Left Termination proof of /tmp/tmp6MlSHN/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]. With regard to the inferred argument filtering the predicates were used in the following modes:
conf_in: (b) (f)
del2_in: (b,f) (f,f)
del_in: (f,b,f) (f,f,b) (f,f,f)
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_a(Y))
conf_in_a(X) → U1_a(X, del2_in_aa(X, Z))
del2_in_aa(X, Y) → U4_aa(X, Y, del_in_aaa(U, X, Z))
del_in_aaa(X, .(X, T), T) → del_out_aaa(X, .(X, T), T)
del_in_aaa(X, .(H, T), .(H, T1)) → U6_aaa(X, H, T, T1, del_in_aaa(X, T, T1))
U6_aaa(X, H, T, T1, del_out_aaa(X, T, T1)) → del_out_aaa(X, .(H, T), .(H, T1))
U4_aa(X, Y, del_out_aaa(U, X, Z)) → U5_aa(X, Y, del_in_aaa(V, Z, Y))
U5_aa(X, Y, del_out_aaa(V, Z, Y)) → del2_out_aa(X, Y)
U1_a(X, del2_out_aa(X, Z)) → U2_a(X, del_in_aaa(U, Y, Z))
U2_a(X, del_out_aaa(U, Y, Z)) → U3_a(X, conf_in_a(Y))
U3_a(X, conf_out_a(Y)) → conf_out_a(X)
U3_g(X, conf_out_a(Y)) → conf_out_g(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_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_a(Y))
conf_in_a(X) → U1_a(X, del2_in_aa(X, Z))
del2_in_aa(X, Y) → U4_aa(X, Y, del_in_aaa(U, X, Z))
del_in_aaa(X, .(X, T), T) → del_out_aaa(X, .(X, T), T)
del_in_aaa(X, .(H, T), .(H, T1)) → U6_aaa(X, H, T, T1, del_in_aaa(X, T, T1))
U6_aaa(X, H, T, T1, del_out_aaa(X, T, T1)) → del_out_aaa(X, .(H, T), .(H, T1))
U4_aa(X, Y, del_out_aaa(U, X, Z)) → U5_aa(X, Y, del_in_aaa(V, Z, Y))
U5_aa(X, Y, del_out_aaa(V, Z, Y)) → del2_out_aa(X, Y)
U1_a(X, del2_out_aa(X, Z)) → U2_a(X, del_in_aaa(U, Y, Z))
U2_a(X, del_out_aaa(U, Y, Z)) → U3_a(X, conf_in_a(Y))
U3_a(X, conf_out_a(Y)) → conf_out_a(X)
U3_g(X, conf_out_a(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_a(Y))
U2_G(X, del_out_aag(U, Y, Z)) → CONF_IN_A(Y)
CONF_IN_A(X) → U1_A(X, del2_in_aa(X, Z))
CONF_IN_A(X) → DEL2_IN_AA(X, Z)
DEL2_IN_AA(X, Y) → U4_AA(X, Y, del_in_aaa(U, X, Z))
DEL2_IN_AA(X, Y) → DEL_IN_AAA(U, X, Z)
DEL_IN_AAA(X, .(H, T), .(H, T1)) → U6_AAA(X, H, T, T1, del_in_aaa(X, T, T1))
DEL_IN_AAA(X, .(H, T), .(H, T1)) → DEL_IN_AAA(X, T, T1)
U4_AA(X, Y, del_out_aaa(U, X, Z)) → U5_AA(X, Y, del_in_aaa(V, Z, Y))
U4_AA(X, Y, del_out_aaa(U, X, Z)) → DEL_IN_AAA(V, Z, Y)
U1_A(X, del2_out_aa(X, Z)) → U2_A(X, del_in_aaa(U, Y, Z))
U1_A(X, del2_out_aa(X, Z)) → DEL_IN_AAA(U, Y, Z)
U2_A(X, del_out_aaa(U, Y, Z)) → U3_A(X, conf_in_a(Y))
U2_A(X, del_out_aaa(U, Y, Z)) → CONF_IN_A(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_a(Y))
conf_in_a(X) → U1_a(X, del2_in_aa(X, Z))
del2_in_aa(X, Y) → U4_aa(X, Y, del_in_aaa(U, X, Z))
del_in_aaa(X, .(X, T), T) → del_out_aaa(X, .(X, T), T)
del_in_aaa(X, .(H, T), .(H, T1)) → U6_aaa(X, H, T, T1, del_in_aaa(X, T, T1))
U6_aaa(X, H, T, T1, del_out_aaa(X, T, T1)) → del_out_aaa(X, .(H, T), .(H, T1))
U4_aa(X, Y, del_out_aaa(U, X, Z)) → U5_aa(X, Y, del_in_aaa(V, Z, Y))
U5_aa(X, Y, del_out_aaa(V, Z, Y)) → del2_out_aa(X, Y)
U1_a(X, del2_out_aa(X, Z)) → U2_a(X, del_in_aaa(U, Y, Z))
U2_a(X, del_out_aaa(U, Y, Z)) → U3_a(X, conf_in_a(Y))
U3_a(X, conf_out_a(Y)) → conf_out_a(X)
U3_g(X, conf_out_a(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) = .(x1, x2)
del_out_aga(x1, x2, x3) = del_out_aga(x1, x3)
U6_aga(x1, x2, x3, x4, x5) = U6_aga(x2, 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
U6_aag(x1, x2, x3, x4, x5) = U6_aag(x5)
U3_g(x1, x2) = U3_g(x2)
conf_in_a(x1) = conf_in_a
U1_a(x1, x2) = U1_a(x2)
del2_in_aa(x1, x2) = del2_in_aa
U4_aa(x1, x2, x3) = U4_aa(x3)
del_in_aaa(x1, x2, x3) = del_in_aaa
del_out_aaa(x1, x2, x3) = del_out_aaa
U6_aaa(x1, x2, x3, x4, x5) = U6_aaa(x5)
U5_aa(x1, x2, x3) = U5_aa(x3)
del2_out_aa(x1, x2) = del2_out_aa
U2_a(x1, x2) = U2_a(x2)
U3_a(x1, x2) = U3_a(x2)
conf_out_a(x1) = conf_out_a
conf_out_g(x1) = conf_out_g
U1_A(x1, x2) = U1_A(x2)
U5_GA(x1, x2, x3) = U5_GA(x3)
U6_AAG(x1, x2, x3, x4, x5) = U6_AAG(x5)
U6_AGA(x1, x2, x3, x4, x5) = U6_AGA(x2, x5)
DEL_IN_AAA(x1, x2, x3) = DEL_IN_AAA
U4_GA(x1, x2, x3) = U4_GA(x3)
U1_G(x1, x2) = U1_G(x2)
U3_G(x1, x2) = U3_G(x2)
U2_G(x1, x2) = U2_G(x2)
U5_AA(x1, x2, x3) = U5_AA(x3)
U3_A(x1, x2) = U3_A(x2)
DEL_IN_AGA(x1, x2, x3) = DEL_IN_AGA(x2)
U2_A(x1, x2) = U2_A(x2)
U6_AAA(x1, x2, x3, x4, x5) = U6_AAA(x5)
DEL_IN_AAG(x1, x2, x3) = DEL_IN_AAG(x3)
DEL2_IN_GA(x1, x2) = DEL2_IN_GA(x1)
U4_AA(x1, x2, x3) = U4_AA(x3)
CONF_IN_A(x1) = CONF_IN_A
DEL2_IN_AA(x1, x2) = DEL2_IN_AA
CONF_IN_G(x1) = CONF_IN_G(x1)

