4.07/1.81	YES
4.07/1.83	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
4.07/1.83	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.07/1.83	
4.07/1.83	
4.07/1.83	Left Termination of the query pattern
4.07/1.83	
4.07/1.83	sum(g,a,g)
4.07/1.83	
4.07/1.83	w.r.t. the given Prolog program could successfully be proven:
4.07/1.83	
4.07/1.83	(0) Prolog
4.07/1.83	(1) PrologToDTProblemTransformerProof [SOUND, 0 ms]
4.07/1.83	(2) TRIPLES
4.07/1.83	(3) TriplesToPiDPProof [SOUND, 7 ms]
4.07/1.83	(4) PiDP
4.07/1.83	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
4.07/1.83	(6) PiDP
4.07/1.83	(7) PiDPToQDPProof [SOUND, 0 ms]
4.07/1.83	(8) QDP
4.07/1.83	(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
4.07/1.83	(10) YES
4.07/1.83	
4.07/1.83	
4.07/1.83	----------------------------------------
4.07/1.83	
4.07/1.83	(0)
4.07/1.83	Obligation:
4.07/1.83	Clauses:
4.07/1.83	
4.07/1.83	sum([], [], []).
4.07/1.83	sum(.(X1, Y1), .(X2, Y2), .(X3, Y3)) :- ','(add(X1, X2, X3), sum(Y1, Y2, Y3)).
4.07/1.83	add(0, X, X).
4.07/1.83	add(s(X), Y, s(Z)) :- add(X, Y, Z).
4.07/1.83	
4.07/1.83	
4.07/1.83	Query: sum(g,a,g)
4.07/1.83	----------------------------------------
4.07/1.83	
4.07/1.83	(1) PrologToDTProblemTransformerProof (SOUND)
4.07/1.83	Built DT problem from termination graph DT10.
4.07/1.83	
4.07/1.83	{
4.07/1.83	    "root": 1,
4.07/1.83	    "program": {
4.07/1.83	        "directives": [],
4.07/1.83	        "clauses": [
4.07/1.83	            [
4.07/1.83	                "(sum ([]) ([]) ([]))",
4.07/1.83	                null
4.07/1.83	            ],
4.07/1.83	            [
4.07/1.83	                "(sum (. X1 Y1) (. X2 Y2) (. X3 Y3))",
4.07/1.83	                "(',' (add X1 X2 X3) (sum Y1 Y2 Y3))"
4.07/1.83	            ],
4.07/1.83	            [
4.07/1.83	                "(add (0) X X)",
4.07/1.83	                null
4.07/1.83	            ],
4.07/1.83	            [
4.07/1.83	                "(add (s X) Y (s Z))",
4.07/1.83	                "(add X Y Z)"
4.07/1.83	            ]
4.07/1.83	        ]
4.07/1.83	    },
4.07/1.83	    "graph": {
4.07/1.83	        "nodes": {
4.07/1.83	            "27": {
4.07/1.83	                "goal": [
4.07/1.83	                    {
4.07/1.83	                        "clause": -1,
4.07/1.83	                        "scope": -1,
4.07/1.83	                        "term": "(true)"
4.07/1.83	                    },
4.07/1.83	                    {
4.07/1.83	                        "clause": 1,
4.07/1.83	                        "scope": 1,
4.07/1.83	                        "term": "(sum ([]) T2 ([]))"
4.07/1.83	                    }
4.07/1.83	                ],
4.07/1.83	                "kb": {
4.07/1.83	                    "nonunifying": [],
4.07/1.83	                    "intvars": {},
4.07/1.83	                    "arithmetic": {
4.07/1.83	                        "type": "PlainIntegerRelationState",
4.07/1.83	                        "relations": []
4.07/1.83	                    },
4.07/1.83	                    "ground": [],
4.07/1.83	                    "free": [],
4.07/1.83	                    "exprvars": []
4.07/1.83	                }
4.07/1.83	            },
4.07/1.83	            "28": {
4.07/1.83	                "goal": [{
4.07/1.83	                    "clause": 1,
4.07/1.83	                    "scope": 1,
4.07/1.83	                    "term": "(sum T1 T2 T3)"
4.07/1.83	                }],
4.07/1.83	                "kb": {
4.07/1.83	                    "nonunifying": [[
4.07/1.83	                        "(sum T1 T2 T3)",
4.07/1.83	                        "(sum ([]) ([]) ([]))"
4.07/1.83	                    ]],
4.07/1.83	                    "intvars": {},
4.07/1.83	                    "arithmetic": {
4.07/1.83	                        "type": "PlainIntegerRelationState",
4.07/1.83	                        "relations": []
4.07/1.83	                    },
