Left Termination of the query pattern append3_in_4(g, g, g, a) w.r.t. the given Prolog program could successfully be proven:



↳ Prolog
  ↳ PrologToPiTRSProof

Clauses:

append([], L, L).
append(.(H, L1), L2, .(H, L3)) :- append(L1, L2, L3).
append3(A, B, C, D) :- ','(append(A, B, E), append(E, C, D)).

Queries:

append3(g,g,g,a).

We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
append3_in: (b,b,b,f)
append_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

append3_in_ggga(A, B, C, D) → U2_ggga(A, B, C, D, append_in_gga(A, B, E))
append_in_gga([], L, L) → append_out_gga([], L, L)
append_in_gga(.(H, L1), L2, .(H, L3)) → U1_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3))
U1_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) → append_out_gga(.(H, L1), L2, .(H, L3))
U2_ggga(A, B, C, D, append_out_gga(A, B, E)) → U3_ggga(A, B, C, D, append_in_gga(E, C, D))
U3_ggga(A, B, C, D, append_out_gga(E, C, D)) → append3_out_ggga(A, B, C, D)

The argument filtering Pi contains the following mapping:
append3_in_ggga(x1, x2, x3, x4)  =  append3_in_ggga(x1, x2, x3)
U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x3, x5)
append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
[]  =  []
append_out_gga(x1, x2, x3)  =  append_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x5)
append3_out_ggga(x1, x2, x3, x4)  =  append3_out_ggga(x4)

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:

append3_in_ggga(A, B, C, D) → U2_ggga(A, B, C, D, append_in_gga(A, B, E))
append_in_gga([], L, L) → append_out_gga([], L, L)
append_in_gga(.(H, L1), L2, .(H, L3)) → U1_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3))
U1_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) → append_out_gga(.(H, L1), L2, .(H, L3))
U2_ggga(A, B, C, D, append_out_gga(A, B, E)) → U3_ggga(A, B, C, D, append_in_gga(E, C, D))
U3_ggga(A, B, C, D, append_out_gga(E, C, D)) → append3_out_ggga(A, B, C, D)

The argument filtering Pi contains the following mapping:
append3_in_ggga(x1, x2, x3, x4)  =  append3_in_ggga(x1, x2, x3)
U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x3, x5)
append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
[]  =  []
append_out_gga(x1, x2, x3)  =  append_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x5)
append3_out_ggga(x1, x2, x3, x4)  =  append3_out_ggga(x4)


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

APPEND3_IN_GGGA(A, B, C, D) → U2_GGGA(A, B, C, D, append_in_gga(A, B, E))
APPEND3_IN_GGGA(A, B, C, D) → APPEND_IN_GGA(A, B, E)
APPEND_IN_GGA(.(H, L1), L2, .(H, L3)) → U1_GGA(H, L1, L2, L3, append_in_gga(L1, L2, L3))
APPEND_IN_GGA(.(H, L1), L2, .(H, L3)) → APPEND_IN_GGA(L1, L2, L3)
U2_GGGA(A, B, C, D, append_out_gga(A, B, E)) → U3_GGGA(A, B, C, D, append_in_gga(E, C, D))
U2_GGGA(A, B, C, D, append_out_gga(A, B, E)) → APPEND_IN_GGA(E, C, D)

The TRS R consists of the following rules:

append3_in_ggga(A, B, C, D) → U2_ggga(A, B, C, D, append_in_gga(A, B, E))
append_in_gga([], L, L) → append_out_gga([], L, L)
append_in_gga(.(H, L1), L2, .(H, L3)) → U1_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3))
U1_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) → append_out_gga(.(H, L1), L2, .(H, L3))
U2_ggga(A, B, C, D, append_out_gga(A, B, E)) → U3_ggga(A, B, C, D, append_in_gga(E, C, D))
U3_ggga(A, B, C, D, append_out_gga(E, C, D)) → append3_out_ggga(A, B, C, D)

The argument filtering Pi contains the following mapping:
append3_in_ggga(x1, x2, x3, x4)  =  append3_in_ggga(x1, x2, x3)
U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x3, x5)
append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
[]  =  []
append_out_gga(x1, x2, x3)  =  append_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x5)
append3_out_ggga(x1, x2, x3, x4)  =  append3_out_ggga(x4)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x5)
APPEND_IN_GGA(x1, x2, x3)  =  APPEND_IN_GGA(x1, x2)
U3_GGGA(x1, x2, x3, x4, x5)  =  U3_GGGA(x5)
U2_GGGA(x1, x2, x3, x4, x5)  =  U2_GGGA(x3, x5)
APPEND3_IN_GGGA(x1, x2, x3, x4)  =  APPEND3_IN_GGGA(x1, x2, x3)

