Left Termination proof of /tmp/tmpdLIp3K/gcd.pl

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

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇒, 187 ms)

↳2 Prolog

↳3 PrologToPiTRSProof (⇒, 0 ms)

↳4 PiTRS

↳5 DependencyPairsProof (⇔, 116 ms)

↳6 PiDP

↳7 DependencyGraphProof (⇔, 15 ms)

↳8 AND

    ↳9 PiDP

        ↳10 UsableRulesProof (⇔, 0 ms)

        ↳11 PiDP

        ↳12 PiDPToQDPProof (⇔, 0 ms)

        ↳13 QDP

        ↳14 QDPSizeChangeProof (⇔, 0 ms)

        ↳15 YES

    ↳16 PiDP

        ↳17 UsableRulesProof (⇔, 0 ms)

        ↳18 PiDP

        ↳19 PiDPToQDPProof (⇒, 0 ms)

        ↳20 QDP

        ↳21 QDPSizeChangeProof (⇔, 0 ms)

        ↳22 YES

    ↳23 PiDP

        ↳24 UsableRulesProof (⇔, 0 ms)

        ↳25 PiDP

        ↳26 PiDPToQDPProof (⇔, 0 ms)

        ↳27 QDP

        ↳28 QDPSizeChangeProof (⇔, 0 ms)

        ↳29 YES

    ↳30 PiDP

        ↳31 UsableRulesProof (⇔, 0 ms)

        ↳32 PiDP

        ↳33 PiDPToQDPProof (⇒, 4 ms)

        ↳34 QDP

        ↳35 Rewriting (⇔, 10 ms)

        ↳36 QDP

        ↳37 UsableRulesProof (⇔, 0 ms)

        ↳38 QDP

        ↳39 QReductionProof (⇔, 0 ms)

        ↳40 QDP

        ↳41 QDPOrderProof (⇔, 156 ms)

        ↳42 QDP

        ↳43 DependencyGraphProof (⇔, 0 ms)

        ↳44 QDP

        ↳45 UsableRulesProof (⇔, 0 ms)

        ↳46 QDP

        ↳47 QReductionProof (⇔, 0 ms)

        ↳48 QDP

        ↳49 QDPOrderProof (⇔, 68 ms)

        ↳50 QDP

        ↳51 DependencyGraphProof (⇔, 0 ms)

        ↳52 TRUE

(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) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

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(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="27: 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="32: 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="66: 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:

Clauses:

leA(s(T33), s(T34)) :- leA(T33, T34).
leA(0, s(T41)).
leA(0, 0).
addB(s(T84), X115, s(T85)) :- addB(T84, X115, T85).
addB(0, T90, T90).
gtC(s(T131), s(T132)) :- gtC(T131, T132).
gtC(s(T137), 0).
addD(T72, X91, T73) :- addB(T72, X91, T73).
gcdE(s(T19), s(T20), T10) :- leA(T19, T20).
gcdE(s(T51), s(T52), T54) :- ','(leA(T51, T52), addD(T51, X55, T52)).
gcdE(s(T51), s(T52), T54) :- ','(leA(T51, T52), ','(addD(T51, T57, T52), gcdE(s(T51), T57, T54))).
gcdE(0, s(T105), s(T105)).
gcdE(0, 0, 0).
gcdE(T115, T116, T118) :- gtC(T115, T116).
gcdE(T144, 0, T144) :- gtC(T144, 0).
gcdE(T152, s(T151), T154) :- ','(gtC(T152, s(T151)), addB(s(T151), X202, T152)).
gcdE(T152, s(T151), T154) :- ','(gtC(T152, s(T151)), ','(addB(s(T151), T157, T152), gcdE(s(T151), T157, T154))).

Query: gcdE(g,g,a)

(3) PrologToPiTRSProof (SOUND transformation)

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

gcdE_in_gga(s(T19), s(T20), T10) → U5_gga(T19, T20, T10, leA_in_gg(T19, T20))
leA_in_gg(s(T33), s(T34)) → U1_gg(T33, T34, leA_in_gg(T33, T34))
leA_in_gg(0, s(T41)) → leA_out_gg(0, s(T41))
leA_in_gg(0, 0) → leA_out_gg(0, 0)
U1_gg(T33, T34, leA_out_gg(T33, T34)) → leA_out_gg(s(T33), s(T34))
U5_gga(T19, T20, T10, leA_out_gg(T19, T20)) → gcdE_out_gga(s(T19), s(T20), T10)
gcdE_in_gga(s(T51), s(T52), T54) → U6_gga(T51, T52, T54, leA_in_gg(T51, T52))
U6_gga(T51, T52, T54, leA_out_gg(T51, T52)) → U7_gga(T51, T52, T54, addD_in_gag(T51, X55, T52))
addD_in_gag(T72, X91, T73) → U4_gag(T72, X91, T73, addB_in_gag(T72, X91, T73))
addB_in_gag(s(T84), X115, s(T85)) → U2_gag(T84, X115, T85, addB_in_gag(T84, X115, T85))
addB_in_gag(0, T90, T90) → addB_out_gag(0, T90, T90)
U2_gag(T84, X115, T85, addB_out_gag(T84, X115, T85)) → addB_out_gag(s(T84), X115, s(T85))
U4_gag(T72, X91, T73, addB_out_gag(T72, X91, T73)) → addD_out_gag(T72, X91, T73)
U7_gga(T51, T52, T54, addD_out_gag(T51, X55, T52)) → gcdE_out_gga(s(T51), s(T52), T54)
U6_gga(T51, T52, T54, leA_out_gg(T51, T52)) → U8_gga(T51, T52, T54, addD_in_gag(T51, T57, T52))
U8_gga(T51, T52, T54, addD_out_gag(T51, T57, T52)) → U9_gga(T51, T52, T54, gcdE_in_gga(s(T51), T57, T54))
gcdE_in_gga(0, s(T105), s(T105)) → gcdE_out_gga(0, s(T105), s(T105))
gcdE_in_gga(0, 0, 0) → gcdE_out_gga(0, 0, 0)
gcdE_in_gga(T115, T116, T118) → U10_gga(T115, T116, T118, gtC_in_gg(T115, T116))
gtC_in_gg(s(T131), s(T132)) → U3_gg(T131, T132, gtC_in_gg(T131, T132))
gtC_in_gg(s(T137), 0) → gtC_out_gg(s(T137), 0)
U3_gg(T131, T132, gtC_out_gg(T131, T132)) → gtC_out_gg(s(T131), s(T132))
U10_gga(T115, T116, T118, gtC_out_gg(T115, T116)) → gcdE_out_gga(T115, T116, T118)
gcdE_in_gga(T144, 0, T144) → U11_gga(T144, gtC_in_gg(T144, 0))
U11_gga(T144, gtC_out_gg(T144, 0)) → gcdE_out_gga(T144, 0, T144)
gcdE_in_gga(T152, s(T151), T154) → U12_gga(T152, T151, T154, gtC_in_gg(T152, s(T151)))
U12_gga(T152, T151, T154, gtC_out_gg(T152, s(T151))) → U13_gga(T152, T151, T154, addB_in_gag(s(T151), X202, T152))
U13_gga(T152, T151, T154, addB_out_gag(s(T151), X202, T152)) → gcdE_out_gga(T152, s(T151), T154)
U12_gga(T152, T151, T154, gtC_out_gg(T152, s(T151))) → U14_gga(T152, T151, T154, addB_in_gag(s(T151), T157, T152))
U14_gga(T152, T151, T154, addB_out_gag(s(T151), T157, T152)) → U15_gga(T152, T151, T154, gcdE_in_gga(s(T151), T157, T154))
U15_gga(T152, T151, T154, gcdE_out_gga(s(T151), T157, T154)) → gcdE_out_gga(T152, s(T151), T154)
U9_gga(T51, T52, T54, gcdE_out_gga(s(T51), T57, T54)) → gcdE_out_gga(s(T51), s(T52), T54)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

