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) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
3 [label="A: flat(T4, T5)\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: 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];
10 [label="10: true | flat(niltree, T5), SCOPE: 1, CLAUSE: 1 | flat(niltree, T5), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: 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];
16 [label="16: flat(niltree, T5), SCOPE: 1, CLAUSE: 1 | flat(niltree, T5), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: flat(niltree, T5), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: 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];
28 [label="28: 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];
30 [label="30: 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];
32 [label="32: flat(T10, T12), SCOPE: 2, CLAUSE: 0\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: 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];
36 [label="36: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: flat(T10, T12), SCOPE: 2, CLAUSE: 1\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: 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];
46 [label="46: flat(T26, T28)\n\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: flat(T10, T12), SCOPE: 2, CLAUSE: 2\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ?_2 | flat(tree(T9, niltree, T10), T5), SCOPE: 1, CLAUSE: 2\n\nT9 is ground\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: 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];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: flat(tree(T9, niltree, T10), T5), SCOPE: 1, CLAUSE: 2\n\nT9 is ground\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: 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];
66 [label="66: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: 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];
69 [label="69: 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];
72 [label="72: 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];
73 [label="73: 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];
76 [label="76: flat(tree(T95, T96, T97), T99)\n\nT95 is ground\nT96 is ground\nT97 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: 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];
83 [label="83: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 84 [arrowhead = none ];
84 -> 4;

85 [label="EVAL with clause\nflat(niltree, nil).\nand substitutionT4 -> niltree,\nT5 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 85 [arrowhead = none ];
85 -> 10;

86 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 86 [arrowhead = none ];
86 -> 13;

87 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 87 [arrowhead = none ];
87 -> 16;

88 [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];
13 -> 88 [arrowhead = none ];
88 -> 24;

89 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 89 [arrowhead = none ];
89 -> 28;

90 [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];
16 -> 90 [arrowhead = none ];
90 -> 18;

91 [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];
18 -> 91 [arrowhead = none ];
91 -> 22;

92 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 92 [arrowhead = none ];
92 -> 30;

93 [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];
28 -> 93 [arrowhead = none ];
93 -> 64;

94 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 94 [arrowhead = none ];
94 -> 66;

95 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
30 -> 95 [arrowhead = none ];
95 -> 32;

96 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
30 -> 96 [arrowhead = none ];
96 -> 33;

97 [label="EVAL with clause\nflat(niltree, nil).\nand substitutionT10 -> niltree,\nT12 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 97 [arrowhead = none ];
97 -> 36;

98 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 98 [arrowhead = none ];
98 -> 37;

99 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
33 -> 99 [arrowhead = none ];
99 -> 42;

100 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
33 -> 100 [arrowhead = none ];
100 -> 43;

101 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 101 [arrowhead = none ];
101 -> 39;

102 [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];
42 -> 102 [arrowhead = none ];
102 -> 46;

103 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 103 [arrowhead = none ];
103 -> 47;

104 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
43 -> 104 [arrowhead = none ];
104 -> 51;

105 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
43 -> 105 [arrowhead = none ];
105 -> 52;

106 [label="INSTANCE with matching:\nT4 -> T26\nT5 -> T28", fontsize=14, style = filled, fillcolor = lightgrey];
46 -> 106 [arrowhead = none , style=dashed];
106 -> 3[style=dashed];

107 [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];
51 -> 107 [arrowhead = none ];
107 -> 56;

108 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 108 [arrowhead = none ];
108 -> 57;

109 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 109 [arrowhead = none ];
109 -> 58;

110 [label="INSTANCE with matching:\nT4 -> tree(T54, T55, tree(T53, T56, T57))\nT5 -> T59", fontsize=14, style = filled, fillcolor = lightgrey];
56 -> 110 [arrowhead = none , style=dashed];
110 -> 3[style=dashed];

111 [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];
58 -> 111 [arrowhead = none ];
111 -> 61;

112 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
64 -> 112 [arrowhead = none ];
112 -> 67;

113 [label="BACKTRACK\nfor clause: flat(niltree, nil)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
67 -> 113 [arrowhead = none ];
113 -> 69;

114 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
69 -> 114 [arrowhead = none ];
114 -> 72;

115 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
69 -> 115 [arrowhead = none ];
115 -> 73;

116 [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];
72 -> 116 [arrowhead = none ];
116 -> 76;

117 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
72 -> 117 [arrowhead = none ];
117 -> 78;

118 [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];
73 -> 118 [arrowhead = none ];
118 -> 82;

