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

(0) Obligation:

Clauses:

append([], Ys, Ys).
append(.(X, Xs), Ys, .(X, Zs)) :- append(Xs, Ys, Zs).
sublist(X, Y) :- ','(append(P, X1, Y), append(X2, X, P)).

Query: sublist(a,g)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: sublist(T1, T2)\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: sublist(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(append(X7, X8, T6), append(X9, T7, X7))\n\nT6 is ground\nX7 is free\nX8 is free\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(append(X7, X8, T6), append(X9, T7, X7)), SCOPE: 2, CLAUSE: 0 | ','(append(X7, X8, T6), append(X9, T7, X7)), SCOPE: 2, CLAUSE: 1\n\nT6 is ground\nX7 is free\nX8 is free\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(append(X7, X8, T6), append(X9, T7, X7)), SCOPE: 2, CLAUSE: 0\n\nT6 is ground\nX7 is free\nX8 is free\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(append(X7, X8, T6), append(X9, T7, X7)), SCOPE: 2, CLAUSE: 1\n\nT6 is ground\nX7 is free\nX8 is free\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: append(X9, T7, [])\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: append(X9, T7, []), SCOPE: 3, CLAUSE: 0 | append(X9, T7, []), SCOPE: 3, CLAUSE: 1\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: append(X9, T7, []), SCOPE: 3, CLAUSE: 0\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: append(X9, T7, []), SCOPE: 3, CLAUSE: 1\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(append(X56, X57, T30), append(X9, T7, .(T29, X56)))\n\nT29 is ground\nT30 is ground\nX9 is free\nX56 is free\nX57 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: append(X56, X57, T30)\n\nT30 is ground\nX56 is free\nX57 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: append(X9, T7, .(T29, T38))\n\nT29 is ground\nT38 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: append(X56, X57, T30), SCOPE: 4, CLAUSE: 0 | append(X56, X57, T30), SCOPE: 4, CLAUSE: 1\n\nT30 is ground\nX56 is free\nX57 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: append(X56, X57, T30), SCOPE: 4, CLAUSE: 0\n\nT30 is ground\nX56 is free\nX57 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: append(X56, X57, T30), SCOPE: 4, CLAUSE: 1\n\nT30 is ground\nX56 is free\nX57 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: append(X93, X94, T51)\n\nT51 is ground\nX93 is free\nX94 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: append(X9, T7, .(T29, T38)), SCOPE: 5, CLAUSE: 0 | append(X9, T7, .(T29, T38)), SCOPE: 5, CLAUSE: 1\n\nT29 is ground\nT38 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: append(X9, T7, .(T29, T38)), SCOPE: 5, CLAUSE: 0\n\nT29 is ground\nT38 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: append(X9, T7, .(T29, T38)), SCOPE: 5, CLAUSE: 1\n\nT29 is ground\nT38 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: append(X123, T86, T85)\n\nT85 is ground\nX123 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: append(X123, T86, T85), SCOPE: 6, CLAUSE: 0 | append(X123, T86, T85), SCOPE: 6, CLAUSE: 1\n\nT85 is ground\nX123 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: append(X123, T86, T85), SCOPE: 6, CLAUSE: 0\n\nT85 is ground\nX123 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: append(X123, T86, T85), SCOPE: 6, CLAUSE: 1\n\nT85 is ground\nX123 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: append(X148, T103, T102)\n\nT102 is ground\nX148 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 41 [arrowhead = none ];
41 -> 2;

42 [label="ONLY EVAL with clause\nsublist(X5, X6) :- ','(append(X7, X8, X6), append(X9, X5, X7)).\nand substitutionT1 -> T7,\nX5 -> T7,\nT2 -> T6,\nX6 -> T6,\nT5 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 42 [arrowhead = none ];
42 -> 3;

43 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 43 [arrowhead = none ];
43 -> 4;

44 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 44 [arrowhead = none ];
44 -> 5;

45 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 45 [arrowhead = none ];
45 -> 6;

46 [label="ONLY EVAL with clause\nappend([], X22, X22).\nand substitutionX7 -> [],\nX8 -> T16,\nX22 -> T16,\nT6 -> T16,\nX23 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 46 [arrowhead = none ];
46 -> 7;

