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

(0) Obligation:

Clauses:

hbal_tree(zero, nil).
hbal_tree(s(zero), t(x, nil, nil)).
hbal_tree(s(s(X)), t(x, L, R)) :- ','(distr(s(X), X, DL, DR), ','(hbal_tree(DL, L), hbal_tree(DR, R))).
distr(D1, X1, D1, D1).
distr(D1, D2, D1, D2).
distr(D1, D2, D2, D1).

Query: hbal_tree(g,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
3 [label="A: hbal_tree(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
6 [label="6: hbal_tree(T1, T2), SCOPE: 1, CLAUSE: 0 | hbal_tree(T1, T2), SCOPE: 1, CLAUSE: 1 | hbal_tree(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: true | hbal_tree(zero, T2), SCOPE: 1, CLAUSE: 1 | hbal_tree(zero, T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: hbal_tree(T1, T2), SCOPE: 1, CLAUSE: 1 | hbal_tree(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nhbal_tree(T1, T2) !=? hbal_tree(zero, nil)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: hbal_tree(zero, T2), SCOPE: 1, CLAUSE: 1 | hbal_tree(zero, T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: hbal_tree(zero, T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: true | hbal_tree(s(zero), T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: hbal_tree(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nhbal_tree(T1, T2) !=? hbal_tree(zero, nil)\nhbal_tree(T1, T2) !=? hbal_tree(s(zero), t(x, nil, nil))", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: hbal_tree(s(zero), T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(distr(s(T6), T6, X14, X15), ','(hbal_tree(X14, T9), hbal_tree(X15, T10)))\n\nT6 is ground\nX14 is free\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(distr(s(T6), T6, X14, X15), ','(hbal_tree(X14, T9), hbal_tree(X15, T10))), SCOPE: 2, CLAUSE: 3 | ','(distr(s(T6), T6, X14, X15), ','(hbal_tree(X14, T9), hbal_tree(X15, T10))), SCOPE: 2, CLAUSE: 4 | ','(distr(s(T6), T6, X14, X15), ','(hbal_tree(X14, T9), hbal_tree(X15, T10))), SCOPE: 2, CLAUSE: 5\n\nT6 is ground\nX14 is free\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: ','(distr(s(T6), T6, X14, X15), ','(hbal_tree(X14, T9), hbal_tree(X15, T10))), SCOPE: 2, CLAUSE: 3\n\nT6 is ground\nX14 is free\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(distr(s(T6), T6, X14, X15), ','(hbal_tree(X14, T9), hbal_tree(X15, T10))), SCOPE: 2, CLAUSE: 4 | ','(distr(s(T6), T6, X14, X15), ','(hbal_tree(X14, T9), hbal_tree(X15, T10))), SCOPE: 2, CLAUSE: 5\n\nT6 is ground\nX14 is free\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="B: ','(hbal_tree(s(T15), T9), hbal_tree(s(T15), T10))\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: hbal_tree(s(T15), T9)\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: hbal_tree(s(T15), T16)\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(distr(s(T6), T6, X14, X15), ','(hbal_tree(X14, T9), hbal_tree(X15, T10))), SCOPE: 2, CLAUSE: 4\n\nT6 is ground\nX14 is free\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(distr(s(T6), T6, X14, X15), ','(hbal_tree(X14, T9), hbal_tree(X15, T10))), SCOPE: 2, CLAUSE: 5\n\nT6 is ground\nX14 is free\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="C: ','(hbal_tree(s(T21), T9), hbal_tree(T21, T10))\n\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: hbal_tree(s(T21), T9)\n\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: hbal_tree(T21, T22)\n\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="D: ','(hbal_tree(T25, T9), hbal_tree(s(T25), T10))\n\nT25 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: hbal_tree(T25, T9)\n\nT25 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: hbal_tree(s(T25), T26)\n\nT25 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 81 [arrowhead = none ];
81 -> 6;

82 [label="EVAL with clause\nhbal_tree(zero, nil).\nand substitutionT1 -> zero,\nT2 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 82 [arrowhead = none ];
82 -> 11;

83 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 83 [arrowhead = none ];
83 -> 14;

84 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 84 [arrowhead = none ];
84 -> 17;

85 [label="EVAL with clause\nhbal_tree(s(zero), t(x, nil, nil)).\nand substitutionT1 -> s(zero),\nT2 -> t(x, nil, nil)", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 85 [arrowhead = none ];
85 -> 28;

86 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 86 [arrowhead = none ];
86 -> 30;

87 [label="BACKTRACK\nfor clause: hbal_tree(s(zero), t(x, nil, nil))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 87 [arrowhead = none ];
87 -> 19;

88 [label="BACKTRACK\nfor clause: hbal_tree(s(s(X)), t(x, L, R)) :- ','(distr(s(X), X, DL, DR), ','(hbal_tree(DL, L), hbal_tree(DR, R)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 88 [arrowhead = none ];
88 -> 23;

89 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 89 [arrowhead = none ];
89 -> 32;

90 [label="EVAL with clause\nhbal_tree(s(s(X11)), t(x, X12, X13)) :- ','(distr(s(X11), X11, X14, X15), ','(hbal_tree(X14, X12), hbal_tree(X15, X13))).\nand substitutionX11 -> T6,\nT1 -> s(s(T6)),\nX12 -> T9,\nX13 -> T10,\nT2 -> t(x, T9, T10),\nT7 -> T9,\nT8 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 90 [arrowhead = none ];
90 -> 38;

91 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 91 [arrowhead = none ];
91 -> 41;

