Left Termination of the query pattern hidden_flatten_in_3(g, a, a) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

hidden_flatten([], L, L).
hidden_flatten(.(.(H, T), L), S, F) :- ','(!, ','(hidden_flatten(L, S, Lf), hidden_flatten(.(H, T), Lf, F))).
hidden_flatten(.(H, T), S, .(H, L)) :- hidden_flatten(T, S, L).

Queries:

hidden_flatten(g,a,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: hidden_flatten(T1, T2, T3)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: hidden_flatten(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | hidden_flatten(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | hidden_flatten(T1, T2, T3), SCOPE: 1, CLAUSE: 2 | ?_1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: hidden_flatten(T1, T2, T3), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: hidden_flatten(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | hidden_flatten(T1, T2, T3), SCOPE: 1, CLAUSE: 2 | ?_1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: \n\nX3 is free", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(!_1, ','(hidden_flatten(T7, T10, X14), hidden_flatten(.(T5, T6), X14, T11))) | hidden_flatten(.(.(T5, T6), T7), T2, T3), SCOPE: 1, CLAUSE: 2 | ?_1\n\nT5 is ground\nT6 is ground\nT7 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: hidden_flatten(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nX9 is free; \nX10 is free; \nX11 is free; \nX12 is free; \nX13 is free; \nhidden_flatten(T1, T2, T3) !=? hidden_flatten(.(.(X9, X10), X11), X12, X13)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(hidden_flatten(T7, T10, X14), hidden_flatten(.(T5, T6), X14, T11))\n\nT5 is ground\nT6 is ground\nT7 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: hidden_flatten(T7, T10, X14)\n\nT7 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: hidden_flatten(.(T5, T6), T12, T13)\n\nT5 is ground\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: hidden_flatten(T7, T10, X14), SCOPE: 2, CLAUSE: 0 | hidden_flatten(T7, T10, X14), SCOPE: 2, CLAUSE: 1 | hidden_flatten(T7, T10, X14), SCOPE: 2, CLAUSE: 2 | ?_2\n\nT7 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: hidden_flatten(T7, T10, X14), SCOPE: 2, CLAUSE: 0\n\nT7 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: hidden_flatten(T7, T10, X14), SCOPE: 2, CLAUSE: 1 | hidden_flatten(T7, T10, X14), SCOPE: 2, CLAUSE: 2 | ?_2\n\nT7 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\nX14 is free\nX17 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(!_2, ','(hidden_flatten(T17, T19, X28), hidden_flatten(.(T15, T16), X28, X29))) | hidden_flatten(.(.(T15, T16), T17), T10, X14), SCOPE: 2, CLAUSE: 2 | ?_2\n\nT15 is ground\nT16 is ground\nT17 is ground\nX14 is free; \nX29 is free; \nX28 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: hidden_flatten(T7, T10, X14), SCOPE: 2, CLAUSE: 2\n\nT7 is ground\nX14 is free; \nX23 is free; \nX24 is free; \nX25 is free; \nX26 is free; \nX27 is free; \nhidden_flatten(T7, T10, X14) !=? hidden_flatten(.(.(X23, X24), X25), X26, X27)", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(hidden_flatten(T17, T19, X28), hidden_flatten(.(T15, T16), X28, X29))\n\nT15 is ground\nT16 is ground\nT17 is ground\nX29 is free; \nX28 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: hidden_flatten(T17, T19, X28)\n\nT17 is ground\nX28 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: hidden_flatten(.(T15, T16), T20, X29)\n\nT15 is ground\nT16 is ground\nX29 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: hidden_flatten(T22, T24, X38)\n\nT22 is ground\nX14 is free; \nX23 is free; \nX24 is free; \nX25 is free; \nX26 is free; \nX27 is free; \nX38 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\nX14 is free\nX23 is free; \nX24 is free; \nX25 is free; \nX26 is free; \nX27 is free; \nX34 is free; \nX35 is free; \nX36 is free; \nX37 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: hidden_flatten(T26, T29, T30)\n\nT26 is ground\nX9 is free; \nX10 is free; \nX11 is free; \nX12 is free; \nX13 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\nX9 is free\nX10 is free; \nX11 is free; \nX12 is free; \nX13 is free; \nX43 is free; \nX44 is free; \nX45 is free; \nX46 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 28 [arrowhead = none ];
28 -> 2;

29 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
2 -> 29 [arrowhead = none ];
29 -> 3;

30 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
2 -> 30 [arrowhead = none ];
30 -> 4;

31 [label="EVAL with clause\nhidden_flatten([], X3, X3).\nand substitution\nT1 -> []\nT2 -> T4\nX3 -> T4\nT3 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 31 [arrowhead = none ];
31 -> 5;

32 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 32 [arrowhead = none ];
32 -> 6;

33 [label="EVAL with clause\nhidden_flatten(.(.(X9, X10), X11), X12, X13) :- ','(!, ','(hidden_flatten(X11, X12, X14), hidden_flatten(.(X9, X10), X14, X13))).\nand substitution\nX9 -> T5\nX10 -> T6\nX11 -> T7\nT1 -> .(.(T5, T6), T7)\nT2 -> T10\nX12 -> T10\nT3 -> T11\nX13 -> T11\nT8 -> T10\nT9 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 33 [arrowhead = none ];
33 -> 8;

34 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 34 [arrowhead = none ];
34 -> 9;

35 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 35 [arrowhead = none ];
35 -> 7;

36 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 36 [arrowhead = none ];
36 -> 10;

37 [label="EVAL with clause\nhidden_flatten(.(X43, X44), X45, .(X43, X46)) :- hidden_flatten(X44, X45, X46).\nand substitution\nX43 -> T25\nX44 -> T26\nT1 -> .(T25, T26)\nT2 -> T29\nX45 -> T29\nX46 -> T30\nT3 -> .(T25, T30)\nT27 -> T29\nT28 -> T30", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 37 [arrowhead = none ];
37 -> 26;

38 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 38 [arrowhead = none ];
38 -> 27;

39 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
10 -> 39 [arrowhead = none ];
39 -> 11;

40 [label="SPLIT 2\nreplacements:\nX14 -> T12\nT11 -> T13", fontsize=14, style = filled, fillcolor = violet];
10 -> 40 [arrowhead = none ];
40 -> 12;

41 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 41 [arrowhead = none ];
41 -> 13;

42 [label="INSTANCE with matching:\nT1 -> .(T5, T6)\nT2 -> T12\nT3 -> T13", fontsize=14, style = filled, fillcolor = lightgrey];
12 -> 42 [arrowhead = none , style=dashed];
42 -> 1[style=dashed];

43 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
13 -> 43 [arrowhead = none ];
43 -> 14;

44 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
13 -> 44 [arrowhead = none ];
44 -> 15;

45 [label="EVAL with clause\nhidden_flatten([], X17, X17).\nand substitution\nT7 -> []\nT10 -> T14\nX17 -> T14\nX14 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 45 [arrowhead = none ];
45 -> 16;

46 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 46 [arrowhead = none ];
46 -> 17;

47 [label="EVAL with clause\nhidden_flatten(.(.(X23, X24), X25), X26, X27) :- ','(!, ','(hidden_flatten(X25, X26, X28), hidden_flatten(.(X23, X24), X28, X27))).\nand substitution\nX23 -> T15\nX24 -> T16\nX25 -> T17\nT7 -> .(.(T15, T16), T17)\nT10 -> T19\nX26 -> T19\nX14 -> X29\nX27 -> X29\nT18 -> T19", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 47 [arrowhead = none ];
47 -> 19;

48 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 48 [arrowhead = none ];
48 -> 20;

49 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 49 [arrowhead = none ];
49 -> 18;

50 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 50 [arrowhead = none ];
50 -> 21;

