16.00/4.97 YES 16.00/5.00 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 16.00/5.00 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 16.00/5.00 16.00/5.00 16.00/5.00 Left Termination of the query pattern 16.00/5.00 16.00/5.00 mergesort(g,a) 16.00/5.00 16.00/5.00 w.r.t. the given Prolog program could successfully be proven: 16.00/5.00 16.00/5.00 (0) Prolog 16.00/5.00 (1) PrologToTRSTransformerProof [SOUND, 17 ms] 16.00/5.00 (2) QTRS 16.00/5.00 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 16.00/5.00 (4) QDP 16.00/5.00 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 16.00/5.00 (6) AND 16.00/5.00 (7) QDP 16.00/5.00 (8) UsableRulesProof [EQUIVALENT, 0 ms] 16.00/5.00 (9) QDP 16.00/5.00 (10) QDPSizeChangeProof [EQUIVALENT, 0 ms] 16.00/5.00 (11) YES 16.00/5.00 (12) QDP 16.00/5.00 (13) UsableRulesProof [EQUIVALENT, 0 ms] 16.00/5.00 (14) QDP 16.00/5.00 (15) QDPSizeChangeProof [EQUIVALENT, 0 ms] 16.00/5.00 (16) YES 16.00/5.00 (17) QDP 16.00/5.00 (18) QDPOrderProof [EQUIVALENT, 12 ms] 16.00/5.00 (19) QDP 16.00/5.00 (20) DependencyGraphProof [EQUIVALENT, 0 ms] 16.00/5.00 (21) TRUE 16.00/5.00 (22) QDP 16.00/5.00 (23) UsableRulesProof [EQUIVALENT, 0 ms] 16.00/5.00 (24) QDP 16.00/5.00 (25) QDPSizeChangeProof [EQUIVALENT, 0 ms] 16.00/5.00 (26) YES 16.00/5.00 (27) QDP 16.00/5.00 (28) QDPOrderProof [EQUIVALENT, 74 ms] 16.00/5.00 (29) QDP 16.00/5.00 (30) DependencyGraphProof [EQUIVALENT, 0 ms] 16.00/5.00 (31) TRUE 16.00/5.00 16.00/5.00 16.00/5.00 ---------------------------------------- 16.00/5.00 16.00/5.00 (0) 16.00/5.00 Obligation: 16.00/5.00 Clauses: 16.00/5.00 16.00/5.00 mergesort([], []). 16.00/5.00 mergesort(.(X, []), .(X, [])). 16.00/5.00 mergesort(.(X, .(Y, Xs)), Ys) :- ','(split(.(X, .(Y, Xs)), X1s, X2s), ','(mergesort(X1s, Y1s), ','(mergesort(X2s, Y2s), merge(Y1s, Y2s, Ys)))). 16.00/5.00 split([], [], []). 16.00/5.00 split(.(X, Xs), .(X, Ys), Zs) :- split(Xs, Zs, Ys). 16.00/5.00 merge([], Xs, Xs). 16.00/5.00 merge(Xs, [], Xs). 16.00/5.00 merge(.(X, Xs), .(Y, Ys), .(X, Zs)) :- ','(le(X, Y), merge(Xs, .(Y, Ys), Zs)). 16.00/5.00 merge(.(X, Xs), .(Y, Ys), .(Y, Zs)) :- ','(gt(X, Y), merge(.(X, Xs), Ys, Zs)). 16.00/5.00 gt(s(X), s(Y)) :- gt(X, Y). 16.00/5.00 gt(s(X), 0). 16.00/5.00 le(s(X), s(Y)) :- le(X, Y). 16.00/5.00 le(0, s(Y)). 16.00/5.00 le(0, 0). 16.00/5.00 16.00/5.00 16.00/5.00 Query: mergesort(g,a) 16.00/5.00 ---------------------------------------- 16.00/5.00 16.00/5.00 (1) PrologToTRSTransformerProof (SOUND) 16.00/5.00 Transformed Prolog program to TRS. 16.00/5.00 16.00/5.00 { 16.00/5.00 "root": 3, 16.00/5.00 "program": { 16.00/5.00 "directives": [], 16.00/5.00 "clauses": [ 16.00/5.00 [ 16.00/5.00 "(mergesort ([]) ([]))", 16.00/5.00 null 16.00/5.00 ], 16.00/5.00 [ 16.00/5.00 "(mergesort (. X ([])) (. X ([])))", 16.00/5.00 null 16.00/5.00 ], 16.00/5.00 [ 16.00/5.00 "(mergesort (. X (. Y Xs)) Ys)", 16.00/5.00 "(',' (split (. X (. Y Xs)) X1s X2s) (',' (mergesort X1s Y1s) (',' (mergesort X2s Y2s) (merge Y1s Y2s Ys))))" 16.00/5.00 ], 16.00/5.00 [ 16.00/5.00 "(split ([]) ([]) ([]))", 16.00/5.00 null 16.00/5.00 ], 16.00/5.00 [ 16.00/5.00 "(split (. X Xs) (. X Ys) Zs)", 16.00/5.00 "(split Xs Zs Ys)" 16.00/5.00 ], 16.00/5.00 [ 16.00/5.00 "(merge ([]) Xs Xs)", 16.00/5.00 null 16.00/5.00 ], 16.00/5.00 [ 16.00/5.00 "(merge Xs ([]) Xs)", 16.00/5.00 null 16.00/5.00 ], 16.00/5.00 [ 16.00/5.00 "(merge (. X Xs) (. Y Ys) (. X Zs))", 16.00/5.00 "(',' (le X Y) (merge Xs (. Y Ys) Zs))" 16.00/5.00 ], 16.00/5.00 [ 16.00/5.00 "(merge (. X Xs) (. Y Ys) (. Y Zs))", 16.00/5.00 "(',' (gt X Y) (merge (. X Xs) Ys Zs))" 16.00/5.00 ], 16.00/5.00 [ 16.00/5.00 "(gt (s X) (s Y))", 16.00/5.00 "(gt X Y)" 16.00/5.00 ], 16.00/5.00 [ 16.00/5.00 "(gt (s X) (0))", 16.00/5.00 null 16.00/5.00 ], 16.00/5.00 [ 16.00/5.00 "(le (s X) (s Y))", 16.00/5.00 "(le X Y)" 16.00/5.00 ], 16.00/5.00 [ 16.00/5.00 "(le (0) (s Y))", 16.00/5.00 null 16.00/5.00 ], 16.00/5.00 [ 16.00/5.00 "(le (0) (0))", 16.00/5.00 null 16.00/5.00 ] 16.00/5.00 ] 16.00/5.00 }, 16.00/5.00 "graph": { 16.00/5.00 "nodes": { 16.00/5.00 "47": { 16.00/5.00 "goal": [{ 16.00/5.00 "clause": 1, 16.00/5.00 "scope": 1, 16.00/5.00 "term": "(mergesort T1 T2)" 16.00/5.00 }], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": ["T1"], 16.00/5.00 "free": [], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "48": { 16.00/5.00 "goal": [{ 16.00/5.00 "clause": 2, 16.00/5.00 "scope": 1, 16.00/5.00 "term": "(mergesort T1 T2)" 16.00/5.00 }], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": ["T1"], 16.00/5.00 "free": [], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "49": { 16.00/5.00 "goal": [{ 16.00/5.00 "clause": -1, 16.00/5.00 "scope": -1, 16.00/5.00 "term": "(true)" 16.00/5.00 }], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": [], 16.00/5.00 "free": [], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "type": "Nodes", 16.00/5.00 "670": { 16.00/5.00 "goal": [], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": [], 16.00/5.00 "free": [], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "671": { 16.00/5.00 "goal": [], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": [], 16.00/5.00 "free": [], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "672": { 16.00/5.00 "goal": [{ 16.00/5.00 "clause": 7, 16.00/5.00 "scope": 5, 16.00/5.00 "term": "(merge T44 T45 T20)" 16.00/5.00 }], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": [ 16.00/5.00 "T44", 16.00/5.00 "T45" 16.00/5.00 ], 16.00/5.00 "free": [], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "673": { 16.00/5.00 "goal": [{ 16.00/5.00 "clause": 8, 16.00/5.00 "scope": 5, 16.00/5.00 "term": "(merge T44 T45 T20)" 16.00/5.00 }], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": [ 16.00/5.00 "T44", 16.00/5.00 "T45" 16.00/5.00 ], 16.00/5.00 "free": [], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "750": { 16.00/5.00 "goal": [{ 16.00/5.00 "clause": -1, 16.00/5.00 "scope": -1, 16.00/5.00 "term": "(',' (gt T119 T121) (merge (. T119 T120) T122 T124))" 16.00/5.00 }], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": [ 16.00/5.00 "T119", 16.00/5.00 "T120", 16.00/5.00 "T121", 16.00/5.00 "T122" 16.00/5.00 ], 16.00/5.00 "free": [], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "751": { 16.00/5.00 "goal": [], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": [], 16.00/5.00 "free": [], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "752": { 16.00/5.00 "goal": [{ 16.00/5.00 "clause": -1, 16.00/5.00 "scope": -1, 16.00/5.00 "term": "(gt T119 T121)" 16.00/5.00 }], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": [ 16.00/5.00 "T119", 16.00/5.00 "T121" 16.00/5.00 ], 16.00/5.00 "free": [], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "753": { 16.00/5.00 "goal": [{ 16.00/5.00 "clause": -1, 16.00/5.00 "scope": -1, 16.00/5.00 "term": "(merge (. T119 T120) T122 T124)" 16.00/5.00 }], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": [ 16.00/5.00 "T119", 16.00/5.00 "T120", 16.00/5.00 "T122" 16.00/5.00 ], 16.00/5.00 "free": [], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "754": { 16.00/5.00 "goal": [ 16.00/5.00 { 16.00/5.00 "clause": 9, 16.00/5.00 "scope": 7, 16.00/5.00 "term": "(gt T119 T121)" 16.00/5.00 }, 16.00/5.00 { 16.00/5.00 "clause": 10, 16.00/5.00 "scope": 7, 16.00/5.00 "term": "(gt T119 T121)" 16.00/5.00 } 16.00/5.00 ], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": [ 16.00/5.00 "T119", 16.00/5.00 "T121" 16.00/5.00 ], 16.00/5.00 "free": [], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "755": { 16.00/5.00 "goal": [{ 16.00/5.00 "clause": 9, 16.00/5.00 "scope": 7, 16.00/5.00 "term": "(gt T119 T121)" 16.00/5.00 }], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": [ 16.00/5.00 "T119", 16.00/5.00 "T121" 16.00/5.00 ], 16.00/5.00 "free": [], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "756": { 16.00/5.00 "goal": [{ 16.00/5.00 "clause": 10, 16.00/5.00 "scope": 7, 16.00/5.00 "term": "(gt T119 T121)" 16.00/5.00 }], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": [ 16.00/5.00 "T119", 16.00/5.00 "T121" 16.00/5.00 ], 16.00/5.00 "free": [], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "757": { 16.00/5.00 "goal": [{ 16.00/5.00 "clause": -1, 16.00/5.00 "scope": -1, 16.00/5.00 "term": "(gt T137 T138)" 16.00/5.00 }], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": [ 16.00/5.00 "T137", 16.00/5.00 "T138" 16.00/5.00 ], 16.00/5.00 "free": [], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "439": { 16.00/5.00 "goal": [{ 16.00/5.00 "clause": -1, 16.00/5.00 "scope": -1, 16.00/5.00 "term": "(split T37 X61 X60)" 16.00/5.00 }], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": ["T37"], 16.00/5.00 "free": [ 16.00/5.00 "X60", 16.00/5.00 "X61" 16.00/5.00 ], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "758": { 16.00/5.00 "goal": [], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": [], 16.00/5.00 "free": [], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "50": { 16.00/5.00 "goal": [], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": [], 16.00/5.00 "free": [], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "759": { 16.00/5.00 "goal": [{ 16.00/5.00 "clause": -1, 16.00/5.00 "scope": -1, 16.00/5.00 "term": "(true)" 16.00/5.00 }], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": [], 16.00/5.00 "free": [], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "51": { 16.00/5.00 "goal": [], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": [], 16.00/5.00 "free": [], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "680": { 16.00/5.00 "goal": [{ 16.00/5.00 "clause": -1, 16.00/5.00 "scope": -1, 16.00/5.00 "term": "(',' (le T78 T80) (merge T79 (. 