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) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: app3_b(T1, T2, T3, T4)\n\nT1 is ground\nT2 is ground\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: 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];
7 [label="7: ','(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];
10 [label="10: ','(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];
12 [label="12: ','(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];
15 [label="15: ','(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];
17 [label="B: app(T9, T18, T13)\n\nT9 is ground\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: 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];
26 [label="26: app(T9, T18, T13), SCOPE: 3, CLAUSE: 2\n\nT9 is ground\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: app(T9, T18, T13), SCOPE: 3, CLAUSE: 3\n\nT9 is ground\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: app(T35, T36, T38)\n\nT35 is ground\nT36 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="C: ','(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];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="D: app(T48, T49, X50)\n\nT48 is ground\nT49 is ground\nX50 is free", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: app(T9, .(T47, T52), T13)\n\nT9 is ground\nT47 is ground\nT52 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: 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];
59 [label="59: app(T48, T49, X50), SCOPE: 4, CLAUSE: 2\n\nT48 is ground\nT49 is ground\nX50 is free", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: app(T48, T49, X50), SCOPE: 4, CLAUSE: 3\n\nT48 is ground\nT49 is ground\nX50 is free", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: app(T67, T68, X74)\n\nT67 is ground\nT68 is ground\nX74 is free", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 73 [arrowhead = none ];
73 -> 4;

74 [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];
4 -> 74 [arrowhead = none ];
74 -> 7;

75 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 75 [arrowhead = none ];
75 -> 10;

76 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 76 [arrowhead = none ];
76 -> 12;

77 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 77 [arrowhead = none ];
77 -> 15;

78 [label="EVAL with clause\napp([], X14, X14).\nand substitutionT10 -> [],\nT11 -> T18,\nX14 -> T18,\nX9 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 78 [arrowhead = none ];
78 -> 17;

79 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 79 [arrowhead = none ];
79 -> 18;

80 [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];
15 -> 80 [arrowhead = none ];
80 -> 47;

81 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 81 [arrowhead = none ];
81 -> 49;

82 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 82 [arrowhead = none ];
82 -> 21;

83 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
21 -> 83 [arrowhead = none ];
83 -> 26;

84 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
21 -> 84 [arrowhead = none ];
84 -> 27;

85 [label="EVAL with clause\napp([], X21, X21).\nand substitutionT9 -> [],\nT18 -> T25,\nX21 -> T25,\nT13 -> T25", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 85 [arrowhead = none ];
85 -> 29;

86 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 86 [arrowhead = none ];
86 -> 32;

87 [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];
27 -> 87 [arrowhead = none ];
87 -> 39;

88 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 88 [arrowhead = none ];
88 -> 43;

89 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 89 [arrowhead = none ];
89 -> 33;

90 [label="INSTANCE with matching:\nT9 -> T35\nT18 -> T36\nT13 -> T38", fontsize=14, style = filled, fillcolor = lightgrey];
39 -> 90 [arrowhead = none , style=dashed];
90 -> 17[style=dashed];

91 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
47 -> 91 [arrowhead = none ];
91 -> 51;

92 [label="SPLIT 2\nnew knowledge:\nT48 is ground\nT49 is ground\nT52 is ground\nreplacements:X50 -> T52", fontsize=14, style = filled, fillcolor = violet];
47 -> 92 [arrowhead = none ];
92 -> 53;

93 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
51 -> 93 [arrowhead = none ];
93 -> 55;

94 [label="INSTANCE with matching:\nT18 -> .(T47, T52)", fontsize=14, style = filled, fillcolor = lightgrey];
53 -> 94 [arrowhead = none , style=dashed];
94 -> 17[style=dashed];

95 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
55 -> 95 [arrowhead = none ];
95 -> 59;

96 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
55 -> 96 [arrowhead = none ];
96 -> 60;

97 [label="EVAL with clause\napp([], X59, X59).\nand substitutionT48 -> [],\nT49 -> T59,\nX59 -> T59,\nX50 -> T59", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 97 [arrowhead = none ];
97 -> 64;

98 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
59 -> 98 [arrowhead = none ];
98 -> 67;

99 [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];
60 -> 99 [arrowhead = none ];
99 -> 70;

