Left Termination proof of /tmp/tmpJyzlfd/flatten.pl

Left Termination of the query pattern flatten(b,f) w.r.t. the given Prolog program could successfully be proven:

↳ PROLOG

  ↳ PrologToPiTRSProof

flatten₂(atom₁(X), .₂(X, {}₀)).
flatten₂(cons₂(atom₁(X), U), .₂(X, Y)) :- flatten₂(U, Y).
flatten₂(cons₂(cons₂(U, V), W), X) :- flatten₂(cons₂(U, cons₂(V, W)), X).

With regard to the inferred argument filtering the predicates were used in the following modes:
flatten₂: (b,f)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

flatten_2_in_ga₂(atom_1₁(X), ._2₂(X, []_0)) -> flatten_2_out_ga₂(atom_1₁(X), ._2₂(X, []_0))
flatten_2_in_ga₂(cons_2₂(atom_1₁(X), U), ._2₂(X, Y)) -> if_flatten_2_in_1_ga₄(X, U, Y, flatten_2_in_ga₂(U, Y))
flatten_2_in_ga₂(cons_2₂(cons_2₂(U, V), W), X) -> if_flatten_2_in_2_ga₅(U, V, W, X, flatten_2_in_ga₂(cons_2₂(U, cons_2₂(V, W)), X))
if_flatten_2_in_2_ga₅(U, V, W, X, flatten_2_out_ga₂(cons_2₂(U, cons_2₂(V, W)), X)) -> flatten_2_out_ga₂(cons_2₂(cons_2₂(U, V), W), X)
if_flatten_2_in_1_ga₄(X, U, Y, flatten_2_out_ga₂(U, Y)) -> flatten_2_out_ga₂(cons_2₂(atom_1₁(X), U), ._2₂(X, Y))

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:

flatten_2_in_ga₂(atom_1₁(X), ._2₂(X, []_0)) -> flatten_2_out_ga₂(atom_1₁(X), ._2₂(X, []_0))
flatten_2_in_ga₂(cons_2₂(atom_1₁(X), U), ._2₂(X, Y)) -> if_flatten_2_in_1_ga₄(X, U, Y, flatten_2_in_ga₂(U, Y))
flatten_2_in_ga₂(cons_2₂(cons_2₂(U, V), W), X) -> if_flatten_2_in_2_ga₅(U, V, W, X, flatten_2_in_ga₂(cons_2₂(U, cons_2₂(V, W)), X))
if_flatten_2_in_2_ga₅(U, V, W, X, flatten_2_out_ga₂(cons_2₂(U, cons_2₂(V, W)), X)) -> flatten_2_out_ga₂(cons_2₂(cons_2₂(U, V), W), X)
if_flatten_2_in_1_ga₄(X, U, Y, flatten_2_out_ga₂(U, Y)) -> flatten_2_out_ga₂(cons_2₂(atom_1₁(X), U), ._2₂(X, Y))

FLATTEN_2_IN_GA₂(cons_2₂(atom_1₁(X), U), ._2₂(X, Y)) -> IF_FLATTEN_2_IN_1_GA₄(X, U, Y, flatten_2_in_ga₂(U, Y))
FLATTEN_2_IN_GA₂(cons_2₂(atom_1₁(X), U), ._2₂(X, Y)) -> FLATTEN_2_IN_GA₂(U, Y)
FLATTEN_2_IN_GA₂(cons_2₂(cons_2₂(U, V), W), X) -> IF_FLATTEN_2_IN_2_GA₅(U, V, W, X, flatten_2_in_ga₂(cons_2₂(U, cons_2₂(V, W)), X))
FLATTEN_2_IN_GA₂(cons_2₂(cons_2₂(U, V), W), X) -> FLATTEN_2_IN_GA₂(cons_2₂(U, cons_2₂(V, W)), X)

The TRS R consists of the following rules:

flatten_2_in_ga₂(atom_1₁(X), ._2₂(X, []_0)) -> flatten_2_out_ga₂(atom_1₁(X), ._2₂(X, []_0))
flatten_2_in_ga₂(cons_2₂(atom_1₁(X), U), ._2₂(X, Y)) -> if_flatten_2_in_1_ga₄(X, U, Y, flatten_2_in_ga₂(U, Y))
flatten_2_in_ga₂(cons_2₂(cons_2₂(U, V), W), X) -> if_flatten_2_in_2_ga₅(U, V, W, X, flatten_2_in_ga₂(cons_2₂(U, cons_2₂(V, W)), X))
if_flatten_2_in_2_ga₅(U, V, W, X, flatten_2_out_ga₂(cons_2₂(U, cons_2₂(V, W)), X)) -> flatten_2_out_ga₂(cons_2₂(cons_2₂(U, V), W), X)
if_flatten_2_in_1_ga₄(X, U, Y, flatten_2_out_ga₂(U, Y)) -> flatten_2_out_ga₂(cons_2₂(atom_1₁(X), U), ._2₂(X, Y))

