Left Termination of the query pattern front_in_2(g, a) 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 PiDPToQDPProof (⇐)
        ↳16 QDP
        ↳17 QDPSizeChangeProof (⇔)
        ↳18 TRUE
  ↳19 PrologToPiTRSProof (⇐)
    ↳20 PiTRS
    ↳21 DependencyPairsProof (⇔)
    ↳22 PiDP
    ↳23 DependencyGraphProof (⇔)
    ↳24 AND
      ↳25 PiDP
        ↳26 UsableRulesProof (⇔)
        ↳27 PiDP
        ↳28 PiDPToQDPProof (⇐)
        ↳29 QDP
        ↳30 QDPSizeChangeProof (⇔)
        ↳31 TRUE
      ↳32 PiDP

(0) Obligation:

Clauses:

front(void, []).
front(tree(X, void, void), .(X, [])).
front(tree(X1, L, R), Xs) :- ','(front(L, Ls), ','(front(R, Rs), app(Ls, Rs, Xs))).
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).

Queries:

front(g,a).

(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:
front_in: (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:

front_in_ga(void, []) → front_out_ga(void, [])
front_in_ga(tree(X, void, void), .(X, [])) → front_out_ga(tree(X, void, void), .(X, []))
front_in_ga(tree(X1, L, R), Xs) → U1_ga(X1, L, R, Xs, front_in_ga(L, Ls))
U1_ga(X1, L, R, Xs, front_out_ga(L, Ls)) → U2_ga(X1, L, R, Xs, Ls, front_in_ga(R, Rs))
U2_ga(X1, L, R, Xs, Ls, front_out_ga(R, Rs)) → U3_ga(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_ga(X1, L, R, Xs, app_out_gga(Ls, Rs, Xs)) → front_out_ga(tree(X1, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_in_ga(x1, x2)  =  front_in_ga(x1)
void  =  void
front_out_ga(x1, x2)  =  front_out_ga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x3, x5)
U2_ga(x1, x2, x3, x4, x5, x6)  =  U2_ga(x5, x6)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
[]  =  []
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

front_in_ga(void, []) → front_out_ga(void, [])
front_in_ga(tree(X, void, void), .(X, [])) → front_out_ga(tree(X, void, void), .(X, []))
front_in_ga(tree(X1, L, R), Xs) → U1_ga(X1, L, R, Xs, front_in_ga(L, Ls))
U1_ga(X1, L, R, Xs, front_out_ga(L, Ls)) → U2_ga(X1, L, R, Xs, Ls, front_in_ga(R, Rs))
U2_ga(X1, L, R, Xs, Ls, front_out_ga(R, Rs)) → U3_ga(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_ga(X1, L, R, Xs, app_out_gga(Ls, Rs, Xs)) → front_out_ga(tree(X1, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_in_ga(x1, x2)  =  front_in_ga(x1)
void  =  void
front_out_ga(x1, x2)  =  front_out_ga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x3, x5)
U2_ga(x1, x2, x3, x4, x5, x6)  =  U2_ga(x5, x6)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
[]  =  []
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x5)

(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:

FRONT_IN_GA(tree(X1, L, R), Xs) → U1_GA(X1, L, R, Xs, front_in_ga(L, Ls))
FRONT_IN_GA(tree(X1, L, R), Xs) → FRONT_IN_GA(L, Ls)
U1_GA(X1, L, R, Xs, front_out_ga(L, Ls)) → U2_GA(X1, L, R, Xs, Ls, front_in_ga(R, Rs))
U1_GA(X1, L, R, Xs, front_out_ga(L, Ls)) → FRONT_IN_GA(R, Rs)
U2_GA(X1, L, R, Xs, Ls, front_out_ga(R, Rs)) → U3_GA(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs))
U2_GA(X1, L, R, Xs, Ls, front_out_ga(R, Rs)) → APP_IN_GGA(Ls, Rs, Xs)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U4_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)

