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


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


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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Left Termination of the query pattern

group(g,g,a)

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

(0) Prolog
(1) PrologToDTProblemTransformerProof [SOUND, 128 ms]
(2) TRIPLES
(3) TriplesToPiDPProof [SOUND, 0 ms]
(4) PiDP
(5) DependencyGraphProof [EQUIVALENT, 0 ms]
(6) AND
    (7) PiDP
        (8) UsableRulesProof [EQUIVALENT, 0 ms]
        (9) PiDP
        (10) PiDPToQDPProof [SOUND, 0 ms]
        (11) QDP
        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (13) YES
    (14) PiDP
        (15) UsableRulesProof [EQUIVALENT, 0 ms]
        (16) PiDP
        (17) PiDPToQDPProof [SOUND, 0 ms]
        (18) QDP
        (19) QDPQMonotonicMRRProof [EQUIVALENT, 28 ms]
        (20) QDP
        (21) DependencyGraphProof [EQUIVALENT, 0 ms]
        (22) TRUE


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

(0)
Obligation:
Clauses:

group([], [], []).
group(G, .(N1, Ns), .(G1, Gs)) :- ','(selectN(N1, G, G1), ','(subtract(G, G1, R), group(R, Ns, Gs))).
selectN(0, X1, []) :- !.
selectN(N, L, .(X, S)) :- ','(>(N, 0), ','(el(X, L, R), ','(is(N1, -(N, 1)), selectN(N1, R, S)))).
el(X, .(X, L), L).
el(X, .(X2, L), R) :- el(X, L, R).


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

(1) PrologToDTProblemTransformerProof (SOUND)
Built DT problem from termination graph DT10.

