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

(0) Obligation:

Clauses:

select(X1, [], X2) :- ','(!, failure(a)).
select(X, Y, Zs) :- ','(head(Y, X), tail(Y, Zs)).
select(X, Y, .(H, Zs)) :- ','(head(Y, H), ','(tail(Y, T), select(X, T, Zs))).
head([], X3).
head(.(H, X4), H).
tail([], []).
tail(.(X5, T), T).
failure(b).

Query: select(g,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\nT1 is ground\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 | select(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(!_1, failure(a)) | select(T6, [], T3), SCOPE: 1, CLAUSE: 1 | select(T6, [], T3), SCOPE: 1, CLAUSE: 2\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: select(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | select(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground\nX8 is free\nX9 is free\nselect(T1, T2, T3) !=? select(X8, [], X9)", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: failure(a)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: failure(a), SCOPE: 2, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(head(T12, T11), tail(T12, T14)) | select(T11, T12, T3), SCOPE: 1, CLAUSE: 2\n\nT11 is ground\nT12 is ground\nX8 is free\nX9 is free\nselect(T11, T12, T3) !=? select(X8, [], X9)", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(head(T12, T11), tail(T12, T14)), SCOPE: 3, CLAUSE: 3 | ','(head(T12, T11), tail(T12, T14)), SCOPE: 3, CLAUSE: 4 | ?_3 | select(T11, T12, T3), SCOPE: 1, CLAUSE: 2\n\nT11 is ground\nT12 is ground\nX8 is free\nX9 is free\nselect(T11, T12, T3) !=? select(X8, [], X9)", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(head(T12, T11), tail(T12, T14)), SCOPE: 3, CLAUSE: 4 | ?_3 | select(T11, T12, T3), SCOPE: 1, CLAUSE: 2\n\nT11 is ground\nT12 is ground\nX8 is free\nX9 is free\nselect(T11, T12, T3) !=? select(X8, [], X9)", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(head(T12, T11), tail(T12, T14)), SCOPE: 3, CLAUSE: 4\n\nT11 is ground\nT12 is ground\nX8 is free\nX9 is free\nselect(T11, T12, T3) !=? select(X8, [], X9)", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ?_3 | select(T11, T12, T3), SCOPE: 1, CLAUSE: 2\n\nT11 is ground\nT12 is ground\nX8 is free\nX9 is free\nselect(T11, T12, T3) !=? select(X8, [], X9)", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: tail(.(T24, T25), T14)\n\nT24 is ground\nT25 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: tail(.(T24, T25), T14), SCOPE: 4, CLAUSE: 5 | tail(.(T24, T25), T14), SCOPE: 4, CLAUSE: 6\n\nT24 is ground\nT25 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: tail(.(T24, T25), T14), SCOPE: 4, CLAUSE: 6\n\nT24 is ground\nT25 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: select(T11, T12, T3), SCOPE: 1, CLAUSE: 2\n\nT11 is ground\nT12 is ground\nX8 is free\nX9 is free\nselect(T11, T12, T3) !=? select(X8, [], X9)", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: ','(head(T41, T44), ','(tail(T41, X46), select(T40, X46, T45)))\n\nT40 is ground\nT41 is ground\nX8 is free\nX9 is free\nX46 is free\nselect(T40, T41, T3) !=? select(X8, [], X9)", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: ','(head(T41, T44), ','(tail(T41, X46), select(T40, X46, T45))), SCOPE: 5, CLAUSE: 3 | ','(head(T41, T44), ','(tail(T41, X46), select(T40, X46, T45))), SCOPE: 5, CLAUSE: 4\n\nT40 is ground\nT41 is ground\nX8 is free\nX9 is free\nX46 is free\nselect(T40, T41, T3) !=? select(X8, [], X9)", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: ','(head(T41, T44), ','(tail(T41, X46), select(T40, X46, T45))), SCOPE: 5, CLAUSE: 4\n\nT40 is ground\nT41 is ground\nX8 is free\nX9 is free\nX46 is free\nselect(T40, T41, T3) !=? select(X8, [], X9)", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: ','(tail(.(T53, T54), X46), select(T40, X46, T55))\n\nT40 is ground\nT53 is ground\nT54 is ground\nX46 is free", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: ','(tail(.(T53, T54), X46), select(T40, X46, T55)), SCOPE: 6, CLAUSE: 5 | ','(tail(.(T53, T54), X46), select(T40, X46, T55)), SCOPE: 6, CLAUSE: 6\n\nT40 is ground\nT53 is ground\nT54 is ground\nX46 is free", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: ','(tail(.(T53, T54), X46), select(T40, X46, T55)), SCOPE: 6, CLAUSE: 6\n\nT40 is ground\nT53 is ground\nT54 is ground\nX46 is free", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: select(T40, T65, T55)\n\nT40 is ground\nT65 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 87 [arrowhead = none ];
87 -> 4;

