3.71/1.79	YES
3.85/1.87	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
3.85/1.87	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.85/1.87	
3.85/1.87	
3.85/1.87	Left Termination of the query pattern
3.85/1.87	
3.85/1.87	p(g,a)
3.85/1.87	
3.85/1.87	w.r.t. the given Prolog program could successfully be proven:
3.85/1.87	
3.85/1.87	(0) Prolog
3.85/1.87	(1) PrologToDTProblemTransformerProof [SOUND, 0 ms]
3.85/1.87	(2) TRIPLES
3.85/1.87	(3) TriplesToPiDPProof [SOUND, 0 ms]
3.85/1.87	(4) PiDP
3.85/1.87	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
3.85/1.87	(6) PiDP
3.85/1.87	(7) PiDPToQDPProof [SOUND, 0 ms]
3.85/1.87	(8) QDP
3.85/1.87	(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.85/1.87	(10) YES
3.85/1.87	
3.85/1.87	
3.85/1.87	----------------------------------------
3.85/1.87	
3.85/1.87	(0)
3.85/1.87	Obligation:
3.85/1.87	Clauses:
3.85/1.87	
3.85/1.87	p(X, X).
3.85/1.87	p(f(X), g(Y)) :- ','(p(f(X), f(Z)), p(Z, g(Y))).
3.85/1.87	
3.85/1.87	
3.85/1.87	Query: p(g,a)
3.85/1.87	----------------------------------------
3.85/1.87	
3.85/1.87	(1) PrologToDTProblemTransformerProof (SOUND)
3.85/1.87	Built DT problem from termination graph DT10.
3.85/1.87	
3.85/1.87	{
3.85/1.87	    "root": 2,
3.85/1.87	    "program": {
3.85/1.87	        "directives": [],
3.85/1.87	        "clauses": [
3.85/1.87	            [
3.85/1.87	                "(p X X)",
3.85/1.87	                null
3.85/1.87	            ],
3.85/1.87	            [
3.85/1.87	                "(p (f X) (g Y))",
3.85/1.87	                "(',' (p (f X) (f Z)) (p Z (g Y)))"
3.85/1.87	            ]
3.85/1.87	        ]
3.85/1.87	    },
3.85/1.87	    "graph": {
3.85/1.87	        "nodes": {
3.85/1.87	            "type": "Nodes",
3.85/1.87	            "120": {
3.85/1.87	                "goal": [{
3.85/1.87	                    "clause": 1,
3.85/1.87	                    "scope": 2,
3.85/1.87	                    "term": "(',' (p (f T7) (f X7)) (p X7 (g T9)))"
3.85/1.87	                }],
3.85/1.87	                "kb": {
3.85/1.87	                    "nonunifying": [],
3.85/1.87	                    "intvars": {},
3.85/1.87	                    "arithmetic": {
3.85/1.87	                        "type": "PlainIntegerRelationState",
3.85/1.87	                        "relations": []
3.85/1.87	                    },
3.85/1.87	                    "ground": ["T7"],
3.85/1.87	                    "free": ["X7"],
3.85/1.87	                    "exprvars": []
3.85/1.87	                }
3.85/1.87	            },
3.85/1.87	            "121": {
3.85/1.87	                "goal": [{
3.85/1.87	                    "clause": -1,
3.85/1.87	                    "scope": -1,
3.85/1.87	                    "term": "(p T22 (g T9))"
3.85/1.87	                }],
3.85/1.87	                "kb": {
3.85/1.87	                    "nonunifying": [],
3.85/1.87	                    "intvars": {},
3.85/1.87	                    "arithmetic": {
3.85/1.87	                        "type": "PlainIntegerRelationState",
3.85/1.87	                        "relations": []
3.85/1.87	                    },
3.85/1.87	                    "ground": ["T22"],
3.85/1.87	                    "free": [],
3.85/1.87	                    "exprvars": []
3.85/1.87	                }
3.85/1.87	            },
3.85/1.87	            "122": {
3.85/1.87	                "goal": [],
3.85/1.87	                "kb": {
3.85/1.87	                    "nonunifying": [],
3.85/1.87	                    "intvars": {},
3.85/1.87	                    "arithmetic": {
3.85/1.87	                        "type": "PlainIntegerRelationState",
3.85/1.87	                        "relations": []
3.85/1.87	                    },
3.85/1.87	                    "ground": [],
3.85/1.87	                    "free": [],
