(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.

(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