Left Termination proof of /tmp/tmp5Ckg4I/NJ5.pl

Left Termination of the query pattern
f(g,g,a)
w.r.t. the given Prolog program could successfully be proven:

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇒, 83 ms)

↳2 Prolog

↳3 PrologToPiTRSProof (⇒, 23 ms)

↳4 PiTRS

↳5 DependencyPairsProof (⇔, 0 ms)

↳6 PiDP

↳7 DependencyGraphProof (⇔, 0 ms)

↳8 PiDP

↳9 UsableRulesProof (⇔, 0 ms)

↳10 PiDP

↳11 PiDPToQDPProof (⇒, 2 ms)

↳12 QDP

↳13 QDPSizeChangeProof (⇔, 0 ms)

↳14 YES

(0) Obligation:

Clauses:

f(RES, [], RES).
f([], .(Head, Tail), RES) :- f(.(Head, Tail), Tail, RES).
f(.(Head, Tail), Y, RES) :- f(Y, Tail, RES).

Query: f(g,g,a)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: f(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: f(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | f(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | f(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | f(T5, [], T3), SCOPE: 1, CLAUSE: 1 | f(T5, [], T3), SCOPE: 1, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: f(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | f(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground\nX2 is free\nf(T1, T2, T3) !=? f(X2, [], X2)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: f(T5, [], T3), SCOPE: 1, CLAUSE: 1 | f(T5, [], T3), SCOPE: 1, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: f(T5, [], T3), SCOPE: 1, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: f([], T10, T12)\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: f([], T10, T12), SCOPE: 2, CLAUSE: 0 | f([], T10, T12), SCOPE: 2, CLAUSE: 1 | f([], T10, T12), SCOPE: 2, CLAUSE: 2\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: f([], T10, T12), SCOPE: 2, CLAUSE: 0\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: f([], T10, T12), SCOPE: 2, CLAUSE: 1 | f([], T10, T12), SCOPE: 2, CLAUSE: 2\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: f([], T10, T12), SCOPE: 2, CLAUSE: 1\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: f([], T10, T12), SCOPE: 2, CLAUSE: 2\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: f(.(T25, T26), T26, T28)\n\nT25 is ground\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: f(.(T36, T37), T37, T39) | f([], .(T36, T37), T3), SCOPE: 1, CLAUSE: 2\n\nT36 is ground\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: f(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground\nX2 is free\nX43 is free\nX44 is free\nX45 is free\nf(T1, T2, T3) !=? f(X2, [], X2)\nf(T1, T2, T3) !=? f([], .(X43, X44), X45)", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: f(.(T36, T37), T37, T39), SCOPE: 3, CLAUSE: 0 | f(.(T36, T37), T37, T39), SCOPE: 3, CLAUSE: 1 | f(.(T36, T37), T37, T39), SCOPE: 3, CLAUSE: 2 | ?_3 | f([], .(T36, T37), T3), SCOPE: 1, CLAUSE: 2\n\nT36 is ground\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: f(.(T36, T37), T37, T39), SCOPE: 3, CLAUSE: 0\n\nT36 is ground\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: f(.(T36, T37), T37, T39), SCOPE: 3, CLAUSE: 1 | f(.