4.66/1.97 YES 4.66/2.00 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 4.66/2.00 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.66/2.00 4.66/2.00 4.66/2.00 Left Termination of the query pattern 4.66/2.00 4.66/2.00 average(g,g,a) 4.66/2.00 4.66/2.00 w.r.t. the given Prolog program could successfully be proven: 4.66/2.00 4.66/2.00 (0) Prolog 4.66/2.00 (1) PrologToDTProblemTransformerProof [SOUND, 40 ms] 4.66/2.00 (2) TRIPLES 4.66/2.00 (3) TriplesToPiDPProof [SOUND, 0 ms] 4.66/2.00 (4) PiDP 4.66/2.00 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 4.66/2.00 (6) PiDP 4.66/2.00 (7) PiDPToQDPProof [SOUND, 1 ms] 4.66/2.00 (8) QDP 4.66/2.00 (9) MRRProof [EQUIVALENT, 0 ms] 4.66/2.00 (10) QDP 4.66/2.00 (11) PisEmptyProof [EQUIVALENT, 0 ms] 4.66/2.00 (12) YES 4.66/2.00 4.66/2.00 4.66/2.00 ---------------------------------------- 4.66/2.00 4.66/2.00 (0) 4.66/2.00 Obligation: 4.66/2.00 Clauses: 4.66/2.00 4.66/2.00 average(0, 0, 0). 4.66/2.00 average(0, s(0), 0). 4.66/2.00 average(0, s(s(0)), s(0)). 4.66/2.00 average(s(X), Y, Z) :- average(X, s(Y), Z). 4.66/2.00 average(X, s(s(s(Y))), s(Z)) :- average(s(X), Y, Z). 4.66/2.00 4.66/2.00 4.66/2.00 Query: average(g,g,a) 4.66/2.00 ---------------------------------------- 4.66/2.00 4.66/2.00 (1) PrologToDTProblemTransformerProof (SOUND) 4.66/2.00 Built DT problem from termination graph DT10. 4.66/2.00 4.66/2.00 { 4.66/2.00 "root": 5, 4.66/2.00 "program": { 4.66/2.00 "directives": [], 4.66/2.00 "clauses": [ 4.66/2.00 [ 4.66/2.00 "(average (0) (0) (0))", 4.66/2.00 null 4.66/2.00 ], 4.66/2.00 [ 4.66/2.00 "(average (0) (s (0)) (0))", 4.66/2.00 null 4.66/2.00 ], 4.66/2.00 [ 4.66/2.00 "(average (0) (s (s (0))) (s (0)))", 4.66/2.00 null 4.66/2.00 ], 4.66/2.00 [ 4.66/2.00 "(average (s X) Y Z)", 4.66/2.00 "(average X (s Y) Z)" 4.66/2.00 ], 4.66/2.00 [ 4.66/2.00 "(average X (s (s (s Y))) (s Z))", 4.66/2.00 "(average (s X) Y Z)" 4.66/2.00 ] 4.66/2.00 ] 4.66/2.00 }, 4.66/2.00 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4.66/2.00 "relations": [] 4.66/2.00 }, 4.66/2.00 "ground": [], 4.66/2.00 "free": [], 4.66/2.00 "exprvars": [] 4.66/2.00 } 4.66/2.00 }, 4.66/2.00 "188": { 4.66/2.00 "goal": [{ 4.66/2.00 "clause": 3, 4.66/2.00 "scope": 2, 4.66/2.00 "term": "(average T7 (s T8) T10)" 4.66/2.00 }], 4.66/2.00 "kb": { 4.66/2.00 "nonunifying": [], 4.66/2.00 "intvars": {}, 4.66/2.00 "arithmetic": { 4.66/2.00 "type": "PlainIntegerRelationState", 4.66/2.00 "relations": [] 4.66/2.00 }, 4.66/2.00 "ground": [ 4.66/2.00 "T7", 4.66/2.00 "T8" 4.66/2.00 ], 4.66/2.00 "free": [], 4.66/2.00 "exprvars": [] 4.66/2.00 } 4.66/2.00 }, 4.66/2.00 "189": { 4.66/2.00 "goal": [ 4.66/2.00 { 4.66/2.00 "clause": 4, 4.66/2.00 "scope": 2, 4.66/2.00 "term": "(average T7 (s T8) T10)" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "clause": -1, 4.66/2.00 "scope": 2, 4.66/2.00 "term": null 