(0) Obligation:
Clauses:
member(X, .(X, X1)).
member(X, .(X2, Xs)) :- member(X, Xs).
Query: member(a,g)
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph DT10.
(2) Obligation:
Triples:
memberA(X1, .(X2, .(X3, X4))) :- memberA(X1, X4).
memberA(X1, .(X2, .(X3, X4))) :- memberA(X1, X4).
Clauses:
membercA(X1, .(X1, X2)).
membercA(X1, .(X2, .(X1, X3))).
membercA(X1, .(X2, .(X3, X4))) :- membercA(X1, X4).
membercA(X1, .(X2, .(X1, X3))).
membercA(X1, .(X2, .(X3, X4))) :- membercA(X1, X4).
Afs:
memberA(x1, x2) = memberA(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:
memberA_in: (f,b)
Transforming
TRIPLES into the following
Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
MEMBERA_IN_AG(X1, .(X2, .(X3, X4))) → U1_AG(X1, X2, X3, X4, memberA_in_ag(X1, X4))
MEMBERA_IN_AG(X1, .(X2, .(X3, X4))) → MEMBERA_IN_AG(X1, X4)
R is empty.
The argument filtering Pi contains the following mapping:
memberA_in_ag(
x1,
x2) =
memberA_in_ag(
x2)
.(
x1,
x2) =
.(
x1,
x2)
MEMBERA_IN_AG(
x1,
x2) =
MEMBERA_IN_AG(
x2)
U1_AG(
x1,
x2,
x3,
x4,
x5) =
U1_AG(
x2,
x3,
x4,
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:
MEMBERA_IN_AG(X1, .(X2, .(X3, X4))) → U1_AG(X1, X2, X3, X4, memberA_in_ag(X1, X4))
MEMBERA_IN_AG(X1, .(X2, .(X3, X4))) → MEMBERA_IN_AG(X1, X4)
R is empty.
The argument filtering Pi contains the following mapping:
memberA_in_ag(
x1,
x2) =
memberA_in_ag(
x2)
.(
x1,
x2) =
.(
x1,
x2)
MEMBERA_IN_AG(
x1,
x2) =
MEMBERA_IN_AG(
x2)
U1_AG(
x1,
x2,
x3,
x4,
x5) =
U1_AG(
x2,
x3,
x4,
x5)
We have to consider all (P,R,Pi)-chains
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.
(6) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
MEMBERA_IN_AG(X1, .(X2, .(X3, X4))) → MEMBERA_IN_AG(X1, X4)
R is empty.
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
MEMBERA_IN_AG(
x1,
x2) =
MEMBERA_IN_AG(
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:
MEMBERA_IN_AG(.(X2, .(X3, X4))) → MEMBERA_IN_AG(X4)
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:
- MEMBERA_IN_AG(.(X2, .(X3, X4))) → MEMBERA_IN_AG(X4)
The graph contains the following edges 1 > 1
(10) YES