Left Termination proof of /tmp/tmpTVOrp2/pl8.3.1a.pl

Left Termination of the query pattern minsort_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

↳ Prolog

  ↳ PrologToPiTRSProof

Clauses:

minsort([], []).
minsort(L, .(X, L1)) :- ','(min1(X, L), ','(remove(X, L, L2), minsort(L2, L1))).
min1(M, .(X, L)) :- min2(X, M, L).
min2(X, X, []).
min2(X, A, .(M, L)) :- ','(min(X, M, B), min2(B, A, L)).
min(X, Y, X) :- le(X, Y).
min(X, Y, Y) :- gt(X, Y).
remove(N, .(N, L), L).
remove(N, .(M, L), .(M, L1)) :- ','(notEq(N, M), remove(N, L, L1)).
gt(s(X), s(Y)) :- gt(X, Y).
gt(s(X), 0).
le(s(X), s(Y)) :- le(X, Y).
le(0, s(Y)).
le(0, 0).
notEq(s(X), s(Y)) :- notEq(X, Y).
notEq(s(X), 0).
notEq(0, s(X)).

Queries:

minsort(g,a).

We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
minsort_in: (b,f)
min1_in: (f,b)
min2_in: (b,f,b)
min_in: (b,b,f)
le_in: (b,b)
gt_in: (b,b)
remove_in: (b,b,f)
notEq_in: (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

minsort_in_ga([], []) → minsort_out_ga([], [])
minsort_in_ga(L, .(X, L1)) → U1_ga(L, X, L1, min1_in_ag(X, L))
min1_in_ag(M, .(X, L)) → U4_ag(M, X, L, min2_in_gag(X, M, L))
min2_in_gag(X, X, []) → min2_out_gag(X, X, [])
min2_in_gag(X, A, .(M, L)) → U5_gag(X, A, M, L, min_in_gga(X, M, B))
min_in_gga(X, Y, X) → U7_gga(X, Y, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U12_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U12_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_gga(X, Y, le_out_gg(X, Y)) → min_out_gga(X, Y, X)
min_in_gga(X, Y, Y) → U8_gga(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U11_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U11_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U8_gga(X, Y, gt_out_gg(X, Y)) → min_out_gga(X, Y, Y)
U5_gag(X, A, M, L, min_out_gga(X, M, B)) → U6_gag(X, A, M, L, min2_in_gag(B, A, L))
U6_gag(X, A, M, L, min2_out_gag(B, A, L)) → min2_out_gag(X, A, .(M, L))
U4_ag(M, X, L, min2_out_gag(X, M, L)) → min1_out_ag(M, .(X, L))
U1_ga(L, X, L1, min1_out_ag(X, L)) → U2_ga(L, X, L1, remove_in_gga(X, L, L2))
remove_in_gga(N, .(N, L), L) → remove_out_gga(N, .(N, L), L)
remove_in_gga(N, .(M, L), .(M, L1)) → U9_gga(N, M, L, L1, notEq_in_gg(N, M))
notEq_in_gg(s(X), s(Y)) → U13_gg(X, Y, notEq_in_gg(X, Y))
notEq_in_gg(s(X), 0) → notEq_out_gg(s(X), 0)
notEq_in_gg(0, s(X)) → notEq_out_gg(0, s(X))
U13_gg(X, Y, notEq_out_gg(X, Y)) → notEq_out_gg(s(X), s(Y))
U9_gga(N, M, L, L1, notEq_out_gg(N, M)) → U10_gga(N, M, L, L1, remove_in_gga(N, L, L1))
U10_gga(N, M, L, L1, remove_out_gga(N, L, L1)) → remove_out_gga(N, .(M, L), .(M, L1))
U2_ga(L, X, L1, remove_out_gga(X, L, L2)) → U3_ga(L, X, L1, minsort_in_ga(L2, L1))
U3_ga(L, X, L1, minsort_out_ga(L2, L1)) → minsort_out_ga(L, .(X, L1))

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:

minsort_in_ga([], []) → minsort_out_ga([], [])
minsort_in_ga(L, .(X, L1)) → U1_ga(L, X, L1, min1_in_ag(X, L))
min1_in_ag(M, .(X, L)) → U4_ag(M, X, L, min2_in_gag(X, M, L))
min2_in_gag(X, X, []) → min2_out_gag(X, X, [])
min2_in_gag(X, A, .(M, L)) → U5_gag(X, A, M, L, min_in_gga(X, M, B))
min_in_gga(X, Y, X) → U7_gga(X, Y, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U12_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U12_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_gga(X, Y, le_out_gg(X, Y)) → min_out_gga(X, Y, X)
min_in_gga(X, Y, Y) → U8_gga(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U11_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U11_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U8_gga(X, Y, gt_out_gg(X, Y)) → min_out_gga(X, Y, Y)
U5_gag(X, A, M, L, min_out_gga(X, M, B)) → U6_gag(X, A, M, L, min2_in_gag(B, A, L))
U6_gag(X, A, M, L, min2_out_gag(B, A, L)) → min2_out_gag(X, A, .(M, L))
U4_ag(M, X, L, min2_out_gag(X, M, L)) → min1_out_ag(M, .(X, L))
U1_ga(L, X, L1, min1_out_ag(X, L)) → U2_ga(L, X, L1, remove_in_gga(X, L, L2))
remove_in_gga(N, .(N, L), L) → remove_out_gga(N, .(N, L), L)
remove_in_gga(N, .(M, L), .(M, L1)) → U9_gga(N, M, L, L1, notEq_in_gg(N, M))
notEq_in_gg(s(X), s(Y)) → U13_gg(X, Y, notEq_in_gg(X, Y))
notEq_in_gg(s(X), 0) → notEq_out_gg(s(X), 0)
notEq_in_gg(0, s(X)) → notEq_out_gg(0, s(X))
U13_gg(X, Y, notEq_out_gg(X, Y)) → notEq_out_gg(s(X), s(Y))
U9_gga(N, M, L, L1, notEq_out_gg(N, M)) → U10_gga(N, M, L, L1, remove_in_gga(N, L, L1))
U10_gga(N, M, L, L1, remove_out_gga(N, L, L1)) → remove_out_gga(N, .(M, L), .(M, L1))
U2_ga(L, X, L1, remove_out_gga(X, L, L2)) → U3_ga(L, X, L1, minsort_in_ga(L2, L1))
U3_ga(L, X, L1, minsort_out_ga(L2, L1)) → minsort_out_ga(L, .(X, L1))

