Left Termination proof of /tmp/tmpkvlxSP/mul1.pl

Left Termination of the query pattern mul_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 Instantiation (⇔)

                                                ↳15 QDP

                                                ↳16 NonTerminationProof (⇔)

                                                ↳17 FALSE

                            ↳18 PiDP

                                ↳19 UsableRulesProof (⇔)

                                    ↳20 PiDP

                                        ↳21 PiDPToQDPProof (⇐)

                                            ↳22 QDP

                                                ↳23 QDPSizeChangeProof (⇔)

                                                    ↳24 TRUE

                            ↳25 PiDP

                                ↳26 UsableRulesProof (⇔)

                                    ↳27 PiDP

                                        ↳28 PiDPToQDPProof (⇐)

                                            ↳29 QDP

                                                ↳30 QDPSizeChangeProof (⇔)

                                                    ↳31 TRUE

    ↳32 PrologToPiTRSProof (⇐)

        ↳33 PiTRS

            ↳34 DependencyPairsProof (⇔)

                ↳35 PiDP

                    ↳36 DependencyGraphProof (⇔)

                        ↳37 AND

                            ↳38 PiDP

                                ↳39 UsableRulesProof (⇔)

                                    ↳40 PiDP

                                        ↳41 PiDPToQDPProof (⇐)

                                            ↳42 QDP

                                                ↳43 Instantiation (⇔)

                                                ↳44 QDP

                                                ↳45 NonTerminationProof (⇔)

                                                ↳46 FALSE

                            ↳47 PiDP

                                ↳48 UsableRulesProof (⇔)

                                    ↳49 PiDP

                                        ↳50 PiDPToQDPProof (⇐)

                                            ↳51 QDP

                                                ↳52 QDPSizeChangeProof (⇔)

                                                    ↳53 TRUE

                            ↳54 PiDP

                                ↳55 UsableRulesProof (⇔)

                                    ↳56 PiDP

                                        ↳57 PiDPToQDPProof (⇐)

                                            ↳58 QDP

                                                ↳59 QDPSizeChangeProof (⇔)

                                                    ↳60 TRUE

(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) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog 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 | ?_1\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(!_1, eq(T6, 0)) | mul(T4, 0, T3), SCOPE: 1, CLAUSE: 1 | ?_1\n\nT4 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\nX4 is free; \nX5 is free; \nmul(T1, T2, T3) !=? mul(X4, 0, X5)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: eq(T6, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: eq(T6, 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\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(p(T9, X14), ','(mul(T8, X14, X15), add(T8, X15, T11)))\n\nT8 is ground\nT9 is ground\nX4 is free; \nX5 is free; \nX14 is free; \nX15 is free; \nmul(T8, T9, T3) !=? mul(X4, 0, X5)", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(p(T9, X14), ','(mul(T8, X14, X15), add(T8, X15, T11))), SCOPE: 3, CLAUSE: 4 | ','(p(T9, X14), ','(mul(T8, X14, X15), add(T8, X15, T11))), SCOPE: 3, CLAUSE: 5\n\nT8 is ground\nT9 is ground\nX4 is free; \nX5 is free; \nX14 is free; \nX15 is free; \nmul(T8, T9, T3) !=? mul(X4, 0, X5)", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(p(T9, X14), ','(mul(T8, X14, X15), add(T8, X15, T11))), SCOPE: 3, CLAUSE: 5\n\nT8 is ground\nT9 is ground\nX4 is free; \nX5 is free; \nX14 is free; \nX15 is free; \nmul(T8, T9, T3) !=? mul(X4, 0, X5)\np(T9, X14) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(mul(T8, T12, X15), add(T8, X15, T11))\n\nT8 is ground\nT12 is ground\nX4 is free; \nX5 is free; \nX14 is free; \nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\nX4 is free\nX5 is free; \nX14 is free; \nX17 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: mul(T8, T12, X15)\n\nT8 is ground\nT12 is ground\nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: add(T8, T13, T11)\n\nT8 is ground\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: mul(T8, T12, X15), SCOPE: 4, CLAUSE: 0 | mul(T8, T12, X15), SCOPE: 4, CLAUSE: 1 | ?_4\n\nT8 is ground\nT12 is ground\nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(!_4, eq(X22, 0)) | mul(T14, 0, X15), SCOPE: 4, CLAUSE: 1 | ?_4\n\nT14 is ground\nX15 is free; \nX22 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: mul(T8, T12, X15), SCOPE: 4, CLAUSE: 1\n\nT8 is ground\nT12 is ground\nX15 is free; \nX20 is free; \nX21 is free; \nmul(T8, T12, X15) !=? mul(X20, 0, X21)", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: eq(X22, 0)\n\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: eq(X22, 0), SCOPE: 5, CLAUSE: 6\n\nX22 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(T16, X32), ','(mul(T15, X32, X33), add(T15, X33, X34)))\n\nT15 is ground\nT16 is ground\nX15 is free; \nX20 is free; \nX21 is free; \nX34 is free; \nX32 is free; \nX33 is free; \nmul(T15, T16, X15) !=? mul(X20, 0, X21)", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(p(T16, X32), ','(mul(T15, X32, X33), add(T15, X33, X34))), SCOPE: 6, CLAUSE: 4 | ','(p(T16, X32), ','(mul(T15, X32, X33), add(T15, X33, X34))), SCOPE: 6, CLAUSE: 5\n\nT15 is ground\nT16 is ground\nX15 is free; \nX20 is free; \nX21 is free; \nX34 is free; \nX32 is free; \nX33 is free; \nmul(T15, T16, X15) !=? mul(X20, 0, X21)", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(p(T16, X32), ','(mul(T15, X32, X33), add(T15, X33, X34))), SCOPE: 6, CLAUSE: 5\n\nT15 is ground\nT16 is ground\nX15 is free; \nX20 is free; \nX21 is free; \nX34 is free; \nX32 is free; \nX33 is free; \nmul(T15, T16, X15) !=? mul(X20, 0, X21)\np(T16, X32) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(mul(T15, T17, X33), add(T15, X33, X34))\n\nT15 is ground\nT17 is ground\nX15 is free; \nX20 is free; \nX21 is free; \nX34 is free; \nX32 is free; \nX33 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\nX15 is free\nX20 is free; \nX21 is free; \nX32 is free; \nX36 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: mul(T15, T17, X33)\n\nT15 is ground\nT17 is ground\nX33 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: add(T15, T18, X34)\n\nT15 is ground\nT18 is ground\nX34 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: add(T15, T18, X34), SCOPE: 7, CLAUSE: 2 | add(T15, T18, X34), SCOPE: 7, CLAUSE: 3 | ?_7\n\nT15 is ground\nT18 is ground\nX34 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(!_7, eq(X41, T19)) | add(T19, 0, X34), SCOPE: 7, CLAUSE: 3 | ?_7\n\nT19 is ground\nX34 is free; \nX41 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: add(T15, T18, X34), SCOPE: 7, CLAUSE: 3\n\nT15 is ground\nT18 is ground\nX34 is free; \nX39 is free; \nX40 is free; \nadd(T15, T18, X34) !=? add(X39, 0, X40)", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: eq(X41, T19)\n\nT19 is ground\nX41 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: eq(X41, T19), SCOPE: 8, CLAUSE: 6\n\nT19 is ground\nX41 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(T22, X51), ','(add(T21, X51, X52), p(X53, X52)))\n\nT21 is ground\nT22 is ground\nX34 is free; \nX39 is free; \nX40 is free; \nX53 is free; \nX51 is free; \nX52 is free; \nadd(T21, T22, X34) !=? add(X39, 0, X40)", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(p(T22, X51), ','(add(T21, X51, X52), p(X53, X52))), SCOPE: 9, CLAUSE: 4 | ','(p(T22, X51), ','(add(T21, X51, X52), p(X53, X52))), SCOPE: 9, CLAUSE: 5\n\nT21 is ground\nT22 is ground\nX34 is free; \nX39 is free; \nX40 is free; \nX53 is free; \nX51 is free; \nX52 is free; \nadd(T21, T22, X34) !=? add(X39, 0, X40)", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(p(T22, X51), ','(add(T21, X51, X52), p(X53, X52))), SCOPE: 9, CLAUSE: 5\n\nT21 is ground\nT22 is ground\nX34 is free; \nX39 is free; \nX40 is free; \nX53 is free; \nX51 is free; \nX52 is free; \nadd(T21, T22, X34) !=? add(X39, 0, X40)\np(T22, X51) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(add(T21, T23, X52), p(X53, X52))\n\nT21 is ground\nT23 is ground\nX34 is free; \nX39 is free; \nX40 is free; \nX53 is free; \nX51 is free; \nX52 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\nX34 is free\nX39 is free; \nX40 is free; \nX51 is free; \nX55 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: add(T21, T23, X52)\n\nT21 is ground\nT23 is ground\nX52 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: p(X53, T24)\n\nT24 is ground\nX53 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: p(X53, T24), SCOPE: 10, CLAUSE: 4 | p(X53, T24), SCOPE: 10, CLAUSE: 5\n\nT24 is ground\nX53 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: true | p(X53, 0), SCOPE: 10, CLAUSE: 5\n\nX53 is free", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: p(X53, T24), SCOPE: 10, CLAUSE: 5\n\nT24 is ground\nX53 is free; \np(X53, T24) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: p(X53, 0), SCOPE: 10, CLAUSE: 5\n\nX53 is free", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: true\n\nX53 is free", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: add(T8, T13, T11), SCOPE: 11, CLAUSE: 2 | add(T8, T13, T11), SCOPE: 11, CLAUSE: 3 | ?_11\n\nT8 is ground\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(!_11, eq(T28, T26)) | add(T26, 0, T11), SCOPE: 11, CLAUSE: 3 | ?_11\n\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: add(T8, T13, T11), SCOPE: 11, CLAUSE: 3\n\nT8 is ground\nT13 is ground\nX64 is free; \nX65 is free; \nadd(T8, T13, T11) !=? add(X64, 0, X65)", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: eq(T28, T26)\n\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: eq(T28, T26), SCOPE: 12, CLAUSE: 6\n\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\nX67 is free", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(p(T31, X74), ','(add(T30, X74, X75), p(T33, X75)))\n\nT30 is ground\nT31 is ground\nX64 is free; \nX65 is free; \nX74 is free; \nX75 is free; \nadd(T30, T31, T11) !=? add(X64, 0, X65)", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(p(T31, X74), ','(add(T30, X74, X75), p(T33, X75))), SCOPE: 13, CLAUSE: 4 | ','(p(T31, X74), ','(add(T30, X74, X75), p(T33, X75))), SCOPE: 13, CLAUSE: 5\n\nT30 is ground\nT31 is ground\nX64 is free; \nX65 is free; \nX74 is free; \nX75 is free; \nadd(T30, T31, T11) !=? add(X64, 0, X65)", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(p(T31, X74), ','(add(T30, X74, X75), p(T33, X75))), SCOPE: 13, CLAUSE: 5\n\nT30 is ground\nT31 is ground\nX64 is free; \nX65 is free; \nX74 is free; \nX75 is free; \nadd(T30, T31, T11) !=? add(X64, 0, X65)\np(T31, X74) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ','(add(T30, T34, X75), p(T33, X75))\n\nT30 is ground\nT34 is ground\nX64 is free; \nX65 is free; \nX74 is free; \nX75 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: \n\nX64 is free\nX65 is free; \nX74 is free; \nX77 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: add(T30, T34, X75)\n\nT30 is ground\nT34 is ground\nX75 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: p(T33, T35)\n\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: p(T33, T35), SCOPE: 14, CLAUSE: 4 | p(T33, T35), SCOPE: 14, CLAUSE: 5\n\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: true | p(T33, 0), SCOPE: 14, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: p(T33, T35), SCOPE: 14, CLAUSE: 5\n\nT35 is ground\np(T33, T35) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: p(T33, 0), SCOPE: 14, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: \n\nX79 is free", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: \n\nX81 is free", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 78 [arrowhead = none ];
78 -> 2;

79 [label="EVAL with clause\nmul(X4, 0, X5) :- ','(!, eq(X5, 0)).\nand substitution\nT1 -> T4\nX4 -> T4\nT2 -> 0\nT3 -> T6\nX5 -> T6\nT5 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 79 [arrowhead = none ];
79 -> 3;

80 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 80 [arrowhead = none ];
80 -> 4;

81 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 81 [arrowhead = none ];
81 -> 5;

82 [label="EVAL with clause\nmul(X11, X12, X13) :- ','(p(X12, X14), ','(mul(X11, X14, X15), add(X11, X15, X13))).\nand substitution\nT1 -> T8\nX11 -> T8\nT2 -> T9\nX12 -> T9\nT3 -> T11\nX13 -> T11\nT10 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 82 [arrowhead = none ];
82 -> 10;

83 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 83 [arrowhead = none ];
83 -> 6;

84 [label="EVAL with clause\neq(X7, X7).\nand substitution\nT6 -> 0\nX7 -> 0\nT7 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 84 [arrowhead = none ];
84 -> 7;

85 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 85 [arrowhead = none ];
85 -> 8;

86 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 86 [arrowhead = none ];
86 -> 9;

87 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 87 [arrowhead = none ];
87 -> 11;

88 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (mul(T8, T9, T3), mul(X4, 0, X5))", fontsize=14, style = filled, fillcolor = white];
11 -> 88 [arrowhead = none ];
88 -> 12;

89 [label="EVAL with clause\np(s(X17), X17).\nand substitution\nX17 -> T12\nT9 -> s(T12)\nX14 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 89 [arrowhead = none ];
89 -> 13;

90 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 90 [arrowhead = none ];
90 -> 14;

91 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
13 -> 91 [arrowhead = none ];
91 -> 15;

92 [label="SPLIT 2\nnew knowledge:\nT13 is ground\nreplacements:\nX15 -> T13", fontsize=14, style = filled, fillcolor = violet];
13 -> 92 [arrowhead = none ];
92 -> 16;

93 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 93 [arrowhead = none ];
93 -> 17;

94 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 94 [arrowhead = none ];
94 -> 53;

95 [label="EVAL with clause\nmul(X20, 0, X21) :- ','(!, eq(X21, 0)).\nand substitution\nT8 -> T14\nX20 -> T14\nT12 -> 0\nX15 -> X22\nX21 -> X22", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 95 [arrowhead = none ];
95 -> 18;

96 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 96 [arrowhead = none ];
96 -> 19;

97 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 97 [arrowhead = none ];
97 -> 20;

98 [label="EVAL with clause\nmul(X29, X30, X31) :- ','(p(X30, X32), ','(mul(X29, X32, X33), add(X29, X33, X31))).\nand substitution\nT8 -> T15\nX29 -> T15\nT12 -> T16\nX30 -> T16\nX15 -> X34\nX31 -> X34", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 98 [arrowhead = none ];
98 -> 24;

99 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 99 [arrowhead = none ];
99 -> 21;

100 [label="EVAL with clause\neq(X24, X24).\nand substitution\nX22 -> 0\nX24 -> 0\nX25 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 100 [arrowhead = none ];
100 -> 22;

101 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 101 [arrowhead = none ];
101 -> 23;

102 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 102 [arrowhead = none ];
102 -> 25;

103 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (mul(T15, T16, X15), mul(X20, 0, X21))", fontsize=14, style = filled, fillcolor = white];
25 -> 103 [arrowhead = none ];
103 -> 26;

104 [label="EVAL with clause\np(s(X36), X36).\nand substitution\nX36 -> T17\nT16 -> s(T17)\nX32 -> T17", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 104 [arrowhead = none ];
104 -> 27;

105 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 105 [arrowhead = none ];
105 -> 28;

106 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
27 -> 106 [arrowhead = none ];
106 -> 29;

107 [label="SPLIT 2\nnew knowledge:\nT18 is ground\nreplacements:\nX33 -> T18", fontsize=14, style = filled, fillcolor = violet];
27 -> 107 [arrowhead = none ];
107 -> 30;

108 [label="INSTANCE with matching:\nT8 -> T15\nT12 -> T17\nX15 -> X33", fontsize=14, style = filled, fillcolor = lightgrey];
29 -> 108 [arrowhead = none , style=dashed];
108 -> 15[style=dashed];

109 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
30 -> 109 [arrowhead = none ];
109 -> 31;

110 [label="EVAL with clause\nadd(X39, 0, X40) :- ','(!, eq(X40, X39)).\nand substitution\nT15 -> T19\nX39 -> T19\nT18 -> 0\nX34 -> X41\nX40 -> X41", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 110 [arrowhead = none ];
110 -> 32;

111 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 111 [arrowhead = none ];
111 -> 33;

112 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 112 [arrowhead = none ];
112 -> 34;

113 [label="EVAL with clause\nadd(X48, X49, X50) :- ','(p(X49, X51), ','(add(X48, X51, X52), p(X50, X52))).\nand substitution\nT15 -> T21\nX48 -> T21\nT18 -> T22\nX49 -> T22\nX34 -> X53\nX50 -> X53", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 113 [arrowhead = none ];
113 -> 38;

114 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 114 [arrowhead = none ];
114 -> 35;

115 [label="EVAL with clause\neq(X43, X43).\nand substitution\nX41 -> T20\nX43 -> T20\nT19 -> T20\nX44 -> T20", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 115 [arrowhead = none ];
115 -> 36;

116 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 116 [arrowhead = none ];
116 -> 37;

117 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
38 -> 117 [arrowhead = none ];
117 -> 39;

118 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (add(T21, T22, X34), add(X39, 0, X40))", fontsize=14, style = filled, fillcolor = white];
39 -> 118 [arrowhead = none ];
118 -> 40;

119 [label="EVAL with clause\np(s(X55), X55).\nand substitution\nX55 -> T23\nT22 -> s(T23)\nX51 -> T23", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 119 [arrowhead = none ];
119 -> 41;

120 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 120 [arrowhead = none ];
120 -> 42;

121 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
41 -> 121 [arrowhead = none ];
121 -> 43;

122 [label="SPLIT 2\nnew knowledge:\nT24 is ground\nreplacements:\nX52 -> T24", fontsize=14, style = filled, fillcolor = violet];
41 -> 122 [arrowhead = none ];
122 -> 44;

123 [label="INSTANCE with matching:\nT15 -> T21\nT18 -> T23\nX34 -> X52", fontsize=14, style = filled, fillcolor = lightgrey];
43 -> 123 [arrowhead = none , style=dashed];
123 -> 30[style=dashed];

124 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
44 -> 124 [arrowhead = none ];
124 -> 45;

125 [label="EVAL with clause\np(0, 0).\nand substitution\nX53 -> 0\nT24 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 125 [arrowhead = none ];
125 -> 46;

126 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 126 [arrowhead = none ];
126 -> 47;

127 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 127 [arrowhead = none ];
127 -> 48;

128 [label="EVAL with clause\np(s(X60), X60).\nand substitution\nX60 -> T25\nX53 -> s(T25)\nT24 -> T25\nX61 -> T25", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 128 [arrowhead = none ];
128 -> 51;

129 [label="EVAL with clause\np(s(X57), X57).\nand substitution\nX57 -> 0\nX53 -> s(0)\nX58 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 129 [arrowhead = none ];
129 -> 49;

130 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 130 [arrowhead = none ];
130 -> 50;

131 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 131 [arrowhead = none ];
131 -> 52;

132 [label="EVAL with clause\nadd(X64, 0, X65) :- ','(!, eq(X65, X64)).\nand substitution\nT8 -> T26\nX64 -> T26\nT13 -> 0\nT11 -> T28\nX65 -> T28\nT27 -> T28", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 132 [arrowhead = none ];
132 -> 54;

133 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 133 [arrowhead = none ];
133 -> 55;

134 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 134 [arrowhead = none ];
134 -> 56;

135 [label="EVAL with clause\nadd(X71, X72, X73) :- ','(p(X72, X74), ','(add(X71, X74, X75), p(X73, X75))).\nand substitution\nT8 -> T30\nX71 -> T30\nT13 -> T31\nX72 -> T31\nT11 -> T33\nX73 -> T33\nT32 -> T33", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 135 [arrowhead = none ];
135 -> 61;

136 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
56 -> 136 [arrowhead = none ];
136 -> 57;

137 [label="EVAL with clause\neq(X67, X67).\nand substitution\nT28 -> T29\nX67 -> T29\nT26 -> T29", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 137 [arrowhead = none ];
137 -> 58;

138 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 138 [arrowhead = none ];
138 -> 59;

139 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
58 -> 139 [arrowhead = none ];
139 -> 60;

140 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
61 -> 140 [arrowhead = none ];
140 -> 62;

141 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (add(T30, T31, T11), add(X64, 0, X65))", fontsize=14, style = filled, fillcolor = white];
62 -> 141 [arrowhead = none ];
141 -> 63;

142 [label="EVAL with clause\np(s(X77), X77).\nand substitution\nX77 -> T34\nT31 -> s(T34)\nX74 -> T34", fontsize=14, style = filled, fillcolor = lightblue];
63 -> 142 [arrowhead = none ];
142 -> 64;

143 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
63 -> 143 [arrowhead = none ];
143 -> 65;

144 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
64 -> 144 [arrowhead = none ];
144 -> 66;

145 [label="SPLIT 2\nnew knowledge:\nT35 is ground\nreplacements:\nX75 -> T35", fontsize=14, style = filled, fillcolor = violet];
64 -> 145 [arrowhead = none ];
145 -> 67;

146 [label="INSTANCE with matching:\nT15 -> T30\nT18 -> T34\nX34 -> X75", fontsize=14, style = filled, fillcolor = lightgrey];
66 -> 146 [arrowhead = none , style=dashed];
146 -> 30[style=dashed];

147 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
67 -> 147 [arrowhead = none ];
147 -> 68;

148 [label="EVAL with clause\np(0, 0).\nand substitution\nT33 -> 0\nT35 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 148 [arrowhead = none ];
148 -> 69;

149 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 149 [arrowhead = none ];
149 -> 70;

150 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
69 -> 150 [arrowhead = none ];
150 -> 71;

151 [label="EVAL with clause\np(s(X81), X81).\nand substitution\nX81 -> T37\nT33 -> s(T37)\nT35 -> T37", fontsize=14, style = filled, fillcolor = lightblue];
70 -> 151 [arrowhead = none ];
151 -> 75;

152 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
70 -> 152 [arrowhead = none ];
152 -> 76;

153 [label="EVAL with clause\np(s(X79), X79).\nand substitution\nX79 -> 0\nT33 -> s(0)\nT36 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
71 -> 153 [arrowhead = none ];
153 -> 72;

154 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
71 -> 154 [arrowhead = none ];
154 -> 73;

155 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
72 -> 155 [arrowhead = none ];
155 -> 74;

156 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
75 -> 156 [arrowhead = none ];
156 -> 77;

}

(2) Obligation:

Clauses:

mul15(T14, 0, 0).
mul15(T15, s(T17), X34) :- mul15(T15, T17, X33).
mul15(T20, s(T17), T20) :- mul15(T20, T17, 0).
mul15(T21, s(T17), X53) :- ','(mul15(T21, T17, s(T23)), add30(T21, T23, X52)).
mul15(T21, s(T17), 0) :- ','(mul15(T21, T17, s(T23)), add30(T21, T23, 0)).
mul15(T21, s(T17), s(0)) :- ','(mul15(T21, T17, s(T23)), add30(T21, T23, 0)).
mul15(T21, s(T17), s(T25)) :- ','(mul15(T21, T17, s(T23)), add30(T21, T23, T25)).
add30(T20, 0, T20).
add30(T21, s(T23), X53) :- add30(T21, T23, X52).
add30(T21, s(T23), 0) :- add30(T21, T23, 0).
add30(T21, s(T23), s(0)) :- add30(T21, T23, 0).
add30(T21, s(T23), s(T25)) :- add30(T21, T23, T25).
mul1(T4, 0, 0).
mul1(T8, s(T12), T11) :- mul15(T8, T12, X15).
mul1(T29, s(T12), T29) :- mul15(T29, T12, 0).
mul1(T30, s(T12), T33) :- ','(mul15(T30, T12, s(T34)), add30(T30, T34, X75)).
mul1(T30, s(T12), 0) :- ','(mul15(T30, T12, s(T34)), add30(T30, T34, 0)).
mul1(T30, s(T12), s(0)) :- ','(mul15(T30, T12, s(T34)), add30(T30, T34, 0)).
mul1(T30, s(T12), s(T37)) :- ','(mul15(T30, T12, s(T34)), add30(T30, T34, T37)).