92 [label="BACKTRACK\nfor clause: hbal_tree(s(s(X)), t(x, L, R)) :- ','(distr(s(X), X, DL, DR), ','(hbal_tree(DL, L), hbal_tree(DR, R)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 92 [arrowhead = none ];
92 -> 34;

93 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
38 -> 93 [arrowhead = none ];
93 -> 43;

94 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
43 -> 94 [arrowhead = none ];
94 -> 48;

95 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
43 -> 95 [arrowhead = none ];
95 -> 49;

96 [label="ONLY EVAL with clause\ndistr(X24, X25, X24, X24).\nand substitutionT6 -> T15,\nX24 -> s(T15),\nX25 -> T15,\nX14 -> s(T15),\nX15 -> s(T15)", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 96 [arrowhead = none ];
96 -> 52;

97 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
49 -> 97 [arrowhead = none ];
97 -> 61;

98 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
49 -> 98 [arrowhead = none ];
98 -> 62;

99 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
52 -> 99 [arrowhead = none ];
99 -> 53;

100 [label="SPLIT 2\nnew knowledge:\nT15 is ground\nT9 is ground\nreplacements:T10 -> T16", fontsize=14, style = filled, fillcolor = violet];
52 -> 100 [arrowhead = none ];
100 -> 54;

101 [label="INSTANCE with matching:\nT1 -> s(T15)\nT2 -> T9", fontsize=14, style = filled, fillcolor = lightgrey];
53 -> 101 [arrowhead = none , style=dashed];
101 -> 3[style=dashed];

102 [label="INSTANCE with matching:\nT1 -> s(T15)\nT2 -> T16", fontsize=14, style = filled, fillcolor = lightgrey];
54 -> 102 [arrowhead = none , style=dashed];
102 -> 3[style=dashed];

103 [label="ONLY EVAL with clause\ndistr(X34, X35, X34, X35).\nand substitutionT6 -> T21,\nX34 -> s(T21),\nX35 -> T21,\nX14 -> s(T21),\nX15 -> T21", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 103 [arrowhead = none ];
103 -> 66;

104 [label="ONLY EVAL with clause\ndistr(X40, X41, X41, X40).\nand substitutionT6 -> T25,\nX40 -> s(T25),\nX41 -> T25,\nX14 -> T25,\nX15 -> s(T25)", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 104 [arrowhead = none ];
104 -> 76;

105 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
66 -> 105 [arrowhead = none ];
105 -> 70;

106 [label="SPLIT 2\nnew knowledge:\nT21 is ground\nT9 is ground\nreplacements:T10 -> T22", fontsize=14, style = filled, fillcolor = violet];
66 -> 106 [arrowhead = none ];
106 -> 71;

107 [label="INSTANCE with matching:\nT1 -> s(T21)\nT2 -> T9", fontsize=14, style = filled, fillcolor = lightgrey];
70 -> 107 [arrowhead = none , style=dashed];
107 -> 3[style=dashed];

108 [label="INSTANCE with matching:\nT1 -> T21\nT2 -> T22", fontsize=14, style = filled, fillcolor = lightgrey];
71 -> 108 [arrowhead = none , style=dashed];
108 -> 3[style=dashed];

109 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
76 -> 109 [arrowhead = none ];
109 -> 79;

110 [label="SPLIT 2\nnew knowledge:\nT25 is ground\nT9 is ground\nreplacements:T10 -> T26", fontsize=14, style = filled, fillcolor = violet];
76 -> 110 [arrowhead = none ];
110 -> 80;

111 [label="INSTANCE with matching:\nT1 -> T25\nT2 -> T9", fontsize=14, style = filled, fillcolor = lightgrey];
79 -> 111 [arrowhead = none , style=dashed];
111 -> 3[style=dashed];

112 [label="INSTANCE with matching:\nT1 -> s(T25)\nT2 -> T26", fontsize=14, style = filled, fillcolor = lightgrey];
80 -> 112 [arrowhead = none , style=dashed];
112 -> 3[style=dashed];

}

(2) Obligation:

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

hbal_treeA_in_ga(zero, nil) → hbal_treeA_out_ga(zero, nil)
hbal_treeA_in_ga(s(zero), t(x, nil, nil)) → hbal_treeA_out_ga(s(zero), t(x, nil, nil))
hbal_treeA_in_ga(s(s(T15)), t(x, T9, T10)) → U1_ga(T15, T9, T10, pB_in_gaa(T15, T9, T10))
pB_in_gaa(T15, T9, T16) → U4_gaa(T15, T9, T16, hbal_treeA_in_ga(s(T15), T9))
hbal_treeA_in_ga(s(s(T21)), t(x, T9, T10)) → U2_ga(T21, T9, T10, pC_in_gaa(T21, T9, T10))
pC_in_gaa(T21, T9, T22) → U6_gaa(T21, T9, T22, hbal_treeA_in_ga(s(T21), T9))
hbal_treeA_in_ga(s(s(T25)), t(x, T9, T10)) → U3_ga(T25, T9, T10, pD_in_gaa(T25, T9, T10))
pD_in_gaa(T25, T9, T26) → U8_gaa(T25, T9, T26, hbal_treeA_in_ga(T25, T9))
U8_gaa(T25, T9, T26, hbal_treeA_out_ga(T25, T9)) → U9_gaa(T25, T9, T26, hbal_treeA_in_ga(s(T25), T26))
U9_gaa(T25, T9, T26, hbal_treeA_out_ga(s(T25), T26)) → pD_out_gaa(T25, T9, T26)
U3_ga(T25, T9, T10, pD_out_gaa(T25, T9, T10)) → hbal_treeA_out_ga(s(s(T25)), t(x, T9, T10))
U6_gaa(T21, T9, T22, hbal_treeA_out_ga(s(T21), T9)) → U7_gaa(T21, T9, T22, hbal_treeA_in_ga(T21, T22))
U7_gaa(T21, T9, T22, hbal_treeA_out_ga(T21, T22)) → pC_out_gaa(T21, T9, T22)
U2_ga(T21, T9, T10, pC_out_gaa(T21, T9, T10)) → hbal_treeA_out_ga(s(s(T21)), t(x, T9, T10))
U4_gaa(T15, T9, T16, hbal_treeA_out_ga(s(T15), T9)) → U5_gaa(T15, T9, T16, hbal_treeA_in_ga(s(T15), T16))
U5_gaa(T15, T9, T16, hbal_treeA_out_ga(s(T15), T16)) → pB_out_gaa(T15, T9, T16)
U1_ga(T15, T9, T10, pB_out_gaa(T15, T9, T10)) → hbal_treeA_out_ga(s(s(T15)), t(x, T9, T10))

