Left Termination proof of /tmp/tmps7P7aA/permutation1-fb.pl

Left Termination of the query pattern perm1(b,f) w.r.t. the given Prolog program could successfully be proven:

↳ PROLOG

  ↳ PrologToPiTRSProof

perm1₂({}₀, {}₀).
perm1₂(Xs, .₂(X, Ys)) :- select₃(X, Xs, Zs), perm1₂(Zs, Ys).
select₃(X, .₂(X, Xs), Xs).
select₃(X, .₂(Y, Xs), .₂(Y, Zs)) :- select₃(X, Xs, Zs).

With regard to the inferred argument filtering the predicates were used in the following modes:
perm1₂: (b,f)
select₃: (f,b,f)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

perm1_2_in_ga₂([]_0, []_0) -> perm1_2_out_ga₂([]_0, []_0)
perm1_2_in_ga₂(Xs, ._2₂(X, Ys)) -> if_perm1_2_in_1_ga₄(Xs, X, Ys, select_3_in_aga₃(X, Xs, Zs))
select_3_in_aga₃(X, ._2₂(X, Xs), Xs) -> select_3_out_aga₃(X, ._2₂(X, Xs), Xs)
select_3_in_aga₃(X, ._2₂(Y, Xs), ._2₂(Y, Zs)) -> if_select_3_in_1_aga₅(X, Y, Xs, Zs, select_3_in_aga₃(X, Xs, Zs))
if_select_3_in_1_aga₅(X, Y, Xs, Zs, select_3_out_aga₃(X, Xs, Zs)) -> select_3_out_aga₃(X, ._2₂(Y, Xs), ._2₂(Y, Zs))
if_perm1_2_in_1_ga₄(Xs, X, Ys, select_3_out_aga₃(X, Xs, Zs)) -> if_perm1_2_in_2_ga₅(Xs, X, Ys, Zs, perm1_2_in_ga₂(Zs, Ys))
if_perm1_2_in_2_ga₅(Xs, X, Ys, Zs, perm1_2_out_ga₂(Zs, Ys)) -> perm1_2_out_ga₂(Xs, ._2₂(X, Ys))

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:

perm1_2_in_ga₂([]_0, []_0) -> perm1_2_out_ga₂([]_0, []_0)
perm1_2_in_ga₂(Xs, ._2₂(X, Ys)) -> if_perm1_2_in_1_ga₄(Xs, X, Ys, select_3_in_aga₃(X, Xs, Zs))
select_3_in_aga₃(X, ._2₂(X, Xs), Xs) -> select_3_out_aga₃(X, ._2₂(X, Xs), Xs)
select_3_in_aga₃(X, ._2₂(Y, Xs), ._2₂(Y, Zs)) -> if_select_3_in_1_aga₅(X, Y, Xs, Zs, select_3_in_aga₃(X, Xs, Zs))
if_select_3_in_1_aga₅(X, Y, Xs, Zs, select_3_out_aga₃(X, Xs, Zs)) -> select_3_out_aga₃(X, ._2₂(Y, Xs), ._2₂(Y, Zs))
if_perm1_2_in_1_ga₄(Xs, X, Ys, select_3_out_aga₃(X, Xs, Zs)) -> if_perm1_2_in_2_ga₅(Xs, X, Ys, Zs, perm1_2_in_ga₂(Zs, Ys))
if_perm1_2_in_2_ga₅(Xs, X, Ys, Zs, perm1_2_out_ga₂(Zs, Ys)) -> perm1_2_out_ga₂(Xs, ._2₂(X, Ys))

PERM1_2_IN_GA₂(Xs, ._2₂(X, Ys)) -> IF_PERM1_2_IN_1_GA₄(Xs, X, Ys, select_3_in_aga₃(X, Xs, Zs))
PERM1_2_IN_GA₂(Xs, ._2₂(X, Ys)) -> SELECT_3_IN_AGA₃(X, Xs, Zs)
SELECT_3_IN_AGA₃(X, ._2₂(Y, Xs), ._2₂(Y, Zs)) -> IF_SELECT_3_IN_1_AGA₅(X, Y, Xs, Zs, select_3_in_aga₃(X, Xs, Zs))
SELECT_3_IN_AGA₃(X, ._2₂(Y, Xs), ._2₂(Y, Zs)) -> SELECT_3_IN_AGA₃(X, Xs, Zs)
IF_PERM1_2_IN_1_GA₄(Xs, X, Ys, select_3_out_aga₃(X, Xs, Zs)) -> IF_PERM1_2_IN_2_GA₅(Xs, X, Ys, Zs, perm1_2_in_ga₂(Zs, Ys))
IF_PERM1_2_IN_1_GA₄(Xs, X, Ys, select_3_out_aga₃(X, Xs, Zs)) -> PERM1_2_IN_GA₂(Zs, Ys)

