Left Termination proof of /tmp/tmpnVloKG/lessleaves.pl

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

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇒, 86 ms)

↳2 Prolog

↳3 PrologToPiTRSProof (⇒, 0 ms)

↳4 PiTRS

↳5 DependencyPairsProof (⇔, 19 ms)

↳6 PiDP

↳7 DependencyGraphProof (⇔, 0 ms)

↳8 AND

    ↳9 PiDP

        ↳10 UsableRulesProof (⇔, 0 ms)

        ↳11 PiDP

        ↳12 PiDPToQDPProof (⇒, 0 ms)

        ↳13 QDP

        ↳14 QDPSizeChangeProof (⇔, 0 ms)

        ↳15 YES

    ↳16 PiDP

        ↳17 UsableRulesProof (⇔, 0 ms)

        ↳18 PiDP

        ↳19 PiDPToQDPProof (⇒, 0 ms)

        ↳20 QDP

        ↳21 UsableRulesReductionPairsProof (⇔, 179 ms)

        ↳22 QDP

        ↳23 MRRProof (⇔, 0 ms)

        ↳24 QDP

        ↳25 PisEmptyProof (⇔, 0 ms)

        ↳26 YES

(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) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: 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="12: ','(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="14: 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:

Clauses:

appendA(nil, T29, T29).
appendA(cons(T36, T37), T38, cons(T36, X47)) :- appendA(T37, T38, X47).
pB(T13, T14, X18, T19) :- appendA(T13, T14, X18).
pB(T13, T14, T22, T19) :- ','(appendA(T13, T14, T22), lessleavesC(T19, T22)).
lessleavesC(nil, cons(T5, T6)).
lessleavesC(cons(nil, T19), cons(T13, T14)) :- pB(T13, T14, X18, T19).
lessleavesC(cons(cons(T51, T52), T53), cons(T13, T14)) :- appendA(T52, T53, X68).
lessleavesC(cons(cons(T51, T52), T53), cons(T13, T14)) :- ','(appendA(T52, T53, T56), pB(T13, T14, X18, cons(T51, T56))).

Query: lessleavesC(g,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:
lessleavesC_in: (b,b)
pB_in: (b,b,f,b)
appendA_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

lessleavesC_in_gg(nil, cons(T5, T6)) → lessleavesC_out_gg(nil, cons(T5, T6))
lessleavesC_in_gg(cons(nil, T19), cons(T13, T14)) → U5_gg(T19, T13, T14, pB_in_ggag(T13, T14, X18, T19))
pB_in_ggag(T13, T14, X18, T19) → U2_ggag(T13, T14, X18, T19, appendA_in_gga(T13, T14, X18))
appendA_in_gga(nil, T29, T29) → appendA_out_gga(nil, T29, T29)
appendA_in_gga(cons(T36, T37), T38, cons(T36, X47)) → U1_gga(T36, T37, T38, X47, appendA_in_gga(T37, T38, X47))
U1_gga(T36, T37, T38, X47, appendA_out_gga(T37, T38, X47)) → appendA_out_gga(cons(T36, T37), T38, cons(T36, X47))
U2_ggag(T13, T14, X18, T19, appendA_out_gga(T13, T14, X18)) → pB_out_ggag(T13, T14, X18, T19)
pB_in_ggag(T13, T14, T22, T19) → U3_ggag(T13, T14, T22, T19, appendA_in_gga(T13, T14, T22))
U3_ggag(T13, T14, T22, T19, appendA_out_gga(T13, T14, T22)) → U4_ggag(T13, T14, T22, T19, lessleavesC_in_gg(T19, T22))
lessleavesC_in_gg(cons(cons(T51, T52), T53), cons(T13, T14)) → U6_gg(T51, T52, T53, T13, T14, appendA_in_gga(T52, T53, X68))
U6_gg(T51, T52, T53, T13, T14, appendA_out_gga(T52, T53, X68)) → lessleavesC_out_gg(cons(cons(T51, T52), T53), cons(T13, T14))
lessleavesC_in_gg(cons(cons(T51, T52), T53), cons(T13, T14)) → U7_gg(T51, T52, T53, T13, T14, appendA_in_gga(T52, T53, T56))
U7_gg(T51, T52, T53, T13, T14, appendA_out_gga(T52, T53, T56)) → U8_gg(T51, T52, T53, T13, T14, pB_in_ggag(T13, T14, X18, cons(T51, T56)))
U8_gg(T51, T52, T53, T13, T14, pB_out_ggag(T13, T14, X18, cons(T51, T56))) → lessleavesC_out_gg(cons(cons(T51, T52), T53), cons(T13, T14))
U4_ggag(T13, T14, T22, T19, lessleavesC_out_gg(T19, T22)) → pB_out_ggag(T13, T14, T22, T19)
U5_gg(T19, T13, T14, pB_out_ggag(T13, T14, X18, T19)) → lessleavesC_out_gg(cons(nil, T19), cons(T13, T14))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