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"kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": ["T1"], 16.00/5.00 "free": [], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "38": { 16.00/5.00 "goal": [{ 16.00/5.00 "clause": 0, 16.00/5.00 "scope": 1, 16.00/5.00 "term": "(mergesort T1 T2)" 16.00/5.00 }], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": ["T1"], 16.00/5.00 "free": [], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "39": { 16.00/5.00 "goal": [ 16.00/5.00 { 16.00/5.00 "clause": 1, 16.00/5.00 "scope": 1, 16.00/5.00 "term": "(mergesort T1 T2)" 16.00/5.00 }, 16.00/5.00 { 16.00/5.00 "clause": 2, 16.00/5.00 "scope": 1, 16.00/5.00 "term": "(mergesort T1 T2)" 16.00/5.00 } 16.00/5.00 ], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": ["T1"], 16.00/5.00 "free": [], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "380": { 16.00/5.00 "goal": [{ 16.00/5.00 "clause": -1, 16.00/5.00 "scope": -1, 16.00/5.00 "term": "(',' (mergesort T21 X24) (',' (mergesort T22 X25) (merge X24 X25 T20)))" 16.00/5.00 }], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": [ 16.00/5.00 "T21", 16.00/5.00 "T22" 16.00/5.00 ], 16.00/5.00 "free": [ 16.00/5.00 "X24", 16.00/5.00 "X25" 16.00/5.00 ], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "383": { 16.00/5.00 "goal": [ 16.00/5.00 { 16.00/5.00 "clause": 3, 16.00/5.00 "scope": 2, 16.00/5.00 "term": "(split (. T16 (. T17 T18)) X22 X23)" 16.00/5.00 }, 16.00/5.00 { 16.00/5.00 "clause": 4, 16.00/5.00 "scope": 2, 16.00/5.00 "term": "(split (. T16 (. T17 T18)) X22 X23)" 16.00/5.00 } 16.00/5.00 ], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": [ 16.00/5.00 "T16", 16.00/5.00 "T17", 16.00/5.00 "T18" 16.00/5.00 ], 16.00/5.00 "free": [ 16.00/5.00 "X22", 16.00/5.00 "X23" 16.00/5.00 ], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "660": { 16.00/5.00 "goal": [{ 16.00/5.00 "clause": -1, 16.00/5.00 "scope": -1, 16.00/5.00 "term": "(merge T44 T45 T20)" 16.00/5.00 }], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": [ 16.00/5.00 "T44", 16.00/5.00 "T45" 16.00/5.00 ], 16.00/5.00 "free": [], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "386": { 16.00/5.00 "goal": [{ 16.00/5.00 "clause": 4, 16.00/5.00 "scope": 2, 16.00/5.00 "term": "(split (. T16 (. T17 T18)) X22 X23)" 16.00/5.00 }], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": [ 16.00/5.00 "T16", 16.00/5.00 "T17", 16.00/5.00 "T18" 16.00/5.00 ], 16.00/5.00 "free": [ 16.00/5.00 "X22", 16.00/5.00 "X23" 16.00/5.00 ], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "661": { 16.00/5.00 "goal": [ 16.00/5.00 { 16.00/5.00 "clause": 5, 16.00/5.00 "scope": 5, 16.00/5.00 "term": "(merge T44 T45 T20)" 16.00/5.00 }, 16.00/5.00 { 16.00/5.00 "clause": 6, 16.00/5.00 "scope": 5, 16.00/5.00 "term": "(merge T44 T45 T20)" 16.00/5.00 }, 16.00/5.00 { 16.00/5.00 "clause": 7, 16.00/5.00 "scope": 5, 16.00/5.00 "term": "(merge T44 T45 T20)" 16.00/5.00 }, 16.00/5.00 { 16.00/5.00 "clause": 8, 16.00/5.00 "scope": 5, 16.00/5.00 "term": "(merge T44 T45 T20)" 16.00/5.00 } 16.00/5.00 ], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": [ 16.00/5.00 "T44", 16.00/5.00 "T45" 16.00/5.00 ], 16.00/5.00 "free": [], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "662": { 16.00/5.00 "goal": [{ 16.00/5.00 "clause": 5, 16.00/5.00 "scope": 5, 16.00/5.00 "term": "(merge T44 T45 T20)" 16.00/5.00 }], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": [ 16.00/5.00 "T44", 16.00/5.00 "T45" 16.00/5.00 ], 16.00/5.00 "free": [], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "663": { 16.00/5.00 "goal": [ 16.00/5.00 { 16.00/5.00 "clause": 6, 16.00/5.00 "scope": 5, 16.00/5.00 "term": "(merge T44 T45 T20)" 16.00/5.00 }, 16.00/5.00 { 16.00/5.00 "clause": 7, 16.00/5.00 "scope": 5, 16.00/5.00 "term": "(merge T44 T45 T20)" 16.00/5.00 }, 16.00/5.00 { 16.00/5.00 "clause": 8, 16.00/5.00 "scope": 5, 16.00/5.00 "term": "(merge T44 T45 T20)" 16.00/5.00 } 16.00/5.00 ], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": [ 16.00/5.00 "T44", 16.00/5.00 "T45" 16.00/5.00 ], 16.00/5.00 "free": [], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "664": { 16.00/5.00 "goal": [{ 16.00/5.00 "clause": -1, 16.00/5.00 "scope": -1, 16.00/5.00 "term": "(true)" 16.00/5.00 }], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": [], 16.00/5.00 "free": [], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "741": { 16.00/5.00 "goal": [], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": [], 16.00/5.00 "free": [], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "665": { 16.00/5.00 "goal": [], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": [], 16.00/5.00 "free": [], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "742": { 16.00/5.00 "goal": [{ 16.00/5.00 "clause": -1, 16.00/5.00 "scope": -1, 16.00/5.00 "term": "(true)" 16.00/5.00 }], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": [], 16.00/5.00 "free": [], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "666": { 16.00/5.00 "goal": [], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": [], 16.00/5.00 "free": [], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "743": { 16.00/5.00 "goal": [], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": [], 16.00/5.00 "free": [], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "667": { 16.00/5.00 "goal": [{ 16.00/5.00 "clause": 6, 16.00/5.00 "scope": 5, 16.00/5.00 "term": "(merge T44 T45 T20)" 16.00/5.00 }], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": [ 16.00/5.00 "T44", 16.00/5.00 "T45" 16.00/5.00 ], 16.00/5.00 "free": [], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "744": { 16.00/5.00 "goal": [], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": [], 16.00/5.00 "free": [], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "668": { 16.00/5.00 "goal": [ 16.00/5.00 { 16.00/5.00 "clause": 7, 16.00/5.00 "scope": 5, 16.00/5.00 "term": "(merge T44 T45 T20)" 16.00/5.00 }, 16.00/5.00 { 16.00/5.00 "clause": 8, 16.00/5.00 "scope": 5, 16.00/5.00 "term": "(merge T44 T45 T20)" 16.00/5.00 } 16.00/5.00 ], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": [ 16.00/5.00 "T44", 16.00/5.00 "T45" 16.00/5.00 ], 16.00/5.00 "free": [], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "669": { 16.00/5.00 "goal": [{ 16.00/5.00 "clause": -1, 16.00/5.00 "scope": -1, 16.00/5.00 "term": "(true)" 16.00/5.00 }], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": [], 16.00/5.00 "free": [], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "41": { 16.00/5.00 "goal": [{ 16.00/5.00 "clause": -1, 16.00/5.00 "scope": -1, 16.00/5.00 "term": "(true)" 16.00/5.00 }], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": [], 16.00/5.00 "free": [], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "42": { 16.00/5.00 "goal": [], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": [], 16.00/5.00 "free": [], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "43": { 16.00/5.00 "goal": [], 16.00/5.00 "kb": { 16.00/5.00 "nonunifying": [], 16.00/5.00 "intvars": {}, 16.00/5.00 "arithmetic": { 16.00/5.00 "type": "PlainIntegerRelationState", 16.00/5.00 "relations": [] 16.00/5.00 }, 16.00/5.00 "ground": [], 16.00/5.00 "free": [], 16.00/5.00 "exprvars": [] 16.00/5.00 } 16.00/5.00 } 16.00/5.00 }, 16.00/5.00 "edges": [ 16.00/5.00 { 16.00/5.00 "from": 3, 16.00/5.00 "to": 36, 16.00/5.00 "label": "CASE" 16.00/5.00 }, 16.00/5.00 { 16.00/5.00 "from": 36, 16.00/5.00 "to": 38, 16.00/5.00 "label": "PARALLEL" 16.00/5.00 }, 16.00/5.00 { 16.00/5.00 "from": 36, 16.00/5.00 "to": 39, 16.00/5.00 "label": "PARALLEL" 16.00/5.00 }, 16.00/5.00 { 16.00/5.00 "from": 38, 16.00/5.00 "to": 41, 16.00/5.00 "label": "EVAL with clause\nmergesort([], []).