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

(0) Obligation:

Clauses:

m(X, 0, Z) :- ','(!, =(Z, X)).
m(0, Y, Z) :- ','(!, =(Z, 0)).
m(X, Y, Z) :- ','(p(X, A), ','(p(Y, B), m(A, B, Z))).
p(0, 0).
p(s(0), 0).
p(s(s(X)), s(Y)) :- p(s(X), Y).
=(X, X).

Query: m(g,g,a)

(1) BuiltinConflictTransformerProof (EQUIVALENT transformation)

Renamed defined predicates conflicting with built-in predicates [PROLOG].

(2) Obligation:

Clauses:

m(X, 0, Z) :- ','(!, user_defined_=(Z, X)).
m(0, Y, Z) :- ','(!, user_defined_=(Z, 0)).
m(X, Y, Z) :- ','(p(X, A), ','(p(Y, B), m(A, B, Z))).
p(0, 0).
p(s(0), 0).
p(s(s(X)), s(Y)) :- p(s(X), Y).
user_defined_=(X, X).

Query: m(g,g,a)

(3) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="B: m(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: m(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | m(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | m(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(!_1, user_defined_=(T8, T6)) | m(T6, 0, T3), SCOPE: 1, CLAUSE: 1 | m(T6, 0, T3), SCOPE: 1, CLAUSE: 2\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: m(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | m(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground\nX3 is free\nX4 is free\nm(T1, T2, T3) !=? m(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: user_defined_=(T8, T6)\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: user_defined_=(T8, T6), SCOPE: 2, CLAUSE: 6\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(!_1, user_defined_=(T16, 0)) | m(0, T14, T3), SCOPE: 1, CLAUSE: 2\n\nT14 is ground\nX3 is free\nX4 is free\nm(0, T14, T3) !=? m(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: m(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground\nX3 is free\nX4 is free\nX10 is free\nX11 is free\nm(T1, T2, T3) !=? m(X3, 0, X4)\nm(T1, T2, T3) !=? m(0, X10, X11)", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: user_defined_=(T16, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: user_defined_=(T16, 0), SCOPE: 3, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(p(T23, X21), ','(p(T24, X22), m(X21, X22, T26)))\n\nT23 is ground\nT24 is ground\nX3 is free\nX4 is free\nX10 is free\nX11 is free\nX21 is free\nX22 is free\nm(T23, T24, T3) !=? m(X3, 0, X4)\nm(T23, T24, T3) !=? m(0, X10, X11)", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(p(T23, X21), ','(p(T24, X22), m(X21, X22, T26))), SCOPE: 4, CLAUSE: 3 | ','(p(T23, X21), ','(p(T24, X22), m(X21, X22, T26))), SCOPE: 4, CLAUSE: 4 | ','(p(T23, X21), ','(p(T24, X22), m(X21, X22, T26))), SCOPE: 4, CLAUSE: 5\n\nT23 is ground\nT24 is ground\nX3 is free\nX4 is free\nX10 is free\nX11 is free\nX21 is free\nX22 is free\nm(T23, T24, T3) !=? m(X3, 0, X4)\nm(T23, T24, T3) !=? m(0, X10, X11)", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(p(T23, X21), ','(p(T24, X22), m(X21, X22, T26))), SCOPE: 4, CLAUSE: 4 | ','(p(T23, X21), ','(p(T24, X22), m(X21, X22, T26))), SCOPE: 4, CLAUSE: 5\n\nT23 is ground\nT24 is ground\nX3 is free\nX4 is free\nX10 is free\nX11 is free\nX21 is free\nX22 is free\nm(T23, T24, T3) !=? m(X3, 0, X4)\nm(T23, T24, T3) !=? m(0, X10, X11)", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(p(T23, X21), ','(p(T24, X22), m(X21, X22, T26))), SCOPE: 4, CLAUSE: 4\n\nT23 is ground\nT24 is ground\nX3 is free\nX4 is free\nX10 is free\nX11 is free\nX21 is free\nX22 is free\nm(T23, T24, T3) !=? m(X3, 0, X4)\nm(T23, T24, T3) !=? m(0, X10, X11)", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(p(T23, X21), ','(p(T24, X22), m(X21, X22, T26))), SCOPE: 4, CLAUSE: 5\n\nT23 is ground\nT24 is ground\nX3 is free\nX4 is free\nX10 is free\nX11 is free\nX21 is free\nX22 is free\nm(T23, T24, T3) !=? m(X3, 0, X4)\nm(T23, T24, T3) !=? m(0, X10, X11)", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(p(T24, X22), m(0, X22, T26))\n\nT24 is ground\nX3 is free\nX4 is free\nX22 is free\nm(s(0), T24, T3) !=? m(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="C: p(T24, X22)\n\nT24 is ground\nX3 is free\nX4 is free\nX22 is free\nm(s(0), T24, T3) !=? m(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: m(0, T27, T26)\n\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: p(T24, X22), SCOPE: 5, CLAUSE: 3 | p(T24, X22), SCOPE: 5, CLAUSE: 4 | p(T24, X22), SCOPE: 5, CLAUSE: 5\n\nT24 is ground\nX3 is free\nX4 is free\nX22 is free\nm(s(0), T24, T3) !