Left Termination of the query pattern gcd_in_3(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).

Queries:

gcd(g,g,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: 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="9: 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(T54), X64, s(T55)), gcd(s(T54), X64, T57))\n\nT54 is ground\nT55 is ground\nX64 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: add(s(T54), X64, s(T55))\n\nT54 is ground\nT55 is ground\nX64 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: gcd(s(T54), T60, T57)\n\nT54 is ground\nT60 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: add(s(T54), X64, s(T55)), SCOPE: 5, CLAUSE: 9 | add(s(T54), X64, s(T55)), SCOPE: 5, CLAUSE: 10\n\nT54 is ground\nT55 is ground\nX64 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: add(s(T54), X64, s(T55)), SCOPE: 5, CLAUSE: 9\n\nT54 is ground\nT55 is ground\nX64 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: add(s(T54), X64, s(T55)), SCOPE: 5, CLAUSE: 10\n\nT54 is ground\nT55 is ground\nX64 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: add(T75, X100, T76)\n\nT75 is ground\nT76 is ground\nX100 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: add(T75, X100, T76), SCOPE: 6, CLAUSE: 9 | add(T75, X100, T76), SCOPE: 6, CLAUSE: 10\n\nT75 is ground\nT76 is ground\nX100 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: add(T75, X100, T76), SCOPE: 6, CLAUSE: 9\n\nT75 is ground\nT76 is ground\nX100 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: add(T75, X100, T76), SCOPE: 6, CLAUSE: 10\n\nT75 is ground\nT76 is ground\nX100 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: add(T87, X124, T88)\n\nT87 is ground\nT88 is ground\nX124 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(T101), T10)\n\nT101 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(T101), T10), SCOPE: 7, CLAUSE: 2 | gcd_le(0, s(T101), T10), SCOPE: 7, CLAUSE: 3\n\nT101 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: gcd_le(0, s(T101), T10), SCOPE: 7, CLAUSE: 2\n\nT101 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: gcd_le(0, s(T101), T10), SCOPE: 7, CLAUSE: 3\n\nT101 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(T118, T119), gcd_le(T119, T118, T121))\n\nT118 is ground\nT119 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: gt(T118, T119)\n\nT118 is ground\nT119 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: gcd_le(T119, T118, T121)\n\nT118 is ground\nT119 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: gt(T118, T119), SCOPE: 9, CLAUSE: 4 | gt(T118, T119), SCOPE: 9, CLAUSE: 5\n\nT118 is ground\nT119 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: gt(T118, T119), SCOPE: 9, CLAUSE: 4\n\nT118 is ground\nT119 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: gt(T118, T119), SCOPE: 9, CLAUSE: 5\n\nT118 is ground\nT119 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: gt(T134, T135)\n\nT134 is ground\nT135 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(T119, T118, T121), SCOPE: 10, CLAUSE: 2 | gcd_le(T119, T118, T121), SCOPE: 10, CLAUSE: 3\n\nT118 is ground\nT119 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: gcd_le(T119, T118, T121), SCOPE: 10, CLAUSE: 2\n\nT118 is ground\nT119 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: gcd_le(T119, T118, T121), SCOPE: 10, CLAUSE: 3\n\nT118 is ground\nT119 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(T154), X222, T155), gcd(s(T154), X222, T157))\n\nT154 is ground\nT155 is ground\nX222 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: add(s(T154), X222, T155)\n\nT154 is ground\nT155 is ground\nX222 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: gcd(s(T154), T160, T157)\n\nT154 is ground\nT160 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 substitution\nT1 -> T7\nX4 -> T7\nT2 -> T8\nX5 -> T8\nT3 -> T10\nX6 -> T10\nT9 -> T10", fontsize=14, style = filled, fillcolor = white];
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 substitution\nX15 -> 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 = lightblue];
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", 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(X31), s(X32)) :- le(X31, X32).\nand substitution\nX31 -> T33\nT19 -> s(T33)\nX32 -> 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 = lightblue];
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(X41)).\nand substitution\nT19 -> 0\nX41 -> 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 = lightblue];
16 -> 107 [arrowhead = none ];
107 -> 19;

108 [label="EVAL with clause\nle(0, 0).\nand substitution\nT19 -> 0\nT20 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 108 [arrowhead = none ];
108 -> 21;

109 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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 = white];
24 -> 112 [arrowhead = none ];
112 -> 25;

113 [label="ONLY EVAL with clause\ngcd_le(s(X61), X62, X63) :- ','(add(s(X61), X64, X62), gcd(s(X61), X64, X63)).\nand substitution\nT19 -> T54\nX61 -> T54\nT20 -> T55\nX62 -> s(T55)\nT10 -> T57\nX63 -> T57\nT56 -> T57", fontsize=14, style = filled, fillcolor = white];
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:\nT60 is ground\nreplacements:\nX64 -> T60", 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(T54)\nT2 -> T60\nT3 -> T57", 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(X97), X98, s(X99)) :- add(X97, X98, X99).\nand substitution\nT54 -> T75\nX97 -> T75\nX64 -> X100\nX98 -> X100\nT55 -> T76\nX99 -> T76", fontsize=14, style = filled, fillcolor = white];
30 -> 120 [arrowhead = none ];
120 -> 32;

121 [label="BACKTRACK\nfor clause: add(0, X, X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
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(X121), X122, s(X123)) :- add(X121, X122, X123).\nand substitution\nX121 -> T87\nT75 -> s(T87)\nX100 -> X124\nX122 -> X124\nX123 -> T88\nT76 -> s(T88)", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 125 [arrowhead = none ];
125 -> 36;

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

127 [label="EVAL with clause\nadd(0, X136, X136).\nand substitution\nT75 -> 0\nX100 -> T93\nX136 -> T93\nT76 -> T93\nX137 -> T93", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 127 [arrowhead = none ];
127 -> 38;

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

129 [label="INSTANCE with matching:\nT75 -> T87\nX100 -> X124\nT76 -> T88", 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(X147)).\nand substitution\nT7 -> 0\nX147 -> T101\nT8 -> s(T101)", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 131 [arrowhead = none ];
131 -> 44;

132 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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, X155, X155).\nand substitution\nT101 -> T108\nX155 -> s(T108)\nT10 -> s(T108)", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 138 [arrowhead = none ];
138 -> 49;

139 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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 = white];
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 substitution\nT7 -> 0\nT8 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 142 [arrowhead = none ];
142 -> 55;

143 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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, X166, X166).\nand substitution\nX166 -> 0\nT10 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 148 [arrowhead = none ];
148 -> 60;

149 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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 = white];
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(X179, X180, X181) :- ','(gt(X179, X180), gcd_le(X180, X179, X181)).\nand substitution\nT7 -> T118\nX179 -> T118\nT8 -> T119\nX180 -> T119\nT3 -> T121\nX181 -> T121\nT120 -> T121", fontsize=14, style = filled, fillcolor = white];
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", 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(X194), s(X195)) :- gt(X194, X195).\nand substitution\nX194 -> T134\nT118 -> s(T134)\nX195 -> T135\nT119 -> s(T135)", fontsize=14, style = filled, fillcolor = lightblue];
69 -> 159 [arrowhead = none ];
159 -> 71;

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

161 [label="EVAL with clause\ngt(s(X202), 0).\nand substitution\nX202 -> T140\nT118 -> s(T140)\nT119 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
70 -> 161 [arrowhead = none ];
161 -> 73;

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

163 [label="INSTANCE with matching:\nT118 -> T134\nT119 -> T135", 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, X210, X210).\nand substitution\nT119 -> 0\nT118 -> T147\nX210 -> T147\nT121 -> T147", fontsize=14, style = filled, fillcolor = lightblue];
77 -> 167 [arrowhead = none ];
167 -> 79;

