Left Termination proof of /tmp/tmpg0X2zT/qsort.pl

Left Termination of the query pattern qs_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

↳ Prolog

  ↳ PrologToPiTRSProof

Clauses:

qs([], []).
qs(cons(X, L), S) :- ','(split(L, X, L1, L2), ','(qs(L1, S1), ','(qs(L2, S2), append(S1, cons(X, S2), S)))).
append([], L, L).
append(cons(X, L1), L2, cons(X, L3)) :- append(L1, L2, L3).
split([], X, [], []).
split(cons(X, L), Y, cons(X, L1), L2) :- ','(less(X, Y), split(L, Y, L1, L2)).
split(cons(X, L), Y, L1, cons(X, L2)) :- ','(geq(X, Y), split(L, Y, L1, L2)).
less(0, s(X)).
less(s(X), s(Y)) :- less(X, Y).
geq(X, X).
geq(s(X), 0).
geq(s(X), s(Y)) :- geq(X, Y).

Queries:

qs(g,a).

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

qs_in_ga([], []) → qs_out_ga([], [])
qs_in_ga(cons(X, L), S) → U1_ga(X, L, S, split_in_ggaa(L, X, L1, L2))
split_in_ggaa([], X, [], []) → split_out_ggaa([], X, [], [])
split_in_ggaa(cons(X, L), Y, cons(X, L1), L2) → U6_ggaa(X, L, Y, L1, L2, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U10_gg(X, Y, less_in_gg(X, Y))
U10_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U6_ggaa(X, L, Y, L1, L2, less_out_gg(X, Y)) → U7_ggaa(X, L, Y, L1, L2, split_in_ggaa(L, Y, L1, L2))
split_in_ggaa(cons(X, L), Y, L1, cons(X, L2)) → U8_ggaa(X, L, Y, L1, L2, geq_in_gg(X, Y))
geq_in_gg(X, X) → geq_out_gg(X, X)
geq_in_gg(s(X), 0) → geq_out_gg(s(X), 0)
geq_in_gg(s(X), s(Y)) → U11_gg(X, Y, geq_in_gg(X, Y))
U11_gg(X, Y, geq_out_gg(X, Y)) → geq_out_gg(s(X), s(Y))
U8_ggaa(X, L, Y, L1, L2, geq_out_gg(X, Y)) → U9_ggaa(X, L, Y, L1, L2, split_in_ggaa(L, Y, L1, L2))
U9_ggaa(X, L, Y, L1, L2, split_out_ggaa(L, Y, L1, L2)) → split_out_ggaa(cons(X, L), Y, L1, cons(X, L2))
U7_ggaa(X, L, Y, L1, L2, split_out_ggaa(L, Y, L1, L2)) → split_out_ggaa(cons(X, L), Y, cons(X, L1), L2)
U1_ga(X, L, S, split_out_ggaa(L, X, L1, L2)) → U2_ga(X, L, S, L2, qs_in_ga(L1, S1))
U2_ga(X, L, S, L2, qs_out_ga(L1, S1)) → U3_ga(X, L, S, S1, qs_in_ga(L2, S2))
U3_ga(X, L, S, S1, qs_out_ga(L2, S2)) → U4_ga(X, L, S, append_in_gga(S1, cons(X, S2), S))
append_in_gga([], L, L) → append_out_gga([], L, L)
append_in_gga(cons(X, L1), L2, cons(X, L3)) → U5_gga(X, L1, L2, L3, append_in_gga(L1, L2, L3))
U5_gga(X, L1, L2, L3, append_out_gga(L1, L2, L3)) → append_out_gga(cons(X, L1), L2, cons(X, L3))
U4_ga(X, L, S, append_out_gga(S1, cons(X, S2), S)) → qs_out_ga(cons(X, L), S)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

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

qs_in_ga([], []) → qs_out_ga([], [])
qs_in_ga(cons(X, L), S) → U1_ga(X, L, S, split_in_ggaa(L, X, L1, L2))
split_in_ggaa([], X, [], []) → split_out_ggaa([], X, [], [])
split_in_ggaa(cons(X, L), Y, cons(X, L1), L2) → U6_ggaa(X, L, Y, L1, L2, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U10_gg(X, Y, less_in_gg(X, Y))
U10_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U6_ggaa(X, L, Y, L1, L2, less_out_gg(X, Y)) → U7_ggaa(X, L, Y, L1, L2, split_in_ggaa(L, Y, L1, L2))
split_in_ggaa(cons(X, L), Y, L1, cons(X, L2)) → U8_ggaa(X, L, Y, L1, L2, geq_in_gg(X, Y))
geq_in_gg(X, X) → geq_out_gg(X, X)
geq_in_gg(s(X), 0) → geq_out_gg(s(X), 0)
geq_in_gg(s(X), s(Y)) → U11_gg(X, Y, geq_in_gg(X, Y))
U11_gg(X, Y, geq_out_gg(X, Y)) → geq_out_gg(s(X), s(Y))
U8_ggaa(X, L, Y, L1, L2, geq_out_gg(X, Y)) → U9_ggaa(X, L, Y, L1, L2, split_in_ggaa(L, Y, L1, L2))
U9_ggaa(X, L, Y, L1, L2, split_out_ggaa(L, Y, L1, L2)) → split_out_ggaa(cons(X, L), Y, L1, cons(X, L2))
U7_ggaa(X, L, Y, L1, L2, split_out_ggaa(L, Y, L1, L2)) → split_out_ggaa(cons(X, L), Y, cons(X, L1), L2)
U1_ga(X, L, S, split_out_ggaa(L, X, L1, L2)) → U2_ga(X, L, S, L2, qs_in_ga(L1, S1))
U2_ga(X, L, S, L2, qs_out_ga(L1, S1)) → U3_ga(X, L, S, S1, qs_in_ga(L2, S2))
U3_ga(X, L, S, S1, qs_out_ga(L2, S2)) → U4_ga(X, L, S, append_in_gga(S1, cons(X, S2), S))
append_in_gga([], L, L) → append_out_gga([], L, L)
append_in_gga(cons(X, L1), L2, cons(X, L3)) → U5_gga(X, L1, L2, L3, append_in_gga(L1, L2, L3))
U5_gga(X, L1, L2, L3, append_out_gga(L1, L2, L3)) → append_out_gga(cons(X, L1), L2, cons(X, L3))
U4_ga(X, L, S, append_out_gga(S1, cons(X, S2), S)) → qs_out_ga(cons(X, L), S)

