Left Termination of the query pattern q_in_3(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(X), X).
q(X, Y, Z) :- m(X, Y, Z).
q(X, Y, Z).
=(X, X).

Queries:

q(g,g,a).

(1) BuiltinConflictTransformerProof (SOUND 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(X), X).
q(X, Y, Z) :- m(X, Y, Z).
q(X, Y, Z).
user_defined_=(X, X).

Queries:

q(g,g,a).

(3) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: q(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: q(T1, T2, T3), SCOPE: 1, CLAUSE: 5 | q(T1, T2, T3), SCOPE: 1, CLAUSE: 6\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: m(T7, T8, T10) | q(T7, T8, T3), SCOPE: 1, CLAUSE: 6\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: m(T7, T8, T10), SCOPE: 2, CLAUSE: 0 | m(T7, T8, T10), SCOPE: 2, CLAUSE: 1 | m(T7, T8, T10), SCOPE: 2, CLAUSE: 2 | ?_2 | q(T7, T8, T3), SCOPE: 1, CLAUSE: 6\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(!_2, user_defined_=(T17, T15)) | m(T15, 0, T10), SCOPE: 2, CLAUSE: 1 | m(T15, 0, T10), SCOPE: 2, CLAUSE: 2 | ?_2 | q(T15, 0, T3), SCOPE: 1, CLAUSE: 6\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: m(T7, T8, T10), SCOPE: 2, CLAUSE: 1 | m(T7, T8, T10), SCOPE: 2, CLAUSE: 2 | ?_2 | q(T7, T8, T3), SCOPE: 1, CLAUSE: 6\n\nT7 is ground\nT8 is ground\nm(T7, T8, T10) !=? m(X11, 0, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: user_defined_=(T17, T15) | q(T15, 0, T3), SCOPE: 1, CLAUSE: 6\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: user_defined_=(T17, T15)\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: q(T15, 0, T3), SCOPE: 1, CLAUSE: 6\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: user_defined_=(T17, T15), SCOPE: 3, CLAUSE: 7\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \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: ','(!_2, user_defined_=(T33, 0)) | m(0, T31, T10), SCOPE: 2, CLAUSE: 2 | ?_2 | q(0, T31, T3), SCOPE: 1, CLAUSE: 6\n\nT31 is ground\nm(0, T31, T10) !=? m(X11, 0, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: m(T7, T8, T10), SCOPE: 2, CLAUSE: 2 | ?_2 | q(T7, T8, T3), SCOPE: 1, CLAUSE: 6\n\nT7 is ground\nT8 is ground\nm(T7, T8, T10) !=? m(X11, 0, X12)\nm(T7, T8, T10) !=? m(0, X32, X33)", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: user_defined_=(T33, 0) | q(0, T31, T3), SCOPE: 1, CLAUSE: 6\n\nT31 is ground\nm(0, T31, T10) !=? m(X11, 0, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: user_defined_=(T33, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: q(0, T31, T3), SCOPE: 1, CLAUSE: 6\n\nT31 is ground\nm(0, T31, T10) !=? m(X11, 0, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: user_defined_=(T33, 0), SCOPE: 4, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", 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: m(T7, T8, T10), SCOPE: 2, CLAUSE: 2\n\nT7 is ground\nT8 is ground\nm(T7, T8, T10) !=? m(X11, 0, X12)\nm(T7, T8, T10) !=? m(0, X32, X33)", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ?_2 | q(T7, T8, T3), SCOPE: 1, CLAUSE: 6\n\nT7 is ground\nT8 is ground\nm(T7, T8, T10) !=? m(X11, 0, X12)\nm(T7, T8, T10) !=? m(0, X32, X33)", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(p(T59, X74), ','(p(T60, X75), m(X74, X75, T62)))\n\nT59 is ground\nT60 is ground\nX74 is free; \nX75 is free; \nm(T59, T60, T10) !=? m(X11, 0, X12)\nm(T59, T60, T10) !=? m(0, X32, X33)", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(p(T59, X74), ','(p(T60, X75), m(X74, X75, T62))), SCOPE: 5, CLAUSE: 3 | ','(p(T59, X74), ','(p(T60, X75), m(X74, X75, T62))), SCOPE: 5, CLAUSE: 4\n\nT59 is ground\nT60 is ground\nX74 is free; \nX75 is free; \nm(T59, T60, T10) !=? m(X11, 0, X12)\nm(T59, T60, T10) !=? m(0, X32, X33)", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(p(T59, X74), ','(p(T60, X75), m(X74, X75, T62))), SCOPE: 5, CLAUSE: 4\n\nT59 is ground\nT60 is ground\nX74 is free; \nX75 is free; \nm(T59, T60, T10) !=? m(X11, 0, X12)\nm(T59, T60, T10) !=? m(0, X32, X33)", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(p(T60, X75), m(T66, X75, T62))\n\nT60 is ground\nT66 is ground\nX75 is free; \nm(s(T66), T60, T10) !=? m(X11, 0, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(p(T60, X75), m(T66, X75, T62)), SCOPE: 6, CLAUSE: 3 | ','(p(T60, X75), m(T66, X75, T62)), SCOPE: 6, CLAUSE: 4\n\nT60 is ground\nT66 is ground\nX75 is free; \nm(s(T66), T60, T10) !=? m(X11, 0, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(p(T60, X75), m(T66, X75, T62)), SCOPE: 6, CLAUSE: 4\n\nT60 is ground\nT66 is ground\nX75 is free; \nm(s(T66), T60, T10) !=? m(X11, 0, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: m(T66, T71, T62)\n\nT66 is ground\nT71 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: m(T66, T71, T62), SCOPE: 7, CLAUSE: 0 | m(T66, T71, T62), SCOPE: 7, CLAUSE: 1 | m(T66, T71, T62), SCOPE: 7, CLAUSE: 2\n\nT66 is ground\nT71 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(!_7, user_defined_=(T80, T78)) | m(T78, 0, T62), SCOPE: 7, CLAUSE: 1 | m(T78, 0, T62), SCOPE: 7, CLAUSE: 2\n\nT78 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: m(T66, T71, T62), SCOPE: 7, CLAUSE: 1 | m(T66, T71, T62), SCOPE: 7, CLAUSE: 2\n\nT66 is ground\nT71 is ground\nm(T66, T71, T62) !=? m(X95, 0, X96)", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: user_defined_=(T80, T78)\n\nT78 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: user_defined_=(T80, T78), SCOPE: 8, CLAUSE: 7\n\nT78 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(!_7, user_defined_=(T90, 0)) | m(0, T88, T62), SCOPE: 7, CLAUSE: 2\n\nT88 is ground\nm(0, T88, T62) !=? m(X95, 0, X96)", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: m(T66, T71, T62), SCOPE: 7, CLAUSE: 2\n\nT66 is ground\nT71 is ground\nm(T66, T71, T62) !=? m(X95, 0, X96)\nm(T66, T71, T62) !=? m(0, X107, X108)", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: user_defined_=(T90, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: user_defined_=(T90, 0), SCOPE: 9, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(p(T102, X129), ','(p(T103, X130), m(X129, X130, T105)))\n\nT102 is ground\nT103 is ground\nX129 is free; \nX130 is free; \nm(T102, T103, T62) !=? m(X95, 0, X96)\nm(T102, T103, T62) !=? m(0, X107, X108)", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(p(T102, X129), ','(p(T103, X130), m(X129, X130, T105))), SCOPE: 10, CLAUSE: 3 | ','(p(T102, X129), ','(p(T103, X130), m(X129, X130, T105))), SCOPE: 10, CLAUSE: 4\n\nT102 is ground\nT103 is ground\nX129 is free; \nX130 is free; \nm(T102, T103, T62) !=? m(X95, 0, X96)\nm(T102, T103, T62) !=? m(0, X107, X108)", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(p(T102, X129), ','(p(T103, X130), m(X129, X130, T105))), SCOPE: 10, CLAUSE: 4\n\nT102 is ground\nT103 is ground\nX129 is free; \nX130 is free; \nm(T102, T103, T62) !=? m(X95, 0, X96)\nm(T102, T103, T62) !=? m(0, X107, X108)", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ','(p(T103, X130), m(T109, X130, T105))\n\nT103 is ground\nT109 is ground\nX130 is free; \nm(s(T109), T103, T62) !=? m(X95, 0, X96)", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(p(T103, X130), m(T109, X130, T105)), SCOPE: 11, CLAUSE: 3 | ','(p(T103, X130), m(T109, X130, T105)), SCOPE: 11, CLAUSE: 4\n\nT103 is ground\nT109 is ground\nX130 is free; \nm(s(T109), T103, T62) !=? m(X95, 0, X96)", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(p(T103, X130), m(T109, X130, T105)), SCOPE: 11, CLAUSE: 4\n\nT103 is ground\nT109 is ground\nX130 is free; \nm(s(T109), T103, T62) !=? m(X95, 0, X96)", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: m(T109, T114, T105)\n\nT109 is ground\nT114 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: q(T7, T8, T3), SCOPE: 1, CLAUSE: 6\n\nT7 is ground\nT8 is ground\nm(T7, T8, T10) !=? m(X11, 0, X12)\nm(T7, T8, T10) !=? m(0, X32, X33)", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 65 [arrowhead = none ];
65 -> 2;

