3.64/1.63	YES
3.64/1.64	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
3.64/1.64	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.64/1.64	
3.64/1.64	
3.64/1.64	Left Termination of the query pattern
3.64/1.64	
3.64/1.64	len(g,a)
3.64/1.64	
3.64/1.64	w.r.t. the given Prolog program could successfully be proven:
3.64/1.64	
3.64/1.64	(0) Prolog
3.64/1.64	(1) PrologToDTProblemTransformerProof [SOUND, 0 ms]
3.64/1.64	(2) TRIPLES
3.64/1.64	(3) TriplesToPiDPProof [SOUND, 0 ms]
3.64/1.64	(4) PiDP
3.64/1.64	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
3.64/1.64	(6) PiDP
3.64/1.64	(7) PiDPToQDPProof [SOUND, 0 ms]
3.64/1.64	(8) QDP
3.64/1.64	(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.64/1.64	(10) YES
3.64/1.64	
3.64/1.64	
3.64/1.64	----------------------------------------
3.64/1.64	
3.64/1.64	(0)
3.64/1.64	Obligation:
3.64/1.64	Clauses:
3.64/1.64	
3.64/1.64	len([], 0).
3.64/1.64	len(.(X1, Ts), s(N)) :- len(Ts, N).
3.64/1.64	
3.64/1.64	
3.64/1.64	Query: len(g,a)
3.64/1.64	----------------------------------------
3.64/1.64	
3.64/1.64	(1) PrologToDTProblemTransformerProof (SOUND)
3.64/1.64	Built DT problem from termination graph DT10.
3.64/1.64	
3.64/1.64	{
3.64/1.64	    "root": 5,
3.64/1.64	    "program": {
3.64/1.64	        "directives": [],
3.64/1.64	        "clauses": [
3.64/1.64	            [
3.64/1.64	                "(len ([]) (0))",
3.64/1.64	                null
3.64/1.64	            ],
3.64/1.64	            [
3.64/1.64	                "(len (. X1 Ts) (s N))",
3.64/1.64	                "(len Ts N)"
3.64/1.64	            ]
3.64/1.64	        ]
3.64/1.64	    },
3.64/1.64	    "graph": {
3.64/1.64	        "nodes": {
3.64/1.64	            "59": {
3.64/1.64	                "goal": [{
3.64/1.64	                    "clause": -1,
3.64/1.64	                    "scope": -1,
3.64/1.64	                    "term": "(len T7 T9)"
3.64/1.64	                }],
3.64/1.64	                "kb": {
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3.64/1.64	                        "relations": []
3.64/1.64	                    },
3.64/1.64	                    "ground": ["T7"],
3.64/1.64	                    "free": [],
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3.64/1.64	                }
3.64/1.64	            },
3.64/1.64	            "type": "Nodes",
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3.64/1.64	                    "clause": -1,
3.64/1.64	                    "scope": -1,
3.64/1.64	                    "term": "(len T1 T2)"
3.64/1.64	                }],
3.64/1.64	                "kb": {
3.64/1.64	                    "nonunifying": [],
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3.64/1.64	                    "arithmetic": {
3.64/1.64	                        "type": "PlainIntegerRelationState",
3.64/1.65	                        "relations": []
3.64/1.65	                    },
3.64/1.65	                    "ground": ["T1"],
3.64/1.65	                    "free": [],
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3.64/1.65	                }
3.64/1.65	            },
3.64/1.65	            "6": {
3.64/1.65	                "goal": [
3.64/1.65	                    {
3.64/1.65	                        "clause": 0,
3.64/1.65	                        "scope": 1,
3.64/1.65	                        "term": "(len T1 T2)"
3.64/1.65	                    },
3.64/1.65	                    {
3.64/1.65	                        "clause": 1,
3.64/1.65	                        "scope": 1,
3.64/1.65	                        "term": "(len T1 T2)"
3.64/1.65	                    }
3.64/1.65	                ],
3.64/1.65	                "kb": {
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3.64/1.65	                        "relations": []
3.64/1.65	                    },
3.64/1.65	                    "ground": ["T1"],
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3.64/1.65	                }
3.64/1.65	            },
3.64/1.65	            "90": {
3.64/1.65	                "goal": [{
3.64/1.65	                    "clause": -1,
3.64/1.65	                    "scope": -1,
3.64/1.65	                    "term": "(true)"
3.64/1.65	                }],
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3.64/1.65	                }
3.64/1.65	            },
3.64/1.65	            "91": {
3.64/1.65	                "goal": [],
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3.64/1.65	                    "nonunifying": [],
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3.64/1.65	                    "arithmetic": {
3.64/1.65	                        "type": "PlainIntegerRelationState",
3.64/1.65	                        "relations": []
3.64/1.65	                    },
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3.64/1.65	                    "free": [],
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3.64/1.65	                }
3.64/1.65	            },
3.64/1.65	            "92": {
3.64/1.65	                "goal": [],
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3.64/1.65	                    "arithmetic": {
3.64/1.65	                        "type": "PlainIntegerRelationState",
3.64/1.65	                        "relations": []
3.64/1.65	                    },
3.64/1.65	                    "ground": [],
3.64/1.65	                    "free": [],
3.64/1.65	                    "exprvars": []
3.64/1.65	                }
3.64/1.65	            },
3.64/1.65	            "60": {
3.64/1.65	                "goal": [],
3.64/1.65	                "kb": {
3.64/1.65	                    "nonunifying": [],
3.64/1.65	                    "intvars": {},
3.64/1.65	                    "arithmetic": {
3.64/1.65	                        "type": "PlainIntegerRelationState",
3.64/1.65	                        "relations": []
3.64/1.65	                    },
3.64/1.65	                    "ground": [],
3.64/1.65	                    "free": [],
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3.64/1.65	                }
3.64/1.65	            },
3.64/1.65	            "93": {
3.64/1.65	                "goal": [{
3.64/1.65	                    "clause": -1,
3.64/1.65	                    "scope": -1,
3.64/1.65	                    "term": "(len T17 T19)"
3.64/1.65	                }],
3.64/1.65	                "kb": {
3.64/1.65	                    "nonunifying": [],
3.64/1.65	                    "intvars": {},
3.64/1.65	                    "arithmetic": {
3.64/1.65	                        "type": "PlainIntegerRelationState",
3.64/1.65	                        "relations": []
3.64/1.65	                    },
3.64/1.65	                    "ground": ["T17"],
3.64/1.65	                    "free": [],
3.64/1.65	                    "exprvars": []
3.64/1.65	                }
3.64/1.65	            },
3.64/1.65	            "94": {
3.64/1.65	                "goal": [],
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3.64/1.65	                    "nonunifying": [],
3.64/1.65	                    "intvars": {},
3.64/1.65	                    "arithmetic": {
3.64/1.65	                        "type": "PlainIntegerRelationState",
3.64/1.65	                        "relations": []
3.64/1.65	                    },
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3.64/1.65	                    "free": [],
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3.64/1.65	                }
3.64/1.65	            },
3.64/1.65	            "73": {
3.64/1.65	                "goal": [
3.64/1.65	                    {
3.64/1.65	                        "clause": 0,
3.64/1.65	                        "scope": 2,
3.64/1.65	                        "term": "(len T7 T9)"
3.64/1.65	                    },
3.64/1.65	                    {
3.64/1.65	                        "clause": 1,
3.64/1.65	                        "scope": 2,
3.64/1.65	                        "term": "(len T7 T9)"
3.64/1.65	                    }
3.64/1.65	                ],
3.64/1.65	                "kb": {
3.64/1.65	                    "nonunifying": [],
3.64/1.65	                    "intvars": {},
3.64/1.65	                    "arithmetic": {
3.64/1.65	                        "type": "PlainIntegerRelationState",
3.64/1.65	                        "relations": []
3.64/1.65	                    },
3.64/1.65	                    "ground": ["T7"],
3.64/1.65	                    "free": [],
3.64/1.65	                    "exprvars": []
3.64/1.65	                }
3.64/1.65	            },
3.64/1.65	            "52": {
3.64/1.65	                "goal": [{
3.64/1.65	                    "clause": 1,
3.64/1.65	                    "scope": 1,
3.64/1.65	                    "term": "(len T1 T2)"
3.64/1.65	                }],
3.64/1.65	                "kb": {
3.64/1.65	                    "nonunifying": [[
3.64/1.65	                        "(len T1 T2)",
