Left Termination proof of /tmp/tmpxaI6OW/ackermann-ioi.pl

Left Termination of the query pattern ackermann_in_3(g, a, g) w.r.t. the given Prolog program could not be shown:

0 Prolog

↳1 PrologToDTProblemTransformerProof (⇐)

↳2 TRIPLES

↳3 TriplesToPiDPProof (⇐)

↳4 PiDP

↳5 DependencyGraphProof (⇔)

↳6 AND

    ↳7 PiDP

        ↳8 UsableRulesProof (⇔)

        ↳9 PiDP

        ↳10 PiDPToQDPProof (⇐)

        ↳11 QDP

        ↳12 QDPSizeChangeProof (⇔)

        ↳13 YES

    ↳14 PiDP

        ↳15 UsableRulesProof (⇔)

        ↳16 PiDP

        ↳17 PiDPToQDPProof (⇐)

        ↳18 QDP

        ↳19 QDPSizeChangeProof (⇔)

        ↳20 YES

    ↳21 PiDP

        ↳22 UsableRulesProof (⇔)

        ↳23 PiDP

        ↳24 PiDPToQDPProof (⇐)

        ↳25 QDP

        ↳26 NonTerminationProof (⇔)

        ↳27 NO

    ↳28 PiDP

        ↳29 UsableRulesProof (⇔)

        ↳30 PiDP

        ↳31 PiDPToQDPProof (⇐)

        ↳32 QDP

        ↳33 QDPSizeChangeProof (⇔)

        ↳34 YES

(0) Obligation:

Clauses:

ackermann(0, N, s(N)).
ackermann(s(M), 0, Val) :- ackermann(M, s(0), Val).
ackermann(s(M), s(N), Val) :- ','(ackermann(s(M), N, Val1), ackermann(M, Val1, Val)).

Queries:

ackermann(g,a,g).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: ackermann(T1, T2, T3)\n\nT1 is ground\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: ackermann(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | ackermann(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | ackermann(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | ackermann(0, T2, s(T5)), SCOPE: 1, CLAUSE: 1 | ackermann(0, T2, s(T5)), SCOPE: 1, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ackermann(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | ackermann(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT3 is ground\nackermann(T1, T2, T3) !=? ackermann(0, X2, s(X2))", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ackermann(0, T2, s(T5)), SCOPE: 1, CLAUSE: 1 | ackermann(0, T2, s(T5)), SCOPE: 1, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ackermann(0, T2, s(T5)), SCOPE: 1, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ackermann(T8, s(0), T9) | ackermann(s(T8), T2, T9), SCOPE: 1, CLAUSE: 2\n\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ackermann(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT3 is ground\nackermann(T1, T2, T3) !=? ackermann(0, X2, s(X2))\nackermann(T1, T2, T3) !=? ackermann(s(X11), 0, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ackermann(T8, s(0), T9), SCOPE: 2, CLAUSE: 0 | ackermann(T8, s(0), T9), SCOPE: 2, CLAUSE: 1 | ackermann(T8, s(0), T9), SCOPE: 2, CLAUSE: 2 | ?_2 | ackermann(s(T8), T2, T9), SCOPE: 1, CLAUSE: 2\n\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ackermann(T8, s(0), T9), SCOPE: 2, CLAUSE: 0\n\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ackermann(T8, s(0), T9), SCOPE: 2, CLAUSE: 1 | ackermann(T8, s(0), T9), SCOPE: 2, CLAUSE: 2 | ?_2 | ackermann(s(T8), T2, T9), SCOPE: 1, CLAUSE: 2\n\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ackermann(T8, s(0), T9), SCOPE: 2, CLAUSE: 2 | ?_2 | ackermann(s(T8), T2, T9), SCOPE: 1, CLAUSE: 2\n\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ackermann(T8, s(0), T9), SCOPE: 2, CLAUSE: 2\n\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ?_2 | ackermann(s(T8), T2, T9), SCOPE: 1, CLAUSE: 2\n\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(ackermann(s(T19), 0, X40), ackermann(T19, X40, T20))\n\nT19 is ground\nT20 is ground\nX40 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ackermann(s(T19), 0, X40)\n\nT19 is ground\nX40 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ackermann(T19, T22, T20)\n\nT19 is ground\nT20 is ground\nT22 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ackermann(s(T19), 0, X40), SCOPE: 3, CLAUSE: 0 | ackermann(s(T19), 0, X40), SCOPE: 3, CLAUSE: 1 | ackermann(s(T19), 0, X40), SCOPE: 3, CLAUSE: 2\n\nT19 is ground\nX40 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ackermann(s(T19), 0, X40), SCOPE: 3, CLAUSE: 1 | ackermann(s(T19), 0, X40), SCOPE: 3, CLAUSE: 2\n\nT19 is ground\nX40 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ackermann(s(T19), 0, X40), SCOPE: 3, CLAUSE: 1\n\nT19 is ground\nX40 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ackermann(s(T19), 0, X40), SCOPE: 3, CLAUSE: 2\n\nT19 is ground\nX40 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ackermann(T28, s(0), X73)\n\nT28 is ground\nX73 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ackermann(T28, s(0), X73), SCOPE: 4, CLAUSE: 0 | ackermann(T28, s(0), X73), SCOPE: 4, CLAUSE: 1 | ackermann(T28, s(0), X73), SCOPE: 4, CLAUSE: 2\n\nT28 is ground\nX73 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ackermann(T28, s(0), X73), SCOPE: 4, CLAUSE: 0\n\nT28 is ground\nX73 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ackermann(T28, s(0), X73), SCOPE: 4, CLAUSE: 1 | ackermann(T28, s(0), X73), SCOPE: 4, CLAUSE: 2\n\nT28 is ground\nX73 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ackermann(T28, s(0), X73), SCOPE: 4, CLAUSE: 2\n\nT28 is ground\nX73 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(ackermann(s(T32), 0, X96), ackermann(T32, X96, X97))\n\nT32 is ground\nX97 is free; \nX96 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ackermann(s(T32), 0, X96)\n\nT32 is ground\nX96 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ackermann(T32, T34, X97)\n\nT32 is ground\nT34 is ground\nX97 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ackermann(T32, T34, X97), SCOPE: 5, CLAUSE: 0 | ackermann(T32, T34, X97), SCOPE: 5, CLAUSE: 1 | ackermann(T32, T34, X97), SCOPE: 5, CLAUSE: 2\n\nT32 is ground\nT34 is ground\nX97 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ackermann(T32, T34, X97), SCOPE: 5, CLAUSE: 0\n\nT32 is ground\nT34 is ground\nX97 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ackermann(T32, T34, X97), SCOPE: 5, CLAUSE: 1 | ackermann(T32, T34, X97), SCOPE: 5, CLAUSE: 2\n\nT32 is ground\nT34 is ground\nX97 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ackermann(T32, T34, X97), SCOPE: 5, CLAUSE: 1\n\nT32 is ground\nT34 is ground\nX97 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ackermann(T32, T34, X97), SCOPE: 5, CLAUSE: 2\n\nT32 is ground\nT34 is ground\nX97 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ackermann(T47, s(0), X133)\n\nT47 is ground\nX133 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(ackermann(s(T52), T53, X150), ackermann(T52, X150, X151))\n\nT52 is ground\nT53 is ground\nX151 is free; \nX150 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: ackermann(s(T52), T53, X150)\n\nT52 is ground\nT53 is ground\nX150 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ackermann(T52, T55, X151)\n\nT52 is ground\nT55 is ground\nX151 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ackermann(s(T8), T2, T9), SCOPE: 1, CLAUSE: 2\n\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(ackermann(s(T68), T71, X182), ackermann(T68, X182, T70))\n\nT68 is ground\nT70 is ground\nX182 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ackermann(s(T68), T71, X182)\n\nT68 is ground\nX182 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ackermann(T68, T73, T70)\n\nT68 is ground\nT70 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ackermann(s(T68), T71, X182), SCOPE: 6, CLAUSE: 0 | ackermann(s(T68), T71, X182), SCOPE: 6, CLAUSE: 1 | ackermann(s(T68), T71, X182), SCOPE: 6, CLAUSE: 2\n\nT68 is ground\nX182 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ackermann(s(T68), T71, X182), SCOPE: 6, CLAUSE: 1 | ackermann(s(T68), T71, X182), SCOPE: 6, CLAUSE: 2\n\nT68 is ground\nX182 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ackermann(s(T68), T71, X182), SCOPE: 6, CLAUSE: 1\n\nT68 is ground\nX182 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ackermann(s(T68), T71, X182), SCOPE: 6, CLAUSE: 2\n\nT68 is ground\nX182 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ackermann(T79, s(0), X211)\n\nT79 is ground\nX211 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ','(ackermann(s(T84), T86, X228), ackermann(T84, X228, X229))\n\nT84 is ground\nX229 is free; \nX228 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: ackermann(s(T84), T86, X228)\n\nT84 is ground\nX228 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: ackermann(T84, T88, X229)\n\nT84 is ground\nX229 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: ackermann(T84, T88, X229), SCOPE: 7, CLAUSE: 0 | ackermann(T84, T88, X229), SCOPE: 7, CLAUSE: 1 | ackermann(T84, T88, X229), SCOPE: 7, CLAUSE: 2\n\nT84 is ground\nX229 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: ackermann(T84, T88, X229), SCOPE: 7, CLAUSE: 0\n\nT84 is ground\nX229 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: ackermann(T84, T88, X229), SCOPE: 7, CLAUSE: 1 | ackermann(T84, T88, X229), SCOPE: 7, CLAUSE: 2\n\nT84 is ground\nX229 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: ackermann(T84, T88, X229), SCOPE: 7, CLAUSE: 1\n\nT84 is ground\nX229 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: ackermann(T84, T88, X229), SCOPE: 7, CLAUSE: 2\n\nT84 is ground\nX229 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: ackermann(T101, s(0), X265)\n\nT101 is ground\nX265 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: ','(ackermann(s(T106), T108, X282), ackermann(T106, X282, X283))\n\nT106 is ground\nX283 is free; \nX282 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: ','(ackermann(s(T115), T118, X300), ackermann(T115, X300, T117))\n\nT115 is ground\nT117 is ground\nX300 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: ','(ackermann(s(T115), T118, X300), ackermann(T115, X300, T117)), SCOPE: 8, CLAUSE: 0 | ','(ackermann(s(T115), T118, X300), ackermann(T115, X300, T117)), SCOPE: 8, CLAUSE: 1 | ','(ackermann(s(T115), T118, X300), ackermann(T115, X300, T117)), SCOPE: 8, CLAUSE: 2\n\nT115 is ground\nT117 is ground\nX300 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: ','(ackermann(s(T115), T118, X300), ackermann(T115, X300, T117)), SCOPE: 8, CLAUSE: 1 | ','(ackermann(s(T115), T118, X300), ackermann(T115, X300, T117)), SCOPE: 8, CLAUSE: 2\n\nT115 is ground\nT117 is ground\nX300 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: ','(ackermann(s(T115), T118, X300), ackermann(T115, X300, T117)), SCOPE: 8, CLAUSE: 1\n\nT115 is ground\nT117 is ground\nX300 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: ','(ackermann(s(T115), T118, X300), ackermann(T115, X300, T117)), SCOPE: 8, CLAUSE: 2\n\nT115 is ground\nT117 is ground\nX300 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: ','(ackermann(T123, s(0), X319), ackermann(T123, X319, T117))\n\nT117 is ground\nT123 is ground\nX319 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: ackermann(T123, s(0), X319)\n\nT123 is ground\nX319 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: ackermann(T123, T124, T117)\n\nT117 is ground\nT123 is ground\nT124 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: ','(ackermann(s(T131), T133, X340), ','(ackermann(T131, X340, X341), ackermann(T131, X341, T117)))\n\nT117 is ground\nT131 is ground\nX341 is free; \nX340 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: ackermann(s(T131), T133, X340)\n\nT131 is ground\nX340 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: ','(ackermann(T131, T135, X341), ackermann(T131, X341, T117))\n\nT117 is ground\nT131 is ground\nX341 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: ackermann(T131, T135, X341)\n\nT131 is ground\nX341 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: ackermann(T131, T139, T117)\n\nT117 is ground\nT131 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 97 [arrowhead = none ];
97 -> 2;

98 [label="EVAL with clause\nackermann(0, X2, s(X2)).\nand substitution\nT1 -> 0\nT2 -> T5\nX2 -> T5\nT3 -> s(T5)", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 98 [arrowhead = none ];
98 -> 3;

99 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 99 [arrowhead = none ];
99 -> 4;

100 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 100 [arrowhead = none ];
100 -> 5;

101 [label="EVAL with clause\nackermann(s(X11), 0, X12) :- ackermann(X11, s(0), X12).\nand substitution\nX11 -> T8\nT1 -> s(T8)\nT2 -> 0\nT3 -> T9\nX12 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 101 [arrowhead = none ];
101 -> 8;

102 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 102 [arrowhead = none ];
102 -> 9;

103 [label="BACKTRACK\nfor clause: ackermann(s(M), 0, Val) :- ackermann(M, s(0), Val)because of non-unification", fontsize=14, style = filled, fillcolor = white];
5 -> 103 [arrowhead = none ];
103 -> 6;

104 [label="BACKTRACK\nfor clause: ackermann(s(M), s(N), Val) :- ','(ackermann(s(M), N, Val1), ackermann(M, Val1, Val))because of non-unification", fontsize=14, style = filled, fillcolor = white];
6 -> 104 [arrowhead = none ];
104 -> 7;

105 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
8 -> 105 [arrowhead = none ];
105 -> 10;

106 [label="EVAL with clause\nackermann(s(X297), s(X298), X299) :- ','(ackermann(s(X297), X298, X300), ackermann(X297, X300, X299)).\nand substitution\nX297 -> T115\nT1 -> s(T115)\nX298 -> T118\nT2 -> s(T118)\nT3 -> T117\nX299 -> T117\nT116 -> T118", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 106 [arrowhead = none ];
106 -> 81;

107 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 107 [arrowhead = none ];
107 -> 82;

108 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 108 [arrowhead = none ];
108 -> 11;

109 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 109 [arrowhead = none ];
109 -> 12;

110 [label="EVAL with clause\nackermann(0, X19, s(X19)).\nand substitution\nT8 -> 0\nX19 -> s(0)\nT9 -> s(s(0))", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 110 [arrowhead = none ];
110 -> 13;

111 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 111 [arrowhead = none ];
111 -> 14;

112 [label="BACKTRACK\nfor clause: ackermann(s(M), 0, Val) :- ackermann(M, s(0), Val)because of non-unification", fontsize=14, style = filled, fillcolor = white];
12 -> 112 [arrowhead = none ];
112 -> 16;

113 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 113 [arrowhead = none ];
113 -> 15;

114 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 114 [arrowhead = none ];
114 -> 17;

115 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 115 [arrowhead = none ];
115 -> 18;

116 [label="EVAL with clause\nackermann(s(X37), s(X38), X39) :- ','(ackermann(s(X37), X38, X40), ackermann(X37, X40, X39)).\nand substitution\nX37 -> T19\nT8 -> s(T19)\nX38 -> 0\nT9 -> T20\nX39 -> T20", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 116 [arrowhead = none ];
116 -> 19;

117 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 117 [arrowhead = none ];
117 -> 20;

118 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 118 [arrowhead = none ];
118 -> 54;

119 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
19 -> 119 [arrowhead = none ];
119 -> 21;

120 [label="SPLIT 2\nnew knowledge:\nT22 is ground\nreplacements:\nX40 -> T22", fontsize=14, style = filled, fillcolor = violet];
19 -> 120 [arrowhead = none ];
120 -> 22;

121 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 121 [arrowhead = none ];
121 -> 23;

122 [label="INSTANCE with matching:\nT1 -> T19\nT2 -> T22\nT3 -> T20", fontsize=14, style = filled, fillcolor = lightgrey];
22 -> 122 [arrowhead = none , style=dashed];
122 -> 1[style=dashed];

123 [label="BACKTRACK\nfor clause: ackermann(0, N, s(N))because of non-unification", fontsize=14, style = filled, fillcolor = white];
23 -> 123 [arrowhead = none ];
123 -> 24;

124 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
24 -> 124 [arrowhead = none ];
124 -> 25;

125 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
24 -> 125 [arrowhead = none ];
125 -> 26;

126 [label="ONLY EVAL with clause\nackermann(s(X71), 0, X72) :- ackermann(X71, s(0), X72).\nand substitution\nT19 -> T28\nX71 -> T28\nX40 -> X73\nX72 -> X73", fontsize=14, style = filled, fillcolor = white];
25 -> 126 [arrowhead = none ];
126 -> 27;

127 [label="BACKTRACK\nfor clause: ackermann(s(M), s(N), Val) :- ','(ackermann(s(M), N, Val1), ackermann(M, Val1, Val))because of non-unification", fontsize=14, style = filled, fillcolor = white];
26 -> 127 [arrowhead = none ];
127 -> 53;

128 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
27 -> 128 [arrowhead = none ];
128 -> 28;

129 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
28 -> 129 [arrowhead = none ];
129 -> 29;

130 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
28 -> 130 [arrowhead = none ];
130 -> 30;

131 [label="EVAL with clause\nackermann(0, X80, s(X80)).\nand substitution\nT28 -> 0\nX80 -> s(0)\nX73 -> s(s(0))", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 131 [arrowhead = none ];
131 -> 31;

132 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 132 [arrowhead = none ];
132 -> 32;

133 [label="BACKTRACK\nfor clause: ackermann(s(M), 0, Val) :- ackermann(M, s(0), Val)because of non-unification", fontsize=14, style = filled, fillcolor = white];
30 -> 133 [arrowhead = none ];
133 -> 34;

134 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 134 [arrowhead = none ];
134 -> 33;

135 [label="EVAL with clause\nackermann(s(X93), s(X94), X95) :- ','(ackermann(s(X93), X94, X96), ackermann(X93, X96, X95)).\nand substitution\nX93 -> T32\nT28 -> s(T32)\nX94 -> 0\nX73 -> X97\nX95 -> X97", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 135 [arrowhead = none ];
135 -> 35;

136 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 136 [arrowhead = none ];
136 -> 36;

137 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
35 -> 137 [arrowhead = none ];
137 -> 37;

138 [label="SPLIT 2\nnew knowledge:\nT34 is ground\nreplacements:\nX96 -> T34", fontsize=14, style = filled, fillcolor = violet];
35 -> 138 [arrowhead = none ];
138 -> 38;

139 [label="INSTANCE with matching:\nT19 -> T32\nX40 -> X96", fontsize=14, style = filled, fillcolor = lightgrey];
37 -> 139 [arrowhead = none , style=dashed];
139 -> 21[style=dashed];

140 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
38 -> 140 [arrowhead = none ];
140 -> 39;

141 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
39 -> 141 [arrowhead = none ];
141 -> 40;

142 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
39 -> 142 [arrowhead = none ];
142 -> 41;

