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

(0) Obligation:

Clauses:

gcd(X, Y, D) :- ','(le(X, Y), gcd_le(X, Y, D)).
gcd(X, Y, D) :- ','(gt(X, Y), gcd_le(Y, X, D)).
gcd_le(0, Y, Y).
gcd_le(s(X), Y, D) :- ','(add(s(X), Z, Y), gcd(s(X), Z, D)).
gt(s(X), s(Y)) :- gt(X, Y).
gt(s(X), 0).
le(s(X), s(Y)) :- le(X, Y).
le(0, s(Y)).
le(0, 0).
add(s(X), Y, s(Z)) :- add(X, Y, Z).
add(0, X, X).

Query: gcd(g,g,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="B: gcd(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: gcd(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | gcd(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(le(T7, T8), gcd_le(T7, T8, T10)) | gcd(T7, T8, T3), SCOPE: 1, CLAUSE: 1\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(le(T7, T8), gcd_le(T7, T8, T10)), SCOPE: 2, CLAUSE: 6 | ','(le(T7, T8), gcd_le(T7, T8, T10)), SCOPE: 2, CLAUSE: 7 | ','(le(T7, T8), gcd_le(T7, T8, T10)), SCOPE: 2, CLAUSE: 8 | ?_2 | gcd(T7, T8, T3), SCOPE: 1, CLAUSE: 1\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(le(T7, T8), gcd_le(T7, T8, T10)), SCOPE: 2, CLAUSE: 6\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(le(T7, T8), gcd_le(T7, T8, T10)), SCOPE: 2, CLAUSE: 7 | ','(le(T7, T8), gcd_le(T7, T8, T10)), SCOPE: 2, CLAUSE: 8 | ?_2 | gcd(T7, T8, T3), SCOPE: 1, CLAUSE: 1\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(le(T19, T20), gcd_le(s(T19), s(T20), T10))\n\nT19 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="A: le(T19, T20)\n\nT19 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: gcd_le(s(T19), s(T20), T10)\n\nT19 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: le(T19, T20), SCOPE: 3, CLAUSE: 6 | le(T19, T20), SCOPE: 3, CLAUSE: 7 | le(T19, T20), SCOPE: 3, CLAUSE: 8\n\nT19 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: le(T19, T20), SCOPE: 3, CLAUSE: 6\n\nT19 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: le(T19, T20), SCOPE: 3, CLAUSE: 7 | le(T19, T20), SCOPE: 3, CLAUSE: 8\n\nT19 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: le(T33, T34)\n\nT33 is ground\nT34 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: le(T19, T20), SCOPE: 3, CLAUSE: 7\n\nT19 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: le(T19, T20), SCOPE: 3, CLAUSE: 8\n\nT19 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: gcd_le(s(T19), s(T20), T10), SCOPE: 4, CLAUSE: 2 | gcd_le(s(T19), s(T20), T10), SCOPE: 4, CLAUSE: 3\n\nT19 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: gcd_le(s(T19), s(T20), T10), SCOPE: 4, CLAUSE: 3\n\nT19 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(add(s(T51), X55, s(T52)), gcd(s(T51), X55, T54))\n\nT51 is ground\nT52 is ground\nX55 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="C: add(s(T51), X55, s(T52))\n\nT51 is ground\nT52 is ground\nX55 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: gcd(s(T51), T57, T54)\n\nT51 is ground\nT57 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: add(s(T51), X55, s(T52)), SCOPE: 5, CLAUSE: 9 | add(s(T51), X55, s(T52)), SCOPE: 5, CLAUSE: 10\n\nT51 is ground\nT52 is ground\nX55 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: add(s(T51), X55, s(T52)), SCOPE: 5, CLAUSE: 9\n\nT51 is ground\nT52 is ground\nX55 is free", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: add(s(T51), X55, s(T52)), SCOPE: 5, CLAUSE: 10\n\nT51 is ground\nT52 is ground\nX55 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="E: add(T72, X91, T73)\n\nT72 is ground\nT73 is ground\nX91 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: add(T72, X91, T73), SCOPE: 6, CLAUSE: 9 | add(T72, X91, T73), SCOPE: 6, CLAUSE: 10\n\nT72 is ground\nT73 is ground\nX91 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: add(T72, X91, T73), SCOPE: 6, CLAUSE: 9\n\nT72 is ground\nT73 is ground\nX91 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: add(T72, X91, T73), SCOPE: 6, CLAUSE: 10\n\nT72 is ground\nT73 is ground\nX91 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: add(T84, X115, T85)\n\nT84 is ground\nT85 is ground\nX115 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \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: ','(le(T7, T8), gcd_le(T7, T8, T10)), SCOPE: 2, CLAUSE: 7\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(le(T7, T8), gcd_le(T7, T8, T10)), SCOPE: 2, CLAUSE: 8 | ?_2 | gcd(T7, T8, T3), SCOPE: 1, CLAUSE: 1\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: gcd_le(0, s(T98), T10)\n\nT98 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: gcd_le(0, s(T98), T10), SCOPE: 7, CLAUSE: 2 | gcd_le(0, s(T98), T10), SCOPE: 7, CLAUSE: 3\n\nT98 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: gcd_le(0, s(T98), T10), SCOPE: 7, CLAUSE: 2\n\nT98 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: gcd_le(0, s(T98), T10), SCOPE: 7, CLAUSE: 3\n\nT98 is ground", 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: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(le(T7, T8), gcd_le(T7, T8, T10)), SCOPE: 2, CLAUSE: 8\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ?_2 | gcd(T7, T8, T3), SCOPE: 1, CLAUSE: 1\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: gcd_le(0, 0, T10)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: gcd_le(0, 0, T10), SCOPE: 8, CLAUSE: 2 | gcd_le(0, 0, T10), SCOPE: 8, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: gcd_le(0, 0, T10), SCOPE: 8, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: gcd_le(0, 0, T10), SCOPE: 8, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: gcd(T7, T8, T3), SCOPE: 1, CLAUSE: 1\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ','(gt(T115, T116), gcd_le(T116, T115, T118))\n\nT115 is ground\nT116 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="D: gt(T115, T116)\n\nT115 is ground\nT116 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: gcd_le(T116, T115, T118)\n\nT115 is ground\nT116 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: gt(T115, T116), SCOPE: 9, CLAUSE: 4 | gt(T115, T116), SCOPE: 9, CLAUSE: 5\n\nT115 is ground\nT116 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: gt(T115, T116), SCOPE: 9, CLAUSE: 4\n\nT115 is ground\nT116 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: gt(T115, T116), SCOPE: 9, CLAUSE: 5\n\nT115 is ground\nT116 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: gt(T131, T132)\n\nT131 is ground\nT132 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: gcd_le(T116, T115, T118), SCOPE: 10, CLAUSE: 2 | gcd_le(T116, T115, T118), SCOPE: 10, CLAUSE: 3\n\nT115 is ground\nT116 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: gcd_le(T116, T115, T118), SCOPE: 10, CLAUSE: 2\n\nT115 is ground\nT116 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: gcd_le(T116, T115, T118), SCOPE: 10, CLAUSE: 3\n\nT115 is ground\nT116 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: ','(add(s(T151), X202, T152), gcd(s(T151), X202, T154))\n\nT151 is ground\nT152 is ground\nX202 is free", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: add(s(T151), X202, T152)\n\nT151 is ground\nT152 is ground\nX202 is free", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: gcd(s(T151), T157, T154)\n\nT151 is ground\nT157 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 86 [arrowhead = none ];
86 -> 2;

87 [label="ONLY EVAL with clause\ngcd(X4, X5, X6) :- ','(le(X4, X5), gcd_le(X4, X5, X6)).\nand substitutionT1 -> T7,\nX4 -> T7,\nT2 -> T8,\nX5 -> T8,\nT3 -> T10,\nX6 -> T10,\nT9 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 87 [arrowhead = none ];
87 -> 3;

88 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 88 [arrowhead = none ];
88 -> 4;

89 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 89 [arrowhead = none ];
89 -> 5;

90 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 90 [arrowhead = none ];
90 -> 6;

91 [label="EVAL with clause\nle(s(X15), s(X16)) :- le(X15, X16).\nand substitutionX15 -> T19,\nT7 -> s(T19),\nX16 -> T20,\nT8 -> s(T20)", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 91 [arrowhead = none ];
91 -> 7;

92 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 92 [arrowhead = none ];
92 -> 8;

93 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 93 [arrowhead = none ];
93 -> 42;

94 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 94 [arrowhead = none ];
94 -> 43;

95 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
7 -> 95 [arrowhead = none ];
95 -> 9;

96 [label="SPLIT 2\nnew knowledge:\nT19 is ground\nT20 is ground", fontsize=14, style = filled, fillcolor = violet];
7 -> 96 [arrowhead = none ];
96 -> 10;

97 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 97 [arrowhead = none ];
97 -> 11;

98 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 98 [arrowhead = none ];
98 -> 24;

99 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 99 [arrowhead = none ];
99 -> 12;

100 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 100 [arrowhead = none ];
100 -> 13;

