Left Termination proof of /tmp/tmpNQ4nJX/confdel.pl

Left Termination of the query pattern conf_in_1(g) w.r.t. the given Prolog program could not be shown:

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

Clauses:

conf(X) :- ','(del2(X, Z), ','(del(U, Y, Z), conf(Y))).
del2(X, Y) :- ','(del(U, X, Z), del(V, Z, Y)).
del(X, .(X, T), T).
del(X, .(H, T), .(H, T1)) :- del(X, T, T1).
s2l(s(X), .(Y, Xs)) :- s2l(X, Xs).
s2l(0, []).
goal(X) :- ','(s2l(X, XS), conf(XS)).

Queries:

conf(g).

We use the technique of [30].Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

conf_in(X) → U1(X, del2_in(X, Z))
del2_in(X, Y) → U4(X, Y, del_in(U, X, Z))
del_in(X, .(H, T), .(H, T1)) → U6(X, H, T, T1, del_in(X, T, T1))
del_in(X, .(X, T), T) → del_out(X, .(X, T), T)
U6(X, H, T, T1, del_out(X, T, T1)) → del_out(X, .(H, T), .(H, T1))
U4(X, Y, del_out(U, X, Z)) → U5(X, Y, del_in(V, Z, Y))
U5(X, Y, del_out(V, Z, Y)) → del2_out(X, Y)
U1(X, del2_out(X, Z)) → U2(X, del_in(U, Y, Z))
U2(X, del_out(U, Y, Z)) → U3(X, conf_in(Y))
U3(X, conf_out(Y)) → conf_out(X)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

  ↳ PrologToPiTRSProof

Pi-finite rewrite system:
The TRS R consists of the following rules:

conf_in(X) → U1(X, del2_in(X, Z))
del2_in(X, Y) → U4(X, Y, del_in(U, X, Z))
del_in(X, .(H, T), .(H, T1)) → U6(X, H, T, T1, del_in(X, T, T1))
del_in(X, .(X, T), T) → del_out(X, .(X, T), T)
U6(X, H, T, T1, del_out(X, T, T1)) → del_out(X, .(H, T), .(H, T1))
U4(X, Y, del_out(U, X, Z)) → U5(X, Y, del_in(V, Z, Y))
U5(X, Y, del_out(V, Z, Y)) → del2_out(X, Y)
U1(X, del2_out(X, Z)) → U2(X, del_in(U, Y, Z))
U2(X, del_out(U, Y, Z)) → U3(X, conf_in(Y))
U3(X, conf_out(Y)) → conf_out(X)

CONF_IN(X) → U1¹(X, del2_in(X, Z))
CONF_IN(X) → DEL2_IN(X, Z)
DEL2_IN(X, Y) → U4¹(X, Y, del_in(U, X, Z))
DEL2_IN(X, Y) → DEL_IN(U, X, Z)
DEL_IN(X, .(H, T), .(H, T1)) → U6¹(X, H, T, T1, del_in(X, T, T1))
DEL_IN(X, .(H, T), .(H, T1)) → DEL_IN(X, T, T1)
U4¹(X, Y, del_out(U, X, Z)) → U5¹(X, Y, del_in(V, Z, Y))
U4¹(X, Y, del_out(U, X, Z)) → DEL_IN(V, Z, Y)
U1¹(X, del2_out(X, Z)) → U2¹(X, del_in(U, Y, Z))
U1¹(X, del2_out(X, Z)) → DEL_IN(U, Y, Z)
U2¹(X, del_out(U, Y, Z)) → U3¹(X, conf_in(Y))
U2¹(X, del_out(U, Y, Z)) → CONF_IN(Y)

The TRS R consists of the following rules:

conf_in(X) → U1(X, del2_in(X, Z))
del2_in(X, Y) → U4(X, Y, del_in(U, X, Z))
del_in(X, .(H, T), .(H, T1)) → U6(X, H, T, T1, del_in(X, T, T1))
del_in(X, .(X, T), T) → del_out(X, .(X, T), T)
U6(X, H, T, T1, del_out(X, T, T1)) → del_out(X, .(H, T), .(H, T1))
U4(X, Y, del_out(U, X, Z)) → U5(X, Y, del_in(V, Z, Y))
U5(X, Y, del_out(V, Z, Y)) → del2_out(X, Y)
U1(X, del2_out(X, Z)) → U2(X, del_in(U, Y, Z))
U2(X, del_out(U, Y, Z)) → U3(X, conf_in(Y))
U3(X, conf_out(Y)) → conf_out(X)

