/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern goal(g) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToIRSwTTransformerProof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, 114 ms] (2) AND (3) IRSwT (4) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (5) TRUE (6) IRSwT (7) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (8) IRSwT (9) IntTRSCompressionProof [EQUIVALENT, 28 ms] (10) IRSwT (11) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (12) IRSwT (13) IRSwTTerminationDigraphProof [EQUIVALENT, 6 ms] (14) IRSwT (15) TempFilterProof [SOUND, 1 ms] (16) IRSwT (17) IRSwTToQDPProof [SOUND, 0 ms] (18) QDP (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] (20) YES ---------------------------------------- (0) Obligation: Clauses: goal(X) :- ','(s2t(X, T), tappend(T, X1, X2)). tappend(nil, Y, Z) :- ','(!, eq(Y, Z)). tappend(T, T1, node(T1, X, T2)) :- ','(left(T, nil), ','(right(T, T2), value(T, X))). tappend(T, T2, node(T1, X, T2)) :- ','(left(T, T1), ','(right(T, nil), value(T, X))). tappend(T, T3, node(U, X, T2)) :- ','(left(T, T1), ','(right(T, T2), ','(value(T, X), tappend(T1, T3, U)))). tappend(T, T1, node(T1, X, U)) :- ','(left(T, T1), ','(right(T, T2), ','(value(T, X), tappend(T2, T3, U)))). s2t(0, L) :- ','(!, eq(L, nil)). s2t(X, node(T, X3, T)) :- ','(p(X, P), s2t(P, T)). s2t(X, node(nil, X4, T)) :- ','(p(X, P), s2t(P, T)). s2t(X, node(T, X5, nil)) :- ','(p(X, P), s2t(P, T)). s2t(X, node(nil, X6, nil)). left(nil, nil). left(node(L, X7, X8), L). right(nil, nil). right(node(X9, X10, R), R). value(nil, nil). value(node(X11, X, X12), X). p(0, 0). p(s(X), X). eq(X, X). Query: goal(g) ---------------------------------------- (1) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 2, "program": { "directives": [], "clauses": [ [ "(goal X)", "(',' (s2t X T) (tappend T X1 X2))" ], [ "(tappend (nil) Y Z)", "(',' (!) (eq Y Z))" ], [ "(tappend T T1 (node T1 X T2))", "(',' (left T (nil)) (',' (right T T2) (value T X)))" ], [ "(tappend T T2 (node T1 X T2))", "(',' (left T T1) (',' (right T (nil)) (value T X)))" ], [ "(tappend T T3 (node U X T2))", "(',' (left T T1) (',' (right T T2) (',' (value T X) (tappend T1 T3 U))))" ], [ "(tappend T T1 (node T1 X U))", "(',' (left T T1) (',' (right T T2) (',' (value T X) (tappend T2 T3 U))))" ], [ "(s2t (0) L)", "(',' (!) (eq L (nil)))" ], [ "(s2t X (node T X3 T))", "(',' (p X P) (s2t P T))" ], [ "(s2t X (node (nil) X4 T))", "(',' (p X P) (s2t P T))" ], [ "(s2t X (node T X5 (nil)))", "(',' (p X P) (s2t P T))" ], [ "(s2t X (node (nil) X6 (nil)))", null ], [ "(left (nil) (nil))", null ], [ "(left (node L X7 X8) L)", null ], [ "(right (nil) (nil))", null ], [ "(right (node X9 X10 R) R)", null ], [ "(value (nil) (nil))", null ], [ "(value (node X11 X X12) X)", null ], [ "(p (0) (0))", null ], [ "(p (s X) X)", null ], [ "(eq X X)", null ] ] }, "graph": { "nodes": { "1461": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1460": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "1459": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, 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"PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1291": { "goal": [{ "clause": 19, "scope": 8, "term": "(eq X180 X181)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X180", "X181" ], "exprvars": [] } }, "582": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "583": { "goal": [{ "clause": 7, "scope": 2, "term": "(s2t T7 X21)" }], "kb": { "nonunifying": [[ "(s2t T7 X21)", "(s2t (0) X32)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T7"], "free": [ "X21", "X32" ], "exprvars": [] } }, "1327": { "goal": [ { "clause": 3, "scope": 7, "term": "(tappend T8 X22 X23)" }, { "clause": 4, "scope": 7, "term": "(tappend T8 X22 X23)" }, { "clause": 5, "scope": 7, "term": "(tappend T8 X22 X23)" } ], "kb": { "nonunifying": [[ "(tappend T8 X22 X23)", "(tappend (nil) X178 X179)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X22", "X23", "X178", "X179" ], "exprvars": [] } }, "584": { "goal": [ { "clause": 8, "scope": 2, "term": "(s2t T7 X21)" }, { "clause": 9, "scope": 2, "term": "(s2t T7 X21)" }, { "clause": 10, "scope": 2, "term": "(s2t T7 X21)" } ], "kb": { "nonunifying": [[ "(s2t T7 X21)", "(s2t (0) X32)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T7"], "free": [ "X21", "X32" ], "exprvars": [] } }, "1326": { "goal": [{ "clause": 2, "scope": 7, "term": "(tappend T8 X22 X23)" }], "kb": { "nonunifying": [[ "(tappend T8 X22 X23)", "(tappend (nil) X178 X179)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X22", "X23", "X178", "X179" ], "exprvars": [] } }, "1447": { "goal": [ { "clause": 13, "scope": 15, "term": "(',' (right (nil) (nil)) (value (nil) X290))" }, { "clause": 14, "scope": 15, "term": "(',' (right (nil) (nil)) (value (nil) X290))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X290"], "exprvars": [] } }, "1722": { "goal": [ { "clause": 11, "scope": 26, "term": "(',' (left T158 X509) (',' (right T158 X507) (',' (value T158 X510) (tappend X507 X508 X511))))" }, { "clause": 12, "scope": 26, "term": "(',' (left T158 X509) (',' (right T158 X507) (',' (value T158 X510) (tappend X507 X508 X511))))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X509", "X510", "X511", "X507", "X508" ], "exprvars": [] } }, "101": { "goal": [{ "clause": -1, "scope": -1, "term": "(s2t T7 X21)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T7"], "free": ["X21"], "exprvars": [] } }, "585": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (p T13 X69) (s2t X69 X70))" }], "kb": { "nonunifying": [[ "(s2t T13 X21)", "(s2t (0) X32)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T13"], "free": [ "X21", "X32", "X70", "X69" ], "exprvars": [] } }, "1446": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "102": { "goal": [{ "clause": -1, "scope": -1, "term": "(tappend T8 X22 X23)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X22", "X23" ], "exprvars": [] } }, "1445": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (right (nil) (nil)) (value (nil) X290))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X290"], "exprvars": [] } }, "1444": { "goal": [{ "clause": 12, "scope": 14, "term": "(',' (left T78 X289) (',' (right T78 (nil)) (value T78 X290)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X289", "X290" ], "exprvars": [] } }, "1443": { "goal": [{ "clause": 11, "scope": 14, "term": "(',' (left T78 X289) (',' (right T78 (nil)) (value T78 X290)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X289", "X290" ], "exprvars": [] } }, "1046": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1442": { "goal": [ { "clause": 11, "scope": 14, "term": "(',' (left T78 X289) (',' (right T78 (nil)) (value T78 X290)))" }, { "clause": 12, "scope": 14, "term": "(',' (left T78 X289) (',' (right T78 (nil)) (value T78 X290)))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X289", "X290" ], "exprvars": [] } }, "1287": { "goal": [{ "clause": -1, "scope": -1, "term": "(eq X180 X181)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X180", "X181" ], "exprvars": [] } }, "1441": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (left T78 X289) (',' (right T78 (nil)) (value T78 X290)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X289", "X290" ], "exprvars": [] } }, "1608": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (left T115 X375) (',' (right T115 X379) (',' (value T115 X378) (tappend X375 X376 X377))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X376", "X377", "X378", "X379", "X375" ], "exprvars": [] } }, "1727": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1726": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (right (nil) X507) (',' (value (nil) X510) (tappend X507 X508 X511)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X510", "X511", "X507", "X508" ], "exprvars": [] } }, "1725": { "goal": [{ "clause": 12, "scope": 26, "term": "(',' (left T158 X509) (',' (right T158 X507) (',' (value T158 X510) (tappend X507 X508 X511))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X509", "X510", "X511", "X507", "X508" ], "exprvars": [] } }, "1724": { "goal": [{ "clause": 11, "scope": 26, "term": "(',' (left T158 X509) (',' (right T158 X507) (',' (value T158 X510) (tappend X507 X508 X511))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X509", "X510", "X511", "X507", "X508" ], "exprvars": [] } } }, "edges": [ { "from": 2, "to": 3, "label": "CASE" }, { "from": 3, "to": 97, "label": "ONLY EVAL with clause\ngoal(X20) :- ','(s2t(X20, X21), tappend(X21, X22, X23)).