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) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="A: sublist(T1, T2)\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
5 [label="5: sublist(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(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];
13 [label="13: ','(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];
14 [label="14: ','(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];
15 [label="15: ','(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];
18 [label="18: append(X9, T7, [])\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: append(X9, T7, []), SCOPE: 3, CLAUSE: 0 | append(X9, T7, []), SCOPE: 3, CLAUSE: 1\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: append(X9, T7, []), SCOPE: 3, CLAUSE: 0\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: append(X9, T7, []), SCOPE: 3, CLAUSE: 1\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="B: ','(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];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="C: append(X56, X57, T30)\n\nT30 is ground\nX56 is free\nX57 is free", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="E: append(X9, T7, .(T29, T38))\n\nT29 is ground\nT38 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: 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];
55 [label="55: append(X56, X57, T30), SCOPE: 4, CLAUSE: 0\n\nT30 is ground\nX56 is free\nX57 is free", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: append(X56, X57, T30), SCOPE: 4, CLAUSE: 1\n\nT30 is ground\nX56 is free\nX57 is free", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: append(X93, X94, T51)\n\nT51 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];
71 [label="71: 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];
72 [label="72: 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];
74 [label="74: 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];
77 [label="77: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="D: append(X123, T86, T85)\n\nT85 is ground\nX123 is free", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: 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];
92 [label="92: append(X123, T86, T85), SCOPE: 6, CLAUSE: 0\n\nT85 is ground\nX123 is free", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: append(X123, T86, T85), SCOPE: 6, CLAUSE: 1\n\nT85 is ground\nX123 is free", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: append(X148, T103, T102)\n\nT102 is ground\nX148 is free", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 99 [arrowhead = none ];
99 -> 5;

100 [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];
5 -> 100 [arrowhead = none ];
100 -> 12;

101 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 101 [arrowhead = none ];
101 -> 13;

102 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
13 -> 102 [arrowhead = none ];
102 -> 14;

103 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
13 -> 103 [arrowhead = none ];
103 -> 15;

104 [label="ONLY EVAL with clause\nappend([], X22, X22).\nand substitutionX7 -> [],\nX8 -> T16,\nX22 -> T16,\nT6 -> T16,\nX23 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 104 [arrowhead = none ];
104 -> 18;

105 [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];
15 -> 105 [arrowhead = none ];
105 -> 41;

106 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 106 [arrowhead = none ];
106 -> 42;

107 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 107 [arrowhead = none ];
107 -> 20;

108 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 108 [arrowhead = none ];
108 -> 21;

109 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 109 [arrowhead = none ];
109 -> 22;

110 [label="EVAL with clause\nappend([], X30, X30).\nand substitutionX9 -> [],\nT7 -> [],\nX30 -> [],\nT23 -> []", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 110 [arrowhead = none ];
110 -> 27;

111 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 111 [arrowhead = none ];
111 -> 28;

112 [label="BACKTRACK\nfor clause: append(.(X, Xs), Ys, .(X, Zs)) :- append(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 112 [arrowhead = none ];
112 -> 35;

113 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 113 [arrowhead = none ];
113 -> 31;

114 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
41 -> 114 [arrowhead = none ];
114 -> 48;

115 [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];
41 -> 115 [arrowhead = none ];
115 -> 49;

116 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
48 -> 116 [arrowhead = none ];
116 -> 53;

117 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
49 -> 117 [arrowhead = none ];
117 -> 71;

118 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
53 -> 118 [arrowhead = none ];
118 -> 55;

119 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
53 -> 119 [arrowhead = none ];
119 -> 57;

120 [label="ONLY EVAL with clause\nappend([], X72, X72).\nand substitutionX56 -> [],\nX57 -> T45,\nX72 -> T45,\nT30 -> T45,\nX73 -> T45", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 120 [arrowhead = none ];
120 -> 59;

121 [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];
57 -> 121 [arrowhead = none ];
121 -> 67;

