3.72/1.76	YES
3.98/1.78	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
3.98/1.78	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.98/1.78	
3.98/1.78	
3.98/1.78	Left Termination of the query pattern
3.98/1.78	
3.98/1.78	sum(g,g,a)
3.98/1.78	
3.98/1.78	w.r.t. the given Prolog program could successfully be proven:
3.98/1.78	
3.98/1.78	(0) Prolog
3.98/1.78	(1) PrologToDTProblemTransformerProof [SOUND, 0 ms]
3.98/1.78	(2) TRIPLES
3.98/1.78	(3) TriplesToPiDPProof [SOUND, 0 ms]
3.98/1.78	(4) PiDP
3.98/1.78	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
3.98/1.78	(6) PiDP
3.98/1.78	(7) PiDPToQDPProof [SOUND, 7 ms]
3.98/1.78	(8) QDP
3.98/1.78	(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.98/1.78	(10) YES
3.98/1.78	
3.98/1.78	
3.98/1.78	----------------------------------------
3.98/1.78	
3.98/1.78	(0)
3.98/1.78	Obligation:
3.98/1.78	Clauses:
3.98/1.78	
3.98/1.78	sum([], [], []).
3.98/1.78	sum(.(X1, Y1), .(X2, Y2), .(X3, Y3)) :- ','(add(X1, X2, X3), sum(Y1, Y2, Y3)).
3.98/1.78	add(0, X, X).
3.98/1.78	add(s(X), Y, s(Z)) :- add(X, Y, Z).
3.98/1.78	
3.98/1.78	
3.98/1.78	Query: sum(g,g,a)
3.98/1.78	----------------------------------------
3.98/1.78	
3.98/1.78	(1) PrologToDTProblemTransformerProof (SOUND)
3.98/1.78	Built DT problem from termination graph DT10.
3.98/1.78	
3.98/1.78	{
3.98/1.78	    "root": 1,
3.98/1.78	    "program": {
3.98/1.78	        "directives": [],
3.98/1.78	        "clauses": [
3.98/1.78	            [
3.98/1.78	                "(sum ([]) ([]) ([]))",
3.98/1.78	                null
3.98/1.78	            ],
3.98/1.78	            [
3.98/1.78	                "(sum (. X1 Y1) (. X2 Y2) (. X3 Y3))",
3.98/1.78	                "(',' (add X1 X2 X3) (sum Y1 Y2 Y3))"
3.98/1.78	            ],
3.98/1.78	            [
3.98/1.78	                "(add (0) X X)",
3.98/1.78	                null
3.98/1.78	            ],
3.98/1.78	            [
3.98/1.78	                "(add (s X) Y (s Z))",
3.98/1.78	                "(add X Y Z)"
3.98/1.78	            ]
3.98/1.78	        ]
3.98/1.78	    },
3.98/1.78	    "graph": {
3.98/1.78	        "nodes": {
3.98/1.78	            "22": {
3.98/1.78	                "goal": [
3.98/1.78	                    {
3.98/1.78	                        "clause": 2,
3.98/1.78	                        "scope": 2,
3.98/1.78	                        "term": "(',' (add T10 T12 T16) (sum T11 T13 T17))"
3.98/1.78	                    },
3.98/1.78	                    {
3.98/1.78	                        "clause": 3,
3.98/1.78	                        "scope": 2,
3.98/1.78	                        "term": "(',' (add T10 T12 T16) (sum T11 T13 T17))"
3.98/1.78	                    }
3.98/1.78	                ],
3.98/1.78	                "kb": {
3.98/1.78	                    "nonunifying": [],
3.98/1.78	                    "intvars": {},
3.98/1.78	                    "arithmetic": {
3.98/1.78	                        "type": "PlainIntegerRelationState",
3.98/1.78	                        "relations": []
3.98/1.78	                    },
3.98/1.78	                    "ground": [
3.98/1.78	                        "T10",
3.98/1.78	                        "T11",
3.98/1.78	                        "T12",
3.98/1.78	                        "T13"
3.98/1.78	                    ],
3.98/1.78	                    "free": [],
3.98/1.78	                    "exprvars": []
3.98/1.78	                }
3.98/1.78	            },
3.98/1.78	            "25": {
3.98/1.78	                "goal": [{
3.98/1.78	                    "clause": 2,
3.98/1.78	                    "scope": 2,
3.98/1.78	                    "term": "(',' (add T10 T12 T16) (sum T11 T13 T17))"
3.98/1.78	                }],
3.98/1.78	                "kb": {
3.98/1.78	                    "nonunifying": [],
3.98/1.78	                    "intvars": {},
3.98/1.78	                    "arithmetic": {