The argument filtering Pi contains the following mapping:
conf_in(x1) = conf_in
U1(x1, x2) = U1(x2)
del2_in(x1, x2) = del2_in
U4(x1, x2, x3) = U4(x3)
del_in(x1, x2, x3) = del_in
.(x1, x2) = .(x2)
U6(x1, x2, x3, x4, x5) = U6(x5)
del_out(x1, x2, x3) = del_out
U5(x1, x2, x3) = U5(x3)
del2_out(x1, x2) = del2_out
U2(x1, x2) = U2(x2)
U3(x1, x2) = U3(x2)
conf_out(x1) = conf_out
U5¹(x1, x2, x3) = U5¹(x3)
DEL_IN(x1, x2, x3) = DEL_IN
U4¹(x1, x2, x3) = U4¹(x3)
U1¹(x1, x2) = U1¹(x2)
CONF_IN(x1) = CONF_IN
U3¹(x1, x2) = U3¹(x2)
U2¹(x1, x2) = U2¹(x2)
U6¹(x1, x2, x3, x4, x5) = U6¹(x5)
DEL2_IN(x1, x2) = DEL2_IN

We have to consider all (P,R,Pi)-chains

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

  ↳ PrologToPiTRSProof

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

CONF_IN(X) → U1¹(X, del2_in(X, Z))
CONF_IN(X) → DEL2_IN(X, Z)
DEL2_IN(X, Y) → U4¹(X, Y, del_in(U, X, Z))
DEL2_IN(X, Y) → DEL_IN(U, X, Z)
DEL_IN(X, .(H, T), .(H, T1)) → U6¹(X, H, T, T1, del_in(X, T, T1))
DEL_IN(X, .(H, T), .(H, T1)) → DEL_IN(X, T, T1)
U4¹(X, Y, del_out(U, X, Z)) → U5¹(X, Y, del_in(V, Z, Y))
U4¹(X, Y, del_out(U, X, Z)) → DEL_IN(V, Z, Y)
U1¹(X, del2_out(X, Z)) → U2¹(X, del_in(U, Y, Z))
U1¹(X, del2_out(X, Z)) → DEL_IN(U, Y, Z)
U2¹(X, del_out(U, Y, Z)) → U3¹(X, conf_in(Y))
U2¹(X, del_out(U, Y, Z)) → CONF_IN(Y)

The TRS R consists of the following rules:

conf_in(X) → U1(X, del2_in(X, Z))
del2_in(X, Y) → U4(X, Y, del_in(U, X, Z))
del_in(X, .(H, T), .(H, T1)) → U6(X, H, T, T1, del_in(X, T, T1))
del_in(X, .(X, T), T) → del_out(X, .(X, T), T)
U6(X, H, T, T1, del_out(X, T, T1)) → del_out(X, .(H, T), .(H, T1))
U4(X, Y, del_out(U, X, Z)) → U5(X, Y, del_in(V, Z, Y))
U5(X, Y, del_out(V, Z, Y)) → del2_out(X, Y)
U1(X, del2_out(X, Z)) → U2(X, del_in(U, Y, Z))
U2(X, del_out(U, Y, Z)) → U3(X, conf_in(Y))
U3(X, conf_out(Y)) → conf_out(X)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

DEL_IN(X, .(H, T), .(H, T1)) → DEL_IN(X, T, T1)

The TRS R consists of the following rules:

conf_in(X) → U1(X, del2_in(X, Z))
del2_in(X, Y) → U4(X, Y, del_in(U, X, Z))
del_in(X, .(H, T), .(H, T1)) → U6(X, H, T, T1, del_in(X, T, T1))
del_in(X, .(X, T), T) → del_out(X, .(X, T), T)
U6(X, H, T, T1, del_out(X, T, T1)) → del_out(X, .(H, T), .(H, T1))
U4(X, Y, del_out(U, X, Z)) → U5(X, Y, del_in(V, Z, Y))
U5(X, Y, del_out(V, Z, Y)) → del2_out(X, Y)
U1(X, del2_out(X, Z)) → U2(X, del_in(U, Y, Z))
U2(X, del_out(U, Y, Z)) → U3(X, conf_in(Y))
U3(X, conf_out(Y)) → conf_out(X)

The argument filtering Pi contains the following mapping:
conf_in(x1) = conf_in
U1(x1, x2) = U1(x2)
del2_in(x1, x2) = del2_in
U4(x1, x2, x3) = U4(x3)
del_in(x1, x2, x3) = del_in
.(x1, x2) = .(x2)
U6(x1, x2, x3, x4, x5) = U6(x5)
del_out(x1, x2, x3) = del_out
U5(x1, x2, x3) = U5(x3)
del2_out(x1, x2) = del2_out
U2(x1, x2) = U2(x2)
U3(x1, x2) = U3(x2)
conf_out(x1) = conf_out
DEL_IN(x1, x2, x3) = DEL_IN

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

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

DEL_IN(X, .(H, T), .(H, T1)) → DEL_IN(X, T, T1)

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

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

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ NonTerminationProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

DEL_IN → DEL_IN

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

DEL_IN → DEL_IN

The TRS R consists of the following rules:none

s = DEL_IN evaluates to t =DEL_IN

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from DEL_IN to DEL_IN.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

  ↳ PrologToPiTRSProof

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

U1¹(X, del2_out(X, Z)) → U2¹(X, del_in(U, Y, Z))
U2¹(X, del_out(U, Y, Z)) → CONF_IN(Y)
CONF_IN(X) → U1¹(X, del2_in(X, Z))

The TRS R consists of the following rules:

conf_in(X) → U1(X, del2_in(X, Z))
del2_in(X, Y) → U4(X, Y, del_in(U, X, Z))
del_in(X, .(H, T), .(H, T1)) → U6(X, H, T, T1, del_in(X, T, T1))
del_in(X, .(X, T), T) → del_out(X, .(X, T), T)
U6(X, H, T, T1, del_out(X, T, T1)) → del_out(X, .(H, T), .(H, T1))
U4(X, Y, del_out(U, X, Z)) → U5(X, Y, del_in(V, Z, Y))
U5(X, Y, del_out(V, Z, Y)) → del2_out(X, Y)
U1(X, del2_out(X, Z)) → U2(X, del_in(U, Y, Z))
U2(X, del_out(U, Y, Z)) → U3(X, conf_in(Y))
U3(X, conf_out(Y)) → conf_out(X)

The argument filtering Pi contains the following mapping:
conf_in(x1) = conf_in
U1(x1, x2) = U1(x2)
del2_in(x1, x2) = del2_in
U4(x1, x2, x3) = U4(x3)
del_in(x1, x2, x3) = del_in
.(x1, x2) = .(x2)
U6(x1, x2, x3, x4, x5) = U6(x5)
del_out(x1, x2, x3) = del_out
U5(x1, x2, x3) = U5(x3)
del2_out(x1, x2) = del2_out
U2(x1, x2) = U2(x2)
U3(x1, x2) = U3(x2)
conf_out(x1) = conf_out
U1¹(x1, x2) = U1¹(x2)
CONF_IN(x1) = CONF_IN
U2¹(x1, x2) = U2¹(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

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

  ↳ PrologToPiTRSProof

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

U1¹(X, del2_out(X, Z)) → U2¹(X, del_in(U, Y, Z))
U2¹(X, del_out(U, Y, Z)) → CONF_IN(Y)
CONF_IN(X) → U1¹(X, del2_in(X, Z))

The TRS R consists of the following rules:

del_in(X, .(H, T), .(H, T1)) → U6(X, H, T, T1, del_in(X, T, T1))
del_in(X, .(X, T), T) → del_out(X, .(X, T), T)
del2_in(X, Y) → U4(X, Y, del_in(U, X, Z))
U6(X, H, T, T1, del_out(X, T, T1)) → del_out(X, .(H, T), .(H, T1))
U4(X, Y, del_out(U, X, Z)) → U5(X, Y, del_in(V, Z, Y))
U5(X, Y, del_out(V, Z, Y)) → del2_out(X, Y)

The argument filtering Pi contains the following mapping:
del2_in(x1, x2) = del2_in
U4(x1, x2, x3) = U4(x3)
del_in(x1, x2, x3) = del_in
.(x1, x2) = .(x2)
U6(x1, x2, x3, x4, x5) = U6(x5)
del_out(x1, x2, x3) = del_out
U5(x1, x2, x3) = U5(x3)
del2_out(x1, x2) = del2_out
U1¹(x1, x2) = U1¹(x2)
CONF_IN(x1) = CONF_IN
U2¹(x1, x2) = U2¹(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

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

  ↳ PrologToPiTRSProof

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

CONF_IN → U1¹(del2_in)
U1¹(del2_out) → U2¹(del_in)
U2¹(del_out) → CONF_IN

The TRS R consists of the following rules:

del_in → U6(del_in)
del_in → del_out
del2_in → U4(del_in)
U6(del_out) → del_out
U4(del_out) → U5(del_in)
U5(del_out) → del2_out

The set Q consists of the following terms:

del_in
del2_in
U6(x0)
U4(x0)
U5(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule U1¹(del2_out) → U2¹(del_in) at position [0] we obtained the following new rules:

U1¹(del2_out) → U2¹(del_out)
U1¹(del2_out) → U2¹(U6(del_in))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

  ↳ PrologToPiTRSProof

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

CONF_IN → U1¹(del2_in)
U1¹(del2_out) → U2¹(del_out)
U2¹(del_out) → CONF_IN
U1¹(del2_out) → U2¹(U6(del_in))

The TRS R consists of the following rules:

del_in → U6(del_in)
del_in → del_out
del2_in → U4(del_in)
U6(del_out) → del_out
U4(del_out) → U5(del_in)
U5(del_out) → del2_out

The set Q consists of the following terms:

del_in
del2_in
U6(x0)
U4(x0)
U5(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule CONF_IN → U1¹(del2_in) at position [0] we obtained the following new rules:

CONF_IN → U1¹(U4(del_in))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

  ↳ PrologToPiTRSProof

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

U1¹(del2_out) → U2¹(del_out)
U2¹(del_out) → CONF_IN
CONF_IN → U1¹(U4(del_in))
U1¹(del2_out) → U2¹(U6(del_in))

The TRS R consists of the following rules:

del_in → U6(del_in)
del_in → del_out
del2_in → U4(del_in)
U6(del_out) → del_out
U4(del_out) → U5(del_in)
U5(del_out) → del2_out

The set Q consists of the following terms:

del_in
del2_in
U6(x0)
U4(x0)
U5(x0)

We have to consider all (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ QReductionProof

  ↳ PrologToPiTRSProof

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

U1¹(del2_out) → U2¹(del_out)
U2¹(del_out) → CONF_IN
CONF_IN → U1¹(U4(del_in))
U1¹(del2_out) → U2¹(U6(del_in))

The TRS R consists of the following rules:

del_in → U6(del_in)
del_in → del_out
U6(del_out) → del_out
U4(del_out) → U5(del_in)
U5(del_out) → del2_out

The set Q consists of the following terms:

del_in
del2_in
U6(x0)
U4(x0)
U5(x0)

We have to consider all (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

del2_in

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ QReductionProof

                                      ↳ QDP

  ↳ PrologToPiTRSProof

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

U1¹(del2_out) → U2¹(del_out)
U2¹(del_out) → CONF_IN
U1¹(del2_out) → U2¹(U6(del_in))
CONF_IN → U1¹(U4(del_in))

The TRS R consists of the following rules:

del_in → U6(del_in)
del_in → del_out
U6(del_out) → del_out
U4(del_out) → U5(del_in)
U5(del_out) → del2_out

The set Q consists of the following terms:

del_in
U6(x0)
U4(x0)
U5(x0)

We have to consider all (P,Q,R)-chains.
We use the technique of [30].Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

conf_in(X) → U1(X, del2_in(X, Z))
del2_in(X, Y) → U4(X, Y, del_in(U, X, Z))
del_in(X, .(H, T), .(H, T1)) → U6(X, H, T, T1, del_in(X, T, T1))
del_in(X, .(X, T), T) → del_out(X, .(X, T), T)
U6(X, H, T, T1, del_out(X, T, T1)) → del_out(X, .(H, T), .(H, T1))
U4(X, Y, del_out(U, X, Z)) → U5(X, Y, del_in(V, Z, Y))
U5(X, Y, del_out(V, Z, Y)) → del2_out(X, Y)
U1(X, del2_out(X, Z)) → U2(X, del_in(U, Y, Z))
U2(X, del_out(U, Y, Z)) → U3(X, conf_in(Y))
U3(X, conf_out(Y)) → conf_out(X)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

Pi-finite rewrite system:
The TRS R consists of the following rules:

conf_in(X) → U1(X, del2_in(X, Z))
del2_in(X, Y) → U4(X, Y, del_in(U, X, Z))
del_in(X, .(H, T), .(H, T1)) → U6(X, H, T, T1, del_in(X, T, T1))
del_in(X, .(X, T), T) → del_out(X, .(X, T), T)
U6(X, H, T, T1, del_out(X, T, T1)) → del_out(X, .(H, T), .(H, T1))
U4(X, Y, del_out(U, X, Z)) → U5(X, Y, del_in(V, Z, Y))
U5(X, Y, del_out(V, Z, Y)) → del2_out(X, Y)
U1(X, del2_out(X, Z)) → U2(X, del_in(U, Y, Z))
U2(X, del_out(U, Y, Z)) → U3(X, conf_in(Y))
U3(X, conf_out(Y)) → conf_out(X)

CONF_IN(X) → U1¹(X, del2_in(X, Z))
CONF_IN(X) → DEL2_IN(X, Z)
DEL2_IN(X, Y) → U4¹(X, Y, del_in(U, X, Z))
DEL2_IN(X, Y) → DEL_IN(U, X, Z)
DEL_IN(X, .(H, T), .(H, T1)) → U6¹(X, H, T, T1, del_in(X, T, T1))
DEL_IN(X, .(H, T), .(H, T1)) → DEL_IN(X, T, T1)
U4¹(X, Y, del_out(U, X, Z)) → U5¹(X, Y, del_in(V, Z, Y))
U4¹(X, Y, del_out(U, X, Z)) → DEL_IN(V, Z, Y)
U1¹(X, del2_out(X, Z)) → U2¹(X, del_in(U, Y, Z))
U1¹(X, del2_out(X, Z)) → DEL_IN(U, Y, Z)
U2¹(X, del_out(U, Y, Z)) → U3¹(X, conf_in(Y))
U2¹(X, del_out(U, Y, Z)) → CONF_IN(Y)

The TRS R consists of the following rules:

conf_in(X) → U1(X, del2_in(X, Z))
del2_in(X, Y) → U4(X, Y, del_in(U, X, Z))
del_in(X, .(H, T), .(H, T1)) → U6(X, H, T, T1, del_in(X, T, T1))
del_in(X, .(X, T), T) → del_out(X, .(X, T), T)
U6(X, H, T, T1, del_out(X, T, T1)) → del_out(X, .(H, T), .(H, T1))
U4(X, Y, del_out(U, X, Z)) → U5(X, Y, del_in(V, Z, Y))
U5(X, Y, del_out(V, Z, Y)) → del2_out(X, Y)
U1(X, del2_out(X, Z)) → U2(X, del_in(U, Y, Z))
U2(X, del_out(U, Y, Z)) → U3(X, conf_in(Y))
U3(X, conf_out(Y)) → conf_out(X)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

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

CONF_IN(X) → U1¹(X, del2_in(X, Z))
CONF_IN(X) → DEL2_IN(X, Z)
DEL2_IN(X, Y) → U4¹(X, Y, del_in(U, X, Z))
DEL2_IN(X, Y) → DEL_IN(U, X, Z)
DEL_IN(X, .(H, T), .(H, T1)) → U6¹(X, H, T, T1, del_in(X, T, T1))
DEL_IN(X, .(H, T), .(H, T1)) → DEL_IN(X, T, T1)
U4¹(X, Y, del_out(U, X, Z)) → U5¹(X, Y, del_in(V, Z, Y))
U4¹(X, Y, del_out(U, X, Z)) → DEL_IN(V, Z, Y)
U1¹(X, del2_out(X, Z)) → U2¹(X, del_in(U, Y, Z))
U1¹(X, del2_out(X, Z)) → DEL_IN(U, Y, Z)
U2¹(X, del_out(U, Y, Z)) → U3¹(X, conf_in(Y))
U2¹(X, del_out(U, Y, Z)) → CONF_IN(Y)

The TRS R consists of the following rules:

conf_in(X) → U1(X, del2_in(X, Z))
del2_in(X, Y) → U4(X, Y, del_in(U, X, Z))
del_in(X, .(H, T), .(H, T1)) → U6(X, H, T, T1, del_in(X, T, T1))
del_in(X, .(X, T), T) → del_out(X, .(X, T), T)
U6(X, H, T, T1, del_out(X, T, T1)) → del_out(X, .(H, T), .(H, T1))
U4(X, Y, del_out(U, X, Z)) → U5(X, Y, del_in(V, Z, Y))
U5(X, Y, del_out(V, Z, Y)) → del2_out(X, Y)
U1(X, del2_out(X, Z)) → U2(X, del_in(U, Y, Z))
U2(X, del_out(U, Y, Z)) → U3(X, conf_in(Y))
U3(X, conf_out(Y)) → conf_out(X)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

DEL_IN(X, .(H, T), .(H, T1)) → DEL_IN(X, T, T1)

The TRS R consists of the following rules:

conf_in(X) → U1(X, del2_in(X, Z))
del2_in(X, Y) → U4(X, Y, del_in(U, X, Z))
del_in(X, .(H, T), .(H, T1)) → U6(X, H, T, T1, del_in(X, T, T1))
del_in(X, .(X, T), T) → del_out(X, .(X, T), T)
U6(X, H, T, T1, del_out(X, T, T1)) → del_out(X, .(H, T), .(H, T1))
U4(X, Y, del_out(U, X, Z)) → U5(X, Y, del_in(V, Z, Y))
U5(X, Y, del_out(V, Z, Y)) → del2_out(X, Y)
U1(X, del2_out(X, Z)) → U2(X, del_in(U, Y, Z))
U2(X, del_out(U, Y, Z)) → U3(X, conf_in(Y))
U3(X, conf_out(Y)) → conf_out(X)

The argument filtering Pi contains the following mapping:
conf_in(x1) = conf_in
U1(x1, x2) = U1(x2)
del2_in(x1, x2) = del2_in
U4(x1, x2, x3) = U4(x3)
del_in(x1, x2, x3) = del_in
.(x1, x2) = .(x2)
U6(x1, x2, x3, x4, x5) = U6(x5)
del_out(x1, x2, x3) = del_out
U5(x1, x2, x3) = U5(x3)
del2_out(x1, x2) = del2_out
U2(x1, x2) = U2(x2)
U3(x1, x2) = U3(x2)
conf_out(x1) = conf_out
DEL_IN(x1, x2, x3) = DEL_IN

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

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

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

DEL_IN(X, .(H, T), .(H, T1)) → DEL_IN(X, T, T1)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ NonTerminationProof

              ↳ PiDP

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

DEL_IN → DEL_IN

The TRS R consists of the following rules:none

s = DEL_IN evaluates to t =DEL_IN

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from DEL_IN to DEL_IN.

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

U1¹(X, del2_out(X, Z)) → U2¹(X, del_in(U, Y, Z))
U2¹(X, del_out(U, Y, Z)) → CONF_IN(Y)
CONF_IN(X) → U1¹(X, del2_in(X, Z))

The TRS R consists of the following rules:

conf_in(X) → U1(X, del2_in(X, Z))
del2_in(X, Y) → U4(X, Y, del_in(U, X, Z))
del_in(X, .(H, T), .(H, T1)) → U6(X, H, T, T1, del_in(X, T, T1))
del_in(X, .(X, T), T) → del_out(X, .(X, T), T)
U6(X, H, T, T1, del_out(X, T, T1)) → del_out(X, .(H, T), .(H, T1))
U4(X, Y, del_out(U, X, Z)) → U5(X, Y, del_in(V, Z, Y))
U5(X, Y, del_out(V, Z, Y)) → del2_out(X, Y)
U1(X, del2_out(X, Z)) → U2(X, del_in(U, Y, Z))
U2(X, del_out(U, Y, Z)) → U3(X, conf_in(Y))
U3(X, conf_out(Y)) → conf_out(X)

The argument filtering Pi contains the following mapping:
conf_in(x1) = conf_in
U1(x1, x2) = U1(x2)
del2_in(x1, x2) = del2_in
U4(x1, x2, x3) = U4(x3)
del_in(x1, x2, x3) = del_in
.(x1, x2) = .(x2)
U6(x1, x2, x3, x4, x5) = U6(x5)
del_out(x1, x2, x3) = del_out
U5(x1, x2, x3) = U5(x3)
del2_out(x1, x2) = del2_out
U2(x1, x2) = U2(x2)
U3(x1, x2) = U3(x2)
conf_out(x1) = conf_out
U1¹(x1, x2) = U1¹(x2)
CONF_IN(x1) = CONF_IN
U2¹(x1, x2) = U2¹(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

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

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

U1¹(X, del2_out(X, Z)) → U2¹(X, del_in(U, Y, Z))
U2¹(X, del_out(U, Y, Z)) → CONF_IN(Y)
CONF_IN(X) → U1¹(X, del2_in(X, Z))

The TRS R consists of the following rules:

del_in(X, .(H, T), .(H, T1)) → U6(X, H, T, T1, del_in(X, T, T1))
del_in(X, .(X, T), T) → del_out(X, .(X, T), T)
del2_in(X, Y) → U4(X, Y, del_in(U, X, Z))
U6(X, H, T, T1, del_out(X, T, T1)) → del_out(X, .(H, T), .(H, T1))
U4(X, Y, del_out(U, X, Z)) → U5(X, Y, del_in(V, Z, Y))
U5(X, Y, del_out(V, Z, Y)) → del2_out(X, Y)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

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

CONF_IN → U1¹(del2_in)
U1¹(del2_out) → U2¹(del_in)
U2¹(del_out) → CONF_IN

The TRS R consists of the following rules:

del_in → U6(del_in)
del_in → del_out
del2_in → U4(del_in)
U6(del_out) → del_out
U4(del_out) → U5(del_in)
U5(del_out) → del2_out

The set Q consists of the following terms:

del_in
del2_in
U6(x0)
U4(x0)
U5(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule U1¹(del2_out) → U2¹(del_in) at position [0] we obtained the following new rules:

U1¹(del2_out) → U2¹(del_out)
U1¹(del2_out) → U2¹(U6(del_in))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

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

CONF_IN → U1¹(del2_in)
U1¹(del2_out) → U2¹(del_out)
U2¹(del_out) → CONF_IN
U1¹(del2_out) → U2¹(U6(del_in))

The TRS R consists of the following rules:

del_in → U6(del_in)
del_in → del_out
del2_in → U4(del_in)
U6(del_out) → del_out
U4(del_out) → U5(del_in)
U5(del_out) → del2_out

The set Q consists of the following terms:

del_in
del2_in
U6(x0)
U4(x0)
U5(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule CONF_IN → U1¹(del2_in) at position [0] we obtained the following new rules:

CONF_IN → U1¹(U4(del_in))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

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

U1¹(del2_out) → U2¹(del_out)
U2¹(del_out) → CONF_IN
CONF_IN → U1¹(U4(del_in))
U1¹(del2_out) → U2¹(U6(del_in))

The TRS R consists of the following rules:

del_in → U6(del_in)
del_in → del_out
del2_in → U4(del_in)
U6(del_out) → del_out
U4(del_out) → U5(del_in)
U5(del_out) → del2_out

The set Q consists of the following terms:

del_in
del2_in
U6(x0)
U4(x0)
U5(x0)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ QReductionProof

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

U1¹(del2_out) → U2¹(del_out)
U2¹(del_out) → CONF_IN
CONF_IN → U1¹(U4(del_in))
U1¹(del2_out) → U2¹(U6(del_in))

The TRS R consists of the following rules:

del_in → U6(del_in)
del_in → del_out
U4(del_out) → U5(del_in)
U5(del_out) → del2_out
U6(del_out) → del_out

The set Q consists of the following terms:

del_in
del2_in
U6(x0)
U4(x0)
U5(x0)

We have to consider all (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

del2_in

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ QReductionProof

                                      ↳ QDP

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

U1¹(del2_out) → U2¹(del_out)
U2¹(del_out) → CONF_IN
U1¹(del2_out) → U2¹(U6(del_in))
CONF_IN → U1¹(U4(del_in))

The TRS R consists of the following rules:

del_in → U6(del_in)
del_in → del_out
U4(del_out) → U5(del_in)
U5(del_out) → del2_out
U6(del_out) → del_out

The set Q consists of the following terms:

del_in
U6(x0)
U4(x0)
U5(x0)

We have to consider all (P,Q,R)-chains.