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

0 Prolog
↳1 PrologToPiTRSProof (⇒, 0 ms)
↳2 PiTRS
↳3 DependencyPairsProof (⇔, 12 ms)
↳4 PiDP
↳5 DependencyGraphProof (⇔, 0 ms)
↳6 AND
    ↳7 PiDP
        ↳8 UsableRulesProof (⇔, 0 ms)
        ↳9 PiDP
        ↳10 PiDPToQDPProof (⇒, 10 ms)
        ↳11 QDP
        ↳12 QDPSizeChangeProof (⇔, 0 ms)
        ↳13 YES
    ↳14 PiDP
        ↳15 UsableRulesProof (⇔, 0 ms)
        ↳16 PiDP
        ↳17 PiDPToQDPProof (⇒, 0 ms)
        ↳18 QDP
        ↳19 QDPSizeChangeProof (⇔, 0 ms)
        ↳20 YES
    ↳21 PiDP
        ↳22 UsableRulesProof (⇔, 0 ms)
        ↳23 PiDP
        ↳24 PiDPToQDPProof (⇒, 0 ms)
        ↳25 QDP
        ↳26 MRRProof (⇔, 27 ms)
        ↳27 QDP
        ↳28 PisEmptyProof (⇔, 0 ms)
        ↳29 YES

(0) Obligation:

Clauses:

perm([], []).
perm(L, .(H, T)) :- ','(append2(V, .(H, U), L), ','(append1(V, U, W), perm(W, T))).
append1([], L, L).
append1(.(H, L1), L2, .(H, L3)) :- append1(L1, L2, L3).
append2([], L, L).
append2(.(H, L1), L2, .(H, L3)) :- append2(L1, L2, L3).

Query: perm(g,a)

(1) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
perm_in: (b,f)
append2_in: (f,f,b)
append1_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(L, .(H, T)) → U1_ga(L, H, T, append2_in_aag(V, .(H, U), L))
append2_in_aag([], L, L) → append2_out_aag([], L, L)
append2_in_aag(.(H, L1), L2, .(H, L3)) → U5_aag(H, L1, L2, L3, append2_in_aag(L1, L2, L3))
U5_aag(H, L1, L2, L3, append2_out_aag(L1, L2, L3)) → append2_out_aag(.(H, L1), L2, .(H, L3))
U1_ga(L, H, T, append2_out_aag(V, .(H, U), L)) → U2_ga(L, H, T, append1_in_gga(V, U, W))
append1_in_gga([], L, L) → append1_out_gga([], L, L)
append1_in_gga(.(H, L1), L2, .(H, L3)) → U4_gga(H, L1, L2, L3, append1_in_gga(L1, L2, L3))
U4_gga(H, L1, L2, L3, append1_out_gga(L1, L2, L3)) → append1_out_gga(.(H, L1), L2, .(H, L3))
U2_ga(L, H, T, append1_out_gga(V, U, W)) → U3_ga(L, H, T, perm_in_ga(W, T))
U3_ga(L, H, T, perm_out_ga(W, T)) → perm_out_ga(L, .(H, T))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
append2_in_aag(x1, x2, x3)  =  append2_in_aag(x3)
.(x1, x2)  =  .(x2)
append2_out_aag(x1, x2, x3)  =  append2_out_aag(x1, x2)
U5_aag(x1, x2, x3, x4, x5)  =  U5_aag(x5)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
append1_in_gga(x1, x2, x3)  =  append1_in_gga(x1, x2)
append1_out_gga(x1, x2, x3)  =  append1_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(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:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(L, .(H, T)) → U1_ga(L, H, T, append2_in_aag(V, .(H, U), L))
append2_in_aag([], L, L) → append2_out_aag([], L, L)
append2_in_aag(.(H, L1), L2, .(H, L3)) → U5_aag(H, L1, L2, L3, append2_in_aag(L1, L2, L3))
U5_aag(H, L1, L2, L3, append2_out_aag(L1, L2, L3)) → append2_out_aag(.(H, L1), L2, .(H, L3))
U1_ga(L, H, T, append2_out_aag(V, .(H, U), L)) → U2_ga(L, H, T, append1_in_gga(V, U, W))
append1_in_gga([], L, L) → append1_out_gga([], L, L)
append1_in_gga(.(H, L1), L2, .(H, L3)) → U4_gga(H, L1, L2, L3, append1_in_gga(L1, L2, L3))
U4_gga(H, L1, L2, L3, append1_out_gga(L1, L2, L3)) → append1_out_gga(.(H, L1), L2, .(H, L3))
U2_ga(L, H, T, append1_out_gga(V, U, W)) → U3_ga(L, H, T, perm_in_ga(W, T))
U3_ga(L, H, T, perm_out_ga(W, T)) → perm_out_ga(L, .(H, T))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
append2_in_aag(x1, x2, x3)  =  append2_in_aag(x3)
.(x1, x2)  =  .(x2)
append2_out_aag(x1, x2, x3)  =  append2_out_aag(x1, x2)
U5_aag(x1, x2, x3, x4, x5)  =  U5_aag(x5)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
append1_in_gga(x1, x2, x3)  =  append1_in_gga(x1, x2)
append1_out_gga(x1, x2, x3)  =  append1_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(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:

PERM_IN_GA(L, .(H, T)) → U1_GA(L, H, T, append2_in_aag(V, .(H, U), L))
PERM_IN_GA(L, .(H, T)) → APPEND2_IN_AAG(V, .(H, U), L)
APPEND2_IN_AAG(.(H, L1), L2, .(H, L3)) → U5_AAG(H, L1, L2, L3, append2_in_aag(L1, L2, L3))
APPEND2_IN_AAG(.(H, L1), L2, .(H, L3)) → APPEND2_IN_AAG(L1, L2, L3)
U1_GA(L, H, T, append2_out_aag(V, .(H, U), L)) → U2_GA(L, H, T, append1_in_gga(V, U, W))
U1_GA(L, H, T, append2_out_aag(V, .(H, U), L)) → APPEND1_IN_GGA(V, U, W)
APPEND1_IN_GGA(.(H, L1), L2, .(H, L3)) → U4_GGA(H, L1, L2, L3, append1_in_gga(L1, L2, L3))
APPEND1_IN_GGA(.(H, L1), L2, .(H, L3)) → APPEND1_IN_GGA(L1, L2, L3)
U2_GA(L, H, T, append1_out_gga(V, U, W)) → U3_GA(L, H, T, perm_in_ga(W, T))
U2_GA(L, H, T, append1_out_gga(V, U, W)) → PERM_IN_GA(W, T)

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(L, .(H, T)) → U1_ga(L, H, T, append2_in_aag(V, .(H, U), L))
append2_in_aag([], L, L) → append2_out_aag([], L, L)
append2_in_aag(.(H, L1), L2, .(H, L3)) → U5_aag(H, L1, L2, L3, append2_in_aag(L1, L2, L3))
U5_aag(H, L1, L2, L3, append2_out_aag(L1, L2, L3)) → append2_out_aag(.(H, L1), L2, .(H, L3))
U1_ga(L, H, T, append2_out_aag(V, .(H, U), L)) → U2_ga(L, H, T, append1_in_gga(V, U, W))
append1_in_gga([], L, L) → append1_out_gga([], L, L)
append1_in_gga(.(H, L1), L2, .(H, L3)) → U4_gga(H, L1, L2, L3, append1_in_gga(L1, L2, L3))
U4_gga(H, L1, L2, L3, append1_out_gga(L1, L2, L3)) → append1_out_gga(.(H, L1), L2, .(H, L3))
U2_ga(L, H, T, append1_out_gga(V, U, W)) → U3_ga(L, H, T, perm_in_ga(W, T))
U3_ga(L, H, T, perm_out_ga(W, T)) → perm_out_ga(L, .(H, T))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
append2_in_aag(x1, x2, x3)  =  append2_in_aag(x3)
.(x1, x2)  =  .(x2)
append2_out_aag(x1, x2, x3)  =  append2_out_aag(x1, x2)
U5_aag(x1, x2, x3, x4, x5)  =  U5_aag(x5)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
append1_in_gga(x1, x2, x3)  =  append1_in_gga(x1, x2)
append1_out_gga(x1, x2, x3)  =  append1_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
APPEND2_IN_AAG(x1, x2, x3)  =  APPEND2_IN_AAG(x3)
U5_AAG(x1, x2, x3, x4, x5)  =  U5_AAG(x5)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)
APPEND1_IN_GGA(x1, x2, x3)  =  APPEND1_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x5)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x4)

