Left Termination proof of /tmp/tmpan1XKJ/slowsort-fb.pl

Left Termination of the query pattern ss_in_2(a, g) w.r.t. the given Prolog program could not be shown:

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

Clauses:

ss(Xs, Ys) :- ','(perm(Xs, Ys), ordered(Ys)).
perm([], []).
perm(Xs, .(X, Ys)) :- ','(app(X1s, .(X, X2s), Xs), ','(app(X1s, X2s, Zs), perm(Zs, Ys))).
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
ordered([]).
ordered(.(X, [])).
ordered(.(X, .(Y, Xs))) :- ','(less(X, s(Y)), ordered(.(Y, Xs))).
less(0, s(X)).
less(s(X), s(Y)) :- less(X, Y).

Queries:

ss(a,g).

We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
ss_in: (f,b)
perm_in: (f,b)
app_in: (f,f,f)
ordered_in: (b)
less_in: (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

ss_in_ag(Xs, Ys) → U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))
U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_ag(Xs, Ys, ordered_in_g(Ys))
ordered_in_g([]) → ordered_out_g([])
ordered_in_g(.(X, [])) → ordered_out_g(.(X, []))
ordered_in_g(.(X, .(Y, Xs))) → U7_g(X, Y, Xs, less_in_gg(X, s(Y)))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U7_g(X, Y, Xs, less_out_gg(X, s(Y))) → U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) → ordered_out_g(.(X, .(Y, Xs)))
U2_ag(Xs, Ys, ordered_out_g(Ys)) → ss_out_ag(Xs, Ys)

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:

ss_in_ag(Xs, Ys) → U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))
U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_ag(Xs, Ys, ordered_in_g(Ys))
ordered_in_g([]) → ordered_out_g([])
ordered_in_g(.(X, [])) → ordered_out_g(.(X, []))
ordered_in_g(.(X, .(Y, Xs))) → U7_g(X, Y, Xs, less_in_gg(X, s(Y)))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U7_g(X, Y, Xs, less_out_gg(X, s(Y))) → U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) → ordered_out_g(.(X, .(Y, Xs)))
U2_ag(Xs, Ys, ordered_out_g(Ys)) → ss_out_ag(Xs, Ys)

SS_IN_AG(Xs, Ys) → U1_AG(Xs, Ys, perm_in_ag(Xs, Ys))
SS_IN_AG(Xs, Ys) → PERM_IN_AG(Xs, Ys)
PERM_IN_AG(Xs, .(X, Ys)) → U3_AG(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
PERM_IN_AG(Xs, .(X, Ys)) → APP_IN_AAA(X1s, .(X, X2s), Xs)
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → U6_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAA(Xs, Ys, Zs)
U3_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_AG(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U3_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → APP_IN_AAA(X1s, X2s, Zs)
U4_AG(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_AG(Xs, X, Ys, perm_in_ag(Zs, Ys))
U4_AG(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → PERM_IN_AG(Zs, Ys)
U1_AG(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_AG(Xs, Ys, ordered_in_g(Ys))
U1_AG(Xs, Ys, perm_out_ag(Xs, Ys)) → ORDERED_IN_G(Ys)
ORDERED_IN_G(.(X, .(Y, Xs))) → U7_G(X, Y, Xs, less_in_gg(X, s(Y)))
ORDERED_IN_G(.(X, .(Y, Xs))) → LESS_IN_GG(X, s(Y))
LESS_IN_GG(s(X), s(Y)) → U9_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U7_G(X, Y, Xs, less_out_gg(X, s(Y))) → U8_G(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U7_G(X, Y, Xs, less_out_gg(X, s(Y))) → ORDERED_IN_G(.(Y, Xs))

The TRS R consists of the following rules:

