Left Termination proof of /tmp/tmpH1g-SP/flat.pl

Left Termination of the query pattern flat_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

↳ Prolog

  ↳ PrologToPiTRSProof

Clauses:

flat(niltree, nil).
flat(tree(X, niltree, T), cons(X, Xs)) :- flat(T, Xs).
flat(tree(X, tree(Y, T1, T2), T3), Xs) :- flat(tree(Y, T1, tree(X, T2, T3)), Xs).

Queries:

flat(g,a).

We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
flat_in: (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

flat_in_ga(niltree, nil) → flat_out_ga(niltree, nil)
flat_in_ga(tree(X, niltree, T), cons(X, Xs)) → U1_ga(X, T, Xs, flat_in_ga(T, Xs))
flat_in_ga(tree(X, tree(Y, T1, T2), T3), Xs) → U2_ga(X, Y, T1, T2, T3, Xs, flat_in_ga(tree(Y, T1, tree(X, T2, T3)), Xs))
U2_ga(X, Y, T1, T2, T3, Xs, flat_out_ga(tree(Y, T1, tree(X, T2, T3)), Xs)) → flat_out_ga(tree(X, tree(Y, T1, T2), T3), Xs)
U1_ga(X, T, Xs, flat_out_ga(T, Xs)) → flat_out_ga(tree(X, niltree, T), cons(X, 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:

flat_in_ga(niltree, nil) → flat_out_ga(niltree, nil)
flat_in_ga(tree(X, niltree, T), cons(X, Xs)) → U1_ga(X, T, Xs, flat_in_ga(T, Xs))
flat_in_ga(tree(X, tree(Y, T1, T2), T3), Xs) → U2_ga(X, Y, T1, T2, T3, Xs, flat_in_ga(tree(Y, T1, tree(X, T2, T3)), Xs))
U2_ga(X, Y, T1, T2, T3, Xs, flat_out_ga(tree(Y, T1, tree(X, T2, T3)), Xs)) → flat_out_ga(tree(X, tree(Y, T1, T2), T3), Xs)
U1_ga(X, T, Xs, flat_out_ga(T, Xs)) → flat_out_ga(tree(X, niltree, T), cons(X, Xs))

The argument filtering Pi contains the following mapping:
flat_in_ga(x1, x2) = flat_in_ga(x1)
niltree = niltree
flat_out_ga(x1, x2) = flat_out_ga(x2)
tree(x1, x2, x3) = tree(x1, x2, x3)
U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4, x5, x6, x7) = U2_ga(x7)
cons(x1, x2) = cons(x1, x2)

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

FLAT_IN_GA(tree(X, niltree, T), cons(X, Xs)) → U1_GA(X, T, Xs, flat_in_ga(T, Xs))
FLAT_IN_GA(tree(X, niltree, T), cons(X, Xs)) → FLAT_IN_GA(T, Xs)
FLAT_IN_GA(tree(X, tree(Y, T1, T2), T3), Xs) → U2_GA(X, Y, T1, T2, T3, Xs, flat_in_ga(tree(Y, T1, tree(X, T2, T3)), Xs))
FLAT_IN_GA(tree(X, tree(Y, T1, T2), T3), Xs) → FLAT_IN_GA(tree(Y, T1, tree(X, T2, T3)), Xs)

The TRS R consists of the following rules:

flat_in_ga(niltree, nil) → flat_out_ga(niltree, nil)
flat_in_ga(tree(X, niltree, T), cons(X, Xs)) → U1_ga(X, T, Xs, flat_in_ga(T, Xs))
flat_in_ga(tree(X, tree(Y, T1, T2), T3), Xs) → U2_ga(X, Y, T1, T2, T3, Xs, flat_in_ga(tree(Y, T1, tree(X, T2, T3)), Xs))
U2_ga(X, Y, T1, T2, T3, Xs, flat_out_ga(tree(Y, T1, tree(X, T2, T3)), Xs)) → flat_out_ga(tree(X, tree(Y, T1, T2), T3), Xs)
U1_ga(X, T, Xs, flat_out_ga(T, Xs)) → flat_out_ga(tree(X, niltree, T), cons(X, Xs))

The argument filtering Pi contains the following mapping:
flat_in_ga(x1, x2) = flat_in_ga(x1)
niltree = niltree
flat_out_ga(x1, x2) = flat_out_ga(x2)
tree(x1, x2, x3) = tree(x1, x2, x3)
U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4, x5, x6, x7) = U2_ga(x7)
cons(x1, x2) = cons(x1, x2)
FLAT_IN_GA(x1, x2) = FLAT_IN_GA(x1)
U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4)
U2_GA(x1, x2, x3, x4, x5, x6, x7) = U2_GA(x7)

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:

FLAT_IN_GA(tree(X, niltree, T), cons(X, Xs)) → U1_GA(X, T, Xs, flat_in_ga(T, Xs))
FLAT_IN_GA(tree(X, niltree, T), cons(X, Xs)) → FLAT_IN_GA(T, Xs)
FLAT_IN_GA(tree(X, tree(Y, T1, T2), T3), Xs) → U2_GA(X, Y, T1, T2, T3, Xs, flat_in_ga(tree(Y, T1, tree(X, T2, T3)), Xs))
FLAT_IN_GA(tree(X, tree(Y, T1, T2), T3), Xs) → FLAT_IN_GA(tree(Y, T1, tree(X, T2, T3)), Xs)

