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

(0) Obligation:

Clauses:

rotate(X, Y) :- ','(append2(A, B, X), append1(B, A, Y)).
append1(.(X, Xs), Ys, .(X, Zs)) :- append1(Xs, Ys, Zs).
append1([], Ys, Ys).
append2(.(X, Xs), Ys, .(X, Zs)) :- append2(Xs, Ys, Zs).
append2([], Ys, Ys).

Query: rotate(g,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: rotate(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
6 [label="6: rotate(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(append2(X5, X6, T5), append1(X6, X5, T7))\n\nT5 is ground\nX5 is free\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(append2(X5, X6, T5), append1(X6, X5, T7)), SCOPE: 2, CLAUSE: 3 | ','(append2(X5, X6, T5), append1(X6, X5, T7)), SCOPE: 2, CLAUSE: 4\n\nT5 is ground\nX5 is free\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(append2(X5, X6, T5), append1(X6, X5, T7)), SCOPE: 2, CLAUSE: 3\n\nT5 is ground\nX5 is free\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(append2(X5, X6, T5), append1(X6, X5, T7)), SCOPE: 2, CLAUSE: 4\n\nT5 is ground\nX5 is free\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="B: ','(append2(X40, X41, T17), append1(X41, .(T16, X40), T7))\n\nT16 is ground\nT17 is ground\nX40 is free\nX41 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="D: append2(X40, X41, T17)\n\nT17 is ground\nX40 is free\nX41 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="E: append1(T21, .(T16, T20), T7)\n\nT16 is ground\nT20 is ground\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: append2(X40, X41, T17), SCOPE: 3, CLAUSE: 3 | append2(X40, X41, T17), SCOPE: 3, CLAUSE: 4\n\nT17 is ground\nX40 is free\nX41 is free", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: append2(X40, X41, T17), SCOPE: 3, CLAUSE: 3\n\nT17 is ground\nX40 is free\nX41 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: append2(X40, X41, T17), SCOPE: 3, CLAUSE: 4\n\nT17 is ground\nX40 is free\nX41 is free", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: append2(X89, X90, T33)\n\nT33 is ground\nX89 is free\nX90 is free", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: append1(T21, .(T16, T20), T7), SCOPE: 4, CLAUSE: 1 | append1(T21, .(T16, T20), T7), SCOPE: 4, CLAUSE: 2\n\nT16 is ground\nT20 is ground\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: append1(T21, .(T16, T20), T7), SCOPE: 4, CLAUSE: 1\n\nT16 is ground\nT20 is ground\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: append1(T21, .(T16, T20), T7), SCOPE: 4, CLAUSE: 2\n\nT16 is ground\nT20 is ground\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: append1(T65, .(T66, T67), T69)\n\nT65 is ground\nT66 is ground\nT67 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="C: append1(T86, [], T7)\n\nT86 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: append1(T86, [], T7), SCOPE: 5, CLAUSE: 1 | append1(T86, [], T7), SCOPE: 5, CLAUSE: 2\n\nT86 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: append1(T86, [], T7), SCOPE: 5, CLAUSE: 1\n\nT86 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: append1(T86, [], T7), SCOPE: 5, CLAUSE: 2\n\nT86 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: append1(T103, [], T105)\n\nT103 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 89 [arrowhead = none ];
89 -> 6;

90 [label="ONLY EVAL with clause\nrotate(X3, X4) :- ','(append2(X5, X6, X3), append1(X6, X5, X4)).\nand substitutionT1 -> T5,\nX3 -> T5,\nT2 -> T7,\nX4 -> T7,\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 90 [arrowhead = none ];
90 -> 7;

91 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 91 [arrowhead = none ];
91 -> 10;

92 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 92 [arrowhead = none ];
92 -> 13;

93 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 93 [arrowhead = none ];
93 -> 15;

94 [label="EVAL with clause\nappend2(.(X35, X36), X37, .(X35, X38)) :- append2(X36, X37, X38).\nand substitutionX35 -> T16,\nX36 -> X40,\nX5 -> .(T16, X40),\nX6 -> X41,\nX37 -> X41,\nX39 -> T16,\nX38 -> T17,\nT5 -> .(T16, T17)", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 94 [arrowhead = none ];
94 -> 19;

95 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 95 [arrowhead = none ];
95 -> 22;

96 [label="ONLY EVAL with clause\nappend2([], X143, X143).\nand substitutionX5 -> [],\nX6 -> T86,\nX143 -> T86,\nT5 -> T86,\nX144 -> T86", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 96 [arrowhead = none ];
96 -> 80;

97 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
19 -> 97 [arrowhead = none ];
97 -> 27;

98 [label="SPLIT 2\nnew knowledge:\nT20 is ground\nT21 is ground\nT17 is ground\nreplacements:X40 -> T20,\nX41 -> T21", fontsize=14, style = filled, fillcolor = violet];
19 -> 98 [arrowhead = none ];
98 -> 28;

99 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
27 -> 99 [arrowhead = none ];
99 -> 34;

100 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
28 -> 100 [arrowhead = none ];
100 -> 64;

101 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
34 -> 101 [arrowhead = none ];
101 -> 39;

102 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
34 -> 102 [arrowhead = none ];
102 -> 40;

