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

(0) Obligation:

Clauses:

rem(X, Y, R) :- ','(notZero(Y), ','(sub(X, Y, Z), rem(Z, Y, R))).
rem(X, Y, X) :- ','(notZero(Y), geq(X, Y)).
sub(s(X), s(Y), Z) :- sub(X, Y, Z).
sub(X, 0, X).
notZero(s(X)).
geq(s(X), s(Y)) :- geq(X, Y).
geq(X, 0).

Query: rem(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: rem(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: rem(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | rem(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(notZero(T8), ','(sub(T7, T8, X7), rem(X7, T8, T10))) | rem(T7, T8, T3), SCOPE: 1, CLAUSE: 1\n\nT7 is ground\nT8 is ground\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(notZero(T8), ','(sub(T7, T8, X7), rem(X7, T8, T10))), SCOPE: 2, CLAUSE: 4 | ?_2 | rem(T7, T8, T3), SCOPE: 1, CLAUSE: 1\n\nT7 is ground\nT8 is ground\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(notZero(T8), ','(sub(T7, T8, X7), rem(X7, T8, T10))), SCOPE: 2, CLAUSE: 4\n\nT7 is ground\nT8 is ground\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ?_2 | rem(T7, T8, T3), SCOPE: 1, CLAUSE: 1\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(sub(T7, s(T15), X7), rem(X7, s(T15), T10))\n\nT7 is ground\nT15 is ground\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: sub(T7, s(T15), X7)\n\nT7 is ground\nT15 is ground\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: rem(T18, s(T15), T10)\n\nT15 is ground\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: sub(T7, s(T15), X7), SCOPE: 3, CLAUSE: 2 | sub(T7, s(T15), X7), SCOPE: 3, CLAUSE: 3\n\nT7 is ground\nT15 is ground\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: sub(T7, s(T15), X7), SCOPE: 3, CLAUSE: 2\n\nT7 is ground\nT15 is ground\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: sub(T7, s(T15), X7), SCOPE: 3, CLAUSE: 3\n\nT7 is ground\nT15 is ground\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: sub(T29, T30, X40)\n\nT29 is ground\nT30 is ground\nX40 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: sub(T29, T30, X40), SCOPE: 4, CLAUSE: 2 | sub(T29, T30, X40), SCOPE: 4, CLAUSE: 3\n\nT29 is ground\nT30 is ground\nX40 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: sub(T29, T30, X40), SCOPE: 4, CLAUSE: 2\n\nT29 is ground\nT30 is ground\nX40 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: sub(T29, T30, X40), SCOPE: 4, CLAUSE: 3\n\nT29 is ground\nT30 is ground\nX40 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: sub(T41, T42, X64)\n\nT41 is ground\nT42 is ground\nX64 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: rem(T7, T8, T3), SCOPE: 1, CLAUSE: 1\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(notZero(T57), geq(T56, T57))\n\nT56 is ground\nT57 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(notZero(T57), geq(T56, T57)), SCOPE: 5, CLAUSE: 4\n\nT56 is ground\nT57 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: geq(T56, s(T62))\n\nT56 is ground\nT62 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: geq(T56, s(T62)), SCOPE: 6, CLAUSE: 5 | geq(T56, s(T62)), SCOPE: 6, CLAUSE: 6\n\nT56 is ground\nT62 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: geq(T56, s(T62)), SCOPE: 6, CLAUSE: 5\n\nT56 is ground\nT62 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: geq(T56, s(T62)), SCOPE: 6, CLAUSE: 6\n\nT56 is ground\nT62 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: geq(T73, T74)\n\nT73 is ground\nT74 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: geq(T73, T74), SCOPE: 7, CLAUSE: 5 | geq(T73, T74), SCOPE: 7, CLAUSE: 6\n\nT73 is ground\nT74 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: geq(T73, T74), SCOPE: 7, CLAUSE: 5\n\nT73 is ground\nT74 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: geq(T73, T74), SCOPE: 7, CLAUSE: 6\n\nT73 is ground\nT74 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: geq(T85, T86)\n\nT85 is ground\nT86 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 45 [arrowhead = none ];
45 -> 2;

46 [label="ONLY EVAL with clause\nrem(X4, X5, X6) :- ','(notZero(X5), ','(sub(X4, X5, X7), rem(X7, X5, X6))).\nand substitutionT1 -> T7,\nX4 -> T7,\nT2 -> T8,\nX5 -> T8,\nT3 -> T10,\nX6 -> T10,\nT9 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 46 [arrowhead = none ];
46 -> 3;

47 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 47 [arrowhead = none ];
47 -> 4;

48 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 48 [arrowhead = none ];
48 -> 5;

49 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 49 [arrowhead = none ];
49 -> 6;

50 [label="EVAL with clause\nnotZero(s(X12)).\nand substitutionX12 -> T15,\nT8 -> s(T15)", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 50 [arrowhead = none ];
50 -> 7;

51 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 51 [arrowhead = none ];
51 -> 8;

52 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 52 [arrowhead = none ];
52 -> 25;

53 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
7 -> 53 [arrowhead = none ];
53 -> 9;

54 [label="SPLIT 2\nnew knowledge:\nT7 is ground\nT15 is ground\nT18 is ground\nreplacements:X7 -> T18", fontsize=14, style = filled, fillcolor = violet];
7 -> 54 [arrowhead = none ];
54 -> 10;

55 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 55 [arrowhead = none ];
55 -> 11;

56 [label="INSTANCE with matching:\nT1 -> T18\nT2 -> s(T15)\nT3 -> T10", fontsize=14, style = filled, fillcolor = lightgrey];
10 -> 56 [arrowhead = none , style=dashed];
56 -> 1[style=dashed];

57 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 57 [arrowhead = none ];
57 -> 12;

58 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 58 [arrowhead = none ];
58 -> 13;

59 [label="EVAL with clause\nsub(s(X37), s(X38), X39) :- sub(X37, X38, X39).\nand substitutionX37 -> T29,\nT7 -> s(T29),\nT15 -> T30,\nX38 -> T30,\nX7 -> X40,\nX39 -> X40", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 59 [arrowhead = none ];
59 -> 14;

60 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 60 [arrowhead = none ];
60 -> 15;

61 [label="BACKTRACK\nfor clause: sub(X, 0, X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 61 [arrowhead = none ];
61 -> 24;

62 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 62 [arrowhead = none ];
62 -> 16;

63 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 63 [arrowhead = none ];
63 -> 17;

64 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 64 [arrowhead = none ];
64 -> 18;

65 [label="EVAL with clause\nsub(s(X61), s(X62), X63) :- sub(X61, X62, X63).\nand substitutionX61 -> T41,\nT29 -> s(T41),\nX62 -> T42,\nT30 -> s(T42),\nX40 -> X64,\nX63 -> X64", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 65 [arrowhead = none ];
65 -> 19;

66 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 66 [arrowhead = none ];
66 -> 20;

67 [label="EVAL with clause\nsub(X71, 0, X71).\nand substitutionT29 -> T47,\nX71 -> T47,\nT30 -> 0,\nX40 -> T47", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 67 [arrowhead = none ];
67 -> 21;

68 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 68 [arrowhead = none ];
68 -> 22;

69 [label="INSTANCE with matching:\nT29 -> T41\nT30 -> T42\nX40 -> X64", fontsize=14, style = filled, fillcolor = lightgrey];
19 -> 69 [arrowhead = none , style=dashed];
69 -> 14[style=dashed];

70 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 70 [arrowhead = none ];
70 -> 23;

71 [label="EVAL with clause\nrem(X81, X82, X81) :- ','(notZero(X82), geq(X81, X82)).\nand substitutionT7 -> T56,\nX81 -> T56,\nT8 -> T57,\nX82 -> T57,\nT3 -> T56", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 71 [arrowhead = none ];
71 -> 26;

72 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 72 [arrowhead = none ];
72 -> 27;

73 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 73 [arrowhead = none ];
73 -> 28;

74 [label="EVAL with clause\nnotZero(s(X87)).\nand substitutionX87 -> T62,\nT57 -> s(T62)", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 74 [arrowhead = none ];
74 -> 29;

75 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 75 [arrowhead = none ];
75 -> 30;

76 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 76 [arrowhead = none ];
76 -> 31;

77 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
31 -> 77 [arrowhead = none ];
77 -> 32;

78 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
31 -> 78 [arrowhead = none ];
78 -> 33;

79 [label="EVAL with clause\ngeq(s(X98), s(X99)) :- geq(X98, X99).\nand substitutionX98 -> T73,\nT56 -> s(T73),\nT62 -> T74,\nX99 -> T74", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 79 [arrowhead = none ];
79 -> 34;

80 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 80 [arrowhead = none ];
80 -> 35;

81 [label="BACKTRACK\nfor clause: geq(X, 0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 81 [arrowhead = none ];
81 -> 44;

82 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 82 [arrowhead = none ];
82 -> 36;

83 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 83 [arrowhead = none ];
83 -> 37;

84 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 84 [arrowhead = none ];
84 -> 38;

85 [label="EVAL with clause\ngeq(s(X110), s(X111)) :- geq(X110, X111).\nand substitutionX110 -> T85,\nT73 -> s(T85),\nX111 -> T86,\nT74 -> s(T86)", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 85 [arrowhead = none ];
85 -> 39;

86 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 86 [arrowhead = none ];
86 -> 40;

87 [label="EVAL with clause\ngeq(X116, 0).\nand substitutionT73 -> T91,\nX116 -> T91,\nT74 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 87 [arrowhead = none ];
87 -> 41;

88 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 88 [arrowhead = none ];
88 -> 42;

89 [label="INSTANCE with matching:\nT73 -> T85\nT74 -> T86", fontsize=14, style = filled, fillcolor = lightgrey];
39 -> 89 [arrowhead = none , style=dashed];
89 -> 34[style=dashed];

90 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 90 [arrowhead = none ];
90 -> 43;

}

(2) Obligation:

Clauses:

subA(s(T41), s(T42), X64) :- subA(T41, T42, X64).
subA(T47, 0, T47).
geqB(s(T85), s(T86)) :- geqB(T85, T86).
geqB(T91, 0).
subC(s(T29), T30, X40) :- subA(T29, T30, X40).
remD(T7, s(T15), T10) :- subC(T7, T15, X7).
remD(T7, s(T15), T10) :- ','(subC(T7, T15, T18), remD(T18, s(T15), T10)).
remD(s(T73), s(T74), s(T73)) :- geqB(T73, T74).

