/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.pl /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern goal(g) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToIRSwTTransformerProof [SOUND, 49 ms] (2) AND (3) IRSwT (4) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (5) TRUE (6) IRSwT (7) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (8) TRUE (9) IRSwT (10) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (11) IRSwT (12) IntTRSCompressionProof [EQUIVALENT, 1 ms] (13) IRSwT (14) IRSFormatTransformerProof [EQUIVALENT, 1 ms] (15) IRSwT (16) IRSwTTerminationDigraphProof [EQUIVALENT, 6 ms] (17) IRSwT (18) TempFilterProof [SOUND, 2 ms] (19) IRSwT (20) IRSwTToQDPProof [SOUND, 0 ms] (21) QDP (22) QDPSizeChangeProof [EQUIVALENT, 0 ms] (23) YES ---------------------------------------- (0) Obligation: Clauses: goal(X) :- ','(s2l(X, Xs), conf(Xs)). conf(X) :- ','(del2(X, Z), ','(del(U, Y, Z), conf(Y))). del2(X, Y) :- ','(del(U, X, Z), del(V, Z, Y)). del(X1, [], X2) :- ','(!, failure(a)). del(H, X, T) :- ','(head(X, H), tail(X, T)). del(X, Y, .(H, T2)) :- ','(head(Y, H), ','(tail(Y, T1), del(X, T1, T2))). s2l(0, L) :- ','(!, eq(L, [])). s2l(X, .(X3, Xs)) :- ','(p(X, P), s2l(P, Xs)). head([], X4). head(.(H, X5), H). tail([], []). tail(.(X6, Xs), Xs). p(0, 0). p(s(X), X). failure(b). eq(X, X). Query: goal(g) ---------------------------------------- (1) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 1, "program": { "directives": [], "clauses": [ [ "(goal X)", "(',' (s2l X Xs) (conf Xs))" ], [ "(conf X)", "(',' (del2 X Z) (',' (del U Y Z) (conf Y)))" ], [ "(del2 X Y)", "(',' (del U X Z) (del V Z Y))" ], [ "(del X1 ([]) X2)", "(',' (!) (failure (a)))" ], [ "(del H X T)", "(',' (head X H) (tail X T))" ], [ "(del X Y (. H T2))", "(',' (head Y H) (',' (tail Y T1) (del X T1 T2)))" ], [ "(s2l (0) L)", "(',' (!) (eq L ([])))" ], [ "(s2l X (. X3 Xs))", "(',' (p X P) (s2l P Xs))" ], [ "(head ([]) X4)", null ], [ "(head (. H X5) H)", null ], [ "(tail ([]) ([]))", null ], [ "(tail (. X6 Xs) Xs)", null ], [ "(p (0) (0))", null ], [ "(p (s X) X)", null ], [ "(failure (b))", null ], [ "(eq X X)", null ] ] }, "graph": { "nodes": { "88": { "goal": [{ "clause": -1, "scope": -1, "term": "(eq X23 ([]))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X23"], "exprvars": [] } }, "89": { "goal": [{ "clause": 15, "scope": 3, "term": "(eq X23 ([]))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X23"], "exprvars": [] } }, "49": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (s2l T6 X13) (conf X13))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T6"], "free": ["X13"], "exprvars": [] } }, "1143": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (tail (. 