MINSORT_IN_GA(L, .(X, L1)) → U1_GA(L, X, L1, min1_in_ag(X, L))
MINSORT_IN_GA(L, .(X, L1)) → MIN1_IN_AG(X, L)
MIN1_IN_AG(M, .(X, L)) → U4_AG(M, X, L, min2_in_gag(X, M, L))
MIN1_IN_AG(M, .(X, L)) → MIN2_IN_GAG(X, M, L)
MIN2_IN_GAG(X, A, .(M, L)) → U5_GAG(X, A, M, L, min_in_gga(X, M, B))
MIN2_IN_GAG(X, A, .(M, L)) → MIN_IN_GGA(X, M, B)
MIN_IN_GGA(X, Y, X) → U7_GGA(X, Y, le_in_gg(X, Y))
MIN_IN_GGA(X, Y, X) → LE_IN_GG(X, Y)
LE_IN_GG(s(X), s(Y)) → U12_GG(X, Y, le_in_gg(X, Y))
LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)
MIN_IN_GGA(X, Y, Y) → U8_GGA(X, Y, gt_in_gg(X, Y))
MIN_IN_GGA(X, Y, Y) → GT_IN_GG(X, Y)
GT_IN_GG(s(X), s(Y)) → U11_GG(X, Y, gt_in_gg(X, Y))
GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)
U5_GAG(X, A, M, L, min_out_gga(X, M, B)) → U6_GAG(X, A, M, L, min2_in_gag(B, A, L))
U5_GAG(X, A, M, L, min_out_gga(X, M, B)) → MIN2_IN_GAG(B, A, L)
U1_GA(L, X, L1, min1_out_ag(X, L)) → U2_GA(L, X, L1, remove_in_gga(X, L, L2))
U1_GA(L, X, L1, min1_out_ag(X, L)) → REMOVE_IN_GGA(X, L, L2)
REMOVE_IN_GGA(N, .(M, L), .(M, L1)) → U9_GGA(N, M, L, L1, notEq_in_gg(N, M))
REMOVE_IN_GGA(N, .(M, L), .(M, L1)) → NOTEQ_IN_GG(N, M)
NOTEQ_IN_GG(s(X), s(Y)) → U13_GG(X, Y, notEq_in_gg(X, Y))
NOTEQ_IN_GG(s(X), s(Y)) → NOTEQ_IN_GG(X, Y)
U9_GGA(N, M, L, L1, notEq_out_gg(N, M)) → U10_GGA(N, M, L, L1, remove_in_gga(N, L, L1))
U9_GGA(N, M, L, L1, notEq_out_gg(N, M)) → REMOVE_IN_GGA(N, L, L1)
U2_GA(L, X, L1, remove_out_gga(X, L, L2)) → U3_GA(L, X, L1, minsort_in_ga(L2, L1))
U2_GA(L, X, L1, remove_out_gga(X, L, L2)) → MINSORT_IN_GA(L2, L1)

The TRS R consists of the following rules:

minsort_in_ga([], []) → minsort_out_ga([], [])
minsort_in_ga(L, .(X, L1)) → U1_ga(L, X, L1, min1_in_ag(X, L))
min1_in_ag(M, .(X, L)) → U4_ag(M, X, L, min2_in_gag(X, M, L))
min2_in_gag(X, X, []) → min2_out_gag(X, X, [])
min2_in_gag(X, A, .(M, L)) → U5_gag(X, A, M, L, min_in_gga(X, M, B))
min_in_gga(X, Y, X) → U7_gga(X, Y, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U12_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U12_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_gga(X, Y, le_out_gg(X, Y)) → min_out_gga(X, Y, X)
min_in_gga(X, Y, Y) → U8_gga(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U11_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U11_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U8_gga(X, Y, gt_out_gg(X, Y)) → min_out_gga(X, Y, Y)
U5_gag(X, A, M, L, min_out_gga(X, M, B)) → U6_gag(X, A, M, L, min2_in_gag(B, A, L))
U6_gag(X, A, M, L, min2_out_gag(B, A, L)) → min2_out_gag(X, A, .(M, L))
U4_ag(M, X, L, min2_out_gag(X, M, L)) → min1_out_ag(M, .(X, L))
U1_ga(L, X, L1, min1_out_ag(X, L)) → U2_ga(L, X, L1, remove_in_gga(X, L, L2))
remove_in_gga(N, .(N, L), L) → remove_out_gga(N, .(N, L), L)
remove_in_gga(N, .(M, L), .(M, L1)) → U9_gga(N, M, L, L1, notEq_in_gg(N, M))
notEq_in_gg(s(X), s(Y)) → U13_gg(X, Y, notEq_in_gg(X, Y))
notEq_in_gg(s(X), 0) → notEq_out_gg(s(X), 0)
notEq_in_gg(0, s(X)) → notEq_out_gg(0, s(X))
U13_gg(X, Y, notEq_out_gg(X, Y)) → notEq_out_gg(s(X), s(Y))
U9_gga(N, M, L, L1, notEq_out_gg(N, M)) → U10_gga(N, M, L, L1, remove_in_gga(N, L, L1))
U10_gga(N, M, L, L1, remove_out_gga(N, L, L1)) → remove_out_gga(N, .(M, L), .(M, L1))
U2_ga(L, X, L1, remove_out_gga(X, L, L2)) → U3_ga(L, X, L1, minsort_in_ga(L2, L1))
U3_ga(L, X, L1, minsort_out_ga(L2, L1)) → minsort_out_ga(L, .(X, L1))

The argument filtering Pi contains the following mapping:
minsort_in_ga(x1, x2) = minsort_in_ga(x1)
[] = []
minsort_out_ga(x1, x2) = minsort_out_ga(x2)
U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4)
min1_in_ag(x1, x2) = min1_in_ag(x2)
.(x1, x2) = .(x1, x2)
U4_ag(x1, x2, x3, x4) = U4_ag(x4)
min2_in_gag(x1, x2, x3) = min2_in_gag(x1, x3)
min2_out_gag(x1, x2, x3) = min2_out_gag(x2)
U5_gag(x1, x2, x3, x4, x5) = U5_gag(x4, x5)
min_in_gga(x1, x2, x3) = min_in_gga(x1, x2)
U7_gga(x1, x2, x3) = U7_gga(x1, x3)
le_in_gg(x1, x2) = le_in_gg(x1, x2)
s(x1) = s(x1)
U12_gg(x1, x2, x3) = U12_gg(x3)
0 = 0
le_out_gg(x1, x2) = le_out_gg
min_out_gga(x1, x2, x3) = min_out_gga(x3)
U8_gga(x1, x2, x3) = U8_gga(x2, x3)
gt_in_gg(x1, x2) = gt_in_gg(x1, x2)
U11_gg(x1, x2, x3) = U11_gg(x3)
gt_out_gg(x1, x2) = gt_out_gg
U6_gag(x1, x2, x3, x4, x5) = U6_gag(x5)
min1_out_ag(x1, x2) = min1_out_ag(x1)
U2_ga(x1, x2, x3, x4) = U2_ga(x2, x4)
remove_in_gga(x1, x2, x3) = remove_in_gga(x1, x2)
remove_out_gga(x1, x2, x3) = remove_out_gga(x3)
U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x2, x3, x5)
notEq_in_gg(x1, x2) = notEq_in_gg(x1, x2)
U13_gg(x1, x2, x3) = U13_gg(x3)
notEq_out_gg(x1, x2) = notEq_out_gg
U10_gga(x1, x2, x3, x4, x5) = U10_gga(x2, x5)
U3_ga(x1, x2, x3, x4) = U3_ga(x2, x4)
MIN1_IN_AG(x1, x2) = MIN1_IN_AG(x2)
MIN_IN_GGA(x1, x2, x3) = MIN_IN_GGA(x1, x2)
U5_GAG(x1, x2, x3, x4, x5) = U5_GAG(x4, x5)
LE_IN_GG(x1, x2) = LE_IN_GG(x1, x2)
U9_GGA(x1, x2, x3, x4, x5) = U9_GGA(x1, x2, x3, x5)
MIN2_IN_GAG(x1, x2, x3) = MIN2_IN_GAG(x1, x3)
U8_GGA(x1, x2, x3) = U8_GGA(x2, x3)
U3_GA(x1, x2, x3, x4) = U3_GA(x2, x4)
U2_GA(x1, x2, x3, x4) = U2_GA(x2, x4)
U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4)
U7_GGA(x1, x2, x3) = U7_GGA(x1, x3)
U12_GG(x1, x2, x3) = U12_GG(x3)
U10_GGA(x1, x2, x3, x4, x5) = U10_GGA(x2, x5)
U6_GAG(x1, x2, x3, x4, x5) = U6_GAG(x5)
U13_GG(x1, x2, x3) = U13_GG(x3)
NOTEQ_IN_GG(x1, x2) = NOTEQ_IN_GG(x1, x2)
MINSORT_IN_GA(x1, x2) = MINSORT_IN_GA(x1)
U4_AG(x1, x2, x3, x4) = U4_AG(x4)
U11_GG(x1, x2, x3) = U11_GG(x3)
REMOVE_IN_GGA(x1, x2, x3) = REMOVE_IN_GGA(x1, x2)
GT_IN_GG(x1, x2) = GT_IN_GG(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:

MINSORT_IN_GA(L, .(X, L1)) → U1_GA(L, X, L1, min1_in_ag(X, L))
MINSORT_IN_GA(L, .(X, L1)) → MIN1_IN_AG(X, L)
MIN1_IN_AG(M, .(X, L)) → U4_AG(M, X, L, min2_in_gag(X, M, L))
MIN1_IN_AG(M, .(X, L)) → MIN2_IN_GAG(X, M, L)
MIN2_IN_GAG(X, A, .(M, L)) → U5_GAG(X, A, M, L, min_in_gga(X, M, B))
MIN2_IN_GAG(X, A, .(M, L)) → MIN_IN_GGA(X, M, B)
MIN_IN_GGA(X, Y, X) → U7_GGA(X, Y, le_in_gg(X, Y))
MIN_IN_GGA(X, Y, X) → LE_IN_GG(X, Y)
LE_IN_GG(s(X), s(Y)) → U12_GG(X, Y, le_in_gg(X, Y))
LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)
MIN_IN_GGA(X, Y, Y) → U8_GGA(X, Y, gt_in_gg(X, Y))
MIN_IN_GGA(X, Y, Y) → GT_IN_GG(X, Y)
GT_IN_GG(s(X), s(Y)) → U11_GG(X, Y, gt_in_gg(X, Y))
GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)
U5_GAG(X, A, M, L, min_out_gga(X, M, B)) → U6_GAG(X, A, M, L, min2_in_gag(B, A, L))
U5_GAG(X, A, M, L, min_out_gga(X, M, B)) → MIN2_IN_GAG(B, A, L)
U1_GA(L, X, L1, min1_out_ag(X, L)) → U2_GA(L, X, L1, remove_in_gga(X, L, L2))
U1_GA(L, X, L1, min1_out_ag(X, L)) → REMOVE_IN_GGA(X, L, L2)
REMOVE_IN_GGA(N, .(M, L), .(M, L1)) → U9_GGA(N, M, L, L1, notEq_in_gg(N, M))
REMOVE_IN_GGA(N, .(M, L), .(M, L1)) → NOTEQ_IN_GG(N, M)
NOTEQ_IN_GG(s(X), s(Y)) → U13_GG(X, Y, notEq_in_gg(X, Y))
NOTEQ_IN_GG(s(X), s(Y)) → NOTEQ_IN_GG(X, Y)
U9_GGA(N, M, L, L1, notEq_out_gg(N, M)) → U10_GGA(N, M, L, L1, remove_in_gga(N, L, L1))
U9_GGA(N, M, L, L1, notEq_out_gg(N, M)) → REMOVE_IN_GGA(N, L, L1)
U2_GA(L, X, L1, remove_out_gga(X, L, L2)) → U3_GA(L, X, L1, minsort_in_ga(L2, L1))
U2_GA(L, X, L1, remove_out_gga(X, L, L2)) → MINSORT_IN_GA(L2, L1)

The TRS R consists of the following rules:

minsort_in_ga([], []) → minsort_out_ga([], [])
minsort_in_ga(L, .(X, L1)) → U1_ga(L, X, L1, min1_in_ag(X, L))
min1_in_ag(M, .(X, L)) → U4_ag(M, X, L, min2_in_gag(X, M, L))
min2_in_gag(X, X, []) → min2_out_gag(X, X, [])
min2_in_gag(X, A, .(M, L)) → U5_gag(X, A, M, L, min_in_gga(X, M, B))
min_in_gga(X, Y, X) → U7_gga(X, Y, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U12_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U12_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_gga(X, Y, le_out_gg(X, Y)) → min_out_gga(X, Y, X)
min_in_gga(X, Y, Y) → U8_gga(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U11_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U11_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U8_gga(X, Y, gt_out_gg(X, Y)) → min_out_gga(X, Y, Y)
U5_gag(X, A, M, L, min_out_gga(X, M, B)) → U6_gag(X, A, M, L, min2_in_gag(B, A, L))
U6_gag(X, A, M, L, min2_out_gag(B, A, L)) → min2_out_gag(X, A, .(M, L))
U4_ag(M, X, L, min2_out_gag(X, M, L)) → min1_out_ag(M, .(X, L))
U1_ga(L, X, L1, min1_out_ag(X, L)) → U2_ga(L, X, L1, remove_in_gga(X, L, L2))
remove_in_gga(N, .(N, L), L) → remove_out_gga(N, .(N, L), L)
remove_in_gga(N, .(M, L), .(M, L1)) → U9_gga(N, M, L, L1, notEq_in_gg(N, M))
notEq_in_gg(s(X), s(Y)) → U13_gg(X, Y, notEq_in_gg(X, Y))
notEq_in_gg(s(X), 0) → notEq_out_gg(s(X), 0)
notEq_in_gg(0, s(X)) → notEq_out_gg(0, s(X))
U13_gg(X, Y, notEq_out_gg(X, Y)) → notEq_out_gg(s(X), s(Y))
U9_gga(N, M, L, L1, notEq_out_gg(N, M)) → U10_gga(N, M, L, L1, remove_in_gga(N, L, L1))
U10_gga(N, M, L, L1, remove_out_gga(N, L, L1)) → remove_out_gga(N, .(M, L), .(M, L1))
U2_ga(L, X, L1, remove_out_gga(X, L, L2)) → U3_ga(L, X, L1, minsort_in_ga(L2, L1))
U3_ga(L, X, L1, minsort_out_ga(L2, L1)) → minsort_out_ga(L, .(X, L1))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

NOTEQ_IN_GG(s(X), s(Y)) → NOTEQ_IN_GG(X, Y)

The TRS R consists of the following rules:

minsort_in_ga([], []) → minsort_out_ga([], [])
minsort_in_ga(L, .(X, L1)) → U1_ga(L, X, L1, min1_in_ag(X, L))
min1_in_ag(M, .(X, L)) → U4_ag(M, X, L, min2_in_gag(X, M, L))
min2_in_gag(X, X, []) → min2_out_gag(X, X, [])
min2_in_gag(X, A, .(M, L)) → U5_gag(X, A, M, L, min_in_gga(X, M, B))
min_in_gga(X, Y, X) → U7_gga(X, Y, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U12_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U12_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_gga(X, Y, le_out_gg(X, Y)) → min_out_gga(X, Y, X)
min_in_gga(X, Y, Y) → U8_gga(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U11_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U11_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U8_gga(X, Y, gt_out_gg(X, Y)) → min_out_gga(X, Y, Y)
U5_gag(X, A, M, L, min_out_gga(X, M, B)) → U6_gag(X, A, M, L, min2_in_gag(B, A, L))
U6_gag(X, A, M, L, min2_out_gag(B, A, L)) → min2_out_gag(X, A, .(M, L))
U4_ag(M, X, L, min2_out_gag(X, M, L)) → min1_out_ag(M, .(X, L))
U1_ga(L, X, L1, min1_out_ag(X, L)) → U2_ga(L, X, L1, remove_in_gga(X, L, L2))
remove_in_gga(N, .(N, L), L) → remove_out_gga(N, .(N, L), L)
remove_in_gga(N, .(M, L), .(M, L1)) → U9_gga(N, M, L, L1, notEq_in_gg(N, M))
notEq_in_gg(s(X), s(Y)) → U13_gg(X, Y, notEq_in_gg(X, Y))
notEq_in_gg(s(X), 0) → notEq_out_gg(s(X), 0)
notEq_in_gg(0, s(X)) → notEq_out_gg(0, s(X))
U13_gg(X, Y, notEq_out_gg(X, Y)) → notEq_out_gg(s(X), s(Y))
U9_gga(N, M, L, L1, notEq_out_gg(N, M)) → U10_gga(N, M, L, L1, remove_in_gga(N, L, L1))
U10_gga(N, M, L, L1, remove_out_gga(N, L, L1)) → remove_out_gga(N, .(M, L), .(M, L1))
U2_ga(L, X, L1, remove_out_gga(X, L, L2)) → U3_ga(L, X, L1, minsort_in_ga(L2, L1))
U3_ga(L, X, L1, minsort_out_ga(L2, L1)) → minsort_out_ga(L, .(X, L1))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

NOTEQ_IN_GG(s(X), s(Y)) → NOTEQ_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

              ↳ PiDP

              ↳ PiDP

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

NOTEQ_IN_GG(s(X), s(Y)) → NOTEQ_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:

NOTEQ_IN_GG(s(X), s(Y)) → NOTEQ_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

              ↳ PiDP

              ↳ PiDP

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

REMOVE_IN_GGA(N, .(M, L), .(M, L1)) → U9_GGA(N, M, L, L1, notEq_in_gg(N, M))
U9_GGA(N, M, L, L1, notEq_out_gg(N, M)) → REMOVE_IN_GGA(N, L, L1)

The TRS R consists of the following rules:

minsort_in_ga([], []) → minsort_out_ga([], [])
minsort_in_ga(L, .(X, L1)) → U1_ga(L, X, L1, min1_in_ag(X, L))
min1_in_ag(M, .(X, L)) → U4_ag(M, X, L, min2_in_gag(X, M, L))
min2_in_gag(X, X, []) → min2_out_gag(X, X, [])
min2_in_gag(X, A, .(M, L)) → U5_gag(X, A, M, L, min_in_gga(X, M, B))
min_in_gga(X, Y, X) → U7_gga(X, Y, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U12_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U12_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_gga(X, Y, le_out_gg(X, Y)) → min_out_gga(X, Y, X)
min_in_gga(X, Y, Y) → U8_gga(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U11_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U11_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U8_gga(X, Y, gt_out_gg(X, Y)) → min_out_gga(X, Y, Y)
U5_gag(X, A, M, L, min_out_gga(X, M, B)) → U6_gag(X, A, M, L, min2_in_gag(B, A, L))
U6_gag(X, A, M, L, min2_out_gag(B, A, L)) → min2_out_gag(X, A, .(M, L))
U4_ag(M, X, L, min2_out_gag(X, M, L)) → min1_out_ag(M, .(X, L))
U1_ga(L, X, L1, min1_out_ag(X, L)) → U2_ga(L, X, L1, remove_in_gga(X, L, L2))
remove_in_gga(N, .(N, L), L) → remove_out_gga(N, .(N, L), L)
remove_in_gga(N, .(M, L), .(M, L1)) → U9_gga(N, M, L, L1, notEq_in_gg(N, M))
notEq_in_gg(s(X), s(Y)) → U13_gg(X, Y, notEq_in_gg(X, Y))
notEq_in_gg(s(X), 0) → notEq_out_gg(s(X), 0)
notEq_in_gg(0, s(X)) → notEq_out_gg(0, s(X))
U13_gg(X, Y, notEq_out_gg(X, Y)) → notEq_out_gg(s(X), s(Y))
U9_gga(N, M, L, L1, notEq_out_gg(N, M)) → U10_gga(N, M, L, L1, remove_in_gga(N, L, L1))
U10_gga(N, M, L, L1, remove_out_gga(N, L, L1)) → remove_out_gga(N, .(M, L), .(M, L1))
U2_ga(L, X, L1, remove_out_gga(X, L, L2)) → U3_ga(L, X, L1, minsort_in_ga(L2, L1))
U3_ga(L, X, L1, minsort_out_ga(L2, L1)) → minsort_out_ga(L, .(X, L1))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

REMOVE_IN_GGA(N, .(M, L), .(M, L1)) → U9_GGA(N, M, L, L1, notEq_in_gg(N, M))
U9_GGA(N, M, L, L1, notEq_out_gg(N, M)) → REMOVE_IN_GGA(N, L, L1)

The TRS R consists of the following rules:

notEq_in_gg(s(X), s(Y)) → U13_gg(X, Y, notEq_in_gg(X, Y))
notEq_in_gg(s(X), 0) → notEq_out_gg(s(X), 0)
notEq_in_gg(0, s(X)) → notEq_out_gg(0, s(X))
U13_gg(X, Y, notEq_out_gg(X, Y)) → notEq_out_gg(s(X), s(Y))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
s(x1) = s(x1)
0 = 0
notEq_in_gg(x1, x2) = notEq_in_gg(x1, x2)
U13_gg(x1, x2, x3) = U13_gg(x3)
notEq_out_gg(x1, x2) = notEq_out_gg
U9_GGA(x1, x2, x3, x4, x5) = U9_GGA(x1, x2, x3, x5)
REMOVE_IN_GGA(x1, x2, x3) = REMOVE_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