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

(4) Obligation:

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

PERM_IN_GA(L, .(H, T)) → U1_GA(L, H, T, append2_in_aag(V, .(H, U), L))
PERM_IN_GA(L, .(H, T)) → APPEND2_IN_AAG(V, .(H, U), L)
APPEND2_IN_AAG(.(H, L1), L2, .(H, L3)) → U5_AAG(H, L1, L2, L3, append2_in_aag(L1, L2, L3))
APPEND2_IN_AAG(.(H, L1), L2, .(H, L3)) → APPEND2_IN_AAG(L1, L2, L3)
U1_GA(L, H, T, append2_out_aag(V, .(H, U), L)) → U2_GA(L, H, T, append1_in_gga(V, U, W))
U1_GA(L, H, T, append2_out_aag(V, .(H, U), L)) → APPEND1_IN_GGA(V, U, W)
APPEND1_IN_GGA(.(H, L1), L2, .(H, L3)) → U4_GGA(H, L1, L2, L3, append1_in_gga(L1, L2, L3))
APPEND1_IN_GGA(.(H, L1), L2, .(H, L3)) → APPEND1_IN_GGA(L1, L2, L3)
U2_GA(L, H, T, append1_out_gga(V, U, W)) → U3_GA(L, H, T, perm_in_ga(W, T))
U2_GA(L, H, T, append1_out_gga(V, U, W)) → PERM_IN_GA(W, T)

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(L, .(H, T)) → U1_ga(L, H, T, append2_in_aag(V, .(H, U), L))
append2_in_aag([], L, L) → append2_out_aag([], L, L)
append2_in_aag(.(H, L1), L2, .(H, L3)) → U5_aag(H, L1, L2, L3, append2_in_aag(L1, L2, L3))
U5_aag(H, L1, L2, L3, append2_out_aag(L1, L2, L3)) → append2_out_aag(.(H, L1), L2, .(H, L3))
U1_ga(L, H, T, append2_out_aag(V, .(H, U), L)) → U2_ga(L, H, T, append1_in_gga(V, U, W))
append1_in_gga([], L, L) → append1_out_gga([], L, L)
append1_in_gga(.(H, L1), L2, .(H, L3)) → U4_gga(H, L1, L2, L3, append1_in_gga(L1, L2, L3))
U4_gga(H, L1, L2, L3, append1_out_gga(L1, L2, L3)) → append1_out_gga(.(H, L1), L2, .(H, L3))
U2_ga(L, H, T, append1_out_gga(V, U, W)) → U3_ga(L, H, T, perm_in_ga(W, T))
U3_ga(L, H, T, perm_out_ga(W, T)) → perm_out_ga(L, .(H, T))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
append2_in_aag(x1, x2, x3)  =  append2_in_aag(x3)
.(x1, x2)  =  .(x2)
append2_out_aag(x1, x2, x3)  =  append2_out_aag(x1, x2)
U5_aag(x1, x2, x3, x4, x5)  =  U5_aag(x5)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
append1_in_gga(x1, x2, x3)  =  append1_in_gga(x1, x2)
append1_out_gga(x1, x2, x3)  =  append1_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
APPEND2_IN_AAG(x1, x2, x3)  =  APPEND2_IN_AAG(x3)
U5_AAG(x1, x2, x3, x4, x5)  =  U5_AAG(x5)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)
APPEND1_IN_GGA(x1, x2, x3)  =  APPEND1_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x5)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x4)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 5 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