lessleavesC_in_gg(nil, cons(T5, T6)) → lessleavesC_out_gg(nil, cons(T5, T6))
lessleavesC_in_gg(cons(nil, T19), cons(T13, T14)) → U5_gg(T19, T13, T14, pB_in_ggag(T13, T14, X18, T19))
pB_in_ggag(T13, T14, X18, T19) → U2_ggag(T13, T14, X18, T19, appendA_in_gga(T13, T14, X18))
appendA_in_gga(nil, T29, T29) → appendA_out_gga(nil, T29, T29)
appendA_in_gga(cons(T36, T37), T38, cons(T36, X47)) → U1_gga(T36, T37, T38, X47, appendA_in_gga(T37, T38, X47))
U1_gga(T36, T37, T38, X47, appendA_out_gga(T37, T38, X47)) → appendA_out_gga(cons(T36, T37), T38, cons(T36, X47))
U2_ggag(T13, T14, X18, T19, appendA_out_gga(T13, T14, X18)) → pB_out_ggag(T13, T14, X18, T19)
pB_in_ggag(T13, T14, T22, T19) → U3_ggag(T13, T14, T22, T19, appendA_in_gga(T13, T14, T22))
U3_ggag(T13, T14, T22, T19, appendA_out_gga(T13, T14, T22)) → U4_ggag(T13, T14, T22, T19, lessleavesC_in_gg(T19, T22))
lessleavesC_in_gg(cons(cons(T51, T52), T53), cons(T13, T14)) → U6_gg(T51, T52, T53, T13, T14, appendA_in_gga(T52, T53, X68))
U6_gg(T51, T52, T53, T13, T14, appendA_out_gga(T52, T53, X68)) → lessleavesC_out_gg(cons(cons(T51, T52), T53), cons(T13, T14))
lessleavesC_in_gg(cons(cons(T51, T52), T53), cons(T13, T14)) → U7_gg(T51, T52, T53, T13, T14, appendA_in_gga(T52, T53, T56))
U7_gg(T51, T52, T53, T13, T14, appendA_out_gga(T52, T53, T56)) → U8_gg(T51, T52, T53, T13, T14, pB_in_ggag(T13, T14, X18, cons(T51, T56)))
U8_gg(T51, T52, T53, T13, T14, pB_out_ggag(T13, T14, X18, cons(T51, T56))) → lessleavesC_out_gg(cons(cons(T51, T52), T53), cons(T13, T14))
U4_ggag(T13, T14, T22, T19, lessleavesC_out_gg(T19, T22)) → pB_out_ggag(T13, T14, T22, T19)
U5_gg(T19, T13, T14, pB_out_ggag(T13, T14, X18, T19)) → lessleavesC_out_gg(cons(nil, T19), cons(T13, T14))

(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:

LESSLEAVESC_IN_GG(cons(nil, T19), cons(T13, T14)) → U5_GG(T19, T13, T14, pB_in_ggag(T13, T14, X18, T19))
LESSLEAVESC_IN_GG(cons(nil, T19), cons(T13, T14)) → PB_IN_GGAG(T13, T14, X18, T19)
PB_IN_GGAG(T13, T14, X18, T19) → U2_GGAG(T13, T14, X18, T19, appendA_in_gga(T13, T14, X18))
PB_IN_GGAG(T13, T14, X18, T19) → APPENDA_IN_GGA(T13, T14, X18)
APPENDA_IN_GGA(cons(T36, T37), T38, cons(T36, X47)) → U1_GGA(T36, T37, T38, X47, appendA_in_gga(T37, T38, X47))
APPENDA_IN_GGA(cons(T36, T37), T38, cons(T36, X47)) → APPENDA_IN_GGA(T37, T38, X47)
PB_IN_GGAG(T13, T14, T22, T19) → U3_GGAG(T13, T14, T22, T19, appendA_in_gga(T13, T14, T22))
U3_GGAG(T13, T14, T22, T19, appendA_out_gga(T13, T14, T22)) → U4_GGAG(T13, T14, T22, T19, lessleavesC_in_gg(T19, T22))
U3_GGAG(T13, T14, T22, T19, appendA_out_gga(T13, T14, T22)) → LESSLEAVESC_IN_GG(T19, T22)
LESSLEAVESC_IN_GG(cons(cons(T51, T52), T53), cons(T13, T14)) → U6_GG(T51, T52, T53, T13, T14, appendA_in_gga(T52, T53, X68))
LESSLEAVESC_IN_GG(cons(cons(T51, T52), T53), cons(T13, T14)) → APPENDA_IN_GGA(T52, T53, X68)
LESSLEAVESC_IN_GG(cons(cons(T51, T52), T53), cons(T13, T14)) → U7_GG(T51, T52, T53, T13, T14, appendA_in_gga(T52, T53, T56))
U7_GG(T51, T52, T53, T13, T14, appendA_out_gga(T52, T53, T56)) → U8_GG(T51, T52, T53, T13, T14, pB_in_ggag(T13, T14, X18, cons(T51, T56)))
U7_GG(T51, T52, T53, T13, T14, appendA_out_gga(T52, T53, T56)) → PB_IN_GGAG(T13, T14, X18, cons(T51, T56))

The TRS R consists of the following rules:

lessleavesC_in_gg(nil, cons(T5, T6)) → lessleavesC_out_gg(nil, cons(T5, T6))
lessleavesC_in_gg(cons(nil, T19), cons(T13, T14)) → U5_gg(T19, T13, T14, pB_in_ggag(T13, T14, X18, T19))
pB_in_ggag(T13, T14, X18, T19) → U2_ggag(T13, T14, X18, T19, appendA_in_gga(T13, T14, X18))
appendA_in_gga(nil, T29, T29) → appendA_out_gga(nil, T29, T29)
appendA_in_gga(cons(T36, T37), T38, cons(T36, X47)) → U1_gga(T36, T37, T38, X47, appendA_in_gga(T37, T38, X47))
U1_gga(T36, T37, T38, X47, appendA_out_gga(T37, T38, X47)) → appendA_out_gga(cons(T36, T37), T38, cons(T36, X47))
U2_ggag(T13, T14, X18, T19, appendA_out_gga(T13, T14, X18)) → pB_out_ggag(T13, T14, X18, T19)
pB_in_ggag(T13, T14, T22, T19) → U3_ggag(T13, T14, T22, T19, appendA_in_gga(T13, T14, T22))
U3_ggag(T13, T14, T22, T19, appendA_out_gga(T13, T14, T22)) → U4_ggag(T13, T14, T22, T19, lessleavesC_in_gg(T19, T22))
lessleavesC_in_gg(cons(cons(T51, T52), T53), cons(T13, T14)) → U6_gg(T51, T52, T53, T13, T14, appendA_in_gga(T52, T53, X68))
U6_gg(T51, T52, T53, T13, T14, appendA_out_gga(T52, T53, X68)) → lessleavesC_out_gg(cons(cons(T51, T52), T53), cons(T13, T14))
lessleavesC_in_gg(cons(cons(T51, T52), T53), cons(T13, T14)) → U7_gg(T51, T52, T53, T13, T14, appendA_in_gga(T52, T53, T56))
U7_gg(T51, T52, T53, T13, T14, appendA_out_gga(T52, T53, T56)) → U8_gg(T51, T52, T53, T13, T14, pB_in_ggag(T13, T14, X18, cons(T51, T56)))
U8_gg(T51, T52, T53, T13, T14, pB_out_ggag(T13, T14, X18, cons(T51, T56))) → lessleavesC_out_gg(cons(cons(T51, T52), T53), cons(T13, T14))
U4_ggag(T13, T14, T22, T19, lessleavesC_out_gg(T19, T22)) → pB_out_ggag(T13, T14, T22, T19)
U5_gg(T19, T13, T14, pB_out_ggag(T13, T14, X18, T19)) → lessleavesC_out_gg(cons(nil, T19), cons(T13, T14))

