(0) Obligation:
Clauses:som3([], Bs, Bs).
som3(As, [], As).
som3(.(A, As), .(B, Bs), .(+(A, B), Cs)) :- som3(As, Bs, Cs).
som4_1(As, Bs, Cs, Ds) :- ','(som3(As, Bs, Es), som3(Es, Cs, Ds)).
som4_2(As, Bs, Cs, Ds) :- ','(som3(Es, Cs, Ds), som3(As, Bs, Es)).
Queries:
som3(g,a,a).
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph.
(2) Obligation:
Triples:som31(.(T13, .(T40, T41)), .(T15, .(T42, T45)), .(+(T13, T15), .(+(T40, T42), T46))) :- som31(T41, T45, T46).
som31(.(T54, .(T81, T82)), .(T56, .(T83, T86)), .(+(T54, T56), .(+(T81, T83), T87))) :- som31(T82, T86, T87).
Clauses:
som3c1([], T5, T5).
som3c1([], [], []).
som3c1(T7, [], T7).
som3c1(.(T13, []), .(T15, T24), .(+(T13, T15), T24)).
som3c1(.(T13, T29), .(T15, []), .(+(T13, T15), T29)).
som3c1(.(T13, .(T40, T41)), .(T15, .(T42, T45)), .(+(T13, T15), .(+(T40, T42), T46))) :- som3c1(T41, T45, T46).
som3c1(.(T54, []), .(T56, T65), .(+(T54, T56), T65)).
som3c1(.(T54, T70), .(T56, []), .(+(T54, T56), T70)).
som3c1(.(T54, .(T81, T82)), .(T56, .(T83, T86)), .(+(T54, T56), .(+(T81, T83), T87))) :- som3c1(T82, T86, T87).
Afs:
som31(x1, x2, x3) = som31(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:som31_in: (b,f,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
SOM31_IN_GAA(.(T13, .(T40, T41)), .(T15, .(T42, T45)), .(+(T13, T15), .(+(T40, T42), T46))) → U1_GAA(T13, T40, T41, T15, T42, T45, T46, som31_in_gaa(T41, T45, T46))
SOM31_IN_GAA(.(T13, .(T40, T41)), .(T15, .(T42, T45)), .(+(T13, T15), .(+(T40, T42), T46))) → SOM31_IN_GAA(T41, T45, T46)
SOM31_IN_GAA(.(T54, .(T81, T82)), .(T56, .(T83, T86)), .(+(T54, T56), .(+(T81, T83), T87))) → U2_GAA(T54, T81, T82, T56, T83, T86, T87, som31_in_gaa(T82, T86, T87))
R is empty.
The argument filtering Pi contains the following mapping:
som31_in_gaa(x1, x2, x3) = som31_in_gaa(x1)
.(x1, x2) = .(x1, x2)
SOM31_IN_GAA(x1, x2, x3) = SOM31_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5, x6, x7, x8) = U1_GAA(x1, x2, x3, x8)
U2_GAA(x1, x2, x3, x4, x5, x6, x7, x8) = U2_GAA(x1, x2, x3, x8)
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:
SOM31_IN_GAA(.(T13, .(T40, T41)), .(T15, .(T42, T45)), .(+(T13, T15), .(+(T40, T42), T46))) → U1_GAA(T13, T40, T41, T15, T42, T45, T46, som31_in_gaa(T41, T45, T46))
SOM31_IN_GAA(.(T13, .(T40, T41)), .(T15, .(T42, T45)), .(+(T13, T15), .(+(T40, T42), T46))) → SOM31_IN_GAA(T41, T45, T46)
SOM31_IN_GAA(.(T54, .(T81, T82)), .(T56, .(T83, T86)), .(+(T54, T56), .(+(T81, T83), T87))) → U2_GAA(T54, T81, T82, T56, T83, T86, T87, som31_in_gaa(T82, T86, T87))
R is empty.
The argument filtering Pi contains the following mapping:
som31_in_gaa(x1, x2, x3) = som31_in_gaa(x1)
.(x1, x2) = .(x1, x2)
SOM31_IN_GAA(x1, x2, x3) = SOM31_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5, x6, x7, x8) = U1_GAA(x1, x2, x3, x8)
U2_GAA(x1, x2, x3, x4, x5, x6, x7, x8) = U2_GAA(x1, x2, x3, x8)
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:
SOM31_IN_GAA(.(T13, .(T40, T41)), .(T15, .(T42, T45)), .(+(T13, T15), .(+(T40, T42), T46))) → SOM31_IN_GAA(T41, T45, T46)
R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
SOM31_IN_GAA(x1, x2, x3) = SOM31_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:
SOM31_IN_GAA(.(T13, .(T40, T41))) → SOM31_IN_GAA(T41)
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:
- SOM31_IN_GAA(.(T13, .(T40, T41))) → SOM31_IN_GAA(T41)
The graph contains the following edges 1 > 1