Left Termination of the query pattern
sublist(g,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(g,g)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
3 [label="A: sublist(T1, T2)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
6 [label="6: sublist(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(append(X7, X8, T6), append(X9, T5, X7))\n\nT5 is ground\nT6 is ground\nX7 is free\nX8 is free\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(append(X7, X8, T6), append(X9, T5, X7)), SCOPE: 2, CLAUSE: 0 | ','(append(X7, X8, T6), append(X9, T5, X7)), SCOPE: 2, CLAUSE: 1\n\nT5 is ground\nT6 is ground\nX7 is free\nX8 is free\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(append(X7, X8, T6), append(X9, T5, X7)), SCOPE: 2, CLAUSE: 0\n\nT5 is ground\nT6 is ground\nX7 is free\nX8 is free\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(append(X7, X8, T6), append(X9, T5, X7)), SCOPE: 2, CLAUSE: 1\n\nT5 is ground\nT6 is ground\nX7 is free\nX8 is free\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: append(X9, T5, [])\n\nT5 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: append(X9, T5, []), SCOPE: 3, CLAUSE: 0 | append(X9, T5, []), SCOPE: 3, CLAUSE: 1\n\nT5 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: append(X9, T5, []), SCOPE: 3, CLAUSE: 0\n\nT5 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: append(X9, T5, []), SCOPE: 3, CLAUSE: 1\n\nT5 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="B: ','(append(X56, X57, T29), append(X9, T5, .(T28, X56)))\n\nT5 is ground\nT28 is ground\nT29 is ground\nX9 is free\nX56 is free\nX57 is free", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="C: append(X56, X57, T29)\n\nT29 is ground\nX56 is free\nX57 is free", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="E: append(X9, T5, .(T28, T37))\n\nT5 is ground\nT28 is ground\nT37 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: append(X56, X57, T29), SCOPE: 4, CLAUSE: 0 | append(X56, X57, T29), SCOPE: 4, CLAUSE: 1\n\nT29 is ground\nX56 is free\nX57 is free", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: append(X56, X57, T29), SCOPE: 4, CLAUSE: 0\n\nT29 is ground\nX56 is free\nX57 is free", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: append(X56, X57, T29), SCOPE: 4, CLAUSE: 1\n\nT29 is ground\nX56 is free\nX57 is free", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: append(X93, X94, T50)\n\nT50 is ground\nX93 is free\nX94 is free", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: append(X9, T5, .(T28, T37)), SCOPE: 5, CLAUSE: 0 | append(X9, T5, .(T28, T37)), SCOPE: 5, CLAUSE: 1\n\nT5 is ground\nT28 is ground\nT37 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: append(X9, T5, .(T28, T37)), SCOPE: 5, CLAUSE: 0\n\nT5 is ground\nT28 is ground\nT37 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: append(X9, T5, .(T28, T37)), SCOPE: 5, CLAUSE: 1\n\nT5 is ground\nT28 is ground\nT37 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="D: append(X123, T81, T83)\n\nT81 is ground\nT83 is ground\nX123 is free", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: append(X123, T81, T83), SCOPE: 6, CLAUSE: 0 | append(X123, T81, T83), SCOPE: 6, CLAUSE: 1\n\nT81 is ground\nT83 is ground\nX123 is free", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: append(X123, T81, T83), SCOPE: 6, CLAUSE: 0\n\nT81 is ground\nT83 is ground\nX123 is free", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: append(X123, T81, T83), SCOPE: 6, CLAUSE: 1\n\nT81 is ground\nT83 is ground\nX123 is free", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: append(X148, T97, T99)\n\nT97 is ground\nT99 is ground\nX148 is free", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 88 [arrowhead = none ];
88 -> 6;

89 [label="ONLY EVAL with clause\nsublist(X5, X6) :- ','(append(X7, X8, X6), append(X9, X5, X7)).\nand substitutionT1 -> T5,\nX5 -> T5,\nT2 -> T6,\nX6 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 89 [arrowhead = none ];
89 -> 8;

90 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
8 -> 90 [arrowhead = none ];
90 -> 10;

91 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 91 [arrowhead = none ];
91 -> 13;

92 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 92 [arrowhead = none ];
92 -> 15;

93 [label="ONLY EVAL with clause\nappend([], X22, X22).\nand substitutionX7 -> [],\nX8 -> T15,\nX22 -> T15,\nT6 -> T15,\nX23 -> T15", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 93 [arrowhead = none ];
93 -> 17;

94 [label="EVAL with clause\nappend(.(X51, X52), X53, .(X51, X54)) :- append(X52, X53, X54).\nand substitutionX51 -> T28,\nX52 -> X56,\nX7 -> .(T28, X56),\nX8 -> X57,\nX53 -> X57,\nX55 -> T28,\nX54 -> T29,\nT6 -> .(T28, T29)", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 94 [arrowhead = none ];
94 -> 43;

