TSTP Solution File: GRA002+3 by ePrincess---1.0
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%------------------------------------------------------------------------------
% File : ePrincess---1.0
% Problem : GRA002+3 : TPTP v8.1.0. Bugfixed v3.2.0.
% Transfm : none
% Format : tptp:raw
% Command : ePrincess-casc -timeout=%d %s
% Computer : n019.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 600s
% DateTime : Sat Jul 16 07:15:31 EDT 2022
% Result : Theorem 208.87s 151.60s
% Output : Proof 215.25s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : GRA002+3 : TPTP v8.1.0. Bugfixed v3.2.0.
% 0.07/0.13 % Command : ePrincess-casc -timeout=%d %s
% 0.14/0.34 % Computer : n019.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 600
% 0.14/0.34 % DateTime : Tue May 31 02:57:55 EDT 2022
% 0.14/0.35 % CPUTime :
% 0.49/0.60 ____ _
% 0.49/0.60 ___ / __ \_____(_)___ ________ __________
% 0.49/0.60 / _ \/ /_/ / ___/ / __ \/ ___/ _ \/ ___/ ___/
% 0.49/0.60 / __/ ____/ / / / / / / /__/ __(__ |__ )
% 0.49/0.60 \___/_/ /_/ /_/_/ /_/\___/\___/____/____/
% 0.49/0.60
% 0.49/0.60 A Theorem Prover for First-Order Logic
% 0.49/0.60 (ePrincess v.1.0)
% 0.49/0.60
% 0.49/0.60 (c) Philipp Rümmer, 2009-2015
% 0.49/0.60 (c) Peter Backeman, 2014-2015
% 0.49/0.60 (contributions by Angelo Brillout, Peter Baumgartner)
% 0.49/0.60 Free software under GNU Lesser General Public License (LGPL).
% 0.49/0.60 Bug reports to peter@backeman.se
% 0.49/0.60
% 0.49/0.60 For more information, visit http://user.uu.se/~petba168/breu/
% 0.49/0.60
% 0.49/0.60 Loading /export/starexec/sandbox/benchmark/theBenchmark.p ...
% 0.70/0.65 Prover 0: Options: -triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMaximal -resolutionMethod=nonUnifying +ignoreQuantifiers -generateTriggers=all
% 1.81/0.99 Prover 0: Preprocessing ...
% 3.14/1.36 Prover 0: Warning: ignoring some quantifiers
% 3.14/1.39 Prover 0: Constructing countermodel ...
% 20.65/5.94 Prover 1: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -resolutionMethod=normal +ignoreQuantifiers -generateTriggers=all
% 21.18/6.08 Prover 1: Preprocessing ...
% 21.96/6.20 Prover 1: Constructing countermodel ...
% 31.57/8.54 Prover 2: Options: +triggersInConjecture +genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -resolutionMethod=nonUnifying +ignoreQuantifiers -generateTriggers=all
% 31.93/8.57 Prover 2: Preprocessing ...
% 32.32/8.67 Prover 2: Warning: ignoring some quantifiers
% 32.32/8.68 Prover 2: Constructing countermodel ...
% 41.42/11.56 Prover 0: stopped
% 41.74/11.76 Prover 3: Options: -triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=maximal -resolutionMethod=nonUnifying +ignoreQuantifiers -generateTriggers=all
% 41.89/11.80 Prover 3: Preprocessing ...
% 41.89/11.83 Prover 3: Warning: ignoring some quantifiers
% 41.89/11.84 Prover 3: Constructing countermodel ...
% 87.02/52.69 Prover 3: stopped
% 87.36/52.89 Prover 4: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -resolutionMethod=nonUnifying +ignoreQuantifiers -generateTriggers=complete
% 87.51/52.93 Prover 4: Preprocessing ...
% 87.87/53.03 Prover 4: Warning: ignoring some quantifiers
% 87.87/53.04 Prover 4: Constructing countermodel ...
% 141.42/96.19 Prover 1: stopped
% 141.71/96.39 Prover 5: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=none +reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allMinimal -resolutionMethod=nonUnifying +ignoreQuantifiers -generateTriggers=all
% 141.71/96.43 Prover 5: Preprocessing ...
% 142.12/96.51 Prover 5: Constructing countermodel ...
% 150.54/103.68 Prover 5: stopped
% 150.78/103.92 Prover 6: Options: +triggersInConjecture +genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -resolutionMethod=normal +ignoreQuantifiers -generateTriggers=all
% 150.88/103.96 Prover 6: Preprocessing ...
% 151.19/104.02 Prover 6: Warning: ignoring some quantifiers
% 151.19/104.03 Prover 6: Constructing countermodel ...
% 166.89/118.77 Prover 2: stopped
% 167.15/118.97 Prover 7: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=none +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=maximalOutermost -resolutionMethod=normal -ignoreQuantifiers -generateTriggers=all
% 167.20/119.00 Prover 7: Preprocessing ...
% 167.42/119.04 Prover 7: Proving ...
% 208.87/151.60 Prover 6: proved (22651ms)
% 208.87/151.60 Prover 4: stopped
% 208.87/151.60 Prover 7: stopped
% 208.87/151.60
% 208.87/151.60 No countermodel exists, formula is valid
% 208.87/151.60 % SZS status Theorem for theBenchmark
% 208.87/151.60
% 208.87/151.60 Generating proof ... Warning: ignoring some quantifiers
% 214.77/153.06 found it (size 80)
% 214.77/153.06
% 214.77/153.06 % SZS output start Proof for theBenchmark
% 214.77/153.06 Assumed formulas after preprocessing and simplification:
% 214.77/153.06 | (0) ? [v0] : ? [v1] : ? [v2] : ? [v3] : ? [v4] : ? [v5] : ? [v6] : ( ~ (v6 = 0) & minus(v4, n1) = v5 & number_of_in(triangles, graph) = v0 & shortest_path(v2, v3, v1) = 0 & length_of(v1) = v4 & less_or_equal(v5, v0) = v6 & complete & ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : ! [v12] : ! [v13] : (v10 = 0 | ~ (path_cons(v11, v13) = v9) | ~ (path(v7, v8, v9) = v10) | ~ (head_of(v11) = v12) | ? [v14] : (( ~ (v14 = v7) & tail_of(v11) = v14) | ( ~ (v14 = 0) & path(v12, v8, v13) = v14) | ( ~ (v14 = 0) & vertex(v8) = v14) | ( ~ (v14 = 0) & vertex(v7) = v14) | ( ~ (v14 = 0) & edge(v11) = v14))) & ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : ! [v12] : ! [v13] : (v10 = 0 | ~ (path(v12, v8, v13) = 0) | ~ (path(v7, v8, v9) = v10) | ~ (head_of(v11) = v12) | ? [v14] : (( ~ (v14 = v9) & path_cons(v11, v13) = v14) | ( ~ (v14 = v7) & tail_of(v11) = v14) | ( ~ (v14 = 0) & vertex(v8) = v14) | ( ~ (v14 = 0) & vertex(v7) = v14) | ( ~ (v14 = 0) & edge(v11) = v14))) & ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : ! [v12] : ! [v13] : ( ~ (shortest_path(v7, v8, v11) = 0) | ~ (tail_of(v9) = v12) | ~ (head_of(v10) = v13) | ? [v14] : (( ~ (v14 = 0) & precedes(v10, v9, v11) = v14 & ! [v15] : ( ~ (tail_of(v15) = v12) | ? [v16] : ( ~ (v16 = v13) & head_of(v15) = v16)) & ! [v15] : ( ~ (head_of(v15) = v13) | ? [v16] : ( ~ (v16 = v12) & tail_of(v15) = v16))) | ( ~ (v14 = 0) & precedes(v9, v10, v11) = v14))) & ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : ! [v12] : ! [v13] : ( ~ (precedes(v13, v11, v7) = 0) | ~ (sequential(v10, v11) = v12) | ~ (path(v8, v9, v7) = 0) | ? [v14] : ((v14 = 0 & precedes(v10, v11, v7) = 0) | ( ~ (v14 = 0) & sequential(v10, v13) = v14) | ( ~ (v14 = 0) & on_path(v11, v7) = v14) | ( ~ (v14 = 0) & on_path(v10, v7) = v14))) & ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : ! [v12] : ! [v13] : ( ~ (sequential(v10, v13) = 0) | ~ (sequential(v10, v11) = v12) | ~ (path(v8, v9, v7) = 0) | ? [v14] : ((v14 = 0 & precedes(v10, v11, v7) = 0) | ( ~ (v14 = 0) & precedes(v13, v11, v7) = v14) | ( ~ (v14 = 0) & on_path(v11, v7) = v14) | ( ~ (v14 = 0) & on_path(v10, v7) = v14))) & ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : ! [v12] : (v12 = 0 | ~ (precedes(v10, v11, v7) = v12) | ~ (path(v8, v9, v7) = 0) | ? [v13] : (( ~ (v13 = 0) & sequential(v10, v11) = v13 & ! [v14] : ( ~ (precedes(v14, v11, v7) = 0) | ? [v15] : ( ~ (v15 = 0) & sequential(v10, v14) = v15)) & ! [v14] : ( ~ (sequential(v10, v14) = 0) | ? [v15] : ( ~ (v15 = 0) & precedes(v14, v11, v7) = v15))) | ( ~ (v13 = 0) & on_path(v11, v7) = v13) | ( ~ (v13 = 0) & on_path(v10, v7) = v13))) & ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : ! [v12] : (v10 = 0 | ~ (path_cons(v11, empty) = v12) | ~ (path(v7, v8, v9) = v10) | ? [v13] : (( ~ (v13 = v7) & tail_of(v11) = v13) | ( ~ (v13 = 0) & vertex(v8) = v13) | ( ~ (v13 = 0) & vertex(v7) = v13) | ( ~ (v13 = 0) & edge(v11) = v13) | (head_of(v11) = v13 & ! [v14] : ( ~ (path_cons(v11, v14) = v9) | ? [v15] : ( ~ (v15 = 0) & path(v13, v8, v14) = v15)) & ! [v14] : ( ~ (path(v13, v8, v14) = 0) | ? [v15] : ( ~ (v15 = v9) & path_cons(v11, v14) = v15)) & ( ~ (v13 = v8) | ~ (v12 = v9))))) & ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : ! [v12] : ( ~ (shortest_path(v7, v8, v11) = 0) | ~ (precedes(v10, v9, v11) = v12) | ? [v13] : ? [v14] : (( ~ (v13 = 0) & precedes(v9, v10, v11) = v13) | ( ~ (v12 = 0) & tail_of(v9) = v13 & head_of(v10) = v14 & ! [v15] : ( ~ (tail_of(v15) = v13) | ? [v16] : ( ~ (v16 = v14) & head_of(v15) = v16)) & ! [v15] : ( ~ (head_of(v15) = v14) | ? [v16] : ( ~ (v16 = v13) & tail_of(v15) = v16))))) & ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : ! [v12] : ( ~ (sequential(v10, v11) = v12) | ~ (path(v8, v9, v7) = 0) | ? [v13] : ? [v14] : ? [v15] : ? [v16] : ? [v17] : ((v14 = 0 & v13 = 0 & on_path(v11, v7) = 0 & on_path(v10, v7) = 0 & ( ~ (v12 = 0) | ( ! [v18] : ( ~ (precedes(v18, v11, v7) = 0) | ? [v19] : ( ~ (v19 = 0) & sequential(v10, v18) = v19)) & ! [v18] : ( ~ (sequential(v10, v18) = 0) | ? [v19] : ( ~ (v19 = 0) & precedes(v18, v11, v7) = v19)))) & (v12 = 0 | (v17 = 0 & v16 = 0 & precedes(v15, v11, v7) = 0 & sequential(v10, v15) = 0))) | ( ~ (v13 = 0) & precedes(v10, v11, v7) = v13))) & ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : ! [v12] : ( ~ (in_path(v11, v9) = v12) | ~ (path(v7, v8, v9) = 0) | ~ (tail_of(v10) = v11) | ? [v13] : ? [v14] : ? [v15] : ((v15 = 0 & v13 = 0 & v12 = 0 & in_path(v14, v9) = 0 & edge(v10) = 0 & head_of(v10) = v14) | ( ~ (v13 = 0) & on_path(v10, v9) = v13))) & ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : ! [v12] : ( ~ (in_path(v11, v9) = v12) | ~ (path(v7, v8, v9) = 0) | ~ (head_of(v10) = v11) | ? [v13] : ? [v14] : ? [v15] : ((v15 = 0 & v13 = 0 & v12 = 0 & in_path(v14, v9) = 0 & tail_of(v10) = v14 & edge(v10) = 0) | ( ~ (v13 = 0) & on_path(v10, v9) = v13))) & ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : (v11 = 0 | ~ (path(v7, v8, v9) = 0) | ~ (vertex(v10) = v11) | ? [v12] : ( ~ (v12 = 0) & in_path(v10, v9) = v12)) & ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : (v10 = 0 | ~ (path(v7, v8, v9) = v10) | ~ (tail_of(v11) = v7) | ? [v12] : ? [v13] : (( ~ (v12 = 0) & vertex(v8) = v12) | ( ~ (v12 = 0) & vertex(v7) = v12) | ( ~ (v12 = 0) & edge(v11) = v12) | (head_of(v11) = v12 & ! [v14] : ( ~ (path_cons(v11, v14) = v9) | ? [v15] : ( ~ (v15 = 0) & path(v12, v8, v14) = v15)) & ! [v14] : ( ~ (path(v12, v8, v14) = 0) | ? [v15] : ( ~ (v15 = v9) & path_cons(v11, v14) = v15)) & ( ~ (v12 = v8) | ( ~ (v13 = v9) & path_cons(v11, empty) = v13))))) & ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : (v10 = 0 | ~ (path(v7, v8, v9) = v10) | ~ (edge(v11) = 0) | ? [v12] : ? [v13] : (( ~ (v12 = v7) & tail_of(v11) = v12) | ( ~ (v12 = 0) & vertex(v8) = v12) | ( ~ (v12 = 0) & vertex(v7) = v12) | (head_of(v11) = v12 & ! [v14] : ( ~ (path_cons(v11, v14) = v9) | ? [v15] : ( ~ (v15 = 0) & path(v12, v8, v14) = v15)) & ! [v14] : ( ~ (path(v12, v8, v14) = 0) | ? [v15] : ( ~ (v15 = v9) & path_cons(v11, v14) = v15)) & ( ~ (v12 = v8) | ( ~ (v13 = v9) & path_cons(v11, empty) = v13))))) & ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : (v10 = 0 | ~ (path(v7, v8, v9) = v10) | ~ (head_of(v11) = v8) | ? [v12] : (( ~ (v12 = v9) & path_cons(v11, empty) = v12) | ( ~ (v12 = v7) & tail_of(v11) = v12) | ( ~ (v12 = 0) & vertex(v8) = v12) | ( ~ (v12 = 0) & vertex(v7) = v12) | ( ~ (v12 = 0) & edge(v11) = v12))) & ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : (v8 = v7 | ~ (triangle(v11, v10, v9) = v8) | ~ (triangle(v11, v10, v9) = v7)) & ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : (v8 = v7 | ~ (shortest_path(v11, v10, v9) = v8) | ~ (shortest_path(v11, v10, v9) = v7)) & ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : (v8 = v7 | ~ (precedes(v11, v10, v9) = v8) | ~ (precedes(v11, v10, v9) = v7)) & ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : (v8 = v7 | ~ (path(v11, v10, v9) = v8) | ~ (path(v11, v10, v9) = v7)) & ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : ( ~ (shortest_path(v7, v8, v11) = 0) | ~ (precedes(v9, v10, v11) = 0) | ? [v12] : ? [v13] : ? [v14] : ( ~ (v14 = 0) & precedes(v10, v9, v11) = v14 & tail_of(v9) = v12 & head_of(v10) = v13 & ! [v15] : ( ~ (tail_of(v15) = v12) | ? [v16] : ( ~ (v16 = v13) & head_of(v15) = v16)) & ! [v15] : ( ~ (head_of(v15) = v13) | ? [v16] : ( ~ (v16 = v12) & tail_of(v15) = v16)))) & ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : ( ~ (shortest_path(v7, v8, v11) = 0) | ~ (precedes(v9, v10, v11) = 0) | ? [v12] : ? [v13] : ((v13 = 0 & triangle(v9, v10, v12) = 0) | ( ~ (v12 = 0) & sequential(v9, v10) = v12))) & ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : ( ~ (shortest_path(v7, v8, v11) = 0) | ~ (sequential(v9, v10) = 0) | ? [v12] : ? [v13] : ((v13 = 0 & triangle(v9, v10, v12) = 0) | ( ~ (v12 = 0) & precedes(v9, v10, v11) = v12))) & ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : ( ~ (precedes(v10, v11, v7) = 0) | ~ (path(v8, v9, v7) = 0) | ? [v12] : ? [v13] : ? [v14] : ? [v15] : (on_path(v11, v7) = 0 & on_path(v10, v7) = 0 & ((v15 = 0 & v14 = 0 & precedes(v13, v11, v7) = 0 & sequential(v10, v13) = 0) | (v12 = 0 & sequential(v10, v11) = 0)) & (( ~ (v12 = 0) & sequential(v10, v11) = v12) | ( ! [v16] : ( ~ (precedes(v16, v11, v7) = 0) | ? [v17] : ( ~ (v17 = 0) & sequential(v10, v16) = v17)) & ! [v16] : ( ~ (sequential(v10, v16) = 0) | ? [v17] : ( ~ (v17 = 0) & precedes(v16, v11, v7) = v17)))))) & ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : ( ~ (sequential(v10, v11) = 0) | ~ (path(v8, v9, v7) = 0) | ? [v12] : ((v12 = 0 & precedes(v10, v11, v7) = 0) | ( ~ (v12 = 0) & on_path(v11, v7) = v12) | ( ~ (v12 = 0) & on_path(v10, v7) = v12))) & ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : ( ~ (path(v7, v8, v9) = 0) | ~ (vertex(v10) = v11) | ? [v12] : ? [v13] : ? [v14] : ((v13 = 0 & on_path(v12, v9) = 0 & ((v14 = v10 & tail_of(v12) = v10) | (v14 = v10 & head_of(v12) = v10))) | ( ~ (v12 = 0) & in_path(v10, v9) = v12))) & ! [v7] : ! [v8] : ! [v9] : ! [v10] : ! [v11] : ( ~ (path(v7, v8, v9) = 0) | ~ (edge(v10) = v11) | ? [v12] : ? [v13] : ? [v14] : ? [v15] : ((v15 = 0 & v13 = 0 & v11 = 0 & in_path(v14, v9) = 0 & in_path(v12, v9) = 0 & tail_of(v10) = v14 & head_of(v10) = v12) | ( ~ (v12 = 0) & on_path(v10, v9) = v12))) & ! [v7] : ! [v8] : ! [v9] : ! [v10] : (v10 = 0 | v8 = v7 | ~ (shortest_path(v7, v8, v9) = v10) | ? [v11] : ? [v12] : ? [v13] : ? [v14] : ? [v15] : ((v13 = 0 & ~ (v15 = 0) & length_of(v12) = v14 & length_of(v9) = v11 & less_or_equal(v11, v14) = v15 & path(v7, v8, v12) = 0) | ( ~ (v11 = 0) & path(v7, v8, v9) = v11))) & ! [v7] : ! [v8] : ! [v9] : ! [v10] : (v10 = 0 | ~ (triangle(v7, v8, v9) = v10) | ? [v11] : (( ~ (v11 = 0) & sequential(v9, v7) = v11) | ( ~ (v11 = 0) & sequential(v8, v9) = v11) | ( ~ (v11 = 0) & sequential(v7, v8) = v11) | ( ~ (v11 = 0) & edge(v9) = v11) | ( ~ (v11 = 0) & edge(v8) = v11) | ( ~ (v11 = 0) & edge(v7) = v11))) & ! [v7] : ! [v8] : ! [v9] : ! [v10] : (v8 = v7 | ~ (minus(v10, v9) = v8) | ~ (minus(v10, v9) = v7)) & ! [v7] : ! [v8] : ! [v9] : ! [v10] : (v8 = v7 | ~ (number_of_in(v10, v9) = v8) | ~ (number_of_in(v10, v9) = v7)) & ! [v7] : ! [v8] : ! [v9] : ! [v10] : (v8 = v7 | ~ (less_or_equal(v10, v9) = v8) | ~ (less_or_equal(v10, v9) = v7)) & ! [v7] : ! [v8] : ! [v9] : ! [v10] : (v8 = v7 | ~ (sequential(v10, v9) = v8) | ~ (sequential(v10, v9) = v7)) & ! [v7] : ! [v8] : ! [v9] : ! [v10] : (v8 = v7 | ~ (on_path(v10, v9) = v8) | ~ (on_path(v10, v9) = v7)) & ! [v7] : ! [v8] : ! [v9] : ! [v10] : (v8 = v7 | ~ (in_path(v10, v9) = v8) | ~ (in_path(v10, v9) = v7)) & ! [v7] : ! [v8] : ! [v9] : ! [v10] : (v8 = v7 | ~ (path_cons(v10, v9) = v8) | ~ (path_cons(v10, v9) = v7)) & ! [v7] : ! [v8] : ! [v9] : ! [v10] : ( ~ (on_path(v10, v9) = 0) | ~ (path(v7, v8, v9) = 0) | ? [v11] : ? [v12] : (in_path(v12, v9) = 0 & in_path(v11, v9) = 0 & tail_of(v10) = v12 & edge(v10) = 0 & head_of(v10) = v11)) & ! [v7] : ! [v8] : ! [v9] : ! [v10] : ( ~ (in_path(v10, v9) = 0) | ~ (path(v7, v8, v9) = 0) | vertex(v10) = 0) & ! [v7] : ! [v8] : ! [v9] : ! [v10] : ( ~ (in_path(v10, v9) = 0) | ~ (path(v7, v8, v9) = 0) | ? [v11] : ? [v12] : (on_path(v11, v9) = 0 & ((v12 = v10 & tail_of(v11) = v10) | (v12 = v10 & head_of(v11) = v10)))) & ! [v7] : ! [v8] : ! [v9] : ! [v10] : ( ~ (path(v7, v8, v9) = v10) | ? [v11] : ((v10 = 0 & ~ (v8 = v7) & length_of(v9) = v11 & ! [v12] : ! [v13] : ( ~ (length_of(v12) = v13) | ? [v14] : ((v14 = 0 & less_or_equal(v11, v13) = 0) | ( ~ (v14 = 0) & path(v7, v8, v12) = v14))) & ! [v12] : ( ~ (path(v7, v8, v12) = 0) | ? [v13] : (length_of(v12) = v13 & less_or_equal(v11, v13) = 0))) | ( ~ (v11 = 0) & shortest_path(v7, v8, v9) = v11))) & ! [v7] : ! [v8] : ! [v9] : (v9 = 0 | v8 = v7 | ~ (sequential(v7, v8) = v9) | ? [v10] : ? [v11] : (( ~ (v11 = v10) & tail_of(v8) = v11 & head_of(v7) = v10) | ( ~ (v10 = 0) & edge(v8) = v10) | ( ~ (v10 = 0) & edge(v7) = v10))) & ! [v7] : ! [v8] : ! [v9] : (v8 = v7 | ~ (length_of(v9) = v8) | ~ (length_of(v9) = v7)) & ! [v7] : ! [v8] : ! [v9] : (v8 = v7 | ~ (path(v7, v8, v9) = 0) | ? [v10] : ? [v11] : ? [v12] : ? [v13] : ? [v14] : ((v12 = 0 & ~ (v14 = 0) & length_of(v11) = v13 & length_of(v9) = v10 & less_or_equal(v10, v13) = v14 & path(v7, v8, v11) = 0) | (v10 = 0 & shortest_path(v7, v8, v9) = 0))) & ! [v7] : ! [v8] : ! [v9] : (v8 = v7 | ~ (vertex(v9) = v8) | ~ (vertex(v9) = v7)) & ! [v7] : ! [v8] : ! [v9] : (v8 = v7 | ~ (tail_of(v9) = v8) | ~ (tail_of(v9) = v7)) & ! [v7] : ! [v8] : ! [v9] : (v8 = v7 | ~ (edge(v9) = v8) | ~ (edge(v9) = v7)) & ! [v7] : ! [v8] : ! [v9] : (v8 = v7 | ~ (head_of(v9) = v8) | ~ (head_of(v9) = v7)) & ! [v7] : ! [v8] : ! [v9] : ( ~ (number_of_in(v7, v8) = v9) | ? [v10] : (number_of_in(v7, graph) = v10 & less_or_equal(v9, v10) = 0)) & ! [v7] : ! [v8] : ! [v9] : ( ~ (triangle(v7, v8, v9) = 0) | (sequential(v9, v7) = 0 & sequential(v8, v9) = 0 & sequential(v7, v8) = 0 & edge(v9) = 0 & edge(v8) = 0 & edge(v7) = 0)) & ! [v7] : ! [v8] : ! [v9] : ( ~ (shortest_path(v7, v8, v9) = 0) | ? [v10] : (length_of(v9) = v10 & path(v7, v8, v9) = 0 & ! [v11] : ! [v12] : ( ~ (length_of(v11) = v12) | ? [v13] : ((v13 = 0 & less_or_equal(v10, v12) = 0) | ( ~ (v13 = 0) & path(v7, v8, v11) = v13))) & ! [v11] : ( ~ (path(v7, v8, v11) = 0) | ? [v12] : (length_of(v11) = v12 & less_or_equal(v10, v12) = 0)))) & ! [v7] : ! [v8] : ! [v9] : ( ~ (path(v8, v9, v7) = 0) | ? [v10] : ? [v11] : ? [v12] : ? [v13] : ? [v14] : ((v14 = 0 & v13 = 0 & v12 = 0 & sequential(v10, v11) = 0 & on_path(v11, v7) = 0 & on_path(v10, v7) = 0 & ! [v15] : ~ (triangle(v10, v11, v15) = 0)) | (v11 = v10 & number_of_in(triangles, v7) = v10 & number_of_in(sequential_pairs, v7) = v10))) & ! [v7] : ! [v8] : ! [v9] : ( ~ (path(v7, v8, v9) = 0) | ? [v10] : ? [v11] : ? [v12] : ? [v13] : ? [v14] : ? [v15] : (tail_of(v10) = v7 & edge(v10) = 0 & head_of(v10) = v11 & ( ~ (v11 = v8) | ( ~ (v12 = v9) & path_cons(v10, empty) = v12) | ( ! [v16] : ( ~ (path_cons(v10, v16) = v9) | ? [v17] : ( ~ (v17 = 0) & path(v8, v8, v16) = v17)) & ! [v16] : ( ~ (path(v8, v8, v16) = 0) | ? [v17] : ( ~ (v17 = v9) & path_cons(v10, v16) = v17)))) & ((v15 = v9 & v14 = 0 & path_cons(v10, v13) = v9 & path(v11, v8, v13) = 0) | (v12 = v9 & v11 = v8 & path_cons(v10, empty) = v9)))) & ! [v7] : ! [v8] : ! [v9] : ( ~ (path(v7, v8, v9) = 0) | ? [v10] : ? [v11] : (minus(v11, n1) = v10 & number_of_in(sequential_pairs, v9) = v10 & length_of(v9) = v11)) & ! [v7] : ! [v8] : ! [v9] : ( ~ (path(v7, v8, v9) = 0) | ? [v10] : (number_of_in(edges, v9) = v10 & length_of(v9) = v10)) & ! [v7] : ! [v8] : ! [v9] : ( ~ (path(v7, v8, v9) = 0) | (vertex(v8) = 0 & vertex(v7) = 0)) & ! [v7] : ! [v8] : (v8 = v7 | ~ (vertex(v8) = 0) | ~ (vertex(v7) = 0) | ? [v9] : ? [v10] : ? [v11] : (tail_of(v9) = v11 & edge(v9) = 0 & head_of(v9) = v10 & ((v11 = v8 & v10 = v7) | (v11 = v7 & v10 = v8)))) & ! [v7] : ! [v8] : ~ (shortest_path(v7, v7, v8) = 0) & ! [v7] : ! [v8] : ( ~ (sequential(v7, v8) = 0) | ? [v9] : (tail_of(v8) = v9 & edge(v8) = 0 & edge(v7) = 0 & head_of(v7) = v9)) & ! [v7] : ! [v8] : ( ~ (tail_of(v7) = v8) | ? [v9] : ? [v10] : ? [v11] : ((v11 = 0 & v10 = 0 & vertex(v9) = 0 & vertex(v8) = 0 & head_of(v7) = v9) | ( ~ (v9 = 0) & edge(v7) = v9))) & ! [v7] : ! [v8] : ( ~ (tail_of(v7) = v8) | ? [v9] : (( ~ (v9 = v8) & head_of(v7) = v9) | ( ~ (v9 = 0) & edge(v7) = v9))) & ! [v7] : ! [v8] : ( ~ (head_of(v7) = v8) | ? [v9] : ? [v10] : ? [v11] : ((v11 = 0 & v9 = 0 & vertex(v10) = 0 & vertex(v8) = 0 & tail_of(v7) = v10) | ( ~ (v9 = 0) & edge(v7) = v9))) & ! [v7] : ! [v8] : ( ~ (head_of(v7) = v8) | ? [v9] : (( ~ (v9 = v8) & tail_of(v7) = v9) | ( ~ (v9 = 0) & edge(v7) = v9))) & ! [v7] : ~ (sequential(v7, v7) = 0) & ! [v7] : ( ~ (edge(v7) = 0) | ? [v8] : ? [v9] : ( ~ (v9 = v8) & tail_of(v7) = v9 & head_of(v7) = v8)) & ! [v7] : ( ~ (edge(v7) = 0) | ? [v8] : ? [v9] : (vertex(v9) = 0 & vertex(v8) = 0 & tail_of(v7) = v9 & head_of(v7) = v8)) & ? [v7] : ? [v8] : ? [v9] : ? [v10] : triangle(v9, v8, v7) = v10 & ? [v7] : ? [v8] : ? [v9] : ? [v10] : shortest_path(v9, v8, v7) = v10 & ? [v7] : ? [v8] : ? [v9] : ? [v10] : precedes(v9, v8, v7) = v10 & ? [v7] : ? [v8] : ? [v9] : ? [v10] : path(v9, v8, v7) = v10 & ? [v7] : ? [v8] : ? [v9] : minus(v8, v7) = v9 & ? [v7] : ? [v8] : ? [v9] : number_of_in(v8, v7) = v9 & ? [v7] : ? [v8] : ? [v9] : less_or_equal(v8, v7) = v9 & ? [v7] : ? [v8] : ? [v9] : sequential(v8, v7) = v9 & ? [v7] : ? [v8] : ? [v9] : on_path(v8, v7) = v9 & ? [v7] : ? [v8] : ? [v9] : in_path(v8, v7) = v9 & ? [v7] : ? [v8] : ? [v9] : path_cons(v8, v7) = v9 & ? [v7] : ? [v8] : length_of(v7) = v8 & ? [v7] : ? [v8] : vertex(v7) = v8 & ? [v7] : ? [v8] : tail_of(v7) = v8 & ? [v7] : ? [v8] : edge(v7) = v8 & ? [v7] : ? [v8] : head_of(v7) = v8)
% 215.05/153.13 | Instantiating (0) with all_0_0_0, all_0_1_1, all_0_2_2, all_0_3_3, all_0_4_4, all_0_5_5, all_0_6_6 yields:
% 215.05/153.13 | (1) ~ (all_0_0_0 = 0) & minus(all_0_2_2, n1) = all_0_1_1 & number_of_in(triangles, graph) = all_0_6_6 & shortest_path(all_0_4_4, all_0_3_3, all_0_5_5) = 0 & length_of(all_0_5_5) = all_0_2_2 & less_or_equal(all_0_1_1, all_0_6_6) = all_0_0_0 & complete & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v3 = 0 | ~ (path_cons(v4, v6) = v2) | ~ (path(v0, v1, v2) = v3) | ~ (head_of(v4) = v5) | ? [v7] : (( ~ (v7 = v0) & tail_of(v4) = v7) | ( ~ (v7 = 0) & path(v5, v1, v6) = v7) | ( ~ (v7 = 0) & vertex(v1) = v7) | ( ~ (v7 = 0) & vertex(v0) = v7) | ( ~ (v7 = 0) & edge(v4) = v7))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v3 = 0 | ~ (path(v5, v1, v6) = 0) | ~ (path(v0, v1, v2) = v3) | ~ (head_of(v4) = v5) | ? [v7] : (( ~ (v7 = v2) & path_cons(v4, v6) = v7) | ( ~ (v7 = v0) & tail_of(v4) = v7) | ( ~ (v7 = 0) & vertex(v1) = v7) | ( ~ (v7 = 0) & vertex(v0) = v7) | ( ~ (v7 = 0) & edge(v4) = v7))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ( ~ (shortest_path(v0, v1, v4) = 0) | ~ (tail_of(v2) = v5) | ~ (head_of(v3) = v6) | ? [v7] : (( ~ (v7 = 0) & precedes(v3, v2, v4) = v7 & ! [v8] : ( ~ (tail_of(v8) = v5) | ? [v9] : ( ~ (v9 = v6) & head_of(v8) = v9)) & ! [v8] : ( ~ (head_of(v8) = v6) | ? [v9] : ( ~ (v9 = v5) & tail_of(v8) = v9))) | ( ~ (v7 = 0) & precedes(v2, v3, v4) = v7))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ( ~ (precedes(v6, v4, v0) = 0) | ~ (sequential(v3, v4) = v5) | ~ (path(v1, v2, v0) = 0) | ? [v7] : ((v7 = 0 & precedes(v3, v4, v0) = 0) | ( ~ (v7 = 0) & sequential(v3, v6) = v7) | ( ~ (v7 = 0) & on_path(v4, v0) = v7) | ( ~ (v7 = 0) & on_path(v3, v0) = v7))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ( ~ (sequential(v3, v6) = 0) | ~ (sequential(v3, v4) = v5) | ~ (path(v1, v2, v0) = 0) | ? [v7] : ((v7 = 0 & precedes(v3, v4, v0) = 0) | ( ~ (v7 = 0) & precedes(v6, v4, v0) = v7) | ( ~ (v7 = 0) & on_path(v4, v0) = v7) | ( ~ (v7 = 0) & on_path(v3, v0) = v7))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | ~ (precedes(v3, v4, v0) = v5) | ~ (path(v1, v2, v0) = 0) | ? [v6] : (( ~ (v6 = 0) & sequential(v3, v4) = v6 & ! [v7] : ( ~ (precedes(v7, v4, v0) = 0) | ? [v8] : ( ~ (v8 = 0) & sequential(v3, v7) = v8)) & ! [v7] : ( ~ (sequential(v3, v7) = 0) | ? [v8] : ( ~ (v8 = 0) & precedes(v7, v4, v0) = v8))) | ( ~ (v6 = 0) & on_path(v4, v0) = v6) | ( ~ (v6 = 0) & on_path(v3, v0) = v6))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v3 = 0 | ~ (path_cons(v4, empty) = v5) | ~ (path(v0, v1, v2) = v3) | ? [v6] : (( ~ (v6 = v0) & tail_of(v4) = v6) | ( ~ (v6 = 0) & vertex(v1) = v6) | ( ~ (v6 = 0) & vertex(v0) = v6) | ( ~ (v6 = 0) & edge(v4) = v6) | (head_of(v4) = v6 & ! [v7] : ( ~ (path_cons(v4, v7) = v2) | ? [v8] : ( ~ (v8 = 0) & path(v6, v1, v7) = v8)) & ! [v7] : ( ~ (path(v6, v1, v7) = 0) | ? [v8] : ( ~ (v8 = v2) & path_cons(v4, v7) = v8)) & ( ~ (v6 = v1) | ~ (v5 = v2))))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ( ~ (shortest_path(v0, v1, v4) = 0) | ~ (precedes(v3, v2, v4) = v5) | ? [v6] : ? [v7] : (( ~ (v6 = 0) & precedes(v2, v3, v4) = v6) | ( ~ (v5 = 0) & tail_of(v2) = v6 & head_of(v3) = v7 & ! [v8] : ( ~ (tail_of(v8) = v6) | ? [v9] : ( ~ (v9 = v7) & head_of(v8) = v9)) & ! [v8] : ( ~ (head_of(v8) = v7) | ? [v9] : ( ~ (v9 = v6) & tail_of(v8) = v9))))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ( ~ (sequential(v3, v4) = v5) | ~ (path(v1, v2, v0) = 0) | ? [v6] : ? [v7] : ? [v8] : ? [v9] : ? [v10] : ((v7 = 0 & v6 = 0 & on_path(v4, v0) = 0 & on_path(v3, v0) = 0 & ( ~ (v5 = 0) | ( ! [v11] : ( ~ (precedes(v11, v4, v0) = 0) | ? [v12] : ( ~ (v12 = 0) & sequential(v3, v11) = v12)) & ! [v11] : ( ~ (sequential(v3, v11) = 0) | ? [v12] : ( ~ (v12 = 0) & precedes(v11, v4, v0) = v12)))) & (v5 = 0 | (v10 = 0 & v9 = 0 & precedes(v8, v4, v0) = 0 & sequential(v3, v8) = 0))) | ( ~ (v6 = 0) & precedes(v3, v4, v0) = v6))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ( ~ (in_path(v4, v2) = v5) | ~ (path(v0, v1, v2) = 0) | ~ (tail_of(v3) = v4) | ? [v6] : ? [v7] : ? [v8] : ((v8 = 0 & v6 = 0 & v5 = 0 & in_path(v7, v2) = 0 & edge(v3) = 0 & head_of(v3) = v7) | ( ~ (v6 = 0) & on_path(v3, v2) = v6))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ( ~ (in_path(v4, v2) = v5) | ~ (path(v0, v1, v2) = 0) | ~ (head_of(v3) = v4) | ? [v6] : ? [v7] : ? [v8] : ((v8 = 0 & v6 = 0 & v5 = 0 & in_path(v7, v2) = 0 & tail_of(v3) = v7 & edge(v3) = 0) | ( ~ (v6 = 0) & on_path(v3, v2) = v6))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v4 = 0 | ~ (path(v0, v1, v2) = 0) | ~ (vertex(v3) = v4) | ? [v5] : ( ~ (v5 = 0) & in_path(v3, v2) = v5)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v3 = 0 | ~ (path(v0, v1, v2) = v3) | ~ (tail_of(v4) = v0) | ? [v5] : ? [v6] : (( ~ (v5 = 0) & vertex(v1) = v5) | ( ~ (v5 = 0) & vertex(v0) = v5) | ( ~ (v5 = 0) & edge(v4) = v5) | (head_of(v4) = v5 & ! [v7] : ( ~ (path_cons(v4, v7) = v2) | ? [v8] : ( ~ (v8 = 0) & path(v5, v1, v7) = v8)) & ! [v7] : ( ~ (path(v5, v1, v7) = 0) | ? [v8] : ( ~ (v8 = v2) & path_cons(v4, v7) = v8)) & ( ~ (v5 = v1) | ( ~ (v6 = v2) & path_cons(v4, empty) = v6))))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v3 = 0 | ~ (path(v0, v1, v2) = v3) | ~ (edge(v4) = 0) | ? [v5] : ? [v6] : (( ~ (v5 = v0) & tail_of(v4) = v5) | ( ~ (v5 = 0) & vertex(v1) = v5) | ( ~ (v5 = 0) & vertex(v0) = v5) | (head_of(v4) = v5 & ! [v7] : ( ~ (path_cons(v4, v7) = v2) | ? [v8] : ( ~ (v8 = 0) & path(v5, v1, v7) = v8)) & ! [v7] : ( ~ (path(v5, v1, v7) = 0) | ? [v8] : ( ~ (v8 = v2) & path_cons(v4, v7) = v8)) & ( ~ (v5 = v1) | ( ~ (v6 = v2) & path_cons(v4, empty) = v6))))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v3 = 0 | ~ (path(v0, v1, v2) = v3) | ~ (head_of(v4) = v1) | ? [v5] : (( ~ (v5 = v2) & path_cons(v4, empty) = v5) | ( ~ (v5 = v0) & tail_of(v4) = v5) | ( ~ (v5 = 0) & vertex(v1) = v5) | ( ~ (v5 = 0) & vertex(v0) = v5) | ( ~ (v5 = 0) & edge(v4) = v5))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v1 = v0 | ~ (triangle(v4, v3, v2) = v1) | ~ (triangle(v4, v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v1 = v0 | ~ (shortest_path(v4, v3, v2) = v1) | ~ (shortest_path(v4, v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v1 = v0 | ~ (precedes(v4, v3, v2) = v1) | ~ (precedes(v4, v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v1 = v0 | ~ (path(v4, v3, v2) = v1) | ~ (path(v4, v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (shortest_path(v0, v1, v4) = 0) | ~ (precedes(v2, v3, v4) = 0) | ? [v5] : ? [v6] : ? [v7] : ( ~ (v7 = 0) & precedes(v3, v2, v4) = v7 & tail_of(v2) = v5 & head_of(v3) = v6 & ! [v8] : ( ~ (tail_of(v8) = v5) | ? [v9] : ( ~ (v9 = v6) & head_of(v8) = v9)) & ! [v8] : ( ~ (head_of(v8) = v6) | ? [v9] : ( ~ (v9 = v5) & tail_of(v8) = v9)))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (shortest_path(v0, v1, v4) = 0) | ~ (precedes(v2, v3, v4) = 0) | ? [v5] : ? [v6] : ((v6 = 0 & triangle(v2, v3, v5) = 0) | ( ~ (v5 = 