3.21/1.61	YES
3.62/1.63	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
3.62/1.63	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.62/1.63	
3.62/1.63	
3.62/1.63	Left Termination of the query pattern
3.62/1.63	
3.62/1.63	goal()
3.62/1.63	
3.62/1.63	w.r.t. the given Prolog program could successfully be proven:
3.62/1.63	
3.62/1.63	(0) Prolog
3.62/1.63	(1) PrologToIRSwTTransformerProof [SOUND, 38 ms]
3.62/1.63	(2) IRSwT
3.62/1.63	(3) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
3.62/1.63	(4) TRUE
3.62/1.63	
3.62/1.63	
3.62/1.63	----------------------------------------
3.62/1.63	
3.62/1.63	(0)
3.62/1.63	Obligation:
3.62/1.63	Clauses:
3.62/1.63	
3.62/1.63	even(s(s(X))) :- even(X).
3.62/1.63	even(0).
3.62/1.63	lte(s(X), s(Y)) :- lte(X, Y).
3.62/1.63	lte(0, Y).
3.62/1.63	goal :- ','(lte(X, s(s(s(s(0))))), even(X)).
3.62/1.63	
3.62/1.63	
3.62/1.63	Query: goal()
3.62/1.63	----------------------------------------
3.62/1.63	
3.62/1.63	(1) PrologToIRSwTTransformerProof (SOUND)
3.62/1.63	Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert
3.62/1.63	
3.62/1.63	{
3.62/1.63	    "root": 1,
3.62/1.63	    "program": {
3.62/1.63	        "directives": [],
3.62/1.63	        "clauses": [
3.62/1.63	            [
3.62/1.63	                "(even (s (s X)))",
3.62/1.63	                "(even X)"
3.62/1.63	            ],
3.62/1.63	            [
3.62/1.63	                "(even (0))",
3.62/1.63	                null
3.62/1.63	            ],
3.62/1.63	            [
3.62/1.63	                "(lte (s X) (s Y))",
3.62/1.63	                "(lte X Y)"
3.62/1.63	            ],
3.62/1.63	            [
3.62/1.63	                "(lte (0) Y)",
3.62/1.63	                null
3.62/1.63	            ],
3.62/1.63	            [
3.62/1.63	                "(goal)",
3.62/1.63	                "(',' (lte X (s (s (s (s (0)))))) (even X))"
3.62/1.63	            ]
3.62/1.63	        ]
3.62/1.63	    },
3.62/1.63	    "graph": {
3.62/1.63	        "nodes": {
3.62/1.63	            "88": {
3.62/1.63	                "goal": [{
3.62/1.63	                    "clause": -1,
3.62/1.63	                    "scope": -1,
3.62/1.63	                    "term": "(true)"
3.62/1.63	                }],
3.62/1.63	                "kb": {
3.62/1.63	                    "nonunifying": [],
3.62/1.63	                    "intvars": {},
3.62/1.63	                    "arithmetic": {
3.62/1.63	                        "type": "PlainIntegerRelationState",
3.62/1.63	                        "relations": []
3.62/1.63	                    },
3.62/1.63	                    "ground": [],
3.62/1.63	                    "free": [],
3.62/1.63	                    "exprvars": []
3.62/1.63	                }
3.62/1.63	            },
3.62/1.63	            "67": {
3.62/1.63	                "goal": [{
3.62/1.63	                    "clause": -1,
3.62/1.63	                    "scope": -1,
3.62/1.63	                    "term": "(lte X80 (s (0)))"
3.62/1.63	                }],
3.62/1.63	                "kb": {
3.62/1.63	                    "nonunifying": [],
3.62/1.63	                    "intvars": {},
3.62/1.63	                    "arithmetic": {
3.62/1.63	                        "type": "PlainIntegerRelationState",
3.62/1.63	                        "relations": []
3.62/1.63	                    },
3.62/1.63	                    "ground": [],
3.62/1.63	                    "free": ["X80"],
3.62/1.63	                    "exprvars": []
3.62/1.63	                }
3.62/1.63	            },
3.62/1.63	            "89": {
3.62/1.63	                "goal": [],
3.62/1.63	                "kb": {
3.62/1.63	                    "nonunifying": [],
3.62/1.63	                    "intvars": {},
3.62/1.63	                    "arithmetic": {
3.62/1.63	                        "type": "PlainIntegerRelationState",
