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(X, 0, Z) :- ','(!, fail).
div(0, Y, Z) :- ','(!, =(Z, 0)).
div(X, Y, s(Z)) :- ','(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).
=(X, X).

Queries:

div(g,g,a).

(1) PredefinedPredicateTransformerProof (SOUND transformation)

Added definitions of predefined predicates [PROLOG].

(2) Obligation:

Clauses:

div(X, 0, Z) :- ','(!, fail).
div(0, Y, Z) :- ','(!, =1(Z, 0)).
div(X, Y, s(Z)) :- ','(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).
=1(X, X).
fail :- fail(b).
fail(a).

Queries:

div(g,g,a).

(3) 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 | div(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, fail) | div(T4, 0, T3), SCOPE: 1, CLAUSE: 1 | div(T4, 0, T3), SCOPE: 1, CLAUSE: 2 | ?_1\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: div(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | div(T1, T2, T3), SCOPE: 1, CLAUSE: 2 | ?_1\n\nT1 is ground\nT2 is ground\nX3 is free; \nX4 is free; \ndiv(T1, T2, T3) !=? div(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: fail\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(!_1, =1(T8, 0)) | div(0, T6, T3), SCOPE: 1, CLAUSE: 2 | ?_1\n\nT6 is ground\nX3 is free; \nX4 is free; \ndiv(0, T6, T3) !=? div(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: div(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground\nX3 is free; \nX4 is free; \nX7 is free; \nX8 is free; \ndiv(T1, T2, T3) !=? div(X3, 0, X4)\ndiv(T1, T2, T3) !=? div(0, X7, X8)", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: =1(T8, 0)\n\nX3 is free\nX4 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: =1(T8, 0), SCOPE: 2, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(minus(T10, T11, X17), div(X17, T11, T13))\n\nT10 is ground\nT11 is ground\nX3 is free; \nX4 is free; \nX7 is free; \nX8 is free; \nX17 is free; \ndiv(T10, T11, T3) !=? div(X3, 0, X4)\ndiv(T10, T11, T3) !=? div(0, X7, X8)", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\nX3 is free\nX4 is free; \nX7 is free; \nX8 is free; \nX14 is free; \nX15 is free; \nX16 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: minus(T10, T11, X17)\n\nT10 is ground\nT11 is ground\nX3 is free; \nX4 is free; \nX7 is free; \nX8 is free; \nX17 is free; \ndiv(T10, T11, T3) !=? div(X3, 0, X4)\ndiv(T10, T11, T3) !=? div(0, X7, X8)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: div(T14, T11, T13)\n\nT11 is ground\nT14 is ground\nX3 is free; \nX4 is free; \nX7 is free; \nX8 is free; \ndiv(T10, T11, T3) !=? div(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: minus(T10, T11, X17), SCOPE: 3, CLAUSE: 3 | minus(T10, T11, X17), SCOPE: 3, CLAUSE: 4 | minus(T10, T11, X17), SCOPE: 3, CLAUSE: 5\n\nT10 is ground\nT11 is ground\nX3 is free; \nX4 is free; \nX7 is free; \nX8 is free; \nX17 is free; \ndiv(T10, T11, T3) !=? div(X3, 0, X4)\ndiv(T10, T11, T3) !=? div(0, X7, X8)", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: minus(T10, T11, X17), SCOPE: 3, CLAUSE: 4 | minus(T10, T11, X17), SCOPE: 3, CLAUSE: 5\n\nT10 is ground\nT11 is ground\nX3 is free; \nX4 is free; \nX7 is free; \nX8 is free; \nX17 is free; \nX19 is free; \ndiv(T10, T11, T3) !=? div(X3, 0, X4)\ndiv(T10, T11, T3) !=? div(0, X7, X8)\nminus(T10, T11, X17) !=? minus(0, X19, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: minus(T10, T11, X17), SCOPE: 3, CLAUSE: 5\n\nT10 is ground\nT11 is ground\nX3 is free; \nX4 is free; \nX7 is free; \nX8 is free; \nX17 is free; \nX19 is free; \nX21 is free; \ndiv(T10, T11, T3) !=? div(X3, 0, X4)\ndiv(T10, T11, T3) !=? div(0, X7, X8)\nminus(T10, T11, X17) !=? minus(0, X19, 0)\nminus(T10, T11, X17) !=? minus(X21, 0, X21)", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: minus(T15, T16, X28)\n\nT15 is ground\nT16 is ground\nX3 is free; \nX4 is free; \nX7 is free; \nX8 is free; \nX17 is free; \nX19 is free; \nX21 is free; \nX28 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\nX3 is free\nX4 is free; \nX7 is free; \nX8 is free; \nX17 is free; \nX19 is free; \nX21 is free; \nX25 is free; \nX26 is free; \nX27 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: minus(T15, T16, X28), SCOPE: 4, CLAUSE: 3 | minus(T15, T16, X28), SCOPE: 4, CLAUSE: 4 | minus(T15, T16, X28), SCOPE: 4, CLAUSE: 5\n\nT15 is ground\nT16 is ground\nX28 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: minus(T15, T16, X28), SCOPE: 4, CLAUSE: 3\n\nT15 is ground\nT16 is ground\nX28 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: minus(T15, T16, X28), SCOPE: 4, CLAUSE: 4 | minus(T15, T16, X28), SCOPE: 4, CLAUSE: 5\n\nT15 is ground\nT16 is ground\nX28 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\nX28 is free\nX31 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: minus(T15, T16, X28), SCOPE: 4, CLAUSE: 4\n\nT15 is ground\nT16 is ground\nX28 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: minus(T15, T16, X28), SCOPE: 4, CLAUSE: 5\n\nT15 is ground\nT16 is ground\nX28 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\nX28 is free\nX34 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: minus(T19, T20, X41)\n\nT19 is ground\nT20 is ground\nX41 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\nX28 is free\nX38 is free; \nX39 is free; \nX40 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: minus(T19, T20, X41), SCOPE: 5, CLAUSE: 3 | minus(T19, T20, X41), SCOPE: 5, CLAUSE: 4 | minus(T19, T20, X41), SCOPE: 5, CLAUSE: 5\n\nT19 is ground\nT20 is ground\nX41 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 37 [arrowhead = none ];
37 -> 2;

