0.11/0.12 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.11/0.13 % Command : do_cvc5 %s %d 0.12/0.33 % Computer : n027.cluster.edu 0.12/0.33 % Model : x86_64 x86_64 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.12/0.33 % Memory : 8042.1875MB 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64 0.12/0.33 % CPULimit : 960 0.12/0.33 % WCLimit : 120 0.12/0.33 % DateTime : Tue Aug 9 05:01:21 EDT 2022 0.12/0.33 % CPUTime : 0.18/0.46 %----Proving TF0_NAR, FOF, or CNF 25.84/26.20 ------- cvc5-fof casc J11 : /export/starexec/sandbox2/benchmark/theBenchmark.p at /export/starexec/sandbox2/benchmark/theBenchmark.p... 25.84/26.20 --- Run --decision=internal --simplification=none --no-inst-no-entail --no-cbqi --full-saturate-quant at 10... 25.84/26.20 --- Run --no-e-matching --full-saturate-quant at 5... 25.84/26.20 --- Run --no-e-matching --enum-inst-sum --full-saturate-quant at 5... 25.84/26.20 --- Run --finite-model-find --uf-ss=no-minimal at 5... 25.84/26.20 --- Run --multi-trigger-when-single --full-saturate-quant at 5... 25.84/26.20 % SZS status Theorem for theBenchmark 25.84/26.20 % SZS output start Proof for theBenchmark 25.84/26.20 (let ((_let_1 (not (=> complete (forall ((V1 $$unsorted) (V2 $$unsorted) (E1 $$unsorted) (E2 $$unsorted) (P $$unsorted)) (=> (and (precedes E1 E2 P) (shortest_path V1 V2 P)) (exists ((E3 $$unsorted)) (and (edge E3) (= (head_of E2) (tail_of E3)) (= (tail_of E1) (head_of E3)))))))))) (let ((_let_2 (forall ((V1 $$unsorted) (V2 $$unsorted) (P $$unsorted) (E $$unsorted)) (=> (and (path V1 V2 P) (on_path E P)) (and (edge E) (in_path (head_of E) P) (in_path (tail_of E) P)))))) (let ((_let_3 (forall ((V1 $$unsorted) (V2 $$unsorted) (E1 $$unsorted) (E2 $$unsorted) (P $$unsorted)) (=> (and (shortest_path V1 V2 P) (precedes E1 E2 P)) (and (not (exists ((E3 $$unsorted)) (and (= (tail_of E1) (tail_of E3)) (= (head_of E3) (head_of E2))))) (not (precedes E2 E1 P))))))) (let ((_let_4 (forall ((P $$unsorted) (V1 $$unsorted) (V2 $$unsorted)) (=> (path V1 V2 P) (forall ((E1 $$unsorted) (E2 $$unsorted)) (=> (precedes E1 E2 P) (and (on_path E2 P) (xor (sequential E1 E2) (exists ((E3 $$unsorted)) (and (precedes E3 E2 P) (sequential E1 E3)))) (on_path E1 P)))))))) (let ((_let_5 (forall ((V1 $$unsorted) (V2 $$unsorted) (SP $$unsorted)) (= (and (path V1 V2 SP) (forall ((P $$unsorted)) (=> (path V1 V2 P) (less_or_equal (length_of SP) (length_of P)))) (not (= V1 V2))) (shortest_path V1 V2 SP))))) (let ((_let_6 (forall ((E $$unsorted)) (=> (edge E) (and (vertex (head_of E)) (vertex (tail_of E))))))) (let ((_let_7 (forall ((P $$unsorted) (V1 $$unsorted) (V2 $$unsorted)) (=> (path V1 V2 P) (forall ((E1 $$unsorted) (E2 $$unsorted)) (=> (and (on_path E2 P) (or (exists ((E3 $$unsorted)) (and (precedes E3 E2 P) (sequential E1 E3))) (sequential E1 E2)) (on_path E1 P)) (precedes E1 E2 P))))))) (let ((_let_8 (=> complete (forall ((V1 $$unsorted) (V2 $$unsorted)) (=> (and (vertex V2) (not (= V1 V2)) (vertex V1)) (exists ((E $$unsorted)) (let ((_let_1 (tail_of E))) (let ((_let_2 (head_of E))) (and (xor (and (= V1 _let_2) (= _let_1 V2)) (and (= _let_2 V2) (= V1 _let_1))) (edge