Left Termination proof of /tmp/tmpTfG5lX/ack.pl

Left Termination of the query pattern ack_in_3(g, g, a) 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 QDPSizeChangeProof (⇔)

                ↳30 YES

            ↳31 PiDP

                ↳32 UsableRulesProof (⇔)

                ↳33 PiDP

                ↳34 PiDPToQDPProof (⇐)

                ↳35 QDP

                ↳36 QDPSizeChangeProof (⇔)

                ↳37 YES

    ↳38 PrologToPiTRSProof (⇐)

        ↳39 PiTRS

        ↳40 DependencyPairsProof (⇔)

        ↳41 PiDP

        ↳42 DependencyGraphProof (⇔)

        ↳43 AND

            ↳44 PiDP

                ↳45 UsableRulesProof (⇔)

                ↳46 PiDP

                ↳47 PiDPToQDPProof (⇐)

                ↳48 QDP

                ↳49 QDPOrderProof (⇔)

                ↳50 QDP

                ↳51 DependencyGraphProof (⇔)

                ↳52 QDP

                ↳53 UsableRulesProof (⇔)

                ↳54 QDP

                ↳55 QReductionProof (⇔)

                ↳56 QDP

                ↳57 NonTerminationProof (⇔)

                ↳58 NO

            ↳59 PiDP

                ↳60 UsableRulesProof (⇔)

                ↳61 PiDP

                ↳62 PiDPToQDPProof (⇐)

                ↳63 QDP

                ↳64 QDPSizeChangeProof (⇔)

                ↳65 YES

            ↳66 PiDP

                ↳67 UsableRulesProof (⇔)

                ↳68 PiDP

                ↳69 PiDPToQDPProof (⇐)

                ↳70 QDP

                ↳71 QDPSizeChangeProof (⇔)

                ↳72 YES

(0) Obligation:

Clauses:

ack(0, N, s(N)).
ack(s(M), 0, A) :- ack(M, s(0), A).
ack(s(M), s(N), A) :- ','(ack(s(M), N, A1), ack(M, A1, A)).

Queries:

ack(g,g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: ack(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: ack(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | ack(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | ack(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | ack(0, T5, T3), SCOPE: 1, CLAUSE: 1 | ack(0, T5, T3), SCOPE: 1, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ack(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | ack(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground\nack(T1, T2, T3) !=? ack(0, X2, s(X2))", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ack(0, T5, T3), SCOPE: 1, CLAUSE: 1 | ack(0, T5, T3), SCOPE: 1, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ack(0, T5, T3), 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: ack(T8, s(0), T10) | ack(s(T8), 0, T3), SCOPE: 1, CLAUSE: 2\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ack(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground\nack(T1, T2, T3) !=? ack(0, X2, s(X2))\nack(T1, T2, T3) !=? ack(s(X11), 0, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ack(T8, s(0), T10), SCOPE: 2, CLAUSE: 0 | ack(T8, s(0), T10), SCOPE: 2, CLAUSE: 1 | ack(T8, s(0), T10), SCOPE: 2, CLAUSE: 2 | ?_2 | ack(s(T8), 0, T3), SCOPE: 1, CLAUSE: 2\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ack(T8, s(0), T10), SCOPE: 2, CLAUSE: 0\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ack(T8, s(0), T10), SCOPE: 2, CLAUSE: 1 | ack(T8, s(0), T10), SCOPE: 2, CLAUSE: 2 | ?_2 | ack(s(T8), 0, T3), SCOPE: 1, CLAUSE: 2\n\nT8 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: ack(T8, s(0), T10), SCOPE: 2, CLAUSE: 2 | ?_2 | ack(s(T8), 0, T3), SCOPE: 1, CLAUSE: 2\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ack(T8, s(0), T10), SCOPE: 2, CLAUSE: 2\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ?_2 | ack(s(T8), 0, T3), SCOPE: 1, CLAUSE: 2\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(ack(s(T20), 0, X40), ack(T20, X40, T22))\n\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: ack(s(T20), 0, X40)\n\nT20 is ground\nX40 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ack(T20, T24, T22)\n\nT20 is ground\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ack(s(T20), 0, X40), SCOPE: 3, CLAUSE: 0 | ack(s(T20), 0, X40), SCOPE: 3, CLAUSE: 1 | ack(s(T20), 0, X40), SCOPE: 3, CLAUSE: 2\n\nT20 is ground\nX40 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ack(s(T20), 0, X40), SCOPE: 3, CLAUSE: 1 | ack(s(T20), 0, X40), SCOPE: 3, CLAUSE: 2\n\nT20 is ground\nX40 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ack(s(T20), 0, X40), SCOPE: 3, CLAUSE: 1\n\nT20 is ground\nX40 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ack(s(T20), 0, X40), SCOPE: 3, CLAUSE: 2\n\nT20 is ground\nX40 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ack(T30, s(0), X73)\n\nT30 is ground\nX73 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ack(T30, s(0), X73), SCOPE: 4, CLAUSE: 0 | ack(T30, s(0), X73), SCOPE: 4, CLAUSE: 1 | ack(T30, s(0), X73), SCOPE: 4, CLAUSE: 2\n\nT30 is ground\nX73 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ack(T30, s(0), X73), SCOPE: 4, CLAUSE: 0\n\nT30 is ground\nX73 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ack(T30, s(0), X73), SCOPE: 4, CLAUSE: 1 | ack(T30, s(0), X73), SCOPE: 4, CLAUSE: 2\n\nT30 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: ack(T30, s(0), X73), SCOPE: 4, CLAUSE: 2\n\nT30 is ground\nX73 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(ack(s(T34), 0, X96), ack(T34, X96, X97))\n\nT34 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: ack(s(T34), 0, X96)\n\nT34 is ground\nX96 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ack(T34, T36, X97)\n\nT34 is ground\nT36 is ground\nX97 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ack(T34, T36, X97), SCOPE: 5, CLAUSE: 0 | ack(T34, T36, X97), SCOPE: 5, CLAUSE: 1 | ack(T34, T36, X97), SCOPE: 5, CLAUSE: 2\n\nT34 is ground\nT36 is ground\nX97 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ack(T34, T36, X97), SCOPE: 5, CLAUSE: 0\n\nT34 is ground\nT36 is ground\nX97 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ack(T34, T36, X97), SCOPE: 5, CLAUSE: 1 | ack(T34, T36, X97), SCOPE: 5, CLAUSE: 2\n\nT34 is ground\nT36 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: ack(T34, T36, X97), SCOPE: 5, CLAUSE: 1\n\nT34 is ground\nT36 is ground\nX97 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ack(T34, T36, X97), SCOPE: 5, CLAUSE: 2\n\nT34 is ground\nT36 is ground\nX97 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ack(T49, s(0), X133)\n\nT49 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: ','(ack(s(T54), T55, X150), ack(T54, X150, X151))\n\nT54 is ground\nT55 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: ack(s(T54), T55, X150)\n\nT54 is ground\nT55 is ground\nX150 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ack(T54, T57, X151)\n\nT54 is ground\nT57 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: ack(s(T8), 0, T3), SCOPE: 1, CLAUSE: 2\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ','(ack(s(T68), T69, X181), ack(T68, X181, T71))\n\nT68 is ground\nT69 is ground\nX181 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(ack(s(T68), T69, X181), ack(T68, X181, T71)), SCOPE: 6, CLAUSE: 0 | ','(ack(s(T68), T69, X181), ack(T68, X181, T71)), SCOPE: 6, CLAUSE: 1 | ','(ack(s(T68), T69, X181), ack(T68, X181, T71)), SCOPE: 6, CLAUSE: 2\n\nT68 is ground\nT69 is ground\nX181 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(ack(s(T68), T69, X181), ack(T68, X181, T71)), SCOPE: 6, CLAUSE: 1 | ','(ack(s(T68), T69, X181), ack(T68, X181, T71)), SCOPE: 6, CLAUSE: 2\n\nT68 is ground\nT69 is ground\nX181 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ','(ack(s(T68), T69, X181), ack(T68, X181, T71)), SCOPE: 6, CLAUSE: 1\n\nT68 is ground\nT69 is ground\nX181 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(ack(s(T68), T69, X181), ack(T68, X181, T71)), SCOPE: 6, CLAUSE: 2\n\nT68 is ground\nT69 is ground\nX181 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(ack(T76, s(0), X200), ack(T76, X200, T71))\n\nT76 is ground\nX200 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ack(T76, s(0), X200)\n\nT76 is ground\nX200 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ack(T76, T77, T71)\n\nT76 is ground\nT77 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: ','(ack(s(T84), T85, X221), ','(ack(T84, X221, X222), ack(T84, X222, T71)))\n\nT84 is ground\nT85 is ground\nX222 is free; \nX221 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: ack(s(T84), T85, X221)\n\nT84 is ground\nT85 is ground\nX221 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: ','(ack(T84, T87, X222), ack(T84, X222, T71))\n\nT84 is ground\nT87 is ground\nX222 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: ack(T84, T87, X222)\n\nT84 is ground\nT87 is ground\nX222 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: ack(T84, T91, T71)\n\nT84 is ground\nT91 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 72 [arrowhead = none ];
72 -> 2;

73 [label="EVAL with clause\nack(0, X2, s(X2)).\nand substitution\nT1 -> 0\nT2 -> T5\nX2 -> T5\nT3 -> s(T5)", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 73 [arrowhead = none ];
73 -> 3;

74 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 74 [arrowhead = none ];
74 -> 4;

75 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 75 [arrowhead = none ];
75 -> 5;

76 [label="EVAL with clause\nack(s(X11), 0, X12) :- ack(X11, s(0), X12).\nand substitution\nX11 -> T8\nT1 -> s(T8)\nT2 -> 0\nT3 -> T10\nX12 -> T10\nT9 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 76 [arrowhead = none ];
76 -> 8;

77 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 77 [arrowhead = none ];
77 -> 9;

78 [label="BACKTRACK\nfor clause: ack(s(M), 0, A) :- ack(M, s(0), A)because of non-unification", fontsize=14, style = filled, fillcolor = white];
5 -> 78 [arrowhead = none ];
78 -> 6;

79 [label="BACKTRACK\nfor clause: ack(s(M), s(N), A) :- ','(ack(s(M), N, A1), ack(M, A1, A))because of non-unification", fontsize=14, style = filled, fillcolor = white];
6 -> 79 [arrowhead = none ];
79 -> 7;

80 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
8 -> 80 [arrowhead = none ];
80 -> 10;

81 [label="EVAL with clause\nack(s(X178), s(X179), X180) :- ','(ack(s(X178), X179, X181), ack(X178, X181, X180)).\nand substitution\nX178 -> T68\nT1 -> s(T68)\nX179 -> T69\nT2 -> s(T69)\nT3 -> T71\nX180 -> T71\nT70 -> T71", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 81 [arrowhead = none ];
81 -> 56;

82 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 82 [arrowhead = none ];
82 -> 57;

83 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 83 [arrowhead = none ];
83 -> 11;

84 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 84 [arrowhead = none ];
84 -> 12;

85 [label="EVAL with clause\nack(0, X19, s(X19)).\nand substitution\nT8 -> 0\nX19 -> s(0)\nT10 -> s(s(0))", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 85 [arrowhead = none ];
85 -> 13;

86 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 86 [arrowhead = none ];
86 -> 14;

87 [label="BACKTRACK\nfor clause: ack(s(M), 0, A) :- ack(M, s(0), A)because of non-unification", fontsize=14, style = filled, fillcolor = white];
12 -> 87 [arrowhead = none ];
87 -> 16;

88 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 88 [arrowhead = none ];
88 -> 15;

89 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 89 [arrowhead = none ];
89 -> 17;

90 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 90 [arrowhead = none ];
90 -> 18;

91 [label="EVAL with clause\nack(s(X37), s(X38), X39) :- ','(ack(s(X37), X38, X40), ack(X37, X40, X39)).\nand substitution\nX37 -> T20\nT8 -> s(T20)\nX38 -> 0\nT10 -> T22\nX39 -> T22\nT21 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 91 [arrowhead = none ];
91 -> 19;

92 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 92 [arrowhead = none ];
92 -> 20;

93 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 93 [arrowhead = none ];
93 -> 54;

94 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
19 -> 94 [arrowhead = none ];
94 -> 21;

95 [label="SPLIT 2\nnew knowledge:\nT24 is ground\nreplacements:\nX40 -> T24", fontsize=14, style = filled, fillcolor = violet];
19 -> 95 [arrowhead = none ];
95 -> 22;

96 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 96 [arrowhead = none ];
96 -> 23;

97 [label="INSTANCE with matching:\nT1 -> T20\nT2 -> T24\nT3 -> T22", fontsize=14, style = filled, fillcolor = lightgrey];
22 -> 97 [arrowhead = none , style=dashed];
97 -> 1[style=dashed];

98 [label="BACKTRACK\nfor clause: ack(0, N, s(N))because of non-unification", fontsize=14, style = filled, fillcolor = white];
23 -> 98 [arrowhead = none ];
98 -> 24;

99 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
24 -> 99 [arrowhead = none ];
99 -> 25;

100 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
24 -> 100 [arrowhead = none ];
100 -> 26;

101 [label="ONLY EVAL with clause\nack(s(X71), 0, X72) :- ack(X71, s(0), X72).\nand substitution\nT20 -> T30\nX71 -> T30\nX40 -> X73\nX72 -> X73", fontsize=14, style = filled, fillcolor = white];
25 -> 101 [arrowhead = none ];
101 -> 27;

102 [label="BACKTRACK\nfor clause: ack(s(M), s(N), A) :- ','(ack(s(M), N, A1), ack(M, A1, A))because of non-unification", fontsize=14, style = filled, fillcolor = white];
26 -> 102 [arrowhead = none ];
102 -> 53;

103 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
27 -> 103 [arrowhead = none ];
103 -> 28;

104 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
28 -> 104 [arrowhead = none ];
104 -> 29;

105 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
28 -> 105 [arrowhead = none ];
105 -> 30;

106 [label="EVAL with clause\nack(0, X80, s(X80)).\nand substitution\nT30 -> 0\nX80 -> s(0)\nX73 -> s(s(0))", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 106 [arrowhead = none ];
106 -> 31;

107 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 107 [arrowhead = none ];
107 -> 32;

108 [label="BACKTRACK\nfor clause: ack(s(M), 0, A) :- ack(M, s(0), A)because of non-unification", fontsize=14, style = filled, fillcolor = white];
30 -> 108 [arrowhead = none ];
108 -> 34;

109 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 109 [arrowhead = none ];
109 -> 33;

110 [label="EVAL with clause\nack(s(X93), s(X94), X95) :- ','(ack(s(X93), X94, X96), ack(X93, X96, X95)).\nand substitution\nX93 -> T34\nT30 -> s(T34)\nX94 -> 0\nX73 -> X97\nX95 -> X97", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 110 [arrowhead = none ];
110 -> 35;

111 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 111 [arrowhead = none ];
111 -> 36;

112 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
35 -> 112 [arrowhead = none ];
112 -> 37;

113 [label="SPLIT 2\nnew knowledge:\nT36 is ground\nreplacements:\nX96 -> T36", fontsize=14, style = filled, fillcolor = violet];
35 -> 113 [arrowhead = none ];
113 -> 38;

114 [label="INSTANCE with matching:\nT20 -> T34\nX40 -> X96", fontsize=14, style = filled, fillcolor = lightgrey];
37 -> 114 [arrowhead = none , style=dashed];
114 -> 21[style=dashed];

115 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
38 -> 115 [arrowhead = none ];
115 -> 39;

116 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
39 -> 116 [arrowhead = none ];
116 -> 40;

117 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
39 -> 117 [arrowhead = none ];
117 -> 41;

118 [label="EVAL with clause\nack(0, X117, s(X117)).\nand substitution\nT34 -> 0\nT36 -> T44\nX117 -> T44\nX97 -> s(T44)", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 118 [arrowhead = none ];
118 -> 42;

119 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 119 [arrowhead = none ];
119 -> 43;

120 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
41 -> 120 [arrowhead = none ];
120 -> 45;

121 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
41 -> 121 [arrowhead = none ];
121 -> 46;

122 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 122 [arrowhead = none ];
122 -> 44;

123 [label="EVAL with clause\nack(s(X131), 0, X132) :- ack(X131, s(0), X132).\nand substitution\nX131 -> T49\nT34 -> s(T49)\nT36 -> 0\nX97 -> X133\nX132 -> X133", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 123 [arrowhead = none ];
123 -> 47;

124 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 124 [arrowhead = none ];
124 -> 48;

125 [label="EVAL with clause\nack(s(X147), s(X148), X149) :- ','(ack(s(X147), X148, X150), ack(X147, X150, X149)).\nand substitution\nX147 -> T54\nT34 -> s(T54)\nX148 -> T55\nT36 -> s(T55)\nX97 -> X151\nX149 -> X151", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 125 [arrowhead = none ];
125 -> 49;

126 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 126 [arrowhead = none ];
126 -> 50;

127 [label="INSTANCE with matching:\nT30 -> T49\nX73 -> X133", fontsize=14, style = filled, fillcolor = lightgrey];
47 -> 127 [arrowhead = none , style=dashed];
127 -> 27[style=dashed];

128 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
49 -> 128 [arrowhead = none ];
128 -> 51;

129 [label="SPLIT 2\nnew knowledge:\nT57 is ground\nreplacements:\nX150 -> T57", fontsize=14, style = filled, fillcolor = violet];
49 -> 129 [arrowhead = none ];
129 -> 52;

130 [label="INSTANCE with matching:\nT34 -> s(T54)\nT36 -> T55\nX97 -> X150", fontsize=14, style = filled, fillcolor = lightgrey];
51 -> 130 [arrowhead = none , style=dashed];
130 -> 38[style=dashed];

131 [label="INSTANCE with matching:\nT34 -> T54\nT36 -> T57\nX97 -> X151", fontsize=14, style = filled, fillcolor = lightgrey];
52 -> 131 [arrowhead = none , style=dashed];
131 -> 38[style=dashed];

132 [label="BACKTRACK\nfor clause: ack(s(M), s(N), A) :- ','(ack(s(M), N, A1), ack(M, A1, A))because of non-unification", fontsize=14, style = filled, fillcolor = white];
54 -> 132 [arrowhead = none ];
132 -> 55;

133 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
56 -> 133 [arrowhead = none ];
133 -> 58;

134 [label="BACKTRACK\nfor clause: ack(0, N, s(N))because of non-unification", fontsize=14, style = filled, fillcolor = white];
58 -> 134 [arrowhead = none ];
134 -> 59;

135 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
59 -> 135 [arrowhead = none ];
135 -> 60;

136 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
59 -> 136 [arrowhead = none ];
136 -> 61;

137 [label="EVAL with clause\nack(s(X198), 0, X199) :- ack(X198, s(0), X199).\nand substitution\nT68 -> T76\nX198 -> T76\nT69 -> 0\nX181 -> X200\nX199 -> X200", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 137 [arrowhead = none ];
137 -> 62;

138 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 138 [arrowhead = none ];
138 -> 63;

139 [label="EVAL with clause\nack(s(X218), s(X219), X220) :- ','(ack(s(X218), X219, X221), ack(X218, X221, X220)).\nand substitution\nT68 -> T84\nX218 -> T84\nX219 -> T85\nT69 -> s(T85)\nX181 -> X222\nX220 -> X222", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 139 [arrowhead = none ];
139 -> 66;

140 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 140 [arrowhead = none ];
140 -> 67;

141 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
62 -> 141 [arrowhead = none ];
141 -> 64;

142 [label="SPLIT 2\nnew knowledge:\nT77 is ground\nreplacements:\nX200 -> T77", fontsize=14, style = filled, fillcolor = violet];
62 -> 142 [arrowhead = none ];
142 -> 65;

143 [label="INSTANCE with matching:\nT30 -> T76\nX73 -> X200", fontsize=14, style = filled, fillcolor = lightgrey];
64 -> 143 [arrowhead = none , style=dashed];
143 -> 27[style=dashed];

144 [label="INSTANCE with matching:\nT1 -> T76\nT2 -> T77\nT3 -> T71", fontsize=14, style = filled, fillcolor = lightgrey];
65 -> 144 [arrowhead = none , style=dashed];
144 -> 1[style=dashed];

145 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
66 -> 145 [arrowhead = none ];
145 -> 68;

146 [label="SPLIT 2\nnew knowledge:\nT87 is ground\nreplacements:\nX221 -> T87", fontsize=14, style = filled, fillcolor = violet];
66 -> 146 [arrowhead = none ];
146 -> 69;

147 [label="INSTANCE with matching:\nT34 -> s(T84)\nT36 -> T85\nX97 -> X221", fontsize=14, style = filled, fillcolor = lightgrey];
68 -> 147 [arrowhead = none , style=dashed];
147 -> 38[style=dashed];

148 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
69 -> 148 [arrowhead = none ];
148 -> 70;

149 [label="SPLIT 2\nnew knowledge:\nT91 is ground\nreplacements:\nX222 -> T91", fontsize=14, style = filled, fillcolor = violet];
69 -> 149 [arrowhead = none ];
149 -> 71;

150 [label="INSTANCE with matching:\nT34 -> T84\nT36 -> T87\nX97 -> X222", fontsize=14, style = filled, fillcolor = lightgrey];
70 -> 150 [arrowhead = none , style=dashed];
150 -> 38[style=dashed];

151 [label="INSTANCE with matching:\nT1 -> T84\nT2 -> T91\nT3 -> T71", fontsize=14, style = filled, fillcolor = lightgrey];
71 -> 151 [arrowhead = none , style=dashed];
151 -> 1[style=dashed];

}

(2) Obligation:

Clauses:

ack21(T30, X73) :- ack27(T30, X73).
ack27(0, s(s(0))).
ack27(s(T34), X97) :- ack21(T34, X96).
ack27(s(T34), X97) :- ','(ack21(T34, T36), ack38(T34, T36, X97)).
ack38(0, T44, s(T44)).
ack38(s(T49), 0, X133) :- ack27(T49, X133).
ack38(s(T54), s(T55), X151) :- ack38(s(T54), T55, X150).
ack38(s(T54), s(T55), X151) :- ','(ack38(s(T54), T55, T57), ack38(T54, T57, X151)).
ack1(0, T5, s(T5)).
ack1(s(0), 0, s(s(0))).
ack1(s(s(T20)), 0, T22) :- ack21(T20, X40).
ack1(s(s(T20)), 0, T22) :- ','(ack21(T20, T24), ack1(T20, T24, T22)).
ack1(s(T76), s(0), T71) :- ack27(T76, X200).
ack1(s(T76), s(0), T71) :- ','(ack27(T76, T77), ack1(T76, T77, T71)).
ack1(s(T84), s(s(T85)), T71) :- ack38(s(T84), T85, X221).
ack1(s(T84), s(s(T85)), T71) :- ','(ack38(s(T84), T85, T87), ack38(T84, T87, X222)).
ack1(s(T84), s(s(T85)), T71) :- ','(ack38(s(T84), T85, T87), ','(ack38(T84, T87, T91), ack1(T84, T91, T71))).

Queries:

ack1(g,g,a).

(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:
ack1_in: (b,b,f) (b,f,f)
ack21_in: (b,f)
ack27_in: (b,f)
ack38_in: (b,f,f) (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

ack1_in_gga(0, T5, s(T5)) → ack1_out_gga(0, T5, s(T5))
ack1_in_gga(s(0), 0, s(s(0))) → ack1_out_gga(s(0), 0, s(s(0)))
ack1_in_gga(s(s(T20)), 0, T22) → U9_gga(T20, T22, ack21_in_ga(T20, X40))
ack21_in_ga(T30, X73) → U1_ga(T30, X73, ack27_in_ga(T30, X73))
ack27_in_ga(0, s(s(0))) → ack27_out_ga(0, s(s(0)))
ack27_in_ga(s(T34), X97) → U2_ga(T34, X97, ack21_in_ga(T34, X96))
U2_ga(T34, X97, ack21_out_ga(T34, X96)) → ack27_out_ga(s(T34), X97)
ack27_in_ga(s(T34), X97) → U3_ga(T34, X97, ack21_in_ga(T34, T36))
U3_ga(T34, X97, ack21_out_ga(T34, T36)) → U4_ga(T34, X97, ack38_in_gaa(T34, T36, X97))
ack38_in_gaa(0, T44, s(T44)) → ack38_out_gaa(0, T44, s(T44))
ack38_in_gaa(s(T49), 0, X133) → U5_gaa(T49, X133, ack27_in_ga(T49, X133))
U5_gaa(T49, X133, ack27_out_ga(T49, X133)) → ack38_out_gaa(s(T49), 0, X133)
ack38_in_gaa(s(T54), s(T55), X151) → U6_gaa(T54, T55, X151, ack38_in_gaa(s(T54), T55, X150))
ack38_in_gaa(s(T54), s(T55), X151) → U7_gaa(T54, T55, X151, ack38_in_gaa(s(T54), T55, T57))
U7_gaa(T54, T55, X151, ack38_out_gaa(s(T54), T55, T57)) → U8_gaa(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
U8_gaa(T54, T55, X151, ack38_out_gaa(T54, T57, X151)) → ack38_out_gaa(s(T54), s(T55), X151)
U6_gaa(T54, T55, X151, ack38_out_gaa(s(T54), T55, X150)) → ack38_out_gaa(s(T54), s(T55), X151)
U4_ga(T34, X97, ack38_out_gaa(T34, T36, X97)) → ack27_out_ga(s(T34), X97)
U1_ga(T30, X73, ack27_out_ga(T30, X73)) → ack21_out_ga(T30, X73)
U9_gga(T20, T22, ack21_out_ga(T20, X40)) → ack1_out_gga(s(s(T20)), 0, T22)
ack1_in_gga(s(s(T20)), 0, T22) → U10_gga(T20, T22, ack21_in_ga(T20, T24))
U10_gga(T20, T22, ack21_out_ga(T20, T24)) → U11_gga(T20, T22, ack1_in_gaa(T20, T24, T22))
ack1_in_gaa(0, T5, s(T5)) → ack1_out_gaa(0, T5, s(T5))
ack1_in_gaa(s(0), 0, s(s(0))) → ack1_out_gaa(s(0), 0, s(s(0)))
ack1_in_gaa(s(s(T20)), 0, T22) → U9_gaa(T20, T22, ack21_in_ga(T20, X40))
U9_gaa(T20, T22, ack21_out_ga(T20, X40)) → ack1_out_gaa(s(s(T20)), 0, T22)
ack1_in_gaa(s(s(T20)), 0, T22) → U10_gaa(T20, T22, ack21_in_ga(T20, T24))
U10_gaa(T20, T22, ack21_out_ga(T20, T24)) → U11_gaa(T20, T22, ack1_in_gaa(T20, T24, T22))
ack1_in_gaa(s(T76), s(0), T71) → U12_gaa(T76, T71, ack27_in_ga(T76, X200))
U12_gaa(T76, T71, ack27_out_ga(T76, X200)) → ack1_out_gaa(s(T76), s(0), T71)
ack1_in_gaa(s(T76), s(0), T71) → U13_gaa(T76, T71, ack27_in_ga(T76, T77))
U13_gaa(T76, T71, ack27_out_ga(T76, T77)) → U14_gaa(T76, T71, ack1_in_gaa(T76, T77, T71))
ack1_in_gaa(s(T84), s(s(T85)), T71) → U15_gaa(T84, T85, T71, ack38_in_gaa(s(T84), T85, X221))
U15_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, X221)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
ack1_in_gaa(s(T84), s(s(T85)), T71) → U16_gaa(T84, T85, T71, ack38_in_gaa(s(T84), T85, T87))
U16_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U17_gaa(T84, T85, T71, ack38_in_gaa(T84, T87, X222))
U17_gaa(T84, T85, T71, ack38_out_gaa(T84, T87, X222)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
U16_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U18_gaa(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_gaa(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → U19_gaa(T84, T85, T71, ack1_in_gaa(T84, T91, T71))
U19_gaa(T84, T85, T71, ack1_out_gaa(T84, T91, T71)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
U14_gaa(T76, T71, ack1_out_gaa(T76, T77, T71)) → ack1_out_gaa(s(T76), s(0), T71)
U11_gaa(T20, T22, ack1_out_gaa(T20, T24, T22)) → ack1_out_gaa(s(s(T20)), 0, T22)
U11_gga(T20, T22, ack1_out_gaa(T20, T24, T22)) → ack1_out_gga(s(s(T20)), 0, T22)
ack1_in_gga(s(T76), s(0), T71) → U12_gga(T76, T71, ack27_in_ga(T76, X200))
U12_gga(T76, T71, ack27_out_ga(T76, X200)) → ack1_out_gga(s(T76), s(0), T71)
ack1_in_gga(s(T76), s(0), T71) → U13_gga(T76, T71, ack27_in_ga(T76, T77))
U13_gga(T76, T71, ack27_out_ga(T76, T77)) → U14_gga(T76, T71, ack1_in_gaa(T76, T77, T71))
U14_gga(T76, T71, ack1_out_gaa(T76, T77, T71)) → ack1_out_gga(s(T76), s(0), T71)
ack1_in_gga(s(T84), s(s(T85)), T71) → U15_gga(T84, T85, T71, ack38_in_gga(s(T84), T85, X221))
ack38_in_gga(0, T44, s(T44)) → ack38_out_gga(0, T44, s(T44))
ack38_in_gga(s(T49), 0, X133) → U5_gga(T49, X133, ack27_in_ga(T49, X133))
U5_gga(T49, X133, ack27_out_ga(T49, X133)) → ack38_out_gga(s(T49), 0, X133)
ack38_in_gga(s(T54), s(T55), X151) → U6_gga(T54, T55, X151, ack38_in_gga(s(T54), T55, X150))
ack38_in_gga(s(T54), s(T55), X151) → U7_gga(T54, T55, X151, ack38_in_gga(s(T54), T55, T57))
U7_gga(T54, T55, X151, ack38_out_gga(s(T54), T55, T57)) → U8_gga(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
U8_gga(T54, T55, X151, ack38_out_gaa(T54, T57, X151)) → ack38_out_gga(s(T54), s(T55), X151)
U6_gga(T54, T55, X151, ack38_out_gga(s(T54), T55, X150)) → ack38_out_gga(s(T54), s(T55), X151)
U15_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, X221)) → ack1_out_gga(s(T84), s(s(T85)), T71)
ack1_in_gga(s(T84), s(s(T85)), T71) → U16_gga(T84, T85, T71, ack38_in_gga(s(T84), T85, T87))
U16_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → U17_gga(T84, T85, T71, ack38_in_gaa(T84, T87, X222))
U17_gga(T84, T85, T71, ack38_out_gaa(T84, T87, X222)) → ack1_out_gga(s(T84), s(s(T85)), T71)
U16_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → U18_gga(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_gga(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → U19_gga(T84, T85, T71, ack1_in_gaa(T84, T91, T71))
U19_gga(T84, T85, T71, ack1_out_gaa(T84, T91, T71)) → ack1_out_gga(s(T84), s(s(T85)), T71)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