122 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
57 -> 122 [arrowhead = none ];
122 -> 68;

123 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
59 -> 123 [arrowhead = none ];
123 -> 60;

124 [label="INSTANCE with matching:\nX56 -> X93\nX57 -> X94\nT30 -> T51", fontsize=14, style = filled, fillcolor = lightgrey];
67 -> 124 [arrowhead = none , style=dashed];
124 -> 48[style=dashed];

125 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
71 -> 125 [arrowhead = none ];
125 -> 72;

126 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
71 -> 126 [arrowhead = none ];
126 -> 74;

127 [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];
72 -> 127 [arrowhead = none ];
127 -> 77;

128 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
72 -> 128 [arrowhead = none ];
128 -> 78;

129 [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];
74 -> 129 [arrowhead = none ];
129 -> 84;

130 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
77 -> 130 [arrowhead = none ];
130 -> 79;

131 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
84 -> 131 [arrowhead = none ];
131 -> 88;

132 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
88 -> 132 [arrowhead = none ];
132 -> 92;

133 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
88 -> 133 [arrowhead = none ];
133 -> 93;

134 [label="EVAL with clause\nappend([], X130, X130).\nand substitutionX123 -> [],\nT86 -> T93,\nX130 -> T93,\nT85 -> T93", fontsize=14, style = filled, fillcolor = lightblue];
92 -> 134 [arrowhead = none ];
134 -> 94;

135 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
92 -> 135 [arrowhead = none ];
135 -> 95;

136 [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];
93 -> 136 [arrowhead = none ];
136 -> 97;

137 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
93 -> 137 [arrowhead = none ];
137 -> 98;

138 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
94 -> 138 [arrowhead = none ];
138 -> 96;

139 [label="INSTANCE with matching:\nX123 -> X148\nT86 -> T103\nT85 -> T102", fontsize=14, style = filled, fillcolor = lightgrey];
97 -> 139 [arrowhead = none , style=dashed];
139 -> 84[style=dashed];

}

(2) Obligation:

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

sublistA_in_ag([], T16) → sublistA_out_ag([], T16)
sublistA_in_ag(T7, .(T29, T30)) → U1_ag(T7, T29, T30, pB_in_aagaag(X56, X57, T30, X9, T7, T29))
pB_in_aagaag(T38, T39, T30, X9, T7, T29) → U5_aagaag(T38, T39, T30, X9, T7, T29, appendC_in_aag(T38, T39, T30))
appendC_in_aag([], T45, T45) → appendC_out_aag([], T45, T45)
appendC_in_aag(.(T50, X93), X94, .(T50, T51)) → U2_aag(T50, X93, X94, T51, appendC_in_aag(X93, X94, T51))
U2_aag(T50, X93, X94, T51, appendC_out_aag(X93, X94, T51)) → appendC_out_aag(.(T50, X93), X94, .(T50, T51))
U5_aagaag(T38, T39, T30, X9, T7, T29, appendC_out_aag(T38, T39, T30)) → U6_aagaag(T38, T39, T30, X9, T7, T29, appendE_in_aagg(X9, T7, T29, T38))
appendE_in_aagg([], .(T72, T73), T72, T73) → appendE_out_aagg([], .(T72, T73), T72, T73)
appendE_in_aagg(.(T84, X123), T86, T84, T85) → U4_aagg(T84, X123, T86, T85, appendD_in_aag(X123, T86, T85))
appendD_in_aag([], T93, T93) → appendD_out_aag([], T93, T93)
appendD_in_aag(.(T101, X148), T103, .(T101, T102)) → U3_aag(T101, X148, T103, T102, appendD_in_aag(X148, T103, T102))
U3_aag(T101, X148, T103, T102, appendD_out_aag(X148, T103, T102)) → appendD_out_aag(.(T101, X148), T103, .(T101, T102))
U4_aagg(T84, X123, T86, T85, appendD_out_aag(X123, T86, T85)) → appendE_out_aagg(.(T84, X123), T86, T84, T85)
U6_aagaag(T38, T39, T30, X9, T7, T29, appendE_out_aagg(X9, T7, T29, T38)) → pB_out_aagaag(T38, T39, T30, X9, T7, T29)
U1_ag(T7, T29, T30, pB_out_aagaag(X56, X57, T30, X9, T7, T29)) → sublistA_out_ag(T7, .(T29, T30))

