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

(0) Obligation:

Clauses:

sameleaves(leaf(L), leaf(L)).
sameleaves(tree(T1, T2), tree(S1, S2)) :- ','(getleave(T1, T2, L, T), ','(getleave(S1, S2, L, S), sameleaves(T, S))).
getleave(leaf(A), C, A, C).
getleave(tree(A, B), C, L, O) :- getleave(A, tree(B, C), L, O).

Query: sameleaves(g,g)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: sameleaves(T3, T4)\n\nT3 is ground\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: sameleaves(T3, T4), SCOPE: 1, CLAUSE: 0 | sameleaves(T3, T4), SCOPE: 1, CLAUSE: 1\n\nT3 is ground\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | sameleaves(leaf(T6), leaf(T6)), SCOPE: 1, CLAUSE: 1\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: sameleaves(T3, T4), SCOPE: 1, CLAUSE: 1\n\nT3 is ground\nT4 is ground\nX2 is free\nsameleaves(T3, T4) !=? sameleaves(leaf(X2), leaf(X2))", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: sameleaves(leaf(T6), leaf(T6)), SCOPE: 1, CLAUSE: 1\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="B: ','(getleave(T11, T12, X15, X16), ','(getleave(T13, T14, X15, X17), sameleaves(X16, X17)))\n\nT11 is ground\nT12 is ground\nT13 is ground\nT14 is ground\nX15 is free\nX16 is free\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(getleave(T11, T12, X15, X16), ','(getleave(T13, T14, X15, X17), sameleaves(X16, X17))), SCOPE: 2, CLAUSE: 2 | ','(getleave(T11, T12, X15, X16), ','(getleave(T13, T14, X15, X17), sameleaves(X16, X17))), SCOPE: 2, CLAUSE: 3\n\nT11 is ground\nT12 is ground\nT13 is ground\nT14 is ground\nX15 is free\nX16 is free\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(getleave(T11, T12, X15, X16), ','(getleave(T13, T14, X15, X17), sameleaves(X16, X17))), SCOPE: 2, CLAUSE: 2\n\nT11 is ground\nT12 is ground\nT13 is ground\nT14 is ground\nX15 is free\nX16 is free\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(getleave(T11, T12, X15, X16), ','(getleave(T13, T14, X15, X17), sameleaves(X16, X17))), SCOPE: 2, CLAUSE: 3\n\nT11 is ground\nT12 is ground\nT13 is ground\nT14 is ground\nX15 is free\nX16 is free\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(getleave(T13, T14, T23, X17), sameleaves(T24, X17))\n\nT13 is ground\nT14 is ground\nT23 is ground\nT24 is ground\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="C: getleave(T13, T14, T23, X17)\n\nT13 is ground\nT14 is ground\nT23 is ground\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: sameleaves(T24, T29)\n\nT24 is ground\nT29 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: getleave(T13, T14, T23, X17), SCOPE: 3, CLAUSE: 2 | getleave(T13, T14, T23, X17), SCOPE: 3, CLAUSE: 3\n\nT13 is ground\nT14 is ground\nT23 is ground\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: getleave(T13, T14, T23, X17), SCOPE: 3, CLAUSE: 2\n\nT13 is ground\nT14 is ground\nT23 is ground\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: getleave(T13, T14, T23, X17), SCOPE: 3, CLAUSE: 3\n\nT13 is ground\nT14 is ground\nT23 is ground\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: getleave(T52, tree(T53, T54), T55, X63)\n\nT52 is ground\nT53 is ground\nT54 is ground\nT55 is ground\nX63 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(getleave(T70, tree(T71, T72), X89, X90), ','(getleave(T13, T14, X89, X17), sameleaves(X90, X17)))\n\nT13 is ground\nT14 is ground\nT70 is ground\nT71 is ground\nT72 is ground\nX17 is free\nX89 is free\nX90 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 26 [arrowhead = none ];
26 -> 2;

27 [label="EVAL with clause\nsameleaves(leaf(X2), leaf(X2)).\nand substitutionX2 -> T6,\nT3 -> leaf(T6),\nT4 -> leaf(T6)", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 27 [arrowhead = none ];
27 -> 3;

28 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 28 [arrowhead = none ];
28 -> 4;

29 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 29 [arrowhead = none ];
29 -> 5;