=? m(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: p(T24, X22), SCOPE: 5, CLAUSE: 4 | p(T24, X22), SCOPE: 5, CLAUSE: 5\n\nT24 is ground\nX3 is free\nX4 is free\nX22 is free\nm(s(0), T24, T3) !=? m(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: p(T24, X22), SCOPE: 5, CLAUSE: 4\n\nT24 is ground\nX3 is free\nX4 is free\nX22 is free\nm(s(0), T24, T3) !=? m(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: p(T24, X22), SCOPE: 5, CLAUSE: 5\n\nT24 is ground\nX3 is free\nX4 is free\nX22 is free\nm(s(0), T24, T3) !=? m(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="A: p(s(T30), X31)\n\nT30 is ground\nX31 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: p(s(T30), X31), SCOPE: 6, CLAUSE: 3 | p(s(T30), X31), SCOPE: 6, CLAUSE: 4 | p(s(T30), X31), SCOPE: 6, CLAUSE: 5\n\nT30 is ground\nX31 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: p(s(T30), X31), SCOPE: 6, CLAUSE: 4 | p(s(T30), X31), SCOPE: 6, CLAUSE: 5\n\nT30 is ground\nX31 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: p(s(T30), X31), SCOPE: 6, CLAUSE: 4\n\nT30 is ground\nX31 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: p(s(T30), X31), SCOPE: 6, CLAUSE: 5\n\nT30 is ground\nX31 is free", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: p(s(T33), X40)\n\nT33 is ground\nX40 is free", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: m(0, T27, T26), SCOPE: 7, CLAUSE: 0 | m(0, T27, T26), SCOPE: 7, CLAUSE: 1 | m(0, T27, T26), SCOPE: 7, CLAUSE: 2\n\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(!_7, user_defined_=(T38, 0)) | m(0, 0, T26), SCOPE: 7, CLAUSE: 1 | m(0, 0, T26), SCOPE: 7, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: m(0, T27, T26), SCOPE: 7, CLAUSE: 1 | m(0, T27, T26), SCOPE: 7, CLAUSE: 2\n\nT27 is ground\nX47 is free\nX48 is free\nm(0, T27, T26) !=? m(X47, 0, X48)", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: user_defined_=(T38, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: user_defined_=(T38, 0), SCOPE: 8, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(!_7, user_defined_=(T49, 0)) | m(0, T47, T26), SCOPE: 7, CLAUSE: 2\n\nT47 is ground\nX47 is free\nX48 is free\nm(0, T47, T26) !=? m(X47, 0, X48)", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: user_defined_=(T49, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: user_defined_=(T49, 0), SCOPE: 9, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(p(s(T55), X69), ','(p(T24, X22), m(s(X69), X22, T26)))\n\nT24 is ground\nT55 is ground\nX3 is free\nX4 is free\nX22 is free\nX69 is free\nm(s(s(T55)), T24, T3) !=? m(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: p(s(T55), X69)\n\nT24 is ground\nT55 is ground\nX3 is free\nX4 is free\nX69 is free\nm(s(s(T55)), T24, T3) !=? m(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(p(T24, X22), m(s(T56), X22, T26))\n\nT24 is ground\nT55 is ground\nT56 is ground\nX3 is free\nX4 is free\nX22 is free\nm(s(s(T55)), T24, T3) !=? m(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="D: p(T24, X22)\n\nT24 is ground\nT55 is ground\nX3 is free\nX4 is free\nX22 is free\nm(s(s(T55)), T24, T3) !=? m(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: m(s(T56), T57, T26)\n\nT56 is ground\nT57 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: p(T24, X22), SCOPE: 10, CLAUSE: 3 | p(T24, X22), SCOPE: 10, CLAUSE: 4 | p(T24, X22), SCOPE: 10, CLAUSE: 5\n\nT24 is ground\nT55 is ground\nX3 is free\nX4 is free\nX22 is free\nm(s(s(T55)), T24, T3) !=? m(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: p(T24, X22), SCOPE: 10, CLAUSE: 4 | p(T24, X22), SCOPE: 10, CLAUSE: 5\n\nT24 is ground\nT55 is ground\nX3 is free\nX4 is free\nX22 is free\nm(s(s(T55)), T24, T3) !=? m(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: p(T24, X22), SCOPE: 10, CLAUSE: 4\n\nT24 is ground\nT55 is ground\nX3 is free\nX4 is free\nX22 is free\nm(s(s(T55)), T24, T3) !=? m(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: p(T24, X22), SCOPE: 10, CLAUSE: 5\n\nT24 is ground\nT55 is ground\nX3 is free\nX4 is free\nX22 is free\nm(s(s(T55)), T24, T3) !=? m(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: p(s(T60), X78)\n\nT60 is ground\nX78 is free", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 73 [arrowhead = none ];
73 -> 2;