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:

APPEND3_IN_GGGA(A, B, C, D) → U2_GGGA(A, B, C, D, append_in_gga(A, B, E))
APPEND3_IN_GGGA(A, B, C, D) → APPEND_IN_GGA(A, B, E)
APPEND_IN_GGA(.(H, L1), L2, .(H, L3)) → U1_GGA(H, L1, L2, L3, append_in_gga(L1, L2, L3))
APPEND_IN_GGA(.(H, L1), L2, .(H, L3)) → APPEND_IN_GGA(L1, L2, L3)
U2_GGGA(A, B, C, D, append_out_gga(A, B, E)) → U3_GGGA(A, B, C, D, append_in_gga(E, C, D))
U2_GGGA(A, B, C, D, append_out_gga(A, B, E)) → APPEND_IN_GGA(E, C, D)

The TRS R consists of the following rules:

append3_in_ggga(A, B, C, D) → U2_ggga(A, B, C, D, append_in_gga(A, B, E))
append_in_gga([], L, L) → append_out_gga([], L, L)
append_in_gga(.(H, L1), L2, .(H, L3)) → U1_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3))
U1_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) → append_out_gga(.(H, L1), L2, .(H, L3))
U2_ggga(A, B, C, D, append_out_gga(A, B, E)) → U3_ggga(A, B, C, D, append_in_gga(E, C, D))
U3_ggga(A, B, C, D, append_out_gga(E, C, D)) → append3_out_ggga(A, B, C, D)

The argument filtering Pi contains the following mapping:
append3_in_ggga(x1, x2, x3, x4)  =  append3_in_ggga(x1, x2, x3)
U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x3, x5)
append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
[]  =  []
append_out_gga(x1, x2, x3)  =  append_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x5)
append3_out_ggga(x1, x2, x3, x4)  =  append3_out_ggga(x4)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x5)
APPEND_IN_GGA(x1, x2, x3)  =  APPEND_IN_GGA(x1, x2)
U3_GGGA(x1, x2, x3, x4, x5)  =  U3_GGGA(x5)
U2_GGGA(x1, x2, x3, x4, x5)  =  U2_GGGA(x3, x5)
APPEND3_IN_GGGA(x1, x2, x3, x4)  =  APPEND3_IN_GGGA(x1, x2, x3)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 1 SCC with 5 less nodes.

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

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

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

The TRS R consists of the following rules:

append3_in_ggga(A, B, C, D) → U2_ggga(A, B, C, D, append_in_gga(A, B, E))
append_in_gga([], L, L) → append_out_gga([], L, L)
append_in_gga(.(H, L1), L2, .(H, L3)) → U1_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3))
U1_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) → append_out_gga(.(H, L1), L2, .(H, L3))
U2_ggga(A, B, C, D, append_out_gga(A, B, E)) → U3_ggga(A, B, C, D, append_in_gga(E, C, D))
U3_ggga(A, B, C, D, append_out_gga(E, C, D)) → append3_out_ggga(A, B, C, D)

The argument filtering Pi contains the following mapping:
append3_in_ggga(x1, x2, x3, x4)  =  append3_in_ggga(x1, x2, x3)
U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x3, x5)
append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
[]  =  []
append_out_gga(x1, x2, x3)  =  append_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x5)
append3_out_ggga(x1, x2, x3, x4)  =  append3_out_ggga(x4)
APPEND_IN_GGA(x1, x2, x3)  =  APPEND_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
            ↳ PiDP
              ↳ UsableRulesProof
                ↳ PiDP
                  ↳ PiDPToQDPProof

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

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

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPEND_IN_GGA(x1, x2, x3)  =  APPEND_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
            ↳ PiDP
              ↳ UsableRulesProof
                ↳ PiDP
                  ↳ PiDPToQDPProof
                    ↳ QDP
                      ↳ QDPSizeChangeProof

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

APPEND_IN_GGA(.(H, L1), L2) → APPEND_IN_GGA(L1, L2)

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: