28 file('/export/starexec/sandbox/benchmark/theBenchmark.p',con)).
% 0.19/0.28 thf(f38,negated_conjecture,(
% 0.19/0.28 ~(mvalid @ (mimplies @ (mdia_s4 @ (mforall_ind @ (^[X0 : mu] : ((f @ X0))))) @ (mexists_ind @ (^[X0 : mu] : ((mdia_s4 @ (f @ X0)))))))),
% 0.19/0.28 inference(negated_conjecture,[status(cth)],[f37])).
% 0.19/0.28 thf(f45,plain,(
% 0.19/0.28 ((^[X0 : ($i > $o), X1 : ($i > $o), X2 : $i] : ((X0 @ X2) | (X1 @ X2))) = mor)),
% 0.19/0.28 inference(rectify,[],[f3])).
% 0.19/0.28 thf(f46,plain,(
% 0.19/0.28 (mor = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: ((Y1 @ Y2) | (Y0 @ Y2))))))))),
% 0.19/0.28 inference(fool_elimination,[],[f45])).
% 0.19/0.28 thf(f47,plain,(
% 0.19/0.28 (mforall_ind = (^[X0 : (mu > $i > $o), X1 : $i] : (! [X2 : mu] : ((exists_in_world @ X2 @ X1) => (X0 @ X2 @ X1)))))),
% 0.19/0.28 inference(rectify,[],[f15])).
% 0.19/0.28 thf(f48,plain,(
% 0.19/0.28 (mforall_ind = (^[Y0 : mu > $i > $o]: ((^[Y1 : $i]: (!! @ mu @ (^[Y2 : mu]: ((exists_in_world @ Y2 @ Y1) => (Y0 @ Y2 @ Y1))))))))),
% 0.19/0.28 inference(fool_elimination,[],[f47])).
% 0.19/0.28 thf(f49,plain,(
% 0.19/0.28 (mimplies = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: (mor @ (mnot @ Y0) @ Y1)))))),
% 0.19/0.28 inference(fool_elimination,[],[f9])).
% 0.19/0.28 thf(f54,plain,(
% 0.19/0.28 (mbox_s4 = (^[X0 : ($i > $o), X1 : $i] : (! [X2 : $i] : (~(rel_s4 @ X1 @ X2) | (X0 @ X2)))))),
% 0.19/0.28 inference(rectify,[],[f32])).
% 0.19/0.28 thf(f55,plain,(
% 0.19/0.28 (mbox_s4 = (^[Y0 : $i > $o]: ((^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((Y0 @ Y2) | (~ (rel_s4 @ Y1 @ Y2)))))))))),
% 0.19/0.28 inference(fool_elimination,[],[f54])).
% 0.19/0.28 thf(f56,plain,(
% 0.19/0.28 ((^[X0 : ($i > $o)] : (! [X1 : $i] : (X0 @ X1))) = mvalid)),
% 0.19/0.28 inference(rectify,[],[f28])).
% 0.19/0.28 thf(f57,plain,(
% 0.19/0.28 (mvalid = (^[Y0 : $i > $o]: (!! @ $i @ (^[Y1 : $i]: (Y0 @ Y1)))))),
% 0.19/0.28 inference(fool_elimination,[],[f56])).
% 0.19/0.28 thf(f58,plain,(
% 0.19/0.28 (mexists_ind = (^[X0 : (mu > $i > $o)] : ((mnot @ (mforall_ind @ (^[X1 : mu] : ((mnot @ (X0 @ X1)))))))))),
% 0.19/0.28 inference(rectify,[],[f16])).
% 0.19/0.28 thf(f59,plain,(
% 0.19/0.28 (mexists_ind = (^[Y0 : mu > $i > $o]: (mnot @ (mforall_ind @ (^[Y1 : mu]: (mnot @ (Y0 @ Y1)))))))),
% 0.19/0.28 inference(fool_elimination,[],[f58])).
