Left Termination of the query pattern div_in_3(g, g, a) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

div(0, Y, 0) :- no(zero(Y)).
div(X, Y, s(Z)) :- ','(no(zero(X)), ','(no(zero(Y)), ','(minus(X, Y, U), div(U, Y, Z)))).
minus(0, Y, 0).
minus(X, 0, X).
minus(s(X), s(Y), Z) :- minus(X, Y, Z).
zero(0).
no(X) :- ','(X, ','(!, failure(a))).
no(X1).
failure(b).

Queries:

div(g,g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: div(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: div(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | div(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: div(T1, T2, T3), SCOPE: 1, CLAUSE: 0\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: div(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: no(zero(T4))\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: \n\nX4 is free", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: no(zero(T4)), SCOPE: 2, CLAUSE: 6 | no(zero(T4)), SCOPE: 2, CLAUSE: 7 | ?_2\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(zero(T5), ','(!_2, failure(a))) | no(zero(T5)), SCOPE: 2, CLAUSE: 7 | ?_2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(zero(T5), ','(!_2, failure(a))), SCOPE: 3, CLAUSE: 5 | no(zero(T5)), SCOPE: 2, CLAUSE: 7 | ?_2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(!_2, failure(a)) | no(zero(0)), SCOPE: 2, CLAUSE: 7 | ?_2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: no(zero(T5)), SCOPE: 2, CLAUSE: 7\n\nT5 is ground\nzero(T5) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: failure(a)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: failure(a), SCOPE: 4, CLAUSE: 8\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(no(zero(T7)), ','(no(zero(T8)), ','(minus(T7, T8, X15), div(X15, T8, T10))))\n\nT7 is ground\nT8 is ground\nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\nX12 is free\nX13 is free; \nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(no(zero(T7)), ','(no(zero(T8)), ','(minus(T7, T8, X15), div(X15, T8, T10)))), SCOPE: 5, CLAUSE: 6 | ','(no(zero(T7)), ','(no(zero(T8)), ','(minus(T7, T8, X15), div(X15, T8, T10)))), SCOPE: 5, CLAUSE: 7 | ?_5\n\nT7 is ground\nT8 is ground\nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(zero(T11), ','(!_5, ','(failure(a), ','(no(zero(T8)), ','(minus(T11, T8, X15), div(X15, T8, T10)))))) | ','(no(zero(T11)), ','(no(zero(T8)), ','(minus(T11, T8, X15), div(X15, T8, T10)))), SCOPE: 5, CLAUSE: 7 | ?_5\n\nT8 is ground\nT11 is ground\nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(zero(T11), ','(!_5, ','(failure(a), ','(no(zero(T8)), ','(minus(T11, T8, X15), div(X15, T8, T10)))))), SCOPE: 6, CLAUSE: 5 | ','(no(zero(T11)), ','(no(zero(T8)), ','(minus(T11, T8, X15), div(X15, T8, T10)))), SCOPE: 5, CLAUSE: 7 | ?_5\n\nT8 is ground\nT11 is ground\nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(!_5, ','(failure(a), ','(no(zero(T8)), ','(minus(0, T8, X15), div(X15, T8, T10))))) | ','(no(zero(0)), ','(no(zero(T8)), ','(minus(0, T8, X15), div(X15, T8, T10)))), SCOPE: 5, CLAUSE: 7 | ?_5\n\nT8 is ground\nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(no(zero(T11)), ','(no(zero(T8)), ','(minus(T11, T8, X15), div(X15, T8, T10)))), SCOPE: 5, CLAUSE: 7\n\nT8 is ground\nT11 is ground\nX15 is free; \nzero(T11) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(failure(a), ','(no(zero(T8)), ','(minus(0, T8, X15), div(X15, T8, T10))))\n\nT8 is ground\nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(failure(a), ','(no(zero(T8)), ','(minus(0, T8, X15), div(X15, T8, T10)))), SCOPE: 7, CLAUSE: 8\n\nT8 is ground\nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(no(zero(T8)), ','(minus(T12, T8, X15), div(X15, T8, T10)))\n\nT8 is ground\nT12 is ground\nX15 is free; \nzero(T12) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(no(zero(T8)), ','(minus(T12, T8, X15), div(X15, T8, T10))), SCOPE: 8, CLAUSE: 6 | ','(no(zero(T8)), ','(minus(T12, T8, X15), div(X15, T8, T10))), SCOPE: 8, CLAUSE: 7 | ?_8\n\nT8 is ground\nT12 is ground\nX15 is free; \nzero(T12) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(zero(T13), ','(!_8, ','(failure(a), ','(minus(T12, T13, X15), div(X15, T13, T10))))) | ','(no(zero(T13)), ','(minus(T12, T13, X15), div(X15, T13, T10))), SCOPE: 8, CLAUSE: 7 | ?