Left Termination of the query pattern
app2(a,g,g)
w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

app1(.(X, Xs), Ys, .(X, Zs)) :- app1(Xs, Ys, Zs).
app1([], Ys, Ys).
app2(.(X, Xs), Ys, .(X, Zs)) :- app2(Xs, Ys, Zs).
app2([], Ys, Ys).

Query: app2(a,g,g)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: app2(T1, T2, T3)\n\nT2 is ground\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
6 [label="6: app2(T1, T2, T3), SCOPE: 1, CLAUSE: 2 | app2(T1, T2, T3), SCOPE: 1, CLAUSE: 3\n\nT2 is ground\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: app2(T12, T10, T11) | app2(T1, T10, .(T8, T11)), SCOPE: 1, CLAUSE: 3\n\nT10 is ground\nT8 is ground\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: app2(T1, T2, T3), SCOPE: 1, CLAUSE: 3\n\nT2 is ground\nT3 is ground\nX5 is free\nX6 is free\nX7 is free\nX8 is free\napp2(T1, T2, T3) !=? app2(.(X5, X6), X7, .(X5, X8))", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: app2(T12, T10, T11), SCOPE: 2, CLAUSE: 2 | app2(T12, T10, T11), SCOPE: 2, CLAUSE: 3 | ?_2 | app2(T1, T10, .(T8, T11)), SCOPE: 1, CLAUSE: 3\n\nT10 is ground\nT8 is ground\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: app2(T12, T10, T11), SCOPE: 2, CLAUSE: 2\n\nT10 is ground\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: app2(T12, T10, T11), SCOPE: 2, CLAUSE: 3 | ?_2 | app2(T1, T10, .(T8, T11)), SCOPE: 1, CLAUSE: 3\n\nT10 is ground\nT8 is ground\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: app2(T33, T31, T32)\n\nT31 is ground\nT32 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: app2(T12, T10, T11), SCOPE: 2, CLAUSE: 3\n\nT10 is ground\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ?_2 | app2(T1, T10, .(T8, T11)), SCOPE: 1, CLAUSE: 3\n\nT10 is ground\nT8 is ground\nT11 is ground", 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: app2(T1, T10, .(T8, T11)), SCOPE: 1, CLAUSE: 3\n\nT10 is ground\nT8 is ground\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 52 [arrowhead = none ];
52 -> 6;

53 [label="EVAL with clause\napp2(.(X5, X6), X7, .(X5, X8)) :- app2(X6, X7, X8).\nand substitutionX5 -> T8,\nX6 -> T12,\nT1 -> .(T8, T12),\nT2 -> T10,\nX7 -> T10,\nX8 -> T11,\nT3 -> .(T8, T11),\nT9 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 53 [arrowhead = none ];
53 -> 11;

54 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 54 [arrowhead = none ];
54 -> 12;

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

56 [label="EVAL with clause\napp2([], X42, X42).\nand substitutionT1 -> [],\nT2 -> T53,\nX42 -> T53,\nT3 -> T53", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 56 [arrowhead = none ];
56 -> 49;

57 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 57 [arrowhead = none ];
57 -> 50;

58 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 58 [arrowhead = none ];
58 -> 18;

59 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 59 [arrowhead = none ];
59 -> 20;

60 [label="EVAL with clause\napp2(.(X25, X26), X27, .(X25, X28)) :- app2(X26, X27, X28).\nand substitutionX25 -> T29,\nX26 -> T33,\nT12 -> .(T29, T33),\nT10 -> T31,\nX27 -> T31,\nX28 -> T32,\nT11 -> .(T29, T32),\nT30 -> T33", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 60 [arrowhead = none ];
60 -> 23;

61 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 61 [arrowhead = none ];
61 -> 24;

62 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 62 [arrowhead = none ];
62 -> 34;

63 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 63 [arrowhead = none ];
63 -> 35;

64 [label="INSTANCE with matching:\nT1 -> T33\nT2 -> T31\nT3 -> T32", fontsize=14, style = filled, fillcolor = lightgrey];
23 -> 64 [arrowhead = none , style=dashed];
64 -> 1[style=dashed];

65 [label="EVAL with clause\napp2([], X37, X37).\nand substitutionT12 -> [],\nT10 -> T42,\nX37 -> T42,\nT11 -> T42", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 65 [arrowhead = none ];
65 -> 36;

66 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 66 [arrowhead = none ];
66 -> 37;

67 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 67 [arrowhead = none ];
67 -> 39;

68 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 68 [arrowhead = none ];
68 -> 38;

69 [label="EVAL with clause\napp2([], X40, X40).\nand substitutionT1 -> [],\nT10 -> .(T50, T51),\nX40 -> .(T50, T51),\nT8 -> T50,\nT11 -> T51,\nT49 -> .(T50, T51)", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 69 [arrowhead = none ];
69 -> 46;

70 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 70 [arrowhead = none ];
70 -> 47;

71 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 71 [arrowhead = none ];
71 -> 48;

72 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 72 [arrowhead = none ];
72 -> 51;

}

(2) Obligation:

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

app2A_in_agg(.(T8, .(T29, T33)), T31, .(T8, .(T29, T32))) → U1_agg(T8, T29, T33, T31, T32, app2A_in_agg(T33, T31, T32))
app2A_in_agg(.(T8, []), T42, .(T8, T42)) → app2A_out_agg(.(T8, []), T42, .(T8, T42))
app2A_in_agg([], .(T50, T51), .(T50, T51)) → app2A_out_agg([], .(T50, T51), .(T50, T51))
app2A_in_agg([], T53, T53) → app2A_out_agg([], T53, T53)
U1_agg(T8, T29, T33, T31, T32, app2A_out_agg(T33, T31, T32)) → app2A_out_agg(.(T8, .(T29, T33)), T31, .(T8, .(T29, T32)))

The argument filtering Pi contains the following mapping:
app2A_in_agg(x1, x2, x3)  =  app2A_in_agg(x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_agg(x1, x2, x3, x4, x5, x6)  =  U1_agg(x1, x2, x4, x5, x6)
app2A_out_agg(x1, x2, x3)  =  app2A_out_agg(x1, x2, x3)

(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:

APP2A_IN_AGG(.(T8, .(T29, T33)), T31, .(T8, .(T29, T32))) → U1_AGG(T8, T29, T33, T31, T32, app2A_in_agg(T33, T31, T32))
APP2A_IN_AGG(.(T8, .(T29, T33)), T31, .(T8, .(T29, T32))) → APP2A_IN_AGG(T33, T31, T32)

The TRS R consists of the following rules:

app2A_in_agg(.(T8, .(T29, T33)), T31, .(T8, .(T29, T32))) → U1_agg(T8, T29, T33, T31, T32, app2A_in_agg(T33, T31, T32))
app2A_in_agg(.(T8, []), T42, .(T8, T42)) → app2A_out_agg(.(T8, []), T42, .(T8, T42))
app2A_in_agg([], .(T50, T51), .(T50, T51)) → app2A_out_agg([], .(T50, T51), .(T50, T51))
app2A_in_agg([], T53, T53) → app2A_out_agg([], T53, T53)
U1_agg(T8, T29, T33, T31, T32, app2A_out_agg(T33, T31, T32)) → app2A_out_agg(.(T8, .(T29, T33)), T31, .(T8, .(T29, T32)))

The argument filtering Pi contains the following mapping:
app2A_in_agg(x1, x2, x3)  =  app2A_in_agg(x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_agg(x1, x2, x3, x4, x5, x6)  =  U1_agg(x1, x2, x4, x5, x6)
app2A_out_agg(x1, x2, x3)  =  app2A_out_agg(x1, x2, x3)
APP2A_IN_AGG(x1, x2, x3)  =  APP2A_IN_AGG(x2, x3)
U1_AGG(x1, x2, x3, x4, x5, x6)  =  U1_AGG(x1, x2, x4, x5, x6)

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

(4) Obligation:

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

APP2A_IN_AGG(.(T8, .(T29, T33)), T31, .(T8, .(T29, T32))) → U1_AGG(T8, T29, T33, T31, T32, app2A_in_agg(T33, T31, T32))
APP2A_IN_AGG(.(T8, .(T29, T33)), T31, .(T8, .(T29, T32))) → APP2A_IN_AGG(T33, T31, T32)

The TRS R consists of the following rules:

app2A_in_agg(.(T8, .(T29, T33)), T31, .(T8, .(T29, T32))) → U1_agg(T8, T29, T33, T31, T32, app2A_in_agg(T33, T31, T32))
app2A_in_agg(.(T8, []), T42, .(T8, T42)) → app2A_out_agg(.(T8, []), T42, .(T8, T42))
app2A_in_agg([], .(T50, T51), .(T50, T51)) → app2A_out_agg([], .(T50, T51), .(T50, T51))
app2A_in_agg([], T53, T53) → app2A_out_agg([], T53, T53)
U1_agg(T8, T29, T33, T31, T32, app2A_out_agg(T33, T31, T32)) → app2A_out_agg(.(T8, .(T29, T33)), T31, .(T8, .(T29, T32)))

The argument filtering Pi contains the following mapping:
app2A_in_agg(x1, x2, x3)  =  app2A_in_agg(x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_agg(x1, x2, x3, x4, x5, x6)  =  U1_agg(x1, x2, x4, x5, x6)
app2A_out_agg(x1, x2, x3)  =  app2A_out_agg(x1, x2, x3)
APP2A_IN_AGG(x1, x2, x3)  =  APP2A_IN_AGG(x2, x3)
U1_AGG(x1, x2, x3, x4, x5, x6)  =  U1_AGG(x1, x2, x4, x5, x6)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.

(6) Obligation:

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

APP2A_IN_AGG(.(T8, .(T29, T33)), T31, .(T8, .(T29, T32))) → APP2A_IN_AGG(T33, T31, T32)

The TRS R consists of the following rules:

app2A_in_agg(.(T8, .(T29, T33)), T31, .(T8, .(T29, T32))) → U1_agg(T8, T29, T33, T31, T32, app2A_in_agg(T33, T31, T32))
app2A_in_agg(.(T8, []), T42, .(T8, T42)) → app2A_out_agg(.(T8, []), T42, .(T8, T42))
app2A_in_agg([], .(T50, T51), .(T50, T51)) → app2A_out_agg([], .(T50, T51), .(T50, T51))
app2A_in_agg([], T53, T53) → app2A_out_agg([], T53, T53)
U1_agg(T8, T29, T33, T31, T32, app2A_out_agg(T33, T31, T32)) → app2A_out_agg(.(T8, .(T29, T33)), T31, .(T8, .(T29, T32)))

The argument filtering Pi contains the following mapping:
app2A_in_agg(x1, x2, x3)  =  app2A_in_agg(x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_agg(x1, x2, x3, x4, x5, x6)  =  U1_agg(x1, x2, x4, x5, x6)
app2A_out_agg(x1, x2, x3)  =  app2A_out_agg(x1, x2, x3)
APP2A_IN_AGG(x1, x2, x3)  =  APP2A_IN_AGG(x2, x3)

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

(7) UsableRulesProof (EQUIVALENT transformation)

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

(8) Obligation:

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

APP2A_IN_AGG(.(T8, .(T29, T33)), T31, .(T8, .(T29, T32))) → APP2A_IN_AGG(T33, T31, T32)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP2A_IN_AGG(x1, x2, x3)  =  APP2A_IN_AGG(x2, x3)

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

(9) PiDPToQDPProof (SOUND transformation)

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

(10) Obligation:

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

APP2A_IN_AGG(T31, .(T8, .(T29, T32))) → APP2A_IN_AGG(T31, T32)

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

(11) 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:

• APP2A_IN_AGG(T31, .(T8, .(T29, T32))) → APP2A_IN_AGG(T31, T32)
  The graph contains the following edges 1 >= 1, 2 > 2

(12) YES