QS_IN_GA(cons(X, L), S) → U1_GA(X, L, S, split_in_ggaa(L, X, L1, L2))
QS_IN_GA(cons(X, L), S) → SPLIT_IN_GGAA(L, X, L1, L2)
SPLIT_IN_GGAA(cons(X, L), Y, cons(X, L1), L2) → U6_GGAA(X, L, Y, L1, L2, less_in_gg(X, Y))
SPLIT_IN_GGAA(cons(X, L), Y, cons(X, L1), L2) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U10_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U6_GGAA(X, L, Y, L1, L2, less_out_gg(X, Y)) → U7_GGAA(X, L, Y, L1, L2, split_in_ggaa(L, Y, L1, L2))
U6_GGAA(X, L, Y, L1, L2, less_out_gg(X, Y)) → SPLIT_IN_GGAA(L, Y, L1, L2)
SPLIT_IN_GGAA(cons(X, L), Y, L1, cons(X, L2)) → U8_GGAA(X, L, Y, L1, L2, geq_in_gg(X, Y))
SPLIT_IN_GGAA(cons(X, L), Y, L1, cons(X, L2)) → GEQ_IN_GG(X, Y)
GEQ_IN_GG(s(X), s(Y)) → U11_GG(X, Y, geq_in_gg(X, Y))
GEQ_IN_GG(s(X), s(Y)) → GEQ_IN_GG(X, Y)
U8_GGAA(X, L, Y, L1, L2, geq_out_gg(X, Y)) → U9_GGAA(X, L, Y, L1, L2, split_in_ggaa(L, Y, L1, L2))
U8_GGAA(X, L, Y, L1, L2, geq_out_gg(X, Y)) → SPLIT_IN_GGAA(L, Y, L1, L2)
U1_GA(X, L, S, split_out_ggaa(L, X, L1, L2)) → U2_GA(X, L, S, L2, qs_in_ga(L1, S1))
U1_GA(X, L, S, split_out_ggaa(L, X, L1, L2)) → QS_IN_GA(L1, S1)
U2_GA(X, L, S, L2, qs_out_ga(L1, S1)) → U3_GA(X, L, S, S1, qs_in_ga(L2, S2))
U2_GA(X, L, S, L2, qs_out_ga(L1, S1)) → QS_IN_GA(L2, S2)
U3_GA(X, L, S, S1, qs_out_ga(L2, S2)) → U4_GA(X, L, S, append_in_gga(S1, cons(X, S2), S))
U3_GA(X, L, S, S1, qs_out_ga(L2, S2)) → APPEND_IN_GGA(S1, cons(X, S2), S)
APPEND_IN_GGA(cons(X, L1), L2, cons(X, L3)) → U5_GGA(X, L1, L2, L3, append_in_gga(L1, L2, L3))
APPEND_IN_GGA(cons(X, L1), L2, cons(X, L3)) → APPEND_IN_GGA(L1, L2, L3)

The TRS R consists of the following rules:

qs_in_ga([], []) → qs_out_ga([], [])
qs_in_ga(cons(X, L), S) → U1_ga(X, L, S, split_in_ggaa(L, X, L1, L2))
split_in_ggaa([], X, [], []) → split_out_ggaa([], X, [], [])
split_in_ggaa(cons(X, L), Y, cons(X, L1), L2) → U6_ggaa(X, L, Y, L1, L2, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U10_gg(X, Y, less_in_gg(X, Y))
U10_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U6_ggaa(X, L, Y, L1, L2, less_out_gg(X, Y)) → U7_ggaa(X, L, Y, L1, L2, split_in_ggaa(L, Y, L1, L2))
split_in_ggaa(cons(X, L), Y, L1, cons(X, L2)) → U8_ggaa(X, L, Y, L1, L2, geq_in_gg(X, Y))
geq_in_gg(X, X) → geq_out_gg(X, X)
geq_in_gg(s(X), 0) → geq_out_gg(s(X), 0)
geq_in_gg(s(X), s(Y)) → U11_gg(X, Y, geq_in_gg(X, Y))
U11_gg(X, Y, geq_out_gg(X, Y)) → geq_out_gg(s(X), s(Y))
U8_ggaa(X, L, Y, L1, L2, geq_out_gg(X, Y)) → U9_ggaa(X, L, Y, L1, L2, split_in_ggaa(L, Y, L1, L2))
U9_ggaa(X, L, Y, L1, L2, split_out_ggaa(L, Y, L1, L2)) → split_out_ggaa(cons(X, L), Y, L1, cons(X, L2))
U7_ggaa(X, L, Y, L1, L2, split_out_ggaa(L, Y, L1, L2)) → split_out_ggaa(cons(X, L), Y, cons(X, L1), L2)
U1_ga(X, L, S, split_out_ggaa(L, X, L1, L2)) → U2_ga(X, L, S, L2, qs_in_ga(L1, S1))
U2_ga(X, L, S, L2, qs_out_ga(L1, S1)) → U3_ga(X, L, S, S1, qs_in_ga(L2, S2))
U3_ga(X, L, S, S1, qs_out_ga(L2, S2)) → U4_ga(X, L, S, append_in_gga(S1, cons(X, S2), S))
append_in_gga([], L, L) → append_out_gga([], L, L)
append_in_gga(cons(X, L1), L2, cons(X, L3)) → U5_gga(X, L1, L2, L3, append_in_gga(L1, L2, L3))
U5_gga(X, L1, L2, L3, append_out_gga(L1, L2, L3)) → append_out_gga(cons(X, L1), L2, cons(X, L3))
U4_ga(X, L, S, append_out_gga(S1, cons(X, S2), S)) → qs_out_ga(cons(X, L), S)

