Left Termination of the query pattern mult_in_3(g, g, a) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

mult(0, Y, 0).
mult(s(X), Y, Z) :- ','(mult(X, Y, Z1), add(Z1, Y, Z)).
add(0, Y, Y).
add(s(X), Y, s(Z)) :- add(X, Y, Z).

Queries:

mult(g,g,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: mult(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: mult(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | mult(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | mult(0, T5, T3), SCOPE: 1, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: mult(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground\nmult(T1, T2, T3) !=? mult(0, X2, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: mult(0, T5, T3), SCOPE: 1, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(mult(T9, T10, X13), add(X13, T10, T12))\n\nT9 is ground\nT10 is ground\nX13 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(mult(T9, T10, X13), add(X13, T10, T12)), SCOPE: 2, CLAUSE: 0 | ','(mult(T9, T10, X13), add(X13, T10, T12)), SCOPE: 2, CLAUSE: 1\n\nT9 is ground\nT10 is ground\nX13 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(mult(T9, T10, X13), add(X13, T10, T12)), SCOPE: 2, CLAUSE: 0\n\nT9 is ground\nT10 is ground\nX13 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(mult(T9, T10, X13), add(X13, T10, T12)), SCOPE: 2, CLAUSE: 1\n\nT9 is ground\nT10 is ground\nX13 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: add(0, T17, T12)\n\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: add(0, T17, T12), SCOPE: 3, CLAUSE: 2 | add(0, T17, T12), SCOPE: 3, CLAUSE: 3\n\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: add(0, T17, T12), SCOPE: 3, CLAUSE: 2\n\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: add(0, T17, T12), SCOPE: 3, CLAUSE: 3\n\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(mult(T29, T30, X46), ','(add(X46, T30, X47), add(X47, T30, T12)))\n\nT29 is ground\nT30 is ground\nX47 is free; \nX46 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: mult(T29, T30, X46)\n\nT29 is ground\nT30 is ground\nX46 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(add(T33, T30, X47), add(X47, T30, T12))\n\nT30 is ground\nT33 is ground\nX47 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: mult(T29, T30, X46), SCOPE: 4, CLAUSE: 0 | mult(T29, T30, X46), SCOPE: 4, CLAUSE: 1\n\nT29 is ground\nT30 is ground\nX46 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: mult(T29, T30, X46), SCOPE: 4, CLAUSE: 0\n\nT29 is ground\nT30 is ground\nX46 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: mult(T29, T30, X46), SCOPE: 4, CLAUSE: 1\n\nT29 is ground\nT30 is ground\nX46 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(mult(T45, T46, X73), add(X73, T46, X74))\n\nT45 is ground\nT46 is ground\nX74 is free; \nX73 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: mult(T45, T46, X73)\n\nT45 is ground\nT46 is ground\nX73 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: add(T49, T46, X74)\n\nT46 is ground\nT49 is ground\nX74 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: add(T49, T46, X74), SCOPE: 5, CLAUSE: 2 | add(T49, T46, X74), SCOPE: 5, CLAUSE: 3\n\nT46 is ground\nT49 is ground\nX74 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: add(T49, T46, X74), SCOPE: 5, CLAUSE: 2\n\nT46 is ground\nT49 is ground\nX74 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: add(T49, T46, X74), SCOPE: 5, CLAUSE: 3\n\nT46 is ground\nT49 is ground\nX74 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: add(T63, T64, X101)\n\nT63 is ground\nT64 is ground\nX101 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: add(T33, T30, X47)\n\nT30 is ground\nT33 is ground\nX47 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: add(T69, T30, T12)\n\nT30 is ground\nT69 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: add(T69, T30, T12), SCOPE: 6, CLAUSE: 2 | add(T69, T30, T12), SCOPE: 6, CLAUSE: 3\n\nT30 is ground\nT69 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: add(T69, T30, T12), SCOPE: 6, CLAUSE: 2\n\nT30 is ground\nT69 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: add(T69, T30, T12), SCOPE: 6, CLAUSE: 3\n\nT30 is ground\nT69 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: add(T85, T86, T88)\n\nT85 is ground\nT86 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 53 [arrowhead = none ];
53 -> 2;

54 [label="EVAL with clause\nmult(0, X2, 0).\nand substitution\nT1 -> 0\nT2 -> T5\nX2 -> T5\nT3 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 54 [arrowhead = none ];
54 -> 3;