100 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
60 -> 100 [arrowhead = none ];
100 -> 72;

101 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
64 -> 101 [arrowhead = none ];
101 -> 68;

102 [label="INSTANCE with matching:\nT48 -> T67\nT49 -> T68\nX50 -> X74", fontsize=14, style = filled, fillcolor = lightgrey];
70 -> 102 [arrowhead = none , style=dashed];
102 -> 51[style=dashed];

}

(2) Obligation:

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

app3_bA_in_ggga(T9, [], T18, T13) → U1_ggga(T9, T18, T13, appB_in_gga(T9, T18, T13))
appB_in_gga([], T25, T25) → appB_out_gga([], T25, T25)
appB_in_gga(.(T34, T35), T36, .(T34, T38)) → U3_gga(T34, T35, T36, T38, appB_in_gga(T35, T36, T38))
U3_gga(T34, T35, T36, T38, appB_out_gga(T35, T36, T38)) → appB_out_gga(.(T34, T35), T36, .(T34, T38))
U1_ggga(T9, T18, T13, appB_out_gga(T9, T18, T13)) → app3_bA_out_ggga(T9, [], T18, T13)
app3_bA_in_ggga(T9, .(T47, T48), T49, T13) → U2_ggga(T9, T47, T48, T49, T13, pC_in_ggagga(T48, T49, X50, T9, T47, T13))
pC_in_ggagga(T48, T49, T52, T9, T47, T13) → U5_ggagga(T48, T49, T52, T9, T47, T13, appD_in_gga(T48, T49, T52))
appD_in_gga([], T59, T59) → appD_out_gga([], T59, T59)
appD_in_gga(.(T66, T67), T68, .(T66, X74)) → U4_gga(T66, T67, T68, X74, appD_in_gga(T67, T68, X74))
U4_gga(T66, T67, T68, X74, appD_out_gga(T67, T68, X74)) → appD_out_gga(.(T66, T67), T68, .(T66, X74))
U5_ggagga(T48, T49, T52, T9, T47, T13, appD_out_gga(T48, T49, T52)) → U6_ggagga(T48, T49, T52, T9, T47, T13, appB_in_gga(T9, .(T47, T52), T13))
U6_ggagga(T48, T49, T52, T9, T47, T13, appB_out_gga(T9, .(T47, T52), T13)) → pC_out_ggagga(T48, T49, T52, T9, T47, T13)
U2_ggga(T9, T47, T48, T49, T13, pC_out_ggagga(T48, T49, X50, T9, T47, T13)) → app3_bA_out_ggga(T9, .(T47, T48), T49, T13)

The argument filtering Pi contains the following mapping:
app3_bA_in_ggga(x1, x2, x3, x4)  =  app3_bA_in_ggga(x1, x2, x3)
[]  =  []
U1_ggga(x1, x2, x3, x4)  =  U1_ggga(x1, x2, x4)
appB_in_gga(x1, x2, x3)  =  appB_in_gga(x1, x2)
appB_out_gga(x1, x2, x3)  =  appB_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x2, x3, x5)
app3_bA_out_ggga(x1, x2, x3, x4)  =  app3_bA_out_ggga(x1, x2, x3, x4)
U2_ggga(x1, x2, x3, x4, x5, x6)  =  U2_ggga(x1, x2, x3, x4, x6)
pC_in_ggagga(x1, x2, x3, x4, x5, x6)  =  pC_in_ggagga(x1, x2, x4, x5)
U5_ggagga(x1, x2, x3, x4, x5, x6, x7)  =  U5_ggagga(x1, x2, x4, x5, x7)
appD_in_gga(x1, x2, x3)  =  appD_in_gga(x1, x2)
appD_out_gga(x1, x2, x3)  =  appD_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
U6_ggagga(x1, x2, x3, x4, x5, x6, x7)  =  U6_ggagga(x1, x2, x3, x4, x5, x7)
pC_out_ggagga(x1, x2, x3, x4, x5, x6)  =  pC_out_ggagga(x1, x2, x3, x4, x5, x6)

(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:

APP3_BA_IN_GGGA(T9, [], T18, T13) → U1_GGGA(T9, T18, T13, appB_in_gga(T9, T18, T13))
APP3_BA_IN_GGGA(T9, [], T18, T13) → APPB_IN_GGA(T9, T18, T13)
APPB_IN_GGA(.(T34, T35), T36, .(T34, T38)) → U3_GGA(T34, T35, T36, T38, appB_in_gga(T35, T36, T38))
APPB_IN_GGA(.(T34, T35), T36, .(T34, T38)) → APPB_IN_GGA(T35, T36, T38)
APP3_BA_IN_GGGA(T9, .(T47, T48), T49, T13) → U2_GGGA(T9, T47, T48, T49, T13, pC_in_ggagga(T48, T49, X50, T9, T47, T13))
APP3_BA_IN_GGGA(T9, .(T47, T48), T49, T13) → PC_IN_GGAGGA(T48, T49, X50, T9, T47, T13)
PC_IN_GGAGGA(T48, T49, T52, T9, T47, T13) → U5_GGAGGA(T48, T49, T52, T9, T47, T13, appD_in_gga(T48, T49, T52))
PC_IN_GGAGGA(T48, T49, T52, T9, T47, T13) → APPD_IN_GGA(T48, T49, T52)
APPD_IN_GGA(.(T66, T67), T68, .(T66, X74)) → U4_GGA(T66, T67, T68, X74, appD_in_gga(T67, T68, X74))
APPD_IN_GGA(.(T66, T67), T68, .(T66, X74)) → APPD_IN_GGA(T67, T68, X74)
U5_GGAGGA(T48, T49, T52, T9, T47, T13, appD_out_gga(T48, T49, T52)) → U6_GGAGGA(T48, T49, T52, T9, T47, T13, appB_in_gga(T9, .(T47, T52), T13))
U5_GGAGGA(T48, T49, T52, T9, T47, T13, appD_out_gga(T48, T49, T52)) → APPB_IN_GGA(T9, .(T47, T52), T13)

The TRS R consists of the following rules:

app3_bA_in_ggga(T9, [], T18, T13) → U1_ggga(T9, T18, T13, appB_in_gga(T9, T18, T13))
appB_in_gga([], T25, T25) → appB_out_gga([], T25, T25)
appB_in_gga(.(T34, T35), T36, .(T34, T38)) → U3_gga(T34, T35, T36, T38, appB_in_gga(T35, T36, T38))
U3_gga(T34, T35, T36, T38, appB_out_gga(T35, T36, T38)) → appB_out_gga(.(T34, T35), T36, .(T34, T38))
U1_ggga(T9, T18, T13, appB_out_gga(T9, T18, T13)) → app3_bA_out_ggga(T9, [], T18, T13)
app3_bA_in_ggga(T9, .(T47, T48), T49, T13) → U2_ggga(T9, T47, T48, T49, T13, pC_in_ggagga(T48, T49, X50, T9, T47, T13))
pC_in_ggagga(T48, T49, T52, T9, T47, T13) → U5_ggagga(T48, T49, T52, T9, T47, T13, appD_in_gga(T48, T49, T52))
appD_in_gga([], T59, T59) → appD_out_gga([], T59, T59)
appD_in_gga(.(T66, T67), T68, .(T66, X74)) → U4_gga(T66, T67, T68, X74, appD_in_gga(T67, T68, X74))
U4_gga(T66, T67, T68, X74, appD_out_gga(T67, T68, X74)) → appD_out_gga(.(T66, T67), T68, .(T66, X74))
U5_ggagga(T48, T49, T52, T9, T47, T13, appD_out_gga(T48, T49, T52)) → U6_ggagga(T48, T49, T52, T9, T47, T13, appB_in_gga(T9, .(T47, T52), T13))
U6_ggagga(T48, T49, T52, T9, T47, T13, appB_out_gga(T9, .(T47, T52), T13)) → pC_out_ggagga(T48, T49, T52, T9, T47, T13)
U2_ggga(T9, T47, T48, T49, T13, pC_out_ggagga(T48, T49, X50, T9, T47, T13)) → app3_bA_out_ggga(T9, .(T47, T48), T49, T13)

