Left Termination proof of /tmp/tmpsvz7JC/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 PrologToPrologProblemTransformerProof (⇐)

↳2 Prolog

    ↳3 PrologToPiTRSProof (⇐)

        ↳4 PiTRS

        ↳5 DependencyPairsProof (⇔)

        ↳6 PiDP

        ↳7 DependencyGraphProof (⇔)

        ↳8 AND

            ↳9 PiDP

                ↳10 UsableRulesProof (⇔)

                ↳11 PiDP

                ↳12 PiDPToQDPProof (⇐)

                ↳13 QDP

                ↳14 QDPOrderProof (⇔)

                ↳15 QDP

                ↳16 DependencyGraphProof (⇔)

                ↳17 QDP

                ↳18 UsableRulesProof (⇔)

                ↳19 QDP

                ↳20 QReductionProof (⇔)

                ↳21 QDP

                ↳22 NonTerminationProof (⇔)

                ↳23 NO

            ↳24 PiDP

                ↳25 UsableRulesProof (⇔)

                ↳26 PiDP

                ↳27 PiDPToQDPProof (⇐)

                ↳28 QDP

                ↳29 QDPOrderProof (⇔)

                ↳30 QDP

                ↳31 DependencyGraphProof (⇔)

                ↳32 QDP

                ↳33 UsableRulesProof (⇔)

                ↳34 QDP

                ↳35 QReductionProof (⇔)

                ↳36 QDP

                ↳37 NonTerminationProof (⇔)

                ↳38 NO

            ↳39 PiDP

                ↳40 UsableRulesProof (⇔)

                ↳41 PiDP

                ↳42 PiDPToQDPProof (⇐)

                ↳43 QDP

                ↳44 QDPSizeChangeProof (⇔)

                ↳45 YES

    ↳46 PrologToPiTRSProof (⇐)

        ↳47 PiTRS

        ↳48 DependencyPairsProof (⇔)

        ↳49 PiDP

        ↳50 DependencyGraphProof (⇔)

        ↳51 AND

            ↳52 PiDP

                ↳53 UsableRulesProof (⇔)

                ↳54 PiDP

                ↳55 PiDPToQDPProof (⇐)

                ↳56 QDP

                ↳57 QDPOrderProof (⇔)

                ↳58 QDP

                ↳59 DependencyGraphProof (⇔)

                ↳60 QDP

                ↳61 QDPOrderProof (⇔)

                ↳62 QDP

                ↳63 DependencyGraphProof (⇔)

                ↳64 QDP

                ↳65 UsableRulesProof (⇔)

                ↳66 QDP

                ↳67 QReductionProof (⇔)

                ↳68 QDP

                ↳69 NonTerminationProof (⇔)

                ↳70 NO

            ↳71 PiDP

                ↳72 UsableRulesProof (⇔)

                ↳73 PiDP

                ↳74 PiDPToQDPProof (⇐)

                ↳75 QDP

                ↳76 QDPOrderProof (⇔)

                ↳77 QDP

                ↳78 DependencyGraphProof (⇔)

                ↳79 QDP

                ↳80 UsableRulesProof (⇔)

                ↳81 QDP

                ↳82 QReductionProof (⇔)

                ↳83 QDP

                ↳84 NonTerminationProof (⇔)

                ↳85 NO

            ↳86 PiDP

                ↳87 UsableRulesProof (⇔)

                ↳88 PiDP

                ↳89 PiDPToQDPProof (⇐)

                ↳90 QDP

                ↳91 QDPSizeChangeProof (⇔)

                ↳92 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) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog 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:

Clauses:

ackermann21(T28, X73) :- ackermann27(T28, X73).
ackermann27(0, s(s(0))).
ackermann27(s(T32), X97) :- ackermann21(T32, X96).
ackermann27(s(T32), X97) :- ','(ackermann21(T32, T34), ackermann38(T32, T34, X97)).
ackermann38(0, T42, s(T42)).
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) :- ','(ackermann38(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) :- ','(ackermann57(T84, T86, T88), ackermann68(T84, T88, X229)).
ackermann68(0, T96, s(T96)).
ackermann68(s(T101), 0, X265) :- ackermann27(T101, X265).
ackermann68(s(T106), s(T108), X283) :- p65(T106, T108, X282, X283).
ackermann1(0, T5, s(T5)).
ackermann1(s(0), 0, s(s(0))).
ackermann1(s(s(T19)), 0, T20) :- ackermann21(T19, X40).
ackermann1(s(s(T19)), 0, T20) :- ','(ackermann21(T19, T22), ackermann1(T19, T22, T20)).
ackermann1(s(T68), s(T71), T70) :- ackermann57(T68, T71, X182).
ackermann1(s(T68), s(T71), T70) :- ','(ackermann57(T68, T71, T73), ackermann1(T68, T73, T70)).
ackermann1(s(T123), s(0), T117) :- ackermann27(T123, X319).
ackermann1(s(T123), s(0), T117) :- ','(ackermann27(T123, T124), ackermann1(T123, T124, T117)).
ackermann1(s(T131), s(s(T133)), T117) :- ackermann57(T131, T133, X340).
ackermann1(s(T131), s(s(T133)), T117) :- ','(ackermann57(T131, T133, T135), ackermann68(T131, T135, X341)).
ackermann1(s(T131), s(s(T133)), T117) :- ','(ackermann57(T131, T133, T135), ','(ackermann68(T131, T135, T139), ackermann1(T131, T139, T117))).

Queries:

ackermann1(g,a,g).

(3) PrologToPiTRSProof (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)
ackermann21_in: (b,f)
ackermann27_in: (b,f)
ackermann38_in: (b,f,f)
ackermann57_in: (b,f,f)
p65_in: (b,f,f,f)
ackermann68_in: (b,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

ackermann1_in_gag(0, T5, s(T5)) → ackermann1_out_gag(0, T5, s(T5))
ackermann1_in_gag(s(0), 0, s(s(0))) → ackermann1_out_gag(s(0), 0, s(s(0)))
ackermann1_in_gag(s(s(T19)), 0, T20) → U16_gag(T19, T20, ackermann21_in_ga(T19, X40))
ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
U16_gag(T19, T20, ackermann21_out_ga(T19, X40)) → ackermann1_out_gag(s(s(T19)), 0, T20)
ackermann1_in_gag(s(s(T19)), 0, T20) → U17_gag(T19, T20, ackermann21_in_ga(T19, T22))
U17_gag(T19, T20, ackermann21_out_ga(T19, T22)) → U18_gag(T19, T20, ackermann1_in_gag(T19, T22, T20))
ackermann1_in_gag(s(T68), s(T71), T70) → U19_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, X182))
ackermann57_in_gaa(T79, 0, X211) → U9_gaa(T79, X211, ackermann27_in_ga(T79, X211))
U9_gaa(T79, X211, ackermann27_out_ga(T79, X211)) → ackermann57_out_gaa(T79, 0, X211)
ackermann57_in_gaa(T84, s(T86), X229) → U10_gaa(T84, 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))
U11_gaaa(T84, T86, X228, X229, ackermann57_out_gaa(T84, T86, X228)) → p65_out_gaaa(T84, T86, X228, X229)
p65_in_gaaa(T84, T86, T88, X229) → U12_gaaa(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U12_gaaa(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_gaaa(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
ackermann68_in_gaa(0, T96, s(T96)) → ackermann68_out_gaa(0, T96, s(T96))
ackermann68_in_gaa(s(T101), 0, X265) → U14_gaa(T101, X265, ackermann27_in_ga(T101, X265))
U14_gaa(T101, X265, ackermann27_out_ga(T101, X265)) → ackermann68_out_gaa(s(T101), 0, X265)
ackermann68_in_gaa(s(T106), s(T108), X283) → U15_gaa(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
U15_gaa(T106, T108, X283, p65_out_gaaa(T106, T108, X282, X283)) → ackermann68_out_gaa(s(T106), s(T108), X283)
U13_gaaa(T84, T86, T88, X229, ackermann68_out_gaa(T84, T88, X229)) → p65_out_gaaa(T84, T86, T88, X229)
U10_gaa(T84, T86, X229, p65_out_gaaa(T84, T86, X228, X229)) → ackermann57_out_gaa(T84, s(T86), X229)
U19_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, X182)) → ackermann1_out_gag(s(T68), s(T71), T70)
ackermann1_in_gag(s(T68), s(T71), T70) → U20_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → U21_gag(T68, T71, T70, ackermann1_in_gag(T68, T73, T70))
ackermann1_in_gag(s(T123), s(0), T117) → U22_gag(T123, T117, ackermann27_in_ga(T123, X319))
U22_gag(T123, T117, ackermann27_out_ga(T123, X319)) → ackermann1_out_gag(s(T123), s(0), T117)
ackermann1_in_gag(s(T123), s(0), T117) → U23_gag(T123, T117, ackermann27_in_ga(T123, T124))
U23_gag(T123, T117, ackermann27_out_ga(T123, T124)) → U24_gag(T123, T117, ackermann1_in_gag(T123, T124, T117))
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U25_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, X340))
U25_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, X340)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U26_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U27_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, X341))
U27_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, X341)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → U29_gag(T131, T133, T117, ackermann1_in_gag(T131, T139, T117))
U29_gag(T131, T133, T117, ackermann1_out_gag(T131, T139, T117)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U24_gag(T123, T117, ackermann1_out_gag(T123, T124, T117)) → ackermann1_out_gag(s(T123), s(0), T117)
U21_gag(T68, T71, T70, ackermann1_out_gag(T68, T73, T70)) → ackermann1_out_gag(s(T68), s(T71), T70)
U18_gag(T19, T20, ackermann1_out_gag(T19, T22, T20)) → ackermann1_out_gag(s(s(T19)), 0, T20)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

ackermann1_in_gag(0, T5, s(T5)) → ackermann1_out_gag(0, T5, s(T5))
ackermann1_in_gag(s(0), 0, s(s(0))) → ackermann1_out_gag(s(0), 0, s(s(0)))
ackermann1_in_gag(s(s(T19)), 0, T20) → U16_gag(T19, T20, ackermann21_in_ga(T19, X40))
ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
U16_gag(T19, T20, ackermann21_out_ga(T19, X40)) → ackermann1_out_gag(s(s(T19)), 0, T20)
ackermann1_in_gag(s(s(T19)), 0, T20) → U17_gag(T19, T20, ackermann21_in_ga(T19, T22))
U17_gag(T19, T20, ackermann21_out_ga(T19, T22)) → U18_gag(T19, T20, ackermann1_in_gag(T19, T22, T20))
ackermann1_in_gag(s(T68), s(T71), T70) → U19_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, X182))
ackermann57_in_gaa(T79, 0, X211) → U9_gaa(T79, X211, ackermann27_in_ga(T79, X211))
U9_gaa(T79, X211, ackermann27_out_ga(T79, X211)) → ackermann57_out_gaa(T79, 0, X211)
ackermann57_in_gaa(T84, s(T86), X229) → U10_gaa(T84, 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))
U11_gaaa(T84, T86, X228, X229, ackermann57_out_gaa(T84, T86, X228)) → p65_out_gaaa(T84, T86, X228, X229)
p65_in_gaaa(T84, T86, T88, X229) → U12_gaaa(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U12_gaaa(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_gaaa(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
ackermann68_in_gaa(0, T96, s(T96)) → ackermann68_out_gaa(0, T96, s(T96))
ackermann68_in_gaa(s(T101), 0, X265) → U14_gaa(T101, X265, ackermann27_in_ga(T101, X265))
U14_gaa(T101, X265, ackermann27_out_ga(T101, X265)) → ackermann68_out_gaa(s(T101), 0, X265)
ackermann68_in_gaa(s(T106), s(T108), X283) → U15_gaa(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
U15_gaa(T106, T108, X283, p65_out_gaaa(T106, T108, X282, X283)) → ackermann68_out_gaa(s(T106), s(T108), X283)
U13_gaaa(T84, T86, T88, X229, ackermann68_out_gaa(T84, T88, X229)) → p65_out_gaaa(T84, T86, T88, X229)
U10_gaa(T84, T86, X229, p65_out_gaaa(T84, T86, X228, X229)) → ackermann57_out_gaa(T84, s(T86), X229)
U19_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, X182)) → ackermann1_out_gag(s(T68), s(T71), T70)
ackermann1_in_gag(s(T68), s(T71), T70) → U20_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → U21_gag(T68, T71, T70, ackermann1_in_gag(T68, T73, T70))
ackermann1_in_gag(s(T123), s(0), T117) → U22_gag(T123, T117, ackermann27_in_ga(T123, X319))
U22_gag(T123, T117, ackermann27_out_ga(T123, X319)) → ackermann1_out_gag(s(T123), s(0), T117)
ackermann1_in_gag(s(T123), s(0), T117) → U23_gag(T123, T117, ackermann27_in_ga(T123, T124))
U23_gag(T123, T117, ackermann27_out_ga(T123, T124)) → U24_gag(T123, T117, ackermann1_in_gag(T123, T124, T117))
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U25_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, X340))
U25_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, X340)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U26_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U27_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, X341))
U27_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, X341)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → U29_gag(T131, T133, T117, ackermann1_in_gag(T131, T139, T117))
U29_gag(T131, T133, T117, ackermann1_out_gag(T131, T139, T117)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U24_gag(T123, T117, ackermann1_out_gag(T123, T124, T117)) → ackermann1_out_gag(s(T123), s(0), T117)
U21_gag(T68, T71, T70, ackermann1_out_gag(T68, T73, T70)) → ackermann1_out_gag(s(T68), s(T71), T70)
U18_gag(T19, T20, ackermann1_out_gag(T19, T22, T20)) → ackermann1_out_gag(s(s(T19)), 0, T20)

(5) DependencyPairsProof (EQUIVALENT transformation)

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

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, ackermann21_in_ga(T32, T34))
U3_GA(T32, X97, ackermann21_out_ga(T32, T34)) → U4_GA(T32, X97, ackermann38_in_gaa(T32, T34, X97))
U3_GA(T32, X97, ackermann21_out_ga(T32, T34)) → ACKERMANN38_IN_GAA(T32, T34, X97)
ACKERMANN38_IN_GAA(s(T47), 0, X133) → U5_GAA(T47, X133, ackermann27_in_ga(T47, X133))
ACKERMANN38_IN_GAA(s(T47), 0, X133) → ACKERMANN27_IN_GA(T47, X133)
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → U6_GAA(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → ACKERMANN38_IN_GAA(s(T52), T53, X150)
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → U7_GAA(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_GAA(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_GAA(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U7_GAA(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → ACKERMANN38_IN_GAA(T52, T55, X151)
ACKERMANN1_IN_GAG(s(s(T19)), 0, T20) → U17_GAG(T19, T20, ackermann21_in_ga(T19, T22))
U17_GAG(T19, T20, ackermann21_out_ga(T19, T22)) → U18_GAG(T19, T20, ackermann1_in_gag(T19, T22, T20))
U17_GAG(T19, T20, ackermann21_out_ga(T19, T22)) → ACKERMANN1_IN_GAG(T19, T22, T20)
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, ackermann57_in_gaa(T84, T86, T88))
U12_GAAA(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_GAAA(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
U12_GAAA(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → ACKERMANN68_IN_GAA(T84, T88, X229)
ACKERMANN68_IN_GAA(s(T101), 0, X265) → U14_GAA(T101, X265, ackermann27_in_ga(T101, X265))
ACKERMANN68_IN_GAA(s(T101), 0, X265) → ACKERMANN27_IN_GA(T101, X265)
ACKERMANN68_IN_GAA(s(T106), s(T108), X283) → U15_GAA(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
ACKERMANN68_IN_GAA(s(T106), s(T108), X283) → P65_IN_GAAA(T106, T108, X282, X283)
ACKERMANN1_IN_GAG(s(T68), s(T71), T70) → U20_GAG(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_GAG(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → U21_GAG(T68, T71, T70, ackermann1_in_gag(T68, T73, T70))
U20_GAG(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → ACKERMANN1_IN_GAG(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, ackermann27_in_ga(T123, T124))
U23_GAG(T123, T117, ackermann27_out_ga(T123, T124)) → U24_GAG(T123, T117, ackermann1_in_gag(T123, T124, T117))
U23_GAG(T123, T117, ackermann27_out_ga(T123, T124)) → ACKERMANN1_IN_GAG(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, ackermann57_in_gaa(T131, T133, T135))
U26_GAG(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U27_GAG(T131, T133, T117, ackermann68_in_gaa(T131, T135, X341))
U26_GAG(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → ACKERMANN68_IN_GAA(T131, T135, X341)
U26_GAG(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_GAG(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_GAG(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → U29_GAG(T131, T133, T117, ackermann1_in_gag(T131, T139, T117))
U28_GAG(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → ACKERMANN1_IN_GAG(T131, T139, T117)

The TRS R consists of the following rules:

ackermann1_in_gag(0, T5, s(T5)) → ackermann1_out_gag(0, T5, s(T5))
ackermann1_in_gag(s(0), 0, s(s(0))) → ackermann1_out_gag(s(0), 0, s(s(0)))
ackermann1_in_gag(s(s(T19)), 0, T20) → U16_gag(T19, T20, ackermann21_in_ga(T19, X40))
ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
U16_gag(T19, T20, ackermann21_out_ga(T19, X40)) → ackermann1_out_gag(s(s(T19)), 0, T20)
ackermann1_in_gag(s(s(T19)), 0, T20) → U17_gag(T19, T20, ackermann21_in_ga(T19, T22))
U17_gag(T19, T20, ackermann21_out_ga(T19, T22)) → U18_gag(T19, T20, ackermann1_in_gag(T19, T22, T20))
ackermann1_in_gag(s(T68), s(T71), T70) → U19_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, X182))
ackermann57_in_gaa(T79, 0, X211) → U9_gaa(T79, X211, ackermann27_in_ga(T79, X211))
U9_gaa(T79, X211, ackermann27_out_ga(T79, X211)) → ackermann57_out_gaa(T79, 0, X211)
ackermann57_in_gaa(T84, s(T86), X229) → U10_gaa(T84, 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))
U11_gaaa(T84, T86, X228, X229, ackermann57_out_gaa(T84, T86, X228)) → p65_out_gaaa(T84, T86, X228, X229)
p65_in_gaaa(T84, T86, T88, X229) → U12_gaaa(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U12_gaaa(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_gaaa(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
ackermann68_in_gaa(0, T96, s(T96)) → ackermann68_out_gaa(0, T96, s(T96))
ackermann68_in_gaa(s(T101), 0, X265) → U14_gaa(T101, X265, ackermann27_in_ga(T101, X265))
U14_gaa(T101, X265, ackermann27_out_ga(T101, X265)) → ackermann68_out_gaa(s(T101), 0, X265)
ackermann68_in_gaa(s(T106), s(T108), X283) → U15_gaa(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
U15_gaa(T106, T108, X283, p65_out_gaaa(T106, T108, X282, X283)) → ackermann68_out_gaa(s(T106), s(T108), X283)
U13_gaaa(T84, T86, T88, X229, ackermann68_out_gaa(T84, T88, X229)) → p65_out_gaaa(T84, T86, T88, X229)
U10_gaa(T84, T86, X229, p65_out_gaaa(T84, T86, X228, X229)) → ackermann57_out_gaa(T84, s(T86), X229)
U19_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, X182)) → ackermann1_out_gag(s(T68), s(T71), T70)
ackermann1_in_gag(s(T68), s(T71), T70) → U20_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → U21_gag(T68, T71, T70, ackermann1_in_gag(T68, T73, T70))
ackermann1_in_gag(s(T123), s(0), T117) → U22_gag(T123, T117, ackermann27_in_ga(T123, X319))
U22_gag(T123, T117, ackermann27_out_ga(T123, X319)) → ackermann1_out_gag(s(T123), s(0), T117)
ackermann1_in_gag(s(T123), s(0), T117) → U23_gag(T123, T117, ackermann27_in_ga(T123, T124))
U23_gag(T123, T117, ackermann27_out_ga(T123, T124)) → U24_gag(T123, T117, ackermann1_in_gag(T123, T124, T117))
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U25_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, X340))
U25_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, X340)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U26_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U27_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, X341))
U27_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, X341)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → U29_gag(T131, T133, T117, ackermann1_in_gag(T131, T139, T117))
U29_gag(T131, T133, T117, ackermann1_out_gag(T131, T139, T117)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U24_gag(T123, T117, ackermann1_out_gag(T123, T124, T117)) → ackermann1_out_gag(s(T123), s(0), T117)
U21_gag(T68, T71, T70, ackermann1_out_gag(T68, T73, T70)) → ackermann1_out_gag(s(T68), s(T71), T70)
U18_gag(T19, T20, ackermann1_out_gag(T19, T22, T20)) → ackermann1_out_gag(s(s(T19)), 0, T20)

The argument filtering Pi contains the following mapping:
ackermann1_in_gag(x1, x2, x3) = ackermann1_in_gag(x1, x3)
0 = 0
s(x1) = s(x1)
ackermann1_out_gag(x1, x2, x3) = ackermann1_out_gag(x1, x2, 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)
ackermann27_out_ga(x1, x2) = ackermann27_out_ga(x1)
U2_ga(x1, x2, x3) = U2_ga(x1, x3)
ackermann21_out_ga(x1, x2) = ackermann21_out_ga(x1)
U3_ga(x1, x2, x3) = U3_ga(x1, x3)
U4_ga(x1, x2, x3) = U4_ga(x1, x3)
ackermann38_in_gaa(x1, x2, x3) = ackermann38_in_gaa(x1)
ackermann38_out_gaa(x1, x2, x3) = ackermann38_out_gaa(x1)
U5_gaa(x1, x2, x3) = U5_gaa(x1, x3)
U6_gaa(x1, x2, x3, x4) = U6_gaa(x1, x4)
U7_gaa(x1, x2, x3, x4) = U7_gaa(x1, x4)
U8_gaa(x1, x2, x3, x4) = U8_gaa(x1, x4)
U17_gag(x1, x2, x3) = U17_gag(x1, x2, x3)
U18_gag(x1, x2, x3) = U18_gag(x1, x2, x3)
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)
ackermann57_out_gaa(x1, x2, x3) = ackermann57_out_gaa(x1, x2)
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)
p65_out_gaaa(x1, x2, x3, x4) = p65_out_gaaa(x1, x2)
U12_gaaa(x1, x2, x3, x4, x5) = U12_gaaa(x1, x5)
U13_gaaa(x1, x2, x3, x4, x5) = U13_gaaa(x1, x2, x5)
ackermann68_in_gaa(x1, x2, x3) = ackermann68_in_gaa(x1)
ackermann68_out_gaa(x1, x2, x3) = ackermann68_out_gaa(x1)
U14_gaa(x1, x2, x3) = U14_gaa(x1, x3)
U15_gaa(x1, x2, x3, x4) = U15_gaa(x1, x4)
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)
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_GAA(x1, x2, x3) = ACKERMANN38_IN_GAA(x1)
U5_GAA(x1, x2, x3) = U5_GAA(x1, x3)
U6_GAA(x1, x2, x3, x4) = U6_GAA(x1, x4)
U7_GAA(x1, x2, x3, x4) = U7_GAA(x1, x4)
U8_GAA(x1, x2, x3, x4) = U8_GAA(x1, x4)
U17_GAG(x1, x2, x3) = U17_GAG(x1, x2, x3)
U18_GAG(x1, x2, x3) = U18_GAG(x1, x2, x3)
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)
ACKERMANN68_IN_GAA(x1, x2, x3) = ACKERMANN68_IN_GAA(x1)
U14_GAA(x1, x2, x3) = U14_GAA(x1, x3)
U15_GAA(x1, x2, x3, x4) = U15_GAA(x1, x4)
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

