13.82/4.37 YES 14.36/4.43 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 14.36/4.43 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 14.36/4.43 14.36/4.43 14.36/4.43 Left Termination of the query pattern 14.36/4.43 14.36/4.43 mergesort(g,a) 14.36/4.43 14.36/4.43 w.r.t. the given Prolog program could successfully be proven: 14.36/4.43 14.36/4.43 (0) Prolog 14.36/4.43 (1) PrologToTRSTransformerProof [SOUND, 48 ms] 14.36/4.43 (2) QTRS 14.36/4.43 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 14.36/4.43 (4) QDP 14.36/4.43 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 14.36/4.43 (6) AND 14.36/4.43 (7) QDP 14.36/4.43 (8) UsableRulesProof [EQUIVALENT, 0 ms] 14.36/4.43 (9) QDP 14.36/4.43 (10) QDPSizeChangeProof [EQUIVALENT, 0 ms] 14.36/4.43 (11) YES 14.36/4.43 (12) QDP 14.36/4.43 (13) UsableRulesProof [EQUIVALENT, 0 ms] 14.36/4.43 (14) QDP 14.36/4.43 (15) QDPSizeChangeProof [EQUIVALENT, 0 ms] 14.36/4.43 (16) YES 14.36/4.43 (17) QDP 14.36/4.43 (18) QDPOrderProof [EQUIVALENT, 0 ms] 14.36/4.43 (19) QDP 14.36/4.43 (20) DependencyGraphProof [EQUIVALENT, 0 ms] 14.36/4.43 (21) TRUE 14.36/4.43 (22) QDP 14.36/4.43 (23) UsableRulesProof [EQUIVALENT, 0 ms] 14.36/4.43 (24) QDP 14.36/4.43 (25) QDPSizeChangeProof [EQUIVALENT, 0 ms] 14.36/4.43 (26) YES 14.36/4.43 (27) QDP 14.36/4.43 (28) QDPOrderProof [EQUIVALENT, 64 ms] 14.36/4.43 (29) QDP 14.36/4.43 (30) DependencyGraphProof [EQUIVALENT, 0 ms] 14.36/4.43 (31) TRUE 14.36/4.43 14.36/4.43 14.36/4.43 ---------------------------------------- 14.36/4.43 14.36/4.43 (0) 14.36/4.43 Obligation: 14.36/4.43 Clauses: 14.36/4.43 14.36/4.43 mergesort([], []). 14.36/4.43 mergesort(.(X, []), .(X, [])). 14.36/4.43 mergesort(.(X, .(Y, Xs)), Ys) :- ','(split(.(X, .(Y, Xs)), X1s, X2s), ','(mergesort(X1s, Y1s), ','(mergesort(X2s, Y2s), merge(Y1s, Y2s, Ys)))). 14.36/4.43 split([], [], []). 14.36/4.43 split(.(X, Xs), .(X, Ys), Zs) :- split(Xs, Zs, Ys). 14.36/4.43 merge([], Xs, Xs). 14.36/4.43 merge(Xs, [], Xs). 14.36/4.43 merge(.(X, Xs), .(Y, Ys), .(X, Zs)) :- ','(le(X, Y), merge(Xs, .(Y, Ys), Zs)). 14.36/4.43 merge(.(X, Xs), .(Y, Ys), .(Y, Zs)) :- ','(gt(X, Y), merge(.(X, Xs), Ys, Zs)). 14.36/4.43 gt(s(X), s(Y)) :- gt(X, Y). 14.36/4.43 gt(s(X), 0). 14.36/4.43 le(s(X), s(Y)) :- le(X, Y). 14.36/4.43 le(0, s(Y)). 14.36/4.43 le(0, 0). 14.36/4.43 14.36/4.43 14.36/4.43 Query: mergesort(g,a) 14.36/4.43 ---------------------------------------- 14.36/4.43 14.36/4.43 (1) PrologToTRSTransformerProof (SOUND) 14.36/4.43 Transformed Prolog program to TRS. 14.36/4.43 14.36/4.43 { 14.36/4.43 "root": 3, 14.36/4.43 "program": { 14.36/4.43 "directives": [], 14.36/4.43 "clauses": [ 14.36/4.43 [ 14.36/4.43 "(mergesort ([]) ([]))", 14.36/4.43 null 14.36/4.43 ], 14.36/4.43 [ 14.36/4.43 "(mergesort (. X ([])) (. X ([])))", 14.36/4.43 null 14.36/4.43 ], 14.36/4.43 [ 14.36/4.43 "(mergesort (. X (. Y Xs)) Ys)", 14.36/4.43 "(',' (split (. X (. Y Xs)) X1s X2s) (',' (mergesort X1s Y1s) (',' (mergesort X2s Y2s) (merge Y1s Y2s Ys))))" 14.36/4.43 ], 14.36/4.43 [ 14.36/4.43 "(split ([]) ([]) ([]))", 14.36/4.43 null 14.36/4.43 ], 14.36/4.43 [ 14.36/4.43 "(split (. X Xs) (. X Ys) Zs)", 14.36/4.43 "(split Xs Zs Ys)" 14.36/4.43 ], 14.36/4.43 [ 14.36/4.43 "(merge ([]) Xs Xs)", 14.36/4.43 null 14.36/4.43 ], 14.36/4.43 [ 14.36/4.43 "(merge Xs ([]) Xs)", 14.36/4.43 null 14.36/4.43 ], 14.36/4.43 [ 14.36/4.43 "(merge (. X Xs) (. Y Ys) (. X Zs))", 14.36/4.43 "(',' (le X Y) (merge Xs (. Y Ys) Zs))" 14.36/4.43 ], 14.36/4.43 [ 14.36/4.43 "(merge (. X Xs) (. Y Ys) (. Y Zs))", 14.36/4.43 "(',' (gt X Y) (merge (. X Xs) Ys Zs))" 14.36/4.43 ], 14.36/4.43 [ 14.36/4.43 "(gt (s X) (s Y))", 14.36/4.43 "(gt X Y)" 14.36/4.43 ], 14.36/4.43 [ 14.36/4.43 "(gt (s X) (0))", 14.36/4.43 null 14.36/4.43 ], 14.36/4.43 [ 14.36/4.43 "(le (s X) (s Y))", 14.36/4.43 "(le X Y)" 14.36/4.43 ], 14.36/4.43 [ 14.36/4.43 "(le (0) (s Y))", 14.36/4.43 null 14.36/4.43 ], 14.36/4.43 [ 14.36/4.43 "(le (0) (0))", 14.36/4.43 null 14.36/4.43 ] 14.36/4.43 ] 14.36/4.43 }, 14.36/4.43 "graph": { 14.36/4.43 "nodes": { 14.36/4.43 "45": { 14.36/4.43 "goal": [{ 14.36/4.43 "clause": -1, 14.36/4.43 "scope": -1, 14.36/4.43 "term": "(',' (split (. T16 (. T17 T18)) X22 X23) (',' (mergesort X22 X24) (',' (mergesort X23 X25) (merge X24 X25 T20))))" 14.36/4.43 }], 14.36/4.43 "kb": { 14.36/4.43 "nonunifying": [], 14.36/4.43 "intvars": {}, 14.36/4.43 "arithmetic": { 14.36/4.43 "type": "PlainIntegerRelationState", 14.36/4.43 "relations": [] 14.36/4.43 }, 14.36/4.43 "ground": [ 14.36/4.43 "T16", 14.36/4.43 "T17", 14.36/4.43 "T18" 14.36/4.43 ], 14.36/4.43 "free": [ 14.36/4.43 "X22", 14.36/4.43 "X23", 14.36/4.43 "X24", 14.36/4.43 "X25" 14.36/4.43 ], 14.36/4.43 "exprvars": [] 14.36/4.43 } 14.36/4.43 }, 14.36/4.43 "46": { 14.36/4.43 "goal": [], 14.36/4.43 "kb": { 14.36/4.43 "nonunifying": [], 14.36/4.43 "intvars": {}, 14.36/4.43 "arithmetic": { 14.36/4.43 "type": "PlainIntegerRelationState", 14.36/4.43 "relations": [] 14.36/4.43 }, 14.36/4.43 "ground": [], 14.36/4.43 "free": [], 14.36/4.43 "exprvars": [] 14.36/4.43 } 14.36/4.43 }, 14.36/4.43 "49": { 14.36/4.43 "goal": [{ 14.36/4.43 "clause": -1, 14.36/4.43 "scope": -1, 14.36/4.43 "term": "(split (. T16 (. T17 T18)) X22 X23)" 14.36/4.43 }], 14.36/4.43 "kb": { 14.36/4.43 "nonunifying": [], 14.36/4.43 "intvars": {}, 14.36/4.43 "arithmetic": { 14.36/4.43 "type": "PlainIntegerRelationState", 14.36/4.43 "relations": [] 14.36/4.43 }, 14.36/4.43 "ground": [ 14.36/4.43 "T16", 14.36/4.43 "T17", 14.36/4.43 "T18" 14.36/4.43 ], 14.36/4.43 "free": [ 14.36/4.43 "X22", 14.36/4.43 "X23" 14.36/4.43 ], 14.36/4.43 "exprvars": [] 14.36/4.43 } 14.36/4.43 }, 14.36/4.43 "type": "Nodes", 14.36/4.43 "352": { 14.36/4.43 "goal": [{ 14.36/4.43 "clause": -1, 14.36/4.43 "scope": -1, 14.36/4.43 "term": "(le T78 T80)" 14.36/4.43 }], 14.36/4.43 "kb": { 14.36/4.43 "nonunifying": [], 14.36/4.43 "intvars": {}, 14.36/4.43 "arithmetic": { 14.36/4.43 "type": "PlainIntegerRelationState", 14.36/4.43 "relations": [] 14.36/4.43 }, 14.36/4.43 "ground": [ 14.36/4.43 "T78", 14.36/4.43 "T80" 14.36/4.43 ], 14.36/4.43 "free": [], 14.36/4.43 "exprvars": [] 14.36/4.43 } 14.36/4.43 }, 14.36/4.43 "353": { 14.36/4.43 "goal": [{ 14.36/4.43 "clause": -1, 14.36/4.43 "scope": -1, 14.36/4.43 "term": "(merge T79 (. 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"PlainIntegerRelationState", 14.36/4.43 "relations": [] 14.36/4.43 }, 14.36/4.43 "ground": [], 14.36/4.43 "free": [], 14.36/4.43 "exprvars": [] 14.36/4.43 } 14.36/4.43 }, 14.36/4.43 "70": { 14.36/4.43 "goal": [], 14.36/4.43 "kb": { 14.36/4.43 "nonunifying": [], 14.36/4.43 "intvars": {}, 14.36/4.43 "arithmetic": { 14.36/4.43 "type": "PlainIntegerRelationState", 14.36/4.43 "relations": [] 14.36/4.43 }, 14.36/4.43 "ground": [], 14.36/4.43 "free": [], 14.36/4.43 "exprvars": [] 14.36/4.43 } 14.36/4.43 }, 14.36/4.43 "339": { 14.36/4.43 "goal": [{ 14.36/4.43 "clause": -1, 14.36/4.43 "scope": -1, 14.36/4.43 "term": "(',' (le T78 T80) (merge T79 (. T80 T81) T83))" 14.36/4.43 }], 14.36/4.43 "kb": { 14.36/4.43 "nonunifying": [], 14.36/4.43 "intvars": {}, 14.36/4.43 "arithmetic": { 14.36/4.43 "type": "PlainIntegerRelationState", 14.36/4.43 "relations": [] 14.36/4.43 }, 14.36/4.43 "ground": [ 14.36/4.43 "T78", 14.36/4.43 "T79", 14.36/4.43 "T80", 14.36/4.43 "T81" 14.36/4.43 ], 14.36/4.43 "free": [], 14.36/4.43 "exprvars": [] 14.36/4.43 } 14.36/4.43 }, 14.36/4.43 "614": { 14.36/4.43 "goal": [{ 14.36/4.43 "clause": 9, 14.36/4.43 "scope": 7, 14.36/4.43 "term": "(gt T119 T121)" 14.36/4.43 }], 14.36/4.43 "kb": { 14.36/4.43 "nonunifying": [], 14.36/4.43 "intvars": {}, 14.36/4.43 "arithmetic": { 14.36/4.43 "type": "PlainIntegerRelationState", 14.36/4.43 "relations": [] 14.36/4.43 }, 14.36/4.43 "ground": [ 14.36/4.43 "T119", 14.36/4.43 "T121" 14.36/4.43 ], 14.36/4.43 "free": [], 14.36/4.43 "exprvars": [] 14.36/4.43 } 14.36/4.43 }, 14.36/4.43 "71": { 14.36/4.43 "goal": [], 14.36/4.43 "kb": { 14.36/4.43 "nonunifying": [], 14.36/4.43 "intvars": {}, 14.36/4.43 "arithmetic": { 14.36/4.43 "type": "PlainIntegerRelationState", 14.36/4.43 "relations": [] 14.36/4.43 }, 14.36/4.43 "ground": [], 14.36/4.43 "free": [], 14.36/4.43 "exprvars": [] 14.36/4.43 } 14.36/4.43 }, 14.36/4.43 "615": { 14.36/4.43 "goal": [{ 14.36/4.43 "clause": 10, 14.36/4.43 "scope": 7, 14.36/4.43 "term": "(gt T119 T121)" 14.36/4.43 }], 14.36/4.43 "kb": { 14.36/4.43 "nonunifying": [], 14.36/4.43 "intvars": {}, 14.36/4.43 "arithmetic": { 14.36/4.43 "type": "PlainIntegerRelationState", 14.36/4.43 "relations": [] 14.36/4.43 }, 14.36/4.43 "ground": [ 14.36/4.43 "T119", 14.36/4.43 "T121" 14.36/4.43 ], 14.36/4.43 "free": [], 14.36/4.43 "exprvars": [] 14.36/4.43 } 14.36/4.43 }, 14.36/4.43 "72": { 14.36/4.43 "goal": [{ 14.36/4.43 "clause": -1, 14.36/4.43 "scope": -1, 14.36/4.43 "term": "(split T43 X79 X78)" 14.36/4.43 }], 14.36/4.43 "kb": { 14.36/4.43 "nonunifying": [], 14.36/4.43 "intvars": {}, 14.36/4.43 "arithmetic": { 14.36/4.43 "type": "PlainIntegerRelationState", 14.36/4.43 "relations": [] 14.36/4.43 }, 14.36/4.43 "ground": ["T43"], 14.36/4.43 "free": [ 14.36/4.43 "X78", 14.36/4.43 "X79" 14.36/4.43 ], 14.36/4.43 "exprvars": [] 14.36/4.43 } 14.36/4.43 }, 14.36/4.43 "73": { 14.36/4.43 "goal": [], 14.36/4.43 "kb": { 14.36/4.43 "nonunifying": [], 14.36/4.43 "intvars": {}, 14.36/4.43 "arithmetic": { 14.36/4.43 "type": "PlainIntegerRelationState", 14.36/4.43 "relations": [] 14.36/4.43 }, 14.36/4.43 "ground": [], 14.36/4.43 "free": [], 14.36/4.43 "exprvars": [] 14.36/4.43 } 14.36/4.43 }, 14.36/4.43 "34": { 14.36/4.43 "goal": [{ 14.36/4.43 "clause": -1, 14.36/4.43 "scope": -1, 14.36/4.43 "term": "(true)" 14.36/4.43 }], 14.36/4.43 "kb": { 14.36/4.43 "nonunifying": [], 14.36/4.43 "intvars": {}, 14.36/4.43 "arithmetic": { 14.36/4.43 "type": "PlainIntegerRelationState", 14.36/4.43 "relations": [] 14.36/4.43 }, 14.36/4.43 "ground": [], 14.36/4.43 "free": [], 14.36/4.43 "exprvars": [] 14.36/4.43 } 14.36/4.43 }, 14.36/4.43 "35": { 14.36/4.43 "goal": [], 14.36/4.43 "kb": { 14.36/4.43 "nonunifying": [], 14.36/4.43 "intvars": {}, 14.36/4.43 "arithmetic": { 14.36/4.43 "type": "PlainIntegerRelationState", 14.36/4.43 "relations": [] 14.36/4.43 }, 14.36/4.43 "ground": [], 14.36/4.43 "free": [], 14.36/4.43 "exprvars": [] 14.36/4.43 } 14.36/4.43 }, 14.36/4.43 "36": { 14.36/4.43 "goal": [], 14.36/4.43 "kb": { 14.36/4.43 "nonunifying": [], 14.36/4.43 "intvars": {}, 14.36/4.43 "arithmetic": { 14.36/4.43 "type": "PlainIntegerRelationState", 14.36/4.43 "relations": [] 14.36/4.43 }, 14.36/4.43 "ground": [], 14.36/4.43 "free": [], 14.36/4.43 "exprvars": [] 14.36/4.43 } 14.36/4.43 }, 14.36/4.43 "37": { 14.36/4.43 "goal": [{ 14.36/4.43 "clause": 1, 14.36/4.43 "scope": 1, 14.36/4.43 "term": "(mergesort T1 T2)" 14.36/4.43 }], 14.36/4.43 "kb": { 14.36/4.43 "nonunifying": [], 14.36/4.43 "intvars": {}, 14.36/4.43 "arithmetic": { 14.36/4.43 "type": "PlainIntegerRelationState", 14.36/4.43 "relations": [] 14.36/4.43 }, 14.36/4.43 "ground": ["T1"], 14.36/4.43 "free": [], 14.36/4.43 "exprvars": [] 14.36/4.43 } 14.36/4.43 }, 14.36/4.43 "38": { 14.36/4.43 "goal": [{ 14.36/4.43 "clause": 2, 14.36/4.43 "scope": 1, 14.36/4.43 "term": "(mergesort T1 T2)" 14.36/4.43 }], 14.36/4.43 "kb": { 14.36/4.43 "nonunifying": [], 14.36/4.43 "intvars": {}, 14.36/4.43 "arithmetic": { 14.36/4.43 "type": "PlainIntegerRelationState", 14.36/4.43 "relations": [] 14.36/4.43 }, 14.36/4.43 "ground": ["T1"], 14.36/4.43 "free": [], 14.36/4.43 "exprvars": [] 14.36/4.43 } 14.36/4.43 }, 14.36/4.43 "380": { 14.36/4.43 "goal": [{ 14.36/4.43 "clause": -1, 14.36/4.43 "scope": -1, 14.36/4.43 "term": "(',' (gt T119 T121) (merge (. T119 T120) T122 T124))" 14.36/4.43 }], 14.36/4.43 "kb": { 14.36/4.43 "nonunifying": [], 14.36/4.43 "intvars": {}, 14.36/4.43 "arithmetic": { 14.36/4.43 "type": "PlainIntegerRelationState", 14.36/4.43 "relations": [] 14.36/4.43 }, 14.36/4.43 "ground": [ 14.36/4.43 "T119", 14.36/4.43 "T120", 14.36/4.43 "T121", 14.36/4.43 "T122" 14.36/4.43 ], 14.36/4.43 "free": [], 14.36/4.43 "exprvars": [] 14.36/4.43 } 14.36/4.43 }, 14.36/4.43 "381": { 14.36/4.43 "goal": [], 14.36/4.43 "kb": { 14.36/4.43 "nonunifying": [], 14.36/4.43 "intvars": {}, 14.36/4.43 "arithmetic": { 14.36/4.43 "type": "PlainIntegerRelationState", 14.36/4.43 "relations": [] 14.36/4.43 }, 14.36/4.43 "ground": [], 14.36/4.43 "free": [], 14.36/4.43 "exprvars": [] 14.36/4.43 } 14.36/4.43 }, 14.36/4.43 "261": { 14.36/4.43 "goal": [{ 14.36/4.43 "clause": 5, 14.36/4.43 "scope": 5, 14.36/4.43 "term": "(merge T44 T45 T20)" 14.36/4.43 }], 14.36/4.43 "kb": { 14.36/4.43 "nonunifying": [], 14.36/4.43 "intvars": {}, 14.36/4.43 "arithmetic": { 14.36/4.43 "type": "PlainIntegerRelationState", 14.36/4.43 "relations": [] 14.36/4.43 }, 14.36/4.43 "ground": [ 14.36/4.43 "T44", 14.36/4.43 "T45" 14.36/4.43 ], 14.36/4.43 "free": [], 14.36/4.43 "exprvars": [] 14.36/4.43 } 14.36/4.43 }, 14.36/4.43 "263": { 14.36/4.43 "goal": [ 14.36/4.43 { 14.36/4.43 "clause": 6, 14.36/4.43 "scope": 5, 14.36/4.43 "term": "(merge T44 T45 T20)" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "clause": 7, 14.36/4.43 "scope": 5, 14.36/4.43 "term": "(merge T44 T45 T20)" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "clause": 8, 14.36/4.43 "scope": 5, 14.36/4.43 "term": "(merge T44 T45 T20)" 14.36/4.43 } 14.36/4.43 ], 14.36/4.43 "kb": { 14.36/4.43 "nonunifying": [], 14.36/4.43 "intvars": {}, 14.36/4.43 "arithmetic": { 14.36/4.43 "type": "PlainIntegerRelationState", 14.36/4.43 "relations": [] 14.36/4.43 }, 14.36/4.43 "ground": [ 14.36/4.43 "T44", 14.36/4.43 "T45" 14.36/4.43 ], 14.36/4.43 "free": [], 14.36/4.43 "exprvars": [] 14.36/4.43 } 14.36/4.43 }, 14.36/4.43 "342": { 14.36/4.43 "goal": [], 14.36/4.43 "kb": { 14.36/4.43 "nonunifying": [], 14.36/4.43 "intvars": {}, 14.36/4.43 "arithmetic": { 14.36/4.43 "type": "PlainIntegerRelationState", 14.36/4.43 "relations": [] 14.36/4.43 }, 14.36/4.43 "ground": [], 14.36/4.43 "free": [], 14.36/4.43 "exprvars": [] 14.36/4.43 } 14.36/4.43 }, 14.36/4.43 "226": { 14.36/4.43 "goal": [ 14.36/4.43 { 14.36/4.43 "clause": 5, 14.36/4.43 "scope": 5, 14.36/4.43 "term": "(merge T44 T45 T20)" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "clause": 6, 14.36/4.43 "scope": 5, 14.36/4.43 "term": "(merge T44 T45 T20)" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "clause": 7, 14.36/4.43 "scope": 5, 14.36/4.43 "term": "(merge T44 T45 T20)" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "clause": 8, 14.36/4.43 "scope": 5, 14.36/4.43 "term": "(merge T44 T45 T20)" 14.36/4.43 } 14.36/4.43 ], 14.36/4.43 "kb": { 14.36/4.43 "nonunifying": [], 14.36/4.43 "intvars": {}, 14.36/4.43 "arithmetic": { 14.36/4.43 "type": "PlainIntegerRelationState", 14.36/4.43 "relations": [] 14.36/4.43 }, 14.36/4.43 "ground": [ 14.36/4.43 "T44", 14.36/4.43 "T45" 14.36/4.43 ], 14.36/4.43 "free": [], 14.36/4.43 "exprvars": [] 14.36/4.43 } 14.36/4.43 }, 14.36/4.43 "40": { 14.36/4.43 "goal": [{ 14.36/4.43 "clause": -1, 14.36/4.43 "scope": -1, 14.36/4.43 "term": "(true)" 14.36/4.43 }], 14.36/4.43 "kb": { 14.36/4.43 "nonunifying": [], 14.36/4.43 "intvars": {}, 14.36/4.43 "arithmetic": { 14.36/4.43 "type": "PlainIntegerRelationState", 14.36/4.43 "relations": [] 14.36/4.43 }, 14.36/4.43 "ground": [], 14.36/4.43 "free": [], 14.36/4.43 "exprvars": [] 14.36/4.43 } 14.36/4.43 }, 14.36/4.43 "41": { 14.36/4.43 "goal": [], 14.36/4.43 "kb": { 14.36/4.43 "nonunifying": [], 14.36/4.43 "intvars": {}, 14.36/4.43 "arithmetic": { 14.36/4.43 "type": "PlainIntegerRelationState", 14.36/4.43 "relations": [] 14.36/4.43 }, 14.36/4.43 "ground": [], 14.36/4.43 "free": [], 14.36/4.43 "exprvars": [] 14.36/4.43 } 14.36/4.43 }, 14.36/4.43 "42": { 14.36/4.43 "goal": [], 14.36/4.43 "kb": { 14.36/4.43 "nonunifying": [], 14.36/4.43 "intvars": {}, 14.36/4.43 "arithmetic": { 14.36/4.43 "type": "PlainIntegerRelationState", 14.36/4.43 "relations": [] 14.36/4.43 }, 14.36/4.43 "ground": [], 14.36/4.43 "free": [], 14.36/4.43 "exprvars": [] 14.36/4.43 } 14.36/4.43 } 14.36/4.43 }, 14.36/4.43 "edges": [ 14.36/4.43 { 14.36/4.43 "from": 3, 14.36/4.43 "to": 4, 14.36/4.43 "label": "CASE" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 4, 14.36/4.43 "to": 9, 14.36/4.43 "label": "PARALLEL" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 4, 14.36/4.43 "to": 10, 14.36/4.43 "label": "PARALLEL" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 9, 14.36/4.43 "to": 34, 14.36/4.43 "label": "EVAL with clause\nmergesort([], []).\nand substitutionT1 -> [],\nT2 -> []" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 9, 14.36/4.43 "to": 35, 14.36/4.43 "label": "EVAL-BACKTRACK" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 10, 14.36/4.43 "to": 37, 14.36/4.43 "label": "PARALLEL" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 10, 14.36/4.43 "to": 38, 14.36/4.43 "label": "PARALLEL" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 34, 14.36/4.43 "to": 36, 14.36/4.43 "label": "SUCCESS" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 37, 14.36/4.43 "to": 40, 14.36/4.43 "label": "EVAL with clause\nmergesort(.(X5, []), .(X5, [])).\nand substitutionX5 -> T7,\nT1 -> .(T7, []),\nT2 -> .(T7, [])" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 37, 14.36/4.43 "to": 41, 14.36/4.43 "label": "EVAL-BACKTRACK" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 38, 14.36/4.43 "to": 45, 14.36/4.43 "label": "EVAL with clause\nmergesort(.(X18, .(X19, X20)), X21) :- ','(split(.(X18, .(X19, X20)), X22, X23), ','(mergesort(X22, X24), ','(mergesort(X23, X25), merge(X24, X25, X21)))).\nand substitutionX18 -> T16,\nX19 -> T17,\nX20 -> T18,\nT1 -> .(T16, .(T17, T18)),\nT2 -> T20,\nX21 -> T20,\nT19 -> T20" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 38, 14.36/4.43 "to": 46, 14.36/4.43 "label": "EVAL-BACKTRACK" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 40, 14.36/4.43 "to": 42, 14.36/4.43 "label": "SUCCESS" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 45, 14.36/4.43 "to": 49, 14.36/4.43 "label": "SPLIT 1" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 45, 14.36/4.43 "to": 50, 14.36/4.43 "label": "SPLIT 2\nnew knowledge:\nT16 is ground\nT17 is ground\nT18 is ground\nT21 is ground\nT22 is ground\nreplacements:X22 -> T21,\nX23 -> T22" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 49, 14.36/4.43 "to": 51, 14.36/4.43 "label": "CASE" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 50, 14.36/4.43 "to": 126, 14.36/4.43 "label": "SPLIT 1" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 50, 14.36/4.43 "to": 127, 14.36/4.43 "label": "SPLIT 2\nnew knowledge:\nT21 is ground\nT44 is ground\nreplacements:X24 -> T44" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 51, 14.36/4.43 "to": 52, 14.36/4.43 "label": "BACKTRACK\nfor clause: split([], [], [])because of non-unification" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 52, 14.36/4.43 "to": 53, 14.36/4.43 "label": "ONLY EVAL with clause\nsplit(.(X38, X39), .(X38, X40), X41) :- split(X39, X41, X40).\nand substitutionT16 -> T29,\nX38 -> T29,\nT17 -> T30,\nT18 -> T31,\nX39 -> .(T30, T31),\nX40 -> X42,\nX22 -> .(T29, X42),\nX23 -> X43,\nX41 -> X43" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 53, 14.36/4.43 "to": 54, 14.36/4.43 "label": "CASE" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 54, 14.36/4.43 "to": 55, 14.36/4.43 "label": "BACKTRACK\nfor clause: split([], [], [])because of non-unification" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 55, 14.36/4.43 "to": 63, 14.36/4.43 "label": "ONLY EVAL with clause\nsplit(.(X56, X57), .(X56, X58), X59) :- split(X57, X59, X58).\nand substitutionT30 -> T36,\nX56 -> T36,\nT31 -> T37,\nX57 -> T37,\nX58 -> X60,\nX43 -> .(T36, X60),\nX42 -> X61,\nX59 -> X61" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 63, 14.36/4.43 "to": 64, 14.36/4.43 "label": "CASE" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 64, 14.36/4.43 "to": 65, 14.36/4.43 "label": "PARALLEL" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 64, 14.36/4.43 "to": 66, 14.36/4.43 "label": "PARALLEL" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 65, 14.36/4.43 "to": 68, 14.36/4.43 "label": "EVAL with clause\nsplit([], [], []).\nand substitutionT37 -> [],\nX61 -> [],\nX60 -> []" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 65, 14.36/4.43 "to": 70, 14.36/4.43 "label": "EVAL-BACKTRACK" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 66, 14.36/4.43 "to": 72, 14.36/4.43 "label": "EVAL with clause\nsplit(.(X74, X75), .(X74, X76), X77) :- split(X75, X77, X76).\nand substitutionX74 -> T42,\nX75 -> T43,\nT37 -> .(T42, T43),\nX76 -> X78,\nX61 -> .(T42, X78),\nX60 -> X79,\nX77 -> X79" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 66, 14.36/4.43 "to": 73, 14.36/4.43 "label": "EVAL-BACKTRACK" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 68, 14.36/4.43 "to": 71, 14.36/4.43 "label": "SUCCESS" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 72, 14.36/4.43 "to": 63, 14.36/4.43 "label": "INSTANCE with matching:\nT37 -> T43\nX61 -> X79\nX60 -> X78" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 126, 14.36/4.43 "to": 3, 14.36/4.43 "label": "INSTANCE with matching:\nT1 -> T21\nT2 -> X24" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 127, 14.36/4.43 "to": 128, 14.36/4.43 "label": "SPLIT 1" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 127, 14.36/4.43 "to": 129, 14.36/4.43 "label": "SPLIT 2\nnew knowledge:\nT22 is ground\nT45 is ground\nreplacements:X25 -> T45" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 128, 14.36/4.43 "to": 3, 14.36/4.43 "label": "INSTANCE with matching:\nT1 -> T22\nT2 -> X25" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 129, 14.36/4.43 "to": 226, 14.36/4.43 "label": "CASE" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 226, 14.36/4.43 "to": 261, 14.36/4.43 "label": "PARALLEL" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 226, 14.36/4.43 "to": 263, 14.36/4.43 "label": "PARALLEL" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 261, 14.36/4.43 "to": 293, 14.36/4.43 "label": "EVAL with clause\nmerge([], X86, X86).\nand substitutionT44 -> [],\nT45 -> T52,\nX86 -> T52,\nT20 -> T52" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 261, 14.36/4.43 "to": 298, 14.36/4.43 "label": "EVAL-BACKTRACK" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 263, 14.36/4.43 "to": 318, 14.36/4.43 "label": "PARALLEL" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 263, 14.36/4.43 "to": 319, 14.36/4.43 "label": "PARALLEL" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 293, 14.36/4.43 "to": 299, 14.36/4.43 "label": "SUCCESS" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 318, 14.36/4.43 "to": 321, 14.36/4.43 "label": "EVAL with clause\nmerge(X91, [], X91).\nand substitutionT44 -> T57,\nX91 -> T57,\nT45 -> [],\nT20 -> T57" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 318, 14.36/4.43 "to": 322, 14.36/4.43 "label": "EVAL-BACKTRACK" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 319, 14.36/4.43 "to": 324, 14.36/4.43 "label": "PARALLEL" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 319, 14.36/4.43 "to": 325, 14.36/4.43 "label": "PARALLEL" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 321, 14.36/4.43 "to": 323, 14.36/4.43 "label": "SUCCESS" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 324, 14.36/4.43 "to": 339, 14.36/4.43 "label": "EVAL with clause\nmerge(.(X112, X113), .(X114, X115), .(X112, X116)) :- ','(le(X112, X114), merge(X113, .(X114, X115), X116)).\nand substitutionX112 -> T78,\nX113 -> T79,\nT44 -> .(T78, T79),\nX114 -> T80,\nX115 -> T81,\nT45 -> .(T80, T81),\nX116 -> T83,\nT20 -> .(T78, T83),\nT82 -> T83" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 324, 14.36/4.43 "to": 342, 14.36/4.43 "label": "EVAL-BACKTRACK" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 325, 14.36/4.43 "to": 380, 14.36/4.43 "label": "EVAL with clause\nmerge(.(X150, X151), .(X152, X153), .(X152, X154)) :- ','(gt(X150, X152), merge(.(X150, X151), X153, X154)).\nand substitutionX150 -> T119,\nX151 -> T120,\nT44 -> .(T119, T120),\nX152 -> T121,\nX153 -> T122,\nT45 -> .(T121, T122),\nX154 -> T124,\nT20 -> .(T121, T124),\nT123 -> T124" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 325, 14.36/4.43 "to": 381, 14.36/4.43 "label": "EVAL-BACKTRACK" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 339, 14.36/4.43 "to": 352, 14.36/4.43 "label": "SPLIT 1" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 339, 14.36/4.43 "to": 353, 14.36/4.43 "label": "SPLIT 2\nnew knowledge:\nT78 is ground\nT80 is ground" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 352, 14.36/4.43 "to": 357, 14.36/4.43 "label": "CASE" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 353, 14.36/4.43 "to": 129, 14.36/4.43 "label": "INSTANCE with matching:\nT44 -> T79\nT45 -> .(T80, T81)\nT20 -> T83" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 357, 14.36/4.43 "to": 358, 14.36/4.43 "label": "PARALLEL" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 357, 14.36/4.43 "to": 359, 14.36/4.43 "label": "PARALLEL" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 358, 14.36/4.43 "to": 360, 14.36/4.43 "label": "EVAL with clause\nle(s(X129), s(X130)) :- le(X129, X130).\nand substitutionX129 -> T96,\nT78 -> s(T96),\nX130 -> T97,\nT80 -> s(T97)" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 358, 14.36/4.43 "to": 361, 14.36/4.43 "label": "EVAL-BACKTRACK" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 359, 14.36/4.43 "to": 362, 14.36/4.43 "label": "PARALLEL" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 359, 14.36/4.43 "to": 363, 14.36/4.43 "label": "PARALLEL" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 360, 14.36/4.43 "to": 352, 14.36/4.43 "label": "INSTANCE with matching:\nT78 -> T96\nT80 -> T97" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 362, 14.36/4.43 "to": 373, 14.36/4.43 "label": "EVAL with clause\nle(0, s(X137)).\nand substitutionT78 -> 0,\nX137 -> T104,\nT80 -> s(T104)" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 362, 14.36/4.43 "to": 374, 14.36/4.43 "label": "EVAL-BACKTRACK" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 363, 14.36/4.43 "to": 376, 14.36/4.43 "label": "EVAL with clause\nle(0, 0).\nand substitutionT78 -> 0,\nT80 -> 0" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 363, 14.36/4.43 "to": 377, 14.36/4.43 "label": "EVAL-BACKTRACK" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 373, 14.36/4.43 "to": 375, 14.36/4.43 "label": "SUCCESS" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 376, 14.36/4.43 "to": 378, 14.36/4.43 "label": "SUCCESS" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 380, 14.36/4.43 "to": 597, 14.36/4.43 "label": "SPLIT 1" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 380, 14.36/4.43 "to": 598, 14.36/4.43 "label": "SPLIT 2\nnew knowledge:\nT119 is ground\nT121 is ground" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 597, 14.36/4.43 "to": 607, 14.36/4.43 "label": "CASE" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 598, 14.36/4.43 "to": 129, 14.36/4.43 "label": "INSTANCE with matching:\nT44 -> .(T119, T120)\nT45 -> T122\nT20 -> T124" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 607, 14.36/4.43 "to": 614, 14.36/4.43 "label": "PARALLEL" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 607, 14.36/4.43 "to": 615, 14.36/4.43 "label": "PARALLEL" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 614, 14.36/4.43 "to": 760, 14.36/4.43 "label": "EVAL with clause\ngt(s(X167), s(X168)) :- gt(X167, X168).\nand substitutionX167 -> T137,\nT119 -> s(T137),\nX168 -> T138,\nT121 -> s(T138)" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 614, 14.36/4.43 "to": 761, 14.36/4.43 "label": "EVAL-BACKTRACK" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 615, 14.36/4.43 "to": 762, 14.36/4.43 "label": "EVAL with clause\ngt(s(X173), 0).\nand substitutionX173 -> T143,\nT119 -> s(T143),\nT121 -> 0" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 615, 14.36/4.43 "to": 763, 14.36/4.43 "label": "EVAL-BACKTRACK" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 760, 14.36/4.43 "to": 597, 14.36/4.43 "label": "INSTANCE with matching:\nT119 -> T137\nT121 -> T138" 14.36/4.43 }, 14.36/4.43 { 14.36/4.43 "from": 762, 14.36/4.43 "to": 764, 14.36/4.43 "label": "SUCCESS" 14.36/4.43 } 14.36/4.43 ], 14.36/4.43 "type": "Graph" 14.36/4.43 } 14.36/4.43 } 14.36/4.43 14.36/4.43 ---------------------------------------- 14.36/4.43 14.36/4.43 (2) 14.36/4.43 Obligation: 14.36/4.43 Q restricted rewrite system: 14.36/4.43 The TRS R consists of the following rules: 14.36/4.43 14.36/4.43 f3_in([]) -> f3_out1([]) 14.36/4.43 f3_in(.(T7, [])) -> f3_out1(.(T7, [])) 14.36/4.43 f3_in(.(T16, .(T17, T18))) -> U1(f45_in(T16, T17, T18), .(T16, .(T17, T18))) 14.36/4.43 U1(f45_out1(X22, X23, X24, X25, T20), .(T16, .(T17, T18))) -> f3_out1(T20) 14.36/4.44 f63_in([]) -> f63_out1([], []) 14.36/4.44 f63_in(.(T42, T43)) -> U2(f63_in(T43), .(T42, T43)) 14.36/4.44 U2(f63_out1(X79, X78), .(T42, T43)) -> f63_out1(.(T42, X78), X79) 14.36/4.44 f129_in([], T52) -> f129_out1(T52) 14.36/4.44 f129_in(T57, []) -> f129_out1(T57) 14.36/4.44 f129_in(.(T78, T79), .(T80, T81)) -> U3(f339_in(T78, T80, T79, T81), .(T78, T79), .(T80, T81)) 14.36/4.44 U3(f339_out1(T83), .(T78, T79), .(T80, T81)) -> f129_out1(.(T78, T83)) 14.36/4.44 f129_in(.(T119, T120), .(T121, T122)) -> U4(f380_in(T119, T121, T120, T122), .(T119, T120), .(T121, T122)) 14.36/4.44 U4(f380_out1(T124), .(T119, T120), .(T121, T122)) -> f129_out1(.(T121, T124)) 14.36/4.44 f352_in(s(T96), s(T97)) -> U5(f352_in(T96, T97), s(T96), s(T97)) 14.36/4.44 U5(f352_out1, s(T96), s(T97)) -> f352_out1 14.36/4.44 f352_in(0, s(T104)) -> f352_out1 14.36/4.44 f352_in(0, 0) -> f352_out1 14.36/4.44 f597_in(s(T137), s(T138)) -> U6(f597_in(T137, T138), s(T137), s(T138)) 14.36/4.44 U6(f597_out1, s(T137), s(T138)) -> f597_out1 14.36/4.44 f597_in(s(T143), 0) -> f597_out1 14.36/4.44 f49_in(T29, T36, T37) -> U7(f63_in(T37), T29, T36, T37) 14.36/4.44 U7(f63_out1(X61, X60), T29, T36, T37) -> f49_out1(.(T29, X61), .(T36, X60)) 14.36/4.44 f45_in(T16, T17, T18) -> U8(f49_in(T16, T17, T18), T16, T17, T18) 14.36/4.44 U8(f49_out1(T21, T22), T16, T17, T18) -> U9(f50_in(T21, T22), T16, T17, T18, T21, T22) 14.36/4.44 U9(f50_out1(X24, X25, T20), T16, T17, T18, T21, T22) -> f45_out1(T21, T22, X24, X25, T20) 14.36/4.44 f50_in(T21, T22) -> U10(f3_in(T21), T21, T22) 14.36/4.44 U10(f3_out1(T44), T21, T22) -> U11(f127_in(T22, T44), T21, T22, T44) 14.36/4.44 U11(f127_out1(X25, T20), T21, T22, T44) -> f50_out1(T44, X25, T20) 14.36/4.44 f127_in(T22, T44) -> U12(f3_in(T22), T22, T44) 14.36/4.44 U12(f3_out1(T45), T22, T44) -> U13(f129_in(T44, T45), T22, T44, T45) 14.36/4.44 U13(f129_out1(T20), T22, T44, T45) -> f127_out1(T45, T20) 14.36/4.44 f339_in(T78, T80, T79, T81) -> U14(f352_in(T78, T80), T78, T80, T79, T81) 14.36/4.44 U14(f352_out1, T78, T80, T79, T81) -> U15(f129_in(T79, .(T80, T81)), T78, T80, T79, T81) 14.36/4.44 U15(f129_out1(T83), T78, T80, T79, T81) -> f339_out1(T83) 14.36/4.44 f380_in(T119, T121, T120, T122) -> U16(f597_in(T119, T121), T119, T121, T120, T122) 14.36/4.44 U16(f597_out1, T119, T121, T120, T122) -> U17(f129_in(.(T119, T120), T122), T119, T121, T120, T122) 14.36/4.44 U17(f129_out1(T124), T119, T121, T120, T122) -> f380_out1(T124) 14.36/4.44 14.36/4.44 Q is empty. 14.36/4.44 14.36/4.44 ---------------------------------------- 14.36/4.44 14.36/4.44 (3) DependencyPairsProof (EQUIVALENT) 14.36/4.44 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 14.36/4.44 ---------------------------------------- 14.36/4.44 14.36/4.44 (4) 14.36/4.44 Obligation: 14.36/4.44 Q DP problem: 14.36/4.44 The TRS P consists of the following rules: 14.36/4.44 14.36/4.44 F3_IN(.(T16, .(T17, T18))) -> U1^1(f45_in(T16, T17, T18), .(T16, .(T17, T18))) 14.36/4.44 F3_IN(.(T16, .(T17, T18))) -> F45_IN(T16, T17, T18) 14.36/4.44 F63_IN(.(T42, T43)) -> U2^1(f63_in(T43), .(T42, T43)) 14.36/4.44 F63_IN(.(T42, T43)) -> F63_IN(T43) 14.36/4.44 F129_IN(.(T78, T79), .(T80, T81)) -> U3^1(f339_in(T78, T80, T79, T81), .(T78, T79), .(T80, T81)) 14.36/4.44 F129_IN(.(T78, T79), .(T80, T81)) -> F339_IN(T78, T80, T79, T81) 14.36/4.44 F129_IN(.(T119, T120), .(T121, T122)) -> U4^1(f380_in(T119, T121, T120, T122), .(T119, T120), .(T121, T122)) 14.36/4.44 F129_IN(.(T119, T120), .(T121, T122)) -> F380_IN(T119, T121, T120, T122) 14.36/4.44 F352_IN(s(T96), s(T97)) -> U5^1(f352_in(T96, T97), s(T96), s(T97)) 14.36/4.44 F352_IN(s(T96), s(T97)) -> F352_IN(T96, T97) 14.36/4.44 F597_IN(s(T137), s(T138)) -> U6^1(f597_in(T137, T138), s(T137), s(T138)) 14.36/4.44 F597_IN(s(T137), s(T138)) -> F597_IN(T137, T138) 14.36/4.44 F49_IN(T29, T36, T37) -> U7^1(f63_in(T37), T29, T36, T37) 14.36/4.44 F49_IN(T29, T36, T37) -> F63_IN(T37) 14.36/4.44 F45_IN(T16, T17, T18) -> U8^1(f49_in(T16, T17, T18), T16, T17, T18) 14.36/4.44 F45_IN(T16, T17, T18) -> F49_IN(T16, T17, T18) 14.36/4.44 U8^1(f49_out1(T21, T22), T16, T17, T18) -> U9^1(f50_in(T21, T22), T16, T17, T18, T21, T22) 14.36/4.44 U8^1(f49_out1(T21, T22), T16, T17, T18) -> F50_IN(T21, T22) 14.36/4.44 F50_IN(T21, T22) -> U10^1(f3_in(T21), T21, T22) 14.36/4.44 F50_IN(T21, T22) -> F3_IN(T21) 14.36/4.44 U10^1(f3_out1(T44), T21, T22) -> U11^1(f127_in(T22, T44), T21, T22, T44) 14.36/4.44 U10^1(f3_out1(T44), T21, T22) -> F127_IN(T22, T44) 14.36/4.44 F127_IN(T22, T44) -> U12^1(f3_in(T22), T22, T44) 14.36/4.44 F127_IN(T22, T44) -> F3_IN(T22) 14.36/4.44 U12^1(f3_out1(T45), T22, T44) -> U13^1(f129_in(T44, T45), T22, T44, T45) 14.36/4.44 U12^1(f3_out1(T45), T22, T44) -> F129_IN(T44, T45) 14.36/4.44 F339_IN(T78, T80, T79, T81) -> U14^1(f352_in(T78, T80), T78, T80, T79, T81) 14.36/4.44 F339_IN(T78, T80, T79, T81) -> F352_IN(T78, T80) 14.36/4.44 U14^1(f352_out1, T78, T80, T79, T81) -> U15^1(f129_in(T79, .(T80, T81)), T78, T80, T79, T81) 14.36/4.44 U14^1(f352_out1, T78, T80, T79, T81) -> F129_IN(T79, .(T80, T81)) 14.36/4.44 F380_IN(T119, T121, T120, T122) -> U16^1(f597_in(T119, T121), T119, T121, T120, T122) 14.36/4.44 F380_IN(T119, T121, T120, T122) -> F597_IN(T119, T121) 14.36/4.44 U16^1(f597_out1, T119, T121, T120, T122) -> U17^1(f129_in(.(T119, T120), T122), T119, T121, T120, T122) 14.36/4.44 U16^1(f597_out1, T119, T121, T120, T122) -> F129_IN(.(T119, T120), T122) 14.36/4.44 14.36/4.44 The TRS R consists of the following rules: 14.36/4.44 14.36/4.44 f3_in([]) -> f3_out1([]) 14.36/4.44 f3_in(.(T7, [])) -> f3_out1(.(T7, [])) 14.36/4.44 f3_in(.(T16, .(T17, T18))) -> U1(f45_in(T16, T17, T18), .(T16, .(T17, T18))) 14.36/4.44 U1(f45_out1(X22, X23, X24, X25, T20), .(T16, .(T17, T18))) -> f3_out1(T20) 14.36/4.44 f63_in([]) -> f63_out1([], []) 14.36/4.44 f63_in(.(T42, T43)) -> U2(f63_in(T43), .(T42, T43)) 14.36/4.44 U2(f63_out1(X79, X78), .(T42, T43)) -> f63_out1(.(T42, X78), X79) 14.36/4.44 f129_in([], T52) -> f129_out1(T52) 14.36/4.44 f129_in(T57, []) -> f129_out1(T57) 14.36/4.44 f129_in(.(T78, T79), .