\nand substitutionT4 -> T7,\nX20 -> T7" }, { "from": 97, "to": 101, "label": "SPLIT 1" }, { "from": 97, "to": 102, "label": "SPLIT 2\nnew knowledge:\nT7 is ground\nreplacements:X21 -> T8" }, { "from": 101, "to": 549, "label": "CASE" }, { "from": 102, "to": 1121, "label": "CASE" }, { "from": 549, "to": 550, "label": "EVAL with clause\ns2t(0, X32) :- ','(!_2, eq(X32, nil)).\nand substitutionT7 -> 0,\nX21 -> X33,\nX32 -> X33" }, { "from": 549, "to": 551, "label": "EVAL-BACKTRACK" }, { "from": 550, "to": 572, "label": "CUT" }, { "from": 551, "to": 583, "label": "PARALLEL" }, { "from": 551, "to": 584, "label": "PARALLEL" }, { "from": 572, "to": 577, "label": "CASE" }, { "from": 577, "to": 581, "label": "ONLY EVAL with clause\neq(X38, X38).\nand substitutionX33 -> nil,\nX38 -> nil,\nX39 -> nil" }, { "from": 581, "to": 582, "label": "SUCCESS" }, { "from": 583, "to": 585, "label": "ONLY EVAL with clause\ns2t(X66, node(X67, X68, X67)) :- ','(p(X66, X69), s2t(X69, X67)).\nand substitutionT7 -> T13,\nX66 -> T13,\nX67 -> X70,\nX68 -> X71,\nX21 -> node(X70, X71, X70)" }, { "from": 584, "to": 872, "label": "PARALLEL" }, { "from": 584, "to": 875, "label": "PARALLEL" }, { "from": 585, "to": 845, "label": "CASE" }, { "from": 845, "to": 850, "label": "BACKTRACK\nfor clause: p(0, 0)\nwith clash: (s2t(T13, X21), s2t(0, X32))" }, { "from": 850, "to": 858, "label": "EVAL with clause\np(s(X76), X76).\nand substitutionX76 -> T16,\nT13 -> s(T16),\nX69 -> T16" }, { "from": 850, "to": 859, "label": "EVAL-BACKTRACK" }, { "from": 858, "to": 101, "label": "INSTANCE with matching:\nT7 -> T16\nX21 -> X70" }, { "from": 872, "to": 937, "label": "ONLY EVAL with clause\ns2t(X105, node(nil, X106, X107)) :- ','(p(X105, X108), s2t(X108, X107)).\nand substitutionT7 -> T21,\nX105 -> T21,\nX106 -> X109,\nX107 -> X110,\nX21 -> node(nil, X109, X110)" }, { "from": 875, "to": 957, "label": "PARALLEL" }, { "from": 875, "to": 960, "label": "PARALLEL" }, { "from": 937, "to": 938, "label": "CASE" }, { "from": 938, "to": 939, "label": "BACKTRACK\nfor clause: p(0, 0)\nwith clash: (s2t(T21, X21), s2t(0, X32))" }, { "from": 939, "to": 940, "label": "EVAL with clause\np(s(X115), X115).\nand substitutionX115 -> T24,\nT21 -> s(T24),\nX108 -> T24" }, { "from": 939, "to": 941, "label": "EVAL-BACKTRACK" }, { "from": 940, "to": 101, "label": "INSTANCE with matching:\nT7 -> T24\nX21 -> X110" }, { "from": 957, "to": 998, "label": "ONLY EVAL with clause\ns2t(X144, node(X145, X146, nil)) :- ','(p(X144, X147), s2t(X147, X145)).\nand substitutionT7 -> T29,\nX144 -> T29,\nX145 -> X148,\nX146 -> X149,\nX21 -> node(X148, X149, nil)" }, { "from": 960, "to": 1112, "label": "ONLY EVAL with clause\ns2t(X163, node(nil, X164, nil)).\nand substitutionT7 -> T35,\nX163 -> T35,\nX164 -> X165,\nX21 -> node(nil, X165, nil)" }, { "from": 998, "to": 1007, "label": "CASE" }, { "from": 1007, "to": 1025, "label": "BACKTRACK\nfor clause: p(0, 0)\nwith clash: (s2t(T29, X21), s2t(0, X32))" }, { "from": 1025, "to": 1041, "label": "EVAL with clause\np(s(X154), X154).\nand substitutionX154 -> T32,\nT29 -> s(T32),\nX147 -> T32" }, { "from": 1025, "to": 1046, "label": "EVAL-BACKTRACK" }, { "from": 1041, "to": 101, "label": "INSTANCE with matching:\nT7 -> T32\nX21 -> X148" }, { "from": 1112, "to": 1114, "label": "SUCCESS" }, { "from": 1121, "to": 1281, "label": "EVAL with clause\ntappend(nil, X178, X179) :- ','(!_7, eq(X178, X179)).