Left Termination of the query pattern overlap_in_2(g, g) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

overlap(Xs, Ys) :- ','(member2(X, Xs), member1(X, Ys)).
has_a_or_b(Xs) :- overlap(Xs, .(a, .(b, []))).
member1(X, .(Y, Xs)) :- member1(X, Xs).
member1(X, .(X, Xs)).
member2(X, .(Y, Xs)) :- member2(X, Xs).
member2(X, .(X, Xs)).

Queries:

overlap(g,g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

p3(X25, .(T15, T16), T6) :- p3(X25, T16, T6).
p3(X41, .(X41, T21), T6) :- member19(X41, T6).
member19(X67, .(T32, T33)) :- member19(X67, T33).
member19(X83, .(X83, T38)).
overlap1(T5, T6) :- p3(X5, T5, T6).

Queries:

overlap1(g,g).

(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:
overlap1_in: (b,b)
p3_in: (f,b,b)
member19_in: (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

overlap1_in_gg(T5, T6) → U4_gg(T5, T6, p3_in_agg(X5, T5, T6))
p3_in_agg(X25, .(T15, T16), T6) → U1_agg(X25, T15, T16, T6, p3_in_agg(X25, T16, T6))
p3_in_agg(X41, .(X41, T21), T6) → U2_agg(X41, T21, T6, member19_in_gg(X41, T6))
member19_in_gg(X67, .(T32, T33)) → U3_gg(X67, T32, T33, member19_in_gg(X67, T33))
member19_in_gg(X83, .(X83, T38)) → member19_out_gg(X83, .(X83, T38))
U3_gg(X67, T32, T33, member19_out_gg(X67, T33)) → member19_out_gg(X67, .(T32, T33))
U2_agg(X41, T21, T6, member19_out_gg(X41, T6)) → p3_out_agg(X41, .(X41, T21), T6)
U1_agg(X25, T15, T16, T6, p3_out_agg(X25, T16, T6)) → p3_out_agg(X25, .(T15, T16), T6)
U4_gg(T5, T6, p3_out_agg(X5, T5, T6)) → overlap1_out_gg(T5, T6)

The argument filtering Pi contains the following mapping:
overlap1_in_gg(x1, x2)  =  overlap1_in_gg(x1, x2)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
p3_in_agg(x1, x2, x3)  =  p3_in_agg(x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_agg(x1, x2, x3, x4, x5)  =  U1_agg(x5)
U2_agg(x1, x2, x3, x4)  =  U2_agg(x1, x4)
member19_in_gg(x1, x2)  =  member19_in_gg(x1, x2)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x4)
member19_out_gg(x1, x2)  =  member19_out_gg
p3_out_agg(x1, x2, x3)  =  p3_out_agg(x1)
overlap1_out_gg(x1, x2)  =  overlap1_out_gg

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

overlap1_in_gg(T5, T6) → U4_gg(T5, T6, p3_in_agg(X5, T5, T6))
p3_in_agg(X25, .(T15, T16), T6) → U1_agg(X25, T15, T16, T6, p3_in_agg(X25, T16, T6))
p3_in_agg(X41, .(X41, T21), T6) → U2_agg(X41, T21, T6, member19_in_gg(X41, T6))
member19_in_gg(X67, .(T32, T33)) → U3_gg(X67, T32, T33, member19_in_gg(X67, T33))
member19_in_gg(X83, .(X83, T38)) → member19_out_gg(X83, .(X83, T38))
U3_gg(X67, T32, T33, member19_out_gg(X67, T33)) → member19_out_gg(X67, .(T32, T33))
U2_agg(X41, T21, T6, member19_out_gg(X41, T6)) → p3_out_agg(X41, .(X41, T21), T6)
U1_agg(X25, T15, T16, T6, p3_out_agg(X25, T16, T6)) → p3_out_agg(X25, .(T15, T16), T6)
U4_gg(T5, T6, p3_out_agg(X5, T5, T6)) → overlap1_out_gg(T5, T6)

The argument filtering Pi contains the following mapping:
overlap1_in_gg(x1, x2)  =  overlap1_in_gg(x1, x2)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
p3_in_agg(x1, x2, x3)  =  p3_in_agg(x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_agg(x1, x2, x3, x4, x5)  =  U1_agg(x5)
U2_agg(x1, x2, x3, x4)  =  U2_agg(x1, x4)
member19_in_gg(x1, x2)  =  member19_in_gg(x1, x2)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x4)
member19_out_gg(x1, x2)  =  member19_out_gg
p3_out_agg(x1, x2, x3)  =  p3_out_agg(x1)
overlap1_out_gg(x1, x2)  =  overlap1_out_gg

(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:

OVERLAP1_IN_GG(T5, T6) → U4_GG(T5, T6, p3_in_agg(X5, T5, T6))
OVERLAP1_IN_GG(T5, T6) → P3_IN_AGG(X5, T5, T6)
P3_IN_AGG(X25, .(T15, T16), T6) → U1_AGG(X25, T15, T16, T6, p3_in_agg(X25, T16, T6))
P3_IN_AGG(X25, .(T15, T16), T6) → P3_IN_AGG(X25, T16, T6)
P3_IN_AGG(X41, .(X41, T21), T6) → U2_AGG(X41, T21, T6, member19_in_gg(X41, T6))
P3_IN_AGG(X41, .(X41, T21), T6) → MEMBER19_IN_GG(X41, T6)
MEMBER19_IN_GG(X67, .(T32, T33)) → U3_GG(X67, T32, T33, member19_in_gg(X67, T33))
MEMBER19_IN_GG(X67, .(T32, T33)) → MEMBER19_IN_GG(X67, T33)

The TRS R consists of the following rules:

overlap1_in_gg(T5, T6) → U4_gg(T5, T6, p3_in_agg(X5, T5, T6))
p3_in_agg(X25, .(T15, T16), T6) → U1_agg(X25, T15, T16, T6, p3_in_agg(X25, T16, T6))
p3_in_agg(X41, .(X41, T21), T6) → U2_agg(X41, T21, T6, member19_in_gg(X41, T6))
member19_in_gg(X67, .(T32, T33)) → U3_gg(X67, T32, T33, member19_in_gg(X67, T33))
member19_in_gg(X83, .(X83, T38)) → member19_out_gg(X83, .(X83, T38))
U3_gg(X67, T32, T33, member19_out_gg(X67, T33)) → member19_out_gg(X67, .(T32, T33))
U2_agg(X41, T21, T6, member19_out_gg(X41, T6)) → p3_out_agg(X41, .(X41, T21), T6)
U1_agg(X25, T15, T16, T6, p3_out_agg(X25, T16, T6)) → p3_out_agg(X25, .(T15, T16), T6)
U4_gg(T5, T6, p3_out_agg(X5, T5, T6)) → overlap1_out_gg(T5, T6)

The argument filtering Pi contains the following mapping:
overlap1_in_gg(x1, x2)  =  overlap1_in_gg(x1, x2)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
p3_in_agg(x1, x2, x3)  =  p3_in_agg(x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_agg(x1, x2, x3, x4, x5)  =  U1_agg(x5)
U2_agg(x1, x2, x3, x4)  =  U2_agg(x1, x4)
member19_in_gg(x1, x2)  =  member19_in_gg(x1, x2)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x4)
member19_out_gg(x1, x2)  =  member19_out_gg
p3_out_agg(x1, x2, x3)  =  p3_out_agg(x1)
overlap1_out_gg(x1, x2)  =  overlap1_out_gg
OVERLAP1_IN_GG(x1, x2)  =  OVERLAP1_IN_GG(x1, x2)
U4_GG(x1, x2, x3)  =  U4_GG(x3)
P3_IN_AGG(x1, x2, x3)  =  P3_IN_AGG(x2, x3)
U1_AGG(x1, x2, x3, x4, x5)  =  U1_AGG(x5)
U2_AGG(x1, x2, x3, x4)  =  U2_AGG(x1, x4)
MEMBER19_IN_GG(x1, x2)  =  MEMBER19_IN_GG(x1, x2)
U3_GG(x1, x2, x3, x4)  =  U3_GG(x4)