0) & sequential(v2, v3) = v5))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (shortest_path(v0, v1, v4) = 0) | ~ (sequential(v2, v3) = 0) | ? [v5] : ? [v6] : ((v6 = 0 & triangle(v2, v3, v5) = 0) | ( ~ (v5 = 0) & precedes(v2, v3, v4) = v5))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (precedes(v3, v4, v0) = 0) | ~ (path(v1, v2, v0) = 0) | ? [v5] : ? [v6] : ? [v7] : ? [v8] : (on_path(v4, v0) = 0 & on_path(v3, v0) = 0 & ((v8 = 0 & v7 = 0 & precedes(v6, v4, v0) = 0 & sequential(v3, v6) = 0) | (v5 = 0 & sequential(v3, v4) = 0)) & (( ~ (v5 = 0) & sequential(v3, v4) = v5) | ( ! [v9] : ( ~ (precedes(v9, v4, v0) = 0) | ? [v10] : ( ~ (v10 = 0) & sequential(v3, v9) = v10)) & ! [v9] : ( ~ (sequential(v3, v9) = 0) | ? [v10] : ( ~ (v10 = 0) & precedes(v9, v4, v0) = v10)))))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (sequential(v3, v4) = 0) | ~ (path(v1, v2, v0) = 0) | ? [v5] : ((v5 = 0 & precedes(v3, v4, v0) = 0) | ( ~ (v5 = 0) & on_path(v4, v0) = v5) | ( ~ (v5 = 0) & on_path(v3, v0) = v5))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (path(v0, v1, v2) = 0) | ~ (vertex(v3) = v4) | ? [v5] : ? [v6] : ? [v7] : ((v6 = 0 & on_path(v5, v2) = 0 & ((v7 = v3 & tail_of(v5) = v3) | (v7 = v3 & head_of(v5) = v3))) | ( ~ (v5 = 0) & in_path(v3, v2) = v5))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (path(v0, v1, v2) = 0) | ~ (edge(v3) = v4) | ? [v5] : ? [v6] : ? [v7] : ? [v8] : ((v8 = 0 & v6 = 0 & v4 = 0 & in_path(v7, v2) = 0 & in_path(v5, v2) = 0 & tail_of(v3) = v7 & head_of(v3) = v5) | ( ~ (v5 = 0) & on_path(v3, v2) = v5))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | v1 = v0 | ~ (shortest_path(v0, v1, v2) = v3) | ? [v4] : ? [v5] : ? [v6] : ? [v7] : ? [v8] : ((v6 = 0 & ~ (v8 = 0) & length_of(v5) = v7 & length_of(v2) = v4 & less_or_equal(v4, v7) = v8 & path(v0, v1, v5) = 0) | ( ~ (v4 = 0) & path(v0, v1, v2) = v4))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (triangle(v0, v1, v2) = v3) | ? [v4] : (( ~ (v4 = 0) & sequential(v2, v0) = v4) | ( ~ (v4 = 0) & sequential(v1, v2) = v4) | ( ~ (v4 = 0) & sequential(v0, v1) = v4) | ( ~ (v4 = 0) & edge(v2) = v4) | ( ~ (v4 = 0) & edge(v1) = v4) | ( ~ (v4 = 0) & edge(v0) = v4))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (minus(v3, v2) = v1) | ~ (minus(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (number_of_in(v3, v2) = v1) | ~ (number_of_in(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (less_or_equal(v3, v2) = v1) | ~ (less_or_equal(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (sequential(v3, v2) = v1) | ~ (sequential(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (on_path(v3, v2) = v1) | ~ (on_path(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (in_path(v3, v2) = v1) | ~ (in_path(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (path_cons(v3, v2) = v1) | ~ (path_cons(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (on_path(v3, v2) = 0) | ~ (path(v0, v1, v2) = 0) | ? [v4] : ? [v5] : (in_path(v5, v2) = 0 & in_path(v4, v2) = 0 & tail_of(v3) = v5 & edge(v3) = 0 & head_of(v3) = v4)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (in_path(v3, v2) = 0) | ~ (path(v0, v1, v2) = 0) | vertex(v3) = 0) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (in_path(v3, v2) = 0) | ~ (path(v0, v1, v2) = 0) | ? [v4] : ? [v5] : (on_path(v4, v2) = 0 & ((v5 = v3 & tail_of(v4) = v3) | (v5 = v3 & head_of(v4) = v3)))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (path(v0, v1, v2) = v3) | ? [v4] : ((v3 = 0 & ~ (v1 = v0) & length_of(v2) = v4 & ! [v5] : ! [v6] : ( ~ (length_of(v5) = v6) | ? [v7] : ((v7 = 0 & less_or_equal(v4, v6) = 0) | ( ~ (v7 = 0) & path(v0, v1, v5) = v7))) & ! [v5] : ( ~ (path(v0, v1, v5) = 0) | ? [v6] : (length_of(v5) = v6 & less_or_equal(v4, v6) = 0))) | ( ~ (v4 = 0) & shortest_path(v0, v1, v2) = v4))) & ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | v1 = v0 | ~ (sequential(v0, v1) = v2) | ? [v3] : ? [v4] : (( ~ (v4 = v3) & tail_of(v1) = v4 & head_of(v0) = v3) | ( ~ (v3 = 0) & edge(v1) = v3) | ( ~ (v3 = 0) & edge(v0) = v3))) & ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (length_of(v2) = v1) | ~ (length_of(v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (path(v0, v1, v2) = 0) | ? [v3] : ? [v4] : ? [v5] : ? [v6] : ? [v7] : ((v5 = 0 & ~ (v7 = 0) & length_of(v4) = v6 & length_of(v2) = v3 & less_or_equal(v3, v6) = v7 & path(v0, v1, v4) = 0) | (v3 = 0 & shortest_path(v0, v1, v2) = 0))) & ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (vertex(v2) = v1) | ~ (vertex(v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (tail_of(v2) = v1) | ~ (tail_of(v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (edge(v2) = v1) | ~ (edge(v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (head_of(v2) = v1) | ~ (head_of(v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (number_of_in(v0, v1) = v2) | ? [v3] : (number_of_in(v0, graph) = v3 & less_or_equal(v2, v3) = 0)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (triangle(v0, v1, v2) = 0) | (sequential(v2, v0) = 0 & sequential(v1, v2) = 0 & sequential(v0, v1) = 0 & edge(v2) = 0 & edge(v1) = 0 & edge(v0) = 0)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (shortest_path(v0, v1, v2) = 0) | ? [v3] : (length_of(v2) = v3 & path(v0, v1, v2) = 0 & ! [v4] : ! [v5] : ( ~ (length_of(v4) = v5) | ? [v6] : ((v6 = 0 & less_or_equal(v3, v5) = 0) | ( ~ (v6 = 0) & path(v0, v1, v4) = v6))) & ! [v4] : ( ~ (path(v0, v1, v4) = 0) | ? [v5] : (length_of(v4) = v5 & less_or_equal(v3, v5) = 0)))) & ! [v0] : ! [v1] : ! [v2] : ( ~ (path(v1, v2, v0) = 0) | ? [v3] : ? [v4] : ? [v5] : ? [v6] : ? [v7] : ((v7 = 0 & v6 = 0 & v5 = 0 & sequential(v3, v4) = 0 & on_path(v4, v0) = 0 & on_path(v3, v0) = 0 & ! [v8] : ~ (triangle(v3, v4, v8) = 0)) | (v4 = v3 & number_of_in(triangles, v0) = v3 & number_of_in(sequential_pairs, v0) = v3))) & ! [v0] : ! [v1] : ! [v2] : ( ~ (path(v0, v1, v2) = 0) | ? [v3] : ? [v4] : ? [v5] : ? [v6] : ? [v7] : ? [v8] : (tail_of(v3) = v0 & edge(v3) = 0 & head_of(v3) = v4 & ( ~ (v4 = v1) | ( ~ (v5 = v2) & path_cons(v3, empty) = v5) | ( ! [v9] : ( ~ (path_cons(v3, v9) = v2) | ? [v10] : ( ~ (v10 = 0) & path(v1, v1, v9) = v10)) & ! [v9] : ( ~ (path(v1, v1, v9) = 0) | ? [v10] : ( ~ (v10 = v2) & path_cons(v3, v9) = v10)))) & ((v8 = v2 & v7 = 0 & path_cons(v3, v6) = v2 & path(v4, v1, v6) = 0) | (v5 = v2 & v4 = v1 & path_cons(v3, empty) = v2)))) & ! [v0] : ! [v1] : ! [v2] : ( ~ (path(v0, v1, v2) = 0) | ? [v3] : ? [v4] : (minus(v4, n1) = v3 & number_of_in(sequential_pairs, v2) = v3 & length_of(v2) = v4)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (path(v0, v1, v2) = 0) | ? [v3] : (number_of_in(edges, v2) = v3 & length_of(v2) = v3)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (path(v0, v1, v2) = 0) | (vertex(v1) = 0 & vertex(v0) = 0)) & ! [v0] : ! [v1] : (v1 = v0 | ~ (vertex(v1) = 0) | ~ (vertex(v0) = 0) | ? [v2] : ? [v3] : ? [v4] : (tail_of(v2) = v4 & edge(v2) = 0 & head_of(v2) = v3 & ((v4 = v1 & v3 = v0) | (v4 = v0 & v3 = v1)))) & ! [v0] : ! [v1] : ~ (shortest_path(v0, v0, v1) = 0) & ! [v0] : ! [v1] : ( ~ (sequential(v0, v1) = 0) | ? [v2] : (tail_of(v1) = v2 & edge(v1) = 0 & edge(v0) = 0 & head_of(v0) = v2)) & ! [v0] : ! [v1] : ( ~ (tail_of(v0) = v1) | ? [v2] : ? [v3] : ? [v4] : ((v4 = 0 & v3 = 0 & vertex(v2) = 0 & vertex(v1) = 0 & head_of(v0) = v2) | ( ~ (v2 = 0) & edge(v0) = v2))) & ! [v0] : ! [v1] : ( ~ (tail_of(v0) = v1) | ? [v2] : (( ~ (v2 = v1) & head_of(v0) = v2) | ( ~ (v2 = 0) & edge(v0) = v2))) & ! [v0] : ! [v1] : ( ~ (head_of(v0) = v1) | ? [v2] : ? [v3] : ? [v4] : ((v4 = 0 & v2 = 0 & vertex(v3) = 0 & vertex(v1) = 0 & tail_of(v0) = v3) | ( ~ (v2 = 0) & edge(v0) = v2))) & ! [v0] : ! [v1] : ( ~ (head_of(v0) = v1) | ? [v2] : (( ~ (v2 = v1) & tail_of(v0) = v2) | ( ~ (v2 = 0) & edge(v0) = v2))) & ! [v0] : ~ (sequential(v0, v0) = 0) & ! [v0] : ( ~ (edge(v0) = 0) | ? [v1] : ? [v2] : ( ~ (v2 = v1) & tail_of(v0) = v2 & head_of(v0) = v1)) & ! [v0] : ( ~ (edge(v0) = 0) | ? [v1] : ? [v2] : (vertex(v2) = 0 & vertex(v1) = 0 & tail_of(v0) = v2 & head_of(v0) = v1)) & ? [v0] : ? [v1] : ? [v2] : ? [v3] : triangle(v2, v1, v0) = v3 & ? [v0] : ? [v1] : ? [v2] : ? [v3] : shortest_path(v2, v1, v0) = v3 & ? [v0] : ? [v1] : ? [v2] : ? [v3] : precedes(v2, v1, v0) = v3 & ? [v0] : ? [v1] : ? [v2] : ? [v3] : path(v2, v1, v0) = v3 & ? [v0] : ? [v1] : ? [v2] : minus(v1, v0) = v2 & ? [v0] : ? [v1] : ? [v2] : number_of_in(v1, v0) = v2 & ? [v0] : ? [v1] : ? [v2] : less_or_equal(v1, v0) = v2 & ? [v0] : ? [v1] : ? [v2] : sequential(v1, v0) = v2 & ? [v0] : ? [v1] : ? [v2] : on_path(v1, v0) = v2 & ? [v0] : ? [v1] : ? [v2] : in_path(v1, v0) = v2 & ? [v0] : ? [v1] : ? [v2] : path_cons(v1, v0) = v2 & ? [v0] : ? [v1] : length_of(v0) = v1 & ? [v0] : ? [v1] : vertex(v0) = v1 & ? [v0] : ? [v1] : tail_of(v0) = v1 & ? [v0] : ? [v1] : edge(v0) = v1 & ? [v0] : ? [v1] : head_of(v0) = v1
% 215.25/153.16 |
% 215.25/153.16 | Applying alpha-rule on (1) yields:
% 215.25/153.16 | (2) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ( ~ (precedes(v6, v4, v0) = 0) | ~ (sequential(v3, v4) = v5) | ~ (path(v1, v2, v0) = 0) | ? [v7] : ((v7 = 0 & precedes(v3, v4, v0) = 0) | ( ~ (v7 = 0) & sequential(v3, v6) = v7) | ( ~ (v7 = 0) & on_path(v4, v0) = v7) | ( ~ (v7 = 0) & on_path(v3, v0) = v7)))
% 215.25/153.16 | (3) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v1 = v0 | ~ (path(v4, v3, v2) = v1) | ~ (path(v4, v3, v2) = v0))