The argument filtering Pi contains the following mapping:
qs_in_ga(x1, x2) = qs_in_ga(x1)
[] = []
qs_out_ga(x1, x2) = qs_out_ga(x2)
cons(x1, x2) = cons(x1, x2)
U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4)
split_in_ggaa(x1, x2, x3, x4) = split_in_ggaa(x1, x2)
split_out_ggaa(x1, x2, x3, x4) = split_out_ggaa(x3, x4)
U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x1, x2, x3, x6)
less_in_gg(x1, x2) = less_in_gg(x1, x2)
0 = 0
s(x1) = s(x1)
less_out_gg(x1, x2) = less_out_gg
U10_gg(x1, x2, x3) = U10_gg(x3)
U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x1, x6)
U8_ggaa(x1, x2, x3, x4, x5, x6) = U8_ggaa(x1, x2, x3, x6)
geq_in_gg(x1, x2) = geq_in_gg(x1, x2)
geq_out_gg(x1, x2) = geq_out_gg
U11_gg(x1, x2, x3) = U11_gg(x3)
U9_ggaa(x1, x2, x3, x4, x5, x6) = U9_ggaa(x1, x6)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5)
U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4) = U4_ga(x4)
append_in_gga(x1, x2, x3) = append_in_gga(x1, x2)
append_out_gga(x1, x2, x3) = append_out_gga(x3)
U5_gga(x1, x2, x3, x4, x5) = U5_gga(x1, x5)
QS_IN_GA(x1, x2) = QS_IN_GA(x1)
U6_GGAA(x1, x2, x3, x4, x5, x6) = U6_GGAA(x1, x2, x3, x6)
GEQ_IN_GG(x1, x2) = GEQ_IN_GG(x1, x2)
U8_GGAA(x1, x2, x3, x4, x5, x6) = U8_GGAA(x1, x2, x3, x6)
U7_GGAA(x1, x2, x3, x4, x5, x6) = U7_GGAA(x1, x6)
U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4)
U9_GGAA(x1, x2, x3, x4, x5, x6) = U9_GGAA(x1, x6)
U5_GGA(x1, x2, x3, x4, x5) = U5_GGA(x1, x5)
LESS_IN_GG(x1, x2) = LESS_IN_GG(x1, x2)
U2_GA(x1, x2, x3, x4, x5) = U2_GA(x1, x4, x5)
U4_GA(x1, x2, x3, x4) = U4_GA(x4)
APPEND_IN_GGA(x1, x2, x3) = APPEND_IN_GGA(x1, x2)
U10_GG(x1, x2, x3) = U10_GG(x3)
U11_GG(x1, x2, x3) = U11_GG(x3)
U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x4, x5)
SPLIT_IN_GGAA(x1, x2, x3, x4) = SPLIT_IN_GGAA(x1, x2)

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

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

QS_IN_GA(cons(X, L), S) → U1_GA(X, L, S, split_in_ggaa(L, X, L1, L2))
QS_IN_GA(cons(X, L), S) → SPLIT_IN_GGAA(L, X, L1, L2)
SPLIT_IN_GGAA(cons(X, L), Y, cons(X, L1), L2) → U6_GGAA(X, L, Y, L1, L2, less_in_gg(X, Y))
SPLIT_IN_GGAA(cons(X, L), Y, cons(X, L1), L2) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U10_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U6_GGAA(X, L, Y, L1, L2, less_out_gg(X, Y)) → U7_GGAA(X, L, Y, L1, L2, split_in_ggaa(L, Y, L1, L2))
U6_GGAA(X, L, Y, L1, L2, less_out_gg(X, Y)) → SPLIT_IN_GGAA(L, Y, L1, L2)
SPLIT_IN_GGAA(cons(X, L), Y, L1, cons(X, L2)) → U8_GGAA(X, L, Y, L1, L2, geq_in_gg(X, Y))
SPLIT_IN_GGAA(cons(X, L), Y, L1, cons(X, L2)) → GEQ_IN_GG(X, Y)
GEQ_IN_GG(s(X), s(Y)) → U11_GG(X, Y, geq_in_gg(X, Y))
GEQ_IN_GG(s(X), s(Y)) → GEQ_IN_GG(X, Y)
U8_GGAA(X, L, Y, L1, L2, geq_out_gg(X, Y)) → U9_GGAA(X, L, Y, L1, L2, split_in_ggaa(L, Y, L1, L2))
U8_GGAA(X, L, Y, L1, L2, geq_out_gg(X, Y)) → SPLIT_IN_GGAA(L, Y, L1, L2)
U1_GA(X, L, S, split_out_ggaa(L, X, L1, L2)) → U2_GA(X, L, S, L2, qs_in_ga(L1, S1))
U1_GA(X, L, S, split_out_ggaa(L, X, L1, L2)) → QS_IN_GA(L1, S1)
U2_GA(X, L, S, L2, qs_out_ga(L1, S1)) → U3_GA(X, L, S, S1, qs_in_ga(L2, S2))
U2_GA(X, L, S, L2, qs_out_ga(L1, S1)) → QS_IN_GA(L2, S2)
U3_GA(X, L, S, S1, qs_out_ga(L2, S2)) → U4_GA(X, L, S, append_in_gga(S1, cons(X, S2), S))
U3_GA(X, L, S, S1, qs_out_ga(L2, S2)) → APPEND_IN_GGA(S1, cons(X, S2), S)
APPEND_IN_GGA(cons(X, L1), L2, cons(X, L3)) → U5_GGA(X, L1, L2, L3, append_in_gga(L1, L2, L3))
APPEND_IN_GGA(cons(X, L1), L2, cons(X, L3)) → APPEND_IN_GGA(L1, L2, L3)

The TRS R consists of the following rules:

qs_in_ga([], []) → qs_out_ga([], [])
qs_in_ga(cons(X, L), S) → U1_ga(X, L, S, split_in_ggaa(L, X, L1, L2))
split_in_ggaa([], X, [], []) → split_out_ggaa([], X, [], [])
split_in_ggaa(cons(X, L), Y, cons(X, L1), L2) → U6_ggaa(X, L, Y, L1, L2, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U10_gg(X, Y, less_in_gg(X, Y))
U10_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U6_ggaa(X, L, Y, L1, L2, less_out_gg(X, Y)) → U7_ggaa(X, L, Y, L1, L2, split_in_ggaa(L, Y, L1, L2))
split_in_ggaa(cons(X, L), Y, L1, cons(X, L2)) → U8_ggaa(X, L, Y, L1, L2, geq_in_gg(X, Y))
geq_in_gg(X, X) → geq_out_gg(X, X)
geq_in_gg(s(X), 0) → geq_out_gg(s(X), 0)
geq_in_gg(s(X), s(Y)) → U11_gg(X, Y, geq_in_gg(X, Y))
U11_gg(X, Y, geq_out_gg(X, Y)) → geq_out_gg(s(X), s(Y))
U8_ggaa(X, L, Y, L1, L2, geq_out_gg(X, Y)) → U9_ggaa(X, L, Y, L1, L2, split_in_ggaa(L, Y, L1, L2))
U9_ggaa(X, L, Y, L1, L2, split_out_ggaa(L, Y, L1, L2)) → split_out_ggaa(cons(X, L), Y, L1, cons(X, L2))
U7_ggaa(X, L, Y, L1, L2, split_out_ggaa(L, Y, L1, L2)) → split_out_ggaa(cons(X, L), Y, cons(X, L1), L2)
U1_ga(X, L, S, split_out_ggaa(L, X, L1, L2)) → U2_ga(X, L, S, L2, qs_in_ga(L1, S1))
U2_ga(X, L, S, L2, qs_out_ga(L1, S1)) → U3_ga(X, L, S, S1, qs_in_ga(L2, S2))
U3_ga(X, L, S, S1, qs_out_ga(L2, S2)) → U4_ga(X, L, S, append_in_gga(S1, cons(X, S2), S))
append_in_gga([], L, L) → append_out_gga([], L, L)
append_in_gga(cons(X, L1), L2, cons(X, L3)) → U5_gga(X, L1, L2, L3, append_in_gga(L1, L2, L3))
U5_gga(X, L1, L2, L3, append_out_gga(L1, L2, L3)) → append_out_gga(cons(X, L1), L2, cons(X, L3))
U4_ga(X, L, S, append_out_gga(S1, cons(X, S2), S)) → qs_out_ga(cons(X, L), S)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