\nand substitutionT1 -> [],\nT2 -> []" 16.00/5.00 }, 16.00/5.00 { 16.00/5.00 "from": 38, 16.00/5.00 "to": 42, 16.00/5.00 "label": "EVAL-BACKTRACK" 16.00/5.00 }, 16.00/5.00 { 16.00/5.00 "from": 39, 16.00/5.00 "to": 47, 16.00/5.00 "label": "PARALLEL" 16.00/5.00 }, 16.00/5.00 { 16.00/5.00 "from": 39, 16.00/5.00 "to": 48, 16.00/5.00 "label": "PARALLEL" 16.00/5.00 }, 16.00/5.00 { 16.00/5.00 "from": 41, 16.00/5.00 "to": 43, 16.00/5.00 "label": "SUCCESS" 16.00/5.00 }, 16.00/5.00 { 16.00/5.00 "from": 47, 16.00/5.00 "to": 49, 16.00/5.00 "label": "EVAL with clause\nmergesort(.(X5, []), .(X5, [])).\nand substitutionX5 -> T7,\nT1 -> .(T7, []),\nT2 -> .(T7, [])" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 47, 16.00/5.01 "to": 50, 16.00/5.01 "label": "EVAL-BACKTRACK" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 48, 16.00/5.01 "to": 369, 16.00/5.01 "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" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 48, 16.00/5.01 "to": 370, 16.00/5.01 "label": "EVAL-BACKTRACK" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 49, 16.00/5.01 "to": 51, 16.00/5.01 "label": "SUCCESS" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 369, 16.00/5.01 "to": 376, 16.00/5.01 "label": "SPLIT 1" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 369, 16.00/5.01 "to": 380, 16.00/5.01 "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" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 376, 16.00/5.01 "to": 383, 16.00/5.01 "label": "CASE" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 380, 16.00/5.01 "to": 647, 16.00/5.01 "label": "SPLIT 1" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 380, 16.00/5.01 "to": 648, 16.00/5.01 "label": "SPLIT 2\nnew knowledge:\nT21 is ground\nT44 is ground\nreplacements:X24 -> T44" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 383, 16.00/5.01 "to": 386, 16.00/5.01 "label": "BACKTRACK\nfor clause: split([], [], [])because of non-unification" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 386, 16.00/5.01 "to": 407, 16.00/5.01 "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" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 407, 16.00/5.01 "to": 413, 16.00/5.01 "label": "CASE" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 413, 16.00/5.01 "to": 415, 16.00/5.01 "label": "BACKTRACK\nfor clause: split([], [], [])because of non-unification" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 415, 16.00/5.01 "to": 439, 16.00/5.01 "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" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 439, 16.00/5.01 "to": 441, 16.00/5.01 "label": "CASE" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 441, 16.00/5.01 "to": 443, 16.00/5.01 "label": "PARALLEL" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 441, 16.00/5.01 "to": 444, 16.00/5.01 "label": "PARALLEL" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 443, 16.00/5.01 "to": 445, 16.00/5.01 "label": "EVAL with clause\nsplit([], [], []).\nand substitutionT37 -> [],\nX61 -> [],\nX60 -> []" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 443, 16.00/5.01 "to": 446, 16.00/5.01 "label": "EVAL-BACKTRACK" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 444, 16.00/5.01 "to": 448, 16.00/5.01 "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" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 444, 16.00/5.01 "to": 449, 16.00/5.01 "label": "EVAL-BACKTRACK" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 445, 16.00/5.01 "to": 447, 16.00/5.01 "label": "SUCCESS" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 448, 16.00/5.01 "to": 439, 16.00/5.01 "label": "INSTANCE with matching:\nT37 -> T43\nX61 -> X79\nX60 -> X78" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 647, 16.00/5.01 "to": 3, 16.00/5.01 "label": "INSTANCE with matching:\nT1 -> T21\nT2 -> X24" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 648, 16.00/5.01 "to": 659, 16.00/5.01 "label": "SPLIT 1" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 648, 16.00/5.01 "to": 660, 16.00/5.01 "label": "SPLIT 2\nnew knowledge:\nT22 is ground\nT45 is ground\nreplacements:X25 -> T45" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 659, 16.00/5.01 "to": 3, 16.00/5.01 "label": "INSTANCE with matching:\nT1 -> T22\nT2 -> X25" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 660, 16.00/5.01 "to": 661, 16.00/5.01 "label": "CASE" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 661, 16.00/5.01 "to": 662, 16.00/5.01 "label": "PARALLEL" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 661, 16.00/5.01 "to": 663, 16.00/5.01 "label": "PARALLEL" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 662, 16.00/5.01 "to": 664, 16.00/5.01 "label": "EVAL with clause\nmerge([], X86, X86).\nand substitutionT44 -> [],\nT45 -> T52,\nX86 -> T52,\nT20 -> T52" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 662, 16.00/5.01 "to": 665, 16.00/5.01 "label": "EVAL-BACKTRACK" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 663, 16.00/5.01 "to": 667, 16.00/5.01 "label": "PARALLEL" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 663, 16.00/5.01 "to": 668, 16.00/5.01 "label": "PARALLEL" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 664, 16.00/5.01 "to": 666, 16.00/5.01 "label": "SUCCESS" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 667, 16.00/5.01 "to": 669, 16.00/5.01 "label": "EVAL with clause\nmerge(X91, [], X91).\nand substitutionT44 -> T57,\nX91 -> T57,\nT45 -> [],\nT20 -> T57" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 667, 16.00/5.01 "to": 670, 16.00/5.01 "label": "EVAL-BACKTRACK" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 668, 16.00/5.01 "to": 672, 16.00/5.01 "label": "PARALLEL" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 668, 16.00/5.01 "to": 673, 16.00/5.01 "label": "PARALLEL" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 669, 16.00/5.01 "to": 671, 16.00/5.01 "label": "SUCCESS" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 672, 16.00/5.01 "to": 680, 16.00/5.01 "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" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 672, 16.00/5.01 "to": 681, 16.00/5.01 "label": "EVAL-BACKTRACK" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 673, 16.00/5.01 "to": 750, 16.00/5.01 "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" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 673, 16.00/5.01 "to": 751, 16.00/5.01 "label": "EVAL-BACKTRACK" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 680, 16.00/5.01 "to": 682, 16.00/5.01 "label": "SPLIT 1" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 680, 16.00/5.01 "to": 683, 16.00/5.01 "label": "SPLIT 2\nnew knowledge:\nT78 is ground\nT80 is ground" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 682, 16.00/5.01 "to": 684, 16.00/5.01 "label": "CASE" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 683, 16.00/5.01 "to": 660, 16.00/5.01 "label": "INSTANCE with matching:\nT44 -> T79\nT45 -> .(T80, T81)\nT20 -> T83" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 684, 16.00/5.01 "to": 685, 16.00/5.01 "label": "PARALLEL" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 684, 16.00/5.01 "to": 686, 16.00/5.01 "label": "PARALLEL" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 685, 16.00/5.01 "to": 687, 16.00/5.01 "label": "EVAL with clause\nle(s(X129), s(X130)) :- le(X129, X130).\nand substitutionX129 -> T96,\nT78 -> s(T96),\nX130 -> T97,\nT80 -> s(T97)" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 685, 16.00/5.01 "to": 688, 16.00/5.01 "label": "EVAL-BACKTRACK" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 686, 16.00/5.01 "to": 735, 16.00/5.01 "label": "PARALLEL" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 686, 16.00/5.01 "to": 736, 16.00/5.01 "label": "PARALLEL" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 687, 16.00/5.01 "to": 682, 16.00/5.01 "label": "INSTANCE with matching:\nT78 -> T96\nT80 -> T97" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 735, 16.00/5.01 "to": 737, 16.00/5.01 "label": "EVAL with clause\nle(0, s(X137)).\nand substitutionT78 -> 0,\nX137 -> T104,\nT80 -> s(T104)" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 735, 16.00/5.01 "to": 739, 16.00/5.01 "label": "EVAL-BACKTRACK" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 736, 16.00/5.01 "to": 742, 16.00/5.01 "label": "EVAL with clause\nle(0, 0).