Entrants' Sample Solutions

CASC-J13

LEO-II 1.7.0
Prover9 1109a
Vampire 5.0

ProoVer 2026

GDV 2.0
GDV-LP 2.0

CASC-J13

LEO-II 1.7.0

Alexander Steen
University of Greifswald, Germany

Sample solution for `SET014^4`

% SZS output start CNFRefutation
thf(tp_complement,type,(complement: (($i>$o)>($i>$o)))).
thf(tp_disjoint,type,(disjoint: (($i>$o)>(($i>$o)>$o)))).
thf(tp_emptyset,type,(emptyset: ($i>$o))).
thf(tp_excl_union,type,(excl_union: (($i>$o)>(($i>$o)>($i>$o))))).
thf(tp_in,type,(in: ($i>(($i>$o)>$o)))).
thf(tp_intersection,type,(intersection: (($i>$o)>(($i>$o)>($i>$o))))).
thf(tp_is_a,type,(is_a: ($i>(($i>$o)>$o)))).
thf(tp_meets,type,(meets: (($i>$o)>(($i>$o)>$o)))).
thf(tp_misses,type,(misses: (($i>$o)>(($i>$o)>$o)))).
thf(tp_sK1_X,type,(sK1_X: ($i>$o))).
thf(tp_sK2_SY0,type,(sK2_SY0: ($i>$o))).
thf(tp_sK3_SY2,type,(sK3_SY2: ($i>$o))).
thf(tp_sK4_SX0,type,(sK4_SX0: $i)).
thf(tp_setminus,type,(setminus: (($i>$o)>(($i>$o)>($i>$o))))).
thf(tp_singleton,type,(singleton: ($i>($i>$o)))).
thf(tp_subset,type,(subset: (($i>$o)>(($i>$o)>$o)))).
thf(tp_union,type,(union: (($i>$o)>(($i>$o)>($i>$o))))).
thf(tp_unord_pair,type,(unord_pair: ($i>($i>($i>$o))))).
thf(complement,definition,(complement = (^[X:($i>$o),U:$i]: (~ (X@U)))), file('/home/mwisnie/Downloads/TPTP-v6.3.0/Problems/SET/SET014^4.p',complement)).
thf(disjoint,definition,(disjoint = (^[X:($i>$o),Y:($i>$o)]: (((intersection@X)@Y) = emptyset))), file('/home/mwisnie/Downloads/TPTP-v6.3.0/Problems/SET/SET014^4.p',disjoint)).
thf(emptyset,definition,(emptyset = (^[X:$i]: $false)), file('/home/mwisnie/Downloads/TPTP-v6.3.0/Problems/SET/SET014^4.p',emptyset)).
thf(excl_union,definition,(excl_union = (^[X:($i>$o),Y:($i>$o),U:$i]: (((X@U) & (~ (Y@U))) | ((~ (X@U)) & (Y@U))))), file('/home/mwisnie/Downloads/TPTP-v6.3.0/Problems/SET/SET014^4.p',excl_union)).
thf(in,definition,(in = (^[X:$i,M:($i>$o)]: (M@X))), file('/home/mwisnie/Downloads/TPTP-v6.3.0/Problems/SET/SET014^4.p',in)).
thf(intersection,definition,(intersection = (^[X:($i>$o),Y:($i>$o),U:$i]: ((X@U) & (Y@U)))), file('/home/mwisnie/Downloads/TPTP-v6.3.0/Problems/SET/SET014^4.p',intersection)).
thf(is_a,definition,(is_a = (^[X:$i,M:($i>$o)]: (M@X))), file('/home/mwisnie/Downloads/TPTP-v6.3.0/Problems/SET/SET014^4.p',is_a)).
thf(meets,definition,(meets = (^[X:($i>$o),Y:($i>$o)]: (?[U:$i]: ((X@U) & (Y@U))))), file('/home/mwisnie/Downloads/TPTP-v6.3.0/Problems/SET/SET014^4.p',meets)).
thf(misses,definition,(misses = (^[X:($i>$o),Y:($i>$o)]: (~ (?[U:$i]: ((X@U) & (Y@U)))))), file('/home/mwisnie/Downloads/TPTP-v6.3.0/Problems/SET/SET014^4.p',misses)).
thf(setminus,definition,(setminus = (^[X:($i>$o),Y:($i>$o),U:$i]: ((X@U) & (~ (Y@U))))), file('/home/mwisnie/Downloads/TPTP-v6.3.0/Problems/SET/SET014^4.p',setminus)).
thf(singleton,definition,(singleton = (^[X:$i,U:$i]: (U = X))), file('/home/mwisnie/Downloads/TPTP-v6.3.0/Problems/SET/SET014^4.p',singleton)).
thf(subset,definition,(subset = (^[X:($i>$o),Y:($i>$o)]: (![U:$i]: ((X@U) => (Y@U))))), file('/home/mwisnie/Downloads/TPTP-v6.3.0/Problems/SET/SET014^4.p',subset)).
thf(union,definition,(union = (^[X:($i>$o),Y:($i>$o),U:$i]: ((X@U) | (Y@U)))), file('/home/mwisnie/Downloads/TPTP-v6.3.0/Problems/SET/SET014^4.p',union)).
thf(unord_pair,definition,(unord_pair = (^[X:$i,Y:$i,U:$i]: ((U = X) | (U = Y)))), file('/home/mwisnie/Downloads/TPTP-v6.3.0/Problems/SET/SET014^4.p',unord_pair)).
thf(1,conjecture,(![X:($i>$o),Y:($i>$o),A:($i>$o)]: ((((subset@X)@A) & ((subset@Y)@A)) => ((subset@((union@X)@Y))@A))), file('/home/mwisnie/Downloads/TPTP-v6.3.0/Problems/SET/SET014^4.p',thm)).
thf(2,negated_conjecture,(((![X:($i>$o),Y:($i>$o),A:($i>$o)]: ((((subset@X)@A) & ((subset@Y)@A)) => ((subset@((union@X)@Y))@A)))=$false)), inference(negate_conjecture,[status(cth)],[1])).
thf(3,plain,(((![SY0:($i>$o),SY1:($i>$o)]: ((((subset@sK1_X)@SY1) & ((subset@SY0)@SY1)) => ((subset@((union@sK1_X)@SY0))@SY1)))=$false)), inference(extcnf_forall_neg,[status(esa)],[2])).
thf(4,plain,(((![SY2:($i>$o)]: ((((subset@sK1_X)@SY2) & ((subset@sK2_SY0)@SY2)) => ((subset@((union@sK1_X)@sK2_SY0))@SY2)))=$false)), inference(extcnf_forall_neg,[status(esa)],[3])).
thf(5,plain,((((((subset@sK1_X)@sK3_SY2) & ((subset@sK2_SY0)@sK3_SY2)) => ((subset@((union@sK1_X)@sK2_SY0))@sK3_SY2))=$false)), inference(extcnf_forall_neg,[status(esa)],[4])).
thf(6,plain,((((subset@sK1_X)@sK3_SY2)=$true)), inference(standard_cnf,[status(thm)],[5])).
thf(7,plain,((((subset@sK2_SY0)@sK3_SY2)=$true)), inference(standard_cnf,[status(thm)],[5])).
thf(8,plain,((((subset@((union@sK1_X)@sK2_SY0))@sK3_SY2)=$false)), inference(standard_cnf,[status(thm)],[5])).
thf(9,plain,(((~ ((subset@((union@sK1_X)@sK2_SY0))@sK3_SY2))=$true)), inference(polarity_switch,[status(thm)],[8])).
thf(10,plain,((((subset@sK2_SY0)@sK3_SY2)=$true)), inference(copy,[status(thm)],[7])).
thf(11,plain,((((subset@sK1_X)@sK3_SY2)=$true)), inference(copy,[status(thm)],[6])).
thf(12,plain,(((~ ((subset@((union@sK1_X)@sK2_SY0))@sK3_SY2))=$true)), inference(copy,[status(thm)],[9])).
thf(13,plain,(((~ (![SX0:$i]: ((~ ((sK1_X@SX0) | (sK2_SY0@SX0))) | (sK3_SY2@SX0))))=$true)), inference(unfold_def,[status(thm)],[12,complement,disjoint,emptyset,excl_union,in,intersection,is_a,meets,misses,setminus,singleton,subset,union,unord_pair])).
thf(14,plain,(((![SX0:$i]: ((~ (sK1_X@SX0)) | (sK3_SY2@SX0)))=$true)), inference(unfold_def,[status(thm)],[11,complement,disjoint,emptyset,excl_union,in,intersection,is_a,meets,misses,setminus,singleton,subset,union,unord_pair])).
thf(15,plain,(((![SX0:$i]: ((~ (sK2_SY0@SX0)) | (sK3_SY2@SX0)))=$true)), inference(unfold_def,[status(thm)],[10,complement,disjoint,emptyset,excl_union,in,intersection,is_a,meets,misses,setminus,singleton,subset,union,unord_pair])).
thf(16,plain,(((![SX0:$i]: ((~ ((sK1_X@SX0) | (sK2_SY0@SX0))) | (sK3_SY2@SX0)))=$false)), inference(extcnf_not_pos,[status(thm)],[13])).
thf(17,plain,(![SV1:$i]: ((((~ (sK1_X@SV1)) | (sK3_SY2@SV1))=$true))), inference(extcnf_forall_pos,[status(thm)],[14])).
thf(18,plain,(![SV2:$i]: ((((~ (sK2_SY0@SV2)) | (sK3_SY2@SV2))=$true))), inference(extcnf_forall_pos,[status(thm)],[15])).
thf(19,plain,((((~ ((sK1_X@sK4_SX0) | (sK2_SY0@sK4_SX0))) | (sK3_SY2@sK4_SX0))=$false)), inference(extcnf_forall_neg,[status(esa)],[16])).
thf(20,plain,(![SV1:$i]: (((~ (sK1_X@SV1))=$true) | ((sK3_SY2@SV1)=$true))), inference(extcnf_or_pos,[status(thm)],[17])).
thf(21,plain,(![SV2:$i]: (((~ (sK2_SY0@SV2))=$true) | ((sK3_SY2@SV2)=$true))), inference(extcnf_or_pos,[status(thm)],[18])).
thf(22,plain,(((~ ((sK1_X@sK4_SX0) | (sK2_SY0@sK4_SX0)))=$false)), inference(extcnf_or_neg,[status(thm)],[19])).
thf(23,plain,(((sK3_SY2@sK4_SX0)=$false)), inference(extcnf_or_neg,[status(thm)],[19])).
thf(24,plain,(![SV1:$i]: (((sK1_X@SV1)=$false) | ((sK3_SY2@SV1)=$true))), inference(extcnf_not_pos,[status(thm)],[20])).
thf(25,plain,(![SV2:$i]: (((sK2_SY0@SV2)=$false) | ((sK3_SY2@SV2)=$true))), inference(extcnf_not_pos,[status(thm)],[21])).
thf(26,plain,((((sK1_X@sK4_SX0) | (sK2_SY0@sK4_SX0))=$true)), inference(extcnf_not_neg,[status(thm)],[22])).
thf(27,plain,(((sK1_X@sK4_SX0)=$true) | ((sK2_SY0@sK4_SX0)=$true)), inference(extcnf_or_pos,[status(thm)],[26])).
thf(28,plain,((($false)=$true)), inference(fo_atp_e,[status(thm)],[23,27,25,24])).
thf(29,plain,($false), inference(solved_all_splits,[solved_all_splits(join,[])],[28])).
% SZS output end CNFRefutation

