Left Termination of the query pattern
app3_b(g,g,g,a)
w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

app3_a(Xs, Ys, Zs, Us) :- ','(app(Xs, Ys, Vs), app(Vs, Zs, Us)).
app3_b(Xs, Ys, Zs, Us) :- ','(app(Ys, Zs, Vs), app(Xs, Vs, Us)).
app([], Ys, Ys).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).

Query: app3_b(g,g,g,a)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: app3_b(T1, T2, T3, T4)\n\nT1 is ground\nT2 is ground\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: app3_b(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(app(T10, T11, X9), app(T9, X9, T13))\n\nT9 is ground\nT10 is ground\nT11 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(app(T10, T11, X9), app(T9, X9, T13)), SCOPE: 2, CLAUSE: 2 | ','(app(T10, T11, X9), app(T9, X9, T13)), SCOPE: 2, CLAUSE: 3\n\nT9 is ground\nT10 is ground\nT11 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(app(T10, T11, X9), app(T9, X9, T13)), SCOPE: 2, CLAUSE: 2\n\nT9 is ground\nT10 is ground\nT11 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(app(T10, T11, X9), app(T9, X9, T13)), SCOPE: 2, CLAUSE: 3\n\nT9 is ground\nT10 is ground\nT11 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: app(T9, T18, T13)\n\nT9 is ground\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: app(T9, T18, T13), SCOPE: 3, CLAUSE: 2 | app(T9, T18, T13), SCOPE: 3, CLAUSE: 3\n\nT9 is ground\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: app(T9, T18, T13), SCOPE: 3, CLAUSE: 2\n\nT9 is ground\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: app(T9, T18, T13), SCOPE: 3, CLAUSE: 3\n\nT9 is ground\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: app(T35, T36, T38)\n\nT35 is ground\nT36 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(app(T48, T49, X50), app(T9, .(T47, X50), T13))\n\nT9 is ground\nT47 is ground\nT48 is ground\nT49 is ground\nX50 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: app(T48, T49, X50)\n\nT48 is ground\nT49 is ground\nX50 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: app(T9, .(T47, T52), T13)\n\nT9 is ground\nT47 is ground\nT52 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: app(T48, T49, X50), SCOPE: 4, CLAUSE: 2 | app(T48, T49, X50), SCOPE: 4, CLAUSE: 3\n\nT48 is ground\nT49 is ground\nX50 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: app(T48, T49, X50), SCOPE: 4, CLAUSE: 2\n\nT48 is ground\nT49 is ground\nX50 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: app(T48, T49, X50), SCOPE: 4, CLAUSE: 3\n\nT48 is ground\nT49 is ground\nX50 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: app(T67, T68, X74)\n\nT67 is ground\nT68 is ground\nX74 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 29 [arrowhead = none ];
29 -> 2;

30 [label="ONLY EVAL with clause\napp3_b(X5, X6, X7, X8) :- ','(app(X6, X7, X9), app(X5, X9, X8)).\nand substitutionT1 -> T9,\nX5 -> T9,\nT2 -> T10,\nX6 -> T10,\nT3 -> T11,\nX7 -> T11,\nT4 -> T13,\nX8 -> T13,\nT12 -> T13", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 30 [arrowhead = none ];
30 -> 3;

31 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 31 [arrowhead = none ];
31 -> 4;

32 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 32 [arrowhead = none ];
32 -> 5;

33 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 33 [arrowhead = none ];
33 -> 6;

34 [label="EVAL with clause\napp([], X14, X14).\nand substitutionT10 -> [],\nT11 -> T18,\nX14 -> T18,\nX9 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 34 [arrowhead = none ];
34 -> 7;

35 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 35 [arrowhead = none ];
35 -> 8;

36 [label="EVAL with clause\napp(.(X46, X47), X48, .(X46, X49)) :- app(X47, X48, X49).\nand substitutionX46 -> T47,\nX47 -> T48,\nT10 -> .(T47, T48),\nT11 -> T49,\nX48 -> T49,\nX49 -> X50,\nX9 -> .(T47, X50)", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 36 [arrowhead = none ];
36 -> 17;