The TRS R consists of the following rules:

perm1_2_in_ga₂([]_0, []_0) -> perm1_2_out_ga₂([]_0, []_0)
perm1_2_in_ga₂(Xs, ._2₂(X, Ys)) -> if_perm1_2_in_1_ga₄(Xs, X, Ys, select_3_in_aga₃(X, Xs, Zs))
select_3_in_aga₃(X, ._2₂(X, Xs), Xs) -> select_3_out_aga₃(X, ._2₂(X, Xs), Xs)
select_3_in_aga₃(X, ._2₂(Y, Xs), ._2₂(Y, Zs)) -> if_select_3_in_1_aga₅(X, Y, Xs, Zs, select_3_in_aga₃(X, Xs, Zs))
if_select_3_in_1_aga₅(X, Y, Xs, Zs, select_3_out_aga₃(X, Xs, Zs)) -> select_3_out_aga₃(X, ._2₂(Y, Xs), ._2₂(Y, Zs))
if_perm1_2_in_1_ga₄(Xs, X, Ys, select_3_out_aga₃(X, Xs, Zs)) -> if_perm1_2_in_2_ga₅(Xs, X, Ys, Zs, perm1_2_in_ga₂(Zs, Ys))
if_perm1_2_in_2_ga₅(Xs, X, Ys, Zs, perm1_2_out_ga₂(Zs, Ys)) -> perm1_2_out_ga₂(Xs, ._2₂(X, Ys))

The argument filtering Pi contains the following mapping:
perm1_2_in_ga₂(x1, x2) = perm1_2_in_ga₁(x1)
[]_0 = []_0
._2₂(x1, x2) = ._2₂(x1, x2)
perm1_2_out_ga₂(x1, x2) = perm1_2_out_ga₁(x2)
if_perm1_2_in_1_ga₄(x1, x2, x3, x4) = if_perm1_2_in_1_ga₁(x4)
select_3_in_aga₃(x1, x2, x3) = select_3_in_aga₁(x2)
select_3_out_aga₃(x1, x2, x3) = select_3_out_aga₂(x1, x3)
if_select_3_in_1_aga₅(x1, x2, x3, x4, x5) = if_select_3_in_1_aga₂(x2, x5)
if_perm1_2_in_2_ga₅(x1, x2, x3, x4, x5) = if_perm1_2_in_2_ga₂(x2, x5)
SELECT_3_IN_AGA₃(x1, x2, x3) = SELECT_3_IN_AGA₁(x2)
IF_PERM1_2_IN_1_GA₄(x1, x2, x3, x4) = IF_PERM1_2_IN_1_GA₁(x4)
IF_PERM1_2_IN_2_GA₅(x1, x2, x3, x4, x5) = IF_PERM1_2_IN_2_GA₂(x2, x5)
IF_SELECT_3_IN_1_AGA₅(x1, x2, x3, x4, x5) = IF_SELECT_3_IN_1_AGA₂(x2, x5)
PERM1_2_IN_GA₂(x1, x2) = PERM1_2_IN_GA₁(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:

PERM1_2_IN_GA₂(Xs, ._2₂(X, Ys)) -> IF_PERM1_2_IN_1_GA₄(Xs, X, Ys, select_3_in_aga₃(X, Xs, Zs))
PERM1_2_IN_GA₂(Xs, ._2₂(X, Ys)) -> SELECT_3_IN_AGA₃(X, Xs, Zs)
SELECT_3_IN_AGA₃(X, ._2₂(Y, Xs), ._2₂(Y, Zs)) -> IF_SELECT_3_IN_1_AGA₅(X, Y, Xs, Zs, select_3_in_aga₃(X, Xs, Zs))
SELECT_3_IN_AGA₃(X, ._2₂(Y, Xs), ._2₂(Y, Zs)) -> SELECT_3_IN_AGA₃(X, Xs, Zs)
IF_PERM1_2_IN_1_GA₄(Xs, X, Ys, select_3_out_aga₃(X, Xs, Zs)) -> IF_PERM1_2_IN_2_GA₅(Xs, X, Ys, Zs, perm1_2_in_ga₂(Zs, Ys))
IF_PERM1_2_IN_1_GA₄(Xs, X, Ys, select_3_out_aga₃(X, Xs, Zs)) -> PERM1_2_IN_GA₂(Zs, Ys)

The TRS R consists of the following rules:

perm1_2_in_ga₂([]_0, []_0) -> perm1_2_out_ga₂([]_0, []_0)
perm1_2_in_ga₂(Xs, ._2₂(X, Ys)) -> if_perm1_2_in_1_ga₄(Xs, X, Ys, select_3_in_aga₃(X, Xs, Zs))
select_3_in_aga₃(X, ._2₂(X, Xs), Xs) -> select_3_out_aga₃(X, ._2₂(X, Xs), Xs)
select_3_in_aga₃(X, ._2₂(Y, Xs), ._2₂(Y, Zs)) -> if_select_3_in_1_aga₅(X, Y, Xs, Zs, select_3_in_aga₃(X, Xs, Zs))
if_select_3_in_1_aga₅(X, Y, Xs, Zs, select_3_out_aga₃(X, Xs, Zs)) -> select_3_out_aga₃(X, ._2₂(Y, Xs), ._2₂(Y, Zs))
if_perm1_2_in_1_ga₄(Xs, X, Ys, select_3_out_aga₃(X, Xs, Zs)) -> if_perm1_2_in_2_ga₅(Xs, X, Ys, Zs, perm1_2_in_ga₂(Zs, Ys))
if_perm1_2_in_2_ga₅(Xs, X, Ys, Zs, perm1_2_out_ga₂(Zs, Ys)) -> perm1_2_out_ga₂(Xs, ._2₂(X, Ys))

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

SELECT_3_IN_AGA₃(X, ._2₂(Y, Xs), ._2₂(Y, Zs)) -> SELECT_3_IN_AGA₃(X, Xs, Zs)

The TRS R consists of the following rules:

perm1_2_in_ga₂([]_0, []_0) -> perm1_2_out_ga₂([]_0, []_0)
perm1_2_in_ga₂(Xs, ._2₂(X, Ys)) -> if_perm1_2_in_1_ga₄(Xs, X, Ys, select_3_in_aga₃(X, Xs, Zs))
select_3_in_aga₃(X, ._2₂(X, Xs), Xs) -> select_3_out_aga₃(X, ._2₂(X, Xs), Xs)
select_3_in_aga₃(X, ._2₂(Y, Xs), ._2₂(Y, Zs)) -> if_select_3_in_1_aga₅(X, Y, Xs, Zs, select_3_in_aga₃(X, Xs, Zs))
if_select_3_in_1_aga₅(X, Y, Xs, Zs, select_3_out_aga₃(X, Xs, Zs)) -> select_3_out_aga₃(X, ._2₂(Y, Xs), ._2₂(Y, Zs))
if_perm1_2_in_1_ga₄(Xs, X, Ys, select_3_out_aga₃(X, Xs, Zs)) -> if_perm1_2_in_2_ga₅(Xs, X, Ys, Zs, perm1_2_in_ga₂(Zs, Ys))
if_perm1_2_in_2_ga₅(Xs, X, Ys, Zs, perm1_2_out_ga₂(Zs, Ys)) -> perm1_2_out_ga₂(Xs, ._2₂(X, Ys))

The argument filtering Pi contains the following mapping:
perm1_2_in_ga₂(x1, x2) = perm1_2_in_ga₁(x1)
[]_0 = []_0
._2₂(x1, x2) = ._2₂(x1, x2)
perm1_2_out_ga₂(x1, x2) = perm1_2_out_ga₁(x2)
if_perm1_2_in_1_ga₄(x1, x2, x3, x4) = if_perm1_2_in_1_ga₁(x4)
select_3_in_aga₃(x1, x2, x3) = select_3_in_aga₁(x2)
select_3_out_aga₃(x1, x2, x3) = select_3_out_aga₂(x1, x3)
if_select_3_in_1_aga₅(x1, x2, x3, x4, x5) = if_select_3_in_1_aga₂(x2, x5)
if_perm1_2_in_2_ga₅(x1, x2, x3, x4, x5) = if_perm1_2_in_2_ga₂(x2, x5)
SELECT_3_IN_AGA₃(x1, x2, x3) = SELECT_3_IN_AGA₁(x2)

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

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

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

SELECT_3_IN_AGA₃(X, ._2₂(Y, Xs), ._2₂(Y, Zs)) -> SELECT_3_IN_AGA₃(X, Xs, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
._2₂(x1, x2) = ._2₂(x1, x2)
SELECT_3_IN_AGA₃(x1, x2, x3) = SELECT_3_IN_AGA₁(x2)

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

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

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

SELECT_3_IN_AGA₁(._2₂(Y, Xs)) -> SELECT_3_IN_AGA₁(Xs)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {SELECT_3_IN_AGA₁}.
By using the subterm criterion together with the size-change analysis 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:

SELECT_3_IN_AGA₁(._2₂(Y, Xs)) -> SELECT_3_IN_AGA₁(Xs)
The graph contains the following edges 1 > 1

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

IF_PERM1_2_IN_1_GA₄(Xs, X, Ys, select_3_out_aga₃(X, Xs, Zs)) -> PERM1_2_IN_GA₂(Zs, Ys)
PERM1_2_IN_GA₂(Xs, ._2₂(X, Ys)) -> IF_PERM1_2_IN_1_GA₄(Xs, X, Ys, select_3_in_aga₃(X, Xs, Zs))

The TRS R consists of the following rules:

perm1_2_in_ga₂([]_0, []_0) -> perm1_2_out_ga₂([]_0, []_0)
perm1_2_in_ga₂(Xs, ._2₂(X, Ys)) -> if_perm1_2_in_1_ga₄(Xs, X, Ys, select_3_in_aga₃(X, Xs, Zs))
select_3_in_aga₃(X, ._2₂(X, Xs), Xs) -> select_3_out_aga₃(X, ._2₂(X, Xs), Xs)
select_3_in_aga₃(X, ._2₂(Y, Xs), ._2₂(Y, Zs)) -> if_select_3_in_1_aga₅(X, Y, Xs, Zs, select_3_in_aga₃(X, Xs, Zs))
if_select_3_in_1_aga₅(X, Y, Xs, Zs, select_3_out_aga₃(X, Xs, Zs)) -> select_3_out_aga₃(X, ._2₂(Y, Xs), ._2₂(Y, Zs))
if_perm1_2_in_1_ga₄(Xs, X, Ys, select_3_out_aga₃(X, Xs, Zs)) -> if_perm1_2_in_2_ga₅(Xs, X, Ys, Zs, perm1_2_in_ga₂(Zs, Ys))
if_perm1_2_in_2_ga₅(Xs, X, Ys, Zs, perm1_2_out_ga₂(Zs, Ys)) -> perm1_2_out_ga₂(Xs, ._2₂(X, Ys))

The argument filtering Pi contains the following mapping:
perm1_2_in_ga₂(x1, x2) = perm1_2_in_ga₁(x1)
[]_0 = []_0
._2₂(x1, x2) = ._2₂(x1, x2)
perm1_2_out_ga₂(x1, x2) = perm1_2_out_ga₁(x2)
if_perm1_2_in_1_ga₄(x1, x2, x3, x4) = if_perm1_2_in_1_ga₁(x4)
select_3_in_aga₃(x1, x2, x3) = select_3_in_aga₁(x2)
select_3_out_aga₃(x1, x2, x3) = select_3_out_aga₂(x1, x3)
if_select_3_in_1_aga₅(x1, x2, x3, x4, x5) = if_select_3_in_1_aga₂(x2, x5)
if_perm1_2_in_2_ga₅(x1, x2, x3, x4, x5) = if_perm1_2_in_2_ga₂(x2, x5)
IF_PERM1_2_IN_1_GA₄(x1, x2, x3, x4) = IF_PERM1_2_IN_1_GA₁(x4)
PERM1_2_IN_GA₂(x1, x2) = PERM1_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

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

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

IF_PERM1_2_IN_1_GA₄(Xs, X, Ys, select_3_out_aga₃(X, Xs, Zs)) -> PERM1_2_IN_GA₂(Zs, Ys)
PERM1_2_IN_GA₂(Xs, ._2₂(X, Ys)) -> IF_PERM1_2_IN_1_GA₄(Xs, X, Ys, select_3_in_aga₃(X, Xs, Zs))

The TRS R consists of the following rules:

select_3_in_aga₃(X, ._2₂(X, Xs), Xs) -> select_3_out_aga₃(X, ._2₂(X, Xs), Xs)
select_3_in_aga₃(X, ._2₂(Y, Xs), ._2₂(Y, Zs)) -> if_select_3_in_1_aga₅(X, Y, Xs, Zs, select_3_in_aga₃(X, Xs, Zs))
if_select_3_in_1_aga₅(X, Y, Xs, Zs, select_3_out_aga₃(X, Xs, Zs)) -> select_3_out_aga₃(X, ._2₂(Y, Xs), ._2₂(Y, Zs))

The argument filtering Pi contains the following mapping:
._2₂(x1, x2) = ._2₂(x1, x2)
select_3_in_aga₃(x1, x2, x3) = select_3_in_aga₁(x2)
select_3_out_aga₃(x1, x2, x3) = select_3_out_aga₂(x1, x3)
if_select_3_in_1_aga₅(x1, x2, x3, x4, x5) = if_select_3_in_1_aga₂(x2, x5)
IF_PERM1_2_IN_1_GA₄(x1, x2, x3, x4) = IF_PERM1_2_IN_1_GA₁(x4)
PERM1_2_IN_GA₂(x1, x2) = PERM1_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

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

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

IF_PERM1_2_IN_1_GA₁(select_3_out_aga₂(X, Zs)) -> PERM1_2_IN_GA₁(Zs)
PERM1_2_IN_GA₁(Xs) -> IF_PERM1_2_IN_1_GA₁(select_3_in_aga₁(Xs))

The TRS R consists of the following rules:

select_3_in_aga₁(._2₂(X, Xs)) -> select_3_out_aga₂(X, Xs)
select_3_in_aga₁(._2₂(Y, Xs)) -> if_select_3_in_1_aga₂(Y, select_3_in_aga₁(Xs))
if_select_3_in_1_aga₂(Y, select_3_out_aga₂(X, Zs)) -> select_3_out_aga₂(X, ._2₂(Y, Zs))

The set Q consists of the following terms:

select_3_in_aga₁(x0)
if_select_3_in_1_aga₂(x0, x1)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {PERM1_2_IN_GA₁, IF_PERM1_2_IN_1_GA₁}.
We used the following order together with the size-change analysis to show that there are no infinite chains for this DP problem.

Order:Polynomial interpretation:

POL(._2₂(x₁, x₂)) = 1 + x₂
POL(select_3_out_aga₂(x₁, x₂)) = 1 + x₂
POL(if_select_3_in_1_aga₂(x₁, x₂)) = 1 + x₂
POL(select_3_in_aga₁(x₁)) = x₁

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

PERM1_2_IN_GA₁(Xs) -> IF_PERM1_2_IN_1_GA₁(select_3_in_aga₁(Xs)) (allowed arguments on rhs = {1})
The graph contains the following edges 1 >= 1
IF_PERM1_2_IN_1_GA₁(select_3_out_aga₂(X, Zs)) -> PERM1_2_IN_GA₁(Zs) (allowed arguments on rhs = {1})
The graph contains the following edges 1 > 1

We oriented the following set of usable rules.

select_3_in_aga₁(._2₂(X, Xs)) -> select_3_out_aga₂(X, Xs)
select_3_in_aga₁(._2₂(Y, Xs)) -> if_select_3_in_1_aga₂(Y, select_3_in_aga₁(Xs))
if_select_3_in_1_aga₂(Y, select_3_out_aga₂(X, Zs)) -> select_3_out_aga₂(X, ._2₂(Y, Zs))