APPEND_IN_GGA(cons(X, L1), L2, cons(X, L3)) → APPEND_IN_GGA(L1, L2, L3)

The TRS R consists of the following rules:

qs_in_ga([], []) → qs_out_ga([], [])
qs_in_ga(cons(X, L), S) → U1_ga(X, L, S, split_in_ggaa(L, X, L1, L2))
split_in_ggaa([], X, [], []) → split_out_ggaa([], X, [], [])
split_in_ggaa(cons(X, L), Y, cons(X, L1), L2) → U6_ggaa(X, L, Y, L1, L2, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U10_gg(X, Y, less_in_gg(X, Y))
U10_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U6_ggaa(X, L, Y, L1, L2, less_out_gg(X, Y)) → U7_ggaa(X, L, Y, L1, L2, split_in_ggaa(L, Y, L1, L2))
split_in_ggaa(cons(X, L), Y, L1, cons(X, L2)) → U8_ggaa(X, L, Y, L1, L2, geq_in_gg(X, Y))
geq_in_gg(X, X) → geq_out_gg(X, X)
geq_in_gg(s(X), 0) → geq_out_gg(s(X), 0)
geq_in_gg(s(X), s(Y)) → U11_gg(X, Y, geq_in_gg(X, Y))
U11_gg(X, Y, geq_out_gg(X, Y)) → geq_out_gg(s(X), s(Y))
U8_ggaa(X, L, Y, L1, L2, geq_out_gg(X, Y)) → U9_ggaa(X, L, Y, L1, L2, split_in_ggaa(L, Y, L1, L2))
U9_ggaa(X, L, Y, L1, L2, split_out_ggaa(L, Y, L1, L2)) → split_out_ggaa(cons(X, L), Y, L1, cons(X, L2))
U7_ggaa(X, L, Y, L1, L2, split_out_ggaa(L, Y, L1, L2)) → split_out_ggaa(cons(X, L), Y, cons(X, L1), L2)
U1_ga(X, L, S, split_out_ggaa(L, X, L1, L2)) → U2_ga(X, L, S, L2, qs_in_ga(L1, S1))
U2_ga(X, L, S, L2, qs_out_ga(L1, S1)) → U3_ga(X, L, S, S1, qs_in_ga(L2, S2))
U3_ga(X, L, S, S1, qs_out_ga(L2, S2)) → U4_ga(X, L, S, append_in_gga(S1, cons(X, S2), S))
append_in_gga([], L, L) → append_out_gga([], L, L)
append_in_gga(cons(X, L1), L2, cons(X, L3)) → U5_gga(X, L1, L2, L3, append_in_gga(L1, L2, L3))
U5_gga(X, L1, L2, L3, append_out_gga(L1, L2, L3)) → append_out_gga(cons(X, L1), L2, cons(X, L3))
U4_ga(X, L, S, append_out_gga(S1, cons(X, S2), S)) → qs_out_ga(cons(X, L), S)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

APPEND_IN_GGA(cons(X, L1), L2, cons(X, L3)) → APPEND_IN_GGA(L1, L2, L3)

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

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

APPEND_IN_GGA(cons(X, L1), L2) → APPEND_IN_GGA(L1, L2)

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:

APPEND_IN_GGA(cons(X, L1), L2) → APPEND_IN_GGA(L1, L2)
The graph contains the following edges 1 > 1, 2 >= 2

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

GEQ_IN_GG(s(X), s(Y)) → GEQ_IN_GG(X, Y)

The TRS R consists of the following rules:

qs_in_ga([], []) → qs_out_ga([], [])
qs_in_ga(cons(X, L), S) → U1_ga(X, L, S, split_in_ggaa(L, X, L1, L2))
split_in_ggaa([], X, [], []) → split_out_ggaa([], X, [], [])
split_in_ggaa(cons(X, L), Y, cons(X, L1), L2) → U6_ggaa(X, L, Y, L1, L2, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U10_gg(X, Y, less_in_gg(X, Y))
U10_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U6_ggaa(X, L, Y, L1, L2, less_out_gg(X, Y)) → U7_ggaa(X, L, Y, L1, L2, split_in_ggaa(L, Y, L1, L2))
split_in_ggaa(cons(X, L), Y, L1, cons(X, L2)) → U8_ggaa(X, L, Y, L1, L2, geq_in_gg(X, Y))
geq_in_gg(X, X) → geq_out_gg(X, X)
geq_in_gg(s(X), 0) → geq_out_gg(s(X), 0)
geq_in_gg(s(X), s(Y)) → U11_gg(X, Y, geq_in_gg(X, Y))
U11_gg(X, Y, geq_out_gg(X, Y)) → geq_out_gg(s(X), s(Y))
U8_ggaa(X, L, Y, L1, L2, geq_out_gg(X, Y)) → U9_ggaa(X, L, Y, L1, L2, split_in_ggaa(L, Y, L1, L2))
U9_ggaa(X, L, Y, L1, L2, split_out_ggaa(L, Y, L1, L2)) → split_out_ggaa(cons(X, L), Y, L1, cons(X, L2))
U7_ggaa(X, L, Y, L1, L2, split_out_ggaa(L, Y, L1, L2)) → split_out_ggaa(cons(X, L), Y, cons(X, L1), L2)
U1_ga(X, L, S, split_out_ggaa(L, X, L1, L2)) → U2_ga(X, L, S, L2, qs_in_ga(L1, S1))
U2_ga(X, L, S, L2, qs_out_ga(L1, S1)) → U3_ga(X, L, S, S1, qs_in_ga(L2, S2))
U3_ga(X, L, S, S1, qs_out_ga(L2, S2)) → U4_ga(X, L, S, append_in_gga(S1, cons(X, S2), S))
append_in_gga([], L, L) → append_out_gga([], L, L)
append_in_gga(cons(X, L1), L2, cons(X, L3)) → U5_gga(X, L1, L2, L3, append_in_gga(L1, L2, L3))
U5_gga(X, L1, L2, L3, append_out_gga(L1, L2, L3)) → append_out_gga(cons(X, L1), L2, cons(X, L3))
U4_ga(X, L, S, append_out_gga(S1, cons(X, S2), S)) → qs_out_ga(cons(X, L), S)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