66 [label="ONLY EVAL with clause\nq(X4, X5, X6) :- m(X4, X5, X6).\nand substitution\nT1 -> T7\nX4 -> T7\nT2 -> T8\nX5 -> T8\nT3 -> T10\nX6 -> T10\nT9 -> T10", fontsize=14, style = filled, fillcolor = white];
2 -> 66 [arrowhead = none ];
66 -> 3;

67 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 67 [arrowhead = none ];
67 -> 4;

68 [label="EVAL with clause\nm(X11, 0, X12) :- ','(!_2, user_defined_=(X12, X11)).\nand substitution\nT7 -> T15\nX11 -> T15\nT8 -> 0\nT10 -> T17\nX12 -> T17\nT16 -> T17", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 68 [arrowhead = none ];
68 -> 5;

69 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 69 [arrowhead = none ];
69 -> 6;

70 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 70 [arrowhead = none ];
70 -> 7;

71 [label="EVAL with clause\nm(0, X32, X33) :- ','(!_2, user_defined_=(X33, 0)).\nand substitution\nT7 -> 0\nT8 -> T31\nX32 -> T31\nT10 -> T33\nX33 -> T33\nT32 -> T33", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 71 [arrowhead = none ];
71 -> 16;

72 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 72 [arrowhead = none ];
72 -> 17;

73 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 73 [arrowhead = none ];
73 -> 8;

74 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 74 [arrowhead = none ];
74 -> 9;

75 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
8 -> 75 [arrowhead = none ];
75 -> 10;

76 [label="ONLY EVAL with clause\nq(X25, X26, X27).\nand substitution\nT15 -> T25\nX25 -> T25\nX26 -> 0\nT3 -> T26\nX27 -> T26", fontsize=14, style = filled, fillcolor = white];
9 -> 76 [arrowhead = none ];
76 -> 14;

77 [label="EVAL with clause\nuser_defined_=(X17, X17).\nand substitution\nT17 -> T20\nX17 -> T20\nT15 -> T20", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 77 [arrowhead = none ];
77 -> 11;

78 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 78 [arrowhead = none ];
78 -> 12;

79 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 79 [arrowhead = none ];
79 -> 13;

80 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 80 [arrowhead = none ];
80 -> 15;

81 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 81 [arrowhead = none ];
81 -> 18;

82 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
17 -> 82 [arrowhead = none ];
82 -> 27;

83 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
17 -> 83 [arrowhead = none ];
83 -> 28;

84 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 84 [arrowhead = none ];
84 -> 19;

85 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 85 [arrowhead = none ];
85 -> 20;

86 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 86 [arrowhead = none ];
86 -> 21;

87 [label="ONLY EVAL with clause\nq(X46, X47, X48).\nand substitution\nX46 -> 0\nT31 -> T41\nX47 -> T41\nT3 -> T42\nX48 -> T42", fontsize=14, style = filled, fillcolor = white];
20 -> 87 [arrowhead = none ];
87 -> 25;

88 [label="EVAL with clause\nuser_defined_=(X38, X38).\nand substitution\nT33 -> 0\nX38 -> 0\nT36 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 88 [arrowhead = none ];
88 -> 22;

89 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 89 [arrowhead = none ];
89 -> 23;

