Left Termination of the query pattern
app3_b(g,g,g,a)
w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

app3_a(Xs, Ys, Zs, Us) :- ','(app(Xs, Ys, Vs), app(Vs, Zs, Us)).
app3_b(Xs, Ys, Zs, Us) :- ','(app(Ys, Zs, Vs), app(Xs, Vs, Us)).
app([], Ys, Ys).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).

Query: app3_b(g,g,g,a)

(1) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
app3_b_in: (b,b,b,f)
app_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

app3_b_in_ggga(Xs, Ys, Zs, Us) → U3_ggga(Xs, Ys, Zs, Us, app_in_gga(Ys, Zs, Vs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U5_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U5_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_ggga(Xs, Ys, Zs, Us, app_out_gga(Ys, Zs, Vs)) → U4_ggga(Xs, Ys, Zs, Us, app_in_gga(Xs, Vs, Us))
U4_ggga(Xs, Ys, Zs, Us, app_out_gga(Xs, Vs, Us)) → app3_b_out_ggga(Xs, Ys, Zs, Us)

The argument filtering Pi contains the following mapping:
app3_b_in_ggga(x1, x2, x3, x4)  =  app3_b_in_ggga(x1, x2, x3)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x2, x3, x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
[]  =  []
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x3, x5)
U4_ggga(x1, x2, x3, x4, x5)  =  U4_ggga(x1, x2, x3, x5)
app3_b_out_ggga(x1, x2, x3, x4)  =  app3_b_out_ggga(x1, x2, x3, x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

app3_b_in_ggga(Xs, Ys, Zs, Us) → U3_ggga(Xs, Ys, Zs, Us, app_in_gga(Ys, Zs, Vs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U5_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U5_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_ggga(Xs, Ys, Zs, Us, app_out_gga(Ys, Zs, Vs)) → U4_ggga(Xs, Ys, Zs, Us, app_in_gga(Xs, Vs, Us))
U4_ggga(Xs, Ys, Zs, Us, app_out_gga(Xs, Vs, Us)) → app3_b_out_ggga(Xs, Ys, Zs, Us)

The argument filtering Pi contains the following mapping:
app3_b_in_ggga(x1, x2, x3, x4)  =  app3_b_in_ggga(x1, x2, x3)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x2, x3, x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
[]  =  []
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x3, x5)
U4_ggga(x1, x2, x3, x4, x5)  =  U4_ggga(x1, x2, x3, x5)
app3_b_out_ggga(x1, x2, x3, x4)  =  app3_b_out_ggga(x1, x2, x3, x4)

(3) 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:

APP3_B_IN_GGGA(Xs, Ys, Zs, Us) → U3_GGGA(Xs, Ys, Zs, Us, app_in_gga(Ys, Zs, Vs))
APP3_B_IN_GGGA(Xs, Ys, Zs, Us) → APP_IN_GGA(Ys, Zs, Vs)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U5_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)
U3_GGGA(Xs, Ys, Zs, Us, app_out_gga(Ys, Zs, Vs)) → U4_GGGA(Xs, Ys, Zs, Us, app_in_gga(Xs, Vs, Us))
U3_GGGA(Xs, Ys, Zs, Us, app_out_gga(Ys, Zs, Vs)) → APP_IN_GGA(Xs, Vs, Us)

The TRS R consists of the following rules:

app3_b_in_ggga(Xs, Ys, Zs, Us) → U3_ggga(Xs, Ys, Zs, Us, app_in_gga(Ys, Zs, Vs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U5_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U5_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_ggga(Xs, Ys, Zs, Us, app_out_gga(Ys, Zs, Vs)) → U4_ggga(Xs, Ys, Zs, Us, app_in_gga(Xs, Vs, Us))
U4_ggga(Xs, Ys, Zs, Us, app_out_gga(Xs, Vs, Us)) → app3_b_out_ggga(Xs, Ys, Zs, Us)