74 [label="EVAL with clause\nm(X3, 0, X4) :- ','(!_1, user_defined_=(X4, X3)).\nand substitutionT1 -> T6,\nX3 -> T6,\nT2 -> 0,\nT3 -> T8,\nX4 -> T8,\nT7 -> T8", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 74 [arrowhead = none ];
74 -> 3;

75 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 75 [arrowhead = none ];
75 -> 4;

76 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 76 [arrowhead = none ];
76 -> 5;

77 [label="EVAL with clause\nm(0, X10, X11) :- ','(!_1, user_defined_=(X11, 0)).\nand substitutionT1 -> 0,\nT2 -> T14,\nX10 -> T14,\nT3 -> T16,\nX11 -> T16,\nT15 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 77 [arrowhead = none ];
77 -> 10;

78 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 78 [arrowhead = none ];
78 -> 11;

79 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 79 [arrowhead = none ];
79 -> 6;

80 [label="EVAL with clause\nuser_defined_=(X7, X7).\nand substitutionT8 -> T11,\nX7 -> T11,\nT6 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 80 [arrowhead = none ];
80 -> 7;

81 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 81 [arrowhead = none ];
81 -> 8;

82 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 82 [arrowhead = none ];
82 -> 9;

83 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 83 [arrowhead = none ];
83 -> 12;

84 [label="ONLY EVAL with clause\nm(X18, X19, X20) :- ','(p(X18, X21), ','(p(X19, X22), m(X21, X22, X20))).\nand substitutionT1 -> T23,\nX18 -> T23,\nT2 -> T24,\nX19 -> T24,\nT3 -> T26,\nX20 -> T26,\nT25 -> T26", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 84 [arrowhead = none ];
84 -> 17;

85 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 85 [arrowhead = none ];
85 -> 13;

86 [label="EVAL with clause\nuser_defined_=(X14, X14).\nand substitutionT16 -> 0,\nX14 -> 0,\nT19 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 86 [arrowhead = none ];
86 -> 14;

87 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 87 [arrowhead = none ];
87 -> 15;

88 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 88 [arrowhead = none ];
88 -> 16;

89 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 89 [arrowhead = none ];
89 -> 18;

90 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (m(T23, T24, T3), m(0, X10, X11))", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 90 [arrowhead = none ];
90 -> 19;

91 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
19 -> 91 [arrowhead = none ];
91 -> 20;

92 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
19 -> 92 [arrowhead = none ];
92 -> 21;

93 [label="EVAL with clause\np(s(0), 0).\nand substitutionT23 -> s(0),\nX21 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 93 [arrowhead = none ];
93 -> 22;

94 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 94 [arrowhead = none ];
94 -> 23;

95 [label="EVAL with clause\np(s(s(X67)), s(X68)) :- p(s(X67), X68).\nand substitutionX67 -> T55,\nT23 -> s(s(T55)),\nX68 -> X69,\nX21 -> s(X69)", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 95 [arrowhead = none ];
95 -> 58;

96 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 96 [arrowhead = none ];
96 -> 59;

97 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
22 -> 97 [arrowhead = none ];
97 -> 24;