The argument filtering Pi contains the following mapping:
hbal_treeA_in_ga(x1, x2)  =  hbal_treeA_in_ga(x1)
zero  =  zero
hbal_treeA_out_ga(x1, x2)  =  hbal_treeA_out_ga(x1, x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
pB_in_gaa(x1, x2, x3)  =  pB_in_gaa(x1)
U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x1, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
pC_in_gaa(x1, x2, x3)  =  pC_in_gaa(x1)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x1, x4)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
pD_in_gaa(x1, x2, x3)  =  pD_in_gaa(x1)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x4)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x2, x4)
pD_out_gaa(x1, x2, x3)  =  pD_out_gaa(x1, x2, x3)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x1, x2, x4)
pC_out_gaa(x1, x2, x3)  =  pC_out_gaa(x1, x2, x3)
U5_gaa(x1, x2, x3, x4)  =  U5_gaa(x1, x2, x4)
pB_out_gaa(x1, x2, x3)  =  pB_out_gaa(x1, x2, x3)
t(x1, x2, x3)  =  t(x1, x2, x3)
x  =  x

(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:

HBAL_TREEA_IN_GA(s(s(T15)), t(x, T9, T10)) → U1_GA(T15, T9, T10, pB_in_gaa(T15, T9, T10))
HBAL_TREEA_IN_GA(s(s(T15)), t(x, T9, T10)) → PB_IN_GAA(T15, T9, T10)
PB_IN_GAA(T15, T9, T16) → U4_GAA(T15, T9, T16, hbal_treeA_in_ga(s(T15), T9))
PB_IN_GAA(T15, T9, T16) → HBAL_TREEA_IN_GA(s(T15), T9)
HBAL_TREEA_IN_GA(s(s(T21)), t(x, T9, T10)) → U2_GA(T21, T9, T10, pC_in_gaa(T21, T9, T10))
HBAL_TREEA_IN_GA(s(s(T21)), t(x, T9, T10)) → PC_IN_GAA(T21, T9, T10)
PC_IN_GAA(T21, T9, T22) → U6_GAA(T21, T9, T22, hbal_treeA_in_ga(s(T21), T9))
PC_IN_GAA(T21, T9, T22) → HBAL_TREEA_IN_GA(s(T21), T9)
HBAL_TREEA_IN_GA(s(s(T25)), t(x, T9, T10)) → U3_GA(T25, T9, T10, pD_in_gaa(T25, T9, T10))
HBAL_TREEA_IN_GA(s(s(T25)), t(x, T9, T10)) → PD_IN_GAA(T25, T9, T10)
PD_IN_GAA(T25, T9, T26) → U8_GAA(T25, T9, T26, hbal_treeA_in_ga(T25, T9))
PD_IN_GAA(T25, T9, T26) → HBAL_TREEA_IN_GA(T25, T9)
U8_GAA(T25, T9, T26, hbal_treeA_out_ga(T25, T9)) → U9_GAA(T25, T9, T26, hbal_treeA_in_ga(s(T25), T26))
U8_GAA(T25, T9, T26, hbal_treeA_out_ga(T25, T9)) → HBAL_TREEA_IN_GA(s(T25), T26)
U6_GAA(T21, T9, T22, hbal_treeA_out_ga(s(T21), T9)) → U7_GAA(T21, T9, T22, hbal_treeA_in_ga(T21, T22))
U6_GAA(T21, T9, T22, hbal_treeA_out_ga(s(T21), T9)) → HBAL_TREEA_IN_GA(T21, T22)
U4_GAA(T15, T9, T16, hbal_treeA_out_ga(s(T15), T9)) → U5_GAA(T15, T9, T16, hbal_treeA_in_ga(s(T15), T16))
U4_GAA(T15, T9, T16, hbal_treeA_out_ga(s(T15), T9)) → HBAL_TREEA_IN_GA(s(T15), T16)