gcdE_in_gga(s(T19), s(T20), T10) → U5_gga(T19, T20, T10, leA_in_gg(T19, T20))
leA_in_gg(s(T33), s(T34)) → U1_gg(T33, T34, leA_in_gg(T33, T34))
leA_in_gg(0, s(T41)) → leA_out_gg(0, s(T41))
leA_in_gg(0, 0) → leA_out_gg(0, 0)
U1_gg(T33, T34, leA_out_gg(T33, T34)) → leA_out_gg(s(T33), s(T34))
U5_gga(T19, T20, T10, leA_out_gg(T19, T20)) → gcdE_out_gga(s(T19), s(T20), T10)
gcdE_in_gga(s(T51), s(T52), T54) → U6_gga(T51, T52, T54, leA_in_gg(T51, T52))
U6_gga(T51, T52, T54, leA_out_gg(T51, T52)) → U7_gga(T51, T52, T54, addD_in_gag(T51, X55, T52))
addD_in_gag(T72, X91, T73) → U4_gag(T72, X91, T73, addB_in_gag(T72, X91, T73))
addB_in_gag(s(T84), X115, s(T85)) → U2_gag(T84, X115, T85, addB_in_gag(T84, X115, T85))
addB_in_gag(0, T90, T90) → addB_out_gag(0, T90, T90)
U2_gag(T84, X115, T85, addB_out_gag(T84, X115, T85)) → addB_out_gag(s(T84), X115, s(T85))
U4_gag(T72, X91, T73, addB_out_gag(T72, X91, T73)) → addD_out_gag(T72, X91, T73)
U7_gga(T51, T52, T54, addD_out_gag(T51, X55, T52)) → gcdE_out_gga(s(T51), s(T52), T54)
U6_gga(T51, T52, T54, leA_out_gg(T51, T52)) → U8_gga(T51, T52, T54, addD_in_gag(T51, T57, T52))
U8_gga(T51, T52, T54, addD_out_gag(T51, T57, T52)) → U9_gga(T51, T52, T54, gcdE_in_gga(s(T51), T57, T54))
gcdE_in_gga(0, s(T105), s(T105)) → gcdE_out_gga(0, s(T105), s(T105))
gcdE_in_gga(0, 0, 0) → gcdE_out_gga(0, 0, 0)
gcdE_in_gga(T115, T116, T118) → U10_gga(T115, T116, T118, gtC_in_gg(T115, T116))
gtC_in_gg(s(T131), s(T132)) → U3_gg(T131, T132, gtC_in_gg(T131, T132))
gtC_in_gg(s(T137), 0) → gtC_out_gg(s(T137), 0)
U3_gg(T131, T132, gtC_out_gg(T131, T132)) → gtC_out_gg(s(T131), s(T132))
U10_gga(T115, T116, T118, gtC_out_gg(T115, T116)) → gcdE_out_gga(T115, T116, T118)
gcdE_in_gga(T144, 0, T144) → U11_gga(T144, gtC_in_gg(T144, 0))
U11_gga(T144, gtC_out_gg(T144, 0)) → gcdE_out_gga(T144, 0, T144)
gcdE_in_gga(T152, s(T151), T154) → U12_gga(T152, T151, T154, gtC_in_gg(T152, s(T151)))
U12_gga(T152, T151, T154, gtC_out_gg(T152, s(T151))) → U13_gga(T152, T151, T154, addB_in_gag(s(T151), X202, T152))
U13_gga(T152, T151, T154, addB_out_gag(s(T151), X202, T152)) → gcdE_out_gga(T152, s(T151), T154)
U12_gga(T152, T151, T154, gtC_out_gg(T152, s(T151))) → U14_gga(T152, T151, T154, addB_in_gag(s(T151), T157, T152))
U14_gga(T152, T151, T154, addB_out_gag(s(T151), T157, T152)) → U15_gga(T152, T151, T154, gcdE_in_gga(s(T151), T157, T154))
U15_gga(T152, T151, T154, gcdE_out_gga(s(T151), T157, T154)) → gcdE_out_gga(T152, s(T151), T154)
U9_gga(T51, T52, T54, gcdE_out_gga(s(T51), T57, T54)) → gcdE_out_gga(s(T51), s(T52), T54)

(5) DependencyPairsProof (EQUIVALENT transformation)

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

GCDE_IN_GGA(s(T19), s(T20), T10) → U5_GGA(T19, T20, T10, leA_in_gg(T19, T20))
GCDE_IN_GGA(s(T19), s(T20), T10) → LEA_IN_GG(T19, T20)
LEA_IN_GG(s(T33), s(T34)) → U1_GG(T33, T34, leA_in_gg(T33, T34))
LEA_IN_GG(s(T33), s(T34)) → LEA_IN_GG(T33, T34)
GCDE_IN_GGA(s(T51), s(T52), T54) → U6_GGA(T51, T52, T54, leA_in_gg(T51, T52))
U6_GGA(T51, T52, T54, leA_out_gg(T51, T52)) → U7_GGA(T51, T52, T54, addD_in_gag(T51, X55, T52))
U6_GGA(T51, T52, T54, leA_out_gg(T51, T52)) → ADDD_IN_GAG(T51, X55, T52)
ADDD_IN_GAG(T72, X91, T73) → U4_GAG(T72, X91, T73, addB_in_gag(T72, X91, T73))
ADDD_IN_GAG(T72, X91, T73) → ADDB_IN_GAG(T72, X91, T73)
ADDB_IN_GAG(s(T84), X115, s(T85)) → U2_GAG(T84, X115, T85, addB_in_gag(T84, X115, T85))
ADDB_IN_GAG(s(T84), X115, s(T85)) → ADDB_IN_GAG(T84, X115, T85)
U6_GGA(T51, T52, T54, leA_out_gg(T51, T52)) → U8_GGA(T51, T52, T54, addD_in_gag(T51, T57, T52))
U8_GGA(T51, T52, T54, addD_out_gag(T51, T57, T52)) → U9_GGA(T51, T52, T54, gcdE_in_gga(s(T51), T57, T54))
U8_GGA(T51, T52, T54, addD_out_gag(T51, T57, T52)) → GCDE_IN_GGA(s(T51), T57, T54)
GCDE_IN_GGA(T115, T116, T118) → U10_GGA(T115, T116, T118, gtC_in_gg(T115, T116))
GCDE_IN_GGA(T115, T116, T118) → GTC_IN_GG(T115, T116)
GTC_IN_GG(s(T131), s(T132)) → U3_GG(T131, T132, gtC_in_gg(T131, T132))
GTC_IN_GG(s(T131), s(T132)) → GTC_IN_GG(T131, T132)
GCDE_IN_GGA(T144, 0, T144) → U11_GGA(T144, gtC_in_gg(T144, 0))
GCDE_IN_GGA(T144, 0, T144) → GTC_IN_GG(T144, 0)
GCDE_IN_GGA(T152, s(T151), T154) → U12_GGA(T152, T151, T154, gtC_in_gg(T152, s(T151)))
GCDE_IN_GGA(T152, s(T151), T154) → GTC_IN_GG(T152, s(T151))
U12_GGA(T152, T151, T154, gtC_out_gg(T152, s(T151))) → U13_GGA(T152, T151, T154, addB_in_gag(s(T151), X202, T152))
U12_GGA(T152, T151, T154, gtC_out_gg(T152, s(T151))) → ADDB_IN_GAG(s(T151), X202, T152)
U12_GGA(T152, T151, T154, gtC_out_gg(T152, s(T151))) → U14_GGA(T152, T151, T154, addB_in_gag(s(T151), T157, T152))
U14_GGA(T152, T151, T154, addB_out_gag(s(T151), T157, T152)) → U15_GGA(T152, T151, T154, gcdE_in_gga(s(T151), T157, T154))
U14_GGA(T152, T151, T154, addB_out_gag(s(T151), T157, T152)) → GCDE_IN_GGA(s(T151), T157, T154)

The TRS R consists of the following rules:

gcdE_in_gga(s(T19), s(T20), T10) → U5_gga(T19, T20, T10, leA_in_gg(T19, T20))
leA_in_gg(s(T33), s(T34)) → U1_gg(T33, T34, leA_in_gg(T33, T34))
leA_in_gg(0, s(T41)) → leA_out_gg(0, s(T41))
leA_in_gg(0, 0) → leA_out_gg(0, 0)
U1_gg(T33, T34, leA_out_gg(T33, T34)) → leA_out_gg(s(T33), s(T34))
U5_gga(T19, T20, T10, leA_out_gg(T19, T20)) → gcdE_out_gga(s(T19), s(T20), T10)
gcdE_in_gga(s(T51), s(T52), T54) → U6_gga(T51, T52, T54, leA_in_gg(T51, T52))
U6_gga(T51, T52, T54, leA_out_gg(T51, T52)) → U7_gga(T51, T52, T54, addD_in_gag(T51, X55, T52))
addD_in_gag(T72, X91, T73) → U4_gag(T72, X91, T73, addB_in_gag(T72, X91, T73))
addB_in_gag(s(T84), X115, s(T85)) → U2_gag(T84, X115, T85, addB_in_gag(T84, X115, T85))
addB_in_gag(0, T90, T90) → addB_out_gag(0, T90, T90)
U2_gag(T84, X115, T85, addB_out_gag(T84, X115, T85)) → addB_out_gag(s(T84), X115, s(T85))
U4_gag(T72, X91, T73, addB_out_gag(T72, X91, T73)) → addD_out_gag(T72, X91, T73)
U7_gga(T51, T52, T54, addD_out_gag(T51, X55, T52)) → gcdE_out_gga(s(T51), s(T52), T54)
U6_gga(T51, T52, T54, leA_out_gg(T51, T52)) → U8_gga(T51, T52, T54, addD_in_gag(T51, T57, T52))
U8_gga(T51, T52, T54, addD_out_gag(T51, T57, T52)) → U9_gga(T51, T52, T54, gcdE_in_gga(s(T51), T57, T54))
gcdE_in_gga(0, s(T105), s(T105)) → gcdE_out_gga(0, s(T105), s(T105))
gcdE_in_gga(0, 0, 0) → gcdE_out_gga(0, 0, 0)
gcdE_in_gga(T115, T116, T118) → U10_gga(T115, T116, T118, gtC_in_gg(T115, T116))
gtC_in_gg(s(T131), s(T132)) → U3_gg(T131, T132, gtC_in_gg(T131, T132))
gtC_in_gg(s(T137), 0) → gtC_out_gg(s(T137), 0)
U3_gg(T131, T132, gtC_out_gg(T131, T132)) → gtC_out_gg(s(T131), s(T132))
U10_gga(T115, T116, T118, gtC_out_gg(T115, T116)) → gcdE_out_gga(T115, T116, T118)
gcdE_in_gga(T144, 0, T144) → U11_gga(T144, gtC_in_gg(T144, 0))
U11_gga(T144, gtC_out_gg(T144, 0)) → gcdE_out_gga(T144, 0, T144)
gcdE_in_gga(T152, s(T151), T154) → U12_gga(T152, T151, T154, gtC_in_gg(T152, s(T151)))
U12_gga(T152, T151, T154, gtC_out_gg(T152, s(T151))) → U13_gga(T152, T151, T154, addB_in_gag(s(T151), X202, T152))
U13_gga(T152, T151, T154, addB_out_gag(s(T151), X202, T152)) → gcdE_out_gga(T152, s(T151), T154)
U12_gga(T152, T151, T154, gtC_out_gg(T152, s(T151))) → U14_gga(T152, T151, T154, addB_in_gag(s(T151), T157, T152))
U14_gga(T152, T151, T154, addB_out_gag(s(T151), T157, T152)) → U15_gga(T152, T151, T154, gcdE_in_gga(s(T151), T157, T154))
U15_gga(T152, T151, T154, gcdE_out_gga(s(T151), T157, T154)) → gcdE_out_gga(T152, s(T151), T154)
U9_gga(T51, T52, T54, gcdE_out_gga(s(T51), T57, T54)) → gcdE_out_gga(s(T51), s(T52), T54)