98 [label="SPLIT 2\nnew knowledge:\nT24 is ground\nT27 is ground\nreplacements:X22 -> T27", fontsize=14, style = filled, fillcolor = violet];
22 -> 98 [arrowhead = none ];
98 -> 25;

99 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 99 [arrowhead = none ];
99 -> 26;

100 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 100 [arrowhead = none ];
100 -> 44;

101 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (m(s(0), T24, T3), m(X3, 0, X4))", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 101 [arrowhead = none ];
101 -> 27;

102 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 102 [arrowhead = none ];
102 -> 28;

103 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 103 [arrowhead = none ];
103 -> 29;

104 [label="EVAL with clause\np(s(0), 0).\nand substitutionT24 -> s(0),\nX22 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 104 [arrowhead = none ];
104 -> 30;

105 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 105 [arrowhead = none ];
105 -> 31;

106 [label="EVAL with clause\np(s(s(X29)), s(X30)) :- p(s(X29), X30).\nand substitutionX29 -> T30,\nT24 -> s(s(T30)),\nX30 -> X31,\nX22 -> s(X31)", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 106 [arrowhead = none ];
106 -> 33;

107 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 107 [arrowhead = none ];
107 -> 34;

108 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 108 [arrowhead = none ];
108 -> 32;

109 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 109 [arrowhead = none ];
109 -> 35;

110 [label="BACKTRACK\nfor clause: p(0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 110 [arrowhead = none ];
110 -> 36;

111 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 111 [arrowhead = none ];
111 -> 37;

112 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 112 [arrowhead = none ];
112 -> 38;

113 [label="EVAL with clause\np(s(0), 0).\nand substitutionT30 -> 0,\nX31 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 113 [arrowhead = none ];
113 -> 39;

114 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 114 [arrowhead = none ];
114 -> 40;

115 [label="EVAL with clause\np(s(s(X38)), s(X39)) :- p(s(X38), X39).\nand substitutionX38 -> T33,\nT30 -> s(T33),\nX39 -> X40,\nX31 -> s(X40)", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 115 [arrowhead = none ];
115 -> 42;

116 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 116 [arrowhead = none ];
116 -> 43;

117 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 117 [arrowhead = none ];
117 -> 41;

118 [label="INSTANCE with matching:\nT30 -> T33\nX31 -> X40", fontsize=14, style = filled, fillcolor = lightgrey];
42 -> 118 [arrowhead = none , style=dashed];
118 -> 33[style=dashed];

119 [label="EVAL with clause\nm(X47, 0, X48) :- ','(!_7, user_defined_=(X48, X47)).\nand substitutionX47 -> 0,\nT27 -> 0,\nT26 -> T38,\nX48 -> T38,\nT37 -> T38", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 119 [arrowhead = none ];
119 -> 45;

120 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 120 [arrowhead = none ];
120 -> 46;

121 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 121 [arrowhead = none ];
121 -> 47;

122 [label="ONLY EVAL with clause\nm(0, X56, X57) :- ','(!_7, user_defined_=(X57, 0)).\nand substitutionT27 -> T47,\nX56 -> T47,\nT26 -> T49,\nX57 -> T49,\nT48 -> T49", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 122 [arrowhead = none ];
122 -> 52;

123 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
47 -> 123 [arrowhead = none ];
123 -> 48;

124 [label="EVAL with clause\nuser_defined_=(X51, X51).\nand substitutionT38 -> 0,\nX51 -> 0,\nT41 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 124 [arrowhead = none ];
124 -> 49;

125 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 125 [arrowhead = none ];
125 -> 50;

126 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 126 [arrowhead = none ];
126 -> 51;

127 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 127 [arrowhead = none ];
127 -> 53;

128 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
53 -> 128 [arrowhead = none ];
128 -> 54;

129 [label="EVAL with clause\nuser_defined_=(X60, X60).\nand substitutionT49 -> 0,\nX60 -> 0,\nT52 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 129 [arrowhead = none ];
129 -> 55;

130 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 130 [arrowhead = none ];
130 -> 56;

131 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
55 -> 131 [arrowhead = none ];
131 -> 57;

132 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
58 -> 132 [arrowhead = none ];
132 -> 60;

133 [label="SPLIT 2\nnew knowledge:\nT55 is ground\nT56 is ground\nreplacements:X69 -> T56", fontsize=14, style = filled, fillcolor = violet];
58 -> 133 [arrowhead = none ];
133 -> 61;

