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

(0) Obligation:

Clauses:

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

Query: sublist(a,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\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
6 [label="6: sublist(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="B: ','(app(X7, X8, T6), app(T7, X9, X8))\n\nT6 is ground\nX7 is free\nX8 is free\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(app(X7, X8, T6), app(T7, X9, X8)), SCOPE: 2, CLAUSE: 1 | ','(app(X7, X8, T6), app(T7, X9, X8)), SCOPE: 2, CLAUSE: 2\n\nT6 is ground\nX7 is free\nX8 is free\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(app(X7, X8, T6), app(T7, X9, X8)), SCOPE: 2, CLAUSE: 1\n\nT6 is ground\nX7 is free\nX8 is free\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(app(X7, X8, T6), app(T7, X9, X8)), SCOPE: 2, CLAUSE: 2\n\nT6 is ground\nX7 is free\nX8 is free\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="C: app(T7, X9, T16)\n\nT16 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: app(T7, X9, T16), SCOPE: 3, CLAUSE: 1 | app(T7, X9, T16), SCOPE: 3, CLAUSE: 2\n\nT16 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: app(T7, X9, T16), SCOPE: 3, CLAUSE: 1\n\nT16 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: app(T7, X9, T16), SCOPE: 3, CLAUSE: 2\n\nT16 is ground\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];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: app(T33, X56, T32)\n\nT32 is ground\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(app(X80, X81, T41), app(T7, X9, X81))\n\nT41 is ground\nX9 is free\nX80 is free\nX81 is free", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 55 [arrowhead = none ];
55 -> 6;

56 [label="ONLY EVAL with clause\nsublist(X5, X6) :- ','(app(X7, X8, X6), app(X5, X9, X8)).\nand substitutionT1 -> T7,\nX5 -> T7,\nT2 -> T6,\nX6 -> T6,\nT5 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 56 [arrowhead = none ];
56 -> 7;

57 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 57 [arrowhead = none ];
57 -> 9;

58 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 58 [arrowhead = none ];
58 -> 12;

59 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 59 [arrowhead = none ];
59 -> 13;

60 [label="ONLY EVAL with clause\napp([], X26, X26).\nand substitutionX7 -> [],\nX8 -> T16,\nX26 -> T16,\nT6 -> T16,\nX27 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 60 [arrowhead = none ];
60 -> 15;

61 [label="EVAL with clause\napp(.(X75, X76), X77, .(X75, X78)) :- app(X76, X77, X78).\nand substitutionX75 -> T40,\nX76 -> X80,\nX7 -> .(T40, X80),\nX8 -> X81,\nX77 -> X81,\nX79 -> T40,\nX78 -> T41,\nT6 -> .(T40, T41)", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 61 [arrowhead = none ];
61 -> 53;

62 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 62 [arrowhead = none ];
62 -> 54;

63 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 63 [arrowhead = none ];
63 -> 19;

64 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
19 -> 64 [arrowhead = none ];
64 -> 20;

65 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
19 -> 65 [arrowhead = none ];
65 -> 21;

66 [label="EVAL with clause\napp([], X40, X40).\nand substitutionT7 -> [],\nX9 -> T23,\nX40 -> T23,\nT16 -> T23,\nX41 -> T23", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 66 [arrowhead = none ];
66 -> 27;

67 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 67 [arrowhead = none ];
67 -> 28;

68 [label="EVAL with clause\napp(.(X52, X53), X54, .(X52, X55)) :- app(X53, X54, X55).\nand substitutionX52 -> T30,\nX53 -> T33,\nT7 -> .(T30, T33),\nX9 -> X56,\nX54 -> X56,\nX55 -> T32,\nT16 -> .(T30, T32),\nT31 -> T33", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 68 [arrowhead = none ];
68 -> 37;

69 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 69 [arrowhead = none ];
69 -> 38;

70 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 70 [arrowhead = none ];
70 -> 29;

71 [label="INSTANCE with matching:\nT7 -> T33\nX9 -> X56\nT16 -> T32", fontsize=14, style = filled, fillcolor = lightgrey];
37 -> 71 [arrowhead = none , style=dashed];
71 -> 15[style=dashed];

