Left Termination proof of /tmp/tmpoCd8ob/inorder.pl

Left Termination of the query pattern inorder(b,f) w.r.t. the given Prolog program could successfully be proven:

↳ PROLOG

  ↳ PrologToPiTRSProof

inorder₂(nil₀, {}₀).
inorder₂(tree₃(L, V, R), I) :- inorder₂(L, LI), inorder₂(R, RI), append₃(LI, .₂(V, RI), I).
append₃({}₀, X, X).
append₃(.₂(X, Xs), Ys, .₂(X, Zs)) :- append₃(Xs, Ys, Zs).

With regard to the inferred argument filtering the predicates were used in the following modes:
inorder₂: (b,f)
append₃: (b,b,f)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

inorder_2_in_ga₂(nil_0, []_0) -> inorder_2_out_ga₂(nil_0, []_0)
inorder_2_in_ga₂(tree_3₃(L, V, R), I) -> if_inorder_2_in_1_ga₅(L, V, R, I, inorder_2_in_ga₂(L, LI))
if_inorder_2_in_1_ga₅(L, V, R, I, inorder_2_out_ga₂(L, LI)) -> if_inorder_2_in_2_ga₆(L, V, R, I, LI, inorder_2_in_ga₂(R, RI))
if_inorder_2_in_2_ga₆(L, V, R, I, LI, inorder_2_out_ga₂(R, RI)) -> if_inorder_2_in_3_ga₇(L, V, R, I, LI, RI, append_3_in_gga₃(LI, ._2₂(V, RI), I))
append_3_in_gga₃([]_0, X, X) -> append_3_out_gga₃([]_0, X, X)
append_3_in_gga₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_append_3_in_1_gga₅(X, Xs, Ys, Zs, append_3_in_gga₃(Xs, Ys, Zs))
if_append_3_in_1_gga₅(X, Xs, Ys, Zs, append_3_out_gga₃(Xs, Ys, Zs)) -> append_3_out_gga₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_inorder_2_in_3_ga₇(L, V, R, I, LI, RI, append_3_out_gga₃(LI, ._2₂(V, RI), I)) -> inorder_2_out_ga₂(tree_3₃(L, V, R), I)

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:

inorder_2_in_ga₂(nil_0, []_0) -> inorder_2_out_ga₂(nil_0, []_0)
inorder_2_in_ga₂(tree_3₃(L, V, R), I) -> if_inorder_2_in_1_ga₅(L, V, R, I, inorder_2_in_ga₂(L, LI))
if_inorder_2_in_1_ga₅(L, V, R, I, inorder_2_out_ga₂(L, LI)) -> if_inorder_2_in_2_ga₆(L, V, R, I, LI, inorder_2_in_ga₂(R, RI))
if_inorder_2_in_2_ga₆(L, V, R, I, LI, inorder_2_out_ga₂(R, RI)) -> if_inorder_2_in_3_ga₇(L, V, R, I, LI, RI, append_3_in_gga₃(LI, ._2₂(V, RI), I))
append_3_in_gga₃([]_0, X, X) -> append_3_out_gga₃([]_0, X, X)
append_3_in_gga₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_append_3_in_1_gga₅(X, Xs, Ys, Zs, append_3_in_gga₃(Xs, Ys, Zs))
if_append_3_in_1_gga₅(X, Xs, Ys, Zs, append_3_out_gga₃(Xs, Ys, Zs)) -> append_3_out_gga₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_inorder_2_in_3_ga₇(L, V, R, I, LI, RI, append_3_out_gga₃(LI, ._2₂(V, RI), I)) -> inorder_2_out_ga₂(tree_3₃(L, V, R), I)