51 [label="EVAL with clause\nhidden_flatten(.(X34, X35), X36, .(X34, X37)) :- hidden_flatten(X35, X36, X37).\nand substitution\nX34 -> T21\nX35 -> T22\nT7 -> .(T21, T22)\nT10 -> T24\nX36 -> T24\nX37 -> X38\nX14 -> .(T21, X38)\nT23 -> T24", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 51 [arrowhead = none ];
51 -> 24;

52 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 52 [arrowhead = none ];
52 -> 25;

53 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
21 -> 53 [arrowhead = none ];
53 -> 22;

54 [label="SPLIT 2\nreplacements:\nX28 -> T20", fontsize=14, style = filled, fillcolor = violet];
21 -> 54 [arrowhead = none ];
54 -> 23;

55 [label="INSTANCE with matching:\nT7 -> T17\nT10 -> T19\nX14 -> X28", fontsize=14, style = filled, fillcolor = lightgrey];
22 -> 55 [arrowhead = none , style=dashed];
55 -> 11[style=dashed];

56 [label="INSTANCE with matching:\nT1 -> .(T15, T16)\nT2 -> T20\nT3 -> X29", fontsize=14, style = filled, fillcolor = lightgrey];
23 -> 56 [arrowhead = none , style=dashed];
56 -> 1[style=dashed];

57 [label="INSTANCE with matching:\nT7 -> T22\nT10 -> T24\nX14 -> X38", fontsize=14, style = filled, fillcolor = lightgrey];
24 -> 57 [arrowhead = none , style=dashed];
57 -> 11[style=dashed];

58 [label="INSTANCE with matching:\nT1 -> T26\nT2 -> T29\nT3 -> T30", fontsize=14, style = filled, fillcolor = lightgrey];
26 -> 58 [arrowhead = none , style=dashed];
58 -> 1[style=dashed];

}

(2) Obligation:

Clauses:

hidden_flatten1([], T4, T4).
hidden_flatten1(.(.(T5, T6), T7), T10, T11) :- hidden_flatten11(T7, T10, X14).
hidden_flatten1(.(.(T5, T6), T7), T10, T13) :- ','(hidden_flatten11(T7, T10, T12), hidden_flatten1(.(T5, T6), T12, T13)).
hidden_flatten1(.(T25, T26), T29, .(T25, T30)) :- hidden_flatten1(T26, T29, T30).
hidden_flatten11([], T14, T14).
hidden_flatten11(.(.(T15, T16), T17), T19, X29) :- hidden_flatten11(T17, T19, X28).
hidden_flatten11(.(.(T15, T16), T17), T19, X29) :- ','(hidden_flatten11(T17, T19, T20), hidden_flatten1(.(T15, T16), T20, X29)).
hidden_flatten11(.(T21, T22), T24, .(T21, X38)) :- hidden_flatten11(T22, T24, X38).

Queries:

hidden_flatten1(g,a,a).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
hidden_flatten1_in: (b,f,f)
hidden_flatten11_in: (b,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

hidden_flatten1_in_gaa([], T4, T4) → hidden_flatten1_out_gaa([], T4, T4)
hidden_flatten1_in_gaa(.(.(T5, T6), T7), T10, T11) → U1_gaa(T5, T6, T7, T10, T11, hidden_flatten11_in_gaa(T7, T10, X14))
hidden_flatten11_in_gaa([], T14, T14) → hidden_flatten11_out_gaa([], T14, T14)
hidden_flatten11_in_gaa(.(.(T15, T16), T17), T19, X29) → U5_gaa(T15, T16, T17, T19, X29, hidden_flatten11_in_gaa(T17, T19, X28))
hidden_flatten11_in_gaa(.(.(T15, T16), T17), T19, X29) → U6_gaa(T15, T16, T17, T19, X29, hidden_flatten11_in_gaa(T17, T19, T20))
hidden_flatten11_in_gaa(.(T21, T22), T24, .(T21, X38)) → U8_gaa(T21, T22, T24, X38, hidden_flatten11_in_gaa(T22, T24, X38))
U8_gaa(T21, T22, T24, X38, hidden_flatten11_out_gaa(T22, T24, X38)) → hidden_flatten11_out_gaa(.(T21, T22), T24, .(T21, X38))
U6_gaa(T15, T16, T17, T19, X29, hidden_flatten11_out_gaa(T17, T19, T20)) → U7_gaa(T15, T16, T17, T19, X29, hidden_flatten1_in_gaa(.(T15, T16), T20, X29))
hidden_flatten1_in_gaa(.(.(T5, T6), T7), T10, T13) → U2_gaa(T5, T6, T7, T10, T13, hidden_flatten11_in_gaa(T7, T10, T12))
U2_gaa(T5, T6, T7, T10, T13, hidden_flatten11_out_gaa(T7, T10, T12)) → U3_gaa(T5, T6, T7, T10, T13, hidden_flatten1_in_gaa(.(T5, T6), T12, T13))
hidden_flatten1_in_gaa(.(T25, T26), T29, .(T25, T30)) → U4_gaa(T25, T26, T29, T30, hidden_flatten1_in_gaa(T26, T29, T30))
U4_gaa(T25, T26, T29, T30, hidden_flatten1_out_gaa(T26, T29, T30)) → hidden_flatten1_out_gaa(.(T25, T26), T29, .(T25, T30))
U3_gaa(T5, T6, T7, T10, T13, hidden_flatten1_out_gaa(.(T5, T6), T12, T13)) → hidden_flatten1_out_gaa(.(.(T5, T6), T7), T10, T13)
U7_gaa(T15, T16, T17, T19, X29, hidden_flatten1_out_gaa(.(T15, T16), T20, X29)) → hidden_flatten11_out_gaa(.(.(T15, T16), T17), T19, X29)
U5_gaa(T15, T16, T17, T19, X29, hidden_flatten11_out_gaa(T17, T19, X28)) → hidden_flatten11_out_gaa(.(.(T15, T16), T17), T19, X29)
U1_gaa(T5, T6, T7, T10, T11, hidden_flatten11_out_gaa(T7, T10, X14)) → hidden_flatten1_out_gaa(.(.(T5, T6), T7), T10, T11)

The argument filtering Pi contains the following mapping:
hidden_flatten1_in_gaa(x1, x2, x3)  =  hidden_flatten1_in_gaa(x1)
[]  =  []
hidden_flatten1_out_gaa(x1, x2, x3)  =  hidden_flatten1_out_gaa
.(x1, x2)  =  .(x1, x2)
U1_gaa(x1, x2, x3, x4, x5, x6)  =  U1_gaa(x6)
hidden_flatten11_in_gaa(x1, x2, x3)  =  hidden_flatten11_in_gaa(x1)
hidden_flatten11_out_gaa(x1, x2, x3)  =  hidden_flatten11_out_gaa
U5_gaa(x1, x2, x3, x4, x5, x6)  =  U5_gaa(x6)
U6_gaa(x1, x2, x3, x4, x5, x6)  =  U6_gaa(x1, x2, x6)
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x5)
U7_gaa(x1, x2, x3, x4, x5, x6)  =  U7_gaa(x6)
U2_gaa(x1, x2, x3, x4, x5, x6)  =  U2_gaa(x1, x2, x6)
U3_gaa(x1, x2, x3, x4, x5, x6)  =  U3_gaa(x6)
U4_gaa(x1, x2, x3, x4, x5)  =  U4_gaa(x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

