Left Termination proof of /tmp/tmpp8wSrV/confdel.pl

Left Termination of the query pattern conf_in_1(g) w.r.t. the given Prolog program could successfully be proven:

↳ Prolog

  ↳ 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]. With regard to the inferred argument filtering the predicates were used in the following modes:
conf_in: (b)
del2_in: (b,f)
del_in: (f,b,f) (f,f,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

conf_in_g(X) → U1_g(X, del2_in_ga(X, Z))
del2_in_ga(X, Y) → U4_ga(X, Y, del_in_aga(U, X, Z))
del_in_aga(X, .(X, T), T) → del_out_aga(X, .(X, T), T)
del_in_aga(X, .(H, T), .(H, T1)) → U6_aga(X, H, T, T1, del_in_aga(X, T, T1))
U6_aga(X, H, T, T1, del_out_aga(X, T, T1)) → del_out_aga(X, .(H, T), .(H, T1))
U4_ga(X, Y, del_out_aga(U, X, Z)) → U5_ga(X, Y, del_in_aga(V, Z, Y))
U5_ga(X, Y, del_out_aga(V, Z, Y)) → del2_out_ga(X, Y)
U1_g(X, del2_out_ga(X, Z)) → U2_g(X, del_in_aag(U, Y, Z))
del_in_aag(X, .(X, T), T) → del_out_aag(X, .(X, T), T)
del_in_aag(X, .(H, T), .(H, T1)) → U6_aag(X, H, T, T1, del_in_aag(X, T, T1))
U6_aag(X, H, T, T1, del_out_aag(X, T, T1)) → del_out_aag(X, .(H, T), .(H, T1))
U2_g(X, del_out_aag(U, Y, Z)) → U3_g(X, conf_in_g(Y))
U3_g(X, conf_out_g(Y)) → conf_out_g(X)

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:

conf_in_g(X) → U1_g(X, del2_in_ga(X, Z))
del2_in_ga(X, Y) → U4_ga(X, Y, del_in_aga(U, X, Z))
del_in_aga(X, .(X, T), T) → del_out_aga(X, .(X, T), T)
del_in_aga(X, .(H, T), .(H, T1)) → U6_aga(X, H, T, T1, del_in_aga(X, T, T1))
U6_aga(X, H, T, T1, del_out_aga(X, T, T1)) → del_out_aga(X, .(H, T), .(H, T1))
U4_ga(X, Y, del_out_aga(U, X, Z)) → U5_ga(X, Y, del_in_aga(V, Z, Y))
U5_ga(X, Y, del_out_aga(V, Z, Y)) → del2_out_ga(X, Y)
U1_g(X, del2_out_ga(X, Z)) → U2_g(X, del_in_aag(U, Y, Z))
del_in_aag(X, .(X, T), T) → del_out_aag(X, .(X, T), T)
del_in_aag(X, .(H, T), .(H, T1)) → U6_aag(X, H, T, T1, del_in_aag(X, T, T1))
U6_aag(X, H, T, T1, del_out_aag(X, T, T1)) → del_out_aag(X, .(H, T), .(H, T1))
U2_g(X, del_out_aag(U, Y, Z)) → U3_g(X, conf_in_g(Y))
U3_g(X, conf_out_g(Y)) → conf_out_g(X)

CONF_IN_G(X) → U1_G(X, del2_in_ga(X, Z))
CONF_IN_G(X) → DEL2_IN_GA(X, Z)
DEL2_IN_GA(X, Y) → U4_GA(X, Y, del_in_aga(U, X, Z))
DEL2_IN_GA(X, Y) → DEL_IN_AGA(U, X, Z)
DEL_IN_AGA(X, .(H, T), .(H, T1)) → U6_AGA(X, H, T, T1, del_in_aga(X, T, T1))
DEL_IN_AGA(X, .(H, T), .(H, T1)) → DEL_IN_AGA(X, T, T1)
U4_GA(X, Y, del_out_aga(U, X, Z)) → U5_GA(X, Y, del_in_aga(V, Z, Y))
U4_GA(X, Y, del_out_aga(U, X, Z)) → DEL_IN_AGA(V, Z, Y)
U1_G(X, del2_out_ga(X, Z)) → U2_G(X, del_in_aag(U, Y, Z))
U1_G(X, del2_out_ga(X, Z)) → DEL_IN_AAG(U, Y, Z)
DEL_IN_AAG(X, .(H, T), .(H, T1)) → U6_AAG(X, H, T, T1, del_in_aag(X, T, T1))
DEL_IN_AAG(X, .(H, T), .(H, T1)) → DEL_IN_AAG(X, T, T1)
U2_G(X, del_out_aag(U, Y, Z)) → U3_G(X, conf_in_g(Y))
U2_G(X, del_out_aag(U, Y, Z)) → CONF_IN_G(Y)