(T80, T81)) -> U3(f339_in(T78, T80, T79, T81), .(T78, T79), .(T80, T81)) 14.36/4.44 U3(f339_out1(T83), .(T78, T79), .(T80, T81)) -> f129_out1(.(T78, T83)) 14.36/4.44 f129_in(.(T119, T120), .(T121, T122)) -> U4(f380_in(T119, T121, T120, T122), .(T119, T120), .(T121, T122)) 14.36/4.44 U4(f380_out1(T124), .(T119, T120), .(T121, T122)) -> f129_out1(.(T121, T124)) 14.36/4.44 f352_in(s(T96), s(T97)) -> U5(f352_in(T96, T97), s(T96), s(T97)) 14.36/4.44 U5(f352_out1, s(T96), s(T97)) -> f352_out1 14.36/4.44 f352_in(0, s(T104)) -> f352_out1 14.36/4.44 f352_in(0, 0) -> f352_out1 14.36/4.44 f597_in(s(T137), s(T138)) -> U6(f597_in(T137, T138), s(T137), s(T138)) 14.36/4.44 U6(f597_out1, s(T137), s(T138)) -> f597_out1 14.36/4.44 f597_in(s(T143), 0) -> f597_out1 14.36/4.44 f49_in(T29, T36, T37) -> U7(f63_in(T37), T29, T36, T37) 14.36/4.44 U7(f63_out1(X61, X60), T29, T36, T37) -> f49_out1(.(T29, X61), .(T36, X60)) 14.36/4.44 f45_in(T16, T17, T18) -> U8(f49_in(T16, T17, T18), T16, T17, T18) 14.36/4.44 U8(f49_out1(T21, T22), T16, T17, T18) -> U9(f50_in(T21, T22), T16, T17, T18, T21, T22) 14.36/4.44 U9(f50_out1(X24, X25, T20), T16, T17, T18, T21, T22) -> f45_out1(T21, T22, X24, X25, T20) 14.36/4.44 f50_in(T21, T22) -> U10(f3_in(T21), T21, T22) 14.36/4.44 U10(f3_out1(T44), T21, T22) -> U11(f127_in(T22, T44), T21, T22, T44) 14.36/4.44 U11(f127_out1(X25, T20), T21, T22, T44) -> f50_out1(T44, X25, T20) 14.36/4.44 f127_in(T22, T44) -> U12(f3_in(T22), T22, T44) 14.36/4.44 U12(f3_out1(T45), T22, T44) -> U13(f129_in(T44, T45), T22, T44, T45) 14.36/4.44 U13(f129_out1(T20), T22, T44, T45) -> f127_out1(T45, T20) 14.36/4.44 f339_in(T78, T80, T79, T81) -> U14(f352_in(T78, T80), T78, T80, T79, T81) 14.36/4.44 U14(f352_out1, T78, T80, T79, T81) -> U15(f129_in(T79, .(T80, T81)), T78, T80, T79, T81) 14.36/4.44 U15(f129_out1(T83), T78, T80, T79, T81) -> f339_out1(T83) 14.36/4.44 f380_in(T119, T121, T120, T122) -> U16(f597_in(T119, T121), T119, T121, T120, T122) 14.36/4.44 U16(f597_out1, T119, T121, T120, T122) -> U17(f129_in(.(T119, T120), T122), T119, T121, T120, T122) 14.36/4.44 U17(f129_out1(T124), T119, T121, T120, T122) -> f380_out1(T124) 14.36/4.44 14.36/4.44 Q is empty. 14.36/4.44 We have to consider all minimal (P,Q,R)-chains. 14.36/4.44 ---------------------------------------- 14.36/4.44 14.36/4.44 (5) DependencyGraphProof (EQUIVALENT) 14.36/4.44 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 5 SCCs with 18 less nodes. 14.36/4.44 ---------------------------------------- 14.36/4.44 14.36/4.44 (6) 14.36/4.44 Complex Obligation (AND) 14.36/4.44 14.36/4.44 ---------------------------------------- 14.36/4.44 14.36/4.44 (7) 14.36/4.44 Obligation: 14.36/4.44 Q DP problem: 14.36/4.44 The TRS P consists of the following rules: 14.36/4.44 14.36/4.44 F597_IN(s(T137), s(T138)) -> F597_IN(T137, T138) 14.36/4.44 14.36/4.44 The TRS R consists of the following rules: 14.36/4.44 14.36/4.44 f3_in([]) -> f3_out1([]) 14.36/4.44 f3_in(.(T7, [])) -> f3_out1(.(T7, [])) 14.36/4.44 f3_in(.(T16, .(T17, T18))) -> U1(f45_in(T16, T17, T18), .(T16, .(T17, T18))) 14.36/4.44 U1(f45_out1(X22, X23, X24, X25, T20), .(T16, .(T17, T18))) -> f3_out1(T20) 14.36/4.44 f63_in([]) -> f63_out1([], []) 14.36/4.44 f63_in(.(T42, T43)) -> U2(f63_in(T43), .(T42, T43)) 14.36/4.44 U2(f63_out1(X79, X78), .(T42, T43)) -> f63_out1(.(T42, X78), X79) 14.36/4.44 f129_in([], T52) -> f129_out1(T52) 14.36/4.44 f129_in(T57, []) -> f129_out1(T57) 14.36/4.44 f129_in(.(T78, T79), .(T80, T81)) -> U3(f339_in(T78, T80, T79, T81), .(T78, T79), .(T80, T81)) 14.36/4.44 U3(f339_out1(T83), .(T78, T79), .(T80, T81)) -> f129_out1(.(T78, T83)) 14.36/4.44 f129_in(.(T119, T120), .(T121, T122)) -> U4(f380_in(T119, T121, T120, T122), .(T119, T120), .(T121, T122)) 14.36/4.44 U4(f380_out1(T124), .(T119, T120), .(T121, T122)) -> f129_out1(.(T121, T124)) 14.36/4.44 f352_in(s(T96), s(T97)) -> U5(f352_in(T96, T97), s(T96), s(T97)) 14.36/4.44 U5(f352_out1, s(T96), s(T97)) -> f352_out1 14.36/4.44 f352_in(0, s(T104)) -> f352_out1 14.36/4.44 f352_in(0, 0) -> f352_out1 14.36/4.44 f597_in(s(T137), s(T138)) -> U6(f597_in(T137, T138), s(T137), s(T138)) 14.36/4.44 U6(f597_out1, s(T137), s(T138)) -> f597_out1 14.36/4.44 f597_in(s(T143), 0) -> f597_out1 14.36/4.44 f49_in(T29, T36, T37) -> U7(f63_in(T37), T29, T36, T37) 14.36/4.44 U7(f63_out1(X61, X60), T29, T36, T37) -> f49_out1(.(T29, X61), .(T36, X60)) 14.36/4.44 f45_in(T16, T17, T18) -> U8(f49_in(T16, T17, T18), T16, T17, T18) 14.36/4.44 U8(f49_out1(T21, T22), T16, T17, T18) -> U9(f50_in(T21, T22), T16, T17, T18, T21, T22) 14.36/4.44 U9(f50_out1(X24, X25, T20), T16, T17, T18, T21, T22) -> f45_out1(T21, T22, X24, X25, T20) 14.36/4.44 f50_in(T21, T22) -> U10(f3_in(T21), T21, T22) 14.36/4.44 U10(f3_out1(T44), T21, T22) -> U11(f127_in(T22, T44), T21, T22, T44) 14.36/4.44 U11(f127_out1(X25, T20), T21, T22, T44) -> f50_out1(T44, X25, T20) 14.36/4.44 f127_in(T22, T44) -> U12(f3_in(T22), T22, T44) 14.36/4.44 U12(f3_out1(T45), T22, T44) -> U13(f129_in(T44, T45), T22, T44, T45) 14.36/4.44 U13(f129_out1(T20), T22, T44, T45) -> f127_out1(T45, T20) 14.36/4.44 f339_in(T78, T80, T79, T81) -> U14(f352_in(T78, T80), T78, T80, T79, T81) 14.36/4.44 U14(f352_out1, T78, T80, T79, T81) -> U15(f129_in(T79, .(T80, T81)), T78, T80, T79, T81) 14.36/4.44 U15(f129_out1(T83), T78, T80, T79, T81) -> f339_out1(T83) 14.36/4.44 f380_in(T119, T121, T120, T122) -> U16(f597_in(T119, T121), T119, T121, T120, T122) 14.36/4.44 U16(f597_out1, T119, T121, T120, T122) -> U17(f129_in(.(T119, T120), T122), T119, T121, T120, T122) 14.36/4.44 U17(f129_out1(T124), T119, T121, T120, T122) -> f380_out1(T124) 14.36/4.44 14.36/4.44 Q is empty. 14.36/4.44 We have to consider all minimal (P,Q,R)-chains. 14.36/4.44 ---------------------------------------- 14.36/4.44 14.36/4.44 (8) UsableRulesProof (EQUIVALENT) 14.36/4.44 We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. 14.36/4.44 ---------------------------------------- 14.36/4.44 14.36/4.44 (9) 14.36/4.44 Obligation: 14.36/4.44 Q DP problem: 14.36/4.44 The TRS P consists of the following rules: 14.36/4.44 14.36/4.44 F597_IN(s(T137), s(T138)) -> F597_IN(T137, T138) 14.36/4.44 14.36/4.44 R is empty. 14.36/4.44 Q is empty. 14.36/4.44 We have to consider all minimal (P,Q,R)-chains. 14.36/4.44 ---------------------------------------- 14.36/4.44 14.36/4.44 (10) QDPSizeChangeProof (EQUIVALENT) 14.36/4.44 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. 14.36/4.44 14.36/4.44 From the DPs we obtained the following set of size-change graphs: 14.36/4.44 *F597_IN(s(T137), s(T138)) -> F597_IN(T137, T138) 14.36/4.44 The graph contains the following edges 1 > 1, 2 > 2 14.36/4.44 14.36/4.44 14.36/4.44 ---------------------------------------- 14.36/4.44 14.36/4.44 (11) 14.36/4.44 YES 14.36/4.44 14.36/4.44 ---------------------------------------- 14.36/4.44 14.36/4.44 (12) 14.36/4.44 Obligation: 14.36/4.44 Q DP problem: 14.36/4.44 The TRS P consists of the following rules: 14.36/4.44 14.36/4.44 F352_IN(s(T96), s(T97)) -> F352_IN(T96, T97) 14.36/4.44 14.36/4.44 The TRS R consists of the following rules: 14.36/4.44 14.36/4.44 f3_in([]) -> f3_out1([]) 14.36/4.44 f3_in(.(T7, [])) -> f3_out1(.(T7, [])) 14.36/4.44 f3_in(.(T16, .(T17, T18))) -> U1(f45_in(T16, T17, T18), .(T16, .(T17, T18))) 14.36/4.44 U1(f45_out1(X22, X23, X24, X25, T20), .(T16, .(T17, T18))) -> f3_out1(T20) 14.36/4.44 f63_in([]) -> f63_out1([], []) 14.36/4.44 f63_in(.(T42, T43)) -> U2(f63_in(T43), .(T42, T43)) 14.36/4.44 U2(f63_out1(X79, X78), .(T42, T43)) -> f63_out1(.(T42, X78), X79) 14.36/4.44 f129_in([], T52) -> f129_out1(T52) 14.36/4.44 f129_in(T57, []) -> f129_out1(T57) 14.36/4.44 f129_in(.(T78, T79), .(T80, T81)) -> U3(f339_in(T78, T80, T79, T81), .(T78, T79), .(T80, T81)) 14.36/4.44 U3(f339_out1(T83), .(T78, T79), .(T80, T81)) -> f129_out1(.(T78, T83)) 14.36/4.44 f129_in(.(T119, T120), .(T121, T122)) -> U4(f380_in(T119, T121, T120, T122), .(T119, T120), .(T121, T122)) 14.36/4.44 U4(f380_out1(T124), .(T119, T120), .(T121, T122)) -> f129_out1(.(T121, T124)) 14.36/4.44 f352_in(s(T96), s(T97)) -> U5(f352_in(T96, T97), s(T96), s(T97)) 14.36/4.44 U5(f352_out1, s(T96), s(T97)) -> f352_out1 14.36/4.44 f352_in(0, s(T104)) -> f352_out1 14.36/4.44 f352_in(0, 0) -> f352_out1 14.36/4.44 f597_in(s(T137), s(T138)) -> U6(f597_in(T137, T138), s(T137), s(T138)) 14.36/4.44 U6(f597_out1, s(T137), s(T138)) -> f597_out1 14.36/4.44 f597_in(s(T143), 0) -> f597_out1 14.36/4.44 f49_in(T29, T36, T37) -> U7(f63_in(T37), T29, T36, T37) 14.36/4.44 U7(f63_out1(X61, X60), T29, T36, T37) -> f49_out1(.(T29, X61), .(T36, X60)) 14.36/4.44 f45_in(T16, T17, T18) -> U8(f49_in(T16, T17, T18), T16, T17, T18) 14.36/4.44 U8(f49_out1(T21, T22), T16, T17, T18) -> U9(f50_in(T21, T22), T16, T17, T18, T21, T22) 14.36/4.44 U9(f50_out1(X24, X25, T20), T16, T17, T18, T21, T22) -> f45_out1(T21, T22, X24, X25, T20) 14.36/4.44 f50_in(T21, T22) -> U10(f3_in(T21), T21, T22) 14.36/4.44 U10(f3_out1(T44), T21, T22) -> U11(f127_in(T22, T44), T21, T22, T44) 14.36/4.44 U11(f127_out1(X25, T20), T21, T22, T44) -> f50_out1(T44, X25, T20) 14.36/4.44 f127_in(T22, T44) -> U12(f3_in(T22), T22, T44) 14.36/4.44 U12(f3_out1(T45), T22, T44) -> U13(f129_in(T44, T45), T22, T44, T45) 14.36/4.44 U13(f129_out1(T20), T22, T44, T45) -> f127_out1(T45, T20) 14.36/4.44 f339_in(T78, T80, T79, T81) -> U14(f352_in(T78, T80), T78, T80, T79, T81) 14.36/4.44 U14(f352_out1, T78, T80, T79, T81) -> U15(f129_in(T79, .(T80, T81)), T78, T80, T79, T81) 14.36/4.44 U15(f129_out1(T83), T78, T80, T79, T81) -> f339_out1(T83) 14.36/4.44 f380_in(T119, T121, T120, T122) -> U16(f597_in(T119, T121), T119, T121, T120, T122) 14.36/4.44 U16(f597_out1, T119, T121, T120, T122) -> U17(f129_in(.(T119, T120), T122), T119, T121, T120, T122) 14.36/4.44 U17(f129_out1(T124), T119, T121, T120, T122) -> f380_out1(T124) 14.36/4.44 14.36/4.44 Q is empty. 14.36/4.44 We have to consider all minimal (P,Q,R)-chains. 14.36/4.44 ---------------------------------------- 14.36/4.44 14.36/4.44 (13) UsableRulesProof (EQUIVALENT) 14.36/4.44 We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. 14.36/4.44 ---------------------------------------- 14.36/4.44 14.36/4.44 (14) 14.36/4.44 Obligation: 14.36/4.44 Q DP problem: 14.36/4.44 The TRS P consists of the following rules: 14.36/4.44 14.36/4.44 F352_IN(s(T96), s(T97)) -> F352_IN(T96, T97) 14.36/4.44 14.36/4.44 R is empty. 14.36/4.44 Q is empty. 14.36/4.44 We have to consider all minimal (P,Q,R)-chains. 14.36/4.44 ---------------------------------------- 14.36/4.44 14.36/4.44 (15) QDPSizeChangeProof (EQUIVALENT) 14.36/4.44 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. 14.36/4.44 14.36/4.44 From the DPs we obtained the following set of size-change graphs: 14.36/4.44 *F352_IN(s(T96), s(T97)) -> F352_IN(T96, T97) 14.36/4.44 The graph contains the following edges 1 > 1, 2 > 2 14.36/4.44 14.36/4.44 14.36/4.44 ---------------------------------------- 14.36/4.44 14.36/4.44 (16) 14.36/4.44 YES 14.36/4.44 14.36/4.44 ---------------------------------------- 14.36/4.44 14.36/4.44 (17) 14.36/4.44 Obligation: 14.36/4.44 Q DP problem: 14.36/4.44 The TRS P consists of the following rules: 14.36/4.44 14.36/4.44 F129_IN(.(T78, T79), .(T80, T81)) -> F339_IN(T78, T80, T79, T81) 14.36/4.44 F339_IN(T78, T80, T79, T81) -> U14^1(f352_in(T78, T80), T78, T80, T79, T81) 14.36/4.44 U14^1(f352_out1, T78, T80, T79, T81) -> F129_IN(T79, .(T80, T81)) 14.36/4.44 F129_IN(.(T119, T120), .(T121, T122)) -> F380_IN(T119, T121, T120, T122) 14.36/4.44 F380_IN(T119, T121, T120, T122) -> U16^1(f597_in(T119, T121), T119, T121, T120, T122) 14.36/4.44 U16^1(f597_out1, T119, T121, T120, T122) -> F129_IN(.(T119, T120), T122) 14.36/4.44 14.36/4.44 The TRS R consists of the following rules: 14.36/4.44 14.36/4.44 f3_in([]) -> f3_out1([]) 14.36/4.44 f3_in(.(T7, [])) -> f3_out1(.(T7, [])) 14.36/4.44 f3_in(.(T16, .(T17, T18))) -> U1(f45_in(T16, T17, T18), .(T16, .(T17, T18))) 14.36/4.44 U1(f45_out1(X22, X23, X24, X25, T20), .(T16, .(T17, T18))) -> f3_out1(T20) 14.36/4.44 f63_in([]) -> f63_out1([], []) 14.36/4.44 f63_in(.(T42, T43)) -> U2(f63_in(T43), .(T42, T43)) 14.36/4.44 U2(f63_out1(X79, X78), .(T42, T43)) -> f63_out1(.(T42, X78), X79) 14.36/4.44 f129_in([], T52) -> f129_out1(T52) 14.36/4.44 f129_in(T57, []) -> f129_out1(T57) 14.36/4.44 f129_in(.(T78, T79), .(T80, T81)) -> U3(f339_in(T78, T80, T79, T81), .(T78, T79), .(T80, T81)) 14.36/4.44 U3(f339_out1(T83), .(T78, T79), .(T80, T81)) -> f129_out1(.(T78, T83)) 14.36/4.44 f129_in(.(T119, T120), .(T121, T122)) -> U4(f380_in(T119, T121, T120, T122), .(T119, T120), .(T121, T122)) 14.36/4.44 U4(f380_out1(T124), .(T119, T120), .(T121, T122)) -> f129_out1(.(T121, T124)) 14.36/4.44 f352_in(s(T96), s(T97)) -> U5(f352_in(T96, T97), s(T96), s(T97)) 14.36/4.44 U5(f352_out1, s(T96), s(T97)) -> f352_out1 14.36/4.44 f352_in(0, s(T104)) -> f352_out1 14.36/4.44 f352_in(0, 0) -> f352_out1 14.36/4.44 f597_in(s(T137), s(T138)) -> U6(f597_in(T137, T138), s(T137), s(T138)) 14.36/4.44 U6(f597_out1, s(T137), s(T138)) -> f597_out1 14.36/4.44 f597_in(s(T143), 0) -> f597_out1 14.36/4.44 f49_in(T29, T36, T37) -> U7(f63_in(T37), T29, T36, T37) 14.36/4.44 U7(f63_out1(X61, X60), T29, T36, T37) -> f49_out1(.(T29, X61), .(T36, X60)) 14.36/4.44 f45_in(T16, T17, T18) -> U8(f49_in(T16, T17, T18), T16, T17, T18) 14.36/4.44 U8(f49_out1(T21, T22), T16, T17, T18) -> U9(f50_in(T21, T22), T16, T17, T18, T21, T22) 14.36/4.44 U9(f50_out1(X24, X25, T20), T16, T17, T18, T21, T22) -> f45_out1(T21, T22, X24, X25, T20) 14.36/4.44 f50_in(T21, T22) -> U10(f3_in(T21), T21, T22) 14.36/4.44 U10(f3_out1(T44), T21, T22) -> U11(f127_in(T22, T44), T21, T22, T44) 14.36/4.44 U11(f127_out1(X25, T20), T21, T22, T44) -> f50_out1(T44, X25, T20) 14.36/4.44 f127_in(T22, T44) -> U12(f3_in(T22), T22, T44) 14.36/4.44 U12(f3_out1(T45), T22, T44) -> U13(f129_in(T44, T45), T22, T44, T45) 14.36/4.44 U13(f129_out1(T20), T22, T44, T45) -> f127_out1(T45, T20) 14.36/4.44 f339_in(T78, T80, T79, T81) -> U14(f352_in(T78, T80), T78, T80, T79, T81) 14.36/4.44 U14(f352_out1, T78, T80, T79, T81) -> U15(f129_in(T79, .(T80, T81)), T78, T80, T79, T81) 14.36/4.44 U15(f129_out1(T83), T78, T80, T79, T81) -> f339_out1(T83) 14.36/4.44 f380_in(T119, T121, T120, T122) -> U16(f597_in(T119, T121), T119, T121, T120, T122) 14.36/4.44 U16(f597_out1, T119, T121, T120, T122) -> U17(f129_in(.(T119, T120), T122), T119, T121, T120, T122) 14.36/4.44 U17(f129_out1(T124), T119, T121, T120, T122) -> f380_out1(T124) 14.36/4.44 14.36/4.44 Q is empty. 14.36/4.44 We have to consider all minimal (P,Q,R)-chains. 14.36/4.44 ---------------------------------------- 14.36/4.44 14.36/4.44 (18) QDPOrderProof (EQUIVALENT) 14.36/4.44 We use the reduction pair processor [LPAR04,JAR06]. 14.36/4.44 14.36/4.44 14.36/4.44 The following pairs can be oriented strictly and are deleted. 14.36/4.44 14.36/4.44 F129_IN(.(T78, T79), .(T80, T81)) -> F339_IN(T78, T80, T79, T81) 14.36/4.44 F129_IN(.(T119, T120), .(T121, T122)) -> F380_IN(T119, T121, T120, T122) 14.36/4.44 U16^1(f597_out1, T119, T121, T120, T122) -> F129_IN(.(T119, T120), T122) 14.36/4.44 The remaining pairs can at least be oriented weakly. 14.36/4.44 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 14.36/4.44 14.36/4.44 POL( U14^1_5(x_1, ..., x_5) ) = x_1 + 2x_3 + 2x_4 + 2x_5 + 2 14.36/4.44 POL( f352_in_2(x_1, x_2) ) = x_1 14.36/4.44 POL( s_1(x_1) ) = 2x_1 + 2 14.36/4.44 POL( U5_3(x_1, ..., x_3) ) = 2 14.36/4.44 POL( 0 ) = 2 14.36/4.44 POL( f352_out1 ) = 2 14.36/4.44 POL( U16^1_5(x_1, ..., x_5) ) = 2x_1 + 2x_2 + 2x_4 + 2x_5 + 2 14.36/4.44 POL( f597_in_2(x_1, x_2) ) = x_2 14.36/4.44 POL( U6_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2} 14.36/4.44 POL( f597_out1 ) = 2 14.36/4.44 POL( F129_IN_2(x_1, x_2) ) = 2x_1 + 2x_2 14.36/4.44 POL( ._2(x_1, x_2) ) = x_1 + x_2 + 2 14.36/4.44 POL( F339_IN_4(x_1, ..., x_4) ) = x_1 + 2x_2 + 2x_3 + 2x_4 + 2 14.36/4.44 POL( F380_IN_4(x_1, ..., x_4) ) = 2x_1 + 2x_2 + 2x_3 + 2x_4 + 2 14.36/4.44 14.36/4.44 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 14.36/4.44 14.36/4.44 f352_in(s(T96), s(T97)) -> U5(f352_in(T96, T97), s(T96), s(T97)) 14.36/4.44 f352_in(0, s(T104)) -> f352_out1 14.36/4.44 f352_in(0, 0) -> f352_out1 14.36/4.44 f597_in(s(T137), s(T138)) -> U6(f597_in(T137, T138), s(T137), s(T138)) 14.36/4.44 f597_in(s(T143), 0) -> f597_out1 14.36/4.44 U5(f352_out1, s(T96), s(T97)) -> f352_out1 14.36/4.44 U6(f597_out1, s(T137), s(T138)) -> f597_out1 14.36/4.44 14.36/4.44 14.36/4.44 ---------------------------------------- 14.36/4.44 14.36/4.44 (19) 14.36/4.44 Obligation: 14.36/4.44 Q DP problem: 14.36/4.44 The TRS P consists of the following rules: 14.36/4.44 14.36/4.44 F339_IN(T78, T80, T79, T81) -> U14^1(f352_in(T78, T80), T78, T80, T79, T81) 14.36/4.44 U14^1(f352_out1, T78, T80, T79, T81) -> F129_IN(T79, .(T80, T81)) 14.36/4.44 F380_IN(T119, T121, T120, T122) -> U16^1(f597_in(T119, T121), T119, T121, T120, T122) 14.36/4.44 14.36/4.44 The TRS R consists of the following rules: 14.36/4.44 14.36/4.44 f3_in([]) -> f3_out1([]) 14.36/4.44 f3_in(.(T7, [])) -> f3_out1(.(T7, [])) 14.36/4.44 f3_in(.(T16, .(T17, T18))) -> U1(f45_in(T16, T17, T18), .(T16, .(T17, T18))) 14.36/4.44 U1(f45_out1(X22, X23, X24, X25, T20), .(T16, .(T17, T18))) -> f3_out1(T20) 14.36/4.44 f63_in([]) -> f63_out1([], []) 14.36/4.44 f63_in(.(T42, T43)) -> U2(f63_in(T43), .(T42, T43)) 14.36/4.44 U2(f63_out1(X79, X78), .(T42, T43)) -> f63_out1(.(T42, X78), X79) 14.36/4.44 f129_in([], T52) -> f129_out1(T52) 14.36/4.44 f129_in(T57, []) -> f129_out1(T57) 14.36/4.44 f129_in(.