143 [label="EVAL with clause\nackermann(0, X117, s(X117)).\nand substitution\nT32 -> 0\nT34 -> T42\nX117 -> T42\nX97 -> s(T42)", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 143 [arrowhead = none ];
143 -> 42;

144 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 144 [arrowhead = none ];
144 -> 43;

145 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
41 -> 145 [arrowhead = none ];
145 -> 45;

146 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
41 -> 146 [arrowhead = none ];
146 -> 46;

147 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 147 [arrowhead = none ];
147 -> 44;

148 [label="EVAL with clause\nackermann(s(X131), 0, X132) :- ackermann(X131, s(0), X132).\nand substitution\nX131 -> T47\nT32 -> s(T47)\nT34 -> 0\nX97 -> X133\nX132 -> X133", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 148 [arrowhead = none ];
148 -> 47;

149 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 149 [arrowhead = none ];
149 -> 48;

150 [label="EVAL with clause\nackermann(s(X147), s(X148), X149) :- ','(ackermann(s(X147), X148, X150), ackermann(X147, X150, X149)).\nand substitution\nX147 -> T52\nT32 -> s(T52)\nX148 -> T53\nT34 -> s(T53)\nX97 -> X151\nX149 -> X151", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 150 [arrowhead = none ];
150 -> 49;

151 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 151 [arrowhead = none ];
151 -> 50;

152 [label="INSTANCE with matching:\nT28 -> T47\nX73 -> X133", fontsize=14, style = filled, fillcolor = lightgrey];
47 -> 152 [arrowhead = none , style=dashed];
152 -> 27[style=dashed];

153 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
49 -> 153 [arrowhead = none ];
153 -> 51;

154 [label="SPLIT 2\nnew knowledge:\nT55 is ground\nreplacements:\nX150 -> T55", fontsize=14, style = filled, fillcolor = violet];
49 -> 154 [arrowhead = none ];
154 -> 52;

155 [label="INSTANCE with matching:\nT32 -> s(T52)\nT34 -> T53\nX97 -> X150", fontsize=14, style = filled, fillcolor = lightgrey];
51 -> 155 [arrowhead = none , style=dashed];
155 -> 38[style=dashed];

156 [label="INSTANCE with matching:\nT32 -> T52\nT34 -> T55\nX97 -> X151", fontsize=14, style = filled, fillcolor = lightgrey];
52 -> 156 [arrowhead = none , style=dashed];
156 -> 38[style=dashed];

157 [label="EVAL with clause\nackermann(s(X179), s(X180), X181) :- ','(ackermann(s(X179), X180, X182), ackermann(X179, X182, X181)).\nand substitution\nT8 -> T68\nX179 -> T68\nX180 -> T71\nT2 -> s(T71)\nT9 -> T70\nX181 -> T70\nT69 -> T71", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 157 [arrowhead = none ];
157 -> 55;

158 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 158 [arrowhead = none ];
158 -> 56;

159 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
55 -> 159 [arrowhead = none ];
159 -> 57;

160 [label="SPLIT 2\nreplacements:\nX182 -> T73", fontsize=14, style = filled, fillcolor = violet];
55 -> 160 [arrowhead = none ];
160 -> 58;

161 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
57 -> 161 [arrowhead = none ];
161 -> 59;

162 [label="INSTANCE with matching:\nT1 -> T68\nT2 -> T73\nT3 -> T70", fontsize=14, style = filled, fillcolor = lightgrey];
58 -> 162 [arrowhead = none , style=dashed];
162 -> 1[style=dashed];

163 [label="BACKTRACK\nfor clause: ackermann(0, N, s(N))because of non-unification", fontsize=14, style = filled, fillcolor = white];
59 -> 163 [arrowhead = none ];
163 -> 60;

164 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
60 -> 164 [arrowhead = none ];
164 -> 61;

165 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
60 -> 165 [arrowhead = none ];
165 -> 62;

166 [label="EVAL with clause\nackermann(s(X209), 0, X210) :- ackermann(X209, s(0), X210).\nand substitution\nT68 -> T79\nX209 -> T79\nT71 -> 0\nX182 -> X211\nX210 -> X211", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 166 [arrowhead = none ];
166 -> 63;

167 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 167 [arrowhead = none ];
167 -> 64;

168 [label="EVAL with clause\nackermann(s(X225), s(X226), X227) :- ','(ackermann(s(X225), X226, X228), ackermann(X225, X228, X227)).\nand substitution\nT68 -> T84\nX225 -> T84\nX226 -> T86\nT71 -> s(T86)\nX182 -> X229\nX227 -> X229\nT85 -> T86", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 168 [arrowhead = none ];
168 -> 65;

169 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 169 [arrowhead = none ];
169 -> 66;

170 [label="INSTANCE with matching:\nT28 -> T79\nX73 -> X211", fontsize=14, style = filled, fillcolor = lightgrey];
63 -> 170 [arrowhead = none , style=dashed];
170 -> 27[style=dashed];

171 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
65 -> 171 [arrowhead = none ];
171 -> 67;

172 [label="SPLIT 2\nreplacements:\nX228 -> T88", fontsize=14, style = filled, fillcolor = violet];
65 -> 172 [arrowhead = none ];
172 -> 68;

173 [label="INSTANCE with matching:\nT68 -> T84\nT71 -> T86\nX182 -> X228", fontsize=14, style = filled, fillcolor = lightgrey];
67 -> 173 [arrowhead = none , style=dashed];
173 -> 57[style=dashed];

174 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
68 -> 174 [arrowhead = none ];
174 -> 69;

175 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
69 -> 175 [arrowhead = none ];
175 -> 70;

176 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
69 -> 176 [arrowhead = none ];
176 -> 71;

177 [label="EVAL with clause\nackermann(0, X249, s(X249)).\nand substitution\nT84 -> 0\nT88 -> T96\nX249 -> T96\nX229 -> s(T96)", fontsize=14, style = filled, fillcolor = lightblue];
70 -> 177 [arrowhead = none ];
177 -> 72;

178 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
70 -> 178 [arrowhead = none ];
178 -> 73;

179 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
71 -> 179 [arrowhead = none ];
179 -> 75;

180 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
71 -> 180 [arrowhead = none ];
180 -> 76;

181 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
72 -> 181 [arrowhead = none ];
181 -> 74;

182 [label="EVAL with clause\nackermann(s(X263), 0, X264) :- ackermann(X263, s(0), X264).\nand substitution\nX263 -> T101\nT84 -> s(T101)\nT88 -> 0\nX229 -> X265\nX264 -> X265", fontsize=14, style = filled, fillcolor = lightblue];
75 -> 182 [arrowhead = none ];
182 -> 77;

183 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
75 -> 183 [arrowhead = none ];
183 -> 78;

184 [label="EVAL with clause\nackermann(s(X279), s(X280), X281) :- ','(ackermann(s(X279), X280, X282), ackermann(X279, X282, X281)).\nand substitution\nX279 -> T106\nT84 -> s(T106)\nX280 -> T108\nT88 -> s(T108)\nX229 -> X283\nX281 -> X283\nT107 -> T108", fontsize=14, style = filled, fillcolor = lightblue];
76 -> 184 [arrowhead = none ];
184 -> 79;

185 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
76 -> 185 [arrowhead = none ];
185 -> 80;

186 [label="INSTANCE with matching:\nT79 -> T101\nX211 -> X265", fontsize=14, style = filled, fillcolor = lightgrey];
77 -> 186 [arrowhead = none , style=dashed];
186 -> 63[style=dashed];

187 [label="INSTANCE with matching:\nT84 -> T106\nT86 -> T108\nX228 -> X282\nX229 -> X283", fontsize=14, style = filled, fillcolor = lightgrey];
79 -> 187 [arrowhead = none , style=dashed];
187 -> 65[style=dashed];

188 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
81 -> 188 [arrowhead = none ];
188 -> 83;

189 [label="BACKTRACK\nfor clause: ackermann(0, N, s(N))because of non-unification", fontsize=14, style = filled, fillcolor = white];
83 -> 189 [arrowhead = none ];
189 -> 84;

190 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
84 -> 190 [arrowhead = none ];
190 -> 85;

191 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
84 -> 191 [arrowhead = none ];
191 -> 86;

192 [label="EVAL with clause\nackermann(s(X317), 0, X318) :- ackermann(X317, s(0), X318).\nand substitution\nT115 -> T123\nX317 -> T123\nT118 -> 0\nX300 -> X319\nX318 -> X319", fontsize=14, style = filled, fillcolor = lightblue];
85 -> 192 [arrowhead = none ];
192 -> 87;

193 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
85 -> 193 [arrowhead = none ];
193 -> 88;

194 [label="EVAL with clause\nackermann(s(X337), s(X338), X339) :- ','(ackermann(s(X337), X338, X340), ackermann(X337, X340, X339)).\nand substitution\nT115 -> T131\nX337 -> T131\nX338 -> T133\nT118 -> s(T133)\nX300 -> X341\nX339 -> X341\nT132 -> T133", fontsize=14, style = filled, fillcolor = lightblue];
86 -> 194 [arrowhead = none ];
194 -> 91;

195 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
86 -> 195 [arrowhead = none ];
195 -> 92;

196 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
87 -> 196 [arrowhead = none ];
196 -> 89;

197 [label="SPLIT 2\nnew knowledge:\nT124 is ground\nreplacements:\nX319 -> T124", fontsize=14, style = filled, fillcolor = violet];
87 -> 197 [arrowhead = none ];
197 -> 90;

198 [label="INSTANCE with matching:\nT79 -> T123\nX211 -> X319", fontsize=14, style = filled, fillcolor = lightgrey];
89 -> 198 [arrowhead = none , style=dashed];
198 -> 63[style=dashed];

199 [label="INSTANCE with matching:\nT1 -> T123\nT2 -> T124\nT3 -> T117", fontsize=14, style = filled, fillcolor = lightgrey];
90 -> 199 [arrowhead = none , style=dashed];
199 -> 1[style=dashed];

200 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
91 -> 200 [arrowhead = none ];
200 -> 93;

201 [label="SPLIT 2\nreplacements:\nX340 -> T135", fontsize=14, style = filled, fillcolor = violet];
91 -> 201 [arrowhead = none ];
201 -> 94;

202 [label="INSTANCE with matching:\nT68 -> T131\nT71 -> T133\nX182 -> X340", fontsize=14, style = filled, fillcolor = lightgrey];
93 -> 202 [arrowhead = none , style=dashed];
202 -> 57[style=dashed];

203 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
94 -> 203 [arrowhead = none ];
203 -> 95;

204 [label="SPLIT 2\nreplacements:\nX341 -> T139", fontsize=14, style = filled, fillcolor = violet];
94 -> 204 [arrowhead = none ];
204 -> 96;

205 [label="INSTANCE with matching:\nT84 -> T131\nT88 -> T135\nX229 -> X341", fontsize=14, style = filled, fillcolor = lightgrey];
95 -> 205 [arrowhead = none , style=dashed];
205 -> 68[style=dashed];

206 [label="INSTANCE with matching:\nT1 -> T131\nT2 -> T139\nT3 -> T117", fontsize=14, style = filled, fillcolor = lightgrey];
96 -> 206 [arrowhead = none , style=dashed];
206 -> 1[style=dashed];

}

(2) Obligation:

Triples:

ackermann21(T28, X73) :- ackermann27(T28, X73).
ackermann27(s(T32), X97) :- ackermann21(T32, X96).
ackermann27(s(T32), X97) :- ','(ackermannc21(T32, T34), ackermann38(T32, T34, X97)).
ackermann38(s(T47), 0, X133) :- ackermann27(T47, X133).
ackermann38(s(T52), s(T53), X151) :- ackermann38(s(T52), T53, X150).
ackermann38(s(T52), s(T53), X151) :- ','(ackermannc38(s(T52), T53, T55), ackermann38(T52, T55, X151)).
ackermann57(T79, 0, X211) :- ackermann27(T79, X211).
ackermann57(T84, s(T86), X229) :- p65(T84, T86, X228, X229).
p65(T84, T86, X228, X229) :- ackermann57(T84, T86, X228).
p65(T84, T86, T88, X229) :- ','(ackermannc57(T84, T86, T88), ackermann68(T84, T88, X229)).
ackermann68(s(T101), 0, X265) :- ackermann27(T101, X265).
ackermann68(s(T106), s(T108), X283) :- p65(T106, T108, X282, X283).
ackermann1(s(s(T19)), 0, T20) :- ackermann21(T19, X40).
ackermann1(s(s(T19)), 0, T20) :- ','(ackermannc21(T19, T22), ackermann1(T19, T22, T20)).
ackermann1(s(T68), s(T71), T70) :- ackermann57(T68, T71, X182).
ackermann1(s(T68), s(T71), T70) :- ','(ackermannc57(T68, T71, T73), ackermann1(T68, T73, T70)).
ackermann1(s(T123), s(0), T117) :- ackermann27(T123, X319).
ackermann1(s(T123), s(0), T117) :- ','(ackermannc27(T123, T124), ackermann1(T123, T124, T117)).
ackermann1(s(T131), s(s(T133)), T117) :- ackermann57(T131, T133, X340).
ackermann1(s(T131), s(s(T133)), T117) :- ','(ackermannc57(T131, T133, T135), ackermann68(T131, T135, X341)).
ackermann1(s(T131), s(s(T133)), T117) :- ','(ackermannc57(T131, T133, T135), ','(ackermannc68(T131, T135, T139), ackermann1(T131, T139, T117))).

Clauses:

ackermannc1(0, T5, s(T5)).
ackermannc1(s(0), 0, s(s(0))).
ackermannc1(s(s(T19)), 0, T20) :- ','(ackermannc21(T19, T22), ackermannc1(T19, T22, T20)).
ackermannc1(s(T68), s(T71), T70) :- ','(ackermannc57(T68, T71, T73), ackermannc1(T68, T73, T70)).
ackermannc1(s(T123), s(0), T117) :- ','(ackermannc27(T123, T124), ackermannc1(T123, T124, T117)).
ackermannc1(s(T131), s(s(T133)), T117) :- ','(ackermannc57(T131, T133, T135), ','(ackermannc68(T131, T135, T139), ackermannc1(T131, T139, T117))).
ackermannc21(T28, X73) :- ackermannc27(T28, X73).
ackermannc27(0, s(s(0))).
ackermannc27(s(T32), X97) :- ','(ackermannc21(T32, T34), ackermannc38(T32, T34, X97)).
ackermannc38(0, T42, s(T42)).
ackermannc38(s(T47), 0, X133) :- ackermannc27(T47, X133).
ackermannc38(s(T52), s(T53), X151) :- ','(ackermannc38(s(T52), T53, T55), ackermannc38(T52, T55, X151)).
ackermannc57(T79, 0, X211) :- ackermannc27(T79, X211).
ackermannc57(T84, s(T86), X229) :- qc65(T84, T86, X228, X229).
qc65(T84, T86, T88, X229) :- ','(ackermannc57(T84, T86, T88), ackermannc68(T84, T88, X229)).
ackermannc68(0, T96, s(T96)).
ackermannc68(s(T101), 0, X265) :- ackermannc27(T101, X265).
ackermannc68(s(T106), s(T108), X283) :- qc65(T106, T108, X282, X283).

Afs:

ackermann1(x1, x2, x3) = ackermann1(x1, x3)

(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:
ackermann1_in: (b,f,b) (b,b,b)
ackermann21_in: (b,f)
ackermann27_in: (b,f)
ackermannc21_in: (b,f)
ackermannc27_in: (b,f)
ackermannc38_in: (b,b,f)
ackermann38_in: (b,b,f)
ackermann57_in: (b,b,f) (b,f,f)
p65_in: (b,b,f,f) (b,f,f,f)
ackermannc57_in: (b,b,f) (b,f,f)
qc65_in: (b,b,f,f) (b,f,f,f)
ackermannc68_in: (b,b,f)
ackermann68_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