95 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 95 [arrowhead = none ];
95 -> 44;

96 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 96 [arrowhead = none ];
96 -> 21;

97 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
21 -> 97 [arrowhead = none ];
97 -> 24;

98 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
21 -> 98 [arrowhead = none ];
98 -> 25;

99 [label="EVAL with clause\nappend([], X30, X30).\nand substitutionX9 -> [],\nT5 -> [],\nX30 -> [],\nT22 -> []", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 99 [arrowhead = none ];
99 -> 28;

100 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 100 [arrowhead = none ];
100 -> 30;

101 [label="BACKTRACK\nfor clause: append(.(X, Xs), Ys, .(X, Zs)) :- append(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 101 [arrowhead = none ];
101 -> 38;

102 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 102 [arrowhead = none ];
102 -> 35;

103 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
43 -> 103 [arrowhead = none ];
103 -> 50;

104 [label="SPLIT 2\nnew knowledge:\nT37 is ground\nT38 is ground\nT29 is ground\nreplacements:X56 -> T37,\nX57 -> T38", fontsize=14, style = filled, fillcolor = violet];
43 -> 104 [arrowhead = none ];
104 -> 51;

105 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
50 -> 105 [arrowhead = none ];
105 -> 53;

106 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
51 -> 106 [arrowhead = none ];
106 -> 69;

107 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
53 -> 107 [arrowhead = none ];
107 -> 59;

108 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
53 -> 108 [arrowhead = none ];
108 -> 60;

109 [label="ONLY EVAL with clause\nappend([], X72, X72).\nand substitutionX56 -> [],\nX57 -> T44,\nX72 -> T44,\nT29 -> T44,\nX73 -> T44", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 109 [arrowhead = none ];
109 -> 63;

110 [label="EVAL with clause\nappend(.(X88, X89), X90, .(X88, X91)) :- append(X89, X90, X91).\nand substitutionX88 -> T49,\nX89 -> X93,\nX56 -> .(T49, X93),\nX57 -> X94,\nX90 -> X94,\nX92 -> T49,\nX91 -> T50,\nT29 -> .(T49, T50)", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 110 [arrowhead = none ];
110 -> 67;

111 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
60 -> 111 [arrowhead = none ];
111 -> 68;

112 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
63 -> 112 [arrowhead = none ];
112 -> 64;

113 [label="INSTANCE with matching:\nX56 -> X93\nX57 -> X94\nT29 -> T50", fontsize=14, style = filled, fillcolor = lightgrey];
67 -> 113 [arrowhead = none , style=dashed];
113 -> 50[style=dashed];

114 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
69 -> 114 [arrowhead = none ];
114 -> 70;

115 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
69 -> 115 [arrowhead = none ];
115 -> 71;

116 [label="EVAL with clause\nappend([], X103, X103).\nand substitutionX9 -> [],\nT5 -> .(T71, T72),\nX103 -> .(T71, T72),\nT28 -> T71,\nT37 -> T72,\nT70 -> .(T71, T72)", fontsize=14, style = filled, fillcolor = lightblue];
70 -> 116 [arrowhead = none ];
116 -> 72;

117 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
70 -> 117 [arrowhead = none ];
117 -> 73;

118 [label="ONLY EVAL with clause\nappend(.(X118, X119), X120, .(X118, X121)) :- append(X119, X120, X121).\nand substitutionX118 -> T82,\nX119 -> X123,\nX9 -> .(T82, X123),\nT5 -> T81,\nX120 -> T81,\nT28 -> T82,\nX122 -> T82,\nT37 -> T83,\nX121 -> T83", fontsize=14, style = filled, fillcolor = lightblue];
71 -> 118 [arrowhead = none ];
118 -> 78;

119 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
72 -> 119 [arrowhead = none ];
119 -> 74;

120 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
78 -> 120 [arrowhead = none ];
120 -> 80;

121 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
80 -> 121 [arrowhead = none ];
121 -> 81;

122 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
80 -> 122 [arrowhead = none ];
122 -> 82;

123 [label="EVAL with clause\nappend([], X130, X130).\nand substitutionX123 -> [],\nT81 -> T90,\nX130 -> T90,\nT83 -> T90", fontsize=14, style = filled, fillcolor = lightblue];
81 -> 123 [arrowhead = none ];
123 -> 83;

124 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
81 -> 124 [arrowhead = none ];
124 -> 84;

125 [label="EVAL with clause\nappend(.(X143, X144), X145, .(X143, X146)) :- append(X144, X145, X146).\nand substitutionX143 -> T98,\nX144 -> X148,\nX123 -> .(T98, X148),\nT81 -> T97,\nX145 -> T97,\nX147 -> T98,\nX146 -> T99,\nT83 -> .(T98, T99)", fontsize=14, style = filled, fillcolor = lightblue];
82 -> 125 [arrowhead = none ];
125 -> 86;

126 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
82 -> 126 [arrowhead = none ];
126 -> 87;

127 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
83 -> 127 [arrowhead = none ];
127 -> 85;

128 [label="INSTANCE with matching:\nX123 -> X148\nT81 -> T97\nT83 -> T99", fontsize=14, style = filled, fillcolor = lightgrey];
86 -> 128 [arrowhead = none , style=dashed];
128 -> 78[style=dashed];

}

(2) Obligation:

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

sublistA_in_gg([], T15) → sublistA_out_gg([], T15)
sublistA_in_gg(T5, .(T28, T29)) → U1_gg(T5, T28, T29, pB_in_aagagg(X56, X57, T29, X9, T5, T28))
pB_in_aagagg(T37, T38, T29, X9, T5, T28) → U5_aagagg(T37, T38, T29, X9, T5, T28, appendC_in_aag(T37, T38, T29))
appendC_in_aag([], T44, T44) → appendC_out_aag([], T44, T44)
appendC_in_aag(.(T49, X93), X94, .(T49, T50)) → U2_aag(T49, X93, X94, T50, appendC_in_aag(X93, X94, T50))
U2_aag(T49, X93, X94, T50, appendC_out_aag(X93, X94, T50)) → appendC_out_aag(.(T49, X93), X94, .(T49, T50))
U5_aagagg(T37, T38, T29, X9, T5, T28, appendC_out_aag(T37, T38, T29)) → U6_aagagg(T37, T38, T29, X9, T5, T28, appendE_in_aggg(X9, T5, T28, T37))
appendE_in_aggg([], .(T71, T72), T71, T72) → appendE_out_aggg([], .(T71, T72), T71, T72)
appendE_in_aggg(.(T82, X123), T81, T82, T83) → U4_aggg(T82, X123, T81, T83, appendD_in_agg(X123, T81, T83))
appendD_in_agg([], T90, T90) → appendD_out_agg([], T90, T90)
appendD_in_agg(.(T98, X148), T97, .(T98, T99)) → U3_agg(T98, X148, T97, T99, appendD_in_agg(X148, T97, T99))
U3_agg(T98, X148, T97, T99, appendD_out_agg(X148, T97, T99)) → appendD_out_agg(.(T98, X148), T97, .(T98, T99))
U4_aggg(T82, X123, T81, T83, appendD_out_agg(X123, T81, T83)) → appendE_out_aggg(.(T82, X123), T81, T82, T83)
U6_aagagg(T37, T38, T29, X9, T5, T28, appendE_out_aggg(X9, T5, T28, T37)) → pB_out_aagagg(T37, T38, T29, X9, T5, T28)
U1_gg(T5, T28, T29, pB_out_aagagg(X56, X57, T29, X9, T5, T28)) → sublistA_out_gg(T5, .(T28, T29))

The argument filtering Pi contains the following mapping:
sublistA_in_gg(x1, x2)  =  sublistA_in_gg(x1, x2)
[]  =  []
sublistA_out_gg(x1, x2)  =  sublistA_out_gg(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_gg(x1, x2, x3, x4)  =  U1_gg(x1, x2, x3, x4)
pB_in_aagagg(x1, x2, x3, x4, x5, x6)  =  pB_in_aagagg(x3, x5, x6)
U5_aagagg(x1, x2, x3, x4, x5, x6, x7)  =  U5_aagagg(x3, x5, x6, x7)
appendC_in_aag(x1, x2, x3)  =  appendC_in_aag(x3)
appendC_out_aag(x1, x2, x3)  =  appendC_out_aag(x1, x2, x3)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x1, x4, x5)
U6_aagagg(x1, x2, x3, x4, x5, x6, x7)  =  U6_aagagg(x1, x2, x3, x5, x6, x7)
appendE_in_aggg(x1, x2, x3, x4)  =  appendE_in_aggg(x2, x3, x4)
appendE_out_aggg(x1, x2, x3, x4)  =  appendE_out_aggg(x1, x2, x3, x4)
U4_aggg(x1, x2, x3, x4, x5)  =  U4_aggg(x1, x3, x4, x5)
appendD_in_agg(x1, x2, x3)  =  appendD_in_agg(x2, x3)
appendD_out_agg(x1, x2, x3)  =  appendD_out_agg(x1, x2, x3)
U3_agg(x1, x2, x3, x4, x5)  =  U3_agg(x1, x3, x4, x5)
pB_out_aagagg(x1, x2, x3, x4, x5, x6)  =  pB_out_aagagg(x1, x2, x3, x4, x5, x6)

(3) 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:

SUBLISTA_IN_GG(T5, .(T28, T29)) → U1_GG(T5, T28, T29, pB_in_aagagg(X56, X57, T29, X9, T5, T28))
SUBLISTA_IN_GG(T5, .(T28, T29)) → PB_IN_AAGAGG(X56, X57, T29, X9, T5, T28)
PB_IN_AAGAGG(T37, T38, T29, X9, T5, T28) → U5_AAGAGG(T37, T38, T29, X9, T5, T28, appendC_in_aag(T37, T38, T29))
PB_IN_AAGAGG(T37, T38, T29, X9, T5, T28) → APPENDC_IN_AAG(T37, T38, T29)
APPENDC_IN_AAG(.(T49, X93), X94, .(T49, T50)) → U2_AAG(T49, X93, X94, T50, appendC_in_aag(X93, X94, T50))
APPENDC_IN_AAG(.(T49, X93), X94, .(T49, T50)) → APPENDC_IN_AAG(X93, X94, T50)
U5_AAGAGG(T37, T38, T29, X9, T5, T28, appendC_out_aag(T37, T38, T29)) → U6_AAGAGG(T37, T38, T29, X9, T5, T28, appendE_in_aggg(X9, T5, T28, T37))
U5_AAGAGG(T37, T38, T29, X9, T5, T28, appendC_out_aag(T37, T38, T29)) → APPENDE_IN_AGGG(X9, T5, T28, T37)
APPENDE_IN_AGGG(.(T82, X123), T81, T82, T83) → U4_AGGG(T82, X123, T81, T83, appendD_in_agg(X123, T81, T83))
APPENDE_IN_AGGG(.(T82, X123), T81, T82, T83) → APPENDD_IN_AGG(X123, T81, T83)
APPENDD_IN_AGG(.(T98, X148), T97, .(T98, T99)) → U3_AGG(T98, X148, T97, T99, appendD_in_agg(X148, T97, T99))
APPENDD_IN_AGG(.(T98, X148), T97, .(T98, T99)) → APPENDD_IN_AGG(X148, T97, T99)

The TRS R consists of the following rules:

sublistA_in_gg([], T15) → sublistA_out_gg([], T15)
sublistA_in_gg(T5, .(T28, T29)) → U1_gg(T5, T28, T29, pB_in_aagagg(X56, X57, T29, X9, T5, T28))
pB_in_aagagg(T37, T38, T29, X9, T5, T28) → U5_aagagg(T37, T38, T29, X9, T5, T28, appendC_in_aag(T37, T38, T29))
appendC_in_aag([], T44, T44) → appendC_out_aag([], T44, T44)
appendC_in_aag(.(T49, X93), X94, .(T49, T50)) → U2_aag(T49, X93, X94, T50, appendC_in_aag(X93, X94, T50))
U2_aag(T49, X93, X94, T50, appendC_out_aag(X93, X94, T50)) → appendC_out_aag(.(T49, X93), X94, .(T49, T50))
U5_aagagg(T37, T38, T29, X9, T5, T28, appendC_out_aag(T37, T38, T29)) → U6_aagagg(T37, T38, T29, X9, T5, T28, appendE_in_aggg(X9, T5, T28, T37))
appendE_in_aggg([], .(T71, T72), T71, T72) → appendE_out_aggg([], .(T71, T72), T71, T72)
appendE_in_aggg(.(T82, X123), T81, T82, T83) → U4_aggg(T82, X123, T81, T83, appendD_in_agg(X123, T81, T83))
appendD_in_agg([], T90, T90) → appendD_out_agg([], T90, T90)
appendD_in_agg(.(T98, X148), T97, .(T98, T99)) → U3_agg(T98, X148, T97, T99, appendD_in_agg(X148, T97, T99))
U3_agg(T98, X148, T97, T99, appendD_out_agg(X148, T97, T99)) → appendD_out_agg(.(T98, X148), T97, .(T98, T99))
U4_aggg(T82, X123, T81, T83, appendD_out_agg(X123, T81, T83)) → appendE_out_aggg(.(T82, X123), T81, T82, T83)
U6_aagagg(T37, T38, T29, X9, T5, T28, appendE_out_aggg(X9, T5, T28, T37)) → pB_out_aagagg(T37, T38, T29, X9, T5, T28)
U1_gg(T5, T28, T29, pB_out_aagagg(X56, X57, T29, X9, T5, T28)) → sublistA_out_gg(T5, .(T28, T29))