ack1_in_gga(0, T5, s(T5)) → ack1_out_gga(0, T5, s(T5))
ack1_in_gga(s(0), 0, s(s(0))) → ack1_out_gga(s(0), 0, s(s(0)))
ack1_in_gga(s(s(T20)), 0, T22) → U9_gga(T20, T22, ack21_in_ga(T20, X40))
ack21_in_ga(T30, X73) → U1_ga(T30, X73, ack27_in_ga(T30, X73))
ack27_in_ga(0, s(s(0))) → ack27_out_ga(0, s(s(0)))
ack27_in_ga(s(T34), X97) → U2_ga(T34, X97, ack21_in_ga(T34, X96))
U2_ga(T34, X97, ack21_out_ga(T34, X96)) → ack27_out_ga(s(T34), X97)
ack27_in_ga(s(T34), X97) → U3_ga(T34, X97, ack21_in_ga(T34, T36))
U3_ga(T34, X97, ack21_out_ga(T34, T36)) → U4_ga(T34, X97, ack38_in_gaa(T34, T36, X97))
ack38_in_gaa(0, T44, s(T44)) → ack38_out_gaa(0, T44, s(T44))
ack38_in_gaa(s(T49), 0, X133) → U5_gaa(T49, X133, ack27_in_ga(T49, X133))
U5_gaa(T49, X133, ack27_out_ga(T49, X133)) → ack38_out_gaa(s(T49), 0, X133)
ack38_in_gaa(s(T54), s(T55), X151) → U6_gaa(T54, T55, X151, ack38_in_gaa(s(T54), T55, X150))
ack38_in_gaa(s(T54), s(T55), X151) → U7_gaa(T54, T55, X151, ack38_in_gaa(s(T54), T55, T57))
U7_gaa(T54, T55, X151, ack38_out_gaa(s(T54), T55, T57)) → U8_gaa(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
U8_gaa(T54, T55, X151, ack38_out_gaa(T54, T57, X151)) → ack38_out_gaa(s(T54), s(T55), X151)
U6_gaa(T54, T55, X151, ack38_out_gaa(s(T54), T55, X150)) → ack38_out_gaa(s(T54), s(T55), X151)
U4_ga(T34, X97, ack38_out_gaa(T34, T36, X97)) → ack27_out_ga(s(T34), X97)
U1_ga(T30, X73, ack27_out_ga(T30, X73)) → ack21_out_ga(T30, X73)
U9_gga(T20, T22, ack21_out_ga(T20, X40)) → ack1_out_gga(s(s(T20)), 0, T22)
ack1_in_gga(s(s(T20)), 0, T22) → U10_gga(T20, T22, ack21_in_ga(T20, T24))
U10_gga(T20, T22, ack21_out_ga(T20, T24)) → U11_gga(T20, T22, ack1_in_gaa(T20, T24, T22))
ack1_in_gaa(0, T5, s(T5)) → ack1_out_gaa(0, T5, s(T5))
ack1_in_gaa(s(0), 0, s(s(0))) → ack1_out_gaa(s(0), 0, s(s(0)))
ack1_in_gaa(s(s(T20)), 0, T22) → U9_gaa(T20, T22, ack21_in_ga(T20, X40))
U9_gaa(T20, T22, ack21_out_ga(T20, X40)) → ack1_out_gaa(s(s(T20)), 0, T22)
ack1_in_gaa(s(s(T20)), 0, T22) → U10_gaa(T20, T22, ack21_in_ga(T20, T24))
U10_gaa(T20, T22, ack21_out_ga(T20, T24)) → U11_gaa(T20, T22, ack1_in_gaa(T20, T24, T22))
ack1_in_gaa(s(T76), s(0), T71) → U12_gaa(T76, T71, ack27_in_ga(T76, X200))
U12_gaa(T76, T71, ack27_out_ga(T76, X200)) → ack1_out_gaa(s(T76), s(0), T71)
ack1_in_gaa(s(T76), s(0), T71) → U13_gaa(T76, T71, ack27_in_ga(T76, T77))
U13_gaa(T76, T71, ack27_out_ga(T76, T77)) → U14_gaa(T76, T71, ack1_in_gaa(T76, T77, T71))
ack1_in_gaa(s(T84), s(s(T85)), T71) → U15_gaa(T84, T85, T71, ack38_in_gaa(s(T84), T85, X221))
U15_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, X221)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
ack1_in_gaa(s(T84), s(s(T85)), T71) → U16_gaa(T84, T85, T71, ack38_in_gaa(s(T84), T85, T87))
U16_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U17_gaa(T84, T85, T71, ack38_in_gaa(T84, T87, X222))
U17_gaa(T84, T85, T71, ack38_out_gaa(T84, T87, X222)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
U16_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U18_gaa(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_gaa(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → U19_gaa(T84, T85, T71, ack1_in_gaa(T84, T91, T71))
U19_gaa(T84, T85, T71, ack1_out_gaa(T84, T91, T71)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
U14_gaa(T76, T71, ack1_out_gaa(T76, T77, T71)) → ack1_out_gaa(s(T76), s(0), T71)
U11_gaa(T20, T22, ack1_out_gaa(T20, T24, T22)) → ack1_out_gaa(s(s(T20)), 0, T22)
U11_gga(T20, T22, ack1_out_gaa(T20, T24, T22)) → ack1_out_gga(s(s(T20)), 0, T22)
ack1_in_gga(s(T76), s(0), T71) → U12_gga(T76, T71, ack27_in_ga(T76, X200))
U12_gga(T76, T71, ack27_out_ga(T76, X200)) → ack1_out_gga(s(T76), s(0), T71)
ack1_in_gga(s(T76), s(0), T71) → U13_gga(T76, T71, ack27_in_ga(T76, T77))
U13_gga(T76, T71, ack27_out_ga(T76, T77)) → U14_gga(T76, T71, ack1_in_gaa(T76, T77, T71))
U14_gga(T76, T71, ack1_out_gaa(T76, T77, T71)) → ack1_out_gga(s(T76), s(0), T71)
ack1_in_gga(s(T84), s(s(T85)), T71) → U15_gga(T84, T85, T71, ack38_in_gga(s(T84), T85, X221))
ack38_in_gga(0, T44, s(T44)) → ack38_out_gga(0, T44, s(T44))
ack38_in_gga(s(T49), 0, X133) → U5_gga(T49, X133, ack27_in_ga(T49, X133))
U5_gga(T49, X133, ack27_out_ga(T49, X133)) → ack38_out_gga(s(T49), 0, X133)
ack38_in_gga(s(T54), s(T55), X151) → U6_gga(T54, T55, X151, ack38_in_gga(s(T54), T55, X150))
ack38_in_gga(s(T54), s(T55), X151) → U7_gga(T54, T55, X151, ack38_in_gga(s(T54), T55, T57))
U7_gga(T54, T55, X151, ack38_out_gga(s(T54), T55, T57)) → U8_gga(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
U8_gga(T54, T55, X151, ack38_out_gaa(T54, T57, X151)) → ack38_out_gga(s(T54), s(T55), X151)
U6_gga(T54, T55, X151, ack38_out_gga(s(T54), T55, X150)) → ack38_out_gga(s(T54), s(T55), X151)
U15_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, X221)) → ack1_out_gga(s(T84), s(s(T85)), T71)
ack1_in_gga(s(T84), s(s(T85)), T71) → U16_gga(T84, T85, T71, ack38_in_gga(s(T84), T85, T87))
U16_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → U17_gga(T84, T85, T71, ack38_in_gaa(T84, T87, X222))
U17_gga(T84, T85, T71, ack38_out_gaa(T84, T87, X222)) → ack1_out_gga(s(T84), s(s(T85)), T71)
U16_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → U18_gga(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_gga(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → U19_gga(T84, T85, T71, ack1_in_gaa(T84, T91, T71))
U19_gga(T84, T85, T71, ack1_out_gaa(T84, T91, T71)) → ack1_out_gga(s(T84), s(s(T85)), T71)

(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:

ACK1_IN_GGA(s(s(T20)), 0, T22) → U9_GGA(T20, T22, ack21_in_ga(T20, X40))
ACK1_IN_GGA(s(s(T20)), 0, T22) → ACK21_IN_GA(T20, X40)
ACK21_IN_GA(T30, X73) → U1_GA(T30, X73, ack27_in_ga(T30, X73))
ACK21_IN_GA(T30, X73) → ACK27_IN_GA(T30, X73)
ACK27_IN_GA(s(T34), X97) → U2_GA(T34, X97, ack21_in_ga(T34, X96))
ACK27_IN_GA(s(T34), X97) → ACK21_IN_GA(T34, X96)
ACK27_IN_GA(s(T34), X97) → U3_GA(T34, X97, ack21_in_ga(T34, T36))
U3_GA(T34, X97, ack21_out_ga(T34, T36)) → U4_GA(T34, X97, ack38_in_gaa(T34, T36, X97))
U3_GA(T34, X97, ack21_out_ga(T34, T36)) → ACK38_IN_GAA(T34, T36, X97)
ACK38_IN_GAA(s(T49), 0, X133) → U5_GAA(T49, X133, ack27_in_ga(T49, X133))
ACK38_IN_GAA(s(T49), 0, X133) → ACK27_IN_GA(T49, X133)
ACK38_IN_GAA(s(T54), s(T55), X151) → U6_GAA(T54, T55, X151, ack38_in_gaa(s(T54), T55, X150))
ACK38_IN_GAA(s(T54), s(T55), X151) → ACK38_IN_GAA(s(T54), T55, X150)
ACK38_IN_GAA(s(T54), s(T55), X151) → U7_GAA(T54, T55, X151, ack38_in_gaa(s(T54), T55, T57))
U7_GAA(T54, T55, X151, ack38_out_gaa(s(T54), T55, T57)) → U8_GAA(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
U7_GAA(T54, T55, X151, ack38_out_gaa(s(T54), T55, T57)) → ACK38_IN_GAA(T54, T57, X151)
ACK1_IN_GGA(s(s(T20)), 0, T22) → U10_GGA(T20, T22, ack21_in_ga(T20, T24))
U10_GGA(T20, T22, ack21_out_ga(T20, T24)) → U11_GGA(T20, T22, ack1_in_gaa(T20, T24, T22))
U10_GGA(T20, T22, ack21_out_ga(T20, T24)) → ACK1_IN_GAA(T20, T24, T22)
ACK1_IN_GAA(s(s(T20)), 0, T22) → U9_GAA(T20, T22, ack21_in_ga(T20, X40))
ACK1_IN_GAA(s(s(T20)), 0, T22) → ACK21_IN_GA(T20, X40)
ACK1_IN_GAA(s(s(T20)), 0, T22) → U10_GAA(T20, T22, ack21_in_ga(T20, T24))
U10_GAA(T20, T22, ack21_out_ga(T20, T24)) → U11_GAA(T20, T22, ack1_in_gaa(T20, T24, T22))
U10_GAA(T20, T22, ack21_out_ga(T20, T24)) → ACK1_IN_GAA(T20, T24, T22)
ACK1_IN_GAA(s(T76), s(0), T71) → U12_GAA(T76, T71, ack27_in_ga(T76, X200))
ACK1_IN_GAA(s(T76), s(0), T71) → ACK27_IN_GA(T76, X200)
ACK1_IN_GAA(s(T76), s(0), T71) → U13_GAA(T76, T71, ack27_in_ga(T76, T77))
U13_GAA(T76, T71, ack27_out_ga(T76, T77)) → U14_GAA(T76, T71, ack1_in_gaa(T76, T77, T71))
U13_GAA(T76, T71, ack27_out_ga(T76, T77)) → ACK1_IN_GAA(T76, T77, T71)
ACK1_IN_GAA(s(T84), s(s(T85)), T71) → U15_GAA(T84, T85, T71, ack38_in_gaa(s(T84), T85, X221))
ACK1_IN_GAA(s(T84), s(s(T85)), T71) → ACK38_IN_GAA(s(T84), T85, X221)
ACK1_IN_GAA(s(T84), s(s(T85)), T71) → U16_GAA(T84, T85, T71, ack38_in_gaa(s(T84), T85, T87))
U16_GAA(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U17_GAA(T84, T85, T71, ack38_in_gaa(T84, T87, X222))
U16_GAA(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → ACK38_IN_GAA(T84, T87, X222)
U16_GAA(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U18_GAA(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_GAA(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → U19_GAA(T84, T85, T71, ack1_in_gaa(T84, T91, T71))
U18_GAA(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → ACK1_IN_GAA(T84, T91, T71)
ACK1_IN_GGA(s(T76), s(0), T71) → U12_GGA(T76, T71, ack27_in_ga(T76, X200))
ACK1_IN_GGA(s(T76), s(0), T71) → ACK27_IN_GA(T76, X200)
ACK1_IN_GGA(s(T76), s(0), T71) → U13_GGA(T76, T71, ack27_in_ga(T76, T77))
U13_GGA(T76, T71, ack27_out_ga(T76, T77)) → U14_GGA(T76, T71, ack1_in_gaa(T76, T77, T71))
U13_GGA(T76, T71, ack27_out_ga(T76, T77)) → ACK1_IN_GAA(T76, T77, T71)
ACK1_IN_GGA(s(T84), s(s(T85)), T71) → U15_GGA(T84, T85, T71, ack38_in_gga(s(T84), T85, X221))
ACK1_IN_GGA(s(T84), s(s(T85)), T71) → ACK38_IN_GGA(s(T84), T85, X221)
ACK38_IN_GGA(s(T49), 0, X133) → U5_GGA(T49, X133, ack27_in_ga(T49, X133))
ACK38_IN_GGA(s(T49), 0, X133) → ACK27_IN_GA(T49, X133)
ACK38_IN_GGA(s(T54), s(T55), X151) → U6_GGA(T54, T55, X151, ack38_in_gga(s(T54), T55, X150))
ACK38_IN_GGA(s(T54), s(T55), X151) → ACK38_IN_GGA(s(T54), T55, X150)
ACK38_IN_GGA(s(T54), s(T55), X151) → U7_GGA(T54, T55, X151, ack38_in_gga(s(T54), T55, T57))
U7_GGA(T54, T55, X151, ack38_out_gga(s(T54), T55, T57)) → U8_GGA(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
U7_GGA(T54, T55, X151, ack38_out_gga(s(T54), T55, T57)) → ACK38_IN_GAA(T54, T57, X151)
ACK1_IN_GGA(s(T84), s(s(T85)), T71) → U16_GGA(T84, T85, T71, ack38_in_gga(s(T84), T85, T87))
U16_GGA(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → U17_GGA(T84, T85, T71, ack38_in_gaa(T84, T87, X222))
U16_GGA(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → ACK38_IN_GAA(T84, T87, X222)
U16_GGA(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → U18_GGA(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_GGA(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → U19_GGA(T84, T85, T71, ack1_in_gaa(T84, T91, T71))
U18_GGA(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → ACK1_IN_GAA(T84, T91, T71)

The TRS R consists of the following rules:

ack1_in_gga(0, T5, s(T5)) → ack1_out_gga(0, T5, s(T5))
ack1_in_gga(s(0), 0, s(s(0))) → ack1_out_gga(s(0), 0, s(s(0)))
ack1_in_gga(s(s(T20)), 0, T22) → U9_gga(T20, T22, ack21_in_ga(T20, X40))
ack21_in_ga(T30, X73) → U1_ga(T30, X73, ack27_in_ga(T30, X73))
ack27_in_ga(0, s(s(0))) → ack27_out_ga(0, s(s(0)))
ack27_in_ga(s(T34), X97) → U2_ga(T34, X97, ack21_in_ga(T34, X96))
U2_ga(T34, X97, ack21_out_ga(T34, X96)) → ack27_out_ga(s(T34), X97)
ack27_in_ga(s(T34), X97) → U3_ga(T34, X97, ack21_in_ga(T34, T36))
U3_ga(T34, X97, ack21_out_ga(T34, T36)) → U4_ga(T34, X97, ack38_in_gaa(T34, T36, X97))
ack38_in_gaa(0, T44, s(T44)) → ack38_out_gaa(0, T44, s(T44))
ack38_in_gaa(s(T49), 0, X133) → U5_gaa(T49, X133, ack27_in_ga(T49, X133))
U5_gaa(T49, X133, ack27_out_ga(T49, X133)) → ack38_out_gaa(s(T49), 0, X133)
ack38_in_gaa(s(T54), s(T55), X151) → U6_gaa(T54, T55, X151, ack38_in_gaa(s(T54), T55, X150))
ack38_in_gaa(s(T54), s(T55), X151) → U7_gaa(T54, T55, X151, ack38_in_gaa(s(T54), T55, T57))
U7_gaa(T54, T55, X151, ack38_out_gaa(s(T54), T55, T57)) → U8_gaa(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
U8_gaa(T54, T55, X151, ack38_out_gaa(T54, T57, X151)) → ack38_out_gaa(s(T54), s(T55), X151)
U6_gaa(T54, T55, X151, ack38_out_gaa(s(T54), T55, X150)) → ack38_out_gaa(s(T54), s(T55), X151)
U4_ga(T34, X97, ack38_out_gaa(T34, T36, X97)) → ack27_out_ga(s(T34), X97)
U1_ga(T30, X73, ack27_out_ga(T30, X73)) → ack21_out_ga(T30, X73)
U9_gga(T20, T22, ack21_out_ga(T20, X40)) → ack1_out_gga(s(s(T20)), 0, T22)
ack1_in_gga(s(s(T20)), 0, T22) → U10_gga(T20, T22, ack21_in_ga(T20, T24))
U10_gga(T20, T22, ack21_out_ga(T20, T24)) → U11_gga(T20, T22, ack1_in_gaa(T20, T24, T22))
ack1_in_gaa(0, T5, s(T5)) → ack1_out_gaa(0, T5, s(T5))
ack1_in_gaa(s(0), 0, s(s(0))) → ack1_out_gaa(s(0), 0, s(s(0)))
ack1_in_gaa(s(s(T20)), 0, T22) → U9_gaa(T20, T22, ack21_in_ga(T20, X40))
U9_gaa(T20, T22, ack21_out_ga(T20, X40)) → ack1_out_gaa(s(s(T20)), 0, T22)
ack1_in_gaa(s(s(T20)), 0, T22) → U10_gaa(T20, T22, ack21_in_ga(T20, T24))
U10_gaa(T20, T22, ack21_out_ga(T20, T24)) → U11_gaa(T20, T22, ack1_in_gaa(T20, T24, T22))
ack1_in_gaa(s(T76), s(0), T71) → U12_gaa(T76, T71, ack27_in_ga(T76, X200))
U12_gaa(T76, T71, ack27_out_ga(T76, X200)) → ack1_out_gaa(s(T76), s(0), T71)
ack1_in_gaa(s(T76), s(0), T71) → U13_gaa(T76, T71, ack27_in_ga(T76, T77))
U13_gaa(T76, T71, ack27_out_ga(T76, T77)) → U14_gaa(T76, T71, ack1_in_gaa(T76, T77, T71))
ack1_in_gaa(s(T84), s(s(T85)), T71) → U15_gaa(T84, T85, T71, ack38_in_gaa(s(T84), T85, X221))
U15_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, X221)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
ack1_in_gaa(s(T84), s(s(T85)), T71) → U16_gaa(T84, T85, T71, ack38_in_gaa(s(T84), T85, T87))
U16_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U17_gaa(T84, T85, T71, ack38_in_gaa(T84, T87, X222))
U17_gaa(T84, T85, T71, ack38_out_gaa(T84, T87, X222)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
U16_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U18_gaa(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_gaa(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → U19_gaa(T84, T85, T71, ack1_in_gaa(T84, T91, T71))
U19_gaa(T84, T85, T71, ack1_out_gaa(T84, T91, T71)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
U14_gaa(T76, T71, ack1_out_gaa(T76, T77, T71)) → ack1_out_gaa(s(T76), s(0), T71)
U11_gaa(T20, T22, ack1_out_gaa(T20, T24, T22)) → ack1_out_gaa(s(s(T20)), 0, T22)
U11_gga(T20, T22, ack1_out_gaa(T20, T24, T22)) → ack1_out_gga(s(s(T20)), 0, T22)
ack1_in_gga(s(T76), s(0), T71) → U12_gga(T76, T71, ack27_in_ga(T76, X200))
U12_gga(T76, T71, ack27_out_ga(T76, X200)) → ack1_out_gga(s(T76), s(0), T71)
ack1_in_gga(s(T76), s(0), T71) → U13_gga(T76, T71, ack27_in_ga(T76, T77))
U13_gga(T76, T71, ack27_out_ga(T76, T77)) → U14_gga(T76, T71, ack1_in_gaa(T76, T77, T71))
U14_gga(T76, T71, ack1_out_gaa(T76, T77, T71)) → ack1_out_gga(s(T76), s(0), T71)
ack1_in_gga(s(T84), s(s(T85)), T71) → U15_gga(T84, T85, T71, ack38_in_gga(s(T84), T85, X221))
ack38_in_gga(0, T44, s(T44)) → ack38_out_gga(0, T44, s(T44))
ack38_in_gga(s(T49), 0, X133) → U5_gga(T49, X133, ack27_in_ga(T49, X133))
U5_gga(T49, X133, ack27_out_ga(T49, X133)) → ack38_out_gga(s(T49), 0, X133)
ack38_in_gga(s(T54), s(T55), X151) → U6_gga(T54, T55, X151, ack38_in_gga(s(T54), T55, X150))
ack38_in_gga(s(T54), s(T55), X151) → U7_gga(T54, T55, X151, ack38_in_gga(s(T54), T55, T57))
U7_gga(T54, T55, X151, ack38_out_gga(s(T54), T55, T57)) → U8_gga(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
U8_gga(T54, T55, X151, ack38_out_gaa(T54, T57, X151)) → ack38_out_gga(s(T54), s(T55), X151)
U6_gga(T54, T55, X151, ack38_out_gga(s(T54), T55, X150)) → ack38_out_gga(s(T54), s(T55), X151)
U15_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, X221)) → ack1_out_gga(s(T84), s(s(T85)), T71)
ack1_in_gga(s(T84), s(s(T85)), T71) → U16_gga(T84, T85, T71, ack38_in_gga(s(T84), T85, T87))
U16_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → U17_gga(T84, T85, T71, ack38_in_gaa(T84, T87, X222))
U17_gga(T84, T85, T71, ack38_out_gaa(T84, T87, X222)) → ack1_out_gga(s(T84), s(s(T85)), T71)
U16_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → U18_gga(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_gga(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → U19_gga(T84, T85, T71, ack1_in_gaa(T84, T91, T71))
U19_gga(T84, T85, T71, ack1_out_gaa(T84, T91, T71)) → ack1_out_gga(s(T84), s(s(T85)), T71)