The argument filtering Pi contains the following mapping:
gcdE_in_gga(x1, x2, x3) = gcdE_in_gga(x1, x2)
s(x1) = s(x1)
U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4)
leA_in_gg(x1, x2) = leA_in_gg(x1, x2)
U1_gg(x1, x2, x3) = U1_gg(x1, x2, x3)
0 = 0
leA_out_gg(x1, x2) = leA_out_gg(x1, x2)
gcdE_out_gga(x1, x2, x3) = gcdE_out_gga(x1, x2)
U6_gga(x1, x2, x3, x4) = U6_gga(x1, x2, x4)
U7_gga(x1, x2, x3, x4) = U7_gga(x1, x2, x4)
addD_in_gag(x1, x2, x3) = addD_in_gag(x1, x3)
U4_gag(x1, x2, x3, x4) = U4_gag(x1, x3, x4)
addB_in_gag(x1, x2, x3) = addB_in_gag(x1, x3)
U2_gag(x1, x2, x3, x4) = U2_gag(x1, x3, x4)
addB_out_gag(x1, x2, x3) = addB_out_gag(x1, x2, x3)
addD_out_gag(x1, x2, x3) = addD_out_gag(x1, x2, x3)
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)
gtC_in_gg(x1, x2) = gtC_in_gg(x1, x2)
U3_gg(x1, x2, x3) = U3_gg(x1, x2, x3)
gtC_out_gg(x1, x2) = gtC_out_gg(x1, x2)
U11_gga(x1, x2) = U11_gga(x1, x2)
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)
U15_gga(x1, x2, x3, x4) = U15_gga(x1, x2, x4)
GCDE_IN_GGA(x1, x2, x3) = GCDE_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4) = U5_GGA(x1, x2, x4)
LEA_IN_GG(x1, x2) = LEA_IN_GG(x1, x2)
U1_GG(x1, x2, x3) = U1_GG(x1, x2, x3)
U6_GGA(x1, x2, x3, x4) = U6_GGA(x1, x2, x4)
U7_GGA(x1, x2, x3, x4) = U7_GGA(x1, x2, x4)
ADDD_IN_GAG(x1, x2, x3) = ADDD_IN_GAG(x1, x3)
U4_GAG(x1, x2, x3, x4) = U4_GAG(x1, x3, x4)
ADDB_IN_GAG(x1, x2, x3) = ADDB_IN_GAG(x1, x3)
U2_GAG(x1, x2, x3, x4) = U2_GAG(x1, x3, 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)
GTC_IN_GG(x1, x2) = GTC_IN_GG(x1, x2)
U3_GG(x1, x2, x3) = U3_GG(x1, x2, x3)
U11_GGA(x1, x2) = U11_GGA(x1, x2)
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)
U15_GGA(x1, x2, x3, x4) = U15_GGA(x1, x2, x4)

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

(6) Obligation:

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

GCDE_IN_GGA(s(T19), s(T20), T10) → U5_GGA(T19, T20, T10, leA_in_gg(T19, T20))
GCDE_IN_GGA(s(T19), s(T20), T10) → LEA_IN_GG(T19, T20)
LEA_IN_GG(s(T33), s(T34)) → U1_GG(T33, T34, leA_in_gg(T33, T34))
LEA_IN_GG(s(T33), s(T34)) → LEA_IN_GG(T33, T34)
GCDE_IN_GGA(s(T51), s(T52), T54) → U6_GGA(T51, T52, T54, leA_in_gg(T51, T52))
U6_GGA(T51, T52, T54, leA_out_gg(T51, T52)) → U7_GGA(T51, T52, T54, addD_in_gag(T51, X55, T52))
U6_GGA(T51, T52, T54, leA_out_gg(T51, T52)) → ADDD_IN_GAG(T51, X55, T52)
ADDD_IN_GAG(T72, X91, T73) → U4_GAG(T72, X91, T73, addB_in_gag(T72, X91, T73))
ADDD_IN_GAG(T72, X91, T73) → ADDB_IN_GAG(T72, X91, T73)
ADDB_IN_GAG(s(T84), X115, s(T85)) → U2_GAG(T84, X115, T85, addB_in_gag(T84, X115, T85))
ADDB_IN_GAG(s(T84), X115, s(T85)) → ADDB_IN_GAG(T84, X115, T85)
U6_GGA(T51, T52, T54, leA_out_gg(T51, T52)) → U8_GGA(T51, T52, T54, addD_in_gag(T51, T57, T52))
U8_GGA(T51, T52, T54, addD_out_gag(T51, T57, T52)) → U9_GGA(T51, T52, T54, gcdE_in_gga(s(T51), T57, T54))
U8_GGA(T51, T52, T54, addD_out_gag(T51, T57, T52)) → GCDE_IN_GGA(s(T51), T57, T54)
GCDE_IN_GGA(T115, T116, T118) → U10_GGA(T115, T116, T118, gtC_in_gg(T115, T116))
GCDE_IN_GGA(T115, T116, T118) → GTC_IN_GG(T115, T116)
GTC_IN_GG(s(T131), s(T132)) → U3_GG(T131, T132, gtC_in_gg(T131, T132))
GTC_IN_GG(s(T131), s(T132)) → GTC_IN_GG(T131, T132)
GCDE_IN_GGA(T144, 0, T144) → U11_GGA(T144, gtC_in_gg(T144, 0))
GCDE_IN_GGA(T144, 0, T144) → GTC_IN_GG(T144, 0)
GCDE_IN_GGA(T152, s(T151), T154) → U12_GGA(T152, T151, T154, gtC_in_gg(T152, s(T151)))
GCDE_IN_GGA(T152, s(T151), T154) → GTC_IN_GG(T152, s(T151))
U12_GGA(T152, T151, T154, gtC_out_gg(T152, s(T151))) → U13_GGA(T152, T151, T154, addB_in_gag(s(T151), X202, T152))
U12_GGA(T152, T151, T154, gtC_out_gg(T152, s(T151))) → ADDB_IN_GAG(s(T151), X202, T152)
U12_GGA(T152, T151, T154, gtC_out_gg(T152, s(T151))) → U14_GGA(T152, T151, T154, addB_in_gag(s(T151), T157, T152))
U14_GGA(T152, T151, T154, addB_out_gag(s(T151), T157, T152)) → U15_GGA(T152, T151, T154, gcdE_in_gga(s(T151), T157, T154))
U14_GGA(T152, T151, T154, addB_out_gag(s(T151), T157, T152)) → GCDE_IN_GGA(s(T151), T157, T154)

The TRS R consists of the following rules:

gcdE_in_gga(s(T19), s(T20), T10) → U5_gga(T19, T20, T10, leA_in_gg(T19, T20))
leA_in_gg(s(T33), s(T34)) → U1_gg(T33, T34, leA_in_gg(T33, T34))
leA_in_gg(0, s(T41)) → leA_out_gg(0, s(T41))
leA_in_gg(0, 0) → leA_out_gg(0, 0)
U1_gg(T33, T34, leA_out_gg(T33, T34)) → leA_out_gg(s(T33), s(T34))
U5_gga(T19, T20, T10, leA_out_gg(T19, T20)) → gcdE_out_gga(s(T19), s(T20), T10)
gcdE_in_gga(s(T51), s(T52), T54) → U6_gga(T51, T52, T54, leA_in_gg(T51, T52))
U6_gga(T51, T52, T54, leA_out_gg(T51, T52)) → U7_gga(T51, T52, T54, addD_in_gag(T51, X55, T52))
addD_in_gag(T72, X91, T73) → U4_gag(T72, X91, T73, addB_in_gag(T72, X91, T73))
addB_in_gag(s(T84), X115, s(T85)) → U2_gag(T84, X115, T85, addB_in_gag(T84, X115, T85))
addB_in_gag(0, T90, T90) → addB_out_gag(0, T90, T90)
U2_gag(T84, X115, T85, addB_out_gag(T84, X115, T85)) → addB_out_gag(s(T84), X115, s(T85))
U4_gag(T72, X91, T73, addB_out_gag(T72, X91, T73)) → addD_out_gag(T72, X91, T73)
U7_gga(T51, T52, T54, addD_out_gag(T51, X55, T52)) → gcdE_out_gga(s(T51), s(T52), T54)
U6_gga(T51, T52, T54, leA_out_gg(T51, T52)) → U8_gga(T51, T52, T54, addD_in_gag(T51, T57, T52))
U8_gga(T51, T52, T54, addD_out_gag(T51, T57, T52)) → U9_gga(T51, T52, T54, gcdE_in_gga(s(T51), T57, T54))
gcdE_in_gga(0, s(T105), s(T105)) → gcdE_out_gga(0, s(T105), s(T105))
gcdE_in_gga(0, 0, 0) → gcdE_out_gga(0, 0, 0)
gcdE_in_gga(T115, T116, T118) → U10_gga(T115, T116, T118, gtC_in_gg(T115, T116))
gtC_in_gg(s(T131), s(T132)) → U3_gg(T131, T132, gtC_in_gg(T131, T132))
gtC_in_gg(s(T137), 0) → gtC_out_gg(s(T137), 0)
U3_gg(T131, T132, gtC_out_gg(T131, T132)) → gtC_out_gg(s(T131), s(T132))
U10_gga(T115, T116, T118, gtC_out_gg(T115, T116)) → gcdE_out_gga(T115, T116, T118)
gcdE_in_gga(T144, 0, T144) → U11_gga(T144, gtC_in_gg(T144, 0))
U11_gga(T144, gtC_out_gg(T144, 0)) → gcdE_out_gga(T144, 0, T144)
gcdE_in_gga(T152, s(T151), T154) → U12_gga(T152, T151, T154, gtC_in_gg(T152, s(T151)))
U12_gga(T152, T151, T154, gtC_out_gg(T152, s(T151))) → U13_gga(T152, T151, T154, addB_in_gag(s(T151), X202, T152))
U13_gga(T152, T151, T154, addB_out_gag(s(T151), X202, T152)) → gcdE_out_gga(T152, s(T151), T154)
U12_gga(T152, T151, T154, gtC_out_gg(T152, s(T151))) → U14_gga(T152, T151, T154, addB_in_gag(s(T151), T157, T152))
U14_gga(T152, T151, T154, addB_out_gag(s(T151), T157, T152)) → U15_gga(T152, T151, T154, gcdE_in_gga(s(T151), T157, T154))
U15_gga(T152, T151, T154, gcdE_out_gga(s(T151), T157, T154)) → gcdE_out_gga(T152, s(T151), T154)
U9_gga(T51, T52, T54, gcdE_out_gga(s(T51), T57, T54)) → gcdE_out_gga(s(T51), s(T52), T54)