APPEND1_IN_GGA(.(H, L1), L2, .(H, L3)) → APPEND1_IN_GGA(L1, L2, L3)

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(L, .(H, T)) → U1_ga(L, H, T, append2_in_aag(V, .(H, U), L))
append2_in_aag([], L, L) → append2_out_aag([], L, L)
append2_in_aag(.(H, L1), L2, .(H, L3)) → U5_aag(H, L1, L2, L3, append2_in_aag(L1, L2, L3))
U5_aag(H, L1, L2, L3, append2_out_aag(L1, L2, L3)) → append2_out_aag(.(H, L1), L2, .(H, L3))
U1_ga(L, H, T, append2_out_aag(V, .(H, U), L)) → U2_ga(L, H, T, append1_in_gga(V, U, W))
append1_in_gga([], L, L) → append1_out_gga([], L, L)
append1_in_gga(.(H, L1), L2, .(H, L3)) → U4_gga(H, L1, L2, L3, append1_in_gga(L1, L2, L3))
U4_gga(H, L1, L2, L3, append1_out_gga(L1, L2, L3)) → append1_out_gga(.(H, L1), L2, .(H, L3))
U2_ga(L, H, T, append1_out_gga(V, U, W)) → U3_ga(L, H, T, perm_in_ga(W, T))
U3_ga(L, H, T, perm_out_ga(W, T)) → perm_out_ga(L, .(H, T))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
append2_in_aag(x1, x2, x3)  =  append2_in_aag(x3)
.(x1, x2)  =  .(x2)
append2_out_aag(x1, x2, x3)  =  append2_out_aag(x1, x2)
U5_aag(x1, x2, x3, x4, x5)  =  U5_aag(x5)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
append1_in_gga(x1, x2, x3)  =  append1_in_gga(x1, x2)
append1_out_gga(x1, x2, x3)  =  append1_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
APPEND1_IN_GGA(x1, x2, x3)  =  APPEND1_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:

APPEND1_IN_GGA(.(H, L1), L2, .(H, L3)) → APPEND1_IN_GGA(L1, L2, L3)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APPEND1_IN_GGA(x1, x2, x3)  =  APPEND1_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:

APPEND1_IN_GGA(.(L1), L2) → APPEND1_IN_GGA(L1, L2)

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:

  • APPEND1_IN_GGA(.(L1), L2) → APPEND1_IN_GGA(L1, L2)
    The graph contains the following edges 1 > 1, 2 >= 2

(13) YES

(14) Obligation:

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

APPEND2_IN_AAG(.(H, L1), L2, .(H, L3)) → APPEND2_IN_AAG(L1, L2, L3)

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(L, .(H, T)) → U1_ga(L, H, T, append2_in_aag(V, .(H, U), L))
append2_in_aag([], L, L) → append2_out_aag([], L, L)
append2_in_aag(.(H, L1), L2, .(H, L3)) → U5_aag(H, L1, L2, L3, append2_in_aag(L1, L2, L3))
U5_aag(H, L1, L2, L3, append2_out_aag(L1, L2, L3)) → append2_out_aag(.(H, L1), L2, .(H, L3))
U1_ga(L, H, T, append2_out_aag(V, .(H, U), L)) → U2_ga(L, H, T, append1_in_gga(V, U, W))
append1_in_gga([], L, L) → append1_out_gga([], L, L)
append1_in_gga(.(H, L1), L2, .(H, L3)) → U4_gga(H, L1, L2, L3, append1_in_gga(L1, L2, L3))
U4_gga(H, L1, L2, L3, append1_out_gga(L1, L2, L3)) → append1_out_gga(.(H, L1), L2, .(H, L3))
U2_ga(L, H, T, append1_out_gga(V, U, W)) → U3_ga(L, H, T, perm_in_ga(W, T))
U3_ga(L, H, T, perm_out_ga(W, T)) → perm_out_ga(L, .(H, T))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
append2_in_aag(x1, x2, x3)  =  append2_in_aag(x3)
.(x1, x2)  =  .(x2)
append2_out_aag(x1, x2, x3)  =  append2_out_aag(x1, x2)
U5_aag(x1, x2, x3, x4, x5)  =  U5_aag(x5)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
append1_in_gga(x1, x2, x3)  =  append1_in_gga(x1, x2)
append1_out_gga(x1, x2, x3)  =  append1_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
APPEND2_IN_AAG(x1, x2, x3)  =  APPEND2_IN_AAG(x3)

