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

(0) Obligation:

Clauses:

permute([], []).
permute(.(X, Y), .(U, V)) :- ','(delete(U, .(X, Y), W), permute(W, V)).
delete(X, .(X, Y), Y).
delete(U, .(X, Y), .(X, Z)) :- delete(U, Y, Z).

Query: permute(g,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: permute(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: permute(T1, T2), SCOPE: 1, CLAUSE: 0 | permute(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | permute([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: permute(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\npermute(T1, T2) !=? permute([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: permute([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(delete(T11, .(T7, T8), X13), permute(X13, T12))\n\nT7 is ground\nT8 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(delete(T11, .(T7, T8), X13), permute(X13, T12)), SCOPE: 2, CLAUSE: 2 | ','(delete(T11, .(T7, T8), X13), permute(X13, T12)), SCOPE: 2, CLAUSE: 3\n\nT7 is ground\nT8 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(delete(T11, .(T7, T8), X13), permute(X13, T12)), SCOPE: 2, CLAUSE: 2\n\nT7 is ground\nT8 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(delete(T11, .(T7, T8), X13), permute(X13, T12)), SCOPE: 2, CLAUSE: 3\n\nT7 is ground\nT8 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: permute(T22, T23)\n\nT22 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(delete(T39, T38, X42), permute(.(T37, X42), T40))\n\nT37 is ground\nT38 is ground\nX42 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="B: delete(T39, T38, X42)\n\nT38 is ground\nX42 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: permute(.(T37, T45), T46)\n\nT37 is ground\nT45 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: delete(T39, T38, X42), SCOPE: 3, CLAUSE: 2 | delete(T39, T38, X42), SCOPE: 3, CLAUSE: 3\n\nT38 is ground\nX42 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: delete(T39, T38, X42), SCOPE: 3, CLAUSE: 2\n\nT38 is ground\nX42 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: delete(T39, T38, X42), SCOPE: 3, CLAUSE: 3\n\nT38 is ground\nX42 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: delete(T70, T69, X75)\n\nT69 is ground\nX75 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 25 [arrowhead = none ];
25 -> 2;

26 [label="EVAL with clause\npermute([], []).\nand substitutionT1 -> [],\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 26 [arrowhead = none ];
26 -> 3;

27 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 27 [arrowhead = none ];
27 -> 4;

28 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 28 [arrowhead = none ];
28 -> 5;

29 [label="EVAL with clause\npermute(.(X9, X10), .(X11, X12)) :- ','(delete(X11, .(X9, X10), X13), permute(X13, X12)).\nand substitutionX9 -> T7,\nX10 -> T8,\nT1 -> .(T7, T8),\nX11 -> T11,\nX12 -> T12,\nT2 -> .(T11, T12),\nT9 -> T11,\nT10 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 29 [arrowhead = none ];
29 -> 7;

30 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 30 [arrowhead = none ];
30 -> 8;

31 [label="BACKTRACK\nfor clause: permute(.(X, Y), .(U, V)) :- ','(delete(U, .(X, Y), W), permute(W, V))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 31 [arrowhead = none ];
31 -> 6;

32 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 32 [arrowhead = none ];
32 -> 9;

33 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 33 [arrowhead = none ];
33 -> 10;

34 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 34 [arrowhead = none ];
34 -> 11;

35 [label="EVAL with clause\ndelete(X22, .(X22, X23), X23).\nand substitutionT11 -> T21,\nX22 -> T21,\nT7 -> T21,\nT8 -> T22,\nX23 -> T22,\nX13 -> T22,\nT12 -> T23", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 35 [arrowhead = none ];
35 -> 12;

36 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 36 [arrowhead = none ];
36 -> 13;

37 [label="ONLY EVAL with clause\ndelete(X38, .(X39, X40), .(X39, X41)) :- delete(X38, X40, X41).\nand substitutionT11 -> T39,\nX38 -> T39,\nT7 -> T37,\nX39 -> T37,\nT8 -> T38,\nX40 -> T38,\nX41 -> X42,\nX13 -> .(T37, X42),\nT36 -> T39,\nT12 -> T40", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 37 [arrowhead = none ];
37 -> 14;

38 [label="INSTANCE with matching:\nT1 -> T22\nT2 -> T23", fontsize=14, style = filled, fillcolor = lightgrey];
12 -> 38 [arrowhead = none , style=dashed];
38 -> 1[style=dashed];

39 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
14 -> 39 [arrowhead = none ];
39 -> 15;

40 [label="SPLIT 2\nnew knowledge:\nT39 is ground\nT38 is ground\nT45 is ground\nreplacements:X42 -> T45,\nT40 -> T46", fontsize=14, style = filled, fillcolor = violet];
14 -> 40 [arrowhead = none ];
40 -> 16;

41 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 41 [arrowhead = none ];
41 -> 17;

42 [label="INSTANCE with matching:\nT1 -> .(T37, T45)\nT2 -> T46", fontsize=14, style = filled, fillcolor = lightgrey];
16 -> 42 [arrowhead = none , style=dashed];
42 -> 1[style=dashed];

43 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
17 -> 43 [arrowhead = none ];
43 -> 18;

44 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
17 -> 44 [arrowhead = none ];
44 -> 19;

45 [label="EVAL with clause\ndelete(X59, .(X59, X60), X60).\nand substitutionT39 -> T59,\nX59 -> T59,\nX60 -> T60,\nT38 -> .(T59, T60),\nX42 -> T60", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 45 [arrowhead = none ];
45 -> 20;

46 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 46 [arrowhead = none ];
46 -> 21;

47 [label="EVAL with clause\ndelete(X71, .(X72, X73), .(X72, X74)) :- delete(X71, X73, X74).\nand substitutionT39 -> T70,\nX71 -> T70,\nX72 -> T68,\nX73 -> T69,\nT38 -> .(T68, T69),\nX74 -> X75,\nX42 -> .(T68, X75),\nT67 -> T70", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 47 [arrowhead = none ];
47 -> 23;

48 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 48 [arrowhead = none ];
48 -> 24;

49 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 49 [arrowhead = none ];
49 -> 22;

50 [label="INSTANCE with matching:\nT39 -> T70\nT38 -> T69\nX42 -> X75", fontsize=14, style = filled, fillcolor = lightgrey];
23 -> 50 [arrowhead = none , style=dashed];
50 -> 15[style=dashed];

}

(2) Obligation:

Triples:

deleteB(X1, .(X2, X3), .(X2, X4)) :- deleteB(X1, X3, X4).
permuteA(.(X1, X2), .(X1, X3)) :- permuteA(X2, X3).
permuteA(.(X1, X2), .(X3, X4)) :- deleteB(X3, X2, X5).
permuteA(.(X1, X2), .(X3, X4)) :- ','(deletecB(X3, X2, X5), permuteA(.(X1, X5), X4)).

Clauses:

permutecA([], []).
permutecA(.(X1, X2), .(X1, X3)) :- permutecA(X2, X3).
permutecA(.(X1, X2), .(X3, X4)) :- ','(deletecB(X3, X2, X5), permutecA(.(X1, X5), X4)).
deletecB(X1, .(X1, X2), X2).
deletecB(X1, .(X2, X3), .(X2, X4)) :- deletecB(X1, X3, X4).

Afs:

permuteA(x1, x2)  =  permuteA(x1)

(3) TriplesToPiDPProof (SOUND transformation)

