Left Termination proof of /tmp/tmpAsCHyO/preorder

Left Termination of the query pattern preorder_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

↳ Prolog

  ↳ PrologToPiTRSProof

Clauses:

preorder(T, Xs) :- preorder_dl(T, -(Xs, [])).
preorder_dl(nil, -(X, X)).
preorder_dl(tree(L, X, R), -(.(X, Xs), Zs)) :- ','(preorder_dl(L, -(Xs, Ys)), preorder_dl(R, -(Ys, Zs))).

Queries:

preorder(g,a).

We use the technique of [30].Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

preorder_in(T, Xs) → U1(T, Xs, preorder_dl_in(T, -(Xs, [])))
preorder_dl_in(tree(L, X, R), -(.(X, Xs), Zs)) → U2(L, X, R, Xs, Zs, preorder_dl_in(L, -(Xs, Ys)))
preorder_dl_in(nil, -(X, X)) → preorder_dl_out(nil, -(X, X))
U2(L, X, R, Xs, Zs, preorder_dl_out(L, -(Xs, Ys))) → U3(L, X, R, Xs, Zs, preorder_dl_in(R, -(Ys, Zs)))
U3(L, X, R, Xs, Zs, preorder_dl_out(R, -(Ys, Zs))) → preorder_dl_out(tree(L, X, R), -(.(X, Xs), Zs))
U1(T, Xs, preorder_dl_out(T, -(Xs, []))) → preorder_out(T, Xs)

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:

preorder_in(T, Xs) → U1(T, Xs, preorder_dl_in(T, -(Xs, [])))
preorder_dl_in(tree(L, X, R), -(.(X, Xs), Zs)) → U2(L, X, R, Xs, Zs, preorder_dl_in(L, -(Xs, Ys)))
preorder_dl_in(nil, -(X, X)) → preorder_dl_out(nil, -(X, X))
U2(L, X, R, Xs, Zs, preorder_dl_out(L, -(Xs, Ys))) → U3(L, X, R, Xs, Zs, preorder_dl_in(R, -(Ys, Zs)))
U3(L, X, R, Xs, Zs, preorder_dl_out(R, -(Ys, Zs))) → preorder_dl_out(tree(L, X, R), -(.(X, Xs), Zs))
U1(T, Xs, preorder_dl_out(T, -(Xs, []))) → preorder_out(T, Xs)

The argument filtering Pi contains the following mapping:
preorder_in(x1, x2) = preorder_in(x1)
U1(x1, x2, x3) = U1(x3)
preorder_dl_in(x1, x2) = preorder_dl_in(x1)
-(x1, x2) = -(x1, x2)
[] = []
tree(x1, x2, x3) = tree(x1, x2, x3)
.(x1, x2) = .(x1, x2)
U2(x1, x2, x3, x4, x5, x6) = U2(x3, x6)
nil = nil
preorder_dl_out(x1, x2) = preorder_dl_out
U3(x1, x2, x3, x4, x5, x6) = U3(x6)
preorder_out(x1, x2) = preorder_out

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

PREORDER_IN(T, Xs) → U1¹(T, Xs, preorder_dl_in(T, -(Xs, [])))
PREORDER_IN(T, Xs) → PREORDER_DL_IN(T, -(Xs, []))
PREORDER_DL_IN(tree(L, X, R), -(.(X, Xs), Zs)) → U2¹(L, X, R, Xs, Zs, preorder_dl_in(L, -(Xs, Ys)))
PREORDER_DL_IN(tree(L, X, R), -(.(X, Xs), Zs)) → PREORDER_DL_IN(L, -(Xs, Ys))
U2¹(L, X, R, Xs, Zs, preorder_dl_out(L, -(Xs, Ys))) → U3¹(L, X, R, Xs, Zs, preorder_dl_in(R, -(Ys, Zs)))
U2¹(L, X, R, Xs, Zs, preorder_dl_out(L, -(Xs, Ys))) → PREORDER_DL_IN(R, -(Ys, Zs))

The TRS R consists of the following rules:

preorder_in(T, Xs) → U1(T, Xs, preorder_dl_in(T, -(Xs, [])))
preorder_dl_in(tree(L, X, R), -(.(X, Xs), Zs)) → U2(L, X, R, Xs, Zs, preorder_dl_in(L, -(Xs, Ys)))
preorder_dl_in(nil, -(X, X)) → preorder_dl_out(nil, -(X, X))
U2(L, X, R, Xs, Zs, preorder_dl_out(L, -(Xs, Ys))) → U3(L, X, R, Xs, Zs, preorder_dl_in(R, -(Ys, Zs)))
U3(L, X, R, Xs, Zs, preorder_dl_out(R, -(Ys, Zs))) → preorder_dl_out(tree(L, X, R), -(.(X, Xs), Zs))
U1(T, Xs, preorder_dl_out(T, -(Xs, []))) → preorder_out(T, Xs)

The argument filtering Pi contains the following mapping:
preorder_in(x1, x2) = preorder_in(x1)
U1(x1, x2, x3) = U1(x3)
preorder_dl_in(x1, x2) = preorder_dl_in(x1)
-(x1, x2) = -(x1, x2)
[] = []
tree(x1, x2, x3) = tree(x1, x2, x3)
.(x1, x2) = .(x1, x2)
U2(x1, x2, x3, x4, x5, x6) = U2(x3, x6)
nil = nil
preorder_dl_out(x1, x2) = preorder_dl_out
U3(x1, x2, x3, x4, x5, x6) = U3(x6)
preorder_out(x1, x2) = preorder_out
PREORDER_IN(x1, x2) = PREORDER_IN(x1)
PREORDER_DL_IN(x1, x2) = PREORDER_DL_IN(x1)
U2¹(x1, x2, x3, x4, x5, x6) = U2¹(x3, x6)
U3¹(x1, x2, x3, x4, x5, x6) = U3¹(x6)
U1¹(x1, x2, x3) = U1¹(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:

PREORDER_IN(T, Xs) → U1¹(T, Xs, preorder_dl_in(T, -(Xs, [])))
PREORDER_IN(T, Xs) → PREORDER_DL_IN(T, -(Xs, []))
PREORDER_DL_IN(tree(L, X, R), -(.(X, Xs), Zs)) → U2¹(L, X, R, Xs, Zs, preorder_dl_in(L, -(Xs, Ys)))
PREORDER_DL_IN(tree(L, X, R), -(.(X, Xs), Zs)) → PREORDER_DL_IN(L, -(Xs, Ys))
U2¹(L, X, R, Xs, Zs, preorder_dl_out(L, -(Xs, Ys))) → U3¹(L, X, R, Xs, Zs, preorder_dl_in(R, -(Ys, Zs)))
U2¹(L, X, R, Xs, Zs, preorder_dl_out(L, -(Xs, Ys))) → PREORDER_DL_IN(R, -(Ys, Zs))

The TRS R consists of the following rules:

preorder_in(T, Xs) → U1(T, Xs, preorder_dl_in(T, -(Xs, [])))
preorder_dl_in(tree(L, X, R), -(.(X, Xs), Zs)) → U2(L, X, R, Xs, Zs, preorder_dl_in(L, -(Xs, Ys)))
preorder_dl_in(nil, -(X, X)) → preorder_dl_out(nil, -(X, X))
U2(L, X, R, Xs, Zs, preorder_dl_out(L, -(Xs, Ys))) → U3(L, X, R, Xs, Zs, preorder_dl_in(R, -(Ys, Zs)))
U3(L, X, R, Xs, Zs, preorder_dl_out(R, -(Ys, Zs))) → preorder_dl_out(tree(L, X, R), -(.(X, Xs), Zs))
U1(T, Xs, preorder_dl_out(T, -(Xs, []))) → preorder_out(T, Xs)

The argument filtering Pi contains the following mapping:
preorder_in(x1, x2) = preorder_in(x1)
U1(x1, x2, x3) = U1(x3)
preorder_dl_in(x1, x2) = preorder_dl_in(x1)
-(x1, x2) = -(x1, x2)
[] = []
tree(x1, x2, x3) = tree(x1, x2, x3)
.(x1, x2) = .(x1, x2)
U2(x1, x2, x3, x4, x5, x6) = U2(x3, x6)
nil = nil
preorder_dl_out(x1, x2) = preorder_dl_out
U3(x1, x2, x3, x4, x5, x6) = U3(x6)
preorder_out(x1, x2) = preorder_out
PREORDER_IN(x1, x2) = PREORDER_IN(x1)
PREORDER_DL_IN(x1, x2) = PREORDER_DL_IN(x1)
U2¹(x1, x2, x3, x4, x5, x6) = U2¹(x3, x6)
U3¹(x1, x2, x3, x4, x5, x6) = U3¹(x6)
U1¹(x1, x2, x3) = U1¹(x3)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 1 SCC with 3 less nodes.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ PiDP

              ↳ UsableRulesProof

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

PREORDER_DL_IN(tree(L, X, R), -(.(X, Xs), Zs)) → PREORDER_DL_IN(L, -(Xs, Ys))
U2¹(L, X, R, Xs, Zs, preorder_dl_out(L, -(Xs, Ys))) → PREORDER_DL_IN(R, -(Ys, Zs))
PREORDER_DL_IN(tree(L, X, R), -(.(X, Xs), Zs)) → U2¹(L, X, R, Xs, Zs, preorder_dl_in(L, -(Xs, Ys)))