(T36, T37), T37, T39), SCOPE: 3, CLAUSE: 2 | ?_3 | f([], .(T36, T37), T3), SCOPE: 1, CLAUSE: 2\n\nT36 is ground\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: f(.(T36, T37), T37, T39), SCOPE: 3, CLAUSE: 2 | ?_3 | f([], .(T36, T37), T3), SCOPE: 1, CLAUSE: 2\n\nT36 is ground\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: f(.(T36, T37), T37, T39), SCOPE: 3, CLAUSE: 2\n\nT36 is ground\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ?_3 | f([], .(T36, T37), T3), SCOPE: 1, CLAUSE: 2\n\nT36 is ground\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: f(T69, T69, T71)\n\nT69 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: f(T69, T69, T71), SCOPE: 4, CLAUSE: 0 | f(T69, T69, T71), SCOPE: 4, CLAUSE: 1 | f(T69, T69, T71), SCOPE: 4, CLAUSE: 2\n\nT69 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: f(T69, T69, T71), SCOPE: 4, CLAUSE: 0\n\nT69 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: f(T69, T69, T71), SCOPE: 4, CLAUSE: 1 | f(T69, T69, T71), SCOPE: 4, CLAUSE: 2\n\nT69 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: f(T69, T69, T71), SCOPE: 4, CLAUSE: 2\n\nT69 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: f(.(T85, T86), T86, T88)\n\nT85 is ground\nT86 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: f([], .(T36, T37), T3), SCOPE: 1, CLAUSE: 2\n\nT36 is ground\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: f(T99, T98, T101)\n\nT97 is ground\nT98 is ground\nT99 is ground\nX2 is free\nf(.(T97, T98), T99, T3) !=? f(X2, [], X2)", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: f(T99, T98, T101), SCOPE: 5, CLAUSE: 0 | f(T99, T98, T101), SCOPE: 5, CLAUSE: 1 | f(T99, T98, T101), SCOPE: 5, CLAUSE: 2\n\nT97 is ground\nT98 is ground\nT99 is ground\nX2 is free\nf(.(T97, T98), T99, T3) !=? f(X2, [], X2)", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: f(T99, T98, T101), SCOPE: 5, CLAUSE: 0\n\nT97 is ground\nT98 is ground\nT99 is ground\nX2 is free\nf(.(T97, T98), T99, T3) !=? f(X2, [], X2)", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: f(T99, T98, T101), SCOPE: 5, CLAUSE: 1 | f(T99, T98, T101), SCOPE: 5, CLAUSE: 2\n\nT97 is ground\nT98 is ground\nT99 is ground\nX2 is free\nf(.(T97, T98), T99, T3) !=? f(X2, [], X2)", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: f(T99, T98, T101), SCOPE: 5, CLAUSE: 1\n\nT97 is ground\nT98 is ground\nT99 is ground\nX2 is free\nf(.(T97, T98), T99, T3) !=? f(X2, [], X2)", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: f(T99, T98, T101), SCOPE: 5, CLAUSE: 2\n\nT97 is ground\nT98 is ground\nT99 is ground\nX2 is free\nf(.(T97, T98), T99, T3) !=? f(X2, [], X2)", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: f(.(T119, T120), T120, T122)\n\nT97 is ground\nT119 is ground\nT120 is ground\nX2 is free\nf(.(T97, .(T119, T120)), [], T3) !=? f(X2, [], X2)", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: f(T137, T136, T139)\n\nT136 is ground\nT137 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 57 [arrowhead = none ];
57 -> 2;

