Left Termination of the query pattern select_in_3(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).

Queries:

select(g,g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

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 | ?_1\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(!_1, failure(a)) | select(T4, [], T3), SCOPE: 1, CLAUSE: 1 | select(T4, [], T3), SCOPE: 1, CLAUSE: 2 | ?_1\n\nT4 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: select(T1, T2, T3), SCOPE: 1, CLAUSE: 1\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];
9 [label="9: 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];
10 [label="10: ','(head(T7, T6), tail(T7, T9))\n\nT6 is ground\nT7 is ground\nX8 is free; \nX9 is free; \nselect(T6, T7, T3) !=? select(X8, [], X9)", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(head(T7, T6), tail(T7, T9)), SCOPE: 3, CLAUSE: 3 | ','(head(T7, T6), tail(T7, T9)), SCOPE: 3, CLAUSE: 4\n\nT6 is ground\nT7 is ground\nX8 is free; \nX9 is free; \nselect(T6, T7, T3) !=? select(X8, [], X9)", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(head(T7, T6), tail(T7, T9)), SCOPE: 3, CLAUSE: 4\n\nT6 is ground\nT7 is ground\nX8 is free; \nX9 is free; \nX20 is free; \nselect(T6, T7, T3) !=? select(X8, [], X9)\nhead(T7, T6) !=? head([], X20)", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: tail(.(T10, T11), T9)\n\nT10 is ground\nT11 is ground\nX8 is free; \nX9 is free; \nX20 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\nX8 is free\nX9 is free; \nX20 is free; \nX23 is free; \nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: tail(.(T10, T11), T9), SCOPE: 4, CLAUSE: 5 | tail(.(T10, T11), T9), SCOPE: 4, CLAUSE: 6\n\nT10 is ground\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: tail(.(T10, T11), T9), SCOPE: 4, CLAUSE: 6\n\nT10 is ground\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\nX27 is free\nX28 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(head(T15, T18), ','(tail(T15, X37), select(T14, X37, T19)))\n\nT14 is ground\nT15 is ground\nX8 is free; \nX9 is free; \nX37 is free; \nselect(T14, T15, T3) !=? select(X8, [], X9)", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\nX8 is free\nX9 is free; \nX33 is free; \nX34 is free; \nX35 is free; \nX36 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(head(T15, T18), ','(tail(T15, X37), select(T14, X37, T19))), SCOPE: 5, CLAUSE: 3 | ','(head(T15, T18), ','(tail(T15, X37), select(T14, X37, T19))), SCOPE: 5, CLAUSE: 4\n\nT14 is ground\nT15 is ground\nX8 is free; \nX9 is free; \nX37 is free; \nselect(T14, T15, T3) !=? select(X8, [], X9)", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(head(T15, T18), ','(tail(T15, X37), select(T14, X37, T19))), SCOPE: 5, CLAUSE: 4\n\nT14 is ground\nT15 is ground\nX8 is free; \nX9 is free; \nX37 is free; \nX39 is free; \nselect(T14, T15, T3) !=? select(X8, [], X9)\nhead(T15, T18) !=? head([], X39)", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(tail(.(T20, T21), X37), select(T14, X37, T22))\n\nT14 is ground\nT20 is ground\nT21 is ground\nX8 is free; \nX9 is free; \nX37 is free; \nX39 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\nX8 is free\nX9 is free; \nX39 is free; \nX42 is free; \nX43 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(tail(.(T20, T21), X37), select(T14, X37, T22)), SCOPE: 6, CLAUSE: 5 | ','(tail(.(T20, T21), X37), select(T14, X37, T22)), SCOPE: 6, CLAUSE: 6\n\nT14 is ground\nT20 is ground\nT21 is ground\nX37 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(tail(.(T20, T21), X37), select(T14, X37, T22)), SCOPE: 6, CLAUSE: 6\n\nT14 is ground\nT20 is ground\nT21 is ground\nX37 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: select(T14, T24, T22)\n\nT14 is ground\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 29 [arrowhead = none ];
29 -> 2;

30 [label="EVAL with clause\nselect(X8, [], X9) :- ','(!, failure(a)).\nand substitution\nT1 -> T4\nX8 -> T4\nT2 -> []\nT3 -> T5\nX9 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 30 [arrowhead = none ];
30 -> 3;

31 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 31 [arrowhead = none ];
31 -> 4;