37 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 37 [arrowhead = none ];
37 -> 18;

38 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 38 [arrowhead = none ];
38 -> 9;

39 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 39 [arrowhead = none ];
39 -> 10;

40 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 40 [arrowhead = none ];
40 -> 11;

41 [label="EVAL with clause\napp([], X21, X21).\nand substitutionT9 -> [],\nT18 -> T25,\nX21 -> T25,\nT13 -> T25", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 41 [arrowhead = none ];
41 -> 12;

42 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 42 [arrowhead = none ];
42 -> 13;

43 [label="EVAL with clause\napp(.(X30, X31), X32, .(X30, X33)) :- app(X31, X32, X33).\nand substitutionX30 -> T34,\nX31 -> T35,\nT9 -> .(T34, T35),\nT18 -> T36,\nX32 -> T36,\nX33 -> T38,\nT13 -> .(T34, T38),\nT37 -> T38", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 43 [arrowhead = none ];
43 -> 15;

44 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 44 [arrowhead = none ];
44 -> 16;

45 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 45 [arrowhead = none ];
45 -> 14;

46 [label="INSTANCE with matching:\nT9 -> T35\nT18 -> T36\nT13 -> T38", fontsize=14, style = filled, fillcolor = lightgrey];
15 -> 46 [arrowhead = none , style=dashed];
46 -> 7[style=dashed];

47 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
17 -> 47 [arrowhead = none ];
47 -> 19;

48 [label="SPLIT 2\nnew knowledge:\nT48 is ground\nT49 is ground\nT52 is ground\nreplacements:X50 -> T52", fontsize=14, style = filled, fillcolor = violet];
17 -> 48 [arrowhead = none ];
48 -> 20;

49 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 49 [arrowhead = none ];
49 -> 21;

50 [label="INSTANCE with matching:\nT18 -> .(T47, T52)", fontsize=14, style = filled, fillcolor = lightgrey];
20 -> 50 [arrowhead = none , style=dashed];
50 -> 7[style=dashed];

51 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
21 -> 51 [arrowhead = none ];
51 -> 22;

52 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
21 -> 52 [arrowhead = none ];
52 -> 23;

53 [label="EVAL with clause\napp([], X59, X59).\nand substitutionT48 -> [],\nT49 -> T59,\nX59 -> T59,\nX50 -> T59", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 53 [arrowhead = none ];
53 -> 24;

54 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 54 [arrowhead = none ];
54 -> 25;

55 [label="EVAL with clause\napp(.(X70, X71), X72, .(X70, X73)) :- app(X71, X72, X73).\nand substitutionX70 -> T66,\nX71 -> T67,\nT48 -> .(T66, T67),\nT49 -> T68,\nX72 -> T68,\nX73 -> X74,\nX50 -> .(T66, X74)", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 55 [arrowhead = none ];
55 -> 27;

56 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 56 [arrowhead = none ];
56 -> 28;

57 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 57 [arrowhead = none ];
57 -> 26;

58 [label="INSTANCE with matching:\nT48 -> T67\nT49 -> T68\nX50 -> X74", fontsize=14, style = filled, fillcolor = lightgrey];
27 -> 58 [arrowhead = none , style=dashed];
58 -> 19[style=dashed];

}

(2) Obligation:

Clauses:

appA([], T25, T25).
appA(.(T34, T35), T36, .(T34, T38)) :- appA(T35, T36, T38).
appB([], T59, T59).
appB(.(T66, T67), T68, .(T66, X74)) :- appB(T67, T68, X74).
app3_bC(T9, [], T18, T13) :- appA(T9, T18, T13).
app3_bC(T9, .(T47, T48), T49, T13) :- appB(T48, T49, X50).
app3_bC(T9, .(T47, T48), T49, T13) :- ','(appB(T48, T49, T52), appA(T9, .(T47, T52), T13)).

