11.31/3.80	YES
11.64/3.85	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
11.64/3.85	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.64/3.85	
11.64/3.85	
11.64/3.85	Left Termination of the query pattern
11.64/3.85	
11.64/3.85	mergesort(g,a)
11.64/3.85	
11.64/3.85	w.r.t. the given Prolog program could successfully be proven:
11.64/3.85	
11.64/3.85	(0) Prolog
11.64/3.85	(1) PrologToDTProblemTransformerProof [SOUND, 0 ms]
11.64/3.85	(2) TRIPLES
11.64/3.85	(3) TriplesToPiDPProof [SOUND, 0 ms]
11.64/3.85	(4) PiDP
11.64/3.85	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
11.64/3.85	(6) AND
11.64/3.85	    (7) PiDP
11.64/3.85	        (8) UsableRulesProof [EQUIVALENT, 0 ms]
11.64/3.85	        (9) PiDP
11.64/3.85	        (10) PiDPToQDPProof [SOUND, 0 ms]
11.64/3.85	        (11) QDP
11.64/3.85	        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.64/3.85	        (13) YES
11.64/3.85	    (14) PiDP
11.64/3.85	        (15) UsableRulesProof [EQUIVALENT, 0 ms]
11.64/3.85	        (16) PiDP
11.64/3.85	        (17) PiDPToQDPProof [SOUND, 0 ms]
11.64/3.85	        (18) QDP
11.64/3.85	        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.64/3.85	        (20) YES
11.64/3.85	    (21) PiDP
11.64/3.85	        (22) PiDPToQDPProof [SOUND, 0 ms]
11.64/3.85	        (23) QDP
11.64/3.85	        (24) TransformationProof [EQUIVALENT, 0 ms]
11.64/3.85	        (25) QDP
11.64/3.85	        (26) QDPQMonotonicMRRProof [EQUIVALENT, 50 ms]
11.64/3.85	        (27) QDP
11.64/3.85	        (28) DependencyGraphProof [EQUIVALENT, 0 ms]
11.64/3.85	        (29) QDP
11.64/3.85	        (30) UsableRulesProof [EQUIVALENT, 0 ms]
11.64/3.85	        (31) QDP
11.64/3.85	        (32) QReductionProof [EQUIVALENT, 0 ms]
11.64/3.85	        (33) QDP
11.64/3.85	        (34) QDPQMonotonicMRRProof [EQUIVALENT, 0 ms]
11.64/3.85	        (35) QDP
11.64/3.85	        (36) DependencyGraphProof [EQUIVALENT, 0 ms]
11.64/3.85	        (37) TRUE
11.64/3.85	
11.64/3.85	
11.64/3.85	----------------------------------------
11.64/3.85	
11.64/3.85	(0)
11.64/3.85	Obligation:
11.64/3.85	Clauses:
11.64/3.85	
11.64/3.85	mergesort([], []).
11.64/3.85	mergesort(.(X, []), .(X, [])).
11.64/3.85	mergesort(.(X, .(Y, Xs)), Ys) :- ','(split(.(X, .(Y, Xs)), X1s, X2s), ','(mergesort(X1s, Y1s), ','(mergesort(X2s, Y2s), merge(Y1s, Y2s, Ys)))).
11.64/3.85	split([], [], []).
11.64/3.85	split(.(X, Xs), .(X, Ys), Zs) :- split(Xs, Zs, Ys).
11.64/3.85	merge([], Xs, Xs).
11.64/3.85	merge(Xs, [], Xs).
11.64/3.85	merge(.(X, Xs), .(Y, Ys), .(X, Zs)) :- ','(=(X, Y), merge(.(X, Xs), Ys, Zs)).
11.64/3.85	
11.64/3.85	
11.64/3.85	Query: mergesort(g,a)
11.64/3.85	----------------------------------------
11.64/3.85	
11.64/3.85	(1) PrologToDTProblemTransformerProof (SOUND)
11.64/3.85	Built DT problem from termination graph DT10.
11.64/3.85	
11.64/3.85	{
11.64/3.85	    "root": 7,
11.64/3.85	    "program": {
11.64/3.85	        "directives": [],
11.64/3.85	        "clauses": [
11.64/3.85	            [
11.64/3.85	                "(mergesort ([]) ([]))",
11.64/3.85	                null
11.64/3.85	            ],
11.64/3.85	            [
11.64/3.85	                "(mergesort (. X ([])) (. X ([])))",
11.64/3.85	                null
11.64/3.85	            ],
11.64/3.85	            [
11.64/3.85	                "(mergesort (. X (. Y Xs)) Ys)",
11.64/3.85	                "(',' (split (. X (. Y Xs)) X1s X2s) (',' (mergesort X1s Y1s) (',' (mergesort X2s Y2s) (merge Y1s Y2s Ys))))"
11.64/3.85	            ],
11.64/3.85	            [
11.64/3.85	                "(split ([]) ([]) ([]))",
11.64/3.85	                null
11.64/3.85	            ],
11.64/3.85	            [
11.64/3.85	                "(split (. X Xs) (. X Ys) Zs)",
11.64/3.85	                "(split Xs Zs Ys)"
11.64/3.85	            ],
11.64/3.85	            [
11.64/3.85	                "(merge ([]) Xs Xs)",
11.64/3.85	                null
11.64/3.85	            ],
11.64/3.85	            [
11.64/3.85	                "(merge Xs ([]) Xs)",
11.64/3.85	                null
11.64/3.85	            ],
11.64/3.85	            [
11.64/3.85	                "(merge (. X Xs) (. Y Ys) (. X Zs))",
11.64/3.85	                "(',' (= X Y) (merge (. X Xs) Ys Zs))"
11.64/3.85	            ]
11.64/3.85	        ]
11.64/3.85	    },
11.64/3.85	    "graph": {
11.64/3.85	        "nodes": {
11.64/3.85	            "46": {
11.64/3.85	                "goal": [
11.64/3.85	                    {
11.64/3.85	                        "clause": -1,
11.64/3.85	                        "scope": -1,
11.64/3.85	                        "term": "(true)"
11.64/3.85	                    },
11.64/3.85	                    {
11.64/3.85	                        "clause": 1,
11.64/3.85	                        "scope": 1,
11.64/3.85	                        "term": "(mergesort ([]) T2)"
11.64/3.85	                    },
11.64/3.85	                    {
11.64/3.85	                        "clause": 2,
11.64/3.85	                        "scope": 1,
11.64/3.85	                        "term": "(mergesort ([]) T2)"
11.64/3.85	                    }
11.64/3.85	                ],
11.64/3.85	                "kb": {
11.64/3.85	                    "nonunifying": [],
11.64/3.85	                    "intvars": {},
11.64/3.85	                    "arithmetic": {
11.64/3.85	                        "type": "PlainIntegerRelationState",
11.64/3.85	                        "relations": []
11.64/3.85	                    },
11.64/3.85	                    "ground": [],
11.64/3.85	                    "free": [],
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11.64/3.85	                }
11.64/3.85	            },
11.64/3.85	            "48": {
11.64/3.85	                "goal": [
11.64/3.85	                    {
11.64/3.85	                        "clause": 1,
11.64/3.85	                        "scope": 1,
11.64/3.85	                        "term": "(mergesort T1 T2)"
11.64/3.85	                    },
11.64/3.85	                    {
11.64/3.85	                        "clause": 2,
11.64/3.85	                        "scope": 1,
11.64/3.85	                        "term": "(mergesort T1 T2)"
11.64/3.85	                    }
11.64/3.85	                ],
11.64/3.85	                "kb": {
11.64/3.85	                    "nonunifying": [[
11.64/3.85	                        "(mergesort T1 T2)",
11.64/3.85	                        "(mergesort ([]) ([]))"
11.64/3.85	                    ]],
11.64/3.85	                    "intvars": {},
11.64/3.85	                    "arithmetic": {
11.64/3.85	                        "type": "PlainIntegerRelationState",
11.64/3.85	                        "relations": []
11.64/3.85	                    },
11.64/3.85	                    "ground": ["T1"],
11.64/3.85	                    "free": [],
11.64/3.85	                    "exprvars": []
11.64/3.85	                }
11.64/3.85	            },
11.64/3.85	            "type": "Nodes",
11.64/3.85	            "393": {
11.64/3.85	                "goal": [{
11.64/3.85	                    "clause": -1,
11.64/3.85	                    "scope": -1,
11.64/3.85	                    "term": "(split (. T22 T23) X41 X40)"
11.64/3.85	                }],
11.64/3.85	                "kb": {
11.64/3.85	                    "nonunifying": [],
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11.64/3.85	                    "arithmetic": {
11.64/3.85	                        "type": "PlainIntegerRelationState",
11.64/3.85	                        "relations": []
11.64/3.85	                    },
11.64/3.85	                    "ground": [
11.64/3.85	                        "T22",
11.64/3.85	                        "T23"
11.64/3.85	                    ],
11.64/3.85	                    "free": [
11.64/3.85	                        "X40",
11.64/3.85	                        "X41"
11.64/3.85	                    ],
11.64/3.85	                    "exprvars": []
11.64/3.85	                }
11.64/3.85	            },
11.64/3.85	            "394": {
11.64/3.85	                "goal": [{
11.64/3.85	                    "clause": -1,
11.64/3.85	                    "scope": -1,
11.64/3.85	                    "term": "(',' (mergesort (. T21 T25) X22) (',' (mergesort T24 X23) (merge X22 X23 T14)))"
11.64/3.85	                }],
11.64/3.85	                "kb": {
11.64/3.85	                    "nonunifying": [],
11.64/3.85	                    "intvars": {},
11.64/3.85	                    "arithmetic": {
11.64/3.85	                        "type": "PlainIntegerRelationState",
11.64/3.85	                        "relations": []
11.64/3.85	                    },
11.64/3.85	                    "ground": [
11.64/3.85	                        "T21",
11.64/3.85	                        "T24",
11.64/3.85	                        "T25"
11.64/3.85	                    ],
11.64/3.85	                    "free": [
11.64/3.85	                        "X22",
11.64/3.85	                        "X23"
11.64/3.85	                    ],
11.64/3.85	                    "exprvars": []
11.64/3.85	                }
11.64/3.85	            },
11.64/3.85	            "471": {
11.64/3.85	                "goal": [],
11.64/3.85	                "kb": {
11.64/3.85	                    "nonunifying": [],
11.64/3.85	                    "intvars": {},
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11.64/3.85	                        "type": "PlainIntegerRelationState",
11.64/3.85	                        "relations": []
11.64/3.85	                    },
11.64/3.85	                    "ground": [],
11.64/3.85	                    "free": [],
11.64/3.85	                    "exprvars": []
11.64/3.85	                }
11.64/3.85	            },
11.64/3.85	            "395": {
11.64/3.85	                "goal": [
11.64/3.85	                    {
11.64/3.85	                        "clause": 3,
11.64/3.85	                        "scope": 3,
11.64/3.85	                        "term": "(split (. T22 T23) X41 X40)"
11.64/3.85	                    },
11.64/3.85	                    {
11.64/3.85	                        "clause": 4,
11.64/3.85	                        "scope": 3,
11.64/3.85	                        "term": "(split (. T22 T23) X41 X40)"
11.64/3.85	                    }
11.64/3.85	                ],
11.64/3.85	                "kb": {
11.64/3.85	                    "nonunifying": [],
11.64/3.85	                    "intvars": {},
11.64/3.85	                    "arithmetic": {
11.64/3.85	                        "type": "PlainIntegerRelationState",
11.64/3.85	                        "relations": []
11.64/3.85	                    },
11.64/3.85	                    "ground": [
11.64/3.85	                        "T22",
11.64/3.85	                        "T23"
11.64/3.85	                    ],
11.64/3.85	                    "free": [
11.64/3.85	                        "X40",
11.64/3.85	                        "X41"
11.64/3.85	                    ],
11.64/3.85	                    "exprvars": []
11.64/3.85	                }
11.64/3.85	            },
11.64/3.85	            "472": {
11.64/3.85	                "goal": [{
11.64/3.85	                    "clause": -1,
11.64/3.85	                    "scope": -1,
11.64/3.85	                    "term": "(merge (. T69 T63) T65 T67)"
11.64/3.85	                }],
11.64/3.85	                "kb": {
11.64/3.85	                    "nonunifying": [],
11.64/3.85	                    "intvars": {},
11.64/3.85	                    "arithmetic": {
11.64/3.85	                        "type": "PlainIntegerRelationState",
11.64/3.85	                        "relations": []
11.64/3.85	                    },
11.64/3.85	                    "ground": [
11.64/3.85	                        "T63",
11.64/3.85	                        "T65",
11.64/3.85	                        "T69"
11.64/3.85	                    ],
11.64/3.85	                    "free": [],
11.64/3.85	                    "exprvars": []
11.64/3.85	                }
11.64/3.85	            },
11.64/3.85	            "396": {
11.64/3.85	                "goal": [{
11.64/3.85	                    "clause": 4,
11.64/3.85	                    "scope": 3,
11.64/3.85	                    "term": "(split (. T22 T23) X41 X40)"
11.64/3.85	                }],
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11.64/3.85	                    "intvars": {},
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11.64/3.85	                        "type": "PlainIntegerRelationState",
11.64/3.85	                        "relations": []
11.64/3.85	                    },
11.64/3.85	                    "ground": [
11.64/3.85	                        "T22",
11.64/3.85	                        "T23"
11.64/3.85	                    ],
11.64/3.85	                    "free": [
11.64/3.85	                        "X40",
11.64/3.85	                        "X41"
11.64/3.85	                    ],
11.64/3.85	                    "exprvars": []
11.64/3.85	                }
11.64/3.85	            },
11.64/3.85	            "473": {
11.64/3.85	                "goal": [],
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11.64/3.85	                    "nonunifying": [],
11.64/3.85	                    "intvars": {},
11.64/3.85	                    "arithmetic": {
11.64/3.85	                        "type": "PlainIntegerRelationState",
11.64/3.85	                        "relations": []
11.64/3.85	                    },
11.64/3.85	                    "ground": [],
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11.64/3.85	                }
11.64/3.85	            },
11.64/3.85	            "397": {
11.64/3.85	                "goal": [{
11.64/3.85	                    "clause": -1,
11.64/3.85	                    "scope": -1,
11.64/3.85	                    "term": "(split T31 X59 X58)"
11.64/3.85	                }],
11.64/3.85	                "kb": {
11.64/3.85	                    "nonunifying": [],
11.64/3.85	                    "intvars": {},
11.64/3.85	                    "arithmetic": {
11.64/3.85	                        "type": "PlainIntegerRelationState",
11.64/3.85	                        "relations": []
11.64/3.85	                    },
11.64/3.85	                    "ground": ["T31"],
11.64/3.85	                    "free": [
11.64/3.85	                        "X58",
11.64/3.85	                        "X59"
11.64/3.85	                    ],
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11.64/3.85	                }
11.64/3.85	            },
11.64/3.85	            "398": {
11.64/3.85	                "goal": [
11.64/3.85	                    {
11.64/3.85	                        "clause": 3,
11.64/3.85	                        "scope": 4,
11.64/3.85	                        "term": "(split T31 X59 X58)"
11.64/3.85	                    },
11.64/3.85	                    {
11.64/3.85	                        "clause": 4,
11.64/3.85	                        "scope": 4,
11.64/3.85	                        "term": "(split T31 X59 X58)"
11.64/3.85	                    }
11.64/3.85	                ],
11.64/3.85	                "kb": {
11.64/3.85	                    "nonunifying": [],
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11.64/3.85	                    "arithmetic": {
11.64/3.85	                        "type": "PlainIntegerRelationState",
11.64/3.85	                        "relations": []
11.64/3.85	                    },
11.64/3.85	                    "ground": ["T31"],
11.64/3.85	                    "free": [
11.64/3.85	                        "X58",
11.64/3.85	                        "X59"
11.64/3.85	                    ],
11.64/3.85	                    "exprvars": []
11.64/3.85	                }
11.64/3.85	            },
11.64/3.85	            "399": {
11.64/3.85	                "goal": [{
11.64/3.85	                    "clause": 3,
11.64/3.85	                    "scope": 4,
11.64/3.85	                    "term": "(split T31 X59 X58)"
11.64/3.85	                }],
11.64/3.85	                "kb": {
11.64/3.85	                    "nonunifying": [],
11.64/3.85	                    "intvars": {},
11.64/3.85	                    "arithmetic": {
11.64/3.85	                        "type": "PlainIntegerRelationState",
11.64/3.85	                        "relations": []
11.64/3.85	                    },
11.64/3.85	                    "ground": ["T31"],
11.64/3.85	                    "free": [
11.64/3.85	                        "X58",
11.64/3.85	                        "X59"
11.64/3.85	                    ],
11.64/3.85	                    "exprvars": []
11.64/3.85	                }
11.64/3.85	            },
11.64/3.85	            "438": {
11.64/3.85	                "goal": [{
11.64/3.85	                    "clause": -1,
11.64/3.85	                    "scope": -1,
11.64/3.85	                    "term": "(mergesort (. T21 T25) X22)"
11.64/3.85	                }],
11.64/3.85	                "kb": {
11.64/3.85	                    "nonunifying": [],
11.64/3.85	                    "intvars": {},
11.64/3.85	                    "arithmetic": {
11.64/3.85	                        "type": "PlainIntegerRelationState",
11.64/3.85	                        "relations": []
11.64/3.85	                    },
11.64/3.85	                    "ground": [
11.64/3.85	                        "T21",
11.64/3.85	                        "T25"
11.64/3.85	                    ],
11.64/3.85	                    "free": ["X22"],
11.64/3.85	                    "exprvars": []
11.64/3.85	                }
11.64/3.85	            },
11.64/3.85	            "439": {
11.64/3.85	                "goal": [{
11.64/3.85	                    "clause": -1,
11.64/3.85	                    "scope": -1,
11.64/3.85	                    "term": "(',' (mergesort T24 X23) (merge T38 X23 T14))"
11.64/3.85	                }],
11.64/3.85	                "kb": {
11.64/3.85	                    "nonunifying": [],
11.64/3.85	                    "intvars": {},
11.64/3.85	                    "arithmetic": {
11.64/3.85	                        "type": "PlainIntegerRelationState",
11.64/3.85	                        "relations": []
11.64/3.85	                    },
11.64/3.85	                    "ground": [
11.64/3.85	                        "T24",
11.64/3.85	                        "T38"
11.64/3.85	                    ],
11.64/3.85	                    "free": ["X23"],
11.64/3.85	                    "exprvars": []
11.64/3.85	                }
11.64/3.85	            },
11.64/3.85	            "241": {
11.64/3.85	                "goal": [
11.64/3.85	                    {
11.64/3.85	                        "clause": 1,
11.64/3.85	                        "scope": 1,
11.64/3.85	                        "term": "(mergesort ([]) T2)"
11.64/3.85	                    },
11.64/3.85	                    {
11.64/3.85	                        "clause": 2,
11.64/3.85	                        "scope": 1,
11.64/3.85	                        "term": "(mergesort ([]) T2)"
11.64/3.85	                    }
11.64/3.85	                ],
11.64/3.85	                "kb": {
11.64/3.85	                    "nonunifying": [],
11.64/3.85	                    "intvars": {},
11.64/3.85	                    "arithmetic": {
11.64/3.85	                        "type": "PlainIntegerRelationState",
11.64/3.85	                        "relations": []
11.64/3.85	                    },
11.64/3.85	                    "ground": [],
11.64/3.85	                    "free": [],
11.64/3.85	                    "exprvars": []
11.64/3.85	                }
11.64/3.85	            },
11.64/3.85	            "242": {
11.64/3.85	                "goal": [{
11.64/3.85	                    "clause": 2,
11.64/3.85	                    "scope": 1,
11.64/3.85	                    "term": "(mergesort ([]) T2)"
11.64/3.85	                }],
11.64/3.85	                "kb": {
11.64/3.85	                    "nonunifying": [],
11.64/3.85	                    "intvars": {},
11.64/3.85	                    "arithmetic": {
11.64/3.85	                        "type": "PlainIntegerRelationState",
11.64/3.85	                        "relations": []
11.64/3.85	                    },
11.64/3.85	                    "ground": [],
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11.64/3.85	                }
11.64/3.85	            },
11.64/3.85	            "243": {
11.64/3.85	                "goal": [],
11.64/3.85	                "kb": {
11.64/3.85	                    "nonunifying": [],
11.64/3.85	                    "intvars": {},
11.64/3.85	                    "arithmetic": {
11.64/3.85	                        "type": "PlainIntegerRelationState",
11.64/3.85	                        "relations": []
11.64/3.85	                    },
11.64/3.85	                    "ground": [],
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11.64/3.85	                }
11.64/3.85	            },
11.64/3.85	            "244": {
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11.64/3.85	                        "T38",
11.64/3.85	                        "T39"
11.64/3.85	                    ],
11.64/3.85	                    "free": [],
11.64/3.85	                    "exprvars": []
11.64/3.85	                }
11.64/3.85	            },
11.64/3.85	            "457": {
11.64/3.85	                "goal": [{
11.64/3.85	                    "clause": 7,
11.64/3.85	                    "scope": 5,
11.64/3.85	                    "term": "(merge T38 T39 T14)"
11.64/3.85	                }],
11.64/3.85	                "kb": {
11.64/3.85	                    "nonunifying": [],
11.64/3.85	                    "intvars": {},
11.64/3.85	                    "arithmetic": {
11.64/3.85	                        "type": "PlainIntegerRelationState",
11.64/3.85	                        "relations": []
11.64/3.85	                    },
11.64/3.85	                    "ground": [
11.64/3.85	                        "T38",
11.64/3.85	                        "T39"
11.64/3.85	                    ],
11.64/3.85	                    "free": [],
11.64/3.85	                    "exprvars": []
11.64/3.85	                }
11.64/3.85	            },
11.64/3.85	            "262": {
11.64/3.85	                "goal": [{
11.64/3.85	                    "clause": -1,
11.64/3.85	                    "scope": -1,
11.64/3.85	                    "term": "(',' (split (. T22 T23) X41 X40) (',' (mergesort (. T21 X40) X22) (',' (mergesort X41 X23) (merge X22 X23 T14))))"
11.64/3.85	                }],
11.64/3.85	                "kb": {
11.64/3.85	                    "nonunifying": [],
11.64/3.85	                    "intvars": {},
11.64/3.85	                    "arithmetic": {
11.64/3.85	                        "type": "PlainIntegerRelationState",
11.64/3.85	                        "relations": []
11.64/3.85	                    },
11.64/3.85	                    "ground": [
11.64/3.85	                        "T21",
11.64/3.85	                        "T22",
11.64/3.85	                        "T23"
11.64/3.85	                    ],
11.64/3.85	                    "free": [
11.64/3.85	                        "X22",
11.64/3.85	                        "X23",
11.64/3.85	                        "X40",
11.64/3.85	                        "X41"
11.64/3.85	                    ],
11.64/3.85	                    "exprvars": []
11.64/3.85	                }
11.64/3.85	            },
11.64/3.85	            "460": {
11.64/3.85	                "goal": [{
11.64/3.85	                    "clause": -1,
11.64/3.85	                    "scope": -1,
11.64/3.85	                    "term": "(true)"
11.64/3.85	                }],
11.64/3.85	                "kb": {
11.64/3.85	                    "nonunifying": [],
11.64/3.85	                    "intvars": {},
11.64/3.85	                    "arithmetic": {
11.64/3.85	                        "type": "PlainIntegerRelationState",
11.64/3.85	                        "relations": []
11.64/3.85	                    },
11.64/3.85	                    "ground": [],
11.64/3.85	                    "free": [],
11.64/3.85	                    "exprvars": []
11.64/3.85	                }
11.64/3.85	            },
11.64/3.85	            "461": {
11.64/3.85	                "goal": [],
11.64/3.85	                "kb": {
11.64/3.85	                    "nonunifying": [],
11.64/3.85	                    "intvars": {},
11.64/3.85	                    "arithmetic": {
11.64/3.85	                        "type": "PlainIntegerRelationState",
11.64/3.85	                        "relations": []
11.64/3.85	                    },
11.64/3.85	                    "ground": [],
11.64/3.85	                    "free": [],
11.64/3.85	                    "exprvars": []
11.64/3.85	                }
11.64/3.85	            },
11.64/3.85	            "462": {
11.64/3.85	                "goal": [],
11.64/3.85	                "kb": {
11.64/3.85	                    "nonunifying": [],
11.64/3.85	                    "intvars": {},
11.64/3.85	                    "arithmetic": {
11.64/3.85	                        "type": "PlainIntegerRelationState",
11.64/3.85	                        "relations": []
11.64/3.85	                    },
11.64/3.85	                    "ground": [],
11.64/3.85	                    "free": [],
11.64/3.85	                    "exprvars": []
11.64/3.85	                }
11.64/3.85	            },
11.64/3.85	            "469": {
11.64/3.85	                "goal": [{
11.64/3.85	                    "clause": -1,
11.64/3.85	                    "scope": -1,
11.64/3.85	                    "term": "(',' (= T62 T64) (merge (. T62 T63) T65 T67))"
11.64/3.85	                }],
11.64/3.85	                "kb": {
11.64/3.85	                    "nonunifying": [],
11.64/3.85	                    "intvars": {},
11.64/3.85	                    "arithmetic": {
11.64/3.85	                        "type": "PlainIntegerRelationState",
