3.71/1.75	YES
3.71/1.77	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
3.71/1.77	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.71/1.77	
3.71/1.77	
3.71/1.77	Left Termination of the query pattern
3.71/1.77	
3.71/1.77	p(g,a)
3.71/1.77	
3.71/1.77	w.r.t. the given Prolog program could successfully be proven:
3.71/1.77	
3.71/1.77	(0) Prolog
3.71/1.77	(1) PrologToDTProblemTransformerProof [SOUND, 0 ms]
3.71/1.77	(2) TRIPLES
3.71/1.77	(3) TriplesToPiDPProof [SOUND, 7 ms]
3.71/1.77	(4) PiDP
3.71/1.77	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
3.71/1.77	(6) PiDP
3.71/1.77	(7) PiDPToQDPProof [SOUND, 0 ms]
3.71/1.77	(8) QDP
3.71/1.77	(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.71/1.77	(10) YES
3.71/1.77	
3.71/1.77	
3.71/1.77	----------------------------------------
3.71/1.77	
3.71/1.77	(0)
3.71/1.77	Obligation:
3.71/1.77	Clauses:
3.71/1.77	
3.71/1.77	p(X, X).
3.71/1.77	p(f(X), g(Y)) :- ','(p(f(X), f(Z)), p(Z, g(W))).
3.71/1.77	
3.71/1.77	
3.71/1.77	Query: p(g,a)
3.71/1.77	----------------------------------------
3.71/1.77	
3.71/1.77	(1) PrologToDTProblemTransformerProof (SOUND)
3.71/1.77	Built DT problem from termination graph DT10.
3.71/1.77	
3.71/1.77	{
3.71/1.77	    "root": 7,
3.71/1.77	    "program": {
3.71/1.77	        "directives": [],
3.71/1.77	        "clauses": [
3.71/1.77	            [
3.71/1.77	                "(p X X)",
3.71/1.77	                null
3.71/1.77	            ],
3.71/1.77	            [
3.71/1.77	                "(p (f X) (g Y))",
3.71/1.77	                "(',' (p (f X) (f Z)) (p Z (g W)))"
3.71/1.77	            ]
3.71/1.77	        ]
3.71/1.77	    },
3.71/1.77	    "graph": {
3.71/1.77	        "nodes": {
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3.71/1.77	                        "clause": 0,
3.71/1.77	                        "scope": 2,
3.71/1.77	                        "term": "(',' (p (f T7) (f X7)) (p X7 (g X8)))"
3.71/1.77	                    },
3.71/1.77	                    {
3.71/1.77	                        "clause": 1,
3.71/1.77	                        "scope": 2,
3.71/1.77	                        "term": "(',' (p (f T7) (f X7)) (p X7 (g X8)))"
3.71/1.77	                    }
3.71/1.77	                ],
3.71/1.77	                "kb": {
3.71/1.77	                    "nonunifying": [],
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3.71/1.77	                    "arithmetic": {
3.71/1.77	                        "type": "PlainIntegerRelationState",
3.71/1.77	                        "relations": []
3.71/1.77	                    },
3.71/1.77	                    "ground": ["T7"],
3.71/1.77	                    "free": [
3.71/1.77	                        "X7",
3.71/1.77	                        "X8"
3.71/1.77	                    ],
3.71/1.77	                    "exprvars": []
3.71/1.77	                }
3.71/1.77	            },
3.71/1.77	            "44": {
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3.71/1.77	                    "clause": 1,
3.71/1.77	                    "scope": 3,
3.71/1.77	                    "term": "(',' (p (f T29) (f X26)) (p X26 (g X27)))"
3.71/1.77	                }],
3.71/1.77	                "kb": {
3.71/1.77	                    "nonunifying": [[
3.71/1.77	                        "(p (f T29) T2)",
3.71/1.77	                        "(p X2 X2)"
3.71/1.77	                    ]],
3.71/1.77	                    "intvars": {},
3.71/1.77	                    "arithmetic": {
3.71/1.77	                        "type": "PlainIntegerRelationState",
3.71/1.77	                        "relations": []
3.71/1.77	                    },
3.71/1.77	                    "ground": ["T29"],
3.71/1.77	                    "free": [
3.71/1.77	                        "X2",
3.71/1.77	                        "X26",
3.71/1.77	                        "X27"
3.71/1.77	                    ],
3.71/1.77	                    "exprvars": []
3.71/1.77	                }
