3.80/1.78	YES
3.80/1.80	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
3.80/1.80	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.80/1.80	
3.80/1.80	
3.80/1.80	Left Termination of the query pattern
3.80/1.80	
3.80/1.80	app3_b(g,g,g,a)
3.80/1.80	
3.80/1.80	w.r.t. the given Prolog program could successfully be proven:
3.80/1.80	
3.80/1.80	(0) Prolog
3.80/1.80	(1) PrologToDTProblemTransformerProof [SOUND, 0 ms]
3.80/1.80	(2) TRIPLES
3.80/1.80	(3) TriplesToPiDPProof [SOUND, 0 ms]
3.80/1.80	(4) PiDP
3.80/1.80	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
3.80/1.80	(6) PiDP
3.80/1.80	(7) UsableRulesProof [EQUIVALENT, 0 ms]
3.80/1.80	(8) PiDP
3.80/1.80	(9) PiDPToQDPProof [SOUND, 0 ms]
3.80/1.80	(10) QDP
3.80/1.80	(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.80/1.80	(12) YES
3.80/1.80	
3.80/1.80	
3.80/1.80	----------------------------------------
3.80/1.80	
3.80/1.80	(0)
3.80/1.80	Obligation:
3.80/1.80	Clauses:
3.80/1.80	
3.80/1.80	app3_a(Xs, Ys, Zs, Us) :- ','(app(Xs, Ys, Vs), app(Vs, Zs, Us)).
3.80/1.80	app3_b(Xs, Ys, Zs, Us) :- ','(app(Ys, Zs, Vs), app(Xs, Vs, Us)).
3.80/1.80	app([], Ys, Ys).
3.80/1.80	app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
3.80/1.80	
3.80/1.80	
3.80/1.80	Query: app3_b(g,g,g,a)
3.80/1.80	----------------------------------------
3.80/1.80	
3.80/1.80	(1) PrologToDTProblemTransformerProof (SOUND)
3.80/1.80	Built DT problem from termination graph DT10.
3.80/1.80	
3.80/1.80	{
3.80/1.80	    "root": 1,
3.80/1.80	    "program": {
3.80/1.80	        "directives": [],
3.80/1.80	        "clauses": [
3.80/1.80	            [
3.80/1.80	                "(app3_a Xs Ys Zs Us)",
3.80/1.80	                "(',' (app Xs Ys Vs) (app Vs Zs Us))"
3.80/1.80	            ],
3.80/1.80	            [
3.80/1.80	                "(app3_b Xs Ys Zs Us)",
3.80/1.80	                "(',' (app Ys Zs Vs) (app Xs Vs Us))"
3.80/1.80	            ],
3.80/1.80	            [
3.80/1.80	                "(app ([]) Ys Ys)",
3.80/1.80	                null
3.80/1.80	            ],
3.80/1.80	            [
3.80/1.80	                "(app (. X Xs) Ys (. X Zs))",
3.80/1.80	                "(app Xs Ys Zs)"
3.80/1.80	            ]
3.80/1.80	        ]
3.80/1.80	    },
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3.80/1.80	                }],
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3.80/1.80	                    },
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3.80/1.80	                        "T9",
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3.80/1.80	                    ],
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3.80/1.80	                }
3.80/1.80	            },
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3.80/1.80	                }],
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3.80/1.80	                    },
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3.80/1.80	                    ],
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3.80/1.80	                "from": 160,
3.80/1.80	                "to": 219,
3.80/1.80	                "label": "EVAL with clause\napp([], X14, X14).\nand substitutionT10 -> [],\nT11 -> T18,\nX14 -> T18,\nX9 -> T18"
3.80/1.80	            },
3.80/1.80	            {
3.80/1.80	                "from": 160,
3.80/1.80	                "to": 220,
3.80/1.80	                "label": "EVAL-BACKTRACK"
3.80/1.80	            },
3.80/1.80	            {
3.80/1.80	                "from": 162,
3.80/1.80	                "to": 261,