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_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_a(Y))
U2_G(X, del_out_aag(U, Y, Z)) → CONF_IN_A(Y)
CONF_IN_A(X) → U1_A(X, del2_in_aa(X, Z))
CONF_IN_A(X) → DEL2_IN_AA(X, Z)
DEL2_IN_AA(X, Y) → U4_AA(X, Y, del_in_aaa(U, X, Z))
DEL2_IN_AA(X, Y) → DEL_IN_AAA(U, X, Z)
DEL_IN_AAA(X, .(H, T), .(H, T1)) → U6_AAA(X, H, T, T1, del_in_aaa(X, T, T1))
DEL_IN_AAA(X, .(H, T), .(H, T1)) → DEL_IN_AAA(X, T, T1)
U4_AA(X, Y, del_out_aaa(U, X, Z)) → U5_AA(X, Y, del_in_aaa(V, Z, Y))
U4_AA(X, Y, del_out_aaa(U, X, Z)) → DEL_IN_AAA(V, Z, Y)
U1_A(X, del2_out_aa(X, Z)) → U2_A(X, del_in_aaa(U, Y, Z))
U1_A(X, del2_out_aa(X, Z)) → DEL_IN_AAA(U, Y, Z)
U2_A(X, del_out_aaa(U, Y, Z)) → U3_A(X, conf_in_a(Y))
U2_A(X, del_out_aaa(U, Y, Z)) → CONF_IN_A(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_a(Y))
conf_in_a(X) → U1_a(X, del2_in_aa(X, Z))
del2_in_aa(X, Y) → U4_aa(X, Y, del_in_aaa(U, X, Z))
del_in_aaa(X, .(X, T), T) → del_out_aaa(X, .(X, T), T)
del_in_aaa(X, .(H, T), .(H, T1)) → U6_aaa(X, H, T, T1, del_in_aaa(X, T, T1))
U6_aaa(X, H, T, T1, del_out_aaa(X, T, T1)) → del_out_aaa(X, .(H, T), .(H, T1))
U4_aa(X, Y, del_out_aaa(U, X, Z)) → U5_aa(X, Y, del_in_aaa(V, Z, Y))
U5_aa(X, Y, del_out_aaa(V, Z, Y)) → del2_out_aa(X, Y)
U1_a(X, del2_out_aa(X, Z)) → U2_a(X, del_in_aaa(U, Y, Z))
U2_a(X, del_out_aaa(U, Y, Z)) → U3_a(X, conf_in_a(Y))
U3_a(X, conf_out_a(Y)) → conf_out_a(X)
U3_g(X, conf_out_a(Y)) → conf_out_g(X)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