3.62/1.63	                        "relations": []
3.62/1.63	                    },
3.62/1.63	                    "ground": [],
3.62/1.63	                    "free": [],
3.62/1.63	                    "exprvars": []
3.62/1.63	                }
3.62/1.63	            },
3.62/1.63	            "46": {
3.62/1.63	                "goal": [{
3.62/1.63	                    "clause": 2,
3.62/1.63	                    "scope": 2,
3.62/1.63	                    "term": "(lte X5 (s (s (s (s (0))))))"
3.62/1.63	                }],
3.62/1.63	                "kb": {
3.62/1.63	                    "nonunifying": [],
3.62/1.63	                    "intvars": {},
3.62/1.63	                    "arithmetic": {
3.62/1.63	                        "type": "PlainIntegerRelationState",
3.62/1.63	                        "relations": []
3.62/1.63	                    },
3.62/1.63	                    "ground": [],
3.62/1.63	                    "free": ["X5"],
3.62/1.63	                    "exprvars": []
3.62/1.63	                }
3.62/1.63	            },
3.62/1.63	            "47": {
3.62/1.63	                "goal": [{
3.62/1.63	                    "clause": 3,
3.62/1.63	                    "scope": 2,
3.62/1.63	                    "term": "(lte X5 (s (s (s (s (0))))))"
3.62/1.63	                }],
3.62/1.63	                "kb": {
3.62/1.63	                    "nonunifying": [],
3.62/1.63	                    "intvars": {},
3.62/1.63	                    "arithmetic": {
3.62/1.63	                        "type": "PlainIntegerRelationState",
3.62/1.63	                        "relations": []
3.62/1.63	                    },
3.62/1.63	                    "ground": [],
3.62/1.63	                    "free": ["X5"],
3.62/1.63	                    "exprvars": []
3.62/1.63	                }
3.62/1.63	            },
3.62/1.63	            "type": "Nodes",
3.62/1.63	            "70": {
3.62/1.63	                "goal": [
3.62/1.63	                    {
3.62/1.63	                        "clause": 2,
3.62/1.63	                        "scope": 5,
3.62/1.63	                        "term": "(lte X80 (s (0)))"
3.62/1.63	                    },
3.62/1.63	                    {
3.62/1.63	                        "clause": 3,
3.62/1.63	                        "scope": 5,
3.62/1.63	                        "term": "(lte X80 (s (0)))"
3.62/1.63	                    }
3.62/1.63	                ],
3.62/1.63	                "kb": {
3.62/1.63	                    "nonunifying": [],
3.62/1.63	                    "intvars": {},
3.62/1.63	                    "arithmetic": {
3.62/1.63	                        "type": "PlainIntegerRelationState",
3.62/1.63	                        "relations": []
3.62/1.63	                    },
3.62/1.63	                    "ground": [],
3.62/1.63	                    "free": ["X80"],
3.62/1.63	                    "exprvars": []
3.62/1.63	                }
3.62/1.63	            },
3.62/1.63	            "92": {
3.62/1.63	                "goal": [{
3.62/1.63	                    "clause": -1,
3.62/1.63	                    "scope": -1,
3.62/1.63	                    "term": "(true)"
3.62/1.63	                }],
3.62/1.63	                "kb": {
3.62/1.63	                    "nonunifying": [],
3.62/1.63	                    "intvars": {},
3.62/1.63	                    "arithmetic": {
3.62/1.63	                        "type": "PlainIntegerRelationState",
3.62/1.63	                        "relations": []
3.62/1.63	                    },
3.62/1.63	                    "ground": [],
3.62/1.63	                    "free": [],
3.62/1.63	                    "exprvars": []
3.62/1.63	                }
3.62/1.63	            },
3.62/1.63	            "93": {
3.62/1.63	                "goal": [],
3.62/1.63	                "kb": {
