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

(0) Obligation:

Clauses:

flat(niltree, nil).
flat(tree(X, niltree, T), cons(X, Xs)) :- flat(T, Xs).
flat(tree(X, tree(Y, T1, T2), T3), Xs) :- flat(tree(Y, T1, tree(X, T2, T3)), Xs).

Query: flat(g,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: flat(T4, T5)\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: flat(T4, T5), SCOPE: 1, CLAUSE: 0 | flat(T4, T5), SCOPE: 1, CLAUSE: 1 | flat(T4, T5), SCOPE: 1, CLAUSE: 2\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | flat(niltree, T5), SCOPE: 1, CLAUSE: 1 | flat(niltree, T5), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: flat(T4, T5), SCOPE: 1, CLAUSE: 1 | flat(T4, T5), SCOPE: 1, CLAUSE: 2\n\nT4 is ground\nflat(T4, T5) !=? flat(niltree, nil)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: flat(niltree, T5), SCOPE: 1, CLAUSE: 1 | flat(niltree, T5), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: flat(niltree, T5), 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(T10, T12) | flat(tree(T9, niltree, T10), T5), SCOPE: 1, CLAUSE: 2\n\nT9 is ground\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: flat(T4, T5), SCOPE: 1, CLAUSE: 2\n\nT4 is ground\nX13 is free\nX14 is free\nX15 is free\nflat(T4, T5) !=? flat(niltree, nil)\nflat(T4, T5) !=? flat(tree(X13, niltree, X14), cons(X13, X15))", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: flat(T10, T12), SCOPE: 2, CLAUSE: 0 | flat(T10, T12), SCOPE: 2, CLAUSE: 1 | flat(T10, T12), SCOPE: 2, CLAUSE: 2 | ?_2 | flat(tree(T9, niltree, T10), T5), SCOPE: 1, CLAUSE: 2\n\nT9 is ground\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: flat(T10, T12), SCOPE: 2, CLAUSE: 0\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: flat(T10, T12), SCOPE: 2, CLAUSE: 1 | flat(T10, T12), SCOPE: 2, CLAUSE: 2 | ?_2 | flat(tree(T9, niltree, T10), T5), SCOPE: 1, CLAUSE: 2\n\nT9 is ground\nT10 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(T10, T12), SCOPE: 2, CLAUSE: 1\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: flat(T10, T12), SCOPE: 2, CLAUSE: 2 | ?_2 | flat(tree(T9, niltree, T10), T5), SCOPE: 1, CLAUSE: 2\n\nT9 is ground\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: flat(T26, T28)\n\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: flat(T10, T12), SCOPE: 2, CLAUSE: 2\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ?_2 | flat(tree(T9, niltree, T10), T5), SCOPE: 1, CLAUSE: 2\n\nT9 is ground\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: flat(tree(T54, T55, tree(T53, T56, T57)), T59)\n\nT53 is ground\nT54 is ground\nT55 is ground\nT56 is ground\nT57 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: flat(tree(T9, niltree, T10), T5), SCOPE: 1, CLAUSE: 2\n\nT9 is ground\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: flat(tree(T68, T69, tree(T67, T70, T71)), T73)\n\nT67 is ground\nT68 is ground\nT69 is ground\nT70 is ground\nT71 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: flat(tree(T68, T69, tree(T67, T70, T71)), T73), SCOPE: 3, CLAUSE: 0 | flat(tree(T68, T69, tree(T67, T70, T71)), T73), SCOPE: 3, CLAUSE: 1 | flat(tree(T68, T69, tree(T67, T70, T71)), T73), SCOPE: 3, CLAUSE: 2\n\nT67 is ground\nT68 is ground\nT69 is ground\nT70 is ground\nT71 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: flat(tree(T68, T69, tree(T67, T70, T71)), T73), SCOPE: 3, CLAUSE: 1 | flat(tree(T68, T69, tree(T67, T70, T71)), T73), SCOPE: 3, CLAUSE: 2\n\nT67 is ground\nT68 is ground\nT69 is ground\nT70 is ground\nT71 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: flat(tree(T68, T69, tree(T67, T70, T71)), T73), SCOPE: 3, CLAUSE: 1\n\nT67 is ground\nT68 is ground\nT69 is ground\nT70 is ground\nT71 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: flat(tree(T68, T69, tree(T67, T70, T71)), T73), SCOPE: 3, CLAUSE: 2\n\nT67 is ground\nT68 is ground\nT69 is ground\nT70 is ground\nT71 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: flat(tree(T95, T96, T97), T99)\n\nT95 is ground\nT96 is ground\nT97 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: flat(tree(T117, T118, tree(T116, T119, tree(T120, T121, T122))), T124)\n\nT116 is ground\nT117 is ground\nT118 is ground\nT119 is ground\nT120 is ground\nT121 is ground\nT122 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(niltree, nil).\nand substitutionT4 -> niltree,\nT5 -> nil", 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(tree(X13, niltree, X14), cons(X13, X15)) :- flat(X14, X15).\nand substitutionX13 -> T9,\nX14 -> T10,\nT4 -> tree(T9, niltree, T10),\nX15 -> T12,\nT5 -> cons(T9, T12),\nT11 -> T12", 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(tree(X, niltree, T), cons(X, Xs)) :- flat(T, Xs)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 42 [arrowhead = none ];
42 -> 6;

