Left Termination proof of /tmp/tmp1ufzi8/giesl97.pl

Left Termination of the query pattern f_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 Narrowing (⇐)

                                                    ↳15 QDP

                                                        ↳16 NonTerminationProof (⇔)

                                                            ↳17 FALSE

                            ↳18 PiDP

                                ↳19 UsableRulesProof (⇔)

                                    ↳20 PiDP

                                        ↳21 PiDPToQDPProof (⇐)

                                            ↳22 QDP

                                                ↳23 Narrowing (⇐)

                                                    ↳24 QDP

                                                        ↳25 NonTerminationProof (⇔)

                                                            ↳26 FALSE

                            ↳27 PiDP

                                ↳28 UsableRulesProof (⇔)

                                    ↳29 PiDP

                                        ↳30 PiDPToQDPProof (⇐)

                                            ↳31 QDP

                                                ↳32 QDPSizeChangeProof (⇔)

                                                    ↳33 TRUE

    ↳34 PrologToPiTRSProof (⇐)

        ↳35 PiTRS

            ↳36 DependencyPairsProof (⇔)

                ↳37 PiDP

                    ↳38 DependencyGraphProof (⇔)

                        ↳39 AND

                            ↳40 PiDP

                                ↳41 UsableRulesProof (⇔)

                                    ↳42 PiDP

                                        ↳43 PiDPToQDPProof (⇐)

                                            ↳44 QDP

                                                ↳45 Narrowing (⇐)

                                                    ↳46 QDP

                                                        ↳47 NonTerminationProof (⇔)

                                                            ↳48 FALSE

                            ↳49 PiDP

                                ↳50 UsableRulesProof (⇔)

                                    ↳51 PiDP

                                        ↳52 PiDPToQDPProof (⇐)

                                            ↳53 QDP

                                                ↳54 Narrowing (⇐)

                                                    ↳55 QDP

                                                        ↳56 NonTerminationProof (⇔)

                                                            ↳57 FALSE

                            ↳58 PiDP

                                ↳59 UsableRulesProof (⇔)

                                    ↳60 PiDP

                                        ↳61 PiDPToQDPProof (⇐)

                                            ↳62 QDP

                                                ↳63 QDPSizeChangeProof (⇔)

                                                    ↳64 TRUE

(0) Obligation:

Clauses:

f(0, Y, Z) :- ','(!, eq(Z, 0)).
f(X, Y, Z) :- ','(p(X, P), ','(f(P, Y, U), f(U, Y, Z))).
p(0, 0).
p(s(X), X).
eq(X, X).

Queries:

f(g,a,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: f(T1, T2, T3)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: f(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | f(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | ?_1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(!_1, eq(T6, 0)) | f(0, T2, T3), SCOPE: 1, CLAUSE: 1 | ?_1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: f(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nX3 is free; \nX4 is free; \nf(T1, T2, T3) !=? f(0, X3, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: eq(T6, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: eq(T6, 0), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(p(T8, X13), ','(f(X13, T11, X14), f(X14, T11, T12)))\n\nT8 is ground\nX3 is free; \nX4 is free; \nX13 is free; \nX14 is free; \nf(T8, T2, T3) !=? f(0, X3, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(p(T8, X13), ','(f(X13, T11, X14), f(X14, T11, T12))), SCOPE: 3, CLAUSE: 2 | ','(p(T8, X13), ','(f(X13, T11, X14), f(X14, T11, T12))), SCOPE: 3, CLAUSE: 3\n\nT8 is ground\nX3 is free; \nX4 is free; \nX13 is free; \nX14 is free; \nf(T8, T2, T3) !=? f(0, X3, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(p(T8, X13), ','(f(X13, T11, X14), f(X14, T11, T12))), SCOPE: 3, CLAUSE: 3\n\nT8 is ground\nX3 is free; \nX4 is free; \nX13 is free; \nX14 is free; \nf(T8, T2, T3) !=? f(0, X3, X4)\np(T8, X13) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(f(T13, T11, X14), f(X14, T11, T12))\n\nT13 is ground\nX3 is free; \nX4 is free; \nX13 is free; \nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\nX3 is free\nX4 is free; \nX13 is free; \nX16 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: f(T13, T11, X14)\n\nT13 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: f(T14, T15, T16)\n\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: f(T13, T11, X14), SCOPE: 4, CLAUSE: 0 | f(T13, T11, X14), SCOPE: 4, CLAUSE: 1 | ?_4\n\nT13 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(!_4, eq(X21, 0)) | f(0, T11, X14), SCOPE: 4, CLAUSE: 1 | ?_4\n\nX14 is free\nX21 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: f(T13, T11, X14), SCOPE: 4, CLAUSE: 1\n\nT13 is ground\nX14 is free; \nX19 is free; \nX20 is free; \nf(T13, T11, X14) !=? f(0, X19, X20)", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: eq(X21, 0)\n\nX21 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: eq(X21, 0), SCOPE: 5, CLAUSE: 4\n\nX21 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(p(T18, X31), ','(f(X31, T20, X32), f(X32, T20, X33)))\n\nT18 is ground\nX14 is free; \nX19 is free; \nX20 is free; \nX33 is free; \nX31 is free; \nX32 is free; \nf(T18, T11, X14) !=? f(0, X19, X20)", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(p(T18, X31), ','(f(X31, T20, X32), f(X32, T20, X33))), SCOPE: 6, CLAUSE: 2 | ','(p(T18, X31), ','(f(X31, T20, X32), f(X32, T20, X33))), SCOPE: 6, CLAUSE: 3\n\nT18 is ground\nX14 is free; \nX19 is free; \nX20 is free; \nX33 is free; \nX31 is free; \nX32 is free; \nf(T18, T11, X14) !=? f(0, X19, X20)", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(p(T18, X31), ','(f(X31, T20, X32), f(X32, T20, X33))), SCOPE: 6, CLAUSE: 3\n\nT18 is ground\nX14 is free; \nX19 is free; \nX20 is free; \nX33 is free; \nX31 is free; \nX32 is free; \nf(T18, T11, X14) !=? f(0, X19, X20)\np(T18, X31) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(f(T21, T20, X32), f(X32, T20, X33))\n\nT21 is ground\nX14 is free; \nX19 is free; \nX20 is free; \nX33 is free; \nX31 is free; \nX32 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\nX14 is free\nX19 is free; \nX20 is free; \nX31 is free; \nX35 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: f(T21, T20, X32)\n\nT21 is ground\nX32 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: f(T22, T23, X33)\n\nT22 is ground\nX33 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 31 [arrowhead = none ];
31 -> 2;

32 [label="EVAL with clause\nf(0, X3, X4) :- ','(!, eq(X4, 0)).\nand substitution\nT1 -> 0\nT2 -> T4\nX3 -> T4\nT3 -> T6\nX4 -> T6\nT5 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 32 [arrowhead = none ];
32 -> 3;

33 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 33 [arrowhead = none ];
33 -> 4;

34 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 34 [arrowhead = none ];
34 -> 5;

35 [label="EVAL with clause\nf(X10, X11, X12) :- ','(p(X10, X13), ','(f(X13, X11, X14), f(X14, X11, X12))).\nand substitution\nT1 -> T8\nX10 -> T8\nT2 -> T11\nX11 -> T11\nT3 -> T12\nX12 -> T12\nT9 -> T11\nT10 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 35 [arrowhead = none ];
35 -> 10;

36 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 36 [arrowhead = none ];
36 -> 6;

37 [label="EVAL with clause\neq(X6, X6).\nand substitution\nT6 -> 0\nX6 -> 0\nT7 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 37 [arrowhead = none ];
37 -> 7;

38 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 38 [arrowhead = none ];
38 -> 8;

39 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 39 [arrowhead = none ];
39 -> 9;

40 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 40 [arrowhead = none ];
40 -> 11;

41 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (f(T8, T2, T3), f(0, X3, X4))", fontsize=14, style = filled, fillcolor = white];
11 -> 41 [arrowhead = none ];
41 -> 12;

42 [label="EVAL with clause\np(s(X16), X16).\nand substitution\nX16 -> T13\nT8 -> s(T13)\nX13 -> T13", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 42 [arrowhead = none ];
42 -> 13;

43 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 43 [arrowhead = none ];
43 -> 14;

44 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
13 -> 44 [arrowhead = none ];
44 -> 15;

45 [label="SPLIT 2\nnew knowledge:\nT14 is ground\nreplacements:\nX14 -> T14\nT11 -> T15\nT12 -> T16", fontsize=14, style = filled, fillcolor = violet];
13 -> 45 [arrowhead = none ];
45 -> 16;

46 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 46 [arrowhead = none ];
46 -> 17;

47 [label="INSTANCE with matching:\nT1 -> T14\nT2 -> T15\nT3 -> T16", fontsize=14, style = filled, fillcolor = lightgrey];
16 -> 47 [arrowhead = none , style=dashed];
47 -> 1[style=dashed];

48 [label="EVAL with clause\nf(0, X19, X20) :- ','(!, eq(X20, 0)).\nand substitution\nT13 -> 0\nT11 -> T17\nX19 -> T17\nX14 -> X21\nX20 -> X21", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 48 [arrowhead = none ];
48 -> 18;

49 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 49 [arrowhead = none ];
49 -> 19;

50 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 50 [arrowhead = none ];
50 -> 20;

51 [label="EVAL with clause\nf(X28, X29, X30) :- ','(p(X28, X31), ','(f(X31, X29, X32), f(X32, X29, X30))).\nand substitution\nT13 -> T18\nX28 -> T18\nT11 -> T20\nX29 -> T20\nX14 -> X33\nX30 -> X33\nT19 -> T20", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 51 [arrowhead = none ];
51 -> 24;

52 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 52 [arrowhead = none ];
52 -> 21;

53 [label="EVAL with clause\neq(X23, X23).\nand substitution\nX21 -> 0\nX23 -> 0\nX24 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 53 [arrowhead = none ];
53 -> 22;

54 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 54 [arrowhead = none ];
54 -> 23;

55 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 55 [arrowhead = none ];
55 -> 25;

56 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (f(T18, T11, X14), f(0, X19, X20))", fontsize=14, style = filled, fillcolor = white];
25 -> 56 [arrowhead = none ];
56 -> 26;

57 [label="EVAL with clause\np(s(X35), X35).\nand substitution\nX35 -> T21\nT18 -> s(T21)\nX31 -> T21", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 57 [arrowhead = none ];
57 -> 27;

58 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 58 [arrowhead = none ];
58 -> 28;

59 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
27 -> 59 [arrowhead = none ];
59 -> 29;

60 [label="SPLIT 2\nnew knowledge:\nT22 is ground\nreplacements:\nX32 -> T22\nT20 -> T23", fontsize=14, style = filled, fillcolor = violet];
27 -> 60 [arrowhead = none ];
60 -> 30;

61 [label="INSTANCE with matching:\nT13 -> T21\nT11 -> T20\nX14 -> X32", fontsize=14, style = filled, fillcolor = lightgrey];
29 -> 61 [arrowhead = none , style=dashed];
61 -> 15[style=dashed];

62 [label="INSTANCE with matching:\nT13 -> T22\nT11 -> T23\nX14 -> X33", fontsize=14, style = filled, fillcolor = lightgrey];
30 -> 62 [arrowhead = none , style=dashed];
62 -> 15[style=dashed];

}

(2) Obligation:

Clauses:

f1(0, T4, 0).
f1(s(T13), T11, T12) :- f15(T13, T11, X14).
f1(s(T13), T15, T16) :- ','(f15(T13, T15, T14), f1(T14, T15, T16)).
f15(0, T17, 0).
f15(s(T21), T20, X33) :- f15(T21, T20, X32).
f15(s(T21), T23, X33) :- ','(f15(T21, T23, T22), f15(T22, T23, X33)).