4.07/1.83	                    "ground": [
4.07/1.83	                        "T1",
4.07/1.83	                        "T3"
4.07/1.83	                    ],
4.07/1.83	                    "free": [],
4.07/1.83	                    "exprvars": []
4.07/1.83	                }
4.07/1.83	            },
4.07/1.83	            "29": {
4.07/1.83	                "goal": [{
4.07/1.83	                    "clause": 1,
4.07/1.83	                    "scope": 1,
4.07/1.83	                    "term": "(sum ([]) T2 ([]))"
4.07/1.83	                }],
4.07/1.83	                "kb": {
4.07/1.83	                    "nonunifying": [],
4.07/1.83	                    "intvars": {},
4.07/1.83	                    "arithmetic": {
4.07/1.83	                        "type": "PlainIntegerRelationState",
4.07/1.83	                        "relations": []
4.07/1.83	                    },
4.07/1.83	                    "ground": [],
4.07/1.83	                    "free": [],
4.07/1.83	                    "exprvars": []
4.07/1.83	                }
4.07/1.83	            },
4.07/1.83	            "type": "Nodes",
4.07/1.83	            "120": {
4.07/1.83	                "goal": [],
4.07/1.83	                "kb": {
4.07/1.83	                    "nonunifying": [],
4.07/1.83	                    "intvars": {},
4.07/1.83	                    "arithmetic": {
4.07/1.83	                        "type": "PlainIntegerRelationState",
4.07/1.83	                        "relations": []
4.07/1.83	                    },
4.07/1.83	                    "ground": [],
4.07/1.83	                    "free": [],
4.07/1.83	                    "exprvars": []
4.07/1.83	                }
4.07/1.83	            },
4.07/1.83	            "121": {
4.07/1.83	                "goal": [
4.07/1.83	                    {
4.07/1.83	                        "clause": 2,
4.07/1.83	                        "scope": 2,
4.07/1.83	                        "term": "(',' (add T10 T16 T14) (sum T11 T17 T15))"
4.07/1.83	                    },
4.07/1.83	                    {
4.07/1.83	                        "clause": 3,
4.07/1.83	                        "scope": 2,
4.07/1.83	                        "term": "(',' (add T10 T16 T14) (sum T11 T17 T15))"
4.07/1.83	                    }
4.07/1.83	                ],
4.07/1.83	                "kb": {
4.07/1.83	                    "nonunifying": [],
4.07/1.83	                    "intvars": {},
4.07/1.83	                    "arithmetic": {
4.07/1.83	                        "type": "PlainIntegerRelationState",
4.07/1.83	                        "relations": []
4.07/1.83	                    },
4.07/1.83	                    "ground": [
4.07/1.83	                        "T10",
4.07/1.83	                        "T11",
4.07/1.83	                        "T14",
4.07/1.83	                        "T15"
4.07/1.83	                    ],
4.07/1.83	                    "free": [],
4.07/1.83	                    "exprvars": []
4.07/1.83	                }
4.07/1.83	            },
4.07/1.83	            "1": {
4.07/1.83	                "goal": [{
4.07/1.83	                    "clause": -1,
4.07/1.83	                    "scope": -1,
4.07/1.83	                    "term": "(sum T1 T2 T3)"
4.07/1.83	                }],
4.07/1.83	                "kb": {
4.07/1.83	                    "nonunifying": [],
4.07/1.83	                    "intvars": {},
4.07/1.83	                    "arithmetic": {
4.07/1.83	                        "type": "PlainIntegerRelationState",
4.07/1.83	                        "relations": []
4.07/1.83	                    },
4.07/1.83	                    "ground": [
4.07/1.83	                        "T1",
4.07/1.83	                        "T3"
4.07/1.83	                    ],
4.07/1.83	                    "free": [],
4.07/1.83	                    "exprvars": []
4.07/1.83	                }
4.07/1.83	            },
4.07/1.83	            "122": {
4.07/1.83	                "goal": [{
4.07/1.83	                    "clause": 2,
4.07/1.83	                    "scope": 2,
4.07/1.83	                    "term": "(',' (add T10 T16 T14) (sum T11 T17 T15))"
4.07/1.83	                }],
4.07/1.83	                "kb": {
4.07/1.83	                    "nonunifying": [],
4.07/1.83	                    "intvars": {},
4.07/1.83	                    "arithmetic": {
4.07/1.83	                        "type": "PlainIntegerRelationState",
4.07/1.83	                        "relations": []
4.07/1.83	                    },
4.07/1.83	                    "ground": [
4.07/1.83	                        "T10",
4.07/1.83	                        "T11",
4.07/1.83	                        "T14",
4.07/1.83	                        "T15"
4.07/1.83	                    ],
4.07/1.83	                    "free": [],
4.07/1.83	                    "exprvars": []
4.07/1.83	                }
4.07/1.83	            },
4.07/1.83	            "221": {
4.07/1.83	                "goal": [{
4.07/1.83	                    "clause": -1,
4.07/1.83	                    "scope": -1,
4.07/1.83	                    "term": "(',' (add T30 T33 T32) (sum T11 T34 T15))"
4.07/1.83	                }],
4.07/1.83	                "kb": {
4.07/1.83	                    "nonunifying": [],
4.07/1.83	                    "intvars": {},
4.07/1.83	                    "arithmetic": {
4.07/1.83	                        "type": "PlainIntegerRelationState",
4.07/1.83	                        "relations": []
4.07/1.83	                    },
4.07/1.83	                    "ground": [
4.07/1.83	                        "T11",
4.07/1.83	                        "T15",
4.07/1.83	                        "T30",
4.07/1.83	                        "T32"
4.07/1.83	                    ],
4.07/1.83	                    "free": [],
4.07/1.83	                    "exprvars": []
4.07/1.83	                }
4.07/1.83	            },
4.07/1.83	            "123": {
4.07/1.83	                "goal": [{
4.07/1.83	                    "clause": 3,
4.07/1.83	                    "scope": 2,
4.07/1.83	                    "term": "(',' (add T10 T16 T14) (sum T11 T17 T15))"
4.07/1.83	                }],
4.07/1.83	                "kb": {
4.07/1.83	                    "nonunifying": [],
4.07/1.83	                    "intvars": {},
4.07/1.83	                    "arithmetic": {
4.07/1.83	                        "type": "PlainIntegerRelationState",
4.07/1.83	                        "relations": []
4.07/1.83	                    },
4.07/1.83	                    "ground": [
4.07/1.83	                        "T10",
4.07/1.83	                        "T11",
4.07/1.83	                        "T14",
4.07/1.83	                        "T15"
4.07/1.83	                    ],
4.07/1.83	                    "free": [],
4.07/1.83	                    "exprvars": []
4.07/1.83	                }
4.07/1.83	            },
4.07/1.83	            "222": {
4.07/1.83	                "goal": [],
4.07/1.83	                "kb": {
4.07/1.83	                    "nonunifying": [],
4.07/1.83	                    "intvars": {},
4.07/1.83	                    "arithmetic": {
4.07/1.83	                        "type": "PlainIntegerRelationState",
4.07/1.83	                        "relations": []
4.07/1.83	                    },
4.07/1.84	                    "ground": [],
4.07/1.84	                    "free": [],
4.07/1.84	                    "exprvars": []
4.07/1.84	                }
4.07/1.84	            },
4.07/1.84	            "3": {
4.07/1.84	                "goal": [
4.07/1.84	                    {
4.07/1.84	                        "clause": 0,
4.07/1.84	                        "scope": 1,
4.07/1.84	                        "term": "(sum T1 T2 T3)"
4.07/1.84	                    },
4.07/1.84	                    {
4.07/1.84	                        "clause": 1,
4.07/1.84	                        "scope": 1,
4.07/1.84	                        "term": "(sum T1 T2 T3)"
4.07/1.84	                    }
4.07/1.84	                ],
4.07/1.84	                "kb": {
4.07/1.84	                    "nonunifying": [],
4.07/1.84	                    "intvars": {},
4.07/1.84	                    "arithmetic": {
4.07/1.84	                        "type": "PlainIntegerRelationState",
4.07/1.84	                        "relations": []
4.07/1.84	                    },
4.07/1.84	                    "ground": [
4.07/1.84	                        "T1",
4.07/1.84	                        "T3"
4.07/1.84	                    ],
4.07/1.84	                    "free": [],
4.07/1.84	                    "exprvars": []
4.07/1.84	                }
4.07/1.84	            },
4.07/1.84	            "126": {
4.07/1.84	                "goal": [{
4.07/1.84	                    "clause": -1,
4.07/1.84	                    "scope": -1,
4.07/1.84	                    "term": "(sum T11 T23 T15)"
4.07/1.84	                }],
4.07/1.84	                "kb": {
4.07/1.84	                    "nonunifying": [],
4.07/1.84	                    "intvars": {},
4.07/1.84	                    "arithmetic": {
4.07/1.84	                        "type": "PlainIntegerRelationState",
4.07/1.84	                        "relations": []
4.07/1.84	                    },
4.07/1.84	                    "ground": [
4.07/1.84	                        "T11",
4.07/1.84	                        "T15"
4.07/1.84	                    ],
4.07/1.84	                    "free": [],
4.07/1.84	                    "exprvars": []
4.07/1.84	                }
4.07/1.84	            },
4.07/1.84	            "127": {
4.07/1.84	                "goal": [],
4.07/1.84	                "kb": {
4.07/1.84	                    "nonunifying": [],
4.07/1.84	                    "intvars": {},
4.07/1.84	                    "arithmetic": {
4.07/1.84	                        "type": "PlainIntegerRelationState",
4.07/1.84	                        "relations": []
4.07/1.84	                    },
4.07/1.84	                    "ground": [],
4.07/1.84	                    "free": [],
4.07/1.84	                    "exprvars": []
4.07/1.84	                }
4.07/1.84	            },
4.07/1.84	            "119": {
4.07/1.84	                "goal": [{
4.07/1.84	                    "clause": -1,
4.07/1.84	                    "scope": -1,
4.07/1.84	                    "term": "(',' (add T10 T16 T14) (sum T11 T17 T15))"
4.07/1.84	                }],
4.07/1.84	                "kb": {
4.07/1.84	                    "nonunifying": [],
4.07/1.84	                    "intvars": {},
4.07/1.84	                    "arithmetic": {
4.07/1.84	                        "type": "PlainIntegerRelationState",
4.07/1.84	                        "relations": []
4.07/1.84	                    },
4.07/1.84	                    "ground": [
4.07/1.84	                        "T10",
4.07/1.84	                        "T11",
4.07/1.84	                        "T14",
4.07/1.84	                        "T15"
4.07/1.84	                    ],
4.07/1.84	                    "free": [],
4.07/1.84	                    "exprvars": []
4.07/1.84	                }
4.07/1.84	            },
4.07/1.84	            "30": {
4.07/1.84	                "goal": [],
4.07/1.84	                "kb": {
4.07/1.84	                    "nonunifying": [],
4.07/1.84	                    "intvars": {},
4.07/1.84	                    "arithmetic": {
4.07/1.84	                        "type": "PlainIntegerRelationState",
4.07/1.84	                        "relations": []
4.07/1.84	                    },
4.07/1.84	                    "ground": [],
4.07/1.84	                    "free": [],
4.07/1.84	                    "exprvars": []
4.07/1.84	                }
4.07/1.84	            }
4.07/1.84	        },
4.07/1.84	        "edges": [
4.07/1.84	            {