% 215.25/153.16 | (4) ! [v0] : ! [v1] : ! [v2] : ( ~ (path(v0, v1, v2) = 0) | ? [v3] : ? [v4] : ? [v5] : ? [v6] : ? [v7] : ? [v8] : (tail_of(v3) = v0 & edge(v3) = 0 & head_of(v3) = v4 & ( ~ (v4 = v1) | ( ~ (v5 = v2) & path_cons(v3, empty) = v5) | ( ! [v9] : ( ~ (path_cons(v3, v9) = v2) | ? [v10] : ( ~ (v10 = 0) & path(v1, v1, v9) = v10)) & ! [v9] : ( ~ (path(v1, v1, v9) = 0) | ? [v10] : ( ~ (v10 = v2) & path_cons(v3, v9) = v10)))) & ((v8 = v2 & v7 = 0 & path_cons(v3, v6) = v2 & path(v4, v1, v6) = 0) | (v5 = v2 & v4 = v1 & path_cons(v3, empty) = v2))))
% 215.25/153.17 | (5) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v3 = 0 | ~ (path(v0, v1, v2) = v3) | ~ (tail_of(v4) = v0) | ? [v5] : ? [v6] : (( ~ (v5 = 0) & vertex(v1) = v5) | ( ~ (v5 = 0) & vertex(v0) = v5) | ( ~ (v5 = 0) & edge(v4) = v5) | (head_of(v4) = v5 & ! [v7] : ( ~ (path_cons(v4, v7) = v2) | ? [v8] : ( ~ (v8 = 0) & path(v5, v1, v7) = v8)) & ! [v7] : ( ~ (path(v5, v1, v7) = 0) | ? [v8] : ( ~ (v8 = v2) & path_cons(v4, v7) = v8)) & ( ~ (v5 = v1) | ( ~ (v6 = v2) & path_cons(v4, empty) = v6)))))
% 215.25/153.17 | (6) less_or_equal(all_0_1_1, all_0_6_6) = all_0_0_0
% 215.25/153.17 | (7) ? [v0] : ? [v1] : ? [v2] : on_path(v1, v0) = v2
% 215.25/153.17 | (8) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (shortest_path(v0, v1, v4) = 0) | ~ (precedes(v2, v3, v4) = 0) | ? [v5] : ? [v6] : ((v6 = 0 & triangle(v2, v3, v5) = 0) | ( ~ (v5 = 0) & sequential(v2, v3) = v5)))
% 215.25/153.17 | (9) ? [v0] : ? [v1] : ? [v2] : path_cons(v1, v0) = v2
% 215.25/153.17 | (10) ! [v0] : ! [v1] : ! [v2] : ( ~ (path(v0, v1, v2) = 0) | (vertex(v1) = 0 & vertex(v0) = 0))
% 215.25/153.17 | (11) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ( ~ (shortest_path(v0, v1, v4) = 0) | ~ (tail_of(v2) = v5) | ~ (head_of(v3) = v6) | ? [v7] : (( ~ (v7 = 0) & precedes(v3, v2, v4) = v7 & ! [v8] : ( ~ (tail_of(v8) = v5) | ? [v9] : ( ~ (v9 = v6) & head_of(v8) = v9)) & ! [v8] : ( ~ (head_of(v8) = v6) | ? [v9] : ( ~ (v9 = v5) & tail_of(v8) = v9))) | ( ~ (v7 = 0) & precedes(v2, v3, v4) = v7)))
% 215.25/153.17 | (12) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v3 = 0 | ~ (path(v0, v1, v2) = v3) | ~ (edge(v4) = 0) | ? [v5] : ? [v6] : (( ~ (v5 = v0) & tail_of(v4) = v5) | ( ~ (v5 = 0) & vertex(v1) = v5) | ( ~ (v5 = 0) & vertex(v0) = v5) | (head_of(v4) = v5 & ! [v7] : ( ~ (path_cons(v4, v7) = v2) | ? [v8] : ( ~ (v8 = 0) & path(v5, v1, v7) = v8)) & ! [v7] : ( ~ (path(v5, v1, v7) = 0) | ? [v8] : ( ~ (v8 = v2) & path_cons(v4, v7) = v8)) & ( ~ (v5 = v1) | ( ~ (v6 = v2) & path_cons(v4, empty) = v6)))))
% 215.25/153.17 | (13) ! [v0] : ! [v1] : (v1 = v0 | ~ (vertex(v1) = 0) | ~ (vertex(v0) = 0) | ? [v2] : ? [v3] : ? [v4] : (tail_of(v2) = v4 & edge(v2) = 0 & head_of(v2) = v3 & ((v4 = v1 & v3 = v0) | (v4 = v0 & v3 = v1))))
% 215.25/153.17 | (14) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (less_or_equal(v3, v2) = v1) | ~ (less_or_equal(v3, v2) = v0))
% 215.25/153.17 | (15) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (path(v0, v1, v2) = 0) | ~ (vertex(v3) = v4) | ? [v5] : ? [v6] : ? [v7] : ((v6 = 0 & on_path(v5, v2) = 0 & ((v7 = v3 & tail_of(v5) = v3) | (v7 = v3 & head_of(v5) = v3))) | ( ~ (v5 = 0) & in_path(v3, v2) = v5)))
% 215.25/153.17 | (16) ! [v0] : ! [v1] : ~ (shortest_path(v0, v0, v1) = 0)
% 215.25/153.17 | (17) ? [v0] : ? [v1] : tail_of(v0) = v1
% 215.25/153.17 | (18) ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | v1 = v0 | ~ (sequential(v0, v1) = v2) | ? [v3] : ? [v4] : (( ~ (v4 = v3) & tail_of(v1) = v4 & head_of(v0) = v3) | ( ~ (v3 = 0) & edge(v1) = v3) | ( ~ (v3 = 0) & edge(v0) = v3)))
% 215.25/153.17 | (19) ? [v0] : ? [v1] : edge(v0) = v1
% 215.25/153.17 | (20) ! [v0] : ! [v1] : ( ~ (head_of(v0) = v1) | ? [v2] : (( ~ (v2 = v1) & tail_of(v0) = v2) | ( ~ (v2 = 0) & edge(v0) = v2)))
% 215.25/153.17 | (21) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (path(v0, v1, v2) = 0) | ? [v3] : ? [v4] : ? [v5] : ? [v6] : ? [v7] : ((v5 = 0 & ~ (v7 = 0) & length_of(v4) = v6 & length_of(v2) = v3 & less_or_equal(v3, v6) = v7 & path(v0, v1, v4) = 0) | (v3 = 0 & shortest_path(v0, v1, v2) = 0)))
% 215.25/153.17 | (22) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (path_cons(v3, v2) = v1) | ~ (path_cons(v3, v2) = v0))
% 215.25/153.17 | (23) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v3 = 0 | ~ (path(v0, v1, v2) = v3) | ~ (head_of(v4) = v1) | ? [v5] : (( ~ (v5 = v2) & path_cons(v4, empty) = v5) | ( ~ (v5 = v0) & tail_of(v4) = v5) | ( ~ (v5 = 0) & vertex(v1) = v5) | ( ~ (v5 = 0) & vertex(v0) = v5) | ( ~ (v5 = 0) & edge(v4) = v5)))
% 215.25/153.17 | (24) ! [v0] : ! [v1] : ( ~ (tail_of(v0) = v1) | ? [v2] : (( ~ (v2 = v1) & head_of(v0) = v2) | ( ~ (v2 = 0) & edge(v0) = v2)))
% 215.25/153.17 | (25) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (minus(v3, v2) = v1) | ~ (minus(v3, v2) = v0))
% 215.25/153.18 | (26) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (path(v0, v1, v2) = v3) | ? [v4] : ((v3 = 0 & ~ (v1 = v0) & length_of(v2) = v4 & ! [v5] : ! [v6] : ( ~ (length_of(v5) = v6) | ? [v7] : ((v7 = 0 & less_or_equal(v4, v6) = 0) | ( ~ (v7 = 0) & path(v0, v1, v5) = v7))) & ! [v5] : ( ~ (path(v0, v1, v5) = 0) | ? [v6] : (length_of(v5) = v6 & less_or_equal(v4, v6) = 0))) | ( ~ (v4 = 0) & shortest_path(v0, v1, v2) = v4)))
% 215.25/153.18 | (27) ? [v0] : ? [v1] : vertex(v0) = v1
% 215.25/153.18 | (28) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (on_path(v3, v2) = v1) | ~ (on_path(v3, v2) = v0))
% 215.25/153.18 | (29) ? [v0] : ? [v1] : ? [v2] : ? [v3] : precedes(v2, v1, v0) = v3
% 215.25/153.18 | (30) ! [v0] : ( ~ (edge(v0) = 0) | ? [v1] : ? [v2] : ( ~ (v2 = v1) & tail_of(v0) = v2 & head_of(v0) = v1))
% 215.25/153.18 | (31) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ( ~ (in_path(v4, v2) = v5) | ~ (path(v0, v1, v2) = 0) | ~ (head_of(v3) = v4) | ? [v6] : ? [v7] : ? [v8] : ((v8 = 0 & v6 = 0 & v5 = 0 & in_path(v7, v2) = 0 & tail_of(v3) = v7 & edge(v3) = 0) | ( ~ (v6 = 0) & on_path(v3, v2) = v6)))
% 215.25/153.18 | (32) ! [v0] : ! [v1] : ( ~ (tail_of(v0) = v1) | ? [v2] : ? [v3] : ? [v4] : ((v4 = 0 & v3 = 0 & vertex(v2) = 0 & vertex(v1) = 0 & head_of(v0) = v2) | ( ~ (v2 = 0) & edge(v0) = v2)))
% 215.25/153.18 | (33) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v5 = 0 | ~ (precedes(v3, v4, v0) = v5) | ~ (path(v1, v2, v0) = 0) | ? [v6] : (( ~ (v6 = 0) & sequential(v3, v4) = v6 & ! [v7] : ( ~ (precedes(v7, v4, v0) = 0) | ? [v8] : ( ~ (v8 = 0) & sequential(v3, v7) = v8)) & ! [v7] : ( ~ (sequential(v3, v7) = 0) | ? [v8] : ( ~ (v8 = 0) & precedes(v7, v4, v0) = v8))) | ( ~ (v6 = 0) & on_path(v4, v0) = v6) | ( ~ (v6 = 0) & on_path(v3, v0) = v6)))
% 215.25/153.18 | (34) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ( ~ (sequential(v3, v6) = 0) | ~ (sequential(v3, v4) = v5) | ~ (path(v1, v2, v0) = 0) | ? [v7] : ((v7 = 0 & precedes(v3, v4, v0) = 0) | ( ~ (v7 = 0) & precedes(v6, v4, v0) = v7) | ( ~ (v7 = 0) & on_path(v4, v0) = v7) | ( ~ (v7 = 0) & on_path(v3, v0) = v7)))
% 215.25/153.18 | (35) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (number_of_in(v3, v2) = v1) | ~ (number_of_in(v3, v2) = v0))
% 215.25/153.18 | (36) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | ~ (triangle(v0, v1, v2) = v3) | ? [v4] : (( ~ (v4 = 0) & sequential(v2, v0) = v4) | ( ~ (v4 = 0) & sequential(v1, v2) = v4) | ( ~ (v4 = 0) & sequential(v0, v1) = v4) | ( ~ (v4 = 0) & edge(v2) = v4) | ( ~ (v4 = 0) & edge(v1) = v4) | ( ~ (v4 = 0) & edge(v0) = v4)))
% 215.25/153.18 | (37) ? [v0] : ? [v1] : ? [v2] : in_path(v1, v0) = v2
% 215.25/153.18 | (38) ! [v0] : ( ~ (edge(v0) = 0) | ? [v1] : ? [v2] : (vertex(v2) = 0 & vertex(v1) = 0 & tail_of(v0) = v2 & head_of(v0) = v1))
% 215.25/153.18 | (39) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ( ~ (in_path(v4, v2) = v5) | ~ (path(v0, v1, v2) = 0) | ~ (tail_of(v3) = v4) | ? [v6] : ? [v7] : ? [v8] : ((v8 = 0 & v6 = 0 & v5 = 0 & in_path(v7, v2) = 0 & edge(v3) = 0 & head_of(v3) = v7) | ( ~ (v6 = 0) & on_path(v3, v2) = v6)))
% 215.25/153.18 | (40) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (path(v0, v1, v2) = 0) | ~ (edge(v3) = v4) | ? [v5] : ? [v6] : ? [v7] : ? [v8] : ((v8 = 0 & v6 = 0 & v4 = 0 & in_path(v7, v2) = 0 & in_path(v5, v2) = 0 & tail_of(v3) = v7 & head_of(v3) = v5) | ( ~ (v5 = 0) & on_path(v3, v2) = v5)))
% 215.25/153.18 | (41) ! [v0] : ! [v1] : ! [v2] : ( ~ (path(v1, v2, v0) = 0) | ? [v3] : ? [v4] : ? [v5] : ? [v6] : ? [v7] : ((v7 = 0 & v6 = 0 & v5 = 0 & sequential(v3, v4) = 0 & on_path(v4, v0) = 0 & on_path(v3, v0) = 0 & ! [v8] : ~ (triangle(v3, v4, v8) = 0)) | (v4 = v3 & number_of_in(triangles, v0) = v3 & number_of_in(sequential_pairs, v0) = v3)))
% 215.25/153.18 | (42) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (sequential(v3, v2) = v1) | ~ (sequential(v3, v2) = v0))
% 215.25/153.18 | (43) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (in_path(v3, v2) = 0) | ~ (path(v0, v1, v2) = 0) | vertex(v3) = 0)
% 215.25/153.18 | (44) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (tail_of(v2) = v1) | ~ (tail_of(v2) = v0))
% 215.25/153.18 | (45) complete
% 215.25/153.18 | (46) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ( ~ (sequential(v3, v4) = v5) | ~ (path(v1, v2, v0) = 0) | ? [v6] : ? [v7] : ? [v8] : ? [v9] : ? [v10] : ((v7 = 0 & v6 = 0 & on_path(v4, v0) = 0 & on_path(v3, v0) = 0 & ( ~ (v5 = 0) | ( ! [v11] : ( ~ (precedes(v11, v4, v0) = 0) | ? [v12] : ( ~ (v12 = 0) & sequential(v3, v11) = v12)) & ! [v11] : ( ~ (sequential(v3, v11) = 0) | ? [v12] : ( ~ (v12 = 0) & precedes(v11, v4, v0) = v12)))) & (v5 = 0 | (v10 = 0 & v9 = 0 & precedes(v8, v4, v0) = 0 & sequential(v3, v8) = 0))) | ( ~ (v6 = 0) & precedes(v3, v4, v0) = v6)))
