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

(0) Obligation:

Clauses:

preorder(T, Xs) :- preorder_dl(T, -(Xs, [])).
preorder_dl(nil, -(X, X)).
preorder_dl(tree(L, X, R), -(.(X, Xs), Zs)) :- ','(preorder_dl(L, -(Xs, Ys)), preorder_dl(R, -(Ys, Zs))).

Query: preorder(g,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: preorder(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
5 [label="5: preorder(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="B: preorder_dl(T5, -(T7, []))\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: preorder_dl(T5, -(T7, [])), SCOPE: 2, CLAUSE: 1 | preorder_dl(T5, -(T7, [])), SCOPE: 2, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: preorder_dl(T5, -(T7, [])), SCOPE: 2, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: preorder_dl(T5, -(T7, [])), SCOPE: 2, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="C: ','(preorder_dl(T21, -(T25, X26)), preorder_dl(T23, -(X26, [])))\n\nT21 is ground\nT23 is ground\nX26 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="D: preorder_dl(T21, -(T25, X26))\n\nT21 is ground\nX26 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: preorder_dl(T23, -(T28, []))\n\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: preorder_dl(T21, -(T25, X26)), SCOPE: 3, CLAUSE: 1 | preorder_dl(T21, -(T25, X26)), SCOPE: 3, CLAUSE: 2\n\nT21 is ground\nX26 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: preorder_dl(T21, -(T25, X26)), SCOPE: 3, CLAUSE: 1\n\nT21 is ground\nX26 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: preorder_dl(T21, -(T25, X26)), SCOPE: 3, CLAUSE: 2\n\nT21 is ground\nX26 is free", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="E: ','(preorder_dl(T44, -(T48, X54)), preorder_dl(T46, -(X54, X55)))\n\nT44 is ground\nT46 is ground\nX55 is free\nX54 is free", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: preorder_dl(T44, -(T48, X54))\n\nT44 is ground\nX54 is free", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: preorder_dl(T46, -(T51, X55))\n\nT46 is ground\nX55 is free", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 58 [arrowhead = none ];
58 -> 5;

59 [label="ONLY EVAL with clause\npreorder(X3, X4) :- preorder_dl(X3, -(X4, [])).\nand substitutionT1 -> T5,\nX3 -> T5,\nT2 -> T7,\nX4 -> T7,\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 59 [arrowhead = none ];
59 -> 7;

60 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 60 [arrowhead = none ];
60 -> 9;

61 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 61 [arrowhead = none ];
61 -> 11;

62 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 62 [arrowhead = none ];
62 -> 12;

63 [label="EVAL with clause\npreorder_dl(nil, -(X9, X9)).\nand substitutionT5 -> nil,\nT7 -> [],\nX9 -> [],\nT12 -> []", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 63 [arrowhead = none ];
63 -> 15;

64 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 64 [arrowhead = none ];
64 -> 17;

65 [label="EVAL with clause\npreorder_dl(tree(X21, X22, X23), -(.(X22, X24), X25)) :- ','(preorder_dl(X21, -(X24, X26)), preorder_dl(X23, -(X26, X25))).\nand substitutionX21 -> T21,\nX22 -> T22,\nX23 -> T23,\nT5 -> tree(T21, T22, T23),\nX24 -> T25,\nT7 -> .(T22, T25),\nX25 -> [],\nT24 -> T25", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 65 [arrowhead = none ];
65 -> 22;

66 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 66 [arrowhead = none ];
66 -> 25;

67 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 67 [arrowhead = none ];
67 -> 21;

68 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
22 -> 68 [arrowhead = none ];
68 -> 29;

69 [label="SPLIT 2\nnew knowledge:\nT21 is ground\nreplacements:X26 -> T28", fontsize=14, style = filled, fillcolor = violet];
22 -> 69 [arrowhead = none ];
69 -> 30;

70 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 70 [arrowhead = none ];
70 -> 36;

71 [label="INSTANCE with matching:\nT5 -> T23\nT7 -> T28", fontsize=14, style = filled, fillcolor = lightgrey];
30 -> 71 [arrowhead = none , style=dashed];
71 -> 7[style=dashed];

72 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 72 [arrowhead = none ];
72 -> 38;

73 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 73 [arrowhead = none ];
73 -> 40;

74 [label="EVAL with clause\npreorder_dl(nil, -(X35, X35)).\nand substitutionT21 -> nil,\nT25 -> T35,\nX35 -> T35,\nX26 -> T35", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 74 [arrowhead = none ];
74 -> 42;

75 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 75 [arrowhead = none ];
75 -> 44;

76 [label="EVAL with clause\npreorder_dl(tree(X49, X50, X51), -(.(X50, X52), X53)) :- ','(preorder_dl(X49, -(X52, X54)), preorder_dl(X51, -(X54, X53))).\nand substitutionX49 -> T44,\nX50 -> T45,\nX51 -> T46,\nT21 -> tree(T44, T45, T46),\nX52 -> T48,\nT25 -> .(T45, T48),\nX26 -> X55,\nX53 -> X55,\nT47 -> T48", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 76 [arrowhead = none ];
76 -> 52;

77 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
40 -> 77 [arrowhead = none ];
77 -> 54;

78 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 78 [arrowhead = none ];
78 -> 46;

79 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
52 -> 79 [arrowhead = none ];
79 -> 56;

80 [label="SPLIT 2\nnew knowledge:\nT44 is ground\nreplacements:X54 -> T51", fontsize=14, style = filled, fillcolor = violet];
52 -> 80 [arrowhead = none ];
80 -> 57;

81 [label="INSTANCE with matching:\nT21 -> T44\nT25 -> T48\nX26 -> X54", fontsize=14, style = filled, fillcolor = lightgrey];
56 -> 81 [arrowhead = none , style=dashed];
81 -> 29[style=dashed];