3.80/1.80	                "label": "EVAL with clause\napp(.(X46, X47), X48, .(X46, X49)) :- app(X47, X48, X49).\nand substitutionX46 -> T47,\nX47 -> T48,\nT10 -> .(T47, T48),\nT11 -> T49,\nX48 -> T49,\nX49 -> X50,\nX9 -> .(T47, X50)"
3.80/1.80	            },
3.80/1.80	            {
3.80/1.80	                "from": 162,
3.80/1.80	                "to": 263,
3.80/1.80	                "label": "EVAL-BACKTRACK"
3.80/1.80	            },
3.80/1.80	            {
3.80/1.80	                "from": 219,
3.80/1.80	                "to": 223,
3.80/1.80	                "label": "CASE"
3.80/1.80	            },
3.80/1.80	            {
3.80/1.80	                "from": 223,
3.80/1.80	                "to": 224,
3.80/1.80	                "label": "PARALLEL"
3.80/1.80	            },
3.80/1.80	            {
3.80/1.80	                "from": 223,
3.80/1.80	                "to": 225,
3.80/1.80	                "label": "PARALLEL"
3.80/1.80	            },
3.80/1.80	            {
3.80/1.80	                "from": 224,
3.80/1.80	                "to": 226,
3.80/1.80	                "label": "EVAL with clause\napp([], X21, X21).\nand substitutionT9 -> [],\nT18 -> T25,\nX21 -> T25,\nT13 -> T25"
3.80/1.80	            },
3.80/1.80	            {
3.80/1.80	                "from": 224,
3.80/1.80	                "to": 227,
3.80/1.80	                "label": "EVAL-BACKTRACK"
3.80/1.80	            },
3.80/1.80	            {
3.80/1.80	                "from": 225,
3.80/1.80	                "to": 233,
3.80/1.80	                "label": "EVAL with clause\napp(.(X30, X31), X32, .(X30, X33)) :- app(X31, X32, X33).\nand substitutionX30 -> T34,\nX31 -> T35,\nT9 -> .(T34, T35),\nT18 -> T36,\nX32 -> T36,\nX33 -> T38,\nT13 -> .(T34, T38),\nT37 -> T38"
3.80/1.80	            },
3.80/1.80	            {
3.80/1.80	                "from": 225,
3.80/1.80	                "to": 236,
3.80/1.80	                "label": "EVAL-BACKTRACK"
3.80/1.80	            },
3.80/1.80	            {
3.80/1.80	                "from": 226,
3.80/1.80	                "to": 228,
3.80/1.80	                "label": "SUCCESS"
3.80/1.80	            },
3.80/1.80	            {
3.80/1.80	                "from": 233,
3.80/1.80	                "to": 219,
3.80/1.80	                "label": "INSTANCE with matching:\nT9 -> T35\nT18 -> T36\nT13 -> T38"
3.80/1.80	            },
3.80/1.80	            {
3.80/1.80	                "from": 261,
3.80/1.80	                "to": 267,
3.80/1.80	                "label": "SPLIT 1"
3.80/1.80	            },
3.80/1.80	            {
3.80/1.80	                "from": 261,
3.80/1.80	                "to": 268,
3.80/1.80	                "label": "SPLIT 2\nnew knowledge:\nT48 is ground\nT49 is ground\nT52 is ground\nreplacements:X50 -> T52"
3.80/1.80	            },
3.80/1.80	            {
3.80/1.80	                "from": 267,
3.80/1.80	                "to": 219,
3.80/1.80	                "label": "INSTANCE with matching:\nT9 -> T48\nT18 -> T49\nT13 -> X50"
3.80/1.80	            },
3.80/1.80	            {
3.80/1.80	                "from": 268,
3.80/1.80	                "to": 219,
3.80/1.80	                "label": "INSTANCE with matching:\nT18 -> .(T47, T52)"
3.80/1.80	            }
3.80/1.80	        ],
3.80/1.80	        "type": "Graph"
3.80/1.80	    }
3.80/1.80	}
3.80/1.80	
3.80/1.80	----------------------------------------
3.80/1.80	
3.80/1.80	(2)
3.80/1.80	Obligation:
3.80/1.80	Triples:
3.80/1.80	
3.80/1.80	appA(.(X1, X2), X3, .(X1, X4)) :- appA(X2, X3, X4).
3.80/1.80	app3_bB(X1, [], X2, X3) :- appA(X1, X2, X3).
3.80/1.80	app3_bB(X1, .(X2, X3), X4, X5) :- appA(X3, X4, X6).
3.80/1.80	app3_bB(X1, .(X2, X3), X4, X5) :- ','(appcA(X3, X4, X6), appA(X1, .(X2, X6), X5)).
3.80/1.80	
3.80/1.80	Clauses:
3.80/1.80	
3.80/1.80	appcA([], X1, X1).
3.80/1.80	appcA(.(X1, X2), X3, .(X1, X4)) :- appcA(X2, X3, X4).
3.80/1.80	
3.80/1.80	Afs:
3.80/1.80	
3.80/1.80	app3_bB(x1, x2, x3, x4)  =  app3_bB(x1, x2, x3)
3.80/1.80	
3.80/1.80	
3.80/1.80	----------------------------------------
3.80/1.80	
3.80/1.80	(3) TriplesToPiDPProof (SOUND)
3.80/1.80	We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
3.80/1.80	
3.80/1.80	app3_bB_in_4: (b,b,b,f)
3.80/1.80	
3.80/1.80	appA_in_3: (b,b,f)
3.80/1.80	
3.80/1.80	appcA_in_3: (b,b,f)
3.80/1.80	
3.80/1.80	Transforming TRIPLES into the following Term Rewriting System:
3.80/1.80	
3.80/1.80	Pi DP problem:
3.80/1.80	The TRS P consists of the following rules:
3.80/1.80	
3.80/1.80	   APP3_BB_IN_GGGA(X1, [], X2, X3) -> U2_GGGA(X1, X2, X3, appA_in_gga(X1, X2, X3))
3.80/1.80	   APP3_BB_IN_GGGA(X1, [], X2, X3) -> APPA_IN_GGA(X1, X2, X3)
3.80/1.80	   APPA_IN_GGA(.(X1, X2), X3, .(X1, X4)) -> U1_GGA(X1, X2, X3, X4, appA_in_gga(X2, X3, X4))
3.80/1.80	   APPA_IN_GGA(.(X1, X2), X3, .(X1, X4)) -> APPA_IN_GGA(X2, X3, X4)
3.80/1.80	   APP3_BB_IN_GGGA(X1, .(X2, X3), X4, X5) -> U3_GGGA(X1, X2, X3, X4, X5, appA_in_gga(X3, X4, X6))
3.80/1.80	   APP3_BB_IN_GGGA(X1, .(X2, X3), X4, X5) -> APPA_IN_GGA(X3, X4, X6)
3.80/1.80	   APP3_BB_IN_GGGA(X1, .(X2, X3), X4, X5) -> U4_GGGA(X1, X2, X3, X4, X5, appcA_in_gga(X3, X4, X6))
3.80/1.80	   U4_GGGA(X1, X2, X3, X4, X5, appcA_out_gga(X3, X4, X6)) -> U5_GGGA(X1, X2, X3, X4, X5, appA_in_gga(X1, .(X2, X6), X5))
3.80/1.80	   U4_GGGA(X1, X2, X3, X4, X5, appcA_out_gga(X3, X4, X6)) -> APPA_IN_GGA(X1, .(X2, X6), X5)
3.80/1.80	
3.80/1.80	The TRS R consists of the following rules:
3.80/1.80	
3.80/1.80	   appcA_in_gga([], X1, X1) -> appcA_out_gga([], X1, X1)
3.80/1.80	   appcA_in_gga(.(X1, X2), X3, .(X1, X4)) -> U7_gga(X1, X2, X3, X4, appcA_in_gga(X2, X3, X4))
