(0) Obligation:
Clauses:select(X, .(X, Xs), Xs).
select(X, .(Y, Xs), .(Y, Zs)) :- select(X, Xs, Zs).
Queries:
select(g,a,a).
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph.
(2) Obligation:
Triples:select1(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41))) :- select1(T36, T40, T41).
select1(T74, .(T51, .(T75, T78)), .(T51, .(T75, T79))) :- select1(T74, T78, T79).
Clauses:
selectc1(T6, .(T6, T7), T7).
selectc1(T26, .(T13, .(T26, T27)), .(T13, T27)).
selectc1(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41))) :- selectc1(T36, T40, T41).
selectc1(T64, .(T51, .(T64, T65)), .(T51, T65)).
selectc1(T74, .(T51, .(T75, T78)), .(T51, .(T75, T79))) :- selectc1(T74, T78, T79).
Afs:
select1(x1, x2, x3) = select1(x1)
(3) TriplesToPiDPProof (SOUND transformation)
We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:select1_in: (b,f,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
SELECT1_IN_GAA(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41))) → U1_GAA(T36, T13, T37, T40, T41, select1_in_gaa(T36, T40, T41))
SELECT1_IN_GAA(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41))) → SELECT1_IN_GAA(T36, T40, T41)
SELECT1_IN_GAA(T74, .(T51, .(T75, T78)), .(T51, .(T75, T79))) → U2_GAA(T74, T51, T75, T78, T79, select1_in_gaa(T74, T78, T79))
R is empty.
The argument filtering Pi contains the following mapping:
select1_in_gaa(x1, x2, x3) = select1_in_gaa(x1)
.(x1, x2) = .(x2)
SELECT1_IN_GAA(x1, x2, x3) = SELECT1_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5, x6) = U1_GAA(x1, x6)
U2_GAA(x1, x2, x3, x4, x5, x6) = U2_GAA(x1, 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:
SELECT1_IN_GAA(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41))) → U1_GAA(T36, T13, T37, T40, T41, select1_in_gaa(T36, T40, T41))
SELECT1_IN_GAA(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41))) → SELECT1_IN_GAA(T36, T40, T41)
SELECT1_IN_GAA(T74, .(T51, .(T75, T78)), .(T51, .(T75, T79))) → U2_GAA(T74, T51, T75, T78, T79, select1_in_gaa(T74, T78, T79))
R is empty.
The argument filtering Pi contains the following mapping:
select1_in_gaa(x1, x2, x3) = select1_in_gaa(x1)
.(x1, x2) = .(x2)
SELECT1_IN_GAA(x1, x2, x3) = SELECT1_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5, x6) = U1_GAA(x1, x6)
U2_GAA(x1, x2, x3, x4, x5, x6) = U2_GAA(x1, 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 2 less nodes.(6) Obligation:
Pi DP problem:The TRS P consists of the following rules:
SELECT1_IN_GAA(T36, .(T13, .(T37, T40)), .(T13, .(T37, T41))) → SELECT1_IN_GAA(T36, T40, T41)
R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
SELECT1_IN_GAA(x1, x2, x3) = SELECT1_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:
SELECT1_IN_GAA(T36) → SELECT1_IN_GAA(T36)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(9) NonTerminationProof (EQUIVALENT transformation)
We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.Found a loop by semiunifying a rule from P directly.
s = SELECT1_IN_GAA(T36) evaluates to t =SELECT1_IN_GAA(T36)
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Matcher: [ ]
- Semiunifier: [ ]
Rewriting sequence
The DP semiunifies directly so there is only one rewrite step from SELECT1_IN_GAA(T36) to SELECT1_IN_GAA(T36).