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 18 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

GTC_IN_GG(s(T131), s(T132)) → GTC_IN_GG(T131, T132)

The TRS R consists of the following rules:

gcdE_in_gga(s(T19), s(T20), T10) → U5_gga(T19, T20, T10, leA_in_gg(T19, T20))
leA_in_gg(s(T33), s(T34)) → U1_gg(T33, T34, leA_in_gg(T33, T34))
leA_in_gg(0, s(T41)) → leA_out_gg(0, s(T41))
leA_in_gg(0, 0) → leA_out_gg(0, 0)
U1_gg(T33, T34, leA_out_gg(T33, T34)) → leA_out_gg(s(T33), s(T34))
U5_gga(T19, T20, T10, leA_out_gg(T19, T20)) → gcdE_out_gga(s(T19), s(T20), T10)
gcdE_in_gga(s(T51), s(T52), T54) → U6_gga(T51, T52, T54, leA_in_gg(T51, T52))
U6_gga(T51, T52, T54, leA_out_gg(T51, T52)) → U7_gga(T51, T52, T54, addD_in_gag(T51, X55, T52))
addD_in_gag(T72, X91, T73) → U4_gag(T72, X91, T73, addB_in_gag(T72, X91, T73))
addB_in_gag(s(T84), X115, s(T85)) → U2_gag(T84, X115, T85, addB_in_gag(T84, X115, T85))
addB_in_gag(0, T90, T90) → addB_out_gag(0, T90, T90)
U2_gag(T84, X115, T85, addB_out_gag(T84, X115, T85)) → addB_out_gag(s(T84), X115, s(T85))
U4_gag(T72, X91, T73, addB_out_gag(T72, X91, T73)) → addD_out_gag(T72, X91, T73)
U7_gga(T51, T52, T54, addD_out_gag(T51, X55, T52)) → gcdE_out_gga(s(T51), s(T52), T54)
U6_gga(T51, T52, T54, leA_out_gg(T51, T52)) → U8_gga(T51, T52, T54, addD_in_gag(T51, T57, T52))
U8_gga(T51, T52, T54, addD_out_gag(T51, T57, T52)) → U9_gga(T51, T52, T54, gcdE_in_gga(s(T51), T57, T54))
gcdE_in_gga(0, s(T105), s(T105)) → gcdE_out_gga(0, s(T105), s(T105))
gcdE_in_gga(0, 0, 0) → gcdE_out_gga(0, 0, 0)
gcdE_in_gga(T115, T116, T118) → U10_gga(T115, T116, T118, gtC_in_gg(T115, T116))
gtC_in_gg(s(T131), s(T132)) → U3_gg(T131, T132, gtC_in_gg(T131, T132))
gtC_in_gg(s(T137), 0) → gtC_out_gg(s(T137), 0)
U3_gg(T131, T132, gtC_out_gg(T131, T132)) → gtC_out_gg(s(T131), s(T132))
U10_gga(T115, T116, T118, gtC_out_gg(T115, T116)) → gcdE_out_gga(T115, T116, T118)
gcdE_in_gga(T144, 0, T144) → U11_gga(T144, gtC_in_gg(T144, 0))
U11_gga(T144, gtC_out_gg(T144, 0)) → gcdE_out_gga(T144, 0, T144)
gcdE_in_gga(T152, s(T151), T154) → U12_gga(T152, T151, T154, gtC_in_gg(T152, s(T151)))
U12_gga(T152, T151, T154, gtC_out_gg(T152, s(T151))) → U13_gga(T152, T151, T154, addB_in_gag(s(T151), X202, T152))
U13_gga(T152, T151, T154, addB_out_gag(s(T151), X202, T152)) → gcdE_out_gga(T152, s(T151), T154)
U12_gga(T152, T151, T154, gtC_out_gg(T152, s(T151))) → U14_gga(T152, T151, T154, addB_in_gag(s(T151), T157, T152))
U14_gga(T152, T151, T154, addB_out_gag(s(T151), T157, T152)) → U15_gga(T152, T151, T154, gcdE_in_gga(s(T151), T157, T154))
U15_gga(T152, T151, T154, gcdE_out_gga(s(T151), T157, T154)) → gcdE_out_gga(T152, s(T151), T154)
U9_gga(T51, T52, T54, gcdE_out_gga(s(T51), T57, T54)) → gcdE_out_gga(s(T51), s(T52), T54)

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

GTC_IN_GG(s(T131), s(T132)) → GTC_IN_GG(T131, T132)

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

(12) PiDPToQDPProof (EQUIVALENT transformation)

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

(13) Obligation:

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

GTC_IN_GG(s(T131), s(T132)) → GTC_IN_GG(T131, T132)

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

(14) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

GTC_IN_GG(s(T131), s(T132)) → GTC_IN_GG(T131, T132)
The graph contains the following edges 1 > 1, 2 > 2

(15) YES

(16) Obligation:

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

ADDB_IN_GAG(s(T84), X115, s(T85)) → ADDB_IN_GAG(T84, X115, T85)

The TRS R consists of the following rules:

gcdE_in_gga(s(T19), s(T20), T10) → U5_gga(T19, T20, T10, leA_in_gg(T19, T20))
leA_in_gg(s(T33), s(T34)) → U1_gg(T33, T34, leA_in_gg(T33, T34))
leA_in_gg(0, s(T41)) → leA_out_gg(0, s(T41))
leA_in_gg(0, 0) → leA_out_gg(0, 0)
U1_gg(T33, T34, leA_out_gg(T33, T34)) → leA_out_gg(s(T33), s(T34))
U5_gga(T19, T20, T10, leA_out_gg(T19, T20)) → gcdE_out_gga(s(T19), s(T20), T10)
gcdE_in_gga(s(T51), s(T52), T54) → U6_gga(T51, T52, T54, leA_in_gg(T51, T52))
U6_gga(T51, T52, T54, leA_out_gg(T51, T52)) → U7_gga(T51, T52, T54, addD_in_gag(T51, X55, T52))
addD_in_gag(T72, X91, T73) → U4_gag(T72, X91, T73, addB_in_gag(T72, X91, T73))
addB_in_gag(s(T84), X115, s(T85)) → U2_gag(T84, X115, T85, addB_in_gag(T84, X115, T85))
addB_in_gag(0, T90, T90) → addB_out_gag(0, T90, T90)
U2_gag(T84, X115, T85, addB_out_gag(T84, X115, T85)) → addB_out_gag(s(T84), X115, s(T85))
U4_gag(T72, X91, T73, addB_out_gag(T72, X91, T73)) → addD_out_gag(T72, X91, T73)
U7_gga(T51, T52, T54, addD_out_gag(T51, X55, T52)) → gcdE_out_gga(s(T51), s(T52), T54)
U6_gga(T51, T52, T54, leA_out_gg(T51, T52)) → U8_gga(T51, T52, T54, addD_in_gag(T51, T57, T52))
U8_gga(T51, T52, T54, addD_out_gag(T51, T57, T52)) → U9_gga(T51, T52, T54, gcdE_in_gga(s(T51), T57, T54))
gcdE_in_gga(0, s(T105), s(T105)) → gcdE_out_gga(0, s(T105), s(T105))
gcdE_in_gga(0, 0, 0) → gcdE_out_gga(0, 0, 0)
gcdE_in_gga(T115, T116, T118) → U10_gga(T115, T116, T118, gtC_in_gg(T115, T116))
gtC_in_gg(s(T131), s(T132)) → U3_gg(T131, T132, gtC_in_gg(T131, T132))
gtC_in_gg(s(T137), 0) → gtC_out_gg(s(T137), 0)
U3_gg(T131, T132, gtC_out_gg(T131, T132)) → gtC_out_gg(s(T131), s(T132))
U10_gga(T115, T116, T118, gtC_out_gg(T115, T116)) → gcdE_out_gga(T115, T116, T118)
gcdE_in_gga(T144, 0, T144) → U11_gga(T144, gtC_in_gg(T144, 0))
U11_gga(T144, gtC_out_gg(T144, 0)) → gcdE_out_gga(T144, 0, T144)
gcdE_in_gga(T152, s(T151), T154) → U12_gga(T152, T151, T154, gtC_in_gg(T152, s(T151)))
U12_gga(T152, T151, T154, gtC_out_gg(T152, s(T151))) → U13_gga(T152, T151, T154, addB_in_gag(s(T151), X202, T152))
U13_gga(T152, T151, T154, addB_out_gag(s(T151), X202, T152)) → gcdE_out_gga(T152, s(T151), T154)
U12_gga(T152, T151, T154, gtC_out_gg(T152, s(T151))) → U14_gga(T152, T151, T154, addB_in_gag(s(T151), T157, T152))
U14_gga(T152, T151, T154, addB_out_gag(s(T151), T157, T152)) → U15_gga(T152, T151, T154, gcdE_in_gga(s(T151), T157, T154))
U15_gga(T152, T151, T154, gcdE_out_gga(s(T151), T157, T154)) → gcdE_out_gga(T152, s(T151), T154)
U9_gga(T51, T52, T54, gcdE_out_gga(s(T51), T57, T54)) → gcdE_out_gga(s(T51), s(T52), T54)