% 0.19/0.28 thf(f62,plain,(
% 0.19/0.28 (mdia_s4 = (^[Y0 : $i > $o]: (mnot @ (mbox_s4 @ (mnot @ Y0)))))),
% 0.19/0.28 inference(fool_elimination,[],[f33])).
% 0.19/0.28 thf(f68,plain,(
% 0.19/0.28 ! [X0 : $i,X1 : $i,X2 : mu] : (((rel_s4 @ X0 @ X1) & (exists_in_world @ X2 @ X0)) => (exists_in_world @ X2 @ X1))),
% 0.19/0.28 inference(rectify,[],[f36])).
% 0.19/0.28 thf(f69,plain,(
% 0.19/0.28 ! [X0 : $i,X1 : $i,X2 : mu] : ((($true = ((exists_in_world @ X2 @ X0))) & ($true = ((rel_s4 @ X0 @ X1)))) => (((exists_in_world @ X2 @ X1)) = $true))),
% 0.19/0.28 inference(fool_elimination,[],[f68])).
% 0.19/0.28 thf(f89,plain,(
% 0.19/0.28 ! [X0 : $i] : ? [X1 : mu] : (exists_in_world @ X1 @ X0)),
% 0.19/0.28 inference(rectify,[],[f14])).
% 0.19/0.28 thf(f90,plain,(
% 0.19/0.28 ! [X0 : $i] : ? [X1 : mu] : (((exists_in_world @ X1 @ X0)) = $true)),
% 0.19/0.28 inference(fool_elimination,[],[f89])).
% 0.19/0.28 thf(f96,plain,(
% 0.19/0.28 ~(mvalid @ (mimplies @ (mdia_s4 @ (mforall_ind @ (^[X0 : mu] : ((f @ X0))))) @ (mexists_ind @ (^[X1 : mu] : ((mdia_s4 @ (f @ X1)))))))),
% 0.19/0.28 inference(rectify,[],[f38])).
% 0.19/0.28 thf(f97,plain,(
% 0.19/0.28 ~ ($true = ((mvalid @ (mimplies @ (mdia_s4 @ (mforall_ind @ (^[Y0 : mu]: (f @ Y0)))) @ (mexists_ind @ (^[Y0 : mu]: (mdia_s4 @ (f @ Y0))))))))),
% 0.19/0.28 inference(fool_elimination,[],[f96])).
% 0.19/0.28 thf(f100,plain,(
% 0.19/0.28 ((^[X0 : ($i > $o), X1 : $i] : (~(X0 @ X1))) = mnot)),
% 0.19/0.28 inference(rectify,[],[f2])).
% 0.19/0.28 thf(f101,plain,(
% 0.19/0.28 (mnot = (^[Y0 : $i > $o]: ((^[Y1 : $i]: (~ (Y0 @ Y1))))))),
% 0.19/0.28 inference(fool_elimination,[],[f100])).
% 0.19/0.28 thf(f102,plain,(
% 0.19/0.28 ($true != ((mvalid @ (mimplies @ (mdia_s4 @ (mforall_ind @ (^[Y0 : mu]: (f @ Y0)))) @ (mexists_ind @ (^[Y0 : mu]: (mdia_s4 @ (f @ Y0))))))))),
% 0.19/0.28 inference(flattening,[],[f97])).
% 0.19/0.28 thf(f103,plain,(
% 0.19/0.28 ! [X0 : $i,X1 : $i,X2 : mu] : ((((exists_in_world @ X2 @ X1)) = $true) | (($true != ((exists_in_world @ X2 @ X0))) | ($true != ((rel_s4 @ X0 @ X1)))))),
% 0.19/0.28 inference(ennf_transformation,[],[f69])).
% 0.19/0.28 thf(f104,plain,(
% 0.19/0.28 ! [X0 : $i,X1 : $i,X2 : mu] : (($true != ((rel_s4 @ X0 @ X1))) | (((exists_in_world @ X2 @ X1)) = $true) | ($true != ((exists_in_world @ X2 @ X0))))),
% 0.19/0.28 inference(flattening,[],[f103])).
