Left Termination of the query pattern mul_in_3(g, g, a) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

mul(X1, 0, Z) :- ','(!, eq(Z, 0)).
mul(X, Y, Z) :- ','(p(Y, P), ','(mul(X, P, V), add(X, V, Z))).
add(X, 0, Z) :- ','(!, eq(Z, X)).
add(X, Y, Z) :- ','(p(Y, V), ','(add(X, V, W), p(Z, W))).
p(0, 0).
p(s(X), X).
eq(X, X).

Queries:

mul(g,g,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: mul(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: mul(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | mul(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(!_1, eq(T8, 0)) | mul(T6, 0, T3), SCOPE: 1, CLAUSE: 1\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: mul(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground\nmul(T1, T2, T3) !=? mul(X4, 0, X5)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: eq(T8, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: eq(T8, 0), SCOPE: 2, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(p(T16, X18), ','(mul(T15, X18, X19), add(T15, X19, T18)))\n\nT15 is ground\nT16 is ground\nX18 is free; \nX19 is free; \nmul(T15, T16, T3) !=? mul(X4, 0, X5)", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(p(T16, X18), ','(mul(T15, X18, X19), add(T15, X19, T18))), SCOPE: 3, CLAUSE: 4 | ','(p(T16, X18), ','(mul(T15, X18, X19), add(T15, X19, T18))), SCOPE: 3, CLAUSE: 5\n\nT15 is ground\nT16 is ground\nX18 is free; \nX19 is free; \nmul(T15, T16, T3) !=? mul(X4, 0, X5)", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(p(T16, X18), ','(mul(T15, X18, X19), add(T15, X19, T18))), SCOPE: 3, CLAUSE: 5\n\nT15 is ground\nT16 is ground\nX18 is free; \nX19 is free; \nmul(T15, T16, T3) !=? mul(X4, 0, X5)", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(mul(T15, T21, X19), add(T15, X19, T18))\n\nT15 is ground\nT21 is ground\nX19 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: mul(T15, T21, X19)\n\nT15 is ground\nT21 is ground\nX19 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: add(T15, T23, T18)\n\nT15 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: mul(T15, T21, X19), SCOPE: 4, CLAUSE: 0 | mul(T15, T21, X19), SCOPE: 4, CLAUSE: 1\n\nT15 is ground\nT21 is ground\nX19 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(!_4, eq(X38, 0)) | mul(T27, 0, X19), SCOPE: 4, CLAUSE: 1\n\nT27 is ground\nX19 is free; \nX38 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: mul(T15, T21, X19), SCOPE: 4, CLAUSE: 1\n\nT15 is ground\nT21 is ground\nX19 is free; \nmul(T15, T21, X19) !=? mul(X36, 0, X37)", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: eq(X38, 0)\n\nX38 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: eq(X38, 0), SCOPE: 5, CLAUSE: 6\n\nX38 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(p(T34, X63), ','(mul(T33, X63, X64), add(T33, X64, X65)))\n\nT33 is ground\nT34 is ground\nX65 is free; \nX63 is free; \nX64 is free; \nmul(T33, T34, X19) !=? mul(X36, 0, X37)", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(p(T34, X63), ','(mul(T33, X63, X64), add(T33, X64, X65))), SCOPE: 6, CLAUSE: 4 | ','(p(T34, X63), ','(mul(T33, X63, X64), add(T33, X64, X65))), SCOPE: 6, CLAUSE: 5\n\nT33 is ground\nT34 is ground\nX65 is free; \nX63 is free; \nX64 is free; \nmul(T33, T34, X19) !=? mul(X36, 0, X37)", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(p(T34, X63), ','(mul(T33, X63, X64), add(T33, X64, X65))), SCOPE: 6, CLAUSE: 5\n\nT33 is ground\nT34 is ground\nX65 is free; \nX63 is free; \nX64 is free; \nmul(T33, T34, X19) !=? mul(X36, 0, X37)", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(mul(T33, T38, X64), add(T33, X64, X65))\n\nT33 is ground\nT38 is ground\nX65 is free; \nX64 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: mul(T33, T38, X64)\n\nT33 is ground\nT38 is ground\nX64 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: add(T33, T40, X65)\n\nT33 is ground\nT40 is ground\nX65 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: add(T33, T40, X65), SCOPE: 7, CLAUSE: 2 | add(T33, T40, X65), SCOPE: 7, CLAUSE: 3\n\nT33 is ground\nT40 is ground\nX65 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(!