ACKERMANN1_IN_GAG(s(s(T19)), 0, T20) → U16_GAG(T19, T20, ackermann21_in_ga(T19, X40))
ACKERMANN1_IN_GAG(s(s(T19)), 0, T20) → ACKERMANN21_IN_GA(T19, X40)
ACKERMANN21_IN_GA(T28, X73) → U1_GA(T28, X73, ackermann27_in_ga(T28, X73))
ACKERMANN21_IN_GA(T28, X73) → ACKERMANN27_IN_GA(T28, X73)
ACKERMANN27_IN_GA(s(T32), X97) → U2_GA(T32, X97, ackermann21_in_ga(T32, X96))
ACKERMANN27_IN_GA(s(T32), X97) → ACKERMANN21_IN_GA(T32, X96)
ACKERMANN27_IN_GA(s(T32), X97) → U3_GA(T32, X97, ackermannc21_in_ga(T32, T34))
U3_GA(T32, X97, ackermannc21_out_ga(T32, T34)) → U4_GA(T32, X97, ackermann38_in_gga(T32, T34, X97))
U3_GA(T32, X97, ackermannc21_out_ga(T32, T34)) → ACKERMANN38_IN_GGA(T32, T34, X97)
ACKERMANN38_IN_GGA(s(T47), 0, X133) → U5_GGA(T47, X133, ackermann27_in_ga(T47, X133))
ACKERMANN38_IN_GGA(s(T47), 0, X133) → ACKERMANN27_IN_GA(T47, X133)
ACKERMANN38_IN_GGA(s(T52), s(T53), X151) → U6_GGA(T52, T53, X151, ackermann38_in_gga(s(T52), T53, X150))
ACKERMANN38_IN_GGA(s(T52), s(T53), X151) → ACKERMANN38_IN_GGA(s(T52), T53, X150)
ACKERMANN38_IN_GGA(s(T52), s(T53), X151) → U7_GGA(T52, T53, X151, ackermannc38_in_gga(s(T52), T53, T55))
U7_GGA(T52, T53, X151, ackermannc38_out_gga(s(T52), T53, T55)) → U8_GGA(T52, T53, X151, ackermann38_in_gga(T52, T55, X151))
U7_GGA(T52, T53, X151, ackermannc38_out_gga(s(T52), T53, T55)) → ACKERMANN38_IN_GGA(T52, T55, X151)
ACKERMANN1_IN_GAG(s(s(T19)), 0, T20) → U17_GAG(T19, T20, ackermannc21_in_ga(T19, T22))
U17_GAG(T19, T20, ackermannc21_out_ga(T19, T22)) → U18_GAG(T19, T20, ackermann1_in_ggg(T19, T22, T20))
U17_GAG(T19, T20, ackermannc21_out_ga(T19, T22)) → ACKERMANN1_IN_GGG(T19, T22, T20)
ACKERMANN1_IN_GGG(s(s(T19)), 0, T20) → U16_GGG(T19, T20, ackermann21_in_ga(T19, X40))
ACKERMANN1_IN_GGG(s(s(T19)), 0, T20) → ACKERMANN21_IN_GA(T19, X40)
ACKERMANN1_IN_GGG(s(s(T19)), 0, T20) → U17_GGG(T19, T20, ackermannc21_in_ga(T19, T22))
U17_GGG(T19, T20, ackermannc21_out_ga(T19, T22)) → U18_GGG(T19, T20, ackermann1_in_ggg(T19, T22, T20))
U17_GGG(T19, T20, ackermannc21_out_ga(T19, T22)) → ACKERMANN1_IN_GGG(T19, T22, T20)
ACKERMANN1_IN_GGG(s(T68), s(T71), T70) → U19_GGG(T68, T71, T70, ackermann57_in_gga(T68, T71, X182))
ACKERMANN1_IN_GGG(s(T68), s(T71), T70) → ACKERMANN57_IN_GGA(T68, T71, X182)
ACKERMANN57_IN_GGA(T79, 0, X211) → U9_GGA(T79, X211, ackermann27_in_ga(T79, X211))
ACKERMANN57_IN_GGA(T79, 0, X211) → ACKERMANN27_IN_GA(T79, X211)
ACKERMANN57_IN_GGA(T84, s(T86), X229) → U10_GGA(T84, T86, X229, p65_in_ggaa(T84, T86, X228, X229))
ACKERMANN57_IN_GGA(T84, s(T86), X229) → P65_IN_GGAA(T84, T86, X228, X229)
P65_IN_GGAA(T84, T86, X228, X229) → U11_GGAA(T84, T86, X228, X229, ackermann57_in_gga(T84, T86, X228))
P65_IN_GGAA(T84, T86, X228, X229) → ACKERMANN57_IN_GGA(T84, T86, X228)
P65_IN_GGAA(T84, T86, T88, X229) → U12_GGAA(T84, T86, T88, X229, ackermannc57_in_gga(T84, T86, T88))
U12_GGAA(T84, T86, T88, X229, ackermannc57_out_gga(T84, T86, T88)) → U13_GGAA(T84, T86, T88, X229, ackermann68_in_gga(T84, T88, X229))
U12_GGAA(T84, T86, T88, X229, ackermannc57_out_gga(T84, T86, T88)) → ACKERMANN68_IN_GGA(T84, T88, X229)
ACKERMANN68_IN_GGA(s(T101), 0, X265) → U14_GGA(T101, X265, ackermann27_in_ga(T101, X265))
ACKERMANN68_IN_GGA(s(T101), 0, X265) → ACKERMANN27_IN_GA(T101, X265)
ACKERMANN68_IN_GGA(s(T106), s(T108), X283) → U15_GGA(T106, T108, X283, p65_in_ggaa(T106, T108, X282, X283))
ACKERMANN68_IN_GGA(s(T106), s(T108), X283) → P65_IN_GGAA(T106, T108, X282, X283)
ACKERMANN1_IN_GGG(s(T68), s(T71), T70) → U20_GGG(T68, T71, T70, ackermannc57_in_gga(T68, T71, T73))
U20_GGG(T68, T71, T70, ackermannc57_out_gga(T68, T71, T73)) → U21_GGG(T68, T71, T70, ackermann1_in_ggg(T68, T73, T70))
U20_GGG(T68, T71, T70, ackermannc57_out_gga(T68, T71, T73)) → ACKERMANN1_IN_GGG(T68, T73, T70)
ACKERMANN1_IN_GGG(s(T123), s(0), T117) → U22_GGG(T123, T117, ackermann27_in_ga(T123, X319))
ACKERMANN1_IN_GGG(s(T123), s(0), T117) → ACKERMANN27_IN_GA(T123, X319)
ACKERMANN1_IN_GGG(s(T123), s(0), T117) → U23_GGG(T123, T117, ackermannc27_in_ga(T123, T124))
U23_GGG(T123, T117, ackermannc27_out_ga(T123, T124)) → U24_GGG(T123, T117, ackermann1_in_ggg(T123, T124, T117))
U23_GGG(T123, T117, ackermannc27_out_ga(T123, T124)) → ACKERMANN1_IN_GGG(T123, T124, T117)
ACKERMANN1_IN_GGG(s(T131), s(s(T133)), T117) → U25_GGG(T131, T133, T117, ackermann57_in_gga(T131, T133, X340))
ACKERMANN1_IN_GGG(s(T131), s(s(T133)), T117) → ACKERMANN57_IN_GGA(T131, T133, X340)
ACKERMANN1_IN_GGG(s(T131), s(s(T133)), T117) → U26_GGG(T131, T133, T117, ackermannc57_in_gga(T131, T133, T135))
U26_GGG(T131, T133, T117, ackermannc57_out_gga(T131, T133, T135)) → U27_GGG(T131, T133, T117, ackermann68_in_gga(T131, T135, X341))
U26_GGG(T131, T133, T117, ackermannc57_out_gga(T131, T133, T135)) → ACKERMANN68_IN_GGA(T131, T135, X341)
U26_GGG(T131, T133, T117, ackermannc57_out_gga(T131, T133, T135)) → U28_GGG(T131, T133, T117, ackermannc68_in_gga(T131, T135, T139))
U28_GGG(T131, T133, T117, ackermannc68_out_gga(T131, T135, T139)) → U29_GGG(T131, T133, T117, ackermann1_in_ggg(T131, T139, T117))
U28_GGG(T131, T133, T117, ackermannc68_out_gga(T131, T135, T139)) → ACKERMANN1_IN_GGG(T131, T139, T117)
ACKERMANN1_IN_GAG(s(T68), s(T71), T70) → U19_GAG(T68, T71, T70, ackermann57_in_gaa(T68, T71, X182))
ACKERMANN1_IN_GAG(s(T68), s(T71), T70) → ACKERMANN57_IN_GAA(T68, T71, X182)
ACKERMANN57_IN_GAA(T79, 0, X211) → U9_GAA(T79, X211, ackermann27_in_ga(T79, X211))
ACKERMANN57_IN_GAA(T79, 0, X211) → ACKERMANN27_IN_GA(T79, X211)
ACKERMANN57_IN_GAA(T84, s(T86), X229) → U10_GAA(T84, T86, X229, p65_in_gaaa(T84, T86, X228, X229))
ACKERMANN57_IN_GAA(T84, s(T86), X229) → P65_IN_GAAA(T84, T86, X228, X229)
P65_IN_GAAA(T84, T86, X228, X229) → U11_GAAA(T84, T86, X228, X229, ackermann57_in_gaa(T84, T86, X228))
P65_IN_GAAA(T84, T86, X228, X229) → ACKERMANN57_IN_GAA(T84, T86, X228)
P65_IN_GAAA(T84, T86, T88, X229) → U12_GAAA(T84, T86, T88, X229, ackermannc57_in_gaa(T84, T86, T88))
U12_GAAA(T84, T86, T88, X229, ackermannc57_out_gaa(T84, T86, T88)) → U13_GAAA(T84, T86, T88, X229, ackermann68_in_gga(T84, T88, X229))
U12_GAAA(T84, T86, T88, X229, ackermannc57_out_gaa(T84, T86, T88)) → ACKERMANN68_IN_GGA(T84, T88, X229)
ACKERMANN1_IN_GAG(s(T68), s(T71), T70) → U20_GAG(T68, T71, T70, ackermannc57_in_gaa(T68, T71, T73))
U20_GAG(T68, T71, T70, ackermannc57_out_gaa(T68, T71, T73)) → U21_GAG(T68, T71, T70, ackermann1_in_ggg(T68, T73, T70))
U20_GAG(T68, T71, T70, ackermannc57_out_gaa(T68, T71, T73)) → ACKERMANN1_IN_GGG(T68, T73, T70)
ACKERMANN1_IN_GAG(s(T123), s(0), T117) → U22_GAG(T123, T117, ackermann27_in_ga(T123, X319))
ACKERMANN1_IN_GAG(s(T123), s(0), T117) → ACKERMANN27_IN_GA(T123, X319)
ACKERMANN1_IN_GAG(s(T123), s(0), T117) → U23_GAG(T123, T117, ackermannc27_in_ga(T123, T124))
U23_GAG(T123, T117, ackermannc27_out_ga(T123, T124)) → U24_GAG(T123, T117, ackermann1_in_ggg(T123, T124, T117))
U23_GAG(T123, T117, ackermannc27_out_ga(T123, T124)) → ACKERMANN1_IN_GGG(T123, T124, T117)
ACKERMANN1_IN_GAG(s(T131), s(s(T133)), T117) → U25_GAG(T131, T133, T117, ackermann57_in_gaa(T131, T133, X340))
ACKERMANN1_IN_GAG(s(T131), s(s(T133)), T117) → ACKERMANN57_IN_GAA(T131, T133, X340)
ACKERMANN1_IN_GAG(s(T131), s(s(T133)), T117) → U26_GAG(T131, T133, T117, ackermannc57_in_gaa(T131, T133, T135))
U26_GAG(T131, T133, T117, ackermannc57_out_gaa(T131, T133, T135)) → U27_GAG(T131, T133, T117, ackermann68_in_gga(T131, T135, X341))
U26_GAG(T131, T133, T117, ackermannc57_out_gaa(T131, T133, T135)) → ACKERMANN68_IN_GGA(T131, T135, X341)
U26_GAG(T131, T133, T117, ackermannc57_out_gaa(T131, T133, T135)) → U28_GAG(T131, T133, T117, ackermannc68_in_gga(T131, T135, T139))
U28_GAG(T131, T133, T117, ackermannc68_out_gga(T131, T135, T139)) → U29_GAG(T131, T133, T117, ackermann1_in_ggg(T131, T139, T117))
U28_GAG(T131, T133, T117, ackermannc68_out_gga(T131, T135, T139)) → ACKERMANN1_IN_GGG(T131, T139, T117)

The TRS R consists of the following rules:

ackermannc21_in_ga(T28, X73) → U40_ga(T28, X73, ackermannc27_in_ga(T28, X73))
ackermannc27_in_ga(0, s(s(0))) → ackermannc27_out_ga(0, s(s(0)))
ackermannc27_in_ga(s(T32), X97) → U41_ga(T32, X97, ackermannc21_in_ga(T32, T34))
U41_ga(T32, X97, ackermannc21_out_ga(T32, T34)) → U42_ga(T32, X97, ackermannc38_in_gga(T32, T34, X97))
ackermannc38_in_gga(0, T42, s(T42)) → ackermannc38_out_gga(0, T42, s(T42))
ackermannc38_in_gga(s(T47), 0, X133) → U43_gga(T47, X133, ackermannc27_in_ga(T47, X133))
U43_gga(T47, X133, ackermannc27_out_ga(T47, X133)) → ackermannc38_out_gga(s(T47), 0, X133)
ackermannc38_in_gga(s(T52), s(T53), X151) → U44_gga(T52, T53, X151, ackermannc38_in_gga(s(T52), T53, T55))
U44_gga(T52, T53, X151, ackermannc38_out_gga(s(T52), T53, T55)) → U45_gga(T52, T53, X151, ackermannc38_in_gga(T52, T55, X151))
U45_gga(T52, T53, X151, ackermannc38_out_gga(T52, T55, X151)) → ackermannc38_out_gga(s(T52), s(T53), X151)
U42_ga(T32, X97, ackermannc38_out_gga(T32, T34, X97)) → ackermannc27_out_ga(s(T32), X97)
U40_ga(T28, X73, ackermannc27_out_ga(T28, X73)) → ackermannc21_out_ga(T28, X73)
ackermannc57_in_gga(T79, 0, X211) → U46_gga(T79, X211, ackermannc27_in_ga(T79, X211))
U46_gga(T79, X211, ackermannc27_out_ga(T79, X211)) → ackermannc57_out_gga(T79, 0, X211)
ackermannc57_in_gga(T84, s(T86), X229) → U47_gga(T84, T86, X229, qc65_in_ggaa(T84, T86, X228, X229))
qc65_in_ggaa(T84, T86, T88, X229) → U48_ggaa(T84, T86, T88, X229, ackermannc57_in_gga(T84, T86, T88))
U48_ggaa(T84, T86, T88, X229, ackermannc57_out_gga(T84, T86, T88)) → U49_ggaa(T84, T86, T88, X229, ackermannc68_in_gga(T84, T88, X229))
ackermannc68_in_gga(0, T96, s(T96)) → ackermannc68_out_gga(0, T96, s(T96))
ackermannc68_in_gga(s(T101), 0, X265) → U50_gga(T101, X265, ackermannc27_in_ga(T101, X265))
U50_gga(T101, X265, ackermannc27_out_ga(T101, X265)) → ackermannc68_out_gga(s(T101), 0, X265)
ackermannc68_in_gga(s(T106), s(T108), X283) → U51_gga(T106, T108, X283, qc65_in_ggaa(T106, T108, X282, X283))
U51_gga(T106, T108, X283, qc65_out_ggaa(T106, T108, X282, X283)) → ackermannc68_out_gga(s(T106), s(T108), X283)
U49_ggaa(T84, T86, T88, X229, ackermannc68_out_gga(T84, T88, X229)) → qc65_out_ggaa(T84, T86, T88, X229)
U47_gga(T84, T86, X229, qc65_out_ggaa(T84, T86, X228, X229)) → ackermannc57_out_gga(T84, s(T86), X229)
ackermannc57_in_gaa(T79, 0, X211) → U46_gaa(T79, X211, ackermannc27_in_ga(T79, X211))
U46_gaa(T79, X211, ackermannc27_out_ga(T79, X211)) → ackermannc57_out_gaa(T79, 0, X211)
ackermannc57_in_gaa(T84, s(T86), X229) → U47_gaa(T84, T86, X229, qc65_in_gaaa(T84, T86, X228, X229))
qc65_in_gaaa(T84, T86, T88, X229) → U48_gaaa(T84, T86, T88, X229, ackermannc57_in_gaa(T84, T86, T88))
U48_gaaa(T84, T86, T88, X229, ackermannc57_out_gaa(T84, T86, T88)) → U49_gaaa(T84, T86, T88, X229, ackermannc68_in_gga(T84, T88, X229))
U49_gaaa(T84, T86, T88, X229, ackermannc68_out_gga(T84, T88, X229)) → qc65_out_gaaa(T84, T86, T88, X229)
U47_gaa(T84, T86, X229, qc65_out_gaaa(T84, T86, X228, X229)) → ackermannc57_out_gaa(T84, s(T86), X229)

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
ackermann21_in_ga(x1, x2) = ackermann21_in_ga(x1)
ackermann27_in_ga(x1, x2) = ackermann27_in_ga(x1)
ackermannc21_in_ga(x1, x2) = ackermannc21_in_ga(x1)
U40_ga(x1, x2, x3) = U40_ga(x1, x3)
ackermannc27_in_ga(x1, x2) = ackermannc27_in_ga(x1)
0 = 0
ackermannc27_out_ga(x1, x2) = ackermannc27_out_ga(x1, x2)
U41_ga(x1, x2, x3) = U41_ga(x1, x3)
ackermannc21_out_ga(x1, x2) = ackermannc21_out_ga(x1, x2)
U42_ga(x1, x2, x3) = U42_ga(x1, x3)
ackermannc38_in_gga(x1, x2, x3) = ackermannc38_in_gga(x1, x2)
ackermannc38_out_gga(x1, x2, x3) = ackermannc38_out_gga(x1, x2, x3)
U43_gga(x1, x2, x3) = U43_gga(x1, x3)
U44_gga(x1, x2, x3, x4) = U44_gga(x1, x2, x4)
U45_gga(x1, x2, x3, x4) = U45_gga(x1, x2, x4)
ackermann38_in_gga(x1, x2, x3) = ackermann38_in_gga(x1, x2)
ackermann1_in_ggg(x1, x2, x3) = ackermann1_in_ggg(x1, x2, x3)
ackermann57_in_gga(x1, x2, x3) = ackermann57_in_gga(x1, x2)
p65_in_ggaa(x1, x2, x3, x4) = p65_in_ggaa(x1, x2)
ackermannc57_in_gga(x1, x2, x3) = ackermannc57_in_gga(x1, x2)
U46_gga(x1, x2, x3) = U46_gga(x1, x3)
ackermannc57_out_gga(x1, x2, x3) = ackermannc57_out_gga(x1, x2, x3)
U47_gga(x1, x2, x3, x4) = U47_gga(x1, x2, x4)
qc65_in_ggaa(x1, x2, x3, x4) = qc65_in_ggaa(x1, x2)
U48_ggaa(x1, x2, x3, x4, x5) = U48_ggaa(x1, x2, x5)
U49_ggaa(x1, x2, x3, x4, x5) = U49_ggaa(x1, x2, x3, x5)
ackermannc68_in_gga(x1, x2, x3) = ackermannc68_in_gga(x1, x2)
ackermannc68_out_gga(x1, x2, x3) = ackermannc68_out_gga(x1, x2, x3)
U50_gga(x1, x2, x3) = U50_gga(x1, x3)
U51_gga(x1, x2, x3, x4) = U51_gga(x1, x2, x4)
qc65_out_ggaa(x1, x2, x3, x4) = qc65_out_ggaa(x1, x2, x3, x4)
ackermann68_in_gga(x1, x2, x3) = ackermann68_in_gga(x1, x2)
ackermann57_in_gaa(x1, x2, x3) = ackermann57_in_gaa(x1)
p65_in_gaaa(x1, x2, x3, x4) = p65_in_gaaa(x1)
ackermannc57_in_gaa(x1, x2, x3) = ackermannc57_in_gaa(x1)
U46_gaa(x1, x2, x3) = U46_gaa(x1, x3)
ackermannc57_out_gaa(x1, x2, x3) = ackermannc57_out_gaa(x1, x2, x3)
U47_gaa(x1, x2, x3, x4) = U47_gaa(x1, x4)
qc65_in_gaaa(x1, x2, x3, x4) = qc65_in_gaaa(x1)
U48_gaaa(x1, x2, x3, x4, x5) = U48_gaaa(x1, x5)
U49_gaaa(x1, x2, x3, x4, x5) = U49_gaaa(x1, x2, x3, x5)
qc65_out_gaaa(x1, x2, x3, x4) = qc65_out_gaaa(x1, x2, x3, x4)
ACKERMANN1_IN_GAG(x1, x2, x3) = ACKERMANN1_IN_GAG(x1, x3)
U16_GAG(x1, x2, x3) = U16_GAG(x1, x2, x3)
ACKERMANN21_IN_GA(x1, x2) = ACKERMANN21_IN_GA(x1)
U1_GA(x1, x2, x3) = U1_GA(x1, x3)
ACKERMANN27_IN_GA(x1, x2) = ACKERMANN27_IN_GA(x1)
U2_GA(x1, x2, x3) = U2_GA(x1, x3)
U3_GA(x1, x2, x3) = U3_GA(x1, x3)
U4_GA(x1, x2, x3) = U4_GA(x1, x3)
ACKERMANN38_IN_GGA(x1, x2, x3) = ACKERMANN38_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3) = U5_GGA(x1, x3)
U6_GGA(x1, x2, x3, x4) = U6_GGA(x1, x2, x4)
U7_GGA(x1, x2, x3, x4) = U7_GGA(x1, x2, x4)
U8_GGA(x1, x2, x3, x4) = U8_GGA(x1, x2, x4)
U17_GAG(x1, x2, x3) = U17_GAG(x1, x2, x3)
U18_GAG(x1, x2, x3) = U18_GAG(x1, x2, x3)
ACKERMANN1_IN_GGG(x1, x2, x3) = ACKERMANN1_IN_GGG(x1, x2, x3)
U16_GGG(x1, x2, x3) = U16_GGG(x1, x2, x3)
U17_GGG(x1, x2, x3) = U17_GGG(x1, x2, x3)
U18_GGG(x1, x2, x3) = U18_GGG(x1, x2, x3)
U19_GGG(x1, x2, x3, x4) = U19_GGG(x1, x2, x3, x4)
ACKERMANN57_IN_GGA(x1, x2, x3) = ACKERMANN57_IN_GGA(x1, x2)
U9_GGA(x1, x2, x3) = U9_GGA(x1, x3)
U10_GGA(x1, x2, x3, x4) = U10_GGA(x1, x2, x4)
P65_IN_GGAA(x1, x2, x3, x4) = P65_IN_GGAA(x1, x2)
U11_GGAA(x1, x2, x3, x4, x5) = U11_GGAA(x1, x2, x5)
U12_GGAA(x1, x2, x3, x4, x5) = U12_GGAA(x1, x2, x5)
U13_GGAA(x1, x2, x3, x4, x5) = U13_GGAA(x1, x2, x5)
ACKERMANN68_IN_GGA(x1, x2, x3) = ACKERMANN68_IN_GGA(x1, x2)
U14_GGA(x1, x2, x3) = U14_GGA(x1, x3)
U15_GGA(x1, x2, x3, x4) = U15_GGA(x1, x2, x4)
U20_GGG(x1, x2, x3, x4) = U20_GGG(x1, x2, x3, x4)
U21_GGG(x1, x2, x3, x4) = U21_GGG(x1, x2, x3, x4)
U22_GGG(x1, x2, x3) = U22_GGG(x1, x2, x3)
U23_GGG(x1, x2, x3) = U23_GGG(x1, x2, x3)
U24_GGG(x1, x2, x3) = U24_GGG(x1, x2, x3)
U25_GGG(x1, x2, x3, x4) = U25_GGG(x1, x2, x3, x4)
U26_GGG(x1, x2, x3, x4) = U26_GGG(x1, x2, x3, x4)
U27_GGG(x1, x2, x3, x4) = U27_GGG(x1, x2, x3, x4)
U28_GGG(x1, x2, x3, x4) = U28_GGG(x1, x2, x3, x4)
U29_GGG(x1, x2, x3, x4) = U29_GGG(x1, x2, x3, x4)
U19_GAG(x1, x2, x3, x4) = U19_GAG(x1, x3, x4)
ACKERMANN57_IN_GAA(x1, x2, x3) = ACKERMANN57_IN_GAA(x1)
U9_GAA(x1, x2, x3) = U9_GAA(x1, x3)
U10_GAA(x1, x2, x3, x4) = U10_GAA(x1, x4)
P65_IN_GAAA(x1, x2, x3, x4) = P65_IN_GAAA(x1)
U11_GAAA(x1, x2, x3, x4, x5) = U11_GAAA(x1, x5)
U12_GAAA(x1, x2, x3, x4, x5) = U12_GAAA(x1, x5)
U13_GAAA(x1, x2, x3, x4, x5) = U13_GAAA(x1, x2, x5)
U20_GAG(x1, x2, x3, x4) = U20_GAG(x1, x3, x4)
U21_GAG(x1, x2, x3, x4) = U21_GAG(x1, x2, x3, x4)
U22_GAG(x1, x2, x3) = U22_GAG(x1, x2, x3)
U23_GAG(x1, x2, x3) = U23_GAG(x1, x2, x3)
U24_GAG(x1, x2, x3) = U24_GAG(x1, x2, x3)
U25_GAG(x1, x2, x3, x4) = U25_GAG(x1, x3, x4)
U26_GAG(x1, x2, x3, x4) = U26_GAG(x1, x3, x4)
U27_GAG(x1, x2, x3, x4) = U27_GAG(x1, x2, x3, x4)
U28_GAG(x1, x2, x3, x4) = U28_GAG(x1, x2, x3, x4)
U29_GAG(x1, x2, x3, x4) = U29_GAG(x1, x2, x3, 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:

ACKERMANN1_IN_GAG(s(s(T19)), 0, T20) → U16_GAG(T19, T20, ackermann21_in_ga(T19, X40))
ACKERMANN1_IN_GAG(s(s(T19)), 0, T20) → ACKERMANN21_IN_GA(T19, X40)
ACKERMANN21_IN_GA(T28, X73) → U1_GA(T28, X73, ackermann27_in_ga(T28, X73))
ACKERMANN21_IN_GA(T28, X73) → ACKERMANN27_IN_GA(T28, X73)
ACKERMANN27_IN_GA(s(T32), X97) → U2_GA(T32, X97, ackermann21_in_ga(T32, X96))
ACKERMANN27_IN_GA(s(T32), X97) → ACKERMANN21_IN_GA(T32, X96)
ACKERMANN27_IN_GA(s(T32), X97) → U3_GA(T32, X97, ackermannc21_in_ga(T32, T34))
U3_GA(T32, X97, ackermannc21_out_ga(T32, T34)) → U4_GA(T32, X97, ackermann38_in_gga(T32, T34, X97))
U3_GA(T32, X97, ackermannc21_out_ga(T32, T34)) → ACKERMANN38_IN_GGA(T32, T34, X97)
ACKERMANN38_IN_GGA(s(T47), 0, X133) → U5_GGA(T47, X133, ackermann27_in_ga(T47, X133))
ACKERMANN38_IN_GGA(s(T47), 0, X133) → ACKERMANN27_IN_GA(T47, X133)
ACKERMANN38_IN_GGA(s(T52), s(T53), X151) → U6_GGA(T52, T53, X151, ackermann38_in_gga(s(T52), T53, X150))
ACKERMANN38_IN_GGA(s(T52), s(T53), X151) → ACKERMANN38_IN_GGA(s(T52), T53, X150)
ACKERMANN38_IN_GGA(s(T52), s(T53), X151) → U7_GGA(T52, T53, X151, ackermannc38_in_gga(s(T52), T53, T55))
U7_GGA(T52, T53, X151, ackermannc38_out_gga(s(T52), T53, T55)) → U8_GGA(T52, T53, X151, ackermann38_in_gga(T52, T55, X151))
U7_GGA(T52, T53, X151, ackermannc38_out_gga(s(T52), T53, T55)) → ACKERMANN38_IN_GGA(T52, T55, X151)
ACKERMANN1_IN_GAG(s(s(T19)), 0, T20) → U17_GAG(T19, T20, ackermannc21_in_ga(T19, T22))
U17_GAG(T19, T20, ackermannc21_out_ga(T19, T22)) → U18_GAG(T19, T20, ackermann1_in_ggg(T19, T22, T20))
U17_GAG(T19, T20, ackermannc21_out_ga(T19, T22)) → ACKERMANN1_IN_GGG(T19, T22, T20)
ACKERMANN1_IN_GGG(s(s(T19)), 0, T20) → U16_GGG(T19, T20, ackermann21_in_ga(T19, X40))
ACKERMANN1_IN_GGG(s(s(T19)), 0, T20) → ACKERMANN21_IN_GA(T19, X40)
ACKERMANN1_IN_GGG(s(s(T19)), 0, T20) → U17_GGG(T19, T20, ackermannc21_in_ga(T19, T22))
U17_GGG(T19, T20, ackermannc21_out_ga(T19, T22)) → U18_GGG(T19, T20, ackermann1_in_ggg(T19, T22, T20))
U17_GGG(T19, T20, ackermannc21_out_ga(T19, T22)) → ACKERMANN1_IN_GGG(T19, T22, T20)
ACKERMANN1_IN_GGG(s(T68), s(T71), T70) → U19_GGG(T68, T71, T70, ackermann57_in_gga(T68, T71, X182))
ACKERMANN1_IN_GGG(s(T68), s(T71), T70) → ACKERMANN57_IN_GGA(T68, T71, X182)
ACKERMANN57_IN_GGA(T79, 0, X211) → U9_GGA(T79, X211, ackermann27_in_ga(T79, X211))
ACKERMANN57_IN_GGA(T79, 0, X211) → ACKERMANN27_IN_GA(T79, X211)
ACKERMANN57_IN_GGA(T84, s(T86), X229) → U10_GGA(T84, T86, X229, p65_in_ggaa(T84, T86, X228, X229))
ACKERMANN57_IN_GGA(T84, s(T86), X229) → P65_IN_GGAA(T84, T86, X228, X229)
P65_IN_GGAA(T84, T86, X228, X229) → U11_GGAA(T84, T86, X228, X229, ackermann57_in_gga(T84, T86, X228))
P65_IN_GGAA(T84, T86, X228, X229) → ACKERMANN57_IN_GGA(T84, T86, X228)
P65_IN_GGAA(T84, T86, T88, X229) → U12_GGAA(T84, T86, T88, X229, ackermannc57_in_gga(T84, T86, T88))
U12_GGAA(T84, T86, T88, X229, ackermannc57_out_gga(T84, T86, T88)) → U13_GGAA(T84, T86, T88, X229, ackermann68_in_gga(T84, T88, X229))
U12_GGAA(T84, T86, T88, X229, ackermannc57_out_gga(T84, T86, T88)) → ACKERMANN68_IN_GGA(T84, T88, X229)
ACKERMANN68_IN_GGA(s(T101), 0, X265) → U14_GGA(T101, X265, ackermann27_in_ga(T101, X265))
ACKERMANN68_IN_GGA(s(T101), 0, X265) → ACKERMANN27_IN_GA(T101, X265)
ACKERMANN68_IN_GGA(s(T106), s(T108), X283) → U15_GGA(T106, T108, X283, p65_in_ggaa(T106, T108, X282, X283))
ACKERMANN68_IN_GGA(s(T106), s(T108), X283) → P65_IN_GGAA(T106, T108, X282, X283)
ACKERMANN1_IN_GGG(s(T68), s(T71), T70) → U20_GGG(T68, T71, T70, ackermannc57_in_gga(T68, T71, T73))
U20_GGG(T68, T71, T70, ackermannc57_out_gga(T68, T71, T73)) → U21_GGG(T68, T71, T70, ackermann1_in_ggg(T68, T73, T70))
U20_GGG(T68, T71, T70, ackermannc57_out_gga(T68, T71, T73)) → ACKERMANN1_IN_GGG(T68, T73, T70)
ACKERMANN1_IN_GGG(s(T123), s(0), T117) → U22_GGG(T123, T117, ackermann27_in_ga(T123, X319))
ACKERMANN1_IN_GGG(s(T123), s(0), T117) → ACKERMANN27_IN_GA(T123, X319)
ACKERMANN1_IN_GGG(s(T123), s(0), T117) → U23_GGG(T123, T117, ackermannc27_in_ga(T123, T124))
U23_GGG(T123, T117, ackermannc27_out_ga(T123, T124)) → U24_GGG(T123, T117, ackermann1_in_ggg(T123, T124, T117))
U23_GGG(T123, T117, ackermannc27_out_ga(T123, T124)) → ACKERMANN1_IN_GGG(T123, T124, T117)
ACKERMANN1_IN_GGG(s(T131), s(s(T133)), T117) → U25_GGG(T131, T133, T117, ackermann57_in_gga(T131, T133, X340))
ACKERMANN1_IN_GGG(s(T131), s(s(T133)), T117) → ACKERMANN57_IN_GGA(T131, T133, X340)
ACKERMANN1_IN_GGG(s(T131), s(s(T133)), T117) → U26_GGG(T131, T133, T117, ackermannc57_in_gga(T131, T133, T135))
U26_GGG(T131, T133, T117, ackermannc57_out_gga(T131, T133, T135)) → U27_GGG(T131, T133, T117, ackermann68_in_gga(T131, T135, X341))
U26_GGG(T131, T133, T117, ackermannc57_out_gga(T131, T133, T135)) → ACKERMANN68_IN_GGA(T131, T135, X341)
U26_GGG(T131, T133, T117, ackermannc57_out_gga(T131, T133, T135)) → U28_GGG(T131, T133, T117, ackermannc68_in_gga(T131, T135, T139))
U28_GGG(T131, T133, T117, ackermannc68_out_gga(T131, T135, T139)) → U29_GGG(T131, T133, T117, ackermann1_in_ggg(T131, T139, T117))
U28_GGG(T131, T133, T117, ackermannc68_out_gga(T131, T135, T139)) → ACKERMANN1_IN_GGG(T131, T139, T117)
ACKERMANN1_IN_GAG(s(T68), s(T71), T70) → U19_GAG(T68, T71, T70, ackermann57_in_gaa(T68, T71, X182))
ACKERMANN1_IN_GAG(s(T68), s(T71), T70) → ACKERMANN57_IN_GAA(T68, T71, X182)
ACKERMANN57_IN_GAA(T79, 0, X211) → U9_GAA(T79, X211, ackermann27_in_ga(T79, X211))
ACKERMANN57_IN_GAA(T79, 0, X211) → ACKERMANN27_IN_GA(T79, X211)
ACKERMANN57_IN_GAA(T84, s(T86), X229) → U10_GAA(T84, T86, X229, p65_in_gaaa(T84, T86, X228, X229))
ACKERMANN57_IN_GAA(T84, s(T86), X229) → P65_IN_GAAA(T84, T86, X228, X229)
P65_IN_GAAA(T84, T86, X228, X229) → U11_GAAA(T84, T86, X228, X229, ackermann57_in_gaa(T84, T86, X228))
P65_IN_GAAA(T84, T86, X228, X229) → ACKERMANN57_IN_GAA(T84, T86, X228)
P65_IN_GAAA(T84, T86, T88, X229) → U12_GAAA(T84, T86, T88, X229, ackermannc57_in_gaa(T84, T86, T88))
U12_GAAA(T84, T86, T88, X229, ackermannc57_out_gaa(T84, T86, T88)) → U13_GAAA(T84, T86, T88, X229, ackermann68_in_gga(T84, T88, X229))
U12_GAAA(T84, T86, T88, X229, ackermannc57_out_gaa(T84, T86, T88)) → ACKERMANN68_IN_GGA(T84, T88, X229)
ACKERMANN1_IN_GAG(s(T68), s(T71), T70) → U20_GAG(T68, T71, T70, ackermannc57_in_gaa(T68, T71, T73))
U20_GAG(T68, T71, T70, ackermannc57_out_gaa(T68, T71, T73)) → U21_GAG(T68, T71, T70, ackermann1_in_ggg(T68, T73, T70))
U20_GAG(T68, T71, T70, ackermannc57_out_gaa(T68, T71, T73)) → ACKERMANN1_IN_GGG(T68, T73, T70)
ACKERMANN1_IN_GAG(s(T123), s(0), T117) → U22_GAG(T123, T117, ackermann27_in_ga(T123, X319))
ACKERMANN1_IN_GAG(s(T123), s(0), T117) → ACKERMANN27_IN_GA(T123, X319)
ACKERMANN1_IN_GAG(s(T123), s(0), T117) → U23_GAG(T123, T117, ackermannc27_in_ga(T123, T124))
U23_GAG(T123, T117, ackermannc27_out_ga(T123, T124)) → U24_GAG(T123, T117, ackermann1_in_ggg(T123, T124, T117))
U23_GAG(T123, T117, ackermannc27_out_ga(T123, T124)) → ACKERMANN1_IN_GGG(T123, T124, T117)
ACKERMANN1_IN_GAG(s(T131), s(s(T133)), T117) → U25_GAG(T131, T133, T117, ackermann57_in_gaa(T131, T133, X340))
ACKERMANN1_IN_GAG(s(T131), s(s(T133)), T117) → ACKERMANN57_IN_GAA(T131, T133, X340)
ACKERMANN1_IN_GAG(s(T131), s(s(T133)), T117) → U26_GAG(T131, T133, T117, ackermannc57_in_gaa(T131, T133, T135))
U26_GAG(T131, T133, T117, ackermannc57_out_gaa(T131, T133, T135)) → U27_GAG(T131, T133, T117, ackermann68_in_gga(T131, T135, X341))
U26_GAG(T131, T133, T117, ackermannc57_out_gaa(T131, T133, T135)) → ACKERMANN68_IN_GGA(T131, T135, X341)
U26_GAG(T131, T133, T117, ackermannc57_out_gaa(T131, T133, T135)) → U28_GAG(T131, T133, T117, ackermannc68_in_gga(T131, T135, T139))
U28_GAG(T131, T133, T117, ackermannc68_out_gga(T131, T135, T139)) → U29_GAG(T131, T133, T117, ackermann1_in_ggg(T131, T139, T117))
U28_GAG(T131, T133, T117, ackermannc68_out_gga(T131, T135, T139)) → ACKERMANN1_IN_GGG(T131, T139, T117)

