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) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: rotate(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: rotate(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(append2(X5, X6, T5), append1(X6, X5, T7))\n\nT5 is ground\nX5 is free\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(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];
5 [label="5: ','(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];
6 [label="6: ','(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];
7 [label="7: ','(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];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: append2(X40, X41, T17)\n\nT17 is ground\nX40 is free\nX41 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: append1(T21, .(T16, T20), T7)\n\nT16 is ground\nT20 is ground\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: 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];
12 [label="12: append2(X40, X41, T17), SCOPE: 3, CLAUSE: 3\n\nT17 is ground\nX40 is free\nX41 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: append2(X40, X41, T17), SCOPE: 3, CLAUSE: 4\n\nT17 is ground\nX40 is free\nX41 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: append2(X89, X90, T33)\n\nT33 is ground\nX89 is free\nX90 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: 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];
19 [label="19: 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];
20 [label="20: 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];
21 [label="21: append1(T65, .(T66, T67), T69)\n\nT65 is ground\nT66 is ground\nT67 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: append1(T86, [], T7)\n\nT86 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: append1(T86, [], T7), SCOPE: 5, CLAUSE: 1 | append1(T86, [], T7), SCOPE: 5, CLAUSE: 2\n\nT86 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: append1(T86, [], T7), SCOPE: 5, CLAUSE: 1\n\nT86 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: append1(T86, [], T7), SCOPE: 5, CLAUSE: 2\n\nT86 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: append1(T103, [], T105)\n\nT103 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 35 [arrowhead = none ];
35 -> 2;

36 [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];
2 -> 36 [arrowhead = none ];
36 -> 3;

37 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 37 [arrowhead = none ];
37 -> 4;

38 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 38 [arrowhead = none ];
38 -> 5;

39 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 39 [arrowhead = none ];
39 -> 6;

40 [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];
5 -> 40 [arrowhead = none ];
40 -> 7;

41 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 41 [arrowhead = none ];
41 -> 8;

42 [label="ONLY EVAL with clause\nappend2([], X143, X143).\nand substitutionX5 -> [],\nX6 -> T86,\nX143 -> T86,\nT5 -> T86,\nX144 -> T86", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 42 [arrowhead = none ];
42 -> 26;

43 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
7 -> 43 [arrowhead = none ];
43 -> 9;

44 [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];
7 -> 44 [arrowhead = none ];
44 -> 10;

45 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 45 [arrowhead = none ];
45 -> 11;

46 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 46 [arrowhead = none ];
46 -> 18;

47 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 47 [arrowhead = none ];
47 -> 12;

48 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 48 [arrowhead = none ];
48 -> 13;

49 [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];
12 -> 49 [arrowhead = none ];
49 -> 14;

50 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 50 [arrowhead = none ];
50 -> 15;

51 [label="ONLY EVAL with clause\nappend2([], X102, X102).\nand substitutionX40 -> [],\nX41 -> T38,\nX102 -> T38,\nT17 -> T38,\nX103 -> T38", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 51 [arrowhead = none ];
51 -> 16;

52 [label="INSTANCE with matching:\nX40 -> X89\nX41 -> X90\nT17 -> T33", fontsize=14, style = filled, fillcolor = lightgrey];
14 -> 52 [arrowhead = none , style=dashed];
52 -> 9[style=dashed];

53 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 53 [arrowhead = none ];
53 -> 17;

54 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 54 [arrowhead = none ];
54 -> 19;

55 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 55 [arrowhead = none ];
55 -> 20;

56 [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];
19 -> 56 [arrowhead = none ];
56 -> 21;

57 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 57 [arrowhead = none ];
57 -> 22;

58 [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];
20 -> 58 [arrowhead = none ];
58 -> 23;

59 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 59 [arrowhead = none ];
59 -> 24;

60 [label="INSTANCE with matching:\nT21 -> T65\nT16 -> T66\nT20 -> T67\nT7 -> T69", fontsize=14, style = filled, fillcolor = lightgrey];
21 -> 60 [arrowhead = none , style=dashed];
60 -> 10[style=dashed];

61 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 61 [arrowhead = none ];
61 -> 25;

62 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 62 [arrowhead = none ];
62 -> 27;

63 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 63 [arrowhead = none ];
63 -> 28;

64 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 64 [arrowhead = none ];
64 -> 29;

65 [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];
28 -> 65 [arrowhead = none ];
65 -> 30;

66 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 66 [arrowhead = none ];
66 -> 31;