3.64/1.65	                        "(len ([]) (0))"
3.64/1.65	                    ]],
3.64/1.65	                    "intvars": {},
3.64/1.65	                    "arithmetic": {
3.64/1.65	                        "type": "PlainIntegerRelationState",
3.64/1.65	                        "relations": []
3.64/1.65	                    },
3.64/1.65	                    "ground": ["T1"],
3.64/1.65	                    "free": [],
3.64/1.65	                    "exprvars": []
3.64/1.65	                }
3.64/1.65	            },
3.64/1.65	            "85": {
3.64/1.65	                "goal": [{
3.64/1.65	                    "clause": 0,
3.64/1.65	                    "scope": 2,
3.64/1.65	                    "term": "(len T7 T9)"
3.64/1.65	                }],
3.64/1.65	                "kb": {
3.64/1.65	                    "nonunifying": [],
3.64/1.65	                    "intvars": {},
3.64/1.65	                    "arithmetic": {
3.64/1.65	                        "type": "PlainIntegerRelationState",
3.64/1.65	                        "relations": []
3.64/1.65	                    },
3.64/1.65	                    "ground": ["T7"],
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3.64/1.65	                }
3.64/1.65	            },
3.64/1.65	            "31": {
3.64/1.65	                "goal": [
3.64/1.65	                    {
3.64/1.65	                        "clause": -1,
3.64/1.65	                        "scope": -1,
3.64/1.65	                        "term": "(true)"
3.64/1.65	                    },
3.64/1.65	                    {
3.64/1.65	                        "clause": 1,
3.64/1.65	                        "scope": 1,
3.64/1.65	                        "term": "(len ([]) T2)"
3.64/1.65	                    }
3.64/1.65	                ],
3.64/1.65	                "kb": {
3.64/1.65	                    "nonunifying": [],
3.64/1.65	                    "intvars": {},
3.64/1.65	                    "arithmetic": {
3.64/1.65	                        "type": "PlainIntegerRelationState",
3.64/1.65	                        "relations": []
3.64/1.65	                    },
3.64/1.65	                    "ground": [],
3.64/1.65	                    "free": [],
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3.64/1.65	                }
3.64/1.65	            },
3.64/1.65	            "53": {
3.64/1.65	                "goal": [{
3.64/1.65	                    "clause": 1,
3.64/1.65	                    "scope": 1,
3.64/1.65	                    "term": "(len ([]) T2)"
3.64/1.65	                }],
3.64/1.65	                "kb": {
3.64/1.65	                    "nonunifying": [],
3.64/1.65	                    "intvars": {},
3.64/1.65	                    "arithmetic": {
3.64/1.65	                        "type": "PlainIntegerRelationState",
3.64/1.65	                        "relations": []
3.64/1.65	                    },
3.64/1.65	                    "ground": [],
3.64/1.65	                    "free": [],
3.64/1.65	                    "exprvars": []
3.64/1.65	                }
3.64/1.65	            },
3.64/1.65	            "86": {
3.64/1.65	                "goal": [{
3.64/1.65	                    "clause": 1,
3.64/1.65	                    "scope": 2,
3.64/1.65	                    "term": "(len T7 T9)"
3.64/1.65	                }],
3.64/1.65	                "kb": {
3.64/1.65	                    "nonunifying": [],
3.64/1.65	                    "intvars": {},
3.64/1.65	                    "arithmetic": {
3.64/1.65	                        "type": "PlainIntegerRelationState",
3.64/1.65	                        "relations": []
3.64/1.65	                    },
3.64/1.65	                    "ground": ["T7"],
3.64/1.65	                    "free": [],
3.64/1.65	                    "exprvars": []
3.64/1.65	                }
3.64/1.65	            },
3.64/1.65	            "54": {
3.64/1.65	                "goal": [],
3.64/1.65	                "kb": {
3.64/1.65	                    "nonunifying": [],
3.64/1.65	                    "intvars": {},
3.64/1.65	                    "arithmetic": {
3.64/1.65	                        "type": "PlainIntegerRelationState",
3.64/1.65	                        "relations": []
3.64/1.65	                    },
3.64/1.65	                    "ground": [],
3.64/1.65	                    "free": [],
3.64/1.65	                    "exprvars": []
3.64/1.65	                }
3.64/1.65	            }
3.64/1.65	        },
3.64/1.65	        "edges": [