3.71/1.77	            },
3.71/1.77	            "34": {
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3.71/1.77	                }],
3.71/1.77	                "kb": {
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3.71/1.77	                        "relations": []
3.71/1.77	                    },
3.71/1.77	                    "ground": ["T7"],
3.71/1.77	                    "free": [
3.71/1.77	                        "X7",
3.71/1.77	                        "X8"
3.71/1.77	                    ],
3.71/1.77	                    "exprvars": []
3.71/1.77	                }
3.71/1.77	            },
3.71/1.77	            "45": {
3.71/1.77	                "goal": [{
3.71/1.77	                    "clause": -1,
3.71/1.77	                    "scope": -1,
3.71/1.77	                    "term": "(p T43 (g X27))"
3.71/1.77	                }],
3.71/1.77	                "kb": {
3.71/1.77	                    "nonunifying": [[
3.71/1.77	                        "(p (f T43) T2)",
3.71/1.77	                        "(p X2 X2)"
3.71/1.77	                    ]],
3.71/1.77	                    "intvars": {},
3.71/1.77	                    "arithmetic": {
3.71/1.77	                        "type": "PlainIntegerRelationState",
3.71/1.77	                        "relations": []
3.71/1.77	                    },
3.71/1.77	                    "ground": ["T43"],
3.71/1.77	                    "free": [
3.71/1.77	                        "X2",
3.71/1.77	                        "X27"
3.71/1.77	                    ],
3.71/1.77	                    "exprvars": []
3.71/1.77	                }
3.71/1.77	            },
3.71/1.77	            "35": {
3.71/1.77	                "goal": [{
3.71/1.77	                    "clause": 1,
3.71/1.77	                    "scope": 2,
3.71/1.77	                    "term": "(',' (p (f T7) (f X7)) (p X7 (g X8)))"
3.71/1.77	                }],
3.71/1.77	                "kb": {
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3.71/1.77	                        "relations": []
3.71/1.77	                    },
3.71/1.77	                    "ground": ["T7"],
3.71/1.77	                    "free": [
3.71/1.77	                        "X7",
3.71/1.77	                        "X8"
3.71/1.77	                    ],
3.71/1.77	                    "exprvars": []
3.71/1.77	                }
3.71/1.77	            },
3.71/1.77	            "46": {
3.71/1.77	                "goal": [],
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3.71/1.77	                    "nonunifying": [],
3.71/1.77	                    "intvars": {},
3.71/1.77	                    "arithmetic": {
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3.71/1.77	                        "relations": []
3.71/1.77	                    },
3.71/1.77	                    "ground": [],
3.71/1.77	                    "free": [],
3.71/1.77	                    "exprvars": []
3.71/1.77	                }
3.71/1.77	            },
3.71/1.77	            "36": {
3.71/1.77	                "goal": [{
3.71/1.77	                    "clause": -1,
3.71/1.77	                    "scope": -1,
3.71/1.77	                    "term": "(p T21 (g X8))"
3.71/1.77	                }],
3.71/1.77	                "kb": {
3.71/1.77	                    "nonunifying": [],
3.71/1.77	                    "intvars": {},
3.71/1.77	                    "arithmetic": {
3.71/1.77	                        "type": "PlainIntegerRelationState",
3.71/1.77	                        "relations": []
3.71/1.77	                    },
3.71/1.77	                    "ground": ["T21"],
3.71/1.77	                    "free": ["X8"],
3.71/1.77	                    "exprvars": []
3.71/1.77	                }
3.71/1.77	            },
3.71/1.77	            "28": {
3.71/1.77	                "goal": [{
3.71/1.77	                    "clause": 1,
3.71/1.77	                    "scope": 1,
3.71/1.77	                    "term": "(p T4 T2)"