\nand substitutionT78 -> 0,\nT80 -> 0" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 736, 16.00/5.01 "to": 743, 16.00/5.01 "label": "EVAL-BACKTRACK" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 737, 16.00/5.01 "to": 741, 16.00/5.01 "label": "SUCCESS" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 742, 16.00/5.01 "to": 744, 16.00/5.01 "label": "SUCCESS" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 750, 16.00/5.01 "to": 752, 16.00/5.01 "label": "SPLIT 1" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 750, 16.00/5.01 "to": 753, 16.00/5.01 "label": "SPLIT 2\nnew knowledge:\nT119 is ground\nT121 is ground" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 752, 16.00/5.01 "to": 754, 16.00/5.01 "label": "CASE" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 753, 16.00/5.01 "to": 660, 16.00/5.01 "label": "INSTANCE with matching:\nT44 -> .(T119, T120)\nT45 -> T122\nT20 -> T124" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 754, 16.00/5.01 "to": 755, 16.00/5.01 "label": "PARALLEL" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 754, 16.00/5.01 "to": 756, 16.00/5.01 "label": "PARALLEL" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 755, 16.00/5.01 "to": 757, 16.00/5.01 "label": "EVAL with clause\ngt(s(X167), s(X168)) :- gt(X167, X168).\nand substitutionX167 -> T137,\nT119 -> s(T137),\nX168 -> T138,\nT121 -> s(T138)" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 755, 16.00/5.01 "to": 758, 16.00/5.01 "label": "EVAL-BACKTRACK" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 756, 16.00/5.01 "to": 759, 16.00/5.01 "label": "EVAL with clause\ngt(s(X173), 0).\nand substitutionX173 -> T143,\nT119 -> s(T143),\nT121 -> 0" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 756, 16.00/5.01 "to": 760, 16.00/5.01 "label": "EVAL-BACKTRACK" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 757, 16.00/5.01 "to": 752, 16.00/5.01 "label": "INSTANCE with matching:\nT119 -> T137\nT121 -> T138" 16.00/5.01 }, 16.00/5.01 { 16.00/5.01 "from": 759, 16.00/5.01 "to": 761, 16.00/5.01 "label": "SUCCESS" 16.00/5.01 } 16.00/5.01 ], 16.00/5.01 "type": "Graph" 16.00/5.01 } 16.00/5.01 } 16.00/5.01 16.00/5.01 ---------------------------------------- 16.00/5.01 16.00/5.01 (2) 16.00/5.01 Obligation: 16.00/5.01 Q restricted rewrite system: 16.00/5.01 The TRS R consists of the following rules: 16.00/5.01 16.00/5.01 f3_in([]) -> f3_out1([]) 16.00/5.01 f3_in(.(T7, [])) -> f3_out1(.(T7, [])) 16.00/5.01 f3_in(.(T16, .(T17, T18))) -> U1(f369_in(T16, T17, T18), .(T16, .(T17, T18))) 16.00/5.01 U1(f369_out1(X22, X23, X24, X25, T20), .(T16, .(T17, T18))) -> f3_out1(T20) 16.00/5.01 f439_in([]) -> f439_out1([], []) 16.00/5.01 f439_in(.(T42, T43)) -> U2(f439_in(T43), .(T42, T43)) 16.00/5.01 U2(f439_out1(X79, X78), .(T42, T43)) -> f439_out1(.(T42, X78), X79) 16.00/5.01 f660_in([], T52) -> f660_out1(T52) 16.00/5.01 f660_in(T57, []) -> f660_out1(T57) 16.00/5.01 f660_in(.(T78, T79), .(T80, T81)) -> U3(f680_in(T78, T80, T79, T81), .(T78, T79), .(T80, T81)) 16.00/5.01 U3(f680_out1(T83), .(T78, T79), .(T80, T81)) -> f660_out1(.(T78, T83)) 16.00/5.01 f660_in(.(T119, T120), .(T121, T122)) -> U4(f750_in(T119, T121, T120, T122), .(T119, T120), .(T121, T122)) 16.00/5.01 U4(f750_out1(T124), .(T119, T120), .(T121, T122)) -> f660_out1(.(T121, T124)) 16.00/5.01 f682_in(s(T96), s(T97)) -> U5(f682_in(T96, T97), s(T96), s(T97)) 16.00/5.01 U5(f682_out1, s(T96), s(T97)) -> f682_out1 16.00/5.01 f682_in(0, s(T104)) -> f682_out1 16.00/5.01 f682_in(0, 0) -> f682_out1 16.00/5.01 f752_in(s(T137), s(T138)) -> U6(f752_in(T137, T138), s(T137), s(T138)) 16.00/5.01 U6(f752_out1, s(T137), s(T138)) -> f752_out1 16.00/5.01 f752_in(s(T143), 0) -> f752_out1 16.00/5.01 f376_in(T29, T36, T37) -> U7(f439_in(T37), T29, T36, T37) 16.00/5.01 U7(f439_out1(X61, X60), T29, T36, T37) -> f376_out1(.(T29, X61), .(T36, X60)) 16.00/5.01 f369_in(T16, T17, T18) -> U8(f376_in(T16, T17, T18), T16, T17, T18) 16.00/5.01 U8(f376_out1(T21, T22), T16, T17, T18) -> U9(f380_in(T21, T22), T16, T17, T18, T21, T22) 16.00/5.01 U9(f380_out1(X24, X25, T20), T16, T17, T18, T21, T22) -> f369_out1(T21, T22, X24, X25, T20) 16.00/5.01 f380_in(T21, T22) -> U10(f3_in(T21), T21, T22) 16.00/5.01 U10(f3_out1(T44), T21, T22) -> U11(f648_in(T22, T44), T21, T22, T44) 16.00/5.01 U11(f648_out1(X25, T20), T21, T22, T44) -> f380_out1(T44, X25, T20) 16.00/5.01 f648_in(T22, T44) -> U12(f3_in(T22), T22, T44) 16.00/5.01 U12(f3_out1(T45), T22, T44) -> U13(f660_in(T44, T45), T22, T44, T45) 16.00/5.01 U13(f660_out1(T20), T22, T44, T45) -> f648_out1(T45, T20) 16.00/5.01 f680_in(T78, T80, T79, T81) -> U14(f682_in(T78, T80), T78, T80, T79, T81) 16.00/5.01 U14(f682_out1, T78, T80, T79, T81) -> U15(f660_in(T79, .(T80, T81)), T78, T80, T79, T81) 16.00/5.01 U15(f660_out1(T83), T78, T80, T79, T81) -> f680_out1(T83) 16.00/5.01 f750_in(T119, T121, T120, T122) -> U16(f752_in(T119, T121), T119, T121, T120, T122) 16.00/5.01 U16(f752_out1, T119, T121, T120, T122) -> U17(f660_in(.(T119, T120), T122), T119, T121, T120, T122) 16.00/5.01 U17(f660_out1(T124), T119, T121, T120, T122) -> f750_out1(T124) 16.00/5.01 16.00/5.01 Q is empty. 16.00/5.01 16.00/5.01 ---------------------------------------- 16.00/5.01 16.00/5.01 (3) DependencyPairsProof (EQUIVALENT) 16.00/5.01 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 16.00/5.01 ---------------------------------------- 16.00/5.01 16.00/5.01 (4) 16.00/5.01 Obligation: 16.00/5.01 Q DP problem: 16.00/5.01 The TRS P consists of the following rules: 16.00/5.01 16.00/5.01 F3_IN(.(T16, .(T17, T18))) -> U1^1(f369_in(T16, T17, T18), .(T16, .(T17, T18))) 16.00/5.01 F3_IN(.(T16, .(T17, T18))) -> F369_IN(T16, T17, T18) 16.00/5.01 F439_IN(.(T42, T43)) -> U2^1(f439_in(T43), .(T42, T43)) 16.00/5.01 F439_IN(.(T42, T43)) -> F439_IN(T43) 16.00/5.01 F660_IN(.(T78, T79), .(T80, T81)) -> U3^1(f680_in(T78, T80, T79, T81), .(T78, T79), .(T80, T81)) 16.00/5.01 F660_IN(.(T78, T79), .(T80, T81)) -> F680_IN(T78, T80, T79, T81) 16.00/5.01 F660_IN(.(T119, T120), .(T121, T122)) -> U4^1(f750_in(T119, T121, T120, T122), .(T119, T120), .(T121, T122)) 16.00/5.01 F660_IN(.(T119, T120), .(T121, T122)) -> F750_IN(T119, T121, T120, T122) 16.00/5.01 F682_IN(s(T96), s(T97)) -> U5^1(f682_in(T96, T97), s(T96), s(T97)) 16.00/5.01 F682_IN(s(T96), s(T97)) -> F682_IN(T96, T97) 16.00/5.01 F752_IN(s(T137), s(T138)) -> U6^1(f752_in(T137, T138), s(T137), s(T138)) 16.00/5.01 F752_IN(s(T137), s(T138)) -> F752_IN(T137, T138) 16.00/5.01 F376_IN(T29, T36, T37) -> U7^1(f439_in(T37), T29, T36, T37) 16.00/5.01 F376_IN(T29, T36, T37) -> F439_IN(T37) 16.00/5.01 F369_IN(T16, T17, T18) -> U8^1(f376_in(T16, T17, T18), T16, T17, T18) 16.00/5.01 F369_IN(T16, T17, T18) -> F376_IN(T16, T17, T18) 16.00/5.01 U8^1(f376_out1(T21, T22), T16, T17, T18) -> U9^1(f380_in(T21, T22), T16, T17, T18, T21, T22) 16.00/5.01 U8^1(f376_out1(T21, T22), T16, T17, T18) -> F380_IN(T21, T22) 16.00/5.01 F380_IN(T21, T22) -> U10^1(f3_in(T21), T21, T22) 16.00/5.01 F380_IN(T21, T22) -> F3_IN(T21) 16.00/5.01 U10^1(f3_out1(T44), T21, T22) -> U11^1(f648_in(T22, T44), T21, T22, T44) 16.00/5.01 U10^1(f3_out1(T44), T21, T22) -> F648_IN(T22, T44) 16.00/5.01 F648_IN(T22, T44) -> U12^1(f3_in(T22), T22, T44) 16.00/5.01 F648_IN(T22, T44) -> F3_IN(T22) 16.00/5.01 U12^1(f3_out1(T45), T22, T44) -> U13^1(f660_in(T44, T45), T22, T44, T45) 16.00/5.01 U12^1(f3_out1(T45), T22, T44) -> F660_IN(T44, T45) 16.00/5.01 F680_IN(T78, T80, T79, T81) -> U14^1(f682_in(T78, T80), T78, T80, T79, T81) 16.00/5.01 F680_IN(T78, T80, T79, T81) -> F682_IN(T78, T80) 16.00/5.01 U14^1(f682_out1, T78, T80, T79, T81) -> U15^1(f660_in(T79, .(T80, T81)), T78, T80, T79, T81) 16.00/5.01 U14^1(f682_out1, T78, T80, T79, T81) -> F660_IN(T79, .(T80, T81)) 16.00/5.01 F750_IN(T119, T121, T120, T122) -> U16^1(f752_in(T119, T121), T119, T121, T120, T122) 16.00/5.01 F750_IN(T119, T121, T120, T122) -> F752_IN(T119, T121) 16.00/5.01 U16^1(f752_out1, T119, T121, T120, T122) -> U17^1(f660_in(.(T119, T120), T122), T119, T121, T120, T122) 16.00/5.01 U16^1(f752_out1, T119, T121, T120, T122) -> F660_IN(.(T119, T120), T122) 16.00/5.01 16.00/5.01 The TRS R consists of the following rules: 16.00/5.01 16.00/5.01 f3_in([]) -> f3_out1([]) 16.00/5.01 f3_in(.(T7, [])) -> f3_out1(.(T7, [])) 16.00/5.01 f3_in(.(T16, .(T17, T18))) -> U1(f369_in(T16, T17, T18), .(T16, .(T17, T18))) 16.00/5.01 U1(f369_out1(X22, X23, X24, X25, T20), .(T16, .(T17, T18))) -> f3_out1(T20) 16.00/5.01 f439_in([]) -> f439_out1([], []) 16.00/5.01 f439_in(.(T42, T43)) -> U2(f439_in(T43), .(T42, T43)) 16.00/5.01 U2(f439_out1(X79, X78), .(T42, T43)) -> f439_out1(.(T42, X78), X79) 16.00/5.01 f660_in([], T52) -> f660_out1(T52) 16.00/5.01 f660_in(T57, []) -> f660_out1(T57) 16.00/5.01 f660_in(.(T78, T79), .(T80, T81)) -> U3(f680_in(T78, T80, T79, T81), .(T78, T79), .(T80, T81)) 16.00/5.01 U3(f680_out1(T83), .(T78, T79), .(T80, T81)) -> f660_out1(.(T78, T83)) 16.00/5.01 f660_in(.(T119, T120), .(T121, T122)) -> U4(f750_in(T119, T121, T120, T122), .(T119, T120), .(T121, T122)) 16.00/5.01 U4(f750_out1(T124), .