3.85/1.87	                    "exprvars": []
3.85/1.87	                }
3.85/1.87	            },
3.85/1.87	            "2": {
3.85/1.87	                "goal": [{
3.85/1.87	                    "clause": -1,
3.85/1.87	                    "scope": -1,
3.85/1.87	                    "term": "(p T1 T2)"
3.85/1.87	                }],
3.85/1.87	                "kb": {
3.85/1.87	                    "nonunifying": [],
3.85/1.87	                    "intvars": {},
3.85/1.87	                    "arithmetic": {
3.85/1.87	                        "type": "PlainIntegerRelationState",
3.85/1.87	                        "relations": []
3.85/1.87	                    },
3.85/1.87	                    "ground": ["T1"],
3.85/1.87	                    "free": [],
3.85/1.87	                    "exprvars": []
3.85/1.87	                }
3.85/1.87	            },
3.85/1.87	            "123": {
3.85/1.87	                "goal": [{
3.85/1.87	                    "clause": -1,
3.85/1.87	                    "scope": -1,
3.85/1.87	                    "term": "(',' (p (f T30) (f X25)) (p X25 (g T32)))"
3.85/1.87	                }],
3.85/1.87	                "kb": {
3.85/1.87	                    "nonunifying": [[
3.85/1.87	                        "(p (f T30) T2)",
3.85/1.87	                        "(p X2 X2)"
3.85/1.87	                    ]],
3.85/1.87	                    "intvars": {},
3.85/1.87	                    "arithmetic": {
3.85/1.87	                        "type": "PlainIntegerRelationState",
3.85/1.87	                        "relations": []
3.85/1.87	                    },
3.85/1.87	                    "ground": ["T30"],
3.85/1.87	                    "free": [
3.85/1.87	                        "X2",
3.85/1.87	                        "X25"
3.85/1.87	                    ],
3.85/1.87	                    "exprvars": []
3.85/1.87	                }
3.85/1.87	            },
3.85/1.87	            "124": {
3.85/1.87	                "goal": [],
3.85/1.87	                "kb": {
3.85/1.87	                    "nonunifying": [],
3.85/1.87	                    "intvars": {},
3.85/1.87	                    "arithmetic": {
3.85/1.87	                        "type": "PlainIntegerRelationState",
3.85/1.87	                        "relations": []
3.85/1.87	                    },
3.85/1.87	                    "ground": [],
3.85/1.87	                    "free": [],
3.85/1.87	                    "exprvars": []
3.85/1.87	                }
3.85/1.87	            },
3.85/1.87	            "125": {
3.85/1.87	                "goal": [
3.85/1.87	                    {
3.85/1.87	                        "clause": 0,
3.85/1.87	                        "scope": 3,
3.85/1.87	                        "term": "(',' (p (f T30) (f X25)) (p X25 (g T32)))"
3.85/1.87	                    },
3.85/1.87	                    {
3.85/1.87	                        "clause": 1,
3.85/1.87	                        "scope": 3,
3.85/1.87	                        "term": "(',' (p (f T30) (f X25)) (p X25 (g T32)))"
3.85/1.87	                    }
3.85/1.87	                ],
3.85/1.87	                "kb": {
3.85/1.87	                    "nonunifying": [[
3.85/1.87	                        "(p (f T30) T2)",
3.85/1.87	                        "(p X2 X2)"
3.85/1.87	                    ]],
3.85/1.87	                    "intvars": {},
3.85/1.87	                    "arithmetic": {
3.85/1.87	                        "type": "PlainIntegerRelationState",
3.85/1.87	                        "relations": []
3.85/1.87	                    },
3.85/1.87	                    "ground": ["T30"],
3.85/1.87	                    "free": [
3.85/1.87	                        "X2",
3.85/1.87	                        "X25"
3.85/1.87	                    ],
3.85/1.87	                    "exprvars": []