ss_in_ag(Xs, Ys) → U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))
U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_ag(Xs, Ys, ordered_in_g(Ys))
ordered_in_g([]) → ordered_out_g([])
ordered_in_g(.(X, [])) → ordered_out_g(.(X, []))
ordered_in_g(.(X, .(Y, Xs))) → U7_g(X, Y, Xs, less_in_gg(X, s(Y)))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U7_g(X, Y, Xs, less_out_gg(X, s(Y))) → U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) → ordered_out_g(.(X, .(Y, Xs)))
U2_ag(Xs, Ys, ordered_out_g(Ys)) → ss_out_ag(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in_ag(x1, x2) = ss_in_ag(x2)
U1_ag(x1, x2, x3) = U1_ag(x2, x3)
perm_in_ag(x1, x2) = perm_in_ag(x2)
[] = []
perm_out_ag(x1, x2) = perm_out_ag
.(x1, x2) = .(x1, x2)
U3_ag(x1, x2, x3, x4) = U3_ag(x3, x4)
app_in_aaa(x1, x2, x3) = app_in_aaa
app_out_aaa(x1, x2, x3) = app_out_aaa
U6_aaa(x1, x2, x3, x4, x5) = U6_aaa(x5)
U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4)
U5_ag(x1, x2, x3, x4) = U5_ag(x4)
U2_ag(x1, x2, x3) = U2_ag(x3)
ordered_in_g(x1) = ordered_in_g(x1)
ordered_out_g(x1) = ordered_out_g
U7_g(x1, x2, x3, x4) = U7_g(x2, x3, x4)
less_in_gg(x1, x2) = less_in_gg(x1, x2)
0 = 0
s(x1) = s(x1)
less_out_gg(x1, x2) = less_out_gg
U9_gg(x1, x2, x3) = U9_gg(x3)
U8_g(x1, x2, x3, x4) = U8_g(x4)
ss_out_ag(x1, x2) = ss_out_ag
APP_IN_AAA(x1, x2, x3) = APP_IN_AAA
U7_G(x1, x2, x3, x4) = U7_G(x2, x3, x4)
PERM_IN_AG(x1, x2) = PERM_IN_AG(x2)
U3_AG(x1, x2, x3, x4) = U3_AG(x3, x4)
U2_AG(x1, x2, x3) = U2_AG(x3)
LESS_IN_GG(x1, x2) = LESS_IN_GG(x1, x2)
ORDERED_IN_G(x1) = ORDERED_IN_G(x1)
U9_GG(x1, x2, x3) = U9_GG(x3)
U6_AAA(x1, x2, x3, x4, x5) = U6_AAA(x5)
U4_AG(x1, x2, x3, x4) = U4_AG(x3, x4)
U8_G(x1, x2, x3, x4) = U8_G(x4)
SS_IN_AG(x1, x2) = SS_IN_AG(x2)
U1_AG(x1, x2, x3) = U1_AG(x2, x3)
U5_AG(x1, x2, x3, x4) = U5_AG(x4)

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:

SS_IN_AG(Xs, Ys) → U1_AG(Xs, Ys, perm_in_ag(Xs, Ys))
SS_IN_AG(Xs, Ys) → PERM_IN_AG(Xs, Ys)
PERM_IN_AG(Xs, .(X, Ys)) → U3_AG(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
PERM_IN_AG(Xs, .(X, Ys)) → APP_IN_AAA(X1s, .(X, X2s), Xs)
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → U6_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAA(Xs, Ys, Zs)
U3_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_AG(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U3_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → APP_IN_AAA(X1s, X2s, Zs)
U4_AG(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_AG(Xs, X, Ys, perm_in_ag(Zs, Ys))
U4_AG(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → PERM_IN_AG(Zs, Ys)
U1_AG(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_AG(Xs, Ys, ordered_in_g(Ys))
U1_AG(Xs, Ys, perm_out_ag(Xs, Ys)) → ORDERED_IN_G(Ys)
ORDERED_IN_G(.(X, .(Y, Xs))) → U7_G(X, Y, Xs, less_in_gg(X, s(Y)))
ORDERED_IN_G(.(X, .(Y, Xs))) → LESS_IN_GG(X, s(Y))
LESS_IN_GG(s(X), s(Y)) → U9_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U7_G(X, Y, Xs, less_out_gg(X, s(Y))) → U8_G(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U7_G(X, Y, Xs, less_out_gg(X, s(Y))) → ORDERED_IN_G(.(Y, Xs))

The TRS R consists of the following rules:

ss_in_ag(Xs, Ys) → U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))
U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_ag(Xs, Ys, ordered_in_g(Ys))
ordered_in_g([]) → ordered_out_g([])
ordered_in_g(.(X, [])) → ordered_out_g(.(X, []))
ordered_in_g(.(X, .(Y, Xs))) → U7_g(X, Y, Xs, less_in_gg(X, s(Y)))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U7_g(X, Y, Xs, less_out_gg(X, s(Y))) → U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) → ordered_out_g(.(X, .(Y, Xs)))
U2_ag(Xs, Ys, ordered_out_g(Ys)) → ss_out_ag(Xs, Ys)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)

The TRS R consists of the following rules:

ss_in_ag(Xs, Ys) → U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))
U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_ag(Xs, Ys, ordered_in_g(Ys))
ordered_in_g([]) → ordered_out_g([])
ordered_in_g(.(X, [])) → ordered_out_g(.(X, []))
ordered_in_g(.(X, .(Y, Xs))) → U7_g(X, Y, Xs, less_in_gg(X, s(Y)))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U7_g(X, Y, Xs, less_out_gg(X, s(Y))) → U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) → ordered_out_g(.(X, .(Y, Xs)))
U2_ag(Xs, Ys, ordered_out_g(Ys)) → ss_out_ag(Xs, Ys)