72 [label="INSTANCE with matching:\nX7 -> X80\nX8 -> X81\nT6 -> T41", fontsize=14, style = filled, fillcolor = lightgrey];
53 -> 72 [arrowhead = none , style=dashed];
72 -> 7[style=dashed];

}

(2) Obligation:

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

sublistA_in_ag(T7, T6) → U1_ag(T7, T6, pB_in_aagaa(X7, X8, T6, T7, X9))
pB_in_aagaa([], T16, T16, T7, X9) → U3_aagaa(T16, T7, X9, appC_in_aag(T7, X9, T16))
appC_in_aag([], T23, T23) → appC_out_aag([], T23, T23)
appC_in_aag(.(T30, T33), X56, .(T30, T32)) → U2_aag(T30, T33, X56, T32, appC_in_aag(T33, X56, T32))
U2_aag(T30, T33, X56, T32, appC_out_aag(T33, X56, T32)) → appC_out_aag(.(T30, T33), X56, .(T30, T32))
U3_aagaa(T16, T7, X9, appC_out_aag(T7, X9, T16)) → pB_out_aagaa([], T16, T16, T7, X9)
pB_in_aagaa(.(T40, X80), X81, .(T40, T41), T7, X9) → U4_aagaa(T40, X80, X81, T41, T7, X9, pB_in_aagaa(X80, X81, T41, T7, X9))
U4_aagaa(T40, X80, X81, T41, T7, X9, pB_out_aagaa(X80, X81, T41, T7, X9)) → pB_out_aagaa(.(T40, X80), X81, .(T40, T41), T7, X9)
U1_ag(T7, T6, pB_out_aagaa(X7, X8, T6, T7, X9)) → sublistA_out_ag(T7, T6)

The argument filtering Pi contains the following mapping:
sublistA_in_ag(x1, x2)  =  sublistA_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
pB_in_aagaa(x1, x2, x3, x4, x5)  =  pB_in_aagaa(x3)
U3_aagaa(x1, x2, x3, x4)  =  U3_aagaa(x1, x4)
appC_in_aag(x1, x2, x3)  =  appC_in_aag(x3)
appC_out_aag(x1, x2, x3)  =  appC_out_aag(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x1, x4, x5)
pB_out_aagaa(x1, x2, x3, x4, x5)  =  pB_out_aagaa(x1, x2, x3, x4, x5)
U4_aagaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_aagaa(x1, x4, x7)
sublistA_out_ag(x1, x2)  =  sublistA_out_ag(x1, x2)
[]  =  []

(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, T6) → U1_AG(T7, T6, pB_in_aagaa(X7, X8, T6, T7, X9))
SUBLISTA_IN_AG(T7, T6) → PB_IN_AAGAA(X7, X8, T6, T7, X9)
PB_IN_AAGAA([], T16, T16, T7, X9) → U3_AAGAA(T16, T7, X9, appC_in_aag(T7, X9, T16))
PB_IN_AAGAA([], T16, T16, T7, X9) → APPC_IN_AAG(T7, X9, T16)
APPC_IN_AAG(.(T30, T33), X56, .(T30, T32)) → U2_AAG(T30, T33, X56, T32, appC_in_aag(T33, X56, T32))
APPC_IN_AAG(.(T30, T33), X56, .(T30, T32)) → APPC_IN_AAG(T33, X56, T32)
PB_IN_AAGAA(.(T40, X80), X81, .(T40, T41), T7, X9) → U4_AAGAA(T40, X80, X81, T41, T7, X9, pB_in_aagaa(X80, X81, T41, T7, X9))
PB_IN_AAGAA(.(T40, X80), X81, .(T40, T41), T7, X9) → PB_IN_AAGAA(X80, X81, T41, T7, X9)