The argument filtering Pi contains the following mapping:
sublistA_in_gg(x1, x2)  =  sublistA_in_gg(x1, x2)
[]  =  []
sublistA_out_gg(x1, x2)  =  sublistA_out_gg(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_gg(x1, x2, x3, x4)  =  U1_gg(x1, x2, x3, x4)
pB_in_aagagg(x1, x2, x3, x4, x5, x6)  =  pB_in_aagagg(x3, x5, x6)
U5_aagagg(x1, x2, x3, x4, x5, x6, x7)  =  U5_aagagg(x3, x5, x6, x7)
appendC_in_aag(x1, x2, x3)  =  appendC_in_aag(x3)
appendC_out_aag(x1, x2, x3)  =  appendC_out_aag(x1, x2, x3)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x1, x4, x5)
U6_aagagg(x1, x2, x3, x4, x5, x6, x7)  =  U6_aagagg(x1, x2, x3, x5, x6, x7)
appendE_in_aggg(x1, x2, x3, x4)  =  appendE_in_aggg(x2, x3, x4)
appendE_out_aggg(x1, x2, x3, x4)  =  appendE_out_aggg(x1, x2, x3, x4)
U4_aggg(x1, x2, x3, x4, x5)  =  U4_aggg(x1, x3, x4, x5)
appendD_in_agg(x1, x2, x3)  =  appendD_in_agg(x2, x3)
appendD_out_agg(x1, x2, x3)  =  appendD_out_agg(x1, x2, x3)
U3_agg(x1, x2, x3, x4, x5)  =  U3_agg(x1, x3, x4, x5)
pB_out_aagagg(x1, x2, x3, x4, x5, x6)  =  pB_out_aagagg(x1, x2, x3, x4, x5, x6)
SUBLISTA_IN_GG(x1, x2)  =  SUBLISTA_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4)  =  U1_GG(x1, x2, x3, x4)
PB_IN_AAGAGG(x1, x2, x3, x4, x5, x6)  =  PB_IN_AAGAGG(x3, x5, x6)
U5_AAGAGG(x1, x2, x3, x4, x5, x6, x7)  =  U5_AAGAGG(x3, x5, x6, x7)
APPENDC_IN_AAG(x1, x2, x3)  =  APPENDC_IN_AAG(x3)
U2_AAG(x1, x2, x3, x4, x5)  =  U2_AAG(x1, x4, x5)
U6_AAGAGG(x1, x2, x3, x4, x5, x6, x7)  =  U6_AAGAGG(x1, x2, x3, x5, x6, x7)
APPENDE_IN_AGGG(x1, x2, x3, x4)  =  APPENDE_IN_AGGG(x2, x3, x4)
U4_AGGG(x1, x2, x3, x4, x5)  =  U4_AGGG(x1, x3, x4, x5)
APPENDD_IN_AGG(x1, x2, x3)  =  APPENDD_IN_AGG(x2, x3)
U3_AGG(x1, x2, x3, x4, x5)  =  U3_AGG(x1, x3, x4, x5)

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

(4) Obligation:

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

SUBLISTA_IN_GG(T5, .(T28, T29)) → U1_GG(T5, T28, T29, pB_in_aagagg(X56, X57, T29, X9, T5, T28))
SUBLISTA_IN_GG(T5, .(T28, T29)) → PB_IN_AAGAGG(X56, X57, T29, X9, T5, T28)
PB_IN_AAGAGG(T37, T38, T29, X9, T5, T28) → U5_AAGAGG(T37, T38, T29, X9, T5, T28, appendC_in_aag(T37, T38, T29))
PB_IN_AAGAGG(T37, T38, T29, X9, T5, T28) → APPENDC_IN_AAG(T37, T38, T29)
APPENDC_IN_AAG(.(T49, X93), X94, .(T49, T50)) → U2_AAG(T49, X93, X94, T50, appendC_in_aag(X93, X94, T50))
APPENDC_IN_AAG(.(T49, X93), X94, .(T49, T50)) → APPENDC_IN_AAG(X93, X94, T50)
U5_AAGAGG(T37, T38, T29, X9, T5, T28, appendC_out_aag(T37, T38, T29)) → U6_AAGAGG(T37, T38, T29, X9, T5, T28, appendE_in_aggg(X9, T5, T28, T37))
U5_AAGAGG(T37, T38, T29, X9, T5, T28, appendC_out_aag(T37, T38, T29)) → APPENDE_IN_AGGG(X9, T5, T28, T37)
APPENDE_IN_AGGG(.(T82, X123), T81, T82, T83) → U4_AGGG(T82, X123, T81, T83, appendD_in_agg(X123, T81, T83))
APPENDE_IN_AGGG(.(T82, X123), T81, T82, T83) → APPENDD_IN_AGG(X123, T81, T83)
APPENDD_IN_AGG(.(T98, X148), T97, .(T98, T99)) → U3_AGG(T98, X148, T97, T99, appendD_in_agg(X148, T97, T99))
APPENDD_IN_AGG(.(T98, X148), T97, .(T98, T99)) → APPENDD_IN_AGG(X148, T97, T99)