101 [label="EVAL with clause\nle(s(X29), s(X30)) :- le(X29, X30).\nand substitutionX29 -> T33,\nT19 -> s(T33),\nX30 -> T34,\nT20 -> s(T34)", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 101 [arrowhead = none ];
101 -> 14;

102 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 102 [arrowhead = none ];
102 -> 15;

103 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
13 -> 103 [arrowhead = none ];
103 -> 16;

104 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
13 -> 104 [arrowhead = none ];
104 -> 17;

105 [label="INSTANCE with matching:\nT19 -> T33\nT20 -> T34", fontsize=14, style = filled, fillcolor = lightgrey];
14 -> 105 [arrowhead = none , style=dashed];
105 -> 9[style=dashed];

106 [label="EVAL with clause\nle(0, s(X37)).\nand substitutionT19 -> 0,\nX37 -> T41,\nT20 -> s(T41)", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 106 [arrowhead = none ];
106 -> 18;

107 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 107 [arrowhead = none ];
107 -> 19;

108 [label="EVAL with clause\nle(0, 0).\nand substitutionT19 -> 0,\nT20 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 108 [arrowhead = none ];
108 -> 21;

109 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 109 [arrowhead = none ];
109 -> 22;

110 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 110 [arrowhead = none ];
110 -> 20;

111 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 111 [arrowhead = none ];
111 -> 23;

112 [label="BACKTRACK\nfor clause: gcd_le(0, Y, Y)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 112 [arrowhead = none ];
112 -> 25;

113 [label="ONLY EVAL with clause\ngcd_le(s(X52), X53, X54) :- ','(add(s(X52), X55, X53), gcd(s(X52), X55, X54)).\nand substitutionT19 -> T51,\nX52 -> T51,\nT20 -> T52,\nX53 -> s(T52),\nT10 -> T54,\nX54 -> T54,\nT53 -> T54", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 113 [arrowhead = none ];
113 -> 26;

114 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
26 -> 114 [arrowhead = none ];
114 -> 27;

115 [label="SPLIT 2\nnew knowledge:\nT51 is ground\nT57 is ground\nT52 is ground\nreplacements:X55 -> T57", fontsize=14, style = filled, fillcolor = violet];
26 -> 115 [arrowhead = none ];
115 -> 28;

116 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
27 -> 116 [arrowhead = none ];
116 -> 29;

117 [label="INSTANCE with matching:\nT1 -> s(T51)\nT2 -> T57\nT3 -> T54", fontsize=14, style = filled, fillcolor = lightgrey];
28 -> 117 [arrowhead = none , style=dashed];
117 -> 1[style=dashed];

118 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
29 -> 118 [arrowhead = none ];
118 -> 30;

119 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
29 -> 119 [arrowhead = none ];
119 -> 31;

120 [label="ONLY EVAL with clause\nadd(s(X88), X89, s(X90)) :- add(X88, X89, X90).\nand substitutionT51 -> T72,\nX88 -> T72,\nX55 -> X91,\nX89 -> X91,\nT52 -> T73,\nX90 -> T73", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 120 [arrowhead = none ];
120 -> 32;

121 [label="BACKTRACK\nfor clause: add(0, X, X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 121 [arrowhead = none ];
121 -> 41;

122 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
32 -> 122 [arrowhead = none ];
122 -> 33;

123 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
33 -> 123 [arrowhead = none ];
123 -> 34;

124 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
33 -> 124 [arrowhead = none ];
124 -> 35;

125 [label="EVAL with clause\nadd(s(X112), X113, s(X114)) :- add(X112, X113, X114).\nand substitutionX112 -> T84,\nT72 -> s(T84),\nX91 -> X115,\nX113 -> X115,\nX114 -> T85,\nT73 -> s(T85)", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 125 [arrowhead = none ];
125 -> 36;

126 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 126 [arrowhead = none ];
126 -> 37;

127 [label="EVAL with clause\nadd(0, X124, X124).\nand substitutionT72 -> 0,\nX91 -> T90,\nX124 -> T90,\nT73 -> T90,\nX125 -> T90", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 127 [arrowhead = none ];
127 -> 38;

128 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 128 [arrowhead = none ];
128 -> 39;

129 [label="INSTANCE with matching:\nT72 -> T84\nX91 -> X115\nT73 -> T85", fontsize=14, style = filled, fillcolor = lightgrey];
36 -> 129 [arrowhead = none , style=dashed];
129 -> 32[style=dashed];

130 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 130 [arrowhead = none ];
130 -> 40;

131 [label="EVAL with clause\nle(0, s(X134)).\nand substitutionT7 -> 0,\nX134 -> T98,\nT8 -> s(T98)", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 131 [arrowhead = none ];
131 -> 44;

132 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 132 [arrowhead = none ];
132 -> 45;

133 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
43 -> 133 [arrowhead = none ];
133 -> 53;

134 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
43 -> 134 [arrowhead = none ];
134 -> 54;

135 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
44 -> 135 [arrowhead = none ];
135 -> 46;

136 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
46 -> 136 [arrowhead = none ];
136 -> 47;

137 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
46 -> 137 [arrowhead = none ];
137 -> 48;

138 [label="EVAL with clause\ngcd_le(0, X141, X141).\nand substitutionT98 -> T105,\nX141 -> s(T105),\nT10 -> s(T105)", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 138 [arrowhead = none ];
138 -> 49;

139 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 139 [arrowhead = none ];
139 -> 50;

140 [label="BACKTRACK\nfor clause: gcd_le(s(X), Y, D) :- ','(add(s(X), Z, Y), gcd(s(X), Z, D))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 140 [arrowhead = none ];
140 -> 52;

141 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 141 [arrowhead = none ];
141 -> 51;

142 [label="EVAL with clause\nle(0, 0).\nand substitutionT7 -> 0,\nT8 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 142 [arrowhead = none ];
142 -> 55;

143 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
53 -> 143 [arrowhead = none ];
143 -> 56;

144 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 144 [arrowhead = none ];
144 -> 64;

145 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
55 -> 145 [arrowhead = none ];
145 -> 57;

146 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
57 -> 146 [arrowhead = none ];
146 -> 58;

147 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
57 -> 147 [arrowhead = none ];
147 -> 59;

148 [label="EVAL with clause\ngcd_le(0, X151, X151).\nand substitutionX151 -> 0,\nT10 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 148 [arrowhead = none ];
148 -> 60;

149 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
58 -> 149 [arrowhead = none ];
149 -> 61;

150 [label="BACKTRACK\nfor clause: gcd_le(s(X), Y, D) :- ','(add(s(X), Z, Y), gcd(s(X), Z, D))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
59 -> 150 [arrowhead = none ];
150 -> 63;

151 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
60 -> 151 [arrowhead = none ];
151 -> 62;

152 [label="ONLY EVAL with clause\ngcd(X163, X164, X165) :- ','(gt(X163, X164), gcd_le(X164, X163, X165)).\nand substitutionT7 -> T115,\nX163 -> T115,\nT8 -> T116,\nX164 -> T116,\nT3 -> T118,\nX165 -> T118,\nT117 -> T118", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 152 [arrowhead = none ];
152 -> 65;

153 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
65 -> 153 [arrowhead = none ];
153 -> 66;

154 [label="SPLIT 2\nnew knowledge:\nT115 is ground\nT116 is ground", fontsize=14, style = filled, fillcolor = violet];
65 -> 154 [arrowhead = none ];
154 -> 67;

155 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
66 -> 155 [arrowhead = none ];
155 -> 68;

156 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
67 -> 156 [arrowhead = none ];
156 -> 76;

157 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
68 -> 157 [arrowhead = none ];
157 -> 69;

158 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
68 -> 158 [arrowhead = none ];
158 -> 70;

159 [label="EVAL with clause\ngt(s(X178), s(X179)) :- gt(X178, X179).\nand substitutionX178 -> T131,\nT115 -> s(T131),\nX179 -> T132,\nT116 -> s(T132)", fontsize=14, style = filled, fillcolor = lightblue];
69 -> 159 [arrowhead = none ];
159 -> 71;

160 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
69 -> 160 [arrowhead = none ];
160 -> 72;

161 [label="EVAL with clause\ngt(s(X184), 0).\nand substitutionX184 -> T137,\nT115 -> s(T137),\nT116 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
70 -> 161 [arrowhead = none ];
161 -> 73;

162 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
70 -> 162 [arrowhead = none ];
162 -> 74;

163 [label="INSTANCE with matching:\nT115 -> T131\nT116 -> T132", fontsize=14, style = filled, fillcolor = lightgrey];
71 -> 163 [arrowhead = none , style=dashed];
163 -> 66[style=dashed];

164 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
73 -> 164 [arrowhead = none ];
164 -> 75;

165 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
76 -> 165 [arrowhead = none ];
165 -> 77;

166 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
76 -> 166 [arrowhead = none ];
166 -> 78;

167 [label="EVAL with clause\ngcd_le(0, X191, X191).\nand substitutionT116 -> 0,\nT115 -> T144,\nX191 -> T144,\nT118 -> T144", fontsize=14, style = filled, fillcolor = lightblue];
77 -> 167 [arrowhead = none ];
167 -> 79;

168 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
77 -> 168 [arrowhead = none ];
168 -> 80;

169 [label="EVAL with clause\ngcd_le(s(X199), X200, X201) :- ','(add(s(X199), X202, X200), gcd(s(X199), X202, X201)).\nand substitutionX199 -> T151,\nT116 -> s(T151),\nT115 -> T152,\nX200 -> T152,\nT118 -> T154,\nX201 -> T154,\nT153 -> T154", fontsize=14, style = filled, fillcolor = lightblue];
78 -> 169 [arrowhead = none ];
169 -> 82;

170 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
78 -> 170 [arrowhead = none ];
170 -> 83;

171 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
79 -> 171 [arrowhead = none ];
171 -> 81;

172 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
82 -> 172 [arrowhead = none ];
172 -> 84;

173 [label="SPLIT 2\nnew knowledge:\nT151 is ground\nT157 is ground\nT152 is ground\nreplacements:X202 -> T157", fontsize=14, style = filled, fillcolor = violet];
82 -> 173 [arrowhead = none ];
173 -> 85;

174 [label="INSTANCE with matching:\nT72 -> s(T151)\nX91 -> X202\nT73 -> T152", fontsize=14, style = filled, fillcolor = lightgrey];
84 -> 174 [arrowhead = none , style=dashed];
174 -> 32[style=dashed];

175 [label="INSTANCE with matching:\nT1 -> s(T151)\nT2 -> T157\nT3 -> T154", fontsize=14, style = filled, fillcolor = lightgrey];
85 -> 175 [arrowhead = none , style=dashed];
175 -> 1[style=dashed];

}

(2) Obligation:

Triples:

leA(s(X1), s(X2)) :- leA(X1, X2).
addE(s(X1), X2, s(X3)) :- addE(X1, X2, X3).
gtD(s(X1), s(X2)) :- gtD(X1, X2).
gcdB(s(X1), s(X2), X3) :- leA(X1, X2).
gcdB(s(X1), s(X2), X3) :- ','(lecA(X1, X2), addE(X1, X4, X2)).
gcdB(s(X1), s(X2), X3) :- ','(lecA(X1, X2), ','(addcC(X1, X4, X2), gcdB(s(X1), X4, X3))).
gcdB(X1, X2, X3) :- gtD(X1, X2).
gcdB(X1, s(X2), X3) :- ','(gtcD(X1, s(X2)), addE(s(X2), X4, X1)).
gcdB(X1, s(X2), X3) :- ','(gtcD(X1, s(X2)), ','(addcE(s(X2), X4, X1), gcdB(s(X2), X4, X3))).

Clauses:

lecA(s(X1), s(X2)) :- lecA(X1, X2).
lecA(0, s(X1)).
lecA(0, 0).
gcdcB(s(X1), s(X2), X3) :- ','(lecA(X1, X2), ','(addcC(X1, X4, X2), gcdcB(s(X1), X4, X3))).
gcdcB(0, s(X1), s(X1)).
gcdcB(0, 0, 0).
gcdcB(X1, 0, X1) :- gtcD(X1, 0).
gcdcB(X1, s(X2), X3) :- ','(gtcD(X1, s(X2)), ','(addcE(s(X2), X4, X1), gcdcB(s(X2), X4, X3))).
addcE(s(X1), X2, s(X3)) :- addcE(X1, X2, X3).
addcE(0, X1, X1).
gtcD(s(X1), s(X2)) :- gtcD(X1, X2).
gtcD(s(X1), 0).
addcC(X1, X2, X3) :- addcE(X1, X2, X3).

Afs:

gcdB(x1, x2, x3)  =  gcdB(x1, x2)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
gcdB_in: (b,b,f)
leA_in: (b,b)
lecA_in: (b,b)
addE_in: (b,f,b)
addcC_in: (b,f,b)
addcE_in: (b,f,b)
gtD_in: (b,b)
gtcD_in: (b,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

GCDB_IN_GGA(s(X1), s(X2), X3) → U4_GGA(X1, X2, X3, leA_in_gg(X1, X2))
GCDB_IN_GGA(s(X1), s(X2), X3) → LEA_IN_GG(X1, X2)
LEA_IN_GG(s(X1), s(X2)) → U1_GG(X1, X2, leA_in_gg(X1, X2))
LEA_IN_GG(s(X1), s(X2)) → LEA_IN_GG(X1, X2)
GCDB_IN_GGA(s(X1), s(X2), X3) → U5_GGA(X1, X2, X3, lecA_in_gg(X1, X2))
U5_GGA(X1, X2, X3, lecA_out_gg(X1, X2)) → U6_GGA(X1, X2, X3, addE_in_gag(X1, X4, X2))
U5_GGA(X1, X2, X3, lecA_out_gg(X1, X2)) → ADDE_IN_GAG(X1, X4, X2)
ADDE_IN_GAG(s(X1), X2, s(X3)) → U2_GAG(X1, X2, X3, addE_in_gag(X1, X2, X3))
ADDE_IN_GAG(s(X1), X2, s(X3)) → ADDE_IN_GAG(X1, X2, X3)
U5_GGA(X1, X2, X3, lecA_out_gg(X1, X2)) → U7_GGA(X1, X2, X3, addcC_in_gag(X1, X4, X2))
U7_GGA(X1, X2, X3, addcC_out_gag(X1, X4, X2)) → U8_GGA(X1, X2, X3, gcdB_in_gga(s(X1), X4, X3))
U7_GGA(X1, X2, X3, addcC_out_gag(X1, X4, X2)) → GCDB_IN_GGA(s(X1), X4, X3)
GCDB_IN_GGA(X1, X2, X3) → U9_GGA(X1, X2, X3, gtD_in_gg(X1, X2))
GCDB_IN_GGA(X1, X2, X3) → GTD_IN_GG(X1, X2)
GTD_IN_GG(s(X1), s(X2)) → U3_GG(X1, X2, gtD_in_gg(X1, X2))
GTD_IN_GG(s(X1), s(X2)) → GTD_IN_GG(X1, X2)
GCDB_IN_GGA(X1, s(X2), X3) → U10_GGA(X1, X2, X3, gtcD_in_gg(X1, s(X2)))
U10_GGA(X1, X2, X3, gtcD_out_gg(X1, s(X2))) → U11_GGA(X1, X2, X3, addE_in_gag(s(X2), X4, X1))
U10_GGA(X1, X2, X3, gtcD_out_gg(X1, s(X2))) → ADDE_IN_GAG(s(X2), X4, X1)
U10_GGA(X1, X2, X3, gtcD_out_gg(X1, s(X2))) → U12_GGA(X1, X2, X3, addcE_in_gag(s(X2), X4, X1))
U12_GGA(X1, X2, X3, addcE_out_gag(s(X2), X4, X1)) → U13_GGA(X1, X2, X3, gcdB_in_gga(s(X2), X4, X3))
U12_GGA(X1, X2, X3, addcE_out_gag(s(X2), X4, X1)) → GCDB_IN_GGA(s(X2), X4, X3)

The TRS R consists of the following rules:

lecA_in_gg(s(X1), s(X2)) → U15_gg(X1, X2, lecA_in_gg(X1, X2))
lecA_in_gg(0, s(X1)) → lecA_out_gg(0, s(X1))
lecA_in_gg(0, 0) → lecA_out_gg(0, 0)
U15_gg(X1, X2, lecA_out_gg(X1, X2)) → lecA_out_gg(s(X1), s(X2))
addcC_in_gag(X1, X2, X3) → U25_gag(X1, X2, X3, addcE_in_gag(X1, X2, X3))
addcE_in_gag(s(X1), X2, s(X3)) → U23_gag(X1, X2, X3, addcE_in_gag(X1, X2, X3))
addcE_in_gag(0, X1, X1) → addcE_out_gag(0, X1, X1)
U23_gag(X1, X2, X3, addcE_out_gag(X1, X2, X3)) → addcE_out_gag(s(X1), X2, s(X3))
U25_gag(X1, X2, X3, addcE_out_gag(X1, X2, X3)) → addcC_out_gag(X1, X2, X3)
gtcD_in_gg(s(X1), s(X2)) → U24_gg(X1, X2, gtcD_in_gg(X1, X2))
gtcD_in_gg(s(X1), 0) → gtcD_out_gg(s(X1), 0)
U24_gg(X1, X2, gtcD_out_gg(X1, X2)) → gtcD_out_gg(s(X1), s(X2))