32 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 32 [arrowhead = none ];
32 -> 5;

33 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 33 [arrowhead = none ];
33 -> 8;

34 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 34 [arrowhead = none ];
34 -> 9;

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 = white];
6 -> 36 [arrowhead = none ];
36 -> 7;

37 [label="EVAL with clause\nselect(X16, X17, X18) :- ','(head(X17, X16), tail(X17, X18)).\nand substitution\nT1 -> T6\nX16 -> T6\nT2 -> T7\nX17 -> T7\nT3 -> T9\nX18 -> T9\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 37 [arrowhead = none ];
37 -> 10;

38 [label="EVAL with clause\nselect(X33, X34, .(X35, X36)) :- ','(head(X34, X35), ','(tail(X34, X37), select(X33, X37, X36))).\nand substitution\nT1 -> T14\nX33 -> T14\nT2 -> T15\nX34 -> T15\nX35 -> T18\nX36 -> T19\nT3 -> .(T18, T19)\nT16 -> T18\nT17 -> T19", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 38 [arrowhead = none ];
38 -> 20;

39 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 39 [arrowhead = none ];
39 -> 21;

40 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 40 [arrowhead = none ];
40 -> 11;

41 [label="BACKTRACK\nfor clause: head([], X3)\nwith clash: (select(T6, T7, T3), select(X8, [], X9))", fontsize=14, style = filled, fillcolor = white];
11 -> 41 [arrowhead = none ];
41 -> 12;

42 [label="EVAL with clause\nhead(.(X23, X24), X23).\nand substitution\nX23 -> T10\nX24 -> T11\nT7 -> .(T10, T11)\nT6 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 42 [arrowhead = none ];
42 -> 13;

43 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 43 [arrowhead = none ];
43 -> 14;

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 = white];
15 -> 45 [arrowhead = none ];
45 -> 16;

46 [label="EVAL with clause\ntail(.(X27, X28), X28).\nand substitution\nT10 -> T12\nX27 -> T12\nT11 -> T13\nX28 -> T13\nT9 -> T13", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 46 [arrowhead = none ];
46 -> 17;

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

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

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

50 [label="BACKTRACK\nfor clause: head([], X3)\nwith clash: (select(T14, T15, T3), select(X8, [], X9))", fontsize=14, style = filled, fillcolor = white];
22 -> 50 [arrowhead = none ];
50 -> 23;

51 [label="EVAL with clause\nhead(.(X42, X43), X42).\nand substitution\nX42 -> T20\nX43 -> T21\nT15 -> .(T20, T21)\nT18 -> T20\nT19 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 51 [arrowhead = none ];
51 -> 24;

52 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 52 [arrowhead = none ];
52 -> 25;

53 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 53 [arrowhead = none ];
53 -> 26;

54 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
26 -> 54 [arrowhead = none ];
54 -> 27;

55 [label="EVAL with clause\ntail(.(X46, X47), X47).\nand substitution\nT20 -> T23\nX46 -> T23\nT21 -> T24\nX47 -> T24\nX37 -> T24", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 55 [arrowhead = none ];
55 -> 28;

56 [label="INSTANCE with matching:\nT1 -> T14\nT2 -> T24\nT3 -> T22", fontsize=14, style = filled, fillcolor = lightgrey];
28 -> 56 [arrowhead = none , style=dashed];
56 -> 1[style=dashed];

}

(2) Obligation:

Clauses:

select1(T12, .(T12, T13), T13).
select1(T14, .(T23, T24), .(T23, T22)) :- select1(T14, T24, T22).

Queries:

select1(g,g,a).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
select1_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

select1_in_gga(T12, .(T12, T13), T13) → select1_out_gga(T12, .(T12, T13), T13)
select1_in_gga(T14, .(T23, T24), .(T23, T22)) → U1_gga(T14, T23, T24, T22, select1_in_gga(T14, T24, T22))
U1_gga(T14, T23, T24, T22, select1_out_gga(T14, T24, T22)) → select1_out_gga(T14, .(T23, T24), .(T23, T22))

The argument filtering Pi contains the following mapping:
select1_in_gga(x1, x2, x3)  =  select1_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
select1_out_gga(x1, x2, x3)  =  select1_out_gga(x1, x2, x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x2, x3, 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:

select1_in_gga(T12, .(T12, T13), T13) → select1_out_gga(T12, .(T12, T13), T13)
select1_in_gga(T14, .(T23, T24), .(T23, T22)) → U1_gga(T14, T23, T24, T22, select1_in_gga(T14, T24, T22))
U1_gga(T14, T23, T24, T22, select1_out_gga(T14, T24, T22)) → select1_out_gga(T14, .(T23, T24), .(T23, T22))

The argument filtering Pi contains the following mapping:
select1_in_gga(x1, x2, x3)  =  select1_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
select1_out_gga(x1, x2, x3)  =  select1_out_gga(x1, x2, x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x2, x3, 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:

SELECT1_IN_GGA(T14, .(T23, T24), .(T23, T22)) → U1_GGA(T14, T23, T24, T22, select1_in_gga(T14, T24, T22))
SELECT1_IN_GGA(T14, .(T23, T24), .(T23, T22)) → SELECT1_IN_GGA(T14, T24, T22)

The TRS R consists of the following rules:

select1_in_gga(T12, .(T12, T13), T13) → select1_out_gga(T12, .(T12, T13), T13)
select1_in_gga(T14, .(T23, T24), .(T23, T22)) → U1_gga(T14, T23, T24, T22, select1_in_gga(T14, T24, T22))
U1_gga(T14, T23, T24, T22, select1_out_gga(T14, T24, T22)) → select1_out_gga(T14, .(T23, T24), .(T23, T22))

The argument filtering Pi contains the following mapping:
select1_in_gga(x1, x2, x3)  =  select1_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
select1_out_gga(x1, x2, x3)  =  select1_out_gga(x1, x2, x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x2, x3, x5)
SELECT1_IN_GGA(x1, x2, x3)  =  SELECT1_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

(6) Obligation:

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

SELECT1_IN_GGA(T14, .(T23, T24), .(T23, T22)) → U1_GGA(T14, T23, T24, T22, select1_in_gga(T14, T24, T22))
SELECT1_IN_GGA(T14, .(T23, T24), .(T23, T22)) → SELECT1_IN_GGA(T14, T24, T22)

The TRS R consists of the following rules:

select1_in_gga(T12, .(T12, T13), T13) → select1_out_gga(T12, .(T12, T13), T13)
select1_in_gga(T14, .(T23, T24), .(T23, T22)) → U1_gga(T14, T23, T24, T22, select1_in_gga(T14, T24, T22))
U1_gga(T14, T23, T24, T22, select1_out_gga(T14, T24, T22)) → select1_out_gga(T14, .(T23, T24), .(T23, T22))

The argument filtering Pi contains the following mapping:
select1_in_gga(x1, x2, x3)  =  select1_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
select1_out_gga(x1, x2, x3)  =  select1_out_gga(x1, x2, x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x2, x3, x5)
SELECT1_IN_GGA(x1, x2, x3)  =  SELECT1_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

(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:

SELECT1_IN_GGA(T14, .(T23, T24), .(T23, T22)) → SELECT1_IN_GGA(T14, T24, T22)

The TRS R consists of the following rules:

select1_in_gga(T12, .(T12, T13), T13) → select1_out_gga(T12, .(T12, T13), T13)
select1_in_gga(T14, .(T23, T24), .(T23, T22)) → U1_gga(T14, T23, T24, T22, select1_in_gga(T14, T24, T22))
U1_gga(T14, T23, T24, T22, select1_out_gga(T14, T24, T22)) → select1_out_gga(T14, .(T23, T24), .(T23, T22))

The argument filtering Pi contains the following mapping:
select1_in_gga(x1, x2, x3)  =  select1_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
select1_out_gga(x1, x2, x3)  =  select1_out_gga(x1, x2, x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x2, x3, x5)
SELECT1_IN_GGA(x1, x2, x3)  =  SELECT1_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:

SELECT1_IN_GGA(T14, .(T23, T24), .(T23, T22)) → SELECT1_IN_GGA(T14, T24, T22)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
SELECT1_IN_GGA(x1, x2, x3)  =  SELECT1_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:

SELECT1_IN_GGA(T14, .(T23, T24)) → SELECT1_IN_GGA(T14, T24)

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:

• SELECT1_IN_GGA(T14, .(T23, T24)) → SELECT1_IN_GGA(T14, T24)
  The graph contains the following edges 1 >= 1, 2 > 2

(14) TRUE