E)))))))))) (let ((_let_9 (forall ((E1 $$unsorted) (E2 $$unsorted)) (= (and (edge E1) (= (head_of E1) (tail_of E2)) (not (= E1 E2)) (edge E2)) (sequential E1 E2))))) (let ((_let_10 (forall ((E3 $$unsorted)) (or (not (precedes E3 skv_5 skv_6)) (not (sequential skv_4 E3)))))) (let ((_let_11 (sequential skv_4 skv_8))) (let ((_let_12 (not _let_11))) (let ((_let_13 (or (not (precedes skv_8 skv_5 skv_6)) _let_12))) (let ((_let_14 (sequential skv_4 skv_5))) (let ((_let_15 (not _let_14))) (let ((_let_16 (not _let_10))) (let ((_let_17 (= _let_16 _let_15))) (let ((_let_18 (on_path skv_4 skv_6))) (let ((_let_19 (on_path skv_5 skv_6))) (let ((_let_20 (and _let_19 _let_17 _let_18))) (let ((_let_21 (precedes skv_4 skv_5 skv_6))) (let ((_let_22 (not _let_21))) (let ((_let_23 (path skv_2 skv_3 skv_6))) (let ((_let_24 (not _let_23))) (let ((_let_25 (or _let_24 _let_22 _let_20))) (let ((_let_26 (forall ((P $$unsorted) (V1 $$unsorted) (V2 $$unsorted) (BOUND_VARIABLE_873 $$unsorted) (BOUND_VARIABLE_871 $$unsorted)) (or (not (path V1 V2 P)) (not (precedes BOUND_VARIABLE_871 BOUND_VARIABLE_873 P)) (and (on_path BOUND_VARIABLE_873 P) (= (not (forall ((E3 $$unsorted)) (or (not (precedes E3 BOUND_VARIABLE_873 P)) (not (sequential BOUND_VARIABLE_871 E3))))) (not (sequential BOUND_VARIABLE_871 BOUND_VARIABLE_873))) (on_path BOUND_VARIABLE_871 P)))))) (let ((_let_27 (EQ_RESOLVE (ASSUME :args (_let_4)) (MACRO_SR_EQ_INTRO :args (_let_4 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_28 (and _let_23 (forall ((P $$unsorted)) (or (not (path skv_2 skv_3 P)) (less_or_equal (length_of skv_6) (length_of P)))) (not (= skv_2 skv_3))))) (let ((_let_29 (shortest_path skv_2 skv_3 skv_6))) (let ((_let_30 (= _let_29 _let_28))) (let ((_let_31 (forall ((V1 $$unsorted) (V2 $$unsorted) (SP $$unsorted)) (= (shortest_path V1 V2 SP) (and (path V1 V2 SP) (forall ((P $$unsorted)) (or (not (path V1 V2 P)) (less_or_equal (length_of SP) (length_of P)))) (not (= V1 V2))))))) (let ((_let_32 (EQ_RESOLVE (ASSUME :args (_let_5)) (MACRO_SR_EQ_INTRO :args (_let_5 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_33 (forall ((E3 $$unsorted)) (or (not (edge E3)) (not (= (tail_of E3) (head_of skv_5))) (not (= (head_of E3) (tail_of skv_4))))))) (let ((_let_34 (not _let_33))) (let ((_let_35 (not _let_29))) (let ((_let_36 (or _let_22 _let_35 _let_34))) (let ((_let_37 (forall ((V1 $$unsorted) (V2 $$unsorted) (E1 $$unsorted) (E2 $$unsorted) (P $$unsorted)) (or (not (precedes E1 E2 P)) (not (shortest_path V1 V2 P)) (not (forall ((E3 $$unsorted)) (or (not (edge E3)) (not (= (tail_of E3) (head_of E2))) (not (= (head_of E3) (tail_of E1)))))))))) (let ((_let_38 (not _let_36))) (let ((_let_39 (MACRO_SR_EQ_INTRO :args (_let_1 SB_DEFAULT SBA_FIXPOINT)))) (let ((_let_40 (ASSUME :args (_let_1)))) (let ((_let_41 (MACRO_SR_PRED_TRANSFORM (NOT_IMPLIES_ELIM1 (EQ_RESOLVE _let_40 _let_39)) :args ((= complete true))))) (let ((_let_42 (EQ_RESOLVE _let_40 (TRANS _let_39 (MACRO_SR_EQ_INTRO _let_41 :args ((not (=> complete _let_37)) SB_DEFAULT SBA_FIXPOINT)))))) (let ((_let_43 (OR))) (let ((_let_44 (not _let_37))) (let ((_let_45 (MACRO_RESOLUTION_TRUST (EQ_RESOLVE (IMPLIES_ELIM (SCOPE (SKOLEMIZE _let_42) :args (_let_44))) (CONG (MACRO_SR_PRED_INTRO :args ((= (not _let_44) _let_37))) (REFL :args (_let_38)) :args _let_43)) _let_42 :args (_let_38 true _let_37)))) (let ((_let_46 (REFL :args (_let_36)))) (let ((_let_47 (MACRO_RESOLUTION_TRUST (REORDERING (EQ_RESOLVE (CNF_OR_NEG :args (_let_36 1)) (CONG _let_46 (MACRO_SR_PRED_INTRO :args ((= (not _let_35) _let_29))) :args _let_43)) :args ((or _let_29 _let_36))) _let_45 :args (_let_29 true _let_36)))) (let ((_let_48 (MACRO_RESOLUTION_TRUST (REORDERING (CNF_AND_POS :args (_let_28 0)) :args ((or _let_23 (not _let_28)))) (MACRO_RESOLUTION_TRUST (REORDERING (CNF_EQUIV_POS1 :args (_let_30)) :args ((or _let_35 _let_28 (not _let_30)))) _let_47 (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (SCOPE (INSTANTIATE _let_32 :args (skv_2 skv_3 skv_6 QUANTIFIERS_INST_E_MATCHING_SIMPLE ((shortest_path V1 V2 SP)))) :args (_let_31))) _let_32 :args (_let_30 false _let_31)) :args (_let_28 false _let_29 false _let_30)) :args (_let_23 false _let_28)))) (let ((_let_49 (MACRO_RESOLUTION_TRUST (REORDERING (EQ_RESOLVE (CNF_OR_NEG :args (_let_36 0)) (CONG _let_46 (MACRO_SR_PRED_INTRO :args ((= (not _let_22) _let_21))) :args _let_43)) :args ((or _let_21 _let_36))) _let_45 :args (_let_21 true _let_36)))) (let ((_let_50 (MACRO_RESOLUTION_TRUST (REORDERING (CNF_OR_POS :args (_let_25)) :args ((or _let_22 _let_24 _let_20 (not _let_25)))) _let_49 _let_48 (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (SCOPE (INSTANTIATE _let_27 :args (skv_6 skv_2 skv_3 skv_5 skv_4 QUANTIFIERS_INST_E_MATCHING ((not (= (path V1 V2 P) false)) (not (= (precedes BOUND_VARIABLE_871 BOUND_VARIABLE_873 P) false))))) :args (_let_26))) _let_27 :args (_let_25 false _let_26)) :args (_let_20 false _let_21 false _let_23 false _let_25)))) (let ((_let_51 (not _let_20))) (let ((_let_52 (edge skv_5))) (let ((_let_53 (= skv_4 skv_5))) (let ((_let_54 (not _let_53))) (let ((_let_55 (head_of skv_4))) (let ((_let_56 (tail_of skv_5))) (let ((_let_57 (edge skv_4))) (let ((_let_58 (and _let_57 (= _let_56 _let_55) _let_54 _let_52))) (let ((_let_59 (= _let_14 _let_58))) (let ((_let_60 (forall ((E1 $$unsorted) (E2 $$unsorted)) (= (sequential E1 E2) (and (edge E1) (= (tail_of E2) (head_of E1)) (not (= E1 E2)) (edge E2)))))) (let ((_let_61 (EQ_RESOLVE (ASSUME :args (_let_9)) (MACRO_SR_EQ_INTRO :args (_let_9 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_62 (_let_60))) (let ((_let_63 ((sequential E1 E2)))) (let ((_let_64 (not _let_58))) (let ((_let_65 (tail_of skv_4))) (let ((_let_66 (vertex _let_65))) (let ((_let_67 (and (vertex _let_55) _let_66))) (let ((_let_68 (not _let_57))) (let ((_let_69 (or _let_68 _let_67))) (let ((_let_70 (head_of skv_5))) (let ((_let_71 (= _let_65 _let_70))) (let ((_let_72 (and _let_52 _let_71 _let_54 _let_57))) (let ((_let_73 (vertex _let_70))) (let ((_let_74 (forall ((E $$unsorted)) (let ((_let_1 (tail_of skv_4))) (let ((_let_2 (tail_of E))) (let ((_let_3 (head_of skv_5))) (let ((_let_4 (head_of E))) (or (= (and (= _let_4 _let_1) (= _let_2 _let_3)) (and (= _let_4 _let_3) (= _let_2 _let_1))) (not (edge E)))))))))) (let ((_let_75 (not _let_74))) (let ((_let_76 (not _let_66))) (let ((_let_77 (not _let_73))) (let ((_let_78 (or _let_77 _let_71 _let_76 _let_75))) (let ((_let_79 (forall ((E $$unsorted)) (or (not (edge E)) (and (vertex (head_of E)) (vertex (tail_of E))))))) (let ((_let_80 (EQ_RESOLVE (ASSUME :args (_let_6)) (MACRO_SR_EQ_INTRO :args (_let_6 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_81 (_let_79))) (let ((_let_82 ((not (= (edge E) false))))) (let ((_let_83 (and _let_52 (in_path _let_70 skv_6) (in_path _let_56 skv_6)))) (let ((_let_84 (not _let_19))) (let ((_let_85 (or _let_24 _let_84 _let_83))) (let ((_let_86 (forall ((V1 $$unsorted) (V2 $$unsorted) (P $$unsorted) (E $$unsorted)) (or (not (path V1 V2 P)) (not (on_path E P)) (and (edge E) (in_path (head_of E) P) (in_path (tail_of E) P)))))) (let ((_let_87 (EQ_RESOLVE (ASSUME :args (_let_2)) (MACRO_SR_EQ_INTRO :args (_let_2 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_88 (MACRO_RESOLUTION_TRUST (REORDERING (CNF_AND_POS :args (_let_20 0)) :args ((or _let_19 _let_51))) _let_50 :args (_let_19 false _let_20)))) (let ((_let_89 (MACRO_RESOLUTION_TRUST (REORDERING (CNF_AND_POS :args (_let_83 0)) :args ((or _let_52 (not _let_83)))) (MACRO_RESOLUTION_TRUST (REORDERING (CNF_OR_POS :args (_let_85)) :args ((or _let_24 _let_84 _let_83 (not _let_85)))) _let_48 _let_88 (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (SCOPE (INSTANTIATE _let_87 :args (skv_2 skv_3 skv_6 skv_5 QUANTIFIERS_INST_E_MATCHING ((not (= (path V1 V2 P) false)) (not (= (on_path E P) false))))) :args (_let_86))) _let_87 :args (_let_85 false _let_86)) :args (_let_83 false _let_23 false _let_19 false _let_85)) :args (_let_52 false _let_83)))) (let ((_let_90 (= _let_65 _let_56))) (let ((_let_91 (precedes skv_5 skv_4 skv_6))) (let ((_let_92 (not _let_91))) (let ((_let_93 (not _let_90))) (let ((_let_94 (and _let_93 _let_92))) (let ((_let_95 (or _let_35 _let_22 _let_94))) (let ((_let_96 (forall ((V1 $$unsorted) (V2 $$unsorted) (E1 $$unsorted) (E2 $$unsorted) (P $$unsorted) (BOUND_VARIABLE_1046 $$unsorted)) (or (not (shortest_path V1 V2 P)) (not (precedes E1 E2 P)) (and (or (not (= (tail_of E1) (tail_of BOUND_VARIABLE_1046))) (not (= (head_of E2) (head_of BOUND_VARIABLE_1046)))) (not (precedes E2 E1 P))))))) (let ((_let_97 (EQ_RESOLVE (ASSUME :args (_let_3)) (MACRO_SR_EQ_INTRO :args (_let_3 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_98 (_let_96))) (let ((_let_99 (tail_of BOUND_VARIABLE_1046))) (let ((_let_100 (not (= (shortest_path V1 V2 P) false)))) (let ((_let_101 (_let_53))) (let ((_let_102 (ASSUME :args _let_101))) (let ((_let_103 (sequential skv_5 skv_4))) (let ((_let_104 (= _let_103 _let_72))) (let ((_let_105 (not _let_72))) (let ((_let_106 (not _let_103))) (let ((_let_107 (and (or _let_92 (not (sequential skv_5 skv_5))) _let_106))) (let ((_let_108 (not _let_18))) (let ((_let_109 (or _let_24 _let_108 _let_107 _let_84 _let_91))) (let ((_let_110 (forall ((P $$unsorted) (V1 $$unsorted) (V2 $$unsorted) (BOUND_VARIABLE_785 $$unsorted) (BOUND_VARIABLE_783 $$unsorted) (BOUND_VARIABLE_781 $$unsorted)) (or (not (path V1 V2 P)) (not (on_path BOUND_VARIABLE_783 P)) (and (or (not (precedes BOUND_VARIABLE_785 BOUND_VARIABLE_783 P)) (not (sequential BOUND_VARIABLE_781 BOUND_VARIABLE_785))) (not (sequential BOUND_VARIABLE_781 BOUND_VARIABLE_783))) (not (on_path BOUND_VARIABLE_781 P)) (precedes BOUND_VARIABLE_781 BOUND_VARIABLE_783 P))))) (let ((_let_111 (EQ_RESOLVE (ASSUME :args (_let_7)) (MACRO_SR_EQ_INTRO :args (_let_7 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_112 (and (or (not (= _let_65 (tail_of skv_7))) (not (= _let_70 (head_of skv_7)))) _let_92))) (let ((_let_113 (or _let_35 _let_22 _let_112))) (let ((_let_114 (not _let_71))) (let ((_let_115 (not _let_52))) (let ((_let_116 (_let_72))) (let ((_let_117 (and _let_73 (vertex _let_56)))) (let ((_let_118 (or _let_115 _let_117))) (let ((_let_119 (edge skv_17))) (let ((_let_120 (not _let_119))) (let ((_let_121 (tail_of skv_17))) (let ((_let_122 (= _let_65 _let_121))) (let ((_let_123 (head_of skv_17))) (let ((_let_124 (= _let_70 _let_123))) (let ((_let_125 (and _let_124 _let_122))) (let ((_let_126 (= _let_70 _let_121))) (let ((_let_127 (= _let_65 _let_123))) (let ((_let_128 (and _let_127 _let_126))) (let ((_let_129 (= _let_128 _let_125))) (let ((_let_130 (or _let_129 _let_120))) (let ((_let_131 (not _let_127))) (let ((_let_132 (not _let_126))) (let ((_let_133 (or _let_120 _let_132 _let_131))) (let ((_let_134 (not _let_124))) (let ((_let_135 (not _let_122))) (let ((_let_136 (or _let_135 _let_134))) (let ((_let_137 (not _let_125))) (let ((_let_138 (and _let_136 _let_92))) (let ((_let_139 (or _let_35 _let_22 _let_138))) (let ((_let_140 (not _let_128))) (let ((_let_141 (_let_33))) (let ((_let_142 (_let_75))) (let ((_let_143 (forall ((V1 $$unsorted) (V2 $$unsorted)) (or (not (vertex V2)) (= V1 V2) (not (vertex V1)) (not (forall ((E $$unsorted)) (let ((_let_1 (tail_of E))) (let ((_let_2 (head_of E))) (or (= (and (= V1 _let_2) (= V2 _let_1)) (and (= V2 _let_2) (= V1 _let_1))) (not (edge E))))))))))) (let ((_let_144 (EQ_RESOLVE (ASSUME :args (_let_8)) (TRANS (MACRO_SR_EQ_INTRO :args (_let_8 SB_DEFAULT SBA_FIXPOINT)) (MACRO_SR_EQ_INTRO _let_41 :args ((=> complete _let_143) SB_DEFAULT SBA_FIXPOINT)))))) (let ((_let_145 (MACRO_RESOLUTION_TRUST (REORDERING (CNF_OR_POS :args (_let_78)) :args ((or _let_71 _let_76 _let_77 _let_75 (not _let_78)))) (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (MACRO_SR_PRED_ELIM (SCOPE (INSTANTIATE _let_144 :args (_let_65 _let_70 QUANTIFIERS_INST_E_MATCHING ((not (= (vertex V2) false)) (not (= (vertex V1) false))))) :args (_let_143)))) _let_144 :args (_let_78 false _let_143)) (MACRO_RESOLUTION_TRUST (EQ_RESOLVE (IMPLIES_ELIM (EQ_RESOLVE (SCOPE (SKOLEMIZE (ASSUME :args _let_142)) :args _let_142) (REWRITE :args ((=> _let_75 (not (or (= (and (= _let_123 _let_65) (= _let_121 _let_70)) (and (= _let_123 _let_70) (= _let_121 _let_65))) _let_120))))))) (CONG (MACRO_SR_PRED_INTRO :args ((= (not _let_75) _let_74))) (REFL :args ((not _let_130))) :args _let_43)) (MACRO_RESOLUTION_TRUST (REORDERING (CNF_OR_POS :args (_let_133)) :args ((or _let_120 _let_131 _let_132 (not _let_133)))) (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (MACRO_SR_PRED_ELIM (SCOPE (INSTANTIATE (ASSUME :args _let_141) :args (skv_17 QUANTIFIERS_INST_CBQI_CONFLICT)) :args _let_141))) (MACRO_RESOLUTION_TRUST (REORDERING (EQ_RESOLVE (CNF_OR_NEG :args (_let_36 2)) (CONG _let_46 (MACRO_SR_PRED_INTRO :args ((= (not _let_34) _let_33))) :args _let_43)) :args ((or _let_33 _let_36))) _let_45 :args (_let_33 true _let_36)) :args (_let_133 false _let_33)) (REORDERING (CNF_AND_POS :args (_let_128 1)) :args ((or _let_126 _let_140))) (REORDERING (CNF_AND_POS :args (_let_128 0)) 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(INSTANTIATE _let_61 :args (skv_4 skv_5 QUANTIFIERS_INST_E_MATCHING_SIMPLE _let_63)) :args _let_62)) _let_61 :args (_let_59 false _let_60)) :args (_let_15 true _let_58 false _let_59)) (MACRO_RESOLUTION_TRUST (REORDERING (CNF_AND_POS :args (_let_20 1)) :args ((or _let_17 _let_51))) _let_50 :args (_let_17 false _let_20)) :args (_let_16 true _let_14 false _let_17)) :args (false false _let_13 true _let_10)) :args (_let_9 _let_8 _let_7 (forall ((V1 $$unsorted) (V2 $$unsorted) (P $$unsorted) (V $$unsorted)) (=> (and (in_path V P) (path V1 V2 P)) (and (exists ((E $$unsorted)) (and (on_path E P) (or (= V (head_of E)) (= (tail_of E) V)))) (vertex V)))) _let_6 _let_5 _let_4 (forall ((E $$unsorted)) (=> (edge E) (not (= (tail_of E) (head_of E))))) (forall ((V1 $$unsorted) (V2 $$unsorted) (P $$unsorted)) (=> (path V1 V2 P) (and (exists ((E $$unsorted)) (and (xor (and (= P (path_cons E empty)) (= (head_of E) V2)) (exists ((TP $$unsorted)) (and (= P (path_cons E TP)) (path (head_of E) V2 TP)))) (= V1 (tail_of E)) (edge E))) (vertex V2) (vertex V1)))) (forall ((V1 $$unsorted) (V2 $$unsorted) (P $$unsorted)) (=> (and (exists ((E $$unsorted)) (and (edge E) (= V1 (tail_of E)) (or (exists ((TP $$unsorted)) (and (path (head_of E) V2 TP) (= P (path_cons E TP)))) (and (= P (path_cons E empty)) (= (head_of E) V2))))) (vertex V2) (vertex V1)) (path V1 V2 P))) _let_3 _let_2 _let_1 (forall ((P $$unsorted) (V1 $$unsorted) (V2 $$unsorted)) (=> (and (path V1 V2 P) (forall ((E1 $$unsorted) (E2 $$unsorted)) (=> (and (on_path E2 P) (sequential E1 E2) (on_path E1 P)) (exists ((E3 $$unsorted)) (triangle E1 E2 E3))))) (= (number_of_in triangles P) (number_of_in sequential_pairs P)))) (forall ((V1 $$unsorted) (V2 $$unsorted) (P $$unsorted)) (=> (path V1 V2 P) (= (number_of_in edges P) (length_of P)))) (forall ((V1 $$unsorted) (V2 $$unsorted) (P $$unsorted)) (=> (path V1 V2 P) (= (minus (length_of P) n1) (number_of_in sequential_pairs P)))) (forall ((V1 $$unsorted) (V2 $$unsorted) (E1 $$unsorted) (E2 $$unsorted) (P $$unsorted)) (=> (and (precedes E1 E2 P) (shortest_path V1 V2 P)) (not (exists ((E3 $$unsorted)) (and (edge E3) (= (tail_of E3) (tail_of E1)) (= (head_of E2) (head_of E3))))))) (forall ((Things $$unsorted) (InThese $$unsorted)) (less_or_equal (number_of_in Things InThese) (number_of_in Things graph))) (forall ((E1 $$unsorted) (E2 $$unsorted) (E3 $$unsorted)) (= (triangle E1 E2 E3) (and (edge E2) (sequential E1 E2) (sequential E2 E3) (sequential E3 E1) (edge E3) (edge E1)))) true)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 25.84/26.20 % SZS output end Proof for theBenchmark 25.84/26.20 EOF