3.51/1.73	YES
3.51/1.75	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
3.51/1.75	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.51/1.75	
3.51/1.75	
3.51/1.75	Left Termination of the query pattern
3.51/1.75	
3.51/1.75	p(g)
3.51/1.75	
3.51/1.75	w.r.t. the given Prolog program could successfully be proven:
3.51/1.75	
3.51/1.75	(0) Prolog
3.51/1.75	(1) PrologToDTProblemTransformerProof [SOUND, 0 ms]
3.51/1.75	(2) TRIPLES
3.51/1.75	(3) TriplesToPiDPProof [SOUND, 8 ms]
3.51/1.75	(4) PiDP
3.51/1.75	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
3.51/1.75	(6) PiDP
3.51/1.75	(7) PiDPToQDPProof [SOUND, 0 ms]
3.51/1.75	(8) QDP
3.51/1.75	(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.51/1.75	(10) YES
3.51/1.75	
3.51/1.75	
3.51/1.75	----------------------------------------
3.51/1.75	
3.51/1.75	(0)
3.51/1.75	Obligation:
3.51/1.75	Clauses:
3.51/1.75	
3.51/1.75	p(0).
3.51/1.75	p(s(X)) :- ','(geq(X, Y), p(Y)).
3.51/1.75	geq(X, X).
3.51/1.75	geq(s(X), Y) :- geq(X, Y).
3.51/1.75	
3.51/1.75	
3.51/1.75	Query: p(g)
3.51/1.75	----------------------------------------
3.51/1.75	
3.51/1.75	(1) PrologToDTProblemTransformerProof (SOUND)
3.51/1.75	Built DT problem from termination graph DT10.
3.51/1.75	
3.51/1.75	{
3.51/1.75	    "root": 5,
3.51/1.75	    "program": {
3.51/1.75	        "directives": [],
3.51/1.75	        "clauses": [
3.51/1.75	            [
3.51/1.75	                "(p (0))",
3.51/1.75	                null
3.51/1.75	            ],
3.51/1.75	            [
3.51/1.75	                "(p (s X))",
3.51/1.75	                "(',' (geq X Y) (p Y))"
3.51/1.75	            ],
3.51/1.75	            [
3.51/1.75	                "(geq X X)",
3.51/1.75	                null
3.51/1.75	            ],
3.51/1.75	            [
3.51/1.75	                "(geq (s X) Y)",
3.51/1.75	                "(geq X Y)"
3.51/1.75	            ]
3.51/1.75	        ]
3.51/1.75	    },
3.51/1.75	    "graph": {
3.51/1.75	        "nodes": {
3.51/1.75	            "160": {
3.51/1.75	                "goal": [{
3.51/1.75	                    "clause": -1,
3.51/1.75	                    "scope": -1,
3.51/1.75	                    "term": "(',' (geq T11 X18) (p X18))"
3.51/1.75	                }],
3.51/1.75	                "kb": {
3.51/1.75	                    "nonunifying": [],
3.51/1.75	                    "intvars": {},
3.51/1.75	                    "arithmetic": {
3.51/1.75	                        "type": "PlainIntegerRelationState",
3.51/1.75	                        "relations": []
3.51/1.75	                    },
3.51/1.75	                    "ground": ["T11"],
3.51/1.75	                    "free": ["X18"],
3.51/1.75	                    "exprvars": []
3.51/1.75	                }
3.51/1.75	            },
3.51/1.75	            "type": "Nodes",
3.51/1.75	            "161": {
3.51/1.75	                "goal": [],
3.51/1.75	                "kb": {
3.51/1.75	                    "nonunifying": [],
3.51/1.75	                    "intvars": {},
3.51/1.75	                    "arithmetic": {
3.51/1.75	                        "type": "PlainIntegerRelationState",
3.51/1.75	                        "relations": []
3.51/1.75	                    },
3.51/1.75	                    "ground": [],
3.51/1.75	                    "free": [],
3.51/1.75	                    "exprvars": []
3.51/1.75	                }
3.51/1.75	            },
3.51/1.75	            "140": {
3.51/1.75	                "goal": [{
3.51/1.75	                    "clause": -1,
3.51/1.75	                    "scope": -1,
3.51/1.75	                    "term": "(p T8)"
3.51/1.75	                }],
3.51/1.75	                "kb": {
3.51/1.75	                    "nonunifying": [],
3.51/1.75	                    "intvars": {},
3.51/1.75	                    "arithmetic": {
3.51/1.75	                        "type": "PlainIntegerRelationState",
3.51/1.75	                        "relations": []
3.51/1.75	                    },
3.51/1.75	                    "ground": ["T8"],
3.51/1.75	                    "free": [],
3.51/1.75	                    "exprvars": []
3.51/1.75	                }
3.51/1.75	            },