The TRS R consists of the following rules:

sublistA_in_ag(T7, T6) → U1_ag(T7, T6, pB_in_aagaa(X7, X8, T6, T7, X9))
pB_in_aagaa([], T16, T16, T7, X9) → U3_aagaa(T16, T7, X9, appC_in_aag(T7, X9, T16))
appC_in_aag([], T23, T23) → appC_out_aag([], T23, T23)
appC_in_aag(.(T30, T33), X56, .(T30, T32)) → U2_aag(T30, T33, X56, T32, appC_in_aag(T33, X56, T32))
U2_aag(T30, T33, X56, T32, appC_out_aag(T33, X56, T32)) → appC_out_aag(.(T30, T33), X56, .(T30, T32))
U3_aagaa(T16, T7, X9, appC_out_aag(T7, X9, T16)) → pB_out_aagaa([], T16, T16, T7, X9)
pB_in_aagaa(.(T40, X80), X81, .(T40, T41), T7, X9) → U4_aagaa(T40, X80, X81, T41, T7, X9, pB_in_aagaa(X80, X81, T41, T7, X9))
U4_aagaa(T40, X80, X81, T41, T7, X9, pB_out_aagaa(X80, X81, T41, T7, X9)) → pB_out_aagaa(.(T40, X80), X81, .(T40, T41), T7, X9)
U1_ag(T7, T6, pB_out_aagaa(X7, X8, T6, T7, X9)) → sublistA_out_ag(T7, T6)

The argument filtering Pi contains the following mapping:
sublistA_in_ag(x1, x2)  =  sublistA_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
pB_in_aagaa(x1, x2, x3, x4, x5)  =  pB_in_aagaa(x3)
U3_aagaa(x1, x2, x3, x4)  =  U3_aagaa(x1, x4)
appC_in_aag(x1, x2, x3)  =  appC_in_aag(x3)
appC_out_aag(x1, x2, x3)  =  appC_out_aag(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x1, x4, x5)
pB_out_aagaa(x1, x2, x3, x4, x5)  =  pB_out_aagaa(x1, x2, x3, x4, x5)
U4_aagaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_aagaa(x1, x4, x7)
sublistA_out_ag(x1, x2)  =  sublistA_out_ag(x1, x2)
[]  =  []
SUBLISTA_IN_AG(x1, x2)  =  SUBLISTA_IN_AG(x2)
U1_AG(x1, x2, x3)  =  U1_AG(x2, x3)
PB_IN_AAGAA(x1, x2, x3, x4, x5)  =  PB_IN_AAGAA(x3)
U3_AAGAA(x1, x2, x3, x4)  =  U3_AAGAA(x1, x4)
APPC_IN_AAG(x1, x2, x3)  =  APPC_IN_AAG(x3)
U2_AAG(x1, x2, x3, x4, x5)  =  U2_AAG(x1, x4, x5)
U4_AAGAA(x1, x2, x3, x4, x5, x6, x7)  =  U4_AAGAA(x1, x4, x7)

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, T6) → U1_AG(T7, T6, pB_in_aagaa(X7, X8, T6, T7, X9))
SUBLISTA_IN_AG(T7, T6) → PB_IN_AAGAA(X7, X8, T6, T7, X9)
PB_IN_AAGAA([], T16, T16, T7, X9) → U3_AAGAA(T16, T7, X9, appC_in_aag(T7, X9, T16))
PB_IN_AAGAA([], T16, T16, T7, X9) → APPC_IN_AAG(T7, X9, T16)
APPC_IN_AAG(.(T30, T33), X56, .(T30, T32)) → U2_AAG(T30, T33, X56, T32, appC_in_aag(T33, X56, T32))
APPC_IN_AAG(.(T30, T33), X56, .(T30, T32)) → APPC_IN_AAG(T33, X56, T32)
PB_IN_AAGAA(.(T40, X80), X81, .(T40, T41), T7, X9) → U4_AAGAA(T40, X80, X81, T41, T7, X9, pB_in_aagaa(X80, X81, T41, T7, X9))
PB_IN_AAGAA(.(T40, X80), X81, .(T40, T41), T7, X9) → PB_IN_AAGAA(X80, X81, T41, T7, X9)

