Left Termination proof of /tmp/tmpEgm0Z6/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]. With regard to the inferred argument filtering the predicates were used in the following modes:
preorder_in: (b,f)
preorder_dl_in: (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

preorder_in_ga(T, Xs) → U1_ga(T, Xs, preorder_dl_in_ga(T, -(Xs, [])))
preorder_dl_in_ga(nil, -(X, X)) → preorder_dl_out_ga(nil, -(X, X))
preorder_dl_in_ga(tree(L, X, R), -(.(X, Xs), Zs)) → U2_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))
U2_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) → U3_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs)))
U3_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(R, -(Ys, Zs))) → preorder_dl_out_ga(tree(L, X, R), -(.(X, Xs), Zs))
U1_ga(T, Xs, preorder_dl_out_ga(T, -(Xs, []))) → preorder_out_ga(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_ga(T, Xs) → U1_ga(T, Xs, preorder_dl_in_ga(T, -(Xs, [])))
preorder_dl_in_ga(nil, -(X, X)) → preorder_dl_out_ga(nil, -(X, X))
preorder_dl_in_ga(tree(L, X, R), -(.(X, Xs), Zs)) → U2_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))
U2_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) → U3_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs)))
U3_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(R, -(Ys, Zs))) → preorder_dl_out_ga(tree(L, X, R), -(.(X, Xs), Zs))
U1_ga(T, Xs, preorder_dl_out_ga(T, -(Xs, []))) → preorder_out_ga(T, Xs)

The argument filtering Pi contains the following mapping:
preorder_in_ga(x1, x2) = preorder_in_ga(x1)
U1_ga(x1, x2, x3) = U1_ga(x3)
preorder_dl_in_ga(x1, x2) = preorder_dl_in_ga(x1)
nil = nil
preorder_dl_out_ga(x1, x2) = preorder_dl_out_ga
tree(x1, x2, x3) = tree(x1, x2, x3)
U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x3, x6)
U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x6)
preorder_out_ga(x1, x2) = preorder_out_ga

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_GA(T, Xs) → U1_GA(T, Xs, preorder_dl_in_ga(T, -(Xs, [])))
PREORDER_IN_GA(T, Xs) → PREORDER_DL_IN_GA(T, -(Xs, []))
PREORDER_DL_IN_GA(tree(L, X, R), -(.(X, Xs), Zs)) → U2_GA(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))
PREORDER_DL_IN_GA(tree(L, X, R), -(.(X, Xs), Zs)) → PREORDER_DL_IN_GA(L, -(Xs, Ys))
U2_GA(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) → U3_GA(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs)))
U2_GA(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) → PREORDER_DL_IN_GA(R, -(Ys, Zs))

The TRS R consists of the following rules:

preorder_in_ga(T, Xs) → U1_ga(T, Xs, preorder_dl_in_ga(T, -(Xs, [])))
preorder_dl_in_ga(nil, -(X, X)) → preorder_dl_out_ga(nil, -(X, X))
preorder_dl_in_ga(tree(L, X, R), -(.(X, Xs), Zs)) → U2_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))
U2_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) → U3_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs)))
U3_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(R, -(Ys, Zs))) → preorder_dl_out_ga(tree(L, X, R), -(.(X, Xs), Zs))
U1_ga(T, Xs, preorder_dl_out_ga(T, -(Xs, []))) → preorder_out_ga(T, Xs)

The argument filtering Pi contains the following mapping:
preorder_in_ga(x1, x2) = preorder_in_ga(x1)
U1_ga(x1, x2, x3) = U1_ga(x3)
preorder_dl_in_ga(x1, x2) = preorder_dl_in_ga(x1)
nil = nil
preorder_dl_out_ga(x1, x2) = preorder_dl_out_ga
tree(x1, x2, x3) = tree(x1, x2, x3)
U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x3, x6)
U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x6)
preorder_out_ga(x1, x2) = preorder_out_ga
U2_GA(x1, x2, x3, x4, x5, x6) = U2_GA(x3, x6)
U3_GA(x1, x2, x3, x4, x5, x6) = U3_GA(x6)
U1_GA(x1, x2, x3) = U1_GA(x3)
PREORDER_IN_GA(x1, x2) = PREORDER_IN_GA(x1)
PREORDER_DL_IN_GA(x1, x2) = PREORDER_DL_IN_GA(x1)

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_GA(T, Xs) → U1_GA(T, Xs, preorder_dl_in_ga(T, -(Xs, [])))
PREORDER_IN_GA(T, Xs) → PREORDER_DL_IN_GA(T, -(Xs, []))
PREORDER_DL_IN_GA(tree(L, X, R), -(.(X, Xs), Zs)) → U2_GA(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))
PREORDER_DL_IN_GA(tree(L, X, R), -(.(X, Xs), Zs)) → PREORDER_DL_IN_GA(L, -(Xs, Ys))
U2_GA(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) → U3_GA(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs)))
U2_GA(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) → PREORDER_DL_IN_GA(R, -(Ys, Zs))

The TRS R consists of the following rules:

preorder_in_ga(T, Xs) → U1_ga(T, Xs, preorder_dl_in_ga(T, -(Xs, [])))
preorder_dl_in_ga(nil, -(X, X)) → preorder_dl_out_ga(nil, -(X, X))
preorder_dl_in_ga(tree(L, X, R), -(.(X, Xs), Zs)) → U2_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))
U2_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) → U3_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs)))
U3_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(R, -(Ys, Zs))) → preorder_dl_out_ga(tree(L, X, R), -(.(X, Xs), Zs))
U1_ga(T, Xs, preorder_dl_out_ga(T, -(Xs, []))) → preorder_out_ga(T, Xs)

