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

(0) Obligation:

Clauses:

append(nil, Y, Y).
append(cons(U, V), Y, cons(U, Z)) :- append(V, Y, Z).
lessleaves(nil, cons(W, Z)).
lessleaves(cons(U, V), cons(W, Z)) :- ','(append(U, V, U1), ','(append(W, Z, W1), lessleaves(U1, W1))).

Query: lessleaves(g,g)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: lessleaves(T1, T2)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: lessleaves(T1, T2), SCOPE: 1, CLAUSE: 2 | lessleaves(T1, T2), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | lessleaves(nil, cons(T5, T6)), SCOPE: 1, CLAUSE: 3\n\nT5 is ground\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: lessleaves(T1, T2), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nT2 is ground\nX3 is free\nX4 is free\nlessleaves(T1, T2) !=? lessleaves(nil, cons(X3, X4))", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: lessleaves(nil, cons(T5, T6)), SCOPE: 1, CLAUSE: 3\n\nT5 is ground\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(append(T11, T12, X17), ','(append(T13, T14, X18), lessleaves(X17, X18)))\n\nT11 is ground\nT12 is ground\nT13 is ground\nT14 is ground\nX17 is free\nX18 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(append(T11, T12, X17), ','(append(T13, T14, X18), lessleaves(X17, X18))), SCOPE: 2, CLAUSE: 0 | ','(append(T11, T12, X17), ','(append(T13, T14, X18), lessleaves(X17, X18))), SCOPE: 2, CLAUSE: 1\n\nT11 is ground\nT12 is ground\nT13 is ground\nT14 is ground\nX17 is free\nX18 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(append(T11, T12, X17), ','(append(T13, T14, X18), lessleaves(X17, X18))), SCOPE: 2, CLAUSE: 0\n\nT11 is ground\nT12 is ground\nT13 is ground\nT14 is ground\nX17 is free\nX18 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(append(T11, T12, X17), ','(append(T13, T14, X18), lessleaves(X17, X18))), SCOPE: 2, CLAUSE: 1\n\nT11 is ground\nT12 is ground\nT13 is ground\nT14 is ground\nX17 is free\nX18 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="B: ','(append(T13, T14, X18), lessleaves(T19, X18))\n\nT13 is ground\nT14 is ground\nT19 is ground\nX18 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="C: append(T13, T14, X18)\n\nT13 is ground\nT14 is ground\nX18 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: lessleaves(T19, T22)\n\nT19 is ground\nT22 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: append(T13, T14, X18), SCOPE: 3, CLAUSE: 0 | append(T13, T14, X18), SCOPE: 3, CLAUSE: 1\n\nT13 is ground\nT14 is ground\nX18 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: append(T13, T14, X18), SCOPE: 3, CLAUSE: 0\n\nT13 is ground\nT14 is ground\nX18 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: append(T13, T14, X18), SCOPE: 3, CLAUSE: 1\n\nT13 is ground\nT14 is ground\nX18 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: append(T37, T38, X47)\n\nT37 is ground\nT38 is ground\nX47 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(append(T52, T53, X68), ','(append(T13, T14, X18), lessleaves(cons(T51, X68), X18)))\n\nT13 is ground\nT14 is ground\nT51 is ground\nT52 is ground\nT53 is ground\nX18 is free\nX68 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: append(T52, T53, X68)\n\nT52 is ground\nT53 is ground\nX68 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(append(T13, T14, X18), lessleaves(cons(T51, T56), X18))\n\nT13 is ground\nT14 is ground\nT51 is ground\nT56 is ground\nX18 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 28 [arrowhead = none ];
28 -> 2;

29 [label="EVAL with clause\nlessleaves(nil, cons(X3, X4)).\nand substitutionT1 -> nil,\nX3 -> T5,\nX4 -> T6,\nT2 -> cons(T5, T6)", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 29 [arrowhead = none ];
29 -> 3;