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

169 [label="EVAL with clause\ngcd_le(s(X219), X220, X221) :- ','(add(s(X219), X222, X220), gcd(s(X219), X222, X221)).\nand substitution\nX219 -> T154\nT119 -> s(T154)\nT118 -> T155\nX220 -> T155\nT121 -> T157\nX221 -> T157\nT156 -> T157", fontsize=14, style = filled, fillcolor = lightblue];
78 -> 169 [arrowhead = none ];
169 -> 82;

170 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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:\nT160 is ground\nreplacements:\nX222 -> T160", fontsize=14, style = filled, fillcolor = violet];
82 -> 173 [arrowhead = none ];
173 -> 85;

174 [label="INSTANCE with matching:\nT75 -> s(T154)\nX100 -> X222\nT76 -> T155", fontsize=14, style = filled, fillcolor = lightgrey];
84 -> 174 [arrowhead = none , style=dashed];
174 -> 32[style=dashed];

175 [label="INSTANCE with matching:\nT1 -> s(T154)\nT2 -> T160\nT3 -> T157", fontsize=14, style = filled, fillcolor = lightgrey];
85 -> 175 [arrowhead = none , style=dashed];
175 -> 1[style=dashed];

}

(2) Obligation:

Triples:

le9(s(T33), s(T34)) :- le9(T33, T34).
add32(s(T87), X124, s(T88)) :- add32(T87, X124, T88).
gt66(s(T134), s(T135)) :- gt66(T134, T135).
gcd1(s(T19), s(T20), T10) :- le9(T19, T20).
gcd1(s(T75), s(T76), T57) :- ','(lec9(T75, T76), add32(T75, X100, T76)).
gcd1(s(T54), s(T55), T57) :- ','(lec9(T54, T55), ','(addc27(T54, T60, T55), gcd1(s(T54), T60, T57))).
gcd1(T118, T119, T121) :- gt66(T118, T119).
gcd1(T155, s(T154), T157) :- ','(gtc66(T155, s(T154)), add32(s(T154), X222, T155)).
gcd1(T155, s(T154), T157) :- ','(gtc66(T155, s(T154)), ','(addc32(s(T154), T160, T155), gcd1(s(T154), T160, T157))).

Clauses:

lec9(s(T33), s(T34)) :- lec9(T33, T34).
lec9(0, s(T41)).
lec9(0, 0).
gcdc1(s(T54), s(T55), T57) :- ','(lec9(T54, T55), ','(addc27(T54, T60, T55), gcdc1(s(T54), T60, T57))).
gcdc1(0, s(T108), s(T108)).
gcdc1(0, 0, 0).
gcdc1(T147, 0, T147) :- gtc66(T147, 0).
gcdc1(T155, s(T154), T157) :- ','(gtc66(T155, s(T154)), ','(addc32(s(T154), T160, T155), gcdc1(s(T154), T160, T157))).
addc32(s(T87), X124, s(T88)) :- addc32(T87, X124, T88).
addc32(0, T93, T93).
gtc66(s(T134), s(T135)) :- gtc66(T134, T135).
gtc66(s(T140), 0).
addc27(T75, X100, T76) :- addc32(T75, X100, T76).

Afs:

gcd1(x1, x2, x3)  =  gcd1(x1, x2)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
gcd1_in: (b,b,f)
le9_in: (b,b)
lec9_in: (b,b)
add32_in: (b,f,b)
addc27_in: (b,f,b)
addc32_in: (b,f,b)
gt66_in: (b,b)
gtc66_in: (b,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

GCD1_IN_GGA(s(T19), s(T20), T10) → U4_GGA(T19, T20, T10, le9_in_gg(T19, T20))
GCD1_IN_GGA(s(T19), s(T20), T10) → LE9_IN_GG(T19, T20)
LE9_IN_GG(s(T33), s(T34)) → U1_GG(T33, T34, le9_in_gg(T33, T34))
LE9_IN_GG(s(T33), s(T34)) → LE9_IN_GG(T33, T34)
GCD1_IN_GGA(s(T75), s(T76), T57) → U5_GGA(T75, T76, T57, lec9_in_gg(T75, T76))
U5_GGA(T75, T76, T57, lec9_out_gg(T75, T76)) → U6_GGA(T75, T76, T57, add32_in_gag(T75, X100, T76))
U5_GGA(T75, T76, T57, lec9_out_gg(T75, T76)) → ADD32_IN_GAG(T75, X100, T76)
ADD32_IN_GAG(s(T87), X124, s(T88)) → U2_GAG(T87, X124, T88, add32_in_gag(T87, X124, T88))
ADD32_IN_GAG(s(T87), X124, s(T88)) → ADD32_IN_GAG(T87, X124, T88)
GCD1_IN_GGA(s(T54), s(T55), T57) → U7_GGA(T54, T55, T57, lec9_in_gg(T54, T55))
U7_GGA(T54, T55, T57, lec9_out_gg(T54, T55)) → U8_GGA(T54, T55, T57, addc27_in_gag(T54, T60, T55))
U8_GGA(T54, T55, T57, addc27_out_gag(T54, T60, T55)) → U9_GGA(T54, T55, T57, gcd1_in_gga(s(T54), T60, T57))
U8_GGA(T54, T55, T57, addc27_out_gag(T54, T60, T55)) → GCD1_IN_GGA(s(T54), T60, T57)
GCD1_IN_GGA(T118, T119, T121) → U10_GGA(T118, T119, T121, gt66_in_gg(T118, T119))
GCD1_IN_GGA(T118, T119, T121) → GT66_IN_GG(T118, T119)
GT66_IN_GG(s(T134), s(T135)) → U3_GG(T134, T135, gt66_in_gg(T134, T135))
GT66_IN_GG(s(T134), s(T135)) → GT66_IN_GG(T134, T135)
GCD1_IN_GGA(T155, s(T154), T157) → U11_GGA(T155, T154, T157, gtc66_in_gg(T155, s(T154)))
U11_GGA(T155, T154, T157, gtc66_out_gg(T155, s(T154))) → U12_GGA(T155, T154, T157, add32_in_gag(s(T154), X222, T155))
U11_GGA(T155, T154, T157, gtc66_out_gg(T155, s(T154))) → ADD32_IN_GAG(s(T154), X222, T155)
U11_GGA(T155, T154, T157, gtc66_out_gg(T155, s(T154))) → U13_GGA(T155, T154, T157, addc32_in_gag(s(T154), T160, T155))
U13_GGA(T155, T154, T157, addc32_out_gag(s(T154), T160, T155)) → U14_GGA(T155, T154, T157, gcd1_in_gga(s(T154), T160, T157))
U13_GGA(T155, T154, T157, addc32_out_gag(s(T154), T160, T155)) → GCD1_IN_GGA(s(T154), T160, T157)

The TRS R consists of the following rules:

lec9_in_gg(s(T33), s(T34)) → U16_gg(T33, T34, lec9_in_gg(T33, T34))
lec9_in_gg(0, s(T41)) → lec9_out_gg(0, s(T41))
lec9_in_gg(0, 0) → lec9_out_gg(0, 0)
U16_gg(T33, T34, lec9_out_gg(T33, T34)) → lec9_out_gg(s(T33), s(T34))
addc27_in_gag(T75, X100, T76) → U26_gag(T75, X100, T76, addc32_in_gag(T75, X100, T76))
addc32_in_gag(s(T87), X124, s(T88)) → U24_gag(T87, X124, T88, addc32_in_gag(T87, X124, T88))
addc32_in_gag(0, T93, T93) → addc32_out_gag(0, T93, T93)
U24_gag(T87, X124, T88, addc32_out_gag(T87, X124, T88)) → addc32_out_gag(s(T87), X124, s(T88))
U26_gag(T75, X100, T76, addc32_out_gag(T75, X100, T76)) → addc27_out_gag(T75, X100, T76)
gtc66_in_gg(s(T134), s(T135)) → U25_gg(T134, T135, gtc66_in_gg(T134, T135))
gtc66_in_gg(s(T140), 0) → gtc66_out_gg(s(T140), 0)
U25_gg(T134, T135, gtc66_out_gg(T134, T135)) → gtc66_out_gg(s(T134), s(T135))

