(0) Obligation:
Clauses:append([], L, L).
append(.(H, L1), L2, .(H, L3)) :- append(L1, L2, L3).
append1([], L, L).
append1(.(H, L1), L2, .(H, L3)) :- append1(L1, L2, L3).
Queries:
append(a,a,a).
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph.
(2) Obligation:
Triples:append11(.(T10, .(T30, T34)), T35, .(T10, .(T30, T36))) :- append11(T34, T35, T36).
append11(.(T43, .(T63, T67)), T68, .(T43, .(T63, T69))) :- append11(T67, T68, T69).
Clauses:
appendc1([], T5, T5).
appendc1(.(T10, []), T21, .(T10, T21)).
appendc1(.(T10, .(T30, T34)), T35, .(T10, .(T30, T36))) :- appendc1(T34, T35, T36).
appendc1(.(T43, []), T54, .(T43, T54)).
appendc1(.(T43, .(T63, T67)), T68, .(T43, .(T63, T69))) :- appendc1(T67, T68, T69).
Afs:
append11(x1, x2, x3) = append11
(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:append11_in: (f,f,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
APPEND11_IN_AAA(.(T10, .(T30, T34)), T35, .(T10, .(T30, T36))) → U1_AAA(T10, T30, T34, T35, T36, append11_in_aaa(T34, T35, T36))
APPEND11_IN_AAA(.(T10, .(T30, T34)), T35, .(T10, .(T30, T36))) → APPEND11_IN_AAA(T34, T35, T36)
APPEND11_IN_AAA(.(T43, .(T63, T67)), T68, .(T43, .(T63, T69))) → U2_AAA(T43, T63, T67, T68, T69, append11_in_aaa(T67, T68, T69))
R is empty.
The argument filtering Pi contains the following mapping:
append11_in_aaa(x1, x2, x3) = append11_in_aaa
.(x1, x2) = .(x2)
APPEND11_IN_AAA(x1, x2, x3) = APPEND11_IN_AAA
U1_AAA(x1, x2, x3, x4, x5, x6) = U1_AAA(x6)
U2_AAA(x1, x2, x3, x4, x5, x6) = U2_AAA(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:
APPEND11_IN_AAA(.(T10, .(T30, T34)), T35, .(T10, .(T30, T36))) → U1_AAA(T10, T30, T34, T35, T36, append11_in_aaa(T34, T35, T36))
APPEND11_IN_AAA(.(T10, .(T30, T34)), T35, .(T10, .(T30, T36))) → APPEND11_IN_AAA(T34, T35, T36)
APPEND11_IN_AAA(.(T43, .(T63, T67)), T68, .(T43, .(T63, T69))) → U2_AAA(T43, T63, T67, T68, T69, append11_in_aaa(T67, T68, T69))
R is empty.
The argument filtering Pi contains the following mapping:
append11_in_aaa(x1, x2, x3) = append11_in_aaa
.(x1, x2) = .(x2)
APPEND11_IN_AAA(x1, x2, x3) = APPEND11_IN_AAA
U1_AAA(x1, x2, x3, x4, x5, x6) = U1_AAA(x6)
U2_AAA(x1, x2, x3, x4, x5, x6) = U2_AAA(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:
APPEND11_IN_AAA(.(T10, .(T30, T34)), T35, .(T10, .(T30, T36))) → APPEND11_IN_AAA(T34, T35, T36)
R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
APPEND11_IN_AAA(x1, x2, x3) = APPEND11_IN_AAA
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:
APPEND11_IN_AAA → APPEND11_IN_AAA
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 = APPEND11_IN_AAA evaluates to t =APPEND11_IN_AAA
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 APPEND11_IN_AAA to APPEND11_IN_AAA.