58 [label="EVAL with clause\nf(X2, [], X2).\nand substitutionT1 -> T5,\nX2 -> T5,\nT2 -> [],\nT3 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 58 [arrowhead = none ];
58 -> 3;

59 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 59 [arrowhead = none ];
59 -> 4;

60 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 60 [arrowhead = none ];
60 -> 5;

61 [label="EVAL with clause\nf([], .(X43, X44), X45) :- f(.(X43, X44), X44, X45).\nand substitutionT1 -> [],\nX43 -> T36,\nX44 -> T37,\nT2 -> .(T36, T37),\nT3 -> T39,\nX45 -> T39,\nT38 -> T39", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 61 [arrowhead = none ];
61 -> 20;

62 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 62 [arrowhead = none ];
62 -> 21;

63 [label="BACKTRACK\nfor clause: f([], .(Head, Tail), RES) :- f(.(Head, Tail), Tail, RES)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 63 [arrowhead = none ];
63 -> 6;

64 [label="EVAL with clause\nf(.(X10, X11), X12, X13) :- f(X12, X11, X13).\nand substitutionX10 -> T9,\nX11 -> T10,\nT5 -> .(T9, T10),\nX12 -> [],\nT3 -> T12,\nX13 -> T12,\nT11 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 64 [arrowhead = none ];
64 -> 7;

65 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 65 [arrowhead = none ];
65 -> 8;

66 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 66 [arrowhead = none ];
66 -> 9;

67 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 67 [arrowhead = none ];
67 -> 10;

68 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 68 [arrowhead = none ];
68 -> 11;

69 [label="EVAL with clause\nf(X18, [], X18).\nand substitutionX18 -> [],\nT10 -> [],\nT12 -> []", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 69 [arrowhead = none ];
69 -> 12;

70 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 70 [arrowhead = none ];
70 -> 13;

71 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 71 [arrowhead = none ];
71 -> 15;

72 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 72 [arrowhead = none ];
72 -> 16;

73 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 73 [arrowhead = none ];
73 -> 14;

74 [label="EVAL with clause\nf([], .(X31, X32), X33) :- f(.(X31, X32), X32, X33).\nand substitutionX31 -> T25,\nX32 -> T26,\nT10 -> .(T25, T26),\nT12 -> T28,\nX33 -> T28,\nT27 -> T28", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 74 [arrowhead = none ];
74 -> 17;

75 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 75 [arrowhead = none ];
75 -> 18;

76 [label="BACKTRACK\nfor clause: f(.(Head, Tail), Y, RES) :- f(Y, Tail, RES)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 76 [arrowhead = none ];
76 -> 19;

77 [label="INSTANCE with matching:\nT1 -> .(T25, T26)\nT2 -> T26\nT3 -> T28", fontsize=14, style = filled, fillcolor = lightgrey];
17 -> 77 [arrowhead = none , style=dashed];
77 -> 1[style=dashed];

78 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 78 [arrowhead = none ];
78 -> 22;

79 [label="EVAL with clause\nf(.(X110, X111), X112, X113) :- f(X112, X111, X113).\nand substitutionX110 -> T97,\nX111 -> T98,\nT1 -> .(T97, T98),\nT2 -> T99,\nX112 -> T99,\nT3 -> T101,\nX113 -> T101,\nT100 -> T101", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 79 [arrowhead = none ];
79 -> 43;

80 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 80 [arrowhead = none ];
80 -> 44;

81 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 81 [arrowhead = none ];
81 -> 23;

82 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 82 [arrowhead = none ];
82 -> 24;

83 [label="EVAL with clause\nf(X50, [], X50).\nand substitutionT36 -> T48,\nT37 -> [],\nX50 -> .(T48, []),\nT49 -> [],\nT39 -> .(T48, [])", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 83 [arrowhead = none ];
83 -> 25;

84 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 84 [arrowhead = none ];
84 -> 26;

85 [label="BACKTRACK\nfor clause: f([], .(Head, Tail), RES) :- f(.(Head, Tail), Tail, RES)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 85 [arrowhead = none ];
85 -> 28;

86 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 86 [arrowhead = none ];
86 -> 27;

87 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
28 -> 87 [arrowhead = none ];
87 -> 29;

88 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
28 -> 88 [arrowhead = none ];
88 -> 30;

89 [label="ONLY EVAL with clause\nf(.(X74, X75), X76, X77) :- f(X76, X75, X77).\nand substitutionT36 -> T68,\nX74 -> T68,\nT37 -> T69,\nX75 -> T69,\nX76 -> T69,\nT39 -> T71,\nX77 -> T71,\nT70 -> T71", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 89 [arrowhead = none ];
89 -> 31;

90 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 90 [arrowhead = none ];
90 -> 41;

91 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 91 [arrowhead = none ];
91 -> 32;

92 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
32 -> 92 [arrowhead = none ];
92 -> 33;

93 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
32 -> 93 [arrowhead = none ];
93 -> 34;

94 [label="EVAL with clause\nf(X84, [], X84).\nand substitutionT69 -> [],\nX84 -> [],\nT78 -> [],\nT71 -> []", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 94 [arrowhead = none ];
94 -> 35;