{
    "root": 1,
    "program": {
        "directives": [],
        "clauses": [
            [
                "(group ([]) ([]) ([]))",
                null
            ],
            [
                "(group G (. N1 Ns) (. G1 Gs))",
                "(',' (selectN N1 G G1) (',' (subtract G G1 R) (group R Ns Gs)))"
            ],
            [
                "(selectN (0) X1 ([]))",
                "(!)"
            ],
            [
                "(selectN N L (. X S))",
                "(',' (> N (0)) (',' (el X L R) (',' (is N1 (- N (1))) (selectN N1 R S))))"
            ],
            [
                "(el X (. X L) L)",
                null
            ],
            [
                "(el X (. X2 L) R)",
                "(el X L R)"
            ]
        ]
    },
    "graph": {
        "nodes": {
            "3002": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": [
                            {
                                "lhs": {
                                    "type": "PlainIntegerConstant",
                                    "value": "0"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            }
                        ]
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": ["T28"]
                }
            },
            "3001": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(el T211 T210 X244)"
                }],
                "kb": {
                    "nonunifying": [[
                        "(selectN T28 (. T65 (. T89 (. T113 (. T137 (. T161 (. T185 (. T209 T210))))))) T14)",
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                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            }
                        ]
                    },
                    "ground": [
                        "T28",
                        "T65",
                        "T89",
                        "T113",
                        "T137",
                        "T161",
                        "T162"
                    ],
                    "free": [
                        "X21",
                        "X186"
                    ],
                    "exprvars": ["T28"]
                }
            },
            "2935": {
                "goal": [{
                    "clause": 4,
                    "scope": 8,
                    "term": "(el T163 T162 X186)"
                }],
                "kb": {
                    "nonunifying": [[
                        "(selectN T28 (. T65 (. T89 (. T113 (. T137 (. T161 T162))))) T14)",
                        "(selectN (0) X21 ([]))"
                    ]],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": [
                            {
                                "lhs": {
                                    "type": "PlainIntegerConstant",
                                    "value": "0"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            },
                            {
                                "lhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "type": "PlainIntegerRelation",
                                "rhs": {
                                    "name": "T28",
                                    "type": "PlainIntegerVariable"
                                },
                                "operation": "<"
                            }
                        ]
                    },
                    "ground": [
                        "T28",
                        "T65",
                        "T89",
                        "T113",
                        "T137",
                        "T161",
                        "T162"
                    ],
                    "free": [
                        "X21",
                        "X186"
                    ],
                    "exprvars": ["T28"]
                }
            }
        },
        "edges": [
            {
                "from": 1,
                "to": 2,
                "label": "CASE"
            },
            {
                "from": 2,
                "to": 12,
                "label": "EVAL with clause\ngroup([], [], []).\nand substitutionT1 -> [],\nT2 -> [],\nT3 -> []"
            },
            {
                "from": 2,
                "to": 14,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 12,
                "to": 23,
                "label": "SUCCESS"
            },
            {
                "from": 14,
                "to": 27,
                "label": "EVAL with clause\ngroup(X13, .(X14, X15), .(X16, X17)) :- ','(selectN(X14, X13, X16), ','(subtract(X13, X16, X18), group(X18, X15, X17))).\nand substitutionT1 -> T9,\nX13 -> T9,\nX14 -> T10,\nX15 -> T11,\nT2 -> .(T10, T11),\nX16 -> T14,\nX17 -> T15,\nT3 -> .(T14, T15),\nT12 -> T14,\nT13 -> T15"
            },
            {
                "from": 14,
                "to": 29,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 23,
                "to": 24,
                "label": "BACKTRACK\nfor clause: group(G, .(N1, Ns), .(G1, Gs)) :- ','(selectN(N1, G, G1), ','(subtract(G, G1, R), group(R, Ns, Gs)))because of non-unification"
            },
            {
                "from": 27,
                "to": 30,
                "label": "CASE"
            },
            {
                "from": 30,
                "to": 48,
                "label": "EVAL with clause\nselectN(0, X21, []) :- !_2.\nand substitutionT10 -> 0,\nT9 -> T18,\nX21 -> T18,\nT14 -> [],\nT15 -> T19"
            },
            {
                "from": 30,
                "to": 49,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 48,
                "to": 50,
                "label": "CUT"
            },
            {
                "from": 49,
                "to": 52,
                "label": "EVAL with clause\nselectN(X32, X33, .(X34, X35)) :- ','(>(X32, 0), ','(el(X34, X33, X36), ','(is(X37, -(X32, 1)), selectN(X37, X36, X35)))).\nand substitutionT10 -> T28,\nX32 -> T28,\nT9 -> T29,\nX33 -> T29,\nX34 -> T32,\nX35 -> T33,\nT14 -> .(T32, T33),\nT30 -> T32,\nT31 -> T33,\nT15 -> T34"
            },
            {
                "from": 49,
                "to": 53,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 50,
                "to": 51,
                "label": "UNDEFINED ERROR"
            },
            {
                "from": 52,
                "to": 54,
                "label": "IS ERROR"
            },
            {
                "from": 52,
                "to": 2425,
                "label": "ARITHCOMP SUCCESS"
            },
            {
                "from": 52,
                "to": 2427,
                "label": "ARITHCOMP FAIL"
            },
            {
                "from": 2425,
                "to": 2684,
                "label": "SPLIT 1"
            },
            {
                "from": 2425,
                "to": 2688,
                "label": "SPLIT 2\nnew knowledge:\nT41 is ground\nT29 is ground\nT39 is ground\nreplacements:X36 -> T39,\nT33 -> T40,\nT32 -> T41,\nT34 -> T42,\nT14 -> T43"
            },
            {
                "from": 2684,
                "to": 2706,
                "label": "CASE"
            },
            {
                "from": 2688,
                "to": 3201,
                "label": "IS ERROR"
            },
