(0) Obligation:
Clauses:
duplicate([], []).
duplicate(X, .(H, .(H, Z))) :- ','(no(empty(X)), ','(head(X, H), ','(tail(X, T), duplicate(T, Z)))).
head([], X1).
head(.(H, X2), H).
tail([], []).
tail(.(X3, T), T).
empty([]).
no(X) :- ','(X, ','(!, failure(a))).
no(X4).
failure(b).
Query: duplicate(g,a)
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph DT10.
(2) Obligation:
Triples:
duplicateA(.(X1, X2), .(X1, .(X1, X3))) :- duplicateA(X2, X3).
Clauses:
duplicatecA([], []).
duplicatecA(.(X1, X2), .(X1, .(X1, X3))) :- duplicatecA(X2, X3).
Afs:
duplicateA(x1, x2) = duplicateA(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:
duplicateA_in: (b,f)
Transforming
TRIPLES into the following
Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
DUPLICATEA_IN_GA(.(X1, X2), .(X1, .(X1, X3))) → U1_GA(X1, X2, X3, duplicateA_in_ga(X2, X3))
DUPLICATEA_IN_GA(.(X1, X2), .(X1, .(X1, X3))) → DUPLICATEA_IN_GA(X2, X3)
R is empty.
The argument filtering Pi contains the following mapping:
duplicateA_in_ga(
x1,
x2) =
duplicateA_in_ga(
x1)
.(
x1,
x2) =
.(
x1,
x2)
DUPLICATEA_IN_GA(
x1,
x2) =
DUPLICATEA_IN_GA(
x1)
U1_GA(
x1,
x2,
x3,
x4) =
U1_GA(
x1,
x2,
x4)
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:
DUPLICATEA_IN_GA(.(X1, X2), .(X1, .(X1, X3))) → U1_GA(X1, X2, X3, duplicateA_in_ga(X2, X3))
DUPLICATEA_IN_GA(.(X1, X2), .(X1, .(X1, X3))) → DUPLICATEA_IN_GA(X2, X3)
R is empty.
The argument filtering Pi contains the following mapping:
duplicateA_in_ga(
x1,
x2) =
duplicateA_in_ga(
x1)
.(
x1,
x2) =
.(
x1,
x2)
DUPLICATEA_IN_GA(
x1,
x2) =
DUPLICATEA_IN_GA(
x1)
U1_GA(
x1,
x2,
x3,
x4) =
U1_GA(
x1,
x2,
x4)
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:
DUPLICATEA_IN_GA(.(X1, X2), .(X1, .(X1, X3))) → DUPLICATEA_IN_GA(X2, X3)
R is empty.
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
DUPLICATEA_IN_GA(
x1,
x2) =
DUPLICATEA_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:
DUPLICATEA_IN_GA(.(X1, X2)) → DUPLICATEA_IN_GA(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:
- DUPLICATEA_IN_GA(.(X1, X2)) → DUPLICATEA_IN_GA(X2)
The graph contains the following edges 1 > 1
(10) YES