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

(0) Obligation:

Clauses:

select(X, .(X, Xs), Xs).
select(X, .(Y, Xs), .(Y, Zs)) :- select(X, Xs, Zs).

Query: select(a,g,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
3 [label="A: select(T1, T2, T3)\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: select(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | select(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: true | select(T1, .(T6, T7), T3), SCOPE: 1, CLAUSE: 1\n\nT6 is ground\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: select(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT2 is ground\nX3 is free\nX4 is free\nselect(T1, T2, T3) !=? select(X3, .(X3, X4), X4)", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: select(T1, .(T6, T7), T3), SCOPE: 1, CLAUSE: 1\n\nT6 is ground\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: select(T16, T14, T17)\n\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: select(T16, T14, T17), SCOPE: 2, CLAUSE: 0 | select(T16, T14, T17), SCOPE: 2, CLAUSE: 1\n\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: select(T16, T14, T17), SCOPE: 2, CLAUSE: 0\n\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: select(T16, T14, T17), SCOPE: 2, CLAUSE: 1\n\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: select(T40, T38, T41)\n\nT38 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: select(T54, T52, T55)\n\nT51 is ground\nT52 is ground\nX3 is free\nX4 is free\nselect(T1, .(T51, T52), T3) !=? select(X3, .(X3, X4), X4)", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: select(T54, T52, T55), SCOPE: 3, CLAUSE: 0 | select(T54, T52, T55), SCOPE: 3, CLAUSE: 1\n\nT51 is ground\nT52 is ground\nX3 is free\nX4 is free\nselect(T1, .(T51, T52), T3) !=? select(X3, .(X3, X4), X4)", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: select(T54, T52, T55), SCOPE: 3, CLAUSE: 0\n\nT51 is ground\nT52 is ground\nX3 is free\nX4 is free\nselect(T1, .(T51, T52), T3) !=? select(X3, .(X3, X4), X4)", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: select(T54, T52, T55), SCOPE: 3, CLAUSE: 1\n\nT51 is ground\nT52 is ground\nX3 is free\nX4 is free\nselect(T1, .(T51, T52), T3) !=? select(X3, .(X3, X4), X4)", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: select(T78, T76, T79)\n\nT51 is ground\nT75 is ground\nT76 is ground\nX3 is free\nX4 is free\nselect(T1, .(T51, .(T75, T76)), T3) !=? select(X3, .(X3, X4), X4)", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 58 [arrowhead = none ];
58 -> 4;

59 [label="EVAL with clause\nselect(X3, .(X3, X4), X4).\nand substitutionT1 -> T6,\nX3 -> T6,\nX4 -> T7,\nT2 -> .(T6, T7),\nT3 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 59 [arrowhead = none ];
59 -> 8;

60 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 60 [arrowhead = none ];
60 -> 10;

61 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 61 [arrowhead = none ];
61 -> 11;

62 [label="EVAL with clause\nselect(X43, .(X44, X45), .(X44, X46)) :- select(X43, X45, X46).\nand substitutionT1 -> T54,\nX43 -> T54,\nX44 -> T51,\nX45 -> T52,\nT2 -> .(T51, T52),\nX46 -> T55,\nT3 -> .(T51, T55),\nT50 -> T54,\nT53 -> T55", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 62 [arrowhead = none ];
62 -> 40;

63 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 63 [arrowhead = none ];
63 -> 41;

64 [label="EVAL with clause\nselect(X9, .(X10, X11), .(X10, X12)) :- select(X9, X11, X12).\nand substitutionT1 -> T16,\nX9 -> T16,\nT6 -> T13,\nX10 -> T13,\nT7 -> T14,\nX11 -> T14,\nX12 -> T17,\nT3 -> .(T13, T17),\nT12 -> T16,\nT15 -> T17", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 64 [arrowhead = none ];
64 -> 15;

65 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 65 [arrowhead = none ];
65 -> 18;

66 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 66 [arrowhead = none ];
66 -> 21;

67 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
21 -> 67 [arrowhead = none ];
67 -> 22;

68 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
21 -> 68 [arrowhead = none ];
68 -> 23;

69 [label="EVAL with clause\nselect(X21, .(X21, X22), X22).\nand substitutionT16 -> T26,\nX21 -> T26,\nX22 -> T27,\nT14 -> .(T26, T27),\nT17 -> T27", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 69 [arrowhead = none ];
69 -> 27;

70 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 70 [arrowhead = none ];
70 -> 28;

71 [label="EVAL with clause\nselect(X31, .(X32, X33), .(X32, X34)) :- select(X31, X33, X34).\nand substitutionT16 -> T40,\nX31 -> T40,\nX32 -> T37,\nX33 -> T38,\nT14 -> .(T37, T38),\nX34 -> T41,\nT17 -> .(T37, T41),\nT36 -> T40,\nT39 -> T41", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 71 [arrowhead = none ];
71 -> 36;

72 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 72 [arrowhead = none ];
72 -> 37;

73 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 73 [arrowhead = none ];
73 -> 30;

74 [label="INSTANCE with matching:\nT1 -> T40\nT2 -> T38\nT3 -> T41", fontsize=14, style = filled, fillcolor = lightgrey];
36 -> 74 [arrowhead = none , style=dashed];
74 -> 3[style=dashed];

75 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
40 -> 75 [arrowhead = none ];
75 -> 42;

76 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
42 -> 76 [arrowhead = none ];
76 -> 43;

77 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
42 -> 77 [arrowhead = none ];
77 -> 49;

78 [label="EVAL with clause\nselect(X55, .(X55, X56), X56).\nand substitutionT54 -> T64,\nX55 -> T64,\nX56 -> T65,\nT52 -> .(T64, T65),\nT55 -> T65", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 78 [arrowhead = none ];
78 -> 53;

79 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 79 [arrowhead = none ];
79 -> 54;

80 [label="EVAL with clause\nselect(X65, .(X66, X67), .(X66, X68)) :- select(X65, X67, X68).\nand substitutionT54 -> T78,\nX65 -> T78,\nX66 -> T75,\nX67 -> T76,\nT52 -> .(T75, T76),\nX68 -> T79,\nT55 -> .(T75, T79),\nT74 -> T78,\nT77 -> T79", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 80 [arrowhead = none ];
80 -> 56;

81 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 81 [arrowhead = none ];
81 -> 57;

82 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
53 -> 82 [arrowhead = none ];
82 -> 55;

83 [label="INSTANCE with matching:\nT1 -> T78\nT2 -> T76\nT3 -> T79", fontsize=14, style = filled, fillcolor = lightgrey];
56 -> 83 [arrowhead = none , style=dashed];
83 -> 3[style=dashed];

}

(2) Obligation:

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

selectA_in_aga(T6, .(T6, T7), T7) → selectA_out_aga(T6, .(T6, T7), T7)
selectA_in_aga(T26, .(T13, .(T26, T27)), .(T13, T27)) → selectA_out_aga(T26, .(T13, .(T26, T27)), .(T13, T27))
selectA_in_aga(T40, .(T13, .(T37, T38)), .(T13, .(T37, T41))) → U1_aga(T40, T13, T37, T38, T41, selectA_in_aga(T40, T38, T41))
selectA_in_aga(T78, .(T51, .(T75, T76)), .(T51, .(T75, T79))) → U2_aga(T78, T51, T75, T76, T79, selectA_in_aga(T78, T76, T79))
U2_aga(T78, T51, T75, T76, T79, selectA_out_aga(T78, T76, T79)) → selectA_out_aga(T78, .(T51, .(T75, T76)), .(T51, .(T75, T79)))
U1_aga(T40, T13, T37, T38, T41, selectA_out_aga(T40, T38, T41)) → selectA_out_aga(T40, .(T13, .(T37, T38)), .(T13, .(T37, T41)))

The argument filtering Pi contains the following mapping:
selectA_in_aga(x1, x2, x3)  =  selectA_in_aga(x2)
.(x1, x2)  =  .(x1, x2)
selectA_out_aga(x1, x2, x3)  =  selectA_out_aga(x1, x2, x3)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x2, x3, x4, x6)
U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x2, x3, x4, x6)

(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:

SELECTA_IN_AGA(T40, .(T13, .(T37, T38)), .(T13, .(T37, T41))) → U1_AGA(T40, T13, T37, T38, T41, selectA_in_aga(T40, T38, T41))
SELECTA_IN_AGA(T40, .(T13, .(T37, T38)), .(T13, .(T37, T41))) → SELECTA_IN_AGA(T40, T38, T41)
SELECTA_IN_AGA(T78, .(T51, .(T75, T76)), .(T51, .(T75, T79))) → U2_AGA(T78, T51, T75, T76, T79, selectA_in_aga(T78, T76, T79))

The TRS R consists of the following rules:

selectA_in_aga(T6, .(T6, T7), T7) → selectA_out_aga(T6, .(T6, T7), T7)
selectA_in_aga(T26, .(T13, .(T26, T27)), .(T13, T27)) → selectA_out_aga(T26, .(T13, .(T26, T27)), .(T13, T27))
selectA_in_aga(T40, .(T13, .(T37, T38)), .(T13, .(T37, T41))) → U1_aga(T40, T13, T37, T38, T41, selectA_in_aga(T40, T38, T41))
selectA_in_aga(T78, .(T51, .(T75, T76)), .(T51, .(T75, T79))) → U2_aga(T78, T51, T75, T76, T79, selectA_in_aga(T78, T76, T79))
U2_aga(T78, T51, T75, T76, T79, selectA_out_aga(T78, T76, T79)) → selectA_out_aga(T78, .(T51, .(T75, T76)), .(T51, .(T75, T79)))
U1_aga(T40, T13, T37, T38, T41, selectA_out_aga(T40, T38, T41)) → selectA_out_aga(T40, .(T13, .(T37, T38)), .(T13, .(T37, T41)))

The argument filtering Pi contains the following mapping:
selectA_in_aga(x1, x2, x3)  =  selectA_in_aga(x2)
.(x1, x2)  =  .(x1, x2)
selectA_out_aga(x1, x2, x3)  =  selectA_out_aga(x1, x2, x3)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x2, x3, x4, x6)
U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x2, x3, x4, x6)
SELECTA_IN_AGA(x1, x2, x3)  =  SELECTA_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5, x6)  =  U1_AGA(x2, x3, x4, x6)
U2_AGA(x1, x2, x3, x4, x5, x6)  =  U2_AGA(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:

SELECTA_IN_AGA(T40, .(T13, .(T37, T38)), .(T13, .(T37, T41))) → U1_AGA(T40, T13, T37, T38, T41, selectA_in_aga(T40, T38, T41))
SELECTA_IN_AGA(T40, .(T13, .(T37, T38)), .(T13, .(T37, T41))) → SELECTA_IN_AGA(T40, T38, T41)
SELECTA_IN_AGA(T78, .(T51, .(T75, T76)), .(T51, .(T75, T79))) → U2_AGA(T78, T51, T75, T76, T79, selectA_in_aga(T78, T76, T79))