Queries:

mul1(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:
mul1_in: (b,b,f)
mul15_in: (b,b,f) (b,b,b)
add30_in: (b,f,f) (b,f,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

mul1_in_gga(T4, 0, 0) → mul1_out_gga(T4, 0, 0)
mul1_in_gga(T8, s(T12), T11) → U15_gga(T8, T12, T11, mul15_in_gga(T8, T12, X15))
mul15_in_gga(T14, 0, 0) → mul15_out_gga(T14, 0, 0)
mul15_in_gga(T15, s(T17), X34) → U1_gga(T15, T17, X34, mul15_in_gga(T15, T17, X33))
mul15_in_gga(T20, s(T17), T20) → U2_gga(T20, T17, mul15_in_ggg(T20, T17, 0))
mul15_in_ggg(T14, 0, 0) → mul15_out_ggg(T14, 0, 0)
mul15_in_ggg(T15, s(T17), X34) → U1_ggg(T15, T17, X34, mul15_in_gga(T15, T17, X33))
mul15_in_gga(T21, s(T17), X53) → U3_gga(T21, T17, X53, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), 0) → U5_gga(T21, T17, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), s(0)) → U7_gga(T21, T17, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), s(T25)) → U9_gga(T21, T17, T25, mul15_in_gga(T21, T17, s(T23)))
U9_gga(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → U10_gga(T21, T17, T25, add30_in_gaa(T21, T23, T25))
add30_in_gaa(T20, 0, T20) → add30_out_gaa(T20, 0, T20)
add30_in_gaa(T21, s(T23), X53) → U11_gaa(T21, T23, X53, add30_in_gaa(T21, T23, X52))
add30_in_gaa(T21, s(T23), 0) → U12_gaa(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T20, 0, T20) → add30_out_gag(T20, 0, T20)
add30_in_gag(T21, s(T23), X53) → U11_gag(T21, T23, X53, add30_in_gaa(T21, T23, X52))
add30_in_gaa(T21, s(T23), s(0)) → U13_gaa(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), 0) → U12_gag(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), s(0)) → U13_gag(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), s(T25)) → U14_gag(T21, T23, T25, add30_in_gag(T21, T23, T25))
U14_gag(T21, T23, T25, add30_out_gag(T21, T23, T25)) → add30_out_gag(T21, s(T23), s(T25))
U13_gag(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gag(T21, s(T23), s(0))
U12_gag(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gag(T21, s(T23), 0)
U13_gaa(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gaa(T21, s(T23), s(0))
add30_in_gaa(T21, s(T23), s(T25)) → U14_gaa(T21, T23, T25, add30_in_gaa(T21, T23, T25))
U14_gaa(T21, T23, T25, add30_out_gaa(T21, T23, T25)) → add30_out_gaa(T21, s(T23), s(T25))
U11_gag(T21, T23, X53, add30_out_gaa(T21, T23, X52)) → add30_out_gag(T21, s(T23), X53)
U12_gaa(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gaa(T21, s(T23), 0)
U11_gaa(T21, T23, X53, add30_out_gaa(T21, T23, X52)) → add30_out_gaa(T21, s(T23), X53)
U10_gga(T21, T17, T25, add30_out_gaa(T21, T23, T25)) → mul15_out_gga(T21, s(T17), s(T25))
U7_gga(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U8_gga(T21, T17, add30_in_gag(T21, T23, 0))
U8_gga(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_gga(T21, s(T17), s(0))
U5_gga(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U6_gga(T21, T17, add30_in_gag(T21, T23, 0))
U6_gga(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_gga(T21, s(T17), 0)
U3_gga(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → U4_gga(T21, T17, X53, add30_in_gaa(T21, T23, X52))
U4_gga(T21, T17, X53, add30_out_gaa(T21, T23, X52)) → mul15_out_gga(T21, s(T17), X53)
U1_ggg(T15, T17, X34, mul15_out_gga(T15, T17, X33)) → mul15_out_ggg(T15, s(T17), X34)
mul15_in_ggg(T20, s(T17), T20) → U2_ggg(T20, T17, mul15_in_ggg(T20, T17, 0))
mul15_in_ggg(T21, s(T17), X53) → U3_ggg(T21, T17, X53, mul15_in_gga(T21, T17, s(T23)))
U3_ggg(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → U4_ggg(T21, T17, X53, add30_in_gaa(T21, T23, X52))
U4_ggg(T21, T17, X53, add30_out_gaa(T21, T23, X52)) → mul15_out_ggg(T21, s(T17), X53)
mul15_in_ggg(T21, s(T17), 0) → U5_ggg(T21, T17, mul15_in_gga(T21, T17, s(T23)))
U5_ggg(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U6_ggg(T21, T17, add30_in_gag(T21, T23, 0))
U6_ggg(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_ggg(T21, s(T17), 0)
mul15_in_ggg(T21, s(T17), s(0)) → U7_ggg(T21, T17, mul15_in_gga(T21, T17, s(T23)))
U7_ggg(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U8_ggg(T21, T17, add30_in_gag(T21, T23, 0))
U8_ggg(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_ggg(T21, s(T17), s(0))
mul15_in_ggg(T21, s(T17), s(T25)) → U9_ggg(T21, T17, T25, mul15_in_gga(T21, T17, s(T23)))
U9_ggg(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → U10_ggg(T21, T17, T25, add30_in_gag(T21, T23, T25))
U10_ggg(T21, T17, T25, add30_out_gag(T21, T23, T25)) → mul15_out_ggg(T21, s(T17), s(T25))
U2_ggg(T20, T17, mul15_out_ggg(T20, T17, 0)) → mul15_out_ggg(T20, s(T17), T20)
U2_gga(T20, T17, mul15_out_ggg(T20, T17, 0)) → mul15_out_gga(T20, s(T17), T20)
U1_gga(T15, T17, X34, mul15_out_gga(T15, T17, X33)) → mul15_out_gga(T15, s(T17), X34)
U15_gga(T8, T12, T11, mul15_out_gga(T8, T12, X15)) → mul1_out_gga(T8, s(T12), T11)
mul1_in_gga(T29, s(T12), T29) → U16_gga(T29, T12, mul15_in_ggg(T29, T12, 0))
U16_gga(T29, T12, mul15_out_ggg(T29, T12, 0)) → mul1_out_gga(T29, s(T12), T29)
mul1_in_gga(T30, s(T12), T33) → U17_gga(T30, T12, T33, mul15_in_gga(T30, T12, s(T34)))
U17_gga(T30, T12, T33, mul15_out_gga(T30, T12, s(T34))) → U18_gga(T30, T12, T33, add30_in_gaa(T30, T34, X75))
U18_gga(T30, T12, T33, add30_out_gaa(T30, T34, X75)) → mul1_out_gga(T30, s(T12), T33)
mul1_in_gga(T30, s(T12), 0) → U19_gga(T30, T12, mul15_in_gga(T30, T12, s(T34)))
U19_gga(T30, T12, mul15_out_gga(T30, T12, s(T34))) → U20_gga(T30, T12, add30_in_gag(T30, T34, 0))
U20_gga(T30, T12, add30_out_gag(T30, T34, 0)) → mul1_out_gga(T30, s(T12), 0)
mul1_in_gga(T30, s(T12), s(0)) → U21_gga(T30, T12, mul15_in_gga(T30, T12, s(T34)))
U21_gga(T30, T12, mul15_out_gga(T30, T12, s(T34))) → U22_gga(T30, T12, add30_in_gag(T30, T34, 0))
U22_gga(T30, T12, add30_out_gag(T30, T34, 0)) → mul1_out_gga(T30, s(T12), s(0))
mul1_in_gga(T30, s(T12), s(T37)) → U23_gga(T30, T12, T37, mul15_in_gga(T30, T12, s(T34)))
U23_gga(T30, T12, T37, mul15_out_gga(T30, T12, s(T34))) → U24_gga(T30, T12, T37, add30_in_gaa(T30, T34, T37))
U24_gga(T30, T12, T37, add30_out_gaa(T30, T34, T37)) → mul1_out_gga(T30, s(T12), s(T37))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

mul1_in_gga(T4, 0, 0) → mul1_out_gga(T4, 0, 0)
mul1_in_gga(T8, s(T12), T11) → U15_gga(T8, T12, T11, mul15_in_gga(T8, T12, X15))
mul15_in_gga(T14, 0, 0) → mul15_out_gga(T14, 0, 0)
mul15_in_gga(T15, s(T17), X34) → U1_gga(T15, T17, X34, mul15_in_gga(T15, T17, X33))
mul15_in_gga(T20, s(T17), T20) → U2_gga(T20, T17, mul15_in_ggg(T20, T17, 0))
mul15_in_ggg(T14, 0, 0) → mul15_out_ggg(T14, 0, 0)
mul15_in_ggg(T15, s(T17), X34) → U1_ggg(T15, T17, X34, mul15_in_gga(T15, T17, X33))
mul15_in_gga(T21, s(T17), X53) → U3_gga(T21, T17, X53, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), 0) → U5_gga(T21, T17, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), s(0)) → U7_gga(T21, T17, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), s(T25)) → U9_gga(T21, T17, T25, mul15_in_gga(T21, T17, s(T23)))
U9_gga(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → U10_gga(T21, T17, T25, add30_in_gaa(T21, T23, T25))
add30_in_gaa(T20, 0, T20) → add30_out_gaa(T20, 0, T20)
add30_in_gaa(T21, s(T23), X53) → U11_gaa(T21, T23, X53, add30_in_gaa(T21, T23, X52))
add30_in_gaa(T21, s(T23), 0) → U12_gaa(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T20, 0, T20) → add30_out_gag(T20, 0, T20)
add30_in_gag(T21, s(T23), X53) → U11_gag(T21, T23, X53, add30_in_gaa(T21, T23, X52))
add30_in_gaa(T21, s(T23), s(0)) → U13_gaa(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), 0) → U12_gag(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), s(0)) → U13_gag(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), s(T25)) → U14_gag(T21, T23, T25, add30_in_gag(T21, T23, T25))
U14_gag(T21, T23, T25, add30_out_gag(T21, T23, T25)) → add30_out_gag(T21, s(T23), s(T25))
U13_gag(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gag(T21, s(T23), s(0))
U12_gag(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gag(T21, s(T23), 0)
U13_gaa(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gaa(T21, s(T23), s(0))
add30_in_gaa(T21, s(T23), s(T25)) → U14_gaa(T21, T23, T25, add30_in_gaa(T21, T23, T25))
U14_gaa(T21, T23, T25, add30_out_gaa(T21, T23, T25)) → add30_out_gaa(T21, s(T23), s(T25))
U11_gag(T21, T23, X53, add30_out_gaa(T21, T23, X52)) → add30_out_gag(T21, s(T23), X53)
U12_gaa(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gaa(T21, s(T23), 0)
U11_gaa(T21, T23, X53, add30_out_gaa(T21, T23, X52)) → add30_out_gaa(T21, s(T23), X53)
U10_gga(T21, T17, T25, add30_out_gaa(T21, T23, T25)) → mul15_out_gga(T21, s(T17), s(T25))
U7_gga(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U8_gga(T21, T17, add30_in_gag(T21, T23, 0))
U8_gga(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_gga(T21, s(T17), s(0))
U5_gga(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U6_gga(T21, T17, add30_in_gag(T21, T23, 0))
U6_gga(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_gga(T21, s(T17), 0)
U3_gga(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → U4_gga(T21, T17, X53, add30_in_gaa(T21, T23, X52))
U4_gga(T21, T17, X53, add30_out_gaa(T21, T23, X52)) → mul15_out_gga(T21, s(T17), X53)
U1_ggg(T15, T17, X34, mul15_out_gga(T15, T17, X33)) → mul15_out_ggg(T15, s(T17), X34)
mul15_in_ggg(T20, s(T17), T20) → U2_ggg(T20, T17, mul15_in_ggg(T20, T17, 0))
mul15_in_ggg(T21, s(T17), X53) → U3_ggg(T21, T17, X53, mul15_in_gga(T21, T17, s(T23)))
U3_ggg(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → U4_ggg(T21, T17, X53, add30_in_gaa(T21, T23, X52))
U4_ggg(T21, T17, X53, add30_out_gaa(T21, T23, X52)) → mul15_out_ggg(T21, s(T17), X53)
mul15_in_ggg(T21, s(T17), 0) → U5_ggg(T21, T17, mul15_in_gga(T21, T17, s(T23)))
U5_ggg(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U6_ggg(T21, T17, add30_in_gag(T21, T23, 0))
U6_ggg(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_ggg(T21, s(T17), 0)
mul15_in_ggg(T21, s(T17), s(0)) → U7_ggg(T21, T17, mul15_in_gga(T21, T17, s(T23)))
U7_ggg(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U8_ggg(T21, T17, add30_in_gag(T21, T23, 0))
U8_ggg(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_ggg(T21, s(T17), s(0))
mul15_in_ggg(T21, s(T17), s(T25)) → U9_ggg(T21, T17, T25, mul15_in_gga(T21, T17, s(T23)))
U9_ggg(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → U10_ggg(T21, T17, T25, add30_in_gag(T21, T23, T25))
U10_ggg(T21, T17, T25, add30_out_gag(T21, T23, T25)) → mul15_out_ggg(T21, s(T17), s(T25))
U2_ggg(T20, T17, mul15_out_ggg(T20, T17, 0)) → mul15_out_ggg(T20, s(T17), T20)
U2_gga(T20, T17, mul15_out_ggg(T20, T17, 0)) → mul15_out_gga(T20, s(T17), T20)
U1_gga(T15, T17, X34, mul15_out_gga(T15, T17, X33)) → mul15_out_gga(T15, s(T17), X34)
U15_gga(T8, T12, T11, mul15_out_gga(T8, T12, X15)) → mul1_out_gga(T8, s(T12), T11)
mul1_in_gga(T29, s(T12), T29) → U16_gga(T29, T12, mul15_in_ggg(T29, T12, 0))
U16_gga(T29, T12, mul15_out_ggg(T29, T12, 0)) → mul1_out_gga(T29, s(T12), T29)
mul1_in_gga(T30, s(T12), T33) → U17_gga(T30, T12, T33, mul15_in_gga(T30, T12, s(T34)))
U17_gga(T30, T12, T33, mul15_out_gga(T30, T12, s(T34))) → U18_gga(T30, T12, T33, add30_in_gaa(T30, T34, X75))
U18_gga(T30, T12, T33, add30_out_gaa(T30, T34, X75)) → mul1_out_gga(T30, s(T12), T33)
mul1_in_gga(T30, s(T12), 0) → U19_gga(T30, T12, mul15_in_gga(T30, T12, s(T34)))
U19_gga(T30, T12, mul15_out_gga(T30, T12, s(T34))) → U20_gga(T30, T12, add30_in_gag(T30, T34, 0))
U20_gga(T30, T12, add30_out_gag(T30, T34, 0)) → mul1_out_gga(T30, s(T12), 0)
mul1_in_gga(T30, s(T12), s(0)) → U21_gga(T30, T12, mul15_in_gga(T30, T12, s(T34)))
U21_gga(T30, T12, mul15_out_gga(T30, T12, s(T34))) → U22_gga(T30, T12, add30_in_gag(T30, T34, 0))
U22_gga(T30, T12, add30_out_gag(T30, T34, 0)) → mul1_out_gga(T30, s(T12), s(0))
mul1_in_gga(T30, s(T12), s(T37)) → U23_gga(T30, T12, T37, mul15_in_gga(T30, T12, s(T34)))
U23_gga(T30, T12, T37, mul15_out_gga(T30, T12, s(T34))) → U24_gga(T30, T12, T37, add30_in_gaa(T30, T34, T37))
U24_gga(T30, T12, T37, add30_out_gaa(T30, T34, T37)) → mul1_out_gga(T30, s(T12), s(T37))

(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:

MUL1_IN_GGA(T8, s(T12), T11) → U15_GGA(T8, T12, T11, mul15_in_gga(T8, T12, X15))
MUL1_IN_GGA(T8, s(T12), T11) → MUL15_IN_GGA(T8, T12, X15)
MUL15_IN_GGA(T15, s(T17), X34) → U1_GGA(T15, T17, X34, mul15_in_gga(T15, T17, X33))
MUL15_IN_GGA(T15, s(T17), X34) → MUL15_IN_GGA(T15, T17, X33)
MUL15_IN_GGA(T20, s(T17), T20) → U2_GGA(T20, T17, mul15_in_ggg(T20, T17, 0))
MUL15_IN_GGA(T20, s(T17), T20) → MUL15_IN_GGG(T20, T17, 0)
MUL15_IN_GGG(T15, s(T17), X34) → U1_GGG(T15, T17, X34, mul15_in_gga(T15, T17, X33))
MUL15_IN_GGG(T15, s(T17), X34) → MUL15_IN_GGA(T15, T17, X33)
MUL15_IN_GGA(T21, s(T17), X53) → U3_GGA(T21, T17, X53, mul15_in_gga(T21, T17, s(T23)))
MUL15_IN_GGA(T21, s(T17), X53) → MUL15_IN_GGA(T21, T17, s(T23))
MUL15_IN_GGA(T21, s(T17), 0) → U5_GGA(T21, T17, mul15_in_gga(T21, T17, s(T23)))
MUL15_IN_GGA(T21, s(T17), 0) → MUL15_IN_GGA(T21, T17, s(T23))
MUL15_IN_GGA(T21, s(T17), s(0)) → U7_GGA(T21, T17, mul15_in_gga(T21, T17, s(T23)))
MUL15_IN_GGA(T21, s(T17), s(0)) → MUL15_IN_GGA(T21, T17, s(T23))
MUL15_IN_GGA(T21, s(T17), s(T25)) → U9_GGA(T21, T17, T25, mul15_in_gga(T21, T17, s(T23)))
MUL15_IN_GGA(T21, s(T17), s(T25)) → MUL15_IN_GGA(T21, T17, s(T23))
U9_GGA(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → U10_GGA(T21, T17, T25, add30_in_gaa(T21, T23, T25))
U9_GGA(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → ADD30_IN_GAA(T21, T23, T25)
ADD30_IN_GAA(T21, s(T23), X53) → U11_GAA(T21, T23, X53, add30_in_gaa(T21, T23, X52))
ADD30_IN_GAA(T21, s(T23), X53) → ADD30_IN_GAA(T21, T23, X52)
ADD30_IN_GAA(T21, s(T23), 0) → U12_GAA(T21, T23, add30_in_gag(T21, T23, 0))
ADD30_IN_GAA(T21, s(T23), 0) → ADD30_IN_GAG(T21, T23, 0)
ADD30_IN_GAG(T21, s(T23), X53) → U11_GAG(T21, T23, X53, add30_in_gaa(T21, T23, X52))
ADD30_IN_GAG(T21, s(T23), X53) → ADD30_IN_GAA(T21, T23, X52)
ADD30_IN_GAA(T21, s(T23), s(0)) → U13_GAA(T21, T23, add30_in_gag(T21, T23, 0))
ADD30_IN_GAA(T21, s(T23), s(0)) → ADD30_IN_GAG(T21, T23, 0)
ADD30_IN_GAG(T21, s(T23), 0) → U12_GAG(T21, T23, add30_in_gag(T21, T23, 0))
ADD30_IN_GAG(T21, s(T23), 0) → ADD30_IN_GAG(T21, T23, 0)
ADD30_IN_GAG(T21, s(T23), s(0)) → U13_GAG(T21, T23, add30_in_gag(T21, T23, 0))
ADD30_IN_GAG(T21, s(T23), s(0)) → ADD30_IN_GAG(T21, T23, 0)
ADD30_IN_GAG(T21, s(T23), s(T25)) → U14_GAG(T21, T23, T25, add30_in_gag(T21, T23, T25))
ADD30_IN_GAG(T21, s(T23), s(T25)) → ADD30_IN_GAG(T21, T23, T25)
ADD30_IN_GAA(T21, s(T23), s(T25)) → U14_GAA(T21, T23, T25, add30_in_gaa(T21, T23, T25))
ADD30_IN_GAA(T21, s(T23), s(T25)) → ADD30_IN_GAA(T21, T23, T25)
U7_GGA(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U8_GGA(T21, T17, add30_in_gag(T21, T23, 0))
U7_GGA(T21, T17, mul15_out_gga(T21, T17, s(T23))) → ADD30_IN_GAG(T21, T23, 0)
U5_GGA(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U6_GGA(T21, T17, add30_in_gag(T21, T23, 0))
U5_GGA(T21, T17, mul15_out_gga(T21, T17, s(T23))) → ADD30_IN_GAG(T21, T23, 0)
U3_GGA(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → U4_GGA(T21, T17, X53, add30_in_gaa(T21, T23, X52))
U3_GGA(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → ADD30_IN_GAA(T21, T23, X52)
MUL15_IN_GGG(T20, s(T17), T20) → U2_GGG(T20, T17, mul15_in_ggg(T20, T17, 0))
MUL15_IN_GGG(T20, s(T17), T20) → MUL15_IN_GGG(T20, T17, 0)
MUL15_IN_GGG(T21, s(T17), X53) → U3_GGG(T21, T17, X53, mul15_in_gga(T21, T17, s(T23)))
MUL15_IN_GGG(T21, s(T17), X53) → MUL15_IN_GGA(T21, T17, s(T23))
U3_GGG(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → U4_GGG(T21, T17, X53, add30_in_gaa(T21, T23, X52))
U3_GGG(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → ADD30_IN_GAA(T21, T23, X52)
MUL15_IN_GGG(T21, s(T17), 0) → U5_GGG(T21, T17, mul15_in_gga(T21, T17, s(T23)))
MUL15_IN_GGG(T21, s(T17), 0) → MUL15_IN_GGA(T21, T17, s(T23))
U5_GGG(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U6_GGG(T21, T17, add30_in_gag(T21, T23, 0))
U5_GGG(T21, T17, mul15_out_gga(T21, T17, s(T23))) → ADD30_IN_GAG(T21, T23, 0)
MUL15_IN_GGG(T21, s(T17), s(0)) → U7_GGG(T21, T17, mul15_in_gga(T21, T17, s(T23)))
MUL15_IN_GGG(T21, s(T17), s(0)) → MUL15_IN_GGA(T21, T17, s(T23))
U7_GGG(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U8_GGG(T21, T17, add30_in_gag(T21, T23, 0))
U7_GGG(T21, T17, mul15_out_gga(T21, T17, s(T23))) → ADD30_IN_GAG(T21, T23, 0)
MUL15_IN_GGG(T21, s(T17), s(T25)) → U9_GGG(T21, T17, T25, mul15_in_gga(T21, T17, s(T23)))
MUL15_IN_GGG(T21, s(T17), s(T25)) → MUL15_IN_GGA(T21, T17, s(T23))
U9_GGG(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → U10_GGG(T21, T17, T25, add30_in_gag(T21, T23, T25))
U9_GGG(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → ADD30_IN_GAG(T21, T23, T25)
MUL1_IN_GGA(T29, s(T12), T29) → U16_GGA(T29, T12, mul15_in_ggg(T29, T12, 0))
MUL1_IN_GGA(T29, s(T12), T29) → MUL15_IN_GGG(T29, T12, 0)
MUL1_IN_GGA(T30, s(T12), T33) → U17_GGA(T30, T12, T33, mul15_in_gga(T30, T12, s(T34)))
MUL1_IN_GGA(T30, s(T12), T33) → MUL15_IN_GGA(T30, T12, s(T34))
U17_GGA(T30, T12, T33, mul15_out_gga(T30, T12, s(T34))) → U18_GGA(T30, T12, T33, add30_in_gaa(T30, T34, X75))
U17_GGA(T30, T12, T33, mul15_out_gga(T30, T12, s(T34))) → ADD30_IN_GAA(T30, T34, X75)
MUL1_IN_GGA(T30, s(T12), 0) → U19_GGA(T30, T12, mul15_in_gga(T30, T12, s(T34)))
MUL1_IN_GGA(T30, s(T12), 0) → MUL15_IN_GGA(T30, T12, s(T34))
U19_GGA(T30, T12, mul15_out_gga(T30, T12, s(T34))) → U20_GGA(T30, T12, add30_in_gag(T30, T34, 0))
U19_GGA(T30, T12, mul15_out_gga(T30, T12, s(T34))) → ADD30_IN_GAG(T30, T34, 0)
MUL1_IN_GGA(T30, s(T12), s(0)) → U21_GGA(T30, T12, mul15_in_gga(T30, T12, s(T34)))
MUL1_IN_GGA(T30, s(T12), s(0)) → MUL15_IN_GGA(T30, T12, s(T34))
U21_GGA(T30, T12, mul15_out_gga(T30, T12, s(T34))) → U22_GGA(T30, T12, add30_in_gag(T30, T34, 0))
U21_GGA(T30, T12, mul15_out_gga(T30, T12, s(T34))) → ADD30_IN_GAG(T30, T34, 0)
MUL1_IN_GGA(T30, s(T12), s(T37)) → U23_GGA(T30, T12, T37, mul15_in_gga(T30, T12, s(T34)))
MUL1_IN_GGA(T30, s(T12), s(T37)) → MUL15_IN_GGA(T30, T12, s(T34))
U23_GGA(T30, T12, T37, mul15_out_gga(T30, T12, s(T34))) → U24_GGA(T30, T12, T37, add30_in_gaa(T30, T34, T37))
U23_GGA(T30, T12, T37, mul15_out_gga(T30, T12, s(T34))) → ADD30_IN_GAA(T30, T34, T37)