3.51/1.75	            "122": {
3.51/1.75	                "goal": [
3.51/1.75	                    {
3.51/1.75	                        "clause": -1,
3.51/1.75	                        "scope": -1,
3.51/1.75	                        "term": "(true)"
3.51/1.75	                    },
3.51/1.75	                    {
3.51/1.75	                        "clause": 1,
3.51/1.75	                        "scope": 1,
3.51/1.75	                        "term": "(p (0))"
3.51/1.75	                    }
3.51/1.75	                ],
3.51/1.75	                "kb": {
3.51/1.75	                    "nonunifying": [],
3.51/1.75	                    "intvars": {},
3.51/1.75	                    "arithmetic": {
3.51/1.75	                        "type": "PlainIntegerRelationState",
3.51/1.75	                        "relations": []
3.51/1.75	                    },
3.51/1.75	                    "ground": [],
3.51/1.75	                    "free": [],
3.51/1.75	                    "exprvars": []
3.51/1.75	                }
3.51/1.75	            },
3.51/1.75	            "123": {
3.51/1.75	                "goal": [{
3.51/1.75	                    "clause": 1,
3.51/1.75	                    "scope": 1,
3.51/1.75	                    "term": "(p T1)"
3.51/1.75	                }],
3.51/1.75	                "kb": {
3.51/1.75	                    "nonunifying": [[
3.51/1.75	                        "(p T1)",
3.51/1.75	                        "(p (0))"
3.51/1.75	                    ]],
3.51/1.75	                    "intvars": {},
3.51/1.75	                    "arithmetic": {
3.51/1.75	                        "type": "PlainIntegerRelationState",
3.51/1.75	                        "relations": []
3.51/1.75	                    },
3.51/1.75	                    "ground": ["T1"],
3.51/1.75	                    "free": [],
3.51/1.75	                    "exprvars": []
3.51/1.75	                }
3.51/1.75	            },
3.51/1.75	            "124": {
3.51/1.75	                "goal": [{
3.51/1.75	                    "clause": 1,
3.51/1.75	                    "scope": 1,
3.51/1.75	                    "term": "(p (0))"
3.51/1.75	                }],
3.51/1.75	                "kb": {
3.51/1.75	                    "nonunifying": [],
3.51/1.75	                    "intvars": {},
3.51/1.75	                    "arithmetic": {
3.51/1.75	                        "type": "PlainIntegerRelationState",
3.51/1.75	                        "relations": []
3.51/1.75	                    },
3.51/1.75	                    "ground": [],
3.51/1.75	                    "free": [],
3.51/1.75	                    "exprvars": []
3.51/1.75	                }
3.51/1.75	            },
3.51/1.75	            "136": {
3.51/1.75	                "goal": [],
3.51/1.75	                "kb": {
3.51/1.75	                    "nonunifying": [],
3.51/1.75	                    "intvars": {},
3.51/1.75	                    "arithmetic": {
3.51/1.75	                        "type": "PlainIntegerRelationState",
3.51/1.75	                        "relations": []
3.51/1.75	                    },
3.51/1.75	                    "ground": [],
3.51/1.75	                    "free": [],
3.51/1.75	                    "exprvars": []
3.51/1.75	                }
3.51/1.75	            },
3.51/1.75	            "5": {
3.51/1.75	                "goal": [{
3.51/1.75	                    "clause": -1,
3.51/1.75	                    "scope": -1,
3.51/1.75	                    "term": "(p T1)"
3.51/1.75	                }],
3.51/1.75	                "kb": {
3.51/1.75	                    "nonunifying": [],
3.51/1.75	                    "intvars": {},
3.51/1.75	                    "arithmetic": {
3.51/1.75	                        "type": "PlainIntegerRelationState",
3.51/1.75	                        "relations": []
3.51/1.75	                    },
3.51/1.75	                    "ground": ["T1"],
3.51/1.75	                    "free": [],
3.51/1.75	                    "exprvars": []
3.51/1.75	                }
3.51/1.75	            },
3.51/1.75	            "126": {
3.51/1.75	                "goal": [],
3.51/1.75	                "kb": {