The argument filtering Pi contains the following mapping:
sublistA_in_ag(x1, x2)  =  sublistA_in_ag(x2)
sublistA_out_ag(x1, x2)  =  sublistA_out_ag(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x2, x3, x4)
pB_in_aagaag(x1, x2, x3, x4, x5, x6)  =  pB_in_aagaag(x3, x6)
U5_aagaag(x1, x2, x3, x4, x5, x6, x7)  =  U5_aagaag(x3, 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_aagaag(x1, x2, x3, x4, x5, x6, x7)  =  U6_aagaag(x1, x2, x3, x6, x7)
appendE_in_aagg(x1, x2, x3, x4)  =  appendE_in_aagg(x3, x4)
appendE_out_aagg(x1, x2, x3, x4)  =  appendE_out_aagg(x1, x2, x3, x4)
U4_aagg(x1, x2, x3, x4, x5)  =  U4_aagg(x1, x4, x5)
appendD_in_aag(x1, x2, x3)  =  appendD_in_aag(x3)
appendD_out_aag(x1, x2, x3)  =  appendD_out_aag(x1, x2, x3)
U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x1, x4, x5)
pB_out_aagaag(x1, x2, x3, x4, x5, x6)  =  pB_out_aagaag(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_AG(T7, .(T29, T30)) → U1_AG(T7, T29, T30, pB_in_aagaag(X56, X57, T30, X9, T7, T29))
SUBLISTA_IN_AG(T7, .(T29, T30)) → PB_IN_AAGAAG(X56, X57, T30, X9, T7, T29)
PB_IN_AAGAAG(T38, T39, T30, X9, T7, T29) → U5_AAGAAG(T38, T39, T30, X9, T7, T29, appendC_in_aag(T38, T39, T30))
PB_IN_AAGAAG(T38, T39, T30, X9, T7, T29) → APPENDC_IN_AAG(T38, T39, T30)
APPENDC_IN_AAG(.(T50, X93), X94, .(T50, T51)) → U2_AAG(T50, X93, X94, T51, appendC_in_aag(X93, X94, T51))
APPENDC_IN_AAG(.(T50, X93), X94, .(T50, T51)) → APPENDC_IN_AAG(X93, X94, T51)
U5_AAGAAG(T38, T39, T30, X9, T7, T29, appendC_out_aag(T38, T39, T30)) → U6_AAGAAG(T38, T39, T30, X9, T7, T29, appendE_in_aagg(X9, T7, T29, T38))
U5_AAGAAG(T38, T39, T30, X9, T7, T29, appendC_out_aag(T38, T39, T30)) → APPENDE_IN_AAGG(X9, T7, T29, T38)
APPENDE_IN_AAGG(.(T84, X123), T86, T84, T85) → U4_AAGG(T84, X123, T86, T85, appendD_in_aag(X123, T86, T85))
APPENDE_IN_AAGG(.(T84, X123), T86, T84, T85) → APPENDD_IN_AAG(X123, T86, T85)
APPENDD_IN_AAG(.(T101, X148), T103, .(T101, T102)) → U3_AAG(T101, X148, T103, T102, appendD_in_aag(X148, T103, T102))
APPENDD_IN_AAG(.(T101, X148), T103, .(T101, T102)) → APPENDD_IN_AAG(X148, T103, T102)