82 [label="INSTANCE with matching:\nT21 -> T46\nT25 -> T51\nX26 -> X55", fontsize=14, style = filled, fillcolor = lightgrey];
57 -> 82 [arrowhead = none , style=dashed];
82 -> 29[style=dashed];

}

(2) Obligation:

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

preorderA_in_ga(T5, T7) → U1_ga(T5, T7, preorder_dlB_in_ga(T5, T7))
preorder_dlB_in_ga(nil, []) → preorder_dlB_out_ga(nil, [])
preorder_dlB_in_ga(tree(T21, T22, T23), .(T22, T25)) → U2_ga(T21, T22, T23, T25, pC_in_gaag(T21, T25, X26, T23))
pC_in_gaag(T21, T25, T28, T23) → U4_gaag(T21, T25, T28, T23, preorder_dlD_in_gaa(T21, T25, T28))
preorder_dlD_in_gaa(nil, T35, T35) → preorder_dlD_out_gaa(nil, T35, T35)
preorder_dlD_in_gaa(tree(T44, T45, T46), .(T45, T48), X55) → U3_gaa(T44, T45, T46, T48, X55, pE_in_gaaga(T44, T48, X54, T46, X55))
pE_in_gaaga(T44, T48, T51, T46, X55) → U6_gaaga(T44, T48, T51, T46, X55, preorder_dlD_in_gaa(T44, T48, T51))
U6_gaaga(T44, T48, T51, T46, X55, preorder_dlD_out_gaa(T44, T48, T51)) → U7_gaaga(T44, T48, T51, T46, X55, preorder_dlD_in_gaa(T46, T51, X55))
U7_gaaga(T44, T48, T51, T46, X55, preorder_dlD_out_gaa(T46, T51, X55)) → pE_out_gaaga(T44, T48, T51, T46, X55)
U3_gaa(T44, T45, T46, T48, X55, pE_out_gaaga(T44, T48, X54, T46, X55)) → preorder_dlD_out_gaa(tree(T44, T45, T46), .(T45, T48), X55)
U4_gaag(T21, T25, T28, T23, preorder_dlD_out_gaa(T21, T25, T28)) → U5_gaag(T21, T25, T28, T23, preorder_dlB_in_ga(T23, T28))
U5_gaag(T21, T25, T28, T23, preorder_dlB_out_ga(T23, T28)) → pC_out_gaag(T21, T25, T28, T23)
U2_ga(T21, T22, T23, T25, pC_out_gaag(T21, T25, X26, T23)) → preorder_dlB_out_ga(tree(T21, T22, T23), .(T22, T25))
U1_ga(T5, T7, preorder_dlB_out_ga(T5, T7)) → preorderA_out_ga(T5, T7)

The argument filtering Pi contains the following mapping:
preorderA_in_ga(x1, x2)  =  preorderA_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
preorder_dlB_in_ga(x1, x2)  =  preorder_dlB_in_ga(x1)
nil  =  nil
preorder_dlB_out_ga(x1, x2)  =  preorder_dlB_out_ga(x1)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
pC_in_gaag(x1, x2, x3, x4)  =  pC_in_gaag(x1, x4)
U4_gaag(x1, x2, x3, x4, x5)  =  U4_gaag(x1, x4, x5)
preorder_dlD_in_gaa(x1, x2, x3)  =  preorder_dlD_in_gaa(x1)
preorder_dlD_out_gaa(x1, x2, x3)  =  preorder_dlD_out_gaa(x1)
U3_gaa(x1, x2, x3, x4, x5, x6)  =  U3_gaa(x1, x2, x3, x6)
pE_in_gaaga(x1, x2, x3, x4, x5)  =  pE_in_gaaga(x1, x4)
U6_gaaga(x1, x2, x3, x4, x5, x6)  =  U6_gaaga(x1, x4, x6)
U7_gaaga(x1, x2, x3, x4, x5, x6)  =  U7_gaaga(x1, x4, x6)
pE_out_gaaga(x1, x2, x3, x4, x5)  =  pE_out_gaaga(x1, x4)
U5_gaag(x1, x2, x3, x4, x5)  =  U5_gaag(x1, x4, x5)
pC_out_gaag(x1, x2, x3, x4)  =  pC_out_gaag(x1, x4)
preorderA_out_ga(x1, x2)  =  preorderA_out_ga(x1)

(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:

PREORDERA_IN_GA(T5, T7) → U1_GA(T5, T7, preorder_dlB_in_ga(T5, T7))
PREORDERA_IN_GA(T5, T7) → PREORDER_DLB_IN_GA(T5, T7)
PREORDER_DLB_IN_GA(tree(T21, T22, T23), .(T22, T25)) → U2_GA(T21, T22, T23, T25, pC_in_gaag(T21, T25, X26, T23))
PREORDER_DLB_IN_GA(tree(T21, T22, T23), .(T22, T25)) → PC_IN_GAAG(T21, T25, X26, T23)
PC_IN_GAAG(T21, T25, T28, T23) → U4_GAAG(T21, T25, T28, T23, preorder_dlD_in_gaa(T21, T25, T28))
PC_IN_GAAG(T21, T25, T28, T23) → PREORDER_DLD_IN_GAA(T21, T25, T28)
PREORDER_DLD_IN_GAA(tree(T44, T45, T46), .(T45, T48), X55) → U3_GAA(T44, T45, T46, T48, X55, pE_in_gaaga(T44, T48, X54, T46, X55))
PREORDER_DLD_IN_GAA(tree(T44, T45, T46), .(T45, T48), X55) → PE_IN_GAAGA(T44, T48, X54, T46, X55)
PE_IN_GAAGA(T44, T48, T51, T46, X55) → U6_GAAGA(T44, T48, T51, T46, X55, preorder_dlD_in_gaa(T44, T48, T51))
PE_IN_GAAGA(T44, T48, T51, T46, X55) → PREORDER_DLD_IN_GAA(T44, T48, T51)
U6_GAAGA(T44, T48, T51, T46, X55, preorder_dlD_out_gaa(T44, T48, T51)) → U7_GAAGA(T44, T48, T51, T46, X55, preorder_dlD_in_gaa(T46, T51, X55))
U6_GAAGA(T44, T48, T51, T46, X55, preorder_dlD_out_gaa(T44, T48, T51)) → PREORDER_DLD_IN_GAA(T46, T51, X55)
U4_GAAG(T21, T25, T28, T23, preorder_dlD_out_gaa(T21, T25, T28)) → U5_GAAG(T21, T25, T28, T23, preorder_dlB_in_ga(T23, T28))
U4_GAAG(T21, T25, T28, T23, preorder_dlD_out_gaa(T21, T25, T28)) → PREORDER_DLB_IN_GA(T23, T28)

