Left Termination proof of /tmp/tmpvqBC3I/inorder.pl

Left Termination of the query pattern inorder_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

↳ Prolog

  ↳ PrologToPiTRSProof

Clauses:

inorder(nil, []).
inorder(tree(L, V, R), I) :- ','(inorder(L, LI), ','(inorder(R, RI), append(LI, .(V, RI), I))).
append([], X, X).
append(.(X, Xs), Ys, .(X, Zs)) :- append(Xs, Ys, Zs).

Queries:

inorder(g,a).

We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
inorder_in: (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:

inorder_in_ga(nil, []) → inorder_out_ga(nil, [])
inorder_in_ga(tree(L, V, R), I) → U1_ga(L, V, R, I, inorder_in_ga(L, LI))
U1_ga(L, V, R, I, inorder_out_ga(L, LI)) → U2_ga(L, V, R, I, LI, inorder_in_ga(R, RI))
U2_ga(L, V, R, I, LI, inorder_out_ga(R, RI)) → U3_ga(L, V, R, I, append_in_gga(LI, .(V, RI), I))
append_in_gga([], X, X) → append_out_gga([], X, X)
append_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, append_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, append_out_gga(Xs, Ys, Zs)) → append_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_ga(L, V, R, I, append_out_gga(LI, .(V, RI), I)) → inorder_out_ga(tree(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_in_ga(nil, []) → inorder_out_ga(nil, [])
inorder_in_ga(tree(L, V, R), I) → U1_ga(L, V, R, I, inorder_in_ga(L, LI))
U1_ga(L, V, R, I, inorder_out_ga(L, LI)) → U2_ga(L, V, R, I, LI, inorder_in_ga(R, RI))
U2_ga(L, V, R, I, LI, inorder_out_ga(R, RI)) → U3_ga(L, V, R, I, append_in_gga(LI, .(V, RI), I))
append_in_gga([], X, X) → append_out_gga([], X, X)
append_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, append_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, append_out_gga(Xs, Ys, Zs)) → append_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_ga(L, V, R, I, append_out_gga(LI, .(V, RI), I)) → inorder_out_ga(tree(L, V, R), I)

The argument filtering Pi contains the following mapping:
inorder_in_ga(x1, x2) = inorder_in_ga(x1)
nil = nil
inorder_out_ga(x1, x2) = inorder_out_ga(x2)
tree(x1, x2, x3) = tree(x1, x2, x3)
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x2, x3, x5)
U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x2, x5, x6)
U3_ga(x1, x2, x3, x4, x5) = U3_ga(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)
U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x5)

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

INORDER_IN_GA(tree(L, V, R), I) → U1_GA(L, V, R, I, inorder_in_ga(L, LI))
INORDER_IN_GA(tree(L, V, R), I) → INORDER_IN_GA(L, LI)
U1_GA(L, V, R, I, inorder_out_ga(L, LI)) → U2_GA(L, V, R, I, LI, inorder_in_ga(R, RI))
U1_GA(L, V, R, I, inorder_out_ga(L, LI)) → INORDER_IN_GA(R, RI)
U2_GA(L, V, R, I, LI, inorder_out_ga(R, RI)) → U3_GA(L, V, R, I, append_in_gga(LI, .(V, RI), I))
U2_GA(L, V, R, I, LI, inorder_out_ga(R, RI)) → APPEND_IN_GGA(LI, .(V, RI), I)
APPEND_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U4_GGA(X, Xs, Ys, Zs, append_in_gga(Xs, Ys, Zs))
APPEND_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APPEND_IN_GGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

inorder_in_ga(nil, []) → inorder_out_ga(nil, [])
inorder_in_ga(tree(L, V, R), I) → U1_ga(L, V, R, I, inorder_in_ga(L, LI))
U1_ga(L, V, R, I, inorder_out_ga(L, LI)) → U2_ga(L, V, R, I, LI, inorder_in_ga(R, RI))
U2_ga(L, V, R, I, LI, inorder_out_ga(R, RI)) → U3_ga(L, V, R, I, append_in_gga(LI, .(V, RI), I))
append_in_gga([], X, X) → append_out_gga([], X, X)
append_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, append_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, append_out_gga(Xs, Ys, Zs)) → append_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_ga(L, V, R, I, append_out_gga(LI, .(V, RI), I)) → inorder_out_ga(tree(L, V, R), I)

