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 PrologToPrologProblemTransformerProof (⇐)
↳2 Prolog
↳3 PrologToPiTRSProof (⇐)
↳4 PiTRS
↳5 DependencyPairsProof (⇔)
↳6 PiDP
↳7 DependencyGraphProof (⇔)
↳8 PiDP
↳9 UsableRulesProof (⇔)
↳10 PiDP
↳11 PiDPToQDPProof (⇐)
↳12 QDP
↳13 QDPSizeChangeProof (⇔)
↳14 YES

(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) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

app21(.(T8, .(T29, T33)), T31, .(T8, .(T29, T32))) :- app21(T33, T31, T32).
app21(.(T8, []), T42, .(T8, T42)).
app21([], .(T50, T51), .(T50, T51)).
app21([], T53, T53).

Queries:

app21(a,g,g).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
app21_in: (f,b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

app21_in_agg(.(T8, .(T29, T33)), T31, .(T8, .(T29, T32))) → U1_agg(T8, T29, T33, T31, T32, app21_in_agg(T33, T31, T32))
app21_in_agg(.(T8, []), T42, .(T8, T42)) → app21_out_agg(.(T8, []), T42, .(T8, T42))
app21_in_agg([], .(T50, T51), .(T50, T51)) → app21_out_agg([], .(T50, T51), .(T50, T51))
app21_in_agg([], T53, T53) → app21_out_agg([], T53, T53)
U1_agg(T8, T29, T33, T31, T32, app21_out_agg(T33, T31, T32)) → app21_out_agg(.(T8, .(T29, T33)), T31, .(T8, .(T29, T32)))

The argument filtering Pi contains the following mapping:
app21_in_agg(x1, x2, x3)  =  app21_in_agg(x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_agg(x1, x2, x3, x4, x5, x6)  =  U1_agg(x1, x2, x6)
app21_out_agg(x1, x2, x3)  =  app21_out_agg(x1)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

app21_in_agg(.(T8, .(T29, T33)), T31, .(T8, .(T29, T32))) → U1_agg(T8, T29, T33, T31, T32, app21_in_agg(T33, T31, T32))
app21_in_agg(.(T8, []), T42, .(T8, T42)) → app21_out_agg(.(T8, []), T42, .(T8, T42))
app21_in_agg([], .(T50, T51), .(T50, T51)) → app21_out_agg([], .(T50, T51), .(T50, T51))
app21_in_agg([], T53, T53) → app21_out_agg([], T53, T53)
U1_agg(T8, T29, T33, T31, T32, app21_out_agg(T33, T31, T32)) → app21_out_agg(.(T8, .(T29, T33)), T31, .(T8, .(T29, T32)))

The argument filtering Pi contains the following mapping:
app21_in_agg(x1, x2, x3)  =  app21_in_agg(x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_agg(x1, x2, x3, x4, x5, x6)  =  U1_agg(x1, x2, x6)
app21_out_agg(x1, x2, x3)  =  app21_out_agg(x1)

(5) 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:

APP21_IN_AGG(.(T8, .(T29, T33)), T31, .(T8, .(T29, T32))) → U1_AGG(T8, T29, T33, T31, T32, app21_in_agg(T33, T31, T32))
APP21_IN_AGG(.(T8, .(T29, T33)), T31, .(T8, .(T29, T32))) → APP21_IN_AGG(T33, T31, T32)

The TRS R consists of the following rules:

app21_in_agg(.(T8, .(T29, T33)), T31, .(T8, .(T29, T32))) → U1_agg(T8, T29, T33, T31, T32, app21_in_agg(T33, T31, T32))
app21_in_agg(.(T8, []), T42, .(T8, T42)) → app21_out_agg(.(T8, []), T42, .(T8, T42))
app21_in_agg([], .(T50, T51), .(T50, T51)) → app21_out_agg([], .(T50, T51), .(T50, T51))
app21_in_agg([], T53, T53) → app21_out_agg([], T53, T53)
U1_agg(T8, T29, T33, T31, T32, app21_out_agg(T33, T31, T32)) → app21_out_agg(.(T8, .(T29, T33)), T31, .(T8, .(T29, T32)))