(6) 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, ackermann21_in_ga(T32, T34))
U3_GA(T32, X97, ackermann21_out_ga(T32, T34)) → U4_GA(T32, X97, ackermann38_in_gaa(T32, T34, X97))
U3_GA(T32, X97, ackermann21_out_ga(T32, T34)) → ACKERMANN38_IN_GAA(T32, T34, X97)
ACKERMANN38_IN_GAA(s(T47), 0, X133) → U5_GAA(T47, X133, ackermann27_in_ga(T47, X133))
ACKERMANN38_IN_GAA(s(T47), 0, X133) → ACKERMANN27_IN_GA(T47, X133)
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → U6_GAA(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → ACKERMANN38_IN_GAA(s(T52), T53, X150)
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → U7_GAA(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_GAA(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_GAA(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U7_GAA(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → ACKERMANN38_IN_GAA(T52, T55, X151)
ACKERMANN1_IN_GAG(s(s(T19)), 0, T20) → U17_GAG(T19, T20, ackermann21_in_ga(T19, T22))
U17_GAG(T19, T20, ackermann21_out_ga(T19, T22)) → U18_GAG(T19, T20, ackermann1_in_gag(T19, T22, T20))
U17_GAG(T19, T20, ackermann21_out_ga(T19, T22)) → ACKERMANN1_IN_GAG(T19, T22, T20)
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, ackermann57_in_gaa(T84, T86, T88))
U12_GAAA(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_GAAA(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
U12_GAAA(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → ACKERMANN68_IN_GAA(T84, T88, X229)
ACKERMANN68_IN_GAA(s(T101), 0, X265) → U14_GAA(T101, X265, ackermann27_in_ga(T101, X265))
ACKERMANN68_IN_GAA(s(T101), 0, X265) → ACKERMANN27_IN_GA(T101, X265)
ACKERMANN68_IN_GAA(s(T106), s(T108), X283) → U15_GAA(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
ACKERMANN68_IN_GAA(s(T106), s(T108), X283) → P65_IN_GAAA(T106, T108, X282, X283)
ACKERMANN1_IN_GAG(s(T68), s(T71), T70) → U20_GAG(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_GAG(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → U21_GAG(T68, T71, T70, ackermann1_in_gag(T68, T73, T70))
U20_GAG(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → ACKERMANN1_IN_GAG(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, ackermann27_in_ga(T123, T124))
U23_GAG(T123, T117, ackermann27_out_ga(T123, T124)) → U24_GAG(T123, T117, ackermann1_in_gag(T123, T124, T117))
U23_GAG(T123, T117, ackermann27_out_ga(T123, T124)) → ACKERMANN1_IN_GAG(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, ackermann57_in_gaa(T131, T133, T135))
U26_GAG(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U27_GAG(T131, T133, T117, ackermann68_in_gaa(T131, T135, X341))
U26_GAG(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → ACKERMANN68_IN_GAA(T131, T135, X341)
U26_GAG(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_GAG(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_GAG(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → U29_GAG(T131, T133, T117, ackermann1_in_gag(T131, T139, T117))
U28_GAG(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → ACKERMANN1_IN_GAG(T131, T139, T117)

The TRS R consists of the following rules:

ackermann1_in_gag(0, T5, s(T5)) → ackermann1_out_gag(0, T5, s(T5))
ackermann1_in_gag(s(0), 0, s(s(0))) → ackermann1_out_gag(s(0), 0, s(s(0)))
ackermann1_in_gag(s(s(T19)), 0, T20) → U16_gag(T19, T20, ackermann21_in_ga(T19, X40))
ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
U16_gag(T19, T20, ackermann21_out_ga(T19, X40)) → ackermann1_out_gag(s(s(T19)), 0, T20)
ackermann1_in_gag(s(s(T19)), 0, T20) → U17_gag(T19, T20, ackermann21_in_ga(T19, T22))
U17_gag(T19, T20, ackermann21_out_ga(T19, T22)) → U18_gag(T19, T20, ackermann1_in_gag(T19, T22, T20))
ackermann1_in_gag(s(T68), s(T71), T70) → U19_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, X182))
ackermann57_in_gaa(T79, 0, X211) → U9_gaa(T79, X211, ackermann27_in_ga(T79, X211))
U9_gaa(T79, X211, ackermann27_out_ga(T79, X211)) → ackermann57_out_gaa(T79, 0, X211)
ackermann57_in_gaa(T84, s(T86), X229) → U10_gaa(T84, 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))
U11_gaaa(T84, T86, X228, X229, ackermann57_out_gaa(T84, T86, X228)) → p65_out_gaaa(T84, T86, X228, X229)
p65_in_gaaa(T84, T86, T88, X229) → U12_gaaa(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U12_gaaa(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_gaaa(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
ackermann68_in_gaa(0, T96, s(T96)) → ackermann68_out_gaa(0, T96, s(T96))
ackermann68_in_gaa(s(T101), 0, X265) → U14_gaa(T101, X265, ackermann27_in_ga(T101, X265))
U14_gaa(T101, X265, ackermann27_out_ga(T101, X265)) → ackermann68_out_gaa(s(T101), 0, X265)
ackermann68_in_gaa(s(T106), s(T108), X283) → U15_gaa(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
U15_gaa(T106, T108, X283, p65_out_gaaa(T106, T108, X282, X283)) → ackermann68_out_gaa(s(T106), s(T108), X283)
U13_gaaa(T84, T86, T88, X229, ackermann68_out_gaa(T84, T88, X229)) → p65_out_gaaa(T84, T86, T88, X229)
U10_gaa(T84, T86, X229, p65_out_gaaa(T84, T86, X228, X229)) → ackermann57_out_gaa(T84, s(T86), X229)
U19_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, X182)) → ackermann1_out_gag(s(T68), s(T71), T70)
ackermann1_in_gag(s(T68), s(T71), T70) → U20_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → U21_gag(T68, T71, T70, ackermann1_in_gag(T68, T73, T70))
ackermann1_in_gag(s(T123), s(0), T117) → U22_gag(T123, T117, ackermann27_in_ga(T123, X319))
U22_gag(T123, T117, ackermann27_out_ga(T123, X319)) → ackermann1_out_gag(s(T123), s(0), T117)
ackermann1_in_gag(s(T123), s(0), T117) → U23_gag(T123, T117, ackermann27_in_ga(T123, T124))
U23_gag(T123, T117, ackermann27_out_ga(T123, T124)) → U24_gag(T123, T117, ackermann1_in_gag(T123, T124, T117))
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U25_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, X340))
U25_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, X340)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U26_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U27_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, X341))
U27_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, X341)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → U29_gag(T131, T133, T117, ackermann1_in_gag(T131, T139, T117))
U29_gag(T131, T133, T117, ackermann1_out_gag(T131, T139, T117)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U24_gag(T123, T117, ackermann1_out_gag(T123, T124, T117)) → ackermann1_out_gag(s(T123), s(0), T117)
U21_gag(T68, T71, T70, ackermann1_out_gag(T68, T73, T70)) → ackermann1_out_gag(s(T68), s(T71), T70)
U18_gag(T19, T20, ackermann1_out_gag(T19, T22, T20)) → ackermann1_out_gag(s(s(T19)), 0, T20)

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 28 less nodes.

(8) Complex Obligation (AND)

(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, ackermann21_in_ga(T32, T34))
U3_GA(T32, X97, ackermann21_out_ga(T32, T34)) → ACKERMANN38_IN_GAA(T32, T34, X97)
ACKERMANN38_IN_GAA(s(T47), 0, X133) → ACKERMANN27_IN_GA(T47, X133)
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → ACKERMANN38_IN_GAA(s(T52), T53, X150)
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → U7_GAA(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_GAA(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → ACKERMANN38_IN_GAA(T52, T55, X151)

The TRS R consists of the following rules:

ackermann1_in_gag(0, T5, s(T5)) → ackermann1_out_gag(0, T5, s(T5))
ackermann1_in_gag(s(0), 0, s(s(0))) → ackermann1_out_gag(s(0), 0, s(s(0)))
ackermann1_in_gag(s(s(T19)), 0, T20) → U16_gag(T19, T20, ackermann21_in_ga(T19, X40))
ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
U16_gag(T19, T20, ackermann21_out_ga(T19, X40)) → ackermann1_out_gag(s(s(T19)), 0, T20)
ackermann1_in_gag(s(s(T19)), 0, T20) → U17_gag(T19, T20, ackermann21_in_ga(T19, T22))
U17_gag(T19, T20, ackermann21_out_ga(T19, T22)) → U18_gag(T19, T20, ackermann1_in_gag(T19, T22, T20))
ackermann1_in_gag(s(T68), s(T71), T70) → U19_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, X182))
ackermann57_in_gaa(T79, 0, X211) → U9_gaa(T79, X211, ackermann27_in_ga(T79, X211))
U9_gaa(T79, X211, ackermann27_out_ga(T79, X211)) → ackermann57_out_gaa(T79, 0, X211)
ackermann57_in_gaa(T84, s(T86), X229) → U10_gaa(T84, 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))
U11_gaaa(T84, T86, X228, X229, ackermann57_out_gaa(T84, T86, X228)) → p65_out_gaaa(T84, T86, X228, X229)
p65_in_gaaa(T84, T86, T88, X229) → U12_gaaa(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U12_gaaa(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_gaaa(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
ackermann68_in_gaa(0, T96, s(T96)) → ackermann68_out_gaa(0, T96, s(T96))
ackermann68_in_gaa(s(T101), 0, X265) → U14_gaa(T101, X265, ackermann27_in_ga(T101, X265))
U14_gaa(T101, X265, ackermann27_out_ga(T101, X265)) → ackermann68_out_gaa(s(T101), 0, X265)
ackermann68_in_gaa(s(T106), s(T108), X283) → U15_gaa(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
U15_gaa(T106, T108, X283, p65_out_gaaa(T106, T108, X282, X283)) → ackermann68_out_gaa(s(T106), s(T108), X283)
U13_gaaa(T84, T86, T88, X229, ackermann68_out_gaa(T84, T88, X229)) → p65_out_gaaa(T84, T86, T88, X229)
U10_gaa(T84, T86, X229, p65_out_gaaa(T84, T86, X228, X229)) → ackermann57_out_gaa(T84, s(T86), X229)
U19_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, X182)) → ackermann1_out_gag(s(T68), s(T71), T70)
ackermann1_in_gag(s(T68), s(T71), T70) → U20_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → U21_gag(T68, T71, T70, ackermann1_in_gag(T68, T73, T70))
ackermann1_in_gag(s(T123), s(0), T117) → U22_gag(T123, T117, ackermann27_in_ga(T123, X319))
U22_gag(T123, T117, ackermann27_out_ga(T123, X319)) → ackermann1_out_gag(s(T123), s(0), T117)
ackermann1_in_gag(s(T123), s(0), T117) → U23_gag(T123, T117, ackermann27_in_ga(T123, T124))
U23_gag(T123, T117, ackermann27_out_ga(T123, T124)) → U24_gag(T123, T117, ackermann1_in_gag(T123, T124, T117))
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U25_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, X340))
U25_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, X340)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U26_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U27_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, X341))
U27_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, X341)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → U29_gag(T131, T133, T117, ackermann1_in_gag(T131, T139, T117))
U29_gag(T131, T133, T117, ackermann1_out_gag(T131, T139, T117)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U24_gag(T123, T117, ackermann1_out_gag(T123, T124, T117)) → ackermann1_out_gag(s(T123), s(0), T117)
U21_gag(T68, T71, T70, ackermann1_out_gag(T68, T73, T70)) → ackermann1_out_gag(s(T68), s(T71), T70)
U18_gag(T19, T20, ackermann1_out_gag(T19, T22, T20)) → ackermann1_out_gag(s(s(T19)), 0, T20)

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

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, ackermann21_in_ga(T32, T34))
U3_GA(T32, X97, ackermann21_out_ga(T32, T34)) → ACKERMANN38_IN_GAA(T32, T34, X97)
ACKERMANN38_IN_GAA(s(T47), 0, X133) → ACKERMANN27_IN_GA(T47, X133)
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → ACKERMANN38_IN_GAA(s(T52), T53, X150)
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → U7_GAA(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_GAA(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → ACKERMANN38_IN_GAA(T52, T55, X151)

The TRS R consists of the following rules:

ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

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, ackermann21_in_ga(T32))
U3_GA(T32, ackermann21_out_ga(T32)) → ACKERMANN38_IN_GAA(T32)
ACKERMANN38_IN_GAA(s(T47)) → ACKERMANN27_IN_GA(T47)
ACKERMANN38_IN_GAA(s(T52)) → ACKERMANN38_IN_GAA(s(T52))
ACKERMANN38_IN_GAA(s(T52)) → U7_GAA(T52, ackermann38_in_gaa(s(T52)))
U7_GAA(T52, ackermann38_out_gaa(s(T52))) → ACKERMANN38_IN_GAA(T52)

The TRS R consists of the following rules:

ackermann21_in_ga(T28) → U1_ga(T28, ackermann27_in_ga(T28))
ackermann38_in_gaa(s(T47)) → U5_gaa(T47, ackermann27_in_ga(T47))
ackermann38_in_gaa(s(T52)) → U6_gaa(T52, ackermann38_in_gaa(s(T52)))
ackermann38_in_gaa(s(T52)) → U7_gaa(T52, ackermann38_in_gaa(s(T52)))
U1_ga(T28, ackermann27_out_ga(T28)) → ackermann21_out_ga(T28)
U5_gaa(T47, ackermann27_out_ga(T47)) → ackermann38_out_gaa(s(T47))
U6_gaa(T52, ackermann38_out_gaa(s(T52))) → ackermann38_out_gaa(s(T52))
U7_gaa(T52, ackermann38_out_gaa(s(T52))) → U8_gaa(T52, ackermann38_in_gaa(T52))
ackermann27_in_ga(0) → ackermann27_out_ga(0)
ackermann27_in_ga(s(T32)) → U2_ga(T32, ackermann21_in_ga(T32))
ackermann27_in_ga(s(T32)) → U3_ga(T32, ackermann21_in_ga(T32))
U8_gaa(T52, ackermann38_out_gaa(T52)) → ackermann38_out_gaa(s(T52))
U2_ga(T32, ackermann21_out_ga(T32)) → ackermann27_out_ga(s(T32))
U3_ga(T32, ackermann21_out_ga(T32)) → U4_ga(T32, ackermann38_in_gaa(T32))
ackermann38_in_gaa(0) → ackermann38_out_gaa(0)
U4_ga(T32, ackermann38_out_gaa(T32)) → ackermann27_out_ga(s(T32))

The set Q consists of the following terms:

ackermann21_in_ga(x0)
ackermann38_in_gaa(x0)
U1_ga(x0, x1)
U5_gaa(x0, x1)
U6_gaa(x0, x1)
U7_gaa(x0, x1)
ackermann27_in_ga(x0)
U8_gaa(x0, x1)
U2_ga(x0, x1)
U3_ga(x0, x1)
U4_ga(x0, x1)

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

(14) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACKERMANN27_IN_GA(s(T32)) → ACKERMANN21_IN_GA(T32)
U3_GA(T32, ackermann21_out_ga(T32)) → ACKERMANN38_IN_GAA(T32)
ACKERMANN38_IN_GAA(s(T47)) → ACKERMANN27_IN_GA(T47)
U7_GAA(T52, ackermann38_out_gaa(s(T52))) → ACKERMANN38_IN_GAA(T52)

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

POL(0) = 1
POL(ACKERMANN21_IN_GA(x₁)) = x₁
POL(ACKERMANN27_IN_GA(x₁)) = x₁
POL(ACKERMANN38_IN_GAA(x₁)) = x₁
POL(U1_ga(x₁, x₂)) = 0
POL(U2_ga(x₁, x₂)) = 0
POL(U3_GA(x₁, x₂)) = 1 + x₁
POL(U3_ga(x₁, x₂)) = 0
POL(U4_ga(x₁, x₂)) = 0
POL(U5_gaa(x₁, x₂)) = 0
POL(U6_gaa(x₁, x₂)) = 0
POL(U7_GAA(x₁, x₂)) = 1 + x₁
POL(U7_gaa(x₁, x₂)) = 0
POL(U8_gaa(x₁, x₂)) = 0
POL(ackermann21_in_ga(x₁)) = 0
POL(ackermann21_out_ga(x₁)) = 0
POL(ackermann27_in_ga(x₁)) = 1 + x₁
POL(ackermann27_out_ga(x₁)) = 1
POL(ackermann38_in_gaa(x₁)) = 1 + x₁
POL(ackermann38_out_gaa(x₁)) = 1
POL(s(x₁)) = 1 + x₁

The following usable rules [FROCOS05] were oriented: none

(15) 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)) → U3_GA(T32, ackermann21_in_ga(T32))
ACKERMANN38_IN_GAA(s(T52)) → ACKERMANN38_IN_GAA(s(T52))
ACKERMANN38_IN_GAA(s(T52)) → U7_GAA(T52, ackermann38_in_gaa(s(T52)))

The TRS R consists of the following rules:

ackermann21_in_ga(T28) → U1_ga(T28, ackermann27_in_ga(T28))
ackermann38_in_gaa(s(T47)) → U5_gaa(T47, ackermann27_in_ga(T47))
ackermann38_in_gaa(s(T52)) → U6_gaa(T52, ackermann38_in_gaa(s(T52)))
ackermann38_in_gaa(s(T52)) → U7_gaa(T52, ackermann38_in_gaa(s(T52)))
U1_ga(T28, ackermann27_out_ga(T28)) → ackermann21_out_ga(T28)
U5_gaa(T47, ackermann27_out_ga(T47)) → ackermann38_out_gaa(s(T47))
U6_gaa(T52, ackermann38_out_gaa(s(T52))) → ackermann38_out_gaa(s(T52))
U7_gaa(T52, ackermann38_out_gaa(s(T52))) → U8_gaa(T52, ackermann38_in_gaa(T52))
ackermann27_in_ga(0) → ackermann27_out_ga(0)
ackermann27_in_ga(s(T32)) → U2_ga(T32, ackermann21_in_ga(T32))
ackermann27_in_ga(s(T32)) → U3_ga(T32, ackermann21_in_ga(T32))
U8_gaa(T52, ackermann38_out_gaa(T52)) → ackermann38_out_gaa(s(T52))
U2_ga(T32, ackermann21_out_ga(T32)) → ackermann27_out_ga(s(T32))
U3_ga(T32, ackermann21_out_ga(T32)) → U4_ga(T32, ackermann38_in_gaa(T32))
ackermann38_in_gaa(0) → ackermann38_out_gaa(0)
U4_ga(T32, ackermann38_out_gaa(T32)) → ackermann27_out_ga(s(T32))

The set Q consists of the following terms:

ackermann21_in_ga(x0)
ackermann38_in_gaa(x0)
U1_ga(x0, x1)
U5_gaa(x0, x1)
U6_gaa(x0, x1)
U7_gaa(x0, x1)
ackermann27_in_ga(x0)
U8_gaa(x0, x1)
U2_ga(x0, x1)
U3_ga(x0, x1)
U4_ga(x0, x1)

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

(16) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

(17) Obligation:

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

ACKERMANN38_IN_GAA(s(T52)) → ACKERMANN38_IN_GAA(s(T52))

The TRS R consists of the following rules:

ackermann21_in_ga(T28) → U1_ga(T28, ackermann27_in_ga(T28))
ackermann38_in_gaa(s(T47)) → U5_gaa(T47, ackermann27_in_ga(T47))
ackermann38_in_gaa(s(T52)) → U6_gaa(T52, ackermann38_in_gaa(s(T52)))
ackermann38_in_gaa(s(T52)) → U7_gaa(T52, ackermann38_in_gaa(s(T52)))
U1_ga(T28, ackermann27_out_ga(T28)) → ackermann21_out_ga(T28)
U5_gaa(T47, ackermann27_out_ga(T47)) → ackermann38_out_gaa(s(T47))
U6_gaa(T52, ackermann38_out_gaa(s(T52))) → ackermann38_out_gaa(s(T52))
U7_gaa(T52, ackermann38_out_gaa(s(T52))) → U8_gaa(T52, ackermann38_in_gaa(T52))
ackermann27_in_ga(0) → ackermann27_out_ga(0)
ackermann27_in_ga(s(T32)) → U2_ga(T32, ackermann21_in_ga(T32))
ackermann27_in_ga(s(T32)) → U3_ga(T32, ackermann21_in_ga(T32))
U8_gaa(T52, ackermann38_out_gaa(T52)) → ackermann38_out_gaa(s(T52))
U2_ga(T32, ackermann21_out_ga(T32)) → ackermann27_out_ga(s(T32))
U3_ga(T32, ackermann21_out_ga(T32)) → U4_ga(T32, ackermann38_in_gaa(T32))
ackermann38_in_gaa(0) → ackermann38_out_gaa(0)
U4_ga(T32, ackermann38_out_gaa(T32)) → ackermann27_out_ga(s(T32))

The set Q consists of the following terms:

ackermann21_in_ga(x0)
ackermann38_in_gaa(x0)
U1_ga(x0, x1)
U5_gaa(x0, x1)
U6_gaa(x0, x1)
U7_gaa(x0, x1)
ackermann27_in_ga(x0)
U8_gaa(x0, x1)
U2_ga(x0, x1)
U3_ga(x0, x1)
U4_ga(x0, x1)

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

(18) UsableRulesProof (EQUIVALENT transformation)

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

(19) Obligation:

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

ACKERMANN38_IN_GAA(s(T52)) → ACKERMANN38_IN_GAA(s(T52))

R is empty.
The set Q consists of the following terms:

ackermann21_in_ga(x0)
ackermann38_in_gaa(x0)
U1_ga(x0, x1)
U5_gaa(x0, x1)
U6_gaa(x0, x1)
U7_gaa(x0, x1)
ackermann27_in_ga(x0)
U8_gaa(x0, x1)
U2_ga(x0, x1)
U3_ga(x0, x1)
U4_ga(x0, x1)

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

(20) QReductionProof (EQUIVALENT transformation)

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

ackermann21_in_ga(x0)
ackermann38_in_gaa(x0)
U1_ga(x0, x1)
U5_gaa(x0, x1)
U6_gaa(x0, x1)
U7_gaa(x0, x1)
ackermann27_in_ga(x0)
U8_gaa(x0, x1)
U2_ga(x0, x1)
U3_ga(x0, x1)
U4_ga(x0, x1)

(21) Obligation:

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

ACKERMANN38_IN_GAA(s(T52)) → ACKERMANN38_IN_GAA(s(T52))

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

(22) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = ACKERMANN38_IN_GAA(s(T52)) evaluates to t =ACKERMANN38_IN_GAA(s(T52))

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

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from ACKERMANN38_IN_GAA(s(T52)) to ACKERMANN38_IN_GAA(s(T52)).

