(0) Obligation:

Clauses:

app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).

Query: app(g,a,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

(2) Obligation:

Triples:

appA(.(X1, .(X2, X3)), X4, .(X1, .(X2, X5))) :- appA(X3, X4, X5).

Clauses:

appcA([], X1, X1).
appcA(.(X1, []), X2, .(X1, X2)).
appcA(.(X1, .(X2, X3)), X4, .(X1, .(X2, X5))) :- appcA(X3, X4, X5).

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)), X4, .(X1, .(X2, X5))) → U1_GAA(X1, X2, X3, X4, X5, appA_in_gaa(X3, X4, X5))
APPA_IN_GAA(.(X1, .(X2, X3)), X4, .(X1, .(X2, X5))) → APPA_IN_GAA(X3, X4, X5)

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, x6)  =  U1_GAA(x1, x2, x3, x6)

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)), X4, .(X1, .(X2, X5))) → U1_GAA(X1, X2, X3, X4, X5, appA_in_gaa(X3, X4, X5))
APPA_IN_GAA(.(X1, .(X2, X3)), X4, .(X1, .(X2, X5))) → APPA_IN_GAA(X3, X4, X5)

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, x6)  =  U1_GAA(x1, x2, x3, x6)

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)), X4, .(X1, .(X2, X5))) → APPA_IN_GAA(X3, X4, X5)

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, X3))) → APPA_IN_GAA(X3)

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, X3))) → APPA_IN_GAA(X3)
    The graph contains the following edges 1 > 1

(10) YES