Left Termination of the query pattern
star(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).

Query: star(g,g)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: 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: true | star(T4, []), SCOPE: 1, CLAUSE: 1\n\nT4 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\nX3 is free\nstar(T1, T2) !=? star(X3, [])", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: star(T4, []), SCOPE: 1, CLAUSE: 1\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(app(T11, X13, T12), star(.(T10, T11), T12))\n\nT10 is ground\nT11 is ground\nT12 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(app(T11, X13, T12), star(.(T10, T11), T12)), SCOPE: 2, CLAUSE: 2 | ','(app(T11, X13, T12), star(.(T10, T11), T12)), SCOPE: 2, CLAUSE: 3\n\nT10 is ground\nT11 is ground\nT12 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(app(T11, X13, T12), star(.(T10, T11), T12)), SCOPE: 2, CLAUSE: 2\n\nT10 is ground\nT11 is ground\nT12 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(app(T11, X13, T12), star(.(T10, T11), T12)), SCOPE: 2, CLAUSE: 3\n\nT10 is ground\nT11 is ground\nT12 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: star(.(T10, []), T17)\n\nT10 is ground\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(app(T27, X40, T28), star(.(T10, .(T26, T27)), .(T26, T28)))\n\nT10 is ground\nT26 is ground\nT27 is ground\nT28 is ground\nX40 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="B: app(T27, X40, T28)\n\nT27 is ground\nT28 is ground\nX40 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: star(.(T10, .(T26, T27)), .(T26, T28))\n\nT10 is ground\nT26 is ground\nT27 is ground\nT28 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: app(T27, X40, T28), SCOPE: 3, CLAUSE: 2 | app(T27, X40, T28), SCOPE: 3, CLAUSE: 3\n\nT27 is ground\nT28 is ground\nX40 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: app(T27, X40, T28), SCOPE: 3, CLAUSE: 2\n\nT27 is ground\nT28 is ground\nX40 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: app(T27, X40, T28), SCOPE: 3, CLAUSE: 3\n\nT27 is ground\nT28 is ground\nX40 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: app(T46, X73, T47)\n\nT46 is ground\nT47 is ground\nX73 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 26 [arrowhead = none ];
26 -> 2;

27 [label="EVAL with clause\nstar(X3, []).\nand substitutionT1 -> T4,\nX3 -> T4,\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 27 [arrowhead = none ];
27 -> 3;

28 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 28 [arrowhead = none ];
28 -> 4;

29 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 29 [arrowhead = none ];
29 -> 5;

30 [label="EVAL with clause\nstar(.(X10, X11), .(X10, X12)) :- ','(app(X11, X13, X12), star(.(X10, X11), X12)).\nand substitutionX10 -> T10,\nX11 -> T11,\nT1 -> .(T10, T11),\nX12 -> T12,\nT2 -> .(T10, T12)", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 30 [arrowhead = none ];
30 -> 7;

31 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 31 [arrowhead = none ];
31 -> 8;

32 [label="BACKTRACK\nfor clause: star(.(X, U), .(X, W)) :- ','(app(U, V, W), star(.(X, U), W))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 32 [arrowhead = none ];
32 -> 6;

33 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 33 [arrowhead = none ];
33 -> 9;

34 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 34 [arrowhead = none ];
34 -> 10;

35 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 35 [arrowhead = none ];
35 -> 11;

36 [label="EVAL with clause\napp([], X22, X22).\nand substitutionT11 -> [],\nX13 -> T17,\nX22 -> T17,\nT12 -> T17,\nX23 -> T17", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 36 [arrowhead = none ];
36 -> 12;

37 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 37 [arrowhead = none ];
37 -> 13;

38 [label="EVAL with clause\napp(.(X36, X37), X38, .(X36, X39)) :- app(X37, X38, X39).\nand substitutionX36 -> T26,\nX37 -> T27,\nT11 -> .(T26, T27),\nX13 -> X40,\nX38 -> X40,\nX39 -> T28,\nT12 -> .(T26, T28)", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 38 [arrowhead = none ];
38 -> 14;

39 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 39 [arrowhead = none ];
39 -> 15;

40 [label="INSTANCE with matching:\nT1 -> .(T10, [])\nT2 -> T17", fontsize=14, style = filled, fillcolor = lightgrey];
12 -> 40 [arrowhead = none , style=dashed];
40 -> 1[style=dashed];

41 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
14 -> 41 [arrowhead = none ];
41 -> 16;

42 [label="SPLIT 2\nnew knowledge:\nT27 is ground\nT31 is ground\nT28 is ground\nreplacements:X40 -> T31", fontsize=14, style = filled, fillcolor = violet];
14 -> 42 [arrowhead = none ];
42 -> 17;

43 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 43 [arrowhead = none ];
43 -> 18;

44 [label="INSTANCE with matching:\nT1 -> .(T10, .(T26, T27))\nT2 -> .(T26, T28)", fontsize=14, style = filled, fillcolor = lightgrey];
17 -> 44 [arrowhead = none , style=dashed];
44 -> 1[style=dashed];

45 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 45 [arrowhead = none ];
45 -> 19;

46 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 46 [arrowhead = none ];
46 -> 20;

47 [label="EVAL with clause\napp([], X57, X57).\nand substitutionT27 -> [],\nX40 -> T38,\nX57 -> T38,\nT28 -> T38,\nX58 -> T38", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 47 [arrowhead = none ];
47 -> 21;

48 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 48 [arrowhead = none ];
48 -> 22;

49 [label="EVAL with clause\napp(.(X69, X70), X71, .(X69, X72)) :- app(X70, X71, X72).\nand substitutionX69 -> T45,\nX70 -> T46,\nT27 -> .(T45, T46),\nX40 -> X73,\nX71 -> X73,\nX72 -> T47,\nT28 -> .(T45, T47)", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 49 [arrowhead = none ];
49 -> 24;

50 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 50 [arrowhead = none ];
50 -> 25;

51 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 51 [arrowhead = none ];
51 -> 23;

52 [label="INSTANCE with matching:\nT27 -> T46\nX40 -> X73\nT28 -> T47", fontsize=14, style = filled, fillcolor = lightgrey];
24 -> 52 [arrowhead = none , style=dashed];
52 -> 16[style=dashed];

}

(2) Obligation:

Triples:

appB(.(X1, X2), X3, .(X1, X4)) :- appB(X2, X3, X4).
starA(.(X1, []), .(X1, X2)) :- starA(.(X1, []), X2).
starA(.(X1, .(X2, X3)), .(X1, .(X2, X4))) :- appB(X3, X5, X4).
starA(.(X1, .(X2, X3)), .(X1, .(X2, X4))) :- ','(appcB(X3, X5, X4), starA(.(X1, .(X2, X3)), .(X2, X4))).

Clauses:

starcA(X1, []).
starcA(.(X1, []), .(X1, X2)) :- starcA(.(X1, []), X2).
starcA(.(X1, .(X2, X3)), .(X1, .(X2, X4))) :- ','(appcB(X3, X5, X4), starcA(.(X1, .(X2, X3)), .(X2, X4))).
appcB([], X1, X1).
appcB(.(X1, X2), X3, .(X1, X4)) :- appcB(X2, X3, X4).