Query: remD(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:
remD_in: (b,b,f)
subC_in: (b,b,f)
subA_in: (b,b,f)
geqB_in: (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

remD_in_gga(T7, s(T15), T10) → U4_gga(T7, T15, T10, subC_in_gga(T7, T15, X7))
subC_in_gga(s(T29), T30, X40) → U3_gga(T29, T30, X40, subA_in_gga(T29, T30, X40))
subA_in_gga(s(T41), s(T42), X64) → U1_gga(T41, T42, X64, subA_in_gga(T41, T42, X64))
subA_in_gga(T47, 0, T47) → subA_out_gga(T47, 0, T47)
U1_gga(T41, T42, X64, subA_out_gga(T41, T42, X64)) → subA_out_gga(s(T41), s(T42), X64)
U3_gga(T29, T30, X40, subA_out_gga(T29, T30, X40)) → subC_out_gga(s(T29), T30, X40)
U4_gga(T7, T15, T10, subC_out_gga(T7, T15, X7)) → remD_out_gga(T7, s(T15), T10)
remD_in_gga(T7, s(T15), T10) → U5_gga(T7, T15, T10, subC_in_gga(T7, T15, T18))
U5_gga(T7, T15, T10, subC_out_gga(T7, T15, T18)) → U6_gga(T7, T15, T10, remD_in_gga(T18, s(T15), T10))
remD_in_gga(s(T73), s(T74), s(T73)) → U7_gga(T73, T74, geqB_in_gg(T73, T74))
geqB_in_gg(s(T85), s(T86)) → U2_gg(T85, T86, geqB_in_gg(T85, T86))
geqB_in_gg(T91, 0) → geqB_out_gg(T91, 0)
U2_gg(T85, T86, geqB_out_gg(T85, T86)) → geqB_out_gg(s(T85), s(T86))
U7_gga(T73, T74, geqB_out_gg(T73, T74)) → remD_out_gga(s(T73), s(T74), s(T73))
U6_gga(T7, T15, T10, remD_out_gga(T18, s(T15), T10)) → remD_out_gga(T7, s(T15), T10)

The argument filtering Pi contains the following mapping:
remD_in_gga(x1, x2, x3)  =  remD_in_gga(x1, x2)
s(x1)  =  s(x1)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
subC_in_gga(x1, x2, x3)  =  subC_in_gga(x1, x2)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
subA_in_gga(x1, x2, x3)  =  subA_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
0  =  0
subA_out_gga(x1, x2, x3)  =  subA_out_gga(x3)
subC_out_gga(x1, x2, x3)  =  subC_out_gga(x3)
remD_out_gga(x1, x2, x3)  =  remD_out_gga
U5_gga(x1, x2, x3, x4)  =  U5_gga(x2, x4)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
U7_gga(x1, x2, x3)  =  U7_gga(x3)
geqB_in_gg(x1, x2)  =  geqB_in_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
geqB_out_gg(x1, x2)  =  geqB_out_gg

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

remD_in_gga(T7, s(T15), T10) → U4_gga(T7, T15, T10, subC_in_gga(T7, T15, X7))
subC_in_gga(s(T29), T30, X40) → U3_gga(T29, T30, X40, subA_in_gga(T29, T30, X40))
subA_in_gga(s(T41), s(T42), X64) → U1_gga(T41, T42, X64, subA_in_gga(T41, T42, X64))
subA_in_gga(T47, 0, T47) → subA_out_gga(T47, 0, T47)
U1_gga(T41, T42, X64, subA_out_gga(T41, T42, X64)) → subA_out_gga(s(T41), s(T42), X64)
U3_gga(T29, T30, X40, subA_out_gga(T29, T30, X40)) → subC_out_gga(s(T29), T30, X40)
U4_gga(T7, T15, T10, subC_out_gga(T7, T15, X7)) → remD_out_gga(T7, s(T15), T10)
remD_in_gga(T7, s(T15), T10) → U5_gga(T7, T15, T10, subC_in_gga(T7, T15, T18))
U5_gga(T7, T15, T10, subC_out_gga(T7, T15, T18)) → U6_gga(T7, T15, T10, remD_in_gga(T18, s(T15), T10))
remD_in_gga(s(T73), s(T74), s(T73)) → U7_gga(T73, T74, geqB_in_gg(T73, T74))
geqB_in_gg(s(T85), s(T86)) → U2_gg(T85, T86, geqB_in_gg(T85, T86))
geqB_in_gg(T91, 0) → geqB_out_gg(T91, 0)
U2_gg(T85, T86, geqB_out_gg(T85, T86)) → geqB_out_gg(s(T85), s(T86))
U7_gga(T73, T74, geqB_out_gg(T73, T74)) → remD_out_gga(s(T73), s(T74), s(T73))
U6_gga(T7, T15, T10, remD_out_gga(T18, s(T15), T10)) → remD_out_gga(T7, s(T15), T10)