38 [label="EVAL with clause\ndiv(X3, 0, X4) :- ','(!, fail).\nand substitution\nT1 -> T4\nX3 -> T4\nT2 -> 0\nT3 -> T5\nX4 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 38 [arrowhead = none ];
38 -> 3;

39 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 39 [arrowhead = none ];
39 -> 4;

40 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 40 [arrowhead = none ];
40 -> 5;

41 [label="EVAL with clause\ndiv(0, X7, X8) :- ','(!, =1(X8, 0)).\nand substitution\nT1 -> 0\nT2 -> T6\nX7 -> T6\nT3 -> T8\nX8 -> T8\nT7 -> T8", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 41 [arrowhead = none ];
41 -> 7;

42 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 42 [arrowhead = none ];
42 -> 8;

43 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 43 [arrowhead = none ];
43 -> 6;

44 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 44 [arrowhead = none ];
44 -> 9;

45 [label="EVAL with clause\ndiv(X14, X15, s(X16)) :- ','(minus(X14, X15, X17), div(X17, X15, X16)).\nand substitution\nT1 -> T10\nX14 -> T10\nT2 -> T11\nX15 -> T11\nX16 -> T13\nT3 -> s(T13)\nT12 -> T13", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 45 [arrowhead = none ];
45 -> 14;

46 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 46 [arrowhead = none ];
46 -> 15;

47 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 47 [arrowhead = none ];
47 -> 10;

48 [label="EVAL with clause\n=1(X10, X10).\nand substitution\nT8 -> 0\nX10 -> 0\nT9 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 48 [arrowhead = none ];
48 -> 11;

49 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 49 [arrowhead = none ];
49 -> 12;

50 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 50 [arrowhead = none ];
50 -> 13;

51 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
14 -> 51 [arrowhead = none ];
51 -> 16;

52 [label="SPLIT 2\nnew knowledge:\nT14 is ground\nreplacements:\nX17 -> T14", fontsize=14, style = filled, fillcolor = violet];
14 -> 52 [arrowhead = none ];
52 -> 17;

53 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 53 [arrowhead = none ];
53 -> 18;

54 [label="INSTANCE with matching:\nT1 -> T14\nT2 -> T11\nT3 -> T13", fontsize=14, style = filled, fillcolor = lightgrey];
17 -> 54 [arrowhead = none , style=dashed];
54 -> 1[style=dashed];

