(0) Obligation:
Clauses:
app1(.(X, Xs), Ys, .(X, Zs)) :- app1(Xs, Ys, Zs).
app1([], Ys, Ys).
app2(.(X, Xs), Ys, .(X, Zs)) :- app2(Xs, Ys, Zs).
app2([], Ys, Ys).
Query: app2(a,g,g)
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph DT10.
(2) Obligation:
Triples:
app2A(.(X1, .(X2, X3)), X4, .(X1, .(X2, X5))) :- app2A(X3, X4, X5).
Clauses:
app2cA(.(X1, .(X2, X3)), X4, .(X1, .(X2, X5))) :- app2cA(X3, X4, X5).
app2cA(.(X1, []), X2, .(X1, X2)).
app2cA([], .(X1, X2), .(X1, X2)).
app2cA([], X1, X1).
Afs:
app2A(x1, x2, x3) = app2A(x2, x3)
(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:
app2A_in: (f,b,b)
Transforming
TRIPLES into the following
Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
APP2A_IN_AGG(.(X1, .(X2, X3)), X4, .(X1, .(X2, X5))) → U1_AGG(X1, X2, X3, X4, X5, app2A_in_agg(X3, X4, X5))
APP2A_IN_AGG(.(X1, .(X2, X3)), X4, .(X1, .(X2, X5))) → APP2A_IN_AGG(X3, X4, X5)
R is empty.
The argument filtering Pi contains the following mapping:
app2A_in_agg(
x1,
x2,
x3) =
app2A_in_agg(
x2,
x3)
.(
x1,
x2) =
.(
x1,
x2)
APP2A_IN_AGG(
x1,
x2,
x3) =
APP2A_IN_AGG(
x2,
x3)
U1_AGG(
x1,
x2,
x3,
x4,
x5,
x6) =
U1_AGG(
x1,
x2,
x4,
x5,
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:
APP2A_IN_AGG(.(X1, .(X2, X3)), X4, .(X1, .(X2, X5))) → U1_AGG(X1, X2, X3, X4, X5, app2A_in_agg(X3, X4, X5))
APP2A_IN_AGG(.(X1, .(X2, X3)), X4, .(X1, .(X2, X5))) → APP2A_IN_AGG(X3, X4, X5)
R is empty.
The argument filtering Pi contains the following mapping:
app2A_in_agg(
x1,
x2,
x3) =
app2A_in_agg(
x2,
x3)
.(
x1,
x2) =
.(
x1,
x2)
APP2A_IN_AGG(
x1,
x2,
x3) =
APP2A_IN_AGG(
x2,
x3)
U1_AGG(
x1,
x2,
x3,
x4,
x5,
x6) =
U1_AGG(
x1,
x2,
x4,
x5,
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:
APP2A_IN_AGG(.(X1, .(X2, X3)), X4, .(X1, .(X2, X5))) → APP2A_IN_AGG(X3, X4, X5)
R is empty.
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
APP2A_IN_AGG(
x1,
x2,
x3) =
APP2A_IN_AGG(
x2,
x3)
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:
APP2A_IN_AGG(X4, .(X1, .(X2, X5))) → APP2A_IN_AGG(X4, X5)
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:
- APP2A_IN_AGG(X4, .(X1, .(X2, X5))) → APP2A_IN_AGG(X4, X5)
The graph contains the following edges 1 >= 1, 2 > 2
(10) YES