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

U9_GGA(N, M, L, notEq_out_gg) → REMOVE_IN_GGA(N, L)
REMOVE_IN_GGA(N, .(M, L)) → U9_GGA(N, M, L, notEq_in_gg(N, M))

The TRS R consists of the following rules:

notEq_in_gg(s(X), s(Y)) → U13_gg(notEq_in_gg(X, Y))
notEq_in_gg(s(X), 0) → notEq_out_gg
notEq_in_gg(0, s(X)) → notEq_out_gg
U13_gg(notEq_out_gg) → notEq_out_gg

The set Q consists of the following terms:

notEq_in_gg(x0, x1)
U13_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:

U9_GGA(N, M, L, notEq_out_gg) → REMOVE_IN_GGA(N, L)
The graph contains the following edges 1 >= 1, 3 >= 2
REMOVE_IN_GGA(N, .(M, L)) → U9_GGA(N, M, L, notEq_in_gg(N, M))
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)

The TRS R consists of the following rules:

minsort_in_ga([], []) → minsort_out_ga([], [])
minsort_in_ga(L, .(X, L1)) → U1_ga(L, X, L1, min1_in_ag(X, L))
min1_in_ag(M, .(X, L)) → U4_ag(M, X, L, min2_in_gag(X, M, L))
min2_in_gag(X, X, []) → min2_out_gag(X, X, [])
min2_in_gag(X, A, .(M, L)) → U5_gag(X, A, M, L, min_in_gga(X, M, B))
min_in_gga(X, Y, X) → U7_gga(X, Y, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U12_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U12_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_gga(X, Y, le_out_gg(X, Y)) → min_out_gga(X, Y, X)
min_in_gga(X, Y, Y) → U8_gga(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U11_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U11_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U8_gga(X, Y, gt_out_gg(X, Y)) → min_out_gga(X, Y, Y)
U5_gag(X, A, M, L, min_out_gga(X, M, B)) → U6_gag(X, A, M, L, min2_in_gag(B, A, L))
U6_gag(X, A, M, L, min2_out_gag(B, A, L)) → min2_out_gag(X, A, .(M, L))
U4_ag(M, X, L, min2_out_gag(X, M, L)) → min1_out_ag(M, .(X, L))
U1_ga(L, X, L1, min1_out_ag(X, L)) → U2_ga(L, X, L1, remove_in_gga(X, L, L2))
remove_in_gga(N, .(N, L), L) → remove_out_gga(N, .(N, L), L)
remove_in_gga(N, .(M, L), .(M, L1)) → U9_gga(N, M, L, L1, notEq_in_gg(N, M))
notEq_in_gg(s(X), s(Y)) → U13_gg(X, Y, notEq_in_gg(X, Y))
notEq_in_gg(s(X), 0) → notEq_out_gg(s(X), 0)
notEq_in_gg(0, s(X)) → notEq_out_gg(0, s(X))
U13_gg(X, Y, notEq_out_gg(X, Y)) → notEq_out_gg(s(X), s(Y))
U9_gga(N, M, L, L1, notEq_out_gg(N, M)) → U10_gga(N, M, L, L1, remove_in_gga(N, L, L1))
U10_gga(N, M, L, L1, remove_out_gga(N, L, L1)) → remove_out_gga(N, .(M, L), .(M, L1))
U2_ga(L, X, L1, remove_out_gga(X, L, L2)) → U3_ga(L, X, L1, minsort_in_ga(L2, L1))
U3_ga(L, X, L1, minsort_out_ga(L2, L1)) → minsort_out_ga(L, .(X, L1))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

GT_IN_GG(s(X), s(Y)) → GT_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

              ↳ PiDP

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

GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)

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

GT_IN_GG(s(X), s(Y)) → GT_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

              ↳ PiDP

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

LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)

The TRS R consists of the following rules:

minsort_in_ga([], []) → minsort_out_ga([], [])
minsort_in_ga(L, .(X, L1)) → U1_ga(L, X, L1, min1_in_ag(X, L))
min1_in_ag(M, .(X, L)) → U4_ag(M, X, L, min2_in_gag(X, M, L))
min2_in_gag(X, X, []) → min2_out_gag(X, X, [])
min2_in_gag(X, A, .(M, L)) → U5_gag(X, A, M, L, min_in_gga(X, M, B))
min_in_gga(X, Y, X) → U7_gga(X, Y, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U12_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U12_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_gga(X, Y, le_out_gg(X, Y)) → min_out_gga(X, Y, X)
min_in_gga(X, Y, Y) → U8_gga(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U11_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U11_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U8_gga(X, Y, gt_out_gg(X, Y)) → min_out_gga(X, Y, Y)
U5_gag(X, A, M, L, min_out_gga(X, M, B)) → U6_gag(X, A, M, L, min2_in_gag(B, A, L))
U6_gag(X, A, M, L, min2_out_gag(B, A, L)) → min2_out_gag(X, A, .(M, L))
U4_ag(M, X, L, min2_out_gag(X, M, L)) → min1_out_ag(M, .(X, L))
U1_ga(L, X, L1, min1_out_ag(X, L)) → U2_ga(L, X, L1, remove_in_gga(X, L, L2))
remove_in_gga(N, .(N, L), L) → remove_out_gga(N, .(N, L), L)
remove_in_gga(N, .(M, L), .(M, L1)) → U9_gga(N, M, L, L1, notEq_in_gg(N, M))
notEq_in_gg(s(X), s(Y)) → U13_gg(X, Y, notEq_in_gg(X, Y))
notEq_in_gg(s(X), 0) → notEq_out_gg(s(X), 0)
notEq_in_gg(0, s(X)) → notEq_out_gg(0, s(X))
U13_gg(X, Y, notEq_out_gg(X, Y)) → notEq_out_gg(s(X), s(Y))
U9_gga(N, M, L, L1, notEq_out_gg(N, M)) → U10_gga(N, M, L, L1, remove_in_gga(N, L, L1))
U10_gga(N, M, L, L1, remove_out_gga(N, L, L1)) → remove_out_gga(N, .(M, L), .(M, L1))
U2_ga(L, X, L1, remove_out_gga(X, L, L2)) → U3_ga(L, X, L1, minsort_in_ga(L2, L1))
U3_ga(L, X, L1, minsort_out_ga(L2, L1)) → minsort_out_ga(L, .(X, L1))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

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

LE_IN_GG(s(X), s(Y)) → LE_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

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

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

LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)

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

LE_IN_GG(s(X), s(Y)) → LE_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

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

MIN2_IN_GAG(X, A, .(M, L)) → U5_GAG(X, A, M, L, min_in_gga(X, M, B))
U5_GAG(X, A, M, L, min_out_gga(X, M, B)) → MIN2_IN_GAG(B, A, L)

