3.39/1.74	YES
3.69/1.76	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
3.69/1.76	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.69/1.76	
3.69/1.76	
3.69/1.76	Left Termination of the query pattern
3.69/1.76	
3.69/1.76	duplicate(g,a)
3.69/1.76	
3.69/1.76	w.r.t. the given Prolog program could successfully be proven:
3.69/1.76	
3.69/1.76	(0) Prolog
3.69/1.76	(1) PrologToDTProblemTransformerProof [SOUND, 0 ms]
3.69/1.76	(2) TRIPLES
3.69/1.76	(3) TriplesToPiDPProof [SOUND, 0 ms]
3.69/1.76	(4) PiDP
3.69/1.76	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
3.69/1.76	(6) PiDP
3.69/1.76	(7) PiDPToQDPProof [SOUND, 0 ms]
3.69/1.76	(8) QDP
3.69/1.76	(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.69/1.76	(10) YES
3.69/1.76	
3.69/1.76	
3.69/1.76	----------------------------------------
3.69/1.76	
3.69/1.76	(0)
3.69/1.76	Obligation:
3.69/1.76	Clauses:
3.69/1.76	
3.69/1.76	duplicate([], []).
3.69/1.76	duplicate(.(X, Y), .(X, .(X, Z))) :- duplicate(Y, Z).
3.69/1.76	
3.69/1.76	
3.69/1.76	Query: duplicate(g,a)
3.69/1.76	----------------------------------------
3.69/1.76	
3.69/1.76	(1) PrologToDTProblemTransformerProof (SOUND)
3.69/1.76	Built DT problem from termination graph DT10.
3.69/1.76	
3.69/1.76	{
3.69/1.76	    "root": 1,
3.69/1.76	    "program": {
3.69/1.76	        "directives": [],
3.69/1.76	        "clauses": [
3.69/1.76	            [
3.69/1.76	                "(duplicate ([]) ([]))",
3.69/1.76	                null
3.69/1.76	            ],
3.69/1.76	            [
3.69/1.76	                "(duplicate (. X Y) (. X (. X Z)))",
3.69/1.76	                "(duplicate Y Z)"
3.69/1.76	            ]
3.69/1.76	        ]
3.69/1.76	    },
3.69/1.76	    "graph": {
3.69/1.76	        "nodes": {
3.69/1.76	            "77": {
3.69/1.76	                "goal": [
3.69/1.76	                    {
3.69/1.76	                        "clause": 0,
3.69/1.76	                        "scope": 1,
3.69/1.76	                        "term": "(duplicate T1 T2)"
3.69/1.76	                    },
3.69/1.76	                    {
3.69/1.76	                        "clause": 1,
3.69/1.76	                        "scope": 1,
3.69/1.76	                        "term": "(duplicate T1 T2)"
3.69/1.76	                    }
3.69/1.76	                ],
3.69/1.76	                "kb": {
3.69/1.76	                    "nonunifying": [],
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3.69/1.76	                        "relations": []
3.69/1.76	                    },
3.69/1.76	                    "ground": ["T1"],
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3.69/1.76	                }
3.69/1.76	            },
3.69/1.76	            "88": {
3.69/1.76	                "goal": [
3.69/1.76	                    {
3.69/1.76	                        "clause": 0,
3.69/1.76	                        "scope": 2,
3.69/1.76	                        "term": "(duplicate T7 T9)"
3.69/1.76	                    },
3.69/1.76	                    {
3.69/1.76	                        "clause": 1,
3.69/1.76	                        "scope": 2,
3.69/1.76	                        "term": "(duplicate T7 T9)"
3.69/1.76	                    }
3.69/1.76	                ],
3.69/1.76	                "kb": {
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3.69/1.76	                    "arithmetic": {
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3.69/1.76	                        "relations": []
3.69/1.76	                    },
3.69/1.76	                    "ground": ["T7"],
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3.69/1.76	                }
3.69/1.76	            },
3.69/1.76	            "78": {
3.69/1.76	                "goal": [
