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

(0) Obligation:

Clauses:

div(X, s(Y), Z) :- div_s(X, Y, Z).
div_s(0, Y, 0).
div_s(s(X), Y, 0) :- lss(X, Y).
div_s(s(X), Y, s(Z)) :- ','(sub(X, Y, R), div_s(R, Y, Z)).
lss(s(X), s(Y)) :- lss(X, Y).
lss(0, s(Y)).
sub(s(X), s(Y), Z) :- sub(X, Y, Z).
sub(X, 0, X).

Query: div(g,g,a)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

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\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: div_s(T7, T8, T10)\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: div_s(T7, T8, T10), SCOPE: 2, CLAUSE: 1 | div_s(T7, T8, T10), SCOPE: 2, CLAUSE: 2 | div_s(T7, T8, T10), SCOPE: 2, CLAUSE: 3\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: div_s(T7, T8, T10), SCOPE: 2, CLAUSE: 1\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: div_s(T7, T8, T10), SCOPE: 2, CLAUSE: 2 | div_s(T7, T8, T10), SCOPE: 2, CLAUSE: 3\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: div_s(T7, T8, T10), SCOPE: 2, CLAUSE: 2\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: div_s(T7, T8, T10), SCOPE: 2, CLAUSE: 3\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: lss(T24, T25)\n\nT24 is ground\nT25 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: lss(T24, T25), SCOPE: 3, CLAUSE: 4 | lss(T24, T25), SCOPE: 3, CLAUSE: 5\n\nT24 is ground\nT25 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: lss(T24, T25), SCOPE: 3, CLAUSE: 4\n\nT24 is ground\nT25 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: lss(T24, T25), SCOPE: 3, CLAUSE: 5\n\nT24 is ground\nT25 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: lss(T36, T37)\n\nT36 is ground\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(sub(T49, T50, X49), div_s(X49, T50, T52))\n\nT49 is ground\nT50 is ground\nX49 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: sub(T49, T50, X49)\n\nT49 is ground\nT50 is ground\nX49 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: div_s(T55, T50, T52)\n\nT50 is ground\nT55 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: sub(T49, T50, X49), SCOPE: 4, CLAUSE: 6 | sub(T49, T50, X49), SCOPE: 4, CLAUSE: 7\n\nT49 is ground\nT50 is ground\nX49 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: sub(T49, T50, X49), SCOPE: 4, CLAUSE: 6\n\nT49 is ground\nT50 is ground\nX49 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: sub(T49, T50, X49), SCOPE: 4, CLAUSE: 7\n\nT49 is ground\nT50 is ground\nX49 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: sub(T66, T67, X77)\n\nT66 is ground\nT67 is ground\nX77 is free", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 35 [arrowhead = none ];
35 -> 2;

36 [label="EVAL with clause\ndiv(X4, s(X5), X6) :- div_s(X4, X5, X6).\nand substitutionT1 -> T7,\nX4 -> T7,\nX5 -> T8,\nT2 -> s(T8),\nT3 -> T10,\nX6 -> T10,\nT9 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 36 [arrowhead = none ];
36 -> 3;

37 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 37 [arrowhead = none ];
37 -> 4;

38 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 38 [arrowhead = none ];
38 -> 5;

39 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 39 [arrowhead = none ];
39 -> 6;

40 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 40 [arrowhead = none ];
40 -> 7;

41 [label="EVAL with clause\ndiv_s(0, X11, 0).\nand substitutionT7 -> 0,\nT8 -> T15,\nX11 -> T15,\nT10 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 41 [arrowhead = none ];
41 -> 8;

42 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 42 [arrowhead = none ];
42 -> 9;

43 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 43 [arrowhead = none ];
43 -> 11;

44 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 44 [arrowhead = none ];
44 -> 12;

45 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 45 [arrowhead = none ];
45 -> 10;

46 [label="EVAL with clause\ndiv_s(s(X20), X21, 0) :- lss(X20, X21).\nand substitutionX20 -> T24,\nT7 -> s(T24),\nT8 -> T25,\nX21 -> T25,\nT10 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 46 [arrowhead = none ];
46 -> 13;

47 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 47 [arrowhead = none ];
47 -> 14;

48 [label="EVAL with clause\ndiv_s(s(X46), X47, s(X48)) :- ','(sub(X46, X47, X49), div_s(X49, X47, X48)).\nand substitutionX46 -> T49,\nT7 -> s(T49),\nT8 -> T50,\nX47 -> T50,\nX48 -> T52,\nT10 -> s(T52),\nT51 -> T52", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 48 [arrowhead = none ];
48 -> 23;

49 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 49 [arrowhead = none ];
49 -> 24;

50 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 50 [arrowhead = none ];
50 -> 15;

51 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
15 -> 51 [arrowhead = none ];
51 -> 16;