119 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
73 -> 119 [arrowhead = none ];
119 -> 83;

120 [label="INSTANCE with matching:\nT4 -> tree(T95, T96, T97)\nT5 -> T99", fontsize=14, style = filled, fillcolor = lightgrey];
76 -> 120 [arrowhead = none , style=dashed];
120 -> 3[style=dashed];

121 [label="INSTANCE with matching:\nT4 -> tree(T117, T118, tree(T116, T119, tree(T120, T121, T122)))\nT5 -> T124", fontsize=14, style = filled, fillcolor = lightgrey];
82 -> 121 [arrowhead = none , style=dashed];
121 -> 3[style=dashed];

}

(2) Obligation:

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

flatA_in_ga(niltree, nil) → flatA_out_ga(niltree, nil)
flatA_in_ga(tree(T9, niltree, niltree), cons(T9, nil)) → flatA_out_ga(tree(T9, niltree, niltree), cons(T9, nil))
flatA_in_ga(tree(T9, niltree, tree(T25, niltree, T26)), cons(T9, cons(T25, T28))) → U1_ga(T9, T25, T26, T28, flatA_in_ga(T26, T28))
flatA_in_ga(tree(T9, niltree, tree(T53, tree(T54, T55, T56), T57)), cons(T9, T59)) → U2_ga(T9, T53, T54, T55, T56, T57, T59, flatA_in_ga(tree(T54, T55, tree(T53, T56, T57)), T59))
flatA_in_ga(tree(T95, tree(T94, niltree, T96), T97), cons(T94, T99)) → U3_ga(T95, T94, T96, T97, T99, flatA_in_ga(tree(T95, T96, T97), T99))
flatA_in_ga(tree(T120, tree(T116, tree(T117, T118, T119), T121), T122), T124) → U4_ga(T120, T116, T117, T118, T119, T121, T122, T124, flatA_in_ga(tree(T117, T118, tree(T116, T119, tree(T120, T121, T122))), T124))
U4_ga(T120, T116, T117, T118, T119, T121, T122, T124, flatA_out_ga(tree(T117, T118, tree(T116, T119, tree(T120, T121, T122))), T124)) → flatA_out_ga(tree(T120, tree(T116, tree(T117, T118, T119), T121), T122), T124)
U3_ga(T95, T94, T96, T97, T99, flatA_out_ga(tree(T95, T96, T97), T99)) → flatA_out_ga(tree(T95, tree(T94, niltree, T96), T97), cons(T94, T99))
U2_ga(T9, T53, T54, T55, T56, T57, T59, flatA_out_ga(tree(T54, T55, tree(T53, T56, T57)), T59)) → flatA_out_ga(tree(T9, niltree, tree(T53, tree(T54, T55, T56), T57)), cons(T9, T59))
U1_ga(T9, T25, T26, T28, flatA_out_ga(T26, T28)) → flatA_out_ga(tree(T9, niltree, tree(T25, niltree, T26)), cons(T9, cons(T25, T28)))

The argument filtering Pi contains the following mapping:
flatA_in_ga(x1, x2)  =  flatA_in_ga(x1)
niltree  =  niltree
flatA_out_ga(x1, x2)  =  flatA_out_ga(x1, x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
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)
cons(x1, x2)  =  cons(x1, x2)

(3) DependencyPairsProof (EQUIVALENT transformation)

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

FLATA_IN_GA(tree(T9, niltree, tree(T25, niltree, T26)), cons(T9, cons(T25, T28))) → U1_GA(T9, T25, T26, T28, flatA_in_ga(T26, T28))
FLATA_IN_GA(tree(T9, niltree, tree(T25, niltree, T26)), cons(T9, cons(T25, T28))) → FLATA_IN_GA(T26, T28)
FLATA_IN_GA(tree(T9, niltree, tree(T53, tree(T54, T55, T56), T57)), cons(T9, T59)) → U2_GA(T9, T53, T54, T55, T56, T57, T59, flatA_in_ga(tree(T54, T55, tree(T53, T56, T57)), T59))
FLATA_IN_GA(tree(T9, niltree, tree(T53, tree(T54, T55, T56), T57)), cons(T9, T59)) → FLATA_IN_GA(tree(T54, T55, tree(T53, T56, T57)), T59)
FLATA_IN_GA(tree(T95, tree(T94, niltree, T96), T97), cons(T94, T99)) → U3_GA(T95, T94, T96, T97, T99, flatA_in_ga(tree(T95, T96, T97), T99))
FLATA_IN_GA(tree(T95, tree(T94, niltree, T96), T97), cons(T94, T99)) → FLATA_IN_GA(tree(T95, T96, T97), T99)
FLATA_IN_GA(tree(T120, tree(T116, tree(T117, T118, T119), T121), T122), T124) → U4_GA(T120, T116, T117, T118, T119, T121, T122, T124, flatA_in_ga(tree(T117, T118, tree(T116, T119, tree(T120, T121, T122))), T124))
FLATA_IN_GA(tree(T120, tree(T116, tree(T117, T118, T119), T121), T122), T124) → FLATA_IN_GA(tree(T117, T118, tree(T116, T119, tree(T120, T121, T122))), T124)