↳ 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:

LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)

R is empty.
Pi is empty.
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

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)

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:

LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
The graph contains the following edges 1 > 1, 2 > 2

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

ORDERED_IN_G(.(X, .(Y, Xs))) → U7_G(X, Y, Xs, less_in_gg(X, s(Y)))
U7_G(X, Y, Xs, less_out_gg(X, s(Y))) → ORDERED_IN_G(.(Y, Xs))

The TRS R consists of the following rules:

ss_in_ag(Xs, Ys) → U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))
U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_ag(Xs, Ys, ordered_in_g(Ys))
ordered_in_g([]) → ordered_out_g([])
ordered_in_g(.(X, [])) → ordered_out_g(.(X, []))
ordered_in_g(.(X, .(Y, Xs))) → U7_g(X, Y, Xs, less_in_gg(X, s(Y)))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U7_g(X, Y, Xs, less_out_gg(X, s(Y))) → U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) → ordered_out_g(.(X, .(Y, Xs)))
U2_ag(Xs, Ys, ordered_out_g(Ys)) → ss_out_ag(Xs, Ys)

↳ 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:

ORDERED_IN_G(.(X, .(Y, Xs))) → U7_G(X, Y, Xs, less_in_gg(X, s(Y)))
U7_G(X, Y, Xs, less_out_gg(X, s(Y))) → ORDERED_IN_G(.(Y, Xs))

The TRS R consists of the following rules:

less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
less_in_gg(x1, x2) = less_in_gg(x1, x2)
0 = 0
s(x1) = s(x1)
less_out_gg(x1, x2) = less_out_gg
U9_gg(x1, x2, x3) = U9_gg(x3)
U7_G(x1, x2, x3, x4) = U7_G(x2, x3, x4)
ORDERED_IN_G(x1) = ORDERED_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

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

ORDERED_IN_G(.(X, .(Y, Xs))) → U7_G(Y, Xs, less_in_gg(X, s(Y)))
U7_G(Y, Xs, less_out_gg) → ORDERED_IN_G(.(Y, Xs))

The TRS R consists of the following rules:

less_in_gg(0, s(X)) → less_out_gg
less_in_gg(s(X), s(Y)) → U9_gg(less_in_gg(X, Y))
U9_gg(less_out_gg) → less_out_gg

The set Q consists of the following terms:

less_in_gg(x0, x1)
U9_gg(x0)

We have to consider all (P,Q,R)-chains.
By using the usable rules with reduction pair processor [15] with a polynomial ordering [25], all dependency pairs and the corresponding usable rules [17] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

U7_G(Y, Xs, less_out_gg) → ORDERED_IN_G(.(Y, Xs))

The following rules are removed from R:

less_in_gg(0, s(X)) → less_out_gg

Used ordering: POLO with Polynomial interpretation [25]:

POL(.(x₁, x₂)) = 2·x₁ + 2·x₂
POL(0) = 2
POL(ORDERED_IN_G(x₁)) = 2 + x₁
POL(U7_G(x₁, x₂, x₃)) = 2 + 2·x₁ + 2·x₂ + x₃
POL(U9_gg(x₁)) = x₁
POL(less_in_gg(x₁, x₂)) = x₁ + 2·x₂
POL(less_out_gg) = 2
POL(s(x₁)) = x₁

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

                          ↳ QDP

                            ↳ DependencyGraphProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

ORDERED_IN_G(.(X, .(Y, Xs))) → U7_G(Y, Xs, less_in_gg(X, s(Y)))

The TRS R consists of the following rules:

less_in_gg(s(X), s(Y)) → U9_gg(less_in_gg(X, Y))
U9_gg(less_out_gg) → less_out_gg

The set Q consists of the following terms:

less_in_gg(x0, x1)
U9_gg(x0)

We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 1 less node.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAA(Xs, Ys, Zs)

The TRS R consists of the following rules:

ss_in_ag(Xs, Ys) → U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))
U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_ag(Xs, Ys, ordered_in_g(Ys))
ordered_in_g([]) → ordered_out_g([])
ordered_in_g(.(X, [])) → ordered_out_g(.(X, []))
ordered_in_g(.(X, .(Y, Xs))) → U7_g(X, Y, Xs, less_in_gg(X, s(Y)))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U7_g(X, Y, Xs, less_out_gg(X, s(Y))) → U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) → ordered_out_g(.(X, .(Y, Xs)))
U2_ag(Xs, Ys, ordered_out_g(Ys)) → ss_out_ag(Xs, Ys)