The argument filtering Pi contains the following mapping:
lessleavesC_in_gg(x1, x2) = lessleavesC_in_gg(x1, x2)
nil = nil
cons(x1, x2) = cons(x1, x2)
lessleavesC_out_gg(x1, x2) = lessleavesC_out_gg
U5_gg(x1, x2, x3, x4) = U5_gg(x4)
pB_in_ggag(x1, x2, x3, x4) = pB_in_ggag(x1, x2, x4)
U2_ggag(x1, x2, x3, x4, x5) = U2_ggag(x5)
appendA_in_gga(x1, x2, x3) = appendA_in_gga(x1, x2)
appendA_out_gga(x1, x2, x3) = appendA_out_gga(x3)
U1_gga(x1, x2, x3, x4, x5) = U1_gga(x1, x5)
pB_out_ggag(x1, x2, x3, x4) = pB_out_ggag(x3)
U3_ggag(x1, x2, x3, x4, x5) = U3_ggag(x4, x5)
U4_ggag(x1, x2, x3, x4, x5) = U4_ggag(x3, x5)
U6_gg(x1, x2, x3, x4, x5, x6) = U6_gg(x6)
U7_gg(x1, x2, x3, x4, x5, x6) = U7_gg(x1, x4, x5, x6)
U8_gg(x1, x2, x3, x4, x5, x6) = U8_gg(x6)
LESSLEAVESC_IN_GG(x1, x2) = LESSLEAVESC_IN_GG(x1, x2)
U5_GG(x1, x2, x3, x4) = U5_GG(x4)
PB_IN_GGAG(x1, x2, x3, x4) = PB_IN_GGAG(x1, x2, x4)
U2_GGAG(x1, x2, x3, x4, x5) = U2_GGAG(x5)
APPENDA_IN_GGA(x1, x2, x3) = APPENDA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5) = U1_GGA(x1, x5)
U3_GGAG(x1, x2, x3, x4, x5) = U3_GGAG(x4, x5)
U4_GGAG(x1, x2, x3, x4, x5) = U4_GGAG(x3, x5)
U6_GG(x1, x2, x3, x4, x5, x6) = U6_GG(x6)
U7_GG(x1, x2, x3, x4, x5, x6) = U7_GG(x1, x4, x5, x6)
U8_GG(x1, x2, x3, x4, x5, x6) = U8_GG(x6)

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

(6) Obligation:

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

LESSLEAVESC_IN_GG(cons(nil, T19), cons(T13, T14)) → U5_GG(T19, T13, T14, pB_in_ggag(T13, T14, X18, T19))
LESSLEAVESC_IN_GG(cons(nil, T19), cons(T13, T14)) → PB_IN_GGAG(T13, T14, X18, T19)
PB_IN_GGAG(T13, T14, X18, T19) → U2_GGAG(T13, T14, X18, T19, appendA_in_gga(T13, T14, X18))
PB_IN_GGAG(T13, T14, X18, T19) → APPENDA_IN_GGA(T13, T14, X18)
APPENDA_IN_GGA(cons(T36, T37), T38, cons(T36, X47)) → U1_GGA(T36, T37, T38, X47, appendA_in_gga(T37, T38, X47))
APPENDA_IN_GGA(cons(T36, T37), T38, cons(T36, X47)) → APPENDA_IN_GGA(T37, T38, X47)
PB_IN_GGAG(T13, T14, T22, T19) → U3_GGAG(T13, T14, T22, T19, appendA_in_gga(T13, T14, T22))
U3_GGAG(T13, T14, T22, T19, appendA_out_gga(T13, T14, T22)) → U4_GGAG(T13, T14, T22, T19, lessleavesC_in_gg(T19, T22))
U3_GGAG(T13, T14, T22, T19, appendA_out_gga(T13, T14, T22)) → LESSLEAVESC_IN_GG(T19, T22)
LESSLEAVESC_IN_GG(cons(cons(T51, T52), T53), cons(T13, T14)) → U6_GG(T51, T52, T53, T13, T14, appendA_in_gga(T52, T53, X68))
LESSLEAVESC_IN_GG(cons(cons(T51, T52), T53), cons(T13, T14)) → APPENDA_IN_GGA(T52, T53, X68)
LESSLEAVESC_IN_GG(cons(cons(T51, T52), T53), cons(T13, T14)) → U7_GG(T51, T52, T53, T13, T14, appendA_in_gga(T52, T53, T56))
U7_GG(T51, T52, T53, T13, T14, appendA_out_gga(T52, T53, T56)) → U8_GG(T51, T52, T53, T13, T14, pB_in_ggag(T13, T14, X18, cons(T51, T56)))
U7_GG(T51, T52, T53, T13, T14, appendA_out_gga(T52, T53, T56)) → PB_IN_GGAG(T13, T14, X18, cons(T51, T56))

