Left Termination of the query pattern
perm1(g,a)
w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

perm1([], []).
perm1(Xs, .(X, Ys)) :- ','(select(X, Xs, Zs), perm1(Zs, Ys)).
select(X, .(X, Xs), Xs).
select(X, .(Y, Xs), .(Y, Zs)) :- select(X, Xs, Zs).

Query: perm1(g,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="A: perm1(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
6 [label="6: perm1(T1, T2), SCOPE: 1, CLAUSE: 0 | perm1(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: true | perm1([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: perm1(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nperm1(T1, T2) !=? perm1([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: perm1([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(select(T7, [], X7), perm1(X7, T8))\n\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(select(T7, [], X7), perm1(X7, T8)), SCOPE: 2, CLAUSE: 2 | ','(select(T7, [], X7), perm1(X7, T8)), SCOPE: 2, CLAUSE: 3\n\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(select(T7, [], X7), perm1(X7, T8)), SCOPE: 2, CLAUSE: 3\n\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(select(T17, T14, X20), perm1(X20, T18))\n\nT14 is ground\nX20 is free\nperm1(T14, T2) !=? perm1([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(select(T17, T14, X20), perm1(X20, T18)), SCOPE: 3, CLAUSE: 2 | ','(select(T17, T14, X20), perm1(X20, T18)), SCOPE: 3, CLAUSE: 3\n\nT14 is ground\nX20 is free\nperm1(T14, T2) !=? perm1([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(select(T17, T14, X20), perm1(X20, T18)), SCOPE: 3, CLAUSE: 2\n\nT14 is ground\nX20 is free\nperm1(T14, T2) !=? perm1([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(select(T17, T14, X20), perm1(X20, T18)), SCOPE: 3, CLAUSE: 3\n\nT14 is ground\nX20 is free\nperm1(T14, T2) !=? perm1([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: perm1(T28, T29)\n\nT28 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="B: ','(select(T39, T38, X45), perm1(.(T37, X45), T40))\n\nT37 is ground\nT38 is ground\nX45 is free", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="C: select(T39, T38, X45)\n\nT38 is ground\nX45 is free", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: perm1(.(T37, T45), T46)\n\nT37 is ground\nT45 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: select(T39, T38, X45), SCOPE: 4, CLAUSE: 2 | select(T39, T38, X45), SCOPE: 4, CLAUSE: 3\n\nT38 is ground\nX45 is free", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: select(T39, T38, X45), SCOPE: 4, CLAUSE: 2\n\nT38 is ground\nX45 is free", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: select(T39, T38, X45), SCOPE: 4, CLAUSE: 3\n\nT38 is ground\nX45 is free", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: select(T70, T69, X78)\n\nT69 is ground\nX78 is free", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 78 [arrowhead = none ];
78 -> 6;

79 [label="EVAL with clause\nperm1([], []).\nand substitutionT1 -> [],\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 79 [arrowhead = none ];
79 -> 11;

80 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 80 [arrowhead = none ];
80 -> 14;

81 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 81 [arrowhead = none ];
81 -> 17;

82 [label="EVAL with clause\nperm1(X17, .(X18, X19)) :- ','(select(X18, X17, X20), perm1(X20, X19)).\nand substitutionT1 -> T14,\nX17 -> T14,\nX18 -> T17,\nX19 -> T18,\nT2 -> .(T17, T18),\nT15 -> T17,\nT16 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 82 [arrowhead = none ];
82 -> 31;

83 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 83 [arrowhead = none ];
83 -> 33;

84 [label="EVAL with clause\nperm1(X4, .(X5, X6)) :- ','(select(X5, X4, X7), perm1(X7, X6)).\nand substitutionX4 -> [],\nX5 -> T7,\nX6 -> T8,\nT2 -> .(T7, T8),\nT5 -> T7,\nT6 -> T8", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 84 [arrowhead = none ];
84 -> 19;

85 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 85 [arrowhead = none ];
85 -> 21;

86 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 86 [arrowhead = none ];
86 -> 25;