3.62/1.63	                    "nonunifying": [],
3.62/1.63	                    "intvars": {},
3.62/1.63	                    "arithmetic": {
3.62/1.63	                        "type": "PlainIntegerRelationState",
3.62/1.63	                        "relations": []
3.62/1.63	                    },
3.62/1.63	                    "ground": [],
3.62/1.63	                    "free": [],
3.62/1.63	                    "exprvars": []
3.62/1.63	                }
3.62/1.63	            },
3.62/1.63	            "50": {
3.62/1.63	                "goal": [{
3.62/1.63	                    "clause": -1,
3.62/1.63	                    "scope": -1,
3.62/1.63	                    "term": "(lte X32 (s (s (s (0)))))"
3.62/1.63	                }],
3.62/1.63	                "kb": {
3.62/1.63	                    "nonunifying": [],
3.62/1.63	                    "intvars": {},
3.62/1.63	                    "arithmetic": {
3.62/1.63	                        "type": "PlainIntegerRelationState",
3.62/1.63	                        "relations": []
3.62/1.63	                    },
3.62/1.63	                    "ground": [],
3.62/1.63	                    "free": ["X32"],
3.62/1.63	                    "exprvars": []
3.62/1.63	                }
3.62/1.63	            },
3.62/1.63	            "94": {
3.62/1.63	                "goal": [{
3.62/1.63	                    "clause": -1,
3.62/1.63	                    "scope": -1,
3.62/1.63	                    "term": "(true)"
3.62/1.63	                }],
3.62/1.63	                "kb": {
3.62/1.63	                    "nonunifying": [],
3.62/1.63	                    "intvars": {},
3.62/1.63	                    "arithmetic": {
3.62/1.63	                        "type": "PlainIntegerRelationState",
3.62/1.63	                        "relations": []
3.62/1.63	                    },
3.62/1.63	                    "ground": [],
3.62/1.63	                    "free": [],
3.62/1.63	                    "exprvars": []
3.62/1.63	                }
3.62/1.63	            },
3.62/1.63	            "73": {
3.62/1.63	                "goal": [{
3.62/1.63	                    "clause": 2,
3.62/1.63	                    "scope": 5,
3.62/1.63	                    "term": "(lte X80 (s (0)))"
3.62/1.63	                }],
3.62/1.63	                "kb": {
3.62/1.63	                    "nonunifying": [],
3.62/1.63	                    "intvars": {},
3.62/1.63	                    "arithmetic": {
3.62/1.63	                        "type": "PlainIntegerRelationState",
3.62/1.63	                        "relations": []
3.62/1.63	                    },
3.62/1.63	                    "ground": [],
3.62/1.63	                    "free": ["X80"],
3.62/1.63	                    "exprvars": []
3.62/1.63	                }
3.62/1.63	            },
3.62/1.63	            "95": {
3.62/1.63	                "goal": [],
3.62/1.63	                "kb": {
3.62/1.63	                    "nonunifying": [],
3.62/1.63	                    "intvars": {},
3.62/1.63	                    "arithmetic": {
3.62/1.63	                        "type": "PlainIntegerRelationState",
3.62/1.63	                        "relations": []
3.62/1.63	                    },
3.62/1.63	                    "ground": [],
3.62/1.63	                    "free": [],
3.62/1.63	                    "exprvars": []
3.62/1.63	                }
3.62/1.63	            },
3.62/1.63	            "74": {
3.62/1.63	                "goal": [{
3.62/1.63	                    "clause": 3,
3.62/1.63	                    "scope": 5,
3.62/1.63	                    "term": "(lte X80 (s (0)))"
3.62/1.63	                }],
3.62/1.63	                "kb": {
3.62/1.63	                    "nonunifying": [],
3.62/1.63	                    "intvars": {},
3.62/1.63	                    "arithmetic": {
3.62/1.63	                        "type": "PlainIntegerRelationState",