47 [label="EVAL with clause\nappend(.(X51, X52), X53, .(X51, X54)) :- append(X52, X53, X54).\nand substitutionX51 -> T29,\nX52 -> X56,\nX7 -> .(T29, X56),\nX8 -> X57,\nX53 -> X57,\nX55 -> T29,\nX54 -> T30,\nT6 -> .(T29, T30)", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 47 [arrowhead = none ];
47 -> 15;

48 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 48 [arrowhead = none ];
48 -> 16;

49 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 49 [arrowhead = none ];
49 -> 8;

50 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 50 [arrowhead = none ];
50 -> 9;

51 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 51 [arrowhead = none ];
51 -> 10;

52 [label="EVAL with clause\nappend([], X30, X30).\nand substitutionX9 -> [],\nT7 -> [],\nX30 -> [],\nT23 -> []", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 52 [arrowhead = none ];
52 -> 11;

53 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 53 [arrowhead = none ];
53 -> 12;

54 [label="BACKTRACK\nfor clause: append(.(X, Xs), Ys, .(X, Zs)) :- append(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 54 [arrowhead = none ];
54 -> 14;

55 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 55 [arrowhead = none ];
55 -> 13;

56 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
15 -> 56 [arrowhead = none ];
56 -> 17;

57 [label="SPLIT 2\nnew knowledge:\nT38 is ground\nT39 is ground\nT30 is ground\nreplacements:X56 -> T38,\nX57 -> T39", fontsize=14, style = filled, fillcolor = violet];
15 -> 57 [arrowhead = none ];
57 -> 18;

58 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 58 [arrowhead = none ];
58 -> 19;

59 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 59 [arrowhead = none ];
59 -> 26;

60 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
19 -> 60 [arrowhead = none ];
60 -> 20;

61 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
19 -> 61 [arrowhead = none ];
61 -> 21;

62 [label="ONLY EVAL with clause\nappend([], X72, X72).\nand substitutionX56 -> [],\nX57 -> T45,\nX72 -> T45,\nT30 -> T45,\nX73 -> T45", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 62 [arrowhead = none ];
62 -> 22;

63 [label="EVAL with clause\nappend(.(X88, X89), X90, .(X88, X91)) :- append(X89, X90, X91).\nand substitutionX88 -> T50,\nX89 -> X93,\nX56 -> .(T50, X93),\nX57 -> X94,\nX90 -> X94,\nX92 -> T50,\nX91 -> T51,\nT30 -> .(T50, T51)", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 63 [arrowhead = none ];
63 -> 24;

64 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 64 [arrowhead = none ];
64 -> 25;

65 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 65 [arrowhead = none ];
65 -> 23;

66 [label="INSTANCE with matching:\nX56 -> X93\nX57 -> X94\nT30 -> T51", fontsize=14, style = filled, fillcolor = lightgrey];
24 -> 66 [arrowhead = none , style=dashed];
66 -> 17[style=dashed];

67 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 67 [arrowhead = none ];
67 -> 27;

68 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 68 [arrowhead = none ];
68 -> 28;

69 [label="EVAL with clause\nappend([], X103, X103).\nand substitutionX9 -> [],\nT7 -> .(T72, T73),\nX103 -> .(T72, T73),\nT29 -> T72,\nT38 -> T73,\nT71 -> .(T72, T73)", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 69 [arrowhead = none ];
69 -> 29;

70 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 70 [arrowhead = none ];
70 -> 30;

71 [label="ONLY EVAL with clause\nappend(.(X118, X119), X120, .(X118, X121)) :- append(X119, X120, X121).\nand substitutionX118 -> T84,\nX119 -> X123,\nX9 -> .(T84, X123),\nT7 -> T86,\nX120 -> T86,\nT29 -> T84,\nX122 -> T84,\nT38 -> T85,\nX121 -> T85,\nT83 -> T86", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 71 [arrowhead = none ];
71 -> 32;

72 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 72 [arrowhead = none ];
72 -> 31;

73 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
32 -> 73 [arrowhead = none ];
73 -> 33;