87 [label="BACKTRACK\nfor clause: select(X, .(X, Xs), Xs)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 87 [arrowhead = none ];
87 -> 27;

88 [label="BACKTRACK\nfor clause: select(X, .(Y, Xs), .(Y, Zs)) :- select(X, Xs, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 88 [arrowhead = none ];
88 -> 29;

89 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 89 [arrowhead = none ];
89 -> 35;

90 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 90 [arrowhead = none ];
90 -> 40;

91 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 91 [arrowhead = none ];
91 -> 41;

92 [label="EVAL with clause\nselect(X29, .(X29, X30), X30).\nand substitutionT17 -> T27,\nX29 -> T27,\nX30 -> T28,\nT14 -> .(T27, T28),\nX20 -> T28,\nT18 -> T29", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 92 [arrowhead = none ];
92 -> 45;

93 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
40 -> 93 [arrowhead = none ];
93 -> 49;

94 [label="EVAL with clause\nselect(X41, .(X42, X43), .(X42, X44)) :- select(X41, X43, X44).\nand substitutionT17 -> T39,\nX41 -> T39,\nX42 -> T37,\nX43 -> T38,\nT14 -> .(T37, T38),\nX44 -> X45,\nX20 -> .(T37, X45),\nT36 -> T39,\nT18 -> T40", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 94 [arrowhead = none ];
94 -> 56;

95 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 95 [arrowhead = none ];
95 -> 63;

96 [label="INSTANCE with matching:\nT1 -> T28\nT2 -> T29", fontsize=14, style = filled, fillcolor = lightgrey];
45 -> 96 [arrowhead = none , style=dashed];
96 -> 2[style=dashed];

97 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
56 -> 97 [arrowhead = none ];
97 -> 68;

98 [label="SPLIT 2\nnew knowledge:\nT39 is ground\nT38 is ground\nT45 is ground\nreplacements:X45 -> T45,\nT40 -> T46", fontsize=14, style = filled, fillcolor = violet];
56 -> 98 [arrowhead = none ];
98 -> 69;

99 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
68 -> 99 [arrowhead = none ];
99 -> 70;

100 [label="INSTANCE with matching:\nT1 -> .(T37, T45)\nT2 -> T46", fontsize=14, style = filled, fillcolor = lightgrey];
69 -> 100 [arrowhead = none , style=dashed];
100 -> 2[style=dashed];

101 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
70 -> 101 [arrowhead = none ];
101 -> 71;

102 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
70 -> 102 [arrowhead = none ];
102 -> 72;

103 [label="EVAL with clause\nselect(X62, .(X62, X63), X63).\nand substitutionT39 -> T59,\nX62 -> T59,\nX63 -> T60,\nT38 -> .(T59, T60),\nX45 -> T60", fontsize=14, style = filled, fillcolor = lightblue];
71 -> 103 [arrowhead = none ];
103 -> 73;

104 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
71 -> 104 [arrowhead = none ];
104 -> 74;

105 [label="EVAL with clause\nselect(X74, .(X75, X76), .(X75, X77)) :- select(X74, X76, X77).\nand substitutionT39 -> T70,\nX74 -> T70,\nX75 -> T68,\nX76 -> T69,\nT38 -> .(T68, T69),\nX77 -> X78,\nX45 -> .(T68, X78),\nT67 -> T70", fontsize=14, style = filled, fillcolor = lightblue];
72 -> 105 [arrowhead = none ];
105 -> 76;

106 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
72 -> 106 [arrowhead = none ];
106 -> 77;

107 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
73 -> 107 [arrowhead = none ];
107 -> 75;

108 [label="INSTANCE with matching:\nT39 -> T70\nT38 -> T69\nX45 -> X78", fontsize=14, style = filled, fillcolor = lightgrey];
76 -> 108 [arrowhead = none , style=dashed];
108 -> 68[style=dashed];

}

(2) Obligation:

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

perm1A_in_ga([], []) → perm1A_out_ga([], [])
perm1A_in_ga(.(T27, T28), .(T27, T29)) → U1_ga(T27, T28, T29, perm1A_in_ga(T28, T29))
perm1A_in_ga(.(T37, T38), .(T39, T40)) → U2_ga(T37, T38, T39, T40, pB_in_agaga(T39, T38, X45, T37, T40))
pB_in_agaga(T39, T38, T45, T37, T46) → U4_agaga(T39, T38, T45, T37, T46, selectC_in_aga(T39, T38, T45))
selectC_in_aga(T59, .(T59, T60), T60) → selectC_out_aga(T59, .(T59, T60), T60)
selectC_in_aga(T70, .(T68, T69), .(T68, X78)) → U3_aga(T70, T68, T69, X78, selectC_in_aga(T70, T69, X78))
U3_aga(T70, T68, T69, X78, selectC_out_aga(T70, T69, X78)) → selectC_out_aga(T70, .(T68, T69), .(T68, X78))
U4_agaga(T39, T38, T45, T37, T46, selectC_out_aga(T39, T38, T45)) → U5_agaga(T39, T38, T45, T37, T46, perm1A_in_ga(.(T37, T45), T46))
U5_agaga(T39, T38, T45, T37, T46, perm1A_out_ga(.(T37, T45), T46)) → pB_out_agaga(T39, T38, T45, T37, T46)
U2_ga(T37, T38, T39, T40, pB_out_agaga(T39, T38, X45, T37, T40)) → perm1A_out_ga(.(T37, T38), .(T39, T40))
U1_ga(T27, T28, T29, perm1A_out_ga(T28, T29)) → perm1A_out_ga(.(T27, T28), .(T27, T29))

The argument filtering Pi contains the following mapping:
perm1A_in_ga(x1, x2)  =  perm1A_in_ga(x1)
[]  =  []
perm1A_out_ga(x1, x2)  =  perm1A_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x5)
pB_in_agaga(x1, x2, x3, x4, x5)  =  pB_in_agaga(x2, x4)
U4_agaga(x1, x2, x3, x4, x5, x6)  =  U4_agaga(x2, x4, x6)
selectC_in_aga(x1, x2, x3)  =  selectC_in_aga(x2)
selectC_out_aga(x1, x2, x3)  =  selectC_out_aga(x1, x2, x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x2, x3, x5)
U5_agaga(x1, x2, x3, x4, x5, x6)  =  U5_agaga(x1, x2, x3, x4, x6)
pB_out_agaga(x1, x2, x3, x4, x5)  =  pB_out_agaga(x1, x2, x3, x4, x5)

(3) 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:

PERM1A_IN_GA(.(T27, T28), .(T27, T29)) → U1_GA(T27, T28, T29, perm1A_in_ga(T28, T29))
PERM1A_IN_GA(.(T27, T28), .(T27, T29)) → PERM1A_IN_GA(T28, T29)
PERM1A_IN_GA(.(T37, T38), .(T39, T40)) → U2_GA(T37, T38, T39, T40, pB_in_agaga(T39, T38, X45, T37, T40))
PERM1A_IN_GA(.(T37, T38), .(T39, T40)) → PB_IN_AGAGA(T39, T38, X45, T37, T40)
PB_IN_AGAGA(T39, T38, T45, T37, T46) → U4_AGAGA(T39, T38, T45, T37, T46, selectC_in_aga(T39, T38, T45))
PB_IN_AGAGA(T39, T38, T45, T37, T46) → SELECTC_IN_AGA(T39, T38, T45)
SELECTC_IN_AGA(T70, .(T68, T69), .(T68, X78)) → U3_AGA(T70, T68, T69, X78, selectC_in_aga(T70, T69, X78))
SELECTC_IN_AGA(T70, .(T68, T69), .(T68, X78)) → SELECTC_IN_AGA(T70, T69, X78)
U4_AGAGA(T39, T38, T45, T37, T46, selectC_out_aga(T39, T38, T45)) → U5_AGAGA(T39, T38, T45, T37, T46, perm1A_in_ga(.(T37, T45), T46))
U4_AGAGA(T39, T38, T45, T37, T46, selectC_out_aga(T39, T38, T45)) → PERM1A_IN_GA(.(T37, T45), T46)