55 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 55 [arrowhead = none ];
55 -> 4;

56 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 56 [arrowhead = none ];
56 -> 5;

57 [label="EVAL with clause\nmult(s(X10), X11, X12) :- ','(mult(X10, X11, X13), add(X13, X11, X12)).\nand substitution\nX10 -> T9\nT1 -> s(T9)\nT2 -> T10\nX11 -> T10\nT3 -> T12\nX12 -> T12\nT11 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 57 [arrowhead = none ];
57 -> 7;

58 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 58 [arrowhead = none ];
58 -> 8;

59 [label="BACKTRACK\nfor clause: mult(s(X), Y, Z) :- ','(mult(X, Y, Z1), add(Z1, Y, Z))because of non-unification", fontsize=14, style = filled, fillcolor = white];
5 -> 59 [arrowhead = none ];
59 -> 6;

60 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 60 [arrowhead = none ];
60 -> 9;

61 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 61 [arrowhead = none ];
61 -> 10;

62 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 62 [arrowhead = none ];
62 -> 11;

63 [label="EVAL with clause\nmult(0, X21, 0).\nand substitution\nT9 -> 0\nT10 -> T17\nX21 -> T17\nX13 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 63 [arrowhead = none ];
63 -> 12;

64 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 64 [arrowhead = none ];
64 -> 13;

65 [label="EVAL with clause\nmult(s(X43), X44, X45) :- ','(mult(X43, X44, X46), add(X46, X44, X45)).\nand substitution\nX43 -> T29\nT9 -> s(T29)\nT10 -> T30\nX44 -> T30\nX13 -> X47\nX45 -> X47", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 65 [arrowhead = none ];
65 -> 21;

66 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 66 [arrowhead = none ];
66 -> 22;

67 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 67 [arrowhead = none ];
67 -> 14;

68 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 68 [arrowhead = none ];
68 -> 15;

69 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 69 [arrowhead = none ];
69 -> 16;

70 [label="EVAL with clause\nadd(0, X29, X29).\nand substitution\nT17 -> T24\nX29 -> T24\nT12 -> T24", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 70 [arrowhead = none ];
70 -> 17;

71 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 71 [arrowhead = none ];
71 -> 18;

72 [label="BACKTRACK\nfor clause: add(s(X), Y, s(Z)) :- add(X, Y, Z)because of non-unification", fontsize=14, style = filled, fillcolor = white];
16 -> 72 [arrowhead = none ];
72 -> 20;

73 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 73 [arrowhead = none ];
73 -> 19;

74 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
21 -> 74 [arrowhead = none ];
74 -> 23;

75 [label="SPLIT 2\nnew knowledge:\nT33 is ground\nreplacements:\nX46 -> T33", fontsize=14, style = filled, fillcolor = violet];
21 -> 75 [arrowhead = none ];
75 -> 24;

76 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 76 [arrowhead = none ];
76 -> 25;

77 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
24 -> 77 [arrowhead = none ];
77 -> 43;

78 [label="SPLIT 2\nnew knowledge:\nT69 is ground\nreplacements:\nX47 -> T69", fontsize=14, style = filled, fillcolor = violet];
24 -> 78 [arrowhead = none ];
78 -> 44;

79 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 79 [arrowhead = none ];
79 -> 26;

80 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 80 [arrowhead = none ];
80 -> 27;

81 [label="EVAL with clause\nmult(0, X59, 0).\nand substitution\nT29 -> 0\nT30 -> T40\nX59 -> T40\nX46 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 81 [arrowhead = none ];
81 -> 28;

82 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 82 [arrowhead = none ];
82 -> 29;

83 [label="EVAL with clause\nmult(s(X70), X71, X72) :- ','(mult(X70, X71, X73), add(X73, X71, X72)).\nand substitution\nX70 -> T45\nT29 -> s(T45)\nT30 -> T46\nX71 -> T46\nX46 -> X74\nX72 -> X74", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 83 [arrowhead = none ];
83 -> 31;

84 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 84 [arrowhead = none ];
84 -> 32;

85 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 85 [arrowhead = none ];
85 -> 30;

86 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
31 -> 86 [arrowhead = none ];
86 -> 33;