The TRS R consists of the following rules:

ackermannc21_in_ga(T28, X73) → U40_ga(T28, X73, ackermannc27_in_ga(T28, X73))
ackermannc27_in_ga(0, s(s(0))) → ackermannc27_out_ga(0, s(s(0)))
ackermannc27_in_ga(s(T32), X97) → U41_ga(T32, X97, ackermannc21_in_ga(T32, T34))
U41_ga(T32, X97, ackermannc21_out_ga(T32, T34)) → U42_ga(T32, X97, ackermannc38_in_gga(T32, T34, X97))
ackermannc38_in_gga(0, T42, s(T42)) → ackermannc38_out_gga(0, T42, s(T42))
ackermannc38_in_gga(s(T47), 0, X133) → U43_gga(T47, X133, ackermannc27_in_ga(T47, X133))
U43_gga(T47, X133, ackermannc27_out_ga(T47, X133)) → ackermannc38_out_gga(s(T47), 0, X133)
ackermannc38_in_gga(s(T52), s(T53), X151) → U44_gga(T52, T53, X151, ackermannc38_in_gga(s(T52), T53, T55))
U44_gga(T52, T53, X151, ackermannc38_out_gga(s(T52), T53, T55)) → U45_gga(T52, T53, X151, ackermannc38_in_gga(T52, T55, X151))
U45_gga(T52, T53, X151, ackermannc38_out_gga(T52, T55, X151)) → ackermannc38_out_gga(s(T52), s(T53), X151)
U42_ga(T32, X97, ackermannc38_out_gga(T32, T34, X97)) → ackermannc27_out_ga(s(T32), X97)
U40_ga(T28, X73, ackermannc27_out_ga(T28, X73)) → ackermannc21_out_ga(T28, X73)
ackermannc57_in_gga(T79, 0, X211) → U46_gga(T79, X211, ackermannc27_in_ga(T79, X211))
U46_gga(T79, X211, ackermannc27_out_ga(T79, X211)) → ackermannc57_out_gga(T79, 0, X211)
ackermannc57_in_gga(T84, s(T86), X229) → U47_gga(T84, T86, X229, qc65_in_ggaa(T84, T86, X228, X229))
qc65_in_ggaa(T84, T86, T88, X229) → U48_ggaa(T84, T86, T88, X229, ackermannc57_in_gga(T84, T86, T88))
U48_ggaa(T84, T86, T88, X229, ackermannc57_out_gga(T84, T86, T88)) → U49_ggaa(T84, T86, T88, X229, ackermannc68_in_gga(T84, T88, X229))
ackermannc68_in_gga(0, T96, s(T96)) → ackermannc68_out_gga(0, T96, s(T96))
ackermannc68_in_gga(s(T101), 0, X265) → U50_gga(T101, X265, ackermannc27_in_ga(T101, X265))
U50_gga(T101, X265, ackermannc27_out_ga(T101, X265)) → ackermannc68_out_gga(s(T101), 0, X265)
ackermannc68_in_gga(s(T106), s(T108), X283) → U51_gga(T106, T108, X283, qc65_in_ggaa(T106, T108, X282, X283))
U51_gga(T106, T108, X283, qc65_out_ggaa(T106, T108, X282, X283)) → ackermannc68_out_gga(s(T106), s(T108), X283)
U49_ggaa(T84, T86, T88, X229, ackermannc68_out_gga(T84, T88, X229)) → qc65_out_ggaa(T84, T86, T88, X229)
U47_gga(T84, T86, X229, qc65_out_ggaa(T84, T86, X228, X229)) → ackermannc57_out_gga(T84, s(T86), X229)
ackermannc57_in_gaa(T79, 0, X211) → U46_gaa(T79, X211, ackermannc27_in_ga(T79, X211))
U46_gaa(T79, X211, ackermannc27_out_ga(T79, X211)) → ackermannc57_out_gaa(T79, 0, X211)
ackermannc57_in_gaa(T84, s(T86), X229) → U47_gaa(T84, T86, X229, qc65_in_gaaa(T84, T86, X228, X229))
qc65_in_gaaa(T84, T86, T88, X229) → U48_gaaa(T84, T86, T88, X229, ackermannc57_in_gaa(T84, T86, T88))
U48_gaaa(T84, T86, T88, X229, ackermannc57_out_gaa(T84, T86, T88)) → U49_gaaa(T84, T86, T88, X229, ackermannc68_in_gga(T84, T88, X229))
U49_gaaa(T84, T86, T88, X229, ackermannc68_out_gga(T84, T88, X229)) → qc65_out_gaaa(T84, T86, T88, X229)
U47_gaa(T84, T86, X229, qc65_out_gaaa(T84, T86, X228, X229)) → ackermannc57_out_gaa(T84, s(T86), X229)

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 58 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

ACKERMANN21_IN_GA(T28, X73) → ACKERMANN27_IN_GA(T28, X73)
ACKERMANN27_IN_GA(s(T32), X97) → ACKERMANN21_IN_GA(T32, X96)
ACKERMANN27_IN_GA(s(T32), X97) → U3_GA(T32, X97, ackermannc21_in_ga(T32, T34))
U3_GA(T32, X97, ackermannc21_out_ga(T32, T34)) → ACKERMANN38_IN_GGA(T32, T34, X97)
ACKERMANN38_IN_GGA(s(T47), 0, X133) → ACKERMANN27_IN_GA(T47, X133)
ACKERMANN38_IN_GGA(s(T52), s(T53), X151) → ACKERMANN38_IN_GGA(s(T52), T53, X150)
ACKERMANN38_IN_GGA(s(T52), s(T53), X151) → U7_GGA(T52, T53, X151, ackermannc38_in_gga(s(T52), T53, T55))
U7_GGA(T52, T53, X151, ackermannc38_out_gga(s(T52), T53, T55)) → ACKERMANN38_IN_GGA(T52, T55, X151)

The TRS R consists of the following rules:

ackermannc21_in_ga(T28, X73) → U40_ga(T28, X73, ackermannc27_in_ga(T28, X73))
ackermannc27_in_ga(0, s(s(0))) → ackermannc27_out_ga(0, s(s(0)))
ackermannc27_in_ga(s(T32), X97) → U41_ga(T32, X97, ackermannc21_in_ga(T32, T34))
U41_ga(T32, X97, ackermannc21_out_ga(T32, T34)) → U42_ga(T32, X97, ackermannc38_in_gga(T32, T34, X97))
ackermannc38_in_gga(0, T42, s(T42)) → ackermannc38_out_gga(0, T42, s(T42))
ackermannc38_in_gga(s(T47), 0, X133) → U43_gga(T47, X133, ackermannc27_in_ga(T47, X133))
U43_gga(T47, X133, ackermannc27_out_ga(T47, X133)) → ackermannc38_out_gga(s(T47), 0, X133)
ackermannc38_in_gga(s(T52), s(T53), X151) → U44_gga(T52, T53, X151, ackermannc38_in_gga(s(T52), T53, T55))
U44_gga(T52, T53, X151, ackermannc38_out_gga(s(T52), T53, T55)) → U45_gga(T52, T53, X151, ackermannc38_in_gga(T52, T55, X151))
U45_gga(T52, T53, X151, ackermannc38_out_gga(T52, T55, X151)) → ackermannc38_out_gga(s(T52), s(T53), X151)
U42_ga(T32, X97, ackermannc38_out_gga(T32, T34, X97)) → ackermannc27_out_ga(s(T32), X97)
U40_ga(T28, X73, ackermannc27_out_ga(T28, X73)) → ackermannc21_out_ga(T28, X73)
ackermannc57_in_gga(T79, 0, X211) → U46_gga(T79, X211, ackermannc27_in_ga(T79, X211))
U46_gga(T79, X211, ackermannc27_out_ga(T79, X211)) → ackermannc57_out_gga(T79, 0, X211)
ackermannc57_in_gga(T84, s(T86), X229) → U47_gga(T84, T86, X229, qc65_in_ggaa(T84, T86, X228, X229))
qc65_in_ggaa(T84, T86, T88, X229) → U48_ggaa(T84, T86, T88, X229, ackermannc57_in_gga(T84, T86, T88))
U48_ggaa(T84, T86, T88, X229, ackermannc57_out_gga(T84, T86, T88)) → U49_ggaa(T84, T86, T88, X229, ackermannc68_in_gga(T84, T88, X229))
ackermannc68_in_gga(0, T96, s(T96)) → ackermannc68_out_gga(0, T96, s(T96))
ackermannc68_in_gga(s(T101), 0, X265) → U50_gga(T101, X265, ackermannc27_in_ga(T101, X265))
U50_gga(T101, X265, ackermannc27_out_ga(T101, X265)) → ackermannc68_out_gga(s(T101), 0, X265)
ackermannc68_in_gga(s(T106), s(T108), X283) → U51_gga(T106, T108, X283, qc65_in_ggaa(T106, T108, X282, X283))
U51_gga(T106, T108, X283, qc65_out_ggaa(T106, T108, X282, X283)) → ackermannc68_out_gga(s(T106), s(T108), X283)
U49_ggaa(T84, T86, T88, X229, ackermannc68_out_gga(T84, T88, X229)) → qc65_out_ggaa(T84, T86, T88, X229)
U47_gga(T84, T86, X229, qc65_out_ggaa(T84, T86, X228, X229)) → ackermannc57_out_gga(T84, s(T86), X229)
ackermannc57_in_gaa(T79, 0, X211) → U46_gaa(T79, X211, ackermannc27_in_ga(T79, X211))
U46_gaa(T79, X211, ackermannc27_out_ga(T79, X211)) → ackermannc57_out_gaa(T79, 0, X211)
ackermannc57_in_gaa(T84, s(T86), X229) → U47_gaa(T84, T86, X229, qc65_in_gaaa(T84, T86, X228, X229))
qc65_in_gaaa(T84, T86, T88, X229) → U48_gaaa(T84, T86, T88, X229, ackermannc57_in_gaa(T84, T86, T88))
U48_gaaa(T84, T86, T88, X229, ackermannc57_out_gaa(T84, T86, T88)) → U49_gaaa(T84, T86, T88, X229, ackermannc68_in_gga(T84, T88, X229))
U49_gaaa(T84, T86, T88, X229, ackermannc68_out_gga(T84, T88, X229)) → qc65_out_gaaa(T84, T86, T88, X229)
U47_gaa(T84, T86, X229, qc65_out_gaaa(T84, T86, X228, X229)) → ackermannc57_out_gaa(T84, s(T86), X229)

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
ackermannc21_in_ga(x1, x2) = ackermannc21_in_ga(x1)
U40_ga(x1, x2, x3) = U40_ga(x1, x3)
ackermannc27_in_ga(x1, x2) = ackermannc27_in_ga(x1)
0 = 0
ackermannc27_out_ga(x1, x2) = ackermannc27_out_ga(x1, x2)
U41_ga(x1, x2, x3) = U41_ga(x1, x3)
ackermannc21_out_ga(x1, x2) = ackermannc21_out_ga(x1, x2)
U42_ga(x1, x2, x3) = U42_ga(x1, x3)
ackermannc38_in_gga(x1, x2, x3) = ackermannc38_in_gga(x1, x2)
ackermannc38_out_gga(x1, x2, x3) = ackermannc38_out_gga(x1, x2, x3)
U43_gga(x1, x2, x3) = U43_gga(x1, x3)
U44_gga(x1, x2, x3, x4) = U44_gga(x1, x2, x4)
U45_gga(x1, x2, x3, x4) = U45_gga(x1, x2, x4)
ackermannc57_in_gga(x1, x2, x3) = ackermannc57_in_gga(x1, x2)
U46_gga(x1, x2, x3) = U46_gga(x1, x3)
ackermannc57_out_gga(x1, x2, x3) = ackermannc57_out_gga(x1, x2, x3)
U47_gga(x1, x2, x3, x4) = U47_gga(x1, x2, x4)
qc65_in_ggaa(x1, x2, x3, x4) = qc65_in_ggaa(x1, x2)
U48_ggaa(x1, x2, x3, x4, x5) = U48_ggaa(x1, x2, x5)
U49_ggaa(x1, x2, x3, x4, x5) = U49_ggaa(x1, x2, x3, x5)
ackermannc68_in_gga(x1, x2, x3) = ackermannc68_in_gga(x1, x2)
ackermannc68_out_gga(x1, x2, x3) = ackermannc68_out_gga(x1, x2, x3)
U50_gga(x1, x2, x3) = U50_gga(x1, x3)
U51_gga(x1, x2, x3, x4) = U51_gga(x1, x2, x4)
qc65_out_ggaa(x1, x2, x3, x4) = qc65_out_ggaa(x1, x2, x3, x4)
ackermannc57_in_gaa(x1, x2, x3) = ackermannc57_in_gaa(x1)
U46_gaa(x1, x2, x3) = U46_gaa(x1, x3)
ackermannc57_out_gaa(x1, x2, x3) = ackermannc57_out_gaa(x1, x2, x3)
U47_gaa(x1, x2, x3, x4) = U47_gaa(x1, x4)
qc65_in_gaaa(x1, x2, x3, x4) = qc65_in_gaaa(x1)
U48_gaaa(x1, x2, x3, x4, x5) = U48_gaaa(x1, x5)
U49_gaaa(x1, x2, x3, x4, x5) = U49_gaaa(x1, x2, x3, x5)
qc65_out_gaaa(x1, x2, x3, x4) = qc65_out_gaaa(x1, x2, x3, x4)
ACKERMANN21_IN_GA(x1, x2) = ACKERMANN21_IN_GA(x1)
ACKERMANN27_IN_GA(x1, x2) = ACKERMANN27_IN_GA(x1)
U3_GA(x1, x2, x3) = U3_GA(x1, x3)
ACKERMANN38_IN_GGA(x1, x2, x3) = ACKERMANN38_IN_GGA(x1, x2)
U7_GGA(x1, x2, x3, x4) = U7_GGA(x1, x2, x4)

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:

ACKERMANN21_IN_GA(T28, X73) → ACKERMANN27_IN_GA(T28, X73)
ACKERMANN27_IN_GA(s(T32), X97) → ACKERMANN21_IN_GA(T32, X96)
ACKERMANN27_IN_GA(s(T32), X97) → U3_GA(T32, X97, ackermannc21_in_ga(T32, T34))
U3_GA(T32, X97, ackermannc21_out_ga(T32, T34)) → ACKERMANN38_IN_GGA(T32, T34, X97)
ACKERMANN38_IN_GGA(s(T47), 0, X133) → ACKERMANN27_IN_GA(T47, X133)
ACKERMANN38_IN_GGA(s(T52), s(T53), X151) → ACKERMANN38_IN_GGA(s(T52), T53, X150)
ACKERMANN38_IN_GGA(s(T52), s(T53), X151) → U7_GGA(T52, T53, X151, ackermannc38_in_gga(s(T52), T53, T55))
U7_GGA(T52, T53, X151, ackermannc38_out_gga(s(T52), T53, T55)) → ACKERMANN38_IN_GGA(T52, T55, X151)

The TRS R consists of the following rules:

ackermannc21_in_ga(T28, X73) → U40_ga(T28, X73, ackermannc27_in_ga(T28, X73))
ackermannc38_in_gga(s(T47), 0, X133) → U43_gga(T47, X133, ackermannc27_in_ga(T47, X133))
ackermannc38_in_gga(s(T52), s(T53), X151) → U44_gga(T52, T53, X151, ackermannc38_in_gga(s(T52), T53, T55))
U40_ga(T28, X73, ackermannc27_out_ga(T28, X73)) → ackermannc21_out_ga(T28, X73)
U43_gga(T47, X133, ackermannc27_out_ga(T47, X133)) → ackermannc38_out_gga(s(T47), 0, X133)
U44_gga(T52, T53, X151, ackermannc38_out_gga(s(T52), T53, T55)) → U45_gga(T52, T53, X151, ackermannc38_in_gga(T52, T55, X151))
ackermannc27_in_ga(0, s(s(0))) → ackermannc27_out_ga(0, s(s(0)))
ackermannc27_in_ga(s(T32), X97) → U41_ga(T32, X97, ackermannc21_in_ga(T32, T34))
U45_gga(T52, T53, X151, ackermannc38_out_gga(T52, T55, X151)) → ackermannc38_out_gga(s(T52), s(T53), X151)
U41_ga(T32, X97, ackermannc21_out_ga(T32, T34)) → U42_ga(T32, X97, ackermannc38_in_gga(T32, T34, X97))
ackermannc38_in_gga(0, T42, s(T42)) → ackermannc38_out_gga(0, T42, s(T42))
U42_ga(T32, X97, ackermannc38_out_gga(T32, T34, X97)) → ackermannc27_out_ga(s(T32), X97)

(10) PiDPToQDPProof (SOUND 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:

ACKERMANN21_IN_GA(T28) → ACKERMANN27_IN_GA(T28)
ACKERMANN27_IN_GA(s(T32)) → ACKERMANN21_IN_GA(T32)
ACKERMANN27_IN_GA(s(T32)) → U3_GA(T32, ackermannc21_in_ga(T32))
U3_GA(T32, ackermannc21_out_ga(T32, T34)) → ACKERMANN38_IN_GGA(T32, T34)
ACKERMANN38_IN_GGA(s(T47), 0) → ACKERMANN27_IN_GA(T47)
ACKERMANN38_IN_GGA(s(T52), s(T53)) → ACKERMANN38_IN_GGA(s(T52), T53)
ACKERMANN38_IN_GGA(s(T52), s(T53)) → U7_GGA(T52, T53, ackermannc38_in_gga(s(T52), T53))
U7_GGA(T52, T53, ackermannc38_out_gga(s(T52), T53, T55)) → ACKERMANN38_IN_GGA(T52, T55)

The TRS R consists of the following rules:

ackermannc21_in_ga(T28) → U40_ga(T28, ackermannc27_in_ga(T28))
ackermannc38_in_gga(s(T47), 0) → U43_gga(T47, ackermannc27_in_ga(T47))
ackermannc38_in_gga(s(T52), s(T53)) → U44_gga(T52, T53, ackermannc38_in_gga(s(T52), T53))
U40_ga(T28, ackermannc27_out_ga(T28, X73)) → ackermannc21_out_ga(T28, X73)
U43_gga(T47, ackermannc27_out_ga(T47, X133)) → ackermannc38_out_gga(s(T47), 0, X133)
U44_gga(T52, T53, ackermannc38_out_gga(s(T52), T53, T55)) → U45_gga(T52, T53, ackermannc38_in_gga(T52, T55))
ackermannc27_in_ga(0) → ackermannc27_out_ga(0, s(s(0)))
ackermannc27_in_ga(s(T32)) → U41_ga(T32, ackermannc21_in_ga(T32))
U45_gga(T52, T53, ackermannc38_out_gga(T52, T55, X151)) → ackermannc38_out_gga(s(T52), s(T53), X151)
U41_ga(T32, ackermannc21_out_ga(T32, T34)) → U42_ga(T32, ackermannc38_in_gga(T32, T34))
ackermannc38_in_gga(0, T42) → ackermannc38_out_gga(0, T42, s(T42))
U42_ga(T32, ackermannc38_out_gga(T32, T34, X97)) → ackermannc27_out_ga(s(T32), X97)

The set Q consists of the following terms:

ackermannc21_in_ga(x0)
ackermannc38_in_gga(x0, x1)
U40_ga(x0, x1)
U43_gga(x0, x1)
U44_gga(x0, x1, x2)
ackermannc27_in_ga(x0)
U45_gga(x0, x1, x2)
U41_ga(x0, x1)
U42_ga(x0, x1)

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:

ACKERMANN27_IN_GA(s(T32)) → ACKERMANN21_IN_GA(T32)
The graph contains the following edges 1 > 1
ACKERMANN27_IN_GA(s(T32)) → U3_GA(T32, ackermannc21_in_ga(T32))
The graph contains the following edges 1 > 1
ACKERMANN21_IN_GA(T28) → ACKERMANN27_IN_GA(T28)
The graph contains the following edges 1 >= 1
ACKERMANN38_IN_GGA(s(T47), 0) → ACKERMANN27_IN_GA(T47)
The graph contains the following edges 1 > 1
U3_GA(T32, ackermannc21_out_ga(T32, T34)) → ACKERMANN38_IN_GGA(T32, T34)
The graph contains the following edges 1 >= 1, 2 > 1, 2 > 2
U7_GGA(T52, T53, ackermannc38_out_gga(s(T52), T53, T55)) → ACKERMANN38_IN_GGA(T52, T55)
The graph contains the following edges 1 >= 1, 3 > 1, 3 > 2
ACKERMANN38_IN_GGA(s(T52), s(T53)) → ACKERMANN38_IN_GGA(s(T52), T53)
The graph contains the following edges 1 >= 1, 2 > 2
ACKERMANN38_IN_GGA(s(T52), s(T53)) → U7_GGA(T52, T53, ackermannc38_in_gga(s(T52), T53))
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:

ACKERMANN57_IN_GGA(T84, s(T86), X229) → P65_IN_GGAA(T84, T86, X228, X229)
P65_IN_GGAA(T84, T86, X228, X229) → ACKERMANN57_IN_GGA(T84, T86, X228)
P65_IN_GGAA(T84, T86, T88, X229) → U12_GGAA(T84, T86, T88, X229, ackermannc57_in_gga(T84, T86, T88))
U12_GGAA(T84, T86, T88, X229, ackermannc57_out_gga(T84, T86, T88)) → ACKERMANN68_IN_GGA(T84, T88, X229)
ACKERMANN68_IN_GGA(s(T106), s(T108), X283) → P65_IN_GGAA(T106, T108, X282, X283)

The TRS R consists of the following rules:

ackermannc21_in_ga(T28, X73) → U40_ga(T28, X73, ackermannc27_in_ga(T28, X73))
ackermannc27_in_ga(0, s(s(0))) → ackermannc27_out_ga(0, s(s(0)))
ackermannc27_in_ga(s(T32), X97) → U41_ga(T32, X97, ackermannc21_in_ga(T32, T34))
U41_ga(T32, X97, ackermannc21_out_ga(T32, T34)) → U42_ga(T32, X97, ackermannc38_in_gga(T32, T34, X97))
ackermannc38_in_gga(0, T42, s(T42)) → ackermannc38_out_gga(0, T42, s(T42))
ackermannc38_in_gga(s(T47), 0, X133) → U43_gga(T47, X133, ackermannc27_in_ga(T47, X133))
U43_gga(T47, X133, ackermannc27_out_ga(T47, X133)) → ackermannc38_out_gga(s(T47), 0, X133)
ackermannc38_in_gga(s(T52), s(T53), X151) → U44_gga(T52, T53, X151, ackermannc38_in_gga(s(T52), T53, T55))
U44_gga(T52, T53, X151, ackermannc38_out_gga(s(T52), T53, T55)) → U45_gga(T52, T53, X151, ackermannc38_in_gga(T52, T55, X151))
U45_gga(T52, T53, X151, ackermannc38_out_gga(T52, T55, X151)) → ackermannc38_out_gga(s(T52), s(T53), X151)
U42_ga(T32, X97, ackermannc38_out_gga(T32, T34, X97)) → ackermannc27_out_ga(s(T32), X97)
U40_ga(T28, X73, ackermannc27_out_ga(T28, X73)) → ackermannc21_out_ga(T28, X73)
ackermannc57_in_gga(T79, 0, X211) → U46_gga(T79, X211, ackermannc27_in_ga(T79, X211))
U46_gga(T79, X211, ackermannc27_out_ga(T79, X211)) → ackermannc57_out_gga(T79, 0, X211)
ackermannc57_in_gga(T84, s(T86), X229) → U47_gga(T84, T86, X229, qc65_in_ggaa(T84, T86, X228, X229))
qc65_in_ggaa(T84, T86, T88, X229) → U48_ggaa(T84, T86, T88, X229, ackermannc57_in_gga(T84, T86, T88))
U48_ggaa(T84, T86, T88, X229, ackermannc57_out_gga(T84, T86, T88)) → U49_ggaa(T84, T86, T88, X229, ackermannc68_in_gga(T84, T88, X229))
ackermannc68_in_gga(0, T96, s(T96)) → ackermannc68_out_gga(0, T96, s(T96))
ackermannc68_in_gga(s(T101), 0, X265) → U50_gga(T101, X265, ackermannc27_in_ga(T101, X265))
U50_gga(T101, X265, ackermannc27_out_ga(T101, X265)) → ackermannc68_out_gga(s(T101), 0, X265)
ackermannc68_in_gga(s(T106), s(T108), X283) → U51_gga(T106, T108, X283, qc65_in_ggaa(T106, T108, X282, X283))
U51_gga(T106, T108, X283, qc65_out_ggaa(T106, T108, X282, X283)) → ackermannc68_out_gga(s(T106), s(T108), X283)
U49_ggaa(T84, T86, T88, X229, ackermannc68_out_gga(T84, T88, X229)) → qc65_out_ggaa(T84, T86, T88, X229)
U47_gga(T84, T86, X229, qc65_out_ggaa(T84, T86, X228, X229)) → ackermannc57_out_gga(T84, s(T86), X229)
ackermannc57_in_gaa(T79, 0, X211) → U46_gaa(T79, X211, ackermannc27_in_ga(T79, X211))
U46_gaa(T79, X211, ackermannc27_out_ga(T79, X211)) → ackermannc57_out_gaa(T79, 0, X211)
ackermannc57_in_gaa(T84, s(T86), X229) → U47_gaa(T84, T86, X229, qc65_in_gaaa(T84, T86, X228, X229))
qc65_in_gaaa(T84, T86, T88, X229) → U48_gaaa(T84, T86, T88, X229, ackermannc57_in_gaa(T84, T86, T88))
U48_gaaa(T84, T86, T88, X229, ackermannc57_out_gaa(T84, T86, T88)) → U49_gaaa(T84, T86, T88, X229, ackermannc68_in_gga(T84, T88, X229))
U49_gaaa(T84, T86, T88, X229, ackermannc68_out_gga(T84, T88, X229)) → qc65_out_gaaa(T84, T86, T88, X229)
U47_gaa(T84, T86, X229, qc65_out_gaaa(T84, T86, X228, X229)) → ackermannc57_out_gaa(T84, s(T86), X229)

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
ackermannc21_in_ga(x1, x2) = ackermannc21_in_ga(x1)
U40_ga(x1, x2, x3) = U40_ga(x1, x3)
ackermannc27_in_ga(x1, x2) = ackermannc27_in_ga(x1)
0 = 0
ackermannc27_out_ga(x1, x2) = ackermannc27_out_ga(x1, x2)
U41_ga(x1, x2, x3) = U41_ga(x1, x3)
ackermannc21_out_ga(x1, x2) = ackermannc21_out_ga(x1, x2)
U42_ga(x1, x2, x3) = U42_ga(x1, x3)
ackermannc38_in_gga(x1, x2, x3) = ackermannc38_in_gga(x1, x2)
ackermannc38_out_gga(x1, x2, x3) = ackermannc38_out_gga(x1, x2, x3)
U43_gga(x1, x2, x3) = U43_gga(x1, x3)
U44_gga(x1, x2, x3, x4) = U44_gga(x1, x2, x4)
U45_gga(x1, x2, x3, x4) = U45_gga(x1, x2, x4)
ackermannc57_in_gga(x1, x2, x3) = ackermannc57_in_gga(x1, x2)
U46_gga(x1, x2, x3) = U46_gga(x1, x3)
ackermannc57_out_gga(x1, x2, x3) = ackermannc57_out_gga(x1, x2, x3)
U47_gga(x1, x2, x3, x4) = U47_gga(x1, x2, x4)
qc65_in_ggaa(x1, x2, x3, x4) = qc65_in_ggaa(x1, x2)
U48_ggaa(x1, x2, x3, x4, x5) = U48_ggaa(x1, x2, x5)
U49_ggaa(x1, x2, x3, x4, x5) = U49_ggaa(x1, x2, x3, x5)
ackermannc68_in_gga(x1, x2, x3) = ackermannc68_in_gga(x1, x2)
ackermannc68_out_gga(x1, x2, x3) = ackermannc68_out_gga(x1, x2, x3)
U50_gga(x1, x2, x3) = U50_gga(x1, x3)
U51_gga(x1, x2, x3, x4) = U51_gga(x1, x2, x4)
qc65_out_ggaa(x1, x2, x3, x4) = qc65_out_ggaa(x1, x2, x3, x4)
ackermannc57_in_gaa(x1, x2, x3) = ackermannc57_in_gaa(x1)
U46_gaa(x1, x2, x3) = U46_gaa(x1, x3)
ackermannc57_out_gaa(x1, x2, x3) = ackermannc57_out_gaa(x1, x2, x3)
U47_gaa(x1, x2, x3, x4) = U47_gaa(x1, x4)
qc65_in_gaaa(x1, x2, x3, x4) = qc65_in_gaaa(x1)
U48_gaaa(x1, x2, x3, x4, x5) = U48_gaaa(x1, x5)
U49_gaaa(x1, x2, x3, x4, x5) = U49_gaaa(x1, x2, x3, x5)
qc65_out_gaaa(x1, x2, x3, x4) = qc65_out_gaaa(x1, x2, x3, x4)
ACKERMANN57_IN_GGA(x1, x2, x3) = ACKERMANN57_IN_GGA(x1, x2)
P65_IN_GGAA(x1, x2, x3, x4) = P65_IN_GGAA(x1, x2)
U12_GGAA(x1, x2, x3, x4, x5) = U12_GGAA(x1, x2, x5)
ACKERMANN68_IN_GGA(x1, x2, x3) = ACKERMANN68_IN_GGA(x1, x2)

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:

ACKERMANN57_IN_GGA(T84, s(T86), X229) → P65_IN_GGAA(T84, T86, X228, X229)
P65_IN_GGAA(T84, T86, X228, X229) → ACKERMANN57_IN_GGA(T84, T86, X228)
P65_IN_GGAA(T84, T86, T88, X229) → U12_GGAA(T84, T86, T88, X229, ackermannc57_in_gga(T84, T86, T88))
U12_GGAA(T84, T86, T88, X229, ackermannc57_out_gga(T84, T86, T88)) → ACKERMANN68_IN_GGA(T84, T88, X229)
ACKERMANN68_IN_GGA(s(T106), s(T108), X283) → P65_IN_GGAA(T106, T108, X282, X283)

The TRS R consists of the following rules:

ackermannc57_in_gga(T79, 0, X211) → U46_gga(T79, X211, ackermannc27_in_ga(T79, X211))
ackermannc57_in_gga(T84, s(T86), X229) → U47_gga(T84, T86, X229, qc65_in_ggaa(T84, T86, X228, X229))
U46_gga(T79, X211, ackermannc27_out_ga(T79, X211)) → ackermannc57_out_gga(T79, 0, X211)
U47_gga(T84, T86, X229, qc65_out_ggaa(T84, T86, X228, X229)) → ackermannc57_out_gga(T84, s(T86), X229)
ackermannc27_in_ga(0, s(s(0))) → ackermannc27_out_ga(0, s(s(0)))
ackermannc27_in_ga(s(T32), X97) → U41_ga(T32, X97, ackermannc21_in_ga(T32, T34))
qc65_in_ggaa(T84, T86, T88, X229) → U48_ggaa(T84, T86, T88, X229, ackermannc57_in_gga(T84, T86, T88))
U41_ga(T32, X97, ackermannc21_out_ga(T32, T34)) → U42_ga(T32, X97, ackermannc38_in_gga(T32, T34, X97))
U48_ggaa(T84, T86, T88, X229, ackermannc57_out_gga(T84, T86, T88)) → U49_ggaa(T84, T86, T88, X229, ackermannc68_in_gga(T84, T88, X229))
ackermannc21_in_ga(T28, X73) → U40_ga(T28, X73, ackermannc27_in_ga(T28, X73))
U42_ga(T32, X97, ackermannc38_out_gga(T32, T34, X97)) → ackermannc27_out_ga(s(T32), X97)
U49_ggaa(T84, T86, T88, X229, ackermannc68_out_gga(T84, T88, X229)) → qc65_out_ggaa(T84, T86, T88, X229)
U40_ga(T28, X73, ackermannc27_out_ga(T28, X73)) → ackermannc21_out_ga(T28, X73)
ackermannc38_in_gga(0, T42, s(T42)) → ackermannc38_out_gga(0, T42, s(T42))
ackermannc38_in_gga(s(T47), 0, X133) → U43_gga(T47, X133, ackermannc27_in_ga(T47, X133))
ackermannc38_in_gga(s(T52), s(T53), X151) → U44_gga(T52, T53, X151, ackermannc38_in_gga(s(T52), T53, T55))
ackermannc68_in_gga(0, T96, s(T96)) → ackermannc68_out_gga(0, T96, s(T96))
ackermannc68_in_gga(s(T101), 0, X265) → U50_gga(T101, X265, ackermannc27_in_ga(T101, X265))
ackermannc68_in_gga(s(T106), s(T108), X283) → U51_gga(T106, T108, X283, qc65_in_ggaa(T106, T108, X282, X283))
U43_gga(T47, X133, ackermannc27_out_ga(T47, X133)) → ackermannc38_out_gga(s(T47), 0, X133)
U44_gga(T52, T53, X151, ackermannc38_out_gga(s(T52), T53, T55)) → U45_gga(T52, T53, X151, ackermannc38_in_gga(T52, T55, X151))
U50_gga(T101, X265, ackermannc27_out_ga(T101, X265)) → ackermannc68_out_gga(s(T101), 0, X265)
U51_gga(T106, T108, X283, qc65_out_ggaa(T106, T108, X282, X283)) → ackermannc68_out_gga(s(T106), s(T108), X283)
U45_gga(T52, T53, X151, ackermannc38_out_gga(T52, T55, X151)) → ackermannc38_out_gga(s(T52), s(T53), X151)

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
ackermannc21_in_ga(x1, x2) = ackermannc21_in_ga(x1)
U40_ga(x1, x2, x3) = U40_ga(x1, x3)
ackermannc27_in_ga(x1, x2) = ackermannc27_in_ga(x1)
0 = 0
ackermannc27_out_ga(x1, x2) = ackermannc27_out_ga(x1, x2)
U41_ga(x1, x2, x3) = U41_ga(x1, x3)
ackermannc21_out_ga(x1, x2) = ackermannc21_out_ga(x1, x2)
U42_ga(x1, x2, x3) = U42_ga(x1, x3)
ackermannc38_in_gga(x1, x2, x3) = ackermannc38_in_gga(x1, x2)
ackermannc38_out_gga(x1, x2, x3) = ackermannc38_out_gga(x1, x2, x3)
U43_gga(x1, x2, x3) = U43_gga(x1, x3)
U44_gga(x1, x2, x3, x4) = U44_gga(x1, x2, x4)
U45_gga(x1, x2, x3, x4) = U45_gga(x1, x2, x4)
ackermannc57_in_gga(x1, x2, x3) = ackermannc57_in_gga(x1, x2)
U46_gga(x1, x2, x3) = U46_gga(x1, x3)
ackermannc57_out_gga(x1, x2, x3) = ackermannc57_out_gga(x1, x2, x3)
U47_gga(x1, x2, x3, x4) = U47_gga(x1, x2, x4)
qc65_in_ggaa(x1, x2, x3, x4) = qc65_in_ggaa(x1, x2)
U48_ggaa(x1, x2, x3, x4, x5) = U48_ggaa(x1, x2, x5)
U49_ggaa(x1, x2, x3, x4, x5) = U49_ggaa(x1, x2, x3, x5)
ackermannc68_in_gga(x1, x2, x3) = ackermannc68_in_gga(x1, x2)
ackermannc68_out_gga(x1, x2, x3) = ackermannc68_out_gga(x1, x2, x3)
U50_gga(x1, x2, x3) = U50_gga(x1, x3)
U51_gga(x1, x2, x3, x4) = U51_gga(x1, x2, x4)
qc65_out_ggaa(x1, x2, x3, x4) = qc65_out_ggaa(x1, x2, x3, x4)
ACKERMANN57_IN_GGA(x1, x2, x3) = ACKERMANN57_IN_GGA(x1, x2)
P65_IN_GGAA(x1, x2, x3, x4) = P65_IN_GGAA(x1, x2)
U12_GGAA(x1, x2, x3, x4, x5) = U12_GGAA(x1, x2, x5)
ACKERMANN68_IN_GGA(x1, x2, x3) = ACKERMANN68_IN_GGA(x1, x2)

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:

ACKERMANN57_IN_GGA(T84, s(T86)) → P65_IN_GGAA(T84, T86)
P65_IN_GGAA(T84, T86) → ACKERMANN57_IN_GGA(T84, T86)
P65_IN_GGAA(T84, T86) → U12_GGAA(T84, T86, ackermannc57_in_gga(T84, T86))
U12_GGAA(T84, T86, ackermannc57_out_gga(T84, T86, T88)) → ACKERMANN68_IN_GGA(T84, T88)
ACKERMANN68_IN_GGA(s(T106), s(T108)) → P65_IN_GGAA(T106, T108)

The TRS R consists of the following rules:

ackermannc57_in_gga(T79, 0) → U46_gga(T79, ackermannc27_in_ga(T79))
ackermannc57_in_gga(T84, s(T86)) → U47_gga(T84, T86, qc65_in_ggaa(T84, T86))
U46_gga(T79, ackermannc27_out_ga(T79, X211)) → ackermannc57_out_gga(T79, 0, X211)
U47_gga(T84, T86, qc65_out_ggaa(T84, T86, X228, X229)) → ackermannc57_out_gga(T84, s(T86), X229)
ackermannc27_in_ga(0) → ackermannc27_out_ga(0, s(s(0)))
ackermannc27_in_ga(s(T32)) → U41_ga(T32, ackermannc21_in_ga(T32))
qc65_in_ggaa(T84, T86) → U48_ggaa(T84, T86, ackermannc57_in_gga(T84, T86))
U41_ga(T32, ackermannc21_out_ga(T32, T34)) → U42_ga(T32, ackermannc38_in_gga(T32, T34))
U48_ggaa(T84, T86, ackermannc57_out_gga(T84, T86, T88)) → U49_ggaa(T84, T86, T88, ackermannc68_in_gga(T84, T88))
ackermannc21_in_ga(T28) → U40_ga(T28, ackermannc27_in_ga(T28))
U42_ga(T32, ackermannc38_out_gga(T32, T34, X97)) → ackermannc27_out_ga(s(T32), X97)
U49_ggaa(T84, T86, T88, ackermannc68_out_gga(T84, T88, X229)) → qc65_out_ggaa(T84, T86, T88, X229)
U40_ga(T28, ackermannc27_out_ga(T28, X73)) → ackermannc21_out_ga(T28, X73)
ackermannc38_in_gga(0, T42) → ackermannc38_out_gga(0, T42, s(T42))
ackermannc38_in_gga(s(T47), 0) → U43_gga(T47, ackermannc27_in_ga(T47))
ackermannc38_in_gga(s(T52), s(T53)) → U44_gga(T52, T53, ackermannc38_in_gga(s(T52), T53))
ackermannc68_in_gga(0, T96) → ackermannc68_out_gga(0, T96, s(T96))
ackermannc68_in_gga(s(T101), 0) → U50_gga(T101, ackermannc27_in_ga(T101))
ackermannc68_in_gga(s(T106), s(T108)) → U51_gga(T106, T108, qc65_in_ggaa(T106, T108))
U43_gga(T47, ackermannc27_out_ga(T47, X133)) → ackermannc38_out_gga(s(T47), 0, X133)
U44_gga(T52, T53, ackermannc38_out_gga(s(T52), T53, T55)) → U45_gga(T52, T53, ackermannc38_in_gga(T52, T55))
U50_gga(T101, ackermannc27_out_ga(T101, X265)) → ackermannc68_out_gga(s(T101), 0, X265)
U51_gga(T106, T108, qc65_out_ggaa(T106, T108, X282, X283)) → ackermannc68_out_gga(s(T106), s(T108), X283)
U45_gga(T52, T53, ackermannc38_out_gga(T52, T55, X151)) → ackermannc38_out_gga(s(T52), s(T53), X151)

The set Q consists of the following terms:

ackermannc57_in_gga(x0, x1)
U46_gga(x0, x1)
U47_gga(x0, x1, x2)
ackermannc27_in_ga(x0)
qc65_in_ggaa(x0, x1)
U41_ga(x0, x1)
U48_ggaa(x0, x1, x2)
ackermannc21_in_ga(x0)
U42_ga(x0, x1)
U49_ggaa(x0, x1, x2, x3)
U40_ga(x0, x1)
ackermannc38_in_gga(x0, x1)
ackermannc68_in_gga(x0, x1)
U43_gga(x0, x1)
U44_gga(x0, x1, x2)
U50_gga(x0, x1)
U51_gga(x0, x1, x2)
U45_gga(x0, x1, x2)

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:

P65_IN_GGAA(T84, T86) → ACKERMANN57_IN_GGA(T84, T86)
The graph contains the following edges 1 >= 1, 2 >= 2
P65_IN_GGAA(T84, T86) → U12_GGAA(T84, T86, ackermannc57_in_gga(T84, T86))
The graph contains the following edges 1 >= 1, 2 >= 2
ACKERMANN57_IN_GGA(T84, s(T86)) → P65_IN_GGAA(T84, T86)
The graph contains the following edges 1 >= 1, 2 > 2
ACKERMANN68_IN_GGA(s(T106), s(T108)) → P65_IN_GGAA(T106, T108)
The graph contains the following edges 1 > 1, 2 > 2
U12_GGAA(T84, T86, ackermannc57_out_gga(T84, T86, T88)) → ACKERMANN68_IN_GGA(T84, T88)
The graph contains the following edges 1 >= 1, 3 > 1, 3 > 2

(20) YES

(21) Obligation:

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

P65_IN_GAAA(T84, T86, X228, X229) → ACKERMANN57_IN_GAA(T84, T86, X228)
ACKERMANN57_IN_GAA(T84, s(T86), X229) → P65_IN_GAAA(T84, T86, X228, X229)

The TRS R consists of the following rules:

ackermannc21_in_ga(T28, X73) → U40_ga(T28, X73, ackermannc27_in_ga(T28, X73))
ackermannc27_in_ga(0, s(s(0))) → ackermannc27_out_ga(0, s(s(0)))
ackermannc27_in_ga(s(T32), X97) → U41_ga(T32, X97, ackermannc21_in_ga(T32, T34))
U41_ga(T32, X97, ackermannc21_out_ga(T32, T34)) → U42_ga(T32, X97, ackermannc38_in_gga(T32, T34, X97))
ackermannc38_in_gga(0, T42, s(T42)) → ackermannc38_out_gga(0, T42, s(T42))
ackermannc38_in_gga(s(T47), 0, X133) → U43_gga(T47, X133, ackermannc27_in_ga(T47, X133))
U43_gga(T47, X133, ackermannc27_out_ga(T47, X133)) → ackermannc38_out_gga(s(T47), 0, X133)
ackermannc38_in_gga(s(T52), s(T53), X151) → U44_gga(T52, T53, X151, ackermannc38_in_gga(s(T52), T53, T55))
U44_gga(T52, T53, X151, ackermannc38_out_gga(s(T52), T53, T55)) → U45_gga(T52, T53, X151, ackermannc38_in_gga(T52, T55, X151))
U45_gga(T52, T53, X151, ackermannc38_out_gga(T52, T55, X151)) → ackermannc38_out_gga(s(T52), s(T53), X151)
U42_ga(T32, X97, ackermannc38_out_gga(T32, T34, X97)) → ackermannc27_out_ga(s(T32), X97)
U40_ga(T28, X73, ackermannc27_out_ga(T28, X73)) → ackermannc21_out_ga(T28, X73)
ackermannc57_in_gga(T79, 0, X211) → U46_gga(T79, X211, ackermannc27_in_ga(T79, X211))
U46_gga(T79, X211, ackermannc27_out_ga(T79, X211)) → ackermannc57_out_gga(T79, 0, X211)
ackermannc57_in_gga(T84, s(T86), X229) → U47_gga(T84, T86, X229, qc65_in_ggaa(T84, T86, X228, X229))
qc65_in_ggaa(T84, T86, T88, X229) → U48_ggaa(T84, T86, T88, X229, ackermannc57_in_gga(T84, T86, T88))
U48_ggaa(T84, T86, T88, X229, ackermannc57_out_gga(T84, T86, T88)) → U49_ggaa(T84, T86, T88, X229, ackermannc68_in_gga(T84, T88, X229))
ackermannc68_in_gga(0, T96, s(T96)) → ackermannc68_out_gga(0, T96, s(T96))
ackermannc68_in_gga(s(T101), 0, X265) → U50_gga(T101, X265, ackermannc27_in_ga(T101, X265))
U50_gga(T101, X265, ackermannc27_out_ga(T101, X265)) → ackermannc68_out_gga(s(T101), 0, X265)
ackermannc68_in_gga(s(T106), s(T108), X283) → U51_gga(T106, T108, X283, qc65_in_ggaa(T106, T108, X282, X283))
U51_gga(T106, T108, X283, qc65_out_ggaa(T106, T108, X282, X283)) → ackermannc68_out_gga(s(T106), s(T108), X283)
U49_ggaa(T84, T86, T88, X229, ackermannc68_out_gga(T84, T88, X229)) → qc65_out_ggaa(T84, T86, T88, X229)
U47_gga(T84, T86, X229, qc65_out_ggaa(T84, T86, X228, X229)) → ackermannc57_out_gga(T84, s(T86), X229)
ackermannc57_in_gaa(T79, 0, X211) → U46_gaa(T79, X211, ackermannc27_in_ga(T79, X211))
U46_gaa(T79, X211, ackermannc27_out_ga(T79, X211)) → ackermannc57_out_gaa(T79, 0, X211)
ackermannc57_in_gaa(T84, s(T86), X229) → U47_gaa(T84, T86, X229, qc65_in_gaaa(T84, T86, X228, X229))
qc65_in_gaaa(T84, T86, T88, X229) → U48_gaaa(T84, T86, T88, X229, ackermannc57_in_gaa(T84, T86, T88))
U48_gaaa(T84, T86, T88, X229, ackermannc57_out_gaa(T84, T86, T88)) → U49_gaaa(T84, T86, T88, X229, ackermannc68_in_gga(T84, T88, X229))
U49_gaaa(T84, T86, T88, X229, ackermannc68_out_gga(T84, T88, X229)) → qc65_out_gaaa(T84, T86, T88, X229)
U47_gaa(T84, T86, X229, qc65_out_gaaa(T84, T86, X228, X229)) → ackermannc57_out_gaa(T84, s(T86), X229)

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
ackermannc21_in_ga(x1, x2) = ackermannc21_in_ga(x1)
U40_ga(x1, x2, x3) = U40_ga(x1, x3)
ackermannc27_in_ga(x1, x2) = ackermannc27_in_ga(x1)
0 = 0
ackermannc27_out_ga(x1, x2) = ackermannc27_out_ga(x1, x2)
U41_ga(x1, x2, x3) = U41_ga(x1, x3)
ackermannc21_out_ga(x1, x2) = ackermannc21_out_ga(x1, x2)
U42_ga(x1, x2, x3) = U42_ga(x1, x3)
ackermannc38_in_gga(x1, x2, x3) = ackermannc38_in_gga(x1, x2)
ackermannc38_out_gga(x1, x2, x3) = ackermannc38_out_gga(x1, x2, x3)
U43_gga(x1, x2, x3) = U43_gga(x1, x3)
U44_gga(x1, x2, x3, x4) = U44_gga(x1, x2, x4)
U45_gga(x1, x2, x3, x4) = U45_gga(x1, x2, x4)
ackermannc57_in_gga(x1, x2, x3) = ackermannc57_in_gga(x1, x2)
U46_gga(x1, x2, x3) = U46_gga(x1, x3)
ackermannc57_out_gga(x1, x2, x3) = ackermannc57_out_gga(x1, x2, x3)
U47_gga(x1, x2, x3, x4) = U47_gga(x1, x2, x4)
qc65_in_ggaa(x1, x2, x3, x4) = qc65_in_ggaa(x1, x2)
U48_ggaa(x1, x2, x3, x4, x5) = U48_ggaa(x1, x2, x5)
U49_ggaa(x1, x2, x3, x4, x5) = U49_ggaa(x1, x2, x3, x5)
ackermannc68_in_gga(x1, x2, x3) = ackermannc68_in_gga(x1, x2)
ackermannc68_out_gga(x1, x2, x3) = ackermannc68_out_gga(x1, x2, x3)
U50_gga(x1, x2, x3) = U50_gga(x1, x3)
U51_gga(x1, x2, x3, x4) = U51_gga(x1, x2, x4)
qc65_out_ggaa(x1, x2, x3, x4) = qc65_out_ggaa(x1, x2, x3, x4)
ackermannc57_in_gaa(x1, x2, x3) = ackermannc57_in_gaa(x1)
U46_gaa(x1, x2, x3) = U46_gaa(x1, x3)
ackermannc57_out_gaa(x1, x2, x3) = ackermannc57_out_gaa(x1, x2, x3)
U47_gaa(x1, x2, x3, x4) = U47_gaa(x1, x4)
qc65_in_gaaa(x1, x2, x3, x4) = qc65_in_gaaa(x1)
U48_gaaa(x1, x2, x3, x4, x5) = U48_gaaa(x1, x5)
U49_gaaa(x1, x2, x3, x4, x5) = U49_gaaa(x1, x2, x3, x5)
qc65_out_gaaa(x1, x2, x3, x4) = qc65_out_gaaa(x1, x2, x3, x4)
ACKERMANN57_IN_GAA(x1, x2, x3) = ACKERMANN57_IN_GAA(x1)
P65_IN_GAAA(x1, x2, x3, x4) = P65_IN_GAAA(x1)

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:

P65_IN_GAAA(T84, T86, X228, X229) → ACKERMANN57_IN_GAA(T84, T86, X228)
ACKERMANN57_IN_GAA(T84, s(T86), X229) → P65_IN_GAAA(T84, T86, X228, X229)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
ACKERMANN57_IN_GAA(x1, x2, x3) = ACKERMANN57_IN_GAA(x1)
P65_IN_GAAA(x1, x2, x3, x4) = P65_IN_GAAA(x1)

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

(24) PiDPToQDPProof (SOUND 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:

P65_IN_GAAA(T84) → ACKERMANN57_IN_GAA(T84)
ACKERMANN57_IN_GAA(T84) → P65_IN_GAAA(T84)

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

(26) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = ACKERMANN57_IN_GAA(T84') evaluates to t =ACKERMANN57_IN_GAA(T84')

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Matcher: [ ]
Semiunifier: [ ]

Rewriting sequence

ACKERMANN57_IN_GAA(T84') → P65_IN_GAAA(T84')
with rule ACKERMANN57_IN_GAA(T84'') → P65_IN_GAAA(T84'') at position [] and matcher [T84'' / T84']

P65_IN_GAAA(T84') → ACKERMANN57_IN_GAA(T84')
with rule P65_IN_GAAA(T84) → ACKERMANN57_IN_GAA(T84)

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence

All these steps are and every following step will be a correct step w.r.t to Q.

(27) NO

(28) Obligation:

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

ACKERMANN1_IN_GGG(s(s(T19)), 0, T20) → U17_GGG(T19, T20, ackermannc21_in_ga(T19, T22))
U17_GGG(T19, T20, ackermannc21_out_ga(T19, T22)) → ACKERMANN1_IN_GGG(T19, T22, T20)
ACKERMANN1_IN_GGG(s(T68), s(T71), T70) → U20_GGG(T68, T71, T70, ackermannc57_in_gga(T68, T71, T73))
U20_GGG(T68, T71, T70, ackermannc57_out_gga(T68, T71, T73)) → ACKERMANN1_IN_GGG(T68, T73, T70)
ACKERMANN1_IN_GGG(s(T123), s(0), T117) → U23_GGG(T123, T117, ackermannc27_in_ga(T123, T124))
U23_GGG(T123, T117, ackermannc27_out_ga(T123, T124)) → ACKERMANN1_IN_GGG(T123, T124, T117)
ACKERMANN1_IN_GGG(s(T131), s(s(T133)), T117) → U26_GGG(T131, T133, T117, ackermannc57_in_gga(T131, T133, T135))
U26_GGG(T131, T133, T117, ackermannc57_out_gga(T131, T133, T135)) → U28_GGG(T131, T133, T117, ackermannc68_in_gga(T131, T135, T139))
U28_GGG(T131, T133, T117, ackermannc68_out_gga(T131, T135, T139)) → ACKERMANN1_IN_GGG(T131, T139, T117)

The TRS R consists of the following rules:

ackermannc21_in_ga(T28, X73) → U40_ga(T28, X73, ackermannc27_in_ga(T28, X73))
ackermannc27_in_ga(0, s(s(0))) → ackermannc27_out_ga(0, s(s(0)))
ackermannc27_in_ga(s(T32), X97) → U41_ga(T32, X97, ackermannc21_in_ga(T32, T34))
U41_ga(T32, X97, ackermannc21_out_ga(T32, T34)) → U42_ga(T32, X97, ackermannc38_in_gga(T32, T34, X97))
ackermannc38_in_gga(0, T42, s(T42)) → ackermannc38_out_gga(0, T42, s(T42))
ackermannc38_in_gga(s(T47), 0, X133) → U43_gga(T47, X133, ackermannc27_in_ga(T47, X133))
U43_gga(T47, X133, ackermannc27_out_ga(T47, X133)) → ackermannc38_out_gga(s(T47), 0, X133)
ackermannc38_in_gga(s(T52), s(T53), X151) → U44_gga(T52, T53, X151, ackermannc38_in_gga(s(T52), T53, T55))
U44_gga(T52, T53, X151, ackermannc38_out_gga(s(T52), T53, T55)) → U45_gga(T52, T53, X151, ackermannc38_in_gga(T52, T55, X151))
U45_gga(T52, T53, X151, ackermannc38_out_gga(T52, T55, X151)) → ackermannc38_out_gga(s(T52), s(T53), X151)
U42_ga(T32, X97, ackermannc38_out_gga(T32, T34, X97)) → ackermannc27_out_ga(s(T32), X97)
U40_ga(T28, X73, ackermannc27_out_ga(T28, X73)) → ackermannc21_out_ga(T28, X73)
ackermannc57_in_gga(T79, 0, X211) → U46_gga(T79, X211, ackermannc27_in_ga(T79, X211))
U46_gga(T79, X211, ackermannc27_out_ga(T79, X211)) → ackermannc57_out_gga(T79, 0, X211)
ackermannc57_in_gga(T84, s(T86), X229) → U47_gga(T84, T86, X229, qc65_in_ggaa(T84, T86, X228, X229))
qc65_in_ggaa(T84, T86, T88, X229) → U48_ggaa(T84, T86, T88, X229, ackermannc57_in_gga(T84, T86, T88))
U48_ggaa(T84, T86, T88, X229, ackermannc57_out_gga(T84, T86, T88)) → U49_ggaa(T84, T86, T88, X229, ackermannc68_in_gga(T84, T88, X229))
ackermannc68_in_gga(0, T96, s(T96)) → ackermannc68_out_gga(0, T96, s(T96))
ackermannc68_in_gga(s(T101), 0, X265) → U50_gga(T101, X265, ackermannc27_in_ga(T101, X265))
U50_gga(T101, X265, ackermannc27_out_ga(T101, X265)) → ackermannc68_out_gga(s(T101), 0, X265)
ackermannc68_in_gga(s(T106), s(T108), X283) → U51_gga(T106, T108, X283, qc65_in_ggaa(T106, T108, X282, X283))
U51_gga(T106, T108, X283, qc65_out_ggaa(T106, T108, X282, X283)) → ackermannc68_out_gga(s(T106), s(T108), X283)
U49_ggaa(T84, T86, T88, X229, ackermannc68_out_gga(T84, T88, X229)) → qc65_out_ggaa(T84, T86, T88, X229)
U47_gga(T84, T86, X229, qc65_out_ggaa(T84, T86, X228, X229)) → ackermannc57_out_gga(T84, s(T86), X229)
ackermannc57_in_gaa(T79, 0, X211) → U46_gaa(T79, X211, ackermannc27_in_ga(T79, X211))
U46_gaa(T79, X211, ackermannc27_out_ga(T79, X211)) → ackermannc57_out_gaa(T79, 0, X211)
ackermannc57_in_gaa(T84, s(T86), X229) → U47_gaa(T84, T86, X229, qc65_in_gaaa(T84, T86, X228, X229))
qc65_in_gaaa(T84, T86, T88, X229) → U48_gaaa(T84, T86, T88, X229, ackermannc57_in_gaa(T84, T86, T88))
U48_gaaa(T84, T86, T88, X229, ackermannc57_out_gaa(T84, T86, T88)) → U49_gaaa(T84, T86, T88, X229, ackermannc68_in_gga(T84, T88, X229))
U49_gaaa(T84, T86, T88, X229, ackermannc68_out_gga(T84, T88, X229)) → qc65_out_gaaa(T84, T86, T88, X229)
U47_gaa(T84, T86, X229, qc65_out_gaaa(T84, T86, X228, X229)) → ackermannc57_out_gaa(T84, s(T86), X229)