The TRS R consists of the following rules:

mul1_in_gga(T4, 0, 0) → mul1_out_gga(T4, 0, 0)
mul1_in_gga(T8, s(T12), T11) → U15_gga(T8, T12, T11, mul15_in_gga(T8, T12, X15))
mul15_in_gga(T14, 0, 0) → mul15_out_gga(T14, 0, 0)
mul15_in_gga(T15, s(T17), X34) → U1_gga(T15, T17, X34, mul15_in_gga(T15, T17, X33))
mul15_in_gga(T20, s(T17), T20) → U2_gga(T20, T17, mul15_in_ggg(T20, T17, 0))
mul15_in_ggg(T14, 0, 0) → mul15_out_ggg(T14, 0, 0)
mul15_in_ggg(T15, s(T17), X34) → U1_ggg(T15, T17, X34, mul15_in_gga(T15, T17, X33))
mul15_in_gga(T21, s(T17), X53) → U3_gga(T21, T17, X53, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), 0) → U5_gga(T21, T17, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), s(0)) → U7_gga(T21, T17, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), s(T25)) → U9_gga(T21, T17, T25, mul15_in_gga(T21, T17, s(T23)))
U9_gga(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → U10_gga(T21, T17, T25, add30_in_gaa(T21, T23, T25))
add30_in_gaa(T20, 0, T20) → add30_out_gaa(T20, 0, T20)
add30_in_gaa(T21, s(T23), X53) → U11_gaa(T21, T23, X53, add30_in_gaa(T21, T23, X52))
add30_in_gaa(T21, s(T23), 0) → U12_gaa(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T20, 0, T20) → add30_out_gag(T20, 0, T20)
add30_in_gag(T21, s(T23), X53) → U11_gag(T21, T23, X53, add30_in_gaa(T21, T23, X52))
add30_in_gaa(T21, s(T23), s(0)) → U13_gaa(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), 0) → U12_gag(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), s(0)) → U13_gag(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), s(T25)) → U14_gag(T21, T23, T25, add30_in_gag(T21, T23, T25))
U14_gag(T21, T23, T25, add30_out_gag(T21, T23, T25)) → add30_out_gag(T21, s(T23), s(T25))
U13_gag(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gag(T21, s(T23), s(0))
U12_gag(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gag(T21, s(T23), 0)
U13_gaa(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gaa(T21, s(T23), s(0))
add30_in_gaa(T21, s(T23), s(T25)) → U14_gaa(T21, T23, T25, add30_in_gaa(T21, T23, T25))
U14_gaa(T21, T23, T25, add30_out_gaa(T21, T23, T25)) → add30_out_gaa(T21, s(T23), s(T25))
U11_gag(T21, T23, X53, add30_out_gaa(T21, T23, X52)) → add30_out_gag(T21, s(T23), X53)
U12_gaa(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gaa(T21, s(T23), 0)
U11_gaa(T21, T23, X53, add30_out_gaa(T21, T23, X52)) → add30_out_gaa(T21, s(T23), X53)
U10_gga(T21, T17, T25, add30_out_gaa(T21, T23, T25)) → mul15_out_gga(T21, s(T17), s(T25))
U7_gga(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U8_gga(T21, T17, add30_in_gag(T21, T23, 0))
U8_gga(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_gga(T21, s(T17), s(0))
U5_gga(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U6_gga(T21, T17, add30_in_gag(T21, T23, 0))
U6_gga(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_gga(T21, s(T17), 0)
U3_gga(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → U4_gga(T21, T17, X53, add30_in_gaa(T21, T23, X52))
U4_gga(T21, T17, X53, add30_out_gaa(T21, T23, X52)) → mul15_out_gga(T21, s(T17), X53)
U1_ggg(T15, T17, X34, mul15_out_gga(T15, T17, X33)) → mul15_out_ggg(T15, s(T17), X34)
mul15_in_ggg(T20, s(T17), T20) → U2_ggg(T20, T17, mul15_in_ggg(T20, T17, 0))
mul15_in_ggg(T21, s(T17), X53) → U3_ggg(T21, T17, X53, mul15_in_gga(T21, T17, s(T23)))
U3_ggg(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → U4_ggg(T21, T17, X53, add30_in_gaa(T21, T23, X52))
U4_ggg(T21, T17, X53, add30_out_gaa(T21, T23, X52)) → mul15_out_ggg(T21, s(T17), X53)
mul15_in_ggg(T21, s(T17), 0) → U5_ggg(T21, T17, mul15_in_gga(T21, T17, s(T23)))
U5_ggg(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U6_ggg(T21, T17, add30_in_gag(T21, T23, 0))
U6_ggg(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_ggg(T21, s(T17), 0)
mul15_in_ggg(T21, s(T17), s(0)) → U7_ggg(T21, T17, mul15_in_gga(T21, T17, s(T23)))
U7_ggg(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U8_ggg(T21, T17, add30_in_gag(T21, T23, 0))
U8_ggg(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_ggg(T21, s(T17), s(0))
mul15_in_ggg(T21, s(T17), s(T25)) → U9_ggg(T21, T17, T25, mul15_in_gga(T21, T17, s(T23)))
U9_ggg(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → U10_ggg(T21, T17, T25, add30_in_gag(T21, T23, T25))
U10_ggg(T21, T17, T25, add30_out_gag(T21, T23, T25)) → mul15_out_ggg(T21, s(T17), s(T25))
U2_ggg(T20, T17, mul15_out_ggg(T20, T17, 0)) → mul15_out_ggg(T20, s(T17), T20)
U2_gga(T20, T17, mul15_out_ggg(T20, T17, 0)) → mul15_out_gga(T20, s(T17), T20)
U1_gga(T15, T17, X34, mul15_out_gga(T15, T17, X33)) → mul15_out_gga(T15, s(T17), X34)
U15_gga(T8, T12, T11, mul15_out_gga(T8, T12, X15)) → mul1_out_gga(T8, s(T12), T11)
mul1_in_gga(T29, s(T12), T29) → U16_gga(T29, T12, mul15_in_ggg(T29, T12, 0))
U16_gga(T29, T12, mul15_out_ggg(T29, T12, 0)) → mul1_out_gga(T29, s(T12), T29)
mul1_in_gga(T30, s(T12), T33) → U17_gga(T30, T12, T33, mul15_in_gga(T30, T12, s(T34)))
U17_gga(T30, T12, T33, mul15_out_gga(T30, T12, s(T34))) → U18_gga(T30, T12, T33, add30_in_gaa(T30, T34, X75))
U18_gga(T30, T12, T33, add30_out_gaa(T30, T34, X75)) → mul1_out_gga(T30, s(T12), T33)
mul1_in_gga(T30, s(T12), 0) → U19_gga(T30, T12, mul15_in_gga(T30, T12, s(T34)))
U19_gga(T30, T12, mul15_out_gga(T30, T12, s(T34))) → U20_gga(T30, T12, add30_in_gag(T30, T34, 0))
U20_gga(T30, T12, add30_out_gag(T30, T34, 0)) → mul1_out_gga(T30, s(T12), 0)
mul1_in_gga(T30, s(T12), s(0)) → U21_gga(T30, T12, mul15_in_gga(T30, T12, s(T34)))
U21_gga(T30, T12, mul15_out_gga(T30, T12, s(T34))) → U22_gga(T30, T12, add30_in_gag(T30, T34, 0))
U22_gga(T30, T12, add30_out_gag(T30, T34, 0)) → mul1_out_gga(T30, s(T12), s(0))
mul1_in_gga(T30, s(T12), s(T37)) → U23_gga(T30, T12, T37, mul15_in_gga(T30, T12, s(T34)))
U23_gga(T30, T12, T37, mul15_out_gga(T30, T12, s(T34))) → U24_gga(T30, T12, T37, add30_in_gaa(T30, T34, T37))
U24_gga(T30, T12, T37, add30_out_gaa(T30, T34, T37)) → mul1_out_gga(T30, s(T12), s(T37))

The argument filtering Pi contains the following mapping:
mul1_in_gga(x1, x2, x3) = mul1_in_gga(x1, x2)
0 = 0
mul1_out_gga(x1, x2, x3) = mul1_out_gga
s(x1) = s(x1)
U15_gga(x1, x2, x3, x4) = U15_gga(x4)
mul15_in_gga(x1, x2, x3) = mul15_in_gga(x1, x2)
mul15_out_gga(x1, x2, x3) = mul15_out_gga
U1_gga(x1, x2, x3, x4) = U1_gga(x4)
U2_gga(x1, x2, x3) = U2_gga(x3)
mul15_in_ggg(x1, x2, x3) = mul15_in_ggg(x1, x2, x3)
mul15_out_ggg(x1, x2, x3) = mul15_out_ggg
U1_ggg(x1, x2, x3, x4) = U1_ggg(x4)
U3_gga(x1, x2, x3, x4) = U3_gga(x1, x4)
U5_gga(x1, x2, x3) = U5_gga(x1, x3)
U7_gga(x1, x2, x3) = U7_gga(x1, x3)
U9_gga(x1, x2, x3, x4) = U9_gga(x1, x4)
U10_gga(x1, x2, x3, x4) = U10_gga(x4)
add30_in_gaa(x1, x2, x3) = add30_in_gaa(x1)
add30_out_gaa(x1, x2, x3) = add30_out_gaa(x2)
U11_gaa(x1, x2, x3, x4) = U11_gaa(x4)
U12_gaa(x1, x2, x3) = U12_gaa(x3)
add30_in_gag(x1, x2, x3) = add30_in_gag(x1, x3)
add30_out_gag(x1, x2, x3) = add30_out_gag(x2)
U11_gag(x1, x2, x3, x4) = U11_gag(x4)
U13_gaa(x1, x2, x3) = U13_gaa(x3)
U12_gag(x1, x2, x3) = U12_gag(x3)
U13_gag(x1, x2, x3) = U13_gag(x3)
U14_gag(x1, x2, x3, x4) = U14_gag(x4)
U14_gaa(x1, x2, x3, x4) = U14_gaa(x4)
U8_gga(x1, x2, x3) = U8_gga(x3)
U6_gga(x1, x2, x3) = U6_gga(x3)
U4_gga(x1, x2, x3, x4) = U4_gga(x4)
U2_ggg(x1, x2, x3) = U2_ggg(x3)
U3_ggg(x1, x2, x3, x4) = U3_ggg(x1, x4)
U4_ggg(x1, x2, x3, x4) = U4_ggg(x4)
U5_ggg(x1, x2, x3) = U5_ggg(x1, x3)
U6_ggg(x1, x2, x3) = U6_ggg(x3)
U7_ggg(x1, x2, x3) = U7_ggg(x1, x3)
U8_ggg(x1, x2, x3) = U8_ggg(x3)
U9_ggg(x1, x2, x3, x4) = U9_ggg(x1, x3, x4)
U10_ggg(x1, x2, x3, x4) = U10_ggg(x4)
U16_gga(x1, x2, x3) = U16_gga(x3)
U17_gga(x1, x2, x3, x4) = U17_gga(x1, x4)
U18_gga(x1, x2, x3, x4) = U18_gga(x4)
U19_gga(x1, x2, x3) = U19_gga(x1, x3)
U20_gga(x1, x2, x3) = U20_gga(x3)
U21_gga(x1, x2, x3) = U21_gga(x1, x3)
U22_gga(x1, x2, x3) = U22_gga(x3)
U23_gga(x1, x2, x3, x4) = U23_gga(x1, x4)
U24_gga(x1, x2, x3, x4) = U24_gga(x4)
MUL1_IN_GGA(x1, x2, x3) = MUL1_IN_GGA(x1, x2)
U15_GGA(x1, x2, x3, x4) = U15_GGA(x4)
MUL15_IN_GGA(x1, x2, x3) = MUL15_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4) = U1_GGA(x4)
U2_GGA(x1, x2, x3) = U2_GGA(x3)
MUL15_IN_GGG(x1, x2, x3) = MUL15_IN_GGG(x1, x2, x3)
U1_GGG(x1, x2, x3, x4) = U1_GGG(x4)
U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x4)
U5_GGA(x1, x2, x3) = U5_GGA(x1, x3)
U7_GGA(x1, x2, x3) = U7_GGA(x1, x3)
U9_GGA(x1, x2, x3, x4) = U9_GGA(x1, x4)
U10_GGA(x1, x2, x3, x4) = U10_GGA(x4)
ADD30_IN_GAA(x1, x2, x3) = ADD30_IN_GAA(x1)
U11_GAA(x1, x2, x3, x4) = U11_GAA(x4)
U12_GAA(x1, x2, x3) = U12_GAA(x3)
ADD30_IN_GAG(x1, x2, x3) = ADD30_IN_GAG(x1, x3)
U11_GAG(x1, x2, x3, x4) = U11_GAG(x4)
U13_GAA(x1, x2, x3) = U13_GAA(x3)
U12_GAG(x1, x2, x3) = U12_GAG(x3)
U13_GAG(x1, x2, x3) = U13_GAG(x3)
U14_GAG(x1, x2, x3, x4) = U14_GAG(x4)
U14_GAA(x1, x2, x3, x4) = U14_GAA(x4)
U8_GGA(x1, x2, x3) = U8_GGA(x3)
U6_GGA(x1, x2, x3) = U6_GGA(x3)
U4_GGA(x1, x2, x3, x4) = U4_GGA(x4)
U2_GGG(x1, x2, x3) = U2_GGG(x3)
U3_GGG(x1, x2, x3, x4) = U3_GGG(x1, x4)
U4_GGG(x1, x2, x3, x4) = U4_GGG(x4)
U5_GGG(x1, x2, x3) = U5_GGG(x1, x3)
U6_GGG(x1, x2, x3) = U6_GGG(x3)
U7_GGG(x1, x2, x3) = U7_GGG(x1, x3)
U8_GGG(x1, x2, x3) = U8_GGG(x3)
U9_GGG(x1, x2, x3, x4) = U9_GGG(x1, x3, x4)
U10_GGG(x1, x2, x3, x4) = U10_GGG(x4)
U16_GGA(x1, x2, x3) = U16_GGA(x3)
U17_GGA(x1, x2, x3, x4) = U17_GGA(x1, x4)
U18_GGA(x1, x2, x3, x4) = U18_GGA(x4)
U19_GGA(x1, x2, x3) = U19_GGA(x1, x3)
U20_GGA(x1, x2, x3) = U20_GGA(x3)
U21_GGA(x1, x2, x3) = U21_GGA(x1, x3)
U22_GGA(x1, x2, x3) = U22_GGA(x3)
U23_GGA(x1, x2, x3, x4) = U23_GGA(x1, x4)
U24_GGA(x1, x2, x3, x4) = U24_GGA(x4)

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

(6) Obligation:

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

MUL1_IN_GGA(T8, s(T12), T11) → U15_GGA(T8, T12, T11, mul15_in_gga(T8, T12, X15))
MUL1_IN_GGA(T8, s(T12), T11) → MUL15_IN_GGA(T8, T12, X15)
MUL15_IN_GGA(T15, s(T17), X34) → U1_GGA(T15, T17, X34, mul15_in_gga(T15, T17, X33))
MUL15_IN_GGA(T15, s(T17), X34) → MUL15_IN_GGA(T15, T17, X33)
MUL15_IN_GGA(T20, s(T17), T20) → U2_GGA(T20, T17, mul15_in_ggg(T20, T17, 0))
MUL15_IN_GGA(T20, s(T17), T20) → MUL15_IN_GGG(T20, T17, 0)
MUL15_IN_GGG(T15, s(T17), X34) → U1_GGG(T15, T17, X34, mul15_in_gga(T15, T17, X33))
MUL15_IN_GGG(T15, s(T17), X34) → MUL15_IN_GGA(T15, T17, X33)
MUL15_IN_GGA(T21, s(T17), X53) → U3_GGA(T21, T17, X53, mul15_in_gga(T21, T17, s(T23)))
MUL15_IN_GGA(T21, s(T17), X53) → MUL15_IN_GGA(T21, T17, s(T23))
MUL15_IN_GGA(T21, s(T17), 0) → U5_GGA(T21, T17, mul15_in_gga(T21, T17, s(T23)))
MUL15_IN_GGA(T21, s(T17), 0) → MUL15_IN_GGA(T21, T17, s(T23))
MUL15_IN_GGA(T21, s(T17), s(0)) → U7_GGA(T21, T17, mul15_in_gga(T21, T17, s(T23)))
MUL15_IN_GGA(T21, s(T17), s(0)) → MUL15_IN_GGA(T21, T17, s(T23))
MUL15_IN_GGA(T21, s(T17), s(T25)) → U9_GGA(T21, T17, T25, mul15_in_gga(T21, T17, s(T23)))
MUL15_IN_GGA(T21, s(T17), s(T25)) → MUL15_IN_GGA(T21, T17, s(T23))
U9_GGA(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → U10_GGA(T21, T17, T25, add30_in_gaa(T21, T23, T25))
U9_GGA(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → ADD30_IN_GAA(T21, T23, T25)
ADD30_IN_GAA(T21, s(T23), X53) → U11_GAA(T21, T23, X53, add30_in_gaa(T21, T23, X52))
ADD30_IN_GAA(T21, s(T23), X53) → ADD30_IN_GAA(T21, T23, X52)
ADD30_IN_GAA(T21, s(T23), 0) → U12_GAA(T21, T23, add30_in_gag(T21, T23, 0))
ADD30_IN_GAA(T21, s(T23), 0) → ADD30_IN_GAG(T21, T23, 0)
ADD30_IN_GAG(T21, s(T23), X53) → U11_GAG(T21, T23, X53, add30_in_gaa(T21, T23, X52))
ADD30_IN_GAG(T21, s(T23), X53) → ADD30_IN_GAA(T21, T23, X52)
ADD30_IN_GAA(T21, s(T23), s(0)) → U13_GAA(T21, T23, add30_in_gag(T21, T23, 0))
ADD30_IN_GAA(T21, s(T23), s(0)) → ADD30_IN_GAG(T21, T23, 0)
ADD30_IN_GAG(T21, s(T23), 0) → U12_GAG(T21, T23, add30_in_gag(T21, T23, 0))
ADD30_IN_GAG(T21, s(T23), 0) → ADD30_IN_GAG(T21, T23, 0)
ADD30_IN_GAG(T21, s(T23), s(0)) → U13_GAG(T21, T23, add30_in_gag(T21, T23, 0))
ADD30_IN_GAG(T21, s(T23), s(0)) → ADD30_IN_GAG(T21, T23, 0)
ADD30_IN_GAG(T21, s(T23), s(T25)) → U14_GAG(T21, T23, T25, add30_in_gag(T21, T23, T25))
ADD30_IN_GAG(T21, s(T23), s(T25)) → ADD30_IN_GAG(T21, T23, T25)
ADD30_IN_GAA(T21, s(T23), s(T25)) → U14_GAA(T21, T23, T25, add30_in_gaa(T21, T23, T25))
ADD30_IN_GAA(T21, s(T23), s(T25)) → ADD30_IN_GAA(T21, T23, T25)
U7_GGA(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U8_GGA(T21, T17, add30_in_gag(T21, T23, 0))
U7_GGA(T21, T17, mul15_out_gga(T21, T17, s(T23))) → ADD30_IN_GAG(T21, T23, 0)
U5_GGA(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U6_GGA(T21, T17, add30_in_gag(T21, T23, 0))
U5_GGA(T21, T17, mul15_out_gga(T21, T17, s(T23))) → ADD30_IN_GAG(T21, T23, 0)
U3_GGA(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → U4_GGA(T21, T17, X53, add30_in_gaa(T21, T23, X52))
U3_GGA(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → ADD30_IN_GAA(T21, T23, X52)
MUL15_IN_GGG(T20, s(T17), T20) → U2_GGG(T20, T17, mul15_in_ggg(T20, T17, 0))
MUL15_IN_GGG(T20, s(T17), T20) → MUL15_IN_GGG(T20, T17, 0)
MUL15_IN_GGG(T21, s(T17), X53) → U3_GGG(T21, T17, X53, mul15_in_gga(T21, T17, s(T23)))
MUL15_IN_GGG(T21, s(T17), X53) → MUL15_IN_GGA(T21, T17, s(T23))
U3_GGG(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → U4_GGG(T21, T17, X53, add30_in_gaa(T21, T23, X52))
U3_GGG(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → ADD30_IN_GAA(T21, T23, X52)
MUL15_IN_GGG(T21, s(T17), 0) → U5_GGG(T21, T17, mul15_in_gga(T21, T17, s(T23)))
MUL15_IN_GGG(T21, s(T17), 0) → MUL15_IN_GGA(T21, T17, s(T23))
U5_GGG(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U6_GGG(T21, T17, add30_in_gag(T21, T23, 0))
U5_GGG(T21, T17, mul15_out_gga(T21, T17, s(T23))) → ADD30_IN_GAG(T21, T23, 0)
MUL15_IN_GGG(T21, s(T17), s(0)) → U7_GGG(T21, T17, mul15_in_gga(T21, T17, s(T23)))
MUL15_IN_GGG(T21, s(T17), s(0)) → MUL15_IN_GGA(T21, T17, s(T23))
U7_GGG(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U8_GGG(T21, T17, add30_in_gag(T21, T23, 0))
U7_GGG(T21, T17, mul15_out_gga(T21, T17, s(T23))) → ADD30_IN_GAG(T21, T23, 0)
MUL15_IN_GGG(T21, s(T17), s(T25)) → U9_GGG(T21, T17, T25, mul15_in_gga(T21, T17, s(T23)))
MUL15_IN_GGG(T21, s(T17), s(T25)) → MUL15_IN_GGA(T21, T17, s(T23))
U9_GGG(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → U10_GGG(T21, T17, T25, add30_in_gag(T21, T23, T25))
U9_GGG(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → ADD30_IN_GAG(T21, T23, T25)
MUL1_IN_GGA(T29, s(T12), T29) → U16_GGA(T29, T12, mul15_in_ggg(T29, T12, 0))
MUL1_IN_GGA(T29, s(T12), T29) → MUL15_IN_GGG(T29, T12, 0)
MUL1_IN_GGA(T30, s(T12), T33) → U17_GGA(T30, T12, T33, mul15_in_gga(T30, T12, s(T34)))
MUL1_IN_GGA(T30, s(T12), T33) → MUL15_IN_GGA(T30, T12, s(T34))
U17_GGA(T30, T12, T33, mul15_out_gga(T30, T12, s(T34))) → U18_GGA(T30, T12, T33, add30_in_gaa(T30, T34, X75))
U17_GGA(T30, T12, T33, mul15_out_gga(T30, T12, s(T34))) → ADD30_IN_GAA(T30, T34, X75)
MUL1_IN_GGA(T30, s(T12), 0) → U19_GGA(T30, T12, mul15_in_gga(T30, T12, s(T34)))
MUL1_IN_GGA(T30, s(T12), 0) → MUL15_IN_GGA(T30, T12, s(T34))
U19_GGA(T30, T12, mul15_out_gga(T30, T12, s(T34))) → U20_GGA(T30, T12, add30_in_gag(T30, T34, 0))
U19_GGA(T30, T12, mul15_out_gga(T30, T12, s(T34))) → ADD30_IN_GAG(T30, T34, 0)
MUL1_IN_GGA(T30, s(T12), s(0)) → U21_GGA(T30, T12, mul15_in_gga(T30, T12, s(T34)))
MUL1_IN_GGA(T30, s(T12), s(0)) → MUL15_IN_GGA(T30, T12, s(T34))
U21_GGA(T30, T12, mul15_out_gga(T30, T12, s(T34))) → U22_GGA(T30, T12, add30_in_gag(T30, T34, 0))
U21_GGA(T30, T12, mul15_out_gga(T30, T12, s(T34))) → ADD30_IN_GAG(T30, T34, 0)
MUL1_IN_GGA(T30, s(T12), s(T37)) → U23_GGA(T30, T12, T37, mul15_in_gga(T30, T12, s(T34)))
MUL1_IN_GGA(T30, s(T12), s(T37)) → MUL15_IN_GGA(T30, T12, s(T34))
U23_GGA(T30, T12, T37, mul15_out_gga(T30, T12, s(T34))) → U24_GGA(T30, T12, T37, add30_in_gaa(T30, T34, T37))
U23_GGA(T30, T12, T37, mul15_out_gga(T30, T12, s(T34))) → ADD30_IN_GAA(T30, T34, T37)

