(0) Obligation:
Clauses:
rev(L, R) :- rev(L, [], R).
rev([], Y, Y).
rev(L, S, R) :- ','(no(empty(L)), ','(head(L, X), ','(tail(L, T), rev(T, .(X, S), R)))).
head([], X1).
head(.(X, X2), X).
tail([], []).
tail(.(X3, Xs), Xs).
empty([]).
no(X) :- ','(X, ','(!, failure(a))).
no(X4).
failure(b).
Queries:
rev(g,a).
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph.
(2) Obligation:
Triples:
rev212(.(T1013, T1014), T965, T966, T968) :- rev212(T1014, T1013, .(T965, T966), T968).
rev1(.(T811, .(T810, .(T809, .(T808, .(T807, .(T806, .(T805, .(T918, T919)))))))), T813) :- rev212(T919, T918, .(T805, .(T806, .(T807, .(T808, .(T809, .(T810, .(T811, []))))))), T813).
Clauses:
revc212([], T949, T950, .(T949, T950)).
revc212(.(T1013, T1014), T965, T966, T968) :- revc212(T1014, T1013, .(T965, T966), T968).
Afs:
rev1(x1, x2) = rev1(x1)
(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:
rev1_in: (b,f)
rev212_in: (b,b,b,f)
Transforming
TRIPLES into the following
Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
REV1_IN_GA(.(T811, .(T810, .(T809, .(T808, .(T807, .(T806, .(T805, .(T918, T919)))))))), T813) → U2_GA(T811, T810, T809, T808, T807, T806, T805, T918, T919, T813, rev212_in_ggga(T919, T918, .(T805, .(T806, .(T807, .(T808, .(T809, .(T810, .(T811, []))))))), T813))
REV1_IN_GA(.(T811, .(T810, .(T809, .(T808, .(T807, .(T806, .(T805, .(T918, T919)))))))), T813) → REV212_IN_GGGA(T919, T918, .(T805, .(T806, .(T807, .(T808, .(T809, .(T810, .(T811, []))))))), T813)
REV212_IN_GGGA(.(T1013, T1014), T965, T966, T968) → U1_GGGA(T1013, T1014, T965, T966, T968, rev212_in_ggga(T1014, T1013, .(T965, T966), T968))
REV212_IN_GGGA(.(T1013, T1014), T965, T966, T968) → REV212_IN_GGGA(T1014, T1013, .(T965, T966), T968)
R is empty.
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
rev212_in_ggga(
x1,
x2,
x3,
x4) =
rev212_in_ggga(
x1,
x2,
x3)
[] =
[]
REV1_IN_GA(
x1,
x2) =
REV1_IN_GA(
x1)
U2_GA(
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x8,
x9,
x10,
x11) =
U2_GA(
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x8,
x9,
x11)
REV212_IN_GGGA(
x1,
x2,
x3,
x4) =
REV212_IN_GGGA(
x1,
x2,
x3)
U1_GGGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U1_GGGA(
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:
REV1_IN_GA(.(T811, .(T810, .(T809, .(T808, .(T807, .(T806, .(T805, .(T918, T919)))))))), T813) → U2_GA(T811, T810, T809, T808, T807, T806, T805, T918, T919, T813, rev212_in_ggga(T919, T918, .(T805, .(T806, .(T807, .(T808, .(T809, .(T810, .(T811, []))))))), T813))
REV1_IN_GA(.(T811, .(T810, .(T809, .(T808, .(T807, .(T806, .(T805, .(T918, T919)))))))), T813) → REV212_IN_GGGA(T919, T918, .(T805, .(T806, .(T807, .(T808, .(T809, .(T810, .(T811, []))))))), T813)
REV212_IN_GGGA(.(T1013, T1014), T965, T966, T968) → U1_GGGA(T1013, T1014, T965, T966, T968, rev212_in_ggga(T1014, T1013, .(T965, T966), T968))
REV212_IN_GGGA(.(T1013, T1014), T965, T966, T968) → REV212_IN_GGGA(T1014, T1013, .(T965, T966), T968)
R is empty.
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
rev212_in_ggga(
x1,
x2,
x3,
x4) =
rev212_in_ggga(
x1,
x2,
x3)
[] =
[]
REV1_IN_GA(
x1,
x2) =
REV1_IN_GA(
x1)
U2_GA(
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x8,
x9,
x10,
x11) =
U2_GA(
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x8,
x9,
x11)
REV212_IN_GGGA(
x1,
x2,
x3,
x4) =
REV212_IN_GGGA(
x1,
x2,
x3)
U1_GGGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U1_GGGA(
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 3 less nodes.
(6) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
REV212_IN_GGGA(.(T1013, T1014), T965, T966, T968) → REV212_IN_GGGA(T1014, T1013, .(T965, T966), T968)
R is empty.
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
REV212_IN_GGGA(
x1,
x2,
x3,
x4) =
REV212_IN_GGGA(
x1,
x2,
x3)
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:
REV212_IN_GGGA(.(T1013, T1014), T965, T966) → REV212_IN_GGGA(T1014, T1013, .(T965, T966))
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:
- REV212_IN_GGGA(.(T1013, T1014), T965, T966) → REV212_IN_GGGA(T1014, T1013, .(T965, T966))
The graph contains the following edges 1 > 1, 1 > 2
(10) YES