3.45/1.68	YES
3.69/1.70	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
3.69/1.70	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.69/1.70	
3.69/1.70	
3.69/1.70	Left Termination of the query pattern
3.69/1.70	
3.69/1.70	even(g)
3.69/1.70	
3.69/1.70	w.r.t. the given Prolog program could successfully be proven:
3.69/1.70	
3.69/1.70	(0) Prolog
3.69/1.70	(1) PrologToDTProblemTransformerProof [SOUND, 0 ms]
3.69/1.70	(2) TRIPLES
3.69/1.70	(3) TriplesToPiDPProof [SOUND, 0 ms]
3.69/1.70	(4) PiDP
3.69/1.70	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
3.69/1.70	(6) PiDP
3.69/1.70	(7) PiDPToQDPProof [EQUIVALENT, 0 ms]
3.69/1.70	(8) QDP
3.69/1.70	(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.69/1.70	(10) YES
3.69/1.70	
3.69/1.70	
3.69/1.70	----------------------------------------
3.69/1.70	
3.69/1.70	(0)
3.69/1.70	Obligation:
3.69/1.70	Clauses:
3.69/1.70	
3.69/1.70	even(0).
3.69/1.70	even(s(s(0))).
3.69/1.70	even(s(s(s(X)))) :- odd(X).
3.69/1.70	odd(s(0)).
3.69/1.70	odd(s(X)) :- even(s(s(X))).
3.69/1.70	
3.69/1.70	
3.69/1.70	Query: even(g)
3.69/1.70	----------------------------------------
3.69/1.70	
3.69/1.70	(1) PrologToDTProblemTransformerProof (SOUND)
3.69/1.70	Built DT problem from termination graph DT10.
3.69/1.70	
3.69/1.70	{
3.69/1.70	    "root": 9,
3.69/1.70	    "program": {
3.69/1.70	        "directives": [],
3.69/1.70	        "clauses": [
3.69/1.70	            [
3.69/1.70	                "(even (0))",
3.69/1.70	                null
3.69/1.70	            ],
3.69/1.70	            [
3.69/1.70	                "(even (s (s (0))))",
3.69/1.70	                null
3.69/1.70	            ],
3.69/1.70	            [
3.69/1.70	                "(even (s (s (s X))))",
3.69/1.70	                "(odd X)"
3.69/1.70	            ],
3.69/1.70	            [
3.69/1.70	                "(odd (s (0)))",
3.69/1.70	                null
3.69/1.70	            ],
3.69/1.70	            [
3.69/1.70	                "(odd (s X))",
3.69/1.70	                "(even (s (s X)))"
3.69/1.70	            ]
3.69/1.70	        ]
3.69/1.70	    },
3.69/1.70	    "graph": {
3.69/1.70	        "nodes": {
3.69/1.70	            "88": {
3.69/1.70	                "goal": [{
3.69/1.70	                    "clause": 2,
3.69/1.70	                    "scope": 1,
3.69/1.70	                    "term": "(even (0))"
3.69/1.70	                }],
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3.69/1.70	                }
3.69/1.70	            },
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3.69/1.70	                }
3.69/1.70	            },
3.69/1.70	            "type": "Nodes",
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3.69/1.70	                    "clause": -1,
3.69/1.70	                    "scope": -1,
3.69/1.70	                    "term": "(true)"
3.69/1.70	                }],
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3.69/1.70	            },
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3.69/1.70	            },
3.69/1.70	            "113": {
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3.69/1.70	                    "clause": -1,
3.69/1.70	                    "scope": -1,
3.69/1.70	                    "term": "(even (s (s T6)))"
3.69/1.70	                }],
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3.69/1.70	                        "relations": []
3.69/1.70	                    },
3.69/1.70	                    "ground": ["T6"],
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3.69/1.70	                }
3.69/1.70	            },
3.69/1.70	            "114": {
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3.69/1.70	                    },
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3.69/1.70	                }
3.69/1.70	            },
3.69/1.70	            "105": {
3.69/1.70	                "goal": [{
3.69/1.70	                    "clause": -1,
3.69/1.70	                    "scope": -1,
3.69/1.70	                    "term": "(odd T3)"
3.69/1.70	                }],
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3.69/1.70	            },
3.69/1.70	            "90": {
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3.69/1.70	                    },