30 [label="EVAL with clause\nsameleaves(tree(X11, X12), tree(X13, X14)) :- ','(getleave(X11, X12, X15, X16), ','(getleave(X13, X14, X15, X17), sameleaves(X16, X17))).\nand substitutionX11 -> T11,\nX12 -> T12,\nT3 -> tree(T11, T12),\nX13 -> T13,\nX14 -> T14,\nT4 -> tree(T13, T14)", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 30 [arrowhead = none ];
30 -> 7;

31 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 31 [arrowhead = none ];
31 -> 8;

32 [label="BACKTRACK\nfor clause: sameleaves(tree(T1, T2), tree(S1, S2)) :- ','(getleave(T1, T2, L, T), ','(getleave(S1, S2, L, S), sameleaves(T, S)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 32 [arrowhead = none ];
32 -> 6;

33 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 33 [arrowhead = none ];
33 -> 9;

34 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 34 [arrowhead = none ];
34 -> 10;

35 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 35 [arrowhead = none ];
35 -> 11;

36 [label="EVAL with clause\ngetleave(leaf(X26), X27, X26, X27).\nand substitutionX26 -> T23,\nT11 -> leaf(T23),\nT12 -> T24,\nX27 -> T24,\nX15 -> T23,\nX16 -> T24", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 36 [arrowhead = none ];
36 -> 12;

37 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 37 [arrowhead = none ];
37 -> 13;

38 [label="EVAL with clause\ngetleave(tree(X84, X85), X86, X87, X88) :- getleave(X84, tree(X85, X86), X87, X88).\nand substitutionX84 -> T70,\nX85 -> T71,\nT11 -> tree(T70, T71),\nT12 -> T72,\nX86 -> T72,\nX15 -> X89,\nX87 -> X89,\nX16 -> X90,\nX88 -> X90", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 38 [arrowhead = none ];
38 -> 24;

39 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 39 [arrowhead = none ];
39 -> 25;

40 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
12 -> 40 [arrowhead = none ];
40 -> 14;

41 [label="SPLIT 2\nnew knowledge:\nT13 is ground\nT14 is ground\nT23 is ground\nT29 is ground\nreplacements:X17 -> T29", fontsize=14, style = filled, fillcolor = violet];
12 -> 41 [arrowhead = none ];
41 -> 15;

42 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 42 [arrowhead = none ];
42 -> 16;

43 [label="INSTANCE with matching:\nT3 -> T24\nT4 -> T29", fontsize=14, style = filled, fillcolor = lightgrey];
15 -> 43 [arrowhead = none , style=dashed];
43 -> 1[style=dashed];

44 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 44 [arrowhead = none ];
44 -> 17;

45 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 45 [arrowhead = none ];
45 -> 18;

46 [label="EVAL with clause\ngetleave(leaf(X44), X45, X44, X45).\nand substitutionX44 -> T42,\nT13 -> leaf(T42),\nT14 -> T43,\nX45 -> T43,\nT23 -> T42,\nX17 -> T43", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 46 [arrowhead = none ];
46 -> 19;

47 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 47 [arrowhead = none ];
47 -> 20;

48 [label="EVAL with clause\ngetleave(tree(X58, X59), X60, X61, X62) :- getleave(X58, tree(X59, X60), X61, X62).\nand substitutionX58 -> T52,\nX59 -> T53,\nT13 -> tree(T52, T53),\nT14 -> T54,\nX60 -> T54,\nT23 -> T55,\nX61 -> T55,\nX17 -> X63,\nX62 -> X63", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 48 [arrowhead = none ];
48 -> 22;

49 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 49 [arrowhead = none ];
49 -> 23;

50 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 50 [arrowhead = none ];
50 -> 21;

51 [label="INSTANCE with matching:\nT13 -> T52\nT14 -> tree(T53, T54)\nT23 -> T55\nX17 -> X63", fontsize=14, style = filled, fillcolor = lightgrey];
22 -> 51 [arrowhead = none , style=dashed];
51 -> 14[style=dashed];

52 [label="INSTANCE with matching:\nT11 -> T70\nT12 -> tree(T71, T72)\nX15 -> X89\nX16 -> X90", fontsize=14, style = filled, fillcolor = lightgrey];
24 -> 52 [arrowhead = none , style=dashed];
52 -> 7[style=dashed];

}

(2) Obligation:

Triples:

getleaveC(tree(X1, X2), X3, X4, X5) :- getleaveC(X1, tree(X2, X3), X4, X5).
pB(leaf(X1), X2, X1, X2, X3, X4, X5) :- getleaveC(X3, X4, X1, X5).
pB(leaf(X1), X2, X1, X2, X3, X4, X5) :- ','(getleavecC(X3, X4, X1, X5), sameleavesA(X2, X5)).
pB(tree(X1, X2), X3, X4, X5, X6, X7, X8) :- pB(X1, tree(X2, X3), X4, X5, X6, X7, X8).
sameleavesA(tree(X1, X2), tree(X3, X4)) :- pB(X1, X2, X5, X6, X3, X4, X7).