The TRS R consists of the following rules:

hbal_treeA_in_ga(zero, nil) → hbal_treeA_out_ga(zero, nil)
hbal_treeA_in_ga(s(zero), t(x, nil, nil)) → hbal_treeA_out_ga(s(zero), t(x, nil, nil))
hbal_treeA_in_ga(s(s(T15)), t(x, T9, T10)) → U1_ga(T15, T9, T10, pB_in_gaa(T15, T9, T10))
pB_in_gaa(T15, T9, T16) → U4_gaa(T15, T9, T16, hbal_treeA_in_ga(s(T15), T9))
hbal_treeA_in_ga(s(s(T21)), t(x, T9, T10)) → U2_ga(T21, T9, T10, pC_in_gaa(T21, T9, T10))
pC_in_gaa(T21, T9, T22) → U6_gaa(T21, T9, T22, hbal_treeA_in_ga(s(T21), T9))
hbal_treeA_in_ga(s(s(T25)), t(x, T9, T10)) → U3_ga(T25, T9, T10, pD_in_gaa(T25, T9, T10))
pD_in_gaa(T25, T9, T26) → U8_gaa(T25, T9, T26, hbal_treeA_in_ga(T25, T9))
U8_gaa(T25, T9, T26, hbal_treeA_out_ga(T25, T9)) → U9_gaa(T25, T9, T26, hbal_treeA_in_ga(s(T25), T26))
U9_gaa(T25, T9, T26, hbal_treeA_out_ga(s(T25), T26)) → pD_out_gaa(T25, T9, T26)
U3_ga(T25, T9, T10, pD_out_gaa(T25, T9, T10)) → hbal_treeA_out_ga(s(s(T25)), t(x, T9, T10))
U6_gaa(T21, T9, T22, hbal_treeA_out_ga(s(T21), T9)) → U7_gaa(T21, T9, T22, hbal_treeA_in_ga(T21, T22))
U7_gaa(T21, T9, T22, hbal_treeA_out_ga(T21, T22)) → pC_out_gaa(T21, T9, T22)
U2_ga(T21, T9, T10, pC_out_gaa(T21, T9, T10)) → hbal_treeA_out_ga(s(s(T21)), t(x, T9, T10))
U4_gaa(T15, T9, T16, hbal_treeA_out_ga(s(T15), T9)) → U5_gaa(T15, T9, T16, hbal_treeA_in_ga(s(T15), T16))
U5_gaa(T15, T9, T16, hbal_treeA_out_ga(s(T15), T16)) → pB_out_gaa(T15, T9, T16)
U1_ga(T15, T9, T10, pB_out_gaa(T15, T9, T10)) → hbal_treeA_out_ga(s(s(T15)), t(x, T9, T10))

The argument filtering Pi contains the following mapping:
hbal_treeA_in_ga(x1, x2)  =  hbal_treeA_in_ga(x1)
zero  =  zero
hbal_treeA_out_ga(x1, x2)  =  hbal_treeA_out_ga(x1, x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
pB_in_gaa(x1, x2, x3)  =  pB_in_gaa(x1)
U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x1, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
pC_in_gaa(x1, x2, x3)  =  pC_in_gaa(x1)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x1, x4)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
pD_in_gaa(x1, x2, x3)  =  pD_in_gaa(x1)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x4)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x2, x4)
pD_out_gaa(x1, x2, x3)  =  pD_out_gaa(x1, x2, x3)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x1, x2, x4)
pC_out_gaa(x1, x2, x3)  =  pC_out_gaa(x1, x2, x3)
U5_gaa(x1, x2, x3, x4)  =  U5_gaa(x1, x2, x4)
pB_out_gaa(x1, x2, x3)  =  pB_out_gaa(x1, x2, x3)
t(x1, x2, x3)  =  t(x1, x2, x3)
x  =  x
HBAL_TREEA_IN_GA(x1, x2)  =  HBAL_TREEA_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
PB_IN_GAA(x1, x2, x3)  =  PB_IN_GAA(x1)
U4_GAA(x1, x2, x3, x4)  =  U4_GAA(x1, x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x4)
PC_IN_GAA(x1, x2, x3)  =  PC_IN_GAA(x1)
U6_GAA(x1, x2, x3, x4)  =  U6_GAA(x1, x4)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)
PD_IN_GAA(x1, x2, x3)  =  PD_IN_GAA(x1)
U8_GAA(x1, x2, x3, x4)  =  U8_GAA(x1, x4)
U9_GAA(x1, x2, x3, x4)  =  U9_GAA(x1, x2, x4)
U7_GAA(x1, x2, x3, x4)  =  U7_GAA(x1, x2, x4)
U5_GAA(x1, x2, x3, x4)  =  U5_GAA(x1, x2, x4)

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

(4) Obligation:

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

HBAL_TREEA_IN_GA(s(s(T15)), t(x, T9, T10)) → U1_GA(T15, T9, T10, pB_in_gaa(T15, T9, T10))
HBAL_TREEA_IN_GA(s(s(T15)), t(x, T9, T10)) → PB_IN_GAA(T15, T9, T10)
PB_IN_GAA(T15, T9, T16) → U4_GAA(T15, T9, T16, hbal_treeA_in_ga(s(T15), T9))
PB_IN_GAA(T15, T9, T16) → HBAL_TREEA_IN_GA(s(T15), T9)
HBAL_TREEA_IN_GA(s(s(T21)), t(x, T9, T10)) → U2_GA(T21, T9, T10, pC_in_gaa(T21, T9, T10))
HBAL_TREEA_IN_GA(s(s(T21)), t(x, T9, T10)) → PC_IN_GAA(T21, T9, T10)
PC_IN_GAA(T21, T9, T22) → U6_GAA(T21, T9, T22, hbal_treeA_in_ga(s(T21), T9))
PC_IN_GAA(T21, T9, T22) → HBAL_TREEA_IN_GA(s(T21), T9)
HBAL_TREEA_IN_GA(s(s(T25)), t(x, T9, T10)) → U3_GA(T25, T9, T10, pD_in_gaa(T25, T9, T10))
HBAL_TREEA_IN_GA(s(s(T25)), t(x, T9, T10)) → PD_IN_GAA(T25, T9, T10)
PD_IN_GAA(T25, T9, T26) → U8_GAA(T25, T9, T26, hbal_treeA_in_ga(T25, T9))
PD_IN_GAA(T25, T9, T26) → HBAL_TREEA_IN_GA(T25, T9)
U8_GAA(T25, T9, T26, hbal_treeA_out_ga(T25, T9)) → U9_GAA(T25, T9, T26, hbal_treeA_in_ga(s(T25), T26))
U8_GAA(T25, T9, T26, hbal_treeA_out_ga(T25, T9)) → HBAL_TREEA_IN_GA(s(T25), T26)
U6_GAA(T21, T9, T22, hbal_treeA_out_ga(s(T21), T9)) → U7_GAA(T21, T9, T22, hbal_treeA_in_ga(T21, T22))
U6_GAA(T21, T9, T22, hbal_treeA_out_ga(s(T21), T9)) → HBAL_TREEA_IN_GA(T21, T22)
U4_GAA(T15, T9, T16, hbal_treeA_out_ga(s(T15), T9)) → U5_GAA(T15, T9, T16, hbal_treeA_in_ga(s(T15), T16))
U4_GAA(T15, T9, T16, hbal_treeA_out_ga(s(T15), T9)) → HBAL_TREEA_IN_GA(s(T15), T16)

The TRS R consists of the following rules:

hbal_treeA_in_ga(zero, nil) → hbal_treeA_out_ga(zero, nil)
hbal_treeA_in_ga(s(zero), t(x, nil, nil)) → hbal_treeA_out_ga(s(zero), t(x, nil, nil))
hbal_treeA_in_ga(s(s(T15)), t(x, T9, T10)) → U1_ga(T15, T9, T10, pB_in_gaa(T15, T9, T10))
pB_in_gaa(T15, T9, T16) → U4_gaa(T15, T9, T16, hbal_treeA_in_ga(s(T15), T9))
hbal_treeA_in_ga(s(s(T21)), t(x, T9, T10)) → U2_ga(T21, T9, T10, pC_in_gaa(T21, T9, T10))
pC_in_gaa(T21, T9, T22) → U6_gaa(T21, T9, T22, hbal_treeA_in_ga(s(T21), T9))
hbal_treeA_in_ga(s(s(T25)), t(x, T9, T10)) → U3_ga(T25, T9, T10, pD_in_gaa(T25, T9, T10))
pD_in_gaa(T25, T9, T26) → U8_gaa(T25, T9, T26, hbal_treeA_in_ga(T25, T9))
U8_gaa(T25, T9, T26, hbal_treeA_out_ga(T25, T9)) → U9_gaa(T25, T9, T26, hbal_treeA_in_ga(s(T25), T26))
U9_gaa(T25, T9, T26, hbal_treeA_out_ga(s(T25), T26)) → pD_out_gaa(T25, T9, T26)
U3_ga(T25, T9, T10, pD_out_gaa(T25, T9, T10)) → hbal_treeA_out_ga(s(s(T25)), t(x, T9, T10))
U6_gaa(T21, T9, T22, hbal_treeA_out_ga(s(T21), T9)) → U7_gaa(T21, T9, T22, hbal_treeA_in_ga(T21, T22))
U7_gaa(T21, T9, T22, hbal_treeA_out_ga(T21, T22)) → pC_out_gaa(T21, T9, T22)
U2_ga(T21, T9, T10, pC_out_gaa(T21, T9, T10)) → hbal_treeA_out_ga(s(s(T21)), t(x, T9, T10))
U4_gaa(T15, T9, T16, hbal_treeA_out_ga(s(T15), T9)) → U5_gaa(T15, T9, T16, hbal_treeA_in_ga(s(T15), T16))
U5_gaa(T15, T9, T16, hbal_treeA_out_ga(s(T15), T16)) → pB_out_gaa(T15, T9, T16)
U1_ga(T15, T9, T10, pB_out_gaa(T15, T9, T10)) → hbal_treeA_out_ga(s(s(T15)), t(x, T9, T10))

The argument filtering Pi contains the following mapping:
hbal_treeA_in_ga(x1, x2)  =  hbal_treeA_in_ga(x1)
zero  =  zero
hbal_treeA_out_ga(x1, x2)  =  hbal_treeA_out_ga(x1, x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
pB_in_gaa(x1, x2, x3)  =  pB_in_gaa(x1)
U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x1, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
pC_in_gaa(x1, x2, x3)  =  pC_in_gaa(x1)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x1, x4)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
pD_in_gaa(x1, x2, x3)  =  pD_in_gaa(x1)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x4)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x2, x4)
pD_out_gaa(x1, x2, x3)  =  pD_out_gaa(x1, x2, x3)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x1, x2, x4)
pC_out_gaa(x1, x2, x3)  =  pC_out_gaa(x1, x2, x3)
U5_gaa(x1, x2, x3, x4)  =  U5_gaa(x1, x2, x4)
pB_out_gaa(x1, x2, x3)  =  pB_out_gaa(x1, x2, x3)
t(x1, x2, x3)  =  t(x1, x2, x3)
x  =  x
HBAL_TREEA_IN_GA(x1, x2)  =  HBAL_TREEA_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
PB_IN_GAA(x1, x2, x3)  =  PB_IN_GAA(x1)
U4_GAA(x1, x2, x3, x4)  =  U4_GAA(x1, x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x4)
PC_IN_GAA(x1, x2, x3)  =  PC_IN_GAA(x1)
U6_GAA(x1, x2, x3, x4)  =  U6_GAA(x1, x4)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)
PD_IN_GAA(x1, x2, x3)  =  PD_IN_GAA(x1)
U8_GAA(x1, x2, x3, x4)  =  U8_GAA(x1, x4)
U9_GAA(x1, x2, x3, x4)  =  U9_GAA(x1, x2, x4)
U7_GAA(x1, x2, x3, x4)  =  U7_GAA(x1, x2, x4)
U5_GAA(x1, x2, x3, x4)  =  U5_GAA(x1, x2, x4)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