The TRS R consists of the following rules:

conf_in_g(X) → U1_g(X, del2_in_ga(X, Z))
del2_in_ga(X, Y) → U4_ga(X, Y, del_in_aga(U, X, Z))
del_in_aga(X, .(X, T), T) → del_out_aga(X, .(X, T), T)
del_in_aga(X, .(H, T), .(H, T1)) → U6_aga(X, H, T, T1, del_in_aga(X, T, T1))
U6_aga(X, H, T, T1, del_out_aga(X, T, T1)) → del_out_aga(X, .(H, T), .(H, T1))
U4_ga(X, Y, del_out_aga(U, X, Z)) → U5_ga(X, Y, del_in_aga(V, Z, Y))
U5_ga(X, Y, del_out_aga(V, Z, Y)) → del2_out_ga(X, Y)
U1_g(X, del2_out_ga(X, Z)) → U2_g(X, del_in_aag(U, Y, Z))
del_in_aag(X, .(X, T), T) → del_out_aag(X, .(X, T), T)
del_in_aag(X, .(H, T), .(H, T1)) → U6_aag(X, H, T, T1, del_in_aag(X, T, T1))
U6_aag(X, H, T, T1, del_out_aag(X, T, T1)) → del_out_aag(X, .(H, T), .(H, T1))
U2_g(X, del_out_aag(U, Y, Z)) → U3_g(X, conf_in_g(Y))
U3_g(X, conf_out_g(Y)) → conf_out_g(X)

The argument filtering Pi contains the following mapping:
conf_in_g(x1) = conf_in_g(x1)
U1_g(x1, x2) = U1_g(x2)
del2_in_ga(x1, x2) = del2_in_ga(x1)
U4_ga(x1, x2, x3) = U4_ga(x3)
del_in_aga(x1, x2, x3) = del_in_aga(x2)
.(x1, x2) = .(x2)
del_out_aga(x1, x2, x3) = del_out_aga(x3)
U6_aga(x1, x2, x3, x4, x5) = U6_aga(x5)
U5_ga(x1, x2, x3) = U5_ga(x3)
del2_out_ga(x1, x2) = del2_out_ga(x2)
U2_g(x1, x2) = U2_g(x2)
del_in_aag(x1, x2, x3) = del_in_aag(x3)
del_out_aag(x1, x2, x3) = del_out_aag(x2)
U6_aag(x1, x2, x3, x4, x5) = U6_aag(x5)
U3_g(x1, x2) = U3_g(x2)
conf_out_g(x1) = conf_out_g
U2_G(x1, x2) = U2_G(x2)
U6_AAG(x1, x2, x3, x4, x5) = U6_AAG(x5)
U5_GA(x1, x2, x3) = U5_GA(x3)
DEL_IN_AGA(x1, x2, x3) = DEL_IN_AGA(x2)
U6_AGA(x1, x2, x3, x4, x5) = U6_AGA(x5)
U4_GA(x1, x2, x3) = U4_GA(x3)
DEL_IN_AAG(x1, x2, x3) = DEL_IN_AAG(x3)
U1_G(x1, x2) = U1_G(x2)
U3_G(x1, x2) = U3_G(x2)
DEL2_IN_GA(x1, x2) = DEL2_IN_GA(x1)
CONF_IN_G(x1) = CONF_IN_G(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:

CONF_IN_G(X) → U1_G(X, del2_in_ga(X, Z))
CONF_IN_G(X) → DEL2_IN_GA(X, Z)
DEL2_IN_GA(X, Y) → U4_GA(X, Y, del_in_aga(U, X, Z))
DEL2_IN_GA(X, Y) → DEL_IN_AGA(U, X, Z)
DEL_IN_AGA(X, .(H, T), .(H, T1)) → U6_AGA(X, H, T, T1, del_in_aga(X, T, T1))
DEL_IN_AGA(X, .(H, T), .(H, T1)) → DEL_IN_AGA(X, T, T1)
U4_GA(X, Y, del_out_aga(U, X, Z)) → U5_GA(X, Y, del_in_aga(V, Z, Y))
U4_GA(X, Y, del_out_aga(U, X, Z)) → DEL_IN_AGA(V, Z, Y)
U1_G(X, del2_out_ga(X, Z)) → U2_G(X, del_in_aag(U, Y, Z))
U1_G(X, del2_out_ga(X, Z)) → DEL_IN_AAG(U, Y, Z)
DEL_IN_AAG(X, .(H, T), .(H, T1)) → U6_AAG(X, H, T, T1, del_in_aag(X, T, T1))
DEL_IN_AAG(X, .(H, T), .(H, T1)) → DEL_IN_AAG(X, T, T1)
U2_G(X, del_out_aag(U, Y, Z)) → U3_G(X, conf_in_g(Y))
U2_G(X, del_out_aag(U, Y, Z)) → CONF_IN_G(Y)

The TRS R consists of the following rules:

conf_in_g(X) → U1_g(X, del2_in_ga(X, Z))
del2_in_ga(X, Y) → U4_ga(X, Y, del_in_aga(U, X, Z))
del_in_aga(X, .(X, T), T) → del_out_aga(X, .(X, T), T)
del_in_aga(X, .(H, T), .(H, T1)) → U6_aga(X, H, T, T1, del_in_aga(X, T, T1))
U6_aga(X, H, T, T1, del_out_aga(X, T, T1)) → del_out_aga(X, .(H, T), .(H, T1))
U4_ga(X, Y, del_out_aga(U, X, Z)) → U5_ga(X, Y, del_in_aga(V, Z, Y))
U5_ga(X, Y, del_out_aga(V, Z, Y)) → del2_out_ga(X, Y)
U1_g(X, del2_out_ga(X, Z)) → U2_g(X, del_in_aag(U, Y, Z))
del_in_aag(X, .(X, T), T) → del_out_aag(X, .(X, T), T)
del_in_aag(X, .(H, T), .(H, T1)) → U6_aag(X, H, T, T1, del_in_aag(X, T, T1))
U6_aag(X, H, T, T1, del_out_aag(X, T, T1)) → del_out_aag(X, .(H, T), .(H, T1))
U2_g(X, del_out_aag(U, Y, Z)) → U3_g(X, conf_in_g(Y))
U3_g(X, conf_out_g(Y)) → conf_out_g(X)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

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

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

The TRS R consists of the following rules:

conf_in_g(X) → U1_g(X, del2_in_ga(X, Z))
del2_in_ga(X, Y) → U4_ga(X, Y, del_in_aga(U, X, Z))
del_in_aga(X, .(X, T), T) → del_out_aga(X, .(X, T), T)
del_in_aga(X, .(H, T), .(H, T1)) → U6_aga(X, H, T, T1, del_in_aga(X, T, T1))
U6_aga(X, H, T, T1, del_out_aga(X, T, T1)) → del_out_aga(X, .(H, T), .(H, T1))
U4_ga(X, Y, del_out_aga(U, X, Z)) → U5_ga(X, Y, del_in_aga(V, Z, Y))
U5_ga(X, Y, del_out_aga(V, Z, Y)) → del2_out_ga(X, Y)
U1_g(X, del2_out_ga(X, Z)) → U2_g(X, del_in_aag(U, Y, Z))
del_in_aag(X, .(X, T), T) → del_out_aag(X, .(X, T), T)
del_in_aag(X, .(H, T), .(H, T1)) → U6_aag(X, H, T, T1, del_in_aag(X, T, T1))
U6_aag(X, H, T, T1, del_out_aag(X, T, T1)) → del_out_aag(X, .(H, T), .(H, T1))
U2_g(X, del_out_aag(U, Y, Z)) → U3_g(X, conf_in_g(Y))
U3_g(X, conf_out_g(Y)) → conf_out_g(X)

