(0) Obligation:
Clauses:sublist(Xs, Ys) :- ','(app(X1, Zs, Ys), app(Xs, X2, Zs)).
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
Queries:
sublist(a,g).
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph.
(2) Obligation:
Triples:app7(.(T30, T33), X57, .(T30, T32)) :- app7(T33, X57, T32).
p3([], T16, T16, T7, X9) :- app7(T7, X9, T16).
p3(.(X84, X85), X86, .(X84, T38), T7, X9) :- p3(X85, X86, T38, T7, X9).
sublist1(T7, T6) :- p3(X7, X8, T6, T7, X9).
Clauses:
appc7([], T23, T23).
appc7(.(T30, T33), X57, .(T30, T32)) :- appc7(T33, X57, T32).
qc3([], T16, T16, T7, X9) :- appc7(T7, X9, T16).
qc3(.(X84, X85), X86, .(X84, T38), T7, X9) :- qc3(X85, X86, T38, T7, X9).
Afs:
sublist1(x1, x2) = sublist1(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:sublist1_in: (f,b)
p3_in: (f,f,b,f,f)
app7_in: (f,f,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
SUBLIST1_IN_AG(T7, T6) → U4_AG(T7, T6, p3_in_aagaa(X7, X8, T6, T7, X9))
SUBLIST1_IN_AG(T7, T6) → P3_IN_AAGAA(X7, X8, T6, T7, X9)
P3_IN_AAGAA([], T16, T16, T7, X9) → U2_AAGAA(T16, T7, X9, app7_in_aag(T7, X9, T16))
P3_IN_AAGAA([], T16, T16, T7, X9) → APP7_IN_AAG(T7, X9, T16)
APP7_IN_AAG(.(T30, T33), X57, .(T30, T32)) → U1_AAG(T30, T33, X57, T32, app7_in_aag(T33, X57, T32))
APP7_IN_AAG(.(T30, T33), X57, .(T30, T32)) → APP7_IN_AAG(T33, X57, T32)
P3_IN_AAGAA(.(X84, X85), X86, .(X84, T38), T7, X9) → U3_AAGAA(X84, X85, X86, T38, T7, X9, p3_in_aagaa(X85, X86, T38, T7, X9))
P3_IN_AAGAA(.(X84, X85), X86, .(X84, T38), T7, X9) → P3_IN_AAGAA(X85, X86, T38, T7, X9)
R is empty.
The argument filtering Pi contains the following mapping:
p3_in_aagaa(x1, x2, x3, x4, x5) = p3_in_aagaa(x3)
app7_in_aag(x1, x2, x3) = app7_in_aag(x3)
.(x1, x2) = .(x1, x2)
[] = []
SUBLIST1_IN_AG(x1, x2) = SUBLIST1_IN_AG(x2)
U4_AG(x1, x2, x3) = U4_AG(x2, x3)
P3_IN_AAGAA(x1, x2, x3, x4, x5) = P3_IN_AAGAA(x3)
U2_AAGAA(x1, x2, x3, x4) = U2_AAGAA(x1, x4)
APP7_IN_AAG(x1, x2, x3) = APP7_IN_AAG(x3)
U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x1, x4, x5)
U3_AAGAA(x1, x2, x3, x4, x5, x6, x7) = U3_AAGAA(x1, x4, 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:
SUBLIST1_IN_AG(T7, T6) → U4_AG(T7, T6, p3_in_aagaa(X7, X8, T6, T7, X9))
SUBLIST1_IN_AG(T7, T6) → P3_IN_AAGAA(X7, X8, T6, T7, X9)
P3_IN_AAGAA([], T16, T16, T7, X9) → U2_AAGAA(T16, T7, X9, app7_in_aag(T7, X9, T16))
P3_IN_AAGAA([], T16, T16, T7, X9) → APP7_IN_AAG(T7, X9, T16)
APP7_IN_AAG(.(T30, T33), X57, .(T30, T32)) → U1_AAG(T30, T33, X57, T32, app7_in_aag(T33, X57, T32))
APP7_IN_AAG(.(T30, T33), X57, .(T30, T32)) → APP7_IN_AAG(T33, X57, T32)
P3_IN_AAGAA(.(X84, X85), X86, .(X84, T38), T7, X9) → U3_AAGAA(X84, X85, X86, T38, T7, X9, p3_in_aagaa(X85, X86, T38, T7, X9))
P3_IN_AAGAA(.(X84, X85), X86, .(X84, T38), T7, X9) → P3_IN_AAGAA(X85, X86, T38, T7, X9)
R is empty.
The argument filtering Pi contains the following mapping:
p3_in_aagaa(x1, x2, x3, x4, x5) = p3_in_aagaa(x3)
app7_in_aag(x1, x2, x3) = app7_in_aag(x3)
.(x1, x2) = .(x1, x2)
[] = []
SUBLIST1_IN_AG(x1, x2) = SUBLIST1_IN_AG(x2)
U4_AG(x1, x2, x3) = U4_AG(x2, x3)
P3_IN_AAGAA(x1, x2, x3, x4, x5) = P3_IN_AAGAA(x3)
U2_AAGAA(x1, x2, x3, x4) = U2_AAGAA(x1, x4)
APP7_IN_AAG(x1, x2, x3) = APP7_IN_AAG(x3)
U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x1, x4, x5)
U3_AAGAA(x1, x2, x3, x4, x5, x6, x7) = U3_AAGAA(x1, x4, x7)
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:
APP7_IN_AAG(.(T30, T33), X57, .(T30, T32)) → APP7_IN_AAG(T33, X57, T32)
R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APP7_IN_AAG(x1, x2, x3) = APP7_IN_AAG(x3)
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:
APP7_IN_AAG(.(T30, T32)) → APP7_IN_AAG(T32)
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:
- APP7_IN_AAG(.(T30, T32)) → APP7_IN_AAG(T32)
The graph contains the following edges 1 > 1
(11) YES
(12) Obligation:
Pi DP problem:The TRS P consists of the following rules:
P3_IN_AAGAA(.(X84, X85), X86, .(X84, T38), T7, X9) → P3_IN_AAGAA(X85, X86, T38, T7, X9)
R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
P3_IN_AAGAA(x1, x2, x3, x4, x5) = P3_IN_AAGAA(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:
P3_IN_AAGAA(.(X84, T38)) → P3_IN_AAGAA(T38)
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:
- P3_IN_AAGAA(.(X84, T38)) → P3_IN_AAGAA(T38)
The graph contains the following edges 1 > 1