Left Termination proof of /tmp/tmpGGZfk2/pl7.2.9.pl

Left Termination of the query pattern mult_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 QDPSizeChangeProof (⇔)

                ↳15 YES

            ↳16 PiDP

                ↳17 UsableRulesProof (⇔)

                ↳18 PiDP

                ↳19 PiDPToQDPProof (⇐)

                ↳20 QDP

                ↳21 NonTerminationProof (⇔)

                ↳22 NO

            ↳23 PiDP

                ↳24 UsableRulesProof (⇔)

                ↳25 PiDP

                ↳26 PiDPToQDPProof (⇐)

                ↳27 QDP

                ↳28 QDPSizeChangeProof (⇔)

                ↳29 YES

    ↳30 PrologToPiTRSProof (⇐)

        ↳31 PiTRS

        ↳32 DependencyPairsProof (⇔)

        ↳33 PiDP

        ↳34 DependencyGraphProof (⇔)

        ↳35 AND

            ↳36 PiDP

                ↳37 UsableRulesProof (⇔)

                ↳38 PiDP

                ↳39 PiDPToQDPProof (⇐)

                ↳40 QDP

                ↳41 QDPSizeChangeProof (⇔)

                ↳42 YES

            ↳43 PiDP

                ↳44 UsableRulesProof (⇔)

                ↳45 PiDP

                ↳46 PiDPToQDPProof (⇐)

                ↳47 QDP

                ↳48 NonTerminationProof (⇔)

                ↳49 NO

            ↳50 PiDP

                ↳51 UsableRulesProof (⇔)

                ↳52 PiDP

                ↳53 PiDPToQDPProof (⇐)

                ↳54 QDP

                ↳55 QDPSizeChangeProof (⇔)

                ↳56 YES

(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) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog 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:

Clauses:

mult23(0, T40, 0).
mult23(s(T45), T46, X74) :- mult23(T45, T46, X73).
mult23(s(T45), T46, X74) :- ','(mult23(T45, T46, T49), add34(T49, T46, X74)).
add34(0, T58, T58).
add34(s(T63), T64, s(X101)) :- add34(T63, T64, X101).
add44(0, T78, T78).
add44(s(T85), T86, s(T88)) :- add44(T85, T86, T88).
mult1(0, T5, 0).
mult1(s(0), T24, T24).
mult1(s(s(T29)), T30, T12) :- mult23(T29, T30, X46).
mult1(s(s(T29)), T30, T12) :- ','(mult23(T29, T30, T33), add34(T33, T30, X47)).
mult1(s(s(T29)), T30, T12) :- ','(mult23(T29, T30, T33), ','(add34(T33, T30, T69), add44(T69, T30, T12))).

Queries:

mult1(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:
mult1_in: (b,b,f)
mult23_in: (b,b,f)
add34_in: (f,b,f)
add44_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

mult1_in_gga(0, T5, 0) → mult1_out_gga(0, T5, 0)
mult1_in_gga(s(0), T24, T24) → mult1_out_gga(s(0), T24, T24)
mult1_in_gga(s(s(T29)), T30, T12) → U6_gga(T29, T30, T12, mult23_in_gga(T29, T30, X46))
mult23_in_gga(0, T40, 0) → mult23_out_gga(0, T40, 0)
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) → U2_gga(T45, T46, X74, mult23_in_gga(T45, T46, T49))
U2_gga(T45, T46, X74, mult23_out_gga(T45, T46, T49)) → U3_gga(T45, T46, X74, add34_in_aga(T49, T46, X74))
add34_in_aga(0, T58, T58) → add34_out_aga(0, T58, T58)
add34_in_aga(s(T63), T64, s(X101)) → U4_aga(T63, T64, X101, add34_in_aga(T63, T64, X101))
U4_aga(T63, T64, X101, add34_out_aga(T63, T64, X101)) → add34_out_aga(s(T63), T64, s(X101))
U3_gga(T45, T46, X74, add34_out_aga(T49, T46, X74)) → mult23_out_gga(s(T45), T46, X74)
U1_gga(T45, T46, X74, mult23_out_gga(T45, T46, X73)) → mult23_out_gga(s(T45), T46, X74)
U6_gga(T29, T30, T12, mult23_out_gga(T29, T30, X46)) → mult1_out_gga(s(s(T29)), T30, T12)
mult1_in_gga(s(s(T29)), T30, T12) → U7_gga(T29, T30, T12, mult23_in_gga(T29, T30, T33))
U7_gga(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → U8_gga(T29, T30, T12, add34_in_aga(T33, T30, X47))
U8_gga(T29, T30, T12, add34_out_aga(T33, T30, X47)) → mult1_out_gga(s(s(T29)), T30, T12)
U7_gga(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → U9_gga(T29, T30, T12, add34_in_aga(T33, T30, T69))
U9_gga(T29, T30, T12, add34_out_aga(T33, T30, T69)) → U10_gga(T29, T30, T12, add44_in_gga(T69, T30, T12))
add44_in_gga(0, T78, T78) → add44_out_gga(0, T78, T78)
add44_in_gga(s(T85), T86, s(T88)) → U5_gga(T85, T86, T88, add44_in_gga(T85, T86, T88))
U5_gga(T85, T86, T88, add44_out_gga(T85, T86, T88)) → add44_out_gga(s(T85), T86, s(T88))
U10_gga(T29, T30, T12, add44_out_gga(T69, T30, T12)) → mult1_out_gga(s(s(T29)), T30, T12)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

mult1_in_gga(0, T5, 0) → mult1_out_gga(0, T5, 0)
mult1_in_gga(s(0), T24, T24) → mult1_out_gga(s(0), T24, T24)
mult1_in_gga(s(s(T29)), T30, T12) → U6_gga(T29, T30, T12, mult23_in_gga(T29, T30, X46))
mult23_in_gga(0, T40, 0) → mult23_out_gga(0, T40, 0)
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) → U2_gga(T45, T46, X74, mult23_in_gga(T45, T46, T49))
U2_gga(T45, T46, X74, mult23_out_gga(T45, T46, T49)) → U3_gga(T45, T46, X74, add34_in_aga(T49, T46, X74))
add34_in_aga(0, T58, T58) → add34_out_aga(0, T58, T58)
add34_in_aga(s(T63), T64, s(X101)) → U4_aga(T63, T64, X101, add34_in_aga(T63, T64, X101))
U4_aga(T63, T64, X101, add34_out_aga(T63, T64, X101)) → add34_out_aga(s(T63), T64, s(X101))
U3_gga(T45, T46, X74, add34_out_aga(T49, T46, X74)) → mult23_out_gga(s(T45), T46, X74)
U1_gga(T45, T46, X74, mult23_out_gga(T45, T46, X73)) → mult23_out_gga(s(T45), T46, X74)
U6_gga(T29, T30, T12, mult23_out_gga(T29, T30, X46)) → mult1_out_gga(s(s(T29)), T30, T12)
mult1_in_gga(s(s(T29)), T30, T12) → U7_gga(T29, T30, T12, mult23_in_gga(T29, T30, T33))
U7_gga(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → U8_gga(T29, T30, T12, add34_in_aga(T33, T30, X47))
U8_gga(T29, T30, T12, add34_out_aga(T33, T30, X47)) → mult1_out_gga(s(s(T29)), T30, T12)
U7_gga(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → U9_gga(T29, T30, T12, add34_in_aga(T33, T30, T69))
U9_gga(T29, T30, T12, add34_out_aga(T33, T30, T69)) → U10_gga(T29, T30, T12, add44_in_gga(T69, T30, T12))
add44_in_gga(0, T78, T78) → add44_out_gga(0, T78, T78)
add44_in_gga(s(T85), T86, s(T88)) → U5_gga(T85, T86, T88, add44_in_gga(T85, T86, T88))
U5_gga(T85, T86, T88, add44_out_gga(T85, T86, T88)) → add44_out_gga(s(T85), T86, s(T88))
U10_gga(T29, T30, T12, add44_out_gga(T69, T30, T12)) → mult1_out_gga(s(s(T29)), T30, T12)

(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:

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, mult23_in_gga(T45, T46, T49))
U2_GGA(T45, T46, X74, mult23_out_gga(T45, T46, T49)) → U3_GGA(T45, T46, X74, add34_in_aga(T49, T46, X74))
U2_GGA(T45, T46, X74, mult23_out_gga(T45, T46, T49)) → ADD34_IN_AGA(T49, T46, X74)
ADD34_IN_AGA(s(T63), T64, s(X101)) → U4_AGA(T63, T64, X101, add34_in_aga(T63, T64, X101))
ADD34_IN_AGA(s(T63), T64, s(X101)) → ADD34_IN_AGA(T63, T64, X101)
MULT1_IN_GGA(s(s(T29)), T30, T12) → U7_GGA(T29, T30, T12, mult23_in_gga(T29, T30, T33))
U7_GGA(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → U8_GGA(T29, T30, T12, add34_in_aga(T33, T30, X47))
U7_GGA(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → ADD34_IN_AGA(T33, T30, X47)
U7_GGA(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → U9_GGA(T29, T30, T12, add34_in_aga(T33, T30, T69))
U9_GGA(T29, T30, T12, add34_out_aga(T33, T30, T69)) → U10_GGA(T29, T30, T12, add44_in_gga(T69, T30, T12))
U9_GGA(T29, T30, T12, add34_out_aga(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:

mult1_in_gga(0, T5, 0) → mult1_out_gga(0, T5, 0)
mult1_in_gga(s(0), T24, T24) → mult1_out_gga(s(0), T24, T24)
mult1_in_gga(s(s(T29)), T30, T12) → U6_gga(T29, T30, T12, mult23_in_gga(T29, T30, X46))
mult23_in_gga(0, T40, 0) → mult23_out_gga(0, T40, 0)
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) → U2_gga(T45, T46, X74, mult23_in_gga(T45, T46, T49))
U2_gga(T45, T46, X74, mult23_out_gga(T45, T46, T49)) → U3_gga(T45, T46, X74, add34_in_aga(T49, T46, X74))
add34_in_aga(0, T58, T58) → add34_out_aga(0, T58, T58)
add34_in_aga(s(T63), T64, s(X101)) → U4_aga(T63, T64, X101, add34_in_aga(T63, T64, X101))
U4_aga(T63, T64, X101, add34_out_aga(T63, T64, X101)) → add34_out_aga(s(T63), T64, s(X101))
U3_gga(T45, T46, X74, add34_out_aga(T49, T46, X74)) → mult23_out_gga(s(T45), T46, X74)
U1_gga(T45, T46, X74, mult23_out_gga(T45, T46, X73)) → mult23_out_gga(s(T45), T46, X74)
U6_gga(T29, T30, T12, mult23_out_gga(T29, T30, X46)) → mult1_out_gga(s(s(T29)), T30, T12)
mult1_in_gga(s(s(T29)), T30, T12) → U7_gga(T29, T30, T12, mult23_in_gga(T29, T30, T33))
U7_gga(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → U8_gga(T29, T30, T12, add34_in_aga(T33, T30, X47))
U8_gga(T29, T30, T12, add34_out_aga(T33, T30, X47)) → mult1_out_gga(s(s(T29)), T30, T12)
U7_gga(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → U9_gga(T29, T30, T12, add34_in_aga(T33, T30, T69))
U9_gga(T29, T30, T12, add34_out_aga(T33, T30, T69)) → U10_gga(T29, T30, T12, add44_in_gga(T69, T30, T12))
add44_in_gga(0, T78, T78) → add44_out_gga(0, T78, T78)
add44_in_gga(s(T85), T86, s(T88)) → U5_gga(T85, T86, T88, add44_in_gga(T85, T86, T88))
U5_gga(T85, T86, T88, add44_out_gga(T85, T86, T88)) → add44_out_gga(s(T85), T86, s(T88))
U10_gga(T29, T30, T12, add44_out_gga(T69, T30, T12)) → mult1_out_gga(s(s(T29)), T30, T12)

The argument filtering Pi contains the following mapping:
mult1_in_gga(x1, x2, x3) = mult1_in_gga(x1, x2)
0 = 0
mult1_out_gga(x1, x2, x3) = mult1_out_gga(x1, x2)
s(x1) = s(x1)
U6_gga(x1, x2, x3, x4) = U6_gga(x1, x2, x4)
mult23_in_gga(x1, x2, x3) = mult23_in_gga(x1, x2)
mult23_out_gga(x1, x2, x3) = mult23_out_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_aga(x1, x2, x3) = add34_in_aga(x2)
add34_out_aga(x1, x2, x3) = add34_out_aga(x1, x2, x3)
U4_aga(x1, x2, x3, x4) = U4_aga(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)
add44_out_gga(x1, x2, x3) = add44_out_gga(x1, x2, x3)
U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4)
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_AGA(x1, x2, x3) = ADD34_IN_AGA(x2)
U4_AGA(x1, x2, x3, x4) = U4_AGA(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

(6) 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, mult23_in_gga(T45, T46, T49))
U2_GGA(T45, T46, X74, mult23_out_gga(T45, T46, T49)) → U3_GGA(T45, T46, X74, add34_in_aga(T49, T46, X74))
U2_GGA(T45, T46, X74, mult23_out_gga(T45, T46, T49)) → ADD34_IN_AGA(T49, T46, X74)
ADD34_IN_AGA(s(T63), T64, s(X101)) → U4_AGA(T63, T64, X101, add34_in_aga(T63, T64, X101))
ADD34_IN_AGA(s(T63), T64, s(X101)) → ADD34_IN_AGA(T63, T64, X101)
MULT1_IN_GGA(s(s(T29)), T30, T12) → U7_GGA(T29, T30, T12, mult23_in_gga(T29, T30, T33))
U7_GGA(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → U8_GGA(T29, T30, T12, add34_in_aga(T33, T30, X47))
U7_GGA(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → ADD34_IN_AGA(T33, T30, X47)
U7_GGA(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → U9_GGA(T29, T30, T12, add34_in_aga(T33, T30, T69))
U9_GGA(T29, T30, T12, add34_out_aga(T33, T30, T69)) → U10_GGA(T29, T30, T12, add44_in_gga(T69, T30, T12))
U9_GGA(T29, T30, T12, add34_out_aga(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:

mult1_in_gga(0, T5, 0) → mult1_out_gga(0, T5, 0)
mult1_in_gga(s(0), T24, T24) → mult1_out_gga(s(0), T24, T24)
mult1_in_gga(s(s(T29)), T30, T12) → U6_gga(T29, T30, T12, mult23_in_gga(T29, T30, X46))
mult23_in_gga(0, T40, 0) → mult23_out_gga(0, T40, 0)
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) → U2_gga(T45, T46, X74, mult23_in_gga(T45, T46, T49))
U2_gga(T45, T46, X74, mult23_out_gga(T45, T46, T49)) → U3_gga(T45, T46, X74, add34_in_aga(T49, T46, X74))
add34_in_aga(0, T58, T58) → add34_out_aga(0, T58, T58)
add34_in_aga(s(T63), T64, s(X101)) → U4_aga(T63, T64, X101, add34_in_aga(T63, T64, X101))
U4_aga(T63, T64, X101, add34_out_aga(T63, T64, X101)) → add34_out_aga(s(T63), T64, s(X101))
U3_gga(T45, T46, X74, add34_out_aga(T49, T46, X74)) → mult23_out_gga(s(T45), T46, X74)
U1_gga(T45, T46, X74, mult23_out_gga(T45, T46, X73)) → mult23_out_gga(s(T45), T46, X74)
U6_gga(T29, T30, T12, mult23_out_gga(T29, T30, X46)) → mult1_out_gga(s(s(T29)), T30, T12)
mult1_in_gga(s(s(T29)), T30, T12) → U7_gga(T29, T30, T12, mult23_in_gga(T29, T30, T33))
U7_gga(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → U8_gga(T29, T30, T12, add34_in_aga(T33, T30, X47))
U8_gga(T29, T30, T12, add34_out_aga(T33, T30, X47)) → mult1_out_gga(s(s(T29)), T30, T12)
U7_gga(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → U9_gga(T29, T30, T12, add34_in_aga(T33, T30, T69))
U9_gga(T29, T30, T12, add34_out_aga(T33, T30, T69)) → U10_gga(T29, T30, T12, add44_in_gga(T69, T30, T12))
add44_in_gga(0, T78, T78) → add44_out_gga(0, T78, T78)
add44_in_gga(s(T85), T86, s(T88)) → U5_gga(T85, T86, T88, add44_in_gga(T85, T86, T88))
U5_gga(T85, T86, T88, add44_out_gga(T85, T86, T88)) → add44_out_gga(s(T85), T86, s(T88))
U10_gga(T29, T30, T12, add44_out_gga(T69, T30, T12)) → mult1_out_gga(s(s(T29)), T30, T12)

(7) DependencyGraphProof (EQUIVALENT transformation)

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

(8) Complex Obligation (AND)

(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)

The TRS R consists of the following rules:

mult1_in_gga(0, T5, 0) → mult1_out_gga(0, T5, 0)
mult1_in_gga(s(0), T24, T24) → mult1_out_gga(s(0), T24, T24)
mult1_in_gga(s(s(T29)), T30, T12) → U6_gga(T29, T30, T12, mult23_in_gga(T29, T30, X46))
mult23_in_gga(0, T40, 0) → mult23_out_gga(0, T40, 0)
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) → U2_gga(T45, T46, X74, mult23_in_gga(T45, T46, T49))
U2_gga(T45, T46, X74, mult23_out_gga(T45, T46, T49)) → U3_gga(T45, T46, X74, add34_in_aga(T49, T46, X74))
add34_in_aga(0, T58, T58) → add34_out_aga(0, T58, T58)
add34_in_aga(s(T63), T64, s(X101)) → U4_aga(T63, T64, X101, add34_in_aga(T63, T64, X101))
U4_aga(T63, T64, X101, add34_out_aga(T63, T64, X101)) → add34_out_aga(s(T63), T64, s(X101))
U3_gga(T45, T46, X74, add34_out_aga(T49, T46, X74)) → mult23_out_gga(s(T45), T46, X74)
U1_gga(T45, T46, X74, mult23_out_gga(T45, T46, X73)) → mult23_out_gga(s(T45), T46, X74)
U6_gga(T29, T30, T12, mult23_out_gga(T29, T30, X46)) → mult1_out_gga(s(s(T29)), T30, T12)
mult1_in_gga(s(s(T29)), T30, T12) → U7_gga(T29, T30, T12, mult23_in_gga(T29, T30, T33))
U7_gga(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → U8_gga(T29, T30, T12, add34_in_aga(T33, T30, X47))
U8_gga(T29, T30, T12, add34_out_aga(T33, T30, X47)) → mult1_out_gga(s(s(T29)), T30, T12)
U7_gga(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → U9_gga(T29, T30, T12, add34_in_aga(T33, T30, T69))
U9_gga(T29, T30, T12, add34_out_aga(T33, T30, T69)) → U10_gga(T29, T30, T12, add44_in_gga(T69, T30, T12))
add44_in_gga(0, T78, T78) → add44_out_gga(0, T78, T78)
add44_in_gga(s(T85), T86, s(T88)) → U5_gga(T85, T86, T88, add44_in_gga(T85, T86, T88))
U5_gga(T85, T86, T88, add44_out_gga(T85, T86, T88)) → add44_out_gga(s(T85), T86, s(T88))
U10_gga(T29, T30, T12, add44_out_gga(T69, T30, T12)) → mult1_out_gga(s(s(T29)), T30, T12)

(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:

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

(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:

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.