3.98/1.78	                        "type": "PlainIntegerRelationState",
3.98/1.78	                        "relations": []
3.98/1.78	                    },
3.98/1.78	                    "ground": [
3.98/1.78	                        "T10",
3.98/1.78	                        "T11",
3.98/1.78	                        "T12",
3.98/1.78	                        "T13"
3.98/1.78	                    ],
3.98/1.78	                    "free": [],
3.98/1.78	                    "exprvars": []
3.98/1.78	                }
3.98/1.78	            },
3.98/1.78	            "15": {
3.98/1.78	                "goal": [
3.98/1.78	                    {
3.98/1.78	                        "clause": -1,
3.98/1.78	                        "scope": -1,
3.98/1.78	                        "term": "(true)"
3.98/1.78	                    },
3.98/1.78	                    {
3.98/1.78	                        "clause": 1,
3.98/1.78	                        "scope": 1,
3.98/1.78	                        "term": "(sum ([]) ([]) T3)"
3.98/1.78	                    }
3.98/1.78	                ],
3.98/1.78	                "kb": {
3.98/1.78	                    "nonunifying": [],
3.98/1.78	                    "intvars": {},
3.98/1.78	                    "arithmetic": {
3.98/1.78	                        "type": "PlainIntegerRelationState",
3.98/1.78	                        "relations": []
3.98/1.78	                    },
3.98/1.78	                    "ground": [],
3.98/1.78	                    "free": [],
3.98/1.78	                    "exprvars": []
3.98/1.78	                }
3.98/1.78	            },
3.98/1.78	            "26": {
3.98/1.78	                "goal": [{
3.98/1.78	                    "clause": 3,
3.98/1.78	                    "scope": 2,
3.98/1.78	                    "term": "(',' (add T10 T12 T16) (sum T11 T13 T17))"
3.98/1.78	                }],
3.98/1.78	                "kb": {
3.98/1.78	                    "nonunifying": [],
3.98/1.78	                    "intvars": {},
3.98/1.78	                    "arithmetic": {
3.98/1.78	                        "type": "PlainIntegerRelationState",
3.98/1.78	                        "relations": []
3.98/1.78	                    },
3.98/1.78	                    "ground": [
3.98/1.78	                        "T10",
3.98/1.78	                        "T11",
3.98/1.78	                        "T12",
3.98/1.78	                        "T13"
3.98/1.78	                    ],
3.98/1.78	                    "free": [],
3.98/1.78	                    "exprvars": []
3.98/1.78	                }
3.98/1.78	            },
3.98/1.78	            "27": {
3.98/1.78	                "goal": [{
3.98/1.78	                    "clause": -1,
3.98/1.78	                    "scope": -1,
3.98/1.78	                    "term": "(sum T11 T13 T23)"
3.98/1.78	                }],
3.98/1.78	                "kb": {
3.98/1.78	                    "nonunifying": [],
3.98/1.78	                    "intvars": {},
3.98/1.78	                    "arithmetic": {
3.98/1.78	                        "type": "PlainIntegerRelationState",
3.98/1.78	                        "relations": []
3.98/1.78	                    },
3.98/1.78	                    "ground": [
3.98/1.78	                        "T11",
3.98/1.78	                        "T13"
3.98/1.78	                    ],
3.98/1.78	                    "free": [],
3.98/1.78	                    "exprvars": []
3.98/1.78	                }
3.98/1.78	            },
3.98/1.78	            "17": {
3.98/1.78	                "goal": [{
3.98/1.78	                    "clause": 1,
3.98/1.78	                    "scope": 1,
3.98/1.78	                    "term": "(sum T1 T2 T3)"
3.98/1.78	                }],
3.98/1.78	                "kb": {
3.98/1.78	                    "nonunifying": [[
3.98/1.78	                        "(sum T1 T2 T3)",
3.98/1.78	                        "(sum ([]) ([]) ([]))"