(T119, T120), .(T121, T122)) -> f660_out1(.(T121, T124)) 16.00/5.01 f682_in(s(T96), s(T97)) -> U5(f682_in(T96, T97), s(T96), s(T97)) 16.00/5.01 U5(f682_out1, s(T96), s(T97)) -> f682_out1 16.00/5.01 f682_in(0, s(T104)) -> f682_out1 16.00/5.01 f682_in(0, 0) -> f682_out1 16.00/5.01 f752_in(s(T137), s(T138)) -> U6(f752_in(T137, T138), s(T137), s(T138)) 16.00/5.01 U6(f752_out1, s(T137), s(T138)) -> f752_out1 16.00/5.01 f752_in(s(T143), 0) -> f752_out1 16.00/5.01 f376_in(T29, T36, T37) -> U7(f439_in(T37), T29, T36, T37) 16.00/5.01 U7(f439_out1(X61, X60), T29, T36, T37) -> f376_out1(.(T29, X61), .(T36, X60)) 16.00/5.01 f369_in(T16, T17, T18) -> U8(f376_in(T16, T17, T18), T16, T17, T18) 16.00/5.01 U8(f376_out1(T21, T22), T16, T17, T18) -> U9(f380_in(T21, T22), T16, T17, T18, T21, T22) 16.00/5.01 U9(f380_out1(X24, X25, T20), T16, T17, T18, T21, T22) -> f369_out1(T21, T22, X24, X25, T20) 16.00/5.01 f380_in(T21, T22) -> U10(f3_in(T21), T21, T22) 16.00/5.01 U10(f3_out1(T44), T21, T22) -> U11(f648_in(T22, T44), T21, T22, T44) 16.00/5.01 U11(f648_out1(X25, T20), T21, T22, T44) -> f380_out1(T44, X25, T20) 16.00/5.01 f648_in(T22, T44) -> U12(f3_in(T22), T22, T44) 16.00/5.01 U12(f3_out1(T45), T22, T44) -> U13(f660_in(T44, T45), T22, T44, T45) 16.00/5.01 U13(f660_out1(T20), T22, T44, T45) -> f648_out1(T45, T20) 16.00/5.01 f680_in(T78, T80, T79, T81) -> U14(f682_in(T78, T80), T78, T80, T79, T81) 16.00/5.01 U14(f682_out1, T78, T80, T79, T81) -> U15(f660_in(T79, .(T80, T81)), T78, T80, T79, T81) 16.00/5.01 U15(f660_out1(T83), T78, T80, T79, T81) -> f680_out1(T83) 16.00/5.01 f750_in(T119, T121, T120, T122) -> U16(f752_in(T119, T121), T119, T121, T120, T122) 16.00/5.01 U16(f752_out1, T119, T121, T120, T122) -> U17(f660_in(.(T119, T120), T122), T119, T121, T120, T122) 16.00/5.01 U17(f660_out1(T124), T119, T121, T120, T122) -> f750_out1(T124) 16.00/5.01 16.00/5.01 Q is empty. 16.00/5.01 We have to consider all minimal (P,Q,R)-chains. 16.00/5.01 ---------------------------------------- 16.00/5.01 16.00/5.01 (5) DependencyGraphProof (EQUIVALENT) 16.00/5.01 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 5 SCCs with 18 less nodes. 16.00/5.01 ---------------------------------------- 16.00/5.01 16.00/5.01 (6) 16.00/5.01 Complex Obligation (AND) 16.00/5.01 16.00/5.01 ---------------------------------------- 16.00/5.01 16.00/5.01 (7) 16.00/5.01 Obligation: 16.00/5.01 Q DP problem: 16.00/5.01 The TRS P consists of the following rules: 16.00/5.01 16.00/5.01 F752_IN(s(T137), s(T138)) -> F752_IN(T137, T138) 16.00/5.01 16.00/5.01 The TRS R consists of the following rules: 16.00/5.01 16.00/5.01 f3_in([]) -> f3_out1([]) 16.00/5.01 f3_in(.(T7, [])) -> f3_out1(.(T7, [])) 16.00/5.01 f3_in(.(T16, .(T17, T18))) -> U1(f369_in(T16, T17, T18), .(T16, .(T17, T18))) 16.00/5.01 U1(f369_out1(X22, X23, X24, X25, T20), .(T16, .(T17, T18))) -> f3_out1(T20) 16.00/5.01 f439_in([]) -> f439_out1([], []) 16.00/5.01 f439_in(.(T42, T43)) -> U2(f439_in(T43), .(T42, T43)) 16.00/5.01 U2(f439_out1(X79, X78), .(T42, T43)) -> f439_out1(.(T42, X78), X79) 16.00/5.01 f660_in([], T52) -> f660_out1(T52) 16.00/5.01 f660_in(T57, []) -> f660_out1(T57) 16.00/5.01 f660_in(.(T78, T79), .(T80, T81)) -> U3(f680_in(T78, T80, T79, T81), .(T78, T79), .(T80, T81)) 16.00/5.01 U3(f680_out1(T83), .(T78, T79), .(T80, T81)) -> f660_out1(.(T78, T83)) 16.00/5.01 f660_in(.(T119, T120), .(T121, T122)) -> U4(f750_in(T119, T121, T120, T122), .(T119, T120), .(T121, T122)) 16.00/5.01 U4(f750_out1(T124), .(T119, T120), .(T121, T122)) -> f660_out1(.(T121, T124)) 16.00/5.01 f682_in(s(T96), s(T97)) -> U5(f682_in(T96, T97), s(T96), s(T97)) 16.00/5.01 U5(f682_out1, s(T96), s(T97)) -> f682_out1 16.00/5.01 f682_in(0, s(T104)) -> f682_out1 16.00/5.01 f682_in(0, 0) -> f682_out1 16.00/5.01 f752_in(s(T137), s(T138)) -> U6(f752_in(T137, T138), s(T137), s(T138)) 16.00/5.01 U6(f752_out1, s(T137), s(T138)) -> f752_out1 16.00/5.01 f752_in(s(T143), 0) -> f752_out1 16.00/5.01 f376_in(T29, T36, T37) -> U7(f439_in(T37), T29, T36, T37) 16.00/5.01 U7(f439_out1(X61, X60), T29, T36, T37) -> f376_out1(.(T29, X61), .(T36, X60)) 16.00/5.01 f369_in(T16, T17, T18) -> U8(f376_in(T16, T17, T18), T16, T17, T18) 16.00/5.01 U8(f376_out1(T21, T22), T16, T17, T18) -> U9(f380_in(T21, T22), T16, T17, T18, T21, T22) 16.00/5.01 U9(f380_out1(X24, X25, T20), T16, T17, T18, T21, T22) -> f369_out1(T21, T22, X24, X25, T20) 16.00/5.01 f380_in(T21, T22) -> U10(f3_in(T21), T21, T22) 16.00/5.01 U10(f3_out1(T44), T21, T22) -> U11(f648_in(T22, T44), T21, T22, T44) 16.00/5.01 U11(f648_out1(X25, T20), T21, T22, T44) -> f380_out1(T44, X25, T20) 16.00/5.01 f648_in(T22, T44) -> U12(f3_in(T22), T22, T44) 16.00/5.01 U12(f3_out1(T45), T22, T44) -> U13(f660_in(T44, T45), T22, T44, T45) 16.00/5.01 U13(f660_out1(T20), T22, T44, T45) -> f648_out1(T45, T20) 16.00/5.01 f680_in(T78, T80, T79, T81) -> U14(f682_in(T78, T80), T78, T80, T79, T81) 16.00/5.01 U14(f682_out1, T78, T80, T79, T81) -> U15(f660_in(T79, .(T80, T81)), T78, T80, T79, T81) 16.00/5.01 U15(f660_out1(T83), T78, T80, T79, T81) -> f680_out1(T83) 16.00/5.01 f750_in(T119, T121, T120, T122) -> U16(f752_in(T119, T121), T119, T121, T120, T122) 16.00/5.01 U16(f752_out1, T119, T121, T120, T122) -> U17(f660_in(.(T119, T120), T122), T119, T121, T120, T122) 16.00/5.01 U17(f660_out1(T124), T119, T121, T120, T122) -> f750_out1(T124) 16.00/5.01 16.00/5.01 Q is empty. 16.00/5.01 We have to consider all minimal (P,Q,R)-chains. 16.00/5.01 ---------------------------------------- 16.00/5.01 16.00/5.01 (8) UsableRulesProof (EQUIVALENT) 16.00/5.01 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. 16.00/5.01 ---------------------------------------- 16.00/5.01 16.00/5.01 (9) 16.00/5.01 Obligation: 16.00/5.01 Q DP problem: 16.00/5.01 The TRS P consists of the following rules: 16.00/5.01 16.00/5.01 F752_IN(s(T137), s(T138)) -> F752_IN(T137, T138) 16.00/5.01 16.00/5.01 R is empty. 16.00/5.01 Q is empty. 16.00/5.01 We have to consider all minimal (P,Q,R)-chains. 16.00/5.01 ---------------------------------------- 16.00/5.01 16.00/5.01 (10) QDPSizeChangeProof (EQUIVALENT) 16.00/5.01 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. 16.00/5.01 16.00/5.01 From the DPs we obtained the following set of size-change graphs: 16.00/5.01 *F752_IN(s(T137), s(T138)) -> F752_IN(T137, T138) 16.00/5.01 The graph contains the following edges 1 > 1, 2 > 2 16.00/5.01 16.00/5.01 16.00/5.01 ---------------------------------------- 16.00/5.01 16.00/5.01 (11) 16.00/5.01 YES 16.00/5.01 16.00/5.01 ---------------------------------------- 16.00/5.01 16.00/5.01 (12) 16.00/5.01 Obligation: 16.00/5.01 Q DP problem: 16.00/5.01 The TRS P consists of the following rules: 16.00/5.01 16.00/5.01 F682_IN(s(T96), s(T97)) -> F682_IN(T96, T97) 16.00/5.01 16.00/5.01 The TRS R consists of the following rules: 16.00/5.01 16.00/5.01 f3_in([]) -> f3_out1([]) 16.00/5.01 f3_in(.(T7, [])) -> f3_out1(.(T7, [])) 16.00/5.01 f3_in(.(T16, .(T17, T18))) -> U1(f369_in(T16, T17, T18), .(T16, .(T17, T18))) 16.00/5.01 U1(f369_out1(X22, X23, X24, X25, T20), .(T16, .(T17, T18))) -> f3_out1(T20) 16.00/5.01 f439_in([]) -> f439_out1([], []) 16.00/5.01 f439_in(.(T42, T43)) -> U2(f439_in(T43), .(T42, T43)) 16.00/5.01 U2(f439_out1(X79, X78), .(T42, T43)) -> f439_out1(.(T42, X78), X79) 16.00/5.01 f660_in([], T52) -> f660_out1(T52) 16.00/5.01 f660_in(T57, []) -> f660_out1(T57) 16.00/5.01 f660_in(.(T78, T79), .(T80, T81)) -> U3(f680_in(T78, T80, T79, T81), .(T78, T79), .(T80, T81)) 16.00/5.01 U3(f680_out1(T83), .(T78, T79), .(T80, T81)) -> f660_out1(.(T78, T83)) 16.00/5.01 f660_in(.(T119, T120), .(T121, T122)) -> U4(f750_in(T119, T121, T120, T122), .(T119, T120), .(T121, T122)) 16.00/5.01 U4(f750_out1(T124), .(T119, T120), .(T121, T122)) -> f660_out1(.(T121, T124)) 16.00/5.01 f682_in(s(T96), s(T97)) -> U5(f682_in(T96, T97), s(T96), s(T97)) 16.00/5.01 U5(f682_out1, s(T96), s(T97)) -> f682_out1 16.00/5.01 f682_in(0, s(T104)) -> f682_out1 16.00/5.01 f682_in(0, 0) -> f682_out1 16.00/5.01 f752_in(s(T137), s(T138)) -> U6(f752_in(T137, T138), s(T137), s(T138)) 16.00/5.01 U6(f752_out1, s(T137), s(T138)) -> f752_out1 16.00/5.01 f752_in(s(T143), 0) -> f752_out1 16.00/5.01 f376_in(T29, T36, T37) -> U7(f439_in(T37), T29, T36, T37) 16.00/5.01 U7(f439_out1(X61, X60), T29, T36, T37) -> f376_out1(.(T29, X61), .