Left Termination of the query pattern star_in_2(g, g) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

star(X1, []).
star(.(X, U), .(X, W)) :- ','(app(U, V, W), star(.(X, U), W)).
app([], L, L).
app(.(X, L), M, .(X, N)) :- app(L, M, N).

Queries:

star(g,g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: star(T1, T2)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: star(T1, T2), SCOPE: 1, CLAUSE: 0 | star(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: star(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: star(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: \n\nX4 is free", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(app(T5, X11, T6), star(.(T4, T5), T6))\n\nT4 is ground\nT5 is ground\nT6 is ground\nX11 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\nX8 is free\nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: app(T5, X11, T6)\n\nT5 is ground\nT6 is ground\nX11 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: star(.(T4, T5), T6)\n\nT4 is ground\nT5 is ground\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: app(T5, X11, T6), SCOPE: 2, CLAUSE: 2 | app(T5, X11, T6), SCOPE: 2, CLAUSE: 3\n\nT5 is ground\nT6 is ground\nX11 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: app(T5, X11, T6), SCOPE: 2, CLAUSE: 2\n\nT5 is ground\nT6 is ground\nX11 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: app(T5, X11, T6), SCOPE: 2, CLAUSE: 3\n\nT5 is ground\nT6 is ground\nX11 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\nX11 is free\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: app(T10, X24, T11)\n\nT10 is ground\nT11 is ground\nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\nX11 is free\nX20 is free; \nX21 is free; \nX22 is free; \nX23 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: star(.(T4, T5), T6), SCOPE: 3, CLAUSE: 0 | star(.(T4, T5), T6), SCOPE: 3, CLAUSE: 1\n\nT4 is ground\nT5 is ground\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: star(.(T4, T5), T6), SCOPE: 3, CLAUSE: 0\n\nT4 is ground\nT5 is ground\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: star(.(T4, T5), T6), SCOPE: 3, CLAUSE: 1\n\nT4 is ground\nT5 is ground\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(app(T15, X34, T16), star(.(T14, T15), T16))\n\nT14 is ground\nT15 is ground\nT16 is ground\nX34 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\nX31 is free\nX32 is free; \nX33 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 28 [arrowhead = none ];
28 -> 2;

29 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
2 -> 29 [arrowhead = none ];
29 -> 3;

30 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
2 -> 30 [arrowhead = none ];
30 -> 4;

31 [label="EVAL with clause\nstar(X4, []).\nand substitution\nT1 -> T3\nX4 -> T3\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 31 [arrowhead = none ];
31 -> 5;

32 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 32 [arrowhead = none ];
32 -> 6;

33 [label="EVAL with clause\nstar(.(X8, X9), .(X8, X10)) :- ','(app(X9, X11, X10), star(.(X8, X9), X10)).\nand substitution\nX8 -> T4\nX9 -> T5\nT1 -> .(T4, T5)\nX10 -> T6\nT2 -> .(T4, T6)", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 33 [arrowhead = none ];
33 -> 8;

34 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 34 [arrowhead = none ];
34 -> 9;

35 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 35 [arrowhead = none ];
35 -> 7;

36 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
8 -> 36 [arrowhead = none ];
36 -> 10;

37 [label="SPLIT 2\nnew knowledge:\nT7 is ground\nreplacements:\nX11 -> T7", fontsize=14, style = filled, fillcolor = violet];
8 -> 37 [arrowhead = none ];
37 -> 11;

38 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 38 [arrowhead = none ];
38 -> 12;

39 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 39 [arrowhead = none ];
39 -> 20;

40 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 40 [arrowhead = none ];
40 -> 13;

41 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 41 [arrowhead = none ];
41 -> 14;

42 [label="EVAL with clause\napp([], X14, X14).\nand substitution\nT5 -> []\nX11 -> T8\nX14 -> T8\nT6 -> T8\nX15 -> T8", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 42 [arrowhead = none ];
42 -> 15;

43 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 43 [arrowhead = none ];
43 -> 16;

44 [label="EVAL with clause\napp(.(X20, X21), X22, .(X20, X23)) :- app(X21, X22, X23).\nand substitution\nX20 -> T9\nX21 -> T10\nT5 -> .(T9, T10)\nX11 -> X24\nX22 -> X24\nX23 -> T11\nT6 -> .(T9, T11)", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 44 [arrowhead = none ];
44 -> 18;

45 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 45 [arrowhead = none ];
45 -> 19;

46 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 46 [arrowhead = none ];
46 -> 17;

47 [label="INSTANCE with matching:\nT5 -> T10\nX11 -> X24\nT6 -> T11", fontsize=14, style = filled, fillcolor = lightgrey];
18 -> 47 [arrowhead = none , style=dashed];
47 -> 10[style=dashed];

48 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 48 [arrowhead = none ];
48 -> 21;

49 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 49 [arrowhead = none ];
49 -> 22;

50 [label="EVAL with clause\nstar(X27, []).\nand substitution\nT4 -> T12\nT5 -> T13\nX27 -> .(T12, T13)\nT6 -> []", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 50 [arrowhead = none ];
50 -> 23;

51 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 51 [arrowhead = none ];
51 -> 24;

