/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.pl /export/starexec/sandbox2/output/output_files


--------------------------------------------------------------------------------


YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Left Termination of the query pattern

rec2(g,a)

w.r.t. the given Prolog program could successfully be proven:

(0) Prolog
(1) PrologToIRSwTTransformerProof [SOUND, 24 ms]
(2) TRUE


----------------------------------------

(0)
Obligation:
Clauses:

rec1(I, Ret) :- ','(=<(I, 0), ','(!, is(Ret, 0))).
rec1(I, Ret) :- ','(!, ','(is(I1, +(I, 1)), ','(rec1(I1, Ret1), ','(rec2(Ret1, Ret2), is(Ret, Ret2))))).
rec2(J, Ret) :- ','(=<(J, 0), ','(!, is(Ret, 0))).
rec2(J, Ret) :- ','(!, ','(is(I1, -(I, 1)), ','(rec2(I1, Ret1), ','(rec1(Ret1, Ret2), is(Ret, Ret2))))).


Query: rec2(g,a)
----------------------------------------

(1) PrologToIRSwTTransformerProof (SOUND)
Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert

{
    "root": 2,
    "program": {
        "directives": [],
        "clauses": [
            [
                "(rec1 I Ret)",
                "(',' (=< I (0)) (',' (!) (is Ret (0))))"
            ],
            [
                "(rec1 I Ret)",
                "(',' (!) (',' (is I1 (+ I (1))) (',' (rec1 I1 Ret1) (',' (rec2 Ret1 Ret2) (is Ret Ret2)))))"
            ],
            [
                "(rec2 J Ret)",
                "(',' (=< J (0)) (',' (!) (is Ret (0))))"
            ],
            [
                "(rec2 J Ret)",
                "(',' (!) (',' (is I1 (- I (1))) (',' (rec2 I1 Ret1) (',' (rec1 Ret1 Ret2) (is Ret Ret2)))))"
            ]
        ]
    },
    "graph": {
        "nodes": {
            "11": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "391": {
                "goal": [
                    {
                        "clause": -1,
                        "scope": -1,
                        "term": "(',' (!_1) (is T10 (0)))"
                    },
                    {
                        "clause": 3,
                        "scope": 1,
                        "term": "(rec2 T8 T2)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": [{
                            "lhs": {
                                "name": "T8",
                                "type": "PlainIntegerVariable"
                            },
                            "type": "PlainIntegerRelation",
                            "rhs": {
                                "type": "PlainIntegerConstant",
                                "value": "0"
                            },
                            "operation": "<="
                        }]
                    },
                    "ground": ["T8"],
                    "free": [],
                    "exprvars": ["T8"]
                }
            },
            "type": "Nodes",
            "392": {
                "goal": [{
                    "clause": 3,
                    "scope": 1,
                    "term": "(rec2 T8 T2)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": [{
                            "lhs": {
                                "name": "T8",
                                "type": "PlainIntegerVariable"
                            },
                            "type": "PlainIntegerRelation",
                            "rhs": {
                                "type": "PlainIntegerConstant",
                                "value": "0"
                            },
                            "operation": ">"
                        }]
                    },
                    "ground": ["T8"],
                    "free": [],
                    "exprvars": ["T8"]
                }
            },
            "393": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(is T10 (0))"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": [{
                            "lhs": {
                                "name": "T8",
                                "type": "PlainIntegerVariable"
                            },
                            "type": "PlainIntegerRelation",
                            "rhs": {
                                "type": "PlainIntegerConstant",
                                "value": "0"
                            },
                            "operation": "<="
                        }]
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": ["T8"]
                }
            },
            "394": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(true)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": [
                            {
                                "lhs": {
                                    "name": "T11",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "type": "PlainIntegerConstant",
                                    "value": "0"
                                },
                                "operation": "="
                            },
                            {
                                "lhs": {
                                    "name": "T8",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "type": "PlainIntegerConstant",
                                    "value": "0"
                                },
                                "operation": "<="
                            }
                        ]
                    },
                    "ground": ["T11"],
                    "free": [],
                    "exprvars": [
                        "T11",
                        "T8"
                    ]
                }
            },
            "2": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(rec2 T1 T2)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T1"],
                    "free": [],
                    "exprvars": []
                }
            },
            "4": {
                "goal": [
                    {
                        "clause": 2,
                        "scope": 1,
                        "term": "(rec2 T1 T2)"
                    },
                    {
                        "clause": 3,
                        "scope": 1,
                        "term": "(rec2 T1 T2)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T1"],
                    "free": [],
                    "exprvars": []
                }
            },
            "458": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": [
                            {
                                "lhs": {
                                    "name": "T11",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "type": "PlainIntegerConstant",
                                    "value": "0"
                                },
                                "operation": "="
                            },
                            {
                                "lhs": {
                                    "name": "T8",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "type": "PlainIntegerConstant",
                                    "value": "0"
                                },
                                "operation": "<="
                            }
                        ]
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": [
                        "T11",
                        "T8"
                    ]
                }
            },
            "603": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(',' (!_1) (',' (is X17 (- X18 (1))) (',' (rec2 X17 X19) (',' (rec1 X19 X20) (is T19 X20)))))"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": [
                            {
                                "lhs": {
                                    "name": "T8",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "type": "PlainIntegerConstant",
                                    "value": "0"
                                },
                                "operation": ">"
                            },
                            {
                                "lhs": {
                                    "type": "PlainIntegerConstant",
                                    "value": "0"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "type": "PlainIntegerConstant",
                                    "value": "0"
                                },
                                "operation": ">"
                            }
                        ]
                    },
                    "ground": [],
                    "free": [
                        "X17",
                        "X18",
                        "X19",
                        "X20"
                    ],
                    "exprvars": [
                        "T17",
                        "T8"
                    ]
                }
            },
            "604": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(',' (is X17 (- X18 (1))) (',' (rec2 X17 X19) (',' (rec1 X19 X20) (is T19 X20))))"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": [
                            {
                                "lhs": {
                                    "name": "T8",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "type": "PlainIntegerConstant",
                                    "value": "0"
                                },
                                "operation": ">"
                            },
                            {
                                "lhs": {
                                    "type": "PlainIntegerConstant",
                                    "value": "0"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "type": "PlainIntegerConstant",
                                    "value": "0"
                                },
                                "operation": ">"
                            }
                        ]
                    },
                    "ground": [],
                    "free": [
                        "X17",
                        "X18",
                        "X19",
                        "X20"
                    ],
                    "exprvars": [
                        "T17",
                        "T8"
                    ]
                }
            },
            "418": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": [{
                            "lhs": {
                                "name": "T8",
                                "type": "PlainIntegerVariable"
                            },
                            "type": "PlainIntegerRelation",
                            "rhs": {
                                "type": "PlainIntegerConstant",
                                "value": "0"
                            },
                            "operation": "<="
                        }]
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": ["T8"]
                }
            },
            "605": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "10": {
                "goal": [
                    {
                        "clause": -1,
                        "scope": -1,
                        "term": "(',' (=< T8 (0)) (',' (!_1) (is T10 (0))))"
                    },
                    {
                        "clause": 3,
                        "scope": 1,
                        "term": "(rec2 T8 T2)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T8"],
                    "free": [],
                    "exprvars": []
                }
            }
        },
        "edges": [
            {
                "from": 2,
                "to": 4,
                "label": "CASE"
            },
            {
                "from": 4,
                "to": 10,
                "label": "ONLY EVAL with clause\nrec2(X5, X6) :- ','(=<(X5, 0), ','(!_1, is(X6, 0))).\nand substitutionT1 -> T8,\nX5 -> T8,\nT2 -> T10,\nX6 -> T10,\nT9 -> T10"
            },
            {
                "from": 10,
                "to": 11,
                "label": "IS ERROR"
            },
            {
                "from": 10,
                "to": 391,
                "label": "ARITHCOMP SUCCESS"
            },
            {
                "from": 10,
                "to": 392,
                "label": "ARITHCOMP FAIL"
            },
            {
                "from": 391,
                "to": 393,
                "label": "CUT"
            },
            {
                "from": 392,
                "to": 603,
                "label": "ONLY EVAL with clause\nrec2(X15, X16) :- ','(!_1, ','(is(X17, -(X18, 1)), ','(rec2(X17, X19), ','(rec1(X19, X20), is(X16, X20))))).\nand substitutionT8 -> T17,\nX15 -> T17,\nT2 -> T19,\nX16 -> T19,\nT18 -> T19"
            },
            {
                "from": 393,
                "to": 394,
                "label": "\nT10 -> T11"
            },
            {
                "from": 393,
                "to": 418,
                "label": "IS FAIL"
            },
            {
                "from": 394,
                "to": 458,
                "label": "SUCCESS"
            },
            {
                "from": 603,
                "to": 604,
                "label": "CUT"
            },
            {
                "from": 604,
                "to": 605,
                "label": "IS ERROR"
            }
        ],
        "type": "Graph"
    }
}

----------------------------------------

(2)
TRUE