(0) Obligation:
Clauses:app([], L, L) :- !.
app(X, Y, .(H, Z)) :- ','(head(X, H), ','(tail(X, T), app(T, Y, Z))).
head([], X1).
head(.(X, X2), X).
tail([], []).
tail(.(X3, Xs), Xs).
Queries:
app(g,a,a).
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph.
(2) Obligation:
Triples:app17(T55, .(T54, T56)) :- app17(T55, T56).
app1([], T22, .(T21, T23)) :- app17(T22, T23).
app1(.(T75, T76), T67, .(T75, T68)) :- app1(T76, T67, T68).
Clauses:
appc17(T30, T30).
appc17(T55, .(T54, T56)) :- appc17(T55, T56).
appc1([], T5, T5).
appc1([], T22, .(T21, T23)) :- appc17(T22, T23).
appc1(.(T75, T76), T67, .(T75, T68)) :- appc1(T76, T67, T68).
Afs:
app1(x1, x2, x3) = app1(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:app1_in: (b,f,f)
app17_in: (f,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
APP1_IN_GAA([], T22, .(T21, T23)) → U2_GAA(T22, T21, T23, app17_in_aa(T22, T23))
APP1_IN_GAA([], T22, .(T21, T23)) → APP17_IN_AA(T22, T23)
APP17_IN_AA(T55, .(T54, T56)) → U1_AA(T55, T54, T56, app17_in_aa(T55, T56))
APP17_IN_AA(T55, .(T54, T56)) → APP17_IN_AA(T55, T56)
APP1_IN_GAA(.(T75, T76), T67, .(T75, T68)) → U3_GAA(T75, T76, T67, T68, app1_in_gaa(T76, T67, T68))
APP1_IN_GAA(.(T75, T76), T67, .(T75, T68)) → APP1_IN_GAA(T76, T67, T68)
R is empty.
The argument filtering Pi contains the following mapping:
app1_in_gaa(x1, x2, x3) = app1_in_gaa(x1)
[] = []
app17_in_aa(x1, x2) = app17_in_aa
.(x1, x2) = .(x2)
APP1_IN_GAA(x1, x2, x3) = APP1_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4) = U2_GAA(x4)
APP17_IN_AA(x1, x2) = APP17_IN_AA
U1_AA(x1, x2, x3, x4) = U1_AA(x4)
U3_GAA(x1, x2, x3, x4, x5) = U3_GAA(x2, 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:
APP1_IN_GAA([], T22, .(T21, T23)) → U2_GAA(T22, T21, T23, app17_in_aa(T22, T23))
APP1_IN_GAA([], T22, .(T21, T23)) → APP17_IN_AA(T22, T23)
APP17_IN_AA(T55, .(T54, T56)) → U1_AA(T55, T54, T56, app17_in_aa(T55, T56))
APP17_IN_AA(T55, .(T54, T56)) → APP17_IN_AA(T55, T56)
APP1_IN_GAA(.(T75, T76), T67, .(T75, T68)) → U3_GAA(T75, T76, T67, T68, app1_in_gaa(T76, T67, T68))
APP1_IN_GAA(.(T75, T76), T67, .(T75, T68)) → APP1_IN_GAA(T76, T67, T68)
R is empty.
The argument filtering Pi contains the following mapping:
app1_in_gaa(x1, x2, x3) = app1_in_gaa(x1)
[] = []
app17_in_aa(x1, x2) = app17_in_aa
.(x1, x2) = .(x2)
APP1_IN_GAA(x1, x2, x3) = APP1_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4) = U2_GAA(x4)
APP17_IN_AA(x1, x2) = APP17_IN_AA
U1_AA(x1, x2, x3, x4) = U1_AA(x4)
U3_GAA(x1, x2, x3, x4, x5) = U3_GAA(x2, 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 4 less nodes.(6) Complex Obligation (AND)
(7) Obligation:
Pi DP problem:The TRS P consists of the following rules:
APP17_IN_AA(T55, .(T54, T56)) → APP17_IN_AA(T55, T56)
R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
APP17_IN_AA(x1, x2) = APP17_IN_AA
We have to consider all (P,R,Pi)-chains
(8) PiDPToQDPProof (SOUND 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:
APP17_IN_AA → APP17_IN_AA
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(10) 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 = APP17_IN_AA evaluates to t =APP17_IN_AA
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 APP17_IN_AA to APP17_IN_AA.
(11) NO
(12) Obligation:
Pi DP problem:The TRS P consists of the following rules:
APP1_IN_GAA(.(T75, T76), T67, .(T75, T68)) → APP1_IN_GAA(T76, T67, T68)
R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
APP1_IN_GAA(x1, x2, x3) = APP1_IN_GAA(x1)
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:
APP1_IN_GAA(.(T76)) → APP1_IN_GAA(T76)
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:
- APP1_IN_GAA(.(T76)) → APP1_IN_GAA(T76)
The graph contains the following edges 1 > 1