The argument filtering Pi contains the following mapping:
conf_in_g(x1) = conf_in_g(x1)
U1_g(x1, x2) = U1_g(x2)
del2_in_ga(x1, x2) = del2_in_ga(x1)
U4_ga(x1, x2, x3) = U4_ga(x3)
del_in_aga(x1, x2, x3) = del_in_aga(x2)
.(x1, x2) = .(x2)
del_out_aga(x1, x2, x3) = del_out_aga(x3)
U6_aga(x1, x2, x3, x4, x5) = U6_aga(x5)
U5_ga(x1, x2, x3) = U5_ga(x3)
del2_out_ga(x1, x2) = del2_out_ga(x2)
U2_g(x1, x2) = U2_g(x2)
del_in_aag(x1, x2, x3) = del_in_aag(x3)
del_out_aag(x1, x2, x3) = del_out_aag(x2)
U6_aag(x1, x2, x3, x4, x5) = U6_aag(x5)
U3_g(x1, x2) = U3_g(x2)
conf_out_g(x1) = conf_out_g
DEL_IN_AAG(x1, x2, x3) = DEL_IN_AAG(x3)

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

              ↳ PiDP

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

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

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

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

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

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

DEL_IN_AAG(.(T1)) → DEL_IN_AAG(T1)

R is empty.
Q is empty.
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:

DEL_IN_AAG(.(T1)) → DEL_IN_AAG(T1)
The graph contains the following edges 1 > 1

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

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

The TRS R consists of the following rules:

conf_in_g(X) → U1_g(X, del2_in_ga(X, Z))
del2_in_ga(X, Y) → U4_ga(X, Y, del_in_aga(U, X, Z))
del_in_aga(X, .(X, T), T) → del_out_aga(X, .(X, T), T)
del_in_aga(X, .(H, T), .(H, T1)) → U6_aga(X, H, T, T1, del_in_aga(X, T, T1))
U6_aga(X, H, T, T1, del_out_aga(X, T, T1)) → del_out_aga(X, .(H, T), .(H, T1))
U4_ga(X, Y, del_out_aga(U, X, Z)) → U5_ga(X, Y, del_in_aga(V, Z, Y))
U5_ga(X, Y, del_out_aga(V, Z, Y)) → del2_out_ga(X, Y)
U1_g(X, del2_out_ga(X, Z)) → U2_g(X, del_in_aag(U, Y, Z))
del_in_aag(X, .(X, T), T) → del_out_aag(X, .(X, T), T)
del_in_aag(X, .(H, T), .(H, T1)) → U6_aag(X, H, T, T1, del_in_aag(X, T, T1))
U6_aag(X, H, T, T1, del_out_aag(X, T, T1)) → del_out_aag(X, .(H, T), .(H, T1))
U2_g(X, del_out_aag(U, Y, Z)) → U3_g(X, conf_in_g(Y))
U3_g(X, conf_out_g(Y)) → conf_out_g(X)

The argument filtering Pi contains the following mapping:
conf_in_g(x1) = conf_in_g(x1)
U1_g(x1, x2) = U1_g(x2)
del2_in_ga(x1, x2) = del2_in_ga(x1)
U4_ga(x1, x2, x3) = U4_ga(x3)
del_in_aga(x1, x2, x3) = del_in_aga(x2)
.(x1, x2) = .(x2)
del_out_aga(x1, x2, x3) = del_out_aga(x3)
U6_aga(x1, x2, x3, x4, x5) = U6_aga(x5)
U5_ga(x1, x2, x3) = U5_ga(x3)
del2_out_ga(x1, x2) = del2_out_ga(x2)
U2_g(x1, x2) = U2_g(x2)
del_in_aag(x1, x2, x3) = del_in_aag(x3)
del_out_aag(x1, x2, x3) = del_out_aag(x2)
U6_aag(x1, x2, x3, x4, x5) = U6_aag(x5)
U3_g(x1, x2) = U3_g(x2)
conf_out_g(x1) = conf_out_g
DEL_IN_AGA(x1, x2, x3) = DEL_IN_AGA(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

              ↳ PiDP

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

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

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

                        ↳ QDPSizeChangeProof

              ↳ PiDP

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

DEL_IN_AGA(.(T)) → DEL_IN_AGA(T)