The TRS R consists of the following rules:

preorderA_in_ga(T5, T7) → U1_ga(T5, T7, preorder_dlB_in_ga(T5, T7))
preorder_dlB_in_ga(nil, []) → preorder_dlB_out_ga(nil, [])
preorder_dlB_in_ga(tree(T21, T22, T23), .(T22, T25)) → U2_ga(T21, T22, T23, T25, pC_in_gaag(T21, T25, X26, T23))
pC_in_gaag(T21, T25, T28, T23) → U4_gaag(T21, T25, T28, T23, preorder_dlD_in_gaa(T21, T25, T28))
preorder_dlD_in_gaa(nil, T35, T35) → preorder_dlD_out_gaa(nil, T35, T35)
preorder_dlD_in_gaa(tree(T44, T45, T46), .(T45, T48), X55) → U3_gaa(T44, T45, T46, T48, X55, pE_in_gaaga(T44, T48, X54, T46, X55))
pE_in_gaaga(T44, T48, T51, T46, X55) → U6_gaaga(T44, T48, T51, T46, X55, preorder_dlD_in_gaa(T44, T48, T51))
U6_gaaga(T44, T48, T51, T46, X55, preorder_dlD_out_gaa(T44, T48, T51)) → U7_gaaga(T44, T48, T51, T46, X55, preorder_dlD_in_gaa(T46, T51, X55))
U7_gaaga(T44, T48, T51, T46, X55, preorder_dlD_out_gaa(T46, T51, X55)) → pE_out_gaaga(T44, T48, T51, T46, X55)
U3_gaa(T44, T45, T46, T48, X55, pE_out_gaaga(T44, T48, X54, T46, X55)) → preorder_dlD_out_gaa(tree(T44, T45, T46), .(T45, T48), X55)
U4_gaag(T21, T25, T28, T23, preorder_dlD_out_gaa(T21, T25, T28)) → U5_gaag(T21, T25, T28, T23, preorder_dlB_in_ga(T23, T28))
U5_gaag(T21, T25, T28, T23, preorder_dlB_out_ga(T23, T28)) → pC_out_gaag(T21, T25, T28, T23)
U2_ga(T21, T22, T23, T25, pC_out_gaag(T21, T25, X26, T23)) → preorder_dlB_out_ga(tree(T21, T22, T23), .(T22, T25))
U1_ga(T5, T7, preorder_dlB_out_ga(T5, T7)) → preorderA_out_ga(T5, T7)

The argument filtering Pi contains the following mapping:
preorderA_in_ga(x1, x2)  =  preorderA_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
preorder_dlB_in_ga(x1, x2)  =  preorder_dlB_in_ga(x1)
nil  =  nil
preorder_dlB_out_ga(x1, x2)  =  preorder_dlB_out_ga(x1)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
pC_in_gaag(x1, x2, x3, x4)  =  pC_in_gaag(x1, x4)
U4_gaag(x1, x2, x3, x4, x5)  =  U4_gaag(x1, x4, x5)
preorder_dlD_in_gaa(x1, x2, x3)  =  preorder_dlD_in_gaa(x1)
preorder_dlD_out_gaa(x1, x2, x3)  =  preorder_dlD_out_gaa(x1)
U3_gaa(x1, x2, x3, x4, x5, x6)  =  U3_gaa(x1, x2, x3, x6)
pE_in_gaaga(x1, x2, x3, x4, x5)  =  pE_in_gaaga(x1, x4)
U6_gaaga(x1, x2, x3, x4, x5, x6)  =  U6_gaaga(x1, x4, x6)
U7_gaaga(x1, x2, x3, x4, x5, x6)  =  U7_gaaga(x1, x4, x6)
pE_out_gaaga(x1, x2, x3, x4, x5)  =  pE_out_gaaga(x1, x4)
U5_gaag(x1, x2, x3, x4, x5)  =  U5_gaag(x1, x4, x5)
pC_out_gaag(x1, x2, x3, x4)  =  pC_out_gaag(x1, x4)
preorderA_out_ga(x1, x2)  =  preorderA_out_ga(x1)
PREORDERA_IN_GA(x1, x2)  =  PREORDERA_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
PREORDER_DLB_IN_GA(x1, x2)  =  PREORDER_DLB_IN_GA(x1)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x3, x5)
PC_IN_GAAG(x1, x2, x3, x4)  =  PC_IN_GAAG(x1, x4)
U4_GAAG(x1, x2, x3, x4, x5)  =  U4_GAAG(x1, x4, x5)
PREORDER_DLD_IN_GAA(x1, x2, x3)  =  PREORDER_DLD_IN_GAA(x1)
U3_GAA(x1, x2, x3, x4, x5, x6)  =  U3_GAA(x1, x2, x3, x6)
PE_IN_GAAGA(x1, x2, x3, x4, x5)  =  PE_IN_GAAGA(x1, x4)
U6_GAAGA(x1, x2, x3, x4, x5, x6)  =  U6_GAAGA(x1, x4, x6)
U7_GAAGA(x1, x2, x3, x4, x5, x6)  =  U7_GAAGA(x1, x4, x6)
U5_GAAG(x1, x2, x3, x4, x5)  =  U5_GAAG(x1, x4, x5)