The TRS R consists of the following rules:

sublistA_in_gg([], T15) → sublistA_out_gg([], T15)
sublistA_in_gg(T5, .(T28, T29)) → U1_gg(T5, T28, T29, pB_in_aagagg(X56, X57, T29, X9, T5, T28))
pB_in_aagagg(T37, T38, T29, X9, T5, T28) → U5_aagagg(T37, T38, T29, X9, T5, T28, appendC_in_aag(T37, T38, T29))
appendC_in_aag([], T44, T44) → appendC_out_aag([], T44, T44)
appendC_in_aag(.(T49, X93), X94, .(T49, T50)) → U2_aag(T49, X93, X94, T50, appendC_in_aag(X93, X94, T50))
U2_aag(T49, X93, X94, T50, appendC_out_aag(X93, X94, T50)) → appendC_out_aag(.(T49, X93), X94, .(T49, T50))
U5_aagagg(T37, T38, T29, X9, T5, T28, appendC_out_aag(T37, T38, T29)) → U6_aagagg(T37, T38, T29, X9, T5, T28, appendE_in_aggg(X9, T5, T28, T37))
appendE_in_aggg([], .(T71, T72), T71, T72) → appendE_out_aggg([], .(T71, T72), T71, T72)
appendE_in_aggg(.(T82, X123), T81, T82, T83) → U4_aggg(T82, X123, T81, T83, appendD_in_agg(X123, T81, T83))
appendD_in_agg([], T90, T90) → appendD_out_agg([], T90, T90)
appendD_in_agg(.(T98, X148), T97, .(T98, T99)) → U3_agg(T98, X148, T97, T99, appendD_in_agg(X148, T97, T99))
U3_agg(T98, X148, T97, T99, appendD_out_agg(X148, T97, T99)) → appendD_out_agg(.(T98, X148), T97, .(T98, T99))
U4_aggg(T82, X123, T81, T83, appendD_out_agg(X123, T81, T83)) → appendE_out_aggg(.(T82, X123), T81, T82, T83)
U6_aagagg(T37, T38, T29, X9, T5, T28, appendE_out_aggg(X9, T5, T28, T37)) → pB_out_aagagg(T37, T38, T29, X9, T5, T28)
U1_gg(T5, T28, T29, pB_out_aagagg(X56, X57, T29, X9, T5, T28)) → sublistA_out_gg(T5, .(T28, T29))

The argument filtering Pi contains the following mapping:
sublistA_in_gg(x1, x2)  =  sublistA_in_gg(x1, x2)
[]  =  []
sublistA_out_gg(x1, x2)  =  sublistA_out_gg(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_gg(x1, x2, x3, x4)  =  U1_gg(x1, x2, x3, x4)
pB_in_aagagg(x1, x2, x3, x4, x5, x6)  =  pB_in_aagagg(x3, x5, x6)
U5_aagagg(x1, x2, x3, x4, x5, x6, x7)  =  U5_aagagg(x3, x5, x6, x7)
appendC_in_aag(x1, x2, x3)  =  appendC_in_aag(x3)
appendC_out_aag(x1, x2, x3)  =  appendC_out_aag(x1, x2, x3)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x1, x4, x5)
U6_aagagg(x1, x2, x3, x4, x5, x6, x7)  =  U6_aagagg(x1, x2, x3, x5, x6, x7)
appendE_in_aggg(x1, x2, x3, x4)  =  appendE_in_aggg(x2, x3, x4)
appendE_out_aggg(x1, x2, x3, x4)  =  appendE_out_aggg(x1, x2, x3, x4)
U4_aggg(x1, x2, x3, x4, x5)  =  U4_aggg(x1, x3, x4, x5)
appendD_in_agg(x1, x2, x3)  =  appendD_in_agg(x2, x3)
appendD_out_agg(x1, x2, x3)  =  appendD_out_agg(x1, x2, x3)
U3_agg(x1, x2, x3, x4, x5)  =  U3_agg(x1, x3, x4, x5)
pB_out_aagagg(x1, x2, x3, x4, x5, x6)  =  pB_out_aagagg(x1, x2, x3, x4, x5, x6)
SUBLISTA_IN_GG(x1, x2)  =  SUBLISTA_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4)  =  U1_GG(x1, x2, x3, x4)
PB_IN_AAGAGG(x1, x2, x3, x4, x5, x6)  =  PB_IN_AAGAGG(x3, x5, x6)
U5_AAGAGG(x1, x2, x3, x4, x5, x6, x7)  =  U5_AAGAGG(x3, x5, x6, x7)
APPENDC_IN_AAG(x1, x2, x3)  =  APPENDC_IN_AAG(x3)
U2_AAG(x1, x2, x3, x4, x5)  =  U2_AAG(x1, x4, x5)
U6_AAGAGG(x1, x2, x3, x4, x5, x6, x7)  =  U6_AAGAGG(x1, x2, x3, x5, x6, x7)
APPENDE_IN_AGGG(x1, x2, x3, x4)  =  APPENDE_IN_AGGG(x2, x3, x4)
U4_AGGG(x1, x2, x3, x4, x5)  =  U4_AGGG(x1, x3, x4, x5)
APPENDD_IN_AGG(x1, x2, x3)  =  APPENDD_IN_AGG(x2, x3)
U3_AGG(x1, x2, x3, x4, x5)  =  U3_AGG(x1, x3, x4, x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 10 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

