(0) Obligation:
Clauses:
app([], Y, Z) :- ','(!, eq(Y, Z)).
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).
eq(X, X).
Query: app(g,a,a)
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph DT10.
(2) Obligation:
Triples:
appA(.(X1, X2), X3, .(X1, X4)) :- appA(X2, X3, X4).
Clauses:
appcA([], X1, X1).
appcA(.(X1, X2), X3, .(X1, X4)) :- appcA(X2, X3, X4).
Afs:
appA(x1, x2, x3) = appA(x1)
(3) TriplesToPiDPProof (SOUND transformation)
We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
appA_in: (b,f,f)
Transforming
TRIPLES into the following
Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
APPA_IN_GAA(.(X1, X2), X3, .(X1, X4)) → U1_GAA(X1, X2, X3, X4, appA_in_gaa(X2, X3, X4))
APPA_IN_GAA(.(X1, X2), X3, .(X1, X4)) → APPA_IN_GAA(X2, X3, X4)
R is empty.
The argument filtering Pi contains the following mapping:
appA_in_gaa(
x1,
x2,
x3) =
appA_in_gaa(
x1)
.(
x1,
x2) =
.(
x1,
x2)
APPA_IN_GAA(
x1,
x2,
x3) =
APPA_IN_GAA(
x1)
U1_GAA(
x1,
x2,
x3,
x4,
x5) =
U1_GAA(
x1,
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:
APPA_IN_GAA(.(X1, X2), X3, .(X1, X4)) → U1_GAA(X1, X2, X3, X4, appA_in_gaa(X2, X3, X4))
APPA_IN_GAA(.(X1, X2), X3, .(X1, X4)) → APPA_IN_GAA(X2, X3, X4)
R is empty.
The argument filtering Pi contains the following mapping:
appA_in_gaa(
x1,
x2,
x3) =
appA_in_gaa(
x1)
.(
x1,
x2) =
.(
x1,
x2)
APPA_IN_GAA(
x1,
x2,
x3) =
APPA_IN_GAA(
x1)
U1_GAA(
x1,
x2,
x3,
x4,
x5) =
U1_GAA(
x1,
x2,
x5)
We have to consider all (P,R,Pi)-chains
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.
(6) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
APPA_IN_GAA(.(X1, X2), X3, .(X1, X4)) → APPA_IN_GAA(X2, X3, X4)
R is empty.
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
APPA_IN_GAA(
x1,
x2,
x3) =
APPA_IN_GAA(
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:
APPA_IN_GAA(.(X1, X2)) → APPA_IN_GAA(X2)
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:
- APPA_IN_GAA(.(X1, X2)) → APPA_IN_GAA(X2)
The graph contains the following edges 1 > 1
(10) YES