3.98/1.78	                    ]],
3.98/1.78	                    "intvars": {},
3.98/1.78	                    "arithmetic": {
3.98/1.78	                        "type": "PlainIntegerRelationState",
3.98/1.78	                        "relations": []
3.98/1.78	                    },
3.98/1.78	                    "ground": [
3.98/1.78	                        "T1",
3.98/1.78	                        "T2"
3.98/1.78	                    ],
3.98/1.78	                    "free": [],
3.98/1.78	                    "exprvars": []
3.98/1.78	                }
3.98/1.78	            },
3.98/1.78	            "28": {
3.98/1.78	                "goal": [],
3.98/1.79	                "kb": {
3.98/1.79	                    "nonunifying": [],
3.98/1.79	                    "intvars": {},
3.98/1.79	                    "arithmetic": {
3.98/1.79	                        "type": "PlainIntegerRelationState",
3.98/1.79	                        "relations": []
3.98/1.79	                    },
3.98/1.79	                    "ground": [],
3.98/1.79	                    "free": [],
3.98/1.79	                    "exprvars": []
3.98/1.79	                }
3.98/1.79	            },
3.98/1.79	            "18": {
3.98/1.79	                "goal": [{
3.98/1.79	                    "clause": 1,
3.98/1.79	                    "scope": 1,
3.98/1.79	                    "term": "(sum ([]) ([]) T3)"
3.98/1.79	                }],
3.98/1.79	                "kb": {
3.98/1.79	                    "nonunifying": [],
3.98/1.79	                    "intvars": {},
3.98/1.79	                    "arithmetic": {
3.98/1.79	                        "type": "PlainIntegerRelationState",
3.98/1.79	                        "relations": []
3.98/1.79	                    },
3.98/1.79	                    "ground": [],
3.98/1.79	                    "free": [],
3.98/1.79	                    "exprvars": []
3.98/1.79	                }
3.98/1.79	            },
3.98/1.79	            "19": {
3.98/1.79	                "goal": [],
3.98/1.79	                "kb": {
3.98/1.79	                    "nonunifying": [],
3.98/1.79	                    "intvars": {},
3.98/1.79	                    "arithmetic": {
3.98/1.79	                        "type": "PlainIntegerRelationState",
3.98/1.79	                        "relations": []
3.98/1.79	                    },
3.98/1.79	                    "ground": [],
3.98/1.79	                    "free": [],
3.98/1.79	                    "exprvars": []
3.98/1.79	                }
3.98/1.79	            },
3.98/1.79	            "type": "Nodes",
3.98/1.79	            "1": {
3.98/1.79	                "goal": [{
3.98/1.79	                    "clause": -1,
3.98/1.79	                    "scope": -1,
3.98/1.79	                    "term": "(sum T1 T2 T3)"
3.98/1.79	                }],
3.98/1.79	                "kb": {
3.98/1.79	                    "nonunifying": [],
3.98/1.79	                    "intvars": {},
3.98/1.79	                    "arithmetic": {
3.98/1.79	                        "type": "PlainIntegerRelationState",
3.98/1.79	                        "relations": []
3.98/1.79	                    },
3.98/1.79	                    "ground": [
3.98/1.79	                        "T1",
3.98/1.79	                        "T2"
3.98/1.79	                    ],
3.98/1.79	                    "free": [],
3.98/1.79	                    "exprvars": []
3.98/1.79	                }
3.98/1.79	            },
3.98/1.79	            "221": {
3.98/1.79	                "goal": [{
3.98/1.79	                    "clause": -1,
3.98/1.79	                    "scope": -1,
3.98/1.79	                    "term": "(',' (add T30 T31 T33) (sum T11 T13 T34))"
3.98/1.79	                }],
3.98/1.79	                "kb": {
3.98/1.79	                    "nonunifying": [],
3.98/1.79	                    "intvars": {},