52 [label="EVAL with clause\nstar(.(X31, X32), .(X31, X33)) :- ','(app(X32, X34, X33), star(.(X31, X32), X33)).\nand substitution\nT4 -> T14\nX31 -> T14\nT5 -> T15\nX32 -> T15\nX33 -> T16\nT6 -> .(T14, T16)", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 52 [arrowhead = none ];
52 -> 26;

53 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 53 [arrowhead = none ];
53 -> 27;

54 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 54 [arrowhead = none ];
54 -> 25;

55 [label="INSTANCE with matching:\nT5 -> T15\nX11 -> X34\nT6 -> T16\nT4 -> T14", fontsize=14, style = filled, fillcolor = lightgrey];
26 -> 55 [arrowhead = none , style=dashed];
55 -> 8[style=dashed];

}

(2) Obligation:

Clauses:

app10([], T8, T8).
app10(.(T9, T10), X24, .(T9, T11)) :- app10(T10, X24, T11).
p8(T5, X11, T6, T4) :- app10(T5, X11, T6).
p8(T13, T7, [], T12) :- app10(T13, T7, []).
p8(T15, T7, .(T14, T16), T14) :- ','(app10(T15, T7, .(T14, T16)), p8(T15, X34, T16, T14)).
star1(T3, []).
star1(.(T4, T5), .(T4, T6)) :- app10(T5, X11, T6).
star1(.(T12, T13), .(T12, [])) :- app10(T13, T7, []).
star1(.(T14, T15), .(T14, .(T14, T16))) :- ','(app10(T15, T7, .(T14, T16)), p8(T15, X34, T16, T14)).

Queries:

star1(g,g).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
star1_in: (b,b)
app10_in: (b,f,b)
p8_in: (b,f,b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

star1_in_gg(T3, []) → star1_out_gg(T3, [])
star1_in_gg(.(T4, T5), .(T4, T6)) → U6_gg(T4, T5, T6, app10_in_gag(T5, X11, T6))
app10_in_gag([], T8, T8) → app10_out_gag([], T8, T8)
app10_in_gag(.(T9, T10), X24, .(T9, T11)) → U1_gag(T9, T10, X24, T11, app10_in_gag(T10, X24, T11))
U1_gag(T9, T10, X24, T11, app10_out_gag(T10, X24, T11)) → app10_out_gag(.(T9, T10), X24, .(T9, T11))
U6_gg(T4, T5, T6, app10_out_gag(T5, X11, T6)) → star1_out_gg(.(T4, T5), .(T4, T6))
star1_in_gg(.(T12, T13), .(T12, [])) → U7_gg(T12, T13, app10_in_gag(T13, T7, []))
U7_gg(T12, T13, app10_out_gag(T13, T7, [])) → star1_out_gg(.(T12, T13), .(T12, []))
star1_in_gg(.(T14, T15), .(T14, .(T14, T16))) → U8_gg(T14, T15, T16, app10_in_gag(T15, T7, .(T14, T16)))
U8_gg(T14, T15, T16, app10_out_gag(T15, T7, .(T14, T16))) → U9_gg(T14, T15, T16, p8_in_gagg(T15, X34, T16, T14))
p8_in_gagg(T5, X11, T6, T4) → U2_gagg(T5, X11, T6, T4, app10_in_gag(T5, X11, T6))
U2_gagg(T5, X11, T6, T4, app10_out_gag(T5, X11, T6)) → p8_out_gagg(T5, X11, T6, T4)
p8_in_gagg(T13, T7, [], T12) → U3_gagg(T13, T7, T12, app10_in_gag(T13, T7, []))
U3_gagg(T13, T7, T12, app10_out_gag(T13, T7, [])) → p8_out_gagg(T13, T7, [], T12)
p8_in_gagg(T15, T7, .(T14, T16), T14) → U4_gagg(T15, T7, T14, T16, app10_in_gag(T15, T7, .(T14, T16)))
U4_gagg(T15, T7, T14, T16, app10_out_gag(T15, T7, .(T14, T16))) → U5_gagg(T15, T7, T14, T16, p8_in_gagg(T15, X34, T16, T14))
U5_gagg(T15, T7, T14, T16, p8_out_gagg(T15, X34, T16, T14)) → p8_out_gagg(T15, T7, .(T14, T16), T14)
U9_gg(T14, T15, T16, p8_out_gagg(T15, X34, T16, T14)) → star1_out_gg(.(T14, T15), .(T14, .(T14, T16)))

The argument filtering Pi contains the following mapping:
star1_in_gg(x1, x2)  =  star1_in_gg(x1, x2)
[]  =  []
star1_out_gg(x1, x2)  =  star1_out_gg
.(x1, x2)  =  .(x1, x2)
U6_gg(x1, x2, x3, x4)  =  U6_gg(x4)
app10_in_gag(x1, x2, x3)  =  app10_in_gag(x1, x3)
app10_out_gag(x1, x2, x3)  =  app10_out_gag(x2)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U8_gg(x1, x2, x3, x4)  =  U8_gg(x1, x2, x3, x4)
U9_gg(x1, x2, x3, x4)  =  U9_gg(x4)
p8_in_gagg(x1, x2, x3, x4)  =  p8_in_gagg(x1, x3, x4)
U2_gagg(x1, x2, x3, x4, x5)  =  U2_gagg(x5)
p8_out_gagg(x1, x2, x3, x4)  =  p8_out_gagg(x2)
U3_gagg(x1, x2, x3, x4)  =  U3_gagg(x4)
U4_gagg(x1, x2, x3, x4, x5)  =  U4_gagg(x1, x3, x4, x5)
U5_gagg(x1, x2, x3, x4, x5)  =  U5_gagg(x2, x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