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

PERMUTEA_IN_GA(.(X1, X2), .(X1, X3)) → U2_GA(X1, X2, X3, permuteA_in_ga(X2, X3))
PERMUTEA_IN_GA(.(X1, X2), .(X1, X3)) → PERMUTEA_IN_GA(X2, X3)
PERMUTEA_IN_GA(.(X1, X2), .(X3, X4)) → U3_GA(X1, X2, X3, X4, deleteB_in_aga(X3, X2, X5))
PERMUTEA_IN_GA(.(X1, X2), .(X3, X4)) → DELETEB_IN_AGA(X3, X2, X5)
DELETEB_IN_AGA(X1, .(X2, X3), .(X2, X4)) → U1_AGA(X1, X2, X3, X4, deleteB_in_aga(X1, X3, X4))
DELETEB_IN_AGA(X1, .(X2, X3), .(X2, X4)) → DELETEB_IN_AGA(X1, X3, X4)
PERMUTEA_IN_GA(.(X1, X2), .(X3, X4)) → U4_GA(X1, X2, X3, X4, deletecB_in_aga(X3, X2, X5))
U4_GA(X1, X2, X3, X4, deletecB_out_aga(X3, X2, X5)) → U5_GA(X1, X2, X3, X4, permuteA_in_ga(.(X1, X5), X4))
U4_GA(X1, X2, X3, X4, deletecB_out_aga(X3, X2, X5)) → PERMUTEA_IN_GA(.(X1, X5), X4)

The TRS R consists of the following rules:

deletecB_in_aga(X1, .(X1, X2), X2) → deletecB_out_aga(X1, .(X1, X2), X2)
deletecB_in_aga(X1, .(X2, X3), .(X2, X4)) → U10_aga(X1, X2, X3, X4, deletecB_in_aga(X1, X3, X4))
U10_aga(X1, X2, X3, X4, deletecB_out_aga(X1, X3, X4)) → deletecB_out_aga(X1, .(X2, X3), .(X2, X4))

The argument filtering Pi contains the following mapping:
permuteA_in_ga(x1, x2)  =  permuteA_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
deleteB_in_aga(x1, x2, x3)  =  deleteB_in_aga(x2)
deletecB_in_aga(x1, x2, x3)  =  deletecB_in_aga(x2)
deletecB_out_aga(x1, x2, x3)  =  deletecB_out_aga(x1, x2, x3)
U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x2, x3, x5)
PERMUTEA_IN_GA(x1, x2)  =  PERMUTEA_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x2, x4)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x5)
DELETEB_IN_AGA(x1, x2, x3)  =  DELETEB_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x2, x3, x5)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x2, x5)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x1, x2, x5)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

PERMUTEA_IN_GA(.(X1, X2), .(X1, X3)) → U2_GA(X1, X2, X3, permuteA_in_ga(X2, X3))
PERMUTEA_IN_GA(.(X1, X2), .(X1, X3)) → PERMUTEA_IN_GA(X2, X3)
PERMUTEA_IN_GA(.(X1, X2), .(X3, X4)) → U3_GA(X1, X2, X3, X4, deleteB_in_aga(X3, X2, X5))
PERMUTEA_IN_GA(.(X1, X2), .(X3, X4)) → DELETEB_IN_AGA(X3, X2, X5)
DELETEB_IN_AGA(X1, .(X2, X3), .(X2, X4)) → U1_AGA(X1, X2, X3, X4, deleteB_in_aga(X1, X3, X4))
DELETEB_IN_AGA(X1, .(X2, X3), .(X2, X4)) → DELETEB_IN_AGA(X1, X3, X4)
PERMUTEA_IN_GA(.(X1, X2), .(X3, X4)) → U4_GA(X1, X2, X3, X4, deletecB_in_aga(X3, X2, X5))
U4_GA(X1, X2, X3, X4, deletecB_out_aga(X3, X2, X5)) → U5_GA(X1, X2, X3, X4, permuteA_in_ga(.(X1, X5), X4))
U4_GA(X1, X2, X3, X4, deletecB_out_aga(X3, X2, X5)) → PERMUTEA_IN_GA(.(X1, X5), X4)

The TRS R consists of the following rules:

deletecB_in_aga(X1, .(X1, X2), X2) → deletecB_out_aga(X1, .(X1, X2), X2)
deletecB_in_aga(X1, .(X2, X3), .(X2, X4)) → U10_aga(X1, X2, X3, X4, deletecB_in_aga(X1, X3, X4))
U10_aga(X1, X2, X3, X4, deletecB_out_aga(X1, X3, X4)) → deletecB_out_aga(X1, .(X2, X3), .(X2, X4))

