Left Termination of the query pattern ack_in_3(g, g, a) w.r.t. the given Prolog program could successfully be proven:

(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) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT 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:

Triples:

ack21(T30, X73) :- ack27(T30, X73).
ack27(s(T34), X97) :- ack21(T34, X96).
ack27(s(T34), X97) :- ','(ackc21(T34, T36), ack38(T34, T36, X97)).
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) :- ','(ackc38(s(T54), T55, T57), ack38(T54, T57, X151)).
ack1(s(s(T20)), 0, T22) :- ack21(T20, X40).
ack1(s(s(T20)), 0, T22) :- ','(ackc21(T20, T24), ack1(T20, T24, T22)).
ack1(s(T76), s(0), T71) :- ack27(T76, X200).
ack1(s(T76), s(0), T71) :- ','(ackc27(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) :- ','(ackc38(s(T84), T85, T87), ack38(T84, T87, X222)).
ack1(s(T84), s(s(T85)), T71) :- ','(ackc38(s(T84), T85, T87), ','(ackc38(T84, T87, T91), ack1(T84, T91, T71))).

Clauses:

ackc1(0, T5, s(T5)).
ackc1(s(0), 0, s(s(0))).
ackc1(s(s(T20)), 0, T22) :- ','(ackc21(T20, T24), ackc1(T20, T24, T22)).
ackc1(s(T76), s(0), T71) :- ','(ackc27(T76, T77), ackc1(T76, T77, T71)).
ackc1(s(T84), s(s(T85)), T71) :- ','(ackc38(s(T84), T85, T87), ','(ackc38(T84, T87, T91), ackc1(T84, T91, T71))).
ackc21(T30, X73) :- ackc27(T30, X73).
ackc27(0, s(s(0))).
ackc27(s(T34), X97) :- ','(ackc21(T34, T36), ackc38(T34, T36, X97)).
ackc38(0, T44, s(T44)).
ackc38(s(T49), 0, X133) :- ackc27(T49, X133).
ackc38(s(T54), s(T55), X151) :- ','(ackc38(s(T54), T55, T57), ackc38(T54, T57, X151)).

Afs:

ack1(x1, x2, x3)  =  ack1(x1, x2)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
ack1_in: (b,b,f)
ack21_in: (b,f)
ack27_in: (b,f)
ackc21_in: (b,f)
ackc27_in: (b,f)
ackc38_in: (b,b,f)
ack38_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
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, ackc21_in_ga(T34, T36))
U3_GA(T34, X97, ackc21_out_ga(T34, T36)) → U4_GA(T34, X97, ack38_in_gga(T34, T36, X97))
U3_GA(T34, X97, ackc21_out_ga(T34, T36)) → ACK38_IN_GGA(T34, T36, X97)
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, ackc38_in_gga(s(T54), T55, T57))
U7_GGA(T54, T55, X151, ackc38_out_gga(s(T54), T55, T57)) → U8_GGA(T54, T55, X151, ack38_in_gga(T54, T57, X151))
U7_GGA(T54, T55, X151, ackc38_out_gga(s(T54), T55, T57)) → ACK38_IN_GGA(T54, T57, X151)
ACK1_IN_GGA(s(s(T20)), 0, T22) → U10_GGA(T20, T22, ackc21_in_ga(T20, T24))
U10_GGA(T20, T22, ackc21_out_ga(T20, T24)) → U11_GGA(T20, T22, ack1_in_gga(T20, T24, T22))
U10_GGA(T20, T22, ackc21_out_ga(T20, T24)) → ACK1_IN_GGA(T20, T24, T22)
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, ackc27_in_ga(T76, T77))
U13_GGA(T76, T71, ackc27_out_ga(T76, T77)) → U14_GGA(T76, T71, ack1_in_gga(T76, T77, T71))
U13_GGA(T76, T71, ackc27_out_ga(T76, T77)) → ACK1_IN_GGA(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)
ACK1_IN_GGA(s(T84), s(s(T85)), T71) → U16_GGA(T84, T85, T71, ackc38_in_gga(s(T84), T85, T87))
U16_GGA(T84, T85, T71, ackc38_out_gga(s(T84), T85, T87)) → U17_GGA(T84, T85, T71, ack38_in_gga(T84, T87, X222))
U16_GGA(T84, T85, T71, ackc38_out_gga(s(T84), T85, T87)) → ACK38_IN_GGA(T84, T87, X222)
U16_GGA(T84, T85, T71, ackc38_out_gga(s(T84), T85, T87)) → U18_GGA(T84, T85, T71, ackc38_in_gga(T84, T87, T91))
U18_GGA(T84, T85, T71, ackc38_out_gga(T84, T87, T91)) → U19_GGA(T84, T85, T71, ack1_in_gga(T84, T91, T71))
U18_GGA(T84, T85, T71, ackc38_out_gga(T84, T87, T91)) → ACK1_IN_GGA(T84, T91, T71)