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3.69/1.70	                }
3.69/1.70	            },
3.69/1.70	            "106": {
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3.69/1.70	                    },
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3.69/1.70	                }
3.69/1.70	            },
3.69/1.70	            "107": {
3.69/1.70	                "goal": [
3.69/1.70	                    {
3.69/1.70	                        "clause": 3,
3.69/1.70	                        "scope": 2,
3.69/1.70	                        "term": "(odd T3)"
3.69/1.70	                    },
3.69/1.70	                    {
3.69/1.70	                        "clause": 4,
3.69/1.70	                        "scope": 2,
3.69/1.70	                        "term": "(odd T3)"
3.69/1.70	                    }
3.69/1.70	                ],
3.69/1.70	                "kb": {
3.69/1.70	                    "nonunifying": [],
3.69/1.70	                    "intvars": {},
3.69/1.70	                    "arithmetic": {
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3.69/1.70	                    },
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3.69/1.70	                }
3.69/1.70	            },
3.69/1.70	            "9": {
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3.69/1.70	                    "clause": -1,
3.69/1.70	                    "scope": -1,
3.69/1.70	                    "term": "(even T1)"
3.69/1.70	                }],
3.69/1.70	                "kb": {
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3.69/1.70	                    },
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3.69/1.70	                }
3.69/1.70	            },
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3.69/1.70	                    "clause": 3,
3.69/1.70	                    "scope": 2,
3.69/1.70	                    "term": "(odd T3)"
3.69/1.70	                }],
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3.69/1.70	            },
3.69/1.70	            "93": {
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3.69/1.70	                        "clause": -1,
3.69/1.70	                        "scope": -1,
3.69/1.70	                        "term": "(true)"
3.69/1.70	                    },
3.69/1.70	                    {
3.69/1.70	                        "clause": 2,
3.69/1.70	                        "scope": 1,
3.69/1.70	                        "term": "(even (s (s (0))))"
3.69/1.70	                    }
3.69/1.70	                ],
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3.69/1.70	                }
3.69/1.70	            },
3.69/1.70	            "109": {
3.69/1.70	                "goal": [{
3.69/1.70	                    "clause": 4,
3.69/1.70	                    "scope": 2,
3.69/1.70	                    "term": "(odd T3)"
3.69/1.70	                }],
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3.69/1.70	                }
3.69/1.70	            },
3.69/1.70	            "50": {
3.69/1.70	                "goal": [
3.69/1.70	                    {
3.69/1.70	                        "clause": -1,
3.69/1.70	                        "scope": -1,
3.69/1.70	                        "term": "(true)"
3.69/1.70	                    },
3.69/1.70	                    {
3.69/1.70	                        "clause": 1,
3.69/1.70	                        "scope": 1,
3.69/1.70	                        "term": "(even (0))"
3.69/1.70	                    },
3.69/1.70	                    {
3.69/1.70	                        "clause": 2,
3.69/1.70	                        "scope": 1,
3.69/1.70	                        "term": "(even (0))"
3.69/1.70	                    }
3.69/1.70	                ],
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3.69/1.70	                }
3.69/1.70	            },
3.69/1.70	            "52": {
3.69/1.70	                "goal": [
3.69/1.70	                    {
3.69/1.70	                        "clause": 1,
3.69/1.70	                        "scope": 1,
3.69/1.70	                        "term": "(even T1)"
3.69/1.70	                    },
3.69/1.70	                    {
3.69/1.70	                        "clause": 2,