The TRS R consists of the following rules:

mul1_in_gga(T4, 0, 0) → mul1_out_gga(T4, 0, 0)
mul1_in_gga(T8, s(T12), T11) → U15_gga(T8, T12, T11, mul15_in_gga(T8, T12, X15))
mul15_in_gga(T14, 0, 0) → mul15_out_gga(T14, 0, 0)
mul15_in_gga(T15, s(T17), X34) → U1_gga(T15, T17, X34, mul15_in_gga(T15, T17, X33))
mul15_in_gga(T20, s(T17), T20) → U2_gga(T20, T17, mul15_in_ggg(T20, T17, 0))
mul15_in_ggg(T14, 0, 0) → mul15_out_ggg(T14, 0, 0)
mul15_in_ggg(T15, s(T17), X34) → U1_ggg(T15, T17, X34, mul15_in_gga(T15, T17, X33))
mul15_in_gga(T21, s(T17), X53) → U3_gga(T21, T17, X53, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), 0) → U5_gga(T21, T17, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), s(0)) → U7_gga(T21, T17, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), s(T25)) → U9_gga(T21, T17, T25, mul15_in_gga(T21, T17, s(T23)))
U9_gga(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → U10_gga(T21, T17, T25, add30_in_gaa(T21, T23, T25))
add30_in_gaa(T20, 0, T20) → add30_out_gaa(T20, 0, T20)
add30_in_gaa(T21, s(T23), X53) → U11_gaa(T21, T23, X53, add30_in_gaa(T21, T23, X52))
add30_in_gaa(T21, s(T23), 0) → U12_gaa(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T20, 0, T20) → add30_out_gag(T20, 0, T20)
add30_in_gag(T21, s(T23), X53) → U11_gag(T21, T23, X53, add30_in_gaa(T21, T23, X52))
add30_in_gaa(T21, s(T23), s(0)) → U13_gaa(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), 0) → U12_gag(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), s(0)) → U13_gag(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), s(T25)) → U14_gag(T21, T23, T25, add30_in_gag(T21, T23, T25))
U14_gag(T21, T23, T25, add30_out_gag(T21, T23, T25)) → add30_out_gag(T21, s(T23), s(T25))
U13_gag(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gag(T21, s(T23), s(0))
U12_gag(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gag(T21, s(T23), 0)
U13_gaa(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gaa(T21, s(T23), s(0))
add30_in_gaa(T21, s(T23), s(T25)) → U14_gaa(T21, T23, T25, add30_in_gaa(T21, T23, T25))
U14_gaa(T21, T23, T25, add30_out_gaa(T21, T23, T25)) → add30_out_gaa(T21, s(T23), s(T25))
U11_gag(T21, T23, X53, add30_out_gaa(T21, T23, X52)) → add30_out_gag(T21, s(T23), X53)
U12_gaa(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gaa(T21, s(T23), 0)
U11_gaa(T21, T23, X53, add30_out_gaa(T21, T23, X52)) → add30_out_gaa(T21, s(T23), X53)
U10_gga(T21, T17, T25, add30_out_gaa(T21, T23, T25)) → mul15_out_gga(T21, s(T17), s(T25))
U7_gga(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U8_gga(T21, T17, add30_in_gag(T21, T23, 0))
U8_gga(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_gga(T21, s(T17), s(0))
U5_gga(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U6_gga(T21, T17, add30_in_gag(T21, T23, 0))
U6_gga(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_gga(T21, s(T17), 0)
U3_gga(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → U4_gga(T21, T17, X53, add30_in_gaa(T21, T23, X52))
U4_gga(T21, T17, X53, add30_out_gaa(T21, T23, X52)) → mul15_out_gga(T21, s(T17), X53)
U1_ggg(T15, T17, X34, mul15_out_gga(T15, T17, X33)) → mul15_out_ggg(T15, s(T17), X34)
mul15_in_ggg(T20, s(T17), T20) → U2_ggg(T20, T17, mul15_in_ggg(T20, T17, 0))
mul15_in_ggg(T21, s(T17), X53) → U3_ggg(T21, T17, X53, mul15_in_gga(T21, T17, s(T23)))
U3_ggg(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → U4_ggg(T21, T17, X53, add30_in_gaa(T21, T23, X52))
U4_ggg(T21, T17, X53, add30_out_gaa(T21, T23, X52)) → mul15_out_ggg(T21, s(T17), X53)
mul15_in_ggg(T21, s(T17), 0) → U5_ggg(T21, T17, mul15_in_gga(T21, T17, s(T23)))
U5_ggg(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U6_ggg(T21, T17, add30_in_gag(T21, T23, 0))
U6_ggg(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_ggg(T21, s(T17), 0)
mul15_in_ggg(T21, s(T17), s(0)) → U7_ggg(T21, T17, mul15_in_gga(T21, T17, s(T23)))
U7_ggg(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U8_ggg(T21, T17, add30_in_gag(T21, T23, 0))
U8_ggg(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_ggg(T21, s(T17), s(0))
mul15_in_ggg(T21, s(T17), s(T25)) → U9_ggg(T21, T17, T25, mul15_in_gga(T21, T17, s(T23)))
U9_ggg(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → U10_ggg(T21, T17, T25, add30_in_gag(T21, T23, T25))
U10_ggg(T21, T17, T25, add30_out_gag(T21, T23, T25)) → mul15_out_ggg(T21, s(T17), s(T25))
U2_ggg(T20, T17, mul15_out_ggg(T20, T17, 0)) → mul15_out_ggg(T20, s(T17), T20)
U2_gga(T20, T17, mul15_out_ggg(T20, T17, 0)) → mul15_out_gga(T20, s(T17), T20)
U1_gga(T15, T17, X34, mul15_out_gga(T15, T17, X33)) → mul15_out_gga(T15, s(T17), X34)
U15_gga(T8, T12, T11, mul15_out_gga(T8, T12, X15)) → mul1_out_gga(T8, s(T12), T11)
mul1_in_gga(T29, s(T12), T29) → U16_gga(T29, T12, mul15_in_ggg(T29, T12, 0))
U16_gga(T29, T12, mul15_out_ggg(T29, T12, 0)) → mul1_out_gga(T29, s(T12), T29)
mul1_in_gga(T30, s(T12), T33) → U17_gga(T30, T12, T33, mul15_in_gga(T30, T12, s(T34)))
U17_gga(T30, T12, T33, mul15_out_gga(T30, T12, s(T34))) → U18_gga(T30, T12, T33, add30_in_gaa(T30, T34, X75))
U18_gga(T30, T12, T33, add30_out_gaa(T30, T34, X75)) → mul1_out_gga(T30, s(T12), T33)
mul1_in_gga(T30, s(T12), 0) → U19_gga(T30, T12, mul15_in_gga(T30, T12, s(T34)))
U19_gga(T30, T12, mul15_out_gga(T30, T12, s(T34))) → U20_gga(T30, T12, add30_in_gag(T30, T34, 0))
U20_gga(T30, T12, add30_out_gag(T30, T34, 0)) → mul1_out_gga(T30, s(T12), 0)
mul1_in_gga(T30, s(T12), s(0)) → U21_gga(T30, T12, mul15_in_gga(T30, T12, s(T34)))
U21_gga(T30, T12, mul15_out_gga(T30, T12, s(T34))) → U22_gga(T30, T12, add30_in_gag(T30, T34, 0))
U22_gga(T30, T12, add30_out_gag(T30, T34, 0)) → mul1_out_gga(T30, s(T12), s(0))
mul1_in_gga(T30, s(T12), s(T37)) → U23_gga(T30, T12, T37, mul15_in_gga(T30, T12, s(T34)))
U23_gga(T30, T12, T37, mul15_out_gga(T30, T12, s(T34))) → U24_gga(T30, T12, T37, add30_in_gaa(T30, T34, T37))
U24_gga(T30, T12, T37, add30_out_gaa(T30, T34, T37)) → mul1_out_gga(T30, s(T12), s(T37))

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 59 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

ADD30_IN_GAA(T21, s(T23), 0) → ADD30_IN_GAG(T21, T23, 0)
ADD30_IN_GAG(T21, s(T23), X53) → ADD30_IN_GAA(T21, T23, X52)
ADD30_IN_GAA(T21, s(T23), X53) → ADD30_IN_GAA(T21, T23, X52)
ADD30_IN_GAA(T21, s(T23), s(0)) → ADD30_IN_GAG(T21, T23, 0)
ADD30_IN_GAG(T21, s(T23), 0) → ADD30_IN_GAG(T21, T23, 0)
ADD30_IN_GAA(T21, s(T23), s(T25)) → ADD30_IN_GAA(T21, T23, T25)

The TRS R consists of the following rules:

mul1_in_gga(T4, 0, 0) → mul1_out_gga(T4, 0, 0)
mul1_in_gga(T8, s(T12), T11) → U15_gga(T8, T12, T11, mul15_in_gga(T8, T12, X15))
mul15_in_gga(T14, 0, 0) → mul15_out_gga(T14, 0, 0)
mul15_in_gga(T15, s(T17), X34) → U1_gga(T15, T17, X34, mul15_in_gga(T15, T17, X33))
mul15_in_gga(T20, s(T17), T20) → U2_gga(T20, T17, mul15_in_ggg(T20, T17, 0))
mul15_in_ggg(T14, 0, 0) → mul15_out_ggg(T14, 0, 0)
mul15_in_ggg(T15, s(T17), X34) → U1_ggg(T15, T17, X34, mul15_in_gga(T15, T17, X33))
mul15_in_gga(T21, s(T17), X53) → U3_gga(T21, T17, X53, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), 0) → U5_gga(T21, T17, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), s(0)) → U7_gga(T21, T17, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), s(T25)) → U9_gga(T21, T17, T25, mul15_in_gga(T21, T17, s(T23)))
U9_gga(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → U10_gga(T21, T17, T25, add30_in_gaa(T21, T23, T25))
add30_in_gaa(T20, 0, T20) → add30_out_gaa(T20, 0, T20)
add30_in_gaa(T21, s(T23), X53) → U11_gaa(T21, T23, X53, add30_in_gaa(T21, T23, X52))
add30_in_gaa(T21, s(T23), 0) → U12_gaa(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T20, 0, T20) → add30_out_gag(T20, 0, T20)
add30_in_gag(T21, s(T23), X53) → U11_gag(T21, T23, X53, add30_in_gaa(T21, T23, X52))
add30_in_gaa(T21, s(T23), s(0)) → U13_gaa(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), 0) → U12_gag(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), s(0)) → U13_gag(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), s(T25)) → U14_gag(T21, T23, T25, add30_in_gag(T21, T23, T25))
U14_gag(T21, T23, T25, add30_out_gag(T21, T23, T25)) → add30_out_gag(T21, s(T23), s(T25))
U13_gag(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gag(T21, s(T23), s(0))
U12_gag(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gag(T21, s(T23), 0)
U13_gaa(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gaa(T21, s(T23), s(0))
add30_in_gaa(T21, s(T23), s(T25)) → U14_gaa(T21, T23, T25, add30_in_gaa(T21, T23, T25))
U14_gaa(T21, T23, T25, add30_out_gaa(T21, T23, T25)) → add30_out_gaa(T21, s(T23), s(T25))
U11_gag(T21, T23, X53, add30_out_gaa(T21, T23, X52)) → add30_out_gag(T21, s(T23), X53)
U12_gaa(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gaa(T21, s(T23), 0)
U11_gaa(T21, T23, X53, add30_out_gaa(T21, T23, X52)) → add30_out_gaa(T21, s(T23), X53)
U10_gga(T21, T17, T25, add30_out_gaa(T21, T23, T25)) → mul15_out_gga(T21, s(T17), s(T25))
U7_gga(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U8_gga(T21, T17, add30_in_gag(T21, T23, 0))
U8_gga(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_gga(T21, s(T17), s(0))
U5_gga(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U6_gga(T21, T17, add30_in_gag(T21, T23, 0))
U6_gga(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_gga(T21, s(T17), 0)
U3_gga(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → U4_gga(T21, T17, X53, add30_in_gaa(T21, T23, X52))
U4_gga(T21, T17, X53, add30_out_gaa(T21, T23, X52)) → mul15_out_gga(T21, s(T17), X53)
U1_ggg(T15, T17, X34, mul15_out_gga(T15, T17, X33)) → mul15_out_ggg(T15, s(T17), X34)
mul15_in_ggg(T20, s(T17), T20) → U2_ggg(T20, T17, mul15_in_ggg(T20, T17, 0))
mul15_in_ggg(T21, s(T17), X53) → U3_ggg(T21, T17, X53, mul15_in_gga(T21, T17, s(T23)))
U3_ggg(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → U4_ggg(T21, T17, X53, add30_in_gaa(T21, T23, X52))
U4_ggg(T21, T17, X53, add30_out_gaa(T21, T23, X52)) → mul15_out_ggg(T21, s(T17), X53)
mul15_in_ggg(T21, s(T17), 0) → U5_ggg(T21, T17, mul15_in_gga(T21, T17, s(T23)))
U5_ggg(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U6_ggg(T21, T17, add30_in_gag(T21, T23, 0))
U6_ggg(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_ggg(T21, s(T17), 0)
mul15_in_ggg(T21, s(T17), s(0)) → U7_ggg(T21, T17, mul15_in_gga(T21, T17, s(T23)))
U7_ggg(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U8_ggg(T21, T17, add30_in_gag(T21, T23, 0))
U8_ggg(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_ggg(T21, s(T17), s(0))
mul15_in_ggg(T21, s(T17), s(T25)) → U9_ggg(T21, T17, T25, mul15_in_gga(T21, T17, s(T23)))
U9_ggg(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → U10_ggg(T21, T17, T25, add30_in_gag(T21, T23, T25))
U10_ggg(T21, T17, T25, add30_out_gag(T21, T23, T25)) → mul15_out_ggg(T21, s(T17), s(T25))
U2_ggg(T20, T17, mul15_out_ggg(T20, T17, 0)) → mul15_out_ggg(T20, s(T17), T20)
U2_gga(T20, T17, mul15_out_ggg(T20, T17, 0)) → mul15_out_gga(T20, s(T17), T20)
U1_gga(T15, T17, X34, mul15_out_gga(T15, T17, X33)) → mul15_out_gga(T15, s(T17), X34)
U15_gga(T8, T12, T11, mul15_out_gga(T8, T12, X15)) → mul1_out_gga(T8, s(T12), T11)
mul1_in_gga(T29, s(T12), T29) → U16_gga(T29, T12, mul15_in_ggg(T29, T12, 0))
U16_gga(T29, T12, mul15_out_ggg(T29, T12, 0)) → mul1_out_gga(T29, s(T12), T29)
mul1_in_gga(T30, s(T12), T33) → U17_gga(T30, T12, T33, mul15_in_gga(T30, T12, s(T34)))
U17_gga(T30, T12, T33, mul15_out_gga(T30, T12, s(T34))) → U18_gga(T30, T12, T33, add30_in_gaa(T30, T34, X75))
U18_gga(T30, T12, T33, add30_out_gaa(T30, T34, X75)) → mul1_out_gga(T30, s(T12), T33)
mul1_in_gga(T30, s(T12), 0) → U19_gga(T30, T12, mul15_in_gga(T30, T12, s(T34)))
U19_gga(T30, T12, mul15_out_gga(T30, T12, s(T34))) → U20_gga(T30, T12, add30_in_gag(T30, T34, 0))
U20_gga(T30, T12, add30_out_gag(T30, T34, 0)) → mul1_out_gga(T30, s(T12), 0)
mul1_in_gga(T30, s(T12), s(0)) → U21_gga(T30, T12, mul15_in_gga(T30, T12, s(T34)))
U21_gga(T30, T12, mul15_out_gga(T30, T12, s(T34))) → U22_gga(T30, T12, add30_in_gag(T30, T34, 0))
U22_gga(T30, T12, add30_out_gag(T30, T34, 0)) → mul1_out_gga(T30, s(T12), s(0))
mul1_in_gga(T30, s(T12), s(T37)) → U23_gga(T30, T12, T37, mul15_in_gga(T30, T12, s(T34)))
U23_gga(T30, T12, T37, mul15_out_gga(T30, T12, s(T34))) → U24_gga(T30, T12, T37, add30_in_gaa(T30, T34, T37))
U24_gga(T30, T12, T37, add30_out_gaa(T30, T34, T37)) → mul1_out_gga(T30, s(T12), s(T37))

(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:

ADD30_IN_GAA(T21, s(T23), 0) → ADD30_IN_GAG(T21, T23, 0)
ADD30_IN_GAG(T21, s(T23), X53) → ADD30_IN_GAA(T21, T23, X52)
ADD30_IN_GAA(T21, s(T23), X53) → ADD30_IN_GAA(T21, T23, X52)
ADD30_IN_GAA(T21, s(T23), s(0)) → ADD30_IN_GAG(T21, T23, 0)
ADD30_IN_GAG(T21, s(T23), 0) → ADD30_IN_GAG(T21, T23, 0)
ADD30_IN_GAA(T21, s(T23), s(T25)) → ADD30_IN_GAA(T21, T23, T25)

R is empty.
The argument filtering Pi contains the following mapping:
0 = 0
s(x1) = s(x1)
ADD30_IN_GAA(x1, x2, x3) = ADD30_IN_GAA(x1)
ADD30_IN_GAG(x1, x2, x3) = ADD30_IN_GAG(x1, x3)

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:

ADD30_IN_GAA(T21) → ADD30_IN_GAG(T21, 0)
ADD30_IN_GAG(T21, X53) → ADD30_IN_GAA(T21)
ADD30_IN_GAA(T21) → ADD30_IN_GAA(T21)
ADD30_IN_GAG(T21, 0) → ADD30_IN_GAG(T21, 0)

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

(14) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule ADD30_IN_GAG(T21, X53) → ADD30_IN_GAA(T21) we obtained the following new rules [LPAR04]:

ADD30_IN_GAG(z0, 0) → ADD30_IN_GAA(z0)

(15) Obligation:

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

ADD30_IN_GAA(T21) → ADD30_IN_GAG(T21, 0)
ADD30_IN_GAA(T21) → ADD30_IN_GAA(T21)
ADD30_IN_GAG(T21, 0) → ADD30_IN_GAG(T21, 0)
ADD30_IN_GAG(z0, 0) → ADD30_IN_GAA(z0)

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

(16) 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 = ADD30_IN_GAA(T21) evaluates to t =ADD30_IN_GAA(T21)

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 ADD30_IN_GAA(T21) to ADD30_IN_GAA(T21).

(17) FALSE

(18) Obligation:

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

ADD30_IN_GAG(T21, s(T23), s(T25)) → ADD30_IN_GAG(T21, T23, T25)

The TRS R consists of the following rules:

mul1_in_gga(T4, 0, 0) → mul1_out_gga(T4, 0, 0)
mul1_in_gga(T8, s(T12), T11) → U15_gga(T8, T12, T11, mul15_in_gga(T8, T12, X15))
mul15_in_gga(T14, 0, 0) → mul15_out_gga(T14, 0, 0)
mul15_in_gga(T15, s(T17), X34) → U1_gga(T15, T17, X34, mul15_in_gga(T15, T17, X33))
mul15_in_gga(T20, s(T17), T20) → U2_gga(T20, T17, mul15_in_ggg(T20, T17, 0))
mul15_in_ggg(T14, 0, 0) → mul15_out_ggg(T14, 0, 0)
mul15_in_ggg(T15, s(T17), X34) → U1_ggg(T15, T17, X34, mul15_in_gga(T15, T17, X33))
mul15_in_gga(T21, s(T17), X53) → U3_gga(T21, T17, X53, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), 0) → U5_gga(T21, T17, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), s(0)) → U7_gga(T21, T17, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), s(T25)) → U9_gga(T21, T17, T25, mul15_in_gga(T21, T17, s(T23)))
U9_gga(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → U10_gga(T21, T17, T25, add30_in_gaa(T21, T23, T25))
add30_in_gaa(T20, 0, T20) → add30_out_gaa(T20, 0, T20)
add30_in_gaa(T21, s(T23), X53) → U11_gaa(T21, T23, X53, add30_in_gaa(T21, T23, X52))
add30_in_gaa(T21, s(T23), 0) → U12_gaa(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T20, 0, T20) → add30_out_gag(T20, 0, T20)
add30_in_gag(T21, s(T23), X53) → U11_gag(T21, T23, X53, add30_in_gaa(T21, T23, X52))
add30_in_gaa(T21, s(T23), s(0)) → U13_gaa(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), 0) → U12_gag(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), s(0)) → U13_gag(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), s(T25)) → U14_gag(T21, T23, T25, add30_in_gag(T21, T23, T25))
U14_gag(T21, T23, T25, add30_out_gag(T21, T23, T25)) → add30_out_gag(T21, s(T23), s(T25))
U13_gag(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gag(T21, s(T23), s(0))
U12_gag(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gag(T21, s(T23), 0)
U13_gaa(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gaa(T21, s(T23), s(0))
add30_in_gaa(T21, s(T23), s(T25)) → U14_gaa(T21, T23, T25, add30_in_gaa(T21, T23, T25))
U14_gaa(T21, T23, T25, add30_out_gaa(T21, T23, T25)) → add30_out_gaa(T21, s(T23), s(T25))
U11_gag(T21, T23, X53, add30_out_gaa(T21, T23, X52)) → add30_out_gag(T21, s(T23), X53)
U12_gaa(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gaa(T21, s(T23), 0)
U11_gaa(T21, T23, X53, add30_out_gaa(T21, T23, X52)) → add30_out_gaa(T21, s(T23), X53)
U10_gga(T21, T17, T25, add30_out_gaa(T21, T23, T25)) → mul15_out_gga(T21, s(T17), s(T25))
U7_gga(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U8_gga(T21, T17, add30_in_gag(T21, T23, 0))
U8_gga(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_gga(T21, s(T17), s(0))
U5_gga(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U6_gga(T21, T17, add30_in_gag(T21, T23, 0))
U6_gga(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_gga(T21, s(T17), 0)
U3_gga(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → U4_gga(T21, T17, X53, add30_in_gaa(T21, T23, X52))
U4_gga(T21, T17, X53, add30_out_gaa(T21, T23, X52)) → mul15_out_gga(T21, s(T17), X53)
U1_ggg(T15, T17, X34, mul15_out_gga(T15, T17, X33)) → mul15_out_ggg(T15, s(T17), X34)
mul15_in_ggg(T20, s(T17), T20) → U2_ggg(T20, T17, mul15_in_ggg(T20, T17, 0))
mul15_in_ggg(T21, s(T17), X53) → U3_ggg(T21, T17, X53, mul15_in_gga(T21, T17, s(T23)))
U3_ggg(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → U4_ggg(T21, T17, X53, add30_in_gaa(T21, T23, X52))
U4_ggg(T21, T17, X53, add30_out_gaa(T21, T23, X52)) → mul15_out_ggg(T21, s(T17), X53)
mul15_in_ggg(T21, s(T17), 0) → U5_ggg(T21, T17, mul15_in_gga(T21, T17, s(T23)))
U5_ggg(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U6_ggg(T21, T17, add30_in_gag(T21, T23, 0))
U6_ggg(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_ggg(T21, s(T17), 0)
mul15_in_ggg(T21, s(T17), s(0)) → U7_ggg(T21, T17, mul15_in_gga(T21, T17, s(T23)))
U7_ggg(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U8_ggg(T21, T17, add30_in_gag(T21, T23, 0))
U8_ggg(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_ggg(T21, s(T17), s(0))
mul15_in_ggg(T21, s(T17), s(T25)) → U9_ggg(T21, T17, T25, mul15_in_gga(T21, T17, s(T23)))
U9_ggg(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → U10_ggg(T21, T17, T25, add30_in_gag(T21, T23, T25))
U10_ggg(T21, T17, T25, add30_out_gag(T21, T23, T25)) → mul15_out_ggg(T21, s(T17), s(T25))
U2_ggg(T20, T17, mul15_out_ggg(T20, T17, 0)) → mul15_out_ggg(T20, s(T17), T20)
U2_gga(T20, T17, mul15_out_ggg(T20, T17, 0)) → mul15_out_gga(T20, s(T17), T20)
U1_gga(T15, T17, X34, mul15_out_gga(T15, T17, X33)) → mul15_out_gga(T15, s(T17), X34)
U15_gga(T8, T12, T11, mul15_out_gga(T8, T12, X15)) → mul1_out_gga(T8, s(T12), T11)
mul1_in_gga(T29, s(T12), T29) → U16_gga(T29, T12, mul15_in_ggg(T29, T12, 0))
U16_gga(T29, T12, mul15_out_ggg(T29, T12, 0)) → mul1_out_gga(T29, s(T12), T29)
mul1_in_gga(T30, s(T12), T33) → U17_gga(T30, T12, T33, mul15_in_gga(T30, T12, s(T34)))
U17_gga(T30, T12, T33, mul15_out_gga(T30, T12, s(T34))) → U18_gga(T30, T12, T33, add30_in_gaa(T30, T34, X75))
U18_gga(T30, T12, T33, add30_out_gaa(T30, T34, X75)) → mul1_out_gga(T30, s(T12), T33)
mul1_in_gga(T30, s(T12), 0) → U19_gga(T30, T12, mul15_in_gga(T30, T12, s(T34)))
U19_gga(T30, T12, mul15_out_gga(T30, T12, s(T34))) → U20_gga(T30, T12, add30_in_gag(T30, T34, 0))
U20_gga(T30, T12, add30_out_gag(T30, T34, 0)) → mul1_out_gga(T30, s(T12), 0)
mul1_in_gga(T30, s(T12), s(0)) → U21_gga(T30, T12, mul15_in_gga(T30, T12, s(T34)))
U21_gga(T30, T12, mul15_out_gga(T30, T12, s(T34))) → U22_gga(T30, T12, add30_in_gag(T30, T34, 0))
U22_gga(T30, T12, add30_out_gag(T30, T34, 0)) → mul1_out_gga(T30, s(T12), s(0))
mul1_in_gga(T30, s(T12), s(T37)) → U23_gga(T30, T12, T37, mul15_in_gga(T30, T12, s(T34)))
U23_gga(T30, T12, T37, mul15_out_gga(T30, T12, s(T34))) → U24_gga(T30, T12, T37, add30_in_gaa(T30, T34, T37))
U24_gga(T30, T12, T37, add30_out_gaa(T30, T34, T37)) → mul1_out_gga(T30, s(T12), s(T37))

(19) UsableRulesProof (EQUIVALENT transformation)

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

(20) Obligation:

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

ADD30_IN_GAG(T21, s(T23), s(T25)) → ADD30_IN_GAG(T21, T23, T25)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
ADD30_IN_GAG(x1, x2, x3) = ADD30_IN_GAG(x1, x3)

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

(21) PiDPToQDPProof (SOUND transformation)

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

(22) Obligation:

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

ADD30_IN_GAG(T21, s(T25)) → ADD30_IN_GAG(T21, T25)

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

(23) 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_GAG(T21, s(T25)) → ADD30_IN_GAG(T21, T25)
The graph contains the following edges 1 >= 1, 2 > 2

(24) TRUE

(25) Obligation:

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

MUL15_IN_GGA(T20, s(T17), T20) → MUL15_IN_GGG(T20, T17, 0)
MUL15_IN_GGG(T15, s(T17), X34) → MUL15_IN_GGA(T15, T17, X33)
MUL15_IN_GGA(T15, s(T17), X34) → MUL15_IN_GGA(T15, T17, X33)
MUL15_IN_GGA(T21, s(T17), X53) → MUL15_IN_GGA(T21, T17, s(T23))
MUL15_IN_GGA(T21, s(T17), s(0)) → MUL15_IN_GGA(T21, T17, s(T23))
MUL15_IN_GGA(T21, s(T17), s(T25)) → MUL15_IN_GGA(T21, T17, s(T23))
MUL15_IN_GGA(T21, s(T17), 0) → MUL15_IN_GGA(T21, T17, s(T23))
MUL15_IN_GGG(T20, s(T17), T20) → MUL15_IN_GGG(T20, T17, 0)
MUL15_IN_GGG(T21, s(T17), X53) → MUL15_IN_GGA(T21, T17, s(T23))
MUL15_IN_GGG(T21, s(T17), 0) → MUL15_IN_GGA(T21, T17, s(T23))

The TRS R consists of the following rules:

mul1_in_gga(T4, 0, 0) → mul1_out_gga(T4, 0, 0)
mul1_in_gga(T8, s(T12), T11) → U15_gga(T8, T12, T11, mul15_in_gga(T8, T12, X15))
mul15_in_gga(T14, 0, 0) → mul15_out_gga(T14, 0, 0)
mul15_in_gga(T15, s(T17), X34) → U1_gga(T15, T17, X34, mul15_in_gga(T15, T17, X33))
mul15_in_gga(T20, s(T17), T20) → U2_gga(T20, T17, mul15_in_ggg(T20, T17, 0))
mul15_in_ggg(T14, 0, 0) → mul15_out_ggg(T14, 0, 0)
mul15_in_ggg(T15, s(T17), X34) → U1_ggg(T15, T17, X34, mul15_in_gga(T15, T17, X33))
mul15_in_gga(T21, s(T17), X53) → U3_gga(T21, T17, X53, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), 0) → U5_gga(T21, T17, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), s(0)) → U7_gga(T21, T17, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), s(T25)) → U9_gga(T21, T17, T25, mul15_in_gga(T21, T17, s(T23)))
U9_gga(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → U10_gga(T21, T17, T25, add30_in_gaa(T21, T23, T25))
add30_in_gaa(T20, 0, T20) → add30_out_gaa(T20, 0, T20)
add30_in_gaa(T21, s(T23), X53) → U11_gaa(T21, T23, X53, add30_in_gaa(T21, T23, X52))
add30_in_gaa(T21, s(T23), 0) → U12_gaa(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T20, 0, T20) → add30_out_gag(T20, 0, T20)
add30_in_gag(T21, s(T23), X53) → U11_gag(T21, T23, X53, add30_in_gaa(T21, T23, X52))
add30_in_gaa(T21, s(T23), s(0)) → U13_gaa(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), 0) → U12_gag(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), s(0)) → U13_gag(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), s(T25)) → U14_gag(T21, T23, T25, add30_in_gag(T21, T23, T25))
U14_gag(T21, T23, T25, add30_out_gag(T21, T23, T25)) → add30_out_gag(T21, s(T23), s(T25))
U13_gag(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gag(T21, s(T23), s(0))
U12_gag(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gag(T21, s(T23), 0)
U13_gaa(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gaa(T21, s(T23), s(0))
add30_in_gaa(T21, s(T23), s(T25)) → U14_gaa(T21, T23, T25, add30_in_gaa(T21, T23, T25))
U14_gaa(T21, T23, T25, add30_out_gaa(T21, T23, T25)) → add30_out_gaa(T21, s(T23), s(T25))
U11_gag(T21, T23, X53, add30_out_gaa(T21, T23, X52)) → add30_out_gag(T21, s(T23), X53)
U12_gaa(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gaa(T21, s(T23), 0)
U11_gaa(T21, T23, X53, add30_out_gaa(T21, T23, X52)) → add30_out_gaa(T21, s(T23), X53)
U10_gga(T21, T17, T25, add30_out_gaa(T21, T23, T25)) → mul15_out_gga(T21, s(T17), s(T25))
U7_gga(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U8_gga(T21, T17, add30_in_gag(T21, T23, 0))
U8_gga(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_gga(T21, s(T17), s(0))
U5_gga(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U6_gga(T21, T17, add30_in_gag(T21, T23, 0))
U6_gga(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_gga(T21, s(T17), 0)
U3_gga(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → U4_gga(T21, T17, X53, add30_in_gaa(T21, T23, X52))
U4_gga(T21, T17, X53, add30_out_gaa(T21, T23, X52)) → mul15_out_gga(T21, s(T17), X53)
U1_ggg(T15, T17, X34, mul15_out_gga(T15, T17, X33)) → mul15_out_ggg(T15, s(T17), X34)
mul15_in_ggg(T20, s(T17), T20) → U2_ggg(T20, T17, mul15_in_ggg(T20, T17, 0))
mul15_in_ggg(T21, s(T17), X53) → U3_ggg(T21, T17, X53, mul15_in_gga(T21, T17, s(T23)))
U3_ggg(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → U4_ggg(T21, T17, X53, add30_in_gaa(T21, T23, X52))
U4_ggg(T21, T17, X53, add30_out_gaa(T21, T23, X52)) → mul15_out_ggg(T21, s(T17), X53)
mul15_in_ggg(T21, s(T17), 0) → U5_ggg(T21, T17, mul15_in_gga(T21, T17, s(T23)))
U5_ggg(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U6_ggg(T21, T17, add30_in_gag(T21, T23, 0))
U6_ggg(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_ggg(T21, s(T17), 0)
mul15_in_ggg(T21, s(T17), s(0)) → U7_ggg(T21, T17, mul15_in_gga(T21, T17, s(T23)))
U7_ggg(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U8_ggg(T21, T17, add30_in_gag(T21, T23, 0))
U8_ggg(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_ggg(T21, s(T17), s(0))
mul15_in_ggg(T21, s(T17), s(T25)) → U9_ggg(T21, T17, T25, mul15_in_gga(T21, T17, s(T23)))
U9_ggg(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → U10_ggg(T21, T17, T25, add30_in_gag(T21, T23, T25))
U10_ggg(T21, T17, T25, add30_out_gag(T21, T23, T25)) → mul15_out_ggg(T21, s(T17), s(T25))
U2_ggg(T20, T17, mul15_out_ggg(T20, T17, 0)) → mul15_out_ggg(T20, s(T17), T20)
U2_gga(T20, T17, mul15_out_ggg(T20, T17, 0)) → mul15_out_gga(T20, s(T17), T20)
U1_gga(T15, T17, X34, mul15_out_gga(T15, T17, X33)) → mul15_out_gga(T15, s(T17), X34)
U15_gga(T8, T12, T11, mul15_out_gga(T8, T12, X15)) → mul1_out_gga(T8, s(T12), T11)
mul1_in_gga(T29, s(T12), T29) → U16_gga(T29, T12, mul15_in_ggg(T29, T12, 0))
U16_gga(T29, T12, mul15_out_ggg(T29, T12, 0)) → mul1_out_gga(T29, s(T12), T29)
mul1_in_gga(T30, s(T12), T33) → U17_gga(T30, T12, T33, mul15_in_gga(T30, T12, s(T34)))
U17_gga(T30, T12, T33, mul15_out_gga(T30, T12, s(T34))) → U18_gga(T30, T12, T33, add30_in_gaa(T30, T34, X75))
U18_gga(T30, T12, T33, add30_out_gaa(T30, T34, X75)) → mul1_out_gga(T30, s(T12), T33)
mul1_in_gga(T30, s(T12), 0) → U19_gga(T30, T12, mul15_in_gga(T30, T12, s(T34)))
U19_gga(T30, T12, mul15_out_gga(T30, T12, s(T34))) → U20_gga(T30, T12, add30_in_gag(T30, T34, 0))
U20_gga(T30, T12, add30_out_gag(T30, T34, 0)) → mul1_out_gga(T30, s(T12), 0)
mul1_in_gga(T30, s(T12), s(0)) → U21_gga(T30, T12, mul15_in_gga(T30, T12, s(T34)))
U21_gga(T30, T12, mul15_out_gga(T30, T12, s(T34))) → U22_gga(T30, T12, add30_in_gag(T30, T34, 0))
U22_gga(T30, T12, add30_out_gag(T30, T34, 0)) → mul1_out_gga(T30, s(T12), s(0))
mul1_in_gga(T30, s(T12), s(T37)) → U23_gga(T30, T12, T37, mul15_in_gga(T30, T12, s(T34)))
U23_gga(T30, T12, T37, mul15_out_gga(T30, T12, s(T34))) → U24_gga(T30, T12, T37, add30_in_gaa(T30, T34, T37))
U24_gga(T30, T12, T37, add30_out_gaa(T30, T34, T37)) → mul1_out_gga(T30, s(T12), s(T37))

(26) UsableRulesProof (EQUIVALENT transformation)

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

(27) Obligation:

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

MUL15_IN_GGA(T20, s(T17), T20) → MUL15_IN_GGG(T20, T17, 0)
MUL15_IN_GGG(T15, s(T17), X34) → MUL15_IN_GGA(T15, T17, X33)
MUL15_IN_GGA(T15, s(T17), X34) → MUL15_IN_GGA(T15, T17, X33)
MUL15_IN_GGA(T21, s(T17), X53) → MUL15_IN_GGA(T21, T17, s(T23))
MUL15_IN_GGA(T21, s(T17), s(0)) → MUL15_IN_GGA(T21, T17, s(T23))
MUL15_IN_GGA(T21, s(T17), s(T25)) → MUL15_IN_GGA(T21, T17, s(T23))
MUL15_IN_GGA(T21, s(T17), 0) → MUL15_IN_GGA(T21, T17, s(T23))
MUL15_IN_GGG(T20, s(T17), T20) → MUL15_IN_GGG(T20, T17, 0)
MUL15_IN_GGG(T21, s(T17), X53) → MUL15_IN_GGA(T21, T17, s(T23))
MUL15_IN_GGG(T21, s(T17), 0) → MUL15_IN_GGA(T21, T17, s(T23))

R is empty.
The argument filtering Pi contains the following mapping:
0 = 0
s(x1) = s(x1)
MUL15_IN_GGA(x1, x2, x3) = MUL15_IN_GGA(x1, x2)
MUL15_IN_GGG(x1, x2, x3) = MUL15_IN_GGG(x1, x2, x3)

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

(28) PiDPToQDPProof (SOUND transformation)

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

(29) Obligation:

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

MUL15_IN_GGA(T20, s(T17)) → MUL15_IN_GGG(T20, T17, 0)
MUL15_IN_GGG(T15, s(T17), X34) → MUL15_IN_GGA(T15, T17)
MUL15_IN_GGA(T15, s(T17)) → MUL15_IN_GGA(T15, T17)
MUL15_IN_GGG(T20, s(T17), T20) → MUL15_IN_GGG(T20, T17, 0)
MUL15_IN_GGG(T21, s(T17), 0) → MUL15_IN_GGA(T21, T17)

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

(30) 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_GGG(T20, s(T17), T20) → MUL15_IN_GGG(T20, T17, 0)
The graph contains the following edges 1 >= 1, 3 >= 1, 2 > 2
MUL15_IN_GGA(T15, s(T17)) → MUL15_IN_GGA(T15, T17)
The graph contains the following edges 1 >= 1, 2 > 2
MUL15_IN_GGA(T20, s(T17)) → MUL15_IN_GGG(T20, T17, 0)
The graph contains the following edges 1 >= 1, 2 > 2
MUL15_IN_GGG(T15, s(T17), X34) → MUL15_IN_GGA(T15, T17)
The graph contains the following edges 1 >= 1, 2 > 2
MUL15_IN_GGG(T21, s(T17), 0) → MUL15_IN_GGA(T21, T17)
The graph contains the following edges 1 >= 1, 2 > 2

(31) TRUE

(32) PrologToPiTRSProof (SOUND transformation)

mul1_in_gga(T4, 0, 0) → mul1_out_gga(T4, 0, 0)
mul1_in_gga(T8, s(T12), T11) → U15_gga(T8, T12, T11, mul15_in_gga(T8, T12, X15))
mul15_in_gga(T14, 0, 0) → mul15_out_gga(T14, 0, 0)
mul15_in_gga(T15, s(T17), X34) → U1_gga(T15, T17, X34, mul15_in_gga(T15, T17, X33))
mul15_in_gga(T20, s(T17), T20) → U2_gga(T20, T17, mul15_in_ggg(T20, T17, 0))
mul15_in_ggg(T14, 0, 0) → mul15_out_ggg(T14, 0, 0)
mul15_in_ggg(T15, s(T17), X34) → U1_ggg(T15, T17, X34, mul15_in_gga(T15, T17, X33))
mul15_in_gga(T21, s(T17), X53) → U3_gga(T21, T17, X53, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), 0) → U5_gga(T21, T17, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), s(0)) → U7_gga(T21, T17, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), s(T25)) → U9_gga(T21, T17, T25, mul15_in_gga(T21, T17, s(T23)))
U9_gga(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → U10_gga(T21, T17, T25, add30_in_gaa(T21, T23, T25))
add30_in_gaa(T20, 0, T20) → add30_out_gaa(T20, 0, T20)
add30_in_gaa(T21, s(T23), X53) → U11_gaa(T21, T23, X53, add30_in_gaa(T21, T23, X52))
add30_in_gaa(T21, s(T23), 0) → U12_gaa(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T20, 0, T20) → add30_out_gag(T20, 0, T20)
add30_in_gag(T21, s(T23), X53) → U11_gag(T21, T23, X53, add30_in_gaa(T21, T23, X52))
add30_in_gaa(T21, s(T23), s(0)) → U13_gaa(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), 0) → U12_gag(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), s(0)) → U13_gag(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), s(T25)) → U14_gag(T21, T23, T25, add30_in_gag(T21, T23, T25))
U14_gag(T21, T23, T25, add30_out_gag(T21, T23, T25)) → add30_out_gag(T21, s(T23), s(T25))
U13_gag(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gag(T21, s(T23), s(0))
U12_gag(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gag(T21, s(T23), 0)
U13_gaa(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gaa(T21, s(T23), s(0))
add30_in_gaa(T21, s(T23), s(T25)) → U14_gaa(T21, T23, T25, add30_in_gaa(T21, T23, T25))
U14_gaa(T21, T23, T25, add30_out_gaa(T21, T23, T25)) → add30_out_gaa(T21, s(T23), s(T25))
U11_gag(T21, T23, X53, add30_out_gaa(T21, T23, X52)) → add30_out_gag(T21, s(T23), X53)
U12_gaa(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gaa(T21, s(T23), 0)
U11_gaa(T21, T23, X53, add30_out_gaa(T21, T23, X52)) → add30_out_gaa(T21, s(T23), X53)
U10_gga(T21, T17, T25, add30_out_gaa(T21, T23, T25)) → mul15_out_gga(T21, s(T17), s(T25))
U7_gga(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U8_gga(T21, T17, add30_in_gag(T21, T23, 0))
U8_gga(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_gga(T21, s(T17), s(0))
U5_gga(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U6_gga(T21, T17, add30_in_gag(T21, T23, 0))
U6_gga(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_gga(T21, s(T17), 0)
U3_gga(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → U4_gga(T21, T17, X53, add30_in_gaa(T21, T23, X52))
U4_gga(T21, T17, X53, add30_out_gaa(T21, T23, X52)) → mul15_out_gga(T21, s(T17), X53)
U1_ggg(T15, T17, X34, mul15_out_gga(T15, T17, X33)) → mul15_out_ggg(T15, s(T17), X34)
mul15_in_ggg(T20, s(T17), T20) → U2_ggg(T20, T17, mul15_in_ggg(T20, T17, 0))
mul15_in_ggg(T21, s(T17), X53) → U3_ggg(T21, T17, X53, mul15_in_gga(T21, T17, s(T23)))
U3_ggg(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → U4_ggg(T21, T17, X53, add30_in_gaa(T21, T23, X52))
U4_ggg(T21, T17, X53, add30_out_gaa(T21, T23, X52)) → mul15_out_ggg(T21, s(T17), X53)
mul15_in_ggg(T21, s(T17), 0) → U5_ggg(T21, T17, mul15_in_gga(T21, T17, s(T23)))
U5_ggg(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U6_ggg(T21, T17, add30_in_gag(T21, T23, 0))
U6_ggg(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_ggg(T21, s(T17), 0)
mul15_in_ggg(T21, s(T17), s(0)) → U7_ggg(T21, T17, mul15_in_gga(T21, T17, s(T23)))
U7_ggg(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U8_ggg(T21, T17, add30_in_gag(T21, T23, 0))
U8_ggg(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_ggg(T21, s(T17), s(0))
mul15_in_ggg(T21, s(T17), s(T25)) → U9_ggg(T21, T17, T25, mul15_in_gga(T21, T17, s(T23)))
U9_ggg(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → U10_ggg(T21, T17, T25, add30_in_gag(T21, T23, T25))
U10_ggg(T21, T17, T25, add30_out_gag(T21, T23, T25)) → mul15_out_ggg(T21, s(T17), s(T25))
U2_ggg(T20, T17, mul15_out_ggg(T20, T17, 0)) → mul15_out_ggg(T20, s(T17), T20)
U2_gga(T20, T17, mul15_out_ggg(T20, T17, 0)) → mul15_out_gga(T20, s(T17), T20)
U1_gga(T15, T17, X34, mul15_out_gga(T15, T17, X33)) → mul15_out_gga(T15, s(T17), X34)
U15_gga(T8, T12, T11, mul15_out_gga(T8, T12, X15)) → mul1_out_gga(T8, s(T12), T11)
mul1_in_gga(T29, s(T12), T29) → U16_gga(T29, T12, mul15_in_ggg(T29, T12, 0))
U16_gga(T29, T12, mul15_out_ggg(T29, T12, 0)) → mul1_out_gga(T29, s(T12), T29)
mul1_in_gga(T30, s(T12), T33) → U17_gga(T30, T12, T33, mul15_in_gga(T30, T12, s(T34)))
U17_gga(T30, T12, T33, mul15_out_gga(T30, T12, s(T34))) → U18_gga(T30, T12, T33, add30_in_gaa(T30, T34, X75))
U18_gga(T30, T12, T33, add30_out_gaa(T30, T34, X75)) → mul1_out_gga(T30, s(T12), T33)
mul1_in_gga(T30, s(T12), 0) → U19_gga(T30, T12, mul15_in_gga(T30, T12, s(T34)))
U19_gga(T30, T12, mul15_out_gga(T30, T12, s(T34))) → U20_gga(T30, T12, add30_in_gag(T30, T34, 0))
U20_gga(T30, T12, add30_out_gag(T30, T34, 0)) → mul1_out_gga(T30, s(T12), 0)
mul1_in_gga(T30, s(T12), s(0)) → U21_gga(T30, T12, mul15_in_gga(T30, T12, s(T34)))
U21_gga(T30, T12, mul15_out_gga(T30, T12, s(T34))) → U22_gga(T30, T12, add30_in_gag(T30, T34, 0))
U22_gga(T30, T12, add30_out_gag(T30, T34, 0)) → mul1_out_gga(T30, s(T12), s(0))
mul1_in_gga(T30, s(T12), s(T37)) → U23_gga(T30, T12, T37, mul15_in_gga(T30, T12, s(T34)))
U23_gga(T30, T12, T37, mul15_out_gga(T30, T12, s(T34))) → U24_gga(T30, T12, T37, add30_in_gaa(T30, T34, T37))
U24_gga(T30, T12, T37, add30_out_gaa(T30, T34, T37)) → mul1_out_gga(T30, s(T12), s(T37))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(33) Obligation:

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

mul1_in_gga(T4, 0, 0) → mul1_out_gga(T4, 0, 0)
mul1_in_gga(T8, s(T12), T11) → U15_gga(T8, T12, T11, mul15_in_gga(T8, T12, X15))
mul15_in_gga(T14, 0, 0) → mul15_out_gga(T14, 0, 0)
mul15_in_gga(T15, s(T17), X34) → U1_gga(T15, T17, X34, mul15_in_gga(T15, T17, X33))
mul15_in_gga(T20, s(T17), T20) → U2_gga(T20, T17, mul15_in_ggg(T20, T17, 0))
mul15_in_ggg(T14, 0, 0) → mul15_out_ggg(T14, 0, 0)
mul15_in_ggg(T15, s(T17), X34) → U1_ggg(T15, T17, X34, mul15_in_gga(T15, T17, X33))
mul15_in_gga(T21, s(T17), X53) → U3_gga(T21, T17, X53, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), 0) → U5_gga(T21, T17, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), s(0)) → U7_gga(T21, T17, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), s(T25)) → U9_gga(T21, T17, T25, mul15_in_gga(T21, T17, s(T23)))
U9_gga(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → U10_gga(T21, T17, T25, add30_in_gaa(T21, T23, T25))
add30_in_gaa(T20, 0, T20) → add30_out_gaa(T20, 0, T20)
add30_in_gaa(T21, s(T23), X53) → U11_gaa(T21, T23, X53, add30_in_gaa(T21, T23, X52))
add30_in_gaa(T21, s(T23), 0) → U12_gaa(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T20, 0, T20) → add30_out_gag(T20, 0, T20)
add30_in_gag(T21, s(T23), X53) → U11_gag(T21, T23, X53, add30_in_gaa(T21, T23, X52))
add30_in_gaa(T21, s(T23), s(0)) → U13_gaa(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), 0) → U12_gag(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), s(0)) → U13_gag(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), s(T25)) → U14_gag(T21, T23, T25, add30_in_gag(T21, T23, T25))
U14_gag(T21, T23, T25, add30_out_gag(T21, T23, T25)) → add30_out_gag(T21, s(T23), s(T25))
U13_gag(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gag(T21, s(T23), s(0))
U12_gag(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gag(T21, s(T23), 0)
U13_gaa(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gaa(T21, s(T23), s(0))
add30_in_gaa(T21, s(T23), s(T25)) → U14_gaa(T21, T23, T25, add30_in_gaa(T21, T23, T25))
U14_gaa(T21, T23, T25, add30_out_gaa(T21, T23, T25)) → add30_out_gaa(T21, s(T23), s(T25))
U11_gag(T21, T23, X53, add30_out_gaa(T21, T23, X52)) → add30_out_gag(T21, s(T23), X53)
U12_gaa(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gaa(T21, s(T23), 0)
U11_gaa(T21, T23, X53, add30_out_gaa(T21, T23, X52)) → add30_out_gaa(T21, s(T23), X53)
U10_gga(T21, T17, T25, add30_out_gaa(T21, T23, T25)) → mul15_out_gga(T21, s(T17), s(T25))
U7_gga(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U8_gga(T21, T17, add30_in_gag(T21, T23, 0))
U8_gga(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_gga(T21, s(T17), s(0))
U5_gga(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U6_gga(T21, T17, add30_in_gag(T21, T23, 0))
U6_gga(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_gga(T21, s(T17), 0)
U3_gga(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → U4_gga(T21, T17, X53, add30_in_gaa(T21, T23, X52))
U4_gga(T21, T17, X53, add30_out_gaa(T21, T23, X52)) → mul15_out_gga(T21, s(T17), X53)
U1_ggg(T15, T17, X34, mul15_out_gga(T15, T17, X33)) → mul15_out_ggg(T15, s(T17), X34)
mul15_in_ggg(T20, s(T17), T20) → U2_ggg(T20, T17, mul15_in_ggg(T20, T17, 0))
mul15_in_ggg(T21, s(T17), X53) → U3_ggg(T21, T17, X53, mul15_in_gga(T21, T17, s(T23)))
U3_ggg(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → U4_ggg(T21, T17, X53, add30_in_gaa(T21, T23, X52))
U4_ggg(T21, T17, X53, add30_out_gaa(T21, T23, X52)) → mul15_out_ggg(T21, s(T17), X53)
mul15_in_ggg(T21, s(T17), 0) → U5_ggg(T21, T17, mul15_in_gga(T21, T17, s(T23)))
U5_ggg(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U6_ggg(T21, T17, add30_in_gag(T21, T23, 0))
U6_ggg(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_ggg(T21, s(T17), 0)
mul15_in_ggg(T21, s(T17), s(0)) → U7_ggg(T21, T17, mul15_in_gga(T21, T17, s(T23)))
U7_ggg(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U8_ggg(T21, T17, add30_in_gag(T21, T23, 0))
U8_ggg(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_ggg(T21, s(T17), s(0))
mul15_in_ggg(T21, s(T17), s(T25)) → U9_ggg(T21, T17, T25, mul15_in_gga(T21, T17, s(T23)))
U9_ggg(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → U10_ggg(T21, T17, T25, add30_in_gag(T21, T23, T25))
U10_ggg(T21, T17, T25, add30_out_gag(T21, T23, T25)) → mul15_out_ggg(T21, s(T17), s(T25))
U2_ggg(T20, T17, mul15_out_ggg(T20, T17, 0)) → mul15_out_ggg(T20, s(T17), T20)
U2_gga(T20, T17, mul15_out_ggg(T20, T17, 0)) → mul15_out_gga(T20, s(T17), T20)
U1_gga(T15, T17, X34, mul15_out_gga(T15, T17, X33)) → mul15_out_gga(T15, s(T17), X34)
U15_gga(T8, T12, T11, mul15_out_gga(T8, T12, X15)) → mul1_out_gga(T8, s(T12), T11)
mul1_in_gga(T29, s(T12), T29) → U16_gga(T29, T12, mul15_in_ggg(T29, T12, 0))
U16_gga(T29, T12, mul15_out_ggg(T29, T12, 0)) → mul1_out_gga(T29, s(T12), T29)
mul1_in_gga(T30, s(T12), T33) → U17_gga(T30, T12, T33, mul15_in_gga(T30, T12, s(T34)))
U17_gga(T30, T12, T33, mul15_out_gga(T30, T12, s(T34))) → U18_gga(T30, T12, T33, add30_in_gaa(T30, T34, X75))
U18_gga(T30, T12, T33, add30_out_gaa(T30, T34, X75)) → mul1_out_gga(T30, s(T12), T33)
mul1_in_gga(T30, s(T12), 0) → U19_gga(T30, T12, mul15_in_gga(T30, T12, s(T34)))
U19_gga(T30, T12, mul15_out_gga(T30, T12, s(T34))) → U20_gga(T30, T12, add30_in_gag(T30, T34, 0))
U20_gga(T30, T12, add30_out_gag(T30, T34, 0)) → mul1_out_gga(T30, s(T12), 0)
mul1_in_gga(T30, s(T12), s(0)) → U21_gga(T30, T12, mul15_in_gga(T30, T12, s(T34)))
U21_gga(T30, T12, mul15_out_gga(T30, T12, s(T34))) → U22_gga(T30, T12, add30_in_gag(T30, T34, 0))
U22_gga(T30, T12, add30_out_gag(T30, T34, 0)) → mul1_out_gga(T30, s(T12), s(0))
mul1_in_gga(T30, s(T12), s(T37)) → U23_gga(T30, T12, T37, mul15_in_gga(T30, T12, s(T34)))
U23_gga(T30, T12, T37, mul15_out_gga(T30, T12, s(T34))) → U24_gga(T30, T12, T37, add30_in_gaa(T30, T34, T37))
U24_gga(T30, T12, T37, add30_out_gaa(T30, T34, T37)) → mul1_out_gga(T30, s(T12), s(T37))