3.71/1.77	                }],
3.71/1.77	                "kb": {
3.71/1.77	                    "nonunifying": [],
3.71/1.77	                    "intvars": {},
3.71/1.77	                    "arithmetic": {
3.71/1.77	                        "type": "PlainIntegerRelationState",
3.71/1.77	                        "relations": []
3.71/1.77	                    },
3.71/1.77	                    "ground": ["T4"],
3.71/1.77	                    "free": [],
3.71/1.77	                    "exprvars": []
3.71/1.77	                }
3.71/1.77	            },
3.71/1.77	            "39": {
3.71/1.77	                "goal": [],
3.71/1.77	                "kb": {
3.71/1.77	                    "nonunifying": [],
3.71/1.77	                    "intvars": {},
3.71/1.77	                    "arithmetic": {
3.71/1.77	                        "type": "PlainIntegerRelationState",
3.71/1.77	                        "relations": []
3.71/1.77	                    },
3.71/1.77	                    "ground": [],
3.71/1.77	                    "free": [],
3.71/1.77	                    "exprvars": []
3.71/1.77	                }
3.71/1.77	            },
3.71/1.77	            "type": "Nodes",
3.71/1.77	            "7": {
3.71/1.77	                "goal": [{
3.71/1.77	                    "clause": -1,
3.71/1.77	                    "scope": -1,
3.71/1.77	                    "term": "(p T1 T2)"
3.71/1.77	                }],
3.71/1.77	                "kb": {
3.71/1.77	                    "nonunifying": [],
3.71/1.77	                    "intvars": {},
3.71/1.77	                    "arithmetic": {
3.71/1.77	                        "type": "PlainIntegerRelationState",
3.71/1.77	                        "relations": []
3.71/1.77	                    },
3.71/1.77	                    "ground": ["T1"],
3.71/1.77	                    "free": [],
3.71/1.77	                    "exprvars": []
3.71/1.77	                }
3.71/1.77	            },
3.71/1.77	            "8": {
3.71/1.77	                "goal": [
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3.71/1.77	                        "clause": 0,
3.71/1.77	                        "scope": 1,
3.71/1.77	                        "term": "(p T1 T2)"
3.71/1.77	                    },
3.71/1.77	                    {
3.71/1.77	                        "clause": 1,
3.71/1.77	                        "scope": 1,
3.71/1.77	                        "term": "(p T1 T2)"
3.71/1.77	                    }
3.71/1.77	                ],
3.71/1.77	                "kb": {
3.71/1.77	                    "nonunifying": [],
3.71/1.77	                    "intvars": {},
3.71/1.77	                    "arithmetic": {
3.71/1.77	                        "type": "PlainIntegerRelationState",
3.71/1.77	                        "relations": []
3.71/1.77	                    },
3.71/1.77	                    "ground": ["T1"],
3.71/1.77	                    "free": [],
3.71/1.77	                    "exprvars": []
3.71/1.77	                }
3.71/1.77	            },
3.71/1.77	            "40": {
3.71/1.77	                "goal": [{
3.71/1.77	                    "clause": -1,
3.71/1.77	                    "scope": -1,
3.71/1.77	                    "term": "(',' (p (f T29) (f X26)) (p X26 (g X27)))"
3.71/1.77	                }],
3.71/1.77	                "kb": {
3.71/1.77	                    "nonunifying": [[
3.71/1.77	                        "(p (f T29) T2)",
3.71/1.77	                        "(p X2 X2)"
3.71/1.77	                    ]],
3.71/1.77	                    "intvars": {},
3.71/1.77	                    "arithmetic": {
3.71/1.77	                        "type": "PlainIntegerRelationState",
3.71/1.77	                        "relations": []
3.71/1.77	                    },
3.71/1.77	                    "ground": ["T29"],
3.71/1.77	                    "free": [
3.71/1.77	                        "X2",
3.71/1.77	                        "X26",
3.71/1.77	                        "X27"