43 [label="BACKTRACK\nfor clause: flat(tree(X, tree(Y, T1, T2), T3), Xs) :- flat(tree(Y, T1, tree(X, T2, T3)), Xs)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(tree(X73, tree(X74, X75, X76), X77), X78) :- flat(tree(X74, X75, tree(X73, X76, X77)), X78).\nand substitutionX73 -> T67,\nX74 -> T68,\nX75 -> T69,\nX76 -> T70,\nX77 -> T71,\nT4 -> tree(T67, tree(T68, T69, T70), T71),\nT5 -> T73,\nX78 -> T73,\nT72 -> T73", 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(niltree, nil).\nand substitutionT10 -> niltree,\nT12 -> nil", 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(tree(X28, niltree, X29), cons(X28, X30)) :- flat(X29, X30).\nand substitutionX28 -> T25,\nX29 -> T26,\nT10 -> tree(T25, niltree, T26),\nX30 -> T28,\nT12 -> cons(T25, T28),\nT27 -> T28", 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:\nT4 -> T26\nT5 -> T28", fontsize=14, style = filled, fillcolor = lightgrey];
18 -> 58 [arrowhead = none , style=dashed];
58 -> 1[style=dashed];

59 [label="EVAL with clause\nflat(tree(X55, tree(X56, X57, X58), X59), X60) :- flat(tree(X56, X57, tree(X55, X58, X59)), X60).\nand substitutionX55 -> T53,\nX56 -> T54,\nX57 -> T55,\nX58 -> T56,\nX59 -> T57,\nT10 -> tree(T53, tree(T54, T55, T56), T57),\nT12 -> T59,\nX60 -> T59,\nT58 -> T59", 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:\nT4 -> tree(T54, T55, tree(T53, T56, T57))\nT5 -> T59", fontsize=14, style = filled, fillcolor = lightgrey];
22 -> 62 [arrowhead = none , style=dashed];
62 -> 1[style=dashed];

63 [label="BACKTRACK\nfor clause: flat(tree(X, tree(Y, T1, T2), T3), Xs) :- flat(tree(Y, T1, tree(X, T2, T3)), Xs)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(niltree, nil)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(tree(X91, niltree, X92), cons(X91, X93)) :- flat(X92, X93).\nand substitutionT68 -> T94,\nX91 -> T94,\nT69 -> niltree,\nT67 -> T95,\nT70 -> T96,\nT71 -> T97,\nX92 -> tree(T95, T96, T97),\nX93 -> T99,\nT73 -> cons(T94, T99),\nT98 -> T99", 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(tree(X106, tree(X107, X108, X109), X110), X111) :- flat(tree(X107, X108, tree(X106, X109, X110)), X111).\nand substitutionT68 -> T116,\nX106 -> T116,\nX107 -> T117,\nX108 -> T118,\nX109 -> T119,\nT69 -> tree(T117, T118, T119),\nT67 -> T120,\nT70 -> T121,\nT71 -> T122,\nX110 -> tree(T120, T121, T122),\nT73 -> T124,\nX111 -> T124,\nT123 -> T124", 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:\nT4 -> tree(T95, T96, T97)\nT5 -> T99", fontsize=14, style = filled, fillcolor = lightgrey];
32 -> 72 [arrowhead = none , style=dashed];
72 -> 1[style=dashed];

73 [label="INSTANCE with matching:\nT4 -> tree(T117, T118, tree(T116, T119, tree(T120, T121, T122)))\nT5 -> T124", fontsize=14, style = filled, fillcolor = lightgrey];
34 -> 73 [arrowhead = none , style=dashed];
73 -> 1[style=dashed];

}

(2) Obligation:

Triples:

flatA(tree(X1, niltree, tree(X2, niltree, X3)), cons(X1, cons(X2, X4))) :- flatA(X3, X4).
flatA(tree(X1, niltree, tree(X2, tree(X3, X4, X5), X6)), cons(X1, X7)) :- flatA(tree(X3, X4, tree(X2, X5, X6)), X7).
flatA(tree(X1, tree(X2, niltree, X3), X4), cons(X2, X5)) :- flatA(tree(X1, X3, X4), X5).
flatA(tree(X1, tree(X2, tree(X3, X4, X5), X6), X7), X8) :- flatA(tree(X3, X4, tree(X2, X5, tree(X1, X6, X7))), X8).