The TRS R consists of the following rules:

minsort_in_ga([], []) → minsort_out_ga([], [])
minsort_in_ga(L, .(X, L1)) → U1_ga(L, X, L1, min1_in_ag(X, L))
min1_in_ag(M, .(X, L)) → U4_ag(M, X, L, min2_in_gag(X, M, L))
min2_in_gag(X, X, []) → min2_out_gag(X, X, [])
min2_in_gag(X, A, .(M, L)) → U5_gag(X, A, M, L, min_in_gga(X, M, B))
min_in_gga(X, Y, X) → U7_gga(X, Y, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U12_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U12_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_gga(X, Y, le_out_gg(X, Y)) → min_out_gga(X, Y, X)
min_in_gga(X, Y, Y) → U8_gga(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U11_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U11_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U8_gga(X, Y, gt_out_gg(X, Y)) → min_out_gga(X, Y, Y)
U5_gag(X, A, M, L, min_out_gga(X, M, B)) → U6_gag(X, A, M, L, min2_in_gag(B, A, L))
U6_gag(X, A, M, L, min2_out_gag(B, A, L)) → min2_out_gag(X, A, .(M, L))
U4_ag(M, X, L, min2_out_gag(X, M, L)) → min1_out_ag(M, .(X, L))
U1_ga(L, X, L1, min1_out_ag(X, L)) → U2_ga(L, X, L1, remove_in_gga(X, L, L2))
remove_in_gga(N, .(N, L), L) → remove_out_gga(N, .(N, L), L)
remove_in_gga(N, .(M, L), .(M, L1)) → U9_gga(N, M, L, L1, notEq_in_gg(N, M))
notEq_in_gg(s(X), s(Y)) → U13_gg(X, Y, notEq_in_gg(X, Y))
notEq_in_gg(s(X), 0) → notEq_out_gg(s(X), 0)
notEq_in_gg(0, s(X)) → notEq_out_gg(0, s(X))
U13_gg(X, Y, notEq_out_gg(X, Y)) → notEq_out_gg(s(X), s(Y))
U9_gga(N, M, L, L1, notEq_out_gg(N, M)) → U10_gga(N, M, L, L1, remove_in_gga(N, L, L1))
U10_gga(N, M, L, L1, remove_out_gga(N, L, L1)) → remove_out_gga(N, .(M, L), .(M, L1))
U2_ga(L, X, L1, remove_out_gga(X, L, L2)) → U3_ga(L, X, L1, minsort_in_ga(L2, L1))
U3_ga(L, X, L1, minsort_out_ga(L2, L1)) → minsort_out_ga(L, .(X, L1))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

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

MIN2_IN_GAG(X, A, .(M, L)) → U5_GAG(X, A, M, L, min_in_gga(X, M, B))
U5_GAG(X, A, M, L, min_out_gga(X, M, B)) → MIN2_IN_GAG(B, A, L)

The TRS R consists of the following rules:

min_in_gga(X, Y, X) → U7_gga(X, Y, le_in_gg(X, Y))
min_in_gga(X, Y, Y) → U8_gga(X, Y, gt_in_gg(X, Y))
U7_gga(X, Y, le_out_gg(X, Y)) → min_out_gga(X, Y, X)
U8_gga(X, Y, gt_out_gg(X, Y)) → min_out_gga(X, Y, Y)
le_in_gg(s(X), s(Y)) → U12_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
gt_in_gg(s(X), s(Y)) → U11_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U12_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U11_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
min_in_gga(x1, x2, x3) = min_in_gga(x1, x2)
U7_gga(x1, x2, x3) = U7_gga(x1, x3)
le_in_gg(x1, x2) = le_in_gg(x1, x2)
s(x1) = s(x1)
U12_gg(x1, x2, x3) = U12_gg(x3)
0 = 0
le_out_gg(x1, x2) = le_out_gg
min_out_gga(x1, x2, x3) = min_out_gga(x3)
U8_gga(x1, x2, x3) = U8_gga(x2, x3)
gt_in_gg(x1, x2) = gt_in_gg(x1, x2)
U11_gg(x1, x2, x3) = U11_gg(x3)
gt_out_gg(x1, x2) = gt_out_gg
U5_GAG(x1, x2, x3, x4, x5) = U5_GAG(x4, x5)
MIN2_IN_GAG(x1, x2, x3) = MIN2_IN_GAG(x1, x3)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

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

MIN2_IN_GAG(X, .(M, L)) → U5_GAG(L, min_in_gga(X, M))
U5_GAG(L, min_out_gga(B)) → MIN2_IN_GAG(B, L)

The TRS R consists of the following rules:

min_in_gga(X, Y) → U7_gga(X, le_in_gg(X, Y))
min_in_gga(X, Y) → U8_gga(Y, gt_in_gg(X, Y))
U7_gga(X, le_out_gg) → min_out_gga(X)
U8_gga(Y, gt_out_gg) → min_out_gga(Y)
le_in_gg(s(X), s(Y)) → U12_gg(le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
gt_in_gg(s(X), s(Y)) → U11_gg(gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg
U12_gg(le_out_gg) → le_out_gg
U11_gg(gt_out_gg) → gt_out_gg

The set Q consists of the following terms:

min_in_gga(x0, x1)
U7_gga(x0, x1)
U8_gga(x0, x1)
le_in_gg(x0, x1)
gt_in_gg(x0, x1)
U12_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:

U5_GAG(L, min_out_gga(B)) → MIN2_IN_GAG(B, L)
The graph contains the following edges 2 > 1, 1 >= 2
MIN2_IN_GAG(X, .(M, L)) → U5_GAG(L, min_in_gga(X, M))
The graph contains the following edges 2 > 1

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

U1_GA(L, X, L1, min1_out_ag(X, L)) → U2_GA(L, X, L1, remove_in_gga(X, L, L2))
U2_GA(L, X, L1, remove_out_gga(X, L, L2)) → MINSORT_IN_GA(L2, L1)
MINSORT_IN_GA(L, .(X, L1)) → U1_GA(L, X, L1, min1_in_ag(X, L))