The TRS R consists of the following rules:

selectA_in_aga(T6, .(T6, T7), T7) → selectA_out_aga(T6, .(T6, T7), T7)
selectA_in_aga(T26, .(T13, .(T26, T27)), .(T13, T27)) → selectA_out_aga(T26, .(T13, .(T26, T27)), .(T13, T27))
selectA_in_aga(T40, .(T13, .(T37, T38)), .(T13, .(T37, T41))) → U1_aga(T40, T13, T37, T38, T41, selectA_in_aga(T40, T38, T41))
selectA_in_aga(T78, .(T51, .(T75, T76)), .(T51, .(T75, T79))) → U2_aga(T78, T51, T75, T76, T79, selectA_in_aga(T78, T76, T79))
U2_aga(T78, T51, T75, T76, T79, selectA_out_aga(T78, T76, T79)) → selectA_out_aga(T78, .(T51, .(T75, T76)), .(T51, .(T75, T79)))
U1_aga(T40, T13, T37, T38, T41, selectA_out_aga(T40, T38, T41)) → selectA_out_aga(T40, .(T13, .(T37, T38)), .(T13, .(T37, T41)))

The argument filtering Pi contains the following mapping:
selectA_in_aga(x1, x2, x3)  =  selectA_in_aga(x2)
.(x1, x2)  =  .(x1, x2)
selectA_out_aga(x1, x2, x3)  =  selectA_out_aga(x1, x2, x3)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x2, x3, x4, x6)
U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x2, x3, x4, x6)
SELECTA_IN_AGA(x1, x2, x3)  =  SELECTA_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5, x6)  =  U1_AGA(x2, x3, x4, x6)
U2_AGA(x1, x2, x3, x4, x5, x6)  =  U2_AGA(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 1 SCC with 2 less nodes.

(6) Obligation:

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

SELECTA_IN_AGA(T40, .(T13, .(T37, T38)), .(T13, .(T37, T41))) → SELECTA_IN_AGA(T40, T38, T41)

The TRS R consists of the following rules:

selectA_in_aga(T6, .(T6, T7), T7) → selectA_out_aga(T6, .(T6, T7), T7)
selectA_in_aga(T26, .(T13, .(T26, T27)), .(T13, T27)) → selectA_out_aga(T26, .(T13, .(T26, T27)), .(T13, T27))
selectA_in_aga(T40, .(T13, .(T37, T38)), .(T13, .(T37, T41))) → U1_aga(T40, T13, T37, T38, T41, selectA_in_aga(T40, T38, T41))
selectA_in_aga(T78, .(T51, .(T75, T76)), .(T51, .(T75, T79))) → U2_aga(T78, T51, T75, T76, T79, selectA_in_aga(T78, T76, T79))
U2_aga(T78, T51, T75, T76, T79, selectA_out_aga(T78, T76, T79)) → selectA_out_aga(T78, .(T51, .(T75, T76)), .(T51, .(T75, T79)))
U1_aga(T40, T13, T37, T38, T41, selectA_out_aga(T40, T38, T41)) → selectA_out_aga(T40, .(T13, .(T37, T38)), .(T13, .(T37, T41)))

The argument filtering Pi contains the following mapping:
selectA_in_aga(x1, x2, x3)  =  selectA_in_aga(x2)
.(x1, x2)  =  .(x1, x2)
selectA_out_aga(x1, x2, x3)  =  selectA_out_aga(x1, x2, x3)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x2, x3, x4, x6)
U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x2, x3, x4, x6)
SELECTA_IN_AGA(x1, x2, x3)  =  SELECTA_IN_AGA(x2)

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

(7) UsableRulesProof (EQUIVALENT transformation)

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

(8) Obligation:

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

SELECTA_IN_AGA(T40, .(T13, .(T37, T38)), .(T13, .(T37, T41))) → SELECTA_IN_AGA(T40, T38, T41)

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

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

(9) PiDPToQDPProof (SOUND transformation)

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

(10) Obligation:

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

SELECTA_IN_AGA(.(T13, .(T37, T38))) → SELECTA_IN_AGA(T38)

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

(11) 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:

• SELECTA_IN_AGA(.(T13, .(T37, T38))) → SELECTA_IN_AGA(T38)
  The graph contains the following edges 1 > 1

(12) YES