3.69/1.76	                    {
3.69/1.76	                        "clause": -1,
3.69/1.76	                        "scope": -1,
3.69/1.76	                        "term": "(true)"
3.69/1.76	                    },
3.69/1.76	                    {
3.69/1.76	                        "clause": 1,
3.69/1.76	                        "scope": 1,
3.69/1.76	                        "term": "(duplicate ([]) T2)"
3.69/1.76	                    }
3.69/1.76	                ],
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3.69/1.76	                        "relations": []
3.69/1.76	                    },
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3.69/1.76	                }
3.69/1.76	            },
3.69/1.76	            "89": {
3.69/1.76	                "goal": [{
3.69/1.76	                    "clause": 0,
3.69/1.76	                    "scope": 2,
3.69/1.76	                    "term": "(duplicate T7 T9)"
3.69/1.76	                }],
3.69/1.76	                "kb": {
3.69/1.76	                    "nonunifying": [],
3.69/1.76	                    "intvars": {},
3.69/1.76	                    "arithmetic": {
3.69/1.76	                        "type": "PlainIntegerRelationState",
3.69/1.76	                        "relations": []
3.69/1.76	                    },
3.69/1.76	                    "ground": ["T7"],
3.69/1.76	                    "free": [],
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3.69/1.76	                }
3.69/1.76	            },
3.69/1.76	            "79": {
3.69/1.76	                "goal": [{
3.69/1.76	                    "clause": 1,
3.69/1.76	                    "scope": 1,
3.69/1.76	                    "term": "(duplicate T1 T2)"
3.69/1.76	                }],
3.69/1.76	                "kb": {
3.69/1.76	                    "nonunifying": [[
3.69/1.76	                        "(duplicate T1 T2)",
3.69/1.76	                        "(duplicate ([]) ([]))"
3.69/1.76	                    ]],
3.69/1.76	                    "intvars": {},
3.69/1.76	                    "arithmetic": {
3.69/1.76	                        "type": "PlainIntegerRelationState",
3.69/1.76	                        "relations": []
3.69/1.76	                    },
3.69/1.76	                    "ground": ["T1"],
3.69/1.76	                    "free": [],
3.69/1.76	                    "exprvars": []
3.69/1.76	                }
3.69/1.76	            },
3.69/1.76	            "type": "Nodes",
3.69/1.76	            "1": {
3.69/1.76	                "goal": [{
3.69/1.76	                    "clause": -1,
3.69/1.76	                    "scope": -1,
3.69/1.76	                    "term": "(duplicate T1 T2)"
3.69/1.76	                }],
3.69/1.76	                "kb": {
3.69/1.76	                    "nonunifying": [],
3.69/1.76	                    "intvars": {},
3.69/1.76	                    "arithmetic": {
3.69/1.76	                        "type": "PlainIntegerRelationState",
3.69/1.76	                        "relations": []
3.69/1.76	                    },
3.69/1.76	                    "ground": ["T1"],
3.69/1.76	                    "free": [],
3.69/1.76	                    "exprvars": []
3.69/1.76	                }
3.69/1.76	            },
3.69/1.76	            "90": {
3.69/1.76	                "goal": [{
3.69/1.76	                    "clause": 1,
3.69/1.76	                    "scope": 2,
3.69/1.76	                    "term": "(duplicate T7 T9)"
3.69/1.76	                }],
3.69/1.76	                "kb": {
3.69/1.76	                    "nonunifying": [],
3.69/1.76	                    "intvars": {},
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3.69/1.76	                        "type": "PlainIntegerRelationState",
