Left Termination of the query pattern reverse_concatenate_in_3(g, g, a) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

concatenate([], L, L).
concatenate(.(X, L1), L2, .(X, L3)) :- concatenate(L1, L2, L3).
member(X, .(X, L)).
member(X, .(Y, L)) :- member(X, L).
reverse(L, L1) :- reverse_concatenate(L, [], L1).
reverse_concatenate([], L, L).
reverse_concatenate(.(X, L1), L2, L3) :- reverse_concatenate(L1, .(X, L2), L3).

Queries:

reverse_concatenate(g,g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

reverse_concatenate1([], T5, T5).
reverse_concatenate1(.(T23, []), T24, .(T23, T24)).
reverse_concatenate1(.(T37, .(T35, T36)), T38, T40) :- reverse_concatenate1(T36, .(T35, .(T37, T38)), T40).

Queries:

reverse_concatenate1(g,g,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:
reverse_concatenate1_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

reverse_concatenate1_in_gga([], T5, T5) → reverse_concatenate1_out_gga([], T5, T5)
reverse_concatenate1_in_gga(.(T23, []), T24, .(T23, T24)) → reverse_concatenate1_out_gga(.(T23, []), T24, .(T23, T24))
reverse_concatenate1_in_gga(.(T37, .(T35, T36)), T38, T40) → U1_gga(T37, T35, T36, T38, T40, reverse_concatenate1_in_gga(T36, .(T35, .(T37, T38)), T40))
U1_gga(T37, T35, T36, T38, T40, reverse_concatenate1_out_gga(T36, .(T35, .(T37, T38)), T40)) → reverse_concatenate1_out_gga(.(T37, .(T35, T36)), T38, T40)

The argument filtering Pi contains the following mapping:
reverse_concatenate1_in_gga(x1, x2, x3)  =  reverse_concatenate1_in_gga(x1, x2)
[]  =  []
reverse_concatenate1_out_gga(x1, x2, x3)  =  reverse_concatenate1_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

reverse_concatenate1_in_gga([], T5, T5) → reverse_concatenate1_out_gga([], T5, T5)
reverse_concatenate1_in_gga(.(T23, []), T24, .(T23, T24)) → reverse_concatenate1_out_gga(.(T23, []), T24, .(T23, T24))
reverse_concatenate1_in_gga(.(T37, .(T35, T36)), T38, T40) → U1_gga(T37, T35, T36, T38, T40, reverse_concatenate1_in_gga(T36, .(T35, .(T37, T38)), T40))
U1_gga(T37, T35, T36, T38, T40, reverse_concatenate1_out_gga(T36, .(T35, .(T37, T38)), T40)) → reverse_concatenate1_out_gga(.(T37, .(T35, T36)), T38, T40)

The argument filtering Pi contains the following mapping:
reverse_concatenate1_in_gga(x1, x2, x3)  =  reverse_concatenate1_in_gga(x1, x2)
[]  =  []
reverse_concatenate1_out_gga(x1, x2, x3)  =  reverse_concatenate1_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)

(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:

REVERSE_CONCATENATE1_IN_GGA(.(T37, .(T35, T36)), T38, T40) → U1_GGA(T37, T35, T36, T38, T40, reverse_concatenate1_in_gga(T36, .(T35, .(T37, T38)), T40))
REVERSE_CONCATENATE1_IN_GGA(.(T37, .(T35, T36)), T38, T40) → REVERSE_CONCATENATE1_IN_GGA(T36, .(T35, .(T37, T38)), T40)

The TRS R consists of the following rules:

reverse_concatenate1_in_gga([], T5, T5) → reverse_concatenate1_out_gga([], T5, T5)
reverse_concatenate1_in_gga(.(T23, []), T24, .(T23, T24)) → reverse_concatenate1_out_gga(.(T23, []), T24, .(T23, T24))
reverse_concatenate1_in_gga(.(T37, .(T35, T36)), T38, T40) → U1_gga(T37, T35, T36, T38, T40, reverse_concatenate1_in_gga(T36, .(T35, .(T37, T38)), T40))
U1_gga(T37, T35, T36, T38, T40, reverse_concatenate1_out_gga(T36, .(T35, .(T37, T38)), T40)) → reverse_concatenate1_out_gga(.(T37, .(T35, T36)), T38, T40)