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

ADDB_IN_GAG(s(T84), X115, s(T85)) → ADDB_IN_GAG(T84, X115, T85)

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

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

ADDB_IN_GAG(s(T84), s(T85)) → ADDB_IN_GAG(T84, T85)

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

(21) 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:

ADDB_IN_GAG(s(T84), s(T85)) → ADDB_IN_GAG(T84, T85)
The graph contains the following edges 1 > 1, 2 > 2

(22) YES

(23) Obligation:

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

LEA_IN_GG(s(T33), s(T34)) → LEA_IN_GG(T33, T34)

The TRS R consists of the following rules:

gcdE_in_gga(s(T19), s(T20), T10) → U5_gga(T19, T20, T10, leA_in_gg(T19, T20))
leA_in_gg(s(T33), s(T34)) → U1_gg(T33, T34, leA_in_gg(T33, T34))
leA_in_gg(0, s(T41)) → leA_out_gg(0, s(T41))
leA_in_gg(0, 0) → leA_out_gg(0, 0)
U1_gg(T33, T34, leA_out_gg(T33, T34)) → leA_out_gg(s(T33), s(T34))
U5_gga(T19, T20, T10, leA_out_gg(T19, T20)) → gcdE_out_gga(s(T19), s(T20), T10)
gcdE_in_gga(s(T51), s(T52), T54) → U6_gga(T51, T52, T54, leA_in_gg(T51, T52))
U6_gga(T51, T52, T54, leA_out_gg(T51, T52)) → U7_gga(T51, T52, T54, addD_in_gag(T51, X55, T52))
addD_in_gag(T72, X91, T73) → U4_gag(T72, X91, T73, addB_in_gag(T72, X91, T73))
addB_in_gag(s(T84), X115, s(T85)) → U2_gag(T84, X115, T85, addB_in_gag(T84, X115, T85))
addB_in_gag(0, T90, T90) → addB_out_gag(0, T90, T90)
U2_gag(T84, X115, T85, addB_out_gag(T84, X115, T85)) → addB_out_gag(s(T84), X115, s(T85))
U4_gag(T72, X91, T73, addB_out_gag(T72, X91, T73)) → addD_out_gag(T72, X91, T73)
U7_gga(T51, T52, T54, addD_out_gag(T51, X55, T52)) → gcdE_out_gga(s(T51), s(T52), T54)
U6_gga(T51, T52, T54, leA_out_gg(T51, T52)) → U8_gga(T51, T52, T54, addD_in_gag(T51, T57, T52))
U8_gga(T51, T52, T54, addD_out_gag(T51, T57, T52)) → U9_gga(T51, T52, T54, gcdE_in_gga(s(T51), T57, T54))
gcdE_in_gga(0, s(T105), s(T105)) → gcdE_out_gga(0, s(T105), s(T105))
gcdE_in_gga(0, 0, 0) → gcdE_out_gga(0, 0, 0)
gcdE_in_gga(T115, T116, T118) → U10_gga(T115, T116, T118, gtC_in_gg(T115, T116))
gtC_in_gg(s(T131), s(T132)) → U3_gg(T131, T132, gtC_in_gg(T131, T132))
gtC_in_gg(s(T137), 0) → gtC_out_gg(s(T137), 0)
U3_gg(T131, T132, gtC_out_gg(T131, T132)) → gtC_out_gg(s(T131), s(T132))
U10_gga(T115, T116, T118, gtC_out_gg(T115, T116)) → gcdE_out_gga(T115, T116, T118)
gcdE_in_gga(T144, 0, T144) → U11_gga(T144, gtC_in_gg(T144, 0))
U11_gga(T144, gtC_out_gg(T144, 0)) → gcdE_out_gga(T144, 0, T144)
gcdE_in_gga(T152, s(T151), T154) → U12_gga(T152, T151, T154, gtC_in_gg(T152, s(T151)))
U12_gga(T152, T151, T154, gtC_out_gg(T152, s(T151))) → U13_gga(T152, T151, T154, addB_in_gag(s(T151), X202, T152))
U13_gga(T152, T151, T154, addB_out_gag(s(T151), X202, T152)) → gcdE_out_gga(T152, s(T151), T154)
U12_gga(T152, T151, T154, gtC_out_gg(T152, s(T151))) → U14_gga(T152, T151, T154, addB_in_gag(s(T151), T157, T152))
U14_gga(T152, T151, T154, addB_out_gag(s(T151), T157, T152)) → U15_gga(T152, T151, T154, gcdE_in_gga(s(T151), T157, T154))
U15_gga(T152, T151, T154, gcdE_out_gga(s(T151), T157, T154)) → gcdE_out_gga(T152, s(T151), T154)
U9_gga(T51, T52, T54, gcdE_out_gga(s(T51), T57, T54)) → gcdE_out_gga(s(T51), s(T52), T54)

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) Obligation:

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

LEA_IN_GG(s(T33), s(T34)) → LEA_IN_GG(T33, T34)

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

(26) PiDPToQDPProof (EQUIVALENT transformation)

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

(27) Obligation:

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

LEA_IN_GG(s(T33), s(T34)) → LEA_IN_GG(T33, T34)

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

(28) 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(T33), s(T34)) → LEA_IN_GG(T33, T34)
The graph contains the following edges 1 > 1, 2 > 2

(29) YES

(30) Obligation:

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

GCDE_IN_GGA(s(T51), s(T52), T54) → U6_GGA(T51, T52, T54, leA_in_gg(T51, T52))
U6_GGA(T51, T52, T54, leA_out_gg(T51, T52)) → U8_GGA(T51, T52, T54, addD_in_gag(T51, T57, T52))
U8_GGA(T51, T52, T54, addD_out_gag(T51, T57, T52)) → GCDE_IN_GGA(s(T51), T57, T54)
GCDE_IN_GGA(T152, s(T151), T154) → U12_GGA(T152, T151, T154, gtC_in_gg(T152, s(T151)))
U12_GGA(T152, T151, T154, gtC_out_gg(T152, s(T151))) → U14_GGA(T152, T151, T154, addB_in_gag(s(T151), T157, T152))
U14_GGA(T152, T151, T154, addB_out_gag(s(T151), T157, T152)) → GCDE_IN_GGA(s(T151), T157, T154)

The TRS R consists of the following rules:

gcdE_in_gga(s(T19), s(T20), T10) → U5_gga(T19, T20, T10, leA_in_gg(T19, T20))
leA_in_gg(s(T33), s(T34)) → U1_gg(T33, T34, leA_in_gg(T33, T34))
leA_in_gg(0, s(T41)) → leA_out_gg(0, s(T41))
leA_in_gg(0, 0) → leA_out_gg(0, 0)
U1_gg(T33, T34, leA_out_gg(T33, T34)) → leA_out_gg(s(T33), s(T34))
U5_gga(T19, T20, T10, leA_out_gg(T19, T20)) → gcdE_out_gga(s(T19), s(T20), T10)
gcdE_in_gga(s(T51), s(T52), T54) → U6_gga(T51, T52, T54, leA_in_gg(T51, T52))
U6_gga(T51, T52, T54, leA_out_gg(T51, T52)) → U7_gga(T51, T52, T54, addD_in_gag(T51, X55, T52))
addD_in_gag(T72, X91, T73) → U4_gag(T72, X91, T73, addB_in_gag(T72, X91, T73))
addB_in_gag(s(T84), X115, s(T85)) → U2_gag(T84, X115, T85, addB_in_gag(T84, X115, T85))
addB_in_gag(0, T90, T90) → addB_out_gag(0, T90, T90)
U2_gag(T84, X115, T85, addB_out_gag(T84, X115, T85)) → addB_out_gag(s(T84), X115, s(T85))
U4_gag(T72, X91, T73, addB_out_gag(T72, X91, T73)) → addD_out_gag(T72, X91, T73)
U7_gga(T51, T52, T54, addD_out_gag(T51, X55, T52)) → gcdE_out_gga(s(T51), s(T52), T54)
U6_gga(T51, T52, T54, leA_out_gg(T51, T52)) → U8_gga(T51, T52, T54, addD_in_gag(T51, T57, T52))
U8_gga(T51, T52, T54, addD_out_gag(T51, T57, T52)) → U9_gga(T51, T52, T54, gcdE_in_gga(s(T51), T57, T54))
gcdE_in_gga(0, s(T105), s(T105)) → gcdE_out_gga(0, s(T105), s(T105))
gcdE_in_gga(0, 0, 0) → gcdE_out_gga(0, 0, 0)
gcdE_in_gga(T115, T116, T118) → U10_gga(T115, T116, T118, gtC_in_gg(T115, T116))
gtC_in_gg(s(T131), s(T132)) → U3_gg(T131, T132, gtC_in_gg(T131, T132))
gtC_in_gg(s(T137), 0) → gtC_out_gg(s(T137), 0)
U3_gg(T131, T132, gtC_out_gg(T131, T132)) → gtC_out_gg(s(T131), s(T132))
U10_gga(T115, T116, T118, gtC_out_gg(T115, T116)) → gcdE_out_gga(T115, T116, T118)
gcdE_in_gga(T144, 0, T144) → U11_gga(T144, gtC_in_gg(T144, 0))
U11_gga(T144, gtC_out_gg(T144, 0)) → gcdE_out_gga(T144, 0, T144)
gcdE_in_gga(T152, s(T151), T154) → U12_gga(T152, T151, T154, gtC_in_gg(T152, s(T151)))
U12_gga(T152, T151, T154, gtC_out_gg(T152, s(T151))) → U13_gga(T152, T151, T154, addB_in_gag(s(T151), X202, T152))
U13_gga(T152, T151, T154, addB_out_gag(s(T151), X202, T152)) → gcdE_out_gga(T152, s(T151), T154)
U12_gga(T152, T151, T154, gtC_out_gg(T152, s(T151))) → U14_gga(T152, T151, T154, addB_in_gag(s(T151), T157, T152))
U14_gga(T152, T151, T154, addB_out_gag(s(T151), T157, T152)) → U15_gga(T152, T151, T154, gcdE_in_gga(s(T151), T157, T154))
U15_gga(T152, T151, T154, gcdE_out_gga(s(T151), T157, T154)) → gcdE_out_gga(T152, s(T151), T154)
U9_gga(T51, T52, T54, gcdE_out_gga(s(T51), T57, T54)) → gcdE_out_gga(s(T51), s(T52), T54)