The argument filtering Pi contains the following mapping:
gcd1_in_gga(x1, x2, x3)  =  gcd1_in_gga(x1, x2)
s(x1)  =  s(x1)
le9_in_gg(x1, x2)  =  le9_in_gg(x1, x2)
lec9_in_gg(x1, x2)  =  lec9_in_gg(x1, x2)
U16_gg(x1, x2, x3)  =  U16_gg(x1, x2, x3)
0  =  0
lec9_out_gg(x1, x2)  =  lec9_out_gg(x1, x2)
add32_in_gag(x1, x2, x3)  =  add32_in_gag(x1, x3)
addc27_in_gag(x1, x2, x3)  =  addc27_in_gag(x1, x3)
U26_gag(x1, x2, x3, x4)  =  U26_gag(x1, x3, x4)
addc32_in_gag(x1, x2, x3)  =  addc32_in_gag(x1, x3)
U24_gag(x1, x2, x3, x4)  =  U24_gag(x1, x3, x4)
addc32_out_gag(x1, x2, x3)  =  addc32_out_gag(x1, x2, x3)
addc27_out_gag(x1, x2, x3)  =  addc27_out_gag(x1, x2, x3)
gt66_in_gg(x1, x2)  =  gt66_in_gg(x1, x2)
gtc66_in_gg(x1, x2)  =  gtc66_in_gg(x1, x2)
U25_gg(x1, x2, x3)  =  U25_gg(x1, x2, x3)
gtc66_out_gg(x1, x2)  =  gtc66_out_gg(x1, x2)
GCD1_IN_GGA(x1, x2, x3)  =  GCD1_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)
LE9_IN_GG(x1, x2)  =  LE9_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)
ADD32_IN_GAG(x1, x2, x3)  =  ADD32_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)
U10_GGA(x1, x2, x3, x4)  =  U10_GGA(x1, x2, x4)
GT66_IN_GG(x1, x2)  =  GT66_IN_GG(x1, x2)
U3_GG(x1, x2, x3)  =  U3_GG(x1, x2, x3)
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)
U14_GGA(x1, x2, x3, x4)  =  U14_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:

GCD1_IN_GGA(s(T19), s(T20), T10) → U4_GGA(T19, T20, T10, le9_in_gg(T19, T20))
GCD1_IN_GGA(s(T19), s(T20), T10) → LE9_IN_GG(T19, T20)
LE9_IN_GG(s(T33), s(T34)) → U1_GG(T33, T34, le9_in_gg(T33, T34))
LE9_IN_GG(s(T33), s(T34)) → LE9_IN_GG(T33, T34)
GCD1_IN_GGA(s(T75), s(T76), T57) → U5_GGA(T75, T76, T57, lec9_in_gg(T75, T76))
U5_GGA(T75, T76, T57, lec9_out_gg(T75, T76)) → U6_GGA(T75, T76, T57, add32_in_gag(T75, X100, T76))
U5_GGA(T75, T76, T57, lec9_out_gg(T75, T76)) → ADD32_IN_GAG(T75, X100, T76)
ADD32_IN_GAG(s(T87), X124, s(T88)) → U2_GAG(T87, X124, T88, add32_in_gag(T87, X124, T88))
ADD32_IN_GAG(s(T87), X124, s(T88)) → ADD32_IN_GAG(T87, X124, T88)
GCD1_IN_GGA(s(T54), s(T55), T57) → U7_GGA(T54, T55, T57, lec9_in_gg(T54, T55))
U7_GGA(T54, T55, T57, lec9_out_gg(T54, T55)) → U8_GGA(T54, T55, T57, addc27_in_gag(T54, T60, T55))
U8_GGA(T54, T55, T57, addc27_out_gag(T54, T60, T55)) → U9_GGA(T54, T55, T57, gcd1_in_gga(s(T54), T60, T57))
U8_GGA(T54, T55, T57, addc27_out_gag(T54, T60, T55)) → GCD1_IN_GGA(s(T54), T60, T57)
GCD1_IN_GGA(T118, T119, T121) → U10_GGA(T118, T119, T121, gt66_in_gg(T118, T119))
GCD1_IN_GGA(T118, T119, T121) → GT66_IN_GG(T118, T119)
GT66_IN_GG(s(T134), s(T135)) → U3_GG(T134, T135, gt66_in_gg(T134, T135))
GT66_IN_GG(s(T134), s(T135)) → GT66_IN_GG(T134, T135)
GCD1_IN_GGA(T155, s(T154), T157) → U11_GGA(T155, T154, T157, gtc66_in_gg(T155, s(T154)))
U11_GGA(T155, T154, T157, gtc66_out_gg(T155, s(T154))) → U12_GGA(T155, T154, T157, add32_in_gag(s(T154), X222, T155))
U11_GGA(T155, T154, T157, gtc66_out_gg(T155, s(T154))) → ADD32_IN_GAG(s(T154), X222, T155)
U11_GGA(T155, T154, T157, gtc66_out_gg(T155, s(T154))) → U13_GGA(T155, T154, T157, addc32_in_gag(s(T154), T160, T155))
U13_GGA(T155, T154, T157, addc32_out_gag(s(T154), T160, T155)) → U14_GGA(T155, T154, T157, gcd1_in_gga(s(T154), T160, T157))
U13_GGA(T155, T154, T157, addc32_out_gag(s(T154), T160, T155)) → GCD1_IN_GGA(s(T154), T160, T157)

The TRS R consists of the following rules:

lec9_in_gg(s(T33), s(T34)) → U16_gg(T33, T34, lec9_in_gg(T33, T34))
lec9_in_gg(0, s(T41)) → lec9_out_gg(0, s(T41))
lec9_in_gg(0, 0) → lec9_out_gg(0, 0)
U16_gg(T33, T34, lec9_out_gg(T33, T34)) → lec9_out_gg(s(T33), s(T34))
addc27_in_gag(T75, X100, T76) → U26_gag(T75, X100, T76, addc32_in_gag(T75, X100, T76))
addc32_in_gag(s(T87), X124, s(T88)) → U24_gag(T87, X124, T88, addc32_in_gag(T87, X124, T88))
addc32_in_gag(0, T93, T93) → addc32_out_gag(0, T93, T93)
U24_gag(T87, X124, T88, addc32_out_gag(T87, X124, T88)) → addc32_out_gag(s(T87), X124, s(T88))
U26_gag(T75, X100, T76, addc32_out_gag(T75, X100, T76)) → addc27_out_gag(T75, X100, T76)
gtc66_in_gg(s(T134), s(T135)) → U25_gg(T134, T135, gtc66_in_gg(T134, T135))
gtc66_in_gg(s(T140), 0) → gtc66_out_gg(s(T140), 0)
U25_gg(T134, T135, gtc66_out_gg(T134, T135)) → gtc66_out_gg(s(T134), s(T135))

