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

(0) Obligation:

Clauses:

flat([], []).
flat(.([], T), R) :- flat(T, R).
flat(.(.(H, T), TT), .(H, R)) :- flat(.(T, TT), R).

Query: flat(g,a)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: flat(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: flat(T1, T2), SCOPE: 1, CLAUSE: 0 | flat(T1, T2), SCOPE: 1, CLAUSE: 1 | flat(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | flat([], T2), SCOPE: 1, CLAUSE: 1 | flat([], T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: flat(T1, T2), SCOPE: 1, CLAUSE: 1 | flat(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nflat(T1, T2) !=? flat([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: flat([], T2), SCOPE: 1, CLAUSE: 1 | flat([], T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: flat([], T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: flat(T5, T7) | flat(.([], T5), T2), SCOPE: 1, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: flat(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nX9 is free\nX10 is free\nflat(T1, T2) !=? flat([], [])\nflat(T1, T2) !=? flat(.([], X9), X10)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: flat(T5, T7), SCOPE: 2, CLAUSE: 0 | flat(T5, T7), SCOPE: 2, CLAUSE: 1 | flat(T5, T7), SCOPE: 2, CLAUSE: 2 | ?_2 | flat(.([], T5), T2), SCOPE: 1, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: flat(T5, T7), SCOPE: 2, CLAUSE: 0\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: flat(T5, T7), SCOPE: 2, CLAUSE: 1 | flat(T5, T7), SCOPE: 2, CLAUSE: 2 | ?_2 | flat(.([], T5), T2), SCOPE: 1, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: flat(T5, T7), SCOPE: 2, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: flat(T5, T7), SCOPE: 2, CLAUSE: 2 | ?_2 | flat(.([], T5), T2), SCOPE: 1, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: flat(T16, T18)\n\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: flat(T5, T7), SCOPE: 2, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ?_2 | flat(.([], T5), T2), SCOPE: 1, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: flat(.(T36, T37), T39)\n\nT36 is ground\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: flat(.([], T5), T2), SCOPE: 1, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: flat(.(T45, T46), T48)\n\nT45 is ground\nT46 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: flat(.(T45, T46), T48), SCOPE: 3, CLAUSE: 0 | flat(.(T45, T46), T48), SCOPE: 3, CLAUSE: 1 | flat(.(T45, T46), T48), SCOPE: 3, CLAUSE: 2\n\nT45 is ground\nT46 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: flat(.(T45, T46), T48), SCOPE: 3, CLAUSE: 1 | flat(.(T45, T46), T48), SCOPE: 3, CLAUSE: 2\n\nT45 is ground\nT46 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: flat(.(T45, T46), T48), SCOPE: 3, CLAUSE: 1\n\nT45 is ground\nT46 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: flat(.(T45, T46), T48), SCOPE: 3, CLAUSE: 2\n\nT45 is ground\nT46 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: flat(T57, T59)\n\nT57 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: flat(.(T69, T70), T72)\n\nT69 is ground\nT70 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 36 [arrowhead = none ];
36 -> 2;

37 [label="EVAL with clause\nflat([], []).\nand substitutionT1 -> [],\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 37 [arrowhead = none ];
37 -> 3;

38 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 38 [arrowhead = none ];
38 -> 4;

39 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 39 [arrowhead = none ];
39 -> 5;

40 [label="EVAL with clause\nflat(.([], X9), X10) :- flat(X9, X10).\nand substitutionX9 -> T5,\nT1 -> .([], T5),\nT2 -> T7,\nX10 -> T7,\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 40 [arrowhead = none ];
40 -> 8;

41 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 41 [arrowhead = none ];
41 -> 9;

42 [label="BACKTRACK\nfor clause: flat(.([], T), R) :- flat(T, R)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 42 [arrowhead = none ];
42 -> 6;

43 [label="BACKTRACK\nfor clause: flat(.(.(H, T), TT), .(H, R)) :- flat(.(T, TT), R)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 43 [arrowhead = none ];
43 -> 7;