The argument filtering Pi contains the following mapping:
gcdB_in_gga(x1, x2, x3)  =  gcdB_in_gga(x1, x2)
s(x1)  =  s(x1)
leA_in_gg(x1, x2)  =  leA_in_gg(x1, x2)
lecA_in_gg(x1, x2)  =  lecA_in_gg(x1, x2)
U15_gg(x1, x2, x3)  =  U15_gg(x1, x2, x3)
0  =  0
lecA_out_gg(x1, x2)  =  lecA_out_gg(x1, x2)
addE_in_gag(x1, x2, x3)  =  addE_in_gag(x1, x3)
addcC_in_gag(x1, x2, x3)  =  addcC_in_gag(x1, x3)
U25_gag(x1, x2, x3, x4)  =  U25_gag(x1, x3, x4)
addcE_in_gag(x1, x2, x3)  =  addcE_in_gag(x1, x3)
U23_gag(x1, x2, x3, x4)  =  U23_gag(x1, x3, x4)
addcE_out_gag(x1, x2, x3)  =  addcE_out_gag(x1, x2, x3)
addcC_out_gag(x1, x2, x3)  =  addcC_out_gag(x1, x2, x3)
gtD_in_gg(x1, x2)  =  gtD_in_gg(x1, x2)
gtcD_in_gg(x1, x2)  =  gtcD_in_gg(x1, x2)
U24_gg(x1, x2, x3)  =  U24_gg(x1, x2, x3)
gtcD_out_gg(x1, x2)  =  gtcD_out_gg(x1, x2)
GCDB_IN_GGA(x1, x2, x3)  =  GCDB_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)
LEA_IN_GG(x1, x2)  =  LEA_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x1, x2, x3)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x1, x2, x4)
U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x1, x2, x4)
ADDE_IN_GAG(x1, x2, x3)  =  ADDE_IN_GAG(x1, x3)
U2_GAG(x1, x2, x3, x4)  =  U2_GAG(x1, x3, x4)
U7_GGA(x1, x2, x3, x4)  =  U7_GGA(x1, x2, x4)
U8_GGA(x1, x2, x3, x4)  =  U8_GGA(x1, x2, x4)
U9_GGA(x1, x2, x3, x4)  =  U9_GGA(x1, x2, x4)
GTD_IN_GG(x1, x2)  =  GTD_IN_GG(x1, x2)
U3_GG(x1, x2, x3)  =  U3_GG(x1, x2, x3)
U10_GGA(x1, x2, x3, x4)  =  U10_GGA(x1, x2, x4)
U11_GGA(x1, x2, x3, x4)  =  U11_GGA(x1, x2, x4)
U12_GGA(x1, x2, x3, x4)  =  U12_GGA(x1, x2, x4)
U13_GGA(x1, x2, x3, x4)  =  U13_GGA(x1, x2, x4)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

GCDB_IN_GGA(s(X1), s(X2), X3) → U4_GGA(X1, X2, X3, leA_in_gg(X1, X2))
GCDB_IN_GGA(s(X1), s(X2), X3) → LEA_IN_GG(X1, X2)
LEA_IN_GG(s(X1), s(X2)) → U1_GG(X1, X2, leA_in_gg(X1, X2))
LEA_IN_GG(s(X1), s(X2)) → LEA_IN_GG(X1, X2)
GCDB_IN_GGA(s(X1), s(X2), X3) → U5_GGA(X1, X2, X3, lecA_in_gg(X1, X2))
U5_GGA(X1, X2, X3, lecA_out_gg(X1, X2)) → U6_GGA(X1, X2, X3, addE_in_gag(X1, X4, X2))
U5_GGA(X1, X2, X3, lecA_out_gg(X1, X2)) → ADDE_IN_GAG(X1, X4, X2)
ADDE_IN_GAG(s(X1), X2, s(X3)) → U2_GAG(X1, X2, X3, addE_in_gag(X1, X2, X3))
ADDE_IN_GAG(s(X1), X2, s(X3)) → ADDE_IN_GAG(X1, X2, X3)
U5_GGA(X1, X2, X3, lecA_out_gg(X1, X2)) → U7_GGA(X1, X2, X3, addcC_in_gag(X1, X4, X2))
U7_GGA(X1, X2, X3, addcC_out_gag(X1, X4, X2)) → U8_GGA(X1, X2, X3, gcdB_in_gga(s(X1), X4, X3))
U7_GGA(X1, X2, X3, addcC_out_gag(X1, X4, X2)) → GCDB_IN_GGA(s(X1), X4, X3)
GCDB_IN_GGA(X1, X2, X3) → U9_GGA(X1, X2, X3, gtD_in_gg(X1, X2))
GCDB_IN_GGA(X1, X2, X3) → GTD_IN_GG(X1, X2)
GTD_IN_GG(s(X1), s(X2)) → U3_GG(X1, X2, gtD_in_gg(X1, X2))
GTD_IN_GG(s(X1), s(X2)) → GTD_IN_GG(X1, X2)
GCDB_IN_GGA(X1, s(X2), X3) → U10_GGA(X1, X2, X3, gtcD_in_gg(X1, s(X2)))
U10_GGA(X1, X2, X3, gtcD_out_gg(X1, s(X2))) → U11_GGA(X1, X2, X3, addE_in_gag(s(X2), X4, X1))
U10_GGA(X1, X2, X3, gtcD_out_gg(X1, s(X2))) → ADDE_IN_GAG(s(X2), X4, X1)
U10_GGA(X1, X2, X3, gtcD_out_gg(X1, s(X2))) → U12_GGA(X1, X2, X3, addcE_in_gag(s(X2), X4, X1))
U12_GGA(X1, X2, X3, addcE_out_gag(s(X2), X4, X1)) → U13_GGA(X1, X2, X3, gcdB_in_gga(s(X2), X4, X3))
U12_GGA(X1, X2, X3, addcE_out_gag(s(X2), X4, X1)) → GCDB_IN_GGA(s(X2), X4, X3)

The TRS R consists of the following rules:

lecA_in_gg(s(X1), s(X2)) → U15_gg(X1, X2, lecA_in_gg(X1, X2))
lecA_in_gg(0, s(X1)) → lecA_out_gg(0, s(X1))
lecA_in_gg(0, 0) → lecA_out_gg(0, 0)
U15_gg(X1, X2, lecA_out_gg(X1, X2)) → lecA_out_gg(s(X1), s(X2))
addcC_in_gag(X1, X2, X3) → U25_gag(X1, X2, X3, addcE_in_gag(X1, X2, X3))
addcE_in_gag(s(X1), X2, s(X3)) → U23_gag(X1, X2, X3, addcE_in_gag(X1, X2, X3))
addcE_in_gag(0, X1, X1) → addcE_out_gag(0, X1, X1)
U23_gag(X1, X2, X3, addcE_out_gag(X1, X2, X3)) → addcE_out_gag(s(X1), X2, s(X3))
U25_gag(X1, X2, X3, addcE_out_gag(X1, X2, X3)) → addcC_out_gag(X1, X2, X3)
gtcD_in_gg(s(X1), s(X2)) → U24_gg(X1, X2, gtcD_in_gg(X1, X2))
gtcD_in_gg(s(X1), 0) → gtcD_out_gg(s(X1), 0)
U24_gg(X1, X2, gtcD_out_gg(X1, X2)) → gtcD_out_gg(s(X1), s(X2))

The argument filtering Pi contains the following mapping:
gcdB_in_gga(x1, x2, x3)  =  gcdB_in_gga(x1, x2)
s(x1)  =  s(x1)
leA_in_gg(x1, x2)  =  leA_in_gg(x1, x2)
lecA_in_gg(x1, x2)  =  lecA_in_gg(x1, x2)
U15_gg(x1, x2, x3)  =  U15_gg(x1, x2, x3)
0  =  0
lecA_out_gg(x1, x2)  =  lecA_out_gg(x1, x2)
addE_in_gag(x1, x2, x3)  =  addE_in_gag(x1, x3)
addcC_in_gag(x1, x2, x3)  =  addcC_in_gag(x1, x3)
U25_gag(x1, x2, x3, x4)  =  U25_gag(x1, x3, x4)
addcE_in_gag(x1, x2, x3)  =  addcE_in_gag(x1, x3)
U23_gag(x1, x2, x3, x4)  =  U23_gag(x1, x3, x4)
addcE_out_gag(x1, x2, x3)  =  addcE_out_gag(x1, x2, x3)
addcC_out_gag(x1, x2, x3)  =  addcC_out_gag(x1, x2, x3)
gtD_in_gg(x1, x2)  =  gtD_in_gg(x1, x2)
gtcD_in_gg(x1, x2)  =  gtcD_in_gg(x1, x2)
U24_gg(x1, x2, x3)  =  U24_gg(x1, x2, x3)
gtcD_out_gg(x1, x2)  =  gtcD_out_gg(x1, x2)
GCDB_IN_GGA(x1, x2, x3)  =  GCDB_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)
LEA_IN_GG(x1, x2)  =  LEA_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x1, x2, x3)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x1, x2, x4)
U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x1, x2, x4)
ADDE_IN_GAG(x1, x2, x3)  =  ADDE_IN_GAG(x1, x3)
U2_GAG(x1, x2, x3, x4)  =  U2_GAG(x1, x3, x4)
U7_GGA(x1, x2, x3, x4)  =  U7_GGA(x1, x2, x4)
U8_GGA(x1, x2, x3, x4)  =  U8_GGA(x1, x2, x4)
U9_GGA(x1, x2, x3, x4)  =  U9_GGA(x1, x2, x4)
GTD_IN_GG(x1, x2)  =  GTD_IN_GG(x1, x2)
U3_GG(x1, x2, x3)  =  U3_GG(x1, x2, x3)
U10_GGA(x1, x2, x3, x4)  =  U10_GGA(x1, x2, x4)
U11_GGA(x1, x2, x3, x4)  =  U11_GGA(x1, x2, x4)
U12_GGA(x1, x2, x3, x4)  =  U12_GGA(x1, x2, x4)
U13_GGA(x1, x2, x3, x4)  =  U13_GGA(x1, x2, x4)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 13 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

GTD_IN_GG(s(X1), s(X2)) → GTD_IN_GG(X1, X2)

The TRS R consists of the following rules:

lecA_in_gg(s(X1), s(X2)) → U15_gg(X1, X2, lecA_in_gg(X1, X2))
lecA_in_gg(0, s(X1)) → lecA_out_gg(0, s(X1))
lecA_in_gg(0, 0) → lecA_out_gg(0, 0)
U15_gg(X1, X2, lecA_out_gg(X1, X2)) → lecA_out_gg(s(X1), s(X2))
addcC_in_gag(X1, X2, X3) → U25_gag(X1, X2, X3, addcE_in_gag(X1, X2, X3))
addcE_in_gag(s(X1), X2, s(X3)) → U23_gag(X1, X2, X3, addcE_in_gag(X1, X2, X3))
addcE_in_gag(0, X1, X1) → addcE_out_gag(0, X1, X1)
U23_gag(X1, X2, X3, addcE_out_gag(X1, X2, X3)) → addcE_out_gag(s(X1), X2, s(X3))
U25_gag(X1, X2, X3, addcE_out_gag(X1, X2, X3)) → addcC_out_gag(X1, X2, X3)
gtcD_in_gg(s(X1), s(X2)) → U24_gg(X1, X2, gtcD_in_gg(X1, X2))
gtcD_in_gg(s(X1), 0) → gtcD_out_gg(s(X1), 0)
U24_gg(X1, X2, gtcD_out_gg(X1, X2)) → gtcD_out_gg(s(X1), s(X2))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
lecA_in_gg(x1, x2)  =  lecA_in_gg(x1, x2)
U15_gg(x1, x2, x3)  =  U15_gg(x1, x2, x3)
0  =  0
lecA_out_gg(x1, x2)  =  lecA_out_gg(x1, x2)
addcC_in_gag(x1, x2, x3)  =  addcC_in_gag(x1, x3)
U25_gag(x1, x2, x3, x4)  =  U25_gag(x1, x3, x4)
addcE_in_gag(x1, x2, x3)  =  addcE_in_gag(x1, x3)
U23_gag(x1, x2, x3, x4)  =  U23_gag(x1, x3, x4)
addcE_out_gag(x1, x2, x3)  =  addcE_out_gag(x1, x2, x3)
addcC_out_gag(x1, x2, x3)  =  addcC_out_gag(x1, x2, x3)
gtcD_in_gg(x1, x2)  =  gtcD_in_gg(x1, x2)
U24_gg(x1, x2, x3)  =  U24_gg(x1, x2, x3)
gtcD_out_gg(x1, x2)  =  gtcD_out_gg(x1, x2)
GTD_IN_GG(x1, x2)  =  GTD_IN_GG(x1, x2)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

GTD_IN_GG(s(X1), s(X2)) → GTD_IN_GG(X1, X2)

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

(10) PiDPToQDPProof (EQUIVALENT transformation)

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

(11) Obligation:

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

GTD_IN_GG(s(X1), s(X2)) → GTD_IN_GG(X1, X2)

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

(12) 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:

• GTD_IN_GG(s(X1), s(X2)) → GTD_IN_GG(X1, X2)
  The graph contains the following edges 1 > 1, 2 > 2

(13) YES

(14) Obligation:

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

ADDE_IN_GAG(s(X1), X2, s(X3)) → ADDE_IN_GAG(X1, X2, X3)

The TRS R consists of the following rules:

lecA_in_gg(s(X1), s(X2)) → U15_gg(X1, X2, lecA_in_gg(X1, X2))
lecA_in_gg(0, s(X1)) → lecA_out_gg(0, s(X1))
lecA_in_gg(0, 0) → lecA_out_gg(0, 0)
U15_gg(X1, X2, lecA_out_gg(X1, X2)) → lecA_out_gg(s(X1), s(X2))
addcC_in_gag(X1, X2, X3) → U25_gag(X1, X2, X3, addcE_in_gag(X1, X2, X3))
addcE_in_gag(s(X1), X2, s(X3)) → U23_gag(X1, X2, X3, addcE_in_gag(X1, X2, X3))
addcE_in_gag(0, X1, X1) → addcE_out_gag(0, X1, X1)
U23_gag(X1, X2, X3, addcE_out_gag(X1, X2, X3)) → addcE_out_gag(s(X1), X2, s(X3))
U25_gag(X1, X2, X3, addcE_out_gag(X1, X2, X3)) → addcC_out_gag(X1, X2, X3)
gtcD_in_gg(s(X1), s(X2)) → U24_gg(X1, X2, gtcD_in_gg(X1, X2))
gtcD_in_gg(s(X1), 0) → gtcD_out_gg(s(X1), 0)
U24_gg(X1, X2, gtcD_out_gg(X1, X2)) → gtcD_out_gg(s(X1), s(X2))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
lecA_in_gg(x1, x2)  =  lecA_in_gg(x1, x2)
U15_gg(x1, x2, x3)  =  U15_gg(x1, x2, x3)
0  =  0
lecA_out_gg(x1, x2)  =  lecA_out_gg(x1, x2)
addcC_in_gag(x1, x2, x3)  =  addcC_in_gag(x1, x3)
U25_gag(x1, x2, x3, x4)  =  U25_gag(x1, x3, x4)
addcE_in_gag(x1, x2, x3)  =  addcE_in_gag(x1, x3)
U23_gag(x1, x2, x3, x4)  =  U23_gag(x1, x3, x4)
addcE_out_gag(x1, x2, x3)  =  addcE_out_gag(x1, x2, x3)
addcC_out_gag(x1, x2, x3)  =  addcC_out_gag(x1, x2, x3)
gtcD_in_gg(x1, x2)  =  gtcD_in_gg(x1, x2)
U24_gg(x1, x2, x3)  =  U24_gg(x1, x2, x3)
gtcD_out_gg(x1, x2)  =  gtcD_out_gg(x1, x2)
ADDE_IN_GAG(x1, x2, x3)  =  ADDE_IN_GAG(x1, x3)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

ADDE_IN_GAG(s(X1), X2, s(X3)) → ADDE_IN_GAG(X1, X2, X3)

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

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:

ADDE_IN_GAG(s(X1), s(X3)) → ADDE_IN_GAG(X1, X3)

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

(19) 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:

• ADDE_IN_GAG(s(X1), s(X3)) → ADDE_IN_GAG(X1, X3)
  The graph contains the following edges 1 > 1, 2 > 2

(20) YES

(21) Obligation:

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

LEA_IN_GG(s(X1), s(X2)) → LEA_IN_GG(X1, X2)

The TRS R consists of the following rules:

lecA_in_gg(s(X1), s(X2)) → U15_gg(X1, X2, lecA_in_gg(X1, X2))
lecA_in_gg(0, s(X1)) → lecA_out_gg(0, s(X1))
lecA_in_gg(0, 0) → lecA_out_gg(0, 0)
U15_gg(X1, X2, lecA_out_gg(X1, X2)) → lecA_out_gg(s(X1), s(X2))
addcC_in_gag(X1, X2, X3) → U25_gag(X1, X2, X3, addcE_in_gag(X1, X2, X3))
addcE_in_gag(s(X1), X2, s(X3)) → U23_gag(X1, X2, X3, addcE_in_gag(X1, X2, X3))
addcE_in_gag(0, X1, X1) → addcE_out_gag(0, X1, X1)
U23_gag(X1, X2, X3, addcE_out_gag(X1, X2, X3)) → addcE_out_gag(s(X1), X2, s(X3))
U25_gag(X1, X2, X3, addcE_out_gag(X1, X2, X3)) → addcC_out_gag(X1, X2, X3)
gtcD_in_gg(s(X1), s(X2)) → U24_gg(X1, X2, gtcD_in_gg(X1, X2))
gtcD_in_gg(s(X1), 0) → gtcD_out_gg(s(X1), 0)
U24_gg(X1, X2, gtcD_out_gg(X1, X2)) → gtcD_out_gg(s(X1), s(X2))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
lecA_in_gg(x1, x2)  =  lecA_in_gg(x1, x2)
U15_gg(x1, x2, x3)  =  U15_gg(x1, x2, x3)
0  =  0
lecA_out_gg(x1, x2)  =  lecA_out_gg(x1, x2)
addcC_in_gag(x1, x2, x3)  =  addcC_in_gag(x1, x3)
U25_gag(x1, x2, x3, x4)  =  U25_gag(x1, x3, x4)
addcE_in_gag(x1, x2, x3)  =  addcE_in_gag(x1, x3)
U23_gag(x1, x2, x3, x4)  =  U23_gag(x1, x3, x4)
addcE_out_gag(x1, x2, x3)  =  addcE_out_gag(x1, x2, x3)
addcC_out_gag(x1, x2, x3)  =  addcC_out_gag(x1, x2, x3)
gtcD_in_gg(x1, x2)  =  gtcD_in_gg(x1, x2)
U24_gg(x1, x2, x3)  =  U24_gg(x1, x2, x3)
gtcD_out_gg(x1, x2)  =  gtcD_out_gg(x1, x2)
LEA_IN_GG(x1, x2)  =  LEA_IN_GG(x1, x2)