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

(6) Obligation:

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

HBAL_TREEA_IN_GA(s(s(T15)), t(x, T9, T10)) → PB_IN_GAA(T15, T9, T10)
PB_IN_GAA(T15, T9, T16) → U4_GAA(T15, T9, T16, hbal_treeA_in_ga(s(T15), T9))
U4_GAA(T15, T9, T16, hbal_treeA_out_ga(s(T15), T9)) → HBAL_TREEA_IN_GA(s(T15), T16)
HBAL_TREEA_IN_GA(s(s(T21)), t(x, T9, T10)) → PC_IN_GAA(T21, T9, T10)
PC_IN_GAA(T21, T9, T22) → U6_GAA(T21, T9, T22, hbal_treeA_in_ga(s(T21), T9))
U6_GAA(T21, T9, T22, hbal_treeA_out_ga(s(T21), T9)) → HBAL_TREEA_IN_GA(T21, T22)
HBAL_TREEA_IN_GA(s(s(T25)), t(x, T9, T10)) → PD_IN_GAA(T25, T9, T10)
PD_IN_GAA(T25, T9, T26) → U8_GAA(T25, T9, T26, hbal_treeA_in_ga(T25, T9))
U8_GAA(T25, T9, T26, hbal_treeA_out_ga(T25, T9)) → HBAL_TREEA_IN_GA(s(T25), T26)
PD_IN_GAA(T25, T9, T26) → HBAL_TREEA_IN_GA(T25, T9)
PC_IN_GAA(T21, T9, T22) → HBAL_TREEA_IN_GA(s(T21), T9)
PB_IN_GAA(T15, T9, T16) → HBAL_TREEA_IN_GA(s(T15), T9)

The TRS R consists of the following rules:

hbal_treeA_in_ga(zero, nil) → hbal_treeA_out_ga(zero, nil)
hbal_treeA_in_ga(s(zero), t(x, nil, nil)) → hbal_treeA_out_ga(s(zero), t(x, nil, nil))
hbal_treeA_in_ga(s(s(T15)), t(x, T9, T10)) → U1_ga(T15, T9, T10, pB_in_gaa(T15, T9, T10))
pB_in_gaa(T15, T9, T16) → U4_gaa(T15, T9, T16, hbal_treeA_in_ga(s(T15), T9))
hbal_treeA_in_ga(s(s(T21)), t(x, T9, T10)) → U2_ga(T21, T9, T10, pC_in_gaa(T21, T9, T10))
pC_in_gaa(T21, T9, T22) → U6_gaa(T21, T9, T22, hbal_treeA_in_ga(s(T21), T9))
hbal_treeA_in_ga(s(s(T25)), t(x, T9, T10)) → U3_ga(T25, T9, T10, pD_in_gaa(T25, T9, T10))
pD_in_gaa(T25, T9, T26) → U8_gaa(T25, T9, T26, hbal_treeA_in_ga(T25, T9))
U8_gaa(T25, T9, T26, hbal_treeA_out_ga(T25, T9)) → U9_gaa(T25, T9, T26, hbal_treeA_in_ga(s(T25), T26))
U9_gaa(T25, T9, T26, hbal_treeA_out_ga(s(T25), T26)) → pD_out_gaa(T25, T9, T26)
U3_ga(T25, T9, T10, pD_out_gaa(T25, T9, T10)) → hbal_treeA_out_ga(s(s(T25)), t(x, T9, T10))
U6_gaa(T21, T9, T22, hbal_treeA_out_ga(s(T21), T9)) → U7_gaa(T21, T9, T22, hbal_treeA_in_ga(T21, T22))
U7_gaa(T21, T9, T22, hbal_treeA_out_ga(T21, T22)) → pC_out_gaa(T21, T9, T22)
U2_ga(T21, T9, T10, pC_out_gaa(T21, T9, T10)) → hbal_treeA_out_ga(s(s(T21)), t(x, T9, T10))
U4_gaa(T15, T9, T16, hbal_treeA_out_ga(s(T15), T9)) → U5_gaa(T15, T9, T16, hbal_treeA_in_ga(s(T15), T16))
U5_gaa(T15, T9, T16, hbal_treeA_out_ga(s(T15), T16)) → pB_out_gaa(T15, T9, T16)
U1_ga(T15, T9, T10, pB_out_gaa(T15, T9, T10)) → hbal_treeA_out_ga(s(s(T15)), t(x, T9, T10))

The argument filtering Pi contains the following mapping:
hbal_treeA_in_ga(x1, x2)  =  hbal_treeA_in_ga(x1)
zero  =  zero
hbal_treeA_out_ga(x1, x2)  =  hbal_treeA_out_ga(x1, x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
pB_in_gaa(x1, x2, x3)  =  pB_in_gaa(x1)
U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x1, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
pC_in_gaa(x1, x2, x3)  =  pC_in_gaa(x1)
U6_gaa(x1, x2, x3, x4)  =  U6_gaa(x1, x4)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
pD_in_gaa(x1, x2, x3)  =  pD_in_gaa(x1)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x4)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x2, x4)
pD_out_gaa(x1, x2, x3)  =  pD_out_gaa(x1, x2, x3)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x1, x2, x4)
pC_out_gaa(x1, x2, x3)  =  pC_out_gaa(x1, x2, x3)
U5_gaa(x1, x2, x3, x4)  =  U5_gaa(x1, x2, x4)
pB_out_gaa(x1, x2, x3)  =  pB_out_gaa(x1, x2, x3)
t(x1, x2, x3)  =  t(x1, x2, x3)
x  =  x
HBAL_TREEA_IN_GA(x1, x2)  =  HBAL_TREEA_IN_GA(x1)
PB_IN_GAA(x1, x2, x3)  =  PB_IN_GAA(x1)
U4_GAA(x1, x2, x3, x4)  =  U4_GAA(x1, x4)
PC_IN_GAA(x1, x2, x3)  =  PC_IN_GAA(x1)
U6_GAA(x1, x2, x3, x4)  =  U6_GAA(x1, x4)
PD_IN_GAA(x1, x2, x3)  =  PD_IN_GAA(x1)
U8_GAA(x1, x2, x3, x4)  =  U8_GAA(x1, x4)

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:

HBAL_TREEA_IN_GA(s(s(T15))) → PB_IN_GAA(T15)
PB_IN_GAA(T15) → U4_GAA(T15, hbal_treeA_in_ga(s(T15)))
U4_GAA(T15, hbal_treeA_out_ga(s(T15), T9)) → HBAL_TREEA_IN_GA(s(T15))
HBAL_TREEA_IN_GA(s(s(T21))) → PC_IN_GAA(T21)
PC_IN_GAA(T21) → U6_GAA(T21, hbal_treeA_in_ga(s(T21)))
U6_GAA(T21, hbal_treeA_out_ga(s(T21), T9)) → HBAL_TREEA_IN_GA(T21)
HBAL_TREEA_IN_GA(s(s(T25))) → PD_IN_GAA(T25)
PD_IN_GAA(T25) → U8_GAA(T25, hbal_treeA_in_ga(T25))
U8_GAA(T25, hbal_treeA_out_ga(T25, T9)) → HBAL_TREEA_IN_GA(s(T25))
PD_IN_GAA(T25) → HBAL_TREEA_IN_GA(T25)
PC_IN_GAA(T21) → HBAL_TREEA_IN_GA(s(T21))
PB_IN_GAA(T15) → HBAL_TREEA_IN_GA(s(T15))

The TRS R consists of the following rules:

hbal_treeA_in_ga(zero) → hbal_treeA_out_ga(zero, nil)
hbal_treeA_in_ga(s(zero)) → hbal_treeA_out_ga(s(zero), t(x, nil, nil))
hbal_treeA_in_ga(s(s(T15))) → U1_ga(T15, pB_in_gaa(T15))
pB_in_gaa(T15) → U4_gaa(T15, hbal_treeA_in_ga(s(T15)))
hbal_treeA_in_ga(s(s(T21))) → U2_ga(T21, pC_in_gaa(T21))
pC_in_gaa(T21) → U6_gaa(T21, hbal_treeA_in_ga(s(T21)))
hbal_treeA_in_ga(s(s(T25))) → U3_ga(T25, pD_in_gaa(T25))
pD_in_gaa(T25) → U8_gaa(T25, hbal_treeA_in_ga(T25))
U8_gaa(T25, hbal_treeA_out_ga(T25, T9)) → U9_gaa(T25, T9, hbal_treeA_in_ga(s(T25)))
U9_gaa(T25, T9, hbal_treeA_out_ga(s(T25), T26)) → pD_out_gaa(T25, T9, T26)
U3_ga(T25, pD_out_gaa(T25, T9, T10)) → hbal_treeA_out_ga(s(s(T25)), t(x, T9, T10))
U6_gaa(T21, hbal_treeA_out_ga(s(T21), T9)) → U7_gaa(T21, T9, hbal_treeA_in_ga(T21))
U7_gaa(T21, T9, hbal_treeA_out_ga(T21, T22)) → pC_out_gaa(T21, T9, T22)
U2_ga(T21, pC_out_gaa(T21, T9, T10)) → hbal_treeA_out_ga(s(s(T21)), t(x, T9, T10))
U4_gaa(T15, hbal_treeA_out_ga(s(T15), T9)) → U5_gaa(T15, T9, hbal_treeA_in_ga(s(T15)))
U5_gaa(T15, T9, hbal_treeA_out_ga(s(T15), T16)) → pB_out_gaa(T15, T9, T16)
U1_ga(T15, pB_out_gaa(T15, T9, T10)) → hbal_treeA_out_ga(s(s(T15)), t(x, T9, T10))

The set Q consists of the following terms:

hbal_treeA_in_ga(x0)
pB_in_gaa(x0)
pC_in_gaa(x0)
pD_in_gaa(x0)
U8_gaa(x0, x1)
U9_gaa(x0, x1, x2)
U3_ga(x0, x1)
U6_gaa(x0, x1)
U7_gaa(x0, x1, x2)
U2_ga(x0, x1)
U4_gaa(x0, x1)
U5_gaa(x0, x1, x2)
U1_ga(x0, x1)

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

(9) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

HBAL_TREEA_IN_GA(s(s(T15))) → PB_IN_GAA(T15)
HBAL_TREEA_IN_GA(s(s(T21))) → PC_IN_GAA(T21)
PC_IN_GAA(T21) → U6_GAA(T21, hbal_treeA_in_ga(s(T21)))
HBAL_TREEA_IN_GA(s(s(T25))) → PD_IN_GAA(T25)
PD_IN_GAA(T25) → HBAL_TREEA_IN_GA(T25)