INORDER_2_IN_GA₂(tree_3₃(L, V, R), I) -> IF_INORDER_2_IN_1_GA₅(L, V, R, I, inorder_2_in_ga₂(L, LI))
INORDER_2_IN_GA₂(tree_3₃(L, V, R), I) -> INORDER_2_IN_GA₂(L, LI)
IF_INORDER_2_IN_1_GA₅(L, V, R, I, inorder_2_out_ga₂(L, LI)) -> IF_INORDER_2_IN_2_GA₆(L, V, R, I, LI, inorder_2_in_ga₂(R, RI))
IF_INORDER_2_IN_1_GA₅(L, V, R, I, inorder_2_out_ga₂(L, LI)) -> INORDER_2_IN_GA₂(R, RI)
IF_INORDER_2_IN_2_GA₆(L, V, R, I, LI, inorder_2_out_ga₂(R, RI)) -> IF_INORDER_2_IN_3_GA₇(L, V, R, I, LI, RI, append_3_in_gga₃(LI, ._2₂(V, RI), I))
IF_INORDER_2_IN_2_GA₆(L, V, R, I, LI, inorder_2_out_ga₂(R, RI)) -> APPEND_3_IN_GGA₃(LI, ._2₂(V, RI), I)
APPEND_3_IN_GGA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> IF_APPEND_3_IN_1_GGA₅(X, Xs, Ys, Zs, append_3_in_gga₃(Xs, Ys, Zs))
APPEND_3_IN_GGA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APPEND_3_IN_GGA₃(Xs, Ys, Zs)

The TRS R consists of the following rules:

inorder_2_in_ga₂(nil_0, []_0) -> inorder_2_out_ga₂(nil_0, []_0)
inorder_2_in_ga₂(tree_3₃(L, V, R), I) -> if_inorder_2_in_1_ga₅(L, V, R, I, inorder_2_in_ga₂(L, LI))
if_inorder_2_in_1_ga₅(L, V, R, I, inorder_2_out_ga₂(L, LI)) -> if_inorder_2_in_2_ga₆(L, V, R, I, LI, inorder_2_in_ga₂(R, RI))
if_inorder_2_in_2_ga₆(L, V, R, I, LI, inorder_2_out_ga₂(R, RI)) -> if_inorder_2_in_3_ga₇(L, V, R, I, LI, RI, append_3_in_gga₃(LI, ._2₂(V, RI), I))
append_3_in_gga₃([]_0, X, X) -> append_3_out_gga₃([]_0, X, X)
append_3_in_gga₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_append_3_in_1_gga₅(X, Xs, Ys, Zs, append_3_in_gga₃(Xs, Ys, Zs))
if_append_3_in_1_gga₅(X, Xs, Ys, Zs, append_3_out_gga₃(Xs, Ys, Zs)) -> append_3_out_gga₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_inorder_2_in_3_ga₇(L, V, R, I, LI, RI, append_3_out_gga₃(LI, ._2₂(V, RI), I)) -> inorder_2_out_ga₂(tree_3₃(L, V, R), I)

The argument filtering Pi contains the following mapping:
inorder_2_in_ga₂(x1, x2) = inorder_2_in_ga₁(x1)
nil_0 = nil_0
[]_0 = []_0
tree_3₃(x1, x2, x3) = tree_3₃(x1, x2, x3)
._2₂(x1, x2) = ._2₂(x1, x2)
inorder_2_out_ga₂(x1, x2) = inorder_2_out_ga₁(x2)
if_inorder_2_in_1_ga₅(x1, x2, x3, x4, x5) = if_inorder_2_in_1_ga₃(x2, x3, x5)
if_inorder_2_in_2_ga₆(x1, x2, x3, x4, x5, x6) = if_inorder_2_in_2_ga₃(x2, x5, x6)
if_inorder_2_in_3_ga₇(x1, x2, x3, x4, x5, x6, x7) = if_inorder_2_in_3_ga₁(x7)
append_3_in_gga₃(x1, x2, x3) = append_3_in_gga₂(x1, x2)
append_3_out_gga₃(x1, x2, x3) = append_3_out_gga₁(x3)
if_append_3_in_1_gga₅(x1, x2, x3, x4, x5) = if_append_3_in_1_gga₂(x1, x5)
INORDER_2_IN_GA₂(x1, x2) = INORDER_2_IN_GA₁(x1)
IF_APPEND_3_IN_1_GGA₅(x1, x2, x3, x4, x5) = IF_APPEND_3_IN_1_GGA₂(x1, x5)
IF_INORDER_2_IN_2_GA₆(x1, x2, x3, x4, x5, x6) = IF_INORDER_2_IN_2_GA₃(x2, x5, x6)
IF_INORDER_2_IN_1_GA₅(x1, x2, x3, x4, x5) = IF_INORDER_2_IN_1_GA₃(x2, x3, x5)
IF_INORDER_2_IN_3_GA₇(x1, x2, x3, x4, x5, x6, x7) = IF_INORDER_2_IN_3_GA₁(x7)
APPEND_3_IN_GGA₃(x1, x2, x3) = APPEND_3_IN_GGA₂(x1, x2)

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:

INORDER_2_IN_GA₂(tree_3₃(L, V, R), I) -> IF_INORDER_2_IN_1_GA₅(L, V, R, I, inorder_2_in_ga₂(L, LI))
INORDER_2_IN_GA₂(tree_3₃(L, V, R), I) -> INORDER_2_IN_GA₂(L, LI)
IF_INORDER_2_IN_1_GA₅(L, V, R, I, inorder_2_out_ga₂(L, LI)) -> IF_INORDER_2_IN_2_GA₆(L, V, R, I, LI, inorder_2_in_ga₂(R, RI))
IF_INORDER_2_IN_1_GA₅(L, V, R, I, inorder_2_out_ga₂(L, LI)) -> INORDER_2_IN_GA₂(R, RI)
IF_INORDER_2_IN_2_GA₆(L, V, R, I, LI, inorder_2_out_ga₂(R, RI)) -> IF_INORDER_2_IN_3_GA₇(L, V, R, I, LI, RI, append_3_in_gga₃(LI, ._2₂(V, RI), I))
IF_INORDER_2_IN_2_GA₆(L, V, R, I, LI, inorder_2_out_ga₂(R, RI)) -> APPEND_3_IN_GGA₃(LI, ._2₂(V, RI), I)
APPEND_3_IN_GGA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> IF_APPEND_3_IN_1_GGA₅(X, Xs, Ys, Zs, append_3_in_gga₃(Xs, Ys, Zs))
APPEND_3_IN_GGA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APPEND_3_IN_GGA₃(Xs, Ys, Zs)

The TRS R consists of the following rules:

inorder_2_in_ga₂(nil_0, []_0) -> inorder_2_out_ga₂(nil_0, []_0)
inorder_2_in_ga₂(tree_3₃(L, V, R), I) -> if_inorder_2_in_1_ga₅(L, V, R, I, inorder_2_in_ga₂(L, LI))
if_inorder_2_in_1_ga₅(L, V, R, I, inorder_2_out_ga₂(L, LI)) -> if_inorder_2_in_2_ga₆(L, V, R, I, LI, inorder_2_in_ga₂(R, RI))
if_inorder_2_in_2_ga₆(L, V, R, I, LI, inorder_2_out_ga₂(R, RI)) -> if_inorder_2_in_3_ga₇(L, V, R, I, LI, RI, append_3_in_gga₃(LI, ._2₂(V, RI), I))
append_3_in_gga₃([]_0, X, X) -> append_3_out_gga₃([]_0, X, X)
append_3_in_gga₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_append_3_in_1_gga₅(X, Xs, Ys, Zs, append_3_in_gga₃(Xs, Ys, Zs))
if_append_3_in_1_gga₅(X, Xs, Ys, Zs, append_3_out_gga₃(Xs, Ys, Zs)) -> append_3_out_gga₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_inorder_2_in_3_ga₇(L, V, R, I, LI, RI, append_3_out_gga₃(LI, ._2₂(V, RI), I)) -> inorder_2_out_ga₂(tree_3₃(L, V, R), I)

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

APPEND_3_IN_GGA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APPEND_3_IN_GGA₃(Xs, Ys, Zs)

The TRS R consists of the following rules:

inorder_2_in_ga₂(nil_0, []_0) -> inorder_2_out_ga₂(nil_0, []_0)
inorder_2_in_ga₂(tree_3₃(L, V, R), I) -> if_inorder_2_in_1_ga₅(L, V, R, I, inorder_2_in_ga₂(L, LI))
if_inorder_2_in_1_ga₅(L, V, R, I, inorder_2_out_ga₂(L, LI)) -> if_inorder_2_in_2_ga₆(L, V, R, I, LI, inorder_2_in_ga₂(R, RI))
if_inorder_2_in_2_ga₆(L, V, R, I, LI, inorder_2_out_ga₂(R, RI)) -> if_inorder_2_in_3_ga₇(L, V, R, I, LI, RI, append_3_in_gga₃(LI, ._2₂(V, RI), I))
append_3_in_gga₃([]_0, X, X) -> append_3_out_gga₃([]_0, X, X)
append_3_in_gga₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_append_3_in_1_gga₅(X, Xs, Ys, Zs, append_3_in_gga₃(Xs, Ys, Zs))
if_append_3_in_1_gga₅(X, Xs, Ys, Zs, append_3_out_gga₃(Xs, Ys, Zs)) -> append_3_out_gga₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_inorder_2_in_3_ga₇(L, V, R, I, LI, RI, append_3_out_gga₃(LI, ._2₂(V, RI), I)) -> inorder_2_out_ga₂(tree_3₃(L, V, R), I)