Afs:

starA(x1, x2)  =  starA(x1, x2)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
starA_in: (b,b)
appB_in: (b,f,b)
appcB_in: (b,f,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

STARA_IN_GG(.(X1, []), .(X1, X2)) → U2_GG(X1, X2, starA_in_gg(.(X1, []), X2))
STARA_IN_GG(.(X1, []), .(X1, X2)) → STARA_IN_GG(.(X1, []), X2)
STARA_IN_GG(.(X1, .(X2, X3)), .(X1, .(X2, X4))) → U3_GG(X1, X2, X3, X4, appB_in_gag(X3, X5, X4))
STARA_IN_GG(.(X1, .(X2, X3)), .(X1, .(X2, X4))) → APPB_IN_GAG(X3, X5, X4)
APPB_IN_GAG(.(X1, X2), X3, .(X1, X4)) → U1_GAG(X1, X2, X3, X4, appB_in_gag(X2, X3, X4))
APPB_IN_GAG(.(X1, X2), X3, .(X1, X4)) → APPB_IN_GAG(X2, X3, X4)
STARA_IN_GG(.(X1, .(X2, X3)), .(X1, .(X2, X4))) → U4_GG(X1, X2, X3, X4, appcB_in_gag(X3, X5, X4))
U4_GG(X1, X2, X3, X4, appcB_out_gag(X3, X5, X4)) → U5_GG(X1, X2, X3, X4, starA_in_gg(.(X1, .(X2, X3)), .(X2, X4)))
U4_GG(X1, X2, X3, X4, appcB_out_gag(X3, X5, X4)) → STARA_IN_GG(.(X1, .(X2, X3)), .(X2, X4))

The TRS R consists of the following rules:

appcB_in_gag([], X1, X1) → appcB_out_gag([], X1, X1)
appcB_in_gag(.(X1, X2), X3, .(X1, X4)) → U10_gag(X1, X2, X3, X4, appcB_in_gag(X2, X3, X4))
U10_gag(X1, X2, X3, X4, appcB_out_gag(X2, X3, X4)) → appcB_out_gag(.(X1, X2), X3, .(X1, X4))

The argument filtering Pi contains the following mapping:
starA_in_gg(x1, x2)  =  starA_in_gg(x1, x2)
.(x1, x2)  =  .(x1, x2)
[]  =  []
appB_in_gag(x1, x2, x3)  =  appB_in_gag(x1, x3)
appcB_in_gag(x1, x2, x3)  =  appcB_in_gag(x1, x3)
appcB_out_gag(x1, x2, x3)  =  appcB_out_gag(x1, x2, x3)
U10_gag(x1, x2, x3, x4, x5)  =  U10_gag(x1, x2, x4, x5)
STARA_IN_GG(x1, x2)  =  STARA_IN_GG(x1, x2)
U2_GG(x1, x2, x3)  =  U2_GG(x1, x2, x3)
U3_GG(x1, x2, x3, x4, x5)  =  U3_GG(x1, x2, x3, x4, x5)
APPB_IN_GAG(x1, x2, x3)  =  APPB_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4, x5)  =  U1_GAG(x1, x2, x4, x5)
U4_GG(x1, x2, x3, x4, x5)  =  U4_GG(x1, x2, x3, x4, x5)
U5_GG(x1, x2, x3, x4, x5)  =  U5_GG(x1, x2, x3, x4, x5)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

STARA_IN_GG(.(X1, []), .(X1, X2)) → U2_GG(X1, X2, starA_in_gg(.(X1, []), X2))
STARA_IN_GG(.(X1, []), .(X1, X2)) → STARA_IN_GG(.(X1, []), X2)
STARA_IN_GG(.(X1, .(X2, X3)), .(X1, .(X2, X4))) → U3_GG(X1, X2, X3, X4, appB_in_gag(X3, X5, X4))
STARA_IN_GG(.(X1, .(X2, X3)), .(X1, .(X2, X4))) → APPB_IN_GAG(X3, X5, X4)
APPB_IN_GAG(.(X1, X2), X3, .(X1, X4)) → U1_GAG(X1, X2, X3, X4, appB_in_gag(X2, X3, X4))
APPB_IN_GAG(.(X1, X2), X3, .(X1, X4)) → APPB_IN_GAG(X2, X3, X4)
STARA_IN_GG(.(X1, .(X2, X3)), .(X1, .(X2, X4))) → U4_GG(X1, X2, X3, X4, appcB_in_gag(X3, X5, X4))
U4_GG(X1, X2, X3, X4, appcB_out_gag(X3, X5, X4)) → U5_GG(X1, X2, X3, X4, starA_in_gg(.(X1, .(X2, X3)), .(X2, X4)))
U4_GG(X1, X2, X3, X4, appcB_out_gag(X3, X5, X4)) → STARA_IN_GG(.(X1, .(X2, X3)), .(X2, X4))