134 [label="INSTANCE with matching:\nT30 -> T55\nX31 -> X69", fontsize=14, style = filled, fillcolor = lightgrey];
60 -> 134 [arrowhead = none , style=dashed];
134 -> 33[style=dashed];

135 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
61 -> 135 [arrowhead = none ];
135 -> 62;

136 [label="SPLIT 2\nnew knowledge:\nT24 is ground\nT57 is ground\nreplacements:X22 -> T57", fontsize=14, style = filled, fillcolor = violet];
61 -> 136 [arrowhead = none ];
136 -> 63;

137 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
62 -> 137 [arrowhead = none ];
137 -> 64;

138 [label="INSTANCE with matching:\nT1 -> s(T56)\nT2 -> T57\nT3 -> T26", fontsize=14, style = filled, fillcolor = lightgrey];
63 -> 138 [arrowhead = none , style=dashed];
138 -> 1[style=dashed];

139 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (m(s(s(T55)), T24, T3), m(X3, 0, X4))", fontsize=14, style = filled, fillcolor = lightgreen];
64 -> 139 [arrowhead = none ];
139 -> 65;

140 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
65 -> 140 [arrowhead = none ];
140 -> 66;

141 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
65 -> 141 [arrowhead = none ];
141 -> 67;

142 [label="EVAL with clause\np(s(0), 0).\nand substitutionT24 -> s(0),\nX22 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
66 -> 142 [arrowhead = none ];
142 -> 68;

143 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
66 -> 143 [arrowhead = none ];
143 -> 69;

144 [label="EVAL with clause\np(s(s(X76)), s(X77)) :- p(s(X76), X77).\nand substitutionX76 -> T60,\nT24 -> s(s(T60)),\nX77 -> X78,\nX22 -> s(X78)", fontsize=14, style = filled, fillcolor = lightblue];
67 -> 144 [arrowhead = none ];
144 -> 71;

145 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
67 -> 145 [arrowhead = none ];
145 -> 72;

146 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
68 -> 146 [arrowhead = none ];
146 -> 70;

147 [label="INSTANCE with matching:\nT30 -> T60\nX31 -> X78", fontsize=14, style = filled, fillcolor = lightgrey];
71 -> 147 [arrowhead = none , style=dashed];
147 -> 33[style=dashed];

}

(4) Obligation:

Triples:

pA(s(X1), s(X2)) :- pA(X1, X2).
mB(s(0), s(s(X1)), X2) :- pA(X1, X3).
mB(s(s(X1)), X2, X3) :- pA(X1, X4).
mB(s(s(X1)), s(s(X2)), X3) :- ','(pcA(X1, X4), pA(X2, X5)).
mB(s(s(X1)), X2, X3) :- ','(pcA(X1, X4), ','(pcD(X2, X5), mB(s(X4), X5, X3))).

Clauses:

pcA(0, 0).
pcA(s(X1), s(X2)) :- pcA(X1, X2).
mcB(X1, 0, X1).
mcB(0, X1, 0).
mcB(s(0), X1, 0) :- pcC(X1, 0).
mcB(s(0), X1, 0) :- pcC(X1, X2).
mcB(s(s(X1)), X2, X3) :- ','(pcA(X1, X4), ','(pcD(X2, X5), mcB(s(X4), X5, X3))).
pcC(s(0), 0).
pcC(s(s(X1)), s(X2)) :- pcA(X1, X2).
pcD(s(0), 0).
pcD(s(s(X1)), s(X2)) :- pcA(X1, X2).

Afs:

mB(x1, x2, x3)  =  mB(x1, x2)

(5) TriplesToPiDPProof (SOUND transformation)

We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
mB_in: (b,b,f)
pA_in: (b,f)
pcA_in: (b,f)
pcD_in: (b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