The argument filtering Pi contains the following mapping:
permuteA_in_ga(x1, x2)  =  permuteA_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
deleteB_in_aga(x1, x2, x3)  =  deleteB_in_aga(x2)
deletecB_in_aga(x1, x2, x3)  =  deletecB_in_aga(x2)
deletecB_out_aga(x1, x2, x3)  =  deletecB_out_aga(x1, x2, x3)
U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x2, x3, x5)
PERMUTEA_IN_GA(x1, x2)  =  PERMUTEA_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x2, x4)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x5)
DELETEB_IN_AGA(x1, x2, x3)  =  DELETEB_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x2, x3, x5)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x2, x5)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x1, x2, x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 5 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

DELETEB_IN_AGA(X1, .(X2, X3), .(X2, X4)) → DELETEB_IN_AGA(X1, X3, X4)

The TRS R consists of the following rules:

deletecB_in_aga(X1, .(X1, X2), X2) → deletecB_out_aga(X1, .(X1, X2), X2)
deletecB_in_aga(X1, .(X2, X3), .(X2, X4)) → U10_aga(X1, X2, X3, X4, deletecB_in_aga(X1, X3, X4))
U10_aga(X1, X2, X3, X4, deletecB_out_aga(X1, X3, X4)) → deletecB_out_aga(X1, .(X2, X3), .(X2, X4))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
deletecB_in_aga(x1, x2, x3)  =  deletecB_in_aga(x2)
deletecB_out_aga(x1, x2, x3)  =  deletecB_out_aga(x1, x2, x3)
U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x2, x3, x5)
DELETEB_IN_AGA(x1, x2, x3)  =  DELETEB_IN_AGA(x2)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

DELETEB_IN_AGA(X1, .(X2, X3), .(X2, X4)) → DELETEB_IN_AGA(X1, X3, X4)

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

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

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

DELETEB_IN_AGA(.(X2, X3)) → DELETEB_IN_AGA(X3)

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

(12) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

• DELETEB_IN_AGA(.(X2, X3)) → DELETEB_IN_AGA(X3)
  The graph contains the following edges 1 > 1

(13) YES

(14) Obligation:

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

PERMUTEA_IN_GA(.(X1, X2), .(X3, X4)) → U4_GA(X1, X2, X3, X4, deletecB_in_aga(X3, X2, X5))
U4_GA(X1, X2, X3, X4, deletecB_out_aga(X3, X2, X5)) → PERMUTEA_IN_GA(.(X1, X5), X4)
PERMUTEA_IN_GA(.(X1, X2), .(X1, X3)) → PERMUTEA_IN_GA(X2, X3)

The TRS R consists of the following rules:

deletecB_in_aga(X1, .(X1, X2), X2) → deletecB_out_aga(X1, .(X1, X2), X2)
deletecB_in_aga(X1, .(X2, X3), .(X2, X4)) → U10_aga(X1, X2, X3, X4, deletecB_in_aga(X1, X3, X4))
U10_aga(X1, X2, X3, X4, deletecB_out_aga(X1, X3, X4)) → deletecB_out_aga(X1, .(X2, X3), .(X2, X4))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
deletecB_in_aga(x1, x2, x3)  =  deletecB_in_aga(x2)
deletecB_out_aga(x1, x2, x3)  =  deletecB_out_aga(x1, x2, x3)
U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x2, x3, x5)
PERMUTEA_IN_GA(x1, x2)  =  PERMUTEA_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x2, x5)

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

(15) PiDPToQDPProof (SOUND transformation)

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

(16) Obligation:

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

PERMUTEA_IN_GA(.(X1, X2)) → U4_GA(X1, X2, deletecB_in_aga(X2))
U4_GA(X1, X2, deletecB_out_aga(X3, X2, X5)) → PERMUTEA_IN_GA(.(X1, X5))
PERMUTEA_IN_GA(.(X1, X2)) → PERMUTEA_IN_GA(X2)