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

DEL_IN_AGA(.(T)) → DEL_IN_AGA(T)
The graph contains the following edges 1 > 1

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

U2_G(X, del_out_aag(U, Y, Z)) → CONF_IN_G(Y)
CONF_IN_G(X) → U1_G(X, del2_in_ga(X, Z))
U1_G(X, del2_out_ga(X, Z)) → U2_G(X, del_in_aag(U, Y, Z))

The TRS R consists of the following rules:

conf_in_g(X) → U1_g(X, del2_in_ga(X, Z))
del2_in_ga(X, Y) → U4_ga(X, Y, del_in_aga(U, X, Z))
del_in_aga(X, .(X, T), T) → del_out_aga(X, .(X, T), T)
del_in_aga(X, .(H, T), .(H, T1)) → U6_aga(X, H, T, T1, del_in_aga(X, T, T1))
U6_aga(X, H, T, T1, del_out_aga(X, T, T1)) → del_out_aga(X, .(H, T), .(H, T1))
U4_ga(X, Y, del_out_aga(U, X, Z)) → U5_ga(X, Y, del_in_aga(V, Z, Y))
U5_ga(X, Y, del_out_aga(V, Z, Y)) → del2_out_ga(X, Y)
U1_g(X, del2_out_ga(X, Z)) → U2_g(X, del_in_aag(U, Y, Z))
del_in_aag(X, .(X, T), T) → del_out_aag(X, .(X, T), T)
del_in_aag(X, .(H, T), .(H, T1)) → U6_aag(X, H, T, T1, del_in_aag(X, T, T1))
U6_aag(X, H, T, T1, del_out_aag(X, T, T1)) → del_out_aag(X, .(H, T), .(H, T1))
U2_g(X, del_out_aag(U, Y, Z)) → U3_g(X, conf_in_g(Y))
U3_g(X, conf_out_g(Y)) → conf_out_g(X)

The argument filtering Pi contains the following mapping:
conf_in_g(x1) = conf_in_g(x1)
U1_g(x1, x2) = U1_g(x2)
del2_in_ga(x1, x2) = del2_in_ga(x1)
U4_ga(x1, x2, x3) = U4_ga(x3)
del_in_aga(x1, x2, x3) = del_in_aga(x2)
.(x1, x2) = .(x2)
del_out_aga(x1, x2, x3) = del_out_aga(x3)
U6_aga(x1, x2, x3, x4, x5) = U6_aga(x5)
U5_ga(x1, x2, x3) = U5_ga(x3)
del2_out_ga(x1, x2) = del2_out_ga(x2)
U2_g(x1, x2) = U2_g(x2)
del_in_aag(x1, x2, x3) = del_in_aag(x3)
del_out_aag(x1, x2, x3) = del_out_aag(x2)
U6_aag(x1, x2, x3, x4, x5) = U6_aag(x5)
U3_g(x1, x2) = U3_g(x2)
conf_out_g(x1) = conf_out_g
U2_G(x1, x2) = U2_G(x2)
U1_G(x1, x2) = U1_G(x2)
CONF_IN_G(x1) = CONF_IN_G(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

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

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

U2_G(X, del_out_aag(U, Y, Z)) → CONF_IN_G(Y)
CONF_IN_G(X) → U1_G(X, del2_in_ga(X, Z))
U1_G(X, del2_out_ga(X, Z)) → U2_G(X, del_in_aag(U, Y, Z))

The TRS R consists of the following rules:

del2_in_ga(X, Y) → U4_ga(X, Y, del_in_aga(U, X, Z))
del_in_aag(X, .(X, T), T) → del_out_aag(X, .(X, T), T)
del_in_aag(X, .(H, T), .(H, T1)) → U6_aag(X, H, T, T1, del_in_aag(X, T, T1))
U4_ga(X, Y, del_out_aga(U, X, Z)) → U5_ga(X, Y, del_in_aga(V, Z, Y))
U6_aag(X, H, T, T1, del_out_aag(X, T, T1)) → del_out_aag(X, .(H, T), .(H, T1))
del_in_aga(X, .(X, T), T) → del_out_aga(X, .(X, T), T)
del_in_aga(X, .(H, T), .(H, T1)) → U6_aga(X, H, T, T1, del_in_aga(X, T, T1))
U5_ga(X, Y, del_out_aga(V, Z, Y)) → del2_out_ga(X, Y)
U6_aga(X, H, T, T1, del_out_aga(X, T, T1)) → del_out_aga(X, .(H, T), .(H, T1))