The argument filtering Pi contains the following mapping:
remD_in_gga(x1, x2, x3)  =  remD_in_gga(x1, x2)
s(x1)  =  s(x1)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
subC_in_gga(x1, x2, x3)  =  subC_in_gga(x1, x2)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
subA_in_gga(x1, x2, x3)  =  subA_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
0  =  0
subA_out_gga(x1, x2, x3)  =  subA_out_gga(x3)
subC_out_gga(x1, x2, x3)  =  subC_out_gga(x3)
remD_out_gga(x1, x2, x3)  =  remD_out_gga
U5_gga(x1, x2, x3, x4)  =  U5_gga(x2, x4)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
U7_gga(x1, x2, x3)  =  U7_gga(x3)
geqB_in_gg(x1, x2)  =  geqB_in_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
geqB_out_gg(x1, x2)  =  geqB_out_gg

(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:

REMD_IN_GGA(T7, s(T15), T10) → U4_GGA(T7, T15, T10, subC_in_gga(T7, T15, X7))
REMD_IN_GGA(T7, s(T15), T10) → SUBC_IN_GGA(T7, T15, X7)
SUBC_IN_GGA(s(T29), T30, X40) → U3_GGA(T29, T30, X40, subA_in_gga(T29, T30, X40))
SUBC_IN_GGA(s(T29), T30, X40) → SUBA_IN_GGA(T29, T30, X40)
SUBA_IN_GGA(s(T41), s(T42), X64) → U1_GGA(T41, T42, X64, subA_in_gga(T41, T42, X64))
SUBA_IN_GGA(s(T41), s(T42), X64) → SUBA_IN_GGA(T41, T42, X64)
REMD_IN_GGA(T7, s(T15), T10) → U5_GGA(T7, T15, T10, subC_in_gga(T7, T15, T18))
U5_GGA(T7, T15, T10, subC_out_gga(T7, T15, T18)) → U6_GGA(T7, T15, T10, remD_in_gga(T18, s(T15), T10))
U5_GGA(T7, T15, T10, subC_out_gga(T7, T15, T18)) → REMD_IN_GGA(T18, s(T15), T10)
REMD_IN_GGA(s(T73), s(T74), s(T73)) → U7_GGA(T73, T74, geqB_in_gg(T73, T74))
REMD_IN_GGA(s(T73), s(T74), s(T73)) → GEQB_IN_GG(T73, T74)
GEQB_IN_GG(s(T85), s(T86)) → U2_GG(T85, T86, geqB_in_gg(T85, T86))
GEQB_IN_GG(s(T85), s(T86)) → GEQB_IN_GG(T85, T86)

The TRS R consists of the following rules:

remD_in_gga(T7, s(T15), T10) → U4_gga(T7, T15, T10, subC_in_gga(T7, T15, X7))
subC_in_gga(s(T29), T30, X40) → U3_gga(T29, T30, X40, subA_in_gga(T29, T30, X40))
subA_in_gga(s(T41), s(T42), X64) → U1_gga(T41, T42, X64, subA_in_gga(T41, T42, X64))
subA_in_gga(T47, 0, T47) → subA_out_gga(T47, 0, T47)
U1_gga(T41, T42, X64, subA_out_gga(T41, T42, X64)) → subA_out_gga(s(T41), s(T42), X64)
U3_gga(T29, T30, X40, subA_out_gga(T29, T30, X40)) → subC_out_gga(s(T29), T30, X40)
U4_gga(T7, T15, T10, subC_out_gga(T7, T15, X7)) → remD_out_gga(T7, s(T15), T10)
remD_in_gga(T7, s(T15), T10) → U5_gga(T7, T15, T10, subC_in_gga(T7, T15, T18))
U5_gga(T7, T15, T10, subC_out_gga(T7, T15, T18)) → U6_gga(T7, T15, T10, remD_in_gga(T18, s(T15), T10))
remD_in_gga(s(T73), s(T74), s(T73)) → U7_gga(T73, T74, geqB_in_gg(T73, T74))
geqB_in_gg(s(T85), s(T86)) → U2_gg(T85, T86, geqB_in_gg(T85, T86))
geqB_in_gg(T91, 0) → geqB_out_gg(T91, 0)
U2_gg(T85, T86, geqB_out_gg(T85, T86)) → geqB_out_gg(s(T85), s(T86))
U7_gga(T73, T74, geqB_out_gg(T73, T74)) → remD_out_gga(s(T73), s(T74), s(T73))
U6_gga(T7, T15, T10, remD_out_gga(T18, s(T15), T10)) → remD_out_gga(T7, s(T15), T10)

