Left Termination proof of /tmp/tmpAKUvqY/app2.pl

Left Termination of the query pattern app_in_3(g, a, a) w.r.t. the given Prolog program could not be shown:

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇐)

↳2 Prolog

    ↳3 PrologToPiTRSProof (⇐)

        ↳4 PiTRS

            ↳5 DependencyPairsProof (⇔)

                ↳6 PiDP

                    ↳7 DependencyGraphProof (⇔)

                        ↳8 AND

                            ↳9 PiDP

                                ↳10 UsableRulesProof (⇔)

                                    ↳11 PiDP

                                        ↳12 PiDPToQDPProof (⇐)

                                            ↳13 QDP

                                                ↳14 NonTerminationProof (⇔)

                                                    ↳15 FALSE

                            ↳16 PiDP

                                ↳17 UsableRulesProof (⇔)

                                    ↳18 PiDP

                                        ↳19 PiDPToQDPProof (⇐)

                                            ↳20 QDP

                                                ↳21 QDPSizeChangeProof (⇔)

                                                    ↳22 TRUE

    ↳23 PrologToPiTRSProof (⇐)

        ↳24 PiTRS

            ↳25 DependencyPairsProof (⇔)

                ↳26 PiDP

                    ↳27 DependencyGraphProof (⇔)

                        ↳28 AND

                            ↳29 PiDP

                                ↳30 UsableRulesProof (⇔)

                                    ↳31 PiDP

                                        ↳32 PiDPToQDPProof (⇐)

                                            ↳33 QDP

                                                ↳34 NonTerminationProof (⇔)

                                                    ↳35 FALSE

                            ↳36 PiDP

                                ↳37 UsableRulesProof (⇔)

                                    ↳38 PiDP

                                        ↳39 PiDPToQDPProof (⇐)

                                            ↳40 QDP

                                                ↳41 QDPSizeChangeProof (⇔)

                                                    ↳42 TRUE

(0) Obligation:

Clauses:

app([], L, L) :- !.
app(X, Y, .(H, Z)) :- ','(head(X, H), ','(tail(X, T), app(T, Y, Z))).
head([], X1).
head(.(X, X2), X).
tail([], []).
tail(.(X3, Xs), Xs).

Queries:

app(g,a,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: app(T1, T2, T3)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: app(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | app(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | ?_1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: !_1 | app([], T2, T3), SCOPE: 1, CLAUSE: 1 | ?_1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: app(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nX5 is free; \napp(T1, T2, T3) !=? app([], X5, X5)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(head(T5, T9), ','(tail(T5, X14), app(X14, T10, T11)))\n\nT5 is ground\nX5 is free; \nX14 is free; \napp(T5, T2, T3) !=? app([], X5, X5)", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\nX5 is free\nX10 is free; \nX11 is free; \nX12 is free; \nX13 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(head(T5, T9), ','(tail(T5, X14), app(X14, T10, T11))), SCOPE: 2, CLAUSE: 2 | ','(head(T5, T9), ','(tail(T5, X14), app(X14, T10, T11))), SCOPE: 2, CLAUSE: 3\n\nT5 is ground\nX5 is free; \nX14 is free; \napp(T5, T2, T3) !=? app([], X5, X5)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(tail([], X14), app(X14, T13, T14)) | ','(head([], T9), ','(tail([], X14), app(X14, T10, T11))), SCOPE: 2, CLAUSE: 3\n\nX5 is free\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(head(T5, T9), ','(tail(T5, X14), app(X14, T10, T11))), SCOPE: 2, CLAUSE: 3\n\nT5 is ground\nX5 is free; \nX14 is free; \nX16 is free; \napp(T5, T2, T3) !=? app([], X5, X5)\nhead(T5, T9) !=? head([], X16)", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(tail([], X14), app(X14, T13, T14)), SCOPE: 3, CLAUSE: 4 | ','(tail([], X14), app(X14, T13, T14)), SCOPE: 3, CLAUSE: 5 | ','(head([], T9), ','(tail([], X14), app(X14, T10, T11))), SCOPE: 2, CLAUSE: 3\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: app([], T13, T14) | ','(tail([], X14), app(X14, T13, T14)), SCOPE: 3, CLAUSE: 5 | ','(head([], T9), ','(tail([], X14), app(X14, T10, T11))), SCOPE: 2, CLAUSE: 3\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: app([], T13, T14)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(tail([], X14), app(X14, T13, T14)), SCOPE: 3, CLAUSE: 5 | ','(head([], T9), ','(tail([], X14), app(X14, T10, T11))), SCOPE: 2, CLAUSE: 3\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: app([], T13, T14), SCOPE: 4, CLAUSE: 0 | app([], T13, T14), SCOPE: 4, CLAUSE: 1 | ?_4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: !_4 | app([], T13, T14), SCOPE: 4, CLAUSE: 1 | ?_4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: app([], T13, T14), SCOPE: 4, CLAUSE: 1\n\nX18 is free\napp([], T13, T14) !=? app([], X18, X18)", 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: ','(head([], T19), ','(tail([], X27), app(X27, T20, T21)))\n\nX18 is free\nX27 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\nX18 is free\nX23 is free; \nX24 is free; \nX25 is free; \nX26 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(head([], T19), ','(tail([], X27), app(X27, T20, T21))), SCOPE: 5, CLAUSE: 2 | ','(head([], T19), ','(tail([], X27), app(X27, T20, T21))), SCOPE: 5, CLAUSE: 3\n\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(tail([], X27), app(X27, T23, T24)) | ','(head([], T19), ','(tail([], X27), app(X27, T20, T21))), SCOPE: 5, CLAUSE: 3\n\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(tail([], X27), app(X27, T23, T24)), SCOPE: 6, CLAUSE: 4 | ','(tail([], X27), app(X27, T23, T24)), SCOPE: 6, CLAUSE: 5 | ','(head([], T19), ','(tail([], X27), app(X27, T20, T21))), SCOPE: 5, CLAUSE: 3\n\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: app([], T23, T24) | ','(tail([], X27), app(X27, T23, T24)), SCOPE: 6, CLAUSE: 5 | ','(head([], T19), ','(tail([], X27), app(X27, T20, T21))), SCOPE: 5, CLAUSE: 3\n\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(head([], T9), ','(tail([], X14), app(X14, T10, T11))), SCOPE: 2, CLAUSE: 3\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(tail(.(T25, T26), X14), app(X14, T27, T28))\n\nT25 is ground\nT26 is ground\nX5 is free; \nX14 is free; \nX16 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\nX5 is free\nX16 is free; \nX36 is free; \nX37 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(tail(.(T25, T26), X14), app(X14, T27, T28)), SCOPE: 7, CLAUSE: 4 | ','(tail(.(T25, T26), X14), app(X14, T27, T28)), SCOPE: 7, CLAUSE: 5\n\nT25 is ground\nT26 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(tail(.(T25, T26), X14), app(X14, T27, T28)), SCOPE: 7, CLAUSE: 5\n\nT25 is ground\nT26 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: app(T30, T27, T28)\n\nT30 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 34 [arrowhead = none ];
34 -> 2;