GEQ_IN_GG(s(X), s(Y)) → GEQ_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

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

GEQ_IN_GG(s(X), s(Y)) → GEQ_IN_GG(X, Y)

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

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ 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:

qs_in_ga([], []) → qs_out_ga([], [])
qs_in_ga(cons(X, L), S) → U1_ga(X, L, S, split_in_ggaa(L, X, L1, L2))
split_in_ggaa([], X, [], []) → split_out_ggaa([], X, [], [])
split_in_ggaa(cons(X, L), Y, cons(X, L1), L2) → U6_ggaa(X, L, Y, L1, L2, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U10_gg(X, Y, less_in_gg(X, Y))
U10_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U6_ggaa(X, L, Y, L1, L2, less_out_gg(X, Y)) → U7_ggaa(X, L, Y, L1, L2, split_in_ggaa(L, Y, L1, L2))
split_in_ggaa(cons(X, L), Y, L1, cons(X, L2)) → U8_ggaa(X, L, Y, L1, L2, geq_in_gg(X, Y))
geq_in_gg(X, X) → geq_out_gg(X, X)
geq_in_gg(s(X), 0) → geq_out_gg(s(X), 0)
geq_in_gg(s(X), s(Y)) → U11_gg(X, Y, geq_in_gg(X, Y))
U11_gg(X, Y, geq_out_gg(X, Y)) → geq_out_gg(s(X), s(Y))
U8_ggaa(X, L, Y, L1, L2, geq_out_gg(X, Y)) → U9_ggaa(X, L, Y, L1, L2, split_in_ggaa(L, Y, L1, L2))
U9_ggaa(X, L, Y, L1, L2, split_out_ggaa(L, Y, L1, L2)) → split_out_ggaa(cons(X, L), Y, L1, cons(X, L2))
U7_ggaa(X, L, Y, L1, L2, split_out_ggaa(L, Y, L1, L2)) → split_out_ggaa(cons(X, L), Y, cons(X, L1), L2)
U1_ga(X, L, S, split_out_ggaa(L, X, L1, L2)) → U2_ga(X, L, S, L2, qs_in_ga(L1, S1))
U2_ga(X, L, S, L2, qs_out_ga(L1, S1)) → U3_ga(X, L, S, S1, qs_in_ga(L2, S2))
U3_ga(X, L, S, S1, qs_out_ga(L2, S2)) → U4_ga(X, L, S, append_in_gga(S1, cons(X, S2), S))
append_in_gga([], L, L) → append_out_gga([], L, L)
append_in_gga(cons(X, L1), L2, cons(X, L3)) → U5_gga(X, L1, L2, L3, append_in_gga(L1, L2, L3))
U5_gga(X, L1, L2, L3, append_out_gga(L1, L2, L3)) → append_out_gga(cons(X, L1), L2, cons(X, L3))
U4_ga(X, L, S, append_out_gga(S1, cons(X, S2), S)) → qs_out_ga(cons(X, L), S)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ 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

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ 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

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

SPLIT_IN_GGAA(cons(X, L), Y, cons(X, L1), L2) → U6_GGAA(X, L, Y, L1, L2, less_in_gg(X, Y))
U8_GGAA(X, L, Y, L1, L2, geq_out_gg(X, Y)) → SPLIT_IN_GGAA(L, Y, L1, L2)
U6_GGAA(X, L, Y, L1, L2, less_out_gg(X, Y)) → SPLIT_IN_GGAA(L, Y, L1, L2)
SPLIT_IN_GGAA(cons(X, L), Y, L1, cons(X, L2)) → U8_GGAA(X, L, Y, L1, L2, geq_in_gg(X, Y))

The TRS R consists of the following rules:

qs_in_ga([], []) → qs_out_ga([], [])
qs_in_ga(cons(X, L), S) → U1_ga(X, L, S, split_in_ggaa(L, X, L1, L2))
split_in_ggaa([], X, [], []) → split_out_ggaa([], X, [], [])
split_in_ggaa(cons(X, L), Y, cons(X, L1), L2) → U6_ggaa(X, L, Y, L1, L2, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U10_gg(X, Y, less_in_gg(X, Y))
U10_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U6_ggaa(X, L, Y, L1, L2, less_out_gg(X, Y)) → U7_ggaa(X, L, Y, L1, L2, split_in_ggaa(L, Y, L1, L2))
split_in_ggaa(cons(X, L), Y, L1, cons(X, L2)) → U8_ggaa(X, L, Y, L1, L2, geq_in_gg(X, Y))
geq_in_gg(X, X) → geq_out_gg(X, X)
geq_in_gg(s(X), 0) → geq_out_gg(s(X), 0)
geq_in_gg(s(X), s(Y)) → U11_gg(X, Y, geq_in_gg(X, Y))
U11_gg(X, Y, geq_out_gg(X, Y)) → geq_out_gg(s(X), s(Y))
U8_ggaa(X, L, Y, L1, L2, geq_out_gg(X, Y)) → U9_ggaa(X, L, Y, L1, L2, split_in_ggaa(L, Y, L1, L2))
U9_ggaa(X, L, Y, L1, L2, split_out_ggaa(L, Y, L1, L2)) → split_out_ggaa(cons(X, L), Y, L1, cons(X, L2))
U7_ggaa(X, L, Y, L1, L2, split_out_ggaa(L, Y, L1, L2)) → split_out_ggaa(cons(X, L), Y, cons(X, L1), L2)
U1_ga(X, L, S, split_out_ggaa(L, X, L1, L2)) → U2_ga(X, L, S, L2, qs_in_ga(L1, S1))
U2_ga(X, L, S, L2, qs_out_ga(L1, S1)) → U3_ga(X, L, S, S1, qs_in_ga(L2, S2))
U3_ga(X, L, S, S1, qs_out_ga(L2, S2)) → U4_ga(X, L, S, append_in_gga(S1, cons(X, S2), S))
append_in_gga([], L, L) → append_out_gga([], L, L)
append_in_gga(cons(X, L1), L2, cons(X, L3)) → U5_gga(X, L1, L2, L3, append_in_gga(L1, L2, L3))
U5_gga(X, L1, L2, L3, append_out_gga(L1, L2, L3)) → append_out_gga(cons(X, L1), L2, cons(X, L3))
U4_ga(X, L, S, append_out_gga(S1, cons(X, S2), S)) → qs_out_ga(cons(X, L), S)