52 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
15 -> 52 [arrowhead = none ];
52 -> 17;

53 [label="EVAL with clause\nlss(s(X32), s(X33)) :- lss(X32, X33).\nand substitutionX32 -> T36,\nT24 -> s(T36),\nX33 -> T37,\nT25 -> s(T37)", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 53 [arrowhead = none ];
53 -> 18;

54 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 54 [arrowhead = none ];
54 -> 19;

55 [label="EVAL with clause\nlss(0, s(X38)).\nand substitutionT24 -> 0,\nX38 -> T42,\nT25 -> s(T42)", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 55 [arrowhead = none ];
55 -> 20;

56 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 56 [arrowhead = none ];
56 -> 21;

57 [label="INSTANCE with matching:\nT24 -> T36\nT25 -> T37", fontsize=14, style = filled, fillcolor = lightgrey];
18 -> 57 [arrowhead = none , style=dashed];
57 -> 13[style=dashed];

58 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 58 [arrowhead = none ];
58 -> 22;

59 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
23 -> 59 [arrowhead = none ];
59 -> 25;

60 [label="SPLIT 2\nnew knowledge:\nT49 is ground\nT50 is ground\nT55 is ground\nreplacements:X49 -> T55", fontsize=14, style = filled, fillcolor = violet];
23 -> 60 [arrowhead = none ];
60 -> 26;

61 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 61 [arrowhead = none ];
61 -> 27;

62 [label="INSTANCE with matching:\nT7 -> T55\nT8 -> T50\nT10 -> T52", fontsize=14, style = filled, fillcolor = lightgrey];
26 -> 62 [arrowhead = none , style=dashed];
62 -> 3[style=dashed];

63 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 63 [arrowhead = none ];
63 -> 28;

64 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 64 [arrowhead = none ];
64 -> 29;

65 [label="EVAL with clause\nsub(s(X74), s(X75), X76) :- sub(X74, X75, X76).\nand substitutionX74 -> T66,\nT49 -> s(T66),\nX75 -> T67,\nT50 -> s(T67),\nX49 -> X77,\nX76 -> X77", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 65 [arrowhead = none ];
65 -> 30;

66 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 66 [arrowhead = none ];
66 -> 31;

67 [label="EVAL with clause\nsub(X84, 0, X84).\nand substitutionT49 -> T72,\nX84 -> T72,\nT50 -> 0,\nX49 -> T72", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 67 [arrowhead = none ];
67 -> 32;

68 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 68 [arrowhead = none ];
68 -> 33;

69 [label="INSTANCE with matching:\nT49 -> T66\nT50 -> T67\nX49 -> X77", fontsize=14, style = filled, fillcolor = lightgrey];
30 -> 69 [arrowhead = none , style=dashed];
69 -> 25[style=dashed];

70 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 70 [arrowhead = none ];
70 -> 34;

}

(2) Obligation:

Clauses:

lssA(s(T36), s(T37)) :- lssA(T36, T37).
lssA(0, s(T42)).
div_sB(0, T15, 0).
div_sB(s(T24), T25, 0) :- lssA(T24, T25).
div_sB(s(T49), T50, s(T52)) :- subC(T49, T50, X49).
div_sB(s(T49), T50, s(T52)) :- ','(subC(T49, T50, T55), div_sB(T55, T50, T52)).
subC(s(T66), s(T67), X77) :- subC(T66, T67, X77).
subC(T72, 0, T72).
divD(T7, s(T8), T10) :- div_sB(T7, T8, T10).

Query: divD(g,g,a)

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
divD_in: (b,b,f)
div_sB_in: (b,b,f)
lssA_in: (b,b)
subC_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

divD_in_gga(T7, s(T8), T10) → U7_gga(T7, T8, T10, div_sB_in_gga(T7, T8, T10))
div_sB_in_gga(0, T15, 0) → div_sB_out_gga(0, T15, 0)
div_sB_in_gga(s(T24), T25, 0) → U2_gga(T24, T25, lssA_in_gg(T24, T25))
lssA_in_gg(s(T36), s(T37)) → U1_gg(T36, T37, lssA_in_gg(T36, T37))
lssA_in_gg(0, s(T42)) → lssA_out_gg(0, s(T42))
U1_gg(T36, T37, lssA_out_gg(T36, T37)) → lssA_out_gg(s(T36), s(T37))
U2_gga(T24, T25, lssA_out_gg(T24, T25)) → div_sB_out_gga(s(T24), T25, 0)
div_sB_in_gga(s(T49), T50, s(T52)) → U3_gga(T49, T50, T52, subC_in_gga(T49, T50, X49))
subC_in_gga(s(T66), s(T67), X77) → U6_gga(T66, T67, X77, subC_in_gga(T66, T67, X77))
subC_in_gga(T72, 0, T72) → subC_out_gga(T72, 0, T72)
U6_gga(T66, T67, X77, subC_out_gga(T66, T67, X77)) → subC_out_gga(s(T66), s(T67), X77)
U3_gga(T49, T50, T52, subC_out_gga(T49, T50, X49)) → div_sB_out_gga(s(T49), T50, s(T52))
div_sB_in_gga(s(T49), T50, s(T52)) → U4_gga(T49, T50, T52, subC_in_gga(T49, T50, T55))
U4_gga(T49, T50, T52, subC_out_gga(T49, T50, T55)) → U5_gga(T49, T50, T52, div_sB_in_gga(T55, T50, T52))
U5_gga(T49, T50, T52, div_sB_out_gga(T55, T50, T52)) → div_sB_out_gga(s(T49), T50, s(T52))
U7_gga(T7, T8, T10, div_sB_out_gga(T7, T8, T10)) → divD_out_gga(T7, s(T8), T10)