Clauses:

sameleavescA(leaf(X1), leaf(X1)).
sameleavescA(tree(X1, X2), tree(X3, X4)) :- qcB(X1, X2, X5, X6, X3, X4, X7).
getleavecC(leaf(X1), X2, X1, X2).
getleavecC(tree(X1, X2), X3, X4, X5) :- getleavecC(X1, tree(X2, X3), X4, X5).
qcB(leaf(X1), X2, X1, X2, X3, X4, X5) :- ','(getleavecC(X3, X4, X1, X5), sameleavescA(X2, X5)).
qcB(tree(X1, X2), X3, X4, X5, X6, X7, X8) :- qcB(X1, tree(X2, X3), X4, X5, X6, X7, X8).

Afs:

sameleavesA(x1, x2)  =  sameleavesA(x1, x2)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
sameleavesA_in: (b,b)
pB_in: (b,b,f,f,b,b,f)
getleaveC_in: (b,b,b,f)
getleavecC_in: (b,b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

SAMELEAVESA_IN_GG(tree(X1, X2), tree(X3, X4)) → U6_GG(X1, X2, X3, X4, pB_in_ggaagga(X1, X2, X5, X6, X3, X4, X7))
SAMELEAVESA_IN_GG(tree(X1, X2), tree(X3, X4)) → PB_IN_GGAAGGA(X1, X2, X5, X6, X3, X4, X7)
PB_IN_GGAAGGA(leaf(X1), X2, X1, X2, X3, X4, X5) → U2_GGAAGGA(X1, X2, X3, X4, X5, getleaveC_in_ggga(X3, X4, X1, X5))
PB_IN_GGAAGGA(leaf(X1), X2, X1, X2, X3, X4, X5) → GETLEAVEC_IN_GGGA(X3, X4, X1, X5)
GETLEAVEC_IN_GGGA(tree(X1, X2), X3, X4, X5) → U1_GGGA(X1, X2, X3, X4, X5, getleaveC_in_ggga(X1, tree(X2, X3), X4, X5))
GETLEAVEC_IN_GGGA(tree(X1, X2), X3, X4, X5) → GETLEAVEC_IN_GGGA(X1, tree(X2, X3), X4, X5)
PB_IN_GGAAGGA(leaf(X1), X2, X1, X2, X3, X4, X5) → U3_GGAAGGA(X1, X2, X3, X4, X5, getleavecC_in_ggga(X3, X4, X1, X5))
U3_GGAAGGA(X1, X2, X3, X4, X5, getleavecC_out_ggga(X3, X4, X1, X5)) → U4_GGAAGGA(X1, X2, X3, X4, X5, sameleavesA_in_gg(X2, X5))
U3_GGAAGGA(X1, X2, X3, X4, X5, getleavecC_out_ggga(X3, X4, X1, X5)) → SAMELEAVESA_IN_GG(X2, X5)
PB_IN_GGAAGGA(tree(X1, X2), X3, X4, X5, X6, X7, X8) → U5_GGAAGGA(X1, X2, X3, X4, X5, X6, X7, X8, pB_in_ggaagga(X1, tree(X2, X3), X4, X5, X6, X7, X8))
PB_IN_GGAAGGA(tree(X1, X2), X3, X4, X5, X6, X7, X8) → PB_IN_GGAAGGA(X1, tree(X2, X3), X4, X5, X6, X7, X8)

The TRS R consists of the following rules:

getleavecC_in_ggga(leaf(X1), X2, X1, X2) → getleavecC_out_ggga(leaf(X1), X2, X1, X2)
getleavecC_in_ggga(tree(X1, X2), X3, X4, X5) → U9_ggga(X1, X2, X3, X4, X5, getleavecC_in_ggga(X1, tree(X2, X3), X4, X5))
U9_ggga(X1, X2, X3, X4, X5, getleavecC_out_ggga(X1, tree(X2, X3), X4, X5)) → getleavecC_out_ggga(tree(X1, X2), X3, X4, X5)