3.51/1.75	                    "nonunifying": [],
3.51/1.75	                    "intvars": {},
3.51/1.75	                    "arithmetic": {
3.51/1.75	                        "type": "PlainIntegerRelationState",
3.51/1.75	                        "relations": []
3.51/1.75	                    },
3.51/1.75	                    "ground": [],
3.51/1.75	                    "free": [],
3.51/1.75	                    "exprvars": []
3.51/1.75	                }
3.51/1.75	            },
3.51/1.75	            "137": {
3.51/1.75	                "goal": [
3.51/1.75	                    {
3.51/1.75	                        "clause": 2,
3.51/1.75	                        "scope": 2,
3.51/1.75	                        "term": "(',' (geq T3 X4) (p X4))"
3.51/1.75	                    },
3.51/1.75	                    {
3.51/1.75	                        "clause": 3,
3.51/1.75	                        "scope": 2,
3.51/1.75	                        "term": "(',' (geq T3 X4) (p X4))"
3.51/1.75	                    }
3.51/1.75	                ],
3.51/1.75	                "kb": {
3.51/1.75	                    "nonunifying": [],
3.51/1.75	                    "intvars": {},
3.51/1.75	                    "arithmetic": {
3.51/1.75	                        "type": "PlainIntegerRelationState",
3.51/1.75	                        "relations": []
3.51/1.75	                    },
3.51/1.75	                    "ground": ["T3"],
3.51/1.75	                    "free": ["X4"],
3.51/1.75	                    "exprvars": []
3.51/1.75	                }
3.51/1.75	            },
3.51/1.75	            "6": {
3.51/1.75	                "goal": [
3.51/1.75	                    {
3.51/1.75	                        "clause": 0,
3.51/1.75	                        "scope": 1,
3.51/1.75	                        "term": "(p T1)"
3.51/1.75	                    },
3.51/1.75	                    {
3.51/1.75	                        "clause": 1,
3.51/1.75	                        "scope": 1,
3.51/1.75	                        "term": "(p T1)"
3.51/1.75	                    }
3.51/1.75	                ],
3.51/1.75	                "kb": {
3.51/1.75	                    "nonunifying": [],
3.51/1.75	                    "intvars": {},
3.51/1.75	                    "arithmetic": {
3.51/1.75	                        "type": "PlainIntegerRelationState",
3.51/1.75	                        "relations": []
3.51/1.75	                    },
3.51/1.75	                    "ground": ["T1"],
3.51/1.75	                    "free": [],
3.51/1.75	                    "exprvars": []
3.51/1.75	                }
3.51/1.75	            },
3.51/1.75	            "138": {
3.51/1.75	                "goal": [{
3.51/1.75	                    "clause": 2,
3.51/1.75	                    "scope": 2,
3.51/1.75	                    "term": "(',' (geq T3 X4) (p X4))"
3.51/1.75	                }],
3.51/1.75	                "kb": {
3.51/1.75	                    "nonunifying": [],
3.51/1.75	                    "intvars": {},
3.51/1.75	                    "arithmetic": {
3.51/1.75	                        "type": "PlainIntegerRelationState",
3.51/1.75	                        "relations": []
3.51/1.75	                    },
3.51/1.75	                    "ground": ["T3"],
3.51/1.75	                    "free": ["X4"],
3.51/1.75	                    "exprvars": []
3.51/1.75	                }
3.51/1.75	            },
3.51/1.75	            "128": {
3.51/1.75	                "goal": [{
3.51/1.75	                    "clause": -1,
3.51/1.75	                    "scope": -1,
3.51/1.75	                    "term": "(',' (geq T3 X4) (p X4))"
3.51/1.75	                }],
3.51/1.75	                "kb": {
3.51/1.75	                    "nonunifying": [],
3.51/1.75	                    "intvars": {},
3.51/1.75	                    "arithmetic": {
3.51/1.75	                        "type": "PlainIntegerRelationState",
3.51/1.75	                        "relations": []
3.51/1.75	                    },
3.51/1.75	                    "ground": ["T3"],
3.51/1.75	                    "free": ["X4"],
3.51/1.75	                    "exprvars": []
3.51/1.75	                }
3.51/1.75	            },
