(0) Obligation:
Clauses:fl([], [], 0).
fl(.(E, X), R, s(Z)) :- ','(append(E, Y, R), fl(X, Y, Z)).
append([], X, X).
append(.(X, Xs), Ys, .(X, Zs)) :- append(Xs, Ys, Zs).
Queries:
fl(g,g,a).
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph.
(2) Obligation:
Triples:p7([], T17, T17, T9, T12) :- fl1(T9, T17, T12).
p7(.(T24, T25), X43, .(T24, T26), T9, T12) :- p7(T25, X43, T26, T9, T12).
fl1(.(T8, T9), T10, s(T12)) :- p7(T8, X13, T10, T9, T12).
Clauses:
flc1([], [], 0).
flc1(.(T8, T9), T10, s(T12)) :- qc7(T8, X13, T10, T9, T12).
qc7([], T17, T17, T9, T12) :- flc1(T9, T17, T12).
qc7(.(T24, T25), X43, .(T24, T26), T9, T12) :- qc7(T25, X43, T26, T9, T12).
Afs:
fl1(x1, x2, x3) = fl1(x1, x2)
(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:fl1_in: (b,b,f)
p7_in: (b,f,b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
FL1_IN_GGA(.(T8, T9), T10, s(T12)) → U3_GGA(T8, T9, T10, T12, p7_in_gagga(T8, X13, T10, T9, T12))
FL1_IN_GGA(.(T8, T9), T10, s(T12)) → P7_IN_GAGGA(T8, X13, T10, T9, T12)
P7_IN_GAGGA([], T17, T17, T9, T12) → U1_GAGGA(T17, T9, T12, fl1_in_gga(T9, T17, T12))
P7_IN_GAGGA([], T17, T17, T9, T12) → FL1_IN_GGA(T9, T17, T12)
P7_IN_GAGGA(.(T24, T25), X43, .(T24, T26), T9, T12) → U2_GAGGA(T24, T25, X43, T26, T9, T12, p7_in_gagga(T25, X43, T26, T9, T12))
P7_IN_GAGGA(.(T24, T25), X43, .(T24, T26), T9, T12) → P7_IN_GAGGA(T25, X43, T26, T9, T12)
R is empty.
The argument filtering Pi contains the following mapping:
fl1_in_gga(x1, x2, x3) = fl1_in_gga(x1, x2)
.(x1, x2) = .(x1, x2)
p7_in_gagga(x1, x2, x3, x4, x5) = p7_in_gagga(x1, x3, x4)
[] = []
s(x1) = s(x1)
FL1_IN_GGA(x1, x2, x3) = FL1_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4, x5) = U3_GGA(x1, x2, x3, x5)
P7_IN_GAGGA(x1, x2, x3, x4, x5) = P7_IN_GAGGA(x1, x3, x4)
U1_GAGGA(x1, x2, x3, x4) = U1_GAGGA(x1, x2, x4)
U2_GAGGA(x1, x2, x3, x4, x5, x6, x7) = U2_GAGGA(x1, x2, x4, x5, x7)
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:
FL1_IN_GGA(.(T8, T9), T10, s(T12)) → U3_GGA(T8, T9, T10, T12, p7_in_gagga(T8, X13, T10, T9, T12))
FL1_IN_GGA(.(T8, T9), T10, s(T12)) → P7_IN_GAGGA(T8, X13, T10, T9, T12)
P7_IN_GAGGA([], T17, T17, T9, T12) → U1_GAGGA(T17, T9, T12, fl1_in_gga(T9, T17, T12))
P7_IN_GAGGA([], T17, T17, T9, T12) → FL1_IN_GGA(T9, T17, T12)
P7_IN_GAGGA(.(T24, T25), X43, .(T24, T26), T9, T12) → U2_GAGGA(T24, T25, X43, T26, T9, T12, p7_in_gagga(T25, X43, T26, T9, T12))
P7_IN_GAGGA(.(T24, T25), X43, .(T24, T26), T9, T12) → P7_IN_GAGGA(T25, X43, T26, T9, T12)
R is empty.
The argument filtering Pi contains the following mapping:
fl1_in_gga(x1, x2, x3) = fl1_in_gga(x1, x2)
.(x1, x2) = .(x1, x2)
p7_in_gagga(x1, x2, x3, x4, x5) = p7_in_gagga(x1, x3, x4)
[] = []
s(x1) = s(x1)
FL1_IN_GGA(x1, x2, x3) = FL1_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4, x5) = U3_GGA(x1, x2, x3, x5)
P7_IN_GAGGA(x1, x2, x3, x4, x5) = P7_IN_GAGGA(x1, x3, x4)
U1_GAGGA(x1, x2, x3, x4) = U1_GAGGA(x1, x2, x4)
U2_GAGGA(x1, x2, x3, x4, x5, x6, x7) = U2_GAGGA(x1, x2, x4, x5, x7)
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:
FL1_IN_GGA(.(T8, T9), T10, s(T12)) → P7_IN_GAGGA(T8, X13, T10, T9, T12)
P7_IN_GAGGA([], T17, T17, T9, T12) → FL1_IN_GGA(T9, T17, T12)
P7_IN_GAGGA(.(T24, T25), X43, .(T24, T26), T9, T12) → P7_IN_GAGGA(T25, X43, T26, T9, T12)
R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
s(x1) = s(x1)
FL1_IN_GGA(x1, x2, x3) = FL1_IN_GGA(x1, x2)
P7_IN_GAGGA(x1, x2, x3, x4, x5) = P7_IN_GAGGA(x1, x3, x4)
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:
FL1_IN_GGA(.(T8, T9), T10) → P7_IN_GAGGA(T8, T10, T9)
P7_IN_GAGGA([], T17, T9) → FL1_IN_GGA(T9, T17)
P7_IN_GAGGA(.(T24, T25), .(T24, T26), T9) → P7_IN_GAGGA(T25, T26, T9)
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:
- P7_IN_GAGGA([], T17, T9) → FL1_IN_GGA(T9, T17)
The graph contains the following edges 3 >= 1, 2 >= 2 - P7_IN_GAGGA(.(T24, T25), .(T24, T26), T9) → P7_IN_GAGGA(T25, T26, T9)
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3 - FL1_IN_GGA(.(T8, T9), T10) → P7_IN_GAGGA(T8, T10, T9)
The graph contains the following edges 1 > 1, 2 >= 2, 1 > 3