The argument filtering Pi contains the following mapping:
app3_bA_in_ggga(x1, x2, x3, x4)  =  app3_bA_in_ggga(x1, x2, x3)
[]  =  []
U1_ggga(x1, x2, x3, x4)  =  U1_ggga(x1, x2, x4)
appB_in_gga(x1, x2, x3)  =  appB_in_gga(x1, x2)
appB_out_gga(x1, x2, x3)  =  appB_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x2, x3, x5)
app3_bA_out_ggga(x1, x2, x3, x4)  =  app3_bA_out_ggga(x1, x2, x3, x4)
U2_ggga(x1, x2, x3, x4, x5, x6)  =  U2_ggga(x1, x2, x3, x4, x6)
pC_in_ggagga(x1, x2, x3, x4, x5, x6)  =  pC_in_ggagga(x1, x2, x4, x5)
U5_ggagga(x1, x2, x3, x4, x5, x6, x7)  =  U5_ggagga(x1, x2, x4, x5, x7)
appD_in_gga(x1, x2, x3)  =  appD_in_gga(x1, x2)
appD_out_gga(x1, x2, x3)  =  appD_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
U6_ggagga(x1, x2, x3, x4, x5, x6, x7)  =  U6_ggagga(x1, x2, x3, x4, x5, x7)
pC_out_ggagga(x1, x2, x3, x4, x5, x6)  =  pC_out_ggagga(x1, x2, x3, x4, x5, x6)
APP3_BA_IN_GGGA(x1, x2, x3, x4)  =  APP3_BA_IN_GGGA(x1, x2, x3)
U1_GGGA(x1, x2, x3, x4)  =  U1_GGGA(x1, x2, x4)
APPB_IN_GGA(x1, x2, x3)  =  APPB_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x1, x2, x3, x5)
U2_GGGA(x1, x2, x3, x4, x5, x6)  =  U2_GGGA(x1, x2, x3, x4, x6)
PC_IN_GGAGGA(x1, x2, x3, x4, x5, x6)  =  PC_IN_GGAGGA(x1, x2, x4, x5)
U5_GGAGGA(x1, x2, x3, x4, x5, x6, x7)  =  U5_GGAGGA(x1, x2, x4, x5, x7)
APPD_IN_GGA(x1, x2, x3)  =  APPD_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x2, x3, x5)
U6_GGAGGA(x1, x2, x3, x4, x5, x6, x7)  =  U6_GGAGGA(x1, x2, x3, x4, x5, x7)

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

(4) Obligation:

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

APP3_BA_IN_GGGA(T9, [], T18, T13) → U1_GGGA(T9, T18, T13, appB_in_gga(T9, T18, T13))
APP3_BA_IN_GGGA(T9, [], T18, T13) → APPB_IN_GGA(T9, T18, T13)
APPB_IN_GGA(.(T34, T35), T36, .(T34, T38)) → U3_GGA(T34, T35, T36, T38, appB_in_gga(T35, T36, T38))
APPB_IN_GGA(.(T34, T35), T36, .(T34, T38)) → APPB_IN_GGA(T35, T36, T38)
APP3_BA_IN_GGGA(T9, .(T47, T48), T49, T13) → U2_GGGA(T9, T47, T48, T49, T13, pC_in_ggagga(T48, T49, X50, T9, T47, T13))
APP3_BA_IN_GGGA(T9, .(T47, T48), T49, T13) → PC_IN_GGAGGA(T48, T49, X50, T9, T47, T13)
PC_IN_GGAGGA(T48, T49, T52, T9, T47, T13) → U5_GGAGGA(T48, T49, T52, T9, T47, T13, appD_in_gga(T48, T49, T52))
PC_IN_GGAGGA(T48, T49, T52, T9, T47, T13) → APPD_IN_GGA(T48, T49, T52)
APPD_IN_GGA(.(T66, T67), T68, .(T66, X74)) → U4_GGA(T66, T67, T68, X74, appD_in_gga(T67, T68, X74))
APPD_IN_GGA(.(T66, T67), T68, .(T66, X74)) → APPD_IN_GGA(T67, T68, X74)
U5_GGAGGA(T48, T49, T52, T9, T47, T13, appD_out_gga(T48, T49, T52)) → U6_GGAGGA(T48, T49, T52, T9, T47, T13, appB_in_gga(T9, .(T47, T52), T13))
U5_GGAGGA(T48, T49, T52, T9, T47, T13, appD_out_gga(T48, T49, T52)) → APPB_IN_GGA(T9, .(T47, T52), T13)