The argument filtering Pi contains the following mapping:
qs_in_ga(x1, x2) = qs_in_ga(x1)
[] = []
qs_out_ga(x1, x2) = qs_out_ga(x2)
cons(x1, x2) = cons(x1, x2)
U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4)
split_in_ggaa(x1, x2, x3, x4) = split_in_ggaa(x1, x2)
split_out_ggaa(x1, x2, x3, x4) = split_out_ggaa(x3, x4)
U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x1, x2, x3, x6)
less_in_gg(x1, x2) = less_in_gg(x1, x2)
0 = 0
s(x1) = s(x1)
less_out_gg(x1, x2) = less_out_gg
U10_gg(x1, x2, x3) = U10_gg(x3)
U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x1, x6)
U8_ggaa(x1, x2, x3, x4, x5, x6) = U8_ggaa(x1, x2, x3, x6)
geq_in_gg(x1, x2) = geq_in_gg(x1, x2)
geq_out_gg(x1, x2) = geq_out_gg
U11_gg(x1, x2, x3) = U11_gg(x3)
U9_ggaa(x1, x2, x3, x4, x5, x6) = U9_ggaa(x1, x6)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5)
U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4) = U4_ga(x4)
append_in_gga(x1, x2, x3) = append_in_gga(x1, x2)
append_out_gga(x1, x2, x3) = append_out_gga(x3)
U5_gga(x1, x2, x3, x4, x5) = U5_gga(x1, x5)
U6_GGAA(x1, x2, x3, x4, x5, x6) = U6_GGAA(x1, x2, x3, x6)
U8_GGAA(x1, x2, x3, x4, x5, x6) = U8_GGAA(x1, x2, x3, x6)
SPLIT_IN_GGAA(x1, x2, x3, x4) = SPLIT_IN_GGAA(x1, x2)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

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

SPLIT_IN_GGAA(cons(X, L), Y, cons(X, L1), L2) → U6_GGAA(X, L, Y, L1, L2, less_in_gg(X, Y))
U8_GGAA(X, L, Y, L1, L2, geq_out_gg(X, Y)) → SPLIT_IN_GGAA(L, Y, L1, L2)
U6_GGAA(X, L, Y, L1, L2, less_out_gg(X, Y)) → SPLIT_IN_GGAA(L, Y, L1, L2)
SPLIT_IN_GGAA(cons(X, L), Y, L1, cons(X, L2)) → U8_GGAA(X, L, Y, L1, L2, geq_in_gg(X, Y))

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)) → U10_gg(X, Y, less_in_gg(X, Y))
geq_in_gg(X, X) → geq_out_gg(X, X)
geq_in_gg(s(X), 0) → geq_out_gg(s(X), 0)
geq_in_gg(s(X), s(Y)) → U11_gg(X, Y, geq_in_gg(X, Y))
U10_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U11_gg(X, Y, geq_out_gg(X, Y)) → geq_out_gg(s(X), s(Y))

The argument filtering Pi contains the following mapping:
cons(x1, x2) = cons(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
U10_gg(x1, x2, x3) = U10_gg(x3)
geq_in_gg(x1, x2) = geq_in_gg(x1, x2)
geq_out_gg(x1, x2) = geq_out_gg
U11_gg(x1, x2, x3) = U11_gg(x3)
U6_GGAA(x1, x2, x3, x4, x5, x6) = U6_GGAA(x1, x2, x3, x6)
U8_GGAA(x1, x2, x3, x4, x5, x6) = U8_GGAA(x1, x2, x3, x6)
SPLIT_IN_GGAA(x1, x2, x3, x4) = SPLIT_IN_GGAA(x1, 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

              ↳ PiDP

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

SPLIT_IN_GGAA(cons(X, L), Y) → U8_GGAA(X, L, Y, geq_in_gg(X, Y))
U6_GGAA(X, L, Y, less_out_gg) → SPLIT_IN_GGAA(L, Y)
SPLIT_IN_GGAA(cons(X, L), Y) → U6_GGAA(X, L, Y, less_in_gg(X, Y))
U8_GGAA(X, L, Y, geq_out_gg) → SPLIT_IN_GGAA(L, Y)

The TRS R consists of the following rules:

less_in_gg(0, s(X)) → less_out_gg
less_in_gg(s(X), s(Y)) → U10_gg(less_in_gg(X, Y))
geq_in_gg(X, X) → geq_out_gg
geq_in_gg(s(X), 0) → geq_out_gg
geq_in_gg(s(X), s(Y)) → U11_gg(geq_in_gg(X, Y))
U10_gg(less_out_gg) → less_out_gg
U11_gg(geq_out_gg) → geq_out_gg

The set Q consists of the following terms:

less_in_gg(x0, x1)
geq_in_gg(x0, x1)
U10_gg(x0)
U11_gg(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:

U6_GGAA(X, L, Y, less_out_gg) → SPLIT_IN_GGAA(L, Y)
The graph contains the following edges 2 >= 1, 3 >= 2
U8_GGAA(X, L, Y, geq_out_gg) → SPLIT_IN_GGAA(L, Y)
The graph contains the following edges 2 >= 1, 3 >= 2
SPLIT_IN_GGAA(cons(X, L), Y) → U8_GGAA(X, L, Y, geq_in_gg(X, Y))
The graph contains the following edges 1 > 1, 1 > 2, 2 >= 3
SPLIT_IN_GGAA(cons(X, L), Y) → U6_GGAA(X, L, Y, less_in_gg(X, Y))
The graph contains the following edges 1 > 1, 1 > 2, 2 >= 3

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ PiDPToQDPProof

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

QS_IN_GA(cons(X, L), S) → U1_GA(X, L, S, split_in_ggaa(L, X, L1, L2))
U1_GA(X, L, S, split_out_ggaa(L, X, L1, L2)) → U2_GA(X, L, S, L2, qs_in_ga(L1, S1))
U2_GA(X, L, S, L2, qs_out_ga(L1, S1)) → QS_IN_GA(L2, S2)
U1_GA(X, L, S, split_out_ggaa(L, X, L1, L2)) → QS_IN_GA(L1, S1)

The TRS R consists of the following rules:

qs_in_ga([], []) → qs_out_ga([], [])
qs_in_ga(cons(X, L), S) → U1_ga(X, L, S, split_in_ggaa(L, X, L1, L2))
split_in_ggaa([], X, [], []) → split_out_ggaa([], X, [], [])
split_in_ggaa(cons(X, L), Y, cons(X, L1), L2) → U6_ggaa(X, L, Y, L1, L2, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U10_gg(X, Y, less_in_gg(X, Y))
U10_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U6_ggaa(X, L, Y, L1, L2, less_out_gg(X, Y)) → U7_ggaa(X, L, Y, L1, L2, split_in_ggaa(L, Y, L1, L2))
split_in_ggaa(cons(X, L), Y, L1, cons(X, L2)) → U8_ggaa(X, L, Y, L1, L2, geq_in_gg(X, Y))
geq_in_gg(X, X) → geq_out_gg(X, X)
geq_in_gg(s(X), 0) → geq_out_gg(s(X), 0)
geq_in_gg(s(X), s(Y)) → U11_gg(X, Y, geq_in_gg(X, Y))
U11_gg(X, Y, geq_out_gg(X, Y)) → geq_out_gg(s(X), s(Y))
U8_ggaa(X, L, Y, L1, L2, geq_out_gg(X, Y)) → U9_ggaa(X, L, Y, L1, L2, split_in_ggaa(L, Y, L1, L2))
U9_ggaa(X, L, Y, L1, L2, split_out_ggaa(L, Y, L1, L2)) → split_out_ggaa(cons(X, L), Y, L1, cons(X, L2))
U7_ggaa(X, L, Y, L1, L2, split_out_ggaa(L, Y, L1, L2)) → split_out_ggaa(cons(X, L), Y, cons(X, L1), L2)
U1_ga(X, L, S, split_out_ggaa(L, X, L1, L2)) → U2_ga(X, L, S, L2, qs_in_ga(L1, S1))
U2_ga(X, L, S, L2, qs_out_ga(L1, S1)) → U3_ga(X, L, S, S1, qs_in_ga(L2, S2))
U3_ga(X, L, S, S1, qs_out_ga(L2, S2)) → U4_ga(X, L, S, append_in_gga(S1, cons(X, S2), S))
append_in_gga([], L, L) → append_out_gga([], L, L)
append_in_gga(cons(X, L1), L2, cons(X, L3)) → U5_gga(X, L1, L2, L3, append_in_gga(L1, L2, L3))
U5_gga(X, L1, L2, L3, append_out_gga(L1, L2, L3)) → append_out_gga(cons(X, L1), L2, cons(X, L3))
U4_ga(X, L, S, append_out_gga(S1, cons(X, S2), S)) → qs_out_ga(cons(X, L), S)

The argument filtering Pi contains the following mapping:
qs_in_ga(x1, x2) = qs_in_ga(x1)
[] = []
qs_out_ga(x1, x2) = qs_out_ga(x2)
cons(x1, x2) = cons(x1, x2)
U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4)
split_in_ggaa(x1, x2, x3, x4) = split_in_ggaa(x1, x2)
split_out_ggaa(x1, x2, x3, x4) = split_out_ggaa(x3, x4)
U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x1, x2, x3, x6)
less_in_gg(x1, x2) = less_in_gg(x1, x2)
0 = 0
s(x1) = s(x1)
less_out_gg(x1, x2) = less_out_gg
U10_gg(x1, x2, x3) = U10_gg(x3)
U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x1, x6)
U8_ggaa(x1, x2, x3, x4, x5, x6) = U8_ggaa(x1, x2, x3, x6)
geq_in_gg(x1, x2) = geq_in_gg(x1, x2)
geq_out_gg(x1, x2) = geq_out_gg
U11_gg(x1, x2, x3) = U11_gg(x3)
U9_ggaa(x1, x2, x3, x4, x5, x6) = U9_ggaa(x1, x6)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5)
U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4) = U4_ga(x4)
append_in_gga(x1, x2, x3) = append_in_gga(x1, x2)
append_out_gga(x1, x2, x3) = append_out_gga(x3)
U5_gga(x1, x2, x3, x4, x5) = U5_gga(x1, x5)
QS_IN_GA(x1, x2) = QS_IN_GA(x1)
U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4)
U2_GA(x1, x2, x3, x4, x5) = U2_GA(x1, x4, x5)

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

              ↳ PiDP

                ↳ PiDPToQDPProof

                  ↳ QDP

                    ↳ QDPOrderProof

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

U2_GA(X, L2, qs_out_ga(S1)) → QS_IN_GA(L2)
QS_IN_GA(cons(X, L)) → U1_GA(X, split_in_ggaa(L, X))
U1_GA(X, split_out_ggaa(L1, L2)) → QS_IN_GA(L1)
U1_GA(X, split_out_ggaa(L1, L2)) → U2_GA(X, L2, qs_in_ga(L1))

The TRS R consists of the following rules:

qs_in_ga([]) → qs_out_ga([])
qs_in_ga(cons(X, L)) → U1_ga(X, split_in_ggaa(L, X))
split_in_ggaa([], X) → split_out_ggaa([], [])
split_in_ggaa(cons(X, L), Y) → U6_ggaa(X, L, Y, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg
less_in_gg(s(X), s(Y)) → U10_gg(less_in_gg(X, Y))
U10_gg(less_out_gg) → less_out_gg
U6_ggaa(X, L, Y, less_out_gg) → U7_ggaa(X, split_in_ggaa(L, Y))
split_in_ggaa(cons(X, L), Y) → U8_ggaa(X, L, Y, geq_in_gg(X, Y))
geq_in_gg(X, X) → geq_out_gg
geq_in_gg(s(X), 0) → geq_out_gg
geq_in_gg(s(X), s(Y)) → U11_gg(geq_in_gg(X, Y))
U11_gg(geq_out_gg) → geq_out_gg
U8_ggaa(X, L, Y, geq_out_gg) → U9_ggaa(X, split_in_ggaa(L, Y))
U9_ggaa(X, split_out_ggaa(L1, L2)) → split_out_ggaa(L1, cons(X, L2))
U7_ggaa(X, split_out_ggaa(L1, L2)) → split_out_ggaa(cons(X, L1), L2)
U1_ga(X, split_out_ggaa(L1, L2)) → U2_ga(X, L2, qs_in_ga(L1))
U2_ga(X, L2, qs_out_ga(S1)) → U3_ga(X, S1, qs_in_ga(L2))
U3_ga(X, S1, qs_out_ga(S2)) → U4_ga(append_in_gga(S1, cons(X, S2)))
append_in_gga([], L) → append_out_gga(L)
append_in_gga(cons(X, L1), L2) → U5_gga(X, append_in_gga(L1, L2))
U5_gga(X, append_out_gga(L3)) → append_out_gga(cons(X, L3))
U4_ga(append_out_gga(S)) → qs_out_ga(S)