11.64/3.85	                        "relations": []
11.64/3.85	                    },
11.64/3.85	                    "ground": [
11.64/3.85	                        "T62",
11.64/3.85	                        "T63",
11.64/3.85	                        "T64",
11.64/3.85	                        "T65"
11.64/3.85	                    ],
11.64/3.85	                    "free": [],
11.64/3.85	                    "exprvars": []
11.64/3.85	                }
11.64/3.85	            }
11.64/3.85	        },
11.64/3.85	        "edges": [
11.64/3.85	            {
11.64/3.85	                "from": 7,
11.64/3.85	                "to": 21,
11.64/3.85	                "label": "CASE"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 21,
11.64/3.85	                "to": 46,
11.64/3.85	                "label": "EVAL with clause\nmergesort([], []).\nand substitutionT1 -> [],\nT2 -> []"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 21,
11.64/3.85	                "to": 48,
11.64/3.85	                "label": "EVAL-BACKTRACK"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 46,
11.64/3.85	                "to": 241,
11.64/3.85	                "label": "SUCCESS"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 48,
11.64/3.85	                "to": 244,
11.64/3.85	                "label": "EVAL with clause\nmergesort(.(X7, []), .(X7, [])).\nand substitutionX7 -> T4,\nT1 -> .(T4, []),\nT2 -> .(T4, [])"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 48,
11.64/3.85	                "to": 246,
11.64/3.85	                "label": "EVAL-BACKTRACK"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 241,
11.64/3.85	                "to": 242,
11.64/3.85	                "label": "BACKTRACK\nfor clause: mergesort(.(X, []), .(X, []))because of non-unification"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 242,
11.64/3.85	                "to": 243,
11.64/3.85	                "label": "BACKTRACK\nfor clause: mergesort(.(X, .(Y, Xs)), Ys) :- ','(split(.(X, .(Y, Xs)), X1s, X2s), ','(mergesort(X1s, Y1s), ','(mergesort(X2s, Y2s), merge(Y1s, Y2s, Ys))))because of non-unification"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 244,
11.64/3.85	                "to": 247,
11.64/3.85	                "label": "SUCCESS"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 246,
11.64/3.85	                "to": 251,
11.64/3.85	                "label": "EVAL with clause\nmergesort(.(X16, .(X17, X18)), X19) :- ','(split(.(X16, .(X17, X18)), X20, X21), ','(mergesort(X20, X22), ','(mergesort(X21, X23), merge(X22, X23, X19)))).\nand substitutionX16 -> T10,\nX17 -> T11,\nX18 -> T12,\nT1 -> .(T10, .(T11, T12)),\nT2 -> T14,\nX19 -> T14,\nT13 -> T14"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 246,
11.64/3.85	                "to": 252,
11.64/3.85	                "label": "EVAL-BACKTRACK"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 247,
11.64/3.85	                "to": 248,
11.64/3.85	                "label": "BACKTRACK\nfor clause: mergesort(.(X, .(Y, Xs)), Ys) :- ','(split(.(X, .(Y, Xs)), X1s, X2s), ','(mergesort(X1s, Y1s), ','(mergesort(X2s, Y2s), merge(Y1s, Y2s, Ys))))because of non-unification"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 251,
11.64/3.85	                "to": 253,
11.64/3.85	                "label": "CASE"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 253,
11.64/3.85	                "to": 254,
11.64/3.85	                "label": "BACKTRACK\nfor clause: split([], [], [])because of non-unification"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 254,
11.64/3.85	                "to": 262,
11.64/3.85	                "label": "ONLY EVAL with clause\nsplit(.(X36, X37), .(X36, X38), X39) :- split(X37, X39, X38).\nand substitutionT10 -> T21,\nX36 -> T21,\nT11 -> T22,\nT12 -> T23,\nX37 -> .(T22, T23),\nX38 -> X40,\nX20 -> .(T21, X40),\nX21 -> X41,\nX39 -> X41"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 262,
11.64/3.85	                "to": 393,
11.64/3.85	                "label": "SPLIT 1"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 262,
11.64/3.85	                "to": 394,
11.64/3.85	                "label": "SPLIT 2\nnew knowledge:\nT22 is ground\nT23 is ground\nT24 is ground\nT25 is ground\nreplacements:X41 -> T24,\nX40 -> T25"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 393,
11.64/3.85	                "to": 395,
11.64/3.85	                "label": "CASE"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 394,
11.64/3.85	                "to": 438,
11.64/3.85	                "label": "SPLIT 1"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 394,
11.64/3.85	                "to": 439,
11.64/3.85	                "label": "SPLIT 2\nnew knowledge:\nT21 is ground\nT25 is ground\nT38 is ground\nreplacements:X22 -> T38"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 395,
11.64/3.85	                "to": 396,
11.64/3.85	                "label": "BACKTRACK\nfor clause: split([], [], [])because of non-unification"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 396,
11.64/3.85	                "to": 397,
11.64/3.85	                "label": "ONLY EVAL with clause\nsplit(.(X54, X55), .(X54, X56), X57) :- split(X55, X57, X56).\nand substitutionT22 -> T30,\nX54 -> T30,\nT23 -> T31,\nX55 -> T31,\nX56 -> X58,\nX41 -> .(T30, X58),\nX40 -> X59,\nX57 -> X59"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 397,
11.64/3.85	                "to": 398,
11.64/3.85	                "label": "CASE"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 398,
11.64/3.85	                "to": 399,
11.64/3.85	                "label": "PARALLEL"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 398,
11.64/3.85	                "to": 400,
11.64/3.85	                "label": "PARALLEL"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 399,
11.64/3.85	                "to": 401,
11.64/3.85	                "label": "EVAL with clause\nsplit([], [], []).\nand substitutionT31 -> [],\nX59 -> [],\nX58 -> []"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 399,
11.64/3.85	                "to": 402,
11.64/3.85	                "label": "EVAL-BACKTRACK"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 400,
11.64/3.85	                "to": 407,
11.64/3.85	                "label": "EVAL with clause\nsplit(.(X72, X73), .(X72, X74), X75) :- split(X73, X75, X74).\nand substitutionX72 -> T36,\nX73 -> T37,\nT31 -> .(T36, T37),\nX74 -> X76,\nX59 -> .(T36, X76),\nX58 -> X77,\nX75 -> X77"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 400,
11.64/3.85	                "to": 410,
11.64/3.85	                "label": "EVAL-BACKTRACK"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 401,
11.64/3.85	                "to": 403,
11.64/3.85	                "label": "SUCCESS"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 407,
11.64/3.85	                "to": 397,
11.64/3.85	                "label": "INSTANCE with matching:\nT31 -> T37\nX59 -> X77\nX58 -> X76"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 438,
11.64/3.85	                "to": 7,
11.64/3.85	                "label": "INSTANCE with matching:\nT1 -> .(T21, T25)\nT2 -> X22"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 439,
11.64/3.85	                "to": 442,
11.64/3.85	                "label": "SPLIT 1"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 439,
11.64/3.85	                "to": 443,
11.64/3.85	                "label": "SPLIT 2\nnew knowledge:\nT24 is ground\nT39 is ground\nreplacements:X23 -> T39"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 442,
11.64/3.85	                "to": 7,
11.64/3.85	                "label": "INSTANCE with matching:\nT1 -> T24\nT2 -> X23"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 443,
11.64/3.85	                "to": 445,
11.64/3.85	                "label": "CASE"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 445,
11.64/3.85	                "to": 447,
11.64/3.85	                "label": "PARALLEL"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 445,
11.64/3.85	                "to": 449,
11.64/3.85	                "label": "PARALLEL"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 447,
11.64/3.85	                "to": 450,
11.64/3.85	                "label": "EVAL with clause\nmerge([], X84, X84).\nand substitutionT38 -> [],\nT39 -> T46,\nX84 -> T46,\nT14 -> T46"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 447,
11.64/3.85	                "to": 451,
11.64/3.85	                "label": "EVAL-BACKTRACK"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 449,
11.64/3.85	                "to": 456,
11.64/3.85	                "label": "PARALLEL"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 449,
11.64/3.85	                "to": 457,
11.64/3.85	                "label": "PARALLEL"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 450,
11.64/3.85	                "to": 452,
11.64/3.85	                "label": "SUCCESS"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 456,
11.64/3.85	                "to": 460,
11.64/3.85	                "label": "EVAL with clause\nmerge(X89, [], X89).\nand substitutionT38 -> T51,\nX89 -> T51,\nT39 -> [],\nT14 -> T51"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 456,
11.64/3.85	                "to": 461,
11.64/3.85	                "label": "EVAL-BACKTRACK"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 457,
11.64/3.85	                "to": 469,
11.64/3.85	                "label": "EVAL with clause\nmerge(.(X100, X101), .(X102, X103), .(X100, X104)) :- ','(=(X100, X102), merge(.(X100, X101), X103, X104)).\nand substitutionX100 -> T62,\nX101 -> T63,\nT38 -> .(T62, T63),\nX102 -> T64,\nX103 -> T65,\nT39 -> .(T64, T65),\nX104 -> T67,\nT14 -> .(T62, T67),\nT66 -> T67"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 457,
11.64/3.85	                "to": 471,
11.64/3.85	                "label": "EVAL-BACKTRACK"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 460,
11.64/3.85	                "to": 462,
11.64/3.85	                "label": "SUCCESS"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 469,
11.64/3.85	                "to": 472,
11.64/3.85	                "label": "UNIFY CASE with substitutionT62 -> T69,\nT64 -> T69"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 469,
11.64/3.85	                "to": 473,
11.64/3.85	                "label": "UNIFY-BACKTRACK"
11.64/3.85	            },
11.64/3.85	            {
11.64/3.85	                "from": 472,
11.64/3.85	                "to": 443,
11.64/3.85	                "label": "INSTANCE with matching:\nT38 -> .(T69, T63)\nT39 -> T65\nT14 -> T67"
11.64/3.85	            }
11.64/3.85	        ],
11.64/3.85	        "type": "Graph"
11.64/3.85	    }
11.64/3.85	}
11.64/3.85	
11.64/3.85	----------------------------------------
11.64/3.85	
11.64/3.85	(2)
11.64/3.85	Obligation:
11.64/3.85	Triples:
11.64/3.85	
11.64/3.85	splitA(.(X1, X2), .(X1, X3), X4) :- splitA(X2, X4, X3).
11.64/3.85	mergeC(.(X1, X2), .(X1, X3), .(X1, X4)) :- mergeC(.(X1, X2), X3, X4).
11.64/3.85	mergesortB(.(X1, .(X2, X3)), X4) :- splitA(X3, X5, X6).
11.64/3.85	mergesortB(.(X1, .(X2, X3)), X4) :- ','(splitcD(X2, X3, X5, X6), mergesortB(.(X1, X6), X7)).
11.64/3.85	mergesortB(.(X1, .(X2, X3)), X4) :- ','(splitcD(X2, X3, X5, X6), ','(mergesortcB(.(X1, X6), X7), mergesortB(X5, X8))).
11.64/3.85	mergesortB(.(X1, .(X2, X3)), X4) :- ','(splitcD(X2, X3, X5, X6), ','(mergesortcB(.(X1, X6), X7), ','(mergesortcB(X5, X8), mergeC(X7, X8, X4)))).
11.64/3.85	
11.64/3.85	Clauses:
11.64/3.85	
11.64/3.85	splitcA([], [], []).
11.64/3.85	splitcA(.(X1, X2), .(X1, X3), X4) :- splitcA(X2, X4, X3).
11.64/3.85	mergesortcB([], []).
11.64/3.85	mergesortcB(.(X1, []), .(X1, [])).
11.64/3.85	mergesortcB(.(X1, .(X2, X3)), X4) :- ','(splitcD(X2, X3, X5, X6), ','(mergesortcB(.(X1, X6), X7), ','(mergesortcB(X5, X8), mergecC(X7, X8, X4)))).
11.64/3.85	mergecC([], X1, X1).
11.64/3.85	mergecC(X1, [], X1).
11.64/3.85	mergecC(.(X1, X2), .(X1, X3), .(X1, X4)) :- mergecC(.(X1, X2), X3, X4).
11.64/3.85	splitcD(X1, X2, .(X1, X3), X4) :- splitcA(X2, X4, X3).
11.64/3.85	
11.64/3.85	Afs:
11.64/3.85	
11.64/3.85	mergesortB(x1, x2)  =  mergesortB(x1)
11.64/3.86	
11.64/3.86	
11.64/3.86	----------------------------------------
11.64/3.86	
11.64/3.86	(3) TriplesToPiDPProof (SOUND)
11.64/3.86	We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
11.64/3.86	
11.64/3.86	mergesortB_in_2: (b,f)
11.64/3.86	
11.64/3.86	splitA_in_3: (b,f,f)
11.64/3.86	
11.64/3.86	splitcD_in_4: (b,b,f,f)
11.64/3.86	
11.64/3.86	splitcA_in_3: (b,f,f)
11.64/3.86	
11.64/3.86	mergesortcB_in_2: (b,f)
11.64/3.86	
11.64/3.86	mergecC_in_3: (b,b,f)
11.64/3.86	
11.64/3.86	mergeC_in_3: (b,b,f)
11.64/3.86	
11.64/3.86	Transforming TRIPLES into the following Term Rewriting System:
11.64/3.86	
11.64/3.86	Pi DP problem:
11.64/3.86	The TRS P consists of the following rules:
11.64/3.86	