3.98/1.79	                    "arithmetic": {
3.98/1.79	                        "type": "PlainIntegerRelationState",
3.98/1.79	                        "relations": []
3.98/1.79	                    },
3.98/1.79	                    "ground": [
3.98/1.79	                        "T11",
3.98/1.79	                        "T13",
3.98/1.79	                        "T30",
3.98/1.79	                        "T31"
3.98/1.79	                    ],
3.98/1.79	                    "free": [],
3.98/1.79	                    "exprvars": []
3.98/1.79	                }
3.98/1.79	            },
3.98/1.79	            "222": {
3.98/1.79	                "goal": [],
3.98/1.79	                "kb": {
3.98/1.79	                    "nonunifying": [],
3.98/1.79	                    "intvars": {},
3.98/1.79	                    "arithmetic": {
3.98/1.79	                        "type": "PlainIntegerRelationState",
3.98/1.79	                        "relations": []
3.98/1.79	                    },
3.98/1.79	                    "ground": [],
3.98/1.79	                    "free": [],
3.98/1.79	                    "exprvars": []
3.98/1.79	                }
3.98/1.79	            },
3.98/1.79	            "4": {
3.98/1.79	                "goal": [
3.98/1.79	                    {
3.98/1.79	                        "clause": 0,
3.98/1.79	                        "scope": 1,
3.98/1.79	                        "term": "(sum T1 T2 T3)"
3.98/1.79	                    },
3.98/1.79	                    {
3.98/1.79	                        "clause": 1,
3.98/1.79	                        "scope": 1,
3.98/1.79	                        "term": "(sum T1 T2 T3)"
3.98/1.79	                    }
3.98/1.79	                ],
3.98/1.79	                "kb": {
3.98/1.79	                    "nonunifying": [],
3.98/1.79	                    "intvars": {},
3.98/1.79	                    "arithmetic": {
3.98/1.79	                        "type": "PlainIntegerRelationState",
3.98/1.79	                        "relations": []
3.98/1.79	                    },
3.98/1.79	                    "ground": [
3.98/1.79	                        "T1",
3.98/1.79	                        "T2"
3.98/1.79	                    ],
3.98/1.79	                    "free": [],
3.98/1.79	                    "exprvars": []
3.98/1.79	                }
3.98/1.79	            },
3.98/1.79	            "20": {
3.98/1.79	                "goal": [{
3.98/1.79	                    "clause": -1,
3.98/1.79	                    "scope": -1,
3.98/1.79	                    "term": "(',' (add T10 T12 T16) (sum T11 T13 T17))"
3.98/1.79	                }],
3.98/1.79	                "kb": {
3.98/1.79	                    "nonunifying": [],
3.98/1.79	                    "intvars": {},
3.98/1.79	                    "arithmetic": {
3.98/1.79	                        "type": "PlainIntegerRelationState",
3.98/1.79	                        "relations": []
3.98/1.79	                    },
3.98/1.79	                    "ground": [
3.98/1.79	                        "T10",
3.98/1.79	                        "T11",
3.98/1.79	                        "T12",
3.98/1.79	                        "T13"
3.98/1.79	                    ],
3.98/1.79	                    "free": [],
3.98/1.79	                    "exprvars": []
3.98/1.79	                }
3.98/1.79	            },
3.98/1.79	            "21": {
3.98/1.79	                "goal": [],
3.98/1.79	                "kb": {
3.98/1.79	                    "nonunifying": [],
3.98/1.79	                    "intvars": {},
3.98/1.79	                    "arithmetic": {
3.98/1.79	                        "type": "PlainIntegerRelationState",
3.98/1.79	                        "relations": []
3.98/1.79	                    },
3.98/1.79	                    "ground": [],
3.98/1.79	                    "free": [],
3.98/1.79	                    "exprvars": []
3.98/1.79	                }