The set Q consists of the following terms:

qs_in_ga(x0)
split_in_ggaa(x0, x1)
less_in_gg(x0, x1)
U10_gg(x0)
U6_ggaa(x0, x1, x2, x3)
geq_in_gg(x0, x1)
U11_gg(x0)
U8_ggaa(x0, x1, x2, x3)
U9_ggaa(x0, x1)
U7_ggaa(x0, x1)
U1_ga(x0, x1)
U2_ga(x0, x1, x2)
U3_ga(x0, x1, x2)
append_in_gga(x0, x1)
U5_gga(x0, x1)
U4_ga(x0)

We have to consider all (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

U1_GA(X, split_out_ggaa(L1, L2)) → QS_IN_GA(L1)
U1_GA(X, split_out_ggaa(L1, L2)) → U2_GA(X, L2, qs_in_ga(L1))

The remaining pairs can at least be oriented weakly.

U2_GA(X, L2, qs_out_ga(S1)) → QS_IN_GA(L2)
QS_IN_GA(cons(X, L)) → U1_GA(X, split_in_ggaa(L, X))

Used ordering: Polynomial interpretation [25]:

POL(0) = 1
POL(QS_IN_GA(x₁)) = x₁
POL(U10_gg(x₁)) = 1
POL(U11_gg(x₁)) = 0
POL(U1_GA(x₁, x₂)) = 1 + x₂
POL(U1_ga(x₁, x₂)) = 0
POL(U2_GA(x₁, x₂, x₃)) = x₂
POL(U2_ga(x₁, x₂, x₃)) = 0
POL(U3_ga(x₁, x₂, x₃)) = 0
POL(U4_ga(x₁)) = 0
POL(U5_gga(x₁, x₂)) = x₂
POL(U6_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₂
POL(U7_ggaa(x₁, x₂)) = 1 + x₂
POL(U8_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₂
POL(U9_ggaa(x₁, x₂)) = 1 + x₂
POL([]) = 0
POL(append_in_gga(x₁, x₂)) = 1 + x₁ + x₂
POL(append_out_gga(x₁)) = 0
POL(cons(x₁, x₂)) = 1 + x₂
POL(geq_in_gg(x₁, x₂)) = 0
POL(geq_out_gg) = 0
POL(less_in_gg(x₁, x₂)) = 1 + x₁
POL(less_out_gg) = 1
POL(qs_in_ga(x₁)) = 0
POL(qs_out_ga(x₁)) = 0
POL(s(x₁)) = 1 + x₁
POL(split_in_ggaa(x₁, x₂)) = x₁
POL(split_out_ggaa(x₁, x₂)) = x₁ + x₂

The following usable rules [17] were oriented:

U7_ggaa(X, split_out_ggaa(L1, L2)) → split_out_ggaa(cons(X, L1), L2)
split_in_ggaa([], X) → split_out_ggaa([], [])
split_in_ggaa(cons(X, L), Y) → U6_ggaa(X, L, Y, less_in_gg(X, Y))
split_in_ggaa(cons(X, L), Y) → U8_ggaa(X, L, Y, geq_in_gg(X, Y))
U8_ggaa(X, L, Y, geq_out_gg) → U9_ggaa(X, split_in_ggaa(L, Y))
U6_ggaa(X, L, Y, less_out_gg) → U7_ggaa(X, split_in_ggaa(L, Y))
U9_ggaa(X, split_out_ggaa(L1, L2)) → split_out_ggaa(L1, cons(X, L2))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ PiDPToQDPProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

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

QS_IN_GA(cons(X, L)) → U1_GA(X, split_in_ggaa(L, X))
U2_GA(X, L2, qs_out_ga(S1)) → QS_IN_GA(L2)

The TRS R consists of the following rules:

qs_in_ga([]) → qs_out_ga([])
qs_in_ga(cons(X, L)) → U1_ga(X, split_in_ggaa(L, X))
split_in_ggaa([], X) → split_out_ggaa([], [])
split_in_ggaa(cons(X, L), Y) → U6_ggaa(X, L, Y, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg
less_in_gg(s(X), s(Y)) → U10_gg(less_in_gg(X, Y))
U10_gg(less_out_gg) → less_out_gg
U6_ggaa(X, L, Y, less_out_gg) → U7_ggaa(X, split_in_ggaa(L, Y))
split_in_ggaa(cons(X, L), Y) → U8_ggaa(X, L, Y, geq_in_gg(X, Y))
geq_in_gg(X, X) → geq_out_gg
geq_in_gg(s(X), 0) → geq_out_gg
geq_in_gg(s(X), s(Y)) → U11_gg(geq_in_gg(X, Y))
U11_gg(geq_out_gg) → geq_out_gg
U8_ggaa(X, L, Y, geq_out_gg) → U9_ggaa(X, split_in_ggaa(L, Y))
U9_ggaa(X, split_out_ggaa(L1, L2)) → split_out_ggaa(L1, cons(X, L2))
U7_ggaa(X, split_out_ggaa(L1, L2)) → split_out_ggaa(cons(X, L1), L2)
U1_ga(X, split_out_ggaa(L1, L2)) → U2_ga(X, L2, qs_in_ga(L1))
U2_ga(X, L2, qs_out_ga(S1)) → U3_ga(X, S1, qs_in_ga(L2))
U3_ga(X, S1, qs_out_ga(S2)) → U4_ga(append_in_gga(S1, cons(X, S2)))
append_in_gga([], L) → append_out_gga(L)
append_in_gga(cons(X, L1), L2) → U5_gga(X, append_in_gga(L1, L2))
U5_gga(X, append_out_gga(L3)) → append_out_gga(cons(X, L3))
U4_ga(append_out_gga(S)) → qs_out_ga(S)

The set Q consists of the following terms:

qs_in_ga(x0)
split_in_ggaa(x0, x1)
less_in_gg(x0, x1)
U10_gg(x0)
U6_ggaa(x0, x1, x2, x3)
geq_in_gg(x0, x1)
U11_gg(x0)
U8_ggaa(x0, x1, x2, x3)
U9_ggaa(x0, x1)
U7_ggaa(x0, x1)
U1_ga(x0, x1)
U2_ga(x0, x1, x2)
U3_ga(x0, x1, x2)
append_in_gga(x0, x1)
U5_gga(x0, x1)
U4_ga(x0)

We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 2 less nodes.