The TRS R consists of the following rules:

ackc21_in_ga(T30, X73) → U28_ga(T30, X73, ackc27_in_ga(T30, X73))
ackc27_in_ga(0, s(s(0))) → ackc27_out_ga(0, s(s(0)))
ackc27_in_ga(s(T34), X97) → U29_ga(T34, X97, ackc21_in_ga(T34, T36))
U29_ga(T34, X97, ackc21_out_ga(T34, T36)) → U30_ga(T34, X97, ackc38_in_gga(T34, T36, X97))
ackc38_in_gga(0, T44, s(T44)) → ackc38_out_gga(0, T44, s(T44))
ackc38_in_gga(s(T49), 0, X133) → U31_gga(T49, X133, ackc27_in_ga(T49, X133))
U31_gga(T49, X133, ackc27_out_ga(T49, X133)) → ackc38_out_gga(s(T49), 0, X133)
ackc38_in_gga(s(T54), s(T55), X151) → U32_gga(T54, T55, X151, ackc38_in_gga(s(T54), T55, T57))
U32_gga(T54, T55, X151, ackc38_out_gga(s(T54), T55, T57)) → U33_gga(T54, T55, X151, ackc38_in_gga(T54, T57, X151))
U33_gga(T54, T55, X151, ackc38_out_gga(T54, T57, X151)) → ackc38_out_gga(s(T54), s(T55), X151)
U30_ga(T34, X97, ackc38_out_gga(T34, T36, X97)) → ackc27_out_ga(s(T34), X97)
U28_ga(T30, X73, ackc27_out_ga(T30, X73)) → ackc21_out_ga(T30, X73)

The argument filtering Pi contains the following mapping:
ack1_in_gga(x1, x2, x3)  =  ack1_in_gga(x1, x2)
s(x1)  =  s(x1)
0  =  0
ack21_in_ga(x1, x2)  =  ack21_in_ga(x1)
ack27_in_ga(x1, x2)  =  ack27_in_ga(x1)
ackc21_in_ga(x1, x2)  =  ackc21_in_ga(x1)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
ackc27_in_ga(x1, x2)  =  ackc27_in_ga(x1)
ackc27_out_ga(x1, x2)  =  ackc27_out_ga(x1, x2)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
ackc21_out_ga(x1, x2)  =  ackc21_out_ga(x1, x2)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
ackc38_in_gga(x1, x2, x3)  =  ackc38_in_gga(x1, x2)
ackc38_out_gga(x1, x2, x3)  =  ackc38_out_gga(x1, x2, x3)
U31_gga(x1, x2, x3)  =  U31_gga(x1, x3)
U32_gga(x1, x2, x3, x4)  =  U32_gga(x1, x2, x4)
U33_gga(x1, x2, x3, x4)  =  U33_gga(x1, x2, x4)
ack38_in_gga(x1, x2, x3)  =  ack38_in_gga(x1, x2)
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_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)
U10_GGA(x1, x2, x3)  =  U10_GGA(x1, x3)
U11_GGA(x1, x2, x3)  =  U11_GGA(x1, x3)
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)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