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T45 T46) X154)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X154"], "exprvars": [] } }, "75": { "goal": [{ "clause": -1, "scope": -1, "term": "(conf T7)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "780": { "goal": [{ "clause": -1, "scope": -1, "term": "(failure (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "781": { "goal": [{ "clause": 14, "scope": 15, "term": "(failure (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "100": { "goal": [{ "clause": 13, "scope": 4, "term": "(',' (p T10 X46) (s2l X46 X48))" }], "kb": { "nonunifying": [[ "(s2l T10 X13)", "(s2l (0) X22)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": [ "X13", "X22", "X48", "X46" ], "exprvars": [] } }, "782": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "740": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "741": { "goal": [ { "clause": 10, "scope": 11, "term": "(tail (. T45 T46) X154)" }, { "clause": 11, "scope": 11, "term": "(tail (. T45 T46) X154)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X154"], "exprvars": [] } }, "742": { "goal": [{ "clause": 11, "scope": 11, "term": "(tail (. T45 T46) X154)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X154"], "exprvars": [] } }, "622": { "goal": [ { "clause": 3, "scope": 7, "term": "(del X104 T30 X105)" }, { "clause": 4, "scope": 7, "term": "(del X104 T30 X105)" }, { "clause": 5, "scope": 7, "term": "(del X104 T30 X105)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X105", "X104" ], "exprvars": [] } }, "744": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "745": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "702": { "goal": [{ "clause": 9, "scope": 9, "term": "(',' (head T38 X153) (tail T38 X154))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X153", "X154" ], "exprvars": [] } }, "309": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (del2 T20 X72) (',' (del X73 X74 X72) (conf X74)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X72", "X73", "X74" ], "exprvars": [] } }, "85": { "goal": [ { "clause": 6, "scope": 2, "term": "(s2l T6 X13)" }, { "clause": 7, "scope": 2, "term": "(s2l T6 X13)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T6"], "free": ["X13"], "exprvars": [] } }, "86": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (!_2) (eq X23 ([])))" }, { "clause": 7, "scope": 2, "term": "(s2l (0) X13)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X13", "X23" ], "exprvars": [] } }, "87": { "goal": [{ "clause": 7, "scope": 2, "term": "(s2l T6 X13)" }], "kb": { "nonunifying": [[ "(s2l T6 X13)", "(s2l (0) X22)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T6"], "free": [ "X13", "X22" ], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 5, "label": "CASE" }, { "from": 5, "to": 49, "label": "ONLY EVAL with clause\ngoal(X12) :- ','(s2l(X12, X13), conf(X13)).\nand substitutionT3 -> T6,\nX12 -> T6" }, { "from": 49, "to": 70, "label": "SPLIT 1" }, { "from": 49, "to": 75, "label": "SPLIT 2\nnew knowledge:\nT6 is ground\nreplacements:X13 -> T7" }, { "from": 70, "to": 85, "label": "CASE" }, { "from": 75, "to": 299, "label": "CASE" }, { "from": 85, "to": 86, "label": "EVAL with clause\ns2l(0, X22) :- ','(!_2, eq(X22, [])).\nand substitutionT6 -> 0,\nX13 -> X23,\nX22 -> X23" }, { "from": 85, "to": 87, "label": "EVAL-BACKTRACK" }, { "from": 86, "to": 88, "label": "CUT" }, { "from": 87, "to": 95, "label": "ONLY EVAL with clause\ns2l(X43, .