APPENDD_IN_AGG(.(T98, X148), T97, .(T98, T99)) → APPENDD_IN_AGG(X148, T97, T99)

The TRS R consists of the following rules:

sublistA_in_gg([], T15) → sublistA_out_gg([], T15)
sublistA_in_gg(T5, .(T28, T29)) → U1_gg(T5, T28, T29, pB_in_aagagg(X56, X57, T29, X9, T5, T28))
pB_in_aagagg(T37, T38, T29, X9, T5, T28) → U5_aagagg(T37, T38, T29, X9, T5, T28, appendC_in_aag(T37, T38, T29))
appendC_in_aag([], T44, T44) → appendC_out_aag([], T44, T44)
appendC_in_aag(.(T49, X93), X94, .(T49, T50)) → U2_aag(T49, X93, X94, T50, appendC_in_aag(X93, X94, T50))
U2_aag(T49, X93, X94, T50, appendC_out_aag(X93, X94, T50)) → appendC_out_aag(.(T49, X93), X94, .(T49, T50))
U5_aagagg(T37, T38, T29, X9, T5, T28, appendC_out_aag(T37, T38, T29)) → U6_aagagg(T37, T38, T29, X9, T5, T28, appendE_in_aggg(X9, T5, T28, T37))
appendE_in_aggg([], .(T71, T72), T71, T72) → appendE_out_aggg([], .(T71, T72), T71, T72)
appendE_in_aggg(.(T82, X123), T81, T82, T83) → U4_aggg(T82, X123, T81, T83, appendD_in_agg(X123, T81, T83))
appendD_in_agg([], T90, T90) → appendD_out_agg([], T90, T90)
appendD_in_agg(.(T98, X148), T97, .(T98, T99)) → U3_agg(T98, X148, T97, T99, appendD_in_agg(X148, T97, T99))
U3_agg(T98, X148, T97, T99, appendD_out_agg(X148, T97, T99)) → appendD_out_agg(.(T98, X148), T97, .(T98, T99))
U4_aggg(T82, X123, T81, T83, appendD_out_agg(X123, T81, T83)) → appendE_out_aggg(.(T82, X123), T81, T82, T83)
U6_aagagg(T37, T38, T29, X9, T5, T28, appendE_out_aggg(X9, T5, T28, T37)) → pB_out_aagagg(T37, T38, T29, X9, T5, T28)
U1_gg(T5, T28, T29, pB_out_aagagg(X56, X57, T29, X9, T5, T28)) → sublistA_out_gg(T5, .(T28, T29))

The argument filtering Pi contains the following mapping:
sublistA_in_gg(x1, x2)  =  sublistA_in_gg(x1, x2)
[]  =  []
sublistA_out_gg(x1, x2)  =  sublistA_out_gg(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_gg(x1, x2, x3, x4)  =  U1_gg(x1, x2, x3, x4)
pB_in_aagagg(x1, x2, x3, x4, x5, x6)  =  pB_in_aagagg(x3, x5, x6)
U5_aagagg(x1, x2, x3, x4, x5, x6, x7)  =  U5_aagagg(x3, x5, x6, x7)
appendC_in_aag(x1, x2, x3)  =  appendC_in_aag(x3)
appendC_out_aag(x1, x2, x3)  =  appendC_out_aag(x1, x2, x3)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x1, x4, x5)
U6_aagagg(x1, x2, x3, x4, x5, x6, x7)  =  U6_aagagg(x1, x2, x3, x5, x6, x7)
appendE_in_aggg(x1, x2, x3, x4)  =  appendE_in_aggg(x2, x3, x4)
appendE_out_aggg(x1, x2, x3, x4)  =  appendE_out_aggg(x1, x2, x3, x4)
U4_aggg(x1, x2, x3, x4, x5)  =  U4_aggg(x1, x3, x4, x5)
appendD_in_agg(x1, x2, x3)  =  appendD_in_agg(x2, x3)
appendD_out_agg(x1, x2, x3)  =  appendD_out_agg(x1, x2, x3)
U3_agg(x1, x2, x3, x4, x5)  =  U3_agg(x1, x3, x4, x5)
pB_out_aagagg(x1, x2, x3, x4, x5, x6)  =  pB_out_aagagg(x1, x2, x3, x4, x5, x6)
APPENDD_IN_AGG(x1, x2, x3)  =  APPENDD_IN_AGG(x2, x3)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

APPENDD_IN_AGG(.(T98, X148), T97, .(T98, T99)) → APPENDD_IN_AGG(X148, T97, T99)

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

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

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

APPENDD_IN_AGG(T97, .(T98, T99)) → APPENDD_IN_AGG(T97, T99)

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

(12) 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:

• APPENDD_IN_AGG(T97, .(T98, T99)) → APPENDD_IN_AGG(T97, T99)
  The graph contains the following edges 1 >= 1, 2 > 2