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
ackermannc21_in_ga(x1, x2) = ackermannc21_in_ga(x1)
U40_ga(x1, x2, x3) = U40_ga(x1, x3)
ackermannc27_in_ga(x1, x2) = ackermannc27_in_ga(x1)
0 = 0
ackermannc27_out_ga(x1, x2) = ackermannc27_out_ga(x1, x2)
U41_ga(x1, x2, x3) = U41_ga(x1, x3)
ackermannc21_out_ga(x1, x2) = ackermannc21_out_ga(x1, x2)
U42_ga(x1, x2, x3) = U42_ga(x1, x3)
ackermannc38_in_gga(x1, x2, x3) = ackermannc38_in_gga(x1, x2)
ackermannc38_out_gga(x1, x2, x3) = ackermannc38_out_gga(x1, x2, x3)
U43_gga(x1, x2, x3) = U43_gga(x1, x3)
U44_gga(x1, x2, x3, x4) = U44_gga(x1, x2, x4)
U45_gga(x1, x2, x3, x4) = U45_gga(x1, x2, x4)
ackermannc57_in_gga(x1, x2, x3) = ackermannc57_in_gga(x1, x2)
U46_gga(x1, x2, x3) = U46_gga(x1, x3)
ackermannc57_out_gga(x1, x2, x3) = ackermannc57_out_gga(x1, x2, x3)
U47_gga(x1, x2, x3, x4) = U47_gga(x1, x2, x4)
qc65_in_ggaa(x1, x2, x3, x4) = qc65_in_ggaa(x1, x2)
U48_ggaa(x1, x2, x3, x4, x5) = U48_ggaa(x1, x2, x5)
U49_ggaa(x1, x2, x3, x4, x5) = U49_ggaa(x1, x2, x3, x5)
ackermannc68_in_gga(x1, x2, x3) = ackermannc68_in_gga(x1, x2)
ackermannc68_out_gga(x1, x2, x3) = ackermannc68_out_gga(x1, x2, x3)
U50_gga(x1, x2, x3) = U50_gga(x1, x3)
U51_gga(x1, x2, x3, x4) = U51_gga(x1, x2, x4)
qc65_out_ggaa(x1, x2, x3, x4) = qc65_out_ggaa(x1, x2, x3, x4)
ackermannc57_in_gaa(x1, x2, x3) = ackermannc57_in_gaa(x1)
U46_gaa(x1, x2, x3) = U46_gaa(x1, x3)
ackermannc57_out_gaa(x1, x2, x3) = ackermannc57_out_gaa(x1, x2, x3)
U47_gaa(x1, x2, x3, x4) = U47_gaa(x1, x4)
qc65_in_gaaa(x1, x2, x3, x4) = qc65_in_gaaa(x1)
U48_gaaa(x1, x2, x3, x4, x5) = U48_gaaa(x1, x5)
U49_gaaa(x1, x2, x3, x4, x5) = U49_gaaa(x1, x2, x3, x5)
qc65_out_gaaa(x1, x2, x3, x4) = qc65_out_gaaa(x1, x2, x3, x4)
ACKERMANN1_IN_GGG(x1, x2, x3) = ACKERMANN1_IN_GGG(x1, x2, x3)
U17_GGG(x1, x2, x3) = U17_GGG(x1, x2, x3)
U20_GGG(x1, x2, x3, x4) = U20_GGG(x1, x2, x3, x4)
U23_GGG(x1, x2, x3) = U23_GGG(x1, x2, x3)
U26_GGG(x1, x2, x3, x4) = U26_GGG(x1, x2, x3, x4)
U28_GGG(x1, x2, x3, x4) = U28_GGG(x1, x2, x3, x4)

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

(29) UsableRulesProof (EQUIVALENT transformation)

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

(30) Obligation:

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

ACKERMANN1_IN_GGG(s(s(T19)), 0, T20) → U17_GGG(T19, T20, ackermannc21_in_ga(T19, T22))
U17_GGG(T19, T20, ackermannc21_out_ga(T19, T22)) → ACKERMANN1_IN_GGG(T19, T22, T20)
ACKERMANN1_IN_GGG(s(T68), s(T71), T70) → U20_GGG(T68, T71, T70, ackermannc57_in_gga(T68, T71, T73))
U20_GGG(T68, T71, T70, ackermannc57_out_gga(T68, T71, T73)) → ACKERMANN1_IN_GGG(T68, T73, T70)
ACKERMANN1_IN_GGG(s(T123), s(0), T117) → U23_GGG(T123, T117, ackermannc27_in_ga(T123, T124))
U23_GGG(T123, T117, ackermannc27_out_ga(T123, T124)) → ACKERMANN1_IN_GGG(T123, T124, T117)
ACKERMANN1_IN_GGG(s(T131), s(s(T133)), T117) → U26_GGG(T131, T133, T117, ackermannc57_in_gga(T131, T133, T135))
U26_GGG(T131, T133, T117, ackermannc57_out_gga(T131, T133, T135)) → U28_GGG(T131, T133, T117, ackermannc68_in_gga(T131, T135, T139))
U28_GGG(T131, T133, T117, ackermannc68_out_gga(T131, T135, T139)) → ACKERMANN1_IN_GGG(T131, T139, T117)

The TRS R consists of the following rules:

ackermannc21_in_ga(T28, X73) → U40_ga(T28, X73, ackermannc27_in_ga(T28, X73))
ackermannc57_in_gga(T79, 0, X211) → U46_gga(T79, X211, ackermannc27_in_ga(T79, X211))
ackermannc57_in_gga(T84, s(T86), X229) → U47_gga(T84, T86, X229, qc65_in_ggaa(T84, T86, X228, X229))
ackermannc27_in_ga(0, s(s(0))) → ackermannc27_out_ga(0, s(s(0)))
ackermannc27_in_ga(s(T32), X97) → U41_ga(T32, X97, ackermannc21_in_ga(T32, T34))
ackermannc68_in_gga(0, T96, s(T96)) → ackermannc68_out_gga(0, T96, s(T96))
ackermannc68_in_gga(s(T101), 0, X265) → U50_gga(T101, X265, ackermannc27_in_ga(T101, X265))
ackermannc68_in_gga(s(T106), s(T108), X283) → U51_gga(T106, T108, X283, qc65_in_ggaa(T106, T108, X282, X283))
U40_ga(T28, X73, ackermannc27_out_ga(T28, X73)) → ackermannc21_out_ga(T28, X73)
U46_gga(T79, X211, ackermannc27_out_ga(T79, X211)) → ackermannc57_out_gga(T79, 0, X211)
U47_gga(T84, T86, X229, qc65_out_ggaa(T84, T86, X228, X229)) → ackermannc57_out_gga(T84, s(T86), X229)
U41_ga(T32, X97, ackermannc21_out_ga(T32, T34)) → U42_ga(T32, X97, ackermannc38_in_gga(T32, T34, X97))
U50_gga(T101, X265, ackermannc27_out_ga(T101, X265)) → ackermannc68_out_gga(s(T101), 0, X265)
U51_gga(T106, T108, X283, qc65_out_ggaa(T106, T108, X282, X283)) → ackermannc68_out_gga(s(T106), s(T108), X283)
qc65_in_ggaa(T84, T86, T88, X229) → U48_ggaa(T84, T86, T88, X229, ackermannc57_in_gga(T84, T86, T88))
U42_ga(T32, X97, ackermannc38_out_gga(T32, T34, X97)) → ackermannc27_out_ga(s(T32), X97)
U48_ggaa(T84, T86, T88, X229, ackermannc57_out_gga(T84, T86, T88)) → U49_ggaa(T84, T86, T88, X229, ackermannc68_in_gga(T84, T88, X229))
ackermannc38_in_gga(0, T42, s(T42)) → ackermannc38_out_gga(0, T42, s(T42))
ackermannc38_in_gga(s(T47), 0, X133) → U43_gga(T47, X133, ackermannc27_in_ga(T47, X133))
ackermannc38_in_gga(s(T52), s(T53), X151) → U44_gga(T52, T53, X151, ackermannc38_in_gga(s(T52), T53, T55))
U49_ggaa(T84, T86, T88, X229, ackermannc68_out_gga(T84, T88, X229)) → qc65_out_ggaa(T84, T86, T88, X229)
U43_gga(T47, X133, ackermannc27_out_ga(T47, X133)) → ackermannc38_out_gga(s(T47), 0, X133)
U44_gga(T52, T53, X151, ackermannc38_out_gga(s(T52), T53, T55)) → U45_gga(T52, T53, X151, ackermannc38_in_gga(T52, T55, X151))
U45_gga(T52, T53, X151, ackermannc38_out_gga(T52, T55, X151)) → ackermannc38_out_gga(s(T52), s(T53), X151)

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
ackermannc21_in_ga(x1, x2) = ackermannc21_in_ga(x1)
U40_ga(x1, x2, x3) = U40_ga(x1, x3)
ackermannc27_in_ga(x1, x2) = ackermannc27_in_ga(x1)
0 = 0
ackermannc27_out_ga(x1, x2) = ackermannc27_out_ga(x1, x2)
U41_ga(x1, x2, x3) = U41_ga(x1, x3)
ackermannc21_out_ga(x1, x2) = ackermannc21_out_ga(x1, x2)
U42_ga(x1, x2, x3) = U42_ga(x1, x3)
ackermannc38_in_gga(x1, x2, x3) = ackermannc38_in_gga(x1, x2)
ackermannc38_out_gga(x1, x2, x3) = ackermannc38_out_gga(x1, x2, x3)
U43_gga(x1, x2, x3) = U43_gga(x1, x3)
U44_gga(x1, x2, x3, x4) = U44_gga(x1, x2, x4)
U45_gga(x1, x2, x3, x4) = U45_gga(x1, x2, x4)
ackermannc57_in_gga(x1, x2, x3) = ackermannc57_in_gga(x1, x2)
U46_gga(x1, x2, x3) = U46_gga(x1, x3)
ackermannc57_out_gga(x1, x2, x3) = ackermannc57_out_gga(x1, x2, x3)
U47_gga(x1, x2, x3, x4) = U47_gga(x1, x2, x4)
qc65_in_ggaa(x1, x2, x3, x4) = qc65_in_ggaa(x1, x2)
U48_ggaa(x1, x2, x3, x4, x5) = U48_ggaa(x1, x2, x5)
U49_ggaa(x1, x2, x3, x4, x5) = U49_ggaa(x1, x2, x3, x5)
ackermannc68_in_gga(x1, x2, x3) = ackermannc68_in_gga(x1, x2)
ackermannc68_out_gga(x1, x2, x3) = ackermannc68_out_gga(x1, x2, x3)
U50_gga(x1, x2, x3) = U50_gga(x1, x3)
U51_gga(x1, x2, x3, x4) = U51_gga(x1, x2, x4)
qc65_out_ggaa(x1, x2, x3, x4) = qc65_out_ggaa(x1, x2, x3, x4)
ACKERMANN1_IN_GGG(x1, x2, x3) = ACKERMANN1_IN_GGG(x1, x2, x3)
U17_GGG(x1, x2, x3) = U17_GGG(x1, x2, x3)
U20_GGG(x1, x2, x3, x4) = U20_GGG(x1, x2, x3, x4)
U23_GGG(x1, x2, x3) = U23_GGG(x1, x2, x3)
U26_GGG(x1, x2, x3, x4) = U26_GGG(x1, x2, x3, x4)
U28_GGG(x1, x2, x3, x4) = U28_GGG(x1, x2, x3, x4)

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

(31) PiDPToQDPProof (SOUND transformation)

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

(32) Obligation:

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

ACKERMANN1_IN_GGG(s(s(T19)), 0, T20) → U17_GGG(T19, T20, ackermannc21_in_ga(T19))
U17_GGG(T19, T20, ackermannc21_out_ga(T19, T22)) → ACKERMANN1_IN_GGG(T19, T22, T20)
ACKERMANN1_IN_GGG(s(T68), s(T71), T70) → U20_GGG(T68, T71, T70, ackermannc57_in_gga(T68, T71))
U20_GGG(T68, T71, T70, ackermannc57_out_gga(T68, T71, T73)) → ACKERMANN1_IN_GGG(T68, T73, T70)
ACKERMANN1_IN_GGG(s(T123), s(0), T117) → U23_GGG(T123, T117, ackermannc27_in_ga(T123))
U23_GGG(T123, T117, ackermannc27_out_ga(T123, T124)) → ACKERMANN1_IN_GGG(T123, T124, T117)
ACKERMANN1_IN_GGG(s(T131), s(s(T133)), T117) → U26_GGG(T131, T133, T117, ackermannc57_in_gga(T131, T133))
U26_GGG(T131, T133, T117, ackermannc57_out_gga(T131, T133, T135)) → U28_GGG(T131, T133, T117, ackermannc68_in_gga(T131, T135))
U28_GGG(T131, T133, T117, ackermannc68_out_gga(T131, T135, T139)) → ACKERMANN1_IN_GGG(T131, T139, T117)

The TRS R consists of the following rules:

ackermannc21_in_ga(T28) → U40_ga(T28, ackermannc27_in_ga(T28))
ackermannc57_in_gga(T79, 0) → U46_gga(T79, ackermannc27_in_ga(T79))
ackermannc57_in_gga(T84, s(T86)) → U47_gga(T84, T86, qc65_in_ggaa(T84, T86))
ackermannc27_in_ga(0) → ackermannc27_out_ga(0, s(s(0)))
ackermannc27_in_ga(s(T32)) → U41_ga(T32, ackermannc21_in_ga(T32))
ackermannc68_in_gga(0, T96) → ackermannc68_out_gga(0, T96, s(T96))
ackermannc68_in_gga(s(T101), 0) → U50_gga(T101, ackermannc27_in_ga(T101))
ackermannc68_in_gga(s(T106), s(T108)) → U51_gga(T106, T108, qc65_in_ggaa(T106, T108))
U40_ga(T28, ackermannc27_out_ga(T28, X73)) → ackermannc21_out_ga(T28, X73)
U46_gga(T79, ackermannc27_out_ga(T79, X211)) → ackermannc57_out_gga(T79, 0, X211)
U47_gga(T84, T86, qc65_out_ggaa(T84, T86, X228, X229)) → ackermannc57_out_gga(T84, s(T86), X229)
U41_ga(T32, ackermannc21_out_ga(T32, T34)) → U42_ga(T32, ackermannc38_in_gga(T32, T34))
U50_gga(T101, ackermannc27_out_ga(T101, X265)) → ackermannc68_out_gga(s(T101), 0, X265)
U51_gga(T106, T108, qc65_out_ggaa(T106, T108, X282, X283)) → ackermannc68_out_gga(s(T106), s(T108), X283)
qc65_in_ggaa(T84, T86) → U48_ggaa(T84, T86, ackermannc57_in_gga(T84, T86))
U42_ga(T32, ackermannc38_out_gga(T32, T34, X97)) → ackermannc27_out_ga(s(T32), X97)
U48_ggaa(T84, T86, ackermannc57_out_gga(T84, T86, T88)) → U49_ggaa(T84, T86, T88, ackermannc68_in_gga(T84, T88))
ackermannc38_in_gga(0, T42) → ackermannc38_out_gga(0, T42, s(T42))
ackermannc38_in_gga(s(T47), 0) → U43_gga(T47, ackermannc27_in_ga(T47))
ackermannc38_in_gga(s(T52), s(T53)) → U44_gga(T52, T53, ackermannc38_in_gga(s(T52), T53))
U49_ggaa(T84, T86, T88, ackermannc68_out_gga(T84, T88, X229)) → qc65_out_ggaa(T84, T86, T88, X229)
U43_gga(T47, ackermannc27_out_ga(T47, X133)) → ackermannc38_out_gga(s(T47), 0, X133)
U44_gga(T52, T53, ackermannc38_out_gga(s(T52), T53, T55)) → U45_gga(T52, T53, ackermannc38_in_gga(T52, T55))
U45_gga(T52, T53, ackermannc38_out_gga(T52, T55, X151)) → ackermannc38_out_gga(s(T52), s(T53), X151)

The set Q consists of the following terms:

ackermannc21_in_ga(x0)
ackermannc57_in_gga(x0, x1)
ackermannc27_in_ga(x0)
ackermannc68_in_gga(x0, x1)
U40_ga(x0, x1)
U46_gga(x0, x1)
U47_gga(x0, x1, x2)
U41_ga(x0, x1)
U50_gga(x0, x1)
U51_gga(x0, x1, x2)
qc65_in_ggaa(x0, x1)
U42_ga(x0, x1)
U48_ggaa(x0, x1, x2)
ackermannc38_in_gga(x0, x1)
U49_ggaa(x0, x1, x2, x3)
U43_gga(x0, x1)
U44_gga(x0, x1, x2)
U45_gga(x0, x1, x2)

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

(33) 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:

U17_GGG(T19, T20, ackermannc21_out_ga(T19, T22)) → ACKERMANN1_IN_GGG(T19, T22, T20)
The graph contains the following edges 1 >= 1, 3 > 1, 3 > 2, 2 >= 3
ACKERMANN1_IN_GGG(s(s(T19)), 0, T20) → U17_GGG(T19, T20, ackermannc21_in_ga(T19))
The graph contains the following edges 1 > 1, 3 >= 2
U26_GGG(T131, T133, T117, ackermannc57_out_gga(T131, T133, T135)) → U28_GGG(T131, T133, T117, ackermannc68_in_gga(T131, T135))
The graph contains the following edges 1 >= 1, 4 > 1, 2 >= 2, 4 > 2, 3 >= 3
U20_GGG(T68, T71, T70, ackermannc57_out_gga(T68, T71, T73)) → ACKERMANN1_IN_GGG(T68, T73, T70)
The graph contains the following edges 1 >= 1, 4 > 1, 4 > 2, 3 >= 3
ACKERMANN1_IN_GGG(s(T68), s(T71), T70) → U20_GGG(T68, T71, T70, ackermannc57_in_gga(T68, T71))
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3
U23_GGG(T123, T117, ackermannc27_out_ga(T123, T124)) → ACKERMANN1_IN_GGG(T123, T124, T117)
The graph contains the following edges 1 >= 1, 3 > 1, 3 > 2, 2 >= 3
U28_GGG(T131, T133, T117, ackermannc68_out_gga(T131, T135, T139)) → ACKERMANN1_IN_GGG(T131, T139, T117)
The graph contains the following edges 1 >= 1, 4 > 1, 4 > 2, 3 >= 3
ACKERMANN1_IN_GGG(s(T123), s(0), T117) → U23_GGG(T123, T117, ackermannc27_in_ga(T123))
The graph contains the following edges 1 > 1, 3 >= 2
ACKERMANN1_IN_GGG(s(T131), s(s(T133)), T117) → U26_GGG(T131, T133, T117, ackermannc57_in_gga(T131, T133))
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3

(0) Obligation:

(1) PrologToDTProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) TriplesToPiDPProof (SOUND transformation)

(4) Obligation:

(5) DependencyGraphProof (EQUIVALENT transformation)

(6) Complex Obligation (AND)

(7) Obligation:

(8) UsableRulesProof (EQUIVALENT transformation)

(9) Obligation:

(10) PiDPToQDPProof (SOUND transformation)

(11) Obligation:

(12) QDPSizeChangeProof (EQUIVALENT transformation)

(13) YES

(14) Obligation:

(15) UsableRulesProof (EQUIVALENT transformation)

(16) Obligation:

(17) PiDPToQDPProof (SOUND transformation)

(18) Obligation:

(19) QDPSizeChangeProof (EQUIVALENT transformation)

(20) YES

(21) Obligation:

(22) UsableRulesProof (EQUIVALENT transformation)

(23) Obligation:

(24) PiDPToQDPProof (SOUND transformation)

(25) Obligation:

(26) NonTerminationProof (EQUIVALENT transformation)

(27) NO

(28) Obligation:

(29) UsableRulesProof (EQUIVALENT transformation)

(30) Obligation:

(31) PiDPToQDPProof (SOUND transformation)

(32) Obligation:

(33) QDPSizeChangeProof (EQUIVALENT transformation)

(34) YES