(T36, X60)) 16.00/5.01 f369_in(T16, T17, T18) -> U8(f376_in(T16, T17, T18), T16, T17, T18) 16.00/5.01 U8(f376_out1(T21, T22), T16, T17, T18) -> U9(f380_in(T21, T22), T16, T17, T18, T21, T22) 16.00/5.01 U9(f380_out1(X24, X25, T20), T16, T17, T18, T21, T22) -> f369_out1(T21, T22, X24, X25, T20) 16.00/5.01 f380_in(T21, T22) -> U10(f3_in(T21), T21, T22) 16.00/5.01 U10(f3_out1(T44), T21, T22) -> U11(f648_in(T22, T44), T21, T22, T44) 16.00/5.01 U11(f648_out1(X25, T20), T21, T22, T44) -> f380_out1(T44, X25, T20) 16.00/5.01 f648_in(T22, T44) -> U12(f3_in(T22), T22, T44) 16.00/5.01 U12(f3_out1(T45), T22, T44) -> U13(f660_in(T44, T45), T22, T44, T45) 16.00/5.01 U13(f660_out1(T20), T22, T44, T45) -> f648_out1(T45, T20) 16.00/5.01 f680_in(T78, T80, T79, T81) -> U14(f682_in(T78, T80), T78, T80, T79, T81) 16.00/5.01 U14(f682_out1, T78, T80, T79, T81) -> U15(f660_in(T79, .(T80, T81)), T78, T80, T79, T81) 16.00/5.01 U15(f660_out1(T83), T78, T80, T79, T81) -> f680_out1(T83) 16.00/5.01 f750_in(T119, T121, T120, T122) -> U16(f752_in(T119, T121), T119, T121, T120, T122) 16.00/5.01 U16(f752_out1, T119, T121, T120, T122) -> U17(f660_in(.(T119, T120), T122), T119, T121, T120, T122) 16.00/5.01 U17(f660_out1(T124), T119, T121, T120, T122) -> f750_out1(T124) 16.00/5.01 16.00/5.01 Q is empty. 16.00/5.01 We have to consider all minimal (P,Q,R)-chains. 16.00/5.01 ---------------------------------------- 16.00/5.01 16.00/5.01 (13) UsableRulesProof (EQUIVALENT) 16.00/5.01 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. 16.00/5.01 ---------------------------------------- 16.00/5.01 16.00/5.01 (14) 16.00/5.01 Obligation: 16.00/5.01 Q DP problem: 16.00/5.01 The TRS P consists of the following rules: 16.00/5.01 16.00/5.01 F682_IN(s(T96), s(T97)) -> F682_IN(T96, T97) 16.00/5.01 16.00/5.01 R is empty. 16.00/5.01 Q is empty. 16.00/5.01 We have to consider all minimal (P,Q,R)-chains. 16.00/5.01 ---------------------------------------- 16.00/5.01 16.00/5.01 (15) QDPSizeChangeProof (EQUIVALENT) 16.00/5.01 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. 16.00/5.01 16.00/5.01 From the DPs we obtained the following set of size-change graphs: 16.00/5.01 *F682_IN(s(T96), s(T97)) -> F682_IN(T96, T97) 16.00/5.01 The graph contains the following edges 1 > 1, 2 > 2 16.00/5.01 16.00/5.01 16.00/5.01 ---------------------------------------- 16.00/5.01 16.00/5.01 (16) 16.00/5.01 YES 16.00/5.01 16.00/5.01 ---------------------------------------- 16.00/5.01 16.00/5.01 (17) 16.00/5.01 Obligation: 16.00/5.01 Q DP problem: 16.00/5.01 The TRS P consists of the following rules: 16.00/5.01 16.00/5.01 F660_IN(.(T78, T79), .(T80, T81)) -> F680_IN(T78, T80, T79, T81) 16.00/5.01 F680_IN(T78, T80, T79, T81) -> U14^1(f682_in(T78, T80), T78, T80, T79, T81) 16.00/5.01 U14^1(f682_out1, T78, T80, T79, T81) -> F660_IN(T79, .(T80, T81)) 16.00/5.01 F660_IN(.(T119, T120), .(T121, T122)) -> F750_IN(T119, T121, T120, T122) 16.00/5.01 F750_IN(T119, T121, T120, T122) -> U16^1(f752_in(T119, T121), T119, T121, T120, T122) 16.00/5.01 U16^1(f752_out1, T119, T121, T120, T122) -> F660_IN(.(T119, T120), T122) 16.00/5.01 16.00/5.01 The TRS R consists of the following rules: 16.00/5.01 16.00/5.01 f3_in([]) -> f3_out1([]) 16.00/5.01 f3_in(.(T7, [])) -> f3_out1(.(T7, [])) 16.00/5.01 f3_in(.(T16, .(T17, T18))) -> U1(f369_in(T16, T17, T18), .(T16, .(T17, T18))) 16.00/5.01 U1(f369_out1(X22, X23, X24, X25, T20), .(T16, .(T17, T18))) -> f3_out1(T20) 16.00/5.01 f439_in([]) -> f439_out1([], []) 16.00/5.01 f439_in(.(T42, T43)) -> U2(f439_in(T43), .(T42, T43)) 16.00/5.01 U2(f439_out1(X79, X78), .(T42, T43)) -> f439_out1(.(T42, X78), X79) 16.00/5.01 f660_in([], T52) -> f660_out1(T52) 16.00/5.01 f660_in(T57, []) -> f660_out1(T57) 16.00/5.01 f660_in(.(T78, T79), .(T80, T81)) -> U3(f680_in(T78, T80, T79, T81), .(T78, T79), .(T80, T81)) 16.00/5.01 U3(f680_out1(T83), .(T78, T79), .(T80, T81)) -> f660_out1(.(T78, T83)) 16.00/5.01 f660_in(.(T119, T120), .(T121, T122)) -> U4(f750_in(T119, T121, T120, T122), .(T119, T120), .(T121, T122)) 16.00/5.01 U4(f750_out1(T124), .(T119, T120), .(T121, T122)) -> f660_out1(.(T121, T124)) 16.00/5.01 f682_in(s(T96), s(T97)) -> U5(f682_in(T96, T97), s(T96), s(T97)) 16.00/5.01 U5(f682_out1, s(T96), s(T97)) -> f682_out1 16.00/5.01 f682_in(0, s(T104)) -> f682_out1 16.00/5.01 f682_in(0, 0) -> f682_out1 16.00/5.01 f752_in(s(T137), s(T138)) -> U6(f752_in(T137, T138), s(T137), s(T138)) 16.00/5.01 U6(f752_out1, s(T137), s(T138)) -> f752_out1 16.00/5.01 f752_in(s(T143), 0) -> f752_out1 16.00/5.01 f376_in(T29, T36, T37) -> U7(f439_in(T37), T29, T36, T37) 16.00/5.01 U7(f439_out1(X61, X60), T29, T36, T37) -> f376_out1(.(T29, X61), .(T36, X60)) 16.00/5.01 f369_in(T16, T17, T18) -> U8(f376_in(T16, T17, T18), T16, T17, T18) 16.00/5.01 U8(f376_out1(T21, T22), T16, T17, T18) -> U9(f380_in(T21, T22), T16, T17, T18, T21, T22) 16.00/5.01 U9(f380_out1(X24, X25, T20), T16, T17, T18, T21, T22) -> f369_out1(T21, T22, X24, X25, T20) 16.00/5.01 f380_in(T21, T22) -> U10(f3_in(T21), T21, T22) 16.00/5.01 U10(f3_out1(T44), T21, T22) -> U11(f648_in(T22, T44), T21, T22, T44) 16.00/5.01 U11(f648_out1(X25, T20), T21, T22, T44) -> f380_out1(T44, X25, T20) 16.00/5.01 f648_in(T22, T44) -> U12(f3_in(T22), T22, T44) 16.00/5.01 U12(f3_out1(T45), T22, T44) -> U13(f660_in(T44, T45), T22, T44, T45) 16.00/5.01 U13(f660_out1(T20), T22, T44, T45) -> f648_out1(T45, T20) 16.00/5.01 f680_in(T78, T80, T79, T81) -> U14(f682_in(T78, T80), T78, T80, T79, T81) 16.00/5.01 U14(f682_out1, T78, T80, T79, T81) -> U15(f660_in(T79, .(T80, T81)), T78, T80, T79, T81) 16.00/5.01 U15(f660_out1(T83), T78, T80, T79, T81) -> f680_out1(T83) 16.00/5.01 f750_in(T119, T121, T120, T122) -> U16(f752_in(T119, T121), T119, T121, T120, T122) 16.00/5.01 U16(f752_out1, T119, T121, T120, T122) -> U17(f660_in(.(T119, T120), T122), T119, T121, T120, T122) 16.00/5.01 U17(f660_out1(T124), T119, T121, T120, T122) -> f750_out1(T124) 16.00/5.01 16.00/5.01 Q is empty. 16.00/5.01 We have to consider all minimal (P,Q,R)-chains. 16.00/5.01 ---------------------------------------- 16.00/5.01 16.00/5.01 (18) QDPOrderProof (EQUIVALENT) 16.00/5.01 We use the reduction pair processor [LPAR04,JAR06]. 16.00/5.01 16.00/5.01 16.00/5.01 The following pairs can be oriented strictly and are deleted. 16.00/5.01 16.00/5.01 F660_IN(.(T78, T79), .(T80, T81)) -> F680_IN(T78, T80, T79, T81) 16.00/5.01 F660_IN(.(T119, T120), .(T121, T122)) -> F750_IN(T119, T121, T120, T122) 16.00/5.01 U16^1(f752_out1, T119, T121, T120, T122) -> F660_IN(.(T119, T120), T122) 16.00/5.01 The remaining pairs can at least be oriented weakly. 16.00/5.01 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 16.00/5.01 16.00/5.01 POL( U14^1_5(x_1, ..., x_5) ) = x_1 + 2x_3 + 2x_4 + 2x_5 + 2 16.00/5.01 POL( f682_in_2(x_1, x_2) ) = x_1 16.00/5.01 POL( s_1(x_1) ) = 2x_1 + 2 16.00/5.01 POL( U5_3(x_1, ..., x_3) ) = 2 16.00/5.01 POL( 0 ) = 2 16.00/5.01 POL( f682_out1 ) = 2 16.00/5.01 POL( U16^1_5(x_1, ..., x_5) ) = 2x_1 + 2x_2 + 2x_4 + 2x_5 + 2 16.00/5.01 POL( f752_in_2(x_1, x_2) ) = x_2 16.00/5.01 POL( U6_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2} 16.00/5.01 POL( f752_out1 ) = 2 16.00/5.01 POL( F660_IN_2(x_1, x_2) ) = 2x_1 + 2x_2 16.00/5.01 POL( ._2(x_1, x_2) ) = x_1 + x_2 + 2 16.00/5.01 POL( F680_IN_4(x_1, ..., x_4) ) = x_1 + 2x_2 + 2x_3 + 2x_4 + 2 16.00/5.01 POL( F750_IN_4(x_1, ..., x_4) ) = 2x_1 + 2x_2 + 2x_3 + 2x_4 + 2 16.00/5.01 16.00/5.01 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 16.00/5.01 16.00/5.01 f682_in(s(T96), s(T97)) -> U5(f682_in(T96, T97), s(T96), s(T97)) 16.00/5.01 f682_in(0, s(T104)) -> f682_out1 16.00/5.01 f682_in(0, 0) -> f682_out1 16.00/5.01 f752_in(s(T137), s(T138)) -> U6(f752_in(T137, T138), s(T137), s(T138)) 16.00/5.01 f752_in(s(T143), 0) -> f752_out1 16.00/5.01 U5(f682_out1, s(T96), s(T97)) -> f682_out1 16.00/5.01 U6(f752_out1, s(T137), s(T138)) -> f752_out1 16.00/5.01 16.00/5.01 16.00/5.01 ---------------------------------------- 16.00/5.01 16.00/5.01 (19) 16.00/5.01 Obligation: 16.00/5.01 Q DP problem: 16.00/5.01 The TRS P consists of the following rules: 16.00/5.01 16.00/5.01 F680_IN(T78, T80, T79, T81) -> U14^1(f682_in(T78, T80), T78, T80, T79, T81) 16.00/5.01 U14^1(f682_out1, T78, T80, T79, T81) -> F660_IN(T79, .(T80, T81)) 16.00/5.01 F750_IN(T119, T121, T120, T122) -> U16^1(f752_in(T119, T121), T119, T121, T120, T122) 16.00/5.01 16.00/5.01 The TRS R consists of the following rules: 16.00/5.01 16.00/5.01 f3_in([]) -> f3_out1([]) 16.00/5.01 f3_in(.(T7, [])) -> f3_out1(.(T7, [])) 16.00/5.01 f3_in(.(T16, .(T17, T18))) -> U1(f369_in(T16, T17, T18), .(T16, .(T17, T18))) 16.00/5.01 U1(f369_out1(X22, X23, X24, X25, T20), .(T16, .(T17, T18))) -> f3_out1(T20) 16.00/5.01 f439_in([]) -> f439_out1([], []) 16.00/5.01 f439_in(.(T42, T43)) -> U2(f439_in(T43), .(T42, T43)) 16.00/5.01 U2(f439_out1(X79, X78), .(T42, T43)) -> f439_out1(.(T42, X78), X79) 16.00/5.01 f660_in([], T52) -> f660_out1(T52) 16.00/5.01 f660_in(T57, []) -> f660_out1(T57) 16.00/5.01 f660_in(.(T78, T79), .(T80, T81)) -> U3(f680_in(T78, T80, T79, T81), .(T78, T79), .