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "clause": 4, 4.66/2.00 "scope": 1, 4.66/2.00 "term": "(average (s T7) T8 T3)" 4.66/2.00 } 4.66/2.00 ], 4.66/2.00 "kb": { 4.66/2.00 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"PlainIntegerRelationState", 4.66/2.00 "relations": [] 4.66/2.00 }, 4.66/2.00 "ground": [ 4.66/2.00 "T1", 4.66/2.00 "T2" 4.66/2.00 ], 4.66/2.00 "free": [], 4.66/2.00 "exprvars": [] 4.66/2.00 } 4.66/2.00 }, 4.66/2.00 "41": { 4.66/2.00 "goal": [ 4.66/2.00 { 4.66/2.00 "clause": 1, 4.66/2.00 "scope": 1, 4.66/2.00 "term": "(average (0) (0) T3)" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "clause": 2, 4.66/2.00 "scope": 1, 4.66/2.00 "term": "(average (0) (0) T3)" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "clause": 3, 4.66/2.00 "scope": 1, 4.66/2.00 "term": "(average (0) (0) T3)" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "clause": 4, 4.66/2.00 "scope": 1, 4.66/2.00 "term": "(average (0) (0) T3)" 4.66/2.00 } 4.66/2.00 ], 4.66/2.00 "kb": { 4.66/2.00 "nonunifying": [], 4.66/2.00 "intvars": {}, 4.66/2.00 "arithmetic": { 4.66/2.00 "type": "PlainIntegerRelationState", 4.66/2.00 "relations": [] 4.66/2.00 }, 4.66/2.00 "ground": [], 4.66/2.00 "free": [], 4.66/2.00 "exprvars": [] 4.66/2.00 } 4.66/2.00 }, 4.66/2.00 "42": { 4.66/2.00 "goal": [ 4.66/2.00 { 4.66/2.00 "clause": 2, 4.66/2.00 "scope": 1, 4.66/2.00 "term": "(average (0) (0) T3)" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "clause": 3, 4.66/2.00 "scope": 1, 4.66/2.00 "term": "(average (0) (0) T3)" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "clause": 4, 4.66/2.00 "scope": 1, 4.66/2.00 "term": "(average (0) (0) T3)" 4.66/2.00 } 4.66/2.00 ], 4.66/2.00 "kb": { 4.66/2.00 "nonunifying": [], 4.66/2.00 "intvars": {}, 4.66/2.00 "arithmetic": { 4.66/2.00 "type": "PlainIntegerRelationState", 4.66/2.00 "relations": [] 4.66/2.00 }, 4.66/2.00 "ground": [], 4.66/2.00 "free": [], 4.66/2.00 "exprvars": [] 4.66/2.00 } 4.66/2.00 } 4.66/2.00 }, 4.66/2.00 "edges": [ 4.66/2.00 { 4.66/2.00 "from": 5, 4.66/2.00 "to": 6, 4.66/2.00 "label": "CASE" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "from": 6, 4.66/2.00 "to": 36, 4.66/2.00 "label": "EVAL with clause\naverage(0, 0, 0).\nand substitutionT1 -> 0,\nT2 -> 0,\nT3 -> 0" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "from": 6, 4.66/2.00 "to": 40, 4.66/2.00 "label": "EVAL-BACKTRACK" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "from": 36, 4.66/2.00 "to": 41, 4.66/2.00 "label": "SUCCESS" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "from": 40, 4.66/2.00 "to": 155, 4.66/2.00 "label": "EVAL with clause\naverage(0, s(0), 0).\nand substitutionT1 -> 0,\nT2 -> s(0),\nT3 -> 0" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "from": 40, 4.66/2.00 "to": 156, 4.66/2.00 "label": "EVAL-BACKTRACK" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "from": 41, 4.66/2.00 "to": 42, 4.66/2.00 "label": "BACKTRACK\nfor clause: average(0, s(0), 0)because of non-unification" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "from": 42, 4.66/2.00 "to": 150, 4.66/2.00 "label": "BACKTRACK\nfor clause: average(0, s(s(0)), s(0))because of non-unification" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "from": 150, 4.66/2.00 "to": 152, 4.66/2.00 "label": "BACKTRACK\nfor clause: average(s(X), Y, Z) :- average(X, s(Y), Z)because of non-unification" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "from": 152, 4.66/2.00 "to": 154, 4.66/2.00 "label": "BACKTRACK\nfor clause: average(X, s(s(s(Y))), s(Z)) :- average(s(X), Y, Z)because of non-unification" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "from": 155, 4.66/2.00 "to": 157, 4.66/2.00 "label": "SUCCESS" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "from": 156, 4.66/2.00 "to": 161, 4.66/2.00 "label": "EVAL with clause\naverage(0, s(s(0)), s(0)).\nand substitutionT1 -> 0,\nT2 -> s(s(0)),\nT3 -> s(0)" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "from": 156, 4.66/2.00 "to": 162, 4.66/2.00 "label": "EVAL-BACKTRACK" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "from": 157, 4.66/2.00 "to": 158, 4.66/2.00 "label": "BACKTRACK\nfor clause: average(0, s(s(0)), s(0))because of non-unification" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "from": 158, 4.66/2.00 "to": 159, 4.66/2.00 "label": "BACKTRACK\nfor clause: average(s(X), Y, Z) :- average(X, s(Y), Z)because of non-unification" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "from": 159, 4.66/2.00 "to": 160, 4.66/2.00 "label": "BACKTRACK\nfor clause: average(X, s(s(s(Y))), s(Z)) :- average(s(X), Y, Z)because of non-unification" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "from": 161, 4.66/2.00 "to": 163, 4.66/2.00 "label": "SUCCESS" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "from": 162, 4.66/2.00 "to": 166, 4.66/2.00 "label": "EVAL with clause\naverage(s(X22), X23, X24) :- average(X22, s(X23), X24).\nand substitutionX22 -> T7,\nT1 -> s(T7),\nT2 -> T8,\nX23 -> T8,\nT3 -> T10,\nX24 -> T10,\nT9 -> T10" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "from": 162, 4.66/2.00 "to": 175, 4.66/2.00 "label": "EVAL-BACKTRACK" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "from": 163, 4.66/2.00 "to": 164, 4.66/2.00 "label": "BACKTRACK\nfor clause: average(s(X), Y, Z) :- average(X, s(Y), Z)because of non-unification" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "from": 164, 4.66/2.00 "to": 165, 4.66/2.00 "label": "BACKTRACK\nfor clause: average(X, s(s(s(Y))), s(Z)) :- average(s(X), Y, Z)because of non-unification" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "from": 166, 4.66/2.00 "to": 176, 4.66/2.00 "label": "CASE" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "from": 175, 4.66/2.00 "to": 199, 4.66/2.00 "label": "EVAL with clause\naverage(X67, s(s(s(X68))), s(X69)) :- average(s(X67), X68, X69).