(0) Obligation:
Clauses:
select(X, .(X, Xs), Xs).
select(X, .(Y, Xs), .(Y, Zs)) :- select(X, Xs, Zs).
Query: select(g,g,a)
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph DT10.
(2) Obligation:
Triples:
selectA(X1, .(X1, .(X2, X3)), .(X1, .(X2, X4))) :- selectA(X1, X3, X4).
selectA(X1, .(X2, .(X3, X4)), .(X2, .(X3, X5))) :- selectA(X1, X4, X5).
Clauses:
selectcA(X1, .(X1, X2), X2).
selectcA(X1, .(X1, .(X1, X2)), .(X1, X2)).
selectcA(X1, .(X1, .(X2, X3)), .(X1, .(X2, X4))) :- selectcA(X1, X3, X4).
selectcA(X1, .(X2, .(X1, X3)), .(X2, X3)).
selectcA(X1, .(X2, .(X3, X4)), .(X2, .(X3, X5))) :- selectcA(X1, X4, X5).
Afs:
selectA(x1, x2, x3) = selectA(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:
selectA_in: (b,b,f)
Transforming
TRIPLES into the following
Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
SELECTA_IN_GGA(X1, .(X1, .(X2, X3)), .(X1, .(X2, X4))) → U1_GGA(X1, X2, X3, X4, selectA_in_gga(X1, X3, X4))
SELECTA_IN_GGA(X1, .(X1, .(X2, X3)), .(X1, .(X2, X4))) → SELECTA_IN_GGA(X1, X3, X4)
SELECTA_IN_GGA(X1, .(X2, .(X3, X4)), .(X2, .(X3, X5))) → U2_GGA(X1, X2, X3, X4, X5, selectA_in_gga(X1, X4, X5))
SELECTA_IN_GGA(X1, .(X2, .(X3, X4)), .(X2, .(X3, X5))) → SELECTA_IN_GGA(X1, X4, X5)
R is empty.
The argument filtering Pi contains the following mapping:
selectA_in_gga(
x1,
x2,
x3) =
selectA_in_gga(
x1,
x2)
.(
x1,
x2) =
.(
x1,
x2)
SELECTA_IN_GGA(
x1,
x2,
x3) =
SELECTA_IN_GGA(
x1,
x2)
U1_GGA(
x1,
x2,
x3,
x4,
x5) =
U1_GGA(
x1,
x2,
x3,
x5)
U2_GGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U2_GGA(
x1,
x2,
x3,
x4,
x6)
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:
SELECTA_IN_GGA(X1, .(X1, .(X2, X3)), .(X1, .(X2, X4))) → U1_GGA(X1, X2, X3, X4, selectA_in_gga(X1, X3, X4))
SELECTA_IN_GGA(X1, .(X1, .(X2, X3)), .(X1, .(X2, X4))) → SELECTA_IN_GGA(X1, X3, X4)
SELECTA_IN_GGA(X1, .(X2, .(X3, X4)), .(X2, .(X3, X5))) → U2_GGA(X1, X2, X3, X4, X5, selectA_in_gga(X1, X4, X5))
SELECTA_IN_GGA(X1, .(X2, .(X3, X4)), .(X2, .(X3, X5))) → SELECTA_IN_GGA(X1, X4, X5)
R is empty.
The argument filtering Pi contains the following mapping:
selectA_in_gga(
x1,
x2,
x3) =
selectA_in_gga(
x1,
x2)
.(
x1,
x2) =
.(
x1,
x2)
SELECTA_IN_GGA(
x1,
x2,
x3) =
SELECTA_IN_GGA(
x1,
x2)
U1_GGA(
x1,
x2,
x3,
x4,
x5) =
U1_GGA(
x1,
x2,
x3,
x5)
U2_GGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U2_GGA(
x1,
x2,
x3,
x4,
x6)
We have to consider all (P,R,Pi)-chains
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes.
(6) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
SELECTA_IN_GGA(X1, .(X2, .(X3, X4)), .(X2, .(X3, X5))) → SELECTA_IN_GGA(X1, X4, X5)
SELECTA_IN_GGA(X1, .(X1, .(X2, X3)), .(X1, .(X2, X4))) → SELECTA_IN_GGA(X1, X3, X4)
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
(7) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
SELECTA_IN_GGA(X1, .(X2, .(X3, X4))) → SELECTA_IN_GGA(X1, X4)
SELECTA_IN_GGA(X1, .(X1, .(X2, X3))) → SELECTA_IN_GGA(X1, X3)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(9) 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(X1, .(X2, .(X3, X4))) → SELECTA_IN_GGA(X1, X4)
The graph contains the following edges 1 >= 1, 2 > 2
- SELECTA_IN_GGA(X1, .(X1, .(X2, X3))) → SELECTA_IN_GGA(X1, X3)
The graph contains the following edges 1 >= 1, 2 > 1, 2 > 2
(10) YES