Left Termination proof of /tmp/tmpMOrmB0/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,b)
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_gg(T, -(Xs, [])))
preorder_dl_in_gg(nil, -(X, X)) → preorder_dl_out_gg(nil, -(X, X))
preorder_dl_in_gg(tree(L, X, R), -(.(X, Xs), Zs)) → U2_gg(L, X, R, Xs, Zs, preorder_dl_in_gg(L, -(Xs, Ys)))
U2_gg(L, X, R, Xs, Zs, preorder_dl_out_gg(L, -(Xs, Ys))) → U3_gg(L, X, R, Xs, Zs, preorder_dl_in_gg(R, -(Ys, Zs)))
U3_gg(L, X, R, Xs, Zs, preorder_dl_out_gg(R, -(Ys, Zs))) → preorder_dl_out_gg(tree(L, X, R), -(.(X, Xs), Zs))
U1_ga(T, Xs, preorder_dl_out_gg(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_gg(T, -(Xs, [])))
preorder_dl_in_gg(nil, -(X, X)) → preorder_dl_out_gg(nil, -(X, X))
preorder_dl_in_gg(tree(L, X, R), -(.(X, Xs), Zs)) → U2_gg(L, X, R, Xs, Zs, preorder_dl_in_gg(L, -(Xs, Ys)))
U2_gg(L, X, R, Xs, Zs, preorder_dl_out_gg(L, -(Xs, Ys))) → U3_gg(L, X, R, Xs, Zs, preorder_dl_in_gg(R, -(Ys, Zs)))
U3_gg(L, X, R, Xs, Zs, preorder_dl_out_gg(R, -(Ys, Zs))) → preorder_dl_out_gg(tree(L, X, R), -(.(X, Xs), Zs))
U1_ga(T, Xs, preorder_dl_out_gg(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_gg(x1, x2) = preorder_dl_in_gg(x1, x2)
-(x1, x2) = -
nil = nil
preorder_dl_out_gg(x1, x2) = preorder_dl_out_gg
tree(x1, x2, x3) = tree(x1, x2, x3)
U2_gg(x1, x2, x3, x4, x5, x6) = U2_gg(x3, x6)
U3_gg(x1, x2, x3, x4, x5, x6) = U3_gg(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_gg(T, -(Xs, [])))
PREORDER_IN_GA(T, Xs) → PREORDER_DL_IN_GG(T, -(Xs, []))
PREORDER_DL_IN_GG(tree(L, X, R), -(.(X, Xs), Zs)) → U2_GG(L, X, R, Xs, Zs, preorder_dl_in_gg(L, -(Xs, Ys)))
PREORDER_DL_IN_GG(tree(L, X, R), -(.(X, Xs), Zs)) → PREORDER_DL_IN_GG(L, -(Xs, Ys))
U2_GG(L, X, R, Xs, Zs, preorder_dl_out_gg(L, -(Xs, Ys))) → U3_GG(L, X, R, Xs, Zs, preorder_dl_in_gg(R, -(Ys, Zs)))
U2_GG(L, X, R, Xs, Zs, preorder_dl_out_gg(L, -(Xs, Ys))) → PREORDER_DL_IN_GG(R, -(Ys, Zs))

The TRS R consists of the following rules:

preorder_in_ga(T, Xs) → U1_ga(T, Xs, preorder_dl_in_gg(T, -(Xs, [])))
preorder_dl_in_gg(nil, -(X, X)) → preorder_dl_out_gg(nil, -(X, X))
preorder_dl_in_gg(tree(L, X, R), -(.(X, Xs), Zs)) → U2_gg(L, X, R, Xs, Zs, preorder_dl_in_gg(L, -(Xs, Ys)))
U2_gg(L, X, R, Xs, Zs, preorder_dl_out_gg(L, -(Xs, Ys))) → U3_gg(L, X, R, Xs, Zs, preorder_dl_in_gg(R, -(Ys, Zs)))
U3_gg(L, X, R, Xs, Zs, preorder_dl_out_gg(R, -(Ys, Zs))) → preorder_dl_out_gg(tree(L, X, R), -(.(X, Xs), Zs))
U1_ga(T, Xs, preorder_dl_out_gg(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_gg(x1, x2) = preorder_dl_in_gg(x1, x2)
-(x1, x2) = -
nil = nil
preorder_dl_out_gg(x1, x2) = preorder_dl_out_gg
tree(x1, x2, x3) = tree(x1, x2, x3)
U2_gg(x1, x2, x3, x4, x5, x6) = U2_gg(x3, x6)
U3_gg(x1, x2, x3, x4, x5, x6) = U3_gg(x6)
preorder_out_ga(x1, x2) = preorder_out_ga
U3_GG(x1, x2, x3, x4, x5, x6) = U3_GG(x6)
U2_GG(x1, x2, x3, x4, x5, x6) = U2_GG(x3, x6)
PREORDER_DL_IN_GG(x1, x2) = PREORDER_DL_IN_GG(x1, x2)
U1_GA(x1, x2, x3) = U1_GA(x3)
PREORDER_IN_GA(x1, x2) = PREORDER_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_gg(T, -(Xs, [])))
PREORDER_IN_GA(T, Xs) → PREORDER_DL_IN_GG(T, -(Xs, []))
PREORDER_DL_IN_GG(tree(L, X, R), -(.(X, Xs), Zs)) → U2_GG(L, X, R, Xs, Zs, preorder_dl_in_gg(L, -(Xs, Ys)))
PREORDER_DL_IN_GG(tree(L, X, R), -(.(X, Xs), Zs)) → PREORDER_DL_IN_GG(L, -(Xs, Ys))
U2_GG(L, X, R, Xs, Zs, preorder_dl_out_gg(L, -(Xs, Ys))) → U3_GG(L, X, R, Xs, Zs, preorder_dl_in_gg(R, -(Ys, Zs)))
U2_GG(L, X, R, Xs, Zs, preorder_dl_out_gg(L, -(Xs, Ys))) → PREORDER_DL_IN_GG(R, -(Ys, Zs))

The TRS R consists of the following rules:

preorder_in_ga(T, Xs) → U1_ga(T, Xs, preorder_dl_in_gg(T, -(Xs, [])))
preorder_dl_in_gg(nil, -(X, X)) → preorder_dl_out_gg(nil, -(X, X))
preorder_dl_in_gg(tree(L, X, R), -(.(X, Xs), Zs)) → U2_gg(L, X, R, Xs, Zs, preorder_dl_in_gg(L, -(Xs, Ys)))
U2_gg(L, X, R, Xs, Zs, preorder_dl_out_gg(L, -(Xs, Ys))) → U3_gg(L, X, R, Xs, Zs, preorder_dl_in_gg(R, -(Ys, Zs)))
U3_gg(L, X, R, Xs, Zs, preorder_dl_out_gg(R, -(Ys, Zs))) → preorder_dl_out_gg(tree(L, X, R), -(.(X, Xs), Zs))
U1_ga(T, Xs, preorder_dl_out_gg(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_gg(x1, x2) = preorder_dl_in_gg(x1, x2)
-(x1, x2) = -
nil = nil
preorder_dl_out_gg(x1, x2) = preorder_dl_out_gg
tree(x1, x2, x3) = tree(x1, x2, x3)
U2_gg(x1, x2, x3, x4, x5, x6) = U2_gg(x3, x6)
U3_gg(x1, x2, x3, x4, x5, x6) = U3_gg(x6)
preorder_out_ga(x1, x2) = preorder_out_ga
U3_GG(x1, x2, x3, x4, x5, x6) = U3_GG(x6)
U2_GG(x1, x2, x3, x4, x5, x6) = U2_GG(x3, x6)
PREORDER_DL_IN_GG(x1, x2) = PREORDER_DL_IN_GG(x1, x2)
U1_GA(x1, x2, x3) = U1_GA(x3)
PREORDER_IN_GA(x1, x2) = PREORDER_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_GG(tree(L, X, R), -(.(X, Xs), Zs)) → U2_GG(L, X, R, Xs, Zs, preorder_dl_in_gg(L, -(Xs, Ys)))
PREORDER_DL_IN_GG(tree(L, X, R), -(.(X, Xs), Zs)) → PREORDER_DL_IN_GG(L, -(Xs, Ys))
U2_GG(L, X, R, Xs, Zs, preorder_dl_out_gg(L, -(Xs, Ys))) → PREORDER_DL_IN_GG(R, -(Ys, Zs))