90 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 90 [arrowhead = none ];
90 -> 24;

91 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 91 [arrowhead = none ];
91 -> 26;

92 [label="ONLY EVAL with clause\nm(X71, X72, X73) :- ','(p(X71, X74), ','(p(X72, X75), m(X74, X75, X73))).\nand substitution\nT7 -> T59\nX71 -> T59\nT8 -> T60\nX72 -> T60\nT10 -> T62\nX73 -> T62\nT61 -> T62", fontsize=14, style = filled, fillcolor = white];
27 -> 92 [arrowhead = none ];
92 -> 29;

93 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 93 [arrowhead = none ];
93 -> 62;

94 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 94 [arrowhead = none ];
94 -> 30;

95 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (m(T59, T60, T10), m(0, X32, X33))", fontsize=14, style = filled, fillcolor = white];
30 -> 95 [arrowhead = none ];
95 -> 31;

96 [label="EVAL with clause\np(s(X81), X81).\nand substitution\nX81 -> T66\nT59 -> s(T66)\nX74 -> T66", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 96 [arrowhead = none ];
96 -> 32;

97 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 97 [arrowhead = none ];
97 -> 33;

98 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
32 -> 98 [arrowhead = none ];
98 -> 34;

99 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (m(s(T66), T60, T10), m(X11, 0, X12))", fontsize=14, style = filled, fillcolor = white];
34 -> 99 [arrowhead = none ];
99 -> 35;

100 [label="EVAL with clause\np(s(X87), X87).\nand substitution\nX87 -> T71\nT60 -> s(T71)\nX75 -> T71", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 100 [arrowhead = none ];
100 -> 36;

101 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 101 [arrowhead = none ];
101 -> 37;

102 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
36 -> 102 [arrowhead = none ];
102 -> 38;

103 [label="EVAL with clause\nm(X95, 0, X96) :- ','(!_7, user_defined_=(X96, X95)).\nand substitution\nT66 -> T78\nX95 -> T78\nT71 -> 0\nT62 -> T80\nX96 -> T80\nT79 -> T80", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 103 [arrowhead = none ];
103 -> 39;

104 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 104 [arrowhead = none ];
104 -> 40;

105 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 105 [arrowhead = none ];
105 -> 41;

106 [label="EVAL with clause\nm(0, X107, X108) :- ','(!_7, user_defined_=(X108, 0)).\nand substitution\nT66 -> 0\nT71 -> T88\nX107 -> T88\nT62 -> T90\nX108 -> T90\nT89 -> T90", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 106 [arrowhead = none ];
106 -> 46;

107 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 107 [arrowhead = none ];
107 -> 47;

108 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
41 -> 108 [arrowhead = none ];
108 -> 42;

109 [label="EVAL with clause\nuser_defined_=(X101, X101).\nand substitution\nT80 -> T83\nX101 -> T83\nT78 -> T83", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 109 [arrowhead = none ];
109 -> 43;

110 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 110 [arrowhead = none ];
110 -> 44;

111 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 111 [arrowhead = none ];
111 -> 45;

112 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 112 [arrowhead = none ];
112 -> 48;

113 [label="ONLY EVAL with clause\nm(X126, X127, X128) :- ','(p(X126, X129), ','(p(X127, X130), m(X129, X130, X128))).\nand substitution\nT66 -> T102\nX126 -> T102\nT71 -> T103\nX127 -> T103\nT62 -> T105\nX128 -> T105\nT104 -> T105", fontsize=14, style = filled, fillcolor = white];
47 -> 113 [arrowhead = none ];
113 -> 53;

114 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
48 -> 114 [arrowhead = none ];
114 -> 49;

115 [label="EVAL with clause\nuser_defined_=(X113, X113).\nand substitution\nT90 -> 0\nX113 -> 0\nT93 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 115 [arrowhead = none ];
115 -> 50;

116 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 116 [arrowhead = none ];
116 -> 51;

117 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 117 [arrowhead = none ];
117 -> 52;

118 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
53 -> 118 [arrowhead = none ];
118 -> 54;

119 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (m(T102, T103, T62), m(0, X107, X108))", fontsize=14, style = filled, fillcolor = white];
54 -> 119 [arrowhead = none ];
119 -> 55;