\nand substitutionT8 -> nil,\nX22 -> X180,\nX178 -> X180,\nX23 -> X181,\nX179 -> X181" }, { "from": 1121, "to": 1284, "label": "EVAL-BACKTRACK" }, { "from": 1281, "to": 1287, "label": "CUT" }, { "from": 1284, "to": 1326, "label": "PARALLEL" }, { "from": 1284, "to": 1327, "label": "PARALLEL" }, { "from": 1287, "to": 1291, "label": "CASE" }, { "from": 1291, "to": 1314, "label": "ONLY EVAL with clause\neq(X186, X186).\nand substitutionX180 -> X187,\nX186 -> X187,\nX181 -> X187" }, { "from": 1314, "to": 1316, "label": "SUCCESS" }, { "from": 1326, "to": 1381, "label": "ONLY EVAL with clause\ntappend(X216, X217, node(X217, X218, X219)) :- ','(left(X216, nil), ','(right(X216, X219), value(X216, X218))).\nand substitutionT8 -> T43,\nX216 -> T43,\nX22 -> X220,\nX217 -> X220,\nX218 -> X221,\nX219 -> X222,\nX23 -> node(X220, X221, X222),\nT42 -> T43" }, { "from": 1327, "to": 1423, "label": "PARALLEL" }, { "from": 1327, "to": 1424, "label": "PARALLEL" }, { "from": 1381, "to": 1382, "label": "CASE" }, { "from": 1382, "to": 1383, "label": "PARALLEL" }, { "from": 1382, "to": 1384, "label": "PARALLEL" }, { "from": 1383, "to": 1385, "label": "EVAL with clause\nleft(nil, nil).\nand substitutionT43 -> nil" }, { "from": 1383, "to": 1386, "label": "EVAL-BACKTRACK" }, { "from": 1384, "to": 1398, "label": "EVAL with clause\nleft(node(X235, X236, X237), X235).\nand substitutionX235 -> nil,\nX236 -> T53,\nX237 -> T54,\nT43 -> node(nil, T53, T54),\nT50 -> nil,\nT51 -> T53,\nT52 -> T54" }, { "from": 1384, "to": 1399, "label": "EVAL-BACKTRACK" }, { "from": 1385, "to": 1387, "label": "CASE" }, { "from": 1387, "to": 1388, "label": "PARALLEL" }, { "from": 1387, "to": 1389, "label": "PARALLEL" }, { "from": 1388, "to": 1390, "label": "ONLY EVAL with clause\nright(nil, nil).\nand substitutionX222 -> nil" }, { "from": 1389, "to": 1397, "label": "BACKTRACK\nfor clause: right(node(X9, X10, R), R)because of non-unification" }, { "from": 1390, "to": 1391, "label": "CASE" }, { "from": 1391, "to": 1392, "label": "PARALLEL" }, { "from": 1391, "to": 1393, "label": "PARALLEL" }, { "from": 1392, "to": 1394, "label": "ONLY EVAL with clause\nvalue(nil, nil).\nand substitutionX221 -> nil" }, { "from": 1393, "to": 1396, "label": "BACKTRACK\nfor clause: value(node(X11, X, X12), X)because of non-unification" }, { "from": 1394, "to": 1395, "label": "SUCCESS" }, { "from": 1398, "to": 1400, "label": "CASE" }, { "from": 1400, "to": 1401, "label": "BACKTRACK\nfor clause: right(nil, nil)because of non-unification" }, { "from": 1401, "to": 1402, "label": "ONLY EVAL with clause\nright(node(X244, X245, X246), X246).\nand substitutionX244 -> nil,\nT53 -> T63,\nX245 -> T63,\nT54 -> T64,\nX246 -> T64,\nX222 -> T64,\nT61 -> T63,\nT62 -> T64" }, { "from": 1402, "to": 1403, "label": "CASE" }, { "from": 1403, "to": 1404, "label": "BACKTRACK\nfor clause: value(nil, nil)because of non-unification" }, { "from": 1404, "to": 1405, "label": "ONLY EVAL with clause\nvalue(node(X253, X254, X255), X254).\nand substitutionX253 -> nil,\nT63 -> T69,\nX254 -> T69,\nT64 -> T70,\nX255 -> T70,\nX221 -> T69" }, { "from": 1405, "to": 1406, "label": "SUCCESS" }, { "from": 1423, "to": 1441, "label": "ONLY EVAL with clause\ntappend(X284, X285, node(X286, X287, X285)) :- ','(left(X284, X286), ','(right(X284, nil), value(X284, X287))).\nand substitutionT8 -> T78,\nX284 -> T78,\nX22 -> X288,\nX285 -> X288,\nX286 -> X289,\nX287 -> X290,\nX23 -> node(X289, X290, X288),\nT77 -> T78" }, { "from": 1424, "to": 1539, "label": "PARALLEL" }, { "from": 1424, "to": 1540, "label": "PARALLEL" }, { "from": 1441, "to": 1442, "label": "CASE" }, { "from": 1442, "to": 1443, "label": "PARALLEL" }, { "from": 1442, "to": 1444, "label": "PARALLEL" }, { "from": 1443, "to": 1445, "label": "EVAL with clause\nleft(nil, nil).\nand substitutionT78 -> nil,\nX289 -> nil" }, { "from": 1443, "to": 1446, "label": "EVAL-BACKTRACK" }, { "from": 1444, "to": 1466, "label": "EVAL with clause\nleft(node(X303, X304, X305), X303).