The argument filtering Pi contains the following mapping:
remD_in_gga(x1, x2, x3)  =  remD_in_gga(x1, x2)
s(x1)  =  s(x1)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
subC_in_gga(x1, x2, x3)  =  subC_in_gga(x1, x2)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
subA_in_gga(x1, x2, x3)  =  subA_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
0  =  0
subA_out_gga(x1, x2, x3)  =  subA_out_gga(x3)
subC_out_gga(x1, x2, x3)  =  subC_out_gga(x3)
remD_out_gga(x1, x2, x3)  =  remD_out_gga
U5_gga(x1, x2, x3, x4)  =  U5_gga(x2, x4)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
U7_gga(x1, x2, x3)  =  U7_gga(x3)
geqB_in_gg(x1, x2)  =  geqB_in_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
geqB_out_gg(x1, x2)  =  geqB_out_gg
REMD_IN_GGA(x1, x2, x3)  =  REMD_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x4)
SUBC_IN_GGA(x1, x2, x3)  =  SUBC_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x4)
SUBA_IN_GGA(x1, x2, x3)  =  SUBA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x4)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x2, x4)
U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x4)
U7_GGA(x1, x2, x3)  =  U7_GGA(x3)
GEQB_IN_GG(x1, x2)  =  GEQB_IN_GG(x1, x2)
U2_GG(x1, x2, x3)  =  U2_GG(x3)

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

(6) Obligation:

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

REMD_IN_GGA(T7, s(T15), T10) → U4_GGA(T7, T15, T10, subC_in_gga(T7, T15, X7))
REMD_IN_GGA(T7, s(T15), T10) → SUBC_IN_GGA(T7, T15, X7)
SUBC_IN_GGA(s(T29), T30, X40) → U3_GGA(T29, T30, X40, subA_in_gga(T29, T30, X40))
SUBC_IN_GGA(s(T29), T30, X40) → SUBA_IN_GGA(T29, T30, X40)
SUBA_IN_GGA(s(T41), s(T42), X64) → U1_GGA(T41, T42, X64, subA_in_gga(T41, T42, X64))
SUBA_IN_GGA(s(T41), s(T42), X64) → SUBA_IN_GGA(T41, T42, X64)
REMD_IN_GGA(T7, s(T15), T10) → U5_GGA(T7, T15, T10, subC_in_gga(T7, T15, T18))
U5_GGA(T7, T15, T10, subC_out_gga(T7, T15, T18)) → U6_GGA(T7, T15, T10, remD_in_gga(T18, s(T15), T10))
U5_GGA(T7, T15, T10, subC_out_gga(T7, T15, T18)) → REMD_IN_GGA(T18, s(T15), T10)
REMD_IN_GGA(s(T73), s(T74), s(T73)) → U7_GGA(T73, T74, geqB_in_gg(T73, T74))
REMD_IN_GGA(s(T73), s(T74), s(T73)) → GEQB_IN_GG(T73, T74)
GEQB_IN_GG(s(T85), s(T86)) → U2_GG(T85, T86, geqB_in_gg(T85, T86))
GEQB_IN_GG(s(T85), s(T86)) → GEQB_IN_GG(T85, T86)

The TRS R consists of the following rules:

remD_in_gga(T7, s(T15), T10) → U4_gga(T7, T15, T10, subC_in_gga(T7, T15, X7))
subC_in_gga(s(T29), T30, X40) → U3_gga(T29, T30, X40, subA_in_gga(T29, T30, X40))
subA_in_gga(s(T41), s(T42), X64) → U1_gga(T41, T42, X64, subA_in_gga(T41, T42, X64))
subA_in_gga(T47, 0, T47) → subA_out_gga(T47, 0, T47)
U1_gga(T41, T42, X64, subA_out_gga(T41, T42, X64)) → subA_out_gga(s(T41), s(T42), X64)
U3_gga(T29, T30, X40, subA_out_gga(T29, T30, X40)) → subC_out_gga(s(T29), T30, X40)
U4_gga(T7, T15, T10, subC_out_gga(T7, T15, X7)) → remD_out_gga(T7, s(T15), T10)
remD_in_gga(T7, s(T15), T10) → U5_gga(T7, T15, T10, subC_in_gga(T7, T15, T18))
U5_gga(T7, T15, T10, subC_out_gga(T7, T15, T18)) → U6_gga(T7, T15, T10, remD_in_gga(T18, s(T15), T10))
remD_in_gga(s(T73), s(T74), s(T73)) → U7_gga(T73, T74, geqB_in_gg(T73, T74))
geqB_in_gg(s(T85), s(T86)) → U2_gg(T85, T86, geqB_in_gg(T85, T86))
geqB_in_gg(T91, 0) → geqB_out_gg(T91, 0)
U2_gg(T85, T86, geqB_out_gg(T85, T86)) → geqB_out_gg(s(T85), s(T86))
U7_gga(T73, T74, geqB_out_gg(T73, T74)) → remD_out_gga(s(T73), s(T74), s(T73))
U6_gga(T7, T15, T10, remD_out_gga(T18, s(T15), T10)) → remD_out_gga(T7, s(T15), T10)

The argument filtering Pi contains the following mapping:
remD_in_gga(x1, x2, x3)  =  remD_in_gga(x1, x2)
s(x1)  =  s(x1)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
subC_in_gga(x1, x2, x3)  =  subC_in_gga(x1, x2)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
subA_in_gga(x1, x2, x3)  =  subA_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
0  =  0
subA_out_gga(x1, x2, x3)  =  subA_out_gga(x3)
subC_out_gga(x1, x2, x3)  =  subC_out_gga(x3)
remD_out_gga(x1, x2, x3)  =  remD_out_gga
U5_gga(x1, x2, x3, x4)  =  U5_gga(x2, x4)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
U7_gga(x1, x2, x3)  =  U7_gga(x3)
geqB_in_gg(x1, x2)  =  geqB_in_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
geqB_out_gg(x1, x2)  =  geqB_out_gg
REMD_IN_GGA(x1, x2, x3)  =  REMD_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x4)
SUBC_IN_GGA(x1, x2, x3)  =  SUBC_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x4)
SUBA_IN_GGA(x1, x2, x3)  =  SUBA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x4)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x2, x4)
U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x4)
U7_GGA(x1, x2, x3)  =  U7_GGA(x3)
GEQB_IN_GG(x1, x2)  =  GEQB_IN_GG(x1, x2)
U2_GG(x1, x2, x3)  =  U2_GG(x3)

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:

GEQB_IN_GG(s(T85), s(T86)) → GEQB_IN_GG(T85, T86)

The TRS R consists of the following rules:

remD_in_gga(T7, s(T15), T10) → U4_gga(T7, T15, T10, subC_in_gga(T7, T15, X7))
subC_in_gga(s(T29), T30, X40) → U3_gga(T29, T30, X40, subA_in_gga(T29, T30, X40))
subA_in_gga(s(T41), s(T42), X64) → U1_gga(T41, T42, X64, subA_in_gga(T41, T42, X64))
subA_in_gga(T47, 0, T47) → subA_out_gga(T47, 0, T47)
U1_gga(T41, T42, X64, subA_out_gga(T41, T42, X64)) → subA_out_gga(s(T41), s(T42), X64)
U3_gga(T29, T30, X40, subA_out_gga(T29, T30, X40)) → subC_out_gga(s(T29), T30, X40)
U4_gga(T7, T15, T10, subC_out_gga(T7, T15, X7)) → remD_out_gga(T7, s(T15), T10)
remD_in_gga(T7, s(T15), T10) → U5_gga(T7, T15, T10, subC_in_gga(T7, T15, T18))
U5_gga(T7, T15, T10, subC_out_gga(T7, T15, T18)) → U6_gga(T7, T15, T10, remD_in_gga(T18, s(T15), T10))
remD_in_gga(s(T73), s(T74), s(T73)) → U7_gga(T73, T74, geqB_in_gg(T73, T74))
geqB_in_gg(s(T85), s(T86)) → U2_gg(T85, T86, geqB_in_gg(T85, T86))
geqB_in_gg(T91, 0) → geqB_out_gg(T91, 0)
U2_gg(T85, T86, geqB_out_gg(T85, T86)) → geqB_out_gg(s(T85), s(T86))
U7_gga(T73, T74, geqB_out_gg(T73, T74)) → remD_out_gga(s(T73), s(T74), s(T73))
U6_gga(T7, T15, T10, remD_out_gga(T18, s(T15), T10)) → remD_out_gga(T7, s(T15), T10)

The argument filtering Pi contains the following mapping:
remD_in_gga(x1, x2, x3)  =  remD_in_gga(x1, x2)
s(x1)  =  s(x1)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
subC_in_gga(x1, x2, x3)  =  subC_in_gga(x1, x2)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
subA_in_gga(x1, x2, x3)  =  subA_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
0  =  0
subA_out_gga(x1, x2, x3)  =  subA_out_gga(x3)
subC_out_gga(x1, x2, x3)  =  subC_out_gga(x3)
remD_out_gga(x1, x2, x3)  =  remD_out_gga
U5_gga(x1, x2, x3, x4)  =  U5_gga(x2, x4)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
U7_gga(x1, x2, x3)  =  U7_gga(x3)
geqB_in_gg(x1, x2)  =  geqB_in_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
geqB_out_gg(x1, x2)  =  geqB_out_gg
GEQB_IN_GG(x1, x2)  =  GEQB_IN_GG(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:

GEQB_IN_GG(s(T85), s(T86)) → GEQB_IN_GG(T85, T86)

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

(12) PiDPToQDPProof (EQUIVALENT 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:

GEQB_IN_GG(s(T85), s(T86)) → GEQB_IN_GG(T85, T86)

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

(14) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

• GEQB_IN_GG(s(T85), s(T86)) → GEQB_IN_GG(T85, T86)
  The graph contains the following edges 1 > 1, 2 > 2

(15) YES

(16) Obligation:

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

SUBA_IN_GGA(s(T41), s(T42), X64) → SUBA_IN_GGA(T41, T42, X64)

The TRS R consists of the following rules:

remD_in_gga(T7, s(T15), T10) → U4_gga(T7, T15, T10, subC_in_gga(T7, T15, X7))
subC_in_gga(s(T29), T30, X40) → U3_gga(T29, T30, X40, subA_in_gga(T29, T30, X40))
subA_in_gga(s(T41), s(T42), X64) → U1_gga(T41, T42, X64, subA_in_gga(T41, T42, X64))
subA_in_gga(T47, 0, T47) → subA_out_gga(T47, 0, T47)
U1_gga(T41, T42, X64, subA_out_gga(T41, T42, X64)) → subA_out_gga(s(T41), s(T42), X64)
U3_gga(T29, T30, X40, subA_out_gga(T29, T30, X40)) → subC_out_gga(s(T29), T30, X40)
U4_gga(T7, T15, T10, subC_out_gga(T7, T15, X7)) → remD_out_gga(T7, s(T15), T10)
remD_in_gga(T7, s(T15), T10) → U5_gga(T7, T15, T10, subC_in_gga(T7, T15, T18))
U5_gga(T7, T15, T10, subC_out_gga(T7, T15, T18)) → U6_gga(T7, T15, T10, remD_in_gga(T18, s(T15), T10))
remD_in_gga(s(T73), s(T74), s(T73)) → U7_gga(T73, T74, geqB_in_gg(T73, T74))
geqB_in_gg(s(T85), s(T86)) → U2_gg(T85, T86, geqB_in_gg(T85, T86))
geqB_in_gg(T91, 0) → geqB_out_gg(T91, 0)
U2_gg(T85, T86, geqB_out_gg(T85, T86)) → geqB_out_gg(s(T85), s(T86))
U7_gga(T73, T74, geqB_out_gg(T73, T74)) → remD_out_gga(s(T73), s(T74), s(T73))
U6_gga(T7, T15, T10, remD_out_gga(T18, s(T15), T10)) → remD_out_gga(T7, s(T15), T10)

The argument filtering Pi contains the following mapping:
remD_in_gga(x1, x2, x3)  =  remD_in_gga(x1, x2)
s(x1)  =  s(x1)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
subC_in_gga(x1, x2, x3)  =  subC_in_gga(x1, x2)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
subA_in_gga(x1, x2, x3)  =  subA_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
0  =  0
subA_out_gga(x1, x2, x3)  =  subA_out_gga(x3)
subC_out_gga(x1, x2, x3)  =  subC_out_gga(x3)
remD_out_gga(x1, x2, x3)  =  remD_out_gga
U5_gga(x1, x2, x3, x4)  =  U5_gga(x2, x4)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
U7_gga(x1, x2, x3)  =  U7_gga(x3)
geqB_in_gg(x1, x2)  =  geqB_in_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
geqB_out_gg(x1, x2)  =  geqB_out_gg
SUBA_IN_GGA(x1, x2, x3)  =  SUBA_IN_GGA(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:

SUBA_IN_GGA(s(T41), s(T42), X64) → SUBA_IN_GGA(T41, T42, X64)

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

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

SUBA_IN_GGA(s(T41), s(T42)) → SUBA_IN_GGA(T41, T42)

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:

• SUBA_IN_GGA(s(T41), s(T42)) → SUBA_IN_GGA(T41, T42)
  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:

REMD_IN_GGA(T7, s(T15), T10) → U5_GGA(T7, T15, T10, subC_in_gga(T7, T15, T18))
U5_GGA(T7, T15, T10, subC_out_gga(T7, T15, T18)) → REMD_IN_GGA(T18, s(T15), T10)

The TRS R consists of the following rules:

remD_in_gga(T7, s(T15), T10) → U4_gga(T7, T15, T10, subC_in_gga(T7, T15, X7))
subC_in_gga(s(T29), T30, X40) → U3_gga(T29, T30, X40, subA_in_gga(T29, T30, X40))
subA_in_gga(s(T41), s(T42), X64) → U1_gga(T41, T42, X64, subA_in_gga(T41, T42, X64))
subA_in_gga(T47, 0, T47) → subA_out_gga(T47, 0, T47)
U1_gga(T41, T42, X64, subA_out_gga(T41, T42, X64)) → subA_out_gga(s(T41), s(T42), X64)
U3_gga(T29, T30, X40, subA_out_gga(T29, T30, X40)) → subC_out_gga(s(T29), T30, X40)
U4_gga(T7, T15, T10, subC_out_gga(T7, T15, X7)) → remD_out_gga(T7, s(T15), T10)
remD_in_gga(T7, s(T15), T10) → U5_gga(T7, T15, T10, subC_in_gga(T7, T15, T18))
U5_gga(T7, T15, T10, subC_out_gga(T7, T15, T18)) → U6_gga(T7, T15, T10, remD_in_gga(T18, s(T15), T10))
remD_in_gga(s(T73), s(T74), s(T73)) → U7_gga(T73, T74, geqB_in_gg(T73, T74))
geqB_in_gg(s(T85), s(T86)) → U2_gg(T85, T86, geqB_in_gg(T85, T86))
geqB_in_gg(T91, 0) → geqB_out_gg(T91, 0)
U2_gg(T85, T86, geqB_out_gg(T85, T86)) → geqB_out_gg(s(T85), s(T86))
U7_gga(T73, T74, geqB_out_gg(T73, T74)) → remD_out_gga(s(T73), s(T74), s(T73))
U6_gga(T7, T15, T10, remD_out_gga(T18, s(T15), T10)) → remD_out_gga(T7, s(T15), T10)

The argument filtering Pi contains the following mapping:
remD_in_gga(x1, x2, x3)  =  remD_in_gga(x1, x2)
s(x1)  =  s(x1)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
subC_in_gga(x1, x2, x3)  =  subC_in_gga(x1, x2)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
subA_in_gga(x1, x2, x3)  =  subA_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
0  =  0
subA_out_gga(x1, x2, x3)  =  subA_out_gga(x3)
subC_out_gga(x1, x2, x3)  =  subC_out_gga(x3)
remD_out_gga(x1, x2, x3)  =  remD_out_gga
U5_gga(x1, x2, x3, x4)  =  U5_gga(x2, x4)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
U7_gga(x1, x2, x3)  =  U7_gga(x3)
geqB_in_gg(x1, x2)  =  geqB_in_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
geqB_out_gg(x1, x2)  =  geqB_out_gg
REMD_IN_GGA(x1, x2, x3)  =  REMD_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4)  =  U5_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:

REMD_IN_GGA(T7, s(T15), T10) → U5_GGA(T7, T15, T10, subC_in_gga(T7, T15, T18))
U5_GGA(T7, T15, T10, subC_out_gga(T7, T15, T18)) → REMD_IN_GGA(T18, s(T15), T10)

The TRS R consists of the following rules:

subC_in_gga(s(T29), T30, X40) → U3_gga(T29, T30, X40, subA_in_gga(T29, T30, X40))
U3_gga(T29, T30, X40, subA_out_gga(T29, T30, X40)) → subC_out_gga(s(T29), T30, X40)
subA_in_gga(s(T41), s(T42), X64) → U1_gga(T41, T42, X64, subA_in_gga(T41, T42, X64))
subA_in_gga(T47, 0, T47) → subA_out_gga(T47, 0, T47)
U1_gga(T41, T42, X64, subA_out_gga(T41, T42, X64)) → subA_out_gga(s(T41), s(T42), X64)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
subC_in_gga(x1, x2, x3)  =  subC_in_gga(x1, x2)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
subA_in_gga(x1, x2, x3)  =  subA_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
0  =  0
subA_out_gga(x1, x2, x3)  =  subA_out_gga(x3)
subC_out_gga(x1, x2, x3)  =  subC_out_gga(x3)
REMD_IN_GGA(x1, x2, x3)  =  REMD_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4)  =  U5_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:

REMD_IN_GGA(T7, s(T15)) → U5_GGA(T15, subC_in_gga(T7, T15))
U5_GGA(T15, subC_out_gga(T18)) → REMD_IN_GGA(T18, s(T15))

The TRS R consists of the following rules:

subC_in_gga(s(T29), T30) → U3_gga(subA_in_gga(T29, T30))
U3_gga(subA_out_gga(X40)) → subC_out_gga(X40)
subA_in_gga(s(T41), s(T42)) → U1_gga(subA_in_gga(T41, T42))
subA_in_gga(T47, 0) → subA_out_gga(T47)
U1_gga(subA_out_gga(X64)) → subA_out_gga(X64)

The set Q consists of the following terms:

subC_in_gga(x0, x1)
U3_gga(x0)
subA_in_gga(x0, x1)
U1_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.

U5_GGA(T15, subC_out_gga(T18)) → REMD_IN_GGA(T18, s(T15))

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

POL(0) = 0   
POL(REMD_IN_GGA(x₁, x₂)) = 1 + x₁   
POL(U1_gga(x₁)) = x₁   
POL(U3_gga(x₁)) = x₁   
POL(U5_GGA(x₁, x₂)) = 1 + x₂   
POL(s(x₁)) = 1 + x₁   
POL(subA_in_gga(x₁, x₂)) = 1 + x₁   
POL(subA_out_gga(x₁)) = 1 + x₁   
POL(subC_in_gga(x₁, x₂)) = x₁   
POL(subC_out_gga(x₁)) = 1 + x₁   

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

subC_in_gga(s(T29), T30) → U3_gga(subA_in_gga(T29, T30))
subA_in_gga(s(T41), s(T42)) → U1_gga(subA_in_gga(T41, T42))
subA_in_gga(T47, 0) → subA_out_gga(T47)
U3_gga(subA_out_gga(X40)) → subC_out_gga(X40)
U1_gga(subA_out_gga(X64)) → subA_out_gga(X64)

(29) Obligation:

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

REMD_IN_GGA(T7, s(T15)) → U5_GGA(T15, subC_in_gga(T7, T15))

The TRS R consists of the following rules:

subC_in_gga(s(T29), T30) → U3_gga(subA_in_gga(T29, T30))
U3_gga(subA_out_gga(X40)) → subC_out_gga(X40)
subA_in_gga(s(T41), s(T42)) → U1_gga(subA_in_gga(T41, T42))
subA_in_gga(T47, 0) → subA_out_gga(T47)
U1_gga(subA_out_gga(X64)) → subA_out_gga(X64)

The set Q consists of the following terms:

subC_in_gga(x0, x1)
U3_gga(x0)
subA_in_gga(x0, x1)
U1_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