55 [label="BACKTRACK\nfor clause: minus(0, Y, 0)\nwith clash: (div(T10, T11, T3), div(0, X7, X8))", fontsize=14, style = filled, fillcolor = white];
18 -> 55 [arrowhead = none ];
55 -> 19;

56 [label="BACKTRACK\nfor clause: minus(X, 0, X)\nwith clash: (div(T10, T11, T3), div(X3, 0, X4))", fontsize=14, style = filled, fillcolor = white];
19 -> 56 [arrowhead = none ];
56 -> 20;

57 [label="EVAL with clause\nminus(s(X25), s(X26), X27) :- minus(X25, X26, X27).\nand substitution\nX25 -> T15\nT10 -> s(T15)\nX26 -> T16\nT11 -> s(T16)\nX17 -> X28\nX27 -> X28", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 57 [arrowhead = none ];
57 -> 21;

58 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 58 [arrowhead = none ];
58 -> 22;

59 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 59 [arrowhead = none ];
59 -> 23;

60 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
23 -> 60 [arrowhead = none ];
60 -> 24;

61 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
23 -> 61 [arrowhead = none ];
61 -> 25;

62 [label="EVAL with clause\nminus(0, X31, 0).\nand substitution\nT15 -> 0\nT16 -> T17\nX31 -> T17\nX28 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 62 [arrowhead = none ];
62 -> 26;

63 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 63 [arrowhead = none ];
63 -> 27;

64 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 64 [arrowhead = none ];
64 -> 29;

65 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 65 [arrowhead = none ];
65 -> 30;

66 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 66 [arrowhead = none ];
66 -> 28;

67 [label="EVAL with clause\nminus(X34, 0, X34).\nand substitution\nT15 -> T18\nX34 -> T18\nT16 -> 0\nX28 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 67 [arrowhead = none ];
67 -> 31;

68 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 68 [arrowhead = none ];
68 -> 32;

69 [label="EVAL with clause\nminus(s(X38), s(X39), X40) :- minus(X38, X39, X40).\nand substitution\nX38 -> T19\nT15 -> s(T19)\nX39 -> T20\nT16 -> s(T20)\nX28 -> X41\nX40 -> X41", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 69 [arrowhead = none ];
69 -> 34;

70 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 70 [arrowhead = none ];
70 -> 35;

71 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 71 [arrowhead = none ];
71 -> 33;

72 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 72 [arrowhead = none ];
72 -> 36;

73 [label="INSTANCE with matching:\nT15 -> T19\nT16 -> T20\nX28 -> X41", fontsize=14, style = filled, fillcolor = lightgrey];
36 -> 73 [arrowhead = none , style=dashed];
73 -> 23[style=dashed];

}

(4) Obligation:

Clauses:

div1(0, T6, 0).
div1(T10, T11, s(T13)) :- minus16(T10, T11, X17).
div1(T10, T11, s(T13)) :- ','(minus16(T10, T11, T14), div1(T14, T11, T13)).
minus23(0, T17, 0).
minus23(T18, 0, T18).
minus23(s(T19), s(T20), X41) :- minus23(T19, T20, X41).
minus16(s(0), s(T17), 0).
minus16(s(T18), s(0), T18).
minus16(s(s(T19)), s(s(T20)), X41) :- minus23(T19, T20, X41).

Queries:

div1(g,g,a).

(5) 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)
minus16_in: (b,b,f)
minus23_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(T10, T11, s(T13)) → U1_gga(T10, T11, T13, minus16_in_gga(T10, T11, X17))
minus16_in_gga(s(0), s(T17), 0) → minus16_out_gga(s(0), s(T17), 0)
minus16_in_gga(s(T18), s(0), T18) → minus16_out_gga(s(T18), s(0), T18)
minus16_in_gga(s(s(T19)), s(s(T20)), X41) → U5_gga(T19, T20, X41, minus23_in_gga(T19, T20, X41))
minus23_in_gga(0, T17, 0) → minus23_out_gga(0, T17, 0)
minus23_in_gga(T18, 0, T18) → minus23_out_gga(T18, 0, T18)
minus23_in_gga(s(T19), s(T20), X41) → U4_gga(T19, T20, X41, minus23_in_gga(T19, T20, X41))
U4_gga(T19, T20, X41, minus23_out_gga(T19, T20, X41)) → minus23_out_gga(s(T19), s(T20), X41)
U5_gga(T19, T20, X41, minus23_out_gga(T19, T20, X41)) → minus16_out_gga(s(s(T19)), s(s(T20)), X41)
U1_gga(T10, T11, T13, minus16_out_gga(T10, T11, X17)) → div1_out_gga(T10, T11, s(T13))
div1_in_gga(T10, T11, s(T13)) → U2_gga(T10, T11, T13, minus16_in_gga(T10, T11, T14))
U2_gga(T10, T11, T13, minus16_out_gga(T10, T11, T14)) → U3_gga(T10, T11, T13, div1_in_gga(T14, T11, T13))
U3_gga(T10, T11, T13, div1_out_gga(T14, T11, T13)) → div1_out_gga(T10, T11, s(T13))