The argument filtering Pi contains the following mapping:
divD_in_gga(x1, x2, x3)  =  divD_in_gga(x1, x2)
s(x1)  =  s(x1)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)
div_sB_in_gga(x1, x2, x3)  =  div_sB_in_gga(x1, x2)
0  =  0
div_sB_out_gga(x1, x2, x3)  =  div_sB_out_gga
U2_gga(x1, x2, x3)  =  U2_gga(x3)
lssA_in_gg(x1, x2)  =  lssA_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
lssA_out_gg(x1, x2)  =  lssA_out_gg
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
subC_in_gga(x1, x2, x3)  =  subC_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
subC_out_gga(x1, x2, x3)  =  subC_out_gga(x3)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x2, x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
divD_out_gga(x1, x2, x3)  =  divD_out_gga

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

divD_in_gga(T7, s(T8), T10) → U7_gga(T7, T8, T10, div_sB_in_gga(T7, T8, T10))
div_sB_in_gga(0, T15, 0) → div_sB_out_gga(0, T15, 0)
div_sB_in_gga(s(T24), T25, 0) → U2_gga(T24, T25, lssA_in_gg(T24, T25))
lssA_in_gg(s(T36), s(T37)) → U1_gg(T36, T37, lssA_in_gg(T36, T37))
lssA_in_gg(0, s(T42)) → lssA_out_gg(0, s(T42))
U1_gg(T36, T37, lssA_out_gg(T36, T37)) → lssA_out_gg(s(T36), s(T37))
U2_gga(T24, T25, lssA_out_gg(T24, T25)) → div_sB_out_gga(s(T24), T25, 0)
div_sB_in_gga(s(T49), T50, s(T52)) → U3_gga(T49, T50, T52, subC_in_gga(T49, T50, X49))
subC_in_gga(s(T66), s(T67), X77) → U6_gga(T66, T67, X77, subC_in_gga(T66, T67, X77))
subC_in_gga(T72, 0, T72) → subC_out_gga(T72, 0, T72)
U6_gga(T66, T67, X77, subC_out_gga(T66, T67, X77)) → subC_out_gga(s(T66), s(T67), X77)
U3_gga(T49, T50, T52, subC_out_gga(T49, T50, X49)) → div_sB_out_gga(s(T49), T50, s(T52))
div_sB_in_gga(s(T49), T50, s(T52)) → U4_gga(T49, T50, T52, subC_in_gga(T49, T50, T55))
U4_gga(T49, T50, T52, subC_out_gga(T49, T50, T55)) → U5_gga(T49, T50, T52, div_sB_in_gga(T55, T50, T52))
U5_gga(T49, T50, T52, div_sB_out_gga(T55, T50, T52)) → div_sB_out_gga(s(T49), T50, s(T52))
U7_gga(T7, T8, T10, div_sB_out_gga(T7, T8, T10)) → divD_out_gga(T7, s(T8), T10)

The argument filtering Pi contains the following mapping:
divD_in_gga(x1, x2, x3)  =  divD_in_gga(x1, x2)
s(x1)  =  s(x1)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)
div_sB_in_gga(x1, x2, x3)  =  div_sB_in_gga(x1, x2)
0  =  0
div_sB_out_gga(x1, x2, x3)  =  div_sB_out_gga
U2_gga(x1, x2, x3)  =  U2_gga(x3)
lssA_in_gg(x1, x2)  =  lssA_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
lssA_out_gg(x1, x2)  =  lssA_out_gg
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
subC_in_gga(x1, x2, x3)  =  subC_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
subC_out_gga(x1, x2, x3)  =  subC_out_gga(x3)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x2, x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
divD_out_gga(x1, x2, x3)  =  divD_out_gga

(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:

DIVD_IN_GGA(T7, s(T8), T10) → U7_GGA(T7, T8, T10, div_sB_in_gga(T7, T8, T10))
DIVD_IN_GGA(T7, s(T8), T10) → DIV_SB_IN_GGA(T7, T8, T10)
DIV_SB_IN_GGA(s(T24), T25, 0) → U2_GGA(T24, T25, lssA_in_gg(T24, T25))
DIV_SB_IN_GGA(s(T24), T25, 0) → LSSA_IN_GG(T24, T25)
LSSA_IN_GG(s(T36), s(T37)) → U1_GG(T36, T37, lssA_in_gg(T36, T37))
LSSA_IN_GG(s(T36), s(T37)) → LSSA_IN_GG(T36, T37)
DIV_SB_IN_GGA(s(T49), T50, s(T52)) → U3_GGA(T49, T50, T52, subC_in_gga(T49, T50, X49))
DIV_SB_IN_GGA(s(T49), T50, s(T52)) → SUBC_IN_GGA(T49, T50, X49)
SUBC_IN_GGA(s(T66), s(T67), X77) → U6_GGA(T66, T67, X77, subC_in_gga(T66, T67, X77))
SUBC_IN_GGA(s(T66), s(T67), X77) → SUBC_IN_GGA(T66, T67, X77)
DIV_SB_IN_GGA(s(T49), T50, s(T52)) → U4_GGA(T49, T50, T52, subC_in_gga(T49, T50, T55))
U4_GGA(T49, T50, T52, subC_out_gga(T49, T50, T55)) → U5_GGA(T49, T50, T52, div_sB_in_gga(T55, T50, T52))
U4_GGA(T49, T50, T52, subC_out_gga(T49, T50, T55)) → DIV_SB_IN_GGA(T55, T50, T52)

The TRS R consists of the following rules:

divD_in_gga(T7, s(T8), T10) → U7_gga(T7, T8, T10, div_sB_in_gga(T7, T8, T10))
div_sB_in_gga(0, T15, 0) → div_sB_out_gga(0, T15, 0)
div_sB_in_gga(s(T24), T25, 0) → U2_gga(T24, T25, lssA_in_gg(T24, T25))
lssA_in_gg(s(T36), s(T37)) → U1_gg(T36, T37, lssA_in_gg(T36, T37))
lssA_in_gg(0, s(T42)) → lssA_out_gg(0, s(T42))
U1_gg(T36, T37, lssA_out_gg(T36, T37)) → lssA_out_gg(s(T36), s(T37))
U2_gga(T24, T25, lssA_out_gg(T24, T25)) → div_sB_out_gga(s(T24), T25, 0)
div_sB_in_gga(s(T49), T50, s(T52)) → U3_gga(T49, T50, T52, subC_in_gga(T49, T50, X49))
subC_in_gga(s(T66), s(T67), X77) → U6_gga(T66, T67, X77, subC_in_gga(T66, T67, X77))
subC_in_gga(T72, 0, T72) → subC_out_gga(T72, 0, T72)
U6_gga(T66, T67, X77, subC_out_gga(T66, T67, X77)) → subC_out_gga(s(T66), s(T67), X77)
U3_gga(T49, T50, T52, subC_out_gga(T49, T50, X49)) → div_sB_out_gga(s(T49), T50, s(T52))
div_sB_in_gga(s(T49), T50, s(T52)) → U4_gga(T49, T50, T52, subC_in_gga(T49, T50, T55))
U4_gga(T49, T50, T52, subC_out_gga(T49, T50, T55)) → U5_gga(T49, T50, T52, div_sB_in_gga(T55, T50, T52))
U5_gga(T49, T50, T52, div_sB_out_gga(T55, T50, T52)) → div_sB_out_gga(s(T49), T50, s(T52))
U7_gga(T7, T8, T10, div_sB_out_gga(T7, T8, T10)) → divD_out_gga(T7, s(T8), T10)

The argument filtering Pi contains the following mapping:
divD_in_gga(x1, x2, x3)  =  divD_in_gga(x1, x2)
s(x1)  =  s(x1)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)
div_sB_in_gga(x1, x2, x3)  =  div_sB_in_gga(x1, x2)
0  =  0
div_sB_out_gga(x1, x2, x3)  =  div_sB_out_gga
U2_gga(x1, x2, x3)  =  U2_gga(x3)
lssA_in_gg(x1, x2)  =  lssA_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
lssA_out_gg(x1, x2)  =  lssA_out_gg
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
subC_in_gga(x1, x2, x3)  =  subC_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
subC_out_gga(x1, x2, x3)  =  subC_out_gga(x3)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x2, x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
divD_out_gga(x1, x2, x3)  =  divD_out_gga
DIVD_IN_GGA(x1, x2, x3)  =  DIVD_IN_GGA(x1, x2)
U7_GGA(x1, x2, x3, x4)  =  U7_GGA(x4)
DIV_SB_IN_GGA(x1, x2, x3)  =  DIV_SB_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3)  =  U2_GGA(x3)
LSSA_IN_GG(x1, x2)  =  LSSA_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x3)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x4)
SUBC_IN_GGA(x1, x2, x3)  =  SUBC_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x4)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x2, x4)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x4)

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

(6) Obligation:

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

DIVD_IN_GGA(T7, s(T8), T10) → U7_GGA(T7, T8, T10, div_sB_in_gga(T7, T8, T10))
DIVD_IN_GGA(T7, s(T8), T10) → DIV_SB_IN_GGA(T7, T8, T10)
DIV_SB_IN_GGA(s(T24), T25, 0) → U2_GGA(T24, T25, lssA_in_gg(T24, T25))
DIV_SB_IN_GGA(s(T24), T25, 0) → LSSA_IN_GG(T24, T25)
LSSA_IN_GG(s(T36), s(T37)) → U1_GG(T36, T37, lssA_in_gg(T36, T37))
LSSA_IN_GG(s(T36), s(T37)) → LSSA_IN_GG(T36, T37)
DIV_SB_IN_GGA(s(T49), T50, s(T52)) → U3_GGA(T49, T50, T52, subC_in_gga(T49, T50, X49))
DIV_SB_IN_GGA(s(T49), T50, s(T52)) → SUBC_IN_GGA(T49, T50, X49)
SUBC_IN_GGA(s(T66), s(T67), X77) → U6_GGA(T66, T67, X77, subC_in_gga(T66, T67, X77))
SUBC_IN_GGA(s(T66), s(T67), X77) → SUBC_IN_GGA(T66, T67, X77)
DIV_SB_IN_GGA(s(T49), T50, s(T52)) → U4_GGA(T49, T50, T52, subC_in_gga(T49, T50, T55))
U4_GGA(T49, T50, T52, subC_out_gga(T49, T50, T55)) → U5_GGA(T49, T50, T52, div_sB_in_gga(T55, T50, T52))
U4_GGA(T49, T50, T52, subC_out_gga(T49, T50, T55)) → DIV_SB_IN_GGA(T55, T50, T52)

The TRS R consists of the following rules:

divD_in_gga(T7, s(T8), T10) → U7_gga(T7, T8, T10, div_sB_in_gga(T7, T8, T10))
div_sB_in_gga(0, T15, 0) → div_sB_out_gga(0, T15, 0)
div_sB_in_gga(s(T24), T25, 0) → U2_gga(T24, T25, lssA_in_gg(T24, T25))
lssA_in_gg(s(T36), s(T37)) → U1_gg(T36, T37, lssA_in_gg(T36, T37))
lssA_in_gg(0, s(T42)) → lssA_out_gg(0, s(T42))
U1_gg(T36, T37, lssA_out_gg(T36, T37)) → lssA_out_gg(s(T36), s(T37))
U2_gga(T24, T25, lssA_out_gg(T24, T25)) → div_sB_out_gga(s(T24), T25, 0)
div_sB_in_gga(s(T49), T50, s(T52)) → U3_gga(T49, T50, T52, subC_in_gga(T49, T50, X49))
subC_in_gga(s(T66), s(T67), X77) → U6_gga(T66, T67, X77, subC_in_gga(T66, T67, X77))
subC_in_gga(T72, 0, T72) → subC_out_gga(T72, 0, T72)
U6_gga(T66, T67, X77, subC_out_gga(T66, T67, X77)) → subC_out_gga(s(T66), s(T67), X77)
U3_gga(T49, T50, T52, subC_out_gga(T49, T50, X49)) → div_sB_out_gga(s(T49), T50, s(T52))
div_sB_in_gga(s(T49), T50, s(T52)) → U4_gga(T49, T50, T52, subC_in_gga(T49, T50, T55))
U4_gga(T49, T50, T52, subC_out_gga(T49, T50, T55)) → U5_gga(T49, T50, T52, div_sB_in_gga(T55, T50, T52))
U5_gga(T49, T50, T52, div_sB_out_gga(T55, T50, T52)) → div_sB_out_gga(s(T49), T50, s(T52))
U7_gga(T7, T8, T10, div_sB_out_gga(T7, T8, T10)) → divD_out_gga(T7, s(T8), T10)

The argument filtering Pi contains the following mapping:
divD_in_gga(x1, x2, x3)  =  divD_in_gga(x1, x2)
s(x1)  =  s(x1)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)
div_sB_in_gga(x1, x2, x3)  =  div_sB_in_gga(x1, x2)
0  =  0
div_sB_out_gga(x1, x2, x3)  =  div_sB_out_gga
U2_gga(x1, x2, x3)  =  U2_gga(x3)
lssA_in_gg(x1, x2)  =  lssA_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
lssA_out_gg(x1, x2)  =  lssA_out_gg
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
subC_in_gga(x1, x2, x3)  =  subC_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
subC_out_gga(x1, x2, x3)  =  subC_out_gga(x3)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x2, x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
divD_out_gga(x1, x2, x3)  =  divD_out_gga
DIVD_IN_GGA(x1, x2, x3)  =  DIVD_IN_GGA(x1, x2)
U7_GGA(x1, x2, x3, x4)  =  U7_GGA(x4)
DIV_SB_IN_GGA(x1, x2, x3)  =  DIV_SB_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3)  =  U2_GGA(x3)
LSSA_IN_GG(x1, x2)  =  LSSA_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x3)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x4)
SUBC_IN_GGA(x1, x2, x3)  =  SUBC_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x4)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x2, x4)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x4)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 9 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

SUBC_IN_GGA(s(T66), s(T67), X77) → SUBC_IN_GGA(T66, T67, X77)

The TRS R consists of the following rules:

divD_in_gga(T7, s(T8), T10) → U7_gga(T7, T8, T10, div_sB_in_gga(T7, T8, T10))
div_sB_in_gga(0, T15, 0) → div_sB_out_gga(0, T15, 0)
div_sB_in_gga(s(T24), T25, 0) → U2_gga(T24, T25, lssA_in_gg(T24, T25))
lssA_in_gg(s(T36), s(T37)) → U1_gg(T36, T37, lssA_in_gg(T36, T37))
lssA_in_gg(0, s(T42)) → lssA_out_gg(0, s(T42))
U1_gg(T36, T37, lssA_out_gg(T36, T37)) → lssA_out_gg(s(T36), s(T37))
U2_gga(T24, T25, lssA_out_gg(T24, T25)) → div_sB_out_gga(s(T24), T25, 0)
div_sB_in_gga(s(T49), T50, s(T52)) → U3_gga(T49, T50, T52, subC_in_gga(T49, T50, X49))
subC_in_gga(s(T66), s(T67), X77) → U6_gga(T66, T67, X77, subC_in_gga(T66, T67, X77))
subC_in_gga(T72, 0, T72) → subC_out_gga(T72, 0, T72)
U6_gga(T66, T67, X77, subC_out_gga(T66, T67, X77)) → subC_out_gga(s(T66), s(T67), X77)
U3_gga(T49, T50, T52, subC_out_gga(T49, T50, X49)) → div_sB_out_gga(s(T49), T50, s(T52))
div_sB_in_gga(s(T49), T50, s(T52)) → U4_gga(T49, T50, T52, subC_in_gga(T49, T50, T55))
U4_gga(T49, T50, T52, subC_out_gga(T49, T50, T55)) → U5_gga(T49, T50, T52, div_sB_in_gga(T55, T50, T52))
U5_gga(T49, T50, T52, div_sB_out_gga(T55, T50, T52)) → div_sB_out_gga(s(T49), T50, s(T52))
U7_gga(T7, T8, T10, div_sB_out_gga(T7, T8, T10)) → divD_out_gga(T7, s(T8), T10)

The argument filtering Pi contains the following mapping:
divD_in_gga(x1, x2, x3)  =  divD_in_gga(x1, x2)
s(x1)  =  s(x1)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)
div_sB_in_gga(x1, x2, x3)  =  div_sB_in_gga(x1, x2)
0  =  0
div_sB_out_gga(x1, x2, x3)  =  div_sB_out_gga
U2_gga(x1, x2, x3)  =  U2_gga(x3)
lssA_in_gg(x1, x2)  =  lssA_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
lssA_out_gg(x1, x2)  =  lssA_out_gg
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
subC_in_gga(x1, x2, x3)  =  subC_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
subC_out_gga(x1, x2, x3)  =  subC_out_gga(x3)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x2, x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
divD_out_gga(x1, x2, x3)  =  divD_out_gga
SUBC_IN_GGA(x1, x2, x3)  =  SUBC_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:

SUBC_IN_GGA(s(T66), s(T67), X77) → SUBC_IN_GGA(T66, T67, X77)

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

SUBC_IN_GGA(s(T66), s(T67)) → SUBC_IN_GGA(T66, T67)

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:

• SUBC_IN_GGA(s(T66), s(T67)) → SUBC_IN_GGA(T66, T67)
  The graph contains the following edges 1 > 1, 2 > 2

(15) YES

(16) Obligation:

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

LSSA_IN_GG(s(T36), s(T37)) → LSSA_IN_GG(T36, T37)

The TRS R consists of the following rules:

divD_in_gga(T7, s(T8), T10) → U7_gga(T7, T8, T10, div_sB_in_gga(T7, T8, T10))
div_sB_in_gga(0, T15, 0) → div_sB_out_gga(0, T15, 0)
div_sB_in_gga(s(T24), T25, 0) → U2_gga(T24, T25, lssA_in_gg(T24, T25))
lssA_in_gg(s(T36), s(T37)) → U1_gg(T36, T37, lssA_in_gg(T36, T37))
lssA_in_gg(0, s(T42)) → lssA_out_gg(0, s(T42))
U1_gg(T36, T37, lssA_out_gg(T36, T37)) → lssA_out_gg(s(T36), s(T37))
U2_gga(T24, T25, lssA_out_gg(T24, T25)) → div_sB_out_gga(s(T24), T25, 0)
div_sB_in_gga(s(T49), T50, s(T52)) → U3_gga(T49, T50, T52, subC_in_gga(T49, T50, X49))
subC_in_gga(s(T66), s(T67), X77) → U6_gga(T66, T67, X77, subC_in_gga(T66, T67, X77))
subC_in_gga(T72, 0, T72) → subC_out_gga(T72, 0, T72)
U6_gga(T66, T67, X77, subC_out_gga(T66, T67, X77)) → subC_out_gga(s(T66), s(T67), X77)
U3_gga(T49, T50, T52, subC_out_gga(T49, T50, X49)) → div_sB_out_gga(s(T49), T50, s(T52))
div_sB_in_gga(s(T49), T50, s(T52)) → U4_gga(T49, T50, T52, subC_in_gga(T49, T50, T55))
U4_gga(T49, T50, T52, subC_out_gga(T49, T50, T55)) → U5_gga(T49, T50, T52, div_sB_in_gga(T55, T50, T52))
U5_gga(T49, T50, T52, div_sB_out_gga(T55, T50, T52)) → div_sB_out_gga(s(T49), T50, s(T52))
U7_gga(T7, T8, T10, div_sB_out_gga(T7, T8, T10)) → divD_out_gga(T7, s(T8), T10)

The argument filtering Pi contains the following mapping:
divD_in_gga(x1, x2, x3)  =  divD_in_gga(x1, x2)
s(x1)  =  s(x1)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)
div_sB_in_gga(x1, x2, x3)  =  div_sB_in_gga(x1, x2)
0  =  0
div_sB_out_gga(x1, x2, x3)  =  div_sB_out_gga
U2_gga(x1, x2, x3)  =  U2_gga(x3)
lssA_in_gg(x1, x2)  =  lssA_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
lssA_out_gg(x1, x2)  =  lssA_out_gg
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
subC_in_gga(x1, x2, x3)  =  subC_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
subC_out_gga(x1, x2, x3)  =  subC_out_gga(x3)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x2, x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
divD_out_gga(x1, x2, x3)  =  divD_out_gga
LSSA_IN_GG(x1, x2)  =  LSSA_IN_GG(x1, x2)

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:

LSSA_IN_GG(s(T36), s(T37)) → LSSA_IN_GG(T36, T37)

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

(19) PiDPToQDPProof (EQUIVALENT 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:

LSSA_IN_GG(s(T36), s(T37)) → LSSA_IN_GG(T36, T37)

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

(21) 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:

• LSSA_IN_GG(s(T36), s(T37)) → LSSA_IN_GG(T36, T37)
  The graph contains the following edges 1 > 1, 2 > 2

(22) YES

(23) Obligation:

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

DIV_SB_IN_GGA(s(T49), T50, s(T52)) → U4_GGA(T49, T50, T52, subC_in_gga(T49, T50, T55))
U4_GGA(T49, T50, T52, subC_out_gga(T49, T50, T55)) → DIV_SB_IN_GGA(T55, T50, T52)

The TRS R consists of the following rules:

divD_in_gga(T7, s(T8), T10) → U7_gga(T7, T8, T10, div_sB_in_gga(T7, T8, T10))
div_sB_in_gga(0, T15, 0) → div_sB_out_gga(0, T15, 0)
div_sB_in_gga(s(T24), T25, 0) → U2_gga(T24, T25, lssA_in_gg(T24, T25))
lssA_in_gg(s(T36), s(T37)) → U1_gg(T36, T37, lssA_in_gg(T36, T37))
lssA_in_gg(0, s(T42)) → lssA_out_gg(0, s(T42))
U1_gg(T36, T37, lssA_out_gg(T36, T37)) → lssA_out_gg(s(T36), s(T37))
U2_gga(T24, T25, lssA_out_gg(T24, T25)) → div_sB_out_gga(s(T24), T25, 0)
div_sB_in_gga(s(T49), T50, s(T52)) → U3_gga(T49, T50, T52, subC_in_gga(T49, T50, X49))
subC_in_gga(s(T66), s(T67), X77) → U6_gga(T66, T67, X77, subC_in_gga(T66, T67, X77))
subC_in_gga(T72, 0, T72) → subC_out_gga(T72, 0, T72)
U6_gga(T66, T67, X77, subC_out_gga(T66, T67, X77)) → subC_out_gga(s(T66), s(T67), X77)
U3_gga(T49, T50, T52, subC_out_gga(T49, T50, X49)) → div_sB_out_gga(s(T49), T50, s(T52))
div_sB_in_gga(s(T49), T50, s(T52)) → U4_gga(T49, T50, T52, subC_in_gga(T49, T50, T55))
U4_gga(T49, T50, T52, subC_out_gga(T49, T50, T55)) → U5_gga(T49, T50, T52, div_sB_in_gga(T55, T50, T52))
U5_gga(T49, T50, T52, div_sB_out_gga(T55, T50, T52)) → div_sB_out_gga(s(T49), T50, s(T52))
U7_gga(T7, T8, T10, div_sB_out_gga(T7, T8, T10)) → divD_out_gga(T7, s(T8), T10)

The argument filtering Pi contains the following mapping:
divD_in_gga(x1, x2, x3)  =  divD_in_gga(x1, x2)
s(x1)  =  s(x1)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)
div_sB_in_gga(x1, x2, x3)  =  div_sB_in_gga(x1, x2)
0  =  0
div_sB_out_gga(x1, x2, x3)  =  div_sB_out_gga
U2_gga(x1, x2, x3)  =  U2_gga(x3)
lssA_in_gg(x1, x2)  =  lssA_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
lssA_out_gg(x1, x2)  =  lssA_out_gg
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
subC_in_gga(x1, x2, x3)  =  subC_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
subC_out_gga(x1, x2, x3)  =  subC_out_gga(x3)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x2, x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
divD_out_gga(x1, x2, x3)  =  divD_out_gga
DIV_SB_IN_GGA(x1, x2, x3)  =  DIV_SB_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x2, x4)

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

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) Obligation:

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

DIV_SB_IN_GGA(s(T49), T50, s(T52)) → U4_GGA(T49, T50, T52, subC_in_gga(T49, T50, T55))
U4_GGA(T49, T50, T52, subC_out_gga(T49, T50, T55)) → DIV_SB_IN_GGA(T55, T50, T52)

The TRS R consists of the following rules:

subC_in_gga(s(T66), s(T67), X77) → U6_gga(T66, T67, X77, subC_in_gga(T66, T67, X77))
subC_in_gga(T72, 0, T72) → subC_out_gga(T72, 0, T72)
U6_gga(T66, T67, X77, subC_out_gga(T66, T67, X77)) → subC_out_gga(s(T66), s(T67), X77)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
0  =  0
subC_in_gga(x1, x2, x3)  =  subC_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
subC_out_gga(x1, x2, x3)  =  subC_out_gga(x3)
DIV_SB_IN_GGA(x1, x2, x3)  =  DIV_SB_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x2, x4)

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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) Obligation:

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

DIV_SB_IN_GGA(s(T49), T50) → U4_GGA(T50, subC_in_gga(T49, T50))
U4_GGA(T50, subC_out_gga(T55)) → DIV_SB_IN_GGA(T55, T50)

The TRS R consists of the following rules:

subC_in_gga(s(T66), s(T67)) → U6_gga(subC_in_gga(T66, T67))
subC_in_gga(T72, 0) → subC_out_gga(T72)
U6_gga(subC_out_gga(X77)) → subC_out_gga(X77)

The set Q consists of the following terms:

subC_in_gga(x0, x1)
U6_gga(x0)

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

(28) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

DIV_SB_IN_GGA(s(T49), T50) → U4_GGA(T50, subC_in_gga(T49, T50))

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

POL(0) = 0   
POL(DIV_SB_IN_GGA(x₁, x₂)) = x₁   
POL(U4_GGA(x₁, x₂)) = x₂   
POL(U6_gga(x₁)) = 1 + x₁   
POL(s(x₁)) = 1 + x₁   
POL(subC_in_gga(x₁, x₂)) = x₁   
POL(subC_out_gga(x₁)) = x₁   

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

subC_in_gga(s(T66), s(T67)) → U6_gga(subC_in_gga(T66, T67))
subC_in_gga(T72, 0) → subC_out_gga(T72)
U6_gga(subC_out_gga(X77)) → subC_out_gga(X77)

(29) Obligation:

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

U4_GGA(T50, subC_out_gga(T55)) → DIV_SB_IN_GGA(T55, T50)

The TRS R consists of the following rules:

subC_in_gga(s(T66), s(T67)) → U6_gga(subC_in_gga(T66, T67))
subC_in_gga(T72, 0) → subC_out_gga(T72)
U6_gga(subC_out_gga(X77)) → subC_out_gga(X77)

The set Q consists of the following terms:

subC_in_gga(x0, x1)
U6_gga(x0)

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

(30) DependencyGraphProof (EQUIVALENT transformation)

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

(31) TRUE