\nand substitutionT1 -> T56,\nX67 -> T56,\nX68 -> T57,\nT2 -> s(s(s(T57))),\nX69 -> T59,\nT3 -> s(T59),\nT58 -> T59" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "from": 175, 4.66/2.00 "to": 200, 4.66/2.00 "label": "EVAL-BACKTRACK" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "from": 176, 4.66/2.00 "to": 177, 4.66/2.00 "label": "BACKTRACK\nfor clause: average(0, 0, 0)because of non-unification" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "from": 177, 4.66/2.00 "to": 178, 4.66/2.00 "label": "PARALLEL" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "from": 177, 4.66/2.00 "to": 179, 4.66/2.00 "label": "PARALLEL" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "from": 178, 4.66/2.00 "to": 180, 4.66/2.00 "label": "EVAL with clause\naverage(0, s(0), 0).\nand substitutionT7 -> 0,\nT8 -> 0,\nT10 -> 0" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "from": 178, 4.66/2.00 "to": 181, 4.66/2.00 "label": "EVAL-BACKTRACK" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "from": 179, 4.66/2.00 "to": 183, 4.66/2.00 "label": "PARALLEL" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "from": 179, 4.66/2.00 "to": 184, 4.66/2.00 "label": "PARALLEL" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "from": 180, 4.66/2.00 "to": 182, 4.66/2.00 "label": "SUCCESS" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "from": 183, 4.66/2.00 "to": 185, 4.66/2.00 "label": "EVAL with clause\naverage(0, s(s(0)), s(0)).\nand substitutionT7 -> 0,\nT8 -> s(0),\nT10 -> s(0)" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "from": 183, 4.66/2.00 "to": 186, 4.66/2.00 "label": "EVAL-BACKTRACK" 4.66/2.00 }, 4.66/2.00 { 4.66/2.00 "from": 184, 4.66/2.01 "to": 188, 4.66/2.01 "label": "PARALLEL" 4.66/2.01 }, 4.66/2.01 { 4.66/2.01 "from": 184, 4.66/2.01 "to": 189, 4.66/2.01 "label": "PARALLEL" 4.66/2.01 }, 4.66/2.01 { 4.66/2.01 "from": 185, 4.66/2.01 "to": 187, 4.66/2.01 "label": "SUCCESS" 4.66/2.01 }, 4.66/2.01 { 4.66/2.01 "from": 188, 4.66/2.01 "to": 190, 4.66/2.01 "label": "EVAL with clause\naverage(s(X37), X38, X39) :- average(X37, s(X38), X39).\nand substitutionX37 -> T23,\nT7 -> s(T23),\nT8 -> T24,\nX38 -> s(T24),\nT10 -> T26,\nX39 -> T26,\nT25 -> T26" 4.66/2.01 }, 4.66/2.01 { 4.66/2.01 "from": 188, 4.66/2.01 "to": 191, 4.66/2.01 "label": "EVAL-BACKTRACK" 4.66/2.01 }, 4.66/2.01 { 4.66/2.01 "from": 189, 4.66/2.01 "to": 192, 4.66/2.01 "label": "PARALLEL" 4.66/2.01 }, 4.66/2.01 { 4.66/2.01 "from": 189, 4.66/2.01 "to": 193, 4.66/2.01 "label": "PARALLEL" 4.66/2.01 }, 4.66/2.01 { 4.66/2.01 "from": 190, 4.66/2.01 "to": 5, 4.66/2.01 "label": "INSTANCE with matching:\nT1 -> T23\nT2 -> s(s(T24))\nT3 -> T26" 4.66/2.01 }, 4.66/2.01 { 4.66/2.01 "from": 192, 4.66/2.01 "to": 194, 4.66/2.01 "label": "EVAL with clause\naverage(X52, s(s(s(X53))), s(X54)) :- average(s(X52), X53, X54).