88 [label="EVAL with clause\nselect(X8, [], X9) :- ','(!_1, failure(a)).\nand substitutionT1 -> T6,\nX8 -> T6,\nT2 -> [],\nT3 -> T7,\nX9 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 88 [arrowhead = none ];
88 -> 9;

89 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 89 [arrowhead = none ];
89 -> 12;

90 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 90 [arrowhead = none ];
90 -> 13;

91 [label="ONLY EVAL with clause\nselect(X13, X14, X15) :- ','(head(X14, X13), tail(X14, X15)).\nand substitutionT1 -> T11,\nX13 -> T11,\nT2 -> T12,\nX14 -> T12,\nT3 -> T14,\nX15 -> T14,\nT13 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 91 [arrowhead = none ];
91 -> 23;

92 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 92 [arrowhead = none ];
92 -> 17;

93 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 93 [arrowhead = none ];
93 -> 20;

94 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 94 [arrowhead = none ];
94 -> 25;

95 [label="BACKTRACK\nfor clause: head([], X3)\nwith clash: (select(T11, T12, T3), select(X8, [], X9))", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 95 [arrowhead = none ];
95 -> 29;

96 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
29 -> 96 [arrowhead = none ];
96 -> 32;

97 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
29 -> 97 [arrowhead = none ];
97 -> 34;

98 [label="EVAL with clause\nhead(.(X25, X26), X25).\nand substitutionX25 -> T24,\nX26 -> T25,\nT12 -> .(T24, T25),\nT11 -> T24", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 98 [arrowhead = none ];
98 -> 49;

99 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 99 [arrowhead = none ];
99 -> 50;

100 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 100 [arrowhead = none ];
100 -> 60;

101 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
49 -> 101 [arrowhead = none ];
101 -> 51;

102 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 102 [arrowhead = none ];
102 -> 52;

103 [label="EVAL with clause\ntail(.(X31, X32), X32).\nand substitutionT24 -> T30,\nX31 -> T30,\nT25 -> T31,\nX32 -> T31,\nT14 -> T31", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 103 [arrowhead = none ];
103 -> 56;

104 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 104 [arrowhead = none ];
104 -> 57;

105 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
56 -> 105 [arrowhead = none ];
105 -> 58;

106 [label="EVAL with clause\nselect(X42, X43, .(X44, X45)) :- ','(head(X43, X44), ','(tail(X43, X46), select(X42, X46, X45))).\nand substitutionT11 -> T40,\nX42 -> T40,\nT12 -> T41,\nX43 -> T41,\nX44 -> T44,\nX45 -> T45,\nT3 -> .(T44, T45),\nT42 -> T44,\nT43 -> T45", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 106 [arrowhead = none ];
106 -> 69;

107 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
60 -> 107 [arrowhead = none ];
107 -> 70;

108 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
69 -> 108 [arrowhead = none ];
108 -> 80;

109 [label="BACKTRACK\nfor clause: head([], X3)\nwith clash: (select(T40, T41, T3), select(X8, [], X9))", fontsize=14, style = filled, fillcolor = lightgreen];
80 -> 109 [arrowhead = none ];
109 -> 81;

110 [label="EVAL with clause\nhead(.(X54, X55), X54).\nand substitutionX54 -> T53,\nX55 -> T54,\nT41 -> .(T53, T54),\nT44 -> T53,\nT45 -> T55", fontsize=14, style = filled, fillcolor = lightblue];
81 -> 110 [arrowhead = none ];
110 -> 82;

111 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
81 -> 111 [arrowhead = none ];
111 -> 83;

112 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
82 -> 112 [arrowhead = none ];
112 -> 84;

113 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
84 -> 113 [arrowhead = none ];
113 -> 85;

114 [label="ONLY EVAL with clause\ntail(.(X64, X65), X65).\nand substitutionT53 -> T64,\nX64 -> T64,\nT54 -> T65,\nX65 -> T65,\nX46 -> T65", fontsize=14, style = filled, fillcolor = lightblue];
85 -> 114 [arrowhead = none ];
114 -> 86;

115 [label="INSTANCE with matching:\nT1 -> T40\nT2 -> T65\nT3 -> T55", fontsize=14, style = filled, fillcolor = lightgrey];
86 -> 115 [arrowhead = none , style=dashed];
115 -> 3[style=dashed];