44 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
8 -> 44 [arrowhead = none ];
44 -> 10;

45 [label="EVAL with clause\nflat(.(.(X49, X50), X51), .(X49, X52)) :- flat(.(X50, X51), X52).\nand substitutionX49 -> T44,\nX50 -> T45,\nX51 -> T46,\nT1 -> .(.(T44, T45), T46),\nX52 -> T48,\nT2 -> .(T44, T48),\nT47 -> T48", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 45 [arrowhead = none ];
45 -> 26;

46 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 46 [arrowhead = none ];
46 -> 27;

47 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 47 [arrowhead = none ];
47 -> 11;

48 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 48 [arrowhead = none ];
48 -> 12;

49 [label="EVAL with clause\nflat([], []).\nand substitutionT5 -> [],\nT7 -> []", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 49 [arrowhead = none ];
49 -> 13;

50 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 50 [arrowhead = none ];
50 -> 14;

51 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 51 [arrowhead = none ];
51 -> 16;

52 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 52 [arrowhead = none ];
52 -> 17;

53 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 53 [arrowhead = none ];
53 -> 15;

54 [label="EVAL with clause\nflat(.([], X19), X20) :- flat(X19, X20).\nand substitutionX19 -> T16,\nT5 -> .([], T16),\nT7 -> T18,\nX20 -> T18,\nT17 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 54 [arrowhead = none ];
54 -> 18;

55 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 55 [arrowhead = none ];
55 -> 19;

56 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
17 -> 56 [arrowhead = none ];
56 -> 20;

57 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
17 -> 57 [arrowhead = none ];
57 -> 21;

58 [label="INSTANCE with matching:\nT1 -> T16\nT2 -> T18", fontsize=14, style = filled, fillcolor = lightgrey];
18 -> 58 [arrowhead = none , style=dashed];
58 -> 1[style=dashed];

59 [label="EVAL with clause\nflat(.(.(X37, X38), X39), .(X37, X40)) :- flat(.(X38, X39), X40).\nand substitutionX37 -> T35,\nX38 -> T36,\nX39 -> T37,\nT5 -> .(.(T35, T36), T37),\nX40 -> T39,\nT7 -> .(T35, T39),\nT38 -> T39", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 59 [arrowhead = none ];
59 -> 22;

60 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 60 [arrowhead = none ];
60 -> 23;

61 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 61 [arrowhead = none ];
61 -> 24;

62 [label="INSTANCE with matching:\nT1 -> .(T36, T37)\nT2 -> T39", fontsize=14, style = filled, fillcolor = lightgrey];
22 -> 62 [arrowhead = none , style=dashed];
62 -> 1[style=dashed];

63 [label="BACKTRACK\nfor clause: flat(.(.(H, T), TT), .(H, R)) :- flat(.(T, TT), R)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 63 [arrowhead = none ];
63 -> 25;

64 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 64 [arrowhead = none ];
64 -> 28;

65 [label="BACKTRACK\nfor clause: flat([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 65 [arrowhead = none ];
65 -> 29;

66 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
29 -> 66 [arrowhead = none ];
66 -> 30;

67 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
29 -> 67 [arrowhead = none ];
67 -> 31;

68 [label="EVAL with clause\nflat(.([], X61), X62) :- flat(X61, X62).\nand substitutionT45 -> [],\nT46 -> T57,\nX61 -> T57,\nT48 -> T59,\nX62 -> T59,\nT58 -> T59", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 68 [arrowhead = none ];
68 -> 32;

69 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 69 [arrowhead = none ];
69 -> 33;

70 [label="EVAL with clause\nflat(.(.(X71, X72), X73), .(X71, X74)) :- flat(.(X72, X73), X74).\nand substitutionX71 -> T68,\nX72 -> T69,\nT45 -> .(T68, T69),\nT46 -> T70,\nX73 -> T70,\nX74 -> T72,\nT48 -> .(T68, T72),\nT71 -> T72", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 70 [arrowhead = none ];
70 -> 34;