3.85/1.87	                }
3.85/1.87	            },
3.85/1.87	            "126": {
3.85/1.87	                "goal": [{
3.85/1.87	                    "clause": 0,
3.85/1.87	                    "scope": 3,
3.85/1.87	                    "term": "(',' (p (f T30) (f X25)) (p X25 (g T32)))"
3.85/1.87	                }],
3.85/1.87	                "kb": {
3.85/1.87	                    "nonunifying": [[
3.85/1.87	                        "(p (f T30) T2)",
3.85/1.87	                        "(p X2 X2)"
3.85/1.87	                    ]],
3.85/1.87	                    "intvars": {},
3.85/1.87	                    "arithmetic": {
3.85/1.87	                        "type": "PlainIntegerRelationState",
3.85/1.87	                        "relations": []
3.85/1.87	                    },
3.85/1.87	                    "ground": ["T30"],
3.85/1.87	                    "free": [
3.85/1.87	                        "X2",
3.85/1.87	                        "X25"
3.85/1.87	                    ],
3.85/1.87	                    "exprvars": []
3.85/1.87	                }
3.85/1.87	            },
3.85/1.87	            "127": {
3.85/1.87	                "goal": [{
3.85/1.87	                    "clause": 1,
3.85/1.87	                    "scope": 3,
3.85/1.87	                    "term": "(',' (p (f T30) (f X25)) (p X25 (g T32)))"
3.85/1.87	                }],
3.85/1.87	                "kb": {
3.85/1.87	                    "nonunifying": [[
3.85/1.87	                        "(p (f T30) T2)",
3.85/1.87	                        "(p X2 X2)"
3.85/1.87	                    ]],
3.85/1.87	                    "intvars": {},
3.85/1.87	                    "arithmetic": {
3.85/1.87	                        "type": "PlainIntegerRelationState",
3.85/1.87	                        "relations": []
3.85/1.87	                    },
3.85/1.87	                    "ground": ["T30"],
3.85/1.87	                    "free": [
3.85/1.87	                        "X2",
3.85/1.87	                        "X25"
3.85/1.87	                    ],
3.85/1.87	                    "exprvars": []
3.85/1.87	                }
3.85/1.87	            },
3.85/1.88	            "90": {
3.85/1.88	                "goal": [
3.85/1.88	                    {
3.85/1.88	                        "clause": 0,
3.85/1.88	                        "scope": 1,
3.85/1.88	                        "term": "(p T1 T2)"
3.85/1.88	                    },
3.85/1.88	                    {
3.85/1.88	                        "clause": 1,
3.85/1.88	                        "scope": 1,
3.85/1.88	                        "term": "(p T1 T2)"
3.85/1.88	                    }
3.85/1.88	                ],
3.85/1.88	                "kb": {
3.85/1.88	                    "nonunifying": [],
3.85/1.88	                    "intvars": {},
3.85/1.88	                    "arithmetic": {
3.85/1.88	                        "type": "PlainIntegerRelationState",
3.85/1.88	                        "relations": []
3.85/1.88	                    },
3.85/1.88	                    "ground": ["T1"],
3.85/1.88	                    "free": [],
3.85/1.88	                    "exprvars": []
3.85/1.88	                }
3.85/1.88	            },
3.85/1.88	            "128": {
3.85/1.88	                "goal": [{
3.85/1.88	                    "clause": -1,
3.85/1.88	                    "scope": -1,
3.85/1.88	                    "term": "(p T45 (g T32))"
3.85/1.88	                }],
3.85/1.88	                "kb": {
3.85/1.88	                    "nonunifying": [[
3.85/1.88	                        "(p (f T45) T2)",
3.85/1.88	                        "(p X2 X2)"
3.85/1.88	                    ]],
3.85/1.88	                    "intvars": {},
3.85/1.88	                    "arithmetic": {
3.85/1.88	                        "type": "PlainIntegerRelationState",