30 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 30 [arrowhead = none ];
30 -> 4;

31 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 31 [arrowhead = none ];
31 -> 5;

32 [label="EVAL with clause\nlessleaves(cons(X13, X14), cons(X15, X16)) :- ','(append(X13, X14, X17), ','(append(X15, X16, X18), lessleaves(X17, X18))).\nand substitutionX13 -> T11,\nX14 -> T12,\nT1 -> cons(T11, T12),\nX15 -> T13,\nX16 -> T14,\nT2 -> cons(T13, T14)", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 32 [arrowhead = none ];
32 -> 7;

33 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 33 [arrowhead = none ];
33 -> 8;

34 [label="BACKTRACK\nfor clause: lessleaves(cons(U, V), cons(W, Z)) :- ','(append(U, V, U1), ','(append(W, Z, W1), lessleaves(U1, W1)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 34 [arrowhead = none ];
34 -> 6;

35 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 35 [arrowhead = none ];
35 -> 9;

36 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 36 [arrowhead = none ];
36 -> 10;

37 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 37 [arrowhead = none ];
37 -> 11;

38 [label="EVAL with clause\nappend(nil, X23, X23).\nand substitutionT11 -> nil,\nT12 -> T19,\nX23 -> T19,\nX17 -> T19", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 38 [arrowhead = none ];
38 -> 12;

39 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 39 [arrowhead = none ];
39 -> 13;

40 [label="EVAL with clause\nappend(cons(X64, X65), X66, cons(X64, X67)) :- append(X65, X66, X67).\nand substitutionX64 -> T51,\nX65 -> T52,\nT11 -> cons(T51, T52),\nT12 -> T53,\nX66 -> T53,\nX67 -> X68,\nX17 -> cons(T51, X68)", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 40 [arrowhead = none ];
40 -> 24;

41 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 41 [arrowhead = none ];
41 -> 25;

42 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
12 -> 42 [arrowhead = none ];
42 -> 14;

43 [label="SPLIT 2\nnew knowledge:\nT13 is ground\nT14 is ground\nT22 is ground\nreplacements:X18 -> T22", fontsize=14, style = filled, fillcolor = violet];
12 -> 43 [arrowhead = none ];
43 -> 15;

44 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 44 [arrowhead = none ];
44 -> 16;

45 [label="INSTANCE with matching:\nT1 -> T19\nT2 -> T22", fontsize=14, style = filled, fillcolor = lightgrey];
15 -> 45 [arrowhead = none , style=dashed];
45 -> 1[style=dashed];

46 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 46 [arrowhead = none ];
46 -> 17;

47 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 47 [arrowhead = none ];
47 -> 18;

48 [label="EVAL with clause\nappend(nil, X32, X32).\nand substitutionT13 -> nil,\nT14 -> T29,\nX32 -> T29,\nX18 -> T29", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 48 [arrowhead = none ];
48 -> 19;

49 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 49 [arrowhead = none ];
49 -> 20;

50 [label="EVAL with clause\nappend(cons(X43, X44), X45, cons(X43, X46)) :- append(X44, X45, X46).\nand substitutionX43 -> T36,\nX44 -> T37,\nT13 -> cons(T36, T37),\nT14 -> T38,\nX45 -> T38,\nX46 -> X47,\nX18 -> cons(T36, X47)", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 50 [arrowhead = none ];
50 -> 22;

51 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 51 [arrowhead = none ];
51 -> 23;

52 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 52 [arrowhead = none ];
52 -> 21;

53 [label="INSTANCE with matching:\nT13 -> T37\nT14 -> T38\nX18 -> X47", fontsize=14, style = filled, fillcolor = lightgrey];
22 -> 53 [arrowhead = none , style=dashed];
53 -> 14[style=dashed];

54 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
24 -> 54 [arrowhead = none ];
54 -> 26;