Query: app3_bC(g,g,g,a)

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
app3_bC_in: (b,b,b,f)
appA_in: (b,b,f)
appB_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

app3_bC_in_ggga(T9, [], T18, T13) → U3_ggga(T9, T18, T13, appA_in_gga(T9, T18, T13))
appA_in_gga([], T25, T25) → appA_out_gga([], T25, T25)
appA_in_gga(.(T34, T35), T36, .(T34, T38)) → U1_gga(T34, T35, T36, T38, appA_in_gga(T35, T36, T38))
U1_gga(T34, T35, T36, T38, appA_out_gga(T35, T36, T38)) → appA_out_gga(.(T34, T35), T36, .(T34, T38))
U3_ggga(T9, T18, T13, appA_out_gga(T9, T18, T13)) → app3_bC_out_ggga(T9, [], T18, T13)
app3_bC_in_ggga(T9, .(T47, T48), T49, T13) → U4_ggga(T9, T47, T48, T49, T13, appB_in_gga(T48, T49, X50))
appB_in_gga([], T59, T59) → appB_out_gga([], T59, T59)
appB_in_gga(.(T66, T67), T68, .(T66, X74)) → U2_gga(T66, T67, T68, X74, appB_in_gga(T67, T68, X74))
U2_gga(T66, T67, T68, X74, appB_out_gga(T67, T68, X74)) → appB_out_gga(.(T66, T67), T68, .(T66, X74))
U4_ggga(T9, T47, T48, T49, T13, appB_out_gga(T48, T49, X50)) → app3_bC_out_ggga(T9, .(T47, T48), T49, T13)
app3_bC_in_ggga(T9, .(T47, T48), T49, T13) → U5_ggga(T9, T47, T48, T49, T13, appB_in_gga(T48, T49, T52))
U5_ggga(T9, T47, T48, T49, T13, appB_out_gga(T48, T49, T52)) → U6_ggga(T9, T47, T48, T49, T13, appA_in_gga(T9, .(T47, T52), T13))
U6_ggga(T9, T47, T48, T49, T13, appA_out_gga(T9, .(T47, T52), T13)) → app3_bC_out_ggga(T9, .(T47, T48), T49, T13)

The argument filtering Pi contains the following mapping:
app3_bC_in_ggga(x1, x2, x3, x4)  =  app3_bC_in_ggga(x1, x2, x3)
[]  =  []
U3_ggga(x1, x2, x3, x4)  =  U3_ggga(x4)
appA_in_gga(x1, x2, x3)  =  appA_in_gga(x1, x2)
appA_out_gga(x1, x2, x3)  =  appA_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
app3_bC_out_ggga(x1, x2, x3, x4)  =  app3_bC_out_ggga
U4_ggga(x1, x2, x3, x4, x5, x6)  =  U4_ggga(x6)
appB_in_gga(x1, x2, x3)  =  appB_in_gga(x1, x2)
appB_out_gga(x1, x2, x3)  =  appB_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
U5_ggga(x1, x2, x3, x4, x5, x6)  =  U5_ggga(x1, x2, x6)
U6_ggga(x1, x2, x3, x4, x5, x6)  =  U6_ggga(x6)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

app3_bC_in_ggga(T9, [], T18, T13) → U3_ggga(T9, T18, T13, appA_in_gga(T9, T18, T13))
appA_in_gga([], T25, T25) → appA_out_gga([], T25, T25)
appA_in_gga(.(T34, T35), T36, .(T34, T38)) → U1_gga(T34, T35, T36, T38, appA_in_gga(T35, T36, T38))
U1_gga(T34, T35, T36, T38, appA_out_gga(T35, T36, T38)) → appA_out_gga(.(T34, T35), T36, .(T34, T38))
U3_ggga(T9, T18, T13, appA_out_gga(T9, T18, T13)) → app3_bC_out_ggga(T9, [], T18, T13)
app3_bC_in_ggga(T9, .(T47, T48), T49, T13) → U4_ggga(T9, T47, T48, T49, T13, appB_in_gga(T48, T49, X50))
appB_in_gga([], T59, T59) → appB_out_gga([], T59, T59)
appB_in_gga(.(T66, T67), T68, .(T66, X74)) → U2_gga(T66, T67, T68, X74, appB_in_gga(T67, T68, X74))
U2_gga(T66, T67, T68, X74, appB_out_gga(T67, T68, X74)) → appB_out_gga(.(T66, T67), T68, .(T66, X74))
U4_ggga(T9, T47, T48, T49, T13, appB_out_gga(T48, T49, X50)) → app3_bC_out_ggga(T9, .(T47, T48), T49, T13)
app3_bC_in_ggga(T9, .(T47, T48), T49, T13) → U5_ggga(T9, T47, T48, T49, T13, appB_in_gga(T48, T49, T52))
U5_ggga(T9, T47, T48, T49, T13, appB_out_gga(T48, T49, T52)) → U6_ggga(T9, T47, T48, T49, T13, appA_in_gga(T9, .(T47, T52), T13))
U6_ggga(T9, T47, T48, T49, T13, appA_out_gga(T9, .(T47, T52), T13)) → app3_bC_out_ggga(T9, .(T47, T48), T49, T13)