4.07/1.84	                "from": 1,
4.07/1.84	                "to": 3,
4.07/1.84	                "label": "CASE"
4.07/1.84	            },
4.07/1.84	            {
4.07/1.84	                "from": 3,
4.07/1.84	                "to": 27,
4.07/1.84	                "label": "EVAL with clause\nsum([], [], []).\nand substitutionT1 -> [],\nT2 -> [],\nT3 -> []"
4.07/1.84	            },
4.07/1.84	            {
4.07/1.84	                "from": 3,
4.07/1.84	                "to": 28,
4.07/1.84	                "label": "EVAL-BACKTRACK"
4.07/1.84	            },
4.07/1.84	            {
4.07/1.84	                "from": 27,
4.07/1.84	                "to": 29,
4.07/1.84	                "label": "SUCCESS"
4.07/1.84	            },
4.07/1.84	            {
4.07/1.84	                "from": 28,
4.07/1.84	                "to": 119,
4.07/1.84	                "label": "EVAL with clause\nsum(.(X16, X17), .(X18, X19), .(X20, X21)) :- ','(add(X16, X18, X20), sum(X17, X19, X21)).\nand substitutionX16 -> T10,\nX17 -> T11,\nT1 -> .(T10, T11),\nX18 -> T16,\nX19 -> T17,\nT2 -> .(T16, T17),\nX20 -> T14,\nX21 -> T15,\nT3 -> .(T14, T15),\nT12 -> T16,\nT13 -> T17"
4.07/1.84	            },
4.07/1.84	            {
4.07/1.84	                "from": 28,
4.07/1.84	                "to": 120,
4.07/1.84	                "label": "EVAL-BACKTRACK"
4.07/1.84	            },
4.07/1.84	            {
4.07/1.84	                "from": 29,
4.07/1.84	                "to": 30,
4.07/1.84	                "label": "BACKTRACK\nfor clause: sum(.(X1, Y1), .(X2, Y2), .(X3, Y3)) :- ','(add(X1, X2, X3), sum(Y1, Y2, Y3))because of non-unification"
4.07/1.84	            },
4.07/1.84	            {
4.07/1.84	                "from": 119,
4.07/1.84	                "to": 121,
4.07/1.84	                "label": "CASE"
4.07/1.84	            },
4.07/1.84	            {
4.07/1.84	                "from": 121,
4.07/1.84	                "to": 122,
4.07/1.84	                "label": "PARALLEL"
4.07/1.84	            },
4.07/1.84	            {
4.07/1.84	                "from": 121,
4.07/1.84	                "to": 123,
4.07/1.84	                "label": "PARALLEL"
4.07/1.84	            },
4.07/1.84	            {
4.07/1.84	                "from": 122,
4.07/1.84	                "to": 126,
4.07/1.84	                "label": "EVAL with clause\nadd(0, X26, X26).\nand substitutionT10 -> 0,\nT16 -> T22,\nX26 -> T22,\nT14 -> T22,\nT17 -> T23"
4.07/1.84	            },
4.07/1.84	            {
4.07/1.84	                "from": 122,
4.07/1.84	                "to": 127,
4.07/1.84	                "label": "EVAL-BACKTRACK"
4.07/1.84	            },
4.07/1.84	            {
4.07/1.84	                "from": 123,
4.07/1.84	                "to": 221,
4.07/1.84	                "label": "EVAL with clause\nadd(s(X33), X34, s(X35)) :- add(X33, X34, X35).\nand substitutionX33 -> T30,\nT10 -> s(T30),\nT16 -> T33,\nX34 -> T33,\nX35 -> T32,\nT14 -> s(T32),\nT31 -> T33,\nT17 -> T34"
4.07/1.84	            },
4.07/1.84	            {
4.07/1.84	                "from": 123,
4.07/1.84	                "to": 222,
4.07/1.84	                "label": "EVAL-BACKTRACK"
4.07/1.84	            },
4.07/1.84	            {
4.07/1.84	                "from": 126,
4.07/1.84	                "to": 1,
4.07/1.84	                "label": "INSTANCE with matching:\nT1 -> T11\nT2 -> T23\nT3 -> T15"
4.07/1.84	            },
4.07/1.84	            {
4.07/1.84	                "from": 221,
4.07/1.84	                "to": 119,
4.07/1.84	                "label": "INSTANCE with matching:\nT10 -> T30\nT16 -> T33\nT14 -> T32\nT17 -> T34"
4.07/1.84	            }
4.07/1.84	        ],
4.07/1.84	        "type": "Graph"
4.07/1.84	    }
4.07/1.84	}
4.07/1.84	
4.07/1.84	----------------------------------------
4.07/1.84	
4.07/1.84	(2)
4.07/1.84	Obligation:
4.07/1.84	Triples:
4.07/1.84	
4.07/1.84	pB(0, X1, X1, X2, X3, X4) :- sumA(X2, X3, X4).
