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

(0) Obligation:

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).

(1) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. 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))

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)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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))

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)

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
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))

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)
MINSORT_IN_GA(x1, x2)  =  MINSORT_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
MIN1_IN_AG(x1, x2)  =  MIN1_IN_AG(x2)
U4_AG(x1, x2, x3, x4)  =  U4_AG(x4)
MIN2_IN_GAG(x1, x2, x3)  =  MIN2_IN_GAG(x1, x3)
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)
U12_GG(x1, x2, x3)  =  U12_GG(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)
U6_GAG(x1, x2, x3, x4, x5)  =  U6_GAG(x5)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x2, x4)
REMOVE_IN_GGA(x1, x2, x3)  =  REMOVE_IN_GGA(x1, x2)
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)
U10_GGA(x1, x2, x3, x4, x5)  =  U10_GGA(x2, x5)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x2, x4)

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

(4) Obligation:

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))

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)
MINSORT_IN_GA(x1, x2)  =  MINSORT_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
MIN1_IN_AG(x1, x2)  =  MIN1_IN_AG(x2)
U4_AG(x1, x2, x3, x4)  =  U4_AG(x4)
MIN2_IN_GAG(x1, x2, x3)  =  MIN2_IN_GAG(x1, x3)
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)
U12_GG(x1, x2, x3)  =  U12_GG(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)
U6_GAG(x1, x2, x3, x4, x5)  =  U6_GAG(x5)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x2, x4)
REMOVE_IN_GGA(x1, x2, x3)  =  REMOVE_IN_GGA(x1, x2)
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)
U10_GGA(x1, x2, x3, x4, x5)  =  U10_GGA(x2, x5)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x2, x4)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 16 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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))

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)
NOTEQ_IN_GG(x1, x2)  =  NOTEQ_IN_GG(x1, x2)

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

(8) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(9) Obligation:

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

(10) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(11) Obligation:

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.

(12) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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

(13) TRUE

(14) Obligation:

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

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

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)
REMOVE_IN_GGA(x1, x2, x3)  =  REMOVE_IN_GGA(x1, x2)
U9_GGA(x1, x2, x3, x4, x5)  =  U9_GGA(x1, x2, x3, x5)

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

(15) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(16) Obligation:

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

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

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
REMOVE_IN_GGA(x1, x2, x3)  =  REMOVE_IN_GGA(x1, x2)
U9_GGA(x1, x2, x3, x4, x5)  =  U9_GGA(x1, x2, x3, x5)

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

(17) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(18) Obligation:

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.

(19) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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:

• 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

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

(20) TRUE

(21) Obligation:

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))

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)
GT_IN_GG(x1, x2)  =  GT_IN_GG(x1, x2)

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

(22) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(23) Obligation:

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

(24) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(25) Obligation:

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

(26) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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:

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

(27) TRUE

(28) Obligation:

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))

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)
LE_IN_GG(x1, x2)  =  LE_IN_GG(x1, x2)

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

(29) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(30) Obligation:

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

(31) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(32) Obligation:

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

(33) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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:

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

(34) TRUE

(35) Obligation:

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

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

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)
MIN2_IN_GAG(x1, x2, x3)  =  MIN2_IN_GAG(x1, x3)
U5_GAG(x1, x2, x3, x4, x5)  =  U5_GAG(x4, x5)

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

(36) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(37) Obligation:

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

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

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
MIN2_IN_GAG(x1, x2, x3)  =  MIN2_IN_GAG(x1, x3)
U5_GAG(x1, x2, x3, x4, x5)  =  U5_GAG(x4, x5)

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

(38) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(39) Obligation:

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

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

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.

(40) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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:

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

• U5_GAG(L, min_out_gga(B)) → MIN2_IN_GAG(B, L)
  The graph contains the following edges 2 > 1, 1 >= 2

(41) TRUE

(42) Obligation:

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))

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)
MINSORT_IN_GA(x1, x2)  =  MINSORT_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x2, x4)

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

(43) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(44) Obligation:

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)
MINSORT_IN_GA(x1, x2)  =  MINSORT_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x2, x4)

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

(45) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(46) Obligation:

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

(47) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

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.
Used ordering: Polynomial interpretation [POLO]:

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)) = 0   
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)) = 0   
POL(U8_gga(x₁, x₂)) = 0   
POL(U9_gga(x₁, x₂, x₃, x₄)) = 1 + x₃   
POL([]) = 0   
POL(gt_in_gg(x₁, x₂)) = x₂   
POL(gt_out_gg) = 0   
POL(le_in_gg(x₁, x₂)) = 0   
POL(le_out_gg) = 0   
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)) = 0   
POL(min_out_gga(x₁)) = 0   
POL(notEq_in_gg(x₁, x₂)) = x₁ + x₂   
POL(notEq_out_gg) = 0   
POL(remove_in_gga(x₁, x₂)) = x₂   
POL(remove_out_gga(x₁)) = 1 + x₁   
POL(s(x₁)) = x₁   

The following usable rules [FROCOS05] 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))
U9_gga(N, M, L, notEq_out_gg) → U10_gga(M, remove_in_gga(N, L))
U10_gga(M, remove_out_gga(L1)) → remove_out_gga(.(M, L1))

(48) Obligation:

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.

(49) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(50) TRUE

(51) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. 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))

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(x1, 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(x2, x3, x4)
min2_in_gag(x1, x2, x3)  =  min2_in_gag(x1, x3)
min2_out_gag(x1, x2, x3)  =  min2_out_gag(x1, x2, x3)
U5_gag(x1, x2, x3, x4, x5)  =  U5_gag(x1, x3, x4, x5)
min_in_gga(x1, x2, x3)  =  min_in_gga(x1, x2)
U7_gga(x1, x2, x3)  =  U7_gga(x1, x2, x3)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
s(x1)  =  s(x1)
U12_gg(x1, x2, x3)  =  U12_gg(x1, x2, x3)
0  =  0
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
min_out_gga(x1, x2, x3)  =  min_out_gga(x1, x2, x3)
U8_gga(x1, x2, x3)  =  U8_gga(x1, x2, x3)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)
gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)
U6_gag(x1, x2, x3, x4, x5)  =  U6_gag(x1, x3, x4, x5)
min1_out_ag(x1, x2)  =  min1_out_ag(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
remove_in_gga(x1, x2, x3)  =  remove_in_gga(x1, x2)
remove_out_gga(x1, x2, x3)  =  remove_out_gga(x1, x2, 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(x1, x2, x3)
notEq_out_gg(x1, x2)  =  notEq_out_gg(x1, x2)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x2, x3, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x2, x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(52) Obligation:

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))

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(x1, 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(x2, x3, x4)
min2_in_gag(x1, x2, x3)  =  min2_in_gag(x1, x3)
min2_out_gag(x1, x2, x3)  =  min2_out_gag(x1, x2, x3)
U5_gag(x1, x2, x3, x4, x5)  =  U5_gag(x1, x3, x4, x5)
min_in_gga(x1, x2, x3)  =  min_in_gga(x1, x2)
U7_gga(x1, x2, x3)  =  U7_gga(x1, x2, x3)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
s(x1)  =  s(x1)
U12_gg(x1, x2, x3)  =  U12_gg(x1, x2, x3)
0  =  0
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
min_out_gga(x1, x2, x3)  =  min_out_gga(x1, x2, x3)
U8_gga(x1, x2, x3)  =  U8_gga(x1, x2, x3)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)
gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)
U6_gag(x1, x2, x3, x4, x5)  =  U6_gag(x1, x3, x4, x5)
min1_out_ag(x1, x2)  =  min1_out_ag(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
remove_in_gga(x1, x2, x3)  =  remove_in_gga(x1, x2)
remove_out_gga(x1, x2, x3)  =  remove_out_gga(x1, x2, 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(x1, x2, x3)
notEq_out_gg(x1, x2)  =  notEq_out_gg(x1, x2)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x2, x3, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x2, x4)

(53) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
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))