(X44, X45)) :- ','(p(X43, X46), s2l(X46, X45)).\nand substitutionT6 -> T10,\nX43 -> T10,\nX44 -> X47,\nX45 -> X48,\nX13 -> .(X47, X48)" }, { "from": 88, "to": 89, "label": "CASE" }, { "from": 89, "to": 90, "label": "ONLY EVAL with clause\neq(X28, X28).\nand substitutionX23 -> [],\nX28 -> [],\nX29 -> []" }, { "from": 90, "to": 91, "label": "SUCCESS" }, { "from": 95, "to": 96, "label": "CASE" }, { "from": 96, "to": 100, "label": "BACKTRACK\nfor clause: p(0, 0)\nwith clash: (s2l(T10, X13), s2l(0, X22))" }, { "from": 100, "to": 162, "label": "EVAL with clause\np(s(X53), X53).\nand substitutionX53 -> T13,\nT10 -> s(T13),\nX46 -> T13" }, { "from": 100, "to": 175, "label": "EVAL-BACKTRACK" }, { "from": 162, "to": 70, "label": "INSTANCE with matching:\nT6 -> T13\nX13 -> X48" }, { "from": 299, "to": 309, "label": "ONLY EVAL with clause\nconf(X71) :- ','(del2(X71, X72), ','(del(X73, X74, X72), conf(X74))).\nand substitutionT7 -> T20,\nX71 -> T20,\nT19 -> T20" }, { "from": 309, "to": 360, "label": "SPLIT 1" }, { "from": 309, "to": 363, "label": "SPLIT 2\nreplacements:X72 -> T22" }, { "from": 360, "to": 408, "label": "CASE" }, { "from": 363, "to": 1152, "label": "SPLIT 1" }, { "from": 363, "to": 1153, "label": "SPLIT 2\nreplacements:X73 -> T75,\nX74 -> T76" }, { "from": 408, "to": 600, "label": "ONLY EVAL with clause\ndel2(X100, X101) :- ','(del(X102, X100, X103), del(X104, X103, X101)).\nand substitutionT20 -> T28,\nX100 -> T28,\nX72 -> X105,\nX101 -> X105,\nT27 -> T28" }, { "from": 600, "to": 601, "label": "SPLIT 1" }, { "from": 600, "to": 602, "label": "SPLIT 2\nreplacements:X102 -> T29,\nX103 -> T30" }, { "from": 601, "to": 602, "label": "INSTANCE with matching:\nX104 -> X102\nT30 -> T28\nX105 -> X103" }, { "from": 602, "to": 622, "label": "CASE" }, { "from": 622, "to": 645, "label": "EVAL with clause\ndel(X126, [], X127) :- ','(!_7, failure(a)).\nand substitutionX104 -> X128,\nX126 -> X128,\nT30 -> [],\nX105 -> X129,\nX127 -> X129" }, { "from": 622, "to": 647, "label": "EVAL-BACKTRACK" }, { "from": 645, "to": 648, "label": "CUT" }, { "from": 647, "to": 657, "label": "PARALLEL" }, { "from": 647, "to": 659, "label": "PARALLEL" }, { "from": 648, "to": 649, "label": "CASE" }, { "from": 649, "to": 650, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 657, "to": 687, "label": "ONLY EVAL with clause\ndel(X150, X151, X152) :- ','(head(X151, X150), tail(X151, X152)).\nand substitutionX104 -> X153,\nX150 -> X153,\nT30 -> T38,\nX151 -> T38,\nX105 -> X154,\nX152 -> X154,\nT37 -> T38" }, { "from": 659, "to": 750, "label": "ONLY EVAL with clause\ndel(X198, X199, .(X200, X201)) :- ','(head(X199, X200), ','(tail(X199, X202), del(X198, X202, X201))).\nand substitutionX104 -> X203,\nX198 -> X203,\nT30 -> T57,\nX199 -> T57,\nX200 -> X204,\nX201 -> X205,\nX105 -> .(X204, X205),\nT56 -> T57" }, { "from": 687, "to": 692, "label": "CASE" }, { "from": 692, "to": 699, "label": "PARALLEL" }, { "from": 692, "to": 702, "label": "PARALLEL" }, { "from": 699, "to": 714, "label": "EVAL with clause\nhead([], X163).