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, ackc21_in_ga(T34, T36))
U3_GA(T34, X97, ackc21_out_ga(T34, T36)) → U4_GA(T34, X97, ack38_in_gga(T34, T36, X97))
U3_GA(T34, X97, ackc21_out_ga(T34, T36)) → ACK38_IN_GGA(T34, T36, X97)
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, ackc38_in_gga(s(T54), T55, T57))
U7_GGA(T54, T55, X151, ackc38_out_gga(s(T54), T55, T57)) → U8_GGA(T54, T55, X151, ack38_in_gga(T54, T57, X151))
U7_GGA(T54, T55, X151, ackc38_out_gga(s(T54), T55, T57)) → ACK38_IN_GGA(T54, T57, X151)
ACK1_IN_GGA(s(s(T20)), 0, T22) → U10_GGA(T20, T22, ackc21_in_ga(T20, T24))
U10_GGA(T20, T22, ackc21_out_ga(T20, T24)) → U11_GGA(T20, T22, ack1_in_gga(T20, T24, T22))
U10_GGA(T20, T22, ackc21_out_ga(T20, T24)) → ACK1_IN_GGA(T20, T24, T22)
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, ackc27_in_ga(T76, T77))
U13_GGA(T76, T71, ackc27_out_ga(T76, T77)) → U14_GGA(T76, T71, ack1_in_gga(T76, T77, T71))
U13_GGA(T76, T71, ackc27_out_ga(T76, T77)) → ACK1_IN_GGA(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)
ACK1_IN_GGA(s(T84), s(s(T85)), T71) → U16_GGA(T84, T85, T71, ackc38_in_gga(s(T84), T85, T87))
U16_GGA(T84, T85, T71, ackc38_out_gga(s(T84), T85, T87)) → U17_GGA(T84, T85, T71, ack38_in_gga(T84, T87, X222))
U16_GGA(T84, T85, T71, ackc38_out_gga(s(T84), T85, T87)) → ACK38_IN_GGA(T84, T87, X222)
U16_GGA(T84, T85, T71, ackc38_out_gga(s(T84), T85, T87)) → U18_GGA(T84, T85, T71, ackc38_in_gga(T84, T87, T91))
U18_GGA(T84, T85, T71, ackc38_out_gga(T84, T87, T91)) → U19_GGA(T84, T85, T71, ack1_in_gga(T84, T91, T71))
U18_GGA(T84, T85, T71, ackc38_out_gga(T84, T87, T91)) → ACK1_IN_GGA(T84, T91, T71)

The TRS R consists of the following rules:

ackc21_in_ga(T30, X73) → U28_ga(T30, X73, ackc27_in_ga(T30, X73))
ackc27_in_ga(0, s(s(0))) → ackc27_out_ga(0, s(s(0)))
ackc27_in_ga(s(T34), X97) → U29_ga(T34, X97, ackc21_in_ga(T34, T36))
U29_ga(T34, X97, ackc21_out_ga(T34, T36)) → U30_ga(T34, X97, ackc38_in_gga(T34, T36, X97))
ackc38_in_gga(0, T44, s(T44)) → ackc38_out_gga(0, T44, s(T44))
ackc38_in_gga(s(T49), 0, X133) → U31_gga(T49, X133, ackc27_in_ga(T49, X133))
U31_gga(T49, X133, ackc27_out_ga(T49, X133)) → ackc38_out_gga(s(T49), 0, X133)
ackc38_in_gga(s(T54), s(T55), X151) → U32_gga(T54, T55, X151, ackc38_in_gga(s(T54), T55, T57))
U32_gga(T54, T55, X151, ackc38_out_gga(s(T54), T55, T57)) → U33_gga(T54, T55, X151, ackc38_in_gga(T54, T57, X151))
U33_gga(T54, T55, X151, ackc38_out_gga(T54, T57, X151)) → ackc38_out_gga(s(T54), s(T55), X151)
U30_ga(T34, X97, ackc38_out_gga(T34, T36, X97)) → ackc27_out_ga(s(T34), X97)
U28_ga(T30, X73, ackc27_out_ga(T30, X73)) → ackc21_out_ga(T30, X73)