The argument filtering Pi contains the following mapping:
inorder_2_in_ga₂(x1, x2) = inorder_2_in_ga₁(x1)
nil_0 = nil_0
[]_0 = []_0
tree_3₃(x1, x2, x3) = tree_3₃(x1, x2, x3)
._2₂(x1, x2) = ._2₂(x1, x2)
inorder_2_out_ga₂(x1, x2) = inorder_2_out_ga₁(x2)
if_inorder_2_in_1_ga₅(x1, x2, x3, x4, x5) = if_inorder_2_in_1_ga₃(x2, x3, x5)
if_inorder_2_in_2_ga₆(x1, x2, x3, x4, x5, x6) = if_inorder_2_in_2_ga₃(x2, x5, x6)
if_inorder_2_in_3_ga₇(x1, x2, x3, x4, x5, x6, x7) = if_inorder_2_in_3_ga₁(x7)
append_3_in_gga₃(x1, x2, x3) = append_3_in_gga₂(x1, x2)
append_3_out_gga₃(x1, x2, x3) = append_3_out_gga₁(x3)
if_append_3_in_1_gga₅(x1, x2, x3, x4, x5) = if_append_3_in_1_gga₂(x1, x5)
APPEND_3_IN_GGA₃(x1, x2, x3) = APPEND_3_IN_GGA₂(x1, x2)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting we can delete all non-usable rules from R.

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

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

APPEND_3_IN_GGA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APPEND_3_IN_GGA₃(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
._2₂(x1, x2) = ._2₂(x1, x2)
APPEND_3_IN_GGA₃(x1, x2, x3) = APPEND_3_IN_GGA₂(x1, x2)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem into ordinary QDP problem by application of Pi.

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

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

APPEND_3_IN_GGA₂(._2₂(X, Xs), Ys) -> APPEND_3_IN_GGA₂(Xs, Ys)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {APPEND_3_IN_GGA₂}.
By using the subterm criterion together with the size-change analysis 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:

APPEND_3_IN_GGA₂(._2₂(X, Xs), Ys) -> APPEND_3_IN_GGA₂(Xs, Ys)
The graph contains the following edges 1 > 1, 2 >= 2

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ PiDPToQDPProof

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

IF_INORDER_2_IN_1_GA₅(L, V, R, I, inorder_2_out_ga₂(L, LI)) -> INORDER_2_IN_GA₂(R, RI)
INORDER_2_IN_GA₂(tree_3₃(L, V, R), I) -> IF_INORDER_2_IN_1_GA₅(L, V, R, I, inorder_2_in_ga₂(L, LI))
INORDER_2_IN_GA₂(tree_3₃(L, V, R), I) -> INORDER_2_IN_GA₂(L, LI)

The TRS R consists of the following rules:

inorder_2_in_ga₂(nil_0, []_0) -> inorder_2_out_ga₂(nil_0, []_0)
inorder_2_in_ga₂(tree_3₃(L, V, R), I) -> if_inorder_2_in_1_ga₅(L, V, R, I, inorder_2_in_ga₂(L, LI))
if_inorder_2_in_1_ga₅(L, V, R, I, inorder_2_out_ga₂(L, LI)) -> if_inorder_2_in_2_ga₆(L, V, R, I, LI, inorder_2_in_ga₂(R, RI))
if_inorder_2_in_2_ga₆(L, V, R, I, LI, inorder_2_out_ga₂(R, RI)) -> if_inorder_2_in_3_ga₇(L, V, R, I, LI, RI, append_3_in_gga₃(LI, ._2₂(V, RI), I))
append_3_in_gga₃([]_0, X, X) -> append_3_out_gga₃([]_0, X, X)
append_3_in_gga₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_append_3_in_1_gga₅(X, Xs, Ys, Zs, append_3_in_gga₃(Xs, Ys, Zs))
if_append_3_in_1_gga₅(X, Xs, Ys, Zs, append_3_out_gga₃(Xs, Ys, Zs)) -> append_3_out_gga₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_inorder_2_in_3_ga₇(L, V, R, I, LI, RI, append_3_out_gga₃(LI, ._2₂(V, RI), I)) -> inorder_2_out_ga₂(tree_3₃(L, V, R), I)