The argument filtering Pi contains the following mapping:
inorder_in_ga(x1, x2) = inorder_in_ga(x1)
nil = nil
inorder_out_ga(x1, x2) = inorder_out_ga(x2)
tree(x1, x2, x3) = tree(x1, x2, x3)
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x2, x3, x5)
U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x2, x5, x6)
U3_ga(x1, x2, x3, x4, x5) = U3_ga(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)
U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x5)
U2_GA(x1, x2, x3, x4, x5, x6) = U2_GA(x2, x5, x6)
INORDER_IN_GA(x1, x2) = INORDER_IN_GA(x1)
APPEND_IN_GGA(x1, x2, x3) = APPEND_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5) = U4_GGA(x1, x5)
U1_GA(x1, x2, x3, x4, x5) = U1_GA(x2, x3, x5)
U3_GA(x1, x2, x3, x4, x5) = U3_GA(x5)

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_IN_GA(tree(L, V, R), I) → U1_GA(L, V, R, I, inorder_in_ga(L, LI))
INORDER_IN_GA(tree(L, V, R), I) → INORDER_IN_GA(L, LI)
U1_GA(L, V, R, I, inorder_out_ga(L, LI)) → U2_GA(L, V, R, I, LI, inorder_in_ga(R, RI))
U1_GA(L, V, R, I, inorder_out_ga(L, LI)) → INORDER_IN_GA(R, RI)
U2_GA(L, V, R, I, LI, inorder_out_ga(R, RI)) → U3_GA(L, V, R, I, append_in_gga(LI, .(V, RI), I))
U2_GA(L, V, R, I, LI, inorder_out_ga(R, RI)) → APPEND_IN_GGA(LI, .(V, RI), I)
APPEND_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U4_GGA(X, Xs, Ys, Zs, append_in_gga(Xs, Ys, Zs))
APPEND_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APPEND_IN_GGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

inorder_in_ga(nil, []) → inorder_out_ga(nil, [])
inorder_in_ga(tree(L, V, R), I) → U1_ga(L, V, R, I, inorder_in_ga(L, LI))
U1_ga(L, V, R, I, inorder_out_ga(L, LI)) → U2_ga(L, V, R, I, LI, inorder_in_ga(R, RI))
U2_ga(L, V, R, I, LI, inorder_out_ga(R, RI)) → U3_ga(L, V, R, I, append_in_gga(LI, .(V, RI), I))
append_in_gga([], X, X) → append_out_gga([], X, X)
append_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, append_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, append_out_gga(Xs, Ys, Zs)) → append_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_ga(L, V, R, I, append_out_gga(LI, .(V, RI), I)) → inorder_out_ga(tree(L, V, R), I)

The argument filtering Pi contains the following mapping:
inorder_in_ga(x1, x2) = inorder_in_ga(x1)
nil = nil
inorder_out_ga(x1, x2) = inorder_out_ga(x2)
tree(x1, x2, x3) = tree(x1, x2, x3)
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x2, x3, x5)
U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x2, x5, x6)
U3_ga(x1, x2, x3, x4, x5) = U3_ga(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)
U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x5)
U2_GA(x1, x2, x3, x4, x5, x6) = U2_GA(x2, x5, x6)
INORDER_IN_GA(x1, x2) = INORDER_IN_GA(x1)
APPEND_IN_GGA(x1, x2, x3) = APPEND_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5) = U4_GGA(x1, x5)
U1_GA(x1, x2, x3, x4, x5) = U1_GA(x2, x3, x5)
U3_GA(x1, x2, x3, x4, x5) = U3_GA(x5)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 2 SCCs with 4 less nodes.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

APPEND_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APPEND_IN_GGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

inorder_in_ga(nil, []) → inorder_out_ga(nil, [])
inorder_in_ga(tree(L, V, R), I) → U1_ga(L, V, R, I, inorder_in_ga(L, LI))
U1_ga(L, V, R, I, inorder_out_ga(L, LI)) → U2_ga(L, V, R, I, LI, inorder_in_ga(R, RI))
U2_ga(L, V, R, I, LI, inorder_out_ga(R, RI)) → U3_ga(L, V, R, I, append_in_gga(LI, .(V, RI), I))
append_in_gga([], X, X) → append_out_gga([], X, X)
append_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, append_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, append_out_gga(Xs, Ys, Zs)) → append_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_ga(L, V, R, I, append_out_gga(LI, .(V, RI), I)) → inorder_out_ga(tree(L, V, R), I)

The argument filtering Pi contains the following mapping:
inorder_in_ga(x1, x2) = inorder_in_ga(x1)
nil = nil
inorder_out_ga(x1, x2) = inorder_out_ga(x2)
tree(x1, x2, x3) = tree(x1, x2, x3)
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x2, x3, x5)
U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x2, x5, x6)
U3_ga(x1, x2, x3, x4, x5) = U3_ga(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)
U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x5)
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

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

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

APPEND_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APPEND_IN_GGA(Xs, Ys, Zs)

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

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

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

APPEND_IN_GGA(.(X, Xs), Ys) → APPEND_IN_GGA(Xs, Ys)

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:

APPEND_IN_GGA(.(X, Xs), Ys) → APPEND_IN_GGA(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:

INORDER_IN_GA(tree(L, V, R), I) → INORDER_IN_GA(L, LI)
U1_GA(L, V, R, I, inorder_out_ga(L, LI)) → INORDER_IN_GA(R, RI)
INORDER_IN_GA(tree(L, V, R), I) → U1_GA(L, V, R, I, inorder_in_ga(L, LI))

The TRS R consists of the following rules:

inorder_in_ga(nil, []) → inorder_out_ga(nil, [])
inorder_in_ga(tree(L, V, R), I) → U1_ga(L, V, R, I, inorder_in_ga(L, LI))
U1_ga(L, V, R, I, inorder_out_ga(L, LI)) → U2_ga(L, V, R, I, LI, inorder_in_ga(R, RI))
U2_ga(L, V, R, I, LI, inorder_out_ga(R, RI)) → U3_ga(L, V, R, I, append_in_gga(LI, .(V, RI), I))
append_in_gga([], X, X) → append_out_gga([], X, X)
append_in_gga(.(X, Xs), Ys, .(X, Zs)) → U4_gga(X, Xs, Ys, Zs, append_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, append_out_gga(Xs, Ys, Zs)) → append_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_ga(L, V, R, I, append_out_gga(LI, .(V, RI), I)) → inorder_out_ga(tree(L, V, R), I)

The argument filtering Pi contains the following mapping:
inorder_in_ga(x1, x2) = inorder_in_ga(x1)
nil = nil
inorder_out_ga(x1, x2) = inorder_out_ga(x2)
tree(x1, x2, x3) = tree(x1, x2, x3)
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x2, x3, x5)
U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x2, x5, x6)
U3_ga(x1, x2, x3, x4, x5) = U3_ga(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)
U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x5)
INORDER_IN_GA(x1, x2) = INORDER_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5) = U1_GA(x2, x3, x5)

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

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ PiDPToQDPProof

                  ↳ QDP

                    ↳ QDPSizeChangeProof

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

U1_GA(V, R, inorder_out_ga(LI)) → INORDER_IN_GA(R)
INORDER_IN_GA(tree(L, V, R)) → U1_GA(V, R, inorder_in_ga(L))
INORDER_IN_GA(tree(L, V, R)) → INORDER_IN_GA(L)

The TRS R consists of the following rules:

inorder_in_ga(nil) → inorder_out_ga([])
inorder_in_ga(tree(L, V, R)) → U1_ga(V, R, inorder_in_ga(L))
U1_ga(V, R, inorder_out_ga(LI)) → U2_ga(V, LI, inorder_in_ga(R))
U2_ga(V, LI, inorder_out_ga(RI)) → U3_ga(append_in_gga(LI, .(V, RI)))
append_in_gga([], X) → append_out_gga(X)
append_in_gga(.(X, Xs), Ys) → U4_gga(X, append_in_gga(Xs, Ys))
U4_gga(X, append_out_gga(Zs)) → append_out_gga(.(X, Zs))
U3_ga(append_out_gga(I)) → inorder_out_ga(I)

The set Q consists of the following terms:

inorder_in_ga(x0)
U1_ga(x0, x1, x2)
U2_ga(x0, x1, x2)
append_in_gga(x0, x1)
U4_gga(x0, x1)
U3_ga(x0)

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:

INORDER_IN_GA(tree(L, V, R)) → INORDER_IN_GA(L)
The graph contains the following edges 1 > 1
INORDER_IN_GA(tree(L, V, R)) → U1_GA(V, R, inorder_in_ga(L))
The graph contains the following edges 1 > 1, 1 > 2
U1_GA(V, R, inorder_out_ga(LI)) → INORDER_IN_GA(R)
The graph contains the following edges 2 >= 1