(T78, T79), .(T80, T81)) -> U3(f339_in(T78, T80, T79, T81), .(T78, T79), .(T80, T81)) 14.36/4.44 U3(f339_out1(T83), .(T78, T79), .(T80, T81)) -> f129_out1(.(T78, T83)) 14.36/4.44 f129_in(.(T119, T120), .(T121, T122)) -> U4(f380_in(T119, T121, T120, T122), .(T119, T120), .(T121, T122)) 14.36/4.44 U4(f380_out1(T124), .(T119, T120), .(T121, T122)) -> f129_out1(.(T121, T124)) 14.36/4.44 f352_in(s(T96), s(T97)) -> U5(f352_in(T96, T97), s(T96), s(T97)) 14.36/4.44 U5(f352_out1, s(T96), s(T97)) -> f352_out1 14.36/4.44 f352_in(0, s(T104)) -> f352_out1 14.36/4.44 f352_in(0, 0) -> f352_out1 14.36/4.44 f597_in(s(T137), s(T138)) -> U6(f597_in(T137, T138), s(T137), s(T138)) 14.36/4.44 U6(f597_out1, s(T137), s(T138)) -> f597_out1 14.36/4.44 f597_in(s(T143), 0) -> f597_out1 14.36/4.44 f49_in(T29, T36, T37) -> U7(f63_in(T37), T29, T36, T37) 14.36/4.44 U7(f63_out1(X61, X60), T29, T36, T37) -> f49_out1(.(T29, X61), .(T36, X60)) 14.36/4.44 f45_in(T16, T17, T18) -> U8(f49_in(T16, T17, T18), T16, T17, T18) 14.36/4.44 U8(f49_out1(T21, T22), T16, T17, T18) -> U9(f50_in(T21, T22), T16, T17, T18, T21, T22) 14.36/4.44 U9(f50_out1(X24, X25, T20), T16, T17, T18, T21, T22) -> f45_out1(T21, T22, X24, X25, T20) 14.36/4.44 f50_in(T21, T22) -> U10(f3_in(T21), T21, T22) 14.36/4.44 U10(f3_out1(T44), T21, T22) -> U11(f127_in(T22, T44), T21, T22, T44) 14.36/4.44 U11(f127_out1(X25, T20), T21, T22, T44) -> f50_out1(T44, X25, T20) 14.36/4.44 f127_in(T22, T44) -> U12(f3_in(T22), T22, T44) 14.36/4.44 U12(f3_out1(T45), T22, T44) -> U13(f129_in(T44, T45), T22, T44, T45) 14.36/4.44 U13(f129_out1(T20), T22, T44, T45) -> f127_out1(T45, T20) 14.36/4.44 f339_in(T78, T80, T79, T81) -> U14(f352_in(T78, T80), T78, T80, T79, T81) 14.36/4.44 U14(f352_out1, T78, T80, T79, T81) -> U15(f129_in(T79, .(T80, T81)), T78, T80, T79, T81) 14.36/4.44 U15(f129_out1(T83), T78, T80, T79, T81) -> f339_out1(T83) 14.36/4.44 f380_in(T119, T121, T120, T122) -> U16(f597_in(T119, T121), T119, T121, T120, T122) 14.36/4.44 U16(f597_out1, T119, T121, T120, T122) -> U17(f129_in(.(T119, T120), T122), T119, T121, T120, T122) 14.36/4.44 U17(f129_out1(T124), T119, T121, T120, T122) -> f380_out1(T124) 14.36/4.44 14.36/4.44 Q is empty. 14.36/4.44 We have to consider all minimal (P,Q,R)-chains. 14.36/4.44 ---------------------------------------- 14.36/4.44 14.36/4.44 (20) DependencyGraphProof (EQUIVALENT) 14.36/4.44 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes. 14.36/4.44 ---------------------------------------- 14.36/4.44 14.36/4.44 (21) 14.36/4.44 TRUE 14.36/4.44 14.36/4.44 ---------------------------------------- 14.36/4.44 14.36/4.44 (22) 14.36/4.44 Obligation: 14.36/4.44 Q DP problem: 14.36/4.44 The TRS P consists of the following rules: 14.36/4.44 14.36/4.44 F63_IN(.(T42, T43)) -> F63_IN(T43) 14.36/4.44 14.36/4.44 The TRS R consists of the following rules: 14.36/4.44 14.36/4.44 f3_in([]) -> f3_out1([]) 14.36/4.44 f3_in(.(T7, [])) -> f3_out1(.(T7, [])) 14.36/4.44 f3_in(.(T16, .(T17, T18))) -> U1(f45_in(T16, T17, T18), .(T16, .(T17, T18))) 14.36/4.44 U1(f45_out1(X22, X23, X24, X25, T20), .(T16, .(T17, T18))) -> f3_out1(T20) 14.36/4.44 f63_in([]) -> f63_out1([], []) 14.36/4.44 f63_in(.(T42, T43)) -> U2(f63_in(T43), .(T42, T43)) 14.36/4.44 U2(f63_out1(X79, X78), .(T42, T43)) -> f63_out1(.(T42, X78), X79) 14.36/4.44 f129_in([], T52) -> f129_out1(T52) 14.36/4.44 f129_in(T57, []) -> f129_out1(T57) 14.36/4.44 f129_in(.(T78, T79), .(T80, T81)) -> U3(f339_in(T78, T80, T79, T81), .(T78, T79), .(T80, T81)) 14.36/4.44 U3(f339_out1(T83), .(T78, T79), .(T80, T81)) -> f129_out1(.(T78, T83)) 14.36/4.44 f129_in(.(T119, T120), .(T121, T122)) -> U4(f380_in(T119, T121, T120, T122), .(T119, T120), .(T121, T122)) 14.36/4.44 U4(f380_out1(T124), .(T119, T120), .(T121, T122)) -> f129_out1(.(T121, T124)) 14.36/4.44 f352_in(s(T96), s(T97)) -> U5(f352_in(T96, T97), s(T96), s(T97)) 14.36/4.44 U5(f352_out1, s(T96), s(T97)) -> f352_out1 14.36/4.44 f352_in(0, s(T104)) -> f352_out1 14.36/4.44 f352_in(0, 0) -> f352_out1 14.36/4.44 f597_in(s(T137), s(T138)) -> U6(f597_in(T137, T138), s(T137), s(T138)) 14.36/4.44 U6(f597_out1, s(T137), s(T138)) -> f597_out1 14.36/4.44 f597_in(s(T143), 0) -> f597_out1 14.36/4.44 f49_in(T29, T36, T37) -> U7(f63_in(T37), T29, T36, T37) 14.36/4.44 U7(f63_out1(X61, X60), T29, T36, T37) -> f49_out1(.(T29, X61), .(T36, X60)) 14.36/4.44 f45_in(T16, T17, T18) -> U8(f49_in(T16, T17, T18), T16, T17, T18) 14.36/4.44 U8(f49_out1(T21, T22), T16, T17, T18) -> U9(f50_in(T21, T22), T16, T17, T18, T21, T22) 14.36/4.44 U9(f50_out1(X24, X25, T20), T16, T17, T18, T21, T22) -> f45_out1(T21, T22, X24, X25, T20) 14.36/4.44 f50_in(T21, T22) -> U10(f3_in(T21), T21, T22) 14.36/4.44 U10(f3_out1(T44), T21, T22) -> U11(f127_in(T22, T44), T21, T22, T44) 14.36/4.44 U11(f127_out1(X25, T20), T21, T22, T44) -> f50_out1(T44, X25, T20) 14.36/4.44 f127_in(T22, T44) -> U12(f3_in(T22), T22, T44) 14.36/4.44 U12(f3_out1(T45), T22, T44) -> U13(f129_in(T44, T45), T22, T44, T45) 14.36/4.44 U13(f129_out1(T20), T22, T44, T45) -> f127_out1(T45, T20) 14.36/4.44 f339_in(T78, T80, T79, T81) -> U14(f352_in(T78, T80), T78, T80, T79, T81) 14.36/4.44 U14(f352_out1, T78, T80, T79, T81) -> U15(f129_in(T79, .(T80, T81)), T78, T80, T79, T81) 14.36/4.44 U15(f129_out1(T83), T78, T80, T79, T81) -> f339_out1(T83) 14.36/4.44 f380_in(T119, T121, T120, T122) -> U16(f597_in(T119, T121), T119, T121, T120, T122) 14.36/4.44 U16(f597_out1, T119, T121, T120, T122) -> U17(f129_in(.(T119, T120), T122), T119, T121, T120, T122) 14.36/4.44 U17(f129_out1(T124), T119, T121, T120, T122) -> f380_out1(T124) 14.36/4.44 14.36/4.44 Q is empty. 14.36/4.44 We have to consider all minimal (P,Q,R)-chains. 14.36/4.44 ---------------------------------------- 14.36/4.44 14.36/4.44 (23) UsableRulesProof (EQUIVALENT) 14.36/4.44 We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. 14.36/4.44 ---------------------------------------- 14.36/4.44 14.36/4.44 (24) 14.36/4.44 Obligation: 14.36/4.44 Q DP problem: 14.36/4.44 The TRS P consists of the following rules: 14.36/4.44 14.36/4.44 F63_IN(.(T42, T43)) -> F63_IN(T43) 14.36/4.44 14.36/4.44 R is empty. 14.36/4.44 Q is empty. 14.36/4.44 We have to consider all minimal (P,Q,R)-chains. 14.36/4.44 ---------------------------------------- 14.36/4.44 14.36/4.44 (25) QDPSizeChangeProof (EQUIVALENT) 14.36/4.44 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. 14.36/4.44 14.36/4.44 From the DPs we obtained the following set of size-change graphs: 14.36/4.44 *F63_IN(.(T42, T43)) -> F63_IN(T43) 14.36/4.44 The graph contains the following edges 1 > 1 14.36/4.44 14.36/4.44 14.36/4.44 ---------------------------------------- 14.36/4.44 14.36/4.44 (26) 14.36/4.44 YES 14.36/4.44 14.36/4.44 ---------------------------------------- 14.36/4.44 14.36/4.44 (27) 14.36/4.44 Obligation: 14.36/4.44 Q DP problem: 14.36/4.44 The TRS P consists of the following rules: 14.36/4.44 14.36/4.44 F3_IN(.(T16, .(T17, T18))) -> F45_IN(T16, T17, T18) 14.36/4.44 F45_IN(T16, T17, T18) -> U8^1(f49_in(T16, T17, T18), T16, T17, T18) 14.36/4.44 U8^1(f49_out1(T21, T22), T16, T17, T18) -> F50_IN(T21, T22) 14.36/4.44 F50_IN(T21, T22) -> U10^1(f3_in(T21), T21, T22) 14.36/4.44 U10^1(f3_out1(T44), T21, T22) -> F127_IN(T22, T44) 14.36/4.44 F127_IN(T22, T44) -> F3_IN(T22) 14.36/4.44 F50_IN(T21, T22) -> F3_IN(T21) 14.36/4.44 14.36/4.44 The TRS R consists of the following rules: 14.36/4.44 14.36/4.44 f3_in([]) -> f3_out1([]) 14.36/4.44 f3_in(.(T7, [])) -> f3_out1(.(T7, [])) 14.36/4.44 f3_in(.(T16, .(T17, T18))) -> U1(f45_in(T16, T17, T18), .(T16, .(T17, T18))) 14.36/4.44 U1(f45_out1(X22, X23, X24, X25, T20), .(T16, .(T17, T18))) -> f3_out1(T20) 14.36/4.44 f63_in([]) -> f63_out1([], []) 14.36/4.44 f63_in(.(T42, T43)) -> U2(f63_in(T43), .(T42, T43)) 14.36/4.44 U2(f63_out1(X79, X78), .(T42, T43)) -> f63_out1(.(T42, X78), X79) 14.36/4.44 f129_in([], T52) -> f129_out1(T52) 14.36/4.44 f129_in(T57, []) -> f129_out1(T57) 14.36/4.44 f129_in(.(T78, T79), .(T80, T81)) -> U3(f339_in(T78, T80, T79, T81), .(T78, T79), .(T80, T81)) 14.36/4.44 U3(f339_out1(T83), .(T78, T79), .(T80, T81)) -> f129_out1(.(T78, T83)) 14.36/4.44 f129_in(.(T119, T120), .(T121, T122)) -> U4(f380_in(T119, T121, T120, T122), .(T119, T120), .(T121, T122)) 14.36/4.44 U4(f380_out1(T124), .(T119, T120), .(T121, T122)) -> f129_out1(.(T121, T124)) 14.36/4.44 f352_in(s(T96), s(T97)) -> U5(f352_in(T96, T97), s(T96), s(T97)) 14.36/4.44 U5(f352_out1, s(T96), s(T97)) -> f352_out1 14.36/4.44 f352_in(0, s(T104)) -> f352_out1 14.36/4.44 f352_in(0, 0) -> f352_out1 14.36/4.44 f597_in(s(T137), s(T138)) -> U6(f597_in(T137, T138), s(T137), s(T138)) 14.36/4.44 U6(f597_out1, s(T137), s(T138)) -> f597_out1 14.36/4.44 f597_in(s(T143), 0) -> f597_out1 14.36/4.44 f49_in(T29, T36, T37) -> U7(f63_in(T37), T29, T36, T37) 14.36/4.44 U7(f63_out1(X61, X60), T29, T36, T37) -> f49_out1(.(T29, X61), .(T36, X60)) 14.36/4.44 f45_in(T16, T17, T18) -> U8(f49_in(T16, T17, T18), T16, T17, T18) 14.36/4.44 U8(f49_out1(T21, T22), T16, T17, T18) -> U9(f50_in(T21, T22), T16, T17, T18, T21, T22) 14.36/4.44 U9(f50_out1(X24, X25, T20), T16, T17, T18, T21, T22) -> f45_out1(T21, T22, X24, X25, T20) 14.36/4.44 f50_in(T21, T22) -> U10(f3_in(T21), T21, T22) 14.36/4.44 U10(f3_out1(T44), T21, T22) -> U11(f127_in(T22, T44), T21, T22, T44) 14.36/4.44 U11(f127_out1(X25, T20), T21, T22, T44) -> f50_out1(T44, X25, T20) 14.36/4.44 f127_in(T22, T44) -> U12(f3_in(T22), T22, T44) 14.36/4.44 U12(f3_out1(T45), T22, T44) -> U13(f129_in(T44, T45), T22, T44, T45) 14.36/4.44 U13(f129_out1(T20), T22, T44, T45) -> f127_out1(T45, T20) 14.36/4.44 f339_in(T78, T80, T79, T81) -> U14(f352_in(T78, T80), T78, T80, T79, T81) 14.36/4.44 U14(f352_out1, T78, T80, T79, T81) -> U15(f129_in(T79, .(T80, T81)), T78, T80, T79, T81) 14.36/4.44 U15(f129_out1(T83), T78, T80, T79, T81) -> f339_out1(T83) 14.36/4.44 f380_in(T119, T121, T120, T122) -> U16(f597_in(T119, T121), T119, T121, T120, T122) 14.36/4.44 U16(f597_out1, T119, T121, T120, T122) -> U17(f129_in(.(T119, T120), T122), T119, T121, T120, T122) 14.36/4.44 U17(f129_out1(T124), T119, T121, T120, T122) -> f380_out1(T124) 14.36/4.44 14.36/4.44 Q is empty. 14.36/4.44 We have to consider all minimal (P,Q,R)-chains. 14.36/4.44 ---------------------------------------- 14.36/4.44 14.36/4.44 (28) QDPOrderProof (EQUIVALENT) 14.36/4.44 We use the reduction pair processor [LPAR04,JAR06]. 14.36/4.44 14.36/4.44 14.36/4.44 The following pairs can be oriented strictly and are deleted. 14.36/4.44 14.36/4.44 F3_IN(.(T16, .(T17, T18))) -> F45_IN(T16, T17, T18) 14.36/4.44 The remaining pairs can at least be oriented weakly. 14.36/4.44 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 14.36/4.44 14.36/4.44 POL( U8^1_4(x_1, ..., x_4) ) = x_1 + 2 14.36/4.44 POL( U10^1_3(x_1, ..., x_3) ) = x_3 14.36/4.44 POL( U8_4(x_1, ..., x_4) ) = 2x_2 + 2 14.36/4.44 POL( f49_in_3(x_1, ..., x_3) ) = 2x_3 14.36/4.44 POL( U7_4(x_1, ..., x_4) ) = 2x_1 14.36/4.44 POL( f63_in_1(x_1) ) = x_1 14.36/4.44 POL( f3_in_1(x_1) ) = 0 14.36/4.44 POL( [] ) = 0 14.36/4.44 POL( f3_out1_1(x_1) ) = max{0, 2x_1 - 2} 14.36/4.44 POL( ._2(x_1, x_2) ) = 2x_2 + 1 14.36/4.44 POL( U1_2(x_1, x_2) ) = 2x_1 + x_2 + 1 14.36/4.44 POL( f45_in_3(x_1, ..., x_3) ) = x_1 + 2x_2 + x_3 + 2 14.36/4.44 POL( f45_out1_5(x_1, ..., x_5) ) = 2x_1 + 2x_2 + 2x_3 + 2x_4 + 1 14.36/4.44 POL( f49_out1_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2} 14.36/4.44 POL( U9_6(x_1, ..., x_6) ) = 2x_5 + 2 14.36/4.44 POL( f50_in_2(x_1, x_2) ) = 0 14.36/4.44 POL( f50_out1_3(x_1, ..., x_3) ) = 2x_1 + x_3 + 2 14.36/4.44 POL( U10_3(x_1, ..., x_3) ) = 2 14.36/4.44 POL( U11_4(x_1, ..., x_4) ) = max{0, 2x_4 - 2} 14.36/4.44 POL( f127_in_2(x_1, x_2) ) = 2x_1 + 1 14.36/4.44 POL( f127_out1_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2 14.36/4.44 POL( U12_3(x_1, ..., x_3) ) = max{0, x_2 - 2} 14.36/4.44 POL( U13_4(x_1, ..., x_4) ) = max{0, x_3 + 2x_4 - 2} 14.36/4.44 POL( f129_in_2(x_1, x_2) ) = 0 14.36/4.44 POL( f63_out1_2(x_1, x_2) ) = x_1 + x_2 14.36/4.44 POL( U2_2(x_1, x_2) ) = 2x_1 + 1 14.36/4.44 POL( f129_out1_1(x_1) ) = 2 14.36/4.44 POL( U3_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2} 14.36/4.44 POL( f339_in_4(x_1, ..., x_4) ) = 2x_2 + 2 14.36/4.44 POL( U4_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + x_3 + 1 14.36/4.44 POL( f380_in_4(x_1, ..., x_4) ) = 2x_1 + 2x_2 + 2x_3 + 2 14.36/4.44 POL( f339_out1_1(x_1) ) = 1 14.36/4.44 POL( U14_5(x_1, ..., x_5) ) = max{0, x_4 + x_5 - 2} 14.36/4.44 POL( f352_in_2(x_1, x_2) ) = 0 14.36/4.44 POL( s_1(x_1) ) = 2x_1 14.36/4.44 POL( U5_3(x_1, ..., x_3) ) = max{0, x_2 - 2} 14.36/4.44 POL( 0 ) = 0 14.36/4.44 POL( f352_out1 ) = 0 14.36/4.44 POL( U15_5(x_1, ..., x_5) ) = max{0, x_2 - 2} 14.36/4.44 POL( f380_out1_1(x_1) ) = 0 14.36/4.44 POL( U16_5(x_1, ..., x_5) ) = max{0, x_4 + 2x_5 - 2} 14.36/4.44 POL( f597_in_2(x_1, x_2) ) = 2x_1 + 2x_2 14.36/4.44 POL( U6_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + 2x_3 - 2} 14.36/4.44 POL( f597_out1 ) = 2 14.36/4.44 POL( U17_5(x_1, ..., x_5) ) = max{0, 2x_3 - 2} 14.36/4.44 POL( F3_IN_1(x_1) ) = x_1 14.36/4.44 POL( F45_IN_3(x_1, ..., x_3) ) = 2x_3 + 2 14.36/4.44 POL( F50_IN_2(x_1, x_2) ) = x_1 + x_2 14.36/4.44 POL( F127_IN_2(x_1, x_2) ) = x_1 14.36/4.44 14.36/4.44 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 14.36/4.44 14.36/4.44 f49_in(T29, T36, T37) -> U7(f63_in(T37), T29, T36, T37) 14.36/4.44 f63_in([]) -> f63_out1([], []) 14.36/4.44 f63_in(.(T42, T43)) -> U2(f63_in(T43), .(T42, T43)) 14.36/4.44 U7(f63_out1(X61, X60), T29, T36, T37) -> f49_out1(.(T29, X61), .(T36, X60)) 14.36/4.44 U2(f63_out1(X79, X78), .(T42, T43)) -> f63_out1(.(T42, X78), X79) 14.36/4.44 14.36/4.44 14.36/4.44 ---------------------------------------- 14.36/4.44 14.36/4.44 (29) 14.36/4.44 Obligation: 14.36/4.44 Q DP problem: 14.36/4.44 The TRS P consists of the following rules: 14.36/4.44 14.36/4.44 F45_IN(T16, T17, T18) -> U8^1(f49_in(T16, T17, T18), T16, T17, T18) 14.36/4.44 U8^1(f49_out1(T21, T22), T16, T17, T18) -> F50_IN(T21, T22) 14.36/4.44 F50_IN(T21, T22) -> U10^1(f3_in(T21), T21, T22) 14.36/4.44 U10^1(f3_out1(T44), T21, T22) -> F127_IN(T22, T44) 14.36/4.44 F127_IN(T22, T44) -> F3_IN(T22) 14.36/4.44 F50_IN(T21, T22) -> F3_IN(T21) 14.36/4.44 14.36/4.44 The TRS R consists of the following rules: 14.36/4.44 14.36/4.44 f3_in([]) -> f3_out1([]) 14.36/4.44 f3_in(.(T7, [])) -> f3_out1(.(T7, [])) 14.36/4.44 f3_in(.(T16, .(T17, T18))) -> U1(f45_in(T16, T17, T18), .(T16, .(T17, T18))) 14.36/4.44 U1(f45_out1(X22, X23, X24, X25, T20), .(T16, .(T17, T18))) -> f3_out1(T20) 14.36/4.44 f63_in([]) -> f63_out1([], []) 14.36/4.44 f63_in(.(T42, T43)) -> U2(f63_in(T43), .(T42, T43)) 14.36/4.44 U2(f63_out1(X79, X78), .(T42, T43)) -> f63_out1(.(T42, X78), X79) 14.36/4.44 f129_in([], T52) -> f129_out1(T52) 14.36/4.44 f129_in(T57, []) -> f129_out1(T57) 14.36/4.44 f129_in(.(T78, T79), .(T80, T81)) -> U3(f339_in(T78, T80, T79, T81), .(T78, T79), .(T80, T81)) 14.36/4.44 U3(f339_out1(T83), .(T78, T79), .(T80, T81)) -> f129_out1(.(T78, T83)) 14.36/4.44 f129_in(.(T119, T120), .(T121, T122)) -> U4(f380_in(T119, T121, T120, T122), .(T119, T120), .(T121, T122)) 14.36/4.44 U4(f380_out1(T124), .(T119, T120), .(T121, T122)) -> f129_out1(.(T121, T124)) 14.36/4.44 f352_in(s(T96), s(T97)) -> U5(f352_in(T96, T97), s(T96), s(T97)) 14.36/4.44 U5(f352_out1, s(T96), s(T97)) -> f352_out1 14.36/4.44 f352_in(0, s(T104)) -> f352_out1 14.36/4.44 f352_in(0, 0) -> f352_out1 14.36/4.44 f597_in(s(T137), s(T138)) -> U6(f597_in(T137, T138), s(T137), s(T138)) 14.36/4.44 U6(f597_out1, s(T137), s(T138)) -> f597_out1 14.36/4.44 f597_in(s(T143), 0) -> f597_out1 14.36/4.44 f49_in(T29, T36, T37) -> U7(f63_in(T37), T29, T36, T37) 14.36/4.44 U7(f63_out1(X61, X60), T29, T36, T37) -> f49_out1(.(T29, X61), .(T36, X60)) 14.36/4.44 f45_in(T16, T17, T18) -> U8(f49_in(T16, T17, T18), T16, T17, T18) 14.36/4.44 U8(f49_out1(T21, T22), T16, T17, T18) -> U9(f50_in(T21, T22), T16, T17, T18, T21, T22) 14.36/4.44 U9(f50_out1(X24, X25, T20), T16, T17, T18, T21, T22) -> f45_out1(T21, T22, X24, X25, T20) 14.36/4.44 f50_in(T21, T22) -> U10(f3_in(T21), T21, T22) 14.36/4.44 U10(f3_out1(T44), T21, T22) -> U11(f127_in(T22, T44), T21, T22, T44) 14.36/4.44 U11(f127_out1(X25, T20), T21, T22, T44) -> f50_out1(T44, X25, T20) 14.36/4.44 f127_in(T22, T44) -> U12(f3_in(T22), T22, T44) 14.36/4.44 U12(f3_out1(T45), T22, T44) -> U13(f129_in(T44, T45), T22, T44, T45) 14.36/4.44 U13(f129_out1(T20), T22, T44, T45) -> f127_out1(T45, T20) 14.36/4.44 f339_in(T78, T80, T79, T81) -> U14(f352_in(T78, T80), T78, T80, T79, T81) 14.36/4.44 U14(f352_out1, T78, T80, T79, T81) -> U15(f129_in(T79, .(T80, T81)), T78, T80, T79, T81) 14.36/4.44 U15(f129_out1(T83), T78, T80, T79, T81) -> f339_out1(T83) 14.36/4.44 f380_in(T119, T121, T120, T122) -> U16(f597_in(T119, T121), T119, T121, T120, T122) 14.36/4.44 U16(f597_out1, T119, T121, T120, T122) -> U17(f129_in(.(T119, T120), T122), T119, T121, T120, T122) 14.36/4.44 U17(f129_out1(T124), T119, T121, T120, T122) -> f380_out1(T124) 14.36/4.44 14.36/4.44 Q is empty. 14.36/4.44 We have to consider all minimal (P,Q,R)-chains. 14.36/4.44 ---------------------------------------- 14.36/4.44 14.36/4.44 (30) DependencyGraphProof (EQUIVALENT) 14.36/4.44 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 6 less nodes. 14.36/4.44 ---------------------------------------- 14.36/4.44 14.36/4.44 (31) 14.36/4.44 TRUE 14.46/4.49 EOF