Queries:

f1(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:
f1_in: (b,f,f) (f,f,f)
f15_in: (b,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:

f1_in_gaa(0, T4, 0) → f1_out_gaa(0, T4, 0)
f1_in_gaa(s(T13), T11, T12) → U1_gaa(T13, T11, T12, f15_in_gaa(T13, T11, X14))
f15_in_gaa(0, T17, 0) → f15_out_gaa(0, T17, 0)
f15_in_gaa(s(T21), T20, X33) → U4_gaa(T21, T20, X33, f15_in_gaa(T21, T20, X32))
f15_in_gaa(s(T21), T23, X33) → U5_gaa(T21, T23, X33, f15_in_gaa(T21, T23, T22))
U5_gaa(T21, T23, X33, f15_out_gaa(T21, T23, T22)) → U6_gaa(T21, T23, X33, f15_in_aaa(T22, T23, X33))
f15_in_aaa(0, T17, 0) → f15_out_aaa(0, T17, 0)
f15_in_aaa(s(T21), T20, X33) → U4_aaa(T21, T20, X33, f15_in_aaa(T21, T20, X32))
f15_in_aaa(s(T21), T23, X33) → U5_aaa(T21, T23, X33, f15_in_aaa(T21, T23, T22))
U5_aaa(T21, T23, X33, f15_out_aaa(T21, T23, T22)) → U6_aaa(T21, T23, X33, f15_in_aaa(T22, T23, X33))
U6_aaa(T21, T23, X33, f15_out_aaa(T22, T23, X33)) → f15_out_aaa(s(T21), T23, X33)
U4_aaa(T21, T20, X33, f15_out_aaa(T21, T20, X32)) → f15_out_aaa(s(T21), T20, X33)
U6_gaa(T21, T23, X33, f15_out_aaa(T22, T23, X33)) → f15_out_gaa(s(T21), T23, X33)
U4_gaa(T21, T20, X33, f15_out_gaa(T21, T20, X32)) → f15_out_gaa(s(T21), T20, X33)
U1_gaa(T13, T11, T12, f15_out_gaa(T13, T11, X14)) → f1_out_gaa(s(T13), T11, T12)
f1_in_gaa(s(T13), T15, T16) → U2_gaa(T13, T15, T16, f15_in_gaa(T13, T15, T14))
U2_gaa(T13, T15, T16, f15_out_gaa(T13, T15, T14)) → U3_gaa(T13, T15, T16, f1_in_aaa(T14, T15, T16))
f1_in_aaa(0, T4, 0) → f1_out_aaa(0, T4, 0)
f1_in_aaa(s(T13), T11, T12) → U1_aaa(T13, T11, T12, f15_in_aaa(T13, T11, X14))
U1_aaa(T13, T11, T12, f15_out_aaa(T13, T11, X14)) → f1_out_aaa(s(T13), T11, T12)
f1_in_aaa(s(T13), T15, T16) → U2_aaa(T13, T15, T16, f15_in_aaa(T13, T15, T14))
U2_aaa(T13, T15, T16, f15_out_aaa(T13, T15, T14)) → U3_aaa(T13, T15, T16, f1_in_aaa(T14, T15, T16))
U3_aaa(T13, T15, T16, f1_out_aaa(T14, T15, T16)) → f1_out_aaa(s(T13), T15, T16)
U3_gaa(T13, T15, T16, f1_out_aaa(T14, T15, T16)) → f1_out_gaa(s(T13), T15, T16)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

f1_in_gaa(0, T4, 0) → f1_out_gaa(0, T4, 0)
f1_in_gaa(s(T13), T11, T12) → U1_gaa(T13, T11, T12, f15_in_gaa(T13, T11, X14))
f15_in_gaa(0, T17, 0) → f15_out_gaa(0, T17, 0)
f15_in_gaa(s(T21), T20, X33) → U4_gaa(T21, T20, X33, f15_in_gaa(T21, T20, X32))
f15_in_gaa(s(T21), T23, X33) → U5_gaa(T21, T23, X33, f15_in_gaa(T21, T23, T22))
U5_gaa(T21, T23, X33, f15_out_gaa(T21, T23, T22)) → U6_gaa(T21, T23, X33, f15_in_aaa(T22, T23, X33))
f15_in_aaa(0, T17, 0) → f15_out_aaa(0, T17, 0)
f15_in_aaa(s(T21), T20, X33) → U4_aaa(T21, T20, X33, f15_in_aaa(T21, T20, X32))
f15_in_aaa(s(T21), T23, X33) → U5_aaa(T21, T23, X33, f15_in_aaa(T21, T23, T22))
U5_aaa(T21, T23, X33, f15_out_aaa(T21, T23, T22)) → U6_aaa(T21, T23, X33, f15_in_aaa(T22, T23, X33))
U6_aaa(T21, T23, X33, f15_out_aaa(T22, T23, X33)) → f15_out_aaa(s(T21), T23, X33)
U4_aaa(T21, T20, X33, f15_out_aaa(T21, T20, X32)) → f15_out_aaa(s(T21), T20, X33)
U6_gaa(T21, T23, X33, f15_out_aaa(T22, T23, X33)) → f15_out_gaa(s(T21), T23, X33)
U4_gaa(T21, T20, X33, f15_out_gaa(T21, T20, X32)) → f15_out_gaa(s(T21), T20, X33)
U1_gaa(T13, T11, T12, f15_out_gaa(T13, T11, X14)) → f1_out_gaa(s(T13), T11, T12)
f1_in_gaa(s(T13), T15, T16) → U2_gaa(T13, T15, T16, f15_in_gaa(T13, T15, T14))
U2_gaa(T13, T15, T16, f15_out_gaa(T13, T15, T14)) → U3_gaa(T13, T15, T16, f1_in_aaa(T14, T15, T16))
f1_in_aaa(0, T4, 0) → f1_out_aaa(0, T4, 0)
f1_in_aaa(s(T13), T11, T12) → U1_aaa(T13, T11, T12, f15_in_aaa(T13, T11, X14))
U1_aaa(T13, T11, T12, f15_out_aaa(T13, T11, X14)) → f1_out_aaa(s(T13), T11, T12)
f1_in_aaa(s(T13), T15, T16) → U2_aaa(T13, T15, T16, f15_in_aaa(T13, T15, T14))
U2_aaa(T13, T15, T16, f15_out_aaa(T13, T15, T14)) → U3_aaa(T13, T15, T16, f1_in_aaa(T14, T15, T16))
U3_aaa(T13, T15, T16, f1_out_aaa(T14, T15, T16)) → f1_out_aaa(s(T13), T15, T16)
U3_gaa(T13, T15, T16, f1_out_aaa(T14, T15, T16)) → f1_out_gaa(s(T13), T15, T16)

(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:

F1_IN_GAA(s(T13), T11, T12) → U1_GAA(T13, T11, T12, f15_in_gaa(T13, T11, X14))
F1_IN_GAA(s(T13), T11, T12) → F15_IN_GAA(T13, T11, X14)
F15_IN_GAA(s(T21), T20, X33) → U4_GAA(T21, T20, X33, f15_in_gaa(T21, T20, X32))
F15_IN_GAA(s(T21), T20, X33) → F15_IN_GAA(T21, T20, X32)
F15_IN_GAA(s(T21), T23, X33) → U5_GAA(T21, T23, X33, f15_in_gaa(T21, T23, T22))
U5_GAA(T21, T23, X33, f15_out_gaa(T21, T23, T22)) → U6_GAA(T21, T23, X33, f15_in_aaa(T22, T23, X33))
U5_GAA(T21, T23, X33, f15_out_gaa(T21, T23, T22)) → F15_IN_AAA(T22, T23, X33)
F15_IN_AAA(s(T21), T20, X33) → U4_AAA(T21, T20, X33, f15_in_aaa(T21, T20, X32))
F15_IN_AAA(s(T21), T20, X33) → F15_IN_AAA(T21, T20, X32)
F15_IN_AAA(s(T21), T23, X33) → U5_AAA(T21, T23, X33, f15_in_aaa(T21, T23, T22))
U5_AAA(T21, T23, X33, f15_out_aaa(T21, T23, T22)) → U6_AAA(T21, T23, X33, f15_in_aaa(T22, T23, X33))
U5_AAA(T21, T23, X33, f15_out_aaa(T21, T23, T22)) → F15_IN_AAA(T22, T23, X33)
F1_IN_GAA(s(T13), T15, T16) → U2_GAA(T13, T15, T16, f15_in_gaa(T13, T15, T14))
U2_GAA(T13, T15, T16, f15_out_gaa(T13, T15, T14)) → U3_GAA(T13, T15, T16, f1_in_aaa(T14, T15, T16))
U2_GAA(T13, T15, T16, f15_out_gaa(T13, T15, T14)) → F1_IN_AAA(T14, T15, T16)
F1_IN_AAA(s(T13), T11, T12) → U1_AAA(T13, T11, T12, f15_in_aaa(T13, T11, X14))
F1_IN_AAA(s(T13), T11, T12) → F15_IN_AAA(T13, T11, X14)
F1_IN_AAA(s(T13), T15, T16) → U2_AAA(T13, T15, T16, f15_in_aaa(T13, T15, T14))
U2_AAA(T13, T15, T16, f15_out_aaa(T13, T15, T14)) → U3_AAA(T13, T15, T16, f1_in_aaa(T14, T15, T16))
U2_AAA(T13, T15, T16, f15_out_aaa(T13, T15, T14)) → F1_IN_AAA(T14, T15, T16)

The TRS R consists of the following rules:

f1_in_gaa(0, T4, 0) → f1_out_gaa(0, T4, 0)
f1_in_gaa(s(T13), T11, T12) → U1_gaa(T13, T11, T12, f15_in_gaa(T13, T11, X14))
f15_in_gaa(0, T17, 0) → f15_out_gaa(0, T17, 0)
f15_in_gaa(s(T21), T20, X33) → U4_gaa(T21, T20, X33, f15_in_gaa(T21, T20, X32))
f15_in_gaa(s(T21), T23, X33) → U5_gaa(T21, T23, X33, f15_in_gaa(T21, T23, T22))
U5_gaa(T21, T23, X33, f15_out_gaa(T21, T23, T22)) → U6_gaa(T21, T23, X33, f15_in_aaa(T22, T23, X33))
f15_in_aaa(0, T17, 0) → f15_out_aaa(0, T17, 0)
f15_in_aaa(s(T21), T20, X33) → U4_aaa(T21, T20, X33, f15_in_aaa(T21, T20, X32))
f15_in_aaa(s(T21), T23, X33) → U5_aaa(T21, T23, X33, f15_in_aaa(T21, T23, T22))
U5_aaa(T21, T23, X33, f15_out_aaa(T21, T23, T22)) → U6_aaa(T21, T23, X33, f15_in_aaa(T22, T23, X33))
U6_aaa(T21, T23, X33, f15_out_aaa(T22, T23, X33)) → f15_out_aaa(s(T21), T23, X33)
U4_aaa(T21, T20, X33, f15_out_aaa(T21, T20, X32)) → f15_out_aaa(s(T21), T20, X33)
U6_gaa(T21, T23, X33, f15_out_aaa(T22, T23, X33)) → f15_out_gaa(s(T21), T23, X33)
U4_gaa(T21, T20, X33, f15_out_gaa(T21, T20, X32)) → f15_out_gaa(s(T21), T20, X33)
U1_gaa(T13, T11, T12, f15_out_gaa(T13, T11, X14)) → f1_out_gaa(s(T13), T11, T12)
f1_in_gaa(s(T13), T15, T16) → U2_gaa(T13, T15, T16, f15_in_gaa(T13, T15, T14))
U2_gaa(T13, T15, T16, f15_out_gaa(T13, T15, T14)) → U3_gaa(T13, T15, T16, f1_in_aaa(T14, T15, T16))
f1_in_aaa(0, T4, 0) → f1_out_aaa(0, T4, 0)
f1_in_aaa(s(T13), T11, T12) → U1_aaa(T13, T11, T12, f15_in_aaa(T13, T11, X14))
U1_aaa(T13, T11, T12, f15_out_aaa(T13, T11, X14)) → f1_out_aaa(s(T13), T11, T12)
f1_in_aaa(s(T13), T15, T16) → U2_aaa(T13, T15, T16, f15_in_aaa(T13, T15, T14))
U2_aaa(T13, T15, T16, f15_out_aaa(T13, T15, T14)) → U3_aaa(T13, T15, T16, f1_in_aaa(T14, T15, T16))
U3_aaa(T13, T15, T16, f1_out_aaa(T14, T15, T16)) → f1_out_aaa(s(T13), T15, T16)
U3_gaa(T13, T15, T16, f1_out_aaa(T14, T15, T16)) → f1_out_gaa(s(T13), T15, T16)