55 [label="SPLIT 2\nnew knowledge:\nT52 is ground\nT53 is ground\nT56 is ground\nreplacements:X68 -> T56", fontsize=14, style = filled, fillcolor = violet];
24 -> 55 [arrowhead = none ];
55 -> 27;

56 [label="INSTANCE with matching:\nT13 -> T52\nT14 -> T53\nX18 -> X68", fontsize=14, style = filled, fillcolor = lightgrey];
26 -> 56 [arrowhead = none , style=dashed];
56 -> 14[style=dashed];

57 [label="INSTANCE with matching:\nT19 -> cons(T51, T56)", fontsize=14, style = filled, fillcolor = lightgrey];
27 -> 57 [arrowhead = none , style=dashed];
57 -> 12[style=dashed];

}

(2) Obligation:

Triples:

appendC(cons(X1, X2), X3, cons(X1, X4)) :- appendC(X2, X3, X4).
pB(X1, X2, X3, X4) :- appendC(X1, X2, X3).
pB(X1, X2, X3, X4) :- ','(appendcC(X1, X2, X3), lessleavesA(X4, X3)).
lessleavesA(cons(nil, X1), cons(X2, X3)) :- pB(X2, X3, X4, X1).
lessleavesA(cons(cons(X1, X2), X3), cons(X4, X5)) :- appendC(X2, X3, X6).
lessleavesA(cons(cons(X1, X2), X3), cons(X4, X5)) :- ','(appendcC(X2, X3, X6), pB(X4, X5, X7, cons(X1, X6))).

Clauses:

lessleavescA(nil, cons(X1, X2)).
lessleavescA(cons(nil, X1), cons(X2, X3)) :- qcB(X2, X3, X4, X1).
lessleavescA(cons(cons(X1, X2), X3), cons(X4, X5)) :- ','(appendcC(X2, X3, X6), qcB(X4, X5, X7, cons(X1, X6))).
appendcC(nil, X1, X1).
appendcC(cons(X1, X2), X3, cons(X1, X4)) :- appendcC(X2, X3, X4).
qcB(X1, X2, X3, X4) :- ','(appendcC(X1, X2, X3), lessleavescA(X4, X3)).

Afs:

lessleavesA(x1, x2)  =  lessleavesA(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:
lessleavesA_in: (b,b)
pB_in: (b,b,f,b)
appendC_in: (b,b,f)
appendcC_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

LESSLEAVESA_IN_GG(cons(nil, X1), cons(X2, X3)) → U5_GG(X1, X2, X3, pB_in_ggag(X2, X3, X4, X1))
LESSLEAVESA_IN_GG(cons(nil, X1), cons(X2, X3)) → PB_IN_GGAG(X2, X3, X4, X1)
PB_IN_GGAG(X1, X2, X3, X4) → U2_GGAG(X1, X2, X3, X4, appendC_in_gga(X1, X2, X3))
PB_IN_GGAG(X1, X2, X3, X4) → APPENDC_IN_GGA(X1, X2, X3)
APPENDC_IN_GGA(cons(X1, X2), X3, cons(X1, X4)) → U1_GGA(X1, X2, X3, X4, appendC_in_gga(X2, X3, X4))
APPENDC_IN_GGA(cons(X1, X2), X3, cons(X1, X4)) → APPENDC_IN_GGA(X2, X3, X4)
PB_IN_GGAG(X1, X2, X3, X4) → U3_GGAG(X1, X2, X3, X4, appendcC_in_gga(X1, X2, X3))
U3_GGAG(X1, X2, X3, X4, appendcC_out_gga(X1, X2, X3)) → U4_GGAG(X1, X2, X3, X4, lessleavesA_in_gg(X4, X3))
U3_GGAG(X1, X2, X3, X4, appendcC_out_gga(X1, X2, X3)) → LESSLEAVESA_IN_GG(X4, X3)
LESSLEAVESA_IN_GG(cons(cons(X1, X2), X3), cons(X4, X5)) → U6_GG(X1, X2, X3, X4, X5, appendC_in_gga(X2, X3, X6))
LESSLEAVESA_IN_GG(cons(cons(X1, X2), X3), cons(X4, X5)) → APPENDC_IN_GGA(X2, X3, X6)
LESSLEAVESA_IN_GG(cons(cons(X1, X2), X3), cons(X4, X5)) → U7_GG(X1, X2, X3, X4, X5, appendcC_in_gga(X2, X3, X6))
U7_GG(X1, X2, X3, X4, X5, appendcC_out_gga(X2, X3, X6)) → U8_GG(X1, X2, X3, X4, X5, pB_in_ggag(X4, X5, X7, cons(X1, X6)))
U7_GG(X1, X2, X3, X4, X5, appendcC_out_gga(X2, X3, X6)) → PB_IN_GGAG(X4, X5, X7, cons(X1, X6))