(13) YES

(14) Obligation:

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

APPENDC_IN_AAG(.(T49, X93), X94, .(T49, T50)) → APPENDC_IN_AAG(X93, X94, T50)

The TRS R consists of the following rules:

sublistA_in_gg([], T15) → sublistA_out_gg([], T15)
sublistA_in_gg(T5, .(T28, T29)) → U1_gg(T5, T28, T29, pB_in_aagagg(X56, X57, T29, X9, T5, T28))
pB_in_aagagg(T37, T38, T29, X9, T5, T28) → U5_aagagg(T37, T38, T29, X9, T5, T28, appendC_in_aag(T37, T38, T29))
appendC_in_aag([], T44, T44) → appendC_out_aag([], T44, T44)
appendC_in_aag(.(T49, X93), X94, .(T49, T50)) → U2_aag(T49, X93, X94, T50, appendC_in_aag(X93, X94, T50))
U2_aag(T49, X93, X94, T50, appendC_out_aag(X93, X94, T50)) → appendC_out_aag(.(T49, X93), X94, .(T49, T50))
U5_aagagg(T37, T38, T29, X9, T5, T28, appendC_out_aag(T37, T38, T29)) → U6_aagagg(T37, T38, T29, X9, T5, T28, appendE_in_aggg(X9, T5, T28, T37))
appendE_in_aggg([], .(T71, T72), T71, T72) → appendE_out_aggg([], .(T71, T72), T71, T72)
appendE_in_aggg(.(T82, X123), T81, T82, T83) → U4_aggg(T82, X123, T81, T83, appendD_in_agg(X123, T81, T83))
appendD_in_agg([], T90, T90) → appendD_out_agg([], T90, T90)
appendD_in_agg(.(T98, X148), T97, .(T98, T99)) → U3_agg(T98, X148, T97, T99, appendD_in_agg(X148, T97, T99))
U3_agg(T98, X148, T97, T99, appendD_out_agg(X148, T97, T99)) → appendD_out_agg(.(T98, X148), T97, .(T98, T99))
U4_aggg(T82, X123, T81, T83, appendD_out_agg(X123, T81, T83)) → appendE_out_aggg(.(T82, X123), T81, T82, T83)
U6_aagagg(T37, T38, T29, X9, T5, T28, appendE_out_aggg(X9, T5, T28, T37)) → pB_out_aagagg(T37, T38, T29, X9, T5, T28)
U1_gg(T5, T28, T29, pB_out_aagagg(X56, X57, T29, X9, T5, T28)) → sublistA_out_gg(T5, .(T28, T29))

The argument filtering Pi contains the following mapping:
sublistA_in_gg(x1, x2)  =  sublistA_in_gg(x1, x2)
[]  =  []
sublistA_out_gg(x1, x2)  =  sublistA_out_gg(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_gg(x1, x2, x3, x4)  =  U1_gg(x1, x2, x3, x4)
pB_in_aagagg(x1, x2, x3, x4, x5, x6)  =  pB_in_aagagg(x3, x5, x6)
U5_aagagg(x1, x2, x3, x4, x5, x6, x7)  =  U5_aagagg(x3, x5, x6, x7)
appendC_in_aag(x1, x2, x3)  =  appendC_in_aag(x3)
appendC_out_aag(x1, x2, x3)  =  appendC_out_aag(x1, x2, x3)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x1, x4, x5)
U6_aagagg(x1, x2, x3, x4, x5, x6, x7)  =  U6_aagagg(x1, x2, x3, x5, x6, x7)
appendE_in_aggg(x1, x2, x3, x4)  =  appendE_in_aggg(x2, x3, x4)
appendE_out_aggg(x1, x2, x3, x4)  =  appendE_out_aggg(x1, x2, x3, x4)
U4_aggg(x1, x2, x3, x4, x5)  =  U4_aggg(x1, x3, x4, x5)
appendD_in_agg(x1, x2, x3)  =  appendD_in_agg(x2, x3)
appendD_out_agg(x1, x2, x3)  =  appendD_out_agg(x1, x2, x3)
U3_agg(x1, x2, x3, x4, x5)  =  U3_agg(x1, x3, x4, x5)
pB_out_aagagg(x1, x2, x3, x4, x5, x6)  =  pB_out_aagagg(x1, x2, x3, x4, x5, x6)
APPENDC_IN_AAG(x1, x2, x3)  =  APPENDC_IN_AAG(x3)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

APPENDC_IN_AAG(.(T49, X93), X94, .(T49, T50)) → APPENDC_IN_AAG(X93, X94, T50)

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

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

APPENDC_IN_AAG(.(T49, T50)) → APPENDC_IN_AAG(T50)

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

(19) 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:

• APPENDC_IN_AAG(.(T49, T50)) → APPENDC_IN_AAG(T50)
  The graph contains the following edges 1 > 1

(20) YES