The argument filtering Pi contains the following mapping:
preorder_in_ga(x1, x2) = preorder_in_ga(x1)
U1_ga(x1, x2, x3) = U1_ga(x3)
preorder_dl_in_ga(x1, x2) = preorder_dl_in_ga(x1)
nil = nil
preorder_dl_out_ga(x1, x2) = preorder_dl_out_ga
tree(x1, x2, x3) = tree(x1, x2, x3)
U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x3, x6)
U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x6)
preorder_out_ga(x1, x2) = preorder_out_ga
U2_GA(x1, x2, x3, x4, x5, x6) = U2_GA(x3, x6)
U3_GA(x1, x2, x3, x4, x5, x6) = U3_GA(x6)
U1_GA(x1, x2, x3) = U1_GA(x3)
PREORDER_IN_GA(x1, x2) = PREORDER_IN_GA(x1)
PREORDER_DL_IN_GA(x1, x2) = PREORDER_DL_IN_GA(x1)

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_GA(tree(L, X, R), -(.(X, Xs), Zs)) → PREORDER_DL_IN_GA(L, -(Xs, Ys))
U2_GA(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) → PREORDER_DL_IN_GA(R, -(Ys, Zs))
PREORDER_DL_IN_GA(tree(L, X, R), -(.(X, Xs), Zs)) → U2_GA(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))

The TRS R consists of the following rules:

preorder_in_ga(T, Xs) → U1_ga(T, Xs, preorder_dl_in_ga(T, -(Xs, [])))
preorder_dl_in_ga(nil, -(X, X)) → preorder_dl_out_ga(nil, -(X, X))
preorder_dl_in_ga(tree(L, X, R), -(.(X, Xs), Zs)) → U2_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))
U2_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) → U3_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs)))
U3_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(R, -(Ys, Zs))) → preorder_dl_out_ga(tree(L, X, R), -(.(X, Xs), Zs))
U1_ga(T, Xs, preorder_dl_out_ga(T, -(Xs, []))) → preorder_out_ga(T, Xs)

The argument filtering Pi contains the following mapping:
preorder_in_ga(x1, x2) = preorder_in_ga(x1)
U1_ga(x1, x2, x3) = U1_ga(x3)
preorder_dl_in_ga(x1, x2) = preorder_dl_in_ga(x1)
nil = nil
preorder_dl_out_ga(x1, x2) = preorder_dl_out_ga
tree(x1, x2, x3) = tree(x1, x2, x3)
U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x3, x6)
U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x6)
preorder_out_ga(x1, x2) = preorder_out_ga
U2_GA(x1, x2, x3, x4, x5, x6) = U2_GA(x3, x6)
PREORDER_DL_IN_GA(x1, x2) = PREORDER_DL_IN_GA(x1)

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_GA(tree(L, X, R), -(.(X, Xs), Zs)) → PREORDER_DL_IN_GA(L, -(Xs, Ys))
U2_GA(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) → PREORDER_DL_IN_GA(R, -(Ys, Zs))
PREORDER_DL_IN_GA(tree(L, X, R), -(.(X, Xs), Zs)) → U2_GA(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))

The TRS R consists of the following rules:

preorder_dl_in_ga(nil, -(X, X)) → preorder_dl_out_ga(nil, -(X, X))
preorder_dl_in_ga(tree(L, X, R), -(.(X, Xs), Zs)) → U2_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))
U2_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) → U3_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs)))
U3_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(R, -(Ys, Zs))) → preorder_dl_out_ga(tree(L, X, R), -(.(X, Xs), Zs))

The argument filtering Pi contains the following mapping:
preorder_dl_in_ga(x1, x2) = preorder_dl_in_ga(x1)
nil = nil
preorder_dl_out_ga(x1, x2) = preorder_dl_out_ga
tree(x1, x2, x3) = tree(x1, x2, x3)
U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x3, x6)
U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x6)
U2_GA(x1, x2, x3, x4, x5, x6) = U2_GA(x3, x6)
PREORDER_DL_IN_GA(x1, x2) = PREORDER_DL_IN_GA(x1)

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_GA(tree(L, X, R)) → PREORDER_DL_IN_GA(L)
U2_GA(R, preorder_dl_out_ga) → PREORDER_DL_IN_GA(R)
PREORDER_DL_IN_GA(tree(L, X, R)) → U2_GA(R, preorder_dl_in_ga(L))

The TRS R consists of the following rules:

preorder_dl_in_ga(nil) → preorder_dl_out_ga
preorder_dl_in_ga(tree(L, X, R)) → U2_ga(R, preorder_dl_in_ga(L))
U2_ga(R, preorder_dl_out_ga) → U3_ga(preorder_dl_in_ga(R))
U3_ga(preorder_dl_out_ga) → preorder_dl_out_ga

The set Q consists of the following terms:

preorder_dl_in_ga(x0)
U2_ga(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:

PREORDER_DL_IN_GA(tree(L, X, R)) → PREORDER_DL_IN_GA(L)
The graph contains the following edges 1 > 1
PREORDER_DL_IN_GA(tree(L, X, R)) → U2_GA(R, preorder_dl_in_ga(L))
The graph contains the following edges 1 > 1
U2_GA(R, preorder_dl_out_ga) → PREORDER_DL_IN_GA(R)
The graph contains the following edges 1 >= 1