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(x1, 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(x2, x3, x4)
min2_in_gag(x1, x2, x3)  =  min2_in_gag(x1, x3)
min2_out_gag(x1, x2, x3)  =  min2_out_gag(x1, x2, x3)
U5_gag(x1, x2, x3, x4, x5)  =  U5_gag(x1, x3, x4, x5)
min_in_gga(x1, x2, x3)  =  min_in_gga(x1, x2)
U7_gga(x1, x2, x3)  =  U7_gga(x1, x2, x3)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
s(x1)  =  s(x1)
U12_gg(x1, x2, x3)  =  U12_gg(x1, x2, x3)
0  =  0
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
min_out_gga(x1, x2, x3)  =  min_out_gga(x1, x2, x3)
U8_gga(x1, x2, x3)  =  U8_gga(x1, x2, x3)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)
gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)
U6_gag(x1, x2, x3, x4, x5)  =  U6_gag(x1, x3, x4, x5)
min1_out_ag(x1, x2)  =  min1_out_ag(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
remove_in_gga(x1, x2, x3)  =  remove_in_gga(x1, x2)
remove_out_gga(x1, x2, x3)  =  remove_out_gga(x1, x2, 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(x1, x2, x3)
notEq_out_gg(x1, x2)  =  notEq_out_gg(x1, x2)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x2, x3, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x2, x4)
MINSORT_IN_GA(x1, x2)  =  MINSORT_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
MIN1_IN_AG(x1, x2)  =  MIN1_IN_AG(x2)
U4_AG(x1, x2, x3, x4)  =  U4_AG(x2, x3, x4)
MIN2_IN_GAG(x1, x2, x3)  =  MIN2_IN_GAG(x1, x3)
U5_GAG(x1, x2, x3, x4, x5)  =  U5_GAG(x1, x3, x4, x5)
MIN_IN_GGA(x1, x2, x3)  =  MIN_IN_GGA(x1, x2)
U7_GGA(x1, x2, x3)  =  U7_GGA(x1, x2, x3)
LE_IN_GG(x1, x2)  =  LE_IN_GG(x1, x2)
U12_GG(x1, x2, x3)  =  U12_GG(x1, x2, x3)
U8_GGA(x1, x2, x3)  =  U8_GGA(x1, x2, x3)
GT_IN_GG(x1, x2)  =  GT_IN_GG(x1, x2)
U11_GG(x1, x2, x3)  =  U11_GG(x1, x2, x3)
U6_GAG(x1, x2, x3, x4, x5)  =  U6_GAG(x1, x3, x4, x5)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x2, x4)
REMOVE_IN_GGA(x1, x2, x3)  =  REMOVE_IN_GGA(x1, x2)
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(x1, x2, x3)
U10_GGA(x1, x2, x3, x4, x5)  =  U10_GGA(x1, x2, x3, x5)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x2, x4)

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

(54) Obligation:

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))

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(x1, 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(x2, x3, x4)
min2_in_gag(x1, x2, x3)  =  min2_in_gag(x1, x3)
min2_out_gag(x1, x2, x3)  =  min2_out_gag(x1, x2, x3)
U5_gag(x1, x2, x3, x4, x5)  =  U5_gag(x1, x3, x4, x5)
min_in_gga(x1, x2, x3)  =  min_in_gga(x1, x2)
U7_gga(x1, x2, x3)  =  U7_gga(x1, x2, x3)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
s(x1)  =  s(x1)
U12_gg(x1, x2, x3)  =  U12_gg(x1, x2, x3)
0  =  0
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
min_out_gga(x1, x2, x3)  =  min_out_gga(x1, x2, x3)
U8_gga(x1, x2, x3)  =  U8_gga(x1, x2, x3)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)
gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)
U6_gag(x1, x2, x3, x4, x5)  =  U6_gag(x1, x3, x4, x5)
min1_out_ag(x1, x2)  =  min1_out_ag(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
remove_in_gga(x1, x2, x3)  =  remove_in_gga(x1, x2)
remove_out_gga(x1, x2, x3)  =  remove_out_gga(x1, x2, 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(x1, x2, x3)
notEq_out_gg(x1, x2)  =  notEq_out_gg(x1, x2)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x2, x3, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x2, x4)
MINSORT_IN_GA(x1, x2)  =  MINSORT_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
MIN1_IN_AG(x1, x2)  =  MIN1_IN_AG(x2)
U4_AG(x1, x2, x3, x4)  =  U4_AG(x2, x3, x4)
MIN2_IN_GAG(x1, x2, x3)  =  MIN2_IN_GAG(x1, x3)
U5_GAG(x1, x2, x3, x4, x5)  =  U5_GAG(x1, x3, x4, x5)
MIN_IN_GGA(x1, x2, x3)  =  MIN_IN_GGA(x1, x2)
U7_GGA(x1, x2, x3)  =  U7_GGA(x1, x2, x3)
LE_IN_GG(x1, x2)  =  LE_IN_GG(x1, x2)
U12_GG(x1, x2, x3)  =  U12_GG(x1, x2, x3)
U8_GGA(x1, x2, x3)  =  U8_GGA(x1, x2, x3)
GT_IN_GG(x1, x2)  =  GT_IN_GG(x1, x2)
U11_GG(x1, x2, x3)  =  U11_GG(x1, x2, x3)
U6_GAG(x1, x2, x3, x4, x5)  =  U6_GAG(x1, x3, x4, x5)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x2, x4)
REMOVE_IN_GGA(x1, x2, x3)  =  REMOVE_IN_GGA(x1, x2)
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(x1, x2, x3)
U10_GGA(x1, x2, x3, x4, x5)  =  U10_GGA(x1, x2, x3, x5)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x2, x4)

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

(55) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 16 less nodes.

(56) Complex Obligation (AND)

(57) Obligation:

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))

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(x1, 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(x2, x3, x4)
min2_in_gag(x1, x2, x3)  =  min2_in_gag(x1, x3)
min2_out_gag(x1, x2, x3)  =  min2_out_gag(x1, x2, x3)
U5_gag(x1, x2, x3, x4, x5)  =  U5_gag(x1, x3, x4, x5)
min_in_gga(x1, x2, x3)  =  min_in_gga(x1, x2)
U7_gga(x1, x2, x3)  =  U7_gga(x1, x2, x3)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
s(x1)  =  s(x1)
U12_gg(x1, x2, x3)  =  U12_gg(x1, x2, x3)
0  =  0
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
min_out_gga(x1, x2, x3)  =  min_out_gga(x1, x2, x3)
U8_gga(x1, x2, x3)  =  U8_gga(x1, x2, x3)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)
gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)
U6_gag(x1, x2, x3, x4, x5)  =  U6_gag(x1, x3, x4, x5)
min1_out_ag(x1, x2)  =  min1_out_ag(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
remove_in_gga(x1, x2, x3)  =  remove_in_gga(x1, x2)
remove_out_gga(x1, x2, x3)  =  remove_out_gga(x1, x2, 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(x1, x2, x3)
notEq_out_gg(x1, x2)  =  notEq_out_gg(x1, x2)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x2, x3, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x2, x4)
NOTEQ_IN_GG(x1, x2)  =  NOTEQ_IN_GG(x1, x2)

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

(58) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(59) Obligation:

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

(60) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(61) Obligation:

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.

(62) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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

(63) TRUE

(64) Obligation:

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

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

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(x1, 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(x2, x3, x4)
min2_in_gag(x1, x2, x3)  =  min2_in_gag(x1, x3)
min2_out_gag(x1, x2, x3)  =  min2_out_gag(x1, x2, x3)
U5_gag(x1, x2, x3, x4, x5)  =  U5_gag(x1, x3, x4, x5)
min_in_gga(x1, x2, x3)  =  min_in_gga(x1, x2)
U7_gga(x1, x2, x3)  =  U7_gga(x1, x2, x3)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
s(x1)  =  s(x1)
U12_gg(x1, x2, x3)  =  U12_gg(x1, x2, x3)
0  =  0
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
min_out_gga(x1, x2, x3)  =  min_out_gga(x1, x2, x3)
U8_gga(x1, x2, x3)  =  U8_gga(x1, x2, x3)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)
gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)
U6_gag(x1, x2, x3, x4, x5)  =  U6_gag(x1, x3, x4, x5)
min1_out_ag(x1, x2)  =  min1_out_ag(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
remove_in_gga(x1, x2, x3)  =  remove_in_gga(x1, x2)
remove_out_gga(x1, x2, x3)  =  remove_out_gga(x1, x2, 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(x1, x2, x3)
notEq_out_gg(x1, x2)  =  notEq_out_gg(x1, x2)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x2, x3, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x2, x4)
REMOVE_IN_GGA(x1, x2, x3)  =  REMOVE_IN_GGA(x1, x2)
U9_GGA(x1, x2, x3, x4, x5)  =  U9_GGA(x1, x2, x3, x5)