\nand substitutionT7 -> T39,\nX52 -> T39,\nX53 -> T40,\nT8 -> s(s(T40)),\nX54 -> T42,\nT10 -> s(T42),\nT41 -> T42" 4.66/2.01 }, 4.66/2.01 { 4.66/2.01 "from": 192, 4.66/2.01 "to": 195, 4.66/2.01 "label": "EVAL-BACKTRACK" 4.66/2.01 }, 4.66/2.01 { 4.66/2.01 "from": 193, 4.66/2.01 "to": 196, 4.66/2.01 "label": "FAILURE" 4.66/2.01 }, 4.66/2.01 { 4.66/2.01 "from": 194, 4.66/2.01 "to": 5, 4.66/2.01 "label": "INSTANCE with matching:\nT1 -> s(T39)\nT2 -> T40\nT3 -> T42" 4.66/2.01 }, 4.66/2.01 { 4.66/2.01 "from": 196, 4.66/2.01 "to": 197, 4.66/2.01 "label": "EVAL with clause\naverage(X61, s(s(s(X62))), s(X63)) :- average(s(X61), X62, X63).\nand substitutionT7 -> T49,\nX61 -> s(T49),\nX62 -> T50,\nT8 -> s(s(s(T50))),\nX63 -> T52,\nT3 -> s(T52),\nT51 -> T52" 4.66/2.01 }, 4.66/2.01 { 4.66/2.01 "from": 196, 4.66/2.01 "to": 198, 4.66/2.01 "label": "EVAL-BACKTRACK" 4.66/2.01 }, 4.66/2.01 { 4.66/2.01 "from": 197, 4.66/2.01 "to": 5, 4.66/2.01 "label": "INSTANCE with matching:\nT1 -> s(s(T49))\nT2 -> T50\nT3 -> T52" 4.66/2.01 }, 4.66/2.01 { 4.66/2.01 "from": 199, 4.66/2.01 "to": 201, 4.66/2.01 "label": "CASE" 4.66/2.01 }, 4.66/2.01 { 4.66/2.01 "from": 201, 4.66/2.01 "to": 202, 4.66/2.01 "label": "BACKTRACK\nfor clause: average(0, 0, 0)because of non-unification" 4.66/2.01 }, 4.66/2.01 { 4.66/2.01 "from": 202, 4.66/2.01 "to": 203, 4.66/2.01 "label": "BACKTRACK\nfor clause: average(0, s(0), 0)because of non-unification" 4.66/2.01 }, 4.66/2.01 { 4.66/2.01 "from": 203, 4.66/2.01 "to": 204, 4.66/2.01 "label": "BACKTRACK\nfor clause: average(0, s(s(0)), s(0))because of non-unification" 4.66/2.01 }, 4.66/2.01 { 4.66/2.01 "from": 204, 4.66/2.01 "to": 205, 4.66/2.01 "label": "PARALLEL" 4.66/2.01 }, 4.66/2.01 { 4.66/2.01 "from": 204, 4.66/2.01 "to": 206, 4.66/2.01 "label": "PARALLEL" 4.66/2.01 }, 4.66/2.01 { 4.66/2.01 "from": 205, 4.66/2.01 "to": 207, 4.66/2.01 "label": "ONLY EVAL with clause\naverage(s(X82), X83, X84) :- average(X82, s(X83), X84).\nand substitutionT56 -> T74,\nX82 -> T74,\nT57 -> T75,\nX83 -> T75,\nT59 -> T77,\nX84 -> T77,\nT76 -> T77" 4.66/2.01 }, 4.66/2.01 { 4.66/2.01 "from": 206, 4.66/2.01 "to": 231, 4.66/2.01 "label": "EVAL with clause\naverage(X91, s(s(s(X92))), s(X93)) :- average(s(X91), X92, X93).\nand substitutionT56 -> T84,\nX91 -> s(T84),\nX92 -> T85,\nT57 -> s(s(s(T85))),\nX93 -> T87,\nT59 -> s(T87),\nT86 -> T87" 4.66/2.01 }, 4.66/2.01 { 4.66/2.01 "from": 206, 4.66/2.01 "to": 232, 4.66/2.01 "label": "EVAL-BACKTRACK" 4.66/2.01 }, 4.66/2.01 { 4.66/2.01 "from": 207, 4.66/2.01 "to": 5, 4.66/2.01 "label": "INSTANCE with matching:\nT1 -> T74\nT2 -> s(T75)\nT3 -> T77" 4.66/2.01 }, 4.66/2.01 { 4.66/2.01 "from": 231, 4.66/2.01 "to": 5, 4.66/2.01 "label": "INSTANCE with matching:\nT1 -> s(s(T84))\nT2 -> T85\nT3 -> T87" 4.66/2.01 } 4.66/2.01 ], 4.66/2.01 "type": "Graph" 4.66/2.01 } 4.66/2.01 } 4.66/2.01 4.66/2.01 ---------------------------------------- 4.66/2.01 4.66/2.01 (2) 4.66/2.01 Obligation: 4.66/2.01 Triples: 4.66/2.01 4.66/2.01 averageA(s(s(X1)), X2, X3) :- averageA(X1, s(s(X2)), X3). 