103 [label="EVAL with clause\nappend2(.(X84, X85), X86, .(X84, X87)) :- append2(X85, X86, X87).\nand substitutionX84 -> T32,\nX85 -> X89,\nX40 -> .(T32, X89),\nX41 -> X90,\nX86 -> X90,\nX88 -> T32,\nX87 -> T33,\nT17 -> .(T32, T33)", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 103 [arrowhead = none ];
103 -> 47;

104 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 104 [arrowhead = none ];
104 -> 48;

105 [label="ONLY EVAL with clause\nappend2([], X102, X102).\nand substitutionX40 -> [],\nX41 -> T38,\nX102 -> T38,\nT17 -> T38,\nX103 -> T38", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 105 [arrowhead = none ];
105 -> 61;

106 [label="INSTANCE with matching:\nX40 -> X89\nX41 -> X90\nT17 -> T33", fontsize=14, style = filled, fillcolor = lightgrey];
47 -> 106 [arrowhead = none , style=dashed];
106 -> 27[style=dashed];

107 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
61 -> 107 [arrowhead = none ];
107 -> 62;

108 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
64 -> 108 [arrowhead = none ];
108 -> 73;

109 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
64 -> 109 [arrowhead = none ];
109 -> 74;

110 [label="EVAL with clause\nappend1(.(X124, X125), X126, .(X124, X127)) :- append1(X125, X126, X127).\nand substitutionX124 -> T64,\nX125 -> T65,\nT21 -> .(T64, T65),\nT16 -> T66,\nT20 -> T67,\nX126 -> .(T66, T67),\nX127 -> T69,\nT7 -> .(T64, T69),\nT68 -> T69", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 110 [arrowhead = none ];
110 -> 75;

111 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
73 -> 111 [arrowhead = none ];
111 -> 76;

112 [label="EVAL with clause\nappend1([], X134, X134).\nand substitutionT21 -> [],\nT16 -> T79,\nT20 -> T80,\nX134 -> .(T79, T80),\nT7 -> .(T79, T80)", fontsize=14, style = filled, fillcolor = lightblue];
74 -> 112 [arrowhead = none ];
112 -> 77;

113 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
74 -> 113 [arrowhead = none ];
113 -> 78;

114 [label="INSTANCE with matching:\nT21 -> T65\nT16 -> T66\nT20 -> T67\nT7 -> T69", fontsize=14, style = filled, fillcolor = lightgrey];
75 -> 114 [arrowhead = none , style=dashed];
114 -> 28[style=dashed];

115 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
77 -> 115 [arrowhead = none ];
115 -> 79;

116 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
80 -> 116 [arrowhead = none ];
116 -> 81;

117 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
81 -> 117 [arrowhead = none ];
117 -> 82;

118 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
81 -> 118 [arrowhead = none ];
118 -> 83;

119 [label="EVAL with clause\nappend1(.(X165, X166), X167, .(X165, X168)) :- append1(X166, X167, X168).\nand substitutionX165 -> T102,\nX166 -> T103,\nT86 -> .(T102, T103),\nX167 -> [],\nX168 -> T105,\nT7 -> .(T102, T105),\nT104 -> T105", fontsize=14, style = filled, fillcolor = lightblue];
82 -> 119 [arrowhead = none ];
119 -> 84;

120 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
82 -> 120 [arrowhead = none ];
120 -> 85;

121 [label="EVAL with clause\nappend1([], X175, X175).\nand substitutionT86 -> [],\nX175 -> [],\nT7 -> []", fontsize=14, style = filled, fillcolor = lightblue];
83 -> 121 [arrowhead = none ];
121 -> 86;

122 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
83 -> 122 [arrowhead = none ];
122 -> 87;

123 [label="INSTANCE with matching:\nT86 -> T103\nT7 -> T105", fontsize=14, style = filled, fillcolor = lightgrey];
84 -> 123 [arrowhead = none , style=dashed];
123 -> 80[style=dashed];

124 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
86 -> 124 [arrowhead = none ];
124 -> 88;

}

(2) Obligation:

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

rotateA_in_ga(.(T16, T17), T7) → U1_ga(T16, T17, T7, pB_in_aagga(X40, X41, T17, T16, T7))
pB_in_aagga(T20, T21, T17, T16, T7) → U6_aagga(T20, T21, T17, T16, T7, append2D_in_aag(T20, T21, T17))
append2D_in_aag(.(T32, X89), X90, .(T32, T33)) → U3_aag(T32, X89, X90, T33, append2D_in_aag(X89, X90, T33))
append2D_in_aag([], T38, T38) → append2D_out_aag([], T38, T38)
U3_aag(T32, X89, X90, T33, append2D_out_aag(X89, X90, T33)) → append2D_out_aag(.(T32, X89), X90, .(T32, T33))
U6_aagga(T20, T21, T17, T16, T7, append2D_out_aag(T20, T21, T17)) → U7_aagga(T20, T21, T17, T16, T7, append1E_in_ggga(T21, T16, T20, T7))
append1E_in_ggga(.(T64, T65), T66, T67, .(T64, T69)) → U4_ggga(T64, T65, T66, T67, T69, append1E_in_ggga(T65, T66, T67, T69))
append1E_in_ggga([], T79, T80, .(T79, T80)) → append1E_out_ggga([], T79, T80, .(T79, T80))
U4_ggga(T64, T65, T66, T67, T69, append1E_out_ggga(T65, T66, T67, T69)) → append1E_out_ggga(.(T64, T65), T66, T67, .(T64, T69))
U7_aagga(T20, T21, T17, T16, T7, append1E_out_ggga(T21, T16, T20, T7)) → pB_out_aagga(T20, T21, T17, T16, T7)
U1_ga(T16, T17, T7, pB_out_aagga(X40, X41, T17, T16, T7)) → rotateA_out_ga(.(T16, T17), T7)
rotateA_in_ga(T86, T7) → U2_ga(T86, T7, append1C_in_ga(T86, T7))
append1C_in_ga(.(T102, T103), .(T102, T105)) → U5_ga(T102, T103, T105, append1C_in_ga(T103, T105))
append1C_in_ga([], []) → append1C_out_ga([], [])
U5_ga(T102, T103, T105, append1C_out_ga(T103, T105)) → append1C_out_ga(.(T102, T103), .(T102, T105))
U2_ga(T86, T7, append1C_out_ga(T86, T7)) → rotateA_out_ga(T86, T7)

The argument filtering Pi contains the following mapping:
rotateA_in_ga(x1, x2)  =  rotateA_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
pB_in_aagga(x1, x2, x3, x4, x5)  =  pB_in_aagga(x3, x4)
U6_aagga(x1, x2, x3, x4, x5, x6)  =  U6_aagga(x3, x4, x6)
append2D_in_aag(x1, x2, x3)  =  append2D_in_aag(x3)
U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x1, x4, x5)
append2D_out_aag(x1, x2, x3)  =  append2D_out_aag(x1, x2, x3)
U7_aagga(x1, x2, x3, x4, x5, x6)  =  U7_aagga(x1, x2, x3, x4, x6)
append1E_in_ggga(x1, x2, x3, x4)  =  append1E_in_ggga(x1, x2, x3)
U4_ggga(x1, x2, x3, x4, x5, x6)  =  U4_ggga(x1, x2, x3, x4, x6)
[]  =  []
append1E_out_ggga(x1, x2, x3, x4)  =  append1E_out_ggga(x1, x2, x3, x4)
pB_out_aagga(x1, x2, x3, x4, x5)  =  pB_out_aagga(x1, x2, x3, x4, x5)
rotateA_out_ga(x1, x2)  =  rotateA_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
append1C_in_ga(x1, x2)  =  append1C_in_ga(x1)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x2, x4)
append1C_out_ga(x1, x2)  =  append1C_out_ga(x1, x2)

(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:

ROTATEA_IN_GA(.(T16, T17), T7) → U1_GA(T16, T17, T7, pB_in_aagga(X40, X41, T17, T16, T7))
ROTATEA_IN_GA(.(T16, T17), T7) → PB_IN_AAGGA(X40, X41, T17, T16, T7)
PB_IN_AAGGA(T20, T21, T17, T16, T7) → U6_AAGGA(T20, T21, T17, T16, T7, append2D_in_aag(T20, T21, T17))
PB_IN_AAGGA(T20, T21, T17, T16, T7) → APPEND2D_IN_AAG(T20, T21, T17)
APPEND2D_IN_AAG(.(T32, X89), X90, .(T32, T33)) → U3_AAG(T32, X89, X90, T33, append2D_in_aag(X89, X90, T33))
APPEND2D_IN_AAG(.(T32, X89), X90, .(T32, T33)) → APPEND2D_IN_AAG(X89, X90, T33)
U6_AAGGA(T20, T21, T17, T16, T7, append2D_out_aag(T20, T21, T17)) → U7_AAGGA(T20, T21, T17, T16, T7, append1E_in_ggga(T21, T16, T20, T7))
U6_AAGGA(T20, T21, T17, T16, T7, append2D_out_aag(T20, T21, T17)) → APPEND1E_IN_GGGA(T21, T16, T20, T7)
APPEND1E_IN_GGGA(.(T64, T65), T66, T67, .(T64, T69)) → U4_GGGA(T64, T65, T66, T67, T69, append1E_in_ggga(T65, T66, T67, T69))
APPEND1E_IN_GGGA(.(T64, T65), T66, T67, .(T64, T69)) → APPEND1E_IN_GGGA(T65, T66, T67, T69)
ROTATEA_IN_GA(T86, T7) → U2_GA(T86, T7, append1C_in_ga(T86, T7))
ROTATEA_IN_GA(T86, T7) → APPEND1C_IN_GA(T86, T7)
APPEND1C_IN_GA(.(T102, T103), .(T102, T105)) → U5_GA(T102, T103, T105, append1C_in_ga(T103, T105))
APPEND1C_IN_GA(.(T102, T103), .(T102, T105)) → APPEND1C_IN_GA(T103, T105)