3.98/1.79	            }
3.98/1.79	        },
3.98/1.79	        "edges": [
3.98/1.79	            {
3.98/1.79	                "from": 1,
3.98/1.79	                "to": 4,
3.98/1.79	                "label": "CASE"
3.98/1.79	            },
3.98/1.79	            {
3.98/1.79	                "from": 4,
3.98/1.79	                "to": 15,
3.98/1.79	                "label": "EVAL with clause\nsum([], [], []).\nand substitutionT1 -> [],\nT2 -> [],\nT3 -> []"
3.98/1.79	            },
3.98/1.79	            {
3.98/1.79	                "from": 4,
3.98/1.79	                "to": 17,
3.98/1.79	                "label": "EVAL-BACKTRACK"
3.98/1.79	            },
3.98/1.79	            {
3.98/1.79	                "from": 15,
3.98/1.79	                "to": 18,
3.98/1.79	                "label": "SUCCESS"
3.98/1.79	            },
3.98/1.79	            {
3.98/1.79	                "from": 17,
3.98/1.79	                "to": 20,
3.98/1.79	                "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 -> T12,\nX19 -> T13,\nT2 -> .(T12, T13),\nX20 -> T16,\nX21 -> T17,\nT3 -> .(T16, T17),\nT14 -> T16,\nT15 -> T17"
3.98/1.79	            },
3.98/1.79	            {
3.98/1.79	                "from": 17,
3.98/1.79	                "to": 21,
3.98/1.79	                "label": "EVAL-BACKTRACK"
3.98/1.79	            },
3.98/1.79	            {
3.98/1.79	                "from": 18,
3.98/1.79	                "to": 19,
3.98/1.79	                "label": "BACKTRACK\nfor clause: sum(.(X1, Y1), .(X2, Y2), .(X3, Y3)) :- ','(add(X1, X2, X3), sum(Y1, Y2, Y3))because of non-unification"
3.98/1.79	            },
3.98/1.79	            {
3.98/1.79	                "from": 20,
3.98/1.79	                "to": 22,
3.98/1.79	                "label": "CASE"
3.98/1.79	            },
3.98/1.79	            {
3.98/1.79	                "from": 22,
3.98/1.79	                "to": 25,
3.98/1.79	                "label": "PARALLEL"
3.98/1.79	            },
3.98/1.79	            {
3.98/1.79	                "from": 22,
3.98/1.79	                "to": 26,
3.98/1.79	                "label": "PARALLEL"
3.98/1.79	            },
3.98/1.79	            {
3.98/1.79	                "from": 25,
3.98/1.79	                "to": 27,
3.98/1.79	                "label": "EVAL with clause\nadd(0, X26, X26).\nand substitutionT10 -> 0,\nT12 -> T22,\nX26 -> T22,\nT16 -> T22,\nT17 -> T23"
3.98/1.79	            },
3.98/1.79	            {
3.98/1.79	                "from": 25,
3.98/1.79	                "to": 28,
3.98/1.79	                "label": "EVAL-BACKTRACK"
3.98/1.79	            },
3.98/1.79	            {
3.98/1.79	                "from": 26,
3.98/1.79	                "to": 221,
3.98/1.79	                "label": "EVAL with clause\nadd(s(X33), X34, s(X35)) :- add(X33, X34, X35).\nand substitutionX33 -> T30,\nT10 -> s(T30),\nT12 -> T31,\nX34 -> T31,\nX35 -> T33,\nT16 -> s(T33),\nT32 -> T33,\nT17 -> T34"
3.98/1.79	            },
3.98/1.79	            {
3.98/1.79	                "from": 26,
3.98/1.79	                "to": 222,
3.98/1.79	                "label": "EVAL-BACKTRACK"
3.98/1.79	            },
3.98/1.79	            {
3.98/1.79	                "from": 27,
3.98/1.79	                "to": 1,
3.98/1.79	                "label": "INSTANCE with matching:\nT1 -> T11\nT2 -> T13\nT3 -> T23"
3.98/1.79	            },
3.98/1.79	            {
3.98/1.79	                "from": 221,
3.98/1.79	                "to": 20,
3.98/1.79	                "label": "INSTANCE with matching:\nT10 -> T30\nT12 -> T31\nT16 -> T33\nT17 -> T34"
3.98/1.79	            }
3.98/1.79	        ],
3.98/1.79	        "type": "Graph"
3.98/1.79	    }
3.98/1.79	}
3.98/1.79	
3.98/1.79	----------------------------------------
3.98/1.79	
3.98/1.79	(2)
3.98/1.79	Obligation:
3.98/1.79	Triples:
3.98/1.79	
3.98/1.79	pB(0, X1, X1, X2, X3, X4) :- sumA(X2, X3, X4).