↳ 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:

APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAA(Xs, Ys, Zs)

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

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ NonTerminationProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

APP_IN_AAA → APP_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:

APP_IN_AAA → APP_IN_AAA

The TRS R consists of the following rules:none

s = APP_IN_AAA evaluates to t =APP_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 APP_IN_AAA to APP_IN_AAA.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

  ↳ PrologToPiTRSProof

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

U4_AG(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → PERM_IN_AG(Zs, Ys)
PERM_IN_AG(Xs, .(X, Ys)) → U3_AG(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
U3_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_AG(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))

The TRS R consists of the following rules:

ss_in_ag(Xs, Ys) → U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))
U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_ag(Xs, Ys, ordered_in_g(Ys))
ordered_in_g([]) → ordered_out_g([])
ordered_in_g(.(X, [])) → ordered_out_g(.(X, []))
ordered_in_g(.(X, .(Y, Xs))) → U7_g(X, Y, Xs, less_in_gg(X, s(Y)))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U7_g(X, Y, Xs, less_out_gg(X, s(Y))) → U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) → ordered_out_g(.(X, .(Y, Xs)))
U2_ag(Xs, Ys, ordered_out_g(Ys)) → ss_out_ag(Xs, Ys)

↳ 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:

U4_AG(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → PERM_IN_AG(Zs, Ys)
PERM_IN_AG(Xs, .(X, Ys)) → U3_AG(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
U3_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_AG(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))

The TRS R consists of the following rules:

app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
[] = []
.(x1, x2) = .(x1, x2)
app_in_aaa(x1, x2, x3) = app_in_aaa
app_out_aaa(x1, x2, x3) = app_out_aaa
U6_aaa(x1, x2, x3, x4, x5) = U6_aaa(x5)
PERM_IN_AG(x1, x2) = PERM_IN_AG(x2)
U3_AG(x1, x2, x3, x4) = U3_AG(x3, x4)
U4_AG(x1, x2, x3, x4) = U4_AG(x3, x4)

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:

PERM_IN_AG(.(X, Ys)) → U3_AG(Ys, app_in_aaa)
U3_AG(Ys, app_out_aaa) → U4_AG(Ys, app_in_aaa)
U4_AG(Ys, app_out_aaa) → PERM_IN_AG(Ys)

The TRS R consists of the following rules:

app_in_aaa → app_out_aaa
app_in_aaa → U6_aaa(app_in_aaa)
U6_aaa(app_out_aaa) → app_out_aaa

The set Q consists of the following terms:

app_in_aaa
U6_aaa(x0)

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:

PERM_IN_AG(.(X, Ys)) → U3_AG(Ys, app_in_aaa)
The graph contains the following edges 1 > 1
U3_AG(Ys, app_out_aaa) → U4_AG(Ys, app_in_aaa)
The graph contains the following edges 1 >= 1
U4_AG(Ys, app_out_aaa) → PERM_IN_AG(Ys)
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:
ss_in: (f,b)
perm_in: (f,b)
app_in: (f,f,f)
ordered_in: (b)
less_in: (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

ss_in_ag(Xs, Ys) → U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))
U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_ag(Xs, Ys, ordered_in_g(Ys))
ordered_in_g([]) → ordered_out_g([])
ordered_in_g(.(X, [])) → ordered_out_g(.(X, []))
ordered_in_g(.(X, .(Y, Xs))) → U7_g(X, Y, Xs, less_in_gg(X, s(Y)))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U7_g(X, Y, Xs, less_out_gg(X, s(Y))) → U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) → ordered_out_g(.(X, .(Y, Xs)))
U2_ag(Xs, Ys, ordered_out_g(Ys)) → ss_out_ag(Xs, Ys)

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:

ss_in_ag(Xs, Ys) → U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))
U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_ag(Xs, Ys, ordered_in_g(Ys))
ordered_in_g([]) → ordered_out_g([])
ordered_in_g(.(X, [])) → ordered_out_g(.(X, []))
ordered_in_g(.(X, .(Y, Xs))) → U7_g(X, Y, Xs, less_in_gg(X, s(Y)))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U7_g(X, Y, Xs, less_out_gg(X, s(Y))) → U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) → ordered_out_g(.(X, .(Y, Xs)))
U2_ag(Xs, Ys, ordered_out_g(Ys)) → ss_out_ag(Xs, Ys)

SS_IN_AG(Xs, Ys) → U1_AG(Xs, Ys, perm_in_ag(Xs, Ys))
SS_IN_AG(Xs, Ys) → PERM_IN_AG(Xs, Ys)
PERM_IN_AG(Xs, .(X, Ys)) → U3_AG(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
PERM_IN_AG(Xs, .(X, Ys)) → APP_IN_AAA(X1s, .(X, X2s), Xs)
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → U6_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAA(Xs, Ys, Zs)
U3_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_AG(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U3_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → APP_IN_AAA(X1s, X2s, Zs)
U4_AG(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_AG(Xs, X, Ys, perm_in_ag(Zs, Ys))
U4_AG(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → PERM_IN_AG(Zs, Ys)
U1_AG(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_AG(Xs, Ys, ordered_in_g(Ys))
U1_AG(Xs, Ys, perm_out_ag(Xs, Ys)) → ORDERED_IN_G(Ys)
ORDERED_IN_G(.(X, .(Y, Xs))) → U7_G(X, Y, Xs, less_in_gg(X, s(Y)))
ORDERED_IN_G(.(X, .(Y, Xs))) → LESS_IN_GG(X, s(Y))
LESS_IN_GG(s(X), s(Y)) → U9_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U7_G(X, Y, Xs, less_out_gg(X, s(Y))) → U8_G(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U7_G(X, Y, Xs, less_out_gg(X, s(Y))) → ORDERED_IN_G(.(Y, Xs))

The TRS R consists of the following rules:

ss_in_ag(Xs, Ys) → U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))
U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_ag(Xs, Ys, ordered_in_g(Ys))
ordered_in_g([]) → ordered_out_g([])
ordered_in_g(.(X, [])) → ordered_out_g(.(X, []))
ordered_in_g(.(X, .(Y, Xs))) → U7_g(X, Y, Xs, less_in_gg(X, s(Y)))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U7_g(X, Y, Xs, less_out_gg(X, s(Y))) → U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) → ordered_out_g(.(X, .(Y, Xs)))
U2_ag(Xs, Ys, ordered_out_g(Ys)) → ss_out_ag(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in_ag(x1, x2) = ss_in_ag(x2)
U1_ag(x1, x2, x3) = U1_ag(x2, x3)
perm_in_ag(x1, x2) = perm_in_ag(x2)
[] = []
perm_out_ag(x1, x2) = perm_out_ag(x2)
.(x1, x2) = .(x1, x2)
U3_ag(x1, x2, x3, x4) = U3_ag(x2, x3, x4)
app_in_aaa(x1, x2, x3) = app_in_aaa
app_out_aaa(x1, x2, x3) = app_out_aaa
U6_aaa(x1, x2, x3, x4, x5) = U6_aaa(x5)
U4_ag(x1, x2, x3, x4) = U4_ag(x2, x3, x4)
U5_ag(x1, x2, x3, x4) = U5_ag(x2, x3, x4)
U2_ag(x1, x2, x3) = U2_ag(x2, x3)
ordered_in_g(x1) = ordered_in_g(x1)
ordered_out_g(x1) = ordered_out_g(x1)
U7_g(x1, x2, x3, x4) = U7_g(x1, x2, x3, x4)
less_in_gg(x1, x2) = less_in_gg(x1, x2)
0 = 0
s(x1) = s(x1)
less_out_gg(x1, x2) = less_out_gg(x1, x2)
U9_gg(x1, x2, x3) = U9_gg(x1, x2, x3)
U8_g(x1, x2, x3, x4) = U8_g(x1, x2, x3, x4)
ss_out_ag(x1, x2) = ss_out_ag(x2)
APP_IN_AAA(x1, x2, x3) = APP_IN_AAA
U7_G(x1, x2, x3, x4) = U7_G(x1, x2, x3, x4)
PERM_IN_AG(x1, x2) = PERM_IN_AG(x2)
U3_AG(x1, x2, x3, x4) = U3_AG(x2, x3, x4)
U2_AG(x1, x2, x3) = U2_AG(x2, x3)
LESS_IN_GG(x1, x2) = LESS_IN_GG(x1, x2)
ORDERED_IN_G(x1) = ORDERED_IN_G(x1)
U9_GG(x1, x2, x3) = U9_GG(x1, x2, x3)
U6_AAA(x1, x2, x3, x4, x5) = U6_AAA(x5)
U4_AG(x1, x2, x3, x4) = U4_AG(x2, x3, x4)
U8_G(x1, x2, x3, x4) = U8_G(x1, x2, x3, x4)
SS_IN_AG(x1, x2) = SS_IN_AG(x2)
U1_AG(x1, x2, x3) = U1_AG(x2, x3)
U5_AG(x1, x2, x3, x4) = U5_AG(x2, x3, x4)

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:

SS_IN_AG(Xs, Ys) → U1_AG(Xs, Ys, perm_in_ag(Xs, Ys))
SS_IN_AG(Xs, Ys) → PERM_IN_AG(Xs, Ys)
PERM_IN_AG(Xs, .(X, Ys)) → U3_AG(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
PERM_IN_AG(Xs, .(X, Ys)) → APP_IN_AAA(X1s, .(X, X2s), Xs)
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → U6_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAA(Xs, Ys, Zs)
U3_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_AG(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U3_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → APP_IN_AAA(X1s, X2s, Zs)
U4_AG(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_AG(Xs, X, Ys, perm_in_ag(Zs, Ys))
U4_AG(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → PERM_IN_AG(Zs, Ys)
U1_AG(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_AG(Xs, Ys, ordered_in_g(Ys))
U1_AG(Xs, Ys, perm_out_ag(Xs, Ys)) → ORDERED_IN_G(Ys)
ORDERED_IN_G(.(X, .(Y, Xs))) → U7_G(X, Y, Xs, less_in_gg(X, s(Y)))
ORDERED_IN_G(.(X, .(Y, Xs))) → LESS_IN_GG(X, s(Y))
LESS_IN_GG(s(X), s(Y)) → U9_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U7_G(X, Y, Xs, less_out_gg(X, s(Y))) → U8_G(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U7_G(X, Y, Xs, less_out_gg(X, s(Y))) → ORDERED_IN_G(.(Y, Xs))

The TRS R consists of the following rules:

ss_in_ag(Xs, Ys) → U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))
U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_ag(Xs, Ys, ordered_in_g(Ys))
ordered_in_g([]) → ordered_out_g([])
ordered_in_g(.(X, [])) → ordered_out_g(.(X, []))
ordered_in_g(.(X, .(Y, Xs))) → U7_g(X, Y, Xs, less_in_gg(X, s(Y)))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U7_g(X, Y, Xs, less_out_gg(X, s(Y))) → U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) → ordered_out_g(.(X, .(Y, Xs)))
U2_ag(Xs, Ys, ordered_out_g(Ys)) → ss_out_ag(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in_ag(x1, x2) = ss_in_ag(x2)
U1_ag(x1, x2, x3) = U1_ag(x2, x3)
perm_in_ag(x1, x2) = perm_in_ag(x2)
[] = []
perm_out_ag(x1, x2) = perm_out_ag(x2)
.(x1, x2) = .(x1, x2)
U3_ag(x1, x2, x3, x4) = U3_ag(x2, x3, x4)
app_in_aaa(x1, x2, x3) = app_in_aaa
app_out_aaa(x1, x2, x3) = app_out_aaa
U6_aaa(x1, x2, x3, x4, x5) = U6_aaa(x5)
U4_ag(x1, x2, x3, x4) = U4_ag(x2, x3, x4)
U5_ag(x1, x2, x3, x4) = U5_ag(x2, x3, x4)
U2_ag(x1, x2, x3) = U2_ag(x2, x3)
ordered_in_g(x1) = ordered_in_g(x1)
ordered_out_g(x1) = ordered_out_g(x1)
U7_g(x1, x2, x3, x4) = U7_g(x1, x2, x3, x4)
less_in_gg(x1, x2) = less_in_gg(x1, x2)
0 = 0
s(x1) = s(x1)
less_out_gg(x1, x2) = less_out_gg(x1, x2)
U9_gg(x1, x2, x3) = U9_gg(x1, x2, x3)
U8_g(x1, x2, x3, x4) = U8_g(x1, x2, x3, x4)
ss_out_ag(x1, x2) = ss_out_ag(x2)
APP_IN_AAA(x1, x2, x3) = APP_IN_AAA
U7_G(x1, x2, x3, x4) = U7_G(x1, x2, x3, x4)
PERM_IN_AG(x1, x2) = PERM_IN_AG(x2)
U3_AG(x1, x2, x3, x4) = U3_AG(x2, x3, x4)
U2_AG(x1, x2, x3) = U2_AG(x2, x3)
LESS_IN_GG(x1, x2) = LESS_IN_GG(x1, x2)
ORDERED_IN_G(x1) = ORDERED_IN_G(x1)
U9_GG(x1, x2, x3) = U9_GG(x1, x2, x3)
U6_AAA(x1, x2, x3, x4, x5) = U6_AAA(x5)
U4_AG(x1, x2, x3, x4) = U4_AG(x2, x3, x4)
U8_G(x1, x2, x3, x4) = U8_G(x1, x2, x3, x4)
SS_IN_AG(x1, x2) = SS_IN_AG(x2)
U1_AG(x1, x2, x3) = U1_AG(x2, x3)
U5_AG(x1, x2, x3, x4) = U5_AG(x2, x3, x4)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 4 SCCs with 11 less nodes.