DEL_IN_AAA(X, .(H, T), .(H, T1)) → DEL_IN_AAA(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_a(Y))
conf_in_a(X) → U1_a(X, del2_in_aa(X, Z))
del2_in_aa(X, Y) → U4_aa(X, Y, del_in_aaa(U, X, Z))
del_in_aaa(X, .(X, T), T) → del_out_aaa(X, .(X, T), T)
del_in_aaa(X, .(H, T), .(H, T1)) → U6_aaa(X, H, T, T1, del_in_aaa(X, T, T1))
U6_aaa(X, H, T, T1, del_out_aaa(X, T, T1)) → del_out_aaa(X, .(H, T), .(H, T1))
U4_aa(X, Y, del_out_aaa(U, X, Z)) → U5_aa(X, Y, del_in_aaa(V, Z, Y))
U5_aa(X, Y, del_out_aaa(V, Z, Y)) → del2_out_aa(X, Y)
U1_a(X, del2_out_aa(X, Z)) → U2_a(X, del_in_aaa(U, Y, Z))
U2_a(X, del_out_aaa(U, Y, Z)) → U3_a(X, conf_in_a(Y))
U3_a(X, conf_out_a(Y)) → conf_out_a(X)
U3_g(X, conf_out_a(Y)) → conf_out_g(X)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

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

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

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

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

DEL_IN_AAA → DEL_IN_AAA

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_AAA → DEL_IN_AAA

The TRS R consists of the following rules:none

s = DEL_IN_AAA evaluates to t =DEL_IN_AAA

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_AAA to DEL_IN_AAA.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

U2_A(X, del_out_aaa(U, Y, Z)) → CONF_IN_A(Y)
CONF_IN_A(X) → U1_A(X, del2_in_aa(X, Z))
U1_A(X, del2_out_aa(X, Z)) → U2_A(X, del_in_aaa(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_a(Y))
conf_in_a(X) → U1_a(X, del2_in_aa(X, Z))
del2_in_aa(X, Y) → U4_aa(X, Y, del_in_aaa(U, X, Z))
del_in_aaa(X, .(X, T), T) → del_out_aaa(X, .(X, T), T)
del_in_aaa(X, .(H, T), .(H, T1)) → U6_aaa(X, H, T, T1, del_in_aaa(X, T, T1))
U6_aaa(X, H, T, T1, del_out_aaa(X, T, T1)) → del_out_aaa(X, .(H, T), .(H, T1))
U4_aa(X, Y, del_out_aaa(U, X, Z)) → U5_aa(X, Y, del_in_aaa(V, Z, Y))
U5_aa(X, Y, del_out_aaa(V, Z, Y)) → del2_out_aa(X, Y)
U1_a(X, del2_out_aa(X, Z)) → U2_a(X, del_in_aaa(U, Y, Z))
U2_a(X, del_out_aaa(U, Y, Z)) → U3_a(X, conf_in_a(Y))
U3_a(X, conf_out_a(Y)) → conf_out_a(X)
U3_g(X, conf_out_a(Y)) → conf_out_g(X)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

U2_A(X, del_out_aaa(U, Y, Z)) → CONF_IN_A(Y)
CONF_IN_A(X) → U1_A(X, del2_in_aa(X, Z))
U1_A(X, del2_out_aa(X, Z)) → U2_A(X, del_in_aaa(U, Y, Z))

The TRS R consists of the following rules:

del2_in_aa(X, Y) → U4_aa(X, Y, del_in_aaa(U, X, Z))
del_in_aaa(X, .(X, T), T) → del_out_aaa(X, .(X, T), T)
del_in_aaa(X, .(H, T), .(H, T1)) → U6_aaa(X, H, T, T1, del_in_aaa(X, T, T1))
U4_aa(X, Y, del_out_aaa(U, X, Z)) → U5_aa(X, Y, del_in_aaa(V, Z, Y))
U6_aaa(X, H, T, T1, del_out_aaa(X, T, T1)) → del_out_aaa(X, .(H, T), .(H, T1))
U5_aa(X, Y, del_out_aaa(V, Z, Y)) → del2_out_aa(X, Y)

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
del2_in_aa(x1, x2) = del2_in_aa
U4_aa(x1, x2, x3) = U4_aa(x3)
del_in_aaa(x1, x2, x3) = del_in_aaa
del_out_aaa(x1, x2, x3) = del_out_aaa
U6_aaa(x1, x2, x3, x4, x5) = U6_aaa(x5)
U5_aa(x1, x2, x3) = U5_aa(x3)
del2_out_aa(x1, x2) = del2_out_aa
U1_A(x1, x2) = U1_A(x2)
U2_A(x1, x2) = U2_A(x2)
CONF_IN_A(x1) = CONF_IN_A

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

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

U1_A(del2_out_aa) → U2_A(del_in_aaa)
U2_A(del_out_aaa) → CONF_IN_A
CONF_IN_A → U1_A(del2_in_aa)

The TRS R consists of the following rules:

del2_in_aa → U4_aa(del_in_aaa)
del_in_aaa → del_out_aaa
del_in_aaa → U6_aaa(del_in_aaa)
U4_aa(del_out_aaa) → U5_aa(del_in_aaa)
U6_aaa(del_out_aaa) → del_out_aaa
U5_aa(del_out_aaa) → del2_out_aa

The set Q consists of the following terms:

del2_in_aa
del_in_aaa
U4_aa(x0)
U6_aaa(x0)
U5_aa(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule CONF_IN_A → U1_A(del2_in_aa) at position [0] we obtained the following new rules:

CONF_IN_A → U1_A(U4_aa(del_in_aaa))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

U1_A(del2_out_aa) → U2_A(del_in_aaa)
U2_A(del_out_aaa) → CONF_IN_A
CONF_IN_A → U1_A(U4_aa(del_in_aaa))

The TRS R consists of the following rules:

del2_in_aa → U4_aa(del_in_aaa)
del_in_aaa → del_out_aaa
del_in_aaa → U6_aaa(del_in_aaa)
U4_aa(del_out_aaa) → U5_aa(del_in_aaa)
U6_aaa(del_out_aaa) → del_out_aaa
U5_aa(del_out_aaa) → del2_out_aa

The set Q consists of the following terms:

del2_in_aa
del_in_aaa
U4_aa(x0)
U6_aaa(x0)
U5_aa(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

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

U1_A(del2_out_aa) → U2_A(del_in_aaa)
U2_A(del_out_aaa) → CONF_IN_A
CONF_IN_A → U1_A(U4_aa(del_in_aaa))

The TRS R consists of the following rules:

del_in_aaa → del_out_aaa
del_in_aaa → U6_aaa(del_in_aaa)
U6_aaa(del_out_aaa) → del_out_aaa
U4_aa(del_out_aaa) → U5_aa(del_in_aaa)
U5_aa(del_out_aaa) → del2_out_aa

The set Q consists of the following terms:

del2_in_aa
del_in_aaa
U4_aa(x0)
U6_aaa(x0)
U5_aa(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_aa

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

U1_A(del2_out_aa) → U2_A(del_in_aaa)
U2_A(del_out_aaa) → CONF_IN_A
CONF_IN_A → U1_A(U4_aa(del_in_aaa))

The TRS R consists of the following rules:

del_in_aaa → del_out_aaa
del_in_aaa → U6_aaa(del_in_aaa)
U6_aaa(del_out_aaa) → del_out_aaa
U4_aa(del_out_aaa) → U5_aa(del_in_aaa)
U5_aa(del_out_aaa) → del2_out_aa

The set Q consists of the following terms:

del_in_aaa
U4_aa(x0)
U6_aaa(x0)
U5_aa(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule U1_A(del2_out_aa) → U2_A(del_in_aaa) at position [0] we obtained the following new rules:

U1_A(del2_out_aa) → U2_A(U6_aaa(del_in_aaa))
U1_A(del2_out_aa) → U2_A(del_out_aaa)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

U1_A(del2_out_aa) → U2_A(U6_aaa(del_in_aaa))
U1_A(del2_out_aa) → U2_A(del_out_aaa)
U2_A(del_out_aaa) → CONF_IN_A
CONF_IN_A → U1_A(U4_aa(del_in_aaa))

The TRS R consists of the following rules:

del_in_aaa → del_out_aaa
del_in_aaa → U6_aaa(del_in_aaa)
U6_aaa(del_out_aaa) → del_out_aaa
U4_aa(del_out_aaa) → U5_aa(del_in_aaa)
U5_aa(del_out_aaa) → del2_out_aa

The set Q consists of the following terms:

del_in_aaa
U4_aa(x0)
U6_aaa(x0)
U5_aa(x0)

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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_a(Y))
conf_in_a(X) → U1_a(X, del2_in_aa(X, Z))
del2_in_aa(X, Y) → U4_aa(X, Y, del_in_aaa(U, X, Z))
del_in_aaa(X, .(X, T), T) → del_out_aaa(X, .(X, T), T)
del_in_aaa(X, .(H, T), .(H, T1)) → U6_aaa(X, H, T, T1, del_in_aaa(X, T, T1))
U6_aaa(X, H, T, T1, del_out_aaa(X, T, T1)) → del_out_aaa(X, .(H, T), .(H, T1))
U4_aa(X, Y, del_out_aaa(U, X, Z)) → U5_aa(X, Y, del_in_aaa(V, Z, Y))
U5_aa(X, Y, del_out_aaa(V, Z, Y)) → del2_out_aa(X, Y)
U1_a(X, del2_out_aa(X, Z)) → U2_a(X, del_in_aaa(U, Y, Z))
U2_a(X, del_out_aaa(U, Y, Z)) → U3_a(X, conf_in_a(Y))
U3_a(X, conf_out_a(Y)) → conf_out_a(X)
U3_g(X, conf_out_a(Y)) → conf_out_g(X)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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) = .(x1, 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

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

  ↳ PrologToPiTRSProof

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_a(Y))
conf_in_a(X) → U1_a(X, del2_in_aa(X, Z))
del2_in_aa(X, Y) → U4_aa(X, Y, del_in_aaa(U, X, Z))
del_in_aaa(X, .(X, T), T) → del_out_aaa(X, .(X, T), T)
del_in_aaa(X, .(H, T), .(H, T1)) → U6_aaa(X, H, T, T1, del_in_aaa(X, T, T1))
U6_aaa(X, H, T, T1, del_out_aaa(X, T, T1)) → del_out_aaa(X, .(H, T), .(H, T1))
U4_aa(X, Y, del_out_aaa(U, X, Z)) → U5_aa(X, Y, del_in_aaa(V, Z, Y))
U5_aa(X, Y, del_out_aaa(V, Z, Y)) → del2_out_aa(X, Y)
U1_a(X, del2_out_aa(X, Z)) → U2_a(X, del_in_aaa(U, Y, Z))
U2_a(X, del_out_aaa(U, Y, Z)) → U3_a(X, conf_in_a(Y))
U3_a(X, conf_out_a(Y)) → conf_out_a(X)
U3_g(X, conf_out_a(Y)) → conf_out_g(X)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

  ↳ PrologToPiTRSProof

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) = .(x1, 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

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

  ↳ PrologToPiTRSProof

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

DEL_IN_AGA(.(H, T)) → DEL_IN_AGA(T)