3.71/1.77	                    ],
3.71/1.77	                    "exprvars": []
3.71/1.77	                }
3.71/1.77	            },
3.71/1.77	            "41": {
3.71/1.77	                "goal": [],
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3.71/1.77	                    "nonunifying": [],
3.71/1.77	                    "intvars": {},
3.71/1.77	                    "arithmetic": {
3.71/1.77	                        "type": "PlainIntegerRelationState",
3.71/1.77	                        "relations": []
3.71/1.77	                    },
3.71/1.77	                    "ground": [],
3.71/1.77	                    "free": [],
3.71/1.77	                    "exprvars": []
3.71/1.77	                }
3.71/1.77	            },
3.71/1.77	            "20": {
3.71/1.77	                "goal": [
3.71/1.77	                    {
3.71/1.77	                        "clause": -1,
3.71/1.77	                        "scope": -1,
3.71/1.77	                        "term": "(true)"
3.71/1.77	                    },
3.71/1.77	                    {
3.71/1.77	                        "clause": 1,
3.71/1.77	                        "scope": 1,
3.71/1.77	                        "term": "(p T4 T2)"
3.71/1.77	                    }
3.71/1.77	                ],
3.71/1.77	                "kb": {
3.71/1.77	                    "nonunifying": [],
3.71/1.77	                    "intvars": {},
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3.71/1.77	                        "type": "PlainIntegerRelationState",
3.71/1.77	                        "relations": []
3.71/1.77	                    },
3.71/1.77	                    "ground": ["T4"],
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3.71/1.77	                    "exprvars": []
3.71/1.77	                }
3.71/1.77	            },
3.71/1.77	            "31": {
3.71/1.77	                "goal": [{
3.71/1.77	                    "clause": -1,
3.71/1.77	                    "scope": -1,
3.71/1.77	                    "term": "(',' (p (f T7) (f X7)) (p X7 (g X8)))"
3.71/1.77	                }],
3.71/1.77	                "kb": {
3.71/1.77	                    "nonunifying": [],
3.71/1.77	                    "intvars": {},
3.71/1.77	                    "arithmetic": {
3.71/1.77	                        "type": "PlainIntegerRelationState",
3.71/1.77	                        "relations": []
3.71/1.77	                    },
3.71/1.77	                    "ground": ["T7"],
3.71/1.77	                    "free": [
3.71/1.77	                        "X7",
3.71/1.77	                        "X8"
3.71/1.77	                    ],
3.71/1.77	                    "exprvars": []
3.71/1.77	                }
3.71/1.77	            },
3.71/1.77	            "42": {
3.71/1.77	                "goal": [
3.71/1.77	                    {
3.71/1.77	                        "clause": 0,
3.71/1.77	                        "scope": 3,
3.71/1.77	                        "term": "(',' (p (f T29) (f X26)) (p X26 (g X27)))"
3.71/1.77	                    },
3.71/1.77	                    {
3.71/1.77	                        "clause": 1,
3.71/1.77	                        "scope": 3,
3.71/1.77	                        "term": "(',' (p (f T29) (f X26)) (p X26 (g X27)))"
3.71/1.77	                    }
3.71/1.77	                ],
3.71/1.77	                "kb": {
3.71/1.77	                    "nonunifying": [[
3.71/1.77	                        "(p (f T29) T2)",
3.71/1.77	                        "(p X2 X2)"
3.71/1.77	                    ]],
3.71/1.77	                    "intvars": {},
3.71/1.77	                    "arithmetic": {
3.71/1.77	                        "type": "PlainIntegerRelationState",
3.71/1.77	                        "relations": []
3.71/1.77	                    },
3.71/1.77	                    "ground": ["T29"],
3.71/1.77	                    "free": [
3.71/1.77	                        "X2",
3.71/1.77	                        "X26",
3.71/1.77	                        "X27"
