Left Termination of the query pattern star_in_2(g, g) w.r.t. the given Prolog program could successfully be proven:

0 Prolog
  ↳1 PrologToPiTRSProof (⇐)
    ↳2 PiTRS
    ↳3 DependencyPairsProof (⇔)
    ↳4 PiDP
    ↳5 DependencyGraphProof (⇔)
    ↳6 AND
      ↳7 PiDP
        ↳8 UsableRulesProof (⇔)
        ↳9 PiDP
        ↳10 PiDPToQDPProof (⇐)
        ↳11 QDP
        ↳12 QDPSizeChangeProof (⇔)
        ↳13 TRUE
      ↳14 PiDP
        ↳15 UsableRulesProof (⇔)
        ↳16 PiDP
        ↳17 PiDPToQDPProof (⇐)
        ↳18 QDP
        ↳19 QDPSizeChangeProof (⇔)
        ↳20 TRUE
  ↳21 PrologToPiTRSProof (⇐)
    ↳22 PiTRS
    ↳23 DependencyPairsProof (⇔)
    ↳24 PiDP
    ↳25 DependencyGraphProof (⇔)
    ↳26 AND
      ↳27 PiDP
        ↳28 UsableRulesProof (⇔)
        ↳29 PiDP
        ↳30 PiDPToQDPProof (⇐)
        ↳31 QDP
        ↳32 QDPSizeChangeProof (⇔)
        ↳33 TRUE
      ↳34 PiDP
        ↳35 UsableRulesProof (⇔)
        ↳36 PiDP
        ↳37 PiDPToQDPProof (⇐)
        ↳38 QDP

(0) Obligation:

Clauses:

star(X1, []).
star(.(X, U), .(X, W)) :- ','(app(U, V, W), star(.(X, U), W)).
app([], L, L).
app(.(X, L), M, .(X, N)) :- app(L, M, N).

Queries:

star(g,g).

(1) PrologToPiTRSProof (SOUND transformation)

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

star_in_gg(X1, []) → star_out_gg(X1, [])
star_in_gg(.(X, U), .(X, W)) → U1_gg(X, U, W, app_in_gag(U, V, W))
app_in_gag([], L, L) → app_out_gag([], L, L)
app_in_gag(.(X, L), M, .(X, N)) → U3_gag(X, L, M, N, app_in_gag(L, M, N))
U3_gag(X, L, M, N, app_out_gag(L, M, N)) → app_out_gag(.(X, L), M, .(X, N))
U1_gg(X, U, W, app_out_gag(U, V, W)) → U2_gg(X, U, W, star_in_gg(.(X, U), W))
U2_gg(X, U, W, star_out_gg(.(X, U), W)) → star_out_gg(.(X, U), .(X, W))

The argument filtering Pi contains the following mapping:
star_in_gg(x1, x2)  =  star_in_gg(x1, x2)
[]  =  []
star_out_gg(x1, x2)  =  star_out_gg
.(x1, x2)  =  .(x1, x2)
U1_gg(x1, x2, x3, x4)  =  U1_gg(x1, x2, x3, x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
app_out_gag(x1, x2, x3)  =  app_out_gag(x2)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U2_gg(x1, x2, x3, x4)  =  U2_gg(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:

star_in_gg(X1, []) → star_out_gg(X1, [])
star_in_gg(.(X, U), .(X, W)) → U1_gg(X, U, W, app_in_gag(U, V, W))
app_in_gag([], L, L) → app_out_gag([], L, L)
app_in_gag(.(X, L), M, .(X, N)) → U3_gag(X, L, M, N, app_in_gag(L, M, N))
U3_gag(X, L, M, N, app_out_gag(L, M, N)) → app_out_gag(.(X, L), M, .(X, N))
U1_gg(X, U, W, app_out_gag(U, V, W)) → U2_gg(X, U, W, star_in_gg(.(X, U), W))
U2_gg(X, U, W, star_out_gg(.(X, U), W)) → star_out_gg(.(X, U), .(X, W))

The argument filtering Pi contains the following mapping:
star_in_gg(x1, x2)  =  star_in_gg(x1, x2)
[]  =  []
star_out_gg(x1, x2)  =  star_out_gg
.(x1, x2)  =  .(x1, x2)
U1_gg(x1, x2, x3, x4)  =  U1_gg(x1, x2, x3, x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
app_out_gag(x1, x2, x3)  =  app_out_gag(x2)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U2_gg(x1, x2, x3, x4)  =  U2_gg(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:

STAR_IN_GG(.(X, U), .(X, W)) → U1_GG(X, U, W, app_in_gag(U, V, W))
STAR_IN_GG(.(X, U), .(X, W)) → APP_IN_GAG(U, V, W)
APP_IN_GAG(.(X, L), M, .(X, N)) → U3_GAG(X, L, M, N, app_in_gag(L, M, N))
APP_IN_GAG(.(X, L), M, .(X, N)) → APP_IN_GAG(L, M, N)
U1_GG(X, U, W, app_out_gag(U, V, W)) → U2_GG(X, U, W, star_in_gg(.(X, U), W))
U1_GG(X, U, W, app_out_gag(U, V, W)) → STAR_IN_GG(.(X, U), W)

The TRS R consists of the following rules:

star_in_gg(X1, []) → star_out_gg(X1, [])
star_in_gg(.(X, U), .(X, W)) → U1_gg(X, U, W, app_in_gag(U, V, W))
app_in_gag([], L, L) → app_out_gag([], L, L)
app_in_gag(.(X, L), M, .(X, N)) → U3_gag(X, L, M, N, app_in_gag(L, M, N))
U3_gag(X, L, M, N, app_out_gag(L, M, N)) → app_out_gag(.(X, L), M, .(X, N))
U1_gg(X, U, W, app_out_gag(U, V, W)) → U2_gg(X, U, W, star_in_gg(.(X, U), W))
U2_gg(X, U, W, star_out_gg(.(X, U), W)) → star_out_gg(.(X, U), .(X, W))

The argument filtering Pi contains the following mapping:
star_in_gg(x1, x2)  =  star_in_gg(x1, x2)
[]  =  []
star_out_gg(x1, x2)  =  star_out_gg
.(x1, x2)  =  .(x1, x2)
U1_gg(x1, x2, x3, x4)  =  U1_gg(x1, x2, x3, x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
app_out_gag(x1, x2, x3)  =  app_out_gag(x2)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
STAR_IN_GG(x1, x2)  =  STAR_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4)  =  U1_GG(x1, x2, x3, x4)
APP_IN_GAG(x1, x2, x3)  =  APP_IN_GAG(x1, x3)
U3_GAG(x1, x2, x3, x4, x5)  =  U3_GAG(x5)
U2_GG(x1, x2, x3, x4)  =  U2_GG(x4)

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

(4) Obligation:

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

STAR_IN_GG(.(X, U), .(X, W)) → U1_GG(X, U, W, app_in_gag(U, V, W))
STAR_IN_GG(.(X, U), .(X, W)) → APP_IN_GAG(U, V, W)
APP_IN_GAG(.(X, L), M, .(X, N)) → U3_GAG(X, L, M, N, app_in_gag(L, M, N))
APP_IN_GAG(.(X, L), M, .(X, N)) → APP_IN_GAG(L, M, N)
U1_GG(X, U, W, app_out_gag(U, V, W)) → U2_GG(X, U, W, star_in_gg(.(X, U), W))
U1_GG(X, U, W, app_out_gag(U, V, W)) → STAR_IN_GG(.(X, U), W)

The TRS R consists of the following rules:

star_in_gg(X1, []) → star_out_gg(X1, [])
star_in_gg(.(X, U), .(X, W)) → U1_gg(X, U, W, app_in_gag(U, V, W))
app_in_gag([], L, L) → app_out_gag([], L, L)
app_in_gag(.(X, L), M, .(X, N)) → U3_gag(X, L, M, N, app_in_gag(L, M, N))
U3_gag(X, L, M, N, app_out_gag(L, M, N)) → app_out_gag(.(X, L), M, .(X, N))
U1_gg(X, U, W, app_out_gag(U, V, W)) → U2_gg(X, U, W, star_in_gg(.(X, U), W))
U2_gg(X, U, W, star_out_gg(.(X, U), W)) → star_out_gg(.(X, U), .(X, W))

The argument filtering Pi contains the following mapping:
star_in_gg(x1, x2)  =  star_in_gg(x1, x2)
[]  =  []
star_out_gg(x1, x2)  =  star_out_gg
.(x1, x2)  =  .(x1, x2)
U1_gg(x1, x2, x3, x4)  =  U1_gg(x1, x2, x3, x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
app_out_gag(x1, x2, x3)  =  app_out_gag(x2)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
STAR_IN_GG(x1, x2)  =  STAR_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4)  =  U1_GG(x1, x2, x3, x4)
APP_IN_GAG(x1, x2, x3)  =  APP_IN_GAG(x1, x3)
U3_GAG(x1, x2, x3, x4, x5)  =  U3_GAG(x5)
U2_GG(x1, x2, x3, x4)  =  U2_GG(x4)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 3 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