            {
                "from": 2688,
                "to": 3208,
                "label": "\nX37 -> T269"
            },
            {
                "from": 2706,
                "to": 2722,
                "label": "PARALLEL"
            },
            {
                "from": 2706,
                "to": 2726,
                "label": "PARALLEL"
            },
            {
                "from": 2722,
                "to": 2751,
                "label": "EVAL with clause\nel(X54, .(X54, X55), X55).\nand substitutionT32 -> T56,\nX54 -> T56,\nX55 -> T57,\nT29 -> .(T56, T57),\nX36 -> T57"
            },
            {
                "from": 2722,
                "to": 2752,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 2726,
                "to": 2785,
                "label": "EVAL with clause\nel(X66, .(X67, X68), X69) :- el(X66, X68, X69).\nand substitutionT32 -> T67,\nX66 -> T67,\nX67 -> T65,\nX68 -> T66,\nT29 -> .(T65, T66),\nX36 -> X70,\nX69 -> X70,\nT64 -> T67"
            },
            {
                "from": 2726,
                "to": 2790,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 2751,
                "to": 2755,
                "label": "SUCCESS"
            },
            {
                "from": 2785,
                "to": 2816,
                "label": "CASE"
            },
            {
                "from": 2816,
                "to": 2829,
                "label": "PARALLEL"
            },
            {
                "from": 2816,
                "to": 2830,
                "label": "PARALLEL"
            },
            {
                "from": 2829,
                "to": 2861,
                "label": "EVAL with clause\nel(X83, .(X83, X84), X84).\nand substitutionT67 -> T80,\nX83 -> T80,\nX84 -> T81,\nT66 -> .(T80, T81),\nX70 -> T81"
            },
            {
                "from": 2829,
                "to": 2862,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 2830,
                "to": 2867,
                "label": "EVAL with clause\nel(X95, .(X96, X97), X98) :- el(X95, X97, X98).\nand substitutionT67 -> T91,\nX95 -> T91,\nX96 -> T89,\nX97 -> T90,\nT66 -> .(T89, T90),\nX70 -> X99,\nX98 -> X99,\nT88 -> T91"
            },
            {
                "from": 2830,
                "to": 2868,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 2861,
                "to": 2863,
                "label": "SUCCESS"
            },
            {
                "from": 2867,
                "to": 2872,
                "label": "CASE"
            },
            {
                "from": 2872,
                "to": 2873,
                "label": "PARALLEL"
            },
            {
                "from": 2872,
                "to": 2874,
                "label": "PARALLEL"
            },
            {
                "from": 2873,
                "to": 2875,
                "label": "EVAL with clause\nel(X112, .(X112, X113), X113).\nand substitutionT91 -> T104,\nX112 -> T104,\nX113 -> T105,\nT90 -> .(T104, T105),\nX99 -> T105"
            },
            {
                "from": 2873,
                "to": 2876,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 2874,
                "to": 2878,
                "label": "EVAL with clause\nel(X124, .(X125, X126), X127) :- el(X124, X126, X127).\nand substitutionT91 -> T115,\nX124 -> T115,\nX125 -> T113,\nX126 -> T114,\nT90 -> .(T113, T114),\nX99 -> X128,\nX127 -> X128,\nT112 -> T115"
            },
            {
                "from": 2874,
                "to": 2879,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 2875,
                "to": 2877,
                "label": "SUCCESS"
            },
            {
                "from": 2878,
                "to": 2881,
                "label": "CASE"
            },
            {
                "from": 2881,
                "to": 2883,
                "label": "PARALLEL"
            },
            {
                "from": 2881,
                "to": 2884,
                "label": "PARALLEL"
            },
            {
                "from": 2883,
                "to": 2896,
                "label": "EVAL with clause\nel(X141, .(X141, X142), X142).\nand substitutionT115 -> T128,\nX141 -> T128,\nX142 -> T129,\nT114 -> .(T128, T129),\nX128 -> T129"
            },
            {
                "from": 2883,
                "to": 2897,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 2884,
                "to": 2901,
                "label": "EVAL with clause\nel(X153, .(X154, X155), X156) :- el(X153, X155, X156).\nand substitutionT115 -> T139,\nX153 -> T139,\nX154 -> T137,\nX155 -> T138,\nT114 -> .(T137, T138),\nX128 -> X157,\nX156 -> X157,\nT136 -> T139"
            },
            {
                "from": 2884,
                "to": 2903,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 2896,
                "to": 2898,
                "label": "SUCCESS"
            },
            {
                "from": 2901,
                "to": 2907,
                "label": "CASE"
            },
            {
                "from": 2907,
                "to": 2909,
                "label": "PARALLEL"
            },
            {
                "from": 2907,
                "to": 2913,
                "label": "PARALLEL"
            },
            {
                "from": 2909,
                "to": 2915,
                "label": "EVAL with clause\nel(X170, .(X170, X171), X171).\nand substitutionT139 -> T152,\nX170 -> T152,\nX171 -> T153,\nT138 -> .(T152, T153),\nX157 -> T153"
            },
            {
                "from": 2909,
                "to": 2916,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 2913,
                "to": 2924,
                "label": "EVAL with clause\nel(X182, .(X183, X184), X185) :- el(X182, X184, X185).\nand substitutionT139 -> T163,\nX182 -> T163,\nX183 -> T161,\nX184 -> T162,\nT138 -> .(T161, T162),\nX157 -> X186,\nX185 -> X186,\nT160 -> T163"
            },
            {
                "from": 2913,
                "to": 2925,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 2915,
                "to": 2917,
                "label": "SUCCESS"
            },
            {
                "from": 2924,
                "to": 2932,
                "label": "CASE"
            },
            {
                "from": 2932,
                "to": 2935,
                "label": "PARALLEL"
            },
            {
                "from": 2932,
                "to": 2936,
                "label": "PARALLEL"
            },
            {
                "from": 2935,
                "to": 2939,
                "label": "EVAL with clause\nel(X199, .(X199, X200), X200).\nand substitutionT163 -> T176,\nX199 -> T176,\nX200 -> T177,\nT162 -> .(T176, T177),\nX186 -> T177"
            },
            {
                "from": 2935,
                "to": 2940,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 2936,
                "to": 2945,
                "label": "EVAL with clause\nel(X211, .(X212, X213), X214) :- el(X211, X213, X214).\nand substitutionT163 -> T187,\nX211 -> T187,\nX212 -> T185,\nX213 -> T186,\nT162 -> .(T185, T186),\nX186 -> X215,\nX214 -> X215,\nT184 -> T187"
            },
            {
                "from": 2936,
                "to": 2946,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 2939,
                "to": 2941,
                "label": "SUCCESS"
            },
            {
                "from": 2945,
                "to": 2995,
                "label": "CASE"
            },
            {
                "from": 2995,
                "to": 2996,
                "label": "PARALLEL"
            },
            {
                "from": 2995,
                "to": 2997,
                "label": "PARALLEL"
            },
            {
                "from": 2996,
                "to": 2998,
                "label": "EVAL with clause\nel(X228, .(X228, X229), X229).\nand substitutionT187 -> T200,\nX228 -> T200,\nX229 -> T201,\nT186 -> .(T200, T201),\nX215 -> T201"
            },
            {
                "from": 2996,
                "to": 2999,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 2997,
                "to": 3001,
                "label": "EVAL with clause\nel(X240, .(X241, X242), X243) :- el(X240, X242, X243).\nand substitutionT187 -> T211,\nX240 -> T211,\nX241 -> T209,\nX242 -> T210,\nT186 -> .(T209, T210),\nX215 -> X244,\nX243 -> X244,\nT208 -> T211"
            },
            {
                "from": 2997,
                "to": 3002,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 2998,
                "to": 3000,
                "label": "SUCCESS"
            },
            {
                "from": 3001,
                "to": 3003,
                "label": "CASE"
            },
            {
                "from": 3003,
                "to": 3006,
                "label": "PARALLEL"
            },
            {
                "from": 3003,
                "to": 3008,
                "label": "PARALLEL"
            },
            {
                "from": 3006,
                "to": 3059,
                "label": "EVAL with clause\nel(X257, .(X257, X258), X258).\nand substitutionT211 -> T224,\nX257 -> T224,\nX258 -> T225,\nT210 -> .(T224, T225),\nX244 -> T225"
            },
            {
                "from": 3006,
                "to": 3060,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 3008,
                "to": 3068,
                "label": "EVAL with clause\nel(X269, .(X270, X271), X272) :- el(X269, X271, X272).\nand substitutionT211 -> T235,\nX269 -> T235,\nX270 -> T233,\nX271 -> T234,\nT210 -> .(T233, T234),\nX244 -> X273,\nX272 -> X273,\nT232 -> T235"
            },
            {
                "from": 3008,
                "to": 3069,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 3059,
                "to": 3061,
                "label": "SUCCESS"
            },
            {
                "from": 3068,
                "to": 3130,
                "label": "GENERALIZATION\nT240 <-- .(T89, .(T113, .(T137, .(T161, .(T185, .(T209, .(T233, T234)))))))\n\nNew Knowledge:\nT240 is ground"
            },
            {
                "from": 3130,
                "to": 3191,
                "label": "CASE"
            },
            {
                "from": 3191,
                "to": 3192,
                "label": "PARALLEL"
            },
            {
                "from": 3191,
                "to": 3193,
                "label": "PARALLEL"
            },
            {
                "from": 3192,
                "to": 3194,
                "label": "EVAL with clause\nel(X290, .(X290, X291), X291).\nand substitutionT235 -> T253,\nX290 -> T253,\nX291 -> T254,\nT234 -> .(T253, T254),\nX273 -> T254"
            },
            {
                "from": 3192,
                "to": 3195,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 3193,
                "to": 3197,
                "label": "EVAL with clause\nel(X302, .(X303, X304), X305) :- el(X302, X304, X305).\nand substitutionT235 -> T264,\nX302 -> T264,\nX303 -> T262,\nX304 -> T263,\nT234 -> .(T262, T263),\nX273 -> X306,\nX305 -> X306,\nT261 -> T264"
            },
            {
                "from": 3193,
                "to": 3198,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 3194,
                "to": 3196,
                "label": "SUCCESS"
            },
            {
                "from": 3197,
                "to": 3130,
                "label": "INSTANCE with matching:\nT235 -> T264\nT234 -> T263\nX273 -> X306"
            },
            {
                "from": 3208,
                "to": 3212,
                "label": "SPLIT 1"
            },
            {
                "from": 3208,
                "to": 3213,
                "label": "SPLIT 2\nnew knowledge:\nT269 is ground\nT39 is ground\nT271 is ground\nreplacements:T40 -> T271,\nT42 -> T272,\nT43 -> T273"
            },
            {
                "from": 3212,
                "to": 3214,
                "label": "CASE"
            },
            {
                "from": 3213,
                "to": 3284,
                "label": "UNDEFINED ERROR"
            },
            {
                "from": 3214,
                "to": 3233,
                "label": "EVAL with clause\nselectN(0, X315, []) :- !_12.\nand substitutionT269 -> 0,\nT39 -> T277,\nX315 -> T277,\nT40 -> []"
            },
            {
                "from": 3214,
                "to": 3234,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 3233,
                "to": 3235,
                "label": "CUT"
            },
            {
                "from": 3234,
                "to": 3259,
                "label": "EVAL with clause\nselectN(X326, X327, .(X328, X329)) :- ','(>(X326, 0), ','(el(X328, X327, X330), ','(is(X331, -(X326, 1)), selectN(X331, X330, X329)))).\nand substitutionT269 -> T286,\nX326 -> T286,\nT39 -> T287,\nX327 -> T287,\nX328 -> T290,\nX329 -> T291,\nT40 -> .(T290, T291),\nT288 -> T290,\nT289 -> T291"
            },
            {
                "from": 3234,
                "to": 3260,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 3235,
                "to": 3236,
                "label": "SUCCESS"
            },
            {
                "from": 3259,
                "to": 3278,
                "label": "ARITHCOMP SUCCESS"
            },
            {
                "from": 3259,
                "to": 3279,
                "label": "ARITHCOMP FAIL"
            },
            {
                "from": 3278,
                "to": 3280,
                "label": "SPLIT 1"
            },
            {
                "from": 3278,
                "to": 3281,
                "label": "SPLIT 2\nnew knowledge:\nT290 is ground\nT287 is ground\nT296 is ground\nreplacements:X330 -> T296,\nT291 -> T297,\nT40 -> T298"
            },
            {
                "from": 3280,
                "to": 3130,
                "label": "INSTANCE with matching:\nT235 -> T290\nT234 -> T287\nX273 -> X330"
            },
            {
                "from": 3281,
                "to": 3282,
                "label": "IS ERROR"
            },
            {
                "from": 3281,
                "to": 3283,
                "label": "\nX331 -> T303"
            },
            {
                "from": 3283,
                "to": 3212,
                "label": "INSTANCE with matching:\nT269 -> T303\nT39 -> T296\nT40 -> T297"
            }
        ],
        "type": "Graph"
    }
}

