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) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: select(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: 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];
3 [label="3: ','(!_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];
4 [label="4: 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];
5 [label="5: failure(a)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: failure(a), SCOPE: 2, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(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];
9 [label="9: ','(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];
10 [label="10: ','(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];
11 [label="11: ','(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];
12 [label="12: ?_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];
13 [label="13: tail(.(T24, T25), T14)\n\nT24 is ground\nT25 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: 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];
16 [label="16: tail(.(T24, T25), T14), SCOPE: 4, CLAUSE: 6\n\nT24 is ground\nT25 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: 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];
21 [label="21: ','(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];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(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];
24 [label="24: ','(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];
25 [label="25: ','(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];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(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];
28 [label="28: ','(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];
29 [label="29: select(T40, T65, T55)\n\nT40 is ground\nT65 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 30 [arrowhead = none ];
30 -> 2;

31 [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];
2 -> 31 [arrowhead = none ];
31 -> 3;

32 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 32 [arrowhead = none ];
32 -> 4;

33 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 33 [arrowhead = none ];
33 -> 5;

34 [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];
4 -> 34 [arrowhead = none ];
34 -> 8;

35 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 35 [arrowhead = none ];
35 -> 6;

36 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 36 [arrowhead = none ];
36 -> 7;

37 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
8 -> 37 [arrowhead = none ];
37 -> 9;

38 [label="BACKTRACK\nfor clause: head([], X3)\nwith clash: (select(T11, T12, T3), select(X8, [], X9))", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 38 [arrowhead = none ];
38 -> 10;

39 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 39 [arrowhead = none ];
39 -> 11;

40 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 40 [arrowhead = none ];
40 -> 12;

41 [label="EVAL with clause\nhead(.(X25, X26), X25).\nand substitutionX25 -> T24,\nX26 -> T25,\nT12 -> .(T24, T25),\nT11 -> T24", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 41 [arrowhead = none ];
41 -> 13;

42 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 42 [arrowhead = none ];
42 -> 14;

43 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 43 [arrowhead = none ];
43 -> 20;

44 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 44 [arrowhead = none ];
44 -> 15;

45 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 45 [arrowhead = none ];
45 -> 16;

46 [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];
16 -> 46 [arrowhead = none ];
46 -> 17;

47 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 47 [arrowhead = none ];
47 -> 18;

48 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 48 [arrowhead = none ];
48 -> 19;

49 [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];
20 -> 49 [arrowhead = none ];
49 -> 21;

50 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 50 [arrowhead = none ];
50 -> 22;

51 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 51 [arrowhead = none ];
51 -> 23;

52 [label="BACKTRACK\nfor clause: head([], X3)\nwith clash: (select(T40, T41, T3), select(X8, [], X9))", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 52 [arrowhead = none ];
52 -> 24;

53 [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];
24 -> 53 [arrowhead = none ];
53 -> 25;

54 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 54 [arrowhead = none ];
54 -> 26;

55 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 55 [arrowhead = none ];
55 -> 27;

56 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 56 [arrowhead = none ];
56 -> 28;

57 [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];
28 -> 57 [arrowhead = none ];
57 -> 29;

58 [label="INSTANCE with matching:\nT1 -> T40\nT2 -> T65\nT3 -> T55", fontsize=14, style = filled, fillcolor = lightgrey];
29 -> 58 [arrowhead = none , style=dashed];
58 -> 1[style=dashed];

}

(2) Obligation:

Clauses:

selectA(T30, .(T30, T31), T31).
selectA(T40, .(T64, T65), .(T64, T55)) :- selectA(T40, T65, T55).

Query: selectA(g,g,a)

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
selectA_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
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(x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x2, x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) 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(x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x2, x5)

(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:

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(x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x2, x5)
SELECTA_IN_GGA(x1, x2, x3)  =  SELECTA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x2, x5)

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

(6) 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(x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x2, x5)
SELECTA_IN_GGA(x1, x2, x3)  =  SELECTA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x2, x5)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

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

(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)

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(x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x2, x5)
SELECTA_IN_GGA(x1, x2, x3)  =  SELECTA_IN_GGA(x1, x2)

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

(9) UsableRulesProof (EQUIVALENT transformation)

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

(10) 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

(11) PiDPToQDPProof (SOUND transformation)

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

(12) 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.

(13) 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

(14) YES