_8\n\nT12 is ground\nT13 is ground\nX15 is free; \nzero(T12) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(zero(T13), ','(!_8, ','(failure(a), ','(minus(T12, T13, X15), div(X15, T13, T10))))), SCOPE: 9, CLAUSE: 5 | ','(no(zero(T13)), ','(minus(T12, T13, X15), div(X15, T13, T10))), SCOPE: 8, CLAUSE: 7 | ?_8\n\nT12 is ground\nT13 is ground\nX15 is free; \nzero(T12) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(!_8, ','(failure(a), ','(minus(T12, 0, X15), div(X15, 0, T10)))) | ','(no(zero(0)), ','(minus(T12, 0, X15), div(X15, 0, T10))), SCOPE: 8, CLAUSE: 7 | ?_8\n\nT12 is ground\nX15 is free; \nzero(T12) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(no(zero(T13)), ','(minus(T12, T13, X15), div(X15, T13, T10))), SCOPE: 8, CLAUSE: 7\n\nT12 is ground\nT13 is ground\nX15 is free; \nzero(T12) !=? zero(0)\nzero(T13) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(failure(a), ','(minus(T12, 0, X15), div(X15, 0, T10)))\n\nT12 is ground\nX15 is free; \nzero(T12) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(failure(a), ','(minus(T12, 0, X15), div(X15, 0, T10))), SCOPE: 10, CLAUSE: 8\n\nT12 is ground\nX15 is free; \nzero(T12) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(minus(T12, T14, X15), div(X15, T14, T10))\n\nT12 is ground\nT14 is ground\nX15 is free; \nzero(T12) !=? zero(0)\nzero(T14) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: minus(T12, T14, X15)\n\nT12 is ground\nT14 is ground\nX15 is free; \nzero(T12) !=? zero(0)\nzero(T14) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: div(T15, T14, T10)\n\nT14 is ground\nT15 is ground\nzero(T14) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: minus(T12, T14, X15), SCOPE: 11, CLAUSE: 2 | minus(T12, T14, X15), SCOPE: 11, CLAUSE: 3 | minus(T12, T14, X15), SCOPE: 11, CLAUSE: 4\n\nT12 is ground\nT14 is ground\nX15 is free; \nzero(T12) !=? zero(0)\nzero(T14) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: minus(T12, T14, X15), SCOPE: 11, CLAUSE: 3 | minus(T12, T14, X15), SCOPE: 11, CLAUSE: 4\n\nT12 is ground\nT14 is ground\nX15 is free; \nX25 is free; \nzero(T12) !=? zero(0)\nzero(T14) !=? zero(0)\nminus(T12, T14, X15) !=? minus(0, X25, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: minus(T12, T14, X15), SCOPE: 11, CLAUSE: 4\n\nT12 is ground\nT14 is ground\nX15 is free; \nX25 is free; \nX27 is free; \nzero(T12) !=? zero(0)\nzero(T14) !=? zero(0)\nminus(T12, T14, X15) !=? minus(0, X25, 0)\nminus(T12, T14, X15) !=? minus(X27, 0, X27)", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: minus(T16, T17, X34)\n\nT16 is ground\nT17 is ground\nX15 is free; \nX25 is free; \nX27 is free; \nX34 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\nX15 is free\nX25 is free; \nX27 is free; \nX31 is free; \nX32 is free; \nX33 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: minus(T16, T17, X34), SCOPE: 12, CLAUSE: 2 | minus(T16, T17, X34), SCOPE: 12, CLAUSE: 3 | minus(T16, T17, X34), SCOPE: 12, CLAUSE: 4\n\nT16 is ground\nT17 is ground\nX34 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: minus(T16, T17, X34), SCOPE: 12, CLAUSE: 2\n\nT16 is ground\nT17 is ground\nX34 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: minus(T16, T17, X34), SCOPE: 12, CLAUSE: 3 | minus(T16, T17, X34), SCOPE: 12, CLAUSE: 4\n\nT16 is ground\nT17 is ground\nX34 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\nX34 is free\nX37 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: minus(T16, T17, X34), SCOPE: 12, CLAUSE: 3\n\nT16 is ground\nT17 is ground\nX34 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: minus(T16, T17, X34), SCOPE: 12, CLAUSE: 4\n\nT16 is ground\nT17 is ground\nX34 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\nX34 is free\nX40 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: minus(T20, T21, X47)\n\nT20 is ground\nT21 is ground\nX47 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\nX34 is free\nX44 is free; \nX45 is free; \nX46 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: minus(T20, T21, X47), SCOPE: 13, CLAUSE: 2 | minus(T20, T21, X47), SCOPE: 13, CLAUSE: 3 | minus(T20, T21, X47), SCOPE: 13, CLAUSE: 4\n\nT20 is ground\nT21 is ground\nX47 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 58 [arrowhead = none ];
58 -> 2;

59 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
2 -> 59 [arrowhead = none ];
59 -> 3;

60 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
2 -> 60 [arrowhead = none ];
60 -> 4;

61 [label="EVAL with clause\ndiv(0, X4, 0) :- no(zero(X4)).\nand substitution\nT1 -> 0\nT2 -> T4\nX4 -> T4\nT3 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 61 [arrowhead = none ];
61 -> 5;

62 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 62 [arrowhead = none ];
62 -> 6;

63 [label="EVAL with clause\ndiv(X12, X13, s(X14)) :- ','(no(zero(X12)), ','(no(zero(X13)), ','(minus(X12, X13, X15), div(X15, X13, X14)))).\nand substitution\nT1 -> T7\nX12 -> T7\nT2 -> T8\nX13 -> T8\nX14 -> T10\nT3 -> s(T10)\nT9 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 63 [arrowhead = none ];
63 -> 17;

64 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 64 [arrowhead = none ];
64 -> 18;

65 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 65 [arrowhead = none ];
65 -> 7;

66 [label="EVAL with clause\nno(X6) :- ','(X6, ','(!, failure(a))).\nand substitution\nT4 -> T5\nX6 -> zero(T5)", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 66 [arrowhead = none ];
66 -> 8;

67 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
8 -> 67 [arrowhead = none ];
67 -> 9;

68 [label="EVAL with clause\nzero(0).\nand substitution\nT5 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 68 [arrowhead = none ];
68 -> 10;

69 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 69 [arrowhead = none ];
69 -> 11;

70 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 70 [arrowhead = none ];
70 -> 12;

71 [label="EVAL with clause\nno(X8).\nand substitution\nT5 -> T6\nX8 -> zero(T6)", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 71 [arrowhead = none ];
71 -> 15;

72 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 72 [arrowhead = none ];
72 -> 13;

73 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
13 -> 73 [arrowhead = none ];
73 -> 14;

74 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 74 [arrowhead = none ];
74 -> 16;

75 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 75 [arrowhead = none ];
75 -> 19;

76 [label="EVAL with clause\nno(X17) :- ','(X17, ','(!, failure(a))).\nand substitution\nT7 -> T11\nX17 -> zero(T11)", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 76 [arrowhead = none ];
76 -> 20;

77 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 77 [arrowhead = none ];
77 -> 21;

78 [label="EVAL with clause\nzero(0).\nand substitution\nT11 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 78 [arrowhead = none ];
78 -> 22;

79 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 79 [arrowhead = none ];
79 -> 23;

80 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 80 [arrowhead = none ];
80 -> 24;

81 [label="EVAL with clause\nno(X19).\nand substitution\nT11 -> T12\nX19 -> zero(T12)", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 81 [arrowhead = none ];
81 -> 27;

82 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 82 [arrowhead = none ];
82 -> 25;

83 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
25 -> 83 [arrowhead = none ];
83 -> 26;

84 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
27 -> 84 [arrowhead = none ];
84 -> 28;

85 [label="EVAL with clause\nno(X21) :- ','(X21, ','(!, failure(a))).\nand substitution\nT8 -> T13\nX21 -> zero(T13)", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 85 [arrowhead = none ];
85 -> 29;

86 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 86 [arrowhead = none ];
86 -> 30;

87 [label="EVAL with clause\nzero(0).\nand substitution\nT13 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 87 [arrowhead = none ];
87 -> 31;

88 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 88 [arrowhead = none ];
88 -> 32;

89 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 89 [arrowhead = none ];
89 -> 33;

90 [label="EVAL with clause\nno(X23).\nand substitution\nT13 -> T14\nX23 -> zero(T14)", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 90 [arrowhead = none ];
90 -> 36;

91 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 91 [arrowhead = none ];
91 -> 34;

92 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
34 -> 92 [arrowhead = none ];
92 -> 35;

93 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
36 -> 93 [arrowhead = none ];
93 -> 37;

94 [label="SPLIT 2\nnew knowledge:\nT15 is ground\nreplacements:\nX15 -> T15", fontsize=14, style = filled, fillcolor = violet];
36 -> 94 [arrowhead = none ];
94 -> 38;

95 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
37 -> 95 [arrowhead = none ];
95 -> 39;

96 [label="INSTANCE with matching:\nT1 -> T15\nT2 -> T14\nT3 -> T10", fontsize=14, style = filled, fillcolor = lightgrey];
38 -> 96 [arrowhead = none , style=dashed];
96 -> 1[style=dashed];

97 [label="BACKTRACK\nfor clause: minus(0, Y, 0)\nwith clash: (zero(T12), zero(0))", fontsize=14, style = filled, fillcolor = white];
39 -> 97 [arrowhead = none ];
97 -> 40;

98 [label="BACKTRACK\nfor clause: minus(X, 0, X)\nwith clash: (zero(T14), zero(0))", fontsize=14, style = filled, fillcolor = white];
40 -> 98 [arrowhead = none ];
98 -> 41;

99 [label="EVAL with clause\nminus(s(X31), s(X32), X33) :- minus(X31, X32, X33).\nand substitution\nX31 -> T16\nT12 -> s(T16)\nX32 -> T17\nT14 -> s(T17)\nX15 -> X34\nX33 -> X34", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 99 [arrowhead = none ];
99 -> 42;

100 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 100 [arrowhead = none ];
100 -> 43;

101 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
42 -> 101 [arrowhead = none ];
101 -> 44;

102 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
44 -> 102 [arrowhead = none ];
102 -> 45;

103 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
44 -> 103 [arrowhead = none ];
103 -> 46;

104 [label="EVAL with clause\nminus(0, X37, 0).\nand substitution\nT16 -> 0\nT17 -> T18\nX37 -> T18\nX34 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 104 [arrowhead = none ];
104 -> 47;

105 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 105 [arrowhead = none ];
105 -> 48;

106 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
46 -> 106 [arrowhead = none ];
106 -> 50;

107 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
46 -> 107 [arrowhead = none ];
107 -> 51;

108 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 108 [arrowhead = none ];
108 -> 49;

109 [label="EVAL with clause\nminus(X40, 0, X40).\nand substitution\nT16 -> T19\nX40 -> T19\nT17 -> 0\nX34 -> T19", fontsize=14, style = filled, fillcolor = lightblue];
50 -> 109 [arrowhead = none ];
109 -> 52;

110 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
50 -> 110 [arrowhead = none ];
110 -> 53;

111 [label="EVAL with clause\nminus(s(X44), s(X45), X46) :- minus(X44, X45, X46).\nand substitution\nX44 -> T20\nT16 -> s(T20)\nX45 -> T21\nT17 -> s(T21)\nX34 -> X47\nX46 -> X47", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 111 [arrowhead = none ];
111 -> 55;

112 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 112 [arrowhead = none ];
112 -> 56;

113 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 113 [arrowhead = none ];
113 -> 54;

114 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
55 -> 114 [arrowhead = none ];
114 -> 57;

115 [label="INSTANCE with matching:\nT16 -> T20\nT17 -> T21\nX34 -> X47", fontsize=14, style = filled, fillcolor = lightgrey];
57 -> 115 [arrowhead = none , style=dashed];
115 -> 44[style=dashed];

}

(2) Obligation:

Clauses:

div1(0, T6, 0).
div1(T12, T14, s(T10)) :- minus37(T12, T14, X15).
div1(T12, T14, s(T10)) :- ','(minus37(T12, T14, T15), div1(T15, T14, T10)).
minus44(0, T18, 0).
minus44(T19, 0, T19).
minus44(s(T20), s(T21), X47) :- minus44(T20, T21, X47).
minus37(s(0), s(T18), 0).
minus37(s(T19), s(0), T19).
minus37(s(s(T20)), s(s(T21)), X47) :- minus44(T20, T21, X47).

Queries:

div1(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:
div1_in: (b,b,f)
minus37_in: (b,b,f)
minus44_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

div1_in_gga(0, T6, 0) → div1_out_gga(0, T6, 0)
div1_in_gga(T12, T14, s(T10)) → U1_gga(T12, T14, T10, minus37_in_gga(T12, T14, X15))
minus37_in_gga(s(0), s(T18), 0) → minus37_out_gga(s(0), s(T18), 0)
minus37_in_gga(s(T19), s(0), T19) → minus37_out_gga(s(T19), s(0), T19)
minus37_in_gga(s(s(T20)), s(s(T21)), X47) → U5_gga(T20, T21, X47, minus44_in_gga(T20, T21, X47))
minus44_in_gga(0, T18, 0) → minus44_out_gga(0, T18, 0)
minus44_in_gga(T19, 0, T19) → minus44_out_gga(T19, 0, T19)
minus44_in_gga(s(T20), s(T21), X47) → U4_gga(T20, T21, X47, minus44_in_gga(T20, T21, X47))
U4_gga(T20, T21, X47, minus44_out_gga(T20, T21, X47)) → minus44_out_gga(s(T20), s(T21), X47)
U5_gga(T20, T21, X47, minus44_out_gga(T20, T21, X47)) → minus37_out_gga(s(s(T20)), s(s(T21)), X47)
U1_gga(T12, T14, T10, minus37_out_gga(T12, T14, X15)) → div1_out_gga(T12, T14, s(T10))
div1_in_gga(T12, T14, s(T10)) → U2_gga(T12, T14, T10, minus37_in_gga(T12, T14, T15))
U2_gga(T12, T14, T10, minus37_out_gga(T12, T14, T15)) → U3_gga(T12, T14, T10, div1_in_gga(T15, T14, T10))
U3_gga(T12, T14, T10, div1_out_gga(T15, T14, T10)) → div1_out_gga(T12, T14, s(T10))

The argument filtering Pi contains the following mapping:
div1_in_gga(x1, x2, x3)  =  div1_in_gga(x1, x2)
0  =  0
div1_out_gga(x1, x2, x3)  =  div1_out_gga
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
minus37_in_gga(x1, x2, x3)  =  minus37_in_gga(x1, x2)
s(x1)  =  s(x1)
minus37_out_gga(x1, x2, x3)  =  minus37_out_gga(x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
minus44_in_gga(x1, x2, x3)  =  minus44_in_gga(x1, x2)
minus44_out_gga(x1, x2, x3)  =  minus44_out_gga(x3)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x2, x4)
U3_gga(x1, x2, x3, x4)  =  U3_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:

div1_in_gga(0, T6, 0) → div1_out_gga(0, T6, 0)
div1_in_gga(T12, T14, s(T10)) → U1_gga(T12, T14, T10, minus37_in_gga(T12, T14, X15))
minus37_in_gga(s(0), s(T18), 0) → minus37_out_gga(s(0), s(T18), 0)
minus37_in_gga(s(T19), s(0), T19) → minus37_out_gga(s(T19), s(0), T19)
minus37_in_gga(s(s(T20)), s(s(T21)), X47) → U5_gga(T20, T21, X47, minus44_in_gga(T20, T21, X47))
minus44_in_gga(0, T18, 0) → minus44_out_gga(0, T18, 0)
minus44_in_gga(T19, 0, T19) → minus44_out_gga(T19, 0, T19)
minus44_in_gga(s(T20), s(T21), X47) → U4_gga(T20, T21, X47, minus44_in_gga(T20, T21, X47))
U4_gga(T20, T21, X47, minus44_out_gga(T20, T21, X47)) → minus44_out_gga(s(T20), s(T21), X47)
U5_gga(T20, T21, X47, minus44_out_gga(T20, T21, X47)) → minus37_out_gga(s(s(T20)), s(s(T21)), X47)
U1_gga(T12, T14, T10, minus37_out_gga(T12, T14, X15)) → div1_out_gga(T12, T14, s(T10))
div1_in_gga(T12, T14, s(T10)) → U2_gga(T12, T14, T10, minus37_in_gga(T12, T14, T15))
U2_gga(T12, T14, T10, minus37_out_gga(T12, T14, T15)) → U3_gga(T12, T14, T10, div1_in_gga(T15, T14, T10))
U3_gga(T12, T14, T10, div1_out_gga(T15, T14, T10)) → div1_out_gga(T12, T14, s(T10))

