(0) Obligation:
Clauses:
select(X, Y, Zs) :- ','(no(empty(Y)), ','(head(Y, X), tail(Y, Zs))).
select(X, Y, Z) :- ','(no(empty(Y)), ','(head(Y, H), ','(head(Z, H), ','(tail(Y, T), ','(tail(Z, Zs), select(X, T, Zs)))))).
head([], X1).
head(.(H, X2), H).
tail([], []).
tail(.(X3, T), T).
empty([]).
no(X) :- ','(X, ','(!, failure(a))).
no(X4).
failure(b).
Query: select(g,g,a)
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph DT10.
(2) Obligation:
Triples:
selectA(X1, .(X2, X3)) :- selectA(X1, X3).
selectB(X1, .(X2, X3), []) :- selectA(X1, X3).
selectB(X1, .(X2, X3), .(X2, X4)) :- selectB(X1, X3, X4).
Clauses:
selectcA(X1, .(X1, [])).
selectcA(X1, .(X2, X3)) :- selectcA(X1, X3).
selectcB(X1, .(X1, X2), X2).
selectcB(X1, .(X2, X3), []) :- selectcA(X1, X3).
selectcB(X1, .(X2, X3), .(X2, X4)) :- selectcB(X1, X3, X4).
Afs:
selectB(x1, x2, x3) = selectB(x1, x2)
(3) TriplesToPiDPProof (SOUND transformation)
We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
selectB_in: (b,b,f)
selectA_in: (b,b)
Transforming
TRIPLES into the following
Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
SELECTB_IN_GGA(X1, .(X2, X3), []) → U2_GGA(X1, X2, X3, selectA_in_gg(X1, X3))
SELECTB_IN_GGA(X1, .(X2, X3), []) → SELECTA_IN_GG(X1, X3)
SELECTA_IN_GG(X1, .(X2, X3)) → U1_GG(X1, X2, X3, selectA_in_gg(X1, X3))
SELECTA_IN_GG(X1, .(X2, X3)) → SELECTA_IN_GG(X1, X3)
SELECTB_IN_GGA(X1, .(X2, X3), .(X2, X4)) → U3_GGA(X1, X2, X3, X4, selectB_in_gga(X1, X3, X4))
SELECTB_IN_GGA(X1, .(X2, X3), .(X2, X4)) → SELECTB_IN_GGA(X1, X3, X4)
R is empty.
The argument filtering Pi contains the following mapping:
selectB_in_gga(
x1,
x2,
x3) =
selectB_in_gga(
x1,
x2)
.(
x1,
x2) =
.(
x1,
x2)
selectA_in_gg(
x1,
x2) =
selectA_in_gg(
x1,
x2)
[] =
[]
SELECTB_IN_GGA(
x1,
x2,
x3) =
SELECTB_IN_GGA(
x1,
x2)
U2_GGA(
x1,
x2,
x3,
x4) =
U2_GGA(
x1,
x2,
x3,
x4)
SELECTA_IN_GG(
x1,
x2) =
SELECTA_IN_GG(
x1,
x2)
U1_GG(
x1,
x2,
x3,
x4) =
U1_GG(
x1,
x2,
x3,
x4)
U3_GGA(
x1,
x2,
x3,
x4,
x5) =
U3_GGA(
x1,
x2,
x3,
x5)
We have to consider all (P,R,Pi)-chains
Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES
(4) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
SELECTB_IN_GGA(X1, .(X2, X3), []) → U2_GGA(X1, X2, X3, selectA_in_gg(X1, X3))
SELECTB_IN_GGA(X1, .(X2, X3), []) → SELECTA_IN_GG(X1, X3)
SELECTA_IN_GG(X1, .(X2, X3)) → U1_GG(X1, X2, X3, selectA_in_gg(X1, X3))
SELECTA_IN_GG(X1, .(X2, X3)) → SELECTA_IN_GG(X1, X3)
SELECTB_IN_GGA(X1, .(X2, X3), .(X2, X4)) → U3_GGA(X1, X2, X3, X4, selectB_in_gga(X1, X3, X4))
SELECTB_IN_GGA(X1, .(X2, X3), .(X2, X4)) → SELECTB_IN_GGA(X1, X3, X4)
R is empty.
The argument filtering Pi contains the following mapping:
selectB_in_gga(
x1,
x2,
x3) =
selectB_in_gga(
x1,
x2)
.(
x1,
x2) =
.(
x1,
x2)
selectA_in_gg(
x1,
x2) =
selectA_in_gg(
x1,
x2)
[] =
[]
SELECTB_IN_GGA(
x1,
x2,
x3) =
SELECTB_IN_GGA(
x1,
x2)
U2_GGA(
x1,
x2,
x3,
x4) =
U2_GGA(
x1,
x2,
x3,
x4)
SELECTA_IN_GG(
x1,
x2) =
SELECTA_IN_GG(
x1,
x2)
U1_GG(
x1,
x2,
x3,
x4) =
U1_GG(
x1,
x2,
x3,
x4)
U3_GGA(
x1,
x2,
x3,
x4,
x5) =
U3_GGA(
x1,
x2,
x3,
x5)
We have to consider all (P,R,Pi)-chains
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 4 less nodes.
(6) Complex Obligation (AND)
(7) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
SELECTA_IN_GG(X1, .(X2, X3)) → SELECTA_IN_GG(X1, X3)
R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains
(8) PiDPToQDPProof (EQUIVALENT transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(9) Obligation:
Q DP problem:
The TRS P consists of the following rules:
SELECTA_IN_GG(X1, .(X2, X3)) → SELECTA_IN_GG(X1, X3)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(10) 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_GG(X1, .(X2, X3)) → SELECTA_IN_GG(X1, X3)
The graph contains the following edges 1 >= 1, 2 > 2
(11) YES
(12) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
SELECTB_IN_GGA(X1, .(X2, X3), .(X2, X4)) → SELECTB_IN_GGA(X1, X3, X4)
R is empty.
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
SELECTB_IN_GGA(
x1,
x2,
x3) =
SELECTB_IN_GGA(
x1,
x2)
We have to consider all (P,R,Pi)-chains
(13) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(14) Obligation:
Q DP problem:
The TRS P consists of the following rules:
SELECTB_IN_GGA(X1, .(X2, X3)) → SELECTB_IN_GGA(X1, X3)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(15) 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:
- SELECTB_IN_GGA(X1, .(X2, X3)) → SELECTB_IN_GGA(X1, X3)
The graph contains the following edges 1 >= 1, 2 > 2
(16) YES