% 0.19/0.28 thf(f105,plain,(
% 0.19/0.28 ! [X0 : $i] : ($true = ((exists_in_world @ (sK0 @ X0) @ X0)))),
% 0.19/0.28 inference(skolemize,[status(esa),new_symbols(skolem,[vAPP]),skolemize(X1,sK0 @ X0)],[f90])).
% 0.19/0.28 thf(f107,plain,(
% 0.19/0.28 ( ! [X2 : mu,X0 : $i,X1 : $i] : (($true != ((exists_in_world @ X2 @ X0))) | (((exists_in_world @ X2 @ X1)) = $true) | ($true != ((rel_s4 @ X0 @ X1)))) )),
% 0.19/0.28 inference(cnf_transformation,[],[f104])).
% 0.19/0.28 thf(f108,plain,(
% 0.19/0.28 (mexists_ind = (^[Y0 : mu > $i > $o]: (mnot @ (mforall_ind @ (^[Y1 : mu]: (mnot @ (Y0 @ Y1)))))))),
% 0.19/0.28 inference(cnf_transformation,[],[f59])).
% 0.19/0.28 thf(f112,plain,(
% 0.19/0.28 (mor = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: ((Y1 @ Y2) | (Y0 @ Y2))))))))),
% 0.19/0.28 inference(cnf_transformation,[],[f46])).
% 0.19/0.28 thf(f113,plain,(
% 0.19/0.28 ($true != ((mvalid @ (mimplies @ (mdia_s4 @ (mforall_ind @ (^[Y0 : mu]: (f @ Y0)))) @ (mexists_ind @ (^[Y0 : mu]: (mdia_s4 @ (f @ Y0))))))))),
% 0.19/0.28 inference(cnf_transformation,[],[f102])).
% 0.19/0.28 thf(f122,plain,(
% 0.19/0.28 (mvalid = (^[Y0 : $i > $o]: (!! @ $i @ (^[Y1 : $i]: (Y0 @ Y1)))))),
% 0.19/0.28 inference(cnf_transformation,[],[f57])).
% 0.19/0.28 thf(f132,plain,(
% 0.19/0.28 (mimplies = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: (mor @ (mnot @ Y0) @ Y1)))))),
% 0.19/0.28 inference(cnf_transformation,[],[f49])).
% 0.19/0.28 thf(f133,plain,(
% 0.19/0.28 (mforall_ind = (^[Y0 : mu > $i > $o]: ((^[Y1 : $i]: (!! @ mu @ (^[Y2 : mu]: ((exists_in_world @ Y2 @ Y1) => (Y0 @ Y2 @ Y1))))))))),
% 0.19/0.28 inference(cnf_transformation,[],[f48])).
% 0.19/0.28 thf(f137,plain,(
% 0.19/0.28 (mbox_s4 = (^[Y0 : $i > $o]: ((^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((Y0 @ Y2) | (~ (rel_s4 @ Y1 @ Y2)))))))))),
% 0.19/0.28 inference(cnf_transformation,[],[f55])).
% 0.19/0.28 thf(f138,plain,(
% 0.19/0.28 (mnot = (^[Y0 : $i > $o]: ((^[Y1 : $i]: (~ (Y0 @ Y1))))))),
% 0.19/0.28 inference(cnf_transformation,[],[f101])).
% 0.19/0.28 thf(f139,plain,(
% 0.19/0.28 (mdia_s4 = (^[Y0 : $i > $o]: (mnot @ (mbox_s4 @ (mnot @ Y0)))))),
% 0.19/0.28 inference(cnf_transformation,[],[f62])).
% 0.19/0.28 thf(f141,plain,(
% 0.19/0.28 ( ! [X0 : $i] : (($true = ((exists_in_world @ (sK0 @ X0) @ X0)))) )),
% 0.19/0.28 inference(cnf_transformation,[],[f105])).
% 0.19/0.28 thf(f147,plain,(
% 0.19/0.28 (mimplies = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i > $o]: ((^[Y3 : $i > $o]: ((^[Y4 : $i]: ((Y3 @ Y4) | (Y2 @ Y4))))))) @ ((^[Y2 : $i > $o]: ((^[Y3 : $i]: (~ (Y2 @ Y3))))) @ Y0) @ Y1)))))),
% 0.19/0.28 inference(definition_unfolding,[],[f132,f112,f138])).