The TRS R consists of the following rules:

perm1A_in_ga([], []) → perm1A_out_ga([], [])
perm1A_in_ga(.(T27, T28), .(T27, T29)) → U1_ga(T27, T28, T29, perm1A_in_ga(T28, T29))
perm1A_in_ga(.(T37, T38), .(T39, T40)) → U2_ga(T37, T38, T39, T40, pB_in_agaga(T39, T38, X45, T37, T40))
pB_in_agaga(T39, T38, T45, T37, T46) → U4_agaga(T39, T38, T45, T37, T46, selectC_in_aga(T39, T38, T45))
selectC_in_aga(T59, .(T59, T60), T60) → selectC_out_aga(T59, .(T59, T60), T60)
selectC_in_aga(T70, .(T68, T69), .(T68, X78)) → U3_aga(T70, T68, T69, X78, selectC_in_aga(T70, T69, X78))
U3_aga(T70, T68, T69, X78, selectC_out_aga(T70, T69, X78)) → selectC_out_aga(T70, .(T68, T69), .(T68, X78))
U4_agaga(T39, T38, T45, T37, T46, selectC_out_aga(T39, T38, T45)) → U5_agaga(T39, T38, T45, T37, T46, perm1A_in_ga(.(T37, T45), T46))
U5_agaga(T39, T38, T45, T37, T46, perm1A_out_ga(.(T37, T45), T46)) → pB_out_agaga(T39, T38, T45, T37, T46)
U2_ga(T37, T38, T39, T40, pB_out_agaga(T39, T38, X45, T37, T40)) → perm1A_out_ga(.(T37, T38), .(T39, T40))
U1_ga(T27, T28, T29, perm1A_out_ga(T28, T29)) → perm1A_out_ga(.(T27, T28), .(T27, T29))

The argument filtering Pi contains the following mapping:
perm1A_in_ga(x1, x2)  =  perm1A_in_ga(x1)
[]  =  []
perm1A_out_ga(x1, x2)  =  perm1A_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x5)
pB_in_agaga(x1, x2, x3, x4, x5)  =  pB_in_agaga(x2, x4)
U4_agaga(x1, x2, x3, x4, x5, x6)  =  U4_agaga(x2, x4, x6)
selectC_in_aga(x1, x2, x3)  =  selectC_in_aga(x2)
selectC_out_aga(x1, x2, x3)  =  selectC_out_aga(x1, x2, x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x2, x3, x5)
U5_agaga(x1, x2, x3, x4, x5, x6)  =  U5_agaga(x1, x2, x3, x4, x6)
pB_out_agaga(x1, x2, x3, x4, x5)  =  pB_out_agaga(x1, x2, x3, x4, x5)
PERM1A_IN_GA(x1, x2)  =  PERM1A_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x5)
PB_IN_AGAGA(x1, x2, x3, x4, x5)  =  PB_IN_AGAGA(x2, x4)
U4_AGAGA(x1, x2, x3, x4, x5, x6)  =  U4_AGAGA(x2, x4, x6)
SELECTC_IN_AGA(x1, x2, x3)  =  SELECTC_IN_AGA(x2)
U3_AGA(x1, x2, x3, x4, x5)  =  U3_AGA(x2, x3, x5)
U5_AGAGA(x1, x2, x3, x4, x5, x6)  =  U5_AGAGA(x1, x2, x3, x4, x6)

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

(4) Obligation:

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

PERM1A_IN_GA(.(T27, T28), .(T27, T29)) → U1_GA(T27, T28, T29, perm1A_in_ga(T28, T29))
PERM1A_IN_GA(.(T27, T28), .(T27, T29)) → PERM1A_IN_GA(T28, T29)
PERM1A_IN_GA(.(T37, T38), .(T39, T40)) → U2_GA(T37, T38, T39, T40, pB_in_agaga(T39, T38, X45, T37, T40))
PERM1A_IN_GA(.(T37, T38), .(T39, T40)) → PB_IN_AGAGA(T39, T38, X45, T37, T40)
PB_IN_AGAGA(T39, T38, T45, T37, T46) → U4_AGAGA(T39, T38, T45, T37, T46, selectC_in_aga(T39, T38, T45))
PB_IN_AGAGA(T39, T38, T45, T37, T46) → SELECTC_IN_AGA(T39, T38, T45)
SELECTC_IN_AGA(T70, .(T68, T69), .(T68, X78)) → U3_AGA(T70, T68, T69, X78, selectC_in_aga(T70, T69, X78))
SELECTC_IN_AGA(T70, .(T68, T69), .(T68, X78)) → SELECTC_IN_AGA(T70, T69, X78)
U4_AGAGA(T39, T38, T45, T37, T46, selectC_out_aga(T39, T38, T45)) → U5_AGAGA(T39, T38, T45, T37, T46, perm1A_in_ga(.(T37, T45), T46))
U4_AGAGA(T39, T38, T45, T37, T46, selectC_out_aga(T39, T38, T45)) → PERM1A_IN_GA(.(T37, T45), T46)

The TRS R consists of the following rules:

perm1A_in_ga([], []) → perm1A_out_ga([], [])
perm1A_in_ga(.(T27, T28), .(T27, T29)) → U1_ga(T27, T28, T29, perm1A_in_ga(T28, T29))
perm1A_in_ga(.(T37, T38), .(T39, T40)) → U2_ga(T37, T38, T39, T40, pB_in_agaga(T39, T38, X45, T37, T40))
pB_in_agaga(T39, T38, T45, T37, T46) → U4_agaga(T39, T38, T45, T37, T46, selectC_in_aga(T39, T38, T45))
selectC_in_aga(T59, .(T59, T60), T60) → selectC_out_aga(T59, .(T59, T60), T60)
selectC_in_aga(T70, .(T68, T69), .(T68, X78)) → U3_aga(T70, T68, T69, X78, selectC_in_aga(T70, T69, X78))
U3_aga(T70, T68, T69, X78, selectC_out_aga(T70, T69, X78)) → selectC_out_aga(T70, .(T68, T69), .(T68, X78))
U4_agaga(T39, T38, T45, T37, T46, selectC_out_aga(T39, T38, T45)) → U5_agaga(T39, T38, T45, T37, T46, perm1A_in_ga(.(T37, T45), T46))
U5_agaga(T39, T38, T45, T37, T46, perm1A_out_ga(.(T37, T45), T46)) → pB_out_agaga(T39, T38, T45, T37, T46)
U2_ga(T37, T38, T39, T40, pB_out_agaga(T39, T38, X45, T37, T40)) → perm1A_out_ga(.(T37, T38), .(T39, T40))
U1_ga(T27, T28, T29, perm1A_out_ga(T28, T29)) → perm1A_out_ga(.(T27, T28), .(T27, T29))

The argument filtering Pi contains the following mapping:
perm1A_in_ga(x1, x2)  =  perm1A_in_ga(x1)
[]  =  []
perm1A_out_ga(x1, x2)  =  perm1A_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x5)
pB_in_agaga(x1, x2, x3, x4, x5)  =  pB_in_agaga(x2, x4)
U4_agaga(x1, x2, x3, x4, x5, x6)  =  U4_agaga(x2, x4, x6)
selectC_in_aga(x1, x2, x3)  =  selectC_in_aga(x2)
selectC_out_aga(x1, x2, x3)  =  selectC_out_aga(x1, x2, x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x2, x3, x5)
U5_agaga(x1, x2, x3, x4, x5, x6)  =  U5_agaga(x1, x2, x3, x4, x6)
pB_out_agaga(x1, x2, x3, x4, x5)  =  pB_out_agaga(x1, x2, x3, x4, x5)
PERM1A_IN_GA(x1, x2)  =  PERM1A_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x5)
PB_IN_AGAGA(x1, x2, x3, x4, x5)  =  PB_IN_AGAGA(x2, x4)
U4_AGAGA(x1, x2, x3, x4, x5, x6)  =  U4_AGAGA(x2, x4, x6)
SELECTC_IN_AGA(x1, x2, x3)  =  SELECTC_IN_AGA(x2)
U3_AGA(x1, x2, x3, x4, x5)  =  U3_AGA(x2, x3, x5)
U5_AGAGA(x1, x2, x3, x4, x5, x6)  =  U5_AGAGA(x1, x2, x3, x4, x6)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

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

(6) Complex Obligation (AND)

(7) Obligation:

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

SELECTC_IN_AGA(T70, .(T68, T69), .(T68, X78)) → SELECTC_IN_AGA(T70, T69, X78)

The TRS R consists of the following rules:

perm1A_in_ga([], []) → perm1A_out_ga([], [])
perm1A_in_ga(.(T27, T28), .(T27, T29)) → U1_ga(T27, T28, T29, perm1A_in_ga(T28, T29))
perm1A_in_ga(.(T37, T38), .(T39, T40)) → U2_ga(T37, T38, T39, T40, pB_in_agaga(T39, T38, X45, T37, T40))
pB_in_agaga(T39, T38, T45, T37, T46) → U4_agaga(T39, T38, T45, T37, T46, selectC_in_aga(T39, T38, T45))
selectC_in_aga(T59, .(T59, T60), T60) → selectC_out_aga(T59, .(T59, T60), T60)
selectC_in_aga(T70, .(T68, T69), .(T68, X78)) → U3_aga(T70, T68, T69, X78, selectC_in_aga(T70, T69, X78))
U3_aga(T70, T68, T69, X78, selectC_out_aga(T70, T69, X78)) → selectC_out_aga(T70, .(T68, T69), .(T68, X78))
U4_agaga(T39, T38, T45, T37, T46, selectC_out_aga(T39, T38, T45)) → U5_agaga(T39, T38, T45, T37, T46, perm1A_in_ga(.(T37, T45), T46))
U5_agaga(T39, T38, T45, T37, T46, perm1A_out_ga(.(T37, T45), T46)) → pB_out_agaga(T39, T38, T45, T37, T46)
U2_ga(T37, T38, T39, T40, pB_out_agaga(T39, T38, X45, T37, T40)) → perm1A_out_ga(.(T37, T38), .(T39, T40))
U1_ga(T27, T28, T29, perm1A_out_ga(T28, T29)) → perm1A_out_ga(.(T27, T28), .(T27, T29))

The argument filtering Pi contains the following mapping:
perm1A_in_ga(x1, x2)  =  perm1A_in_ga(x1)
[]  =  []
perm1A_out_ga(x1, x2)  =  perm1A_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x5)
pB_in_agaga(x1, x2, x3, x4, x5)  =  pB_in_agaga(x2, x4)
U4_agaga(x1, x2, x3, x4, x5, x6)  =  U4_agaga(x2, x4, x6)
selectC_in_aga(x1, x2, x3)  =  selectC_in_aga(x2)
selectC_out_aga(x1, x2, x3)  =  selectC_out_aga(x1, x2, x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x2, x3, x5)
U5_agaga(x1, x2, x3, x4, x5, x6)  =  U5_agaga(x1, x2, x3, x4, x6)
pB_out_agaga(x1, x2, x3, x4, x5)  =  pB_out_agaga(x1, x2, x3, x4, x5)
SELECTC_IN_AGA(x1, x2, x3)  =  SELECTC_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:

SELECTC_IN_AGA(T70, .(T68, T69), .(T68, X78)) → SELECTC_IN_AGA(T70, T69, X78)

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

SELECTC_IN_AGA(.(T68, T69)) → SELECTC_IN_AGA(T69)

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:

• SELECTC_IN_AGA(.(T68, T69)) → SELECTC_IN_AGA(T69)
  The graph contains the following edges 1 > 1

(13) YES

(14) Obligation:

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

PERM1A_IN_GA(.(T37, T38), .(T39, T40)) → PB_IN_AGAGA(T39, T38, X45, T37, T40)
PB_IN_AGAGA(T39, T38, T45, T37, T46) → U4_AGAGA(T39, T38, T45, T37, T46, selectC_in_aga(T39, T38, T45))
U4_AGAGA(T39, T38, T45, T37, T46, selectC_out_aga(T39, T38, T45)) → PERM1A_IN_GA(.(T37, T45), T46)
PERM1A_IN_GA(.(T27, T28), .(T27, T29)) → PERM1A_IN_GA(T28, T29)

The TRS R consists of the following rules:

perm1A_in_ga([], []) → perm1A_out_ga([], [])
perm1A_in_ga(.(T27, T28), .(T27, T29)) → U1_ga(T27, T28, T29, perm1A_in_ga(T28, T29))
perm1A_in_ga(.(T37, T38), .(T39, T40)) → U2_ga(T37, T38, T39, T40, pB_in_agaga(T39, T38, X45, T37, T40))
pB_in_agaga(T39, T38, T45, T37, T46) → U4_agaga(T39, T38, T45, T37, T46, selectC_in_aga(T39, T38, T45))
selectC_in_aga(T59, .(T59, T60), T60) → selectC_out_aga(T59, .(T59, T60), T60)
selectC_in_aga(T70, .(T68, T69), .(T68, X78)) → U3_aga(T70, T68, T69, X78, selectC_in_aga(T70, T69, X78))
U3_aga(T70, T68, T69, X78, selectC_out_aga(T70, T69, X78)) → selectC_out_aga(T70, .(T68, T69), .(T68, X78))
U4_agaga(T39, T38, T45, T37, T46, selectC_out_aga(T39, T38, T45)) → U5_agaga(T39, T38, T45, T37, T46, perm1A_in_ga(.(T37, T45), T46))
U5_agaga(T39, T38, T45, T37, T46, perm1A_out_ga(.(T37, T45), T46)) → pB_out_agaga(T39, T38, T45, T37, T46)
U2_ga(T37, T38, T39, T40, pB_out_agaga(T39, T38, X45, T37, T40)) → perm1A_out_ga(.(T37, T38), .(T39, T40))
U1_ga(T27, T28, T29, perm1A_out_ga(T28, T29)) → perm1A_out_ga(.(T27, T28), .(T27, T29))

The argument filtering Pi contains the following mapping:
perm1A_in_ga(x1, x2)  =  perm1A_in_ga(x1)
[]  =  []
perm1A_out_ga(x1, x2)  =  perm1A_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x5)
pB_in_agaga(x1, x2, x3, x4, x5)  =  pB_in_agaga(x2, x4)
U4_agaga(x1, x2, x3, x4, x5, x6)  =  U4_agaga(x2, x4, x6)
selectC_in_aga(x1, x2, x3)  =  selectC_in_aga(x2)
selectC_out_aga(x1, x2, x3)  =  selectC_out_aga(x1, x2, x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x2, x3, x5)
U5_agaga(x1, x2, x3, x4, x5, x6)  =  U5_agaga(x1, x2, x3, x4, x6)
pB_out_agaga(x1, x2, x3, x4, x5)  =  pB_out_agaga(x1, x2, x3, x4, x5)
PERM1A_IN_GA(x1, x2)  =  PERM1A_IN_GA(x1)
PB_IN_AGAGA(x1, x2, x3, x4, x5)  =  PB_IN_AGAGA(x2, x4)
U4_AGAGA(x1, x2, x3, x4, x5, x6)  =  U4_AGAGA(x2, x4, x6)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

PERM1A_IN_GA(.(T37, T38), .(T39, T40)) → PB_IN_AGAGA(T39, T38, X45, T37, T40)
PB_IN_AGAGA(T39, T38, T45, T37, T46) → U4_AGAGA(T39, T38, T45, T37, T46, selectC_in_aga(T39, T38, T45))
U4_AGAGA(T39, T38, T45, T37, T46, selectC_out_aga(T39, T38, T45)) → PERM1A_IN_GA(.(T37, T45), T46)
PERM1A_IN_GA(.(T27, T28), .(T27, T29)) → PERM1A_IN_GA(T28, T29)