67 [label="EVAL with clause\nappend1([], X175, X175).\nand substitutionT86 -> [],\nX175 -> [],\nT7 -> []", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 67 [arrowhead = none ];
67 -> 32;

68 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 68 [arrowhead = none ];
68 -> 33;

69 [label="INSTANCE with matching:\nT86 -> T103\nT7 -> T105", fontsize=14, style = filled, fillcolor = lightgrey];
30 -> 69 [arrowhead = none , style=dashed];
69 -> 26[style=dashed];

70 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 70 [arrowhead = none ];
70 -> 34;

}

(2) Obligation:

Clauses:

append2A(.(T32, X89), X90, .(T32, T33)) :- append2A(X89, X90, T33).
append2A([], T38, T38).
append1B(.(T64, T65), T66, T67, .(T64, T69)) :- append1B(T65, T66, T67, T69).
append1B([], T79, T80, .(T79, T80)).
append1C(.(T102, T103), .(T102, T105)) :- append1C(T103, T105).
append1C([], []).
rotateD(.(T16, T17), T7) :- append2A(X40, X41, T17).
rotateD(.(T16, T17), T7) :- ','(append2A(T20, T21, T17), append1B(T21, T16, T20, T7)).
rotateD(T86, T7) :- append1C(T86, T7).

Query: rotateD(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:
rotateD_in: (b,f)
append2A_in: (f,f,b)
append1B_in: (b,b,b,f)
append1C_in: (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

rotateD_in_ga(.(T16, T17), T7) → U4_ga(T16, T17, T7, append2A_in_aag(X40, X41, T17))
append2A_in_aag(.(T32, X89), X90, .(T32, T33)) → U1_aag(T32, X89, X90, T33, append2A_in_aag(X89, X90, T33))
append2A_in_aag([], T38, T38) → append2A_out_aag([], T38, T38)
U1_aag(T32, X89, X90, T33, append2A_out_aag(X89, X90, T33)) → append2A_out_aag(.(T32, X89), X90, .(T32, T33))
U4_ga(T16, T17, T7, append2A_out_aag(X40, X41, T17)) → rotateD_out_ga(.(T16, T17), T7)
rotateD_in_ga(.(T16, T17), T7) → U5_ga(T16, T17, T7, append2A_in_aag(T20, T21, T17))
U5_ga(T16, T17, T7, append2A_out_aag(T20, T21, T17)) → U6_ga(T16, T17, T7, append1B_in_ggga(T21, T16, T20, T7))
append1B_in_ggga(.(T64, T65), T66, T67, .(T64, T69)) → U2_ggga(T64, T65, T66, T67, T69, append1B_in_ggga(T65, T66, T67, T69))
append1B_in_ggga([], T79, T80, .(T79, T80)) → append1B_out_ggga([], T79, T80, .(T79, T80))
U2_ggga(T64, T65, T66, T67, T69, append1B_out_ggga(T65, T66, T67, T69)) → append1B_out_ggga(.(T64, T65), T66, T67, .(T64, T69))
U6_ga(T16, T17, T7, append1B_out_ggga(T21, T16, T20, T7)) → rotateD_out_ga(.(T16, T17), T7)
rotateD_in_ga(T86, T7) → U7_ga(T86, T7, append1C_in_ga(T86, T7))
append1C_in_ga(.(T102, T103), .(T102, T105)) → U3_ga(T102, T103, T105, append1C_in_ga(T103, T105))
append1C_in_ga([], []) → append1C_out_ga([], [])
U3_ga(T102, T103, T105, append1C_out_ga(T103, T105)) → append1C_out_ga(.(T102, T103), .(T102, T105))
U7_ga(T86, T7, append1C_out_ga(T86, T7)) → rotateD_out_ga(T86, T7)

The argument filtering Pi contains the following mapping:
rotateD_in_ga(x1, x2)  =  rotateD_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
append2A_in_aag(x1, x2, x3)  =  append2A_in_aag(x3)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x1, x5)
append2A_out_aag(x1, x2, x3)  =  append2A_out_aag(x1, x2)
rotateD_out_ga(x1, x2)  =  rotateD_out_ga
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x4)
append1B_in_ggga(x1, x2, x3, x4)  =  append1B_in_ggga(x1, x2, x3)
U2_ggga(x1, x2, x3, x4, x5, x6)  =  U2_ggga(x1, x6)
[]  =  []
append1B_out_ggga(x1, x2, x3, x4)  =  append1B_out_ggga(x4)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
append1C_in_ga(x1, x2)  =  append1C_in_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
append1C_out_ga(x1, x2)  =  append1C_out_ga(x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

rotateD_in_ga(.(T16, T17), T7) → U4_ga(T16, T17, T7, append2A_in_aag(X40, X41, T17))
append2A_in_aag(.(T32, X89), X90, .(T32, T33)) → U1_aag(T32, X89, X90, T33, append2A_in_aag(X89, X90, T33))
append2A_in_aag([], T38, T38) → append2A_out_aag([], T38, T38)
U1_aag(T32, X89, X90, T33, append2A_out_aag(X89, X90, T33)) → append2A_out_aag(.(T32, X89), X90, .(T32, T33))
U4_ga(T16, T17, T7, append2A_out_aag(X40, X41, T17)) → rotateD_out_ga(.(T16, T17), T7)
rotateD_in_ga(.(T16, T17), T7) → U5_ga(T16, T17, T7, append2A_in_aag(T20, T21, T17))
U5_ga(T16, T17, T7, append2A_out_aag(T20, T21, T17)) → U6_ga(T16, T17, T7, append1B_in_ggga(T21, T16, T20, T7))
append1B_in_ggga(.(T64, T65), T66, T67, .(T64, T69)) → U2_ggga(T64, T65, T66, T67, T69, append1B_in_ggga(T65, T66, T67, T69))
append1B_in_ggga([], T79, T80, .(T79, T80)) → append1B_out_ggga([], T79, T80, .(T79, T80))
U2_ggga(T64, T65, T66, T67, T69, append1B_out_ggga(T65, T66, T67, T69)) → append1B_out_ggga(.(T64, T65), T66, T67, .(T64, T69))
U6_ga(T16, T17, T7, append1B_out_ggga(T21, T16, T20, T7)) → rotateD_out_ga(.(T16, T17), T7)
rotateD_in_ga(T86, T7) → U7_ga(T86, T7, append1C_in_ga(T86, T7))
append1C_in_ga(.(T102, T103), .(T102, T105)) → U3_ga(T102, T103, T105, append1C_in_ga(T103, T105))
append1C_in_ga([], []) → append1C_out_ga([], [])
U3_ga(T102, T103, T105, append1C_out_ga(T103, T105)) → append1C_out_ga(.(T102, T103), .(T102, T105))
U7_ga(T86, T7, append1C_out_ga(T86, T7)) → rotateD_out_ga(T86, T7)

The argument filtering Pi contains the following mapping:
rotateD_in_ga(x1, x2)  =  rotateD_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
append2A_in_aag(x1, x2, x3)  =  append2A_in_aag(x3)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x1, x5)
append2A_out_aag(x1, x2, x3)  =  append2A_out_aag(x1, x2)
rotateD_out_ga(x1, x2)  =  rotateD_out_ga
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x4)
append1B_in_ggga(x1, x2, x3, x4)  =  append1B_in_ggga(x1, x2, x3)
U2_ggga(x1, x2, x3, x4, x5, x6)  =  U2_ggga(x1, x6)
[]  =  []
append1B_out_ggga(x1, x2, x3, x4)  =  append1B_out_ggga(x4)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
append1C_in_ga(x1, x2)  =  append1C_in_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
append1C_out_ga(x1, x2)  =  append1C_out_ga(x2)

(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:

ROTATED_IN_GA(.(T16, T17), T7) → U4_GA(T16, T17, T7, append2A_in_aag(X40, X41, T17))
ROTATED_IN_GA(.(T16, T17), T7) → APPEND2A_IN_AAG(X40, X41, T17)
APPEND2A_IN_AAG(.(T32, X89), X90, .(T32, T33)) → U1_AAG(T32, X89, X90, T33, append2A_in_aag(X89, X90, T33))
APPEND2A_IN_AAG(.(T32, X89), X90, .(T32, T33)) → APPEND2A_IN_AAG(X89, X90, T33)
ROTATED_IN_GA(.(T16, T17), T7) → U5_GA(T16, T17, T7, append2A_in_aag(T20, T21, T17))
U5_GA(T16, T17, T7, append2A_out_aag(T20, T21, T17)) → U6_GA(T16, T17, T7, append1B_in_ggga(T21, T16, T20, T7))
U5_GA(T16, T17, T7, append2A_out_aag(T20, T21, T17)) → APPEND1B_IN_GGGA(T21, T16, T20, T7)
APPEND1B_IN_GGGA(.(T64, T65), T66, T67, .(T64, T69)) → U2_GGGA(T64, T65, T66, T67, T69, append1B_in_ggga(T65, T66, T67, T69))
APPEND1B_IN_GGGA(.(T64, T65), T66, T67, .(T64, T69)) → APPEND1B_IN_GGGA(T65, T66, T67, T69)
ROTATED_IN_GA(T86, T7) → U7_GA(T86, T7, append1C_in_ga(T86, T7))
ROTATED_IN_GA(T86, T7) → APPEND1C_IN_GA(T86, T7)
APPEND1C_IN_GA(.(T102, T103), .(T102, T105)) → U3_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:

rotateD_in_ga(.(T16, T17), T7) → U4_ga(T16, T17, T7, append2A_in_aag(X40, X41, T17))
append2A_in_aag(.(T32, X89), X90, .(T32, T33)) → U1_aag(T32, X89, X90, T33, append2A_in_aag(X89, X90, T33))
append2A_in_aag([], T38, T38) → append2A_out_aag([], T38, T38)
U1_aag(T32, X89, X90, T33, append2A_out_aag(X89, X90, T33)) → append2A_out_aag(.(T32, X89), X90, .(T32, T33))
U4_ga(T16, T17, T7, append2A_out_aag(X40, X41, T17)) → rotateD_out_ga(.(T16, T17), T7)
rotateD_in_ga(.(T16, T17), T7) → U5_ga(T16, T17, T7, append2A_in_aag(T20, T21, T17))
U5_ga(T16, T17, T7, append2A_out_aag(T20, T21, T17)) → U6_ga(T16, T17, T7, append1B_in_ggga(T21, T16, T20, T7))
append1B_in_ggga(.(T64, T65), T66, T67, .(T64, T69)) → U2_ggga(T64, T65, T66, T67, T69, append1B_in_ggga(T65, T66, T67, T69))
append1B_in_ggga([], T79, T80, .(T79, T80)) → append1B_out_ggga([], T79, T80, .(T79, T80))
U2_ggga(T64, T65, T66, T67, T69, append1B_out_ggga(T65, T66, T67, T69)) → append1B_out_ggga(.(T64, T65), T66, T67, .(T64, T69))
U6_ga(T16, T17, T7, append1B_out_ggga(T21, T16, T20, T7)) → rotateD_out_ga(.(T16, T17), T7)
rotateD_in_ga(T86, T7) → U7_ga(T86, T7, append1C_in_ga(T86, T7))
append1C_in_ga(.(T102, T103), .(T102, T105)) → U3_ga(T102, T103, T105, append1C_in_ga(T103, T105))
append1C_in_ga([], []) → append1C_out_ga([], [])
U3_ga(T102, T103, T105, append1C_out_ga(T103, T105)) → append1C_out_ga(.(T102, T103), .(T102, T105))
U7_ga(T86, T7, append1C_out_ga(T86, T7)) → rotateD_out_ga(T86, T7)

The argument filtering Pi contains the following mapping:
rotateD_in_ga(x1, x2)  =  rotateD_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
append2A_in_aag(x1, x2, x3)  =  append2A_in_aag(x3)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x1, x5)
append2A_out_aag(x1, x2, x3)  =  append2A_out_aag(x1, x2)
rotateD_out_ga(x1, x2)  =  rotateD_out_ga
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x4)
append1B_in_ggga(x1, x2, x3, x4)  =  append1B_in_ggga(x1, x2, x3)
U2_ggga(x1, x2, x3, x4, x5, x6)  =  U2_ggga(x1, x6)
[]  =  []
append1B_out_ggga(x1, x2, x3, x4)  =  append1B_out_ggga(x4)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
append1C_in_ga(x1, x2)  =  append1C_in_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
append1C_out_ga(x1, x2)  =  append1C_out_ga(x2)
ROTATED_IN_GA(x1, x2)  =  ROTATED_IN_GA(x1)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x4)
APPEND2A_IN_AAG(x1, x2, x3)  =  APPEND2A_IN_AAG(x3)
U1_AAG(x1, x2, x3, x4, x5)  =  U1_AAG(x1, x5)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x1, x4)
U6_GA(x1, x2, x3, x4)  =  U6_GA(x4)
APPEND1B_IN_GGGA(x1, x2, x3, x4)  =  APPEND1B_IN_GGGA(x1, x2, x3)
U2_GGGA(x1, x2, x3, x4, x5, x6)  =  U2_GGGA(x1, x6)
U7_GA(x1, x2, x3)  =  U7_GA(x3)
APPEND1C_IN_GA(x1, x2)  =  APPEND1C_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)