\nand substitutionX303 -> T88,\nX304 -> T89,\nX305 -> T90,\nT78 -> node(T88, T89, T90),\nX289 -> T88,\nT85 -> T88,\nT86 -> T89,\nT87 -> T90" }, { "from": 1444, "to": 1468, "label": "EVAL-BACKTRACK" }, { "from": 1445, "to": 1447, "label": "CASE" }, { "from": 1447, "to": 1450, "label": "PARALLEL" }, { "from": 1447, "to": 1451, "label": "PARALLEL" }, { "from": 1450, "to": 1454, "label": "ONLY EVAL with clause\nright(nil, nil).\nand substitution" }, { "from": 1451, "to": 1461, "label": "BACKTRACK\nfor clause: right(node(X9, X10, R), R)because of non-unification" }, { "from": 1454, "to": 1455, "label": "CASE" }, { "from": 1455, "to": 1456, "label": "PARALLEL" }, { "from": 1455, "to": 1457, "label": "PARALLEL" }, { "from": 1456, "to": 1458, "label": "ONLY EVAL with clause\nvalue(nil, nil).\nand substitutionX290 -> nil" }, { "from": 1457, "to": 1460, "label": "BACKTRACK\nfor clause: value(node(X11, X, X12), X)because of non-unification" }, { "from": 1458, "to": 1459, "label": "SUCCESS" }, { "from": 1466, "to": 1469, "label": "CASE" }, { "from": 1469, "to": 1470, "label": "BACKTRACK\nfor clause: right(nil, nil)because of non-unification" }, { "from": 1470, "to": 1473, "label": "EVAL with clause\nright(node(X312, X313, X314), X314).\nand substitutionT88 -> T100,\nX312 -> T100,\nT89 -> T101,\nX313 -> T101,\nT90 -> nil,\nX314 -> nil,\nT99 -> nil,\nT97 -> T100,\nT98 -> T101" }, { "from": 1470, "to": 1474, "label": "EVAL-BACKTRACK" }, { "from": 1473, "to": 1475, "label": "CASE" }, { "from": 1475, "to": 1476, "label": "BACKTRACK\nfor clause: value(nil, nil)because of non-unification" }, { "from": 1476, "to": 1484, "label": "ONLY EVAL with clause\nvalue(node(X321, X322, X323), X322).\nand substitutionT100 -> T106,\nX321 -> T106,\nT101 -> T107,\nX322 -> T107,\nX323 -> nil,\nX290 -> T107" }, { "from": 1484, "to": 1485, "label": "SUCCESS" }, { "from": 1539, "to": 1608, "label": "ONLY EVAL with clause\ntappend(X370, X371, node(X372, X373, X374)) :- ','(left(X370, X375), ','(right(X370, X374), ','(value(X370, X373), tappend(X375, X371, X372)))).\nand substitutionT8 -> T115,\nX370 -> T115,\nX22 -> X376,\nX371 -> X376,\nX372 -> X377,\nX373 -> X378,\nX374 -> X379,\nX23 -> node(X377, X378, X379),\nT114 -> T115" }, { "from": 1540, "to": 1719, "label": "ONLY EVAL with clause\ntappend(X503, X504, node(X504, X505, X506)) :- ','(left(X503, X504), ','(right(X503, X507), ','(value(X503, X505), tappend(X507, X508, X506)))).\nand substitutionT8 -> T158,\nX503 -> T158,\nX22 -> X509,\nX504 -> X509,\nX505 -> X510,\nX506 -> X511,\nX23 -> node(X509, X510, X511),\nT157 -> T158" }, { "from": 1608, "to": 1611, "label": "CASE" }, { "from": 1611, "to": 1612, "label": "PARALLEL" }, { "from": 1611, "to": 1613, "label": "PARALLEL" }, { "from": 1612, "to": 1614, "label": "EVAL with clause\nleft(nil, nil).\nand substitutionT115 -> nil,\nX375 -> nil" }, { "from": 1612, "to": 1615, "label": "EVAL-BACKTRACK" }, { "from": 1613, "to": 1654, "label": "EVAL with clause\nleft(node(X442, X443, X444), X442).\nand substitutionX442 -> T125,\nX443 -> T126,\nX444 -> T127,\nT115 -> node(T125, T126, T127),\nX375 -> T125,\nT122 -> T125,\nT123 -> T126,\nT124 -> T127" }, { "from": 1613, "to": 1655, "label": "EVAL-BACKTRACK" }, { "from": 1614, "to": 1616, "label": "CASE" }, { "from": 1616, "to": 1621, "label": "PARALLEL" }, { "from": 1616, "to": 1622, "label": "PARALLEL" }, { "from": 1621, "to": 1623, "label": "ONLY EVAL with clause\nright(nil, nil).\nand substitutionX379 -> nil" }, { "from": 1622, "to": 1653, "label": "BACKTRACK\nfor clause: right(node(X9, X10, R), R)because of non-unification" }, { "from": 1623, "to": 1633, "label": "CASE" }, { "from": 1633, "to": 1637, "label": "PARALLEL" }, { "from": 1633, "to": 1638, "label": "PARALLEL" }, { "from": 1637, "to": 1639, "label": "ONLY EVAL with clause\nvalue(nil, nil).\nand substitutionX378 -> nil" }, { "from": 1638, "to": 1652, "label": "BACKTRACK\nfor clause: value(node(X11, X, X12), X)because of non-unification" }, { "from": 1639, "to": 1644, "label": "CASE" }, { "from": 1644, "to": 1647, "label": "ONLY EVAL with clause\ntappend(nil, X420, X421) :- ','(!_22, eq(X420, X421)).\nand substitutionX376 -> X422,\nX420 -> X422,\nX377 -> X423,\nX421 -> X423" }, { "from": 1647, "to": 1648, "label": "CUT" }, { "from": 1648, "to": 1649, "label": "CASE" }, { "from": 1649, "to": 1650, "label": "ONLY EVAL with clause\neq(X428, X428).\nand substitutionX422 -> X429,\nX428 -> X429,\nX423 -> X429" }, { "from": 1650, "to": 1651, "label": "SUCCESS" }, { "from": 1654, "to": 1656, "label": "CASE" }, { "from": 1656, "to": 1657, "label": "BACKTRACK\nfor clause: right(nil, nil)because of non-unification" }, { "from": 1657, "to": 1659, "label": "ONLY EVAL with clause\nright(node(X459, X460, X461), X461).\nand substitutionT125 -> T140,\nX459 -> T140,\nT126 -> T141,\nX460 -> T141,\nT127 -> T142,\nX461 -> T142,\nX379 -> T142,\nT137 -> T140,\nT138 -> T141,\nT139 -> T142" }, { "from": 1659, "to": 1660, "label": "CASE" }, { "from": 1660, "to": 1661, "label": "BACKTRACK\nfor clause: value(nil, nil)because of non-unification" }, { "from": 1661, "to": 1675, "label": "ONLY EVAL with clause\nvalue(node(X476, X477, X478), X477).\nand substitutionT140 -> T153,\nX476 -> T153,\nT141 -> T151,\nX477 -> T151,\nT142 -> T152,\nX478 -> T152,\nX378 -> T151,\nT150 -> T153" }, { "from": 1675, "to": 102, "label": "INSTANCE with matching:\nT8 -> T153\nX22 -> X376\nX23 -> X377" }, { "from": 1719, "to": 1722, "label": "CASE" }, { "from": 1722, "to": 1724, "label": "PARALLEL" }, { "from": 1722, "to": 1725, "label": "PARALLEL" }, { "from": 1724, "to": 1726, "label": "EVAL with clause\nleft(nil, nil).\nand substitutionT158 -> nil,\nX509 -> nil" }, { "from": 1724, "to": 1727, "label": "EVAL-BACKTRACK" }, { "from": 1725, "to": 1761, "label": "EVAL with clause\nleft(node(X556, X557, X558), X556).\nand substitutionX556 -> T168,\nX557 -> T169,\nX558 -> T170,\nT158 -> node(T168, T169, T170),\nX509 -> T168,\nT165 -> T168,\nT166 -> T169,\nT167 -> T170" }, { "from": 1725, "to": 1762, "label": "EVAL-BACKTRACK" }, { "from": 1726, "to": 1730, "label": "CASE" }, { "from": 1730, "to": 1731, "label": "PARALLEL" }, { "from": 1730, "to": 1732, "label": "PARALLEL" }, { "from": 1731, "to": 1739, "label": "ONLY EVAL with clause\nright(nil, nil).\nand substitutionX507 -> nil" }, { "from": 1732, "to": 1759, "label": "BACKTRACK\nfor clause: right(node(X9, X10, R), R)because of non-unification" }, { "from": 1739, "to": 1745, "label": "CASE" }, { "from": 1745, "to": 1750, "label": "PARALLEL" }, { "from": 1745, "to": 1751, "label": "PARALLEL" }, { "from": 1750, "to": 1753, "label": "ONLY EVAL with clause\nvalue(nil, nil).\nand substitutionX510 -> nil" }, { "from": 1751, "to": 1756, "label": "BACKTRACK\nfor clause: value(node(X11, X, X12), X)because of non-unification" }, { "from": 1753, "to": 1639, "label": "INSTANCE with matching:\nX376 -> X508\nX377 -> X511" }, { "from": 1761, "to": 1764, "label": "CASE" }, { "from": 1764, "to": 1765, "label": "BACKTRACK\nfor clause: right(nil, nil)because of non-unification" }, { "from": 1765, "to": 1772, "label": "ONLY EVAL with clause\nright(node(X573, X574, X575), X575).\nand substitutionT168 -> T183,\nX573 -> T183,\nT169 -> T184,\nX574 -> T184,\nT170 -> T185,\nX575 -> T185,\nX507 -> T185,\nT180 -> T183,\nT181 -> T184,\nT182 -> T185" }, { "from": 1772, "to": 1773, "label": "CASE" }, { "from": 1773, "to": 1774, "label": "BACKTRACK\nfor clause: value(nil, nil)because of non-unification" }, { "from": 1774, "to": 1782, "label": "ONLY EVAL with clause\nvalue(node(X590, X591, X592), X591).\nand substitutionT183 -> T193,\nX590 -> T193,\nT184 -> T194,\nX591 -> T194,\nT185 -> T196,\nX592 -> T196,\nX510 -> T194,\nT195 -> T196" }, { "from": 1782, "to": 102, "label": "INSTANCE with matching:\nT8 -> T196\nX22 -> X508\nX23 -> X511" } ], "type": "Graph" } } ---------------------------------------- (2) Complex Obligation (AND) ---------------------------------------- (3) Obligation: Rules: f1654_in -> f1656_in :|: TRUE f1656_out -> f1654_out :|: TRUE f1725_in -> f1762_in :|: TRUE f1761_out -> f1725_out :|: TRUE f1725_in -> f1761_in :|: TRUE f1762_out -> f1725_out :|: TRUE f1424_in -> f1540_in :|: TRUE f1424_in -> f1539_in :|: TRUE f1539_out -> f1424_out :|: TRUE f1540_out -> f1424_out :|: TRUE f1773_out -> f1772_out :|: TRUE f1772_in -> f1773_in :|: TRUE f1611_out -> f1608_out :|: TRUE f1608_in -> f1611_in :|: TRUE f1327_in -> f1424_in :|: TRUE f1424_out -> f1327_out :|: TRUE f1423_out -> f1327_out :|: TRUE f1327_in -> f1423_in :|: TRUE f1774_in -> f1782_in :|: TRUE f1782_out -> f1774_out :|: TRUE f1719_in -> f1722_in :|: TRUE f1722_out -> f1719_out :|: TRUE f1656_in -> f1657_in :|: TRUE f1657_out -> f1656_out :|: TRUE f1326_out -> f1284_out :|: TRUE f1327_out -> f1284_out :|: TRUE f1284_in -> f1327_in :|: TRUE f1284_in -> f1326_in :|: TRUE f1611_in -> f1612_in :|: TRUE f1613_out -> f1611_out :|: TRUE f1612_out -> f1611_out :|: TRUE f1611_in -> f1613_in :|: TRUE f1281_out -> f1121_out :|: TRUE f1121_in -> f1281_in :|: TRUE f1121_in -> f1284_in :|: TRUE f1284_out -> f1121_out :|: TRUE f1661_in -> f1675_in :|: TRUE f1675_out -> f1661_out :|: TRUE f1657_in -> f1659_in :|: TRUE f1659_out -> f1657_out :|: TRUE f1722_in -> f1725_in :|: TRUE f1722_in -> f1724_in :|: TRUE f1724_out -> f1722_out :|: TRUE f1725_out -> f1722_out :|: TRUE f1660_out -> f1659_out :|: TRUE f1659_in -> f1660_in :|: TRUE f1761_in -> f1764_in :|: TRUE f1764_out -> f1761_out :|: TRUE f1774_out -> f1773_out :|: TRUE f1773_in -> f1774_in :|: TRUE f1660_in -> f1661_in :|: TRUE f1661_out -> f1660_out :|: TRUE f1765_in -> f1772_in :|: TRUE f1772_out -> f1765_out :|: TRUE f1719_out -> f1540_out :|: TRUE f1540_in -> f1719_in :|: TRUE f102_in -> f1121_in :|: TRUE f1121_out -> f102_out :|: TRUE f102_out -> f1675_out :|: TRUE f1675_in -> f102_in :|: TRUE f1764_in -> f1765_in :|: TRUE f1765_out -> f1764_out :|: TRUE f1613_in -> f1655_in :|: TRUE f1654_out -> f1613_out :|: TRUE f1613_in -> f1654_in :|: TRUE f1655_out -> f1613_out :|: TRUE f1608_out -> f1539_out :|: TRUE f1539_in -> f1608_in :|: TRUE f1782_in -> f102_in :|: TRUE f102_out -> f1782_out :|: TRUE f2_in(T4) -> f3_in(T4) :|: TRUE f3_out(x) -> f2_out(x) :|: TRUE f97_out(T7) -> f3_out(T7) :|: TRUE f3_in(x1) -> f97_in(x1) :|: TRUE f97_in(x2) -> f101_in(x2) :|: TRUE f101_out(x3) -> f102_in :|: TRUE f102_out -> f97_out(x4) :|: TRUE Start term: f2_in(T4) ---------------------------------------- (4) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (5) TRUE ---------------------------------------- (6) Obligation: Rules: f939_in(T21) -> f941_in :|: TRUE f940_out(T24) -> f939_out(s(T24)) :|: TRUE f941_out -> f939_out(x) :|: TRUE f939_in(s(x1)) -> f940_in(x1) :|: TRUE f845_out(T13) -> f585_out(T13) :|: TRUE f585_in(x2) -> f845_in(x2) :|: TRUE f937_in(x3) -> f938_in(x3) :|: TRUE f938_out(x4) -> f937_out(x4) :|: TRUE f584_out(T7) -> f551_out(T7) :|: TRUE f551_in(x5) -> f584_in(x5) :|: TRUE f583_out(x6) -> f551_out(x6) :|: TRUE f551_in(x7) -> f583_in(x7) :|: TRUE f872_in(x8) -> f937_in(x8) :|: TRUE f937_out(x9) -> f872_out(x9) :|: TRUE f957_out(x10) -> f875_out(x10) :|: TRUE f875_in(x11) -> f957_in(x11) :|: TRUE f875_in(x12) -> f960_in(x12) :|: TRUE f960_out(x13) -> f875_out(x13) :|: TRUE f850_out(x14) -> f845_out(x14) :|: TRUE f845_in(x15) -> f850_in(x15) :|: TRUE f585_out(x16) -> f583_out(x16) :|: TRUE f583_in(x17) -> f585_in(x17) :|: TRUE f1025_in(T29) -> f1046_in :|: TRUE f1025_in(s(T32)) -> f1041_in(T32) :|: TRUE f1046_out -> f1025_out(x18) :|: TRUE f1041_out(x19) -> f1025_out(s(x19)) :|: TRUE f1025_out(x20) -> f1007_out(x20) :|: TRUE f1007_in(x21) -> f1025_in(x21) :|: TRUE f875_out(x22) -> f584_out(x22) :|: TRUE f584_in(x23) -> f872_in(x23) :|: TRUE f872_out(x24) -> f584_out(x24) :|: TRUE f584_in(x25) -> f875_in(x25) :|: TRUE f939_out(x26) -> f938_out(x26) :|: TRUE f938_in(x27) -> f939_in(x27) :|: TRUE f101_out(T16) -> f858_out(T16) :|: TRUE f858_in(x28) -> f101_in(x28) :|: TRUE f858_out(x29) -> f850_out(s(x29)) :|: TRUE f850_in(x30) -> f859_in :|: TRUE f859_out -> f850_out(x31) :|: TRUE f850_in(s(x32)) -> f858_in(x32) :|: TRUE f957_in(x33) -> f998_in(x33) :|: TRUE f998_out(x34) -> f957_out(x34) :|: TRUE f1041_in(x35) -> f101_in(x35) :|: TRUE f101_out(x36) -> f1041_out(x36) :|: TRUE f940_in(x37) -> f101_in(x37) :|: TRUE f101_out(x38) -> f940_out(x38) :|: TRUE f1007_out(x39) -> f998_out(x39) :|: TRUE f998_in(x40) -> f1007_in(x40) :|: TRUE f101_in(x41) -> f549_in(x41) :|: TRUE f549_out(x42) -> f101_out(x42) :|: TRUE f550_out -> f549_out(0) :|: TRUE f551_out(x43) -> f549_out(x43) :|: TRUE f549_in(x44) -> f551_in(x44) :|: TRUE f549_in(0) -> f550_in :|: TRUE f2_in(T4) -> f3_in(T4) :|: TRUE f3_out(x45) -> f2_out(x45) :|: TRUE f97_out(x46) -> f3_out(x46) :|: TRUE f3_in(x47) -> f97_in(x47) :|: TRUE f97_in(x48) -> f101_in(x48) :|: TRUE f101_out(x49) -> f102_in :|: TRUE f102_out -> f97_out(x50) :|: TRUE Start term: f2_in(T4) ---------------------------------------- (7) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f939_in(s(x1)) -> f940_in(x1) :|: TRUE f585_in(x2) -> f845_in(x2) :|: TRUE f937_in(x3) -> f938_in(x3) :|: TRUE f551_in(x5) -> f584_in(x5) :|: TRUE f551_in(x7) -> f583_in(x7) :|: TRUE f872_in(x8) -> f937_in(x8) :|: TRUE f875_in(x11) -> f957_in(x11) :|: TRUE f845_in(x15) -> f850_in(x15) :|: TRUE f583_in(x17) -> f585_in(x17) :|: TRUE f1025_in(s(T32)) -> f1041_in(T32) :|: TRUE f1007_in(x21) -> f1025_in(x21) :|: TRUE f584_in(x23) -> f872_in(x23) :|: TRUE f584_in(x25) -> f875_in(x25) :|: TRUE f938_in(x27) -> f939_in(x27) :|: TRUE f858_in(x28) -> f101_in(x28) :|: TRUE f850_in(s(x32)) -> f858_in(x32) :|: TRUE f957_in(x33) -> f998_in(x33) :|: TRUE f1041_in(x35) -> f101_in(x35) :|: TRUE f940_in(x37) -> f101_in(x37) :|: TRUE f998_in(x40) -> f1007_in(x40) :|: TRUE f101_in(x41) -> f549_in(x41) :|: TRUE f549_in(x44) -> f551_in(x44) :|: TRUE ---------------------------------------- (8) Obligation: Rules: f939_in(s(x1)) -> f940_in(x1) :|: TRUE f585_in(x2) -> f845_in(x2) :|: TRUE f937_in(x3) -> f938_in(x3) :|: TRUE f551_in(x5) -> f584_in(x5) :|: TRUE f551_in(x7) -> f583_in(x7) :|: TRUE f872_in(x8) -> f937_in(x8) :|: TRUE f875_in(x11) -> f957_in(x11) :|: TRUE f845_in(x15) -> f850_in(x15) :|: TRUE f583_in(x17) -> f585_in(x17) :|: TRUE f1025_in(s(T32)) -> f1041_in(T32) :|: TRUE f1007_in(x21) -> f1025_in(x21) :|: TRUE f584_in(x23) -> f872_in(x23) :|: TRUE f584_in(x25) -> f875_in(x25) :|: TRUE f938_in(x27) -> f939_in(x27) :|: TRUE f858_in(x28) -> f101_in(x28) :|: TRUE f850_in(s(x32)) -> f858_in(x32) :|: TRUE f957_in(x33) -> f998_in(x33) :|: TRUE f1041_in(x35) -> f101_in(x35) :|: TRUE f940_in(x37) -> f101_in(x37) :|: TRUE f998_in(x40) -> f1007_in(x40) :|: TRUE f101_in(x41) -> f549_in(x41) :|: TRUE f549_in(x44) -> f551_in(x44) :|: TRUE ---------------------------------------- (9) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (10) Obligation: Rules: f551_in(s(x32:0)) -> f551_in(x32:0) :|: TRUE ---------------------------------------- (11) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (12) Obligation: Rules: f551_in(s(x32:0)) -> f551_in(x32:0) :|: TRUE ---------------------------------------- (13) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f551_in(s(x32:0)) -> f551_in(x32:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (14) Obligation: Termination digraph: Nodes: (1) f551_in(s(x32:0)) -> f551_in(x32:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (15) TempFilterProof (SOUND) Used the following sort dictionary for filtering: f551_in(VARIABLE) s(VARIABLE) Removed predefined arithmetic. ---------------------------------------- (16) Obligation: Rules: f551_in(s(x32:0)) -> f551_in(x32:0) ---------------------------------------- (17) IRSwTToQDPProof (SOUND) Removed the integers and created a QDP-Problem. ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: f551_in(s(x32:0)) -> f551_in(x32:0) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (19) 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: *f551_in(s(x32:0)) -> f551_in(x32:0) The graph contains the following edges 1 > 1 ---------------------------------------- (20) YES