The argument filtering Pi contains the following mapping:
app3_bC_in_ggga(x1, x2, x3, x4)  =  app3_bC_in_ggga(x1, x2, x3)
[]  =  []
U3_ggga(x1, x2, x3, x4)  =  U3_ggga(x4)
appA_in_gga(x1, x2, x3)  =  appA_in_gga(x1, x2)
appA_out_gga(x1, x2, x3)  =  appA_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
app3_bC_out_ggga(x1, x2, x3, x4)  =  app3_bC_out_ggga
U4_ggga(x1, x2, x3, x4, x5, x6)  =  U4_ggga(x6)
appB_in_gga(x1, x2, x3)  =  appB_in_gga(x1, x2)
appB_out_gga(x1, x2, x3)  =  appB_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
U5_ggga(x1, x2, x3, x4, x5, x6)  =  U5_ggga(x1, x2, x6)
U6_ggga(x1, x2, x3, x4, x5, x6)  =  U6_ggga(x6)

(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:

APP3_BC_IN_GGGA(T9, [], T18, T13) → U3_GGGA(T9, T18, T13, appA_in_gga(T9, T18, T13))
APP3_BC_IN_GGGA(T9, [], T18, T13) → APPA_IN_GGA(T9, T18, T13)
APPA_IN_GGA(.(T34, T35), T36, .(T34, T38)) → U1_GGA(T34, T35, T36, T38, appA_in_gga(T35, T36, T38))
APPA_IN_GGA(.(T34, T35), T36, .(T34, T38)) → APPA_IN_GGA(T35, T36, T38)
APP3_BC_IN_GGGA(T9, .(T47, T48), T49, T13) → U4_GGGA(T9, T47, T48, T49, T13, appB_in_gga(T48, T49, X50))
APP3_BC_IN_GGGA(T9, .(T47, T48), T49, T13) → APPB_IN_GGA(T48, T49, X50)
APPB_IN_GGA(.(T66, T67), T68, .(T66, X74)) → U2_GGA(T66, T67, T68, X74, appB_in_gga(T67, T68, X74))
APPB_IN_GGA(.(T66, T67), T68, .(T66, X74)) → APPB_IN_GGA(T67, T68, X74)
APP3_BC_IN_GGGA(T9, .(T47, T48), T49, T13) → U5_GGGA(T9, T47, T48, T49, T13, appB_in_gga(T48, T49, T52))
U5_GGGA(T9, T47, T48, T49, T13, appB_out_gga(T48, T49, T52)) → U6_GGGA(T9, T47, T48, T49, T13, appA_in_gga(T9, .(T47, T52), T13))
U5_GGGA(T9, T47, T48, T49, T13, appB_out_gga(T48, T49, T52)) → APPA_IN_GGA(T9, .(T47, T52), T13)