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

(6) Obligation:

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

ROTATED_IN_GA(.(T16, T17), T7) → U4_GA(T16, T17, T7, append2A_in_aag(X40, X41, T17))
ROTATED_IN_GA(.(T16, T17), T7) → APPEND2A_IN_AAG(X40, X41, T17)
APPEND2A_IN_AAG(.(T32, X89), X90, .(T32, T33)) → U1_AAG(T32, X89, X90, T33, append2A_in_aag(X89, X90, T33))
APPEND2A_IN_AAG(.(T32, X89), X90, .(T32, T33)) → APPEND2A_IN_AAG(X89, X90, T33)
ROTATED_IN_GA(.(T16, T17), T7) → U5_GA(T16, T17, T7, append2A_in_aag(T20, T21, T17))
U5_GA(T16, T17, T7, append2A_out_aag(T20, T21, T17)) → U6_GA(T16, T17, T7, append1B_in_ggga(T21, T16, T20, T7))
U5_GA(T16, T17, T7, append2A_out_aag(T20, T21, T17)) → APPEND1B_IN_GGGA(T21, T16, T20, T7)
APPEND1B_IN_GGGA(.(T64, T65), T66, T67, .(T64, T69)) → U2_GGGA(T64, T65, T66, T67, T69, append1B_in_ggga(T65, T66, T67, T69))
APPEND1B_IN_GGGA(.(T64, T65), T66, T67, .(T64, T69)) → APPEND1B_IN_GGGA(T65, T66, T67, T69)
ROTATED_IN_GA(T86, T7) → U7_GA(T86, T7, append1C_in_ga(T86, T7))
ROTATED_IN_GA(T86, T7) → APPEND1C_IN_GA(T86, T7)
APPEND1C_IN_GA(.(T102, T103), .(T102, T105)) → U3_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:

rotateD_in_ga(.(T16, T17), T7) → U4_ga(T16, T17, T7, append2A_in_aag(X40, X41, T17))
append2A_in_aag(.(T32, X89), X90, .(T32, T33)) → U1_aag(T32, X89, X90, T33, append2A_in_aag(X89, X90, T33))
append2A_in_aag([], T38, T38) → append2A_out_aag([], T38, T38)
U1_aag(T32, X89, X90, T33, append2A_out_aag(X89, X90, T33)) → append2A_out_aag(.(T32, X89), X90, .(T32, T33))
U4_ga(T16, T17, T7, append2A_out_aag(X40, X41, T17)) → rotateD_out_ga(.(T16, T17), T7)
rotateD_in_ga(.(T16, T17), T7) → U5_ga(T16, T17, T7, append2A_in_aag(T20, T21, T17))
U5_ga(T16, T17, T7, append2A_out_aag(T20, T21, T17)) → U6_ga(T16, T17, T7, append1B_in_ggga(T21, T16, T20, T7))
append1B_in_ggga(.(T64, T65), T66, T67, .(T64, T69)) → U2_ggga(T64, T65, T66, T67, T69, append1B_in_ggga(T65, T66, T67, T69))
append1B_in_ggga([], T79, T80, .(T79, T80)) → append1B_out_ggga([], T79, T80, .(T79, T80))
U2_ggga(T64, T65, T66, T67, T69, append1B_out_ggga(T65, T66, T67, T69)) → append1B_out_ggga(.(T64, T65), T66, T67, .(T64, T69))
U6_ga(T16, T17, T7, append1B_out_ggga(T21, T16, T20, T7)) → rotateD_out_ga(.(T16, T17), T7)
rotateD_in_ga(T86, T7) → U7_ga(T86, T7, append1C_in_ga(T86, T7))
append1C_in_ga(.(T102, T103), .(T102, T105)) → U3_ga(T102, T103, T105, append1C_in_ga(T103, T105))
append1C_in_ga([], []) → append1C_out_ga([], [])
U3_ga(T102, T103, T105, append1C_out_ga(T103, T105)) → append1C_out_ga(.(T102, T103), .(T102, T105))
U7_ga(T86, T7, append1C_out_ga(T86, T7)) → rotateD_out_ga(T86, T7)