95 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 95 [arrowhead = none ];
95 -> 36;

96 [label="BACKTRACK\nfor clause: f([], .(Head, Tail), RES) :- f(.(Head, Tail), Tail, RES)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 96 [arrowhead = none ];
96 -> 38;

97 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 97 [arrowhead = none ];
97 -> 37;

98 [label="EVAL with clause\nf(.(X96, X97), X98, X99) :- f(X98, X97, X99).\nand substitutionX96 -> T85,\nX97 -> T86,\nT69 -> .(T85, T86),\nX98 -> .(T85, T86),\nT71 -> T88,\nX99 -> T88,\nT87 -> T88", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 98 [arrowhead = none ];
98 -> 39;

99 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 99 [arrowhead = none ];
99 -> 40;

100 [label="INSTANCE with matching:\nT1 -> .(T85, T86)\nT2 -> T86\nT3 -> T88", fontsize=14, style = filled, fillcolor = lightgrey];
39 -> 100 [arrowhead = none , style=dashed];
100 -> 1[style=dashed];

101 [label="BACKTRACK\nfor clause: f(.(Head, Tail), Y, RES) :- f(Y, Tail, RES)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 101 [arrowhead = none ];
101 -> 42;

102 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
43 -> 102 [arrowhead = none ];
102 -> 45;

103 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 103 [arrowhead = none ];
103 -> 46;

104 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 104 [arrowhead = none ];
104 -> 47;

105 [label="EVAL with clause\nf(X118, [], X118).\nand substitutionT99 -> T106,\nX118 -> T106,\nT98 -> [],\nT101 -> T106", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 105 [arrowhead = none ];
105 -> 48;

106 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 106 [arrowhead = none ];
106 -> 49;

107 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
47 -> 107 [arrowhead = none ];
107 -> 51;

108 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
47 -> 108 [arrowhead = none ];
108 -> 52;

109 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 109 [arrowhead = none ];
109 -> 50;

110 [label="EVAL with clause\nf([], .(X131, X132), X133) :- f(.(X131, X132), X132, X133).\nand substitutionT99 -> [],\nX131 -> T119,\nX132 -> T120,\nT98 -> .(T119, T120),\nT101 -> T122,\nX133 -> T122,\nT121 -> T122", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 110 [arrowhead = none ];
110 -> 53;

111 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 111 [arrowhead = none ];
111 -> 54;

112 [label="EVAL with clause\nf(.(X144, X145), X146, X147) :- f(X146, X145, X147).\nand substitutionX144 -> T135,\nX145 -> T136,\nT99 -> .(T135, T136),\nT98 -> T137,\nX146 -> T137,\nT101 -> T139,\nX147 -> T139,\nT138 -> T139", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 112 [arrowhead = none ];
112 -> 55;

113 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 113 [arrowhead = none ];
113 -> 56;

114 [label="INSTANCE with matching:\nT1 -> .(T119, T120)\nT2 -> T120\nT3 -> T122", fontsize=14, style = filled, fillcolor = lightgrey];
53 -> 114 [arrowhead = none , style=dashed];
114 -> 1[style=dashed];

115 [label="INSTANCE with matching:\nT1 -> T137\nT2 -> T136\nT3 -> T139", fontsize=14, style = filled, fillcolor = lightgrey];
55 -> 115 [arrowhead = none , style=dashed];
115 -> 1[style=dashed];

}

(2) Obligation:

Clauses:

fA(T5, [], T5).
fA(.(T9, []), [], []).
fA(.(T9, .(T25, T26)), [], T28) :- fA(.(T25, T26), T26, T28).
fA([], .(T48, []), .(T48, [])).
fA([], .(T68, []), []).
fA([], .(T68, .(T85, T86)), T88) :- fA(.(T85, T86), T86, T88).
fA(.(T97, []), T106, T106).
fA(.(T97, .(T119, T120)), [], T122) :- fA(.(T119, T120), T120, T122).
fA(.(T97, T137), .(T135, T136), T139) :- fA(T137, T136, T139).

Query: fA(g,g,a)

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
fA_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

fA_in_gga(T5, [], T5) → fA_out_gga(T5, [], T5)
fA_in_gga(.(T9, []), [], []) → fA_out_gga(.(T9, []), [], [])
fA_in_gga(.(T9, .(T25, T26)), [], T28) → U1_gga(T9, T25, T26, T28, fA_in_gga(.(T25, T26), T26, T28))
fA_in_gga([], .(T48, []), .(T48, [])) → fA_out_gga([], .(T48, []), .(T48, []))
fA_in_gga([], .(T68, []), []) → fA_out_gga([], .(T68, []), [])
fA_in_gga([], .(T68, .(T85, T86)), T88) → U2_gga(T68, T85, T86, T88, fA_in_gga(.(T85, T86), T86, T88))
fA_in_gga(.(T97, []), T106, T106) → fA_out_gga(.(T97, []), T106, T106)
fA_in_gga(.(T97, .(T119, T120)), [], T122) → U3_gga(T97, T119, T120, T122, fA_in_gga(.(T119, T120), T120, T122))
fA_in_gga(.(T97, T137), .(T135, T136), T139) → U4_gga(T97, T137, T135, T136, T139, fA_in_gga(T137, T136, T139))
U4_gga(T97, T137, T135, T136, T139, fA_out_gga(T137, T136, T139)) → fA_out_gga(.(T97, T137), .(T135, T136), T139)
U3_gga(T97, T119, T120, T122, fA_out_gga(.(T119, T120), T120, T122)) → fA_out_gga(.(T97, .(T119, T120)), [], T122)
U2_gga(T68, T85, T86, T88, fA_out_gga(.(T85, T86), T86, T88)) → fA_out_gga([], .(T68, .(T85, T86)), T88)
U1_gga(T9, T25, T26, T28, fA_out_gga(.(T25, T26), T26, T28)) → fA_out_gga(.(T9, .(T25, T26)), [], T28)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

fA_in_gga(T5, [], T5) → fA_out_gga(T5, [], T5)
fA_in_gga(.(T9, []), [], []) → fA_out_gga(.(T9, []), [], [])
fA_in_gga(.(T9, .(T25, T26)), [], T28) → U1_gga(T9, T25, T26, T28, fA_in_gga(.(T25, T26), T26, T28))
fA_in_gga([], .(T48, []), .(T48, [])) → fA_out_gga([], .(T48, []), .(T48, []))
fA_in_gga([], .(T68, []), []) → fA_out_gga([], .(T68, []), [])
fA_in_gga([], .(T68, .(T85, T86)), T88) → U2_gga(T68, T85, T86, T88, fA_in_gga(.(T85, T86), T86, T88))
fA_in_gga(.(T97, []), T106, T106) → fA_out_gga(.(T97, []), T106, T106)
fA_in_gga(.(T97, .(T119, T120)), [], T122) → U3_gga(T97, T119, T120, T122, fA_in_gga(.(T119, T120), T120, T122))
fA_in_gga(.(T97, T137), .(T135, T136), T139) → U4_gga(T97, T137, T135, T136, T139, fA_in_gga(T137, T136, T139))
U4_gga(T97, T137, T135, T136, T139, fA_out_gga(T137, T136, T139)) → fA_out_gga(.(T97, T137), .(T135, T136), T139)
U3_gga(T97, T119, T120, T122, fA_out_gga(.(T119, T120), T120, T122)) → fA_out_gga(.(T97, .(T119, T120)), [], T122)
U2_gga(T68, T85, T86, T88, fA_out_gga(.(T85, T86), T86, T88)) → fA_out_gga([], .(T68, .(T85, T86)), T88)
U1_gga(T9, T25, T26, T28, fA_out_gga(.(T25, T26), T26, T28)) → fA_out_gga(.(T9, .(T25, T26)), [], T28)