The TRS R consists of the following rules:

lessleavesC_in_gg(nil, cons(T5, T6)) → lessleavesC_out_gg(nil, cons(T5, T6))
lessleavesC_in_gg(cons(nil, T19), cons(T13, T14)) → U5_gg(T19, T13, T14, pB_in_ggag(T13, T14, X18, T19))
pB_in_ggag(T13, T14, X18, T19) → U2_ggag(T13, T14, X18, T19, appendA_in_gga(T13, T14, X18))
appendA_in_gga(nil, T29, T29) → appendA_out_gga(nil, T29, T29)
appendA_in_gga(cons(T36, T37), T38, cons(T36, X47)) → U1_gga(T36, T37, T38, X47, appendA_in_gga(T37, T38, X47))
U1_gga(T36, T37, T38, X47, appendA_out_gga(T37, T38, X47)) → appendA_out_gga(cons(T36, T37), T38, cons(T36, X47))
U2_ggag(T13, T14, X18, T19, appendA_out_gga(T13, T14, X18)) → pB_out_ggag(T13, T14, X18, T19)
pB_in_ggag(T13, T14, T22, T19) → U3_ggag(T13, T14, T22, T19, appendA_in_gga(T13, T14, T22))
U3_ggag(T13, T14, T22, T19, appendA_out_gga(T13, T14, T22)) → U4_ggag(T13, T14, T22, T19, lessleavesC_in_gg(T19, T22))
lessleavesC_in_gg(cons(cons(T51, T52), T53), cons(T13, T14)) → U6_gg(T51, T52, T53, T13, T14, appendA_in_gga(T52, T53, X68))
U6_gg(T51, T52, T53, T13, T14, appendA_out_gga(T52, T53, X68)) → lessleavesC_out_gg(cons(cons(T51, T52), T53), cons(T13, T14))
lessleavesC_in_gg(cons(cons(T51, T52), T53), cons(T13, T14)) → U7_gg(T51, T52, T53, T13, T14, appendA_in_gga(T52, T53, T56))
U7_gg(T51, T52, T53, T13, T14, appendA_out_gga(T52, T53, T56)) → U8_gg(T51, T52, T53, T13, T14, pB_in_ggag(T13, T14, X18, cons(T51, T56)))
U8_gg(T51, T52, T53, T13, T14, pB_out_ggag(T13, T14, X18, cons(T51, T56))) → lessleavesC_out_gg(cons(cons(T51, T52), T53), cons(T13, T14))
U4_ggag(T13, T14, T22, T19, lessleavesC_out_gg(T19, T22)) → pB_out_ggag(T13, T14, T22, T19)
U5_gg(T19, T13, T14, pB_out_ggag(T13, T14, X18, T19)) → lessleavesC_out_gg(cons(nil, T19), cons(T13, T14))

(7) DependencyGraphProof (EQUIVALENT transformation)

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

(8) Complex Obligation (AND)

(9) Obligation:

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

APPENDA_IN_GGA(cons(T36, T37), T38, cons(T36, X47)) → APPENDA_IN_GGA(T37, T38, X47)

The TRS R consists of the following rules:

lessleavesC_in_gg(nil, cons(T5, T6)) → lessleavesC_out_gg(nil, cons(T5, T6))
lessleavesC_in_gg(cons(nil, T19), cons(T13, T14)) → U5_gg(T19, T13, T14, pB_in_ggag(T13, T14, X18, T19))
pB_in_ggag(T13, T14, X18, T19) → U2_ggag(T13, T14, X18, T19, appendA_in_gga(T13, T14, X18))
appendA_in_gga(nil, T29, T29) → appendA_out_gga(nil, T29, T29)
appendA_in_gga(cons(T36, T37), T38, cons(T36, X47)) → U1_gga(T36, T37, T38, X47, appendA_in_gga(T37, T38, X47))
U1_gga(T36, T37, T38, X47, appendA_out_gga(T37, T38, X47)) → appendA_out_gga(cons(T36, T37), T38, cons(T36, X47))
U2_ggag(T13, T14, X18, T19, appendA_out_gga(T13, T14, X18)) → pB_out_ggag(T13, T14, X18, T19)
pB_in_ggag(T13, T14, T22, T19) → U3_ggag(T13, T14, T22, T19, appendA_in_gga(T13, T14, T22))
U3_ggag(T13, T14, T22, T19, appendA_out_gga(T13, T14, T22)) → U4_ggag(T13, T14, T22, T19, lessleavesC_in_gg(T19, T22))
lessleavesC_in_gg(cons(cons(T51, T52), T53), cons(T13, T14)) → U6_gg(T51, T52, T53, T13, T14, appendA_in_gga(T52, T53, X68))
U6_gg(T51, T52, T53, T13, T14, appendA_out_gga(T52, T53, X68)) → lessleavesC_out_gg(cons(cons(T51, T52), T53), cons(T13, T14))
lessleavesC_in_gg(cons(cons(T51, T52), T53), cons(T13, T14)) → U7_gg(T51, T52, T53, T13, T14, appendA_in_gga(T52, T53, T56))
U7_gg(T51, T52, T53, T13, T14, appendA_out_gga(T52, T53, T56)) → U8_gg(T51, T52, T53, T13, T14, pB_in_ggag(T13, T14, X18, cons(T51, T56)))
U8_gg(T51, T52, T53, T13, T14, pB_out_ggag(T13, T14, X18, cons(T51, T56))) → lessleavesC_out_gg(cons(cons(T51, T52), T53), cons(T13, T14))
U4_ggag(T13, T14, T22, T19, lessleavesC_out_gg(T19, T22)) → pB_out_ggag(T13, T14, T22, T19)
U5_gg(T19, T13, T14, pB_out_ggag(T13, T14, X18, T19)) → lessleavesC_out_gg(cons(nil, T19), cons(T13, T14))

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

APPENDA_IN_GGA(cons(T36, T37), T38, cons(T36, X47)) → APPENDA_IN_GGA(T37, T38, X47)

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

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

APPENDA_IN_GGA(cons(T36, T37), T38) → APPENDA_IN_GGA(T37, T38)

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

(14) 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:

APPENDA_IN_GGA(cons(T36, T37), T38) → APPENDA_IN_GGA(T37, T38)
The graph contains the following edges 1 > 1, 2 >= 2

(15) YES

(16) Obligation:

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

LESSLEAVESC_IN_GG(cons(nil, T19), cons(T13, T14)) → PB_IN_GGAG(T13, T14, X18, T19)
PB_IN_GGAG(T13, T14, T22, T19) → U3_GGAG(T13, T14, T22, T19, appendA_in_gga(T13, T14, T22))
U3_GGAG(T13, T14, T22, T19, appendA_out_gga(T13, T14, T22)) → LESSLEAVESC_IN_GG(T19, T22)
LESSLEAVESC_IN_GG(cons(cons(T51, T52), T53), cons(T13, T14)) → U7_GG(T51, T52, T53, T13, T14, appendA_in_gga(T52, T53, T56))
U7_GG(T51, T52, T53, T13, T14, appendA_out_gga(T52, T53, T56)) → PB_IN_GGAG(T13, T14, X18, cons(T51, T56))

The TRS R consists of the following rules:

lessleavesC_in_gg(nil, cons(T5, T6)) → lessleavesC_out_gg(nil, cons(T5, T6))
lessleavesC_in_gg(cons(nil, T19), cons(T13, T14)) → U5_gg(T19, T13, T14, pB_in_ggag(T13, T14, X18, T19))
pB_in_ggag(T13, T14, X18, T19) → U2_ggag(T13, T14, X18, T19, appendA_in_gga(T13, T14, X18))
appendA_in_gga(nil, T29, T29) → appendA_out_gga(nil, T29, T29)
appendA_in_gga(cons(T36, T37), T38, cons(T36, X47)) → U1_gga(T36, T37, T38, X47, appendA_in_gga(T37, T38, X47))
U1_gga(T36, T37, T38, X47, appendA_out_gga(T37, T38, X47)) → appendA_out_gga(cons(T36, T37), T38, cons(T36, X47))
U2_ggag(T13, T14, X18, T19, appendA_out_gga(T13, T14, X18)) → pB_out_ggag(T13, T14, X18, T19)
pB_in_ggag(T13, T14, T22, T19) → U3_ggag(T13, T14, T22, T19, appendA_in_gga(T13, T14, T22))
U3_ggag(T13, T14, T22, T19, appendA_out_gga(T13, T14, T22)) → U4_ggag(T13, T14, T22, T19, lessleavesC_in_gg(T19, T22))
lessleavesC_in_gg(cons(cons(T51, T52), T53), cons(T13, T14)) → U6_gg(T51, T52, T53, T13, T14, appendA_in_gga(T52, T53, X68))
U6_gg(T51, T52, T53, T13, T14, appendA_out_gga(T52, T53, X68)) → lessleavesC_out_gg(cons(cons(T51, T52), T53), cons(T13, T14))
lessleavesC_in_gg(cons(cons(T51, T52), T53), cons(T13, T14)) → U7_gg(T51, T52, T53, T13, T14, appendA_in_gga(T52, T53, T56))
U7_gg(T51, T52, T53, T13, T14, appendA_out_gga(T52, T53, T56)) → U8_gg(T51, T52, T53, T13, T14, pB_in_ggag(T13, T14, X18, cons(T51, T56)))
U8_gg(T51, T52, T53, T13, T14, pB_out_ggag(T13, T14, X18, cons(T51, T56))) → lessleavesC_out_gg(cons(cons(T51, T52), T53), cons(T13, T14))
U4_ggag(T13, T14, T22, T19, lessleavesC_out_gg(T19, T22)) → pB_out_ggag(T13, T14, T22, T19)
U5_gg(T19, T13, T14, pB_out_ggag(T13, T14, X18, T19)) → lessleavesC_out_gg(cons(nil, T19), cons(T13, T14))

The argument filtering Pi contains the following mapping:
lessleavesC_in_gg(x1, x2) = lessleavesC_in_gg(x1, x2)
nil = nil
cons(x1, x2) = cons(x1, x2)
lessleavesC_out_gg(x1, x2) = lessleavesC_out_gg
U5_gg(x1, x2, x3, x4) = U5_gg(x4)
pB_in_ggag(x1, x2, x3, x4) = pB_in_ggag(x1, x2, x4)
U2_ggag(x1, x2, x3, x4, x5) = U2_ggag(x5)
appendA_in_gga(x1, x2, x3) = appendA_in_gga(x1, x2)
appendA_out_gga(x1, x2, x3) = appendA_out_gga(x3)
U1_gga(x1, x2, x3, x4, x5) = U1_gga(x1, x5)
pB_out_ggag(x1, x2, x3, x4) = pB_out_ggag(x3)
U3_ggag(x1, x2, x3, x4, x5) = U3_ggag(x4, x5)
U4_ggag(x1, x2, x3, x4, x5) = U4_ggag(x3, x5)
U6_gg(x1, x2, x3, x4, x5, x6) = U6_gg(x6)
U7_gg(x1, x2, x3, x4, x5, x6) = U7_gg(x1, x4, x5, x6)
U8_gg(x1, x2, x3, x4, x5, x6) = U8_gg(x6)
LESSLEAVESC_IN_GG(x1, x2) = LESSLEAVESC_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(x4, x5)
U7_GG(x1, x2, x3, x4, x5, x6) = U7_GG(x1, x4, x5, x6)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

LESSLEAVESC_IN_GG(cons(nil, T19), cons(T13, T14)) → PB_IN_GGAG(T13, T14, X18, T19)
PB_IN_GGAG(T13, T14, T22, T19) → U3_GGAG(T13, T14, T22, T19, appendA_in_gga(T13, T14, T22))
U3_GGAG(T13, T14, T22, T19, appendA_out_gga(T13, T14, T22)) → LESSLEAVESC_IN_GG(T19, T22)
LESSLEAVESC_IN_GG(cons(cons(T51, T52), T53), cons(T13, T14)) → U7_GG(T51, T52, T53, T13, T14, appendA_in_gga(T52, T53, T56))
U7_GG(T51, T52, T53, T13, T14, appendA_out_gga(T52, T53, T56)) → PB_IN_GGAG(T13, T14, X18, cons(T51, T56))

The TRS R consists of the following rules:

appendA_in_gga(nil, T29, T29) → appendA_out_gga(nil, T29, T29)
appendA_in_gga(cons(T36, T37), T38, cons(T36, X47)) → U1_gga(T36, T37, T38, X47, appendA_in_gga(T37, T38, X47))
U1_gga(T36, T37, T38, X47, appendA_out_gga(T37, T38, X47)) → appendA_out_gga(cons(T36, T37), T38, cons(T36, X47))

The argument filtering Pi contains the following mapping:
nil = nil
cons(x1, x2) = cons(x1, x2)
appendA_in_gga(x1, x2, x3) = appendA_in_gga(x1, x2)
appendA_out_gga(x1, x2, x3) = appendA_out_gga(x3)
U1_gga(x1, x2, x3, x4, x5) = U1_gga(x1, x5)
LESSLEAVESC_IN_GG(x1, x2) = LESSLEAVESC_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(x4, x5)
U7_GG(x1, x2, x3, x4, x5, x6) = U7_GG(x1, x4, x5, x6)

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

LESSLEAVESC_IN_GG(cons(nil, T19), cons(T13, T14)) → PB_IN_GGAG(T13, T14, T19)
PB_IN_GGAG(T13, T14, T19) → U3_GGAG(T19, appendA_in_gga(T13, T14))
U3_GGAG(T19, appendA_out_gga(T22)) → LESSLEAVESC_IN_GG(T19, T22)
LESSLEAVESC_IN_GG(cons(cons(T51, T52), T53), cons(T13, T14)) → U7_GG(T51, T13, T14, appendA_in_gga(T52, T53))
U7_GG(T51, T13, T14, appendA_out_gga(T56)) → PB_IN_GGAG(T13, T14, cons(T51, T56))