MB_IN_GGA(s(0), s(s(X1)), X2) → U2_GGA(X1, X2, pA_in_ga(X1, X3))
MB_IN_GGA(s(0), s(s(X1)), X2) → PA_IN_GA(X1, X3)
PA_IN_GA(s(X1), s(X2)) → U1_GA(X1, X2, pA_in_ga(X1, X2))
PA_IN_GA(s(X1), s(X2)) → PA_IN_GA(X1, X2)
MB_IN_GGA(s(s(X1)), X2, X3) → U3_GGA(X1, X2, X3, pA_in_ga(X1, X4))
MB_IN_GGA(s(s(X1)), X2, X3) → PA_IN_GA(X1, X4)
MB_IN_GGA(s(s(X1)), s(s(X2)), X3) → U4_GGA(X1, X2, X3, pcA_in_ga(X1, X4))
U4_GGA(X1, X2, X3, pcA_out_ga(X1, X4)) → U5_GGA(X1, X2, X3, pA_in_ga(X2, X5))
U4_GGA(X1, X2, X3, pcA_out_ga(X1, X4)) → PA_IN_GA(X2, X5)
MB_IN_GGA(s(s(X1)), X2, X3) → U6_GGA(X1, X2, X3, pcA_in_ga(X1, X4))
U6_GGA(X1, X2, X3, pcA_out_ga(X1, X4)) → U7_GGA(X1, X2, X3, X4, pcD_in_ga(X2, X5))
U7_GGA(X1, X2, X3, X4, pcD_out_ga(X2, X5)) → U8_GGA(X1, X2, X3, mB_in_gga(s(X4), X5, X3))
U7_GGA(X1, X2, X3, X4, pcD_out_ga(X2, X5)) → MB_IN_GGA(s(X4), X5, X3)

The TRS R consists of the following rules:

pcA_in_ga(0, 0) → pcA_out_ga(0, 0)
pcA_in_ga(s(X1), s(X2)) → U10_ga(X1, X2, pcA_in_ga(X1, X2))
U10_ga(X1, X2, pcA_out_ga(X1, X2)) → pcA_out_ga(s(X1), s(X2))
pcD_in_ga(s(0), 0) → pcD_out_ga(s(0), 0)
pcD_in_ga(s(s(X1)), s(X2)) → U17_ga(X1, X2, pcA_in_ga(X1, X2))
U17_ga(X1, X2, pcA_out_ga(X1, X2)) → pcD_out_ga(s(s(X1)), s(X2))

The argument filtering Pi contains the following mapping:
mB_in_gga(x1, x2, x3)  =  mB_in_gga(x1, x2)
s(x1)  =  s(x1)
0  =  0
pA_in_ga(x1, x2)  =  pA_in_ga(x1)
pcA_in_ga(x1, x2)  =  pcA_in_ga(x1)
pcA_out_ga(x1, x2)  =  pcA_out_ga(x1, x2)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
pcD_in_ga(x1, x2)  =  pcD_in_ga(x1)
pcD_out_ga(x1, x2)  =  pcD_out_ga(x1, x2)
U17_ga(x1, x2, x3)  =  U17_ga(x1, x3)
MB_IN_GGA(x1, x2, x3)  =  MB_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3)  =  U2_GGA(x1, x3)
PA_IN_GA(x1, x2)  =  PA_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x1, x2, x4)
U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x1, x2, x4)
U7_GGA(x1, x2, x3, x4, x5)  =  U7_GGA(x1, x2, x4, x5)
U8_GGA(x1, x2, x3, x4)  =  U8_GGA(x1, x2, x4)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(6) Obligation:

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

MB_IN_GGA(s(0), s(s(X1)), X2) → U2_GGA(X1, X2, pA_in_ga(X1, X3))
MB_IN_GGA(s(0), s(s(X1)), X2) → PA_IN_GA(X1, X3)
PA_IN_GA(s(X1), s(X2)) → U1_GA(X1, X2, pA_in_ga(X1, X2))
PA_IN_GA(s(X1), s(X2)) → PA_IN_GA(X1, X2)
MB_IN_GGA(s(s(X1)), X2, X3) → U3_GGA(X1, X2, X3, pA_in_ga(X1, X4))
MB_IN_GGA(s(s(X1)), X2, X3) → PA_IN_GA(X1, X4)
MB_IN_GGA(s(s(X1)), s(s(X2)), X3) → U4_GGA(X1, X2, X3, pcA_in_ga(X1, X4))
U4_GGA(X1, X2, X3, pcA_out_ga(X1, X4)) → U5_GGA(X1, X2, X3, pA_in_ga(X2, X5))
U4_GGA(X1, X2, X3, pcA_out_ga(X1, X4)) → PA_IN_GA(X2, X5)
MB_IN_GGA(s(s(X1)), X2, X3) → U6_GGA(X1, X2, X3, pcA_in_ga(X1, X4))
U6_GGA(X1, X2, X3, pcA_out_ga(X1, X4)) → U7_GGA(X1, X2, X3, X4, pcD_in_ga(X2, X5))
U7_GGA(X1, X2, X3, X4, pcD_out_ga(X2, X5)) → U8_GGA(X1, X2, X3, mB_in_gga(s(X4), X5, X3))
U7_GGA(X1, X2, X3, X4, pcD_out_ga(X2, X5)) → MB_IN_GGA(s(X4), X5, X3)