(23) NO

(24) Obligation:

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

ACKERMANN57_IN_GAA(T84, s(T86), X229) → P65_IN_GAAA(T84, T86, X228, X229)
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, ackermann57_in_gaa(T84, T86, T88))
U12_GAAA(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → ACKERMANN68_IN_GAA(T84, T88, X229)
ACKERMANN68_IN_GAA(s(T106), s(T108), X283) → P65_IN_GAAA(T106, T108, X282, X283)

The TRS R consists of the following rules:

ackermann1_in_gag(0, T5, s(T5)) → ackermann1_out_gag(0, T5, s(T5))
ackermann1_in_gag(s(0), 0, s(s(0))) → ackermann1_out_gag(s(0), 0, s(s(0)))
ackermann1_in_gag(s(s(T19)), 0, T20) → U16_gag(T19, T20, ackermann21_in_ga(T19, X40))
ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
U16_gag(T19, T20, ackermann21_out_ga(T19, X40)) → ackermann1_out_gag(s(s(T19)), 0, T20)
ackermann1_in_gag(s(s(T19)), 0, T20) → U17_gag(T19, T20, ackermann21_in_ga(T19, T22))
U17_gag(T19, T20, ackermann21_out_ga(T19, T22)) → U18_gag(T19, T20, ackermann1_in_gag(T19, T22, T20))
ackermann1_in_gag(s(T68), s(T71), T70) → U19_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, X182))
ackermann57_in_gaa(T79, 0, X211) → U9_gaa(T79, X211, ackermann27_in_ga(T79, X211))
U9_gaa(T79, X211, ackermann27_out_ga(T79, X211)) → ackermann57_out_gaa(T79, 0, X211)
ackermann57_in_gaa(T84, s(T86), X229) → U10_gaa(T84, 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))
U11_gaaa(T84, T86, X228, X229, ackermann57_out_gaa(T84, T86, X228)) → p65_out_gaaa(T84, T86, X228, X229)
p65_in_gaaa(T84, T86, T88, X229) → U12_gaaa(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U12_gaaa(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_gaaa(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
ackermann68_in_gaa(0, T96, s(T96)) → ackermann68_out_gaa(0, T96, s(T96))
ackermann68_in_gaa(s(T101), 0, X265) → U14_gaa(T101, X265, ackermann27_in_ga(T101, X265))
U14_gaa(T101, X265, ackermann27_out_ga(T101, X265)) → ackermann68_out_gaa(s(T101), 0, X265)
ackermann68_in_gaa(s(T106), s(T108), X283) → U15_gaa(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
U15_gaa(T106, T108, X283, p65_out_gaaa(T106, T108, X282, X283)) → ackermann68_out_gaa(s(T106), s(T108), X283)
U13_gaaa(T84, T86, T88, X229, ackermann68_out_gaa(T84, T88, X229)) → p65_out_gaaa(T84, T86, T88, X229)
U10_gaa(T84, T86, X229, p65_out_gaaa(T84, T86, X228, X229)) → ackermann57_out_gaa(T84, s(T86), X229)
U19_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, X182)) → ackermann1_out_gag(s(T68), s(T71), T70)
ackermann1_in_gag(s(T68), s(T71), T70) → U20_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → U21_gag(T68, T71, T70, ackermann1_in_gag(T68, T73, T70))
ackermann1_in_gag(s(T123), s(0), T117) → U22_gag(T123, T117, ackermann27_in_ga(T123, X319))
U22_gag(T123, T117, ackermann27_out_ga(T123, X319)) → ackermann1_out_gag(s(T123), s(0), T117)
ackermann1_in_gag(s(T123), s(0), T117) → U23_gag(T123, T117, ackermann27_in_ga(T123, T124))
U23_gag(T123, T117, ackermann27_out_ga(T123, T124)) → U24_gag(T123, T117, ackermann1_in_gag(T123, T124, T117))
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U25_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, X340))
U25_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, X340)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U26_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U27_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, X341))
U27_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, X341)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → U29_gag(T131, T133, T117, ackermann1_in_gag(T131, T139, T117))
U29_gag(T131, T133, T117, ackermann1_out_gag(T131, T139, T117)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U24_gag(T123, T117, ackermann1_out_gag(T123, T124, T117)) → ackermann1_out_gag(s(T123), s(0), T117)
U21_gag(T68, T71, T70, ackermann1_out_gag(T68, T73, T70)) → ackermann1_out_gag(s(T68), s(T71), T70)
U18_gag(T19, T20, ackermann1_out_gag(T19, T22, T20)) → ackermann1_out_gag(s(s(T19)), 0, T20)

(25) UsableRulesProof (EQUIVALENT transformation)

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

(26) Obligation:

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

ACKERMANN57_IN_GAA(T84, s(T86), X229) → P65_IN_GAAA(T84, T86, X228, X229)
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, ackermann57_in_gaa(T84, T86, T88))
U12_GAAA(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → ACKERMANN68_IN_GAA(T84, T88, X229)
ACKERMANN68_IN_GAA(s(T106), s(T108), X283) → P65_IN_GAAA(T106, T108, X282, X283)

The TRS R consists of the following rules:

ackermann57_in_gaa(T79, 0, X211) → U9_gaa(T79, 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))
U9_gaa(T79, X211, ackermann27_out_ga(T79, X211)) → ackermann57_out_gaa(T79, 0, X211)
U10_gaa(T84, T86, X229, p65_out_gaaa(T84, T86, X228, X229)) → ackermann57_out_gaa(T84, s(T86), X229)
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
p65_in_gaaa(T84, T86, X228, X229) → U11_gaaa(T84, T86, X228, X229, ackermann57_in_gaa(T84, T86, X228))
p65_in_gaaa(T84, T86, T88, X229) → U12_gaaa(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
U11_gaaa(T84, T86, X228, X229, ackermann57_out_gaa(T84, T86, X228)) → p65_out_gaaa(T84, T86, X228, X229)
U12_gaaa(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_gaaa(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)
U13_gaaa(T84, T86, T88, X229, ackermann68_out_gaa(T84, T88, X229)) → p65_out_gaaa(T84, T86, T88, X229)
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
ackermann68_in_gaa(0, T96, s(T96)) → ackermann68_out_gaa(0, T96, s(T96))
ackermann68_in_gaa(s(T101), 0, X265) → U14_gaa(T101, X265, ackermann27_in_ga(T101, X265))
ackermann68_in_gaa(s(T106), s(T108), X283) → U15_gaa(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U14_gaa(T101, X265, ackermann27_out_ga(T101, X265)) → ackermann68_out_gaa(s(T101), 0, X265)
U15_gaa(T106, T108, X283, p65_out_gaaa(T106, T108, X282, X283)) → ackermann68_out_gaa(s(T106), s(T108), X283)
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)

The argument filtering Pi contains the following mapping:
0 = 0
s(x1) = s(x1)
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)
ackermann27_out_ga(x1, x2) = ackermann27_out_ga(x1)
U2_ga(x1, x2, x3) = U2_ga(x1, x3)
ackermann21_out_ga(x1, x2) = ackermann21_out_ga(x1)
U3_ga(x1, x2, x3) = U3_ga(x1, x3)
U4_ga(x1, x2, x3) = U4_ga(x1, x3)
ackermann38_in_gaa(x1, x2, x3) = ackermann38_in_gaa(x1)
ackermann38_out_gaa(x1, x2, x3) = ackermann38_out_gaa(x1)
U5_gaa(x1, x2, x3) = U5_gaa(x1, x3)
U6_gaa(x1, x2, x3, x4) = U6_gaa(x1, x4)
U7_gaa(x1, x2, x3, x4) = U7_gaa(x1, x4)
U8_gaa(x1, x2, x3, x4) = U8_gaa(x1, x4)
ackermann57_in_gaa(x1, x2, x3) = ackermann57_in_gaa(x1)
U9_gaa(x1, x2, x3) = U9_gaa(x1, x3)
ackermann57_out_gaa(x1, x2, x3) = ackermann57_out_gaa(x1, x2)
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)
p65_out_gaaa(x1, x2, x3, x4) = p65_out_gaaa(x1, x2)
U12_gaaa(x1, x2, x3, x4, x5) = U12_gaaa(x1, x5)
U13_gaaa(x1, x2, x3, x4, x5) = U13_gaaa(x1, x2, x5)
ackermann68_in_gaa(x1, x2, x3) = ackermann68_in_gaa(x1)
ackermann68_out_gaa(x1, x2, x3) = ackermann68_out_gaa(x1)
U14_gaa(x1, x2, x3) = U14_gaa(x1, x3)
U15_gaa(x1, x2, x3, x4) = U15_gaa(x1, x4)
ACKERMANN57_IN_GAA(x1, x2, x3) = ACKERMANN57_IN_GAA(x1)
P65_IN_GAAA(x1, x2, x3, x4) = P65_IN_GAAA(x1)
U12_GAAA(x1, x2, x3, x4, x5) = U12_GAAA(x1, x5)
ACKERMANN68_IN_GAA(x1, x2, x3) = ACKERMANN68_IN_GAA(x1)

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

(27) PiDPToQDPProof (SOUND transformation)

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

(28) Obligation:

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

ACKERMANN57_IN_GAA(T84) → P65_IN_GAAA(T84)
P65_IN_GAAA(T84) → ACKERMANN57_IN_GAA(T84)
P65_IN_GAAA(T84) → U12_GAAA(T84, ackermann57_in_gaa(T84))
U12_GAAA(T84, ackermann57_out_gaa(T84, T86)) → ACKERMANN68_IN_GAA(T84)
ACKERMANN68_IN_GAA(s(T106)) → P65_IN_GAAA(T106)

The TRS R consists of the following rules:

ackermann57_in_gaa(T79) → U9_gaa(T79, ackermann27_in_ga(T79))
ackermann57_in_gaa(T84) → U10_gaa(T84, p65_in_gaaa(T84))
U9_gaa(T79, ackermann27_out_ga(T79)) → ackermann57_out_gaa(T79, 0)
U10_gaa(T84, p65_out_gaaa(T84, T86)) → ackermann57_out_gaa(T84, s(T86))
ackermann27_in_ga(0) → ackermann27_out_ga(0)
ackermann27_in_ga(s(T32)) → U2_ga(T32, ackermann21_in_ga(T32))
ackermann27_in_ga(s(T32)) → U3_ga(T32, ackermann21_in_ga(T32))
p65_in_gaaa(T84) → U11_gaaa(T84, ackermann57_in_gaa(T84))
p65_in_gaaa(T84) → U12_gaaa(T84, ackermann57_in_gaa(T84))
U2_ga(T32, ackermann21_out_ga(T32)) → ackermann27_out_ga(s(T32))
U3_ga(T32, ackermann21_out_ga(T32)) → U4_ga(T32, ackermann38_in_gaa(T32))
U11_gaaa(T84, ackermann57_out_gaa(T84, T86)) → p65_out_gaaa(T84, T86)
U12_gaaa(T84, ackermann57_out_gaa(T84, T86)) → U13_gaaa(T84, T86, ackermann68_in_gaa(T84))
ackermann21_in_ga(T28) → U1_ga(T28, ackermann27_in_ga(T28))
U4_ga(T32, ackermann38_out_gaa(T32)) → ackermann27_out_ga(s(T32))
U13_gaaa(T84, T86, ackermann68_out_gaa(T84)) → p65_out_gaaa(T84, T86)
U1_ga(T28, ackermann27_out_ga(T28)) → ackermann21_out_ga(T28)
ackermann38_in_gaa(0) → ackermann38_out_gaa(0)
ackermann38_in_gaa(s(T47)) → U5_gaa(T47, ackermann27_in_ga(T47))
ackermann38_in_gaa(s(T52)) → U6_gaa(T52, ackermann38_in_gaa(s(T52)))
ackermann38_in_gaa(s(T52)) → U7_gaa(T52, ackermann38_in_gaa(s(T52)))
ackermann68_in_gaa(0) → ackermann68_out_gaa(0)
ackermann68_in_gaa(s(T101)) → U14_gaa(T101, ackermann27_in_ga(T101))
ackermann68_in_gaa(s(T106)) → U15_gaa(T106, p65_in_gaaa(T106))
U5_gaa(T47, ackermann27_out_ga(T47)) → ackermann38_out_gaa(s(T47))
U6_gaa(T52, ackermann38_out_gaa(s(T52))) → ackermann38_out_gaa(s(T52))
U7_gaa(T52, ackermann38_out_gaa(s(T52))) → U8_gaa(T52, ackermann38_in_gaa(T52))
U14_gaa(T101, ackermann27_out_ga(T101)) → ackermann68_out_gaa(s(T101))
U15_gaa(T106, p65_out_gaaa(T106, T108)) → ackermann68_out_gaa(s(T106))
U8_gaa(T52, ackermann38_out_gaa(T52)) → ackermann38_out_gaa(s(T52))

The set Q consists of the following terms:

ackermann57_in_gaa(x0)
U9_gaa(x0, x1)
U10_gaa(x0, x1)
ackermann27_in_ga(x0)
p65_in_gaaa(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
U11_gaaa(x0, x1)
U12_gaaa(x0, x1)
ackermann21_in_ga(x0)
U4_ga(x0, x1)
U13_gaaa(x0, x1, x2)
U1_ga(x0, x1)
ackermann38_in_gaa(x0)
ackermann68_in_gaa(x0)
U5_gaa(x0, x1)
U6_gaa(x0, x1)
U7_gaa(x0, x1)
U14_gaa(x0, x1)
U15_gaa(x0, x1)
U8_gaa(x0, x1)

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

(29) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

P65_IN_GAAA(T84) → U12_GAAA(T84, ackermann57_in_gaa(T84))

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

POL(0) = 0
POL(ACKERMANN57_IN_GAA(x₁)) = 1 + x₁
POL(ACKERMANN68_IN_GAA(x₁)) = x₁
POL(P65_IN_GAAA(x₁)) = 1 + x₁
POL(U10_gaa(x₁, x₂)) = 0
POL(U11_gaaa(x₁, x₂)) = 0
POL(U12_GAAA(x₁, x₂)) = x₁
POL(U12_gaaa(x₁, x₂)) = 0
POL(U13_gaaa(x₁, x₂, x₃)) = 0
POL(U14_gaa(x₁, x₂)) = 1
POL(U15_gaa(x₁, x₂)) = 0
POL(U1_ga(x₁, x₂)) = 0
POL(U2_ga(x₁, x₂)) = 0
POL(U3_ga(x₁, x₂)) = 0
POL(U4_ga(x₁, x₂)) = 0
POL(U5_gaa(x₁, x₂)) = 0
POL(U6_gaa(x₁, x₂)) = 0
POL(U7_gaa(x₁, x₂)) = 0
POL(U8_gaa(x₁, x₂)) = 0
POL(U9_gaa(x₁, x₂)) = 0
POL(ackermann21_in_ga(x₁)) = 0
POL(ackermann21_out_ga(x₁)) = 0
POL(ackermann27_in_ga(x₁)) = 0
POL(ackermann27_out_ga(x₁)) = 0
POL(ackermann38_in_gaa(x₁)) = 0
POL(ackermann38_out_gaa(x₁)) = 0
POL(ackermann57_in_gaa(x₁)) = 0
POL(ackermann57_out_gaa(x₁, x₂)) = 0
POL(ackermann68_in_gaa(x₁)) = x₁
POL(ackermann68_out_gaa(x₁)) = 0
POL(p65_in_gaaa(x₁)) = 0
POL(p65_out_gaaa(x₁, x₂)) = 0
POL(s(x₁)) = 1 + x₁

The following usable rules [FROCOS05] were oriented: none

(30) Obligation:

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

ACKERMANN57_IN_GAA(T84) → P65_IN_GAAA(T84)
P65_IN_GAAA(T84) → ACKERMANN57_IN_GAA(T84)
U12_GAAA(T84, ackermann57_out_gaa(T84, T86)) → ACKERMANN68_IN_GAA(T84)
ACKERMANN68_IN_GAA(s(T106)) → P65_IN_GAAA(T106)

The TRS R consists of the following rules:

ackermann57_in_gaa(T79) → U9_gaa(T79, ackermann27_in_ga(T79))
ackermann57_in_gaa(T84) → U10_gaa(T84, p65_in_gaaa(T84))
U9_gaa(T79, ackermann27_out_ga(T79)) → ackermann57_out_gaa(T79, 0)
U10_gaa(T84, p65_out_gaaa(T84, T86)) → ackermann57_out_gaa(T84, s(T86))
ackermann27_in_ga(0) → ackermann27_out_ga(0)
ackermann27_in_ga(s(T32)) → U2_ga(T32, ackermann21_in_ga(T32))
ackermann27_in_ga(s(T32)) → U3_ga(T32, ackermann21_in_ga(T32))
p65_in_gaaa(T84) → U11_gaaa(T84, ackermann57_in_gaa(T84))
p65_in_gaaa(T84) → U12_gaaa(T84, ackermann57_in_gaa(T84))
U2_ga(T32, ackermann21_out_ga(T32)) → ackermann27_out_ga(s(T32))
U3_ga(T32, ackermann21_out_ga(T32)) → U4_ga(T32, ackermann38_in_gaa(T32))
U11_gaaa(T84, ackermann57_out_gaa(T84, T86)) → p65_out_gaaa(T84, T86)
U12_gaaa(T84, ackermann57_out_gaa(T84, T86)) → U13_gaaa(T84, T86, ackermann68_in_gaa(T84))
ackermann21_in_ga(T28) → U1_ga(T28, ackermann27_in_ga(T28))
U4_ga(T32, ackermann38_out_gaa(T32)) → ackermann27_out_ga(s(T32))
U13_gaaa(T84, T86, ackermann68_out_gaa(T84)) → p65_out_gaaa(T84, T86)
U1_ga(T28, ackermann27_out_ga(T28)) → ackermann21_out_ga(T28)
ackermann38_in_gaa(0) → ackermann38_out_gaa(0)
ackermann38_in_gaa(s(T47)) → U5_gaa(T47, ackermann27_in_ga(T47))
ackermann38_in_gaa(s(T52)) → U6_gaa(T52, ackermann38_in_gaa(s(T52)))
ackermann38_in_gaa(s(T52)) → U7_gaa(T52, ackermann38_in_gaa(s(T52)))
ackermann68_in_gaa(0) → ackermann68_out_gaa(0)
ackermann68_in_gaa(s(T101)) → U14_gaa(T101, ackermann27_in_ga(T101))
ackermann68_in_gaa(s(T106)) → U15_gaa(T106, p65_in_gaaa(T106))
U5_gaa(T47, ackermann27_out_ga(T47)) → ackermann38_out_gaa(s(T47))
U6_gaa(T52, ackermann38_out_gaa(s(T52))) → ackermann38_out_gaa(s(T52))
U7_gaa(T52, ackermann38_out_gaa(s(T52))) → U8_gaa(T52, ackermann38_in_gaa(T52))
U14_gaa(T101, ackermann27_out_ga(T101)) → ackermann68_out_gaa(s(T101))
U15_gaa(T106, p65_out_gaaa(T106, T108)) → ackermann68_out_gaa(s(T106))
U8_gaa(T52, ackermann38_out_gaa(T52)) → ackermann38_out_gaa(s(T52))

The set Q consists of the following terms:

ackermann57_in_gaa(x0)
U9_gaa(x0, x1)
U10_gaa(x0, x1)
ackermann27_in_ga(x0)
p65_in_gaaa(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
U11_gaaa(x0, x1)
U12_gaaa(x0, x1)
ackermann21_in_ga(x0)
U4_ga(x0, x1)
U13_gaaa(x0, x1, x2)
U1_ga(x0, x1)
ackermann38_in_gaa(x0)
ackermann68_in_gaa(x0)
U5_gaa(x0, x1)
U6_gaa(x0, x1)
U7_gaa(x0, x1)
U14_gaa(x0, x1)
U15_gaa(x0, x1)
U8_gaa(x0, x1)

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

(31) DependencyGraphProof (EQUIVALENT transformation)

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

(32) 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)

The TRS R consists of the following rules:

ackermann57_in_gaa(T79) → U9_gaa(T79, ackermann27_in_ga(T79))
ackermann57_in_gaa(T84) → U10_gaa(T84, p65_in_gaaa(T84))
U9_gaa(T79, ackermann27_out_ga(T79)) → ackermann57_out_gaa(T79, 0)
U10_gaa(T84, p65_out_gaaa(T84, T86)) → ackermann57_out_gaa(T84, s(T86))
ackermann27_in_ga(0) → ackermann27_out_ga(0)
ackermann27_in_ga(s(T32)) → U2_ga(T32, ackermann21_in_ga(T32))
ackermann27_in_ga(s(T32)) → U3_ga(T32, ackermann21_in_ga(T32))
p65_in_gaaa(T84) → U11_gaaa(T84, ackermann57_in_gaa(T84))
p65_in_gaaa(T84) → U12_gaaa(T84, ackermann57_in_gaa(T84))
U2_ga(T32, ackermann21_out_ga(T32)) → ackermann27_out_ga(s(T32))
U3_ga(T32, ackermann21_out_ga(T32)) → U4_ga(T32, ackermann38_in_gaa(T32))
U11_gaaa(T84, ackermann57_out_gaa(T84, T86)) → p65_out_gaaa(T84, T86)
U12_gaaa(T84, ackermann57_out_gaa(T84, T86)) → U13_gaaa(T84, T86, ackermann68_in_gaa(T84))
ackermann21_in_ga(T28) → U1_ga(T28, ackermann27_in_ga(T28))
U4_ga(T32, ackermann38_out_gaa(T32)) → ackermann27_out_ga(s(T32))
U13_gaaa(T84, T86, ackermann68_out_gaa(T84)) → p65_out_gaaa(T84, T86)
U1_ga(T28, ackermann27_out_ga(T28)) → ackermann21_out_ga(T28)
ackermann38_in_gaa(0) → ackermann38_out_gaa(0)
ackermann38_in_gaa(s(T47)) → U5_gaa(T47, ackermann27_in_ga(T47))
ackermann38_in_gaa(s(T52)) → U6_gaa(T52, ackermann38_in_gaa(s(T52)))
ackermann38_in_gaa(s(T52)) → U7_gaa(T52, ackermann38_in_gaa(s(T52)))
ackermann68_in_gaa(0) → ackermann68_out_gaa(0)
ackermann68_in_gaa(s(T101)) → U14_gaa(T101, ackermann27_in_ga(T101))
ackermann68_in_gaa(s(T106)) → U15_gaa(T106, p65_in_gaaa(T106))
U5_gaa(T47, ackermann27_out_ga(T47)) → ackermann38_out_gaa(s(T47))
U6_gaa(T52, ackermann38_out_gaa(s(T52))) → ackermann38_out_gaa(s(T52))
U7_gaa(T52, ackermann38_out_gaa(s(T52))) → U8_gaa(T52, ackermann38_in_gaa(T52))
U14_gaa(T101, ackermann27_out_ga(T101)) → ackermann68_out_gaa(s(T101))
U15_gaa(T106, p65_out_gaaa(T106, T108)) → ackermann68_out_gaa(s(T106))
U8_gaa(T52, ackermann38_out_gaa(T52)) → ackermann38_out_gaa(s(T52))

The set Q consists of the following terms:

ackermann57_in_gaa(x0)
U9_gaa(x0, x1)
U10_gaa(x0, x1)
ackermann27_in_ga(x0)
p65_in_gaaa(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
U11_gaaa(x0, x1)
U12_gaaa(x0, x1)
ackermann21_in_ga(x0)
U4_ga(x0, x1)
U13_gaaa(x0, x1, x2)
U1_ga(x0, x1)
ackermann38_in_gaa(x0)
ackermann68_in_gaa(x0)
U5_gaa(x0, x1)
U6_gaa(x0, x1)
U7_gaa(x0, x1)
U14_gaa(x0, x1)
U15_gaa(x0, x1)
U8_gaa(x0, x1)

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

(33) UsableRulesProof (EQUIVALENT transformation)

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

(34) Obligation:

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

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

R is empty.
The set Q consists of the following terms:

ackermann57_in_gaa(x0)
U9_gaa(x0, x1)
U10_gaa(x0, x1)
ackermann27_in_ga(x0)
p65_in_gaaa(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
U11_gaaa(x0, x1)
U12_gaaa(x0, x1)
ackermann21_in_ga(x0)
U4_ga(x0, x1)
U13_gaaa(x0, x1, x2)
U1_ga(x0, x1)
ackermann38_in_gaa(x0)
ackermann68_in_gaa(x0)
U5_gaa(x0, x1)
U6_gaa(x0, x1)
U7_gaa(x0, x1)
U14_gaa(x0, x1)
U15_gaa(x0, x1)
U8_gaa(x0, x1)

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

(35) QReductionProof (EQUIVALENT transformation)

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

ackermann57_in_gaa(x0)
U9_gaa(x0, x1)
U10_gaa(x0, x1)
ackermann27_in_ga(x0)
p65_in_gaaa(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
U11_gaaa(x0, x1)
U12_gaaa(x0, x1)
ackermann21_in_ga(x0)
U4_ga(x0, x1)
U13_gaaa(x0, x1, x2)
U1_ga(x0, x1)
ackermann38_in_gaa(x0)
ackermann68_in_gaa(x0)
U5_gaa(x0, x1)
U6_gaa(x0, x1)
U7_gaa(x0, x1)
U14_gaa(x0, x1)
U15_gaa(x0, x1)
U8_gaa(x0, x1)

(36) 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.

(37) 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:

Semiunifier: [ ]
Matcher: [ ]

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.

(38) NO

(39) Obligation:

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

ACKERMANN1_IN_GAG(s(s(T19)), 0, T20) → U17_GAG(T19, T20, ackermann21_in_ga(T19, T22))
U17_GAG(T19, T20, ackermann21_out_ga(T19, T22)) → ACKERMANN1_IN_GAG(T19, T22, T20)
ACKERMANN1_IN_GAG(s(T68), s(T71), T70) → U20_GAG(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_GAG(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → ACKERMANN1_IN_GAG(T68, T73, T70)
ACKERMANN1_IN_GAG(s(T123), s(0), T117) → U23_GAG(T123, T117, ackermann27_in_ga(T123, T124))
U23_GAG(T123, T117, ackermann27_out_ga(T123, T124)) → ACKERMANN1_IN_GAG(T123, T124, T117)
ACKERMANN1_IN_GAG(s(T131), s(s(T133)), T117) → U26_GAG(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_GAG(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_GAG(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_GAG(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → ACKERMANN1_IN_GAG(T131, T139, T117)

The TRS R consists of the following rules:

ackermann1_in_gag(0, T5, s(T5)) → ackermann1_out_gag(0, T5, s(T5))
ackermann1_in_gag(s(0), 0, s(s(0))) → ackermann1_out_gag(s(0), 0, s(s(0)))
ackermann1_in_gag(s(s(T19)), 0, T20) → U16_gag(T19, T20, ackermann21_in_ga(T19, X40))
ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
U16_gag(T19, T20, ackermann21_out_ga(T19, X40)) → ackermann1_out_gag(s(s(T19)), 0, T20)
ackermann1_in_gag(s(s(T19)), 0, T20) → U17_gag(T19, T20, ackermann21_in_ga(T19, T22))
U17_gag(T19, T20, ackermann21_out_ga(T19, T22)) → U18_gag(T19, T20, ackermann1_in_gag(T19, T22, T20))
ackermann1_in_gag(s(T68), s(T71), T70) → U19_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, X182))
ackermann57_in_gaa(T79, 0, X211) → U9_gaa(T79, X211, ackermann27_in_ga(T79, X211))
U9_gaa(T79, X211, ackermann27_out_ga(T79, X211)) → ackermann57_out_gaa(T79, 0, X211)
ackermann57_in_gaa(T84, s(T86), X229) → U10_gaa(T84, 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))
U11_gaaa(T84, T86, X228, X229, ackermann57_out_gaa(T84, T86, X228)) → p65_out_gaaa(T84, T86, X228, X229)
p65_in_gaaa(T84, T86, T88, X229) → U12_gaaa(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U12_gaaa(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_gaaa(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
ackermann68_in_gaa(0, T96, s(T96)) → ackermann68_out_gaa(0, T96, s(T96))
ackermann68_in_gaa(s(T101), 0, X265) → U14_gaa(T101, X265, ackermann27_in_ga(T101, X265))
U14_gaa(T101, X265, ackermann27_out_ga(T101, X265)) → ackermann68_out_gaa(s(T101), 0, X265)
ackermann68_in_gaa(s(T106), s(T108), X283) → U15_gaa(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
U15_gaa(T106, T108, X283, p65_out_gaaa(T106, T108, X282, X283)) → ackermann68_out_gaa(s(T106), s(T108), X283)
U13_gaaa(T84, T86, T88, X229, ackermann68_out_gaa(T84, T88, X229)) → p65_out_gaaa(T84, T86, T88, X229)
U10_gaa(T84, T86, X229, p65_out_gaaa(T84, T86, X228, X229)) → ackermann57_out_gaa(T84, s(T86), X229)
U19_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, X182)) → ackermann1_out_gag(s(T68), s(T71), T70)
ackermann1_in_gag(s(T68), s(T71), T70) → U20_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → U21_gag(T68, T71, T70, ackermann1_in_gag(T68, T73, T70))
ackermann1_in_gag(s(T123), s(0), T117) → U22_gag(T123, T117, ackermann27_in_ga(T123, X319))
U22_gag(T123, T117, ackermann27_out_ga(T123, X319)) → ackermann1_out_gag(s(T123), s(0), T117)
ackermann1_in_gag(s(T123), s(0), T117) → U23_gag(T123, T117, ackermann27_in_ga(T123, T124))
U23_gag(T123, T117, ackermann27_out_ga(T123, T124)) → U24_gag(T123, T117, ackermann1_in_gag(T123, T124, T117))
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U25_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, X340))
U25_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, X340)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U26_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U27_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, X341))
U27_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, X341)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → U29_gag(T131, T133, T117, ackermann1_in_gag(T131, T139, T117))
U29_gag(T131, T133, T117, ackermann1_out_gag(T131, T139, T117)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U24_gag(T123, T117, ackermann1_out_gag(T123, T124, T117)) → ackermann1_out_gag(s(T123), s(0), T117)
U21_gag(T68, T71, T70, ackermann1_out_gag(T68, T73, T70)) → ackermann1_out_gag(s(T68), s(T71), T70)
U18_gag(T19, T20, ackermann1_out_gag(T19, T22, T20)) → ackermann1_out_gag(s(s(T19)), 0, T20)

(40) UsableRulesProof (EQUIVALENT transformation)

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

(41) Obligation:

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

ACKERMANN1_IN_GAG(s(s(T19)), 0, T20) → U17_GAG(T19, T20, ackermann21_in_ga(T19, T22))
U17_GAG(T19, T20, ackermann21_out_ga(T19, T22)) → ACKERMANN1_IN_GAG(T19, T22, T20)
ACKERMANN1_IN_GAG(s(T68), s(T71), T70) → U20_GAG(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_GAG(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → ACKERMANN1_IN_GAG(T68, T73, T70)
ACKERMANN1_IN_GAG(s(T123), s(0), T117) → U23_GAG(T123, T117, ackermann27_in_ga(T123, T124))
U23_GAG(T123, T117, ackermann27_out_ga(T123, T124)) → ACKERMANN1_IN_GAG(T123, T124, T117)
ACKERMANN1_IN_GAG(s(T131), s(s(T133)), T117) → U26_GAG(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_GAG(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_GAG(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_GAG(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → ACKERMANN1_IN_GAG(T131, T139, T117)

The TRS R consists of the following rules:

ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
ackermann57_in_gaa(T79, 0, X211) → U9_gaa(T79, 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))
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
ackermann68_in_gaa(0, T96, s(T96)) → ackermann68_out_gaa(0, T96, s(T96))
ackermann68_in_gaa(s(T101), 0, X265) → U14_gaa(T101, X265, ackermann27_in_ga(T101, X265))
ackermann68_in_gaa(s(T106), s(T108), X283) → U15_gaa(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
U9_gaa(T79, X211, ackermann27_out_ga(T79, X211)) → ackermann57_out_gaa(T79, 0, X211)
U10_gaa(T84, T86, X229, p65_out_gaaa(T84, T86, X228, X229)) → ackermann57_out_gaa(T84, s(T86), X229)
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
U14_gaa(T101, X265, ackermann27_out_ga(T101, X265)) → ackermann68_out_gaa(s(T101), 0, X265)
U15_gaa(T106, T108, X283, p65_out_gaaa(T106, T108, X282, X283)) → ackermann68_out_gaa(s(T106), s(T108), X283)
p65_in_gaaa(T84, T86, X228, X229) → U11_gaaa(T84, T86, X228, X229, ackermann57_in_gaa(T84, T86, X228))
p65_in_gaaa(T84, T86, T88, X229) → U12_gaaa(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)
U11_gaaa(T84, T86, X228, X229, ackermann57_out_gaa(T84, T86, X228)) → p65_out_gaaa(T84, T86, X228, X229)
U12_gaaa(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_gaaa(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U13_gaaa(T84, T86, T88, X229, ackermann68_out_gaa(T84, T88, X229)) → p65_out_gaaa(T84, T86, T88, X229)
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)

The argument filtering Pi contains the following mapping:
0 = 0
s(x1) = s(x1)
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)
ackermann27_out_ga(x1, x2) = ackermann27_out_ga(x1)
U2_ga(x1, x2, x3) = U2_ga(x1, x3)
ackermann21_out_ga(x1, x2) = ackermann21_out_ga(x1)
U3_ga(x1, x2, x3) = U3_ga(x1, x3)
U4_ga(x1, x2, x3) = U4_ga(x1, x3)
ackermann38_in_gaa(x1, x2, x3) = ackermann38_in_gaa(x1)
ackermann38_out_gaa(x1, x2, x3) = ackermann38_out_gaa(x1)
U5_gaa(x1, x2, x3) = U5_gaa(x1, x3)
U6_gaa(x1, x2, x3, x4) = U6_gaa(x1, x4)
U7_gaa(x1, x2, x3, x4) = U7_gaa(x1, x4)
U8_gaa(x1, x2, x3, x4) = U8_gaa(x1, x4)
ackermann57_in_gaa(x1, x2, x3) = ackermann57_in_gaa(x1)
U9_gaa(x1, x2, x3) = U9_gaa(x1, x3)
ackermann57_out_gaa(x1, x2, x3) = ackermann57_out_gaa(x1, x2)
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)
p65_out_gaaa(x1, x2, x3, x4) = p65_out_gaaa(x1, x2)
U12_gaaa(x1, x2, x3, x4, x5) = U12_gaaa(x1, x5)
U13_gaaa(x1, x2, x3, x4, x5) = U13_gaaa(x1, x2, x5)
ackermann68_in_gaa(x1, x2, x3) = ackermann68_in_gaa(x1)
ackermann68_out_gaa(x1, x2, x3) = ackermann68_out_gaa(x1)
U14_gaa(x1, x2, x3) = U14_gaa(x1, x3)
U15_gaa(x1, x2, x3, x4) = U15_gaa(x1, x4)
ACKERMANN1_IN_GAG(x1, x2, x3) = ACKERMANN1_IN_GAG(x1, x3)
U17_GAG(x1, x2, x3) = U17_GAG(x1, x2, x3)
U20_GAG(x1, x2, x3, x4) = U20_GAG(x1, x3, x4)
U23_GAG(x1, x2, x3) = U23_GAG(x1, x2, x3)
U26_GAG(x1, x2, x3, x4) = U26_GAG(x1, x3, x4)
U28_GAG(x1, x2, x3, x4) = U28_GAG(x1, x2, x3, x4)

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

(42) PiDPToQDPProof (SOUND transformation)

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

(43) Obligation:

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

ACKERMANN1_IN_GAG(s(s(T19)), T20) → U17_GAG(T19, T20, ackermann21_in_ga(T19))
U17_GAG(T19, T20, ackermann21_out_ga(T19)) → ACKERMANN1_IN_GAG(T19, T20)
ACKERMANN1_IN_GAG(s(T68), T70) → U20_GAG(T68, T70, ackermann57_in_gaa(T68))
U20_GAG(T68, T70, ackermann57_out_gaa(T68, T71)) → ACKERMANN1_IN_GAG(T68, T70)
ACKERMANN1_IN_GAG(s(T123), T117) → U23_GAG(T123, T117, ackermann27_in_ga(T123))
U23_GAG(T123, T117, ackermann27_out_ga(T123)) → ACKERMANN1_IN_GAG(T123, T117)
ACKERMANN1_IN_GAG(s(T131), T117) → U26_GAG(T131, T117, ackermann57_in_gaa(T131))
U26_GAG(T131, T117, ackermann57_out_gaa(T131, T133)) → U28_GAG(T131, T133, T117, ackermann68_in_gaa(T131))
U28_GAG(T131, T133, T117, ackermann68_out_gaa(T131)) → ACKERMANN1_IN_GAG(T131, T117)

The TRS R consists of the following rules:

ackermann21_in_ga(T28) → U1_ga(T28, ackermann27_in_ga(T28))
ackermann57_in_gaa(T79) → U9_gaa(T79, ackermann27_in_ga(T79))
ackermann57_in_gaa(T84) → U10_gaa(T84, p65_in_gaaa(T84))
ackermann27_in_ga(0) → ackermann27_out_ga(0)
ackermann27_in_ga(s(T32)) → U2_ga(T32, ackermann21_in_ga(T32))
ackermann27_in_ga(s(T32)) → U3_ga(T32, ackermann21_in_ga(T32))
ackermann68_in_gaa(0) → ackermann68_out_gaa(0)
ackermann68_in_gaa(s(T101)) → U14_gaa(T101, ackermann27_in_ga(T101))
ackermann68_in_gaa(s(T106)) → U15_gaa(T106, p65_in_gaaa(T106))
U1_ga(T28, ackermann27_out_ga(T28)) → ackermann21_out_ga(T28)
U9_gaa(T79, ackermann27_out_ga(T79)) → ackermann57_out_gaa(T79, 0)
U10_gaa(T84, p65_out_gaaa(T84, T86)) → ackermann57_out_gaa(T84, s(T86))
U2_ga(T32, ackermann21_out_ga(T32)) → ackermann27_out_ga(s(T32))
U3_ga(T32, ackermann21_out_ga(T32)) → U4_ga(T32, ackermann38_in_gaa(T32))
U14_gaa(T101, ackermann27_out_ga(T101)) → ackermann68_out_gaa(s(T101))
U15_gaa(T106, p65_out_gaaa(T106, T108)) → ackermann68_out_gaa(s(T106))
p65_in_gaaa(T84) → U11_gaaa(T84, ackermann57_in_gaa(T84))
p65_in_gaaa(T84) → U12_gaaa(T84, ackermann57_in_gaa(T84))
U4_ga(T32, ackermann38_out_gaa(T32)) → ackermann27_out_ga(s(T32))
U11_gaaa(T84, ackermann57_out_gaa(T84, T86)) → p65_out_gaaa(T84, T86)
U12_gaaa(T84, ackermann57_out_gaa(T84, T86)) → U13_gaaa(T84, T86, ackermann68_in_gaa(T84))
ackermann38_in_gaa(0) → ackermann38_out_gaa(0)
ackermann38_in_gaa(s(T47)) → U5_gaa(T47, ackermann27_in_ga(T47))
ackermann38_in_gaa(s(T52)) → U6_gaa(T52, ackermann38_in_gaa(s(T52)))
ackermann38_in_gaa(s(T52)) → U7_gaa(T52, ackermann38_in_gaa(s(T52)))
U13_gaaa(T84, T86, ackermann68_out_gaa(T84)) → p65_out_gaaa(T84, T86)
U5_gaa(T47, ackermann27_out_ga(T47)) → ackermann38_out_gaa(s(T47))
U6_gaa(T52, ackermann38_out_gaa(s(T52))) → ackermann38_out_gaa(s(T52))
U7_gaa(T52, ackermann38_out_gaa(s(T52))) → U8_gaa(T52, ackermann38_in_gaa(T52))
U8_gaa(T52, ackermann38_out_gaa(T52)) → ackermann38_out_gaa(s(T52))

The set Q consists of the following terms:

ackermann21_in_ga(x0)
ackermann57_in_gaa(x0)
ackermann27_in_ga(x0)
ackermann68_in_gaa(x0)
U1_ga(x0, x1)
U9_gaa(x0, x1)
U10_gaa(x0, x1)
U2_ga(x0, x1)
U3_ga(x0, x1)
U14_gaa(x0, x1)
U15_gaa(x0, x1)
p65_in_gaaa(x0)
U4_ga(x0, x1)
U11_gaaa(x0, x1)
U12_gaaa(x0, x1)
ackermann38_in_gaa(x0)
U13_gaaa(x0, x1, x2)
U5_gaa(x0, x1)
U6_gaa(x0, x1)
U7_gaa(x0, x1)
U8_gaa(x0, x1)

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

(44) 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_GAG(T19, T20, ackermann21_out_ga(T19)) → ACKERMANN1_IN_GAG(T19, T20)
The graph contains the following edges 1 >= 1, 3 > 1, 2 >= 2
ACKERMANN1_IN_GAG(s(s(T19)), T20) → U17_GAG(T19, T20, ackermann21_in_ga(T19))
The graph contains the following edges 1 > 1, 2 >= 2
U26_GAG(T131, T117, ackermann57_out_gaa(T131, T133)) → U28_GAG(T131, T133, T117, ackermann68_in_gaa(T131))
The graph contains the following edges 1 >= 1, 3 > 1, 3 > 2, 2 >= 3
U20_GAG(T68, T70, ackermann57_out_gaa(T68, T71)) → ACKERMANN1_IN_GAG(T68, T70)
The graph contains the following edges 1 >= 1, 3 > 1, 2 >= 2
ACKERMANN1_IN_GAG(s(T68), T70) → U20_GAG(T68, T70, ackermann57_in_gaa(T68))
The graph contains the following edges 1 > 1, 2 >= 2
U23_GAG(T123, T117, ackermann27_out_ga(T123)) → ACKERMANN1_IN_GAG(T123, T117)
The graph contains the following edges 1 >= 1, 3 > 1, 2 >= 2
U28_GAG(T131, T133, T117, ackermann68_out_gaa(T131)) → ACKERMANN1_IN_GAG(T131, T117)
The graph contains the following edges 1 >= 1, 4 > 1, 3 >= 2
ACKERMANN1_IN_GAG(s(T123), T117) → U23_GAG(T123, T117, ackermann27_in_ga(T123))
The graph contains the following edges 1 > 1, 2 >= 2
ACKERMANN1_IN_GAG(s(T131), T117) → U26_GAG(T131, T117, ackermann57_in_gaa(T131))
The graph contains the following edges 1 > 1, 2 >= 2

(45) YES

(46) PrologToPiTRSProof (SOUND transformation)

ackermann1_in_gag(0, T5, s(T5)) → ackermann1_out_gag(0, T5, s(T5))
ackermann1_in_gag(s(0), 0, s(s(0))) → ackermann1_out_gag(s(0), 0, s(s(0)))
ackermann1_in_gag(s(s(T19)), 0, T20) → U16_gag(T19, T20, ackermann21_in_ga(T19, X40))
ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
U16_gag(T19, T20, ackermann21_out_ga(T19, X40)) → ackermann1_out_gag(s(s(T19)), 0, T20)
ackermann1_in_gag(s(s(T19)), 0, T20) → U17_gag(T19, T20, ackermann21_in_ga(T19, T22))
U17_gag(T19, T20, ackermann21_out_ga(T19, T22)) → U18_gag(T19, T20, ackermann1_in_gag(T19, T22, T20))
ackermann1_in_gag(s(T68), s(T71), T70) → U19_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, X182))
ackermann57_in_gaa(T79, 0, X211) → U9_gaa(T79, X211, ackermann27_in_ga(T79, X211))
U9_gaa(T79, X211, ackermann27_out_ga(T79, X211)) → ackermann57_out_gaa(T79, 0, X211)
ackermann57_in_gaa(T84, s(T86), X229) → U10_gaa(T84, 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))
U11_gaaa(T84, T86, X228, X229, ackermann57_out_gaa(T84, T86, X228)) → p65_out_gaaa(T84, T86, X228, X229)
p65_in_gaaa(T84, T86, T88, X229) → U12_gaaa(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U12_gaaa(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_gaaa(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
ackermann68_in_gaa(0, T96, s(T96)) → ackermann68_out_gaa(0, T96, s(T96))
ackermann68_in_gaa(s(T101), 0, X265) → U14_gaa(T101, X265, ackermann27_in_ga(T101, X265))
U14_gaa(T101, X265, ackermann27_out_ga(T101, X265)) → ackermann68_out_gaa(s(T101), 0, X265)
ackermann68_in_gaa(s(T106), s(T108), X283) → U15_gaa(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
U15_gaa(T106, T108, X283, p65_out_gaaa(T106, T108, X282, X283)) → ackermann68_out_gaa(s(T106), s(T108), X283)
U13_gaaa(T84, T86, T88, X229, ackermann68_out_gaa(T84, T88, X229)) → p65_out_gaaa(T84, T86, T88, X229)
U10_gaa(T84, T86, X229, p65_out_gaaa(T84, T86, X228, X229)) → ackermann57_out_gaa(T84, s(T86), X229)
U19_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, X182)) → ackermann1_out_gag(s(T68), s(T71), T70)
ackermann1_in_gag(s(T68), s(T71), T70) → U20_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → U21_gag(T68, T71, T70, ackermann1_in_gag(T68, T73, T70))
ackermann1_in_gag(s(T123), s(0), T117) → U22_gag(T123, T117, ackermann27_in_ga(T123, X319))
U22_gag(T123, T117, ackermann27_out_ga(T123, X319)) → ackermann1_out_gag(s(T123), s(0), T117)
ackermann1_in_gag(s(T123), s(0), T117) → U23_gag(T123, T117, ackermann27_in_ga(T123, T124))
U23_gag(T123, T117, ackermann27_out_ga(T123, T124)) → U24_gag(T123, T117, ackermann1_in_gag(T123, T124, T117))
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U25_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, X340))
U25_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, X340)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U26_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U27_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, X341))
U27_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, X341)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → U29_gag(T131, T133, T117, ackermann1_in_gag(T131, T139, T117))
U29_gag(T131, T133, T117, ackermann1_out_gag(T131, T139, T117)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U24_gag(T123, T117, ackermann1_out_gag(T123, T124, T117)) → ackermann1_out_gag(s(T123), s(0), T117)
U21_gag(T68, T71, T70, ackermann1_out_gag(T68, T73, T70)) → ackermann1_out_gag(s(T68), s(T71), T70)
U18_gag(T19, T20, ackermann1_out_gag(T19, T22, T20)) → ackermann1_out_gag(s(s(T19)), 0, T20)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(47) Obligation:

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

ackermann1_in_gag(0, T5, s(T5)) → ackermann1_out_gag(0, T5, s(T5))
ackermann1_in_gag(s(0), 0, s(s(0))) → ackermann1_out_gag(s(0), 0, s(s(0)))
ackermann1_in_gag(s(s(T19)), 0, T20) → U16_gag(T19, T20, ackermann21_in_ga(T19, X40))
ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
U16_gag(T19, T20, ackermann21_out_ga(T19, X40)) → ackermann1_out_gag(s(s(T19)), 0, T20)
ackermann1_in_gag(s(s(T19)), 0, T20) → U17_gag(T19, T20, ackermann21_in_ga(T19, T22))
U17_gag(T19, T20, ackermann21_out_ga(T19, T22)) → U18_gag(T19, T20, ackermann1_in_gag(T19, T22, T20))
ackermann1_in_gag(s(T68), s(T71), T70) → U19_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, X182))
ackermann57_in_gaa(T79, 0, X211) → U9_gaa(T79, X211, ackermann27_in_ga(T79, X211))
U9_gaa(T79, X211, ackermann27_out_ga(T79, X211)) → ackermann57_out_gaa(T79, 0, X211)
ackermann57_in_gaa(T84, s(T86), X229) → U10_gaa(T84, 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))
U11_gaaa(T84, T86, X228, X229, ackermann57_out_gaa(T84, T86, X228)) → p65_out_gaaa(T84, T86, X228, X229)
p65_in_gaaa(T84, T86, T88, X229) → U12_gaaa(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U12_gaaa(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_gaaa(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
ackermann68_in_gaa(0, T96, s(T96)) → ackermann68_out_gaa(0, T96, s(T96))
ackermann68_in_gaa(s(T101), 0, X265) → U14_gaa(T101, X265, ackermann27_in_ga(T101, X265))
U14_gaa(T101, X265, ackermann27_out_ga(T101, X265)) → ackermann68_out_gaa(s(T101), 0, X265)
ackermann68_in_gaa(s(T106), s(T108), X283) → U15_gaa(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
U15_gaa(T106, T108, X283, p65_out_gaaa(T106, T108, X282, X283)) → ackermann68_out_gaa(s(T106), s(T108), X283)
U13_gaaa(T84, T86, T88, X229, ackermann68_out_gaa(T84, T88, X229)) → p65_out_gaaa(T84, T86, T88, X229)
U10_gaa(T84, T86, X229, p65_out_gaaa(T84, T86, X228, X229)) → ackermann57_out_gaa(T84, s(T86), X229)
U19_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, X182)) → ackermann1_out_gag(s(T68), s(T71), T70)
ackermann1_in_gag(s(T68), s(T71), T70) → U20_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → U21_gag(T68, T71, T70, ackermann1_in_gag(T68, T73, T70))
ackermann1_in_gag(s(T123), s(0), T117) → U22_gag(T123, T117, ackermann27_in_ga(T123, X319))
U22_gag(T123, T117, ackermann27_out_ga(T123, X319)) → ackermann1_out_gag(s(T123), s(0), T117)
ackermann1_in_gag(s(T123), s(0), T117) → U23_gag(T123, T117, ackermann27_in_ga(T123, T124))
U23_gag(T123, T117, ackermann27_out_ga(T123, T124)) → U24_gag(T123, T117, ackermann1_in_gag(T123, T124, T117))
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U25_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, X340))
U25_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, X340)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U26_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U27_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, X341))
U27_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, X341)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → U29_gag(T131, T133, T117, ackermann1_in_gag(T131, T139, T117))
U29_gag(T131, T133, T117, ackermann1_out_gag(T131, T139, T117)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U24_gag(T123, T117, ackermann1_out_gag(T123, T124, T117)) → ackermann1_out_gag(s(T123), s(0), T117)
U21_gag(T68, T71, T70, ackermann1_out_gag(T68, T73, T70)) → ackermann1_out_gag(s(T68), s(T71), T70)
U18_gag(T19, T20, ackermann1_out_gag(T19, T22, T20)) → ackermann1_out_gag(s(s(T19)), 0, T20)

(48) DependencyPairsProof (EQUIVALENT transformation)

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

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, ackermann21_in_ga(T32, T34))
U3_GA(T32, X97, ackermann21_out_ga(T32, T34)) → U4_GA(T32, X97, ackermann38_in_gaa(T32, T34, X97))
U3_GA(T32, X97, ackermann21_out_ga(T32, T34)) → ACKERMANN38_IN_GAA(T32, T34, X97)
ACKERMANN38_IN_GAA(s(T47), 0, X133) → U5_GAA(T47, X133, ackermann27_in_ga(T47, X133))
ACKERMANN38_IN_GAA(s(T47), 0, X133) → ACKERMANN27_IN_GA(T47, X133)
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → U6_GAA(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → ACKERMANN38_IN_GAA(s(T52), T53, X150)
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → U7_GAA(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_GAA(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_GAA(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U7_GAA(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → ACKERMANN38_IN_GAA(T52, T55, X151)
ACKERMANN1_IN_GAG(s(s(T19)), 0, T20) → U17_GAG(T19, T20, ackermann21_in_ga(T19, T22))
U17_GAG(T19, T20, ackermann21_out_ga(T19, T22)) → U18_GAG(T19, T20, ackermann1_in_gag(T19, T22, T20))
U17_GAG(T19, T20, ackermann21_out_ga(T19, T22)) → ACKERMANN1_IN_GAG(T19, T22, T20)
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, ackermann57_in_gaa(T84, T86, T88))
U12_GAAA(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_GAAA(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
U12_GAAA(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → ACKERMANN68_IN_GAA(T84, T88, X229)
ACKERMANN68_IN_GAA(s(T101), 0, X265) → U14_GAA(T101, X265, ackermann27_in_ga(T101, X265))
ACKERMANN68_IN_GAA(s(T101), 0, X265) → ACKERMANN27_IN_GA(T101, X265)
ACKERMANN68_IN_GAA(s(T106), s(T108), X283) → U15_GAA(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
ACKERMANN68_IN_GAA(s(T106), s(T108), X283) → P65_IN_GAAA(T106, T108, X282, X283)
ACKERMANN1_IN_GAG(s(T68), s(T71), T70) → U20_GAG(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_GAG(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → U21_GAG(T68, T71, T70, ackermann1_in_gag(T68, T73, T70))
U20_GAG(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → ACKERMANN1_IN_GAG(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, ackermann27_in_ga(T123, T124))
U23_GAG(T123, T117, ackermann27_out_ga(T123, T124)) → U24_GAG(T123, T117, ackermann1_in_gag(T123, T124, T117))
U23_GAG(T123, T117, ackermann27_out_ga(T123, T124)) → ACKERMANN1_IN_GAG(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, ackermann57_in_gaa(T131, T133, T135))
U26_GAG(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U27_GAG(T131, T133, T117, ackermann68_in_gaa(T131, T135, X341))
U26_GAG(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → ACKERMANN68_IN_GAA(T131, T135, X341)
U26_GAG(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_GAG(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_GAG(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → U29_GAG(T131, T133, T117, ackermann1_in_gag(T131, T139, T117))
U28_GAG(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → ACKERMANN1_IN_GAG(T131, T139, T117)

The TRS R consists of the following rules:

ackermann1_in_gag(0, T5, s(T5)) → ackermann1_out_gag(0, T5, s(T5))
ackermann1_in_gag(s(0), 0, s(s(0))) → ackermann1_out_gag(s(0), 0, s(s(0)))
ackermann1_in_gag(s(s(T19)), 0, T20) → U16_gag(T19, T20, ackermann21_in_ga(T19, X40))
ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
U16_gag(T19, T20, ackermann21_out_ga(T19, X40)) → ackermann1_out_gag(s(s(T19)), 0, T20)
ackermann1_in_gag(s(s(T19)), 0, T20) → U17_gag(T19, T20, ackermann21_in_ga(T19, T22))
U17_gag(T19, T20, ackermann21_out_ga(T19, T22)) → U18_gag(T19, T20, ackermann1_in_gag(T19, T22, T20))
ackermann1_in_gag(s(T68), s(T71), T70) → U19_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, X182))
ackermann57_in_gaa(T79, 0, X211) → U9_gaa(T79, X211, ackermann27_in_ga(T79, X211))
U9_gaa(T79, X211, ackermann27_out_ga(T79, X211)) → ackermann57_out_gaa(T79, 0, X211)
ackermann57_in_gaa(T84, s(T86), X229) → U10_gaa(T84, 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))
U11_gaaa(T84, T86, X228, X229, ackermann57_out_gaa(T84, T86, X228)) → p65_out_gaaa(T84, T86, X228, X229)
p65_in_gaaa(T84, T86, T88, X229) → U12_gaaa(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U12_gaaa(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_gaaa(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
ackermann68_in_gaa(0, T96, s(T96)) → ackermann68_out_gaa(0, T96, s(T96))
ackermann68_in_gaa(s(T101), 0, X265) → U14_gaa(T101, X265, ackermann27_in_ga(T101, X265))
U14_gaa(T101, X265, ackermann27_out_ga(T101, X265)) → ackermann68_out_gaa(s(T101), 0, X265)
ackermann68_in_gaa(s(T106), s(T108), X283) → U15_gaa(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
U15_gaa(T106, T108, X283, p65_out_gaaa(T106, T108, X282, X283)) → ackermann68_out_gaa(s(T106), s(T108), X283)
U13_gaaa(T84, T86, T88, X229, ackermann68_out_gaa(T84, T88, X229)) → p65_out_gaaa(T84, T86, T88, X229)
U10_gaa(T84, T86, X229, p65_out_gaaa(T84, T86, X228, X229)) → ackermann57_out_gaa(T84, s(T86), X229)
U19_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, X182)) → ackermann1_out_gag(s(T68), s(T71), T70)
ackermann1_in_gag(s(T68), s(T71), T70) → U20_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → U21_gag(T68, T71, T70, ackermann1_in_gag(T68, T73, T70))
ackermann1_in_gag(s(T123), s(0), T117) → U22_gag(T123, T117, ackermann27_in_ga(T123, X319))
U22_gag(T123, T117, ackermann27_out_ga(T123, X319)) → ackermann1_out_gag(s(T123), s(0), T117)
ackermann1_in_gag(s(T123), s(0), T117) → U23_gag(T123, T117, ackermann27_in_ga(T123, T124))
U23_gag(T123, T117, ackermann27_out_ga(T123, T124)) → U24_gag(T123, T117, ackermann1_in_gag(T123, T124, T117))
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U25_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, X340))
U25_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, X340)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U26_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U27_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, X341))
U27_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, X341)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → U29_gag(T131, T133, T117, ackermann1_in_gag(T131, T139, T117))
U29_gag(T131, T133, T117, ackermann1_out_gag(T131, T139, T117)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U24_gag(T123, T117, ackermann1_out_gag(T123, T124, T117)) → ackermann1_out_gag(s(T123), s(0), T117)
U21_gag(T68, T71, T70, ackermann1_out_gag(T68, T73, T70)) → ackermann1_out_gag(s(T68), s(T71), T70)
U18_gag(T19, T20, ackermann1_out_gag(T19, T22, T20)) → ackermann1_out_gag(s(s(T19)), 0, T20)

The argument filtering Pi contains the following mapping:
ackermann1_in_gag(x1, x2, x3) = ackermann1_in_gag(x1, x3)
0 = 0
s(x1) = s(x1)
ackermann1_out_gag(x1, x2, x3) = ackermann1_out_gag(x2)
U16_gag(x1, x2, x3) = U16_gag(x3)
ackermann21_in_ga(x1, x2) = ackermann21_in_ga(x1)
U1_ga(x1, x2, x3) = U1_ga(x3)
ackermann27_in_ga(x1, x2) = ackermann27_in_ga(x1)
ackermann27_out_ga(x1, x2) = ackermann27_out_ga
U2_ga(x1, x2, x3) = U2_ga(x3)
ackermann21_out_ga(x1, x2) = ackermann21_out_ga
U3_ga(x1, x2, x3) = U3_ga(x1, x3)
U4_ga(x1, x2, x3) = U4_ga(x3)
ackermann38_in_gaa(x1, x2, x3) = ackermann38_in_gaa(x1)
ackermann38_out_gaa(x1, x2, x3) = ackermann38_out_gaa
U5_gaa(x1, x2, x3) = U5_gaa(x3)
U6_gaa(x1, x2, x3, x4) = U6_gaa(x4)
U7_gaa(x1, x2, x3, x4) = U7_gaa(x1, x4)
U8_gaa(x1, x2, x3, x4) = U8_gaa(x4)
U17_gag(x1, x2, x3) = U17_gag(x1, x2, x3)
U18_gag(x1, x2, x3) = U18_gag(x3)
U19_gag(x1, x2, x3, x4) = U19_gag(x4)
ackermann57_in_gaa(x1, x2, x3) = ackermann57_in_gaa(x1)
U9_gaa(x1, x2, x3) = U9_gaa(x3)
ackermann57_out_gaa(x1, x2, x3) = ackermann57_out_gaa(x2)
U10_gaa(x1, x2, x3, x4) = U10_gaa(x4)
p65_in_gaaa(x1, x2, x3, x4) = p65_in_gaaa(x1)
U11_gaaa(x1, x2, x3, x4, x5) = U11_gaaa(x5)
p65_out_gaaa(x1, x2, x3, x4) = p65_out_gaaa(x2)
U12_gaaa(x1, x2, x3, x4, x5) = U12_gaaa(x1, x5)
U13_gaaa(x1, x2, x3, x4, x5) = U13_gaaa(x2, x5)
ackermann68_in_gaa(x1, x2, x3) = ackermann68_in_gaa(x1)
ackermann68_out_gaa(x1, x2, x3) = ackermann68_out_gaa
U14_gaa(x1, x2, x3) = U14_gaa(x3)
U15_gaa(x1, x2, x3, x4) = U15_gaa(x4)
U20_gag(x1, x2, x3, x4) = U20_gag(x1, x3, x4)
U21_gag(x1, x2, x3, x4) = U21_gag(x2, x4)
U22_gag(x1, x2, x3) = U22_gag(x3)
U23_gag(x1, x2, x3) = U23_gag(x1, x2, x3)
U24_gag(x1, x2, x3) = U24_gag(x3)
U25_gag(x1, x2, x3, x4) = U25_gag(x4)
U26_gag(x1, x2, x3, x4) = U26_gag(x1, x3, x4)
U27_gag(x1, x2, x3, x4) = U27_gag(x2, x4)
U28_gag(x1, x2, x3, x4) = U28_gag(x1, x2, x3, x4)
U29_gag(x1, x2, x3, x4) = U29_gag(x2, x4)
ACKERMANN1_IN_GAG(x1, x2, x3) = ACKERMANN1_IN_GAG(x1, x3)
U16_GAG(x1, x2, x3) = U16_GAG(x3)
ACKERMANN21_IN_GA(x1, x2) = ACKERMANN21_IN_GA(x1)
U1_GA(x1, x2, x3) = U1_GA(x3)
ACKERMANN27_IN_GA(x1, x2) = ACKERMANN27_IN_GA(x1)
U2_GA(x1, x2, x3) = U2_GA(x3)
U3_GA(x1, x2, x3) = U3_GA(x1, x3)
U4_GA(x1, x2, x3) = U4_GA(x3)
ACKERMANN38_IN_GAA(x1, x2, x3) = ACKERMANN38_IN_GAA(x1)
U5_GAA(x1, x2, x3) = U5_GAA(x3)
U6_GAA(x1, x2, x3, x4) = U6_GAA(x4)
U7_GAA(x1, x2, x3, x4) = U7_GAA(x1, x4)
U8_GAA(x1, x2, x3, x4) = U8_GAA(x4)
U17_GAG(x1, x2, x3) = U17_GAG(x1, x2, x3)
U18_GAG(x1, x2, x3) = U18_GAG(x3)
U19_GAG(x1, x2, x3, x4) = U19_GAG(x4)
ACKERMANN57_IN_GAA(x1, x2, x3) = ACKERMANN57_IN_GAA(x1)
U9_GAA(x1, x2, x3) = U9_GAA(x3)
U10_GAA(x1, x2, x3, x4) = U10_GAA(x4)
P65_IN_GAAA(x1, x2, x3, x4) = P65_IN_GAAA(x1)
U11_GAAA(x1, x2, x3, x4, x5) = U11_GAAA(x5)
U12_GAAA(x1, x2, x3, x4, x5) = U12_GAAA(x1, x5)
U13_GAAA(x1, x2, x3, x4, x5) = U13_GAAA(x2, x5)
ACKERMANN68_IN_GAA(x1, x2, x3) = ACKERMANN68_IN_GAA(x1)
U14_GAA(x1, x2, x3) = U14_GAA(x3)
U15_GAA(x1, x2, x3, x4) = U15_GAA(x4)
U20_GAG(x1, x2, x3, x4) = U20_GAG(x1, x3, x4)
U21_GAG(x1, x2, x3, x4) = U21_GAG(x2, x4)
U22_GAG(x1, x2, x3) = U22_GAG(x3)
U23_GAG(x1, x2, x3) = U23_GAG(x1, x2, x3)
U24_GAG(x1, x2, x3) = U24_GAG(x3)
U25_GAG(x1, x2, x3, x4) = U25_GAG(x4)
U26_GAG(x1, x2, x3, x4) = U26_GAG(x1, x3, x4)
U27_GAG(x1, x2, x3, x4) = U27_GAG(x2, x4)
U28_GAG(x1, x2, x3, x4) = U28_GAG(x1, x2, x3, x4)
U29_GAG(x1, x2, x3, x4) = U29_GAG(x2, x4)

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

(49) 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, ackermann21_in_ga(T32, T34))
U3_GA(T32, X97, ackermann21_out_ga(T32, T34)) → U4_GA(T32, X97, ackermann38_in_gaa(T32, T34, X97))
U3_GA(T32, X97, ackermann21_out_ga(T32, T34)) → ACKERMANN38_IN_GAA(T32, T34, X97)
ACKERMANN38_IN_GAA(s(T47), 0, X133) → U5_GAA(T47, X133, ackermann27_in_ga(T47, X133))
ACKERMANN38_IN_GAA(s(T47), 0, X133) → ACKERMANN27_IN_GA(T47, X133)
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → U6_GAA(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → ACKERMANN38_IN_GAA(s(T52), T53, X150)
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → U7_GAA(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_GAA(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_GAA(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U7_GAA(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → ACKERMANN38_IN_GAA(T52, T55, X151)
ACKERMANN1_IN_GAG(s(s(T19)), 0, T20) → U17_GAG(T19, T20, ackermann21_in_ga(T19, T22))
U17_GAG(T19, T20, ackermann21_out_ga(T19, T22)) → U18_GAG(T19, T20, ackermann1_in_gag(T19, T22, T20))
U17_GAG(T19, T20, ackermann21_out_ga(T19, T22)) → ACKERMANN1_IN_GAG(T19, T22, T20)
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, ackermann57_in_gaa(T84, T86, T88))
U12_GAAA(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_GAAA(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
U12_GAAA(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → ACKERMANN68_IN_GAA(T84, T88, X229)
ACKERMANN68_IN_GAA(s(T101), 0, X265) → U14_GAA(T101, X265, ackermann27_in_ga(T101, X265))
ACKERMANN68_IN_GAA(s(T101), 0, X265) → ACKERMANN27_IN_GA(T101, X265)
ACKERMANN68_IN_GAA(s(T106), s(T108), X283) → U15_GAA(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
ACKERMANN68_IN_GAA(s(T106), s(T108), X283) → P65_IN_GAAA(T106, T108, X282, X283)
ACKERMANN1_IN_GAG(s(T68), s(T71), T70) → U20_GAG(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_GAG(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → U21_GAG(T68, T71, T70, ackermann1_in_gag(T68, T73, T70))
U20_GAG(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → ACKERMANN1_IN_GAG(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, ackermann27_in_ga(T123, T124))
U23_GAG(T123, T117, ackermann27_out_ga(T123, T124)) → U24_GAG(T123, T117, ackermann1_in_gag(T123, T124, T117))
U23_GAG(T123, T117, ackermann27_out_ga(T123, T124)) → ACKERMANN1_IN_GAG(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, ackermann57_in_gaa(T131, T133, T135))
U26_GAG(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U27_GAG(T131, T133, T117, ackermann68_in_gaa(T131, T135, X341))
U26_GAG(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → ACKERMANN68_IN_GAA(T131, T135, X341)
U26_GAG(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_GAG(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_GAG(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → U29_GAG(T131, T133, T117, ackermann1_in_gag(T131, T139, T117))
U28_GAG(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → ACKERMANN1_IN_GAG(T131, T139, T117)

The TRS R consists of the following rules:

ackermann1_in_gag(0, T5, s(T5)) → ackermann1_out_gag(0, T5, s(T5))
ackermann1_in_gag(s(0), 0, s(s(0))) → ackermann1_out_gag(s(0), 0, s(s(0)))
ackermann1_in_gag(s(s(T19)), 0, T20) → U16_gag(T19, T20, ackermann21_in_ga(T19, X40))
ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
U16_gag(T19, T20, ackermann21_out_ga(T19, X40)) → ackermann1_out_gag(s(s(T19)), 0, T20)
ackermann1_in_gag(s(s(T19)), 0, T20) → U17_gag(T19, T20, ackermann21_in_ga(T19, T22))
U17_gag(T19, T20, ackermann21_out_ga(T19, T22)) → U18_gag(T19, T20, ackermann1_in_gag(T19, T22, T20))
ackermann1_in_gag(s(T68), s(T71), T70) → U19_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, X182))
ackermann57_in_gaa(T79, 0, X211) → U9_gaa(T79, X211, ackermann27_in_ga(T79, X211))
U9_gaa(T79, X211, ackermann27_out_ga(T79, X211)) → ackermann57_out_gaa(T79, 0, X211)
ackermann57_in_gaa(T84, s(T86), X229) → U10_gaa(T84, 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))
U11_gaaa(T84, T86, X228, X229, ackermann57_out_gaa(T84, T86, X228)) → p65_out_gaaa(T84, T86, X228, X229)
p65_in_gaaa(T84, T86, T88, X229) → U12_gaaa(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U12_gaaa(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_gaaa(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
ackermann68_in_gaa(0, T96, s(T96)) → ackermann68_out_gaa(0, T96, s(T96))
ackermann68_in_gaa(s(T101), 0, X265) → U14_gaa(T101, X265, ackermann27_in_ga(T101, X265))
U14_gaa(T101, X265, ackermann27_out_ga(T101, X265)) → ackermann68_out_gaa(s(T101), 0, X265)
ackermann68_in_gaa(s(T106), s(T108), X283) → U15_gaa(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
U15_gaa(T106, T108, X283, p65_out_gaaa(T106, T108, X282, X283)) → ackermann68_out_gaa(s(T106), s(T108), X283)
U13_gaaa(T84, T86, T88, X229, ackermann68_out_gaa(T84, T88, X229)) → p65_out_gaaa(T84, T86, T88, X229)
U10_gaa(T84, T86, X229, p65_out_gaaa(T84, T86, X228, X229)) → ackermann57_out_gaa(T84, s(T86), X229)
U19_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, X182)) → ackermann1_out_gag(s(T68), s(T71), T70)
ackermann1_in_gag(s(T68), s(T71), T70) → U20_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → U21_gag(T68, T71, T70, ackermann1_in_gag(T68, T73, T70))
ackermann1_in_gag(s(T123), s(0), T117) → U22_gag(T123, T117, ackermann27_in_ga(T123, X319))
U22_gag(T123, T117, ackermann27_out_ga(T123, X319)) → ackermann1_out_gag(s(T123), s(0), T117)
ackermann1_in_gag(s(T123), s(0), T117) → U23_gag(T123, T117, ackermann27_in_ga(T123, T124))
U23_gag(T123, T117, ackermann27_out_ga(T123, T124)) → U24_gag(T123, T117, ackermann1_in_gag(T123, T124, T117))
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U25_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, X340))
U25_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, X340)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U26_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U27_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, X341))
U27_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, X341)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → U29_gag(T131, T133, T117, ackermann1_in_gag(T131, T139, T117))
U29_gag(T131, T133, T117, ackermann1_out_gag(T131, T139, T117)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U24_gag(T123, T117, ackermann1_out_gag(T123, T124, T117)) → ackermann1_out_gag(s(T123), s(0), T117)
U21_gag(T68, T71, T70, ackermann1_out_gag(T68, T73, T70)) → ackermann1_out_gag(s(T68), s(T71), T70)
U18_gag(T19, T20, ackermann1_out_gag(T19, T22, T20)) → ackermann1_out_gag(s(s(T19)), 0, T20)

The argument filtering Pi contains the following mapping:
ackermann1_in_gag(x1, x2, x3) = ackermann1_in_gag(x1, x3)
0 = 0
s(x1) = s(x1)
ackermann1_out_gag(x1, x2, x3) = ackermann1_out_gag(x2)
U16_gag(x1, x2, x3) = U16_gag(x3)
ackermann21_in_ga(x1, x2) = ackermann21_in_ga(x1)
U1_ga(x1, x2, x3) = U1_ga(x3)
ackermann27_in_ga(x1, x2) = ackermann27_in_ga(x1)
ackermann27_out_ga(x1, x2) = ackermann27_out_ga
U2_ga(x1, x2, x3) = U2_ga(x3)
ackermann21_out_ga(x1, x2) = ackermann21_out_ga
U3_ga(x1, x2, x3) = U3_ga(x1, x3)
U4_ga(x1, x2, x3) = U4_ga(x3)
ackermann38_in_gaa(x1, x2, x3) = ackermann38_in_gaa(x1)
ackermann38_out_gaa(x1, x2, x3) = ackermann38_out_gaa
U5_gaa(x1, x2, x3) = U5_gaa(x3)
U6_gaa(x1, x2, x3, x4) = U6_gaa(x4)
U7_gaa(x1, x2, x3, x4) = U7_gaa(x1, x4)
U8_gaa(x1, x2, x3, x4) = U8_gaa(x4)
U17_gag(x1, x2, x3) = U17_gag(x1, x2, x3)
U18_gag(x1, x2, x3) = U18_gag(x3)
U19_gag(x1, x2, x3, x4) = U19_gag(x4)
ackermann57_in_gaa(x1, x2, x3) = ackermann57_in_gaa(x1)
U9_gaa(x1, x2, x3) = U9_gaa(x3)
ackermann57_out_gaa(x1, x2, x3) = ackermann57_out_gaa(x2)
U10_gaa(x1, x2, x3, x4) = U10_gaa(x4)
p65_in_gaaa(x1, x2, x3, x4) = p65_in_gaaa(x1)
U11_gaaa(x1, x2, x3, x4, x5) = U11_gaaa(x5)
p65_out_gaaa(x1, x2, x3, x4) = p65_out_gaaa(x2)
U12_gaaa(x1, x2, x3, x4, x5) = U12_gaaa(x1, x5)
U13_gaaa(x1, x2, x3, x4, x5) = U13_gaaa(x2, x5)
ackermann68_in_gaa(x1, x2, x3) = ackermann68_in_gaa(x1)
ackermann68_out_gaa(x1, x2, x3) = ackermann68_out_gaa
U14_gaa(x1, x2, x3) = U14_gaa(x3)
U15_gaa(x1, x2, x3, x4) = U15_gaa(x4)
U20_gag(x1, x2, x3, x4) = U20_gag(x1, x3, x4)
U21_gag(x1, x2, x3, x4) = U21_gag(x2, x4)
U22_gag(x1, x2, x3) = U22_gag(x3)
U23_gag(x1, x2, x3) = U23_gag(x1, x2, x3)
U24_gag(x1, x2, x3) = U24_gag(x3)
U25_gag(x1, x2, x3, x4) = U25_gag(x4)
U26_gag(x1, x2, x3, x4) = U26_gag(x1, x3, x4)
U27_gag(x1, x2, x3, x4) = U27_gag(x2, x4)
U28_gag(x1, x2, x3, x4) = U28_gag(x1, x2, x3, x4)
U29_gag(x1, x2, x3, x4) = U29_gag(x2, x4)
ACKERMANN1_IN_GAG(x1, x2, x3) = ACKERMANN1_IN_GAG(x1, x3)
U16_GAG(x1, x2, x3) = U16_GAG(x3)
ACKERMANN21_IN_GA(x1, x2) = ACKERMANN21_IN_GA(x1)
U1_GA(x1, x2, x3) = U1_GA(x3)
ACKERMANN27_IN_GA(x1, x2) = ACKERMANN27_IN_GA(x1)
U2_GA(x1, x2, x3) = U2_GA(x3)
U3_GA(x1, x2, x3) = U3_GA(x1, x3)
U4_GA(x1, x2, x3) = U4_GA(x3)
ACKERMANN38_IN_GAA(x1, x2, x3) = ACKERMANN38_IN_GAA(x1)
U5_GAA(x1, x2, x3) = U5_GAA(x3)
U6_GAA(x1, x2, x3, x4) = U6_GAA(x4)
U7_GAA(x1, x2, x3, x4) = U7_GAA(x1, x4)
U8_GAA(x1, x2, x3, x4) = U8_GAA(x4)
U17_GAG(x1, x2, x3) = U17_GAG(x1, x2, x3)
U18_GAG(x1, x2, x3) = U18_GAG(x3)
U19_GAG(x1, x2, x3, x4) = U19_GAG(x4)
ACKERMANN57_IN_GAA(x1, x2, x3) = ACKERMANN57_IN_GAA(x1)
U9_GAA(x1, x2, x3) = U9_GAA(x3)
U10_GAA(x1, x2, x3, x4) = U10_GAA(x4)
P65_IN_GAAA(x1, x2, x3, x4) = P65_IN_GAAA(x1)
U11_GAAA(x1, x2, x3, x4, x5) = U11_GAAA(x5)
U12_GAAA(x1, x2, x3, x4, x5) = U12_GAAA(x1, x5)
U13_GAAA(x1, x2, x3, x4, x5) = U13_GAAA(x2, x5)
ACKERMANN68_IN_GAA(x1, x2, x3) = ACKERMANN68_IN_GAA(x1)
U14_GAA(x1, x2, x3) = U14_GAA(x3)
U15_GAA(x1, x2, x3, x4) = U15_GAA(x4)
U20_GAG(x1, x2, x3, x4) = U20_GAG(x1, x3, x4)
U21_GAG(x1, x2, x3, x4) = U21_GAG(x2, x4)
U22_GAG(x1, x2, x3) = U22_GAG(x3)
U23_GAG(x1, x2, x3) = U23_GAG(x1, x2, x3)
U24_GAG(x1, x2, x3) = U24_GAG(x3)
U25_GAG(x1, x2, x3, x4) = U25_GAG(x4)
U26_GAG(x1, x2, x3, x4) = U26_GAG(x1, x3, x4)
U27_GAG(x1, x2, x3, x4) = U27_GAG(x2, x4)
U28_GAG(x1, x2, x3, x4) = U28_GAG(x1, x2, x3, x4)
U29_GAG(x1, x2, x3, x4) = U29_GAG(x2, x4)

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

(50) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 28 less nodes.

(51) Complex Obligation (AND)

(52) 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, ackermann21_in_ga(T32, T34))
U3_GA(T32, X97, ackermann21_out_ga(T32, T34)) → ACKERMANN38_IN_GAA(T32, T34, X97)
ACKERMANN38_IN_GAA(s(T47), 0, X133) → ACKERMANN27_IN_GA(T47, X133)
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → ACKERMANN38_IN_GAA(s(T52), T53, X150)
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → U7_GAA(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_GAA(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → ACKERMANN38_IN_GAA(T52, T55, X151)

The TRS R consists of the following rules:

ackermann1_in_gag(0, T5, s(T5)) → ackermann1_out_gag(0, T5, s(T5))
ackermann1_in_gag(s(0), 0, s(s(0))) → ackermann1_out_gag(s(0), 0, s(s(0)))
ackermann1_in_gag(s(s(T19)), 0, T20) → U16_gag(T19, T20, ackermann21_in_ga(T19, X40))
ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
U16_gag(T19, T20, ackermann21_out_ga(T19, X40)) → ackermann1_out_gag(s(s(T19)), 0, T20)
ackermann1_in_gag(s(s(T19)), 0, T20) → U17_gag(T19, T20, ackermann21_in_ga(T19, T22))
U17_gag(T19, T20, ackermann21_out_ga(T19, T22)) → U18_gag(T19, T20, ackermann1_in_gag(T19, T22, T20))
ackermann1_in_gag(s(T68), s(T71), T70) → U19_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, X182))
ackermann57_in_gaa(T79, 0, X211) → U9_gaa(T79, X211, ackermann27_in_ga(T79, X211))
U9_gaa(T79, X211, ackermann27_out_ga(T79, X211)) → ackermann57_out_gaa(T79, 0, X211)
ackermann57_in_gaa(T84, s(T86), X229) → U10_gaa(T84, 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))
U11_gaaa(T84, T86, X228, X229, ackermann57_out_gaa(T84, T86, X228)) → p65_out_gaaa(T84, T86, X228, X229)
p65_in_gaaa(T84, T86, T88, X229) → U12_gaaa(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U12_gaaa(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_gaaa(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
ackermann68_in_gaa(0, T96, s(T96)) → ackermann68_out_gaa(0, T96, s(T96))
ackermann68_in_gaa(s(T101), 0, X265) → U14_gaa(T101, X265, ackermann27_in_ga(T101, X265))
U14_gaa(T101, X265, ackermann27_out_ga(T101, X265)) → ackermann68_out_gaa(s(T101), 0, X265)
ackermann68_in_gaa(s(T106), s(T108), X283) → U15_gaa(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
U15_gaa(T106, T108, X283, p65_out_gaaa(T106, T108, X282, X283)) → ackermann68_out_gaa(s(T106), s(T108), X283)
U13_gaaa(T84, T86, T88, X229, ackermann68_out_gaa(T84, T88, X229)) → p65_out_gaaa(T84, T86, T88, X229)
U10_gaa(T84, T86, X229, p65_out_gaaa(T84, T86, X228, X229)) → ackermann57_out_gaa(T84, s(T86), X229)
U19_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, X182)) → ackermann1_out_gag(s(T68), s(T71), T70)
ackermann1_in_gag(s(T68), s(T71), T70) → U20_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → U21_gag(T68, T71, T70, ackermann1_in_gag(T68, T73, T70))
ackermann1_in_gag(s(T123), s(0), T117) → U22_gag(T123, T117, ackermann27_in_ga(T123, X319))
U22_gag(T123, T117, ackermann27_out_ga(T123, X319)) → ackermann1_out_gag(s(T123), s(0), T117)
ackermann1_in_gag(s(T123), s(0), T117) → U23_gag(T123, T117, ackermann27_in_ga(T123, T124))
U23_gag(T123, T117, ackermann27_out_ga(T123, T124)) → U24_gag(T123, T117, ackermann1_in_gag(T123, T124, T117))
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U25_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, X340))
U25_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, X340)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U26_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U27_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, X341))
U27_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, X341)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → U29_gag(T131, T133, T117, ackermann1_in_gag(T131, T139, T117))
U29_gag(T131, T133, T117, ackermann1_out_gag(T131, T139, T117)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U24_gag(T123, T117, ackermann1_out_gag(T123, T124, T117)) → ackermann1_out_gag(s(T123), s(0), T117)
U21_gag(T68, T71, T70, ackermann1_out_gag(T68, T73, T70)) → ackermann1_out_gag(s(T68), s(T71), T70)
U18_gag(T19, T20, ackermann1_out_gag(T19, T22, T20)) → ackermann1_out_gag(s(s(T19)), 0, T20)

(53) UsableRulesProof (EQUIVALENT transformation)

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

(54) 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, ackermann21_in_ga(T32, T34))
U3_GA(T32, X97, ackermann21_out_ga(T32, T34)) → ACKERMANN38_IN_GAA(T32, T34, X97)
ACKERMANN38_IN_GAA(s(T47), 0, X133) → ACKERMANN27_IN_GA(T47, X133)
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → ACKERMANN38_IN_GAA(s(T52), T53, X150)
ACKERMANN38_IN_GAA(s(T52), s(T53), X151) → U7_GAA(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_GAA(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → ACKERMANN38_IN_GAA(T52, T55, X151)

The TRS R consists of the following rules:

ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)

(55) PiDPToQDPProof (SOUND transformation)

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

(56) 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, ackermann21_in_ga(T32))
U3_GA(T32, ackermann21_out_ga) → ACKERMANN38_IN_GAA(T32)
ACKERMANN38_IN_GAA(s(T47)) → ACKERMANN27_IN_GA(T47)
ACKERMANN38_IN_GAA(s(T52)) → ACKERMANN38_IN_GAA(s(T52))
ACKERMANN38_IN_GAA(s(T52)) → U7_GAA(T52, ackermann38_in_gaa(s(T52)))
U7_GAA(T52, ackermann38_out_gaa) → ACKERMANN38_IN_GAA(T52)

The TRS R consists of the following rules:

ackermann21_in_ga(T28) → U1_ga(ackermann27_in_ga(T28))
ackermann38_in_gaa(s(T47)) → U5_gaa(ackermann27_in_ga(T47))
ackermann38_in_gaa(s(T52)) → U6_gaa(ackermann38_in_gaa(s(T52)))
ackermann38_in_gaa(s(T52)) → U7_gaa(T52, ackermann38_in_gaa(s(T52)))
U1_ga(ackermann27_out_ga) → ackermann21_out_ga
U5_gaa(ackermann27_out_ga) → ackermann38_out_gaa
U6_gaa(ackermann38_out_gaa) → ackermann38_out_gaa
U7_gaa(T52, ackermann38_out_gaa) → U8_gaa(ackermann38_in_gaa(T52))
ackermann27_in_ga(0) → ackermann27_out_ga
ackermann27_in_ga(s(T32)) → U2_ga(ackermann21_in_ga(T32))
ackermann27_in_ga(s(T32)) → U3_ga(T32, ackermann21_in_ga(T32))
U8_gaa(ackermann38_out_gaa) → ackermann38_out_gaa
U2_ga(ackermann21_out_ga) → ackermann27_out_ga
U3_ga(T32, ackermann21_out_ga) → U4_ga(ackermann38_in_gaa(T32))
ackermann38_in_gaa(0) → ackermann38_out_gaa
U4_ga(ackermann38_out_gaa) → ackermann27_out_ga

The set Q consists of the following terms:

ackermann21_in_ga(x0)
ackermann38_in_gaa(x0)
U1_ga(x0)
U5_gaa(x0)
U6_gaa(x0)
U7_gaa(x0, x1)
ackermann27_in_ga(x0)
U8_gaa(x0)
U2_ga(x0)
U3_ga(x0, x1)
U4_ga(x0)

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

(57) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACKERMANN27_IN_GA(s(T32)) → ACKERMANN21_IN_GA(T32)
ACKERMANN27_IN_GA(s(T32)) → U3_GA(T32, ackermann21_in_ga(T32))

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

POL(ACKERMANN21_IN_GA(x₁)) = 0 +
[ 1, 1 ]

· x₁

POL(ACKERMANN27_IN_GA(x₁)) = 0 +
[ 1, 1 ]

· x₁

POL(s(x₁)) =
/ 1 \

\ 0 /

+
/ 1 0 \

\ 1 1 /

· x₁

POL(U3_GA(x₁, x₂)) = 0 +
[ 0, 1 ]

· x₁ +
[ 0, 0 ]

· x₂

POL(ackermann21_in_ga(x₁)) =
/ 0 \

\ 0 /

+
/ 0 1 \

\ 0 0 /

· x₁

POL(ackermann21_out_ga) =
/ 1 \

\ 1 /

POL(ACKERMANN38_IN_GAA(x₁)) = 0 +
[ 0, 1 ]

· x₁

POL(U7_GAA(x₁, x₂)) = 0 +
[ 0, 1 ]

· x₁ +
[ 0, 0 ]

· x₂

POL(ackermann38_in_gaa(x₁)) =
/ 1 \

\ 1 /

+
/ 1 0 \

\ 1 0 /

· x₁

POL(ackermann38_out_gaa) =
/ 1 \

\ 1 /

POL(U1_ga(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(ackermann27_in_ga(x₁)) =
/ 1 \

\ 0 /

+
/ 1 0 \

\ 0 0 /

· x₁

POL(U5_gaa(x₁)) =
/ 1 \

\ 0 /

+
/ 1 0 \

\ 0 0 /

· x₁

POL(U6_gaa(x₁)) =
/ 1 \

\ 0 /

+
/ 1 1 \

\ 1 0 /

· x₁

POL(U7_gaa(x₁, x₂)) =
/ 1 \

\ 1 /

+
/ 1 1 \

\ 1 0 /

· x₁ +
/ 1 0 \

\ 1 1 /

· x₂

POL(U2_ga(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(ackermann27_out_ga) =
/ 1 \

\ 0 /

POL(0) =
/ 0 \

\ 0 /

POL(U3_ga(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 1 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(U4_ga(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 1 1 /

· x₁

POL(U8_gaa(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

The following usable rules [FROCOS05] were oriented: none

(58) Obligation:

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

ACKERMANN21_IN_GA(T28) → ACKERMANN27_IN_GA(T28)
U3_GA(T32, ackermann21_out_ga) → ACKERMANN38_IN_GAA(T32)
ACKERMANN38_IN_GAA(s(T47)) → ACKERMANN27_IN_GA(T47)
ACKERMANN38_IN_GAA(s(T52)) → ACKERMANN38_IN_GAA(s(T52))
ACKERMANN38_IN_GAA(s(T52)) → U7_GAA(T52, ackermann38_in_gaa(s(T52)))
U7_GAA(T52, ackermann38_out_gaa) → ACKERMANN38_IN_GAA(T52)

The TRS R consists of the following rules:

ackermann21_in_ga(T28) → U1_ga(ackermann27_in_ga(T28))
ackermann38_in_gaa(s(T47)) → U5_gaa(ackermann27_in_ga(T47))
ackermann38_in_gaa(s(T52)) → U6_gaa(ackermann38_in_gaa(s(T52)))
ackermann38_in_gaa(s(T52)) → U7_gaa(T52, ackermann38_in_gaa(s(T52)))
U1_ga(ackermann27_out_ga) → ackermann21_out_ga
U5_gaa(ackermann27_out_ga) → ackermann38_out_gaa
U6_gaa(ackermann38_out_gaa) → ackermann38_out_gaa
U7_gaa(T52, ackermann38_out_gaa) → U8_gaa(ackermann38_in_gaa(T52))
ackermann27_in_ga(0) → ackermann27_out_ga
ackermann27_in_ga(s(T32)) → U2_ga(ackermann21_in_ga(T32))
ackermann27_in_ga(s(T32)) → U3_ga(T32, ackermann21_in_ga(T32))
U8_gaa(ackermann38_out_gaa) → ackermann38_out_gaa
U2_ga(ackermann21_out_ga) → ackermann27_out_ga
U3_ga(T32, ackermann21_out_ga) → U4_ga(ackermann38_in_gaa(T32))
ackermann38_in_gaa(0) → ackermann38_out_gaa
U4_ga(ackermann38_out_gaa) → ackermann27_out_ga

The set Q consists of the following terms:

ackermann21_in_ga(x0)
ackermann38_in_gaa(x0)
U1_ga(x0)
U5_gaa(x0)
U6_gaa(x0)
U7_gaa(x0, x1)
ackermann27_in_ga(x0)
U8_gaa(x0)
U2_ga(x0)
U3_ga(x0, x1)
U4_ga(x0)

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

(59) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

(60) Obligation:

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

ACKERMANN38_IN_GAA(s(T52)) → U7_GAA(T52, ackermann38_in_gaa(s(T52)))
U7_GAA(T52, ackermann38_out_gaa) → ACKERMANN38_IN_GAA(T52)
ACKERMANN38_IN_GAA(s(T52)) → ACKERMANN38_IN_GAA(s(T52))

The TRS R consists of the following rules:

ackermann21_in_ga(T28) → U1_ga(ackermann27_in_ga(T28))
ackermann38_in_gaa(s(T47)) → U5_gaa(ackermann27_in_ga(T47))
ackermann38_in_gaa(s(T52)) → U6_gaa(ackermann38_in_gaa(s(T52)))
ackermann38_in_gaa(s(T52)) → U7_gaa(T52, ackermann38_in_gaa(s(T52)))
U1_ga(ackermann27_out_ga) → ackermann21_out_ga
U5_gaa(ackermann27_out_ga) → ackermann38_out_gaa
U6_gaa(ackermann38_out_gaa) → ackermann38_out_gaa
U7_gaa(T52, ackermann38_out_gaa) → U8_gaa(ackermann38_in_gaa(T52))
ackermann27_in_ga(0) → ackermann27_out_ga
ackermann27_in_ga(s(T32)) → U2_ga(ackermann21_in_ga(T32))
ackermann27_in_ga(s(T32)) → U3_ga(T32, ackermann21_in_ga(T32))
U8_gaa(ackermann38_out_gaa) → ackermann38_out_gaa
U2_ga(ackermann21_out_ga) → ackermann27_out_ga
U3_ga(T32, ackermann21_out_ga) → U4_ga(ackermann38_in_gaa(T32))
ackermann38_in_gaa(0) → ackermann38_out_gaa
U4_ga(ackermann38_out_gaa) → ackermann27_out_ga

The set Q consists of the following terms:

ackermann21_in_ga(x0)
ackermann38_in_gaa(x0)
U1_ga(x0)
U5_gaa(x0)
U6_gaa(x0)
U7_gaa(x0, x1)
ackermann27_in_ga(x0)
U8_gaa(x0)
U2_ga(x0)
U3_ga(x0, x1)
U4_ga(x0)

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

(61) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U7_GAA(T52, ackermann38_out_gaa) → ACKERMANN38_IN_GAA(T52)

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

POL(0) = 1
POL(ACKERMANN38_IN_GAA(x₁)) = x₁
POL(U1_ga(x₁)) = 0
POL(U2_ga(x₁)) = 0
POL(U3_ga(x₁, x₂)) = 0
POL(U4_ga(x₁)) = 0
POL(U5_gaa(x₁)) = 0
POL(U6_gaa(x₁)) = 0
POL(U7_GAA(x₁, x₂)) = 1 + x₁
POL(U7_gaa(x₁, x₂)) = 0
POL(U8_gaa(x₁)) = 0
POL(ackermann21_in_ga(x₁)) = 0
POL(ackermann21_out_ga) = 0
POL(ackermann27_in_ga(x₁)) = 1 + x₁
POL(ackermann27_out_ga) = 1
POL(ackermann38_in_gaa(x₁)) = 1 + x₁
POL(ackermann38_out_gaa) = 1
POL(s(x₁)) = 1 + x₁

The following usable rules [FROCOS05] were oriented: none

(62) Obligation:

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

ACKERMANN38_IN_GAA(s(T52)) → U7_GAA(T52, ackermann38_in_gaa(s(T52)))
ACKERMANN38_IN_GAA(s(T52)) → ACKERMANN38_IN_GAA(s(T52))

The TRS R consists of the following rules:

ackermann21_in_ga(T28) → U1_ga(ackermann27_in_ga(T28))
ackermann38_in_gaa(s(T47)) → U5_gaa(ackermann27_in_ga(T47))
ackermann38_in_gaa(s(T52)) → U6_gaa(ackermann38_in_gaa(s(T52)))
ackermann38_in_gaa(s(T52)) → U7_gaa(T52, ackermann38_in_gaa(s(T52)))
U1_ga(ackermann27_out_ga) → ackermann21_out_ga
U5_gaa(ackermann27_out_ga) → ackermann38_out_gaa
U6_gaa(ackermann38_out_gaa) → ackermann38_out_gaa
U7_gaa(T52, ackermann38_out_gaa) → U8_gaa(ackermann38_in_gaa(T52))
ackermann27_in_ga(0) → ackermann27_out_ga
ackermann27_in_ga(s(T32)) → U2_ga(ackermann21_in_ga(T32))
ackermann27_in_ga(s(T32)) → U3_ga(T32, ackermann21_in_ga(T32))
U8_gaa(ackermann38_out_gaa) → ackermann38_out_gaa
U2_ga(ackermann21_out_ga) → ackermann27_out_ga
U3_ga(T32, ackermann21_out_ga) → U4_ga(ackermann38_in_gaa(T32))
ackermann38_in_gaa(0) → ackermann38_out_gaa
U4_ga(ackermann38_out_gaa) → ackermann27_out_ga

The set Q consists of the following terms:

ackermann21_in_ga(x0)
ackermann38_in_gaa(x0)
U1_ga(x0)
U5_gaa(x0)
U6_gaa(x0)
U7_gaa(x0, x1)
ackermann27_in_ga(x0)
U8_gaa(x0)
U2_ga(x0)
U3_ga(x0, x1)
U4_ga(x0)

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

(63) DependencyGraphProof (EQUIVALENT transformation)

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

(64) Obligation:

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

ACKERMANN38_IN_GAA(s(T52)) → ACKERMANN38_IN_GAA(s(T52))

The TRS R consists of the following rules:

ackermann21_in_ga(T28) → U1_ga(ackermann27_in_ga(T28))
ackermann38_in_gaa(s(T47)) → U5_gaa(ackermann27_in_ga(T47))
ackermann38_in_gaa(s(T52)) → U6_gaa(ackermann38_in_gaa(s(T52)))
ackermann38_in_gaa(s(T52)) → U7_gaa(T52, ackermann38_in_gaa(s(T52)))
U1_ga(ackermann27_out_ga) → ackermann21_out_ga
U5_gaa(ackermann27_out_ga) → ackermann38_out_gaa
U6_gaa(ackermann38_out_gaa) → ackermann38_out_gaa
U7_gaa(T52, ackermann38_out_gaa) → U8_gaa(ackermann38_in_gaa(T52))
ackermann27_in_ga(0) → ackermann27_out_ga
ackermann27_in_ga(s(T32)) → U2_ga(ackermann21_in_ga(T32))
ackermann27_in_ga(s(T32)) → U3_ga(T32, ackermann21_in_ga(T32))
U8_gaa(ackermann38_out_gaa) → ackermann38_out_gaa
U2_ga(ackermann21_out_ga) → ackermann27_out_ga
U3_ga(T32, ackermann21_out_ga) → U4_ga(ackermann38_in_gaa(T32))
ackermann38_in_gaa(0) → ackermann38_out_gaa
U4_ga(ackermann38_out_gaa) → ackermann27_out_ga

The set Q consists of the following terms:

ackermann21_in_ga(x0)
ackermann38_in_gaa(x0)
U1_ga(x0)
U5_gaa(x0)
U6_gaa(x0)
U7_gaa(x0, x1)
ackermann27_in_ga(x0)
U8_gaa(x0)
U2_ga(x0)
U3_ga(x0, x1)
U4_ga(x0)

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

(65) UsableRulesProof (EQUIVALENT transformation)

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

(66) Obligation:

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

ACKERMANN38_IN_GAA(s(T52)) → ACKERMANN38_IN_GAA(s(T52))

R is empty.
The set Q consists of the following terms:

ackermann21_in_ga(x0)
ackermann38_in_gaa(x0)
U1_ga(x0)
U5_gaa(x0)
U6_gaa(x0)
U7_gaa(x0, x1)
ackermann27_in_ga(x0)
U8_gaa(x0)
U2_ga(x0)
U3_ga(x0, x1)
U4_ga(x0)

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

(67) QReductionProof (EQUIVALENT transformation)

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

ackermann21_in_ga(x0)
ackermann38_in_gaa(x0)
U1_ga(x0)
U5_gaa(x0)
U6_gaa(x0)
U7_gaa(x0, x1)
ackermann27_in_ga(x0)
U8_gaa(x0)
U2_ga(x0)
U3_ga(x0, x1)
U4_ga(x0)

(68) Obligation:

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

ACKERMANN38_IN_GAA(s(T52)) → ACKERMANN38_IN_GAA(s(T52))

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

(69) NonTerminationProof (EQUIVALENT transformation)

Matcher: [ ]
Semiunifier: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from ACKERMANN38_IN_GAA(s(T52)) to ACKERMANN38_IN_GAA(s(T52)).

(70) NO

(71) Obligation:

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

ACKERMANN57_IN_GAA(T84, s(T86), X229) → P65_IN_GAAA(T84, T86, X228, X229)
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, ackermann57_in_gaa(T84, T86, T88))
U12_GAAA(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → ACKERMANN68_IN_GAA(T84, T88, X229)
ACKERMANN68_IN_GAA(s(T106), s(T108), X283) → P65_IN_GAAA(T106, T108, X282, X283)

The TRS R consists of the following rules:

ackermann1_in_gag(0, T5, s(T5)) → ackermann1_out_gag(0, T5, s(T5))
ackermann1_in_gag(s(0), 0, s(s(0))) → ackermann1_out_gag(s(0), 0, s(s(0)))
ackermann1_in_gag(s(s(T19)), 0, T20) → U16_gag(T19, T20, ackermann21_in_ga(T19, X40))
ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
U16_gag(T19, T20, ackermann21_out_ga(T19, X40)) → ackermann1_out_gag(s(s(T19)), 0, T20)
ackermann1_in_gag(s(s(T19)), 0, T20) → U17_gag(T19, T20, ackermann21_in_ga(T19, T22))
U17_gag(T19, T20, ackermann21_out_ga(T19, T22)) → U18_gag(T19, T20, ackermann1_in_gag(T19, T22, T20))
ackermann1_in_gag(s(T68), s(T71), T70) → U19_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, X182))
ackermann57_in_gaa(T79, 0, X211) → U9_gaa(T79, X211, ackermann27_in_ga(T79, X211))
U9_gaa(T79, X211, ackermann27_out_ga(T79, X211)) → ackermann57_out_gaa(T79, 0, X211)
ackermann57_in_gaa(T84, s(T86), X229) → U10_gaa(T84, 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))
U11_gaaa(T84, T86, X228, X229, ackermann57_out_gaa(T84, T86, X228)) → p65_out_gaaa(T84, T86, X228, X229)
p65_in_gaaa(T84, T86, T88, X229) → U12_gaaa(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U12_gaaa(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_gaaa(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
ackermann68_in_gaa(0, T96, s(T96)) → ackermann68_out_gaa(0, T96, s(T96))
ackermann68_in_gaa(s(T101), 0, X265) → U14_gaa(T101, X265, ackermann27_in_ga(T101, X265))
U14_gaa(T101, X265, ackermann27_out_ga(T101, X265)) → ackermann68_out_gaa(s(T101), 0, X265)
ackermann68_in_gaa(s(T106), s(T108), X283) → U15_gaa(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
U15_gaa(T106, T108, X283, p65_out_gaaa(T106, T108, X282, X283)) → ackermann68_out_gaa(s(T106), s(T108), X283)
U13_gaaa(T84, T86, T88, X229, ackermann68_out_gaa(T84, T88, X229)) → p65_out_gaaa(T84, T86, T88, X229)
U10_gaa(T84, T86, X229, p65_out_gaaa(T84, T86, X228, X229)) → ackermann57_out_gaa(T84, s(T86), X229)
U19_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, X182)) → ackermann1_out_gag(s(T68), s(T71), T70)
ackermann1_in_gag(s(T68), s(T71), T70) → U20_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → U21_gag(T68, T71, T70, ackermann1_in_gag(T68, T73, T70))
ackermann1_in_gag(s(T123), s(0), T117) → U22_gag(T123, T117, ackermann27_in_ga(T123, X319))
U22_gag(T123, T117, ackermann27_out_ga(T123, X319)) → ackermann1_out_gag(s(T123), s(0), T117)
ackermann1_in_gag(s(T123), s(0), T117) → U23_gag(T123, T117, ackermann27_in_ga(T123, T124))
U23_gag(T123, T117, ackermann27_out_ga(T123, T124)) → U24_gag(T123, T117, ackermann1_in_gag(T123, T124, T117))
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U25_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, X340))
U25_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, X340)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U26_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U27_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, X341))
U27_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, X341)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → U29_gag(T131, T133, T117, ackermann1_in_gag(T131, T139, T117))
U29_gag(T131, T133, T117, ackermann1_out_gag(T131, T139, T117)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U24_gag(T123, T117, ackermann1_out_gag(T123, T124, T117)) → ackermann1_out_gag(s(T123), s(0), T117)
U21_gag(T68, T71, T70, ackermann1_out_gag(T68, T73, T70)) → ackermann1_out_gag(s(T68), s(T71), T70)
U18_gag(T19, T20, ackermann1_out_gag(T19, T22, T20)) → ackermann1_out_gag(s(s(T19)), 0, T20)

(72) UsableRulesProof (EQUIVALENT transformation)

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

(73) Obligation:

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

ACKERMANN57_IN_GAA(T84, s(T86), X229) → P65_IN_GAAA(T84, T86, X228, X229)
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, ackermann57_in_gaa(T84, T86, T88))
U12_GAAA(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → ACKERMANN68_IN_GAA(T84, T88, X229)
ACKERMANN68_IN_GAA(s(T106), s(T108), X283) → P65_IN_GAAA(T106, T108, X282, X283)

The TRS R consists of the following rules:

ackermann57_in_gaa(T79, 0, X211) → U9_gaa(T79, 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))
U9_gaa(T79, X211, ackermann27_out_ga(T79, X211)) → ackermann57_out_gaa(T79, 0, X211)
U10_gaa(T84, T86, X229, p65_out_gaaa(T84, T86, X228, X229)) → ackermann57_out_gaa(T84, s(T86), X229)
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
p65_in_gaaa(T84, T86, X228, X229) → U11_gaaa(T84, T86, X228, X229, ackermann57_in_gaa(T84, T86, X228))
p65_in_gaaa(T84, T86, T88, X229) → U12_gaaa(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
U11_gaaa(T84, T86, X228, X229, ackermann57_out_gaa(T84, T86, X228)) → p65_out_gaaa(T84, T86, X228, X229)
U12_gaaa(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_gaaa(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)
U13_gaaa(T84, T86, T88, X229, ackermann68_out_gaa(T84, T88, X229)) → p65_out_gaaa(T84, T86, T88, X229)
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
ackermann68_in_gaa(0, T96, s(T96)) → ackermann68_out_gaa(0, T96, s(T96))
ackermann68_in_gaa(s(T101), 0, X265) → U14_gaa(T101, X265, ackermann27_in_ga(T101, X265))
ackermann68_in_gaa(s(T106), s(T108), X283) → U15_gaa(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U14_gaa(T101, X265, ackermann27_out_ga(T101, X265)) → ackermann68_out_gaa(s(T101), 0, X265)
U15_gaa(T106, T108, X283, p65_out_gaaa(T106, T108, X282, X283)) → ackermann68_out_gaa(s(T106), s(T108), X283)
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)

The argument filtering Pi contains the following mapping:
0 = 0
s(x1) = s(x1)
ackermann21_in_ga(x1, x2) = ackermann21_in_ga(x1)
U1_ga(x1, x2, x3) = U1_ga(x3)
ackermann27_in_ga(x1, x2) = ackermann27_in_ga(x1)
ackermann27_out_ga(x1, x2) = ackermann27_out_ga
U2_ga(x1, x2, x3) = U2_ga(x3)
ackermann21_out_ga(x1, x2) = ackermann21_out_ga
U3_ga(x1, x2, x3) = U3_ga(x1, x3)
U4_ga(x1, x2, x3) = U4_ga(x3)
ackermann38_in_gaa(x1, x2, x3) = ackermann38_in_gaa(x1)
ackermann38_out_gaa(x1, x2, x3) = ackermann38_out_gaa
U5_gaa(x1, x2, x3) = U5_gaa(x3)
U6_gaa(x1, x2, x3, x4) = U6_gaa(x4)
U7_gaa(x1, x2, x3, x4) = U7_gaa(x1, x4)
U8_gaa(x1, x2, x3, x4) = U8_gaa(x4)
ackermann57_in_gaa(x1, x2, x3) = ackermann57_in_gaa(x1)
U9_gaa(x1, x2, x3) = U9_gaa(x3)
ackermann57_out_gaa(x1, x2, x3) = ackermann57_out_gaa(x2)
U10_gaa(x1, x2, x3, x4) = U10_gaa(x4)
p65_in_gaaa(x1, x2, x3, x4) = p65_in_gaaa(x1)
U11_gaaa(x1, x2, x3, x4, x5) = U11_gaaa(x5)
p65_out_gaaa(x1, x2, x3, x4) = p65_out_gaaa(x2)
U12_gaaa(x1, x2, x3, x4, x5) = U12_gaaa(x1, x5)
U13_gaaa(x1, x2, x3, x4, x5) = U13_gaaa(x2, x5)
ackermann68_in_gaa(x1, x2, x3) = ackermann68_in_gaa(x1)
ackermann68_out_gaa(x1, x2, x3) = ackermann68_out_gaa
U14_gaa(x1, x2, x3) = U14_gaa(x3)
U15_gaa(x1, x2, x3, x4) = U15_gaa(x4)
ACKERMANN57_IN_GAA(x1, x2, x3) = ACKERMANN57_IN_GAA(x1)
P65_IN_GAAA(x1, x2, x3, x4) = P65_IN_GAAA(x1)
U12_GAAA(x1, x2, x3, x4, x5) = U12_GAAA(x1, x5)
ACKERMANN68_IN_GAA(x1, x2, x3) = ACKERMANN68_IN_GAA(x1)

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

(74) PiDPToQDPProof (SOUND transformation)

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

(75) Obligation:

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

ACKERMANN57_IN_GAA(T84) → P65_IN_GAAA(T84)
P65_IN_GAAA(T84) → ACKERMANN57_IN_GAA(T84)
P65_IN_GAAA(T84) → U12_GAAA(T84, ackermann57_in_gaa(T84))
U12_GAAA(T84, ackermann57_out_gaa(T86)) → ACKERMANN68_IN_GAA(T84)
ACKERMANN68_IN_GAA(s(T106)) → P65_IN_GAAA(T106)

The TRS R consists of the following rules:

ackermann57_in_gaa(T79) → U9_gaa(ackermann27_in_ga(T79))
ackermann57_in_gaa(T84) → U10_gaa(p65_in_gaaa(T84))
U9_gaa(ackermann27_out_ga) → ackermann57_out_gaa(0)
U10_gaa(p65_out_gaaa(T86)) → ackermann57_out_gaa(s(T86))
ackermann27_in_ga(0) → ackermann27_out_ga
ackermann27_in_ga(s(T32)) → U2_ga(ackermann21_in_ga(T32))
ackermann27_in_ga(s(T32)) → U3_ga(T32, ackermann21_in_ga(T32))
p65_in_gaaa(T84) → U11_gaaa(ackermann57_in_gaa(T84))
p65_in_gaaa(T84) → U12_gaaa(T84, ackermann57_in_gaa(T84))
U2_ga(ackermann21_out_ga) → ackermann27_out_ga
U3_ga(T32, ackermann21_out_ga) → U4_ga(ackermann38_in_gaa(T32))
U11_gaaa(ackermann57_out_gaa(T86)) → p65_out_gaaa(T86)
U12_gaaa(T84, ackermann57_out_gaa(T86)) → U13_gaaa(T86, ackermann68_in_gaa(T84))
ackermann21_in_ga(T28) → U1_ga(ackermann27_in_ga(T28))
U4_ga(ackermann38_out_gaa) → ackermann27_out_ga
U13_gaaa(T86, ackermann68_out_gaa) → p65_out_gaaa(T86)
U1_ga(ackermann27_out_ga) → ackermann21_out_ga
ackermann38_in_gaa(0) → ackermann38_out_gaa
ackermann38_in_gaa(s(T47)) → U5_gaa(ackermann27_in_ga(T47))
ackermann38_in_gaa(s(T52)) → U6_gaa(ackermann38_in_gaa(s(T52)))
ackermann38_in_gaa(s(T52)) → U7_gaa(T52, ackermann38_in_gaa(s(T52)))
ackermann68_in_gaa(0) → ackermann68_out_gaa
ackermann68_in_gaa(s(T101)) → U14_gaa(ackermann27_in_ga(T101))
ackermann68_in_gaa(s(T106)) → U15_gaa(p65_in_gaaa(T106))
U5_gaa(ackermann27_out_ga) → ackermann38_out_gaa
U6_gaa(ackermann38_out_gaa) → ackermann38_out_gaa
U7_gaa(T52, ackermann38_out_gaa) → U8_gaa(ackermann38_in_gaa(T52))
U14_gaa(ackermann27_out_ga) → ackermann68_out_gaa
U15_gaa(p65_out_gaaa(T108)) → ackermann68_out_gaa
U8_gaa(ackermann38_out_gaa) → ackermann38_out_gaa

The set Q consists of the following terms:

ackermann57_in_gaa(x0)
U9_gaa(x0)
U10_gaa(x0)
ackermann27_in_ga(x0)
p65_in_gaaa(x0)
U2_ga(x0)
U3_ga(x0, x1)
U11_gaaa(x0)
U12_gaaa(x0, x1)
ackermann21_in_ga(x0)
U4_ga(x0)
U13_gaaa(x0, x1)
U1_ga(x0)
ackermann38_in_gaa(x0)
ackermann68_in_gaa(x0)
U5_gaa(x0)
U6_gaa(x0)
U7_gaa(x0, x1)
U14_gaa(x0)
U15_gaa(x0)
U8_gaa(x0)

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

(76) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

P65_IN_GAAA(T84) → U12_GAAA(T84, ackermann57_in_gaa(T84))

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

POL(0) = 0
POL(ACKERMANN57_IN_GAA(x₁)) = 1 + x₁
POL(ACKERMANN68_IN_GAA(x₁)) = x₁
POL(P65_IN_GAAA(x₁)) = 1 + x₁
POL(U10_gaa(x₁)) = 0
POL(U11_gaaa(x₁)) = 0
POL(U12_GAAA(x₁, x₂)) = x₁
POL(U12_gaaa(x₁, x₂)) = 0
POL(U13_gaaa(x₁, x₂)) = 0
POL(U14_gaa(x₁)) = 0
POL(U15_gaa(x₁)) = 0
POL(U1_ga(x₁)) = 0
POL(U2_ga(x₁)) = 0
POL(U3_ga(x₁, x₂)) = 0
POL(U4_ga(x₁)) = 0
POL(U5_gaa(x₁)) = 0
POL(U6_gaa(x₁)) = 0
POL(U7_gaa(x₁, x₂)) = 0
POL(U8_gaa(x₁)) = 0
POL(U9_gaa(x₁)) = 0
POL(ackermann21_in_ga(x₁)) = 0
POL(ackermann21_out_ga) = 0
POL(ackermann27_in_ga(x₁)) = 0
POL(ackermann27_out_ga) = 0
POL(ackermann38_in_gaa(x₁)) = 0
POL(ackermann38_out_gaa) = 0
POL(ackermann57_in_gaa(x₁)) = 0
POL(ackermann57_out_gaa(x₁)) = 0
POL(ackermann68_in_gaa(x₁)) = 1 + x₁
POL(ackermann68_out_gaa) = 0
POL(p65_in_gaaa(x₁)) = 0
POL(p65_out_gaaa(x₁)) = 0
POL(s(x₁)) = 1 + x₁

The following usable rules [FROCOS05] were oriented: none

(77) Obligation:

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

ACKERMANN57_IN_GAA(T84) → P65_IN_GAAA(T84)
P65_IN_GAAA(T84) → ACKERMANN57_IN_GAA(T84)
U12_GAAA(T84, ackermann57_out_gaa(T86)) → ACKERMANN68_IN_GAA(T84)
ACKERMANN68_IN_GAA(s(T106)) → P65_IN_GAAA(T106)

The TRS R consists of the following rules:

ackermann57_in_gaa(T79) → U9_gaa(ackermann27_in_ga(T79))
ackermann57_in_gaa(T84) → U10_gaa(p65_in_gaaa(T84))
U9_gaa(ackermann27_out_ga) → ackermann57_out_gaa(0)
U10_gaa(p65_out_gaaa(T86)) → ackermann57_out_gaa(s(T86))
ackermann27_in_ga(0) → ackermann27_out_ga
ackermann27_in_ga(s(T32)) → U2_ga(ackermann21_in_ga(T32))
ackermann27_in_ga(s(T32)) → U3_ga(T32, ackermann21_in_ga(T32))
p65_in_gaaa(T84) → U11_gaaa(ackermann57_in_gaa(T84))
p65_in_gaaa(T84) → U12_gaaa(T84, ackermann57_in_gaa(T84))
U2_ga(ackermann21_out_ga) → ackermann27_out_ga
U3_ga(T32, ackermann21_out_ga) → U4_ga(ackermann38_in_gaa(T32))
U11_gaaa(ackermann57_out_gaa(T86)) → p65_out_gaaa(T86)
U12_gaaa(T84, ackermann57_out_gaa(T86)) → U13_gaaa(T86, ackermann68_in_gaa(T84))
ackermann21_in_ga(T28) → U1_ga(ackermann27_in_ga(T28))
U4_ga(ackermann38_out_gaa) → ackermann27_out_ga
U13_gaaa(T86, ackermann68_out_gaa) → p65_out_gaaa(T86)
U1_ga(ackermann27_out_ga) → ackermann21_out_ga
ackermann38_in_gaa(0) → ackermann38_out_gaa
ackermann38_in_gaa(s(T47)) → U5_gaa(ackermann27_in_ga(T47))
ackermann38_in_gaa(s(T52)) → U6_gaa(ackermann38_in_gaa(s(T52)))
ackermann38_in_gaa(s(T52)) → U7_gaa(T52, ackermann38_in_gaa(s(T52)))
ackermann68_in_gaa(0) → ackermann68_out_gaa
ackermann68_in_gaa(s(T101)) → U14_gaa(ackermann27_in_ga(T101))
ackermann68_in_gaa(s(T106)) → U15_gaa(p65_in_gaaa(T106))
U5_gaa(ackermann27_out_ga) → ackermann38_out_gaa
U6_gaa(ackermann38_out_gaa) → ackermann38_out_gaa
U7_gaa(T52, ackermann38_out_gaa) → U8_gaa(ackermann38_in_gaa(T52))
U14_gaa(ackermann27_out_ga) → ackermann68_out_gaa
U15_gaa(p65_out_gaaa(T108)) → ackermann68_out_gaa
U8_gaa(ackermann38_out_gaa) → ackermann38_out_gaa

The set Q consists of the following terms:

ackermann57_in_gaa(x0)
U9_gaa(x0)
U10_gaa(x0)
ackermann27_in_ga(x0)
p65_in_gaaa(x0)
U2_ga(x0)
U3_ga(x0, x1)
U11_gaaa(x0)
U12_gaaa(x0, x1)
ackermann21_in_ga(x0)
U4_ga(x0)
U13_gaaa(x0, x1)
U1_ga(x0)
ackermann38_in_gaa(x0)
ackermann68_in_gaa(x0)
U5_gaa(x0)
U6_gaa(x0)
U7_gaa(x0, x1)
U14_gaa(x0)
U15_gaa(x0)
U8_gaa(x0)

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

(78) DependencyGraphProof (EQUIVALENT transformation)

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

(79) 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)