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

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

We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
conf_in: (b) (f)
del2_in: (b,f) (f,f)
del_in: (f,b,f) (f,f,b) (f,f,f)
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_a(Y))
conf_in_a(X) → U1_a(X, del2_in_aa(X, Z))
del2_in_aa(X, Y) → U4_aa(X, Y, del_in_aaa(U, X, Z))
del_in_aaa(X, .(X, T), T) → del_out_aaa(X, .(X, T), T)
del_in_aaa(X, .(H, T), .(H, T1)) → U6_aaa(X, H, T, T1, del_in_aaa(X, T, T1))
U6_aaa(X, H, T, T1, del_out_aaa(X, T, T1)) → del_out_aaa(X, .(H, T), .(H, T1))
U4_aa(X, Y, del_out_aaa(U, X, Z)) → U5_aa(X, Y, del_in_aaa(V, Z, Y))
U5_aa(X, Y, del_out_aaa(V, Z, Y)) → del2_out_aa(X, Y)
U1_a(X, del2_out_aa(X, Z)) → U2_a(X, del_in_aaa(U, Y, Z))
U2_a(X, del_out_aaa(U, Y, Z)) → U3_a(X, conf_in_a(Y))
U3_a(X, conf_out_a(Y)) → conf_out_a(X)
U3_g(X, conf_out_a(Y)) → conf_out_g(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_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_a(Y))
conf_in_a(X) → U1_a(X, del2_in_aa(X, Z))
del2_in_aa(X, Y) → U4_aa(X, Y, del_in_aaa(U, X, Z))
del_in_aaa(X, .(X, T), T) → del_out_aaa(X, .(X, T), T)
del_in_aaa(X, .(H, T), .(H, T1)) → U6_aaa(X, H, T, T1, del_in_aaa(X, T, T1))
U6_aaa(X, H, T, T1, del_out_aaa(X, T, T1)) → del_out_aaa(X, .(H, T), .(H, T1))
U4_aa(X, Y, del_out_aaa(U, X, Z)) → U5_aa(X, Y, del_in_aaa(V, Z, Y))
U5_aa(X, Y, del_out_aaa(V, Z, Y)) → del2_out_aa(X, Y)
U1_a(X, del2_out_aa(X, Z)) → U2_a(X, del_in_aaa(U, Y, Z))
U2_a(X, del_out_aaa(U, Y, Z)) → U3_a(X, conf_in_a(Y))
U3_a(X, conf_out_a(Y)) → conf_out_a(X)
U3_g(X, conf_out_a(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_a(Y))
U2_G(X, del_out_aag(U, Y, Z)) → CONF_IN_A(Y)
CONF_IN_A(X) → U1_A(X, del2_in_aa(X, Z))
CONF_IN_A(X) → DEL2_IN_AA(X, Z)
DEL2_IN_AA(X, Y) → U4_AA(X, Y, del_in_aaa(U, X, Z))
DEL2_IN_AA(X, Y) → DEL_IN_AAA(U, X, Z)
DEL_IN_AAA(X, .(H, T), .(H, T1)) → U6_AAA(X, H, T, T1, del_in_aaa(X, T, T1))
DEL_IN_AAA(X, .(H, T), .(H, T1)) → DEL_IN_AAA(X, T, T1)
U4_AA(X, Y, del_out_aaa(U, X, Z)) → U5_AA(X, Y, del_in_aaa(V, Z, Y))
U4_AA(X, Y, del_out_aaa(U, X, Z)) → DEL_IN_AAA(V, Z, Y)
U1_A(X, del2_out_aa(X, Z)) → U2_A(X, del_in_aaa(U, Y, Z))
U1_A(X, del2_out_aa(X, Z)) → DEL_IN_AAA(U, Y, Z)
U2_A(X, del_out_aaa(U, Y, Z)) → U3_A(X, conf_in_a(Y))
U2_A(X, del_out_aaa(U, Y, Z)) → CONF_IN_A(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_a(Y))
conf_in_a(X) → U1_a(X, del2_in_aa(X, Z))
del2_in_aa(X, Y) → U4_aa(X, Y, del_in_aaa(U, X, Z))
del_in_aaa(X, .(X, T), T) → del_out_aaa(X, .(X, T), T)
del_in_aaa(X, .(H, T), .(H, T1)) → U6_aaa(X, H, T, T1, del_in_aaa(X, T, T1))
U6_aaa(X, H, T, T1, del_out_aaa(X, T, T1)) → del_out_aaa(X, .(H, T), .(H, T1))
U4_aa(X, Y, del_out_aaa(U, X, Z)) → U5_aa(X, Y, del_in_aaa(V, Z, Y))
U5_aa(X, Y, del_out_aaa(V, Z, Y)) → del2_out_aa(X, Y)
U1_a(X, del2_out_aa(X, Z)) → U2_a(X, del_in_aaa(U, Y, Z))
U2_a(X, del_out_aaa(U, Y, Z)) → U3_a(X, conf_in_a(Y))
U3_a(X, conf_out_a(Y)) → conf_out_a(X)
U3_g(X, conf_out_a(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(x1, x2)
del2_in_ga(x1, x2) = del2_in_ga(x1)
U4_ga(x1, x2, x3) = U4_ga(x1, x3)
del_in_aga(x1, x2, x3) = del_in_aga(x2)
.(x1, x2) = .(x1, x2)
del_out_aga(x1, x2, x3) = del_out_aga(x1, x2, x3)
U6_aga(x1, x2, x3, x4, x5) = U6_aga(x2, x3, x5)
U5_ga(x1, x2, x3) = U5_ga(x1, x3)
del2_out_ga(x1, x2) = del2_out_ga(x1, x2)
U2_g(x1, x2) = U2_g(x1, x2)
del_in_aag(x1, x2, x3) = del_in_aag(x3)
del_out_aag(x1, x2, x3) = del_out_aag(x3)
U6_aag(x1, x2, x3, x4, x5) = U6_aag(x2, x4, x5)
U3_g(x1, x2) = U3_g(x1, x2)
conf_in_a(x1) = conf_in_a
U1_a(x1, x2) = U1_a(x2)
del2_in_aa(x1, x2) = del2_in_aa
U4_aa(x1, x2, x3) = U4_aa(x3)
del_in_aaa(x1, x2, x3) = del_in_aaa
del_out_aaa(x1, x2, x3) = del_out_aaa
U6_aaa(x1, x2, x3, x4, x5) = U6_aaa(x5)
U5_aa(x1, x2, x3) = U5_aa(x3)
del2_out_aa(x1, x2) = del2_out_aa
U2_a(x1, x2) = U2_a(x2)
U3_a(x1, x2) = U3_a(x2)
conf_out_a(x1) = conf_out_a
conf_out_g(x1) = conf_out_g(x1)
U1_A(x1, x2) = U1_A(x2)
U5_GA(x1, x2, x3) = U5_GA(x1, x3)
U6_AAG(x1, x2, x3, x4, x5) = U6_AAG(x2, x4, x5)
U6_AGA(x1, x2, x3, x4, x5) = U6_AGA(x2, x3, x5)
DEL_IN_AAA(x1, x2, x3) = DEL_IN_AAA
U4_GA(x1, x2, x3) = U4_GA(x1, x3)
U1_G(x1, x2) = U1_G(x1, x2)
U3_G(x1, x2) = U3_G(x1, x2)
U2_G(x1, x2) = U2_G(x1, x2)
U5_AA(x1, x2, x3) = U5_AA(x3)
U3_A(x1, x2) = U3_A(x2)
DEL_IN_AGA(x1, x2, x3) = DEL_IN_AGA(x2)
U2_A(x1, x2) = U2_A(x2)
U6_AAA(x1, x2, x3, x4, x5) = U6_AAA(x5)
DEL_IN_AAG(x1, x2, x3) = DEL_IN_AAG(x3)
DEL2_IN_GA(x1, x2) = DEL2_IN_GA(x1)
U4_AA(x1, x2, x3) = U4_AA(x3)
CONF_IN_A(x1) = CONF_IN_A
DEL2_IN_AA(x1, x2) = DEL2_IN_AA
CONF_IN_G(x1) = CONF_IN_G(x1)

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ 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_a(Y))
U2_G(X, del_out_aag(U, Y, Z)) → CONF_IN_A(Y)
CONF_IN_A(X) → U1_A(X, del2_in_aa(X, Z))
CONF_IN_A(X) → DEL2_IN_AA(X, Z)
DEL2_IN_AA(X, Y) → U4_AA(X, Y, del_in_aaa(U, X, Z))
DEL2_IN_AA(X, Y) → DEL_IN_AAA(U, X, Z)
DEL_IN_AAA(X, .(H, T), .(H, T1)) → U6_AAA(X, H, T, T1, del_in_aaa(X, T, T1))
DEL_IN_AAA(X, .(H, T), .(H, T1)) → DEL_IN_AAA(X, T, T1)
U4_AA(X, Y, del_out_aaa(U, X, Z)) → U5_AA(X, Y, del_in_aaa(V, Z, Y))
U4_AA(X, Y, del_out_aaa(U, X, Z)) → DEL_IN_AAA(V, Z, Y)
U1_A(X, del2_out_aa(X, Z)) → U2_A(X, del_in_aaa(U, Y, Z))
U1_A(X, del2_out_aa(X, Z)) → DEL_IN_AAA(U, Y, Z)
U2_A(X, del_out_aaa(U, Y, Z)) → U3_A(X, conf_in_a(Y))
U2_A(X, del_out_aaa(U, Y, Z)) → CONF_IN_A(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_a(Y))
conf_in_a(X) → U1_a(X, del2_in_aa(X, Z))
del2_in_aa(X, Y) → U4_aa(X, Y, del_in_aaa(U, X, Z))
del_in_aaa(X, .(X, T), T) → del_out_aaa(X, .(X, T), T)
del_in_aaa(X, .(H, T), .(H, T1)) → U6_aaa(X, H, T, T1, del_in_aaa(X, T, T1))
U6_aaa(X, H, T, T1, del_out_aaa(X, T, T1)) → del_out_aaa(X, .(H, T), .(H, T1))
U4_aa(X, Y, del_out_aaa(U, X, Z)) → U5_aa(X, Y, del_in_aaa(V, Z, Y))
U5_aa(X, Y, del_out_aaa(V, Z, Y)) → del2_out_aa(X, Y)
U1_a(X, del2_out_aa(X, Z)) → U2_a(X, del_in_aaa(U, Y, Z))
U2_a(X, del_out_aaa(U, Y, Z)) → U3_a(X, conf_in_a(Y))
U3_a(X, conf_out_a(Y)) → conf_out_a(X)
U3_g(X, conf_out_a(Y)) → conf_out_g(X)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