The argument filtering Pi contains the following mapping:
rotateD_in_ga(x1, x2)  =  rotateD_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
append2A_in_aag(x1, x2, x3)  =  append2A_in_aag(x3)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x1, x5)
append2A_out_aag(x1, x2, x3)  =  append2A_out_aag(x1, x2)
rotateD_out_ga(x1, x2)  =  rotateD_out_ga
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x4)
append1B_in_ggga(x1, x2, x3, x4)  =  append1B_in_ggga(x1, x2, x3)
U2_ggga(x1, x2, x3, x4, x5, x6)  =  U2_ggga(x1, x6)
[]  =  []
append1B_out_ggga(x1, x2, x3, x4)  =  append1B_out_ggga(x4)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
append1C_in_ga(x1, x2)  =  append1C_in_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
append1C_out_ga(x1, x2)  =  append1C_out_ga(x2)
ROTATED_IN_GA(x1, x2)  =  ROTATED_IN_GA(x1)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x4)
APPEND2A_IN_AAG(x1, x2, x3)  =  APPEND2A_IN_AAG(x3)
U1_AAG(x1, x2, x3, x4, x5)  =  U1_AAG(x1, x5)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x1, x4)
U6_GA(x1, x2, x3, x4)  =  U6_GA(x4)
APPEND1B_IN_GGGA(x1, x2, x3, x4)  =  APPEND1B_IN_GGGA(x1, x2, x3)
U2_GGGA(x1, x2, x3, x4, x5, x6)  =  U2_GGGA(x1, x6)
U7_GA(x1, x2, x3)  =  U7_GA(x3)
APPEND1C_IN_GA(x1, x2)  =  APPEND1C_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 10 less nodes.

(8) Complex Obligation (AND)

(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)

The TRS R consists of the following rules:

rotateD_in_ga(.(T16, T17), T7) → U4_ga(T16, T17, T7, append2A_in_aag(X40, X41, T17))
append2A_in_aag(.(T32, X89), X90, .(T32, T33)) → U1_aag(T32, X89, X90, T33, append2A_in_aag(X89, X90, T33))
append2A_in_aag([], T38, T38) → append2A_out_aag([], T38, T38)
U1_aag(T32, X89, X90, T33, append2A_out_aag(X89, X90, T33)) → append2A_out_aag(.(T32, X89), X90, .(T32, T33))
U4_ga(T16, T17, T7, append2A_out_aag(X40, X41, T17)) → rotateD_out_ga(.(T16, T17), T7)
rotateD_in_ga(.(T16, T17), T7) → U5_ga(T16, T17, T7, append2A_in_aag(T20, T21, T17))
U5_ga(T16, T17, T7, append2A_out_aag(T20, T21, T17)) → U6_ga(T16, T17, T7, append1B_in_ggga(T21, T16, T20, T7))
append1B_in_ggga(.(T64, T65), T66, T67, .(T64, T69)) → U2_ggga(T64, T65, T66, T67, T69, append1B_in_ggga(T65, T66, T67, T69))
append1B_in_ggga([], T79, T80, .(T79, T80)) → append1B_out_ggga([], T79, T80, .(T79, T80))
U2_ggga(T64, T65, T66, T67, T69, append1B_out_ggga(T65, T66, T67, T69)) → append1B_out_ggga(.(T64, T65), T66, T67, .(T64, T69))
U6_ga(T16, T17, T7, append1B_out_ggga(T21, T16, T20, T7)) → rotateD_out_ga(.(T16, T17), T7)
rotateD_in_ga(T86, T7) → U7_ga(T86, T7, append1C_in_ga(T86, T7))
append1C_in_ga(.(T102, T103), .(T102, T105)) → U3_ga(T102, T103, T105, append1C_in_ga(T103, T105))
append1C_in_ga([], []) → append1C_out_ga([], [])
U3_ga(T102, T103, T105, append1C_out_ga(T103, T105)) → append1C_out_ga(.(T102, T103), .(T102, T105))
U7_ga(T86, T7, append1C_out_ga(T86, T7)) → rotateD_out_ga(T86, T7)

The argument filtering Pi contains the following mapping:
rotateD_in_ga(x1, x2)  =  rotateD_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
append2A_in_aag(x1, x2, x3)  =  append2A_in_aag(x3)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x1, x5)
append2A_out_aag(x1, x2, x3)  =  append2A_out_aag(x1, x2)
rotateD_out_ga(x1, x2)  =  rotateD_out_ga
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x4)
append1B_in_ggga(x1, x2, x3, x4)  =  append1B_in_ggga(x1, x2, x3)
U2_ggga(x1, x2, x3, x4, x5, x6)  =  U2_ggga(x1, x6)
[]  =  []
append1B_out_ggga(x1, x2, x3, x4)  =  append1B_out_ggga(x4)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
append1C_in_ga(x1, x2)  =  append1C_in_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
append1C_out_ga(x1, x2)  =  append1C_out_ga(x2)
APPEND1C_IN_GA(x1, x2)  =  APPEND1C_IN_GA(x1)

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:

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

(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:

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

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:

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

(15) YES

(16) Obligation:

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

APPEND1B_IN_GGGA(.(T64, T65), T66, T67, .(T64, T69)) → APPEND1B_IN_GGGA(T65, T66, T67, T69)

The TRS R consists of the following rules:

rotateD_in_ga(.(T16, T17), T7) → U4_ga(T16, T17, T7, append2A_in_aag(X40, X41, T17))
append2A_in_aag(.(T32, X89), X90, .(T32, T33)) → U1_aag(T32, X89, X90, T33, append2A_in_aag(X89, X90, T33))
append2A_in_aag([], T38, T38) → append2A_out_aag([], T38, T38)
U1_aag(T32, X89, X90, T33, append2A_out_aag(X89, X90, T33)) → append2A_out_aag(.(T32, X89), X90, .(T32, T33))
U4_ga(T16, T17, T7, append2A_out_aag(X40, X41, T17)) → rotateD_out_ga(.(T16, T17), T7)
rotateD_in_ga(.(T16, T17), T7) → U5_ga(T16, T17, T7, append2A_in_aag(T20, T21, T17))
U5_ga(T16, T17, T7, append2A_out_aag(T20, T21, T17)) → U6_ga(T16, T17, T7, append1B_in_ggga(T21, T16, T20, T7))
append1B_in_ggga(.(T64, T65), T66, T67, .(T64, T69)) → U2_ggga(T64, T65, T66, T67, T69, append1B_in_ggga(T65, T66, T67, T69))
append1B_in_ggga([], T79, T80, .(T79, T80)) → append1B_out_ggga([], T79, T80, .(T79, T80))
U2_ggga(T64, T65, T66, T67, T69, append1B_out_ggga(T65, T66, T67, T69)) → append1B_out_ggga(.(T64, T65), T66, T67, .(T64, T69))
U6_ga(T16, T17, T7, append1B_out_ggga(T21, T16, T20, T7)) → rotateD_out_ga(.(T16, T17), T7)
rotateD_in_ga(T86, T7) → U7_ga(T86, T7, append1C_in_ga(T86, T7))
append1C_in_ga(.(T102, T103), .(T102, T105)) → U3_ga(T102, T103, T105, append1C_in_ga(T103, T105))
append1C_in_ga([], []) → append1C_out_ga([], [])
U3_ga(T102, T103, T105, append1C_out_ga(T103, T105)) → append1C_out_ga(.(T102, T103), .(T102, T105))
U7_ga(T86, T7, append1C_out_ga(T86, T7)) → rotateD_out_ga(T86, T7)

The argument filtering Pi contains the following mapping:
rotateD_in_ga(x1, x2)  =  rotateD_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
append2A_in_aag(x1, x2, x3)  =  append2A_in_aag(x3)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x1, x5)
append2A_out_aag(x1, x2, x3)  =  append2A_out_aag(x1, x2)
rotateD_out_ga(x1, x2)  =  rotateD_out_ga
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x4)
append1B_in_ggga(x1, x2, x3, x4)  =  append1B_in_ggga(x1, x2, x3)
U2_ggga(x1, x2, x3, x4, x5, x6)  =  U2_ggga(x1, x6)
[]  =  []
append1B_out_ggga(x1, x2, x3, x4)  =  append1B_out_ggga(x4)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
append1C_in_ga(x1, x2)  =  append1C_in_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
append1C_out_ga(x1, x2)  =  append1C_out_ga(x2)
APPEND1B_IN_GGGA(x1, x2, x3, x4)  =  APPEND1B_IN_GGGA(x1, x2, x3)

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:

APPEND1B_IN_GGGA(.(T64, T65), T66, T67, .(T64, T69)) → APPEND1B_IN_GGGA(T65, T66, T67, T69)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPEND1B_IN_GGGA(x1, x2, x3, x4)  =  APPEND1B_IN_GGGA(x1, x2, x3)