87 [label="SPLIT 2\nnew knowledge:\nT49 is ground\nreplacements:\nX73 -> T49", fontsize=14, style = filled, fillcolor = violet];
31 -> 87 [arrowhead = none ];
87 -> 34;

88 [label="INSTANCE with matching:\nT29 -> T45\nT30 -> T46\nX46 -> X73", fontsize=14, style = filled, fillcolor = lightgrey];
33 -> 88 [arrowhead = none , style=dashed];
88 -> 23[style=dashed];

89 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 89 [arrowhead = none ];
89 -> 35;

90 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 90 [arrowhead = none ];
90 -> 36;

91 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 91 [arrowhead = none ];
91 -> 37;

92 [label="EVAL with clause\nadd(0, X88, X88).\nand substitution\nT49 -> 0\nT46 -> T58\nX88 -> T58\nX74 -> T58", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 92 [arrowhead = none ];
92 -> 38;

93 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 93 [arrowhead = none ];
93 -> 39;

94 [label="EVAL with clause\nadd(s(X98), X99, s(X100)) :- add(X98, X99, X100).\nand substitution\nX98 -> T63\nT49 -> s(T63)\nT46 -> T64\nX99 -> T64\nX100 -> X101\nX74 -> s(X101)", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 94 [arrowhead = none ];
94 -> 41;

95 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 95 [arrowhead = none ];
95 -> 42;

96 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 96 [arrowhead = none ];
96 -> 40;

97 [label="INSTANCE with matching:\nT49 -> T63\nT46 -> T64\nX74 -> X101", fontsize=14, style = filled, fillcolor = lightgrey];
41 -> 97 [arrowhead = none , style=dashed];
97 -> 34[style=dashed];

98 [label="INSTANCE with matching:\nT49 -> T33\nT46 -> T30\nX74 -> X47", fontsize=14, style = filled, fillcolor = lightgrey];
43 -> 98 [arrowhead = none , style=dashed];
98 -> 34[style=dashed];

99 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
44 -> 99 [arrowhead = none ];
99 -> 45;

100 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 100 [arrowhead = none ];
100 -> 46;

101 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 101 [arrowhead = none ];
101 -> 47;

102 [label="EVAL with clause\nadd(0, X117, X117).\nand substitution\nT69 -> 0\nT30 -> T78\nX117 -> T78\nT12 -> T78", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 102 [arrowhead = none ];
102 -> 48;

103 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 103 [arrowhead = none ];
103 -> 49;

104 [label="EVAL with clause\nadd(s(X125), X126, s(X127)) :- add(X125, X126, X127).\nand substitution\nX125 -> T85\nT69 -> s(T85)\nT30 -> T86\nX126 -> T86\nX127 -> T88\nT12 -> s(T88)\nT87 -> T88", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 104 [arrowhead = none ];
104 -> 51;

105 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 105 [arrowhead = none ];
105 -> 52;

106 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 106 [arrowhead = none ];
106 -> 50;

107 [label="INSTANCE with matching:\nT69 -> T85\nT30 -> T86\nT12 -> T88", fontsize=14, style = filled, fillcolor = lightgrey];
51 -> 107 [arrowhead = none , style=dashed];
107 -> 44[style=dashed];

}

(2) Obligation:

Triples:

mult23(s(T45), T46, X74) :- mult23(T45, T46, X73).
mult23(s(T45), T46, X74) :- ','(multc23(T45, T46, T49), add34(T49, T46, X74)).
add34(s(T63), T64, s(X101)) :- add34(T63, T64, X101).
add44(s(T85), T86, s(T88)) :- add44(T85, T86, T88).
mult1(s(s(T29)), T30, T12) :- mult23(T29, T30, X46).
mult1(s(s(T29)), T30, T12) :- ','(multc23(T29, T30, T33), add34(T33, T30, X47)).
mult1(s(s(T29)), T30, T12) :- ','(multc23(T29, T30, T33), ','(addc34(T33, T30, T69), add44(T69, T30, T12))).

Clauses:

multc23(0, T40, 0).
multc23(s(T45), T46, X74) :- ','(multc23(T45, T46, T49), addc34(T49, T46, X74)).
addc34(0, T58, T58).
addc34(s(T63), T64, s(X101)) :- addc34(T63, T64, X101).
addc44(0, T78, T78).
addc44(s(T85), T86, s(T88)) :- addc44(T85, T86, T88).