74 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
33 -> 74 [arrowhead = none ];
74 -> 34;

75 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
33 -> 75 [arrowhead = none ];
75 -> 35;

76 [label="EVAL with clause\nappend([], X130, X130).\nand substitutionX123 -> [],\nT86 -> T93,\nX130 -> T93,\nT85 -> T93", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 76 [arrowhead = none ];
76 -> 36;

77 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 77 [arrowhead = none ];
77 -> 37;

78 [label="EVAL with clause\nappend(.(X143, X144), X145, .(X143, X146)) :- append(X144, X145, X146).\nand substitutionX143 -> T101,\nX144 -> X148,\nX123 -> .(T101, X148),\nT86 -> T103,\nX145 -> T103,\nX147 -> T101,\nX146 -> T102,\nT85 -> .(T101, T102),\nT100 -> T103", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 78 [arrowhead = none ];
78 -> 39;

79 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 79 [arrowhead = none ];
79 -> 40;

80 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 80 [arrowhead = none ];
80 -> 38;

81 [label="INSTANCE with matching:\nX123 -> X148\nT86 -> T103\nT85 -> T102", fontsize=14, style = filled, fillcolor = lightgrey];
39 -> 81 [arrowhead = none , style=dashed];
81 -> 32[style=dashed];

}

(2) Obligation:

Clauses:

appendA([], T45, T45).
appendA(.(T50, X93), X94, .(T50, T51)) :- appendA(X93, X94, T51).
appendB([], T93, T93).
appendB(.(T101, X148), T103, .(T101, T102)) :- appendB(X148, T103, T102).
sublistC([], T16).
sublistC(T7, .(T29, T30)) :- appendA(X56, X57, T30).
sublistC(.(T72, T73), .(T72, T30)) :- appendA(T73, T39, T30).
sublistC(T86, .(T84, T30)) :- ','(appendA(T85, T39, T30), appendB(X123, T86, T85)).

Query: sublistC(a,g)

(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:
sublistC_in: (f,b)
appendA_in: (f,f,b)
appendB_in: (f,f,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

sublistC_in_ag([], T16) → sublistC_out_ag([], T16)
sublistC_in_ag(T7, .(T29, T30)) → U3_ag(T7, T29, T30, appendA_in_aag(X56, X57, T30))
appendA_in_aag([], T45, T45) → appendA_out_aag([], T45, T45)
appendA_in_aag(.(T50, X93), X94, .(T50, T51)) → U1_aag(T50, X93, X94, T51, appendA_in_aag(X93, X94, T51))
U1_aag(T50, X93, X94, T51, appendA_out_aag(X93, X94, T51)) → appendA_out_aag(.(T50, X93), X94, .(T50, T51))
U3_ag(T7, T29, T30, appendA_out_aag(X56, X57, T30)) → sublistC_out_ag(T7, .(T29, T30))
sublistC_in_ag(.(T72, T73), .(T72, T30)) → U4_ag(T72, T73, T30, appendA_in_aag(T73, T39, T30))
U4_ag(T72, T73, T30, appendA_out_aag(T73, T39, T30)) → sublistC_out_ag(.(T72, T73), .(T72, T30))
sublistC_in_ag(T86, .(T84, T30)) → U5_ag(T86, T84, T30, appendA_in_aag(T85, T39, T30))
U5_ag(T86, T84, T30, appendA_out_aag(T85, T39, T30)) → U6_ag(T86, T84, T30, appendB_in_aag(X123, T86, T85))
appendB_in_aag([], T93, T93) → appendB_out_aag([], T93, T93)
appendB_in_aag(.(T101, X148), T103, .(T101, T102)) → U2_aag(T101, X148, T103, T102, appendB_in_aag(X148, T103, T102))
U2_aag(T101, X148, T103, T102, appendB_out_aag(X148, T103, T102)) → appendB_out_aag(.(T101, X148), T103, .(T101, T102))
U6_ag(T86, T84, T30, appendB_out_aag(X123, T86, T85)) → sublistC_out_ag(T86, .(T84, T30))

The argument filtering Pi contains the following mapping:
sublistC_in_ag(x1, x2)  =  sublistC_in_ag(x2)
sublistC_out_ag(x1, x2)  =  sublistC_out_ag
.(x1, x2)  =  .(x1, x2)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x4)
appendA_in_aag(x1, x2, x3)  =  appendA_in_aag(x3)
appendA_out_aag(x1, x2, x3)  =  appendA_out_aag(x1, x2)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x1, x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
U6_ag(x1, x2, x3, x4)  =  U6_ag(x4)
appendB_in_aag(x1, x2, x3)  =  appendB_in_aag(x3)
appendB_out_aag(x1, x2, x3)  =  appendB_out_aag(x1, x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x1, x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

sublistC_in_ag([], T16) → sublistC_out_ag([], T16)
sublistC_in_ag(T7, .(T29, T30)) → U3_ag(T7, T29, T30, appendA_in_aag(X56, X57, T30))
appendA_in_aag([], T45, T45) → appendA_out_aag([], T45, T45)
appendA_in_aag(.(T50, X93), X94, .(T50, T51)) → U1_aag(T50, X93, X94, T51, appendA_in_aag(X93, X94, T51))
U1_aag(T50, X93, X94, T51, appendA_out_aag(X93, X94, T51)) → appendA_out_aag(.(T50, X93), X94, .(T50, T51))
U3_ag(T7, T29, T30, appendA_out_aag(X56, X57, T30)) → sublistC_out_ag(T7, .(T29, T30))
sublistC_in_ag(.(T72, T73), .(T72, T30)) → U4_ag(T72, T73, T30, appendA_in_aag(T73, T39, T30))
U4_ag(T72, T73, T30, appendA_out_aag(T73, T39, T30)) → sublistC_out_ag(.(T72, T73), .(T72, T30))
sublistC_in_ag(T86, .(T84, T30)) → U5_ag(T86, T84, T30, appendA_in_aag(T85, T39, T30))
U5_ag(T86, T84, T30, appendA_out_aag(T85, T39, T30)) → U6_ag(T86, T84, T30, appendB_in_aag(X123, T86, T85))
appendB_in_aag([], T93, T93) → appendB_out_aag([], T93, T93)
appendB_in_aag(.(T101, X148), T103, .(T101, T102)) → U2_aag(T101, X148, T103, T102, appendB_in_aag(X148, T103, T102))
U2_aag(T101, X148, T103, T102, appendB_out_aag(X148, T103, T102)) → appendB_out_aag(.(T101, X148), T103, .(T101, T102))
U6_ag(T86, T84, T30, appendB_out_aag(X123, T86, T85)) → sublistC_out_ag(T86, .(T84, T30))

The argument filtering Pi contains the following mapping:
sublistC_in_ag(x1, x2)  =  sublistC_in_ag(x2)
sublistC_out_ag(x1, x2)  =  sublistC_out_ag
.(x1, x2)  =  .(x1, x2)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x4)
appendA_in_aag(x1, x2, x3)  =  appendA_in_aag(x3)
appendA_out_aag(x1, x2, x3)  =  appendA_out_aag(x1, x2)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x1, x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
U6_ag(x1, x2, x3, x4)  =  U6_ag(x4)
appendB_in_aag(x1, x2, x3)  =  appendB_in_aag(x3)
appendB_out_aag(x1, x2, x3)  =  appendB_out_aag(x1, x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x1, x5)

(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:

SUBLISTC_IN_AG(T7, .(T29, T30)) → U3_AG(T7, T29, T30, appendA_in_aag(X56, X57, T30))
SUBLISTC_IN_AG(T7, .(T29, T30)) → APPENDA_IN_AAG(X56, X57, T30)
APPENDA_IN_AAG(.(T50, X93), X94, .(T50, T51)) → U1_AAG(T50, X93, X94, T51, appendA_in_aag(X93, X94, T51))
APPENDA_IN_AAG(.(T50, X93), X94, .(T50, T51)) → APPENDA_IN_AAG(X93, X94, T51)
SUBLISTC_IN_AG(.(T72, T73), .(T72, T30)) → U4_AG(T72, T73, T30, appendA_in_aag(T73, T39, T30))
SUBLISTC_IN_AG(.(T72, T73), .(T72, T30)) → APPENDA_IN_AAG(T73, T39, T30)
SUBLISTC_IN_AG(T86, .(T84, T30)) → U5_AG(T86, T84, T30, appendA_in_aag(T85, T39, T30))
U5_AG(T86, T84, T30, appendA_out_aag(T85, T39, T30)) → U6_AG(T86, T84, T30, appendB_in_aag(X123, T86, T85))
U5_AG(T86, T84, T30, appendA_out_aag(T85, T39, T30)) → APPENDB_IN_AAG(X123, T86, T85)
APPENDB_IN_AAG(.(T101, X148), T103, .(T101, T102)) → U2_AAG(T101, X148, T103, T102, appendB_in_aag(X148, T103, T102))
APPENDB_IN_AAG(.(T101, X148), T103, .(T101, T102)) → APPENDB_IN_AAG(X148, T103, T102)

The TRS R consists of the following rules:

sublistC_in_ag([], T16) → sublistC_out_ag([], T16)
sublistC_in_ag(T7, .(T29, T30)) → U3_ag(T7, T29, T30, appendA_in_aag(X56, X57, T30))
appendA_in_aag([], T45, T45) → appendA_out_aag([], T45, T45)
appendA_in_aag(.(T50, X93), X94, .(T50, T51)) → U1_aag(T50, X93, X94, T51, appendA_in_aag(X93, X94, T51))
U1_aag(T50, X93, X94, T51, appendA_out_aag(X93, X94, T51)) → appendA_out_aag(.(T50, X93), X94, .(T50, T51))
U3_ag(T7, T29, T30, appendA_out_aag(X56, X57, T30)) → sublistC_out_ag(T7, .(T29, T30))
sublistC_in_ag(.(T72, T73), .(T72, T30)) → U4_ag(T72, T73, T30, appendA_in_aag(T73, T39, T30))
U4_ag(T72, T73, T30, appendA_out_aag(T73, T39, T30)) → sublistC_out_ag(.(T72, T73), .(T72, T30))
sublistC_in_ag(T86, .(T84, T30)) → U5_ag(T86, T84, T30, appendA_in_aag(T85, T39, T30))
U5_ag(T86, T84, T30, appendA_out_aag(T85, T39, T30)) → U6_ag(T86, T84, T30, appendB_in_aag(X123, T86, T85))
appendB_in_aag([], T93, T93) → appendB_out_aag([], T93, T93)
appendB_in_aag(.(T101, X148), T103, .(T101, T102)) → U2_aag(T101, X148, T103, T102, appendB_in_aag(X148, T103, T102))
U2_aag(T101, X148, T103, T102, appendB_out_aag(X148, T103, T102)) → appendB_out_aag(.(T101, X148), T103, .(T101, T102))
U6_ag(T86, T84, T30, appendB_out_aag(X123, T86, T85)) → sublistC_out_ag(T86, .(T84, T30))

The argument filtering Pi contains the following mapping:
sublistC_in_ag(x1, x2)  =  sublistC_in_ag(x2)
sublistC_out_ag(x1, x2)  =  sublistC_out_ag
.(x1, x2)  =  .(x1, x2)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x4)
appendA_in_aag(x1, x2, x3)  =  appendA_in_aag(x3)
appendA_out_aag(x1, x2, x3)  =  appendA_out_aag(x1, x2)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x1, x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
U6_ag(x1, x2, x3, x4)  =  U6_ag(x4)
appendB_in_aag(x1, x2, x3)  =  appendB_in_aag(x3)
appendB_out_aag(x1, x2, x3)  =  appendB_out_aag(x1, x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x1, x5)
SUBLISTC_IN_AG(x1, x2)  =  SUBLISTC_IN_AG(x2)
U3_AG(x1, x2, x3, x4)  =  U3_AG(x4)
APPENDA_IN_AAG(x1, x2, x3)  =  APPENDA_IN_AAG(x3)
U1_AAG(x1, x2, x3, x4, x5)  =  U1_AAG(x1, x5)
U4_AG(x1, x2, x3, x4)  =  U4_AG(x4)
U5_AG(x1, x2, x3, x4)  =  U5_AG(x4)
U6_AG(x1, x2, x3, x4)  =  U6_AG(x4)
APPENDB_IN_AAG(x1, x2, x3)  =  APPENDB_IN_AAG(x3)
U2_AAG(x1, x2, x3, x4, x5)  =  U2_AAG(x1, x5)

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

(6) Obligation:

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

SUBLISTC_IN_AG(T7, .(T29, T30)) → U3_AG(T7, T29, T30, appendA_in_aag(X56, X57, T30))
SUBLISTC_IN_AG(T7, .(T29, T30)) → APPENDA_IN_AAG(X56, X57, T30)
APPENDA_IN_AAG(.(T50, X93), X94, .(T50, T51)) → U1_AAG(T50, X93, X94, T51, appendA_in_aag(X93, X94, T51))
APPENDA_IN_AAG(.(T50, X93), X94, .(T50, T51)) → APPENDA_IN_AAG(X93, X94, T51)
SUBLISTC_IN_AG(.(T72, T73), .(T72, T30)) → U4_AG(T72, T73, T30, appendA_in_aag(T73, T39, T30))
SUBLISTC_IN_AG(.(T72, T73), .(T72, T30)) → APPENDA_IN_AAG(T73, T39, T30)
SUBLISTC_IN_AG(T86, .(T84, T30)) → U5_AG(T86, T84, T30, appendA_in_aag(T85, T39, T30))
U5_AG(T86, T84, T30, appendA_out_aag(T85, T39, T30)) → U6_AG(T86, T84, T30, appendB_in_aag(X123, T86, T85))
U5_AG(T86, T84, T30, appendA_out_aag(T85, T39, T30)) → APPENDB_IN_AAG(X123, T86, T85)
APPENDB_IN_AAG(.(T101, X148), T103, .(T101, T102)) → U2_AAG(T101, X148, T103, T102, appendB_in_aag(X148, T103, T102))
APPENDB_IN_AAG(.(T101, X148), T103, .(T101, T102)) → APPENDB_IN_AAG(X148, T103, T102)

The TRS R consists of the following rules:

sublistC_in_ag([], T16) → sublistC_out_ag([], T16)
sublistC_in_ag(T7, .(T29, T30)) → U3_ag(T7, T29, T30, appendA_in_aag(X56, X57, T30))
appendA_in_aag([], T45, T45) → appendA_out_aag([], T45, T45)
appendA_in_aag(.(T50, X93), X94, .(T50, T51)) → U1_aag(T50, X93, X94, T51, appendA_in_aag(X93, X94, T51))
U1_aag(T50, X93, X94, T51, appendA_out_aag(X93, X94, T51)) → appendA_out_aag(.(T50, X93), X94, .(T50, T51))
U3_ag(T7, T29, T30, appendA_out_aag(X56, X57, T30)) → sublistC_out_ag(T7, .(T29, T30))
sublistC_in_ag(.(T72, T73), .(T72, T30)) → U4_ag(T72, T73, T30, appendA_in_aag(T73, T39, T30))
U4_ag(T72, T73, T30, appendA_out_aag(T73, T39, T30)) → sublistC_out_ag(.(T72, T73), .(T72, T30))
sublistC_in_ag(T86, .(T84, T30)) → U5_ag(T86, T84, T30, appendA_in_aag(T85, T39, T30))
U5_ag(T86, T84, T30, appendA_out_aag(T85, T39, T30)) → U6_ag(T86, T84, T30, appendB_in_aag(X123, T86, T85))
appendB_in_aag([], T93, T93) → appendB_out_aag([], T93, T93)
appendB_in_aag(.(T101, X148), T103, .(T101, T102)) → U2_aag(T101, X148, T103, T102, appendB_in_aag(X148, T103, T102))
U2_aag(T101, X148, T103, T102, appendB_out_aag(X148, T103, T102)) → appendB_out_aag(.(T101, X148), T103, .(T101, T102))
U6_ag(T86, T84, T30, appendB_out_aag(X123, T86, T85)) → sublistC_out_ag(T86, .(T84, T30))