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

(4) Obligation:

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

PREORDERA_IN_GA(T5, T7) → U1_GA(T5, T7, preorder_dlB_in_ga(T5, T7))
PREORDERA_IN_GA(T5, T7) → PREORDER_DLB_IN_GA(T5, T7)
PREORDER_DLB_IN_GA(tree(T21, T22, T23), .(T22, T25)) → U2_GA(T21, T22, T23, T25, pC_in_gaag(T21, T25, X26, T23))
PREORDER_DLB_IN_GA(tree(T21, T22, T23), .(T22, T25)) → PC_IN_GAAG(T21, T25, X26, T23)
PC_IN_GAAG(T21, T25, T28, T23) → U4_GAAG(T21, T25, T28, T23, preorder_dlD_in_gaa(T21, T25, T28))
PC_IN_GAAG(T21, T25, T28, T23) → PREORDER_DLD_IN_GAA(T21, T25, T28)
PREORDER_DLD_IN_GAA(tree(T44, T45, T46), .(T45, T48), X55) → U3_GAA(T44, T45, T46, T48, X55, pE_in_gaaga(T44, T48, X54, T46, X55))
PREORDER_DLD_IN_GAA(tree(T44, T45, T46), .(T45, T48), X55) → PE_IN_GAAGA(T44, T48, X54, T46, X55)
PE_IN_GAAGA(T44, T48, T51, T46, X55) → U6_GAAGA(T44, T48, T51, T46, X55, preorder_dlD_in_gaa(T44, T48, T51))
PE_IN_GAAGA(T44, T48, T51, T46, X55) → PREORDER_DLD_IN_GAA(T44, T48, T51)
U6_GAAGA(T44, T48, T51, T46, X55, preorder_dlD_out_gaa(T44, T48, T51)) → U7_GAAGA(T44, T48, T51, T46, X55, preorder_dlD_in_gaa(T46, T51, X55))
U6_GAAGA(T44, T48, T51, T46, X55, preorder_dlD_out_gaa(T44, T48, T51)) → PREORDER_DLD_IN_GAA(T46, T51, X55)
U4_GAAG(T21, T25, T28, T23, preorder_dlD_out_gaa(T21, T25, T28)) → U5_GAAG(T21, T25, T28, T23, preorder_dlB_in_ga(T23, T28))
U4_GAAG(T21, T25, T28, T23, preorder_dlD_out_gaa(T21, T25, T28)) → PREORDER_DLB_IN_GA(T23, T28)

The TRS R consists of the following rules:

preorderA_in_ga(T5, T7) → U1_ga(T5, T7, preorder_dlB_in_ga(T5, T7))
preorder_dlB_in_ga(nil, []) → preorder_dlB_out_ga(nil, [])
preorder_dlB_in_ga(tree(T21, T22, T23), .(T22, T25)) → U2_ga(T21, T22, T23, T25, pC_in_gaag(T21, T25, X26, T23))
pC_in_gaag(T21, T25, T28, T23) → U4_gaag(T21, T25, T28, T23, preorder_dlD_in_gaa(T21, T25, T28))
preorder_dlD_in_gaa(nil, T35, T35) → preorder_dlD_out_gaa(nil, T35, T35)
preorder_dlD_in_gaa(tree(T44, T45, T46), .(T45, T48), X55) → U3_gaa(T44, T45, T46, T48, X55, pE_in_gaaga(T44, T48, X54, T46, X55))
pE_in_gaaga(T44, T48, T51, T46, X55) → U6_gaaga(T44, T48, T51, T46, X55, preorder_dlD_in_gaa(T44, T48, T51))
U6_gaaga(T44, T48, T51, T46, X55, preorder_dlD_out_gaa(T44, T48, T51)) → U7_gaaga(T44, T48, T51, T46, X55, preorder_dlD_in_gaa(T46, T51, X55))
U7_gaaga(T44, T48, T51, T46, X55, preorder_dlD_out_gaa(T46, T51, X55)) → pE_out_gaaga(T44, T48, T51, T46, X55)
U3_gaa(T44, T45, T46, T48, X55, pE_out_gaaga(T44, T48, X54, T46, X55)) → preorder_dlD_out_gaa(tree(T44, T45, T46), .(T45, T48), X55)
U4_gaag(T21, T25, T28, T23, preorder_dlD_out_gaa(T21, T25, T28)) → U5_gaag(T21, T25, T28, T23, preorder_dlB_in_ga(T23, T28))
U5_gaag(T21, T25, T28, T23, preorder_dlB_out_ga(T23, T28)) → pC_out_gaag(T21, T25, T28, T23)
U2_ga(T21, T22, T23, T25, pC_out_gaag(T21, T25, X26, T23)) → preorder_dlB_out_ga(tree(T21, T22, T23), .(T22, T25))
U1_ga(T5, T7, preorder_dlB_out_ga(T5, T7)) → preorderA_out_ga(T5, T7)