3.64/1.65	            {
3.64/1.65	                "from": 5,
3.64/1.65	                "to": 6,
3.64/1.65	                "label": "CASE"
3.64/1.65	            },
3.64/1.65	            {
3.64/1.65	                "from": 6,
3.64/1.65	                "to": 31,
3.64/1.65	                "label": "EVAL with clause\nlen([], 0).\nand substitutionT1 -> [],\nT2 -> 0"
3.64/1.65	            },
3.64/1.65	            {
3.64/1.65	                "from": 6,
3.64/1.65	                "to": 52,
3.64/1.65	                "label": "EVAL-BACKTRACK"
3.64/1.65	            },
3.64/1.65	            {
3.64/1.65	                "from": 31,
3.64/1.65	                "to": 53,
3.64/1.65	                "label": "SUCCESS"
3.64/1.65	            },
3.64/1.65	            {
3.64/1.65	                "from": 52,
3.64/1.65	                "to": 59,
3.64/1.65	                "label": "EVAL with clause\nlen(.(X8, X9), s(X10)) :- len(X9, X10).\nand substitutionX8 -> T6,\nX9 -> T7,\nT1 -> .(T6, T7),\nX10 -> T9,\nT2 -> s(T9),\nT8 -> T9"
3.64/1.65	            },
3.64/1.65	            {
3.64/1.65	                "from": 52,
3.64/1.65	                "to": 60,
3.64/1.65	                "label": "EVAL-BACKTRACK"
3.64/1.65	            },
3.64/1.65	            {
3.64/1.65	                "from": 53,
3.64/1.65	                "to": 54,
3.64/1.65	                "label": "BACKTRACK\nfor clause: len(.(X1, Ts), s(N)) :- len(Ts, N)because of non-unification"
3.64/1.65	            },
3.64/1.65	            {
3.64/1.65	                "from": 59,
3.64/1.65	                "to": 73,
3.64/1.65	                "label": "CASE"
3.64/1.65	            },
3.64/1.65	            {
3.64/1.65	                "from": 73,
3.64/1.65	                "to": 85,
3.64/1.65	                "label": "PARALLEL"
3.64/1.65	            },
3.64/1.65	            {
3.64/1.65	                "from": 73,
3.64/1.65	                "to": 86,
3.64/1.65	                "label": "PARALLEL"
3.64/1.65	            },
3.64/1.65	            {
3.64/1.65	                "from": 85,
3.64/1.65	                "to": 90,
3.64/1.65	                "label": "EVAL with clause\nlen([], 0).\nand substitutionT7 -> [],\nT9 -> 0"
3.64/1.65	            },
3.64/1.65	            {
3.64/1.65	                "from": 85,
3.64/1.65	                "to": 91,
3.64/1.65	                "label": "EVAL-BACKTRACK"
3.64/1.65	            },
3.64/1.65	            {
3.64/1.65	                "from": 86,
3.64/1.65	                "to": 93,
3.64/1.65	                "label": "EVAL with clause\nlen(.(X17, X18), s(X19)) :- len(X18, X19).\nand substitutionX17 -> T16,\nX18 -> T17,\nT7 -> .(T16, T17),\nX19 -> T19,\nT9 -> s(T19),\nT18 -> T19"
3.64/1.65	            },
3.64/1.65	            {
3.64/1.65	                "from": 86,
3.64/1.65	                "to": 94,
3.64/1.65	                "label": "EVAL-BACKTRACK"
3.64/1.65	            },
3.64/1.65	            {
3.64/1.65	                "from": 90,
3.64/1.65	                "to": 92,
3.64/1.65	                "label": "SUCCESS"
3.64/1.65	            },
3.64/1.65	            {
3.64/1.65	                "from": 93,
3.64/1.65	                "to": 5,
3.64/1.65	                "label": "INSTANCE with matching:\nT1 -> T17\nT2 -> T19"
3.64/1.65	            }
3.64/1.65	        ],
3.64/1.65	        "type": "Graph"
3.64/1.65	    }
3.64/1.65	}
3.64/1.65	
3.64/1.65	----------------------------------------
3.64/1.65	
3.64/1.65	(2)
3.64/1.65	Obligation:
3.64/1.65	Triples:
3.64/1.65	
3.64/1.65	lenA(.(X1, .(X2, X3)), s(s(X4))) :- lenA(X3, X4).
3.64/1.65	
3.64/1.65	Clauses:
3.64/1.65	
3.64/1.65	lencA([], 0).
3.64/1.65	lencA(.(X1, []), s(0)).
3.64/1.65	lencA(.(X1, .(X2, X3)), s(s(X4))) :- lencA(X3, X4).
3.64/1.65	
3.64/1.65	Afs:
3.64/1.65	
3.64/1.65	lenA(x1, x2)  =  lenA(x1)
3.64/1.65	
3.64/1.65	
3.64/1.65	----------------------------------------
3.64/1.65	
3.64/1.65	(3) TriplesToPiDPProof (SOUND)
3.64/1.65	We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
3.64/1.65	
3.64/1.65	lenA_in_2: (b,f)
3.64/1.65	
3.64/1.65	Transforming TRIPLES into the following Term Rewriting System:
3.64/1.65	
3.64/1.65	Pi DP problem:
3.64/1.65	The TRS P consists of the following rules:
3.64/1.65	
3.64/1.65	   LENA_IN_GA(.(X1, .(X2, X3)), s(s(X4))) -> U1_GA(X1, X2, X3, X4, lenA_in_ga(X3, X4))
3.64/1.65	   LENA_IN_GA(.(X1, .(X2, X3)), s(s(X4))) -> LENA_IN_GA(X3, X4)
3.64/1.65	
3.64/1.65	R is empty.
