Left Termination proof of /tmp/tmp

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

0 Prolog

↳1 PrologToPiTRSViaGraphTransformerProof (⇒, 54 ms)

↳2 PiTRS

↳3 DependencyPairsProof (⇔, 6 ms)

↳4 PiDP

↳5 DependencyGraphProof (⇔, 0 ms)

↳6 PiDP

↳7 PiDPToQDPProof (⇔, 15 ms)

↳8 QDP

↳9 QDPSizeChangeProof (⇔, 0 ms)

↳10 YES

(0) Obligation:

Clauses:

bin_tree(void).
bin_tree(T) :- ','(no(empty(T)), ','(left(T, L), ','(right(T, R), ','(bin_tree(L), bin_tree(R))))).
left(void, void).
left(tree(X1, L, X2), L).
right(void, void).
right(tree(X3, X4, R), R).
empty(void).
no(X) :- ','(X, ','(!, failure(a))).
no(X5).
failure(b).

Query: bin_tree(g)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: bin_tree(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
5 [label="5: bin_tree(T1), SCOPE: 1, CLAUSE: 0 | bin_tree(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: true | bin_tree(void), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: bin_tree(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nbin_tree(T1) !=? bin_tree(void)", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: bin_tree(void), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(no(empty(void)), ','(left(void, X8), ','(right(void, X9), ','(bin_tree(X8), bin_tree(X9)))))\n\nX8 is free\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(no(empty(void)), ','(left(void, X8), ','(right(void, X9), ','(bin_tree(X8), bin_tree(X9))))), SCOPE: 2, CLAUSE: 7 | ','(no(empty(void)), ','(left(void, X8), ','(right(void, X9), ','(bin_tree(X8), bin_tree(X9))))), SCOPE: 2, CLAUSE: 8\n\nX8 is free\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(','(call(empty(void)), ','(!_2, failure(a))), ','(left(void, X8), ','(right(void, X9), ','(bin_tree(X8), bin_tree(X9))))) | ','(no(empty(void)), ','(left(void, X8), ','(right(void, X9), ','(bin_tree(X8), bin_tree(X9))))), SCOPE: 2, CLAUSE: 8\n\nX8 is free\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(empty(void), ','(','(!_2, failure(a)), ','(left(void, X8), ','(right(void, X9), ','(bin_tree(X8), bin_tree(X9)))))) | ?_3 | ','(no(empty(void)), ','(left(void, X8), ','(right(void, X9), ','(bin_tree(X8), bin_tree(X9))))), SCOPE: 2, CLAUSE: 8\n\nX8 is free\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(empty(void), ','(','(!_2, failure(a)), ','(left(void, X8), ','(right(void, X9), ','(bin_tree(X8), bin_tree(X9)))))), SCOPE: 4, CLAUSE: 6 | ?_4 | ?_3 | ','(no(empty(void)), ','(left(void, X8), ','(right(void, X9), ','(bin_tree(X8), bin_tree(X9))))), SCOPE: 2, CLAUSE: 8\n\nX8 is free\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(','(!_2, failure(a)), ','(left(void, X8), ','(right(void, X9), ','(bin_tree(X8), bin_tree(X9))))) | ?_4 | ?_3 | ','(no(empty(void)), ','(left(void, X8), ','(right(void, X9), ','(bin_tree(X8), bin_tree(X9))))), SCOPE: 2, CLAUSE: 8\n\nX8 is free\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(failure(a), ','(left(void, X8), ','(right(void, X9), ','(bin_tree(X8), bin_tree(X9)))))\n\nX8 is free\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(failure(a), ','(left(void, X8), ','(right(void, X9), ','(bin_tree(X8), bin_tree(X9))))), SCOPE: 5, CLAUSE: 9\n\nX8 is free\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(no(empty(T3)), ','(left(T3, X15), ','(right(T3, X16), ','(bin_tree(X15), bin_tree(X16)))))\n\nT3 is ground\nX15 is free\nX16 is free\nbin_tree(T3) !=? bin_tree(void)", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(no(empty(T3)), ','(left(T3, X15), ','(right(T3, X16), ','(bin_tree(X15), bin_tree(X16))))), SCOPE: 6, CLAUSE: 7 | ','(no(empty(T3)), ','(left(T3, X15), ','(right(T3, X16), ','(bin_tree(X15), bin_tree(X16))))), SCOPE: 6, CLAUSE: 8\n\nT3 is ground\nX15 is free\nX16 is free\nbin_tree(T3) !