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

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

LEA_IN_GG(s(X1), s(X2)) → LEA_IN_GG(X1, X2)

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

(24) PiDPToQDPProof (EQUIVALENT transformation)

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

(25) Obligation:

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

LEA_IN_GG(s(X1), s(X2)) → LEA_IN_GG(X1, X2)

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

(26) 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:

• LEA_IN_GG(s(X1), s(X2)) → LEA_IN_GG(X1, X2)
  The graph contains the following edges 1 > 1, 2 > 2

(27) YES

(28) Obligation:

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

GCDB_IN_GGA(s(X1), s(X2), X3) → U5_GGA(X1, X2, X3, lecA_in_gg(X1, X2))
U5_GGA(X1, X2, X3, lecA_out_gg(X1, X2)) → U7_GGA(X1, X2, X3, addcC_in_gag(X1, X4, X2))
U7_GGA(X1, X2, X3, addcC_out_gag(X1, X4, X2)) → GCDB_IN_GGA(s(X1), X4, X3)
GCDB_IN_GGA(X1, s(X2), X3) → U10_GGA(X1, X2, X3, gtcD_in_gg(X1, s(X2)))
U10_GGA(X1, X2, X3, gtcD_out_gg(X1, s(X2))) → U12_GGA(X1, X2, X3, addcE_in_gag(s(X2), X4, X1))
U12_GGA(X1, X2, X3, addcE_out_gag(s(X2), X4, X1)) → GCDB_IN_GGA(s(X2), X4, X3)

The TRS R consists of the following rules:

lecA_in_gg(s(X1), s(X2)) → U15_gg(X1, X2, lecA_in_gg(X1, X2))
lecA_in_gg(0, s(X1)) → lecA_out_gg(0, s(X1))
lecA_in_gg(0, 0) → lecA_out_gg(0, 0)
U15_gg(X1, X2, lecA_out_gg(X1, X2)) → lecA_out_gg(s(X1), s(X2))
addcC_in_gag(X1, X2, X3) → U25_gag(X1, X2, X3, addcE_in_gag(X1, X2, X3))
addcE_in_gag(s(X1), X2, s(X3)) → U23_gag(X1, X2, X3, addcE_in_gag(X1, X2, X3))
addcE_in_gag(0, X1, X1) → addcE_out_gag(0, X1, X1)
U23_gag(X1, X2, X3, addcE_out_gag(X1, X2, X3)) → addcE_out_gag(s(X1), X2, s(X3))
U25_gag(X1, X2, X3, addcE_out_gag(X1, X2, X3)) → addcC_out_gag(X1, X2, X3)
gtcD_in_gg(s(X1), s(X2)) → U24_gg(X1, X2, gtcD_in_gg(X1, X2))
gtcD_in_gg(s(X1), 0) → gtcD_out_gg(s(X1), 0)
U24_gg(X1, X2, gtcD_out_gg(X1, X2)) → gtcD_out_gg(s(X1), s(X2))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
lecA_in_gg(x1, x2)  =  lecA_in_gg(x1, x2)
U15_gg(x1, x2, x3)  =  U15_gg(x1, x2, x3)
0  =  0
lecA_out_gg(x1, x2)  =  lecA_out_gg(x1, x2)
addcC_in_gag(x1, x2, x3)  =  addcC_in_gag(x1, x3)
U25_gag(x1, x2, x3, x4)  =  U25_gag(x1, x3, x4)
addcE_in_gag(x1, x2, x3)  =  addcE_in_gag(x1, x3)
U23_gag(x1, x2, x3, x4)  =  U23_gag(x1, x3, x4)
addcE_out_gag(x1, x2, x3)  =  addcE_out_gag(x1, x2, x3)
addcC_out_gag(x1, x2, x3)  =  addcC_out_gag(x1, x2, x3)
gtcD_in_gg(x1, x2)  =  gtcD_in_gg(x1, x2)
U24_gg(x1, x2, x3)  =  U24_gg(x1, x2, x3)
gtcD_out_gg(x1, x2)  =  gtcD_out_gg(x1, x2)
GCDB_IN_GGA(x1, x2, x3)  =  GCDB_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x1, x2, x4)
U7_GGA(x1, x2, x3, x4)  =  U7_GGA(x1, x2, x4)
U10_GGA(x1, x2, x3, x4)  =  U10_GGA(x1, x2, x4)
U12_GGA(x1, x2, x3, x4)  =  U12_GGA(x1, x2, x4)

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

(29) PiDPToQDPProof (SOUND transformation)

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

(30) Obligation:

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

GCDB_IN_GGA(s(X1), s(X2)) → U5_GGA(X1, X2, lecA_in_gg(X1, X2))
U5_GGA(X1, X2, lecA_out_gg(X1, X2)) → U7_GGA(X1, X2, addcC_in_gag(X1, X2))
U7_GGA(X1, X2, addcC_out_gag(X1, X4, X2)) → GCDB_IN_GGA(s(X1), X4)
GCDB_IN_GGA(X1, s(X2)) → U10_GGA(X1, X2, gtcD_in_gg(X1, s(X2)))
U10_GGA(X1, X2, gtcD_out_gg(X1, s(X2))) → U12_GGA(X1, X2, addcE_in_gag(s(X2), X1))
U12_GGA(X1, X2, addcE_out_gag(s(X2), X4, X1)) → GCDB_IN_GGA(s(X2), X4)

The TRS R consists of the following rules:

lecA_in_gg(s(X1), s(X2)) → U15_gg(X1, X2, lecA_in_gg(X1, X2))
lecA_in_gg(0, s(X1)) → lecA_out_gg(0, s(X1))
lecA_in_gg(0, 0) → lecA_out_gg(0, 0)
U15_gg(X1, X2, lecA_out_gg(X1, X2)) → lecA_out_gg(s(X1), s(X2))
addcC_in_gag(X1, X3) → U25_gag(X1, X3, addcE_in_gag(X1, X3))
addcE_in_gag(s(X1), s(X3)) → U23_gag(X1, X3, addcE_in_gag(X1, X3))
addcE_in_gag(0, X1) → addcE_out_gag(0, X1, X1)
U23_gag(X1, X3, addcE_out_gag(X1, X2, X3)) → addcE_out_gag(s(X1), X2, s(X3))
U25_gag(X1, X3, addcE_out_gag(X1, X2, X3)) → addcC_out_gag(X1, X2, X3)
gtcD_in_gg(s(X1), s(X2)) → U24_gg(X1, X2, gtcD_in_gg(X1, X2))
gtcD_in_gg(s(X1), 0) → gtcD_out_gg(s(X1), 0)
U24_gg(X1, X2, gtcD_out_gg(X1, X2)) → gtcD_out_gg(s(X1), s(X2))

The set Q consists of the following terms:

lecA_in_gg(x0, x1)
U15_gg(x0, x1, x2)
addcC_in_gag(x0, x1)
addcE_in_gag(x0, x1)
U23_gag(x0, x1, x2)
U25_gag(x0, x1, x2)
gtcD_in_gg(x0, x1)
U24_gg(x0, x1, x2)

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

(31) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U5_GGA(X1, X2, lecA_out_gg(X1, X2)) → U7_GGA(X1, X2, addcC_in_gag(X1, X2)) at position [2] we obtained the following new rules [LPAR04]:

U5_GGA(X1, X2, lecA_out_gg(X1, X2)) → U7_GGA(X1, X2, U25_gag(X1, X2, addcE_in_gag(X1, X2)))

(32) Obligation:

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

GCDB_IN_GGA(s(X1), s(X2)) → U5_GGA(X1, X2, lecA_in_gg(X1, X2))
U7_GGA(X1, X2, addcC_out_gag(X1, X4, X2)) → GCDB_IN_GGA(s(X1), X4)
GCDB_IN_GGA(X1, s(X2)) → U10_GGA(X1, X2, gtcD_in_gg(X1, s(X2)))
U10_GGA(X1, X2, gtcD_out_gg(X1, s(X2))) → U12_GGA(X1, X2, addcE_in_gag(s(X2), X1))
U12_GGA(X1, X2, addcE_out_gag(s(X2), X4, X1)) → GCDB_IN_GGA(s(X2), X4)
U5_GGA(X1, X2, lecA_out_gg(X1, X2)) → U7_GGA(X1, X2, U25_gag(X1, X2, addcE_in_gag(X1, X2)))

The TRS R consists of the following rules:

lecA_in_gg(s(X1), s(X2)) → U15_gg(X1, X2, lecA_in_gg(X1, X2))
lecA_in_gg(0, s(X1)) → lecA_out_gg(0, s(X1))
lecA_in_gg(0, 0) → lecA_out_gg(0, 0)
U15_gg(X1, X2, lecA_out_gg(X1, X2)) → lecA_out_gg(s(X1), s(X2))
addcC_in_gag(X1, X3) → U25_gag(X1, X3, addcE_in_gag(X1, X3))
addcE_in_gag(s(X1), s(X3)) → U23_gag(X1, X3, addcE_in_gag(X1, X3))
addcE_in_gag(0, X1) → addcE_out_gag(0, X1, X1)
U23_gag(X1, X3, addcE_out_gag(X1, X2, X3)) → addcE_out_gag(s(X1), X2, s(X3))
U25_gag(X1, X3, addcE_out_gag(X1, X2, X3)) → addcC_out_gag(X1, X2, X3)
gtcD_in_gg(s(X1), s(X2)) → U24_gg(X1, X2, gtcD_in_gg(X1, X2))
gtcD_in_gg(s(X1), 0) → gtcD_out_gg(s(X1), 0)
U24_gg(X1, X2, gtcD_out_gg(X1, X2)) → gtcD_out_gg(s(X1), s(X2))