3.85/1.88	                        "relations": []
3.85/1.88	                    },
3.85/1.88	                    "ground": ["T45"],
3.85/1.88	                    "free": ["X2"],
3.85/1.88	                    "exprvars": []
3.85/1.88	                }
3.85/1.88	            },
3.85/1.88	            "91": {
3.85/1.88	                "goal": [
3.85/1.88	                    {
3.85/1.88	                        "clause": -1,
3.85/1.88	                        "scope": -1,
3.85/1.88	                        "term": "(true)"
3.85/1.88	                    },
3.85/1.88	                    {
3.85/1.88	                        "clause": 1,
3.85/1.88	                        "scope": 1,
3.85/1.88	                        "term": "(p T4 T2)"
3.85/1.88	                    }
3.85/1.88	                ],
3.85/1.88	                "kb": {
3.85/1.88	                    "nonunifying": [],
3.85/1.88	                    "intvars": {},
3.85/1.88	                    "arithmetic": {
3.85/1.88	                        "type": "PlainIntegerRelationState",
3.85/1.88	                        "relations": []
3.85/1.88	                    },
3.85/1.88	                    "ground": ["T4"],
3.85/1.88	                    "free": [],
3.85/1.88	                    "exprvars": []
3.85/1.88	                }
3.85/1.88	            },
3.85/1.88	            "129": {
3.85/1.88	                "goal": [],
3.85/1.88	                "kb": {
3.85/1.88	                    "nonunifying": [],
3.85/1.88	                    "intvars": {},
3.85/1.88	                    "arithmetic": {
3.85/1.88	                        "type": "PlainIntegerRelationState",
3.85/1.88	                        "relations": []
3.85/1.88	                    },
3.85/1.88	                    "ground": [],
3.85/1.88	                    "free": [],
3.85/1.88	                    "exprvars": []
3.85/1.88	                }
3.85/1.88	            },
3.85/1.88	            "92": {
3.85/1.88	                "goal": [{
3.85/1.88	                    "clause": 1,
3.85/1.88	                    "scope": 1,
3.85/1.88	                    "term": "(p T1 T2)"
3.85/1.88	                }],
3.85/1.88	                "kb": {
3.85/1.88	                    "nonunifying": [[
3.85/1.88	                        "(p T1 T2)",
3.85/1.88	                        "(p X2 X2)"
3.85/1.88	                    ]],
3.85/1.88	                    "intvars": {},
3.85/1.88	                    "arithmetic": {
3.85/1.88	                        "type": "PlainIntegerRelationState",
3.85/1.88	                        "relations": []
3.85/1.88	                    },
3.85/1.88	                    "ground": ["T1"],
3.85/1.88	                    "free": ["X2"],
3.85/1.88	                    "exprvars": []
3.85/1.88	                }
3.85/1.88	            },
3.85/1.88	            "119": {
3.85/1.88	                "goal": [{
3.85/1.88	                    "clause": 0,
3.85/1.88	                    "scope": 2,
3.85/1.88	                    "term": "(',' (p (f T7) (f X7)) (p X7 (g T9)))"
3.85/1.88	                }],
3.85/1.88	                "kb": {
3.85/1.88	                    "nonunifying": [],
3.85/1.88	                    "intvars": {},
3.85/1.88	                    "arithmetic": {
3.85/1.88	                        "type": "PlainIntegerRelationState",
3.85/1.88	                        "relations": []
3.85/1.88	                    },
3.85/1.88	                    "ground": ["T7"],
3.85/1.88	                    "free": ["X7"],
3.85/1.88	                    "exprvars": []
3.85/1.88	                }
3.85/1.88	            },
3.85/1.88	            "93": {
3.85/1.88	                "goal": [{
3.85/1.88	                    "clause": 1,
3.85/1.88	                    "scope": 1,
3.85/1.88	                    "term": "(p T4 T2)"