3.69/1.70	                        "scope": 1,
3.69/1.70	                        "term": "(even T1)"
3.69/1.70	                    }
3.69/1.70	                ],
3.69/1.70	                "kb": {
3.69/1.70	                    "nonunifying": [[
3.69/1.70	                        "(even T1)",
3.69/1.70	                        "(even (0))"
3.69/1.70	                    ]],
3.69/1.70	                    "intvars": {},
3.69/1.70	                    "arithmetic": {
3.69/1.70	                        "type": "PlainIntegerRelationState",
3.69/1.70	                        "relations": []
3.69/1.70	                    },
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3.69/1.70	                }
3.69/1.70	            },
3.69/1.70	            "96": {
3.69/1.70	                "goal": [{
3.69/1.70	                    "clause": 2,
3.69/1.70	                    "scope": 1,
3.69/1.70	                    "term": "(even T1)"
3.69/1.70	                }],
3.69/1.70	                "kb": {
3.69/1.70	                    "nonunifying": [
3.69/1.70	                        [
3.69/1.70	                            "(even T1)",
3.69/1.70	                            "(even (0))"
3.69/1.70	                        ],
3.69/1.70	                        [
3.69/1.70	                            "(even T1)",
3.69/1.70	                            "(even (s (s (0))))"
3.69/1.70	                        ]
3.69/1.70	                    ],
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3.69/1.70	                }
3.69/1.70	            },
3.69/1.70	            "10": {
3.69/1.70	                "goal": [
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3.69/1.70	                        "clause": 0,
3.69/1.70	                        "scope": 1,
3.69/1.70	                        "term": "(even T1)"
3.69/1.70	                    },
3.69/1.70	                    {
3.69/1.70	                        "clause": 1,
3.69/1.70	                        "scope": 1,
3.69/1.70	                        "term": "(even T1)"
3.69/1.70	                    },
3.69/1.70	                    {
3.69/1.70	                        "clause": 2,
3.69/1.70	                        "scope": 1,
3.69/1.70	                        "term": "(even T1)"
3.69/1.70	                    }
3.69/1.70	                ],
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3.69/1.70	                }
3.69/1.70	            },
3.69/1.70	            "87": {
3.69/1.70	                "goal": [
3.69/1.70	                    {
3.69/1.70	                        "clause": 1,
3.69/1.70	                        "scope": 1,
3.69/1.70	                        "term": "(even (0))"
3.69/1.70	                    },
3.69/1.70	                    {
3.69/1.70	                        "clause": 2,
3.69/1.70	                        "scope": 1,
3.69/1.70	                        "term": "(even (0))"
3.69/1.70	                    }
3.69/1.70	                ],
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3.69/1.70	                        "relations": []
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3.69/1.70	                }
3.69/1.70	            },
3.69/1.70	            "98": {
3.69/1.70	                "goal": [{
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3.69/1.70	                    "scope": 1,
3.69/1.70	                    "term": "(even (s (s (0))))"
3.69/1.70	                }],
3.69/1.70	                "kb": {
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3.69/1.70	                        "relations": []
3.69/1.70	                    },
3.69/1.70	                    "ground": [],
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3.69/1.70	                }
3.69/1.70	            }
3.69/1.70	        },
3.69/1.70	        "edges": [
3.69/1.70	            {
3.69/1.70	                "from": 9,
3.69/1.70	                "to": 10,
3.69/1.70	                "label": "CASE"
3.69/1.70	            },
3.69/1.70	            {
3.69/1.70	                "from": 10,
3.69/1.70	                "to": 50,
3.69/1.70	                "label": "EVAL with clause\neven(0).\nand substitutionT1 -> 0"
3.69/1.70	            },
3.69/1.70	            {
3.69/1.70	                "from": 10,
3.69/1.70	                "to": 52,
3.69/1.70	                "label": "EVAL-BACKTRACK"
3.69/1.70	            },
3.69/1.70	            {
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3.69/1.70	                "to": 87,
3.69/1.70	                "label": "SUCCESS"
3.69/1.70	            },
3.69/1.70	            {
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3.69/1.70	                "to": 93,
3.69/1.70	                "label": "EVAL with clause\neven(s(s(0))).\nand substitutionT1 -> s(s(0))"
3.69/1.70	            },
3.69/1.70	            {
3.69/1.70	                "from": 52,
3.69/1.70	                "to": 96,
3.69/1.70	                "label": "EVAL-BACKTRACK"
3.69/1.70	            },
3.69/1.70	            {
3.69/1.70	                "from": 87,
3.69/1.70	                "to": 88,
3.69/1.70	                "label": "BACKTRACK\nfor clause: even(s(s(0)))because of non-unification"
3.69/1.70	            },
3.69/1.70	            {
3.69/1.70	                "from": 88,
3.69/1.70	                "to": 90,
3.69/1.70	                "label": "BACKTRACK\nfor clause: even(s(s(s(X)))) :- odd(X)because of non-unification"
3.69/1.70	            },
3.69/1.70	            {
3.69/1.70	                "from": 93,
3.69/1.70	                "to": 98,
3.69/1.70	                "label": "SUCCESS"
3.69/1.70	            },
3.69/1.70	            {
3.69/1.70	                "from": 96,
3.69/1.70	                "to": 105,
3.69/1.70	                "label": "EVAL with clause\neven(s(s(s(X4)))) :- odd(X4).\nand substitutionX4 -> T3,\nT1 -> s(s(s(T3)))"
3.69/1.70	            },
3.69/1.70	            {
3.69/1.70	                "from": 96,
3.69/1.70	                "to": 106,
3.69/1.70	                "label": "EVAL-BACKTRACK"
3.69/1.70	            },
3.69/1.70	            {
3.69/1.70	                "from": 98,
3.69/1.70	                "to": 99,
3.69/1.70	                "label": "BACKTRACK\nfor clause: even(s(s(s(X)))) :- odd(X)because of non-unification"
3.69/1.70	            },
3.69/1.70	            {
3.69/1.70	                "from": 105,
3.69/1.70	                "to": 107,
3.69/1.70	                "label": "CASE"
3.69/1.70	            },
3.69/1.70	            {
3.69/1.70	                "from": 107,
3.69/1.70	                "to": 108,
3.69/1.70	                "label": "PARALLEL"
3.69/1.70	            },
3.69/1.70	            {
3.69/1.70	                "from": 107,
3.69/1.70	                "to": 109,
3.69/1.70	                "label": "PARALLEL"
3.69/1.70	            },
3.69/1.70	            {
3.69/1.70	                "from": 108,
3.69/1.70	                "to": 110,
3.69/1.70	                "label": "EVAL with clause\nodd(s(0)).\nand substitutionT3 -> s(0)"
3.69/1.70	            },
3.69/1.70	            {
3.69/1.70	                "from": 108,
3.69/1.70	                "to": 111,
3.69/1.70	                "label": "EVAL-BACKTRACK"
3.69/1.70	            },
3.69/1.70	            {
3.69/1.70	                "from": 109,
3.69/1.70	                "to": 113,
3.69/1.70	                "label": "EVAL with clause\nodd(s(X7)) :- even(s(s(X7))).\nand substitutionX7 -> T6,\nT3 -> s(T6)"
3.69/1.70	            },
3.69/1.70	            {
3.69/1.70	                "from": 109,
3.69/1.70	                "to": 114,
3.69/1.70	                "label": "EVAL-BACKTRACK"
3.69/1.70	            },
3.69/1.70	            {
3.69/1.70	                "from": 110,
3.69/1.70	                "to": 112,
3.69/1.70	                "label": "SUCCESS"
3.69/1.70	            },
3.69/1.70	            {
3.69/1.70	                "from": 113,
3.69/1.70	                "to": 9,
3.69/1.70	                "label": "INSTANCE with matching:\nT1 -> s(s(T6))"
3.69/1.70	            }
3.69/1.70	        ],
3.69/1.70	        "type": "Graph"
3.69/1.70	    }
3.69/1.70	}
3.69/1.70	
3.69/1.70	----------------------------------------
3.69/1.70	
3.69/1.70	(2)
3.69/1.70	Obligation:
3.69/1.70	Triples:
3.69/1.70	
3.69/1.70	evenA(s(s(s(s(X1))))) :- evenA(s(s(X1))).
3.69/1.70	
3.69/1.70	Clauses:
3.69/1.70	
3.69/1.70	evencA(0).
3.69/1.70	evencA(s(s(0))).
3.69/1.70	evencA(s(s(s(s(0))))).