(T80, T81)) 16.00/5.01 U3(f680_out1(T83), .(T78, T79), .(T80, T81)) -> f660_out1(.(T78, T83)) 16.00/5.01 f660_in(.(T119, T120), .(T121, T122)) -> U4(f750_in(T119, T121, T120, T122), .(T119, T120), .(T121, T122)) 16.00/5.01 U4(f750_out1(T124), .(T119, T120), .(T121, T122)) -> f660_out1(.(T121, T124)) 16.00/5.01 f682_in(s(T96), s(T97)) -> U5(f682_in(T96, T97), s(T96), s(T97)) 16.00/5.01 U5(f682_out1, s(T96), s(T97)) -> f682_out1 16.00/5.01 f682_in(0, s(T104)) -> f682_out1 16.00/5.01 f682_in(0, 0) -> f682_out1 16.00/5.01 f752_in(s(T137), s(T138)) -> U6(f752_in(T137, T138), s(T137), s(T138)) 16.00/5.01 U6(f752_out1, s(T137), s(T138)) -> f752_out1 16.00/5.01 f752_in(s(T143), 0) -> f752_out1 16.00/5.01 f376_in(T29, T36, T37) -> U7(f439_in(T37), T29, T36, T37) 16.00/5.01 U7(f439_out1(X61, X60), T29, T36, T37) -> f376_out1(.(T29, X61), .(T36, X60)) 16.00/5.01 f369_in(T16, T17, T18) -> U8(f376_in(T16, T17, T18), T16, T17, T18) 16.00/5.01 U8(f376_out1(T21, T22), T16, T17, T18) -> U9(f380_in(T21, T22), T16, T17, T18, T21, T22) 16.00/5.01 U9(f380_out1(X24, X25, T20), T16, T17, T18, T21, T22) -> f369_out1(T21, T22, X24, X25, T20) 16.00/5.01 f380_in(T21, T22) -> U10(f3_in(T21), T21, T22) 16.00/5.01 U10(f3_out1(T44), T21, T22) -> U11(f648_in(T22, T44), T21, T22, T44) 16.00/5.01 U11(f648_out1(X25, T20), T21, T22, T44) -> f380_out1(T44, X25, T20) 16.00/5.01 f648_in(T22, T44) -> U12(f3_in(T22), T22, T44) 16.00/5.01 U12(f3_out1(T45), T22, T44) -> U13(f660_in(T44, T45), T22, T44, T45) 16.00/5.01 U13(f660_out1(T20), T22, T44, T45) -> f648_out1(T45, T20) 16.00/5.01 f680_in(T78, T80, T79, T81) -> U14(f682_in(T78, T80), T78, T80, T79, T81) 16.00/5.01 U14(f682_out1, T78, T80, T79, T81) -> U15(f660_in(T79, .(T80, T81)), T78, T80, T79, T81) 16.00/5.01 U15(f660_out1(T83), T78, T80, T79, T81) -> f680_out1(T83) 16.00/5.01 f750_in(T119, T121, T120, T122) -> U16(f752_in(T119, T121), T119, T121, T120, T122) 16.00/5.01 U16(f752_out1, T119, T121, T120, T122) -> U17(f660_in(.(T119, T120), T122), T119, T121, T120, T122) 16.00/5.01 U17(f660_out1(T124), T119, T121, T120, T122) -> f750_out1(T124) 16.00/5.01 16.00/5.01 Q is empty. 16.00/5.01 We have to consider all minimal (P,Q,R)-chains. 16.00/5.01 ---------------------------------------- 16.00/5.01 16.00/5.01 (20) DependencyGraphProof (EQUIVALENT) 16.00/5.01 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes. 16.00/5.01 ---------------------------------------- 16.00/5.01 16.00/5.01 (21) 16.00/5.01 TRUE 16.00/5.01 16.00/5.01 ---------------------------------------- 16.00/5.01 16.00/5.01 (22) 16.00/5.01 Obligation: 16.00/5.01 Q DP problem: 16.00/5.01 The TRS P consists of the following rules: 16.00/5.01 16.00/5.01 F439_IN(.(T42, T43)) -> F439_IN(T43) 16.00/5.01 16.00/5.01 The TRS R consists of the following rules: 16.00/5.01 16.00/5.01 f3_in([]) -> f3_out1([]) 16.00/5.01 f3_in(.(T7, [])) -> f3_out1(.(T7, [])) 16.00/5.01 f3_in(.(T16, .(T17, T18))) -> U1(f369_in(T16, T17, T18), .(T16, .(T17, T18))) 16.00/5.01 U1(f369_out1(X22, X23, X24, X25, T20), .(T16, .(T17, T18))) -> f3_out1(T20) 16.00/5.01 f439_in([]) -> f439_out1([], []) 16.00/5.01 f439_in(.(T42, T43)) -> U2(f439_in(T43), .(T42, T43)) 16.00/5.01 U2(f439_out1(X79, X78), .(T42, T43)) -> f439_out1(.(T42, X78), X79) 16.00/5.01 f660_in([], T52) -> f660_out1(T52) 16.00/5.01 f660_in(T57, []) -> f660_out1(T57) 16.00/5.01 f660_in(.(T78, T79), .(T80, T81)) -> U3(f680_in(T78, T80, T79, T81), .(T78, T79), .(T80, T81)) 16.00/5.01 U3(f680_out1(T83), .(T78, T79), .(T80, T81)) -> f660_out1(.(T78, T83)) 16.00/5.01 f660_in(.(T119, T120), .(T121, T122)) -> U4(f750_in(T119, T121, T120, T122), .(T119, T120), .(T121, T122)) 16.00/5.01 U4(f750_out1(T124), .(T119, T120), .(T121, T122)) -> f660_out1(.(T121, T124)) 16.00/5.01 f682_in(s(T96), s(T97)) -> U5(f682_in(T96, T97), s(T96), s(T97)) 16.00/5.01 U5(f682_out1, s(T96), s(T97)) -> f682_out1 16.00/5.01 f682_in(0, s(T104)) -> f682_out1 16.00/5.01 f682_in(0, 0) -> f682_out1 16.00/5.01 f752_in(s(T137), s(T138)) -> U6(f752_in(T137, T138), s(T137), s(T138)) 16.00/5.01 U6(f752_out1, s(T137), s(T138)) -> f752_out1 16.00/5.01 f752_in(s(T143), 0) -> f752_out1 16.00/5.01 f376_in(T29, T36, T37) -> U7(f439_in(T37), T29, T36, T37) 16.00/5.01 U7(f439_out1(X61, X60), T29, T36, T37) -> f376_out1(.(T29, X61), .(T36, X60)) 16.00/5.01 f369_in(T16, T17, T18) -> U8(f376_in(T16, T17, T18), T16, T17, T18) 16.00/5.01 U8(f376_out1(T21, T22), T16, T17, T18) -> U9(f380_in(T21, T22), T16, T17, T18, T21, T22) 16.00/5.01 U9(f380_out1(X24, X25, T20), T16, T17, T18, T21, T22) -> f369_out1(T21, T22, X24, X25, T20) 16.00/5.01 f380_in(T21, T22) -> U10(f3_in(T21), T21, T22) 16.00/5.01 U10(f3_out1(T44), T21, T22) -> U11(f648_in(T22, T44), T21, T22, T44) 16.00/5.01 U11(f648_out1(X25, T20), T21, T22, T44) -> f380_out1(T44, X25, T20) 16.00/5.01 f648_in(T22, T44) -> U12(f3_in(T22), T22, T44) 16.00/5.01 U12(f3_out1(T45), T22, T44) -> U13(f660_in(T44, T45), T22, T44, T45) 16.00/5.01 U13(f660_out1(T20), T22, T44, T45) -> f648_out1(T45, T20) 16.00/5.01 f680_in(T78, T80, T79, T81) -> U14(f682_in(T78, T80), T78, T80, T79, T81) 16.00/5.01 U14(f682_out1, T78, T80, T79, T81) -> U15(f660_in(T79, .(T80, T81)), T78, T80, T79, T81) 16.00/5.01 U15(f660_out1(T83), T78, T80, T79, T81) -> f680_out1(T83) 16.00/5.01 f750_in(T119, T121, T120, T122) -> U16(f752_in(T119, T121), T119, T121, T120, T122) 16.00/5.01 U16(f752_out1, T119, T121, T120, T122) -> U17(f660_in(.(T119, T120), T122), T119, T121, T120, T122) 16.00/5.01 U17(f660_out1(T124), T119, T121, T120, T122) -> f750_out1(T124) 16.00/5.01 16.00/5.01 Q is empty. 16.00/5.01 We have to consider all minimal (P,Q,R)-chains. 16.00/5.01 ---------------------------------------- 16.00/5.01 16.00/5.01 (23) UsableRulesProof (EQUIVALENT) 16.00/5.01 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. 16.00/5.01 ---------------------------------------- 16.00/5.01 16.00/5.01 (24) 16.00/5.01 Obligation: 16.00/5.01 Q DP problem: 16.00/5.01 The TRS P consists of the following rules: 16.00/5.01 16.00/5.01 F439_IN(.(T42, T43)) -> F439_IN(T43) 16.00/5.01 16.00/5.01 R is empty. 16.00/5.01 Q is empty. 16.00/5.01 We have to consider all minimal (P,Q,R)-chains. 16.00/5.01 ---------------------------------------- 16.00/5.01 16.00/5.01 (25) QDPSizeChangeProof (EQUIVALENT) 16.00/5.01 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. 16.00/5.01 16.00/5.01 From the DPs we obtained the following set of size-change graphs: 16.00/5.01 *F439_IN(.(T42, T43)) -> F439_IN(T43) 16.00/5.01 The graph contains the following edges 1 > 1 16.00/5.01 16.00/5.01 16.00/5.01 ---------------------------------------- 16.00/5.01 16.00/5.01 (26) 16.00/5.01 YES 16.00/5.01 16.00/5.01 ---------------------------------------- 16.00/5.01 16.00/5.01 (27) 16.00/5.01 Obligation: 16.00/5.01 Q DP problem: 16.00/5.01 The TRS P consists of the following rules: 16.00/5.01 16.00/5.01 F3_IN(.(T16, .(T17, T18))) -> F369_IN(T16, T17, T18) 16.00/5.01 F369_IN(T16, T17, T18) -> U8^1(f376_in(T16, T17, T18), T16, T17, T18) 16.00/5.01 U8^1(f376_out1(T21, T22), T16, T17, T18) -> F380_IN(T21, T22) 16.00/5.01 F380_IN(T21, T22) -> U10^1(f3_in(T21), T21, T22) 16.00/5.01 U10^1(f3_out1(T44), T21, T22) -> F648_IN(T22, T44) 16.00/5.01 F648_IN(T22, T44) -> F3_IN(T22) 16.00/5.01 F380_IN(T21, T22) -> F3_IN(T21) 16.00/5.01 16.00/5.01 The TRS R consists of the following rules: 16.00/5.01 16.00/5.01 f3_in([]) -> f3_out1([]) 16.00/5.01 f3_in(.(T7, [])) -> f3_out1(.(T7, [])) 16.00/5.01 f3_in(.(T16, .(T17, T18))) -> U1(f369_in(T16, T17, T18), .(T16, .(T17, T18))) 16.00/5.01 U1(f369_out1(X22, X23, X24, X25, T20), .(T16, .(T17, T18))) -> f3_out1(T20) 16.00/5.01 f439_in([]) -> f439_out1([], []) 16.00/5.01 f439_in(.(T42, T43)) -> U2(f439_in(T43), .(T42, T43)) 16.00/5.01 U2(f439_out1(X79, X78), .(T42, T43)) -> f439_out1(.(T42, X78), X79) 16.00/5.01 f660_in([], T52) -> f660_out1(T52) 16.00/5.01 f660_in(T57, []) -> f660_out1(T57) 16.00/5.01 f660_in(.(T78, T79), .(T80, T81)) -> U3(f680_in(T78, T80, T79, T81), .(T78, T79), .(T80, T81)) 16.00/5.01 U3(f680_out1(T83), .(T78, T79), .(T80, T81)) -> f660_out1(.(T78, T83)) 16.00/5.01 f660_in(.(T119, T120), .(T121, T122)) -> U4(f750_in(T119, T121, T120, T122), .(T119, T120), .(T121, T122)) 16.00/5.01 U4(f750_out1(T124), .(T119, T120), .(T121, T122)) -> f660_out1(.(T121, T124)) 16.00/5.01 f682_in(s(T96), s(T97)) -> U5(f682_in(T96, T97), s(T96), s(T97)) 16.00/5.01 U5(f682_out1, s(T96), s(T97)) -> f682_out1 16.00/5.01 f682_in(0, s(T104)) -> f682_out1 16.00/5.01 f682_in(0, 0) -> f682_out1 16.00/5.01 f752_in(s(T137), s(T138)) -> U6(f752_in(T137, T138), s(T137), s(T138)) 16.00/5.01 U6(f752_out1, s(T137), s(T138)) -> f752_out1 16.00/5.01 f752_in(s(T143), 0) -> f752_out1 16.00/5.01 f376_in(T29, T36, T37) -> U7(f439_in(T37), T29, T36, T37) 16.00/5.01 U7(f439_out1(X61, X60), T29, T36, T37) -> f376_out1(.