3.71/1.77	                    ],
3.71/1.77	                    "exprvars": []
3.71/1.77	                }
3.71/1.77	            },
3.71/1.77	            "21": {
3.71/1.77	                "goal": [{
3.71/1.77	                    "clause": 1,
3.71/1.77	                    "scope": 1,
3.71/1.77	                    "term": "(p T1 T2)"
3.71/1.77	                }],
3.71/1.77	                "kb": {
3.71/1.77	                    "nonunifying": [[
3.71/1.77	                        "(p T1 T2)",
3.71/1.77	                        "(p X2 X2)"
3.71/1.77	                    ]],
3.71/1.77	                    "intvars": {},
3.71/1.77	                    "arithmetic": {
3.71/1.77	                        "type": "PlainIntegerRelationState",
3.71/1.77	                        "relations": []
3.71/1.77	                    },
3.71/1.77	                    "ground": ["T1"],
3.71/1.77	                    "free": ["X2"],
3.71/1.77	                    "exprvars": []
3.71/1.77	                }
3.71/1.77	            },
3.71/1.77	            "32": {
3.71/1.77	                "goal": [],
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3.71/1.77	                    },
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3.71/1.77	                }
3.71/1.77	            },
3.71/1.77	            "43": {
3.71/1.77	                "goal": [{
3.71/1.77	                    "clause": 0,
3.71/1.77	                    "scope": 3,
3.71/1.77	                    "term": "(',' (p (f T29) (f X26)) (p X26 (g X27)))"
3.71/1.77	                }],
3.71/1.77	                "kb": {
3.71/1.77	                    "nonunifying": [[
3.71/1.77	                        "(p (f T29) T2)",
3.71/1.77	                        "(p X2 X2)"
3.71/1.77	                    ]],
3.71/1.77	                    "intvars": {},
3.71/1.77	                    "arithmetic": {
3.71/1.77	                        "type": "PlainIntegerRelationState",
3.71/1.77	                        "relations": []
3.71/1.77	                    },
3.71/1.77	                    "ground": ["T29"],
3.71/1.77	                    "free": [
3.71/1.77	                        "X2",
3.71/1.77	                        "X26",
3.71/1.77	                        "X27"
3.71/1.77	                    ],
3.71/1.77	                    "exprvars": []
3.71/1.77	                }
3.71/1.77	            }
3.71/1.77	        },
3.71/1.77	        "edges": [
3.71/1.77	            {
3.71/1.77	                "from": 7,
3.71/1.77	                "to": 8,
3.71/1.77	                "label": "CASE"
3.71/1.77	            },
3.71/1.77	            {
3.71/1.77	                "from": 8,
3.71/1.77	                "to": 20,
3.71/1.77	                "label": "EVAL with clause\np(X2, X2).\nand substitutionT1 -> T4,\nX2 -> T4,\nT2 -> T4"
3.71/1.77	            },
3.71/1.77	            {
3.71/1.77	                "from": 8,
3.71/1.77	                "to": 21,
3.71/1.77	                "label": "EVAL-BACKTRACK"
3.71/1.77	            },
3.71/1.77	            {
3.71/1.77	                "from": 20,
3.71/1.77	                "to": 28,
3.71/1.77	                "label": "SUCCESS"
3.71/1.77	            },
3.71/1.77	            {
3.71/1.77	                "from": 21,
3.71/1.77	                "to": 40,
3.71/1.77	                "label": "EVAL with clause\np(f(X24), g(X25)) :- ','(p(f(X24), f(X26)), p(X26, g(X27))).\nand substitutionX24 -> T29,\nT1 -> f(T29),\nX25 -> T30,\nT2 -> g(T30)"
3.71/1.77	            },
3.71/1.77	            {
3.71/1.77	                "from": 21,
3.71/1.77	                "to": 41,
3.71/1.77	                "label": "EVAL-BACKTRACK"
3.71/1.77	            },
3.71/1.77	            {
3.71/1.77	                "from": 28,
3.71/1.77	                "to": 31,
3.71/1.77	                "label": "EVAL with clause\np(f(X5), g(X6)) :- ','(p(f(X5), f(X7)), p(X7, g(X8))).\nand substitutionX5 -> T7,\nT4 -> f(T7),\nX6 -> T8,\nT2 -> g(T8)"