The TRS R consists of the following rules:

pcA_in_ga(0, 0) → pcA_out_ga(0, 0)
pcA_in_ga(s(X1), s(X2)) → U10_ga(X1, X2, pcA_in_ga(X1, X2))
U10_ga(X1, X2, pcA_out_ga(X1, X2)) → pcA_out_ga(s(X1), s(X2))
pcD_in_ga(s(0), 0) → pcD_out_ga(s(0), 0)
pcD_in_ga(s(s(X1)), s(X2)) → U17_ga(X1, X2, pcA_in_ga(X1, X2))
U17_ga(X1, X2, pcA_out_ga(X1, X2)) → pcD_out_ga(s(s(X1)), s(X2))

The argument filtering Pi contains the following mapping:
mB_in_gga(x1, x2, x3)  =  mB_in_gga(x1, x2)
s(x1)  =  s(x1)
0  =  0
pA_in_ga(x1, x2)  =  pA_in_ga(x1)
pcA_in_ga(x1, x2)  =  pcA_in_ga(x1)
pcA_out_ga(x1, x2)  =  pcA_out_ga(x1, x2)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
pcD_in_ga(x1, x2)  =  pcD_in_ga(x1)
pcD_out_ga(x1, x2)  =  pcD_out_ga(x1, x2)
U17_ga(x1, x2, x3)  =  U17_ga(x1, x3)
MB_IN_GGA(x1, x2, x3)  =  MB_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3)  =  U2_GGA(x1, x3)
PA_IN_GA(x1, x2)  =  PA_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x1, x2, x4)
U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x1, x2, x4)
U7_GGA(x1, x2, x3, x4, x5)  =  U7_GGA(x1, x2, x4, x5)
U8_GGA(x1, x2, x3, x4)  =  U8_GGA(x1, x2, x4)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 9 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

PA_IN_GA(s(X1), s(X2)) → PA_IN_GA(X1, X2)

The TRS R consists of the following rules:

pcA_in_ga(0, 0) → pcA_out_ga(0, 0)
pcA_in_ga(s(X1), s(X2)) → U10_ga(X1, X2, pcA_in_ga(X1, X2))
U10_ga(X1, X2, pcA_out_ga(X1, X2)) → pcA_out_ga(s(X1), s(X2))
pcD_in_ga(s(0), 0) → pcD_out_ga(s(0), 0)
pcD_in_ga(s(s(X1)), s(X2)) → U17_ga(X1, X2, pcA_in_ga(X1, X2))
U17_ga(X1, X2, pcA_out_ga(X1, X2)) → pcD_out_ga(s(s(X1)), s(X2))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
0  =  0
pcA_in_ga(x1, x2)  =  pcA_in_ga(x1)
pcA_out_ga(x1, x2)  =  pcA_out_ga(x1, x2)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
pcD_in_ga(x1, x2)  =  pcD_in_ga(x1)
pcD_out_ga(x1, x2)  =  pcD_out_ga(x1, x2)
U17_ga(x1, x2, x3)  =  U17_ga(x1, x3)
PA_IN_GA(x1, x2)  =  PA_IN_GA(x1)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

PA_IN_GA(s(X1), s(X2)) → PA_IN_GA(X1, X2)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
PA_IN_GA(x1, x2)  =  PA_IN_GA(x1)

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

PA_IN_GA(s(X1)) → PA_IN_GA(X1)

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

(14) 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:

• PA_IN_GA(s(X1)) → PA_IN_GA(X1)
  The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

MB_IN_GGA(s(s(X1)), X2, X3) → U6_GGA(X1, X2, X3, pcA_in_ga(X1, X4))
U6_GGA(X1, X2, X3, pcA_out_ga(X1, X4)) → U7_GGA(X1, X2, X3, X4, pcD_in_ga(X2, X5))
U7_GGA(X1, X2, X3, X4, pcD_out_ga(X2, X5)) → MB_IN_GGA(s(X4), X5, X3)