(T29, X61), .(T36, X60)) 16.00/5.01 f369_in(T16, T17, T18) -> U8(f376_in(T16, T17, T18), T16, T17, T18) 16.00/5.01 U8(f376_out1(T21, T22), T16, T17, T18) -> U9(f380_in(T21, T22), T16, T17, T18, T21, T22) 16.00/5.01 U9(f380_out1(X24, X25, T20), T16, T17, T18, T21, T22) -> f369_out1(T21, T22, X24, X25, T20) 16.00/5.01 f380_in(T21, T22) -> U10(f3_in(T21), T21, T22) 16.00/5.01 U10(f3_out1(T44), T21, T22) -> U11(f648_in(T22, T44), T21, T22, T44) 16.00/5.01 U11(f648_out1(X25, T20), T21, T22, T44) -> f380_out1(T44, X25, T20) 16.00/5.01 f648_in(T22, T44) -> U12(f3_in(T22), T22, T44) 16.00/5.01 U12(f3_out1(T45), T22, T44) -> U13(f660_in(T44, T45), T22, T44, T45) 16.00/5.01 U13(f660_out1(T20), T22, T44, T45) -> f648_out1(T45, T20) 16.00/5.01 f680_in(T78, T80, T79, T81) -> U14(f682_in(T78, T80), T78, T80, T79, T81) 16.00/5.01 U14(f682_out1, T78, T80, T79, T81) -> U15(f660_in(T79, .(T80, T81)), T78, T80, T79, T81) 16.00/5.01 U15(f660_out1(T83), T78, T80, T79, T81) -> f680_out1(T83) 16.00/5.01 f750_in(T119, T121, T120, T122) -> U16(f752_in(T119, T121), T119, T121, T120, T122) 16.00/5.01 U16(f752_out1, T119, T121, T120, T122) -> U17(f660_in(.(T119, T120), T122), T119, T121, T120, T122) 16.00/5.01 U17(f660_out1(T124), T119, T121, T120, T122) -> f750_out1(T124) 16.00/5.01 16.00/5.01 Q is empty. 16.00/5.01 We have to consider all minimal (P,Q,R)-chains. 16.00/5.01 ---------------------------------------- 16.00/5.01 16.00/5.01 (28) QDPOrderProof (EQUIVALENT) 16.00/5.01 We use the reduction pair processor [LPAR04,JAR06]. 16.00/5.01 16.00/5.01 16.00/5.01 The following pairs can be oriented strictly and are deleted. 16.00/5.01 16.00/5.01 F3_IN(.(T16, .(T17, T18))) -> F369_IN(T16, T17, T18) 16.00/5.01 The remaining pairs can at least be oriented weakly. 16.00/5.01 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 16.00/5.01 16.00/5.01 POL( U8^1_4(x_1, ..., x_4) ) = x_1 + 2 16.00/5.01 POL( U10^1_3(x_1, ..., x_3) ) = x_3 16.00/5.01 POL( U8_4(x_1, ..., x_4) ) = 2x_2 + 2 16.00/5.01 POL( f376_in_3(x_1, ..., x_3) ) = 2x_3 16.00/5.01 POL( U7_4(x_1, ..., x_4) ) = 2x_1 16.00/5.01 POL( f439_in_1(x_1) ) = x_1 16.00/5.01 POL( f3_in_1(x_1) ) = 0 16.00/5.01 POL( [] ) = 0 16.00/5.01 POL( f3_out1_1(x_1) ) = max{0, 2x_1 - 2} 16.00/5.01 POL( ._2(x_1, x_2) ) = 2x_2 + 1 16.00/5.01 POL( U1_2(x_1, x_2) ) = 2x_1 + x_2 + 1 16.00/5.01 POL( f369_in_3(x_1, ..., x_3) ) = x_1 + 2x_2 + x_3 + 2 16.00/5.01 POL( f369_out1_5(x_1, ..., x_5) ) = 2x_1 + 2x_2 + 2x_3 + 2x_4 + 1 16.00/5.01 POL( f376_out1_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2} 16.00/5.01 POL( U9_6(x_1, ..., x_6) ) = 2x_5 + 2 16.00/5.01 POL( f380_in_2(x_1, x_2) ) = 0 16.00/5.01 POL( f380_out1_3(x_1, ..., x_3) ) = 2x_1 + x_3 + 2 16.00/5.01 POL( U10_3(x_1, ..., x_3) ) = 2 16.00/5.01 POL( U11_4(x_1, ..., x_4) ) = max{0, 2x_4 - 2} 16.00/5.01 POL( f648_in_2(x_1, x_2) ) = 2x_1 + 1 16.00/5.01 POL( f648_out1_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2 16.00/5.01 POL( U12_3(x_1, ..., x_3) ) = max{0, x_2 - 2} 16.00/5.01 POL( U13_4(x_1, ..., x_4) ) = max{0, x_3 + 2x_4 - 2} 16.00/5.01 POL( f660_in_2(x_1, x_2) ) = 0 16.00/5.01 POL( f439_out1_2(x_1, x_2) ) = x_1 + x_2 16.00/5.01 POL( U2_2(x_1, x_2) ) = 2x_1 + 1 16.00/5.01 POL( f660_out1_1(x_1) ) = 2 16.00/5.01 POL( U3_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2} 16.00/5.01 POL( f680_in_4(x_1, ..., x_4) ) = 2x_2 + 2 16.00/5.01 POL( U4_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + x_3 + 1 16.00/5.01 POL( f750_in_4(x_1, ..., x_4) ) = 2x_1 + 2x_2 + 2x_3 + 2 16.00/5.01 POL( f680_out1_1(x_1) ) = 1 16.00/5.01 POL( U14_5(x_1, ..., x_5) ) = max{0, x_4 + x_5 - 2} 16.00/5.01 POL( f682_in_2(x_1, x_2) ) = 0 16.00/5.01 POL( s_1(x_1) ) = 2x_1 16.00/5.01 POL( U5_3(x_1, ..., x_3) ) = max{0, x_2 - 2} 16.00/5.01 POL( 0 ) = 0 16.00/5.01 POL( f682_out1 ) = 0 16.00/5.01 POL( U15_5(x_1, ..., x_5) ) = max{0, x_2 - 2} 16.00/5.01 POL( f750_out1_1(x_1) ) = 0 16.00/5.01 POL( U16_5(x_1, ..., x_5) ) = max{0, x_4 + 2x_5 - 2} 16.00/5.01 POL( f752_in_2(x_1, x_2) ) = 2x_1 + 2x_2 16.00/5.01 POL( U6_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + 2x_3 - 2} 16.00/5.01 POL( f752_out1 ) = 2 16.00/5.01 POL( U17_5(x_1, ..., x_5) ) = max{0, 2x_3 - 2} 16.00/5.01 POL( F3_IN_1(x_1) ) = x_1 16.00/5.01 POL( F369_IN_3(x_1, ..., x_3) ) = 2x_3 + 2 16.00/5.01 POL( F380_IN_2(x_1, x_2) ) = x_1 + x_2 16.00/5.01 POL( F648_IN_2(x_1, x_2) ) = x_1 16.00/5.01 16.00/5.01 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 16.00/5.01 16.00/5.01 f376_in(T29, T36, T37) -> U7(f439_in(T37), T29, T36, T37) 16.00/5.01 f439_in([]) -> f439_out1([], []) 16.00/5.01 f439_in(.(T42, T43)) -> U2(f439_in(T43), .(T42, T43)) 16.00/5.01 U7(f439_out1(X61, X60), T29, T36, T37) -> f376_out1(.(T29, X61), .(T36, X60)) 16.00/5.01 U2(f439_out1(X79, X78), .(T42, T43)) -> f439_out1(.(T42, X78), X79) 16.00/5.01 16.00/5.01 16.00/5.01 ---------------------------------------- 16.00/5.01 16.00/5.01 (29) 16.00/5.01 Obligation: 16.00/5.01 Q DP problem: 16.00/5.01 The TRS P consists of the following rules: 16.00/5.01 16.00/5.01 F369_IN(T16, T17, T18) -> U8^1(f376_in(T16, T17, T18), T16, T17, T18) 16.00/5.01 U8^1(f376_out1(T21, T22), T16, T17, T18) -> F380_IN(T21, T22) 16.00/5.01 F380_IN(T21, T22) -> U10^1(f3_in(T21), T21, T22) 16.00/5.01 U10^1(f3_out1(T44), T21, T22) -> F648_IN(T22, T44) 16.00/5.01 F648_IN(T22, T44) -> F3_IN(T22) 16.00/5.01 F380_IN(T21, T22) -> F3_IN(T21) 16.00/5.01 16.00/5.01 The TRS R consists of the following rules: 16.00/5.01 16.00/5.01 f3_in([]) -> f3_out1([]) 16.00/5.01 f3_in(.(T7, [])) -> f3_out1(.(T7, [])) 16.00/5.01 f3_in(.(T16, .(T17, T18))) -> U1(f369_in(T16, T17, T18), .(T16, .(T17, T18))) 16.00/5.01 U1(f369_out1(X22, X23, X24, X25, T20), .(T16, .(T17, T18))) -> f3_out1(T20) 16.00/5.01 f439_in([]) -> f439_out1([], []) 16.00/5.01 f439_in(.(T42, T43)) -> U2(f439_in(T43), .(T42, T43)) 16.00/5.01 U2(f439_out1(X79, X78), .(T42, T43)) -> f439_out1(.(T42, X78), X79) 16.00/5.01 f660_in([], T52) -> f660_out1(T52) 16.00/5.01 f660_in(T57, []) -> f660_out1(T57) 16.00/5.01 f660_in(.(T78, T79), .(T80, T81)) -> U3(f680_in(T78, T80, T79, T81), .(T78, T79), .(T80, T81)) 16.00/5.01 U3(f680_out1(T83), .(T78, T79), .(T80, T81)) -> f660_out1(.(T78, T83)) 16.00/5.01 f660_in(.(T119, T120), .(T121, T122)) -> U4(f750_in(T119, T121, T120, T122), .(T119, T120), .(T121, T122)) 16.00/5.01 U4(f750_out1(T124), .(T119, T120), .(T121, T122)) -> f660_out1(.(T121, T124)) 16.00/5.01 f682_in(s(T96), s(T97)) -> U5(f682_in(T96, T97), s(T96), s(T97)) 16.00/5.01 U5(f682_out1, s(T96), s(T97)) -> f682_out1 16.00/5.01 f682_in(0, s(T104)) -> f682_out1 16.00/5.01 f682_in(0, 0) -> f682_out1 16.00/5.01 f752_in(s(T137), s(T138)) -> U6(f752_in(T137, T138), s(T137), s(T138)) 16.00/5.01 U6(f752_out1, s(T137), s(T138)) -> f752_out1 16.00/5.01 f752_in(s(T143), 0) -> f752_out1 16.00/5.01 f376_in(T29, T36, T37) -> U7(f439_in(T37), T29, T36, T37) 16.00/5.01 U7(f439_out1(X61, X60), T29, T36, T37) -> f376_out1(.(T29, X61), .(T36, X60)) 16.00/5.01 f369_in(T16, T17, T18) -> U8(f376_in(T16, T17, T18), T16, T17, T18) 16.00/5.01 U8(f376_out1(T21, T22), T16, T17, T18) -> U9(f380_in(T21, T22), T16, T17, T18, T21, T22) 16.00/5.01 U9(f380_out1(X24, X25, T20), T16, T17, T18, T21, T22) -> f369_out1(T21, T22, X24, X25, T20) 16.00/5.01 f380_in(T21, T22) -> U10(f3_in(T21), T21, T22) 16.00/5.01 U10(f3_out1(T44), T21, T22) -> U11(f648_in(T22, T44), T21, T22, T44) 16.00/5.01 U11(f648_out1(X25, T20), T21, T22, T44) -> f380_out1(T44, X25, T20) 16.00/5.01 f648_in(T22, T44) -> U12(f3_in(T22), T22, T44) 16.00/5.01 U12(f3_out1(T45), T22, T44) -> U13(f660_in(T44, T45), T22, T44, T45) 16.00/5.01 U13(f660_out1(T20), T22, T44, T45) -> f648_out1(T45, T20) 16.00/5.01 f680_in(T78, T80, T79, T81) -> U14(f682_in(T78, T80), T78, T80, T79, T81) 16.00/5.01 U14(f682_out1, T78, T80, T79, T81) -> U15(f660_in(T79, .(T80, T81)), T78, T80, T79, T81) 16.00/5.01 U15(f660_out1(T83), T78, T80, T79, T81) -> f680_out1(T83) 16.00/5.01 f750_in(T119, T121, T120, T122) -> U16(f752_in(T119, T121), T119, T121, T120, T122) 16.00/5.01 U16(f752_out1, T119, T121, T120, T122) -> U17(f660_in(.(T119, T120), T122), T119, T121, T120, T122) 16.00/5.01 U17(f660_out1(T124), T119, T121, T120, T122) -> f750_out1(T124) 16.00/5.01 16.00/5.01 Q is empty. 16.00/5.01 We have to consider all minimal (P,Q,R)-chains. 16.00/5.01 ---------------------------------------- 16.00/5.01 16.00/5.01 (30) DependencyGraphProof (EQUIVALENT) 16.00/5.01 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 6 less nodes. 16.00/5.01 ---------------------------------------- 16.00/5.01 16.00/5.01 (31) 16.00/5.01 TRUE 16.28/5.06 EOF