\nand substitutionT38 -> [],\nX153 -> X164,\nX163 -> X164" }, { "from": 699, "to": 717, "label": "EVAL-BACKTRACK" }, { "from": 702, "to": 739, "label": "EVAL with clause\nhead(.(X171, X172), X171).\nand substitutionX171 -> T45,\nX172 -> T46,\nT38 -> .(T45, T46),\nX153 -> T45,\nT43 -> T45,\nT44 -> T46" }, { "from": 702, "to": 740, "label": "EVAL-BACKTRACK" }, { "from": 714, "to": 720, "label": "CASE" }, { "from": 720, "to": 724, "label": "PARALLEL" }, { "from": 720, "to": 725, "label": "PARALLEL" }, { "from": 724, "to": 731, "label": "ONLY EVAL with clause\ntail([], []).\nand substitutionX154 -> []" }, { "from": 725, "to": 736, "label": "BACKTRACK\nfor clause: tail(.(X6, Xs), Xs)because of non-unification" }, { "from": 731, "to": 734, "label": "SUCCESS" }, { "from": 739, "to": 741, "label": "CASE" }, { "from": 741, "to": 742, "label": "BACKTRACK\nfor clause: tail([], [])because of non-unification" }, { "from": 742, "to": 744, "label": "ONLY EVAL with clause\ntail(.(X177, X178), X178).\nand substitutionT45 -> T51,\nX177 -> T51,\nT46 -> T52,\nX178 -> T52,\nX154 -> T52" }, { "from": 744, "to": 745, "label": "SUCCESS" }, { "from": 750, "to": 752, "label": "CASE" }, { "from": 752, "to": 756, "label": "PARALLEL" }, { "from": 752, "to": 758, "label": "PARALLEL" }, { "from": 756, "to": 759, "label": "EVAL with clause\nhead([], X218).\nand substitutionT57 -> [],\nX204 -> X219,\nX218 -> X219" }, { "from": 756, "to": 760, "label": "EVAL-BACKTRACK" }, { "from": 758, "to": 1143, "label": "EVAL with clause\nhead(.(X254, X255), X254).\nand substitutionX254 -> T64,\nX255 -> T65,\nT57 -> .(T64, T65),\nX204 -> T64,\nT62 -> T64,\nT63 -> T65" }, { "from": 758, "to": 1144, "label": "EVAL-BACKTRACK" }, { "from": 759, "to": 763, "label": "CASE" }, { "from": 763, "to": 764, "label": "PARALLEL" }, { "from": 763, "to": 765, "label": "PARALLEL" }, { "from": 764, "to": 767, "label": "ONLY EVAL with clause\ntail([], []).\nand substitutionX202 -> []" }, { "from": 765, "to": 921, "label": "BACKTRACK\nfor clause: tail(.(X6, Xs), Xs)because of non-unification" }, { "from": 767, "to": 772, "label": "CASE" }, { "from": 772, "to": 779, "label": "ONLY EVAL with clause\ndel(X244, [], X245) :- ','(!_14, failure(a)).\nand substitutionX203 -> X246,\nX244 -> X246,\nX205 -> X247,\nX245 -> X247" }, { "from": 779, "to": 780, "label": "CUT" }, { "from": 780, "to": 781, "label": "CASE" }, { "from": 781, "to": 782, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 1143, "to": 1146, "label": "CASE" }, { "from": 1146, "to": 1148, "label": "BACKTRACK\nfor clause: tail([], [])because of non-unification" }, { "from": 1148, "to": 1151, "label": "ONLY EVAL with clause\ntail(.(X268, X269), X269).\nand substitutionT64 -> T71,\nX268 -> T71,\nT65 -> T73,\nX269 -> T73,\nX202 -> T73,\nT72 -> T73" }, { "from": 1151, "to": 602, "label": "INSTANCE with matching:\nX104 -> X203\nT30 -> T73\nX105 -> X205" }, { "from": 1152, "to": 1154, "label": "CASE" }, { "from": 1153, "to": 75, "label": "INSTANCE with matching:\nT7 -> T76" }, { "from": 1154, "to": 1155, "label": "ONLY EVAL with clause\ndel(X286, [], X287) :- ','(!