(0) Obligation:
Clauses:p(X) :- ','(l(X), q(X)).
q(.(A, [])).
r(1).
l([]).
l(.(H, T)) :- ','(r(H), l(T)).
Queries:
p(a).
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph.
(2) Obligation:
Triples:l16(.(1, T23)) :- l16(T23).
p1(.(1, T13)) :- l16(T13).
Clauses:
lc16([]).
lc16(.(1, T23)) :- lc16(T23).
Afs:
p1(x1) = p1
(3) TriplesToPiDPProof (SOUND transformation)
We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:p1_in: (f)
l16_in: (f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
P1_IN_A(.(1, T13)) → U2_A(T13, l16_in_a(T13))
P1_IN_A(.(1, T13)) → L16_IN_A(T13)
L16_IN_A(.(1, T23)) → U1_A(T23, l16_in_a(T23))
L16_IN_A(.(1, T23)) → L16_IN_A(T23)
R is empty.
The argument filtering Pi contains the following mapping:
l16_in_a(x1) = l16_in_a
.(x1, x2) = .(x1, x2)
1 = 1
P1_IN_A(x1) = P1_IN_A
U2_A(x1, x2) = U2_A(x2)
L16_IN_A(x1) = L16_IN_A
U1_A(x1, x2) = U1_A(x2)
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:
P1_IN_A(.(1, T13)) → U2_A(T13, l16_in_a(T13))
P1_IN_A(.(1, T13)) → L16_IN_A(T13)
L16_IN_A(.(1, T23)) → U1_A(T23, l16_in_a(T23))
L16_IN_A(.(1, T23)) → L16_IN_A(T23)
R is empty.
The argument filtering Pi contains the following mapping:
l16_in_a(x1) = l16_in_a
.(x1, x2) = .(x1, x2)
1 = 1
P1_IN_A(x1) = P1_IN_A
U2_A(x1, x2) = U2_A(x2)
L16_IN_A(x1) = L16_IN_A
U1_A(x1, x2) = U1_A(x2)
We have to consider all (P,R,Pi)-chains
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.(6) Obligation:
Pi DP problem:The TRS P consists of the following rules:
L16_IN_A(.(1, T23)) → L16_IN_A(T23)
R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
1 = 1
L16_IN_A(x1) = L16_IN_A
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:
L16_IN_A → L16_IN_A
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(9) NonTerminationProof (EQUIVALENT transformation)
We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.Found a loop by semiunifying a rule from P directly.
s = L16_IN_A evaluates to t =L16_IN_A
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Semiunifier: [ ]
- Matcher: [ ]
Rewriting sequence
The DP semiunifies directly so there is only one rewrite step from L16_IN_A to L16_IN_A.