71 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 71 [arrowhead = none ];
71 -> 35;

72 [label="INSTANCE with matching:\nT1 -> T57\nT2 -> T59", fontsize=14, style = filled, fillcolor = lightgrey];
32 -> 72 [arrowhead = none , style=dashed];
72 -> 1[style=dashed];

73 [label="INSTANCE with matching:\nT1 -> .(T69, T70)\nT2 -> T72", fontsize=14, style = filled, fillcolor = lightgrey];
34 -> 73 [arrowhead = none , style=dashed];
73 -> 1[style=dashed];

}

(2) Obligation:

Clauses:

flatA([], []).
flatA(.([], []), []).
flatA(.([], .([], T16)), T18) :- flatA(T16, T18).
flatA(.([], .(.(T35, T36), T37)), .(T35, T39)) :- flatA(.(T36, T37), T39).
flatA(.(.(T44, []), T57), .(T44, T59)) :- flatA(T57, T59).
flatA(.(.(T44, .(T68, T69)), T70), .(T44, .(T68, T72))) :- flatA(.(T69, T70), T72).

Query: flatA(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:
flatA_in: (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

flatA_in_ga([], []) → flatA_out_ga([], [])
flatA_in_ga(.([], []), []) → flatA_out_ga(.([], []), [])
flatA_in_ga(.([], .([], T16)), T18) → U1_ga(T16, T18, flatA_in_ga(T16, T18))
flatA_in_ga(.([], .(.(T35, T36), T37)), .(T35, T39)) → U2_ga(T35, T36, T37, T39, flatA_in_ga(.(T36, T37), T39))
flatA_in_ga(.(.(T44, []), T57), .(T44, T59)) → U3_ga(T44, T57, T59, flatA_in_ga(T57, T59))
flatA_in_ga(.(.(T44, .(T68, T69)), T70), .(T44, .(T68, T72))) → U4_ga(T44, T68, T69, T70, T72, flatA_in_ga(.(T69, T70), T72))
U4_ga(T44, T68, T69, T70, T72, flatA_out_ga(.(T69, T70), T72)) → flatA_out_ga(.(.(T44, .(T68, T69)), T70), .(T44, .(T68, T72)))
U3_ga(T44, T57, T59, flatA_out_ga(T57, T59)) → flatA_out_ga(.(.(T44, []), T57), .(T44, T59))
U2_ga(T35, T36, T37, T39, flatA_out_ga(.(T36, T37), T39)) → flatA_out_ga(.([], .(.(T35, T36), T37)), .(T35, T39))
U1_ga(T16, T18, flatA_out_ga(T16, T18)) → flatA_out_ga(.([], .([], T16)), T18)

The argument filtering Pi contains the following mapping:
flatA_in_ga(x1, x2)  =  flatA_in_ga(x1)
[]  =  []
flatA_out_ga(x1, x2)  =  flatA_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x1, x2, x6)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

flatA_in_ga([], []) → flatA_out_ga([], [])
flatA_in_ga(.([], []), []) → flatA_out_ga(.([], []), [])
flatA_in_ga(.([], .([], T16)), T18) → U1_ga(T16, T18, flatA_in_ga(T16, T18))
flatA_in_ga(.([], .(.(T35, T36), T37)), .(T35, T39)) → U2_ga(T35, T36, T37, T39, flatA_in_ga(.(T36, T37), T39))
flatA_in_ga(.(.(T44, []), T57), .(T44, T59)) → U3_ga(T44, T57, T59, flatA_in_ga(T57, T59))
flatA_in_ga(.(.(T44, .(T68, T69)), T70), .(T44, .(T68, T72))) → U4_ga(T44, T68, T69, T70, T72, flatA_in_ga(.(T69, T70), T72))
U4_ga(T44, T68, T69, T70, T72, flatA_out_ga(.(T69, T70), T72)) → flatA_out_ga(.(.(T44, .(T68, T69)), T70), .(T44, .(T68, T72)))
U3_ga(T44, T57, T59, flatA_out_ga(T57, T59)) → flatA_out_ga(.(.(T44, []), T57), .(T44, T59))
U2_ga(T35, T36, T37, T39, flatA_out_ga(.(T36, T37), T39)) → flatA_out_ga(.([], .(.(T35, T36), T37)), .(T35, T39))
U1_ga(T16, T18, flatA_out_ga(T16, T18)) → flatA_out_ga(.([], .([], T16)), T18)

The argument filtering Pi contains the following mapping:
flatA_in_ga(x1, x2)  =  flatA_in_ga(x1)
[]  =  []
flatA_out_ga(x1, x2)  =  flatA_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x1, x2, x6)

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

FLATA_IN_GA(.([], .([], T16)), T18) → U1_GA(T16, T18, flatA_in_ga(T16, T18))
FLATA_IN_GA(.([], .([], T16)), T18) → FLATA_IN_GA(T16, T18)
FLATA_IN_GA(.([], .(.(T35, T36), T37)), .(T35, T39)) → U2_GA(T35, T36, T37, T39, flatA_in_ga(.(T36, T37), T39))
FLATA_IN_GA(.([], .(.(T35, T36), T37)), .(T35, T39)) → FLATA_IN_GA(.(T36, T37), T39)
FLATA_IN_GA(.(.(T44, []), T57), .(T44, T59)) → U3_GA(T44, T57, T59, flatA_in_ga(T57, T59))
FLATA_IN_GA(.(.(T44, []), T57), .(T44, T59)) → FLATA_IN_GA(T57, T59)
FLATA_IN_GA(.(.(T44, .(T68, T69)), T70), .(T44, .(T68, T72))) → U4_GA(T44, T68, T69, T70, T72, flatA_in_ga(.(T69, T70), T72))
FLATA_IN_GA(.(.(T44, .(T68, T69)), T70), .(T44, .(T68, T72))) → FLATA_IN_GA(.(T69, T70), T72)

The TRS R consists of the following rules:

flatA_in_ga([], []) → flatA_out_ga([], [])
flatA_in_ga(.([], []), []) → flatA_out_ga(.([], []), [])
flatA_in_ga(.([], .([], T16)), T18) → U1_ga(T16, T18, flatA_in_ga(T16, T18))
flatA_in_ga(.([], .(.(T35, T36), T37)), .(T35, T39)) → U2_ga(T35, T36, T37, T39, flatA_in_ga(.(T36, T37), T39))
flatA_in_ga(.(.(T44, []), T57), .(T44, T59)) → U3_ga(T44, T57, T59, flatA_in_ga(T57, T59))
flatA_in_ga(.(.(T44, .(T68, T69)), T70), .(T44, .(T68, T72))) → U4_ga(T44, T68, T69, T70, T72, flatA_in_ga(.(T69, T70), T72))
U4_ga(T44, T68, T69, T70, T72, flatA_out_ga(.(T69, T70), T72)) → flatA_out_ga(.(.(T44, .(T68, T69)), T70), .(T44, .(T68, T72)))
U3_ga(T44, T57, T59, flatA_out_ga(T57, T59)) → flatA_out_ga(.(.(T44, []), T57), .(T44, T59))
U2_ga(T35, T36, T37, T39, flatA_out_ga(.(T36, T37), T39)) → flatA_out_ga(.([], .(.(T35, T36), T37)), .(T35, T39))
U1_ga(T16, T18, flatA_out_ga(T16, T18)) → flatA_out_ga(.([], .([], T16)), T18)

The argument filtering Pi contains the following mapping:
flatA_in_ga(x1, x2)  =  flatA_in_ga(x1)
[]  =  []
flatA_out_ga(x1, x2)  =  flatA_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x1, x2, x6)
FLATA_IN_GA(x1, x2)  =  FLATA_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x5)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)
U4_GA(x1, x2, x3, x4, x5, x6)  =  U4_GA(x1, x2, x6)

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

(6) Obligation:

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

FLATA_IN_GA(.([], .([], T16)), T18) → U1_GA(T16, T18, flatA_in_ga(T16, T18))
FLATA_IN_GA(.([], .([], T16)), T18) → FLATA_IN_GA(T16, T18)
FLATA_IN_GA(.([], .(.(T35, T36), T37)), .(T35, T39)) → U2_GA(T35, T36, T37, T39, flatA_in_ga(.(T36, T37), T39))
FLATA_IN_GA(.([], .(.(T35, T36), T37)), .(T35, T39)) → FLATA_IN_GA(.(T36, T37), T39)
FLATA_IN_GA(.(.(T44, []), T57), .(T44, T59)) → U3_GA(T44, T57, T59, flatA_in_ga(T57, T59))
FLATA_IN_GA(.(.(T44, []), T57), .(T44, T59)) → FLATA_IN_GA(T57, T59)
FLATA_IN_GA(.(.(T44, .(T68, T69)), T70), .(T44, .(T68, T72))) → U4_GA(T44, T68, T69, T70, T72, flatA_in_ga(.(T69, T70), T72))
FLATA_IN_GA(.(.(T44, .(T68, T69)), T70), .(T44, .(T68, T72))) → FLATA_IN_GA(.(T69, T70), T72)

The TRS R consists of the following rules:

flatA_in_ga([], []) → flatA_out_ga([], [])
flatA_in_ga(.([], []), []) → flatA_out_ga(.([], []), [])
flatA_in_ga(.([], .([], T16)), T18) → U1_ga(T16, T18, flatA_in_ga(T16, T18))
flatA_in_ga(.([], .(.(T35, T36), T37)), .(T35, T39)) → U2_ga(T35, T36, T37, T39, flatA_in_ga(.(T36, T37), T39))
flatA_in_ga(.(.(T44, []), T57), .(T44, T59)) → U3_ga(T44, T57, T59, flatA_in_ga(T57, T59))
flatA_in_ga(.(.(T44, .(T68, T69)), T70), .(T44, .(T68, T72))) → U4_ga(T44, T68, T69, T70, T72, flatA_in_ga(.(T69, T70), T72))
U4_ga(T44, T68, T69, T70, T72, flatA_out_ga(.(T69, T70), T72)) → flatA_out_ga(.(.(T44, .(T68, T69)), T70), .(T44, .(T68, T72)))
U3_ga(T44, T57, T59, flatA_out_ga(T57, T59)) → flatA_out_ga(.(.(T44, []), T57), .(T44, T59))
U2_ga(T35, T36, T37, T39, flatA_out_ga(.(T36, T37), T39)) → flatA_out_ga(.([], .(.(T35, T36), T37)), .(T35, T39))
U1_ga(T16, T18, flatA_out_ga(T16, T18)) → flatA_out_ga(.([], .([], T16)), T18)

The argument filtering Pi contains the following mapping:
flatA_in_ga(x1, x2)  =  flatA_in_ga(x1)
[]  =  []
flatA_out_ga(x1, x2)  =  flatA_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x1, x2, x6)
FLATA_IN_GA(x1, x2)  =  FLATA_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x5)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)
U4_GA(x1, x2, x3, x4, x5, x6)  =  U4_GA(x1, x2, x6)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 4 less nodes.

(8) Obligation:

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

FLATA_IN_GA(.([], .(.(T35, T36), T37)), .(T35, T39)) → FLATA_IN_GA(.(T36, T37), T39)
FLATA_IN_GA(.([], .([], T16)), T18) → FLATA_IN_GA(T16, T18)
FLATA_IN_GA(.(.(T44, []), T57), .(T44, T59)) → FLATA_IN_GA(T57, T59)
FLATA_IN_GA(.(.(T44, .(T68, T69)), T70), .(T44, .(T68, T72))) → FLATA_IN_GA(.(T69, T70), T72)

The TRS R consists of the following rules:

flatA_in_ga([], []) → flatA_out_ga([], [])
flatA_in_ga(.([], []), []) → flatA_out_ga(.([], []), [])
flatA_in_ga(.([], .([], T16)), T18) → U1_ga(T16, T18, flatA_in_ga(T16, T18))
flatA_in_ga(.([], .(.(T35, T36), T37)), .(T35, T39)) → U2_ga(T35, T36, T37, T39, flatA_in_ga(.(T36, T37), T39))
flatA_in_ga(.(.(T44, []), T57), .(T44, T59)) → U3_ga(T44, T57, T59, flatA_in_ga(T57, T59))
flatA_in_ga(.(.(T44, .(T68, T69)), T70), .(T44, .(T68, T72))) → U4_ga(T44, T68, T69, T70, T72, flatA_in_ga(.(T69, T70), T72))
U4_ga(T44, T68, T69, T70, T72, flatA_out_ga(.(T69, T70), T72)) → flatA_out_ga(.(.(T44, .(T68, T69)), T70), .(T44, .(T68, T72)))
U3_ga(T44, T57, T59, flatA_out_ga(T57, T59)) → flatA_out_ga(.(.(T44, []), T57), .(T44, T59))
U2_ga(T35, T36, T37, T39, flatA_out_ga(.(T36, T37), T39)) → flatA_out_ga(.([], .(.(T35, T36), T37)), .(T35, T39))
U1_ga(T16, T18, flatA_out_ga(T16, T18)) → flatA_out_ga(.([], .([], T16)), T18)

The argument filtering Pi contains the following mapping:
flatA_in_ga(x1, x2)  =  flatA_in_ga(x1)
[]  =  []
flatA_out_ga(x1, x2)  =  flatA_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x1, x2, x6)
FLATA_IN_GA(x1, x2)  =  FLATA_IN_GA(x1)

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

(9) UsableRulesProof (EQUIVALENT transformation)

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

(10) Obligation:

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

FLATA_IN_GA(.([], .(.(T35, T36), T37)), .(T35, T39)) → FLATA_IN_GA(.(T36, T37), T39)
FLATA_IN_GA(.([], .([], T16)), T18) → FLATA_IN_GA(T16, T18)
FLATA_IN_GA(.(.(T44, []), T57), .(T44, T59)) → FLATA_IN_GA(T57, T59)
FLATA_IN_GA(.(.(T44, .(T68, T69)), T70), .(T44, .(T68, T72))) → FLATA_IN_GA(.(T69, T70), T72)

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

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

(11) PiDPToQDPProof (SOUND transformation)

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

(12) Obligation:

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

FLATA_IN_GA(.([], .(.(T35, T36), T37))) → FLATA_IN_GA(.(T36, T37))
FLATA_IN_GA(.([], .([], T16))) → FLATA_IN_GA(T16)
FLATA_IN_GA(.(.(T44, []), T57)) → FLATA_IN_GA(T57)
FLATA_IN_GA(.(.(T44, .(T68, T69)), T70)) → FLATA_IN_GA(.(T69, T70))

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

(13) 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.

The following dependency pairs can be deleted:

FLATA_IN_GA(.([], .(.(T35, T36), T37))) → FLATA_IN_GA(.(T36, T37))
FLATA_IN_GA(.([], .([], T16))) → FLATA_IN_GA(T16)
FLATA_IN_GA(.(.(T44, []), T57)) → FLATA_IN_GA(T57)

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 2·x₁ + 2·x₂   
POL(FLATA_IN_GA(x₁)) = 2·x₁   
POL([]) = 0   

(14) Obligation:

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

FLATA_IN_GA(.(.(T44, .(T68, T69)), T70)) → FLATA_IN_GA(.(T69, T70))

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

(15) 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:

FLATA_IN_GA(.(.(T44, .(T68, T69)), T70)) → FLATA_IN_GA(.(T69, T70))

Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 1 + 2·x₁ + 2·x₂   
POL(FLATA_IN_GA(x₁)) = 2·x₁   

(16) Obligation:

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

(17) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(18) YES