star1_in_gg(T3, []) → star1_out_gg(T3, [])
star1_in_gg(.(T4, T5), .(T4, T6)) → U6_gg(T4, T5, T6, app10_in_gag(T5, X11, T6))
app10_in_gag([], T8, T8) → app10_out_gag([], T8, T8)
app10_in_gag(.(T9, T10), X24, .(T9, T11)) → U1_gag(T9, T10, X24, T11, app10_in_gag(T10, X24, T11))
U1_gag(T9, T10, X24, T11, app10_out_gag(T10, X24, T11)) → app10_out_gag(.(T9, T10), X24, .(T9, T11))
U6_gg(T4, T5, T6, app10_out_gag(T5, X11, T6)) → star1_out_gg(.(T4, T5), .(T4, T6))
star1_in_gg(.(T12, T13), .(T12, [])) → U7_gg(T12, T13, app10_in_gag(T13, T7, []))
U7_gg(T12, T13, app10_out_gag(T13, T7, [])) → star1_out_gg(.(T12, T13), .(T12, []))
star1_in_gg(.(T14, T15), .(T14, .(T14, T16))) → U8_gg(T14, T15, T16, app10_in_gag(T15, T7, .(T14, T16)))
U8_gg(T14, T15, T16, app10_out_gag(T15, T7, .(T14, T16))) → U9_gg(T14, T15, T16, p8_in_gagg(T15, X34, T16, T14))
p8_in_gagg(T5, X11, T6, T4) → U2_gagg(T5, X11, T6, T4, app10_in_gag(T5, X11, T6))
U2_gagg(T5, X11, T6, T4, app10_out_gag(T5, X11, T6)) → p8_out_gagg(T5, X11, T6, T4)
p8_in_gagg(T13, T7, [], T12) → U3_gagg(T13, T7, T12, app10_in_gag(T13, T7, []))
U3_gagg(T13, T7, T12, app10_out_gag(T13, T7, [])) → p8_out_gagg(T13, T7, [], T12)
p8_in_gagg(T15, T7, .(T14, T16), T14) → U4_gagg(T15, T7, T14, T16, app10_in_gag(T15, T7, .(T14, T16)))
U4_gagg(T15, T7, T14, T16, app10_out_gag(T15, T7, .(T14, T16))) → U5_gagg(T15, T7, T14, T16, p8_in_gagg(T15, X34, T16, T14))
U5_gagg(T15, T7, T14, T16, p8_out_gagg(T15, X34, T16, T14)) → p8_out_gagg(T15, T7, .(T14, T16), T14)
U9_gg(T14, T15, T16, p8_out_gagg(T15, X34, T16, T14)) → star1_out_gg(.(T14, T15), .(T14, .(T14, T16)))

The argument filtering Pi contains the following mapping:
star1_in_gg(x1, x2)  =  star1_in_gg(x1, x2)
[]  =  []
star1_out_gg(x1, x2)  =  star1_out_gg
.(x1, x2)  =  .(x1, x2)
U6_gg(x1, x2, x3, x4)  =  U6_gg(x4)
app10_in_gag(x1, x2, x3)  =  app10_in_gag(x1, x3)
app10_out_gag(x1, x2, x3)  =  app10_out_gag(x2)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U8_gg(x1, x2, x3, x4)  =  U8_gg(x1, x2, x3, x4)
U9_gg(x1, x2, x3, x4)  =  U9_gg(x4)
p8_in_gagg(x1, x2, x3, x4)  =  p8_in_gagg(x1, x3, x4)
U2_gagg(x1, x2, x3, x4, x5)  =  U2_gagg(x5)
p8_out_gagg(x1, x2, x3, x4)  =  p8_out_gagg(x2)
U3_gagg(x1, x2, x3, x4)  =  U3_gagg(x4)
U4_gagg(x1, x2, x3, x4, x5)  =  U4_gagg(x1, x3, x4, x5)
U5_gagg(x1, x2, x3, x4, x5)  =  U5_gagg(x2, x5)