The argument filtering Pi contains the following mapping:
gcd1_in_gga(x1, x2, x3)  =  gcd1_in_gga(x1, x2)
s(x1)  =  s(x1)
le9_in_gg(x1, x2)  =  le9_in_gg(x1, x2)
lec9_in_gg(x1, x2)  =  lec9_in_gg(x1, x2)
U16_gg(x1, x2, x3)  =  U16_gg(x1, x2, x3)
0  =  0
lec9_out_gg(x1, x2)  =  lec9_out_gg(x1, x2)
add32_in_gag(x1, x2, x3)  =  add32_in_gag(x1, x3)
addc27_in_gag(x1, x2, x3)  =  addc27_in_gag(x1, x3)
U26_gag(x1, x2, x3, x4)  =  U26_gag(x1, x3, x4)
addc32_in_gag(x1, x2, x3)  =  addc32_in_gag(x1, x3)
U24_gag(x1, x2, x3, x4)  =  U24_gag(x1, x3, x4)
addc32_out_gag(x1, x2, x3)  =  addc32_out_gag(x1, x2, x3)
addc27_out_gag(x1, x2, x3)  =  addc27_out_gag(x1, x2, x3)
gt66_in_gg(x1, x2)  =  gt66_in_gg(x1, x2)
gtc66_in_gg(x1, x2)  =  gtc66_in_gg(x1, x2)
U25_gg(x1, x2, x3)  =  U25_gg(x1, x2, x3)
gtc66_out_gg(x1, x2)  =  gtc66_out_gg(x1, x2)
GCD1_IN_GGA(x1, x2, x3)  =  GCD1_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)
LE9_IN_GG(x1, x2)  =  LE9_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)
ADD32_IN_GAG(x1, x2, x3)  =  ADD32_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)
U10_GGA(x1, x2, x3, x4)  =  U10_GGA(x1, x2, x4)
GT66_IN_GG(x1, x2)  =  GT66_IN_GG(x1, x2)
U3_GG(x1, x2, x3)  =  U3_GG(x1, x2, x3)
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)
U14_GGA(x1, x2, x3, x4)  =  U14_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 14 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

GT66_IN_GG(s(T134), s(T135)) → GT66_IN_GG(T134, T135)

The TRS R consists of the following rules:

lec9_in_gg(s(T33), s(T34)) → U16_gg(T33, T34, lec9_in_gg(T33, T34))
lec9_in_gg(0, s(T41)) → lec9_out_gg(0, s(T41))
lec9_in_gg(0, 0) → lec9_out_gg(0, 0)
U16_gg(T33, T34, lec9_out_gg(T33, T34)) → lec9_out_gg(s(T33), s(T34))
addc27_in_gag(T75, X100, T76) → U26_gag(T75, X100, T76, addc32_in_gag(T75, X100, T76))
addc32_in_gag(s(T87), X124, s(T88)) → U24_gag(T87, X124, T88, addc32_in_gag(T87, X124, T88))
addc32_in_gag(0, T93, T93) → addc32_out_gag(0, T93, T93)
U24_gag(T87, X124, T88, addc32_out_gag(T87, X124, T88)) → addc32_out_gag(s(T87), X124, s(T88))
U26_gag(T75, X100, T76, addc32_out_gag(T75, X100, T76)) → addc27_out_gag(T75, X100, T76)
gtc66_in_gg(s(T134), s(T135)) → U25_gg(T134, T135, gtc66_in_gg(T134, T135))
gtc66_in_gg(s(T140), 0) → gtc66_out_gg(s(T140), 0)
U25_gg(T134, T135, gtc66_out_gg(T134, T135)) → gtc66_out_gg(s(T134), s(T135))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
lec9_in_gg(x1, x2)  =  lec9_in_gg(x1, x2)
U16_gg(x1, x2, x3)  =  U16_gg(x1, x2, x3)
0  =  0
lec9_out_gg(x1, x2)  =  lec9_out_gg(x1, x2)
addc27_in_gag(x1, x2, x3)  =  addc27_in_gag(x1, x3)
U26_gag(x1, x2, x3, x4)  =  U26_gag(x1, x3, x4)
addc32_in_gag(x1, x2, x3)  =  addc32_in_gag(x1, x3)
U24_gag(x1, x2, x3, x4)  =  U24_gag(x1, x3, x4)
addc32_out_gag(x1, x2, x3)  =  addc32_out_gag(x1, x2, x3)
addc27_out_gag(x1, x2, x3)  =  addc27_out_gag(x1, x2, x3)
gtc66_in_gg(x1, x2)  =  gtc66_in_gg(x1, x2)
U25_gg(x1, x2, x3)  =  U25_gg(x1, x2, x3)
gtc66_out_gg(x1, x2)  =  gtc66_out_gg(x1, x2)
GT66_IN_GG(x1, x2)  =  GT66_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:

GT66_IN_GG(s(T134), s(T135)) → GT66_IN_GG(T134, T135)

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:

GT66_IN_GG(s(T134), s(T135)) → GT66_IN_GG(T134, T135)

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:

• GT66_IN_GG(s(T134), s(T135)) → GT66_IN_GG(T134, T135)
  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:

ADD32_IN_GAG(s(T87), X124, s(T88)) → ADD32_IN_GAG(T87, X124, T88)

The TRS R consists of the following rules:

lec9_in_gg(s(T33), s(T34)) → U16_gg(T33, T34, lec9_in_gg(T33, T34))
lec9_in_gg(0, s(T41)) → lec9_out_gg(0, s(T41))
lec9_in_gg(0, 0) → lec9_out_gg(0, 0)
U16_gg(T33, T34, lec9_out_gg(T33, T34)) → lec9_out_gg(s(T33), s(T34))
addc27_in_gag(T75, X100, T76) → U26_gag(T75, X100, T76, addc32_in_gag(T75, X100, T76))
addc32_in_gag(s(T87), X124, s(T88)) → U24_gag(T87, X124, T88, addc32_in_gag(T87, X124, T88))
addc32_in_gag(0, T93, T93) → addc32_out_gag(0, T93, T93)
U24_gag(T87, X124, T88, addc32_out_gag(T87, X124, T88)) → addc32_out_gag(s(T87), X124, s(T88))
U26_gag(T75, X100, T76, addc32_out_gag(T75, X100, T76)) → addc27_out_gag(T75, X100, T76)
gtc66_in_gg(s(T134), s(T135)) → U25_gg(T134, T135, gtc66_in_gg(T134, T135))
gtc66_in_gg(s(T140), 0) → gtc66_out_gg(s(T140), 0)
U25_gg(T134, T135, gtc66_out_gg(T134, T135)) → gtc66_out_gg(s(T134), s(T135))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
lec9_in_gg(x1, x2)  =  lec9_in_gg(x1, x2)
U16_gg(x1, x2, x3)  =  U16_gg(x1, x2, x3)
0  =  0
lec9_out_gg(x1, x2)  =  lec9_out_gg(x1, x2)
addc27_in_gag(x1, x2, x3)  =  addc27_in_gag(x1, x3)
U26_gag(x1, x2, x3, x4)  =  U26_gag(x1, x3, x4)
addc32_in_gag(x1, x2, x3)  =  addc32_in_gag(x1, x3)
U24_gag(x1, x2, x3, x4)  =  U24_gag(x1, x3, x4)
addc32_out_gag(x1, x2, x3)  =  addc32_out_gag(x1, x2, x3)
addc27_out_gag(x1, x2, x3)  =  addc27_out_gag(x1, x2, x3)
gtc66_in_gg(x1, x2)  =  gtc66_in_gg(x1, x2)
U25_gg(x1, x2, x3)  =  U25_gg(x1, x2, x3)
gtc66_out_gg(x1, x2)  =  gtc66_out_gg(x1, x2)
ADD32_IN_GAG(x1, x2, x3)  =  ADD32_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:

ADD32_IN_GAG(s(T87), X124, s(T88)) → ADD32_IN_GAG(T87, X124, T88)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
ADD32_IN_GAG(x1, x2, x3)  =  ADD32_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:

ADD32_IN_GAG(s(T87), s(T88)) → ADD32_IN_GAG(T87, T88)

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:

• ADD32_IN_GAG(s(T87), s(T88)) → ADD32_IN_GAG(T87, T88)
  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:

LE9_IN_GG(s(T33), s(T34)) → LE9_IN_GG(T33, T34)

The TRS R consists of the following rules:

lec9_in_gg(s(T33), s(T34)) → U16_gg(T33, T34, lec9_in_gg(T33, T34))
lec9_in_gg(0, s(T41)) → lec9_out_gg(0, s(T41))
lec9_in_gg(0, 0) → lec9_out_gg(0, 0)
U16_gg(T33, T34, lec9_out_gg(T33, T34)) → lec9_out_gg(s(T33), s(T34))
addc27_in_gag(T75, X100, T76) → U26_gag(T75, X100, T76, addc32_in_gag(T75, X100, T76))
addc32_in_gag(s(T87), X124, s(T88)) → U24_gag(T87, X124, T88, addc32_in_gag(T87, X124, T88))
addc32_in_gag(0, T93, T93) → addc32_out_gag(0, T93, T93)
U24_gag(T87, X124, T88, addc32_out_gag(T87, X124, T88)) → addc32_out_gag(s(T87), X124, s(T88))
U26_gag(T75, X100, T76, addc32_out_gag(T75, X100, T76)) → addc27_out_gag(T75, X100, T76)
gtc66_in_gg(s(T134), s(T135)) → U25_gg(T134, T135, gtc66_in_gg(T134, T135))
gtc66_in_gg(s(T140), 0) → gtc66_out_gg(s(T140), 0)
U25_gg(T134, T135, gtc66_out_gg(T134, T135)) → gtc66_out_gg(s(T134), s(T135))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
lec9_in_gg(x1, x2)  =  lec9_in_gg(x1, x2)
U16_gg(x1, x2, x3)  =  U16_gg(x1, x2, x3)
0  =  0
lec9_out_gg(x1, x2)  =  lec9_out_gg(x1, x2)
addc27_in_gag(x1, x2, x3)  =  addc27_in_gag(x1, x3)
U26_gag(x1, x2, x3, x4)  =  U26_gag(x1, x3, x4)
addc32_in_gag(x1, x2, x3)  =  addc32_in_gag(x1, x3)
U24_gag(x1, x2, x3, x4)  =  U24_gag(x1, x3, x4)
addc32_out_gag(x1, x2, x3)  =  addc32_out_gag(x1, x2, x3)
addc27_out_gag(x1, x2, x3)  =  addc27_out_gag(x1, x2, x3)
gtc66_in_gg(x1, x2)  =  gtc66_in_gg(x1, x2)
U25_gg(x1, x2, x3)  =  U25_gg(x1, x2, x3)
gtc66_out_gg(x1, x2)  =  gtc66_out_gg(x1, x2)
LE9_IN_GG(x1, x2)  =  LE9_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:

LE9_IN_GG(s(T33), s(T34)) → LE9_IN_GG(T33, T34)

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:

LE9_IN_GG(s(T33), s(T34)) → LE9_IN_GG(T33, T34)

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:

• LE9_IN_GG(s(T33), s(T34)) → LE9_IN_GG(T33, T34)
  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:

GCD1_IN_GGA(s(T54), s(T55), T57) → U7_GGA(T54, T55, T57, lec9_in_gg(T54, T55))
U7_GGA(T54, T55, T57, lec9_out_gg(T54, T55)) → U8_GGA(T54, T55, T57, addc27_in_gag(T54, T60, T55))
U8_GGA(T54, T55, T57, addc27_out_gag(T54, T60, T55)) → GCD1_IN_GGA(s(T54), T60, T57)
GCD1_IN_GGA(T155, s(T154), T157) → U11_GGA(T155, T154, T157, gtc66_in_gg(T155, s(T154)))
U11_GGA(T155, T154, T157, gtc66_out_gg(T155, s(T154))) → U13_GGA(T155, T154, T157, addc32_in_gag(s(T154), T160, T155))
U13_GGA(T155, T154, T157, addc32_out_gag(s(T154), T160, T155)) → GCD1_IN_GGA(s(T154), T160, T157)

The TRS R consists of the following rules:

lec9_in_gg(s(T33), s(T34)) → U16_gg(T33, T34, lec9_in_gg(T33, T34))
lec9_in_gg(0, s(T41)) → lec9_out_gg(0, s(T41))
lec9_in_gg(0, 0) → lec9_out_gg(0, 0)
U16_gg(T33, T34, lec9_out_gg(T33, T34)) → lec9_out_gg(s(T33), s(T34))
addc27_in_gag(T75, X100, T76) → U26_gag(T75, X100, T76, addc32_in_gag(T75, X100, T76))
addc32_in_gag(s(T87), X124, s(T88)) → U24_gag(T87, X124, T88, addc32_in_gag(T87, X124, T88))
addc32_in_gag(0, T93, T93) → addc32_out_gag(0, T93, T93)
U24_gag(T87, X124, T88, addc32_out_gag(T87, X124, T88)) → addc32_out_gag(s(T87), X124, s(T88))
U26_gag(T75, X100, T76, addc32_out_gag(T75, X100, T76)) → addc27_out_gag(T75, X100, T76)
gtc66_in_gg(s(T134), s(T135)) → U25_gg(T134, T135, gtc66_in_gg(T134, T135))
gtc66_in_gg(s(T140), 0) → gtc66_out_gg(s(T140), 0)
U25_gg(T134, T135, gtc66_out_gg(T134, T135)) → gtc66_out_gg(s(T134), s(T135))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
lec9_in_gg(x1, x2)  =  lec9_in_gg(x1, x2)
U16_gg(x1, x2, x3)  =  U16_gg(x1, x2, x3)
0  =  0
lec9_out_gg(x1, x2)  =  lec9_out_gg(x1, x2)
addc27_in_gag(x1, x2, x3)  =  addc27_in_gag(x1, x3)
U26_gag(x1, x2, x3, x4)  =  U26_gag(x1, x3, x4)
addc32_in_gag(x1, x2, x3)  =  addc32_in_gag(x1, x3)
U24_gag(x1, x2, x3, x4)  =  U24_gag(x1, x3, x4)
addc32_out_gag(x1, x2, x3)  =  addc32_out_gag(x1, x2, x3)
addc27_out_gag(x1, x2, x3)  =  addc27_out_gag(x1, x2, x3)
gtc66_in_gg(x1, x2)  =  gtc66_in_gg(x1, x2)
U25_gg(x1, x2, x3)  =  U25_gg(x1, x2, x3)
gtc66_out_gg(x1, x2)  =  gtc66_out_gg(x1, x2)
GCD1_IN_GGA(x1, x2, x3)  =  GCD1_IN_GGA(x1, x2)
U7_GGA(x1, x2, x3, x4)  =  U7_GGA(x1, x2, x4)
U8_GGA(x1, x2, x3, x4)  =  U8_GGA(x1, x2, x4)
U11_GGA(x1, x2, x3, x4)  =  U11_GGA(x1, x2, x4)
U13_GGA(x1, x2, x3, x4)  =  U13_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:

GCD1_IN_GGA(s(T54), s(T55)) → U7_GGA(T54, T55, lec9_in_gg(T54, T55))
U7_GGA(T54, T55, lec9_out_gg(T54, T55)) → U8_GGA(T54, T55, addc27_in_gag(T54, T55))
U8_GGA(T54, T55, addc27_out_gag(T54, T60, T55)) → GCD1_IN_GGA(s(T54), T60)
GCD1_IN_GGA(T155, s(T154)) → U11_GGA(T155, T154, gtc66_in_gg(T155, s(T154)))
U11_GGA(T155, T154, gtc66_out_gg(T155, s(T154))) → U13_GGA(T155, T154, addc32_in_gag(s(T154), T155))
U13_GGA(T155, T154, addc32_out_gag(s(T154), T160, T155)) → GCD1_IN_GGA(s(T154), T160)

The TRS R consists of the following rules:

lec9_in_gg(s(T33), s(T34)) → U16_gg(T33, T34, lec9_in_gg(T33, T34))
lec9_in_gg(0, s(T41)) → lec9_out_gg(0, s(T41))
lec9_in_gg(0, 0) → lec9_out_gg(0, 0)
U16_gg(T33, T34, lec9_out_gg(T33, T34)) → lec9_out_gg(s(T33), s(T34))
addc27_in_gag(T75, T76) → U26_gag(T75, T76, addc32_in_gag(T75, T76))
addc32_in_gag(s(T87), s(T88)) → U24_gag(T87, T88, addc32_in_gag(T87, T88))
addc32_in_gag(0, T93) → addc32_out_gag(0, T93, T93)
U24_gag(T87, T88, addc32_out_gag(T87, X124, T88)) → addc32_out_gag(s(T87), X124, s(T88))
U26_gag(T75, T76, addc32_out_gag(T75, X100, T76)) → addc27_out_gag(T75, X100, T76)
gtc66_in_gg(s(T134), s(T135)) → U25_gg(T134, T135, gtc66_in_gg(T134, T135))
gtc66_in_gg(s(T140), 0) → gtc66_out_gg(s(T140), 0)
U25_gg(T134, T135, gtc66_out_gg(T134, T135)) → gtc66_out_gg(s(T134), s(T135))

The set Q consists of the following terms:

lec9_in_gg(x0, x1)
U16_gg(x0, x1, x2)
addc27_in_gag(x0, x1)
addc32_in_gag(x0, x1)
U24_gag(x0, x1, x2)
U26_gag(x0, x1, x2)
gtc66_in_gg(x0, x1)
U25_gg(x0, x1, x2)

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

(31) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U7_GGA(T54, T55, lec9_out_gg(T54, T55)) → U8_GGA(T54, T55, addc27_in_gag(T54, T55)) at position [2] we obtained the following new rules [LPAR04]:

U7_GGA(T54, T55, lec9_out_gg(T54, T55)) → U8_GGA(T54, T55, U26_gag(T54, T55, addc32_in_gag(T54, T55)))

(32) Obligation:

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

GCD1_IN_GGA(s(T54), s(T55)) → U7_GGA(T54, T55, lec9_in_gg(T54, T55))
U8_GGA(T54, T55, addc27_out_gag(T54, T60, T55)) → GCD1_IN_GGA(s(T54), T60)
GCD1_IN_GGA(T155, s(T154)) → U11_GGA(T155, T154, gtc66_in_gg(T155, s(T154)))
U11_GGA(T155, T154, gtc66_out_gg(T155, s(T154))) → U13_GGA(T155, T154, addc32_in_gag(s(T154), T155))
U13_GGA(T155, T154, addc32_out_gag(s(T154), T160, T155)) → GCD1_IN_GGA(s(T154), T160)
U7_GGA(T54, T55, lec9_out_gg(T54, T55)) → U8_GGA(T54, T55, U26_gag(T54, T55, addc32_in_gag(T54, T55)))

The TRS R consists of the following rules:

lec9_in_gg(s(T33), s(T34)) → U16_gg(T33, T34, lec9_in_gg(T33, T34))
lec9_in_gg(0, s(T41)) → lec9_out_gg(0, s(T41))
lec9_in_gg(0, 0) → lec9_out_gg(0, 0)
U16_gg(T33, T34, lec9_out_gg(T33, T34)) → lec9_out_gg(s(T33), s(T34))
addc27_in_gag(T75, T76) → U26_gag(T75, T76, addc32_in_gag(T75, T76))
addc32_in_gag(s(T87), s(T88)) → U24_gag(T87, T88, addc32_in_gag(T87, T88))
addc32_in_gag(0, T93) → addc32_out_gag(0, T93, T93)
U24_gag(T87, T88, addc32_out_gag(T87, X124, T88)) → addc32_out_gag(s(T87), X124, s(T88))
U26_gag(T75, T76, addc32_out_gag(T75, X100, T76)) → addc27_out_gag(T75, X100, T76)
gtc66_in_gg(s(T134), s(T135)) → U25_gg(T134, T135, gtc66_in_gg(T134, T135))
gtc66_in_gg(s(T140), 0) → gtc66_out_gg(s(T140), 0)
U25_gg(T134, T135, gtc66_out_gg(T134, T135)) → gtc66_out_gg(s(T134), s(T135))

The set Q consists of the following terms:

lec9_in_gg(x0, x1)
U16_gg(x0, x1, x2)
addc27_in_gag(x0, x1)
addc32_in_gag(x0, x1)
U24_gag(x0, x1, x2)
U26_gag(x0, x1, x2)
gtc66_in_gg(x0, x1)
U25_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:

GCD1_IN_GGA(s(T54), s(T55)) → U7_GGA(T54, T55, lec9_in_gg(T54, T55))
U8_GGA(T54, T55, addc27_out_gag(T54, T60, T55)) → GCD1_IN_GGA(s(T54), T60)
GCD1_IN_GGA(T155, s(T154)) → U11_GGA(T155, T154, gtc66_in_gg(T155, s(T154)))
U11_GGA(T155, T154, gtc66_out_gg(T155, s(T154))) → U13_GGA(T155, T154, addc32_in_gag(s(T154), T155))
U13_GGA(T155, T154, addc32_out_gag(s(T154), T160, T155)) → GCD1_IN_GGA(s(T154), T160)
U7_GGA(T54, T55, lec9_out_gg(T54, T55)) → U8_GGA(T54, T55, U26_gag(T54, T55, addc32_in_gag(T54, T55)))

The TRS R consists of the following rules:

addc32_in_gag(s(T87), s(T88)) → U24_gag(T87, T88, addc32_in_gag(T87, T88))
addc32_in_gag(0, T93) → addc32_out_gag(0, T93, T93)
U26_gag(T75, T76, addc32_out_gag(T75, X100, T76)) → addc27_out_gag(T75, X100, T76)
U24_gag(T87, T88, addc32_out_gag(T87, X124, T88)) → addc32_out_gag(s(T87), X124, s(T88))
gtc66_in_gg(s(T134), s(T135)) → U25_gg(T134, T135, gtc66_in_gg(T134, T135))
gtc66_in_gg(s(T140), 0) → gtc66_out_gg(s(T140), 0)
U25_gg(T134, T135, gtc66_out_gg(T134, T135)) → gtc66_out_gg(s(T134), s(T135))
lec9_in_gg(s(T33), s(T34)) → U16_gg(T33, T34, lec9_in_gg(T33, T34))
lec9_in_gg(0, s(T41)) → lec9_out_gg(0, s(T41))
lec9_in_gg(0, 0) → lec9_out_gg(0, 0)
U16_gg(T33, T34, lec9_out_gg(T33, T34)) → lec9_out_gg(s(T33), s(T34))