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

(2)
Obligation:
Triples:

elA(X1, .(X2, X3), X4) :- elA(X1, X3, X4).
selectNB(X1, X2, .(X3, X4)) :- elA(X3, X2, X5).
selectNB(X1, X2, .(X3, X4)) :- ','(elcA(X3, X2, X5), selectNB(X6, X5, X4)).
groupD(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), .(X10, X11), .(.(X12, X13), X14)) :- elA(X12, X9, X15).
groupD(X1, .(X2, X3), .(.(X4, X5), X6)) :- ','(elcC(X4, X1, X7), selectNB(X8, X7, X5)).

Clauses:

elcA(X1, .(X1, X2), X2).
elcA(X1, .(X2, X3), X4) :- elcA(X1, X3, X4).
selectNcB(0, X1, []).
selectNcB(X1, X2, .(X3, X4)) :- ','(elcA(X3, X2, X5), selectNcB(X6, X5, X4)).
elcC(X1, .(X1, X2), X2).
elcC(X1, .(X2, .(X1, X3)), X3).
elcC(X1, .(X2, .(X3, .(X1, X4))), X4).
elcC(X1, .(X2, .(X3, .(X4, .(X1, X5)))), X5).
elcC(X1, .(X2, .(X3, .(X4, .(X5, .(X1, X6))))), X6).
elcC(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X1, X7)))))), X7).
elcC(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X1, X8))))))), X8).
elcC(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, .(X1, X9)))))))), X9).
elcC(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, .(X9, X10)))))))), X11) :- elcA(X1, X10, X11).

Afs:

groupD(x1, x2, x3)  =  groupD(x1, x2)


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

(3) TriplesToPiDPProof (SOUND)
We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:

groupD_in_3: (b,b,f)

elA_in_3: (f,b,f)

elcC_in_3: (f,b,f)

elcA_in_3: (f,b,f)

selectNB_in_3: (f,b,f)

Transforming TRIPLES into the following Term Rewriting System:

Pi DP problem:
The TRS P consists of the following rules:

   GROUPD_IN_GGA(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), .(X10, X11), .(.(X12, X13), X14)) -> U5_GGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, elA_in_aga(X12, X9, X15))
   GROUPD_IN_GGA(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), .(X10, X11), .(.(X12, X13), X14)) -> ELA_IN_AGA(X12, X9, X15)
   ELA_IN_AGA(X1, .(X2, X3), X4) -> U1_AGA(X1, X2, X3, X4, elA_in_aga(X1, X3, X4))
   ELA_IN_AGA(X1, .(X2, X3), X4) -> ELA_IN_AGA(X1, X3, X4)
   GROUPD_IN_GGA(X1, .(X2, X3), .(.(X4, X5), X6)) -> U6_GGA(X1, X2, X3, X4, X5, X6, elcC_in_aga(X4, X1, X7))
   U6_GGA(X1, X2, X3, X4, X5, X6, elcC_out_aga(X4, X1, X7)) -> U7_GGA(X1, X2, X3, X4, X5, X6, selectNB_in_aga(X8, X7, X5))
   U6_GGA(X1, X2, X3, X4, X5, X6, elcC_out_aga(X4, X1, X7)) -> SELECTNB_IN_AGA(X8, X7, X5)
   SELECTNB_IN_AGA(X1, X2, .(X3, X4)) -> U2_AGA(X1, X2, X3, X4, elA_in_aga(X3, X2, X5))
   SELECTNB_IN_AGA(X1, X2, .(X3, X4)) -> ELA_IN_AGA(X3, X2, X5)
   SELECTNB_IN_AGA(X1, X2, .(X3, X4)) -> U3_AGA(X1, X2, X3, X4, elcA_in_aga(X3, X2, X5))
   U3_AGA(X1, X2, X3, X4, elcA_out_aga(X3, X2, X5)) -> U4_AGA(X1, X2, X3, X4, selectNB_in_aga(X6, X5, X4))
   U3_AGA(X1, X2, X3, X4, elcA_out_aga(X3, X2, X5)) -> SELECTNB_IN_AGA(X6, X5, X4)

The TRS R consists of the following rules:

   elcC_in_aga(X1, .(X1, X2), X2) -> elcC_out_aga(X1, .(X1, X2), X2)
   elcC_in_aga(X1, .(X2, .(X1, X3)), X3) -> elcC_out_aga(X1, .(X2, .(X1, X3)), X3)
   elcC_in_aga(X1, .(X2, .(X3, .(X1, X4))), X4) -> elcC_out_aga(X1, .(X2, .(X3, .(X1, X4))), X4)
   elcC_in_aga(X1, .(X2, .(X3, .(X4, .(X1, X5)))), X5) -> elcC_out_aga(X1, .(X2, .(X3, .(X4, .(X1, X5)))), X5)
   elcC_in_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X1, X6))))), X6) -> elcC_out_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X1, X6))))), X6)
   elcC_in_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X1, X7)))))), X7) -> elcC_out_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X1, X7)))))), X7)
   elcC_in_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X1, X8))))))), X8) -> elcC_out_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X1, X8))))))), X8)
   elcC_in_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, .(X1, X9)))))))), X9) -> elcC_out_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, .(X1, X9)))))))), X9)
   elcC_in_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, .(X9, X10)))))))), X11) -> U12_aga(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, elcA_in_aga(X1, X10, X11))
   elcA_in_aga(X1, .(X1, X2), X2) -> elcA_out_aga(X1, .(X1, X2), X2)
   elcA_in_aga(X1, .(X2, X3), X4) -> U9_aga(X1, X2, X3, X4, elcA_in_aga(X1, X3, X4))
   U9_aga(X1, X2, X3, X4, elcA_out_aga(X1, X3, X4)) -> elcA_out_aga(X1, .(X2, X3), X4)
   U12_aga(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, elcA_out_aga(X1, X10, X11)) -> elcC_out_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, .(X9, X10)))))))), X11)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)

elA_in_aga(x1, x2, x3)  =  elA_in_aga(x2)

elcC_in_aga(x1, x2, x3)  =  elcC_in_aga(x2)

elcC_out_aga(x1, x2, x3)  =  elcC_out_aga(x2, x3)

U12_aga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12)  =  U12_aga(x10, x12)

elcA_in_aga(x1, x2, x3)  =  elcA_in_aga(x2)

elcA_out_aga(x1, x2, x3)  =  elcA_out_aga(x2, x3)

U9_aga(x1, x2, x3, x4, x5)  =  U9_aga(x3, x5)

selectNB_in_aga(x1, x2, x3)  =  selectNB_in_aga(x2)

GROUPD_IN_GGA(x1, x2, x3)  =  GROUPD_IN_GGA(x1, x2)

U5_GGA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15)  =  U5_GGA(x9, x11, x15)

ELA_IN_AGA(x1, x2, x3)  =  ELA_IN_AGA(x2)

U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x3, x5)

U6_GGA(x1, x2, x3, x4, x5, x6, x7)  =  U6_GGA(x1, x3, x7)

U7_GGA(x1, x2, x3, x4, x5, x6, x7)  =  U7_GGA(x1, x3, x7)

SELECTNB_IN_AGA(x1, x2, x3)  =  SELECTNB_IN_AGA(x2)

U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x2, x5)

U3_AGA(x1, x2, x3, x4, x5)  =  U3_AGA(x2, x5)

U4_AGA(x1, x2, x3, x4, x5)  =  U4_AGA(x2, x5)


We have to consider all (P,R,Pi)-chains


Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES



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

(4)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   GROUPD_IN_GGA(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), .(X10, X11), .(.(X12, X13), X14)) -> U5_GGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, elA_in_aga(X12, X9, X15))
   GROUPD_IN_GGA(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), .(X10, X11), .(.(X12, X13), X14)) -> ELA_IN_AGA(X12, X9, X15)
   ELA_IN_AGA(X1, .(X2, X3), X4) -> U1_AGA(X1, X2, X3, X4, elA_in_aga(X1, X3, X4))
   ELA_IN_AGA(X1, .(X2, X3), X4) -> ELA_IN_AGA(X1, X3, X4)
   GROUPD_IN_GGA(X1, .(X2, X3), .(.(X4, X5), X6)) -> U6_GGA(X1, X2, X3, X4, X5, X6, elcC_in_aga(X4, X1, X7))
   U6_GGA(X1, X2, X3, X4, X5, X6, elcC_out_aga(X4, X1, X7)) -> U7_GGA(X1, X2, X3, X4, X5, X6, selectNB_in_aga(X8, X7, X5))
   U6_GGA(X1, X2, X3, X4, X5, X6, elcC_out_aga(X4, X1, X7)) -> SELECTNB_IN_AGA(X8, X7, X5)
   SELECTNB_IN_AGA(X1, X2, .(X3, X4)) -> U2_AGA(X1, X2, X3, X4, elA_in_aga(X3, X2, X5))
   SELECTNB_IN_AGA(X1, X2, .(X3, X4)) -> ELA_IN_AGA(X3, X2, X5)
   SELECTNB_IN_AGA(X1, X2, .(X3, X4)) -> U3_AGA(X1, X2, X3, X4, elcA_in_aga(X3, X2, X5))
   U3_AGA(X1, X2, X3, X4, elcA_out_aga(X3, X2, X5)) -> U4_AGA(X1, X2, X3, X4, selectNB_in_aga(X6, X5, X4))
   U3_AGA(X1, X2, X3, X4, elcA_out_aga(X3, X2, X5)) -> SELECTNB_IN_AGA(X6, X5, X4)

The TRS R consists of the following rules:

   elcC_in_aga(X1, .(X1, X2), X2) -> elcC_out_aga(X1, .(X1, X2), X2)
   elcC_in_aga(X1, .(X2, .(X1, X3)), X3) -> elcC_out_aga(X1, .(X2, .(X1, X3)), X3)
   elcC_in_aga(X1, .(X2, .(X3, .(X1, X4))), X4) -> elcC_out_aga(X1, .(X2, .(X3, .(X1, X4))), X4)
   elcC_in_aga(X1, .(X2, .(X3, .(X4, .(X1, X5)))), X5) -> elcC_out_aga(X1, .(X2, .(X3, .(X4, .(X1, X5)))), X5)
   elcC_in_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X1, X6))))), X6) -> elcC_out_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X1, X6))))), X6)
   elcC_in_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X1, X7)))))), X7) -> elcC_out_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X1, X7)))))), X7)
   elcC_in_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X1, X8))))))), X8) -> elcC_out_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X1, X8))))))), X8)
   elcC_in_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, .(X1, X9)))))))), X9) -> elcC_out_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, .(X1, X9)))))))), X9)
   elcC_in_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, .(X9, X10)))))))), X11) -> U12_aga(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, elcA_in_aga(X1, X10, X11))
   elcA_in_aga(X1, .(X1, X2), X2) -> elcA_out_aga(X1, .(X1, X2), X2)
   elcA_in_aga(X1, .(X2, X3), X4) -> U9_aga(X1, X2, X3, X4, elcA_in_aga(X1, X3, X4))
   U9_aga(X1, X2, X3, X4, elcA_out_aga(X1, X3, X4)) -> elcA_out_aga(X1, .(X2, X3), X4)
   U12_aga(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, elcA_out_aga(X1, X10, X11)) -> elcC_out_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, .(X9, X10)))))))), X11)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)

elA_in_aga(x1, x2, x3)  =  elA_in_aga(x2)

elcC_in_aga(x1, x2, x3)  =  elcC_in_aga(x2)

elcC_out_aga(x1, x2, x3)  =  elcC_out_aga(x2, x3)

U12_aga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12)  =  U12_aga(x10, x12)

elcA_in_aga(x1, x2, x3)  =  elcA_in_aga(x2)

elcA_out_aga(x1, x2, x3)  =  elcA_out_aga(x2, x3)

U9_aga(x1, x2, x3, x4, x5)  =  U9_aga(x3, x5)

selectNB_in_aga(x1, x2, x3)  =  selectNB_in_aga(x2)

GROUPD_IN_GGA(x1, x2, x3)  =  GROUPD_IN_GGA(x1, x2)

U5_GGA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15)  =  U5_GGA(x9, x11, x15)

ELA_IN_AGA(x1, x2, x3)  =  ELA_IN_AGA(x2)

U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x3, x5)

U6_GGA(x1, x2, x3, x4, x5, x6, x7)  =  U6_GGA(x1, x3, x7)

U7_GGA(x1, x2, x3, x4, x5, x6, x7)  =  U7_GGA(x1, x3, x7)

SELECTNB_IN_AGA(x1, x2, x3)  =  SELECTNB_IN_AGA(x2)

U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x2, x5)

U3_AGA(x1, x2, x3, x4, x5)  =  U3_AGA(x2, x5)

U4_AGA(x1, x2, x3, x4, x5)  =  U4_AGA(x2, x5)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(5) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 9 less nodes.
----------------------------------------

(6)
Complex Obligation (AND)

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

(7)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   ELA_IN_AGA(X1, .(X2, X3), X4) -> ELA_IN_AGA(X1, X3, X4)

The TRS R consists of the following rules:

   elcC_in_aga(X1, .(X1, X2), X2) -> elcC_out_aga(X1, .(X1, X2), X2)
   elcC_in_aga(X1, .(X2, .(X1, X3)), X3) -> elcC_out_aga(X1, .(X2, .(X1, X3)), X3)
   elcC_in_aga(X1, .(X2, .(X3, .(X1, X4))), X4) -> elcC_out_aga(X1, .(X2, .(X3, .(X1, X4))), X4)
   elcC_in_aga(X1, .(X2, .(X3, .(X4, .(X1, X5)))), X5) -> elcC_out_aga(X1, .(X2, .(X3, .(X4, .(X1, X5)))), X5)
   elcC_in_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X1, X6))))), X6) -> elcC_out_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X1, X6))))), X6)
   elcC_in_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X1, X7)))))), X7) -> elcC_out_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X1, X7)))))), X7)
   elcC_in_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X1, X8))))))), X8) -> elcC_out_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X1, X8))))))), X8)
   elcC_in_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, .(X1, X9)))))))), X9) -> elcC_out_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, .(X1, X9)))))))), X9)
   elcC_in_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, .(X9, X10)))))))), X11) -> U12_aga(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, elcA_in_aga(X1, X10, X11))
   elcA_in_aga(X1, .(X1, X2), X2) -> elcA_out_aga(X1, .(X1, X2), X2)
   elcA_in_aga(X1, .(X2, X3), X4) -> U9_aga(X1, X2, X3, X4, elcA_in_aga(X1, X3, X4))
   U9_aga(X1, X2, X3, X4, elcA_out_aga(X1, X3, X4)) -> elcA_out_aga(X1, .(X2, X3), X4)
   U12_aga(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, elcA_out_aga(X1, X10, X11)) -> elcC_out_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, .(X9, X10)))))))), X11)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)

elcC_in_aga(x1, x2, x3)  =  elcC_in_aga(x2)

elcC_out_aga(x1, x2, x3)  =  elcC_out_aga(x2, x3)

U12_aga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12)  =  U12_aga(x10, x12)

elcA_in_aga(x1, x2, x3)  =  elcA_in_aga(x2)

elcA_out_aga(x1, x2, x3)  =  elcA_out_aga(x2, x3)

U9_aga(x1, x2, x3, x4, x5)  =  U9_aga(x3, x5)

ELA_IN_AGA(x1, x2, x3)  =  ELA_IN_AGA(x2)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(8) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(9)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   ELA_IN_AGA(X1, .(X2, X3), X4) -> ELA_IN_AGA(X1, X3, X4)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)

ELA_IN_AGA(x1, x2, x3)  =  ELA_IN_AGA(x2)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(10) PiDPToQDPProof (SOUND)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(11)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ELA_IN_AGA(.(X3)) -> ELA_IN_AGA(X3)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(12) QDPSizeChangeProof (EQUIVALENT)
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. 

From the DPs we obtained the following set of size-change graphs:
*ELA_IN_AGA(.(X3)) -> ELA_IN_AGA(X3)
The graph contains the following edges 1 > 1


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

(13)
YES

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

(14)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   SELECTNB_IN_AGA(X1, X2, .(X3, X4)) -> U3_AGA(X1, X2, X3, X4, elcA_in_aga(X3, X2, X5))
   U3_AGA(X1, X2, X3, X4, elcA_out_aga(X3, X2, X5)) -> SELECTNB_IN_AGA(X6, X5, X4)