The TRS R consists of the following rules:

front_in_ga(void, []) → front_out_ga(void, [])
front_in_ga(tree(X, void, void), .(X, [])) → front_out_ga(tree(X, void, void), .(X, []))
front_in_ga(tree(X1, L, R), Xs) → U1_ga(X1, L, R, Xs, front_in_ga(L, Ls))
U1_ga(X1, L, R, Xs, front_out_ga(L, Ls)) → U2_ga(X1, L, R, Xs, Ls, front_in_ga(R, Rs))
U2_ga(X1, L, R, Xs, Ls, front_out_ga(R, Rs)) → U3_ga(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_ga(X1, L, R, Xs, app_out_gga(Ls, Rs, Xs)) → front_out_ga(tree(X1, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_in_ga(x1, x2)  =  front_in_ga(x1)
void  =  void
front_out_ga(x1, x2)  =  front_out_ga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x3, x5)
U2_ga(x1, x2, x3, x4, x5, x6)  =  U2_ga(x5, x6)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
[]  =  []
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x5)
FRONT_IN_GA(x1, x2)  =  FRONT_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x3, x5)
U2_GA(x1, x2, x3, x4, x5, x6)  =  U2_GA(x5, x6)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x5)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x5)

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

(4) Obligation:

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

FRONT_IN_GA(tree(X1, L, R), Xs) → U1_GA(X1, L, R, Xs, front_in_ga(L, Ls))
FRONT_IN_GA(tree(X1, L, R), Xs) → FRONT_IN_GA(L, Ls)
U1_GA(X1, L, R, Xs, front_out_ga(L, Ls)) → U2_GA(X1, L, R, Xs, Ls, front_in_ga(R, Rs))
U1_GA(X1, L, R, Xs, front_out_ga(L, Ls)) → FRONT_IN_GA(R, Rs)
U2_GA(X1, L, R, Xs, Ls, front_out_ga(R, Rs)) → U3_GA(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs))
U2_GA(X1, L, R, Xs, Ls, front_out_ga(R, Rs)) → APP_IN_GGA(Ls, Rs, Xs)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U4_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)

The TRS R consists of the following rules:

front_in_ga(void, []) → front_out_ga(void, [])
front_in_ga(tree(X, void, void), .(X, [])) → front_out_ga(tree(X, void, void), .(X, []))
front_in_ga(tree(X1, L, R), Xs) → U1_ga(X1, L, R, Xs, front_in_ga(L, Ls))
U1_ga(X1, L, R, Xs, front_out_ga(L, Ls)) → U2_ga(X1, L, R, Xs, Ls, front_in_ga(R, Rs))
U2_ga(X1, L, R, Xs, Ls, front_out_ga(R, Rs)) → U3_ga(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_ga(X1, L, R, Xs, app_out_gga(Ls, Rs, Xs)) → front_out_ga(tree(X1, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_in_ga(x1, x2)  =  front_in_ga(x1)
void  =  void
front_out_ga(x1, x2)  =  front_out_ga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x3, x5)
U2_ga(x1, x2, x3, x4, x5, x6)  =  U2_ga(x5, x6)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
[]  =  []
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x5)
FRONT_IN_GA(x1, x2)  =  FRONT_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x3, x5)
U2_GA(x1, x2, x3, x4, x5, x6)  =  U2_GA(x5, x6)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x5)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 4 less nodes.

(6) Complex Obligation (AND)

(7) 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:

front_in_ga(void, []) → front_out_ga(void, [])
front_in_ga(tree(X, void, void), .(X, [])) → front_out_ga(tree(X, void, void), .(X, []))
front_in_ga(tree(X1, L, R), Xs) → U1_ga(X1, L, R, Xs, front_in_ga(L, Ls))
U1_ga(X1, L, R, Xs, front_out_ga(L, Ls)) → U2_ga(X1, L, R, Xs, Ls, front_in_ga(R, Rs))
U2_ga(X1, L, R, Xs, Ls, front_out_ga(R, Rs)) → U3_ga(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_ga(X1, L, R, Xs, app_out_gga(Ls, Rs, Xs)) → front_out_ga(tree(X1, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_in_ga(x1, x2)  =  front_in_ga(x1)
void  =  void
front_out_ga(x1, x2)  =  front_out_ga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x3, x5)
U2_ga(x1, x2, x3, x4, x5, x6)  =  U2_ga(x5, x6)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
[]  =  []
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x5)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)

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_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