3.80/1.80	   U7_gga(X1, X2, X3, X4, appcA_out_gga(X2, X3, X4)) -> appcA_out_gga(.(X1, X2), X3, .(X1, X4))
3.80/1.80	
3.80/1.80	The argument filtering Pi contains the following mapping:
3.80/1.80	[]  =  []
3.80/1.80	
3.80/1.80	appA_in_gga(x1, x2, x3)  =  appA_in_gga(x1, x2)
3.80/1.80	
3.80/1.80	.(x1, x2)  =  .(x1, x2)
3.80/1.80	
3.80/1.80	appcA_in_gga(x1, x2, x3)  =  appcA_in_gga(x1, x2)
3.80/1.80	
3.80/1.80	appcA_out_gga(x1, x2, x3)  =  appcA_out_gga(x1, x2, x3)
3.80/1.80	
3.80/1.80	U7_gga(x1, x2, x3, x4, x5)  =  U7_gga(x1, x2, x3, x5)
3.80/1.80	
3.80/1.80	APP3_BB_IN_GGGA(x1, x2, x3, x4)  =  APP3_BB_IN_GGGA(x1, x2, x3)
3.80/1.80	
3.80/1.80	U2_GGGA(x1, x2, x3, x4)  =  U2_GGGA(x1, x2, x4)
3.80/1.80	
3.80/1.80	APPA_IN_GGA(x1, x2, x3)  =  APPA_IN_GGA(x1, x2)
3.80/1.80	
3.80/1.80	U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x2, x3, x5)
3.80/1.80	
3.80/1.80	U3_GGGA(x1, x2, x3, x4, x5, x6)  =  U3_GGGA(x1, x2, x3, x4, x6)
3.80/1.80	
3.80/1.80	U4_GGGA(x1, x2, x3, x4, x5, x6)  =  U4_GGGA(x1, x2, x3, x4, x6)
3.80/1.80	
3.80/1.80	U5_GGGA(x1, x2, x3, x4, x5, x6)  =  U5_GGGA(x1, x2, x3, x4, x6)
3.80/1.80	
3.80/1.80	
3.80/1.80	We have to consider all (P,R,Pi)-chains
3.80/1.80	
3.80/1.80	
3.80/1.80	Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES
3.80/1.80	
3.80/1.80	
3.80/1.80	
3.80/1.80	----------------------------------------
3.80/1.80	
3.80/1.80	(4)
3.80/1.80	Obligation:
3.80/1.80	Pi DP problem:
3.80/1.80	The TRS P consists of the following rules:
3.80/1.80	
3.80/1.80	   APP3_BB_IN_GGGA(X1, [], X2, X3) -> U2_GGGA(X1, X2, X3, appA_in_gga(X1, X2, X3))
3.80/1.80	   APP3_BB_IN_GGGA(X1, [], X2, X3) -> APPA_IN_GGA(X1, X2, X3)
3.80/1.80	   APPA_IN_GGA(.(X1, X2), X3, .(X1, X4)) -> U1_GGA(X1, X2, X3, X4, appA_in_gga(X2, X3, X4))
3.80/1.80	   APPA_IN_GGA(.(X1, X2), X3, .(X1, X4)) -> APPA_IN_GGA(X2, X3, X4)
3.80/1.80	   APP3_BB_IN_GGGA(X1, .(X2, X3), X4, X5) -> U3_GGGA(X1, X2, X3, X4, X5, appA_in_gga(X3, X4, X6))
3.80/1.80	   APP3_BB_IN_GGGA(X1, .(X2, X3), X4, X5) -> APPA_IN_GGA(X3, X4, X6)
3.80/1.80	   APP3_BB_IN_GGGA(X1, .(X2, X3), X4, X5) -> U4_GGGA(X1, X2, X3, X4, X5, appcA_in_gga(X3, X4, X6))
3.80/1.80	   U4_GGGA(X1, X2, X3, X4, X5, appcA_out_gga(X3, X4, X6)) -> U5_GGGA(X1, X2, X3, X4, X5, appA_in_gga(X1, .(X2, X6), X5))
3.80/1.80	   U4_GGGA(X1, X2, X3, X4, X5, appcA_out_gga(X3, X4, X6)) -> APPA_IN_GGA(X1, .(X2, X6), X5)
3.80/1.80	
3.80/1.80	The TRS R consists of the following rules:
3.80/1.80	
3.80/1.80	   appcA_in_gga([], X1, X1) -> appcA_out_gga([], X1, X1)
3.80/1.80	   appcA_in_gga(.(X1, X2), X3, .(X1, X4)) -> U7_gga(X1, X2, X3, X4, appcA_in_gga(X2, X3, X4))
3.80/1.80	   U7_gga(X1, X2, X3, X4, appcA_out_gga(X2, X3, X4)) -> appcA_out_gga(.(X1, X2), X3, .(X1, X4))
3.80/1.80	
3.80/1.80	The argument filtering Pi contains the following mapping:
3.80/1.80	[]  =  []
3.80/1.80	
3.80/1.80	appA_in_gga(x1, x2, x3)  =  appA_in_gga(x1, x2)
3.80/1.80	
3.80/1.80	.(x1, x2)  =  .(x1, x2)
3.80/1.80	
3.80/1.80	appcA_in_gga(x1, x2, x3)  =  appcA_in_gga(x1, x2)
3.80/1.80	
3.80/1.80	appcA_out_gga(x1, x2, x3)  =  appcA_out_gga(x1, x2, x3)
3.80/1.80	
3.80/1.80	U7_gga(x1, x2, x3, x4, x5)  =  U7_gga(x1, x2, x3, x5)
3.80/1.80	
3.80/1.80	APP3_BB_IN_GGGA(x1, x2, x3, x4)  =  APP3_BB_IN_GGGA(x1, x2, x3)
3.80/1.80	
3.80/1.80	U2_GGGA(x1, x2, x3, x4)  =  U2_GGGA(x1, x2, x4)
3.80/1.80	
3.80/1.80	APPA_IN_GGA(x1, x2, x3)  =  APPA_IN_GGA(x1, x2)
3.80/1.80	
3.80/1.80	U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x2, x3, x5)
3.80/1.80	
3.80/1.80	U3_GGGA(x1, x2, x3, x4, x5, x6)  =  U3_GGGA(x1, x2, x3, x4, x6)
3.80/1.80	
3.80/1.80	U4_GGGA(x1, x2, x3, x4, x5, x6)  =  U4_GGGA(x1, x2, x3, x4, x6)
3.80/1.80	
3.80/1.80	U5_GGGA(x1, x2, x3, x4, x5, x6)  =  U5_GGGA(x1, x2, x3, x4, x6)
3.80/1.80	
3.80/1.80	
3.80/1.80	We have to consider all (P,R,Pi)-chains
3.80/1.80	----------------------------------------
3.80/1.80	
3.80/1.80	(5) DependencyGraphProof (EQUIVALENT)
3.80/1.80	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 8 less nodes.