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

FA_IN_GGA(.(T9, .(T25, T26)), [], T28) → U1_GGA(T9, T25, T26, T28, fA_in_gga(.(T25, T26), T26, T28))
FA_IN_GGA(.(T9, .(T25, T26)), [], T28) → FA_IN_GGA(.(T25, T26), T26, T28)
FA_IN_GGA([], .(T68, .(T85, T86)), T88) → U2_GGA(T68, T85, T86, T88, fA_in_gga(.(T85, T86), T86, T88))
FA_IN_GGA([], .(T68, .(T85, T86)), T88) → FA_IN_GGA(.(T85, T86), T86, T88)
FA_IN_GGA(.(T97, .(T119, T120)), [], T122) → U3_GGA(T97, T119, T120, T122, fA_in_gga(.(T119, T120), T120, T122))
FA_IN_GGA(.(T97, T137), .(T135, T136), T139) → U4_GGA(T97, T137, T135, T136, T139, fA_in_gga(T137, T136, T139))
FA_IN_GGA(.(T97, T137), .(T135, T136), T139) → FA_IN_GGA(T137, T136, T139)

The TRS R consists of the following rules:

fA_in_gga(T5, [], T5) → fA_out_gga(T5, [], T5)
fA_in_gga(.(T9, []), [], []) → fA_out_gga(.(T9, []), [], [])
fA_in_gga(.(T9, .(T25, T26)), [], T28) → U1_gga(T9, T25, T26, T28, fA_in_gga(.(T25, T26), T26, T28))
fA_in_gga([], .(T48, []), .(T48, [])) → fA_out_gga([], .(T48, []), .(T48, []))
fA_in_gga([], .(T68, []), []) → fA_out_gga([], .(T68, []), [])
fA_in_gga([], .(T68, .(T85, T86)), T88) → U2_gga(T68, T85, T86, T88, fA_in_gga(.(T85, T86), T86, T88))
fA_in_gga(.(T97, []), T106, T106) → fA_out_gga(.(T97, []), T106, T106)
fA_in_gga(.(T97, .(T119, T120)), [], T122) → U3_gga(T97, T119, T120, T122, fA_in_gga(.(T119, T120), T120, T122))
fA_in_gga(.(T97, T137), .(T135, T136), T139) → U4_gga(T97, T137, T135, T136, T139, fA_in_gga(T137, T136, T139))
U4_gga(T97, T137, T135, T136, T139, fA_out_gga(T137, T136, T139)) → fA_out_gga(.(T97, T137), .(T135, T136), T139)
U3_gga(T97, T119, T120, T122, fA_out_gga(.(T119, T120), T120, T122)) → fA_out_gga(.(T97, .(T119, T120)), [], T122)
U2_gga(T68, T85, T86, T88, fA_out_gga(.(T85, T86), T86, T88)) → fA_out_gga([], .(T68, .(T85, T86)), T88)
U1_gga(T9, T25, T26, T28, fA_out_gga(.(T25, T26), T26, T28)) → fA_out_gga(.(T9, .(T25, T26)), [], T28)

The argument filtering Pi contains the following mapping:
fA_in_gga(x1, x2, x3) = fA_in_gga(x1, x2)
[] = []
fA_out_gga(x1, x2, x3) = fA_out_gga(x3)
.(x1, x2) = .(x1, x2)
U1_gga(x1, x2, x3, x4, x5) = U1_gga(x5)
U2_gga(x1, x2, x3, x4, x5) = U2_gga(x5)
U3_gga(x1, x2, x3, x4, x5) = U3_gga(x5)
U4_gga(x1, x2, x3, x4, x5, x6) = U4_gga(x6)
FA_IN_GGA(x1, x2, x3) = FA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5) = U1_GGA(x5)
U2_GGA(x1, x2, x3, x4, x5) = U2_GGA(x5)
U3_GGA(x1, x2, x3, x4, x5) = U3_GGA(x5)
U4_GGA(x1, x2, x3, x4, x5, x6) = U4_GGA(x6)

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

(6) Obligation:

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

FA_IN_GGA(.(T9, .(T25, T26)), [], T28) → U1_GGA(T9, T25, T26, T28, fA_in_gga(.(T25, T26), T26, T28))
FA_IN_GGA(.(T9, .(T25, T26)), [], T28) → FA_IN_GGA(.(T25, T26), T26, T28)
FA_IN_GGA([], .(T68, .(T85, T86)), T88) → U2_GGA(T68, T85, T86, T88, fA_in_gga(.(T85, T86), T86, T88))
FA_IN_GGA([], .(T68, .(T85, T86)), T88) → FA_IN_GGA(.(T85, T86), T86, T88)
FA_IN_GGA(.(T97, .(T119, T120)), [], T122) → U3_GGA(T97, T119, T120, T122, fA_in_gga(.(T119, T120), T120, T122))
FA_IN_GGA(.(T97, T137), .(T135, T136), T139) → U4_GGA(T97, T137, T135, T136, T139, fA_in_gga(T137, T136, T139))
FA_IN_GGA(.(T97, T137), .(T135, T136), T139) → FA_IN_GGA(T137, T136, T139)

The TRS R consists of the following rules:

fA_in_gga(T5, [], T5) → fA_out_gga(T5, [], T5)
fA_in_gga(.(T9, []), [], []) → fA_out_gga(.(T9, []), [], [])
fA_in_gga(.(T9, .(T25, T26)), [], T28) → U1_gga(T9, T25, T26, T28, fA_in_gga(.(T25, T26), T26, T28))
fA_in_gga([], .(T48, []), .(T48, [])) → fA_out_gga([], .(T48, []), .(T48, []))
fA_in_gga([], .(T68, []), []) → fA_out_gga([], .(T68, []), [])
fA_in_gga([], .(T68, .(T85, T86)), T88) → U2_gga(T68, T85, T86, T88, fA_in_gga(.(T85, T86), T86, T88))
fA_in_gga(.(T97, []), T106, T106) → fA_out_gga(.(T97, []), T106, T106)
fA_in_gga(.(T97, .(T119, T120)), [], T122) → U3_gga(T97, T119, T120, T122, fA_in_gga(.(T119, T120), T120, T122))
fA_in_gga(.(T97, T137), .(T135, T136), T139) → U4_gga(T97, T137, T135, T136, T139, fA_in_gga(T137, T136, T139))
U4_gga(T97, T137, T135, T136, T139, fA_out_gga(T137, T136, T139)) → fA_out_gga(.(T97, T137), .(T135, T136), T139)
U3_gga(T97, T119, T120, T122, fA_out_gga(.(T119, T120), T120, T122)) → fA_out_gga(.(T97, .(T119, T120)), [], T122)
U2_gga(T68, T85, T86, T88, fA_out_gga(.(T85, T86), T86, T88)) → fA_out_gga([], .(T68, .(T85, T86)), T88)
U1_gga(T9, T25, T26, T28, fA_out_gga(.(T25, T26), T26, T28)) → fA_out_gga(.(T9, .(T25, T26)), [], T28)

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 4 less nodes.

(8) Obligation:

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

FA_IN_GGA(.(T97, T137), .(T135, T136), T139) → FA_IN_GGA(T137, T136, T139)
FA_IN_GGA(.(T9, .(T25, T26)), [], T28) → FA_IN_GGA(.(T25, T26), T26, T28)
FA_IN_GGA([], .(T68, .(T85, T86)), T88) → FA_IN_GGA(.(T85, T86), T86, T88)