DEL_IN_AAA(X, .(H, T), .(H, T1)) → DEL_IN_AAA(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_a(Y))
conf_in_a(X) → U1_a(X, del2_in_aa(X, Z))
del2_in_aa(X, Y) → U4_aa(X, Y, del_in_aaa(U, X, Z))
del_in_aaa(X, .(X, T), T) → del_out_aaa(X, .(X, T), T)
del_in_aaa(X, .(H, T), .(H, T1)) → U6_aaa(X, H, T, T1, del_in_aaa(X, T, T1))
U6_aaa(X, H, T, T1, del_out_aaa(X, T, T1)) → del_out_aaa(X, .(H, T), .(H, T1))
U4_aa(X, Y, del_out_aaa(U, X, Z)) → U5_aa(X, Y, del_in_aaa(V, Z, Y))
U5_aa(X, Y, del_out_aaa(V, Z, Y)) → del2_out_aa(X, Y)
U1_a(X, del2_out_aa(X, Z)) → U2_a(X, del_in_aaa(U, Y, Z))
U2_a(X, del_out_aaa(U, Y, Z)) → U3_a(X, conf_in_a(Y))
U3_a(X, conf_out_a(Y)) → conf_out_a(X)
U3_g(X, conf_out_a(Y)) → conf_out_g(X)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ NonTerminationProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

DEL_IN_AAA → DEL_IN_AAA

The TRS R consists of the following rules:none

s = DEL_IN_AAA evaluates to t =DEL_IN_AAA

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_AAA to DEL_IN_AAA.

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

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

U2_A(X, del_out_aaa(U, Y, Z)) → CONF_IN_A(Y)
CONF_IN_A(X) → U1_A(X, del2_in_aa(X, Z))
U1_A(X, del2_out_aa(X, Z)) → U2_A(X, del_in_aaa(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_a(Y))
conf_in_a(X) → U1_a(X, del2_in_aa(X, Z))
del2_in_aa(X, Y) → U4_aa(X, Y, del_in_aaa(U, X, Z))
del_in_aaa(X, .(X, T), T) → del_out_aaa(X, .(X, T), T)
del_in_aaa(X, .(H, T), .(H, T1)) → U6_aaa(X, H, T, T1, del_in_aaa(X, T, T1))
U6_aaa(X, H, T, T1, del_out_aaa(X, T, T1)) → del_out_aaa(X, .(H, T), .(H, T1))
U4_aa(X, Y, del_out_aaa(U, X, Z)) → U5_aa(X, Y, del_in_aaa(V, Z, Y))
U5_aa(X, Y, del_out_aaa(V, Z, Y)) → del2_out_aa(X, Y)
U1_a(X, del2_out_aa(X, Z)) → U2_a(X, del_in_aaa(U, Y, Z))
U2_a(X, del_out_aaa(U, Y, Z)) → U3_a(X, conf_in_a(Y))
U3_a(X, conf_out_a(Y)) → conf_out_a(X)
U3_g(X, conf_out_a(Y)) → conf_out_g(X)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

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

U2_A(X, del_out_aaa(U, Y, Z)) → CONF_IN_A(Y)
CONF_IN_A(X) → U1_A(X, del2_in_aa(X, Z))
U1_A(X, del2_out_aa(X, Z)) → U2_A(X, del_in_aaa(U, Y, Z))

The TRS R consists of the following rules:

del2_in_aa(X, Y) → U4_aa(X, Y, del_in_aaa(U, X, Z))
del_in_aaa(X, .(X, T), T) → del_out_aaa(X, .(X, T), T)
del_in_aaa(X, .(H, T), .(H, T1)) → U6_aaa(X, H, T, T1, del_in_aaa(X, T, T1))
U4_aa(X, Y, del_out_aaa(U, X, Z)) → U5_aa(X, Y, del_in_aaa(V, Z, Y))
U6_aaa(X, H, T, T1, del_out_aaa(X, T, T1)) → del_out_aaa(X, .(H, T), .(H, T1))
U5_aa(X, Y, del_out_aaa(V, Z, Y)) → del2_out_aa(X, Y)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

              ↳ PiDP

              ↳ PiDP

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

U1_A(del2_out_aa) → U2_A(del_in_aaa)
U2_A(del_out_aaa) → CONF_IN_A
CONF_IN_A → U1_A(del2_in_aa)

The TRS R consists of the following rules:

del2_in_aa → U4_aa(del_in_aaa)
del_in_aaa → del_out_aaa
del_in_aaa → U6_aaa(del_in_aaa)
U4_aa(del_out_aaa) → U5_aa(del_in_aaa)
U6_aaa(del_out_aaa) → del_out_aaa
U5_aa(del_out_aaa) → del2_out_aa

The set Q consists of the following terms:

del2_in_aa
del_in_aaa
U4_aa(x0)
U6_aaa(x0)
U5_aa(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule CONF_IN_A → U1_A(del2_in_aa) at position [0] we obtained the following new rules:

CONF_IN_A → U1_A(U4_aa(del_in_aaa))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

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

U1_A(del2_out_aa) → U2_A(del_in_aaa)
U2_A(del_out_aaa) → CONF_IN_A
CONF_IN_A → U1_A(U4_aa(del_in_aaa))

The TRS R consists of the following rules:

del2_in_aa → U4_aa(del_in_aaa)
del_in_aaa → del_out_aaa
del_in_aaa → U6_aaa(del_in_aaa)
U4_aa(del_out_aaa) → U5_aa(del_in_aaa)
U6_aaa(del_out_aaa) → del_out_aaa
U5_aa(del_out_aaa) → del2_out_aa

The set Q consists of the following terms:

del2_in_aa
del_in_aaa
U4_aa(x0)
U6_aaa(x0)
U5_aa(x0)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

              ↳ PiDP

              ↳ PiDP

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

U1_A(del2_out_aa) → U2_A(del_in_aaa)
U2_A(del_out_aaa) → CONF_IN_A
CONF_IN_A → U1_A(U4_aa(del_in_aaa))

The TRS R consists of the following rules:

del_in_aaa → del_out_aaa
del_in_aaa → U6_aaa(del_in_aaa)
U6_aaa(del_out_aaa) → del_out_aaa
U4_aa(del_out_aaa) → U5_aa(del_in_aaa)
U5_aa(del_out_aaa) → del2_out_aa

The set Q consists of the following terms:

del2_in_aa
del_in_aaa
U4_aa(x0)
U6_aaa(x0)
U5_aa(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_aa

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

              ↳ PiDP

              ↳ PiDP

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

U1_A(del2_out_aa) → U2_A(del_in_aaa)
U2_A(del_out_aaa) → CONF_IN_A
CONF_IN_A → U1_A(U4_aa(del_in_aaa))

The TRS R consists of the following rules:

del_in_aaa → del_out_aaa
del_in_aaa → U6_aaa(del_in_aaa)
U6_aaa(del_out_aaa) → del_out_aaa
U4_aa(del_out_aaa) → U5_aa(del_in_aaa)
U5_aa(del_out_aaa) → del2_out_aa

The set Q consists of the following terms:

del_in_aaa
U4_aa(x0)
U6_aaa(x0)
U5_aa(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule U1_A(del2_out_aa) → U2_A(del_in_aaa) at position [0] we obtained the following new rules:

U1_A(del2_out_aa) → U2_A(U6_aaa(del_in_aaa))
U1_A(del2_out_aa) → U2_A(del_out_aaa)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

              ↳ PiDP

              ↳ PiDP

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

U1_A(del2_out_aa) → U2_A(U6_aaa(del_in_aaa))
U1_A(del2_out_aa) → U2_A(del_out_aaa)
U2_A(del_out_aaa) → CONF_IN_A
CONF_IN_A → U1_A(U4_aa(del_in_aaa))

The TRS R consists of the following rules:

del_in_aaa → del_out_aaa
del_in_aaa → U6_aaa(del_in_aaa)
U6_aaa(del_out_aaa) → del_out_aaa
U4_aa(del_out_aaa) → U5_aa(del_in_aaa)
U5_aa(del_out_aaa) → del2_out_aa

The set Q consists of the following terms:

del_in_aaa
U4_aa(x0)
U6_aaa(x0)
U5_aa(x0)

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ 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_a(Y))
conf_in_a(X) → U1_a(X, del2_in_aa(X, Z))
del2_in_aa(X, Y) → U4_aa(X, Y, del_in_aaa(U, X, Z))
del_in_aaa(X, .(X, T), T) → del_out_aaa(X, .(X, T), T)
del_in_aaa(X, .(H, T), .(H, T1)) → U6_aaa(X, H, T, T1, del_in_aaa(X, T, T1))
U6_aaa(X, H, T, T1, del_out_aaa(X, T, T1)) → del_out_aaa(X, .(H, T), .(H, T1))
U4_aa(X, Y, del_out_aaa(U, X, Z)) → U5_aa(X, Y, del_in_aaa(V, Z, Y))
U5_aa(X, Y, del_out_aaa(V, Z, Y)) → del2_out_aa(X, Y)
U1_a(X, del2_out_aa(X, Z)) → U2_a(X, del_in_aaa(U, Y, Z))
U2_a(X, del_out_aaa(U, Y, Z)) → U3_a(X, conf_in_a(Y))
U3_a(X, conf_out_a(Y)) → conf_out_a(X)
U3_g(X, conf_out_a(Y)) → conf_out_g(X)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ 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)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

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

DEL_IN_AAG(.(H, T1)) → DEL_IN_AAG(T1)

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

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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_a(Y))
conf_in_a(X) → U1_a(X, del2_in_aa(X, Z))
del2_in_aa(X, Y) → U4_aa(X, Y, del_in_aaa(U, X, Z))
del_in_aaa(X, .(X, T), T) → del_out_aaa(X, .(X, T), T)
del_in_aaa(X, .(H, T), .(H, T1)) → U6_aaa(X, H, T, T1, del_in_aaa(X, T, T1))
U6_aaa(X, H, T, T1, del_out_aaa(X, T, T1)) → del_out_aaa(X, .(H, T), .(H, T1))
U4_aa(X, Y, del_out_aaa(U, X, Z)) → U5_aa(X, Y, del_in_aaa(V, Z, Y))
U5_aa(X, Y, del_out_aaa(V, Z, Y)) → del2_out_aa(X, Y)
U1_a(X, del2_out_aa(X, Z)) → U2_a(X, del_in_aaa(U, Y, Z))
U2_a(X, del_out_aaa(U, Y, Z)) → U3_a(X, conf_in_a(Y))
U3_a(X, conf_out_a(Y)) → conf_out_a(X)
U3_g(X, conf_out_a(Y)) → conf_out_g(X)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

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)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

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

DEL_IN_AGA(.(H, T)) → DEL_IN_AGA(T)

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

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