The set Q consists of the following terms:

lec9_in_gg(x0, x1)
U16_gg(x0, x1, x2)
addc27_in_gag(x0, x1)
addc32_in_gag(x0, x1)
U24_gag(x0, x1, x2)
U26_gag(x0, x1, x2)
gtc66_in_gg(x0, x1)
U25_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].

addc27_in_gag(x0, x1)

(36) Obligation:

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

GCD1_IN_GGA(s(T54), s(T55)) → U7_GGA(T54, T55, lec9_in_gg(T54, T55))
U8_GGA(T54, T55, addc27_out_gag(T54, T60, T55)) → GCD1_IN_GGA(s(T54), T60)
GCD1_IN_GGA(T155, s(T154)) → U11_GGA(T155, T154, gtc66_in_gg(T155, s(T154)))
U11_GGA(T155, T154, gtc66_out_gg(T155, s(T154))) → U13_GGA(T155, T154, addc32_in_gag(s(T154), T155))
U13_GGA(T155, T154, addc32_out_gag(s(T154), T160, T155)) → GCD1_IN_GGA(s(T154), T160)
U7_GGA(T54, T55, lec9_out_gg(T54, T55)) → U8_GGA(T54, T55, U26_gag(T54, T55, addc32_in_gag(T54, T55)))

The TRS R consists of the following rules:

addc32_in_gag(s(T87), s(T88)) → U24_gag(T87, T88, addc32_in_gag(T87, T88))
addc32_in_gag(0, T93) → addc32_out_gag(0, T93, T93)
U26_gag(T75, T76, addc32_out_gag(T75, X100, T76)) → addc27_out_gag(T75, X100, T76)
U24_gag(T87, T88, addc32_out_gag(T87, X124, T88)) → addc32_out_gag(s(T87), X124, s(T88))
gtc66_in_gg(s(T134), s(T135)) → U25_gg(T134, T135, gtc66_in_gg(T134, T135))
gtc66_in_gg(s(T140), 0) → gtc66_out_gg(s(T140), 0)
U25_gg(T134, T135, gtc66_out_gg(T134, T135)) → gtc66_out_gg(s(T134), s(T135))
lec9_in_gg(s(T33), s(T34)) → U16_gg(T33, T34, lec9_in_gg(T33, T34))
lec9_in_gg(0, s(T41)) → lec9_out_gg(0, s(T41))
lec9_in_gg(0, 0) → lec9_out_gg(0, 0)
U16_gg(T33, T34, lec9_out_gg(T33, T34)) → lec9_out_gg(s(T33), s(T34))

The set Q consists of the following terms:

lec9_in_gg(x0, x1)
U16_gg(x0, x1, x2)
addc32_in_gag(x0, x1)
U24_gag(x0, x1, x2)
U26_gag(x0, x1, x2)
gtc66_in_gg(x0, x1)
U25_gg(x0, x1, x2)

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

(37) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11_GGA(T155, T154, gtc66_out_gg(T155, s(T154))) → U13_GGA(T155, T154, addc32_in_gag(s(T154), T155))

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

POL(0) = 0   
POL(GCD1_IN_GGA(x₁, x₂)) = x₁   
POL(U11_GGA(x₁, x₂, x₃)) = x₃   
POL(U13_GGA(x₁, x₂, x₃)) = 1 + x₂   
POL(U16_gg(x₁, x₂, x₃)) = 0   
POL(U24_gag(x₁, x₂, x₃)) = 0   
POL(U25_gg(x₁, x₂, x₃)) = 1 + x₃   
POL(U26_gag(x₁, x₂, x₃)) = 0   
POL(U7_GGA(x₁, x₂, x₃)) = 1 + x₁   
POL(U8_GGA(x₁, x₂, x₃)) = 1 + x₁   
POL(addc27_out_gag(x₁, x₂, x₃)) = 0   
POL(addc32_in_gag(x₁, x₂)) = 0   
POL(addc32_out_gag(x₁, x₂, x₃)) = 0   
POL(gtc66_in_gg(x₁, x₂)) = x₁   
POL(gtc66_out_gg(x₁, x₂)) = 1 + x₂   
POL(lec9_in_gg(x₁, x₂)) = 0   
POL(lec9_out_gg(x₁, x₂)) = 0   
POL(s(x₁)) = 1 + x₁   

The following usable rules [FROCOS05] were oriented:

gtc66_in_gg(s(T134), s(T135)) → U25_gg(T134, T135, gtc66_in_gg(T134, T135))
gtc66_in_gg(s(T140), 0) → gtc66_out_gg(s(T140), 0)
U25_gg(T134, T135, gtc66_out_gg(T134, T135)) → gtc66_out_gg(s(T134), s(T135))

(38) Obligation:

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

GCD1_IN_GGA(s(T54), s(T55)) → U7_GGA(T54, T55, lec9_in_gg(T54, T55))
U8_GGA(T54, T55, addc27_out_gag(T54, T60, T55)) → GCD1_IN_GGA(s(T54), T60)
GCD1_IN_GGA(T155, s(T154)) → U11_GGA(T155, T154, gtc66_in_gg(T155, s(T154)))
U13_GGA(T155, T154, addc32_out_gag(s(T154), T160, T155)) → GCD1_IN_GGA(s(T154), T160)
U7_GGA(T54, T55, lec9_out_gg(T54, T55)) → U8_GGA(T54, T55, U26_gag(T54, T55, addc32_in_gag(T54, T55)))

The TRS R consists of the following rules:

addc32_in_gag(s(T87), s(T88)) → U24_gag(T87, T88, addc32_in_gag(T87, T88))
addc32_in_gag(0, T93) → addc32_out_gag(0, T93, T93)
U26_gag(T75, T76, addc32_out_gag(T75, X100, T76)) → addc27_out_gag(T75, X100, T76)
U24_gag(T87, T88, addc32_out_gag(T87, X124, T88)) → addc32_out_gag(s(T87), X124, s(T88))
gtc66_in_gg(s(T134), s(T135)) → U25_gg(T134, T135, gtc66_in_gg(T134, T135))
gtc66_in_gg(s(T140), 0) → gtc66_out_gg(s(T140), 0)
U25_gg(T134, T135, gtc66_out_gg(T134, T135)) → gtc66_out_gg(s(T134), s(T135))
lec9_in_gg(s(T33), s(T34)) → U16_gg(T33, T34, lec9_in_gg(T33, T34))
lec9_in_gg(0, s(T41)) → lec9_out_gg(0, s(T41))
lec9_in_gg(0, 0) → lec9_out_gg(0, 0)
U16_gg(T33, T34, lec9_out_gg(T33, T34)) → lec9_out_gg(s(T33), s(T34))

The set Q consists of the following terms:

lec9_in_gg(x0, x1)
U16_gg(x0, x1, x2)
addc32_in_gag(x0, x1)
U24_gag(x0, x1, x2)
U26_gag(x0, x1, x2)
gtc66_in_gg(x0, x1)
U25_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:

U7_GGA(T54, T55, lec9_out_gg(T54, T55)) → U8_GGA(T54, T55, U26_gag(T54, T55, addc32_in_gag(T54, T55)))
U8_GGA(T54, T55, addc27_out_gag(T54, T60, T55)) → GCD1_IN_GGA(s(T54), T60)
GCD1_IN_GGA(s(T54), s(T55)) → U7_GGA(T54, T55, lec9_in_gg(T54, T55))

The TRS R consists of the following rules:

addc32_in_gag(s(T87), s(T88)) → U24_gag(T87, T88, addc32_in_gag(T87, T88))
addc32_in_gag(0, T93) → addc32_out_gag(0, T93, T93)
U26_gag(T75, T76, addc32_out_gag(T75, X100, T76)) → addc27_out_gag(T75, X100, T76)
U24_gag(T87, T88, addc32_out_gag(T87, X124, T88)) → addc32_out_gag(s(T87), X124, s(T88))
gtc66_in_gg(s(T134), s(T135)) → U25_gg(T134, T135, gtc66_in_gg(T134, T135))
gtc66_in_gg(s(T140), 0) → gtc66_out_gg(s(T140), 0)
U25_gg(T134, T135, gtc66_out_gg(T134, T135)) → gtc66_out_gg(s(T134), s(T135))
lec9_in_gg(s(T33), s(T34)) → U16_gg(T33, T34, lec9_in_gg(T33, T34))
lec9_in_gg(0, s(T41)) → lec9_out_gg(0, s(T41))
lec9_in_gg(0, 0) → lec9_out_gg(0, 0)
U16_gg(T33, T34, lec9_out_gg(T33, T34)) → lec9_out_gg(s(T33), s(T34))

The set Q consists of the following terms:

lec9_in_gg(x0, x1)
U16_gg(x0, x1, x2)
addc32_in_gag(x0, x1)
U24_gag(x0, x1, x2)
U26_gag(x0, x1, x2)
gtc66_in_gg(x0, x1)
U25_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:

U7_GGA(T54, T55, lec9_out_gg(T54, T55)) → U8_GGA(T54, T55, U26_gag(T54, T55, addc32_in_gag(T54, T55)))
U8_GGA(T54, T55, addc27_out_gag(T54, T60, T55)) → GCD1_IN_GGA(s(T54), T60)
GCD1_IN_GGA(s(T54), s(T55)) → U7_GGA(T54, T55, lec9_in_gg(T54, T55))

The TRS R consists of the following rules:

lec9_in_gg(s(T33), s(T34)) → U16_gg(T33, T34, lec9_in_gg(T33, T34))
lec9_in_gg(0, s(T41)) → lec9_out_gg(0, s(T41))
lec9_in_gg(0, 0) → lec9_out_gg(0, 0)
U16_gg(T33, T34, lec9_out_gg(T33, T34)) → lec9_out_gg(s(T33), s(T34))
addc32_in_gag(s(T87), s(T88)) → U24_gag(T87, T88, addc32_in_gag(T87, T88))
addc32_in_gag(0, T93) → addc32_out_gag(0, T93, T93)
U26_gag(T75, T76, addc32_out_gag(T75, X100, T76)) → addc27_out_gag(T75, X100, T76)
U24_gag(T87, T88, addc32_out_gag(T87, X124, T88)) → addc32_out_gag(s(T87), X124, s(T88))

The set Q consists of the following terms:

lec9_in_gg(x0, x1)
U16_gg(x0, x1, x2)
addc32_in_gag(x0, x1)
U24_gag(x0, x1, x2)
U26_gag(x0, x1, x2)
gtc66_in_gg(x0, x1)
U25_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].

gtc66_in_gg(x0, x1)
U25_gg(x0, x1, x2)

(44) Obligation:

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

U7_GGA(T54, T55, lec9_out_gg(T54, T55)) → U8_GGA(T54, T55, U26_gag(T54, T55, addc32_in_gag(T54, T55)))
U8_GGA(T54, T55, addc27_out_gag(T54, T60, T55)) → GCD1_IN_GGA(s(T54), T60)
GCD1_IN_GGA(s(T54), s(T55)) → U7_GGA(T54, T55, lec9_in_gg(T54, T55))

The TRS R consists of the following rules:

lec9_in_gg(s(T33), s(T34)) → U16_gg(T33, T34, lec9_in_gg(T33, T34))
lec9_in_gg(0, s(T41)) → lec9_out_gg(0, s(T41))
lec9_in_gg(0, 0) → lec9_out_gg(0, 0)
U16_gg(T33, T34, lec9_out_gg(T33, T34)) → lec9_out_gg(s(T33), s(T34))
addc32_in_gag(s(T87), s(T88)) → U24_gag(T87, T88, addc32_in_gag(T87, T88))
addc32_in_gag(0, T93) → addc32_out_gag(0, T93, T93)
U26_gag(T75, T76, addc32_out_gag(T75, X100, T76)) → addc27_out_gag(T75, X100, T76)
U24_gag(T87, T88, addc32_out_gag(T87, X124, T88)) → addc32_out_gag(s(T87), X124, s(T88))

The set Q consists of the following terms:

lec9_in_gg(x0, x1)
U16_gg(x0, x1, x2)
addc32_in_gag(x0, x1)
U24_gag(x0, x1, x2)
U26_gag(x0, x1, x2)

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

(45) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

GCD1_IN_GGA(s(T54), s(T55)) → U7_GGA(T54, T55, lec9_in_gg(T54, T55))

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

POL(0) = 0   
POL(GCD1_IN_GGA(x₁, x₂)) = x₂   
POL(U16_gg(x₁, x₂, x₃)) = 0   
POL(U24_gag(x₁, x₂, x₃)) = x₃   
POL(U26_gag(x₁, x₂, x₃)) = x₃   
POL(U7_GGA(x₁, x₂, x₃)) = x₂   
POL(U8_GGA(x₁, x₂, x₃)) = x₃   
POL(addc27_out_gag(x₁, x₂, x₃)) = x₂   
POL(addc32_in_gag(x₁, x₂)) = x₂   
POL(addc32_out_gag(x₁, x₂, x₃)) = x₂   
POL(lec9_in_gg(x₁, x₂)) = 0   
POL(lec9_out_gg(x₁, x₂)) = 0   
POL(s(x₁)) = 1 + x₁   

The following usable rules [FROCOS05] were oriented:

addc32_in_gag(s(T87), s(T88)) → U24_gag(T87, T88, addc32_in_gag(T87, T88))
addc32_in_gag(0, T93) → addc32_out_gag(0, T93, T93)
U26_gag(T75, T76, addc32_out_gag(T75, X100, T76)) → addc27_out_gag(T75, X100, T76)
U24_gag(T87, T88, addc32_out_gag(T87, X124, T88)) → addc32_out_gag(s(T87), X124, s(T88))

(46) Obligation:

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

U7_GGA(T54, T55, lec9_out_gg(T54, T55)) → U8_GGA(T54, T55, U26_gag(T54, T55, addc32_in_gag(T54, T55)))
U8_GGA(T54, T55, addc27_out_gag(T54, T60, T55)) → GCD1_IN_GGA(s(T54), T60)

The TRS R consists of the following rules:

lec9_in_gg(s(T33), s(T34)) → U16_gg(T33, T34, lec9_in_gg(T33, T34))
lec9_in_gg(0, s(T41)) → lec9_out_gg(0, s(T41))
lec9_in_gg(0, 0) → lec9_out_gg(0, 0)
U16_gg(T33, T34, lec9_out_gg(T33, T34)) → lec9_out_gg(s(T33), s(T34))
addc32_in_gag(s(T87), s(T88)) → U24_gag(T87, T88, addc32_in_gag(T87, T88))
addc32_in_gag(0, T93) → addc32_out_gag(0, T93, T93)
U26_gag(T75, T76, addc32_out_gag(T75, X100, T76)) → addc27_out_gag(T75, X100, T76)
U24_gag(T87, T88, addc32_out_gag(T87, X124, T88)) → addc32_out_gag(s(T87), X124, s(T88))

The set Q consists of the following terms:

lec9_in_gg(x0, x1)
U16_gg(x0, x1, x2)
addc32_in_gag(x0, x1)
U24_gag(x0, x1, x2)
U26_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