3.71/1.77	            },
3.71/1.77	            {
3.71/1.77	                "from": 28,
3.71/1.77	                "to": 32,
3.71/1.77	                "label": "EVAL-BACKTRACK"
3.71/1.77	            },
3.71/1.77	            {
3.71/1.77	                "from": 31,
3.71/1.77	                "to": 33,
3.71/1.77	                "label": "CASE"
3.71/1.77	            },
3.71/1.77	            {
3.71/1.77	                "from": 33,
3.71/1.77	                "to": 34,
3.71/1.77	                "label": "PARALLEL"
3.71/1.77	            },
3.71/1.77	            {
3.71/1.77	                "from": 33,
3.71/1.77	                "to": 35,
3.71/1.77	                "label": "PARALLEL"
3.71/1.77	            },
3.71/1.77	            {
3.71/1.77	                "from": 34,
3.71/1.77	                "to": 36,
3.71/1.77	                "label": "ONLY EVAL with clause\np(X17, X17).\nand substitutionT7 -> T21,\nX17 -> f(T21),\nX7 -> T21"
3.71/1.77	            },
3.71/1.77	            {
3.71/1.77	                "from": 35,
3.71/1.77	                "to": 39,
3.71/1.77	                "label": "BACKTRACK\nfor clause: p(f(X), g(Y)) :- ','(p(f(X), f(Z)), p(Z, g(W)))because of non-unification"
3.71/1.77	            },
3.71/1.77	            {
3.71/1.77	                "from": 36,
3.71/1.77	                "to": 7,
3.71/1.77	                "label": "INSTANCE with matching:\nT1 -> T21\nT2 -> g(X8)"
3.71/1.77	            },
3.71/1.77	            {
3.71/1.77	                "from": 40,
3.71/1.77	                "to": 42,
3.71/1.77	                "label": "CASE"
3.71/1.77	            },
3.71/1.77	            {
3.71/1.77	                "from": 42,
3.71/1.77	                "to": 43,
3.71/1.77	                "label": "PARALLEL"
3.71/1.77	            },
3.71/1.77	            {
3.71/1.77	                "from": 42,
3.71/1.77	                "to": 44,
3.71/1.77	                "label": "PARALLEL"
3.71/1.77	            },
3.71/1.77	            {
3.71/1.77	                "from": 43,
3.71/1.77	                "to": 45,
3.71/1.77	                "label": "ONLY EVAL with clause\np(X36, X36).\nand substitutionT29 -> T43,\nX36 -> f(T43),\nX26 -> T43"
3.71/1.77	            },
3.71/1.77	            {
3.71/1.77	                "from": 44,
3.71/1.77	                "to": 46,
3.71/1.77	                "label": "BACKTRACK\nfor clause: p(f(X), g(Y)) :- ','(p(f(X), f(Z)), p(Z, g(W)))because of non-unification"
3.71/1.77	            },
3.71/1.77	            {
3.71/1.77	                "from": 45,
3.71/1.77	                "to": 7,
3.71/1.77	                "label": "INSTANCE with matching:\nT1 -> T43\nT2 -> g(X27)"
3.71/1.77	            }
3.71/1.77	        ],
3.71/1.77	        "type": "Graph"
3.71/1.77	    }
3.71/1.77	}
3.71/1.77	
3.71/1.77	----------------------------------------
3.71/1.77	
3.71/1.77	(2)
3.71/1.77	Obligation:
3.71/1.77	Triples:
3.71/1.77	
3.71/1.77	pA(f(X1), g(X2)) :- pA(X1, g(X3)).
3.71/1.77	pA(f(X1), g(X2)) :- pA(X1, g(X3)).
3.71/1.77	
3.71/1.77	Clauses:
3.71/1.77	
3.71/1.77	pcA(X1, X1).
3.71/1.77	pcA(f(X1), g(X2)) :- pcA(X1, g(X3)).
3.71/1.77	pcA(f(X1), g(X2)) :- pcA(X1, g(X3)).
3.71/1.77	
3.71/1.77	Afs:
3.71/1.77	
3.71/1.77	pA(x1, x2)  =  pA(x1)
3.71/1.77	
3.71/1.77	
3.71/1.77	----------------------------------------
3.71/1.77	
3.71/1.77	(3) TriplesToPiDPProof (SOUND)
3.71/1.77	We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
3.71/1.77	
3.71/1.77	pA_in_2: (b,f) (b,b)
3.71/1.77	
3.71/1.77	Transforming TRIPLES into the following Term Rewriting System:
3.71/1.77	
3.71/1.77	Pi DP problem:
3.71/1.77	The TRS P consists of the following rules:
3.71/1.77	
3.71/1.77	   PA_IN_GA(f(X1), g(X2)) -> U1_GA(X1, X2, pA_in_gg(X1, g(X3)))
3.71/1.77	   PA_IN_GA(f(X1), g(X2)) -> PA_IN_GG(X1, g(X3))
3.71/1.77	   PA_IN_GG(f(X1), g(X2)) -> U1_GG(X1, X2, pA_in_gg(X1, g(X3)))
3.71/1.77	   PA_IN_GG(f(X1), g(X2)) -> PA_IN_GG(X1, g(X3))
3.71/1.77	
3.71/1.77	R is empty.