4.66/2.01 averageA(s(X1), s(s(X2)), s(X3)) :- averageA(s(X1), X2, X3). 4.66/2.01 averageA(s(X1), s(s(s(X2))), s(X3)) :- averageA(s(s(X1)), X2, X3). 4.66/2.01 averageA(X1, s(s(s(X2))), s(X3)) :- averageA(X1, s(X2), X3). 4.66/2.01 averageA(X1, s(s(s(s(s(s(X2)))))), s(s(X3))) :- averageA(s(s(X1)), X2, X3). 4.66/2.01 4.66/2.01 Clauses: 4.66/2.01 4.66/2.01 averagecA(0, 0, 0). 4.66/2.01 averagecA(0, s(0), 0). 4.66/2.01 averagecA(0, s(s(0)), s(0)). 4.66/2.01 averagecA(s(0), 0, 0). 4.66/2.01 averagecA(s(0), s(0), s(0)). 4.66/2.01 averagecA(s(s(X1)), X2, X3) :- averagecA(X1, s(s(X2)), X3). 4.66/2.01 averagecA(s(X1), s(s(X2)), s(X3)) :- averagecA(s(X1), X2, X3). 4.66/2.01 averagecA(s(X1), s(s(s(X2))), s(X3)) :- averagecA(s(s(X1)), X2, X3). 4.66/2.01 averagecA(X1, s(s(s(X2))), s(X3)) :- averagecA(X1, s(X2), X3). 4.66/2.01 averagecA(X1, s(s(s(s(s(s(X2)))))), s(s(X3))) :- averagecA(s(s(X1)), X2, X3). 4.66/2.01 4.66/2.01 Afs: 4.66/2.01 4.66/2.01 averageA(x1, x2, x3) = averageA(x1, x2) 4.66/2.01 4.66/2.01 4.66/2.01 ---------------------------------------- 4.66/2.01 4.66/2.01 (3) TriplesToPiDPProof (SOUND) 4.66/2.01 We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: 4.66/2.01 4.66/2.01 averageA_in_3: (b,b,f) 4.66/2.01 4.66/2.01 Transforming TRIPLES into the following Term Rewriting System: 4.66/2.01 4.66/2.01 Pi DP problem: 4.66/2.01 The TRS P consists of the following rules: 4.66/2.01 4.66/2.01 AVERAGEA_IN_GGA(s(s(X1)), X2, X3) -> U1_GGA(X1, X2, X3, averageA_in_gga(X1, s(s(X2)), X3)) 4.66/2.01 AVERAGEA_IN_GGA(s(s(X1)), X2, X3) -> AVERAGEA_IN_GGA(X1, s(s(X2)), X3) 4.66/2.01 AVERAGEA_IN_GGA(s(X1), s(s(X2)), s(X3)) -> U2_GGA(X1, X2, X3, averageA_in_gga(s(X1), X2, X3)) 4.66/2.01 AVERAGEA_IN_GGA(s(X1), s(s(X2)), s(X3)) -> AVERAGEA_IN_GGA(s(X1), X2, X3) 4.66/2.01 AVERAGEA_IN_GGA(s(X1), s(s(s(X2))), s(X3)) -> U3_GGA(X1, X2, X3, averageA_in_gga(s(s(X1)), X2, X3)) 4.66/2.01 AVERAGEA_IN_GGA(s(X1), s(s(s(X2))), s(X3)) -> AVERAGEA_IN_GGA(s(s(X1)), X2, X3) 4.66/2.01 AVERAGEA_IN_GGA(X1, s(s(s(X2))), s(X3)) -> U4_GGA(X1, X2, X3, averageA_in_gga(X1, s(X2), X3)) 4.66/2.01 AVERAGEA_IN_GGA(X1, s(s(s(X2))), s(X3)) -> AVERAGEA_IN_GGA(X1, s(X2), X3) 4.66/2.01 AVERAGEA_IN_GGA(X1, s(s(s(s(s(s(X2)))))), s(s(X3))) -> U5_GGA(X1, X2, X3, averageA_in_gga(s(s(X1)), X2, X3)) 4.66/2.01 AVERAGEA_IN_GGA(X1, s(s(s(s(s(s(X2)))))), s(s(X3))) -> AVERAGEA_IN_GGA(s(s(X1)), X2, X3) 4.66/2.01 4.66/2.01 R is empty. 