11.64/3.86	   MERGESORTB_IN_GA(.(X1, .(X2, X3)), X4) -> U3_GA(X1, X2, X3, X4, splitA_in_gaa(X3, X5, X6))
11.64/3.86	   MERGESORTB_IN_GA(.(X1, .(X2, X3)), X4) -> SPLITA_IN_GAA(X3, X5, X6)
11.64/3.86	   SPLITA_IN_GAA(.(X1, X2), .(X1, X3), X4) -> U1_GAA(X1, X2, X3, X4, splitA_in_gaa(X2, X4, X3))
11.64/3.86	   SPLITA_IN_GAA(.(X1, X2), .(X1, X3), X4) -> SPLITA_IN_GAA(X2, X4, X3)
11.64/3.86	   MERGESORTB_IN_GA(.(X1, .(X2, X3)), X4) -> U4_GA(X1, X2, X3, X4, splitcD_in_ggaa(X2, X3, X5, X6))
11.64/3.86	   U4_GA(X1, X2, X3, X4, splitcD_out_ggaa(X2, X3, X5, X6)) -> U5_GA(X1, X2, X3, X4, mergesortB_in_ga(.(X1, X6), X7))
11.64/3.86	   U4_GA(X1, X2, X3, X4, splitcD_out_ggaa(X2, X3, X5, X6)) -> MERGESORTB_IN_GA(.(X1, X6), X7)
11.64/3.86	   U4_GA(X1, X2, X3, X4, splitcD_out_ggaa(X2, X3, X5, X6)) -> U6_GA(X1, X2, X3, X4, X5, mergesortcB_in_ga(.(X1, X6), X7))
11.64/3.86	   U6_GA(X1, X2, X3, X4, X5, mergesortcB_out_ga(.(X1, X6), X7)) -> U7_GA(X1, X2, X3, X4, mergesortB_in_ga(X5, X8))
11.64/3.86	   U6_GA(X1, X2, X3, X4, X5, mergesortcB_out_ga(.(X1, X6), X7)) -> MERGESORTB_IN_GA(X5, X8)
11.64/3.86	   U6_GA(X1, X2, X3, X4, X5, mergesortcB_out_ga(.(X1, X6), X7)) -> U8_GA(X1, X2, X3, X4, X7, mergesortcB_in_ga(X5, X8))
11.64/3.86	   U8_GA(X1, X2, X3, X4, X7, mergesortcB_out_ga(X5, X8)) -> U9_GA(X1, X2, X3, X4, mergeC_in_gga(X7, X8, X4))
11.64/3.86	   U8_GA(X1, X2, X3, X4, X7, mergesortcB_out_ga(X5, X8)) -> MERGEC_IN_GGA(X7, X8, X4)
11.64/3.86	   MERGEC_IN_GGA(.(X1, X2), .(X1, X3), .(X1, X4)) -> U2_GGA(X1, X2, X3, X4, mergeC_in_gga(.(X1, X2), X3, X4))
11.64/3.86	   MERGEC_IN_GGA(.(X1, X2), .(X1, X3), .(X1, X4)) -> MERGEC_IN_GGA(.(X1, X2), X3, X4)
11.64/3.86	
11.64/3.86	The TRS R consists of the following rules:
11.64/3.86	
11.64/3.86	   splitcD_in_ggaa(X1, X2, .(X1, X3), X4) -> U17_ggaa(X1, X2, X3, X4, splitcA_in_gaa(X2, X4, X3))
11.64/3.86	   splitcA_in_gaa([], [], []) -> splitcA_out_gaa([], [], [])
11.64/3.86	   splitcA_in_gaa(.(X1, X2), .(X1, X3), X4) -> U11_gaa(X1, X2, X3, X4, splitcA_in_gaa(X2, X4, X3))
11.64/3.86	   U11_gaa(X1, X2, X3, X4, splitcA_out_gaa(X2, X4, X3)) -> splitcA_out_gaa(.(X1, X2), .(X1, X3), X4)
11.64/3.86	   U17_ggaa(X1, X2, X3, X4, splitcA_out_gaa(X2, X4, X3)) -> splitcD_out_ggaa(X1, X2, .(X1, X3), X4)
11.64/3.86	   mergesortcB_in_ga([], []) -> mergesortcB_out_ga([], [])
11.64/3.86	   mergesortcB_in_ga(.(X1, []), .(X1, [])) -> mergesortcB_out_ga(.(X1, []), .(X1, []))
11.64/3.86	   mergesortcB_in_ga(.(X1, .(X2, X3)), X4) -> U12_ga(X1, X2, X3, X4, splitcD_in_ggaa(X2, X3, X5, X6))
11.64/3.86	   U12_ga(X1, X2, X3, X4, splitcD_out_ggaa(X2, X3, X5, X6)) -> U13_ga(X1, X2, X3, X4, X5, X6, mergesortcB_in_ga(.(X1, X6), X7))
11.64/3.86	   U13_ga(X1, X2, X3, X4, X5, X6, mergesortcB_out_ga(.(X1, X6), X7)) -> U14_ga(X1, X2, X3, X4, X5, X6, X7, mergesortcB_in_ga(X5, X8))
11.64/3.86	   U14_ga(X1, X2, X3, X4, X5, X6, X7, mergesortcB_out_ga(X5, X8)) -> U15_ga(X1, X2, X3, X4, mergecC_in_gga(X7, X8, X4))
11.64/3.86	   mergecC_in_gga([], X1, X1) -> mergecC_out_gga([], X1, X1)
11.64/3.86	   mergecC_in_gga(X1, [], X1) -> mergecC_out_gga(X1, [], X1)
11.64/3.86	   mergecC_in_gga(.(X1, X2), .(X1, X3), .(X1, X4)) -> U16_gga(X1, X2, X3, X4, mergecC_in_gga(.(X1, X2), X3, X4))
11.64/3.86	   U16_gga(X1, X2, X3, X4, mergecC_out_gga(.(X1, X2), X3, X4)) -> mergecC_out_gga(.(X1, X2), .(X1, X3), .(X1, X4))
11.64/3.86	   U15_ga(X1, X2, X3, X4, mergecC_out_gga(X7, X8, X4)) -> mergesortcB_out_ga(.(X1, .(X2, X3)), X4)
11.64/3.86	
11.64/3.86	The argument filtering Pi contains the following mapping:
11.64/3.86	mergesortB_in_ga(x1, x2)  =  mergesortB_in_ga(x1)
11.64/3.86	
11.64/3.86	.(x1, x2)  =  .(x1, x2)
11.64/3.86	
11.64/3.86	splitA_in_gaa(x1, x2, x3)  =  splitA_in_gaa(x1)
11.64/3.86	
11.64/3.86	splitcD_in_ggaa(x1, x2, x3, x4)  =  splitcD_in_ggaa(x1, x2)
11.64/3.86	
11.64/3.86	U17_ggaa(x1, x2, x3, x4, x5)  =  U17_ggaa(x1, x2, x5)
11.64/3.86	
11.64/3.86	splitcA_in_gaa(x1, x2, x3)  =  splitcA_in_gaa(x1)
11.64/3.86	
11.64/3.86	[]  =  []
11.64/3.86	
11.64/3.86	splitcA_out_gaa(x1, x2, x3)  =  splitcA_out_gaa(x1, x2, x3)
11.64/3.86	
11.64/3.86	U11_gaa(x1, x2, x3, x4, x5)  =  U11_gaa(x1, x2, x5)
11.64/3.86	
11.64/3.86	splitcD_out_ggaa(x1, x2, x3, x4)  =  splitcD_out_ggaa(x1, x2, x3, x4)
11.64/3.86	
11.64/3.86	mergesortcB_in_ga(x1, x2)  =  mergesortcB_in_ga(x1)
11.64/3.86	
11.64/3.86	mergesortcB_out_ga(x1, x2)  =  mergesortcB_out_ga(x1, x2)
11.64/3.86	
11.64/3.86	U12_ga(x1, x2, x3, x4, x5)  =  U12_ga(x1, x2, x3, x5)
11.64/3.86	
11.64/3.86	U13_ga(x1, x2, x3, x4, x5, x6, x7)  =  U13_ga(x1, x2, x3, x5, x7)
11.64/3.86	
11.64/3.86	U14_ga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_ga(x1, x2, x3, x7, x8)
11.64/3.86	
11.64/3.86	U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x3, x5)
11.64/3.86	
11.64/3.86	mergecC_in_gga(x1, x2, x3)  =  mergecC_in_gga(x1, x2)
11.64/3.86	
11.64/3.86	mergecC_out_gga(x1, x2, x3)  =  mergecC_out_gga(x1, x2, x3)
11.64/3.86	
11.64/3.86	U16_gga(x1, x2, x3, x4, x5)  =  U16_gga(x1, x2, x3, x5)
11.64/3.86	
11.64/3.86	mergeC_in_gga(x1, x2, x3)  =  mergeC_in_gga(x1, x2)
11.64/3.86	
11.64/3.86	MERGESORTB_IN_GA(x1, x2)  =  MERGESORTB_IN_GA(x1)
11.64/3.86	
11.64/3.86	U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x3, x5)
11.64/3.86	
11.64/3.86	SPLITA_IN_GAA(x1, x2, x3)  =  SPLITA_IN_GAA(x1)
11.64/3.86	
11.64/3.86	U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x1, x2, x5)
11.64/3.86	
11.64/3.86	U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x2, x3, x5)
11.64/3.86	
11.64/3.86	U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x1, x2, x3, x5)
11.64/3.86	
11.64/3.86	U6_GA(x1, x2, x3, x4, x5, x6)  =  U6_GA(x1, x2, x3, x5, x6)
11.64/3.86	
11.64/3.86	U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x1, x2, x3, x5)
11.64/3.86	
11.64/3.86	U8_GA(x1, x2, x3, x4, x5, x6)  =  U8_GA(x1, x2, x3, x5, x6)
11.64/3.86	
11.64/3.86	U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x1, x2, x3, x5)
11.64/3.86	
11.64/3.86	MERGEC_IN_GGA(x1, x2, x3)  =  MERGEC_IN_GGA(x1, x2)
11.64/3.86	
11.64/3.86	U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x1, x2, x3, x5)
11.64/3.86	
11.64/3.86	
11.64/3.86	We have to consider all (P,R,Pi)-chains
11.64/3.86	
11.64/3.86	
11.64/3.86	Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES
11.64/3.86	
11.64/3.86	
11.64/3.86	
11.64/3.86	----------------------------------------
11.64/3.86	
11.64/3.86	(4)
11.64/3.86	Obligation:
11.64/3.86	Pi DP problem:
11.64/3.86	The TRS P consists of the following rules:
11.64/3.86	
11.64/3.86	   MERGESORTB_IN_GA(.(X1, .(X2, X3)), X4) -> U3_GA(X1, X2, X3, X4, splitA_in_gaa(X3, X5, X6))
11.64/3.86	   MERGESORTB_IN_GA(.(X1, .(X2, X3)), X4) -> SPLITA_IN_GAA(X3, X5, X6)
11.64/3.86	   SPLITA_IN_GAA(.(X1, X2), .(X1, X3), X4) -> U1_GAA(X1, X2, X3, X4, splitA_in_gaa(X2, X4, X3))
11.64/3.86	   SPLITA_IN_GAA(.(X1, X2), .(X1, X3), X4) -> SPLITA_IN_GAA(X2, X4, X3)
11.64/3.86	   MERGESORTB_IN_GA(.(X1, .(X2, X3)), X4) -> U4_GA(X1, X2, X3, X4, splitcD_in_ggaa(X2, X3, X5, X6))
11.64/3.86	   U4_GA(X1, X2, X3, X4, splitcD_out_ggaa(X2, X3, X5, X6)) -> U5_GA(X1, X2, X3, X4, mergesortB_in_ga(.(X1, X6), X7))
11.64/3.86	   U4_GA(X1, X2, X3, X4, splitcD_out_ggaa(X2, X3, X5, X6)) -> MERGESORTB_IN_GA(.(X1, X6), X7)
11.64/3.86	   U4_GA(X1, X2, X3, X4, splitcD_out_ggaa(X2, X3, X5, X6)) -> U6_GA(X1, X2, X3, X4, X5, mergesortcB_in_ga(.(X1, X6), X7))
11.64/3.86	   U6_GA(X1, X2, X3, X4, X5, mergesortcB_out_ga(.(X1, X6), X7)) -> U7_GA(X1, X2, X3, X4, mergesortB_in_ga(X5, X8))
11.64/3.86	   U6_GA(X1, X2, X3, X4, X5, mergesortcB_out_ga(.(X1, X6), X7)) -> MERGESORTB_IN_GA(X5, X8)
11.64/3.86	   U6_GA(X1, X2, X3, X4, X5, mergesortcB_out_ga(.(X1, X6), X7)) -> U8_GA(X1, X2, X3, X4, X7, mergesortcB_in_ga(X5, X8))
11.64/3.86	   U8_GA(X1, X2, X3, X4, X7, mergesortcB_out_ga(X5, X8)) -> U9_GA(X1, X2, X3, X4, mergeC_in_gga(X7, X8, X4))
11.64/3.86	   U8_GA(X1, X2, X3, X4, X7, mergesortcB_out_ga(X5, X8)) -> MERGEC_IN_GGA(X7, X8, X4)
11.64/3.86	   MERGEC_IN_GGA(.(X1, X2), .(X1, X3), .(X1, X4)) -> U2_GGA(X1, X2, X3, X4, mergeC_in_gga(.(X1, X2), X3, X4))
11.64/3.86	   MERGEC_IN_GGA(.(X1, X2), .(X1, X3), .(X1, X4)) -> MERGEC_IN_GGA(.(X1, X2), X3, X4)
11.64/3.86	
11.64/3.86	The TRS R consists of the following rules:
11.64/3.86	
11.64/3.86	   splitcD_in_ggaa(X1, X2, .(X1, X3), X4) -> U17_ggaa(X1, X2, X3, X4, splitcA_in_gaa(X2, X4, X3))
11.64/3.86	   splitcA_in_gaa([], [], []) -> splitcA_out_gaa([], [], [])
11.64/3.86	   splitcA_in_gaa(.(X1, X2), .(X1, X3), X4) -> U11_gaa(X1, X2, X3, X4, splitcA_in_gaa(X2, X4, X3))
11.64/3.86	   U11_gaa(X1, X2, X3, X4, splitcA_out_gaa(X2, X4, X3)) -> splitcA_out_gaa(.(X1, X2), .(X1, X3), X4)
11.64/3.86	   U17_ggaa(X1, X2, X3, X4, splitcA_out_gaa(X2, X4, X3)) -> splitcD_out_ggaa(X1, X2, .(X1, X3), X4)
11.64/3.86	   mergesortcB_in_ga([], []) -> mergesortcB_out_ga([], [])
11.64/3.86	   mergesortcB_in_ga(.(X1, []), .(X1, [])) -> mergesortcB_out_ga(.(X1, []), .(X1, []))
11.64/3.86	   mergesortcB_in_ga(.(X1, .(X2, X3)), X4) -> U12_ga(X1, X2, X3, X4, splitcD_in_ggaa(X2, X3, X5, X6))
11.64/3.86	   U12_ga(X1, X2, X3, X4, splitcD_out_ggaa(X2, X3, X5, X6)) -> U13_ga(X1, X2, X3, X4, X5, X6, mergesortcB_in_ga(.(X1, X6), X7))
11.64/3.86	   U13_ga(X1, X2, X3, X4, X5, X6, mergesortcB_out_ga(.(X1, X6), X7)) -> U14_ga(X1, X2, X3, X4, X5, X6, X7, mergesortcB_in_ga(X5, X8))
11.64/3.86	   U14_ga(X1, X2, X3, X4, X5, X6, X7, mergesortcB_out_ga(X5, X8)) -> U15_ga(X1, X2, X3, X4, mergecC_in_gga(X7, X8, X4))
11.64/3.86	   mergecC_in_gga([], X1, X1) -> mergecC_out_gga([], X1, X1)
11.64/3.86	   mergecC_in_gga(X1, [], X1) -> mergecC_out_gga(X1, [], X1)
11.64/3.86	   mergecC_in_gga(.(X1, X2), .(X1, X3), .(X1, X4)) -> U16_gga(X1, X2, X3, X4, mergecC_in_gga(.(X1, X2), X3, X4))
11.64/3.86	   U16_gga(X1, X2, X3, X4, mergecC_out_gga(.(X1, X2), X3, X4)) -> mergecC_out_gga(.(X1, X2), .(X1, X3), .(X1, X4))
11.64/3.86	   U15_ga(X1, X2, X3, X4, mergecC_out_gga(X7, X8, X4)) -> mergesortcB_out_ga(.(X1, .(X2, X3)), X4)
11.64/3.86	
11.64/3.86	The argument filtering Pi contains the following mapping:
11.64/3.86	mergesortB_in_ga(x1, x2)  =  mergesortB_in_ga(x1)
11.64/3.86	
11.64/3.86	.(x1, x2)  =  .(x1, x2)
11.64/3.86	
11.64/3.86	splitA_in_gaa(x1, x2, x3)  =  splitA_in_gaa(x1)
11.64/3.86	
11.64/3.86	splitcD_in_ggaa(x1, x2, x3, x4)  =  splitcD_in_ggaa(x1, x2)
11.64/3.86	
11.64/3.86	U17_ggaa(x1, x2, x3, x4, x5)  =  U17_ggaa(x1, x2, x5)
11.64/3.86	
11.64/3.86	splitcA_in_gaa(x1, x2, x3)  =  splitcA_in_gaa(x1)
11.64/3.86	
11.64/3.86	[]  =  []
11.64/3.86	
11.64/3.86	splitcA_out_gaa(x1, x2, x3)  =  splitcA_out_gaa(x1, x2, x3)
11.64/3.86	
11.64/3.86	U11_gaa(x1, x2, x3, x4, x5)  =  U11_gaa(x1, x2, x5)
11.64/3.86	
11.64/3.86	splitcD_out_ggaa(x1, x2, x3, x4)  =  splitcD_out_ggaa(x1, x2, x3, x4)
11.64/3.86	
11.64/3.86	mergesortcB_in_ga(x1, x2)  =  mergesortcB_in_ga(x1)
11.64/3.86	
11.64/3.86	mergesortcB_out_ga(x1, x2)  =  mergesortcB_out_ga(x1, x2)
11.64/3.86	
11.64/3.86	U12_ga(x1, x2, x3, x4, x5)  =  U12_ga(x1, x2, x3, x5)
11.64/3.86	
11.64/3.86	U13_ga(x1, x2, x3, x4, x5, x6, x7)  =  U13_ga(x1, x2, x3, x5, x7)
11.64/3.86	
11.64/3.86	U14_ga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_ga(x1, x2, x3, x7, x8)
11.64/3.86	
11.64/3.86	U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x3, x5)
11.64/3.86	
11.64/3.86	mergecC_in_gga(x1, x2, x3)  =  mergecC_in_gga(x1, x2)
11.64/3.86	
11.64/3.86	mergecC_out_gga(x1, x2, x3)  =  mergecC_out_gga(x1, x2, x3)
11.64/3.86	
11.64/3.86	U16_gga(x1, x2, x3, x4, x5)  =  U16_gga(x1, x2, x3, x5)
11.64/3.86	
11.64/3.86	mergeC_in_gga(x1, x2, x3)  =  mergeC_in_gga(x1, x2)
11.64/3.86	
11.64/3.86	MERGESORTB_IN_GA(x1, x2)  =  MERGESORTB_IN_GA(x1)
11.64/3.86	
11.64/3.86	U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x3, x5)
11.64/3.86	
11.64/3.86	SPLITA_IN_GAA(x1, x2, x3)  =  SPLITA_IN_GAA(x1)
11.64/3.86	
11.64/3.86	U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x1, x2, x5)
11.64/3.86	
11.64/3.86	U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x2, x3, x5)
11.64/3.86	
11.64/3.86	U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x1, x2, x3, x5)
11.64/3.86	
11.64/3.86	U6_GA(x1, x2, x3, x4, x5, x6)  =  U6_GA(x1, x2, x3, x5, x6)
11.64/3.86	
11.64/3.86	U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x1, x2, x3, x5)
11.64/3.86	
11.64/3.86	U8_GA(x1, x2, x3, x4, x5, x6)  =  U8_GA(x1, x2, x3, x5, x6)
11.64/3.86	
11.64/3.86	U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x1, x2, x3, x5)
11.64/3.86	
11.64/3.86	MERGEC_IN_GGA(x1, x2, x3)  =  MERGEC_IN_GGA(x1, x2)
11.64/3.86	
11.64/3.86	U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x1, x2, x3, x5)
11.64/3.86	
11.64/3.86	
11.64/3.86	We have to consider all (P,R,Pi)-chains
11.64/3.86	----------------------------------------
11.64/3.86	
11.64/3.86	(5) DependencyGraphProof (EQUIVALENT)
11.64/3.86	The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 9 less nodes.