The TRS R consists of the following rules:

sublistA_in_ag([], T16) → sublistA_out_ag([], T16)
sublistA_in_ag(T7, .(T29, T30)) → U1_ag(T7, T29, T30, pB_in_aagaag(X56, X57, T30, X9, T7, T29))
pB_in_aagaag(T38, T39, T30, X9, T7, T29) → U5_aagaag(T38, T39, T30, X9, T7, T29, appendC_in_aag(T38, T39, T30))
appendC_in_aag([], T45, T45) → appendC_out_aag([], T45, T45)
appendC_in_aag(.(T50, X93), X94, .(T50, T51)) → U2_aag(T50, X93, X94, T51, appendC_in_aag(X93, X94, T51))
U2_aag(T50, X93, X94, T51, appendC_out_aag(X93, X94, T51)) → appendC_out_aag(.(T50, X93), X94, .(T50, T51))
U5_aagaag(T38, T39, T30, X9, T7, T29, appendC_out_aag(T38, T39, T30)) → U6_aagaag(T38, T39, T30, X9, T7, T29, appendE_in_aagg(X9, T7, T29, T38))
appendE_in_aagg([], .(T72, T73), T72, T73) → appendE_out_aagg([], .(T72, T73), T72, T73)
appendE_in_aagg(.(T84, X123), T86, T84, T85) → U4_aagg(T84, X123, T86, T85, appendD_in_aag(X123, T86, T85))
appendD_in_aag([], T93, T93) → appendD_out_aag([], T93, T93)
appendD_in_aag(.(T101, X148), T103, .(T101, T102)) → U3_aag(T101, X148, T103, T102, appendD_in_aag(X148, T103, T102))
U3_aag(T101, X148, T103, T102, appendD_out_aag(X148, T103, T102)) → appendD_out_aag(.(T101, X148), T103, .(T101, T102))
U4_aagg(T84, X123, T86, T85, appendD_out_aag(X123, T86, T85)) → appendE_out_aagg(.(T84, X123), T86, T84, T85)
U6_aagaag(T38, T39, T30, X9, T7, T29, appendE_out_aagg(X9, T7, T29, T38)) → pB_out_aagaag(T38, T39, T30, X9, T7, T29)
U1_ag(T7, T29, T30, pB_out_aagaag(X56, X57, T30, X9, T7, T29)) → sublistA_out_ag(T7, .(T29, T30))

The argument filtering Pi contains the following mapping:
sublistA_in_ag(x1, x2)  =  sublistA_in_ag(x2)
sublistA_out_ag(x1, x2)  =  sublistA_out_ag(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x2, x3, x4)
pB_in_aagaag(x1, x2, x3, x4, x5, x6)  =  pB_in_aagaag(x3, x6)
U5_aagaag(x1, x2, x3, x4, x5, x6, x7)  =  U5_aagaag(x3, 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_aagaag(x1, x2, x3, x4, x5, x6, x7)  =  U6_aagaag(x1, x2, x3, x6, x7)
appendE_in_aagg(x1, x2, x3, x4)  =  appendE_in_aagg(x3, x4)
appendE_out_aagg(x1, x2, x3, x4)  =  appendE_out_aagg(x1, x2, x3, x4)
U4_aagg(x1, x2, x3, x4, x5)  =  U4_aagg(x1, x4, x5)
appendD_in_aag(x1, x2, x3)  =  appendD_in_aag(x3)
appendD_out_aag(x1, x2, x3)  =  appendD_out_aag(x1, x2, x3)
U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x1, x4, x5)
pB_out_aagaag(x1, x2, x3, x4, x5, x6)  =  pB_out_aagaag(x1, x2, x3, x4, x5, x6)
SUBLISTA_IN_AG(x1, x2)  =  SUBLISTA_IN_AG(x2)
U1_AG(x1, x2, x3, x4)  =  U1_AG(x2, x3, x4)
PB_IN_AAGAAG(x1, x2, x3, x4, x5, x6)  =  PB_IN_AAGAAG(x3, x6)
U5_AAGAAG(x1, x2, x3, x4, x5, x6, x7)  =  U5_AAGAAG(x3, 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_AAGAAG(x1, x2, x3, x4, x5, x6, x7)  =  U6_AAGAAG(x1, x2, x3, x6, x7)
APPENDE_IN_AAGG(x1, x2, x3, x4)  =  APPENDE_IN_AAGG(x3, x4)
U4_AAGG(x1, x2, x3, x4, x5)  =  U4_AAGG(x1, x4, x5)
APPENDD_IN_AAG(x1, x2, x3)  =  APPENDD_IN_AAG(x3)
U3_AAG(x1, x2, x3, x4, x5)  =  U3_AAG(x1, 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_AG(T7, .(T29, T30)) → U1_AG(T7, T29, T30, pB_in_aagaag(X56, X57, T30, X9, T7, T29))
SUBLISTA_IN_AG(T7, .(T29, T30)) → PB_IN_AAGAAG(X56, X57, T30, X9, T7, T29)
PB_IN_AAGAAG(T38, T39, T30, X9, T7, T29) → U5_AAGAAG(T38, T39, T30, X9, T7, T29, appendC_in_aag(T38, T39, T30))
PB_IN_AAGAAG(T38, T39, T30, X9, T7, T29) → APPENDC_IN_AAG(T38, T39, T30)
APPENDC_IN_AAG(.(T50, X93), X94, .(T50, T51)) → U2_AAG(T50, X93, X94, T51, appendC_in_aag(X93, X94, T51))
APPENDC_IN_AAG(.(T50, X93), X94, .(T50, T51)) → APPENDC_IN_AAG(X93, X94, T51)
U5_AAGAAG(T38, T39, T30, X9, T7, T29, appendC_out_aag(T38, T39, T30)) → U6_AAGAAG(T38, T39, T30, X9, T7, T29, appendE_in_aagg(X9, T7, T29, T38))
U5_AAGAAG(T38, T39, T30, X9, T7, T29, appendC_out_aag(T38, T39, T30)) → APPENDE_IN_AAGG(X9, T7, T29, T38)
APPENDE_IN_AAGG(.(T84, X123), T86, T84, T85) → U4_AAGG(T84, X123, T86, T85, appendD_in_aag(X123, T86, T85))
APPENDE_IN_AAGG(.(T84, X123), T86, T84, T85) → APPENDD_IN_AAG(X123, T86, T85)
APPENDD_IN_AAG(.(T101, X148), T103, .(T101, T102)) → U3_AAG(T101, X148, T103, T102, appendD_in_aag(X148, T103, T102))
APPENDD_IN_AAG(.(T101, X148), T103, .(T101, T102)) → APPENDD_IN_AAG(X148, T103, T102)

The TRS R consists of the following rules:

sublistA_in_ag([], T16) → sublistA_out_ag([], T16)
sublistA_in_ag(T7, .(T29, T30)) → U1_ag(T7, T29, T30, pB_in_aagaag(X56, X57, T30, X9, T7, T29))
pB_in_aagaag(T38, T39, T30, X9, T7, T29) → U5_aagaag(T38, T39, T30, X9, T7, T29, appendC_in_aag(T38, T39, T30))
appendC_in_aag([], T45, T45) → appendC_out_aag([], T45, T45)
appendC_in_aag(.(T50, X93), X94, .(T50, T51)) → U2_aag(T50, X93, X94, T51, appendC_in_aag(X93, X94, T51))
U2_aag(T50, X93, X94, T51, appendC_out_aag(X93, X94, T51)) → appendC_out_aag(.(T50, X93), X94, .(T50, T51))
U5_aagaag(T38, T39, T30, X9, T7, T29, appendC_out_aag(T38, T39, T30)) → U6_aagaag(T38, T39, T30, X9, T7, T29, appendE_in_aagg(X9, T7, T29, T38))
appendE_in_aagg([], .(T72, T73), T72, T73) → appendE_out_aagg([], .(T72, T73), T72, T73)
appendE_in_aagg(.(T84, X123), T86, T84, T85) → U4_aagg(T84, X123, T86, T85, appendD_in_aag(X123, T86, T85))
appendD_in_aag([], T93, T93) → appendD_out_aag([], T93, T93)
appendD_in_aag(.(T101, X148), T103, .(T101, T102)) → U3_aag(T101, X148, T103, T102, appendD_in_aag(X148, T103, T102))
U3_aag(T101, X148, T103, T102, appendD_out_aag(X148, T103, T102)) → appendD_out_aag(.(T101, X148), T103, .(T101, T102))
U4_aagg(T84, X123, T86, T85, appendD_out_aag(X123, T86, T85)) → appendE_out_aagg(.(T84, X123), T86, T84, T85)
U6_aagaag(T38, T39, T30, X9, T7, T29, appendE_out_aagg(X9, T7, T29, T38)) → pB_out_aagaag(T38, T39, T30, X9, T7, T29)
U1_ag(T7, T29, T30, pB_out_aagaag(X56, X57, T30, X9, T7, T29)) → sublistA_out_ag(T7, .(T29, T30))