The argument filtering Pi contains the following mapping:
ack1_in_gga(x1, x2, x3) = ack1_in_gga(x1, x2)
0 = 0
ack1_out_gga(x1, x2, x3) = ack1_out_gga(x1, x2)
s(x1) = s(x1)
U9_gga(x1, x2, x3) = U9_gga(x1, x3)
ack21_in_ga(x1, x2) = ack21_in_ga(x1)
U1_ga(x1, x2, x3) = U1_ga(x1, x3)
ack27_in_ga(x1, x2) = ack27_in_ga(x1)
ack27_out_ga(x1, x2) = ack27_out_ga(x1)
U2_ga(x1, x2, x3) = U2_ga(x1, x3)
ack21_out_ga(x1, x2) = ack21_out_ga(x1)
U3_ga(x1, x2, x3) = U3_ga(x1, x3)
U4_ga(x1, x2, x3) = U4_ga(x1, x3)
ack38_in_gga(x1, x2, x3) = ack38_in_gga(x1, x2)
ack38_out_gga(x1, x2, x3) = ack38_out_gga(x1, x2)
U5_gga(x1, x2, x3) = U5_gga(x1, x3)
U6_gga(x1, x2, x3, x4) = U6_gga(x1, x2, x4)
U7_gga(x1, x2, x3, x4) = U7_gga(x1, x2, x4)
U8_gga(x1, x2, x3, x4) = U8_gga(x1, x2, x4)
ack38_in_gaa(x1, x2, x3) = ack38_in_gaa(x1)
ack38_out_gaa(x1, x2, x3) = ack38_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)
U10_gga(x1, x2, x3) = U10_gga(x1, x3)
U11_gga(x1, x2, x3) = U11_gga(x1, x3)
ack1_in_gaa(x1, x2, x3) = ack1_in_gaa(x1)
ack1_out_gaa(x1, x2, x3) = ack1_out_gaa(x1)
U9_gaa(x1, x2, x3) = U9_gaa(x1, x3)
U10_gaa(x1, x2, x3) = U10_gaa(x1, x3)
U11_gaa(x1, x2, x3) = U11_gaa(x1, x3)
U12_gaa(x1, x2, x3) = U12_gaa(x1, x3)
U13_gaa(x1, x2, x3) = U13_gaa(x1, x3)
U14_gaa(x1, x2, x3) = U14_gaa(x1, x3)
U15_gaa(x1, x2, x3, x4) = U15_gaa(x1, x4)
U16_gaa(x1, x2, x3, x4) = U16_gaa(x1, x4)
U17_gaa(x1, x2, x3, x4) = U17_gaa(x1, x4)
U18_gaa(x1, x2, x3, x4) = U18_gaa(x1, x4)
U19_gaa(x1, x2, x3, x4) = U19_gaa(x1, x4)
U12_gga(x1, x2, x3) = U12_gga(x1, x3)
U13_gga(x1, x2, x3) = U13_gga(x1, x3)
U14_gga(x1, x2, x3) = U14_gga(x1, x3)
U15_gga(x1, x2, x3, x4) = U15_gga(x1, x2, x4)
U16_gga(x1, x2, x3, x4) = U16_gga(x1, x2, x4)
U17_gga(x1, x2, x3, x4) = U17_gga(x1, x2, x4)
U18_gga(x1, x2, x3, x4) = U18_gga(x1, x2, x4)
U19_gga(x1, x2, x3, x4) = U19_gga(x1, x2, x4)
ACK1_IN_GGA(x1, x2, x3) = ACK1_IN_GGA(x1, x2)
U9_GGA(x1, x2, x3) = U9_GGA(x1, x3)
ACK21_IN_GA(x1, x2) = ACK21_IN_GA(x1)
U1_GA(x1, x2, x3) = U1_GA(x1, x3)
ACK27_IN_GA(x1, x2) = ACK27_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)
ACK38_IN_GAA(x1, x2, x3) = ACK38_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)
U10_GGA(x1, x2, x3) = U10_GGA(x1, x3)
U11_GGA(x1, x2, x3) = U11_GGA(x1, x3)
ACK1_IN_GAA(x1, x2, x3) = ACK1_IN_GAA(x1)
U9_GAA(x1, x2, x3) = U9_GAA(x1, x3)
U10_GAA(x1, x2, x3) = U10_GAA(x1, x3)
U11_GAA(x1, x2, x3) = U11_GAA(x1, x3)
U12_GAA(x1, x2, x3) = U12_GAA(x1, x3)
U13_GAA(x1, x2, x3) = U13_GAA(x1, x3)
U14_GAA(x1, x2, x3) = U14_GAA(x1, x3)
U15_GAA(x1, x2, x3, x4) = U15_GAA(x1, x4)
U16_GAA(x1, x2, x3, x4) = U16_GAA(x1, x4)
U17_GAA(x1, x2, x3, x4) = U17_GAA(x1, x4)
U18_GAA(x1, x2, x3, x4) = U18_GAA(x1, x4)
U19_GAA(x1, x2, x3, x4) = U19_GAA(x1, x4)
U12_GGA(x1, x2, x3) = U12_GGA(x1, x3)
U13_GGA(x1, x2, x3) = U13_GGA(x1, x3)
U14_GGA(x1, x2, x3) = U14_GGA(x1, x3)
U15_GGA(x1, x2, x3, x4) = U15_GGA(x1, x2, x4)
ACK38_IN_GGA(x1, x2, x3) = ACK38_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3) = U5_GGA(x1, x3)
U6_GGA(x1, x2, x3, x4) = U6_GGA(x1, x2, x4)
U7_GGA(x1, x2, x3, x4) = U7_GGA(x1, x2, x4)
U8_GGA(x1, x2, x3, x4) = U8_GGA(x1, x2, x4)
U16_GGA(x1, x2, x3, x4) = U16_GGA(x1, x2, x4)
U17_GGA(x1, x2, x3, x4) = U17_GGA(x1, x2, x4)
U18_GGA(x1, x2, x3, x4) = U18_GGA(x1, x2, x4)
U19_GGA(x1, x2, x3, x4) = U19_GGA(x1, x2, x4)

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

(6) Obligation:

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

ACK1_IN_GGA(s(s(T20)), 0, T22) → U9_GGA(T20, T22, ack21_in_ga(T20, X40))
ACK1_IN_GGA(s(s(T20)), 0, T22) → ACK21_IN_GA(T20, X40)
ACK21_IN_GA(T30, X73) → U1_GA(T30, X73, ack27_in_ga(T30, X73))
ACK21_IN_GA(T30, X73) → ACK27_IN_GA(T30, X73)
ACK27_IN_GA(s(T34), X97) → U2_GA(T34, X97, ack21_in_ga(T34, X96))
ACK27_IN_GA(s(T34), X97) → ACK21_IN_GA(T34, X96)
ACK27_IN_GA(s(T34), X97) → U3_GA(T34, X97, ack21_in_ga(T34, T36))
U3_GA(T34, X97, ack21_out_ga(T34, T36)) → U4_GA(T34, X97, ack38_in_gaa(T34, T36, X97))
U3_GA(T34, X97, ack21_out_ga(T34, T36)) → ACK38_IN_GAA(T34, T36, X97)
ACK38_IN_GAA(s(T49), 0, X133) → U5_GAA(T49, X133, ack27_in_ga(T49, X133))
ACK38_IN_GAA(s(T49), 0, X133) → ACK27_IN_GA(T49, X133)
ACK38_IN_GAA(s(T54), s(T55), X151) → U6_GAA(T54, T55, X151, ack38_in_gaa(s(T54), T55, X150))
ACK38_IN_GAA(s(T54), s(T55), X151) → ACK38_IN_GAA(s(T54), T55, X150)
ACK38_IN_GAA(s(T54), s(T55), X151) → U7_GAA(T54, T55, X151, ack38_in_gaa(s(T54), T55, T57))
U7_GAA(T54, T55, X151, ack38_out_gaa(s(T54), T55, T57)) → U8_GAA(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
U7_GAA(T54, T55, X151, ack38_out_gaa(s(T54), T55, T57)) → ACK38_IN_GAA(T54, T57, X151)
ACK1_IN_GGA(s(s(T20)), 0, T22) → U10_GGA(T20, T22, ack21_in_ga(T20, T24))
U10_GGA(T20, T22, ack21_out_ga(T20, T24)) → U11_GGA(T20, T22, ack1_in_gaa(T20, T24, T22))
U10_GGA(T20, T22, ack21_out_ga(T20, T24)) → ACK1_IN_GAA(T20, T24, T22)
ACK1_IN_GAA(s(s(T20)), 0, T22) → U9_GAA(T20, T22, ack21_in_ga(T20, X40))
ACK1_IN_GAA(s(s(T20)), 0, T22) → ACK21_IN_GA(T20, X40)
ACK1_IN_GAA(s(s(T20)), 0, T22) → U10_GAA(T20, T22, ack21_in_ga(T20, T24))
U10_GAA(T20, T22, ack21_out_ga(T20, T24)) → U11_GAA(T20, T22, ack1_in_gaa(T20, T24, T22))
U10_GAA(T20, T22, ack21_out_ga(T20, T24)) → ACK1_IN_GAA(T20, T24, T22)
ACK1_IN_GAA(s(T76), s(0), T71) → U12_GAA(T76, T71, ack27_in_ga(T76, X200))
ACK1_IN_GAA(s(T76), s(0), T71) → ACK27_IN_GA(T76, X200)
ACK1_IN_GAA(s(T76), s(0), T71) → U13_GAA(T76, T71, ack27_in_ga(T76, T77))
U13_GAA(T76, T71, ack27_out_ga(T76, T77)) → U14_GAA(T76, T71, ack1_in_gaa(T76, T77, T71))
U13_GAA(T76, T71, ack27_out_ga(T76, T77)) → ACK1_IN_GAA(T76, T77, T71)
ACK1_IN_GAA(s(T84), s(s(T85)), T71) → U15_GAA(T84, T85, T71, ack38_in_gaa(s(T84), T85, X221))
ACK1_IN_GAA(s(T84), s(s(T85)), T71) → ACK38_IN_GAA(s(T84), T85, X221)
ACK1_IN_GAA(s(T84), s(s(T85)), T71) → U16_GAA(T84, T85, T71, ack38_in_gaa(s(T84), T85, T87))
U16_GAA(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U17_GAA(T84, T85, T71, ack38_in_gaa(T84, T87, X222))
U16_GAA(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → ACK38_IN_GAA(T84, T87, X222)
U16_GAA(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U18_GAA(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_GAA(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → U19_GAA(T84, T85, T71, ack1_in_gaa(T84, T91, T71))
U18_GAA(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → ACK1_IN_GAA(T84, T91, T71)
ACK1_IN_GGA(s(T76), s(0), T71) → U12_GGA(T76, T71, ack27_in_ga(T76, X200))
ACK1_IN_GGA(s(T76), s(0), T71) → ACK27_IN_GA(T76, X200)
ACK1_IN_GGA(s(T76), s(0), T71) → U13_GGA(T76, T71, ack27_in_ga(T76, T77))
U13_GGA(T76, T71, ack27_out_ga(T76, T77)) → U14_GGA(T76, T71, ack1_in_gaa(T76, T77, T71))
U13_GGA(T76, T71, ack27_out_ga(T76, T77)) → ACK1_IN_GAA(T76, T77, T71)
ACK1_IN_GGA(s(T84), s(s(T85)), T71) → U15_GGA(T84, T85, T71, ack38_in_gga(s(T84), T85, X221))
ACK1_IN_GGA(s(T84), s(s(T85)), T71) → ACK38_IN_GGA(s(T84), T85, X221)
ACK38_IN_GGA(s(T49), 0, X133) → U5_GGA(T49, X133, ack27_in_ga(T49, X133))
ACK38_IN_GGA(s(T49), 0, X133) → ACK27_IN_GA(T49, X133)
ACK38_IN_GGA(s(T54), s(T55), X151) → U6_GGA(T54, T55, X151, ack38_in_gga(s(T54), T55, X150))
ACK38_IN_GGA(s(T54), s(T55), X151) → ACK38_IN_GGA(s(T54), T55, X150)
ACK38_IN_GGA(s(T54), s(T55), X151) → U7_GGA(T54, T55, X151, ack38_in_gga(s(T54), T55, T57))
U7_GGA(T54, T55, X151, ack38_out_gga(s(T54), T55, T57)) → U8_GGA(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
U7_GGA(T54, T55, X151, ack38_out_gga(s(T54), T55, T57)) → ACK38_IN_GAA(T54, T57, X151)
ACK1_IN_GGA(s(T84), s(s(T85)), T71) → U16_GGA(T84, T85, T71, ack38_in_gga(s(T84), T85, T87))
U16_GGA(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → U17_GGA(T84, T85, T71, ack38_in_gaa(T84, T87, X222))
U16_GGA(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → ACK38_IN_GAA(T84, T87, X222)
U16_GGA(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → U18_GGA(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_GGA(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → U19_GGA(T84, T85, T71, ack1_in_gaa(T84, T91, T71))
U18_GGA(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → ACK1_IN_GAA(T84, T91, T71)

The TRS R consists of the following rules:

ack1_in_gga(0, T5, s(T5)) → ack1_out_gga(0, T5, s(T5))
ack1_in_gga(s(0), 0, s(s(0))) → ack1_out_gga(s(0), 0, s(s(0)))
ack1_in_gga(s(s(T20)), 0, T22) → U9_gga(T20, T22, ack21_in_ga(T20, X40))
ack21_in_ga(T30, X73) → U1_ga(T30, X73, ack27_in_ga(T30, X73))
ack27_in_ga(0, s(s(0))) → ack27_out_ga(0, s(s(0)))
ack27_in_ga(s(T34), X97) → U2_ga(T34, X97, ack21_in_ga(T34, X96))
U2_ga(T34, X97, ack21_out_ga(T34, X96)) → ack27_out_ga(s(T34), X97)
ack27_in_ga(s(T34), X97) → U3_ga(T34, X97, ack21_in_ga(T34, T36))
U3_ga(T34, X97, ack21_out_ga(T34, T36)) → U4_ga(T34, X97, ack38_in_gaa(T34, T36, X97))
ack38_in_gaa(0, T44, s(T44)) → ack38_out_gaa(0, T44, s(T44))
ack38_in_gaa(s(T49), 0, X133) → U5_gaa(T49, X133, ack27_in_ga(T49, X133))
U5_gaa(T49, X133, ack27_out_ga(T49, X133)) → ack38_out_gaa(s(T49), 0, X133)
ack38_in_gaa(s(T54), s(T55), X151) → U6_gaa(T54, T55, X151, ack38_in_gaa(s(T54), T55, X150))
ack38_in_gaa(s(T54), s(T55), X151) → U7_gaa(T54, T55, X151, ack38_in_gaa(s(T54), T55, T57))
U7_gaa(T54, T55, X151, ack38_out_gaa(s(T54), T55, T57)) → U8_gaa(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
U8_gaa(T54, T55, X151, ack38_out_gaa(T54, T57, X151)) → ack38_out_gaa(s(T54), s(T55), X151)
U6_gaa(T54, T55, X151, ack38_out_gaa(s(T54), T55, X150)) → ack38_out_gaa(s(T54), s(T55), X151)
U4_ga(T34, X97, ack38_out_gaa(T34, T36, X97)) → ack27_out_ga(s(T34), X97)
U1_ga(T30, X73, ack27_out_ga(T30, X73)) → ack21_out_ga(T30, X73)
U9_gga(T20, T22, ack21_out_ga(T20, X40)) → ack1_out_gga(s(s(T20)), 0, T22)
ack1_in_gga(s(s(T20)), 0, T22) → U10_gga(T20, T22, ack21_in_ga(T20, T24))
U10_gga(T20, T22, ack21_out_ga(T20, T24)) → U11_gga(T20, T22, ack1_in_gaa(T20, T24, T22))
ack1_in_gaa(0, T5, s(T5)) → ack1_out_gaa(0, T5, s(T5))
ack1_in_gaa(s(0), 0, s(s(0))) → ack1_out_gaa(s(0), 0, s(s(0)))
ack1_in_gaa(s(s(T20)), 0, T22) → U9_gaa(T20, T22, ack21_in_ga(T20, X40))
U9_gaa(T20, T22, ack21_out_ga(T20, X40)) → ack1_out_gaa(s(s(T20)), 0, T22)
ack1_in_gaa(s(s(T20)), 0, T22) → U10_gaa(T20, T22, ack21_in_ga(T20, T24))
U10_gaa(T20, T22, ack21_out_ga(T20, T24)) → U11_gaa(T20, T22, ack1_in_gaa(T20, T24, T22))
ack1_in_gaa(s(T76), s(0), T71) → U12_gaa(T76, T71, ack27_in_ga(T76, X200))
U12_gaa(T76, T71, ack27_out_ga(T76, X200)) → ack1_out_gaa(s(T76), s(0), T71)
ack1_in_gaa(s(T76), s(0), T71) → U13_gaa(T76, T71, ack27_in_ga(T76, T77))
U13_gaa(T76, T71, ack27_out_ga(T76, T77)) → U14_gaa(T76, T71, ack1_in_gaa(T76, T77, T71))
ack1_in_gaa(s(T84), s(s(T85)), T71) → U15_gaa(T84, T85, T71, ack38_in_gaa(s(T84), T85, X221))
U15_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, X221)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
ack1_in_gaa(s(T84), s(s(T85)), T71) → U16_gaa(T84, T85, T71, ack38_in_gaa(s(T84), T85, T87))
U16_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U17_gaa(T84, T85, T71, ack38_in_gaa(T84, T87, X222))
U17_gaa(T84, T85, T71, ack38_out_gaa(T84, T87, X222)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
U16_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U18_gaa(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_gaa(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → U19_gaa(T84, T85, T71, ack1_in_gaa(T84, T91, T71))
U19_gaa(T84, T85, T71, ack1_out_gaa(T84, T91, T71)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
U14_gaa(T76, T71, ack1_out_gaa(T76, T77, T71)) → ack1_out_gaa(s(T76), s(0), T71)
U11_gaa(T20, T22, ack1_out_gaa(T20, T24, T22)) → ack1_out_gaa(s(s(T20)), 0, T22)
U11_gga(T20, T22, ack1_out_gaa(T20, T24, T22)) → ack1_out_gga(s(s(T20)), 0, T22)
ack1_in_gga(s(T76), s(0), T71) → U12_gga(T76, T71, ack27_in_ga(T76, X200))
U12_gga(T76, T71, ack27_out_ga(T76, X200)) → ack1_out_gga(s(T76), s(0), T71)
ack1_in_gga(s(T76), s(0), T71) → U13_gga(T76, T71, ack27_in_ga(T76, T77))
U13_gga(T76, T71, ack27_out_ga(T76, T77)) → U14_gga(T76, T71, ack1_in_gaa(T76, T77, T71))
U14_gga(T76, T71, ack1_out_gaa(T76, T77, T71)) → ack1_out_gga(s(T76), s(0), T71)
ack1_in_gga(s(T84), s(s(T85)), T71) → U15_gga(T84, T85, T71, ack38_in_gga(s(T84), T85, X221))
ack38_in_gga(0, T44, s(T44)) → ack38_out_gga(0, T44, s(T44))
ack38_in_gga(s(T49), 0, X133) → U5_gga(T49, X133, ack27_in_ga(T49, X133))
U5_gga(T49, X133, ack27_out_ga(T49, X133)) → ack38_out_gga(s(T49), 0, X133)
ack38_in_gga(s(T54), s(T55), X151) → U6_gga(T54, T55, X151, ack38_in_gga(s(T54), T55, X150))
ack38_in_gga(s(T54), s(T55), X151) → U7_gga(T54, T55, X151, ack38_in_gga(s(T54), T55, T57))
U7_gga(T54, T55, X151, ack38_out_gga(s(T54), T55, T57)) → U8_gga(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
U8_gga(T54, T55, X151, ack38_out_gaa(T54, T57, X151)) → ack38_out_gga(s(T54), s(T55), X151)
U6_gga(T54, T55, X151, ack38_out_gga(s(T54), T55, X150)) → ack38_out_gga(s(T54), s(T55), X151)
U15_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, X221)) → ack1_out_gga(s(T84), s(s(T85)), T71)
ack1_in_gga(s(T84), s(s(T85)), T71) → U16_gga(T84, T85, T71, ack38_in_gga(s(T84), T85, T87))
U16_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → U17_gga(T84, T85, T71, ack38_in_gaa(T84, T87, X222))
U17_gga(T84, T85, T71, ack38_out_gaa(T84, T87, X222)) → ack1_out_gga(s(T84), s(s(T85)), T71)
U16_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → U18_gga(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_gga(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → U19_gga(T84, T85, T71, ack1_in_gaa(T84, T91, T71))
U19_gga(T84, T85, T71, ack1_out_gaa(T84, T91, T71)) → ack1_out_gga(s(T84), s(s(T85)), T71)

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 41 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

ACK21_IN_GA(T30, X73) → ACK27_IN_GA(T30, X73)
ACK27_IN_GA(s(T34), X97) → ACK21_IN_GA(T34, X96)
ACK27_IN_GA(s(T34), X97) → U3_GA(T34, X97, ack21_in_ga(T34, T36))
U3_GA(T34, X97, ack21_out_ga(T34, T36)) → ACK38_IN_GAA(T34, T36, X97)
ACK38_IN_GAA(s(T49), 0, X133) → ACK27_IN_GA(T49, X133)
ACK38_IN_GAA(s(T54), s(T55), X151) → ACK38_IN_GAA(s(T54), T55, X150)
ACK38_IN_GAA(s(T54), s(T55), X151) → U7_GAA(T54, T55, X151, ack38_in_gaa(s(T54), T55, T57))
U7_GAA(T54, T55, X151, ack38_out_gaa(s(T54), T55, T57)) → ACK38_IN_GAA(T54, T57, X151)

The TRS R consists of the following rules:

ack1_in_gga(0, T5, s(T5)) → ack1_out_gga(0, T5, s(T5))
ack1_in_gga(s(0), 0, s(s(0))) → ack1_out_gga(s(0), 0, s(s(0)))
ack1_in_gga(s(s(T20)), 0, T22) → U9_gga(T20, T22, ack21_in_ga(T20, X40))
ack21_in_ga(T30, X73) → U1_ga(T30, X73, ack27_in_ga(T30, X73))
ack27_in_ga(0, s(s(0))) → ack27_out_ga(0, s(s(0)))
ack27_in_ga(s(T34), X97) → U2_ga(T34, X97, ack21_in_ga(T34, X96))
U2_ga(T34, X97, ack21_out_ga(T34, X96)) → ack27_out_ga(s(T34), X97)
ack27_in_ga(s(T34), X97) → U3_ga(T34, X97, ack21_in_ga(T34, T36))
U3_ga(T34, X97, ack21_out_ga(T34, T36)) → U4_ga(T34, X97, ack38_in_gaa(T34, T36, X97))
ack38_in_gaa(0, T44, s(T44)) → ack38_out_gaa(0, T44, s(T44))
ack38_in_gaa(s(T49), 0, X133) → U5_gaa(T49, X133, ack27_in_ga(T49, X133))
U5_gaa(T49, X133, ack27_out_ga(T49, X133)) → ack38_out_gaa(s(T49), 0, X133)
ack38_in_gaa(s(T54), s(T55), X151) → U6_gaa(T54, T55, X151, ack38_in_gaa(s(T54), T55, X150))
ack38_in_gaa(s(T54), s(T55), X151) → U7_gaa(T54, T55, X151, ack38_in_gaa(s(T54), T55, T57))
U7_gaa(T54, T55, X151, ack38_out_gaa(s(T54), T55, T57)) → U8_gaa(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
U8_gaa(T54, T55, X151, ack38_out_gaa(T54, T57, X151)) → ack38_out_gaa(s(T54), s(T55), X151)
U6_gaa(T54, T55, X151, ack38_out_gaa(s(T54), T55, X150)) → ack38_out_gaa(s(T54), s(T55), X151)
U4_ga(T34, X97, ack38_out_gaa(T34, T36, X97)) → ack27_out_ga(s(T34), X97)
U1_ga(T30, X73, ack27_out_ga(T30, X73)) → ack21_out_ga(T30, X73)
U9_gga(T20, T22, ack21_out_ga(T20, X40)) → ack1_out_gga(s(s(T20)), 0, T22)
ack1_in_gga(s(s(T20)), 0, T22) → U10_gga(T20, T22, ack21_in_ga(T20, T24))
U10_gga(T20, T22, ack21_out_ga(T20, T24)) → U11_gga(T20, T22, ack1_in_gaa(T20, T24, T22))
ack1_in_gaa(0, T5, s(T5)) → ack1_out_gaa(0, T5, s(T5))
ack1_in_gaa(s(0), 0, s(s(0))) → ack1_out_gaa(s(0), 0, s(s(0)))
ack1_in_gaa(s(s(T20)), 0, T22) → U9_gaa(T20, T22, ack21_in_ga(T20, X40))
U9_gaa(T20, T22, ack21_out_ga(T20, X40)) → ack1_out_gaa(s(s(T20)), 0, T22)
ack1_in_gaa(s(s(T20)), 0, T22) → U10_gaa(T20, T22, ack21_in_ga(T20, T24))
U10_gaa(T20, T22, ack21_out_ga(T20, T24)) → U11_gaa(T20, T22, ack1_in_gaa(T20, T24, T22))
ack1_in_gaa(s(T76), s(0), T71) → U12_gaa(T76, T71, ack27_in_ga(T76, X200))
U12_gaa(T76, T71, ack27_out_ga(T76, X200)) → ack1_out_gaa(s(T76), s(0), T71)
ack1_in_gaa(s(T76), s(0), T71) → U13_gaa(T76, T71, ack27_in_ga(T76, T77))
U13_gaa(T76, T71, ack27_out_ga(T76, T77)) → U14_gaa(T76, T71, ack1_in_gaa(T76, T77, T71))
ack1_in_gaa(s(T84), s(s(T85)), T71) → U15_gaa(T84, T85, T71, ack38_in_gaa(s(T84), T85, X221))
U15_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, X221)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
ack1_in_gaa(s(T84), s(s(T85)), T71) → U16_gaa(T84, T85, T71, ack38_in_gaa(s(T84), T85, T87))
U16_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U17_gaa(T84, T85, T71, ack38_in_gaa(T84, T87, X222))
U17_gaa(T84, T85, T71, ack38_out_gaa(T84, T87, X222)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
U16_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U18_gaa(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_gaa(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → U19_gaa(T84, T85, T71, ack1_in_gaa(T84, T91, T71))
U19_gaa(T84, T85, T71, ack1_out_gaa(T84, T91, T71)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
U14_gaa(T76, T71, ack1_out_gaa(T76, T77, T71)) → ack1_out_gaa(s(T76), s(0), T71)
U11_gaa(T20, T22, ack1_out_gaa(T20, T24, T22)) → ack1_out_gaa(s(s(T20)), 0, T22)
U11_gga(T20, T22, ack1_out_gaa(T20, T24, T22)) → ack1_out_gga(s(s(T20)), 0, T22)
ack1_in_gga(s(T76), s(0), T71) → U12_gga(T76, T71, ack27_in_ga(T76, X200))
U12_gga(T76, T71, ack27_out_ga(T76, X200)) → ack1_out_gga(s(T76), s(0), T71)
ack1_in_gga(s(T76), s(0), T71) → U13_gga(T76, T71, ack27_in_ga(T76, T77))
U13_gga(T76, T71, ack27_out_ga(T76, T77)) → U14_gga(T76, T71, ack1_in_gaa(T76, T77, T71))
U14_gga(T76, T71, ack1_out_gaa(T76, T77, T71)) → ack1_out_gga(s(T76), s(0), T71)
ack1_in_gga(s(T84), s(s(T85)), T71) → U15_gga(T84, T85, T71, ack38_in_gga(s(T84), T85, X221))
ack38_in_gga(0, T44, s(T44)) → ack38_out_gga(0, T44, s(T44))
ack38_in_gga(s(T49), 0, X133) → U5_gga(T49, X133, ack27_in_ga(T49, X133))
U5_gga(T49, X133, ack27_out_ga(T49, X133)) → ack38_out_gga(s(T49), 0, X133)
ack38_in_gga(s(T54), s(T55), X151) → U6_gga(T54, T55, X151, ack38_in_gga(s(T54), T55, X150))
ack38_in_gga(s(T54), s(T55), X151) → U7_gga(T54, T55, X151, ack38_in_gga(s(T54), T55, T57))
U7_gga(T54, T55, X151, ack38_out_gga(s(T54), T55, T57)) → U8_gga(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
U8_gga(T54, T55, X151, ack38_out_gaa(T54, T57, X151)) → ack38_out_gga(s(T54), s(T55), X151)
U6_gga(T54, T55, X151, ack38_out_gga(s(T54), T55, X150)) → ack38_out_gga(s(T54), s(T55), X151)
U15_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, X221)) → ack1_out_gga(s(T84), s(s(T85)), T71)
ack1_in_gga(s(T84), s(s(T85)), T71) → U16_gga(T84, T85, T71, ack38_in_gga(s(T84), T85, T87))
U16_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → U17_gga(T84, T85, T71, ack38_in_gaa(T84, T87, X222))
U17_gga(T84, T85, T71, ack38_out_gaa(T84, T87, X222)) → ack1_out_gga(s(T84), s(s(T85)), T71)
U16_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → U18_gga(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_gga(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → U19_gga(T84, T85, T71, ack1_in_gaa(T84, T91, T71))
U19_gga(T84, T85, T71, ack1_out_gaa(T84, T91, T71)) → ack1_out_gga(s(T84), s(s(T85)), T71)

(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:

ACK21_IN_GA(T30, X73) → ACK27_IN_GA(T30, X73)
ACK27_IN_GA(s(T34), X97) → ACK21_IN_GA(T34, X96)
ACK27_IN_GA(s(T34), X97) → U3_GA(T34, X97, ack21_in_ga(T34, T36))
U3_GA(T34, X97, ack21_out_ga(T34, T36)) → ACK38_IN_GAA(T34, T36, X97)
ACK38_IN_GAA(s(T49), 0, X133) → ACK27_IN_GA(T49, X133)
ACK38_IN_GAA(s(T54), s(T55), X151) → ACK38_IN_GAA(s(T54), T55, X150)
ACK38_IN_GAA(s(T54), s(T55), X151) → U7_GAA(T54, T55, X151, ack38_in_gaa(s(T54), T55, T57))
U7_GAA(T54, T55, X151, ack38_out_gaa(s(T54), T55, T57)) → ACK38_IN_GAA(T54, T57, X151)

The TRS R consists of the following rules:

ack21_in_ga(T30, X73) → U1_ga(T30, X73, ack27_in_ga(T30, X73))
ack38_in_gaa(s(T49), 0, X133) → U5_gaa(T49, X133, ack27_in_ga(T49, X133))
ack38_in_gaa(s(T54), s(T55), X151) → U6_gaa(T54, T55, X151, ack38_in_gaa(s(T54), T55, X150))
ack38_in_gaa(s(T54), s(T55), X151) → U7_gaa(T54, T55, X151, ack38_in_gaa(s(T54), T55, T57))
U1_ga(T30, X73, ack27_out_ga(T30, X73)) → ack21_out_ga(T30, X73)
U5_gaa(T49, X133, ack27_out_ga(T49, X133)) → ack38_out_gaa(s(T49), 0, X133)
U6_gaa(T54, T55, X151, ack38_out_gaa(s(T54), T55, X150)) → ack38_out_gaa(s(T54), s(T55), X151)
U7_gaa(T54, T55, X151, ack38_out_gaa(s(T54), T55, T57)) → U8_gaa(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
ack27_in_ga(0, s(s(0))) → ack27_out_ga(0, s(s(0)))
ack27_in_ga(s(T34), X97) → U2_ga(T34, X97, ack21_in_ga(T34, X96))
ack27_in_ga(s(T34), X97) → U3_ga(T34, X97, ack21_in_ga(T34, T36))
U8_gaa(T54, T55, X151, ack38_out_gaa(T54, T57, X151)) → ack38_out_gaa(s(T54), s(T55), X151)
U2_ga(T34, X97, ack21_out_ga(T34, X96)) → ack27_out_ga(s(T34), X97)
U3_ga(T34, X97, ack21_out_ga(T34, T36)) → U4_ga(T34, X97, ack38_in_gaa(T34, T36, X97))
ack38_in_gaa(0, T44, s(T44)) → ack38_out_gaa(0, T44, s(T44))
U4_ga(T34, X97, ack38_out_gaa(T34, T36, X97)) → ack27_out_ga(s(T34), X97)

The argument filtering Pi contains the following mapping:
0 = 0
s(x1) = s(x1)
ack21_in_ga(x1, x2) = ack21_in_ga(x1)
U1_ga(x1, x2, x3) = U1_ga(x1, x3)
ack27_in_ga(x1, x2) = ack27_in_ga(x1)
ack27_out_ga(x1, x2) = ack27_out_ga(x1)
U2_ga(x1, x2, x3) = U2_ga(x1, x3)
ack21_out_ga(x1, x2) = ack21_out_ga(x1)
U3_ga(x1, x2, x3) = U3_ga(x1, x3)
U4_ga(x1, x2, x3) = U4_ga(x1, x3)
ack38_in_gaa(x1, x2, x3) = ack38_in_gaa(x1)
ack38_out_gaa(x1, x2, x3) = ack38_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)
ACK21_IN_GA(x1, x2) = ACK21_IN_GA(x1)
ACK27_IN_GA(x1, x2) = ACK27_IN_GA(x1)
U3_GA(x1, x2, x3) = U3_GA(x1, x3)
ACK38_IN_GAA(x1, x2, x3) = ACK38_IN_GAA(x1)
U7_GAA(x1, x2, x3, x4) = U7_GAA(x1, x4)

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

(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:

ACK21_IN_GA(T30) → ACK27_IN_GA(T30)
ACK27_IN_GA(s(T34)) → ACK21_IN_GA(T34)
ACK27_IN_GA(s(T34)) → U3_GA(T34, ack21_in_ga(T34))
U3_GA(T34, ack21_out_ga(T34)) → ACK38_IN_GAA(T34)
ACK38_IN_GAA(s(T49)) → ACK27_IN_GA(T49)
ACK38_IN_GAA(s(T54)) → ACK38_IN_GAA(s(T54))
ACK38_IN_GAA(s(T54)) → U7_GAA(T54, ack38_in_gaa(s(T54)))
U7_GAA(T54, ack38_out_gaa(s(T54))) → ACK38_IN_GAA(T54)

The TRS R consists of the following rules:

ack21_in_ga(T30) → U1_ga(T30, ack27_in_ga(T30))
ack38_in_gaa(s(T49)) → U5_gaa(T49, ack27_in_ga(T49))
ack38_in_gaa(s(T54)) → U6_gaa(T54, ack38_in_gaa(s(T54)))
ack38_in_gaa(s(T54)) → U7_gaa(T54, ack38_in_gaa(s(T54)))
U1_ga(T30, ack27_out_ga(T30)) → ack21_out_ga(T30)
U5_gaa(T49, ack27_out_ga(T49)) → ack38_out_gaa(s(T49))
U6_gaa(T54, ack38_out_gaa(s(T54))) → ack38_out_gaa(s(T54))
U7_gaa(T54, ack38_out_gaa(s(T54))) → U8_gaa(T54, ack38_in_gaa(T54))
ack27_in_ga(0) → ack27_out_ga(0)
ack27_in_ga(s(T34)) → U2_ga(T34, ack21_in_ga(T34))
ack27_in_ga(s(T34)) → U3_ga(T34, ack21_in_ga(T34))
U8_gaa(T54, ack38_out_gaa(T54)) → ack38_out_gaa(s(T54))
U2_ga(T34, ack21_out_ga(T34)) → ack27_out_ga(s(T34))
U3_ga(T34, ack21_out_ga(T34)) → U4_ga(T34, ack38_in_gaa(T34))
ack38_in_gaa(0) → ack38_out_gaa(0)
U4_ga(T34, ack38_out_gaa(T34)) → ack27_out_ga(s(T34))

The set Q consists of the following terms:

ack21_in_ga(x0)
ack38_in_gaa(x0)
U1_ga(x0, x1)
U5_gaa(x0, x1)
U6_gaa(x0, x1)
U7_gaa(x0, x1)
ack27_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.

ACK27_IN_GA(s(T34)) → ACK21_IN_GA(T34)
U3_GA(T34, ack21_out_ga(T34)) → ACK38_IN_GAA(T34)
ACK38_IN_GAA(s(T49)) → ACK27_IN_GA(T49)
U7_GAA(T54, ack38_out_gaa(s(T54))) → ACK38_IN_GAA(T54)

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

POL(0) = 1
POL(ACK21_IN_GA(x₁)) = x₁
POL(ACK27_IN_GA(x₁)) = x₁
POL(ACK38_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(ack21_in_ga(x₁)) = 0
POL(ack21_out_ga(x₁)) = 0
POL(ack27_in_ga(x₁)) = 1 + x₁
POL(ack27_out_ga(x₁)) = 1
POL(ack38_in_gaa(x₁)) = 1 + x₁
POL(ack38_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:

ACK21_IN_GA(T30) → ACK27_IN_GA(T30)
ACK27_IN_GA(s(T34)) → U3_GA(T34, ack21_in_ga(T34))
ACK38_IN_GAA(s(T54)) → ACK38_IN_GAA(s(T54))
ACK38_IN_GAA(s(T54)) → U7_GAA(T54, ack38_in_gaa(s(T54)))

The TRS R consists of the following rules:

ack21_in_ga(T30) → U1_ga(T30, ack27_in_ga(T30))
ack38_in_gaa(s(T49)) → U5_gaa(T49, ack27_in_ga(T49))
ack38_in_gaa(s(T54)) → U6_gaa(T54, ack38_in_gaa(s(T54)))
ack38_in_gaa(s(T54)) → U7_gaa(T54, ack38_in_gaa(s(T54)))
U1_ga(T30, ack27_out_ga(T30)) → ack21_out_ga(T30)
U5_gaa(T49, ack27_out_ga(T49)) → ack38_out_gaa(s(T49))
U6_gaa(T54, ack38_out_gaa(s(T54))) → ack38_out_gaa(s(T54))
U7_gaa(T54, ack38_out_gaa(s(T54))) → U8_gaa(T54, ack38_in_gaa(T54))
ack27_in_ga(0) → ack27_out_ga(0)
ack27_in_ga(s(T34)) → U2_ga(T34, ack21_in_ga(T34))
ack27_in_ga(s(T34)) → U3_ga(T34, ack21_in_ga(T34))
U8_gaa(T54, ack38_out_gaa(T54)) → ack38_out_gaa(s(T54))
U2_ga(T34, ack21_out_ga(T34)) → ack27_out_ga(s(T34))
U3_ga(T34, ack21_out_ga(T34)) → U4_ga(T34, ack38_in_gaa(T34))
ack38_in_gaa(0) → ack38_out_gaa(0)
U4_ga(T34, ack38_out_gaa(T34)) → ack27_out_ga(s(T34))

The set Q consists of the following terms:

ack21_in_ga(x0)
ack38_in_gaa(x0)
U1_ga(x0, x1)
U5_gaa(x0, x1)
U6_gaa(x0, x1)
U7_gaa(x0, x1)
ack27_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:

ACK38_IN_GAA(s(T54)) → ACK38_IN_GAA(s(T54))

The TRS R consists of the following rules:

ack21_in_ga(T30) → U1_ga(T30, ack27_in_ga(T30))
ack38_in_gaa(s(T49)) → U5_gaa(T49, ack27_in_ga(T49))
ack38_in_gaa(s(T54)) → U6_gaa(T54, ack38_in_gaa(s(T54)))
ack38_in_gaa(s(T54)) → U7_gaa(T54, ack38_in_gaa(s(T54)))
U1_ga(T30, ack27_out_ga(T30)) → ack21_out_ga(T30)
U5_gaa(T49, ack27_out_ga(T49)) → ack38_out_gaa(s(T49))
U6_gaa(T54, ack38_out_gaa(s(T54))) → ack38_out_gaa(s(T54))
U7_gaa(T54, ack38_out_gaa(s(T54))) → U8_gaa(T54, ack38_in_gaa(T54))
ack27_in_ga(0) → ack27_out_ga(0)
ack27_in_ga(s(T34)) → U2_ga(T34, ack21_in_ga(T34))
ack27_in_ga(s(T34)) → U3_ga(T34, ack21_in_ga(T34))
U8_gaa(T54, ack38_out_gaa(T54)) → ack38_out_gaa(s(T54))
U2_ga(T34, ack21_out_ga(T34)) → ack27_out_ga(s(T34))
U3_ga(T34, ack21_out_ga(T34)) → U4_ga(T34, ack38_in_gaa(T34))
ack38_in_gaa(0) → ack38_out_gaa(0)
U4_ga(T34, ack38_out_gaa(T34)) → ack27_out_ga(s(T34))

The set Q consists of the following terms:

ack21_in_ga(x0)
ack38_in_gaa(x0)
U1_ga(x0, x1)
U5_gaa(x0, x1)
U6_gaa(x0, x1)
U7_gaa(x0, x1)
ack27_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:

ACK38_IN_GAA(s(T54)) → ACK38_IN_GAA(s(T54))

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

ack21_in_ga(x0)
ack38_in_gaa(x0)
U1_ga(x0, x1)
U5_gaa(x0, x1)
U6_gaa(x0, x1)
U7_gaa(x0, x1)
ack27_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].

ack21_in_ga(x0)
ack38_in_gaa(x0)
U1_ga(x0, x1)
U5_gaa(x0, x1)
U6_gaa(x0, x1)
U7_gaa(x0, x1)
ack27_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:

ACK38_IN_GAA(s(T54)) → ACK38_IN_GAA(s(T54))

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 = ACK38_IN_GAA(s(T54)) evaluates to t =ACK38_IN_GAA(s(T54))

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 ACK38_IN_GAA(s(T54)) to ACK38_IN_GAA(s(T54)).

(23) NO

(24) Obligation:

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

ACK38_IN_GGA(s(T54), s(T55), X151) → ACK38_IN_GGA(s(T54), T55, X150)

The TRS R consists of the following rules:

ack1_in_gga(0, T5, s(T5)) → ack1_out_gga(0, T5, s(T5))
ack1_in_gga(s(0), 0, s(s(0))) → ack1_out_gga(s(0), 0, s(s(0)))
ack1_in_gga(s(s(T20)), 0, T22) → U9_gga(T20, T22, ack21_in_ga(T20, X40))
ack21_in_ga(T30, X73) → U1_ga(T30, X73, ack27_in_ga(T30, X73))
ack27_in_ga(0, s(s(0))) → ack27_out_ga(0, s(s(0)))
ack27_in_ga(s(T34), X97) → U2_ga(T34, X97, ack21_in_ga(T34, X96))
U2_ga(T34, X97, ack21_out_ga(T34, X96)) → ack27_out_ga(s(T34), X97)
ack27_in_ga(s(T34), X97) → U3_ga(T34, X97, ack21_in_ga(T34, T36))
U3_ga(T34, X97, ack21_out_ga(T34, T36)) → U4_ga(T34, X97, ack38_in_gaa(T34, T36, X97))
ack38_in_gaa(0, T44, s(T44)) → ack38_out_gaa(0, T44, s(T44))
ack38_in_gaa(s(T49), 0, X133) → U5_gaa(T49, X133, ack27_in_ga(T49, X133))
U5_gaa(T49, X133, ack27_out_ga(T49, X133)) → ack38_out_gaa(s(T49), 0, X133)
ack38_in_gaa(s(T54), s(T55), X151) → U6_gaa(T54, T55, X151, ack38_in_gaa(s(T54), T55, X150))
ack38_in_gaa(s(T54), s(T55), X151) → U7_gaa(T54, T55, X151, ack38_in_gaa(s(T54), T55, T57))
U7_gaa(T54, T55, X151, ack38_out_gaa(s(T54), T55, T57)) → U8_gaa(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
U8_gaa(T54, T55, X151, ack38_out_gaa(T54, T57, X151)) → ack38_out_gaa(s(T54), s(T55), X151)
U6_gaa(T54, T55, X151, ack38_out_gaa(s(T54), T55, X150)) → ack38_out_gaa(s(T54), s(T55), X151)
U4_ga(T34, X97, ack38_out_gaa(T34, T36, X97)) → ack27_out_ga(s(T34), X97)
U1_ga(T30, X73, ack27_out_ga(T30, X73)) → ack21_out_ga(T30, X73)
U9_gga(T20, T22, ack21_out_ga(T20, X40)) → ack1_out_gga(s(s(T20)), 0, T22)
ack1_in_gga(s(s(T20)), 0, T22) → U10_gga(T20, T22, ack21_in_ga(T20, T24))
U10_gga(T20, T22, ack21_out_ga(T20, T24)) → U11_gga(T20, T22, ack1_in_gaa(T20, T24, T22))
ack1_in_gaa(0, T5, s(T5)) → ack1_out_gaa(0, T5, s(T5))
ack1_in_gaa(s(0), 0, s(s(0))) → ack1_out_gaa(s(0), 0, s(s(0)))
ack1_in_gaa(s(s(T20)), 0, T22) → U9_gaa(T20, T22, ack21_in_ga(T20, X40))
U9_gaa(T20, T22, ack21_out_ga(T20, X40)) → ack1_out_gaa(s(s(T20)), 0, T22)
ack1_in_gaa(s(s(T20)), 0, T22) → U10_gaa(T20, T22, ack21_in_ga(T20, T24))
U10_gaa(T20, T22, ack21_out_ga(T20, T24)) → U11_gaa(T20, T22, ack1_in_gaa(T20, T24, T22))
ack1_in_gaa(s(T76), s(0), T71) → U12_gaa(T76, T71, ack27_in_ga(T76, X200))
U12_gaa(T76, T71, ack27_out_ga(T76, X200)) → ack1_out_gaa(s(T76), s(0), T71)
ack1_in_gaa(s(T76), s(0), T71) → U13_gaa(T76, T71, ack27_in_ga(T76, T77))
U13_gaa(T76, T71, ack27_out_ga(T76, T77)) → U14_gaa(T76, T71, ack1_in_gaa(T76, T77, T71))
ack1_in_gaa(s(T84), s(s(T85)), T71) → U15_gaa(T84, T85, T71, ack38_in_gaa(s(T84), T85, X221))
U15_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, X221)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
ack1_in_gaa(s(T84), s(s(T85)), T71) → U16_gaa(T84, T85, T71, ack38_in_gaa(s(T84), T85, T87))
U16_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U17_gaa(T84, T85, T71, ack38_in_gaa(T84, T87, X222))
U17_gaa(T84, T85, T71, ack38_out_gaa(T84, T87, X222)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
U16_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U18_gaa(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_gaa(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → U19_gaa(T84, T85, T71, ack1_in_gaa(T84, T91, T71))
U19_gaa(T84, T85, T71, ack1_out_gaa(T84, T91, T71)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
U14_gaa(T76, T71, ack1_out_gaa(T76, T77, T71)) → ack1_out_gaa(s(T76), s(0), T71)
U11_gaa(T20, T22, ack1_out_gaa(T20, T24, T22)) → ack1_out_gaa(s(s(T20)), 0, T22)
U11_gga(T20, T22, ack1_out_gaa(T20, T24, T22)) → ack1_out_gga(s(s(T20)), 0, T22)
ack1_in_gga(s(T76), s(0), T71) → U12_gga(T76, T71, ack27_in_ga(T76, X200))
U12_gga(T76, T71, ack27_out_ga(T76, X200)) → ack1_out_gga(s(T76), s(0), T71)
ack1_in_gga(s(T76), s(0), T71) → U13_gga(T76, T71, ack27_in_ga(T76, T77))
U13_gga(T76, T71, ack27_out_ga(T76, T77)) → U14_gga(T76, T71, ack1_in_gaa(T76, T77, T71))
U14_gga(T76, T71, ack1_out_gaa(T76, T77, T71)) → ack1_out_gga(s(T76), s(0), T71)
ack1_in_gga(s(T84), s(s(T85)), T71) → U15_gga(T84, T85, T71, ack38_in_gga(s(T84), T85, X221))
ack38_in_gga(0, T44, s(T44)) → ack38_out_gga(0, T44, s(T44))
ack38_in_gga(s(T49), 0, X133) → U5_gga(T49, X133, ack27_in_ga(T49, X133))
U5_gga(T49, X133, ack27_out_ga(T49, X133)) → ack38_out_gga(s(T49), 0, X133)
ack38_in_gga(s(T54), s(T55), X151) → U6_gga(T54, T55, X151, ack38_in_gga(s(T54), T55, X150))
ack38_in_gga(s(T54), s(T55), X151) → U7_gga(T54, T55, X151, ack38_in_gga(s(T54), T55, T57))
U7_gga(T54, T55, X151, ack38_out_gga(s(T54), T55, T57)) → U8_gga(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
U8_gga(T54, T55, X151, ack38_out_gaa(T54, T57, X151)) → ack38_out_gga(s(T54), s(T55), X151)
U6_gga(T54, T55, X151, ack38_out_gga(s(T54), T55, X150)) → ack38_out_gga(s(T54), s(T55), X151)
U15_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, X221)) → ack1_out_gga(s(T84), s(s(T85)), T71)
ack1_in_gga(s(T84), s(s(T85)), T71) → U16_gga(T84, T85, T71, ack38_in_gga(s(T84), T85, T87))
U16_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → U17_gga(T84, T85, T71, ack38_in_gaa(T84, T87, X222))
U17_gga(T84, T85, T71, ack38_out_gaa(T84, T87, X222)) → ack1_out_gga(s(T84), s(s(T85)), T71)
U16_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → U18_gga(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_gga(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → U19_gga(T84, T85, T71, ack1_in_gaa(T84, T91, T71))
U19_gga(T84, T85, T71, ack1_out_gaa(T84, T91, T71)) → ack1_out_gga(s(T84), s(s(T85)), T71)

(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:

ACK38_IN_GGA(s(T54), s(T55), X151) → ACK38_IN_GGA(s(T54), T55, X150)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
ACK38_IN_GGA(x1, x2, x3) = ACK38_IN_GGA(x1, x2)

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:

ACK38_IN_GGA(s(T54), s(T55)) → ACK38_IN_GGA(s(T54), T55)

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

(29) 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:

ACK38_IN_GGA(s(T54), s(T55)) → ACK38_IN_GGA(s(T54), T55)
The graph contains the following edges 1 >= 1, 2 > 2

(30) YES

(31) Obligation:

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

ACK1_IN_GAA(s(s(T20)), 0, T22) → U10_GAA(T20, T22, ack21_in_ga(T20, T24))
U10_GAA(T20, T22, ack21_out_ga(T20, T24)) → ACK1_IN_GAA(T20, T24, T22)
ACK1_IN_GAA(s(T76), s(0), T71) → U13_GAA(T76, T71, ack27_in_ga(T76, T77))
U13_GAA(T76, T71, ack27_out_ga(T76, T77)) → ACK1_IN_GAA(T76, T77, T71)
ACK1_IN_GAA(s(T84), s(s(T85)), T71) → U16_GAA(T84, T85, T71, ack38_in_gaa(s(T84), T85, T87))
U16_GAA(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U18_GAA(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_GAA(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → ACK1_IN_GAA(T84, T91, T71)

The TRS R consists of the following rules:

ack1_in_gga(0, T5, s(T5)) → ack1_out_gga(0, T5, s(T5))
ack1_in_gga(s(0), 0, s(s(0))) → ack1_out_gga(s(0), 0, s(s(0)))
ack1_in_gga(s(s(T20)), 0, T22) → U9_gga(T20, T22, ack21_in_ga(T20, X40))
ack21_in_ga(T30, X73) → U1_ga(T30, X73, ack27_in_ga(T30, X73))
ack27_in_ga(0, s(s(0))) → ack27_out_ga(0, s(s(0)))
ack27_in_ga(s(T34), X97) → U2_ga(T34, X97, ack21_in_ga(T34, X96))
U2_ga(T34, X97, ack21_out_ga(T34, X96)) → ack27_out_ga(s(T34), X97)
ack27_in_ga(s(T34), X97) → U3_ga(T34, X97, ack21_in_ga(T34, T36))
U3_ga(T34, X97, ack21_out_ga(T34, T36)) → U4_ga(T34, X97, ack38_in_gaa(T34, T36, X97))
ack38_in_gaa(0, T44, s(T44)) → ack38_out_gaa(0, T44, s(T44))
ack38_in_gaa(s(T49), 0, X133) → U5_gaa(T49, X133, ack27_in_ga(T49, X133))
U5_gaa(T49, X133, ack27_out_ga(T49, X133)) → ack38_out_gaa(s(T49), 0, X133)
ack38_in_gaa(s(T54), s(T55), X151) → U6_gaa(T54, T55, X151, ack38_in_gaa(s(T54), T55, X150))
ack38_in_gaa(s(T54), s(T55), X151) → U7_gaa(T54, T55, X151, ack38_in_gaa(s(T54), T55, T57))
U7_gaa(T54, T55, X151, ack38_out_gaa(s(T54), T55, T57)) → U8_gaa(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
U8_gaa(T54, T55, X151, ack38_out_gaa(T54, T57, X151)) → ack38_out_gaa(s(T54), s(T55), X151)
U6_gaa(T54, T55, X151, ack38_out_gaa(s(T54), T55, X150)) → ack38_out_gaa(s(T54), s(T55), X151)
U4_ga(T34, X97, ack38_out_gaa(T34, T36, X97)) → ack27_out_ga(s(T34), X97)
U1_ga(T30, X73, ack27_out_ga(T30, X73)) → ack21_out_ga(T30, X73)
U9_gga(T20, T22, ack21_out_ga(T20, X40)) → ack1_out_gga(s(s(T20)), 0, T22)
ack1_in_gga(s(s(T20)), 0, T22) → U10_gga(T20, T22, ack21_in_ga(T20, T24))
U10_gga(T20, T22, ack21_out_ga(T20, T24)) → U11_gga(T20, T22, ack1_in_gaa(T20, T24, T22))
ack1_in_gaa(0, T5, s(T5)) → ack1_out_gaa(0, T5, s(T5))
ack1_in_gaa(s(0), 0, s(s(0))) → ack1_out_gaa(s(0), 0, s(s(0)))
ack1_in_gaa(s(s(T20)), 0, T22) → U9_gaa(T20, T22, ack21_in_ga(T20, X40))
U9_gaa(T20, T22, ack21_out_ga(T20, X40)) → ack1_out_gaa(s(s(T20)), 0, T22)
ack1_in_gaa(s(s(T20)), 0, T22) → U10_gaa(T20, T22, ack21_in_ga(T20, T24))
U10_gaa(T20, T22, ack21_out_ga(T20, T24)) → U11_gaa(T20, T22, ack1_in_gaa(T20, T24, T22))
ack1_in_gaa(s(T76), s(0), T71) → U12_gaa(T76, T71, ack27_in_ga(T76, X200))
U12_gaa(T76, T71, ack27_out_ga(T76, X200)) → ack1_out_gaa(s(T76), s(0), T71)
ack1_in_gaa(s(T76), s(0), T71) → U13_gaa(T76, T71, ack27_in_ga(T76, T77))
U13_gaa(T76, T71, ack27_out_ga(T76, T77)) → U14_gaa(T76, T71, ack1_in_gaa(T76, T77, T71))
ack1_in_gaa(s(T84), s(s(T85)), T71) → U15_gaa(T84, T85, T71, ack38_in_gaa(s(T84), T85, X221))
U15_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, X221)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
ack1_in_gaa(s(T84), s(s(T85)), T71) → U16_gaa(T84, T85, T71, ack38_in_gaa(s(T84), T85, T87))
U16_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U17_gaa(T84, T85, T71, ack38_in_gaa(T84, T87, X222))
U17_gaa(T84, T85, T71, ack38_out_gaa(T84, T87, X222)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
U16_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U18_gaa(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_gaa(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → U19_gaa(T84, T85, T71, ack1_in_gaa(T84, T91, T71))
U19_gaa(T84, T85, T71, ack1_out_gaa(T84, T91, T71)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
U14_gaa(T76, T71, ack1_out_gaa(T76, T77, T71)) → ack1_out_gaa(s(T76), s(0), T71)
U11_gaa(T20, T22, ack1_out_gaa(T20, T24, T22)) → ack1_out_gaa(s(s(T20)), 0, T22)
U11_gga(T20, T22, ack1_out_gaa(T20, T24, T22)) → ack1_out_gga(s(s(T20)), 0, T22)
ack1_in_gga(s(T76), s(0), T71) → U12_gga(T76, T71, ack27_in_ga(T76, X200))
U12_gga(T76, T71, ack27_out_ga(T76, X200)) → ack1_out_gga(s(T76), s(0), T71)
ack1_in_gga(s(T76), s(0), T71) → U13_gga(T76, T71, ack27_in_ga(T76, T77))
U13_gga(T76, T71, ack27_out_ga(T76, T77)) → U14_gga(T76, T71, ack1_in_gaa(T76, T77, T71))
U14_gga(T76, T71, ack1_out_gaa(T76, T77, T71)) → ack1_out_gga(s(T76), s(0), T71)
ack1_in_gga(s(T84), s(s(T85)), T71) → U15_gga(T84, T85, T71, ack38_in_gga(s(T84), T85, X221))
ack38_in_gga(0, T44, s(T44)) → ack38_out_gga(0, T44, s(T44))
ack38_in_gga(s(T49), 0, X133) → U5_gga(T49, X133, ack27_in_ga(T49, X133))
U5_gga(T49, X133, ack27_out_ga(T49, X133)) → ack38_out_gga(s(T49), 0, X133)
ack38_in_gga(s(T54), s(T55), X151) → U6_gga(T54, T55, X151, ack38_in_gga(s(T54), T55, X150))
ack38_in_gga(s(T54), s(T55), X151) → U7_gga(T54, T55, X151, ack38_in_gga(s(T54), T55, T57))
U7_gga(T54, T55, X151, ack38_out_gga(s(T54), T55, T57)) → U8_gga(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
U8_gga(T54, T55, X151, ack38_out_gaa(T54, T57, X151)) → ack38_out_gga(s(T54), s(T55), X151)
U6_gga(T54, T55, X151, ack38_out_gga(s(T54), T55, X150)) → ack38_out_gga(s(T54), s(T55), X151)
U15_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, X221)) → ack1_out_gga(s(T84), s(s(T85)), T71)
ack1_in_gga(s(T84), s(s(T85)), T71) → U16_gga(T84, T85, T71, ack38_in_gga(s(T84), T85, T87))
U16_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → U17_gga(T84, T85, T71, ack38_in_gaa(T84, T87, X222))
U17_gga(T84, T85, T71, ack38_out_gaa(T84, T87, X222)) → ack1_out_gga(s(T84), s(s(T85)), T71)
U16_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → U18_gga(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_gga(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → U19_gga(T84, T85, T71, ack1_in_gaa(T84, T91, T71))
U19_gga(T84, T85, T71, ack1_out_gaa(T84, T91, T71)) → ack1_out_gga(s(T84), s(s(T85)), T71)