3.64/1.65	The argument filtering Pi contains the following mapping:
3.64/1.65	lenA_in_ga(x1, x2)  =  lenA_in_ga(x1)
3.64/1.65	
3.64/1.65	.(x1, x2)  =  .(x1, x2)
3.64/1.65	
3.64/1.65	s(x1)  =  s(x1)
3.64/1.65	
3.64/1.65	LENA_IN_GA(x1, x2)  =  LENA_IN_GA(x1)
3.64/1.65	
3.64/1.65	U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x3, x5)
3.64/1.65	
3.64/1.65	
3.64/1.65	We have to consider all (P,R,Pi)-chains
3.64/1.65	
3.64/1.65	
3.64/1.65	Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES
3.64/1.65	
3.64/1.65	
3.64/1.65	
3.64/1.65	----------------------------------------
3.64/1.65	
3.64/1.65	(4)
3.64/1.65	Obligation:
3.64/1.65	Pi DP problem:
3.64/1.65	The TRS P consists of the following rules:
3.64/1.65	
3.64/1.65	   LENA_IN_GA(.(X1, .(X2, X3)), s(s(X4))) -> U1_GA(X1, X2, X3, X4, lenA_in_ga(X3, X4))
3.64/1.65	   LENA_IN_GA(.(X1, .(X2, X3)), s(s(X4))) -> LENA_IN_GA(X3, X4)
3.64/1.65	
3.64/1.65	R is empty.
3.64/1.65	The argument filtering Pi contains the following mapping:
3.64/1.65	lenA_in_ga(x1, x2)  =  lenA_in_ga(x1)
3.64/1.65	
3.64/1.65	.(x1, x2)  =  .(x1, x2)
3.64/1.65	
3.64/1.65	s(x1)  =  s(x1)
3.64/1.65	
3.64/1.65	LENA_IN_GA(x1, x2)  =  LENA_IN_GA(x1)
3.64/1.65	
3.64/1.65	U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x3, x5)
3.64/1.65	
3.64/1.65	
3.64/1.65	We have to consider all (P,R,Pi)-chains
3.64/1.65	----------------------------------------
3.64/1.65	
3.64/1.65	(5) DependencyGraphProof (EQUIVALENT)
3.64/1.65	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.
3.64/1.65	----------------------------------------
3.64/1.65	
3.64/1.65	(6)
3.64/1.65	Obligation:
3.64/1.65	Pi DP problem:
3.64/1.65	The TRS P consists of the following rules:
3.64/1.65	
3.64/1.65	   LENA_IN_GA(.(X1, .(X2, X3)), s(s(X4))) -> LENA_IN_GA(X3, X4)
3.64/1.65	
3.64/1.65	R is empty.
3.64/1.65	The argument filtering Pi contains the following mapping:
3.64/1.65	.(x1, x2)  =  .(x1, x2)
3.64/1.65	
3.64/1.65	s(x1)  =  s(x1)
3.64/1.65	
3.64/1.65	LENA_IN_GA(x1, x2)  =  LENA_IN_GA(x1)
3.64/1.65	
3.64/1.65	
3.64/1.65	We have to consider all (P,R,Pi)-chains
3.64/1.65	----------------------------------------
3.64/1.65	
3.64/1.65	(7) PiDPToQDPProof (SOUND)
3.64/1.65	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
3.64/1.65	----------------------------------------
3.64/1.65	
3.64/1.65	(8)
3.64/1.65	Obligation:
3.64/1.65	Q DP problem:
3.64/1.65	The TRS P consists of the following rules:
3.64/1.65	
3.64/1.65	   LENA_IN_GA(.(X1, .(X2, X3))) -> LENA_IN_GA(X3)
3.64/1.65	
3.64/1.65	R is empty.
3.64/1.65	Q is empty.
3.64/1.65	We have to consider all (P,Q,R)-chains.
3.64/1.65	----------------------------------------
3.64/1.65	
3.64/1.65	(9) QDPSizeChangeProof (EQUIVALENT)
3.64/1.65	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
3.64/1.65	
3.64/1.65	From the DPs we obtained the following set of size-change graphs:
3.64/1.65	*LENA_IN_GA(.(X1, .(X2, X3))) -> LENA_IN_GA(X3)
3.64/1.65	The graph contains the following edges 1 > 1
3.64/1.65	
3.64/1.65	
3.64/1.65	----------------------------------------
3.64/1.65	
3.64/1.65	(10)
3.64/1.65	YES
3.77/1.70	EOF