Entrants' Sample Solutions

Prover9 1109a
Vampire 5.0

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)).

Entrants' Sample Solutions

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

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`