Left Termination of the query pattern perm_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 UsableRulesProof (⇔)
        ↳16 PiDP
        ↳17 PiDPToQDPProof (⇐)
        ↳18 QDP
        ↳19 QDPSizeChangeProof (⇔)
        ↳20 TRUE
      ↳21 PiDP
        ↳22 UsableRulesProof (⇔)
        ↳23 PiDP
        ↳24 PiDPToQDPProof (⇐)
        ↳25 QDP
  ↳26 PrologToPiTRSProof (⇐)
    ↳27 PiTRS
    ↳28 DependencyPairsProof (⇔)
    ↳29 PiDP
    ↳30 DependencyGraphProof (⇔)
    ↳31 AND
      ↳32 PiDP
        ↳33 UsableRulesProof (⇔)
        ↳34 PiDP
        ↳35 PiDPToQDPProof (⇐)
        ↳36 QDP
        ↳37 QDPSizeChangeProof (⇔)
        ↳38 TRUE
      ↳39 PiDP
        ↳40 UsableRulesProof (⇔)
        ↳41 PiDP
        ↳42 PiDPToQDPProof (⇐)
        ↳43 QDP
        ↳44 QDPSizeChangeProof (⇔)
        ↳45 TRUE
      ↳46 PiDP
        ↳47 UsableRulesProof (⇔)
        ↳48 PiDP
        ↳49 PiDPToQDPProof (⇐)
        ↳50 QDP
        ↳51 MRRProof (⇔)
        ↳52 QDP
        ↳53 PisEmptyProof (⇔)
        ↳54 TRUE

(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).

Queries:

perm(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:
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(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, 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, x3)
U5_aag(x1, x2, x3, x4, x5)  =  U5_aag(x4, x5)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
append1_in_gga(x1, x2, x3)  =  append1_in_gga(x1, x2)
append1_out_gga(x1, x2, x3)  =  append1_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, x3, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, 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(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, 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, x3)
U5_aag(x1, x2, x3, x4, x5)  =  U5_aag(x4, x5)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
append1_in_gga(x1, x2, x3)  =  append1_in_gga(x1, x2)
append1_out_gga(x1, x2, x3)  =  append1_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, x3, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, 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(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, 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, x3)
U5_aag(x1, x2, x3, x4, x5)  =  U5_aag(x4, x5)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
append1_in_gga(x1, x2, x3)  =  append1_in_gga(x1, x2)
append1_out_gga(x1, x2, x3)  =  append1_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, x3, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
APPEND2_IN_AAG(x1, x2, x3)  =  APPEND2_IN_AAG(x3)
U5_AAG(x1, x2, x3, x4, x5)  =  U5_AAG(x4, x5)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x4)
APPEND1_IN_GGA(x1, x2, x3)  =  APPEND1_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x2, x3, x5)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, 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(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, 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, x3)
U5_aag(x1, x2, x3, x4, x5)  =  U5_aag(x4, x5)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
append1_in_gga(x1, x2, x3)  =  append1_in_gga(x1, x2)
append1_out_gga(x1, x2, x3)  =  append1_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, x3, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
APPEND2_IN_AAG(x1, x2, x3)  =  APPEND2_IN_AAG(x3)
U5_AAG(x1, x2, x3, x4, x5)  =  U5_AAG(x4, x5)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x4)
APPEND1_IN_GGA(x1, x2, x3)  =  APPEND1_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x2, x3, x5)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, 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(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, 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, x3)
U5_aag(x1, x2, x3, x4, x5)  =  U5_aag(x4, x5)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
append1_in_gga(x1, x2, x3)  =  append1_in_gga(x1, x2)
append1_out_gga(x1, x2, x3)  =  append1_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, x3, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, 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) TRUE

(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(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, 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, x3)
U5_aag(x1, x2, x3, x4, x5)  =  U5_aag(x4, x5)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
append1_in_gga(x1, x2, x3)  =  append1_in_gga(x1, x2)
append1_out_gga(x1, x2, x3)  =  append1_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, x3, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, 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) TRUE

(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(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, 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, x3)
U5_aag(x1, x2, x3, x4, x5)  =  U5_aag(x4, x5)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
append1_in_gga(x1, x2, x3)  =  append1_in_gga(x1, x2)
append1_out_gga(x1, x2, x3)  =  append1_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, x3, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, 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, x3)
U5_aag(x1, x2, x3, x4, x5)  =  U5_aag(x4, x5)
append1_in_gga(x1, x2, x3)  =  append1_in_gga(x1, x2)
append1_out_gga(x1, x2, x3)  =  append1_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, x3, x5)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, 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(L, append2_out_aag(V, .(U), L)) → U2_GA(L, append1_in_gga(V, U))
U2_GA(L, append1_out_gga(V, U, W)) → PERM_IN_GA(W)
PERM_IN_GA(L) → U1_GA(L, append2_in_aag(L))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

append1_in_gga(x0, x1)
append2_in_aag(x0)
U4_gga(x0, x1, x2)
U5_aag(x0, x1)

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

(26) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. 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

(27) 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)

(28) 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

(29) 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

(30) DependencyGraphProof (EQUIVALENT transformation)

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

(31) Complex Obligation (AND)

(32) 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

(33) UsableRulesProof (EQUIVALENT transformation)

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

(34) 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

(35) PiDPToQDPProof (SOUND transformation)

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

(36) 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.

(37) 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

(38) TRUE

(39) 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

(40) UsableRulesProof (EQUIVALENT transformation)

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

(41) 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

(42) PiDPToQDPProof (SOUND transformation)

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

(43) 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.

(44) 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

(45) TRUE

(46) 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

(47) UsableRulesProof (EQUIVALENT transformation)

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

(48) 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

(49) PiDPToQDPProof (SOUND transformation)

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

(50) 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.

(51) 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)
append2_in_aag(L) → append2_out_aag([], L)

Used ordering: Polynomial interpretation [POLO]:

POL(.(x1)) = 5 + x1   
POL(PERM_IN_GA(x1)) = 2 + x1   
POL(U1_GA(x1)) = x1   
POL(U2_GA(x1)) = x1   
POL(U4_gga(x1)) = 5 + x1   
POL(U5_aag(x1)) = 5 + x1   
POL([]) = 0   
POL(append1_in_gga(x1, x2)) = 4 + x1 + x2   
POL(append1_out_gga(x1)) = 3 + x1   
POL(append2_in_aag(x1)) = 1 + x1   
POL(append2_out_aag(x1, x2)) = x1 + x2   

(52) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

append1_in_gga(.(L1), L2) → U4_gga(append1_in_gga(L1, L2))
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.

(53) PisEmptyProof (EQUIVALENT transformation)

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

(54) TRUE