The argument filtering Pi contains the following mapping:
div1_in_gga(x1, x2, x3)  =  div1_in_gga(x1, x2)
0  =  0
div1_out_gga(x1, x2, x3)  =  div1_out_gga
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
minus37_in_gga(x1, x2, x3)  =  minus37_in_gga(x1, x2)
s(x1)  =  s(x1)
minus37_out_gga(x1, x2, x3)  =  minus37_out_gga(x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
minus44_in_gga(x1, x2, x3)  =  minus44_in_gga(x1, x2)
minus44_out_gga(x1, x2, x3)  =  minus44_out_gga(x3)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x2, x4)
U3_gga(x1, x2, x3, x4)  =  U3_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:

DIV1_IN_GGA(T12, T14, s(T10)) → U1_GGA(T12, T14, T10, minus37_in_gga(T12, T14, X15))
DIV1_IN_GGA(T12, T14, s(T10)) → MINUS37_IN_GGA(T12, T14, X15)
MINUS37_IN_GGA(s(s(T20)), s(s(T21)), X47) → U5_GGA(T20, T21, X47, minus44_in_gga(T20, T21, X47))
MINUS37_IN_GGA(s(s(T20)), s(s(T21)), X47) → MINUS44_IN_GGA(T20, T21, X47)
MINUS44_IN_GGA(s(T20), s(T21), X47) → U4_GGA(T20, T21, X47, minus44_in_gga(T20, T21, X47))
MINUS44_IN_GGA(s(T20), s(T21), X47) → MINUS44_IN_GGA(T20, T21, X47)
DIV1_IN_GGA(T12, T14, s(T10)) → U2_GGA(T12, T14, T10, minus37_in_gga(T12, T14, T15))
U2_GGA(T12, T14, T10, minus37_out_gga(T12, T14, T15)) → U3_GGA(T12, T14, T10, div1_in_gga(T15, T14, T10))
U2_GGA(T12, T14, T10, minus37_out_gga(T12, T14, T15)) → DIV1_IN_GGA(T15, T14, T10)

The TRS R consists of the following rules:

div1_in_gga(0, T6, 0) → div1_out_gga(0, T6, 0)
div1_in_gga(T12, T14, s(T10)) → U1_gga(T12, T14, T10, minus37_in_gga(T12, T14, X15))
minus37_in_gga(s(0), s(T18), 0) → minus37_out_gga(s(0), s(T18), 0)
minus37_in_gga(s(T19), s(0), T19) → minus37_out_gga(s(T19), s(0), T19)
minus37_in_gga(s(s(T20)), s(s(T21)), X47) → U5_gga(T20, T21, X47, minus44_in_gga(T20, T21, X47))
minus44_in_gga(0, T18, 0) → minus44_out_gga(0, T18, 0)
minus44_in_gga(T19, 0, T19) → minus44_out_gga(T19, 0, T19)
minus44_in_gga(s(T20), s(T21), X47) → U4_gga(T20, T21, X47, minus44_in_gga(T20, T21, X47))
U4_gga(T20, T21, X47, minus44_out_gga(T20, T21, X47)) → minus44_out_gga(s(T20), s(T21), X47)
U5_gga(T20, T21, X47, minus44_out_gga(T20, T21, X47)) → minus37_out_gga(s(s(T20)), s(s(T21)), X47)
U1_gga(T12, T14, T10, minus37_out_gga(T12, T14, X15)) → div1_out_gga(T12, T14, s(T10))
div1_in_gga(T12, T14, s(T10)) → U2_gga(T12, T14, T10, minus37_in_gga(T12, T14, T15))
U2_gga(T12, T14, T10, minus37_out_gga(T12, T14, T15)) → U3_gga(T12, T14, T10, div1_in_gga(T15, T14, T10))
U3_gga(T12, T14, T10, div1_out_gga(T15, T14, T10)) → div1_out_gga(T12, T14, s(T10))

The argument filtering Pi contains the following mapping:
div1_in_gga(x1, x2, x3)  =  div1_in_gga(x1, x2)
0  =  0
div1_out_gga(x1, x2, x3)  =  div1_out_gga
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
minus37_in_gga(x1, x2, x3)  =  minus37_in_gga(x1, x2)
s(x1)  =  s(x1)
minus37_out_gga(x1, x2, x3)  =  minus37_out_gga(x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
minus44_in_gga(x1, x2, x3)  =  minus44_in_gga(x1, x2)
minus44_out_gga(x1, x2, x3)  =  minus44_out_gga(x3)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x2, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
DIV1_IN_GGA(x1, x2, x3)  =  DIV1_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x4)
MINUS37_IN_GGA(x1, x2, x3)  =  MINUS37_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x4)
MINUS44_IN_GGA(x1, x2, x3)  =  MINUS44_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x4)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x2, x4)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x4)