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(HBAL_TREEA_IN_GA(x₁)) = x₁   
POL(PB_IN_GAA(x₁)) = 1 + x₁   
POL(PC_IN_GAA(x₁)) = 1 + x₁   
POL(PD_IN_GAA(x₁)) = 1 + x₁   
POL(U1_ga(x₁, x₂)) = 0   
POL(U2_ga(x₁, x₂)) = 0   
POL(U3_ga(x₁, x₂)) = 0   
POL(U4_GAA(x₁, x₂)) = 1 + x₁   
POL(U4_gaa(x₁, x₂)) = 0   
POL(U5_gaa(x₁, x₂, x₃)) = 0   
POL(U6_GAA(x₁, x₂)) = x₁   
POL(U6_gaa(x₁, x₂)) = 0   
POL(U7_gaa(x₁, x₂, x₃)) = 0   
POL(U8_GAA(x₁, x₂)) = 1 + x₁   
POL(U8_gaa(x₁, x₂)) = 0   
POL(U9_gaa(x₁, x₂, x₃)) = 0   
POL(hbal_treeA_in_ga(x₁)) = 0   
POL(hbal_treeA_out_ga(x₁, x₂)) = 0   
POL(nil) = 0   
POL(pB_in_gaa(x₁)) = 0   
POL(pB_out_gaa(x₁, x₂, x₃)) = 0   
POL(pC_in_gaa(x₁)) = 0   
POL(pC_out_gaa(x₁, x₂, x₃)) = 0   
POL(pD_in_gaa(x₁)) = 0   
POL(pD_out_gaa(x₁, x₂, x₃)) = 0   
POL(s(x₁)) = 1 + x₁   
POL(t(x₁, x₂, x₃)) = 0   
POL(x) = 0   
POL(zero) = 0   

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none

(10) Obligation:

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

PB_IN_GAA(T15) → U4_GAA(T15, hbal_treeA_in_ga(s(T15)))
U4_GAA(T15, hbal_treeA_out_ga(s(T15), T9)) → HBAL_TREEA_IN_GA(s(T15))
U6_GAA(T21, hbal_treeA_out_ga(s(T21), T9)) → HBAL_TREEA_IN_GA(T21)
PD_IN_GAA(T25) → U8_GAA(T25, hbal_treeA_in_ga(T25))
U8_GAA(T25, hbal_treeA_out_ga(T25, T9)) → HBAL_TREEA_IN_GA(s(T25))
PC_IN_GAA(T21) → HBAL_TREEA_IN_GA(s(T21))
PB_IN_GAA(T15) → HBAL_TREEA_IN_GA(s(T15))

The TRS R consists of the following rules:

hbal_treeA_in_ga(zero) → hbal_treeA_out_ga(zero, nil)
hbal_treeA_in_ga(s(zero)) → hbal_treeA_out_ga(s(zero), t(x, nil, nil))
hbal_treeA_in_ga(s(s(T15))) → U1_ga(T15, pB_in_gaa(T15))
pB_in_gaa(T15) → U4_gaa(T15, hbal_treeA_in_ga(s(T15)))
hbal_treeA_in_ga(s(s(T21))) → U2_ga(T21, pC_in_gaa(T21))
pC_in_gaa(T21) → U6_gaa(T21, hbal_treeA_in_ga(s(T21)))
hbal_treeA_in_ga(s(s(T25))) → U3_ga(T25, pD_in_gaa(T25))
pD_in_gaa(T25) → U8_gaa(T25, hbal_treeA_in_ga(T25))
U8_gaa(T25, hbal_treeA_out_ga(T25, T9)) → U9_gaa(T25, T9, hbal_treeA_in_ga(s(T25)))
U9_gaa(T25, T9, hbal_treeA_out_ga(s(T25), T26)) → pD_out_gaa(T25, T9, T26)
U3_ga(T25, pD_out_gaa(T25, T9, T10)) → hbal_treeA_out_ga(s(s(T25)), t(x, T9, T10))
U6_gaa(T21, hbal_treeA_out_ga(s(T21), T9)) → U7_gaa(T21, T9, hbal_treeA_in_ga(T21))
U7_gaa(T21, T9, hbal_treeA_out_ga(T21, T22)) → pC_out_gaa(T21, T9, T22)
U2_ga(T21, pC_out_gaa(T21, T9, T10)) → hbal_treeA_out_ga(s(s(T21)), t(x, T9, T10))
U4_gaa(T15, hbal_treeA_out_ga(s(T15), T9)) → U5_gaa(T15, T9, hbal_treeA_in_ga(s(T15)))
U5_gaa(T15, T9, hbal_treeA_out_ga(s(T15), T16)) → pB_out_gaa(T15, T9, T16)
U1_ga(T15, pB_out_gaa(T15, T9, T10)) → hbal_treeA_out_ga(s(s(T15)), t(x, T9, T10))

The set Q consists of the following terms:

hbal_treeA_in_ga(x0)
pB_in_gaa(x0)
pC_in_gaa(x0)
pD_in_gaa(x0)
U8_gaa(x0, x1)
U9_gaa(x0, x1, x2)
U3_ga(x0, x1)
U6_gaa(x0, x1)
U7_gaa(x0, x1, x2)
U2_ga(x0, x1)
U4_gaa(x0, x1)
U5_gaa(x0, x1, x2)
U1_ga(x0, x1)

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

(11) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 7 less nodes.

(12) TRUE