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

(65) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(66) Obligation:

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

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

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(x1, x2, x3)
notEq_out_gg(x1, x2)  =  notEq_out_gg(x1, x2)
REMOVE_IN_GGA(x1, x2, x3)  =  REMOVE_IN_GGA(x1, x2)
U9_GGA(x1, x2, x3, x4, x5)  =  U9_GGA(x1, x2, x3, x5)

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

(67) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(68) Obligation:

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

U9_GGA(N, M, L, notEq_out_gg(N, M)) → 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(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 set Q consists of the following terms:

notEq_in_gg(x0, x1)
U13_gg(x0, x1, x2)

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

(69) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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:

• 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

• U9_GGA(N, M, L, notEq_out_gg(N, M)) → REMOVE_IN_GGA(N, L)
  The graph contains the following edges 1 >= 1, 4 > 1, 3 >= 2

(70) TRUE

(71) Obligation:

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))

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(x1, 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(x2, x3, x4)
min2_in_gag(x1, x2, x3)  =  min2_in_gag(x1, x3)
min2_out_gag(x1, x2, x3)  =  min2_out_gag(x1, x2, x3)
U5_gag(x1, x2, x3, x4, x5)  =  U5_gag(x1, x3, x4, x5)
min_in_gga(x1, x2, x3)  =  min_in_gga(x1, x2)
U7_gga(x1, x2, x3)  =  U7_gga(x1, x2, x3)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
s(x1)  =  s(x1)
U12_gg(x1, x2, x3)  =  U12_gg(x1, x2, x3)
0  =  0
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
min_out_gga(x1, x2, x3)  =  min_out_gga(x1, x2, x3)
U8_gga(x1, x2, x3)  =  U8_gga(x1, x2, x3)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)
gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)
U6_gag(x1, x2, x3, x4, x5)  =  U6_gag(x1, x3, x4, x5)
min1_out_ag(x1, x2)  =  min1_out_ag(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
remove_in_gga(x1, x2, x3)  =  remove_in_gga(x1, x2)
remove_out_gga(x1, x2, x3)  =  remove_out_gga(x1, x2, 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(x1, x2, x3)
notEq_out_gg(x1, x2)  =  notEq_out_gg(x1, x2)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x2, x3, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x2, x4)
GT_IN_GG(x1, x2)  =  GT_IN_GG(x1, x2)

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

(72) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(73) Obligation:

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

(74) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(75) Obligation:

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

(76) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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:

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

(77) TRUE

(78) Obligation:

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))

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(x1, 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(x2, x3, x4)
min2_in_gag(x1, x2, x3)  =  min2_in_gag(x1, x3)
min2_out_gag(x1, x2, x3)  =  min2_out_gag(x1, x2, x3)
U5_gag(x1, x2, x3, x4, x5)  =  U5_gag(x1, x3, x4, x5)
min_in_gga(x1, x2, x3)  =  min_in_gga(x1, x2)
U7_gga(x1, x2, x3)  =  U7_gga(x1, x2, x3)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
s(x1)  =  s(x1)
U12_gg(x1, x2, x3)  =  U12_gg(x1, x2, x3)
0  =  0
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
min_out_gga(x1, x2, x3)  =  min_out_gga(x1, x2, x3)
U8_gga(x1, x2, x3)  =  U8_gga(x1, x2, x3)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)
gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)
U6_gag(x1, x2, x3, x4, x5)  =  U6_gag(x1, x3, x4, x5)
min1_out_ag(x1, x2)  =  min1_out_ag(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
remove_in_gga(x1, x2, x3)  =  remove_in_gga(x1, x2)
remove_out_gga(x1, x2, x3)  =  remove_out_gga(x1, x2, 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(x1, x2, x3)
notEq_out_gg(x1, x2)  =  notEq_out_gg(x1, x2)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x2, x3, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x2, x4)
LE_IN_GG(x1, x2)  =  LE_IN_GG(x1, x2)