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

(6) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

OVERLAP1_IN_GG(T5, T6) → U4_GG(T5, T6, p3_in_agg(X5, T5, T6))
OVERLAP1_IN_GG(T5, T6) → P3_IN_AGG(X5, T5, T6)
P3_IN_AGG(X25, .(T15, T16), T6) → U1_AGG(X25, T15, T16, T6, p3_in_agg(X25, T16, T6))
P3_IN_AGG(X25, .(T15, T16), T6) → P3_IN_AGG(X25, T16, T6)
P3_IN_AGG(X41, .(X41, T21), T6) → U2_AGG(X41, T21, T6, member19_in_gg(X41, T6))
P3_IN_AGG(X41, .(X41, T21), T6) → MEMBER19_IN_GG(X41, T6)
MEMBER19_IN_GG(X67, .(T32, T33)) → U3_GG(X67, T32, T33, member19_in_gg(X67, T33))
MEMBER19_IN_GG(X67, .(T32, T33)) → MEMBER19_IN_GG(X67, T33)

The TRS R consists of the following rules:

overlap1_in_gg(T5, T6) → U4_gg(T5, T6, p3_in_agg(X5, T5, T6))
p3_in_agg(X25, .(T15, T16), T6) → U1_agg(X25, T15, T16, T6, p3_in_agg(X25, T16, T6))
p3_in_agg(X41, .(X41, T21), T6) → U2_agg(X41, T21, T6, member19_in_gg(X41, T6))
member19_in_gg(X67, .(T32, T33)) → U3_gg(X67, T32, T33, member19_in_gg(X67, T33))
member19_in_gg(X83, .(X83, T38)) → member19_out_gg(X83, .(X83, T38))
U3_gg(X67, T32, T33, member19_out_gg(X67, T33)) → member19_out_gg(X67, .(T32, T33))
U2_agg(X41, T21, T6, member19_out_gg(X41, T6)) → p3_out_agg(X41, .(X41, T21), T6)
U1_agg(X25, T15, T16, T6, p3_out_agg(X25, T16, T6)) → p3_out_agg(X25, .(T15, T16), T6)
U4_gg(T5, T6, p3_out_agg(X5, T5, T6)) → overlap1_out_gg(T5, T6)

The argument filtering Pi contains the following mapping:
overlap1_in_gg(x1, x2)  =  overlap1_in_gg(x1, x2)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
p3_in_agg(x1, x2, x3)  =  p3_in_agg(x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_agg(x1, x2, x3, x4, x5)  =  U1_agg(x5)
U2_agg(x1, x2, x3, x4)  =  U2_agg(x1, x4)
member19_in_gg(x1, x2)  =  member19_in_gg(x1, x2)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x4)
member19_out_gg(x1, x2)  =  member19_out_gg
p3_out_agg(x1, x2, x3)  =  p3_out_agg(x1)
overlap1_out_gg(x1, x2)  =  overlap1_out_gg
OVERLAP1_IN_GG(x1, x2)  =  OVERLAP1_IN_GG(x1, x2)
U4_GG(x1, x2, x3)  =  U4_GG(x3)
P3_IN_AGG(x1, x2, x3)  =  P3_IN_AGG(x2, x3)
U1_AGG(x1, x2, x3, x4, x5)  =  U1_AGG(x5)
U2_AGG(x1, x2, x3, x4)  =  U2_AGG(x1, x4)
MEMBER19_IN_GG(x1, x2)  =  MEMBER19_IN_GG(x1, x2)
U3_GG(x1, x2, x3, x4)  =  U3_GG(x4)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MEMBER19_IN_GG(X67, .(T32, T33)) → MEMBER19_IN_GG(X67, T33)

The TRS R consists of the following rules:

overlap1_in_gg(T5, T6) → U4_gg(T5, T6, p3_in_agg(X5, T5, T6))
p3_in_agg(X25, .(T15, T16), T6) → U1_agg(X25, T15, T16, T6, p3_in_agg(X25, T16, T6))
p3_in_agg(X41, .(X41, T21), T6) → U2_agg(X41, T21, T6, member19_in_gg(X41, T6))
member19_in_gg(X67, .(T32, T33)) → U3_gg(X67, T32, T33, member19_in_gg(X67, T33))
member19_in_gg(X83, .(X83, T38)) → member19_out_gg(X83, .(X83, T38))
U3_gg(X67, T32, T33, member19_out_gg(X67, T33)) → member19_out_gg(X67, .(T32, T33))
U2_agg(X41, T21, T6, member19_out_gg(X41, T6)) → p3_out_agg(X41, .(X41, T21), T6)
U1_agg(X25, T15, T16, T6, p3_out_agg(X25, T16, T6)) → p3_out_agg(X25, .(T15, T16), T6)
U4_gg(T5, T6, p3_out_agg(X5, T5, T6)) → overlap1_out_gg(T5, T6)

