Left Termination of the query pattern select_in_3(g, g, a) w.r.t. the given Prolog program could successfully be proven:

0 Prolog
↳1 PrologToPrologProblemTransformerProof (⇐)
↳2 Prolog
↳3 PrologToPiTRSProof (⇐)
↳4 PiTRS
↳5 DependencyPairsProof (⇔)
↳6 PiDP
↳7 DependencyGraphProof (⇔)
↳8 PiDP
↳9 UsableRulesProof (⇔)
↳10 PiDP
↳11 PiDPToQDPProof (⇐)
↳12 QDP
↳13 QDPSizeChangeProof (⇔)
↳14 YES

(0) Obligation:

Clauses:

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

Queries:

select(g,g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

select1(T6, .(T6, T7), T7).
select1(T23, .(T23, .(T23, T24)), .(T23, T24)).
select1(T33, .(T33, .(T34, T35)), .(T33, .(T34, T37))) :- select1(T33, T35, T37).
select1(T59, .(T47, .(T59, T60)), .(T47, T60)).
select1(T69, .(T47, .(T70, T71)), .(T47, .(T70, T73))) :- select1(T69, T71, T73).

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(T6, .(T6, T7), T7) → select1_out_gga(T6, .(T6, T7), T7)
select1_in_gga(T23, .(T23, .(T23, T24)), .(T23, T24)) → select1_out_gga(T23, .(T23, .(T23, T24)), .(T23, T24))
select1_in_gga(T33, .(T33, .(T34, T35)), .(T33, .(T34, T37))) → U1_gga(T33, T34, T35, T37, select1_in_gga(T33, T35, T37))
select1_in_gga(T59, .(T47, .(T59, T60)), .(T47, T60)) → select1_out_gga(T59, .(T47, .(T59, T60)), .(T47, T60))
select1_in_gga(T69, .(T47, .(T70, T71)), .(T47, .(T70, T73))) → U2_gga(T69, T47, T70, T71, T73, select1_in_gga(T69, T71, T73))
U2_gga(T69, T47, T70, T71, T73, select1_out_gga(T69, T71, T73)) → select1_out_gga(T69, .(T47, .(T70, T71)), .(T47, .(T70, T73)))
U1_gga(T33, T34, T35, T37, select1_out_gga(T33, T35, T37)) → select1_out_gga(T33, .(T33, .(T34, T35)), .(T33, .(T34, T37)))

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(x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x2, x5)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x2, x3, x6)

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(T6, .(T6, T7), T7) → select1_out_gga(T6, .(T6, T7), T7)
select1_in_gga(T23, .(T23, .(T23, T24)), .(T23, T24)) → select1_out_gga(T23, .(T23, .(T23, T24)), .(T23, T24))
select1_in_gga(T33, .(T33, .(T34, T35)), .(T33, .(T34, T37))) → U1_gga(T33, T34, T35, T37, select1_in_gga(T33, T35, T37))
select1_in_gga(T59, .(T47, .(T59, T60)), .(T47, T60)) → select1_out_gga(T59, .(T47, .(T59, T60)), .(T47, T60))
select1_in_gga(T69, .(T47, .(T70, T71)), .(T47, .(T70, T73))) → U2_gga(T69, T47, T70, T71, T73, select1_in_gga(T69, T71, T73))
U2_gga(T69, T47, T70, T71, T73, select1_out_gga(T69, T71, T73)) → select1_out_gga(T69, .(T47, .(T70, T71)), .(T47, .(T70, T73)))
U1_gga(T33, T34, T35, T37, select1_out_gga(T33, T35, T37)) → select1_out_gga(T33, .(T33, .(T34, T35)), .(T33, .(T34, T37)))

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(x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x2, x5)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x2, x3, x6)

(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(T33, .(T33, .(T34, T35)), .(T33, .(T34, T37))) → U1_GGA(T33, T34, T35, T37, select1_in_gga(T33, T35, T37))
SELECT1_IN_GGA(T33, .(T33, .(T34, T35)), .(T33, .(T34, T37))) → SELECT1_IN_GGA(T33, T35, T37)
SELECT1_IN_GGA(T69, .(T47, .(T70, T71)), .(T47, .(T70, T73))) → U2_GGA(T69, T47, T70, T71, T73, select1_in_gga(T69, T71, T73))
SELECT1_IN_GGA(T69, .(T47, .(T70, T71)), .(T47, .(T70, T73))) → SELECT1_IN_GGA(T69, T71, T73)

The TRS R consists of the following rules:

select1_in_gga(T6, .(T6, T7), T7) → select1_out_gga(T6, .(T6, T7), T7)
select1_in_gga(T23, .(T23, .(T23, T24)), .(T23, T24)) → select1_out_gga(T23, .(T23, .(T23, T24)), .(T23, T24))
select1_in_gga(T33, .(T33, .(T34, T35)), .(T33, .(T34, T37))) → U1_gga(T33, T34, T35, T37, select1_in_gga(T33, T35, T37))
select1_in_gga(T59, .(T47, .(T59, T60)), .(T47, T60)) → select1_out_gga(T59, .(T47, .(T59, T60)), .(T47, T60))
select1_in_gga(T69, .(T47, .(T70, T71)), .(T47, .(T70, T73))) → U2_gga(T69, T47, T70, T71, T73, select1_in_gga(T69, T71, T73))
U2_gga(T69, T47, T70, T71, T73, select1_out_gga(T69, T71, T73)) → select1_out_gga(T69, .(T47, .(T70, T71)), .(T47, .(T70, T73)))
U1_gga(T33, T34, T35, T37, select1_out_gga(T33, T35, T37)) → select1_out_gga(T33, .(T33, .(T34, T35)), .(T33, .(T34, T37)))

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(x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x2, x5)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x2, x3, x6)
SELECT1_IN_GGA(x1, x2, x3)  =  SELECT1_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x2, x5)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x2, x3, x6)

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(T33, .(T33, .(T34, T35)), .(T33, .(T34, T37))) → U1_GGA(T33, T34, T35, T37, select1_in_gga(T33, T35, T37))
SELECT1_IN_GGA(T33, .(T33, .(T34, T35)), .(T33, .(T34, T37))) → SELECT1_IN_GGA(T33, T35, T37)
SELECT1_IN_GGA(T69, .(T47, .(T70, T71)), .(T47, .(T70, T73))) → U2_GGA(T69, T47, T70, T71, T73, select1_in_gga(T69, T71, T73))
SELECT1_IN_GGA(T69, .(T47, .(T70, T71)), .(T47, .(T70, T73))) → SELECT1_IN_GGA(T69, T71, T73)

The TRS R consists of the following rules:

select1_in_gga(T6, .(T6, T7), T7) → select1_out_gga(T6, .(T6, T7), T7)
select1_in_gga(T23, .(T23, .(T23, T24)), .(T23, T24)) → select1_out_gga(T23, .(T23, .(T23, T24)), .(T23, T24))
select1_in_gga(T33, .(T33, .(T34, T35)), .(T33, .(T34, T37))) → U1_gga(T33, T34, T35, T37, select1_in_gga(T33, T35, T37))
select1_in_gga(T59, .(T47, .(T59, T60)), .(T47, T60)) → select1_out_gga(T59, .(T47, .(T59, T60)), .(T47, T60))
select1_in_gga(T69, .(T47, .(T70, T71)), .(T47, .(T70, T73))) → U2_gga(T69, T47, T70, T71, T73, select1_in_gga(T69, T71, T73))
U2_gga(T69, T47, T70, T71, T73, select1_out_gga(T69, T71, T73)) → select1_out_gga(T69, .(T47, .(T70, T71)), .(T47, .(T70, T73)))
U1_gga(T33, T34, T35, T37, select1_out_gga(T33, T35, T37)) → select1_out_gga(T33, .(T33, .(T34, T35)), .(T33, .(T34, T37)))

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(x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x2, x5)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x2, x3, x6)
SELECT1_IN_GGA(x1, x2, x3)  =  SELECT1_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x2, x5)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x2, x3, x6)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes.

(8) Obligation:

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

SELECT1_IN_GGA(T69, .(T47, .(T70, T71)), .(T47, .(T70, T73))) → SELECT1_IN_GGA(T69, T71, T73)
SELECT1_IN_GGA(T33, .(T33, .(T34, T35)), .(T33, .(T34, T37))) → SELECT1_IN_GGA(T33, T35, T37)

The TRS R consists of the following rules:

select1_in_gga(T6, .(T6, T7), T7) → select1_out_gga(T6, .(T6, T7), T7)
select1_in_gga(T23, .(T23, .(T23, T24)), .(T23, T24)) → select1_out_gga(T23, .(T23, .(T23, T24)), .(T23, T24))
select1_in_gga(T33, .(T33, .(T34, T35)), .(T33, .(T34, T37))) → U1_gga(T33, T34, T35, T37, select1_in_gga(T33, T35, T37))
select1_in_gga(T59, .(T47, .(T59, T60)), .(T47, T60)) → select1_out_gga(T59, .(T47, .(T59, T60)), .(T47, T60))
select1_in_gga(T69, .(T47, .(T70, T71)), .(T47, .(T70, T73))) → U2_gga(T69, T47, T70, T71, T73, select1_in_gga(T69, T71, T73))
U2_gga(T69, T47, T70, T71, T73, select1_out_gga(T69, T71, T73)) → select1_out_gga(T69, .(T47, .(T70, T71)), .(T47, .(T70, T73)))
U1_gga(T33, T34, T35, T37, select1_out_gga(T33, T35, T37)) → select1_out_gga(T33, .(T33, .(T34, T35)), .(T33, .(T34, T37)))

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(x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x2, x5)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x2, x3, x6)
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(T69, .(T47, .(T70, T71)), .(T47, .(T70, T73))) → SELECT1_IN_GGA(T69, T71, T73)
SELECT1_IN_GGA(T33, .(T33, .(T34, T35)), .(T33, .(T34, T37))) → SELECT1_IN_GGA(T33, T35, T37)

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(T69, .(T47, .(T70, T71))) → SELECT1_IN_GGA(T69, T71)
SELECT1_IN_GGA(T33, .(T33, .(T34, T35))) → SELECT1_IN_GGA(T33, T35)

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(T69, .(T47, .(T70, T71))) → SELECT1_IN_GGA(T69, T71)
    The graph contains the following edges 1 >= 1, 2 > 2

  • SELECT1_IN_GGA(T33, .(T33, .(T34, T35))) → SELECT1_IN_GGA(T33, T35)
    The graph contains the following edges 1 >= 1, 2 > 1, 2 > 2

(14) YES