The TRS R consists of the following rules:

flat_in_ga(niltree, nil) → flat_out_ga(niltree, nil)
flat_in_ga(tree(X, niltree, T), cons(X, Xs)) → U1_ga(X, T, Xs, flat_in_ga(T, Xs))
flat_in_ga(tree(X, tree(Y, T1, T2), T3), Xs) → U2_ga(X, Y, T1, T2, T3, Xs, flat_in_ga(tree(Y, T1, tree(X, T2, T3)), Xs))
U2_ga(X, Y, T1, T2, T3, Xs, flat_out_ga(tree(Y, T1, tree(X, T2, T3)), Xs)) → flat_out_ga(tree(X, tree(Y, T1, T2), T3), Xs)
U1_ga(X, T, Xs, flat_out_ga(T, Xs)) → flat_out_ga(tree(X, niltree, T), cons(X, Xs))

The argument filtering Pi contains the following mapping:
flat_in_ga(x1, x2) = flat_in_ga(x1)
niltree = niltree
flat_out_ga(x1, x2) = flat_out_ga(x2)
tree(x1, x2, x3) = tree(x1, x2, x3)
U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4, x5, x6, x7) = U2_ga(x7)
cons(x1, x2) = cons(x1, x2)
FLAT_IN_GA(x1, x2) = FLAT_IN_GA(x1)
U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4)
U2_GA(x1, x2, x3, x4, x5, x6, x7) = U2_GA(x7)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 1 SCC with 2 less nodes.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ PiDP

              ↳ UsableRulesProof

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

FLAT_IN_GA(tree(X, tree(Y, T1, T2), T3), Xs) → FLAT_IN_GA(tree(Y, T1, tree(X, T2, T3)), Xs)
FLAT_IN_GA(tree(X, niltree, T), cons(X, Xs)) → FLAT_IN_GA(T, Xs)

The TRS R consists of the following rules:

flat_in_ga(niltree, nil) → flat_out_ga(niltree, nil)
flat_in_ga(tree(X, niltree, T), cons(X, Xs)) → U1_ga(X, T, Xs, flat_in_ga(T, Xs))
flat_in_ga(tree(X, tree(Y, T1, T2), T3), Xs) → U2_ga(X, Y, T1, T2, T3, Xs, flat_in_ga(tree(Y, T1, tree(X, T2, T3)), Xs))
U2_ga(X, Y, T1, T2, T3, Xs, flat_out_ga(tree(Y, T1, tree(X, T2, T3)), Xs)) → flat_out_ga(tree(X, tree(Y, T1, T2), T3), Xs)
U1_ga(X, T, Xs, flat_out_ga(T, Xs)) → flat_out_ga(tree(X, niltree, T), cons(X, Xs))

The argument filtering Pi contains the following mapping:
flat_in_ga(x1, x2) = flat_in_ga(x1)
niltree = niltree
flat_out_ga(x1, x2) = flat_out_ga(x2)
tree(x1, x2, x3) = tree(x1, x2, x3)
U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4, x5, x6, x7) = U2_ga(x7)
cons(x1, x2) = cons(x1, x2)
FLAT_IN_GA(x1, x2) = FLAT_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:

FLAT_IN_GA(tree(X, tree(Y, T1, T2), T3), Xs) → FLAT_IN_GA(tree(Y, T1, tree(X, T2, T3)), Xs)
FLAT_IN_GA(tree(X, niltree, T), cons(X, Xs)) → FLAT_IN_GA(T, Xs)

R is empty.
The argument filtering Pi contains the following mapping:
niltree = niltree
tree(x1, x2, x3) = tree(x1, x2, x3)
cons(x1, x2) = cons(x1, x2)
FLAT_IN_GA(x1, x2) = FLAT_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

                      ↳ UsableRulesReductionPairsProof

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

FLAT_IN_GA(tree(X, niltree, T)) → FLAT_IN_GA(T)
FLAT_IN_GA(tree(X, tree(Y, T1, T2), T3)) → FLAT_IN_GA(tree(Y, T1, tree(X, T2, T3)))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the usable rules with reduction pair processor [15] with a polynomial ordering [25], all dependency pairs and the corresponding usable rules [17] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

FLAT_IN_GA(tree(X, niltree, T)) → FLAT_IN_GA(T)
FLAT_IN_GA(tree(X, tree(Y, T1, T2), T3)) → FLAT_IN_GA(tree(Y, T1, tree(X, T2, T3)))

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [25]:

POL(FLAT_IN_GA(x₁)) = 2·x₁
POL(niltree) = 0
POL(tree(x₁, x₂, x₃)) = 2 + 2·x₁ + 2·x₂ + x₃

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ PiDP

              ↳ UsableRulesProof

                ↳ PiDP

                  ↳ PiDPToQDPProof

                    ↳ QDP

                      ↳ UsableRulesReductionPairsProof

                        ↳ QDP

                          ↳ PisEmptyProof

Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.