The TRS R consists of the following rules:

minsort_in_ga([], []) → minsort_out_ga([], [])
minsort_in_ga(L, .(X, L1)) → U1_ga(L, X, L1, min1_in_ag(X, L))
min1_in_ag(M, .(X, L)) → U4_ag(M, X, L, min2_in_gag(X, M, L))
min2_in_gag(X, X, []) → min2_out_gag(X, X, [])
min2_in_gag(X, A, .(M, L)) → U5_gag(X, A, M, L, min_in_gga(X, M, B))
min_in_gga(X, Y, X) → U7_gga(X, Y, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U12_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
U12_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_gga(X, Y, le_out_gg(X, Y)) → min_out_gga(X, Y, X)
min_in_gga(X, Y, Y) → U8_gga(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U11_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U11_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U8_gga(X, Y, gt_out_gg(X, Y)) → min_out_gga(X, Y, Y)
U5_gag(X, A, M, L, min_out_gga(X, M, B)) → U6_gag(X, A, M, L, min2_in_gag(B, A, L))
U6_gag(X, A, M, L, min2_out_gag(B, A, L)) → min2_out_gag(X, A, .(M, L))
U4_ag(M, X, L, min2_out_gag(X, M, L)) → min1_out_ag(M, .(X, L))
U1_ga(L, X, L1, min1_out_ag(X, L)) → U2_ga(L, X, L1, remove_in_gga(X, L, L2))
remove_in_gga(N, .(N, L), L) → remove_out_gga(N, .(N, L), L)
remove_in_gga(N, .(M, L), .(M, L1)) → U9_gga(N, M, L, L1, notEq_in_gg(N, M))
notEq_in_gg(s(X), s(Y)) → U13_gg(X, Y, notEq_in_gg(X, Y))
notEq_in_gg(s(X), 0) → notEq_out_gg(s(X), 0)
notEq_in_gg(0, s(X)) → notEq_out_gg(0, s(X))
U13_gg(X, Y, notEq_out_gg(X, Y)) → notEq_out_gg(s(X), s(Y))
U9_gga(N, M, L, L1, notEq_out_gg(N, M)) → U10_gga(N, M, L, L1, remove_in_gga(N, L, L1))
U10_gga(N, M, L, L1, remove_out_gga(N, L, L1)) → remove_out_gga(N, .(M, L), .(M, L1))
U2_ga(L, X, L1, remove_out_gga(X, L, L2)) → U3_ga(L, X, L1, minsort_in_ga(L2, L1))
U3_ga(L, X, L1, minsort_out_ga(L2, L1)) → minsort_out_ga(L, .(X, L1))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

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

U1_GA(L, X, L1, min1_out_ag(X, L)) → U2_GA(L, X, L1, remove_in_gga(X, L, L2))
U2_GA(L, X, L1, remove_out_gga(X, L, L2)) → MINSORT_IN_GA(L2, L1)
MINSORT_IN_GA(L, .(X, L1)) → U1_GA(L, X, L1, min1_in_ag(X, L))

The TRS R consists of the following rules:

remove_in_gga(N, .(N, L), L) → remove_out_gga(N, .(N, L), L)
remove_in_gga(N, .(M, L), .(M, L1)) → U9_gga(N, M, L, L1, notEq_in_gg(N, M))
min1_in_ag(M, .(X, L)) → U4_ag(M, X, L, min2_in_gag(X, M, L))
U9_gga(N, M, L, L1, notEq_out_gg(N, M)) → U10_gga(N, M, L, L1, remove_in_gga(N, L, L1))
U4_ag(M, X, L, min2_out_gag(X, M, L)) → min1_out_ag(M, .(X, L))
notEq_in_gg(s(X), s(Y)) → U13_gg(X, Y, notEq_in_gg(X, Y))
notEq_in_gg(s(X), 0) → notEq_out_gg(s(X), 0)
notEq_in_gg(0, s(X)) → notEq_out_gg(0, s(X))
U10_gga(N, M, L, L1, remove_out_gga(N, L, L1)) → remove_out_gga(N, .(M, L), .(M, L1))
min2_in_gag(X, X, []) → min2_out_gag(X, X, [])
min2_in_gag(X, A, .(M, L)) → U5_gag(X, A, M, L, min_in_gga(X, M, B))
U13_gg(X, Y, notEq_out_gg(X, Y)) → notEq_out_gg(s(X), s(Y))
U5_gag(X, A, M, L, min_out_gga(X, M, B)) → U6_gag(X, A, M, L, min2_in_gag(B, A, L))
min_in_gga(X, Y, X) → U7_gga(X, Y, le_in_gg(X, Y))
min_in_gga(X, Y, Y) → U8_gga(X, Y, gt_in_gg(X, Y))
U6_gag(X, A, M, L, min2_out_gag(B, A, L)) → min2_out_gag(X, A, .(M, L))
U7_gga(X, Y, le_out_gg(X, Y)) → min_out_gga(X, Y, X)
U8_gga(X, Y, gt_out_gg(X, Y)) → min_out_gga(X, Y, Y)
le_in_gg(s(X), s(Y)) → U12_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg(0, s(Y))
le_in_gg(0, 0) → le_out_gg(0, 0)
gt_in_gg(s(X), s(Y)) → U11_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg(s(X), 0)
U12_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U11_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))

The argument filtering Pi contains the following mapping:
[] = []
min1_in_ag(x1, x2) = min1_in_ag(x2)
.(x1, x2) = .(x1, x2)
U4_ag(x1, x2, x3, x4) = U4_ag(x4)
min2_in_gag(x1, x2, x3) = min2_in_gag(x1, x3)
min2_out_gag(x1, x2, x3) = min2_out_gag(x2)
U5_gag(x1, x2, x3, x4, x5) = U5_gag(x4, x5)
min_in_gga(x1, x2, x3) = min_in_gga(x1, x2)
U7_gga(x1, x2, x3) = U7_gga(x1, x3)
le_in_gg(x1, x2) = le_in_gg(x1, x2)
s(x1) = s(x1)
U12_gg(x1, x2, x3) = U12_gg(x3)
0 = 0
le_out_gg(x1, x2) = le_out_gg
min_out_gga(x1, x2, x3) = min_out_gga(x3)
U8_gga(x1, x2, x3) = U8_gga(x2, x3)
gt_in_gg(x1, x2) = gt_in_gg(x1, x2)
U11_gg(x1, x2, x3) = U11_gg(x3)
gt_out_gg(x1, x2) = gt_out_gg
U6_gag(x1, x2, x3, x4, x5) = U6_gag(x5)
min1_out_ag(x1, x2) = min1_out_ag(x1)
remove_in_gga(x1, x2, x3) = remove_in_gga(x1, x2)
remove_out_gga(x1, x2, x3) = remove_out_gga(x3)
U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x2, x3, x5)
notEq_in_gg(x1, x2) = notEq_in_gg(x1, x2)
U13_gg(x1, x2, x3) = U13_gg(x3)
notEq_out_gg(x1, x2) = notEq_out_gg
U10_gga(x1, x2, x3, x4, x5) = U10_gga(x2, x5)
U2_GA(x1, x2, x3, x4) = U2_GA(x2, x4)
U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4)
MINSORT_IN_GA(x1, x2) = MINSORT_IN_GA(x1)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPOrderProof

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

U1_GA(L, min1_out_ag(X)) → U2_GA(X, remove_in_gga(X, L))
MINSORT_IN_GA(L) → U1_GA(L, min1_in_ag(L))
U2_GA(X, remove_out_gga(L2)) → MINSORT_IN_GA(L2)