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:

APPEND1B_IN_GGGA(.(T64, T65), T66, T67) → APPEND1B_IN_GGGA(T65, T66, T67)

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:

• APPEND1B_IN_GGGA(.(T64, T65), T66, T67) → APPEND1B_IN_GGGA(T65, T66, T67)
  The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

(22) YES

(23) Obligation:

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

APPEND2A_IN_AAG(.(T32, X89), X90, .(T32, T33)) → APPEND2A_IN_AAG(X89, X90, T33)

The TRS R consists of the following rules:

rotateD_in_ga(.(T16, T17), T7) → U4_ga(T16, T17, T7, append2A_in_aag(X40, X41, T17))
append2A_in_aag(.(T32, X89), X90, .(T32, T33)) → U1_aag(T32, X89, X90, T33, append2A_in_aag(X89, X90, T33))
append2A_in_aag([], T38, T38) → append2A_out_aag([], T38, T38)
U1_aag(T32, X89, X90, T33, append2A_out_aag(X89, X90, T33)) → append2A_out_aag(.(T32, X89), X90, .(T32, T33))
U4_ga(T16, T17, T7, append2A_out_aag(X40, X41, T17)) → rotateD_out_ga(.(T16, T17), T7)
rotateD_in_ga(.(T16, T17), T7) → U5_ga(T16, T17, T7, append2A_in_aag(T20, T21, T17))
U5_ga(T16, T17, T7, append2A_out_aag(T20, T21, T17)) → U6_ga(T16, T17, T7, append1B_in_ggga(T21, T16, T20, T7))
append1B_in_ggga(.(T64, T65), T66, T67, .(T64, T69)) → U2_ggga(T64, T65, T66, T67, T69, append1B_in_ggga(T65, T66, T67, T69))
append1B_in_ggga([], T79, T80, .(T79, T80)) → append1B_out_ggga([], T79, T80, .(T79, T80))
U2_ggga(T64, T65, T66, T67, T69, append1B_out_ggga(T65, T66, T67, T69)) → append1B_out_ggga(.(T64, T65), T66, T67, .(T64, T69))
U6_ga(T16, T17, T7, append1B_out_ggga(T21, T16, T20, T7)) → rotateD_out_ga(.(T16, T17), T7)
rotateD_in_ga(T86, T7) → U7_ga(T86, T7, append1C_in_ga(T86, T7))
append1C_in_ga(.(T102, T103), .(T102, T105)) → U3_ga(T102, T103, T105, append1C_in_ga(T103, T105))
append1C_in_ga([], []) → append1C_out_ga([], [])
U3_ga(T102, T103, T105, append1C_out_ga(T103, T105)) → append1C_out_ga(.(T102, T103), .(T102, T105))
U7_ga(T86, T7, append1C_out_ga(T86, T7)) → rotateD_out_ga(T86, T7)

The argument filtering Pi contains the following mapping:
rotateD_in_ga(x1, x2)  =  rotateD_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
append2A_in_aag(x1, x2, x3)  =  append2A_in_aag(x3)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x1, x5)
append2A_out_aag(x1, x2, x3)  =  append2A_out_aag(x1, x2)
rotateD_out_ga(x1, x2)  =  rotateD_out_ga
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x4)
append1B_in_ggga(x1, x2, x3, x4)  =  append1B_in_ggga(x1, x2, x3)
U2_ggga(x1, x2, x3, x4, x5, x6)  =  U2_ggga(x1, x6)
[]  =  []
append1B_out_ggga(x1, x2, x3, x4)  =  append1B_out_ggga(x4)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
append1C_in_ga(x1, x2)  =  append1C_in_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
append1C_out_ga(x1, x2)  =  append1C_out_ga(x2)
APPEND2A_IN_AAG(x1, x2, x3)  =  APPEND2A_IN_AAG(x3)

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

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) Obligation:

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

APPEND2A_IN_AAG(.(T32, X89), X90, .(T32, T33)) → APPEND2A_IN_AAG(X89, X90, T33)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPEND2A_IN_AAG(x1, x2, x3)  =  APPEND2A_IN_AAG(x3)

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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) Obligation:

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

APPEND2A_IN_AAG(.(T32, T33)) → APPEND2A_IN_AAG(T33)

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

(28) 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:

• APPEND2A_IN_AAG(.(T32, T33)) → APPEND2A_IN_AAG(T33)
  The graph contains the following edges 1 > 1

(29) YES