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:

APPEND2_IN_AAG(.(H, L1), L2, .(H, L3)) → APPEND2_IN_AAG(L1, L2, L3)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APPEND2_IN_AAG(x1, x2, x3)  =  APPEND2_IN_AAG(x3)

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:

APPEND2_IN_AAG(.(L3)) → APPEND2_IN_AAG(L3)

R is empty.
Q is empty.
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:

  • APPEND2_IN_AAG(.(L3)) → APPEND2_IN_AAG(L3)
    The graph contains the following edges 1 > 1

(20) YES

(21) Obligation:

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

U1_GA(L, H, T, append2_out_aag(V, .(H, U), L)) → U2_GA(L, H, T, append1_in_gga(V, U, W))
U2_GA(L, H, T, append1_out_gga(V, U, W)) → PERM_IN_GA(W, T)
PERM_IN_GA(L, .(H, T)) → U1_GA(L, H, T, append2_in_aag(V, .(H, U), L))

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(L, .(H, T)) → U1_ga(L, H, T, append2_in_aag(V, .(H, U), L))
append2_in_aag([], L, L) → append2_out_aag([], L, L)
append2_in_aag(.(H, L1), L2, .(H, L3)) → U5_aag(H, L1, L2, L3, append2_in_aag(L1, L2, L3))
U5_aag(H, L1, L2, L3, append2_out_aag(L1, L2, L3)) → append2_out_aag(.(H, L1), L2, .(H, L3))
U1_ga(L, H, T, append2_out_aag(V, .(H, U), L)) → U2_ga(L, H, T, append1_in_gga(V, U, W))
append1_in_gga([], L, L) → append1_out_gga([], L, L)
append1_in_gga(.(H, L1), L2, .(H, L3)) → U4_gga(H, L1, L2, L3, append1_in_gga(L1, L2, L3))
U4_gga(H, L1, L2, L3, append1_out_gga(L1, L2, L3)) → append1_out_gga(.(H, L1), L2, .(H, L3))
U2_ga(L, H, T, append1_out_gga(V, U, W)) → U3_ga(L, H, T, perm_in_ga(W, T))
U3_ga(L, H, T, perm_out_ga(W, T)) → perm_out_ga(L, .(H, T))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
append2_in_aag(x1, x2, x3)  =  append2_in_aag(x3)
.(x1, x2)  =  .(x2)
append2_out_aag(x1, x2, x3)  =  append2_out_aag(x1, x2)
U5_aag(x1, x2, x3, x4, x5)  =  U5_aag(x5)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
append1_in_gga(x1, x2, x3)  =  append1_in_gga(x1, x2)
append1_out_gga(x1, x2, x3)  =  append1_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

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

U1_GA(L, H, T, append2_out_aag(V, .(H, U), L)) → U2_GA(L, H, T, append1_in_gga(V, U, W))
U2_GA(L, H, T, append1_out_gga(V, U, W)) → PERM_IN_GA(W, T)
PERM_IN_GA(L, .(H, T)) → U1_GA(L, H, T, append2_in_aag(V, .(H, U), L))

The TRS R consists of the following rules:

append1_in_gga([], L, L) → append1_out_gga([], L, L)
append1_in_gga(.(H, L1), L2, .(H, L3)) → U4_gga(H, L1, L2, L3, append1_in_gga(L1, L2, L3))
append2_in_aag([], L, L) → append2_out_aag([], L, L)
append2_in_aag(.(H, L1), L2, .(H, L3)) → U5_aag(H, L1, L2, L3, append2_in_aag(L1, L2, L3))
U4_gga(H, L1, L2, L3, append1_out_gga(L1, L2, L3)) → append1_out_gga(.(H, L1), L2, .(H, L3))
U5_aag(H, L1, L2, L3, append2_out_aag(L1, L2, L3)) → append2_out_aag(.(H, L1), L2, .(H, L3))

The argument filtering Pi contains the following mapping:
[]  =  []
append2_in_aag(x1, x2, x3)  =  append2_in_aag(x3)
.(x1, x2)  =  .(x2)
append2_out_aag(x1, x2, x3)  =  append2_out_aag(x1, x2)
U5_aag(x1, x2, x3, x4, x5)  =  U5_aag(x5)
append1_in_gga(x1, x2, x3)  =  append1_in_gga(x1, x2)
append1_out_gga(x1, x2, x3)  =  append1_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)

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

(24) PiDPToQDPProof (SOUND transformation)

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

(25) Obligation:

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

U1_GA(append2_out_aag(V, .(U))) → U2_GA(append1_in_gga(V, U))
U2_GA(append1_out_gga(W)) → PERM_IN_GA(W)
PERM_IN_GA(L) → U1_GA(append2_in_aag(L))

The TRS R consists of the following rules:

append1_in_gga([], L) → append1_out_gga(L)
append1_in_gga(.(L1), L2) → U4_gga(append1_in_gga(L1, L2))
append2_in_aag(L) → append2_out_aag([], L)
append2_in_aag(.(L3)) → U5_aag(append2_in_aag(L3))
U4_gga(append1_out_gga(L3)) → append1_out_gga(.(L3))
U5_aag(append2_out_aag(L1, L2)) → append2_out_aag(.(L1), L2)

The set Q consists of the following terms:

append1_in_gga(x0, x1)
append2_in_aag(x0)
U4_gga(x0)
U5_aag(x0)

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

(26) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

U1_GA(append2_out_aag(V, .(U))) → U2_GA(append1_in_gga(V, U))
U2_GA(append1_out_gga(W)) → PERM_IN_GA(W)
PERM_IN_GA(L) → U1_GA(append2_in_aag(L))

Strictly oriented rules of the TRS R:

append1_in_gga([], L) → append1_out_gga(L)
append1_in_gga(.(L1), L2) → U4_gga(append1_in_gga(L1, L2))
append2_in_aag(L) → append2_out_aag([], L)
append2_in_aag(.(L3)) → U5_aag(append2_in_aag(L3))
U4_gga(append1_out_gga(L3)) → append1_out_gga(.(L3))
U5_aag(append2_out_aag(L1, L2)) → append2_out_aag(.(L1), L2)

Used ordering: Knuth-Bendix order [KBO] with precedence:
append1ingga2 > [] > PERMINGA1 > .1 > U2GA1 > append2inaag1 > U1GA1 > U5aag1 > append2outaag2 > U4gga1 > append1outgga1

and weight map:

[]=2
append1_out_gga_1=6
._1=7
U4_gga_1=7
append2_in_aag_1=2
U5_aag_1=7
U1_GA_1=3
U2_GA_1=1
PERM_IN_GA_1=6
append1_in_gga_2=8
append2_out_aag_2=0

The variable weight is 1

(27) Obligation:

Q DP problem:
P is empty.
R is empty.
The set Q consists of the following terms:

append1_in_gga(x0, x1)
append2_in_aag(x0)
U4_gga(x0)
U5_aag(x0)

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

(28) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(29) YES