% 0.19/0.28 thf(f152,plain,(
% 0.19/0.28 (mexists_ind = (^[Y0 : mu > $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (~ (Y1 @ Y2))))) @ ((^[Y1 : mu > $i > $o]: ((^[Y2 : $i]: (!! @ mu @ (^[Y3 : mu]: ((exists_in_world @ Y3 @ Y2) => (Y1 @ Y3 @ Y2))))))) @ (^[Y1 : mu]: ((^[Y2 : $i > $o]: ((^[Y3 : $i]: (~ (Y2 @ Y3))))) @ (Y0 @ Y1)))))))),
% 0.19/0.28 inference(definition_unfolding,[],[f108,f138,f133,f138])).
% 0.19/0.28 thf(f154,plain,(
% 0.19/0.28 (mdia_s4 = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (~ (Y1 @ Y2))))) @ ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: ((Y1 @ Y3) | (~ (rel_s4 @ Y2 @ Y3)))))))) @ ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (~ (Y1 @ Y2))))) @ Y0)))))),
% 0.19/0.28 inference(definition_unfolding,[],[f139,f138,f137,f138])).
% 0.19/0.28 thf(f155,plain,(
% 0.19/0.28 ($true != (((^[Y0 : $i > $o]: (!! @ $i @ (^[Y1 : $i]: (Y0 @ Y1)))) @ ((^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i > $o]: ((^[Y3 : $i > $o]: ((^[Y4 : $i]: ((Y3 @ Y4) | (Y2 @ Y4))))))) @ ((^[Y2 : $i > $o]: ((^[Y3 : $i]: (~ (Y2 @ Y3))))) @ Y0) @ Y1)))) @ ((^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (~ (Y1 @ Y2))))) @ ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: ((Y1 @ Y3) | (~ (rel_s4 @ Y2 @ Y3)))))))) @ ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (~ (Y1 @ Y2))))) @ Y0)))) @ ((^[Y0 : mu > $i > $o]: ((^[Y1 : $i]: (!! @ mu @ (^[Y2 : mu]: ((exists_in_world @ Y2 @ Y1) => (Y0 @ Y2 @ Y1))))))) @ (^[Y0 : mu]: (f @ Y0)))) @ ((^[Y0 : mu > $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (~ (Y1 @ Y2))))) @ ((^[Y1 : mu > $i > $o]: ((^[Y2 : $i]: (!! @ mu @ (^[Y3 : mu]: ((exists_in_world @ Y3 @ Y2) => (Y1 @ Y3 @ Y2))))))) @ (^[Y1 : mu]: ((^[Y2 : $i > $o]: ((^[Y3 : $i]: (~ (Y2 @ Y3))))) @ (Y0 @ Y1)))))) @ (^[Y0 : mu]: ((^[Y1 : $i > $o]: ((^[Y2 : $i > $o]: ((^[Y3 : $i]: (~ (Y2 @ Y3))))) @ ((^[Y2 : $i > $o]: ((^[Y3 : $i]: (!! @ $i @ (^[Y4 : $i]: ((Y2 @ Y4) | (~ (rel_s4 @ Y3 @ Y4)))))))) @ ((^[Y2 : $i > $o]: ((^[Y3 : $i]: (~ (Y2 @ Y3))))) @ Y1)))) @ (f @ Y0))))))))),
% 0.19/0.28 inference(definition_unfolding,[],[f113,f122,f147,f154,f133,f152,f154])).
% 0.19/0.28 thf(f158,plain,(
% 0.19/0.28 ($true != ((!! @ $i @ (^[Y0 : $i]: ((~ (!! @ mu @ (^[Y1 : mu]: ((exists_in_world @ Y1 @ Y0) => (~ (~ (!! @ $i @ (^[Y2 : $i]: ((~ (f @ Y1 @ Y2)) | (~ (rel_s4 @ Y0 @ Y2))))))))))) | (~ (~ (!! @ $i @ (^[Y1 : $i]: ((~ (!! @ mu @ (^[Y2 : mu]: ((exists_in_world @ Y2 @ Y1) => (f @ Y2 @ Y1))))) | (~ (rel_s4 @ Y0 @ Y1))))))))))))),
% 0.19/0.28 inference(beta-eta_normalization,[],[f155])).
% 0.19/0.28 thf(f159,plain,(
% 0.19/0.28 ($false = (((^[Y0 : $i]: ((~ (!! @ mu @ (^[Y1 : mu]: ((exists_in_world @ Y1 @ Y0) => (~ (~ (!! @ $i @ (^[Y2 : $i]: ((~ (f @ Y1 @ Y2)) | (~ (rel_s4 @ Y0 @ Y2))))))))))) | (~ (~ (!! @ $i @ (^[Y1 : $i]: ((~ (!! @ mu @ (^[Y2 : mu]: ((exists_in_world @ Y2 @ Y1) => (f @ Y2 @ Y1))))) | (~ (rel_s4 @ Y0 @ Y1))))))))) @ sK1)))),
% 0.19/0.28 inference(sigma_proxy_clausification,[],[f158])).
% 0.19/0.28 thf(f160,plain,(
% 0.19/0.28 ($false = (((~ (!! @ mu @ (^[Y0 : mu]: ((exists_in_world @ Y0 @ sK1) => (~ (~ (!! @ $i @ (^[Y1 : $i]: ((~ (f @ Y0 @ Y1)) | (~ (rel_s4 @ sK1 @ Y1))))))))))) | (~ (~ (!! @ $i @ (^[Y0 : $i]: ((~ (!! @ mu @ (^[Y1 : mu]: ((exists_in_world @ Y1 @ Y0) => (f @ Y1 @ Y0))))) | (~ (rel_s4 @ sK1 @ Y0))))))))))),
% 0.19/0.28 inference(beta-eta_normalization,[],[f159])).
% 0.19/0.28 thf(f161,plain,(
% 0.19/0.28 ($false = ((~ (~ (!! @ $i @ (^[Y0 : $i]: ((~ (!! @ mu @ (^[Y1 : mu]: ((exists_in_world @ Y1 @ Y0) => (f @ Y1 @ Y0))))) | (~ (rel_s4 @ sK1 @ Y0)))))))))),
% 0.19/0.28 inference(or_proxy_clausification,[],[f160])).
% 0.19/0.28 thf(f162,plain,(
% 0.19/0.28 ($false = ((~ (!! @ mu @ (^[Y0 : mu]: ((exists_in_world @ Y0 @ sK1) => (~ (~ (!! @ $i @ (^[Y1 : $i]: ((~ (f @ Y0 @ Y1)) | (~ (rel_s4 @ sK1 @ Y1)))))))))))))),
% 0.19/0.28 inference(or_proxy_clausification,[],[f160])).
% 0.19/0.28 thf(f163,plain,(
% 0.19/0.28 ($true = ((!! @ mu @ (^[Y0 : mu]: ((exists_in_world @ Y0 @ sK1) => (~ (~ (!! @ $i @ (^[Y1 : $i]: ((~ (f @ Y0 @ Y1)) | (~ (rel_s4 @ sK1 @ Y1))))))))))))),
% 0.19/0.28 inference(not_proxy_clausification,[],[f162])).
% 0.19/0.28 thf(f164,plain,(
% 0.19/0.28 ( ! [X1 : mu] : (($true = (((^[Y0 : mu]: ((exists_in_world @ Y0 @ sK1) => (~ (~ (!! @ $i @ (^[Y1 : $i]: ((~ (f @ Y0 @ Y1)) | (~ (rel_s4 @ sK1 @ Y1))))))))) @ X1)))) )),
% 0.19/0.28 inference(pi_proxy_clausification,[],[f163])).
% 0.19/0.28 thf(f165,plain,(
% 0.19/0.28 ( ! [X1 : mu] : (($true = (((exists_in_world @ X1 @ sK1) => (~ (~ (!! @ $i @ (^[Y0 : $i]: ((~ (f @ X1 @ Y0)) | (~ (rel_s4 @ sK1 @ Y0))))))))))) )),
% 0.19/0.28 inference(beta-eta_normalization,[],[f164])).
% 0.19/0.28 thf(f166,plain,(
% 0.19/0.28 ( ! [X1 : mu] : (($false = ((exists_in_world @ X1 @ sK1))) | ($true = ((~ (~ (!! @ $i @ (^[Y0 : $i]: ((~ (f @ X1 @ Y0)) | (~ (rel_s4 @ sK1 @ Y0)))))))))) )),
% 0.19/0.28 inference(imp_proxy_clausification,[],[f165])).
% 0.19/0.28 thf(f167,plain,(
% 0.19/0.28 ( ! [X1 : mu] : (($false = ((exists_in_world @ X1 @ sK1))) | ($false = ((~ (!! @ $i @ (^[Y0 : $i]: ((~ (f @ X1 @ Y0)) | (~ (rel_s4 @ sK1 @ Y0))))))))) )),
% 0.19/0.28 inference(not_proxy_clausification,[],[f166])).
% 0.19/0.28 thf(f168,plain,(
% 0.19/0.28 ( ! [X1 : mu] : (($true = ((!! @ $i @ (^[Y0 : $i]: ((~ (f @ X1 @ Y0)) | (~ (rel_s4 @ sK1 @ Y0))))))) | ($false = ((exists_in_world @ X1 @ sK1)))) )),
% 0.19/0.28 inference(not_proxy_clausification,[],[f167])).
% 0.19/0.28 thf(f169,plain,(
% 0.19/0.28 ( ! [X2 : $i,X1 : mu] : (($false = ((exists_in_world @ X1 @ sK1))) | ($true = (((^[Y0 : $i]: ((~ (f @ X1 @ Y0)) | (~ (rel_s4 @ sK1 @ Y0)))) @ X2)))) )),
% 0.19/0.28 inference(pi_proxy_clausification,[],[f168])).
% 0.19/0.28 thf(f170,plain,(
% 0.19/0.28 ( ! [X2 : $i,X1 : mu] : (($true = (((~ (f @ X1 @ X2)) | (~ (rel_s4 @ sK1 @ X2))))) | ($false = ((exists_in_world @ X1 @ sK1)))) )),
% 0.19/0.28 inference(beta-eta_normalization,[],[f169])).
% 0.19/0.28 thf(f171,plain,(
% 0.19/0.28 ( ! [X2 : $i,X1 : mu] : (($true = ((~ (f @ X1 @ X2)))) | ($true = ((~ (rel_s4 @ sK1 @ X2)))) | ($false = ((exists_in_world @ X1 @ sK1)))) )),
% 0.19/0.28 inference(or_proxy_clausification,[],[f170])).
% 0.19/0.28 thf(f172,plain,(
% 0.19/0.28 ( ! [X2 : $i,X1 : mu] : (($false = ((f @ X1 @ X2))) | ($false = ((exists_in_world @ X1 @ sK1))) | ($true = ((~ (rel_s4 @ sK1 @ X2))))) )),
% 0.19/0.28 inference(not_proxy_clausification,[],[f171])).
% 0.19/0.28 thf(f173,plain,(
% 0.19/0.28 ( ! [X2 : $i,X1 : mu] : (($false = ((f @ X1 @ X2))) | ($false = ((rel_s4 @ sK1 @ X2))) | ($false = ((exists_in_world @ X1 @ sK1)))) )),
% 0.19/0.28 inference(not_proxy_clausification,[],[f172])).
% 0.19/0.28 thf(f174,plain,(
% 0.19/0.28 ($true = ((~ (!! @ $i @ (^[Y0 : $i]: ((~ (!! @ mu @ (^[Y1 : mu]: ((exists_in_world @ Y1 @ Y0) => (f @ Y1 @ Y0))))) | (~ (rel_s4 @ sK1 @ Y0))))))))),
% 0.19/0.28 inference(not_proxy_clausification,[],[f161])).
% 0.19/0.28 thf(f175,plain,(
% 0.19/0.28 ($false = ((!! @ $i @ (^[Y0 : $i]: ((~ (!! @ mu @ (^[Y1 : mu]: ((exists_in_world @ Y1 @ Y0) => (f @ Y1 @ Y0))))) | (~ (rel_s4 @ sK1 @ Y0)))))))),
% 0.19/0.28 inference(not_proxy_clausification,[],[f174])).
% 0.19/0.28 thf(f176,plain,(
% 0.19/0.28 ($false = (((^[Y0 : $i]: ((~ (!! @ mu @ (^[Y1 : mu]: ((exists_in_world @ Y1 @ Y0) => (f @ Y1 @ Y0))))) | (~ (rel_s4 @ sK1 @ Y0)))) @ sK2)))),
% 0.19/0.28 inference(sigma_proxy_clausification,[],[f175])).
% 0.19/0.28 thf(f177,plain,(
% 0.19/0.28 ($false = (((~ (!! @ mu @ (^[Y0 : mu]: ((exists_in_world @ Y0 @ sK2) => (f @ Y0 @ sK2))))) | (~ (rel_s4 @ sK1 @ sK2)))))),
% 0.19/0.28 inference(beta-eta_normalization,[],[f176])).
% 0.19/0.28 thf(f178,plain,(
% 0.19/0.28 ($false = ((~ (rel_s4 @ sK1 @ sK2))))),
% 0.19/0.28 inference(or_proxy_clausification,[],[f177])).
% 0.19/0.28 thf(f179,plain,(
% 0.19/0.28 ($false = ((~ (!! @ mu @ (^[Y0 : mu]: ((exists_in_world @ Y0 @ sK2) => (f @ Y0 @ sK2)))))))),
% 0.19/0.28 inference(or_proxy_clausification,[],[f177])).
% 0.19/0.28 thf(f180,plain,(
% 0.19/0.28 ($true = ((!! @ mu @ (^[Y0 : mu]: ((exists_in_world @ Y0 @ sK2) => (f @ Y0 @ sK2))))))),
% 0.19/0.28 inference(not_proxy_clausification,[],[f179])).
% 0.19/0.28 thf(f181,plain,(
% 0.19/0.28 ( ! [X1 : mu] : (($true = (((^[Y0 : mu]: ((exists_in_world @ Y0 @ sK2) => (f @ Y0 @ sK2))) @ X1)))) )),
% 0.19/0.28 inference(pi_proxy_clausification,[],[f180])).
% 0.19/0.28 thf(f182,plain,(
% 0.19/0.28 ( ! [X1 : mu] : (($true = (((exists_in_world @ X1 @ sK2) => (f @ X1 @ sK2))))) )),
% 0.19/0.28 inference(beta-eta_normalization,[],[f181])).
% 0.19/0.28 thf(f183,plain,(
% 0.19/0.28 ( ! [X1 : mu] : (($true = ((f @ X1 @ sK2))) | ($false = ((exists_in_world @ X1 @ sK2)))) )),
% 0.19/0.28 inference(imp_proxy_clausification,[],[f182])).
% 0.19/0.28 thf(f184,plain,(
% 0.19/0.28 ($true = ((rel_s4 @ sK1 @ sK2)))),
% 0.19/0.28 inference(not_proxy_clausification,[],[f178])).
% 0.19/0.28 thf(f201,plain,(
% 0.19/0.28 ( ! [X0 : mu] : (($false = ((rel_s4 @ sK1 @ sK2))) | ($false = ((exists_in_world @ X0 @ sK2))) | ($false = $true) | ($false = ((exists_in_world @ X0 @ sK1)))) )),
% 0.19/0.28 inference(constrained_superposition,[],[f183,f173])).
% 0.19/0.28 thf(f203,plain,(
% 0.19/0.28 ( ! [X0 : mu] : (($false = ((exists_in_world @ X0 @ sK1))) | ($false = ((exists_in_world @ X0 @ sK2))) | ($false = ((rel_s4 @ sK1 @ sK2)))) )),
% 0.19/0.28 inference(trivial_inequality_removal,[],[f201])).
% 0.19/0.28 thf(f205,plain,(
% 0.19/0.28 ( ! [X0 : mu] : (($false = ((exists_in_world @ X0 @ sK1))) | ($false = ((exists_in_world @ X0 @ sK2))) | ($false = $true)) )),
% 0.19/0.28 inference(forward_demodulation,[],[f203,f184])).
% 0.19/0.28 thf(f206,plain,(
% 0.19/0.28 ( ! [X0 : mu] : (($false = ((exists_in_world @ X0 @ sK2))) | ($false = ((exists_in_world @ X0 @ sK1)))) )),
% 0.19/0.28 inference(trivial_inequality_removal,[],[f205])).
% 0.19/0.28 thf(f222,plain,(
% 0.19/0.28 ( ! [X0 : $i,X1 : $i] : (($true != $true) | (((exists_in_world @ (sK0 @ X0) @ X1)) = $true) | ($true != ((rel_s4 @ X0 @ X1)))) )),
% 0.19/0.28 inference(constrained_superposition,[],[f107,f141])).
% 0.19/0.28 thf(f224,plain,(
% 0.19/0.28 ( ! [X0 : $i,X1 : $i] : (($true != ((rel_s4 @ X0 @ X1))) | (((exists_in_world @ (sK0 @ X0) @ X1)) = $true)) )),
% 0.19/0.28 inference(trivial_inequality_removal,[],[f222])).
% 0.19/0.28 thf(f225,plain,(
% 0.19/0.28 ($true = ((exists_in_world @ (sK0 @ sK1) @ sK2))) | ($true != $true)),
% 0.19/0.28 inference(constrained_superposition,[],[f224,f184])).
% 0.19/0.28 thf(f228,plain,(
% 0.19/0.28 ($true = ((exists_in_world @ (sK0 @ sK1) @ sK2)))),
% 0.19/0.28 inference(trivial_inequality_removal,[],[f225])).
% 0.19/0.28 thf(f230,plain,(
% 0.19/0.28 ($false = ((exists_in_world @ (sK0 @ sK1) @ sK1))) | ($false = $true)),
% 0.19/0.28 inference(constrained_superposition,[],[f228,f206])).
% 0.19/0.28 thf(f234,plain,(
% 0.19/0.28 ($false = ((exists_in_world @ (sK0 @ sK1) @ sK1)))),
% 0.19/0.28 inference(trivial_inequality_removal,[],[f230])).
% 0.19/0.28 thf(f236,plain,(
% 0.19/0.28 ($false = $true)),
% 0.19/0.28 inference(forward_demodulation,[],[f234,f141])).
% 0.19/0.28 thf(f237,plain,(
% 0.19/0.28 $false),
% 0.19/0.28 inference(trivial_inequality_removal,[],[f236])).
% 0.19/0.28 % SZS output end Proof for theBenchmark
% 0.19/0.28 % (3159321)------------------------------
% 0.19/0.28 % (3159321)Version: Vampire 5.0.1 (Release build, commit 5ef7c2677 on 2026-07-16 16:54:09 +0200)
% 0.19/0.28 % (3159321)Linked with Z3 4.14.0.0 3c47fd96cf5645d0c42b2c819d9e9a84380aa721 z3-4.8.4-9178-g3c47fd96c
% 0.19/0.28 % (3159321)CaDiCaL version: 2.1.3
% 0.19/0.28 % (3159321)Termination reason: Refutation
% 0.19/0.28 % (3159321)Time elapsed: 0.006 s
% 0.19/0.28 % (3159321)Peak memory usage: 12 MB
% 0.19/0.28 % (3159321)Instructions burned: 19 (million)
% 0.19/0.28 % (3159316)Success in time 0.039 s
% 0.19/0.28 % Vampire exiting
%------------------------------------------------------------------------------