The TRS R consists of the following rules:

flatA_in_ga(niltree, nil) → flatA_out_ga(niltree, nil)
flatA_in_ga(tree(T9, niltree, niltree), cons(T9, nil)) → flatA_out_ga(tree(T9, niltree, niltree), cons(T9, nil))
flatA_in_ga(tree(T9, niltree, tree(T25, niltree, T26)), cons(T9, cons(T25, T28))) → U1_ga(T9, T25, T26, T28, flatA_in_ga(T26, T28))
flatA_in_ga(tree(T9, niltree, tree(T53, tree(T54, T55, T56), T57)), cons(T9, T59)) → U2_ga(T9, T53, T54, T55, T56, T57, T59, flatA_in_ga(tree(T54, T55, tree(T53, T56, T57)), T59))
flatA_in_ga(tree(T95, tree(T94, niltree, T96), T97), cons(T94, T99)) → U3_ga(T95, T94, T96, T97, T99, flatA_in_ga(tree(T95, T96, T97), T99))
flatA_in_ga(tree(T120, tree(T116, tree(T117, T118, T119), T121), T122), T124) → U4_ga(T120, T116, T117, T118, T119, T121, T122, T124, flatA_in_ga(tree(T117, T118, tree(T116, T119, tree(T120, T121, T122))), T124))
U4_ga(T120, T116, T117, T118, T119, T121, T122, T124, flatA_out_ga(tree(T117, T118, tree(T116, T119, tree(T120, T121, T122))), T124)) → flatA_out_ga(tree(T120, tree(T116, tree(T117, T118, T119), T121), T122), T124)
U3_ga(T95, T94, T96, T97, T99, flatA_out_ga(tree(T95, T96, T97), T99)) → flatA_out_ga(tree(T95, tree(T94, niltree, T96), T97), cons(T94, T99))
U2_ga(T9, T53, T54, T55, T56, T57, T59, flatA_out_ga(tree(T54, T55, tree(T53, T56, T57)), T59)) → flatA_out_ga(tree(T9, niltree, tree(T53, tree(T54, T55, T56), T57)), cons(T9, T59))
U1_ga(T9, T25, T26, T28, flatA_out_ga(T26, T28)) → flatA_out_ga(tree(T9, niltree, tree(T25, niltree, T26)), cons(T9, cons(T25, T28)))

The argument filtering Pi contains the following mapping:
flatA_in_ga(x1, x2)  =  flatA_in_ga(x1)
niltree  =  niltree
flatA_out_ga(x1, x2)  =  flatA_out_ga(x1, x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
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)
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

(4) Obligation:

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

FLATA_IN_GA(tree(T9, niltree, tree(T25, niltree, T26)), cons(T9, cons(T25, T28))) → U1_GA(T9, T25, T26, T28, flatA_in_ga(T26, T28))
FLATA_IN_GA(tree(T9, niltree, tree(T25, niltree, T26)), cons(T9, cons(T25, T28))) → FLATA_IN_GA(T26, T28)
FLATA_IN_GA(tree(T9, niltree, tree(T53, tree(T54, T55, T56), T57)), cons(T9, T59)) → U2_GA(T9, T53, T54, T55, T56, T57, T59, flatA_in_ga(tree(T54, T55, tree(T53, T56, T57)), T59))
FLATA_IN_GA(tree(T9, niltree, tree(T53, tree(T54, T55, T56), T57)), cons(T9, T59)) → FLATA_IN_GA(tree(T54, T55, tree(T53, T56, T57)), T59)
FLATA_IN_GA(tree(T95, tree(T94, niltree, T96), T97), cons(T94, T99)) → U3_GA(T95, T94, T96, T97, T99, flatA_in_ga(tree(T95, T96, T97), T99))
FLATA_IN_GA(tree(T95, tree(T94, niltree, T96), T97), cons(T94, T99)) → FLATA_IN_GA(tree(T95, T96, T97), T99)
FLATA_IN_GA(tree(T120, tree(T116, tree(T117, T118, T119), T121), T122), T124) → U4_GA(T120, T116, T117, T118, T119, T121, T122, T124, flatA_in_ga(tree(T117, T118, tree(T116, T119, tree(T120, T121, T122))), T124))
FLATA_IN_GA(tree(T120, tree(T116, tree(T117, T118, T119), T121), T122), T124) → FLATA_IN_GA(tree(T117, T118, tree(T116, T119, tree(T120, T121, T122))), T124)

The TRS R consists of the following rules:

flatA_in_ga(niltree, nil) → flatA_out_ga(niltree, nil)
flatA_in_ga(tree(T9, niltree, niltree), cons(T9, nil)) → flatA_out_ga(tree(T9, niltree, niltree), cons(T9, nil))
flatA_in_ga(tree(T9, niltree, tree(T25, niltree, T26)), cons(T9, cons(T25, T28))) → U1_ga(T9, T25, T26, T28, flatA_in_ga(T26, T28))
flatA_in_ga(tree(T9, niltree, tree(T53, tree(T54, T55, T56), T57)), cons(T9, T59)) → U2_ga(T9, T53, T54, T55, T56, T57, T59, flatA_in_ga(tree(T54, T55, tree(T53, T56, T57)), T59))
flatA_in_ga(tree(T95, tree(T94, niltree, T96), T97), cons(T94, T99)) → U3_ga(T95, T94, T96, T97, T99, flatA_in_ga(tree(T95, T96, T97), T99))
flatA_in_ga(tree(T120, tree(T116, tree(T117, T118, T119), T121), T122), T124) → U4_ga(T120, T116, T117, T118, T119, T121, T122, T124, flatA_in_ga(tree(T117, T118, tree(T116, T119, tree(T120, T121, T122))), T124))
U4_ga(T120, T116, T117, T118, T119, T121, T122, T124, flatA_out_ga(tree(T117, T118, tree(T116, T119, tree(T120, T121, T122))), T124)) → flatA_out_ga(tree(T120, tree(T116, tree(T117, T118, T119), T121), T122), T124)
U3_ga(T95, T94, T96, T97, T99, flatA_out_ga(tree(T95, T96, T97), T99)) → flatA_out_ga(tree(T95, tree(T94, niltree, T96), T97), cons(T94, T99))
U2_ga(T9, T53, T54, T55, T56, T57, T59, flatA_out_ga(tree(T54, T55, tree(T53, T56, T57)), T59)) → flatA_out_ga(tree(T9, niltree, tree(T53, tree(T54, T55, T56), T57)), cons(T9, T59))
U1_ga(T9, T25, T26, T28, flatA_out_ga(T26, T28)) → flatA_out_ga(tree(T9, niltree, tree(T25, niltree, T26)), cons(T9, cons(T25, T28)))

The argument filtering Pi contains the following mapping:
flatA_in_ga(x1, x2)  =  flatA_in_ga(x1)
niltree  =  niltree
flatA_out_ga(x1, x2)  =  flatA_out_ga(x1, x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
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)
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(T9, niltree, tree(T53, tree(T54, T55, T56), T57)), cons(T9, T59)) → FLATA_IN_GA(tree(T54, T55, tree(T53, T56, T57)), T59)
FLATA_IN_GA(tree(T9, niltree, tree(T25, niltree, T26)), cons(T9, cons(T25, T28))) → FLATA_IN_GA(T26, T28)
FLATA_IN_GA(tree(T95, tree(T94, niltree, T96), T97), cons(T94, T99)) → FLATA_IN_GA(tree(T95, T96, T97), T99)
FLATA_IN_GA(tree(T120, tree(T116, tree(T117, T118, T119), T121), T122), T124) → FLATA_IN_GA(tree(T117, T118, tree(T116, T119, tree(T120, T121, T122))), T124)