(31) UsableRulesProof (EQUIVALENT transformation)

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

(32) Obligation:

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

GCDE_IN_GGA(s(T51), s(T52), T54) → U6_GGA(T51, T52, T54, leA_in_gg(T51, T52))
U6_GGA(T51, T52, T54, leA_out_gg(T51, T52)) → U8_GGA(T51, T52, T54, addD_in_gag(T51, T57, T52))
U8_GGA(T51, T52, T54, addD_out_gag(T51, T57, T52)) → GCDE_IN_GGA(s(T51), T57, T54)
GCDE_IN_GGA(T152, s(T151), T154) → U12_GGA(T152, T151, T154, gtC_in_gg(T152, s(T151)))
U12_GGA(T152, T151, T154, gtC_out_gg(T152, s(T151))) → U14_GGA(T152, T151, T154, addB_in_gag(s(T151), T157, T152))
U14_GGA(T152, T151, T154, addB_out_gag(s(T151), T157, T152)) → GCDE_IN_GGA(s(T151), T157, T154)

The TRS R consists of the following rules:

leA_in_gg(s(T33), s(T34)) → U1_gg(T33, T34, leA_in_gg(T33, T34))
leA_in_gg(0, s(T41)) → leA_out_gg(0, s(T41))
leA_in_gg(0, 0) → leA_out_gg(0, 0)
addD_in_gag(T72, X91, T73) → U4_gag(T72, X91, T73, addB_in_gag(T72, X91, T73))
gtC_in_gg(s(T131), s(T132)) → U3_gg(T131, T132, gtC_in_gg(T131, T132))
addB_in_gag(s(T84), X115, s(T85)) → U2_gag(T84, X115, T85, addB_in_gag(T84, X115, T85))
U1_gg(T33, T34, leA_out_gg(T33, T34)) → leA_out_gg(s(T33), s(T34))
U4_gag(T72, X91, T73, addB_out_gag(T72, X91, T73)) → addD_out_gag(T72, X91, T73)
U3_gg(T131, T132, gtC_out_gg(T131, T132)) → gtC_out_gg(s(T131), s(T132))
U2_gag(T84, X115, T85, addB_out_gag(T84, X115, T85)) → addB_out_gag(s(T84), X115, s(T85))
addB_in_gag(0, T90, T90) → addB_out_gag(0, T90, T90)
gtC_in_gg(s(T137), 0) → gtC_out_gg(s(T137), 0)

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
leA_in_gg(x1, x2) = leA_in_gg(x1, x2)
U1_gg(x1, x2, x3) = U1_gg(x1, x2, x3)
0 = 0
leA_out_gg(x1, x2) = leA_out_gg(x1, x2)
addD_in_gag(x1, x2, x3) = addD_in_gag(x1, x3)
U4_gag(x1, x2, x3, x4) = U4_gag(x1, x3, x4)
addB_in_gag(x1, x2, x3) = addB_in_gag(x1, x3)
U2_gag(x1, x2, x3, x4) = U2_gag(x1, x3, x4)
addB_out_gag(x1, x2, x3) = addB_out_gag(x1, x2, x3)
addD_out_gag(x1, x2, x3) = addD_out_gag(x1, x2, x3)
gtC_in_gg(x1, x2) = gtC_in_gg(x1, x2)
U3_gg(x1, x2, x3) = U3_gg(x1, x2, x3)
gtC_out_gg(x1, x2) = gtC_out_gg(x1, x2)
GCDE_IN_GGA(x1, x2, x3) = GCDE_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4) = U6_GGA(x1, x2, x4)
U8_GGA(x1, x2, x3, x4) = U8_GGA(x1, x2, x4)
U12_GGA(x1, x2, x3, x4) = U12_GGA(x1, x2, x4)
U14_GGA(x1, x2, x3, x4) = U14_GGA(x1, x2, x4)

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

(33) PiDPToQDPProof (SOUND transformation)

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

(34) Obligation:

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

GCDE_IN_GGA(s(T51), s(T52)) → U6_GGA(T51, T52, leA_in_gg(T51, T52))
U6_GGA(T51, T52, leA_out_gg(T51, T52)) → U8_GGA(T51, T52, addD_in_gag(T51, T52))
U8_GGA(T51, T52, addD_out_gag(T51, T57, T52)) → GCDE_IN_GGA(s(T51), T57)
GCDE_IN_GGA(T152, s(T151)) → U12_GGA(T152, T151, gtC_in_gg(T152, s(T151)))
U12_GGA(T152, T151, gtC_out_gg(T152, s(T151))) → U14_GGA(T152, T151, addB_in_gag(s(T151), T152))
U14_GGA(T152, T151, addB_out_gag(s(T151), T157, T152)) → GCDE_IN_GGA(s(T151), T157)

The TRS R consists of the following rules:

leA_in_gg(s(T33), s(T34)) → U1_gg(T33, T34, leA_in_gg(T33, T34))
leA_in_gg(0, s(T41)) → leA_out_gg(0, s(T41))
leA_in_gg(0, 0) → leA_out_gg(0, 0)
addD_in_gag(T72, T73) → U4_gag(T72, T73, addB_in_gag(T72, T73))
gtC_in_gg(s(T131), s(T132)) → U3_gg(T131, T132, gtC_in_gg(T131, T132))
addB_in_gag(s(T84), s(T85)) → U2_gag(T84, T85, addB_in_gag(T84, T85))
U1_gg(T33, T34, leA_out_gg(T33, T34)) → leA_out_gg(s(T33), s(T34))
U4_gag(T72, T73, addB_out_gag(T72, X91, T73)) → addD_out_gag(T72, X91, T73)
U3_gg(T131, T132, gtC_out_gg(T131, T132)) → gtC_out_gg(s(T131), s(T132))
U2_gag(T84, T85, addB_out_gag(T84, X115, T85)) → addB_out_gag(s(T84), X115, s(T85))
addB_in_gag(0, T90) → addB_out_gag(0, T90, T90)
gtC_in_gg(s(T137), 0) → gtC_out_gg(s(T137), 0)

The set Q consists of the following terms:

leA_in_gg(x0, x1)
addD_in_gag(x0, x1)
gtC_in_gg(x0, x1)
addB_in_gag(x0, x1)
U1_gg(x0, x1, x2)
U4_gag(x0, x1, x2)
U3_gg(x0, x1, x2)
U2_gag(x0, x1, x2)

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

(35) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U6_GGA(T51, T52, leA_out_gg(T51, T52)) → U8_GGA(T51, T52, addD_in_gag(T51, T52)) at position [2] we obtained the following new rules [LPAR04]:

U6_GGA(T51, T52, leA_out_gg(T51, T52)) → U8_GGA(T51, T52, U4_gag(T51, T52, addB_in_gag(T51, T52)))

(36) Obligation:

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

GCDE_IN_GGA(s(T51), s(T52)) → U6_GGA(T51, T52, leA_in_gg(T51, T52))
U8_GGA(T51, T52, addD_out_gag(T51, T57, T52)) → GCDE_IN_GGA(s(T51), T57)
GCDE_IN_GGA(T152, s(T151)) → U12_GGA(T152, T151, gtC_in_gg(T152, s(T151)))
U12_GGA(T152, T151, gtC_out_gg(T152, s(T151))) → U14_GGA(T152, T151, addB_in_gag(s(T151), T152))
U14_GGA(T152, T151, addB_out_gag(s(T151), T157, T152)) → GCDE_IN_GGA(s(T151), T157)
U6_GGA(T51, T52, leA_out_gg(T51, T52)) → U8_GGA(T51, T52, U4_gag(T51, T52, addB_in_gag(T51, T52)))

The TRS R consists of the following rules:

leA_in_gg(s(T33), s(T34)) → U1_gg(T33, T34, leA_in_gg(T33, T34))
leA_in_gg(0, s(T41)) → leA_out_gg(0, s(T41))
leA_in_gg(0, 0) → leA_out_gg(0, 0)
addD_in_gag(T72, T73) → U4_gag(T72, T73, addB_in_gag(T72, T73))
gtC_in_gg(s(T131), s(T132)) → U3_gg(T131, T132, gtC_in_gg(T131, T132))
addB_in_gag(s(T84), s(T85)) → U2_gag(T84, T85, addB_in_gag(T84, T85))
U1_gg(T33, T34, leA_out_gg(T33, T34)) → leA_out_gg(s(T33), s(T34))
U4_gag(T72, T73, addB_out_gag(T72, X91, T73)) → addD_out_gag(T72, X91, T73)
U3_gg(T131, T132, gtC_out_gg(T131, T132)) → gtC_out_gg(s(T131), s(T132))
U2_gag(T84, T85, addB_out_gag(T84, X115, T85)) → addB_out_gag(s(T84), X115, s(T85))
addB_in_gag(0, T90) → addB_out_gag(0, T90, T90)
gtC_in_gg(s(T137), 0) → gtC_out_gg(s(T137), 0)

The set Q consists of the following terms:

leA_in_gg(x0, x1)
addD_in_gag(x0, x1)
gtC_in_gg(x0, x1)
addB_in_gag(x0, x1)
U1_gg(x0, x1, x2)
U4_gag(x0, x1, x2)
U3_gg(x0, x1, x2)
U2_gag(x0, x1, x2)

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

(37) 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.

(38) Obligation:

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

GCDE_IN_GGA(s(T51), s(T52)) → U6_GGA(T51, T52, leA_in_gg(T51, T52))
U8_GGA(T51, T52, addD_out_gag(T51, T57, T52)) → GCDE_IN_GGA(s(T51), T57)
GCDE_IN_GGA(T152, s(T151)) → U12_GGA(T152, T151, gtC_in_gg(T152, s(T151)))
U12_GGA(T152, T151, gtC_out_gg(T152, s(T151))) → U14_GGA(T152, T151, addB_in_gag(s(T151), T152))
U14_GGA(T152, T151, addB_out_gag(s(T151), T157, T152)) → GCDE_IN_GGA(s(T151), T157)
U6_GGA(T51, T52, leA_out_gg(T51, T52)) → U8_GGA(T51, T52, U4_gag(T51, T52, addB_in_gag(T51, T52)))

The TRS R consists of the following rules:

addB_in_gag(s(T84), s(T85)) → U2_gag(T84, T85, addB_in_gag(T84, T85))
addB_in_gag(0, T90) → addB_out_gag(0, T90, T90)
U4_gag(T72, T73, addB_out_gag(T72, X91, T73)) → addD_out_gag(T72, X91, T73)
U2_gag(T84, T85, addB_out_gag(T84, X115, T85)) → addB_out_gag(s(T84), X115, s(T85))
gtC_in_gg(s(T131), s(T132)) → U3_gg(T131, T132, gtC_in_gg(T131, T132))
gtC_in_gg(s(T137), 0) → gtC_out_gg(s(T137), 0)
U3_gg(T131, T132, gtC_out_gg(T131, T132)) → gtC_out_gg(s(T131), s(T132))
leA_in_gg(s(T33), s(T34)) → U1_gg(T33, T34, leA_in_gg(T33, T34))
leA_in_gg(0, s(T41)) → leA_out_gg(0, s(T41))
leA_in_gg(0, 0) → leA_out_gg(0, 0)
U1_gg(T33, T34, leA_out_gg(T33, T34)) → leA_out_gg(s(T33), s(T34))

The set Q consists of the following terms:

leA_in_gg(x0, x1)
addD_in_gag(x0, x1)
gtC_in_gg(x0, x1)
addB_in_gag(x0, x1)
U1_gg(x0, x1, x2)
U4_gag(x0, x1, x2)
U3_gg(x0, x1, x2)
U2_gag(x0, x1, x2)

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

(39) 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].

addD_in_gag(x0, x1)

(40) Obligation:

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

GCDE_IN_GGA(s(T51), s(T52)) → U6_GGA(T51, T52, leA_in_gg(T51, T52))
U8_GGA(T51, T52, addD_out_gag(T51, T57, T52)) → GCDE_IN_GGA(s(T51), T57)
GCDE_IN_GGA(T152, s(T151)) → U12_GGA(T152, T151, gtC_in_gg(T152, s(T151)))
U12_GGA(T152, T151, gtC_out_gg(T152, s(T151))) → U14_GGA(T152, T151, addB_in_gag(s(T151), T152))
U14_GGA(T152, T151, addB_out_gag(s(T151), T157, T152)) → GCDE_IN_GGA(s(T151), T157)
U6_GGA(T51, T52, leA_out_gg(T51, T52)) → U8_GGA(T51, T52, U4_gag(T51, T52, addB_in_gag(T51, T52)))

The TRS R consists of the following rules:

addB_in_gag(s(T84), s(T85)) → U2_gag(T84, T85, addB_in_gag(T84, T85))
addB_in_gag(0, T90) → addB_out_gag(0, T90, T90)
U4_gag(T72, T73, addB_out_gag(T72, X91, T73)) → addD_out_gag(T72, X91, T73)
U2_gag(T84, T85, addB_out_gag(T84, X115, T85)) → addB_out_gag(s(T84), X115, s(T85))
gtC_in_gg(s(T131), s(T132)) → U3_gg(T131, T132, gtC_in_gg(T131, T132))
gtC_in_gg(s(T137), 0) → gtC_out_gg(s(T137), 0)
U3_gg(T131, T132, gtC_out_gg(T131, T132)) → gtC_out_gg(s(T131), s(T132))
leA_in_gg(s(T33), s(T34)) → U1_gg(T33, T34, leA_in_gg(T33, T34))
leA_in_gg(0, s(T41)) → leA_out_gg(0, s(T41))
leA_in_gg(0, 0) → leA_out_gg(0, 0)
U1_gg(T33, T34, leA_out_gg(T33, T34)) → leA_out_gg(s(T33), s(T34))

The set Q consists of the following terms:

leA_in_gg(x0, x1)
gtC_in_gg(x0, x1)
addB_in_gag(x0, x1)
U1_gg(x0, x1, x2)
U4_gag(x0, x1, x2)
U3_gg(x0, x1, x2)
U2_gag(x0, x1, x2)

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

(41) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

U12_GGA(T152, T151, gtC_out_gg(T152, s(T151))) → U14_GGA(T152, T151, addB_in_gag(s(T151), T152))

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

POL(0) = 0
POL(GCDE_IN_GGA(x₁, x₂)) = x₁
POL(U12_GGA(x₁, x₂, x₃)) = x₃
POL(U14_GGA(x₁, x₂, x₃)) = 1 + x₂
POL(U1_gg(x₁, x₂, x₃)) = 0
POL(U2_gag(x₁, x₂, x₃)) = 0
POL(U3_gg(x₁, x₂, x₃)) = 1 + x₃
POL(U4_gag(x₁, x₂, x₃)) = 0
POL(U6_GGA(x₁, x₂, x₃)) = 1 + x₁
POL(U8_GGA(x₁, x₂, x₃)) = 1 + x₁
POL(addB_in_gag(x₁, x₂)) = 0
POL(addB_out_gag(x₁, x₂, x₃)) = 0
POL(addD_out_gag(x₁, x₂, x₃)) = 0
POL(gtC_in_gg(x₁, x₂)) = x₁
POL(gtC_out_gg(x₁, x₂)) = 1 + x₂
POL(leA_in_gg(x₁, x₂)) = 0
POL(leA_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:

gtC_in_gg(s(T131), s(T132)) → U3_gg(T131, T132, gtC_in_gg(T131, T132))
gtC_in_gg(s(T137), 0) → gtC_out_gg(s(T137), 0)
U3_gg(T131, T132, gtC_out_gg(T131, T132)) → gtC_out_gg(s(T131), s(T132))

(42) Obligation:

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

GCDE_IN_GGA(s(T51), s(T52)) → U6_GGA(T51, T52, leA_in_gg(T51, T52))
U8_GGA(T51, T52, addD_out_gag(T51, T57, T52)) → GCDE_IN_GGA(s(T51), T57)
GCDE_IN_GGA(T152, s(T151)) → U12_GGA(T152, T151, gtC_in_gg(T152, s(T151)))
U14_GGA(T152, T151, addB_out_gag(s(T151), T157, T152)) → GCDE_IN_GGA(s(T151), T157)
U6_GGA(T51, T52, leA_out_gg(T51, T52)) → U8_GGA(T51, T52, U4_gag(T51, T52, addB_in_gag(T51, T52)))

The TRS R consists of the following rules:

addB_in_gag(s(T84), s(T85)) → U2_gag(T84, T85, addB_in_gag(T84, T85))
addB_in_gag(0, T90) → addB_out_gag(0, T90, T90)
U4_gag(T72, T73, addB_out_gag(T72, X91, T73)) → addD_out_gag(T72, X91, T73)
U2_gag(T84, T85, addB_out_gag(T84, X115, T85)) → addB_out_gag(s(T84), X115, s(T85))
gtC_in_gg(s(T131), s(T132)) → U3_gg(T131, T132, gtC_in_gg(T131, T132))
gtC_in_gg(s(T137), 0) → gtC_out_gg(s(T137), 0)
U3_gg(T131, T132, gtC_out_gg(T131, T132)) → gtC_out_gg(s(T131), s(T132))
leA_in_gg(s(T33), s(T34)) → U1_gg(T33, T34, leA_in_gg(T33, T34))
leA_in_gg(0, s(T41)) → leA_out_gg(0, s(T41))
leA_in_gg(0, 0) → leA_out_gg(0, 0)
U1_gg(T33, T34, leA_out_gg(T33, T34)) → leA_out_gg(s(T33), s(T34))

The set Q consists of the following terms:

leA_in_gg(x0, x1)
gtC_in_gg(x0, x1)
addB_in_gag(x0, x1)
U1_gg(x0, x1, x2)
U4_gag(x0, x1, x2)
U3_gg(x0, x1, x2)
U2_gag(x0, x1, x2)

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

(43) DependencyGraphProof (EQUIVALENT transformation)

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

(44) Obligation:

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

U6_GGA(T51, T52, leA_out_gg(T51, T52)) → U8_GGA(T51, T52, U4_gag(T51, T52, addB_in_gag(T51, T52)))
U8_GGA(T51, T52, addD_out_gag(T51, T57, T52)) → GCDE_IN_GGA(s(T51), T57)
GCDE_IN_GGA(s(T51), s(T52)) → U6_GGA(T51, T52, leA_in_gg(T51, T52))

The TRS R consists of the following rules:

addB_in_gag(s(T84), s(T85)) → U2_gag(T84, T85, addB_in_gag(T84, T85))
addB_in_gag(0, T90) → addB_out_gag(0, T90, T90)
U4_gag(T72, T73, addB_out_gag(T72, X91, T73)) → addD_out_gag(T72, X91, T73)
U2_gag(T84, T85, addB_out_gag(T84, X115, T85)) → addB_out_gag(s(T84), X115, s(T85))
gtC_in_gg(s(T131), s(T132)) → U3_gg(T131, T132, gtC_in_gg(T131, T132))
gtC_in_gg(s(T137), 0) → gtC_out_gg(s(T137), 0)
U3_gg(T131, T132, gtC_out_gg(T131, T132)) → gtC_out_gg(s(T131), s(T132))
leA_in_gg(s(T33), s(T34)) → U1_gg(T33, T34, leA_in_gg(T33, T34))
leA_in_gg(0, s(T41)) → leA_out_gg(0, s(T41))
leA_in_gg(0, 0) → leA_out_gg(0, 0)
U1_gg(T33, T34, leA_out_gg(T33, T34)) → leA_out_gg(s(T33), s(T34))

The set Q consists of the following terms:

leA_in_gg(x0, x1)
gtC_in_gg(x0, x1)
addB_in_gag(x0, x1)
U1_gg(x0, x1, x2)
U4_gag(x0, x1, x2)
U3_gg(x0, x1, x2)
U2_gag(x0, x1, x2)

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

(45) 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.

(46) Obligation:

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

U6_GGA(T51, T52, leA_out_gg(T51, T52)) → U8_GGA(T51, T52, U4_gag(T51, T52, addB_in_gag(T51, T52)))
U8_GGA(T51, T52, addD_out_gag(T51, T57, T52)) → GCDE_IN_GGA(s(T51), T57)
GCDE_IN_GGA(s(T51), s(T52)) → U6_GGA(T51, T52, leA_in_gg(T51, T52))

The TRS R consists of the following rules:

leA_in_gg(s(T33), s(T34)) → U1_gg(T33, T34, leA_in_gg(T33, T34))
leA_in_gg(0, s(T41)) → leA_out_gg(0, s(T41))
leA_in_gg(0, 0) → leA_out_gg(0, 0)
U1_gg(T33, T34, leA_out_gg(T33, T34)) → leA_out_gg(s(T33), s(T34))
addB_in_gag(s(T84), s(T85)) → U2_gag(T84, T85, addB_in_gag(T84, T85))
addB_in_gag(0, T90) → addB_out_gag(0, T90, T90)
U4_gag(T72, T73, addB_out_gag(T72, X91, T73)) → addD_out_gag(T72, X91, T73)
U2_gag(T84, T85, addB_out_gag(T84, X115, T85)) → addB_out_gag(s(T84), X115, s(T85))

The set Q consists of the following terms:

leA_in_gg(x0, x1)
gtC_in_gg(x0, x1)
addB_in_gag(x0, x1)
U1_gg(x0, x1, x2)
U4_gag(x0, x1, x2)
U3_gg(x0, x1, x2)
U2_gag(x0, x1, x2)

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

(47) 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].

gtC_in_gg(x0, x1)
U3_gg(x0, x1, x2)

(48) Obligation:

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

U6_GGA(T51, T52, leA_out_gg(T51, T52)) → U8_GGA(T51, T52, U4_gag(T51, T52, addB_in_gag(T51, T52)))
U8_GGA(T51, T52, addD_out_gag(T51, T57, T52)) → GCDE_IN_GGA(s(T51), T57)
GCDE_IN_GGA(s(T51), s(T52)) → U6_GGA(T51, T52, leA_in_gg(T51, T52))

The TRS R consists of the following rules:

leA_in_gg(s(T33), s(T34)) → U1_gg(T33, T34, leA_in_gg(T33, T34))
leA_in_gg(0, s(T41)) → leA_out_gg(0, s(T41))
leA_in_gg(0, 0) → leA_out_gg(0, 0)
U1_gg(T33, T34, leA_out_gg(T33, T34)) → leA_out_gg(s(T33), s(T34))
addB_in_gag(s(T84), s(T85)) → U2_gag(T84, T85, addB_in_gag(T84, T85))
addB_in_gag(0, T90) → addB_out_gag(0, T90, T90)
U4_gag(T72, T73, addB_out_gag(T72, X91, T73)) → addD_out_gag(T72, X91, T73)
U2_gag(T84, T85, addB_out_gag(T84, X115, T85)) → addB_out_gag(s(T84), X115, s(T85))

The set Q consists of the following terms:

leA_in_gg(x0, x1)
addB_in_gag(x0, x1)
U1_gg(x0, x1, x2)
U4_gag(x0, x1, x2)
U2_gag(x0, x1, x2)

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

(49) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

GCDE_IN_GGA(s(T51), s(T52)) → U6_GGA(T51, T52, leA_in_gg(T51, T52))

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

POL(0) = 0
POL(GCDE_IN_GGA(x₁, x₂)) = x₂
POL(U1_gg(x₁, x₂, x₃)) = 0
POL(U2_gag(x₁, x₂, x₃)) = x₃
POL(U4_gag(x₁, x₂, x₃)) = x₃
POL(U6_GGA(x₁, x₂, x₃)) = x₂
POL(U8_GGA(x₁, x₂, x₃)) = x₃
POL(addB_in_gag(x₁, x₂)) = x₂
POL(addB_out_gag(x₁, x₂, x₃)) = x₂
POL(addD_out_gag(x₁, x₂, x₃)) = x₂
POL(leA_in_gg(x₁, x₂)) = 0
POL(leA_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:

addB_in_gag(s(T84), s(T85)) → U2_gag(T84, T85, addB_in_gag(T84, T85))
addB_in_gag(0, T90) → addB_out_gag(0, T90, T90)
U4_gag(T72, T73, addB_out_gag(T72, X91, T73)) → addD_out_gag(T72, X91, T73)
U2_gag(T84, T85, addB_out_gag(T84, X115, T85)) → addB_out_gag(s(T84), X115, s(T85))

(50) Obligation:

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

U6_GGA(T51, T52, leA_out_gg(T51, T52)) → U8_GGA(T51, T52, U4_gag(T51, T52, addB_in_gag(T51, T52)))
U8_GGA(T51, T52, addD_out_gag(T51, T57, T52)) → GCDE_IN_GGA(s(T51), T57)

The TRS R consists of the following rules:

leA_in_gg(s(T33), s(T34)) → U1_gg(T33, T34, leA_in_gg(T33, T34))
leA_in_gg(0, s(T41)) → leA_out_gg(0, s(T41))
leA_in_gg(0, 0) → leA_out_gg(0, 0)
U1_gg(T33, T34, leA_out_gg(T33, T34)) → leA_out_gg(s(T33), s(T34))
addB_in_gag(s(T84), s(T85)) → U2_gag(T84, T85, addB_in_gag(T84, T85))
addB_in_gag(0, T90) → addB_out_gag(0, T90, T90)
U4_gag(T72, T73, addB_out_gag(T72, X91, T73)) → addD_out_gag(T72, X91, T73)
U2_gag(T84, T85, addB_out_gag(T84, X115, T85)) → addB_out_gag(s(T84), X115, s(T85))

The set Q consists of the following terms:

leA_in_gg(x0, x1)
addB_in_gag(x0, x1)
U1_gg(x0, x1, x2)
U4_gag(x0, x1, x2)
U2_gag(x0, x1, x2)

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

(51) DependencyGraphProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) PrologToPiTRSProof (SOUND transformation)

(4) Obligation:

(5) DependencyPairsProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Complex Obligation (AND)

(9) Obligation:

(10) UsableRulesProof (EQUIVALENT transformation)

(11) Obligation:

(12) PiDPToQDPProof (EQUIVALENT transformation)

(13) Obligation:

(14) QDPSizeChangeProof (EQUIVALENT transformation)

(15) YES

(16) Obligation:

(17) UsableRulesProof (EQUIVALENT transformation)

(18) Obligation:

(19) PiDPToQDPProof (SOUND transformation)

(20) Obligation:

(21) QDPSizeChangeProof (EQUIVALENT transformation)

(22) YES

(23) Obligation:

(24) UsableRulesProof (EQUIVALENT transformation)

(25) Obligation:

(26) PiDPToQDPProof (EQUIVALENT transformation)

(27) Obligation:

(28) QDPSizeChangeProof (EQUIVALENT transformation)

(29) YES

(30) Obligation:

(31) UsableRulesProof (EQUIVALENT transformation)

(32) Obligation:

(33) PiDPToQDPProof (SOUND transformation)

(34) Obligation:

(35) Rewriting (EQUIVALENT transformation)

(36) Obligation:

(37) UsableRulesProof (EQUIVALENT transformation)

(38) Obligation:

(39) QReductionProof (EQUIVALENT transformation)

(40) Obligation:

(41) QDPOrderProof (EQUIVALENT transformation)

(42) Obligation:

(43) DependencyGraphProof (EQUIVALENT transformation)

(44) Obligation:

(45) UsableRulesProof (EQUIVALENT transformation)

(46) Obligation:

(47) QReductionProof (EQUIVALENT transformation)

(48) Obligation:

(49) QDPOrderProof (EQUIVALENT transformation)

(50) Obligation:

(51) DependencyGraphProof (EQUIVALENT transformation)

(52) TRUE