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)

The TRS R consists of the following rules:

ss_in_ag(Xs, Ys) → U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))
U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_ag(Xs, Ys, ordered_in_g(Ys))
ordered_in_g([]) → ordered_out_g([])
ordered_in_g(.(X, [])) → ordered_out_g(.(X, []))
ordered_in_g(.(X, .(Y, Xs))) → U7_g(X, Y, Xs, less_in_gg(X, s(Y)))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U7_g(X, Y, Xs, less_out_gg(X, s(Y))) → U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) → ordered_out_g(.(X, .(Y, Xs)))
U2_ag(Xs, Ys, ordered_out_g(Ys)) → ss_out_ag(Xs, Ys)

↳ 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:

LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)

R is empty.
Pi is empty.
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

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)

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

LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
The graph contains the following edges 1 > 1, 2 > 2

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

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

ORDERED_IN_G(.(X, .(Y, Xs))) → U7_G(X, Y, Xs, less_in_gg(X, s(Y)))
U7_G(X, Y, Xs, less_out_gg(X, s(Y))) → ORDERED_IN_G(.(Y, Xs))

The TRS R consists of the following rules:

ss_in_ag(Xs, Ys) → U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))
U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_ag(Xs, Ys, ordered_in_g(Ys))
ordered_in_g([]) → ordered_out_g([])
ordered_in_g(.(X, [])) → ordered_out_g(.(X, []))
ordered_in_g(.(X, .(Y, Xs))) → U7_g(X, Y, Xs, less_in_gg(X, s(Y)))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U7_g(X, Y, Xs, less_out_gg(X, s(Y))) → U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) → ordered_out_g(.(X, .(Y, Xs)))
U2_ag(Xs, Ys, ordered_out_g(Ys)) → ss_out_ag(Xs, Ys)

↳ 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:

ORDERED_IN_G(.(X, .(Y, Xs))) → U7_G(X, Y, Xs, less_in_gg(X, s(Y)))
U7_G(X, Y, Xs, less_out_gg(X, s(Y))) → ORDERED_IN_G(.(Y, Xs))

The TRS R consists of the following rules:

less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))

Pi is empty.
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

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ RuleRemovalProof

              ↳ PiDP

              ↳ PiDP

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

ORDERED_IN_G(.(X, .(Y, Xs))) → U7_G(X, Y, Xs, less_in_gg(X, s(Y)))
U7_G(X, Y, Xs, less_out_gg(X, s(Y))) → ORDERED_IN_G(.(Y, Xs))

The TRS R consists of the following rules:

less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))

The set Q consists of the following terms:

less_in_gg(x0, x1)
U9_gg(x0, x1, x2)

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 dependency pairs:

U7_G(X, Y, Xs, less_out_gg(X, s(Y))) → ORDERED_IN_G(.(Y, Xs))

Strictly oriented rules of the TRS R:

less_in_gg(0, s(X)) → less_out_gg(0, s(X))

Used ordering: POLO with Polynomial interpretation [25]:

POL(.(x₁, x₂)) = 2·x₁ + x₂
POL(0) = 2
POL(ORDERED_IN_G(x₁)) = 2·x₁
POL(U7_G(x₁, x₂, x₃, x₄)) = x₁ + 2·x₂ + 2·x₃ + x₄
POL(U9_gg(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(less_in_gg(x₁, x₂)) = 2·x₁ + x₂
POL(less_out_gg(x₁, x₂)) = 1 + x₁ + x₂
POL(s(x₁)) = 2·x₁

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ DependencyGraphProof

              ↳ PiDP

              ↳ PiDP

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

ORDERED_IN_G(.(X, .(Y, Xs))) → U7_G(X, Y, Xs, less_in_gg(X, s(Y)))

The TRS R consists of the following rules:

less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))

The set Q consists of the following terms:

less_in_gg(x0, x1)
U9_gg(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 1 less node.

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAA(Xs, Ys, Zs)

The TRS R consists of the following rules:

ss_in_ag(Xs, Ys) → U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))
U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_ag(Xs, Ys, ordered_in_g(Ys))
ordered_in_g([]) → ordered_out_g([])
ordered_in_g(.(X, [])) → ordered_out_g(.(X, []))
ordered_in_g(.(X, .(Y, Xs))) → U7_g(X, Y, Xs, less_in_gg(X, s(Y)))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U7_g(X, Y, Xs, less_out_gg(X, s(Y))) → U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) → ordered_out_g(.(X, .(Y, Xs)))
U2_ag(Xs, Ys, ordered_out_g(Ys)) → ss_out_ag(Xs, Ys)

↳ 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:

APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAA(Xs, Ys, Zs)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ NonTerminationProof

              ↳ PiDP

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

APP_IN_AAA → APP_IN_AAA

The TRS R consists of the following rules:none

s = APP_IN_AAA evaluates to t =APP_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 APP_IN_AAA to APP_IN_AAA.