% 215.25/153.19 | (47) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ( ~ (shortest_path(v0, v1, v4) = 0) | ~ (precedes(v3, v2, v4) = v5) | ? [v6] : ? [v7] : (( ~ (v6 = 0) & precedes(v2, v3, v4) = v6) | ( ~ (v5 = 0) & tail_of(v2) = v6 & head_of(v3) = v7 & ! [v8] : ( ~ (tail_of(v8) = v6) | ? [v9] : ( ~ (v9 = v7) & head_of(v8) = v9)) & ! [v8] : ( ~ (head_of(v8) = v7) | ? [v9] : ( ~ (v9 = v6) & tail_of(v8) = v9)))))
% 215.25/153.19 | (48) ? [v0] : ? [v1] : ? [v2] : ? [v3] : shortest_path(v2, v1, v0) = v3
% 215.25/153.19 | (49) ! [v0] : ! [v1] : ! [v2] : ( ~ (triangle(v0, v1, v2) = 0) | (sequential(v2, v0) = 0 & sequential(v1, v2) = 0 & sequential(v0, v1) = 0 & edge(v2) = 0 & edge(v1) = 0 & edge(v0) = 0))
% 215.25/153.19 | (50) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (precedes(v3, v4, v0) = 0) | ~ (path(v1, v2, v0) = 0) | ? [v5] : ? [v6] : ? [v7] : ? [v8] : (on_path(v4, v0) = 0 & on_path(v3, v0) = 0 & ((v8 = 0 & v7 = 0 & precedes(v6, v4, v0) = 0 & sequential(v3, v6) = 0) | (v5 = 0 & sequential(v3, v4) = 0)) & (( ~ (v5 = 0) & sequential(v3, v4) = v5) | ( ! [v9] : ( ~ (precedes(v9, v4, v0) = 0) | ? [v10] : ( ~ (v10 = 0) & sequential(v3, v9) = v10)) & ! [v9] : ( ~ (sequential(v3, v9) = 0) | ? [v10] : ( ~ (v10 = 0) & precedes(v9, v4, v0) = v10))))))
% 215.25/153.19 | (51) ? [v0] : ? [v1] : length_of(v0) = v1
% 215.25/153.19 | (52) ? [v0] : ? [v1] : ? [v2] : ? [v3] : path(v2, v1, v0) = v3
% 215.25/153.19 | (53) ? [v0] : ? [v1] : ? [v2] : sequential(v1, v0) = v2
% 215.25/153.19 | (54) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v1 = v0 | ~ (triangle(v4, v3, v2) = v1) | ~ (triangle(v4, v3, v2) = v0))
% 215.25/153.19 | (55) number_of_in(triangles, graph) = all_0_6_6
% 215.25/153.19 | (56) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (edge(v2) = v1) | ~ (edge(v2) = v0))
% 215.25/153.19 | (57) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v3 = 0 | ~ (path_cons(v4, empty) = v5) | ~ (path(v0, v1, v2) = v3) | ? [v6] : (( ~ (v6 = v0) & tail_of(v4) = v6) | ( ~ (v6 = 0) & vertex(v1) = v6) | ( ~ (v6 = 0) & vertex(v0) = v6) | ( ~ (v6 = 0) & edge(v4) = v6) | (head_of(v4) = v6 & ! [v7] : ( ~ (path_cons(v4, v7) = v2) | ? [v8] : ( ~ (v8 = 0) & path(v6, v1, v7) = v8)) & ! [v7] : ( ~ (path(v6, v1, v7) = 0) | ? [v8] : ( ~ (v8 = v2) & path_cons(v4, v7) = v8)) & ( ~ (v6 = v1) | ~ (v5 = v2)))))
% 215.25/153.19 | (58) minus(all_0_2_2, n1) = all_0_1_1
% 215.25/153.19 | (59) length_of(all_0_5_5) = all_0_2_2
% 215.25/153.19 | (60) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v1 = v0 | ~ (precedes(v4, v3, v2) = v1) | ~ (precedes(v4, v3, v2) = v0))
% 215.25/153.19 | (61) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (shortest_path(v0, v1, v4) = 0) | ~ (precedes(v2, v3, v4) = 0) | ? [v5] : ? [v6] : ? [v7] : ( ~ (v7 = 0) & precedes(v3, v2, v4) = v7 & tail_of(v2) = v5 & head_of(v3) = v6 & ! [v8] : ( ~ (tail_of(v8) = v5) | ? [v9] : ( ~ (v9 = v6) & head_of(v8) = v9)) & ! [v8] : ( ~ (head_of(v8) = v6) | ? [v9] : ( ~ (v9 = v5) & tail_of(v8) = v9))))
% 215.25/153.19 | (62) ! [v0] : ! [v1] : ! [v2] : ( ~ (path(v0, v1, v2) = 0) | ? [v3] : ? [v4] : (minus(v4, n1) = v3 & number_of_in(sequential_pairs, v2) = v3 & length_of(v2) = v4))
% 215.25/153.19 | (63) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v3 = 0 | ~ (path_cons(v4, v6) = v2) | ~ (path(v0, v1, v2) = v3) | ~ (head_of(v4) = v5) | ? [v7] : (( ~ (v7 = v0) & tail_of(v4) = v7) | ( ~ (v7 = 0) & path(v5, v1, v6) = v7) | ( ~ (v7 = 0) & vertex(v1) = v7) | ( ~ (v7 = 0) & vertex(v0) = v7) | ( ~ (v7 = 0) & edge(v4) = v7)))
% 215.25/153.19 | (64) ? [v0] : ? [v1] : ? [v2] : less_or_equal(v1, v0) = v2
% 215.25/153.19 | (65) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (sequential(v3, v4) = 0) | ~ (path(v1, v2, v0) = 0) | ? [v5] : ((v5 = 0 & precedes(v3, v4, v0) = 0) | ( ~ (v5 = 0) & on_path(v4, v0) = v5) | ( ~ (v5 = 0) & on_path(v3, v0) = v5)))
% 215.25/153.19 | (66) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (in_path(v3, v2) = v1) | ~ (in_path(v3, v2) = v0))
% 215.25/153.19 | (67) ! [v0] : ! [v1] : ! [v2] : ( ~ (path(v0, v1, v2) = 0) | ? [v3] : (number_of_in(edges, v2) = v3 & length_of(v2) = v3))
% 215.25/153.19 | (68) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (head_of(v2) = v1) | ~ (head_of(v2) = v0))
% 215.25/153.19 | (69) ! [v0] : ! [v1] : ! [v2] : ( ~ (number_of_in(v0, v1) = v2) | ? [v3] : (number_of_in(v0, graph) = v3 & less_or_equal(v2, v3) = 0))
% 215.25/153.19 | (70) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (in_path(v3, v2) = 0) | ~ (path(v0, v1, v2) = 0) | ? [v4] : ? [v5] : (on_path(v4, v2) = 0 & ((v5 = v3 & tail_of(v4) = v3) | (v5 = v3 & head_of(v4) = v3))))
% 215.25/153.19 | (71) ? [v0] : ? [v1] : ? [v2] : minus(v1, v0) = v2
% 215.25/153.19 | (72) ! [v0] : ~ (sequential(v0, v0) = 0)
% 215.25/153.20 | (73) ! [v0] : ! [v1] : ( ~ (head_of(v0) = v1) | ? [v2] : ? [v3] : ? [v4] : ((v4 = 0 & v2 = 0 & vertex(v3) = 0 & vertex(v1) = 0 & tail_of(v0) = v3) | ( ~ (v2 = 0) & edge(v0) = v2)))
% 215.25/153.20 | (74) ? [v0] : ? [v1] : ? [v2] : ? [v3] : triangle(v2, v1, v0) = v3
% 215.25/153.20 | (75) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : (v3 = 0 | ~ (path(v5, v1, v6) = 0) | ~ (path(v0, v1, v2) = v3) | ~ (head_of(v4) = v5) | ? [v7] : (( ~ (v7 = v2) & path_cons(v4, v6) = v7) | ( ~ (v7 = v0) & tail_of(v4) = v7) | ( ~ (v7 = 0) & vertex(v1) = v7) | ( ~ (v7 = 0) & vertex(v0) = v7) | ( ~ (v7 = 0) & edge(v4) = v7)))
% 215.25/153.20 | (76) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (length_of(v2) = v1) | ~ (length_of(v2) = v0))
% 215.25/153.20 | (77) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (on_path(v3, v2) = 0) | ~ (path(v0, v1, v2) = 0) | ? [v4] : ? [v5] : (in_path(v5, v2) = 0 & in_path(v4, v2) = 0 & tail_of(v3) = v5 & edge(v3) = 0 & head_of(v3) = v4))
% 215.25/153.20 | (78) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v1 = v0 | ~ (shortest_path(v4, v3, v2) = v1) | ~ (shortest_path(v4, v3, v2) = v0))
% 215.25/153.20 | (79) ! [v0] : ! [v1] : ( ~ (sequential(v0, v1) = 0) | ? [v2] : (tail_of(v1) = v2 & edge(v1) = 0 & edge(v0) = 0 & head_of(v0) = v2))
% 215.25/153.20 | (80) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = 0 | v1 = v0 | ~ (shortest_path(v0, v1, v2) = v3) | ? [v4] : ? [v5] : ? [v6] : ? [v7] : ? [v8] : ((v6 = 0 & ~ (v8 = 0) & length_of(v5) = v7 & length_of(v2) = v4 & less_or_equal(v4, v7) = v8 & path(v0, v1, v5) = 0) | ( ~ (v4 = 0) & path(v0, v1, v2) = v4)))
% 215.25/153.20 | (81) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (vertex(v2) = v1) | ~ (vertex(v2) = v0))
% 215.25/153.20 | (82) ? [v0] : ? [v1] : ? [v2] : number_of_in(v1, v0) = v2
% 215.25/153.20 | (83) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (shortest_path(v0, v1, v4) = 0) | ~ (sequential(v2, v3) = 0) | ? [v5] : ? [v6] : ((v6 = 0 & triangle(v2, v3, v5) = 0) | ( ~ (v5 = 0) & precedes(v2, v3, v4) = v5)))
% 215.25/153.20 | (84) ! [v0] : ! [v1] : ! [v2] : ( ~ (shortest_path(v0, v1, v2) = 0) | ? [v3] : (length_of(v2) = v3 & path(v0, v1, v2) = 0 & ! [v4] : ! [v5] : ( ~ (length_of(v4) = v5) | ? [v6] : ((v6 = 0 & less_or_equal(v3, v5) = 0) | ( ~ (v6 = 0) & path(v0, v1, v4) = v6))) & ! [v4] : ( ~ (path(v0, v1, v4) = 0) | ? [v5] : (length_of(v4) = v5 & less_or_equal(v3, v5) = 0))))
% 215.25/153.20 | (85) ~ (all_0_0_0 = 0)
% 215.25/153.20 | (86) shortest_path(all_0_4_4, all_0_3_3, all_0_5_5) = 0
% 215.25/153.20 | (87) ? [v0] : ? [v1] : head_of(v0) = v1
% 215.25/153.20 | (88) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : (v4 = 0 | ~ (path(v0, v1, v2) = 0) | ~ (vertex(v3) = v4) | ? [v5] : ( ~ (v5 = 0) & in_path(v3, v2) = v5))
% 215.25/153.20 |
% 215.25/153.20 | Instantiating formula (69) with all_0_6_6, graph, triangles and discharging atoms number_of_in(triangles, graph) = all_0_6_6, yields:
% 215.25/153.20 | (89) ? [v0] : (number_of_in(triangles, graph) = v0 & less_or_equal(all_0_6_6, v0) = 0)
% 215.25/153.20 |
% 215.25/153.20 | Instantiating formula (84) with all_0_5_5, all_0_3_3, all_0_4_4 and discharging atoms shortest_path(all_0_4_4, all_0_3_3, all_0_5_5) = 0, yields:
% 215.25/153.20 | (90) ? [v0] : (length_of(all_0_5_5) = v0 & path(all_0_4_4, all_0_3_3, all_0_5_5) = 0 & ! [v1] : ! [v2] : ( ~ (length_of(v1) = v2) | ? [v3] : ((v3 = 0 & less_or_equal(v0, v2) = 0) | ( ~ (v3 = 0) & path(all_0_4_4, all_0_3_3, v1) = v3))) & ! [v1] : ( ~ (path(all_0_4_4, all_0_3_3, v1) = 0) | ? [v2] : (length_of(v1) = v2 & less_or_equal(v0, v2) = 0)))
% 215.25/153.20 |
% 215.25/153.20 | Instantiating (90) with all_40_0_54 yields:
% 215.25/153.20 | (91) length_of(all_0_5_5) = all_40_0_54 & path(all_0_4_4, all_0_3_3, all_0_5_5) = 0 & ! [v0] : ! [v1] : ( ~ (length_of(v0) = v1) | ? [v2] : ((v2 = 0 & less_or_equal(all_40_0_54, v1) = 0) | ( ~ (v2 = 0) & path(all_0_4_4, all_0_3_3, v0) = v2))) & ! [v0] : ( ~ (path(all_0_4_4, all_0_3_3, v0) = 0) | ? [v1] : (length_of(v0) = v1 & less_or_equal(all_40_0_54, v1) = 0))
% 215.25/153.20 |
% 215.25/153.20 | Applying alpha-rule on (91) yields:
% 215.25/153.20 | (92) length_of(all_0_5_5) = all_40_0_54
% 215.25/153.20 | (93) path(all_0_4_4, all_0_3_3, all_0_5_5) = 0
% 215.25/153.21 | (94) ! [v0] : ! [v1] : ( ~ (length_of(v0) = v1) | ? [v2] : ((v2 = 0 & less_or_equal(all_40_0_54, v1) = 0) | ( ~ (v2 = 0) & path(all_0_4_4, all_0_3_3, v0) = v2)))
% 215.25/153.21 | (95) ! [v0] : ( ~ (path(all_0_4_4, all_0_3_3, v0) = 0) | ? [v1] : (length_of(v0) = v1 & less_or_equal(all_40_0_54, v1) = 0))
% 215.25/153.21 |
% 215.25/153.21 | Instantiating (89) with all_43_0_55 yields:
% 215.25/153.21 | (96) number_of_in(triangles, graph) = all_43_0_55 & less_or_equal(all_0_6_6, all_43_0_55) = 0
