(0) Obligation:

Clauses:

app([], Y, Y).
app(X, Y, .(H, Z)) :- ','(no(empty(X)), ','(head(X, H), ','(tail(X, T), app(T, Y, Z)))).
head([], X1).
head(.(X, X2), X).
tail([], []).
tail(.(X3, Xs), Xs).
empty([]).
no(X) :- ','(X, ','(!, failure(a))).
no(X4).
failure(b).

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