The TRS R consists of the following rules:

   elcC_in_aga(X1, .(X1, X2), X2) -> elcC_out_aga(X1, .(X1, X2), X2)
   elcC_in_aga(X1, .(X2, .(X1, X3)), X3) -> elcC_out_aga(X1, .(X2, .(X1, X3)), X3)
   elcC_in_aga(X1, .(X2, .(X3, .(X1, X4))), X4) -> elcC_out_aga(X1, .(X2, .(X3, .(X1, X4))), X4)
   elcC_in_aga(X1, .(X2, .(X3, .(X4, .(X1, X5)))), X5) -> elcC_out_aga(X1, .(X2, .(X3, .(X4, .(X1, X5)))), X5)
   elcC_in_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X1, X6))))), X6) -> elcC_out_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X1, X6))))), X6)
   elcC_in_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X1, X7)))))), X7) -> elcC_out_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X1, X7)))))), X7)
   elcC_in_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X1, X8))))))), X8) -> elcC_out_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X1, X8))))))), X8)
   elcC_in_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, .(X1, X9)))))))), X9) -> elcC_out_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, .(X1, X9)))))))), X9)
   elcC_in_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, .(X9, X10)))))))), X11) -> U12_aga(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, elcA_in_aga(X1, X10, X11))
   elcA_in_aga(X1, .(X1, X2), X2) -> elcA_out_aga(X1, .(X1, X2), X2)
   elcA_in_aga(X1, .(X2, X3), X4) -> U9_aga(X1, X2, X3, X4, elcA_in_aga(X1, X3, X4))
   U9_aga(X1, X2, X3, X4, elcA_out_aga(X1, X3, X4)) -> elcA_out_aga(X1, .(X2, X3), X4)
   U12_aga(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, elcA_out_aga(X1, X10, X11)) -> elcC_out_aga(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, .(X9, X10)))))))), X11)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)

elcC_in_aga(x1, x2, x3)  =  elcC_in_aga(x2)

elcC_out_aga(x1, x2, x3)  =  elcC_out_aga(x2, x3)

U12_aga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12)  =  U12_aga(x10, x12)

elcA_in_aga(x1, x2, x3)  =  elcA_in_aga(x2)

elcA_out_aga(x1, x2, x3)  =  elcA_out_aga(x2, x3)

U9_aga(x1, x2, x3, x4, x5)  =  U9_aga(x3, x5)

SELECTNB_IN_AGA(x1, x2, x3)  =  SELECTNB_IN_AGA(x2)

U3_AGA(x1, x2, x3, x4, x5)  =  U3_AGA(x2, x5)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(15) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(16)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   SELECTNB_IN_AGA(X1, X2, .(X3, X4)) -> U3_AGA(X1, X2, X3, X4, elcA_in_aga(X3, X2, X5))
   U3_AGA(X1, X2, X3, X4, elcA_out_aga(X3, X2, X5)) -> SELECTNB_IN_AGA(X6, X5, X4)

The TRS R consists of the following rules:

   elcA_in_aga(X1, .(X1, X2), X2) -> elcA_out_aga(X1, .(X1, X2), X2)
   elcA_in_aga(X1, .(X2, X3), X4) -> U9_aga(X1, X2, X3, X4, elcA_in_aga(X1, X3, X4))
   U9_aga(X1, X2, X3, X4, elcA_out_aga(X1, X3, X4)) -> elcA_out_aga(X1, .(X2, X3), X4)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)

elcA_in_aga(x1, x2, x3)  =  elcA_in_aga(x2)

elcA_out_aga(x1, x2, x3)  =  elcA_out_aga(x2, x3)

U9_aga(x1, x2, x3, x4, x5)  =  U9_aga(x3, x5)

SELECTNB_IN_AGA(x1, x2, x3)  =  SELECTNB_IN_AGA(x2)

U3_AGA(x1, x2, x3, x4, x5)  =  U3_AGA(x2, x5)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(17) PiDPToQDPProof (SOUND)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(18)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   SELECTNB_IN_AGA(X2) -> U3_AGA(X2, elcA_in_aga(X2))
   U3_AGA(X2, elcA_out_aga(X2, X5)) -> SELECTNB_IN_AGA(X5)

The TRS R consists of the following rules:

   elcA_in_aga(.(X2)) -> elcA_out_aga(.(X2), X2)
   elcA_in_aga(.(X3)) -> U9_aga(X3, elcA_in_aga(X3))
   U9_aga(X3, elcA_out_aga(X3, X4)) -> elcA_out_aga(.(X3), X4)

The set Q consists of the following terms:

   elcA_in_aga(x0)
   U9_aga(x0, x1)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(19) QDPQMonotonicMRRProof (EQUIVALENT)
By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain.


Strictly oriented rules of the TRS R:

   elcA_in_aga(.(X2)) -> elcA_out_aga(.(X2), X2)
   elcA_in_aga(.(X3)) -> U9_aga(X3, elcA_in_aga(X3))

Used ordering: Polynomial interpretation [POLO]:

   POL(.(x_1)) = 1 + 2*x_1
   POL(SELECTNB_IN_AGA(x_1)) = 2*x_1
   POL(U3_AGA(x_1, x_2)) = 2*x_2
   POL(U9_aga(x_1, x_2)) = 2*x_2
   POL(elcA_in_aga(x_1)) = x_1
   POL(elcA_out_aga(x_1, x_2)) = x_2


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

(20)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   SELECTNB_IN_AGA(X2) -> U3_AGA(X2, elcA_in_aga(X2))
   U3_AGA(X2, elcA_out_aga(X2, X5)) -> SELECTNB_IN_AGA(X5)

The TRS R consists of the following rules:

   U9_aga(X3, elcA_out_aga(X3, X4)) -> elcA_out_aga(.(X3), X4)

The set Q consists of the following terms:

   elcA_in_aga(x0)
   U9_aga(x0, x1)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(21) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
----------------------------------------

(22)
TRUE