% 215.25/153.21 |
% 215.25/153.21 | Applying alpha-rule on (96) yields:
% 215.25/153.21 | (97) number_of_in(triangles, graph) = all_43_0_55
% 215.25/153.21 | (98) less_or_equal(all_0_6_6, all_43_0_55) = 0
% 215.25/153.21 |
% 215.25/153.21 | Instantiating formula (35) with triangles, graph, all_43_0_55, all_0_6_6 and discharging atoms number_of_in(triangles, graph) = all_43_0_55, number_of_in(triangles, graph) = all_0_6_6, yields:
% 215.25/153.21 | (99) all_43_0_55 = all_0_6_6
% 215.25/153.21 |
% 215.25/153.21 | Instantiating formula (76) with all_0_5_5, all_40_0_54, all_0_2_2 and discharging atoms length_of(all_0_5_5) = all_40_0_54, length_of(all_0_5_5) = all_0_2_2, yields:
% 215.25/153.21 | (100) all_40_0_54 = all_0_2_2
% 215.25/153.21 |
% 215.25/153.21 | From (99) and (97) follows:
% 215.25/153.21 | (55) number_of_in(triangles, graph) = all_0_6_6
% 215.25/153.21 |
% 215.25/153.21 | From (100) and (92) follows:
% 215.25/153.21 | (59) length_of(all_0_5_5) = all_0_2_2
% 215.25/153.21 |
% 215.25/153.21 | Instantiating formula (41) with all_0_3_3, all_0_4_4, all_0_5_5 and discharging atoms path(all_0_4_4, all_0_3_3, all_0_5_5) = 0, yields:
% 215.25/153.21 | (103) ? [v0] : ? [v1] : ? [v2] : ? [v3] : ? [v4] : ((v4 = 0 & v3 = 0 & v2 = 0 & sequential(v0, v1) = 0 & on_path(v1, all_0_5_5) = 0 & on_path(v0, all_0_5_5) = 0 & ! [v5] : ~ (triangle(v0, v1, v5) = 0)) | (v1 = v0 & number_of_in(triangles, all_0_5_5) = v0 & number_of_in(sequential_pairs, all_0_5_5) = v0))
% 215.25/153.21 |
% 215.25/153.21 | Instantiating formula (62) with all_0_5_5, all_0_3_3, all_0_4_4 and discharging atoms path(all_0_4_4, all_0_3_3, all_0_5_5) = 0, yields:
% 215.25/153.21 | (104) ? [v0] : ? [v1] : (minus(v1, n1) = v0 & number_of_in(sequential_pairs, all_0_5_5) = v0 & length_of(all_0_5_5) = v1)
% 215.25/153.21 |
% 215.25/153.21 | Instantiating formula (67) with all_0_5_5, all_0_3_3, all_0_4_4 and discharging atoms path(all_0_4_4, all_0_3_3, all_0_5_5) = 0, yields:
% 215.25/153.21 | (105) ? [v0] : (number_of_in(edges, all_0_5_5) = v0 & length_of(all_0_5_5) = v0)
% 215.25/153.21 |
% 215.25/153.21 | Instantiating formula (26) with 0, all_0_5_5, all_0_3_3, all_0_4_4 and discharging atoms path(all_0_4_4, all_0_3_3, all_0_5_5) = 0, yields:
% 215.25/153.21 | (106) ? [v0] : (( ~ (v0 = 0) & shortest_path(all_0_4_4, all_0_3_3, all_0_5_5) = v0) | ( ~ (all_0_3_3 = all_0_4_4) & length_of(all_0_5_5) = v0 & ! [v1] : ! [v2] : ( ~ (length_of(v1) = v2) | ? [v3] : ((v3 = 0 & less_or_equal(v0, v2) = 0) | ( ~ (v3 = 0) & path(all_0_4_4, all_0_3_3, v1) = v3))) & ! [v1] : ( ~ (path(all_0_4_4, all_0_3_3, v1) = 0) | ? [v2] : (length_of(v1) = v2 & less_or_equal(v0, v2) = 0))))
% 215.25/153.21 |
% 215.25/153.21 | Instantiating (105) with all_60_0_57 yields:
% 215.25/153.21 | (107) number_of_in(edges, all_0_5_5) = all_60_0_57 & length_of(all_0_5_5) = all_60_0_57
% 215.25/153.21 |
% 215.25/153.21 | Applying alpha-rule on (107) yields:
% 215.25/153.21 | (108) number_of_in(edges, all_0_5_5) = all_60_0_57
% 215.25/153.21 | (109) length_of(all_0_5_5) = all_60_0_57
% 215.25/153.21 |
% 215.25/153.21 | Instantiating (104) with all_62_0_58, all_62_1_59 yields:
% 215.25/153.21 | (110) minus(all_62_0_58, n1) = all_62_1_59 & number_of_in(sequential_pairs, all_0_5_5) = all_62_1_59 & length_of(all_0_5_5) = all_62_0_58
% 215.25/153.21 |
% 215.25/153.21 | Applying alpha-rule on (110) yields:
% 215.25/153.21 | (111) minus(all_62_0_58, n1) = all_62_1_59
% 215.25/153.21 | (112) number_of_in(sequential_pairs, all_0_5_5) = all_62_1_59
% 215.25/153.21 | (113) length_of(all_0_5_5) = all_62_0_58
% 215.25/153.21 |
% 215.25/153.21 | Instantiating (103) with all_66_0_66, all_66_1_67, all_66_2_68, all_66_3_69, all_66_4_70 yields:
% 215.25/153.21 | (114) (all_66_0_66 = 0 & all_66_1_67 = 0 & all_66_2_68 = 0 & sequential(all_66_4_70, all_66_3_69) = 0 & on_path(all_66_3_69, all_0_5_5) = 0 & on_path(all_66_4_70, all_0_5_5) = 0 & ! [v0] : ~ (triangle(all_66_4_70, all_66_3_69, v0) = 0)) | (all_66_3_69 = all_66_4_70 & number_of_in(triangles, all_0_5_5) = all_66_4_70 & number_of_in(sequential_pairs, all_0_5_5) = all_66_4_70)
% 215.25/153.21 |
% 215.25/153.21 | Instantiating (106) with all_67_0_71 yields:
% 215.25/153.21 | (115) ( ~ (all_67_0_71 = 0) & shortest_path(all_0_4_4, all_0_3_3, all_0_5_5) = all_67_0_71) | ( ~ (all_0_3_3 = all_0_4_4) & length_of(all_0_5_5) = all_67_0_71 & ! [v0] : ! [v1] : ( ~ (length_of(v0) = v1) | ? [v2] : ((v2 = 0 & less_or_equal(all_67_0_71, v1) = 0) | ( ~ (v2 = 0) & path(all_0_4_4, all_0_3_3, v0) = v2))) & ! [v0] : ( ~ (path(all_0_4_4, all_0_3_3, v0) = 0) | ? [v1] : (length_of(v0) = v1 & less_or_equal(all_67_0_71, v1) = 0)))
% 215.25/153.21 |
% 215.25/153.21 +-Applying beta-rule and splitting (115), into two cases.
% 215.25/153.21 |-Branch one:
% 215.25/153.21 | (116) ~ (all_67_0_71 = 0) & shortest_path(all_0_4_4, all_0_3_3, all_0_5_5) = all_67_0_71
% 215.25/153.21 |
% 215.25/153.21 | Applying alpha-rule on (116) yields:
% 215.25/153.21 | (117) ~ (all_67_0_71 = 0)
% 215.25/153.21 | (118) shortest_path(all_0_4_4, all_0_3_3, all_0_5_5) = all_67_0_71
% 215.25/153.21 |
% 215.25/153.21 | Instantiating formula (78) with all_0_4_4, all_0_3_3, all_0_5_5, all_67_0_71, 0 and discharging atoms shortest_path(all_0_4_4, all_0_3_3, all_0_5_5) = all_67_0_71, shortest_path(all_0_4_4, all_0_3_3, all_0_5_5) = 0, yields:
% 215.25/153.21 | (119) all_67_0_71 = 0
% 215.25/153.21 |
% 215.25/153.21 | Equations (119) can reduce 117 to:
% 215.25/153.21 | (120) $false
% 215.25/153.21 |
% 215.25/153.21 |-The branch is then unsatisfiable
% 215.25/153.21 |-Branch two:
% 215.25/153.21 | (121) ~ (all_0_3_3 = all_0_4_4) & length_of(all_0_5_5) = all_67_0_71 & ! [v0] : ! [v1] : ( ~ (length_of(v0) = v1) | ? [v2] : ((v2 = 0 & less_or_equal(all_67_0_71, v1) = 0) | ( ~ (v2 = 0) & path(all_0_4_4, all_0_3_3, v0) = v2))) & ! [v0] : ( ~ (path(all_0_4_4, all_0_3_3, v0) = 0) | ? [v1] : (length_of(v0) = v1 & less_or_equal(all_67_0_71, v1) = 0))
% 215.25/153.21 |
% 215.25/153.21 | Applying alpha-rule on (121) yields:
% 215.25/153.21 | (122) ~ (all_0_3_3 = all_0_4_4)
% 215.25/153.21 | (123) length_of(all_0_5_5) = all_67_0_71
% 215.25/153.21 | (124) ! [v0] : ! [v1] : ( ~ (length_of(v0) = v1) | ? [v2] : ((v2 = 0 & less_or_equal(all_67_0_71, v1) = 0) | ( ~ (v2 = 0) & path(all_0_4_4, all_0_3_3, v0) = v2)))
% 215.25/153.21 | (125) ! [v0] : ( ~ (path(all_0_4_4, all_0_3_3, v0) = 0) | ? [v1] : (length_of(v0) = v1 & less_or_equal(all_67_0_71, v1) = 0))
% 215.25/153.21 |
% 215.25/153.21 | Instantiating formula (125) with all_0_5_5 and discharging atoms path(all_0_4_4, all_0_3_3, all_0_5_5) = 0, yields:
% 215.25/153.21 | (126) ? [v0] : (length_of(all_0_5_5) = v0 & less_or_equal(all_67_0_71, v0) = 0)
% 215.25/153.21 |
% 215.25/153.21 | Instantiating (126) with all_73_0_72 yields:
% 215.25/153.21 | (127) length_of(all_0_5_5) = all_73_0_72 & less_or_equal(all_67_0_71, all_73_0_72) = 0
% 215.25/153.22 |
% 215.25/153.22 | Applying alpha-rule on (127) yields:
% 215.25/153.22 | (128) length_of(all_0_5_5) = all_73_0_72
% 215.25/153.22 | (129) less_or_equal(all_67_0_71, all_73_0_72) = 0
% 215.25/153.22 |
% 215.25/153.22 | Instantiating formula (76) with all_0_5_5, all_67_0_71, all_0_2_2 and discharging atoms length_of(all_0_5_5) = all_67_0_71, length_of(all_0_5_5) = all_0_2_2, yields:
% 215.25/153.22 | (130) all_67_0_71 = all_0_2_2
% 215.25/153.22 |
% 215.25/153.22 | Instantiating formula (76) with all_0_5_5, all_62_0_58, all_73_0_72 and discharging atoms length_of(all_0_5_5) = all_73_0_72, length_of(all_0_5_5) = all_62_0_58, yields:
% 215.25/153.22 | (131) all_73_0_72 = all_62_0_58
% 215.25/153.22 |
% 215.25/153.22 | Instantiating formula (76) with all_0_5_5, all_62_0_58, all_67_0_71 and discharging atoms length_of(all_0_5_5) = all_67_0_71, length_of(all_0_5_5) = all_62_0_58, yields:
% 215.25/153.22 | (132) all_67_0_71 = all_62_0_58
% 215.25/153.22 |
% 215.25/153.22 | Instantiating formula (76) with all_0_5_5, all_60_0_57, all_73_0_72 and discharging atoms length_of(all_0_5_5) = all_73_0_72, length_of(all_0_5_5) = all_60_0_57, yields:
% 215.25/153.22 | (133) all_73_0_72 = all_60_0_57
% 215.25/153.22 |
% 215.25/153.22 | Combining equations (131,133) yields a new equation:
% 215.25/153.22 | (134) all_62_0_58 = all_60_0_57
% 215.25/153.22 |
% 215.25/153.22 | Simplifying 134 yields:
% 215.25/153.22 | (135) all_62_0_58 = all_60_0_57
% 215.25/153.22 |
% 215.25/153.22 | Combining equations (132,130) yields a new equation:
% 215.25/153.22 | (136) all_62_0_58 = all_0_2_2
% 215.25/153.22 |
% 215.25/153.22 | Simplifying 136 yields:
% 215.25/153.22 | (137) all_62_0_58 = all_0_2_2
% 215.25/153.22 |
% 215.25/153.22 | Combining equations (137,135) yields a new equation:
% 215.25/153.22 | (138) all_60_0_57 = all_0_2_2
% 215.25/153.22 |
% 215.25/153.22 | Combining equations (138,135) yields a new equation:
% 215.25/153.22 | (137) all_62_0_58 = all_0_2_2
% 215.25/153.22 |
% 215.25/153.22 | From (137) and (111) follows:
% 215.25/153.22 | (140) minus(all_0_2_2, n1) = all_62_1_59
% 215.25/153.22 |
% 215.25/153.22 | Instantiating formula (25) with all_0_2_2, n1, all_62_1_59, all_0_1_1 and discharging atoms minus(all_0_2_2, n1) = all_62_1_59, minus(all_0_2_2, n1) = all_0_1_1, yields:
% 215.25/153.22 | (141) all_62_1_59 = all_0_1_1
% 215.25/153.22 |
% 215.25/153.22 | From (141) and (112) follows:
% 215.25/153.22 | (142) number_of_in(sequential_pairs, all_0_5_5) = all_0_1_1
% 215.25/153.22 |
% 215.25/153.22 +-Applying beta-rule and splitting (114), into two cases.