_17, failure(a)).\nand substitutionX73 -> X288,\nX286 -> X288,\nX74 -> [],\nT22 -> T80,\nX287 -> T80" }, { "from": 1155, "to": 1156, "label": "CUT" }, { "from": 1156, "to": 1157, "label": "CASE" }, { "from": 1157, "to": 1158, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" } ], "type": "Graph" } } ---------------------------------------- (2) Complex Obligation (AND) ---------------------------------------- (3) Obligation: Rules: f1146_out -> f1143_out :|: TRUE f1143_in -> f1146_in :|: TRUE f758_in -> f1143_in :|: TRUE f1143_out -> f758_out :|: TRUE f1144_out -> f758_out :|: TRUE f758_in -> f1144_in :|: TRUE f1151_in -> f602_in :|: TRUE f602_out -> f1151_out :|: TRUE f1148_in -> f1151_in :|: TRUE f1151_out -> f1148_out :|: TRUE f622_in -> f645_in :|: TRUE f647_out -> f622_out :|: TRUE f645_out -> f622_out :|: TRUE f622_in -> f647_in :|: TRUE f758_out -> f752_out :|: TRUE f756_out -> f752_out :|: TRUE f752_in -> f756_in :|: TRUE f752_in -> f758_in :|: TRUE f752_out -> f750_out :|: TRUE f750_in -> f752_in :|: TRUE f1146_in -> f1148_in :|: TRUE f1148_out -> f1146_out :|: TRUE f602_in -> f622_in :|: TRUE f622_out -> f602_out :|: TRUE f647_in -> f657_in :|: TRUE f659_out -> f647_out :|: TRUE f657_out -> f647_out :|: TRUE f647_in -> f659_in :|: TRUE f750_out -> f659_out :|: TRUE f659_in -> f750_in :|: TRUE f1_in(T3) -> f5_in(T3) :|: TRUE f5_out(x) -> f1_out(x) :|: TRUE f49_out(T6) -> f5_out(T6) :|: TRUE f5_in(x1) -> f49_in(x1) :|: TRUE f49_in(x2) -> f70_in(x2) :|: TRUE f70_out(x3) -> f75_in :|: TRUE f75_out -> f49_out(x4) :|: TRUE f75_in -> f299_in :|: TRUE f299_out -> f75_out :|: TRUE f299_in -> f309_in :|: TRUE f309_out -> f299_out :|: TRUE f309_in -> f360_in :|: TRUE f360_out -> f363_in :|: TRUE f363_out -> f309_out :|: TRUE f408_out -> f360_out :|: TRUE f360_in -> f408_in :|: TRUE f408_in -> f600_in :|: TRUE f600_out -> f408_out :|: TRUE f602_out -> f600_out :|: TRUE f601_out -> f602_in :|: TRUE f600_in -> f601_in :|: TRUE f602_out -> f601_out :|: TRUE f601_in -> f602_in :|: TRUE Start term: f1_in(T3) ---------------------------------------- (4) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (5) TRUE ---------------------------------------- (6) Obligation: Rules: f649_out -> f648_out :|: TRUE f648_in -> f649_in :|: TRUE f744_in -> f744_out :|: TRUE f75_out -> f1153_out :|: TRUE f1153_in -> f75_in :|: TRUE f1153_out -> f363_out :|: TRUE f1152_out -> f1153_in :|: TRUE f363_in -> f1152_in :|: TRUE f692_out -> f687_out :|: TRUE f687_in -> f692_in :|: TRUE f720_in -> f725_in :|: TRUE f725_out -> f720_out :|: TRUE f724_out -> f720_out :|: TRUE f720_in -> f724_in :|: TRUE f765_out -> f763_out :|: TRUE f764_out -> f763_out :|: TRUE f763_in -> f764_in :|: TRUE f763_in -> f765_in :|: TRUE f758_out -> f752_out :|: TRUE f756_out -> f752_out :|: TRUE f752_in -> f756_in :|: TRUE f752_in -> f758_in :|: TRUE f645_in -> f648_in :|: TRUE f648_out -> f645_out :|: TRUE f725_in -> f736_in :|: TRUE f736_out -> f725_out :|: TRUE f731_in -> f731_out :|: TRUE f75_in -> f299_in :|: TRUE f299_out -> f75_out :|: TRUE f779_in -> f780_in :|: TRUE