11.64/3.86	----------------------------------------
11.64/3.86	
11.64/3.86	(6)
11.64/3.86	Complex Obligation (AND)
11.64/3.86	
11.64/3.86	----------------------------------------
11.64/3.86	
11.64/3.86	(7)
11.64/3.86	Obligation:
11.64/3.86	Pi DP problem:
11.64/3.86	The TRS P consists of the following rules:
11.64/3.86	
11.64/3.86	   MERGEC_IN_GGA(.(X1, X2), .(X1, X3), .(X1, X4)) -> MERGEC_IN_GGA(.(X1, X2), X3, X4)
11.64/3.86	
11.64/3.86	The TRS R consists of the following rules:
11.64/3.86	
11.64/3.86	   splitcD_in_ggaa(X1, X2, .(X1, X3), X4) -> U17_ggaa(X1, X2, X3, X4, splitcA_in_gaa(X2, X4, X3))
11.64/3.86	   splitcA_in_gaa([], [], []) -> splitcA_out_gaa([], [], [])
11.64/3.86	   splitcA_in_gaa(.(X1, X2), .(X1, X3), X4) -> U11_gaa(X1, X2, X3, X4, splitcA_in_gaa(X2, X4, X3))
11.64/3.86	   U11_gaa(X1, X2, X3, X4, splitcA_out_gaa(X2, X4, X3)) -> splitcA_out_gaa(.(X1, X2), .(X1, X3), X4)
11.64/3.86	   U17_ggaa(X1, X2, X3, X4, splitcA_out_gaa(X2, X4, X3)) -> splitcD_out_ggaa(X1, X2, .(X1, X3), X4)
11.64/3.86	   mergesortcB_in_ga([], []) -> mergesortcB_out_ga([], [])
11.64/3.86	   mergesortcB_in_ga(.(X1, []), .(X1, [])) -> mergesortcB_out_ga(.(X1, []), .(X1, []))
11.64/3.86	   mergesortcB_in_ga(.(X1, .(X2, X3)), X4) -> U12_ga(X1, X2, X3, X4, splitcD_in_ggaa(X2, X3, X5, X6))
11.64/3.86	   U12_ga(X1, X2, X3, X4, splitcD_out_ggaa(X2, X3, X5, X6)) -> U13_ga(X1, X2, X3, X4, X5, X6, mergesortcB_in_ga(.(X1, X6), X7))
11.64/3.86	   U13_ga(X1, X2, X3, X4, X5, X6, mergesortcB_out_ga(.(X1, X6), X7)) -> U14_ga(X1, X2, X3, X4, X5, X6, X7, mergesortcB_in_ga(X5, X8))
11.64/3.86	   U14_ga(X1, X2, X3, X4, X5, X6, X7, mergesortcB_out_ga(X5, X8)) -> U15_ga(X1, X2, X3, X4, mergecC_in_gga(X7, X8, X4))
11.64/3.86	   mergecC_in_gga([], X1, X1) -> mergecC_out_gga([], X1, X1)
11.64/3.86	   mergecC_in_gga(X1, [], X1) -> mergecC_out_gga(X1, [], X1)
11.64/3.86	   mergecC_in_gga(.(X1, X2), .(X1, X3), .(X1, X4)) -> U16_gga(X1, X2, X3, X4, mergecC_in_gga(.(X1, X2), X3, X4))
11.64/3.86	   U16_gga(X1, X2, X3, X4, mergecC_out_gga(.(X1, X2), X3, X4)) -> mergecC_out_gga(.(X1, X2), .(X1, X3), .(X1, X4))
11.64/3.86	   U15_ga(X1, X2, X3, X4, mergecC_out_gga(X7, X8, X4)) -> mergesortcB_out_ga(.(X1, .(X2, X3)), X4)
11.64/3.86	
11.64/3.86	The argument filtering Pi contains the following mapping:
11.64/3.86	.(x1, x2)  =  .(x1, x2)
11.64/3.86	
11.64/3.86	splitcD_in_ggaa(x1, x2, x3, x4)  =  splitcD_in_ggaa(x1, x2)
11.64/3.86	
11.64/3.86	U17_ggaa(x1, x2, x3, x4, x5)  =  U17_ggaa(x1, x2, x5)
11.64/3.86	
11.64/3.86	splitcA_in_gaa(x1, x2, x3)  =  splitcA_in_gaa(x1)
11.64/3.86	
11.64/3.86	[]  =  []
11.64/3.86	
11.64/3.86	splitcA_out_gaa(x1, x2, x3)  =  splitcA_out_gaa(x1, x2, x3)
11.64/3.86	
11.64/3.86	U11_gaa(x1, x2, x3, x4, x5)  =  U11_gaa(x1, x2, x5)
11.64/3.86	
11.64/3.86	splitcD_out_ggaa(x1, x2, x3, x4)  =  splitcD_out_ggaa(x1, x2, x3, x4)
11.64/3.86	
11.64/3.86	mergesortcB_in_ga(x1, x2)  =  mergesortcB_in_ga(x1)
11.64/3.86	
11.64/3.86	mergesortcB_out_ga(x1, x2)  =  mergesortcB_out_ga(x1, x2)
11.64/3.86	
11.64/3.86	U12_ga(x1, x2, x3, x4, x5)  =  U12_ga(x1, x2, x3, x5)
11.64/3.86	
11.64/3.86	U13_ga(x1, x2, x3, x4, x5, x6, x7)  =  U13_ga(x1, x2, x3, x5, x7)
11.64/3.86	
11.64/3.86	U14_ga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_ga(x1, x2, x3, x7, x8)
11.64/3.86	
11.64/3.86	U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x3, x5)
11.64/3.86	
11.64/3.86	mergecC_in_gga(x1, x2, x3)  =  mergecC_in_gga(x1, x2)
11.64/3.86	
11.64/3.86	mergecC_out_gga(x1, x2, x3)  =  mergecC_out_gga(x1, x2, x3)
11.64/3.86	
11.64/3.86	U16_gga(x1, x2, x3, x4, x5)  =  U16_gga(x1, x2, x3, x5)
11.64/3.86	
11.64/3.86	MERGEC_IN_GGA(x1, x2, x3)  =  MERGEC_IN_GGA(x1, x2)
11.64/3.86	
11.64/3.86	
11.64/3.86	We have to consider all (P,R,Pi)-chains
11.64/3.86	----------------------------------------
11.64/3.86	
11.64/3.86	(8) UsableRulesProof (EQUIVALENT)
11.64/3.86	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
11.64/3.86	----------------------------------------
11.64/3.86	
11.64/3.86	(9)
11.64/3.86	Obligation:
11.64/3.86	Pi DP problem:
11.64/3.86	The TRS P consists of the following rules:
11.64/3.86	
11.64/3.86	   MERGEC_IN_GGA(.(X1, X2), .(X1, X3), .(X1, X4)) -> MERGEC_IN_GGA(.(X1, X2), X3, X4)
11.64/3.86	
11.64/3.86	R is empty.
11.64/3.86	The argument filtering Pi contains the following mapping:
11.64/3.86	.(x1, x2)  =  .(x1, x2)
11.64/3.86	
11.64/3.86	MERGEC_IN_GGA(x1, x2, x3)  =  MERGEC_IN_GGA(x1, x2)
11.64/3.86	
11.64/3.86	
11.64/3.86	We have to consider all (P,R,Pi)-chains
11.64/3.86	----------------------------------------
11.64/3.86	
11.64/3.86	(10) PiDPToQDPProof (SOUND)
11.64/3.86	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
11.64/3.86	----------------------------------------
11.64/3.86	
11.64/3.86	(11)
11.64/3.86	Obligation:
11.64/3.86	Q DP problem:
11.64/3.86	The TRS P consists of the following rules:
11.64/3.86	
11.64/3.86	   MERGEC_IN_GGA(.(X1, X2), .(X1, X3)) -> MERGEC_IN_GGA(.(X1, X2), X3)
11.64/3.86	
11.64/3.86	R is empty.
11.64/3.86	Q is empty.
11.64/3.86	We have to consider all (P,Q,R)-chains.
11.64/3.86	----------------------------------------
11.64/3.86	
11.64/3.86	(12) QDPSizeChangeProof (EQUIVALENT)
11.64/3.86	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. 
