Left Termination of the query pattern m_in_3(g, g, a) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

m(X, 0, Z) :- ','(!, =(Z, X)).
m(0, Y, Z) :- ','(!, =(Z, 0)).
m(X, Y, Z) :- ','(p(X, A), ','(p(Y, B), m(A, B, Z))).
p(0, 0).
p(s(0), 0).
p(s(s(X)), s(Y)) :- p(s(X), Y).
=(X, X).

Queries:

m(g,g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: m(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: m(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | m(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | m(T1, T2, T3), SCOPE: 1, CLAUSE: 2 | ?_1\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(!_1, =(T6, T4)) | m(T4, 0, T3), SCOPE: 1, CLAUSE: 1 | m(T4, 0, T3), SCOPE: 1, CLAUSE: 2 | ?_1\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: m(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | m(T1, T2, T3), SCOPE: 1, CLAUSE: 2 | ?_1\n\nT1 is ground\nT2 is ground\nX3 is free; \nX4 is free; \nm(T1, T2, T3) !=? m(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: =(T6, T4)\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: =(T6, T4), SCOPE: 2, CLAUSE: 6\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(!_1, =(T10, 0)) | m(0, T8, T3), SCOPE: 1, CLAUSE: 2 | ?_1\n\nT8 is ground\nX3 is free; \nX4 is free; \nm(0, T8, T3) !=? m(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: m(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground\nX3 is free; \nX4 is free; \nX9 is free; \nX10 is free; \nm(T1, T2, T3) !=? m(X3, 0, X4)\nm(T1, T2, T3) !=? m(0, X9, X10)", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: =(T10, 0)\n\nX3 is free\nX4 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: =(T10, 0), SCOPE: 3, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(p(T12, X19), ','(p(T13, X20), m(X19, X20, T15)))\n\nT12 is ground\nT13 is ground\nX3 is free; \nX4 is free; \nX9 is free; \nX10 is free; \nX19 is free; \nX20 is free; \nm(T12, T13, T3) !=? m(X3, 0, X4)\nm(T12, T13, T3) !=? m(0, X9, X10)", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: p(T12, X19)\n\nT12 is ground\nX3 is free; \nX4 is free; \nX9 is free; \nX10 is free; \nX19 is free; \nm(T12, T13, T3) !=? m(0, X9, X10)", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(p(T13, X20), m(T16, X20, T15))\n\nT13 is ground\nT16 is ground\nX3 is free; \nX4 is free; \nX9 is free; \nX10 is free; \nX20 is free; \nm(T12, T13, T3) !=? m(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: p(T12, X19), SCOPE: 4, CLAUSE: 3 | p(T12, X19), SCOPE: 4, CLAUSE: 4 | p(T12, X19), SCOPE: 4, CLAUSE: 5\n\nT12 is ground\nX9 is free; \nX10 is free; \nX19 is free; \nm(T12, T13, T3) !=? m(0, X9, X10)", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: p(T12, X19), SCOPE: 4, CLAUSE: 4 | p(T12, X19), SCOPE: 4, CLAUSE: 5\n\nT12 is ground\nX9 is free; \nX10 is free; \nX19 is free; \nm(T12, T13, T3) !=? m(0, X9, X10)\np(T12, X19) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: p(T12, X19), SCOPE: 4, CLAUSE: 4\n\nT12 is ground\nX9 is free; \nX10 is free; \nX19 is free; \nm(T12, T13, T3) !=? m(0, X9, X10)\np(T12, X19) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: p(T12, X19), SCOPE: 4, CLAUSE: 5\n\nT12 is ground\nX9 is free; \nX10 is free; \nX19 is free; \nm(T12, T13, T3) !=? m(0, X9, X10)\np(T12, X19) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: true\n\nX9 is free\nX10 is free; \nX19 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\nX9 is free\nX10 is free; \nX19 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: p(s(T17), X25)\n\nT17 is ground\nX9 is free; \nX10 is free; \nX19 is free; \nX25 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\nX9 is free\nX10 is free; \nX19 is free; \nX23 is free; \nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: p(s(T17), X25), SCOPE: 5, CLAUSE: 3 | p(s(T17), X25), SCOPE: 5, CLAUSE: 4 | p(s(T17), X25), SCOPE: 5, CLAUSE: 5\n\nT17 is ground\nX25 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: p(s(T17), X25), SCOPE: 5, CLAUSE: 4 | p(s(T17), X25), SCOPE: 5, CLAUSE: 5\n\nT17 is ground\nX25 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: p(s(T17), X25), SCOPE: 5, CLAUSE: 4\n\nT17 is ground\nX25 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: p(s(T17), X25), SCOPE: 5, CLAUSE: 5\n\nT17 is ground\nX25 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\nX25 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: p(s(T18), X30)\n\nT18 is ground\nX30 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\nX25 is free\nX28 is free; \nX29 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: p(s(T18), X30), SCOPE: 6, CLAUSE: 3 | p(s(T18), X30), SCOPE: 6, CLAUSE: 4 | p(s(T18), X30), SCOPE: 6, CLAUSE: 5\n\nT18 is ground\nX30 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: p(s(T18), X30), SCOPE: 6, CLAUSE: 4 | p(s(T18), X30), SCOPE: 6, CLAUSE: 5\n\nT18 is ground\nX30 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: p(T13, X20)\n\nT13 is ground\nX3 is free; \nX4 is free; \nX20 is free; \nm(T12, T13, T3) !=? m(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: m(T16, T19, T15)\n\nT16 is ground\nT19 is ground\nX3 is free; \nX4 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: p(T13, X20), SCOPE: 7, CLAUSE: 3 | p(T13, X20), SCOPE: 7, CLAUSE: 4 | p(T13, X20), SCOPE: 7, CLAUSE: 5\n\nT13 is ground\nX3 is free; \nX4 is free; \nX20 is free; \nm(T12, T13, T3) !=? m(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: p(T13, X20), SCOPE: 7, CLAUSE: 4 | p(T13, X20), SCOPE: 7, CLAUSE: 5\n\nT13 is ground\nX3 is free; \nX4 is free; \nX20 is free; \nm(T12, T13, T3) !=? m(X3, 0, X4)\np(T13, X20) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: p(T13, X20), SCOPE: 7, CLAUSE: 4\n\nT13 is ground\nX3 is free; \nX4 is free; \nX20 is free; \nm(T12, T13, T3) !=? m(X3, 0, X4)\np(T13, X20) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: p(T13, X20), SCOPE: 7, CLAUSE: 5\n\nT13 is ground\nX3 is free; \nX4 is free; \nX20 is free; \nm(T12, T13, T3) !=? m(X3, 0, X4)\np(T13, X20) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: true\n\nX3 is free\nX4 is free; \nX20 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\nX3 is free\nX4 is free; \nX20 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: p(s(T20), X35)\n\nT20 is ground\nX3 is free; \nX4 is free; \nX20 is free; \nX35 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\nX3 is free\nX4 is free; \nX20 is free; \nX33 is free; \nX34 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 51 [arrowhead = none ];
51 -> 2;

52 [label="EVAL with clause\nm(X3, 0, X4) :- ','(!, =(X4, X3)).\nand substitution\nT1 -> T4\nX3 -> T4\nT2 -> 0\nT3 -> T6\nX4 -> T6\nT5 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 52 [arrowhead = none ];
52 -> 3;

53 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 53 [arrowhead = none ];
53 -> 4;

54 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 54 [arrowhead = none ];
54 -> 5;

55 [label="EVAL with clause\nm(0, X9, X10) :- ','(!, =(X10, 0)).\nand substitution\nT1 -> 0\nT2 -> T8\nX9 -> T8\nT3 -> T10\nX10 -> T10\nT9 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 55 [arrowhead = none ];
55 -> 10;

56 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 56 [arrowhead = none ];
56 -> 11;

57 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 57 [arrowhead = none ];
57 -> 6;

58 [label="EVAL with clause\n=(X6, X6).\nand substitution\nT6 -> T7\nX6 -> T7\nT4 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 58 [arrowhead = none ];
58 -> 7;

59 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 59 [arrowhead = none ];
59 -> 8;

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

61 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 61 [arrowhead = none ];
61 -> 12;

62 [label="EVAL with clause\nm(X16, X17, X18) :- ','(p(X16, X19), ','(p(X17, X20), m(X19, X20, X18))).\nand substitution\nT1 -> T12\nX16 -> T12\nT2 -> T13\nX17 -> T13\nT3 -> T15\nX18 -> T15\nT14 -> T15", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 62 [arrowhead = none ];
62 -> 17;

63 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 63 [arrowhead = none ];
63 -> 13;

64 [label="EVAL with clause\n=(X12, X12).\nand substitution\nT10 -> 0\nX12 -> 0\nT11 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 64 [arrowhead = none ];
64 -> 14;

65 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 65 [arrowhead = none ];
65 -> 15;

66 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 66 [arrowhead = none ];
66 -> 16;

67 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
17 -> 67 [arrowhead = none ];
67 -> 18;

68 [label="SPLIT 2\nnew knowledge:\nT16 is ground\nreplacements:\nX19 -> T16", fontsize=14, style = filled, fillcolor = violet];
17 -> 68 [arrowhead = none ];
68 -> 19;

69 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 69 [arrowhead = none ];
69 -> 20;

70 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
19 -> 70 [arrowhead = none ];
70 -> 40;

71 [label="SPLIT 2\nnew knowledge:\nT19 is ground\nreplacements:\nX20 -> T19", fontsize=14, style = filled, fillcolor = violet];
19 -> 71 [arrowhead = none ];
71 -> 41;

72 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (m(T12, T13, T3), m(0, X9, X10))", fontsize=14, style = filled, fillcolor = white];
20 -> 72 [arrowhead = none ];
72 -> 21;

73 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
21 -> 73 [arrowhead = none ];
73 -> 22;

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

75 [label="EVAL with clause\np(s(0), 0).\nand substitution\nT12 -> s(0)\nX19 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 75 [arrowhead = none ];
75 -> 24;

76 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 76 [arrowhead = none ];
76 -> 25;

77 [label="EVAL with clause\np(s(s(X23)), s(X24)) :- p(s(X23), X24).\nand substitution\nX23 -> T17\nT12 -> s(s(T17))\nX24 -> X25\nX19 -> s(X25)", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 77 [arrowhead = none ];
77 -> 27;

78 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 78 [arrowhead = none ];
78 -> 28;

79 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 79 [arrowhead = none ];
79 -> 26;

80 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
27 -> 80 [arrowhead = none ];
80 -> 29;

81 [label="BACKTRACK\nfor clause: p(0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
29 -> 81 [arrowhead = none ];
81 -> 30;

82 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
30 -> 82 [arrowhead = none ];
82 -> 31;

83 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
30 -> 83 [arrowhead = none ];
83 -> 32;

84 [label="EVAL with clause\np(s(0), 0).\nand substitution\nT17 -> 0\nX25 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 84 [arrowhead = none ];
84 -> 33;

85 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 85 [arrowhead = none ];
85 -> 34;

86 [label="EVAL with clause\np(s(s(X28)), s(X29)) :- p(s(X28), X29).\nand substitution\nX28 -> T18\nT17 -> s(T18)\nX29 -> X30\nX25 -> s(X30)", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 86 [arrowhead = none ];
86 -> 36;

87 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 87 [arrowhead = none ];
87 -> 37;

88 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 88 [arrowhead = none ];
88 -> 35;

89 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
36 -> 89 [arrowhead = none ];
89 -> 38;

90 [label="BACKTRACK\nfor clause: p(0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
38 -> 90 [arrowhead = none ];
90 -> 39;

91 [label="INSTANCE with matching:\nT17 -> T18\nX25 -> X30", fontsize=14, style = filled, fillcolor = lightgrey];
39 -> 91 [arrowhead = none , style=dashed];
91 -> 30[style=dashed];

92 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
40 -> 92 [arrowhead = none ];
92 -> 42;

93 [label="INSTANCE with matching:\nT1 -> T16\nT2 -> T19\nT3 -> T15", fontsize=14, style = filled, fillcolor = lightgrey];
41 -> 93 [arrowhead = none , style=dashed];
93 -> 1[style=dashed];

94 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (m(T12, T13, T3), m(X3, 0, X4))", fontsize=14, style = filled, fillcolor = white];
42 -> 94 [arrowhead = none ];
94 -> 43;

95 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
43 -> 95 [arrowhead = none ];
95 -> 44;

96 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
43 -> 96 [arrowhead = none ];
96 -> 45;

97 [label="EVAL with clause\np(s(0), 0).\nand substitution\nT13 -> s(0)\nX20 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 97 [arrowhead = none ];
97 -> 46;

98 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 98 [arrowhead = none ];
98 -> 47;

99 [label="EVAL with clause\np(s(s(X33)), s(X34)) :- p(s(X33), X34).\nand substitution\nX33 -> T20\nT13 -> s(s(T20))\nX34 -> X35\nX20 -> s(X35)", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 99 [arrowhead = none ];
99 -> 49;

100 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 100 [arrowhead = none ];
100 -> 50;

101 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 101 [arrowhead = none ];
101 -> 48;

102 [label="INSTANCE with matching:\nT17 -> T20\nX25 -> X35\nX9 -> X3\nX10 -> X4\nX19 -> X20", fontsize=14, style = filled, fillcolor = lightgrey];
49 -> 102 [arrowhead = none , style=dashed];
102 -> 27[style=dashed];

}

(2) Obligation:

Clauses:

p30(0, 0).
p30(s(T18), s(X30)) :- p30(T18, X30).
m1(T7, 0, T7).
m1(0, T8, 0).
m1(T12, T13, T15) :- p18(T12, X19).
m1(T12, T13, T15) :- ','(p18(T12, T16), p40(T13, X20)).
m1(T12, T13, T15) :- ','(p18(T12, T16), ','(p40(T13, T19), m1(T16, T19, T15))).
p27(0, 0).
p27(s(T18), s(X30)) :- p30(T18, X30).
p18(s(0), 0).
p18(s(s(0)), s(0)).
p18(s(s(s(T18))), s(s(X30))) :- p30(T18, X30).
p40(s(0), 0).
p40(s(s(T20)), s(X35)) :- p27(T20, X35).

Queries:

m1(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:
m1_in: (b,b,f)
p18_in: (b,f)
p30_in: (b,f)
p40_in: (b,f)
p27_in: (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

m1_in_gga(T7, 0, T7) → m1_out_gga(T7, 0, T7)
m1_in_gga(0, T8, 0) → m1_out_gga(0, T8, 0)
m1_in_gga(T12, T13, T15) → U2_gga(T12, T13, T15, p18_in_ga(T12, X19))
p18_in_ga(s(0), 0) → p18_out_ga(s(0), 0)
p18_in_ga(s(s(0)), s(0)) → p18_out_ga(s(s(0)), s(0))
p18_in_ga(s(s(s(T18))), s(s(X30))) → U8_ga(T18, X30, p30_in_ga(T18, X30))
p30_in_ga(0, 0) → p30_out_ga(0, 0)
p30_in_ga(s(T18), s(X30)) → U1_ga(T18, X30, p30_in_ga(T18, X30))
U1_ga(T18, X30, p30_out_ga(T18, X30)) → p30_out_ga(s(T18), s(X30))
U8_ga(T18, X30, p30_out_ga(T18, X30)) → p18_out_ga(s(s(s(T18))), s(s(X30)))
U2_gga(T12, T13, T15, p18_out_ga(T12, X19)) → m1_out_gga(T12, T13, T15)
m1_in_gga(T12, T13, T15) → U3_gga(T12, T13, T15, p18_in_ga(T12, T16))
U3_gga(T12, T13, T15, p18_out_ga(T12, T16)) → U4_gga(T12, T13, T15, p40_in_ga(T13, X20))
p40_in_ga(s(0), 0) → p40_out_ga(s(0), 0)
p40_in_ga(s(s(T20)), s(X35)) → U9_ga(T20, X35, p27_in_ga(T20, X35))
p27_in_ga(0, 0) → p27_out_ga(0, 0)
p27_in_ga(s(T18), s(X30)) → U7_ga(T18, X30, p30_in_ga(T18, X30))
U7_ga(T18, X30, p30_out_ga(T18, X30)) → p27_out_ga(s(T18), s(X30))
U9_ga(T20, X35, p27_out_ga(T20, X35)) → p40_out_ga(s(s(T20)), s(X35))
U4_gga(T12, T13, T15, p40_out_ga(T13, X20)) → m1_out_gga(T12, T13, T15)
U3_gga(T12, T13, T15, p18_out_ga(T12, T16)) → U5_gga(T12, T13, T15, T16, p40_in_ga(T13, T19))
U5_gga(T12, T13, T15, T16, p40_out_ga(T13, T19)) → U6_gga(T12, T13, T15, m1_in_gga(T16, T19, T15))
U6_gga(T12, T13, T15, m1_out_gga(T16, T19, T15)) → m1_out_gga(T12, T13, T15)

The argument filtering Pi contains the following mapping:
m1_in_gga(x1, x2, x3)  =  m1_in_gga(x1, x2)
0  =  0
m1_out_gga(x1, x2, x3)  =  m1_out_gga
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
p18_in_ga(x1, x2)  =  p18_in_ga(x1)
s(x1)  =  s(x1)
p18_out_ga(x1, x2)  =  p18_out_ga(x2)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
p30_in_ga(x1, x2)  =  p30_in_ga(x1)
p30_out_ga(x1, x2)  =  p30_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x2, x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
p40_in_ga(x1, x2)  =  p40_in_ga(x1)
p40_out_ga(x1, x2)  =  p40_out_ga(x2)
U9_ga(x1, x2, x3)  =  U9_ga(x3)
p27_in_ga(x1, x2)  =  p27_in_ga(x1)
p27_out_ga(x1, x2)  =  p27_out_ga(x2)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x4, x5)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

m1_in_gga(T7, 0, T7) → m1_out_gga(T7, 0, T7)
m1_in_gga(0, T8, 0) → m1_out_gga(0, T8, 0)
m1_in_gga(T12, T13, T15) → U2_gga(T12, T13, T15, p18_in_ga(T12, X19))
p18_in_ga(s(0), 0) → p18_out_ga(s(0), 0)
p18_in_ga(s(s(0)), s(0)) → p18_out_ga(s(s(0)), s(0))
p18_in_ga(s(s(s(T18))), s(s(X30))) → U8_ga(T18, X30, p30_in_ga(T18, X30))
p30_in_ga(0, 0) → p30_out_ga(0, 0)
p30_in_ga(s(T18), s(X30)) → U1_ga(T18, X30, p30_in_ga(T18, X30))
U1_ga(T18, X30, p30_out_ga(T18, X30)) → p30_out_ga(s(T18), s(X30))
U8_ga(T18, X30, p30_out_ga(T18, X30)) → p18_out_ga(s(s(s(T18))), s(s(X30)))
U2_gga(T12, T13, T15, p18_out_ga(T12, X19)) → m1_out_gga(T12, T13, T15)
m1_in_gga(T12, T13, T15) → U3_gga(T12, T13, T15, p18_in_ga(T12, T16))
U3_gga(T12, T13, T15, p18_out_ga(T12, T16)) → U4_gga(T12, T13, T15, p40_in_ga(T13, X20))
p40_in_ga(s(0), 0) → p40_out_ga(s(0), 0)
p40_in_ga(s(s(T20)), s(X35)) → U9_ga(T20, X35, p27_in_ga(T20, X35))
p27_in_ga(0, 0) → p27_out_ga(0, 0)
p27_in_ga(s(T18), s(X30)) → U7_ga(T18, X30, p30_in_ga(T18, X30))
U7_ga(T18, X30, p30_out_ga(T18, X30)) → p27_out_ga(s(T18), s(X30))
U9_ga(T20, X35, p27_out_ga(T20, X35)) → p40_out_ga(s(s(T20)), s(X35))
U4_gga(T12, T13, T15, p40_out_ga(T13, X20)) → m1_out_gga(T12, T13, T15)
U3_gga(T12, T13, T15, p18_out_ga(T12, T16)) → U5_gga(T12, T13, T15, T16, p40_in_ga(T13, T19))
U5_gga(T12, T13, T15, T16, p40_out_ga(T13, T19)) → U6_gga(T12, T13, T15, m1_in_gga(T16, T19, T15))
U6_gga(T12, T13, T15, m1_out_gga(T16, T19, T15)) → m1_out_gga(T12, T13, T15)

The argument filtering Pi contains the following mapping:
m1_in_gga(x1, x2, x3)  =  m1_in_gga(x1, x2)
0  =  0
m1_out_gga(x1, x2, x3)  =  m1_out_gga
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
p18_in_ga(x1, x2)  =  p18_in_ga(x1)
s(x1)  =  s(x1)
p18_out_ga(x1, x2)  =  p18_out_ga(x2)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
p30_in_ga(x1, x2)  =  p30_in_ga(x1)
p30_out_ga(x1, x2)  =  p30_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x2, x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
p40_in_ga(x1, x2)  =  p40_in_ga(x1)
p40_out_ga(x1, x2)  =  p40_out_ga(x2)
U9_ga(x1, x2, x3)  =  U9_ga(x3)
p27_in_ga(x1, x2)  =  p27_in_ga(x1)
p27_out_ga(x1, x2)  =  p27_out_ga(x2)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x4, x5)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)

(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:

M1_IN_GGA(T12, T13, T15) → U2_GGA(T12, T13, T15, p18_in_ga(T12, X19))
M1_IN_GGA(T12, T13, T15) → P18_IN_GA(T12, X19)
P18_IN_GA(s(s(s(T18))), s(s(X30))) → U8_GA(T18, X30, p30_in_ga(T18, X30))
P18_IN_GA(s(s(s(T18))), s(s(X30))) → P30_IN_GA(T18, X30)
P30_IN_GA(s(T18), s(X30)) → U1_GA(T18, X30, p30_in_ga(T18, X30))
P30_IN_GA(s(T18), s(X30)) → P30_IN_GA(T18, X30)
M1_IN_GGA(T12, T13, T15) → U3_GGA(T12, T13, T15, p18_in_ga(T12, T16))
U3_GGA(T12, T13, T15, p18_out_ga(T12, T16)) → U4_GGA(T12, T13, T15, p40_in_ga(T13, X20))
U3_GGA(T12, T13, T15, p18_out_ga(T12, T16)) → P40_IN_GA(T13, X20)
P40_IN_GA(s(s(T20)), s(X35)) → U9_GA(T20, X35, p27_in_ga(T20, X35))
P40_IN_GA(s(s(T20)), s(X35)) → P27_IN_GA(T20, X35)
P27_IN_GA(s(T18), s(X30)) → U7_GA(T18, X30, p30_in_ga(T18, X30))
P27_IN_GA(s(T18), s(X30)) → P30_IN_GA(T18, X30)
U3_GGA(T12, T13, T15, p18_out_ga(T12, T16)) → U5_GGA(T12, T13, T15, T16, p40_in_ga(T13, T19))
U5_GGA(T12, T13, T15, T16, p40_out_ga(T13, T19)) → U6_GGA(T12, T13, T15, m1_in_gga(T16, T19, T15))
U5_GGA(T12, T13, T15, T16, p40_out_ga(T13, T19)) → M1_IN_GGA(T16, T19, T15)

The TRS R consists of the following rules:

m1_in_gga(T7, 0, T7) → m1_out_gga(T7, 0, T7)
m1_in_gga(0, T8, 0) → m1_out_gga(0, T8, 0)
m1_in_gga(T12, T13, T15) → U2_gga(T12, T13, T15, p18_in_ga(T12, X19))
p18_in_ga(s(0), 0) → p18_out_ga(s(0), 0)
p18_in_ga(s(s(0)), s(0)) → p18_out_ga(s(s(0)), s(0))
p18_in_ga(s(s(s(T18))), s(s(X30))) → U8_ga(T18, X30, p30_in_ga(T18, X30))
p30_in_ga(0, 0) → p30_out_ga(0, 0)
p30_in_ga(s(T18), s(X30)) → U1_ga(T18, X30, p30_in_ga(T18, X30))
U1_ga(T18, X30, p30_out_ga(T18, X30)) → p30_out_ga(s(T18), s(X30))
U8_ga(T18, X30, p30_out_ga(T18, X30)) → p18_out_ga(s(s(s(T18))), s(s(X30)))
U2_gga(T12, T13, T15, p18_out_ga(T12, X19)) → m1_out_gga(T12, T13, T15)
m1_in_gga(T12, T13, T15) → U3_gga(T12, T13, T15, p18_in_ga(T12, T16))
U3_gga(T12, T13, T15, p18_out_ga(T12, T16)) → U4_gga(T12, T13, T15, p40_in_ga(T13, X20))
p40_in_ga(s(0), 0) → p40_out_ga(s(0), 0)
p40_in_ga(s(s(T20)), s(X35)) → U9_ga(T20, X35, p27_in_ga(T20, X35))
p27_in_ga(0, 0) → p27_out_ga(0, 0)
p27_in_ga(s(T18), s(X30)) → U7_ga(T18, X30, p30_in_ga(T18, X30))
U7_ga(T18, X30, p30_out_ga(T18, X30)) → p27_out_ga(s(T18), s(X30))
U9_ga(T20, X35, p27_out_ga(T20, X35)) → p40_out_ga(s(s(T20)), s(X35))
U4_gga(T12, T13, T15, p40_out_ga(T13, X20)) → m1_out_gga(T12, T13, T15)
U3_gga(T12, T13, T15, p18_out_ga(T12, T16)) → U5_gga(T12, T13, T15, T16, p40_in_ga(T13, T19))
U5_gga(T12, T13, T15, T16, p40_out_ga(T13, T19)) → U6_gga(T12, T13, T15, m1_in_gga(T16, T19, T15))
U6_gga(T12, T13, T15, m1_out_gga(T16, T19, T15)) → m1_out_gga(T12, T13, T15)

The argument filtering Pi contains the following mapping:
m1_in_gga(x1, x2, x3)  =  m1_in_gga(x1, x2)
0  =  0
m1_out_gga(x1, x2, x3)  =  m1_out_gga
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
p18_in_ga(x1, x2)  =  p18_in_ga(x1)
s(x1)  =  s(x1)
p18_out_ga(x1, x2)  =  p18_out_ga(x2)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
p30_in_ga(x1, x2)  =  p30_in_ga(x1)
p30_out_ga(x1, x2)  =  p30_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x2, x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
p40_in_ga(x1, x2)  =  p40_in_ga(x1)
p40_out_ga(x1, x2)  =  p40_out_ga(x2)
U9_ga(x1, x2, x3)  =  U9_ga(x3)
p27_in_ga(x1, x2)  =  p27_in_ga(x1)
p27_out_ga(x1, x2)  =  p27_out_ga(x2)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x4, x5)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
M1_IN_GGA(x1, x2, x3)  =  M1_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x4)
P18_IN_GA(x1, x2)  =  P18_IN_GA(x1)
U8_GA(x1, x2, x3)  =  U8_GA(x3)
P30_IN_GA(x1, x2)  =  P30_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x2, x4)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x4)
P40_IN_GA(x1, x2)  =  P40_IN_GA(x1)
U9_GA(x1, x2, x3)  =  U9_GA(x3)
P27_IN_GA(x1, x2)  =  P27_IN_GA(x1)
U7_GA(x1, x2, x3)  =  U7_GA(x3)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x4, x5)
U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x4)

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

(6) Obligation:

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

M1_IN_GGA(T12, T13, T15) → U2_GGA(T12, T13, T15, p18_in_ga(T12, X19))
M1_IN_GGA(T12, T13, T15) → P18_IN_GA(T12, X19)
P18_IN_GA(s(s(s(T18))), s(s(X30))) → U8_GA(T18, X30, p30_in_ga(T18, X30))
P18_IN_GA(s(s(s(T18))), s(s(X30))) → P30_IN_GA(T18, X30)
P30_IN_GA(s(T18), s(X30)) → U1_GA(T18, X30, p30_in_ga(T18, X30))
P30_IN_GA(s(T18), s(X30)) → P30_IN_GA(T18, X30)
M1_IN_GGA(T12, T13, T15) → U3_GGA(T12, T13, T15, p18_in_ga(T12, T16))
U3_GGA(T12, T13, T15, p18_out_ga(T12, T16)) → U4_GGA(T12, T13, T15, p40_in_ga(T13, X20))
U3_GGA(T12, T13, T15, p18_out_ga(T12, T16)) → P40_IN_GA(T13, X20)
P40_IN_GA(s(s(T20)), s(X35)) → U9_GA(T20, X35, p27_in_ga(T20, X35))
P40_IN_GA(s(s(T20)), s(X35)) → P27_IN_GA(T20, X35)
P27_IN_GA(s(T18), s(X30)) → U7_GA(T18, X30, p30_in_ga(T18, X30))
P27_IN_GA(s(T18), s(X30)) → P30_IN_GA(T18, X30)
U3_GGA(T12, T13, T15, p18_out_ga(T12, T16)) → U5_GGA(T12, T13, T15, T16, p40_in_ga(T13, T19))
U5_GGA(T12, T13, T15, T16, p40_out_ga(T13, T19)) → U6_GGA(T12, T13, T15, m1_in_gga(T16, T19, T15))
U5_GGA(T12, T13, T15, T16, p40_out_ga(T13, T19)) → M1_IN_GGA(T16, T19, T15)

The TRS R consists of the following rules:

m1_in_gga(T7, 0, T7) → m1_out_gga(T7, 0, T7)
m1_in_gga(0, T8, 0) → m1_out_gga(0, T8, 0)
m1_in_gga(T12, T13, T15) → U2_gga(T12, T13, T15, p18_in_ga(T12, X19))
p18_in_ga(s(0), 0) → p18_out_ga(s(0), 0)
p18_in_ga(s(s(0)), s(0)) → p18_out_ga(s(s(0)), s(0))
p18_in_ga(s(s(s(T18))), s(s(X30))) → U8_ga(T18, X30, p30_in_ga(T18, X30))
p30_in_ga(0, 0) → p30_out_ga(0, 0)
p30_in_ga(s(T18), s(X30)) → U1_ga(T18, X30, p30_in_ga(T18, X30))
U1_ga(T18, X30, p30_out_ga(T18, X30)) → p30_out_ga(s(T18), s(X30))
U8_ga(T18, X30, p30_out_ga(T18, X30)) → p18_out_ga(s(s(s(T18))), s(s(X30)))
U2_gga(T12, T13, T15, p18_out_ga(T12, X19)) → m1_out_gga(T12, T13, T15)
m1_in_gga(T12, T13, T15) → U3_gga(T12, T13, T15, p18_in_ga(T12, T16))
U3_gga(T12, T13, T15, p18_out_ga(T12, T16)) → U4_gga(T12, T13, T15, p40_in_ga(T13, X20))
p40_in_ga(s(0), 0) → p40_out_ga(s(0), 0)
p40_in_ga(s(s(T20)), s(X35)) → U9_ga(T20, X35, p27_in_ga(T20, X35))
p27_in_ga(0, 0) → p27_out_ga(0, 0)
p27_in_ga(s(T18), s(X30)) → U7_ga(T18, X30, p30_in_ga(T18, X30))
U7_ga(T18, X30, p30_out_ga(T18, X30)) → p27_out_ga(s(T18), s(X30))
U9_ga(T20, X35, p27_out_ga(T20, X35)) → p40_out_ga(s(s(T20)), s(X35))
U4_gga(T12, T13, T15, p40_out_ga(T13, X20)) → m1_out_gga(T12, T13, T15)
U3_gga(T12, T13, T15, p18_out_ga(T12, T16)) → U5_gga(T12, T13, T15, T16, p40_in_ga(T13, T19))
U5_gga(T12, T13, T15, T16, p40_out_ga(T13, T19)) → U6_gga(T12, T13, T15, m1_in_gga(T16, T19, T15))
U6_gga(T12, T13, T15, m1_out_gga(T16, T19, T15)) → m1_out_gga(T12, T13, T15)

The argument filtering Pi contains the following mapping:
m1_in_gga(x1, x2, x3)  =  m1_in_gga(x1, x2)
0  =  0
m1_out_gga(x1, x2, x3)  =  m1_out_gga
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
p18_in_ga(x1, x2)  =  p18_in_ga(x1)
s(x1)  =  s(x1)
p18_out_ga(x1, x2)  =  p18_out_ga(x2)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
p30_in_ga(x1, x2)  =  p30_in_ga(x1)
p30_out_ga(x1, x2)  =  p30_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x2, x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
p40_in_ga(x1, x2)  =  p40_in_ga(x1)
p40_out_ga(x1, x2)  =  p40_out_ga(x2)
U9_ga(x1, x2, x3)  =  U9_ga(x3)
p27_in_ga(x1, x2)  =  p27_in_ga(x1)
p27_out_ga(x1, x2)  =  p27_out_ga(x2)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x4, x5)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
M1_IN_GGA(x1, x2, x3)  =  M1_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x4)
P18_IN_GA(x1, x2)  =  P18_IN_GA(x1)
U8_GA(x1, x2, x3)  =  U8_GA(x3)
P30_IN_GA(x1, x2)  =  P30_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x2, x4)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x4)
P40_IN_GA(x1, x2)  =  P40_IN_GA(x1)
U9_GA(x1, x2, x3)  =  U9_GA(x3)
P27_IN_GA(x1, x2)  =  P27_IN_GA(x1)
U7_GA(x1, x2, x3)  =  U7_GA(x3)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x4, x5)
U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x4)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 12 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

P30_IN_GA(s(T18), s(X30)) → P30_IN_GA(T18, X30)

The TRS R consists of the following rules:

m1_in_gga(T7, 0, T7) → m1_out_gga(T7, 0, T7)
m1_in_gga(0, T8, 0) → m1_out_gga(0, T8, 0)
m1_in_gga(T12, T13, T15) → U2_gga(T12, T13, T15, p18_in_ga(T12, X19))
p18_in_ga(s(0), 0) → p18_out_ga(s(0), 0)
p18_in_ga(s(s(0)), s(0)) → p18_out_ga(s(s(0)), s(0))
p18_in_ga(s(s(s(T18))), s(s(X30))) → U8_ga(T18, X30, p30_in_ga(T18, X30))
p30_in_ga(0, 0) → p30_out_ga(0, 0)
p30_in_ga(s(T18), s(X30)) → U1_ga(T18, X30, p30_in_ga(T18, X30))
U1_ga(T18, X30, p30_out_ga(T18, X30)) → p30_out_ga(s(T18), s(X30))
U8_ga(T18, X30, p30_out_ga(T18, X30)) → p18_out_ga(s(s(s(T18))), s(s(X30)))
U2_gga(T12, T13, T15, p18_out_ga(T12, X19)) → m1_out_gga(T12, T13, T15)
m1_in_gga(T12, T13, T15) → U3_gga(T12, T13, T15, p18_in_ga(T12, T16))
U3_gga(T12, T13, T15, p18_out_ga(T12, T16)) → U4_gga(T12, T13, T15, p40_in_ga(T13, X20))
p40_in_ga(s(0), 0) → p40_out_ga(s(0), 0)
p40_in_ga(s(s(T20)), s(X35)) → U9_ga(T20, X35, p27_in_ga(T20, X35))
p27_in_ga(0, 0) → p27_out_ga(0, 0)
p27_in_ga(s(T18), s(X30)) → U7_ga(T18, X30, p30_in_ga(T18, X30))
U7_ga(T18, X30, p30_out_ga(T18, X30)) → p27_out_ga(s(T18), s(X30))
U9_ga(T20, X35, p27_out_ga(T20, X35)) → p40_out_ga(s(s(T20)), s(X35))
U4_gga(T12, T13, T15, p40_out_ga(T13, X20)) → m1_out_gga(T12, T13, T15)
U3_gga(T12, T13, T15, p18_out_ga(T12, T16)) → U5_gga(T12, T13, T15, T16, p40_in_ga(T13, T19))
U5_gga(T12, T13, T15, T16, p40_out_ga(T13, T19)) → U6_gga(T12, T13, T15, m1_in_gga(T16, T19, T15))
U6_gga(T12, T13, T15, m1_out_gga(T16, T19, T15)) → m1_out_gga(T12, T13, T15)

The argument filtering Pi contains the following mapping:
m1_in_gga(x1, x2, x3)  =  m1_in_gga(x1, x2)
0  =  0
m1_out_gga(x1, x2, x3)  =  m1_out_gga
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
p18_in_ga(x1, x2)  =  p18_in_ga(x1)
s(x1)  =  s(x1)
p18_out_ga(x1, x2)  =  p18_out_ga(x2)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
p30_in_ga(x1, x2)  =  p30_in_ga(x1)
p30_out_ga(x1, x2)  =  p30_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x2, x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
p40_in_ga(x1, x2)  =  p40_in_ga(x1)
p40_out_ga(x1, x2)  =  p40_out_ga(x2)
U9_ga(x1, x2, x3)  =  U9_ga(x3)
p27_in_ga(x1, x2)  =  p27_in_ga(x1)
p27_out_ga(x1, x2)  =  p27_out_ga(x2)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x4, x5)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
P30_IN_GA(x1, x2)  =  P30_IN_GA(x1)

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

(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:

P30_IN_GA(s(T18), s(X30)) → P30_IN_GA(T18, X30)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
P30_IN_GA(x1, x2)  =  P30_IN_GA(x1)

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:

P30_IN_GA(s(T18)) → P30_IN_GA(T18)

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:

• P30_IN_GA(s(T18)) → P30_IN_GA(T18)
  The graph contains the following edges 1 > 1

(15) TRUE

(16) Obligation:

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

M1_IN_GGA(T12, T13, T15) → U3_GGA(T12, T13, T15, p18_in_ga(T12, T16))
U3_GGA(T12, T13, T15, p18_out_ga(T12, T16)) → U5_GGA(T12, T13, T15, T16, p40_in_ga(T13, T19))
U5_GGA(T12, T13, T15, T16, p40_out_ga(T13, T19)) → M1_IN_GGA(T16, T19, T15)

The TRS R consists of the following rules:

m1_in_gga(T7, 0, T7) → m1_out_gga(T7, 0, T7)
m1_in_gga(0, T8, 0) → m1_out_gga(0, T8, 0)
m1_in_gga(T12, T13, T15) → U2_gga(T12, T13, T15, p18_in_ga(T12, X19))
p18_in_ga(s(0), 0) → p18_out_ga(s(0), 0)
p18_in_ga(s(s(0)), s(0)) → p18_out_ga(s(s(0)), s(0))
p18_in_ga(s(s(s(T18))), s(s(X30))) → U8_ga(T18, X30, p30_in_ga(T18, X30))
p30_in_ga(0, 0) → p30_out_ga(0, 0)
p30_in_ga(s(T18), s(X30)) → U1_ga(T18, X30, p30_in_ga(T18, X30))
U1_ga(T18, X30, p30_out_ga(T18, X30)) → p30_out_ga(s(T18), s(X30))
U8_ga(T18, X30, p30_out_ga(T18, X30)) → p18_out_ga(s(s(s(T18))), s(s(X30)))
U2_gga(T12, T13, T15, p18_out_ga(T12, X19)) → m1_out_gga(T12, T13, T15)
m1_in_gga(T12, T13, T15) → U3_gga(T12, T13, T15, p18_in_ga(T12, T16))
U3_gga(T12, T13, T15, p18_out_ga(T12, T16)) → U4_gga(T12, T13, T15, p40_in_ga(T13, X20))
p40_in_ga(s(0), 0) → p40_out_ga(s(0), 0)
p40_in_ga(s(s(T20)), s(X35)) → U9_ga(T20, X35, p27_in_ga(T20, X35))
p27_in_ga(0, 0) → p27_out_ga(0, 0)
p27_in_ga(s(T18), s(X30)) → U7_ga(T18, X30, p30_in_ga(T18, X30))
U7_ga(T18, X30, p30_out_ga(T18, X30)) → p27_out_ga(s(T18), s(X30))
U9_ga(T20, X35, p27_out_ga(T20, X35)) → p40_out_ga(s(s(T20)), s(X35))
U4_gga(T12, T13, T15, p40_out_ga(T13, X20)) → m1_out_gga(T12, T13, T15)
U3_gga(T12, T13, T15, p18_out_ga(T12, T16)) → U5_gga(T12, T13, T15, T16, p40_in_ga(T13, T19))
U5_gga(T12, T13, T15, T16, p40_out_ga(T13, T19)) → U6_gga(T12, T13, T15, m1_in_gga(T16, T19, T15))
U6_gga(T12, T13, T15, m1_out_gga(T16, T19, T15)) → m1_out_gga(T12, T13, T15)

The argument filtering Pi contains the following mapping:
m1_in_gga(x1, x2, x3)  =  m1_in_gga(x1, x2)
0  =  0
m1_out_gga(x1, x2, x3)  =  m1_out_gga
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
p18_in_ga(x1, x2)  =  p18_in_ga(x1)
s(x1)  =  s(x1)
p18_out_ga(x1, x2)  =  p18_out_ga(x2)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
p30_in_ga(x1, x2)  =  p30_in_ga(x1)
p30_out_ga(x1, x2)  =  p30_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x2, x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
p40_in_ga(x1, x2)  =  p40_in_ga(x1)
p40_out_ga(x1, x2)  =  p40_out_ga(x2)
U9_ga(x1, x2, x3)  =  U9_ga(x3)
p27_in_ga(x1, x2)  =  p27_in_ga(x1)
p27_out_ga(x1, x2)  =  p27_out_ga(x2)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x4, x5)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
M1_IN_GGA(x1, x2, x3)  =  M1_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x2, x4)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x4, x5)

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

(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:

M1_IN_GGA(T12, T13, T15) → U3_GGA(T12, T13, T15, p18_in_ga(T12, T16))
U3_GGA(T12, T13, T15, p18_out_ga(T12, T16)) → U5_GGA(T12, T13, T15, T16, p40_in_ga(T13, T19))
U5_GGA(T12, T13, T15, T16, p40_out_ga(T13, T19)) → M1_IN_GGA(T16, T19, T15)

The TRS R consists of the following rules:

p18_in_ga(s(0), 0) → p18_out_ga(s(0), 0)
p18_in_ga(s(s(0)), s(0)) → p18_out_ga(s(s(0)), s(0))
p18_in_ga(s(s(s(T18))), s(s(X30))) → U8_ga(T18, X30, p30_in_ga(T18, X30))
p40_in_ga(s(0), 0) → p40_out_ga(s(0), 0)
p40_in_ga(s(s(T20)), s(X35)) → U9_ga(T20, X35, p27_in_ga(T20, X35))
U8_ga(T18, X30, p30_out_ga(T18, X30)) → p18_out_ga(s(s(s(T18))), s(s(X30)))
U9_ga(T20, X35, p27_out_ga(T20, X35)) → p40_out_ga(s(s(T20)), s(X35))
p30_in_ga(0, 0) → p30_out_ga(0, 0)
p30_in_ga(s(T18), s(X30)) → U1_ga(T18, X30, p30_in_ga(T18, X30))
p27_in_ga(0, 0) → p27_out_ga(0, 0)
p27_in_ga(s(T18), s(X30)) → U7_ga(T18, X30, p30_in_ga(T18, X30))
U1_ga(T18, X30, p30_out_ga(T18, X30)) → p30_out_ga(s(T18), s(X30))
U7_ga(T18, X30, p30_out_ga(T18, X30)) → p27_out_ga(s(T18), s(X30))

The argument filtering Pi contains the following mapping:
0  =  0
p18_in_ga(x1, x2)  =  p18_in_ga(x1)
s(x1)  =  s(x1)
p18_out_ga(x1, x2)  =  p18_out_ga(x2)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
p30_in_ga(x1, x2)  =  p30_in_ga(x1)
p30_out_ga(x1, x2)  =  p30_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
p40_in_ga(x1, x2)  =  p40_in_ga(x1)
p40_out_ga(x1, x2)  =  p40_out_ga(x2)
U9_ga(x1, x2, x3)  =  U9_ga(x3)
p27_in_ga(x1, x2)  =  p27_in_ga(x1)
p27_out_ga(x1, x2)  =  p27_out_ga(x2)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
M1_IN_GGA(x1, x2, x3)  =  M1_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x2, x4)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x4, x5)

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:

M1_IN_GGA(T12, T13) → U3_GGA(T13, p18_in_ga(T12))
U3_GGA(T13, p18_out_ga(T16)) → U5_GGA(T16, p40_in_ga(T13))
U5_GGA(T16, p40_out_ga(T19)) → M1_IN_GGA(T16, T19)

The TRS R consists of the following rules:

p18_in_ga(s(0)) → p18_out_ga(0)
p18_in_ga(s(s(0))) → p18_out_ga(s(0))
p18_in_ga(s(s(s(T18)))) → U8_ga(p30_in_ga(T18))
p40_in_ga(s(0)) → p40_out_ga(0)
p40_in_ga(s(s(T20))) → U9_ga(p27_in_ga(T20))
U8_ga(p30_out_ga(X30)) → p18_out_ga(s(s(X30)))
U9_ga(p27_out_ga(X35)) → p40_out_ga(s(X35))
p30_in_ga(0) → p30_out_ga(0)
p30_in_ga(s(T18)) → U1_ga(p30_in_ga(T18))
p27_in_ga(0) → p27_out_ga(0)
p27_in_ga(s(T18)) → U7_ga(p30_in_ga(T18))
U1_ga(p30_out_ga(X30)) → p30_out_ga(s(X30))
U7_ga(p30_out_ga(X30)) → p27_out_ga(s(X30))

The set Q consists of the following terms:

p18_in_ga(x0)
p40_in_ga(x0)
U8_ga(x0)
U9_ga(x0)
p30_in_ga(x0)
p27_in_ga(x0)
U1_ga(x0)
U7_ga(x0)

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

(21) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

M1_IN_GGA(T12, T13) → U3_GGA(T13, p18_in_ga(T12))
U3_GGA(T13, p18_out_ga(T16)) → U5_GGA(T16, p40_in_ga(T13))
U5_GGA(T16, p40_out_ga(T19)) → M1_IN_GGA(T16, T19)

Strictly oriented rules of the TRS R:

p18_in_ga(s(0)) → p18_out_ga(0)
p18_in_ga(s(s(0))) → p18_out_ga(s(0))
p18_in_ga(s(s(s(T18)))) → U8_ga(p30_in_ga(T18))
p40_in_ga(s(0)) → p40_out_ga(0)
p40_in_ga(s(s(T20))) → U9_ga(p27_in_ga(T20))
U8_ga(p30_out_ga(X30)) → p18_out_ga(s(s(X30)))
U9_ga(p27_out_ga(X35)) → p40_out_ga(s(X35))
p30_in_ga(0) → p30_out_ga(0)
p27_in_ga(0) → p27_out_ga(0)
p27_in_ga(s(T18)) → U7_ga(p30_in_ga(T18))
U7_ga(p30_out_ga(X30)) → p27_out_ga(s(X30))

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(M1_IN_GGA(x₁, x₂)) = 1 + x₁ + x₂   
POL(U1_ga(x₁)) = 6 + x₁   
POL(U3_GGA(x₁, x₂)) = x₁ + x₂   
POL(U5_GGA(x₁, x₂)) = 1 + x₁ + x₂   
POL(U7_ga(x₁)) = 6 + x₁   
POL(U8_ga(x₁)) = 14 + x₁   
POL(U9_ga(x₁)) = 8 + x₁   
POL(p18_in_ga(x₁)) = x₁   
POL(p18_out_ga(x₁)) = 2 + x₁   
POL(p27_in_ga(x₁)) = 3 + x₁   
POL(p27_out_ga(x₁)) = x₁   
POL(p30_in_ga(x₁)) = 2 + x₁   
POL(p30_out_ga(x₁)) = 1 + x₁   
POL(p40_in_ga(x₁)) = x₁   
POL(p40_out_ga(x₁)) = 1 + x₁   
POL(s(x₁)) = 6 + x₁   

(22) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

p30_in_ga(s(T18)) → U1_ga(p30_in_ga(T18))
U1_ga(p30_out_ga(X30)) → p30_out_ga(s(X30))

The set Q consists of the following terms:

p18_in_ga(x0)
p40_in_ga(x0)
U8_ga(x0)
U9_ga(x0)
p30_in_ga(x0)
p27_in_ga(x0)
U1_ga(x0)
U7_ga(x0)

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

(23) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(24) TRUE