f780_out -> f779_out :|: TRUE f781_in -> f782_in :|: TRUE f782_out -> f781_out :|: TRUE f741_in -> f742_in :|: TRUE f742_out -> f741_out :|: TRUE f758_in -> f1143_in :|: TRUE f1143_out -> f758_out :|: TRUE f1144_out -> f758_out :|: TRUE f758_in -> f1144_in :|: TRUE f1148_in -> f1151_in :|: TRUE f1151_out -> f1148_out :|: TRUE f1146_out -> f1143_out :|: TRUE f1143_in -> f1146_in :|: TRUE f622_in -> f645_in :|: TRUE f647_out -> f622_out :|: TRUE f645_out -> f622_out :|: TRUE f622_in -> f647_in :|: TRUE f1156_in -> f1157_in :|: TRUE f1157_out -> f1156_out :|: TRUE f739_out -> f702_out :|: TRUE f702_in -> f739_in :|: TRUE f702_in -> f740_in :|: TRUE f740_out -> f702_out :|: TRUE f742_in -> f744_in :|: TRUE f744_out -> f742_out :|: TRUE f408_in -> f600_in :|: TRUE f600_out -> f408_out :|: TRUE f309_in -> f360_in :|: TRUE f360_out -> f363_in :|: TRUE f363_out -> f309_out :|: TRUE f408_out -> f360_out :|: TRUE f360_in -> f408_in :|: TRUE f741_out -> f739_out :|: TRUE f739_in -> f741_in :|: TRUE f1157_in -> f1158_in :|: TRUE f1158_out -> f1157_out :|: TRUE f602_out -> f600_out :|: TRUE f601_out -> f602_in :|: TRUE f600_in -> f601_in :|: TRUE f1151_in -> f602_in :|: TRUE f602_out -> f1151_out :|: TRUE f657_in -> f687_in :|: TRUE f687_out -> f657_out :|: TRUE f780_in -> f781_in :|: TRUE f781_out -> f780_out :|: TRUE f714_in -> f720_in :|: TRUE f720_out -> f714_out :|: TRUE f1146_in -> f1148_in :|: TRUE f1148_out -> f1146_out :|: TRUE f750_out -> f659_out :|: TRUE f659_in -> f750_in :|: TRUE f767_out -> f764_out :|: TRUE f764_in -> f767_in :|: TRUE f779_out -> f772_out :|: TRUE f772_in -> f779_in :|: TRUE f724_in -> f731_in :|: TRUE f731_out -> f724_out :|: TRUE f921_out -> f765_out :|: TRUE f765_in -> f921_in :|: TRUE f752_out -> f750_out :|: TRUE f750_in -> f752_in :|: TRUE f767_in -> f772_in :|: TRUE f772_out -> f767_out :|: TRUE f647_in -> f657_in :|: TRUE f659_out -> f647_out :|: TRUE f657_out -> f647_out :|: TRUE f647_in -> f659_in :|: TRUE f756_in -> f759_in :|: TRUE f760_out -> f756_out :|: TRUE f759_out -> f756_out :|: TRUE f756_in -> f760_in :|: TRUE f699_in -> f717_in :|: TRUE f714_out -> f699_out :|: TRUE f699_in -> f714_in :|: TRUE f717_out -> f699_out :|: TRUE f602_out -> f601_out :|: TRUE f601_in -> f602_in :|: TRUE f1155_in -> f1156_in :|: TRUE f1156_out -> f1155_out :|: TRUE f299_in -> f309_in :|: TRUE f309_out -> f299_out :|: TRUE f702_out -> f692_out :|: TRUE f692_in -> f699_in :|: TRUE f699_out -> f692_out :|: TRUE f692_in -> f702_in :|: TRUE f1154_in -> f1155_in :|: TRUE f1155_out -> f1154_out :|: TRUE f763_out -> f759_out :|: TRUE f759_in -> f763_in :|: TRUE f650_out -> f649_out :|: TRUE f649_in -> f650_in :|: TRUE f602_in -> f622_in :|: TRUE f622_out -> f602_out :|: TRUE f1152_in -> f1154_in :|: TRUE f1154_out -> f1152_out :|: TRUE f1_in(T3) -> f5_in(T3) :|: TRUE f5_out(x) -> f1_out(x) :|: TRUE f49_out(T6) -> f5_out(T6) :|: TRUE f5_in(x1) -> f49_in(x1) :|: TRUE f49_in(x2) -> f70_in(x2) :|: TRUE f70_out(x3) -> f75_in :|: TRUE f75_out -> f49_out(x4) :|: TRUE Start term: f1_in(T3) ---------------------------------------- (7) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (8) TRUE ---------------------------------------- (9) Obligation: Rules: f85_out(T6) -> f70_out(T6) :|: TRUE f70_in(x) -> f85_in(x) :|: TRUE f100_in(s(T13)) -> f162_in(T13) :|: TRUE f175_out -> f100_out(T10) :|: TRUE f100_in(x1) -> f175_in :|: TRUE f162_out(x2) -> f100_out(s(x2)) :|: TRUE f95_out(x3) -> f87_out(x3) :|: TRUE f87_in(x4) -> f95_in(x4) :|: TRUE f96_in(x5) -> f100_in(x5) :|: TRUE f100_out(x6) -> f96_out(x6) :|: TRUE f70_out(x7) -> f162_out(x7) :|: TRUE f162_in(x8) -> f70_in(x8) :|: TRUE f95_in(x9) -> f96_in(x9) :|: TRUE f96_out(x10) -> f95_out(x10) :|: TRUE f85_in(0) -> f86_in :|: TRUE f85_in(x11) -> f87_in(x11) :|: TRUE f87_out(x12) -> f85_out(x12) :|: TRUE f86_out -> f85_out(0) :|: TRUE f1_in(T3) -> f5_in(T3) :|: TRUE f5_out(x13) -> f1_out(x13) :|: TRUE f49_out(x14) -> f5_out(x14) :|: TRUE f5_in(x15) -> f49_in(x15) :|: TRUE f49_in(x16) -> f70_in(x16) :|: TRUE f70_out(x17) -> f75_in :|: TRUE f75_out -> f49_out(x18) :|: TRUE Start term: f1_in(T3) ---------------------------------------- (10) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f70_in(x) -> f85_in(x) :|: TRUE f100_in(s(T13)) -> f162_in(T13) :|: TRUE f87_in(x4) -> f95_in(x4) :|: TRUE f96_in(x5) -> f100_in(x5) :|: TRUE f162_in(x8) -> f70_in(x8) :|: TRUE f95_in(x9) -> f96_in(x9) :|: TRUE f85_in(x11) -> f87_in(x11) :|: TRUE ---------------------------------------- (11) Obligation: Rules: f70_in(x) -> f85_in(x) :|: TRUE f100_in(s(T13)) -> f162_in(T13) :|: TRUE f87_in(x4) -> f95_in(x4) :|: TRUE f96_in(x5) -> f100_in(x5) :|: TRUE f162_in(x8) -> f70_in(x8) :|: TRUE f95_in(x9) -> f96_in(x9) :|: TRUE f85_in(x11) -> f87_in(x11) :|: TRUE ---------------------------------------- (12) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (13) Obligation: Rules: f87_in(s(T13:0)) -> f87_in(T13:0) :|: TRUE ---------------------------------------- (14) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (15) Obligation: Rules: f87_in(s(T13:0)) -> f87_in(T13:0) :|: TRUE ---------------------------------------- (16) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f87_in(s(T13:0)) -> f87_in(T13:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (17) Obligation: Termination digraph: Nodes: (1) f87_in(s(T13:0)) -> f87_in(T13:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (18) TempFilterProof (SOUND) Used the following sort dictionary for filtering: f87_in(VARIABLE) s(VARIABLE) Removed predefined arithmetic. ---------------------------------------- (19) Obligation: Rules: f87_in(s(T13:0)) -> f87_in(T13:0) ---------------------------------------- (20) IRSwTToQDPProof (SOUND) Removed the integers and created a QDP-Problem. ---------------------------------------- (21) Obligation: Q DP problem: The TRS P consists of the following rules: f87_in(s(T13:0)) -> f87_in(T13:0) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (22) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *f87_in(s(T13:0)) -> f87_in(T13:0) The graph contains the following edges 1 > 1 ---------------------------------------- (23) YES