The TRS R consists of the following rules:

appcB_in_gag([], X1, X1) → appcB_out_gag([], X1, X1)
appcB_in_gag(.(X1, X2), X3, .(X1, X4)) → U10_gag(X1, X2, X3, X4, appcB_in_gag(X2, X3, X4))
U10_gag(X1, X2, X3, X4, appcB_out_gag(X2, X3, X4)) → appcB_out_gag(.(X1, X2), X3, .(X1, X4))

The argument filtering Pi contains the following mapping:
starA_in_gg(x1, x2)  =  starA_in_gg(x1, x2)
.(x1, x2)  =  .(x1, x2)
[]  =  []
appB_in_gag(x1, x2, x3)  =  appB_in_gag(x1, x3)
appcB_in_gag(x1, x2, x3)  =  appcB_in_gag(x1, x3)
appcB_out_gag(x1, x2, x3)  =  appcB_out_gag(x1, x2, x3)
U10_gag(x1, x2, x3, x4, x5)  =  U10_gag(x1, x2, x4, x5)
STARA_IN_GG(x1, x2)  =  STARA_IN_GG(x1, x2)
U2_GG(x1, x2, x3)  =  U2_GG(x1, x2, x3)
U3_GG(x1, x2, x3, x4, x5)  =  U3_GG(x1, x2, x3, x4, x5)
APPB_IN_GAG(x1, x2, x3)  =  APPB_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4, x5)  =  U1_GAG(x1, x2, x4, x5)
U4_GG(x1, x2, x3, x4, x5)  =  U4_GG(x1, x2, x3, x4, x5)
U5_GG(x1, x2, x3, x4, x5)  =  U5_GG(x1, x2, x3, x4, x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 5 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

APPB_IN_GAG(.(X1, X2), X3, .(X1, X4)) → APPB_IN_GAG(X2, X3, X4)

The TRS R consists of the following rules:

appcB_in_gag([], X1, X1) → appcB_out_gag([], X1, X1)
appcB_in_gag(.(X1, X2), X3, .(X1, X4)) → U10_gag(X1, X2, X3, X4, appcB_in_gag(X2, X3, X4))
U10_gag(X1, X2, X3, X4, appcB_out_gag(X2, X3, X4)) → appcB_out_gag(.(X1, X2), X3, .(X1, X4))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
appcB_in_gag(x1, x2, x3)  =  appcB_in_gag(x1, x3)
appcB_out_gag(x1, x2, x3)  =  appcB_out_gag(x1, x2, x3)
U10_gag(x1, x2, x3, x4, x5)  =  U10_gag(x1, x2, x4, x5)
APPB_IN_GAG(x1, x2, x3)  =  APPB_IN_GAG(x1, 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:

APPB_IN_GAG(.(X1, X2), X3, .(X1, X4)) → APPB_IN_GAG(X2, X3, X4)

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

APPB_IN_GAG(.(X1, X2), .(X1, X4)) → APPB_IN_GAG(X2, X4)

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:

• APPB_IN_GAG(.(X1, X2), .(X1, X4)) → APPB_IN_GAG(X2, X4)
  The graph contains the following edges 1 > 1, 2 > 2

(13) YES

(14) Obligation:

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

STARA_IN_GG(.(X1, .(X2, X3)), .(X1, .(X2, X4))) → U4_GG(X1, X2, X3, X4, appcB_in_gag(X3, X5, X4))
U4_GG(X1, X2, X3, X4, appcB_out_gag(X3, X5, X4)) → STARA_IN_GG(.(X1, .(X2, X3)), .(X2, X4))