3.69/1.76	                        "relations": []
3.69/1.76	                    },
3.69/1.76	                    "ground": ["T7"],
3.69/1.76	                    "free": [],
3.69/1.76	                    "exprvars": []
3.69/1.76	                }
3.69/1.76	            },
3.69/1.76	            "80": {
3.69/1.76	                "goal": [{
3.69/1.76	                    "clause": 1,
3.69/1.76	                    "scope": 1,
3.69/1.76	                    "term": "(duplicate ([]) T2)"
3.69/1.76	                }],
3.69/1.76	                "kb": {
3.69/1.76	                    "nonunifying": [],
3.69/1.76	                    "intvars": {},
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3.69/1.76	                        "type": "PlainIntegerRelationState",
3.69/1.76	                        "relations": []
3.69/1.76	                    },
3.69/1.76	                    "ground": [],
3.69/1.76	                    "free": [],
3.69/1.76	                    "exprvars": []
3.69/1.76	                }
3.69/1.76	            },
3.69/1.76	            "91": {
3.69/1.76	                "goal": [{
3.69/1.76	                    "clause": -1,
3.69/1.76	                    "scope": -1,
3.69/1.76	                    "term": "(true)"
3.69/1.76	                }],
3.69/1.76	                "kb": {
3.69/1.76	                    "nonunifying": [],
3.69/1.76	                    "intvars": {},
3.69/1.76	                    "arithmetic": {
3.69/1.76	                        "type": "PlainIntegerRelationState",
3.69/1.76	                        "relations": []
3.69/1.76	                    },
3.69/1.76	                    "ground": [],
3.69/1.76	                    "free": [],
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3.69/1.76	                }
3.69/1.76	            },
3.69/1.76	            "81": {
3.69/1.76	                "goal": [],
3.69/1.76	                "kb": {
3.69/1.76	                    "nonunifying": [],
3.69/1.76	                    "intvars": {},
3.69/1.76	                    "arithmetic": {
3.69/1.76	                        "type": "PlainIntegerRelationState",
3.69/1.76	                        "relations": []
3.69/1.76	                    },
3.69/1.76	                    "ground": [],
3.69/1.76	                    "free": [],
3.69/1.76	                    "exprvars": []
3.69/1.76	                }
3.69/1.76	            },
3.69/1.76	            "92": {
3.69/1.76	                "goal": [],
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3.69/1.76	                    "nonunifying": [],
3.69/1.76	                    "intvars": {},
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3.69/1.76	                        "type": "PlainIntegerRelationState",
3.69/1.76	                        "relations": []
3.69/1.76	                    },
3.69/1.76	                    "ground": [],
3.69/1.76	                    "free": [],
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3.69/1.76	                }
3.69/1.76	            },
3.69/1.76	            "93": {
3.69/1.76	                "goal": [],
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3.69/1.76	                    "nonunifying": [],
3.69/1.76	                    "intvars": {},
3.69/1.76	                    "arithmetic": {
3.69/1.76	                        "type": "PlainIntegerRelationState",
3.69/1.76	                        "relations": []
3.69/1.76	                    },
3.69/1.76	                    "ground": [],
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3.69/1.76	                }
3.69/1.76	            },
3.69/1.76	            "94": {
3.69/1.76	                "goal": [{
3.69/1.76	                    "clause": -1,