(34) 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:

MUL1_IN_GGA(T8, s(T12), T11) → U15_GGA(T8, T12, T11, mul15_in_gga(T8, T12, X15))
MUL1_IN_GGA(T8, s(T12), T11) → MUL15_IN_GGA(T8, T12, X15)
MUL15_IN_GGA(T15, s(T17), X34) → U1_GGA(T15, T17, X34, mul15_in_gga(T15, T17, X33))
MUL15_IN_GGA(T15, s(T17), X34) → MUL15_IN_GGA(T15, T17, X33)
MUL15_IN_GGA(T20, s(T17), T20) → U2_GGA(T20, T17, mul15_in_ggg(T20, T17, 0))
MUL15_IN_GGA(T20, s(T17), T20) → MUL15_IN_GGG(T20, T17, 0)
MUL15_IN_GGG(T15, s(T17), X34) → U1_GGG(T15, T17, X34, mul15_in_gga(T15, T17, X33))
MUL15_IN_GGG(T15, s(T17), X34) → MUL15_IN_GGA(T15, T17, X33)
MUL15_IN_GGA(T21, s(T17), X53) → U3_GGA(T21, T17, X53, mul15_in_gga(T21, T17, s(T23)))
MUL15_IN_GGA(T21, s(T17), X53) → MUL15_IN_GGA(T21, T17, s(T23))
MUL15_IN_GGA(T21, s(T17), 0) → U5_GGA(T21, T17, mul15_in_gga(T21, T17, s(T23)))
MUL15_IN_GGA(T21, s(T17), 0) → MUL15_IN_GGA(T21, T17, s(T23))
MUL15_IN_GGA(T21, s(T17), s(0)) → U7_GGA(T21, T17, mul15_in_gga(T21, T17, s(T23)))
MUL15_IN_GGA(T21, s(T17), s(0)) → MUL15_IN_GGA(T21, T17, s(T23))
MUL15_IN_GGA(T21, s(T17), s(T25)) → U9_GGA(T21, T17, T25, mul15_in_gga(T21, T17, s(T23)))
MUL15_IN_GGA(T21, s(T17), s(T25)) → MUL15_IN_GGA(T21, T17, s(T23))
U9_GGA(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → U10_GGA(T21, T17, T25, add30_in_gaa(T21, T23, T25))
U9_GGA(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → ADD30_IN_GAA(T21, T23, T25)
ADD30_IN_GAA(T21, s(T23), X53) → U11_GAA(T21, T23, X53, add30_in_gaa(T21, T23, X52))
ADD30_IN_GAA(T21, s(T23), X53) → ADD30_IN_GAA(T21, T23, X52)
ADD30_IN_GAA(T21, s(T23), 0) → U12_GAA(T21, T23, add30_in_gag(T21, T23, 0))
ADD30_IN_GAA(T21, s(T23), 0) → ADD30_IN_GAG(T21, T23, 0)
ADD30_IN_GAG(T21, s(T23), X53) → U11_GAG(T21, T23, X53, add30_in_gaa(T21, T23, X52))
ADD30_IN_GAG(T21, s(T23), X53) → ADD30_IN_GAA(T21, T23, X52)
ADD30_IN_GAA(T21, s(T23), s(0)) → U13_GAA(T21, T23, add30_in_gag(T21, T23, 0))
ADD30_IN_GAA(T21, s(T23), s(0)) → ADD30_IN_GAG(T21, T23, 0)
ADD30_IN_GAG(T21, s(T23), 0) → U12_GAG(T21, T23, add30_in_gag(T21, T23, 0))
ADD30_IN_GAG(T21, s(T23), 0) → ADD30_IN_GAG(T21, T23, 0)
ADD30_IN_GAG(T21, s(T23), s(0)) → U13_GAG(T21, T23, add30_in_gag(T21, T23, 0))
ADD30_IN_GAG(T21, s(T23), s(0)) → ADD30_IN_GAG(T21, T23, 0)
ADD30_IN_GAG(T21, s(T23), s(T25)) → U14_GAG(T21, T23, T25, add30_in_gag(T21, T23, T25))
ADD30_IN_GAG(T21, s(T23), s(T25)) → ADD30_IN_GAG(T21, T23, T25)
ADD30_IN_GAA(T21, s(T23), s(T25)) → U14_GAA(T21, T23, T25, add30_in_gaa(T21, T23, T25))
ADD30_IN_GAA(T21, s(T23), s(T25)) → ADD30_IN_GAA(T21, T23, T25)
U7_GGA(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U8_GGA(T21, T17, add30_in_gag(T21, T23, 0))
U7_GGA(T21, T17, mul15_out_gga(T21, T17, s(T23))) → ADD30_IN_GAG(T21, T23, 0)
U5_GGA(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U6_GGA(T21, T17, add30_in_gag(T21, T23, 0))
U5_GGA(T21, T17, mul15_out_gga(T21, T17, s(T23))) → ADD30_IN_GAG(T21, T23, 0)
U3_GGA(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → U4_GGA(T21, T17, X53, add30_in_gaa(T21, T23, X52))
U3_GGA(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → ADD30_IN_GAA(T21, T23, X52)
MUL15_IN_GGG(T20, s(T17), T20) → U2_GGG(T20, T17, mul15_in_ggg(T20, T17, 0))
MUL15_IN_GGG(T20, s(T17), T20) → MUL15_IN_GGG(T20, T17, 0)
MUL15_IN_GGG(T21, s(T17), X53) → U3_GGG(T21, T17, X53, mul15_in_gga(T21, T17, s(T23)))
MUL15_IN_GGG(T21, s(T17), X53) → MUL15_IN_GGA(T21, T17, s(T23))
U3_GGG(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → U4_GGG(T21, T17, X53, add30_in_gaa(T21, T23, X52))
U3_GGG(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → ADD30_IN_GAA(T21, T23, X52)
MUL15_IN_GGG(T21, s(T17), 0) → U5_GGG(T21, T17, mul15_in_gga(T21, T17, s(T23)))
MUL15_IN_GGG(T21, s(T17), 0) → MUL15_IN_GGA(T21, T17, s(T23))
U5_GGG(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U6_GGG(T21, T17, add30_in_gag(T21, T23, 0))
U5_GGG(T21, T17, mul15_out_gga(T21, T17, s(T23))) → ADD30_IN_GAG(T21, T23, 0)
MUL15_IN_GGG(T21, s(T17), s(0)) → U7_GGG(T21, T17, mul15_in_gga(T21, T17, s(T23)))
MUL15_IN_GGG(T21, s(T17), s(0)) → MUL15_IN_GGA(T21, T17, s(T23))
U7_GGG(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U8_GGG(T21, T17, add30_in_gag(T21, T23, 0))
U7_GGG(T21, T17, mul15_out_gga(T21, T17, s(T23))) → ADD30_IN_GAG(T21, T23, 0)
MUL15_IN_GGG(T21, s(T17), s(T25)) → U9_GGG(T21, T17, T25, mul15_in_gga(T21, T17, s(T23)))
MUL15_IN_GGG(T21, s(T17), s(T25)) → MUL15_IN_GGA(T21, T17, s(T23))
U9_GGG(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → U10_GGG(T21, T17, T25, add30_in_gag(T21, T23, T25))
U9_GGG(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → ADD30_IN_GAG(T21, T23, T25)
MUL1_IN_GGA(T29, s(T12), T29) → U16_GGA(T29, T12, mul15_in_ggg(T29, T12, 0))
MUL1_IN_GGA(T29, s(T12), T29) → MUL15_IN_GGG(T29, T12, 0)
MUL1_IN_GGA(T30, s(T12), T33) → U17_GGA(T30, T12, T33, mul15_in_gga(T30, T12, s(T34)))
MUL1_IN_GGA(T30, s(T12), T33) → MUL15_IN_GGA(T30, T12, s(T34))
U17_GGA(T30, T12, T33, mul15_out_gga(T30, T12, s(T34))) → U18_GGA(T30, T12, T33, add30_in_gaa(T30, T34, X75))
U17_GGA(T30, T12, T33, mul15_out_gga(T30, T12, s(T34))) → ADD30_IN_GAA(T30, T34, X75)
MUL1_IN_GGA(T30, s(T12), 0) → U19_GGA(T30, T12, mul15_in_gga(T30, T12, s(T34)))
MUL1_IN_GGA(T30, s(T12), 0) → MUL15_IN_GGA(T30, T12, s(T34))
U19_GGA(T30, T12, mul15_out_gga(T30, T12, s(T34))) → U20_GGA(T30, T12, add30_in_gag(T30, T34, 0))
U19_GGA(T30, T12, mul15_out_gga(T30, T12, s(T34))) → ADD30_IN_GAG(T30, T34, 0)
MUL1_IN_GGA(T30, s(T12), s(0)) → U21_GGA(T30, T12, mul15_in_gga(T30, T12, s(T34)))
MUL1_IN_GGA(T30, s(T12), s(0)) → MUL15_IN_GGA(T30, T12, s(T34))
U21_GGA(T30, T12, mul15_out_gga(T30, T12, s(T34))) → U22_GGA(T30, T12, add30_in_gag(T30, T34, 0))
U21_GGA(T30, T12, mul15_out_gga(T30, T12, s(T34))) → ADD30_IN_GAG(T30, T34, 0)
MUL1_IN_GGA(T30, s(T12), s(T37)) → U23_GGA(T30, T12, T37, mul15_in_gga(T30, T12, s(T34)))
MUL1_IN_GGA(T30, s(T12), s(T37)) → MUL15_IN_GGA(T30, T12, s(T34))
U23_GGA(T30, T12, T37, mul15_out_gga(T30, T12, s(T34))) → U24_GGA(T30, T12, T37, add30_in_gaa(T30, T34, T37))
U23_GGA(T30, T12, T37, mul15_out_gga(T30, T12, s(T34))) → ADD30_IN_GAA(T30, T34, T37)

The TRS R consists of the following rules:

mul1_in_gga(T4, 0, 0) → mul1_out_gga(T4, 0, 0)
mul1_in_gga(T8, s(T12), T11) → U15_gga(T8, T12, T11, mul15_in_gga(T8, T12, X15))
mul15_in_gga(T14, 0, 0) → mul15_out_gga(T14, 0, 0)
mul15_in_gga(T15, s(T17), X34) → U1_gga(T15, T17, X34, mul15_in_gga(T15, T17, X33))
mul15_in_gga(T20, s(T17), T20) → U2_gga(T20, T17, mul15_in_ggg(T20, T17, 0))
mul15_in_ggg(T14, 0, 0) → mul15_out_ggg(T14, 0, 0)
mul15_in_ggg(T15, s(T17), X34) → U1_ggg(T15, T17, X34, mul15_in_gga(T15, T17, X33))
mul15_in_gga(T21, s(T17), X53) → U3_gga(T21, T17, X53, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), 0) → U5_gga(T21, T17, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), s(0)) → U7_gga(T21, T17, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), s(T25)) → U9_gga(T21, T17, T25, mul15_in_gga(T21, T17, s(T23)))
U9_gga(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → U10_gga(T21, T17, T25, add30_in_gaa(T21, T23, T25))
add30_in_gaa(T20, 0, T20) → add30_out_gaa(T20, 0, T20)
add30_in_gaa(T21, s(T23), X53) → U11_gaa(T21, T23, X53, add30_in_gaa(T21, T23, X52))
add30_in_gaa(T21, s(T23), 0) → U12_gaa(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T20, 0, T20) → add30_out_gag(T20, 0, T20)
add30_in_gag(T21, s(T23), X53) → U11_gag(T21, T23, X53, add30_in_gaa(T21, T23, X52))
add30_in_gaa(T21, s(T23), s(0)) → U13_gaa(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), 0) → U12_gag(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), s(0)) → U13_gag(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), s(T25)) → U14_gag(T21, T23, T25, add30_in_gag(T21, T23, T25))
U14_gag(T21, T23, T25, add30_out_gag(T21, T23, T25)) → add30_out_gag(T21, s(T23), s(T25))
U13_gag(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gag(T21, s(T23), s(0))
U12_gag(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gag(T21, s(T23), 0)
U13_gaa(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gaa(T21, s(T23), s(0))
add30_in_gaa(T21, s(T23), s(T25)) → U14_gaa(T21, T23, T25, add30_in_gaa(T21, T23, T25))
U14_gaa(T21, T23, T25, add30_out_gaa(T21, T23, T25)) → add30_out_gaa(T21, s(T23), s(T25))
U11_gag(T21, T23, X53, add30_out_gaa(T21, T23, X52)) → add30_out_gag(T21, s(T23), X53)
U12_gaa(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gaa(T21, s(T23), 0)
U11_gaa(T21, T23, X53, add30_out_gaa(T21, T23, X52)) → add30_out_gaa(T21, s(T23), X53)
U10_gga(T21, T17, T25, add30_out_gaa(T21, T23, T25)) → mul15_out_gga(T21, s(T17), s(T25))
U7_gga(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U8_gga(T21, T17, add30_in_gag(T21, T23, 0))
U8_gga(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_gga(T21, s(T17), s(0))
U5_gga(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U6_gga(T21, T17, add30_in_gag(T21, T23, 0))
U6_gga(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_gga(T21, s(T17), 0)
U3_gga(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → U4_gga(T21, T17, X53, add30_in_gaa(T21, T23, X52))
U4_gga(T21, T17, X53, add30_out_gaa(T21, T23, X52)) → mul15_out_gga(T21, s(T17), X53)
U1_ggg(T15, T17, X34, mul15_out_gga(T15, T17, X33)) → mul15_out_ggg(T15, s(T17), X34)
mul15_in_ggg(T20, s(T17), T20) → U2_ggg(T20, T17, mul15_in_ggg(T20, T17, 0))
mul15_in_ggg(T21, s(T17), X53) → U3_ggg(T21, T17, X53, mul15_in_gga(T21, T17, s(T23)))
U3_ggg(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → U4_ggg(T21, T17, X53, add30_in_gaa(T21, T23, X52))
U4_ggg(T21, T17, X53, add30_out_gaa(T21, T23, X52)) → mul15_out_ggg(T21, s(T17), X53)
mul15_in_ggg(T21, s(T17), 0) → U5_ggg(T21, T17, mul15_in_gga(T21, T17, s(T23)))
U5_ggg(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U6_ggg(T21, T17, add30_in_gag(T21, T23, 0))
U6_ggg(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_ggg(T21, s(T17), 0)
mul15_in_ggg(T21, s(T17), s(0)) → U7_ggg(T21, T17, mul15_in_gga(T21, T17, s(T23)))
U7_ggg(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U8_ggg(T21, T17, add30_in_gag(T21, T23, 0))
U8_ggg(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_ggg(T21, s(T17), s(0))
mul15_in_ggg(T21, s(T17), s(T25)) → U9_ggg(T21, T17, T25, mul15_in_gga(T21, T17, s(T23)))
U9_ggg(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → U10_ggg(T21, T17, T25, add30_in_gag(T21, T23, T25))
U10_ggg(T21, T17, T25, add30_out_gag(T21, T23, T25)) → mul15_out_ggg(T21, s(T17), s(T25))
U2_ggg(T20, T17, mul15_out_ggg(T20, T17, 0)) → mul15_out_ggg(T20, s(T17), T20)
U2_gga(T20, T17, mul15_out_ggg(T20, T17, 0)) → mul15_out_gga(T20, s(T17), T20)
U1_gga(T15, T17, X34, mul15_out_gga(T15, T17, X33)) → mul15_out_gga(T15, s(T17), X34)
U15_gga(T8, T12, T11, mul15_out_gga(T8, T12, X15)) → mul1_out_gga(T8, s(T12), T11)
mul1_in_gga(T29, s(T12), T29) → U16_gga(T29, T12, mul15_in_ggg(T29, T12, 0))
U16_gga(T29, T12, mul15_out_ggg(T29, T12, 0)) → mul1_out_gga(T29, s(T12), T29)
mul1_in_gga(T30, s(T12), T33) → U17_gga(T30, T12, T33, mul15_in_gga(T30, T12, s(T34)))
U17_gga(T30, T12, T33, mul15_out_gga(T30, T12, s(T34))) → U18_gga(T30, T12, T33, add30_in_gaa(T30, T34, X75))
U18_gga(T30, T12, T33, add30_out_gaa(T30, T34, X75)) → mul1_out_gga(T30, s(T12), T33)
mul1_in_gga(T30, s(T12), 0) → U19_gga(T30, T12, mul15_in_gga(T30, T12, s(T34)))
U19_gga(T30, T12, mul15_out_gga(T30, T12, s(T34))) → U20_gga(T30, T12, add30_in_gag(T30, T34, 0))
U20_gga(T30, T12, add30_out_gag(T30, T34, 0)) → mul1_out_gga(T30, s(T12), 0)
mul1_in_gga(T30, s(T12), s(0)) → U21_gga(T30, T12, mul15_in_gga(T30, T12, s(T34)))
U21_gga(T30, T12, mul15_out_gga(T30, T12, s(T34))) → U22_gga(T30, T12, add30_in_gag(T30, T34, 0))
U22_gga(T30, T12, add30_out_gag(T30, T34, 0)) → mul1_out_gga(T30, s(T12), s(0))
mul1_in_gga(T30, s(T12), s(T37)) → U23_gga(T30, T12, T37, mul15_in_gga(T30, T12, s(T34)))
U23_gga(T30, T12, T37, mul15_out_gga(T30, T12, s(T34))) → U24_gga(T30, T12, T37, add30_in_gaa(T30, T34, T37))
U24_gga(T30, T12, T37, add30_out_gaa(T30, T34, T37)) → mul1_out_gga(T30, s(T12), s(T37))

The argument filtering Pi contains the following mapping:
mul1_in_gga(x1, x2, x3) = mul1_in_gga(x1, x2)
0 = 0
mul1_out_gga(x1, x2, x3) = mul1_out_gga(x1, x2)
s(x1) = s(x1)
U15_gga(x1, x2, x3, x4) = U15_gga(x1, x2, x4)
mul15_in_gga(x1, x2, x3) = mul15_in_gga(x1, x2)
mul15_out_gga(x1, x2, x3) = mul15_out_gga(x1, x2)
U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4)
U2_gga(x1, x2, x3) = U2_gga(x1, x2, x3)
mul15_in_ggg(x1, x2, x3) = mul15_in_ggg(x1, x2, x3)
mul15_out_ggg(x1, x2, x3) = mul15_out_ggg(x1, x2, x3)
U1_ggg(x1, x2, x3, x4) = U1_ggg(x1, x2, x3, x4)
U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4)
U5_gga(x1, x2, x3) = U5_gga(x1, x2, x3)
U7_gga(x1, x2, x3) = U7_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)
add30_in_gaa(x1, x2, x3) = add30_in_gaa(x1)
add30_out_gaa(x1, x2, x3) = add30_out_gaa(x1, x2)
U11_gaa(x1, x2, x3, x4) = U11_gaa(x1, x4)
U12_gaa(x1, x2, x3) = U12_gaa(x1, x3)
add30_in_gag(x1, x2, x3) = add30_in_gag(x1, x3)
add30_out_gag(x1, x2, x3) = add30_out_gag(x1, x2, x3)
U11_gag(x1, x2, x3, x4) = U11_gag(x1, x3, x4)
U13_gaa(x1, x2, x3) = U13_gaa(x1, x3)
U12_gag(x1, x2, x3) = U12_gag(x1, x3)
U13_gag(x1, x2, x3) = U13_gag(x1, x3)
U14_gag(x1, x2, x3, x4) = U14_gag(x1, x3, x4)
U14_gaa(x1, x2, x3, x4) = U14_gaa(x1, x4)
U8_gga(x1, x2, x3) = U8_gga(x1, x2, x3)
U6_gga(x1, x2, x3) = U6_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4)
U2_ggg(x1, x2, x3) = U2_ggg(x1, x2, x3)
U3_ggg(x1, x2, x3, x4) = U3_ggg(x1, x2, x3, x4)
U4_ggg(x1, x2, x3, x4) = U4_ggg(x1, x2, x3, x4)
U5_ggg(x1, x2, x3) = U5_ggg(x1, x2, x3)
U6_ggg(x1, x2, x3) = U6_ggg(x1, x2, x3)
U7_ggg(x1, x2, x3) = U7_ggg(x1, x2, x3)
U8_ggg(x1, x2, x3) = U8_ggg(x1, x2, x3)
U9_ggg(x1, x2, x3, x4) = U9_ggg(x1, x2, x3, x4)
U10_ggg(x1, x2, x3, x4) = U10_ggg(x1, x2, x3, x4)
U16_gga(x1, x2, x3) = U16_gga(x1, x2, x3)
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) = U19_gga(x1, x2, x3)
U20_gga(x1, x2, x3) = U20_gga(x1, x2, x3)
U21_gga(x1, x2, x3) = U21_gga(x1, x2, x3)
U22_gga(x1, x2, x3) = U22_gga(x1, x2, x3)
U23_gga(x1, x2, x3, x4) = U23_gga(x1, x2, x4)
U24_gga(x1, x2, x3, x4) = U24_gga(x1, x2, x4)
MUL1_IN_GGA(x1, x2, x3) = MUL1_IN_GGA(x1, x2)
U15_GGA(x1, x2, x3, x4) = U15_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) = U2_GGA(x1, x2, x3)
MUL15_IN_GGG(x1, x2, x3) = MUL15_IN_GGG(x1, x2, x3)
U1_GGG(x1, x2, x3, x4) = U1_GGG(x1, x2, x3, x4)
U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4)
U5_GGA(x1, x2, x3) = U5_GGA(x1, x2, x3)
U7_GGA(x1, x2, x3) = U7_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)
ADD30_IN_GAA(x1, x2, x3) = ADD30_IN_GAA(x1)
U11_GAA(x1, x2, x3, x4) = U11_GAA(x1, x4)
U12_GAA(x1, x2, x3) = U12_GAA(x1, x3)
ADD30_IN_GAG(x1, x2, x3) = ADD30_IN_GAG(x1, x3)
U11_GAG(x1, x2, x3, x4) = U11_GAG(x1, x3, x4)
U13_GAA(x1, x2, x3) = U13_GAA(x1, x3)
U12_GAG(x1, x2, x3) = U12_GAG(x1, x3)
U13_GAG(x1, x2, x3) = U13_GAG(x1, x3)
U14_GAG(x1, x2, x3, x4) = U14_GAG(x1, x3, x4)
U14_GAA(x1, x2, x3, x4) = U14_GAA(x1, x4)
U8_GGA(x1, x2, x3) = U8_GGA(x1, x2, x3)
U6_GGA(x1, x2, x3) = U6_GGA(x1, x2, x3)
U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4)
U2_GGG(x1, x2, x3) = U2_GGG(x1, x2, x3)
U3_GGG(x1, x2, x3, x4) = U3_GGG(x1, x2, x3, x4)
U4_GGG(x1, x2, x3, x4) = U4_GGG(x1, x2, x3, x4)
U5_GGG(x1, x2, x3) = U5_GGG(x1, x2, x3)
U6_GGG(x1, x2, x3) = U6_GGG(x1, x2, x3)
U7_GGG(x1, x2, x3) = U7_GGG(x1, x2, x3)
U8_GGG(x1, x2, x3) = U8_GGG(x1, x2, x3)
U9_GGG(x1, x2, x3, x4) = U9_GGG(x1, x2, x3, x4)
U10_GGG(x1, x2, x3, x4) = U10_GGG(x1, x2, x3, x4)
U16_GGA(x1, x2, x3) = U16_GGA(x1, x2, x3)
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) = U19_GGA(x1, x2, x3)
U20_GGA(x1, x2, x3) = U20_GGA(x1, x2, x3)
U21_GGA(x1, x2, x3) = U21_GGA(x1, x2, x3)
U22_GGA(x1, x2, x3) = U22_GGA(x1, x2, x3)
U23_GGA(x1, x2, x3, x4) = U23_GGA(x1, x2, x4)
U24_GGA(x1, x2, x3, x4) = U24_GGA(x1, x2, x4)