hidden_flatten1_in_gaa([], T4, T4) → hidden_flatten1_out_gaa([], T4, T4)
hidden_flatten1_in_gaa(.(.(T5, T6), T7), T10, T11) → U1_gaa(T5, T6, T7, T10, T11, hidden_flatten11_in_gaa(T7, T10, X14))
hidden_flatten11_in_gaa([], T14, T14) → hidden_flatten11_out_gaa([], T14, T14)
hidden_flatten11_in_gaa(.(.(T15, T16), T17), T19, X29) → U5_gaa(T15, T16, T17, T19, X29, hidden_flatten11_in_gaa(T17, T19, X28))
hidden_flatten11_in_gaa(.(.(T15, T16), T17), T19, X29) → U6_gaa(T15, T16, T17, T19, X29, hidden_flatten11_in_gaa(T17, T19, T20))
hidden_flatten11_in_gaa(.(T21, T22), T24, .(T21, X38)) → U8_gaa(T21, T22, T24, X38, hidden_flatten11_in_gaa(T22, T24, X38))
U8_gaa(T21, T22, T24, X38, hidden_flatten11_out_gaa(T22, T24, X38)) → hidden_flatten11_out_gaa(.(T21, T22), T24, .(T21, X38))
U6_gaa(T15, T16, T17, T19, X29, hidden_flatten11_out_gaa(T17, T19, T20)) → U7_gaa(T15, T16, T17, T19, X29, hidden_flatten1_in_gaa(.(T15, T16), T20, X29))
hidden_flatten1_in_gaa(.(.(T5, T6), T7), T10, T13) → U2_gaa(T5, T6, T7, T10, T13, hidden_flatten11_in_gaa(T7, T10, T12))
U2_gaa(T5, T6, T7, T10, T13, hidden_flatten11_out_gaa(T7, T10, T12)) → U3_gaa(T5, T6, T7, T10, T13, hidden_flatten1_in_gaa(.(T5, T6), T12, T13))
hidden_flatten1_in_gaa(.(T25, T26), T29, .(T25, T30)) → U4_gaa(T25, T26, T29, T30, hidden_flatten1_in_gaa(T26, T29, T30))
U4_gaa(T25, T26, T29, T30, hidden_flatten1_out_gaa(T26, T29, T30)) → hidden_flatten1_out_gaa(.(T25, T26), T29, .(T25, T30))
U3_gaa(T5, T6, T7, T10, T13, hidden_flatten1_out_gaa(.(T5, T6), T12, T13)) → hidden_flatten1_out_gaa(.(.(T5, T6), T7), T10, T13)
U7_gaa(T15, T16, T17, T19, X29, hidden_flatten1_out_gaa(.(T15, T16), T20, X29)) → hidden_flatten11_out_gaa(.(.(T15, T16), T17), T19, X29)
U5_gaa(T15, T16, T17, T19, X29, hidden_flatten11_out_gaa(T17, T19, X28)) → hidden_flatten11_out_gaa(.(.(T15, T16), T17), T19, X29)
U1_gaa(T5, T6, T7, T10, T11, hidden_flatten11_out_gaa(T7, T10, X14)) → hidden_flatten1_out_gaa(.(.(T5, T6), T7), T10, T11)

The argument filtering Pi contains the following mapping:
hidden_flatten1_in_gaa(x1, x2, x3)  =  hidden_flatten1_in_gaa(x1)
[]  =  []
hidden_flatten1_out_gaa(x1, x2, x3)  =  hidden_flatten1_out_gaa
.(x1, x2)  =  .(x1, x2)
U1_gaa(x1, x2, x3, x4, x5, x6)  =  U1_gaa(x6)
hidden_flatten11_in_gaa(x1, x2, x3)  =  hidden_flatten11_in_gaa(x1)
hidden_flatten11_out_gaa(x1, x2, x3)  =  hidden_flatten11_out_gaa
U5_gaa(x1, x2, x3, x4, x5, x6)  =  U5_gaa(x6)
U6_gaa(x1, x2, x3, x4, x5, x6)  =  U6_gaa(x1, x2, x6)
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x5)
U7_gaa(x1, x2, x3, x4, x5, x6)  =  U7_gaa(x6)
U2_gaa(x1, x2, x3, x4, x5, x6)  =  U2_gaa(x1, x2, x6)
U3_gaa(x1, x2, x3, x4, x5, x6)  =  U3_gaa(x6)
U4_gaa(x1, x2, x3, x4, x5)  =  U4_gaa(x5)

(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:

HIDDEN_FLATTEN1_IN_GAA(.(.(T5, T6), T7), T10, T11) → U1_GAA(T5, T6, T7, T10, T11, hidden_flatten11_in_gaa(T7, T10, X14))
HIDDEN_FLATTEN1_IN_GAA(.(.(T5, T6), T7), T10, T11) → HIDDEN_FLATTEN11_IN_GAA(T7, T10, X14)
HIDDEN_FLATTEN11_IN_GAA(.(.(T15, T16), T17), T19, X29) → U5_GAA(T15, T16, T17, T19, X29, hidden_flatten11_in_gaa(T17, T19, X28))
HIDDEN_FLATTEN11_IN_GAA(.(.(T15, T16), T17), T19, X29) → HIDDEN_FLATTEN11_IN_GAA(T17, T19, X28)
HIDDEN_FLATTEN11_IN_GAA(.(.(T15, T16), T17), T19, X29) → U6_GAA(T15, T16, T17, T19, X29, hidden_flatten11_in_gaa(T17, T19, T20))
HIDDEN_FLATTEN11_IN_GAA(.(T21, T22), T24, .(T21, X38)) → U8_GAA(T21, T22, T24, X38, hidden_flatten11_in_gaa(T22, T24, X38))
HIDDEN_FLATTEN11_IN_GAA(.(T21, T22), T24, .(T21, X38)) → HIDDEN_FLATTEN11_IN_GAA(T22, T24, X38)
U6_GAA(T15, T16, T17, T19, X29, hidden_flatten11_out_gaa(T17, T19, T20)) → U7_GAA(T15, T16, T17, T19, X29, hidden_flatten1_in_gaa(.(T15, T16), T20, X29))
U6_GAA(T15, T16, T17, T19, X29, hidden_flatten11_out_gaa(T17, T19, T20)) → HIDDEN_FLATTEN1_IN_GAA(.(T15, T16), T20, X29)
HIDDEN_FLATTEN1_IN_GAA(.(.(T5, T6), T7), T10, T13) → U2_GAA(T5, T6, T7, T10, T13, hidden_flatten11_in_gaa(T7, T10, T12))
U2_GAA(T5, T6, T7, T10, T13, hidden_flatten11_out_gaa(T7, T10, T12)) → U3_GAA(T5, T6, T7, T10, T13, hidden_flatten1_in_gaa(.(T5, T6), T12, T13))
U2_GAA(T5, T6, T7, T10, T13, hidden_flatten11_out_gaa(T7, T10, T12)) → HIDDEN_FLATTEN1_IN_GAA(.(T5, T6), T12, T13)
HIDDEN_FLATTEN1_IN_GAA(.(T25, T26), T29, .(T25, T30)) → U4_GAA(T25, T26, T29, T30, hidden_flatten1_in_gaa(T26, T29, T30))
HIDDEN_FLATTEN1_IN_GAA(.(T25, T26), T29, .(T25, T30)) → HIDDEN_FLATTEN1_IN_GAA(T26, T29, T30)