The TRS R consists of the following rules:

pcA_in_ga(0, 0) → pcA_out_ga(0, 0)
pcA_in_ga(s(X1), s(X2)) → U10_ga(X1, X2, pcA_in_ga(X1, X2))
U10_ga(X1, X2, pcA_out_ga(X1, X2)) → pcA_out_ga(s(X1), s(X2))
pcD_in_ga(s(0), 0) → pcD_out_ga(s(0), 0)
pcD_in_ga(s(s(X1)), s(X2)) → U17_ga(X1, X2, pcA_in_ga(X1, X2))
U17_ga(X1, X2, pcA_out_ga(X1, X2)) → pcD_out_ga(s(s(X1)), s(X2))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
0  =  0
pcA_in_ga(x1, x2)  =  pcA_in_ga(x1)
pcA_out_ga(x1, x2)  =  pcA_out_ga(x1, x2)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
pcD_in_ga(x1, x2)  =  pcD_in_ga(x1)
pcD_out_ga(x1, x2)  =  pcD_out_ga(x1, x2)
U17_ga(x1, x2, x3)  =  U17_ga(x1, x3)
MB_IN_GGA(x1, x2, x3)  =  MB_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x1, x2, x4)
U7_GGA(x1, x2, x3, x4, x5)  =  U7_GGA(x1, x2, x4, x5)

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

MB_IN_GGA(s(s(X1)), X2) → U6_GGA(X1, X2, pcA_in_ga(X1))
U6_GGA(X1, X2, pcA_out_ga(X1, X4)) → U7_GGA(X1, X2, X4, pcD_in_ga(X2))
U7_GGA(X1, X2, X4, pcD_out_ga(X2, X5)) → MB_IN_GGA(s(X4), X5)

The TRS R consists of the following rules:

pcA_in_ga(0) → pcA_out_ga(0, 0)
pcA_in_ga(s(X1)) → U10_ga(X1, pcA_in_ga(X1))
U10_ga(X1, pcA_out_ga(X1, X2)) → pcA_out_ga(s(X1), s(X2))
pcD_in_ga(s(0)) → pcD_out_ga(s(0), 0)
pcD_in_ga(s(s(X1))) → U17_ga(X1, pcA_in_ga(X1))
U17_ga(X1, pcA_out_ga(X1, X2)) → pcD_out_ga(s(s(X1)), s(X2))

The set Q consists of the following terms:

pcA_in_ga(x0)
U10_ga(x0, x1)
pcD_in_ga(x0)
U17_ga(x0, x1)

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

(19) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

MB_IN_GGA(s(s(X1)), X2) → U6_GGA(X1, X2, pcA_in_ga(X1))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(MB_IN_GGA(x₁, x₂)) = x₁   
POL(U10_ga(x₁, x₂)) = 1 + x₂   
POL(U17_ga(x₁, x₂)) = 0   
POL(U6_GGA(x₁, x₂, x₃)) = 1 + x₃   
POL(U7_GGA(x₁, x₂, x₃, x₄)) = 1 + x₃   
POL(pcA_in_ga(x₁)) = x₁   
POL(pcA_out_ga(x₁, x₂)) = x₂   
POL(pcD_in_ga(x₁)) = 0   
POL(pcD_out_ga(x₁, x₂)) = 0   
POL(s(x₁)) = 1 + x₁   

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

pcA_in_ga(0) → pcA_out_ga(0, 0)
pcA_in_ga(s(X1)) → U10_ga(X1, pcA_in_ga(X1))
U10_ga(X1, pcA_out_ga(X1, X2)) → pcA_out_ga(s(X1), s(X2))

(20) Obligation:

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

U6_GGA(X1, X2, pcA_out_ga(X1, X4)) → U7_GGA(X1, X2, X4, pcD_in_ga(X2))
U7_GGA(X1, X2, X4, pcD_out_ga(X2, X5)) → MB_IN_GGA(s(X4), X5)

The TRS R consists of the following rules:

pcA_in_ga(0) → pcA_out_ga(0, 0)
pcA_in_ga(s(X1)) → U10_ga(X1, pcA_in_ga(X1))
U10_ga(X1, pcA_out_ga(X1, X2)) → pcA_out_ga(s(X1), s(X2))
pcD_in_ga(s(0)) → pcD_out_ga(s(0), 0)
pcD_in_ga(s(s(X1))) → U17_ga(X1, pcA_in_ga(X1))
U17_ga(X1, pcA_out_ga(X1, X2)) → pcD_out_ga(s(s(X1)), s(X2))

The set Q consists of the following terms:

pcA_in_ga(x0)
U10_ga(x0, x1)
pcD_in_ga(x0)
U17_ga(x0, x1)

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

(21) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(22) TRUE