35 [label="EVAL with clause\napp([], X5, X5) :- !.\nand substitution\nT1 -> []\nT2 -> T4\nX5 -> T4\nT3 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 35 [arrowhead = none ];
35 -> 3;

36 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 36 [arrowhead = none ];
36 -> 4;

37 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 37 [arrowhead = none ];
37 -> 5;

38 [label="EVAL with clause\napp(X10, X11, .(X12, X13)) :- ','(head(X10, X12), ','(tail(X10, X14), app(X14, X11, X13))).\nand substitution\nT1 -> T5\nX10 -> T5\nT2 -> T10\nX11 -> T10\nX12 -> T9\nX13 -> T11\nT3 -> .(T9, T11)\nT7 -> T9\nT6 -> T10\nT8 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 38 [arrowhead = none ];
38 -> 7;

39 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 39 [arrowhead = none ];
39 -> 8;

40 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 40 [arrowhead = none ];
40 -> 6;

41 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 41 [arrowhead = none ];
41 -> 9;

42 [label="EVAL with clause\nhead([], X16).\nand substitution\nT5 -> []\nT9 -> T12\nX16 -> T12\nT10 -> T13\nT11 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 42 [arrowhead = none ];
42 -> 10;

43 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 43 [arrowhead = none ];
43 -> 11;

44 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 44 [arrowhead = none ];
44 -> 12;

45 [label="EVAL with clause\nhead(.(X36, X37), X36).\nand substitution\nX36 -> T25\nX37 -> T26\nT5 -> .(T25, T26)\nT9 -> T25\nT10 -> T27\nT11 -> T28", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 45 [arrowhead = none ];
45 -> 29;

46 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 46 [arrowhead = none ];
46 -> 30;

47 [label="EVAL with clause\ntail([], []).\nand substitution\nX14 -> []", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 47 [arrowhead = none ];
47 -> 13;

48 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
13 -> 48 [arrowhead = none ];
48 -> 14;

49 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
13 -> 49 [arrowhead = none ];
49 -> 15;

50 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 50 [arrowhead = none ];
50 -> 16;

51 [label="BACKTRACK\nfor clause: tail(.(X3, Xs), Xs)because of non-unification", fontsize=14, style = filled, fillcolor = white];
15 -> 51 [arrowhead = none ];
51 -> 27;

52 [label="EVAL with clause\napp([], X18, X18) :- !.\nand substitution\nT13 -> T15\nX18 -> T15\nT14 -> T15", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 52 [arrowhead = none ];
52 -> 17;

53 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 53 [arrowhead = none ];
53 -> 18;

54 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 54 [arrowhead = none ];
54 -> 19;

55 [label="EVAL with clause\napp(X23, X24, .(X25, X26)) :- ','(head(X23, X25), ','(tail(X23, X27), app(X27, X24, X26))).\nand substitution\nX23 -> []\nT13 -> T20\nX24 -> T20\nX25 -> T19\nX26 -> T21\nT14 -> .(T19, T21)\nT17 -> T19\nT16 -> T20\nT18 -> T21", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 55 [arrowhead = none ];
55 -> 21;

56 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 56 [arrowhead = none ];
56 -> 22;

57 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 57 [arrowhead = none ];
57 -> 20;

58 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 58 [arrowhead = none ];
58 -> 23;

59 [label="EVAL with clause\nhead([], X29).\nand substitution\nT19 -> T22\nX29 -> T22\nT20 -> T23\nT21 -> T24", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 59 [arrowhead = none ];
59 -> 24;

60 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 60 [arrowhead = none ];
60 -> 25;

61 [label="EVAL with clause\ntail([], []).\nand substitution\nX27 -> []", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 61 [arrowhead = none ];
61 -> 26;

62 [label="INSTANCE with matching:\nT13 -> T23\nT14 -> T24\nX14 -> X27\nT9 -> T19\nT10 -> T20\nT11 -> T21", fontsize=14, style = filled, fillcolor = lightgrey];
26 -> 62 [arrowhead = none , style=dashed];
62 -> 13[style=dashed];