3.71/1.77	The argument filtering Pi contains the following mapping:
3.71/1.77	f(x1)  =  f(x1)
3.71/1.77	
3.71/1.77	pA_in_gg(x1, x2)  =  pA_in_gg(x1, x2)
3.71/1.77	
3.71/1.77	g(x1)  =  g
3.71/1.77	
3.71/1.77	PA_IN_GA(x1, x2)  =  PA_IN_GA(x1)
3.71/1.77	
3.71/1.77	U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
3.71/1.77	
3.71/1.77	PA_IN_GG(x1, x2)  =  PA_IN_GG(x1, x2)
3.71/1.77	
3.71/1.77	U1_GG(x1, x2, x3)  =  U1_GG(x1, x3)
3.71/1.77	
3.71/1.77	
3.71/1.77	We have to consider all (P,R,Pi)-chains
3.71/1.77	
3.71/1.77	
3.71/1.77	Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES
3.71/1.77	
3.71/1.77	
3.71/1.77	
3.71/1.77	----------------------------------------
3.71/1.77	
3.71/1.77	(4)
3.71/1.77	Obligation:
3.71/1.77	Pi DP problem:
3.71/1.77	The TRS P consists of the following rules:
3.71/1.77	
3.71/1.77	   PA_IN_GA(f(X1), g(X2)) -> U1_GA(X1, X2, pA_in_gg(X1, g(X3)))
3.71/1.77	   PA_IN_GA(f(X1), g(X2)) -> PA_IN_GG(X1, g(X3))
3.71/1.77	   PA_IN_GG(f(X1), g(X2)) -> U1_GG(X1, X2, pA_in_gg(X1, g(X3)))
3.71/1.77	   PA_IN_GG(f(X1), g(X2)) -> PA_IN_GG(X1, g(X3))
3.71/1.77	
3.71/1.77	R is empty.
3.71/1.77	The argument filtering Pi contains the following mapping:
3.71/1.77	f(x1)  =  f(x1)
3.71/1.77	
3.71/1.77	pA_in_gg(x1, x2)  =  pA_in_gg(x1, x2)
3.71/1.77	
3.71/1.77	g(x1)  =  g
3.71/1.77	
3.71/1.77	PA_IN_GA(x1, x2)  =  PA_IN_GA(x1)
3.71/1.77	
3.71/1.77	U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
3.71/1.77	
3.71/1.77	PA_IN_GG(x1, x2)  =  PA_IN_GG(x1, x2)
3.71/1.77	
3.71/1.77	U1_GG(x1, x2, x3)  =  U1_GG(x1, x3)
3.71/1.77	
3.71/1.77	
3.71/1.77	We have to consider all (P,R,Pi)-chains
3.71/1.77	----------------------------------------
3.71/1.77	
3.71/1.77	(5) DependencyGraphProof (EQUIVALENT)
3.71/1.77	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.
3.71/1.77	----------------------------------------
3.71/1.77	
3.71/1.77	(6)
3.71/1.77	Obligation:
3.71/1.77	Pi DP problem:
3.71/1.77	The TRS P consists of the following rules:
3.71/1.77	
3.71/1.77	   PA_IN_GG(f(X1), g(X2)) -> PA_IN_GG(X1, g(X3))
3.71/1.77	
3.71/1.77	R is empty.
3.71/1.77	The argument filtering Pi contains the following mapping:
3.71/1.77	f(x1)  =  f(x1)
3.71/1.77	
3.71/1.77	g(x1)  =  g
3.71/1.77	
3.71/1.77	PA_IN_GG(x1, x2)  =  PA_IN_GG(x1, x2)
3.71/1.77	
3.71/1.77	
3.71/1.77	We have to consider all (P,R,Pi)-chains
3.71/1.77	----------------------------------------
3.71/1.77	
3.71/1.77	(7) PiDPToQDPProof (SOUND)
3.71/1.77	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
3.71/1.77	----------------------------------------
3.71/1.77	
3.71/1.77	(8)
3.71/1.77	Obligation:
3.71/1.77	Q DP problem:
3.71/1.77	The TRS P consists of the following rules:
3.71/1.77	
3.71/1.77	   PA_IN_GG(f(X1), g) -> PA_IN_GG(X1, g)
3.71/1.77	
3.71/1.77	R is empty.
3.71/1.77	Q is empty.
3.71/1.77	We have to consider all (P,Q,R)-chains.
3.71/1.77	----------------------------------------
3.71/1.77	
3.71/1.77	(9) QDPSizeChangeProof (EQUIVALENT)
3.71/1.77	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. 
3.71/1.77	
3.71/1.77	From the DPs we obtained the following set of size-change graphs:
3.71/1.77	*PA_IN_GG(f(X1), g) -> PA_IN_GG(X1, g)
3.71/1.77	The graph contains the following edges 1 > 1, 2 >= 2
3.71/1.77	
3.71/1.77	
3.71/1.77	----------------------------------------
3.71/1.77	
3.71/1.77	(10)
3.71/1.77	YES
3.71/1.80	EOF