The argument filtering Pi contains the following mapping:
sublistA_in_ag(x1, x2)  =  sublistA_in_ag(x2)
sublistA_out_ag(x1, x2)  =  sublistA_out_ag(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x2, x3, x4)
pB_in_aagaag(x1, x2, x3, x4, x5, x6)  =  pB_in_aagaag(x3, x6)
U5_aagaag(x1, x2, x3, x4, x5, x6, x7)  =  U5_aagaag(x3, 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_aagaag(x1, x2, x3, x4, x5, x6, x7)  =  U6_aagaag(x1, x2, x3, x6, x7)
appendE_in_aagg(x1, x2, x3, x4)  =  appendE_in_aagg(x3, x4)
appendE_out_aagg(x1, x2, x3, x4)  =  appendE_out_aagg(x1, x2, x3, x4)
U4_aagg(x1, x2, x3, x4, x5)  =  U4_aagg(x1, x4, x5)
appendD_in_aag(x1, x2, x3)  =  appendD_in_aag(x3)
appendD_out_aag(x1, x2, x3)  =  appendD_out_aag(x1, x2, x3)
U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x1, x4, x5)
pB_out_aagaag(x1, x2, x3, x4, x5, x6)  =  pB_out_aagaag(x1, x2, x3, x4, x5, x6)
SUBLISTA_IN_AG(x1, x2)  =  SUBLISTA_IN_AG(x2)
U1_AG(x1, x2, x3, x4)  =  U1_AG(x2, x3, x4)
PB_IN_AAGAAG(x1, x2, x3, x4, x5, x6)  =  PB_IN_AAGAAG(x3, x6)
U5_AAGAAG(x1, x2, x3, x4, x5, x6, x7)  =  U5_AAGAAG(x3, 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_AAGAAG(x1, x2, x3, x4, x5, x6, x7)  =  U6_AAGAAG(x1, x2, x3, x6, x7)
APPENDE_IN_AAGG(x1, x2, x3, x4)  =  APPENDE_IN_AAGG(x3, x4)
U4_AAGG(x1, x2, x3, x4, x5)  =  U4_AAGG(x1, x4, x5)
APPENDD_IN_AAG(x1, x2, x3)  =  APPENDD_IN_AAG(x3)
U3_AAG(x1, x2, x3, x4, x5)  =  U3_AAG(x1, 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_AAG(.(T101, X148), T103, .(T101, T102)) → APPENDD_IN_AAG(X148, T103, T102)