(32) UsableRulesProof (EQUIVALENT transformation)

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

(33) Obligation:

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

ACK1_IN_GAA(s(s(T20)), 0, T22) → U10_GAA(T20, T22, ack21_in_ga(T20, T24))
U10_GAA(T20, T22, ack21_out_ga(T20, T24)) → ACK1_IN_GAA(T20, T24, T22)
ACK1_IN_GAA(s(T76), s(0), T71) → U13_GAA(T76, T71, ack27_in_ga(T76, T77))
U13_GAA(T76, T71, ack27_out_ga(T76, T77)) → ACK1_IN_GAA(T76, T77, T71)
ACK1_IN_GAA(s(T84), s(s(T85)), T71) → U16_GAA(T84, T85, T71, ack38_in_gaa(s(T84), T85, T87))
U16_GAA(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U18_GAA(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_GAA(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → ACK1_IN_GAA(T84, T91, T71)

The TRS R consists of the following rules:

ack21_in_ga(T30, X73) → U1_ga(T30, X73, ack27_in_ga(T30, X73))
ack27_in_ga(0, s(s(0))) → ack27_out_ga(0, s(s(0)))
ack27_in_ga(s(T34), X97) → U2_ga(T34, X97, ack21_in_ga(T34, X96))
ack27_in_ga(s(T34), X97) → U3_ga(T34, X97, ack21_in_ga(T34, T36))
ack38_in_gaa(s(T49), 0, X133) → U5_gaa(T49, X133, ack27_in_ga(T49, X133))
ack38_in_gaa(s(T54), s(T55), X151) → U6_gaa(T54, T55, X151, ack38_in_gaa(s(T54), T55, X150))
ack38_in_gaa(s(T54), s(T55), X151) → U7_gaa(T54, T55, X151, ack38_in_gaa(s(T54), T55, T57))
ack38_in_gaa(0, T44, s(T44)) → ack38_out_gaa(0, T44, s(T44))
U1_ga(T30, X73, ack27_out_ga(T30, X73)) → ack21_out_ga(T30, X73)
U2_ga(T34, X97, ack21_out_ga(T34, X96)) → ack27_out_ga(s(T34), X97)
U3_ga(T34, X97, ack21_out_ga(T34, T36)) → U4_ga(T34, X97, ack38_in_gaa(T34, T36, X97))
U5_gaa(T49, X133, ack27_out_ga(T49, X133)) → ack38_out_gaa(s(T49), 0, X133)
U6_gaa(T54, T55, X151, ack38_out_gaa(s(T54), T55, X150)) → ack38_out_gaa(s(T54), s(T55), X151)
U7_gaa(T54, T55, X151, ack38_out_gaa(s(T54), T55, T57)) → U8_gaa(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
U4_ga(T34, X97, ack38_out_gaa(T34, T36, X97)) → ack27_out_ga(s(T34), X97)
U8_gaa(T54, T55, X151, ack38_out_gaa(T54, T57, X151)) → ack38_out_gaa(s(T54), s(T55), X151)

The argument filtering Pi contains the following mapping:
0 = 0
s(x1) = s(x1)
ack21_in_ga(x1, x2) = ack21_in_ga(x1)
U1_ga(x1, x2, x3) = U1_ga(x1, x3)
ack27_in_ga(x1, x2) = ack27_in_ga(x1)
ack27_out_ga(x1, x2) = ack27_out_ga(x1)
U2_ga(x1, x2, x3) = U2_ga(x1, x3)
ack21_out_ga(x1, x2) = ack21_out_ga(x1)
U3_ga(x1, x2, x3) = U3_ga(x1, x3)
U4_ga(x1, x2, x3) = U4_ga(x1, x3)
ack38_in_gaa(x1, x2, x3) = ack38_in_gaa(x1)
ack38_out_gaa(x1, x2, x3) = ack38_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)
ACK1_IN_GAA(x1, x2, x3) = ACK1_IN_GAA(x1)
U10_GAA(x1, x2, x3) = U10_GAA(x1, x3)
U13_GAA(x1, x2, x3) = U13_GAA(x1, x3)
U16_GAA(x1, x2, x3, x4) = U16_GAA(x1, x4)
U18_GAA(x1, x2, x3, x4) = U18_GAA(x1, x4)

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

(34) PiDPToQDPProof (SOUND transformation)

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

(35) Obligation:

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

ACK1_IN_GAA(s(s(T20))) → U10_GAA(T20, ack21_in_ga(T20))
U10_GAA(T20, ack21_out_ga(T20)) → ACK1_IN_GAA(T20)
ACK1_IN_GAA(s(T76)) → U13_GAA(T76, ack27_in_ga(T76))
U13_GAA(T76, ack27_out_ga(T76)) → ACK1_IN_GAA(T76)
ACK1_IN_GAA(s(T84)) → U16_GAA(T84, ack38_in_gaa(s(T84)))
U16_GAA(T84, ack38_out_gaa(s(T84))) → U18_GAA(T84, ack38_in_gaa(T84))
U18_GAA(T84, ack38_out_gaa(T84)) → ACK1_IN_GAA(T84)

The TRS R consists of the following rules:

ack21_in_ga(T30) → U1_ga(T30, ack27_in_ga(T30))
ack27_in_ga(0) → ack27_out_ga(0)
ack27_in_ga(s(T34)) → U2_ga(T34, ack21_in_ga(T34))
ack27_in_ga(s(T34)) → U3_ga(T34, ack21_in_ga(T34))
ack38_in_gaa(s(T49)) → U5_gaa(T49, ack27_in_ga(T49))
ack38_in_gaa(s(T54)) → U6_gaa(T54, ack38_in_gaa(s(T54)))
ack38_in_gaa(s(T54)) → U7_gaa(T54, ack38_in_gaa(s(T54)))
ack38_in_gaa(0) → ack38_out_gaa(0)
U1_ga(T30, ack27_out_ga(T30)) → ack21_out_ga(T30)
U2_ga(T34, ack21_out_ga(T34)) → ack27_out_ga(s(T34))
U3_ga(T34, ack21_out_ga(T34)) → U4_ga(T34, ack38_in_gaa(T34))
U5_gaa(T49, ack27_out_ga(T49)) → ack38_out_gaa(s(T49))
U6_gaa(T54, ack38_out_gaa(s(T54))) → ack38_out_gaa(s(T54))
U7_gaa(T54, ack38_out_gaa(s(T54))) → U8_gaa(T54, ack38_in_gaa(T54))
U4_ga(T34, ack38_out_gaa(T34)) → ack27_out_ga(s(T34))
U8_gaa(T54, ack38_out_gaa(T54)) → ack38_out_gaa(s(T54))

The set Q consists of the following terms:

ack21_in_ga(x0)
ack27_in_ga(x0)
ack38_in_gaa(x0)
U1_ga(x0, x1)
U2_ga(x0, x1)
U3_ga(x0, x1)
U5_gaa(x0, x1)
U6_gaa(x0, x1)
U7_gaa(x0, x1)
U4_ga(x0, x1)
U8_gaa(x0, x1)

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

(36) 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:

U10_GAA(T20, ack21_out_ga(T20)) → ACK1_IN_GAA(T20)
The graph contains the following edges 1 >= 1, 2 > 1
ACK1_IN_GAA(s(s(T20))) → U10_GAA(T20, ack21_in_ga(T20))
The graph contains the following edges 1 > 1
U16_GAA(T84, ack38_out_gaa(s(T84))) → U18_GAA(T84, ack38_in_gaa(T84))
The graph contains the following edges 1 >= 1, 2 > 1
U13_GAA(T76, ack27_out_ga(T76)) → ACK1_IN_GAA(T76)
The graph contains the following edges 1 >= 1, 2 > 1
U18_GAA(T84, ack38_out_gaa(T84)) → ACK1_IN_GAA(T84)
The graph contains the following edges 1 >= 1, 2 > 1
ACK1_IN_GAA(s(T76)) → U13_GAA(T76, ack27_in_ga(T76))
The graph contains the following edges 1 > 1
ACK1_IN_GAA(s(T84)) → U16_GAA(T84, ack38_in_gaa(s(T84)))
The graph contains the following edges 1 > 1

(37) YES

(38) PrologToPiTRSProof (SOUND transformation)

ack1_in_gga(0, T5, s(T5)) → ack1_out_gga(0, T5, s(T5))
ack1_in_gga(s(0), 0, s(s(0))) → ack1_out_gga(s(0), 0, s(s(0)))
ack1_in_gga(s(s(T20)), 0, T22) → U9_gga(T20, T22, ack21_in_ga(T20, X40))
ack21_in_ga(T30, X73) → U1_ga(T30, X73, ack27_in_ga(T30, X73))
ack27_in_ga(0, s(s(0))) → ack27_out_ga(0, s(s(0)))
ack27_in_ga(s(T34), X97) → U2_ga(T34, X97, ack21_in_ga(T34, X96))
U2_ga(T34, X97, ack21_out_ga(T34, X96)) → ack27_out_ga(s(T34), X97)
ack27_in_ga(s(T34), X97) → U3_ga(T34, X97, ack21_in_ga(T34, T36))
U3_ga(T34, X97, ack21_out_ga(T34, T36)) → U4_ga(T34, X97, ack38_in_gaa(T34, T36, X97))
ack38_in_gaa(0, T44, s(T44)) → ack38_out_gaa(0, T44, s(T44))
ack38_in_gaa(s(T49), 0, X133) → U5_gaa(T49, X133, ack27_in_ga(T49, X133))
U5_gaa(T49, X133, ack27_out_ga(T49, X133)) → ack38_out_gaa(s(T49), 0, X133)
ack38_in_gaa(s(T54), s(T55), X151) → U6_gaa(T54, T55, X151, ack38_in_gaa(s(T54), T55, X150))
ack38_in_gaa(s(T54), s(T55), X151) → U7_gaa(T54, T55, X151, ack38_in_gaa(s(T54), T55, T57))
U7_gaa(T54, T55, X151, ack38_out_gaa(s(T54), T55, T57)) → U8_gaa(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
U8_gaa(T54, T55, X151, ack38_out_gaa(T54, T57, X151)) → ack38_out_gaa(s(T54), s(T55), X151)
U6_gaa(T54, T55, X151, ack38_out_gaa(s(T54), T55, X150)) → ack38_out_gaa(s(T54), s(T55), X151)
U4_ga(T34, X97, ack38_out_gaa(T34, T36, X97)) → ack27_out_ga(s(T34), X97)
U1_ga(T30, X73, ack27_out_ga(T30, X73)) → ack21_out_ga(T30, X73)
U9_gga(T20, T22, ack21_out_ga(T20, X40)) → ack1_out_gga(s(s(T20)), 0, T22)
ack1_in_gga(s(s(T20)), 0, T22) → U10_gga(T20, T22, ack21_in_ga(T20, T24))
U10_gga(T20, T22, ack21_out_ga(T20, T24)) → U11_gga(T20, T22, ack1_in_gaa(T20, T24, T22))
ack1_in_gaa(0, T5, s(T5)) → ack1_out_gaa(0, T5, s(T5))
ack1_in_gaa(s(0), 0, s(s(0))) → ack1_out_gaa(s(0), 0, s(s(0)))
ack1_in_gaa(s(s(T20)), 0, T22) → U9_gaa(T20, T22, ack21_in_ga(T20, X40))
U9_gaa(T20, T22, ack21_out_ga(T20, X40)) → ack1_out_gaa(s(s(T20)), 0, T22)
ack1_in_gaa(s(s(T20)), 0, T22) → U10_gaa(T20, T22, ack21_in_ga(T20, T24))
U10_gaa(T20, T22, ack21_out_ga(T20, T24)) → U11_gaa(T20, T22, ack1_in_gaa(T20, T24, T22))
ack1_in_gaa(s(T76), s(0), T71) → U12_gaa(T76, T71, ack27_in_ga(T76, X200))
U12_gaa(T76, T71, ack27_out_ga(T76, X200)) → ack1_out_gaa(s(T76), s(0), T71)
ack1_in_gaa(s(T76), s(0), T71) → U13_gaa(T76, T71, ack27_in_ga(T76, T77))
U13_gaa(T76, T71, ack27_out_ga(T76, T77)) → U14_gaa(T76, T71, ack1_in_gaa(T76, T77, T71))
ack1_in_gaa(s(T84), s(s(T85)), T71) → U15_gaa(T84, T85, T71, ack38_in_gaa(s(T84), T85, X221))
U15_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, X221)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
ack1_in_gaa(s(T84), s(s(T85)), T71) → U16_gaa(T84, T85, T71, ack38_in_gaa(s(T84), T85, T87))
U16_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U17_gaa(T84, T85, T71, ack38_in_gaa(T84, T87, X222))
U17_gaa(T84, T85, T71, ack38_out_gaa(T84, T87, X222)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
U16_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U18_gaa(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_gaa(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → U19_gaa(T84, T85, T71, ack1_in_gaa(T84, T91, T71))
U19_gaa(T84, T85, T71, ack1_out_gaa(T84, T91, T71)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
U14_gaa(T76, T71, ack1_out_gaa(T76, T77, T71)) → ack1_out_gaa(s(T76), s(0), T71)
U11_gaa(T20, T22, ack1_out_gaa(T20, T24, T22)) → ack1_out_gaa(s(s(T20)), 0, T22)
U11_gga(T20, T22, ack1_out_gaa(T20, T24, T22)) → ack1_out_gga(s(s(T20)), 0, T22)
ack1_in_gga(s(T76), s(0), T71) → U12_gga(T76, T71, ack27_in_ga(T76, X200))
U12_gga(T76, T71, ack27_out_ga(T76, X200)) → ack1_out_gga(s(T76), s(0), T71)
ack1_in_gga(s(T76), s(0), T71) → U13_gga(T76, T71, ack27_in_ga(T76, T77))
U13_gga(T76, T71, ack27_out_ga(T76, T77)) → U14_gga(T76, T71, ack1_in_gaa(T76, T77, T71))
U14_gga(T76, T71, ack1_out_gaa(T76, T77, T71)) → ack1_out_gga(s(T76), s(0), T71)
ack1_in_gga(s(T84), s(s(T85)), T71) → U15_gga(T84, T85, T71, ack38_in_gga(s(T84), T85, X221))
ack38_in_gga(0, T44, s(T44)) → ack38_out_gga(0, T44, s(T44))
ack38_in_gga(s(T49), 0, X133) → U5_gga(T49, X133, ack27_in_ga(T49, X133))
U5_gga(T49, X133, ack27_out_ga(T49, X133)) → ack38_out_gga(s(T49), 0, X133)
ack38_in_gga(s(T54), s(T55), X151) → U6_gga(T54, T55, X151, ack38_in_gga(s(T54), T55, X150))
ack38_in_gga(s(T54), s(T55), X151) → U7_gga(T54, T55, X151, ack38_in_gga(s(T54), T55, T57))
U7_gga(T54, T55, X151, ack38_out_gga(s(T54), T55, T57)) → U8_gga(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
U8_gga(T54, T55, X151, ack38_out_gaa(T54, T57, X151)) → ack38_out_gga(s(T54), s(T55), X151)
U6_gga(T54, T55, X151, ack38_out_gga(s(T54), T55, X150)) → ack38_out_gga(s(T54), s(T55), X151)
U15_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, X221)) → ack1_out_gga(s(T84), s(s(T85)), T71)
ack1_in_gga(s(T84), s(s(T85)), T71) → U16_gga(T84, T85, T71, ack38_in_gga(s(T84), T85, T87))
U16_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → U17_gga(T84, T85, T71, ack38_in_gaa(T84, T87, X222))
U17_gga(T84, T85, T71, ack38_out_gaa(T84, T87, X222)) → ack1_out_gga(s(T84), s(s(T85)), T71)
U16_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → U18_gga(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_gga(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → U19_gga(T84, T85, T71, ack1_in_gaa(T84, T91, T71))
U19_gga(T84, T85, T71, ack1_out_gaa(T84, T91, T71)) → ack1_out_gga(s(T84), s(s(T85)), T71)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(39) Obligation:

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

ack1_in_gga(0, T5, s(T5)) → ack1_out_gga(0, T5, s(T5))
ack1_in_gga(s(0), 0, s(s(0))) → ack1_out_gga(s(0), 0, s(s(0)))
ack1_in_gga(s(s(T20)), 0, T22) → U9_gga(T20, T22, ack21_in_ga(T20, X40))
ack21_in_ga(T30, X73) → U1_ga(T30, X73, ack27_in_ga(T30, X73))
ack27_in_ga(0, s(s(0))) → ack27_out_ga(0, s(s(0)))
ack27_in_ga(s(T34), X97) → U2_ga(T34, X97, ack21_in_ga(T34, X96))
U2_ga(T34, X97, ack21_out_ga(T34, X96)) → ack27_out_ga(s(T34), X97)
ack27_in_ga(s(T34), X97) → U3_ga(T34, X97, ack21_in_ga(T34, T36))
U3_ga(T34, X97, ack21_out_ga(T34, T36)) → U4_ga(T34, X97, ack38_in_gaa(T34, T36, X97))
ack38_in_gaa(0, T44, s(T44)) → ack38_out_gaa(0, T44, s(T44))
ack38_in_gaa(s(T49), 0, X133) → U5_gaa(T49, X133, ack27_in_ga(T49, X133))
U5_gaa(T49, X133, ack27_out_ga(T49, X133)) → ack38_out_gaa(s(T49), 0, X133)
ack38_in_gaa(s(T54), s(T55), X151) → U6_gaa(T54, T55, X151, ack38_in_gaa(s(T54), T55, X150))
ack38_in_gaa(s(T54), s(T55), X151) → U7_gaa(T54, T55, X151, ack38_in_gaa(s(T54), T55, T57))
U7_gaa(T54, T55, X151, ack38_out_gaa(s(T54), T55, T57)) → U8_gaa(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
U8_gaa(T54, T55, X151, ack38_out_gaa(T54, T57, X151)) → ack38_out_gaa(s(T54), s(T55), X151)
U6_gaa(T54, T55, X151, ack38_out_gaa(s(T54), T55, X150)) → ack38_out_gaa(s(T54), s(T55), X151)
U4_ga(T34, X97, ack38_out_gaa(T34, T36, X97)) → ack27_out_ga(s(T34), X97)
U1_ga(T30, X73, ack27_out_ga(T30, X73)) → ack21_out_ga(T30, X73)
U9_gga(T20, T22, ack21_out_ga(T20, X40)) → ack1_out_gga(s(s(T20)), 0, T22)
ack1_in_gga(s(s(T20)), 0, T22) → U10_gga(T20, T22, ack21_in_ga(T20, T24))
U10_gga(T20, T22, ack21_out_ga(T20, T24)) → U11_gga(T20, T22, ack1_in_gaa(T20, T24, T22))
ack1_in_gaa(0, T5, s(T5)) → ack1_out_gaa(0, T5, s(T5))
ack1_in_gaa(s(0), 0, s(s(0))) → ack1_out_gaa(s(0), 0, s(s(0)))
ack1_in_gaa(s(s(T20)), 0, T22) → U9_gaa(T20, T22, ack21_in_ga(T20, X40))
U9_gaa(T20, T22, ack21_out_ga(T20, X40)) → ack1_out_gaa(s(s(T20)), 0, T22)
ack1_in_gaa(s(s(T20)), 0, T22) → U10_gaa(T20, T22, ack21_in_ga(T20, T24))
U10_gaa(T20, T22, ack21_out_ga(T20, T24)) → U11_gaa(T20, T22, ack1_in_gaa(T20, T24, T22))
ack1_in_gaa(s(T76), s(0), T71) → U12_gaa(T76, T71, ack27_in_ga(T76, X200))
U12_gaa(T76, T71, ack27_out_ga(T76, X200)) → ack1_out_gaa(s(T76), s(0), T71)
ack1_in_gaa(s(T76), s(0), T71) → U13_gaa(T76, T71, ack27_in_ga(T76, T77))
U13_gaa(T76, T71, ack27_out_ga(T76, T77)) → U14_gaa(T76, T71, ack1_in_gaa(T76, T77, T71))
ack1_in_gaa(s(T84), s(s(T85)), T71) → U15_gaa(T84, T85, T71, ack38_in_gaa(s(T84), T85, X221))
U15_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, X221)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
ack1_in_gaa(s(T84), s(s(T85)), T71) → U16_gaa(T84, T85, T71, ack38_in_gaa(s(T84), T85, T87))
U16_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U17_gaa(T84, T85, T71, ack38_in_gaa(T84, T87, X222))
U17_gaa(T84, T85, T71, ack38_out_gaa(T84, T87, X222)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
U16_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U18_gaa(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_gaa(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → U19_gaa(T84, T85, T71, ack1_in_gaa(T84, T91, T71))
U19_gaa(T84, T85, T71, ack1_out_gaa(T84, T91, T71)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
U14_gaa(T76, T71, ack1_out_gaa(T76, T77, T71)) → ack1_out_gaa(s(T76), s(0), T71)
U11_gaa(T20, T22, ack1_out_gaa(T20, T24, T22)) → ack1_out_gaa(s(s(T20)), 0, T22)
U11_gga(T20, T22, ack1_out_gaa(T20, T24, T22)) → ack1_out_gga(s(s(T20)), 0, T22)
ack1_in_gga(s(T76), s(0), T71) → U12_gga(T76, T71, ack27_in_ga(T76, X200))
U12_gga(T76, T71, ack27_out_ga(T76, X200)) → ack1_out_gga(s(T76), s(0), T71)
ack1_in_gga(s(T76), s(0), T71) → U13_gga(T76, T71, ack27_in_ga(T76, T77))
U13_gga(T76, T71, ack27_out_ga(T76, T77)) → U14_gga(T76, T71, ack1_in_gaa(T76, T77, T71))
U14_gga(T76, T71, ack1_out_gaa(T76, T77, T71)) → ack1_out_gga(s(T76), s(0), T71)
ack1_in_gga(s(T84), s(s(T85)), T71) → U15_gga(T84, T85, T71, ack38_in_gga(s(T84), T85, X221))
ack38_in_gga(0, T44, s(T44)) → ack38_out_gga(0, T44, s(T44))
ack38_in_gga(s(T49), 0, X133) → U5_gga(T49, X133, ack27_in_ga(T49, X133))
U5_gga(T49, X133, ack27_out_ga(T49, X133)) → ack38_out_gga(s(T49), 0, X133)
ack38_in_gga(s(T54), s(T55), X151) → U6_gga(T54, T55, X151, ack38_in_gga(s(T54), T55, X150))
ack38_in_gga(s(T54), s(T55), X151) → U7_gga(T54, T55, X151, ack38_in_gga(s(T54), T55, T57))
U7_gga(T54, T55, X151, ack38_out_gga(s(T54), T55, T57)) → U8_gga(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
U8_gga(T54, T55, X151, ack38_out_gaa(T54, T57, X151)) → ack38_out_gga(s(T54), s(T55), X151)
U6_gga(T54, T55, X151, ack38_out_gga(s(T54), T55, X150)) → ack38_out_gga(s(T54), s(T55), X151)
U15_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, X221)) → ack1_out_gga(s(T84), s(s(T85)), T71)
ack1_in_gga(s(T84), s(s(T85)), T71) → U16_gga(T84, T85, T71, ack38_in_gga(s(T84), T85, T87))
U16_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → U17_gga(T84, T85, T71, ack38_in_gaa(T84, T87, X222))
U17_gga(T84, T85, T71, ack38_out_gaa(T84, T87, X222)) → ack1_out_gga(s(T84), s(s(T85)), T71)
U16_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → U18_gga(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_gga(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → U19_gga(T84, T85, T71, ack1_in_gaa(T84, T91, T71))
U19_gga(T84, T85, T71, ack1_out_gaa(T84, T91, T71)) → ack1_out_gga(s(T84), s(s(T85)), T71)

(40) 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:

ACK1_IN_GGA(s(s(T20)), 0, T22) → U9_GGA(T20, T22, ack21_in_ga(T20, X40))
ACK1_IN_GGA(s(s(T20)), 0, T22) → ACK21_IN_GA(T20, X40)
ACK21_IN_GA(T30, X73) → U1_GA(T30, X73, ack27_in_ga(T30, X73))
ACK21_IN_GA(T30, X73) → ACK27_IN_GA(T30, X73)
ACK27_IN_GA(s(T34), X97) → U2_GA(T34, X97, ack21_in_ga(T34, X96))
ACK27_IN_GA(s(T34), X97) → ACK21_IN_GA(T34, X96)
ACK27_IN_GA(s(T34), X97) → U3_GA(T34, X97, ack21_in_ga(T34, T36))
U3_GA(T34, X97, ack21_out_ga(T34, T36)) → U4_GA(T34, X97, ack38_in_gaa(T34, T36, X97))
U3_GA(T34, X97, ack21_out_ga(T34, T36)) → ACK38_IN_GAA(T34, T36, X97)
ACK38_IN_GAA(s(T49), 0, X133) → U5_GAA(T49, X133, ack27_in_ga(T49, X133))
ACK38_IN_GAA(s(T49), 0, X133) → ACK27_IN_GA(T49, X133)
ACK38_IN_GAA(s(T54), s(T55), X151) → U6_GAA(T54, T55, X151, ack38_in_gaa(s(T54), T55, X150))
ACK38_IN_GAA(s(T54), s(T55), X151) → ACK38_IN_GAA(s(T54), T55, X150)
ACK38_IN_GAA(s(T54), s(T55), X151) → U7_GAA(T54, T55, X151, ack38_in_gaa(s(T54), T55, T57))
U7_GAA(T54, T55, X151, ack38_out_gaa(s(T54), T55, T57)) → U8_GAA(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
U7_GAA(T54, T55, X151, ack38_out_gaa(s(T54), T55, T57)) → ACK38_IN_GAA(T54, T57, X151)
ACK1_IN_GGA(s(s(T20)), 0, T22) → U10_GGA(T20, T22, ack21_in_ga(T20, T24))
U10_GGA(T20, T22, ack21_out_ga(T20, T24)) → U11_GGA(T20, T22, ack1_in_gaa(T20, T24, T22))
U10_GGA(T20, T22, ack21_out_ga(T20, T24)) → ACK1_IN_GAA(T20, T24, T22)
ACK1_IN_GAA(s(s(T20)), 0, T22) → U9_GAA(T20, T22, ack21_in_ga(T20, X40))
ACK1_IN_GAA(s(s(T20)), 0, T22) → ACK21_IN_GA(T20, X40)
ACK1_IN_GAA(s(s(T20)), 0, T22) → U10_GAA(T20, T22, ack21_in_ga(T20, T24))
U10_GAA(T20, T22, ack21_out_ga(T20, T24)) → U11_GAA(T20, T22, ack1_in_gaa(T20, T24, T22))
U10_GAA(T20, T22, ack21_out_ga(T20, T24)) → ACK1_IN_GAA(T20, T24, T22)
ACK1_IN_GAA(s(T76), s(0), T71) → U12_GAA(T76, T71, ack27_in_ga(T76, X200))
ACK1_IN_GAA(s(T76), s(0), T71) → ACK27_IN_GA(T76, X200)
ACK1_IN_GAA(s(T76), s(0), T71) → U13_GAA(T76, T71, ack27_in_ga(T76, T77))
U13_GAA(T76, T71, ack27_out_ga(T76, T77)) → U14_GAA(T76, T71, ack1_in_gaa(T76, T77, T71))
U13_GAA(T76, T71, ack27_out_ga(T76, T77)) → ACK1_IN_GAA(T76, T77, T71)
ACK1_IN_GAA(s(T84), s(s(T85)), T71) → U15_GAA(T84, T85, T71, ack38_in_gaa(s(T84), T85, X221))
ACK1_IN_GAA(s(T84), s(s(T85)), T71) → ACK38_IN_GAA(s(T84), T85, X221)
ACK1_IN_GAA(s(T84), s(s(T85)), T71) → U16_GAA(T84, T85, T71, ack38_in_gaa(s(T84), T85, T87))
U16_GAA(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U17_GAA(T84, T85, T71, ack38_in_gaa(T84, T87, X222))
U16_GAA(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → ACK38_IN_GAA(T84, T87, X222)
U16_GAA(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U18_GAA(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_GAA(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → U19_GAA(T84, T85, T71, ack1_in_gaa(T84, T91, T71))
U18_GAA(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → ACK1_IN_GAA(T84, T91, T71)
ACK1_IN_GGA(s(T76), s(0), T71) → U12_GGA(T76, T71, ack27_in_ga(T76, X200))
ACK1_IN_GGA(s(T76), s(0), T71) → ACK27_IN_GA(T76, X200)
ACK1_IN_GGA(s(T76), s(0), T71) → U13_GGA(T76, T71, ack27_in_ga(T76, T77))
U13_GGA(T76, T71, ack27_out_ga(T76, T77)) → U14_GGA(T76, T71, ack1_in_gaa(T76, T77, T71))
U13_GGA(T76, T71, ack27_out_ga(T76, T77)) → ACK1_IN_GAA(T76, T77, T71)
ACK1_IN_GGA(s(T84), s(s(T85)), T71) → U15_GGA(T84, T85, T71, ack38_in_gga(s(T84), T85, X221))
ACK1_IN_GGA(s(T84), s(s(T85)), T71) → ACK38_IN_GGA(s(T84), T85, X221)
ACK38_IN_GGA(s(T49), 0, X133) → U5_GGA(T49, X133, ack27_in_ga(T49, X133))
ACK38_IN_GGA(s(T49), 0, X133) → ACK27_IN_GA(T49, X133)
ACK38_IN_GGA(s(T54), s(T55), X151) → U6_GGA(T54, T55, X151, ack38_in_gga(s(T54), T55, X150))
ACK38_IN_GGA(s(T54), s(T55), X151) → ACK38_IN_GGA(s(T54), T55, X150)
ACK38_IN_GGA(s(T54), s(T55), X151) → U7_GGA(T54, T55, X151, ack38_in_gga(s(T54), T55, T57))
U7_GGA(T54, T55, X151, ack38_out_gga(s(T54), T55, T57)) → U8_GGA(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
U7_GGA(T54, T55, X151, ack38_out_gga(s(T54), T55, T57)) → ACK38_IN_GAA(T54, T57, X151)
ACK1_IN_GGA(s(T84), s(s(T85)), T71) → U16_GGA(T84, T85, T71, ack38_in_gga(s(T84), T85, T87))
U16_GGA(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → U17_GGA(T84, T85, T71, ack38_in_gaa(T84, T87, X222))
U16_GGA(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → ACK38_IN_GAA(T84, T87, X222)
U16_GGA(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → U18_GGA(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_GGA(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → U19_GGA(T84, T85, T71, ack1_in_gaa(T84, T91, T71))
U18_GGA(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → ACK1_IN_GAA(T84, T91, T71)

The TRS R consists of the following rules:

ack1_in_gga(0, T5, s(T5)) → ack1_out_gga(0, T5, s(T5))
ack1_in_gga(s(0), 0, s(s(0))) → ack1_out_gga(s(0), 0, s(s(0)))
ack1_in_gga(s(s(T20)), 0, T22) → U9_gga(T20, T22, ack21_in_ga(T20, X40))
ack21_in_ga(T30, X73) → U1_ga(T30, X73, ack27_in_ga(T30, X73))
ack27_in_ga(0, s(s(0))) → ack27_out_ga(0, s(s(0)))
ack27_in_ga(s(T34), X97) → U2_ga(T34, X97, ack21_in_ga(T34, X96))
U2_ga(T34, X97, ack21_out_ga(T34, X96)) → ack27_out_ga(s(T34), X97)
ack27_in_ga(s(T34), X97) → U3_ga(T34, X97, ack21_in_ga(T34, T36))
U3_ga(T34, X97, ack21_out_ga(T34, T36)) → U4_ga(T34, X97, ack38_in_gaa(T34, T36, X97))
ack38_in_gaa(0, T44, s(T44)) → ack38_out_gaa(0, T44, s(T44))
ack38_in_gaa(s(T49), 0, X133) → U5_gaa(T49, X133, ack27_in_ga(T49, X133))
U5_gaa(T49, X133, ack27_out_ga(T49, X133)) → ack38_out_gaa(s(T49), 0, X133)
ack38_in_gaa(s(T54), s(T55), X151) → U6_gaa(T54, T55, X151, ack38_in_gaa(s(T54), T55, X150))
ack38_in_gaa(s(T54), s(T55), X151) → U7_gaa(T54, T55, X151, ack38_in_gaa(s(T54), T55, T57))
U7_gaa(T54, T55, X151, ack38_out_gaa(s(T54), T55, T57)) → U8_gaa(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
U8_gaa(T54, T55, X151, ack38_out_gaa(T54, T57, X151)) → ack38_out_gaa(s(T54), s(T55), X151)
U6_gaa(T54, T55, X151, ack38_out_gaa(s(T54), T55, X150)) → ack38_out_gaa(s(T54), s(T55), X151)
U4_ga(T34, X97, ack38_out_gaa(T34, T36, X97)) → ack27_out_ga(s(T34), X97)
U1_ga(T30, X73, ack27_out_ga(T30, X73)) → ack21_out_ga(T30, X73)
U9_gga(T20, T22, ack21_out_ga(T20, X40)) → ack1_out_gga(s(s(T20)), 0, T22)
ack1_in_gga(s(s(T20)), 0, T22) → U10_gga(T20, T22, ack21_in_ga(T20, T24))
U10_gga(T20, T22, ack21_out_ga(T20, T24)) → U11_gga(T20, T22, ack1_in_gaa(T20, T24, T22))
ack1_in_gaa(0, T5, s(T5)) → ack1_out_gaa(0, T5, s(T5))
ack1_in_gaa(s(0), 0, s(s(0))) → ack1_out_gaa(s(0), 0, s(s(0)))
ack1_in_gaa(s(s(T20)), 0, T22) → U9_gaa(T20, T22, ack21_in_ga(T20, X40))
U9_gaa(T20, T22, ack21_out_ga(T20, X40)) → ack1_out_gaa(s(s(T20)), 0, T22)
ack1_in_gaa(s(s(T20)), 0, T22) → U10_gaa(T20, T22, ack21_in_ga(T20, T24))
U10_gaa(T20, T22, ack21_out_ga(T20, T24)) → U11_gaa(T20, T22, ack1_in_gaa(T20, T24, T22))
ack1_in_gaa(s(T76), s(0), T71) → U12_gaa(T76, T71, ack27_in_ga(T76, X200))
U12_gaa(T76, T71, ack27_out_ga(T76, X200)) → ack1_out_gaa(s(T76), s(0), T71)
ack1_in_gaa(s(T76), s(0), T71) → U13_gaa(T76, T71, ack27_in_ga(T76, T77))
U13_gaa(T76, T71, ack27_out_ga(T76, T77)) → U14_gaa(T76, T71, ack1_in_gaa(T76, T77, T71))
ack1_in_gaa(s(T84), s(s(T85)), T71) → U15_gaa(T84, T85, T71, ack38_in_gaa(s(T84), T85, X221))
U15_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, X221)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
ack1_in_gaa(s(T84), s(s(T85)), T71) → U16_gaa(T84, T85, T71, ack38_in_gaa(s(T84), T85, T87))
U16_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U17_gaa(T84, T85, T71, ack38_in_gaa(T84, T87, X222))
U17_gaa(T84, T85, T71, ack38_out_gaa(T84, T87, X222)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
U16_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U18_gaa(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_gaa(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → U19_gaa(T84, T85, T71, ack1_in_gaa(T84, T91, T71))
U19_gaa(T84, T85, T71, ack1_out_gaa(T84, T91, T71)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
U14_gaa(T76, T71, ack1_out_gaa(T76, T77, T71)) → ack1_out_gaa(s(T76), s(0), T71)
U11_gaa(T20, T22, ack1_out_gaa(T20, T24, T22)) → ack1_out_gaa(s(s(T20)), 0, T22)
U11_gga(T20, T22, ack1_out_gaa(T20, T24, T22)) → ack1_out_gga(s(s(T20)), 0, T22)
ack1_in_gga(s(T76), s(0), T71) → U12_gga(T76, T71, ack27_in_ga(T76, X200))
U12_gga(T76, T71, ack27_out_ga(T76, X200)) → ack1_out_gga(s(T76), s(0), T71)
ack1_in_gga(s(T76), s(0), T71) → U13_gga(T76, T71, ack27_in_ga(T76, T77))
U13_gga(T76, T71, ack27_out_ga(T76, T77)) → U14_gga(T76, T71, ack1_in_gaa(T76, T77, T71))
U14_gga(T76, T71, ack1_out_gaa(T76, T77, T71)) → ack1_out_gga(s(T76), s(0), T71)
ack1_in_gga(s(T84), s(s(T85)), T71) → U15_gga(T84, T85, T71, ack38_in_gga(s(T84), T85, X221))
ack38_in_gga(0, T44, s(T44)) → ack38_out_gga(0, T44, s(T44))
ack38_in_gga(s(T49), 0, X133) → U5_gga(T49, X133, ack27_in_ga(T49, X133))
U5_gga(T49, X133, ack27_out_ga(T49, X133)) → ack38_out_gga(s(T49), 0, X133)
ack38_in_gga(s(T54), s(T55), X151) → U6_gga(T54, T55, X151, ack38_in_gga(s(T54), T55, X150))
ack38_in_gga(s(T54), s(T55), X151) → U7_gga(T54, T55, X151, ack38_in_gga(s(T54), T55, T57))
U7_gga(T54, T55, X151, ack38_out_gga(s(T54), T55, T57)) → U8_gga(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
U8_gga(T54, T55, X151, ack38_out_gaa(T54, T57, X151)) → ack38_out_gga(s(T54), s(T55), X151)
U6_gga(T54, T55, X151, ack38_out_gga(s(T54), T55, X150)) → ack38_out_gga(s(T54), s(T55), X151)
U15_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, X221)) → ack1_out_gga(s(T84), s(s(T85)), T71)
ack1_in_gga(s(T84), s(s(T85)), T71) → U16_gga(T84, T85, T71, ack38_in_gga(s(T84), T85, T87))
U16_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → U17_gga(T84, T85, T71, ack38_in_gaa(T84, T87, X222))
U17_gga(T84, T85, T71, ack38_out_gaa(T84, T87, X222)) → ack1_out_gga(s(T84), s(s(T85)), T71)
U16_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → U18_gga(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_gga(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → U19_gga(T84, T85, T71, ack1_in_gaa(T84, T91, T71))
U19_gga(T84, T85, T71, ack1_out_gaa(T84, T91, T71)) → ack1_out_gga(s(T84), s(s(T85)), T71)

The argument filtering Pi contains the following mapping:
ack1_in_gga(x1, x2, x3) = ack1_in_gga(x1, x2)
0 = 0
ack1_out_gga(x1, x2, x3) = ack1_out_gga
s(x1) = s(x1)
U9_gga(x1, x2, x3) = U9_gga(x3)
ack21_in_ga(x1, x2) = ack21_in_ga(x1)
U1_ga(x1, x2, x3) = U1_ga(x3)
ack27_in_ga(x1, x2) = ack27_in_ga(x1)
ack27_out_ga(x1, x2) = ack27_out_ga
U2_ga(x1, x2, x3) = U2_ga(x3)
ack21_out_ga(x1, x2) = ack21_out_ga
U3_ga(x1, x2, x3) = U3_ga(x1, x3)
U4_ga(x1, x2, x3) = U4_ga(x3)
ack38_in_gga(x1, x2, x3) = ack38_in_gga(x1, x2)
ack38_out_gga(x1, x2, x3) = ack38_out_gga
U5_gga(x1, x2, x3) = U5_gga(x3)
U6_gga(x1, x2, x3, x4) = U6_gga(x4)
U7_gga(x1, x2, x3, x4) = U7_gga(x1, x4)
U8_gga(x1, x2, x3, x4) = U8_gga(x4)
ack38_in_gaa(x1, x2, x3) = ack38_in_gaa(x1)
ack38_out_gaa(x1, x2, x3) = ack38_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)
U10_gga(x1, x2, x3) = U10_gga(x1, x3)
U11_gga(x1, x2, x3) = U11_gga(x3)
ack1_in_gaa(x1, x2, x3) = ack1_in_gaa(x1)
ack1_out_gaa(x1, x2, x3) = ack1_out_gaa
U9_gaa(x1, x2, x3) = U9_gaa(x3)
U10_gaa(x1, x2, x3) = U10_gaa(x1, x3)
U11_gaa(x1, x2, x3) = U11_gaa(x3)
U12_gaa(x1, x2, x3) = U12_gaa(x3)
U13_gaa(x1, x2, x3) = U13_gaa(x1, x3)
U14_gaa(x1, x2, x3) = U14_gaa(x3)
U15_gaa(x1, x2, x3, x4) = U15_gaa(x4)
U16_gaa(x1, x2, x3, x4) = U16_gaa(x1, x4)
U17_gaa(x1, x2, x3, x4) = U17_gaa(x4)
U18_gaa(x1, x2, x3, x4) = U18_gaa(x1, x4)
U19_gaa(x1, x2, x3, x4) = U19_gaa(x4)
U12_gga(x1, x2, x3) = U12_gga(x3)
U13_gga(x1, x2, x3) = U13_gga(x1, x3)
U14_gga(x1, x2, x3) = U14_gga(x3)
U15_gga(x1, x2, x3, x4) = U15_gga(x4)
U16_gga(x1, x2, x3, x4) = U16_gga(x1, x4)
U17_gga(x1, x2, x3, x4) = U17_gga(x4)
U18_gga(x1, x2, x3, x4) = U18_gga(x1, x4)
U19_gga(x1, x2, x3, x4) = U19_gga(x4)
ACK1_IN_GGA(x1, x2, x3) = ACK1_IN_GGA(x1, x2)
U9_GGA(x1, x2, x3) = U9_GGA(x3)
ACK21_IN_GA(x1, x2) = ACK21_IN_GA(x1)
U1_GA(x1, x2, x3) = U1_GA(x3)
ACK27_IN_GA(x1, x2) = ACK27_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)
ACK38_IN_GAA(x1, x2, x3) = ACK38_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)
U10_GGA(x1, x2, x3) = U10_GGA(x1, x3)
U11_GGA(x1, x2, x3) = U11_GGA(x3)
ACK1_IN_GAA(x1, x2, x3) = ACK1_IN_GAA(x1)
U9_GAA(x1, x2, x3) = U9_GAA(x3)
U10_GAA(x1, x2, x3) = U10_GAA(x1, x3)
U11_GAA(x1, x2, x3) = U11_GAA(x3)
U12_GAA(x1, x2, x3) = U12_GAA(x3)
U13_GAA(x1, x2, x3) = U13_GAA(x1, x3)
U14_GAA(x1, x2, x3) = U14_GAA(x3)
U15_GAA(x1, x2, x3, x4) = U15_GAA(x4)
U16_GAA(x1, x2, x3, x4) = U16_GAA(x1, x4)
U17_GAA(x1, x2, x3, x4) = U17_GAA(x4)
U18_GAA(x1, x2, x3, x4) = U18_GAA(x1, x4)
U19_GAA(x1, x2, x3, x4) = U19_GAA(x4)
U12_GGA(x1, x2, x3) = U12_GGA(x3)
U13_GGA(x1, x2, x3) = U13_GGA(x1, x3)
U14_GGA(x1, x2, x3) = U14_GGA(x3)
U15_GGA(x1, x2, x3, x4) = U15_GGA(x4)
ACK38_IN_GGA(x1, x2, x3) = ACK38_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3) = U5_GGA(x3)
U6_GGA(x1, x2, x3, x4) = U6_GGA(x4)
U7_GGA(x1, x2, x3, x4) = U7_GGA(x1, x4)
U8_GGA(x1, x2, x3, x4) = U8_GGA(x4)
U16_GGA(x1, x2, x3, x4) = U16_GGA(x1, x4)
U17_GGA(x1, x2, x3, x4) = U17_GGA(x4)
U18_GGA(x1, x2, x3, x4) = U18_GGA(x1, x4)
U19_GGA(x1, x2, x3, x4) = U19_GGA(x4)

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

(41) Obligation:

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

ACK1_IN_GGA(s(s(T20)), 0, T22) → U9_GGA(T20, T22, ack21_in_ga(T20, X40))
ACK1_IN_GGA(s(s(T20)), 0, T22) → ACK21_IN_GA(T20, X40)
ACK21_IN_GA(T30, X73) → U1_GA(T30, X73, ack27_in_ga(T30, X73))
ACK21_IN_GA(T30, X73) → ACK27_IN_GA(T30, X73)
ACK27_IN_GA(s(T34), X97) → U2_GA(T34, X97, ack21_in_ga(T34, X96))
ACK27_IN_GA(s(T34), X97) → ACK21_IN_GA(T34, X96)
ACK27_IN_GA(s(T34), X97) → U3_GA(T34, X97, ack21_in_ga(T34, T36))
U3_GA(T34, X97, ack21_out_ga(T34, T36)) → U4_GA(T34, X97, ack38_in_gaa(T34, T36, X97))
U3_GA(T34, X97, ack21_out_ga(T34, T36)) → ACK38_IN_GAA(T34, T36, X97)
ACK38_IN_GAA(s(T49), 0, X133) → U5_GAA(T49, X133, ack27_in_ga(T49, X133))
ACK38_IN_GAA(s(T49), 0, X133) → ACK27_IN_GA(T49, X133)
ACK38_IN_GAA(s(T54), s(T55), X151) → U6_GAA(T54, T55, X151, ack38_in_gaa(s(T54), T55, X150))
ACK38_IN_GAA(s(T54), s(T55), X151) → ACK38_IN_GAA(s(T54), T55, X150)
ACK38_IN_GAA(s(T54), s(T55), X151) → U7_GAA(T54, T55, X151, ack38_in_gaa(s(T54), T55, T57))
U7_GAA(T54, T55, X151, ack38_out_gaa(s(T54), T55, T57)) → U8_GAA(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
U7_GAA(T54, T55, X151, ack38_out_gaa(s(T54), T55, T57)) → ACK38_IN_GAA(T54, T57, X151)
ACK1_IN_GGA(s(s(T20)), 0, T22) → U10_GGA(T20, T22, ack21_in_ga(T20, T24))
U10_GGA(T20, T22, ack21_out_ga(T20, T24)) → U11_GGA(T20, T22, ack1_in_gaa(T20, T24, T22))
U10_GGA(T20, T22, ack21_out_ga(T20, T24)) → ACK1_IN_GAA(T20, T24, T22)
ACK1_IN_GAA(s(s(T20)), 0, T22) → U9_GAA(T20, T22, ack21_in_ga(T20, X40))
ACK1_IN_GAA(s(s(T20)), 0, T22) → ACK21_IN_GA(T20, X40)
ACK1_IN_GAA(s(s(T20)), 0, T22) → U10_GAA(T20, T22, ack21_in_ga(T20, T24))
U10_GAA(T20, T22, ack21_out_ga(T20, T24)) → U11_GAA(T20, T22, ack1_in_gaa(T20, T24, T22))
U10_GAA(T20, T22, ack21_out_ga(T20, T24)) → ACK1_IN_GAA(T20, T24, T22)
ACK1_IN_GAA(s(T76), s(0), T71) → U12_GAA(T76, T71, ack27_in_ga(T76, X200))
ACK1_IN_GAA(s(T76), s(0), T71) → ACK27_IN_GA(T76, X200)
ACK1_IN_GAA(s(T76), s(0), T71) → U13_GAA(T76, T71, ack27_in_ga(T76, T77))
U13_GAA(T76, T71, ack27_out_ga(T76, T77)) → U14_GAA(T76, T71, ack1_in_gaa(T76, T77, T71))
U13_GAA(T76, T71, ack27_out_ga(T76, T77)) → ACK1_IN_GAA(T76, T77, T71)
ACK1_IN_GAA(s(T84), s(s(T85)), T71) → U15_GAA(T84, T85, T71, ack38_in_gaa(s(T84), T85, X221))
ACK1_IN_GAA(s(T84), s(s(T85)), T71) → ACK38_IN_GAA(s(T84), T85, X221)
ACK1_IN_GAA(s(T84), s(s(T85)), T71) → U16_GAA(T84, T85, T71, ack38_in_gaa(s(T84), T85, T87))
U16_GAA(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U17_GAA(T84, T85, T71, ack38_in_gaa(T84, T87, X222))
U16_GAA(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → ACK38_IN_GAA(T84, T87, X222)
U16_GAA(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U18_GAA(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_GAA(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → U19_GAA(T84, T85, T71, ack1_in_gaa(T84, T91, T71))
U18_GAA(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → ACK1_IN_GAA(T84, T91, T71)
ACK1_IN_GGA(s(T76), s(0), T71) → U12_GGA(T76, T71, ack27_in_ga(T76, X200))
ACK1_IN_GGA(s(T76), s(0), T71) → ACK27_IN_GA(T76, X200)
ACK1_IN_GGA(s(T76), s(0), T71) → U13_GGA(T76, T71, ack27_in_ga(T76, T77))
U13_GGA(T76, T71, ack27_out_ga(T76, T77)) → U14_GGA(T76, T71, ack1_in_gaa(T76, T77, T71))
U13_GGA(T76, T71, ack27_out_ga(T76, T77)) → ACK1_IN_GAA(T76, T77, T71)
ACK1_IN_GGA(s(T84), s(s(T85)), T71) → U15_GGA(T84, T85, T71, ack38_in_gga(s(T84), T85, X221))
ACK1_IN_GGA(s(T84), s(s(T85)), T71) → ACK38_IN_GGA(s(T84), T85, X221)
ACK38_IN_GGA(s(T49), 0, X133) → U5_GGA(T49, X133, ack27_in_ga(T49, X133))
ACK38_IN_GGA(s(T49), 0, X133) → ACK27_IN_GA(T49, X133)
ACK38_IN_GGA(s(T54), s(T55), X151) → U6_GGA(T54, T55, X151, ack38_in_gga(s(T54), T55, X150))
ACK38_IN_GGA(s(T54), s(T55), X151) → ACK38_IN_GGA(s(T54), T55, X150)
ACK38_IN_GGA(s(T54), s(T55), X151) → U7_GGA(T54, T55, X151, ack38_in_gga(s(T54), T55, T57))
U7_GGA(T54, T55, X151, ack38_out_gga(s(T54), T55, T57)) → U8_GGA(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
U7_GGA(T54, T55, X151, ack38_out_gga(s(T54), T55, T57)) → ACK38_IN_GAA(T54, T57, X151)
ACK1_IN_GGA(s(T84), s(s(T85)), T71) → U16_GGA(T84, T85, T71, ack38_in_gga(s(T84), T85, T87))
U16_GGA(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → U17_GGA(T84, T85, T71, ack38_in_gaa(T84, T87, X222))
U16_GGA(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → ACK38_IN_GAA(T84, T87, X222)
U16_GGA(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → U18_GGA(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_GGA(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → U19_GGA(T84, T85, T71, ack1_in_gaa(T84, T91, T71))
U18_GGA(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → ACK1_IN_GAA(T84, T91, T71)