The TRS R consists of the following rules:

app3_bA_in_ggga(T9, [], T18, T13) → U1_ggga(T9, T18, T13, appB_in_gga(T9, T18, T13))
appB_in_gga([], T25, T25) → appB_out_gga([], T25, T25)
appB_in_gga(.(T34, T35), T36, .(T34, T38)) → U3_gga(T34, T35, T36, T38, appB_in_gga(T35, T36, T38))
U3_gga(T34, T35, T36, T38, appB_out_gga(T35, T36, T38)) → appB_out_gga(.(T34, T35), T36, .(T34, T38))
U1_ggga(T9, T18, T13, appB_out_gga(T9, T18, T13)) → app3_bA_out_ggga(T9, [], T18, T13)
app3_bA_in_ggga(T9, .(T47, T48), T49, T13) → U2_ggga(T9, T47, T48, T49, T13, pC_in_ggagga(T48, T49, X50, T9, T47, T13))
pC_in_ggagga(T48, T49, T52, T9, T47, T13) → U5_ggagga(T48, T49, T52, T9, T47, T13, appD_in_gga(T48, T49, T52))
appD_in_gga([], T59, T59) → appD_out_gga([], T59, T59)
appD_in_gga(.(T66, T67), T68, .(T66, X74)) → U4_gga(T66, T67, T68, X74, appD_in_gga(T67, T68, X74))
U4_gga(T66, T67, T68, X74, appD_out_gga(T67, T68, X74)) → appD_out_gga(.(T66, T67), T68, .(T66, X74))
U5_ggagga(T48, T49, T52, T9, T47, T13, appD_out_gga(T48, T49, T52)) → U6_ggagga(T48, T49, T52, T9, T47, T13, appB_in_gga(T9, .(T47, T52), T13))
U6_ggagga(T48, T49, T52, T9, T47, T13, appB_out_gga(T9, .(T47, T52), T13)) → pC_out_ggagga(T48, T49, T52, T9, T47, T13)
U2_ggga(T9, T47, T48, T49, T13, pC_out_ggagga(T48, T49, X50, T9, T47, T13)) → app3_bA_out_ggga(T9, .(T47, T48), T49, T13)

The argument filtering Pi contains the following mapping:
app3_bA_in_ggga(x1, x2, x3, x4)  =  app3_bA_in_ggga(x1, x2, x3)
[]  =  []
U1_ggga(x1, x2, x3, x4)  =  U1_ggga(x1, x2, x4)
appB_in_gga(x1, x2, x3)  =  appB_in_gga(x1, x2)
appB_out_gga(x1, x2, x3)  =  appB_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x2, x3, x5)
app3_bA_out_ggga(x1, x2, x3, x4)  =  app3_bA_out_ggga(x1, x2, x3, x4)
U2_ggga(x1, x2, x3, x4, x5, x6)  =  U2_ggga(x1, x2, x3, x4, x6)
pC_in_ggagga(x1, x2, x3, x4, x5, x6)  =  pC_in_ggagga(x1, x2, x4, x5)
U5_ggagga(x1, x2, x3, x4, x5, x6, x7)  =  U5_ggagga(x1, x2, x4, x5, x7)
appD_in_gga(x1, x2, x3)  =  appD_in_gga(x1, x2)
appD_out_gga(x1, x2, x3)  =  appD_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
U6_ggagga(x1, x2, x3, x4, x5, x6, x7)  =  U6_ggagga(x1, x2, x3, x4, x5, x7)
pC_out_ggagga(x1, x2, x3, x4, x5, x6)  =  pC_out_ggagga(x1, x2, x3, x4, x5, x6)
APP3_BA_IN_GGGA(x1, x2, x3, x4)  =  APP3_BA_IN_GGGA(x1, x2, x3)
U1_GGGA(x1, x2, x3, x4)  =  U1_GGGA(x1, x2, x4)
APPB_IN_GGA(x1, x2, x3)  =  APPB_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x1, x2, x3, x5)
U2_GGGA(x1, x2, x3, x4, x5, x6)  =  U2_GGGA(x1, x2, x3, x4, x6)
PC_IN_GGAGGA(x1, x2, x3, x4, x5, x6)  =  PC_IN_GGAGGA(x1, x2, x4, x5)
U5_GGAGGA(x1, x2, x3, x4, x5, x6, x7)  =  U5_GGAGGA(x1, x2, x4, x5, x7)
APPD_IN_GGA(x1, x2, x3)  =  APPD_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x2, x3, x5)
U6_GGAGGA(x1, x2, x3, x4, x5, x6, x7)  =  U6_GGAGGA(x1, x2, x3, x4, x5, x7)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 10 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