11.64/3.86	
11.64/3.86	From the DPs we obtained the following set of size-change graphs:
11.64/3.86	*MERGEC_IN_GGA(.(X1, X2), .(X1, X3)) -> MERGEC_IN_GGA(.(X1, X2), X3)
11.64/3.86	The graph contains the following edges 1 >= 1, 2 > 2
11.64/3.86	
11.64/3.86	
11.64/3.86	----------------------------------------
11.64/3.86	
11.64/3.86	(13)
11.64/3.86	YES
11.64/3.86	
11.64/3.86	----------------------------------------
11.64/3.86	
11.64/3.86	(14)
11.64/3.86	Obligation:
11.64/3.86	Pi DP problem:
11.64/3.86	The TRS P consists of the following rules:
11.64/3.86	
11.64/3.86	   SPLITA_IN_GAA(.(X1, X2), .(X1, X3), X4) -> SPLITA_IN_GAA(X2, X4, X3)
11.64/3.86	
11.64/3.86	The TRS R consists of the following rules:
11.64/3.86	
11.64/3.86	   splitcD_in_ggaa(X1, X2, .(X1, X3), X4) -> U17_ggaa(X1, X2, X3, X4, splitcA_in_gaa(X2, X4, X3))
11.64/3.86	   splitcA_in_gaa([], [], []) -> splitcA_out_gaa([], [], [])
11.64/3.86	   splitcA_in_gaa(.(X1, X2), .(X1, X3), X4) -> U11_gaa(X1, X2, X3, X4, splitcA_in_gaa(X2, X4, X3))
11.64/3.86	   U11_gaa(X1, X2, X3, X4, splitcA_out_gaa(X2, X4, X3)) -> splitcA_out_gaa(.(X1, X2), .(X1, X3), X4)
11.64/3.86	   U17_ggaa(X1, X2, X3, X4, splitcA_out_gaa(X2, X4, X3)) -> splitcD_out_ggaa(X1, X2, .(X1, X3), X4)
11.64/3.86	   mergesortcB_in_ga([], []) -> mergesortcB_out_ga([], [])
11.64/3.86	   mergesortcB_in_ga(.(X1, []), .(X1, [])) -> mergesortcB_out_ga(.(X1, []), .(X1, []))
11.64/3.86	   mergesortcB_in_ga(.(X1, .(X2, X3)), X4) -> U12_ga(X1, X2, X3, X4, splitcD_in_ggaa(X2, X3, X5, X6))
11.64/3.86	   U12_ga(X1, X2, X3, X4, splitcD_out_ggaa(X2, X3, X5, X6)) -> U13_ga(X1, X2, X3, X4, X5, X6, mergesortcB_in_ga(.(X1, X6), X7))
11.64/3.86	   U13_ga(X1, X2, X3, X4, X5, X6, mergesortcB_out_ga(.(X1, X6), X7)) -> U14_ga(X1, X2, X3, X4, X5, X6, X7, mergesortcB_in_ga(X5, X8))
11.64/3.86	   U14_ga(X1, X2, X3, X4, X5, X6, X7, mergesortcB_out_ga(X5, X8)) -> U15_ga(X1, X2, X3, X4, mergecC_in_gga(X7, X8, X4))
11.64/3.86	   mergecC_in_gga([], X1, X1) -> mergecC_out_gga([], X1, X1)
11.64/3.86	   mergecC_in_gga(X1, [], X1) -> mergecC_out_gga(X1, [], X1)
11.64/3.86	   mergecC_in_gga(.(X1, X2), .(X1, X3), .(X1, X4)) -> U16_gga(X1, X2, X3, X4, mergecC_in_gga(.(X1, X2), X3, X4))
11.64/3.86	   U16_gga(X1, X2, X3, X4, mergecC_out_gga(.(X1, X2), X3, X4)) -> mergecC_out_gga(.(X1, X2), .(X1, X3), .(X1, X4))
11.64/3.86	   U15_ga(X1, X2, X3, X4, mergecC_out_gga(X7, X8, X4)) -> mergesortcB_out_ga(.(X1, .(X2, X3)), X4)
11.64/3.86	
11.64/3.86	The argument filtering Pi contains the following mapping:
11.64/3.86	.(x1, x2)  =  .(x1, x2)
11.64/3.86	
11.64/3.86	splitcD_in_ggaa(x1, x2, x3, x4)  =  splitcD_in_ggaa(x1, x2)
11.64/3.86	
11.64/3.86	U17_ggaa(x1, x2, x3, x4, x5)  =  U17_ggaa(x1, x2, x5)
11.64/3.86	
11.64/3.86	splitcA_in_gaa(x1, x2, x3)  =  splitcA_in_gaa(x1)
11.64/3.86	
11.64/3.86	[]  =  []
11.64/3.86	
11.64/3.86	splitcA_out_gaa(x1, x2, x3)  =  splitcA_out_gaa(x1, x2, x3)
11.64/3.86	
11.64/3.86	U11_gaa(x1, x2, x3, x4, x5)  =  U11_gaa(x1, x2, x5)
11.64/3.86	
11.64/3.86	splitcD_out_ggaa(x1, x2, x3, x4)  =  splitcD_out_ggaa(x1, x2, x3, x4)
11.64/3.86	
11.64/3.86	mergesortcB_in_ga(x1, x2)  =  mergesortcB_in_ga(x1)
11.64/3.86	
11.64/3.86	mergesortcB_out_ga(x1, x2)  =  mergesortcB_out_ga(x1, x2)
11.64/3.86	
11.64/3.86	U12_ga(x1, x2, x3, x4, x5)  =  U12_ga(x1, x2, x3, x5)
11.64/3.86	
11.64/3.86	U13_ga(x1, x2, x3, x4, x5, x6, x7)  =  U13_ga(x1, x2, x3, x5, x7)
11.64/3.86	
11.64/3.86	U14_ga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_ga(x1, x2, x3, x7, x8)
11.64/3.86	
11.64/3.86	U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x3, x5)
11.64/3.86	
11.64/3.86	mergecC_in_gga(x1, x2, x3)  =  mergecC_in_gga(x1, x2)
11.64/3.86	
11.64/3.86	mergecC_out_gga(x1, x2, x3)  =  mergecC_out_gga(x1, x2, x3)
11.64/3.86	
11.64/3.86	U16_gga(x1, x2, x3, x4, x5)  =  U16_gga(x1, x2, x3, x5)
11.64/3.86	
11.64/3.86	SPLITA_IN_GAA(x1, x2, x3)  =  SPLITA_IN_GAA(x1)
11.64/3.86	
11.64/3.86	
11.64/3.86	We have to consider all (P,R,Pi)-chains
11.64/3.86	----------------------------------------
11.64/3.86	
11.64/3.86	(15) UsableRulesProof (EQUIVALENT)
11.64/3.86	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
11.64/3.86	----------------------------------------
11.64/3.86	
11.64/3.86	(16)
11.64/3.86	Obligation:
11.64/3.86	Pi DP problem:
11.64/3.86	The TRS P consists of the following rules:
11.64/3.86	
11.64/3.86	   SPLITA_IN_GAA(.(X1, X2), .(X1, X3), X4) -> SPLITA_IN_GAA(X2, X4, X3)
11.64/3.86	
11.64/3.86	R is empty.
11.64/3.86	The argument filtering Pi contains the following mapping:
11.64/3.86	.(x1, x2)  =  .(x1, x2)
11.64/3.86	
11.64/3.86	SPLITA_IN_GAA(x1, x2, x3)  =  SPLITA_IN_GAA(x1)
11.64/3.86	
11.64/3.86	
11.64/3.86	We have to consider all (P,R,Pi)-chains
11.64/3.86	----------------------------------------
11.64/3.86	
11.64/3.86	(17) PiDPToQDPProof (SOUND)
11.64/3.86	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
11.64/3.86	----------------------------------------
11.64/3.86	
11.64/3.86	(18)
11.64/3.86	Obligation:
11.64/3.86	Q DP problem:
11.64/3.86	The TRS P consists of the following rules:
11.64/3.86	
11.64/3.86	   SPLITA_IN_GAA(.(X1, X2)) -> SPLITA_IN_GAA(X2)
11.64/3.86	
11.64/3.86	R is empty.
11.64/3.86	Q is empty.
11.64/3.86	We have to consider all (P,Q,R)-chains.
11.64/3.86	----------------------------------------
11.64/3.86	
11.64/3.86	(19) QDPSizeChangeProof (EQUIVALENT)
11.64/3.86	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. 
11.64/3.86	
11.64/3.86	From the DPs we obtained the following set of size-change graphs:
11.64/3.86	*SPLITA_IN_GAA(.(X1, X2)) -> SPLITA_IN_GAA(X2)
11.64/3.86	The graph contains the following edges 1 > 1
11.64/3.86	
11.64/3.86	
11.64/3.86	----------------------------------------
11.64/3.86	
11.64/3.86	(20)
11.64/3.86	YES
11.64/3.86	
11.64/3.86	----------------------------------------
11.64/3.86	
11.64/3.86	(21)
11.64/3.86	Obligation:
11.64/3.86	Pi DP problem:
11.64/3.86	The TRS P consists of the following rules:
11.64/3.86	
11.64/3.86	   MERGESORTB_IN_GA(.(X1, .(X2, X3)), X4) -> U4_GA(X1, X2, X3, X4, splitcD_in_ggaa(X2, X3, X5, X6))
11.64/3.86	   U4_GA(X1, X2, X3, X4, splitcD_out_ggaa(X2, X3, X5, X6)) -> MERGESORTB_IN_GA(.(X1, X6), X7)
11.64/3.86	   U4_GA(X1, X2, X3, X4, splitcD_out_ggaa(X2, X3, X5, X6)) -> U6_GA(X1, X2, X3, X4, X5, mergesortcB_in_ga(.(X1, X6), X7))
11.64/3.86	   U6_GA(X1, X2, X3, X4, X5, mergesortcB_out_ga(.(X1, X6), X7)) -> MERGESORTB_IN_GA(X5, X8)
11.64/3.86	
11.64/3.86	The TRS R consists of the following rules:
11.64/3.86	
11.64/3.86	   splitcD_in_ggaa(X1, X2, .(X1, X3), X4) -> U17_ggaa(X1, X2, X3, X4, splitcA_in_gaa(X2, X4, X3))
11.64/3.86	   splitcA_in_gaa([], [], []) -> splitcA_out_gaa([], [], [])
11.64/3.86	   splitcA_in_gaa(.(X1, X2), .(X1, X3), X4) -> U11_gaa(X1, X2, X3, X4, splitcA_in_gaa(X2, X4, X3))
11.64/3.86	   U11_gaa(X1, X2, X3, X4, splitcA_out_gaa(X2, X4, X3)) -> splitcA_out_gaa(.(X1, X2), .(X1, X3), X4)
11.64/3.86	   U17_ggaa(X1, X2, X3, X4, splitcA_out_gaa(X2, X4, X3)) -> splitcD_out_ggaa(X1, X2, .(X1, X3), X4)
11.64/3.86	   mergesortcB_in_ga([], []) -> mergesortcB_out_ga([], [])
11.64/3.86	   mergesortcB_in_ga(.(X1, []), .(X1, [])) -> mergesortcB_out_ga(.(X1, []), .(X1, []))
11.64/3.86	   mergesortcB_in_ga(.(X1, .(X2, X3)), X4) -> U12_ga(X1, X2, X3, X4, splitcD_in_ggaa(X2, X3, X5, X6))
11.64/3.86	   U12_ga(X1, X2, X3, X4, splitcD_out_ggaa(X2, X3, X5, X6)) -> U13_ga(X1, X2, X3, X4, X5, X6, mergesortcB_in_ga(.(X1, X6), X7))
11.64/3.86	   U13_ga(X1, X2, X3, X4, X5, X6, mergesortcB_out_ga(.(X1, X6), X7)) -> U14_ga(X1, X2, X3, X4, X5, X6, X7, mergesortcB_in_ga(X5, X8))
11.64/3.86	   U14_ga(X1, X2, X3, X4, X5, X6, X7, mergesortcB_out_ga(X5, X8)) -> U15_ga(X1, X2, X3, X4, mergecC_in_gga(X7, X8, X4))
11.64/3.86	   mergecC_in_gga([], X1, X1) -> mergecC_out_gga([], X1, X1)
11.64/3.86	   mergecC_in_gga(X1, [], X1) -> mergecC_out_gga(X1, [], X1)
11.64/3.86	   mergecC_in_gga(.(X1, X2), .(X1, X3), .(X1, X4)) -> U16_gga(X1, X2, X3, X4, mergecC_in_gga(.(X1, X2), X3, X4))
11.64/3.86	   U16_gga(X1, X2, X3, X4, mergecC_out_gga(.(X1, X2), X3, X4)) -> mergecC_out_gga(.(X1, X2), .(X1, X3), .(X1, X4))
11.64/3.86	   U15_ga(X1, X2, X3, X4, mergecC_out_gga(X7, X8, X4)) -> mergesortcB_out_ga(.(X1, .(X2, X3)), X4)
11.64/3.86	
11.64/3.86	The argument filtering Pi contains the following mapping:
11.64/3.86	.(x1, x2)  =  .(x1, x2)
11.64/3.86	
11.64/3.86	splitcD_in_ggaa(x1, x2, x3, x4)  =  splitcD_in_ggaa(x1, x2)
11.64/3.86	
11.64/3.86	U17_ggaa(x1, x2, x3, x4, x5)  =  U17_ggaa(x1, x2, x5)
11.64/3.86	
11.64/3.86	splitcA_in_gaa(x1, x2, x3)  =  splitcA_in_gaa(x1)
11.64/3.86	
11.64/3.86	[]  =  []
11.64/3.86	
11.64/3.86	splitcA_out_gaa(x1, x2, x3)  =  splitcA_out_gaa(x1, x2, x3)
11.64/3.86	
11.64/3.86	U11_gaa(x1, x2, x3, x4, x5)  =  U11_gaa(x1, x2, x5)
11.64/3.86	
11.64/3.86	splitcD_out_ggaa(x1, x2, x3, x4)  =  splitcD_out_ggaa(x1, x2, x3, x4)
11.64/3.86	
11.64/3.86	mergesortcB_in_ga(x1, x2)  =  mergesortcB_in_ga(x1)
11.64/3.86	
11.64/3.86	mergesortcB_out_ga(x1, x2)  =  mergesortcB_out_ga(x1, x2)
11.64/3.86	
11.64/3.86	U12_ga(x1, x2, x3, x4, x5)  =  U12_ga(x1, x2, x3, x5)
11.64/3.86	
11.64/3.86	U13_ga(x1, x2, x3, x4, x5, x6, x7)  =  U13_ga(x1, x2, x3, x5, x7)
11.64/3.86	
11.64/3.86	U14_ga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_ga(x1, x2, x3, x7, x8)
11.64/3.86	
11.64/3.86	U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x3, x5)
11.64/3.86	
11.64/3.86	mergecC_in_gga(x1, x2, x3)  =  mergecC_in_gga(x1, x2)
11.64/3.86	
11.64/3.86	mergecC_out_gga(x1, x2, x3)  =  mergecC_out_gga(x1, x2, x3)
11.64/3.86	
11.64/3.86	U16_gga(x1, x2, x3, x4, x5)  =  U16_gga(x1, x2, x3, x5)
11.64/3.86	
11.64/3.86	MERGESORTB_IN_GA(x1, x2)  =  MERGESORTB_IN_GA(x1)
11.64/3.86	
11.64/3.86	U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x2, x3, x5)
11.64/3.86	
11.64/3.86	U6_GA(x1, x2, x3, x4, x5, x6)  =  U6_GA(x1, x2, x3, x5, x6)
11.64/3.86	
11.64/3.86	
11.64/3.86	We have to consider all (P,R,Pi)-chains
11.64/3.86	----------------------------------------
11.64/3.86	
11.64/3.86	(22) PiDPToQDPProof (SOUND)
11.64/3.86	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
11.64/3.86	----------------------------------------
11.64/3.86	
11.64/3.86	(23)
11.64/3.86	Obligation:
11.64/3.86	Q DP problem:
11.64/3.86	The TRS P consists of the following rules:
11.64/3.86	
11.64/3.86	   MERGESORTB_IN_GA(.(X1, .(X2, X3))) -> U4_GA(X1, X2, X3, splitcD_in_ggaa(X2, X3))
11.64/3.86	   U4_GA(X1, X2, X3, splitcD_out_ggaa(X2, X3, X5, X6)) -> MERGESORTB_IN_GA(.(X1, X6))
11.64/3.86	   U4_GA(X1, X2, X3, splitcD_out_ggaa(X2, X3, X5, X6)) -> U6_GA(X1, X2, X3, X5, mergesortcB_in_ga(.(X1, X6)))
11.64/3.86	   U6_GA(X1, X2, X3, X5, mergesortcB_out_ga(.(X1, X6), X7)) -> MERGESORTB_IN_GA(X5)
11.64/3.86	
11.64/3.86	The TRS R consists of the following rules:
11.64/3.86	
11.64/3.86	   splitcD_in_ggaa(X1, X2) -> U17_ggaa(X1, X2, splitcA_in_gaa(X2))
11.64/3.86	   splitcA_in_gaa([]) -> splitcA_out_gaa([], [], [])
11.64/3.86	   splitcA_in_gaa(.(X1, X2)) -> U11_gaa(X1, X2, splitcA_in_gaa(X2))
11.64/3.86	   U11_gaa(X1, X2, splitcA_out_gaa(X2, X4, X3)) -> splitcA_out_gaa(.(X1, X2), .(X1, X3), X4)
11.64/3.86	   U17_ggaa(X1, X2, splitcA_out_gaa(X2, X4, X3)) -> splitcD_out_ggaa(X1, X2, .(X1, X3), X4)
11.64/3.86	   mergesortcB_in_ga([]) -> mergesortcB_out_ga([], [])
11.64/3.86	   mergesortcB_in_ga(.(X1, [])) -> mergesortcB_out_ga(.(X1, []), .(X1, []))
11.64/3.86	   mergesortcB_in_ga(.(X1, .(X2, X3))) -> U12_ga(X1, X2, X3, splitcD_in_ggaa(X2, X3))
11.64/3.86	   U12_ga(X1, X2, X3, splitcD_out_ggaa(X2, X3, X5, X6)) -> U13_ga(X1, X2, X3, X5, mergesortcB_in_ga(.(X1, X6)))
11.64/3.86	   U13_ga(X1, X2, X3, X5, mergesortcB_out_ga(.(X1, X6), X7)) -> U14_ga(X1, X2, X3, X7, mergesortcB_in_ga(X5))
11.64/3.86	   U14_ga(X1, X2, X3, X7, mergesortcB_out_ga(X5, X8)) -> U15_ga(X1, X2, X3, mergecC_in_gga(X7, X8))
11.64/3.86	   mergecC_in_gga([], X1) -> mergecC_out_gga([], X1, X1)
11.64/3.86	   mergecC_in_gga(X1, []) -> mergecC_out_gga(X1, [], X1)
11.64/3.86	   mergecC_in_gga(.(X1, X2), .(X1, X3)) -> U16_gga(X1, X2, X3, mergecC_in_gga(.(X1, X2), X3))
11.64/3.86	   U16_gga(X1, X2, X3, mergecC_out_gga(.(X1, X2), X3, X4)) -> mergecC_out_gga(.(X1, X2), .(X1, X3), .(X1, X4))
11.64/3.86	   U15_ga(X1, X2, X3, mergecC_out_gga(X7, X8, X4)) -> mergesortcB_out_ga(.(X1, .(X2, X3)), X4)
11.64/3.86	
11.64/3.86	The set Q consists of the following terms:
11.64/3.86	
11.64/3.86	   splitcD_in_ggaa(x0, x1)
11.64/3.86	   splitcA_in_gaa(x0)
11.64/3.86	   U11_gaa(x0, x1, x2)
11.64/3.86	   U17_ggaa(x0, x1, x2)
11.64/3.86	   mergesortcB_in_ga(x0)
11.64/3.86	   U12_ga(x0, x1, x2, x3)
11.64/3.86	   U13_ga(x0, x1, x2, x3, x4)
11.64/3.86	   U14_ga(x0, x1, x2, x3, x4)
11.64/3.86	   mergecC_in_gga(x0, x1)
11.64/3.86	   U16_gga(x0, x1, x2, x3)
11.64/3.86	   U15_ga(x0, x1, x2, x3)
11.64/3.86	
11.64/3.86	We have to consider all (P,Q,R)-chains.
11.64/3.86	----------------------------------------
11.64/3.86	
11.64/3.86	(24) TransformationProof (EQUIVALENT)
11.64/3.86	By rewriting [LPAR04] the rule MERGESORTB_IN_GA(.(X1, .(X2, X3))) -> U4_GA(X1, X2, X3, splitcD_in_ggaa(X2, X3)) at position [3] we obtained the following new rules [LPAR04]:
11.64/3.86	
11.64/3.86	   (MERGESORTB_IN_GA(.(X1, .(X2, X3))) -> U4_GA(X1, X2, X3, U17_ggaa(X2, X3, splitcA_in_gaa(X3))),MERGESORTB_IN_GA(.(X1, .(X2, X3))) -> U4_GA(X1, X2, X3, U17_ggaa(X2, X3, splitcA_in_gaa(X3))))
11.64/3.86	
11.64/3.86	
11.64/3.86	----------------------------------------
11.64/3.86	
11.64/3.86	(25)
11.64/3.86	Obligation:
11.64/3.86	Q DP problem:
11.64/3.86	The TRS P consists of the following rules:
11.64/3.86	
11.64/3.86	   U4_GA(X1, X2, X3, splitcD_out_ggaa(X2, X3, X5, X6)) -> MERGESORTB_IN_GA(.(X1, X6))
11.64/3.86	   U4_GA(X1, X2, X3, splitcD_out_ggaa(X2, X3, X5, X6)) -> U6_GA(X1, X2, X3, X5, mergesortcB_in_ga(.(X1, X6)))
11.64/3.86	   U6_GA(X1, X2, X3, X5, mergesortcB_out_ga(.(X1, X6), X7)) -> MERGESORTB_IN_GA(X5)
11.64/3.86	   MERGESORTB_IN_GA(.(X1, .(X2, X3))) -> U4_GA(X1, X2, X3, U17_ggaa(X2, X3, splitcA_in_gaa(X3)))
11.64/3.86	
11.64/3.86	The TRS R consists of the following rules:
11.64/3.86	
11.64/3.86	   splitcD_in_ggaa(X1, X2) -> U17_ggaa(X1, X2, splitcA_in_gaa(X2))
11.64/3.86	   splitcA_in_gaa([]) -> splitcA_out_gaa([], [], [])
11.64/3.86	   splitcA_in_gaa(.(X1, X2)) -> U11_gaa(X1, X2, splitcA_in_gaa(X2))
11.64/3.86	   U11_gaa(X1, X2, splitcA_out_gaa(X2, X4, X3)) -> splitcA_out_gaa(.(X1, X2), .(X1, X3), X4)
11.64/3.86	   U17_ggaa(X1, X2, splitcA_out_gaa(X2, X4, X3)) -> splitcD_out_ggaa(X1, X2, .(X1, X3), X4)
11.64/3.86	   mergesortcB_in_ga([]) -> mergesortcB_out_ga([], [])
11.64/3.86	   mergesortcB_in_ga(.(X1, [])) -> mergesortcB_out_ga(.(X1, []), .(X1, []))
11.64/3.86	   mergesortcB_in_ga(.(X1, .(X2, X3))) -> U12_ga(X1, X2, X3, splitcD_in_ggaa(X2, X3))
11.64/3.86	   U12_ga(X1, X2, X3, splitcD_out_ggaa(X2, X3, X5, X6)) -> U13_ga(X1, X2, X3, X5, mergesortcB_in_ga(.(X1, X6)))
11.64/3.86	   U13_ga(X1, X2, X3, X5, mergesortcB_out_ga(.(X1, X6), X7)) -> U14_ga(X1, X2, X3, X7, mergesortcB_in_ga(X5))
11.64/3.86	   U14_ga(X1, X2, X3, X7, mergesortcB_out_ga(X5, X8)) -> U15_ga(X1, X2, X3, mergecC_in_gga(X7, X8))
11.64/3.86	   mergecC_in_gga([], X1) -> mergecC_out_gga([], X1, X1)
11.64/3.86	   mergecC_in_gga(X1, []) -> mergecC_out_gga(X1, [], X1)
11.64/3.86	   mergecC_in_gga(.(X1, X2), .(X1, X3)) -> U16_gga(X1, X2, X3, mergecC_in_gga(.(X1, X2), X3))
11.64/3.86	   U16_gga(X1, X2, X3, mergecC_out_gga(.(X1, X2), X3, X4)) -> mergecC_out_gga(.(X1, X2), .(X1, X3), .(X1, X4))
11.64/3.86	   U15_ga(X1, X2, X3, mergecC_out_gga(X7, X8, X4)) -> mergesortcB_out_ga(.(X1, .(X2, X3)), X4)
11.64/3.86	
11.64/3.86	The set Q consists of the following terms:
11.64/3.86	
11.64/3.86	   splitcD_in_ggaa(x0, x1)
11.64/3.86	   splitcA_in_gaa(x0)
11.64/3.86	   U11_gaa(x0, x1, x2)
11.64/3.86	   U17_ggaa(x0, x1, x2)
11.64/3.86	   mergesortcB_in_ga(x0)
11.64/3.86	   U12_ga(x0, x1, x2, x3)
11.64/3.86	   U13_ga(x0, x1, x2, x3, x4)
11.64/3.86	   U14_ga(x0, x1, x2, x3, x4)
11.64/3.86	   mergecC_in_gga(x0, x1)
11.64/3.86	   U16_gga(x0, x1, x2, x3)
11.64/3.86	   U15_ga(x0, x1, x2, x3)
11.64/3.86	
11.64/3.86	We have to consider all (P,Q,R)-chains.
11.64/3.86	----------------------------------------
11.64/3.86	
11.64/3.86	(26) QDPQMonotonicMRRProof (EQUIVALENT)
11.64/3.86	By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain.
11.64/3.86	
11.64/3.86	Strictly oriented dependency pairs:
11.64/3.86	
11.64/3.86	   U6_GA(X1, X2, X3, X5, mergesortcB_out_ga(.(X1, X6), X7)) -> MERGESORTB_IN_GA(X5)
11.64/3.86	
11.64/3.86	
11.64/3.86	Used ordering: Polynomial interpretation [POLO]:
11.64/3.86	
11.64/3.86	   POL(.(x_1, x_2)) = 2 + 2*x_2
11.64/3.86	   POL(MERGESORTB_IN_GA(x_1)) = x_1
11.64/3.86	   POL(U11_gaa(x_1, x_2, x_3)) = 1 + 2*x_3
11.64/3.86	   POL(U12_ga(x_1, x_2, x_3, x_4)) = 0
11.64/3.86	   POL(U13_ga(x_1, x_2, x_3, x_4, x_5)) = 0
11.64/3.86	   POL(U14_ga(x_1, x_2, x_3, x_4, x_5)) = 0
11.64/3.86	   POL(U15_ga(x_1, x_2, x_3, x_4)) = 0
11.64/3.86	   POL(U16_gga(x_1, x_2, x_3, x_4)) = 0
11.64/3.86	   POL(U17_ggaa(x_1, x_2, x_3)) = 2*x_3
11.64/3.86	   POL(U4_GA(x_1, x_2, x_3, x_4)) = 2 + 2*x_4
11.64/3.86	   POL(U6_GA(x_1, x_2, x_3, x_4, x_5)) = 2 + 2*x_4
11.64/3.86	   POL([]) = 0
11.64/3.86	   POL(mergecC_in_gga(x_1, x_2)) = 0
11.64/3.86	   POL(mergecC_out_gga(x_1, x_2, x_3)) = 0
11.64/3.86	   POL(mergesortcB_in_ga(x_1)) = 0
11.64/3.86	   POL(mergesortcB_out_ga(x_1, x_2)) = 0
11.64/3.86	   POL(splitcA_in_gaa(x_1)) = 1 + x_1
11.64/3.86	   POL(splitcA_out_gaa(x_1, x_2, x_3)) = 1 + x_2 + 2*x_3
11.64/3.86	   POL(splitcD_in_ggaa(x_1, x_2)) = 2 + 2*x_2
11.64/3.86	   POL(splitcD_out_ggaa(x_1, x_2, x_3, x_4)) = x_3 + x_4
11.64/3.86	
11.64/3.86	
11.64/3.86	----------------------------------------
11.64/3.86	
11.64/3.86	(27)
11.64/3.86	Obligation:
11.64/3.86	Q DP problem:
11.64/3.86	The TRS P consists of the following rules:
11.64/3.86	
11.64/3.86	   U4_GA(X1, X2, X3, splitcD_out_ggaa(X2, X3, X5, X6)) -> MERGESORTB_IN_GA(.(X1, X6))
11.64/3.86	   U4_GA(X1, X2, X3, splitcD_out_ggaa(X2, X3, X5, X6)) -> U6_GA(X1, X2, X3, X5, mergesortcB_in_ga(.(X1, X6)))
11.64/3.86	   MERGESORTB_IN_GA(.(X1, .(X2, X3))) -> U4_GA(X1, X2, X3, U17_ggaa(X2, X3, splitcA_in_gaa(X3)))
11.64/3.86	
11.64/3.86	The TRS R consists of the following rules:
11.64/3.86	
11.64/3.86	   splitcD_in_ggaa(X1, X2) -> U17_ggaa(X1, X2, splitcA_in_gaa(X2))
11.64/3.86	   splitcA_in_gaa([]) -> splitcA_out_gaa([], [], [])
11.64/3.86	   splitcA_in_gaa(.(X1, X2)) -> U11_gaa(X1, X2, splitcA_in_gaa(X2))
11.64/3.86	   U11_gaa(X1, X2, splitcA_out_gaa(X2, X4, X3)) -> splitcA_out_gaa(.(X1, X2), .(X1, X3), X4)
11.64/3.86	   U17_ggaa(X1, X2, splitcA_out_gaa(X2, X4, X3)) -> splitcD_out_ggaa(X1, X2, .(X1, X3), X4)
11.64/3.86	   mergesortcB_in_ga([]) -> mergesortcB_out_ga([], [])
11.64/3.86	   mergesortcB_in_ga(.(X1, [])) -> mergesortcB_out_ga(.(X1, []), .(X1, []))
11.64/3.86	   mergesortcB_in_ga(.(X1, .(X2, X3))) -> U12_ga(X1, X2, X3, splitcD_in_ggaa(X2, X3))
11.64/3.86	   U12_ga(X1, X2, X3, splitcD_out_ggaa(X2, X3, X5, X6)) -> U13_ga(X1, X2, X3, X5, mergesortcB_in_ga(.(X1, X6)))
11.64/3.86	   U13_ga(X1, X2, X3, X5, mergesortcB_out_ga(.(X1, X6), X7)) -> U14_ga(X1, X2, X3, X7, mergesortcB_in_ga(X5))
11.64/3.86	   U14_ga(X1, X2, X3, X7, mergesortcB_out_ga(X5, X8)) -> U15_ga(X1, X2, X3, mergecC_in_gga(X7, X8))
11.64/3.86	   mergecC_in_gga([], X1) -> mergecC_out_gga([], X1, X1)
11.64/3.86	   mergecC_in_gga(X1, []) -> mergecC_out_gga(X1, [], X1)
11.64/3.86	   mergecC_in_gga(.(X1, X2), .(X1, X3)) -> U16_gga(X1, X2, X3, mergecC_in_gga(.(X1, X2), X3))
11.64/3.86	   U16_gga(X1, X2, X3, mergecC_out_gga(.(X1, X2), X3, X4)) -> mergecC_out_gga(.(X1, X2), .(X1, X3), .(X1, X4))
11.64/3.86	   U15_ga(X1, X2, X3, mergecC_out_gga(X7, X8, X4)) -> mergesortcB_out_ga(.(X1, .(X2, X3)), X4)
11.64/3.86	
11.64/3.86	The set Q consists of the following terms:
11.64/3.86	
11.64/3.86	   splitcD_in_ggaa(x0, x1)
11.64/3.86	   splitcA_in_gaa(x0)
11.64/3.86	   U11_gaa(x0, x1, x2)
11.64/3.86	   U17_ggaa(x0, x1, x2)
11.64/3.86	   mergesortcB_in_ga(x0)
11.64/3.86	   U12_ga(x0, x1, x2, x3)
11.64/3.86	   U13_ga(x0, x1, x2, x3, x4)
11.64/3.86	   U14_ga(x0, x1, x2, x3, x4)
11.64/3.86	   mergecC_in_gga(x0, x1)
11.64/3.86	   U16_gga(x0, x1, x2, x3)
11.64/3.86	   U15_ga(x0, x1, x2, x3)
11.64/3.86	
11.64/3.86	We have to consider all (P,Q,R)-chains.
