Left Termination of the query pattern app2_in_3(a, 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 PiDP
    ↳7 UsableRulesProof (⇔)
    ↳8 PiDP
    ↳9 PiDPToQDPProof (⇐)
    ↳10 QDP
  ↳11 PrologToPiTRSProof (⇐)
    ↳12 PiTRS
    ↳13 DependencyPairsProof (⇔)
    ↳14 PiDP
    ↳15 DependencyGraphProof (⇔)
    ↳16 PiDP
    ↳17 UsableRulesProof (⇔)
    ↳18 PiDP
    ↳19 PiDPToQDPProof (⇐)
    ↳20 QDP
    ↳21 QDPSizeChangeProof (⇔)
    ↳22 TRUE

(0) Obligation:

Clauses:

app1(.(X, Xs), Ys, .(X, Zs)) :- app1(Xs, Ys, Zs).
app1([], Ys, Ys).
app2(.(X, Xs), Ys, .(X, Zs)) :- app2(Xs, Ys, Zs).
app2([], Ys, Ys).

Queries:

app2(a,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:
app2_in: (f,b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

app2_in_agg(.(X, Xs), Ys, .(X, Zs)) → U2_agg(X, Xs, Ys, Zs, app2_in_agg(Xs, Ys, Zs))
app2_in_agg([], Ys, Ys) → app2_out_agg([], Ys, Ys)
U2_agg(X, Xs, Ys, Zs, app2_out_agg(Xs, Ys, Zs)) → app2_out_agg(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
app2_in_agg(x1, x2, x3)  =  app2_in_agg(x2, x3)
.(x1, x2)  =  .(x1, x2)
U2_agg(x1, x2, x3, x4, x5)  =  U2_agg(x1, x3, x4, x5)
app2_out_agg(x1, x2, x3)  =  app2_out_agg(x1, x2, x3)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

app2_in_agg(.(X, Xs), Ys, .(X, Zs)) → U2_agg(X, Xs, Ys, Zs, app2_in_agg(Xs, Ys, Zs))
app2_in_agg([], Ys, Ys) → app2_out_agg([], Ys, Ys)
U2_agg(X, Xs, Ys, Zs, app2_out_agg(Xs, Ys, Zs)) → app2_out_agg(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
app2_in_agg(x1, x2, x3)  =  app2_in_agg(x2, x3)
.(x1, x2)  =  .(x1, x2)
U2_agg(x1, x2, x3, x4, x5)  =  U2_agg(x1, x3, x4, x5)
app2_out_agg(x1, x2, x3)  =  app2_out_agg(x1, x2, x3)

(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:

APP2_IN_AGG(.(X, Xs), Ys, .(X, Zs)) → U2_AGG(X, Xs, Ys, Zs, app2_in_agg(Xs, Ys, Zs))
APP2_IN_AGG(.(X, Xs), Ys, .(X, Zs)) → APP2_IN_AGG(Xs, Ys, Zs)

The TRS R consists of the following rules:

app2_in_agg(.(X, Xs), Ys, .(X, Zs)) → U2_agg(X, Xs, Ys, Zs, app2_in_agg(Xs, Ys, Zs))
app2_in_agg([], Ys, Ys) → app2_out_agg([], Ys, Ys)
U2_agg(X, Xs, Ys, Zs, app2_out_agg(Xs, Ys, Zs)) → app2_out_agg(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
app2_in_agg(x1, x2, x3)  =  app2_in_agg(x2, x3)
.(x1, x2)  =  .(x1, x2)
U2_agg(x1, x2, x3, x4, x5)  =  U2_agg(x1, x3, x4, x5)
app2_out_agg(x1, x2, x3)  =  app2_out_agg(x1, x2, x3)
APP2_IN_AGG(x1, x2, x3)  =  APP2_IN_AGG(x2, x3)
U2_AGG(x1, x2, x3, x4, x5)  =  U2_AGG(x1, x3, x4, x5)

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

(4) Obligation:

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

APP2_IN_AGG(.(X, Xs), Ys, .(X, Zs)) → U2_AGG(X, Xs, Ys, Zs, app2_in_agg(Xs, Ys, Zs))
APP2_IN_AGG(.(X, Xs), Ys, .(X, Zs)) → APP2_IN_AGG(Xs, Ys, Zs)

The TRS R consists of the following rules:

app2_in_agg(.(X, Xs), Ys, .(X, Zs)) → U2_agg(X, Xs, Ys, Zs, app2_in_agg(Xs, Ys, Zs))
app2_in_agg([], Ys, Ys) → app2_out_agg([], Ys, Ys)
U2_agg(X, Xs, Ys, Zs, app2_out_agg(Xs, Ys, Zs)) → app2_out_agg(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
app2_in_agg(x1, x2, x3)  =  app2_in_agg(x2, x3)
.(x1, x2)  =  .(x1, x2)
U2_agg(x1, x2, x3, x4, x5)  =  U2_agg(x1, x3, x4, x5)
app2_out_agg(x1, x2, x3)  =  app2_out_agg(x1, x2, x3)
APP2_IN_AGG(x1, x2, x3)  =  APP2_IN_AGG(x2, x3)
U2_AGG(x1, x2, x3, x4, x5)  =  U2_AGG(x1, x3, x4, 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 1 less node.

(6) Obligation:

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

APP2_IN_AGG(.(X, Xs), Ys, .(X, Zs)) → APP2_IN_AGG(Xs, Ys, Zs)

The TRS R consists of the following rules:

app2_in_agg(.(X, Xs), Ys, .(X, Zs)) → U2_agg(X, Xs, Ys, Zs, app2_in_agg(Xs, Ys, Zs))
app2_in_agg([], Ys, Ys) → app2_out_agg([], Ys, Ys)
U2_agg(X, Xs, Ys, Zs, app2_out_agg(Xs, Ys, Zs)) → app2_out_agg(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
app2_in_agg(x1, x2, x3)  =  app2_in_agg(x2, x3)
.(x1, x2)  =  .(x1, x2)
U2_agg(x1, x2, x3, x4, x5)  =  U2_agg(x1, x3, x4, x5)
app2_out_agg(x1, x2, x3)  =  app2_out_agg(x1, x2, x3)
APP2_IN_AGG(x1, x2, x3)  =  APP2_IN_AGG(x2, x3)

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:

APP2_IN_AGG(.(X, Xs), Ys, .(X, Zs)) → APP2_IN_AGG(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP2_IN_AGG(x1, x2, x3)  =  APP2_IN_AGG(x2, x3)

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:

APP2_IN_AGG(Ys, .(X, Zs)) → APP2_IN_AGG(Ys, Zs)

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

(11) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
app2_in: (f,b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

app2_in_agg(.(X, Xs), Ys, .(X, Zs)) → U2_agg(X, Xs, Ys, Zs, app2_in_agg(Xs, Ys, Zs))
app2_in_agg([], Ys, Ys) → app2_out_agg([], Ys, Ys)
U2_agg(X, Xs, Ys, Zs, app2_out_agg(Xs, Ys, Zs)) → app2_out_agg(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
app2_in_agg(x1, x2, x3)  =  app2_in_agg(x2, x3)
.(x1, x2)  =  .(x1, x2)
U2_agg(x1, x2, x3, x4, x5)  =  U2_agg(x1, x5)
app2_out_agg(x1, x2, x3)  =  app2_out_agg(x1)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(12) Obligation:

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

app2_in_agg(.(X, Xs), Ys, .(X, Zs)) → U2_agg(X, Xs, Ys, Zs, app2_in_agg(Xs, Ys, Zs))
app2_in_agg([], Ys, Ys) → app2_out_agg([], Ys, Ys)
U2_agg(X, Xs, Ys, Zs, app2_out_agg(Xs, Ys, Zs)) → app2_out_agg(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
app2_in_agg(x1, x2, x3)  =  app2_in_agg(x2, x3)
.(x1, x2)  =  .(x1, x2)
U2_agg(x1, x2, x3, x4, x5)  =  U2_agg(x1, x5)
app2_out_agg(x1, x2, x3)  =  app2_out_agg(x1)

(13) 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:

APP2_IN_AGG(.(X, Xs), Ys, .(X, Zs)) → U2_AGG(X, Xs, Ys, Zs, app2_in_agg(Xs, Ys, Zs))
APP2_IN_AGG(.(X, Xs), Ys, .(X, Zs)) → APP2_IN_AGG(Xs, Ys, Zs)

The TRS R consists of the following rules:

app2_in_agg(.(X, Xs), Ys, .(X, Zs)) → U2_agg(X, Xs, Ys, Zs, app2_in_agg(Xs, Ys, Zs))
app2_in_agg([], Ys, Ys) → app2_out_agg([], Ys, Ys)
U2_agg(X, Xs, Ys, Zs, app2_out_agg(Xs, Ys, Zs)) → app2_out_agg(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
app2_in_agg(x1, x2, x3)  =  app2_in_agg(x2, x3)
.(x1, x2)  =  .(x1, x2)
U2_agg(x1, x2, x3, x4, x5)  =  U2_agg(x1, x5)
app2_out_agg(x1, x2, x3)  =  app2_out_agg(x1)
APP2_IN_AGG(x1, x2, x3)  =  APP2_IN_AGG(x2, x3)
U2_AGG(x1, x2, x3, x4, x5)  =  U2_AGG(x1, x5)

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

(14) Obligation:

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

APP2_IN_AGG(.(X, Xs), Ys, .(X, Zs)) → U2_AGG(X, Xs, Ys, Zs, app2_in_agg(Xs, Ys, Zs))
APP2_IN_AGG(.(X, Xs), Ys, .(X, Zs)) → APP2_IN_AGG(Xs, Ys, Zs)

The TRS R consists of the following rules:

app2_in_agg(.(X, Xs), Ys, .(X, Zs)) → U2_agg(X, Xs, Ys, Zs, app2_in_agg(Xs, Ys, Zs))
app2_in_agg([], Ys, Ys) → app2_out_agg([], Ys, Ys)
U2_agg(X, Xs, Ys, Zs, app2_out_agg(Xs, Ys, Zs)) → app2_out_agg(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
app2_in_agg(x1, x2, x3)  =  app2_in_agg(x2, x3)
.(x1, x2)  =  .(x1, x2)
U2_agg(x1, x2, x3, x4, x5)  =  U2_agg(x1, x5)
app2_out_agg(x1, x2, x3)  =  app2_out_agg(x1)
APP2_IN_AGG(x1, x2, x3)  =  APP2_IN_AGG(x2, x3)
U2_AGG(x1, x2, x3, x4, x5)  =  U2_AGG(x1, x5)

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

(15) DependencyGraphProof (EQUIVALENT transformation)

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

(16) Obligation:

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

APP2_IN_AGG(.(X, Xs), Ys, .(X, Zs)) → APP2_IN_AGG(Xs, Ys, Zs)

The TRS R consists of the following rules:

app2_in_agg(.(X, Xs), Ys, .(X, Zs)) → U2_agg(X, Xs, Ys, Zs, app2_in_agg(Xs, Ys, Zs))
app2_in_agg([], Ys, Ys) → app2_out_agg([], Ys, Ys)
U2_agg(X, Xs, Ys, Zs, app2_out_agg(Xs, Ys, Zs)) → app2_out_agg(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
app2_in_agg(x1, x2, x3)  =  app2_in_agg(x2, x3)
.(x1, x2)  =  .(x1, x2)
U2_agg(x1, x2, x3, x4, x5)  =  U2_agg(x1, x5)
app2_out_agg(x1, x2, x3)  =  app2_out_agg(x1)
APP2_IN_AGG(x1, x2, x3)  =  APP2_IN_AGG(x2, x3)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

APP2_IN_AGG(.(X, Xs), Ys, .(X, Zs)) → APP2_IN_AGG(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP2_IN_AGG(x1, x2, x3)  =  APP2_IN_AGG(x2, x3)

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

APP2_IN_AGG(Ys, .(X, Zs)) → APP2_IN_AGG(Ys, Zs)

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

(21) 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:

  • APP2_IN_AGG(Ys, .(X, Zs)) → APP2_IN_AGG(Ys, Zs)
    The graph contains the following edges 1 >= 1, 2 > 2

(22) TRUE