4.66/2.01 The argument filtering Pi contains the following mapping: 4.66/2.01 averageA_in_gga(x1, x2, x3) = averageA_in_gga(x1, x2) 4.66/2.01 4.66/2.01 s(x1) = s(x1) 4.66/2.01 4.66/2.01 AVERAGEA_IN_GGA(x1, x2, x3) = AVERAGEA_IN_GGA(x1, x2) 4.66/2.01 4.66/2.01 U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x2, x4) 4.66/2.01 4.66/2.01 U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x2, x4) 4.66/2.01 4.66/2.01 U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4) 4.66/2.01 4.66/2.01 U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4) 4.66/2.01 4.66/2.01 U5_GGA(x1, x2, x3, x4) = U5_GGA(x1, x2, x4) 4.66/2.01 4.66/2.01 4.66/2.01 We have to consider all (P,R,Pi)-chains 4.66/2.01 4.66/2.01 4.66/2.01 Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES 4.66/2.01 4.66/2.01 4.66/2.01 4.66/2.01 ---------------------------------------- 4.66/2.01 4.66/2.01 (4) 4.66/2.01 Obligation: 4.66/2.01 Pi DP problem: 4.66/2.01 The TRS P consists of the following rules: 4.66/2.01 4.66/2.01 AVERAGEA_IN_GGA(s(s(X1)), X2, X3) -> U1_GGA(X1, X2, X3, averageA_in_gga(X1, s(s(X2)), X3)) 4.66/2.01 AVERAGEA_IN_GGA(s(s(X1)), X2, X3) -> AVERAGEA_IN_GGA(X1, s(s(X2)), X3) 4.66/2.01 AVERAGEA_IN_GGA(s(X1), s(s(X2)), s(X3)) -> U2_GGA(X1, X2, X3, averageA_in_gga(s(X1), X2, X3)) 4.66/2.01 AVERAGEA_IN_GGA(s(X1), s(s(X2)), s(X3)) -> AVERAGEA_IN_GGA(s(X1), X2, X3) 4.66/2.01 AVERAGEA_IN_GGA(s(X1), s(s(s(X2))), s(X3)) -> U3_GGA(X1, X2, X3, averageA_in_gga(s(s(X1)), X2, X3)) 4.66/2.01 AVERAGEA_IN_GGA(s(X1), s(s(s(X2))), s(X3)) -> AVERAGEA_IN_GGA(s(s(X1)), X2, X3) 4.66/2.01 AVERAGEA_IN_GGA(X1, s(s(s(X2))), s(X3)) -> U4_GGA(X1, X2, X3, averageA_in_gga(X1, s(X2), X3)) 4.66/2.01 AVERAGEA_IN_GGA(X1, s(s(s(X2))), s(X3)) -> AVERAGEA_IN_GGA(X1, s(X2), X3) 4.66/2.01 AVERAGEA_IN_GGA(X1, s(s(s(s(s(s(X2)))))), s(s(X3))) -> U5_GGA(X1, X2, X3, averageA_in_gga(s(s(X1)), X2, X3)) 4.66/2.01 AVERAGEA_IN_GGA(X1, s(s(s(s(s(s(X2)))))), s(s(X3))) -> AVERAGEA_IN_GGA(s(s(X1)), X2, X3) 4.66/2.01 4.66/2.01 R is empty. 4.66/2.01 The argument filtering Pi contains the following mapping: 4.66/2.01 averageA_in_gga(x1, x2, x3) = averageA_in_gga(x1, x2) 4.66/2.01 4.66/2.01 s(x1) = s(x1) 4.66/2.01 4.66/2.01 AVERAGEA_IN_GGA(x1, x2, x3) = AVERAGEA_IN_GGA(x1, x2) 4.66/2.01 4.66/2.01 U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x2, x4) 4.66/2.01 4.66/2.01 U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x2, x4) 4.66/2.01 4.66/2.01 U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4) 4.66/2.01 4.66/2.01 U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4) 4.66/2.01 4.66/2.01 U5_GGA(x1, x2, x3, x4) = U5_GGA(x1, x2, x4) 4.66/2.01 4.66/2.01 4.66/2.01 We have to consider all (P,R,Pi)-chains 4.66/2.01 ---------------------------------------- 4.66/2.01 4.66/2.01 (5) DependencyGraphProof (EQUIVALENT) 4.66/2.01 The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 5 less nodes. 