% 215.25/153.22 |-Branch one:
% 215.25/153.22 | (143) all_66_0_66 = 0 & all_66_1_67 = 0 & all_66_2_68 = 0 & sequential(all_66_4_70, all_66_3_69) = 0 & on_path(all_66_3_69, all_0_5_5) = 0 & on_path(all_66_4_70, all_0_5_5) = 0 & ! [v0] : ~ (triangle(all_66_4_70, all_66_3_69, v0) = 0)
% 215.25/153.22 |
% 215.25/153.22 | Applying alpha-rule on (143) yields:
% 215.25/153.22 | (144) all_66_0_66 = 0
% 215.25/153.22 | (145) all_66_1_67 = 0
% 215.25/153.22 | (146) all_66_2_68 = 0
% 215.25/153.22 | (147) on_path(all_66_4_70, all_0_5_5) = 0
% 215.25/153.22 | (148) ! [v0] : ~ (triangle(all_66_4_70, all_66_3_69, v0) = 0)
% 215.25/153.22 | (149) on_path(all_66_3_69, all_0_5_5) = 0
% 215.25/153.22 | (150) sequential(all_66_4_70, all_66_3_69) = 0
% 215.25/153.22 |
% 215.25/153.22 | Instantiating formula (83) with all_0_5_5, all_66_3_69, all_66_4_70, all_0_3_3, all_0_4_4 and discharging atoms shortest_path(all_0_4_4, all_0_3_3, all_0_5_5) = 0, sequential(all_66_4_70, all_66_3_69) = 0, yields:
% 215.25/153.22 | (151) ? [v0] : ? [v1] : ((v1 = 0 & triangle(all_66_4_70, all_66_3_69, v0) = 0) | ( ~ (v0 = 0) & precedes(all_66_4_70, all_66_3_69, all_0_5_5) = v0))
% 215.25/153.22 |
% 215.25/153.22 | Instantiating formula (65) with all_66_3_69, all_66_4_70, all_0_3_3, all_0_4_4, all_0_5_5 and discharging atoms sequential(all_66_4_70, all_66_3_69) = 0, path(all_0_4_4, all_0_3_3, all_0_5_5) = 0, yields:
% 215.25/153.22 | (152) ? [v0] : ((v0 = 0 & precedes(all_66_4_70, all_66_3_69, all_0_5_5) = 0) | ( ~ (v0 = 0) & on_path(all_66_3_69, all_0_5_5) = v0) | ( ~ (v0 = 0) & on_path(all_66_4_70, all_0_5_5) = v0))
% 215.25/153.22 |
% 215.25/153.22 | Instantiating (152) with all_285_0_193 yields:
% 215.25/153.22 | (153) (all_285_0_193 = 0 & precedes(all_66_4_70, all_66_3_69, all_0_5_5) = 0) | ( ~ (all_285_0_193 = 0) & on_path(all_66_3_69, all_0_5_5) = all_285_0_193) | ( ~ (all_285_0_193 = 0) & on_path(all_66_4_70, all_0_5_5) = all_285_0_193)
% 215.25/153.22 |
% 215.25/153.22 | Instantiating (151) with all_286_0_194, all_286_1_195 yields:
% 215.25/153.22 | (154) (all_286_0_194 = 0 & triangle(all_66_4_70, all_66_3_69, all_286_1_195) = 0) | ( ~ (all_286_1_195 = 0) & precedes(all_66_4_70, all_66_3_69, all_0_5_5) = all_286_1_195)
% 215.25/153.22 |
% 215.25/153.22 +-Applying beta-rule and splitting (153), into two cases.
% 215.25/153.22 |-Branch one:
% 215.25/153.22 | (155) (all_285_0_193 = 0 & precedes(all_66_4_70, all_66_3_69, all_0_5_5) = 0) | ( ~ (all_285_0_193 = 0) & on_path(all_66_3_69, all_0_5_5) = all_285_0_193)
% 215.25/153.22 |
% 215.25/153.22 +-Applying beta-rule and splitting (155), into two cases.
% 215.25/153.22 |-Branch one:
% 215.25/153.22 | (156) all_285_0_193 = 0 & precedes(all_66_4_70, all_66_3_69, all_0_5_5) = 0
% 215.25/153.22 |
% 215.25/153.22 | Applying alpha-rule on (156) yields:
% 215.25/153.22 | (157) all_285_0_193 = 0
% 215.25/153.22 | (158) precedes(all_66_4_70, all_66_3_69, all_0_5_5) = 0
% 215.25/153.22 |
% 215.25/153.22 +-Applying beta-rule and splitting (154), into two cases.
% 215.25/153.22 |-Branch one:
% 215.25/153.22 | (159) all_286_0_194 = 0 & triangle(all_66_4_70, all_66_3_69, all_286_1_195) = 0
% 215.25/153.22 |
% 215.25/153.22 | Applying alpha-rule on (159) yields:
% 215.25/153.22 | (160) all_286_0_194 = 0
% 215.25/153.22 | (161) triangle(all_66_4_70, all_66_3_69, all_286_1_195) = 0
% 215.25/153.22 |
% 215.25/153.22 | Instantiating formula (148) with all_286_1_195 and discharging atoms triangle(all_66_4_70, all_66_3_69, all_286_1_195) = 0, yields:
% 215.25/153.22 | (162) $false
% 215.25/153.22 |
% 215.25/153.22 |-The branch is then unsatisfiable
% 215.25/153.22 |-Branch two:
% 215.25/153.22 | (163) ~ (all_286_1_195 = 0) & precedes(all_66_4_70, all_66_3_69, all_0_5_5) = all_286_1_195
% 215.25/153.22 |
% 215.25/153.22 | Applying alpha-rule on (163) yields:
% 215.25/153.22 | (164) ~ (all_286_1_195 = 0)
% 215.25/153.22 | (165) precedes(all_66_4_70, all_66_3_69, all_0_5_5) = all_286_1_195
% 215.25/153.22 |
% 215.25/153.22 | Instantiating formula (60) with all_66_4_70, all_66_3_69, all_0_5_5, 0, all_286_1_195 and discharging atoms precedes(all_66_4_70, all_66_3_69, all_0_5_5) = all_286_1_195, precedes(all_66_4_70, all_66_3_69, all_0_5_5) = 0, yields:
% 215.25/153.22 | (166) all_286_1_195 = 0
% 215.25/153.22 |
% 215.25/153.22 | Equations (166) can reduce 164 to:
% 215.25/153.22 | (120) $false
% 215.25/153.22 |
% 215.25/153.22 |-The branch is then unsatisfiable
% 215.25/153.22 |-Branch two:
% 215.25/153.22 | (168) ~ (all_285_0_193 = 0) & on_path(all_66_3_69, all_0_5_5) = all_285_0_193
% 215.25/153.22 |
% 215.25/153.22 | Applying alpha-rule on (168) yields:
% 215.25/153.22 | (169) ~ (all_285_0_193 = 0)
% 215.25/153.22 | (170) on_path(all_66_3_69, all_0_5_5) = all_285_0_193
% 215.25/153.22 |
% 215.25/153.22 | Instantiating formula (28) with all_66_3_69, all_0_5_5, all_285_0_193, 0 and discharging atoms on_path(all_66_3_69, all_0_5_5) = all_285_0_193, on_path(all_66_3_69, all_0_5_5) = 0, yields:
% 215.25/153.22 | (157) all_285_0_193 = 0
% 215.25/153.22 |
% 215.25/153.22 | Equations (157) can reduce 169 to:
% 215.25/153.22 | (120) $false
% 215.25/153.22 |
% 215.25/153.22 |-The branch is then unsatisfiable
% 215.25/153.22 |-Branch two:
% 215.25/153.22 | (173) ~ (all_285_0_193 = 0) & on_path(all_66_4_70, all_0_5_5) = all_285_0_193
% 215.25/153.22 |
% 215.25/153.22 | Applying alpha-rule on (173) yields:
% 215.25/153.22 | (169) ~ (all_285_0_193 = 0)
% 215.25/153.22 | (175) on_path(all_66_4_70, all_0_5_5) = all_285_0_193
% 215.25/153.22 |
% 215.25/153.22 | Instantiating formula (28) with all_66_4_70, all_0_5_5, all_285_0_193, 0 and discharging atoms on_path(all_66_4_70, all_0_5_5) = all_285_0_193, on_path(all_66_4_70, all_0_5_5) = 0, yields:
% 215.25/153.22 | (157) all_285_0_193 = 0
% 215.25/153.22 |
% 215.25/153.23 | Equations (157) can reduce 169 to:
% 215.25/153.23 | (120) $false
% 215.25/153.23 |
% 215.25/153.23 |-The branch is then unsatisfiable
% 215.25/153.23 |-Branch two:
% 215.25/153.23 | (178) all_66_3_69 = all_66_4_70 & number_of_in(triangles, all_0_5_5) = all_66_4_70 & number_of_in(sequential_pairs, all_0_5_5) = all_66_4_70
% 215.25/153.23 |
% 215.25/153.23 | Applying alpha-rule on (178) yields:
% 215.25/153.23 | (179) all_66_3_69 = all_66_4_70
% 215.25/153.23 | (180) number_of_in(triangles, all_0_5_5) = all_66_4_70
% 215.25/153.23 | (181) number_of_in(sequential_pairs, all_0_5_5) = all_66_4_70
% 215.25/153.23 |
% 215.25/153.23 | Instantiating formula (35) with sequential_pairs, all_0_5_5, all_66_4_70, all_0_1_1 and discharging atoms number_of_in(sequential_pairs, all_0_5_5) = all_66_4_70, number_of_in(sequential_pairs, all_0_5_5) = all_0_1_1, yields:
% 215.25/153.23 | (182) all_66_4_70 = all_0_1_1
% 215.25/153.23 |
% 215.25/153.23 | From (182) and (180) follows:
% 215.25/153.23 | (183) number_of_in(triangles, all_0_5_5) = all_0_1_1
% 215.25/153.23 |
% 215.25/153.23 | Instantiating formula (69) with all_0_1_1, all_0_5_5, triangles and discharging atoms number_of_in(triangles, all_0_5_5) = all_0_1_1, yields:
% 215.25/153.23 | (184) ? [v0] : (number_of_in(triangles, graph) = v0 & less_or_equal(all_0_1_1, v0) = 0)
% 215.25/153.23 |
% 215.25/153.23 | Instantiating (184) with all_309_0_633 yields:
% 215.25/153.23 | (185) number_of_in(triangles, graph) = all_309_0_633 & less_or_equal(all_0_1_1, all_309_0_633) = 0
% 215.25/153.23 |
% 215.25/153.23 | Applying alpha-rule on (185) yields:
% 215.25/153.23 | (186) number_of_in(triangles, graph) = all_309_0_633
% 215.25/153.23 | (187) less_or_equal(all_0_1_1, all_309_0_633) = 0
% 215.25/153.23 |
% 215.25/153.23 | Instantiating formula (35) with triangles, graph, all_309_0_633, all_0_6_6 and discharging atoms number_of_in(triangles, graph) = all_309_0_633, number_of_in(triangles, graph) = all_0_6_6, yields:
% 215.25/153.23 | (188) all_309_0_633 = all_0_6_6
% 215.25/153.23 |
% 215.25/153.23 | From (188) and (187) follows:
% 215.25/153.23 | (189) less_or_equal(all_0_1_1, all_0_6_6) = 0
% 215.25/153.23 |
% 215.25/153.23 | Instantiating formula (14) with all_0_1_1, all_0_6_6, 0, all_0_0_0 and discharging atoms less_or_equal(all_0_1_1, all_0_6_6) = all_0_0_0, less_or_equal(all_0_1_1, all_0_6_6) = 0, yields:
% 215.25/153.23 | (190) all_0_0_0 = 0
% 215.25/153.23 |
% 215.25/153.23 | Equations (190) can reduce 85 to:
% 215.25/153.23 | (120) $false
% 215.25/153.23 |
% 215.25/153.23 |-The branch is then unsatisfiable
% 215.25/153.23 % SZS output end Proof for theBenchmark
% 215.25/153.23
% 215.25/153.23 152615ms
%------------------------------------------------------------------------------