The argument filtering Pi contains the following mapping:
inorder_2_in_ga₂(x1, x2) = inorder_2_in_ga₁(x1)
nil_0 = nil_0
[]_0 = []_0
tree_3₃(x1, x2, x3) = tree_3₃(x1, x2, x3)
._2₂(x1, x2) = ._2₂(x1, x2)
inorder_2_out_ga₂(x1, x2) = inorder_2_out_ga₁(x2)
if_inorder_2_in_1_ga₅(x1, x2, x3, x4, x5) = if_inorder_2_in_1_ga₃(x2, x3, x5)
if_inorder_2_in_2_ga₆(x1, x2, x3, x4, x5, x6) = if_inorder_2_in_2_ga₃(x2, x5, x6)
if_inorder_2_in_3_ga₇(x1, x2, x3, x4, x5, x6, x7) = if_inorder_2_in_3_ga₁(x7)
append_3_in_gga₃(x1, x2, x3) = append_3_in_gga₂(x1, x2)
append_3_out_gga₃(x1, x2, x3) = append_3_out_gga₁(x3)
if_append_3_in_1_gga₅(x1, x2, x3, x4, x5) = if_append_3_in_1_gga₂(x1, x5)
INORDER_2_IN_GA₂(x1, x2) = INORDER_2_IN_GA₁(x1)
IF_INORDER_2_IN_1_GA₅(x1, x2, x3, x4, x5) = IF_INORDER_2_IN_1_GA₃(x2, x3, x5)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem into ordinary QDP problem by application of Pi.

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ PiDPToQDPProof

                  ↳ QDP

                    ↳ QDPSizeChangeProof

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

IF_INORDER_2_IN_1_GA₃(V, R, inorder_2_out_ga₁(LI)) -> INORDER_2_IN_GA₁(R)
INORDER_2_IN_GA₁(tree_3₃(L, V, R)) -> IF_INORDER_2_IN_1_GA₃(V, R, inorder_2_in_ga₁(L))
INORDER_2_IN_GA₁(tree_3₃(L, V, R)) -> INORDER_2_IN_GA₁(L)

The TRS R consists of the following rules:

inorder_2_in_ga₁(nil_0) -> inorder_2_out_ga₁([]_0)
inorder_2_in_ga₁(tree_3₃(L, V, R)) -> if_inorder_2_in_1_ga₃(V, R, inorder_2_in_ga₁(L))
if_inorder_2_in_1_ga₃(V, R, inorder_2_out_ga₁(LI)) -> if_inorder_2_in_2_ga₃(V, LI, inorder_2_in_ga₁(R))
if_inorder_2_in_2_ga₃(V, LI, inorder_2_out_ga₁(RI)) -> if_inorder_2_in_3_ga₁(append_3_in_gga₂(LI, ._2₂(V, RI)))
append_3_in_gga₂([]_0, X) -> append_3_out_gga₁(X)
append_3_in_gga₂(._2₂(X, Xs), Ys) -> if_append_3_in_1_gga₂(X, append_3_in_gga₂(Xs, Ys))
if_append_3_in_1_gga₂(X, append_3_out_gga₁(Zs)) -> append_3_out_gga₁(._2₂(X, Zs))
if_inorder_2_in_3_ga₁(append_3_out_gga₁(I)) -> inorder_2_out_ga₁(I)

The set Q consists of the following terms:

inorder_2_in_ga₁(x0)
if_inorder_2_in_1_ga₃(x0, x1, x2)
if_inorder_2_in_2_ga₃(x0, x1, x2)
append_3_in_gga₂(x0, x1)
if_append_3_in_1_gga₂(x0, x1)
if_inorder_2_in_3_ga₁(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {INORDER_2_IN_GA₁, IF_INORDER_2_IN_1_GA₃}.
By using the subterm criterion together with the size-change analysis 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:

INORDER_2_IN_GA₁(tree_3₃(L, V, R)) -> IF_INORDER_2_IN_1_GA₃(V, R, inorder_2_in_ga₁(L))
The graph contains the following edges 1 > 1, 1 > 2
INORDER_2_IN_GA₁(tree_3₃(L, V, R)) -> INORDER_2_IN_GA₁(L)
The graph contains the following edges 1 > 1
IF_INORDER_2_IN_1_GA₃(V, R, inorder_2_out_ga₁(LI)) -> INORDER_2_IN_GA₁(R)
The graph contains the following edges 2 >= 1