(0) Obligation:
Clauses:
overlap(Xs, Ys) :- ','(member(X, Xs), member(X, Ys)).
member(X, Y) :- ','(no(empty(Y)), head(Y, X)).
member(X, Y) :- ','(no(empty(Y)), ','(tail(Y, T), member(X, T))).
head([], X1).
head(.(H, X2), H).
tail([], []).
tail(.(X3, T), T).
empty([]).
no(X) :- ','(X, ','(!, failure(a))).
no(X4).
failure(b).
Query: overlap(g,g)
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph DT10.
(2) Obligation:
Triples:
memberA(X1, .(X2, X3)) :- memberA(X1, X3).
pB(X1, .(X1, X2), X3) :- memberA(X1, X3).
pB(X1, .(X2, X3), X4) :- pB(X1, X3, X4).
overlapC(X1, X2) :- pB(X3, X1, X2).
Clauses:
membercA(X1, .(X1, X2)).
membercA(X1, .(X2, X3)) :- membercA(X1, X3).
qcB(X1, .(X1, X2), X3) :- membercA(X1, X3).
qcB(X1, .(X2, X3), X4) :- qcB(X1, X3, X4).
Afs:
overlapC(x1, x2) = overlapC(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:
overlapC_in: (b,b)
pB_in: (f,b,b)
memberA_in: (b,b)
Transforming
TRIPLES into the following
Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
OVERLAPC_IN_GG(X1, X2) → U4_GG(X1, X2, pB_in_agg(X3, X1, X2))
OVERLAPC_IN_GG(X1, X2) → PB_IN_AGG(X3, X1, X2)
PB_IN_AGG(X1, .(X1, X2), X3) → U2_AGG(X1, X2, X3, memberA_in_gg(X1, X3))
PB_IN_AGG(X1, .(X1, X2), X3) → MEMBERA_IN_GG(X1, X3)
MEMBERA_IN_GG(X1, .(X2, X3)) → U1_GG(X1, X2, X3, memberA_in_gg(X1, X3))
MEMBERA_IN_GG(X1, .(X2, X3)) → MEMBERA_IN_GG(X1, X3)
PB_IN_AGG(X1, .(X2, X3), X4) → U3_AGG(X1, X2, X3, X4, pB_in_agg(X1, X3, X4))
PB_IN_AGG(X1, .(X2, X3), X4) → PB_IN_AGG(X1, X3, X4)
R is empty.
The argument filtering Pi contains the following mapping:
pB_in_agg(
x1,
x2,
x3) =
pB_in_agg(
x2,
x3)
.(
x1,
x2) =
.(
x1,
x2)
memberA_in_gg(
x1,
x2) =
memberA_in_gg(
x1,
x2)
OVERLAPC_IN_GG(
x1,
x2) =
OVERLAPC_IN_GG(
x1,
x2)
U4_GG(
x1,
x2,
x3) =
U4_GG(
x1,
x2,
x3)
PB_IN_AGG(
x1,
x2,
x3) =
PB_IN_AGG(
x2,
x3)
U2_AGG(
x1,
x2,
x3,
x4) =
U2_AGG(
x1,
x2,
x3,
x4)
MEMBERA_IN_GG(
x1,
x2) =
MEMBERA_IN_GG(
x1,
x2)
U1_GG(
x1,
x2,
x3,
x4) =
U1_GG(
x1,
x2,
x3,
x4)
U3_AGG(
x1,
x2,
x3,
x4,
x5) =
U3_AGG(
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:
OVERLAPC_IN_GG(X1, X2) → U4_GG(X1, X2, pB_in_agg(X3, X1, X2))
OVERLAPC_IN_GG(X1, X2) → PB_IN_AGG(X3, X1, X2)
PB_IN_AGG(X1, .(X1, X2), X3) → U2_AGG(X1, X2, X3, memberA_in_gg(X1, X3))
PB_IN_AGG(X1, .(X1, X2), X3) → MEMBERA_IN_GG(X1, X3)
MEMBERA_IN_GG(X1, .(X2, X3)) → U1_GG(X1, X2, X3, memberA_in_gg(X1, X3))
MEMBERA_IN_GG(X1, .(X2, X3)) → MEMBERA_IN_GG(X1, X3)
PB_IN_AGG(X1, .(X2, X3), X4) → U3_AGG(X1, X2, X3, X4, pB_in_agg(X1, X3, X4))
PB_IN_AGG(X1, .(X2, X3), X4) → PB_IN_AGG(X1, X3, X4)
R is empty.
The argument filtering Pi contains the following mapping:
pB_in_agg(
x1,
x2,
x3) =
pB_in_agg(
x2,
x3)
.(
x1,
x2) =
.(
x1,
x2)
memberA_in_gg(
x1,
x2) =
memberA_in_gg(
x1,
x2)
OVERLAPC_IN_GG(
x1,
x2) =
OVERLAPC_IN_GG(
x1,
x2)
U4_GG(
x1,
x2,
x3) =
U4_GG(
x1,
x2,
x3)
PB_IN_AGG(
x1,
x2,
x3) =
PB_IN_AGG(
x2,
x3)
U2_AGG(
x1,
x2,
x3,
x4) =
U2_AGG(
x1,
x2,
x3,
x4)
MEMBERA_IN_GG(
x1,
x2) =
MEMBERA_IN_GG(
x1,
x2)
U1_GG(
x1,
x2,
x3,
x4) =
U1_GG(
x1,
x2,
x3,
x4)
U3_AGG(
x1,
x2,
x3,
x4,
x5) =
U3_AGG(
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 2 SCCs with 6 less nodes.
(6) Complex Obligation (AND)
(7) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
MEMBERA_IN_GG(X1, .(X2, X3)) → MEMBERA_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:
MEMBERA_IN_GG(X1, .(X2, X3)) → MEMBERA_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:
- MEMBERA_IN_GG(X1, .(X2, X3)) → MEMBERA_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:
PB_IN_AGG(X1, .(X2, X3), X4) → PB_IN_AGG(X1, X3, X4)
R is empty.
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
PB_IN_AGG(
x1,
x2,
x3) =
PB_IN_AGG(
x2,
x3)
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:
PB_IN_AGG(.(X2, X3), X4) → PB_IN_AGG(X3, X4)
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:
- PB_IN_AGG(.(X2, X3), X4) → PB_IN_AGG(X3, X4)
The graph contains the following edges 1 > 1, 2 >= 2
(16) YES