The argument filtering Pi contains the following mapping:
sublistC_in_ag(x1, x2)  =  sublistC_in_ag(x2)
sublistC_out_ag(x1, x2)  =  sublistC_out_ag
.(x1, x2)  =  .(x1, x2)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x4)
appendA_in_aag(x1, x2, x3)  =  appendA_in_aag(x3)
appendA_out_aag(x1, x2, x3)  =  appendA_out_aag(x1, x2)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x1, x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
U6_ag(x1, x2, x3, x4)  =  U6_ag(x4)
appendB_in_aag(x1, x2, x3)  =  appendB_in_aag(x3)
appendB_out_aag(x1, x2, x3)  =  appendB_out_aag(x1, x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x1, x5)
SUBLISTC_IN_AG(x1, x2)  =  SUBLISTC_IN_AG(x2)
U3_AG(x1, x2, x3, x4)  =  U3_AG(x4)
APPENDA_IN_AAG(x1, x2, x3)  =  APPENDA_IN_AAG(x3)
U1_AAG(x1, x2, x3, x4, x5)  =  U1_AAG(x1, x5)
U4_AG(x1, x2, x3, x4)  =  U4_AG(x4)
U5_AG(x1, x2, x3, x4)  =  U5_AG(x4)
U6_AG(x1, x2, x3, x4)  =  U6_AG(x4)
APPENDB_IN_AAG(x1, x2, x3)  =  APPENDB_IN_AAG(x3)
U2_AAG(x1, x2, x3, x4, x5)  =  U2_AAG(x1, x5)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 9 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

APPENDB_IN_AAG(.(T101, X148), T103, .(T101, T102)) → APPENDB_IN_AAG(X148, T103, T102)

The TRS R consists of the following rules:

sublistC_in_ag([], T16) → sublistC_out_ag([], T16)
sublistC_in_ag(T7, .(T29, T30)) → U3_ag(T7, T29, T30, appendA_in_aag(X56, X57, T30))
appendA_in_aag([], T45, T45) → appendA_out_aag([], T45, T45)
appendA_in_aag(.(T50, X93), X94, .(T50, T51)) → U1_aag(T50, X93, X94, T51, appendA_in_aag(X93, X94, T51))
U1_aag(T50, X93, X94, T51, appendA_out_aag(X93, X94, T51)) → appendA_out_aag(.(T50, X93), X94, .(T50, T51))
U3_ag(T7, T29, T30, appendA_out_aag(X56, X57, T30)) → sublistC_out_ag(T7, .(T29, T30))
sublistC_in_ag(.(T72, T73), .(T72, T30)) → U4_ag(T72, T73, T30, appendA_in_aag(T73, T39, T30))
U4_ag(T72, T73, T30, appendA_out_aag(T73, T39, T30)) → sublistC_out_ag(.(T72, T73), .(T72, T30))
sublistC_in_ag(T86, .(T84, T30)) → U5_ag(T86, T84, T30, appendA_in_aag(T85, T39, T30))
U5_ag(T86, T84, T30, appendA_out_aag(T85, T39, T30)) → U6_ag(T86, T84, T30, appendB_in_aag(X123, T86, T85))
appendB_in_aag([], T93, T93) → appendB_out_aag([], T93, T93)
appendB_in_aag(.(T101, X148), T103, .(T101, T102)) → U2_aag(T101, X148, T103, T102, appendB_in_aag(X148, T103, T102))
U2_aag(T101, X148, T103, T102, appendB_out_aag(X148, T103, T102)) → appendB_out_aag(.(T101, X148), T103, .(T101, T102))
U6_ag(T86, T84, T30, appendB_out_aag(X123, T86, T85)) → sublistC_out_ag(T86, .(T84, T30))

The argument filtering Pi contains the following mapping:
sublistC_in_ag(x1, x2)  =  sublistC_in_ag(x2)
sublistC_out_ag(x1, x2)  =  sublistC_out_ag
.(x1, x2)  =  .(x1, x2)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x4)
appendA_in_aag(x1, x2, x3)  =  appendA_in_aag(x3)
appendA_out_aag(x1, x2, x3)  =  appendA_out_aag(x1, x2)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x1, x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
U6_ag(x1, x2, x3, x4)  =  U6_ag(x4)
appendB_in_aag(x1, x2, x3)  =  appendB_in_aag(x3)
appendB_out_aag(x1, x2, x3)  =  appendB_out_aag(x1, x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x1, x5)
APPENDB_IN_AAG(x1, x2, x3)  =  APPENDB_IN_AAG(x3)

