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

(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) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: bin_tree(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: bin_tree(T1), SCOPE: 1, CLAUSE: 0 | bin_tree(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | bin_tree(void), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: bin_tree(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nbin_tree(T1) !=? bin_tree(void)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: bin_tree(void), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(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];
7 [label="7: ','(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];
8 [label="8: ','(','(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];
9 [label="9: ','(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];
10 [label="10: ','(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];
11 [label="11: ','(','(!_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];
12 [label="12: ','(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];
13 [label="13: ','(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];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(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];
16 [label="16: ','(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];
17 [label="17: ','(','(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];
18 [label="18: ','(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];
19 [label="19: ','(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];
20 [label="20: ?_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];
21 [label="21: ?_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];
22 [label="22: ','(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];
23 [label="23: ','(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];
24 [label="24: ','(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];
25 [label="25: ','(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];
26 [label="26: ','(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];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(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];
29 [label="29: ','(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];
30 [label="30: ','(bin_tree(T26), bin_tree(T27))\n\nT26 is ground\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: bin_tree(T26)\n\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: bin_tree(T27)\n\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 33 [arrowhead = none ];
33 -> 2;

34 [label="EVAL with clause\nbin_tree(void).\nand substitutionT1 -> void", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 34 [arrowhead = none ];
34 -> 3;

35 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 35 [arrowhead = none ];
35 -> 4;

36 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 36 [arrowhead = none ];
36 -> 5;

37 [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];
4 -> 37 [arrowhead = none ];
37 -> 15;

38 [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];
5 -> 38 [arrowhead = none ];
38 -> 6;

39 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
6 -> 39 [arrowhead = none ];
39 -> 7;

40 [label="ONLY EVAL with clause\nno(X12) :- ','(call(X12), ','(!_2, failure(a))).\nand substitutionX12 -> empty(void)", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 40 [arrowhead = none ];
40 -> 8;

41 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
8 -> 41 [arrowhead = none ];
41 -> 9;

42 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 42 [arrowhead = none ];
42 -> 10;

43 [label="ONLY EVAL with clause\nempty(void).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 43 [arrowhead = none ];
43 -> 11;

44 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 44 [arrowhead = none ];
44 -> 12;

45 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 45 [arrowhead = none ];
45 -> 13;

46 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 46 [arrowhead = none ];
46 -> 14;

47 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 47 [arrowhead = none ];
47 -> 16;

48 [label="ONLY EVAL with clause\nno(X19) :- ','(call(X19), ','(!_6, failure(a))).\nand substitutionT3 -> T6,\nX19 -> empty(T6)", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 48 [arrowhead = none ];
48 -> 17;

49 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 49 [arrowhead = none ];
49 -> 18;

50 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 50 [arrowhead = none ];
50 -> 19;

51 [label="BACKTRACK\nfor clause: empty(void)\nwith clash: (bin_tree(T6), bin_tree(void))", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 51 [arrowhead = none ];
51 -> 20;

52 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 52 [arrowhead = none ];
52 -> 21;

53 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 53 [arrowhead = none ];
53 -> 22;

54 [label="ONLY EVAL with clause\nno(X22).\nand substitutionT6 -> T9,\nX22 -> empty(T9)", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 54 [arrowhead = none ];
54 -> 23;

55 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 55 [arrowhead = none ];
55 -> 24;

56 [label="BACKTRACK\nfor clause: left(void, void)\nwith clash: (bin_tree(T9), bin_tree(void))", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 56 [arrowhead = none ];
56 -> 25;

57 [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];
25 -> 57 [arrowhead = none ];
57 -> 26;

58 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 58 [arrowhead = none ];
58 -> 27;

59 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 59 [arrowhead = none ];
59 -> 28;

60 [label="BACKTRACK\nfor clause: right(void, void)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 60 [arrowhead = none ];
60 -> 29;

61 [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];
29 -> 61 [arrowhead = none ];
61 -> 30;

62 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
30 -> 62 [arrowhead = none ];
62 -> 31;

63 [label="SPLIT 2\nnew knowledge:\nT26 is ground", fontsize=14, style = filled, fillcolor = violet];
30 -> 63 [arrowhead = none ];
63 -> 32;