11.64/3.86	----------------------------------------
11.64/3.86	
11.64/3.86	(28) DependencyGraphProof (EQUIVALENT)
11.64/3.86	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
11.64/3.86	----------------------------------------
11.64/3.86	
11.64/3.86	(29)
11.64/3.86	Obligation:
11.64/3.86	Q DP problem:
11.64/3.86	The TRS P consists of the following rules:
11.64/3.86	
11.64/3.86	   MERGESORTB_IN_GA(.(X1, .(X2, X3))) -> U4_GA(X1, X2, X3, U17_ggaa(X2, X3, splitcA_in_gaa(X3)))
11.64/3.86	   U4_GA(X1, X2, X3, splitcD_out_ggaa(X2, X3, X5, X6)) -> MERGESORTB_IN_GA(.(X1, X6))
11.64/3.86	
11.64/3.86	The TRS R consists of the following rules:
11.64/3.86	
11.64/3.86	   splitcD_in_ggaa(X1, X2) -> U17_ggaa(X1, X2, splitcA_in_gaa(X2))
11.64/3.86	   splitcA_in_gaa([]) -> splitcA_out_gaa([], [], [])
11.64/3.86	   splitcA_in_gaa(.(X1, X2)) -> U11_gaa(X1, X2, splitcA_in_gaa(X2))
11.64/3.86	   U11_gaa(X1, X2, splitcA_out_gaa(X2, X4, X3)) -> splitcA_out_gaa(.(X1, X2), .(X1, X3), X4)
11.64/3.86	   U17_ggaa(X1, X2, splitcA_out_gaa(X2, X4, X3)) -> splitcD_out_ggaa(X1, X2, .(X1, X3), X4)
11.64/3.86	   mergesortcB_in_ga([]) -> mergesortcB_out_ga([], [])
11.64/3.86	   mergesortcB_in_ga(.(X1, [])) -> mergesortcB_out_ga(.(X1, []), .(X1, []))
11.64/3.86	   mergesortcB_in_ga(.(X1, .(X2, X3))) -> U12_ga(X1, X2, X3, splitcD_in_ggaa(X2, X3))
11.64/3.86	   U12_ga(X1, X2, X3, splitcD_out_ggaa(X2, X3, X5, X6)) -> U13_ga(X1, X2, X3, X5, mergesortcB_in_ga(.(X1, X6)))
11.64/3.86	   U13_ga(X1, X2, X3, X5, mergesortcB_out_ga(.(X1, X6), X7)) -> U14_ga(X1, X2, X3, X7, mergesortcB_in_ga(X5))
11.64/3.86	   U14_ga(X1, X2, X3, X7, mergesortcB_out_ga(X5, X8)) -> U15_ga(X1, X2, X3, mergecC_in_gga(X7, X8))
11.64/3.86	   mergecC_in_gga([], X1) -> mergecC_out_gga([], X1, X1)
11.64/3.86	   mergecC_in_gga(X1, []) -> mergecC_out_gga(X1, [], X1)
11.64/3.86	   mergecC_in_gga(.(X1, X2), .(X1, X3)) -> U16_gga(X1, X2, X3, mergecC_in_gga(.(X1, X2), X3))
11.64/3.86	   U16_gga(X1, X2, X3, mergecC_out_gga(.(X1, X2), X3, X4)) -> mergecC_out_gga(.(X1, X2), .(X1, X3), .(X1, X4))
11.64/3.86	   U15_ga(X1, X2, X3, mergecC_out_gga(X7, X8, X4)) -> mergesortcB_out_ga(.(X1, .(X2, X3)), X4)
11.64/3.86	
11.64/3.86	The set Q consists of the following terms:
11.64/3.86	
11.64/3.86	   splitcD_in_ggaa(x0, x1)
11.64/3.86	   splitcA_in_gaa(x0)
11.64/3.86	   U11_gaa(x0, x1, x2)
11.64/3.86	   U17_ggaa(x0, x1, x2)
11.64/3.86	   mergesortcB_in_ga(x0)
11.64/3.86	   U12_ga(x0, x1, x2, x3)
11.64/3.86	   U13_ga(x0, x1, x2, x3, x4)
11.64/3.86	   U14_ga(x0, x1, x2, x3, x4)
11.64/3.86	   mergecC_in_gga(x0, x1)
11.64/3.86	   U16_gga(x0, x1, x2, x3)
11.64/3.86	   U15_ga(x0, x1, x2, x3)
11.64/3.86	
11.64/3.86	We have to consider all (P,Q,R)-chains.
11.64/3.86	----------------------------------------
11.64/3.86	
11.64/3.86	(30) UsableRulesProof (EQUIVALENT)
11.64/3.86	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
11.64/3.86	----------------------------------------
11.64/3.86	
11.64/3.86	(31)
11.64/3.86	Obligation:
11.64/3.86	Q DP problem:
11.64/3.86	The TRS P consists of the following rules:
11.64/3.86	
11.64/3.86	   MERGESORTB_IN_GA(.(X1, .(X2, X3))) -> U4_GA(X1, X2, X3, U17_ggaa(X2, X3, splitcA_in_gaa(X3)))
11.64/3.86	   U4_GA(X1, X2, X3, splitcD_out_ggaa(X2, X3, X5, X6)) -> MERGESORTB_IN_GA(.(X1, X6))
11.64/3.86	
11.64/3.86	The TRS R consists of the following rules:
11.64/3.86	
11.64/3.86	   splitcA_in_gaa([]) -> splitcA_out_gaa([], [], [])
11.64/3.86	   splitcA_in_gaa(.(X1, X2)) -> U11_gaa(X1, X2, splitcA_in_gaa(X2))
11.64/3.86	   U17_ggaa(X1, X2, splitcA_out_gaa(X2, X4, X3)) -> splitcD_out_ggaa(X1, X2, .(X1, X3), X4)
11.64/3.86	   U11_gaa(X1, X2, splitcA_out_gaa(X2, X4, X3)) -> splitcA_out_gaa(.(X1, X2), .(X1, X3), X4)
11.64/3.86	
11.64/3.86	The set Q consists of the following terms:
11.64/3.86	
11.64/3.86	   splitcD_in_ggaa(x0, x1)
11.64/3.86	   splitcA_in_gaa(x0)
11.64/3.86	   U11_gaa(x0, x1, x2)
11.64/3.86	   U17_ggaa(x0, x1, x2)
11.64/3.86	   mergesortcB_in_ga(x0)
11.64/3.86	   U12_ga(x0, x1, x2, x3)
11.64/3.86	   U13_ga(x0, x1, x2, x3, x4)
11.64/3.86	   U14_ga(x0, x1, x2, x3, x4)
11.64/3.86	   mergecC_in_gga(x0, x1)
11.64/3.86	   U16_gga(x0, x1, x2, x3)
11.64/3.86	   U15_ga(x0, x1, x2, x3)
11.64/3.86	
11.64/3.86	We have to consider all (P,Q,R)-chains.
11.64/3.86	----------------------------------------
11.64/3.86	
11.64/3.86	(32) QReductionProof (EQUIVALENT)
11.64/3.86	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
11.64/3.86	
11.64/3.86	   splitcD_in_ggaa(x0, x1)
11.64/3.86	   mergesortcB_in_ga(x0)
11.64/3.86	   U12_ga(x0, x1, x2, x3)
11.64/3.86	   U13_ga(x0, x1, x2, x3, x4)
11.64/3.86	   U14_ga(x0, x1, x2, x3, x4)
11.64/3.86	   mergecC_in_gga(x0, x1)
11.64/3.86	   U16_gga(x0, x1, x2, x3)
11.64/3.86	   U15_ga(x0, x1, x2, x3)
11.64/3.86	
11.64/3.86	
11.64/3.86	----------------------------------------
11.64/3.86	
11.64/3.86	(33)
11.64/3.86	Obligation:
11.64/3.86	Q DP problem:
11.64/3.86	The TRS P consists of the following rules:
11.64/3.86	
11.64/3.86	   MERGESORTB_IN_GA(.(X1, .(X2, X3))) -> U4_GA(X1, X2, X3, U17_ggaa(X2, X3, splitcA_in_gaa(X3)))
11.64/3.86	   U4_GA(X1, X2, X3, splitcD_out_ggaa(X2, X3, X5, X6)) -> MERGESORTB_IN_GA(.(X1, X6))
11.64/3.86	
11.64/3.86	The TRS R consists of the following rules:
11.64/3.86	
11.64/3.86	   splitcA_in_gaa([]) -> splitcA_out_gaa([], [], [])
11.64/3.86	   splitcA_in_gaa(.(X1, X2)) -> U11_gaa(X1, X2, splitcA_in_gaa(X2))
11.64/3.86	   U17_ggaa(X1, X2, splitcA_out_gaa(X2, X4, X3)) -> splitcD_out_ggaa(X1, X2, .(X1, X3), X4)
11.64/3.86	   U11_gaa(X1, X2, splitcA_out_gaa(X2, X4, X3)) -> splitcA_out_gaa(.(X1, X2), .(X1, X3), X4)
11.64/3.86	
11.64/3.86	The set Q consists of the following terms:
11.64/3.86	
11.64/3.86	   splitcA_in_gaa(x0)
11.64/3.86	   U11_gaa(x0, x1, x2)
11.64/3.86	   U17_ggaa(x0, x1, x2)
11.64/3.86	
11.64/3.86	We have to consider all (P,Q,R)-chains.
11.64/3.86	----------------------------------------
11.64/3.86	
11.64/3.86	(34) QDPQMonotonicMRRProof (EQUIVALENT)
11.64/3.86	By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain.
11.64/3.86	
11.64/3.86	Strictly oriented dependency pairs:
11.64/3.86	
11.64/3.86	   MERGESORTB_IN_GA(.(X1, .(X2, X3))) -> U4_GA(X1, X2, X3, U17_ggaa(X2, X3, splitcA_in_gaa(X3)))
11.64/3.86	
11.64/3.86	Strictly oriented rules of the TRS R:
11.64/3.86	
11.64/3.86	   splitcA_in_gaa([]) -> splitcA_out_gaa([], [], [])
11.64/3.86	
11.64/3.86	Used ordering: Polynomial interpretation [POLO]:
11.64/3.86	
11.64/3.86	   POL(.(x_1, x_2)) = 1 + x_2
11.64/3.86	   POL(MERGESORTB_IN_GA(x_1)) = 1 + 2*x_1
11.64/3.86	   POL(U11_gaa(x_1, x_2, x_3)) = 2 + x_3
11.64/3.86	   POL(U17_ggaa(x_1, x_2, x_3)) = 1 + x_3
11.64/3.86	   POL(U4_GA(x_1, x_2, x_3, x_4)) = 1 + x_4
11.64/3.86	   POL([]) = 0
11.64/3.86	   POL(splitcA_in_gaa(x_1)) = 2 + 2*x_1
11.64/3.86	   POL(splitcA_out_gaa(x_1, x_2, x_3)) = 1 + 2*x_2 + 2*x_3
11.64/3.86	   POL(splitcD_out_ggaa(x_1, x_2, x_3, x_4)) = 2 + 2*x_4
11.64/3.86	
11.64/3.86	
11.64/3.86	----------------------------------------
11.64/3.86	
11.64/3.86	(35)
11.64/3.86	Obligation:
11.64/3.86	Q DP problem:
11.64/3.86	The TRS P consists of the following rules:
11.64/3.86	
11.64/3.86	   U4_GA(X1, X2, X3, splitcD_out_ggaa(X2, X3, X5, X6)) -> MERGESORTB_IN_GA(.(X1, X6))
11.64/3.86	
11.64/3.86	The TRS R consists of the following rules:
11.64/3.86	
11.64/3.86	   splitcA_in_gaa(.(X1, X2)) -> U11_gaa(X1, X2, splitcA_in_gaa(X2))
11.64/3.86	   U17_ggaa(X1, X2, splitcA_out_gaa(X2, X4, X3)) -> splitcD_out_ggaa(X1, X2, .(X1, X3), X4)
11.64/3.86	   U11_gaa(X1, X2, splitcA_out_gaa(X2, X4, X3)) -> splitcA_out_gaa(.(X1, X2), .(X1, X3), X4)
11.64/3.86	
11.64/3.86	The set Q consists of the following terms:
11.64/3.86	
11.64/3.86	   splitcA_in_gaa(x0)
11.64/3.86	   U11_gaa(x0, x1, x2)
11.64/3.86	   U17_ggaa(x0, x1, x2)
11.64/3.86	
11.64/3.86	We have to consider all (P,Q,R)-chains.
11.64/3.86	----------------------------------------
11.64/3.86	
11.64/3.86	(36) DependencyGraphProof (EQUIVALENT)
11.64/3.86	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
11.64/3.86	----------------------------------------
11.64/3.86	
11.64/3.86	(37)
11.64/3.86	TRUE
11.78/3.93	EOF