3.51/1.75	            "139": {
3.51/1.75	                "goal": [{
3.51/1.75	                    "clause": 3,
3.51/1.75	                    "scope": 2,
3.51/1.75	                    "term": "(',' (geq T3 X4) (p X4))"
3.51/1.75	                }],
3.51/1.75	                "kb": {
3.51/1.75	                    "nonunifying": [],
3.51/1.75	                    "intvars": {},
3.51/1.75	                    "arithmetic": {
3.51/1.75	                        "type": "PlainIntegerRelationState",
3.51/1.75	                        "relations": []
3.51/1.75	                    },
3.51/1.75	                    "ground": ["T3"],
3.51/1.75	                    "free": ["X4"],
3.51/1.75	                    "exprvars": []
3.51/1.75	                }
3.51/1.75	            }
3.51/1.75	        },
3.51/1.75	        "edges": [
3.51/1.75	            {
3.51/1.75	                "from": 5,
3.51/1.75	                "to": 6,
3.51/1.75	                "label": "CASE"
3.51/1.75	            },
3.51/1.75	            {
3.51/1.75	                "from": 6,
3.51/1.75	                "to": 122,
3.51/1.75	                "label": "EVAL with clause\np(0).\nand substitutionT1 -> 0"
3.51/1.75	            },
3.51/1.75	            {
3.51/1.75	                "from": 6,
3.51/1.75	                "to": 123,
3.51/1.75	                "label": "EVAL-BACKTRACK"
3.51/1.75	            },
3.51/1.75	            {
3.51/1.75	                "from": 122,
3.51/1.75	                "to": 124,
3.51/1.75	                "label": "SUCCESS"
3.51/1.75	            },
3.51/1.75	            {
3.51/1.75	                "from": 123,
3.51/1.75	                "to": 128,
3.51/1.75	                "label": "EVAL with clause\np(s(X3)) :- ','(geq(X3, X4), p(X4)).\nand substitutionX3 -> T3,\nT1 -> s(T3)"
3.51/1.75	            },
3.51/1.75	            {
3.51/1.75	                "from": 123,
3.51/1.75	                "to": 136,
3.51/1.75	                "label": "EVAL-BACKTRACK"
3.51/1.75	            },
3.51/1.75	            {
3.51/1.75	                "from": 124,
3.51/1.75	                "to": 126,
3.51/1.75	                "label": "BACKTRACK\nfor clause: p(s(X)) :- ','(geq(X, Y), p(Y))because of non-unification"
3.51/1.75	            },
3.51/1.75	            {
3.51/1.75	                "from": 128,
3.51/1.75	                "to": 137,
3.51/1.75	                "label": "CASE"
3.51/1.75	            },
3.51/1.75	            {
3.51/1.75	                "from": 137,
3.51/1.75	                "to": 138,
3.51/1.75	                "label": "PARALLEL"
3.51/1.75	            },
3.51/1.75	            {
3.51/1.75	                "from": 137,
3.51/1.75	                "to": 139,
3.51/1.75	                "label": "PARALLEL"
3.51/1.75	            },
3.51/1.75	            {
3.51/1.75	                "from": 138,
3.51/1.75	                "to": 140,
3.51/1.75	                "label": "ONLY EVAL with clause\ngeq(X9, X9).\nand substitutionT3 -> T8,\nX9 -> T8,\nX4 -> T8"
3.51/1.75	            },
3.51/1.75	            {
3.51/1.75	                "from": 139,
3.51/1.75	                "to": 160,
3.51/1.75	                "label": "EVAL with clause\ngeq(s(X16), X17) :- geq(X16, X17).\nand substitutionX16 -> T11,\nT3 -> s(T11),\nX4 -> X18,\nX17 -> X18"
3.51/1.75	            },
3.51/1.75	            {
3.51/1.75	                "from": 139,
3.51/1.75	                "to": 161,
3.51/1.75	                "label": "EVAL-BACKTRACK"
3.51/1.75	            },
3.51/1.75	            {
3.51/1.75	                "from": 140,
3.51/1.75	                "to": 5,
3.51/1.75	                "label": "INSTANCE with matching:\nT1 -> T8"
3.51/1.75	            },
3.51/1.75	            {
3.51/1.75	                "from": 160,
3.51/1.75	                "to": 128,
3.51/1.75	                "label": "INSTANCE with matching:\nT3 -> T11\nX4 -> X18"
3.51/1.75	            }
3.51/1.75	        ],
3.51/1.75	        "type": "Graph"
3.51/1.75	    }
3.51/1.75	}
3.51/1.75	
3.51/1.75	----------------------------------------
3.51/1.75	
3.51/1.75	(2)
3.51/1.75	Obligation:
3.51/1.75	Triples:
3.51/1.75	
3.51/1.75	pB(X1, X1) :- pA(X1).