64 [label="INSTANCE with matching:\nT1 -> T26", fontsize=14, style = filled, fillcolor = lightgrey];
31 -> 64 [arrowhead = none , style=dashed];
64 -> 1[style=dashed];

65 [label="INSTANCE with matching:\nT1 -> T27", fontsize=14, style = filled, fillcolor = lightgrey];
32 -> 65 [arrowhead = none , style=dashed];
65 -> 1[style=dashed];

}

(2) Obligation:

Clauses:

bin_treeA(void).
bin_treeA(tree(T25, T26, T27)) :- bin_treeA(T26).
bin_treeA(tree(T25, T26, T27)) :- ','(bin_treeA(T26), bin_treeA(T27)).

Query: bin_treeA(g)

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
bin_treeA_in: (b)
Transforming Prolog into the following Term Rewriting System:
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, bin_treeA_in_g(T26))
U1_g(T25, T26, T27, bin_treeA_out_g(T26)) → bin_treeA_out_g(tree(T25, T26, T27))
U1_g(T25, T26, T27, bin_treeA_out_g(T26)) → U2_g(T25, T26, T27, bin_treeA_in_g(T27))
U2_g(T25, T26, T27, bin_treeA_out_g(T27)) → bin_treeA_out_g(tree(T25, T26, T27))

The argument filtering Pi contains the following mapping:
bin_treeA_in_g(x1)  =  bin_treeA_in_g(x1)
void  =  void
bin_treeA_out_g(x1)  =  bin_treeA_out_g
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_g(x1, x2, x3, x4)  =  U1_g(x3, x4)
U2_g(x1, x2, x3, x4)  =  U2_g(x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) 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, bin_treeA_in_g(T26))
U1_g(T25, T26, T27, bin_treeA_out_g(T26)) → bin_treeA_out_g(tree(T25, T26, T27))
U1_g(T25, T26, T27, bin_treeA_out_g(T26)) → U2_g(T25, T26, T27, bin_treeA_in_g(T27))
U2_g(T25, T26, T27, bin_treeA_out_g(T27)) → bin_treeA_out_g(tree(T25, T26, T27))

The argument filtering Pi contains the following mapping:
bin_treeA_in_g(x1)  =  bin_treeA_in_g(x1)
void  =  void
bin_treeA_out_g(x1)  =  bin_treeA_out_g
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_g(x1, x2, x3, x4)  =  U1_g(x3, x4)
U2_g(x1, x2, x3, x4)  =  U2_g(x4)

(5) DependencyPairsProof (EQUIVALENT transformation)

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

BIN_TREEA_IN_G(tree(T25, T26, T27)) → U1_G(T25, T26, T27, bin_treeA_in_g(T26))
BIN_TREEA_IN_G(tree(T25, T26, T27)) → BIN_TREEA_IN_G(T26)
U1_G(T25, T26, T27, bin_treeA_out_g(T26)) → U2_G(T25, T26, T27, bin_treeA_in_g(T27))
U1_G(T25, 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, bin_treeA_in_g(T26))
U1_g(T25, T26, T27, bin_treeA_out_g(T26)) → bin_treeA_out_g(tree(T25, T26, T27))
U1_g(T25, T26, T27, bin_treeA_out_g(T26)) → U2_g(T25, T26, T27, bin_treeA_in_g(T27))
U2_g(T25, T26, T27, bin_treeA_out_g(T27)) → bin_treeA_out_g(tree(T25, T26, T27))

The argument filtering Pi contains the following mapping:
bin_treeA_in_g(x1)  =  bin_treeA_in_g(x1)
void  =  void
bin_treeA_out_g(x1)  =  bin_treeA_out_g
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_g(x1, x2, x3, x4)  =  U1_g(x3, x4)
U2_g(x1, x2, x3, x4)  =  U2_g(x4)
BIN_TREEA_IN_G(x1)  =  BIN_TREEA_IN_G(x1)
U1_G(x1, x2, x3, x4)  =  U1_G(x3, x4)
U2_G(x1, x2, x3, x4)  =  U2_G(x4)