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)
minus16_in_gga(x1, x2, x3)  =  minus16_in_gga(x1, x2)
s(x1)  =  s(x1)
minus16_out_gga(x1, x2, x3)  =  minus16_out_gga(x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
minus23_in_gga(x1, x2, x3)  =  minus23_in_gga(x1, x2)
minus23_out_gga(x1, x2, x3)  =  minus23_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

(6) 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(T10, T11, s(T13)) → U1_gga(T10, T11, T13, minus16_in_gga(T10, T11, X17))
minus16_in_gga(s(0), s(T17), 0) → minus16_out_gga(s(0), s(T17), 0)
minus16_in_gga(s(T18), s(0), T18) → minus16_out_gga(s(T18), s(0), T18)
minus16_in_gga(s(s(T19)), s(s(T20)), X41) → U5_gga(T19, T20, X41, minus23_in_gga(T19, T20, X41))
minus23_in_gga(0, T17, 0) → minus23_out_gga(0, T17, 0)
minus23_in_gga(T18, 0, T18) → minus23_out_gga(T18, 0, T18)
minus23_in_gga(s(T19), s(T20), X41) → U4_gga(T19, T20, X41, minus23_in_gga(T19, T20, X41))
U4_gga(T19, T20, X41, minus23_out_gga(T19, T20, X41)) → minus23_out_gga(s(T19), s(T20), X41)
U5_gga(T19, T20, X41, minus23_out_gga(T19, T20, X41)) → minus16_out_gga(s(s(T19)), s(s(T20)), X41)
U1_gga(T10, T11, T13, minus16_out_gga(T10, T11, X17)) → div1_out_gga(T10, T11, s(T13))
div1_in_gga(T10, T11, s(T13)) → U2_gga(T10, T11, T13, minus16_in_gga(T10, T11, T14))
U2_gga(T10, T11, T13, minus16_out_gga(T10, T11, T14)) → U3_gga(T10, T11, T13, div1_in_gga(T14, T11, T13))
U3_gga(T10, T11, T13, div1_out_gga(T14, T11, T13)) → div1_out_gga(T10, T11, s(T13))

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)
minus16_in_gga(x1, x2, x3)  =  minus16_in_gga(x1, x2)
s(x1)  =  s(x1)
minus16_out_gga(x1, x2, x3)  =  minus16_out_gga(x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
minus23_in_gga(x1, x2, x3)  =  minus23_in_gga(x1, x2)
minus23_out_gga(x1, x2, x3)  =  minus23_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)

(7) 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(T10, T11, s(T13)) → U1_GGA(T10, T11, T13, minus16_in_gga(T10, T11, X17))
DIV1_IN_GGA(T10, T11, s(T13)) → MINUS16_IN_GGA(T10, T11, X17)
MINUS16_IN_GGA(s(s(T19)), s(s(T20)), X41) → U5_GGA(T19, T20, X41, minus23_in_gga(T19, T20, X41))
MINUS16_IN_GGA(s(s(T19)), s(s(T20)), X41) → MINUS23_IN_GGA(T19, T20, X41)
MINUS23_IN_GGA(s(T19), s(T20), X41) → U4_GGA(T19, T20, X41, minus23_in_gga(T19, T20, X41))
MINUS23_IN_GGA(s(T19), s(T20), X41) → MINUS23_IN_GGA(T19, T20, X41)
DIV1_IN_GGA(T10, T11, s(T13)) → U2_GGA(T10, T11, T13, minus16_in_gga(T10, T11, T14))
U2_GGA(T10, T11, T13, minus16_out_gga(T10, T11, T14)) → U3_GGA(T10, T11, T13, div1_in_gga(T14, T11, T13))
U2_GGA(T10, T11, T13, minus16_out_gga(T10, T11, T14)) → DIV1_IN_GGA(T14, T11, T13)