4.07/1.84	pB(s(X1), X2, s(X3), X4, X5, X6) :- pB(X1, X2, X3, X4, X5, X6).
4.07/1.84	sumA(.(X1, X2), .(X3, X4), .(X5, X6)) :- pB(X1, X3, X5, X2, X4, X6).
4.07/1.84	
4.07/1.84	Clauses:
4.07/1.84	
4.07/1.84	sumcA([], [], []).
4.07/1.84	sumcA(.(X1, X2), .(X3, X4), .(X5, X6)) :- qcB(X1, X3, X5, X2, X4, X6).
4.07/1.84	qcB(0, X1, X1, X2, X3, X4) :- sumcA(X2, X3, X4).
4.07/1.84	qcB(s(X1), X2, s(X3), X4, X5, X6) :- qcB(X1, X2, X3, X4, X5, X6).
4.07/1.84	
4.07/1.84	Afs:
4.07/1.84	
4.07/1.84	sumA(x1, x2, x3)  =  sumA(x1, x3)
4.07/1.84	
4.07/1.84	
4.07/1.84	----------------------------------------
4.07/1.84	
4.07/1.84	(3) TriplesToPiDPProof (SOUND)
4.07/1.84	We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
4.07/1.84	
4.07/1.84	sumA_in_3: (b,f,b)
4.07/1.84	
4.07/1.84	pB_in_6: (b,f,b,b,f,b)
4.07/1.84	
4.07/1.84	Transforming TRIPLES into the following Term Rewriting System:
4.07/1.84	
4.07/1.84	Pi DP problem:
4.07/1.84	The TRS P consists of the following rules:
4.07/1.84	
4.07/1.84	   SUMA_IN_GAG(.(X1, X2), .(X3, X4), .(X5, X6)) -> U3_GAG(X1, X2, X3, X4, X5, X6, pB_in_gaggag(X1, X3, X5, X2, X4, X6))
4.07/1.84	   SUMA_IN_GAG(.(X1, X2), .(X3, X4), .(X5, X6)) -> PB_IN_GAGGAG(X1, X3, X5, X2, X4, X6)
4.07/1.84	   PB_IN_GAGGAG(0, X1, X1, X2, X3, X4) -> U1_GAGGAG(X1, X2, X3, X4, sumA_in_gag(X2, X3, X4))
4.07/1.84	   PB_IN_GAGGAG(0, X1, X1, X2, X3, X4) -> SUMA_IN_GAG(X2, X3, X4)
4.07/1.84	   PB_IN_GAGGAG(s(X1), X2, s(X3), X4, X5, X6) -> U2_GAGGAG(X1, X2, X3, X4, X5, X6, pB_in_gaggag(X1, X2, X3, X4, X5, X6))
4.07/1.84	   PB_IN_GAGGAG(s(X1), X2, s(X3), X4, X5, X6) -> PB_IN_GAGGAG(X1, X2, X3, X4, X5, X6)
4.07/1.84	
4.07/1.84	R is empty.