63 [label="BACKTRACK\nfor clause: head(.(X, X2), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
27 -> 63 [arrowhead = none ];
63 -> 28;

64 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 64 [arrowhead = none ];
64 -> 31;

65 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
31 -> 65 [arrowhead = none ];
65 -> 32;

66 [label="EVAL with clause\ntail(.(X40, X41), X41).\nand substitution\nT25 -> T29\nX40 -> T29\nT26 -> T30\nX41 -> T30\nX14 -> T30", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 66 [arrowhead = none ];
66 -> 33;

67 [label="INSTANCE with matching:\nT1 -> T30\nT2 -> T27\nT3 -> T28", fontsize=14, style = filled, fillcolor = lightgrey];
33 -> 67 [arrowhead = none , style=dashed];
67 -> 1[style=dashed];

}

(2) Obligation:

Clauses:

app13(T15, T15, X14, T9, T10, T11).
app13(T23, .(T22, T24), X14, T9, T10, T11) :- app13(T23, T24, X27, T19, T20, T21).
app1([], T4, T4).
app1([], T15, .(T12, T15)).
app1([], T23, .(T12, .(T22, T24))) :- app13(T23, T24, X27, T19, T20, T21).
app1(.(T29, T30), T27, .(T29, T28)) :- app1(T30, T27, T28).

Queries:

app1(g,a,a).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
app1_in: (b,f,f)
app13_in: (f,f,f,f,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

app1_in_gaa([], T4, T4) → app1_out_gaa([], T4, T4)
app1_in_gaa([], T15, .(T12, T15)) → app1_out_gaa([], T15, .(T12, T15))
app1_in_gaa([], T23, .(T12, .(T22, T24))) → U2_gaa(T23, T12, T22, T24, app13_in_aaaaaa(T23, T24, X27, T19, T20, T21))
app13_in_aaaaaa(T15, T15, X14, T9, T10, T11) → app13_out_aaaaaa(T15, T15, X14, T9, T10, T11)
app13_in_aaaaaa(T23, .(T22, T24), X14, T9, T10, T11) → U1_aaaaaa(T23, T22, T24, X14, T9, T10, T11, app13_in_aaaaaa(T23, T24, X27, T19, T20, T21))
U1_aaaaaa(T23, T22, T24, X14, T9, T10, T11, app13_out_aaaaaa(T23, T24, X27, T19, T20, T21)) → app13_out_aaaaaa(T23, .(T22, T24), X14, T9, T10, T11)
U2_gaa(T23, T12, T22, T24, app13_out_aaaaaa(T23, T24, X27, T19, T20, T21)) → app1_out_gaa([], T23, .(T12, .(T22, T24)))
app1_in_gaa(.(T29, T30), T27, .(T29, T28)) → U3_gaa(T29, T30, T27, T28, app1_in_gaa(T30, T27, T28))
U3_gaa(T29, T30, T27, T28, app1_out_gaa(T30, T27, T28)) → app1_out_gaa(.(T29, T30), T27, .(T29, T28))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

app1_in_gaa([], T4, T4) → app1_out_gaa([], T4, T4)
app1_in_gaa([], T15, .(T12, T15)) → app1_out_gaa([], T15, .(T12, T15))
app1_in_gaa([], T23, .(T12, .(T22, T24))) → U2_gaa(T23, T12, T22, T24, app13_in_aaaaaa(T23, T24, X27, T19, T20, T21))
app13_in_aaaaaa(T15, T15, X14, T9, T10, T11) → app13_out_aaaaaa(T15, T15, X14, T9, T10, T11)
app13_in_aaaaaa(T23, .(T22, T24), X14, T9, T10, T11) → U1_aaaaaa(T23, T22, T24, X14, T9, T10, T11, app13_in_aaaaaa(T23, T24, X27, T19, T20, T21))
U1_aaaaaa(T23, T22, T24, X14, T9, T10, T11, app13_out_aaaaaa(T23, T24, X27, T19, T20, T21)) → app13_out_aaaaaa(T23, .(T22, T24), X14, T9, T10, T11)
U2_gaa(T23, T12, T22, T24, app13_out_aaaaaa(T23, T24, X27, T19, T20, T21)) → app1_out_gaa([], T23, .(T12, .(T22, T24)))
app1_in_gaa(.(T29, T30), T27, .(T29, T28)) → U3_gaa(T29, T30, T27, T28, app1_in_gaa(T30, T27, T28))
U3_gaa(T29, T30, T27, T28, app1_out_gaa(T30, T27, T28)) → app1_out_gaa(.(T29, T30), T27, .(T29, T28))

(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:

APP1_IN_GAA([], T23, .(T12, .(T22, T24))) → U2_GAA(T23, T12, T22, T24, app13_in_aaaaaa(T23, T24, X27, T19, T20, T21))
APP1_IN_GAA([], T23, .(T12, .(T22, T24))) → APP13_IN_AAAAAA(T23, T24, X27, T19, T20, T21)
APP13_IN_AAAAAA(T23, .(T22, T24), X14, T9, T10, T11) → U1_AAAAAA(T23, T22, T24, X14, T9, T10, T11, app13_in_aaaaaa(T23, T24, X27, T19, T20, T21))
APP13_IN_AAAAAA(T23, .(T22, T24), X14, T9, T10, T11) → APP13_IN_AAAAAA(T23, T24, X27, T19, T20, T21)
APP1_IN_GAA(.(T29, T30), T27, .(T29, T28)) → U3_GAA(T29, T30, T27, T28, app1_in_gaa(T30, T27, T28))
APP1_IN_GAA(.(T29, T30), T27, .(T29, T28)) → APP1_IN_GAA(T30, T27, T28)