The TRS R consists of the following rules:

hidden_flatten1_in_gaa([], T4, T4) → hidden_flatten1_out_gaa([], T4, T4)
hidden_flatten1_in_gaa(.(.(T5, T6), T7), T10, T11) → U1_gaa(T5, T6, T7, T10, T11, hidden_flatten11_in_gaa(T7, T10, X14))
hidden_flatten11_in_gaa([], T14, T14) → hidden_flatten11_out_gaa([], T14, T14)
hidden_flatten11_in_gaa(.(.(T15, T16), T17), T19, X29) → U5_gaa(T15, T16, T17, T19, X29, hidden_flatten11_in_gaa(T17, T19, X28))
hidden_flatten11_in_gaa(.(.(T15, T16), T17), T19, X29) → U6_gaa(T15, T16, T17, T19, X29, hidden_flatten11_in_gaa(T17, T19, T20))
hidden_flatten11_in_gaa(.(T21, T22), T24, .(T21, X38)) → U8_gaa(T21, T22, T24, X38, hidden_flatten11_in_gaa(T22, T24, X38))
U8_gaa(T21, T22, T24, X38, hidden_flatten11_out_gaa(T22, T24, X38)) → hidden_flatten11_out_gaa(.(T21, T22), T24, .(T21, X38))
U6_gaa(T15, T16, T17, T19, X29, hidden_flatten11_out_gaa(T17, T19, T20)) → U7_gaa(T15, T16, T17, T19, X29, hidden_flatten1_in_gaa(.(T15, T16), T20, X29))
hidden_flatten1_in_gaa(.(.(T5, T6), T7), T10, T13) → U2_gaa(T5, T6, T7, T10, T13, hidden_flatten11_in_gaa(T7, T10, T12))
U2_gaa(T5, T6, T7, T10, T13, hidden_flatten11_out_gaa(T7, T10, T12)) → U3_gaa(T5, T6, T7, T10, T13, hidden_flatten1_in_gaa(.(T5, T6), T12, T13))
hidden_flatten1_in_gaa(.(T25, T26), T29, .(T25, T30)) → U4_gaa(T25, T26, T29, T30, hidden_flatten1_in_gaa(T26, T29, T30))
U4_gaa(T25, T26, T29, T30, hidden_flatten1_out_gaa(T26, T29, T30)) → hidden_flatten1_out_gaa(.(T25, T26), T29, .(T25, T30))
U3_gaa(T5, T6, T7, T10, T13, hidden_flatten1_out_gaa(.(T5, T6), T12, T13)) → hidden_flatten1_out_gaa(.(.(T5, T6), T7), T10, T13)
U7_gaa(T15, T16, T17, T19, X29, hidden_flatten1_out_gaa(.(T15, T16), T20, X29)) → hidden_flatten11_out_gaa(.(.(T15, T16), T17), T19, X29)
U5_gaa(T15, T16, T17, T19, X29, hidden_flatten11_out_gaa(T17, T19, X28)) → hidden_flatten11_out_gaa(.(.(T15, T16), T17), T19, X29)
U1_gaa(T5, T6, T7, T10, T11, hidden_flatten11_out_gaa(T7, T10, X14)) → hidden_flatten1_out_gaa(.(.(T5, T6), T7), T10, T11)

The argument filtering Pi contains the following mapping:
hidden_flatten1_in_gaa(x1, x2, x3)  =  hidden_flatten1_in_gaa(x1)
[]  =  []
hidden_flatten1_out_gaa(x1, x2, x3)  =  hidden_flatten1_out_gaa
.(x1, x2)  =  .(x1, x2)
U1_gaa(x1, x2, x3, x4, x5, x6)  =  U1_gaa(x6)
hidden_flatten11_in_gaa(x1, x2, x3)  =  hidden_flatten11_in_gaa(x1)
hidden_flatten11_out_gaa(x1, x2, x3)  =  hidden_flatten11_out_gaa
U5_gaa(x1, x2, x3, x4, x5, x6)  =  U5_gaa(x6)
U6_gaa(x1, x2, x3, x4, x5, x6)  =  U6_gaa(x1, x2, x6)
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x5)
U7_gaa(x1, x2, x3, x4, x5, x6)  =  U7_gaa(x6)
U2_gaa(x1, x2, x3, x4, x5, x6)  =  U2_gaa(x1, x2, x6)
U3_gaa(x1, x2, x3, x4, x5, x6)  =  U3_gaa(x6)
U4_gaa(x1, x2, x3, x4, x5)  =  U4_gaa(x5)
HIDDEN_FLATTEN1_IN_GAA(x1, x2, x3)  =  HIDDEN_FLATTEN1_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5, x6)  =  U1_GAA(x6)
HIDDEN_FLATTEN11_IN_GAA(x1, x2, x3)  =  HIDDEN_FLATTEN11_IN_GAA(x1)
U5_GAA(x1, x2, x3, x4, x5, x6)  =  U5_GAA(x6)
U6_GAA(x1, x2, x3, x4, x5, x6)  =  U6_GAA(x1, x2, x6)
U8_GAA(x1, x2, x3, x4, x5)  =  U8_GAA(x5)
U7_GAA(x1, x2, x3, x4, x5, x6)  =  U7_GAA(x6)
U2_GAA(x1, x2, x3, x4, x5, x6)  =  U2_GAA(x1, x2, x6)
U3_GAA(x1, x2, x3, x4, x5, x6)  =  U3_GAA(x6)
U4_GAA(x1, x2, x3, x4, x5)  =  U4_GAA(x5)

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

(6) Obligation:

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

HIDDEN_FLATTEN1_IN_GAA(.(.(T5, T6), T7), T10, T11) → U1_GAA(T5, T6, T7, T10, T11, hidden_flatten11_in_gaa(T7, T10, X14))
HIDDEN_FLATTEN1_IN_GAA(.(.(T5, T6), T7), T10, T11) → HIDDEN_FLATTEN11_IN_GAA(T7, T10, X14)
HIDDEN_FLATTEN11_IN_GAA(.(.(T15, T16), T17), T19, X29) → U5_GAA(T15, T16, T17, T19, X29, hidden_flatten11_in_gaa(T17, T19, X28))
HIDDEN_FLATTEN11_IN_GAA(.(.(T15, T16), T17), T19, X29) → HIDDEN_FLATTEN11_IN_GAA(T17, T19, X28)
HIDDEN_FLATTEN11_IN_GAA(.(.(T15, T16), T17), T19, X29) → U6_GAA(T15, T16, T17, T19, X29, hidden_flatten11_in_gaa(T17, T19, T20))
HIDDEN_FLATTEN11_IN_GAA(.(T21, T22), T24, .(T21, X38)) → U8_GAA(T21, T22, T24, X38, hidden_flatten11_in_gaa(T22, T24, X38))
HIDDEN_FLATTEN11_IN_GAA(.(T21, T22), T24, .(T21, X38)) → HIDDEN_FLATTEN11_IN_GAA(T22, T24, X38)
U6_GAA(T15, T16, T17, T19, X29, hidden_flatten11_out_gaa(T17, T19, T20)) → U7_GAA(T15, T16, T17, T19, X29, hidden_flatten1_in_gaa(.(T15, T16), T20, X29))
U6_GAA(T15, T16, T17, T19, X29, hidden_flatten11_out_gaa(T17, T19, T20)) → HIDDEN_FLATTEN1_IN_GAA(.(T15, T16), T20, X29)
HIDDEN_FLATTEN1_IN_GAA(.(.(T5, T6), T7), T10, T13) → U2_GAA(T5, T6, T7, T10, T13, hidden_flatten11_in_gaa(T7, T10, T12))
U2_GAA(T5, T6, T7, T10, T13, hidden_flatten11_out_gaa(T7, T10, T12)) → U3_GAA(T5, T6, T7, T10, T13, hidden_flatten1_in_gaa(.(T5, T6), T12, T13))
U2_GAA(T5, T6, T7, T10, T13, hidden_flatten11_out_gaa(T7, T10, T12)) → HIDDEN_FLATTEN1_IN_GAA(.(T5, T6), T12, T13)
HIDDEN_FLATTEN1_IN_GAA(.(T25, T26), T29, .(T25, T30)) → U4_GAA(T25, T26, T29, T30, hidden_flatten1_in_gaa(T26, T29, T30))
HIDDEN_FLATTEN1_IN_GAA(.(T25, T26), T29, .(T25, T30)) → HIDDEN_FLATTEN1_IN_GAA(T26, T29, T30)

The TRS R consists of the following rules:

hidden_flatten1_in_gaa([], T4, T4) → hidden_flatten1_out_gaa([], T4, T4)
hidden_flatten1_in_gaa(.(.(T5, T6), T7), T10, T11) → U1_gaa(T5, T6, T7, T10, T11, hidden_flatten11_in_gaa(T7, T10, X14))
hidden_flatten11_in_gaa([], T14, T14) → hidden_flatten11_out_gaa([], T14, T14)
hidden_flatten11_in_gaa(.(.(T15, T16), T17), T19, X29) → U5_gaa(T15, T16, T17, T19, X29, hidden_flatten11_in_gaa(T17, T19, X28))
hidden_flatten11_in_gaa(.(.(T15, T16), T17), T19, X29) → U6_gaa(T15, T16, T17, T19, X29, hidden_flatten11_in_gaa(T17, T19, T20))
hidden_flatten11_in_gaa(.(T21, T22), T24, .(T21, X38)) → U8_gaa(T21, T22, T24, X38, hidden_flatten11_in_gaa(T22, T24, X38))
U8_gaa(T21, T22, T24, X38, hidden_flatten11_out_gaa(T22, T24, X38)) → hidden_flatten11_out_gaa(.(T21, T22), T24, .(T21, X38))
U6_gaa(T15, T16, T17, T19, X29, hidden_flatten11_out_gaa(T17, T19, T20)) → U7_gaa(T15, T16, T17, T19, X29, hidden_flatten1_in_gaa(.(T15, T16), T20, X29))
hidden_flatten1_in_gaa(.(.(T5, T6), T7), T10, T13) → U2_gaa(T5, T6, T7, T10, T13, hidden_flatten11_in_gaa(T7, T10, T12))
U2_gaa(T5, T6, T7, T10, T13, hidden_flatten11_out_gaa(T7, T10, T12)) → U3_gaa(T5, T6, T7, T10, T13, hidden_flatten1_in_gaa(.(T5, T6), T12, T13))
hidden_flatten1_in_gaa(.(T25, T26), T29, .(T25, T30)) → U4_gaa(T25, T26, T29, T30, hidden_flatten1_in_gaa(T26, T29, T30))
U4_gaa(T25, T26, T29, T30, hidden_flatten1_out_gaa(T26, T29, T30)) → hidden_flatten1_out_gaa(.(T25, T26), T29, .(T25, T30))
U3_gaa(T5, T6, T7, T10, T13, hidden_flatten1_out_gaa(.(T5, T6), T12, T13)) → hidden_flatten1_out_gaa(.(.(T5, T6), T7), T10, T13)
U7_gaa(T15, T16, T17, T19, X29, hidden_flatten1_out_gaa(.(T15, T16), T20, X29)) → hidden_flatten11_out_gaa(.(.(T15, T16), T17), T19, X29)
U5_gaa(T15, T16, T17, T19, X29, hidden_flatten11_out_gaa(T17, T19, X28)) → hidden_flatten11_out_gaa(.(.(T15, T16), T17), T19, X29)
U1_gaa(T5, T6, T7, T10, T11, hidden_flatten11_out_gaa(T7, T10, X14)) → hidden_flatten1_out_gaa(.(.(T5, T6), T7), T10, T11)

The argument filtering Pi contains the following mapping:
hidden_flatten1_in_gaa(x1, x2, x3)  =  hidden_flatten1_in_gaa(x1)
[]  =  []
hidden_flatten1_out_gaa(x1, x2, x3)  =  hidden_flatten1_out_gaa
.(x1, x2)  =  .(x1, x2)
U1_gaa(x1, x2, x3, x4, x5, x6)  =  U1_gaa(x6)
hidden_flatten11_in_gaa(x1, x2, x3)  =  hidden_flatten11_in_gaa(x1)
hidden_flatten11_out_gaa(x1, x2, x3)  =  hidden_flatten11_out_gaa
U5_gaa(x1, x2, x3, x4, x5, x6)  =  U5_gaa(x6)
U6_gaa(x1, x2, x3, x4, x5, x6)  =  U6_gaa(x1, x2, x6)
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x5)
U7_gaa(x1, x2, x3, x4, x5, x6)  =  U7_gaa(x6)
U2_gaa(x1, x2, x3, x4, x5, x6)  =  U2_gaa(x1, x2, x6)
U3_gaa(x1, x2, x3, x4, x5, x6)  =  U3_gaa(x6)
U4_gaa(x1, x2, x3, x4, x5)  =  U4_gaa(x5)
HIDDEN_FLATTEN1_IN_GAA(x1, x2, x3)  =  HIDDEN_FLATTEN1_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5, x6)  =  U1_GAA(x6)
HIDDEN_FLATTEN11_IN_GAA(x1, x2, x3)  =  HIDDEN_FLATTEN11_IN_GAA(x1)
U5_GAA(x1, x2, x3, x4, x5, x6)  =  U5_GAA(x6)
U6_GAA(x1, x2, x3, x4, x5, x6)  =  U6_GAA(x1, x2, x6)
U8_GAA(x1, x2, x3, x4, x5)  =  U8_GAA(x5)
U7_GAA(x1, x2, x3, x4, x5, x6)  =  U7_GAA(x6)
U2_GAA(x1, x2, x3, x4, x5, x6)  =  U2_GAA(x1, x2, x6)
U3_GAA(x1, x2, x3, x4, x5, x6)  =  U3_GAA(x6)
U4_GAA(x1, x2, x3, x4, x5)  =  U4_GAA(x5)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 6 less nodes.

(8) Obligation:

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

HIDDEN_FLATTEN1_IN_GAA(.(.(T5, T6), T7), T10, T11) → HIDDEN_FLATTEN11_IN_GAA(T7, T10, X14)
HIDDEN_FLATTEN11_IN_GAA(.(.(T15, T16), T17), T19, X29) → HIDDEN_FLATTEN11_IN_GAA(T17, T19, X28)
HIDDEN_FLATTEN11_IN_GAA(.(.(T15, T16), T17), T19, X29) → U6_GAA(T15, T16, T17, T19, X29, hidden_flatten11_in_gaa(T17, T19, T20))
U6_GAA(T15, T16, T17, T19, X29, hidden_flatten11_out_gaa(T17, T19, T20)) → HIDDEN_FLATTEN1_IN_GAA(.(T15, T16), T20, X29)
HIDDEN_FLATTEN1_IN_GAA(.(.(T5, T6), T7), T10, T13) → U2_GAA(T5, T6, T7, T10, T13, hidden_flatten11_in_gaa(T7, T10, T12))
U2_GAA(T5, T6, T7, T10, T13, hidden_flatten11_out_gaa(T7, T10, T12)) → HIDDEN_FLATTEN1_IN_GAA(.(T5, T6), T12, T13)
HIDDEN_FLATTEN1_IN_GAA(.(T25, T26), T29, .(T25, T30)) → HIDDEN_FLATTEN1_IN_GAA(T26, T29, T30)
HIDDEN_FLATTEN11_IN_GAA(.(T21, T22), T24, .(T21, X38)) → HIDDEN_FLATTEN11_IN_GAA(T22, T24, X38)