APP_IN_GAG(.(X, L), M, .(X, N)) → APP_IN_GAG(L, M, N)

The TRS R consists of the following rules:

star_in_gg(X1, []) → star_out_gg(X1, [])
star_in_gg(.(X, U), .(X, W)) → U1_gg(X, U, W, app_in_gag(U, V, W))
app_in_gag([], L, L) → app_out_gag([], L, L)
app_in_gag(.(X, L), M, .(X, N)) → U3_gag(X, L, M, N, app_in_gag(L, M, N))
U3_gag(X, L, M, N, app_out_gag(L, M, N)) → app_out_gag(.(X, L), M, .(X, N))
U1_gg(X, U, W, app_out_gag(U, V, W)) → U2_gg(X, U, W, star_in_gg(.(X, U), W))
U2_gg(X, U, W, star_out_gg(.(X, U), W)) → star_out_gg(.(X, U), .(X, W))

The argument filtering Pi contains the following mapping:
star_in_gg(x1, x2)  =  star_in_gg(x1, x2)
[]  =  []
star_out_gg(x1, x2)  =  star_out_gg
.(x1, x2)  =  .(x1, x2)
U1_gg(x1, x2, x3, x4)  =  U1_gg(x1, x2, x3, x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
app_out_gag(x1, x2, x3)  =  app_out_gag(x2)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
APP_IN_GAG(x1, x2, x3)  =  APP_IN_GAG(x1, x3)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

APP_IN_GAG(.(X, L), M, .(X, N)) → APP_IN_GAG(L, M, N)

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

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

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

APP_IN_GAG(.(X, L), .(X, N)) → APP_IN_GAG(L, N)

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

(12) 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_GAG(.(X, L), .(X, N)) → APP_IN_GAG(L, N)
    The graph contains the following edges 1 > 1, 2 > 2

(13) TRUE

(14) Obligation:

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

U1_GG(X, U, W, app_out_gag(U, V, W)) → STAR_IN_GG(.(X, U), W)
STAR_IN_GG(.(X, U), .(X, W)) → U1_GG(X, U, W, app_in_gag(U, V, W))

The TRS R consists of the following rules:

star_in_gg(X1, []) → star_out_gg(X1, [])
star_in_gg(.(X, U), .(X, W)) → U1_gg(X, U, W, app_in_gag(U, V, W))
app_in_gag([], L, L) → app_out_gag([], L, L)
app_in_gag(.(X, L), M, .(X, N)) → U3_gag(X, L, M, N, app_in_gag(L, M, N))
U3_gag(X, L, M, N, app_out_gag(L, M, N)) → app_out_gag(.(X, L), M, .(X, N))
U1_gg(X, U, W, app_out_gag(U, V, W)) → U2_gg(X, U, W, star_in_gg(.(X, U), W))
U2_gg(X, U, W, star_out_gg(.(X, U), W)) → star_out_gg(.(X, U), .(X, W))

The argument filtering Pi contains the following mapping:
star_in_gg(x1, x2)  =  star_in_gg(x1, x2)
[]  =  []
star_out_gg(x1, x2)  =  star_out_gg
.(x1, x2)  =  .(x1, x2)
U1_gg(x1, x2, x3, x4)  =  U1_gg(x1, x2, x3, x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
app_out_gag(x1, x2, x3)  =  app_out_gag(x2)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
STAR_IN_GG(x1, x2)  =  STAR_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4)  =  U1_GG(x1, x2, x3, x4)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

U1_GG(X, U, W, app_out_gag(U, V, W)) → STAR_IN_GG(.(X, U), W)
STAR_IN_GG(.(X, U), .(X, W)) → U1_GG(X, U, W, app_in_gag(U, V, W))

The TRS R consists of the following rules:

app_in_gag([], L, L) → app_out_gag([], L, L)
app_in_gag(.(X, L), M, .(X, N)) → U3_gag(X, L, M, N, app_in_gag(L, M, N))
U3_gag(X, L, M, N, app_out_gag(L, M, N)) → app_out_gag(.(X, L), M, .(X, N))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
app_out_gag(x1, x2, x3)  =  app_out_gag(x2)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
STAR_IN_GG(x1, x2)  =  STAR_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4)  =  U1_GG(x1, x2, x3, x4)