The TRS R consists of the following rules:

preorder_in_ga(T, Xs) → U1_ga(T, Xs, preorder_dl_in_gg(T, -(Xs, [])))
preorder_dl_in_gg(nil, -(X, X)) → preorder_dl_out_gg(nil, -(X, X))
preorder_dl_in_gg(tree(L, X, R), -(.(X, Xs), Zs)) → U2_gg(L, X, R, Xs, Zs, preorder_dl_in_gg(L, -(Xs, Ys)))
U2_gg(L, X, R, Xs, Zs, preorder_dl_out_gg(L, -(Xs, Ys))) → U3_gg(L, X, R, Xs, Zs, preorder_dl_in_gg(R, -(Ys, Zs)))
U3_gg(L, X, R, Xs, Zs, preorder_dl_out_gg(R, -(Ys, Zs))) → preorder_dl_out_gg(tree(L, X, R), -(.(X, Xs), Zs))
U1_ga(T, Xs, preorder_dl_out_gg(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_gg(x1, x2) = preorder_dl_in_gg(x1, x2)
-(x1, x2) = -
nil = nil
preorder_dl_out_gg(x1, x2) = preorder_dl_out_gg
tree(x1, x2, x3) = tree(x1, x2, x3)
U2_gg(x1, x2, x3, x4, x5, x6) = U2_gg(x3, x6)
U3_gg(x1, x2, x3, x4, x5, x6) = U3_gg(x6)
preorder_out_ga(x1, x2) = preorder_out_ga
U2_GG(x1, x2, x3, x4, x5, x6) = U2_GG(x3, x6)
PREORDER_DL_IN_GG(x1, x2) = PREORDER_DL_IN_GG(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:

PREORDER_DL_IN_GG(tree(L, X, R), -(.(X, Xs), Zs)) → U2_GG(L, X, R, Xs, Zs, preorder_dl_in_gg(L, -(Xs, Ys)))
PREORDER_DL_IN_GG(tree(L, X, R), -(.(X, Xs), Zs)) → PREORDER_DL_IN_GG(L, -(Xs, Ys))
U2_GG(L, X, R, Xs, Zs, preorder_dl_out_gg(L, -(Xs, Ys))) → PREORDER_DL_IN_GG(R, -(Ys, Zs))

The TRS R consists of the following rules:

preorder_dl_in_gg(nil, -(X, X)) → preorder_dl_out_gg(nil, -(X, X))
preorder_dl_in_gg(tree(L, X, R), -(.(X, Xs), Zs)) → U2_gg(L, X, R, Xs, Zs, preorder_dl_in_gg(L, -(Xs, Ys)))
U2_gg(L, X, R, Xs, Zs, preorder_dl_out_gg(L, -(Xs, Ys))) → U3_gg(L, X, R, Xs, Zs, preorder_dl_in_gg(R, -(Ys, Zs)))
U3_gg(L, X, R, Xs, Zs, preorder_dl_out_gg(R, -(Ys, Zs))) → preorder_dl_out_gg(tree(L, X, R), -(.(X, Xs), Zs))

The argument filtering Pi contains the following mapping:
preorder_dl_in_gg(x1, x2) = preorder_dl_in_gg(x1, x2)
-(x1, x2) = -
nil = nil
preorder_dl_out_gg(x1, x2) = preorder_dl_out_gg
tree(x1, x2, x3) = tree(x1, x2, x3)
U2_gg(x1, x2, x3, x4, x5, x6) = U2_gg(x3, x6)
U3_gg(x1, x2, x3, x4, x5, x6) = U3_gg(x6)
U2_GG(x1, x2, x3, x4, x5, x6) = U2_GG(x3, x6)
PREORDER_DL_IN_GG(x1, x2) = PREORDER_DL_IN_GG(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:

PREORDER_DL_IN_GG(tree(L, X, R), -) → U2_GG(R, preorder_dl_in_gg(L, -))
PREORDER_DL_IN_GG(tree(L, X, R), -) → PREORDER_DL_IN_GG(L, -)
U2_GG(R, preorder_dl_out_gg) → PREORDER_DL_IN_GG(R, -)

The TRS R consists of the following rules:

preorder_dl_in_gg(nil, -) → preorder_dl_out_gg
preorder_dl_in_gg(tree(L, X, R), -) → U2_gg(R, preorder_dl_in_gg(L, -))
U2_gg(R, preorder_dl_out_gg) → U3_gg(preorder_dl_in_gg(R, -))
U3_gg(preorder_dl_out_gg) → preorder_dl_out_gg

The set Q consists of the following terms:

preorder_dl_in_gg(x0, x1)
U2_gg(x0, x1)
U3_gg(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:

U2_GG(R, preorder_dl_out_gg) → PREORDER_DL_IN_GG(R, -)
The graph contains the following edges 1 >= 1
PREORDER_DL_IN_GG(tree(L, X, R), -) → PREORDER_DL_IN_GG(L, -)
The graph contains the following edges 1 > 1, 2 >= 2
PREORDER_DL_IN_GG(tree(L, X, R), -) → U2_GG(R, preorder_dl_in_gg(L, -))
The graph contains the following edges 1 > 1