Afs:

mult1(x1, x2, x3)  =  mult1(x1, x2)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
mult1_in: (b,b,f)
mult23_in: (b,b,f)
multc23_in: (b,b,f)
addc34_in: (b,b,f)
add34_in: (b,b,f)
add44_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

MULT1_IN_GGA(s(s(T29)), T30, T12) → U6_GGA(T29, T30, T12, mult23_in_gga(T29, T30, X46))
MULT1_IN_GGA(s(s(T29)), T30, T12) → MULT23_IN_GGA(T29, T30, X46)
MULT23_IN_GGA(s(T45), T46, X74) → U1_GGA(T45, T46, X74, mult23_in_gga(T45, T46, X73))
MULT23_IN_GGA(s(T45), T46, X74) → MULT23_IN_GGA(T45, T46, X73)
MULT23_IN_GGA(s(T45), T46, X74) → U2_GGA(T45, T46, X74, multc23_in_gga(T45, T46, T49))
U2_GGA(T45, T46, X74, multc23_out_gga(T45, T46, T49)) → U3_GGA(T45, T46, X74, add34_in_gga(T49, T46, X74))
U2_GGA(T45, T46, X74, multc23_out_gga(T45, T46, T49)) → ADD34_IN_GGA(T49, T46, X74)
ADD34_IN_GGA(s(T63), T64, s(X101)) → U4_GGA(T63, T64, X101, add34_in_gga(T63, T64, X101))
ADD34_IN_GGA(s(T63), T64, s(X101)) → ADD34_IN_GGA(T63, T64, X101)
MULT1_IN_GGA(s(s(T29)), T30, T12) → U7_GGA(T29, T30, T12, multc23_in_gga(T29, T30, T33))
U7_GGA(T29, T30, T12, multc23_out_gga(T29, T30, T33)) → U8_GGA(T29, T30, T12, add34_in_gga(T33, T30, X47))
U7_GGA(T29, T30, T12, multc23_out_gga(T29, T30, T33)) → ADD34_IN_GGA(T33, T30, X47)
U7_GGA(T29, T30, T12, multc23_out_gga(T29, T30, T33)) → U9_GGA(T29, T30, T12, addc34_in_gga(T33, T30, T69))
U9_GGA(T29, T30, T12, addc34_out_gga(T33, T30, T69)) → U10_GGA(T29, T30, T12, add44_in_gga(T69, T30, T12))
U9_GGA(T29, T30, T12, addc34_out_gga(T33, T30, T69)) → ADD44_IN_GGA(T69, T30, T12)
ADD44_IN_GGA(s(T85), T86, s(T88)) → U5_GGA(T85, T86, T88, add44_in_gga(T85, T86, T88))
ADD44_IN_GGA(s(T85), T86, s(T88)) → ADD44_IN_GGA(T85, T86, T88)

The TRS R consists of the following rules:

multc23_in_gga(0, T40, 0) → multc23_out_gga(0, T40, 0)
multc23_in_gga(s(T45), T46, X74) → U12_gga(T45, T46, X74, multc23_in_gga(T45, T46, T49))
U12_gga(T45, T46, X74, multc23_out_gga(T45, T46, T49)) → U13_gga(T45, T46, X74, addc34_in_gga(T49, T46, X74))
addc34_in_gga(0, T58, T58) → addc34_out_gga(0, T58, T58)
addc34_in_gga(s(T63), T64, s(X101)) → U14_gga(T63, T64, X101, addc34_in_gga(T63, T64, X101))
U14_gga(T63, T64, X101, addc34_out_gga(T63, T64, X101)) → addc34_out_gga(s(T63), T64, s(X101))
U13_gga(T45, T46, X74, addc34_out_gga(T49, T46, X74)) → multc23_out_gga(s(T45), T46, X74)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
mult23_in_gga(x1, x2, x3)  =  mult23_in_gga(x1, x2)
multc23_in_gga(x1, x2, x3)  =  multc23_in_gga(x1, x2)
0  =  0
multc23_out_gga(x1, x2, x3)  =  multc23_out_gga(x1, x2, x3)
U12_gga(x1, x2, x3, x4)  =  U12_gga(x1, x2, x4)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
addc34_in_gga(x1, x2, x3)  =  addc34_in_gga(x1, x2)
addc34_out_gga(x1, x2, x3)  =  addc34_out_gga(x1, x2, x3)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x1, x2, x4)
add34_in_gga(x1, x2, x3)  =  add34_in_gga(x1, x2)
add44_in_gga(x1, x2, x3)  =  add44_in_gga(x1, x2)
MULT1_IN_GGA(x1, x2, x3)  =  MULT1_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x1, x2, x4)
MULT23_IN_GGA(x1, x2, x3)  =  MULT23_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x1, x2, x4)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)
ADD34_IN_GGA(x1, x2, x3)  =  ADD34_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)
U7_GGA(x1, x2, x3, x4)  =  U7_GGA(x1, x2, x4)
U8_GGA(x1, x2, x3, x4)  =  U8_GGA(x1, x2, x4)
U9_GGA(x1, x2, x3, x4)  =  U9_GGA(x1, x2, x4)
U10_GGA(x1, x2, x3, x4)  =  U10_GGA(x1, x2, x4)
ADD44_IN_GGA(x1, x2, x3)  =  ADD44_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x1, x2, x4)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