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

U1_GG(X, U, W, app_out_gag(V)) → STAR_IN_GG(.(X, U), W)
STAR_IN_GG(.(X, U), .(X, W)) → U1_GG(X, U, W, app_in_gag(U, W))

The TRS R consists of the following rules:

app_in_gag([], L) → app_out_gag(L)
app_in_gag(.(X, L), .(X, N)) → U3_gag(app_in_gag(L, N))
U3_gag(app_out_gag(M)) → app_out_gag(M)

The set Q consists of the following terms:

app_in_gag(x0, x1)
U3_gag(x0)

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

(19) 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:

  • STAR_IN_GG(.(X, U), .(X, W)) → U1_GG(X, U, W, app_in_gag(U, W))
    The graph contains the following edges 1 > 1, 2 > 1, 1 > 2, 2 > 3

  • U1_GG(X, U, W, app_out_gag(V)) → STAR_IN_GG(.(X, U), W)
    The graph contains the following edges 3 >= 2

(20) TRUE

(21) PrologToPiTRSProof (SOUND transformation)

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

star_in_gg(X1, []) → star_out_gg(X1, [])
star_in_gg(.(X, U), .(X, W)) → U1_gg(X, U, W, app_in_gag(U, V, W))
app_in_gag([], L, L) → app_out_gag([], L, L)
app_in_gag(.(X, L), M, .(X, N)) → U3_gag(X, L, M, N, app_in_gag(L, M, N))
U3_gag(X, L, M, N, app_out_gag(L, M, N)) → app_out_gag(.(X, L), M, .(X, N))
U1_gg(X, U, W, app_out_gag(U, V, W)) → U2_gg(X, U, W, star_in_gg(.(X, U), W))
U2_gg(X, U, W, star_out_gg(.(X, U), W)) → star_out_gg(.(X, U), .(X, W))

The argument filtering Pi contains the following mapping:
star_in_gg(x1, x2)  =  star_in_gg(x1, x2)
[]  =  []
star_out_gg(x1, x2)  =  star_out_gg(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_gg(x1, x2, x3, x4)  =  U1_gg(x1, x2, x3, x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x2, x4, x5)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x1, x2, x3, x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(22) Obligation:

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

star_in_gg(X1, []) → star_out_gg(X1, [])
star_in_gg(.(X, U), .(X, W)) → U1_gg(X, U, W, app_in_gag(U, V, W))
app_in_gag([], L, L) → app_out_gag([], L, L)
app_in_gag(.(X, L), M, .(X, N)) → U3_gag(X, L, M, N, app_in_gag(L, M, N))
U3_gag(X, L, M, N, app_out_gag(L, M, N)) → app_out_gag(.(X, L), M, .(X, N))
U1_gg(X, U, W, app_out_gag(U, V, W)) → U2_gg(X, U, W, star_in_gg(.(X, U), W))
U2_gg(X, U, W, star_out_gg(.(X, U), W)) → star_out_gg(.(X, U), .(X, W))

The argument filtering Pi contains the following mapping:
star_in_gg(x1, x2)  =  star_in_gg(x1, x2)
[]  =  []
star_out_gg(x1, x2)  =  star_out_gg(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_gg(x1, x2, x3, x4)  =  U1_gg(x1, x2, x3, x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x2, x4, x5)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x1, x2, x3, x4)

(23) 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:

STAR_IN_GG(.(X, U), .(X, W)) → U1_GG(X, U, W, app_in_gag(U, V, W))
STAR_IN_GG(.(X, U), .(X, W)) → APP_IN_GAG(U, V, W)
APP_IN_GAG(.(X, L), M, .(X, N)) → U3_GAG(X, L, M, N, app_in_gag(L, M, N))
APP_IN_GAG(.(X, L), M, .(X, N)) → APP_IN_GAG(L, M, N)
U1_GG(X, U, W, app_out_gag(U, V, W)) → U2_GG(X, U, W, star_in_gg(.(X, U), W))
U1_GG(X, U, W, app_out_gag(U, V, W)) → STAR_IN_GG(.(X, U), W)