The TRS R consists of the following rules:

sublistA_in_ag([], T16) → sublistA_out_ag([], T16)
sublistA_in_ag(T7, .(T29, T30)) → U1_ag(T7, T29, T30, pB_in_aagaag(X56, X57, T30, X9, T7, T29))
pB_in_aagaag(T38, T39, T30, X9, T7, T29) → U5_aagaag(T38, T39, T30, X9, T7, T29, appendC_in_aag(T38, T39, T30))
appendC_in_aag([], T45, T45) → appendC_out_aag([], T45, T45)
appendC_in_aag(.(T50, X93), X94, .(T50, T51)) → U2_aag(T50, X93, X94, T51, appendC_in_aag(X93, X94, T51))
U2_aag(T50, X93, X94, T51, appendC_out_aag(X93, X94, T51)) → appendC_out_aag(.(T50, X93), X94, .(T50, T51))
U5_aagaag(T38, T39, T30, X9, T7, T29, appendC_out_aag(T38, T39, T30)) → U6_aagaag(T38, T39, T30, X9, T7, T29, appendE_in_aagg(X9, T7, T29, T38))
appendE_in_aagg([], .(T72, T73), T72, T73) → appendE_out_aagg([], .(T72, T73), T72, T73)
appendE_in_aagg(.(T84, X123), T86, T84, T85) → U4_aagg(T84, X123, T86, T85, appendD_in_aag(X123, T86, T85))
appendD_in_aag([], T93, T93) → appendD_out_aag([], T93, T93)
appendD_in_aag(.(T101, X148), T103, .(T101, T102)) → U3_aag(T101, X148, T103, T102, appendD_in_aag(X148, T103, T102))
U3_aag(T101, X148, T103, T102, appendD_out_aag(X148, T103, T102)) → appendD_out_aag(.(T101, X148), T103, .(T101, T102))
U4_aagg(T84, X123, T86, T85, appendD_out_aag(X123, T86, T85)) → appendE_out_aagg(.(T84, X123), T86, T84, T85)
U6_aagaag(T38, T39, T30, X9, T7, T29, appendE_out_aagg(X9, T7, T29, T38)) → pB_out_aagaag(T38, T39, T30, X9, T7, T29)
U1_ag(T7, T29, T30, pB_out_aagaag(X56, X57, T30, X9, T7, T29)) → sublistA_out_ag(T7, .(T29, T30))

The argument filtering Pi contains the following mapping:
sublistA_in_ag(x1, x2)  =  sublistA_in_ag(x2)
sublistA_out_ag(x1, x2)  =  sublistA_out_ag(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x2, x3, x4)
pB_in_aagaag(x1, x2, x3, x4, x5, x6)  =  pB_in_aagaag(x3, x6)
U5_aagaag(x1, x2, x3, x4, x5, x6, x7)  =  U5_aagaag(x3, 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_aagaag(x1, x2, x3, x4, x5, x6, x7)  =  U6_aagaag(x1, x2, x3, x6, x7)
appendE_in_aagg(x1, x2, x3, x4)  =  appendE_in_aagg(x3, x4)
appendE_out_aagg(x1, x2, x3, x4)  =  appendE_out_aagg(x1, x2, x3, x4)
U4_aagg(x1, x2, x3, x4, x5)  =  U4_aagg(x1, x4, x5)
appendD_in_aag(x1, x2, x3)  =  appendD_in_aag(x3)
appendD_out_aag(x1, x2, x3)  =  appendD_out_aag(x1, x2, x3)
U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x1, x4, x5)
pB_out_aagaag(x1, x2, x3, x4, x5, x6)  =  pB_out_aagaag(x1, x2, x3, x4, x5, x6)
APPENDD_IN_AAG(x1, x2, x3)  =  APPENDD_IN_AAG(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_AAG(.(T101, X148), T103, .(T101, T102)) → APPENDD_IN_AAG(X148, T103, T102)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPENDD_IN_AAG(x1, x2, x3)  =  APPENDD_IN_AAG(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_AAG(.(T101, T102)) → APPENDD_IN_AAG(T102)

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_AAG(.(T101, T102)) → APPENDD_IN_AAG(T102)
  The graph contains the following edges 1 > 1