The argument filtering Pi contains the following mapping:
flatten_2_in_ga₂(x1, x2) = flatten_2_in_ga₁(x1)
atom_1₁(x1) = atom_1₁(x1)
._2₂(x1, x2) = ._2₂(x1, x2)
[]_0 = []_0
cons_2₂(x1, x2) = cons_2₂(x1, x2)
flatten_2_out_ga₂(x1, x2) = flatten_2_out_ga₁(x2)
if_flatten_2_in_1_ga₄(x1, x2, x3, x4) = if_flatten_2_in_1_ga₂(x1, x4)
if_flatten_2_in_2_ga₅(x1, x2, x3, x4, x5) = if_flatten_2_in_2_ga₁(x5)
FLATTEN_2_IN_GA₂(x1, x2) = FLATTEN_2_IN_GA₁(x1)
IF_FLATTEN_2_IN_2_GA₅(x1, x2, x3, x4, x5) = IF_FLATTEN_2_IN_2_GA₁(x5)
IF_FLATTEN_2_IN_1_GA₄(x1, x2, x3, x4) = IF_FLATTEN_2_IN_1_GA₂(x1, x4)

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:

FLATTEN_2_IN_GA₂(cons_2₂(atom_1₁(X), U), ._2₂(X, Y)) -> IF_FLATTEN_2_IN_1_GA₄(X, U, Y, flatten_2_in_ga₂(U, Y))
FLATTEN_2_IN_GA₂(cons_2₂(atom_1₁(X), U), ._2₂(X, Y)) -> FLATTEN_2_IN_GA₂(U, Y)
FLATTEN_2_IN_GA₂(cons_2₂(cons_2₂(U, V), W), X) -> IF_FLATTEN_2_IN_2_GA₅(U, V, W, X, flatten_2_in_ga₂(cons_2₂(U, cons_2₂(V, W)), X))
FLATTEN_2_IN_GA₂(cons_2₂(cons_2₂(U, V), W), X) -> FLATTEN_2_IN_GA₂(cons_2₂(U, cons_2₂(V, W)), X)

The TRS R consists of the following rules:

flatten_2_in_ga₂(atom_1₁(X), ._2₂(X, []_0)) -> flatten_2_out_ga₂(atom_1₁(X), ._2₂(X, []_0))
flatten_2_in_ga₂(cons_2₂(atom_1₁(X), U), ._2₂(X, Y)) -> if_flatten_2_in_1_ga₄(X, U, Y, flatten_2_in_ga₂(U, Y))
flatten_2_in_ga₂(cons_2₂(cons_2₂(U, V), W), X) -> if_flatten_2_in_2_ga₅(U, V, W, X, flatten_2_in_ga₂(cons_2₂(U, cons_2₂(V, W)), X))
if_flatten_2_in_2_ga₅(U, V, W, X, flatten_2_out_ga₂(cons_2₂(U, cons_2₂(V, W)), X)) -> flatten_2_out_ga₂(cons_2₂(cons_2₂(U, V), W), X)
if_flatten_2_in_1_ga₄(X, U, Y, flatten_2_out_ga₂(U, Y)) -> flatten_2_out_ga₂(cons_2₂(atom_1₁(X), U), ._2₂(X, Y))

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ PiDP

              ↳ UsableRulesProof

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

FLATTEN_2_IN_GA₂(cons_2₂(atom_1₁(X), U), ._2₂(X, Y)) -> FLATTEN_2_IN_GA₂(U, Y)
FLATTEN_2_IN_GA₂(cons_2₂(cons_2₂(U, V), W), X) -> FLATTEN_2_IN_GA₂(cons_2₂(U, cons_2₂(V, W)), X)

The TRS R consists of the following rules:

flatten_2_in_ga₂(atom_1₁(X), ._2₂(X, []_0)) -> flatten_2_out_ga₂(atom_1₁(X), ._2₂(X, []_0))
flatten_2_in_ga₂(cons_2₂(atom_1₁(X), U), ._2₂(X, Y)) -> if_flatten_2_in_1_ga₄(X, U, Y, flatten_2_in_ga₂(U, Y))
flatten_2_in_ga₂(cons_2₂(cons_2₂(U, V), W), X) -> if_flatten_2_in_2_ga₅(U, V, W, X, flatten_2_in_ga₂(cons_2₂(U, cons_2₂(V, W)), X))
if_flatten_2_in_2_ga₅(U, V, W, X, flatten_2_out_ga₂(cons_2₂(U, cons_2₂(V, W)), X)) -> flatten_2_out_ga₂(cons_2₂(cons_2₂(U, V), W), X)
if_flatten_2_in_1_ga₄(X, U, Y, flatten_2_out_ga₂(U, Y)) -> flatten_2_out_ga₂(cons_2₂(atom_1₁(X), U), ._2₂(X, Y))

The argument filtering Pi contains the following mapping:
flatten_2_in_ga₂(x1, x2) = flatten_2_in_ga₁(x1)
atom_1₁(x1) = atom_1₁(x1)
._2₂(x1, x2) = ._2₂(x1, x2)
[]_0 = []_0
cons_2₂(x1, x2) = cons_2₂(x1, x2)
flatten_2_out_ga₂(x1, x2) = flatten_2_out_ga₁(x2)
if_flatten_2_in_1_ga₄(x1, x2, x3, x4) = if_flatten_2_in_1_ga₂(x1, x4)
if_flatten_2_in_2_ga₅(x1, x2, x3, x4, x5) = if_flatten_2_in_2_ga₁(x5)
FLATTEN_2_IN_GA₂(x1, x2) = FLATTEN_2_IN_GA₁(x1)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting 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:

FLATTEN_2_IN_GA₂(cons_2₂(atom_1₁(X), U), ._2₂(X, Y)) -> FLATTEN_2_IN_GA₂(U, Y)
FLATTEN_2_IN_GA₂(cons_2₂(cons_2₂(U, V), W), X) -> FLATTEN_2_IN_GA₂(cons_2₂(U, cons_2₂(V, W)), X)

R is empty.
The argument filtering Pi contains the following mapping:
atom_1₁(x1) = atom_1₁(x1)
._2₂(x1, x2) = ._2₂(x1, x2)
cons_2₂(x1, x2) = cons_2₂(x1, x2)
FLATTEN_2_IN_GA₂(x1, x2) = FLATTEN_2_IN_GA₁(x1)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem into ordinary QDP problem by application of Pi.

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ PiDP

              ↳ UsableRulesProof

                ↳ PiDP

                  ↳ PiDPToQDPProof

                    ↳ QDP

                      ↳ RuleRemovalProof

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

FLATTEN_2_IN_GA₁(cons_2₂(atom_1₁(X), U)) -> FLATTEN_2_IN_GA₁(U)
FLATTEN_2_IN_GA₁(cons_2₂(cons_2₂(U, V), W)) -> FLATTEN_2_IN_GA₁(cons_2₂(U, cons_2₂(V, W)))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {FLATTEN_2_IN_GA₁}.
By using a polynomial ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

FLATTEN_2_IN_GA₁(cons_2₂(atom_1₁(X), U)) -> FLATTEN_2_IN_GA₁(U)

Used ordering: POLO with Polynomial interpretation:

POL(FLATTEN_2_IN_GA₁(x₁)) = 1 + x₁
POL(atom_1₁(x₁)) = 1 + x₁
POL(cons_2₂(x₁, x₂)) = x₁ + x₂

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ PiDP

              ↳ UsableRulesProof

                ↳ PiDP

                  ↳ PiDPToQDPProof

                    ↳ QDP

                      ↳ RuleRemovalProof

                        ↳ QDP

                          ↳ QDPSizeChangeProof

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

FLATTEN_2_IN_GA₁(cons_2₂(cons_2₂(U, V), W)) -> FLATTEN_2_IN_GA₁(cons_2₂(U, cons_2₂(V, W)))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {FLATTEN_2_IN_GA₁}.
We used the following order and afs together with the size-change analysis to show that there are no infinite chains for this DP problem.

Order:Homeomorphic Embedding Order

AFS:
cons_2₂(x1, x2) = cons_2₁(x1)

From the DPs we obtained the following set of size-change graphs:

FLATTEN_2_IN_GA₁(cons_2₂(cons_2₂(U, V), W)) -> FLATTEN_2_IN_GA₁(cons_2₂(U, cons_2₂(V, W))) (allowed arguments on rhs = {1})
The graph contains the following edges 1 > 1

We oriented the following set of usable rules. none