3.51/1.75	pB(s(X1), X2) :- pB(X1, X2).
3.51/1.75	pA(s(X1)) :- pB(X1, X2).
3.51/1.75	
3.51/1.75	Clauses:
3.51/1.75	
3.51/1.75	pcA(0).
3.51/1.75	pcA(s(X1)) :- qcB(X1, X2).
3.51/1.75	qcB(X1, X1) :- pcA(X1).
3.51/1.75	qcB(s(X1), X2) :- qcB(X1, X2).
3.51/1.75	
3.51/1.75	Afs:
3.51/1.75	
3.51/1.75	pA(x1)  =  pA(x1)
3.51/1.75	
3.51/1.75	
3.51/1.75	----------------------------------------
3.51/1.75	
3.51/1.75	(3) TriplesToPiDPProof (SOUND)
3.51/1.75	We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
3.51/1.75	
3.51/1.75	pA_in_1: (b)
3.51/1.75	
3.51/1.75	pB_in_2: (b,f)
3.51/1.75	
3.51/1.75	Transforming TRIPLES into the following Term Rewriting System:
3.51/1.75	
3.51/1.75	Pi DP problem:
3.51/1.75	The TRS P consists of the following rules:
3.51/1.75	
3.51/1.75	   PA_IN_G(s(X1)) -> U3_G(X1, pB_in_ga(X1, X2))
3.51/1.75	   PA_IN_G(s(X1)) -> PB_IN_GA(X1, X2)
3.51/1.75	   PB_IN_GA(X1, X1) -> U1_GA(X1, pA_in_g(X1))
3.51/1.75	   PB_IN_GA(X1, X1) -> PA_IN_G(X1)
3.51/1.75	   PB_IN_GA(s(X1), X2) -> U2_GA(X1, X2, pB_in_ga(X1, X2))
3.51/1.75	   PB_IN_GA(s(X1), X2) -> PB_IN_GA(X1, X2)
3.51/1.75	
3.51/1.75	R is empty.
3.51/1.75	The argument filtering Pi contains the following mapping:
3.51/1.75	pA_in_g(x1)  =  pA_in_g(x1)
3.51/1.75	
3.51/1.75	s(x1)  =  s(x1)
3.51/1.75	
3.51/1.75	pB_in_ga(x1, x2)  =  pB_in_ga(x1)
3.51/1.75	
3.51/1.75	PA_IN_G(x1)  =  PA_IN_G(x1)
3.51/1.75	
3.51/1.75	U3_G(x1, x2)  =  U3_G(x1, x2)
3.51/1.75	
3.51/1.75	PB_IN_GA(x1, x2)  =  PB_IN_GA(x1)
3.51/1.75	
3.51/1.75	U1_GA(x1, x2)  =  U1_GA(x1, x2)
3.51/1.75	
3.51/1.75	U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
3.51/1.75	
3.51/1.75	
3.51/1.75	We have to consider all (P,R,Pi)-chains
3.51/1.75	
3.51/1.75	
3.51/1.75	Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES
3.51/1.75	
3.51/1.75	
3.51/1.75	
3.51/1.75	----------------------------------------
3.51/1.75	
3.51/1.75	(4)
3.51/1.75	Obligation:
3.51/1.75	Pi DP problem:
3.51/1.75	The TRS P consists of the following rules:
3.51/1.75	
3.51/1.75	   PA_IN_G(s(X1)) -> U3_G(X1, pB_in_ga(X1, X2))
3.51/1.75	   PA_IN_G(s(X1)) -> PB_IN_GA(X1, X2)
3.51/1.75	   PB_IN_GA(X1, X1) -> U1_GA(X1, pA_in_g(X1))
3.51/1.75	   PB_IN_GA(X1, X1) -> PA_IN_G(X1)
3.51/1.75	   PB_IN_GA(s(X1), X2) -> U2_GA(X1, X2, pB_in_ga(X1, X2))
3.51/1.75	   PB_IN_GA(s(X1), X2) -> PB_IN_GA(X1, X2)
3.51/1.75	
3.51/1.75	R is empty.