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

STAR1_IN_GG(.(T4, T5), .(T4, T6)) → U6_GG(T4, T5, T6, app10_in_gag(T5, X11, T6))
STAR1_IN_GG(.(T4, T5), .(T4, T6)) → APP10_IN_GAG(T5, X11, T6)
APP10_IN_GAG(.(T9, T10), X24, .(T9, T11)) → U1_GAG(T9, T10, X24, T11, app10_in_gag(T10, X24, T11))
APP10_IN_GAG(.(T9, T10), X24, .(T9, T11)) → APP10_IN_GAG(T10, X24, T11)
STAR1_IN_GG(.(T12, T13), .(T12, [])) → U7_GG(T12, T13, app10_in_gag(T13, T7, []))
STAR1_IN_GG(.(T12, T13), .(T12, [])) → APP10_IN_GAG(T13, T7, [])
STAR1_IN_GG(.(T14, T15), .(T14, .(T14, T16))) → U8_GG(T14, T15, T16, app10_in_gag(T15, T7, .(T14, T16)))
STAR1_IN_GG(.(T14, T15), .(T14, .(T14, T16))) → APP10_IN_GAG(T15, T7, .(T14, T16))
U8_GG(T14, T15, T16, app10_out_gag(T15, T7, .(T14, T16))) → U9_GG(T14, T15, T16, p8_in_gagg(T15, X34, T16, T14))
U8_GG(T14, T15, T16, app10_out_gag(T15, T7, .(T14, T16))) → P8_IN_GAGG(T15, X34, T16, T14)
P8_IN_GAGG(T5, X11, T6, T4) → U2_GAGG(T5, X11, T6, T4, app10_in_gag(T5, X11, T6))
P8_IN_GAGG(T5, X11, T6, T4) → APP10_IN_GAG(T5, X11, T6)
P8_IN_GAGG(T13, T7, [], T12) → U3_GAGG(T13, T7, T12, app10_in_gag(T13, T7, []))
P8_IN_GAGG(T13, T7, [], T12) → APP10_IN_GAG(T13, T7, [])
P8_IN_GAGG(T15, T7, .(T14, T16), T14) → U4_GAGG(T15, T7, T14, T16, app10_in_gag(T15, T7, .(T14, T16)))
P8_IN_GAGG(T15, T7, .(T14, T16), T14) → APP10_IN_GAG(T15, T7, .(T14, T16))
U4_GAGG(T15, T7, T14, T16, app10_out_gag(T15, T7, .(T14, T16))) → U5_GAGG(T15, T7, T14, T16, p8_in_gagg(T15, X34, T16, T14))
U4_GAGG(T15, T7, T14, T16, app10_out_gag(T15, T7, .(T14, T16))) → P8_IN_GAGG(T15, X34, T16, T14)

The TRS R consists of the following rules:

star1_in_gg(T3, []) → star1_out_gg(T3, [])
star1_in_gg(.(T4, T5), .(T4, T6)) → U6_gg(T4, T5, T6, app10_in_gag(T5, X11, T6))
app10_in_gag([], T8, T8) → app10_out_gag([], T8, T8)
app10_in_gag(.(T9, T10), X24, .(T9, T11)) → U1_gag(T9, T10, X24, T11, app10_in_gag(T10, X24, T11))
U1_gag(T9, T10, X24, T11, app10_out_gag(T10, X24, T11)) → app10_out_gag(.(T9, T10), X24, .(T9, T11))
U6_gg(T4, T5, T6, app10_out_gag(T5, X11, T6)) → star1_out_gg(.(T4, T5), .(T4, T6))
star1_in_gg(.(T12, T13), .(T12, [])) → U7_gg(T12, T13, app10_in_gag(T13, T7, []))
U7_gg(T12, T13, app10_out_gag(T13, T7, [])) → star1_out_gg(.(T12, T13), .(T12, []))
star1_in_gg(.(T14, T15), .(T14, .(T14, T16))) → U8_gg(T14, T15, T16, app10_in_gag(T15, T7, .(T14, T16)))
U8_gg(T14, T15, T16, app10_out_gag(T15, T7, .(T14, T16))) → U9_gg(T14, T15, T16, p8_in_gagg(T15, X34, T16, T14))
p8_in_gagg(T5, X11, T6, T4) → U2_gagg(T5, X11, T6, T4, app10_in_gag(T5, X11, T6))
U2_gagg(T5, X11, T6, T4, app10_out_gag(T5, X11, T6)) → p8_out_gagg(T5, X11, T6, T4)
p8_in_gagg(T13, T7, [], T12) → U3_gagg(T13, T7, T12, app10_in_gag(T13, T7, []))
U3_gagg(T13, T7, T12, app10_out_gag(T13, T7, [])) → p8_out_gagg(T13, T7, [], T12)
p8_in_gagg(T15, T7, .(T14, T16), T14) → U4_gagg(T15, T7, T14, T16, app10_in_gag(T15, T7, .(T14, T16)))
U4_gagg(T15, T7, T14, T16, app10_out_gag(T15, T7, .(T14, T16))) → U5_gagg(T15, T7, T14, T16, p8_in_gagg(T15, X34, T16, T14))
U5_gagg(T15, T7, T14, T16, p8_out_gagg(T15, X34, T16, T14)) → p8_out_gagg(T15, T7, .(T14, T16), T14)
U9_gg(T14, T15, T16, p8_out_gagg(T15, X34, T16, T14)) → star1_out_gg(.(T14, T15), .(T14, .(T14, T16)))