The TRS R consists of the following rules:

app1_in_gaa([], T4, T4) → app1_out_gaa([], T4, T4)
app1_in_gaa([], T15, .(T12, T15)) → app1_out_gaa([], T15, .(T12, T15))
app1_in_gaa([], T23, .(T12, .(T22, T24))) → U2_gaa(T23, T12, T22, T24, app13_in_aaaaaa(T23, T24, X27, T19, T20, T21))
app13_in_aaaaaa(T15, T15, X14, T9, T10, T11) → app13_out_aaaaaa(T15, T15, X14, T9, T10, T11)
app13_in_aaaaaa(T23, .(T22, T24), X14, T9, T10, T11) → U1_aaaaaa(T23, T22, T24, X14, T9, T10, T11, app13_in_aaaaaa(T23, T24, X27, T19, T20, T21))
U1_aaaaaa(T23, T22, T24, X14, T9, T10, T11, app13_out_aaaaaa(T23, T24, X27, T19, T20, T21)) → app13_out_aaaaaa(T23, .(T22, T24), X14, T9, T10, T11)
U2_gaa(T23, T12, T22, T24, app13_out_aaaaaa(T23, T24, X27, T19, T20, T21)) → app1_out_gaa([], T23, .(T12, .(T22, T24)))
app1_in_gaa(.(T29, T30), T27, .(T29, T28)) → U3_gaa(T29, T30, T27, T28, app1_in_gaa(T30, T27, T28))
U3_gaa(T29, T30, T27, T28, app1_out_gaa(T30, T27, T28)) → app1_out_gaa(.(T29, T30), T27, .(T29, T28))

The argument filtering Pi contains the following mapping:
app1_in_gaa(x1, x2, x3) = app1_in_gaa(x1)
[] = []
app1_out_gaa(x1, x2, x3) = app1_out_gaa
U2_gaa(x1, x2, x3, x4, x5) = U2_gaa(x5)
app13_in_aaaaaa(x1, x2, x3, x4, x5, x6) = app13_in_aaaaaa
app13_out_aaaaaa(x1, x2, x3, x4, x5, x6) = app13_out_aaaaaa
U1_aaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U1_aaaaaa(x8)
.(x1, x2) = .(x1, x2)
U3_gaa(x1, x2, x3, x4, x5) = U3_gaa(x5)
APP1_IN_GAA(x1, x2, x3) = APP1_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5) = U2_GAA(x5)
APP13_IN_AAAAAA(x1, x2, x3, x4, x5, x6) = APP13_IN_AAAAAA
U1_AAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U1_AAAAAA(x8)
U3_GAA(x1, x2, x3, x4, x5) = U3_GAA(x5)

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

(6) Obligation:

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

APP1_IN_GAA([], T23, .(T12, .(T22, T24))) → U2_GAA(T23, T12, T22, T24, app13_in_aaaaaa(T23, T24, X27, T19, T20, T21))
APP1_IN_GAA([], T23, .(T12, .(T22, T24))) → APP13_IN_AAAAAA(T23, T24, X27, T19, T20, T21)
APP13_IN_AAAAAA(T23, .(T22, T24), X14, T9, T10, T11) → U1_AAAAAA(T23, T22, T24, X14, T9, T10, T11, app13_in_aaaaaa(T23, T24, X27, T19, T20, T21))
APP13_IN_AAAAAA(T23, .(T22, T24), X14, T9, T10, T11) → APP13_IN_AAAAAA(T23, T24, X27, T19, T20, T21)
APP1_IN_GAA(.(T29, T30), T27, .(T29, T28)) → U3_GAA(T29, T30, T27, T28, app1_in_gaa(T30, T27, T28))
APP1_IN_GAA(.(T29, T30), T27, .(T29, T28)) → APP1_IN_GAA(T30, T27, T28)

The TRS R consists of the following rules:

app1_in_gaa([], T4, T4) → app1_out_gaa([], T4, T4)
app1_in_gaa([], T15, .(T12, T15)) → app1_out_gaa([], T15, .(T12, T15))
app1_in_gaa([], T23, .(T12, .(T22, T24))) → U2_gaa(T23, T12, T22, T24, app13_in_aaaaaa(T23, T24, X27, T19, T20, T21))
app13_in_aaaaaa(T15, T15, X14, T9, T10, T11) → app13_out_aaaaaa(T15, T15, X14, T9, T10, T11)
app13_in_aaaaaa(T23, .(T22, T24), X14, T9, T10, T11) → U1_aaaaaa(T23, T22, T24, X14, T9, T10, T11, app13_in_aaaaaa(T23, T24, X27, T19, T20, T21))
U1_aaaaaa(T23, T22, T24, X14, T9, T10, T11, app13_out_aaaaaa(T23, T24, X27, T19, T20, T21)) → app13_out_aaaaaa(T23, .(T22, T24), X14, T9, T10, T11)
U2_gaa(T23, T12, T22, T24, app13_out_aaaaaa(T23, T24, X27, T19, T20, T21)) → app1_out_gaa([], T23, .(T12, .(T22, T24)))
app1_in_gaa(.(T29, T30), T27, .(T29, T28)) → U3_gaa(T29, T30, T27, T28, app1_in_gaa(T30, T27, T28))
U3_gaa(T29, T30, T27, T28, app1_out_gaa(T30, T27, T28)) → app1_out_gaa(.(T29, T30), T27, .(T29, T28))

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 4 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

APP13_IN_AAAAAA(T23, .(T22, T24), X14, T9, T10, T11) → APP13_IN_AAAAAA(T23, T24, X27, T19, T20, T21)

The TRS R consists of the following rules:

app1_in_gaa([], T4, T4) → app1_out_gaa([], T4, T4)
app1_in_gaa([], T15, .(T12, T15)) → app1_out_gaa([], T15, .(T12, T15))
app1_in_gaa([], T23, .(T12, .(T22, T24))) → U2_gaa(T23, T12, T22, T24, app13_in_aaaaaa(T23, T24, X27, T19, T20, T21))
app13_in_aaaaaa(T15, T15, X14, T9, T10, T11) → app13_out_aaaaaa(T15, T15, X14, T9, T10, T11)
app13_in_aaaaaa(T23, .(T22, T24), X14, T9, T10, T11) → U1_aaaaaa(T23, T22, T24, X14, T9, T10, T11, app13_in_aaaaaa(T23, T24, X27, T19, T20, T21))
U1_aaaaaa(T23, T22, T24, X14, T9, T10, T11, app13_out_aaaaaa(T23, T24, X27, T19, T20, T21)) → app13_out_aaaaaa(T23, .(T22, T24), X14, T9, T10, T11)
U2_gaa(T23, T12, T22, T24, app13_out_aaaaaa(T23, T24, X27, T19, T20, T21)) → app1_out_gaa([], T23, .(T12, .(T22, T24)))
app1_in_gaa(.(T29, T30), T27, .(T29, T28)) → U3_gaa(T29, T30, T27, T28, app1_in_gaa(T30, T27, T28))
U3_gaa(T29, T30, T27, T28, app1_out_gaa(T30, T27, T28)) → app1_out_gaa(.(T29, T30), T27, .(T29, T28))

The argument filtering Pi contains the following mapping:
app1_in_gaa(x1, x2, x3) = app1_in_gaa(x1)
[] = []
app1_out_gaa(x1, x2, x3) = app1_out_gaa
U2_gaa(x1, x2, x3, x4, x5) = U2_gaa(x5)
app13_in_aaaaaa(x1, x2, x3, x4, x5, x6) = app13_in_aaaaaa
app13_out_aaaaaa(x1, x2, x3, x4, x5, x6) = app13_out_aaaaaa
U1_aaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U1_aaaaaa(x8)
.(x1, x2) = .(x1, x2)
U3_gaa(x1, x2, x3, x4, x5) = U3_gaa(x5)
APP13_IN_AAAAAA(x1, x2, x3, x4, x5, x6) = APP13_IN_AAAAAA

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

(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:

APP13_IN_AAAAAA(T23, .(T22, T24), X14, T9, T10, T11) → APP13_IN_AAAAAA(T23, T24, X27, T19, T20, T21)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APP13_IN_AAAAAA(x1, x2, x3, x4, x5, x6) = APP13_IN_AAAAAA

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:

APP13_IN_AAAAAA → APP13_IN_AAAAAA

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

(14) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = APP13_IN_AAAAAA evaluates to t =APP13_IN_AAAAAA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Matcher: [ ]
Semiunifier: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APP13_IN_AAAAAA to APP13_IN_AAAAAA.

(15) FALSE

(16) Obligation:

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

APP1_IN_GAA(.(T29, T30), T27, .(T29, T28)) → APP1_IN_GAA(T30, T27, T28)

The TRS R consists of the following rules:

app1_in_gaa([], T4, T4) → app1_out_gaa([], T4, T4)
app1_in_gaa([], T15, .(T12, T15)) → app1_out_gaa([], T15, .(T12, T15))
app1_in_gaa([], T23, .(T12, .(T22, T24))) → U2_gaa(T23, T12, T22, T24, app13_in_aaaaaa(T23, T24, X27, T19, T20, T21))
app13_in_aaaaaa(T15, T15, X14, T9, T10, T11) → app13_out_aaaaaa(T15, T15, X14, T9, T10, T11)
app13_in_aaaaaa(T23, .(T22, T24), X14, T9, T10, T11) → U1_aaaaaa(T23, T22, T24, X14, T9, T10, T11, app13_in_aaaaaa(T23, T24, X27, T19, T20, T21))
U1_aaaaaa(T23, T22, T24, X14, T9, T10, T11, app13_out_aaaaaa(T23, T24, X27, T19, T20, T21)) → app13_out_aaaaaa(T23, .(T22, T24), X14, T9, T10, T11)
U2_gaa(T23, T12, T22, T24, app13_out_aaaaaa(T23, T24, X27, T19, T20, T21)) → app1_out_gaa([], T23, .(T12, .(T22, T24)))
app1_in_gaa(.(T29, T30), T27, .(T29, T28)) → U3_gaa(T29, T30, T27, T28, app1_in_gaa(T30, T27, T28))
U3_gaa(T29, T30, T27, T28, app1_out_gaa(T30, T27, T28)) → app1_out_gaa(.(T29, T30), T27, .(T29, T28))

(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:

APP1_IN_GAA(.(T29, T30), T27, .(T29, T28)) → APP1_IN_GAA(T30, T27, T28)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APP1_IN_GAA(x1, x2, x3) = APP1_IN_GAA(x1)

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:

APP1_IN_GAA(.(T29, T30)) → APP1_IN_GAA(T30)

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

(21) 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:

APP1_IN_GAA(.(T29, T30)) → APP1_IN_GAA(T30)
The graph contains the following edges 1 > 1

(22) TRUE

(23) PrologToPiTRSProof (SOUND transformation)

app1_in_gaa([], T4, T4) → app1_out_gaa([], T4, T4)
app1_in_gaa([], T15, .(T12, T15)) → app1_out_gaa([], T15, .(T12, T15))
app1_in_gaa([], T23, .(T12, .(T22, T24))) → U2_gaa(T23, T12, T22, T24, app13_in_aaaaaa(T23, T24, X27, T19, T20, T21))
app13_in_aaaaaa(T15, T15, X14, T9, T10, T11) → app13_out_aaaaaa(T15, T15, X14, T9, T10, T11)
app13_in_aaaaaa(T23, .(T22, T24), X14, T9, T10, T11) → U1_aaaaaa(T23, T22, T24, X14, T9, T10, T11, app13_in_aaaaaa(T23, T24, X27, T19, T20, T21))
U1_aaaaaa(T23, T22, T24, X14, T9, T10, T11, app13_out_aaaaaa(T23, T24, X27, T19, T20, T21)) → app13_out_aaaaaa(T23, .(T22, T24), X14, T9, T10, T11)
U2_gaa(T23, T12, T22, T24, app13_out_aaaaaa(T23, T24, X27, T19, T20, T21)) → app1_out_gaa([], T23, .(T12, .(T22, T24)))
app1_in_gaa(.(T29, T30), T27, .(T29, T28)) → U3_gaa(T29, T30, T27, T28, app1_in_gaa(T30, T27, T28))
U3_gaa(T29, T30, T27, T28, app1_out_gaa(T30, T27, T28)) → app1_out_gaa(.(T29, T30), T27, .(T29, T28))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(24) Obligation:

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

app1_in_gaa([], T4, T4) → app1_out_gaa([], T4, T4)
app1_in_gaa([], T15, .(T12, T15)) → app1_out_gaa([], T15, .(T12, T15))
app1_in_gaa([], T23, .(T12, .(T22, T24))) → U2_gaa(T23, T12, T22, T24, app13_in_aaaaaa(T23, T24, X27, T19, T20, T21))
app13_in_aaaaaa(T15, T15, X14, T9, T10, T11) → app13_out_aaaaaa(T15, T15, X14, T9, T10, T11)
app13_in_aaaaaa(T23, .(T22, T24), X14, T9, T10, T11) → U1_aaaaaa(T23, T22, T24, X14, T9, T10, T11, app13_in_aaaaaa(T23, T24, X27, T19, T20, T21))
U1_aaaaaa(T23, T22, T24, X14, T9, T10, T11, app13_out_aaaaaa(T23, T24, X27, T19, T20, T21)) → app13_out_aaaaaa(T23, .(T22, T24), X14, T9, T10, T11)
U2_gaa(T23, T12, T22, T24, app13_out_aaaaaa(T23, T24, X27, T19, T20, T21)) → app1_out_gaa([], T23, .(T12, .(T22, T24)))
app1_in_gaa(.(T29, T30), T27, .(T29, T28)) → U3_gaa(T29, T30, T27, T28, app1_in_gaa(T30, T27, T28))
U3_gaa(T29, T30, T27, T28, app1_out_gaa(T30, T27, T28)) → app1_out_gaa(.(T29, T30), T27, .(T29, T28))

(25) 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:

APP1_IN_GAA([], T23, .(T12, .(T22, T24))) → U2_GAA(T23, T12, T22, T24, app13_in_aaaaaa(T23, T24, X27, T19, T20, T21))
APP1_IN_GAA([], T23, .(T12, .(T22, T24))) → APP13_IN_AAAAAA(T23, T24, X27, T19, T20, T21)
APP13_IN_AAAAAA(T23, .(T22, T24), X14, T9, T10, T11) → U1_AAAAAA(T23, T22, T24, X14, T9, T10, T11, app13_in_aaaaaa(T23, T24, X27, T19, T20, T21))
APP13_IN_AAAAAA(T23, .(T22, T24), X14, T9, T10, T11) → APP13_IN_AAAAAA(T23, T24, X27, T19, T20, T21)
APP1_IN_GAA(.(T29, T30), T27, .(T29, T28)) → U3_GAA(T29, T30, T27, T28, app1_in_gaa(T30, T27, T28))
APP1_IN_GAA(.(T29, T30), T27, .(T29, T28)) → APP1_IN_GAA(T30, T27, T28)

The TRS R consists of the following rules:

app1_in_gaa([], T4, T4) → app1_out_gaa([], T4, T4)
app1_in_gaa([], T15, .(T12, T15)) → app1_out_gaa([], T15, .(T12, T15))
app1_in_gaa([], T23, .(T12, .(T22, T24))) → U2_gaa(T23, T12, T22, T24, app13_in_aaaaaa(T23, T24, X27, T19, T20, T21))
app13_in_aaaaaa(T15, T15, X14, T9, T10, T11) → app13_out_aaaaaa(T15, T15, X14, T9, T10, T11)
app13_in_aaaaaa(T23, .(T22, T24), X14, T9, T10, T11) → U1_aaaaaa(T23, T22, T24, X14, T9, T10, T11, app13_in_aaaaaa(T23, T24, X27, T19, T20, T21))
U1_aaaaaa(T23, T22, T24, X14, T9, T10, T11, app13_out_aaaaaa(T23, T24, X27, T19, T20, T21)) → app13_out_aaaaaa(T23, .(T22, T24), X14, T9, T10, T11)
U2_gaa(T23, T12, T22, T24, app13_out_aaaaaa(T23, T24, X27, T19, T20, T21)) → app1_out_gaa([], T23, .(T12, .(T22, T24)))
app1_in_gaa(.(T29, T30), T27, .(T29, T28)) → U3_gaa(T29, T30, T27, T28, app1_in_gaa(T30, T27, T28))
U3_gaa(T29, T30, T27, T28, app1_out_gaa(T30, T27, T28)) → app1_out_gaa(.(T29, T30), T27, .(T29, T28))

The argument filtering Pi contains the following mapping:
app1_in_gaa(x1, x2, x3) = app1_in_gaa(x1)
[] = []
app1_out_gaa(x1, x2, x3) = app1_out_gaa(x1)
U2_gaa(x1, x2, x3, x4, x5) = U2_gaa(x5)
app13_in_aaaaaa(x1, x2, x3, x4, x5, x6) = app13_in_aaaaaa
app13_out_aaaaaa(x1, x2, x3, x4, x5, x6) = app13_out_aaaaaa
U1_aaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U1_aaaaaa(x8)
.(x1, x2) = .(x1, x2)
U3_gaa(x1, x2, x3, x4, x5) = U3_gaa(x1, x2, x5)
APP1_IN_GAA(x1, x2, x3) = APP1_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5) = U2_GAA(x5)
APP13_IN_AAAAAA(x1, x2, x3, x4, x5, x6) = APP13_IN_AAAAAA
U1_AAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U1_AAAAAA(x8)
U3_GAA(x1, x2, x3, x4, x5) = U3_GAA(x1, x2, x5)

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

(26) Obligation:

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

APP1_IN_GAA([], T23, .(T12, .(T22, T24))) → U2_GAA(T23, T12, T22, T24, app13_in_aaaaaa(T23, T24, X27, T19, T20, T21))
APP1_IN_GAA([], T23, .(T12, .(T22, T24))) → APP13_IN_AAAAAA(T23, T24, X27, T19, T20, T21)
APP13_IN_AAAAAA(T23, .(T22, T24), X14, T9, T10, T11) → U1_AAAAAA(T23, T22, T24, X14, T9, T10, T11, app13_in_aaaaaa(T23, T24, X27, T19, T20, T21))
APP13_IN_AAAAAA(T23, .(T22, T24), X14, T9, T10, T11) → APP13_IN_AAAAAA(T23, T24, X27, T19, T20, T21)
APP1_IN_GAA(.(T29, T30), T27, .(T29, T28)) → U3_GAA(T29, T30, T27, T28, app1_in_gaa(T30, T27, T28))
APP1_IN_GAA(.(T29, T30), T27, .(T29, T28)) → APP1_IN_GAA(T30, T27, T28)

The TRS R consists of the following rules:

app1_in_gaa([], T4, T4) → app1_out_gaa([], T4, T4)
app1_in_gaa([], T15, .(T12, T15)) → app1_out_gaa([], T15, .(T12, T15))
app1_in_gaa([], T23, .(T12, .(T22, T24))) → U2_gaa(T23, T12, T22, T24, app13_in_aaaaaa(T23, T24, X27, T19, T20, T21))
app13_in_aaaaaa(T15, T15, X14, T9, T10, T11) → app13_out_aaaaaa(T15, T15, X14, T9, T10, T11)
app13_in_aaaaaa(T23, .(T22, T24), X14, T9, T10, T11) → U1_aaaaaa(T23, T22, T24, X14, T9, T10, T11, app13_in_aaaaaa(T23, T24, X27, T19, T20, T21))
U1_aaaaaa(T23, T22, T24, X14, T9, T10, T11, app13_out_aaaaaa(T23, T24, X27, T19, T20, T21)) → app13_out_aaaaaa(T23, .(T22, T24), X14, T9, T10, T11)
U2_gaa(T23, T12, T22, T24, app13_out_aaaaaa(T23, T24, X27, T19, T20, T21)) → app1_out_gaa([], T23, .(T12, .(T22, T24)))
app1_in_gaa(.(T29, T30), T27, .(T29, T28)) → U3_gaa(T29, T30, T27, T28, app1_in_gaa(T30, T27, T28))
U3_gaa(T29, T30, T27, T28, app1_out_gaa(T30, T27, T28)) → app1_out_gaa(.(T29, T30), T27, .(T29, T28))

(27) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 4 less nodes.

(28) Complex Obligation (AND)

(29) Obligation:

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

APP13_IN_AAAAAA(T23, .(T22, T24), X14, T9, T10, T11) → APP13_IN_AAAAAA(T23, T24, X27, T19, T20, T21)

The TRS R consists of the following rules:

app1_in_gaa([], T4, T4) → app1_out_gaa([], T4, T4)
app1_in_gaa([], T15, .(T12, T15)) → app1_out_gaa([], T15, .(T12, T15))
app1_in_gaa([], T23, .(T12, .(T22, T24))) → U2_gaa(T23, T12, T22, T24, app13_in_aaaaaa(T23, T24, X27, T19, T20, T21))
app13_in_aaaaaa(T15, T15, X14, T9, T10, T11) → app13_out_aaaaaa(T15, T15, X14, T9, T10, T11)
app13_in_aaaaaa(T23, .(T22, T24), X14, T9, T10, T11) → U1_aaaaaa(T23, T22, T24, X14, T9, T10, T11, app13_in_aaaaaa(T23, T24, X27, T19, T20, T21))
U1_aaaaaa(T23, T22, T24, X14, T9, T10, T11, app13_out_aaaaaa(T23, T24, X27, T19, T20, T21)) → app13_out_aaaaaa(T23, .(T22, T24), X14, T9, T10, T11)
U2_gaa(T23, T12, T22, T24, app13_out_aaaaaa(T23, T24, X27, T19, T20, T21)) → app1_out_gaa([], T23, .(T12, .(T22, T24)))
app1_in_gaa(.(T29, T30), T27, .(T29, T28)) → U3_gaa(T29, T30, T27, T28, app1_in_gaa(T30, T27, T28))
U3_gaa(T29, T30, T27, T28, app1_out_gaa(T30, T27, T28)) → app1_out_gaa(.(T29, T30), T27, .(T29, T28))

The argument filtering Pi contains the following mapping:
app1_in_gaa(x1, x2, x3) = app1_in_gaa(x1)
[] = []
app1_out_gaa(x1, x2, x3) = app1_out_gaa(x1)
U2_gaa(x1, x2, x3, x4, x5) = U2_gaa(x5)
app13_in_aaaaaa(x1, x2, x3, x4, x5, x6) = app13_in_aaaaaa
app13_out_aaaaaa(x1, x2, x3, x4, x5, x6) = app13_out_aaaaaa
U1_aaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U1_aaaaaa(x8)
.(x1, x2) = .(x1, x2)
U3_gaa(x1, x2, x3, x4, x5) = U3_gaa(x1, x2, x5)
APP13_IN_AAAAAA(x1, x2, x3, x4, x5, x6) = APP13_IN_AAAAAA

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

(30) UsableRulesProof (EQUIVALENT transformation)

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

(31) Obligation:

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

APP13_IN_AAAAAA(T23, .(T22, T24), X14, T9, T10, T11) → APP13_IN_AAAAAA(T23, T24, X27, T19, T20, T21)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APP13_IN_AAAAAA(x1, x2, x3, x4, x5, x6) = APP13_IN_AAAAAA

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

(32) PiDPToQDPProof (SOUND transformation)

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

(33) Obligation:

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

APP13_IN_AAAAAA → APP13_IN_AAAAAA

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

(34) NonTerminationProof (EQUIVALENT transformation)

Matcher: [ ]
Semiunifier: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APP13_IN_AAAAAA to APP13_IN_AAAAAA.

(35) FALSE

(36) Obligation:

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

APP1_IN_GAA(.(T29, T30), T27, .(T29, T28)) → APP1_IN_GAA(T30, T27, T28)

The TRS R consists of the following rules:

app1_in_gaa([], T4, T4) → app1_out_gaa([], T4, T4)
app1_in_gaa([], T15, .(T12, T15)) → app1_out_gaa([], T15, .(T12, T15))
app1_in_gaa([], T23, .(T12, .(T22, T24))) → U2_gaa(T23, T12, T22, T24, app13_in_aaaaaa(T23, T24, X27, T19, T20, T21))
app13_in_aaaaaa(T15, T15, X14, T9, T10, T11) → app13_out_aaaaaa(T15, T15, X14, T9, T10, T11)
app13_in_aaaaaa(T23, .(T22, T24), X14, T9, T10, T11) → U1_aaaaaa(T23, T22, T24, X14, T9, T10, T11, app13_in_aaaaaa(T23, T24, X27, T19, T20, T21))
U1_aaaaaa(T23, T22, T24, X14, T9, T10, T11, app13_out_aaaaaa(T23, T24, X27, T19, T20, T21)) → app13_out_aaaaaa(T23, .(T22, T24), X14, T9, T10, T11)
U2_gaa(T23, T12, T22, T24, app13_out_aaaaaa(T23, T24, X27, T19, T20, T21)) → app1_out_gaa([], T23, .(T12, .(T22, T24)))
app1_in_gaa(.(T29, T30), T27, .(T29, T28)) → U3_gaa(T29, T30, T27, T28, app1_in_gaa(T30, T27, T28))
U3_gaa(T29, T30, T27, T28, app1_out_gaa(T30, T27, T28)) → app1_out_gaa(.(T29, T30), T27, .(T29, T28))

(37) UsableRulesProof (EQUIVALENT transformation)

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

(38) Obligation:

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

APP1_IN_GAA(.(T29, T30), T27, .(T29, T28)) → APP1_IN_GAA(T30, T27, T28)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APP1_IN_GAA(x1, x2, x3) = APP1_IN_GAA(x1)

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

(39) PiDPToQDPProof (SOUND transformation)

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

(40) Obligation:

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

APP1_IN_GAA(.(T29, T30)) → APP1_IN_GAA(T30)

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

(41) 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:

APP1_IN_GAA(.(T29, T30)) → APP1_IN_GAA(T30)
The graph contains the following edges 1 > 1

(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) NonTerminationProof (EQUIVALENT transformation)

(15) FALSE

(16) Obligation:

(17) UsableRulesProof (EQUIVALENT transformation)

(18) Obligation:

(19) PiDPToQDPProof (SOUND transformation)

(20) Obligation:

(21) QDPSizeChangeProof (EQUIVALENT transformation)

(22) TRUE

(23) PrologToPiTRSProof (SOUND transformation)

(24) Obligation:

(25) DependencyPairsProof (EQUIVALENT transformation)

(26) Obligation:

(27) DependencyGraphProof (EQUIVALENT transformation)

(28) Complex Obligation (AND)

(29) Obligation:

(30) UsableRulesProof (EQUIVALENT transformation)

(31) Obligation:

(32) PiDPToQDPProof (SOUND transformation)

(33) Obligation:

(34) NonTerminationProof (EQUIVALENT transformation)

(35) FALSE

(36) Obligation:

(37) UsableRulesProof (EQUIVALENT transformation)

(38) Obligation:

(39) PiDPToQDPProof (SOUND transformation)

(40) Obligation:

(41) QDPSizeChangeProof (EQUIVALENT transformation)

(42) TRUE