The TRS R consists of the following rules:

hidden_flatten1_in_gaa([], T4, T4) → hidden_flatten1_out_gaa([], T4, T4)
hidden_flatten1_in_gaa(.(.(T5, T6), T7), T10, T11) → U1_gaa(T5, T6, T7, T10, T11, hidden_flatten11_in_gaa(T7, T10, X14))
hidden_flatten11_in_gaa([], T14, T14) → hidden_flatten11_out_gaa([], T14, T14)
hidden_flatten11_in_gaa(.(.(T15, T16), T17), T19, X29) → U5_gaa(T15, T16, T17, T19, X29, hidden_flatten11_in_gaa(T17, T19, X28))
hidden_flatten11_in_gaa(.(.(T15, T16), T17), T19, X29) → U6_gaa(T15, T16, T17, T19, X29, hidden_flatten11_in_gaa(T17, T19, T20))
hidden_flatten11_in_gaa(.(T21, T22), T24, .(T21, X38)) → U8_gaa(T21, T22, T24, X38, hidden_flatten11_in_gaa(T22, T24, X38))
U8_gaa(T21, T22, T24, X38, hidden_flatten11_out_gaa(T22, T24, X38)) → hidden_flatten11_out_gaa(.(T21, T22), T24, .(T21, X38))
U6_gaa(T15, T16, T17, T19, X29, hidden_flatten11_out_gaa(T17, T19, T20)) → U7_gaa(T15, T16, T17, T19, X29, hidden_flatten1_in_gaa(.(T15, T16), T20, X29))
hidden_flatten1_in_gaa(.(.(T5, T6), T7), T10, T13) → U2_gaa(T5, T6, T7, T10, T13, hidden_flatten11_in_gaa(T7, T10, T12))
U2_gaa(T5, T6, T7, T10, T13, hidden_flatten11_out_gaa(T7, T10, T12)) → U3_gaa(T5, T6, T7, T10, T13, hidden_flatten1_in_gaa(.(T5, T6), T12, T13))
hidden_flatten1_in_gaa(.(T25, T26), T29, .(T25, T30)) → U4_gaa(T25, T26, T29, T30, hidden_flatten1_in_gaa(T26, T29, T30))
U4_gaa(T25, T26, T29, T30, hidden_flatten1_out_gaa(T26, T29, T30)) → hidden_flatten1_out_gaa(.(T25, T26), T29, .(T25, T30))
U3_gaa(T5, T6, T7, T10, T13, hidden_flatten1_out_gaa(.(T5, T6), T12, T13)) → hidden_flatten1_out_gaa(.(.(T5, T6), T7), T10, T13)
U7_gaa(T15, T16, T17, T19, X29, hidden_flatten1_out_gaa(.(T15, T16), T20, X29)) → hidden_flatten11_out_gaa(.(.(T15, T16), T17), T19, X29)
U5_gaa(T15, T16, T17, T19, X29, hidden_flatten11_out_gaa(T17, T19, X28)) → hidden_flatten11_out_gaa(.(.(T15, T16), T17), T19, X29)
U1_gaa(T5, T6, T7, T10, T11, hidden_flatten11_out_gaa(T7, T10, X14)) → hidden_flatten1_out_gaa(.(.(T5, T6), T7), T10, T11)

The argument filtering Pi contains the following mapping:
hidden_flatten1_in_gaa(x1, x2, x3)  =  hidden_flatten1_in_gaa(x1)
[]  =  []
hidden_flatten1_out_gaa(x1, x2, x3)  =  hidden_flatten1_out_gaa
.(x1, x2)  =  .(x1, x2)
U1_gaa(x1, x2, x3, x4, x5, x6)  =  U1_gaa(x6)
hidden_flatten11_in_gaa(x1, x2, x3)  =  hidden_flatten11_in_gaa(x1)
hidden_flatten11_out_gaa(x1, x2, x3)  =  hidden_flatten11_out_gaa
U5_gaa(x1, x2, x3, x4, x5, x6)  =  U5_gaa(x6)
U6_gaa(x1, x2, x3, x4, x5, x6)  =  U6_gaa(x1, x2, x6)
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x5)
U7_gaa(x1, x2, x3, x4, x5, x6)  =  U7_gaa(x6)
U2_gaa(x1, x2, x3, x4, x5, x6)  =  U2_gaa(x1, x2, x6)
U3_gaa(x1, x2, x3, x4, x5, x6)  =  U3_gaa(x6)
U4_gaa(x1, x2, x3, x4, x5)  =  U4_gaa(x5)
HIDDEN_FLATTEN1_IN_GAA(x1, x2, x3)  =  HIDDEN_FLATTEN1_IN_GAA(x1)
HIDDEN_FLATTEN11_IN_GAA(x1, x2, x3)  =  HIDDEN_FLATTEN11_IN_GAA(x1)
U6_GAA(x1, x2, x3, x4, x5, x6)  =  U6_GAA(x1, x2, x6)
U2_GAA(x1, x2, x3, x4, x5, x6)  =  U2_GAA(x1, x2, x6)

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

(9) PiDPToQDPProof (SOUND transformation)

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

(10) Obligation:

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

HIDDEN_FLATTEN1_IN_GAA(.(.(T5, T6), T7)) → HIDDEN_FLATTEN11_IN_GAA(T7)
HIDDEN_FLATTEN11_IN_GAA(.(.(T15, T16), T17)) → HIDDEN_FLATTEN11_IN_GAA(T17)
HIDDEN_FLATTEN11_IN_GAA(.(.(T15, T16), T17)) → U6_GAA(T15, T16, hidden_flatten11_in_gaa(T17))
U6_GAA(T15, T16, hidden_flatten11_out_gaa) → HIDDEN_FLATTEN1_IN_GAA(.(T15, T16))
HIDDEN_FLATTEN1_IN_GAA(.(.(T5, T6), T7)) → U2_GAA(T5, T6, hidden_flatten11_in_gaa(T7))
U2_GAA(T5, T6, hidden_flatten11_out_gaa) → HIDDEN_FLATTEN1_IN_GAA(.(T5, T6))
HIDDEN_FLATTEN1_IN_GAA(.(T25, T26)) → HIDDEN_FLATTEN1_IN_GAA(T26)
HIDDEN_FLATTEN11_IN_GAA(.(T21, T22)) → HIDDEN_FLATTEN11_IN_GAA(T22)

The TRS R consists of the following rules:

hidden_flatten1_in_gaa([]) → hidden_flatten1_out_gaa
hidden_flatten1_in_gaa(.(.(T5, T6), T7)) → U1_gaa(hidden_flatten11_in_gaa(T7))
hidden_flatten11_in_gaa([]) → hidden_flatten11_out_gaa
hidden_flatten11_in_gaa(.(.(T15, T16), T17)) → U5_gaa(hidden_flatten11_in_gaa(T17))
hidden_flatten11_in_gaa(.(.(T15, T16), T17)) → U6_gaa(T15, T16, hidden_flatten11_in_gaa(T17))
hidden_flatten11_in_gaa(.(T21, T22)) → U8_gaa(hidden_flatten11_in_gaa(T22))
U8_gaa(hidden_flatten11_out_gaa) → hidden_flatten11_out_gaa
U6_gaa(T15, T16, hidden_flatten11_out_gaa) → U7_gaa(hidden_flatten1_in_gaa(.(T15, T16)))
hidden_flatten1_in_gaa(.(.(T5, T6), T7)) → U2_gaa(T5, T6, hidden_flatten11_in_gaa(T7))
U2_gaa(T5, T6, hidden_flatten11_out_gaa) → U3_gaa(hidden_flatten1_in_gaa(.(T5, T6)))
hidden_flatten1_in_gaa(.(T25, T26)) → U4_gaa(hidden_flatten1_in_gaa(T26))
U4_gaa(hidden_flatten1_out_gaa) → hidden_flatten1_out_gaa
U3_gaa(hidden_flatten1_out_gaa) → hidden_flatten1_out_gaa
U7_gaa(hidden_flatten1_out_gaa) → hidden_flatten11_out_gaa
U5_gaa(hidden_flatten11_out_gaa) → hidden_flatten11_out_gaa
U1_gaa(hidden_flatten11_out_gaa) → hidden_flatten1_out_gaa

The set Q consists of the following terms:

hidden_flatten1_in_gaa(x0)
hidden_flatten11_in_gaa(x0)
U8_gaa(x0)
U6_gaa(x0, x1, x2)
U2_gaa(x0, x1, x2)
U4_gaa(x0)
U3_gaa(x0)
U7_gaa(x0)
U5_gaa(x0)
U1_gaa(x0)

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

(11) UsableRulesReductionPairsProof (EQUIVALENT transformation)

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

No dependency pairs are removed.

The following rules are removed from R:

hidden_flatten11_in_gaa([]) → hidden_flatten11_out_gaa
hidden_flatten1_in_gaa([]) → hidden_flatten1_out_gaa

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 2·x₁ + x₂   
POL(HIDDEN_FLATTEN11_IN_GAA(x₁)) = x₁   
POL(HIDDEN_FLATTEN1_IN_GAA(x₁)) = x₁   
POL(U1_gaa(x₁)) = x₁   
POL(U2_GAA(x₁, x₂, x₃)) = 2·x₁ + x₂ + x₃   
POL(U2_gaa(x₁, x₂, x₃)) = 2·x₁ + x₂ + x₃   
POL(U3_gaa(x₁)) = x₁   
POL(U4_gaa(x₁)) = x₁   
POL(U5_gaa(x₁)) = x₁   
POL(U6_GAA(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + x₃   
POL(U6_gaa(x₁, x₂, x₃)) = 2·x₁ + x₂ + x₃   
POL(U7_gaa(x₁)) = x₁   
POL(U8_gaa(x₁)) = x₁   
POL([]) = 1   
POL(hidden_flatten11_in_gaa(x₁)) = x₁   
POL(hidden_flatten11_out_gaa) = 0   
POL(hidden_flatten1_in_gaa(x₁)) = x₁   
POL(hidden_flatten1_out_gaa) = 0   

(12) Obligation:

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

HIDDEN_FLATTEN1_IN_GAA(.(.(T5, T6), T7)) → HIDDEN_FLATTEN11_IN_GAA(T7)
HIDDEN_FLATTEN11_IN_GAA(.(.(T15, T16), T17)) → HIDDEN_FLATTEN11_IN_GAA(T17)
HIDDEN_FLATTEN11_IN_GAA(.(.(T15, T16), T17)) → U6_GAA(T15, T16, hidden_flatten11_in_gaa(T17))
U6_GAA(T15, T16, hidden_flatten11_out_gaa) → HIDDEN_FLATTEN1_IN_GAA(.(T15, T16))
HIDDEN_FLATTEN1_IN_GAA(.(.(T5, T6), T7)) → U2_GAA(T5, T6, hidden_flatten11_in_gaa(T7))
U2_GAA(T5, T6, hidden_flatten11_out_gaa) → HIDDEN_FLATTEN1_IN_GAA(.(T5, T6))
HIDDEN_FLATTEN1_IN_GAA(.(T25, T26)) → HIDDEN_FLATTEN1_IN_GAA(T26)
HIDDEN_FLATTEN11_IN_GAA(.(T21, T22)) → HIDDEN_FLATTEN11_IN_GAA(T22)

The TRS R consists of the following rules:

hidden_flatten11_in_gaa(.(.(T15, T16), T17)) → U5_gaa(hidden_flatten11_in_gaa(T17))
hidden_flatten11_in_gaa(.(.(T15, T16), T17)) → U6_gaa(T15, T16, hidden_flatten11_in_gaa(T17))
hidden_flatten11_in_gaa(.(T21, T22)) → U8_gaa(hidden_flatten11_in_gaa(T22))
U8_gaa(hidden_flatten11_out_gaa) → hidden_flatten11_out_gaa
U6_gaa(T15, T16, hidden_flatten11_out_gaa) → U7_gaa(hidden_flatten1_in_gaa(.(T15, T16)))
hidden_flatten1_in_gaa(.(.(T5, T6), T7)) → U1_gaa(hidden_flatten11_in_gaa(T7))
hidden_flatten1_in_gaa(.(.(T5, T6), T7)) → U2_gaa(T5, T6, hidden_flatten11_in_gaa(T7))
hidden_flatten1_in_gaa(.(T25, T26)) → U4_gaa(hidden_flatten1_in_gaa(T26))
U7_gaa(hidden_flatten1_out_gaa) → hidden_flatten11_out_gaa
U4_gaa(hidden_flatten1_out_gaa) → hidden_flatten1_out_gaa
U2_gaa(T5, T6, hidden_flatten11_out_gaa) → U3_gaa(hidden_flatten1_in_gaa(.(T5, T6)))
U3_gaa(hidden_flatten1_out_gaa) → hidden_flatten1_out_gaa
U1_gaa(hidden_flatten11_out_gaa) → hidden_flatten1_out_gaa
U5_gaa(hidden_flatten11_out_gaa) → hidden_flatten11_out_gaa

The set Q consists of the following terms:

hidden_flatten1_in_gaa(x0)
hidden_flatten11_in_gaa(x0)
U8_gaa(x0)
U6_gaa(x0, x1, x2)
U2_gaa(x0, x1, x2)
U4_gaa(x0)
U3_gaa(x0)
U7_gaa(x0)
U5_gaa(x0)
U1_gaa(x0)

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

(13) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

HIDDEN_FLATTEN1_IN_GAA(.(.(T5, T6), T7)) → HIDDEN_FLATTEN11_IN_GAA(T7)
HIDDEN_FLATTEN11_IN_GAA(.(.(T15, T16), T17)) → U6_GAA(T15, T16, hidden_flatten11_in_gaa(T17))
U6_GAA(T15, T16, hidden_flatten11_out_gaa) → HIDDEN_FLATTEN1_IN_GAA(.(T15, T16))
HIDDEN_FLATTEN1_IN_GAA(.(.(T5, T6), T7)) → U2_GAA(T5, T6, hidden_flatten11_in_gaa(T7))
U2_GAA(T5, T6, hidden_flatten11_out_gaa) → HIDDEN_FLATTEN1_IN_GAA(.(T5, T6))

Strictly oriented rules of the TRS R:

U6_gaa(T15, T16, hidden_flatten11_out_gaa) → U7_gaa(hidden_flatten1_in_gaa(.(T15, T16)))
hidden_flatten1_in_gaa(.(.(T5, T6), T7)) → U1_gaa(hidden_flatten11_in_gaa(T7))
hidden_flatten1_in_gaa(.(.(T5, T6), T7)) → U2_gaa(T5, T6, hidden_flatten11_in_gaa(T7))
U7_gaa(hidden_flatten1_out_gaa) → hidden_flatten11_out_gaa
U2_gaa(T5, T6, hidden_flatten11_out_gaa) → U3_gaa(hidden_flatten1_in_gaa(.(T5, T6)))
U3_gaa(hidden_flatten1_out_gaa) → hidden_flatten1_out_gaa
U1_gaa(hidden_flatten11_out_gaa) → hidden_flatten1_out_gaa

Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = x₁ + x₂   
POL(HIDDEN_FLATTEN11_IN_GAA(x₁)) = 1 + x₁   
POL(HIDDEN_FLATTEN1_IN_GAA(x₁)) = 2 + x₁   
POL(U1_gaa(x₁)) = x₁   
POL(U2_GAA(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃   
POL(U2_gaa(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(U3_gaa(x₁)) = 1 + x₁   
POL(U4_gaa(x₁)) = x₁   
POL(U5_gaa(x₁)) = x₁   
POL(U6_GAA(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(U6_gaa(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(U7_gaa(x₁)) = 2 + x₁   
POL(U8_gaa(x₁)) = x₁   
POL(hidden_flatten11_in_gaa(x₁)) = x₁   
POL(hidden_flatten11_out_gaa) = 4   
POL(hidden_flatten1_in_gaa(x₁)) = 1 + x₁   
POL(hidden_flatten1_out_gaa) = 3   

(14) Obligation:

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

HIDDEN_FLATTEN11_IN_GAA(.(.(T15, T16), T17)) → HIDDEN_FLATTEN11_IN_GAA(T17)
HIDDEN_FLATTEN1_IN_GAA(.(T25, T26)) → HIDDEN_FLATTEN1_IN_GAA(T26)
HIDDEN_FLATTEN11_IN_GAA(.(T21, T22)) → HIDDEN_FLATTEN11_IN_GAA(T22)

The TRS R consists of the following rules:

hidden_flatten11_in_gaa(.(.(T15, T16), T17)) → U5_gaa(hidden_flatten11_in_gaa(T17))
hidden_flatten11_in_gaa(.(.(T15, T16), T17)) → U6_gaa(T15, T16, hidden_flatten11_in_gaa(T17))
hidden_flatten11_in_gaa(.(T21, T22)) → U8_gaa(hidden_flatten11_in_gaa(T22))
U8_gaa(hidden_flatten11_out_gaa) → hidden_flatten11_out_gaa
hidden_flatten1_in_gaa(.(T25, T26)) → U4_gaa(hidden_flatten1_in_gaa(T26))
U4_gaa(hidden_flatten1_out_gaa) → hidden_flatten1_out_gaa
U5_gaa(hidden_flatten11_out_gaa) → hidden_flatten11_out_gaa

The set Q consists of the following terms:

hidden_flatten1_in_gaa(x0)
hidden_flatten11_in_gaa(x0)
U8_gaa(x0)
U6_gaa(x0, x1, x2)
U2_gaa(x0, x1, x2)
U4_gaa(x0)
U3_gaa(x0)
U7_gaa(x0)
U5_gaa(x0)
U1_gaa(x0)

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

(15) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.

(16) Complex Obligation (AND)

(17) Obligation:

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

HIDDEN_FLATTEN1_IN_GAA(.(T25, T26)) → HIDDEN_FLATTEN1_IN_GAA(T26)

The TRS R consists of the following rules:

hidden_flatten11_in_gaa(.(.(T15, T16), T17)) → U5_gaa(hidden_flatten11_in_gaa(T17))
hidden_flatten11_in_gaa(.(.(T15, T16), T17)) → U6_gaa(T15, T16, hidden_flatten11_in_gaa(T17))
hidden_flatten11_in_gaa(.(T21, T22)) → U8_gaa(hidden_flatten11_in_gaa(T22))
U8_gaa(hidden_flatten11_out_gaa) → hidden_flatten11_out_gaa
hidden_flatten1_in_gaa(.(T25, T26)) → U4_gaa(hidden_flatten1_in_gaa(T26))
U4_gaa(hidden_flatten1_out_gaa) → hidden_flatten1_out_gaa
U5_gaa(hidden_flatten11_out_gaa) → hidden_flatten11_out_gaa

The set Q consists of the following terms:

hidden_flatten1_in_gaa(x0)
hidden_flatten11_in_gaa(x0)
U8_gaa(x0)
U6_gaa(x0, x1, x2)
U2_gaa(x0, x1, x2)
U4_gaa(x0)
U3_gaa(x0)
U7_gaa(x0)
U5_gaa(x0)
U1_gaa(x0)

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

(18) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(19) Obligation:

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

HIDDEN_FLATTEN1_IN_GAA(.(T25, T26)) → HIDDEN_FLATTEN1_IN_GAA(T26)

R is empty.
The set Q consists of the following terms:

hidden_flatten1_in_gaa(x0)
hidden_flatten11_in_gaa(x0)
U8_gaa(x0)
U6_gaa(x0, x1, x2)
U2_gaa(x0, x1, x2)
U4_gaa(x0)
U3_gaa(x0)
U7_gaa(x0)
U5_gaa(x0)
U1_gaa(x0)

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

(20) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

hidden_flatten1_in_gaa(x0)
hidden_flatten11_in_gaa(x0)
U8_gaa(x0)
U6_gaa(x0, x1, x2)
U2_gaa(x0, x1, x2)
U4_gaa(x0)
U3_gaa(x0)
U7_gaa(x0)
U5_gaa(x0)
U1_gaa(x0)

(21) Obligation:

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

HIDDEN_FLATTEN1_IN_GAA(.(T25, T26)) → HIDDEN_FLATTEN1_IN_GAA(T26)

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

(22) 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:

• HIDDEN_FLATTEN1_IN_GAA(.(T25, T26)) → HIDDEN_FLATTEN1_IN_GAA(T26)
  The graph contains the following edges 1 > 1

(23) TRUE

(24) Obligation:

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

HIDDEN_FLATTEN11_IN_GAA(.(T21, T22)) → HIDDEN_FLATTEN11_IN_GAA(T22)
HIDDEN_FLATTEN11_IN_GAA(.(.(T15, T16), T17)) → HIDDEN_FLATTEN11_IN_GAA(T17)

The TRS R consists of the following rules:

hidden_flatten11_in_gaa(.(.(T15, T16), T17)) → U5_gaa(hidden_flatten11_in_gaa(T17))
hidden_flatten11_in_gaa(.(.(T15, T16), T17)) → U6_gaa(T15, T16, hidden_flatten11_in_gaa(T17))
hidden_flatten11_in_gaa(.(T21, T22)) → U8_gaa(hidden_flatten11_in_gaa(T22))
U8_gaa(hidden_flatten11_out_gaa) → hidden_flatten11_out_gaa
hidden_flatten1_in_gaa(.(T25, T26)) → U4_gaa(hidden_flatten1_in_gaa(T26))
U4_gaa(hidden_flatten1_out_gaa) → hidden_flatten1_out_gaa
U5_gaa(hidden_flatten11_out_gaa) → hidden_flatten11_out_gaa

The set Q consists of the following terms:

hidden_flatten1_in_gaa(x0)
hidden_flatten11_in_gaa(x0)
U8_gaa(x0)
U6_gaa(x0, x1, x2)
U2_gaa(x0, x1, x2)
U4_gaa(x0)
U3_gaa(x0)
U7_gaa(x0)
U5_gaa(x0)
U1_gaa(x0)

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

(25) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(26) Obligation:

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

HIDDEN_FLATTEN11_IN_GAA(.(T21, T22)) → HIDDEN_FLATTEN11_IN_GAA(T22)
HIDDEN_FLATTEN11_IN_GAA(.(.(T15, T16), T17)) → HIDDEN_FLATTEN11_IN_GAA(T17)

R is empty.
The set Q consists of the following terms:

hidden_flatten1_in_gaa(x0)
hidden_flatten11_in_gaa(x0)
U8_gaa(x0)
U6_gaa(x0, x1, x2)
U2_gaa(x0, x1, x2)
U4_gaa(x0)
U3_gaa(x0)
U7_gaa(x0)
U5_gaa(x0)
U1_gaa(x0)

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

(27) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

hidden_flatten1_in_gaa(x0)
hidden_flatten11_in_gaa(x0)
U8_gaa(x0)
U6_gaa(x0, x1, x2)
U2_gaa(x0, x1, x2)
U4_gaa(x0)
U3_gaa(x0)
U7_gaa(x0)
U5_gaa(x0)
U1_gaa(x0)

(28) Obligation:

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

HIDDEN_FLATTEN11_IN_GAA(.(T21, T22)) → HIDDEN_FLATTEN11_IN_GAA(T22)
HIDDEN_FLATTEN11_IN_GAA(.(.(T15, T16), T17)) → HIDDEN_FLATTEN11_IN_GAA(T17)

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

(29) 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:

• HIDDEN_FLATTEN11_IN_GAA(.(T21, T22)) → HIDDEN_FLATTEN11_IN_GAA(T22)
  The graph contains the following edges 1 > 1

• HIDDEN_FLATTEN11_IN_GAA(.(.(T15, T16), T17)) → HIDDEN_FLATTEN11_IN_GAA(T17)
  The graph contains the following edges 1 > 1

(30) TRUE