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

(35) Obligation:

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

MUL1_IN_GGA(T8, s(T12), T11) → U15_GGA(T8, T12, T11, mul15_in_gga(T8, T12, X15))
MUL1_IN_GGA(T8, s(T12), T11) → MUL15_IN_GGA(T8, T12, X15)
MUL15_IN_GGA(T15, s(T17), X34) → U1_GGA(T15, T17, X34, mul15_in_gga(T15, T17, X33))
MUL15_IN_GGA(T15, s(T17), X34) → MUL15_IN_GGA(T15, T17, X33)
MUL15_IN_GGA(T20, s(T17), T20) → U2_GGA(T20, T17, mul15_in_ggg(T20, T17, 0))
MUL15_IN_GGA(T20, s(T17), T20) → MUL15_IN_GGG(T20, T17, 0)
MUL15_IN_GGG(T15, s(T17), X34) → U1_GGG(T15, T17, X34, mul15_in_gga(T15, T17, X33))
MUL15_IN_GGG(T15, s(T17), X34) → MUL15_IN_GGA(T15, T17, X33)
MUL15_IN_GGA(T21, s(T17), X53) → U3_GGA(T21, T17, X53, mul15_in_gga(T21, T17, s(T23)))
MUL15_IN_GGA(T21, s(T17), X53) → MUL15_IN_GGA(T21, T17, s(T23))
MUL15_IN_GGA(T21, s(T17), 0) → U5_GGA(T21, T17, mul15_in_gga(T21, T17, s(T23)))
MUL15_IN_GGA(T21, s(T17), 0) → MUL15_IN_GGA(T21, T17, s(T23))
MUL15_IN_GGA(T21, s(T17), s(0)) → U7_GGA(T21, T17, mul15_in_gga(T21, T17, s(T23)))
MUL15_IN_GGA(T21, s(T17), s(0)) → MUL15_IN_GGA(T21, T17, s(T23))
MUL15_IN_GGA(T21, s(T17), s(T25)) → U9_GGA(T21, T17, T25, mul15_in_gga(T21, T17, s(T23)))
MUL15_IN_GGA(T21, s(T17), s(T25)) → MUL15_IN_GGA(T21, T17, s(T23))
U9_GGA(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → U10_GGA(T21, T17, T25, add30_in_gaa(T21, T23, T25))
U9_GGA(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → ADD30_IN_GAA(T21, T23, T25)
ADD30_IN_GAA(T21, s(T23), X53) → U11_GAA(T21, T23, X53, add30_in_gaa(T21, T23, X52))
ADD30_IN_GAA(T21, s(T23), X53) → ADD30_IN_GAA(T21, T23, X52)
ADD30_IN_GAA(T21, s(T23), 0) → U12_GAA(T21, T23, add30_in_gag(T21, T23, 0))
ADD30_IN_GAA(T21, s(T23), 0) → ADD30_IN_GAG(T21, T23, 0)
ADD30_IN_GAG(T21, s(T23), X53) → U11_GAG(T21, T23, X53, add30_in_gaa(T21, T23, X52))
ADD30_IN_GAG(T21, s(T23), X53) → ADD30_IN_GAA(T21, T23, X52)
ADD30_IN_GAA(T21, s(T23), s(0)) → U13_GAA(T21, T23, add30_in_gag(T21, T23, 0))
ADD30_IN_GAA(T21, s(T23), s(0)) → ADD30_IN_GAG(T21, T23, 0)
ADD30_IN_GAG(T21, s(T23), 0) → U12_GAG(T21, T23, add30_in_gag(T21, T23, 0))
ADD30_IN_GAG(T21, s(T23), 0) → ADD30_IN_GAG(T21, T23, 0)
ADD30_IN_GAG(T21, s(T23), s(0)) → U13_GAG(T21, T23, add30_in_gag(T21, T23, 0))
ADD30_IN_GAG(T21, s(T23), s(0)) → ADD30_IN_GAG(T21, T23, 0)
ADD30_IN_GAG(T21, s(T23), s(T25)) → U14_GAG(T21, T23, T25, add30_in_gag(T21, T23, T25))
ADD30_IN_GAG(T21, s(T23), s(T25)) → ADD30_IN_GAG(T21, T23, T25)
ADD30_IN_GAA(T21, s(T23), s(T25)) → U14_GAA(T21, T23, T25, add30_in_gaa(T21, T23, T25))
ADD30_IN_GAA(T21, s(T23), s(T25)) → ADD30_IN_GAA(T21, T23, T25)
U7_GGA(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U8_GGA(T21, T17, add30_in_gag(T21, T23, 0))
U7_GGA(T21, T17, mul15_out_gga(T21, T17, s(T23))) → ADD30_IN_GAG(T21, T23, 0)
U5_GGA(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U6_GGA(T21, T17, add30_in_gag(T21, T23, 0))
U5_GGA(T21, T17, mul15_out_gga(T21, T17, s(T23))) → ADD30_IN_GAG(T21, T23, 0)
U3_GGA(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → U4_GGA(T21, T17, X53, add30_in_gaa(T21, T23, X52))
U3_GGA(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → ADD30_IN_GAA(T21, T23, X52)
MUL15_IN_GGG(T20, s(T17), T20) → U2_GGG(T20, T17, mul15_in_ggg(T20, T17, 0))
MUL15_IN_GGG(T20, s(T17), T20) → MUL15_IN_GGG(T20, T17, 0)
MUL15_IN_GGG(T21, s(T17), X53) → U3_GGG(T21, T17, X53, mul15_in_gga(T21, T17, s(T23)))
MUL15_IN_GGG(T21, s(T17), X53) → MUL15_IN_GGA(T21, T17, s(T23))
U3_GGG(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → U4_GGG(T21, T17, X53, add30_in_gaa(T21, T23, X52))
U3_GGG(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → ADD30_IN_GAA(T21, T23, X52)
MUL15_IN_GGG(T21, s(T17), 0) → U5_GGG(T21, T17, mul15_in_gga(T21, T17, s(T23)))
MUL15_IN_GGG(T21, s(T17), 0) → MUL15_IN_GGA(T21, T17, s(T23))
U5_GGG(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U6_GGG(T21, T17, add30_in_gag(T21, T23, 0))
U5_GGG(T21, T17, mul15_out_gga(T21, T17, s(T23))) → ADD30_IN_GAG(T21, T23, 0)
MUL15_IN_GGG(T21, s(T17), s(0)) → U7_GGG(T21, T17, mul15_in_gga(T21, T17, s(T23)))
MUL15_IN_GGG(T21, s(T17), s(0)) → MUL15_IN_GGA(T21, T17, s(T23))
U7_GGG(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U8_GGG(T21, T17, add30_in_gag(T21, T23, 0))
U7_GGG(T21, T17, mul15_out_gga(T21, T17, s(T23))) → ADD30_IN_GAG(T21, T23, 0)
MUL15_IN_GGG(T21, s(T17), s(T25)) → U9_GGG(T21, T17, T25, mul15_in_gga(T21, T17, s(T23)))
MUL15_IN_GGG(T21, s(T17), s(T25)) → MUL15_IN_GGA(T21, T17, s(T23))
U9_GGG(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → U10_GGG(T21, T17, T25, add30_in_gag(T21, T23, T25))
U9_GGG(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → ADD30_IN_GAG(T21, T23, T25)
MUL1_IN_GGA(T29, s(T12), T29) → U16_GGA(T29, T12, mul15_in_ggg(T29, T12, 0))
MUL1_IN_GGA(T29, s(T12), T29) → MUL15_IN_GGG(T29, T12, 0)
MUL1_IN_GGA(T30, s(T12), T33) → U17_GGA(T30, T12, T33, mul15_in_gga(T30, T12, s(T34)))
MUL1_IN_GGA(T30, s(T12), T33) → MUL15_IN_GGA(T30, T12, s(T34))
U17_GGA(T30, T12, T33, mul15_out_gga(T30, T12, s(T34))) → U18_GGA(T30, T12, T33, add30_in_gaa(T30, T34, X75))
U17_GGA(T30, T12, T33, mul15_out_gga(T30, T12, s(T34))) → ADD30_IN_GAA(T30, T34, X75)
MUL1_IN_GGA(T30, s(T12), 0) → U19_GGA(T30, T12, mul15_in_gga(T30, T12, s(T34)))
MUL1_IN_GGA(T30, s(T12), 0) → MUL15_IN_GGA(T30, T12, s(T34))
U19_GGA(T30, T12, mul15_out_gga(T30, T12, s(T34))) → U20_GGA(T30, T12, add30_in_gag(T30, T34, 0))
U19_GGA(T30, T12, mul15_out_gga(T30, T12, s(T34))) → ADD30_IN_GAG(T30, T34, 0)
MUL1_IN_GGA(T30, s(T12), s(0)) → U21_GGA(T30, T12, mul15_in_gga(T30, T12, s(T34)))
MUL1_IN_GGA(T30, s(T12), s(0)) → MUL15_IN_GGA(T30, T12, s(T34))
U21_GGA(T30, T12, mul15_out_gga(T30, T12, s(T34))) → U22_GGA(T30, T12, add30_in_gag(T30, T34, 0))
U21_GGA(T30, T12, mul15_out_gga(T30, T12, s(T34))) → ADD30_IN_GAG(T30, T34, 0)
MUL1_IN_GGA(T30, s(T12), s(T37)) → U23_GGA(T30, T12, T37, mul15_in_gga(T30, T12, s(T34)))
MUL1_IN_GGA(T30, s(T12), s(T37)) → MUL15_IN_GGA(T30, T12, s(T34))
U23_GGA(T30, T12, T37, mul15_out_gga(T30, T12, s(T34))) → U24_GGA(T30, T12, T37, add30_in_gaa(T30, T34, T37))
U23_GGA(T30, T12, T37, mul15_out_gga(T30, T12, s(T34))) → ADD30_IN_GAA(T30, T34, T37)

The TRS R consists of the following rules:

mul1_in_gga(T4, 0, 0) → mul1_out_gga(T4, 0, 0)
mul1_in_gga(T8, s(T12), T11) → U15_gga(T8, T12, T11, mul15_in_gga(T8, T12, X15))
mul15_in_gga(T14, 0, 0) → mul15_out_gga(T14, 0, 0)
mul15_in_gga(T15, s(T17), X34) → U1_gga(T15, T17, X34, mul15_in_gga(T15, T17, X33))
mul15_in_gga(T20, s(T17), T20) → U2_gga(T20, T17, mul15_in_ggg(T20, T17, 0))
mul15_in_ggg(T14, 0, 0) → mul15_out_ggg(T14, 0, 0)
mul15_in_ggg(T15, s(T17), X34) → U1_ggg(T15, T17, X34, mul15_in_gga(T15, T17, X33))
mul15_in_gga(T21, s(T17), X53) → U3_gga(T21, T17, X53, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), 0) → U5_gga(T21, T17, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), s(0)) → U7_gga(T21, T17, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), s(T25)) → U9_gga(T21, T17, T25, mul15_in_gga(T21, T17, s(T23)))
U9_gga(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → U10_gga(T21, T17, T25, add30_in_gaa(T21, T23, T25))
add30_in_gaa(T20, 0, T20) → add30_out_gaa(T20, 0, T20)
add30_in_gaa(T21, s(T23), X53) → U11_gaa(T21, T23, X53, add30_in_gaa(T21, T23, X52))
add30_in_gaa(T21, s(T23), 0) → U12_gaa(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T20, 0, T20) → add30_out_gag(T20, 0, T20)
add30_in_gag(T21, s(T23), X53) → U11_gag(T21, T23, X53, add30_in_gaa(T21, T23, X52))
add30_in_gaa(T21, s(T23), s(0)) → U13_gaa(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), 0) → U12_gag(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), s(0)) → U13_gag(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), s(T25)) → U14_gag(T21, T23, T25, add30_in_gag(T21, T23, T25))
U14_gag(T21, T23, T25, add30_out_gag(T21, T23, T25)) → add30_out_gag(T21, s(T23), s(T25))
U13_gag(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gag(T21, s(T23), s(0))
U12_gag(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gag(T21, s(T23), 0)
U13_gaa(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gaa(T21, s(T23), s(0))
add30_in_gaa(T21, s(T23), s(T25)) → U14_gaa(T21, T23, T25, add30_in_gaa(T21, T23, T25))
U14_gaa(T21, T23, T25, add30_out_gaa(T21, T23, T25)) → add30_out_gaa(T21, s(T23), s(T25))
U11_gag(T21, T23, X53, add30_out_gaa(T21, T23, X52)) → add30_out_gag(T21, s(T23), X53)
U12_gaa(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gaa(T21, s(T23), 0)
U11_gaa(T21, T23, X53, add30_out_gaa(T21, T23, X52)) → add30_out_gaa(T21, s(T23), X53)
U10_gga(T21, T17, T25, add30_out_gaa(T21, T23, T25)) → mul15_out_gga(T21, s(T17), s(T25))
U7_gga(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U8_gga(T21, T17, add30_in_gag(T21, T23, 0))
U8_gga(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_gga(T21, s(T17), s(0))
U5_gga(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U6_gga(T21, T17, add30_in_gag(T21, T23, 0))
U6_gga(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_gga(T21, s(T17), 0)
U3_gga(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → U4_gga(T21, T17, X53, add30_in_gaa(T21, T23, X52))
U4_gga(T21, T17, X53, add30_out_gaa(T21, T23, X52)) → mul15_out_gga(T21, s(T17), X53)
U1_ggg(T15, T17, X34, mul15_out_gga(T15, T17, X33)) → mul15_out_ggg(T15, s(T17), X34)
mul15_in_ggg(T20, s(T17), T20) → U2_ggg(T20, T17, mul15_in_ggg(T20, T17, 0))
mul15_in_ggg(T21, s(T17), X53) → U3_ggg(T21, T17, X53, mul15_in_gga(T21, T17, s(T23)))
U3_ggg(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → U4_ggg(T21, T17, X53, add30_in_gaa(T21, T23, X52))
U4_ggg(T21, T17, X53, add30_out_gaa(T21, T23, X52)) → mul15_out_ggg(T21, s(T17), X53)
mul15_in_ggg(T21, s(T17), 0) → U5_ggg(T21, T17, mul15_in_gga(T21, T17, s(T23)))
U5_ggg(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U6_ggg(T21, T17, add30_in_gag(T21, T23, 0))
U6_ggg(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_ggg(T21, s(T17), 0)
mul15_in_ggg(T21, s(T17), s(0)) → U7_ggg(T21, T17, mul15_in_gga(T21, T17, s(T23)))
U7_ggg(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U8_ggg(T21, T17, add30_in_gag(T21, T23, 0))
U8_ggg(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_ggg(T21, s(T17), s(0))
mul15_in_ggg(T21, s(T17), s(T25)) → U9_ggg(T21, T17, T25, mul15_in_gga(T21, T17, s(T23)))
U9_ggg(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → U10_ggg(T21, T17, T25, add30_in_gag(T21, T23, T25))
U10_ggg(T21, T17, T25, add30_out_gag(T21, T23, T25)) → mul15_out_ggg(T21, s(T17), s(T25))
U2_ggg(T20, T17, mul15_out_ggg(T20, T17, 0)) → mul15_out_ggg(T20, s(T17), T20)
U2_gga(T20, T17, mul15_out_ggg(T20, T17, 0)) → mul15_out_gga(T20, s(T17), T20)
U1_gga(T15, T17, X34, mul15_out_gga(T15, T17, X33)) → mul15_out_gga(T15, s(T17), X34)
U15_gga(T8, T12, T11, mul15_out_gga(T8, T12, X15)) → mul1_out_gga(T8, s(T12), T11)
mul1_in_gga(T29, s(T12), T29) → U16_gga(T29, T12, mul15_in_ggg(T29, T12, 0))
U16_gga(T29, T12, mul15_out_ggg(T29, T12, 0)) → mul1_out_gga(T29, s(T12), T29)
mul1_in_gga(T30, s(T12), T33) → U17_gga(T30, T12, T33, mul15_in_gga(T30, T12, s(T34)))
U17_gga(T30, T12, T33, mul15_out_gga(T30, T12, s(T34))) → U18_gga(T30, T12, T33, add30_in_gaa(T30, T34, X75))
U18_gga(T30, T12, T33, add30_out_gaa(T30, T34, X75)) → mul1_out_gga(T30, s(T12), T33)
mul1_in_gga(T30, s(T12), 0) → U19_gga(T30, T12, mul15_in_gga(T30, T12, s(T34)))
U19_gga(T30, T12, mul15_out_gga(T30, T12, s(T34))) → U20_gga(T30, T12, add30_in_gag(T30, T34, 0))
U20_gga(T30, T12, add30_out_gag(T30, T34, 0)) → mul1_out_gga(T30, s(T12), 0)
mul1_in_gga(T30, s(T12), s(0)) → U21_gga(T30, T12, mul15_in_gga(T30, T12, s(T34)))
U21_gga(T30, T12, mul15_out_gga(T30, T12, s(T34))) → U22_gga(T30, T12, add30_in_gag(T30, T34, 0))
U22_gga(T30, T12, add30_out_gag(T30, T34, 0)) → mul1_out_gga(T30, s(T12), s(0))
mul1_in_gga(T30, s(T12), s(T37)) → U23_gga(T30, T12, T37, mul15_in_gga(T30, T12, s(T34)))
U23_gga(T30, T12, T37, mul15_out_gga(T30, T12, s(T34))) → U24_gga(T30, T12, T37, add30_in_gaa(T30, T34, T37))
U24_gga(T30, T12, T37, add30_out_gaa(T30, T34, T37)) → mul1_out_gga(T30, s(T12), s(T37))

(36) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 59 less nodes.

(37) Complex Obligation (AND)

(38) Obligation:

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

ADD30_IN_GAA(T21, s(T23), 0) → ADD30_IN_GAG(T21, T23, 0)
ADD30_IN_GAG(T21, s(T23), X53) → ADD30_IN_GAA(T21, T23, X52)
ADD30_IN_GAA(T21, s(T23), X53) → ADD30_IN_GAA(T21, T23, X52)
ADD30_IN_GAA(T21, s(T23), s(0)) → ADD30_IN_GAG(T21, T23, 0)
ADD30_IN_GAG(T21, s(T23), 0) → ADD30_IN_GAG(T21, T23, 0)
ADD30_IN_GAA(T21, s(T23), s(T25)) → ADD30_IN_GAA(T21, T23, T25)

The TRS R consists of the following rules:

mul1_in_gga(T4, 0, 0) → mul1_out_gga(T4, 0, 0)
mul1_in_gga(T8, s(T12), T11) → U15_gga(T8, T12, T11, mul15_in_gga(T8, T12, X15))
mul15_in_gga(T14, 0, 0) → mul15_out_gga(T14, 0, 0)
mul15_in_gga(T15, s(T17), X34) → U1_gga(T15, T17, X34, mul15_in_gga(T15, T17, X33))
mul15_in_gga(T20, s(T17), T20) → U2_gga(T20, T17, mul15_in_ggg(T20, T17, 0))
mul15_in_ggg(T14, 0, 0) → mul15_out_ggg(T14, 0, 0)
mul15_in_ggg(T15, s(T17), X34) → U1_ggg(T15, T17, X34, mul15_in_gga(T15, T17, X33))
mul15_in_gga(T21, s(T17), X53) → U3_gga(T21, T17, X53, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), 0) → U5_gga(T21, T17, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), s(0)) → U7_gga(T21, T17, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), s(T25)) → U9_gga(T21, T17, T25, mul15_in_gga(T21, T17, s(T23)))
U9_gga(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → U10_gga(T21, T17, T25, add30_in_gaa(T21, T23, T25))
add30_in_gaa(T20, 0, T20) → add30_out_gaa(T20, 0, T20)
add30_in_gaa(T21, s(T23), X53) → U11_gaa(T21, T23, X53, add30_in_gaa(T21, T23, X52))
add30_in_gaa(T21, s(T23), 0) → U12_gaa(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T20, 0, T20) → add30_out_gag(T20, 0, T20)
add30_in_gag(T21, s(T23), X53) → U11_gag(T21, T23, X53, add30_in_gaa(T21, T23, X52))
add30_in_gaa(T21, s(T23), s(0)) → U13_gaa(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), 0) → U12_gag(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), s(0)) → U13_gag(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), s(T25)) → U14_gag(T21, T23, T25, add30_in_gag(T21, T23, T25))
U14_gag(T21, T23, T25, add30_out_gag(T21, T23, T25)) → add30_out_gag(T21, s(T23), s(T25))
U13_gag(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gag(T21, s(T23), s(0))
U12_gag(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gag(T21, s(T23), 0)
U13_gaa(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gaa(T21, s(T23), s(0))
add30_in_gaa(T21, s(T23), s(T25)) → U14_gaa(T21, T23, T25, add30_in_gaa(T21, T23, T25))
U14_gaa(T21, T23, T25, add30_out_gaa(T21, T23, T25)) → add30_out_gaa(T21, s(T23), s(T25))
U11_gag(T21, T23, X53, add30_out_gaa(T21, T23, X52)) → add30_out_gag(T21, s(T23), X53)
U12_gaa(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gaa(T21, s(T23), 0)
U11_gaa(T21, T23, X53, add30_out_gaa(T21, T23, X52)) → add30_out_gaa(T21, s(T23), X53)
U10_gga(T21, T17, T25, add30_out_gaa(T21, T23, T25)) → mul15_out_gga(T21, s(T17), s(T25))
U7_gga(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U8_gga(T21, T17, add30_in_gag(T21, T23, 0))
U8_gga(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_gga(T21, s(T17), s(0))
U5_gga(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U6_gga(T21, T17, add30_in_gag(T21, T23, 0))
U6_gga(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_gga(T21, s(T17), 0)
U3_gga(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → U4_gga(T21, T17, X53, add30_in_gaa(T21, T23, X52))
U4_gga(T21, T17, X53, add30_out_gaa(T21, T23, X52)) → mul15_out_gga(T21, s(T17), X53)
U1_ggg(T15, T17, X34, mul15_out_gga(T15, T17, X33)) → mul15_out_ggg(T15, s(T17), X34)
mul15_in_ggg(T20, s(T17), T20) → U2_ggg(T20, T17, mul15_in_ggg(T20, T17, 0))
mul15_in_ggg(T21, s(T17), X53) → U3_ggg(T21, T17, X53, mul15_in_gga(T21, T17, s(T23)))
U3_ggg(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → U4_ggg(T21, T17, X53, add30_in_gaa(T21, T23, X52))
U4_ggg(T21, T17, X53, add30_out_gaa(T21, T23, X52)) → mul15_out_ggg(T21, s(T17), X53)
mul15_in_ggg(T21, s(T17), 0) → U5_ggg(T21, T17, mul15_in_gga(T21, T17, s(T23)))
U5_ggg(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U6_ggg(T21, T17, add30_in_gag(T21, T23, 0))
U6_ggg(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_ggg(T21, s(T17), 0)
mul15_in_ggg(T21, s(T17), s(0)) → U7_ggg(T21, T17, mul15_in_gga(T21, T17, s(T23)))
U7_ggg(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U8_ggg(T21, T17, add30_in_gag(T21, T23, 0))
U8_ggg(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_ggg(T21, s(T17), s(0))
mul15_in_ggg(T21, s(T17), s(T25)) → U9_ggg(T21, T17, T25, mul15_in_gga(T21, T17, s(T23)))
U9_ggg(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → U10_ggg(T21, T17, T25, add30_in_gag(T21, T23, T25))
U10_ggg(T21, T17, T25, add30_out_gag(T21, T23, T25)) → mul15_out_ggg(T21, s(T17), s(T25))
U2_ggg(T20, T17, mul15_out_ggg(T20, T17, 0)) → mul15_out_ggg(T20, s(T17), T20)
U2_gga(T20, T17, mul15_out_ggg(T20, T17, 0)) → mul15_out_gga(T20, s(T17), T20)
U1_gga(T15, T17, X34, mul15_out_gga(T15, T17, X33)) → mul15_out_gga(T15, s(T17), X34)
U15_gga(T8, T12, T11, mul15_out_gga(T8, T12, X15)) → mul1_out_gga(T8, s(T12), T11)
mul1_in_gga(T29, s(T12), T29) → U16_gga(T29, T12, mul15_in_ggg(T29, T12, 0))
U16_gga(T29, T12, mul15_out_ggg(T29, T12, 0)) → mul1_out_gga(T29, s(T12), T29)
mul1_in_gga(T30, s(T12), T33) → U17_gga(T30, T12, T33, mul15_in_gga(T30, T12, s(T34)))
U17_gga(T30, T12, T33, mul15_out_gga(T30, T12, s(T34))) → U18_gga(T30, T12, T33, add30_in_gaa(T30, T34, X75))
U18_gga(T30, T12, T33, add30_out_gaa(T30, T34, X75)) → mul1_out_gga(T30, s(T12), T33)
mul1_in_gga(T30, s(T12), 0) → U19_gga(T30, T12, mul15_in_gga(T30, T12, s(T34)))
U19_gga(T30, T12, mul15_out_gga(T30, T12, s(T34))) → U20_gga(T30, T12, add30_in_gag(T30, T34, 0))
U20_gga(T30, T12, add30_out_gag(T30, T34, 0)) → mul1_out_gga(T30, s(T12), 0)
mul1_in_gga(T30, s(T12), s(0)) → U21_gga(T30, T12, mul15_in_gga(T30, T12, s(T34)))
U21_gga(T30, T12, mul15_out_gga(T30, T12, s(T34))) → U22_gga(T30, T12, add30_in_gag(T30, T34, 0))
U22_gga(T30, T12, add30_out_gag(T30, T34, 0)) → mul1_out_gga(T30, s(T12), s(0))
mul1_in_gga(T30, s(T12), s(T37)) → U23_gga(T30, T12, T37, mul15_in_gga(T30, T12, s(T34)))
U23_gga(T30, T12, T37, mul15_out_gga(T30, T12, s(T34))) → U24_gga(T30, T12, T37, add30_in_gaa(T30, T34, T37))
U24_gga(T30, T12, T37, add30_out_gaa(T30, T34, T37)) → mul1_out_gga(T30, s(T12), s(T37))

(39) UsableRulesProof (EQUIVALENT transformation)

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

(40) Obligation:

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

ADD30_IN_GAA(T21, s(T23), 0) → ADD30_IN_GAG(T21, T23, 0)
ADD30_IN_GAG(T21, s(T23), X53) → ADD30_IN_GAA(T21, T23, X52)
ADD30_IN_GAA(T21, s(T23), X53) → ADD30_IN_GAA(T21, T23, X52)
ADD30_IN_GAA(T21, s(T23), s(0)) → ADD30_IN_GAG(T21, T23, 0)
ADD30_IN_GAG(T21, s(T23), 0) → ADD30_IN_GAG(T21, T23, 0)
ADD30_IN_GAA(T21, s(T23), s(T25)) → ADD30_IN_GAA(T21, T23, T25)

(41) PiDPToQDPProof (SOUND transformation)

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

(42) Obligation:

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

ADD30_IN_GAA(T21) → ADD30_IN_GAG(T21, 0)
ADD30_IN_GAG(T21, X53) → ADD30_IN_GAA(T21)
ADD30_IN_GAA(T21) → ADD30_IN_GAA(T21)
ADD30_IN_GAG(T21, 0) → ADD30_IN_GAG(T21, 0)

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

(43) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule ADD30_IN_GAG(T21, X53) → ADD30_IN_GAA(T21) we obtained the following new rules [LPAR04]:

ADD30_IN_GAG(z0, 0) → ADD30_IN_GAA(z0)

(44) Obligation:

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

ADD30_IN_GAA(T21) → ADD30_IN_GAG(T21, 0)
ADD30_IN_GAA(T21) → ADD30_IN_GAA(T21)
ADD30_IN_GAG(T21, 0) → ADD30_IN_GAG(T21, 0)
ADD30_IN_GAG(z0, 0) → ADD30_IN_GAA(z0)

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

(45) NonTerminationProof (EQUIVALENT transformation)

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from ADD30_IN_GAA(T21) to ADD30_IN_GAA(T21).