The argument filtering Pi contains the following mapping:
app21_in_agg(x1, x2, x3)  =  app21_in_agg(x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_agg(x1, x2, x3, x4, x5, x6)  =  U1_agg(x1, x2, x6)
app21_out_agg(x1, x2, x3)  =  app21_out_agg(x1)
APP21_IN_AGG(x1, x2, x3)  =  APP21_IN_AGG(x2, x3)
U1_AGG(x1, x2, x3, x4, x5, x6)  =  U1_AGG(x1, x2, x6)

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

(6) Obligation:

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

APP21_IN_AGG(.(T8, .(T29, T33)), T31, .(T8, .(T29, T32))) → U1_AGG(T8, T29, T33, T31, T32, app21_in_agg(T33, T31, T32))
APP21_IN_AGG(.(T8, .(T29, T33)), T31, .(T8, .(T29, T32))) → APP21_IN_AGG(T33, T31, T32)

The TRS R consists of the following rules:

app21_in_agg(.(T8, .(T29, T33)), T31, .(T8, .(T29, T32))) → U1_agg(T8, T29, T33, T31, T32, app21_in_agg(T33, T31, T32))
app21_in_agg(.(T8, []), T42, .(T8, T42)) → app21_out_agg(.(T8, []), T42, .(T8, T42))
app21_in_agg([], .(T50, T51), .(T50, T51)) → app21_out_agg([], .(T50, T51), .(T50, T51))
app21_in_agg([], T53, T53) → app21_out_agg([], T53, T53)
U1_agg(T8, T29, T33, T31, T32, app21_out_agg(T33, T31, T32)) → app21_out_agg(.(T8, .(T29, T33)), T31, .(T8, .(T29, T32)))

The argument filtering Pi contains the following mapping:
app21_in_agg(x1, x2, x3)  =  app21_in_agg(x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_agg(x1, x2, x3, x4, x5, x6)  =  U1_agg(x1, x2, x6)
app21_out_agg(x1, x2, x3)  =  app21_out_agg(x1)
APP21_IN_AGG(x1, x2, x3)  =  APP21_IN_AGG(x2, x3)
U1_AGG(x1, x2, x3, x4, x5, x6)  =  U1_AGG(x1, x2, x6)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

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

(8) Obligation:

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

APP21_IN_AGG(.(T8, .(T29, T33)), T31, .(T8, .(T29, T32))) → APP21_IN_AGG(T33, T31, T32)

The TRS R consists of the following rules:

app21_in_agg(.(T8, .(T29, T33)), T31, .(T8, .(T29, T32))) → U1_agg(T8, T29, T33, T31, T32, app21_in_agg(T33, T31, T32))
app21_in_agg(.(T8, []), T42, .(T8, T42)) → app21_out_agg(.(T8, []), T42, .(T8, T42))
app21_in_agg([], .(T50, T51), .(T50, T51)) → app21_out_agg([], .(T50, T51), .(T50, T51))
app21_in_agg([], T53, T53) → app21_out_agg([], T53, T53)
U1_agg(T8, T29, T33, T31, T32, app21_out_agg(T33, T31, T32)) → app21_out_agg(.(T8, .(T29, T33)), T31, .(T8, .(T29, T32)))

The argument filtering Pi contains the following mapping:
app21_in_agg(x1, x2, x3)  =  app21_in_agg(x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_agg(x1, x2, x3, x4, x5, x6)  =  U1_agg(x1, x2, x6)
app21_out_agg(x1, x2, x3)  =  app21_out_agg(x1)
APP21_IN_AGG(x1, x2, x3)  =  APP21_IN_AGG(x2, x3)

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

(9) UsableRulesProof (EQUIVALENT transformation)

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

(10) Obligation:

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

APP21_IN_AGG(.(T8, .(T29, T33)), T31, .(T8, .(T29, T32))) → APP21_IN_AGG(T33, T31, T32)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP21_IN_AGG(x1, x2, x3)  =  APP21_IN_AGG(x2, x3)

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

(11) PiDPToQDPProof (SOUND transformation)

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

(12) Obligation:

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

APP21_IN_AGG(T31, .(T8, .(T29, T32))) → APP21_IN_AGG(T31, T32)

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

(13) 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:

  • APP21_IN_AGG(T31, .(T8, .(T29, T32))) → APP21_IN_AGG(T31, T32)
    The graph contains the following edges 1 >= 1, 2 > 2

(14) YES