Prover9 1109a

William McCune, Bob Veroff
University of New Mexico, USA

Solution for `SEU140+2`

8 (all A all B (subset(A,B) <-> (all C (in(C,A) -> in(C,B))))) # label(d3_tarski) # label(axiom) # label(non_clause).  [assumption].
26 (all A all B (disjoint(A,B) -> disjoint(B,A))) # label(symmetry_r1_xboole_0) # label(axiom) # label(non_clause).  [assumption].
42 (all A all B (-(-disjoint(A,B) & (all C -(in(C,A) & in(C,B)))) & -((exists C (in(C,A) & in(C,B))) & disjoint(A,B)))) # label(t3_xboole_0) # label(lemma) # label(non_clause).  [assumption].
55 -(all A all B all C (subset(A,B) & disjoint(B,C) -> disjoint(A,C))) # label(t63_xboole_1) # label(negated_conjecture) # label(non_clause).  [assumption].
60 subset(c3,c4) # label(t63_xboole_1) # label(negated_conjecture).  [clausify(55)].
61 disjoint(c4,c5) # label(t63_xboole_1) # label(negated_conjecture).  [clausify(55)].
75 disjoint(A,B) | in(f7(A,B),A) # label(t3_xboole_0) # label(lemma).  [clausify(42)].
76 disjoint(A,B) | in(f7(A,B),B) # label(t3_xboole_0) # label(lemma).  [clausify(42)].
92 -disjoint(c3,c5) # label(t63_xboole_1) # label(negated_conjecture).  [clausify(55)].
101 -in(A,B) | -in(A,C) | -disjoint(B,C) # label(t3_xboole_0) # label(lemma).  [clausify(42)].
109 -disjoint(A,B) | disjoint(B,A) # label(symmetry_r1_xboole_0) # label(axiom).  [clausify(26)].
123 -subset(A,B) | -in(C,A) | in(C,B) # label(d3_tarski) # label(axiom).  [clausify(8)].
273 -disjoint(c5,c3).  [ur(109,b,92,a)].
300 -in(A,c3) | in(A,c4).  [resolve(123,a,60,a)].
959 in(f7(c5,c3),c3).  [resolve(273,a,76,a)].
960 in(f7(c5,c3),c5).  [resolve(273,a,75,a)].
1084 -in(f7(c5,c3),c4).  [ur(101,b,960,a,c,61,a)].
1292 $F.  [resolve(300,a,959,a),unit_del(a,1084)].

Vampire 5.0

Michael Rawson
University of Southampton, United Kongdom

Notes regarding saturations

Vampire can testify (counter)-satisfiability of a given problem by finitely saturating the corresponding preprocessed clause set (using a complete version of a calculus). It then reports SZS Status Satisfiable. As supporting evidence, Vampire prints two artefacts: The saturated clause set itself between SZS output start Saturation and SZS output end Saturation, and a section of "Definitions and Model Updates". Among the preprocessing steps used by Vampire in order to transform an arbitrary first-order problem into the CNF on which saturation starts are some steps (we call them interferences) which only preserve model existence, but not all models, or which modify the signature. Each of these steps comes with a model-theoretic argument of the form: "If you give me a model of the post-step F, this is what you must do to get a model of pre-step F". The "Definitions and Model Updates" section lists these transformations in the order in which they should be applied to the model of the final CNF (that just got saturated) in order to arrive at a model of the original input problem. These transformations are implemented in Vampire already to work on finite models found by its finite model finder, but since the model represented by finite saturations is only implicit, we do our best to at least report what transformations have been recorded and should be played back. Here is an explanation for the transformations implemented (so far):

Eliminated symbols are defined in terms of their original definitions. E.g. for predicates:

for all inputs,
    define r(X0,X1) := p(X0,b)

or for functions (constants):

for all inputs,
    define a := aa

Partially eliminated definitions (e.g., when Vampire turns

![X:ta] : (~qq(X) <=> ~(q(X) & (?[X:ta] : q(X))))

into

![X:ta] : (~qq(X) <= ~(q(X) & (?[X:ta] : q(X))))

because all the remaining occurrences of qq are positive.) may require conditional atom flips:

for all groundings,
    whenever ? [X1 : ta] : q(X1) & q(X0) is true, set qq(X0) to true

Here "for all groundings" suggests a Herbrand model, but the meaning is the same as "for all inputs".

We also have globally flip the polarity of every occurrence of a predicate, e.g., '$ki_accessible', which means: "negate the whole truth table for the corresponding predicate" and complements the optional random_polarities preprocessing step ("As part of preprocessing, randomly (though consistently) flip polarities of non-equality predicates in the whole CNF.").

To complement blocked clause elimination (https://easychair.org/publications/paper/Bl2v/open) we may introduce a more complicated version of a conditional flip. E.g.,
```
for all groundings, until fixed point,
    whenever teach(X1,X0,cs) | ~'$ki_accessible'('$ki_local_world',X1) | ~teach('$ki_local_world',X0,psych) | ~'$ki_exists_in_world_$i'('$ki_local_world',X0) is false, set teach(X1,X0,cs) to true 
```
The "until fixed point" bit is necessary and explained in Lemma 4 of the BCE paper. In a nutshell, this means: if you go over your groundings and flip something, you have to then restart and go from the beginning again. In the finite model building version of this (which actually does the changes as described) we use a do-while loop (https://github.com/vprover/vampire/blob/0dce69acaa3452831bf8f851485da9266aa02b57/FMB/FiniteModelMultiSorted.cpp#L1093)

Solution for `SET014^4`

thf(func_def_0, type, in: $i > ($i > $o) > $o).
thf(func_def_2, type, is_a: $i > ($i > $o) > $o).
thf(func_def_3, type, emptyset: $i > $o).
thf(func_def_4, type, unord_pair: $i > $i > $i > $o).
thf(func_def_5, type, singleton: $i > $i > $o).
thf(func_def_6, type, union: ($i > $o) > ($i > $o) > $i > $o).
thf(func_def_7, type, excl_union: ($i > $o) > ($i > $o) > $i > $o).
thf(func_def_8, type, intersection: ($i > $o) > ($i > $o) > $i > $o).
thf(func_def_9, type, setminus: ($i > $o) > ($i > $o) > $i > $o).
thf(func_def_10, type, complement: ($i > $o) > $i > $o).
thf(func_def_11, type, disjoint: ($i > $o) > ($i > $o) > $o).
thf(func_def_12, type, subset: ($i > $o) > ($i > $o) > $o).
thf(func_def_13, type, meets: ($i > $o) > ($i > $o) > $o).
thf(func_def_14, type, misses: ($i > $o) > ($i > $o) > $o).
thf(func_def_28, type, sK0: $i > $o).
thf(func_def_29, type, sK1: $i > $o).
thf(func_def_30, type, sK2: $i > $o).
thf(f113,plain,(
  $false),
  inference(avatar_sat_refutation,[],[f92,f106,f112])).
thf(f112,plain,(
  ~spl3_1),
  inference(avatar_contradiction_clause,[],[f111])).
thf(f111,plain,(
  $false | ~spl3_1),
  inference(trivial_inequality_removal,[],[f107])).
thf(f107,plain,(
  ($true = $false) | ~spl3_1),
  inference(superposition,[],[f87,f96])).
thf(f96,plain,(
  ($false = (sK2 @ sK4))),
  inference(trivial_inequality_removal,[],[f94])).
thf(f94,plain,(
  ($true = $false) | ($false = (sK2 @ sK4))),
  inference(superposition,[],[f79,f73])).
thf(f73,plain,(
  ((sK1 @ sK4) = $false)),
  inference(binary_proxy_clausification,[],[f72])).
thf(f72,plain,(
  ((((sK2 @ sK4) | (sK0 @ sK4)) => (sK1 @ sK4)) = $false)),
  inference(beta_eta_normalization,[],[f71])).
thf(f71,plain,(
  ($false = ((^[Y0 : $i]: (((sK2 @ Y0) | (sK0 @ Y0)) => (sK1 @ Y0))) @ sK4))),
  inference(sigma_clausification,[],[f70])).
thf(f70,plain,(
  ($true != (!! @ $i @ (^[Y0 : $i]: (((sK2 @ Y0) | (sK0 @ Y0)) => (sK1 @ Y0)))))),
  inference(beta_eta_normalization,[],[f67])).
thf(f67,plain,(
  ($true != ((^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: (!! @ $i @ (^[Y2 : $i]: ((Y0 @ Y2) => (Y1 @ Y2))))))) @ ((^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: ((Y0 @ Y2) | (Y1 @ Y2))))))) @ sK2 @ sK0) @ sK1))),
  inference(definition_unfolding,[],[f59,f52,f60])).
thf(f60,plain,(
  (union = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: ((Y0 @ Y2) | (Y1 @ Y2))))))))),
  inference(cnf_transformation,[],[f28])).
thf(f28,plain,(
  (union = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: ((Y0 @ Y2) | (Y1 @ Y2))))))))),
  inference(fool_elimination,[],[f27])).
thf(f27,plain,(
  ((^[X0 : $i > $o, X1 : $i > $o, X2 : $i] : ((X1 @ X2) | (X0 @ X2))) = union)),
  inference(rectify,[],[f6])).
thf(f6,axiom,(
  ((^[X0 : $i > $o, X2 : $i > $o, X3 : $i] : ((X2 @ X3) | (X0 @ X3))) = union)),
  file('Problems/SET/SET014^4.p',union)).
thf(f52,plain,(
  (subset = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: (!! @ $i @ (^[Y2 : $i]: ((Y0 @ Y2) => (Y1 @ Y2))))))))),
  inference(cnf_transformation,[],[f36])).
thf(f36,plain,(
  (subset = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: (!! @ $i @ (^[Y2 : $i]: ((Y0 @ Y2) => (Y1 @ Y2))))))))),
  inference(fool_elimination,[],[f35])).
thf(f35,plain,(
  (subset = (^[X0 : $i > $o, X1 : $i > $o] : (! [X2] : ((X0 @ X2) => (X1 @ X2)))))),
  inference(rectify,[],[f12])).
thf(f12,axiom,(
  (subset = (^[X0 : $i > $o, X2 : $i > $o] : (! [X3] : ((X0 @ X3) => (X2 @ X3)))))),
  file('Problems/SET/SET014^4.p',subset)).
thf(f59,plain,(
  ((subset @ (union @ sK2 @ sK0) @ sK1) != $true)),
  inference(cnf_transformation,[],[f48])).
thf(f48,plain,(
  ((subset @ (union @ sK2 @ sK0) @ sK1) != $true) & ($true = (subset @ sK0 @ sK1)) & ($true = (subset @ sK2 @ sK1))),
  inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2])],[f46,f47])).
thf(f47,plain,(
  ? [X0 : $i > $o,X1 : $i > $o,X2 : $i > $o] : (($true != (subset @ (union @ X2 @ X0) @ X1)) & ($true = (subset @ X0 @ X1)) & ($true = (subset @ X2 @ X1))) => (((subset @ (union @ sK2 @ sK0) @ sK1) != $true) & ($true = (subset @ sK0 
@ sK1)) & ($true = (subset @ sK2 @ sK1)))),
  introduced(choice_axiom,[])).
thf(f46,plain,(
  ? [X0 : $i > $o,X1 : $i > $o,X2 : $i > $o] : (($true != (subset @ (union @ X2 @ X0) @ X1)) & ($true = (subset @ X0 @ X1)) & ($true = (subset @ X2 @ X1)))),
  inference(flattening,[],[f45])).
thf(f45,plain,(
  ? [X0 : $i > $o,X1 : $i > $o,X2 : $i > $o] : (($true != (subset @ (union @ X2 @ X0) @ X1)) & (($true = (subset @ X2 @ X1)) & ($true = (subset @ X0 @ X1))))),
  inference(ennf_transformation,[],[f30])).
thf(f30,plain,(
  ~! [X0 : $i > $o,X1 : $i > $o,X2 : $i > $o] : ((($true = (subset @ X2 @ X1)) & ($true = (subset @ X0 @ X1))) => ($true = (subset @ (union @ X2 @ X0) @ X1)))),
  inference(fool_elimination,[],[f29])).
thf(f29,plain,(
  ~! [X0 : $i > $o,X1 : $i > $o,X2 : $i > $o] : (((subset @ X2 @ X1) & (subset @ X0 @ X1)) => (subset @ (union @ X2 @ X0) @ X1))),
  inference(rectify,[],[f16])).
thf(f16,negated_conjecture,(
  ~! [X2 : $i > $o,X4 : $i > $o,X0 : $i > $o] : (((subset @ X0 @ X4) & (subset @ X2 @ X4)) => (subset @ (union @ X0 @ X2) @ X4))),
  inference(negated_conjecture,[],[f15])).
thf(f15,conjecture,(
  ! [X2 : $i > $o,X4 : $i > $o,X0 : $i > $o] : (((subset @ X0 @ X4) & (subset @ X2 @ X4)) => (subset @ (union @ X0 @ X2) @ X4))),
  file('Problems/SET/SET014^4.p',thm)).
thf(f79,plain,(
  ( ! [X1 : $i] : (($true = (sK1 @ X1)) | ((sK2 @ X1) = $false)) )),
  inference(binary_proxy_clausification,[],[f78])).
thf(f78,plain,(
  ( ! [X1 : $i] : (($true = ((sK2 @ X1) => (sK1 @ X1)))) )),
  inference(beta_eta_normalization,[],[f77])).
thf(f77,plain,(
  ( ! [X1 : $i] : (($true = ((^[Y0 : $i]: ((sK2 @ Y0) => (sK1 @ Y0))) @ X1))) )),
  inference(pi_clausification,[],[f76])).
thf(f76,plain,(
  ($true = (!! @ $i @ (^[Y0 : $i]: ((sK2 @ Y0) => (sK1 @ Y0)))))),
  inference(beta_eta_normalization,[],[f69])).
thf(f69,plain,(
  ($true = ((^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: (!! @ $i @ (^[Y2 : $i]: ((Y0 @ Y2) => (Y1 @ Y2))))))) @ sK2 @ sK1))),
  inference(definition_unfolding,[],[f57,f52])).
thf(f57,plain,(
  ($true = (subset @ sK2 @ sK1))),
  inference(cnf_transformation,[],[f48])).
thf(f87,plain,(
  ($true = (sK2 @ sK4)) | ~spl3_1),
  inference(avatar_component_clause,[],[f85])).
thf(f85,plain,(
  spl3_1 <=> ($true = (sK2 @ sK4))),
  introduced(avatar_definition,[new_symbols(naming,[spl3_1])])).
thf(f106,plain,(
  ~spl3_2),
  inference(avatar_contradiction_clause,[],[f105])).
thf(f105,plain,(
  $false | ~spl3_2),
  inference(trivial_inequality_removal,[],[f101])).
thf(f101,plain,(
  ($true = $false) | ~spl3_2),
  inference(superposition,[],[f100,f91])).
thf(f91,plain,(
  ($true = (sK0 @ sK4)) | ~spl3_2),
  inference(avatar_component_clause,[],[f89])).
thf(f89,plain,(
  spl3_2 <=> ($true = (sK0 @ sK4))),
  introduced(avatar_definition,[new_symbols(naming,[spl3_2])])).
thf(f100,plain,(
  ((sK0 @ sK4) = $false)),
  inference(trivial_inequality_removal,[],[f97])).
thf(f97,plain,(
  ($true = $false) | ((sK0 @ sK4) = $false)),
  inference(superposition,[],[f83,f73])).
thf(f83,plain,(
  ( ! [X1 : $i] : (($true = (sK1 @ X1)) | ($false = (sK0 @ X1))) )),
  inference(binary_proxy_clausification,[],[f82])).
thf(f82,plain,(
  ( ! [X1 : $i] : (($true = ((sK0 @ X1) => (sK1 @ X1)))) )),
  inference(beta_eta_normalization,[],[f81])).
thf(f81,plain,(
  ( ! [X1 : $i] : (($true = ((^[Y0 : $i]: ((sK0 @ Y0) => (sK1 @ Y0))) @ X1))) )),
  inference(pi_clausification,[],[f80])).
thf(f80,plain,(
  ($true = (!! @ $i @ (^[Y0 : $i]: ((sK0 @ Y0) => (sK1 @ Y0)))))),
  inference(beta_eta_normalization,[],[f68])).
thf(f68,plain,(
  ($true = ((^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: (!! @ $i @ (^[Y2 : $i]: ((Y0 @ Y2) => (Y1 @ Y2))))))) @ sK0 @ sK1))),
  inference(definition_unfolding,[],[f58,f52])).
thf(f58,plain,(
  ($true = (subset @ sK0 @ sK1))),
  inference(cnf_transformation,[],[f48])).
thf(f92,plain,(
  spl3_1 | spl3_2),
  inference(avatar_split_clause,[],[f75,f89,f85])).
thf(f75,plain,(
  ($true = (sK2 @ sK4)) | ($true = (sK0 @ sK4))),
  inference(binary_proxy_clausification,[],[f74])).
thf(f74,plain,(
  ($true = ((sK2 @ sK4) | (sK0 @ sK4)))),
  inference(binary_proxy_clausification,[],[f72])).

Solution for `DAT013_1`

% SUCCESS: Derivation has unique formula names
% SUCCESS: All derived formulae have parents and inference information
% WARNING: Took the first false root f43 as the single derivation root
% SUCCESS: Derivation is acyclic
% SUCCESS: Derivation looks like a refutation
% SUCCESS: Negated conjecture f4 is a leaf or CTH from a conjecture
% SUCCESS: Assumptions are propagated
% SUCCESS: Assumptions are discharged
%  NOTICE: Took the conjecture f3 as the proved formula
% CPUTIME: 0.06
% SUCCESS: Verified
% SZS status Verified

tff(type_def_5, type, array: $tType).
tff(func_def_0, type, read: (array * $int) > $int).
tff(func_def_1, type, write: (array * $int * $int) > array).
tff(func_def_5, type, sK0: array).
tff(func_def_6, type, sK1: $int).
tff(func_def_7, type, sK2: $int).
tff(func_def_8, type, sK3: $int).
tff(func_def_9, type, -1: $int > $int).
tff(f43,plain,(
  $false),
  inference(avatar_sat_refutation,[],[f35,f38,f42])).
tff(f42,plain,(
  ~spl4_1),
  inference(avatar_contradiction_clause,[],[f41])).
tff(f41,plain,(
  $false | ~spl4_1),
  inference(alasca_normalization,[],[f40])).
tff(f40,plain,(
  $greater($sum($sum($uminus(sK3),$sum(-2,sK3)),-1),0) | ~spl4_1),
  inference(alasca_fourier_motzkin,[],[f30,f20])).
tff(f20,plain,(
  $greater($sum(-1(sK1),$sum(sK3,-2)),0)),
  inference(alasca_normalization,[],[f16])).
tff(f16,plain,(
  ~$less(sK3,$sum(sK1,3))),
  inference(cnf_transformation,[],[f12])).
tff(f12,plain,(
  (~$less(0,read(sK0,sK3)) & ~$less(sK2,sK3) & ~$less(sK3,$sum(sK1,3))) & ! [X4 : $int] : ($less(0,read(sK0,X4)) | $less(sK2,X4) | $less(X4,sK1))),
  inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2,sK3])],[f9,f11,f10])).
tff(f10,plain,(
  ? [X0 : array,X1 : $int,X2 : $int] : (? [X3 : $int] : (~$less(0,read(X0,X3)) & ~$less(X2,X3) & ~$less(X3,$sum(X1,3))) & ! [X4 : $int] : ($less(0,read(X0,X4)) | $less(X2,X4) | $less(X4,X1))) => (? [X3 : $int] : (~$less(0,read(sK0,X3)) & ~
$less(sK2,X3) & ~$less(X3,$sum(sK1,3))) & ! [X4 : $int] : ($less(0,read(sK0,X4)) | $less(sK2,X4) | $less(X4,sK1)))),
  introduced(definition,[],[choice_axiom])).
tff(f11,plain,(
  ? [X3 : $int] : (~$less(0,read(sK0,X3)) & ~$less(sK2,X3) & ~$less(X3,$sum(sK1,3))) => (~$less(0,read(sK0,sK3)) & ~$less(sK2,sK3) & ~$less(sK3,$sum(sK1,3)))),
  introduced(definition,[],[choice_axiom])).
tff(f9,plain,(
  ? [X0 : array,X1 : $int,X2 : $int] : (? [X3 : $int] : (~$less(0,read(X0,X3)) & ~$less(X2,X3) & ~$less(X3,$sum(X1,3))) & ! [X4 : $int] : ($less(0,read(X0,X4)) | $less(X2,X4) | $less(X4,X1)))),
  inference(rectify,[],[f8])).
tff(f8,plain,(
  ? [X0 : array,X1 : $int,X2 : $int] : (? [X4 : $int] : (~$less(0,read(X0,X4)) & ~$less(X2,X4) & ~$less(X4,$sum(X1,3))) & ! [X3 : $int] : ($less(0,read(X0,X3)) | $less(X2,X3) | $less(X3,X1)))),
  inference(flattening,[],[f7])).
tff(f7,plain,(
  ? [X0 : array,X1 : $int,X2 : $int] : (? [X4 : $int] : (~$less(0,read(X0,X4)) & (~$less(X2,X4) & ~$less(X4,$sum(X1,3)))) & ! [X3 : $int] : ($less(0,read(X0,X3)) | ($less(X2,X3) | $less(X3,X1))))),
  inference(ennf_transformation,[],[f5])).
tff(f5,plain,(
  ~! [X0 : array,X1 : $int,X2 : $int] : (! [X3 : $int] : ((~$less(X2,X3) & ~$less(X3,X1)) => $less(0,read(X0,X3))) => ! [X4 : $int] : ((~$less(X2,X4) & ~$less(X4,$sum(X1,3))) => $less(0,read(X0,X4))))),
  inference(theory_normalization,[],[f4])).
tff(f4,negated_conjecture,(
  ~! [X0 : array,X1 : $int,X2 : $int] : (! [X3 : $int] : (($lesseq(X3,X2) & $lesseq(X1,X3)) => $greater(read(X0,X3),0)) => ! [X4 : $int] : (($lesseq(X4,X2) & $lesseq($sum(X1,3),X4)) => $greater(read(X0,X4),0)))),
  inference(negated_conjecture,[status(cth)],[f3])).
tff(f3,conjecture,(
  ! [X0 : array,X1 : $int,X2 : $int] : (! [X3 : $int] : (($lesseq(X3,X2) & $lesseq(X1,X3)) => $greater(read(X0,X3),0)) => ! [X4 : $int] : (($lesseq(X4,X2) & $lesseq($sum(X1,3),X4)) => $greater(read(X0,X4),0)))),
  file('Problems/DAT/DAT013_1.p',unknown)).
tff(f30,plain,(
  $greater($sum(sK1,-1(sK3)),0) | ~spl4_1),
  inference(avatar_component_clause,[],[f28])).
tff(f28,definition,(
  spl4_1 <=> $greater($sum(sK1,-1(sK3)),0)),
  introduced(definition,[new_symbols(naming,[spl4_1])],[avatar_definition])).
tff(f38,plain,(
  ~spl4_2),
  inference(avatar_contradiction_clause,[],[f37])).
tff(f37,plain,(
  $false | ~spl4_2),
  inference(alasca_normalization,[],[f36])).
tff(f36,plain,(
  $greater($sum($sum($sum(1,$uminus(sK3)),sK3),-1),0) | ~spl4_2),
  inference(alasca_fourier_motzkin,[],[f21,f34])).
tff(f34,plain,(
  $greater($sum(-1(sK2),sK3),0) | ~spl4_2),
  inference(avatar_component_clause,[],[f32])).
tff(f32,definition,(
  spl4_2 <=> $greater($sum(-1(sK2),sK3),0)),
  introduced(definition,[new_symbols(naming,[spl4_2])],[avatar_definition])).
tff(f21,plain,(
  $greater($sum(sK2,$sum(-1(sK3),1)),0)),
  inference(alasca_normalization,[],[f17])).
tff(f17,plain,(
  ~$less(sK2,sK3)),
  inference(cnf_transformation,[],[f12])).
tff(f35,plain,(
  spl4_1 | spl4_2),
  inference(avatar_split_clause,[],[f26,f32,f28])).
tff(f26,plain,(
  $greater($sum(-1(sK2),sK3),0) | $greater($sum(sK1,-1(sK3)),0)),
  inference(alasca_normalization,[],[f25])).
tff(f25,plain,(
  $greater($sum(sK3,-1(sK2)),0) | $greater($sum(-1(sK3),sK1),0) | $greater($sum(1,-1),0)),
  inference(alasca_fourier_motzkin,[],[f19,f22])).
tff(f22,plain,(
  $greater($sum(1,-1(read(sK0,sK3))),0)),
  inference(alasca_normalization,[],[f18])).
tff(f18,plain,(
  ~$less(0,read(sK0,sK3))),
  inference(cnf_transformation,[],[f12])).
tff(f19,plain,(
  ( ! [X4 : $int] : ($greater(read(sK0,X4),0) | $greater($sum(X4,-1(sK2)),0) | $greater($sum(-1(X4),sK1),0)) )),
  inference(alasca_normalization,[],[f15])).
tff(f15,plain,(
  ( ! [X4 : $int] : ($less(0,read(sK0,X4)) | $less(sK2,X4) | $less(X4,sK1)) )),
  inference(cnf_transformation,[],[f12])).

Saturation for `DAT335_2`

% (3727607)# SZS output start Saturation.
cnf(u30,negated_conjecture,
    '$ki_accessible'('$ki_local_world',X1) | teach(X1,X0,cs) | ~teach('$ki_local_world',X0,psych) | ~'$ki_exists_in_world_$i'('$ki_local_world',X0)).

cnf(u25,axiom,
    teach(X0,sue,psych) | '$ki_accessible'('$ki_local_world',X0)).

cnf(u24,axiom,
    ~'$ki_accessible'(X0,X0)).

cnf(u27,axiom,
    teach(X0,sK1(X0),cs) | '$ki_accessible'('$ki_local_world',X0)).

cnf(u16,axiom,
    '$ki_exists_in_world_$i'(X0,X1)).

cnf(u26,axiom,
    teach(X0,mary,psych) | '$ki_accessible'('$ki_local_world',X0)).

cnf(u29,axiom,
    teach(X0,john,math) | '$ki_accessible'('$ki_local_world',X0)).

cnf(u31,negated_conjecture,
    teach('$ki_local_world',X0,cs) | ~teach('$ki_local_world',X0,psych) | ~'$ki_exists_in_world_$i'('$ki_local_world',X0)).

% (3727607)# SZS output end Saturation.
% (3727607)# SZS output start Definitions and Model Updates.
globally flip the polarity of every occurrence of predicate "'$ki_accessible'"
% (3727607)# SZS output end Definitions and Model Updates.

Finite Model for `DAT335_2`

tff('declare_$i1',type,'fmb_$i_1':$i).
tff('finite_domain_$i',axiom,
      ! [X:$i] : (
         X = 'fmb_$i_1'
      ) ).

tff('declare_$ki_world',type,'$ki_world':$tType).
tff('declare_$ki_world1',type,'fmb_$ki_world_1':'$ki_world').
tff('finite_domain_$ki_world',axiom,
      ! [X:'$ki_world'] : (
         X = 'fmb_$ki_world_1'
      ) ).

tff('declare_$ki_local_world',type,'$ki_local_world':'$ki_world').
tff('$ki_local_world_definition',axiom,'$ki_local_world' = 'fmb_$ki_world_1').
tff(declare_cs,type,cs:$i).
tff(cs_definition,axiom,cs = 'fmb_$i_1').
tff(declare_sue,type,sue:$i).
tff(sue_definition,axiom,sue = 'fmb_$i_1').
tff(declare_mary,type,mary:$i).
tff(mary_definition,axiom,mary = 'fmb_$i_1').
tff(declare_john,type,john:$i).
tff(john_definition,axiom,john = 'fmb_$i_1').
tff(declare_math,type,math:$i).
tff(math_definition,axiom,math = 'fmb_$i_1').
tff(declare_psych,type,psych:$i).
tff(psych_definition,axiom,psych = 'fmb_$i_1').
tff('declare_$ki_accessible',type,'$ki_accessible': ('$ki_world' * '$ki_world') > $o).
tff('predicate_$ki_accessible',axiom,
           '$ki_accessible'('fmb_$ki_world_1','fmb_$ki_world_1')

).

tff(declare_teach,type,teach: ('$ki_world' * $i * $i) > $o).
tff(predicate_teach,axiom,
           teach('fmb_$ki_world_1','fmb_$i_1','fmb_$i_1')

).

tff('declare_$ki_exists_in_world_$i',type,'$ki_exists_in_world_$i': ('$ki_world' * $i) > $o).
tff('predicate_$ki_exists_in_world_$i',axiom,
           '$ki_exists_in_world_$i'('fmb_$ki_world_1','fmb_$i_1')

).

Saturation for `SWW469_10`

% # SZS output start Saturation.
cnf(u15,hypothesis,
    sK0 != sK1).

cnf(u14,hypothesis,
    hoare_1310879719gleton).

% # SZS output end Saturation.
% # SZS output start Definitions and Model Updates.
for all inputs,
    define induct_false := $false
for all inputs,
    define induct_true := $true
for all groundings,
    whenever ? [X0 : state,X1 : state] : X0 != X1 is true, set hoare_1310879719gleton to true
% # SZS output end Definitions and Model Updates.

Finite Model for `SWW469_10`

tff('declare_$i1',type,'fmb_$i_1':$i).
tff('finite_domain_$i',axiom,
      ! [X:$i] : (
         X = 'fmb_$i_1'
      ) ).

tff(declare_state,type,state:$tType).
tff(declare_state1,type,fmb_state_1:state).
tff(declare_state2,type,fmb_state_2:state).
tff(finite_domain_state,axiom,
      ! [X:state] : (
         X = fmb_state_1 | X = fmb_state_2
      ) ).

tff(distinct_domain_state,axiom,
         fmb_state_1 != fmb_state_2
).

tff(declare_induct_false,type,induct_false: $o).
tff(induct_false_definition,axiom,~induct_false).
tff(declare_induct_true,type,induct_true: $o).
tff(induct_true_definition,axiom,induct_true).
tff(declare_hoare_1310879719gleton,type,hoare_1310879719gleton: $o).
tff(hoare_1310879719gleton_definition,axiom,hoare_1310879719gleton).

Solution for `SEU140+2`

% SUCCESS: Derivation has unique formula names
% SUCCESS: All derived formulae have parents and inference information
% WARNING: Took the first false root f1401 as the single derivation root
% SUCCESS: Derivation is acyclic
% SUCCESS: Derivation looks like a refutation
% SUCCESS: Negated conjecture f52 is a leaf or CTH from a conjecture
% SUCCESS: Assumptions are propagated
% SUCCESS: Assumptions are discharged
%  NOTICE: Took the conjecture f51 as the proved formula
% CPUTIME: 0.07
% SUCCESS: Verified
% SZS status Verified

fof(f1401,plain,(
  $false),
  inference(subsumption_resolution,[],[f1400,f210])).
fof(f210,plain,(
  ~disjoint(sK10,sK12)),
  inference(cnf_transformation,[],[f134])).
fof(f134,plain,(
  ~disjoint(sK10,sK12) & disjoint(sK11,sK12) & subset(sK10,sK11)),
  inference(skolemisation,[status(esa),new_symbols(skolem,[sK10,sK11,sK12])],[f88,f133])).
fof(f133,plain,(
  ? [X0,X1,X2] : (~disjoint(X0,X2) & disjoint(X1,X2) & subset(X0,X1)) => (~disjoint(sK10,sK12) & disjoint(sK11,sK12) & subset(sK10,sK11))),
  introduced(definition,[],[choice_axiom])).
fof(f88,plain,(
  ? [X0,X1,X2] : (~disjoint(X0,X2) & disjoint(X1,X2) & subset(X0,X1))),
  inference(flattening,[],[f87])).
fof(f87,plain,(
  ? [X0,X1,X2] : (~disjoint(X0,X2) & (disjoint(X1,X2) & subset(X0,X1)))),
  inference(ennf_transformation,[],[f52])).
fof(f52,negated_conjecture,(
  ~! [X0,X1,X2] : ((disjoint(X1,X2) & subset(X0,X1)) => disjoint(X0,X2))),
  inference(negated_conjecture,[status(cth)],[f51])).
fof(f51,conjecture,(
  ! [X0,X1,X2] : ((disjoint(X1,X2) & subset(X0,X1)) => disjoint(X0,X2))),
  file('Problems/SEU/SEU140+2.p',unknown)).
fof(f1400,plain,(
  disjoint(sK10,sK12)),
  inference(resolution,[],[f1383,f179])).
fof(f179,plain,(
  ( ! [X0,X1] : (~disjoint(X0,X1) | disjoint(X1,X0)) )),
  inference(cnf_transformation,[],[f72])).
fof(f72,plain,(
  ! [X0,X1] : (disjoint(X1,X0) | ~disjoint(X0,X1))),
  inference(ennf_transformation,[],[f27])).
fof(f27,axiom,(
  ! [X0,X1] : (disjoint(X0,X1) => disjoint(X1,X0))),
  file('Problems/SEU/SEU140+2.p',unknown)).
fof(f1383,plain,(
  disjoint(sK12,sK10)),
  inference(duplicate_literal_removal,[],[f1380])).
fof(f1380,plain,(
  disjoint(sK12,sK10) | disjoint(sK12,sK10)),
  inference(resolution,[],[f510,f402])).
fof(f402,plain,(
  ( ! [X0] : (in(sK8(X0,sK10),sK11) | disjoint(X0,sK10)) )),
  inference(resolution,[],[f389,f198])).
fof(f198,plain,(
  ( ! [X0,X1] : (in(sK8(X0,X1),X1) | disjoint(X0,X1)) )),
  inference(cnf_transformation,[],[f130])).
fof(f130,plain,(
  ! [X0,X1] : ((~disjoint(X0,X1) | ! [X2] : (~in(X2,X1) | ~in(X2,X0))) & ((in(sK8(X0,X1),X1) & in(sK8(X0,X1),X0)) | disjoint(X0,X1)))),
  inference(skolemisation,[status(esa),new_symbols(skolem,[sK8])],[f82,f129])).
fof(f129,plain,(
  ! [X0,X1] : (? [X3] : (in(X3,X1) & in(X3,X0)) => (in(sK8(X0,X1),X1) & in(sK8(X0,X1),X0)))),
  introduced(definition,[],[choice_axiom])).
fof(f82,plain,(
  ! [X0,X1] : ((~disjoint(X0,X1) | ! [X2] : (~in(X2,X1) | ~in(X2,X0))) & (? [X3] : (in(X3,X1) & in(X3,X0)) | disjoint(X0,X1)))),
  inference(ennf_transformation,[],[f62])).
fof(f62,plain,(
  ! [X0,X1] : (~(disjoint(X0,X1) & ? [X2] : (in(X2,X1) & in(X2,X0))) & ~(! [X3] : ~(in(X3,X1) & in(X3,X0)) & ~disjoint(X0,X1)))),
  inference(rectify,[],[f43])).
fof(f43,axiom,(
  ! [X0,X1] : (~(disjoint(X0,X1) & ? [X2] : (in(X2,X1) & in(X2,X0))) & ~(! [X2] : ~(in(X2,X1) & in(X2,X0)) & ~disjoint(X0,X1)))),
  file('Problems/SEU/SEU140+2.p',unknown)).
fof(f389,plain,(
  ( ! [X0] : (~in(X0,sK10) | in(X0,sK11)) )),
  inference(superposition,[],[f237,f320])).
fof(f320,plain,(
  sK11 = set_union2(sK10,sK11)),
  inference(resolution,[],[f180,f208])).
fof(f208,plain,(
  subset(sK10,sK11)),
  inference(cnf_transformation,[],[f134])).
fof(f180,plain,(
  ( ! [X0,X1] : (~subset(X0,X1) | set_union2(X0,X1) = X1) )),
  inference(cnf_transformation,[],[f73])).
fof(f73,plain,(
  ! [X0,X1] : (set_union2(X0,X1) = X1 | ~subset(X0,X1))),
  inference(ennf_transformation,[],[f28])).
fof(f28,axiom,(
  ! [X0,X1] : (subset(X0,X1) => set_union2(X0,X1) = X1)),
  file('Problems/SEU/SEU140+2.p',unknown)).
fof(f237,plain,(
  ( ! [X0,X1,X4] : (in(X4,set_union2(X0,X1)) | ~in(X4,X0)) )),
  inference(equality_resolution,[],[f145])).
fof(f145,plain,(
  ( ! [X2,X0,X1,X4] : (in(X4,X2) | ~in(X4,X0) | set_union2(X0,X1) != X2) )),
  inference(cnf_transformation,[],[f104])).
fof(f104,plain,(
  ! [X0,X1,X2] : ((set_union2(X0,X1) = X2 | (((~in(sK1(X0,X1,X2),X1) & ~in(sK1(X0,X1,X2),X0)) | ~in(sK1(X0,X1,X2),X2)) & (in(sK1(X0,X1,X2),X1) | in(sK1(X0,X1,X2),X0) | in(sK1(X0,X1,X2),X2)))) & (! [X4] : ((in(X4,X2) | (~in(X4,X1) & ~in(X4,X0))) & (in(X4,X1) | in(X4,X0) | ~in(X4,X2))) | set_union2(X0,X1) != X2))),
  inference(skolemisation,[status(esa),new_symbols(skolem,[sK1])],[f102,f103])).
fof(f103,plain,(
  ! [X0,X1,X2] : (? [X3] : (((~in(X3,X1) & ~in(X3,X0)) | ~in(X3,X2)) & (in(X3,X1) | in(X3,X0) | in(X3,X2))) => (((~in(sK1(X0,X1,X2),X1) & ~in(sK1(X0,X1,X2),X0)) | ~in(sK1(X0,X1,X2),X2)) & (in(sK1(X0,X1,X2),X1) | in(sK1(X0,X1,X2),X0) | in(sK1(X0,X1,X2),X2))))),
  introduced(definition,[],[choice_axiom])).
fof(f102,plain,(
  ! [X0,X1,X2] : ((set_union2(X0,X1) = X2 | ? [X3] : (((~in(X3,X1) & ~in(X3,X0)) | ~in(X3,X2)) & (in(X3,X1) | in(X3,X0) | in(X3,X2)))) & (! [X4] : ((in(X4,X2) | (~in(X4,X1) & ~in(X4,X0))) & (in(X4,X1) | in(X4,X0) | ~in(X4,X2))) | set_union2(X0,X1) != X2))),
  inference(rectify,[],[f101])).
fof(f101,plain,(
  ! [X0,X1,X2] : ((set_union2(X0,X1) = X2 | ? [X3] : (((~in(X3,X1) & ~in(X3,X0)) | ~in(X3,X2)) & (in(X3,X1) | in(X3,X0) | in(X3,X2)))) & (! [X3] : ((in(X3,X2) | (~in(X3,X1) & ~in(X3,X0))) & (in(X3,X1) | in(X3,X0) | ~in(X3,X2))) | set_union2(X0,X1) != X2))),
  inference(flattening,[],[f100])).
fof(f100,plain,(
  ! [X0,X1,X2] : ((set_union2(X0,X1) = X2 | ? [X3] : (((~in(X3,X1) & ~in(X3,X0)) | ~in(X3,X2)) & ((in(X3,X1) | in(X3,X0)) | in(X3,X2)))) & (! [X3] : ((in(X3,X2) | (~in(X3,X1) & ~in(X3,X0))) & ((in(X3,X1) | in(X3,X0)) | ~in(X3,X2))) | set_union2(X0,X1) != X2))),
  inference(nnf_transformation,[],[f7])).
fof(f7,axiom,(
  ! [X0,X1,X2] : (set_union2(X0,X1) = X2 <=> ! [X3] : (in(X3,X2) <=> (in(X3,X1) | in(X3,X0))))),
  file('Problems/SEU/SEU140+2.p',unknown)).
fof(f510,plain,(
  ( ! [X0] : (~in(sK8(sK12,X0),sK11) | disjoint(sK12,X0)) )),
  inference(resolution,[],[f454,f197])).
fof(f197,plain,(
  ( ! [X0,X1] : (in(sK8(X0,X1),X0) | disjoint(X0,X1)) )),
  inference(cnf_transformation,[],[f130])).
fof(f454,plain,(
  ( ! [X0] : (~in(X0,sK12) | ~in(X0,sK11)) )),
  inference(resolution,[],[f199,f271])).
fof(f271,plain,(
  disjoint(sK12,sK11)),
  inference(resolution,[],[f179,f209])).
fof(f209,plain,(
  disjoint(sK11,sK12)),
  inference(cnf_transformation,[],[f134])).
fof(f199,plain,(
  ( ! [X2,X0,X1] : (~disjoint(X0,X1) | ~in(X2,X1) | ~in(X2,X0)) )),
  inference(cnf_transformation,[],[f130])).
% SZS output end Proof for SEU140+2

Solution for `BOO001-1`

% SUCCESS: Derivation has unique formula names
% SUCCESS: All derived formulae have parents and inference information
% WARNING: Took the first false root f295 as the single derivation root
% SUCCESS: Derivation is acyclic
% SUCCESS: Derivation looks like a refutation
% SUCCESS: Negated conjecture f6 is a leaf or CTH from a conjecture
% SUCCESS: Assumptions are propagated
% SUCCESS: Assumptions are discharged
% WARNING: Took the negated conjecture f6 as the proved formula
% CPUTIME: 0.06
% SUCCESS: Verified
% SZS status Verified

fof(f295,plain,(
  $false),
  inference(trivial_inequality_removal,[],[f289])).
fof(f289,plain,(
  a != a),
  inference(superposition,[],[f6,f224])).
fof(f224,plain,(
  ( ! [X0] : (inverse(inverse(X0)) = X0) )),
  inference(superposition,[],[f5,f158])).
fof(f158,plain,(
  ( ! [X0,X1] : (multiply(X1,inverse(X1),X0) = X0) )),
  inference(forward_demodulation,[],[f146,f25])).
fof(f25,plain,(
  ( ! [X2,X0] : (multiply(X0,X2,X0) = X0) )),
  inference(forward_demodulation,[],[f22,f3])).
fof(f3,axiom,(
  ( ! [X2,X3] : (multiply(X2,X2,X3) = X2) )),
  file('Problems/BOO/BOO001-1.p',unknown)).
fof(f22,plain,(
  ( ! [X2,X3,X0,X1] : (multiply(X0,X0,multiply(X1,X2,X3)) = multiply(X0,X2,X0)) )),
  inference(forward_demodulation,[],[f13,f3])).
fof(f13,plain,(
  ( ! [X2,X3,X0,X1] : (multiply(X0,X0,multiply(X1,X2,X3)) = multiply(multiply(X0,X0,X1),X2,X0)) )),
  inference(superposition,[],[f1,f3])).
fof(f1,axiom,(
  ( ! [X2,X3,X0,X1,X4] : (multiply(multiply(X0,X1,X2),X3,multiply(X0,X1,X4)) = multiply(X0,X1,multiply(X2,X3,X4))) )),
  file('Problems/BOO/BOO001-1.p',unknown)).
fof(f146,plain,(
  ( ! [X0,X1] : (multiply(X1,inverse(X1),multiply(X0,X1,X0)) = X0) )),
  inference(superposition,[],[f7,f119])).
fof(f119,plain,(
  ( ! [X3,X0,X1] : (multiply(X3,X1,multiply(X0,inverse(X1),X3)) = X3) )),
  inference(forward_demodulation,[],[f118,f88])).
fof(f88,plain,(
  ( ! [X2,X3,X0,X1,X4] : (multiply(X0,X1,multiply(X2,X3,X0)) = multiply(X0,X1,multiply(X2,X3,multiply(X4,X0,inverse(X1))))) )),
  inference(forward_demodulation,[],[f86,f28])).
fof(f28,plain,(
  ( ! [X2,X3,X0,X1] : (multiply(multiply(X0,X1,X2),X3,X0) = multiply(X0,X1,multiply(X2,X3,X0))) )),
  inference(superposition,[],[f1,f25])).
fof(f86,plain,(
  ( ! [X2,X3,X0,X1,X4] : (multiply(multiply(X0,X1,X2),X3,X0) = multiply(X0,X1,multiply(X2,X3,multiply(X4,X0,inverse(X1))))) )),
  inference(superposition,[],[f1,f70])).
fof(f70,plain,(
  ( ! [X2,X0,X1] : (multiply(X0,X2,multiply(X1,X0,inverse(X2))) = X0) )),
  inference(forward_demodulation,[],[f38,f2])).
fof(f2,axiom,(
  ( ! [X2,X3] : (multiply(X3,X2,X2) = X2) )),
  file('Problems/BOO/BOO001-1.p',unknown)).
fof(f38,plain,(
  ( ! [X2,X0,X1] : (multiply(X1,X0,X0) = multiply(X0,X2,multiply(X1,X0,inverse(X2)))) )),
  inference(superposition,[],[f7,f5])).
fof(f118,plain,(
  ( ! [X2,X3,X0,X1] : (multiply(X3,X1,multiply(X0,inverse(X1),multiply(X2,X3,inverse(X1)))) = X3) )),
  inference(superposition,[],[f70,f12])).
fof(f12,plain,(
  ( ! [X2,X3,X0,X1] : (multiply(X1,X0,multiply(X2,X3,X0)) = multiply(multiply(X1,X0,X2),X3,X0)) )),
  inference(superposition,[],[f1,f2])).
fof(f7,plain,(
  ( ! [X2,X3,X0,X1] : (multiply(X1,X0,multiply(X0,X2,X3)) = multiply(X0,X2,multiply(X1,X0,X3))) )),
  inference(superposition,[],[f1,f2])).
fof(f5,axiom,(
  ( ! [X2,X3] : (multiply(X2,X3,inverse(X3)) = X2) )),
  file('Problems/BOO/BOO001-1.p',unknown)).
fof(f6,negated_conjecture,(
  a != inverse(inverse(a))),
  file('Problems/BOO/BOO001-1.p',unknown)).