The set Q consists of the following terms:

lecA_in_gg(x0, x1)
U15_gg(x0, x1, x2)
addcC_in_gag(x0, x1)
addcE_in_gag(x0, x1)
U23_gag(x0, x1, x2)
U25_gag(x0, x1, x2)
gtcD_in_gg(x0, x1)
U24_gg(x0, x1, x2)

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

(33) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(34) Obligation:

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

GCDB_IN_GGA(s(X1), s(X2)) → U5_GGA(X1, X2, lecA_in_gg(X1, X2))
U7_GGA(X1, X2, addcC_out_gag(X1, X4, X2)) → GCDB_IN_GGA(s(X1), X4)
GCDB_IN_GGA(X1, s(X2)) → U10_GGA(X1, X2, gtcD_in_gg(X1, s(X2)))
U10_GGA(X1, X2, gtcD_out_gg(X1, s(X2))) → U12_GGA(X1, X2, addcE_in_gag(s(X2), X1))
U12_GGA(X1, X2, addcE_out_gag(s(X2), X4, X1)) → GCDB_IN_GGA(s(X2), X4)
U5_GGA(X1, X2, lecA_out_gg(X1, X2)) → U7_GGA(X1, X2, U25_gag(X1, X2, addcE_in_gag(X1, X2)))

The TRS R consists of the following rules:

addcE_in_gag(s(X1), s(X3)) → U23_gag(X1, X3, addcE_in_gag(X1, X3))
addcE_in_gag(0, X1) → addcE_out_gag(0, X1, X1)
U25_gag(X1, X3, addcE_out_gag(X1, X2, X3)) → addcC_out_gag(X1, X2, X3)
U23_gag(X1, X3, addcE_out_gag(X1, X2, X3)) → addcE_out_gag(s(X1), X2, s(X3))
gtcD_in_gg(s(X1), s(X2)) → U24_gg(X1, X2, gtcD_in_gg(X1, X2))
gtcD_in_gg(s(X1), 0) → gtcD_out_gg(s(X1), 0)
U24_gg(X1, X2, gtcD_out_gg(X1, X2)) → gtcD_out_gg(s(X1), s(X2))
lecA_in_gg(s(X1), s(X2)) → U15_gg(X1, X2, lecA_in_gg(X1, X2))
lecA_in_gg(0, s(X1)) → lecA_out_gg(0, s(X1))
lecA_in_gg(0, 0) → lecA_out_gg(0, 0)
U15_gg(X1, X2, lecA_out_gg(X1, X2)) → lecA_out_gg(s(X1), s(X2))

The set Q consists of the following terms:

lecA_in_gg(x0, x1)
U15_gg(x0, x1, x2)
addcC_in_gag(x0, x1)
addcE_in_gag(x0, x1)
U23_gag(x0, x1, x2)
U25_gag(x0, x1, x2)
gtcD_in_gg(x0, x1)
U24_gg(x0, x1, x2)

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

(35) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

addcC_in_gag(x0, x1)

(36) Obligation:

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

GCDB_IN_GGA(s(X1), s(X2)) → U5_GGA(X1, X2, lecA_in_gg(X1, X2))
U7_GGA(X1, X2, addcC_out_gag(X1, X4, X2)) → GCDB_IN_GGA(s(X1), X4)
GCDB_IN_GGA(X1, s(X2)) → U10_GGA(X1, X2, gtcD_in_gg(X1, s(X2)))
U10_GGA(X1, X2, gtcD_out_gg(X1, s(X2))) → U12_GGA(X1, X2, addcE_in_gag(s(X2), X1))
U12_GGA(X1, X2, addcE_out_gag(s(X2), X4, X1)) → GCDB_IN_GGA(s(X2), X4)
U5_GGA(X1, X2, lecA_out_gg(X1, X2)) → U7_GGA(X1, X2, U25_gag(X1, X2, addcE_in_gag(X1, X2)))

The TRS R consists of the following rules:

addcE_in_gag(s(X1), s(X3)) → U23_gag(X1, X3, addcE_in_gag(X1, X3))
addcE_in_gag(0, X1) → addcE_out_gag(0, X1, X1)
U25_gag(X1, X3, addcE_out_gag(X1, X2, X3)) → addcC_out_gag(X1, X2, X3)
U23_gag(X1, X3, addcE_out_gag(X1, X2, X3)) → addcE_out_gag(s(X1), X2, s(X3))
gtcD_in_gg(s(X1), s(X2)) → U24_gg(X1, X2, gtcD_in_gg(X1, X2))
gtcD_in_gg(s(X1), 0) → gtcD_out_gg(s(X1), 0)
U24_gg(X1, X2, gtcD_out_gg(X1, X2)) → gtcD_out_gg(s(X1), s(X2))
lecA_in_gg(s(X1), s(X2)) → U15_gg(X1, X2, lecA_in_gg(X1, X2))
lecA_in_gg(0, s(X1)) → lecA_out_gg(0, s(X1))
lecA_in_gg(0, 0) → lecA_out_gg(0, 0)
U15_gg(X1, X2, lecA_out_gg(X1, X2)) → lecA_out_gg(s(X1), s(X2))

The set Q consists of the following terms:

lecA_in_gg(x0, x1)
U15_gg(x0, x1, x2)
addcE_in_gag(x0, x1)
U23_gag(x0, x1, x2)
U25_gag(x0, x1, x2)
gtcD_in_gg(x0, x1)
U24_gg(x0, x1, x2)

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

(37) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

U10_GGA(X1, X2, gtcD_out_gg(X1, s(X2))) → U12_GGA(X1, X2, addcE_in_gag(s(X2), X1))

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

POL(0) = 0   
POL(GCDB_IN_GGA(x₁, x₂)) = x₁   
POL(U10_GGA(x₁, x₂, x₃)) = x₃   
POL(U12_GGA(x₁, x₂, x₃)) = 1 + x₂   
POL(U15_gg(x₁, x₂, x₃)) = 0   
POL(U23_gag(x₁, x₂, x₃)) = 0   
POL(U24_gg(x₁, x₂, x₃)) = 1 + x₃   
POL(U25_gag(x₁, x₂, x₃)) = 0   
POL(U5_GGA(x₁, x₂, x₃)) = 1 + x₁   
POL(U7_GGA(x₁, x₂, x₃)) = 1 + x₁   
POL(addcC_out_gag(x₁, x₂, x₃)) = 0   
POL(addcE_in_gag(x₁, x₂)) = 0   
POL(addcE_out_gag(x₁, x₂, x₃)) = 0   
POL(gtcD_in_gg(x₁, x₂)) = x₁   
POL(gtcD_out_gg(x₁, x₂)) = 1 + x₂   
POL(lecA_in_gg(x₁, x₂)) = 0   
POL(lecA_out_gg(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:

gtcD_in_gg(s(X1), s(X2)) → U24_gg(X1, X2, gtcD_in_gg(X1, X2))
gtcD_in_gg(s(X1), 0) → gtcD_out_gg(s(X1), 0)
U24_gg(X1, X2, gtcD_out_gg(X1, X2)) → gtcD_out_gg(s(X1), s(X2))

(38) Obligation:

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

GCDB_IN_GGA(s(X1), s(X2)) → U5_GGA(X1, X2, lecA_in_gg(X1, X2))
U7_GGA(X1, X2, addcC_out_gag(X1, X4, X2)) → GCDB_IN_GGA(s(X1), X4)
GCDB_IN_GGA(X1, s(X2)) → U10_GGA(X1, X2, gtcD_in_gg(X1, s(X2)))
U12_GGA(X1, X2, addcE_out_gag(s(X2), X4, X1)) → GCDB_IN_GGA(s(X2), X4)
U5_GGA(X1, X2, lecA_out_gg(X1, X2)) → U7_GGA(X1, X2, U25_gag(X1, X2, addcE_in_gag(X1, X2)))

The TRS R consists of the following rules:

addcE_in_gag(s(X1), s(X3)) → U23_gag(X1, X3, addcE_in_gag(X1, X3))
addcE_in_gag(0, X1) → addcE_out_gag(0, X1, X1)
U25_gag(X1, X3, addcE_out_gag(X1, X2, X3)) → addcC_out_gag(X1, X2, X3)
U23_gag(X1, X3, addcE_out_gag(X1, X2, X3)) → addcE_out_gag(s(X1), X2, s(X3))
gtcD_in_gg(s(X1), s(X2)) → U24_gg(X1, X2, gtcD_in_gg(X1, X2))
gtcD_in_gg(s(X1), 0) → gtcD_out_gg(s(X1), 0)
U24_gg(X1, X2, gtcD_out_gg(X1, X2)) → gtcD_out_gg(s(X1), s(X2))
lecA_in_gg(s(X1), s(X2)) → U15_gg(X1, X2, lecA_in_gg(X1, X2))
lecA_in_gg(0, s(X1)) → lecA_out_gg(0, s(X1))
lecA_in_gg(0, 0) → lecA_out_gg(0, 0)
U15_gg(X1, X2, lecA_out_gg(X1, X2)) → lecA_out_gg(s(X1), s(X2))