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

The TRS R consists of the following rules:

app3_bA_in_ggga(T9, [], T18, T13) → U1_ggga(T9, T18, T13, appB_in_gga(T9, T18, T13))
appB_in_gga([], T25, T25) → appB_out_gga([], T25, T25)
appB_in_gga(.(T34, T35), T36, .(T34, T38)) → U3_gga(T34, T35, T36, T38, appB_in_gga(T35, T36, T38))
U3_gga(T34, T35, T36, T38, appB_out_gga(T35, T36, T38)) → appB_out_gga(.(T34, T35), T36, .(T34, T38))
U1_ggga(T9, T18, T13, appB_out_gga(T9, T18, T13)) → app3_bA_out_ggga(T9, [], T18, T13)
app3_bA_in_ggga(T9, .(T47, T48), T49, T13) → U2_ggga(T9, T47, T48, T49, T13, pC_in_ggagga(T48, T49, X50, T9, T47, T13))
pC_in_ggagga(T48, T49, T52, T9, T47, T13) → U5_ggagga(T48, T49, T52, T9, T47, T13, appD_in_gga(T48, T49, T52))
appD_in_gga([], T59, T59) → appD_out_gga([], T59, T59)
appD_in_gga(.(T66, T67), T68, .(T66, X74)) → U4_gga(T66, T67, T68, X74, appD_in_gga(T67, T68, X74))
U4_gga(T66, T67, T68, X74, appD_out_gga(T67, T68, X74)) → appD_out_gga(.(T66, T67), T68, .(T66, X74))
U5_ggagga(T48, T49, T52, T9, T47, T13, appD_out_gga(T48, T49, T52)) → U6_ggagga(T48, T49, T52, T9, T47, T13, appB_in_gga(T9, .(T47, T52), T13))
U6_ggagga(T48, T49, T52, T9, T47, T13, appB_out_gga(T9, .(T47, T52), T13)) → pC_out_ggagga(T48, T49, T52, T9, T47, T13)
U2_ggga(T9, T47, T48, T49, T13, pC_out_ggagga(T48, T49, X50, T9, T47, T13)) → app3_bA_out_ggga(T9, .(T47, T48), T49, T13)

The argument filtering Pi contains the following mapping:
app3_bA_in_ggga(x1, x2, x3, x4)  =  app3_bA_in_ggga(x1, x2, x3)
[]  =  []
U1_ggga(x1, x2, x3, x4)  =  U1_ggga(x1, x2, x4)
appB_in_gga(x1, x2, x3)  =  appB_in_gga(x1, x2)
appB_out_gga(x1, x2, x3)  =  appB_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x2, x3, x5)
app3_bA_out_ggga(x1, x2, x3, x4)  =  app3_bA_out_ggga(x1, x2, x3, x4)
U2_ggga(x1, x2, x3, x4, x5, x6)  =  U2_ggga(x1, x2, x3, x4, x6)
pC_in_ggagga(x1, x2, x3, x4, x5, x6)  =  pC_in_ggagga(x1, x2, x4, x5)
U5_ggagga(x1, x2, x3, x4, x5, x6, x7)  =  U5_ggagga(x1, x2, x4, x5, x7)
appD_in_gga(x1, x2, x3)  =  appD_in_gga(x1, x2)
appD_out_gga(x1, x2, x3)  =  appD_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
U6_ggagga(x1, x2, x3, x4, x5, x6, x7)  =  U6_ggagga(x1, x2, x3, x4, x5, x7)
pC_out_ggagga(x1, x2, x3, x4, x5, x6)  =  pC_out_ggagga(x1, x2, x3, x4, x5, x6)
APPD_IN_GGA(x1, x2, x3)  =  APPD_IN_GGA(x1, x2)