3.85/1.88	                }],
3.85/1.88	                "kb": {
3.85/1.88	                    "nonunifying": [],
3.85/1.88	                    "intvars": {},
3.85/1.88	                    "arithmetic": {
3.85/1.88	                        "type": "PlainIntegerRelationState",
3.85/1.88	                        "relations": []
3.85/1.88	                    },
3.85/1.88	                    "ground": ["T4"],
3.85/1.88	                    "free": [],
3.85/1.88	                    "exprvars": []
3.85/1.88	                }
3.85/1.88	            },
3.85/1.88	            "94": {
3.85/1.88	                "goal": [{
3.85/1.88	                    "clause": -1,
3.85/1.88	                    "scope": -1,
3.85/1.88	                    "term": "(',' (p (f T7) (f X7)) (p X7 (g T9)))"
3.85/1.88	                }],
3.85/1.88	                "kb": {
3.85/1.88	                    "nonunifying": [],
3.85/1.88	                    "intvars": {},
3.85/1.88	                    "arithmetic": {
3.85/1.88	                        "type": "PlainIntegerRelationState",
3.85/1.88	                        "relations": []
3.85/1.88	                    },
3.85/1.88	                    "ground": ["T7"],
3.85/1.88	                    "free": ["X7"],
3.85/1.88	                    "exprvars": []
3.85/1.88	                }
3.85/1.88	            },
3.85/1.88	            "95": {
3.85/1.88	                "goal": [],
3.85/1.88	                "kb": {
3.85/1.88	                    "nonunifying": [],
3.85/1.88	                    "intvars": {},
3.85/1.88	                    "arithmetic": {
3.85/1.88	                        "type": "PlainIntegerRelationState",
3.85/1.88	                        "relations": []
3.85/1.88	                    },
3.85/1.88	                    "ground": [],
3.85/1.88	                    "free": [],
3.85/1.88	                    "exprvars": []
3.85/1.88	                }
3.85/1.88	            },
3.85/1.88	            "96": {
3.85/1.88	                "goal": [
3.85/1.88	                    {
3.85/1.88	                        "clause": 0,
3.85/1.88	                        "scope": 2,
3.85/1.88	                        "term": "(',' (p (f T7) (f X7)) (p X7 (g T9)))"
3.85/1.88	                    },
3.85/1.88	                    {
3.85/1.88	                        "clause": 1,
3.85/1.88	                        "scope": 2,
3.85/1.88	                        "term": "(',' (p (f T7) (f X7)) (p X7 (g T9)))"
3.85/1.88	                    }
3.85/1.88	                ],
3.85/1.88	                "kb": {
3.85/1.88	                    "nonunifying": [],
3.85/1.88	                    "intvars": {},
3.85/1.88	                    "arithmetic": {
3.85/1.88	                        "type": "PlainIntegerRelationState",
3.85/1.88	                        "relations": []
3.85/1.88	                    },
3.85/1.88	                    "ground": ["T7"],
3.85/1.88	                    "free": ["X7"],
3.85/1.88	                    "exprvars": []
3.85/1.88	                }
3.85/1.88	            }
3.85/1.88	        },
3.85/1.88	        "edges": [
3.85/1.88	            {
3.85/1.88	                "from": 2,
3.85/1.88	                "to": 90,
3.85/1.88	                "label": "CASE"
3.85/1.88	            },
3.85/1.88	            {
3.85/1.88	                "from": 90,
3.85/1.88	                "to": 91,
3.85/1.88	                "label": "EVAL with clause\np(X2, X2).\nand substitutionT1 -> T4,\nX2 -> T4,\nT2 -> T4"
3.85/1.88	            },
3.85/1.88	            {
3.85/1.88	                "from": 90,
3.85/1.88	                "to": 92,
3.85/1.88	                "label": "EVAL-BACKTRACK"
3.85/1.88	            },
3.85/1.88	            {
3.85/1.88	                "from": 91,
3.85/1.88	                "to": 93,
3.85/1.88	                "label": "SUCCESS"