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

U4_AG(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → PERM_IN_AG(Zs, Ys)
PERM_IN_AG(Xs, .(X, Ys)) → U3_AG(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
U3_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_AG(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))

The TRS R consists of the following rules:

ss_in_ag(Xs, Ys) → U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))
U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_ag(Xs, Ys, ordered_in_g(Ys))
ordered_in_g([]) → ordered_out_g([])
ordered_in_g(.(X, [])) → ordered_out_g(.(X, []))
ordered_in_g(.(X, .(Y, Xs))) → U7_g(X, Y, Xs, less_in_gg(X, s(Y)))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U7_g(X, Y, Xs, less_out_gg(X, s(Y))) → U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) → ordered_out_g(.(X, .(Y, Xs)))
U2_ag(Xs, Ys, ordered_out_g(Ys)) → ss_out_ag(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in_ag(x1, x2) = ss_in_ag(x2)
U1_ag(x1, x2, x3) = U1_ag(x2, x3)
perm_in_ag(x1, x2) = perm_in_ag(x2)
[] = []
perm_out_ag(x1, x2) = perm_out_ag(x2)
.(x1, x2) = .(x1, x2)
U3_ag(x1, x2, x3, x4) = U3_ag(x2, x3, x4)
app_in_aaa(x1, x2, x3) = app_in_aaa
app_out_aaa(x1, x2, x3) = app_out_aaa
U6_aaa(x1, x2, x3, x4, x5) = U6_aaa(x5)
U4_ag(x1, x2, x3, x4) = U4_ag(x2, x3, x4)
U5_ag(x1, x2, x3, x4) = U5_ag(x2, x3, x4)
U2_ag(x1, x2, x3) = U2_ag(x2, x3)
ordered_in_g(x1) = ordered_in_g(x1)
ordered_out_g(x1) = ordered_out_g(x1)
U7_g(x1, x2, x3, x4) = U7_g(x1, x2, x3, x4)
less_in_gg(x1, x2) = less_in_gg(x1, x2)
0 = 0
s(x1) = s(x1)
less_out_gg(x1, x2) = less_out_gg(x1, x2)
U9_gg(x1, x2, x3) = U9_gg(x1, x2, x3)
U8_g(x1, x2, x3, x4) = U8_g(x1, x2, x3, x4)
ss_out_ag(x1, x2) = ss_out_ag(x2)
PERM_IN_AG(x1, x2) = PERM_IN_AG(x2)
U3_AG(x1, x2, x3, x4) = U3_AG(x2, x3, x4)
U4_AG(x1, x2, x3, x4) = U4_AG(x2, x3, x4)

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

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

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

U4_AG(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → PERM_IN_AG(Zs, Ys)
PERM_IN_AG(Xs, .(X, Ys)) → U3_AG(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
U3_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_AG(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))

The TRS R consists of the following rules:

app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
[] = []
.(x1, x2) = .(x1, x2)
app_in_aaa(x1, x2, x3) = app_in_aaa
app_out_aaa(x1, x2, x3) = app_out_aaa
U6_aaa(x1, x2, x3, x4, x5) = U6_aaa(x5)
PERM_IN_AG(x1, x2) = PERM_IN_AG(x2)
U3_AG(x1, x2, x3, x4) = U3_AG(x2, x3, x4)
U4_AG(x1, x2, x3, x4) = U4_AG(x2, x3, x4)

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

  ↳ 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:

U4_AG(X, Ys, app_out_aaa) → PERM_IN_AG(Ys)
PERM_IN_AG(.(X, Ys)) → U3_AG(X, Ys, app_in_aaa)
U3_AG(X, Ys, app_out_aaa) → U4_AG(X, Ys, app_in_aaa)

The TRS R consists of the following rules:

app_in_aaa → app_out_aaa
app_in_aaa → U6_aaa(app_in_aaa)
U6_aaa(app_out_aaa) → app_out_aaa

The set Q consists of the following terms:

app_in_aaa
U6_aaa(x0)

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:

PERM_IN_AG(.(X, Ys)) → U3_AG(X, Ys, app_in_aaa)
The graph contains the following edges 1 > 1, 1 > 2
U3_AG(X, Ys, app_out_aaa) → U4_AG(X, Ys, app_in_aaa)
The graph contains the following edges 1 >= 1, 2 >= 2
U4_AG(X, Ys, app_out_aaa) → PERM_IN_AG(Ys)
The graph contains the following edges 2 >= 1