The TRS R consists of the following rules:

appendcC_in_gga(nil, X1, X1) → appendcC_out_gga(nil, X1, X1)
appendcC_in_gga(cons(X1, X2), X3, cons(X1, X4)) → U13_gga(X1, X2, X3, X4, appendcC_in_gga(X2, X3, X4))
U13_gga(X1, X2, X3, X4, appendcC_out_gga(X2, X3, X4)) → appendcC_out_gga(cons(X1, X2), X3, cons(X1, X4))

The argument filtering Pi contains the following mapping:
lessleavesA_in_gg(x1, x2)  =  lessleavesA_in_gg(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
pB_in_ggag(x1, x2, x3, x4)  =  pB_in_ggag(x1, x2, x4)
appendC_in_gga(x1, x2, x3)  =  appendC_in_gga(x1, x2)
appendcC_in_gga(x1, x2, x3)  =  appendcC_in_gga(x1, x2)
appendcC_out_gga(x1, x2, x3)  =  appendcC_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3, x4, x5)  =  U13_gga(x1, x2, x3, x5)
LESSLEAVESA_IN_GG(x1, x2)  =  LESSLEAVESA_IN_GG(x1, x2)
U5_GG(x1, x2, x3, x4)  =  U5_GG(x1, x2, x3, x4)
PB_IN_GGAG(x1, x2, x3, x4)  =  PB_IN_GGAG(x1, x2, x4)
U2_GGAG(x1, x2, x3, x4, x5)  =  U2_GGAG(x1, x2, x4, x5)
APPENDC_IN_GGA(x1, x2, x3)  =  APPENDC_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x2, x3, x5)
U3_GGAG(x1, x2, x3, x4, x5)  =  U3_GGAG(x1, x2, x4, x5)
U4_GGAG(x1, x2, x3, x4, x5)  =  U4_GGAG(x1, x2, x3, x4, x5)
U6_GG(x1, x2, x3, x4, x5, x6)  =  U6_GG(x1, x2, x3, x4, x5, x6)
U7_GG(x1, x2, x3, x4, x5, x6)  =  U7_GG(x1, x2, x3, x4, x5, x6)
U8_GG(x1, x2, x3, x4, x5, x6)  =  U8_GG(x1, x2, x3, x4, x5, x6)

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:

LESSLEAVESA_IN_GG(cons(nil, X1), cons(X2, X3)) → U5_GG(X1, X2, X3, pB_in_ggag(X2, X3, X4, X1))
LESSLEAVESA_IN_GG(cons(nil, X1), cons(X2, X3)) → PB_IN_GGAG(X2, X3, X4, X1)
PB_IN_GGAG(X1, X2, X3, X4) → U2_GGAG(X1, X2, X3, X4, appendC_in_gga(X1, X2, X3))
PB_IN_GGAG(X1, X2, X3, X4) → APPENDC_IN_GGA(X1, X2, X3)
APPENDC_IN_GGA(cons(X1, X2), X3, cons(X1, X4)) → U1_GGA(X1, X2, X3, X4, appendC_in_gga(X2, X3, X4))
APPENDC_IN_GGA(cons(X1, X2), X3, cons(X1, X4)) → APPENDC_IN_GGA(X2, X3, X4)
PB_IN_GGAG(X1, X2, X3, X4) → U3_GGAG(X1, X2, X3, X4, appendcC_in_gga(X1, X2, X3))
U3_GGAG(X1, X2, X3, X4, appendcC_out_gga(X1, X2, X3)) → U4_GGAG(X1, X2, X3, X4, lessleavesA_in_gg(X4, X3))
U3_GGAG(X1, X2, X3, X4, appendcC_out_gga(X1, X2, X3)) → LESSLEAVESA_IN_GG(X4, X3)
LESSLEAVESA_IN_GG(cons(cons(X1, X2), X3), cons(X4, X5)) → U6_GG(X1, X2, X3, X4, X5, appendC_in_gga(X2, X3, X6))
LESSLEAVESA_IN_GG(cons(cons(X1, X2), X3), cons(X4, X5)) → APPENDC_IN_GGA(X2, X3, X6)
LESSLEAVESA_IN_GG(cons(cons(X1, X2), X3), cons(X4, X5)) → U7_GG(X1, X2, X3, X4, X5, appendcC_in_gga(X2, X3, X6))
U7_GG(X1, X2, X3, X4, X5, appendcC_out_gga(X2, X3, X6)) → U8_GG(X1, X2, X3, X4, X5, pB_in_ggag(X4, X5, X7, cons(X1, X6)))
U7_GG(X1, X2, X3, X4, X5, appendcC_out_gga(X2, X3, X6)) → PB_IN_GGAG(X4, X5, X7, cons(X1, X6))

The TRS R consists of the following rules:

appendcC_in_gga(nil, X1, X1) → appendcC_out_gga(nil, X1, X1)
appendcC_in_gga(cons(X1, X2), X3, cons(X1, X4)) → U13_gga(X1, X2, X3, X4, appendcC_in_gga(X2, X3, X4))
U13_gga(X1, X2, X3, X4, appendcC_out_gga(X2, X3, X4)) → appendcC_out_gga(cons(X1, X2), X3, cons(X1, X4))

The argument filtering Pi contains the following mapping:
lessleavesA_in_gg(x1, x2)  =  lessleavesA_in_gg(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
pB_in_ggag(x1, x2, x3, x4)  =  pB_in_ggag(x1, x2, x4)
appendC_in_gga(x1, x2, x3)  =  appendC_in_gga(x1, x2)
appendcC_in_gga(x1, x2, x3)  =  appendcC_in_gga(x1, x2)
appendcC_out_gga(x1, x2, x3)  =  appendcC_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3, x4, x5)  =  U13_gga(x1, x2, x3, x5)
LESSLEAVESA_IN_GG(x1, x2)  =  LESSLEAVESA_IN_GG(x1, x2)
U5_GG(x1, x2, x3, x4)  =  U5_GG(x1, x2, x3, x4)
PB_IN_GGAG(x1, x2, x3, x4)  =  PB_IN_GGAG(x1, x2, x4)
U2_GGAG(x1, x2, x3, x4, x5)  =  U2_GGAG(x1, x2, x4, x5)
APPENDC_IN_GGA(x1, x2, x3)  =  APPENDC_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x2, x3, x5)
U3_GGAG(x1, x2, x3, x4, x5)  =  U3_GGAG(x1, x2, x4, x5)
U4_GGAG(x1, x2, x3, x4, x5)  =  U4_GGAG(x1, x2, x3, x4, x5)
U6_GG(x1, x2, x3, x4, x5, x6)  =  U6_GG(x1, x2, x3, x4, x5, x6)
U7_GG(x1, x2, x3, x4, x5, x6)  =  U7_GG(x1, x2, x3, x4, x5, x6)
U8_GG(x1, x2, x3, x4, x5, x6)  =  U8_GG(x1, x2, x3, x4, x5, x6)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 8 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