Clauses:

flatcA(niltree, nil).
flatcA(tree(X1, niltree, niltree), cons(X1, nil)).
flatcA(tree(X1, niltree, tree(X2, niltree, X3)), cons(X1, cons(X2, X4))) :- flatcA(X3, X4).
flatcA(tree(X1, niltree, tree(X2, tree(X3, X4, X5), X6)), cons(X1, X7)) :- flatcA(tree(X3, X4, tree(X2, X5, X6)), X7).
flatcA(tree(X1, tree(X2, niltree, X3), X4), cons(X2, X5)) :- flatcA(tree(X1, X3, X4), X5).
flatcA(tree(X1, tree(X2, tree(X3, X4, X5), X6), X7), X8) :- flatcA(tree(X3, X4, tree(X2, X5, tree(X1, X6, X7))), X8).

Afs:

flatA(x1, x2)  =  flatA(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
flatA_in: (b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

FLATA_IN_GA(tree(X1, niltree, tree(X2, niltree, X3)), cons(X1, cons(X2, X4))) → U1_GA(X1, X2, X3, X4, flatA_in_ga(X3, X4))
FLATA_IN_GA(tree(X1, niltree, tree(X2, niltree, X3)), cons(X1, cons(X2, X4))) → FLATA_IN_GA(X3, X4)
FLATA_IN_GA(tree(X1, niltree, tree(X2, tree(X3, X4, X5), X6)), cons(X1, X7)) → U2_GA(X1, X2, X3, X4, X5, X6, X7, flatA_in_ga(tree(X3, X4, tree(X2, X5, X6)), X7))
FLATA_IN_GA(tree(X1, niltree, tree(X2, tree(X3, X4, X5), X6)), cons(X1, X7)) → FLATA_IN_GA(tree(X3, X4, tree(X2, X5, X6)), X7)
FLATA_IN_GA(tree(X1, tree(X2, niltree, X3), X4), cons(X2, X5)) → U3_GA(X1, X2, X3, X4, X5, flatA_in_ga(tree(X1, X3, X4), X5))
FLATA_IN_GA(tree(X1, tree(X2, niltree, X3), X4), cons(X2, X5)) → FLATA_IN_GA(tree(X1, X3, X4), X5)
FLATA_IN_GA(tree(X1, tree(X2, tree(X3, X4, X5), X6), X7), X8) → U4_GA(X1, X2, X3, X4, X5, X6, X7, X8, flatA_in_ga(tree(X3, X4, tree(X2, X5, tree(X1, X6, X7))), X8))
FLATA_IN_GA(tree(X1, tree(X2, tree(X3, X4, X5), X6), X7), X8) → FLATA_IN_GA(tree(X3, X4, tree(X2, X5, tree(X1, X6, X7))), X8)

R is empty.
The argument filtering Pi contains the following mapping:
flatA_in_ga(x1, x2)  =  flatA_in_ga(x1)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
niltree  =  niltree
cons(x1, x2)  =  cons(x1, x2)
FLATA_IN_GA(x1, x2)  =  FLATA_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x3, x5)
U2_GA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_GA(x1, x2, x3, x4, x5, x6, x8)
U3_GA(x1, x2, x3, x4, x5, x6)  =  U3_GA(x1, x2, x3, x4, x6)
U4_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U4_GA(x1, x2, x3, x4, x5, x6, x7, x9)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

FLATA_IN_GA(tree(X1, niltree, tree(X2, niltree, X3)), cons(X1, cons(X2, X4))) → U1_GA(X1, X2, X3, X4, flatA_in_ga(X3, X4))
FLATA_IN_GA(tree(X1, niltree, tree(X2, niltree, X3)), cons(X1, cons(X2, X4))) → FLATA_IN_GA(X3, X4)
FLATA_IN_GA(tree(X1, niltree, tree(X2, tree(X3, X4, X5), X6)), cons(X1, X7)) → U2_GA(X1, X2, X3, X4, X5, X6, X7, flatA_in_ga(tree(X3, X4, tree(X2, X5, X6)), X7))
FLATA_IN_GA(tree(X1, niltree, tree(X2, tree(X3, X4, X5), X6)), cons(X1, X7)) → FLATA_IN_GA(tree(X3, X4, tree(X2, X5, X6)), X7)
FLATA_IN_GA(tree(X1, tree(X2, niltree, X3), X4), cons(X2, X5)) → U3_GA(X1, X2, X3, X4, X5, flatA_in_ga(tree(X1, X3, X4), X5))
FLATA_IN_GA(tree(X1, tree(X2, niltree, X3), X4), cons(X2, X5)) → FLATA_IN_GA(tree(X1, X3, X4), X5)
FLATA_IN_GA(tree(X1, tree(X2, tree(X3, X4, X5), X6), X7), X8) → U4_GA(X1, X2, X3, X4, X5, X6, X7, X8, flatA_in_ga(tree(X3, X4, tree(X2, X5, tree(X1, X6, X7))), X8))
FLATA_IN_GA(tree(X1, tree(X2, tree(X3, X4, X5), X6), X7), X8) → FLATA_IN_GA(tree(X3, X4, tree(X2, X5, tree(X1, X6, X7))), X8)

R is empty.
The argument filtering Pi contains the following mapping:
flatA_in_ga(x1, x2)  =  flatA_in_ga(x1)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
niltree  =  niltree
cons(x1, x2)  =  cons(x1, x2)
FLATA_IN_GA(x1, x2)  =  FLATA_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x3, x5)
U2_GA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_GA(x1, x2, x3, x4, x5, x6, x8)
U3_GA(x1, x2, x3, x4, x5, x6)  =  U3_GA(x1, x2, x3, x4, x6)
U4_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U4_GA(x1, x2, x3, x4, x5, x6, x7, x9)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

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

(6) Obligation:

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

FLATA_IN_GA(tree(X1, niltree, tree(X2, tree(X3, X4, X5), X6)), cons(X1, X7)) → FLATA_IN_GA(tree(X3, X4, tree(X2, X5, X6)), X7)
FLATA_IN_GA(tree(X1, niltree, tree(X2, niltree, X3)), cons(X1, cons(X2, X4))) → FLATA_IN_GA(X3, X4)
FLATA_IN_GA(tree(X1, tree(X2, niltree, X3), X4), cons(X2, X5)) → FLATA_IN_GA(tree(X1, X3, X4), X5)
FLATA_IN_GA(tree(X1, tree(X2, tree(X3, X4, X5), X6), X7), X8) → FLATA_IN_GA(tree(X3, X4, tree(X2, X5, tree(X1, X6, X7))), X8)

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

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

(7) PiDPToQDPProof (SOUND transformation)

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

(8) Obligation:

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

FLATA_IN_GA(tree(X1, niltree, tree(X2, tree(X3, X4, X5), X6))) → FLATA_IN_GA(tree(X3, X4, tree(X2, X5, X6)))
FLATA_IN_GA(tree(X1, niltree, tree(X2, niltree, X3))) → FLATA_IN_GA(X3)
FLATA_IN_GA(tree(X1, tree(X2, niltree, X3), X4)) → FLATA_IN_GA(tree(X1, X3, X4))
FLATA_IN_GA(tree(X1, tree(X2, tree(X3, X4, X5), X6), X7)) → FLATA_IN_GA(tree(X3, X4, tree(X2, X5, tree(X1, X6, X7))))

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

(9) 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(tree(X1, niltree, tree(X2, tree(X3, X4, X5), X6))) → FLATA_IN_GA(tree(X3, X4, tree(X2, X5, X6)))
FLATA_IN_GA(tree(X1, niltree, tree(X2, niltree, X3))) → FLATA_IN_GA(X3)
FLATA_IN_GA(tree(X1, tree(X2, niltree, X3), X4)) → FLATA_IN_GA(tree(X1, X3, X4))
FLATA_IN_GA(tree(X1, tree(X2, tree(X3, X4, X5), X6), X7)) → FLATA_IN_GA(tree(X3, X4, tree(X2, X5, tree(X1, X6, X7))))

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(FLATA_IN_GA(x₁)) = 2·x₁   
POL(niltree) = 0   
POL(tree(x₁, x₂, x₃)) = 2 + 2·x₁ + 2·x₂ + x₃   

(10) Obligation:

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

(11) PisEmptyProof (EQUIVALENT transformation)

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

(12) YES