The TRS R consists of the following rules:

appendA_in_gga(nil, T29) → appendA_out_gga(T29)
appendA_in_gga(cons(T36, T37), T38) → U1_gga(T36, appendA_in_gga(T37, T38))
U1_gga(T36, appendA_out_gga(X47)) → appendA_out_gga(cons(T36, X47))

The set Q consists of the following terms:

appendA_in_gga(x0, x1)
U1_gga(x0, x1)

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

(21) UsableRulesReductionPairsProof (EQUIVALENT transformation)

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

LESSLEAVESC_IN_GG(cons(nil, T19), cons(T13, T14)) → PB_IN_GGAG(T13, T14, T19)

The following rules are removed from R:

appendA_in_gga(nil, T29) → appendA_out_gga(T29)

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(LESSLEAVESC_IN_GG(x₁, x₂)) = x₁ + x₂
POL(PB_IN_GGAG(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U1_gga(x₁, x₂)) = x₁ + x₂
POL(U3_GGAG(x₁, x₂)) = x₁ + x₂
POL(U7_GG(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(appendA_in_gga(x₁, x₂)) = x₁ + x₂
POL(appendA_out_gga(x₁)) = x₁
POL(cons(x₁, x₂)) = x₁ + x₂
POL(nil) = 0

(22) Obligation:

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

PB_IN_GGAG(T13, T14, T19) → U3_GGAG(T19, appendA_in_gga(T13, T14))
U3_GGAG(T19, appendA_out_gga(T22)) → LESSLEAVESC_IN_GG(T19, T22)
LESSLEAVESC_IN_GG(cons(cons(T51, T52), T53), cons(T13, T14)) → U7_GG(T51, T13, T14, appendA_in_gga(T52, T53))
U7_GG(T51, T13, T14, appendA_out_gga(T56)) → PB_IN_GGAG(T13, T14, cons(T51, T56))

The TRS R consists of the following rules:

appendA_in_gga(cons(T36, T37), T38) → U1_gga(T36, appendA_in_gga(T37, T38))
U1_gga(T36, appendA_out_gga(X47)) → appendA_out_gga(cons(T36, X47))

The set Q consists of the following terms:

appendA_in_gga(x0, x1)
U1_gga(x0, x1)

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

(23) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

PB_IN_GGAG(T13, T14, T19) → U3_GGAG(T19, appendA_in_gga(T13, T14))
U3_GGAG(T19, appendA_out_gga(T22)) → LESSLEAVESC_IN_GG(T19, T22)
LESSLEAVESC_IN_GG(cons(cons(T51, T52), T53), cons(T13, T14)) → U7_GG(T51, T13, T14, appendA_in_gga(T52, T53))
U7_GG(T51, T13, T14, appendA_out_gga(T56)) → PB_IN_GGAG(T13, T14, cons(T51, T56))

Strictly oriented rules of the TRS R:

appendA_in_gga(cons(T36, T37), T38) → U1_gga(T36, appendA_in_gga(T37, T38))
U1_gga(T36, appendA_out_gga(X47)) → appendA_out_gga(cons(T36, X47))

Used ordering: Knuth-Bendix order [KBO] with precedence:

appendA_{in_gga₂} > cons₂ > U7_GG₄ > PB_{IN_GGAG₃} > U3_GGAG₂ > LESSLEAVESC_{IN_GG₂} > U1_gga₂ > appendA_{out_gga₁}

and weight map:

appendA_out_gga_1=1
cons_2=0
U1_gga_2=0
PB_IN_GGAG_3=1
U3_GGAG_2=1
appendA_in_gga_2=0
LESSLEAVESC_IN_GG_2=2
U7_GG_4=0

The variable weight is 1

(24) Obligation:

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

appendA_in_gga(x0, x1)
U1_gga(x0, x1)

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

(25) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(0) Obligation:

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) PrologToPiTRSProof (SOUND transformation)

(4) Obligation:

(5) DependencyPairsProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Complex Obligation (AND)

(9) Obligation:

(10) UsableRulesProof (EQUIVALENT transformation)

(11) Obligation:

(12) PiDPToQDPProof (SOUND transformation)

(13) Obligation:

(14) QDPSizeChangeProof (EQUIVALENT transformation)

(15) YES

(16) Obligation:

(17) UsableRulesProof (EQUIVALENT transformation)

(18) Obligation:

(19) PiDPToQDPProof (SOUND transformation)

(20) Obligation:

(21) UsableRulesReductionPairsProof (EQUIVALENT transformation)

(22) Obligation:

(23) MRRProof (EQUIVALENT transformation)

(24) Obligation:

(25) PisEmptyProof (EQUIVALENT transformation)

(26) YES