The argument filtering Pi contains the following mapping:
sameleavesA_in_gg(x1, x2)  =  sameleavesA_in_gg(x1, x2)
tree(x1, x2)  =  tree(x1, x2)
pB_in_ggaagga(x1, x2, x3, x4, x5, x6, x7)  =  pB_in_ggaagga(x1, x2, x5, x6)
leaf(x1)  =  leaf(x1)
getleaveC_in_ggga(x1, x2, x3, x4)  =  getleaveC_in_ggga(x1, x2, x3)
getleavecC_in_ggga(x1, x2, x3, x4)  =  getleavecC_in_ggga(x1, x2, x3)
getleavecC_out_ggga(x1, x2, x3, x4)  =  getleavecC_out_ggga(x1, x2, x3, x4)
U9_ggga(x1, x2, x3, x4, x5, x6)  =  U9_ggga(x1, x2, x3, x4, x6)
SAMELEAVESA_IN_GG(x1, x2)  =  SAMELEAVESA_IN_GG(x1, x2)
U6_GG(x1, x2, x3, x4, x5)  =  U6_GG(x1, x2, x3, x4, x5)
PB_IN_GGAAGGA(x1, x2, x3, x4, x5, x6, x7)  =  PB_IN_GGAAGGA(x1, x2, x5, x6)
U2_GGAAGGA(x1, x2, x3, x4, x5, x6)  =  U2_GGAAGGA(x1, x2, x3, x4, x6)
GETLEAVEC_IN_GGGA(x1, x2, x3, x4)  =  GETLEAVEC_IN_GGGA(x1, x2, x3)
U1_GGGA(x1, x2, x3, x4, x5, x6)  =  U1_GGGA(x1, x2, x3, x4, x6)
U3_GGAAGGA(x1, x2, x3, x4, x5, x6)  =  U3_GGAAGGA(x1, x2, x3, x4, x6)
U4_GGAAGGA(x1, x2, x3, x4, x5, x6)  =  U4_GGAAGGA(x1, x2, x3, x4, x5, x6)
U5_GGAAGGA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U5_GGAAGGA(x1, x2, x3, x6, x7, x9)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

SAMELEAVESA_IN_GG(tree(X1, X2), tree(X3, X4)) → U6_GG(X1, X2, X3, X4, pB_in_ggaagga(X1, X2, X5, X6, X3, X4, X7))
SAMELEAVESA_IN_GG(tree(X1, X2), tree(X3, X4)) → PB_IN_GGAAGGA(X1, X2, X5, X6, X3, X4, X7)
PB_IN_GGAAGGA(leaf(X1), X2, X1, X2, X3, X4, X5) → U2_GGAAGGA(X1, X2, X3, X4, X5, getleaveC_in_ggga(X3, X4, X1, X5))
PB_IN_GGAAGGA(leaf(X1), X2, X1, X2, X3, X4, X5) → GETLEAVEC_IN_GGGA(X3, X4, X1, X5)
GETLEAVEC_IN_GGGA(tree(X1, X2), X3, X4, X5) → U1_GGGA(X1, X2, X3, X4, X5, getleaveC_in_ggga(X1, tree(X2, X3), X4, X5))
GETLEAVEC_IN_GGGA(tree(X1, X2), X3, X4, X5) → GETLEAVEC_IN_GGGA(X1, tree(X2, X3), X4, X5)
PB_IN_GGAAGGA(leaf(X1), X2, X1, X2, X3, X4, X5) → U3_GGAAGGA(X1, X2, X3, X4, X5, getleavecC_in_ggga(X3, X4, X1, X5))
U3_GGAAGGA(X1, X2, X3, X4, X5, getleavecC_out_ggga(X3, X4, X1, X5)) → U4_GGAAGGA(X1, X2, X3, X4, X5, sameleavesA_in_gg(X2, X5))
U3_GGAAGGA(X1, X2, X3, X4, X5, getleavecC_out_ggga(X3, X4, X1, X5)) → SAMELEAVESA_IN_GG(X2, X5)
PB_IN_GGAAGGA(tree(X1, X2), X3, X4, X5, X6, X7, X8) → U5_GGAAGGA(X1, X2, X3, X4, X5, X6, X7, X8, pB_in_ggaagga(X1, tree(X2, X3), X4, X5, X6, X7, X8))
PB_IN_GGAAGGA(tree(X1, X2), X3, X4, X5, X6, X7, X8) → PB_IN_GGAAGGA(X1, tree(X2, X3), X4, X5, X6, X7, X8)