The argument filtering Pi contains the following mapping:
preorderA_in_ga(x1, x2)  =  preorderA_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
preorder_dlB_in_ga(x1, x2)  =  preorder_dlB_in_ga(x1)
nil  =  nil
preorder_dlB_out_ga(x1, x2)  =  preorder_dlB_out_ga(x1)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
pC_in_gaag(x1, x2, x3, x4)  =  pC_in_gaag(x1, x4)
U4_gaag(x1, x2, x3, x4, x5)  =  U4_gaag(x1, x4, x5)
preorder_dlD_in_gaa(x1, x2, x3)  =  preorder_dlD_in_gaa(x1)
preorder_dlD_out_gaa(x1, x2, x3)  =  preorder_dlD_out_gaa(x1)
U3_gaa(x1, x2, x3, x4, x5, x6)  =  U3_gaa(x1, x2, x3, x6)
pE_in_gaaga(x1, x2, x3, x4, x5)  =  pE_in_gaaga(x1, x4)
U6_gaaga(x1, x2, x3, x4, x5, x6)  =  U6_gaaga(x1, x4, x6)
U7_gaaga(x1, x2, x3, x4, x5, x6)  =  U7_gaaga(x1, x4, x6)
pE_out_gaaga(x1, x2, x3, x4, x5)  =  pE_out_gaaga(x1, x4)
U5_gaag(x1, x2, x3, x4, x5)  =  U5_gaag(x1, x4, x5)
pC_out_gaag(x1, x2, x3, x4)  =  pC_out_gaag(x1, x4)
preorderA_out_ga(x1, x2)  =  preorderA_out_ga(x1)
PREORDERA_IN_GA(x1, x2)  =  PREORDERA_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
PREORDER_DLB_IN_GA(x1, x2)  =  PREORDER_DLB_IN_GA(x1)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x3, x5)
PC_IN_GAAG(x1, x2, x3, x4)  =  PC_IN_GAAG(x1, x4)
U4_GAAG(x1, x2, x3, x4, x5)  =  U4_GAAG(x1, x4, x5)
PREORDER_DLD_IN_GAA(x1, x2, x3)  =  PREORDER_DLD_IN_GAA(x1)
U3_GAA(x1, x2, x3, x4, x5, x6)  =  U3_GAA(x1, x2, x3, x6)
PE_IN_GAAGA(x1, x2, x3, x4, x5)  =  PE_IN_GAAGA(x1, x4)
U6_GAAGA(x1, x2, x3, x4, x5, x6)  =  U6_GAAGA(x1, x4, x6)
U7_GAAGA(x1, x2, x3, x4, x5, x6)  =  U7_GAAGA(x1, x4, x6)
U5_GAAG(x1, x2, x3, x4, x5)  =  U5_GAAG(x1, x4, x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 7 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

PREORDER_DLD_IN_GAA(tree(T44, T45, T46), .(T45, T48), X55) → PE_IN_GAAGA(T44, T48, X54, T46, X55)
PE_IN_GAAGA(T44, T48, T51, T46, X55) → U6_GAAGA(T44, T48, T51, T46, X55, preorder_dlD_in_gaa(T44, T48, T51))
U6_GAAGA(T44, T48, T51, T46, X55, preorder_dlD_out_gaa(T44, T48, T51)) → PREORDER_DLD_IN_GAA(T46, T51, X55)
PE_IN_GAAGA(T44, T48, T51, T46, X55) → PREORDER_DLD_IN_GAA(T44, T48, T51)

The TRS R consists of the following rules:

preorderA_in_ga(T5, T7) → U1_ga(T5, T7, preorder_dlB_in_ga(T5, T7))
preorder_dlB_in_ga(nil, []) → preorder_dlB_out_ga(nil, [])
preorder_dlB_in_ga(tree(T21, T22, T23), .(T22, T25)) → U2_ga(T21, T22, T23, T25, pC_in_gaag(T21, T25, X26, T23))
pC_in_gaag(T21, T25, T28, T23) → U4_gaag(T21, T25, T28, T23, preorder_dlD_in_gaa(T21, T25, T28))
preorder_dlD_in_gaa(nil, T35, T35) → preorder_dlD_out_gaa(nil, T35, T35)
preorder_dlD_in_gaa(tree(T44, T45, T46), .(T45, T48), X55) → U3_gaa(T44, T45, T46, T48, X55, pE_in_gaaga(T44, T48, X54, T46, X55))
pE_in_gaaga(T44, T48, T51, T46, X55) → U6_gaaga(T44, T48, T51, T46, X55, preorder_dlD_in_gaa(T44, T48, T51))
U6_gaaga(T44, T48, T51, T46, X55, preorder_dlD_out_gaa(T44, T48, T51)) → U7_gaaga(T44, T48, T51, T46, X55, preorder_dlD_in_gaa(T46, T51, X55))
U7_gaaga(T44, T48, T51, T46, X55, preorder_dlD_out_gaa(T46, T51, X55)) → pE_out_gaaga(T44, T48, T51, T46, X55)
U3_gaa(T44, T45, T46, T48, X55, pE_out_gaaga(T44, T48, X54, T46, X55)) → preorder_dlD_out_gaa(tree(T44, T45, T46), .(T45, T48), X55)
U4_gaag(T21, T25, T28, T23, preorder_dlD_out_gaa(T21, T25, T28)) → U5_gaag(T21, T25, T28, T23, preorder_dlB_in_ga(T23, T28))
U5_gaag(T21, T25, T28, T23, preorder_dlB_out_ga(T23, T28)) → pC_out_gaag(T21, T25, T28, T23)
U2_ga(T21, T22, T23, T25, pC_out_gaag(T21, T25, X26, T23)) → preorder_dlB_out_ga(tree(T21, T22, T23), .(T22, T25))
U1_ga(T5, T7, preorder_dlB_out_ga(T5, T7)) → preorderA_out_ga(T5, T7)

