(0) Obligation:
Clauses:
append(X, Y, Z) :- ','(=(X, []), ','(!, =(Y, Z))).
append(X, Y, Z) :- ','(=(X, .(H, Xs)), ','(!, ','(=(Z, .(H, Zs)), append(Xs, Y, Zs)))).
=(X, X).
Query: append(g,a,a)
(1) BuiltinConflictTransformerProof (EQUIVALENT transformation)
Renamed defined predicates conflicting with built-in predicates [PROLOG].
(2) Obligation:
Clauses:
append(X, Y, Z) :- ','(user_defined_=(X, []), ','(!, user_defined_=(Y, Z))).
append(X, Y, Z) :- ','(user_defined_=(X, .(H, Xs)), ','(!, ','(user_defined_=(Z, .(H, Zs)), append(Xs, Y, Zs)))).
user_defined_=(X, X).
Query: append(g,a,a)
(3) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph DT10.
(4) Obligation:
Triples:
appendA(.(X1, X2), X3, .(X1, X4)) :- appendA(X2, X3, X4).
Clauses:
appendcA([], X1, X1).
appendcA(.(X1, X2), X3, .(X1, X4)) :- appendcA(X2, X3, X4).
Afs:
appendA(x1, x2, x3) = appendA(x1)
(5) TriplesToPiDPProof (SOUND transformation)
We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
appendA_in: (b,f,f)
Transforming
TRIPLES into the following
Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
APPENDA_IN_GAA(.(X1, X2), X3, .(X1, X4)) → U1_GAA(X1, X2, X3, X4, appendA_in_gaa(X2, X3, X4))
APPENDA_IN_GAA(.(X1, X2), X3, .(X1, X4)) → APPENDA_IN_GAA(X2, X3, X4)
R is empty.
The argument filtering Pi contains the following mapping:
appendA_in_gaa(
x1,
x2,
x3) =
appendA_in_gaa(
x1)
.(
x1,
x2) =
.(
x1,
x2)
APPENDA_IN_GAA(
x1,
x2,
x3) =
APPENDA_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
(6) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
APPENDA_IN_GAA(.(X1, X2), X3, .(X1, X4)) → U1_GAA(X1, X2, X3, X4, appendA_in_gaa(X2, X3, X4))
APPENDA_IN_GAA(.(X1, X2), X3, .(X1, X4)) → APPENDA_IN_GAA(X2, X3, X4)
R is empty.
The argument filtering Pi contains the following mapping:
appendA_in_gaa(
x1,
x2,
x3) =
appendA_in_gaa(
x1)
.(
x1,
x2) =
.(
x1,
x2)
APPENDA_IN_GAA(
x1,
x2,
x3) =
APPENDA_IN_GAA(
x1)
U1_GAA(
x1,
x2,
x3,
x4,
x5) =
U1_GAA(
x1,
x2,
x5)
We have to consider all (P,R,Pi)-chains
(7) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.
(8) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
APPENDA_IN_GAA(.(X1, X2), X3, .(X1, X4)) → APPENDA_IN_GAA(X2, X3, X4)
R is empty.
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
APPENDA_IN_GAA(
x1,
x2,
x3) =
APPENDA_IN_GAA(
x1)
We have to consider all (P,R,Pi)-chains
(9) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
APPENDA_IN_GAA(.(X1, X2)) → APPENDA_IN_GAA(X2)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(11) 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:
- APPENDA_IN_GAA(.(X1, X2)) → APPENDA_IN_GAA(X2)
The graph contains the following edges 1 > 1
(12) YES