The TRS R consists of the following rules:

sublistA_in_ag(T7, T6) → U1_ag(T7, T6, pB_in_aagaa(X7, X8, T6, T7, X9))
pB_in_aagaa([], T16, T16, T7, X9) → U3_aagaa(T16, T7, X9, appC_in_aag(T7, X9, T16))
appC_in_aag([], T23, T23) → appC_out_aag([], T23, T23)
appC_in_aag(.(T30, T33), X56, .(T30, T32)) → U2_aag(T30, T33, X56, T32, appC_in_aag(T33, X56, T32))
U2_aag(T30, T33, X56, T32, appC_out_aag(T33, X56, T32)) → appC_out_aag(.(T30, T33), X56, .(T30, T32))
U3_aagaa(T16, T7, X9, appC_out_aag(T7, X9, T16)) → pB_out_aagaa([], T16, T16, T7, X9)
pB_in_aagaa(.(T40, X80), X81, .(T40, T41), T7, X9) → U4_aagaa(T40, X80, X81, T41, T7, X9, pB_in_aagaa(X80, X81, T41, T7, X9))
U4_aagaa(T40, X80, X81, T41, T7, X9, pB_out_aagaa(X80, X81, T41, T7, X9)) → pB_out_aagaa(.(T40, X80), X81, .(T40, T41), T7, X9)
U1_ag(T7, T6, pB_out_aagaa(X7, X8, T6, T7, X9)) → sublistA_out_ag(T7, T6)

The argument filtering Pi contains the following mapping:
sublistA_in_ag(x1, x2)  =  sublistA_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
pB_in_aagaa(x1, x2, x3, x4, x5)  =  pB_in_aagaa(x3)
U3_aagaa(x1, x2, x3, x4)  =  U3_aagaa(x1, x4)
appC_in_aag(x1, x2, x3)  =  appC_in_aag(x3)
appC_out_aag(x1, x2, x3)  =  appC_out_aag(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x1, x4, x5)
pB_out_aagaa(x1, x2, x3, x4, x5)  =  pB_out_aagaa(x1, x2, x3, x4, x5)
U4_aagaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_aagaa(x1, x4, x7)
sublistA_out_ag(x1, x2)  =  sublistA_out_ag(x1, x2)
[]  =  []
SUBLISTA_IN_AG(x1, x2)  =  SUBLISTA_IN_AG(x2)
U1_AG(x1, x2, x3)  =  U1_AG(x2, x3)
PB_IN_AAGAA(x1, x2, x3, x4, x5)  =  PB_IN_AAGAA(x3)
U3_AAGAA(x1, x2, x3, x4)  =  U3_AAGAA(x1, x4)
APPC_IN_AAG(x1, x2, x3)  =  APPC_IN_AAG(x3)
U2_AAG(x1, x2, x3, x4, x5)  =  U2_AAG(x1, x4, x5)
U4_AAGAA(x1, x2, x3, x4, x5, x6, x7)  =  U4_AAGAA(x1, x4, x7)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

APPC_IN_AAG(.(T30, T33), X56, .(T30, T32)) → APPC_IN_AAG(T33, X56, T32)

The TRS R consists of the following rules:

sublistA_in_ag(T7, T6) → U1_ag(T7, T6, pB_in_aagaa(X7, X8, T6, T7, X9))
pB_in_aagaa([], T16, T16, T7, X9) → U3_aagaa(T16, T7, X9, appC_in_aag(T7, X9, T16))
appC_in_aag([], T23, T23) → appC_out_aag([], T23, T23)
appC_in_aag(.(T30, T33), X56, .(T30, T32)) → U2_aag(T30, T33, X56, T32, appC_in_aag(T33, X56, T32))
U2_aag(T30, T33, X56, T32, appC_out_aag(T33, X56, T32)) → appC_out_aag(.(T30, T33), X56, .(T30, T32))
U3_aagaa(T16, T7, X9, appC_out_aag(T7, X9, T16)) → pB_out_aagaa([], T16, T16, T7, X9)
pB_in_aagaa(.(T40, X80), X81, .(T40, T41), T7, X9) → U4_aagaa(T40, X80, X81, T41, T7, X9, pB_in_aagaa(X80, X81, T41, T7, X9))
U4_aagaa(T40, X80, X81, T41, T7, X9, pB_out_aagaa(X80, X81, T41, T7, X9)) → pB_out_aagaa(.(T40, X80), X81, .(T40, T41), T7, X9)
U1_ag(T7, T6, pB_out_aagaa(X7, X8, T6, T7, X9)) → sublistA_out_ag(T7, T6)

The argument filtering Pi contains the following mapping:
sublistA_in_ag(x1, x2)  =  sublistA_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
pB_in_aagaa(x1, x2, x3, x4, x5)  =  pB_in_aagaa(x3)
U3_aagaa(x1, x2, x3, x4)  =  U3_aagaa(x1, x4)
appC_in_aag(x1, x2, x3)  =  appC_in_aag(x3)
appC_out_aag(x1, x2, x3)  =  appC_out_aag(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x1, x4, x5)
pB_out_aagaa(x1, x2, x3, x4, x5)  =  pB_out_aagaa(x1, x2, x3, x4, x5)
U4_aagaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_aagaa(x1, x4, x7)
sublistA_out_ag(x1, x2)  =  sublistA_out_ag(x1, x2)
[]  =  []
APPC_IN_AAG(x1, x2, x3)  =  APPC_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:

APPC_IN_AAG(.(T30, T33), X56, .(T30, T32)) → APPC_IN_AAG(T33, X56, T32)

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

APPC_IN_AAG(.(T30, T32)) → APPC_IN_AAG(T32)

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:

• APPC_IN_AAG(.(T30, T32)) → APPC_IN_AAG(T32)
  The graph contains the following edges 1 > 1

(13) YES

(14) Obligation:

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

PB_IN_AAGAA(.(T40, X80), X81, .(T40, T41), T7, X9) → PB_IN_AAGAA(X80, X81, T41, T7, X9)

The TRS R consists of the following rules:

sublistA_in_ag(T7, T6) → U1_ag(T7, T6, pB_in_aagaa(X7, X8, T6, T7, X9))
pB_in_aagaa([], T16, T16, T7, X9) → U3_aagaa(T16, T7, X9, appC_in_aag(T7, X9, T16))
appC_in_aag([], T23, T23) → appC_out_aag([], T23, T23)
appC_in_aag(.(T30, T33), X56, .(T30, T32)) → U2_aag(T30, T33, X56, T32, appC_in_aag(T33, X56, T32))
U2_aag(T30, T33, X56, T32, appC_out_aag(T33, X56, T32)) → appC_out_aag(.(T30, T33), X56, .(T30, T32))
U3_aagaa(T16, T7, X9, appC_out_aag(T7, X9, T16)) → pB_out_aagaa([], T16, T16, T7, X9)
pB_in_aagaa(.(T40, X80), X81, .(T40, T41), T7, X9) → U4_aagaa(T40, X80, X81, T41, T7, X9, pB_in_aagaa(X80, X81, T41, T7, X9))
U4_aagaa(T40, X80, X81, T41, T7, X9, pB_out_aagaa(X80, X81, T41, T7, X9)) → pB_out_aagaa(.(T40, X80), X81, .(T40, T41), T7, X9)
U1_ag(T7, T6, pB_out_aagaa(X7, X8, T6, T7, X9)) → sublistA_out_ag(T7, T6)

The argument filtering Pi contains the following mapping:
sublistA_in_ag(x1, x2)  =  sublistA_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
pB_in_aagaa(x1, x2, x3, x4, x5)  =  pB_in_aagaa(x3)
U3_aagaa(x1, x2, x3, x4)  =  U3_aagaa(x1, x4)
appC_in_aag(x1, x2, x3)  =  appC_in_aag(x3)
appC_out_aag(x1, x2, x3)  =  appC_out_aag(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x1, x4, x5)
pB_out_aagaa(x1, x2, x3, x4, x5)  =  pB_out_aagaa(x1, x2, x3, x4, x5)
U4_aagaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_aagaa(x1, x4, x7)
sublistA_out_ag(x1, x2)  =  sublistA_out_ag(x1, x2)
[]  =  []
PB_IN_AAGAA(x1, x2, x3, x4, x5)  =  PB_IN_AAGAA(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:

PB_IN_AAGAA(.(T40, X80), X81, .(T40, T41), T7, X9) → PB_IN_AAGAA(X80, X81, T41, T7, X9)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
PB_IN_AAGAA(x1, x2, x3, x4, x5)  =  PB_IN_AAGAA(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:

PB_IN_AAGAA(.(T40, T41)) → PB_IN_AAGAA(T41)

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:

• PB_IN_AAGAA(.(T40, T41)) → PB_IN_AAGAA(T41)
  The graph contains the following edges 1 > 1

(20) YES