3.69/1.76	                    "scope": -1,
3.69/1.76	                    "term": "(duplicate T17 T19)"
3.69/1.76	                }],
3.69/1.76	                "kb": {
3.69/1.76	                    "nonunifying": [],
3.69/1.76	                    "intvars": {},
3.69/1.76	                    "arithmetic": {
3.69/1.76	                        "type": "PlainIntegerRelationState",
3.69/1.76	                        "relations": []
3.69/1.76	                    },
3.69/1.76	                    "ground": ["T17"],
3.69/1.76	                    "free": [],
3.69/1.76	                    "exprvars": []
3.69/1.76	                }
3.69/1.76	            },
3.69/1.76	            "95": {
3.69/1.76	                "goal": [],
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3.69/1.76	                    "nonunifying": [],
3.69/1.76	                    "intvars": {},
3.69/1.76	                    "arithmetic": {
3.69/1.76	                        "type": "PlainIntegerRelationState",
3.69/1.76	                        "relations": []
3.69/1.76	                    },
3.69/1.76	                    "ground": [],
3.69/1.76	                    "free": [],
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3.69/1.76	                }
3.69/1.76	            },
3.69/1.76	            "86": {
3.69/1.76	                "goal": [{
3.69/1.76	                    "clause": -1,
3.69/1.76	                    "scope": -1,
3.69/1.76	                    "term": "(duplicate T7 T9)"
3.69/1.76	                }],
3.69/1.76	                "kb": {
3.69/1.76	                    "nonunifying": [],
3.69/1.76	                    "intvars": {},
3.69/1.76	                    "arithmetic": {
3.69/1.76	                        "type": "PlainIntegerRelationState",
3.69/1.76	                        "relations": []
3.69/1.76	                    },
3.69/1.76	                    "ground": ["T7"],
3.69/1.76	                    "free": [],
3.69/1.76	                    "exprvars": []
3.69/1.76	                }
3.69/1.76	            },
3.69/1.76	            "87": {
3.69/1.76	                "goal": [],
3.69/1.76	                "kb": {
3.69/1.76	                    "nonunifying": [],
3.69/1.76	                    "intvars": {},
3.69/1.76	                    "arithmetic": {
3.69/1.76	                        "type": "PlainIntegerRelationState",
3.69/1.76	                        "relations": []
3.69/1.76	                    },
3.69/1.76	                    "ground": [],
3.69/1.76	                    "free": [],
3.69/1.76	                    "exprvars": []
3.69/1.76	                }
3.69/1.76	            }
3.69/1.76	        },
3.69/1.76	        "edges": [
3.69/1.76	            {
3.69/1.76	                "from": 1,
3.69/1.76	                "to": 77,
3.69/1.76	                "label": "CASE"
3.69/1.76	            },
3.69/1.76	            {
3.69/1.76	                "from": 77,
3.69/1.76	                "to": 78,
3.69/1.76	                "label": "EVAL with clause\nduplicate([], []).\nand substitutionT1 -> [],\nT2 -> []"
3.69/1.76	            },
3.69/1.76	            {
3.69/1.76	                "from": 77,
3.69/1.76	                "to": 79,
3.69/1.76	                "label": "EVAL-BACKTRACK"
3.69/1.76	            },
3.69/1.76	            {
3.69/1.76	                "from": 78,
3.69/1.76	                "to": 80,
3.69/1.76	                "label": "SUCCESS"
3.69/1.76	            },
3.69/1.76	            {
3.69/1.76	                "from": 79,
3.69/1.76	                "to": 86,
3.69/1.76	                "label": "EVAL with clause\nduplicate(.(X7, X8), .(X7, .(X7, X9))) :- duplicate(X8, X9).\nand substitutionX7 -> T6,\nX8 -> T7,\nT1 -> .(T6, T7),\nX9 -> T9,\nT2 -> .(T6, .(T6, T9)),\nT8 -> T9"