3.80/1.80	----------------------------------------
3.80/1.80	
3.80/1.80	(6)
3.80/1.80	Obligation:
3.80/1.80	Pi DP problem:
3.80/1.80	The TRS P consists of the following rules:
3.80/1.80	
3.80/1.80	   APPA_IN_GGA(.(X1, X2), X3, .(X1, X4)) -> APPA_IN_GGA(X2, X3, X4)
3.80/1.80	
3.80/1.80	The TRS R consists of the following rules:
3.80/1.80	
3.80/1.80	   appcA_in_gga([], X1, X1) -> appcA_out_gga([], X1, X1)
3.80/1.80	   appcA_in_gga(.(X1, X2), X3, .(X1, X4)) -> U7_gga(X1, X2, X3, X4, appcA_in_gga(X2, X3, X4))
3.80/1.80	   U7_gga(X1, X2, X3, X4, appcA_out_gga(X2, X3, X4)) -> appcA_out_gga(.(X1, X2), X3, .(X1, X4))
3.80/1.80	
3.80/1.80	The argument filtering Pi contains the following mapping:
3.80/1.80	[]  =  []
3.80/1.80	
3.80/1.80	.(x1, x2)  =  .(x1, x2)
3.80/1.80	
3.80/1.80	appcA_in_gga(x1, x2, x3)  =  appcA_in_gga(x1, x2)
3.80/1.80	
3.80/1.80	appcA_out_gga(x1, x2, x3)  =  appcA_out_gga(x1, x2, x3)
3.80/1.80	
3.80/1.80	U7_gga(x1, x2, x3, x4, x5)  =  U7_gga(x1, x2, x3, x5)
3.80/1.80	
3.80/1.80	APPA_IN_GGA(x1, x2, x3)  =  APPA_IN_GGA(x1, x2)
3.80/1.80	
3.80/1.80	
3.80/1.80	We have to consider all (P,R,Pi)-chains
3.80/1.80	----------------------------------------
3.80/1.80	
3.80/1.80	(7) UsableRulesProof (EQUIVALENT)
3.80/1.80	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
3.80/1.80	----------------------------------------
3.80/1.80	
3.80/1.80	(8)
3.80/1.80	Obligation:
3.80/1.80	Pi DP problem:
3.80/1.80	The TRS P consists of the following rules:
3.80/1.80	
3.80/1.80	   APPA_IN_GGA(.(X1, X2), X3, .(X1, X4)) -> APPA_IN_GGA(X2, X3, X4)
3.80/1.80	
3.80/1.80	R is empty.
3.80/1.80	The argument filtering Pi contains the following mapping:
3.80/1.80	.(x1, x2)  =  .(x1, x2)
3.80/1.80	
3.80/1.80	APPA_IN_GGA(x1, x2, x3)  =  APPA_IN_GGA(x1, x2)
3.80/1.80	
3.80/1.80	
3.80/1.80	We have to consider all (P,R,Pi)-chains
3.80/1.80	----------------------------------------
3.80/1.80	
3.80/1.80	(9) PiDPToQDPProof (SOUND)
3.80/1.80	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
3.80/1.80	----------------------------------------
3.80/1.80	
3.80/1.80	(10)
3.80/1.80	Obligation:
3.80/1.80	Q DP problem:
3.80/1.80	The TRS P consists of the following rules:
3.80/1.80	
3.80/1.80	   APPA_IN_GGA(.(X1, X2), X3) -> APPA_IN_GGA(X2, X3)
3.80/1.80	
3.80/1.80	R is empty.
3.80/1.80	Q is empty.
3.80/1.80	We have to consider all (P,Q,R)-chains.
3.80/1.80	----------------------------------------
3.80/1.80	
3.80/1.80	(11) QDPSizeChangeProof (EQUIVALENT)
3.80/1.80	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. 
3.80/1.80	
3.80/1.80	From the DPs we obtained the following set of size-change graphs:
3.80/1.80	*APPA_IN_GGA(.(X1, X2), X3) -> APPA_IN_GGA(X2, X3)
3.80/1.80	The graph contains the following edges 1 > 1, 2 >= 2
3.80/1.80	
3.80/1.80	
3.80/1.80	----------------------------------------
3.80/1.80	
3.80/1.80	(12)
3.80/1.80	YES
3.80/1.81	EOF