3.65/1.65	YES
3.65/1.67	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
3.65/1.67	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.65/1.67	
3.65/1.67	
3.65/1.67	Left Termination of the query pattern
3.65/1.67	
3.65/1.67	even(g)
3.65/1.67	
3.65/1.67	w.r.t. the given Prolog program could successfully be proven:
3.65/1.67	
3.65/1.67	(0) Prolog
3.65/1.67	(1) PrologToDTProblemTransformerProof [SOUND, 0 ms]
3.65/1.67	(2) TRIPLES
3.65/1.67	(3) TriplesToPiDPProof [SOUND, 0 ms]
3.65/1.67	(4) PiDP
3.65/1.67	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
3.65/1.67	(6) PiDP
3.65/1.67	(7) PiDPToQDPProof [EQUIVALENT, 0 ms]
3.65/1.67	(8) QDP
3.65/1.67	(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.65/1.67	(10) YES
3.65/1.67	
3.65/1.67	
3.65/1.67	----------------------------------------
3.65/1.67	
3.65/1.67	(0)
3.65/1.67	Obligation:
3.65/1.67	Clauses:
3.65/1.67	
3.65/1.67	even(0).
3.65/1.67	even(s(X)) :- odd(X).
3.65/1.67	odd(s(X)) :- even(X).
3.65/1.67	
3.65/1.67	
3.65/1.67	Query: even(g)
3.65/1.67	----------------------------------------
3.65/1.67	
3.65/1.67	(1) PrologToDTProblemTransformerProof (SOUND)
3.65/1.67	Built DT problem from termination graph DT10.
3.65/1.67	
3.65/1.67	{
3.65/1.67	    "root": 7,
3.65/1.67	    "program": {
3.65/1.67	        "directives": [],
3.65/1.67	        "clauses": [
3.65/1.67	            [
3.65/1.67	                "(even (0))",
3.65/1.67	                null
3.65/1.67	            ],
3.65/1.67	            [
3.65/1.67	                "(even (s X))",
3.65/1.67	                "(odd X)"
3.65/1.67	            ],
3.65/1.67	            [
3.65/1.67	                "(odd (s X))",
3.65/1.67	                "(even X)"
3.65/1.67	            ]
3.65/1.67	        ]
3.65/1.67	    },
3.65/1.67	    "graph": {
3.65/1.67	        "nodes": {
3.65/1.67	            "88": {
3.65/1.67	                "goal": [{
3.65/1.67	                    "clause": 1,
3.65/1.67	                    "scope": 1,
3.65/1.67	                    "term": "(even (0))"
3.65/1.67	                }],
3.65/1.67	                "kb": {
3.65/1.67	                    "nonunifying": [],
3.65/1.67	                    "intvars": {},
3.65/1.67	                    "arithmetic": {
3.65/1.67	                        "type": "PlainIntegerRelationState",
3.65/1.67	                        "relations": []
3.65/1.67	                    },
3.65/1.67	                    "ground": [],
3.65/1.67	                    "free": [],
3.65/1.67	                    "exprvars": []
3.65/1.67	                }
3.65/1.67	            },
3.65/1.67	            "89": {
3.65/1.67	                "goal": [],
3.65/1.67	                "kb": {
3.65/1.67	                    "nonunifying": [],
3.65/1.67	                    "intvars": {},
3.65/1.67	                    "arithmetic": {
3.65/1.67	                        "type": "PlainIntegerRelationState",
3.65/1.67	                        "relations": []
3.65/1.67	                    },
3.65/1.67	                    "ground": [],
3.65/1.67	                    "free": [],
3.65/1.67	                    "exprvars": []
3.65/1.67	                }
3.65/1.67	            },
3.65/1.67	            "14": {
3.65/1.67	                "goal": [
3.65/1.67	                    {
3.65/1.67	                        "clause": 0,
3.65/1.67	                        "scope": 1,
3.65/1.67	                        "term": "(even T1)"
3.65/1.67	                    },
3.65/1.67	                    {
3.65/1.67	                        "clause": 1,
3.65/1.67	                        "scope": 1,
3.65/1.67	                        "term": "(even T1)"
3.65/1.67	                    }
3.65/1.67	                ],
3.65/1.67	                "kb": {
3.65/1.67	                    "nonunifying": [],
3.65/1.67	                    "intvars": {},
3.65/1.67	                    "arithmetic": {
3.65/1.67	                        "type": "PlainIntegerRelationState",
3.65/1.67	                        "relations": []
3.65/1.67	                    },
3.65/1.67	                    "ground": ["T1"],
3.65/1.67	                    "free": [],
3.65/1.67	                    "exprvars": []
3.65/1.67	                }
3.65/1.67	            },
3.65/1.67	            "7": {
3.65/1.67	                "goal": [{
3.65/1.67	                    "clause": -1,
3.65/1.67	                    "scope": -1,
3.65/1.68	                    "term": "(even T1)"
3.65/1.68	                }],
3.65/1.68	                "kb": {
3.65/1.68	                    "nonunifying": [],
3.65/1.68	                    "intvars": {},
3.65/1.68	                    "arithmetic": {
3.65/1.68	                        "type": "PlainIntegerRelationState",
3.65/1.68	                        "relations": []
3.65/1.68	                    },
3.65/1.68	                    "ground": ["T1"],
3.65/1.68	                    "free": [],
3.65/1.68	                    "exprvars": []
3.65/1.68	                }
3.65/1.68	            },
3.65/1.68	            "90": {
3.65/1.68	                "goal": [{
3.65/1.68	                    "clause": -1,
3.65/1.68	                    "scope": -1,
3.65/1.68	                    "term": "(odd T3)"
3.65/1.68	                }],
3.65/1.68	                "kb": {
3.65/1.68	                    "nonunifying": [],
3.65/1.68	                    "intvars": {},
3.65/1.68	                    "arithmetic": {
3.65/1.68	                        "type": "PlainIntegerRelationState",
3.65/1.68	                        "relations": []
3.65/1.68	                    },
3.65/1.68	                    "ground": ["T3"],
3.65/1.68	                    "free": [],
3.65/1.68	                    "exprvars": []
3.65/1.68	                }
3.65/1.68	            },
3.65/1.68	            "91": {
3.65/1.68	                "goal": [],
3.65/1.68	                "kb": {
3.65/1.68	                    "nonunifying": [],
3.65/1.68	                    "intvars": {},
3.65/1.68	                    "arithmetic": {
3.65/1.68	                        "type": "PlainIntegerRelationState",
3.65/1.68	                        "relations": []
3.65/1.68	                    },
3.65/1.68	                    "ground": [],
3.65/1.68	                    "free": [],
3.65/1.68	                    "exprvars": []
3.65/1.68	                }
3.65/1.68	            },
3.65/1.68	            "92": {
3.65/1.68	                "goal": [{
3.65/1.68	                    "clause": 2,
3.65/1.68	                    "scope": 2,
3.65/1.68	                    "term": "(odd T3)"
3.65/1.68	                }],
3.65/1.68	                "kb": {
3.65/1.68	                    "nonunifying": [],
3.65/1.68	                    "intvars": {},
3.65/1.68	                    "arithmetic": {
3.65/1.68	                        "type": "PlainIntegerRelationState",
3.65/1.68	                        "relations": []
3.65/1.68	                    },
3.65/1.68	                    "ground": ["T3"],
3.65/1.68	                    "free": [],
3.65/1.68	                    "exprvars": []
3.65/1.68	                }
3.65/1.68	            },
3.65/1.68	            "93": {
3.65/1.68	                "goal": [{
3.65/1.68	                    "clause": -1,
3.65/1.68	                    "scope": -1,
3.65/1.68	                    "term": "(even T6)"
3.65/1.68	                }],
3.65/1.68	                "kb": {
3.65/1.68	                    "nonunifying": [],
3.65/1.68	                    "intvars": {},
3.65/1.68	                    "arithmetic": {
3.65/1.68	                        "type": "PlainIntegerRelationState",
3.65/1.68	                        "relations": []
3.65/1.68	                    },
3.65/1.68	                    "ground": ["T6"],
3.65/1.68	                    "free": [],
3.65/1.68	                    "exprvars": []
3.65/1.68	                }
3.65/1.68	            },
3.65/1.68	            "94": {
3.65/1.68	                "goal": [],
3.65/1.68	                "kb": {
3.65/1.68	                    "nonunifying": [],
3.65/1.68	                    "intvars": {},
3.65/1.68	                    "arithmetic": {
3.65/1.68	                        "type": "PlainIntegerRelationState",
3.65/1.68	                        "relations": []
3.65/1.68	                    },
3.65/1.68	                    "ground": [],
3.65/1.68	                    "free": [],
3.65/1.68	                    "exprvars": []
3.65/1.68	                }
3.65/1.68	            },
3.65/1.68	            "type": "Nodes",
3.65/1.68	            "86": {
3.65/1.68	                "goal": [
3.65/1.68	                    {
3.65/1.68	                        "clause": -1,
3.65/1.68	                        "scope": -1,
3.65/1.68	                        "term": "(true)"
3.65/1.68	                    },
3.65/1.68	                    {
3.65/1.68	                        "clause": 1,
3.65/1.68	                        "scope": 1,
3.65/1.68	                        "term": "(even (0))"
3.65/1.68	                    }
3.65/1.68	                ],
3.65/1.68	                "kb": {
3.65/1.68	                    "nonunifying": [],
3.65/1.68	                    "intvars": {},
3.65/1.68	                    "arithmetic": {
3.65/1.68	                        "type": "PlainIntegerRelationState",
3.65/1.68	                        "relations": []
3.65/1.68	                    },
3.65/1.68	                    "ground": [],
3.65/1.68	                    "free": [],
3.65/1.68	                    "exprvars": []
3.65/1.68	                }
3.65/1.68	            },
3.65/1.68	            "87": {
3.65/1.68	                "goal": [{
3.65/1.68	                    "clause": 1,
3.65/1.68	                    "scope": 1,
3.65/1.68	                    "term": "(even T1)"
3.65/1.68	                }],
3.65/1.68	                "kb": {
3.65/1.68	                    "nonunifying": [[
3.65/1.68	                        "(even T1)",
3.65/1.68	                        "(even (0))"
3.65/1.68	                    ]],
3.65/1.68	                    "intvars": {},
3.65/1.68	                    "arithmetic": {
3.65/1.68	                        "type": "PlainIntegerRelationState",
3.65/1.68	                        "relations": []
3.65/1.68	                    },
3.65/1.68	                    "ground": ["T1"],
3.65/1.68	                    "free": [],
3.65/1.68	                    "exprvars": []
3.65/1.68	                }
3.65/1.68	            }
3.65/1.68	        },
3.65/1.68	        "edges": [
3.65/1.68	            {
3.65/1.68	                "from": 7,
3.65/1.68	                "to": 14,
3.65/1.68	                "label": "CASE"
3.65/1.68	            },
3.65/1.68	            {
3.65/1.68	                "from": 14,
3.65/1.68	                "to": 86,
3.65/1.68	                "label": "EVAL with clause\neven(0).\nand substitutionT1 -> 0"
3.65/1.68	            },
3.65/1.68	            {
3.65/1.68	                "from": 14,
3.65/1.68	                "to": 87,
3.65/1.68	                "label": "EVAL-BACKTRACK"
3.65/1.68	            },
3.65/1.68	            {
3.65/1.68	                "from": 86,
3.65/1.68	                "to": 88,
3.65/1.68	                "label": "SUCCESS"
3.65/1.68	            },
3.65/1.68	            {
3.65/1.68	                "from": 87,
3.65/1.68	                "to": 90,
3.65/1.68	                "label": "EVAL with clause\neven(s(X3)) :- odd(X3).\nand substitutionX3 -> T3,\nT1 -> s(T3)"
3.65/1.68	            },
3.65/1.68	            {
3.65/1.68	                "from": 87,
3.65/1.68	                "to": 91,
3.65/1.68	                "label": "EVAL-BACKTRACK"
3.65/1.68	            },
3.65/1.68	            {
3.65/1.68	                "from": 88,
3.65/1.68	                "to": 89,
3.65/1.68	                "label": "BACKTRACK\nfor clause: even(s(X)) :- odd(X)because of non-unification"
3.65/1.68	            },
3.65/1.68	            {
3.65/1.68	                "from": 90,
3.65/1.68	                "to": 92,
3.65/1.68	                "label": "CASE"
3.65/1.68	            },
3.65/1.68	            {
3.65/1.68	                "from": 92,
3.65/1.68	                "to": 93,
3.65/1.68	                "label": "EVAL with clause\nodd(s(X6)) :- even(X6).\nand substitutionX6 -> T6,\nT3 -> s(T6)"
3.65/1.68	            },
3.65/1.68	            {
3.65/1.68	                "from": 92,
3.65/1.68	                "to": 94,
3.65/1.68	                "label": "EVAL-BACKTRACK"
3.65/1.68	            },
3.65/1.68	            {
3.65/1.68	                "from": 93,
3.65/1.68	                "to": 7,
3.65/1.68	                "label": "INSTANCE with matching:\nT1 -> T6"
3.65/1.68	            }
3.65/1.68	        ],
3.65/1.68	        "type": "Graph"
3.65/1.68	    }
3.65/1.68	}
3.65/1.68	
3.65/1.68	----------------------------------------
3.65/1.68	
3.65/1.68	(2)
3.65/1.68	Obligation:
3.65/1.68	Triples:
3.65/1.68	
3.65/1.68	evenA(s(s(X1))) :- evenA(X1).