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

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_GGA(.(X, Xs), Ys) → APP_IN_GGA(Xs, Ys)
    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_GA(X1, L, R, Xs, front_out_ga(L, Ls)) → FRONT_IN_GA(R, Rs)
FRONT_IN_GA(tree(X1, L, R), Xs) → U1_GA(X1, L, R, Xs, front_in_ga(L, Ls))
FRONT_IN_GA(tree(X1, L, R), Xs) → FRONT_IN_GA(L, Ls)

The TRS R consists of the following rules:

front_in_ga(void, []) → front_out_ga(void, [])
front_in_ga(tree(X, void, void), .(X, [])) → front_out_ga(tree(X, void, void), .(X, []))
front_in_ga(tree(X1, L, R), Xs) → U1_ga(X1, L, R, Xs, front_in_ga(L, Ls))
U1_ga(X1, L, R, Xs, front_out_ga(L, Ls)) → U2_ga(X1, L, R, Xs, Ls, front_in_ga(R, Rs))
U2_ga(X1, L, R, Xs, Ls, front_out_ga(R, Rs)) → U3_ga(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_ga(X1, L, R, Xs, app_out_gga(Ls, Rs, Xs)) → front_out_ga(tree(X1, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_in_ga(x1, x2)  =  front_in_ga(x1)
void  =  void
front_out_ga(x1, x2)  =  front_out_ga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x3, x5)
U2_ga(x1, x2, x3, x4, x5, x6)  =  U2_ga(x5, x6)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
[]  =  []
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x5)
FRONT_IN_GA(x1, x2)  =  FRONT_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x3, x5)

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

(15) PiDPToQDPProof (SOUND transformation)

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

(16) Obligation:

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

U1_GA(R, front_out_ga(Ls)) → FRONT_IN_GA(R)
FRONT_IN_GA(tree(X1, L, R)) → U1_GA(R, front_in_ga(L))
FRONT_IN_GA(tree(X1, L, R)) → FRONT_IN_GA(L)

The TRS R consists of the following rules:

front_in_ga(void) → front_out_ga([])
front_in_ga(tree(X, void, void)) → front_out_ga(.(X, []))
front_in_ga(tree(X1, L, R)) → U1_ga(R, front_in_ga(L))
U1_ga(R, front_out_ga(Ls)) → U2_ga(Ls, front_in_ga(R))
U2_ga(Ls, front_out_ga(Rs)) → U3_ga(app_in_gga(Ls, Rs))
app_in_gga([], X) → app_out_gga(X)
app_in_gga(.(X, Xs), Ys) → U4_gga(X, app_in_gga(Xs, Ys))
U4_gga(X, app_out_gga(Zs)) → app_out_gga(.(X, Zs))
U3_ga(app_out_gga(Xs)) → front_out_ga(Xs)

The set Q consists of the following terms:

front_in_ga(x0)
U1_ga(x0, x1)
U2_ga(x0, x1)
app_in_gga(x0, x1)
U4_gga(x0, x1)
U3_ga(x0)

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

(17) 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:

  • FRONT_IN_GA(tree(X1, L, R)) → U1_GA(R, front_in_ga(L))
    The graph contains the following edges 1 > 1

  • FRONT_IN_GA(tree(X1, L, R)) → FRONT_IN_GA(L)
    The graph contains the following edges 1 > 1

  • U1_GA(R, front_out_ga(Ls)) → FRONT_IN_GA(R)
    The graph contains the following edges 1 >= 1

(18) TRUE

(19) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
front_in: (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:

front_in_ga(void, []) → front_out_ga(void, [])
front_in_ga(tree(X, void, void), .(X, [])) → front_out_ga(tree(X, void, void), .(X, []))
front_in_ga(tree(X1, L, R), Xs) → U1_ga(X1, L, R, Xs, front_in_ga(L, Ls))
U1_ga(X1, L, R, Xs, front_out_ga(L, Ls)) → U2_ga(X1, L, R, Xs, Ls, front_in_ga(R, Rs))
U2_ga(X1, L, R, Xs, Ls, front_out_ga(R, Rs)) → U3_ga(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_ga(X1, L, R, Xs, app_out_gga(Ls, Rs, Xs)) → front_out_ga(tree(X1, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_in_ga(x1, x2)  =  front_in_ga(x1)
void  =  void
front_out_ga(x1, x2)  =  front_out_ga(x1, x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
U2_ga(x1, x2, x3, x4, x5, x6)  =  U2_ga(x1, x2, x3, x5, x6)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(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)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(20) Obligation:

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

front_in_ga(void, []) → front_out_ga(void, [])
front_in_ga(tree(X, void, void), .(X, [])) → front_out_ga(tree(X, void, void), .(X, []))
front_in_ga(tree(X1, L, R), Xs) → U1_ga(X1, L, R, Xs, front_in_ga(L, Ls))
U1_ga(X1, L, R, Xs, front_out_ga(L, Ls)) → U2_ga(X1, L, R, Xs, Ls, front_in_ga(R, Rs))
U2_ga(X1, L, R, Xs, Ls, front_out_ga(R, Rs)) → U3_ga(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_ga(X1, L, R, Xs, app_out_gga(Ls, Rs, Xs)) → front_out_ga(tree(X1, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_in_ga(x1, x2)  =  front_in_ga(x1)
void  =  void
front_out_ga(x1, x2)  =  front_out_ga(x1, x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
U2_ga(x1, x2, x3, x4, x5, x6)  =  U2_ga(x1, x2, x3, x5, x6)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(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)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)

(21) 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:

FRONT_IN_GA(tree(X1, L, R), Xs) → U1_GA(X1, L, R, Xs, front_in_ga(L, Ls))
FRONT_IN_GA(tree(X1, L, R), Xs) → FRONT_IN_GA(L, Ls)
U1_GA(X1, L, R, Xs, front_out_ga(L, Ls)) → U2_GA(X1, L, R, Xs, Ls, front_in_ga(R, Rs))
U1_GA(X1, L, R, Xs, front_out_ga(L, Ls)) → FRONT_IN_GA(R, Rs)
U2_GA(X1, L, R, Xs, Ls, front_out_ga(R, Rs)) → U3_GA(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs))
U2_GA(X1, L, R, Xs, Ls, front_out_ga(R, Rs)) → APP_IN_GGA(Ls, Rs, Xs)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U4_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)

The TRS R consists of the following rules:

front_in_ga(void, []) → front_out_ga(void, [])
front_in_ga(tree(X, void, void), .(X, [])) → front_out_ga(tree(X, void, void), .(X, []))
front_in_ga(tree(X1, L, R), Xs) → U1_ga(X1, L, R, Xs, front_in_ga(L, Ls))
U1_ga(X1, L, R, Xs, front_out_ga(L, Ls)) → U2_ga(X1, L, R, Xs, Ls, front_in_ga(R, Rs))
U2_ga(X1, L, R, Xs, Ls, front_out_ga(R, Rs)) → U3_ga(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_ga(X1, L, R, Xs, app_out_gga(Ls, Rs, Xs)) → front_out_ga(tree(X1, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_in_ga(x1, x2)  =  front_in_ga(x1)
void  =  void
front_out_ga(x1, x2)  =  front_out_ga(x1, x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
U2_ga(x1, x2, x3, x4, x5, x6)  =  U2_ga(x1, x2, x3, x5, x6)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(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)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
FRONT_IN_GA(x1, x2)  =  FRONT_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x3, x5)
U2_GA(x1, x2, x3, x4, x5, x6)  =  U2_GA(x1, x2, x3, x5, x6)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x3, x5)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x2, x3, x5)

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

(22) Obligation:

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

FRONT_IN_GA(tree(X1, L, R), Xs) → U1_GA(X1, L, R, Xs, front_in_ga(L, Ls))
FRONT_IN_GA(tree(X1, L, R), Xs) → FRONT_IN_GA(L, Ls)
U1_GA(X1, L, R, Xs, front_out_ga(L, Ls)) → U2_GA(X1, L, R, Xs, Ls, front_in_ga(R, Rs))
U1_GA(X1, L, R, Xs, front_out_ga(L, Ls)) → FRONT_IN_GA(R, Rs)
U2_GA(X1, L, R, Xs, Ls, front_out_ga(R, Rs)) → U3_GA(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs))
U2_GA(X1, L, R, Xs, Ls, front_out_ga(R, Rs)) → APP_IN_GGA(Ls, Rs, Xs)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U4_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)

