(0) Obligation:
Clauses:append1([], X, X).
append1(.(X, Y), U, .(X, Z)) :- append1(Y, U, Z).
append2([], X, X).
append2(.(X, Y), U, .(X, Z)) :- append2(Y, U, Z).
append3([], X, X).
append3(.(X, Y), U, .(X, Z)) :- append3(Y, U, Z).
Queries:
append3(a,a,g).
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph.
(2) Obligation:
Triples:append31(.(T10, .(T29, T33)), T34, .(T10, .(T29, T32))) :- append31(T33, T34, T32).
append31(.(T41, .(T60, T64)), T65, .(T41, .(T60, T63))) :- append31(T64, T65, T63).
Clauses:
append3c1([], T5, T5).
append3c1(.(T10, []), T20, .(T10, T20)).
append3c1(.(T10, .(T29, T33)), T34, .(T10, .(T29, T32))) :- append3c1(T33, T34, T32).
append3c1(.(T41, []), T51, .(T41, T51)).
append3c1(.(T41, .(T60, T64)), T65, .(T41, .(T60, T63))) :- append3c1(T64, T65, T63).
Afs:
append31(x1, x2, x3) = append31(x3)
(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:append31_in: (f,f,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
APPEND31_IN_AAG(.(T10, .(T29, T33)), T34, .(T10, .(T29, T32))) → U1_AAG(T10, T29, T33, T34, T32, append31_in_aag(T33, T34, T32))
APPEND31_IN_AAG(.(T10, .(T29, T33)), T34, .(T10, .(T29, T32))) → APPEND31_IN_AAG(T33, T34, T32)
APPEND31_IN_AAG(.(T41, .(T60, T64)), T65, .(T41, .(T60, T63))) → U2_AAG(T41, T60, T64, T65, T63, append31_in_aag(T64, T65, T63))
R is empty.
The argument filtering Pi contains the following mapping:
append31_in_aag(x1, x2, x3) = append31_in_aag(x3)
.(x1, x2) = .(x1, x2)
APPEND31_IN_AAG(x1, x2, x3) = APPEND31_IN_AAG(x3)
U1_AAG(x1, x2, x3, x4, x5, x6) = U1_AAG(x1, x2, x5, x6)
U2_AAG(x1, x2, x3, x4, x5, x6) = U2_AAG(x1, x2, 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:
APPEND31_IN_AAG(.(T10, .(T29, T33)), T34, .(T10, .(T29, T32))) → U1_AAG(T10, T29, T33, T34, T32, append31_in_aag(T33, T34, T32))
APPEND31_IN_AAG(.(T10, .(T29, T33)), T34, .(T10, .(T29, T32))) → APPEND31_IN_AAG(T33, T34, T32)
APPEND31_IN_AAG(.(T41, .(T60, T64)), T65, .(T41, .(T60, T63))) → U2_AAG(T41, T60, T64, T65, T63, append31_in_aag(T64, T65, T63))
R is empty.
The argument filtering Pi contains the following mapping:
append31_in_aag(x1, x2, x3) = append31_in_aag(x3)
.(x1, x2) = .(x1, x2)
APPEND31_IN_AAG(x1, x2, x3) = APPEND31_IN_AAG(x3)
U1_AAG(x1, x2, x3, x4, x5, x6) = U1_AAG(x1, x2, x5, x6)
U2_AAG(x1, x2, x3, x4, x5, x6) = U2_AAG(x1, x2, 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 2 less nodes.(6) Obligation:
Pi DP problem:The TRS P consists of the following rules:
APPEND31_IN_AAG(.(T10, .(T29, T33)), T34, .(T10, .(T29, T32))) → APPEND31_IN_AAG(T33, T34, T32)
R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APPEND31_IN_AAG(x1, x2, x3) = APPEND31_IN_AAG(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:
APPEND31_IN_AAG(.(T10, .(T29, T32))) → APPEND31_IN_AAG(T32)
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:
- APPEND31_IN_AAG(.(T10, .(T29, T32))) → APPEND31_IN_AAG(T32)
The graph contains the following edges 1 > 1