The TRS R consists of the following rules:

fA_in_gga(T5, [], T5) → fA_out_gga(T5, [], T5)
fA_in_gga(.(T9, []), [], []) → fA_out_gga(.(T9, []), [], [])
fA_in_gga(.(T9, .(T25, T26)), [], T28) → U1_gga(T9, T25, T26, T28, fA_in_gga(.(T25, T26), T26, T28))
fA_in_gga([], .(T48, []), .(T48, [])) → fA_out_gga([], .(T48, []), .(T48, []))
fA_in_gga([], .(T68, []), []) → fA_out_gga([], .(T68, []), [])
fA_in_gga([], .(T68, .(T85, T86)), T88) → U2_gga(T68, T85, T86, T88, fA_in_gga(.(T85, T86), T86, T88))
fA_in_gga(.(T97, []), T106, T106) → fA_out_gga(.(T97, []), T106, T106)
fA_in_gga(.(T97, .(T119, T120)), [], T122) → U3_gga(T97, T119, T120, T122, fA_in_gga(.(T119, T120), T120, T122))
fA_in_gga(.(T97, T137), .(T135, T136), T139) → U4_gga(T97, T137, T135, T136, T139, fA_in_gga(T137, T136, T139))
U4_gga(T97, T137, T135, T136, T139, fA_out_gga(T137, T136, T139)) → fA_out_gga(.(T97, T137), .(T135, T136), T139)
U3_gga(T97, T119, T120, T122, fA_out_gga(.(T119, T120), T120, T122)) → fA_out_gga(.(T97, .(T119, T120)), [], T122)
U2_gga(T68, T85, T86, T88, fA_out_gga(.(T85, T86), T86, T88)) → fA_out_gga([], .(T68, .(T85, T86)), T88)
U1_gga(T9, T25, T26, T28, fA_out_gga(.(T25, T26), T26, T28)) → fA_out_gga(.(T9, .(T25, T26)), [], T28)

(9) UsableRulesProof (EQUIVALENT transformation)

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

(10) Obligation:

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

FA_IN_GGA(.(T97, T137), .(T135, T136), T139) → FA_IN_GGA(T137, T136, T139)
FA_IN_GGA(.(T9, .(T25, T26)), [], T28) → FA_IN_GGA(.(T25, T26), T26, T28)
FA_IN_GGA([], .(T68, .(T85, T86)), T88) → FA_IN_GGA(.(T85, T86), T86, T88)

R is empty.
The argument filtering Pi contains the following mapping:
[] = []
.(x1, x2) = .(x1, x2)
FA_IN_GGA(x1, x2, x3) = FA_IN_GGA(x1, x2)

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

(11) PiDPToQDPProof (SOUND transformation)

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

(12) Obligation:

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

FA_IN_GGA(.(T97, T137), .(T135, T136)) → FA_IN_GGA(T137, T136)
FA_IN_GGA(.(T9, .(T25, T26)), []) → FA_IN_GGA(.(T25, T26), T26)
FA_IN_GGA([], .(T68, .(T85, T86))) → FA_IN_GGA(.(T85, T86), T86)

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

(13) QDPSizeChangeProof (EQUIVALENT transformation)

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:

FA_IN_GGA(.(T97, T137), .(T135, T136)) → FA_IN_GGA(T137, T136)
The graph contains the following edges 1 > 1, 2 > 2
FA_IN_GGA(.(T9, .(T25, T26)), []) → FA_IN_GGA(.(T25, T26), T26)
The graph contains the following edges 1 > 1, 1 > 2
FA_IN_GGA([], .(T68, .(T85, T86))) → FA_IN_GGA(.(T85, T86), T86)
The graph contains the following edges 2 > 1, 2 > 2

(0) Obligation:

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) PrologToPiTRSProof (SOUND transformation)

(4) Obligation:

(5) DependencyPairsProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Obligation:

(9) UsableRulesProof (EQUIVALENT transformation)

(10) Obligation:

(11) PiDPToQDPProof (SOUND transformation)

(12) Obligation:

(13) QDPSizeChangeProof (EQUIVALENT transformation)

(14) YES