The argument filtering Pi contains the following mapping:
preorderA_in_ga(x1, x2)  =  preorderA_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
preorder_dlB_in_ga(x1, x2)  =  preorder_dlB_in_ga(x1)
nil  =  nil
preorder_dlB_out_ga(x1, x2)  =  preorder_dlB_out_ga(x1)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
pC_in_gaag(x1, x2, x3, x4)  =  pC_in_gaag(x1, x4)
U4_gaag(x1, x2, x3, x4, x5)  =  U4_gaag(x1, x4, x5)
preorder_dlD_in_gaa(x1, x2, x3)  =  preorder_dlD_in_gaa(x1)
preorder_dlD_out_gaa(x1, x2, x3)  =  preorder_dlD_out_gaa(x1)
U3_gaa(x1, x2, x3, x4, x5, x6)  =  U3_gaa(x1, x2, x3, x6)
pE_in_gaaga(x1, x2, x3, x4, x5)  =  pE_in_gaaga(x1, x4)
U6_gaaga(x1, x2, x3, x4, x5, x6)  =  U6_gaaga(x1, x4, x6)
U7_gaaga(x1, x2, x3, x4, x5, x6)  =  U7_gaaga(x1, x4, x6)
pE_out_gaaga(x1, x2, x3, x4, x5)  =  pE_out_gaaga(x1, x4)
U5_gaag(x1, x2, x3, x4, x5)  =  U5_gaag(x1, x4, x5)
pC_out_gaag(x1, x2, x3, x4)  =  pC_out_gaag(x1, x4)
preorderA_out_ga(x1, x2)  =  preorderA_out_ga(x1)
PREORDER_DLD_IN_GAA(x1, x2, x3)  =  PREORDER_DLD_IN_GAA(x1)
PE_IN_GAAGA(x1, x2, x3, x4, x5)  =  PE_IN_GAAGA(x1, x4)
U6_GAAGA(x1, x2, x3, x4, x5, x6)  =  U6_GAAGA(x1, x4, x6)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

PREORDER_DLD_IN_GAA(tree(T44, T45, T46), .(T45, T48), X55) → PE_IN_GAAGA(T44, T48, X54, T46, X55)
PE_IN_GAAGA(T44, T48, T51, T46, X55) → U6_GAAGA(T44, T48, T51, T46, X55, preorder_dlD_in_gaa(T44, T48, T51))
U6_GAAGA(T44, T48, T51, T46, X55, preorder_dlD_out_gaa(T44, T48, T51)) → PREORDER_DLD_IN_GAA(T46, T51, X55)
PE_IN_GAAGA(T44, T48, T51, T46, X55) → PREORDER_DLD_IN_GAA(T44, T48, T51)

The TRS R consists of the following rules:

preorder_dlD_in_gaa(nil, T35, T35) → preorder_dlD_out_gaa(nil, T35, T35)
preorder_dlD_in_gaa(tree(T44, T45, T46), .(T45, T48), X55) → U3_gaa(T44, T45, T46, T48, X55, pE_in_gaaga(T44, T48, X54, T46, X55))
U3_gaa(T44, T45, T46, T48, X55, pE_out_gaaga(T44, T48, X54, T46, X55)) → preorder_dlD_out_gaa(tree(T44, T45, T46), .(T45, T48), X55)
pE_in_gaaga(T44, T48, T51, T46, X55) → U6_gaaga(T44, T48, T51, T46, X55, preorder_dlD_in_gaa(T44, T48, T51))
U6_gaaga(T44, T48, T51, T46, X55, preorder_dlD_out_gaa(T44, T48, T51)) → U7_gaaga(T44, T48, T51, T46, X55, preorder_dlD_in_gaa(T46, T51, X55))
U7_gaaga(T44, T48, T51, T46, X55, preorder_dlD_out_gaa(T46, T51, X55)) → pE_out_gaaga(T44, T48, T51, T46, X55)

The argument filtering Pi contains the following mapping:
nil  =  nil
tree(x1, x2, x3)  =  tree(x1, x2, x3)
preorder_dlD_in_gaa(x1, x2, x3)  =  preorder_dlD_in_gaa(x1)
preorder_dlD_out_gaa(x1, x2, x3)  =  preorder_dlD_out_gaa(x1)
U3_gaa(x1, x2, x3, x4, x5, x6)  =  U3_gaa(x1, x2, x3, x6)
pE_in_gaaga(x1, x2, x3, x4, x5)  =  pE_in_gaaga(x1, x4)
U6_gaaga(x1, x2, x3, x4, x5, x6)  =  U6_gaaga(x1, x4, x6)
U7_gaaga(x1, x2, x3, x4, x5, x6)  =  U7_gaaga(x1, x4, x6)
pE_out_gaaga(x1, x2, x3, x4, x5)  =  pE_out_gaaga(x1, x4)
PREORDER_DLD_IN_GAA(x1, x2, x3)  =  PREORDER_DLD_IN_GAA(x1)
PE_IN_GAAGA(x1, x2, x3, x4, x5)  =  PE_IN_GAAGA(x1, x4)
U6_GAAGA(x1, x2, x3, x4, x5, x6)  =  U6_GAAGA(x1, x4, x6)

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

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

PREORDER_DLD_IN_GAA(tree(T44, T45, T46)) → PE_IN_GAAGA(T44, T46)
PE_IN_GAAGA(T44, T46) → U6_GAAGA(T44, T46, preorder_dlD_in_gaa(T44))
U6_GAAGA(T44, T46, preorder_dlD_out_gaa(T44)) → PREORDER_DLD_IN_GAA(T46)
PE_IN_GAAGA(T44, T46) → PREORDER_DLD_IN_GAA(T44)