The TRS R consists of the following rules:

star_in_gg(X1, []) → star_out_gg(X1, [])
star_in_gg(.(X, U), .(X, W)) → U1_gg(X, U, W, app_in_gag(U, V, W))
app_in_gag([], L, L) → app_out_gag([], L, L)
app_in_gag(.(X, L), M, .(X, N)) → U3_gag(X, L, M, N, app_in_gag(L, M, N))
U3_gag(X, L, M, N, app_out_gag(L, M, N)) → app_out_gag(.(X, L), M, .(X, N))
U1_gg(X, U, W, app_out_gag(U, V, W)) → U2_gg(X, U, W, star_in_gg(.(X, U), W))
U2_gg(X, U, W, star_out_gg(.(X, U), W)) → star_out_gg(.(X, U), .(X, W))

The argument filtering Pi contains the following mapping:
star_in_gg(x1, x2)  =  star_in_gg(x1, x2)
[]  =  []
star_out_gg(x1, x2)  =  star_out_gg(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_gg(x1, x2, x3, x4)  =  U1_gg(x1, x2, x3, x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x2, x4, x5)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x1, x2, x3, x4)
STAR_IN_GG(x1, x2)  =  STAR_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4)  =  U1_GG(x1, x2, x3, x4)
APP_IN_GAG(x1, x2, x3)  =  APP_IN_GAG(x1, x3)
U3_GAG(x1, x2, x3, x4, x5)  =  U3_GAG(x1, x2, x4, x5)
U2_GG(x1, x2, x3, x4)  =  U2_GG(x1, x2, x3, x4)

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

(24) Obligation:

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

STAR_IN_GG(.(X, U), .(X, W)) → U1_GG(X, U, W, app_in_gag(U, V, W))
STAR_IN_GG(.(X, U), .(X, W)) → APP_IN_GAG(U, V, W)
APP_IN_GAG(.(X, L), M, .(X, N)) → U3_GAG(X, L, M, N, app_in_gag(L, M, N))
APP_IN_GAG(.(X, L), M, .(X, N)) → APP_IN_GAG(L, M, N)
U1_GG(X, U, W, app_out_gag(U, V, W)) → U2_GG(X, U, W, star_in_gg(.(X, U), W))
U1_GG(X, U, W, app_out_gag(U, V, W)) → STAR_IN_GG(.(X, U), W)

The TRS R consists of the following rules:

star_in_gg(X1, []) → star_out_gg(X1, [])
star_in_gg(.(X, U), .(X, W)) → U1_gg(X, U, W, app_in_gag(U, V, W))
app_in_gag([], L, L) → app_out_gag([], L, L)
app_in_gag(.(X, L), M, .(X, N)) → U3_gag(X, L, M, N, app_in_gag(L, M, N))
U3_gag(X, L, M, N, app_out_gag(L, M, N)) → app_out_gag(.(X, L), M, .(X, N))
U1_gg(X, U, W, app_out_gag(U, V, W)) → U2_gg(X, U, W, star_in_gg(.(X, U), W))
U2_gg(X, U, W, star_out_gg(.(X, U), W)) → star_out_gg(.(X, U), .(X, W))

The argument filtering Pi contains the following mapping:
star_in_gg(x1, x2)  =  star_in_gg(x1, x2)
[]  =  []
star_out_gg(x1, x2)  =  star_out_gg(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_gg(x1, x2, x3, x4)  =  U1_gg(x1, x2, x3, x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x2, x4, x5)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x1, x2, x3, x4)
STAR_IN_GG(x1, x2)  =  STAR_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4)  =  U1_GG(x1, x2, x3, x4)
APP_IN_GAG(x1, x2, x3)  =  APP_IN_GAG(x1, x3)
U3_GAG(x1, x2, x3, x4, x5)  =  U3_GAG(x1, x2, x4, x5)
U2_GG(x1, x2, x3, x4)  =  U2_GG(x1, x2, x3, x4)

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

(25) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 3 less nodes.

(26) Complex Obligation (AND)

(27) Obligation:

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

APP_IN_GAG(.(X, L), M, .(X, N)) → APP_IN_GAG(L, M, N)

The TRS R consists of the following rules:

star_in_gg(X1, []) → star_out_gg(X1, [])
star_in_gg(.(X, U), .(X, W)) → U1_gg(X, U, W, app_in_gag(U, V, W))
app_in_gag([], L, L) → app_out_gag([], L, L)
app_in_gag(.(X, L), M, .(X, N)) → U3_gag(X, L, M, N, app_in_gag(L, M, N))
U3_gag(X, L, M, N, app_out_gag(L, M, N)) → app_out_gag(.(X, L), M, .(X, N))
U1_gg(X, U, W, app_out_gag(U, V, W)) → U2_gg(X, U, W, star_in_gg(.(X, U), W))
U2_gg(X, U, W, star_out_gg(.(X, U), W)) → star_out_gg(.(X, U), .(X, W))

The argument filtering Pi contains the following mapping:
star_in_gg(x1, x2)  =  star_in_gg(x1, x2)
[]  =  []
star_out_gg(x1, x2)  =  star_out_gg(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_gg(x1, x2, x3, x4)  =  U1_gg(x1, x2, x3, x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x2, x4, x5)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x1, x2, x3, x4)
APP_IN_GAG(x1, x2, x3)  =  APP_IN_GAG(x1, x3)

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

(28) UsableRulesProof (EQUIVALENT transformation)

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

(29) Obligation:

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

APP_IN_GAG(.(X, L), M, .(X, N)) → APP_IN_GAG(L, M, N)

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

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

(30) PiDPToQDPProof (SOUND transformation)

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

(31) Obligation:

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

APP_IN_GAG(.(X, L), .(X, N)) → APP_IN_GAG(L, N)

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

(32) 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_GAG(.(X, L), .(X, N)) → APP_IN_GAG(L, N)
    The graph contains the following edges 1 > 1, 2 > 2

(33) TRUE

(34) Obligation:

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

U1_GG(X, U, W, app_out_gag(U, V, W)) → STAR_IN_GG(.(X, U), W)
STAR_IN_GG(.(X, U), .(X, W)) → U1_GG(X, U, W, app_in_gag(U, V, W))

The TRS R consists of the following rules:

star_in_gg(X1, []) → star_out_gg(X1, [])
star_in_gg(.(X, U), .(X, W)) → U1_gg(X, U, W, app_in_gag(U, V, W))
app_in_gag([], L, L) → app_out_gag([], L, L)
app_in_gag(.(X, L), M, .(X, N)) → U3_gag(X, L, M, N, app_in_gag(L, M, N))
U3_gag(X, L, M, N, app_out_gag(L, M, N)) → app_out_gag(.(X, L), M, .(X, N))
U1_gg(X, U, W, app_out_gag(U, V, W)) → U2_gg(X, U, W, star_in_gg(.(X, U), W))
U2_gg(X, U, W, star_out_gg(.(X, U), W)) → star_out_gg(.(X, U), .(X, W))

The argument filtering Pi contains the following mapping:
star_in_gg(x1, x2)  =  star_in_gg(x1, x2)
[]  =  []
star_out_gg(x1, x2)  =  star_out_gg(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_gg(x1, x2, x3, x4)  =  U1_gg(x1, x2, x3, x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x2, x4, x5)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x1, x2, x3, x4)
STAR_IN_GG(x1, x2)  =  STAR_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4)  =  U1_GG(x1, x2, x3, x4)

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

(35) UsableRulesProof (EQUIVALENT transformation)

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

(36) Obligation:

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

U1_GG(X, U, W, app_out_gag(U, V, W)) → STAR_IN_GG(.(X, U), W)
STAR_IN_GG(.(X, U), .(X, W)) → U1_GG(X, U, W, app_in_gag(U, V, W))

The TRS R consists of the following rules:

app_in_gag([], L, L) → app_out_gag([], L, L)
app_in_gag(.(X, L), M, .(X, N)) → U3_gag(X, L, M, N, app_in_gag(L, M, N))
U3_gag(X, L, M, N, app_out_gag(L, M, N)) → app_out_gag(.(X, L), M, .(X, N))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x2, x4, x5)
STAR_IN_GG(x1, x2)  =  STAR_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4)  =  U1_GG(x1, x2, x3, x4)

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

(37) PiDPToQDPProof (SOUND transformation)

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

(38) Obligation:

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

U1_GG(X, U, W, app_out_gag(U, V, W)) → STAR_IN_GG(.(X, U), W)
STAR_IN_GG(.(X, U), .(X, W)) → U1_GG(X, U, W, app_in_gag(U, W))

The TRS R consists of the following rules:

app_in_gag([], L) → app_out_gag([], L, L)
app_in_gag(.(X, L), .(X, N)) → U3_gag(X, L, N, app_in_gag(L, N))
U3_gag(X, L, N, app_out_gag(L, M, N)) → app_out_gag(.(X, L), M, .(X, N))

The set Q consists of the following terms:

app_in_gag(x0, x1)
U3_gag(x0, x1, x2, x3)

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