The TRS R consists of the following rules:

remove_in_gga(N, .(N, L)) → remove_out_gga(L)
remove_in_gga(N, .(M, L)) → U9_gga(N, M, L, notEq_in_gg(N, M))
min1_in_ag(.(X, L)) → U4_ag(min2_in_gag(X, L))
U9_gga(N, M, L, notEq_out_gg) → U10_gga(M, remove_in_gga(N, L))
U4_ag(min2_out_gag(M)) → min1_out_ag(M)
notEq_in_gg(s(X), s(Y)) → U13_gg(notEq_in_gg(X, Y))
notEq_in_gg(s(X), 0) → notEq_out_gg
notEq_in_gg(0, s(X)) → notEq_out_gg
U10_gga(M, remove_out_gga(L1)) → remove_out_gga(.(M, L1))
min2_in_gag(X, []) → min2_out_gag(X)
min2_in_gag(X, .(M, L)) → U5_gag(L, min_in_gga(X, M))
U13_gg(notEq_out_gg) → notEq_out_gg
U5_gag(L, min_out_gga(B)) → U6_gag(min2_in_gag(B, L))
min_in_gga(X, Y) → U7_gga(X, le_in_gg(X, Y))
min_in_gga(X, Y) → U8_gga(Y, gt_in_gg(X, Y))
U6_gag(min2_out_gag(A)) → min2_out_gag(A)
U7_gga(X, le_out_gg) → min_out_gga(X)
U8_gga(Y, gt_out_gg) → min_out_gga(Y)
le_in_gg(s(X), s(Y)) → U12_gg(le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
gt_in_gg(s(X), s(Y)) → U11_gg(gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg
U12_gg(le_out_gg) → le_out_gg
U11_gg(gt_out_gg) → gt_out_gg

The set Q consists of the following terms:

remove_in_gga(x0, x1)
min1_in_ag(x0)
U9_gga(x0, x1, x2, x3)
U4_ag(x0)
notEq_in_gg(x0, x1)
U10_gga(x0, x1)
min2_in_gag(x0, x1)
U13_gg(x0)
U5_gag(x0, x1)
min_in_gga(x0, x1)
U6_gag(x0)
U7_gga(x0, x1)
U8_gga(x0, x1)
le_in_gg(x0, x1)
gt_in_gg(x0, x1)
U12_gg(x0)
U11_gg(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.

U2_GA(X, remove_out_gga(L2)) → MINSORT_IN_GA(L2)

The remaining pairs can at least be oriented weakly.

U1_GA(L, min1_out_ag(X)) → U2_GA(X, remove_in_gga(X, L))
MINSORT_IN_GA(L) → U1_GA(L, min1_in_ag(L))

Used ordering: Polynomial interpretation [25]:

POL(.(x₁, x₂)) = 1 + x₂
POL(0) = 0
POL(MINSORT_IN_GA(x₁)) = x₁
POL(U10_gga(x₁, x₂)) = 1 + x₂
POL(U11_gg(x₁)) = 0
POL(U12_gg(x₁)) = 1
POL(U13_gg(x₁)) = 0
POL(U1_GA(x₁, x₂)) = x₁
POL(U2_GA(x₁, x₂)) = x₂
POL(U4_ag(x₁)) = 0
POL(U5_gag(x₁, x₂)) = 0
POL(U6_gag(x₁)) = 0
POL(U7_gga(x₁, x₂)) = 1
POL(U8_gga(x₁, x₂)) = 0
POL(U9_gga(x₁, x₂, x₃, x₄)) = 1 + x₃
POL([]) = 0
POL(gt_in_gg(x₁, x₂)) = 0
POL(gt_out_gg) = 0
POL(le_in_gg(x₁, x₂)) = 1 + x₁ + x₂
POL(le_out_gg) = 1
POL(min1_in_ag(x₁)) = 0
POL(min1_out_ag(x₁)) = 0
POL(min2_in_gag(x₁, x₂)) = 0
POL(min2_out_gag(x₁)) = 0
POL(min_in_gga(x₁, x₂)) = 1 + x₁ + x₂
POL(min_out_gga(x₁)) = 0
POL(notEq_in_gg(x₁, x₂)) = 0
POL(notEq_out_gg) = 0
POL(remove_in_gga(x₁, x₂)) = x₂
POL(remove_out_gga(x₁)) = 1 + x₁
POL(s(x₁)) = 1 + x₁

The following usable rules [17] were oriented:

remove_in_gga(N, .(N, L)) → remove_out_gga(L)
remove_in_gga(N, .(M, L)) → U9_gga(N, M, L, notEq_in_gg(N, M))
U10_gga(M, remove_out_gga(L1)) → remove_out_gga(.(M, L1))
U9_gga(N, M, L, notEq_out_gg) → U10_gga(M, remove_in_gga(N, L))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

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

U1_GA(L, min1_out_ag(X)) → U2_GA(X, remove_in_gga(X, L))
MINSORT_IN_GA(L) → U1_GA(L, min1_in_ag(L))

The TRS R consists of the following rules:

remove_in_gga(N, .(N, L)) → remove_out_gga(L)
remove_in_gga(N, .(M, L)) → U9_gga(N, M, L, notEq_in_gg(N, M))
min1_in_ag(.(X, L)) → U4_ag(min2_in_gag(X, L))
U9_gga(N, M, L, notEq_out_gg) → U10_gga(M, remove_in_gga(N, L))
U4_ag(min2_out_gag(M)) → min1_out_ag(M)
notEq_in_gg(s(X), s(Y)) → U13_gg(notEq_in_gg(X, Y))
notEq_in_gg(s(X), 0) → notEq_out_gg
notEq_in_gg(0, s(X)) → notEq_out_gg
U10_gga(M, remove_out_gga(L1)) → remove_out_gga(.(M, L1))
min2_in_gag(X, []) → min2_out_gag(X)
min2_in_gag(X, .(M, L)) → U5_gag(L, min_in_gga(X, M))
U13_gg(notEq_out_gg) → notEq_out_gg
U5_gag(L, min_out_gga(B)) → U6_gag(min2_in_gag(B, L))
min_in_gga(X, Y) → U7_gga(X, le_in_gg(X, Y))
min_in_gga(X, Y) → U8_gga(Y, gt_in_gg(X, Y))
U6_gag(min2_out_gag(A)) → min2_out_gag(A)
U7_gga(X, le_out_gg) → min_out_gga(X)
U8_gga(Y, gt_out_gg) → min_out_gga(Y)
le_in_gg(s(X), s(Y)) → U12_gg(le_in_gg(X, Y))
le_in_gg(0, s(Y)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
gt_in_gg(s(X), s(Y)) → U11_gg(gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg
U12_gg(le_out_gg) → le_out_gg
U11_gg(gt_out_gg) → gt_out_gg

The set Q consists of the following terms:

remove_in_gga(x0, x1)
min1_in_ag(x0)
U9_gga(x0, x1, x2, x3)
U4_ag(x0)
notEq_in_gg(x0, x1)
U10_gga(x0, x1)
min2_in_gag(x0, x1)
U13_gg(x0)
U5_gag(x0, x1)
min_in_gga(x0, x1)
U6_gag(x0)
U7_gga(x0, x1)
U8_gga(x0, x1)
le_in_gg(x0, x1)
gt_in_gg(x0, x1)
U12_gg(x0)
U11_gg(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.