4.66/2.01 ---------------------------------------- 4.66/2.01 4.66/2.01 (6) 4.66/2.01 Obligation: 4.66/2.01 Pi DP problem: 4.66/2.01 The TRS P consists of the following rules: 4.66/2.01 4.66/2.01 AVERAGEA_IN_GGA(s(X1), s(s(X2)), s(X3)) -> AVERAGEA_IN_GGA(s(X1), X2, X3) 4.66/2.01 AVERAGEA_IN_GGA(s(s(X1)), X2, X3) -> AVERAGEA_IN_GGA(X1, s(s(X2)), X3) 4.66/2.01 AVERAGEA_IN_GGA(s(X1), s(s(s(X2))), s(X3)) -> AVERAGEA_IN_GGA(s(s(X1)), X2, X3) 4.66/2.01 AVERAGEA_IN_GGA(X1, s(s(s(X2))), s(X3)) -> AVERAGEA_IN_GGA(X1, s(X2), X3) 4.66/2.01 AVERAGEA_IN_GGA(X1, s(s(s(s(s(s(X2)))))), s(s(X3))) -> AVERAGEA_IN_GGA(s(s(X1)), X2, X3) 4.66/2.01 4.66/2.01 R is empty. 4.66/2.01 The argument filtering Pi contains the following mapping: 4.66/2.01 s(x1) = s(x1) 4.66/2.01 4.66/2.01 AVERAGEA_IN_GGA(x1, x2, x3) = AVERAGEA_IN_GGA(x1, x2) 4.66/2.01 4.66/2.01 4.66/2.01 We have to consider all (P,R,Pi)-chains 4.66/2.01 ---------------------------------------- 4.66/2.01 4.66/2.01 (7) PiDPToQDPProof (SOUND) 4.66/2.01 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 4.66/2.01 ---------------------------------------- 4.66/2.01 4.66/2.01 (8) 4.66/2.01 Obligation: 4.66/2.01 Q DP problem: 4.66/2.01 The TRS P consists of the following rules: 4.66/2.01 4.66/2.01 AVERAGEA_IN_GGA(s(X1), s(s(X2))) -> AVERAGEA_IN_GGA(s(X1), X2) 4.66/2.01 AVERAGEA_IN_GGA(s(s(X1)), X2) -> AVERAGEA_IN_GGA(X1, s(s(X2))) 4.66/2.01 AVERAGEA_IN_GGA(s(X1), s(s(s(X2)))) -> AVERAGEA_IN_GGA(s(s(X1)), X2) 4.66/2.01 AVERAGEA_IN_GGA(X1, s(s(s(X2)))) -> AVERAGEA_IN_GGA(X1, s(X2)) 4.66/2.01 AVERAGEA_IN_GGA(X1, s(s(s(s(s(s(X2))))))) -> AVERAGEA_IN_GGA(s(s(X1)), X2) 4.66/2.01 4.66/2.01 R is empty. 4.66/2.01 Q is empty. 4.66/2.01 We have to consider all (P,Q,R)-chains. 4.66/2.01 ---------------------------------------- 4.66/2.01 4.66/2.01 (9) MRRProof (EQUIVALENT) 4.66/2.01 By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. 4.66/2.01 4.66/2.01 Strictly oriented dependency pairs: 4.66/2.01 4.66/2.01 AVERAGEA_IN_GGA(s(X1), s(s(X2))) -> AVERAGEA_IN_GGA(s(X1), X2) 4.66/2.01 AVERAGEA_IN_GGA(s(s(X1)), X2) -> AVERAGEA_IN_GGA(X1, s(s(X2))) 4.66/2.01 AVERAGEA_IN_GGA(s(X1), s(s(s(X2)))) -> AVERAGEA_IN_GGA(s(s(X1)), X2) 4.66/2.01 AVERAGEA_IN_GGA(X1, s(s(s(X2)))) -> AVERAGEA_IN_GGA(X1, s(X2)) 4.66/2.01 AVERAGEA_IN_GGA(X1, s(s(s(s(s(s(X2))))))) -> AVERAGEA_IN_GGA(s(s(X1)), X2) 4.66/2.01 4.66/2.01 4.66/2.01 Used ordering: Knuth-Bendix order [KBO] with precedence:s_1 > AVERAGEA_IN_GGA_2 4.66/2.01 4.66/2.01 and weight map: 4.66/2.01 4.66/2.01 s_1=1 4.66/2.01 AVERAGEA_IN_GGA_2=0 4.66/2.01 4.66/2.01 The variable weight is 1 4.66/2.01 4.66/2.01 ---------------------------------------- 4.66/2.01 4.66/2.01 (10) 4.66/2.01 Obligation: 4.66/2.01 Q DP problem: 4.66/2.01 P is empty. 4.66/2.01 R is empty. 4.66/2.01 Q is empty. 4.66/2.01 We have to consider all (P,Q,R)-chains. 4.66/2.01 ---------------------------------------- 4.66/2.01 4.66/2.01 (11) PisEmptyProof (EQUIVALENT) 4.66/2.01 The TRS P is empty. Hence, there is no (P,Q,R) chain. 4.66/2.01 ---------------------------------------- 4.66/2.01 4.66/2.01 (12) 4.66/2.01 YES 4.66/2.03 EOF