The TRS R consists of the following rules:

div1_in_gga(0, T6, 0) → div1_out_gga(0, T6, 0)
div1_in_gga(T10, T11, s(T13)) → U1_gga(T10, T11, T13, minus16_in_gga(T10, T11, X17))
minus16_in_gga(s(0), s(T17), 0) → minus16_out_gga(s(0), s(T17), 0)
minus16_in_gga(s(T18), s(0), T18) → minus16_out_gga(s(T18), s(0), T18)
minus16_in_gga(s(s(T19)), s(s(T20)), X41) → U5_gga(T19, T20, X41, minus23_in_gga(T19, T20, X41))
minus23_in_gga(0, T17, 0) → minus23_out_gga(0, T17, 0)
minus23_in_gga(T18, 0, T18) → minus23_out_gga(T18, 0, T18)
minus23_in_gga(s(T19), s(T20), X41) → U4_gga(T19, T20, X41, minus23_in_gga(T19, T20, X41))
U4_gga(T19, T20, X41, minus23_out_gga(T19, T20, X41)) → minus23_out_gga(s(T19), s(T20), X41)
U5_gga(T19, T20, X41, minus23_out_gga(T19, T20, X41)) → minus16_out_gga(s(s(T19)), s(s(T20)), X41)
U1_gga(T10, T11, T13, minus16_out_gga(T10, T11, X17)) → div1_out_gga(T10, T11, s(T13))
div1_in_gga(T10, T11, s(T13)) → U2_gga(T10, T11, T13, minus16_in_gga(T10, T11, T14))
U2_gga(T10, T11, T13, minus16_out_gga(T10, T11, T14)) → U3_gga(T10, T11, T13, div1_in_gga(T14, T11, T13))
U3_gga(T10, T11, T13, div1_out_gga(T14, T11, T13)) → div1_out_gga(T10, T11, s(T13))

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)
minus16_in_gga(x1, x2, x3)  =  minus16_in_gga(x1, x2)
s(x1)  =  s(x1)
minus16_out_gga(x1, x2, x3)  =  minus16_out_gga(x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
minus23_in_gga(x1, x2, x3)  =  minus23_in_gga(x1, x2)
minus23_out_gga(x1, x2, x3)  =  minus23_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)
MINUS16_IN_GGA(x1, x2, x3)  =  MINUS16_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x4)
MINUS23_IN_GGA(x1, x2, x3)  =  MINUS23_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

(8) Obligation:

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

DIV1_IN_GGA(T10, T11, s(T13)) → U1_GGA(T10, T11, T13, minus16_in_gga(T10, T11, X17))
DIV1_IN_GGA(T10, T11, s(T13)) → MINUS16_IN_GGA(T10, T11, X17)
MINUS16_IN_GGA(s(s(T19)), s(s(T20)), X41) → U5_GGA(T19, T20, X41, minus23_in_gga(T19, T20, X41))
MINUS16_IN_GGA(s(s(T19)), s(s(T20)), X41) → MINUS23_IN_GGA(T19, T20, X41)
MINUS23_IN_GGA(s(T19), s(T20), X41) → U4_GGA(T19, T20, X41, minus23_in_gga(T19, T20, X41))
MINUS23_IN_GGA(s(T19), s(T20), X41) → MINUS23_IN_GGA(T19, T20, X41)
DIV1_IN_GGA(T10, T11, s(T13)) → U2_GGA(T10, T11, T13, minus16_in_gga(T10, T11, T14))
U2_GGA(T10, T11, T13, minus16_out_gga(T10, T11, T14)) → U3_GGA(T10, T11, T13, div1_in_gga(T14, T11, T13))
U2_GGA(T10, T11, T13, minus16_out_gga(T10, T11, T14)) → DIV1_IN_GGA(T14, T11, T13)