ProoVer 2026

GDV 2.0

Geoff Sutcliffe
University of Miami, USA

Verification of `correct_proof.p`

 NOTICE: Reading the derivation file correct_proof.p
 NOTICE: Took problem file name Problems/correct_problem.p from annotated formula a1
 NOTICE: Starting verification processes
 RESULT: SOT_0HqbAN - Paradox---4.0 says Satisfiable - CPU = 0.00
SUCCESS: Leaf axiom(_like) formulae are satisfiable
 NOTICE: Reading the problem file Problems/correct_problem.p
SUCCESS: Derivation has unique formula names
SUCCESS: All derived formulae have parents and inference information
SUCCESS: All new_symbols are really new
 NOTICE: Took the first false root 'f1' as the single derivation root
SUCCESS: Derivation is acyclic
SUCCESS: Derivation looks like a refutation
SUCCESS: Assumptions are propagated
SUCCESS: Assumptions are discharged
 NOTICE: Took the conjecture c as the proved formula
 RESULT: SOT_XO7UhS - Paradox---4.0 says Satisfiable - CPU = 0.00
SUCCESS: Problem axiom(_like) formulae are satisfiable
SUCCESS: Leaf a1 is a copy of a1 (from the problem)
SUCCESS: Leaf c is a copy of c (from the problem)
SUCCESS: Leaves are verified
 NOTICE: Making CTH test of 's1' from 'c' a CEQ test
 RESULT: SOT_OnyF3a - E---3.3.0 says Theorem - CPU = 0.00
SUCCESS: 's1' is a cth of 'c' (forwards ceq)
 RESULT: SOT_r7_vgC - E---3.3.0 says Theorem - CPU = 0.01
SUCCESS: 'c' is a thm of 'neg_s1' (backwards ceq)
SUCCESS: s1 is a ceq of c
 RESULT: SOT_qWQTeb - E---3.3.0 says ContradictoryAxioms - CPU = 0.01
SUCCESS: 'f1' is a thm of 's1 a1'
SUCCESS: Derived formulae are verified
SUCCESS: Verified
% SZS status VerifiedGood

Verification of `evil_proof.p`

 NOTICE: Reading the derivation file evil_proof.p
 NOTICE: Took problem file name Problems/evil_problem.p from annotated formula a1
 NOTICE: Starting verification processes
SUCCESS: Leaf axiom(_like) formulae are satisfiable
 NOTICE: Reading the problem file Problems/evil_problem.p
SUCCESS: Derivation has unique formula names
SUCCESS: All derived formulae have parents and inference information
SUCCESS: All new_symbols are really new
 NOTICE: Took the first false root 'f1' as the single derivation root
SUCCESS: Derivation is acyclic
SUCCESS: Derivation looks like a refutation
SUCCESS: Assumptions are propagated
SUCCESS: Assumptions are discharged
 NOTICE: Took the conjecture c as the proved formula
SUCCESS: Problem axiom(_like) formulae are satisfiable
SUCCESS: Leaf a1 is a copy of a1 (from the problem)
SUCCESS: Leaf c is a copy of c (from the problem)
SUCCESS: Leaves are verified
 NOTICE: Making CTH test of 's1' from 'c' a CEQ test
 RESULT: SOT_ZBthDX - E---3.3.0 says Theorem - CPU = 0.01
SUCCESS: 's1' is a cth of 'c' (forwards ceq)
 RESULT: SOT_dGAhHU - E---3.3.0 says CounterSatisfiable - CPU = 0.00
FAILURE: 'c' fails to be a thm of 'neg_s1' (backwards ceq)
FAILURE: s1 fails in the backward direction to be a ceq of c
FAILURE: Failed verification
% SZS status VerifiedBad

Verification of `timeout_proof.p`

 NOTICE: Reading the derivation file timeout_proof.p
 NOTICE: Took problem file name Problems/timeout_problem.p from annotated formula a01
 NOTICE: Starting verification processes
SUCCESS: Leaf axiom(_like) formulae are satisfiable
 NOTICE: Reading the problem file Problems/timeout_problem.p
SUCCESS: Derivation has unique formula names
SUCCESS: All derived formulae have parents and inference information
SUCCESS: All new_symbols are really new
 NOTICE: Took the first false root 'zfalse' as the single derivation root
SUCCESS: Derivation is acyclic
WARNING: Refutation has non-false root 's1'
SUCCESS: Assumptions are propagated
SUCCESS: Assumptions are discharged
 NOTICE: Took the conjecture c as the proved formula
SUCCESS: Problem axiom(_like) formulae are satisfiable
SUCCESS: Leaf a01 is a copy of a01 (from the problem)
SUCCESS: Leaf c is a copy of c (from the problem)
SUCCESS: Leaves are verified
 NOTICE: Making CTH test of 's1' from 'c' a CEQ test
 RESULT: SOT_EWOQEc - E---3.3.0 says Theorem - CPU = 0.01
SUCCESS: 's1' is a cth of 'c' (forwards ceq)
 RESULT: SOT_Hrw0V_ - E---3.3.0 says Theorem - CPU = 0.02
SUCCESS: 'c' is a thm of 'neg_s1' (backwards ceq)
SUCCESS: s1 is a ceq of c
 RESULT: SOT_6VMIYo - E---3.3.0 says Timeout - CPU = 0.90
FAILURE: 'sk0' fails to be a thm of 'a01' (forwards esa)
 RESULT: SOT_N23sRJ - E---3.3.0 says Timeout - CPU = 30.21
FAILURE: 'a01' fails to be a thm of 'sk0' (backwards esa)
FAILURE: sk0 fails in both directions to be a esa of a01
 NOTICE: Not verified : 2 not verified steps
% SZS status Unknown : 2 not verified steps

GDV-LP 2.0

Frédéric Blanqui
ENS Paris-Saclay, INRIA, France

Entrants' Sample Solutions

CASC-J13

ProoVer 2026

CASC-J13

LEO-II 1.7.0

Sample solution for SET014^4

Prover9 1109a

Solution for SEU140+2

Vampire 5.0

Notes regarding saturations

Solution for SET014^4

Solution for DAT013_1

Saturation for DAT335_2

Finite Model for DAT335_2

Saturation for SWW469_10

Finite Model for SWW469_10

Solution for SEU140+2

Solution for BOO001-1

ProoVer 2026

GDV 2.0

Verification of correct_proof.p

Verification of evil_proof.p

Verification of timeout_proof.p

GDV-LP 2.0

Verification of correct_proof.p

Verification of evil_proof.p

Verification of timeout_proof.p

Sample solution for `SET014^4`

Solution for `SEU140+2`

Solution for `SET014^4`

Solution for `DAT013_1`

Saturation for `DAT335_2`

Finite Model for `DAT335_2`

Saturation for `SWW469_10`

Finite Model for `SWW469_10`

Solution for `SEU140+2`

Solution for `BOO001-1`

Verification of `correct_proof.p`

Verification of `evil_proof.p`

Verification of `timeout_proof.p`

Verification of `correct_proof.p`

Verification of `evil_proof.p`

Verification of `timeout_proof.p`