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

(6) 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, bin_treeA_in_g(T26))
BIN_TREEA_IN_G(tree(T25, T26, T27)) → BIN_TREEA_IN_G(T26)
U1_G(T25, T26, T27, bin_treeA_out_g(T26)) → U2_G(T25, T26, T27, bin_treeA_in_g(T27))
U1_G(T25, 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, bin_treeA_in_g(T26))
U1_g(T25, T26, T27, bin_treeA_out_g(T26)) → bin_treeA_out_g(tree(T25, T26, T27))
U1_g(T25, T26, T27, bin_treeA_out_g(T26)) → U2_g(T25, T26, T27, bin_treeA_in_g(T27))
U2_g(T25, T26, T27, bin_treeA_out_g(T27)) → bin_treeA_out_g(tree(T25, T26, T27))

The argument filtering Pi contains the following mapping:
bin_treeA_in_g(x1)  =  bin_treeA_in_g(x1)
void  =  void
bin_treeA_out_g(x1)  =  bin_treeA_out_g
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_g(x1, x2, x3, x4)  =  U1_g(x3, x4)
U2_g(x1, x2, x3, x4)  =  U2_g(x4)
BIN_TREEA_IN_G(x1)  =  BIN_TREEA_IN_G(x1)
U1_G(x1, x2, x3, x4)  =  U1_G(x3, x4)
U2_G(x1, x2, x3, x4)  =  U2_G(x4)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

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

(8) Obligation:

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

U1_G(T25, T26, T27, bin_treeA_out_g(T26)) → BIN_TREEA_IN_G(T27)
BIN_TREEA_IN_G(tree(T25, T26, T27)) → U1_G(T25, T26, T27, bin_treeA_in_g(T26))
BIN_TREEA_IN_G(tree(T25, 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, bin_treeA_in_g(T26))
U1_g(T25, T26, T27, bin_treeA_out_g(T26)) → bin_treeA_out_g(tree(T25, T26, T27))
U1_g(T25, T26, T27, bin_treeA_out_g(T26)) → U2_g(T25, T26, T27, bin_treeA_in_g(T27))
U2_g(T25, T26, T27, bin_treeA_out_g(T27)) → bin_treeA_out_g(tree(T25, T26, T27))

The argument filtering Pi contains the following mapping:
bin_treeA_in_g(x1)  =  bin_treeA_in_g(x1)
void  =  void
bin_treeA_out_g(x1)  =  bin_treeA_out_g
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_g(x1, x2, x3, x4)  =  U1_g(x3, x4)
U2_g(x1, x2, x3, x4)  =  U2_g(x4)
BIN_TREEA_IN_G(x1)  =  BIN_TREEA_IN_G(x1)
U1_G(x1, x2, x3, x4)  =  U1_G(x3, x4)

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:

U1_G(T27, bin_treeA_out_g) → BIN_TREEA_IN_G(T27)
BIN_TREEA_IN_G(tree(T25, T26, T27)) → U1_G(T27, bin_treeA_in_g(T26))
BIN_TREEA_IN_G(tree(T25, T26, T27)) → BIN_TREEA_IN_G(T26)

The TRS R consists of the following rules:

bin_treeA_in_g(void) → bin_treeA_out_g
bin_treeA_in_g(tree(T25, T26, T27)) → U1_g(T27, bin_treeA_in_g(T26))
U1_g(T27, bin_treeA_out_g) → bin_treeA_out_g
U1_g(T27, bin_treeA_out_g) → U2_g(bin_treeA_in_g(T27))
U2_g(bin_treeA_out_g) → bin_treeA_out_g

The set Q consists of the following terms:

bin_treeA_in_g(x0)
U1_g(x0, x1)
U2_g(x0)

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

(11) 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:

• BIN_TREEA_IN_G(tree(T25, T26, T27)) → U1_G(T27, bin_treeA_in_g(T26))
  The graph contains the following edges 1 > 1

• BIN_TREEA_IN_G(tree(T25, T26, T27)) → BIN_TREEA_IN_G(T26)
  The graph contains the following edges 1 > 1

• U1_G(T27, bin_treeA_out_g) → BIN_TREEA_IN_G(T27)
  The graph contains the following edges 1 >= 1

(12) YES