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

(79) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(80) Obligation:

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

(81) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(82) Obligation:

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

(83) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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:

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

(84) TRUE

(85) Obligation:

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

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

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(x1, 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(x2, x3, x4)
min2_in_gag(x1, x2, x3)  =  min2_in_gag(x1, x3)
min2_out_gag(x1, x2, x3)  =  min2_out_gag(x1, x2, x3)
U5_gag(x1, x2, x3, x4, x5)  =  U5_gag(x1, x3, x4, x5)
min_in_gga(x1, x2, x3)  =  min_in_gga(x1, x2)
U7_gga(x1, x2, x3)  =  U7_gga(x1, x2, x3)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
s(x1)  =  s(x1)
U12_gg(x1, x2, x3)  =  U12_gg(x1, x2, x3)
0  =  0
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
min_out_gga(x1, x2, x3)  =  min_out_gga(x1, x2, x3)
U8_gga(x1, x2, x3)  =  U8_gga(x1, x2, x3)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)
gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)
U6_gag(x1, x2, x3, x4, x5)  =  U6_gag(x1, x3, x4, x5)
min1_out_ag(x1, x2)  =  min1_out_ag(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
remove_in_gga(x1, x2, x3)  =  remove_in_gga(x1, x2)
remove_out_gga(x1, x2, x3)  =  remove_out_gga(x1, x2, 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(x1, x2, x3)
notEq_out_gg(x1, x2)  =  notEq_out_gg(x1, x2)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x2, x3, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x2, x4)
MIN2_IN_GAG(x1, x2, x3)  =  MIN2_IN_GAG(x1, x3)
U5_GAG(x1, x2, x3, x4, x5)  =  U5_GAG(x1, x3, x4, x5)

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

(86) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(87) Obligation:

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

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

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, x2, x3)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
s(x1)  =  s(x1)
U12_gg(x1, x2, x3)  =  U12_gg(x1, x2, x3)
0  =  0
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
min_out_gga(x1, x2, x3)  =  min_out_gga(x1, x2, x3)
U8_gga(x1, x2, x3)  =  U8_gga(x1, x2, x3)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)
gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)
MIN2_IN_GAG(x1, x2, x3)  =  MIN2_IN_GAG(x1, x3)
U5_GAG(x1, x2, x3, x4, x5)  =  U5_GAG(x1, x3, x4, x5)

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

(88) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(89) Obligation:

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

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

The TRS R consists of the following rules:

min_in_gga(X, Y) → U7_gga(X, Y, le_in_gg(X, Y))
min_in_gga(X, 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 set Q consists of the following terms:

min_in_gga(x0, x1)
U7_gga(x0, x1, x2)
U8_gga(x0, x1, x2)
le_in_gg(x0, x1)
gt_in_gg(x0, x1)
U12_gg(x0, x1, x2)
U11_gg(x0, x1, x2)

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

(90) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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:

• MIN2_IN_GAG(X, .(M, L)) → U5_GAG(X, M, L, min_in_gga(X, M))
  The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3

• U5_GAG(X, M, L, min_out_gga(X, M, B)) → MIN2_IN_GAG(B, L)
  The graph contains the following edges 4 > 1, 3 >= 2

(91) TRUE

(92) Obligation:

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))