3.65/1.68	
3.65/1.68	Clauses:
3.65/1.68	
3.65/1.68	evencA(0).
3.65/1.68	evencA(s(s(X1))) :- evencA(X1).
3.65/1.68	
3.65/1.68	Afs:
3.65/1.68	
3.65/1.68	evenA(x1)  =  evenA(x1)
3.65/1.68	
3.65/1.68	
3.65/1.68	----------------------------------------
3.65/1.68	
3.65/1.68	(3) TriplesToPiDPProof (SOUND)
3.65/1.68	We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
3.65/1.68	
3.65/1.68	evenA_in_1: (b)
3.65/1.68	
3.65/1.68	Transforming TRIPLES into the following Term Rewriting System:
3.65/1.68	
3.65/1.68	Pi DP problem:
3.65/1.68	The TRS P consists of the following rules:
3.65/1.68	
3.65/1.68	   EVENA_IN_G(s(s(X1))) -> U1_G(X1, evenA_in_g(X1))
3.65/1.68	   EVENA_IN_G(s(s(X1))) -> EVENA_IN_G(X1)
3.65/1.68	
3.65/1.68	R is empty.
3.65/1.68	Pi is empty.
3.65/1.68	We have to consider all (P,R,Pi)-chains
3.65/1.68	
3.65/1.68	
3.65/1.68	Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES
3.65/1.68	
3.65/1.68	
3.65/1.68	
3.65/1.68	----------------------------------------
3.65/1.68	
3.65/1.68	(4)
3.65/1.68	Obligation:
3.65/1.68	Pi DP problem:
3.65/1.68	The TRS P consists of the following rules:
3.65/1.68	
3.65/1.68	   EVENA_IN_G(s(s(X1))) -> U1_G(X1, evenA_in_g(X1))
3.65/1.68	   EVENA_IN_G(s(s(X1))) -> EVENA_IN_G(X1)
3.65/1.68	
3.65/1.68	R is empty.
3.65/1.68	Pi is empty.
3.65/1.68	We have to consider all (P,R,Pi)-chains
3.65/1.68	----------------------------------------
3.65/1.68	
3.65/1.68	(5) DependencyGraphProof (EQUIVALENT)
3.65/1.68	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.
3.65/1.68	----------------------------------------
3.65/1.68	
3.65/1.68	(6)
3.65/1.68	Obligation:
3.65/1.68	Pi DP problem:
3.65/1.68	The TRS P consists of the following rules:
3.65/1.68	
3.65/1.68	   EVENA_IN_G(s(s(X1))) -> EVENA_IN_G(X1)
3.65/1.68	
3.65/1.68	R is empty.
3.65/1.68	Pi is empty.
3.65/1.68	We have to consider all (P,R,Pi)-chains
3.65/1.68	----------------------------------------
3.65/1.68	
3.65/1.68	(7) PiDPToQDPProof (EQUIVALENT)
3.65/1.68	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
3.65/1.68	----------------------------------------
3.65/1.68	
3.65/1.68	(8)
3.65/1.68	Obligation:
3.65/1.68	Q DP problem:
3.65/1.68	The TRS P consists of the following rules:
3.65/1.68	
3.65/1.68	   EVENA_IN_G(s(s(X1))) -> EVENA_IN_G(X1)
3.65/1.68	
3.65/1.68	R is empty.
3.65/1.68	Q is empty.
3.65/1.68	We have to consider all (P,Q,R)-chains.
3.65/1.68	----------------------------------------
3.65/1.68	
3.65/1.68	(9) QDPSizeChangeProof (EQUIVALENT)
3.65/1.68	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. 
3.65/1.68	
3.65/1.68	From the DPs we obtained the following set of size-change graphs:
3.65/1.68	*EVENA_IN_G(s(s(X1))) -> EVENA_IN_G(X1)
3.65/1.68	The graph contains the following edges 1 > 1
3.65/1.68	
3.65/1.68	
3.65/1.68	----------------------------------------
3.65/1.68	
3.65/1.68	(10)
3.65/1.68	YES
3.80/1.72	EOF