The argument filtering Pi contains the following mapping:
ack1_in_gga(x1, x2, x3)  =  ack1_in_gga(x1, x2)
s(x1)  =  s(x1)
0  =  0
ack21_in_ga(x1, x2)  =  ack21_in_ga(x1)
ack27_in_ga(x1, x2)  =  ack27_in_ga(x1)
ackc21_in_ga(x1, x2)  =  ackc21_in_ga(x1)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
ackc27_in_ga(x1, x2)  =  ackc27_in_ga(x1)
ackc27_out_ga(x1, x2)  =  ackc27_out_ga(x1, x2)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
ackc21_out_ga(x1, x2)  =  ackc21_out_ga(x1, x2)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
ackc38_in_gga(x1, x2, x3)  =  ackc38_in_gga(x1, x2)
ackc38_out_gga(x1, x2, x3)  =  ackc38_out_gga(x1, x2, x3)
U31_gga(x1, x2, x3)  =  U31_gga(x1, x3)
U32_gga(x1, x2, x3, x4)  =  U32_gga(x1, x2, x4)
U33_gga(x1, x2, x3, x4)  =  U33_gga(x1, x2, x4)
ack38_in_gga(x1, x2, x3)  =  ack38_in_gga(x1, x2)
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_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)
U10_GGA(x1, x2, x3)  =  U10_GGA(x1, x3)
U11_GGA(x1, x2, x3)  =  U11_GGA(x1, x3)
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)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 17 less nodes.

(6) Complex Obligation (AND)

(7) 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, ackc21_in_ga(T34, T36))
U3_GA(T34, X97, ackc21_out_ga(T34, T36)) → ACK38_IN_GGA(T34, T36, X97)
ACK38_IN_GGA(s(T49), 0, X133) → ACK27_IN_GA(T49, X133)
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, ackc38_in_gga(s(T54), T55, T57))
U7_GGA(T54, T55, X151, ackc38_out_gga(s(T54), T55, T57)) → ACK38_IN_GGA(T54, T57, X151)

The TRS R consists of the following rules:

ackc21_in_ga(T30, X73) → U28_ga(T30, X73, ackc27_in_ga(T30, X73))
ackc27_in_ga(0, s(s(0))) → ackc27_out_ga(0, s(s(0)))
ackc27_in_ga(s(T34), X97) → U29_ga(T34, X97, ackc21_in_ga(T34, T36))
U29_ga(T34, X97, ackc21_out_ga(T34, T36)) → U30_ga(T34, X97, ackc38_in_gga(T34, T36, X97))
ackc38_in_gga(0, T44, s(T44)) → ackc38_out_gga(0, T44, s(T44))
ackc38_in_gga(s(T49), 0, X133) → U31_gga(T49, X133, ackc27_in_ga(T49, X133))
U31_gga(T49, X133, ackc27_out_ga(T49, X133)) → ackc38_out_gga(s(T49), 0, X133)
ackc38_in_gga(s(T54), s(T55), X151) → U32_gga(T54, T55, X151, ackc38_in_gga(s(T54), T55, T57))
U32_gga(T54, T55, X151, ackc38_out_gga(s(T54), T55, T57)) → U33_gga(T54, T55, X151, ackc38_in_gga(T54, T57, X151))
U33_gga(T54, T55, X151, ackc38_out_gga(T54, T57, X151)) → ackc38_out_gga(s(T54), s(T55), X151)
U30_ga(T34, X97, ackc38_out_gga(T34, T36, X97)) → ackc27_out_ga(s(T34), X97)
U28_ga(T30, X73, ackc27_out_ga(T30, X73)) → ackc21_out_ga(T30, X73)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
0  =  0
ackc21_in_ga(x1, x2)  =  ackc21_in_ga(x1)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
ackc27_in_ga(x1, x2)  =  ackc27_in_ga(x1)
ackc27_out_ga(x1, x2)  =  ackc27_out_ga(x1, x2)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
ackc21_out_ga(x1, x2)  =  ackc21_out_ga(x1, x2)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
ackc38_in_gga(x1, x2, x3)  =  ackc38_in_gga(x1, x2)
ackc38_out_gga(x1, x2, x3)  =  ackc38_out_gga(x1, x2, x3)
U31_gga(x1, x2, x3)  =  U31_gga(x1, x3)
U32_gga(x1, x2, x3, x4)  =  U32_gga(x1, x2, x4)
U33_gga(x1, x2, x3, x4)  =  U33_gga(x1, x2, 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_GGA(x1, x2, x3)  =  ACK38_IN_GGA(x1, x2)
U7_GGA(x1, x2, x3, x4)  =  U7_GGA(x1, x2, x4)