}

(2) Obligation:

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

selectA_in_gga(T30, .(T30, T31), T31) → selectA_out_gga(T30, .(T30, T31), T31)
selectA_in_gga(T40, .(T64, T65), .(T64, T55)) → U1_gga(T40, T64, T65, T55, selectA_in_gga(T40, T65, T55))
U1_gga(T40, T64, T65, T55, selectA_out_gga(T40, T65, T55)) → selectA_out_gga(T40, .(T64, T65), .(T64, T55))

The argument filtering Pi contains the following mapping:
selectA_in_gga(x1, x2, x3)  =  selectA_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
selectA_out_gga(x1, x2, x3)  =  selectA_out_gga(x1, x2, x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x2, x3, 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:

SELECTA_IN_GGA(T40, .(T64, T65), .(T64, T55)) → U1_GGA(T40, T64, T65, T55, selectA_in_gga(T40, T65, T55))
SELECTA_IN_GGA(T40, .(T64, T65), .(T64, T55)) → SELECTA_IN_GGA(T40, T65, T55)

The TRS R consists of the following rules:

selectA_in_gga(T30, .(T30, T31), T31) → selectA_out_gga(T30, .(T30, T31), T31)
selectA_in_gga(T40, .(T64, T65), .(T64, T55)) → U1_gga(T40, T64, T65, T55, selectA_in_gga(T40, T65, T55))
U1_gga(T40, T64, T65, T55, selectA_out_gga(T40, T65, T55)) → selectA_out_gga(T40, .(T64, T65), .(T64, T55))

The argument filtering Pi contains the following mapping:
selectA_in_gga(x1, x2, x3)  =  selectA_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
selectA_out_gga(x1, x2, x3)  =  selectA_out_gga(x1, x2, x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x2, x3, x5)
SELECTA_IN_GGA(x1, x2, x3)  =  SELECTA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x2, x3, x5)

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

(4) Obligation:

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

SELECTA_IN_GGA(T40, .(T64, T65), .(T64, T55)) → U1_GGA(T40, T64, T65, T55, selectA_in_gga(T40, T65, T55))
SELECTA_IN_GGA(T40, .(T64, T65), .(T64, T55)) → SELECTA_IN_GGA(T40, T65, T55)

The TRS R consists of the following rules:

selectA_in_gga(T30, .(T30, T31), T31) → selectA_out_gga(T30, .(T30, T31), T31)
selectA_in_gga(T40, .(T64, T65), .(T64, T55)) → U1_gga(T40, T64, T65, T55, selectA_in_gga(T40, T65, T55))
U1_gga(T40, T64, T65, T55, selectA_out_gga(T40, T65, T55)) → selectA_out_gga(T40, .(T64, T65), .(T64, T55))

The argument filtering Pi contains the following mapping:
selectA_in_gga(x1, x2, x3)  =  selectA_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
selectA_out_gga(x1, x2, x3)  =  selectA_out_gga(x1, x2, x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x2, x3, x5)
SELECTA_IN_GGA(x1, x2, x3)  =  SELECTA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x2, x3, x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.

(6) Obligation:

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

SELECTA_IN_GGA(T40, .(T64, T65), .(T64, T55)) → SELECTA_IN_GGA(T40, T65, T55)

The TRS R consists of the following rules:

selectA_in_gga(T30, .(T30, T31), T31) → selectA_out_gga(T30, .(T30, T31), T31)
selectA_in_gga(T40, .(T64, T65), .(T64, T55)) → U1_gga(T40, T64, T65, T55, selectA_in_gga(T40, T65, T55))
U1_gga(T40, T64, T65, T55, selectA_out_gga(T40, T65, T55)) → selectA_out_gga(T40, .(T64, T65), .(T64, T55))

The argument filtering Pi contains the following mapping:
selectA_in_gga(x1, x2, x3)  =  selectA_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
selectA_out_gga(x1, x2, x3)  =  selectA_out_gga(x1, x2, x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x2, x3, x5)
SELECTA_IN_GGA(x1, x2, x3)  =  SELECTA_IN_GGA(x1, 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_GGA(T40, .(T64, T65), .(T64, T55)) → SELECTA_IN_GGA(T40, T65, T55)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
SELECTA_IN_GGA(x1, x2, x3)  =  SELECTA_IN_GGA(x1, 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_GGA(T40, .(T64, T65)) → SELECTA_IN_GGA(T40, T65)

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_GGA(T40, .(T64, T65)) → SELECTA_IN_GGA(T40, T65)
  The graph contains the following edges 1 >= 1, 2 > 2

(12) YES