3.98/1.79	pB(s(X1), X2, s(X3), X4, X5, X6) :- pB(X1, X2, X3, X4, X5, X6).
3.98/1.79	sumA(.(X1, X2), .(X3, X4), .(X5, X6)) :- pB(X1, X3, X5, X2, X4, X6).
3.98/1.79	
3.98/1.79	Clauses:
3.98/1.79	
3.98/1.79	sumcA([], [], []).
3.98/1.79	sumcA(.(X1, X2), .(X3, X4), .(X5, X6)) :- qcB(X1, X3, X5, X2, X4, X6).
3.98/1.79	qcB(0, X1, X1, X2, X3, X4) :- sumcA(X2, X3, X4).
3.98/1.79	qcB(s(X1), X2, s(X3), X4, X5, X6) :- qcB(X1, X2, X3, X4, X5, X6).
3.98/1.79	
3.98/1.79	Afs:
3.98/1.79	
3.98/1.79	sumA(x1, x2, x3)  =  sumA(x1, x2)
3.98/1.79	
3.98/1.79	
3.98/1.79	----------------------------------------
3.98/1.79	
3.98/1.79	(3) TriplesToPiDPProof (SOUND)
3.98/1.79	We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
3.98/1.79	
3.98/1.79	sumA_in_3: (b,b,f)
3.98/1.79	
3.98/1.79	pB_in_6: (b,b,f,b,b,f)
3.98/1.79	
3.98/1.79	Transforming TRIPLES into the following Term Rewriting System:
3.98/1.79	
3.98/1.79	Pi DP problem:
3.98/1.79	The TRS P consists of the following rules:
3.98/1.79	
3.98/1.79	   SUMA_IN_GGA(.(X1, X2), .(X3, X4), .(X5, X6)) -> U3_GGA(X1, X2, X3, X4, X5, X6, pB_in_ggagga(X1, X3, X5, X2, X4, X6))
3.98/1.79	   SUMA_IN_GGA(.(X1, X2), .(X3, X4), .(X5, X6)) -> PB_IN_GGAGGA(X1, X3, X5, X2, X4, X6)
3.98/1.79	   PB_IN_GGAGGA(0, X1, X1, X2, X3, X4) -> U1_GGAGGA(X1, X2, X3, X4, sumA_in_gga(X2, X3, X4))
3.98/1.79	   PB_IN_GGAGGA(0, X1, X1, X2, X3, X4) -> SUMA_IN_GGA(X2, X3, X4)
3.98/1.79	   PB_IN_GGAGGA(s(X1), X2, s(X3), X4, X5, X6) -> U2_GGAGGA(X1, X2, X3, X4, X5, X6, pB_in_ggagga(X1, X2, X3, X4, X5, X6))
3.98/1.79	   PB_IN_GGAGGA(s(X1), X2, s(X3), X4, X5, X6) -> PB_IN_GGAGGA(X1, X2, X3, X4, X5, X6)
3.98/1.79	
3.98/1.79	R is empty.