The TRS R consists of the following rules:

ack1_in_gga(0, T5, s(T5)) → ack1_out_gga(0, T5, s(T5))
ack1_in_gga(s(0), 0, s(s(0))) → ack1_out_gga(s(0), 0, s(s(0)))
ack1_in_gga(s(s(T20)), 0, T22) → U9_gga(T20, T22, ack21_in_ga(T20, X40))
ack21_in_ga(T30, X73) → U1_ga(T30, X73, ack27_in_ga(T30, X73))
ack27_in_ga(0, s(s(0))) → ack27_out_ga(0, s(s(0)))
ack27_in_ga(s(T34), X97) → U2_ga(T34, X97, ack21_in_ga(T34, X96))
U2_ga(T34, X97, ack21_out_ga(T34, X96)) → ack27_out_ga(s(T34), X97)
ack27_in_ga(s(T34), X97) → U3_ga(T34, X97, ack21_in_ga(T34, T36))
U3_ga(T34, X97, ack21_out_ga(T34, T36)) → U4_ga(T34, X97, ack38_in_gaa(T34, T36, X97))
ack38_in_gaa(0, T44, s(T44)) → ack38_out_gaa(0, T44, s(T44))
ack38_in_gaa(s(T49), 0, X133) → U5_gaa(T49, X133, ack27_in_ga(T49, X133))
U5_gaa(T49, X133, ack27_out_ga(T49, X133)) → ack38_out_gaa(s(T49), 0, X133)
ack38_in_gaa(s(T54), s(T55), X151) → U6_gaa(T54, T55, X151, ack38_in_gaa(s(T54), T55, X150))
ack38_in_gaa(s(T54), s(T55), X151) → U7_gaa(T54, T55, X151, ack38_in_gaa(s(T54), T55, T57))
U7_gaa(T54, T55, X151, ack38_out_gaa(s(T54), T55, T57)) → U8_gaa(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
U8_gaa(T54, T55, X151, ack38_out_gaa(T54, T57, X151)) → ack38_out_gaa(s(T54), s(T55), X151)
U6_gaa(T54, T55, X151, ack38_out_gaa(s(T54), T55, X150)) → ack38_out_gaa(s(T54), s(T55), X151)
U4_ga(T34, X97, ack38_out_gaa(T34, T36, X97)) → ack27_out_ga(s(T34), X97)
U1_ga(T30, X73, ack27_out_ga(T30, X73)) → ack21_out_ga(T30, X73)
U9_gga(T20, T22, ack21_out_ga(T20, X40)) → ack1_out_gga(s(s(T20)), 0, T22)
ack1_in_gga(s(s(T20)), 0, T22) → U10_gga(T20, T22, ack21_in_ga(T20, T24))
U10_gga(T20, T22, ack21_out_ga(T20, T24)) → U11_gga(T20, T22, ack1_in_gaa(T20, T24, T22))
ack1_in_gaa(0, T5, s(T5)) → ack1_out_gaa(0, T5, s(T5))
ack1_in_gaa(s(0), 0, s(s(0))) → ack1_out_gaa(s(0), 0, s(s(0)))
ack1_in_gaa(s(s(T20)), 0, T22) → U9_gaa(T20, T22, ack21_in_ga(T20, X40))
U9_gaa(T20, T22, ack21_out_ga(T20, X40)) → ack1_out_gaa(s(s(T20)), 0, T22)
ack1_in_gaa(s(s(T20)), 0, T22) → U10_gaa(T20, T22, ack21_in_ga(T20, T24))
U10_gaa(T20, T22, ack21_out_ga(T20, T24)) → U11_gaa(T20, T22, ack1_in_gaa(T20, T24, T22))
ack1_in_gaa(s(T76), s(0), T71) → U12_gaa(T76, T71, ack27_in_ga(T76, X200))
U12_gaa(T76, T71, ack27_out_ga(T76, X200)) → ack1_out_gaa(s(T76), s(0), T71)
ack1_in_gaa(s(T76), s(0), T71) → U13_gaa(T76, T71, ack27_in_ga(T76, T77))
U13_gaa(T76, T71, ack27_out_ga(T76, T77)) → U14_gaa(T76, T71, ack1_in_gaa(T76, T77, T71))
ack1_in_gaa(s(T84), s(s(T85)), T71) → U15_gaa(T84, T85, T71, ack38_in_gaa(s(T84), T85, X221))
U15_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, X221)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
ack1_in_gaa(s(T84), s(s(T85)), T71) → U16_gaa(T84, T85, T71, ack38_in_gaa(s(T84), T85, T87))
U16_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U17_gaa(T84, T85, T71, ack38_in_gaa(T84, T87, X222))
U17_gaa(T84, T85, T71, ack38_out_gaa(T84, T87, X222)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
U16_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U18_gaa(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_gaa(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → U19_gaa(T84, T85, T71, ack1_in_gaa(T84, T91, T71))
U19_gaa(T84, T85, T71, ack1_out_gaa(T84, T91, T71)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
U14_gaa(T76, T71, ack1_out_gaa(T76, T77, T71)) → ack1_out_gaa(s(T76), s(0), T71)
U11_gaa(T20, T22, ack1_out_gaa(T20, T24, T22)) → ack1_out_gaa(s(s(T20)), 0, T22)
U11_gga(T20, T22, ack1_out_gaa(T20, T24, T22)) → ack1_out_gga(s(s(T20)), 0, T22)
ack1_in_gga(s(T76), s(0), T71) → U12_gga(T76, T71, ack27_in_ga(T76, X200))
U12_gga(T76, T71, ack27_out_ga(T76, X200)) → ack1_out_gga(s(T76), s(0), T71)
ack1_in_gga(s(T76), s(0), T71) → U13_gga(T76, T71, ack27_in_ga(T76, T77))
U13_gga(T76, T71, ack27_out_ga(T76, T77)) → U14_gga(T76, T71, ack1_in_gaa(T76, T77, T71))
U14_gga(T76, T71, ack1_out_gaa(T76, T77, T71)) → ack1_out_gga(s(T76), s(0), T71)
ack1_in_gga(s(T84), s(s(T85)), T71) → U15_gga(T84, T85, T71, ack38_in_gga(s(T84), T85, X221))
ack38_in_gga(0, T44, s(T44)) → ack38_out_gga(0, T44, s(T44))
ack38_in_gga(s(T49), 0, X133) → U5_gga(T49, X133, ack27_in_ga(T49, X133))
U5_gga(T49, X133, ack27_out_ga(T49, X133)) → ack38_out_gga(s(T49), 0, X133)
ack38_in_gga(s(T54), s(T55), X151) → U6_gga(T54, T55, X151, ack38_in_gga(s(T54), T55, X150))
ack38_in_gga(s(T54), s(T55), X151) → U7_gga(T54, T55, X151, ack38_in_gga(s(T54), T55, T57))
U7_gga(T54, T55, X151, ack38_out_gga(s(T54), T55, T57)) → U8_gga(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
U8_gga(T54, T55, X151, ack38_out_gaa(T54, T57, X151)) → ack38_out_gga(s(T54), s(T55), X151)
U6_gga(T54, T55, X151, ack38_out_gga(s(T54), T55, X150)) → ack38_out_gga(s(T54), s(T55), X151)
U15_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, X221)) → ack1_out_gga(s(T84), s(s(T85)), T71)
ack1_in_gga(s(T84), s(s(T85)), T71) → U16_gga(T84, T85, T71, ack38_in_gga(s(T84), T85, T87))
U16_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → U17_gga(T84, T85, T71, ack38_in_gaa(T84, T87, X222))
U17_gga(T84, T85, T71, ack38_out_gaa(T84, T87, X222)) → ack1_out_gga(s(T84), s(s(T85)), T71)
U16_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → U18_gga(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_gga(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → U19_gga(T84, T85, T71, ack1_in_gaa(T84, T91, T71))
U19_gga(T84, T85, T71, ack1_out_gaa(T84, T91, T71)) → ack1_out_gga(s(T84), s(s(T85)), T71)

(42) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 41 less nodes.

(43) Complex Obligation (AND)

(44) Obligation:

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

ACK21_IN_GA(T30, X73) → ACK27_IN_GA(T30, X73)
ACK27_IN_GA(s(T34), X97) → ACK21_IN_GA(T34, X96)
ACK27_IN_GA(s(T34), X97) → U3_GA(T34, X97, ack21_in_ga(T34, T36))
U3_GA(T34, X97, ack21_out_ga(T34, T36)) → ACK38_IN_GAA(T34, T36, X97)
ACK38_IN_GAA(s(T49), 0, X133) → ACK27_IN_GA(T49, X133)
ACK38_IN_GAA(s(T54), s(T55), X151) → ACK38_IN_GAA(s(T54), T55, X150)
ACK38_IN_GAA(s(T54), s(T55), X151) → U7_GAA(T54, T55, X151, ack38_in_gaa(s(T54), T55, T57))
U7_GAA(T54, T55, X151, ack38_out_gaa(s(T54), T55, T57)) → ACK38_IN_GAA(T54, T57, X151)

The TRS R consists of the following rules:

ack1_in_gga(0, T5, s(T5)) → ack1_out_gga(0, T5, s(T5))
ack1_in_gga(s(0), 0, s(s(0))) → ack1_out_gga(s(0), 0, s(s(0)))
ack1_in_gga(s(s(T20)), 0, T22) → U9_gga(T20, T22, ack21_in_ga(T20, X40))
ack21_in_ga(T30, X73) → U1_ga(T30, X73, ack27_in_ga(T30, X73))
ack27_in_ga(0, s(s(0))) → ack27_out_ga(0, s(s(0)))
ack27_in_ga(s(T34), X97) → U2_ga(T34, X97, ack21_in_ga(T34, X96))
U2_ga(T34, X97, ack21_out_ga(T34, X96)) → ack27_out_ga(s(T34), X97)
ack27_in_ga(s(T34), X97) → U3_ga(T34, X97, ack21_in_ga(T34, T36))
U3_ga(T34, X97, ack21_out_ga(T34, T36)) → U4_ga(T34, X97, ack38_in_gaa(T34, T36, X97))
ack38_in_gaa(0, T44, s(T44)) → ack38_out_gaa(0, T44, s(T44))
ack38_in_gaa(s(T49), 0, X133) → U5_gaa(T49, X133, ack27_in_ga(T49, X133))
U5_gaa(T49, X133, ack27_out_ga(T49, X133)) → ack38_out_gaa(s(T49), 0, X133)
ack38_in_gaa(s(T54), s(T55), X151) → U6_gaa(T54, T55, X151, ack38_in_gaa(s(T54), T55, X150))
ack38_in_gaa(s(T54), s(T55), X151) → U7_gaa(T54, T55, X151, ack38_in_gaa(s(T54), T55, T57))
U7_gaa(T54, T55, X151, ack38_out_gaa(s(T54), T55, T57)) → U8_gaa(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
U8_gaa(T54, T55, X151, ack38_out_gaa(T54, T57, X151)) → ack38_out_gaa(s(T54), s(T55), X151)
U6_gaa(T54, T55, X151, ack38_out_gaa(s(T54), T55, X150)) → ack38_out_gaa(s(T54), s(T55), X151)
U4_ga(T34, X97, ack38_out_gaa(T34, T36, X97)) → ack27_out_ga(s(T34), X97)
U1_ga(T30, X73, ack27_out_ga(T30, X73)) → ack21_out_ga(T30, X73)
U9_gga(T20, T22, ack21_out_ga(T20, X40)) → ack1_out_gga(s(s(T20)), 0, T22)
ack1_in_gga(s(s(T20)), 0, T22) → U10_gga(T20, T22, ack21_in_ga(T20, T24))
U10_gga(T20, T22, ack21_out_ga(T20, T24)) → U11_gga(T20, T22, ack1_in_gaa(T20, T24, T22))
ack1_in_gaa(0, T5, s(T5)) → ack1_out_gaa(0, T5, s(T5))
ack1_in_gaa(s(0), 0, s(s(0))) → ack1_out_gaa(s(0), 0, s(s(0)))
ack1_in_gaa(s(s(T20)), 0, T22) → U9_gaa(T20, T22, ack21_in_ga(T20, X40))
U9_gaa(T20, T22, ack21_out_ga(T20, X40)) → ack1_out_gaa(s(s(T20)), 0, T22)
ack1_in_gaa(s(s(T20)), 0, T22) → U10_gaa(T20, T22, ack21_in_ga(T20, T24))
U10_gaa(T20, T22, ack21_out_ga(T20, T24)) → U11_gaa(T20, T22, ack1_in_gaa(T20, T24, T22))
ack1_in_gaa(s(T76), s(0), T71) → U12_gaa(T76, T71, ack27_in_ga(T76, X200))
U12_gaa(T76, T71, ack27_out_ga(T76, X200)) → ack1_out_gaa(s(T76), s(0), T71)
ack1_in_gaa(s(T76), s(0), T71) → U13_gaa(T76, T71, ack27_in_ga(T76, T77))
U13_gaa(T76, T71, ack27_out_ga(T76, T77)) → U14_gaa(T76, T71, ack1_in_gaa(T76, T77, T71))
ack1_in_gaa(s(T84), s(s(T85)), T71) → U15_gaa(T84, T85, T71, ack38_in_gaa(s(T84), T85, X221))
U15_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, X221)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
ack1_in_gaa(s(T84), s(s(T85)), T71) → U16_gaa(T84, T85, T71, ack38_in_gaa(s(T84), T85, T87))
U16_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U17_gaa(T84, T85, T71, ack38_in_gaa(T84, T87, X222))
U17_gaa(T84, T85, T71, ack38_out_gaa(T84, T87, X222)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
U16_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U18_gaa(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_gaa(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → U19_gaa(T84, T85, T71, ack1_in_gaa(T84, T91, T71))
U19_gaa(T84, T85, T71, ack1_out_gaa(T84, T91, T71)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
U14_gaa(T76, T71, ack1_out_gaa(T76, T77, T71)) → ack1_out_gaa(s(T76), s(0), T71)
U11_gaa(T20, T22, ack1_out_gaa(T20, T24, T22)) → ack1_out_gaa(s(s(T20)), 0, T22)
U11_gga(T20, T22, ack1_out_gaa(T20, T24, T22)) → ack1_out_gga(s(s(T20)), 0, T22)
ack1_in_gga(s(T76), s(0), T71) → U12_gga(T76, T71, ack27_in_ga(T76, X200))
U12_gga(T76, T71, ack27_out_ga(T76, X200)) → ack1_out_gga(s(T76), s(0), T71)
ack1_in_gga(s(T76), s(0), T71) → U13_gga(T76, T71, ack27_in_ga(T76, T77))
U13_gga(T76, T71, ack27_out_ga(T76, T77)) → U14_gga(T76, T71, ack1_in_gaa(T76, T77, T71))
U14_gga(T76, T71, ack1_out_gaa(T76, T77, T71)) → ack1_out_gga(s(T76), s(0), T71)
ack1_in_gga(s(T84), s(s(T85)), T71) → U15_gga(T84, T85, T71, ack38_in_gga(s(T84), T85, X221))
ack38_in_gga(0, T44, s(T44)) → ack38_out_gga(0, T44, s(T44))
ack38_in_gga(s(T49), 0, X133) → U5_gga(T49, X133, ack27_in_ga(T49, X133))
U5_gga(T49, X133, ack27_out_ga(T49, X133)) → ack38_out_gga(s(T49), 0, X133)
ack38_in_gga(s(T54), s(T55), X151) → U6_gga(T54, T55, X151, ack38_in_gga(s(T54), T55, X150))
ack38_in_gga(s(T54), s(T55), X151) → U7_gga(T54, T55, X151, ack38_in_gga(s(T54), T55, T57))
U7_gga(T54, T55, X151, ack38_out_gga(s(T54), T55, T57)) → U8_gga(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
U8_gga(T54, T55, X151, ack38_out_gaa(T54, T57, X151)) → ack38_out_gga(s(T54), s(T55), X151)
U6_gga(T54, T55, X151, ack38_out_gga(s(T54), T55, X150)) → ack38_out_gga(s(T54), s(T55), X151)
U15_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, X221)) → ack1_out_gga(s(T84), s(s(T85)), T71)
ack1_in_gga(s(T84), s(s(T85)), T71) → U16_gga(T84, T85, T71, ack38_in_gga(s(T84), T85, T87))
U16_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → U17_gga(T84, T85, T71, ack38_in_gaa(T84, T87, X222))
U17_gga(T84, T85, T71, ack38_out_gaa(T84, T87, X222)) → ack1_out_gga(s(T84), s(s(T85)), T71)
U16_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → U18_gga(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_gga(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → U19_gga(T84, T85, T71, ack1_in_gaa(T84, T91, T71))
U19_gga(T84, T85, T71, ack1_out_gaa(T84, T91, T71)) → ack1_out_gga(s(T84), s(s(T85)), T71)

(45) UsableRulesProof (EQUIVALENT transformation)

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

(46) Obligation:

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

ACK21_IN_GA(T30, X73) → ACK27_IN_GA(T30, X73)
ACK27_IN_GA(s(T34), X97) → ACK21_IN_GA(T34, X96)
ACK27_IN_GA(s(T34), X97) → U3_GA(T34, X97, ack21_in_ga(T34, T36))
U3_GA(T34, X97, ack21_out_ga(T34, T36)) → ACK38_IN_GAA(T34, T36, X97)
ACK38_IN_GAA(s(T49), 0, X133) → ACK27_IN_GA(T49, X133)
ACK38_IN_GAA(s(T54), s(T55), X151) → ACK38_IN_GAA(s(T54), T55, X150)
ACK38_IN_GAA(s(T54), s(T55), X151) → U7_GAA(T54, T55, X151, ack38_in_gaa(s(T54), T55, T57))
U7_GAA(T54, T55, X151, ack38_out_gaa(s(T54), T55, T57)) → ACK38_IN_GAA(T54, T57, X151)

The TRS R consists of the following rules:

ack21_in_ga(T30, X73) → U1_ga(T30, X73, ack27_in_ga(T30, X73))
ack38_in_gaa(s(T49), 0, X133) → U5_gaa(T49, X133, ack27_in_ga(T49, X133))
ack38_in_gaa(s(T54), s(T55), X151) → U6_gaa(T54, T55, X151, ack38_in_gaa(s(T54), T55, X150))
ack38_in_gaa(s(T54), s(T55), X151) → U7_gaa(T54, T55, X151, ack38_in_gaa(s(T54), T55, T57))
U1_ga(T30, X73, ack27_out_ga(T30, X73)) → ack21_out_ga(T30, X73)
U5_gaa(T49, X133, ack27_out_ga(T49, X133)) → ack38_out_gaa(s(T49), 0, X133)
U6_gaa(T54, T55, X151, ack38_out_gaa(s(T54), T55, X150)) → ack38_out_gaa(s(T54), s(T55), X151)
U7_gaa(T54, T55, X151, ack38_out_gaa(s(T54), T55, T57)) → U8_gaa(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
ack27_in_ga(0, s(s(0))) → ack27_out_ga(0, s(s(0)))
ack27_in_ga(s(T34), X97) → U2_ga(T34, X97, ack21_in_ga(T34, X96))
ack27_in_ga(s(T34), X97) → U3_ga(T34, X97, ack21_in_ga(T34, T36))
U8_gaa(T54, T55, X151, ack38_out_gaa(T54, T57, X151)) → ack38_out_gaa(s(T54), s(T55), X151)
U2_ga(T34, X97, ack21_out_ga(T34, X96)) → ack27_out_ga(s(T34), X97)
U3_ga(T34, X97, ack21_out_ga(T34, T36)) → U4_ga(T34, X97, ack38_in_gaa(T34, T36, X97))
ack38_in_gaa(0, T44, s(T44)) → ack38_out_gaa(0, T44, s(T44))
U4_ga(T34, X97, ack38_out_gaa(T34, T36, X97)) → ack27_out_ga(s(T34), X97)

The argument filtering Pi contains the following mapping:
0 = 0
s(x1) = s(x1)
ack21_in_ga(x1, x2) = ack21_in_ga(x1)
U1_ga(x1, x2, x3) = U1_ga(x3)
ack27_in_ga(x1, x2) = ack27_in_ga(x1)
ack27_out_ga(x1, x2) = ack27_out_ga
U2_ga(x1, x2, x3) = U2_ga(x3)
ack21_out_ga(x1, x2) = ack21_out_ga
U3_ga(x1, x2, x3) = U3_ga(x1, x3)
U4_ga(x1, x2, x3) = U4_ga(x3)
ack38_in_gaa(x1, x2, x3) = ack38_in_gaa(x1)
ack38_out_gaa(x1, x2, x3) = ack38_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)
ACK21_IN_GA(x1, x2) = ACK21_IN_GA(x1)
ACK27_IN_GA(x1, x2) = ACK27_IN_GA(x1)
U3_GA(x1, x2, x3) = U3_GA(x1, x3)
ACK38_IN_GAA(x1, x2, x3) = ACK38_IN_GAA(x1)
U7_GAA(x1, x2, x3, x4) = U7_GAA(x1, x4)

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

(47) PiDPToQDPProof (SOUND transformation)

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

(48) Obligation:

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

ACK21_IN_GA(T30) → ACK27_IN_GA(T30)
ACK27_IN_GA(s(T34)) → ACK21_IN_GA(T34)
ACK27_IN_GA(s(T34)) → U3_GA(T34, ack21_in_ga(T34))
U3_GA(T34, ack21_out_ga) → ACK38_IN_GAA(T34)
ACK38_IN_GAA(s(T49)) → ACK27_IN_GA(T49)
ACK38_IN_GAA(s(T54)) → ACK38_IN_GAA(s(T54))
ACK38_IN_GAA(s(T54)) → U7_GAA(T54, ack38_in_gaa(s(T54)))
U7_GAA(T54, ack38_out_gaa) → ACK38_IN_GAA(T54)

The TRS R consists of the following rules:

ack21_in_ga(T30) → U1_ga(ack27_in_ga(T30))
ack38_in_gaa(s(T49)) → U5_gaa(ack27_in_ga(T49))
ack38_in_gaa(s(T54)) → U6_gaa(ack38_in_gaa(s(T54)))
ack38_in_gaa(s(T54)) → U7_gaa(T54, ack38_in_gaa(s(T54)))
U1_ga(ack27_out_ga) → ack21_out_ga
U5_gaa(ack27_out_ga) → ack38_out_gaa
U6_gaa(ack38_out_gaa) → ack38_out_gaa
U7_gaa(T54, ack38_out_gaa) → U8_gaa(ack38_in_gaa(T54))
ack27_in_ga(0) → ack27_out_ga
ack27_in_ga(s(T34)) → U2_ga(ack21_in_ga(T34))
ack27_in_ga(s(T34)) → U3_ga(T34, ack21_in_ga(T34))
U8_gaa(ack38_out_gaa) → ack38_out_gaa
U2_ga(ack21_out_ga) → ack27_out_ga
U3_ga(T34, ack21_out_ga) → U4_ga(ack38_in_gaa(T34))
ack38_in_gaa(0) → ack38_out_gaa
U4_ga(ack38_out_gaa) → ack27_out_ga

The set Q consists of the following terms:

ack21_in_ga(x0)
ack38_in_gaa(x0)
U1_ga(x0)
U5_gaa(x0)
U6_gaa(x0)
U7_gaa(x0, x1)
ack27_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.

(49) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACK27_IN_GA(s(T34)) → ACK21_IN_GA(T34)
U3_GA(T34, ack21_out_ga) → ACK38_IN_GAA(T34)
ACK38_IN_GAA(s(T49)) → ACK27_IN_GA(T49)
U7_GAA(T54, ack38_out_gaa) → ACK38_IN_GAA(T54)

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

POL(0) = 1
POL(ACK21_IN_GA(x₁)) = x₁
POL(ACK27_IN_GA(x₁)) = x₁
POL(ACK38_IN_GAA(x₁)) = x₁
POL(U1_ga(x₁)) = 0
POL(U2_ga(x₁)) = 0
POL(U3_GA(x₁, x₂)) = 1 + x₁
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(ack21_in_ga(x₁)) = 0
POL(ack21_out_ga) = 0
POL(ack27_in_ga(x₁)) = 1 + x₁
POL(ack27_out_ga) = 1
POL(ack38_in_gaa(x₁)) = 1 + x₁
POL(ack38_out_gaa) = 1
POL(s(x₁)) = 1 + x₁

The following usable rules [FROCOS05] were oriented: none

(50) Obligation:

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

ACK21_IN_GA(T30) → ACK27_IN_GA(T30)
ACK27_IN_GA(s(T34)) → U3_GA(T34, ack21_in_ga(T34))
ACK38_IN_GAA(s(T54)) → ACK38_IN_GAA(s(T54))
ACK38_IN_GAA(s(T54)) → U7_GAA(T54, ack38_in_gaa(s(T54)))

The TRS R consists of the following rules:

ack21_in_ga(T30) → U1_ga(ack27_in_ga(T30))
ack38_in_gaa(s(T49)) → U5_gaa(ack27_in_ga(T49))
ack38_in_gaa(s(T54)) → U6_gaa(ack38_in_gaa(s(T54)))
ack38_in_gaa(s(T54)) → U7_gaa(T54, ack38_in_gaa(s(T54)))
U1_ga(ack27_out_ga) → ack21_out_ga
U5_gaa(ack27_out_ga) → ack38_out_gaa
U6_gaa(ack38_out_gaa) → ack38_out_gaa
U7_gaa(T54, ack38_out_gaa) → U8_gaa(ack38_in_gaa(T54))
ack27_in_ga(0) → ack27_out_ga
ack27_in_ga(s(T34)) → U2_ga(ack21_in_ga(T34))
ack27_in_ga(s(T34)) → U3_ga(T34, ack21_in_ga(T34))
U8_gaa(ack38_out_gaa) → ack38_out_gaa
U2_ga(ack21_out_ga) → ack27_out_ga
U3_ga(T34, ack21_out_ga) → U4_ga(ack38_in_gaa(T34))
ack38_in_gaa(0) → ack38_out_gaa
U4_ga(ack38_out_gaa) → ack27_out_ga

The set Q consists of the following terms:

ack21_in_ga(x0)
ack38_in_gaa(x0)
U1_ga(x0)
U5_gaa(x0)
U6_gaa(x0)
U7_gaa(x0, x1)
ack27_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.

(51) DependencyGraphProof (EQUIVALENT transformation)

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

(52) Obligation:

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

ACK38_IN_GAA(s(T54)) → ACK38_IN_GAA(s(T54))

The TRS R consists of the following rules:

ack21_in_ga(T30) → U1_ga(ack27_in_ga(T30))
ack38_in_gaa(s(T49)) → U5_gaa(ack27_in_ga(T49))
ack38_in_gaa(s(T54)) → U6_gaa(ack38_in_gaa(s(T54)))
ack38_in_gaa(s(T54)) → U7_gaa(T54, ack38_in_gaa(s(T54)))
U1_ga(ack27_out_ga) → ack21_out_ga
U5_gaa(ack27_out_ga) → ack38_out_gaa
U6_gaa(ack38_out_gaa) → ack38_out_gaa
U7_gaa(T54, ack38_out_gaa) → U8_gaa(ack38_in_gaa(T54))
ack27_in_ga(0) → ack27_out_ga
ack27_in_ga(s(T34)) → U2_ga(ack21_in_ga(T34))
ack27_in_ga(s(T34)) → U3_ga(T34, ack21_in_ga(T34))
U8_gaa(ack38_out_gaa) → ack38_out_gaa
U2_ga(ack21_out_ga) → ack27_out_ga
U3_ga(T34, ack21_out_ga) → U4_ga(ack38_in_gaa(T34))
ack38_in_gaa(0) → ack38_out_gaa
U4_ga(ack38_out_gaa) → ack27_out_ga

The set Q consists of the following terms:

ack21_in_ga(x0)
ack38_in_gaa(x0)
U1_ga(x0)
U5_gaa(x0)
U6_gaa(x0)
U7_gaa(x0, x1)
ack27_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.

(53) 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.

(54) Obligation:

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

ACK38_IN_GAA(s(T54)) → ACK38_IN_GAA(s(T54))

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

ack21_in_ga(x0)
ack38_in_gaa(x0)
U1_ga(x0)
U5_gaa(x0)
U6_gaa(x0)
U7_gaa(x0, x1)
ack27_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.

(55) 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].

ack21_in_ga(x0)
ack38_in_gaa(x0)
U1_ga(x0)
U5_gaa(x0)
U6_gaa(x0)
U7_gaa(x0, x1)
ack27_in_ga(x0)
U8_gaa(x0)
U2_ga(x0)
U3_ga(x0, x1)
U4_ga(x0)

(56) Obligation:

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

ACK38_IN_GAA(s(T54)) → ACK38_IN_GAA(s(T54))

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

(57) NonTerminationProof (EQUIVALENT transformation)

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from ACK38_IN_GAA(s(T54)) to ACK38_IN_GAA(s(T54)).