The argument filtering Pi contains the following mapping:
app3_b_in_ggga(x1, x2, x3, x4)  =  app3_b_in_ggga(x1, x2, x3)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x2, x3, x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
[]  =  []
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x3, x5)
U4_ggga(x1, x2, x3, x4, x5)  =  U4_ggga(x1, x2, x3, x5)
app3_b_out_ggga(x1, x2, x3, x4)  =  app3_b_out_ggga(x1, x2, x3, x4)
APP3_B_IN_GGGA(x1, x2, x3, x4)  =  APP3_B_IN_GGGA(x1, x2, x3)
U3_GGGA(x1, x2, x3, x4, x5)  =  U3_GGGA(x1, x2, x3, x5)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x2, x3, x5)
U4_GGGA(x1, x2, x3, x4, x5)  =  U4_GGGA(x1, x2, x3, x5)

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

(4) Obligation:

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

APP3_B_IN_GGGA(Xs, Ys, Zs, Us) → U3_GGGA(Xs, Ys, Zs, Us, app_in_gga(Ys, Zs, Vs))
APP3_B_IN_GGGA(Xs, Ys, Zs, Us) → APP_IN_GGA(Ys, Zs, Vs)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U5_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)
U3_GGGA(Xs, Ys, Zs, Us, app_out_gga(Ys, Zs, Vs)) → U4_GGGA(Xs, Ys, Zs, Us, app_in_gga(Xs, Vs, Us))
U3_GGGA(Xs, Ys, Zs, Us, app_out_gga(Ys, Zs, Vs)) → APP_IN_GGA(Xs, Vs, Us)

The TRS R consists of the following rules:

app3_b_in_ggga(Xs, Ys, Zs, Us) → U3_ggga(Xs, Ys, Zs, Us, app_in_gga(Ys, Zs, Vs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U5_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U5_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_ggga(Xs, Ys, Zs, Us, app_out_gga(Ys, Zs, Vs)) → U4_ggga(Xs, Ys, Zs, Us, app_in_gga(Xs, Vs, Us))
U4_ggga(Xs, Ys, Zs, Us, app_out_gga(Xs, Vs, Us)) → app3_b_out_ggga(Xs, Ys, Zs, Us)

The argument filtering Pi contains the following mapping:
app3_b_in_ggga(x1, x2, x3, x4)  =  app3_b_in_ggga(x1, x2, x3)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x2, x3, x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
[]  =  []
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x3, x5)
U4_ggga(x1, x2, x3, x4, x5)  =  U4_ggga(x1, x2, x3, x5)
app3_b_out_ggga(x1, x2, x3, x4)  =  app3_b_out_ggga(x1, x2, x3, x4)
APP3_B_IN_GGGA(x1, x2, x3, x4)  =  APP3_B_IN_GGGA(x1, x2, x3)
U3_GGGA(x1, x2, x3, x4, x5)  =  U3_GGGA(x1, x2, x3, x5)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x2, x3, x5)
U4_GGGA(x1, x2, x3, x4, x5)  =  U4_GGGA(x1, x2, x3, x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 5 less nodes.

(6) Obligation:

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

APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

app3_b_in_ggga(Xs, Ys, Zs, Us) → U3_ggga(Xs, Ys, Zs, Us, app_in_gga(Ys, Zs, Vs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U5_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U5_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_ggga(Xs, Ys, Zs, Us, app_out_gga(Ys, Zs, Vs)) → U4_ggga(Xs, Ys, Zs, Us, app_in_gga(Xs, Vs, Us))
U4_ggga(Xs, Ys, Zs, Us, app_out_gga(Xs, Vs, Us)) → app3_b_out_ggga(Xs, Ys, Zs, Us)

The argument filtering Pi contains the following mapping:
app3_b_in_ggga(x1, x2, x3, x4)  =  app3_b_in_ggga(x1, x2, x3)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x1, x2, x3, x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
[]  =  []
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x3, x5)
U4_ggga(x1, x2, x3, x4, x5)  =  U4_ggga(x1, x2, x3, x5)
app3_b_out_ggga(x1, x2, x3, x4)  =  app3_b_out_ggga(x1, x2, x3, x4)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)

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

(7) UsableRulesProof (EQUIVALENT transformation)

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

(8) Obligation:

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

APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)

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

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

(9) PiDPToQDPProof (SOUND transformation)

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

(10) Obligation:

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

APP_IN_GGA(.(X, Xs), Ys) → APP_IN_GGA(Xs, Ys)

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

(11) 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:

• APP_IN_GGA(.(X, Xs), Ys) → APP_IN_GGA(Xs, Ys)
  The graph contains the following edges 1 > 1, 2 >= 2

(12) YES