3.85/1.88	            },
3.85/1.88	            {
3.85/1.88	                "from": 92,
3.85/1.88	                "to": 123,
3.85/1.88	                "label": "EVAL with clause\np(f(X23), g(X24)) :- ','(p(f(X23), f(X25)), p(X25, g(X24))).\nand substitutionX23 -> T30,\nT1 -> f(T30),\nX24 -> T32,\nT2 -> g(T32),\nT31 -> T32"
3.85/1.88	            },
3.85/1.88	            {
3.85/1.88	                "from": 92,
3.85/1.88	                "to": 124,
3.85/1.88	                "label": "EVAL-BACKTRACK"
3.85/1.88	            },
3.85/1.88	            {
3.85/1.88	                "from": 93,
3.85/1.88	                "to": 94,
3.85/1.88	                "label": "EVAL with clause\np(f(X5), g(X6)) :- ','(p(f(X5), f(X7)), p(X7, g(X6))).\nand substitutionX5 -> T7,\nT4 -> f(T7),\nX6 -> T9,\nT2 -> g(T9),\nT8 -> T9"
3.85/1.88	            },
3.85/1.88	            {
3.85/1.88	                "from": 93,
3.85/1.88	                "to": 95,
3.85/1.88	                "label": "EVAL-BACKTRACK"
3.85/1.88	            },
3.85/1.88	            {
3.85/1.88	                "from": 94,
3.85/1.88	                "to": 96,
3.85/1.88	                "label": "CASE"
3.85/1.88	            },
3.85/1.88	            {
3.85/1.88	                "from": 96,
3.85/1.88	                "to": 119,
3.85/1.88	                "label": "PARALLEL"
3.85/1.88	            },
3.85/1.88	            {
3.85/1.88	                "from": 96,
3.85/1.88	                "to": 120,
3.85/1.88	                "label": "PARALLEL"
3.85/1.88	            },
3.85/1.88	            {
3.85/1.88	                "from": 119,
3.85/1.88	                "to": 121,
3.85/1.88	                "label": "ONLY EVAL with clause\np(X16, X16).\nand substitutionT7 -> T22,\nX16 -> f(T22),\nX7 -> T22"
3.85/1.88	            },
3.85/1.88	            {
3.85/1.88	                "from": 120,
3.85/1.88	                "to": 122,
3.85/1.88	                "label": "BACKTRACK\nfor clause: p(f(X), g(Y)) :- ','(p(f(X), f(Z)), p(Z, g(Y)))because of non-unification"
3.85/1.88	            },
3.85/1.88	            {
3.85/1.88	                "from": 121,
3.85/1.88	                "to": 2,
3.85/1.88	                "label": "INSTANCE with matching:\nT1 -> T22\nT2 -> g(T9)"
3.85/1.88	            },
3.85/1.88	            {
3.85/1.88	                "from": 123,
3.85/1.88	                "to": 125,
3.85/1.88	                "label": "CASE"
3.85/1.88	            },
3.85/1.88	            {
3.85/1.88	                "from": 125,
3.85/1.88	                "to": 126,
3.85/1.88	                "label": "PARALLEL"
3.85/1.88	            },
3.85/1.88	            {
3.85/1.88	                "from": 125,
3.85/1.88	                "to": 127,
3.85/1.88	                "label": "PARALLEL"
3.85/1.88	            },
3.85/1.88	            {
3.85/1.88	                "from": 126,
3.85/1.88	                "to": 128,
3.85/1.88	                "label": "ONLY EVAL with clause\np(X34, X34).\nand substitutionT30 -> T45,\nX34 -> f(T45),\nX25 -> T45"
3.85/1.88	            },
3.85/1.88	            {
3.85/1.88	                "from": 127,
3.85/1.88	                "to": 129,
3.85/1.88	                "label": "BACKTRACK\nfor clause: p(f(X), g(Y)) :- ','(p(f(X), f(Z)), p(Z, g(Y)))because of non-unification"
3.85/1.88	            },
3.85/1.88	            {
3.85/1.88	                "from": 128,
3.85/1.88	                "to": 2,
3.85/1.88	                "label": "INSTANCE with matching:\nT1 -> T45\nT2 -> g(T32)"
3.85/1.88	            }
3.85/1.88	        ],
3.85/1.88	        "type": "Graph"
3.85/1.88	    }
3.85/1.88	}
3.85/1.88	
3.85/1.88	----------------------------------------
3.85/1.88	
3.85/1.88	(2)
3.85/1.88	Obligation:
3.85/1.88	Triples:
3.85/1.88	
3.85/1.88	pA(f(X1), g(X2)) :- pA(X1, g(X2)).