The argument filtering Pi contains the following mapping:
del2_in_ga(x1, x2) = del2_in_ga(x1)
U4_ga(x1, x2, x3) = U4_ga(x3)
del_in_aga(x1, x2, x3) = del_in_aga(x2)
.(x1, x2) = .(x2)
del_out_aga(x1, x2, x3) = del_out_aga(x3)
U6_aga(x1, x2, x3, x4, x5) = U6_aga(x5)
U5_ga(x1, x2, x3) = U5_ga(x3)
del2_out_ga(x1, x2) = del2_out_ga(x2)
del_in_aag(x1, x2, x3) = del_in_aag(x3)
del_out_aag(x1, x2, x3) = del_out_aag(x2)
U6_aag(x1, x2, x3, x4, x5) = U6_aag(x5)
U2_G(x1, x2) = U2_G(x2)
U1_G(x1, x2) = U1_G(x2)
CONF_IN_G(x1) = CONF_IN_G(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

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ RuleRemovalProof

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

CONF_IN_G(X) → U1_G(del2_in_ga(X))
U2_G(del_out_aag(Y)) → CONF_IN_G(Y)
U1_G(del2_out_ga(Z)) → U2_G(del_in_aag(Z))

The TRS R consists of the following rules:

del2_in_ga(X) → U4_ga(del_in_aga(X))
del_in_aag(T) → del_out_aag(.(T))
del_in_aag(.(T1)) → U6_aag(del_in_aag(T1))
U4_ga(del_out_aga(Z)) → U5_ga(del_in_aga(Z))
U6_aag(del_out_aag(T)) → del_out_aag(.(T))
del_in_aga(.(T)) → del_out_aga(T)
del_in_aga(.(T)) → U6_aga(del_in_aga(T))
U5_ga(del_out_aga(Y)) → del2_out_ga(Y)
U6_aga(del_out_aga(T1)) → del_out_aga(.(T1))

The set Q consists of the following terms:

del2_in_ga(x0)
del_in_aag(x0)
U4_ga(x0)
U6_aag(x0)
del_in_aga(x0)
U5_ga(x0)
U6_aga(x0)

We have to consider all (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.

Strictly oriented rules of the TRS R:

U4_ga(del_out_aga(Z)) → U5_ga(del_in_aga(Z))

Used ordering: POLO with Polynomial interpretation [25]:

POL(.(x₁)) = 1 + x₁
POL(CONF_IN_G(x₁)) = 2·x₁
POL(U1_G(x₁)) = 2·x₁
POL(U2_G(x₁)) = x₁
POL(U4_ga(x₁)) = x₁
POL(U5_ga(x₁)) = x₁
POL(U6_aag(x₁)) = 2 + x₁
POL(U6_aga(x₁)) = 1 + x₁
POL(del2_in_ga(x₁)) = x₁
POL(del2_out_ga(x₁)) = 1 + x₁
POL(del_in_aag(x₁)) = 2 + 2·x₁
POL(del_in_aga(x₁)) = x₁
POL(del_out_aag(x₁)) = 2·x₁
POL(del_out_aga(x₁)) = 1 + x₁

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ DependencyGraphProof

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

CONF_IN_G(X) → U1_G(del2_in_ga(X))
U2_G(del_out_aag(Y)) → CONF_IN_G(Y)
U1_G(del2_out_ga(Z)) → U2_G(del_in_aag(Z))

The TRS R consists of the following rules:

del2_in_ga(X) → U4_ga(del_in_aga(X))
del_in_aag(T) → del_out_aag(.(T))
del_in_aag(.(T1)) → U6_aag(del_in_aag(T1))
U6_aag(del_out_aag(T)) → del_out_aag(.(T))
del_in_aga(.(T)) → del_out_aga(T)
del_in_aga(.(T)) → U6_aga(del_in_aga(T))
U5_ga(del_out_aga(Y)) → del2_out_ga(Y)
U6_aga(del_out_aga(T1)) → del_out_aga(.(T1))

The set Q consists of the following terms:

del2_in_ga(x0)
del_in_aag(x0)
U4_ga(x0)
U6_aag(x0)
del_in_aga(x0)
U5_ga(x0)
U6_aga(x0)

We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 3 less nodes.