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

(8) PiDPToQDPProof (SOUND transformation)

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

(9) 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, ackc21_in_ga(T34))
U3_GA(T34, ackc21_out_ga(T34, T36)) → ACK38_IN_GGA(T34, T36)
ACK38_IN_GGA(s(T49), 0) → ACK27_IN_GA(T49)
ACK38_IN_GGA(s(T54), s(T55)) → ACK38_IN_GGA(s(T54), T55)
ACK38_IN_GGA(s(T54), s(T55)) → U7_GGA(T54, T55, ackc38_in_gga(s(T54), T55))
U7_GGA(T54, T55, ackc38_out_gga(s(T54), T55, T57)) → ACK38_IN_GGA(T54, T57)

The TRS R consists of the following rules:

ackc21_in_ga(T30) → U28_ga(T30, ackc27_in_ga(T30))
ackc27_in_ga(0) → ackc27_out_ga(0, s(s(0)))
ackc27_in_ga(s(T34)) → U29_ga(T34, ackc21_in_ga(T34))
U29_ga(T34, ackc21_out_ga(T34, T36)) → U30_ga(T34, ackc38_in_gga(T34, T36))
ackc38_in_gga(0, T44) → ackc38_out_gga(0, T44, s(T44))
ackc38_in_gga(s(T49), 0) → U31_gga(T49, ackc27_in_ga(T49))
U31_gga(T49, ackc27_out_ga(T49, X133)) → ackc38_out_gga(s(T49), 0, X133)
ackc38_in_gga(s(T54), s(T55)) → U32_gga(T54, T55, ackc38_in_gga(s(T54), T55))
U32_gga(T54, T55, ackc38_out_gga(s(T54), T55, T57)) → U33_gga(T54, T55, ackc38_in_gga(T54, T57))
U33_gga(T54, T55, ackc38_out_gga(T54, T57, X151)) → ackc38_out_gga(s(T54), s(T55), X151)
U30_ga(T34, ackc38_out_gga(T34, T36, X97)) → ackc27_out_ga(s(T34), X97)
U28_ga(T30, ackc27_out_ga(T30, X73)) → ackc21_out_ga(T30, X73)

The set Q consists of the following terms:

ackc21_in_ga(x0)
ackc27_in_ga(x0)
U29_ga(x0, x1)
ackc38_in_gga(x0, x1)
U31_gga(x0, x1)
U32_gga(x0, x1, x2)
U33_gga(x0, x1, x2)
U30_ga(x0, x1)
U28_ga(x0, x1)

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

(10) 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:

• ACK27_IN_GA(s(T34)) → ACK21_IN_GA(T34)
  The graph contains the following edges 1 > 1

• ACK27_IN_GA(s(T34)) → U3_GA(T34, ackc21_in_ga(T34))
  The graph contains the following edges 1 > 1

• ACK21_IN_GA(T30) → ACK27_IN_GA(T30)
  The graph contains the following edges 1 >= 1

• ACK38_IN_GGA(s(T49), 0) → ACK27_IN_GA(T49)
  The graph contains the following edges 1 > 1

• U3_GA(T34, ackc21_out_ga(T34, T36)) → ACK38_IN_GGA(T34, T36)
  The graph contains the following edges 1 >= 1, 2 > 1, 2 > 2

• U7_GGA(T54, T55, ackc38_out_gga(s(T54), T55, T57)) → ACK38_IN_GGA(T54, T57)
  The graph contains the following edges 1 >= 1, 3 > 1, 3 > 2

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

• ACK38_IN_GGA(s(T54), s(T55)) → U7_GGA(T54, T55, ackc38_in_gga(s(T54), T55))
  The graph contains the following edges 1 > 1, 2 > 2

(11) YES

(12) Obligation:

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

ACK1_IN_GGA(s(s(T20)), 0, T22) → U10_GGA(T20, T22, ackc21_in_ga(T20, T24))
U10_GGA(T20, T22, ackc21_out_ga(T20, T24)) → ACK1_IN_GGA(T20, T24, T22)
ACK1_IN_GGA(s(T76), s(0), T71) → U13_GGA(T76, T71, ackc27_in_ga(T76, T77))
U13_GGA(T76, T71, ackc27_out_ga(T76, T77)) → ACK1_IN_GGA(T76, T77, T71)
ACK1_IN_GGA(s(T84), s(s(T85)), T71) → U16_GGA(T84, T85, T71, ackc38_in_gga(s(T84), T85, T87))
U16_GGA(T84, T85, T71, ackc38_out_gga(s(T84), T85, T87)) → U18_GGA(T84, T85, T71, ackc38_in_gga(T84, T87, T91))
U18_GGA(T84, T85, T71, ackc38_out_gga(T84, T87, T91)) → ACK1_IN_GGA(T84, T91, T71)

The TRS R consists of the following rules:

ackc21_in_ga(T30, X73) → U28_ga(T30, X73, ackc27_in_ga(T30, X73))
ackc27_in_ga(0, s(s(0))) → ackc27_out_ga(0, s(s(0)))
ackc27_in_ga(s(T34), X97) → U29_ga(T34, X97, ackc21_in_ga(T34, T36))
U29_ga(T34, X97, ackc21_out_ga(T34, T36)) → U30_ga(T34, X97, ackc38_in_gga(T34, T36, X97))
ackc38_in_gga(0, T44, s(T44)) → ackc38_out_gga(0, T44, s(T44))
ackc38_in_gga(s(T49), 0, X133) → U31_gga(T49, X133, ackc27_in_ga(T49, X133))
U31_gga(T49, X133, ackc27_out_ga(T49, X133)) → ackc38_out_gga(s(T49), 0, X133)
ackc38_in_gga(s(T54), s(T55), X151) → U32_gga(T54, T55, X151, ackc38_in_gga(s(T54), T55, T57))
U32_gga(T54, T55, X151, ackc38_out_gga(s(T54), T55, T57)) → U33_gga(T54, T55, X151, ackc38_in_gga(T54, T57, X151))
U33_gga(T54, T55, X151, ackc38_out_gga(T54, T57, X151)) → ackc38_out_gga(s(T54), s(T55), X151)
U30_ga(T34, X97, ackc38_out_gga(T34, T36, X97)) → ackc27_out_ga(s(T34), X97)
U28_ga(T30, X73, ackc27_out_ga(T30, X73)) → ackc21_out_ga(T30, X73)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
0  =  0
ackc21_in_ga(x1, x2)  =  ackc21_in_ga(x1)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
ackc27_in_ga(x1, x2)  =  ackc27_in_ga(x1)
ackc27_out_ga(x1, x2)  =  ackc27_out_ga(x1, x2)
U29_ga(x1, x2, x3)  =  U29_ga(x1, x3)
ackc21_out_ga(x1, x2)  =  ackc21_out_ga(x1, x2)
U30_ga(x1, x2, x3)  =  U30_ga(x1, x3)
ackc38_in_gga(x1, x2, x3)  =  ackc38_in_gga(x1, x2)
ackc38_out_gga(x1, x2, x3)  =  ackc38_out_gga(x1, x2, x3)
U31_gga(x1, x2, x3)  =  U31_gga(x1, x3)
U32_gga(x1, x2, x3, x4)  =  U32_gga(x1, x2, x4)
U33_gga(x1, x2, x3, x4)  =  U33_gga(x1, x2, x4)
ACK1_IN_GGA(x1, x2, x3)  =  ACK1_IN_GGA(x1, x2)
U10_GGA(x1, x2, x3)  =  U10_GGA(x1, x3)
U13_GGA(x1, x2, x3)  =  U13_GGA(x1, x3)
U16_GGA(x1, x2, x3, x4)  =  U16_GGA(x1, x2, x4)
U18_GGA(x1, x2, x3, x4)  =  U18_GGA(x1, x2, x4)

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

(13) PiDPToQDPProof (SOUND transformation)

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

(14) Obligation:

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

ACK1_IN_GGA(s(s(T20)), 0) → U10_GGA(T20, ackc21_in_ga(T20))
U10_GGA(T20, ackc21_out_ga(T20, T24)) → ACK1_IN_GGA(T20, T24)
ACK1_IN_GGA(s(T76), s(0)) → U13_GGA(T76, ackc27_in_ga(T76))
U13_GGA(T76, ackc27_out_ga(T76, T77)) → ACK1_IN_GGA(T76, T77)
ACK1_IN_GGA(s(T84), s(s(T85))) → U16_GGA(T84, T85, ackc38_in_gga(s(T84), T85))
U16_GGA(T84, T85, ackc38_out_gga(s(T84), T85, T87)) → U18_GGA(T84, T85, ackc38_in_gga(T84, T87))
U18_GGA(T84, T85, ackc38_out_gga(T84, T87, T91)) → ACK1_IN_GGA(T84, T91)

The TRS R consists of the following rules:

ackc21_in_ga(T30) → U28_ga(T30, ackc27_in_ga(T30))
ackc27_in_ga(0) → ackc27_out_ga(0, s(s(0)))
ackc27_in_ga(s(T34)) → U29_ga(T34, ackc21_in_ga(T34))
U29_ga(T34, ackc21_out_ga(T34, T36)) → U30_ga(T34, ackc38_in_gga(T34, T36))
ackc38_in_gga(0, T44) → ackc38_out_gga(0, T44, s(T44))
ackc38_in_gga(s(T49), 0) → U31_gga(T49, ackc27_in_ga(T49))
U31_gga(T49, ackc27_out_ga(T49, X133)) → ackc38_out_gga(s(T49), 0, X133)
ackc38_in_gga(s(T54), s(T55)) → U32_gga(T54, T55, ackc38_in_gga(s(T54), T55))
U32_gga(T54, T55, ackc38_out_gga(s(T54), T55, T57)) → U33_gga(T54, T55, ackc38_in_gga(T54, T57))
U33_gga(T54, T55, ackc38_out_gga(T54, T57, X151)) → ackc38_out_gga(s(T54), s(T55), X151)
U30_ga(T34, ackc38_out_gga(T34, T36, X97)) → ackc27_out_ga(s(T34), X97)
U28_ga(T30, ackc27_out_ga(T30, X73)) → ackc21_out_ga(T30, X73)

The set Q consists of the following terms:

ackc21_in_ga(x0)
ackc27_in_ga(x0)
U29_ga(x0, x1)
ackc38_in_gga(x0, x1)
U31_gga(x0, x1)
U32_gga(x0, x1, x2)
U33_gga(x0, x1, x2)
U30_ga(x0, x1)
U28_ga(x0, x1)

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

(15) 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_GGA(T20, ackc21_out_ga(T20, T24)) → ACK1_IN_GGA(T20, T24)
  The graph contains the following edges 1 >= 1, 2 > 1, 2 > 2

• ACK1_IN_GGA(s(s(T20)), 0) → U10_GGA(T20, ackc21_in_ga(T20))
  The graph contains the following edges 1 > 1

• U16_GGA(T84, T85, ackc38_out_gga(s(T84), T85, T87)) → U18_GGA(T84, T85, ackc38_in_gga(T84, T87))
  The graph contains the following edges 1 >= 1, 3 > 1, 2 >= 2, 3 > 2

• U13_GGA(T76, ackc27_out_ga(T76, T77)) → ACK1_IN_GGA(T76, T77)
  The graph contains the following edges 1 >= 1, 2 > 1, 2 > 2

• U18_GGA(T84, T85, ackc38_out_gga(T84, T87, T91)) → ACK1_IN_GGA(T84, T91)
  The graph contains the following edges 1 >= 1, 3 > 1, 3 > 2

• ACK1_IN_GGA(s(T76), s(0)) → U13_GGA(T76, ackc27_in_ga(T76))
  The graph contains the following edges 1 > 1

• ACK1_IN_GGA(s(T84), s(s(T85))) → U16_GGA(T84, T85, ackc38_in_gga(s(T84), T85))
  The graph contains the following edges 1 > 1, 2 > 2

(16) YES