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:

APPENDB_IN_AAG(.(T101, X148), T103, .(T101, T102)) → APPENDB_IN_AAG(X148, T103, T102)

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

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:

APPENDB_IN_AAG(.(T101, T102)) → APPENDB_IN_AAG(T102)

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:

• APPENDB_IN_AAG(.(T101, T102)) → APPENDB_IN_AAG(T102)
  The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

APPENDA_IN_AAG(.(T50, X93), X94, .(T50, T51)) → APPENDA_IN_AAG(X93, X94, T51)

The TRS R consists of the following rules:

sublistC_in_ag([], T16) → sublistC_out_ag([], T16)
sublistC_in_ag(T7, .(T29, T30)) → U3_ag(T7, T29, T30, appendA_in_aag(X56, X57, T30))
appendA_in_aag([], T45, T45) → appendA_out_aag([], T45, T45)
appendA_in_aag(.(T50, X93), X94, .(T50, T51)) → U1_aag(T50, X93, X94, T51, appendA_in_aag(X93, X94, T51))
U1_aag(T50, X93, X94, T51, appendA_out_aag(X93, X94, T51)) → appendA_out_aag(.(T50, X93), X94, .(T50, T51))
U3_ag(T7, T29, T30, appendA_out_aag(X56, X57, T30)) → sublistC_out_ag(T7, .(T29, T30))
sublistC_in_ag(.(T72, T73), .(T72, T30)) → U4_ag(T72, T73, T30, appendA_in_aag(T73, T39, T30))
U4_ag(T72, T73, T30, appendA_out_aag(T73, T39, T30)) → sublistC_out_ag(.(T72, T73), .(T72, T30))
sublistC_in_ag(T86, .(T84, T30)) → U5_ag(T86, T84, T30, appendA_in_aag(T85, T39, T30))
U5_ag(T86, T84, T30, appendA_out_aag(T85, T39, T30)) → U6_ag(T86, T84, T30, appendB_in_aag(X123, T86, T85))
appendB_in_aag([], T93, T93) → appendB_out_aag([], T93, T93)
appendB_in_aag(.(T101, X148), T103, .(T101, T102)) → U2_aag(T101, X148, T103, T102, appendB_in_aag(X148, T103, T102))
U2_aag(T101, X148, T103, T102, appendB_out_aag(X148, T103, T102)) → appendB_out_aag(.(T101, X148), T103, .(T101, T102))
U6_ag(T86, T84, T30, appendB_out_aag(X123, T86, T85)) → sublistC_out_ag(T86, .(T84, T30))

The argument filtering Pi contains the following mapping:
sublistC_in_ag(x1, x2)  =  sublistC_in_ag(x2)
sublistC_out_ag(x1, x2)  =  sublistC_out_ag
.(x1, x2)  =  .(x1, x2)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x4)
appendA_in_aag(x1, x2, x3)  =  appendA_in_aag(x3)
appendA_out_aag(x1, x2, x3)  =  appendA_out_aag(x1, x2)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x1, x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
U6_ag(x1, x2, x3, x4)  =  U6_ag(x4)
appendB_in_aag(x1, x2, x3)  =  appendB_in_aag(x3)
appendB_out_aag(x1, x2, x3)  =  appendB_out_aag(x1, x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x1, x5)
APPENDA_IN_AAG(x1, x2, x3)  =  APPENDA_IN_AAG(x3)

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:

APPENDA_IN_AAG(.(T50, X93), X94, .(T50, T51)) → APPENDA_IN_AAG(X93, X94, T51)

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

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:

APPENDA_IN_AAG(.(T50, T51)) → APPENDA_IN_AAG(T51)

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:

• APPENDA_IN_AAG(.(T50, T51)) → APPENDA_IN_AAG(T51)
  The graph contains the following edges 1 > 1

(22) YES