MULT1_IN_GGA(s(s(T29)), T30, T12) → U6_GGA(T29, T30, T12, mult23_in_gga(T29, T30, X46))
MULT1_IN_GGA(s(s(T29)), T30, T12) → MULT23_IN_GGA(T29, T30, X46)
MULT23_IN_GGA(s(T45), T46, X74) → U1_GGA(T45, T46, X74, mult23_in_gga(T45, T46, X73))
MULT23_IN_GGA(s(T45), T46, X74) → MULT23_IN_GGA(T45, T46, X73)
MULT23_IN_GGA(s(T45), T46, X74) → U2_GGA(T45, T46, X74, multc23_in_gga(T45, T46, T49))
U2_GGA(T45, T46, X74, multc23_out_gga(T45, T46, T49)) → U3_GGA(T45, T46, X74, add34_in_gga(T49, T46, X74))
U2_GGA(T45, T46, X74, multc23_out_gga(T45, T46, T49)) → ADD34_IN_GGA(T49, T46, X74)
ADD34_IN_GGA(s(T63), T64, s(X101)) → U4_GGA(T63, T64, X101, add34_in_gga(T63, T64, X101))
ADD34_IN_GGA(s(T63), T64, s(X101)) → ADD34_IN_GGA(T63, T64, X101)
MULT1_IN_GGA(s(s(T29)), T30, T12) → U7_GGA(T29, T30, T12, multc23_in_gga(T29, T30, T33))
U7_GGA(T29, T30, T12, multc23_out_gga(T29, T30, T33)) → U8_GGA(T29, T30, T12, add34_in_gga(T33, T30, X47))
U7_GGA(T29, T30, T12, multc23_out_gga(T29, T30, T33)) → ADD34_IN_GGA(T33, T30, X47)
U7_GGA(T29, T30, T12, multc23_out_gga(T29, T30, T33)) → U9_GGA(T29, T30, T12, addc34_in_gga(T33, T30, T69))
U9_GGA(T29, T30, T12, addc34_out_gga(T33, T30, T69)) → U10_GGA(T29, T30, T12, add44_in_gga(T69, T30, T12))
U9_GGA(T29, T30, T12, addc34_out_gga(T33, T30, T69)) → ADD44_IN_GGA(T69, T30, T12)
ADD44_IN_GGA(s(T85), T86, s(T88)) → U5_GGA(T85, T86, T88, add44_in_gga(T85, T86, T88))
ADD44_IN_GGA(s(T85), T86, s(T88)) → ADD44_IN_GGA(T85, T86, T88)

The TRS R consists of the following rules:

multc23_in_gga(0, T40, 0) → multc23_out_gga(0, T40, 0)
multc23_in_gga(s(T45), T46, X74) → U12_gga(T45, T46, X74, multc23_in_gga(T45, T46, T49))
U12_gga(T45, T46, X74, multc23_out_gga(T45, T46, T49)) → U13_gga(T45, T46, X74, addc34_in_gga(T49, T46, X74))
addc34_in_gga(0, T58, T58) → addc34_out_gga(0, T58, T58)
addc34_in_gga(s(T63), T64, s(X101)) → U14_gga(T63, T64, X101, addc34_in_gga(T63, T64, X101))
U14_gga(T63, T64, X101, addc34_out_gga(T63, T64, X101)) → addc34_out_gga(s(T63), T64, s(X101))
U13_gga(T45, T46, X74, addc34_out_gga(T49, T46, X74)) → multc23_out_gga(s(T45), T46, X74)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
mult23_in_gga(x1, x2, x3)  =  mult23_in_gga(x1, x2)
multc23_in_gga(x1, x2, x3)  =  multc23_in_gga(x1, x2)
0  =  0
multc23_out_gga(x1, x2, x3)  =  multc23_out_gga(x1, x2, x3)
U12_gga(x1, x2, x3, x4)  =  U12_gga(x1, x2, x4)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
addc34_in_gga(x1, x2, x3)  =  addc34_in_gga(x1, x2)
addc34_out_gga(x1, x2, x3)  =  addc34_out_gga(x1, x2, x3)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x1, x2, x4)
add34_in_gga(x1, x2, x3)  =  add34_in_gga(x1, x2)
add44_in_gga(x1, x2, x3)  =  add44_in_gga(x1, x2)
MULT1_IN_GGA(x1, x2, x3)  =  MULT1_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x1, x2, x4)
MULT23_IN_GGA(x1, x2, x3)  =  MULT23_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x1, x2, x4)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)
ADD34_IN_GGA(x1, x2, x3)  =  ADD34_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)
U7_GGA(x1, x2, x3, x4)  =  U7_GGA(x1, x2, x4)
U8_GGA(x1, x2, x3, x4)  =  U8_GGA(x1, x2, x4)
U9_GGA(x1, x2, x3, x4)  =  U9_GGA(x1, x2, x4)
U10_GGA(x1, x2, x3, x4)  =  U10_GGA(x1, x2, x4)
ADD44_IN_GGA(x1, x2, x3)  =  ADD44_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x1, x2, x4)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 14 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

ADD44_IN_GGA(s(T85), T86, s(T88)) → ADD44_IN_GGA(T85, T86, T88)

The TRS R consists of the following rules:

multc23_in_gga(0, T40, 0) → multc23_out_gga(0, T40, 0)
multc23_in_gga(s(T45), T46, X74) → U12_gga(T45, T46, X74, multc23_in_gga(T45, T46, T49))
U12_gga(T45, T46, X74, multc23_out_gga(T45, T46, T49)) → U13_gga(T45, T46, X74, addc34_in_gga(T49, T46, X74))
addc34_in_gga(0, T58, T58) → addc34_out_gga(0, T58, T58)
addc34_in_gga(s(T63), T64, s(X101)) → U14_gga(T63, T64, X101, addc34_in_gga(T63, T64, X101))
U14_gga(T63, T64, X101, addc34_out_gga(T63, T64, X101)) → addc34_out_gga(s(T63), T64, s(X101))
U13_gga(T45, T46, X74, addc34_out_gga(T49, T46, X74)) → multc23_out_gga(s(T45), T46, X74)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
multc23_in_gga(x1, x2, x3)  =  multc23_in_gga(x1, x2)
0  =  0
multc23_out_gga(x1, x2, x3)  =  multc23_out_gga(x1, x2, x3)
U12_gga(x1, x2, x3, x4)  =  U12_gga(x1, x2, x4)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
addc34_in_gga(x1, x2, x3)  =  addc34_in_gga(x1, x2)
addc34_out_gga(x1, x2, x3)  =  addc34_out_gga(x1, x2, x3)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x1, x2, x4)
ADD44_IN_GGA(x1, x2, x3)  =  ADD44_IN_GGA(x1, x2)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

ADD44_IN_GGA(s(T85), T86, s(T88)) → ADD44_IN_GGA(T85, T86, T88)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
ADD44_IN_GGA(x1, x2, x3)  =  ADD44_IN_GGA(x1, x2)

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

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

ADD44_IN_GGA(s(T85), T86) → ADD44_IN_GGA(T85, T86)

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

(12) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• ADD44_IN_GGA(s(T85), T86) → ADD44_IN_GGA(T85, T86)
  The graph contains the following edges 1 > 1, 2 >= 2

(13) YES

(14) Obligation:

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

ADD34_IN_GGA(s(T63), T64, s(X101)) → ADD34_IN_GGA(T63, T64, X101)

The TRS R consists of the following rules:

multc23_in_gga(0, T40, 0) → multc23_out_gga(0, T40, 0)
multc23_in_gga(s(T45), T46, X74) → U12_gga(T45, T46, X74, multc23_in_gga(T45, T46, T49))
U12_gga(T45, T46, X74, multc23_out_gga(T45, T46, T49)) → U13_gga(T45, T46, X74, addc34_in_gga(T49, T46, X74))
addc34_in_gga(0, T58, T58) → addc34_out_gga(0, T58, T58)
addc34_in_gga(s(T63), T64, s(X101)) → U14_gga(T63, T64, X101, addc34_in_gga(T63, T64, X101))
U14_gga(T63, T64, X101, addc34_out_gga(T63, T64, X101)) → addc34_out_gga(s(T63), T64, s(X101))
U13_gga(T45, T46, X74, addc34_out_gga(T49, T46, X74)) → multc23_out_gga(s(T45), T46, X74)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
multc23_in_gga(x1, x2, x3)  =  multc23_in_gga(x1, x2)
0  =  0
multc23_out_gga(x1, x2, x3)  =  multc23_out_gga(x1, x2, x3)
U12_gga(x1, x2, x3, x4)  =  U12_gga(x1, x2, x4)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
addc34_in_gga(x1, x2, x3)  =  addc34_in_gga(x1, x2)
addc34_out_gga(x1, x2, x3)  =  addc34_out_gga(x1, x2, x3)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x1, x2, x4)
ADD34_IN_GGA(x1, x2, x3)  =  ADD34_IN_GGA(x1, x2)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

ADD34_IN_GGA(s(T63), T64, s(X101)) → ADD34_IN_GGA(T63, T64, X101)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
ADD34_IN_GGA(x1, x2, x3)  =  ADD34_IN_GGA(x1, x2)

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

ADD34_IN_GGA(s(T63), T64) → ADD34_IN_GGA(T63, T64)

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

(19) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• ADD34_IN_GGA(s(T63), T64) → ADD34_IN_GGA(T63, T64)
  The graph contains the following edges 1 > 1, 2 >= 2

(20) YES

(21) Obligation:

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

MULT23_IN_GGA(s(T45), T46, X74) → MULT23_IN_GGA(T45, T46, X73)

The TRS R consists of the following rules:

multc23_in_gga(0, T40, 0) → multc23_out_gga(0, T40, 0)
multc23_in_gga(s(T45), T46, X74) → U12_gga(T45, T46, X74, multc23_in_gga(T45, T46, T49))
U12_gga(T45, T46, X74, multc23_out_gga(T45, T46, T49)) → U13_gga(T45, T46, X74, addc34_in_gga(T49, T46, X74))
addc34_in_gga(0, T58, T58) → addc34_out_gga(0, T58, T58)
addc34_in_gga(s(T63), T64, s(X101)) → U14_gga(T63, T64, X101, addc34_in_gga(T63, T64, X101))
U14_gga(T63, T64, X101, addc34_out_gga(T63, T64, X101)) → addc34_out_gga(s(T63), T64, s(X101))
U13_gga(T45, T46, X74, addc34_out_gga(T49, T46, X74)) → multc23_out_gga(s(T45), T46, X74)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
multc23_in_gga(x1, x2, x3)  =  multc23_in_gga(x1, x2)
0  =  0
multc23_out_gga(x1, x2, x3)  =  multc23_out_gga(x1, x2, x3)
U12_gga(x1, x2, x3, x4)  =  U12_gga(x1, x2, x4)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
addc34_in_gga(x1, x2, x3)  =  addc34_in_gga(x1, x2)
addc34_out_gga(x1, x2, x3)  =  addc34_out_gga(x1, x2, x3)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x1, x2, x4)
MULT23_IN_GGA(x1, x2, x3)  =  MULT23_IN_GGA(x1, x2)

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

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

MULT23_IN_GGA(s(T45), T46, X74) → MULT23_IN_GGA(T45, T46, X73)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
MULT23_IN_GGA(x1, x2, x3)  =  MULT23_IN_GGA(x1, x2)

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

(24) PiDPToQDPProof (SOUND transformation)

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

(25) Obligation:

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

MULT23_IN_GGA(s(T45), T46) → MULT23_IN_GGA(T45, T46)

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

(26) 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:

• MULT23_IN_GGA(s(T45), T46) → MULT23_IN_GGA(T45, T46)
  The graph contains the following edges 1 > 1, 2 >= 2

(27) YES