(0) Obligation:
Clauses:fold(X, [], Z) :- ','(!, eq(X, Z)).
fold(X, Y, Z) :- ','(head(Y, H), ','(tail(Y, T), ','(myop(X, H, V), fold(V, T, Z)))).
myop(a, b, a).
head([], X1).
head(.(H, X2), H).
tail([], []).
tail(.(X3, T), T).
eq(X, X).
Queries:
fold(g,g,a).
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph.
(2) Obligation:
Triples:fold19(.(b, T62), T47) :- fold19(T62, T47).
fold1(a, .(b, T32), T18) :- fold19(T32, T18).
Clauses:
foldc19([], a).
foldc19(.(b, T62), T47) :- foldc19(T62, T47).
Afs:
fold1(x1, x2, x3) = fold1(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:fold1_in: (b,b,f)
fold19_in: (b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
FOLD1_IN_GGA(a, .(b, T32), T18) → U2_GGA(T32, T18, fold19_in_ga(T32, T18))
FOLD1_IN_GGA(a, .(b, T32), T18) → FOLD19_IN_GA(T32, T18)
FOLD19_IN_GA(.(b, T62), T47) → U1_GA(T62, T47, fold19_in_ga(T62, T47))
FOLD19_IN_GA(.(b, T62), T47) → FOLD19_IN_GA(T62, T47)
R is empty.
The argument filtering Pi contains the following mapping:
a = a
.(x1, x2) = .(x1, x2)
b = b
fold19_in_ga(x1, x2) = fold19_in_ga(x1)
FOLD1_IN_GGA(x1, x2, x3) = FOLD1_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3) = U2_GGA(x1, x3)
FOLD19_IN_GA(x1, x2) = FOLD19_IN_GA(x1)
U1_GA(x1, x2, x3) = U1_GA(x1, x3)
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:
FOLD1_IN_GGA(a, .(b, T32), T18) → U2_GGA(T32, T18, fold19_in_ga(T32, T18))
FOLD1_IN_GGA(a, .(b, T32), T18) → FOLD19_IN_GA(T32, T18)
FOLD19_IN_GA(.(b, T62), T47) → U1_GA(T62, T47, fold19_in_ga(T62, T47))
FOLD19_IN_GA(.(b, T62), T47) → FOLD19_IN_GA(T62, T47)
R is empty.
The argument filtering Pi contains the following mapping:
a = a
.(x1, x2) = .(x1, x2)
b = b
fold19_in_ga(x1, x2) = fold19_in_ga(x1)
FOLD1_IN_GGA(x1, x2, x3) = FOLD1_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3) = U2_GGA(x1, x3)
FOLD19_IN_GA(x1, x2) = FOLD19_IN_GA(x1)
U1_GA(x1, x2, x3) = U1_GA(x1, x3)
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:
FOLD19_IN_GA(.(b, T62), T47) → FOLD19_IN_GA(T62, T47)
R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
b = b
FOLD19_IN_GA(x1, x2) = FOLD19_IN_GA(x1)
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:
FOLD19_IN_GA(.(b, T62)) → FOLD19_IN_GA(T62)
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:
- FOLD19_IN_GA(.(b, T62)) → FOLD19_IN_GA(T62)
The graph contains the following edges 1 > 1