3.98/1.79	The argument filtering Pi contains the following mapping:
3.98/1.79	sumA_in_gga(x1, x2, x3)  =  sumA_in_gga(x1, x2)
3.98/1.79	
3.98/1.79	.(x1, x2)  =  .(x1, x2)
3.98/1.79	
3.98/1.79	pB_in_ggagga(x1, x2, x3, x4, x5, x6)  =  pB_in_ggagga(x1, x2, x4, x5)
3.98/1.79	
3.98/1.79	0  =  0
3.98/1.79	
3.98/1.79	s(x1)  =  s(x1)
3.98/1.79	
3.98/1.79	SUMA_IN_GGA(x1, x2, x3)  =  SUMA_IN_GGA(x1, x2)
3.98/1.79	
3.98/1.79	U3_GGA(x1, x2, x3, x4, x5, x6, x7)  =  U3_GGA(x1, x2, x3, x4, x7)
3.98/1.79	
3.98/1.79	PB_IN_GGAGGA(x1, x2, x3, x4, x5, x6)  =  PB_IN_GGAGGA(x1, x2, x4, x5)
3.98/1.79	
3.98/1.79	U1_GGAGGA(x1, x2, x3, x4, x5)  =  U1_GGAGGA(x1, x2, x3, x5)
3.98/1.79	
3.98/1.79	U2_GGAGGA(x1, x2, x3, x4, x5, x6, x7)  =  U2_GGAGGA(x1, x2, x4, x5, x7)
3.98/1.79	
3.98/1.79	
3.98/1.79	We have to consider all (P,R,Pi)-chains
3.98/1.79	
3.98/1.79	
3.98/1.79	Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES
3.98/1.79	
3.98/1.79	
3.98/1.79	
3.98/1.79	----------------------------------------
3.98/1.79	
3.98/1.79	(4)
3.98/1.79	Obligation:
3.98/1.79	Pi DP problem:
3.98/1.79	The TRS P consists of the following rules:
3.98/1.79	
3.98/1.79	   SUMA_IN_GGA(.(X1, X2), .(X3, X4), .(X5, X6)) -> U3_GGA(X1, X2, X3, X4, X5, X6, pB_in_ggagga(X1, X3, X5, X2, X4, X6))
3.98/1.79	   SUMA_IN_GGA(.(X1, X2), .(X3, X4), .(X5, X6)) -> PB_IN_GGAGGA(X1, X3, X5, X2, X4, X6)
3.98/1.79	   PB_IN_GGAGGA(0, X1, X1, X2, X3, X4) -> U1_GGAGGA(X1, X2, X3, X4, sumA_in_gga(X2, X3, X4))
3.98/1.79	   PB_IN_GGAGGA(0, X1, X1, X2, X3, X4) -> SUMA_IN_GGA(X2, X3, X4)
3.98/1.79	   PB_IN_GGAGGA(s(X1), X2, s(X3), X4, X5, X6) -> U2_GGAGGA(X1, X2, X3, X4, X5, X6, pB_in_ggagga(X1, X2, X3, X4, X5, X6))
3.98/1.79	   PB_IN_GGAGGA(s(X1), X2, s(X3), X4, X5, X6) -> PB_IN_GGAGGA(X1, X2, X3, X4, X5, X6)
3.98/1.79	
3.98/1.79	R is empty.
3.98/1.79	The argument filtering Pi contains the following mapping:
3.98/1.79	sumA_in_gga(x1, x2, x3)  =  sumA_in_gga(x1, x2)
3.98/1.79	
3.98/1.79	.(x1, x2)  =  .(x1, x2)
3.98/1.79	
3.98/1.79	pB_in_ggagga(x1, x2, x3, x4, x5, x6)  =  pB_in_ggagga(x1, x2, x4, x5)
3.98/1.79	
3.98/1.79	0  =  0
3.98/1.79	
3.98/1.79	s(x1)  =  s(x1)
3.98/1.79	
3.98/1.79	SUMA_IN_GGA(x1, x2, x3)  =  SUMA_IN_GGA(x1, x2)
3.98/1.79	
3.98/1.79	U3_GGA(x1, x2, x3, x4, x5, x6, x7)  =  U3_GGA(x1, x2, x3, x4, x7)
3.98/1.79	
3.98/1.79	PB_IN_GGAGGA(x1, x2, x3, x4, x5, x6)  =  PB_IN_GGAGGA(x1, x2, x4, x5)
3.98/1.79	
3.98/1.79	U1_GGAGGA(x1, x2, x3, x4, x5)  =  U1_GGAGGA(x1, x2, x3, x5)
3.98/1.79	
3.98/1.79	U2_GGAGGA(x1, x2, x3, x4, x5, x6, x7)  =  U2_GGAGGA(x1, x2, x4, x5, x7)
3.98/1.79	
3.98/1.79	
3.98/1.79	We have to consider all (P,R,Pi)-chains
3.98/1.79	----------------------------------------
3.98/1.79	
3.98/1.79	(5) DependencyGraphProof (EQUIVALENT)
3.98/1.79	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.