3.62/1.63	                        "relations": []
3.62/1.63	                    },
3.62/1.63	                    "ground": [],
3.62/1.63	                    "free": ["X80"],
3.62/1.63	                    "exprvars": []
3.62/1.63	                }
3.62/1.63	            },
3.62/1.63	            "96": {
3.62/1.63	                "goal": [{
3.62/1.63	                    "clause": -1,
3.62/1.63	                    "scope": -1,
3.62/1.63	                    "term": "(true)"
3.62/1.63	                }],
3.62/1.63	                "kb": {
3.62/1.63	                    "nonunifying": [],
3.62/1.63	                    "intvars": {},
3.62/1.63	                    "arithmetic": {
3.62/1.63	                        "type": "PlainIntegerRelationState",
3.62/1.63	                        "relations": []
3.62/1.63	                    },
3.62/1.63	                    "ground": [],
3.62/1.63	                    "free": [],
3.62/1.63	                    "exprvars": []
3.62/1.63	                }
3.62/1.63	            },
3.62/1.63	            "53": {
3.62/1.63	                "goal": [
3.62/1.63	                    {
3.62/1.63	                        "clause": 2,
3.62/1.63	                        "scope": 3,
3.62/1.63	                        "term": "(lte X32 (s (s (s (0)))))"
3.62/1.63	                    },
3.62/1.63	                    {
3.62/1.63	                        "clause": 3,
3.62/1.63	                        "scope": 3,
3.62/1.63	                        "term": "(lte X32 (s (s (s (0)))))"
3.62/1.63	                    }
3.62/1.63	                ],
3.62/1.63	                "kb": {
3.62/1.63	                    "nonunifying": [],
3.62/1.63	                    "intvars": {},
3.62/1.63	                    "arithmetic": {
3.62/1.63	                        "type": "PlainIntegerRelationState",
3.62/1.63	                        "relations": []
3.62/1.63	                    },
3.62/1.63	                    "ground": [],
3.62/1.63	                    "free": ["X32"],
3.62/1.63	                    "exprvars": []
3.62/1.63	                }
3.62/1.63	            },
3.62/1.63	            "97": {
3.62/1.63	                "goal": [],
3.62/1.63	                "kb": {
3.62/1.63	                    "nonunifying": [],
3.62/1.63	                    "intvars": {},
3.62/1.63	                    "arithmetic": {
3.62/1.63	                        "type": "PlainIntegerRelationState",
3.62/1.63	                        "relations": []
3.62/1.63	                    },
3.62/1.63	                    "ground": [],
3.62/1.63	                    "free": [],
3.62/1.63	                    "exprvars": []
3.62/1.63	                }
3.62/1.63	            },
3.62/1.63	            "54": {
3.62/1.63	                "goal": [{
3.62/1.63	                    "clause": 2,
3.62/1.63	                    "scope": 3,
3.62/1.63	                    "term": "(lte X32 (s (s (s (0)))))"
3.62/1.63	                }],
3.62/1.63	                "kb": {
3.62/1.63	                    "nonunifying": [],
3.62/1.63	                    "intvars": {},
3.62/1.63	                    "arithmetic": {
3.62/1.63	                        "type": "PlainIntegerRelationState",
3.62/1.63	                        "relations": []
3.62/1.63	                    },
3.62/1.63	                    "ground": [],
3.62/1.63	                    "free": ["X32"],
3.62/1.63	                    "exprvars": []
3.62/1.63	                }
3.62/1.63	            },
3.62/1.63	            "98": {
3.62/1.63	                "goal": [
3.62/1.63	                    {
3.62/1.63	                        "clause": 0,
3.62/1.63	                        "scope": 7,
3.62/1.63	                        "term": "(even T1)"
3.62/1.63	                    },
3.62/1.63	                    {
3.62/1.63	                        "clause": 1,
3.62/1.63	                        "scope": 7,
3.62/1.63	                        "term": "(even T1)"
3.62/1.63	                    }
3.62/1.63	                ],
3.62/1.63	                "kb": {
3.62/1.63	                    "nonunifying": [],
3.62/1.63	                    "intvars": {},
3.62/1.63	                    "arithmetic": {
3.62/1.63	                        "type": "PlainIntegerRelationState",
3.62/1.63	                        "relations": []
3.62/1.63	                    },
3.62/1.63	                    "ground": ["T1"],
3.62/1.63	                    "free": [],
3.62/1.63	                    "exprvars": []
3.62/1.63	                }
3.62/1.63	            },
3.62/1.63	            "55": {
3.62/1.63	                "goal": [{
3.62/1.63	                    "clause": 3,
3.62/1.63	                    "scope": 3,
3.62/1.63	                    "term": "(lte X32 (s (s (s (0)))))"
3.62/1.63	                }],
3.62/1.63	                "kb": {
3.62/1.63	                    "nonunifying": [],
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3.62/1.63	                        "type": "PlainIntegerRelationState",
3.62/1.63	                        "relations": []
3.62/1.63	                    },
3.62/1.63	                    "ground": [],
3.62/1.63	                    "free": ["X32"],
3.62/1.63	                    "exprvars": []
3.62/1.63	                }
3.62/1.63	            },
3.62/1.63	            "99": {
3.62/1.63	                "goal": [{
3.62/1.63	                    "clause": 0,
3.62/1.63	                    "scope": 7,
3.62/1.63	                    "term": "(even T1)"
3.62/1.63	                }],
3.62/1.63	                "kb": {
3.62/1.63	                    "nonunifying": [],
3.62/1.63	                    "intvars": {},
3.62/1.63	                    "arithmetic": {
3.62/1.63	                        "type": "PlainIntegerRelationState",
3.62/1.63	                        "relations": []
3.62/1.63	                    },
3.62/1.63	                    "ground": ["T1"],
3.62/1.63	                    "free": [],
3.62/1.63	                    "exprvars": []
3.62/1.63	                }
3.62/1.63	            },
3.62/1.63	            "78": {
3.62/1.63	                "goal": [{
3.62/1.63	                    "clause": -1,
3.62/1.63	                    "scope": -1,
3.62/1.63	                    "term": "(lte X104 (0))"
3.62/1.63	                }],
3.62/1.63	                "kb": {
3.62/1.63	                    "nonunifying": [],
3.62/1.63	                    "intvars": {},
3.62/1.63	                    "arithmetic": {
3.62/1.63	                        "type": "PlainIntegerRelationState",
3.62/1.63	                        "relations": []
3.62/1.63	                    },
3.62/1.63	                    "ground": [],
3.62/1.63	                    "free": ["X104"],
3.62/1.63	                    "exprvars": []
3.62/1.63	                }
3.62/1.63	            },
3.62/1.63	            "58": {
3.62/1.63	                "goal": [{
3.62/1.63	                    "clause": -1,
3.62/1.63	                    "scope": -1,
3.62/1.63	                    "term": "(lte X56 (s (s (0))))"
3.62/1.63	                }],
3.62/1.63	                "kb": {
3.62/1.63	                    "nonunifying": [],
3.62/1.63	                    "intvars": {},
3.62/1.63	                    "arithmetic": {
3.62/1.63	                        "type": "PlainIntegerRelationState",
3.62/1.63	                        "relations": []
3.62/1.63	                    },
3.62/1.63	                    "ground": [],
3.62/1.63	                    "free": ["X56"],
3.62/1.63	                    "exprvars": []
3.62/1.63	                }
3.62/1.63	            },
3.62/1.63	            "38": {