3.51/1.75	The argument filtering Pi contains the following mapping:
3.51/1.75	pA_in_g(x1)  =  pA_in_g(x1)
3.51/1.75	
3.51/1.75	s(x1)  =  s(x1)
3.51/1.75	
3.51/1.75	pB_in_ga(x1, x2)  =  pB_in_ga(x1)
3.51/1.75	
3.51/1.75	PA_IN_G(x1)  =  PA_IN_G(x1)
3.51/1.75	
3.51/1.75	U3_G(x1, x2)  =  U3_G(x1, x2)
3.51/1.75	
3.51/1.75	PB_IN_GA(x1, x2)  =  PB_IN_GA(x1)
3.51/1.75	
3.51/1.75	U1_GA(x1, x2)  =  U1_GA(x1, x2)
3.51/1.75	
3.51/1.75	U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
3.51/1.75	
3.51/1.75	
3.51/1.75	We have to consider all (P,R,Pi)-chains
3.51/1.75	----------------------------------------
3.51/1.75	
3.51/1.75	(5) DependencyGraphProof (EQUIVALENT)
3.51/1.75	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.
3.51/1.75	----------------------------------------
3.51/1.76	
3.51/1.76	(6)
3.51/1.76	Obligation:
3.51/1.76	Pi DP problem:
3.51/1.76	The TRS P consists of the following rules:
3.51/1.76	
3.51/1.76	   PA_IN_G(s(X1)) -> PB_IN_GA(X1, X2)
3.51/1.76	   PB_IN_GA(X1, X1) -> PA_IN_G(X1)
3.51/1.76	   PB_IN_GA(s(X1), X2) -> PB_IN_GA(X1, X2)
3.51/1.76	
3.51/1.76	R is empty.
3.51/1.76	The argument filtering Pi contains the following mapping:
3.51/1.76	s(x1)  =  s(x1)
3.51/1.76	
3.51/1.76	PA_IN_G(x1)  =  PA_IN_G(x1)
3.51/1.76	
3.51/1.76	PB_IN_GA(x1, x2)  =  PB_IN_GA(x1)
3.51/1.76	
3.51/1.76	
3.51/1.76	We have to consider all (P,R,Pi)-chains
3.51/1.76	----------------------------------------
3.51/1.76	
3.51/1.76	(7) PiDPToQDPProof (SOUND)
3.51/1.76	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
3.51/1.76	----------------------------------------
3.51/1.76	
3.51/1.76	(8)
3.51/1.76	Obligation:
3.51/1.76	Q DP problem:
3.51/1.76	The TRS P consists of the following rules:
3.51/1.76	
3.51/1.76	   PA_IN_G(s(X1)) -> PB_IN_GA(X1)
3.51/1.76	   PB_IN_GA(X1) -> PA_IN_G(X1)
3.51/1.76	   PB_IN_GA(s(X1)) -> PB_IN_GA(X1)
3.51/1.76	
3.51/1.76	R is empty.
3.51/1.76	Q is empty.
3.51/1.76	We have to consider all (P,Q,R)-chains.
3.51/1.76	----------------------------------------
3.51/1.76	
3.51/1.76	(9) QDPSizeChangeProof (EQUIVALENT)
3.51/1.76	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. 
3.51/1.76	
3.51/1.76	From the DPs we obtained the following set of size-change graphs:
3.51/1.76	*PB_IN_GA(X1) -> PA_IN_G(X1)
3.51/1.76	The graph contains the following edges 1 >= 1
3.51/1.76	
3.51/1.76	
3.51/1.76	*PB_IN_GA(s(X1)) -> PB_IN_GA(X1)
3.51/1.76	The graph contains the following edges 1 > 1
3.51/1.76	
3.51/1.76	
3.51/1.76	*PA_IN_G(s(X1)) -> PB_IN_GA(X1)
3.51/1.76	The graph contains the following edges 1 > 1
3.51/1.76	
3.51/1.76	
3.51/1.76	----------------------------------------
3.51/1.76	
3.51/1.76	(10)
3.51/1.76	YES
3.51/1.78	EOF