The TRS R consists of the following rules:

flatA_in_ga(niltree, nil) → flatA_out_ga(niltree, nil)
flatA_in_ga(tree(T9, niltree, niltree), cons(T9, nil)) → flatA_out_ga(tree(T9, niltree, niltree), cons(T9, nil))
flatA_in_ga(tree(T9, niltree, tree(T25, niltree, T26)), cons(T9, cons(T25, T28))) → U1_ga(T9, T25, T26, T28, flatA_in_ga(T26, T28))
flatA_in_ga(tree(T9, niltree, tree(T53, tree(T54, T55, T56), T57)), cons(T9, T59)) → U2_ga(T9, T53, T54, T55, T56, T57, T59, flatA_in_ga(tree(T54, T55, tree(T53, T56, T57)), T59))
flatA_in_ga(tree(T95, tree(T94, niltree, T96), T97), cons(T94, T99)) → U3_ga(T95, T94, T96, T97, T99, flatA_in_ga(tree(T95, T96, T97), T99))
flatA_in_ga(tree(T120, tree(T116, tree(T117, T118, T119), T121), T122), T124) → U4_ga(T120, T116, T117, T118, T119, T121, T122, T124, flatA_in_ga(tree(T117, T118, tree(T116, T119, tree(T120, T121, T122))), T124))
U4_ga(T120, T116, T117, T118, T119, T121, T122, T124, flatA_out_ga(tree(T117, T118, tree(T116, T119, tree(T120, T121, T122))), T124)) → flatA_out_ga(tree(T120, tree(T116, tree(T117, T118, T119), T121), T122), T124)
U3_ga(T95, T94, T96, T97, T99, flatA_out_ga(tree(T95, T96, T97), T99)) → flatA_out_ga(tree(T95, tree(T94, niltree, T96), T97), cons(T94, T99))
U2_ga(T9, T53, T54, T55, T56, T57, T59, flatA_out_ga(tree(T54, T55, tree(T53, T56, T57)), T59)) → flatA_out_ga(tree(T9, niltree, tree(T53, tree(T54, T55, T56), T57)), cons(T9, T59))
U1_ga(T9, T25, T26, T28, flatA_out_ga(T26, T28)) → flatA_out_ga(tree(T9, niltree, tree(T25, niltree, T26)), cons(T9, cons(T25, T28)))

The argument filtering Pi contains the following mapping:
flatA_in_ga(x1, x2)  =  flatA_in_ga(x1)
niltree  =  niltree
flatA_out_ga(x1, x2)  =  flatA_out_ga(x1, x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
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)
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) UsableRulesProof (EQUIVALENT transformation)

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

(8) Obligation:

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

FLATA_IN_GA(tree(T9, niltree, tree(T53, tree(T54, T55, T56), T57)), cons(T9, T59)) → FLATA_IN_GA(tree(T54, T55, tree(T53, T56, T57)), T59)
FLATA_IN_GA(tree(T9, niltree, tree(T25, niltree, T26)), cons(T9, cons(T25, T28))) → FLATA_IN_GA(T26, T28)
FLATA_IN_GA(tree(T95, tree(T94, niltree, T96), T97), cons(T94, T99)) → FLATA_IN_GA(tree(T95, T96, T97), T99)
FLATA_IN_GA(tree(T120, tree(T116, tree(T117, T118, T119), T121), T122), T124) → FLATA_IN_GA(tree(T117, T118, tree(T116, T119, tree(T120, T121, T122))), T124)

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

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:

FLATA_IN_GA(tree(T9, niltree, tree(T53, tree(T54, T55, T56), T57))) → FLATA_IN_GA(tree(T54, T55, tree(T53, T56, T57)))
FLATA_IN_GA(tree(T9, niltree, tree(T25, niltree, T26))) → FLATA_IN_GA(T26)
FLATA_IN_GA(tree(T95, tree(T94, niltree, T96), T97)) → FLATA_IN_GA(tree(T95, T96, T97))
FLATA_IN_GA(tree(T120, tree(T116, tree(T117, T118, T119), T121), T122)) → FLATA_IN_GA(tree(T117, T118, tree(T116, T119, tree(T120, T121, T122))))

R is empty.
Q is empty.
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.

The following dependency pairs can be deleted:

FLATA_IN_GA(tree(T9, niltree, tree(T53, tree(T54, T55, T56), T57))) → FLATA_IN_GA(tree(T54, T55, tree(T53, T56, T57)))
FLATA_IN_GA(tree(T9, niltree, tree(T25, niltree, T26))) → FLATA_IN_GA(T26)
FLATA_IN_GA(tree(T95, tree(T94, niltree, T96), T97)) → FLATA_IN_GA(tree(T95, T96, T97))
FLATA_IN_GA(tree(T120, tree(T116, tree(T117, T118, T119), T121), T122)) → FLATA_IN_GA(tree(T117, T118, tree(T116, T119, tree(T120, T121, T122))))

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   

(12) Obligation:

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

(13) PisEmptyProof (EQUIVALENT transformation)

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

(14) YES