The TRS R consists of the following rules:

preorder_dlD_in_gaa(nil) → preorder_dlD_out_gaa(nil)
preorder_dlD_in_gaa(tree(T44, T45, T46)) → U3_gaa(T44, T45, T46, pE_in_gaaga(T44, T46))
U3_gaa(T44, T45, T46, pE_out_gaaga(T44, T46)) → preorder_dlD_out_gaa(tree(T44, T45, T46))
pE_in_gaaga(T44, T46) → U6_gaaga(T44, T46, preorder_dlD_in_gaa(T44))
U6_gaaga(T44, T46, preorder_dlD_out_gaa(T44)) → U7_gaaga(T44, T46, preorder_dlD_in_gaa(T46))
U7_gaaga(T44, T46, preorder_dlD_out_gaa(T46)) → pE_out_gaaga(T44, T46)

The set Q consists of the following terms:

preorder_dlD_in_gaa(x0)
U3_gaa(x0, x1, x2, x3)
pE_in_gaaga(x0, x1)
U6_gaaga(x0, x1, x2)
U7_gaaga(x0, x1, x2)

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

(12) 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:

• PE_IN_GAAGA(T44, T46) → PREORDER_DLD_IN_GAA(T44)
  The graph contains the following edges 1 >= 1

• PE_IN_GAAGA(T44, T46) → U6_GAAGA(T44, T46, preorder_dlD_in_gaa(T44))
  The graph contains the following edges 1 >= 1, 2 >= 2

• U6_GAAGA(T44, T46, preorder_dlD_out_gaa(T44)) → PREORDER_DLD_IN_GAA(T46)
  The graph contains the following edges 2 >= 1

• PREORDER_DLD_IN_GAA(tree(T44, T45, T46)) → PE_IN_GAAGA(T44, T46)
  The graph contains the following edges 1 > 1, 1 > 2

(13) YES

(14) Obligation:

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

PREORDER_DLB_IN_GA(tree(T21, T22, T23), .(T22, T25)) → PC_IN_GAAG(T21, T25, X26, T23)
PC_IN_GAAG(T21, T25, T28, T23) → U4_GAAG(T21, T25, T28, T23, preorder_dlD_in_gaa(T21, T25, T28))
U4_GAAG(T21, T25, T28, T23, preorder_dlD_out_gaa(T21, T25, T28)) → PREORDER_DLB_IN_GA(T23, T28)

The TRS R consists of the following rules:

preorderA_in_ga(T5, T7) → U1_ga(T5, T7, preorder_dlB_in_ga(T5, T7))
preorder_dlB_in_ga(nil, []) → preorder_dlB_out_ga(nil, [])
preorder_dlB_in_ga(tree(T21, T22, T23), .(T22, T25)) → U2_ga(T21, T22, T23, T25, pC_in_gaag(T21, T25, X26, T23))
pC_in_gaag(T21, T25, T28, T23) → U4_gaag(T21, T25, T28, T23, preorder_dlD_in_gaa(T21, T25, T28))
preorder_dlD_in_gaa(nil, T35, T35) → preorder_dlD_out_gaa(nil, T35, T35)
preorder_dlD_in_gaa(tree(T44, T45, T46), .(T45, T48), X55) → U3_gaa(T44, T45, T46, T48, X55, pE_in_gaaga(T44, T48, X54, T46, X55))
pE_in_gaaga(T44, T48, T51, T46, X55) → U6_gaaga(T44, T48, T51, T46, X55, preorder_dlD_in_gaa(T44, T48, T51))
U6_gaaga(T44, T48, T51, T46, X55, preorder_dlD_out_gaa(T44, T48, T51)) → U7_gaaga(T44, T48, T51, T46, X55, preorder_dlD_in_gaa(T46, T51, X55))
U7_gaaga(T44, T48, T51, T46, X55, preorder_dlD_out_gaa(T46, T51, X55)) → pE_out_gaaga(T44, T48, T51, T46, X55)
U3_gaa(T44, T45, T46, T48, X55, pE_out_gaaga(T44, T48, X54, T46, X55)) → preorder_dlD_out_gaa(tree(T44, T45, T46), .(T45, T48), X55)
U4_gaag(T21, T25, T28, T23, preorder_dlD_out_gaa(T21, T25, T28)) → U5_gaag(T21, T25, T28, T23, preorder_dlB_in_ga(T23, T28))
U5_gaag(T21, T25, T28, T23, preorder_dlB_out_ga(T23, T28)) → pC_out_gaag(T21, T25, T28, T23)
U2_ga(T21, T22, T23, T25, pC_out_gaag(T21, T25, X26, T23)) → preorder_dlB_out_ga(tree(T21, T22, T23), .(T22, T25))
U1_ga(T5, T7, preorder_dlB_out_ga(T5, T7)) → preorderA_out_ga(T5, T7)

The argument filtering Pi contains the following mapping:
preorderA_in_ga(x1, x2)  =  preorderA_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
preorder_dlB_in_ga(x1, x2)  =  preorder_dlB_in_ga(x1)
nil  =  nil
preorder_dlB_out_ga(x1, x2)  =  preorder_dlB_out_ga(x1)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
pC_in_gaag(x1, x2, x3, x4)  =  pC_in_gaag(x1, x4)
U4_gaag(x1, x2, x3, x4, x5)  =  U4_gaag(x1, x4, x5)
preorder_dlD_in_gaa(x1, x2, x3)  =  preorder_dlD_in_gaa(x1)
preorder_dlD_out_gaa(x1, x2, x3)  =  preorder_dlD_out_gaa(x1)
U3_gaa(x1, x2, x3, x4, x5, x6)  =  U3_gaa(x1, x2, x3, x6)
pE_in_gaaga(x1, x2, x3, x4, x5)  =  pE_in_gaaga(x1, x4)
U6_gaaga(x1, x2, x3, x4, x5, x6)  =  U6_gaaga(x1, x4, x6)
U7_gaaga(x1, x2, x3, x4, x5, x6)  =  U7_gaaga(x1, x4, x6)
pE_out_gaaga(x1, x2, x3, x4, x5)  =  pE_out_gaaga(x1, x4)
U5_gaag(x1, x2, x3, x4, x5)  =  U5_gaag(x1, x4, x5)
pC_out_gaag(x1, x2, x3, x4)  =  pC_out_gaag(x1, x4)
preorderA_out_ga(x1, x2)  =  preorderA_out_ga(x1)
PREORDER_DLB_IN_GA(x1, x2)  =  PREORDER_DLB_IN_GA(x1)
PC_IN_GAAG(x1, x2, x3, x4)  =  PC_IN_GAAG(x1, x4)
U4_GAAG(x1, x2, x3, x4, x5)  =  U4_GAAG(x1, x4, x5)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