3.85/1.88	pA(f(X1), g(X2)) :- pA(X1, g(X2)).
3.85/1.88	
3.85/1.88	Clauses:
3.85/1.88	
3.85/1.88	pcA(X1, X1).
3.85/1.88	pcA(f(X1), g(X2)) :- pcA(X1, g(X2)).
3.85/1.88	pcA(f(X1), g(X2)) :- pcA(X1, g(X2)).
3.85/1.88	
3.85/1.88	Afs:
3.85/1.88	
3.85/1.88	pA(x1, x2)  =  pA(x1)
3.85/1.88	
3.85/1.88	
3.85/1.88	----------------------------------------
3.85/1.88	
3.85/1.88	(3) TriplesToPiDPProof (SOUND)
3.85/1.88	We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
3.85/1.88	
3.85/1.88	pA_in_2: (b,f) (b,b)
3.85/1.88	
3.85/1.88	Transforming TRIPLES into the following Term Rewriting System:
3.85/1.88	
3.85/1.88	Pi DP problem:
3.85/1.88	The TRS P consists of the following rules:
3.85/1.88	
3.85/1.88	   PA_IN_GA(f(X1), g(X2)) -> U1_GA(X1, X2, pA_in_gg(X1, g(X2)))
3.85/1.88	   PA_IN_GA(f(X1), g(X2)) -> PA_IN_GG(X1, g(X2))
3.85/1.88	   PA_IN_GG(f(X1), g(X2)) -> U1_GG(X1, X2, pA_in_gg(X1, g(X2)))
3.85/1.88	   PA_IN_GG(f(X1), g(X2)) -> PA_IN_GG(X1, g(X2))
3.85/1.88	
3.85/1.88	R is empty.
3.85/1.88	The argument filtering Pi contains the following mapping:
3.85/1.88	f(x1)  =  f(x1)
3.85/1.88	
3.85/1.88	pA_in_gg(x1, x2)  =  pA_in_gg(x1, x2)
3.85/1.88	
3.85/1.88	g(x1)  =  g
3.85/1.88	
3.85/1.88	PA_IN_GA(x1, x2)  =  PA_IN_GA(x1)
3.85/1.88	
3.85/1.88	U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
3.85/1.88	
3.85/1.88	PA_IN_GG(x1, x2)  =  PA_IN_GG(x1, x2)
3.85/1.88	
3.85/1.88	U1_GG(x1, x2, x3)  =  U1_GG(x1, x3)
3.85/1.88	
3.85/1.88	
3.85/1.88	We have to consider all (P,R,Pi)-chains
3.85/1.88	
3.85/1.88	
3.85/1.88	Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES
3.85/1.88	
3.85/1.88	
3.85/1.88	
3.85/1.88	----------------------------------------
3.85/1.88	
3.85/1.88	(4)
3.85/1.88	Obligation:
3.85/1.88	Pi DP problem:
3.85/1.88	The TRS P consists of the following rules:
3.85/1.88	
3.85/1.88	   PA_IN_GA(f(X1), g(X2)) -> U1_GA(X1, X2, pA_in_gg(X1, g(X2)))
3.85/1.88	   PA_IN_GA(f(X1), g(X2)) -> PA_IN_GG(X1, g(X2))
3.85/1.88	   PA_IN_GG(f(X1), g(X2)) -> U1_GG(X1, X2, pA_in_gg(X1, g(X2)))
3.85/1.88	   PA_IN_GG(f(X1), g(X2)) -> PA_IN_GG(X1, g(X2))
3.85/1.88	
3.85/1.88	R is empty.
3.85/1.88	The argument filtering Pi contains the following mapping:
3.85/1.88	f(x1)  =  f(x1)
3.85/1.88	
3.85/1.88	pA_in_gg(x1, x2)  =  pA_in_gg(x1, x2)
3.85/1.88	
3.85/1.88	g(x1)  =  g
3.85/1.88	
3.85/1.88	PA_IN_GA(x1, x2)  =  PA_IN_GA(x1)
3.85/1.88	
3.85/1.88	U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
3.85/1.88	
3.85/1.88	PA_IN_GG(x1, x2)  =  PA_IN_GG(x1, x2)
3.85/1.88	
3.85/1.88	U1_GG(x1, x2, x3)  =  U1_GG(x1, x3)
3.85/1.88	
3.85/1.88	
3.85/1.88	We have to consider all (P,R,Pi)-chains
3.85/1.88	----------------------------------------
3.85/1.88	
3.85/1.88	(5) DependencyGraphProof (EQUIVALENT)
3.85/1.88	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.
3.85/1.88	----------------------------------------
3.85/1.88	
3.85/1.88	(6)
3.85/1.88	Obligation:
3.85/1.88	Pi DP problem:
3.85/1.88	The TRS P consists of the following rules:
3.85/1.88	
3.85/1.88	   PA_IN_GG(f(X1), g(X2)) -> PA_IN_GG(X1, g(X2))
3.85/1.88	
3.85/1.88	R is empty.
3.85/1.88	The argument filtering Pi contains the following mapping:
3.85/1.88	f(x1)  =  f(x1)
3.85/1.88	
3.85/1.88	g(x1)  =  g
3.85/1.88	
3.85/1.88	PA_IN_GG(x1, x2)  =  PA_IN_GG(x1, x2)
3.85/1.88	
3.85/1.88	
3.85/1.88	We have to consider all (P,R,Pi)-chains
3.85/1.88	----------------------------------------
3.85/1.88	
3.85/1.88	(7) PiDPToQDPProof (SOUND)
3.85/1.88	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
3.85/1.88	----------------------------------------
3.85/1.88	
3.85/1.88	(8)
3.85/1.88	Obligation:
3.85/1.88	Q DP problem:
3.85/1.88	The TRS P consists of the following rules:
3.85/1.88	
3.85/1.88	   PA_IN_GG(f(X1), g) -> PA_IN_GG(X1, g)
3.85/1.88	
3.85/1.88	R is empty.
3.85/1.88	Q is empty.
3.85/1.88	We have to consider all (P,Q,R)-chains.
3.85/1.88	----------------------------------------
3.85/1.88	
3.85/1.88	(9) QDPSizeChangeProof (EQUIVALENT)
3.85/1.88	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. 
3.85/1.88	
3.85/1.88	From the DPs we obtained the following set of size-change graphs:
3.85/1.88	*PA_IN_GG(f(X1), g) -> PA_IN_GG(X1, g)
3.85/1.88	The graph contains the following edges 1 > 1, 2 >= 2
3.85/1.88	
3.85/1.88	
3.85/1.88	----------------------------------------
3.85/1.88	
3.85/1.88	(10)
3.85/1.88	YES
3.85/1.90	EOF