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:

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

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

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:

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

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:

• APPD_IN_GGA(.(T66, T67), T68) → APPD_IN_GGA(T67, T68)
  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:

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

The TRS R consists of the following rules:

app3_bA_in_ggga(T9, [], T18, T13) → U1_ggga(T9, T18, T13, appB_in_gga(T9, T18, T13))
appB_in_gga([], T25, T25) → appB_out_gga([], T25, T25)
appB_in_gga(.(T34, T35), T36, .(T34, T38)) → U3_gga(T34, T35, T36, T38, appB_in_gga(T35, T36, T38))
U3_gga(T34, T35, T36, T38, appB_out_gga(T35, T36, T38)) → appB_out_gga(.(T34, T35), T36, .(T34, T38))
U1_ggga(T9, T18, T13, appB_out_gga(T9, T18, T13)) → app3_bA_out_ggga(T9, [], T18, T13)
app3_bA_in_ggga(T9, .(T47, T48), T49, T13) → U2_ggga(T9, T47, T48, T49, T13, pC_in_ggagga(T48, T49, X50, T9, T47, T13))
pC_in_ggagga(T48, T49, T52, T9, T47, T13) → U5_ggagga(T48, T49, T52, T9, T47, T13, appD_in_gga(T48, T49, T52))
appD_in_gga([], T59, T59) → appD_out_gga([], T59, T59)
appD_in_gga(.(T66, T67), T68, .(T66, X74)) → U4_gga(T66, T67, T68, X74, appD_in_gga(T67, T68, X74))
U4_gga(T66, T67, T68, X74, appD_out_gga(T67, T68, X74)) → appD_out_gga(.(T66, T67), T68, .(T66, X74))
U5_ggagga(T48, T49, T52, T9, T47, T13, appD_out_gga(T48, T49, T52)) → U6_ggagga(T48, T49, T52, T9, T47, T13, appB_in_gga(T9, .(T47, T52), T13))
U6_ggagga(T48, T49, T52, T9, T47, T13, appB_out_gga(T9, .(T47, T52), T13)) → pC_out_ggagga(T48, T49, T52, T9, T47, T13)
U2_ggga(T9, T47, T48, T49, T13, pC_out_ggagga(T48, T49, X50, T9, T47, T13)) → app3_bA_out_ggga(T9, .(T47, T48), T49, T13)

The argument filtering Pi contains the following mapping:
app3_bA_in_ggga(x1, x2, x3, x4)  =  app3_bA_in_ggga(x1, x2, x3)
[]  =  []
U1_ggga(x1, x2, x3, x4)  =  U1_ggga(x1, x2, x4)
appB_in_gga(x1, x2, x3)  =  appB_in_gga(x1, x2)
appB_out_gga(x1, x2, x3)  =  appB_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x2, x3, x5)
app3_bA_out_ggga(x1, x2, x3, x4)  =  app3_bA_out_ggga(x1, x2, x3, x4)
U2_ggga(x1, x2, x3, x4, x5, x6)  =  U2_ggga(x1, x2, x3, x4, x6)
pC_in_ggagga(x1, x2, x3, x4, x5, x6)  =  pC_in_ggagga(x1, x2, x4, x5)
U5_ggagga(x1, x2, x3, x4, x5, x6, x7)  =  U5_ggagga(x1, x2, x4, x5, x7)
appD_in_gga(x1, x2, x3)  =  appD_in_gga(x1, x2)
appD_out_gga(x1, x2, x3)  =  appD_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
U6_ggagga(x1, x2, x3, x4, x5, x6, x7)  =  U6_ggagga(x1, x2, x3, x4, x5, x7)
pC_out_ggagga(x1, x2, x3, x4, x5, x6)  =  pC_out_ggagga(x1, x2, x3, x4, x5, x6)
APPB_IN_GGA(x1, x2, x3)  =  APPB_IN_GGA(x1, x2)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

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

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

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

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

(19) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

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

(20) YES