_7, eq(X90, T45)) | add(T45, 0, X65), SCOPE: 7, CLAUSE: 3\n\nT45 is ground\nX65 is free; \nX90 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: add(T33, T40, X65), SCOPE: 7, CLAUSE: 3\n\nT33 is ground\nT40 is ground\nX65 is free; \nadd(T33, T40, X65) !=? add(X88, 0, X89)", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: eq(X90, T45)\n\nT45 is ground\nX90 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: eq(X90, T45), SCOPE: 8, CLAUSE: 6\n\nT45 is ground\nX90 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(p(T55, X115), ','(add(T54, X115, X116), p(X117, X116)))\n\nT54 is ground\nT55 is ground\nX117 is free; \nX115 is free; \nX116 is free; \nadd(T54, T55, X65) !=? add(X88, 0, X89)", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(p(T55, X115), ','(add(T54, X115, X116), p(X117, X116))), SCOPE: 9, CLAUSE: 4 | ','(p(T55, X115), ','(add(T54, X115, X116), p(X117, X116))), SCOPE: 9, CLAUSE: 5\n\nT54 is ground\nT55 is ground\nX117 is free; \nX115 is free; \nX116 is free; \nadd(T54, T55, X65) !=? add(X88, 0, X89)", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(p(T55, X115), ','(add(T54, X115, X116), p(X117, X116))), SCOPE: 9, CLAUSE: 5\n\nT54 is ground\nT55 is ground\nX117 is free; \nX115 is free; \nX116 is free; \nadd(T54, T55, X65) !=? add(X88, 0, X89)", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(add(T54, T59, X116), p(X117, X116))\n\nT54 is ground\nT59 is ground\nX117 is free; \nX116 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: add(T54, T59, X116)\n\nT54 is ground\nT59 is ground\nX116 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: p(X117, T61)\n\nT61 is ground\nX117 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: p(X117, T61), SCOPE: 10, CLAUSE: 4 | p(X117, T61), SCOPE: 10, CLAUSE: 5\n\nT61 is ground\nX117 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: p(X117, T61), SCOPE: 10, CLAUSE: 4\n\nT61 is ground\nX117 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: p(X117, T61), SCOPE: 10, CLAUSE: 5\n\nT61 is ground\nX117 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: add(T15, T23, T18), SCOPE: 11, CLAUSE: 2 | add(T15, T23, T18), SCOPE: 11, CLAUSE: 3\n\nT15 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(!_11, eq(T74, T72)) | add(T72, 0, T18), SCOPE: 11, CLAUSE: 3\n\nT72 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: add(T15, T23, T18), SCOPE: 11, CLAUSE: 3\n\nT15 is ground\nT23 is ground\nadd(T15, T23, T18) !=? add(X143, 0, X144)", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: eq(T74, T72)\n\nT72 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: eq(T74, T72), SCOPE: 12, CLAUSE: 6\n\nT72 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(p(T87, X165), ','(add(T86, X165, X166), p(T89, X166)))\n\nT86 is ground\nT87 is ground\nX165 is free; \nX166 is free; \nadd(T86, T87, T18) !=? add(X143, 0, X144)", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(p(T87, X165), ','(add(T86, X165, X166), p(T89, X166))), SCOPE: 13, CLAUSE: 4 | ','(p(T87, X165), ','(add(T86, X165, X166), p(T89, X166))), SCOPE: 13, CLAUSE: 5\n\nT86 is ground\nT87 is ground\nX165 is free; \nX166 is free; \nadd(T86, T87, T18) !=? add(X143, 0, X144)", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(p(T87, X165), ','(add(T86, X165, X166), p(T89, X166))), SCOPE: 13, CLAUSE: 5\n\nT86 is ground\nT87 is ground\nX165 is free; \nX166 is free; \nadd(T86, T87, T18) !=? add(X143, 0, X144)", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ','(add(T86, T93, X166), p(T89, X166))\n\nT86 is ground\nT93 is ground\nX166 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: add(T86, T93, X166)\n\nT86 is ground\nT93 is ground\nX166 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: p(T89, T95)\n\nT95 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: p(T89, T95), SCOPE: 14, CLAUSE: 4 | p(T89, T95), SCOPE: 14, CLAUSE: 5\n\nT95 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: p(T89, T95), SCOPE: 14, CLAUSE: 4\n\nT95 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: p(T89, T95), SCOPE: 14, CLAUSE: 5\n\nT95 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 77 [arrowhead = none ];
77 -> 2;

78 [label="EVAL with clause\nmul(X4, 0, X5) :- ','(!_1, eq(X5, 0)).\nand substitution\nT1 -> T6\nX4 -> T6\nT2 -> 0\nT3 -> T8\nX5 -> T8\nT7 -> T8", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 78 [arrowhead = none ];
78 -> 3;

79 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 79 [arrowhead = none ];
79 -> 4;

80 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 80 [arrowhead = none ];
80 -> 5;

81 [label="ONLY EVAL with clause\nmul(X15, X16, X17) :- ','(p(X16, X18), ','(mul(X15, X18, X19), add(X15, X19, X17))).\nand substitution\nT1 -> T15\nX15 -> T15\nT2 -> T16\nX16 -> T16\nT3 -> T18\nX17 -> T18\nT17 -> T18", fontsize=14, style = filled, fillcolor = white];
4 -> 81 [arrowhead = none ];
81 -> 10;

82 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 82 [arrowhead = none ];
82 -> 6;

83 [label="EVAL with clause\neq(X10, X10).\nand substitution\nT8 -> 0\nX10 -> 0\nT11 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 83 [arrowhead = none ];
83 -> 7;

84 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 84 [arrowhead = none ];
84 -> 8;

85 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 85 [arrowhead = none ];
85 -> 9;

86 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 86 [arrowhead = none ];
86 -> 11;

87 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (mul(T15, T16, T3), mul(X4, 0, X5))", fontsize=14, style = filled, fillcolor = white];
11 -> 87 [arrowhead = none ];
87 -> 12;

88 [label="EVAL with clause\np(s(X22), X22).\nand substitution\nX22 -> T21\nT16 -> s(T21)\nX18 -> T21", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 88 [arrowhead = none ];
88 -> 13;

89 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 89 [arrowhead = none ];
89 -> 14;

90 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
13 -> 90 [arrowhead = none ];
90 -> 15;

91 [label="SPLIT 2\nnew knowledge:\nT23 is ground\nreplacements:\nX19 -> T23", fontsize=14, style = filled, fillcolor = violet];
13 -> 91 [arrowhead = none ];
91 -> 16;

92 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 92 [arrowhead = none ];
92 -> 17;

93 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 93 [arrowhead = none ];
93 -> 53;

94 [label="EVAL with clause\nmul(X36, 0, X37) :- ','(!_4, eq(X37, 0)).\nand substitution\nT15 -> T27\nX36 -> T27\nT21 -> 0\nX19 -> X38\nX37 -> X38", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 94 [arrowhead = none ];
94 -> 18;

95 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 95 [arrowhead = none ];
95 -> 19;

96 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 96 [arrowhead = none ];
96 -> 20;

97 [label="ONLY EVAL with clause\nmul(X60, X61, X62) :- ','(p(X61, X63), ','(mul(X60, X63, X64), add(X60, X64, X62))).\nand substitution\nT15 -> T33\nX60 -> T33\nT21 -> T34\nX61 -> T34\nX19 -> X65\nX62 -> X65", fontsize=14, style = filled, fillcolor = white];
19 -> 97 [arrowhead = none ];
97 -> 24;

98 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 98 [arrowhead = none ];
98 -> 21;

99 [label="ONLY EVAL with clause\neq(X45, X45).\nand substitution\nX38 -> 0\nX45 -> 0\nX46 -> 0", fontsize=14, style = filled, fillcolor = white];
21 -> 99 [arrowhead = none ];
99 -> 22;

100 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 100 [arrowhead = none ];
100 -> 23;

101 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 101 [arrowhead = none ];
101 -> 25;

102 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (mul(T33, T34, X19), mul(X36, 0, X37))", fontsize=14, style = filled, fillcolor = white];
25 -> 102 [arrowhead = none ];
102 -> 26;

103 [label="EVAL with clause\np(s(X71), X71).\nand substitution\nX71 -> T38\nT34 -> s(T38)\nX63 -> T38", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 103 [arrowhead = none ];
103 -> 27;

104 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 104 [arrowhead = none ];
104 -> 28;

105 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
27 -> 105 [arrowhead = none ];
105 -> 29;

106 [label="SPLIT 2\nnew knowledge:\nT40 is ground\nreplacements:\nX64 -> T40", fontsize=14, style = filled, fillcolor = violet];
27 -> 106 [arrowhead = none ];
106 -> 30;

107 [label="INSTANCE with matching:\nT15 -> T33\nT21 -> T38\nX19 -> X64", fontsize=14, style = filled, fillcolor = lightgrey];
29 -> 107 [arrowhead = none , style=dashed];
107 -> 15[style=dashed];

108 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
30 -> 108 [arrowhead = none ];
108 -> 31;

109 [label="EVAL with clause\nadd(X88, 0, X89) :- ','(!_7, eq(X89, X88)).\nand substitution\nT33 -> T45\nX88 -> T45\nT40 -> 0\nX65 -> X90\nX89 -> X90", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 109 [arrowhead = none ];
109 -> 32;

110 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 110 [arrowhead = none ];
110 -> 33;

111 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 111 [arrowhead = none ];
111 -> 34;

112 [label="ONLY EVAL with clause\nadd(X112, X113, X114) :- ','(p(X113, X115), ','(add(X112, X115, X116), p(X114, X116))).\nand substitution\nT33 -> T54\nX112 -> T54\nT40 -> T55\nX113 -> T55\nX65 -> X117\nX114 -> X117", fontsize=14, style = filled, fillcolor = white];
33 -> 112 [arrowhead = none ];
112 -> 38;

113 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 113 [arrowhead = none ];
113 -> 35;

114 [label="ONLY EVAL with clause\neq(X97, X97).\nand substitution\nX90 -> T48\nX97 -> T48\nT45 -> T48\nX98 -> T48", fontsize=14, style = filled, fillcolor = white];
35 -> 114 [arrowhead = none ];
114 -> 36;

115 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 115 [arrowhead = none ];
115 -> 37;

116 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
38 -> 116 [arrowhead = none ];
116 -> 39;

117 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (add(T54, T55, X65), add(X88, 0, X89))", fontsize=14, style = filled, fillcolor = white];
39 -> 117 [arrowhead = none ];
117 -> 40;

118 [label="EVAL with clause\np(s(X123), X123).\nand substitution\nX123 -> T59\nT55 -> s(T59)\nX115 -> T59", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 118 [arrowhead = none ];
118 -> 41;

119 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 119 [arrowhead = none ];
119 -> 42;

120 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
41 -> 120 [arrowhead = none ];
120 -> 43;

121 [label="SPLIT 2\nnew knowledge:\nT61 is ground\nreplacements:\nX116 -> T61", fontsize=14, style = filled, fillcolor = violet];
41 -> 121 [arrowhead = none ];
121 -> 44;

122 [label="INSTANCE with matching:\nT33 -> T54\nT40 -> T59\nX65 -> X116", fontsize=14, style = filled, fillcolor = lightgrey];
43 -> 122 [arrowhead = none , style=dashed];
122 -> 30[style=dashed];

123 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
44 -> 123 [arrowhead = none ];
123 -> 45;

124 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 124 [arrowhead = none ];
124 -> 46;

125 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 125 [arrowhead = none ];
125 -> 47;

126 [label="EVAL with clause\np(0, 0).\nand substitution\nX117 -> 0\nT61 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 126 [arrowhead = none ];
126 -> 48;

127 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 127 [arrowhead = none ];
127 -> 49;

128 [label="ONLY EVAL with clause\np(s(X135), X135).\nand substitution\nX135 -> T65\nX117 -> s(T65)\nT61 -> T65\nX136 -> T65", fontsize=14, style = filled, fillcolor = white];
47 -> 128 [arrowhead = none ];
128 -> 51;

129 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 129 [arrowhead = none ];
129 -> 50;

130 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 130 [arrowhead = none ];
130 -> 52;

131 [label="EVAL with clause\nadd(X143, 0, X144) :- ','(!_11, eq(X144, X143)).\nand substitution\nT15 -> T72\nX143 -> T72\nT23 -> 0\nT18 -> T74\nX144 -> T74\nT73 -> T74", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 131 [arrowhead = none ];
131 -> 54;

132 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 132 [arrowhead = none ];
132 -> 55;

133 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 133 [arrowhead = none ];
133 -> 56;

134 [label="ONLY EVAL with clause\nadd(X162, X163, X164) :- ','(p(X163, X165), ','(add(X162, X165, X166), p(X164, X166))).\nand substitution\nT15 -> T86\nX162 -> T86\nT23 -> T87\nX163 -> T87\nT18 -> T89\nX164 -> T89\nT88 -> T89", fontsize=14, style = filled, fillcolor = white];
55 -> 134 [arrowhead = none ];
134 -> 61;

135 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
56 -> 135 [arrowhead = none ];
135 -> 57;

136 [label="EVAL with clause\neq(X149, X149).\nand substitution\nT74 -> T77\nX149 -> T77\nT72 -> T77", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 136 [arrowhead = none ];
136 -> 58;

137 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 137 [arrowhead = none ];
137 -> 59;

138 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
58 -> 138 [arrowhead = none ];
138 -> 60;

139 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
61 -> 139 [arrowhead = none ];
139 -> 62;

140 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (add(T86, T87, T18), add(X143, 0, X144))", fontsize=14, style = filled, fillcolor = white];
62 -> 140 [arrowhead = none ];
140 -> 63;

141 [label="EVAL with clause\np(s(X172), X172).\nand substitution\nX172 -> T93\nT87 -> s(T93)\nX165 -> T93", fontsize=14, style = filled, fillcolor = lightblue];
63 -> 141 [arrowhead = none ];
141 -> 64;

142 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
63 -> 142 [arrowhead = none ];
142 -> 65;

143 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
64 -> 143 [arrowhead = none ];
143 -> 66;

144 [label="SPLIT 2\nnew knowledge:\nT95 is ground\nreplacements:\nX166 -> T95", fontsize=14, style = filled, fillcolor = violet];
64 -> 144 [arrowhead = none ];
144 -> 67;

145 [label="INSTANCE with matching:\nT33 -> T86\nT40 -> T93\nX65 -> X166", fontsize=14, style = filled, fillcolor = lightgrey];
66 -> 145 [arrowhead = none , style=dashed];
145 -> 30[style=dashed];

146 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
67 -> 146 [arrowhead = none ];
146 -> 68;

147 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
68 -> 147 [arrowhead = none ];
147 -> 69;

148 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
68 -> 148 [arrowhead = none ];
148 -> 70;

149 [label="EVAL with clause\np(0, 0).\nand substitution\nT89 -> 0\nT95 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
69 -> 149 [arrowhead = none ];
149 -> 71;

150 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
69 -> 150 [arrowhead = none ];
150 -> 72;

151 [label="EVAL with clause\np(s(X182), X182).\nand substitution\nX182 -> T99\nT89 -> s(T99)\nT95 -> T99", fontsize=14, style = filled, fillcolor = lightblue];
70 -> 151 [arrowhead = none ];
151 -> 74;

152 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
70 -> 152 [arrowhead = none ];
152 -> 75;

153 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
71 -> 153 [arrowhead = none ];
153 -> 73;

154 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
74 -> 154 [arrowhead = none ];
154 -> 76;

}

(2) Obligation:

Triples:

mul15(T33, s(T38), X65) :- mul15(T33, T38, X64).
mul15(T33, s(T38), X65) :- ','(mulc15(T33, T38, T40), add30(T33, T40, X65)).
add30(T54, s(T59), X117) :- add30(T54, T59, X116).
mul1(T15, s(T21), T18) :- mul15(T15, T21, X19).
mul1(T86, s(T21), T89) :- ','(mulc15(T86, T21, s(T93)), add30(T86, T93, X166)).

Clauses:

mulc15(T27, 0, 0).
mulc15(T33, s(T38), X65) :- ','(mulc15(T33, T38, T40), addc30(T33, T40, X65)).
addc30(T48, 0, T48).
addc30(T54, s(T59), 0) :- addc30(T54, T59, 0).
addc30(T54, s(T59), s(T65)) :- addc30(T54, T59, T65).

Afs:

mul1(x1, x2, x3)  =  mul1(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:
mul1_in: (b,b,f)
mul15_in: (b,b,f)
mulc15_in: (b,b,f)
addc30_in: (b,b,f) (b,b,b)
add30_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

MUL1_IN_GGA(T15, s(T21), T18) → U5_GGA(T15, T21, T18, mul15_in_gga(T15, T21, X19))
MUL1_IN_GGA(T15, s(T21), T18) → MUL15_IN_GGA(T15, T21, X19)
MUL15_IN_GGA(T33, s(T38), X65) → U1_GGA(T33, T38, X65, mul15_in_gga(T33, T38, X64))
MUL15_IN_GGA(T33, s(T38), X65) → MUL15_IN_GGA(T33, T38, X64)
MUL15_IN_GGA(T33, s(T38), X65) → U2_GGA(T33, T38, X65, mulc15_in_gga(T33, T38, T40))
U2_GGA(T33, T38, X65, mulc15_out_gga(T33, T38, T40)) → U3_GGA(T33, T38, X65, add30_in_gga(T33, T40, X65))
U2_GGA(T33, T38, X65, mulc15_out_gga(T33, T38, T40)) → ADD30_IN_GGA(T33, T40, X65)
ADD30_IN_GGA(T54, s(T59), X117) → U4_GGA(T54, T59, X117, add30_in_gga(T54, T59, X116))
ADD30_IN_GGA(T54, s(T59), X117) → ADD30_IN_GGA(T54, T59, X116)
MUL1_IN_GGA(T86, s(T21), T89) → U6_GGA(T86, T21, T89, mulc15_in_gga(T86, T21, s(T93)))
U6_GGA(T86, T21, T89, mulc15_out_gga(T86, T21, s(T93))) → U7_GGA(T86, T21, T89, add30_in_gga(T86, T93, X166))
U6_GGA(T86, T21, T89, mulc15_out_gga(T86, T21, s(T93))) → ADD30_IN_GGA(T86, T93, X166)

The TRS R consists of the following rules:

mulc15_in_gga(T27, 0, 0) → mulc15_out_gga(T27, 0, 0)
mulc15_in_gga(T33, s(T38), X65) → U9_gga(T33, T38, X65, mulc15_in_gga(T33, T38, T40))
U9_gga(T33, T38, X65, mulc15_out_gga(T33, T38, T40)) → U10_gga(T33, T38, X65, addc30_in_gga(T33, T40, X65))
addc30_in_gga(T48, 0, T48) → addc30_out_gga(T48, 0, T48)
addc30_in_gga(T54, s(T59), 0) → U11_gga(T54, T59, addc30_in_ggg(T54, T59, 0))
addc30_in_ggg(T48, 0, T48) → addc30_out_ggg(T48, 0, T48)
addc30_in_ggg(T54, s(T59), 0) → U11_ggg(T54, T59, addc30_in_ggg(T54, T59, 0))
addc30_in_ggg(T54, s(T59), s(T65)) → U12_ggg(T54, T59, T65, addc30_in_ggg(T54, T59, T65))
U12_ggg(T54, T59, T65, addc30_out_ggg(T54, T59, T65)) → addc30_out_ggg(T54, s(T59), s(T65))
U11_ggg(T54, T59, addc30_out_ggg(T54, T59, 0)) → addc30_out_ggg(T54, s(T59), 0)
U11_gga(T54, T59, addc30_out_ggg(T54, T59, 0)) → addc30_out_gga(T54, s(T59), 0)
addc30_in_gga(T54, s(T59), s(T65)) → U12_gga(T54, T59, T65, addc30_in_gga(T54, T59, T65))
U12_gga(T54, T59, T65, addc30_out_gga(T54, T59, T65)) → addc30_out_gga(T54, s(T59), s(T65))
U10_gga(T33, T38, X65, addc30_out_gga(T33, T40, X65)) → mulc15_out_gga(T33, s(T38), X65)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
mul15_in_gga(x1, x2, x3)  =  mul15_in_gga(x1, x2)
mulc15_in_gga(x1, x2, x3)  =  mulc15_in_gga(x1, x2)
0  =  0
mulc15_out_gga(x1, x2, x3)  =  mulc15_out_gga(x1, x2, x3)
U9_gga(x1, x2, x3, x4)  =  U9_gga(x1, x2, x4)
U10_gga(x1, x2, x3, x4)  =  U10_gga(x1, x2, x4)
addc30_in_gga(x1, x2, x3)  =  addc30_in_gga(x1, x2)
addc30_out_gga(x1, x2, x3)  =  addc30_out_gga(x1, x2, x3)
U11_gga(x1, x2, x3)  =  U11_gga(x1, x2, x3)
addc30_in_ggg(x1, x2, x3)  =  addc30_in_ggg(x1, x2, x3)
addc30_out_ggg(x1, x2, x3)  =  addc30_out_ggg(x1, x2, x3)
U11_ggg(x1, x2, x3)  =  U11_ggg(x1, x2, x3)
U12_ggg(x1, x2, x3, x4)  =  U12_ggg(x1, x2, x3, x4)
U12_gga(x1, x2, x3, x4)  =  U12_gga(x1, x2, x4)
add30_in_gga(x1, x2, x3)  =  add30_in_gga(x1, x2)
MUL1_IN_GGA(x1, x2, x3)  =  MUL1_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x1, x2, x4)
MUL15_IN_GGA(x1, x2, x3)  =  MUL15_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x1, x2, x4)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)
ADD30_IN_GGA(x1, x2, x3)  =  ADD30_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)
U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x1, x2, x4)
U7_GGA(x1, x2, x3, x4)  =  U7_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:

MUL1_IN_GGA(T15, s(T21), T18) → U5_GGA(T15, T21, T18, mul15_in_gga(T15, T21, X19))
MUL1_IN_GGA(T15, s(T21), T18) → MUL15_IN_GGA(T15, T21, X19)
MUL15_IN_GGA(T33, s(T38), X65) → U1_GGA(T33, T38, X65, mul15_in_gga(T33, T38, X64))
MUL15_IN_GGA(T33, s(T38), X65) → MUL15_IN_GGA(T33, T38, X64)
MUL15_IN_GGA(T33, s(T38), X65) → U2_GGA(T33, T38, X65, mulc15_in_gga(T33, T38, T40))
U2_GGA(T33, T38, X65, mulc15_out_gga(T33, T38, T40)) → U3_GGA(T33, T38, X65, add30_in_gga(T33, T40, X65))
U2_GGA(T33, T38, X65, mulc15_out_gga(T33, T38, T40)) → ADD30_IN_GGA(T33, T40, X65)
ADD30_IN_GGA(T54, s(T59), X117) → U4_GGA(T54, T59, X117, add30_in_gga(T54, T59, X116))
ADD30_IN_GGA(T54, s(T59), X117) → ADD30_IN_GGA(T54, T59, X116)
MUL1_IN_GGA(T86, s(T21), T89) → U6_GGA(T86, T21, T89, mulc15_in_gga(T86, T21, s(T93)))
U6_GGA(T86, T21, T89, mulc15_out_gga(T86, T21, s(T93))) → U7_GGA(T86, T21, T89, add30_in_gga(T86, T93, X166))
U6_GGA(T86, T21, T89, mulc15_out_gga(T86, T21, s(T93))) → ADD30_IN_GGA(T86, T93, X166)

The TRS R consists of the following rules:

mulc15_in_gga(T27, 0, 0) → mulc15_out_gga(T27, 0, 0)
mulc15_in_gga(T33, s(T38), X65) → U9_gga(T33, T38, X65, mulc15_in_gga(T33, T38, T40))
U9_gga(T33, T38, X65, mulc15_out_gga(T33, T38, T40)) → U10_gga(T33, T38, X65, addc30_in_gga(T33, T40, X65))
addc30_in_gga(T48, 0, T48) → addc30_out_gga(T48, 0, T48)
addc30_in_gga(T54, s(T59), 0) → U11_gga(T54, T59, addc30_in_ggg(T54, T59, 0))
addc30_in_ggg(T48, 0, T48) → addc30_out_ggg(T48, 0, T48)
addc30_in_ggg(T54, s(T59), 0) → U11_ggg(T54, T59, addc30_in_ggg(T54, T59, 0))
addc30_in_ggg(T54, s(T59), s(T65)) → U12_ggg(T54, T59, T65, addc30_in_ggg(T54, T59, T65))
U12_ggg(T54, T59, T65, addc30_out_ggg(T54, T59, T65)) → addc30_out_ggg(T54, s(T59), s(T65))
U11_ggg(T54, T59, addc30_out_ggg(T54, T59, 0)) → addc30_out_ggg(T54, s(T59), 0)
U11_gga(T54, T59, addc30_out_ggg(T54, T59, 0)) → addc30_out_gga(T54, s(T59), 0)
addc30_in_gga(T54, s(T59), s(T65)) → U12_gga(T54, T59, T65, addc30_in_gga(T54, T59, T65))
U12_gga(T54, T59, T65, addc30_out_gga(T54, T59, T65)) → addc30_out_gga(T54, s(T59), s(T65))
U10_gga(T33, T38, X65, addc30_out_gga(T33, T40, X65)) → mulc15_out_gga(T33, s(T38), X65)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
mul15_in_gga(x1, x2, x3)  =  mul15_in_gga(x1, x2)
mulc15_in_gga(x1, x2, x3)  =  mulc15_in_gga(x1, x2)
0  =  0
mulc15_out_gga(x1, x2, x3)  =  mulc15_out_gga(x1, x2, x3)
U9_gga(x1, x2, x3, x4)  =  U9_gga(x1, x2, x4)
U10_gga(x1, x2, x3, x4)  =  U10_gga(x1, x2, x4)
addc30_in_gga(x1, x2, x3)  =  addc30_in_gga(x1, x2)
addc30_out_gga(x1, x2, x3)  =  addc30_out_gga(x1, x2, x3)
U11_gga(x1, x2, x3)  =  U11_gga(x1, x2, x3)
addc30_in_ggg(x1, x2, x3)  =  addc30_in_ggg(x1, x2, x3)
addc30_out_ggg(x1, x2, x3)  =  addc30_out_ggg(x1, x2, x3)
U11_ggg(x1, x2, x3)  =  U11_ggg(x1, x2, x3)
U12_ggg(x1, x2, x3, x4)  =  U12_ggg(x1, x2, x3, x4)
U12_gga(x1, x2, x3, x4)  =  U12_gga(x1, x2, x4)
add30_in_gga(x1, x2, x3)  =  add30_in_gga(x1, x2)
MUL1_IN_GGA(x1, x2, x3)  =  MUL1_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x1, x2, x4)
MUL15_IN_GGA(x1, x2, x3)  =  MUL15_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x1, x2, x4)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)
ADD30_IN_GGA(x1, x2, x3)  =  ADD30_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)
U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x1, x2, x4)
U7_GGA(x1, x2, x3, x4)  =  U7_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 10 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

ADD30_IN_GGA(T54, s(T59), X117) → ADD30_IN_GGA(T54, T59, X116)

The TRS R consists of the following rules:

mulc15_in_gga(T27, 0, 0) → mulc15_out_gga(T27, 0, 0)
mulc15_in_gga(T33, s(T38), X65) → U9_gga(T33, T38, X65, mulc15_in_gga(T33, T38, T40))
U9_gga(T33, T38, X65, mulc15_out_gga(T33, T38, T40)) → U10_gga(T33, T38, X65, addc30_in_gga(T33, T40, X65))
addc30_in_gga(T48, 0, T48) → addc30_out_gga(T48, 0, T48)
addc30_in_gga(T54, s(T59), 0) → U11_gga(T54, T59, addc30_in_ggg(T54, T59, 0))
addc30_in_ggg(T48, 0, T48) → addc30_out_ggg(T48, 0, T48)
addc30_in_ggg(T54, s(T59), 0) → U11_ggg(T54, T59, addc30_in_ggg(T54, T59, 0))
addc30_in_ggg(T54, s(T59), s(T65)) → U12_ggg(T54, T59, T65, addc30_in_ggg(T54, T59, T65))
U12_ggg(T54, T59, T65, addc30_out_ggg(T54, T59, T65)) → addc30_out_ggg(T54, s(T59), s(T65))
U11_ggg(T54, T59, addc30_out_ggg(T54, T59, 0)) → addc30_out_ggg(T54, s(T59), 0)
U11_gga(T54, T59, addc30_out_ggg(T54, T59, 0)) → addc30_out_gga(T54, s(T59), 0)
addc30_in_gga(T54, s(T59), s(T65)) → U12_gga(T54, T59, T65, addc30_in_gga(T54, T59, T65))
U12_gga(T54, T59, T65, addc30_out_gga(T54, T59, T65)) → addc30_out_gga(T54, s(T59), s(T65))
U10_gga(T33, T38, X65, addc30_out_gga(T33, T40, X65)) → mulc15_out_gga(T33, s(T38), X65)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
mulc15_in_gga(x1, x2, x3)  =  mulc15_in_gga(x1, x2)
0  =  0
mulc15_out_gga(x1, x2, x3)  =  mulc15_out_gga(x1, x2, x3)
U9_gga(x1, x2, x3, x4)  =  U9_gga(x1, x2, x4)
U10_gga(x1, x2, x3, x4)  =  U10_gga(x1, x2, x4)
addc30_in_gga(x1, x2, x3)  =  addc30_in_gga(x1, x2)
addc30_out_gga(x1, x2, x3)  =  addc30_out_gga(x1, x2, x3)
U11_gga(x1, x2, x3)  =  U11_gga(x1, x2, x3)
addc30_in_ggg(x1, x2, x3)  =  addc30_in_ggg(x1, x2, x3)
addc30_out_ggg(x1, x2, x3)  =  addc30_out_ggg(x1, x2, x3)
U11_ggg(x1, x2, x3)  =  U11_ggg(x1, x2, x3)
U12_ggg(x1, x2, x3, x4)  =  U12_ggg(x1, x2, x3, x4)
U12_gga(x1, x2, x3, x4)  =  U12_gga(x1, x2, x4)
ADD30_IN_GGA(x1, x2, x3)  =  ADD30_IN_GGA(x1, x2)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