3.62/1.63	                "goal": [{
3.62/1.63	                    "clause": -1,
3.62/1.63	                    "scope": -1,
3.62/1.63	                    "term": "(lte X5 (s (s (s (s (0))))))"
3.62/1.64	                }],
3.62/1.64	                "kb": {
3.62/1.64	                    "nonunifying": [],
3.62/1.64	                    "intvars": {},
3.62/1.64	                    "arithmetic": {
3.62/1.64	                        "type": "PlainIntegerRelationState",
3.62/1.64	                        "relations": []
3.62/1.64	                    },
3.62/1.64	                    "ground": [],
3.62/1.64	                    "free": ["X5"],
3.62/1.64	                    "exprvars": []
3.62/1.64	                }
3.62/1.64	            },
3.62/1.64	            "39": {
3.62/1.64	                "goal": [{
3.62/1.64	                    "clause": -1,
3.62/1.64	                    "scope": -1,
3.62/1.64	                    "term": "(even T1)"
3.62/1.64	                }],
3.62/1.64	                "kb": {
3.62/1.64	                    "nonunifying": [],
3.62/1.64	                    "intvars": {},
3.62/1.64	                    "arithmetic": {
3.62/1.64	                        "type": "PlainIntegerRelationState",
3.62/1.64	                        "relations": []
3.62/1.64	                    },
3.62/1.64	                    "ground": ["T1"],
3.62/1.64	                    "free": [],
3.62/1.64	                    "exprvars": []
3.62/1.64	                }
3.62/1.64	            },
3.62/1.64	            "1": {
3.62/1.64	                "goal": [{
3.62/1.64	                    "clause": -1,
3.62/1.64	                    "scope": -1,
3.62/1.64	                    "term": "(goal)"
3.62/1.64	                }],
3.62/1.64	                "kb": {
3.62/1.64	                    "nonunifying": [],
3.62/1.64	                    "intvars": {},
3.62/1.64	                    "arithmetic": {
3.62/1.64	                        "type": "PlainIntegerRelationState",
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3.62/1.64	                "from": 101,
3.62/1.64	                "to": 39,
3.62/1.64	                "label": "INSTANCE with matching:\nT1 -> T7"
3.62/1.64	            },
3.62/1.64	            {
3.62/1.64	                "from": 103,
3.62/1.64	                "to": 105,
3.62/1.64	                "label": "SUCCESS"
3.62/1.64	            }
3.62/1.64	        ],
3.62/1.64	        "type": "Graph"
3.62/1.64	    }
3.62/1.64	}
3.62/1.64	
3.62/1.64	----------------------------------------
3.62/1.64	
3.62/1.64	(2)
3.62/1.64	Obligation:
3.62/1.64	Rules:
3.62/1.64	f98_in(T1) -> f100_in(T1) :|: TRUE
3.62/1.64	f98_in(x) -> f99_in(x) :|: TRUE
3.62/1.64	f100_out(x1) -> f98_out(x1) :|: TRUE
3.62/1.64	f99_out(x2) -> f98_out(x2) :|: TRUE
3.62/1.64	f101_in(T7) -> f39_in(T7) :|: TRUE
3.62/1.64	f39_out(x3) -> f101_out(x3) :|: TRUE
3.62/1.64	f101_out(x4) -> f99_out(s(s(x4))) :|: TRUE
3.62/1.64	f102_out -> f99_out(x5) :|: TRUE
3.62/1.64	f99_in(s(s(x6))) -> f101_in(x6) :|: TRUE
3.62/1.64	f99_in(x7) -> f102_in :|: TRUE
3.62/1.64	f98_out(x8) -> f39_out(x8) :|: TRUE
3.62/1.64	f39_in(x9) -> f98_in(x9) :|: TRUE
3.62/1.64	f1_in -> f8_in :|: TRUE
3.62/1.64	f8_out -> f1_out :|: TRUE
3.62/1.64	f8_in -> f9_in :|: TRUE
3.62/1.64	f9_out -> f8_out :|: TRUE
3.62/1.64	f38_out -> f39_in(x10) :|: TRUE
3.62/1.64	f9_in -> f38_in :|: TRUE
3.62/1.64	f39_out(x11) -> f9_out :|: TRUE
3.62/1.64	Start term: f1_in
3.62/1.64	
3.62/1.64	----------------------------------------
3.62/1.64	
3.62/1.64	(3) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
3.62/1.64	Constructed simple dependency graph.
3.62/1.64	
3.62/1.64	Simplified to the following IRSwTs:
3.62/1.64	
3.62/1.64	
3.62/1.64	----------------------------------------
3.62/1.64	
3.62/1.64	(4)
3.62/1.64	TRUE
3.62/1.69	EOF