The TRS R consists of the following rules:

app3_bC_in_ggga(T9, [], T18, T13) → U3_ggga(T9, T18, T13, appA_in_gga(T9, T18, T13))
appA_in_gga([], T25, T25) → appA_out_gga([], T25, T25)
appA_in_gga(.(T34, T35), T36, .(T34, T38)) → U1_gga(T34, T35, T36, T38, appA_in_gga(T35, T36, T38))
U1_gga(T34, T35, T36, T38, appA_out_gga(T35, T36, T38)) → appA_out_gga(.(T34, T35), T36, .(T34, T38))
U3_ggga(T9, T18, T13, appA_out_gga(T9, T18, T13)) → app3_bC_out_ggga(T9, [], T18, T13)
app3_bC_in_ggga(T9, .(T47, T48), T49, T13) → U4_ggga(T9, T47, T48, T49, T13, appB_in_gga(T48, T49, X50))
appB_in_gga([], T59, T59) → appB_out_gga([], T59, T59)
appB_in_gga(.(T66, T67), T68, .(T66, X74)) → U2_gga(T66, T67, T68, X74, appB_in_gga(T67, T68, X74))
U2_gga(T66, T67, T68, X74, appB_out_gga(T67, T68, X74)) → appB_out_gga(.(T66, T67), T68, .(T66, X74))
U4_ggga(T9, T47, T48, T49, T13, appB_out_gga(T48, T49, X50)) → app3_bC_out_ggga(T9, .(T47, T48), T49, T13)
app3_bC_in_ggga(T9, .(T47, T48), T49, T13) → U5_ggga(T9, T47, T48, T49, T13, appB_in_gga(T48, T49, T52))
U5_ggga(T9, T47, T48, T49, T13, appB_out_gga(T48, T49, T52)) → U6_ggga(T9, T47, T48, T49, T13, appA_in_gga(T9, .(T47, T52), T13))
U6_ggga(T9, T47, T48, T49, T13, appA_out_gga(T9, .(T47, T52), T13)) → app3_bC_out_ggga(T9, .(T47, T48), T49, T13)

The argument filtering Pi contains the following mapping:
app3_bC_in_ggga(x1, x2, x3, x4)  =  app3_bC_in_ggga(x1, x2, x3)
[]  =  []
U3_ggga(x1, x2, x3, x4)  =  U3_ggga(x4)
appA_in_gga(x1, x2, x3)  =  appA_in_gga(x1, x2)
appA_out_gga(x1, x2, x3)  =  appA_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
app3_bC_out_ggga(x1, x2, x3, x4)  =  app3_bC_out_ggga
U4_ggga(x1, x2, x3, x4, x5, x6)  =  U4_ggga(x6)
appB_in_gga(x1, x2, x3)  =  appB_in_gga(x1, x2)
appB_out_gga(x1, x2, x3)  =  appB_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
U5_ggga(x1, x2, x3, x4, x5, x6)  =  U5_ggga(x1, x2, x6)
U6_ggga(x1, x2, x3, x4, x5, x6)  =  U6_ggga(x6)
APP3_BC_IN_GGGA(x1, x2, x3, x4)  =  APP3_BC_IN_GGGA(x1, x2, x3)
U3_GGGA(x1, x2, x3, x4)  =  U3_GGGA(x4)
APPA_IN_GGA(x1, x2, x3)  =  APPA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x5)
U4_GGGA(x1, x2, x3, x4, x5, x6)  =  U4_GGGA(x6)
APPB_IN_GGA(x1, x2, x3)  =  APPB_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x1, x5)
U5_GGGA(x1, x2, x3, x4, x5, x6)  =  U5_GGGA(x1, x2, x6)
U6_GGGA(x1, x2, x3, x4, x5, x6)  =  U6_GGGA(x6)

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

(6) Obligation:

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

APP3_BC_IN_GGGA(T9, [], T18, T13) → U3_GGGA(T9, T18, T13, appA_in_gga(T9, T18, T13))
APP3_BC_IN_GGGA(T9, [], T18, T13) → APPA_IN_GGA(T9, T18, T13)
APPA_IN_GGA(.(T34, T35), T36, .(T34, T38)) → U1_GGA(T34, T35, T36, T38, appA_in_gga(T35, T36, T38))
APPA_IN_GGA(.(T34, T35), T36, .(T34, T38)) → APPA_IN_GGA(T35, T36, T38)
APP3_BC_IN_GGGA(T9, .(T47, T48), T49, T13) → U4_GGGA(T9, T47, T48, T49, T13, appB_in_gga(T48, T49, X50))
APP3_BC_IN_GGGA(T9, .(T47, T48), T49, T13) → APPB_IN_GGA(T48, T49, X50)
APPB_IN_GGA(.(T66, T67), T68, .(T66, X74)) → U2_GGA(T66, T67, T68, X74, appB_in_gga(T67, T68, X74))
APPB_IN_GGA(.(T66, T67), T68, .(T66, X74)) → APPB_IN_GGA(T67, T68, X74)
APP3_BC_IN_GGGA(T9, .(T47, T48), T49, T13) → U5_GGGA(T9, T47, T48, T49, T13, appB_in_gga(T48, T49, T52))
U5_GGGA(T9, T47, T48, T49, T13, appB_out_gga(T48, T49, T52)) → U6_GGGA(T9, T47, T48, T49, T13, appA_in_gga(T9, .(T47, T52), T13))
U5_GGGA(T9, T47, T48, T49, T13, appB_out_gga(T48, T49, T52)) → APPA_IN_GGA(T9, .(T47, T52), T13)

The TRS R consists of the following rules:

app3_bC_in_ggga(T9, [], T18, T13) → U3_ggga(T9, T18, T13, appA_in_gga(T9, T18, T13))
appA_in_gga([], T25, T25) → appA_out_gga([], T25, T25)
appA_in_gga(.(T34, T35), T36, .(T34, T38)) → U1_gga(T34, T35, T36, T38, appA_in_gga(T35, T36, T38))
U1_gga(T34, T35, T36, T38, appA_out_gga(T35, T36, T38)) → appA_out_gga(.(T34, T35), T36, .(T34, T38))
U3_ggga(T9, T18, T13, appA_out_gga(T9, T18, T13)) → app3_bC_out_ggga(T9, [], T18, T13)
app3_bC_in_ggga(T9, .(T47, T48), T49, T13) → U4_ggga(T9, T47, T48, T49, T13, appB_in_gga(T48, T49, X50))
appB_in_gga([], T59, T59) → appB_out_gga([], T59, T59)
appB_in_gga(.(T66, T67), T68, .(T66, X74)) → U2_gga(T66, T67, T68, X74, appB_in_gga(T67, T68, X74))
U2_gga(T66, T67, T68, X74, appB_out_gga(T67, T68, X74)) → appB_out_gga(.(T66, T67), T68, .(T66, X74))
U4_ggga(T9, T47, T48, T49, T13, appB_out_gga(T48, T49, X50)) → app3_bC_out_ggga(T9, .(T47, T48), T49, T13)
app3_bC_in_ggga(T9, .(T47, T48), T49, T13) → U5_ggga(T9, T47, T48, T49, T13, appB_in_gga(T48, T49, T52))
U5_ggga(T9, T47, T48, T49, T13, appB_out_gga(T48, T49, T52)) → U6_ggga(T9, T47, T48, T49, T13, appA_in_gga(T9, .(T47, T52), T13))
U6_ggga(T9, T47, T48, T49, T13, appA_out_gga(T9, .(T47, T52), T13)) → app3_bC_out_ggga(T9, .(T47, T48), T49, T13)

The argument filtering Pi contains the following mapping:
app3_bC_in_ggga(x1, x2, x3, x4)  =  app3_bC_in_ggga(x1, x2, x3)
[]  =  []
U3_ggga(x1, x2, x3, x4)  =  U3_ggga(x4)
appA_in_gga(x1, x2, x3)  =  appA_in_gga(x1, x2)
appA_out_gga(x1, x2, x3)  =  appA_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
app3_bC_out_ggga(x1, x2, x3, x4)  =  app3_bC_out_ggga
U4_ggga(x1, x2, x3, x4, x5, x6)  =  U4_ggga(x6)
appB_in_gga(x1, x2, x3)  =  appB_in_gga(x1, x2)
appB_out_gga(x1, x2, x3)  =  appB_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
U5_ggga(x1, x2, x3, x4, x5, x6)  =  U5_ggga(x1, x2, x6)
U6_ggga(x1, x2, x3, x4, x5, x6)  =  U6_ggga(x6)
APP3_BC_IN_GGGA(x1, x2, x3, x4)  =  APP3_BC_IN_GGGA(x1, x2, x3)
U3_GGGA(x1, x2, x3, x4)  =  U3_GGGA(x4)
APPA_IN_GGA(x1, x2, x3)  =  APPA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x5)
U4_GGGA(x1, x2, x3, x4, x5, x6)  =  U4_GGGA(x6)
APPB_IN_GGA(x1, x2, x3)  =  APPB_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x1, x5)
U5_GGGA(x1, x2, x3, x4, x5, x6)  =  U5_GGGA(x1, x2, x6)
U6_GGGA(x1, x2, x3, x4, x5, x6)  =  U6_GGGA(x6)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 9 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