4.07/1.84	The argument filtering Pi contains the following mapping:
4.07/1.84	sumA_in_gag(x1, x2, x3)  =  sumA_in_gag(x1, x3)
4.07/1.84	
4.07/1.84	.(x1, x2)  =  .(x1, x2)
4.07/1.84	
4.07/1.84	pB_in_gaggag(x1, x2, x3, x4, x5, x6)  =  pB_in_gaggag(x1, x3, x4, x6)
4.07/1.84	
4.07/1.84	0  =  0
4.07/1.84	
4.07/1.84	s(x1)  =  s(x1)
4.07/1.84	
4.07/1.84	SUMA_IN_GAG(x1, x2, x3)  =  SUMA_IN_GAG(x1, x3)
4.07/1.84	
4.07/1.84	U3_GAG(x1, x2, x3, x4, x5, x6, x7)  =  U3_GAG(x1, x2, x5, x6, x7)
4.07/1.84	
4.07/1.84	PB_IN_GAGGAG(x1, x2, x3, x4, x5, x6)  =  PB_IN_GAGGAG(x1, x3, x4, x6)
4.07/1.84	
4.07/1.84	U1_GAGGAG(x1, x2, x3, x4, x5)  =  U1_GAGGAG(x1, x2, x4, x5)
4.07/1.84	
4.07/1.84	U2_GAGGAG(x1, x2, x3, x4, x5, x6, x7)  =  U2_GAGGAG(x1, x3, x4, x6, x7)
4.07/1.84	
4.07/1.84	
4.07/1.84	We have to consider all (P,R,Pi)-chains
4.07/1.84	
4.07/1.84	
4.07/1.84	Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES
4.07/1.84	
4.07/1.84	
4.07/1.84	
4.07/1.84	----------------------------------------
4.07/1.84	
4.07/1.84	(4)
4.07/1.84	Obligation:
4.07/1.84	Pi DP problem:
4.07/1.84	The TRS P consists of the following rules:
4.07/1.84	
4.07/1.84	   SUMA_IN_GAG(.(X1, X2), .(X3, X4), .(X5, X6)) -> U3_GAG(X1, X2, X3, X4, X5, X6, pB_in_gaggag(X1, X3, X5, X2, X4, X6))
4.07/1.84	   SUMA_IN_GAG(.(X1, X2), .(X3, X4), .(X5, X6)) -> PB_IN_GAGGAG(X1, X3, X5, X2, X4, X6)
4.07/1.84	   PB_IN_GAGGAG(0, X1, X1, X2, X3, X4) -> U1_GAGGAG(X1, X2, X3, X4, sumA_in_gag(X2, X3, X4))
4.07/1.84	   PB_IN_GAGGAG(0, X1, X1, X2, X3, X4) -> SUMA_IN_GAG(X2, X3, X4)
4.07/1.84	   PB_IN_GAGGAG(s(X1), X2, s(X3), X4, X5, X6) -> U2_GAGGAG(X1, X2, X3, X4, X5, X6, pB_in_gaggag(X1, X2, X3, X4, X5, X6))
4.07/1.84	   PB_IN_GAGGAG(s(X1), X2, s(X3), X4, X5, X6) -> PB_IN_GAGGAG(X1, X2, X3, X4, X5, X6)
4.07/1.84	
4.07/1.84	R is empty.
4.07/1.84	The argument filtering Pi contains the following mapping:
4.07/1.84	sumA_in_gag(x1, x2, x3)  =  sumA_in_gag(x1, x3)
4.07/1.84	
4.07/1.84	.(x1, x2)  =  .(x1, x2)
4.07/1.84	
4.07/1.84	pB_in_gaggag(x1, x2, x3, x4, x5, x6)  =  pB_in_gaggag(x1, x3, x4, x6)
4.07/1.84	
4.07/1.84	0  =  0
4.07/1.84	
4.07/1.84	s(x1)  =  s(x1)
4.07/1.84	
4.07/1.84	SUMA_IN_GAG(x1, x2, x3)  =  SUMA_IN_GAG(x1, x3)
4.07/1.84	
4.07/1.84	U3_GAG(x1, x2, x3, x4, x5, x6, x7)  =  U3_GAG(x1, x2, x5, x6, x7)
4.07/1.84	
4.07/1.84	PB_IN_GAGGAG(x1, x2, x3, x4, x5, x6)  =  PB_IN_GAGGAG(x1, x3, x4, x6)
4.07/1.84	
4.07/1.84	U1_GAGGAG(x1, x2, x3, x4, x5)  =  U1_GAGGAG(x1, x2, x4, x5)
4.07/1.84	
4.07/1.84	U2_GAGGAG(x1, x2, x3, x4, x5, x6, x7)  =  U2_GAGGAG(x1, x3, x4, x6, x7)
4.07/1.84	
4.07/1.84	
4.07/1.84	We have to consider all (P,R,Pi)-chains
4.07/1.84	----------------------------------------
4.07/1.84	
4.07/1.84	(5) DependencyGraphProof (EQUIVALENT)
4.07/1.84	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.