APPENDC_IN_GGA(cons(X1, X2), X3, cons(X1, X4)) → APPENDC_IN_GGA(X2, X3, X4)

The TRS R consists of the following rules:

appendcC_in_gga(nil, X1, X1) → appendcC_out_gga(nil, X1, X1)
appendcC_in_gga(cons(X1, X2), X3, cons(X1, X4)) → U13_gga(X1, X2, X3, X4, appendcC_in_gga(X2, X3, X4))
U13_gga(X1, X2, X3, X4, appendcC_out_gga(X2, X3, X4)) → appendcC_out_gga(cons(X1, X2), X3, cons(X1, X4))

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
appendcC_in_gga(x1, x2, x3)  =  appendcC_in_gga(x1, x2)
appendcC_out_gga(x1, x2, x3)  =  appendcC_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3, x4, x5)  =  U13_gga(x1, x2, x3, x5)
APPENDC_IN_GGA(x1, x2, x3)  =  APPENDC_IN_GGA(x1, x2)

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:

APPENDC_IN_GGA(cons(X1, X2), X3, cons(X1, X4)) → APPENDC_IN_GGA(X2, X3, X4)

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

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:

APPENDC_IN_GGA(cons(X1, X2), X3) → APPENDC_IN_GGA(X2, X3)

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:

• APPENDC_IN_GGA(cons(X1, X2), X3) → APPENDC_IN_GGA(X2, X3)
  The graph contains the following edges 1 > 1, 2 >= 2

(13) YES

(14) Obligation:

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

LESSLEAVESA_IN_GG(cons(nil, X1), cons(X2, X3)) → PB_IN_GGAG(X2, X3, X4, X1)
PB_IN_GGAG(X1, X2, X3, X4) → U3_GGAG(X1, X2, X3, X4, appendcC_in_gga(X1, X2, X3))
U3_GGAG(X1, X2, X3, X4, appendcC_out_gga(X1, X2, X3)) → LESSLEAVESA_IN_GG(X4, X3)
LESSLEAVESA_IN_GG(cons(cons(X1, X2), X3), cons(X4, X5)) → U7_GG(X1, X2, X3, X4, X5, appendcC_in_gga(X2, X3, X6))
U7_GG(X1, X2, X3, X4, X5, appendcC_out_gga(X2, X3, X6)) → PB_IN_GGAG(X4, X5, X7, cons(X1, X6))

The TRS R consists of the following rules:

appendcC_in_gga(nil, X1, X1) → appendcC_out_gga(nil, X1, X1)
appendcC_in_gga(cons(X1, X2), X3, cons(X1, X4)) → U13_gga(X1, X2, X3, X4, appendcC_in_gga(X2, X3, X4))
U13_gga(X1, X2, X3, X4, appendcC_out_gga(X2, X3, X4)) → appendcC_out_gga(cons(X1, X2), X3, cons(X1, X4))

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
appendcC_in_gga(x1, x2, x3)  =  appendcC_in_gga(x1, x2)
appendcC_out_gga(x1, x2, x3)  =  appendcC_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3, x4, x5)  =  U13_gga(x1, x2, x3, x5)
LESSLEAVESA_IN_GG(x1, x2)  =  LESSLEAVESA_IN_GG(x1, x2)
PB_IN_GGAG(x1, x2, x3, x4)  =  PB_IN_GGAG(x1, x2, x4)
U3_GGAG(x1, x2, x3, x4, x5)  =  U3_GGAG(x1, x2, x4, x5)
U7_GG(x1, x2, x3, x4, x5, x6)  =  U7_GG(x1, x2, x3, x4, x5, 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:

LESSLEAVESA_IN_GG(cons(nil, X1), cons(X2, X3)) → PB_IN_GGAG(X2, X3, X1)
PB_IN_GGAG(X1, X2, X4) → U3_GGAG(X1, X2, X4, appendcC_in_gga(X1, X2))
U3_GGAG(X1, X2, X4, appendcC_out_gga(X1, X2, X3)) → LESSLEAVESA_IN_GG(X4, X3)
LESSLEAVESA_IN_GG(cons(cons(X1, X2), X3), cons(X4, X5)) → U7_GG(X1, X2, X3, X4, X5, appendcC_in_gga(X2, X3))
U7_GG(X1, X2, X3, X4, X5, appendcC_out_gga(X2, X3, X6)) → PB_IN_GGAG(X4, X5, cons(X1, X6))

The TRS R consists of the following rules:

appendcC_in_gga(nil, X1) → appendcC_out_gga(nil, X1, X1)
appendcC_in_gga(cons(X1, X2), X3) → U13_gga(X1, X2, X3, appendcC_in_gga(X2, X3))
U13_gga(X1, X2, X3, appendcC_out_gga(X2, X3, X4)) → appendcC_out_gga(cons(X1, X2), X3, cons(X1, X4))

The set Q consists of the following terms:

appendcC_in_gga(x0, x1)
U13_gga(x0, x1, x2, x3)

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.

LESSLEAVESA_IN_GG(cons(nil, X1), cons(X2, X3)) → PB_IN_GGAG(X2, X3, X1)
LESSLEAVESA_IN_GG(cons(cons(X1, X2), X3), cons(X4, X5)) → U7_GG(X1, X2, X3, X4, X5, appendcC_in_gga(X2, X3))

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

POL(LESSLEAVESA_IN_GG(x₁, x₂)) = 1 + x₂   
POL(PB_IN_GGAG(x₁, x₂, x₃)) = x₁ + x₂   
POL(U13_gga(x₁, x₂, x₃, x₄)) = x₁ + x₄   
POL(U3_GGAG(x₁, x₂, x₃, x₄)) = x₄   
POL(U7_GG(x₁, x₂, x₃, x₄, x₅, x₆)) = x₄ + x₅   
POL(appendcC_in_gga(x₁, x₂)) = x₁ + x₂   
POL(appendcC_out_gga(x₁, x₂, x₃)) = 1 + x₃   
POL(cons(x₁, x₂)) = x₁ + x₂   
POL(nil) = 1   

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

appendcC_in_gga(nil, X1) → appendcC_out_gga(nil, X1, X1)
appendcC_in_gga(cons(X1, X2), X3) → U13_gga(X1, X2, X3, appendcC_in_gga(X2, X3))
U13_gga(X1, X2, X3, appendcC_out_gga(X2, X3, X4)) → appendcC_out_gga(cons(X1, X2), X3, cons(X1, X4))

(18) Obligation:

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

PB_IN_GGAG(X1, X2, X4) → U3_GGAG(X1, X2, X4, appendcC_in_gga(X1, X2))
U3_GGAG(X1, X2, X4, appendcC_out_gga(X1, X2, X3)) → LESSLEAVESA_IN_GG(X4, X3)
U7_GG(X1, X2, X3, X4, X5, appendcC_out_gga(X2, X3, X6)) → PB_IN_GGAG(X4, X5, cons(X1, X6))

The TRS R consists of the following rules:

appendcC_in_gga(nil, X1) → appendcC_out_gga(nil, X1, X1)
appendcC_in_gga(cons(X1, X2), X3) → U13_gga(X1, X2, X3, appendcC_in_gga(X2, X3))
U13_gga(X1, X2, X3, appendcC_out_gga(X2, X3, X4)) → appendcC_out_gga(cons(X1, X2), X3, cons(X1, X4))

The set Q consists of the following terms:

appendcC_in_gga(x0, x1)
U13_gga(x0, x1, x2, x3)

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

(19) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.

(20) TRUE