3.98/1.79	----------------------------------------
3.98/1.79	
3.98/1.79	(6)
3.98/1.79	Obligation:
3.98/1.79	Pi DP problem:
3.98/1.79	The TRS P consists of the following rules:
3.98/1.79	
3.98/1.79	   SUMA_IN_GGA(.(X1, X2), .(X3, X4), .(X5, X6)) -> PB_IN_GGAGGA(X1, X3, X5, X2, X4, X6)
3.98/1.79	   PB_IN_GGAGGA(0, X1, X1, X2, X3, X4) -> SUMA_IN_GGA(X2, X3, X4)
3.98/1.79	   PB_IN_GGAGGA(s(X1), X2, s(X3), X4, X5, X6) -> PB_IN_GGAGGA(X1, X2, X3, X4, X5, X6)
3.98/1.79	
3.98/1.79	R is empty.
3.98/1.79	The argument filtering Pi contains the following mapping:
3.98/1.79	.(x1, x2)  =  .(x1, x2)
3.98/1.79	
3.98/1.79	0  =  0
3.98/1.79	
3.98/1.79	s(x1)  =  s(x1)
3.98/1.79	
3.98/1.79	SUMA_IN_GGA(x1, x2, x3)  =  SUMA_IN_GGA(x1, x2)
3.98/1.79	
3.98/1.79	PB_IN_GGAGGA(x1, x2, x3, x4, x5, x6)  =  PB_IN_GGAGGA(x1, x2, x4, x5)
3.98/1.79	
3.98/1.79	
3.98/1.79	We have to consider all (P,R,Pi)-chains
3.98/1.79	----------------------------------------
3.98/1.79	
3.98/1.79	(7) PiDPToQDPProof (SOUND)
3.98/1.79	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
3.98/1.79	----------------------------------------
3.98/1.79	
3.98/1.79	(8)
3.98/1.79	Obligation:
3.98/1.79	Q DP problem:
3.98/1.79	The TRS P consists of the following rules:
3.98/1.79	
3.98/1.79	   SUMA_IN_GGA(.(X1, X2), .(X3, X4)) -> PB_IN_GGAGGA(X1, X3, X2, X4)
3.98/1.79	   PB_IN_GGAGGA(0, X1, X2, X3) -> SUMA_IN_GGA(X2, X3)
3.98/1.79	   PB_IN_GGAGGA(s(X1), X2, X4, X5) -> PB_IN_GGAGGA(X1, X2, X4, X5)
3.98/1.79	
3.98/1.79	R is empty.
3.98/1.79	Q is empty.
3.98/1.79	We have to consider all (P,Q,R)-chains.
3.98/1.79	----------------------------------------
3.98/1.79	
3.98/1.79	(9) QDPSizeChangeProof (EQUIVALENT)
3.98/1.79	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. 
3.98/1.79	
3.98/1.79	From the DPs we obtained the following set of size-change graphs:
3.98/1.79	*PB_IN_GGAGGA(0, X1, X2, X3) -> SUMA_IN_GGA(X2, X3)
3.98/1.79	The graph contains the following edges 3 >= 1, 4 >= 2
3.98/1.79	
3.98/1.79	
3.98/1.79	*PB_IN_GGAGGA(s(X1), X2, X4, X5) -> PB_IN_GGAGGA(X1, X2, X4, X5)
3.98/1.79	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4
3.98/1.79	
3.98/1.79	
3.98/1.79	*SUMA_IN_GGA(.(X1, X2), .(X3, X4)) -> PB_IN_GGAGGA(X1, X3, X2, X4)
3.98/1.79	The graph contains the following edges 1 > 1, 2 > 2, 1 > 3, 2 > 4
3.98/1.79	
3.98/1.79	
3.98/1.79	----------------------------------------
3.98/1.79	
3.98/1.79	(10)
3.98/1.79	YES
3.98/1.80	EOF