The argument filtering Pi contains the following mapping:
f1_in_gaa(x1, x2, x3) = f1_in_gaa(x1)
0 = 0
f1_out_gaa(x1, x2, x3) = f1_out_gaa
s(x1) = s(x1)
U1_gaa(x1, x2, x3, x4) = U1_gaa(x4)
f15_in_gaa(x1, x2, x3) = f15_in_gaa(x1)
f15_out_gaa(x1, x2, x3) = f15_out_gaa
U4_gaa(x1, x2, x3, x4) = U4_gaa(x4)
U5_gaa(x1, x2, x3, x4) = U5_gaa(x4)
U6_gaa(x1, x2, x3, x4) = U6_gaa(x4)
f15_in_aaa(x1, x2, x3) = f15_in_aaa
f15_out_aaa(x1, x2, x3) = f15_out_aaa(x1)
U4_aaa(x1, x2, x3, x4) = U4_aaa(x4)
U5_aaa(x1, x2, x3, x4) = U5_aaa(x4)
U6_aaa(x1, x2, x3, x4) = U6_aaa(x1, x4)
U2_gaa(x1, x2, x3, x4) = U2_gaa(x4)
U3_gaa(x1, x2, x3, x4) = U3_gaa(x4)
f1_in_aaa(x1, x2, x3) = f1_in_aaa
f1_out_aaa(x1, x2, x3) = f1_out_aaa(x1)
U1_aaa(x1, x2, x3, x4) = U1_aaa(x4)
U2_aaa(x1, x2, x3, x4) = U2_aaa(x4)
U3_aaa(x1, x2, x3, x4) = U3_aaa(x1, x4)
F1_IN_GAA(x1, x2, x3) = F1_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4) = U1_GAA(x4)
F15_IN_GAA(x1, x2, x3) = F15_IN_GAA(x1)
U4_GAA(x1, x2, x3, x4) = U4_GAA(x4)
U5_GAA(x1, x2, x3, x4) = U5_GAA(x4)
U6_GAA(x1, x2, x3, x4) = U6_GAA(x4)
F15_IN_AAA(x1, x2, x3) = F15_IN_AAA
U4_AAA(x1, x2, x3, x4) = U4_AAA(x4)
U5_AAA(x1, x2, x3, x4) = U5_AAA(x4)
U6_AAA(x1, x2, x3, x4) = U6_AAA(x1, x4)
U2_GAA(x1, x2, x3, x4) = U2_GAA(x4)
U3_GAA(x1, x2, x3, x4) = U3_GAA(x4)
F1_IN_AAA(x1, x2, x3) = F1_IN_AAA
U1_AAA(x1, x2, x3, x4) = U1_AAA(x4)
U2_AAA(x1, x2, x3, x4) = U2_AAA(x4)
U3_AAA(x1, x2, x3, x4) = U3_AAA(x1, x4)

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

(6) Obligation:

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

F1_IN_GAA(s(T13), T11, T12) → U1_GAA(T13, T11, T12, f15_in_gaa(T13, T11, X14))
F1_IN_GAA(s(T13), T11, T12) → F15_IN_GAA(T13, T11, X14)
F15_IN_GAA(s(T21), T20, X33) → U4_GAA(T21, T20, X33, f15_in_gaa(T21, T20, X32))
F15_IN_GAA(s(T21), T20, X33) → F15_IN_GAA(T21, T20, X32)
F15_IN_GAA(s(T21), T23, X33) → U5_GAA(T21, T23, X33, f15_in_gaa(T21, T23, T22))
U5_GAA(T21, T23, X33, f15_out_gaa(T21, T23, T22)) → U6_GAA(T21, T23, X33, f15_in_aaa(T22, T23, X33))
U5_GAA(T21, T23, X33, f15_out_gaa(T21, T23, T22)) → F15_IN_AAA(T22, T23, X33)
F15_IN_AAA(s(T21), T20, X33) → U4_AAA(T21, T20, X33, f15_in_aaa(T21, T20, X32))
F15_IN_AAA(s(T21), T20, X33) → F15_IN_AAA(T21, T20, X32)
F15_IN_AAA(s(T21), T23, X33) → U5_AAA(T21, T23, X33, f15_in_aaa(T21, T23, T22))
U5_AAA(T21, T23, X33, f15_out_aaa(T21, T23, T22)) → U6_AAA(T21, T23, X33, f15_in_aaa(T22, T23, X33))
U5_AAA(T21, T23, X33, f15_out_aaa(T21, T23, T22)) → F15_IN_AAA(T22, T23, X33)
F1_IN_GAA(s(T13), T15, T16) → U2_GAA(T13, T15, T16, f15_in_gaa(T13, T15, T14))
U2_GAA(T13, T15, T16, f15_out_gaa(T13, T15, T14)) → U3_GAA(T13, T15, T16, f1_in_aaa(T14, T15, T16))
U2_GAA(T13, T15, T16, f15_out_gaa(T13, T15, T14)) → F1_IN_AAA(T14, T15, T16)
F1_IN_AAA(s(T13), T11, T12) → U1_AAA(T13, T11, T12, f15_in_aaa(T13, T11, X14))
F1_IN_AAA(s(T13), T11, T12) → F15_IN_AAA(T13, T11, X14)
F1_IN_AAA(s(T13), T15, T16) → U2_AAA(T13, T15, T16, f15_in_aaa(T13, T15, T14))
U2_AAA(T13, T15, T16, f15_out_aaa(T13, T15, T14)) → U3_AAA(T13, T15, T16, f1_in_aaa(T14, T15, T16))
U2_AAA(T13, T15, T16, f15_out_aaa(T13, T15, T14)) → F1_IN_AAA(T14, T15, T16)

The TRS R consists of the following rules:

f1_in_gaa(0, T4, 0) → f1_out_gaa(0, T4, 0)
f1_in_gaa(s(T13), T11, T12) → U1_gaa(T13, T11, T12, f15_in_gaa(T13, T11, X14))
f15_in_gaa(0, T17, 0) → f15_out_gaa(0, T17, 0)
f15_in_gaa(s(T21), T20, X33) → U4_gaa(T21, T20, X33, f15_in_gaa(T21, T20, X32))
f15_in_gaa(s(T21), T23, X33) → U5_gaa(T21, T23, X33, f15_in_gaa(T21, T23, T22))
U5_gaa(T21, T23, X33, f15_out_gaa(T21, T23, T22)) → U6_gaa(T21, T23, X33, f15_in_aaa(T22, T23, X33))
f15_in_aaa(0, T17, 0) → f15_out_aaa(0, T17, 0)
f15_in_aaa(s(T21), T20, X33) → U4_aaa(T21, T20, X33, f15_in_aaa(T21, T20, X32))
f15_in_aaa(s(T21), T23, X33) → U5_aaa(T21, T23, X33, f15_in_aaa(T21, T23, T22))
U5_aaa(T21, T23, X33, f15_out_aaa(T21, T23, T22)) → U6_aaa(T21, T23, X33, f15_in_aaa(T22, T23, X33))
U6_aaa(T21, T23, X33, f15_out_aaa(T22, T23, X33)) → f15_out_aaa(s(T21), T23, X33)
U4_aaa(T21, T20, X33, f15_out_aaa(T21, T20, X32)) → f15_out_aaa(s(T21), T20, X33)
U6_gaa(T21, T23, X33, f15_out_aaa(T22, T23, X33)) → f15_out_gaa(s(T21), T23, X33)
U4_gaa(T21, T20, X33, f15_out_gaa(T21, T20, X32)) → f15_out_gaa(s(T21), T20, X33)
U1_gaa(T13, T11, T12, f15_out_gaa(T13, T11, X14)) → f1_out_gaa(s(T13), T11, T12)
f1_in_gaa(s(T13), T15, T16) → U2_gaa(T13, T15, T16, f15_in_gaa(T13, T15, T14))
U2_gaa(T13, T15, T16, f15_out_gaa(T13, T15, T14)) → U3_gaa(T13, T15, T16, f1_in_aaa(T14, T15, T16))
f1_in_aaa(0, T4, 0) → f1_out_aaa(0, T4, 0)
f1_in_aaa(s(T13), T11, T12) → U1_aaa(T13, T11, T12, f15_in_aaa(T13, T11, X14))
U1_aaa(T13, T11, T12, f15_out_aaa(T13, T11, X14)) → f1_out_aaa(s(T13), T11, T12)
f1_in_aaa(s(T13), T15, T16) → U2_aaa(T13, T15, T16, f15_in_aaa(T13, T15, T14))
U2_aaa(T13, T15, T16, f15_out_aaa(T13, T15, T14)) → U3_aaa(T13, T15, T16, f1_in_aaa(T14, T15, T16))
U3_aaa(T13, T15, T16, f1_out_aaa(T14, T15, T16)) → f1_out_aaa(s(T13), T15, T16)
U3_gaa(T13, T15, T16, f1_out_aaa(T14, T15, T16)) → f1_out_gaa(s(T13), T15, T16)

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 14 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

F15_IN_AAA(s(T21), T23, X33) → U5_AAA(T21, T23, X33, f15_in_aaa(T21, T23, T22))
U5_AAA(T21, T23, X33, f15_out_aaa(T21, T23, T22)) → F15_IN_AAA(T22, T23, X33)
F15_IN_AAA(s(T21), T20, X33) → F15_IN_AAA(T21, T20, X32)

The TRS R consists of the following rules:

f1_in_gaa(0, T4, 0) → f1_out_gaa(0, T4, 0)
f1_in_gaa(s(T13), T11, T12) → U1_gaa(T13, T11, T12, f15_in_gaa(T13, T11, X14))
f15_in_gaa(0, T17, 0) → f15_out_gaa(0, T17, 0)
f15_in_gaa(s(T21), T20, X33) → U4_gaa(T21, T20, X33, f15_in_gaa(T21, T20, X32))
f15_in_gaa(s(T21), T23, X33) → U5_gaa(T21, T23, X33, f15_in_gaa(T21, T23, T22))
U5_gaa(T21, T23, X33, f15_out_gaa(T21, T23, T22)) → U6_gaa(T21, T23, X33, f15_in_aaa(T22, T23, X33))
f15_in_aaa(0, T17, 0) → f15_out_aaa(0, T17, 0)
f15_in_aaa(s(T21), T20, X33) → U4_aaa(T21, T20, X33, f15_in_aaa(T21, T20, X32))
f15_in_aaa(s(T21), T23, X33) → U5_aaa(T21, T23, X33, f15_in_aaa(T21, T23, T22))
U5_aaa(T21, T23, X33, f15_out_aaa(T21, T23, T22)) → U6_aaa(T21, T23, X33, f15_in_aaa(T22, T23, X33))
U6_aaa(T21, T23, X33, f15_out_aaa(T22, T23, X33)) → f15_out_aaa(s(T21), T23, X33)
U4_aaa(T21, T20, X33, f15_out_aaa(T21, T20, X32)) → f15_out_aaa(s(T21), T20, X33)
U6_gaa(T21, T23, X33, f15_out_aaa(T22, T23, X33)) → f15_out_gaa(s(T21), T23, X33)
U4_gaa(T21, T20, X33, f15_out_gaa(T21, T20, X32)) → f15_out_gaa(s(T21), T20, X33)
U1_gaa(T13, T11, T12, f15_out_gaa(T13, T11, X14)) → f1_out_gaa(s(T13), T11, T12)
f1_in_gaa(s(T13), T15, T16) → U2_gaa(T13, T15, T16, f15_in_gaa(T13, T15, T14))
U2_gaa(T13, T15, T16, f15_out_gaa(T13, T15, T14)) → U3_gaa(T13, T15, T16, f1_in_aaa(T14, T15, T16))
f1_in_aaa(0, T4, 0) → f1_out_aaa(0, T4, 0)
f1_in_aaa(s(T13), T11, T12) → U1_aaa(T13, T11, T12, f15_in_aaa(T13, T11, X14))
U1_aaa(T13, T11, T12, f15_out_aaa(T13, T11, X14)) → f1_out_aaa(s(T13), T11, T12)
f1_in_aaa(s(T13), T15, T16) → U2_aaa(T13, T15, T16, f15_in_aaa(T13, T15, T14))
U2_aaa(T13, T15, T16, f15_out_aaa(T13, T15, T14)) → U3_aaa(T13, T15, T16, f1_in_aaa(T14, T15, T16))
U3_aaa(T13, T15, T16, f1_out_aaa(T14, T15, T16)) → f1_out_aaa(s(T13), T15, T16)
U3_gaa(T13, T15, T16, f1_out_aaa(T14, T15, T16)) → f1_out_gaa(s(T13), T15, T16)

