Left Termination of the query pattern som3_in_3(g, a, a) w.r.t. the given Prolog program could successfully be proven:

0 Prolog
↳1 PrologToPrologProblemTransformerProof (⇐)
↳2 Prolog
↳3 PrologToPiTRSProof (⇐)
↳4 PiTRS
↳5 DependencyPairsProof (⇔)
↳6 PiDP
↳7 DependencyGraphProof (⇔)
↳8 PiDP
↳9 UsableRulesProof (⇔)
↳10 PiDP
↳11 PiDPToQDPProof (⇐)
↳12 QDP
↳13 QDPSizeChangeProof (⇔)
↳14 YES

(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) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

som31([], T5, T5).
som31([], [], []).
som31(T7, [], T7).
som31(.(T13, []), .(T15, T24), .(+(T13, T15), T24)).
som31(.(T13, T29), .(T15, []), .(+(T13, T15), T29)).
som31(.(T13, .(T40, T41)), .(T15, .(T42, T45)), .(+(T13, T15), .(+(T40, T42), T46))) :- som31(T41, T45, T46).
som31(.(T54, []), .(T56, T65), .(+(T54, T56), T65)).
som31(.(T54, T70), .(T56, []), .(+(T54, T56), T70)).
som31(.(T54, .(T81, T82)), .(T56, .(T83, T86)), .(+(T54, T56), .(+(T81, T83), T87))) :- som31(T82, T86, T87).

Queries:

som31(g,a,a).

(3) PrologToPiTRSProof (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 Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

som31_in_gaa([], T5, T5) → som31_out_gaa([], T5, T5)
som31_in_gaa([], [], []) → som31_out_gaa([], [], [])
som31_in_gaa(T7, [], T7) → som31_out_gaa(T7, [], T7)
som31_in_gaa(.(T13, []), .(T15, T24), .(+(T13, T15), T24)) → som31_out_gaa(.(T13, []), .(T15, T24), .(+(T13, T15), T24))
som31_in_gaa(.(T13, T29), .(T15, []), .(+(T13, T15), T29)) → som31_out_gaa(.(T13, T29), .(T15, []), .(+(T13, T15), T29))
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(.(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))
U2_gaa(T54, T81, T82, T56, T83, T86, T87, som31_out_gaa(T82, T86, T87)) → som31_out_gaa(.(T54, .(T81, T82)), .(T56, .(T83, T86)), .(+(T54, T56), .(+(T81, T83), T87)))
U1_gaa(T13, T40, T41, T15, T42, T45, T46, som31_out_gaa(T41, T45, T46)) → som31_out_gaa(.(T13, .(T40, T41)), .(T15, .(T42, T45)), .(+(T13, T15), .(+(T40, T42), T46)))

The argument filtering Pi contains the following mapping:
som31_in_gaa(x1, x2, x3)  =  som31_in_gaa(x1)
[]  =  []
som31_out_gaa(x1, x2, x3)  =  som31_out_gaa
.(x1, x2)  =  .(x1, x2)
U1_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U1_gaa(x8)
U2_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_gaa(x8)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

som31_in_gaa([], T5, T5) → som31_out_gaa([], T5, T5)
som31_in_gaa([], [], []) → som31_out_gaa([], [], [])
som31_in_gaa(T7, [], T7) → som31_out_gaa(T7, [], T7)
som31_in_gaa(.(T13, []), .(T15, T24), .(+(T13, T15), T24)) → som31_out_gaa(.(T13, []), .(T15, T24), .(+(T13, T15), T24))
som31_in_gaa(.(T13, T29), .(T15, []), .(+(T13, T15), T29)) → som31_out_gaa(.(T13, T29), .(T15, []), .(+(T13, T15), T29))
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(.(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))
U2_gaa(T54, T81, T82, T56, T83, T86, T87, som31_out_gaa(T82, T86, T87)) → som31_out_gaa(.(T54, .(T81, T82)), .(T56, .(T83, T86)), .(+(T54, T56), .(+(T81, T83), T87)))
U1_gaa(T13, T40, T41, T15, T42, T45, T46, som31_out_gaa(T41, T45, T46)) → som31_out_gaa(.(T13, .(T40, T41)), .(T15, .(T42, T45)), .(+(T13, T15), .(+(T40, T42), T46)))

The argument filtering Pi contains the following mapping:
som31_in_gaa(x1, x2, x3)  =  som31_in_gaa(x1)
[]  =  []
som31_out_gaa(x1, x2, x3)  =  som31_out_gaa
.(x1, x2)  =  .(x1, x2)
U1_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U1_gaa(x8)
U2_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_gaa(x8)

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
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))