120 [label="EVAL with clause\np(s(X136), X136).\nand substitution\nX136 -> T109\nT102 -> s(T109)\nX129 -> T109", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 120 [arrowhead = none ];
120 -> 56;

121 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 121 [arrowhead = none ];
121 -> 57;

122 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
56 -> 122 [arrowhead = none ];
122 -> 58;

123 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (m(s(T109), T103, T62), m(X95, 0, X96))", fontsize=14, style = filled, fillcolor = white];
58 -> 123 [arrowhead = none ];
123 -> 59;

124 [label="EVAL with clause\np(s(X142), X142).\nand substitution\nX142 -> T114\nT103 -> s(T114)\nX130 -> T114", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 124 [arrowhead = none ];
124 -> 60;

125 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 125 [arrowhead = none ];
125 -> 61;

126 [label="INSTANCE with matching:\nT66 -> T109\nT71 -> T114\nT62 -> T105", fontsize=14, style = filled, fillcolor = lightgrey];
60 -> 126 [arrowhead = none , style=dashed];
126 -> 36[style=dashed];

127 [label="ONLY EVAL with clause\nq(X152, X153, X154).\nand substitution\nT7 -> T123\nX152 -> T123\nT8 -> T124\nX153 -> T124\nT3 -> T125\nX154 -> T125", fontsize=14, style = filled, fillcolor = white];
62 -> 127 [arrowhead = none ];
127 -> 63;

128 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
63 -> 128 [arrowhead = none ];
128 -> 64;

}

(4) Obligation:

Triples:

m36(s(T109), s(T114), T105) :- m36(T109, T114, T105).
q1(s(T66), s(T71), T62) :- m36(T66, T71, T62).

Clauses:

mc36(T83, 0, T83).
mc36(0, T88, 0).
mc36(s(T109), s(T114), T105) :- mc36(T109, T114, T105).

Afs:

q1(x1, x2, x3)  =  q1(x1, x2)

(5) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
q1_in: (b,b,f)
m36_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

Q1_IN_GGA(s(T66), s(T71), T62) → U2_GGA(T66, T71, T62, m36_in_gga(T66, T71, T62))
Q1_IN_GGA(s(T66), s(T71), T62) → M36_IN_GGA(T66, T71, T62)
M36_IN_GGA(s(T109), s(T114), T105) → U1_GGA(T109, T114, T105, m36_in_gga(T109, T114, T105))
M36_IN_GGA(s(T109), s(T114), T105) → M36_IN_GGA(T109, T114, T105)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
m36_in_gga(x1, x2, x3)  =  m36_in_gga(x1, x2)
Q1_IN_GGA(x1, x2, x3)  =  Q1_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
M36_IN_GGA(x1, x2, x3)  =  M36_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_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:

Q1_IN_GGA(s(T66), s(T71), T62) → U2_GGA(T66, T71, T62, m36_in_gga(T66, T71, T62))
Q1_IN_GGA(s(T66), s(T71), T62) → M36_IN_GGA(T66, T71, T62)
M36_IN_GGA(s(T109), s(T114), T105) → U1_GGA(T109, T114, T105, m36_in_gga(T109, T114, T105))
M36_IN_GGA(s(T109), s(T114), T105) → M36_IN_GGA(T109, T114, T105)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
m36_in_gga(x1, x2, x3)  =  m36_in_gga(x1, x2)
Q1_IN_GGA(x1, x2, x3)  =  Q1_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
M36_IN_GGA(x1, x2, x3)  =  M36_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_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 1 SCC with 3 less nodes.

(8) Obligation:

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

M36_IN_GGA(s(T109), s(T114), T105) → M36_IN_GGA(T109, T114, T105)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
M36_IN_GGA(x1, x2, x3)  =  M36_IN_GGA(x1, x2)

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

(9) PiDPToQDPProof (SOUND transformation)

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

(10) Obligation:

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

M36_IN_GGA(s(T109), s(T114)) → M36_IN_GGA(T109, T114)

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

(11) 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:

• M36_IN_GGA(s(T109), s(T114)) → M36_IN_GGA(T109, T114)
  The graph contains the following edges 1 > 1, 2 > 2

(12) YES