(14) 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

(15) YES

(16) Obligation:

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

ADD34_IN_AGA(s(T63), T64, s(X101)) → ADD34_IN_AGA(T63, T64, X101)

The TRS R consists of the following rules:

mult1_in_gga(0, T5, 0) → mult1_out_gga(0, T5, 0)
mult1_in_gga(s(0), T24, T24) → mult1_out_gga(s(0), T24, T24)
mult1_in_gga(s(s(T29)), T30, T12) → U6_gga(T29, T30, T12, mult23_in_gga(T29, T30, X46))
mult23_in_gga(0, T40, 0) → mult23_out_gga(0, T40, 0)
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) → U2_gga(T45, T46, X74, mult23_in_gga(T45, T46, T49))
U2_gga(T45, T46, X74, mult23_out_gga(T45, T46, T49)) → U3_gga(T45, T46, X74, add34_in_aga(T49, T46, X74))
add34_in_aga(0, T58, T58) → add34_out_aga(0, T58, T58)
add34_in_aga(s(T63), T64, s(X101)) → U4_aga(T63, T64, X101, add34_in_aga(T63, T64, X101))
U4_aga(T63, T64, X101, add34_out_aga(T63, T64, X101)) → add34_out_aga(s(T63), T64, s(X101))
U3_gga(T45, T46, X74, add34_out_aga(T49, T46, X74)) → mult23_out_gga(s(T45), T46, X74)
U1_gga(T45, T46, X74, mult23_out_gga(T45, T46, X73)) → mult23_out_gga(s(T45), T46, X74)
U6_gga(T29, T30, T12, mult23_out_gga(T29, T30, X46)) → mult1_out_gga(s(s(T29)), T30, T12)
mult1_in_gga(s(s(T29)), T30, T12) → U7_gga(T29, T30, T12, mult23_in_gga(T29, T30, T33))
U7_gga(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → U8_gga(T29, T30, T12, add34_in_aga(T33, T30, X47))
U8_gga(T29, T30, T12, add34_out_aga(T33, T30, X47)) → mult1_out_gga(s(s(T29)), T30, T12)
U7_gga(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → U9_gga(T29, T30, T12, add34_in_aga(T33, T30, T69))
U9_gga(T29, T30, T12, add34_out_aga(T33, T30, T69)) → U10_gga(T29, T30, T12, add44_in_gga(T69, T30, T12))
add44_in_gga(0, T78, T78) → add44_out_gga(0, T78, T78)
add44_in_gga(s(T85), T86, s(T88)) → U5_gga(T85, T86, T88, add44_in_gga(T85, T86, T88))
U5_gga(T85, T86, T88, add44_out_gga(T85, T86, T88)) → add44_out_gga(s(T85), T86, s(T88))
U10_gga(T29, T30, T12, add44_out_gga(T69, T30, T12)) → mult1_out_gga(s(s(T29)), T30, T12)

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

ADD34_IN_AGA(s(T63), T64, s(X101)) → ADD34_IN_AGA(T63, T64, X101)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
ADD34_IN_AGA(x1, x2, x3) = ADD34_IN_AGA(x2)

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

ADD34_IN_AGA(T64) → ADD34_IN_AGA(T64)

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

(21) 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 = ADD34_IN_AGA(T64) evaluates to t =ADD34_IN_AGA(T64)

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 ADD34_IN_AGA(T64) to ADD34_IN_AGA(T64).

(22) NO

(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)

The TRS R consists of the following rules:

mult1_in_gga(0, T5, 0) → mult1_out_gga(0, T5, 0)
mult1_in_gga(s(0), T24, T24) → mult1_out_gga(s(0), T24, T24)
mult1_in_gga(s(s(T29)), T30, T12) → U6_gga(T29, T30, T12, mult23_in_gga(T29, T30, X46))
mult23_in_gga(0, T40, 0) → mult23_out_gga(0, T40, 0)
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) → U2_gga(T45, T46, X74, mult23_in_gga(T45, T46, T49))
U2_gga(T45, T46, X74, mult23_out_gga(T45, T46, T49)) → U3_gga(T45, T46, X74, add34_in_aga(T49, T46, X74))
add34_in_aga(0, T58, T58) → add34_out_aga(0, T58, T58)
add34_in_aga(s(T63), T64, s(X101)) → U4_aga(T63, T64, X101, add34_in_aga(T63, T64, X101))
U4_aga(T63, T64, X101, add34_out_aga(T63, T64, X101)) → add34_out_aga(s(T63), T64, s(X101))
U3_gga(T45, T46, X74, add34_out_aga(T49, T46, X74)) → mult23_out_gga(s(T45), T46, X74)
U1_gga(T45, T46, X74, mult23_out_gga(T45, T46, X73)) → mult23_out_gga(s(T45), T46, X74)
U6_gga(T29, T30, T12, mult23_out_gga(T29, T30, X46)) → mult1_out_gga(s(s(T29)), T30, T12)
mult1_in_gga(s(s(T29)), T30, T12) → U7_gga(T29, T30, T12, mult23_in_gga(T29, T30, T33))
U7_gga(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → U8_gga(T29, T30, T12, add34_in_aga(T33, T30, X47))
U8_gga(T29, T30, T12, add34_out_aga(T33, T30, X47)) → mult1_out_gga(s(s(T29)), T30, T12)
U7_gga(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → U9_gga(T29, T30, T12, add34_in_aga(T33, T30, T69))
U9_gga(T29, T30, T12, add34_out_aga(T33, T30, T69)) → U10_gga(T29, T30, T12, add44_in_gga(T69, T30, T12))
add44_in_gga(0, T78, T78) → add44_out_gga(0, T78, T78)
add44_in_gga(s(T85), T86, s(T88)) → U5_gga(T85, T86, T88, add44_in_gga(T85, T86, T88))
U5_gga(T85, T86, T88, add44_out_gga(T85, T86, T88)) → add44_out_gga(s(T85), T86, s(T88))
U10_gga(T29, T30, T12, add44_out_gga(T69, T30, T12)) → mult1_out_gga(s(s(T29)), T30, T12)

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) 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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) 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.

(28) 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

(29) YES

(30) PrologToPiTRSProof (SOUND transformation)

mult1_in_gga(0, T5, 0) → mult1_out_gga(0, T5, 0)
mult1_in_gga(s(0), T24, T24) → mult1_out_gga(s(0), T24, T24)
mult1_in_gga(s(s(T29)), T30, T12) → U6_gga(T29, T30, T12, mult23_in_gga(T29, T30, X46))
mult23_in_gga(0, T40, 0) → mult23_out_gga(0, T40, 0)
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) → U2_gga(T45, T46, X74, mult23_in_gga(T45, T46, T49))
U2_gga(T45, T46, X74, mult23_out_gga(T45, T46, T49)) → U3_gga(T45, T46, X74, add34_in_aga(T49, T46, X74))
add34_in_aga(0, T58, T58) → add34_out_aga(0, T58, T58)
add34_in_aga(s(T63), T64, s(X101)) → U4_aga(T63, T64, X101, add34_in_aga(T63, T64, X101))
U4_aga(T63, T64, X101, add34_out_aga(T63, T64, X101)) → add34_out_aga(s(T63), T64, s(X101))
U3_gga(T45, T46, X74, add34_out_aga(T49, T46, X74)) → mult23_out_gga(s(T45), T46, X74)
U1_gga(T45, T46, X74, mult23_out_gga(T45, T46, X73)) → mult23_out_gga(s(T45), T46, X74)
U6_gga(T29, T30, T12, mult23_out_gga(T29, T30, X46)) → mult1_out_gga(s(s(T29)), T30, T12)
mult1_in_gga(s(s(T29)), T30, T12) → U7_gga(T29, T30, T12, mult23_in_gga(T29, T30, T33))
U7_gga(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → U8_gga(T29, T30, T12, add34_in_aga(T33, T30, X47))
U8_gga(T29, T30, T12, add34_out_aga(T33, T30, X47)) → mult1_out_gga(s(s(T29)), T30, T12)
U7_gga(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → U9_gga(T29, T30, T12, add34_in_aga(T33, T30, T69))
U9_gga(T29, T30, T12, add34_out_aga(T33, T30, T69)) → U10_gga(T29, T30, T12, add44_in_gga(T69, T30, T12))
add44_in_gga(0, T78, T78) → add44_out_gga(0, T78, T78)
add44_in_gga(s(T85), T86, s(T88)) → U5_gga(T85, T86, T88, add44_in_gga(T85, T86, T88))
U5_gga(T85, T86, T88, add44_out_gga(T85, T86, T88)) → add44_out_gga(s(T85), T86, s(T88))
U10_gga(T29, T30, T12, add44_out_gga(T69, T30, T12)) → mult1_out_gga(s(s(T29)), T30, T12)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(31) Obligation:

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

mult1_in_gga(0, T5, 0) → mult1_out_gga(0, T5, 0)
mult1_in_gga(s(0), T24, T24) → mult1_out_gga(s(0), T24, T24)
mult1_in_gga(s(s(T29)), T30, T12) → U6_gga(T29, T30, T12, mult23_in_gga(T29, T30, X46))
mult23_in_gga(0, T40, 0) → mult23_out_gga(0, T40, 0)
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) → U2_gga(T45, T46, X74, mult23_in_gga(T45, T46, T49))
U2_gga(T45, T46, X74, mult23_out_gga(T45, T46, T49)) → U3_gga(T45, T46, X74, add34_in_aga(T49, T46, X74))
add34_in_aga(0, T58, T58) → add34_out_aga(0, T58, T58)
add34_in_aga(s(T63), T64, s(X101)) → U4_aga(T63, T64, X101, add34_in_aga(T63, T64, X101))
U4_aga(T63, T64, X101, add34_out_aga(T63, T64, X101)) → add34_out_aga(s(T63), T64, s(X101))
U3_gga(T45, T46, X74, add34_out_aga(T49, T46, X74)) → mult23_out_gga(s(T45), T46, X74)
U1_gga(T45, T46, X74, mult23_out_gga(T45, T46, X73)) → mult23_out_gga(s(T45), T46, X74)
U6_gga(T29, T30, T12, mult23_out_gga(T29, T30, X46)) → mult1_out_gga(s(s(T29)), T30, T12)
mult1_in_gga(s(s(T29)), T30, T12) → U7_gga(T29, T30, T12, mult23_in_gga(T29, T30, T33))
U7_gga(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → U8_gga(T29, T30, T12, add34_in_aga(T33, T30, X47))
U8_gga(T29, T30, T12, add34_out_aga(T33, T30, X47)) → mult1_out_gga(s(s(T29)), T30, T12)
U7_gga(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → U9_gga(T29, T30, T12, add34_in_aga(T33, T30, T69))
U9_gga(T29, T30, T12, add34_out_aga(T33, T30, T69)) → U10_gga(T29, T30, T12, add44_in_gga(T69, T30, T12))
add44_in_gga(0, T78, T78) → add44_out_gga(0, T78, T78)
add44_in_gga(s(T85), T86, s(T88)) → U5_gga(T85, T86, T88, add44_in_gga(T85, T86, T88))
U5_gga(T85, T86, T88, add44_out_gga(T85, T86, T88)) → add44_out_gga(s(T85), T86, s(T88))
U10_gga(T29, T30, T12, add44_out_gga(T69, T30, T12)) → mult1_out_gga(s(s(T29)), T30, T12)

(32) 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:

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, mult23_in_gga(T45, T46, T49))
U2_GGA(T45, T46, X74, mult23_out_gga(T45, T46, T49)) → U3_GGA(T45, T46, X74, add34_in_aga(T49, T46, X74))
U2_GGA(T45, T46, X74, mult23_out_gga(T45, T46, T49)) → ADD34_IN_AGA(T49, T46, X74)
ADD34_IN_AGA(s(T63), T64, s(X101)) → U4_AGA(T63, T64, X101, add34_in_aga(T63, T64, X101))
ADD34_IN_AGA(s(T63), T64, s(X101)) → ADD34_IN_AGA(T63, T64, X101)
MULT1_IN_GGA(s(s(T29)), T30, T12) → U7_GGA(T29, T30, T12, mult23_in_gga(T29, T30, T33))
U7_GGA(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → U8_GGA(T29, T30, T12, add34_in_aga(T33, T30, X47))
U7_GGA(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → ADD34_IN_AGA(T33, T30, X47)
U7_GGA(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → U9_GGA(T29, T30, T12, add34_in_aga(T33, T30, T69))
U9_GGA(T29, T30, T12, add34_out_aga(T33, T30, T69)) → U10_GGA(T29, T30, T12, add44_in_gga(T69, T30, T12))
U9_GGA(T29, T30, T12, add34_out_aga(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:

mult1_in_gga(0, T5, 0) → mult1_out_gga(0, T5, 0)
mult1_in_gga(s(0), T24, T24) → mult1_out_gga(s(0), T24, T24)
mult1_in_gga(s(s(T29)), T30, T12) → U6_gga(T29, T30, T12, mult23_in_gga(T29, T30, X46))
mult23_in_gga(0, T40, 0) → mult23_out_gga(0, T40, 0)
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) → U2_gga(T45, T46, X74, mult23_in_gga(T45, T46, T49))
U2_gga(T45, T46, X74, mult23_out_gga(T45, T46, T49)) → U3_gga(T45, T46, X74, add34_in_aga(T49, T46, X74))
add34_in_aga(0, T58, T58) → add34_out_aga(0, T58, T58)
add34_in_aga(s(T63), T64, s(X101)) → U4_aga(T63, T64, X101, add34_in_aga(T63, T64, X101))
U4_aga(T63, T64, X101, add34_out_aga(T63, T64, X101)) → add34_out_aga(s(T63), T64, s(X101))
U3_gga(T45, T46, X74, add34_out_aga(T49, T46, X74)) → mult23_out_gga(s(T45), T46, X74)
U1_gga(T45, T46, X74, mult23_out_gga(T45, T46, X73)) → mult23_out_gga(s(T45), T46, X74)
U6_gga(T29, T30, T12, mult23_out_gga(T29, T30, X46)) → mult1_out_gga(s(s(T29)), T30, T12)
mult1_in_gga(s(s(T29)), T30, T12) → U7_gga(T29, T30, T12, mult23_in_gga(T29, T30, T33))
U7_gga(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → U8_gga(T29, T30, T12, add34_in_aga(T33, T30, X47))
U8_gga(T29, T30, T12, add34_out_aga(T33, T30, X47)) → mult1_out_gga(s(s(T29)), T30, T12)
U7_gga(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → U9_gga(T29, T30, T12, add34_in_aga(T33, T30, T69))
U9_gga(T29, T30, T12, add34_out_aga(T33, T30, T69)) → U10_gga(T29, T30, T12, add44_in_gga(T69, T30, T12))
add44_in_gga(0, T78, T78) → add44_out_gga(0, T78, T78)
add44_in_gga(s(T85), T86, s(T88)) → U5_gga(T85, T86, T88, add44_in_gga(T85, T86, T88))
U5_gga(T85, T86, T88, add44_out_gga(T85, T86, T88)) → add44_out_gga(s(T85), T86, s(T88))
U10_gga(T29, T30, T12, add44_out_gga(T69, T30, T12)) → mult1_out_gga(s(s(T29)), T30, T12)

The argument filtering Pi contains the following mapping:
mult1_in_gga(x1, x2, x3) = mult1_in_gga(x1, x2)
0 = 0
mult1_out_gga(x1, x2, x3) = mult1_out_gga
s(x1) = s(x1)
U6_gga(x1, x2, x3, x4) = U6_gga(x4)
mult23_in_gga(x1, x2, x3) = mult23_in_gga(x1, x2)
mult23_out_gga(x1, x2, x3) = mult23_out_gga
U1_gga(x1, x2, x3, x4) = U1_gga(x4)
U2_gga(x1, x2, x3, x4) = U2_gga(x2, x4)
U3_gga(x1, x2, x3, x4) = U3_gga(x4)
add34_in_aga(x1, x2, x3) = add34_in_aga(x2)
add34_out_aga(x1, x2, x3) = add34_out_aga(x1, x3)
U4_aga(x1, x2, x3, x4) = U4_aga(x4)
U7_gga(x1, x2, x3, x4) = U7_gga(x2, x4)
U8_gga(x1, x2, x3, x4) = U8_gga(x4)
U9_gga(x1, x2, x3, x4) = U9_gga(x2, x4)
U10_gga(x1, x2, x3, x4) = U10_gga(x4)
add44_in_gga(x1, x2, x3) = add44_in_gga(x1, x2)
add44_out_gga(x1, x2, x3) = add44_out_gga(x3)
U5_gga(x1, x2, x3, x4) = U5_gga(x4)
MULT1_IN_GGA(x1, x2, x3) = MULT1_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4) = U6_GGA(x4)
MULT23_IN_GGA(x1, x2, x3) = MULT23_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4) = U1_GGA(x4)
U2_GGA(x1, x2, x3, x4) = U2_GGA(x2, x4)
U3_GGA(x1, x2, x3, x4) = U3_GGA(x4)
ADD34_IN_AGA(x1, x2, x3) = ADD34_IN_AGA(x2)
U4_AGA(x1, x2, x3, x4) = U4_AGA(x4)
U7_GGA(x1, x2, x3, x4) = U7_GGA(x2, x4)
U8_GGA(x1, x2, x3, x4) = U8_GGA(x4)
U9_GGA(x1, x2, x3, x4) = U9_GGA(x2, x4)
U10_GGA(x1, x2, x3, x4) = U10_GGA(x4)
ADD44_IN_GGA(x1, x2, x3) = ADD44_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4) = U5_GGA(x4)

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

(33) 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, mult23_in_gga(T45, T46, T49))
U2_GGA(T45, T46, X74, mult23_out_gga(T45, T46, T49)) → U3_GGA(T45, T46, X74, add34_in_aga(T49, T46, X74))
U2_GGA(T45, T46, X74, mult23_out_gga(T45, T46, T49)) → ADD34_IN_AGA(T49, T46, X74)
ADD34_IN_AGA(s(T63), T64, s(X101)) → U4_AGA(T63, T64, X101, add34_in_aga(T63, T64, X101))
ADD34_IN_AGA(s(T63), T64, s(X101)) → ADD34_IN_AGA(T63, T64, X101)
MULT1_IN_GGA(s(s(T29)), T30, T12) → U7_GGA(T29, T30, T12, mult23_in_gga(T29, T30, T33))
U7_GGA(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → U8_GGA(T29, T30, T12, add34_in_aga(T33, T30, X47))
U7_GGA(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → ADD34_IN_AGA(T33, T30, X47)
U7_GGA(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → U9_GGA(T29, T30, T12, add34_in_aga(T33, T30, T69))
U9_GGA(T29, T30, T12, add34_out_aga(T33, T30, T69)) → U10_GGA(T29, T30, T12, add44_in_gga(T69, T30, T12))
U9_GGA(T29, T30, T12, add34_out_aga(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:

mult1_in_gga(0, T5, 0) → mult1_out_gga(0, T5, 0)
mult1_in_gga(s(0), T24, T24) → mult1_out_gga(s(0), T24, T24)
mult1_in_gga(s(s(T29)), T30, T12) → U6_gga(T29, T30, T12, mult23_in_gga(T29, T30, X46))
mult23_in_gga(0, T40, 0) → mult23_out_gga(0, T40, 0)
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) → U2_gga(T45, T46, X74, mult23_in_gga(T45, T46, T49))
U2_gga(T45, T46, X74, mult23_out_gga(T45, T46, T49)) → U3_gga(T45, T46, X74, add34_in_aga(T49, T46, X74))
add34_in_aga(0, T58, T58) → add34_out_aga(0, T58, T58)
add34_in_aga(s(T63), T64, s(X101)) → U4_aga(T63, T64, X101, add34_in_aga(T63, T64, X101))
U4_aga(T63, T64, X101, add34_out_aga(T63, T64, X101)) → add34_out_aga(s(T63), T64, s(X101))
U3_gga(T45, T46, X74, add34_out_aga(T49, T46, X74)) → mult23_out_gga(s(T45), T46, X74)
U1_gga(T45, T46, X74, mult23_out_gga(T45, T46, X73)) → mult23_out_gga(s(T45), T46, X74)
U6_gga(T29, T30, T12, mult23_out_gga(T29, T30, X46)) → mult1_out_gga(s(s(T29)), T30, T12)
mult1_in_gga(s(s(T29)), T30, T12) → U7_gga(T29, T30, T12, mult23_in_gga(T29, T30, T33))
U7_gga(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → U8_gga(T29, T30, T12, add34_in_aga(T33, T30, X47))
U8_gga(T29, T30, T12, add34_out_aga(T33, T30, X47)) → mult1_out_gga(s(s(T29)), T30, T12)
U7_gga(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → U9_gga(T29, T30, T12, add34_in_aga(T33, T30, T69))
U9_gga(T29, T30, T12, add34_out_aga(T33, T30, T69)) → U10_gga(T29, T30, T12, add44_in_gga(T69, T30, T12))
add44_in_gga(0, T78, T78) → add44_out_gga(0, T78, T78)
add44_in_gga(s(T85), T86, s(T88)) → U5_gga(T85, T86, T88, add44_in_gga(T85, T86, T88))
U5_gga(T85, T86, T88, add44_out_gga(T85, T86, T88)) → add44_out_gga(s(T85), T86, s(T88))
U10_gga(T29, T30, T12, add44_out_gga(T69, T30, T12)) → mult1_out_gga(s(s(T29)), T30, T12)

(34) DependencyGraphProof (EQUIVALENT transformation)

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

(35) Complex Obligation (AND)

(36) 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:

mult1_in_gga(0, T5, 0) → mult1_out_gga(0, T5, 0)
mult1_in_gga(s(0), T24, T24) → mult1_out_gga(s(0), T24, T24)
mult1_in_gga(s(s(T29)), T30, T12) → U6_gga(T29, T30, T12, mult23_in_gga(T29, T30, X46))
mult23_in_gga(0, T40, 0) → mult23_out_gga(0, T40, 0)
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) → U2_gga(T45, T46, X74, mult23_in_gga(T45, T46, T49))
U2_gga(T45, T46, X74, mult23_out_gga(T45, T46, T49)) → U3_gga(T45, T46, X74, add34_in_aga(T49, T46, X74))
add34_in_aga(0, T58, T58) → add34_out_aga(0, T58, T58)
add34_in_aga(s(T63), T64, s(X101)) → U4_aga(T63, T64, X101, add34_in_aga(T63, T64, X101))
U4_aga(T63, T64, X101, add34_out_aga(T63, T64, X101)) → add34_out_aga(s(T63), T64, s(X101))
U3_gga(T45, T46, X74, add34_out_aga(T49, T46, X74)) → mult23_out_gga(s(T45), T46, X74)
U1_gga(T45, T46, X74, mult23_out_gga(T45, T46, X73)) → mult23_out_gga(s(T45), T46, X74)
U6_gga(T29, T30, T12, mult23_out_gga(T29, T30, X46)) → mult1_out_gga(s(s(T29)), T30, T12)
mult1_in_gga(s(s(T29)), T30, T12) → U7_gga(T29, T30, T12, mult23_in_gga(T29, T30, T33))
U7_gga(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → U8_gga(T29, T30, T12, add34_in_aga(T33, T30, X47))
U8_gga(T29, T30, T12, add34_out_aga(T33, T30, X47)) → mult1_out_gga(s(s(T29)), T30, T12)
U7_gga(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → U9_gga(T29, T30, T12, add34_in_aga(T33, T30, T69))
U9_gga(T29, T30, T12, add34_out_aga(T33, T30, T69)) → U10_gga(T29, T30, T12, add44_in_gga(T69, T30, T12))
add44_in_gga(0, T78, T78) → add44_out_gga(0, T78, T78)
add44_in_gga(s(T85), T86, s(T88)) → U5_gga(T85, T86, T88, add44_in_gga(T85, T86, T88))
U5_gga(T85, T86, T88, add44_out_gga(T85, T86, T88)) → add44_out_gga(s(T85), T86, s(T88))
U10_gga(T29, T30, T12, add44_out_gga(T69, T30, T12)) → mult1_out_gga(s(s(T29)), T30, T12)

(37) UsableRulesProof (EQUIVALENT transformation)

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

(38) 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

(39) PiDPToQDPProof (SOUND transformation)

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

(40) 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.

(41) 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

(42) YES

(43) Obligation:

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

ADD34_IN_AGA(s(T63), T64, s(X101)) → ADD34_IN_AGA(T63, T64, X101)

The TRS R consists of the following rules:

mult1_in_gga(0, T5, 0) → mult1_out_gga(0, T5, 0)
mult1_in_gga(s(0), T24, T24) → mult1_out_gga(s(0), T24, T24)
mult1_in_gga(s(s(T29)), T30, T12) → U6_gga(T29, T30, T12, mult23_in_gga(T29, T30, X46))
mult23_in_gga(0, T40, 0) → mult23_out_gga(0, T40, 0)
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) → U2_gga(T45, T46, X74, mult23_in_gga(T45, T46, T49))
U2_gga(T45, T46, X74, mult23_out_gga(T45, T46, T49)) → U3_gga(T45, T46, X74, add34_in_aga(T49, T46, X74))
add34_in_aga(0, T58, T58) → add34_out_aga(0, T58, T58)
add34_in_aga(s(T63), T64, s(X101)) → U4_aga(T63, T64, X101, add34_in_aga(T63, T64, X101))
U4_aga(T63, T64, X101, add34_out_aga(T63, T64, X101)) → add34_out_aga(s(T63), T64, s(X101))
U3_gga(T45, T46, X74, add34_out_aga(T49, T46, X74)) → mult23_out_gga(s(T45), T46, X74)
U1_gga(T45, T46, X74, mult23_out_gga(T45, T46, X73)) → mult23_out_gga(s(T45), T46, X74)
U6_gga(T29, T30, T12, mult23_out_gga(T29, T30, X46)) → mult1_out_gga(s(s(T29)), T30, T12)
mult1_in_gga(s(s(T29)), T30, T12) → U7_gga(T29, T30, T12, mult23_in_gga(T29, T30, T33))
U7_gga(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → U8_gga(T29, T30, T12, add34_in_aga(T33, T30, X47))
U8_gga(T29, T30, T12, add34_out_aga(T33, T30, X47)) → mult1_out_gga(s(s(T29)), T30, T12)
U7_gga(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → U9_gga(T29, T30, T12, add34_in_aga(T33, T30, T69))
U9_gga(T29, T30, T12, add34_out_aga(T33, T30, T69)) → U10_gga(T29, T30, T12, add44_in_gga(T69, T30, T12))
add44_in_gga(0, T78, T78) → add44_out_gga(0, T78, T78)
add44_in_gga(s(T85), T86, s(T88)) → U5_gga(T85, T86, T88, add44_in_gga(T85, T86, T88))
U5_gga(T85, T86, T88, add44_out_gga(T85, T86, T88)) → add44_out_gga(s(T85), T86, s(T88))
U10_gga(T29, T30, T12, add44_out_gga(T69, T30, T12)) → mult1_out_gga(s(s(T29)), T30, T12)

(44) UsableRulesProof (EQUIVALENT transformation)

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

(45) Obligation:

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

ADD34_IN_AGA(s(T63), T64, s(X101)) → ADD34_IN_AGA(T63, T64, X101)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
ADD34_IN_AGA(x1, x2, x3) = ADD34_IN_AGA(x2)

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

(46) PiDPToQDPProof (SOUND transformation)

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

(47) Obligation:

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

ADD34_IN_AGA(T64) → ADD34_IN_AGA(T64)

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

(48) NonTerminationProof (EQUIVALENT transformation)

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from ADD34_IN_AGA(T64) to ADD34_IN_AGA(T64).

(49) NO

(50) 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:

mult1_in_gga(0, T5, 0) → mult1_out_gga(0, T5, 0)
mult1_in_gga(s(0), T24, T24) → mult1_out_gga(s(0), T24, T24)
mult1_in_gga(s(s(T29)), T30, T12) → U6_gga(T29, T30, T12, mult23_in_gga(T29, T30, X46))
mult23_in_gga(0, T40, 0) → mult23_out_gga(0, T40, 0)
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) → U2_gga(T45, T46, X74, mult23_in_gga(T45, T46, T49))
U2_gga(T45, T46, X74, mult23_out_gga(T45, T46, T49)) → U3_gga(T45, T46, X74, add34_in_aga(T49, T46, X74))
add34_in_aga(0, T58, T58) → add34_out_aga(0, T58, T58)
add34_in_aga(s(T63), T64, s(X101)) → U4_aga(T63, T64, X101, add34_in_aga(T63, T64, X101))
U4_aga(T63, T64, X101, add34_out_aga(T63, T64, X101)) → add34_out_aga(s(T63), T64, s(X101))
U3_gga(T45, T46, X74, add34_out_aga(T49, T46, X74)) → mult23_out_gga(s(T45), T46, X74)
U1_gga(T45, T46, X74, mult23_out_gga(T45, T46, X73)) → mult23_out_gga(s(T45), T46, X74)
U6_gga(T29, T30, T12, mult23_out_gga(T29, T30, X46)) → mult1_out_gga(s(s(T29)), T30, T12)
mult1_in_gga(s(s(T29)), T30, T12) → U7_gga(T29, T30, T12, mult23_in_gga(T29, T30, T33))
U7_gga(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → U8_gga(T29, T30, T12, add34_in_aga(T33, T30, X47))
U8_gga(T29, T30, T12, add34_out_aga(T33, T30, X47)) → mult1_out_gga(s(s(T29)), T30, T12)
U7_gga(T29, T30, T12, mult23_out_gga(T29, T30, T33)) → U9_gga(T29, T30, T12, add34_in_aga(T33, T30, T69))
U9_gga(T29, T30, T12, add34_out_aga(T33, T30, T69)) → U10_gga(T29, T30, T12, add44_in_gga(T69, T30, T12))
add44_in_gga(0, T78, T78) → add44_out_gga(0, T78, T78)
add44_in_gga(s(T85), T86, s(T88)) → U5_gga(T85, T86, T88, add44_in_gga(T85, T86, T88))
U5_gga(T85, T86, T88, add44_out_gga(T85, T86, T88)) → add44_out_gga(s(T85), T86, s(T88))
U10_gga(T29, T30, T12, add44_out_gga(T69, T30, T12)) → mult1_out_gga(s(s(T29)), T30, T12)

(51) UsableRulesProof (EQUIVALENT transformation)

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

(52) 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

(53) PiDPToQDPProof (SOUND transformation)

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

(54) 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.

(55) 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

(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) QDPSizeChangeProof (EQUIVALENT transformation)

(15) YES

(16) Obligation:

(17) UsableRulesProof (EQUIVALENT transformation)

(18) Obligation:

(19) PiDPToQDPProof (SOUND transformation)

(20) Obligation:

(21) NonTerminationProof (EQUIVALENT transformation)

(22) NO

(23) Obligation:

(24) UsableRulesProof (EQUIVALENT transformation)

(25) Obligation:

(26) PiDPToQDPProof (SOUND transformation)

(27) Obligation:

(28) QDPSizeChangeProof (EQUIVALENT transformation)

(29) YES

(30) PrologToPiTRSProof (SOUND transformation)

(31) Obligation:

(32) DependencyPairsProof (EQUIVALENT transformation)

(33) Obligation:

(34) DependencyGraphProof (EQUIVALENT transformation)

(35) Complex Obligation (AND)

(36) Obligation:

(37) UsableRulesProof (EQUIVALENT transformation)

(38) Obligation:

(39) PiDPToQDPProof (SOUND transformation)

(40) Obligation:

(41) QDPSizeChangeProof (EQUIVALENT transformation)

(42) YES

(43) Obligation:

(44) UsableRulesProof (EQUIVALENT transformation)

(45) Obligation:

(46) PiDPToQDPProof (SOUND transformation)

(47) Obligation:

(48) NonTerminationProof (EQUIVALENT transformation)

(49) NO

(50) Obligation:

(51) UsableRulesProof (EQUIVALENT transformation)

(52) Obligation:

(53) PiDPToQDPProof (SOUND transformation)

(54) Obligation:

(55) QDPSizeChangeProof (EQUIVALENT transformation)

(56) YES