The TRS R consists of the following rules:

selectC_in_aga(T59, .(T59, T60), T60) → selectC_out_aga(T59, .(T59, T60), T60)
selectC_in_aga(T70, .(T68, T69), .(T68, X78)) → U3_aga(T70, T68, T69, X78, selectC_in_aga(T70, T69, X78))
U3_aga(T70, T68, T69, X78, selectC_out_aga(T70, T69, X78)) → selectC_out_aga(T70, .(T68, T69), .(T68, X78))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
selectC_in_aga(x1, x2, x3)  =  selectC_in_aga(x2)
selectC_out_aga(x1, x2, x3)  =  selectC_out_aga(x1, x2, x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x2, x3, x5)
PERM1A_IN_GA(x1, x2)  =  PERM1A_IN_GA(x1)
PB_IN_AGAGA(x1, x2, x3, x4, x5)  =  PB_IN_AGAGA(x2, x4)
U4_AGAGA(x1, x2, x3, x4, x5, x6)  =  U4_AGAGA(x2, x4, x6)

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

PERM1A_IN_GA(.(T37, T38)) → PB_IN_AGAGA(T38, T37)
PB_IN_AGAGA(T38, T37) → U4_AGAGA(T38, T37, selectC_in_aga(T38))
U4_AGAGA(T38, T37, selectC_out_aga(T39, T38, T45)) → PERM1A_IN_GA(.(T37, T45))
PERM1A_IN_GA(.(T27, T28)) → PERM1A_IN_GA(T28)

The TRS R consists of the following rules:

selectC_in_aga(.(T59, T60)) → selectC_out_aga(T59, .(T59, T60), T60)
selectC_in_aga(.(T68, T69)) → U3_aga(T68, T69, selectC_in_aga(T69))
U3_aga(T68, T69, selectC_out_aga(T70, T69, X78)) → selectC_out_aga(T70, .(T68, T69), .(T68, X78))

The set Q consists of the following terms:

selectC_in_aga(x0)
U3_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.

PB_IN_AGAGA(T38, T37) → U4_AGAGA(T38, T37, selectC_in_aga(T38))
PERM1A_IN_GA(.(T27, T28)) → PERM1A_IN_GA(T28)

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

POL(.(x₁, x₂)) = 1 + x₂   
POL(PB_IN_AGAGA(x₁, x₂)) = 1 + x₁   
POL(PERM1A_IN_GA(x₁)) = x₁   
POL(U3_aga(x₁, x₂, x₃)) = 1 + x₃   
POL(U4_AGAGA(x₁, x₂, x₃)) = x₃   
POL(selectC_in_aga(x₁)) = x₁   
POL(selectC_out_aga(x₁, x₂, x₃)) = 1 + x₃   

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

selectC_in_aga(.(T59, T60)) → selectC_out_aga(T59, .(T59, T60), T60)
selectC_in_aga(.(T68, T69)) → U3_aga(T68, T69, selectC_in_aga(T69))
U3_aga(T68, T69, selectC_out_aga(T70, T69, X78)) → selectC_out_aga(T70, .(T68, T69), .(T68, X78))

(20) Obligation:

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

PERM1A_IN_GA(.(T37, T38)) → PB_IN_AGAGA(T38, T37)
U4_AGAGA(T38, T37, selectC_out_aga(T39, T38, T45)) → PERM1A_IN_GA(.(T37, T45))

The TRS R consists of the following rules:

selectC_in_aga(.(T59, T60)) → selectC_out_aga(T59, .(T59, T60), T60)
selectC_in_aga(.(T68, T69)) → U3_aga(T68, T69, selectC_in_aga(T69))
U3_aga(T68, T69, selectC_out_aga(T70, T69, X78)) → selectC_out_aga(T70, .(T68, T69), .(T68, X78))

The set Q consists of the following terms:

selectC_in_aga(x0)
U3_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 2 less nodes.

(22) TRUE