3.69/1.76	            },
3.69/1.76	            {
3.69/1.76	                "from": 79,
3.69/1.76	                "to": 87,
3.69/1.76	                "label": "EVAL-BACKTRACK"
3.69/1.76	            },
3.69/1.76	            {
3.69/1.76	                "from": 80,
3.69/1.76	                "to": 81,
3.69/1.76	                "label": "BACKTRACK\nfor clause: duplicate(.(X, Y), .(X, .(X, Z))) :- duplicate(Y, Z)because of non-unification"
3.69/1.76	            },
3.69/1.76	            {
3.69/1.76	                "from": 86,
3.69/1.76	                "to": 88,
3.69/1.76	                "label": "CASE"
3.69/1.76	            },
3.69/1.76	            {
3.69/1.76	                "from": 88,
3.69/1.76	                "to": 89,
3.69/1.76	                "label": "PARALLEL"
3.69/1.76	            },
3.69/1.76	            {
3.69/1.76	                "from": 88,
3.69/1.76	                "to": 90,
3.69/1.76	                "label": "PARALLEL"
3.69/1.76	            },
3.69/1.76	            {
3.69/1.76	                "from": 89,
3.69/1.76	                "to": 91,
3.69/1.76	                "label": "EVAL with clause\nduplicate([], []).\nand substitutionT7 -> [],\nT9 -> []"
3.69/1.76	            },
3.69/1.76	            {
3.69/1.76	                "from": 89,
3.69/1.76	                "to": 92,
3.69/1.76	                "label": "EVAL-BACKTRACK"
3.69/1.76	            },
3.69/1.76	            {
3.69/1.76	                "from": 90,
3.69/1.76	                "to": 94,
3.69/1.76	                "label": "EVAL with clause\nduplicate(.(X16, X17), .(X16, .(X16, X18))) :- duplicate(X17, X18).\nand substitutionX16 -> T16,\nX17 -> T17,\nT7 -> .(T16, T17),\nX18 -> T19,\nT9 -> .(T16, .(T16, T19)),\nT18 -> T19"
3.69/1.76	            },
3.69/1.76	            {
3.69/1.76	                "from": 90,
3.69/1.76	                "to": 95,
3.69/1.76	                "label": "EVAL-BACKTRACK"
3.69/1.76	            },
3.69/1.76	            {
3.69/1.76	                "from": 91,
3.69/1.76	                "to": 93,
3.69/1.76	                "label": "SUCCESS"
3.69/1.76	            },
3.69/1.76	            {
3.69/1.76	                "from": 94,
3.69/1.76	                "to": 1,
3.69/1.76	                "label": "INSTANCE with matching:\nT1 -> T17\nT2 -> T19"
3.69/1.76	            }
3.69/1.76	        ],
3.69/1.76	        "type": "Graph"
3.69/1.76	    }
3.69/1.76	}
3.69/1.76	
3.69/1.76	----------------------------------------
3.69/1.76	
3.69/1.76	(2)
3.69/1.76	Obligation:
3.69/1.76	Triples:
3.69/1.76	
3.69/1.76	duplicateA(.(X1, .(X2, X3)), .(X1, .(X1, .(X2, .(X2, X4))))) :- duplicateA(X3, X4).
3.69/1.76	
3.69/1.76	Clauses:
3.69/1.76	
3.69/1.76	duplicatecA([], []).
3.69/1.76	duplicatecA(.(X1, []), .(X1, .(X1, []))).
3.69/1.76	duplicatecA(.(X1, .(X2, X3)), .(X1, .(X1, .(X2, .(X2, X4))))) :- duplicatecA(X3, X4).
3.69/1.76	
3.69/1.76	Afs:
3.69/1.76	
3.69/1.76	duplicateA(x1, x2)  =  duplicateA(x1)
3.69/1.76	
3.69/1.76	
3.69/1.76	----------------------------------------
3.69/1.76	
3.69/1.76	(3) TriplesToPiDPProof (SOUND)
3.69/1.76	We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
3.69/1.76	
3.69/1.76	duplicateA_in_2: (b,f)
3.69/1.76	
3.69/1.76	Transforming TRIPLES into the following Term Rewriting System:
3.69/1.76	
3.69/1.76	Pi DP problem:
3.69/1.76	The TRS P consists of the following rules:
3.69/1.76	
3.69/1.76	   DUPLICATEA_IN_GA(.(X1, .(X2, X3)), .(X1, .(X1, .(X2, .(X2, X4))))) -> U1_GA(X1, X2, X3, X4, duplicateA_in_ga(X3, X4))
3.69/1.76	   DUPLICATEA_IN_GA(.(X1, .(X2, X3)), .(X1, .(X1, .(X2, .(X2, X4))))) -> DUPLICATEA_IN_GA(X3, X4)
3.69/1.76	
3.69/1.76	R is empty.