The TRS R consists of the following rules:

appcB_in_gag([], X1, X1) → appcB_out_gag([], X1, X1)
appcB_in_gag(.(X1, X2), X3, .(X1, X4)) → U10_gag(X1, X2, X3, X4, appcB_in_gag(X2, X3, X4))
U10_gag(X1, X2, X3, X4, appcB_out_gag(X2, X3, X4)) → appcB_out_gag(.(X1, X2), X3, .(X1, X4))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
appcB_in_gag(x1, x2, x3)  =  appcB_in_gag(x1, x3)
appcB_out_gag(x1, x2, x3)  =  appcB_out_gag(x1, x2, x3)
U10_gag(x1, x2, x3, x4, x5)  =  U10_gag(x1, x2, x4, x5)
STARA_IN_GG(x1, x2)  =  STARA_IN_GG(x1, x2)
U4_GG(x1, x2, x3, x4, x5)  =  U4_GG(x1, x2, x3, x4, x5)

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

(15) PiDPToQDPProof (SOUND transformation)

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

(16) Obligation:

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

STARA_IN_GG(.(X1, .(X2, X3)), .(X1, .(X2, X4))) → U4_GG(X1, X2, X3, X4, appcB_in_gag(X3, X4))
U4_GG(X1, X2, X3, X4, appcB_out_gag(X3, X5, X4)) → STARA_IN_GG(.(X1, .(X2, X3)), .(X2, X4))

The TRS R consists of the following rules:

appcB_in_gag([], X1) → appcB_out_gag([], X1, X1)
appcB_in_gag(.(X1, X2), .(X1, X4)) → U10_gag(X1, X2, X4, appcB_in_gag(X2, X4))
U10_gag(X1, X2, X4, appcB_out_gag(X2, X3, X4)) → appcB_out_gag(.(X1, X2), X3, .(X1, X4))

The set Q consists of the following terms:

appcB_in_gag(x0, x1)
U10_gag(x0, x1, x2, x3)

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

(17) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule STARA_IN_GG(.(X1, .(X2, X3)), .(X1, .(X2, X4))) → U4_GG(X1, X2, X3, X4, appcB_in_gag(X3, X4)) we obtained the following new rules [LPAR04]:

STARA_IN_GG(.(z0, .(z0, z2)), .(z0, .(z0, x3))) → U4_GG(z0, z0, z2, x3, appcB_in_gag(z2, x3))

(18) Obligation:

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

U4_GG(X1, X2, X3, X4, appcB_out_gag(X3, X5, X4)) → STARA_IN_GG(.(X1, .(X2, X3)), .(X2, X4))
STARA_IN_GG(.(z0, .(z0, z2)), .(z0, .(z0, x3))) → U4_GG(z0, z0, z2, x3, appcB_in_gag(z2, x3))

The TRS R consists of the following rules:

appcB_in_gag([], X1) → appcB_out_gag([], X1, X1)
appcB_in_gag(.(X1, X2), .(X1, X4)) → U10_gag(X1, X2, X4, appcB_in_gag(X2, X4))
U10_gag(X1, X2, X4, appcB_out_gag(X2, X3, X4)) → appcB_out_gag(.(X1, X2), X3, .(X1, X4))

The set Q consists of the following terms:

appcB_in_gag(x0, x1)
U10_gag(x0, x1, x2, x3)

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

(19) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U4_GG(X1, X2, X3, X4, appcB_out_gag(X3, X5, X4)) → STARA_IN_GG(.(X1, .(X2, X3)), .(X2, X4)) we obtained the following new rules [LPAR04]:

U4_GG(z0, z0, z1, z2, appcB_out_gag(z1, x4, z2)) → STARA_IN_GG(.(z0, .(z0, z1)), .(z0, z2))

(20) Obligation:

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

STARA_IN_GG(.(z0, .(z0, z2)), .(z0, .(z0, x3))) → U4_GG(z0, z0, z2, x3, appcB_in_gag(z2, x3))
U4_GG(z0, z0, z1, z2, appcB_out_gag(z1, x4, z2)) → STARA_IN_GG(.(z0, .(z0, z1)), .(z0, z2))

The TRS R consists of the following rules:

appcB_in_gag([], X1) → appcB_out_gag([], X1, X1)
appcB_in_gag(.(X1, X2), .(X1, X4)) → U10_gag(X1, X2, X4, appcB_in_gag(X2, X4))
U10_gag(X1, X2, X4, appcB_out_gag(X2, X3, X4)) → appcB_out_gag(.(X1, X2), X3, .(X1, X4))

The set Q consists of the following terms:

appcB_in_gag(x0, x1)
U10_gag(x0, x1, x2, x3)

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

(21) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

STARA_IN_GG(.(z0, .(z0, z2)), .(z0, .(z0, x3))) → U4_GG(z0, z0, z2, x3, appcB_in_gag(z2, x3))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 1 + x₁ + x₂   
POL(STARA_IN_GG(x₁, x₂)) = x₂   
POL(U10_gag(x₁, x₂, x₃, x₄)) = 0   
POL(U4_GG(x₁, x₂, x₃, x₄, x₅)) = 1 + x₁ + x₄   
POL([]) = 0   
POL(appcB_in_gag(x₁, x₂)) = 0   
POL(appcB_out_gag(x₁, x₂, x₃)) = 0   

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none

(22) Obligation:

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

U4_GG(z0, z0, z1, z2, appcB_out_gag(z1, x4, z2)) → STARA_IN_GG(.(z0, .(z0, z1)), .(z0, z2))

The TRS R consists of the following rules:

appcB_in_gag([], X1) → appcB_out_gag([], X1, X1)
appcB_in_gag(.(X1, X2), .(X1, X4)) → U10_gag(X1, X2, X4, appcB_in_gag(X2, X4))
U10_gag(X1, X2, X4, appcB_out_gag(X2, X3, X4)) → appcB_out_gag(.(X1, X2), X3, .(X1, X4))

The set Q consists of the following terms:

appcB_in_gag(x0, x1)
U10_gag(x0, x1, x2, x3)

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

(23) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(24) TRUE

(25) Obligation:

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

STARA_IN_GG(.(X1, []), .(X1, X2)) → STARA_IN_GG(.(X1, []), X2)

The TRS R consists of the following rules:

appcB_in_gag([], X1, X1) → appcB_out_gag([], X1, X1)
appcB_in_gag(.(X1, X2), X3, .(X1, X4)) → U10_gag(X1, X2, X3, X4, appcB_in_gag(X2, X3, X4))
U10_gag(X1, X2, X3, X4, appcB_out_gag(X2, X3, X4)) → appcB_out_gag(.(X1, X2), X3, .(X1, X4))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
appcB_in_gag(x1, x2, x3)  =  appcB_in_gag(x1, x3)
appcB_out_gag(x1, x2, x3)  =  appcB_out_gag(x1, x2, x3)
U10_gag(x1, x2, x3, x4, x5)  =  U10_gag(x1, x2, x4, x5)
STARA_IN_GG(x1, x2)  =  STARA_IN_GG(x1, x2)

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

(26) UsableRulesProof (EQUIVALENT transformation)

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

(27) Obligation:

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

STARA_IN_GG(.(X1, []), .(X1, X2)) → STARA_IN_GG(.(X1, []), X2)

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

(28) PiDPToQDPProof (EQUIVALENT transformation)

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

(29) Obligation:

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

STARA_IN_GG(.(X1, []), .(X1, X2)) → STARA_IN_GG(.(X1, []), X2)

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

(30) 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:

• STARA_IN_GG(.(X1, []), .(X1, X2)) → STARA_IN_GG(.(X1, []), X2)
  The graph contains the following edges 1 >= 1, 2 > 2

(31) YES