The TRS R consists of the following rules:

preorder_in(T, Xs) → U1(T, Xs, preorder_dl_in(T, -(Xs, [])))
preorder_dl_in(tree(L, X, R), -(.(X, Xs), Zs)) → U2(L, X, R, Xs, Zs, preorder_dl_in(L, -(Xs, Ys)))
preorder_dl_in(nil, -(X, X)) → preorder_dl_out(nil, -(X, X))
U2(L, X, R, Xs, Zs, preorder_dl_out(L, -(Xs, Ys))) → U3(L, X, R, Xs, Zs, preorder_dl_in(R, -(Ys, Zs)))
U3(L, X, R, Xs, Zs, preorder_dl_out(R, -(Ys, Zs))) → preorder_dl_out(tree(L, X, R), -(.(X, Xs), Zs))
U1(T, Xs, preorder_dl_out(T, -(Xs, []))) → preorder_out(T, Xs)

The argument filtering Pi contains the following mapping:
preorder_in(x1, x2) = preorder_in(x1)
U1(x1, x2, x3) = U1(x3)
preorder_dl_in(x1, x2) = preorder_dl_in(x1)
-(x1, x2) = -(x1, x2)
[] = []
tree(x1, x2, x3) = tree(x1, x2, x3)
.(x1, x2) = .(x1, x2)
U2(x1, x2, x3, x4, x5, x6) = U2(x3, x6)
nil = nil
preorder_dl_out(x1, x2) = preorder_dl_out
U3(x1, x2, x3, x4, x5, x6) = U3(x6)
preorder_out(x1, x2) = preorder_out
PREORDER_DL_IN(x1, x2) = PREORDER_DL_IN(x1)
U2¹(x1, x2, x3, x4, x5, x6) = U2¹(x3, x6)

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:

PREORDER_DL_IN(tree(L, X, R), -(.(X, Xs), Zs)) → PREORDER_DL_IN(L, -(Xs, Ys))
U2¹(L, X, R, Xs, Zs, preorder_dl_out(L, -(Xs, Ys))) → PREORDER_DL_IN(R, -(Ys, Zs))
PREORDER_DL_IN(tree(L, X, R), -(.(X, Xs), Zs)) → U2¹(L, X, R, Xs, Zs, preorder_dl_in(L, -(Xs, Ys)))

The TRS R consists of the following rules:

preorder_dl_in(tree(L, X, R), -(.(X, Xs), Zs)) → U2(L, X, R, Xs, Zs, preorder_dl_in(L, -(Xs, Ys)))
preorder_dl_in(nil, -(X, X)) → preorder_dl_out(nil, -(X, X))
U2(L, X, R, Xs, Zs, preorder_dl_out(L, -(Xs, Ys))) → U3(L, X, R, Xs, Zs, preorder_dl_in(R, -(Ys, Zs)))
U3(L, X, R, Xs, Zs, preorder_dl_out(R, -(Ys, Zs))) → preorder_dl_out(tree(L, X, R), -(.(X, Xs), Zs))

The argument filtering Pi contains the following mapping:
preorder_dl_in(x1, x2) = preorder_dl_in(x1)
-(x1, x2) = -(x1, x2)
tree(x1, x2, x3) = tree(x1, x2, x3)
.(x1, x2) = .(x1, x2)
U2(x1, x2, x3, x4, x5, x6) = U2(x3, x6)
nil = nil
preorder_dl_out(x1, x2) = preorder_dl_out
U3(x1, x2, x3, x4, x5, x6) = U3(x6)
PREORDER_DL_IN(x1, x2) = PREORDER_DL_IN(x1)
U2¹(x1, x2, x3, x4, x5, x6) = U2¹(x3, x6)

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:

PREORDER_DL_IN(tree(L, X, R)) → U2¹(R, preorder_dl_in(L))
U2¹(R, preorder_dl_out) → PREORDER_DL_IN(R)
PREORDER_DL_IN(tree(L, X, R)) → PREORDER_DL_IN(L)

The TRS R consists of the following rules:

preorder_dl_in(tree(L, X, R)) → U2(R, preorder_dl_in(L))
preorder_dl_in(nil) → preorder_dl_out
U2(R, preorder_dl_out) → U3(preorder_dl_in(R))
U3(preorder_dl_out) → preorder_dl_out

The set Q consists of the following terms:

preorder_dl_in(x0)
U2(x0, x1)
U3(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:

PREORDER_DL_IN(tree(L, X, R)) → PREORDER_DL_IN(L)
The graph contains the following edges 1 > 1
PREORDER_DL_IN(tree(L, X, R)) → U2¹(R, preorder_dl_in(L))
The graph contains the following edges 1 > 1
U2¹(R, preorder_dl_out) → PREORDER_DL_IN(R)
The graph contains the following edges 1 >= 1