The argument filtering Pi contains the following mapping:
star1_in_gg(x1, x2)  =  star1_in_gg(x1, x2)
[]  =  []
star1_out_gg(x1, x2)  =  star1_out_gg
.(x1, x2)  =  .(x1, x2)
U6_gg(x1, x2, x3, x4)  =  U6_gg(x4)
app10_in_gag(x1, x2, x3)  =  app10_in_gag(x1, x3)
app10_out_gag(x1, x2, x3)  =  app10_out_gag(x2)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U8_gg(x1, x2, x3, x4)  =  U8_gg(x1, x2, x3, x4)
U9_gg(x1, x2, x3, x4)  =  U9_gg(x4)
p8_in_gagg(x1, x2, x3, x4)  =  p8_in_gagg(x1, x3, x4)
U2_gagg(x1, x2, x3, x4, x5)  =  U2_gagg(x5)
p8_out_gagg(x1, x2, x3, x4)  =  p8_out_gagg(x2)
U3_gagg(x1, x2, x3, x4)  =  U3_gagg(x4)
U4_gagg(x1, x2, x3, x4, x5)  =  U4_gagg(x1, x3, x4, x5)
U5_gagg(x1, x2, x3, x4, x5)  =  U5_gagg(x2, x5)
STAR1_IN_GG(x1, x2)  =  STAR1_IN_GG(x1, x2)
U6_GG(x1, x2, x3, x4)  =  U6_GG(x4)
APP10_IN_GAG(x1, x2, x3)  =  APP10_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4, x5)  =  U1_GAG(x5)
U7_GG(x1, x2, x3)  =  U7_GG(x3)
U8_GG(x1, x2, x3, x4)  =  U8_GG(x1, x2, x3, x4)
U9_GG(x1, x2, x3, x4)  =  U9_GG(x4)
P8_IN_GAGG(x1, x2, x3, x4)  =  P8_IN_GAGG(x1, x3, x4)
U2_GAGG(x1, x2, x3, x4, x5)  =  U2_GAGG(x5)
U3_GAGG(x1, x2, x3, x4)  =  U3_GAGG(x4)
U4_GAGG(x1, x2, x3, x4, x5)  =  U4_GAGG(x1, x3, x4, x5)
U5_GAGG(x1, x2, x3, x4, x5)  =  U5_GAGG(x2, x5)

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

(6) Obligation:

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

STAR1_IN_GG(.(T4, T5), .(T4, T6)) → U6_GG(T4, T5, T6, app10_in_gag(T5, X11, T6))
STAR1_IN_GG(.(T4, T5), .(T4, T6)) → APP10_IN_GAG(T5, X11, T6)
APP10_IN_GAG(.(T9, T10), X24, .(T9, T11)) → U1_GAG(T9, T10, X24, T11, app10_in_gag(T10, X24, T11))
APP10_IN_GAG(.(T9, T10), X24, .(T9, T11)) → APP10_IN_GAG(T10, X24, T11)
STAR1_IN_GG(.(T12, T13), .(T12, [])) → U7_GG(T12, T13, app10_in_gag(T13, T7, []))
STAR1_IN_GG(.(T12, T13), .(T12, [])) → APP10_IN_GAG(T13, T7, [])
STAR1_IN_GG(.(T14, T15), .(T14, .(T14, T16))) → U8_GG(T14, T15, T16, app10_in_gag(T15, T7, .(T14, T16)))
STAR1_IN_GG(.(T14, T15), .(T14, .(T14, T16))) → APP10_IN_GAG(T15, T7, .(T14, T16))
U8_GG(T14, T15, T16, app10_out_gag(T15, T7, .(T14, T16))) → U9_GG(T14, T15, T16, p8_in_gagg(T15, X34, T16, T14))
U8_GG(T14, T15, T16, app10_out_gag(T15, T7, .(T14, T16))) → P8_IN_GAGG(T15, X34, T16, T14)
P8_IN_GAGG(T5, X11, T6, T4) → U2_GAGG(T5, X11, T6, T4, app10_in_gag(T5, X11, T6))
P8_IN_GAGG(T5, X11, T6, T4) → APP10_IN_GAG(T5, X11, T6)
P8_IN_GAGG(T13, T7, [], T12) → U3_GAGG(T13, T7, T12, app10_in_gag(T13, T7, []))
P8_IN_GAGG(T13, T7, [], T12) → APP10_IN_GAG(T13, T7, [])
P8_IN_GAGG(T15, T7, .(T14, T16), T14) → U4_GAGG(T15, T7, T14, T16, app10_in_gag(T15, T7, .(T14, T16)))
P8_IN_GAGG(T15, T7, .(T14, T16), T14) → APP10_IN_GAG(T15, T7, .(T14, T16))
U4_GAGG(T15, T7, T14, T16, app10_out_gag(T15, T7, .(T14, T16))) → U5_GAGG(T15, T7, T14, T16, p8_in_gagg(T15, X34, T16, T14))
U4_GAGG(T15, T7, T14, T16, app10_out_gag(T15, T7, .(T14, T16))) → P8_IN_GAGG(T15, X34, T16, T14)

The TRS R consists of the following rules:

star1_in_gg(T3, []) → star1_out_gg(T3, [])
star1_in_gg(.(T4, T5), .(T4, T6)) → U6_gg(T4, T5, T6, app10_in_gag(T5, X11, T6))
app10_in_gag([], T8, T8) → app10_out_gag([], T8, T8)
app10_in_gag(.(T9, T10), X24, .(T9, T11)) → U1_gag(T9, T10, X24, T11, app10_in_gag(T10, X24, T11))
U1_gag(T9, T10, X24, T11, app10_out_gag(T10, X24, T11)) → app10_out_gag(.(T9, T10), X24, .(T9, T11))
U6_gg(T4, T5, T6, app10_out_gag(T5, X11, T6)) → star1_out_gg(.(T4, T5), .(T4, T6))
star1_in_gg(.(T12, T13), .(T12, [])) → U7_gg(T12, T13, app10_in_gag(T13, T7, []))
U7_gg(T12, T13, app10_out_gag(T13, T7, [])) → star1_out_gg(.(T12, T13), .(T12, []))
star1_in_gg(.(T14, T15), .(T14, .(T14, T16))) → U8_gg(T14, T15, T16, app10_in_gag(T15, T7, .(T14, T16)))
U8_gg(T14, T15, T16, app10_out_gag(T15, T7, .(T14, T16))) → U9_gg(T14, T15, T16, p8_in_gagg(T15, X34, T16, T14))
p8_in_gagg(T5, X11, T6, T4) → U2_gagg(T5, X11, T6, T4, app10_in_gag(T5, X11, T6))
U2_gagg(T5, X11, T6, T4, app10_out_gag(T5, X11, T6)) → p8_out_gagg(T5, X11, T6, T4)
p8_in_gagg(T13, T7, [], T12) → U3_gagg(T13, T7, T12, app10_in_gag(T13, T7, []))
U3_gagg(T13, T7, T12, app10_out_gag(T13, T7, [])) → p8_out_gagg(T13, T7, [], T12)
p8_in_gagg(T15, T7, .(T14, T16), T14) → U4_gagg(T15, T7, T14, T16, app10_in_gag(T15, T7, .(T14, T16)))
U4_gagg(T15, T7, T14, T16, app10_out_gag(T15, T7, .(T14, T16))) → U5_gagg(T15, T7, T14, T16, p8_in_gagg(T15, X34, T16, T14))
U5_gagg(T15, T7, T14, T16, p8_out_gagg(T15, X34, T16, T14)) → p8_out_gagg(T15, T7, .(T14, T16), T14)
U9_gg(T14, T15, T16, p8_out_gagg(T15, X34, T16, T14)) → star1_out_gg(.(T14, T15), .(T14, .(T14, T16)))

The argument filtering Pi contains the following mapping:
star1_in_gg(x1, x2)  =  star1_in_gg(x1, x2)
[]  =  []
star1_out_gg(x1, x2)  =  star1_out_gg
.(x1, x2)  =  .(x1, x2)
U6_gg(x1, x2, x3, x4)  =  U6_gg(x4)
app10_in_gag(x1, x2, x3)  =  app10_in_gag(x1, x3)
app10_out_gag(x1, x2, x3)  =  app10_out_gag(x2)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U8_gg(x1, x2, x3, x4)  =  U8_gg(x1, x2, x3, x4)
U9_gg(x1, x2, x3, x4)  =  U9_gg(x4)
p8_in_gagg(x1, x2, x3, x4)  =  p8_in_gagg(x1, x3, x4)
U2_gagg(x1, x2, x3, x4, x5)  =  U2_gagg(x5)
p8_out_gagg(x1, x2, x3, x4)  =  p8_out_gagg(x2)
U3_gagg(x1, x2, x3, x4)  =  U3_gagg(x4)
U4_gagg(x1, x2, x3, x4, x5)  =  U4_gagg(x1, x3, x4, x5)
U5_gagg(x1, x2, x3, x4, x5)  =  U5_gagg(x2, x5)
STAR1_IN_GG(x1, x2)  =  STAR1_IN_GG(x1, x2)
U6_GG(x1, x2, x3, x4)  =  U6_GG(x4)
APP10_IN_GAG(x1, x2, x3)  =  APP10_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4, x5)  =  U1_GAG(x5)
U7_GG(x1, x2, x3)  =  U7_GG(x3)
U8_GG(x1, x2, x3, x4)  =  U8_GG(x1, x2, x3, x4)
U9_GG(x1, x2, x3, x4)  =  U9_GG(x4)
P8_IN_GAGG(x1, x2, x3, x4)  =  P8_IN_GAGG(x1, x3, x4)
U2_GAGG(x1, x2, x3, x4, x5)  =  U2_GAGG(x5)
U3_GAGG(x1, x2, x3, x4)  =  U3_GAGG(x4)
U4_GAGG(x1, x2, x3, x4, x5)  =  U4_GAGG(x1, x3, x4, x5)
U5_GAGG(x1, x2, x3, x4, x5)  =  U5_GAGG(x2, x5)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 15 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

APP10_IN_GAG(.(T9, T10), X24, .(T9, T11)) → APP10_IN_GAG(T10, X24, T11)

The TRS R consists of the following rules:

star1_in_gg(T3, []) → star1_out_gg(T3, [])
star1_in_gg(.(T4, T5), .(T4, T6)) → U6_gg(T4, T5, T6, app10_in_gag(T5, X11, T6))
app10_in_gag([], T8, T8) → app10_out_gag([], T8, T8)
app10_in_gag(.(T9, T10), X24, .(T9, T11)) → U1_gag(T9, T10, X24, T11, app10_in_gag(T10, X24, T11))
U1_gag(T9, T10, X24, T11, app10_out_gag(T10, X24, T11)) → app10_out_gag(.(T9, T10), X24, .(T9, T11))
U6_gg(T4, T5, T6, app10_out_gag(T5, X11, T6)) → star1_out_gg(.(T4, T5), .(T4, T6))
star1_in_gg(.(T12, T13), .(T12, [])) → U7_gg(T12, T13, app10_in_gag(T13, T7, []))
U7_gg(T12, T13, app10_out_gag(T13, T7, [])) → star1_out_gg(.(T12, T13), .(T12, []))
star1_in_gg(.(T14, T15), .(T14, .(T14, T16))) → U8_gg(T14, T15, T16, app10_in_gag(T15, T7, .(T14, T16)))
U8_gg(T14, T15, T16, app10_out_gag(T15, T7, .(T14, T16))) → U9_gg(T14, T15, T16, p8_in_gagg(T15, X34, T16, T14))
p8_in_gagg(T5, X11, T6, T4) → U2_gagg(T5, X11, T6, T4, app10_in_gag(T5, X11, T6))
U2_gagg(T5, X11, T6, T4, app10_out_gag(T5, X11, T6)) → p8_out_gagg(T5, X11, T6, T4)
p8_in_gagg(T13, T7, [], T12) → U3_gagg(T13, T7, T12, app10_in_gag(T13, T7, []))
U3_gagg(T13, T7, T12, app10_out_gag(T13, T7, [])) → p8_out_gagg(T13, T7, [], T12)
p8_in_gagg(T15, T7, .(T14, T16), T14) → U4_gagg(T15, T7, T14, T16, app10_in_gag(T15, T7, .(T14, T16)))
U4_gagg(T15, T7, T14, T16, app10_out_gag(T15, T7, .(T14, T16))) → U5_gagg(T15, T7, T14, T16, p8_in_gagg(T15, X34, T16, T14))
U5_gagg(T15, T7, T14, T16, p8_out_gagg(T15, X34, T16, T14)) → p8_out_gagg(T15, T7, .(T14, T16), T14)
U9_gg(T14, T15, T16, p8_out_gagg(T15, X34, T16, T14)) → star1_out_gg(.(T14, T15), .(T14, .(T14, T16)))

The argument filtering Pi contains the following mapping:
star1_in_gg(x1, x2)  =  star1_in_gg(x1, x2)
[]  =  []
star1_out_gg(x1, x2)  =  star1_out_gg
.(x1, x2)  =  .(x1, x2)
U6_gg(x1, x2, x3, x4)  =  U6_gg(x4)
app10_in_gag(x1, x2, x3)  =  app10_in_gag(x1, x3)
app10_out_gag(x1, x2, x3)  =  app10_out_gag(x2)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U8_gg(x1, x2, x3, x4)  =  U8_gg(x1, x2, x3, x4)
U9_gg(x1, x2, x3, x4)  =  U9_gg(x4)
p8_in_gagg(x1, x2, x3, x4)  =  p8_in_gagg(x1, x3, x4)
U2_gagg(x1, x2, x3, x4, x5)  =  U2_gagg(x5)
p8_out_gagg(x1, x2, x3, x4)  =  p8_out_gagg(x2)
U3_gagg(x1, x2, x3, x4)  =  U3_gagg(x4)
U4_gagg(x1, x2, x3, x4, x5)  =  U4_gagg(x1, x3, x4, x5)
U5_gagg(x1, x2, x3, x4, x5)  =  U5_gagg(x2, x5)
APP10_IN_GAG(x1, x2, x3)  =  APP10_IN_GAG(x1, x3)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

APP10_IN_GAG(.(T9, T10), X24, .(T9, T11)) → APP10_IN_GAG(T10, X24, T11)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP10_IN_GAG(x1, x2, x3)  =  APP10_IN_GAG(x1, x3)

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

APP10_IN_GAG(.(T9, T10), .(T9, T11)) → APP10_IN_GAG(T10, T11)

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

(14) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

• APP10_IN_GAG(.(T9, T10), .(T9, T11)) → APP10_IN_GAG(T10, T11)
  The graph contains the following edges 1 > 1, 2 > 2

(15) TRUE

(16) Obligation:

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

P8_IN_GAGG(T15, T7, .(T14, T16), T14) → U4_GAGG(T15, T7, T14, T16, app10_in_gag(T15, T7, .(T14, T16)))
U4_GAGG(T15, T7, T14, T16, app10_out_gag(T15, T7, .(T14, T16))) → P8_IN_GAGG(T15, X34, T16, T14)

The TRS R consists of the following rules:

star1_in_gg(T3, []) → star1_out_gg(T3, [])
star1_in_gg(.(T4, T5), .(T4, T6)) → U6_gg(T4, T5, T6, app10_in_gag(T5, X11, T6))
app10_in_gag([], T8, T8) → app10_out_gag([], T8, T8)
app10_in_gag(.(T9, T10), X24, .(T9, T11)) → U1_gag(T9, T10, X24, T11, app10_in_gag(T10, X24, T11))
U1_gag(T9, T10, X24, T11, app10_out_gag(T10, X24, T11)) → app10_out_gag(.(T9, T10), X24, .(T9, T11))
U6_gg(T4, T5, T6, app10_out_gag(T5, X11, T6)) → star1_out_gg(.(T4, T5), .(T4, T6))
star1_in_gg(.(T12, T13), .(T12, [])) → U7_gg(T12, T13, app10_in_gag(T13, T7, []))
U7_gg(T12, T13, app10_out_gag(T13, T7, [])) → star1_out_gg(.(T12, T13), .(T12, []))
star1_in_gg(.(T14, T15), .(T14, .(T14, T16))) → U8_gg(T14, T15, T16, app10_in_gag(T15, T7, .(T14, T16)))
U8_gg(T14, T15, T16, app10_out_gag(T15, T7, .(T14, T16))) → U9_gg(T14, T15, T16, p8_in_gagg(T15, X34, T16, T14))
p8_in_gagg(T5, X11, T6, T4) → U2_gagg(T5, X11, T6, T4, app10_in_gag(T5, X11, T6))
U2_gagg(T5, X11, T6, T4, app10_out_gag(T5, X11, T6)) → p8_out_gagg(T5, X11, T6, T4)
p8_in_gagg(T13, T7, [], T12) → U3_gagg(T13, T7, T12, app10_in_gag(T13, T7, []))
U3_gagg(T13, T7, T12, app10_out_gag(T13, T7, [])) → p8_out_gagg(T13, T7, [], T12)
p8_in_gagg(T15, T7, .(T14, T16), T14) → U4_gagg(T15, T7, T14, T16, app10_in_gag(T15, T7, .(T14, T16)))
U4_gagg(T15, T7, T14, T16, app10_out_gag(T15, T7, .(T14, T16))) → U5_gagg(T15, T7, T14, T16, p8_in_gagg(T15, X34, T16, T14))
U5_gagg(T15, T7, T14, T16, p8_out_gagg(T15, X34, T16, T14)) → p8_out_gagg(T15, T7, .(T14, T16), T14)
U9_gg(T14, T15, T16, p8_out_gagg(T15, X34, T16, T14)) → star1_out_gg(.(T14, T15), .(T14, .(T14, T16)))

The argument filtering Pi contains the following mapping:
star1_in_gg(x1, x2)  =  star1_in_gg(x1, x2)
[]  =  []
star1_out_gg(x1, x2)  =  star1_out_gg
.(x1, x2)  =  .(x1, x2)
U6_gg(x1, x2, x3, x4)  =  U6_gg(x4)
app10_in_gag(x1, x2, x3)  =  app10_in_gag(x1, x3)
app10_out_gag(x1, x2, x3)  =  app10_out_gag(x2)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U8_gg(x1, x2, x3, x4)  =  U8_gg(x1, x2, x3, x4)
U9_gg(x1, x2, x3, x4)  =  U9_gg(x4)
p8_in_gagg(x1, x2, x3, x4)  =  p8_in_gagg(x1, x3, x4)
U2_gagg(x1, x2, x3, x4, x5)  =  U2_gagg(x5)
p8_out_gagg(x1, x2, x3, x4)  =  p8_out_gagg(x2)
U3_gagg(x1, x2, x3, x4)  =  U3_gagg(x4)
U4_gagg(x1, x2, x3, x4, x5)  =  U4_gagg(x1, x3, x4, x5)
U5_gagg(x1, x2, x3, x4, x5)  =  U5_gagg(x2, x5)
P8_IN_GAGG(x1, x2, x3, x4)  =  P8_IN_GAGG(x1, x3, x4)
U4_GAGG(x1, x2, x3, x4, x5)  =  U4_GAGG(x1, x3, x4, x5)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

P8_IN_GAGG(T15, T7, .(T14, T16), T14) → U4_GAGG(T15, T7, T14, T16, app10_in_gag(T15, T7, .(T14, T16)))
U4_GAGG(T15, T7, T14, T16, app10_out_gag(T15, T7, .(T14, T16))) → P8_IN_GAGG(T15, X34, T16, T14)

The TRS R consists of the following rules:

app10_in_gag([], T8, T8) → app10_out_gag([], T8, T8)
app10_in_gag(.(T9, T10), X24, .(T9, T11)) → U1_gag(T9, T10, X24, T11, app10_in_gag(T10, X24, T11))
U1_gag(T9, T10, X24, T11, app10_out_gag(T10, X24, T11)) → app10_out_gag(.(T9, T10), X24, .(T9, T11))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
app10_in_gag(x1, x2, x3)  =  app10_in_gag(x1, x3)
app10_out_gag(x1, x2, x3)  =  app10_out_gag(x2)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
P8_IN_GAGG(x1, x2, x3, x4)  =  P8_IN_GAGG(x1, x3, x4)
U4_GAGG(x1, x2, x3, x4, x5)  =  U4_GAGG(x1, x3, x4, x5)

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

P8_IN_GAGG(T15, .(T14, T16), T14) → U4_GAGG(T15, T14, T16, app10_in_gag(T15, .(T14, T16)))
U4_GAGG(T15, T14, T16, app10_out_gag(T7)) → P8_IN_GAGG(T15, T16, T14)

The TRS R consists of the following rules:

app10_in_gag([], T8) → app10_out_gag(T8)
app10_in_gag(.(T9, T10), .(T9, T11)) → U1_gag(app10_in_gag(T10, T11))
U1_gag(app10_out_gag(X24)) → app10_out_gag(X24)

The set Q consists of the following terms:

app10_in_gag(x0, x1)
U1_gag(x0)

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

(21) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

• U4_GAGG(T15, T14, T16, app10_out_gag(T7)) → P8_IN_GAGG(T15, T16, T14)
  The graph contains the following edges 1 >= 1, 3 >= 2, 2 >= 3

• P8_IN_GAGG(T15, .(T14, T16), T14) → U4_GAGG(T15, T14, T16, app10_in_gag(T15, .(T14, T16)))
  The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 2, 2 > 3

(22) TRUE