APPB_IN_GGA(.(T66, T67), T68, .(T66, X74)) → APPB_IN_GGA(T67, T68, X74)

The TRS R consists of the following rules:

app3_bC_in_ggga(T9, [], T18, T13) → U3_ggga(T9, T18, T13, appA_in_gga(T9, T18, T13))
appA_in_gga([], T25, T25) → appA_out_gga([], T25, T25)
appA_in_gga(.(T34, T35), T36, .(T34, T38)) → U1_gga(T34, T35, T36, T38, appA_in_gga(T35, T36, T38))
U1_gga(T34, T35, T36, T38, appA_out_gga(T35, T36, T38)) → appA_out_gga(.(T34, T35), T36, .(T34, T38))
U3_ggga(T9, T18, T13, appA_out_gga(T9, T18, T13)) → app3_bC_out_ggga(T9, [], T18, T13)
app3_bC_in_ggga(T9, .(T47, T48), T49, T13) → U4_ggga(T9, T47, T48, T49, T13, appB_in_gga(T48, T49, X50))
appB_in_gga([], T59, T59) → appB_out_gga([], T59, T59)
appB_in_gga(.(T66, T67), T68, .(T66, X74)) → U2_gga(T66, T67, T68, X74, appB_in_gga(T67, T68, X74))
U2_gga(T66, T67, T68, X74, appB_out_gga(T67, T68, X74)) → appB_out_gga(.(T66, T67), T68, .(T66, X74))
U4_ggga(T9, T47, T48, T49, T13, appB_out_gga(T48, T49, X50)) → app3_bC_out_ggga(T9, .(T47, T48), T49, T13)
app3_bC_in_ggga(T9, .(T47, T48), T49, T13) → U5_ggga(T9, T47, T48, T49, T13, appB_in_gga(T48, T49, T52))
U5_ggga(T9, T47, T48, T49, T13, appB_out_gga(T48, T49, T52)) → U6_ggga(T9, T47, T48, T49, T13, appA_in_gga(T9, .(T47, T52), T13))
U6_ggga(T9, T47, T48, T49, T13, appA_out_gga(T9, .(T47, T52), T13)) → app3_bC_out_ggga(T9, .(T47, T48), T49, T13)

The argument filtering Pi contains the following mapping:
app3_bC_in_ggga(x1, x2, x3, x4)  =  app3_bC_in_ggga(x1, x2, x3)
[]  =  []
U3_ggga(x1, x2, x3, x4)  =  U3_ggga(x4)
appA_in_gga(x1, x2, x3)  =  appA_in_gga(x1, x2)
appA_out_gga(x1, x2, x3)  =  appA_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
app3_bC_out_ggga(x1, x2, x3, x4)  =  app3_bC_out_ggga
U4_ggga(x1, x2, x3, x4, x5, x6)  =  U4_ggga(x6)
appB_in_gga(x1, x2, x3)  =  appB_in_gga(x1, x2)
appB_out_gga(x1, x2, x3)  =  appB_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
U5_ggga(x1, x2, x3, x4, x5, x6)  =  U5_ggga(x1, x2, x6)
U6_ggga(x1, x2, x3, x4, x5, x6)  =  U6_ggga(x6)
APPB_IN_GGA(x1, x2, x3)  =  APPB_IN_GGA(x1, x2)

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:

APPB_IN_GGA(.(T66, T67), T68, .(T66, X74)) → APPB_IN_GGA(T67, T68, X74)

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

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:

APPB_IN_GGA(.(T66, T67), T68) → APPB_IN_GGA(T67, T68)

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:

• APPB_IN_GGA(.(T66, T67), T68) → APPB_IN_GGA(T67, T68)
  The graph contains the following edges 1 > 1, 2 >= 2

(15) YES

(16) Obligation:

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

APPA_IN_GGA(.(T34, T35), T36, .(T34, T38)) → APPA_IN_GGA(T35, T36, T38)

The TRS R consists of the following rules:

app3_bC_in_ggga(T9, [], T18, T13) → U3_ggga(T9, T18, T13, appA_in_gga(T9, T18, T13))
appA_in_gga([], T25, T25) → appA_out_gga([], T25, T25)
appA_in_gga(.(T34, T35), T36, .(T34, T38)) → U1_gga(T34, T35, T36, T38, appA_in_gga(T35, T36, T38))
U1_gga(T34, T35, T36, T38, appA_out_gga(T35, T36, T38)) → appA_out_gga(.(T34, T35), T36, .(T34, T38))
U3_ggga(T9, T18, T13, appA_out_gga(T9, T18, T13)) → app3_bC_out_ggga(T9, [], T18, T13)
app3_bC_in_ggga(T9, .(T47, T48), T49, T13) → U4_ggga(T9, T47, T48, T49, T13, appB_in_gga(T48, T49, X50))
appB_in_gga([], T59, T59) → appB_out_gga([], T59, T59)
appB_in_gga(.(T66, T67), T68, .(T66, X74)) → U2_gga(T66, T67, T68, X74, appB_in_gga(T67, T68, X74))
U2_gga(T66, T67, T68, X74, appB_out_gga(T67, T68, X74)) → appB_out_gga(.(T66, T67), T68, .(T66, X74))
U4_ggga(T9, T47, T48, T49, T13, appB_out_gga(T48, T49, X50)) → app3_bC_out_ggga(T9, .(T47, T48), T49, T13)
app3_bC_in_ggga(T9, .(T47, T48), T49, T13) → U5_ggga(T9, T47, T48, T49, T13, appB_in_gga(T48, T49, T52))
U5_ggga(T9, T47, T48, T49, T13, appB_out_gga(T48, T49, T52)) → U6_ggga(T9, T47, T48, T49, T13, appA_in_gga(T9, .(T47, T52), T13))
U6_ggga(T9, T47, T48, T49, T13, appA_out_gga(T9, .(T47, T52), T13)) → app3_bC_out_ggga(T9, .(T47, T48), T49, T13)

The argument filtering Pi contains the following mapping:
app3_bC_in_ggga(x1, x2, x3, x4)  =  app3_bC_in_ggga(x1, x2, x3)
[]  =  []
U3_ggga(x1, x2, x3, x4)  =  U3_ggga(x4)
appA_in_gga(x1, x2, x3)  =  appA_in_gga(x1, x2)
appA_out_gga(x1, x2, x3)  =  appA_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
app3_bC_out_ggga(x1, x2, x3, x4)  =  app3_bC_out_ggga
U4_ggga(x1, x2, x3, x4, x5, x6)  =  U4_ggga(x6)
appB_in_gga(x1, x2, x3)  =  appB_in_gga(x1, x2)
appB_out_gga(x1, x2, x3)  =  appB_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x5)
U5_ggga(x1, x2, x3, x4, x5, x6)  =  U5_ggga(x1, x2, x6)
U6_ggga(x1, x2, x3, x4, x5, x6)  =  U6_ggga(x6)
APPA_IN_GGA(x1, x2, x3)  =  APPA_IN_GGA(x1, x2)

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:

APPA_IN_GGA(.(T34, T35), T36, .(T34, T38)) → APPA_IN_GGA(T35, T36, T38)

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

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:

APPA_IN_GGA(.(T34, T35), T36) → APPA_IN_GGA(T35, T36)

R is empty.
Q is empty.
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:

• APPA_IN_GGA(.(T34, T35), T36) → APPA_IN_GGA(T35, T36)
  The graph contains the following edges 1 > 1, 2 >= 2

(22) YES