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

(6) Obligation:

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

DIV1_IN_GGA(T12, T14, s(T10)) → U1_GGA(T12, T14, T10, minus37_in_gga(T12, T14, X15))
DIV1_IN_GGA(T12, T14, s(T10)) → MINUS37_IN_GGA(T12, T14, X15)
MINUS37_IN_GGA(s(s(T20)), s(s(T21)), X47) → U5_GGA(T20, T21, X47, minus44_in_gga(T20, T21, X47))
MINUS37_IN_GGA(s(s(T20)), s(s(T21)), X47) → MINUS44_IN_GGA(T20, T21, X47)
MINUS44_IN_GGA(s(T20), s(T21), X47) → U4_GGA(T20, T21, X47, minus44_in_gga(T20, T21, X47))
MINUS44_IN_GGA(s(T20), s(T21), X47) → MINUS44_IN_GGA(T20, T21, X47)
DIV1_IN_GGA(T12, T14, s(T10)) → U2_GGA(T12, T14, T10, minus37_in_gga(T12, T14, T15))
U2_GGA(T12, T14, T10, minus37_out_gga(T12, T14, T15)) → U3_GGA(T12, T14, T10, div1_in_gga(T15, T14, T10))
U2_GGA(T12, T14, T10, minus37_out_gga(T12, T14, T15)) → DIV1_IN_GGA(T15, T14, T10)

The TRS R consists of the following rules:

div1_in_gga(0, T6, 0) → div1_out_gga(0, T6, 0)
div1_in_gga(T12, T14, s(T10)) → U1_gga(T12, T14, T10, minus37_in_gga(T12, T14, X15))
minus37_in_gga(s(0), s(T18), 0) → minus37_out_gga(s(0), s(T18), 0)
minus37_in_gga(s(T19), s(0), T19) → minus37_out_gga(s(T19), s(0), T19)
minus37_in_gga(s(s(T20)), s(s(T21)), X47) → U5_gga(T20, T21, X47, minus44_in_gga(T20, T21, X47))
minus44_in_gga(0, T18, 0) → minus44_out_gga(0, T18, 0)
minus44_in_gga(T19, 0, T19) → minus44_out_gga(T19, 0, T19)
minus44_in_gga(s(T20), s(T21), X47) → U4_gga(T20, T21, X47, minus44_in_gga(T20, T21, X47))
U4_gga(T20, T21, X47, minus44_out_gga(T20, T21, X47)) → minus44_out_gga(s(T20), s(T21), X47)
U5_gga(T20, T21, X47, minus44_out_gga(T20, T21, X47)) → minus37_out_gga(s(s(T20)), s(s(T21)), X47)
U1_gga(T12, T14, T10, minus37_out_gga(T12, T14, X15)) → div1_out_gga(T12, T14, s(T10))
div1_in_gga(T12, T14, s(T10)) → U2_gga(T12, T14, T10, minus37_in_gga(T12, T14, T15))
U2_gga(T12, T14, T10, minus37_out_gga(T12, T14, T15)) → U3_gga(T12, T14, T10, div1_in_gga(T15, T14, T10))
U3_gga(T12, T14, T10, div1_out_gga(T15, T14, T10)) → div1_out_gga(T12, T14, s(T10))

The argument filtering Pi contains the following mapping:
div1_in_gga(x1, x2, x3)  =  div1_in_gga(x1, x2)
0  =  0
div1_out_gga(x1, x2, x3)  =  div1_out_gga
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
minus37_in_gga(x1, x2, x3)  =  minus37_in_gga(x1, x2)
s(x1)  =  s(x1)
minus37_out_gga(x1, x2, x3)  =  minus37_out_gga(x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
minus44_in_gga(x1, x2, x3)  =  minus44_in_gga(x1, x2)
minus44_out_gga(x1, x2, x3)  =  minus44_out_gga(x3)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x2, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
DIV1_IN_GGA(x1, x2, x3)  =  DIV1_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x4)
MINUS37_IN_GGA(x1, x2, x3)  =  MINUS37_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x4)
MINUS44_IN_GGA(x1, x2, x3)  =  MINUS44_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x4)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x2, x4)
U3_GGA(x1, x2, x3, x4)  =  U3_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 6 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

MINUS44_IN_GGA(s(T20), s(T21), X47) → MINUS44_IN_GGA(T20, T21, X47)

The TRS R consists of the following rules:

div1_in_gga(0, T6, 0) → div1_out_gga(0, T6, 0)
div1_in_gga(T12, T14, s(T10)) → U1_gga(T12, T14, T10, minus37_in_gga(T12, T14, X15))
minus37_in_gga(s(0), s(T18), 0) → minus37_out_gga(s(0), s(T18), 0)
minus37_in_gga(s(T19), s(0), T19) → minus37_out_gga(s(T19), s(0), T19)
minus37_in_gga(s(s(T20)), s(s(T21)), X47) → U5_gga(T20, T21, X47, minus44_in_gga(T20, T21, X47))
minus44_in_gga(0, T18, 0) → minus44_out_gga(0, T18, 0)
minus44_in_gga(T19, 0, T19) → minus44_out_gga(T19, 0, T19)
minus44_in_gga(s(T20), s(T21), X47) → U4_gga(T20, T21, X47, minus44_in_gga(T20, T21, X47))
U4_gga(T20, T21, X47, minus44_out_gga(T20, T21, X47)) → minus44_out_gga(s(T20), s(T21), X47)
U5_gga(T20, T21, X47, minus44_out_gga(T20, T21, X47)) → minus37_out_gga(s(s(T20)), s(s(T21)), X47)
U1_gga(T12, T14, T10, minus37_out_gga(T12, T14, X15)) → div1_out_gga(T12, T14, s(T10))
div1_in_gga(T12, T14, s(T10)) → U2_gga(T12, T14, T10, minus37_in_gga(T12, T14, T15))
U2_gga(T12, T14, T10, minus37_out_gga(T12, T14, T15)) → U3_gga(T12, T14, T10, div1_in_gga(T15, T14, T10))
U3_gga(T12, T14, T10, div1_out_gga(T15, T14, T10)) → div1_out_gga(T12, T14, s(T10))

The argument filtering Pi contains the following mapping:
div1_in_gga(x1, x2, x3)  =  div1_in_gga(x1, x2)
0  =  0
div1_out_gga(x1, x2, x3)  =  div1_out_gga
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
minus37_in_gga(x1, x2, x3)  =  minus37_in_gga(x1, x2)
s(x1)  =  s(x1)
minus37_out_gga(x1, x2, x3)  =  minus37_out_gga(x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
minus44_in_gga(x1, x2, x3)  =  minus44_in_gga(x1, x2)
minus44_out_gga(x1, x2, x3)  =  minus44_out_gga(x3)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x2, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
MINUS44_IN_GGA(x1, x2, x3)  =  MINUS44_IN_GGA(x1, x2)

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:

MINUS44_IN_GGA(s(T20), s(T21), X47) → MINUS44_IN_GGA(T20, T21, X47)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
MINUS44_IN_GGA(x1, x2, x3)  =  MINUS44_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:

MINUS44_IN_GGA(s(T20), s(T21)) → MINUS44_IN_GGA(T20, T21)

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:

• MINUS44_IN_GGA(s(T20), s(T21)) → MINUS44_IN_GGA(T20, T21)
  The graph contains the following edges 1 > 1, 2 > 2

(15) TRUE

(16) Obligation:

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

DIV1_IN_GGA(T12, T14, s(T10)) → U2_GGA(T12, T14, T10, minus37_in_gga(T12, T14, T15))
U2_GGA(T12, T14, T10, minus37_out_gga(T12, T14, T15)) → DIV1_IN_GGA(T15, T14, T10)

The TRS R consists of the following rules:

div1_in_gga(0, T6, 0) → div1_out_gga(0, T6, 0)
div1_in_gga(T12, T14, s(T10)) → U1_gga(T12, T14, T10, minus37_in_gga(T12, T14, X15))
minus37_in_gga(s(0), s(T18), 0) → minus37_out_gga(s(0), s(T18), 0)
minus37_in_gga(s(T19), s(0), T19) → minus37_out_gga(s(T19), s(0), T19)
minus37_in_gga(s(s(T20)), s(s(T21)), X47) → U5_gga(T20, T21, X47, minus44_in_gga(T20, T21, X47))
minus44_in_gga(0, T18, 0) → minus44_out_gga(0, T18, 0)
minus44_in_gga(T19, 0, T19) → minus44_out_gga(T19, 0, T19)
minus44_in_gga(s(T20), s(T21), X47) → U4_gga(T20, T21, X47, minus44_in_gga(T20, T21, X47))
U4_gga(T20, T21, X47, minus44_out_gga(T20, T21, X47)) → minus44_out_gga(s(T20), s(T21), X47)
U5_gga(T20, T21, X47, minus44_out_gga(T20, T21, X47)) → minus37_out_gga(s(s(T20)), s(s(T21)), X47)
U1_gga(T12, T14, T10, minus37_out_gga(T12, T14, X15)) → div1_out_gga(T12, T14, s(T10))
div1_in_gga(T12, T14, s(T10)) → U2_gga(T12, T14, T10, minus37_in_gga(T12, T14, T15))
U2_gga(T12, T14, T10, minus37_out_gga(T12, T14, T15)) → U3_gga(T12, T14, T10, div1_in_gga(T15, T14, T10))
U3_gga(T12, T14, T10, div1_out_gga(T15, T14, T10)) → div1_out_gga(T12, T14, s(T10))

The argument filtering Pi contains the following mapping:
div1_in_gga(x1, x2, x3)  =  div1_in_gga(x1, x2)
0  =  0
div1_out_gga(x1, x2, x3)  =  div1_out_gga
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
minus37_in_gga(x1, x2, x3)  =  minus37_in_gga(x1, x2)
s(x1)  =  s(x1)
minus37_out_gga(x1, x2, x3)  =  minus37_out_gga(x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
minus44_in_gga(x1, x2, x3)  =  minus44_in_gga(x1, x2)
minus44_out_gga(x1, x2, x3)  =  minus44_out_gga(x3)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x2, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
DIV1_IN_GGA(x1, x2, x3)  =  DIV1_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x2, x4)

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:

DIV1_IN_GGA(T12, T14, s(T10)) → U2_GGA(T12, T14, T10, minus37_in_gga(T12, T14, T15))
U2_GGA(T12, T14, T10, minus37_out_gga(T12, T14, T15)) → DIV1_IN_GGA(T15, T14, T10)

The TRS R consists of the following rules:

minus37_in_gga(s(0), s(T18), 0) → minus37_out_gga(s(0), s(T18), 0)
minus37_in_gga(s(T19), s(0), T19) → minus37_out_gga(s(T19), s(0), T19)
minus37_in_gga(s(s(T20)), s(s(T21)), X47) → U5_gga(T20, T21, X47, minus44_in_gga(T20, T21, X47))
U5_gga(T20, T21, X47, minus44_out_gga(T20, T21, X47)) → minus37_out_gga(s(s(T20)), s(s(T21)), X47)
minus44_in_gga(0, T18, 0) → minus44_out_gga(0, T18, 0)
minus44_in_gga(T19, 0, T19) → minus44_out_gga(T19, 0, T19)
minus44_in_gga(s(T20), s(T21), X47) → U4_gga(T20, T21, X47, minus44_in_gga(T20, T21, X47))
U4_gga(T20, T21, X47, minus44_out_gga(T20, T21, X47)) → minus44_out_gga(s(T20), s(T21), X47)

The argument filtering Pi contains the following mapping:
0  =  0
minus37_in_gga(x1, x2, x3)  =  minus37_in_gga(x1, x2)
s(x1)  =  s(x1)
minus37_out_gga(x1, x2, x3)  =  minus37_out_gga(x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
minus44_in_gga(x1, x2, x3)  =  minus44_in_gga(x1, x2)
minus44_out_gga(x1, x2, x3)  =  minus44_out_gga(x3)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
DIV1_IN_GGA(x1, x2, x3)  =  DIV1_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x2, x4)

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:

DIV1_IN_GGA(T12, T14) → U2_GGA(T14, minus37_in_gga(T12, T14))
U2_GGA(T14, minus37_out_gga(T15)) → DIV1_IN_GGA(T15, T14)

The TRS R consists of the following rules:

minus37_in_gga(s(0), s(T18)) → minus37_out_gga(0)
minus37_in_gga(s(T19), s(0)) → minus37_out_gga(T19)
minus37_in_gga(s(s(T20)), s(s(T21))) → U5_gga(minus44_in_gga(T20, T21))
U5_gga(minus44_out_gga(X47)) → minus37_out_gga(X47)
minus44_in_gga(0, T18) → minus44_out_gga(0)
minus44_in_gga(T19, 0) → minus44_out_gga(T19)
minus44_in_gga(s(T20), s(T21)) → U4_gga(minus44_in_gga(T20, T21))
U4_gga(minus44_out_gga(X47)) → minus44_out_gga(X47)

The set Q consists of the following terms:

minus37_in_gga(x0, x1)
U5_gga(x0)
minus44_in_gga(x0, x1)
U4_gga(x0)

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

(21) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U2_GGA(T14, minus37_out_gga(T15)) → DIV1_IN_GGA(T15, T14)

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(DIV1_IN_GGA(x₁, x₂)) = x₁   
POL(U2_GGA(x₁, x₂)) = x₂   
POL(U4_gga(x₁)) = 1 + x₁   
POL(U5_gga(x₁)) = x₁   
POL(minus37_in_gga(x₁, x₂)) = x₁   
POL(minus37_out_gga(x₁)) = 1 + x₁   
POL(minus44_in_gga(x₁, x₂)) = 1 + x₁   
POL(minus44_out_gga(x₁)) = 1 + x₁   
POL(s(x₁)) = 1 + x₁   

The following usable rules [FROCOS05] were oriented:

minus37_in_gga(s(0), s(T18)) → minus37_out_gga(0)
minus37_in_gga(s(T19), s(0)) → minus37_out_gga(T19)
minus37_in_gga(s(s(T20)), s(s(T21))) → U5_gga(minus44_in_gga(T20, T21))
minus44_in_gga(0, T18) → minus44_out_gga(0)
minus44_in_gga(T19, 0) → minus44_out_gga(T19)
minus44_in_gga(s(T20), s(T21)) → U4_gga(minus44_in_gga(T20, T21))
U5_gga(minus44_out_gga(X47)) → minus37_out_gga(X47)
U4_gga(minus44_out_gga(X47)) → minus44_out_gga(X47)

(22) Obligation:

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

DIV1_IN_GGA(T12, T14) → U2_GGA(T14, minus37_in_gga(T12, T14))

The TRS R consists of the following rules:

minus37_in_gga(s(0), s(T18)) → minus37_out_gga(0)
minus37_in_gga(s(T19), s(0)) → minus37_out_gga(T19)
minus37_in_gga(s(s(T20)), s(s(T21))) → U5_gga(minus44_in_gga(T20, T21))
U5_gga(minus44_out_gga(X47)) → minus37_out_gga(X47)
minus44_in_gga(0, T18) → minus44_out_gga(0)
minus44_in_gga(T19, 0) → minus44_out_gga(T19)
minus44_in_gga(s(T20), s(T21)) → U4_gga(minus44_in_gga(T20, T21))
U4_gga(minus44_out_gga(X47)) → minus44_out_gga(X47)

The set Q consists of the following terms:

minus37_in_gga(x0, x1)
U5_gga(x0)
minus44_in_gga(x0, x1)
U4_gga(x0)

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

(23) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(24) TRUE