The TRS R consists of the following rules:

ackermann57_in_gaa(T79) → U9_gaa(ackermann27_in_ga(T79))
ackermann57_in_gaa(T84) → U10_gaa(p65_in_gaaa(T84))
U9_gaa(ackermann27_out_ga) → ackermann57_out_gaa(0)
U10_gaa(p65_out_gaaa(T86)) → ackermann57_out_gaa(s(T86))
ackermann27_in_ga(0) → ackermann27_out_ga
ackermann27_in_ga(s(T32)) → U2_ga(ackermann21_in_ga(T32))
ackermann27_in_ga(s(T32)) → U3_ga(T32, ackermann21_in_ga(T32))
p65_in_gaaa(T84) → U11_gaaa(ackermann57_in_gaa(T84))
p65_in_gaaa(T84) → U12_gaaa(T84, ackermann57_in_gaa(T84))
U2_ga(ackermann21_out_ga) → ackermann27_out_ga
U3_ga(T32, ackermann21_out_ga) → U4_ga(ackermann38_in_gaa(T32))
U11_gaaa(ackermann57_out_gaa(T86)) → p65_out_gaaa(T86)
U12_gaaa(T84, ackermann57_out_gaa(T86)) → U13_gaaa(T86, ackermann68_in_gaa(T84))
ackermann21_in_ga(T28) → U1_ga(ackermann27_in_ga(T28))
U4_ga(ackermann38_out_gaa) → ackermann27_out_ga
U13_gaaa(T86, ackermann68_out_gaa) → p65_out_gaaa(T86)
U1_ga(ackermann27_out_ga) → ackermann21_out_ga
ackermann38_in_gaa(0) → ackermann38_out_gaa
ackermann38_in_gaa(s(T47)) → U5_gaa(ackermann27_in_ga(T47))
ackermann38_in_gaa(s(T52)) → U6_gaa(ackermann38_in_gaa(s(T52)))
ackermann38_in_gaa(s(T52)) → U7_gaa(T52, ackermann38_in_gaa(s(T52)))
ackermann68_in_gaa(0) → ackermann68_out_gaa
ackermann68_in_gaa(s(T101)) → U14_gaa(ackermann27_in_ga(T101))
ackermann68_in_gaa(s(T106)) → U15_gaa(p65_in_gaaa(T106))
U5_gaa(ackermann27_out_ga) → ackermann38_out_gaa
U6_gaa(ackermann38_out_gaa) → ackermann38_out_gaa
U7_gaa(T52, ackermann38_out_gaa) → U8_gaa(ackermann38_in_gaa(T52))
U14_gaa(ackermann27_out_ga) → ackermann68_out_gaa
U15_gaa(p65_out_gaaa(T108)) → ackermann68_out_gaa
U8_gaa(ackermann38_out_gaa) → ackermann38_out_gaa

The set Q consists of the following terms:

ackermann57_in_gaa(x0)
U9_gaa(x0)
U10_gaa(x0)
ackermann27_in_ga(x0)
p65_in_gaaa(x0)
U2_ga(x0)
U3_ga(x0, x1)
U11_gaaa(x0)
U12_gaaa(x0, x1)
ackermann21_in_ga(x0)
U4_ga(x0)
U13_gaaa(x0, x1)
U1_ga(x0)
ackermann38_in_gaa(x0)
ackermann68_in_gaa(x0)
U5_gaa(x0)
U6_gaa(x0)
U7_gaa(x0, x1)
U14_gaa(x0)
U15_gaa(x0)
U8_gaa(x0)

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

(80) UsableRulesProof (EQUIVALENT transformation)

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

(81) 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.
The set Q consists of the following terms:

ackermann57_in_gaa(x0)
U9_gaa(x0)
U10_gaa(x0)
ackermann27_in_ga(x0)
p65_in_gaaa(x0)
U2_ga(x0)
U3_ga(x0, x1)
U11_gaaa(x0)
U12_gaaa(x0, x1)
ackermann21_in_ga(x0)
U4_ga(x0)
U13_gaaa(x0, x1)
U1_ga(x0)
ackermann38_in_gaa(x0)
ackermann68_in_gaa(x0)
U5_gaa(x0)
U6_gaa(x0)
U7_gaa(x0, x1)
U14_gaa(x0)
U15_gaa(x0)
U8_gaa(x0)

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

(82) QReductionProof (EQUIVALENT transformation)

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

ackermann57_in_gaa(x0)
U9_gaa(x0)
U10_gaa(x0)
ackermann27_in_ga(x0)
p65_in_gaaa(x0)
U2_ga(x0)
U3_ga(x0, x1)
U11_gaaa(x0)
U12_gaaa(x0, x1)
ackermann21_in_ga(x0)
U4_ga(x0)
U13_gaaa(x0, x1)
U1_ga(x0)
ackermann38_in_gaa(x0)
ackermann68_in_gaa(x0)
U5_gaa(x0)
U6_gaa(x0)
U7_gaa(x0, x1)
U14_gaa(x0)
U15_gaa(x0)
U8_gaa(x0)

(83) 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.

(84) NonTerminationProof (EQUIVALENT transformation)

Matcher: [ ]
Semiunifier: [ ]

(85) NO

(86) Obligation:

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

ACKERMANN1_IN_GAG(s(s(T19)), 0, T20) → U17_GAG(T19, T20, ackermann21_in_ga(T19, T22))
U17_GAG(T19, T20, ackermann21_out_ga(T19, T22)) → ACKERMANN1_IN_GAG(T19, T22, T20)
ACKERMANN1_IN_GAG(s(T68), s(T71), T70) → U20_GAG(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_GAG(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → ACKERMANN1_IN_GAG(T68, T73, T70)
ACKERMANN1_IN_GAG(s(T123), s(0), T117) → U23_GAG(T123, T117, ackermann27_in_ga(T123, T124))
U23_GAG(T123, T117, ackermann27_out_ga(T123, T124)) → ACKERMANN1_IN_GAG(T123, T124, T117)
ACKERMANN1_IN_GAG(s(T131), s(s(T133)), T117) → U26_GAG(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_GAG(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_GAG(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_GAG(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → ACKERMANN1_IN_GAG(T131, T139, T117)

The TRS R consists of the following rules:

ackermann1_in_gag(0, T5, s(T5)) → ackermann1_out_gag(0, T5, s(T5))
ackermann1_in_gag(s(0), 0, s(s(0))) → ackermann1_out_gag(s(0), 0, s(s(0)))
ackermann1_in_gag(s(s(T19)), 0, T20) → U16_gag(T19, T20, ackermann21_in_ga(T19, X40))
ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
U16_gag(T19, T20, ackermann21_out_ga(T19, X40)) → ackermann1_out_gag(s(s(T19)), 0, T20)
ackermann1_in_gag(s(s(T19)), 0, T20) → U17_gag(T19, T20, ackermann21_in_ga(T19, T22))
U17_gag(T19, T20, ackermann21_out_ga(T19, T22)) → U18_gag(T19, T20, ackermann1_in_gag(T19, T22, T20))
ackermann1_in_gag(s(T68), s(T71), T70) → U19_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, X182))
ackermann57_in_gaa(T79, 0, X211) → U9_gaa(T79, X211, ackermann27_in_ga(T79, X211))
U9_gaa(T79, X211, ackermann27_out_ga(T79, X211)) → ackermann57_out_gaa(T79, 0, X211)
ackermann57_in_gaa(T84, s(T86), X229) → U10_gaa(T84, 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))
U11_gaaa(T84, T86, X228, X229, ackermann57_out_gaa(T84, T86, X228)) → p65_out_gaaa(T84, T86, X228, X229)
p65_in_gaaa(T84, T86, T88, X229) → U12_gaaa(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U12_gaaa(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_gaaa(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
ackermann68_in_gaa(0, T96, s(T96)) → ackermann68_out_gaa(0, T96, s(T96))
ackermann68_in_gaa(s(T101), 0, X265) → U14_gaa(T101, X265, ackermann27_in_ga(T101, X265))
U14_gaa(T101, X265, ackermann27_out_ga(T101, X265)) → ackermann68_out_gaa(s(T101), 0, X265)
ackermann68_in_gaa(s(T106), s(T108), X283) → U15_gaa(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
U15_gaa(T106, T108, X283, p65_out_gaaa(T106, T108, X282, X283)) → ackermann68_out_gaa(s(T106), s(T108), X283)
U13_gaaa(T84, T86, T88, X229, ackermann68_out_gaa(T84, T88, X229)) → p65_out_gaaa(T84, T86, T88, X229)
U10_gaa(T84, T86, X229, p65_out_gaaa(T84, T86, X228, X229)) → ackermann57_out_gaa(T84, s(T86), X229)
U19_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, X182)) → ackermann1_out_gag(s(T68), s(T71), T70)
ackermann1_in_gag(s(T68), s(T71), T70) → U20_gag(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_gag(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → U21_gag(T68, T71, T70, ackermann1_in_gag(T68, T73, T70))
ackermann1_in_gag(s(T123), s(0), T117) → U22_gag(T123, T117, ackermann27_in_ga(T123, X319))
U22_gag(T123, T117, ackermann27_out_ga(T123, X319)) → ackermann1_out_gag(s(T123), s(0), T117)
ackermann1_in_gag(s(T123), s(0), T117) → U23_gag(T123, T117, ackermann27_in_ga(T123, T124))
U23_gag(T123, T117, ackermann27_out_ga(T123, T124)) → U24_gag(T123, T117, ackermann1_in_gag(T123, T124, T117))
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U25_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, X340))
U25_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, X340)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
ackermann1_in_gag(s(T131), s(s(T133)), T117) → U26_gag(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U27_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, X341))
U27_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, X341)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U26_gag(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_gag(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_gag(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → U29_gag(T131, T133, T117, ackermann1_in_gag(T131, T139, T117))
U29_gag(T131, T133, T117, ackermann1_out_gag(T131, T139, T117)) → ackermann1_out_gag(s(T131), s(s(T133)), T117)
U24_gag(T123, T117, ackermann1_out_gag(T123, T124, T117)) → ackermann1_out_gag(s(T123), s(0), T117)
U21_gag(T68, T71, T70, ackermann1_out_gag(T68, T73, T70)) → ackermann1_out_gag(s(T68), s(T71), T70)
U18_gag(T19, T20, ackermann1_out_gag(T19, T22, T20)) → ackermann1_out_gag(s(s(T19)), 0, T20)

(87) UsableRulesProof (EQUIVALENT transformation)

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

(88) Obligation:

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

ACKERMANN1_IN_GAG(s(s(T19)), 0, T20) → U17_GAG(T19, T20, ackermann21_in_ga(T19, T22))
U17_GAG(T19, T20, ackermann21_out_ga(T19, T22)) → ACKERMANN1_IN_GAG(T19, T22, T20)
ACKERMANN1_IN_GAG(s(T68), s(T71), T70) → U20_GAG(T68, T71, T70, ackermann57_in_gaa(T68, T71, T73))
U20_GAG(T68, T71, T70, ackermann57_out_gaa(T68, T71, T73)) → ACKERMANN1_IN_GAG(T68, T73, T70)
ACKERMANN1_IN_GAG(s(T123), s(0), T117) → U23_GAG(T123, T117, ackermann27_in_ga(T123, T124))
U23_GAG(T123, T117, ackermann27_out_ga(T123, T124)) → ACKERMANN1_IN_GAG(T123, T124, T117)
ACKERMANN1_IN_GAG(s(T131), s(s(T133)), T117) → U26_GAG(T131, T133, T117, ackermann57_in_gaa(T131, T133, T135))
U26_GAG(T131, T133, T117, ackermann57_out_gaa(T131, T133, T135)) → U28_GAG(T131, T133, T117, ackermann68_in_gaa(T131, T135, T139))
U28_GAG(T131, T133, T117, ackermann68_out_gaa(T131, T135, T139)) → ACKERMANN1_IN_GAG(T131, T139, T117)

The TRS R consists of the following rules:

ackermann21_in_ga(T28, X73) → U1_ga(T28, X73, ackermann27_in_ga(T28, X73))
ackermann57_in_gaa(T79, 0, X211) → U9_gaa(T79, 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))
ackermann27_in_ga(0, s(s(0))) → ackermann27_out_ga(0, s(s(0)))
ackermann27_in_ga(s(T32), X97) → U2_ga(T32, X97, ackermann21_in_ga(T32, X96))
ackermann27_in_ga(s(T32), X97) → U3_ga(T32, X97, ackermann21_in_ga(T32, T34))
ackermann68_in_gaa(0, T96, s(T96)) → ackermann68_out_gaa(0, T96, s(T96))
ackermann68_in_gaa(s(T101), 0, X265) → U14_gaa(T101, X265, ackermann27_in_ga(T101, X265))
ackermann68_in_gaa(s(T106), s(T108), X283) → U15_gaa(T106, T108, X283, p65_in_gaaa(T106, T108, X282, X283))
U1_ga(T28, X73, ackermann27_out_ga(T28, X73)) → ackermann21_out_ga(T28, X73)
U9_gaa(T79, X211, ackermann27_out_ga(T79, X211)) → ackermann57_out_gaa(T79, 0, X211)
U10_gaa(T84, T86, X229, p65_out_gaaa(T84, T86, X228, X229)) → ackermann57_out_gaa(T84, s(T86), X229)
U2_ga(T32, X97, ackermann21_out_ga(T32, X96)) → ackermann27_out_ga(s(T32), X97)
U3_ga(T32, X97, ackermann21_out_ga(T32, T34)) → U4_ga(T32, X97, ackermann38_in_gaa(T32, T34, X97))
U14_gaa(T101, X265, ackermann27_out_ga(T101, X265)) → ackermann68_out_gaa(s(T101), 0, X265)
U15_gaa(T106, T108, X283, p65_out_gaaa(T106, T108, X282, X283)) → ackermann68_out_gaa(s(T106), s(T108), X283)
p65_in_gaaa(T84, T86, X228, X229) → U11_gaaa(T84, T86, X228, X229, ackermann57_in_gaa(T84, T86, X228))
p65_in_gaaa(T84, T86, T88, X229) → U12_gaaa(T84, T86, T88, X229, ackermann57_in_gaa(T84, T86, T88))
U4_ga(T32, X97, ackermann38_out_gaa(T32, T34, X97)) → ackermann27_out_ga(s(T32), X97)
U11_gaaa(T84, T86, X228, X229, ackermann57_out_gaa(T84, T86, X228)) → p65_out_gaaa(T84, T86, X228, X229)
U12_gaaa(T84, T86, T88, X229, ackermann57_out_gaa(T84, T86, T88)) → U13_gaaa(T84, T86, T88, X229, ackermann68_in_gaa(T84, T88, X229))
ackermann38_in_gaa(0, T42, s(T42)) → ackermann38_out_gaa(0, T42, s(T42))
ackermann38_in_gaa(s(T47), 0, X133) → U5_gaa(T47, X133, ackermann27_in_ga(T47, X133))
ackermann38_in_gaa(s(T52), s(T53), X151) → U6_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, X150))
ackermann38_in_gaa(s(T52), s(T53), X151) → U7_gaa(T52, T53, X151, ackermann38_in_gaa(s(T52), T53, T55))
U13_gaaa(T84, T86, T88, X229, ackermann68_out_gaa(T84, T88, X229)) → p65_out_gaaa(T84, T86, T88, X229)
U5_gaa(T47, X133, ackermann27_out_ga(T47, X133)) → ackermann38_out_gaa(s(T47), 0, X133)
U6_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, X150)) → ackermann38_out_gaa(s(T52), s(T53), X151)
U7_gaa(T52, T53, X151, ackermann38_out_gaa(s(T52), T53, T55)) → U8_gaa(T52, T53, X151, ackermann38_in_gaa(T52, T55, X151))
U8_gaa(T52, T53, X151, ackermann38_out_gaa(T52, T55, X151)) → ackermann38_out_gaa(s(T52), s(T53), X151)

The argument filtering Pi contains the following mapping:
0 = 0
s(x1) = s(x1)
ackermann21_in_ga(x1, x2) = ackermann21_in_ga(x1)
U1_ga(x1, x2, x3) = U1_ga(x3)
ackermann27_in_ga(x1, x2) = ackermann27_in_ga(x1)
ackermann27_out_ga(x1, x2) = ackermann27_out_ga
U2_ga(x1, x2, x3) = U2_ga(x3)
ackermann21_out_ga(x1, x2) = ackermann21_out_ga
U3_ga(x1, x2, x3) = U3_ga(x1, x3)
U4_ga(x1, x2, x3) = U4_ga(x3)
ackermann38_in_gaa(x1, x2, x3) = ackermann38_in_gaa(x1)
ackermann38_out_gaa(x1, x2, x3) = ackermann38_out_gaa
U5_gaa(x1, x2, x3) = U5_gaa(x3)
U6_gaa(x1, x2, x3, x4) = U6_gaa(x4)
U7_gaa(x1, x2, x3, x4) = U7_gaa(x1, x4)
U8_gaa(x1, x2, x3, x4) = U8_gaa(x4)
ackermann57_in_gaa(x1, x2, x3) = ackermann57_in_gaa(x1)
U9_gaa(x1, x2, x3) = U9_gaa(x3)
ackermann57_out_gaa(x1, x2, x3) = ackermann57_out_gaa(x2)
U10_gaa(x1, x2, x3, x4) = U10_gaa(x4)
p65_in_gaaa(x1, x2, x3, x4) = p65_in_gaaa(x1)
U11_gaaa(x1, x2, x3, x4, x5) = U11_gaaa(x5)
p65_out_gaaa(x1, x2, x3, x4) = p65_out_gaaa(x2)
U12_gaaa(x1, x2, x3, x4, x5) = U12_gaaa(x1, x5)
U13_gaaa(x1, x2, x3, x4, x5) = U13_gaaa(x2, x5)
ackermann68_in_gaa(x1, x2, x3) = ackermann68_in_gaa(x1)
ackermann68_out_gaa(x1, x2, x3) = ackermann68_out_gaa
U14_gaa(x1, x2, x3) = U14_gaa(x3)
U15_gaa(x1, x2, x3, x4) = U15_gaa(x4)
ACKERMANN1_IN_GAG(x1, x2, x3) = ACKERMANN1_IN_GAG(x1, x3)
U17_GAG(x1, x2, x3) = U17_GAG(x1, x2, x3)
U20_GAG(x1, x2, x3, x4) = U20_GAG(x1, x3, x4)
U23_GAG(x1, x2, x3) = U23_GAG(x1, x2, x3)
U26_GAG(x1, x2, x3, x4) = U26_GAG(x1, x3, x4)
U28_GAG(x1, x2, x3, x4) = U28_GAG(x1, x2, x3, x4)

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

(89) PiDPToQDPProof (SOUND transformation)

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

(90) Obligation:

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

ACKERMANN1_IN_GAG(s(s(T19)), T20) → U17_GAG(T19, T20, ackermann21_in_ga(T19))
U17_GAG(T19, T20, ackermann21_out_ga) → ACKERMANN1_IN_GAG(T19, T20)
ACKERMANN1_IN_GAG(s(T68), T70) → U20_GAG(T68, T70, ackermann57_in_gaa(T68))
U20_GAG(T68, T70, ackermann57_out_gaa(T71)) → ACKERMANN1_IN_GAG(T68, T70)
ACKERMANN1_IN_GAG(s(T123), T117) → U23_GAG(T123, T117, ackermann27_in_ga(T123))
U23_GAG(T123, T117, ackermann27_out_ga) → ACKERMANN1_IN_GAG(T123, T117)
ACKERMANN1_IN_GAG(s(T131), T117) → U26_GAG(T131, T117, ackermann57_in_gaa(T131))
U26_GAG(T131, T117, ackermann57_out_gaa(T133)) → U28_GAG(T131, T133, T117, ackermann68_in_gaa(T131))
U28_GAG(T131, T133, T117, ackermann68_out_gaa) → ACKERMANN1_IN_GAG(T131, T117)

The TRS R consists of the following rules:

ackermann21_in_ga(T28) → U1_ga(ackermann27_in_ga(T28))
ackermann57_in_gaa(T79) → U9_gaa(ackermann27_in_ga(T79))
ackermann57_in_gaa(T84) → U10_gaa(p65_in_gaaa(T84))
ackermann27_in_ga(0) → ackermann27_out_ga
ackermann27_in_ga(s(T32)) → U2_ga(ackermann21_in_ga(T32))
ackermann27_in_ga(s(T32)) → U3_ga(T32, ackermann21_in_ga(T32))
ackermann68_in_gaa(0) → ackermann68_out_gaa
ackermann68_in_gaa(s(T101)) → U14_gaa(ackermann27_in_ga(T101))
ackermann68_in_gaa(s(T106)) → U15_gaa(p65_in_gaaa(T106))
U1_ga(ackermann27_out_ga) → ackermann21_out_ga
U9_gaa(ackermann27_out_ga) → ackermann57_out_gaa(0)
U10_gaa(p65_out_gaaa(T86)) → ackermann57_out_gaa(s(T86))
U2_ga(ackermann21_out_ga) → ackermann27_out_ga
U3_ga(T32, ackermann21_out_ga) → U4_ga(ackermann38_in_gaa(T32))
U14_gaa(ackermann27_out_ga) → ackermann68_out_gaa
U15_gaa(p65_out_gaaa(T108)) → ackermann68_out_gaa
p65_in_gaaa(T84) → U11_gaaa(ackermann57_in_gaa(T84))
p65_in_gaaa(T84) → U12_gaaa(T84, ackermann57_in_gaa(T84))
U4_ga(ackermann38_out_gaa) → ackermann27_out_ga
U11_gaaa(ackermann57_out_gaa(T86)) → p65_out_gaaa(T86)
U12_gaaa(T84, ackermann57_out_gaa(T86)) → U13_gaaa(T86, ackermann68_in_gaa(T84))
ackermann38_in_gaa(0) → ackermann38_out_gaa
ackermann38_in_gaa(s(T47)) → U5_gaa(ackermann27_in_ga(T47))
ackermann38_in_gaa(s(T52)) → U6_gaa(ackermann38_in_gaa(s(T52)))
ackermann38_in_gaa(s(T52)) → U7_gaa(T52, ackermann38_in_gaa(s(T52)))
U13_gaaa(T86, ackermann68_out_gaa) → p65_out_gaaa(T86)
U5_gaa(ackermann27_out_ga) → ackermann38_out_gaa
U6_gaa(ackermann38_out_gaa) → ackermann38_out_gaa
U7_gaa(T52, ackermann38_out_gaa) → U8_gaa(ackermann38_in_gaa(T52))
U8_gaa(ackermann38_out_gaa) → ackermann38_out_gaa

The set Q consists of the following terms:

ackermann21_in_ga(x0)
ackermann57_in_gaa(x0)
ackermann27_in_ga(x0)
ackermann68_in_gaa(x0)
U1_ga(x0)
U9_gaa(x0)
U10_gaa(x0)
U2_ga(x0)
U3_ga(x0, x1)
U14_gaa(x0)
U15_gaa(x0)
p65_in_gaaa(x0)
U4_ga(x0)
U11_gaaa(x0)
U12_gaaa(x0, x1)
ackermann38_in_gaa(x0)
U13_gaaa(x0, x1)
U5_gaa(x0)
U6_gaa(x0)
U7_gaa(x0, x1)
U8_gaa(x0)

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

(91) 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_GAG(T19, T20, ackermann21_out_ga) → ACKERMANN1_IN_GAG(T19, T20)
The graph contains the following edges 1 >= 1, 2 >= 2
ACKERMANN1_IN_GAG(s(s(T19)), T20) → U17_GAG(T19, T20, ackermann21_in_ga(T19))
The graph contains the following edges 1 > 1, 2 >= 2
U26_GAG(T131, T117, ackermann57_out_gaa(T133)) → U28_GAG(T131, T133, T117, ackermann68_in_gaa(T131))
The graph contains the following edges 1 >= 1, 3 > 2, 2 >= 3
U20_GAG(T68, T70, ackermann57_out_gaa(T71)) → ACKERMANN1_IN_GAG(T68, T70)
The graph contains the following edges 1 >= 1, 2 >= 2
ACKERMANN1_IN_GAG(s(T68), T70) → U20_GAG(T68, T70, ackermann57_in_gaa(T68))
The graph contains the following edges 1 > 1, 2 >= 2
U23_GAG(T123, T117, ackermann27_out_ga) → ACKERMANN1_IN_GAG(T123, T117)
The graph contains the following edges 1 >= 1, 2 >= 2
U28_GAG(T131, T133, T117, ackermann68_out_gaa) → ACKERMANN1_IN_GAG(T131, T117)
The graph contains the following edges 1 >= 1, 3 >= 2
ACKERMANN1_IN_GAG(s(T123), T117) → U23_GAG(T123, T117, ackermann27_in_ga(T123))
The graph contains the following edges 1 > 1, 2 >= 2
ACKERMANN1_IN_GAG(s(T131), T117) → U26_GAG(T131, T117, ackermann57_in_gaa(T131))
The graph contains the following edges 1 > 1, 2 >= 2