Left Termination of the query pattern front_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:



↳ Prolog
  ↳ PrologToPiTRSProof

Clauses:

front(void, []).
front(tree(X, void, void), .(X, [])).
front(tree(X, 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).

We use the technique of [30]. 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(X, L, R), Xs) → U1_ga(X, L, R, Xs, front_in_ga(L, Ls))
U1_ga(X, L, R, Xs, front_out_ga(L, Ls)) → U2_ga(X, L, R, Xs, Ls, front_in_ga(R, Rs))
U2_ga(X, L, R, Xs, Ls, front_out_ga(R, Rs)) → U3_ga(X, 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(X, L, R, Xs, app_out_gga(Ls, Rs, Xs)) → front_out_ga(tree(X, 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



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof

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(X, L, R), Xs) → U1_ga(X, L, R, Xs, front_in_ga(L, Ls))
U1_ga(X, L, R, Xs, front_out_ga(L, Ls)) → U2_ga(X, L, R, Xs, Ls, front_in_ga(R, Rs))
U2_ga(X, L, R, Xs, Ls, front_out_ga(R, Rs)) → U3_ga(X, 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(X, L, R, Xs, app_out_gga(Ls, Rs, Xs)) → front_out_ga(tree(X, 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)


Using Dependency Pairs [1,30] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

FRONT_IN_GA(tree(X, L, R), Xs) → U1_GA(X, L, R, Xs, front_in_ga(L, Ls))
FRONT_IN_GA(tree(X, L, R), Xs) → FRONT_IN_GA(L, Ls)
U1_GA(X, L, R, Xs, front_out_ga(L, Ls)) → U2_GA(X, L, R, Xs, Ls, front_in_ga(R, Rs))
U1_GA(X, L, R, Xs, front_out_ga(L, Ls)) → FRONT_IN_GA(R, Rs)
U2_GA(X, L, R, Xs, Ls, front_out_ga(R, Rs)) → U3_GA(X, L, R, Xs, app_in_gga(Ls, Rs, Xs))
U2_GA(X, 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(X, L, R), Xs) → U1_ga(X, L, R, Xs, front_in_ga(L, Ls))
U1_ga(X, L, R, Xs, front_out_ga(L, Ls)) → U2_ga(X, L, R, Xs, Ls, front_in_ga(R, Rs))
U2_ga(X, L, R, Xs, Ls, front_out_ga(R, Rs)) → U3_ga(X, 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(X, L, R, Xs, app_out_gga(Ls, Rs, Xs)) → front_out_ga(tree(X, 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)
U2_GA(x1, x2, x3, x4, x5, x6)  =  U2_GA(x5, x6)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x5)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x3, x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x5)
FRONT_IN_GA(x1, x2)  =  FRONT_IN_GA(x1)

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

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof

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

FRONT_IN_GA(tree(X, L, R), Xs) → U1_GA(X, L, R, Xs, front_in_ga(L, Ls))
FRONT_IN_GA(tree(X, L, R), Xs) → FRONT_IN_GA(L, Ls)
U1_GA(X, L, R, Xs, front_out_ga(L, Ls)) → U2_GA(X, L, R, Xs, Ls, front_in_ga(R, Rs))
U1_GA(X, L, R, Xs, front_out_ga(L, Ls)) → FRONT_IN_GA(R, Rs)
U2_GA(X, L, R, Xs, Ls, front_out_ga(R, Rs)) → U3_GA(X, L, R, Xs, app_in_gga(Ls, Rs, Xs))
U2_GA(X, 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(X, L, R), Xs) → U1_ga(X, L, R, Xs, front_in_ga(L, Ls))
U1_ga(X, L, R, Xs, front_out_ga(L, Ls)) → U2_ga(X, L, R, Xs, Ls, front_in_ga(R, Rs))
U2_ga(X, L, R, Xs, Ls, front_out_ga(R, Rs)) → U3_ga(X, 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(X, L, R, Xs, app_out_gga(Ls, Rs, Xs)) → front_out_ga(tree(X, 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)
U2_GA(x1, x2, x3, x4, x5, x6)  =  U2_GA(x5, x6)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x5)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x3, x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x5)
FRONT_IN_GA(x1, x2)  =  FRONT_IN_GA(x1)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 2 SCCs with 4 less nodes.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP

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(X, L, R), Xs) → U1_ga(X, L, R, Xs, front_in_ga(L, Ls))
U1_ga(X, L, R, Xs, front_out_ga(L, Ls)) → U2_ga(X, L, R, Xs, Ls, front_in_ga(R, Rs))
U2_ga(X, L, R, Xs, Ls, front_out_ga(R, Rs)) → U3_ga(X, 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(X, L, R, Xs, app_out_gga(Ls, Rs, Xs)) → front_out_ga(tree(X, 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
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP

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
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP

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.
By using the subterm criterion [20] together with the size-change analysis [32] 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: 



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ PiDPToQDPProof

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

FRONT_IN_GA(tree(X, L, R), Xs) → FRONT_IN_GA(L, Ls)
U1_GA(X, L, R, Xs, front_out_ga(L, Ls)) → FRONT_IN_GA(R, Rs)
FRONT_IN_GA(tree(X, L, R), Xs) → U1_GA(X, 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(X, L, R), Xs) → U1_ga(X, L, R, Xs, front_in_ga(L, Ls))
U1_ga(X, L, R, Xs, front_out_ga(L, Ls)) → U2_ga(X, L, R, Xs, Ls, front_in_ga(R, Rs))
U2_ga(X, L, R, Xs, Ls, front_out_ga(R, Rs)) → U3_ga(X, 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(X, L, R, Xs, app_out_gga(Ls, Rs, Xs)) → front_out_ga(tree(X, 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)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x3, x5)
FRONT_IN_GA(x1, x2)  =  FRONT_IN_GA(x1)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ PiDPToQDPProof
                  ↳ QDP
                    ↳ QDPSizeChangeProof

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

FRONT_IN_GA(tree(X, L, R)) → FRONT_IN_GA(L)
U1_GA(R, front_out_ga(Ls)) → FRONT_IN_GA(R)
FRONT_IN_GA(tree(X, L, R)) → U1_GA(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(X, 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.
By using the subterm criterion [20] together with the size-change analysis [32] 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: