(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.

(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