=? bin_tree(void)", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(','(call(empty(T6)), ','(!_6, failure(a))), ','(left(T6, X15), ','(right(T6, X16), ','(bin_tree(X15), bin_tree(X16))))) | ','(no(empty(T6)), ','(left(T6, X15), ','(right(T6, X16), ','(bin_tree(X15), bin_tree(X16))))), SCOPE: 6, CLAUSE: 8\n\nT6 is ground\nX15 is free\nX16 is free\nbin_tree(T6) !=? bin_tree(void)", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(empty(T6), ','(','(!_6, failure(a)), ','(left(T6, X15), ','(right(T6, X16), ','(bin_tree(X15), bin_tree(X16)))))) | ?_7 | ','(no(empty(T6)), ','(left(T6, X15), ','(right(T6, X16), ','(bin_tree(X15), bin_tree(X16))))), SCOPE: 6, CLAUSE: 8\n\nT6 is ground\nX15 is free\nX16 is free\nbin_tree(T6) !=? bin_tree(void)", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(empty(T6), ','(','(!_6, failure(a)), ','(left(T6, X15), ','(right(T6, X16), ','(bin_tree(X15), bin_tree(X16)))))), SCOPE: 8, CLAUSE: 6 | ?_8 | ?_7 | ','(no(empty(T6)), ','(left(T6, X15), ','(right(T6, X16), ','(bin_tree(X15), bin_tree(X16))))), SCOPE: 6, CLAUSE: 8\n\nT6 is ground\nX15 is free\nX16 is free\nbin_tree(T6) !=? bin_tree(void)", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ?_8 | ?_7 | ','(no(empty(T6)), ','(left(T6, X15), ','(right(T6, X16), ','(bin_tree(X15), bin_tree(X16))))), SCOPE: 6, CLAUSE: 8\n\nT6 is ground\nX15 is free\nX16 is free\nbin_tree(T6) !=? bin_tree(void)", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ?_7 | ','(no(empty(T6)), ','(left(T6, X15), ','(right(T6, X16), ','(bin_tree(X15), bin_tree(X16))))), SCOPE: 6, CLAUSE: 8\n\nT6 is ground\nX15 is free\nX16 is free\nbin_tree(T6) !=? bin_tree(void)", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: ','(no(empty(T6)), ','(left(T6, X15), ','(right(T6, X16), ','(bin_tree(X15), bin_tree(X16))))), SCOPE: 6, CLAUSE: 8\n\nT6 is ground\nX15 is free\nX16 is free\nbin_tree(T6) !=? bin_tree(void)", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(left(T9, X15), ','(right(T9, X16), ','(bin_tree(X15), bin_tree(X16))))\n\nT9 is ground\nX15 is free\nX16 is free\nbin_tree(T9) !=? bin_tree(void)", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ','(left(T9, X15), ','(right(T9, X16), ','(bin_tree(X15), bin_tree(X16)))), SCOPE: 9, CLAUSE: 2 | ','(left(T9, X15), ','(right(T9, X16), ','(bin_tree(X15), bin_tree(X16)))), SCOPE: 9, CLAUSE: 3\n\nT9 is ground\nX15 is free\nX16 is free\nbin_tree(T9) !=? bin_tree(void)", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ','(left(T9, X15), ','(right(T9, X16), ','(bin_tree(X15), bin_tree(X16)))), SCOPE: 9, CLAUSE: 3\n\nT9 is ground\nX15 is free\nX16 is free\nbin_tree(T9) !=? bin_tree(void)", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: ','(right(tree(T16, T17, T18), X16), ','(bin_tree(T17), bin_tree(X16)))\n\nT16 is ground\nT17 is ground\nT18 is ground\nX16 is free", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: ','(right(tree(T16, T17, T18), X16), ','(bin_tree(T17), bin_tree(X16))), SCOPE: 10, CLAUSE: 4 | ','(right(tree(T16, T17, T18), X16), ','(bin_tree(T17), bin_tree(X16))), SCOPE: 10, CLAUSE: 5\n\nT16 is ground\nT17 is ground\nT18 is ground\nX16 is free", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: ','(right(tree(T16, T17, T18), X16), ','(bin_tree(T17), bin_tree(X16))), SCOPE: 10, CLAUSE: 5\n\nT16 is ground\nT17 is ground\nT18 is ground\nX16 is free", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="B: ','(bin_tree(T26), bin_tree(T27))\n\nT26 is ground\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: bin_tree(T26)\n\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: bin_tree(T27)\n\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 87 [arrowhead = none ];
87 -> 5;

88 [label="EVAL with clause\nbin_tree(void).\nand substitutionT1 -> void", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 88 [arrowhead = none ];
88 -> 13;

89 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 89 [arrowhead = none ];
89 -> 14;

90 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 90 [arrowhead = none ];
90 -> 15;

91 [label="ONLY EVAL with clause\nbin_tree(X14) :- ','(no(empty(X14)), ','(left(X14, X15), ','(right(X14, X16), ','(bin_tree(X15), bin_tree(X16))))).\nand substitutionT1 -> T3,\nX14 -> T3", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 91 [arrowhead = none ];
91 -> 25;

92 [label="ONLY EVAL with clause\nbin_tree(X7) :- ','(no(empty(X7)), ','(left(X7, X8), ','(right(X7, X9), ','(bin_tree(X8), bin_tree(X9))))).\nand substitutionX7 -> void", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 92 [arrowhead = none ];
92 -> 16;

93 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 93 [arrowhead = none ];
93 -> 17;

94 [label="ONLY EVAL with clause\nno(X12) :- ','(call(X12), ','(!_2, failure(a))).\nand substitutionX12 -> empty(void)", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 94 [arrowhead = none ];
94 -> 18;

95 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 95 [arrowhead = none ];
95 -> 19;

96 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 96 [arrowhead = none ];
96 -> 20;

97 [label="ONLY EVAL with clause\nempty(void).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 97 [arrowhead = none ];
97 -> 21;

98 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 98 [arrowhead = none ];
98 -> 22;

99 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
22 -> 99 [arrowhead = none ];
99 -> 23;

100 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 100 [arrowhead = none ];
100 -> 24;

101 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 101 [arrowhead = none ];
101 -> 27;

102 [label="ONLY EVAL with clause\nno(X19) :- ','(call(X19), ','(!_6, failure(a))).\nand substitutionT3 -> T6,\nX19 -> empty(T6)", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 102 [arrowhead = none ];
102 -> 35;

103 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
35 -> 103 [arrowhead = none ];
103 -> 37;

104 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
37 -> 104 [arrowhead = none ];
104 -> 39;

105 [label="BACKTRACK\nfor clause: empty(void)\nwith clash: (bin_tree(T6), bin_tree(void))", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 105 [arrowhead = none ];
105 -> 42;

106 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 106 [arrowhead = none ];
106 -> 43;

107 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 107 [arrowhead = none ];
107 -> 51;

108 [label="ONLY EVAL with clause\nno(X22).\nand substitutionT6 -> T9,\nX22 -> empty(T9)", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 108 [arrowhead = none ];
108 -> 58;

109 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
58 -> 109 [arrowhead = none ];
109 -> 64;

110 [label="BACKTRACK\nfor clause: left(void, void)\nwith clash: (bin_tree(T9), bin_tree(void))", fontsize=14, style = filled, fillcolor = lightgreen];
64 -> 110 [arrowhead = none ];
110 -> 65;

111 [label="EVAL with clause\nleft(tree(X29, X30, X31), X30).\nand substitutionX29 -> T16,\nX30 -> T17,\nX31 -> T18,\nT9 -> tree(T16, T17, T18),\nX15 -> T17", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 111 [arrowhead = none ];
111 -> 75;

112 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
65 -> 112 [arrowhead = none ];
112 -> 76;

113 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
75 -> 113 [arrowhead = none ];
113 -> 77;

114 [label="BACKTRACK\nfor clause: right(void, void)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
77 -> 114 [arrowhead = none ];
114 -> 78;