The TRS R consists of the following rules:

div1_in_gga(0, T6, 0) → div1_out_gga(0, T6, 0)
div1_in_gga(T10, T11, s(T13)) → U1_gga(T10, T11, T13, minus16_in_gga(T10, T11, X17))
minus16_in_gga(s(0), s(T17), 0) → minus16_out_gga(s(0), s(T17), 0)
minus16_in_gga(s(T18), s(0), T18) → minus16_out_gga(s(T18), s(0), T18)
minus16_in_gga(s(s(T19)), s(s(T20)), X41) → U5_gga(T19, T20, X41, minus23_in_gga(T19, T20, X41))
minus23_in_gga(0, T17, 0) → minus23_out_gga(0, T17, 0)
minus23_in_gga(T18, 0, T18) → minus23_out_gga(T18, 0, T18)
minus23_in_gga(s(T19), s(T20), X41) → U4_gga(T19, T20, X41, minus23_in_gga(T19, T20, X41))
U4_gga(T19, T20, X41, minus23_out_gga(T19, T20, X41)) → minus23_out_gga(s(T19), s(T20), X41)
U5_gga(T19, T20, X41, minus23_out_gga(T19, T20, X41)) → minus16_out_gga(s(s(T19)), s(s(T20)), X41)
U1_gga(T10, T11, T13, minus16_out_gga(T10, T11, X17)) → div1_out_gga(T10, T11, s(T13))
div1_in_gga(T10, T11, s(T13)) → U2_gga(T10, T11, T13, minus16_in_gga(T10, T11, T14))
U2_gga(T10, T11, T13, minus16_out_gga(T10, T11, T14)) → U3_gga(T10, T11, T13, div1_in_gga(T14, T11, T13))
U3_gga(T10, T11, T13, div1_out_gga(T14, T11, T13)) → div1_out_gga(T10, T11, s(T13))

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)
minus16_in_gga(x1, x2, x3)  =  minus16_in_gga(x1, x2)
s(x1)  =  s(x1)
minus16_out_gga(x1, x2, x3)  =  minus16_out_gga(x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
minus23_in_gga(x1, x2, x3)  =  minus23_in_gga(x1, x2)
minus23_out_gga(x1, x2, x3)  =  minus23_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)
MINUS16_IN_GGA(x1, x2, x3)  =  MINUS16_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x4)
MINUS23_IN_GGA(x1, x2, x3)  =  MINUS23_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

(9) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes.

(10) Complex Obligation (AND)

(11) Obligation:

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

MINUS23_IN_GGA(s(T19), s(T20), X41) → MINUS23_IN_GGA(T19, T20, X41)

The TRS R consists of the following rules:

div1_in_gga(0, T6, 0) → div1_out_gga(0, T6, 0)
div1_in_gga(T10, T11, s(T13)) → U1_gga(T10, T11, T13, minus16_in_gga(T10, T11, X17))
minus16_in_gga(s(0), s(T17), 0) → minus16_out_gga(s(0), s(T17), 0)
minus16_in_gga(s(T18), s(0), T18) → minus16_out_gga(s(T18), s(0), T18)
minus16_in_gga(s(s(T19)), s(s(T20)), X41) → U5_gga(T19, T20, X41, minus23_in_gga(T19, T20, X41))
minus23_in_gga(0, T17, 0) → minus23_out_gga(0, T17, 0)
minus23_in_gga(T18, 0, T18) → minus23_out_gga(T18, 0, T18)
minus23_in_gga(s(T19), s(T20), X41) → U4_gga(T19, T20, X41, minus23_in_gga(T19, T20, X41))
U4_gga(T19, T20, X41, minus23_out_gga(T19, T20, X41)) → minus23_out_gga(s(T19), s(T20), X41)
U5_gga(T19, T20, X41, minus23_out_gga(T19, T20, X41)) → minus16_out_gga(s(s(T19)), s(s(T20)), X41)
U1_gga(T10, T11, T13, minus16_out_gga(T10, T11, X17)) → div1_out_gga(T10, T11, s(T13))
div1_in_gga(T10, T11, s(T13)) → U2_gga(T10, T11, T13, minus16_in_gga(T10, T11, T14))
U2_gga(T10, T11, T13, minus16_out_gga(T10, T11, T14)) → U3_gga(T10, T11, T13, div1_in_gga(T14, T11, T13))
U3_gga(T10, T11, T13, div1_out_gga(T14, T11, T13)) → div1_out_gga(T10, T11, s(T13))

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)
minus16_in_gga(x1, x2, x3)  =  minus16_in_gga(x1, x2)
s(x1)  =  s(x1)
minus16_out_gga(x1, x2, x3)  =  minus16_out_gga(x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
minus23_in_gga(x1, x2, x3)  =  minus23_in_gga(x1, x2)
minus23_out_gga(x1, x2, x3)  =  minus23_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)
MINUS23_IN_GGA(x1, x2, x3)  =  MINUS23_IN_GGA(x1, x2)