The TRS R consists of the following rules:

deletecB_in_aga(.(X1, X2)) → deletecB_out_aga(X1, .(X1, X2), X2)
deletecB_in_aga(.(X2, X3)) → U10_aga(X2, X3, deletecB_in_aga(X3))
U10_aga(X2, X3, deletecB_out_aga(X1, X3, X4)) → deletecB_out_aga(X1, .(X2, X3), .(X2, X4))

The set Q consists of the following terms:

deletecB_in_aga(x0)
U10_aga(x0, x1, x2)

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

(17) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

PERMUTEA_IN_GA(.(X1, X2)) → PERMUTEA_IN_GA(X2)

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

POL(.(x₁, x₂)) = 1 + x₂   
POL(PERMUTEA_IN_GA(x₁)) = 1 + x₁   
POL(U10_aga(x₁, x₂, x₃)) = 1 + x₃   
POL(U4_GA(x₁, x₂, x₃)) = 1 + x₃   
POL(deletecB_in_aga(x₁)) = 1 + x₁   
POL(deletecB_out_aga(x₁, x₂, x₃)) = 1 + x₃   

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

deletecB_in_aga(.(X1, X2)) → deletecB_out_aga(X1, .(X1, X2), X2)
deletecB_in_aga(.(X2, X3)) → U10_aga(X2, X3, deletecB_in_aga(X3))
U10_aga(X2, X3, deletecB_out_aga(X1, X3, X4)) → deletecB_out_aga(X1, .(X2, X3), .(X2, X4))

(18) Obligation:

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

PERMUTEA_IN_GA(.(X1, X2)) → U4_GA(X1, X2, deletecB_in_aga(X2))
U4_GA(X1, X2, deletecB_out_aga(X3, X2, X5)) → PERMUTEA_IN_GA(.(X1, X5))

The TRS R consists of the following rules:

deletecB_in_aga(.(X1, X2)) → deletecB_out_aga(X1, .(X1, X2), X2)
deletecB_in_aga(.(X2, X3)) → U10_aga(X2, X3, deletecB_in_aga(X3))
U10_aga(X2, X3, deletecB_out_aga(X1, X3, X4)) → deletecB_out_aga(X1, .(X2, X3), .(X2, X4))

The set Q consists of the following terms:

deletecB_in_aga(x0)
U10_aga(x0, x1, x2)

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

(19) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

PERMUTEA_IN_GA(.(X1, X2)) → U4_GA(X1, X2, deletecB_in_aga(X2))

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

POL(.(x₁, x₂)) = 1 + x₂   
POL(PERMUTEA_IN_GA(x₁)) = x₁   
POL(U10_aga(x₁, x₂, x₃)) = 1 + x₃   
POL(U4_GA(x₁, x₂, x₃)) = x₃   
POL(deletecB_in_aga(x₁)) = x₁   
POL(deletecB_out_aga(x₁, x₂, x₃)) = 1 + x₃   

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

deletecB_in_aga(.(X1, X2)) → deletecB_out_aga(X1, .(X1, X2), X2)
deletecB_in_aga(.(X2, X3)) → U10_aga(X2, X3, deletecB_in_aga(X3))
U10_aga(X2, X3, deletecB_out_aga(X1, X3, X4)) → deletecB_out_aga(X1, .(X2, X3), .(X2, X4))

(20) Obligation:

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

U4_GA(X1, X2, deletecB_out_aga(X3, X2, X5)) → PERMUTEA_IN_GA(.(X1, X5))

The TRS R consists of the following rules:

deletecB_in_aga(.(X1, X2)) → deletecB_out_aga(X1, .(X1, X2), X2)
deletecB_in_aga(.(X2, X3)) → U10_aga(X2, X3, deletecB_in_aga(X3))
U10_aga(X2, X3, deletecB_out_aga(X1, X3, X4)) → deletecB_out_aga(X1, .(X2, X3), .(X2, X4))

The set Q consists of the following terms:

deletecB_in_aga(x0)
U10_aga(x0, x1, x2)

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

(21) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(22) TRUE