115 [label="ONLY EVAL with clause\nright(tree(X38, X39, X40), X40).\nand substitutionT16 -> T25,\nX38 -> T25,\nT17 -> T26,\nX39 -> T26,\nT18 -> T27,\nX40 -> T27,\nX16 -> T27", fontsize=14, style = filled, fillcolor = lightblue];
78 -> 115 [arrowhead = none ];
115 -> 84;

116 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
84 -> 116 [arrowhead = none ];
116 -> 85;

117 [label="SPLIT 2\nnew knowledge:\nT26 is ground", fontsize=14, style = filled, fillcolor = violet];
84 -> 117 [arrowhead = none ];
117 -> 86;

118 [label="INSTANCE with matching:\nT1 -> T26", fontsize=14, style = filled, fillcolor = lightgrey];
85 -> 118 [arrowhead = none , style=dashed];
118 -> 1[style=dashed];

119 [label="INSTANCE with matching:\nT1 -> T27", fontsize=14, style = filled, fillcolor = lightgrey];
86 -> 119 [arrowhead = none , style=dashed];
119 -> 1[style=dashed];

}

(2) Obligation:

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

bin_treeA_in_g(void) → bin_treeA_out_g(void)
bin_treeA_in_g(tree(T25, T26, T27)) → U1_g(T25, T26, T27, pB_in_gg(T26, T27))
pB_in_gg(T26, T27) → U2_gg(T26, T27, bin_treeA_in_g(T26))
U2_gg(T26, T27, bin_treeA_out_g(T26)) → U3_gg(T26, T27, bin_treeA_in_g(T27))
U3_gg(T26, T27, bin_treeA_out_g(T27)) → pB_out_gg(T26, T27)
U1_g(T25, T26, T27, pB_out_gg(T26, T27)) → bin_treeA_out_g(tree(T25, T26, T27))

Pi is empty.

(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:

BIN_TREEA_IN_G(tree(T25, T26, T27)) → U1_G(T25, T26, T27, pB_in_gg(T26, T27))
BIN_TREEA_IN_G(tree(T25, T26, T27)) → PB_IN_GG(T26, T27)
PB_IN_GG(T26, T27) → U2_GG(T26, T27, bin_treeA_in_g(T26))
PB_IN_GG(T26, T27) → BIN_TREEA_IN_G(T26)
U2_GG(T26, T27, bin_treeA_out_g(T26)) → U3_GG(T26, T27, bin_treeA_in_g(T27))
U2_GG(T26, T27, bin_treeA_out_g(T26)) → BIN_TREEA_IN_G(T27)

The TRS R consists of the following rules:

bin_treeA_in_g(void) → bin_treeA_out_g(void)
bin_treeA_in_g(tree(T25, T26, T27)) → U1_g(T25, T26, T27, pB_in_gg(T26, T27))
pB_in_gg(T26, T27) → U2_gg(T26, T27, bin_treeA_in_g(T26))
U2_gg(T26, T27, bin_treeA_out_g(T26)) → U3_gg(T26, T27, bin_treeA_in_g(T27))
U3_gg(T26, T27, bin_treeA_out_g(T27)) → pB_out_gg(T26, T27)
U1_g(T25, T26, T27, pB_out_gg(T26, T27)) → bin_treeA_out_g(tree(T25, T26, T27))

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

(4) Obligation:

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

BIN_TREEA_IN_G(tree(T25, T26, T27)) → U1_G(T25, T26, T27, pB_in_gg(T26, T27))
BIN_TREEA_IN_G(tree(T25, T26, T27)) → PB_IN_GG(T26, T27)
PB_IN_GG(T26, T27) → U2_GG(T26, T27, bin_treeA_in_g(T26))
PB_IN_GG(T26, T27) → BIN_TREEA_IN_G(T26)
U2_GG(T26, T27, bin_treeA_out_g(T26)) → U3_GG(T26, T27, bin_treeA_in_g(T27))
U2_GG(T26, T27, bin_treeA_out_g(T26)) → BIN_TREEA_IN_G(T27)

The TRS R consists of the following rules:

bin_treeA_in_g(void) → bin_treeA_out_g(void)
bin_treeA_in_g(tree(T25, T26, T27)) → U1_g(T25, T26, T27, pB_in_gg(T26, T27))
pB_in_gg(T26, T27) → U2_gg(T26, T27, bin_treeA_in_g(T26))
U2_gg(T26, T27, bin_treeA_out_g(T26)) → U3_gg(T26, T27, bin_treeA_in_g(T27))
U3_gg(T26, T27, bin_treeA_out_g(T27)) → pB_out_gg(T26, T27)
U1_g(T25, T26, T27, pB_out_gg(T26, T27)) → bin_treeA_out_g(tree(T25, T26, T27))

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes.

(6) Obligation:

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

BIN_TREEA_IN_G(tree(T25, T26, T27)) → PB_IN_GG(T26, T27)
PB_IN_GG(T26, T27) → U2_GG(T26, T27, bin_treeA_in_g(T26))
U2_GG(T26, T27, bin_treeA_out_g(T26)) → BIN_TREEA_IN_G(T27)
PB_IN_GG(T26, T27) → BIN_TREEA_IN_G(T26)

The TRS R consists of the following rules:

bin_treeA_in_g(void) → bin_treeA_out_g(void)
bin_treeA_in_g(tree(T25, T26, T27)) → U1_g(T25, T26, T27, pB_in_gg(T26, T27))
pB_in_gg(T26, T27) → U2_gg(T26, T27, bin_treeA_in_g(T26))
U2_gg(T26, T27, bin_treeA_out_g(T26)) → U3_gg(T26, T27, bin_treeA_in_g(T27))
U3_gg(T26, T27, bin_treeA_out_g(T27)) → pB_out_gg(T26, T27)
U1_g(T25, T26, T27, pB_out_gg(T26, T27)) → bin_treeA_out_g(tree(T25, T26, T27))

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

(7) PiDPToQDPProof (EQUIVALENT 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:

BIN_TREEA_IN_G(tree(T25, T26, T27)) → PB_IN_GG(T26, T27)
PB_IN_GG(T26, T27) → U2_GG(T26, T27, bin_treeA_in_g(T26))
U2_GG(T26, T27, bin_treeA_out_g(T26)) → BIN_TREEA_IN_G(T27)
PB_IN_GG(T26, T27) → BIN_TREEA_IN_G(T26)

The TRS R consists of the following rules:

bin_treeA_in_g(void) → bin_treeA_out_g(void)
bin_treeA_in_g(tree(T25, T26, T27)) → U1_g(T25, T26, T27, pB_in_gg(T26, T27))
pB_in_gg(T26, T27) → U2_gg(T26, T27, bin_treeA_in_g(T26))
U2_gg(T26, T27, bin_treeA_out_g(T26)) → U3_gg(T26, T27, bin_treeA_in_g(T27))
U3_gg(T26, T27, bin_treeA_out_g(T27)) → pB_out_gg(T26, T27)
U1_g(T25, T26, T27, pB_out_gg(T26, T27)) → bin_treeA_out_g(tree(T25, T26, T27))

The set Q consists of the following terms:

bin_treeA_in_g(x0)
pB_in_gg(x0, x1)
U2_gg(x0, x1, x2)
U3_gg(x0, x1, x2)
U1_g(x0, x1, x2, x3)

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

(9) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

PB_IN_GG(T26, T27) → BIN_TREEA_IN_G(T26)
The graph contains the following edges 1 >= 1
PB_IN_GG(T26, T27) → U2_GG(T26, T27, bin_treeA_in_g(T26))
The graph contains the following edges 1 >= 1, 2 >= 2
U2_GG(T26, T27, bin_treeA_out_g(T26)) → BIN_TREEA_IN_G(T27)
The graph contains the following edges 2 >= 1
BIN_TREEA_IN_G(tree(T25, T26, T27)) → PB_IN_GG(T26, T27)
The graph contains the following edges 1 > 1, 1 > 2

(0) Obligation:

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

(2) Obligation:

(3) DependencyPairsProof (EQUIVALENT transformation)

(4) Obligation:

(5) DependencyGraphProof (EQUIVALENT transformation)

(6) Obligation:

(7) PiDPToQDPProof (EQUIVALENT transformation)

(8) Obligation:

(9) QDPSizeChangeProof (EQUIVALENT transformation)

(10) YES