The TRS R consists of the following rules:

som31_in_gaa([], T5, T5) → som31_out_gaa([], T5, T5)
som31_in_gaa([], [], []) → som31_out_gaa([], [], [])
som31_in_gaa(T7, [], T7) → som31_out_gaa(T7, [], T7)
som31_in_gaa(.(T13, []), .(T15, T24), .(+(T13, T15), T24)) → som31_out_gaa(.(T13, []), .(T15, T24), .(+(T13, T15), T24))
som31_in_gaa(.(T13, T29), .(T15, []), .(+(T13, T15), T29)) → som31_out_gaa(.(T13, T29), .(T15, []), .(+(T13, T15), T29))
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(.(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))
U2_gaa(T54, T81, T82, T56, T83, T86, T87, som31_out_gaa(T82, T86, T87)) → som31_out_gaa(.(T54, .(T81, T82)), .(T56, .(T83, T86)), .(+(T54, T56), .(+(T81, T83), T87)))
U1_gaa(T13, T40, T41, T15, T42, T45, T46, som31_out_gaa(T41, T45, T46)) → som31_out_gaa(.(T13, .(T40, T41)), .(T15, .(T42, T45)), .(+(T13, T15), .(+(T40, T42), T46)))

The argument filtering Pi contains the following mapping:
som31_in_gaa(x1, x2, x3)  =  som31_in_gaa(x1)
[]  =  []
som31_out_gaa(x1, x2, x3)  =  som31_out_gaa
.(x1, x2)  =  .(x1, x2)
U1_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U1_gaa(x8)
U2_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_gaa(x8)
SOM31_IN_GAA(x1, x2, x3)  =  SOM31_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U1_GAA(x8)
U2_GAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_GAA(x8)

We have to consider all (P,R,Pi)-chains

(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))) → 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))

The TRS R consists of the following rules:

som31_in_gaa([], T5, T5) → som31_out_gaa([], T5, T5)
som31_in_gaa([], [], []) → som31_out_gaa([], [], [])
som31_in_gaa(T7, [], T7) → som31_out_gaa(T7, [], T7)
som31_in_gaa(.(T13, []), .(T15, T24), .(+(T13, T15), T24)) → som31_out_gaa(.(T13, []), .(T15, T24), .(+(T13, T15), T24))
som31_in_gaa(.(T13, T29), .(T15, []), .(+(T13, T15), T29)) → som31_out_gaa(.(T13, T29), .(T15, []), .(+(T13, T15), T29))
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(.(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))
U2_gaa(T54, T81, T82, T56, T83, T86, T87, som31_out_gaa(T82, T86, T87)) → som31_out_gaa(.(T54, .(T81, T82)), .(T56, .(T83, T86)), .(+(T54, T56), .(+(T81, T83), T87)))
U1_gaa(T13, T40, T41, T15, T42, T45, T46, som31_out_gaa(T41, T45, T46)) → som31_out_gaa(.(T13, .(T40, T41)), .(T15, .(T42, T45)), .(+(T13, T15), .(+(T40, T42), T46)))

The argument filtering Pi contains the following mapping:
som31_in_gaa(x1, x2, x3)  =  som31_in_gaa(x1)
[]  =  []
som31_out_gaa(x1, x2, x3)  =  som31_out_gaa
.(x1, x2)  =  .(x1, x2)
U1_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U1_gaa(x8)
U2_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_gaa(x8)
SOM31_IN_GAA(x1, x2, x3)  =  SOM31_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U1_GAA(x8)
U2_GAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_GAA(x8)

We have to consider all (P,R,Pi)-chains

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes.

(8) 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)

The TRS R consists of the following rules:

som31_in_gaa([], T5, T5) → som31_out_gaa([], T5, T5)
som31_in_gaa([], [], []) → som31_out_gaa([], [], [])
som31_in_gaa(T7, [], T7) → som31_out_gaa(T7, [], T7)
som31_in_gaa(.(T13, []), .(T15, T24), .(+(T13, T15), T24)) → som31_out_gaa(.(T13, []), .(T15, T24), .(+(T13, T15), T24))
som31_in_gaa(.(T13, T29), .(T15, []), .(+(T13, T15), T29)) → som31_out_gaa(.(T13, T29), .(T15, []), .(+(T13, T15), T29))
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(.(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))
U2_gaa(T54, T81, T82, T56, T83, T86, T87, som31_out_gaa(T82, T86, T87)) → som31_out_gaa(.(T54, .(T81, T82)), .(T56, .(T83, T86)), .(+(T54, T56), .(+(T81, T83), T87)))
U1_gaa(T13, T40, T41, T15, T42, T45, T46, som31_out_gaa(T41, T45, T46)) → som31_out_gaa(.(T13, .(T40, T41)), .(T15, .(T42, T45)), .(+(T13, T15), .(+(T40, T42), T46)))

The argument filtering Pi contains the following mapping:
som31_in_gaa(x1, x2, x3)  =  som31_in_gaa(x1)
[]  =  []
som31_out_gaa(x1, x2, x3)  =  som31_out_gaa
.(x1, x2)  =  .(x1, x2)
U1_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U1_gaa(x8)
U2_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_gaa(x8)
SOM31_IN_GAA(x1, x2, x3)  =  SOM31_IN_GAA(x1)

We have to consider all (P,R,Pi)-chains

(9) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(10) 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

(11) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(12) 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.

(13) 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

(14) YES