The TRS R consists of the following rules:

front_in_ga(void, []) → front_out_ga(void, [])
front_in_ga(tree(X, void, void), .(X, [])) → front_out_ga(tree(X, void, void), .(X, []))
front_in_ga(tree(X1, L, R), Xs) → U1_ga(X1, L, R, Xs, front_in_ga(L, Ls))
U1_ga(X1, L, R, Xs, front_out_ga(L, Ls)) → U2_ga(X1, L, R, Xs, Ls, front_in_ga(R, Rs))
U2_ga(X1, L, R, Xs, Ls, front_out_ga(R, Rs)) → U3_ga(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_ga(X1, L, R, Xs, app_out_gga(Ls, Rs, Xs)) → front_out_ga(tree(X1, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_in_ga(x1, x2)  =  front_in_ga(x1)
void  =  void
front_out_ga(x1, x2)  =  front_out_ga(x1, x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
U2_ga(x1, x2, x3, x4, x5, x6)  =  U2_ga(x1, x2, x3, x5, x6)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(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)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
FRONT_IN_GA(x1, x2)  =  FRONT_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x3, x5)
U2_GA(x1, x2, x3, x4, x5, x6)  =  U2_GA(x1, x2, x3, x5, x6)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x3, x5)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x2, x3, x5)

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

(23) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 4 less nodes.

(24) Complex Obligation (AND)

(25) 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:

front_in_ga(void, []) → front_out_ga(void, [])
front_in_ga(tree(X, void, void), .(X, [])) → front_out_ga(tree(X, void, void), .(X, []))
front_in_ga(tree(X1, L, R), Xs) → U1_ga(X1, L, R, Xs, front_in_ga(L, Ls))
U1_ga(X1, L, R, Xs, front_out_ga(L, Ls)) → U2_ga(X1, L, R, Xs, Ls, front_in_ga(R, Rs))
U2_ga(X1, L, R, Xs, Ls, front_out_ga(R, Rs)) → U3_ga(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_ga(X1, L, R, Xs, app_out_gga(Ls, Rs, Xs)) → front_out_ga(tree(X1, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_in_ga(x1, x2)  =  front_in_ga(x1)
void  =  void
front_out_ga(x1, x2)  =  front_out_ga(x1, x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
U2_ga(x1, x2, x3, x4, x5, x6)  =  U2_ga(x1, x2, x3, x5, x6)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(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)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)

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

(26) UsableRulesProof (EQUIVALENT transformation)

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

(27) 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

(28) PiDPToQDPProof (SOUND transformation)

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

(29) 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.

(30) 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

(31) TRUE

(32) Obligation:

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

U1_GA(X1, L, R, Xs, front_out_ga(L, Ls)) → FRONT_IN_GA(R, Rs)
FRONT_IN_GA(tree(X1, L, R), Xs) → U1_GA(X1, L, R, Xs, front_in_ga(L, Ls))
FRONT_IN_GA(tree(X1, L, R), Xs) → FRONT_IN_GA(L, Ls)

The TRS R consists of the following rules:

front_in_ga(void, []) → front_out_ga(void, [])
front_in_ga(tree(X, void, void), .(X, [])) → front_out_ga(tree(X, void, void), .(X, []))
front_in_ga(tree(X1, L, R), Xs) → U1_ga(X1, L, R, Xs, front_in_ga(L, Ls))
U1_ga(X1, L, R, Xs, front_out_ga(L, Ls)) → U2_ga(X1, L, R, Xs, Ls, front_in_ga(R, Rs))
U2_ga(X1, L, R, Xs, Ls, front_out_ga(R, Rs)) → U3_ga(X1, L, R, Xs, app_in_gga(Ls, Rs, Xs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_ga(X1, L, R, Xs, app_out_gga(Ls, Rs, Xs)) → front_out_ga(tree(X1, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_in_ga(x1, x2)  =  front_in_ga(x1)
void  =  void
front_out_ga(x1, x2)  =  front_out_ga(x1, x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
U2_ga(x1, x2, x3, x4, x5, x6)  =  U2_ga(x1, x2, x3, x5, x6)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(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)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
FRONT_IN_GA(x1, x2)  =  FRONT_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x3, x5)

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