YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Left Termination of the query pattern

flat(g,a)

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

(0) Prolog
(1) PrologToTRSTransformerProof [SOUND, 0 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 97 ms]
(4) QTRS
(5) RisEmptyProof [EQUIVALENT, 0 ms]
(6) YES


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

(0)
Obligation:
Clauses:

flat([], []).
flat(.([], T), R) :- flat(T, R).
flat(.(.(H, T), TT), .(H, R)) :- flat(.(T, TT), R).


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

(1) PrologToTRSTransformerProof (SOUND)
Transformed Prolog program to TRS.

{
    "root": 1,
    "program": {
        "directives": [],
        "clauses": [
            [
                "(flat ([]) ([]))",
                null
            ],
            [
                "(flat (. ([]) T) R)",
                "(flat T R)"
            ],
            [
                "(flat (. (. H T) TT) (. H R))",
                "(flat (. T TT) R)"
            ]
        ]
    },
    "graph": {
        "nodes": {
            "77": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "23": {
                "goal": [{
                    "clause": 1,
                    "scope": 1,
                    "term": "(flat T1 T2)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T1"],
                    "free": [],
                    "exprvars": []
                }
            },
            "26": {
                "goal": [{
                    "clause": 2,
                    "scope": 1,
                    "term": "(flat T1 T2)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T1"],
                    "free": [],
                    "exprvars": []
                }
            },
            "17": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(true)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "28": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(flat T11 T13)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T11"],
                    "free": [],
                    "exprvars": []
                }
            },
            "type": "Nodes",
            "1": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(flat T1 T2)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T1"],
                    "free": [],
                    "exprvars": []
                }
            },
            "5": {
                "goal": [
                    {
                        "clause": 0,
                        "scope": 1,
                        "term": "(flat T1 T2)"
                    },
                    {
                        "clause": 1,
                        "scope": 1,
                        "term": "(flat T1 T2)"
                    },
                    {
                        "clause": 2,
                        "scope": 1,
                        "term": "(flat T1 T2)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T1"],
                    "free": [],
                    "exprvars": []
                }
            },
            "8": {
                "goal": [{
                    "clause": 0,
                    "scope": 1,
                    "term": "(flat T1 T2)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T1"],
                    "free": [],
                    "exprvars": []
                }
            },
            "9": {
                "goal": [
                    {
                        "clause": 1,
                        "scope": 1,
                        "term": "(flat T1 T2)"
                    },
                    {
                        "clause": 2,
                        "scope": 1,
                        "term": "(flat T1 T2)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T1"],
                    "free": [],
                    "exprvars": []
                }
            },
            "30": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "20": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "21": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "76": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(flat (. T23 T24) T26)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T23",
                        "T24"
                    ],
                    "free": [],
                    "exprvars": []
                }
            }
        },
        "edges": [
            {
                "from": 1,
                "to": 5,
                "label": "CASE"
            },
            {
                "from": 5,
                "to": 8,
                "label": "PARALLEL"
            },
            {
                "from": 5,
                "to": 9,
                "label": "PARALLEL"
            },
            {
                "from": 8,
                "to": 17,
                "label": "EVAL with clause\nflat([], []).\nand substitutionT1 -> [],\nT2 -> []"
            },
            {
                "from": 8,
                "to": 20,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 9,
                "to": 23,
                "label": "PARALLEL"
            },
            {
                "from": 9,
                "to": 26,
                "label": "PARALLEL"
            },
            {
                "from": 17,
                "to": 21,
                "label": "SUCCESS"
            },
            {
                "from": 23,
                "to": 28,
                "label": "EVAL with clause\nflat(.([], X9), X10) :- flat(X9, X10).\nand substitutionX9 -> T11,\nT1 -> .([], T11),\nT2 -> T13,\nX10 -> T13,\nT12 -> T13"
            },
            {
                "from": 23,
                "to": 30,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 26,
                "to": 76,
                "label": "EVAL with clause\nflat(.(.(X19, X20), X21), .(X19, X22)) :- flat(.(X20, X21), X22).\nand substitutionX19 -> T22,\nX20 -> T23,\nX21 -> T24,\nT1 -> .(.(T22, T23), T24),\nX22 -> T26,\nT2 -> .(T22, T26),\nT25 -> T26"
            },
            {
                "from": 26,
                "to": 77,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 28,
                "to": 1,
                "label": "INSTANCE with matching:\nT1 -> T11\nT2 -> T13"
            },
            {
                "from": 76,
                "to": 1,
                "label": "INSTANCE with matching:\nT1 -> .(T23, T24)\nT2 -> T26"
            }
        ],
        "type": "Graph"
    }
}

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

(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   f1_in([]) -> f1_out1([])
   f1_in(.([], T11)) -> U1(f1_in(T11), .([], T11))
   U1(f1_out1(T13), .([], T11)) -> f1_out1(T13)
   f1_in(.(.(T22, T23), T24)) -> U2(f1_in(.(T23, T24)), .(.(T22, T23), T24))
   U2(f1_out1(T26), .(.(T22, T23), T24)) -> f1_out1(.(T22, T26))

Q is empty.

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

(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
f1_in/1(YES)
[]/0)
f1_out1/1)YES(
./2(YES,YES)
U1/2(YES,YES)
U2/2(YES,YES)

Quasi precedence:
[f1_in_1, []] > [._2, U2_2] > U1_2


Status:
f1_in_1: [1]
[]: multiset status
._2: multiset status
U1_2: multiset status
U2_2: multiset status

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   f1_in([]) -> f1_out1([])
   f1_in(.([], T11)) -> U1(f1_in(T11), .([], T11))
   U1(f1_out1(T13), .([], T11)) -> f1_out1(T13)
   f1_in(.(.(T22, T23), T24)) -> U2(f1_in(.(T23, T24)), .(.(T22, T23), T24))
   U2(f1_out1(T26), .(.(T22, T23), T24)) -> f1_out1(.(T22, T26))




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

(4)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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

(5) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
----------------------------------------

(6)
YES