The TRS R consists of the following rules:

getleavecC_in_ggga(leaf(X1), X2, X1, X2) → getleavecC_out_ggga(leaf(X1), X2, X1, X2)
getleavecC_in_ggga(tree(X1, X2), X3, X4, X5) → U9_ggga(X1, X2, X3, X4, X5, getleavecC_in_ggga(X1, tree(X2, X3), X4, X5))
U9_ggga(X1, X2, X3, X4, X5, getleavecC_out_ggga(X1, tree(X2, X3), X4, X5)) → getleavecC_out_ggga(tree(X1, X2), X3, X4, X5)

The argument filtering Pi contains the following mapping:
sameleavesA_in_gg(x1, x2)  =  sameleavesA_in_gg(x1, x2)
tree(x1, x2)  =  tree(x1, x2)
pB_in_ggaagga(x1, x2, x3, x4, x5, x6, x7)  =  pB_in_ggaagga(x1, x2, x5, x6)
leaf(x1)  =  leaf(x1)
getleaveC_in_ggga(x1, x2, x3, x4)  =  getleaveC_in_ggga(x1, x2, x3)
getleavecC_in_ggga(x1, x2, x3, x4)  =  getleavecC_in_ggga(x1, x2, x3)
getleavecC_out_ggga(x1, x2, x3, x4)  =  getleavecC_out_ggga(x1, x2, x3, x4)
U9_ggga(x1, x2, x3, x4, x5, x6)  =  U9_ggga(x1, x2, x3, x4, x6)
SAMELEAVESA_IN_GG(x1, x2)  =  SAMELEAVESA_IN_GG(x1, x2)
U6_GG(x1, x2, x3, x4, x5)  =  U6_GG(x1, x2, x3, x4, x5)
PB_IN_GGAAGGA(x1, x2, x3, x4, x5, x6, x7)  =  PB_IN_GGAAGGA(x1, x2, x5, x6)
U2_GGAAGGA(x1, x2, x3, x4, x5, x6)  =  U2_GGAAGGA(x1, x2, x3, x4, x6)
GETLEAVEC_IN_GGGA(x1, x2, x3, x4)  =  GETLEAVEC_IN_GGGA(x1, x2, x3)
U1_GGGA(x1, x2, x3, x4, x5, x6)  =  U1_GGGA(x1, x2, x3, x4, x6)
U3_GGAAGGA(x1, x2, x3, x4, x5, x6)  =  U3_GGAAGGA(x1, x2, x3, x4, x6)
U4_GGAAGGA(x1, x2, x3, x4, x5, x6)  =  U4_GGAAGGA(x1, x2, x3, x4, x5, x6)
U5_GGAAGGA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U5_GGAAGGA(x1, x2, x3, x6, x7, x9)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

GETLEAVEC_IN_GGGA(tree(X1, X2), X3, X4, X5) → GETLEAVEC_IN_GGGA(X1, tree(X2, X3), X4, X5)

The TRS R consists of the following rules:

getleavecC_in_ggga(leaf(X1), X2, X1, X2) → getleavecC_out_ggga(leaf(X1), X2, X1, X2)
getleavecC_in_ggga(tree(X1, X2), X3, X4, X5) → U9_ggga(X1, X2, X3, X4, X5, getleavecC_in_ggga(X1, tree(X2, X3), X4, X5))
U9_ggga(X1, X2, X3, X4, X5, getleavecC_out_ggga(X1, tree(X2, X3), X4, X5)) → getleavecC_out_ggga(tree(X1, X2), X3, X4, X5)

The argument filtering Pi contains the following mapping:
tree(x1, x2)  =  tree(x1, x2)
leaf(x1)  =  leaf(x1)
getleavecC_in_ggga(x1, x2, x3, x4)  =  getleavecC_in_ggga(x1, x2, x3)
getleavecC_out_ggga(x1, x2, x3, x4)  =  getleavecC_out_ggga(x1, x2, x3, x4)
U9_ggga(x1, x2, x3, x4, x5, x6)  =  U9_ggga(x1, x2, x3, x4, x6)
GETLEAVEC_IN_GGGA(x1, x2, x3, x4)  =  GETLEAVEC_IN_GGGA(x1, x2, x3)

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:

GETLEAVEC_IN_GGGA(tree(X1, X2), X3, X4, X5) → GETLEAVEC_IN_GGGA(X1, tree(X2, X3), X4, X5)

R is empty.
The argument filtering Pi contains the following mapping:
tree(x1, x2)  =  tree(x1, x2)
GETLEAVEC_IN_GGGA(x1, x2, x3, x4)  =  GETLEAVEC_IN_GGGA(x1, x2, x3)

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:

GETLEAVEC_IN_GGGA(tree(X1, X2), X3, X4) → GETLEAVEC_IN_GGGA(X1, tree(X2, X3), X4)

R is empty.
Q is empty.
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:

• GETLEAVEC_IN_GGGA(tree(X1, X2), X3, X4) → GETLEAVEC_IN_GGGA(X1, tree(X2, X3), X4)
  The graph contains the following edges 1 > 1, 3 >= 3

(13) YES

(14) Obligation:

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

SAMELEAVESA_IN_GG(tree(X1, X2), tree(X3, X4)) → PB_IN_GGAAGGA(X1, X2, X5, X6, X3, X4, X7)
PB_IN_GGAAGGA(leaf(X1), X2, X1, X2, X3, X4, X5) → U3_GGAAGGA(X1, X2, X3, X4, X5, getleavecC_in_ggga(X3, X4, X1, X5))
U3_GGAAGGA(X1, X2, X3, X4, X5, getleavecC_out_ggga(X3, X4, X1, X5)) → SAMELEAVESA_IN_GG(X2, X5)
PB_IN_GGAAGGA(tree(X1, X2), X3, X4, X5, X6, X7, X8) → PB_IN_GGAAGGA(X1, tree(X2, X3), X4, X5, X6, X7, X8)

The TRS R consists of the following rules:

getleavecC_in_ggga(leaf(X1), X2, X1, X2) → getleavecC_out_ggga(leaf(X1), X2, X1, X2)
getleavecC_in_ggga(tree(X1, X2), X3, X4, X5) → U9_ggga(X1, X2, X3, X4, X5, getleavecC_in_ggga(X1, tree(X2, X3), X4, X5))
U9_ggga(X1, X2, X3, X4, X5, getleavecC_out_ggga(X1, tree(X2, X3), X4, X5)) → getleavecC_out_ggga(tree(X1, X2), X3, X4, X5)

The argument filtering Pi contains the following mapping:
tree(x1, x2)  =  tree(x1, x2)
leaf(x1)  =  leaf(x1)
getleavecC_in_ggga(x1, x2, x3, x4)  =  getleavecC_in_ggga(x1, x2, x3)
getleavecC_out_ggga(x1, x2, x3, x4)  =  getleavecC_out_ggga(x1, x2, x3, x4)
U9_ggga(x1, x2, x3, x4, x5, x6)  =  U9_ggga(x1, x2, x3, x4, x6)
SAMELEAVESA_IN_GG(x1, x2)  =  SAMELEAVESA_IN_GG(x1, x2)
PB_IN_GGAAGGA(x1, x2, x3, x4, x5, x6, x7)  =  PB_IN_GGAAGGA(x1, x2, x5, x6)
U3_GGAAGGA(x1, x2, x3, x4, x5, x6)  =  U3_GGAAGGA(x1, x2, x3, x4, x6)

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

(15) PiDPToQDPProof (SOUND transformation)

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

(16) Obligation:

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

SAMELEAVESA_IN_GG(tree(X1, X2), tree(X3, X4)) → PB_IN_GGAAGGA(X1, X2, X3, X4)
PB_IN_GGAAGGA(leaf(X1), X2, X3, X4) → U3_GGAAGGA(X1, X2, X3, X4, getleavecC_in_ggga(X3, X4, X1))
U3_GGAAGGA(X1, X2, X3, X4, getleavecC_out_ggga(X3, X4, X1, X5)) → SAMELEAVESA_IN_GG(X2, X5)
PB_IN_GGAAGGA(tree(X1, X2), X3, X6, X7) → PB_IN_GGAAGGA(X1, tree(X2, X3), X6, X7)

The TRS R consists of the following rules:

getleavecC_in_ggga(leaf(X1), X2, X1) → getleavecC_out_ggga(leaf(X1), X2, X1, X2)
getleavecC_in_ggga(tree(X1, X2), X3, X4) → U9_ggga(X1, X2, X3, X4, getleavecC_in_ggga(X1, tree(X2, X3), X4))
U9_ggga(X1, X2, X3, X4, getleavecC_out_ggga(X1, tree(X2, X3), X4, X5)) → getleavecC_out_ggga(tree(X1, X2), X3, X4, X5)

The set Q consists of the following terms:

getleavecC_in_ggga(x0, x1, x2)
U9_ggga(x0, x1, x2, x3, x4)

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

(17) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

SAMELEAVESA_IN_GG(tree(X1, X2), tree(X3, X4)) → PB_IN_GGAAGGA(X1, X2, X3, X4)

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

POL(PB_IN_GGAAGGA(x₁, x₂, x₃, x₄)) = x₁ + x₂   
POL(SAMELEAVESA_IN_GG(x₁, x₂)) = x₁   
POL(U3_GGAAGGA(x₁, x₂, x₃, x₄, x₅)) = x₂   
POL(U9_ggga(x₁, x₂, x₃, x₄, x₅)) = 0   
POL(getleavecC_in_ggga(x₁, x₂, x₃)) = 0   
POL(getleavecC_out_ggga(x₁, x₂, x₃, x₄)) = 0   
POL(leaf(x₁)) = 0   
POL(tree(x₁, x₂)) = 1 + x₁ + x₂   

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

(18) Obligation:

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

PB_IN_GGAAGGA(leaf(X1), X2, X3, X4) → U3_GGAAGGA(X1, X2, X3, X4, getleavecC_in_ggga(X3, X4, X1))
U3_GGAAGGA(X1, X2, X3, X4, getleavecC_out_ggga(X3, X4, X1, X5)) → SAMELEAVESA_IN_GG(X2, X5)
PB_IN_GGAAGGA(tree(X1, X2), X3, X6, X7) → PB_IN_GGAAGGA(X1, tree(X2, X3), X6, X7)

The TRS R consists of the following rules:

getleavecC_in_ggga(leaf(X1), X2, X1) → getleavecC_out_ggga(leaf(X1), X2, X1, X2)
getleavecC_in_ggga(tree(X1, X2), X3, X4) → U9_ggga(X1, X2, X3, X4, getleavecC_in_ggga(X1, tree(X2, X3), X4))
U9_ggga(X1, X2, X3, X4, getleavecC_out_ggga(X1, tree(X2, X3), X4, X5)) → getleavecC_out_ggga(tree(X1, X2), X3, X4, X5)

The set Q consists of the following terms:

getleavecC_in_ggga(x0, x1, x2)
U9_ggga(x0, x1, x2, x3, x4)

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

(19) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(20) Obligation:

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

PB_IN_GGAAGGA(tree(X1, X2), X3, X6, X7) → PB_IN_GGAAGGA(X1, tree(X2, X3), X6, X7)

The TRS R consists of the following rules:

getleavecC_in_ggga(leaf(X1), X2, X1) → getleavecC_out_ggga(leaf(X1), X2, X1, X2)
getleavecC_in_ggga(tree(X1, X2), X3, X4) → U9_ggga(X1, X2, X3, X4, getleavecC_in_ggga(X1, tree(X2, X3), X4))
U9_ggga(X1, X2, X3, X4, getleavecC_out_ggga(X1, tree(X2, X3), X4, X5)) → getleavecC_out_ggga(tree(X1, X2), X3, X4, X5)

The set Q consists of the following terms:

getleavecC_in_ggga(x0, x1, x2)
U9_ggga(x0, x1, x2, x3, x4)

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

(21) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(22) Obligation:

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

PB_IN_GGAAGGA(tree(X1, X2), X3, X6, X7) → PB_IN_GGAAGGA(X1, tree(X2, X3), X6, X7)

R is empty.
The set Q consists of the following terms:

getleavecC_in_ggga(x0, x1, x2)
U9_ggga(x0, x1, x2, x3, x4)

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

(23) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

getleavecC_in_ggga(x0, x1, x2)
U9_ggga(x0, x1, x2, x3, x4)

(24) Obligation:

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

PB_IN_GGAAGGA(tree(X1, X2), X3, X6, X7) → PB_IN_GGAAGGA(X1, tree(X2, X3), X6, X7)

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

(25) 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_GGAAGGA(tree(X1, X2), X3, X6, X7) → PB_IN_GGAAGGA(X1, tree(X2, X3), X6, X7)
  The graph contains the following edges 1 > 1, 3 >= 3, 4 >= 4

(26) YES