ADD30_IN_GGA(T54, s(T59), X117) → ADD30_IN_GGA(T54, T59, X116)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
ADD30_IN_GGA(x1, x2, x3)  =  ADD30_IN_GGA(x1, x2)

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

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

ADD30_IN_GGA(T54, s(T59)) → ADD30_IN_GGA(T54, T59)

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

(12) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• ADD30_IN_GGA(T54, s(T59)) → ADD30_IN_GGA(T54, T59)
  The graph contains the following edges 1 >= 1, 2 > 2

(13) YES

(14) Obligation:

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

MUL15_IN_GGA(T33, s(T38), X65) → MUL15_IN_GGA(T33, T38, X64)

The TRS R consists of the following rules:

mulc15_in_gga(T27, 0, 0) → mulc15_out_gga(T27, 0, 0)
mulc15_in_gga(T33, s(T38), X65) → U9_gga(T33, T38, X65, mulc15_in_gga(T33, T38, T40))
U9_gga(T33, T38, X65, mulc15_out_gga(T33, T38, T40)) → U10_gga(T33, T38, X65, addc30_in_gga(T33, T40, X65))
addc30_in_gga(T48, 0, T48) → addc30_out_gga(T48, 0, T48)
addc30_in_gga(T54, s(T59), 0) → U11_gga(T54, T59, addc30_in_ggg(T54, T59, 0))
addc30_in_ggg(T48, 0, T48) → addc30_out_ggg(T48, 0, T48)
addc30_in_ggg(T54, s(T59), 0) → U11_ggg(T54, T59, addc30_in_ggg(T54, T59, 0))
addc30_in_ggg(T54, s(T59), s(T65)) → U12_ggg(T54, T59, T65, addc30_in_ggg(T54, T59, T65))
U12_ggg(T54, T59, T65, addc30_out_ggg(T54, T59, T65)) → addc30_out_ggg(T54, s(T59), s(T65))
U11_ggg(T54, T59, addc30_out_ggg(T54, T59, 0)) → addc30_out_ggg(T54, s(T59), 0)
U11_gga(T54, T59, addc30_out_ggg(T54, T59, 0)) → addc30_out_gga(T54, s(T59), 0)
addc30_in_gga(T54, s(T59), s(T65)) → U12_gga(T54, T59, T65, addc30_in_gga(T54, T59, T65))
U12_gga(T54, T59, T65, addc30_out_gga(T54, T59, T65)) → addc30_out_gga(T54, s(T59), s(T65))
U10_gga(T33, T38, X65, addc30_out_gga(T33, T40, X65)) → mulc15_out_gga(T33, s(T38), X65)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
mulc15_in_gga(x1, x2, x3)  =  mulc15_in_gga(x1, x2)
0  =  0
mulc15_out_gga(x1, x2, x3)  =  mulc15_out_gga(x1, x2, x3)
U9_gga(x1, x2, x3, x4)  =  U9_gga(x1, x2, x4)
U10_gga(x1, x2, x3, x4)  =  U10_gga(x1, x2, x4)
addc30_in_gga(x1, x2, x3)  =  addc30_in_gga(x1, x2)
addc30_out_gga(x1, x2, x3)  =  addc30_out_gga(x1, x2, x3)
U11_gga(x1, x2, x3)  =  U11_gga(x1, x2, x3)
addc30_in_ggg(x1, x2, x3)  =  addc30_in_ggg(x1, x2, x3)
addc30_out_ggg(x1, x2, x3)  =  addc30_out_ggg(x1, x2, x3)
U11_ggg(x1, x2, x3)  =  U11_ggg(x1, x2, x3)
U12_ggg(x1, x2, x3, x4)  =  U12_ggg(x1, x2, x3, x4)
U12_gga(x1, x2, x3, x4)  =  U12_gga(x1, x2, x4)
MUL15_IN_GGA(x1, x2, x3)  =  MUL15_IN_GGA(x1, x2)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

MUL15_IN_GGA(T33, s(T38), X65) → MUL15_IN_GGA(T33, T38, X64)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
MUL15_IN_GGA(x1, x2, x3)  =  MUL15_IN_GGA(x1, x2)

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

MUL15_IN_GGA(T33, s(T38)) → MUL15_IN_GGA(T33, T38)

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

(19) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• MUL15_IN_GGA(T33, s(T38)) → MUL15_IN_GGA(T33, T38)
  The graph contains the following edges 1 >= 1, 2 > 2

(20) YES