The argument filtering Pi contains the following mapping:
overlap1_in_gg(x1, x2)  =  overlap1_in_gg(x1, x2)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
p3_in_agg(x1, x2, x3)  =  p3_in_agg(x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_agg(x1, x2, x3, x4, x5)  =  U1_agg(x5)
U2_agg(x1, x2, x3, x4)  =  U2_agg(x1, x4)
member19_in_gg(x1, x2)  =  member19_in_gg(x1, x2)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x4)
member19_out_gg(x1, x2)  =  member19_out_gg
p3_out_agg(x1, x2, x3)  =  p3_out_agg(x1)
overlap1_out_gg(x1, x2)  =  overlap1_out_gg
MEMBER19_IN_GG(x1, x2)  =  MEMBER19_IN_GG(x1, x2)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MEMBER19_IN_GG(X67, .(T32, T33)) → MEMBER19_IN_GG(X67, T33)

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

(12) PiDPToQDPProof (EQUIVALENT transformation)

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

(13) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MEMBER19_IN_GG(X67, .(T32, T33)) → MEMBER19_IN_GG(X67, T33)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(14) 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:

• MEMBER19_IN_GG(X67, .(T32, T33)) → MEMBER19_IN_GG(X67, T33)
  The graph contains the following edges 1 >= 1, 2 > 2

(15) YES

(16) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

P3_IN_AGG(X25, .(T15, T16), T6) → P3_IN_AGG(X25, T16, T6)

The TRS R consists of the following rules:

overlap1_in_gg(T5, T6) → U4_gg(T5, T6, p3_in_agg(X5, T5, T6))
p3_in_agg(X25, .(T15, T16), T6) → U1_agg(X25, T15, T16, T6, p3_in_agg(X25, T16, T6))
p3_in_agg(X41, .(X41, T21), T6) → U2_agg(X41, T21, T6, member19_in_gg(X41, T6))
member19_in_gg(X67, .(T32, T33)) → U3_gg(X67, T32, T33, member19_in_gg(X67, T33))
member19_in_gg(X83, .(X83, T38)) → member19_out_gg(X83, .(X83, T38))
U3_gg(X67, T32, T33, member19_out_gg(X67, T33)) → member19_out_gg(X67, .(T32, T33))
U2_agg(X41, T21, T6, member19_out_gg(X41, T6)) → p3_out_agg(X41, .(X41, T21), T6)
U1_agg(X25, T15, T16, T6, p3_out_agg(X25, T16, T6)) → p3_out_agg(X25, .(T15, T16), T6)
U4_gg(T5, T6, p3_out_agg(X5, T5, T6)) → overlap1_out_gg(T5, T6)

The argument filtering Pi contains the following mapping:
overlap1_in_gg(x1, x2)  =  overlap1_in_gg(x1, x2)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
p3_in_agg(x1, x2, x3)  =  p3_in_agg(x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_agg(x1, x2, x3, x4, x5)  =  U1_agg(x5)
U2_agg(x1, x2, x3, x4)  =  U2_agg(x1, x4)
member19_in_gg(x1, x2)  =  member19_in_gg(x1, x2)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x4)
member19_out_gg(x1, x2)  =  member19_out_gg
p3_out_agg(x1, x2, x3)  =  p3_out_agg(x1)
overlap1_out_gg(x1, x2)  =  overlap1_out_gg
P3_IN_AGG(x1, x2, x3)  =  P3_IN_AGG(x2, x3)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

P3_IN_AGG(X25, .(T15, T16), T6) → P3_IN_AGG(X25, T16, T6)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
P3_IN_AGG(x1, x2, x3)  =  P3_IN_AGG(x2, x3)

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

Q DP problem:
The TRS P consists of the following rules:

P3_IN_AGG(.(T15, T16), T6) → P3_IN_AGG(T16, T6)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(21) 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:

• P3_IN_AGG(.(T15, T16), T6) → P3_IN_AGG(T16, T6)
  The graph contains the following edges 1 > 1, 2 >= 2

(22) YES