The TRS R consists of the following rules:

rotateA_in_ga(.(T16, T17), T7) → U1_ga(T16, T17, T7, pB_in_aagga(X40, X41, T17, T16, T7))
pB_in_aagga(T20, T21, T17, T16, T7) → U6_aagga(T20, T21, T17, T16, T7, append2D_in_aag(T20, T21, T17))
append2D_in_aag(.(T32, X89), X90, .(T32, T33)) → U3_aag(T32, X89, X90, T33, append2D_in_aag(X89, X90, T33))
append2D_in_aag([], T38, T38) → append2D_out_aag([], T38, T38)
U3_aag(T32, X89, X90, T33, append2D_out_aag(X89, X90, T33)) → append2D_out_aag(.(T32, X89), X90, .(T32, T33))
U6_aagga(T20, T21, T17, T16, T7, append2D_out_aag(T20, T21, T17)) → U7_aagga(T20, T21, T17, T16, T7, append1E_in_ggga(T21, T16, T20, T7))
append1E_in_ggga(.(T64, T65), T66, T67, .(T64, T69)) → U4_ggga(T64, T65, T66, T67, T69, append1E_in_ggga(T65, T66, T67, T69))
append1E_in_ggga([], T79, T80, .(T79, T80)) → append1E_out_ggga([], T79, T80, .(T79, T80))
U4_ggga(T64, T65, T66, T67, T69, append1E_out_ggga(T65, T66, T67, T69)) → append1E_out_ggga(.(T64, T65), T66, T67, .(T64, T69))
U7_aagga(T20, T21, T17, T16, T7, append1E_out_ggga(T21, T16, T20, T7)) → pB_out_aagga(T20, T21, T17, T16, T7)
U1_ga(T16, T17, T7, pB_out_aagga(X40, X41, T17, T16, T7)) → rotateA_out_ga(.(T16, T17), T7)
rotateA_in_ga(T86, T7) → U2_ga(T86, T7, append1C_in_ga(T86, T7))
append1C_in_ga(.(T102, T103), .(T102, T105)) → U5_ga(T102, T103, T105, append1C_in_ga(T103, T105))
append1C_in_ga([], []) → append1C_out_ga([], [])
U5_ga(T102, T103, T105, append1C_out_ga(T103, T105)) → append1C_out_ga(.(T102, T103), .(T102, T105))
U2_ga(T86, T7, append1C_out_ga(T86, T7)) → rotateA_out_ga(T86, T7)

The argument filtering Pi contains the following mapping:
rotateA_in_ga(x1, x2)  =  rotateA_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
pB_in_aagga(x1, x2, x3, x4, x5)  =  pB_in_aagga(x3, x4)
U6_aagga(x1, x2, x3, x4, x5, x6)  =  U6_aagga(x3, x4, x6)
append2D_in_aag(x1, x2, x3)  =  append2D_in_aag(x3)
U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x1, x4, x5)
append2D_out_aag(x1, x2, x3)  =  append2D_out_aag(x1, x2, x3)
U7_aagga(x1, x2, x3, x4, x5, x6)  =  U7_aagga(x1, x2, x3, x4, x6)
append1E_in_ggga(x1, x2, x3, x4)  =  append1E_in_ggga(x1, x2, x3)
U4_ggga(x1, x2, x3, x4, x5, x6)  =  U4_ggga(x1, x2, x3, x4, x6)
[]  =  []
append1E_out_ggga(x1, x2, x3, x4)  =  append1E_out_ggga(x1, x2, x3, x4)
pB_out_aagga(x1, x2, x3, x4, x5)  =  pB_out_aagga(x1, x2, x3, x4, x5)
rotateA_out_ga(x1, x2)  =  rotateA_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
append1C_in_ga(x1, x2)  =  append1C_in_ga(x1)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x2, x4)
append1C_out_ga(x1, x2)  =  append1C_out_ga(x1, x2)
ROTATEA_IN_GA(x1, x2)  =  ROTATEA_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)
PB_IN_AAGGA(x1, x2, x3, x4, x5)  =  PB_IN_AAGGA(x3, x4)
U6_AAGGA(x1, x2, x3, x4, x5, x6)  =  U6_AAGGA(x3, x4, x6)
APPEND2D_IN_AAG(x1, x2, x3)  =  APPEND2D_IN_AAG(x3)
U3_AAG(x1, x2, x3, x4, x5)  =  U3_AAG(x1, x4, x5)
U7_AAGGA(x1, x2, x3, x4, x5, x6)  =  U7_AAGGA(x1, x2, x3, x4, x6)
APPEND1E_IN_GGGA(x1, x2, x3, x4)  =  APPEND1E_IN_GGGA(x1, x2, x3)
U4_GGGA(x1, x2, x3, x4, x5, x6)  =  U4_GGGA(x1, x2, x3, x4, x6)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
APPEND1C_IN_GA(x1, x2)  =  APPEND1C_IN_GA(x1)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x1, x2, x4)

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

(4) Obligation:

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

ROTATEA_IN_GA(.(T16, T17), T7) → U1_GA(T16, T17, T7, pB_in_aagga(X40, X41, T17, T16, T7))
ROTATEA_IN_GA(.(T16, T17), T7) → PB_IN_AAGGA(X40, X41, T17, T16, T7)
PB_IN_AAGGA(T20, T21, T17, T16, T7) → U6_AAGGA(T20, T21, T17, T16, T7, append2D_in_aag(T20, T21, T17))
PB_IN_AAGGA(T20, T21, T17, T16, T7) → APPEND2D_IN_AAG(T20, T21, T17)
APPEND2D_IN_AAG(.(T32, X89), X90, .(T32, T33)) → U3_AAG(T32, X89, X90, T33, append2D_in_aag(X89, X90, T33))
APPEND2D_IN_AAG(.(T32, X89), X90, .(T32, T33)) → APPEND2D_IN_AAG(X89, X90, T33)
U6_AAGGA(T20, T21, T17, T16, T7, append2D_out_aag(T20, T21, T17)) → U7_AAGGA(T20, T21, T17, T16, T7, append1E_in_ggga(T21, T16, T20, T7))
U6_AAGGA(T20, T21, T17, T16, T7, append2D_out_aag(T20, T21, T17)) → APPEND1E_IN_GGGA(T21, T16, T20, T7)
APPEND1E_IN_GGGA(.(T64, T65), T66, T67, .(T64, T69)) → U4_GGGA(T64, T65, T66, T67, T69, append1E_in_ggga(T65, T66, T67, T69))
APPEND1E_IN_GGGA(.(T64, T65), T66, T67, .(T64, T69)) → APPEND1E_IN_GGGA(T65, T66, T67, T69)
ROTATEA_IN_GA(T86, T7) → U2_GA(T86, T7, append1C_in_ga(T86, T7))
ROTATEA_IN_GA(T86, T7) → APPEND1C_IN_GA(T86, T7)
APPEND1C_IN_GA(.(T102, T103), .(T102, T105)) → U5_GA(T102, T103, T105, append1C_in_ga(T103, T105))
APPEND1C_IN_GA(.(T102, T103), .(T102, T105)) → APPEND1C_IN_GA(T103, T105)

The TRS R consists of the following rules:

rotateA_in_ga(.(T16, T17), T7) → U1_ga(T16, T17, T7, pB_in_aagga(X40, X41, T17, T16, T7))
pB_in_aagga(T20, T21, T17, T16, T7) → U6_aagga(T20, T21, T17, T16, T7, append2D_in_aag(T20, T21, T17))
append2D_in_aag(.(T32, X89), X90, .(T32, T33)) → U3_aag(T32, X89, X90, T33, append2D_in_aag(X89, X90, T33))
append2D_in_aag([], T38, T38) → append2D_out_aag([], T38, T38)
U3_aag(T32, X89, X90, T33, append2D_out_aag(X89, X90, T33)) → append2D_out_aag(.(T32, X89), X90, .(T32, T33))
U6_aagga(T20, T21, T17, T16, T7, append2D_out_aag(T20, T21, T17)) → U7_aagga(T20, T21, T17, T16, T7, append1E_in_ggga(T21, T16, T20, T7))
append1E_in_ggga(.(T64, T65), T66, T67, .(T64, T69)) → U4_ggga(T64, T65, T66, T67, T69, append1E_in_ggga(T65, T66, T67, T69))
append1E_in_ggga([], T79, T80, .(T79, T80)) → append1E_out_ggga([], T79, T80, .(T79, T80))
U4_ggga(T64, T65, T66, T67, T69, append1E_out_ggga(T65, T66, T67, T69)) → append1E_out_ggga(.(T64, T65), T66, T67, .(T64, T69))
U7_aagga(T20, T21, T17, T16, T7, append1E_out_ggga(T21, T16, T20, T7)) → pB_out_aagga(T20, T21, T17, T16, T7)
U1_ga(T16, T17, T7, pB_out_aagga(X40, X41, T17, T16, T7)) → rotateA_out_ga(.(T16, T17), T7)
rotateA_in_ga(T86, T7) → U2_ga(T86, T7, append1C_in_ga(T86, T7))
append1C_in_ga(.(T102, T103), .(T102, T105)) → U5_ga(T102, T103, T105, append1C_in_ga(T103, T105))
append1C_in_ga([], []) → append1C_out_ga([], [])
U5_ga(T102, T103, T105, append1C_out_ga(T103, T105)) → append1C_out_ga(.(T102, T103), .(T102, T105))
U2_ga(T86, T7, append1C_out_ga(T86, T7)) → rotateA_out_ga(T86, T7)