(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:

F15_IN_AAA(s(T21), T23, X33) → U5_AAA(T21, T23, X33, f15_in_aaa(T21, T23, T22))
U5_AAA(T21, T23, X33, f15_out_aaa(T21, T23, T22)) → F15_IN_AAA(T22, T23, X33)
F15_IN_AAA(s(T21), T20, X33) → F15_IN_AAA(T21, T20, X32)

The TRS R consists of the following rules:

f15_in_aaa(0, T17, 0) → f15_out_aaa(0, T17, 0)
f15_in_aaa(s(T21), T20, X33) → U4_aaa(T21, T20, X33, f15_in_aaa(T21, T20, X32))
f15_in_aaa(s(T21), T23, X33) → U5_aaa(T21, T23, X33, f15_in_aaa(T21, T23, T22))
U4_aaa(T21, T20, X33, f15_out_aaa(T21, T20, X32)) → f15_out_aaa(s(T21), T20, X33)
U5_aaa(T21, T23, X33, f15_out_aaa(T21, T23, T22)) → U6_aaa(T21, T23, X33, f15_in_aaa(T22, T23, X33))
U6_aaa(T21, T23, X33, f15_out_aaa(T22, T23, X33)) → f15_out_aaa(s(T21), T23, X33)

The argument filtering Pi contains the following mapping:
0 = 0
s(x1) = s(x1)
f15_in_aaa(x1, x2, x3) = f15_in_aaa
f15_out_aaa(x1, x2, x3) = f15_out_aaa(x1)
U4_aaa(x1, x2, x3, x4) = U4_aaa(x4)
U5_aaa(x1, x2, x3, x4) = U5_aaa(x4)
U6_aaa(x1, x2, x3, x4) = U6_aaa(x1, x4)
F15_IN_AAA(x1, x2, x3) = F15_IN_AAA
U5_AAA(x1, x2, x3, x4) = U5_AAA(x4)

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:

F15_IN_AAA → U5_AAA(f15_in_aaa)
U5_AAA(f15_out_aaa(T21)) → F15_IN_AAA
F15_IN_AAA → F15_IN_AAA

The TRS R consists of the following rules:

f15_in_aaa → f15_out_aaa(0)
f15_in_aaa → U4_aaa(f15_in_aaa)
f15_in_aaa → U5_aaa(f15_in_aaa)
U4_aaa(f15_out_aaa(T21)) → f15_out_aaa(s(T21))
U5_aaa(f15_out_aaa(T21)) → U6_aaa(T21, f15_in_aaa)
U6_aaa(T21, f15_out_aaa(T22)) → f15_out_aaa(s(T21))

The set Q consists of the following terms:

f15_in_aaa
U4_aaa(x0)
U5_aaa(x0)
U6_aaa(x0, x1)

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

(14) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule F15_IN_AAA → U5_AAA(f15_in_aaa) at position [0] we obtained the following new rules [LPAR04]:

F15_IN_AAA → U5_AAA(f15_out_aaa(0))
F15_IN_AAA → U5_AAA(U4_aaa(f15_in_aaa))
F15_IN_AAA → U5_AAA(U5_aaa(f15_in_aaa))

(15) Obligation:

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

U5_AAA(f15_out_aaa(T21)) → F15_IN_AAA
F15_IN_AAA → F15_IN_AAA
F15_IN_AAA → U5_AAA(f15_out_aaa(0))
F15_IN_AAA → U5_AAA(U4_aaa(f15_in_aaa))
F15_IN_AAA → U5_AAA(U5_aaa(f15_in_aaa))

The TRS R consists of the following rules:

f15_in_aaa → f15_out_aaa(0)
f15_in_aaa → U4_aaa(f15_in_aaa)
f15_in_aaa → U5_aaa(f15_in_aaa)
U4_aaa(f15_out_aaa(T21)) → f15_out_aaa(s(T21))
U5_aaa(f15_out_aaa(T21)) → U6_aaa(T21, f15_in_aaa)
U6_aaa(T21, f15_out_aaa(T22)) → f15_out_aaa(s(T21))

The set Q consists of the following terms:

f15_in_aaa
U4_aaa(x0)
U5_aaa(x0)
U6_aaa(x0, x1)

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

(16) 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 = F15_IN_AAA evaluates to t =F15_IN_AAA

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

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from F15_IN_AAA to F15_IN_AAA.

(17) FALSE

(18) Obligation:

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

F1_IN_AAA(s(T13), T15, T16) → U2_AAA(T13, T15, T16, f15_in_aaa(T13, T15, T14))
U2_AAA(T13, T15, T16, f15_out_aaa(T13, T15, T14)) → F1_IN_AAA(T14, T15, T16)

The TRS R consists of the following rules:

f1_in_gaa(0, T4, 0) → f1_out_gaa(0, T4, 0)
f1_in_gaa(s(T13), T11, T12) → U1_gaa(T13, T11, T12, f15_in_gaa(T13, T11, X14))
f15_in_gaa(0, T17, 0) → f15_out_gaa(0, T17, 0)
f15_in_gaa(s(T21), T20, X33) → U4_gaa(T21, T20, X33, f15_in_gaa(T21, T20, X32))
f15_in_gaa(s(T21), T23, X33) → U5_gaa(T21, T23, X33, f15_in_gaa(T21, T23, T22))
U5_gaa(T21, T23, X33, f15_out_gaa(T21, T23, T22)) → U6_gaa(T21, T23, X33, f15_in_aaa(T22, T23, X33))
f15_in_aaa(0, T17, 0) → f15_out_aaa(0, T17, 0)
f15_in_aaa(s(T21), T20, X33) → U4_aaa(T21, T20, X33, f15_in_aaa(T21, T20, X32))
f15_in_aaa(s(T21), T23, X33) → U5_aaa(T21, T23, X33, f15_in_aaa(T21, T23, T22))
U5_aaa(T21, T23, X33, f15_out_aaa(T21, T23, T22)) → U6_aaa(T21, T23, X33, f15_in_aaa(T22, T23, X33))
U6_aaa(T21, T23, X33, f15_out_aaa(T22, T23, X33)) → f15_out_aaa(s(T21), T23, X33)
U4_aaa(T21, T20, X33, f15_out_aaa(T21, T20, X32)) → f15_out_aaa(s(T21), T20, X33)
U6_gaa(T21, T23, X33, f15_out_aaa(T22, T23, X33)) → f15_out_gaa(s(T21), T23, X33)
U4_gaa(T21, T20, X33, f15_out_gaa(T21, T20, X32)) → f15_out_gaa(s(T21), T20, X33)
U1_gaa(T13, T11, T12, f15_out_gaa(T13, T11, X14)) → f1_out_gaa(s(T13), T11, T12)
f1_in_gaa(s(T13), T15, T16) → U2_gaa(T13, T15, T16, f15_in_gaa(T13, T15, T14))
U2_gaa(T13, T15, T16, f15_out_gaa(T13, T15, T14)) → U3_gaa(T13, T15, T16, f1_in_aaa(T14, T15, T16))
f1_in_aaa(0, T4, 0) → f1_out_aaa(0, T4, 0)
f1_in_aaa(s(T13), T11, T12) → U1_aaa(T13, T11, T12, f15_in_aaa(T13, T11, X14))
U1_aaa(T13, T11, T12, f15_out_aaa(T13, T11, X14)) → f1_out_aaa(s(T13), T11, T12)
f1_in_aaa(s(T13), T15, T16) → U2_aaa(T13, T15, T16, f15_in_aaa(T13, T15, T14))
U2_aaa(T13, T15, T16, f15_out_aaa(T13, T15, T14)) → U3_aaa(T13, T15, T16, f1_in_aaa(T14, T15, T16))
U3_aaa(T13, T15, T16, f1_out_aaa(T14, T15, T16)) → f1_out_aaa(s(T13), T15, T16)
U3_gaa(T13, T15, T16, f1_out_aaa(T14, T15, T16)) → f1_out_gaa(s(T13), T15, T16)

(19) UsableRulesProof (EQUIVALENT transformation)

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

(20) Obligation:

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

F1_IN_AAA(s(T13), T15, T16) → U2_AAA(T13, T15, T16, f15_in_aaa(T13, T15, T14))
U2_AAA(T13, T15, T16, f15_out_aaa(T13, T15, T14)) → F1_IN_AAA(T14, T15, T16)

The TRS R consists of the following rules:

f15_in_aaa(0, T17, 0) → f15_out_aaa(0, T17, 0)
f15_in_aaa(s(T21), T20, X33) → U4_aaa(T21, T20, X33, f15_in_aaa(T21, T20, X32))
f15_in_aaa(s(T21), T23, X33) → U5_aaa(T21, T23, X33, f15_in_aaa(T21, T23, T22))
U4_aaa(T21, T20, X33, f15_out_aaa(T21, T20, X32)) → f15_out_aaa(s(T21), T20, X33)
U5_aaa(T21, T23, X33, f15_out_aaa(T21, T23, T22)) → U6_aaa(T21, T23, X33, f15_in_aaa(T22, T23, X33))
U6_aaa(T21, T23, X33, f15_out_aaa(T22, T23, X33)) → f15_out_aaa(s(T21), T23, X33)

The argument filtering Pi contains the following mapping:
0 = 0
s(x1) = s(x1)
f15_in_aaa(x1, x2, x3) = f15_in_aaa
f15_out_aaa(x1, x2, x3) = f15_out_aaa(x1)
U4_aaa(x1, x2, x3, x4) = U4_aaa(x4)
U5_aaa(x1, x2, x3, x4) = U5_aaa(x4)
U6_aaa(x1, x2, x3, x4) = U6_aaa(x1, x4)
F1_IN_AAA(x1, x2, x3) = F1_IN_AAA
U2_AAA(x1, x2, x3, x4) = U2_AAA(x4)

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

(21) PiDPToQDPProof (SOUND transformation)

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

(22) Obligation:

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

F1_IN_AAA → U2_AAA(f15_in_aaa)
U2_AAA(f15_out_aaa(T13)) → F1_IN_AAA

The TRS R consists of the following rules:

f15_in_aaa → f15_out_aaa(0)
f15_in_aaa → U4_aaa(f15_in_aaa)
f15_in_aaa → U5_aaa(f15_in_aaa)
U4_aaa(f15_out_aaa(T21)) → f15_out_aaa(s(T21))
U5_aaa(f15_out_aaa(T21)) → U6_aaa(T21, f15_in_aaa)
U6_aaa(T21, f15_out_aaa(T22)) → f15_out_aaa(s(T21))

The set Q consists of the following terms:

f15_in_aaa
U4_aaa(x0)
U5_aaa(x0)
U6_aaa(x0, x1)

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

(23) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule F1_IN_AAA → U2_AAA(f15_in_aaa) at position [0] we obtained the following new rules [LPAR04]:

F1_IN_AAA → U2_AAA(f15_out_aaa(0))
F1_IN_AAA → U2_AAA(U4_aaa(f15_in_aaa))
F1_IN_AAA → U2_AAA(U5_aaa(f15_in_aaa))

(24) Obligation:

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

U2_AAA(f15_out_aaa(T13)) → F1_IN_AAA
F1_IN_AAA → U2_AAA(f15_out_aaa(0))
F1_IN_AAA → U2_AAA(U4_aaa(f15_in_aaa))
F1_IN_AAA → U2_AAA(U5_aaa(f15_in_aaa))

The TRS R consists of the following rules:

f15_in_aaa → f15_out_aaa(0)
f15_in_aaa → U4_aaa(f15_in_aaa)
f15_in_aaa → U5_aaa(f15_in_aaa)
U4_aaa(f15_out_aaa(T21)) → f15_out_aaa(s(T21))
U5_aaa(f15_out_aaa(T21)) → U6_aaa(T21, f15_in_aaa)
U6_aaa(T21, f15_out_aaa(T22)) → f15_out_aaa(s(T21))

The set Q consists of the following terms:

f15_in_aaa
U4_aaa(x0)
U5_aaa(x0)
U6_aaa(x0, x1)

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

(25) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = F1_IN_AAA evaluates to t =F1_IN_AAA

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

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

F1_IN_AAA → U2_AAA(f15_out_aaa(0))
with rule F1_IN_AAA → U2_AAA(f15_out_aaa(0)) at position [] and matcher [ ]

U2_AAA(f15_out_aaa(0)) → F1_IN_AAA
with rule U2_AAA(f15_out_aaa(T13)) → F1_IN_AAA

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence

All these steps are and every following step will be a correct step w.r.t to Q.

(26) FALSE

(27) Obligation:

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

F15_IN_GAA(s(T21), T20, X33) → F15_IN_GAA(T21, T20, X32)

The TRS R consists of the following rules:

f1_in_gaa(0, T4, 0) → f1_out_gaa(0, T4, 0)
f1_in_gaa(s(T13), T11, T12) → U1_gaa(T13, T11, T12, f15_in_gaa(T13, T11, X14))
f15_in_gaa(0, T17, 0) → f15_out_gaa(0, T17, 0)
f15_in_gaa(s(T21), T20, X33) → U4_gaa(T21, T20, X33, f15_in_gaa(T21, T20, X32))
f15_in_gaa(s(T21), T23, X33) → U5_gaa(T21, T23, X33, f15_in_gaa(T21, T23, T22))
U5_gaa(T21, T23, X33, f15_out_gaa(T21, T23, T22)) → U6_gaa(T21, T23, X33, f15_in_aaa(T22, T23, X33))
f15_in_aaa(0, T17, 0) → f15_out_aaa(0, T17, 0)
f15_in_aaa(s(T21), T20, X33) → U4_aaa(T21, T20, X33, f15_in_aaa(T21, T20, X32))
f15_in_aaa(s(T21), T23, X33) → U5_aaa(T21, T23, X33, f15_in_aaa(T21, T23, T22))
U5_aaa(T21, T23, X33, f15_out_aaa(T21, T23, T22)) → U6_aaa(T21, T23, X33, f15_in_aaa(T22, T23, X33))
U6_aaa(T21, T23, X33, f15_out_aaa(T22, T23, X33)) → f15_out_aaa(s(T21), T23, X33)
U4_aaa(T21, T20, X33, f15_out_aaa(T21, T20, X32)) → f15_out_aaa(s(T21), T20, X33)
U6_gaa(T21, T23, X33, f15_out_aaa(T22, T23, X33)) → f15_out_gaa(s(T21), T23, X33)
U4_gaa(T21, T20, X33, f15_out_gaa(T21, T20, X32)) → f15_out_gaa(s(T21), T20, X33)
U1_gaa(T13, T11, T12, f15_out_gaa(T13, T11, X14)) → f1_out_gaa(s(T13), T11, T12)
f1_in_gaa(s(T13), T15, T16) → U2_gaa(T13, T15, T16, f15_in_gaa(T13, T15, T14))
U2_gaa(T13, T15, T16, f15_out_gaa(T13, T15, T14)) → U3_gaa(T13, T15, T16, f1_in_aaa(T14, T15, T16))
f1_in_aaa(0, T4, 0) → f1_out_aaa(0, T4, 0)
f1_in_aaa(s(T13), T11, T12) → U1_aaa(T13, T11, T12, f15_in_aaa(T13, T11, X14))
U1_aaa(T13, T11, T12, f15_out_aaa(T13, T11, X14)) → f1_out_aaa(s(T13), T11, T12)
f1_in_aaa(s(T13), T15, T16) → U2_aaa(T13, T15, T16, f15_in_aaa(T13, T15, T14))
U2_aaa(T13, T15, T16, f15_out_aaa(T13, T15, T14)) → U3_aaa(T13, T15, T16, f1_in_aaa(T14, T15, T16))
U3_aaa(T13, T15, T16, f1_out_aaa(T14, T15, T16)) → f1_out_aaa(s(T13), T15, T16)
U3_gaa(T13, T15, T16, f1_out_aaa(T14, T15, T16)) → f1_out_gaa(s(T13), T15, T16)

(28) UsableRulesProof (EQUIVALENT transformation)

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

(29) Obligation:

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

F15_IN_GAA(s(T21), T20, X33) → F15_IN_GAA(T21, T20, X32)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
F15_IN_GAA(x1, x2, x3) = F15_IN_GAA(x1)

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

(30) PiDPToQDPProof (SOUND transformation)

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

(31) Obligation:

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

F15_IN_GAA(s(T21)) → F15_IN_GAA(T21)

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

(32) 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:

F15_IN_GAA(s(T21)) → F15_IN_GAA(T21)
The graph contains the following edges 1 > 1

(33) TRUE

(34) PrologToPiTRSProof (SOUND transformation)

f1_in_gaa(0, T4, 0) → f1_out_gaa(0, T4, 0)
f1_in_gaa(s(T13), T11, T12) → U1_gaa(T13, T11, T12, f15_in_gaa(T13, T11, X14))
f15_in_gaa(0, T17, 0) → f15_out_gaa(0, T17, 0)
f15_in_gaa(s(T21), T20, X33) → U4_gaa(T21, T20, X33, f15_in_gaa(T21, T20, X32))
f15_in_gaa(s(T21), T23, X33) → U5_gaa(T21, T23, X33, f15_in_gaa(T21, T23, T22))
U5_gaa(T21, T23, X33, f15_out_gaa(T21, T23, T22)) → U6_gaa(T21, T23, X33, f15_in_aaa(T22, T23, X33))
f15_in_aaa(0, T17, 0) → f15_out_aaa(0, T17, 0)
f15_in_aaa(s(T21), T20, X33) → U4_aaa(T21, T20, X33, f15_in_aaa(T21, T20, X32))
f15_in_aaa(s(T21), T23, X33) → U5_aaa(T21, T23, X33, f15_in_aaa(T21, T23, T22))
U5_aaa(T21, T23, X33, f15_out_aaa(T21, T23, T22)) → U6_aaa(T21, T23, X33, f15_in_aaa(T22, T23, X33))
U6_aaa(T21, T23, X33, f15_out_aaa(T22, T23, X33)) → f15_out_aaa(s(T21), T23, X33)
U4_aaa(T21, T20, X33, f15_out_aaa(T21, T20, X32)) → f15_out_aaa(s(T21), T20, X33)
U6_gaa(T21, T23, X33, f15_out_aaa(T22, T23, X33)) → f15_out_gaa(s(T21), T23, X33)
U4_gaa(T21, T20, X33, f15_out_gaa(T21, T20, X32)) → f15_out_gaa(s(T21), T20, X33)
U1_gaa(T13, T11, T12, f15_out_gaa(T13, T11, X14)) → f1_out_gaa(s(T13), T11, T12)
f1_in_gaa(s(T13), T15, T16) → U2_gaa(T13, T15, T16, f15_in_gaa(T13, T15, T14))
U2_gaa(T13, T15, T16, f15_out_gaa(T13, T15, T14)) → U3_gaa(T13, T15, T16, f1_in_aaa(T14, T15, T16))
f1_in_aaa(0, T4, 0) → f1_out_aaa(0, T4, 0)
f1_in_aaa(s(T13), T11, T12) → U1_aaa(T13, T11, T12, f15_in_aaa(T13, T11, X14))
U1_aaa(T13, T11, T12, f15_out_aaa(T13, T11, X14)) → f1_out_aaa(s(T13), T11, T12)
f1_in_aaa(s(T13), T15, T16) → U2_aaa(T13, T15, T16, f15_in_aaa(T13, T15, T14))
U2_aaa(T13, T15, T16, f15_out_aaa(T13, T15, T14)) → U3_aaa(T13, T15, T16, f1_in_aaa(T14, T15, T16))
U3_aaa(T13, T15, T16, f1_out_aaa(T14, T15, T16)) → f1_out_aaa(s(T13), T15, T16)
U3_gaa(T13, T15, T16, f1_out_aaa(T14, T15, T16)) → f1_out_gaa(s(T13), T15, T16)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(35) Obligation:

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

f1_in_gaa(0, T4, 0) → f1_out_gaa(0, T4, 0)
f1_in_gaa(s(T13), T11, T12) → U1_gaa(T13, T11, T12, f15_in_gaa(T13, T11, X14))
f15_in_gaa(0, T17, 0) → f15_out_gaa(0, T17, 0)
f15_in_gaa(s(T21), T20, X33) → U4_gaa(T21, T20, X33, f15_in_gaa(T21, T20, X32))
f15_in_gaa(s(T21), T23, X33) → U5_gaa(T21, T23, X33, f15_in_gaa(T21, T23, T22))
U5_gaa(T21, T23, X33, f15_out_gaa(T21, T23, T22)) → U6_gaa(T21, T23, X33, f15_in_aaa(T22, T23, X33))
f15_in_aaa(0, T17, 0) → f15_out_aaa(0, T17, 0)
f15_in_aaa(s(T21), T20, X33) → U4_aaa(T21, T20, X33, f15_in_aaa(T21, T20, X32))
f15_in_aaa(s(T21), T23, X33) → U5_aaa(T21, T23, X33, f15_in_aaa(T21, T23, T22))
U5_aaa(T21, T23, X33, f15_out_aaa(T21, T23, T22)) → U6_aaa(T21, T23, X33, f15_in_aaa(T22, T23, X33))
U6_aaa(T21, T23, X33, f15_out_aaa(T22, T23, X33)) → f15_out_aaa(s(T21), T23, X33)
U4_aaa(T21, T20, X33, f15_out_aaa(T21, T20, X32)) → f15_out_aaa(s(T21), T20, X33)
U6_gaa(T21, T23, X33, f15_out_aaa(T22, T23, X33)) → f15_out_gaa(s(T21), T23, X33)
U4_gaa(T21, T20, X33, f15_out_gaa(T21, T20, X32)) → f15_out_gaa(s(T21), T20, X33)
U1_gaa(T13, T11, T12, f15_out_gaa(T13, T11, X14)) → f1_out_gaa(s(T13), T11, T12)
f1_in_gaa(s(T13), T15, T16) → U2_gaa(T13, T15, T16, f15_in_gaa(T13, T15, T14))
U2_gaa(T13, T15, T16, f15_out_gaa(T13, T15, T14)) → U3_gaa(T13, T15, T16, f1_in_aaa(T14, T15, T16))
f1_in_aaa(0, T4, 0) → f1_out_aaa(0, T4, 0)
f1_in_aaa(s(T13), T11, T12) → U1_aaa(T13, T11, T12, f15_in_aaa(T13, T11, X14))
U1_aaa(T13, T11, T12, f15_out_aaa(T13, T11, X14)) → f1_out_aaa(s(T13), T11, T12)
f1_in_aaa(s(T13), T15, T16) → U2_aaa(T13, T15, T16, f15_in_aaa(T13, T15, T14))
U2_aaa(T13, T15, T16, f15_out_aaa(T13, T15, T14)) → U3_aaa(T13, T15, T16, f1_in_aaa(T14, T15, T16))
U3_aaa(T13, T15, T16, f1_out_aaa(T14, T15, T16)) → f1_out_aaa(s(T13), T15, T16)
U3_gaa(T13, T15, T16, f1_out_aaa(T14, T15, T16)) → f1_out_gaa(s(T13), T15, T16)

(36) 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:

F1_IN_GAA(s(T13), T11, T12) → U1_GAA(T13, T11, T12, f15_in_gaa(T13, T11, X14))
F1_IN_GAA(s(T13), T11, T12) → F15_IN_GAA(T13, T11, X14)
F15_IN_GAA(s(T21), T20, X33) → U4_GAA(T21, T20, X33, f15_in_gaa(T21, T20, X32))
F15_IN_GAA(s(T21), T20, X33) → F15_IN_GAA(T21, T20, X32)
F15_IN_GAA(s(T21), T23, X33) → U5_GAA(T21, T23, X33, f15_in_gaa(T21, T23, T22))
U5_GAA(T21, T23, X33, f15_out_gaa(T21, T23, T22)) → U6_GAA(T21, T23, X33, f15_in_aaa(T22, T23, X33))
U5_GAA(T21, T23, X33, f15_out_gaa(T21, T23, T22)) → F15_IN_AAA(T22, T23, X33)
F15_IN_AAA(s(T21), T20, X33) → U4_AAA(T21, T20, X33, f15_in_aaa(T21, T20, X32))
F15_IN_AAA(s(T21), T20, X33) → F15_IN_AAA(T21, T20, X32)
F15_IN_AAA(s(T21), T23, X33) → U5_AAA(T21, T23, X33, f15_in_aaa(T21, T23, T22))
U5_AAA(T21, T23, X33, f15_out_aaa(T21, T23, T22)) → U6_AAA(T21, T23, X33, f15_in_aaa(T22, T23, X33))
U5_AAA(T21, T23, X33, f15_out_aaa(T21, T23, T22)) → F15_IN_AAA(T22, T23, X33)
F1_IN_GAA(s(T13), T15, T16) → U2_GAA(T13, T15, T16, f15_in_gaa(T13, T15, T14))
U2_GAA(T13, T15, T16, f15_out_gaa(T13, T15, T14)) → U3_GAA(T13, T15, T16, f1_in_aaa(T14, T15, T16))
U2_GAA(T13, T15, T16, f15_out_gaa(T13, T15, T14)) → F1_IN_AAA(T14, T15, T16)
F1_IN_AAA(s(T13), T11, T12) → U1_AAA(T13, T11, T12, f15_in_aaa(T13, T11, X14))
F1_IN_AAA(s(T13), T11, T12) → F15_IN_AAA(T13, T11, X14)
F1_IN_AAA(s(T13), T15, T16) → U2_AAA(T13, T15, T16, f15_in_aaa(T13, T15, T14))
U2_AAA(T13, T15, T16, f15_out_aaa(T13, T15, T14)) → U3_AAA(T13, T15, T16, f1_in_aaa(T14, T15, T16))
U2_AAA(T13, T15, T16, f15_out_aaa(T13, T15, T14)) → F1_IN_AAA(T14, T15, T16)

The TRS R consists of the following rules:

f1_in_gaa(0, T4, 0) → f1_out_gaa(0, T4, 0)
f1_in_gaa(s(T13), T11, T12) → U1_gaa(T13, T11, T12, f15_in_gaa(T13, T11, X14))
f15_in_gaa(0, T17, 0) → f15_out_gaa(0, T17, 0)
f15_in_gaa(s(T21), T20, X33) → U4_gaa(T21, T20, X33, f15_in_gaa(T21, T20, X32))
f15_in_gaa(s(T21), T23, X33) → U5_gaa(T21, T23, X33, f15_in_gaa(T21, T23, T22))
U5_gaa(T21, T23, X33, f15_out_gaa(T21, T23, T22)) → U6_gaa(T21, T23, X33, f15_in_aaa(T22, T23, X33))
f15_in_aaa(0, T17, 0) → f15_out_aaa(0, T17, 0)
f15_in_aaa(s(T21), T20, X33) → U4_aaa(T21, T20, X33, f15_in_aaa(T21, T20, X32))
f15_in_aaa(s(T21), T23, X33) → U5_aaa(T21, T23, X33, f15_in_aaa(T21, T23, T22))
U5_aaa(T21, T23, X33, f15_out_aaa(T21, T23, T22)) → U6_aaa(T21, T23, X33, f15_in_aaa(T22, T23, X33))
U6_aaa(T21, T23, X33, f15_out_aaa(T22, T23, X33)) → f15_out_aaa(s(T21), T23, X33)
U4_aaa(T21, T20, X33, f15_out_aaa(T21, T20, X32)) → f15_out_aaa(s(T21), T20, X33)
U6_gaa(T21, T23, X33, f15_out_aaa(T22, T23, X33)) → f15_out_gaa(s(T21), T23, X33)
U4_gaa(T21, T20, X33, f15_out_gaa(T21, T20, X32)) → f15_out_gaa(s(T21), T20, X33)
U1_gaa(T13, T11, T12, f15_out_gaa(T13, T11, X14)) → f1_out_gaa(s(T13), T11, T12)
f1_in_gaa(s(T13), T15, T16) → U2_gaa(T13, T15, T16, f15_in_gaa(T13, T15, T14))
U2_gaa(T13, T15, T16, f15_out_gaa(T13, T15, T14)) → U3_gaa(T13, T15, T16, f1_in_aaa(T14, T15, T16))
f1_in_aaa(0, T4, 0) → f1_out_aaa(0, T4, 0)
f1_in_aaa(s(T13), T11, T12) → U1_aaa(T13, T11, T12, f15_in_aaa(T13, T11, X14))
U1_aaa(T13, T11, T12, f15_out_aaa(T13, T11, X14)) → f1_out_aaa(s(T13), T11, T12)
f1_in_aaa(s(T13), T15, T16) → U2_aaa(T13, T15, T16, f15_in_aaa(T13, T15, T14))
U2_aaa(T13, T15, T16, f15_out_aaa(T13, T15, T14)) → U3_aaa(T13, T15, T16, f1_in_aaa(T14, T15, T16))
U3_aaa(T13, T15, T16, f1_out_aaa(T14, T15, T16)) → f1_out_aaa(s(T13), T15, T16)
U3_gaa(T13, T15, T16, f1_out_aaa(T14, T15, T16)) → f1_out_gaa(s(T13), T15, T16)

The argument filtering Pi contains the following mapping:
f1_in_gaa(x1, x2, x3) = f1_in_gaa(x1)
0 = 0
f1_out_gaa(x1, x2, x3) = f1_out_gaa(x1)
s(x1) = s(x1)
U1_gaa(x1, x2, x3, x4) = U1_gaa(x1, x4)
f15_in_gaa(x1, x2, x3) = f15_in_gaa(x1)
f15_out_gaa(x1, x2, x3) = f15_out_gaa(x1)
U4_gaa(x1, x2, x3, x4) = U4_gaa(x1, x4)
U5_gaa(x1, x2, x3, x4) = U5_gaa(x1, x4)
U6_gaa(x1, x2, x3, x4) = U6_gaa(x1, x4)
f15_in_aaa(x1, x2, x3) = f15_in_aaa
f15_out_aaa(x1, x2, x3) = f15_out_aaa(x1)
U4_aaa(x1, x2, x3, x4) = U4_aaa(x4)
U5_aaa(x1, x2, x3, x4) = U5_aaa(x4)
U6_aaa(x1, x2, x3, x4) = U6_aaa(x1, x4)
U2_gaa(x1, x2, x3, x4) = U2_gaa(x1, x4)
U3_gaa(x1, x2, x3, x4) = U3_gaa(x1, x4)
f1_in_aaa(x1, x2, x3) = f1_in_aaa
f1_out_aaa(x1, x2, x3) = f1_out_aaa(x1)
U1_aaa(x1, x2, x3, x4) = U1_aaa(x4)
U2_aaa(x1, x2, x3, x4) = U2_aaa(x4)
U3_aaa(x1, x2, x3, x4) = U3_aaa(x1, x4)
F1_IN_GAA(x1, x2, x3) = F1_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4) = U1_GAA(x1, x4)
F15_IN_GAA(x1, x2, x3) = F15_IN_GAA(x1)
U4_GAA(x1, x2, x3, x4) = U4_GAA(x1, x4)
U5_GAA(x1, x2, x3, x4) = U5_GAA(x1, x4)
U6_GAA(x1, x2, x3, x4) = U6_GAA(x1, x4)
F15_IN_AAA(x1, x2, x3) = F15_IN_AAA
U4_AAA(x1, x2, x3, x4) = U4_AAA(x4)
U5_AAA(x1, x2, x3, x4) = U5_AAA(x4)
U6_AAA(x1, x2, x3, x4) = U6_AAA(x1, x4)
U2_GAA(x1, x2, x3, x4) = U2_GAA(x1, x4)
U3_GAA(x1, x2, x3, x4) = U3_GAA(x1, x4)
F1_IN_AAA(x1, x2, x3) = F1_IN_AAA
U1_AAA(x1, x2, x3, x4) = U1_AAA(x4)
U2_AAA(x1, x2, x3, x4) = U2_AAA(x4)
U3_AAA(x1, x2, x3, x4) = U3_AAA(x1, x4)

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

(37) Obligation:

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

F1_IN_GAA(s(T13), T11, T12) → U1_GAA(T13, T11, T12, f15_in_gaa(T13, T11, X14))
F1_IN_GAA(s(T13), T11, T12) → F15_IN_GAA(T13, T11, X14)
F15_IN_GAA(s(T21), T20, X33) → U4_GAA(T21, T20, X33, f15_in_gaa(T21, T20, X32))
F15_IN_GAA(s(T21), T20, X33) → F15_IN_GAA(T21, T20, X32)
F15_IN_GAA(s(T21), T23, X33) → U5_GAA(T21, T23, X33, f15_in_gaa(T21, T23, T22))
U5_GAA(T21, T23, X33, f15_out_gaa(T21, T23, T22)) → U6_GAA(T21, T23, X33, f15_in_aaa(T22, T23, X33))
U5_GAA(T21, T23, X33, f15_out_gaa(T21, T23, T22)) → F15_IN_AAA(T22, T23, X33)
F15_IN_AAA(s(T21), T20, X33) → U4_AAA(T21, T20, X33, f15_in_aaa(T21, T20, X32))
F15_IN_AAA(s(T21), T20, X33) → F15_IN_AAA(T21, T20, X32)
F15_IN_AAA(s(T21), T23, X33) → U5_AAA(T21, T23, X33, f15_in_aaa(T21, T23, T22))
U5_AAA(T21, T23, X33, f15_out_aaa(T21, T23, T22)) → U6_AAA(T21, T23, X33, f15_in_aaa(T22, T23, X33))
U5_AAA(T21, T23, X33, f15_out_aaa(T21, T23, T22)) → F15_IN_AAA(T22, T23, X33)
F1_IN_GAA(s(T13), T15, T16) → U2_GAA(T13, T15, T16, f15_in_gaa(T13, T15, T14))
U2_GAA(T13, T15, T16, f15_out_gaa(T13, T15, T14)) → U3_GAA(T13, T15, T16, f1_in_aaa(T14, T15, T16))
U2_GAA(T13, T15, T16, f15_out_gaa(T13, T15, T14)) → F1_IN_AAA(T14, T15, T16)
F1_IN_AAA(s(T13), T11, T12) → U1_AAA(T13, T11, T12, f15_in_aaa(T13, T11, X14))
F1_IN_AAA(s(T13), T11, T12) → F15_IN_AAA(T13, T11, X14)
F1_IN_AAA(s(T13), T15, T16) → U2_AAA(T13, T15, T16, f15_in_aaa(T13, T15, T14))
U2_AAA(T13, T15, T16, f15_out_aaa(T13, T15, T14)) → U3_AAA(T13, T15, T16, f1_in_aaa(T14, T15, T16))
U2_AAA(T13, T15, T16, f15_out_aaa(T13, T15, T14)) → F1_IN_AAA(T14, T15, T16)

The TRS R consists of the following rules:

f1_in_gaa(0, T4, 0) → f1_out_gaa(0, T4, 0)
f1_in_gaa(s(T13), T11, T12) → U1_gaa(T13, T11, T12, f15_in_gaa(T13, T11, X14))
f15_in_gaa(0, T17, 0) → f15_out_gaa(0, T17, 0)
f15_in_gaa(s(T21), T20, X33) → U4_gaa(T21, T20, X33, f15_in_gaa(T21, T20, X32))
f15_in_gaa(s(T21), T23, X33) → U5_gaa(T21, T23, X33, f15_in_gaa(T21, T23, T22))
U5_gaa(T21, T23, X33, f15_out_gaa(T21, T23, T22)) → U6_gaa(T21, T23, X33, f15_in_aaa(T22, T23, X33))
f15_in_aaa(0, T17, 0) → f15_out_aaa(0, T17, 0)
f15_in_aaa(s(T21), T20, X33) → U4_aaa(T21, T20, X33, f15_in_aaa(T21, T20, X32))
f15_in_aaa(s(T21), T23, X33) → U5_aaa(T21, T23, X33, f15_in_aaa(T21, T23, T22))
U5_aaa(T21, T23, X33, f15_out_aaa(T21, T23, T22)) → U6_aaa(T21, T23, X33, f15_in_aaa(T22, T23, X33))
U6_aaa(T21, T23, X33, f15_out_aaa(T22, T23, X33)) → f15_out_aaa(s(T21), T23, X33)
U4_aaa(T21, T20, X33, f15_out_aaa(T21, T20, X32)) → f15_out_aaa(s(T21), T20, X33)
U6_gaa(T21, T23, X33, f15_out_aaa(T22, T23, X33)) → f15_out_gaa(s(T21), T23, X33)
U4_gaa(T21, T20, X33, f15_out_gaa(T21, T20, X32)) → f15_out_gaa(s(T21), T20, X33)
U1_gaa(T13, T11, T12, f15_out_gaa(T13, T11, X14)) → f1_out_gaa(s(T13), T11, T12)
f1_in_gaa(s(T13), T15, T16) → U2_gaa(T13, T15, T16, f15_in_gaa(T13, T15, T14))
U2_gaa(T13, T15, T16, f15_out_gaa(T13, T15, T14)) → U3_gaa(T13, T15, T16, f1_in_aaa(T14, T15, T16))
f1_in_aaa(0, T4, 0) → f1_out_aaa(0, T4, 0)
f1_in_aaa(s(T13), T11, T12) → U1_aaa(T13, T11, T12, f15_in_aaa(T13, T11, X14))
U1_aaa(T13, T11, T12, f15_out_aaa(T13, T11, X14)) → f1_out_aaa(s(T13), T11, T12)
f1_in_aaa(s(T13), T15, T16) → U2_aaa(T13, T15, T16, f15_in_aaa(T13, T15, T14))
U2_aaa(T13, T15, T16, f15_out_aaa(T13, T15, T14)) → U3_aaa(T13, T15, T16, f1_in_aaa(T14, T15, T16))
U3_aaa(T13, T15, T16, f1_out_aaa(T14, T15, T16)) → f1_out_aaa(s(T13), T15, T16)
U3_gaa(T13, T15, T16, f1_out_aaa(T14, T15, T16)) → f1_out_gaa(s(T13), T15, T16)

(38) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 14 less nodes.

(39) Complex Obligation (AND)

(40) Obligation:

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

F15_IN_AAA(s(T21), T23, X33) → U5_AAA(T21, T23, X33, f15_in_aaa(T21, T23, T22))
U5_AAA(T21, T23, X33, f15_out_aaa(T21, T23, T22)) → F15_IN_AAA(T22, T23, X33)
F15_IN_AAA(s(T21), T20, X33) → F15_IN_AAA(T21, T20, X32)

The TRS R consists of the following rules:

f1_in_gaa(0, T4, 0) → f1_out_gaa(0, T4, 0)
f1_in_gaa(s(T13), T11, T12) → U1_gaa(T13, T11, T12, f15_in_gaa(T13, T11, X14))
f15_in_gaa(0, T17, 0) → f15_out_gaa(0, T17, 0)
f15_in_gaa(s(T21), T20, X33) → U4_gaa(T21, T20, X33, f15_in_gaa(T21, T20, X32))
f15_in_gaa(s(T21), T23, X33) → U5_gaa(T21, T23, X33, f15_in_gaa(T21, T23, T22))
U5_gaa(T21, T23, X33, f15_out_gaa(T21, T23, T22)) → U6_gaa(T21, T23, X33, f15_in_aaa(T22, T23, X33))
f15_in_aaa(0, T17, 0) → f15_out_aaa(0, T17, 0)
f15_in_aaa(s(T21), T20, X33) → U4_aaa(T21, T20, X33, f15_in_aaa(T21, T20, X32))
f15_in_aaa(s(T21), T23, X33) → U5_aaa(T21, T23, X33, f15_in_aaa(T21, T23, T22))
U5_aaa(T21, T23, X33, f15_out_aaa(T21, T23, T22)) → U6_aaa(T21, T23, X33, f15_in_aaa(T22, T23, X33))
U6_aaa(T21, T23, X33, f15_out_aaa(T22, T23, X33)) → f15_out_aaa(s(T21), T23, X33)
U4_aaa(T21, T20, X33, f15_out_aaa(T21, T20, X32)) → f15_out_aaa(s(T21), T20, X33)
U6_gaa(T21, T23, X33, f15_out_aaa(T22, T23, X33)) → f15_out_gaa(s(T21), T23, X33)
U4_gaa(T21, T20, X33, f15_out_gaa(T21, T20, X32)) → f15_out_gaa(s(T21), T20, X33)
U1_gaa(T13, T11, T12, f15_out_gaa(T13, T11, X14)) → f1_out_gaa(s(T13), T11, T12)
f1_in_gaa(s(T13), T15, T16) → U2_gaa(T13, T15, T16, f15_in_gaa(T13, T15, T14))
U2_gaa(T13, T15, T16, f15_out_gaa(T13, T15, T14)) → U3_gaa(T13, T15, T16, f1_in_aaa(T14, T15, T16))
f1_in_aaa(0, T4, 0) → f1_out_aaa(0, T4, 0)
f1_in_aaa(s(T13), T11, T12) → U1_aaa(T13, T11, T12, f15_in_aaa(T13, T11, X14))
U1_aaa(T13, T11, T12, f15_out_aaa(T13, T11, X14)) → f1_out_aaa(s(T13), T11, T12)
f1_in_aaa(s(T13), T15, T16) → U2_aaa(T13, T15, T16, f15_in_aaa(T13, T15, T14))
U2_aaa(T13, T15, T16, f15_out_aaa(T13, T15, T14)) → U3_aaa(T13, T15, T16, f1_in_aaa(T14, T15, T16))
U3_aaa(T13, T15, T16, f1_out_aaa(T14, T15, T16)) → f1_out_aaa(s(T13), T15, T16)
U3_gaa(T13, T15, T16, f1_out_aaa(T14, T15, T16)) → f1_out_gaa(s(T13), T15, T16)

(41) UsableRulesProof (EQUIVALENT transformation)

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

(42) Obligation:

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

F15_IN_AAA(s(T21), T23, X33) → U5_AAA(T21, T23, X33, f15_in_aaa(T21, T23, T22))
U5_AAA(T21, T23, X33, f15_out_aaa(T21, T23, T22)) → F15_IN_AAA(T22, T23, X33)
F15_IN_AAA(s(T21), T20, X33) → F15_IN_AAA(T21, T20, X32)

The TRS R consists of the following rules:

f15_in_aaa(0, T17, 0) → f15_out_aaa(0, T17, 0)
f15_in_aaa(s(T21), T20, X33) → U4_aaa(T21, T20, X33, f15_in_aaa(T21, T20, X32))
f15_in_aaa(s(T21), T23, X33) → U5_aaa(T21, T23, X33, f15_in_aaa(T21, T23, T22))
U4_aaa(T21, T20, X33, f15_out_aaa(T21, T20, X32)) → f15_out_aaa(s(T21), T20, X33)
U5_aaa(T21, T23, X33, f15_out_aaa(T21, T23, T22)) → U6_aaa(T21, T23, X33, f15_in_aaa(T22, T23, X33))
U6_aaa(T21, T23, X33, f15_out_aaa(T22, T23, X33)) → f15_out_aaa(s(T21), T23, X33)

(43) PiDPToQDPProof (SOUND transformation)

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

(44) Obligation:

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

F15_IN_AAA → U5_AAA(f15_in_aaa)
U5_AAA(f15_out_aaa(T21)) → F15_IN_AAA
F15_IN_AAA → F15_IN_AAA

The TRS R consists of the following rules:

f15_in_aaa → f15_out_aaa(0)
f15_in_aaa → U4_aaa(f15_in_aaa)
f15_in_aaa → U5_aaa(f15_in_aaa)
U4_aaa(f15_out_aaa(T21)) → f15_out_aaa(s(T21))
U5_aaa(f15_out_aaa(T21)) → U6_aaa(T21, f15_in_aaa)
U6_aaa(T21, f15_out_aaa(T22)) → f15_out_aaa(s(T21))

The set Q consists of the following terms:

f15_in_aaa
U4_aaa(x0)
U5_aaa(x0)
U6_aaa(x0, x1)

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

(45) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule F15_IN_AAA → U5_AAA(f15_in_aaa) at position [0] we obtained the following new rules [LPAR04]:

F15_IN_AAA → U5_AAA(f15_out_aaa(0))
F15_IN_AAA → U5_AAA(U4_aaa(f15_in_aaa))
F15_IN_AAA → U5_AAA(U5_aaa(f15_in_aaa))

(46) Obligation:

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

U5_AAA(f15_out_aaa(T21)) → F15_IN_AAA
F15_IN_AAA → F15_IN_AAA
F15_IN_AAA → U5_AAA(f15_out_aaa(0))
F15_IN_AAA → U5_AAA(U4_aaa(f15_in_aaa))
F15_IN_AAA → U5_AAA(U5_aaa(f15_in_aaa))

The TRS R consists of the following rules:

f15_in_aaa → f15_out_aaa(0)
f15_in_aaa → U4_aaa(f15_in_aaa)
f15_in_aaa → U5_aaa(f15_in_aaa)
U4_aaa(f15_out_aaa(T21)) → f15_out_aaa(s(T21))
U5_aaa(f15_out_aaa(T21)) → U6_aaa(T21, f15_in_aaa)
U6_aaa(T21, f15_out_aaa(T22)) → f15_out_aaa(s(T21))

The set Q consists of the following terms:

f15_in_aaa
U4_aaa(x0)
U5_aaa(x0)
U6_aaa(x0, x1)

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

(47) NonTerminationProof (EQUIVALENT transformation)

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from F15_IN_AAA to F15_IN_AAA.

(48) FALSE

(49) Obligation:

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

F1_IN_AAA(s(T13), T15, T16) → U2_AAA(T13, T15, T16, f15_in_aaa(T13, T15, T14))
U2_AAA(T13, T15, T16, f15_out_aaa(T13, T15, T14)) → F1_IN_AAA(T14, T15, T16)

The TRS R consists of the following rules:

f1_in_gaa(0, T4, 0) → f1_out_gaa(0, T4, 0)
f1_in_gaa(s(T13), T11, T12) → U1_gaa(T13, T11, T12, f15_in_gaa(T13, T11, X14))
f15_in_gaa(0, T17, 0) → f15_out_gaa(0, T17, 0)
f15_in_gaa(s(T21), T20, X33) → U4_gaa(T21, T20, X33, f15_in_gaa(T21, T20, X32))
f15_in_gaa(s(T21), T23, X33) → U5_gaa(T21, T23, X33, f15_in_gaa(T21, T23, T22))
U5_gaa(T21, T23, X33, f15_out_gaa(T21, T23, T22)) → U6_gaa(T21, T23, X33, f15_in_aaa(T22, T23, X33))
f15_in_aaa(0, T17, 0) → f15_out_aaa(0, T17, 0)
f15_in_aaa(s(T21), T20, X33) → U4_aaa(T21, T20, X33, f15_in_aaa(T21, T20, X32))
f15_in_aaa(s(T21), T23, X33) → U5_aaa(T21, T23, X33, f15_in_aaa(T21, T23, T22))
U5_aaa(T21, T23, X33, f15_out_aaa(T21, T23, T22)) → U6_aaa(T21, T23, X33, f15_in_aaa(T22, T23, X33))
U6_aaa(T21, T23, X33, f15_out_aaa(T22, T23, X33)) → f15_out_aaa(s(T21), T23, X33)
U4_aaa(T21, T20, X33, f15_out_aaa(T21, T20, X32)) → f15_out_aaa(s(T21), T20, X33)
U6_gaa(T21, T23, X33, f15_out_aaa(T22, T23, X33)) → f15_out_gaa(s(T21), T23, X33)
U4_gaa(T21, T20, X33, f15_out_gaa(T21, T20, X32)) → f15_out_gaa(s(T21), T20, X33)
U1_gaa(T13, T11, T12, f15_out_gaa(T13, T11, X14)) → f1_out_gaa(s(T13), T11, T12)
f1_in_gaa(s(T13), T15, T16) → U2_gaa(T13, T15, T16, f15_in_gaa(T13, T15, T14))
U2_gaa(T13, T15, T16, f15_out_gaa(T13, T15, T14)) → U3_gaa(T13, T15, T16, f1_in_aaa(T14, T15, T16))
f1_in_aaa(0, T4, 0) → f1_out_aaa(0, T4, 0)
f1_in_aaa(s(T13), T11, T12) → U1_aaa(T13, T11, T12, f15_in_aaa(T13, T11, X14))
U1_aaa(T13, T11, T12, f15_out_aaa(T13, T11, X14)) → f1_out_aaa(s(T13), T11, T12)
f1_in_aaa(s(T13), T15, T16) → U2_aaa(T13, T15, T16, f15_in_aaa(T13, T15, T14))
U2_aaa(T13, T15, T16, f15_out_aaa(T13, T15, T14)) → U3_aaa(T13, T15, T16, f1_in_aaa(T14, T15, T16))
U3_aaa(T13, T15, T16, f1_out_aaa(T14, T15, T16)) → f1_out_aaa(s(T13), T15, T16)
U3_gaa(T13, T15, T16, f1_out_aaa(T14, T15, T16)) → f1_out_gaa(s(T13), T15, T16)

(50) UsableRulesProof (EQUIVALENT transformation)

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

(51) Obligation:

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

F1_IN_AAA(s(T13), T15, T16) → U2_AAA(T13, T15, T16, f15_in_aaa(T13, T15, T14))
U2_AAA(T13, T15, T16, f15_out_aaa(T13, T15, T14)) → F1_IN_AAA(T14, T15, T16)

The TRS R consists of the following rules:

f15_in_aaa(0, T17, 0) → f15_out_aaa(0, T17, 0)
f15_in_aaa(s(T21), T20, X33) → U4_aaa(T21, T20, X33, f15_in_aaa(T21, T20, X32))
f15_in_aaa(s(T21), T23, X33) → U5_aaa(T21, T23, X33, f15_in_aaa(T21, T23, T22))
U4_aaa(T21, T20, X33, f15_out_aaa(T21, T20, X32)) → f15_out_aaa(s(T21), T20, X33)
U5_aaa(T21, T23, X33, f15_out_aaa(T21, T23, T22)) → U6_aaa(T21, T23, X33, f15_in_aaa(T22, T23, X33))
U6_aaa(T21, T23, X33, f15_out_aaa(T22, T23, X33)) → f15_out_aaa(s(T21), T23, X33)

The argument filtering Pi contains the following mapping:
0 = 0
s(x1) = s(x1)
f15_in_aaa(x1, x2, x3) = f15_in_aaa
f15_out_aaa(x1, x2, x3) = f15_out_aaa(x1)
U4_aaa(x1, x2, x3, x4) = U4_aaa(x4)
U5_aaa(x1, x2, x3, x4) = U5_aaa(x4)
U6_aaa(x1, x2, x3, x4) = U6_aaa(x1, x4)
F1_IN_AAA(x1, x2, x3) = F1_IN_AAA
U2_AAA(x1, x2, x3, x4) = U2_AAA(x4)

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

(52) PiDPToQDPProof (SOUND transformation)

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

(53) Obligation:

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

F1_IN_AAA → U2_AAA(f15_in_aaa)
U2_AAA(f15_out_aaa(T13)) → F1_IN_AAA

The TRS R consists of the following rules:

f15_in_aaa → f15_out_aaa(0)
f15_in_aaa → U4_aaa(f15_in_aaa)
f15_in_aaa → U5_aaa(f15_in_aaa)
U4_aaa(f15_out_aaa(T21)) → f15_out_aaa(s(T21))
U5_aaa(f15_out_aaa(T21)) → U6_aaa(T21, f15_in_aaa)
U6_aaa(T21, f15_out_aaa(T22)) → f15_out_aaa(s(T21))

The set Q consists of the following terms:

f15_in_aaa
U4_aaa(x0)
U5_aaa(x0)
U6_aaa(x0, x1)

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

(54) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule F1_IN_AAA → U2_AAA(f15_in_aaa) at position [0] we obtained the following new rules [LPAR04]:

F1_IN_AAA → U2_AAA(f15_out_aaa(0))
F1_IN_AAA → U2_AAA(U4_aaa(f15_in_aaa))
F1_IN_AAA → U2_AAA(U5_aaa(f15_in_aaa))

(55) Obligation:

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

U2_AAA(f15_out_aaa(T13)) → F1_IN_AAA
F1_IN_AAA → U2_AAA(f15_out_aaa(0))
F1_IN_AAA → U2_AAA(U4_aaa(f15_in_aaa))
F1_IN_AAA → U2_AAA(U5_aaa(f15_in_aaa))

The TRS R consists of the following rules:

f15_in_aaa → f15_out_aaa(0)
f15_in_aaa → U4_aaa(f15_in_aaa)
f15_in_aaa → U5_aaa(f15_in_aaa)
U4_aaa(f15_out_aaa(T21)) → f15_out_aaa(s(T21))
U5_aaa(f15_out_aaa(T21)) → U6_aaa(T21, f15_in_aaa)
U6_aaa(T21, f15_out_aaa(T22)) → f15_out_aaa(s(T21))

The set Q consists of the following terms:

f15_in_aaa
U4_aaa(x0)
U5_aaa(x0)
U6_aaa(x0, x1)

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

(56) NonTerminationProof (EQUIVALENT transformation)

Semiunifier: [ ]
Matcher: [ ]

(57) FALSE

(58) Obligation:

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

F15_IN_GAA(s(T21), T20, X33) → F15_IN_GAA(T21, T20, X32)

The TRS R consists of the following rules:

f1_in_gaa(0, T4, 0) → f1_out_gaa(0, T4, 0)
f1_in_gaa(s(T13), T11, T12) → U1_gaa(T13, T11, T12, f15_in_gaa(T13, T11, X14))
f15_in_gaa(0, T17, 0) → f15_out_gaa(0, T17, 0)
f15_in_gaa(s(T21), T20, X33) → U4_gaa(T21, T20, X33, f15_in_gaa(T21, T20, X32))
f15_in_gaa(s(T21), T23, X33) → U5_gaa(T21, T23, X33, f15_in_gaa(T21, T23, T22))
U5_gaa(T21, T23, X33, f15_out_gaa(T21, T23, T22)) → U6_gaa(T21, T23, X33, f15_in_aaa(T22, T23, X33))
f15_in_aaa(0, T17, 0) → f15_out_aaa(0, T17, 0)
f15_in_aaa(s(T21), T20, X33) → U4_aaa(T21, T20, X33, f15_in_aaa(T21, T20, X32))
f15_in_aaa(s(T21), T23, X33) → U5_aaa(T21, T23, X33, f15_in_aaa(T21, T23, T22))
U5_aaa(T21, T23, X33, f15_out_aaa(T21, T23, T22)) → U6_aaa(T21, T23, X33, f15_in_aaa(T22, T23, X33))
U6_aaa(T21, T23, X33, f15_out_aaa(T22, T23, X33)) → f15_out_aaa(s(T21), T23, X33)
U4_aaa(T21, T20, X33, f15_out_aaa(T21, T20, X32)) → f15_out_aaa(s(T21), T20, X33)
U6_gaa(T21, T23, X33, f15_out_aaa(T22, T23, X33)) → f15_out_gaa(s(T21), T23, X33)
U4_gaa(T21, T20, X33, f15_out_gaa(T21, T20, X32)) → f15_out_gaa(s(T21), T20, X33)
U1_gaa(T13, T11, T12, f15_out_gaa(T13, T11, X14)) → f1_out_gaa(s(T13), T11, T12)
f1_in_gaa(s(T13), T15, T16) → U2_gaa(T13, T15, T16, f15_in_gaa(T13, T15, T14))
U2_gaa(T13, T15, T16, f15_out_gaa(T13, T15, T14)) → U3_gaa(T13, T15, T16, f1_in_aaa(T14, T15, T16))
f1_in_aaa(0, T4, 0) → f1_out_aaa(0, T4, 0)
f1_in_aaa(s(T13), T11, T12) → U1_aaa(T13, T11, T12, f15_in_aaa(T13, T11, X14))
U1_aaa(T13, T11, T12, f15_out_aaa(T13, T11, X14)) → f1_out_aaa(s(T13), T11, T12)
f1_in_aaa(s(T13), T15, T16) → U2_aaa(T13, T15, T16, f15_in_aaa(T13, T15, T14))
U2_aaa(T13, T15, T16, f15_out_aaa(T13, T15, T14)) → U3_aaa(T13, T15, T16, f1_in_aaa(T14, T15, T16))
U3_aaa(T13, T15, T16, f1_out_aaa(T14, T15, T16)) → f1_out_aaa(s(T13), T15, T16)
U3_gaa(T13, T15, T16, f1_out_aaa(T14, T15, T16)) → f1_out_gaa(s(T13), T15, T16)

(59) UsableRulesProof (EQUIVALENT transformation)

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

(60) Obligation:

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

F15_IN_GAA(s(T21), T20, X33) → F15_IN_GAA(T21, T20, X32)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
F15_IN_GAA(x1, x2, x3) = F15_IN_GAA(x1)

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

(61) PiDPToQDPProof (SOUND transformation)

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

(62) Obligation:

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

F15_IN_GAA(s(T21)) → F15_IN_GAA(T21)

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

(63) 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:

F15_IN_GAA(s(T21)) → F15_IN_GAA(T21)
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) Narrowing (SOUND transformation)

(15) Obligation:

(16) NonTerminationProof (EQUIVALENT transformation)

(17) FALSE

(18) Obligation:

(19) UsableRulesProof (EQUIVALENT transformation)

(20) Obligation:

(21) PiDPToQDPProof (SOUND transformation)

(22) Obligation:

(23) Narrowing (SOUND transformation)

(24) Obligation:

(25) NonTerminationProof (EQUIVALENT transformation)

(26) FALSE

(27) Obligation:

(28) UsableRulesProof (EQUIVALENT transformation)

(29) Obligation:

(30) PiDPToQDPProof (SOUND transformation)

(31) Obligation:

(32) QDPSizeChangeProof (EQUIVALENT transformation)

(33) TRUE

(34) PrologToPiTRSProof (SOUND transformation)

(35) Obligation:

(36) DependencyPairsProof (EQUIVALENT transformation)

(37) Obligation:

(38) DependencyGraphProof (EQUIVALENT transformation)

(39) Complex Obligation (AND)

(40) Obligation:

(41) UsableRulesProof (EQUIVALENT transformation)

(42) Obligation:

(43) PiDPToQDPProof (SOUND transformation)

(44) Obligation:

(45) Narrowing (SOUND transformation)

(46) Obligation:

(47) NonTerminationProof (EQUIVALENT transformation)

(48) FALSE

(49) Obligation:

(50) UsableRulesProof (EQUIVALENT transformation)

(51) Obligation:

(52) PiDPToQDPProof (SOUND transformation)

(53) Obligation:

(54) Narrowing (SOUND transformation)

(55) Obligation:

(56) NonTerminationProof (EQUIVALENT transformation)

(57) FALSE

(58) Obligation:

(59) UsableRulesProof (EQUIVALENT transformation)

(60) Obligation:

(61) PiDPToQDPProof (SOUND transformation)

(62) Obligation:

(63) QDPSizeChangeProof (EQUIVALENT transformation)

(64) TRUE