The set Q consists of the following terms:

lecA_in_gg(x0, x1)
U15_gg(x0, x1, x2)
addcE_in_gag(x0, x1)
U23_gag(x0, x1, x2)
U25_gag(x0, x1, x2)
gtcD_in_gg(x0, x1)
U24_gg(x0, x1, x2)

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

(39) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(40) Obligation:

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

U5_GGA(X1, X2, lecA_out_gg(X1, X2)) → U7_GGA(X1, X2, U25_gag(X1, X2, addcE_in_gag(X1, X2)))
U7_GGA(X1, X2, addcC_out_gag(X1, X4, X2)) → GCDB_IN_GGA(s(X1), X4)
GCDB_IN_GGA(s(X1), s(X2)) → U5_GGA(X1, X2, lecA_in_gg(X1, X2))

The TRS R consists of the following rules:

addcE_in_gag(s(X1), s(X3)) → U23_gag(X1, X3, addcE_in_gag(X1, X3))
addcE_in_gag(0, X1) → addcE_out_gag(0, X1, X1)
U25_gag(X1, X3, addcE_out_gag(X1, X2, X3)) → addcC_out_gag(X1, X2, X3)
U23_gag(X1, X3, addcE_out_gag(X1, X2, X3)) → addcE_out_gag(s(X1), X2, s(X3))
gtcD_in_gg(s(X1), s(X2)) → U24_gg(X1, X2, gtcD_in_gg(X1, X2))
gtcD_in_gg(s(X1), 0) → gtcD_out_gg(s(X1), 0)
U24_gg(X1, X2, gtcD_out_gg(X1, X2)) → gtcD_out_gg(s(X1), s(X2))
lecA_in_gg(s(X1), s(X2)) → U15_gg(X1, X2, lecA_in_gg(X1, X2))
lecA_in_gg(0, s(X1)) → lecA_out_gg(0, s(X1))
lecA_in_gg(0, 0) → lecA_out_gg(0, 0)
U15_gg(X1, X2, lecA_out_gg(X1, X2)) → lecA_out_gg(s(X1), s(X2))

The set Q consists of the following terms:

lecA_in_gg(x0, x1)
U15_gg(x0, x1, x2)
addcE_in_gag(x0, x1)
U23_gag(x0, x1, x2)
U25_gag(x0, x1, x2)
gtcD_in_gg(x0, x1)
U24_gg(x0, x1, x2)

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

(41) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(42) Obligation:

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

U5_GGA(X1, X2, lecA_out_gg(X1, X2)) → U7_GGA(X1, X2, U25_gag(X1, X2, addcE_in_gag(X1, X2)))
U7_GGA(X1, X2, addcC_out_gag(X1, X4, X2)) → GCDB_IN_GGA(s(X1), X4)
GCDB_IN_GGA(s(X1), s(X2)) → U5_GGA(X1, X2, lecA_in_gg(X1, X2))

The TRS R consists of the following rules:

lecA_in_gg(s(X1), s(X2)) → U15_gg(X1, X2, lecA_in_gg(X1, X2))
lecA_in_gg(0, s(X1)) → lecA_out_gg(0, s(X1))
lecA_in_gg(0, 0) → lecA_out_gg(0, 0)
U15_gg(X1, X2, lecA_out_gg(X1, X2)) → lecA_out_gg(s(X1), s(X2))
addcE_in_gag(s(X1), s(X3)) → U23_gag(X1, X3, addcE_in_gag(X1, X3))
addcE_in_gag(0, X1) → addcE_out_gag(0, X1, X1)
U25_gag(X1, X3, addcE_out_gag(X1, X2, X3)) → addcC_out_gag(X1, X2, X3)
U23_gag(X1, X3, addcE_out_gag(X1, X2, X3)) → addcE_out_gag(s(X1), X2, s(X3))

The set Q consists of the following terms:

lecA_in_gg(x0, x1)
U15_gg(x0, x1, x2)
addcE_in_gag(x0, x1)
U23_gag(x0, x1, x2)
U25_gag(x0, x1, x2)
gtcD_in_gg(x0, x1)
U24_gg(x0, x1, x2)

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

(43) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

gtcD_in_gg(x0, x1)
U24_gg(x0, x1, x2)

(44) Obligation:

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

U5_GGA(X1, X2, lecA_out_gg(X1, X2)) → U7_GGA(X1, X2, U25_gag(X1, X2, addcE_in_gag(X1, X2)))
U7_GGA(X1, X2, addcC_out_gag(X1, X4, X2)) → GCDB_IN_GGA(s(X1), X4)
GCDB_IN_GGA(s(X1), s(X2)) → U5_GGA(X1, X2, lecA_in_gg(X1, X2))

The TRS R consists of the following rules:

lecA_in_gg(s(X1), s(X2)) → U15_gg(X1, X2, lecA_in_gg(X1, X2))
lecA_in_gg(0, s(X1)) → lecA_out_gg(0, s(X1))
lecA_in_gg(0, 0) → lecA_out_gg(0, 0)
U15_gg(X1, X2, lecA_out_gg(X1, X2)) → lecA_out_gg(s(X1), s(X2))
addcE_in_gag(s(X1), s(X3)) → U23_gag(X1, X3, addcE_in_gag(X1, X3))
addcE_in_gag(0, X1) → addcE_out_gag(0, X1, X1)
U25_gag(X1, X3, addcE_out_gag(X1, X2, X3)) → addcC_out_gag(X1, X2, X3)
U23_gag(X1, X3, addcE_out_gag(X1, X2, X3)) → addcE_out_gag(s(X1), X2, s(X3))

The set Q consists of the following terms:

lecA_in_gg(x0, x1)
U15_gg(x0, x1, x2)
addcE_in_gag(x0, x1)
U23_gag(x0, x1, x2)
U25_gag(x0, x1, x2)

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

(45) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

GCDB_IN_GGA(s(X1), s(X2)) → U5_GGA(X1, X2, lecA_in_gg(X1, X2))

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

POL(0) = 0   
POL(GCDB_IN_GGA(x₁, x₂)) = x₂   
POL(U15_gg(x₁, x₂, x₃)) = 0   
POL(U23_gag(x₁, x₂, x₃)) = x₃   
POL(U25_gag(x₁, x₂, x₃)) = x₃   
POL(U5_GGA(x₁, x₂, x₃)) = x₂   
POL(U7_GGA(x₁, x₂, x₃)) = x₃   
POL(addcC_out_gag(x₁, x₂, x₃)) = x₂   
POL(addcE_in_gag(x₁, x₂)) = x₂   
POL(addcE_out_gag(x₁, x₂, x₃)) = x₂   
POL(lecA_in_gg(x₁, x₂)) = 0   
POL(lecA_out_gg(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:

addcE_in_gag(s(X1), s(X3)) → U23_gag(X1, X3, addcE_in_gag(X1, X3))
addcE_in_gag(0, X1) → addcE_out_gag(0, X1, X1)
U25_gag(X1, X3, addcE_out_gag(X1, X2, X3)) → addcC_out_gag(X1, X2, X3)
U23_gag(X1, X3, addcE_out_gag(X1, X2, X3)) → addcE_out_gag(s(X1), X2, s(X3))

(46) Obligation:

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

U5_GGA(X1, X2, lecA_out_gg(X1, X2)) → U7_GGA(X1, X2, U25_gag(X1, X2, addcE_in_gag(X1, X2)))
U7_GGA(X1, X2, addcC_out_gag(X1, X4, X2)) → GCDB_IN_GGA(s(X1), X4)

The TRS R consists of the following rules:

lecA_in_gg(s(X1), s(X2)) → U15_gg(X1, X2, lecA_in_gg(X1, X2))
lecA_in_gg(0, s(X1)) → lecA_out_gg(0, s(X1))
lecA_in_gg(0, 0) → lecA_out_gg(0, 0)
U15_gg(X1, X2, lecA_out_gg(X1, X2)) → lecA_out_gg(s(X1), s(X2))
addcE_in_gag(s(X1), s(X3)) → U23_gag(X1, X3, addcE_in_gag(X1, X3))
addcE_in_gag(0, X1) → addcE_out_gag(0, X1, X1)
U25_gag(X1, X3, addcE_out_gag(X1, X2, X3)) → addcC_out_gag(X1, X2, X3)
U23_gag(X1, X3, addcE_out_gag(X1, X2, X3)) → addcE_out_gag(s(X1), X2, s(X3))

The set Q consists of the following terms:

lecA_in_gg(x0, x1)
U15_gg(x0, x1, x2)
addcE_in_gag(x0, x1)
U23_gag(x0, x1, x2)
U25_gag(x0, x1, x2)

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

(47) DependencyGraphProof (EQUIVALENT transformation)

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

(48) TRUE