The argument filtering Pi contains the following mapping:
rotateA_in_ga(x1, x2)  =  rotateA_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
pB_in_aagga(x1, x2, x3, x4, x5)  =  pB_in_aagga(x3, x4)
U6_aagga(x1, x2, x3, x4, x5, x6)  =  U6_aagga(x3, x4, x6)
append2D_in_aag(x1, x2, x3)  =  append2D_in_aag(x3)
U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x1, x4, x5)
append2D_out_aag(x1, x2, x3)  =  append2D_out_aag(x1, x2, x3)
U7_aagga(x1, x2, x3, x4, x5, x6)  =  U7_aagga(x1, x2, x3, x4, x6)
append1E_in_ggga(x1, x2, x3, x4)  =  append1E_in_ggga(x1, x2, x3)
U4_ggga(x1, x2, x3, x4, x5, x6)  =  U4_ggga(x1, x2, x3, x4, x6)
[]  =  []
append1E_out_ggga(x1, x2, x3, x4)  =  append1E_out_ggga(x1, x2, x3, x4)
pB_out_aagga(x1, x2, x3, x4, x5)  =  pB_out_aagga(x1, x2, x3, x4, x5)
rotateA_out_ga(x1, x2)  =  rotateA_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
append1C_in_ga(x1, x2)  =  append1C_in_ga(x1)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x2, x4)
append1C_out_ga(x1, x2)  =  append1C_out_ga(x1, x2)
ROTATEA_IN_GA(x1, x2)  =  ROTATEA_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)
PB_IN_AAGGA(x1, x2, x3, x4, x5)  =  PB_IN_AAGGA(x3, x4)
U6_AAGGA(x1, x2, x3, x4, x5, x6)  =  U6_AAGGA(x3, x4, x6)
APPEND2D_IN_AAG(x1, x2, x3)  =  APPEND2D_IN_AAG(x3)
U3_AAG(x1, x2, x3, x4, x5)  =  U3_AAG(x1, x4, x5)
U7_AAGGA(x1, x2, x3, x4, x5, x6)  =  U7_AAGGA(x1, x2, x3, x4, x6)
APPEND1E_IN_GGGA(x1, x2, x3, x4)  =  APPEND1E_IN_GGGA(x1, x2, x3)
U4_GGGA(x1, x2, x3, x4, x5, x6)  =  U4_GGGA(x1, x2, x3, x4, x6)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
APPEND1C_IN_GA(x1, x2)  =  APPEND1C_IN_GA(x1)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x1, x2, x4)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 11 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

APPEND1C_IN_GA(.(T102, T103), .(T102, T105)) → APPEND1C_IN_GA(T103, T105)

The TRS R consists of the following rules:

rotateA_in_ga(.(T16, T17), T7) → U1_ga(T16, T17, T7, pB_in_aagga(X40, X41, T17, T16, T7))
pB_in_aagga(T20, T21, T17, T16, T7) → U6_aagga(T20, T21, T17, T16, T7, append2D_in_aag(T20, T21, T17))
append2D_in_aag(.(T32, X89), X90, .(T32, T33)) → U3_aag(T32, X89, X90, T33, append2D_in_aag(X89, X90, T33))
append2D_in_aag([], T38, T38) → append2D_out_aag([], T38, T38)
U3_aag(T32, X89, X90, T33, append2D_out_aag(X89, X90, T33)) → append2D_out_aag(.(T32, X89), X90, .(T32, T33))
U6_aagga(T20, T21, T17, T16, T7, append2D_out_aag(T20, T21, T17)) → U7_aagga(T20, T21, T17, T16, T7, append1E_in_ggga(T21, T16, T20, T7))
append1E_in_ggga(.(T64, T65), T66, T67, .(T64, T69)) → U4_ggga(T64, T65, T66, T67, T69, append1E_in_ggga(T65, T66, T67, T69))
append1E_in_ggga([], T79, T80, .(T79, T80)) → append1E_out_ggga([], T79, T80, .(T79, T80))
U4_ggga(T64, T65, T66, T67, T69, append1E_out_ggga(T65, T66, T67, T69)) → append1E_out_ggga(.(T64, T65), T66, T67, .(T64, T69))
U7_aagga(T20, T21, T17, T16, T7, append1E_out_ggga(T21, T16, T20, T7)) → pB_out_aagga(T20, T21, T17, T16, T7)
U1_ga(T16, T17, T7, pB_out_aagga(X40, X41, T17, T16, T7)) → rotateA_out_ga(.(T16, T17), T7)
rotateA_in_ga(T86, T7) → U2_ga(T86, T7, append1C_in_ga(T86, T7))
append1C_in_ga(.(T102, T103), .(T102, T105)) → U5_ga(T102, T103, T105, append1C_in_ga(T103, T105))
append1C_in_ga([], []) → append1C_out_ga([], [])
U5_ga(T102, T103, T105, append1C_out_ga(T103, T105)) → append1C_out_ga(.(T102, T103), .(T102, T105))
U2_ga(T86, T7, append1C_out_ga(T86, T7)) → rotateA_out_ga(T86, T7)

The argument filtering Pi contains the following mapping:
rotateA_in_ga(x1, x2)  =  rotateA_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
pB_in_aagga(x1, x2, x3, x4, x5)  =  pB_in_aagga(x3, x4)
U6_aagga(x1, x2, x3, x4, x5, x6)  =  U6_aagga(x3, x4, x6)
append2D_in_aag(x1, x2, x3)  =  append2D_in_aag(x3)
U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x1, x4, x5)
append2D_out_aag(x1, x2, x3)  =  append2D_out_aag(x1, x2, x3)
U7_aagga(x1, x2, x3, x4, x5, x6)  =  U7_aagga(x1, x2, x3, x4, x6)
append1E_in_ggga(x1, x2, x3, x4)  =  append1E_in_ggga(x1, x2, x3)
U4_ggga(x1, x2, x3, x4, x5, x6)  =  U4_ggga(x1, x2, x3, x4, x6)
[]  =  []
append1E_out_ggga(x1, x2, x3, x4)  =  append1E_out_ggga(x1, x2, x3, x4)
pB_out_aagga(x1, x2, x3, x4, x5)  =  pB_out_aagga(x1, x2, x3, x4, x5)
rotateA_out_ga(x1, x2)  =  rotateA_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
append1C_in_ga(x1, x2)  =  append1C_in_ga(x1)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x2, x4)
append1C_out_ga(x1, x2)  =  append1C_out_ga(x1, x2)
APPEND1C_IN_GA(x1, x2)  =  APPEND1C_IN_GA(x1)

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:

APPEND1C_IN_GA(.(T102, T103), .(T102, T105)) → APPEND1C_IN_GA(T103, T105)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPEND1C_IN_GA(x1, x2)  =  APPEND1C_IN_GA(x1)

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:

APPEND1C_IN_GA(.(T102, T103)) → APPEND1C_IN_GA(T103)

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:

• APPEND1C_IN_GA(.(T102, T103)) → APPEND1C_IN_GA(T103)
  The graph contains the following edges 1 > 1

(13) YES

(14) Obligation:

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

APPEND1E_IN_GGGA(.(T64, T65), T66, T67, .(T64, T69)) → APPEND1E_IN_GGGA(T65, T66, T67, T69)

The TRS R consists of the following rules:

rotateA_in_ga(.(T16, T17), T7) → U1_ga(T16, T17, T7, pB_in_aagga(X40, X41, T17, T16, T7))
pB_in_aagga(T20, T21, T17, T16, T7) → U6_aagga(T20, T21, T17, T16, T7, append2D_in_aag(T20, T21, T17))
append2D_in_aag(.(T32, X89), X90, .(T32, T33)) → U3_aag(T32, X89, X90, T33, append2D_in_aag(X89, X90, T33))
append2D_in_aag([], T38, T38) → append2D_out_aag([], T38, T38)
U3_aag(T32, X89, X90, T33, append2D_out_aag(X89, X90, T33)) → append2D_out_aag(.(T32, X89), X90, .(T32, T33))
U6_aagga(T20, T21, T17, T16, T7, append2D_out_aag(T20, T21, T17)) → U7_aagga(T20, T21, T17, T16, T7, append1E_in_ggga(T21, T16, T20, T7))
append1E_in_ggga(.(T64, T65), T66, T67, .(T64, T69)) → U4_ggga(T64, T65, T66, T67, T69, append1E_in_ggga(T65, T66, T67, T69))
append1E_in_ggga([], T79, T80, .(T79, T80)) → append1E_out_ggga([], T79, T80, .(T79, T80))
U4_ggga(T64, T65, T66, T67, T69, append1E_out_ggga(T65, T66, T67, T69)) → append1E_out_ggga(.(T64, T65), T66, T67, .(T64, T69))
U7_aagga(T20, T21, T17, T16, T7, append1E_out_ggga(T21, T16, T20, T7)) → pB_out_aagga(T20, T21, T17, T16, T7)
U1_ga(T16, T17, T7, pB_out_aagga(X40, X41, T17, T16, T7)) → rotateA_out_ga(.(T16, T17), T7)
rotateA_in_ga(T86, T7) → U2_ga(T86, T7, append1C_in_ga(T86, T7))
append1C_in_ga(.(T102, T103), .(T102, T105)) → U5_ga(T102, T103, T105, append1C_in_ga(T103, T105))
append1C_in_ga([], []) → append1C_out_ga([], [])
U5_ga(T102, T103, T105, append1C_out_ga(T103, T105)) → append1C_out_ga(.(T102, T103), .(T102, T105))
U2_ga(T86, T7, append1C_out_ga(T86, T7)) → rotateA_out_ga(T86, T7)

The argument filtering Pi contains the following mapping:
rotateA_in_ga(x1, x2)  =  rotateA_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
pB_in_aagga(x1, x2, x3, x4, x5)  =  pB_in_aagga(x3, x4)
U6_aagga(x1, x2, x3, x4, x5, x6)  =  U6_aagga(x3, x4, x6)
append2D_in_aag(x1, x2, x3)  =  append2D_in_aag(x3)
U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x1, x4, x5)
append2D_out_aag(x1, x2, x3)  =  append2D_out_aag(x1, x2, x3)
U7_aagga(x1, x2, x3, x4, x5, x6)  =  U7_aagga(x1, x2, x3, x4, x6)
append1E_in_ggga(x1, x2, x3, x4)  =  append1E_in_ggga(x1, x2, x3)
U4_ggga(x1, x2, x3, x4, x5, x6)  =  U4_ggga(x1, x2, x3, x4, x6)
[]  =  []
append1E_out_ggga(x1, x2, x3, x4)  =  append1E_out_ggga(x1, x2, x3, x4)
pB_out_aagga(x1, x2, x3, x4, x5)  =  pB_out_aagga(x1, x2, x3, x4, x5)
rotateA_out_ga(x1, x2)  =  rotateA_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
append1C_in_ga(x1, x2)  =  append1C_in_ga(x1)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x2, x4)
append1C_out_ga(x1, x2)  =  append1C_out_ga(x1, x2)
APPEND1E_IN_GGGA(x1, x2, x3, x4)  =  APPEND1E_IN_GGGA(x1, x2, x3)

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:

APPEND1E_IN_GGGA(.(T64, T65), T66, T67, .(T64, T69)) → APPEND1E_IN_GGGA(T65, T66, T67, T69)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPEND1E_IN_GGGA(x1, x2, x3, x4)  =  APPEND1E_IN_GGGA(x1, x2, x3)

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:

APPEND1E_IN_GGGA(.(T64, T65), T66, T67) → APPEND1E_IN_GGGA(T65, T66, T67)

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:

• APPEND1E_IN_GGGA(.(T64, T65), T66, T67) → APPEND1E_IN_GGGA(T65, T66, T67)
  The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

(20) YES

(21) Obligation:

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

APPEND2D_IN_AAG(.(T32, X89), X90, .(T32, T33)) → APPEND2D_IN_AAG(X89, X90, T33)

The TRS R consists of the following rules:

rotateA_in_ga(.(T16, T17), T7) → U1_ga(T16, T17, T7, pB_in_aagga(X40, X41, T17, T16, T7))
pB_in_aagga(T20, T21, T17, T16, T7) → U6_aagga(T20, T21, T17, T16, T7, append2D_in_aag(T20, T21, T17))
append2D_in_aag(.(T32, X89), X90, .(T32, T33)) → U3_aag(T32, X89, X90, T33, append2D_in_aag(X89, X90, T33))
append2D_in_aag([], T38, T38) → append2D_out_aag([], T38, T38)
U3_aag(T32, X89, X90, T33, append2D_out_aag(X89, X90, T33)) → append2D_out_aag(.(T32, X89), X90, .(T32, T33))
U6_aagga(T20, T21, T17, T16, T7, append2D_out_aag(T20, T21, T17)) → U7_aagga(T20, T21, T17, T16, T7, append1E_in_ggga(T21, T16, T20, T7))
append1E_in_ggga(.(T64, T65), T66, T67, .(T64, T69)) → U4_ggga(T64, T65, T66, T67, T69, append1E_in_ggga(T65, T66, T67, T69))
append1E_in_ggga([], T79, T80, .(T79, T80)) → append1E_out_ggga([], T79, T80, .(T79, T80))
U4_ggga(T64, T65, T66, T67, T69, append1E_out_ggga(T65, T66, T67, T69)) → append1E_out_ggga(.(T64, T65), T66, T67, .(T64, T69))
U7_aagga(T20, T21, T17, T16, T7, append1E_out_ggga(T21, T16, T20, T7)) → pB_out_aagga(T20, T21, T17, T16, T7)
U1_ga(T16, T17, T7, pB_out_aagga(X40, X41, T17, T16, T7)) → rotateA_out_ga(.(T16, T17), T7)
rotateA_in_ga(T86, T7) → U2_ga(T86, T7, append1C_in_ga(T86, T7))
append1C_in_ga(.(T102, T103), .(T102, T105)) → U5_ga(T102, T103, T105, append1C_in_ga(T103, T105))
append1C_in_ga([], []) → append1C_out_ga([], [])
U5_ga(T102, T103, T105, append1C_out_ga(T103, T105)) → append1C_out_ga(.(T102, T103), .(T102, T105))
U2_ga(T86, T7, append1C_out_ga(T86, T7)) → rotateA_out_ga(T86, T7)

The argument filtering Pi contains the following mapping:
rotateA_in_ga(x1, x2)  =  rotateA_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
pB_in_aagga(x1, x2, x3, x4, x5)  =  pB_in_aagga(x3, x4)
U6_aagga(x1, x2, x3, x4, x5, x6)  =  U6_aagga(x3, x4, x6)
append2D_in_aag(x1, x2, x3)  =  append2D_in_aag(x3)
U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x1, x4, x5)
append2D_out_aag(x1, x2, x3)  =  append2D_out_aag(x1, x2, x3)
U7_aagga(x1, x2, x3, x4, x5, x6)  =  U7_aagga(x1, x2, x3, x4, x6)
append1E_in_ggga(x1, x2, x3, x4)  =  append1E_in_ggga(x1, x2, x3)
U4_ggga(x1, x2, x3, x4, x5, x6)  =  U4_ggga(x1, x2, x3, x4, x6)
[]  =  []
append1E_out_ggga(x1, x2, x3, x4)  =  append1E_out_ggga(x1, x2, x3, x4)
pB_out_aagga(x1, x2, x3, x4, x5)  =  pB_out_aagga(x1, x2, x3, x4, x5)
rotateA_out_ga(x1, x2)  =  rotateA_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
append1C_in_ga(x1, x2)  =  append1C_in_ga(x1)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x2, x4)
append1C_out_ga(x1, x2)  =  append1C_out_ga(x1, x2)
APPEND2D_IN_AAG(x1, x2, x3)  =  APPEND2D_IN_AAG(x3)

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

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

APPEND2D_IN_AAG(.(T32, X89), X90, .(T32, T33)) → APPEND2D_IN_AAG(X89, X90, T33)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPEND2D_IN_AAG(x1, x2, x3)  =  APPEND2D_IN_AAG(x3)

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

(24) PiDPToQDPProof (SOUND transformation)

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

(25) Obligation:

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

APPEND2D_IN_AAG(.(T32, T33)) → APPEND2D_IN_AAG(T33)

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

(26) 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:

• APPEND2D_IN_AAG(.(T32, T33)) → APPEND2D_IN_AAG(T33)
  The graph contains the following edges 1 > 1

(27) YES