PREORDER_DLB_IN_GA(tree(T21, T22, T23), .(T22, T25)) → PC_IN_GAAG(T21, T25, X26, T23)
PC_IN_GAAG(T21, T25, T28, T23) → U4_GAAG(T21, T25, T28, T23, preorder_dlD_in_gaa(T21, T25, T28))
U4_GAAG(T21, T25, T28, T23, preorder_dlD_out_gaa(T21, T25, T28)) → PREORDER_DLB_IN_GA(T23, T28)

The TRS R consists of the following rules:

preorder_dlD_in_gaa(nil, T35, T35) → preorder_dlD_out_gaa(nil, T35, T35)
preorder_dlD_in_gaa(tree(T44, T45, T46), .(T45, T48), X55) → U3_gaa(T44, T45, T46, T48, X55, pE_in_gaaga(T44, T48, X54, T46, X55))
U3_gaa(T44, T45, T46, T48, X55, pE_out_gaaga(T44, T48, X54, T46, X55)) → preorder_dlD_out_gaa(tree(T44, T45, T46), .(T45, T48), X55)
pE_in_gaaga(T44, T48, T51, T46, X55) → U6_gaaga(T44, T48, T51, T46, X55, preorder_dlD_in_gaa(T44, T48, T51))
U6_gaaga(T44, T48, T51, T46, X55, preorder_dlD_out_gaa(T44, T48, T51)) → U7_gaaga(T44, T48, T51, T46, X55, preorder_dlD_in_gaa(T46, T51, X55))
U7_gaaga(T44, T48, T51, T46, X55, preorder_dlD_out_gaa(T46, T51, X55)) → pE_out_gaaga(T44, T48, T51, T46, X55)

The argument filtering Pi contains the following mapping:
nil  =  nil
tree(x1, x2, x3)  =  tree(x1, x2, x3)
preorder_dlD_in_gaa(x1, x2, x3)  =  preorder_dlD_in_gaa(x1)
preorder_dlD_out_gaa(x1, x2, x3)  =  preorder_dlD_out_gaa(x1)
U3_gaa(x1, x2, x3, x4, x5, x6)  =  U3_gaa(x1, x2, x3, x6)
pE_in_gaaga(x1, x2, x3, x4, x5)  =  pE_in_gaaga(x1, x4)
U6_gaaga(x1, x2, x3, x4, x5, x6)  =  U6_gaaga(x1, x4, x6)
U7_gaaga(x1, x2, x3, x4, x5, x6)  =  U7_gaaga(x1, x4, x6)
pE_out_gaaga(x1, x2, x3, x4, x5)  =  pE_out_gaaga(x1, x4)
PREORDER_DLB_IN_GA(x1, x2)  =  PREORDER_DLB_IN_GA(x1)
PC_IN_GAAG(x1, x2, x3, x4)  =  PC_IN_GAAG(x1, x4)
U4_GAAG(x1, x2, x3, x4, x5)  =  U4_GAAG(x1, x4, x5)

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

PREORDER_DLB_IN_GA(tree(T21, T22, T23)) → PC_IN_GAAG(T21, T23)
PC_IN_GAAG(T21, T23) → U4_GAAG(T21, T23, preorder_dlD_in_gaa(T21))
U4_GAAG(T21, T23, preorder_dlD_out_gaa(T21)) → PREORDER_DLB_IN_GA(T23)

The TRS R consists of the following rules:

preorder_dlD_in_gaa(nil) → preorder_dlD_out_gaa(nil)
preorder_dlD_in_gaa(tree(T44, T45, T46)) → U3_gaa(T44, T45, T46, pE_in_gaaga(T44, T46))
U3_gaa(T44, T45, T46, pE_out_gaaga(T44, T46)) → preorder_dlD_out_gaa(tree(T44, T45, T46))
pE_in_gaaga(T44, T46) → U6_gaaga(T44, T46, preorder_dlD_in_gaa(T44))
U6_gaaga(T44, T46, preorder_dlD_out_gaa(T44)) → U7_gaaga(T44, T46, preorder_dlD_in_gaa(T46))
U7_gaaga(T44, T46, preorder_dlD_out_gaa(T46)) → pE_out_gaaga(T44, T46)

The set Q consists of the following terms:

preorder_dlD_in_gaa(x0)
U3_gaa(x0, x1, x2, x3)
pE_in_gaaga(x0, x1)
U6_gaaga(x0, x1, x2)
U7_gaaga(x0, x1, x2)

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

(19) 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:

• PC_IN_GAAG(T21, T23) → U4_GAAG(T21, T23, preorder_dlD_in_gaa(T21))
  The graph contains the following edges 1 >= 1, 2 >= 2

• U4_GAAG(T21, T23, preorder_dlD_out_gaa(T21)) → PREORDER_DLB_IN_GA(T23)
  The graph contains the following edges 2 >= 1

• PREORDER_DLB_IN_GA(tree(T21, T22, T23)) → PC_IN_GAAG(T21, T23)
  The graph contains the following edges 1 > 1, 1 > 2

(20) YES