(46) FALSE

(47) Obligation:

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

ADD30_IN_GAG(T21, s(T23), s(T25)) → ADD30_IN_GAG(T21, T23, T25)

The TRS R consists of the following rules:

mul1_in_gga(T4, 0, 0) → mul1_out_gga(T4, 0, 0)
mul1_in_gga(T8, s(T12), T11) → U15_gga(T8, T12, T11, mul15_in_gga(T8, T12, X15))
mul15_in_gga(T14, 0, 0) → mul15_out_gga(T14, 0, 0)
mul15_in_gga(T15, s(T17), X34) → U1_gga(T15, T17, X34, mul15_in_gga(T15, T17, X33))
mul15_in_gga(T20, s(T17), T20) → U2_gga(T20, T17, mul15_in_ggg(T20, T17, 0))
mul15_in_ggg(T14, 0, 0) → mul15_out_ggg(T14, 0, 0)
mul15_in_ggg(T15, s(T17), X34) → U1_ggg(T15, T17, X34, mul15_in_gga(T15, T17, X33))
mul15_in_gga(T21, s(T17), X53) → U3_gga(T21, T17, X53, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), 0) → U5_gga(T21, T17, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), s(0)) → U7_gga(T21, T17, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), s(T25)) → U9_gga(T21, T17, T25, mul15_in_gga(T21, T17, s(T23)))
U9_gga(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → U10_gga(T21, T17, T25, add30_in_gaa(T21, T23, T25))
add30_in_gaa(T20, 0, T20) → add30_out_gaa(T20, 0, T20)
add30_in_gaa(T21, s(T23), X53) → U11_gaa(T21, T23, X53, add30_in_gaa(T21, T23, X52))
add30_in_gaa(T21, s(T23), 0) → U12_gaa(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T20, 0, T20) → add30_out_gag(T20, 0, T20)
add30_in_gag(T21, s(T23), X53) → U11_gag(T21, T23, X53, add30_in_gaa(T21, T23, X52))
add30_in_gaa(T21, s(T23), s(0)) → U13_gaa(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), 0) → U12_gag(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), s(0)) → U13_gag(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), s(T25)) → U14_gag(T21, T23, T25, add30_in_gag(T21, T23, T25))
U14_gag(T21, T23, T25, add30_out_gag(T21, T23, T25)) → add30_out_gag(T21, s(T23), s(T25))
U13_gag(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gag(T21, s(T23), s(0))
U12_gag(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gag(T21, s(T23), 0)
U13_gaa(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gaa(T21, s(T23), s(0))
add30_in_gaa(T21, s(T23), s(T25)) → U14_gaa(T21, T23, T25, add30_in_gaa(T21, T23, T25))
U14_gaa(T21, T23, T25, add30_out_gaa(T21, T23, T25)) → add30_out_gaa(T21, s(T23), s(T25))
U11_gag(T21, T23, X53, add30_out_gaa(T21, T23, X52)) → add30_out_gag(T21, s(T23), X53)
U12_gaa(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gaa(T21, s(T23), 0)
U11_gaa(T21, T23, X53, add30_out_gaa(T21, T23, X52)) → add30_out_gaa(T21, s(T23), X53)
U10_gga(T21, T17, T25, add30_out_gaa(T21, T23, T25)) → mul15_out_gga(T21, s(T17), s(T25))
U7_gga(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U8_gga(T21, T17, add30_in_gag(T21, T23, 0))
U8_gga(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_gga(T21, s(T17), s(0))
U5_gga(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U6_gga(T21, T17, add30_in_gag(T21, T23, 0))
U6_gga(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_gga(T21, s(T17), 0)
U3_gga(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → U4_gga(T21, T17, X53, add30_in_gaa(T21, T23, X52))
U4_gga(T21, T17, X53, add30_out_gaa(T21, T23, X52)) → mul15_out_gga(T21, s(T17), X53)
U1_ggg(T15, T17, X34, mul15_out_gga(T15, T17, X33)) → mul15_out_ggg(T15, s(T17), X34)
mul15_in_ggg(T20, s(T17), T20) → U2_ggg(T20, T17, mul15_in_ggg(T20, T17, 0))
mul15_in_ggg(T21, s(T17), X53) → U3_ggg(T21, T17, X53, mul15_in_gga(T21, T17, s(T23)))
U3_ggg(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → U4_ggg(T21, T17, X53, add30_in_gaa(T21, T23, X52))
U4_ggg(T21, T17, X53, add30_out_gaa(T21, T23, X52)) → mul15_out_ggg(T21, s(T17), X53)
mul15_in_ggg(T21, s(T17), 0) → U5_ggg(T21, T17, mul15_in_gga(T21, T17, s(T23)))
U5_ggg(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U6_ggg(T21, T17, add30_in_gag(T21, T23, 0))
U6_ggg(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_ggg(T21, s(T17), 0)
mul15_in_ggg(T21, s(T17), s(0)) → U7_ggg(T21, T17, mul15_in_gga(T21, T17, s(T23)))
U7_ggg(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U8_ggg(T21, T17, add30_in_gag(T21, T23, 0))
U8_ggg(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_ggg(T21, s(T17), s(0))
mul15_in_ggg(T21, s(T17), s(T25)) → U9_ggg(T21, T17, T25, mul15_in_gga(T21, T17, s(T23)))
U9_ggg(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → U10_ggg(T21, T17, T25, add30_in_gag(T21, T23, T25))
U10_ggg(T21, T17, T25, add30_out_gag(T21, T23, T25)) → mul15_out_ggg(T21, s(T17), s(T25))
U2_ggg(T20, T17, mul15_out_ggg(T20, T17, 0)) → mul15_out_ggg(T20, s(T17), T20)
U2_gga(T20, T17, mul15_out_ggg(T20, T17, 0)) → mul15_out_gga(T20, s(T17), T20)
U1_gga(T15, T17, X34, mul15_out_gga(T15, T17, X33)) → mul15_out_gga(T15, s(T17), X34)
U15_gga(T8, T12, T11, mul15_out_gga(T8, T12, X15)) → mul1_out_gga(T8, s(T12), T11)
mul1_in_gga(T29, s(T12), T29) → U16_gga(T29, T12, mul15_in_ggg(T29, T12, 0))
U16_gga(T29, T12, mul15_out_ggg(T29, T12, 0)) → mul1_out_gga(T29, s(T12), T29)
mul1_in_gga(T30, s(T12), T33) → U17_gga(T30, T12, T33, mul15_in_gga(T30, T12, s(T34)))
U17_gga(T30, T12, T33, mul15_out_gga(T30, T12, s(T34))) → U18_gga(T30, T12, T33, add30_in_gaa(T30, T34, X75))
U18_gga(T30, T12, T33, add30_out_gaa(T30, T34, X75)) → mul1_out_gga(T30, s(T12), T33)
mul1_in_gga(T30, s(T12), 0) → U19_gga(T30, T12, mul15_in_gga(T30, T12, s(T34)))
U19_gga(T30, T12, mul15_out_gga(T30, T12, s(T34))) → U20_gga(T30, T12, add30_in_gag(T30, T34, 0))
U20_gga(T30, T12, add30_out_gag(T30, T34, 0)) → mul1_out_gga(T30, s(T12), 0)
mul1_in_gga(T30, s(T12), s(0)) → U21_gga(T30, T12, mul15_in_gga(T30, T12, s(T34)))
U21_gga(T30, T12, mul15_out_gga(T30, T12, s(T34))) → U22_gga(T30, T12, add30_in_gag(T30, T34, 0))
U22_gga(T30, T12, add30_out_gag(T30, T34, 0)) → mul1_out_gga(T30, s(T12), s(0))
mul1_in_gga(T30, s(T12), s(T37)) → U23_gga(T30, T12, T37, mul15_in_gga(T30, T12, s(T34)))
U23_gga(T30, T12, T37, mul15_out_gga(T30, T12, s(T34))) → U24_gga(T30, T12, T37, add30_in_gaa(T30, T34, T37))
U24_gga(T30, T12, T37, add30_out_gaa(T30, T34, T37)) → mul1_out_gga(T30, s(T12), s(T37))

(48) UsableRulesProof (EQUIVALENT transformation)

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

(49) Obligation:

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

ADD30_IN_GAG(T21, s(T23), s(T25)) → ADD30_IN_GAG(T21, T23, T25)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
ADD30_IN_GAG(x1, x2, x3) = ADD30_IN_GAG(x1, x3)

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

(50) PiDPToQDPProof (SOUND transformation)

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

(51) Obligation:

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

ADD30_IN_GAG(T21, s(T25)) → ADD30_IN_GAG(T21, T25)

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

(52) 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_GAG(T21, s(T25)) → ADD30_IN_GAG(T21, T25)
The graph contains the following edges 1 >= 1, 2 > 2

(53) TRUE

(54) Obligation:

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

MUL15_IN_GGA(T20, s(T17), T20) → MUL15_IN_GGG(T20, T17, 0)
MUL15_IN_GGG(T15, s(T17), X34) → MUL15_IN_GGA(T15, T17, X33)
MUL15_IN_GGA(T15, s(T17), X34) → MUL15_IN_GGA(T15, T17, X33)
MUL15_IN_GGA(T21, s(T17), X53) → MUL15_IN_GGA(T21, T17, s(T23))
MUL15_IN_GGA(T21, s(T17), s(0)) → MUL15_IN_GGA(T21, T17, s(T23))
MUL15_IN_GGA(T21, s(T17), s(T25)) → MUL15_IN_GGA(T21, T17, s(T23))
MUL15_IN_GGA(T21, s(T17), 0) → MUL15_IN_GGA(T21, T17, s(T23))
MUL15_IN_GGG(T20, s(T17), T20) → MUL15_IN_GGG(T20, T17, 0)
MUL15_IN_GGG(T21, s(T17), X53) → MUL15_IN_GGA(T21, T17, s(T23))
MUL15_IN_GGG(T21, s(T17), 0) → MUL15_IN_GGA(T21, T17, s(T23))

The TRS R consists of the following rules:

mul1_in_gga(T4, 0, 0) → mul1_out_gga(T4, 0, 0)
mul1_in_gga(T8, s(T12), T11) → U15_gga(T8, T12, T11, mul15_in_gga(T8, T12, X15))
mul15_in_gga(T14, 0, 0) → mul15_out_gga(T14, 0, 0)
mul15_in_gga(T15, s(T17), X34) → U1_gga(T15, T17, X34, mul15_in_gga(T15, T17, X33))
mul15_in_gga(T20, s(T17), T20) → U2_gga(T20, T17, mul15_in_ggg(T20, T17, 0))
mul15_in_ggg(T14, 0, 0) → mul15_out_ggg(T14, 0, 0)
mul15_in_ggg(T15, s(T17), X34) → U1_ggg(T15, T17, X34, mul15_in_gga(T15, T17, X33))
mul15_in_gga(T21, s(T17), X53) → U3_gga(T21, T17, X53, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), 0) → U5_gga(T21, T17, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), s(0)) → U7_gga(T21, T17, mul15_in_gga(T21, T17, s(T23)))
mul15_in_gga(T21, s(T17), s(T25)) → U9_gga(T21, T17, T25, mul15_in_gga(T21, T17, s(T23)))
U9_gga(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → U10_gga(T21, T17, T25, add30_in_gaa(T21, T23, T25))
add30_in_gaa(T20, 0, T20) → add30_out_gaa(T20, 0, T20)
add30_in_gaa(T21, s(T23), X53) → U11_gaa(T21, T23, X53, add30_in_gaa(T21, T23, X52))
add30_in_gaa(T21, s(T23), 0) → U12_gaa(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T20, 0, T20) → add30_out_gag(T20, 0, T20)
add30_in_gag(T21, s(T23), X53) → U11_gag(T21, T23, X53, add30_in_gaa(T21, T23, X52))
add30_in_gaa(T21, s(T23), s(0)) → U13_gaa(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), 0) → U12_gag(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), s(0)) → U13_gag(T21, T23, add30_in_gag(T21, T23, 0))
add30_in_gag(T21, s(T23), s(T25)) → U14_gag(T21, T23, T25, add30_in_gag(T21, T23, T25))
U14_gag(T21, T23, T25, add30_out_gag(T21, T23, T25)) → add30_out_gag(T21, s(T23), s(T25))
U13_gag(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gag(T21, s(T23), s(0))
U12_gag(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gag(T21, s(T23), 0)
U13_gaa(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gaa(T21, s(T23), s(0))
add30_in_gaa(T21, s(T23), s(T25)) → U14_gaa(T21, T23, T25, add30_in_gaa(T21, T23, T25))
U14_gaa(T21, T23, T25, add30_out_gaa(T21, T23, T25)) → add30_out_gaa(T21, s(T23), s(T25))
U11_gag(T21, T23, X53, add30_out_gaa(T21, T23, X52)) → add30_out_gag(T21, s(T23), X53)
U12_gaa(T21, T23, add30_out_gag(T21, T23, 0)) → add30_out_gaa(T21, s(T23), 0)
U11_gaa(T21, T23, X53, add30_out_gaa(T21, T23, X52)) → add30_out_gaa(T21, s(T23), X53)
U10_gga(T21, T17, T25, add30_out_gaa(T21, T23, T25)) → mul15_out_gga(T21, s(T17), s(T25))
U7_gga(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U8_gga(T21, T17, add30_in_gag(T21, T23, 0))
U8_gga(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_gga(T21, s(T17), s(0))
U5_gga(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U6_gga(T21, T17, add30_in_gag(T21, T23, 0))
U6_gga(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_gga(T21, s(T17), 0)
U3_gga(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → U4_gga(T21, T17, X53, add30_in_gaa(T21, T23, X52))
U4_gga(T21, T17, X53, add30_out_gaa(T21, T23, X52)) → mul15_out_gga(T21, s(T17), X53)
U1_ggg(T15, T17, X34, mul15_out_gga(T15, T17, X33)) → mul15_out_ggg(T15, s(T17), X34)
mul15_in_ggg(T20, s(T17), T20) → U2_ggg(T20, T17, mul15_in_ggg(T20, T17, 0))
mul15_in_ggg(T21, s(T17), X53) → U3_ggg(T21, T17, X53, mul15_in_gga(T21, T17, s(T23)))
U3_ggg(T21, T17, X53, mul15_out_gga(T21, T17, s(T23))) → U4_ggg(T21, T17, X53, add30_in_gaa(T21, T23, X52))
U4_ggg(T21, T17, X53, add30_out_gaa(T21, T23, X52)) → mul15_out_ggg(T21, s(T17), X53)
mul15_in_ggg(T21, s(T17), 0) → U5_ggg(T21, T17, mul15_in_gga(T21, T17, s(T23)))
U5_ggg(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U6_ggg(T21, T17, add30_in_gag(T21, T23, 0))
U6_ggg(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_ggg(T21, s(T17), 0)
mul15_in_ggg(T21, s(T17), s(0)) → U7_ggg(T21, T17, mul15_in_gga(T21, T17, s(T23)))
U7_ggg(T21, T17, mul15_out_gga(T21, T17, s(T23))) → U8_ggg(T21, T17, add30_in_gag(T21, T23, 0))
U8_ggg(T21, T17, add30_out_gag(T21, T23, 0)) → mul15_out_ggg(T21, s(T17), s(0))
mul15_in_ggg(T21, s(T17), s(T25)) → U9_ggg(T21, T17, T25, mul15_in_gga(T21, T17, s(T23)))
U9_ggg(T21, T17, T25, mul15_out_gga(T21, T17, s(T23))) → U10_ggg(T21, T17, T25, add30_in_gag(T21, T23, T25))
U10_ggg(T21, T17, T25, add30_out_gag(T21, T23, T25)) → mul15_out_ggg(T21, s(T17), s(T25))
U2_ggg(T20, T17, mul15_out_ggg(T20, T17, 0)) → mul15_out_ggg(T20, s(T17), T20)
U2_gga(T20, T17, mul15_out_ggg(T20, T17, 0)) → mul15_out_gga(T20, s(T17), T20)
U1_gga(T15, T17, X34, mul15_out_gga(T15, T17, X33)) → mul15_out_gga(T15, s(T17), X34)
U15_gga(T8, T12, T11, mul15_out_gga(T8, T12, X15)) → mul1_out_gga(T8, s(T12), T11)
mul1_in_gga(T29, s(T12), T29) → U16_gga(T29, T12, mul15_in_ggg(T29, T12, 0))
U16_gga(T29, T12, mul15_out_ggg(T29, T12, 0)) → mul1_out_gga(T29, s(T12), T29)
mul1_in_gga(T30, s(T12), T33) → U17_gga(T30, T12, T33, mul15_in_gga(T30, T12, s(T34)))
U17_gga(T30, T12, T33, mul15_out_gga(T30, T12, s(T34))) → U18_gga(T30, T12, T33, add30_in_gaa(T30, T34, X75))
U18_gga(T30, T12, T33, add30_out_gaa(T30, T34, X75)) → mul1_out_gga(T30, s(T12), T33)
mul1_in_gga(T30, s(T12), 0) → U19_gga(T30, T12, mul15_in_gga(T30, T12, s(T34)))
U19_gga(T30, T12, mul15_out_gga(T30, T12, s(T34))) → U20_gga(T30, T12, add30_in_gag(T30, T34, 0))
U20_gga(T30, T12, add30_out_gag(T30, T34, 0)) → mul1_out_gga(T30, s(T12), 0)
mul1_in_gga(T30, s(T12), s(0)) → U21_gga(T30, T12, mul15_in_gga(T30, T12, s(T34)))
U21_gga(T30, T12, mul15_out_gga(T30, T12, s(T34))) → U22_gga(T30, T12, add30_in_gag(T30, T34, 0))
U22_gga(T30, T12, add30_out_gag(T30, T34, 0)) → mul1_out_gga(T30, s(T12), s(0))
mul1_in_gga(T30, s(T12), s(T37)) → U23_gga(T30, T12, T37, mul15_in_gga(T30, T12, s(T34)))
U23_gga(T30, T12, T37, mul15_out_gga(T30, T12, s(T34))) → U24_gga(T30, T12, T37, add30_in_gaa(T30, T34, T37))
U24_gga(T30, T12, T37, add30_out_gaa(T30, T34, T37)) → mul1_out_gga(T30, s(T12), s(T37))

(55) UsableRulesProof (EQUIVALENT transformation)

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

(56) Obligation:

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

MUL15_IN_GGA(T20, s(T17), T20) → MUL15_IN_GGG(T20, T17, 0)
MUL15_IN_GGG(T15, s(T17), X34) → MUL15_IN_GGA(T15, T17, X33)
MUL15_IN_GGA(T15, s(T17), X34) → MUL15_IN_GGA(T15, T17, X33)
MUL15_IN_GGA(T21, s(T17), X53) → MUL15_IN_GGA(T21, T17, s(T23))
MUL15_IN_GGA(T21, s(T17), s(0)) → MUL15_IN_GGA(T21, T17, s(T23))
MUL15_IN_GGA(T21, s(T17), s(T25)) → MUL15_IN_GGA(T21, T17, s(T23))
MUL15_IN_GGA(T21, s(T17), 0) → MUL15_IN_GGA(T21, T17, s(T23))
MUL15_IN_GGG(T20, s(T17), T20) → MUL15_IN_GGG(T20, T17, 0)
MUL15_IN_GGG(T21, s(T17), X53) → MUL15_IN_GGA(T21, T17, s(T23))
MUL15_IN_GGG(T21, s(T17), 0) → MUL15_IN_GGA(T21, T17, s(T23))

(57) PiDPToQDPProof (SOUND transformation)

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

(58) Obligation:

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

MUL15_IN_GGA(T20, s(T17)) → MUL15_IN_GGG(T20, T17, 0)
MUL15_IN_GGG(T15, s(T17), X34) → MUL15_IN_GGA(T15, T17)
MUL15_IN_GGA(T15, s(T17)) → MUL15_IN_GGA(T15, T17)
MUL15_IN_GGG(T20, s(T17), T20) → MUL15_IN_GGG(T20, T17, 0)
MUL15_IN_GGG(T21, s(T17), 0) → MUL15_IN_GGA(T21, T17)

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

(59) 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_GGG(T20, s(T17), T20) → MUL15_IN_GGG(T20, T17, 0)
The graph contains the following edges 1 >= 1, 3 >= 1, 2 > 2
MUL15_IN_GGA(T15, s(T17)) → MUL15_IN_GGA(T15, T17)
The graph contains the following edges 1 >= 1, 2 > 2
MUL15_IN_GGA(T20, s(T17)) → MUL15_IN_GGG(T20, T17, 0)
The graph contains the following edges 1 >= 1, 2 > 2
MUL15_IN_GGG(T15, s(T17), X34) → MUL15_IN_GGA(T15, T17)
The graph contains the following edges 1 >= 1, 2 > 2
MUL15_IN_GGG(T21, s(T17), 0) → MUL15_IN_GGA(T21, T17)
The graph contains the following edges 1 >= 1, 2 > 2

(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) Instantiation (EQUIVALENT transformation)

(15) Obligation:

(16) NonTerminationProof (EQUIVALENT transformation)

(17) FALSE

(18) Obligation:

(19) UsableRulesProof (EQUIVALENT transformation)

(20) Obligation:

(21) PiDPToQDPProof (SOUND transformation)

(22) Obligation:

(23) QDPSizeChangeProof (EQUIVALENT transformation)

(24) TRUE

(25) Obligation:

(26) UsableRulesProof (EQUIVALENT transformation)

(27) Obligation:

(28) PiDPToQDPProof (SOUND transformation)

(29) Obligation:

(30) QDPSizeChangeProof (EQUIVALENT transformation)

(31) TRUE

(32) PrologToPiTRSProof (SOUND transformation)

(33) Obligation:

(34) DependencyPairsProof (EQUIVALENT transformation)

(35) Obligation:

(36) DependencyGraphProof (EQUIVALENT transformation)

(37) Complex Obligation (AND)

(38) Obligation:

(39) UsableRulesProof (EQUIVALENT transformation)

(40) Obligation:

(41) PiDPToQDPProof (SOUND transformation)

(42) Obligation:

(43) Instantiation (EQUIVALENT transformation)

(44) Obligation:

(45) NonTerminationProof (EQUIVALENT transformation)

(46) FALSE

(47) Obligation:

(48) UsableRulesProof (EQUIVALENT transformation)

(49) Obligation:

(50) PiDPToQDPProof (SOUND transformation)

(51) Obligation:

(52) QDPSizeChangeProof (EQUIVALENT transformation)

(53) TRUE

(54) Obligation:

(55) UsableRulesProof (EQUIVALENT transformation)

(56) Obligation:

(57) PiDPToQDPProof (SOUND transformation)

(58) Obligation:

(59) QDPSizeChangeProof (EQUIVALENT transformation)

(60) TRUE