(58) NO

(59) Obligation:

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

ACK38_IN_GGA(s(T54), s(T55), X151) → ACK38_IN_GGA(s(T54), T55, X150)

The TRS R consists of the following rules:

ack1_in_gga(0, T5, s(T5)) → ack1_out_gga(0, T5, s(T5))
ack1_in_gga(s(0), 0, s(s(0))) → ack1_out_gga(s(0), 0, s(s(0)))
ack1_in_gga(s(s(T20)), 0, T22) → U9_gga(T20, T22, ack21_in_ga(T20, X40))
ack21_in_ga(T30, X73) → U1_ga(T30, X73, ack27_in_ga(T30, X73))
ack27_in_ga(0, s(s(0))) → ack27_out_ga(0, s(s(0)))
ack27_in_ga(s(T34), X97) → U2_ga(T34, X97, ack21_in_ga(T34, X96))
U2_ga(T34, X97, ack21_out_ga(T34, X96)) → ack27_out_ga(s(T34), X97)
ack27_in_ga(s(T34), X97) → U3_ga(T34, X97, ack21_in_ga(T34, T36))
U3_ga(T34, X97, ack21_out_ga(T34, T36)) → U4_ga(T34, X97, ack38_in_gaa(T34, T36, X97))
ack38_in_gaa(0, T44, s(T44)) → ack38_out_gaa(0, T44, s(T44))
ack38_in_gaa(s(T49), 0, X133) → U5_gaa(T49, X133, ack27_in_ga(T49, X133))
U5_gaa(T49, X133, ack27_out_ga(T49, X133)) → ack38_out_gaa(s(T49), 0, X133)
ack38_in_gaa(s(T54), s(T55), X151) → U6_gaa(T54, T55, X151, ack38_in_gaa(s(T54), T55, X150))
ack38_in_gaa(s(T54), s(T55), X151) → U7_gaa(T54, T55, X151, ack38_in_gaa(s(T54), T55, T57))
U7_gaa(T54, T55, X151, ack38_out_gaa(s(T54), T55, T57)) → U8_gaa(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
U8_gaa(T54, T55, X151, ack38_out_gaa(T54, T57, X151)) → ack38_out_gaa(s(T54), s(T55), X151)
U6_gaa(T54, T55, X151, ack38_out_gaa(s(T54), T55, X150)) → ack38_out_gaa(s(T54), s(T55), X151)
U4_ga(T34, X97, ack38_out_gaa(T34, T36, X97)) → ack27_out_ga(s(T34), X97)
U1_ga(T30, X73, ack27_out_ga(T30, X73)) → ack21_out_ga(T30, X73)
U9_gga(T20, T22, ack21_out_ga(T20, X40)) → ack1_out_gga(s(s(T20)), 0, T22)
ack1_in_gga(s(s(T20)), 0, T22) → U10_gga(T20, T22, ack21_in_ga(T20, T24))
U10_gga(T20, T22, ack21_out_ga(T20, T24)) → U11_gga(T20, T22, ack1_in_gaa(T20, T24, T22))
ack1_in_gaa(0, T5, s(T5)) → ack1_out_gaa(0, T5, s(T5))
ack1_in_gaa(s(0), 0, s(s(0))) → ack1_out_gaa(s(0), 0, s(s(0)))
ack1_in_gaa(s(s(T20)), 0, T22) → U9_gaa(T20, T22, ack21_in_ga(T20, X40))
U9_gaa(T20, T22, ack21_out_ga(T20, X40)) → ack1_out_gaa(s(s(T20)), 0, T22)
ack1_in_gaa(s(s(T20)), 0, T22) → U10_gaa(T20, T22, ack21_in_ga(T20, T24))
U10_gaa(T20, T22, ack21_out_ga(T20, T24)) → U11_gaa(T20, T22, ack1_in_gaa(T20, T24, T22))
ack1_in_gaa(s(T76), s(0), T71) → U12_gaa(T76, T71, ack27_in_ga(T76, X200))
U12_gaa(T76, T71, ack27_out_ga(T76, X200)) → ack1_out_gaa(s(T76), s(0), T71)
ack1_in_gaa(s(T76), s(0), T71) → U13_gaa(T76, T71, ack27_in_ga(T76, T77))
U13_gaa(T76, T71, ack27_out_ga(T76, T77)) → U14_gaa(T76, T71, ack1_in_gaa(T76, T77, T71))
ack1_in_gaa(s(T84), s(s(T85)), T71) → U15_gaa(T84, T85, T71, ack38_in_gaa(s(T84), T85, X221))
U15_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, X221)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
ack1_in_gaa(s(T84), s(s(T85)), T71) → U16_gaa(T84, T85, T71, ack38_in_gaa(s(T84), T85, T87))
U16_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U17_gaa(T84, T85, T71, ack38_in_gaa(T84, T87, X222))
U17_gaa(T84, T85, T71, ack38_out_gaa(T84, T87, X222)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
U16_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U18_gaa(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_gaa(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → U19_gaa(T84, T85, T71, ack1_in_gaa(T84, T91, T71))
U19_gaa(T84, T85, T71, ack1_out_gaa(T84, T91, T71)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
U14_gaa(T76, T71, ack1_out_gaa(T76, T77, T71)) → ack1_out_gaa(s(T76), s(0), T71)
U11_gaa(T20, T22, ack1_out_gaa(T20, T24, T22)) → ack1_out_gaa(s(s(T20)), 0, T22)
U11_gga(T20, T22, ack1_out_gaa(T20, T24, T22)) → ack1_out_gga(s(s(T20)), 0, T22)
ack1_in_gga(s(T76), s(0), T71) → U12_gga(T76, T71, ack27_in_ga(T76, X200))
U12_gga(T76, T71, ack27_out_ga(T76, X200)) → ack1_out_gga(s(T76), s(0), T71)
ack1_in_gga(s(T76), s(0), T71) → U13_gga(T76, T71, ack27_in_ga(T76, T77))
U13_gga(T76, T71, ack27_out_ga(T76, T77)) → U14_gga(T76, T71, ack1_in_gaa(T76, T77, T71))
U14_gga(T76, T71, ack1_out_gaa(T76, T77, T71)) → ack1_out_gga(s(T76), s(0), T71)
ack1_in_gga(s(T84), s(s(T85)), T71) → U15_gga(T84, T85, T71, ack38_in_gga(s(T84), T85, X221))
ack38_in_gga(0, T44, s(T44)) → ack38_out_gga(0, T44, s(T44))
ack38_in_gga(s(T49), 0, X133) → U5_gga(T49, X133, ack27_in_ga(T49, X133))
U5_gga(T49, X133, ack27_out_ga(T49, X133)) → ack38_out_gga(s(T49), 0, X133)
ack38_in_gga(s(T54), s(T55), X151) → U6_gga(T54, T55, X151, ack38_in_gga(s(T54), T55, X150))
ack38_in_gga(s(T54), s(T55), X151) → U7_gga(T54, T55, X151, ack38_in_gga(s(T54), T55, T57))
U7_gga(T54, T55, X151, ack38_out_gga(s(T54), T55, T57)) → U8_gga(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
U8_gga(T54, T55, X151, ack38_out_gaa(T54, T57, X151)) → ack38_out_gga(s(T54), s(T55), X151)
U6_gga(T54, T55, X151, ack38_out_gga(s(T54), T55, X150)) → ack38_out_gga(s(T54), s(T55), X151)
U15_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, X221)) → ack1_out_gga(s(T84), s(s(T85)), T71)
ack1_in_gga(s(T84), s(s(T85)), T71) → U16_gga(T84, T85, T71, ack38_in_gga(s(T84), T85, T87))
U16_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → U17_gga(T84, T85, T71, ack38_in_gaa(T84, T87, X222))
U17_gga(T84, T85, T71, ack38_out_gaa(T84, T87, X222)) → ack1_out_gga(s(T84), s(s(T85)), T71)
U16_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → U18_gga(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_gga(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → U19_gga(T84, T85, T71, ack1_in_gaa(T84, T91, T71))
U19_gga(T84, T85, T71, ack1_out_gaa(T84, T91, T71)) → ack1_out_gga(s(T84), s(s(T85)), T71)

(60) UsableRulesProof (EQUIVALENT transformation)

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

(61) Obligation:

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

ACK38_IN_GGA(s(T54), s(T55), X151) → ACK38_IN_GGA(s(T54), T55, X150)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
ACK38_IN_GGA(x1, x2, x3) = ACK38_IN_GGA(x1, x2)

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

(62) PiDPToQDPProof (SOUND transformation)

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

(63) Obligation:

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

ACK38_IN_GGA(s(T54), s(T55)) → ACK38_IN_GGA(s(T54), T55)

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

(64) 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:

ACK38_IN_GGA(s(T54), s(T55)) → ACK38_IN_GGA(s(T54), T55)
The graph contains the following edges 1 >= 1, 2 > 2

(65) YES

(66) Obligation:

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

ACK1_IN_GAA(s(s(T20)), 0, T22) → U10_GAA(T20, T22, ack21_in_ga(T20, T24))
U10_GAA(T20, T22, ack21_out_ga(T20, T24)) → ACK1_IN_GAA(T20, T24, T22)
ACK1_IN_GAA(s(T76), s(0), T71) → U13_GAA(T76, T71, ack27_in_ga(T76, T77))
U13_GAA(T76, T71, ack27_out_ga(T76, T77)) → ACK1_IN_GAA(T76, T77, T71)
ACK1_IN_GAA(s(T84), s(s(T85)), T71) → U16_GAA(T84, T85, T71, ack38_in_gaa(s(T84), T85, T87))
U16_GAA(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U18_GAA(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_GAA(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → ACK1_IN_GAA(T84, T91, T71)

The TRS R consists of the following rules:

ack1_in_gga(0, T5, s(T5)) → ack1_out_gga(0, T5, s(T5))
ack1_in_gga(s(0), 0, s(s(0))) → ack1_out_gga(s(0), 0, s(s(0)))
ack1_in_gga(s(s(T20)), 0, T22) → U9_gga(T20, T22, ack21_in_ga(T20, X40))
ack21_in_ga(T30, X73) → U1_ga(T30, X73, ack27_in_ga(T30, X73))
ack27_in_ga(0, s(s(0))) → ack27_out_ga(0, s(s(0)))
ack27_in_ga(s(T34), X97) → U2_ga(T34, X97, ack21_in_ga(T34, X96))
U2_ga(T34, X97, ack21_out_ga(T34, X96)) → ack27_out_ga(s(T34), X97)
ack27_in_ga(s(T34), X97) → U3_ga(T34, X97, ack21_in_ga(T34, T36))
U3_ga(T34, X97, ack21_out_ga(T34, T36)) → U4_ga(T34, X97, ack38_in_gaa(T34, T36, X97))
ack38_in_gaa(0, T44, s(T44)) → ack38_out_gaa(0, T44, s(T44))
ack38_in_gaa(s(T49), 0, X133) → U5_gaa(T49, X133, ack27_in_ga(T49, X133))
U5_gaa(T49, X133, ack27_out_ga(T49, X133)) → ack38_out_gaa(s(T49), 0, X133)
ack38_in_gaa(s(T54), s(T55), X151) → U6_gaa(T54, T55, X151, ack38_in_gaa(s(T54), T55, X150))
ack38_in_gaa(s(T54), s(T55), X151) → U7_gaa(T54, T55, X151, ack38_in_gaa(s(T54), T55, T57))
U7_gaa(T54, T55, X151, ack38_out_gaa(s(T54), T55, T57)) → U8_gaa(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
U8_gaa(T54, T55, X151, ack38_out_gaa(T54, T57, X151)) → ack38_out_gaa(s(T54), s(T55), X151)
U6_gaa(T54, T55, X151, ack38_out_gaa(s(T54), T55, X150)) → ack38_out_gaa(s(T54), s(T55), X151)
U4_ga(T34, X97, ack38_out_gaa(T34, T36, X97)) → ack27_out_ga(s(T34), X97)
U1_ga(T30, X73, ack27_out_ga(T30, X73)) → ack21_out_ga(T30, X73)
U9_gga(T20, T22, ack21_out_ga(T20, X40)) → ack1_out_gga(s(s(T20)), 0, T22)
ack1_in_gga(s(s(T20)), 0, T22) → U10_gga(T20, T22, ack21_in_ga(T20, T24))
U10_gga(T20, T22, ack21_out_ga(T20, T24)) → U11_gga(T20, T22, ack1_in_gaa(T20, T24, T22))
ack1_in_gaa(0, T5, s(T5)) → ack1_out_gaa(0, T5, s(T5))
ack1_in_gaa(s(0), 0, s(s(0))) → ack1_out_gaa(s(0), 0, s(s(0)))
ack1_in_gaa(s(s(T20)), 0, T22) → U9_gaa(T20, T22, ack21_in_ga(T20, X40))
U9_gaa(T20, T22, ack21_out_ga(T20, X40)) → ack1_out_gaa(s(s(T20)), 0, T22)
ack1_in_gaa(s(s(T20)), 0, T22) → U10_gaa(T20, T22, ack21_in_ga(T20, T24))
U10_gaa(T20, T22, ack21_out_ga(T20, T24)) → U11_gaa(T20, T22, ack1_in_gaa(T20, T24, T22))
ack1_in_gaa(s(T76), s(0), T71) → U12_gaa(T76, T71, ack27_in_ga(T76, X200))
U12_gaa(T76, T71, ack27_out_ga(T76, X200)) → ack1_out_gaa(s(T76), s(0), T71)
ack1_in_gaa(s(T76), s(0), T71) → U13_gaa(T76, T71, ack27_in_ga(T76, T77))
U13_gaa(T76, T71, ack27_out_ga(T76, T77)) → U14_gaa(T76, T71, ack1_in_gaa(T76, T77, T71))
ack1_in_gaa(s(T84), s(s(T85)), T71) → U15_gaa(T84, T85, T71, ack38_in_gaa(s(T84), T85, X221))
U15_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, X221)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
ack1_in_gaa(s(T84), s(s(T85)), T71) → U16_gaa(T84, T85, T71, ack38_in_gaa(s(T84), T85, T87))
U16_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U17_gaa(T84, T85, T71, ack38_in_gaa(T84, T87, X222))
U17_gaa(T84, T85, T71, ack38_out_gaa(T84, T87, X222)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
U16_gaa(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U18_gaa(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_gaa(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → U19_gaa(T84, T85, T71, ack1_in_gaa(T84, T91, T71))
U19_gaa(T84, T85, T71, ack1_out_gaa(T84, T91, T71)) → ack1_out_gaa(s(T84), s(s(T85)), T71)
U14_gaa(T76, T71, ack1_out_gaa(T76, T77, T71)) → ack1_out_gaa(s(T76), s(0), T71)
U11_gaa(T20, T22, ack1_out_gaa(T20, T24, T22)) → ack1_out_gaa(s(s(T20)), 0, T22)
U11_gga(T20, T22, ack1_out_gaa(T20, T24, T22)) → ack1_out_gga(s(s(T20)), 0, T22)
ack1_in_gga(s(T76), s(0), T71) → U12_gga(T76, T71, ack27_in_ga(T76, X200))
U12_gga(T76, T71, ack27_out_ga(T76, X200)) → ack1_out_gga(s(T76), s(0), T71)
ack1_in_gga(s(T76), s(0), T71) → U13_gga(T76, T71, ack27_in_ga(T76, T77))
U13_gga(T76, T71, ack27_out_ga(T76, T77)) → U14_gga(T76, T71, ack1_in_gaa(T76, T77, T71))
U14_gga(T76, T71, ack1_out_gaa(T76, T77, T71)) → ack1_out_gga(s(T76), s(0), T71)
ack1_in_gga(s(T84), s(s(T85)), T71) → U15_gga(T84, T85, T71, ack38_in_gga(s(T84), T85, X221))
ack38_in_gga(0, T44, s(T44)) → ack38_out_gga(0, T44, s(T44))
ack38_in_gga(s(T49), 0, X133) → U5_gga(T49, X133, ack27_in_ga(T49, X133))
U5_gga(T49, X133, ack27_out_ga(T49, X133)) → ack38_out_gga(s(T49), 0, X133)
ack38_in_gga(s(T54), s(T55), X151) → U6_gga(T54, T55, X151, ack38_in_gga(s(T54), T55, X150))
ack38_in_gga(s(T54), s(T55), X151) → U7_gga(T54, T55, X151, ack38_in_gga(s(T54), T55, T57))
U7_gga(T54, T55, X151, ack38_out_gga(s(T54), T55, T57)) → U8_gga(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
U8_gga(T54, T55, X151, ack38_out_gaa(T54, T57, X151)) → ack38_out_gga(s(T54), s(T55), X151)
U6_gga(T54, T55, X151, ack38_out_gga(s(T54), T55, X150)) → ack38_out_gga(s(T54), s(T55), X151)
U15_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, X221)) → ack1_out_gga(s(T84), s(s(T85)), T71)
ack1_in_gga(s(T84), s(s(T85)), T71) → U16_gga(T84, T85, T71, ack38_in_gga(s(T84), T85, T87))
U16_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → U17_gga(T84, T85, T71, ack38_in_gaa(T84, T87, X222))
U17_gga(T84, T85, T71, ack38_out_gaa(T84, T87, X222)) → ack1_out_gga(s(T84), s(s(T85)), T71)
U16_gga(T84, T85, T71, ack38_out_gga(s(T84), T85, T87)) → U18_gga(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_gga(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → U19_gga(T84, T85, T71, ack1_in_gaa(T84, T91, T71))
U19_gga(T84, T85, T71, ack1_out_gaa(T84, T91, T71)) → ack1_out_gga(s(T84), s(s(T85)), T71)

(67) UsableRulesProof (EQUIVALENT transformation)

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

(68) Obligation:

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

ACK1_IN_GAA(s(s(T20)), 0, T22) → U10_GAA(T20, T22, ack21_in_ga(T20, T24))
U10_GAA(T20, T22, ack21_out_ga(T20, T24)) → ACK1_IN_GAA(T20, T24, T22)
ACK1_IN_GAA(s(T76), s(0), T71) → U13_GAA(T76, T71, ack27_in_ga(T76, T77))
U13_GAA(T76, T71, ack27_out_ga(T76, T77)) → ACK1_IN_GAA(T76, T77, T71)
ACK1_IN_GAA(s(T84), s(s(T85)), T71) → U16_GAA(T84, T85, T71, ack38_in_gaa(s(T84), T85, T87))
U16_GAA(T84, T85, T71, ack38_out_gaa(s(T84), T85, T87)) → U18_GAA(T84, T85, T71, ack38_in_gaa(T84, T87, T91))
U18_GAA(T84, T85, T71, ack38_out_gaa(T84, T87, T91)) → ACK1_IN_GAA(T84, T91, T71)

The TRS R consists of the following rules:

ack21_in_ga(T30, X73) → U1_ga(T30, X73, ack27_in_ga(T30, X73))
ack27_in_ga(0, s(s(0))) → ack27_out_ga(0, s(s(0)))
ack27_in_ga(s(T34), X97) → U2_ga(T34, X97, ack21_in_ga(T34, X96))
ack27_in_ga(s(T34), X97) → U3_ga(T34, X97, ack21_in_ga(T34, T36))
ack38_in_gaa(s(T49), 0, X133) → U5_gaa(T49, X133, ack27_in_ga(T49, X133))
ack38_in_gaa(s(T54), s(T55), X151) → U6_gaa(T54, T55, X151, ack38_in_gaa(s(T54), T55, X150))
ack38_in_gaa(s(T54), s(T55), X151) → U7_gaa(T54, T55, X151, ack38_in_gaa(s(T54), T55, T57))
ack38_in_gaa(0, T44, s(T44)) → ack38_out_gaa(0, T44, s(T44))
U1_ga(T30, X73, ack27_out_ga(T30, X73)) → ack21_out_ga(T30, X73)
U2_ga(T34, X97, ack21_out_ga(T34, X96)) → ack27_out_ga(s(T34), X97)
U3_ga(T34, X97, ack21_out_ga(T34, T36)) → U4_ga(T34, X97, ack38_in_gaa(T34, T36, X97))
U5_gaa(T49, X133, ack27_out_ga(T49, X133)) → ack38_out_gaa(s(T49), 0, X133)
U6_gaa(T54, T55, X151, ack38_out_gaa(s(T54), T55, X150)) → ack38_out_gaa(s(T54), s(T55), X151)
U7_gaa(T54, T55, X151, ack38_out_gaa(s(T54), T55, T57)) → U8_gaa(T54, T55, X151, ack38_in_gaa(T54, T57, X151))
U4_ga(T34, X97, ack38_out_gaa(T34, T36, X97)) → ack27_out_ga(s(T34), X97)
U8_gaa(T54, T55, X151, ack38_out_gaa(T54, T57, X151)) → ack38_out_gaa(s(T54), s(T55), X151)

The argument filtering Pi contains the following mapping:
0 = 0
s(x1) = s(x1)
ack21_in_ga(x1, x2) = ack21_in_ga(x1)
U1_ga(x1, x2, x3) = U1_ga(x3)
ack27_in_ga(x1, x2) = ack27_in_ga(x1)
ack27_out_ga(x1, x2) = ack27_out_ga
U2_ga(x1, x2, x3) = U2_ga(x3)
ack21_out_ga(x1, x2) = ack21_out_ga
U3_ga(x1, x2, x3) = U3_ga(x1, x3)
U4_ga(x1, x2, x3) = U4_ga(x3)
ack38_in_gaa(x1, x2, x3) = ack38_in_gaa(x1)
ack38_out_gaa(x1, x2, x3) = ack38_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)
ACK1_IN_GAA(x1, x2, x3) = ACK1_IN_GAA(x1)
U10_GAA(x1, x2, x3) = U10_GAA(x1, x3)
U13_GAA(x1, x2, x3) = U13_GAA(x1, x3)
U16_GAA(x1, x2, x3, x4) = U16_GAA(x1, x4)
U18_GAA(x1, x2, x3, x4) = U18_GAA(x1, x4)

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

(69) PiDPToQDPProof (SOUND transformation)

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

(70) Obligation:

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

ACK1_IN_GAA(s(s(T20))) → U10_GAA(T20, ack21_in_ga(T20))
U10_GAA(T20, ack21_out_ga) → ACK1_IN_GAA(T20)
ACK1_IN_GAA(s(T76)) → U13_GAA(T76, ack27_in_ga(T76))
U13_GAA(T76, ack27_out_ga) → ACK1_IN_GAA(T76)
ACK1_IN_GAA(s(T84)) → U16_GAA(T84, ack38_in_gaa(s(T84)))
U16_GAA(T84, ack38_out_gaa) → U18_GAA(T84, ack38_in_gaa(T84))
U18_GAA(T84, ack38_out_gaa) → ACK1_IN_GAA(T84)

The TRS R consists of the following rules:

ack21_in_ga(T30) → U1_ga(ack27_in_ga(T30))
ack27_in_ga(0) → ack27_out_ga
ack27_in_ga(s(T34)) → U2_ga(ack21_in_ga(T34))
ack27_in_ga(s(T34)) → U3_ga(T34, ack21_in_ga(T34))
ack38_in_gaa(s(T49)) → U5_gaa(ack27_in_ga(T49))
ack38_in_gaa(s(T54)) → U6_gaa(ack38_in_gaa(s(T54)))
ack38_in_gaa(s(T54)) → U7_gaa(T54, ack38_in_gaa(s(T54)))
ack38_in_gaa(0) → ack38_out_gaa
U1_ga(ack27_out_ga) → ack21_out_ga
U2_ga(ack21_out_ga) → ack27_out_ga
U3_ga(T34, ack21_out_ga) → U4_ga(ack38_in_gaa(T34))
U5_gaa(ack27_out_ga) → ack38_out_gaa
U6_gaa(ack38_out_gaa) → ack38_out_gaa
U7_gaa(T54, ack38_out_gaa) → U8_gaa(ack38_in_gaa(T54))
U4_ga(ack38_out_gaa) → ack27_out_ga
U8_gaa(ack38_out_gaa) → ack38_out_gaa

The set Q consists of the following terms:

ack21_in_ga(x0)
ack27_in_ga(x0)
ack38_in_gaa(x0)
U1_ga(x0)
U2_ga(x0)
U3_ga(x0, x1)
U5_gaa(x0)
U6_gaa(x0)
U7_gaa(x0, x1)
U4_ga(x0)
U8_gaa(x0)

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

(71) 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:

U10_GAA(T20, ack21_out_ga) → ACK1_IN_GAA(T20)
The graph contains the following edges 1 >= 1
ACK1_IN_GAA(s(s(T20))) → U10_GAA(T20, ack21_in_ga(T20))
The graph contains the following edges 1 > 1
U16_GAA(T84, ack38_out_gaa) → U18_GAA(T84, ack38_in_gaa(T84))
The graph contains the following edges 1 >= 1
U13_GAA(T76, ack27_out_ga) → ACK1_IN_GAA(T76)
The graph contains the following edges 1 >= 1
U18_GAA(T84, ack38_out_gaa) → ACK1_IN_GAA(T84)
The graph contains the following edges 1 >= 1
ACK1_IN_GAA(s(T76)) → U13_GAA(T76, ack27_in_ga(T76))
The graph contains the following edges 1 > 1
ACK1_IN_GAA(s(T84)) → U16_GAA(T84, ack38_in_gaa(s(T84)))
The graph contains the following edges 1 > 1

(0) Obligation:

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) PrologToPiTRSProof (SOUND transformation)

(4) Obligation:

(5) DependencyPairsProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Complex Obligation (AND)

(9) Obligation:

(10) UsableRulesProof (EQUIVALENT transformation)

(11) Obligation:

(12) PiDPToQDPProof (SOUND transformation)

(13) Obligation:

(14) QDPOrderProof (EQUIVALENT transformation)

(15) Obligation:

(16) DependencyGraphProof (EQUIVALENT transformation)

(17) Obligation:

(18) UsableRulesProof (EQUIVALENT transformation)

(19) Obligation:

(20) QReductionProof (EQUIVALENT transformation)

(21) Obligation:

(22) NonTerminationProof (EQUIVALENT transformation)

(23) NO

(24) Obligation:

(25) UsableRulesProof (EQUIVALENT transformation)

(26) Obligation:

(27) PiDPToQDPProof (SOUND transformation)

(28) Obligation:

(29) QDPSizeChangeProof (EQUIVALENT transformation)

(30) YES

(31) Obligation:

(32) UsableRulesProof (EQUIVALENT transformation)

(33) Obligation:

(34) PiDPToQDPProof (SOUND transformation)

(35) Obligation:

(36) QDPSizeChangeProof (EQUIVALENT transformation)

(37) YES

(38) PrologToPiTRSProof (SOUND transformation)

(39) Obligation:

(40) DependencyPairsProof (EQUIVALENT transformation)

(41) Obligation:

(42) DependencyGraphProof (EQUIVALENT transformation)

(43) Complex Obligation (AND)

(44) Obligation:

(45) UsableRulesProof (EQUIVALENT transformation)

(46) Obligation:

(47) PiDPToQDPProof (SOUND transformation)

(48) Obligation:

(49) QDPOrderProof (EQUIVALENT transformation)

(50) Obligation:

(51) DependencyGraphProof (EQUIVALENT transformation)

(52) Obligation:

(53) UsableRulesProof (EQUIVALENT transformation)

(54) Obligation:

(55) QReductionProof (EQUIVALENT transformation)

(56) Obligation:

(57) NonTerminationProof (EQUIVALENT transformation)

(58) NO

(59) Obligation:

(60) UsableRulesProof (EQUIVALENT transformation)

(61) Obligation:

(62) PiDPToQDPProof (SOUND transformation)

(63) Obligation:

(64) QDPSizeChangeProof (EQUIVALENT transformation)

(65) YES

(66) Obligation:

(67) UsableRulesProof (EQUIVALENT transformation)

(68) Obligation:

(69) PiDPToQDPProof (SOUND transformation)

(70) Obligation:

(71) QDPSizeChangeProof (EQUIVALENT transformation)

(72) YES