(13) YES

(14) Obligation:

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

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

The TRS R consists of the following rules:

sublistA_in_ag([], T16) → sublistA_out_ag([], T16)
sublistA_in_ag(T7, .(T29, T30)) → U1_ag(T7, T29, T30, pB_in_aagaag(X56, X57, T30, X9, T7, T29))
pB_in_aagaag(T38, T39, T30, X9, T7, T29) → U5_aagaag(T38, T39, T30, X9, T7, T29, appendC_in_aag(T38, T39, T30))
appendC_in_aag([], T45, T45) → appendC_out_aag([], T45, T45)
appendC_in_aag(.(T50, X93), X94, .(T50, T51)) → U2_aag(T50, X93, X94, T51, appendC_in_aag(X93, X94, T51))
U2_aag(T50, X93, X94, T51, appendC_out_aag(X93, X94, T51)) → appendC_out_aag(.(T50, X93), X94, .(T50, T51))
U5_aagaag(T38, T39, T30, X9, T7, T29, appendC_out_aag(T38, T39, T30)) → U6_aagaag(T38, T39, T30, X9, T7, T29, appendE_in_aagg(X9, T7, T29, T38))
appendE_in_aagg([], .(T72, T73), T72, T73) → appendE_out_aagg([], .(T72, T73), T72, T73)
appendE_in_aagg(.(T84, X123), T86, T84, T85) → U4_aagg(T84, X123, T86, T85, appendD_in_aag(X123, T86, T85))
appendD_in_aag([], T93, T93) → appendD_out_aag([], T93, T93)
appendD_in_aag(.(T101, X148), T103, .(T101, T102)) → U3_aag(T101, X148, T103, T102, appendD_in_aag(X148, T103, T102))
U3_aag(T101, X148, T103, T102, appendD_out_aag(X148, T103, T102)) → appendD_out_aag(.(T101, X148), T103, .(T101, T102))
U4_aagg(T84, X123, T86, T85, appendD_out_aag(X123, T86, T85)) → appendE_out_aagg(.(T84, X123), T86, T84, T85)
U6_aagaag(T38, T39, T30, X9, T7, T29, appendE_out_aagg(X9, T7, T29, T38)) → pB_out_aagaag(T38, T39, T30, X9, T7, T29)
U1_ag(T7, T29, T30, pB_out_aagaag(X56, X57, T30, X9, T7, T29)) → sublistA_out_ag(T7, .(T29, T30))

The argument filtering Pi contains the following mapping:
sublistA_in_ag(x1, x2)  =  sublistA_in_ag(x2)
sublistA_out_ag(x1, x2)  =  sublistA_out_ag(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x2, x3, x4)
pB_in_aagaag(x1, x2, x3, x4, x5, x6)  =  pB_in_aagaag(x3, x6)
U5_aagaag(x1, x2, x3, x4, x5, x6, x7)  =  U5_aagaag(x3, 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_aagaag(x1, x2, x3, x4, x5, x6, x7)  =  U6_aagaag(x1, x2, x3, x6, x7)
appendE_in_aagg(x1, x2, x3, x4)  =  appendE_in_aagg(x3, x4)
appendE_out_aagg(x1, x2, x3, x4)  =  appendE_out_aagg(x1, x2, x3, x4)
U4_aagg(x1, x2, x3, x4, x5)  =  U4_aagg(x1, x4, x5)
appendD_in_aag(x1, x2, x3)  =  appendD_in_aag(x3)
appendD_out_aag(x1, x2, x3)  =  appendD_out_aag(x1, x2, x3)
U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x1, x4, x5)
pB_out_aagaag(x1, x2, x3, x4, x5, x6)  =  pB_out_aagaag(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(.(T50, X93), X94, .(T50, T51)) → APPENDC_IN_AAG(X93, X94, T51)

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(.(T50, T51)) → APPENDC_IN_AAG(T51)

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

(20) YES