3.69/1.76	The argument filtering Pi contains the following mapping:
3.69/1.76	duplicateA_in_ga(x1, x2)  =  duplicateA_in_ga(x1)
3.69/1.76	
3.69/1.76	.(x1, x2)  =  .(x1, x2)
3.69/1.76	
3.69/1.76	DUPLICATEA_IN_GA(x1, x2)  =  DUPLICATEA_IN_GA(x1)
3.69/1.76	
3.69/1.76	U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x3, x5)
3.69/1.76	
3.69/1.76	
3.69/1.76	We have to consider all (P,R,Pi)-chains
3.69/1.76	
3.69/1.76	
3.69/1.76	Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES
3.69/1.76	
3.69/1.76	
3.69/1.76	
3.69/1.76	----------------------------------------
3.69/1.76	
3.69/1.76	(4)
3.69/1.76	Obligation:
3.69/1.76	Pi DP problem:
3.69/1.76	The TRS P consists of the following rules:
3.69/1.76	
3.69/1.76	   DUPLICATEA_IN_GA(.(X1, .(X2, X3)), .(X1, .(X1, .(X2, .(X2, X4))))) -> U1_GA(X1, X2, X3, X4, duplicateA_in_ga(X3, X4))
3.69/1.76	   DUPLICATEA_IN_GA(.(X1, .(X2, X3)), .(X1, .(X1, .(X2, .(X2, X4))))) -> DUPLICATEA_IN_GA(X3, X4)
3.69/1.76	
3.69/1.76	R is empty.
3.69/1.76	The argument filtering Pi contains the following mapping:
3.69/1.76	duplicateA_in_ga(x1, x2)  =  duplicateA_in_ga(x1)
3.69/1.76	
3.69/1.76	.(x1, x2)  =  .(x1, x2)
3.69/1.76	
3.69/1.76	DUPLICATEA_IN_GA(x1, x2)  =  DUPLICATEA_IN_GA(x1)
3.69/1.76	
3.69/1.76	U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x3, x5)
3.69/1.76	
3.69/1.76	
3.69/1.76	We have to consider all (P,R,Pi)-chains
3.69/1.76	----------------------------------------
3.69/1.76	
3.69/1.76	(5) DependencyGraphProof (EQUIVALENT)
3.69/1.76	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.
3.69/1.76	----------------------------------------
3.69/1.76	
3.69/1.76	(6)
3.69/1.76	Obligation:
3.69/1.76	Pi DP problem:
3.69/1.76	The TRS P consists of the following rules:
3.69/1.76	
3.69/1.76	   DUPLICATEA_IN_GA(.(X1, .(X2, X3)), .(X1, .(X1, .(X2, .(X2, X4))))) -> DUPLICATEA_IN_GA(X3, X4)
3.69/1.76	
3.69/1.76	R is empty.
3.69/1.76	The argument filtering Pi contains the following mapping:
3.69/1.76	.(x1, x2)  =  .(x1, x2)
3.69/1.76	
3.69/1.76	DUPLICATEA_IN_GA(x1, x2)  =  DUPLICATEA_IN_GA(x1)
3.69/1.76	
3.69/1.76	
3.69/1.76	We have to consider all (P,R,Pi)-chains
3.69/1.76	----------------------------------------
3.69/1.76	
3.69/1.76	(7) PiDPToQDPProof (SOUND)
3.69/1.76	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
3.69/1.76	----------------------------------------
3.69/1.76	
3.69/1.76	(8)
3.69/1.76	Obligation:
3.69/1.76	Q DP problem:
3.69/1.76	The TRS P consists of the following rules:
3.69/1.76	
3.69/1.76	   DUPLICATEA_IN_GA(.(X1, .(X2, X3))) -> DUPLICATEA_IN_GA(X3)
3.69/1.76	
3.69/1.76	R is empty.
3.69/1.76	Q is empty.
3.69/1.76	We have to consider all (P,Q,R)-chains.
3.69/1.76	----------------------------------------
3.69/1.76	
3.69/1.76	(9) QDPSizeChangeProof (EQUIVALENT)
3.69/1.76	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. 
3.69/1.76	
3.69/1.76	From the DPs we obtained the following set of size-change graphs:
3.69/1.76	*DUPLICATEA_IN_GA(.(X1, .(X2, X3))) -> DUPLICATEA_IN_GA(X3)
3.69/1.76	The graph contains the following edges 1 > 1
3.69/1.76	
3.69/1.76	
3.69/1.76	----------------------------------------
3.69/1.76	
3.69/1.76	(10)
3.69/1.76	YES
3.76/1.80	EOF