4.07/1.84	----------------------------------------
4.07/1.84	
4.07/1.84	(6)
4.07/1.84	Obligation:
4.07/1.84	Pi DP problem:
4.07/1.84	The TRS P consists of the following rules:
4.07/1.84	
4.07/1.84	   SUMA_IN_GAG(.(X1, X2), .(X3, X4), .(X5, X6)) -> PB_IN_GAGGAG(X1, X3, X5, X2, X4, X6)
4.07/1.84	   PB_IN_GAGGAG(0, X1, X1, X2, X3, X4) -> SUMA_IN_GAG(X2, X3, X4)
4.07/1.84	   PB_IN_GAGGAG(s(X1), X2, s(X3), X4, X5, X6) -> PB_IN_GAGGAG(X1, X2, X3, X4, X5, X6)
4.07/1.84	
4.07/1.84	R is empty.
4.07/1.84	The argument filtering Pi contains the following mapping:
4.07/1.84	.(x1, x2)  =  .(x1, x2)
4.07/1.84	
4.07/1.84	0  =  0
4.07/1.84	
4.07/1.84	s(x1)  =  s(x1)
4.07/1.84	
4.07/1.84	SUMA_IN_GAG(x1, x2, x3)  =  SUMA_IN_GAG(x1, x3)
4.07/1.84	
4.07/1.84	PB_IN_GAGGAG(x1, x2, x3, x4, x5, x6)  =  PB_IN_GAGGAG(x1, x3, x4, x6)
4.07/1.84	
4.07/1.84	
4.07/1.84	We have to consider all (P,R,Pi)-chains
4.07/1.84	----------------------------------------
4.07/1.84	
4.07/1.84	(7) PiDPToQDPProof (SOUND)
4.07/1.84	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
4.07/1.84	----------------------------------------
4.07/1.84	
4.07/1.84	(8)
4.07/1.84	Obligation:
4.07/1.84	Q DP problem:
4.07/1.84	The TRS P consists of the following rules:
4.07/1.84	
4.07/1.84	   SUMA_IN_GAG(.(X1, X2), .(X5, X6)) -> PB_IN_GAGGAG(X1, X5, X2, X6)
4.07/1.84	   PB_IN_GAGGAG(0, X1, X2, X4) -> SUMA_IN_GAG(X2, X4)
4.07/1.84	   PB_IN_GAGGAG(s(X1), s(X3), X4, X6) -> PB_IN_GAGGAG(X1, X3, X4, X6)
4.07/1.84	
4.07/1.84	R is empty.
4.07/1.84	Q is empty.
4.07/1.84	We have to consider all (P,Q,R)-chains.
4.07/1.84	----------------------------------------
4.07/1.84	
4.07/1.84	(9) QDPSizeChangeProof (EQUIVALENT)
4.07/1.84	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. 
4.07/1.84	
4.07/1.84	From the DPs we obtained the following set of size-change graphs:
4.07/1.84	*PB_IN_GAGGAG(0, X1, X2, X4) -> SUMA_IN_GAG(X2, X4)
4.07/1.84	The graph contains the following edges 3 >= 1, 4 >= 2
4.07/1.84	
4.07/1.84	
4.07/1.84	*PB_IN_GAGGAG(s(X1), s(X3), X4, X6) -> PB_IN_GAGGAG(X1, X3, X4, X6)
4.07/1.84	The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3, 4 >= 4
4.07/1.84	
4.07/1.84	
4.07/1.84	*SUMA_IN_GAG(.(X1, X2), .(X5, X6)) -> PB_IN_GAGGAG(X1, X5, X2, X6)
4.07/1.84	The graph contains the following edges 1 > 1, 2 > 2, 1 > 3, 2 > 4
4.07/1.84	
4.07/1.84	
4.07/1.84	----------------------------------------
4.07/1.84	
4.07/1.84	(10)
4.07/1.84	YES
4.07/1.87	EOF