The argument filtering Pi contains the following mapping:
reverse_concatenate1_in_gga(x1, x2, x3)  =  reverse_concatenate1_in_gga(x1, x2)
[]  =  []
reverse_concatenate1_out_gga(x1, x2, x3)  =  reverse_concatenate1_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
REVERSE_CONCATENATE1_IN_GGA(x1, x2, x3)  =  REVERSE_CONCATENATE1_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5, x6)  =  U1_GGA(x6)

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

(6) Obligation:

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

REVERSE_CONCATENATE1_IN_GGA(.(T37, .(T35, T36)), T38, T40) → U1_GGA(T37, T35, T36, T38, T40, reverse_concatenate1_in_gga(T36, .(T35, .(T37, T38)), T40))
REVERSE_CONCATENATE1_IN_GGA(.(T37, .(T35, T36)), T38, T40) → REVERSE_CONCATENATE1_IN_GGA(T36, .(T35, .(T37, T38)), T40)

The TRS R consists of the following rules:

reverse_concatenate1_in_gga([], T5, T5) → reverse_concatenate1_out_gga([], T5, T5)
reverse_concatenate1_in_gga(.(T23, []), T24, .(T23, T24)) → reverse_concatenate1_out_gga(.(T23, []), T24, .(T23, T24))
reverse_concatenate1_in_gga(.(T37, .(T35, T36)), T38, T40) → U1_gga(T37, T35, T36, T38, T40, reverse_concatenate1_in_gga(T36, .(T35, .(T37, T38)), T40))
U1_gga(T37, T35, T36, T38, T40, reverse_concatenate1_out_gga(T36, .(T35, .(T37, T38)), T40)) → reverse_concatenate1_out_gga(.(T37, .(T35, T36)), T38, T40)

The argument filtering Pi contains the following mapping:
reverse_concatenate1_in_gga(x1, x2, x3)  =  reverse_concatenate1_in_gga(x1, x2)
[]  =  []
reverse_concatenate1_out_gga(x1, x2, x3)  =  reverse_concatenate1_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
REVERSE_CONCATENATE1_IN_GGA(x1, x2, x3)  =  REVERSE_CONCATENATE1_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5, x6)  =  U1_GGA(x6)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

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

(8) Obligation:

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

REVERSE_CONCATENATE1_IN_GGA(.(T37, .(T35, T36)), T38, T40) → REVERSE_CONCATENATE1_IN_GGA(T36, .(T35, .(T37, T38)), T40)

The TRS R consists of the following rules:

reverse_concatenate1_in_gga([], T5, T5) → reverse_concatenate1_out_gga([], T5, T5)
reverse_concatenate1_in_gga(.(T23, []), T24, .(T23, T24)) → reverse_concatenate1_out_gga(.(T23, []), T24, .(T23, T24))
reverse_concatenate1_in_gga(.(T37, .(T35, T36)), T38, T40) → U1_gga(T37, T35, T36, T38, T40, reverse_concatenate1_in_gga(T36, .(T35, .(T37, T38)), T40))
U1_gga(T37, T35, T36, T38, T40, reverse_concatenate1_out_gga(T36, .(T35, .(T37, T38)), T40)) → reverse_concatenate1_out_gga(.(T37, .(T35, T36)), T38, T40)

The argument filtering Pi contains the following mapping:
reverse_concatenate1_in_gga(x1, x2, x3)  =  reverse_concatenate1_in_gga(x1, x2)
[]  =  []
reverse_concatenate1_out_gga(x1, x2, x3)  =  reverse_concatenate1_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
REVERSE_CONCATENATE1_IN_GGA(x1, x2, x3)  =  REVERSE_CONCATENATE1_IN_GGA(x1, x2)

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:

REVERSE_CONCATENATE1_IN_GGA(.(T37, .(T35, T36)), T38, T40) → REVERSE_CONCATENATE1_IN_GGA(T36, .(T35, .(T37, T38)), T40)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
REVERSE_CONCATENATE1_IN_GGA(x1, x2, x3)  =  REVERSE_CONCATENATE1_IN_GGA(x1, x2)

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:

REVERSE_CONCATENATE1_IN_GGA(.(T37, .(T35, T36)), T38) → REVERSE_CONCATENATE1_IN_GGA(T36, .(T35, .(T37, T38)))

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:

• REVERSE_CONCATENATE1_IN_GGA(.(T37, .(T35, T36)), T38) → REVERSE_CONCATENATE1_IN_GGA(T36, .(T35, .(T37, T38)))
  The graph contains the following edges 1 > 1

(14) YES