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

(12) UsableRulesProof (EQUIVALENT transformation)

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

(13) Obligation:

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

MINUS23_IN_GGA(s(T19), s(T20), X41) → MINUS23_IN_GGA(T19, T20, X41)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
MINUS23_IN_GGA(x1, x2, x3)  =  MINUS23_IN_GGA(x1, x2)

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

(14) PiDPToQDPProof (SOUND transformation)

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

(15) Obligation:

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

MINUS23_IN_GGA(s(T19), s(T20)) → MINUS23_IN_GGA(T19, T20)

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

(16) 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:

• MINUS23_IN_GGA(s(T19), s(T20)) → MINUS23_IN_GGA(T19, T20)
  The graph contains the following edges 1 > 1, 2 > 2

(17) TRUE

(18) Obligation:

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

DIV1_IN_GGA(T10, T11, s(T13)) → U2_GGA(T10, T11, T13, minus16_in_gga(T10, T11, T14))
U2_GGA(T10, T11, T13, minus16_out_gga(T10, T11, T14)) → DIV1_IN_GGA(T14, T11, T13)

The TRS R consists of the following rules:

div1_in_gga(0, T6, 0) → div1_out_gga(0, T6, 0)
div1_in_gga(T10, T11, s(T13)) → U1_gga(T10, T11, T13, minus16_in_gga(T10, T11, X17))
minus16_in_gga(s(0), s(T17), 0) → minus16_out_gga(s(0), s(T17), 0)
minus16_in_gga(s(T18), s(0), T18) → minus16_out_gga(s(T18), s(0), T18)
minus16_in_gga(s(s(T19)), s(s(T20)), X41) → U5_gga(T19, T20, X41, minus23_in_gga(T19, T20, X41))
minus23_in_gga(0, T17, 0) → minus23_out_gga(0, T17, 0)
minus23_in_gga(T18, 0, T18) → minus23_out_gga(T18, 0, T18)
minus23_in_gga(s(T19), s(T20), X41) → U4_gga(T19, T20, X41, minus23_in_gga(T19, T20, X41))
U4_gga(T19, T20, X41, minus23_out_gga(T19, T20, X41)) → minus23_out_gga(s(T19), s(T20), X41)
U5_gga(T19, T20, X41, minus23_out_gga(T19, T20, X41)) → minus16_out_gga(s(s(T19)), s(s(T20)), X41)
U1_gga(T10, T11, T13, minus16_out_gga(T10, T11, X17)) → div1_out_gga(T10, T11, s(T13))
div1_in_gga(T10, T11, s(T13)) → U2_gga(T10, T11, T13, minus16_in_gga(T10, T11, T14))
U2_gga(T10, T11, T13, minus16_out_gga(T10, T11, T14)) → U3_gga(T10, T11, T13, div1_in_gga(T14, T11, T13))
U3_gga(T10, T11, T13, div1_out_gga(T14, T11, T13)) → div1_out_gga(T10, T11, s(T13))

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)
minus16_in_gga(x1, x2, x3)  =  minus16_in_gga(x1, x2)
s(x1)  =  s(x1)
minus16_out_gga(x1, x2, x3)  =  minus16_out_gga(x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
minus23_in_gga(x1, x2, x3)  =  minus23_in_gga(x1, x2)
minus23_out_gga(x1, x2, x3)  =  minus23_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

(19) UsableRulesProof (EQUIVALENT transformation)

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

(20) Obligation:

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

DIV1_IN_GGA(T10, T11, s(T13)) → U2_GGA(T10, T11, T13, minus16_in_gga(T10, T11, T14))
U2_GGA(T10, T11, T13, minus16_out_gga(T10, T11, T14)) → DIV1_IN_GGA(T14, T11, T13)

The TRS R consists of the following rules:

minus16_in_gga(s(0), s(T17), 0) → minus16_out_gga(s(0), s(T17), 0)
minus16_in_gga(s(T18), s(0), T18) → minus16_out_gga(s(T18), s(0), T18)
minus16_in_gga(s(s(T19)), s(s(T20)), X41) → U5_gga(T19, T20, X41, minus23_in_gga(T19, T20, X41))
U5_gga(T19, T20, X41, minus23_out_gga(T19, T20, X41)) → minus16_out_gga(s(s(T19)), s(s(T20)), X41)
minus23_in_gga(0, T17, 0) → minus23_out_gga(0, T17, 0)
minus23_in_gga(T18, 0, T18) → minus23_out_gga(T18, 0, T18)
minus23_in_gga(s(T19), s(T20), X41) → U4_gga(T19, T20, X41, minus23_in_gga(T19, T20, X41))
U4_gga(T19, T20, X41, minus23_out_gga(T19, T20, X41)) → minus23_out_gga(s(T19), s(T20), X41)