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(x1, 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(x2, x3, x4)
min2_in_gag(x1, x2, x3)  =  min2_in_gag(x1, x3)
min2_out_gag(x1, x2, x3)  =  min2_out_gag(x1, x2, x3)
U5_gag(x1, x2, x3, x4, x5)  =  U5_gag(x1, x3, x4, x5)
min_in_gga(x1, x2, x3)  =  min_in_gga(x1, x2)
U7_gga(x1, x2, x3)  =  U7_gga(x1, x2, x3)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
s(x1)  =  s(x1)
U12_gg(x1, x2, x3)  =  U12_gg(x1, x2, x3)
0  =  0
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
min_out_gga(x1, x2, x3)  =  min_out_gga(x1, x2, x3)
U8_gga(x1, x2, x3)  =  U8_gga(x1, x2, x3)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)
gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)
U6_gag(x1, x2, x3, x4, x5)  =  U6_gag(x1, x3, x4, x5)
min1_out_ag(x1, x2)  =  min1_out_ag(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
remove_in_gga(x1, x2, x3)  =  remove_in_gga(x1, x2)
remove_out_gga(x1, x2, x3)  =  remove_out_gga(x1, x2, 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(x1, x2, x3)
notEq_out_gg(x1, x2)  =  notEq_out_gg(x1, x2)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x2, x3, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x2, x4)
MINSORT_IN_GA(x1, x2)  =  MINSORT_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x2, x4)

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

(93) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(94) Obligation:

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(x2, x3, x4)
min2_in_gag(x1, x2, x3)  =  min2_in_gag(x1, x3)
min2_out_gag(x1, x2, x3)  =  min2_out_gag(x1, x2, x3)
U5_gag(x1, x2, x3, x4, x5)  =  U5_gag(x1, x3, x4, x5)
min_in_gga(x1, x2, x3)  =  min_in_gga(x1, x2)
U7_gga(x1, x2, x3)  =  U7_gga(x1, x2, x3)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
s(x1)  =  s(x1)
U12_gg(x1, x2, x3)  =  U12_gg(x1, x2, x3)
0  =  0
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
min_out_gga(x1, x2, x3)  =  min_out_gga(x1, x2, x3)
U8_gga(x1, x2, x3)  =  U8_gga(x1, x2, x3)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)
gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)
U6_gag(x1, x2, x3, x4, x5)  =  U6_gag(x1, x3, x4, x5)
min1_out_ag(x1, x2)  =  min1_out_ag(x1, x2)
remove_in_gga(x1, x2, x3)  =  remove_in_gga(x1, x2)
remove_out_gga(x1, x2, x3)  =  remove_out_gga(x1, x2, 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(x1, x2, x3)
notEq_out_gg(x1, x2)  =  notEq_out_gg(x1, x2)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x2, x3, x5)
MINSORT_IN_GA(x1, x2)  =  MINSORT_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x2, x4)

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

(95) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(96) Obligation:

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

U1_GA(L, min1_out_ag(X, L)) → U2_GA(L, X, remove_in_gga(X, L))
U2_GA(L, X, remove_out_gga(X, L, L2)) → MINSORT_IN_GA(L2)
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(N, .(N, L), L)
remove_in_gga(N, .(M, L)) → U9_gga(N, M, L, notEq_in_gg(N, M))
min1_in_ag(.(X, L)) → U4_ag(X, L, min2_in_gag(X, L))
U9_gga(N, M, L, notEq_out_gg(N, M)) → U10_gga(N, M, L, remove_in_gga(N, L))
U4_ag(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, remove_out_gga(N, L, L1)) → remove_out_gga(N, .(M, L), .(M, L1))
min2_in_gag(X, []) → min2_out_gag(X, X, [])
min2_in_gag(X, .(M, L)) → U5_gag(X, M, L, min_in_gga(X, M))
U13_gg(X, Y, notEq_out_gg(X, Y)) → notEq_out_gg(s(X), s(Y))
U5_gag(X, M, L, min_out_gga(X, M, B)) → U6_gag(X, M, L, min2_in_gag(B, L))
min_in_gga(X, Y) → U7_gga(X, Y, le_in_gg(X, Y))
min_in_gga(X, Y) → U8_gga(X, Y, gt_in_gg(X, Y))
U6_gag(X, 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 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, x1, x2)
notEq_in_gg(x0, x1)
U10_gga(x0, x1, x2, x3)
min2_in_gag(x0, x1)
U13_gg(x0, x1, x2)
U5_gag(x0, x1, x2, x3)
min_in_gga(x0, x1)
U6_gag(x0, x1, x2, x3)
U7_gga(x0, x1, x2)
U8_gga(x0, x1, x2)
le_in_gg(x0, x1)
gt_in_gg(x0, x1)
U12_gg(x0, x1, x2)
U11_gg(x0, x1, x2)

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