3.69/1.70	evencA(s(s(s(s(X1))))) :- evencA(s(s(X1))).
3.69/1.70	
3.69/1.70	Afs:
3.69/1.70	
3.69/1.70	evenA(x1)  =  evenA(x1)
3.69/1.70	
3.69/1.70	
3.69/1.70	----------------------------------------
3.69/1.70	
3.69/1.70	(3) TriplesToPiDPProof (SOUND)
3.69/1.70	We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
3.69/1.70	
3.69/1.70	evenA_in_1: (b)
3.69/1.70	
3.69/1.70	Transforming TRIPLES into the following Term Rewriting System:
3.69/1.70	
3.69/1.70	Pi DP problem:
3.69/1.70	The TRS P consists of the following rules:
3.69/1.70	
3.69/1.70	   EVENA_IN_G(s(s(s(s(X1))))) -> U1_G(X1, evenA_in_g(s(s(X1))))
3.69/1.70	   EVENA_IN_G(s(s(s(s(X1))))) -> EVENA_IN_G(s(s(X1)))
3.69/1.70	
3.69/1.70	R is empty.
3.69/1.70	Pi is empty.
3.69/1.70	We have to consider all (P,R,Pi)-chains
3.69/1.70	
3.69/1.70	
3.69/1.70	Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES
3.69/1.70	
3.69/1.70	
3.69/1.70	
3.69/1.70	----------------------------------------
3.69/1.70	
3.69/1.70	(4)
3.69/1.70	Obligation:
3.69/1.70	Pi DP problem:
3.69/1.70	The TRS P consists of the following rules:
3.69/1.70	
3.69/1.70	   EVENA_IN_G(s(s(s(s(X1))))) -> U1_G(X1, evenA_in_g(s(s(X1))))
3.69/1.70	   EVENA_IN_G(s(s(s(s(X1))))) -> EVENA_IN_G(s(s(X1)))
3.69/1.70	
3.69/1.70	R is empty.
3.69/1.70	Pi is empty.
3.69/1.70	We have to consider all (P,R,Pi)-chains
3.69/1.70	----------------------------------------
3.69/1.70	
3.69/1.70	(5) DependencyGraphProof (EQUIVALENT)
3.69/1.70	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.
3.69/1.70	----------------------------------------
3.69/1.70	
3.69/1.70	(6)
3.69/1.70	Obligation:
3.69/1.70	Pi DP problem:
3.69/1.70	The TRS P consists of the following rules:
3.69/1.70	
3.69/1.70	   EVENA_IN_G(s(s(s(s(X1))))) -> EVENA_IN_G(s(s(X1)))
3.69/1.70	
3.69/1.70	R is empty.
3.69/1.70	Pi is empty.
3.69/1.70	We have to consider all (P,R,Pi)-chains
3.69/1.70	----------------------------------------
3.69/1.70	
3.69/1.70	(7) PiDPToQDPProof (EQUIVALENT)
3.69/1.70	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
3.69/1.70	----------------------------------------
3.69/1.70	
3.69/1.70	(8)
3.69/1.70	Obligation:
3.69/1.70	Q DP problem:
3.69/1.70	The TRS P consists of the following rules:
3.69/1.70	
3.69/1.70	   EVENA_IN_G(s(s(s(s(X1))))) -> EVENA_IN_G(s(s(X1)))
3.69/1.70	
3.69/1.70	R is empty.
3.69/1.70	Q is empty.
3.69/1.70	We have to consider all (P,Q,R)-chains.
3.69/1.70	----------------------------------------
3.69/1.70	
3.69/1.70	(9) QDPSizeChangeProof (EQUIVALENT)
3.69/1.70	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. 
3.69/1.70	
3.69/1.70	From the DPs we obtained the following set of size-change graphs:
3.69/1.70	*EVENA_IN_G(s(s(s(s(X1))))) -> EVENA_IN_G(s(s(X1)))
3.69/1.70	The graph contains the following edges 1 > 1
3.69/1.70	
3.69/1.70	
3.69/1.70	----------------------------------------
3.69/1.70	
3.69/1.70	(10)
3.69/1.70	YES
3.69/1.71	EOF