The argument filtering Pi contains the following mapping:
0  =  0
minus16_in_gga(x1, x2, x3)  =  minus16_in_gga(x1, x2)
s(x1)  =  s(x1)
minus16_out_gga(x1, x2, x3)  =  minus16_out_gga(x3)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
minus23_in_gga(x1, x2, x3)  =  minus23_in_gga(x1, x2)
minus23_out_gga(x1, x2, x3)  =  minus23_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

(21) PiDPToQDPProof (SOUND transformation)

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

(22) Obligation:

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

DIV1_IN_GGA(T10, T11) → U2_GGA(T11, minus16_in_gga(T10, T11))
U2_GGA(T11, minus16_out_gga(T14)) → DIV1_IN_GGA(T14, T11)

The TRS R consists of the following rules:

minus16_in_gga(s(0), s(T17)) → minus16_out_gga(0)
minus16_in_gga(s(T18), s(0)) → minus16_out_gga(T18)
minus16_in_gga(s(s(T19)), s(s(T20))) → U5_gga(minus23_in_gga(T19, T20))
U5_gga(minus23_out_gga(X41)) → minus16_out_gga(X41)
minus23_in_gga(0, T17) → minus23_out_gga(0)
minus23_in_gga(T18, 0) → minus23_out_gga(T18)
minus23_in_gga(s(T19), s(T20)) → U4_gga(minus23_in_gga(T19, T20))
U4_gga(minus23_out_gga(X41)) → minus23_out_gga(X41)

The set Q consists of the following terms:

minus16_in_gga(x0, x1)
U5_gga(x0)
minus23_in_gga(x0, x1)
U4_gga(x0)

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

(23) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U2_GGA(T11, minus16_out_gga(T14)) → DIV1_IN_GGA(T14, T11)

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(minus16_in_gga(x₁, x₂)) = x₁   
POL(minus16_out_gga(x₁)) = 1 + x₁   
POL(minus23_in_gga(x₁, x₂)) = 1 + x₁   
POL(minus23_out_gga(x₁)) = 1 + x₁   
POL(s(x₁)) = 1 + x₁   

The following usable rules [FROCOS05] were oriented:

minus16_in_gga(s(0), s(T17)) → minus16_out_gga(0)
minus16_in_gga(s(T18), s(0)) → minus16_out_gga(T18)
minus16_in_gga(s(s(T19)), s(s(T20))) → U5_gga(minus23_in_gga(T19, T20))
minus23_in_gga(0, T17) → minus23_out_gga(0)
minus23_in_gga(T18, 0) → minus23_out_gga(T18)
minus23_in_gga(s(T19), s(T20)) → U4_gga(minus23_in_gga(T19, T20))
U5_gga(minus23_out_gga(X41)) → minus16_out_gga(X41)
U4_gga(minus23_out_gga(X41)) → minus23_out_gga(X41)

(24) Obligation:

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

DIV1_IN_GGA(T10, T11) → U2_GGA(T11, minus16_in_gga(T10, T11))

The TRS R consists of the following rules:

minus16_in_gga(s(0), s(T17)) → minus16_out_gga(0)
minus16_in_gga(s(T18), s(0)) → minus16_out_gga(T18)
minus16_in_gga(s(s(T19)), s(s(T20))) → U5_gga(minus23_in_gga(T19, T20))
U5_gga(minus23_out_gga(X41)) → minus16_out_gga(X41)
minus23_in_gga(0, T17) → minus23_out_gga(0)
minus23_in_gga(T18, 0) → minus23_out_gga(T18)
minus23_in_gga(s(T19), s(T20)) → U4_gga(minus23_in_gga(T19, T20))
U4_gga(minus23_out_gga(X41)) → minus23_out_gga(X41)

The set Q consists of the following terms:

minus16_in_gga(x0, x1)
U5_gga(x0)
minus23_in_gga(x0, x1)
U4_gga(x0)

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

(25) DependencyGraphProof (EQUIVALENT transformation)

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

(26) TRUE