(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, [], []).
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, [], []) → remove_out_gga(N, [], [])
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
(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, [], []) → remove_out_gga(N, [], [])
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)
(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, [], []) → remove_out_gga(N, [], [])
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
(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, [], []) → remove_out_gga(N, [], [])
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
(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, [], []) → remove_out_gga(N, [], [])
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
(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, [], []) → remove_out_gga(N, [], [])
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
(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(
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
(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(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.
(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(N, M)) → REMOVE_IN_GGA(N, L)
The graph contains the following edges 1 >= 1, 4 > 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, [], []) → remove_out_gga(N, [], [])
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
(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, [], []) → remove_out_gga(N, [], [])
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
(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, [], []) → remove_out_gga(N, [], [])
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
(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,
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
(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(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.
(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(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
(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, [], []) → remove_out_gga(N, [], [])
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
(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, [], []) → remove_out_gga(N, [], [])
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
(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, 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, []) → remove_out_gga(N, [], [])
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.
(47) Narrowing (SOUND transformation)
By narrowing [LPAR04] the rule
U1_GA(
L,
min1_out_ag(
X,
L)) →
U2_GA(
L,
X,
remove_in_gga(
X,
L)) at position [2] we obtained the following new rules [LPAR04]:
U1_GA([], min1_out_ag(x0, [])) → U2_GA([], x0, remove_out_gga(x0, [], []))
U1_GA(.(x0, x1), min1_out_ag(x0, .(x0, x1))) → U2_GA(.(x0, x1), x0, remove_out_gga(x0, .(x0, x1), x1))
U1_GA(.(x1, x2), min1_out_ag(x0, .(x1, x2))) → U2_GA(.(x1, x2), x0, U9_gga(x0, x1, x2, notEq_in_gg(x0, x1)))
(48) Obligation:
Q DP problem:
The TRS P consists of the following rules:
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))
U1_GA([], min1_out_ag(x0, [])) → U2_GA([], x0, remove_out_gga(x0, [], []))
U1_GA(.(x0, x1), min1_out_ag(x0, .(x0, x1))) → U2_GA(.(x0, x1), x0, remove_out_gga(x0, .(x0, x1), x1))
U1_GA(.(x1, x2), min1_out_ag(x0, .(x1, x2))) → U2_GA(.(x1, x2), x0, U9_gga(x0, x1, x2, notEq_in_gg(x0, x1)))
The TRS R consists of the following rules:
remove_in_gga(N, []) → remove_out_gga(N, [], [])
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.
(49) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
(50) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MINSORT_IN_GA(L) → U1_GA(L, min1_in_ag(L))
U1_GA(.(x0, x1), min1_out_ag(x0, .(x0, x1))) → U2_GA(.(x0, x1), x0, remove_out_gga(x0, .(x0, x1), x1))
U2_GA(L, X, remove_out_gga(X, L, L2)) → MINSORT_IN_GA(L2)
U1_GA(.(x1, x2), min1_out_ag(x0, .(x1, x2))) → U2_GA(.(x1, x2), x0, U9_gga(x0, x1, x2, notEq_in_gg(x0, x1)))
The TRS R consists of the following rules:
remove_in_gga(N, []) → remove_out_gga(N, [], [])
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.
(51) Narrowing (SOUND transformation)
By narrowing [LPAR04] the rule
MINSORT_IN_GA(
L) →
U1_GA(
L,
min1_in_ag(
L)) at position [1] we obtained the following new rules [LPAR04]:
MINSORT_IN_GA(.(x0, x1)) → U1_GA(.(x0, x1), U4_ag(x0, x1, min2_in_gag(x0, x1)))
(52) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U1_GA(.(x0, x1), min1_out_ag(x0, .(x0, x1))) → U2_GA(.(x0, x1), x0, remove_out_gga(x0, .(x0, x1), x1))
U2_GA(L, X, remove_out_gga(X, L, L2)) → MINSORT_IN_GA(L2)
U1_GA(.(x1, x2), min1_out_ag(x0, .(x1, x2))) → U2_GA(.(x1, x2), x0, U9_gga(x0, x1, x2, notEq_in_gg(x0, x1)))
MINSORT_IN_GA(.(x0, x1)) → U1_GA(.(x0, x1), U4_ag(x0, x1, min2_in_gag(x0, x1)))
The TRS R consists of the following rules:
remove_in_gga(N, []) → remove_out_gga(N, [], [])
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.
(53) UsableRulesProof (EQUIVALENT transformation)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
(54) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U1_GA(.(x0, x1), min1_out_ag(x0, .(x0, x1))) → U2_GA(.(x0, x1), x0, remove_out_gga(x0, .(x0, x1), x1))
U2_GA(L, X, remove_out_gga(X, L, L2)) → MINSORT_IN_GA(L2)
U1_GA(.(x1, x2), min1_out_ag(x0, .(x1, x2))) → U2_GA(.(x1, x2), x0, U9_gga(x0, x1, x2, notEq_in_gg(x0, x1)))
MINSORT_IN_GA(.(x0, x1)) → U1_GA(.(x0, x1), U4_ag(x0, x1, min2_in_gag(x0, x1)))
The TRS R consists of the following rules:
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))
U4_ag(X, L, min2_out_gag(X, M, L)) → min1_out_ag(M, .(X, 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))
U5_gag(X, M, L, min_out_gga(X, M, B)) → U6_gag(X, M, L, min2_in_gag(B, L))
U6_gag(X, M, L, min2_out_gag(B, A, L)) → min2_out_gag(X, A, .(M, L))
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)
U8_gga(X, Y, gt_out_gg(X, Y)) → min_out_gga(X, Y, Y)
U11_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(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)
U7_gga(X, Y, le_out_gg(X, Y)) → min_out_gga(X, Y, X)
U12_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
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))
U9_gga(N, M, L, notEq_out_gg(N, M)) → U10_gga(N, M, L, remove_in_gga(N, L))
remove_in_gga(N, []) → remove_out_gga(N, [], [])
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))
U10_gga(N, M, L, remove_out_gga(N, L, L1)) → remove_out_gga(N, .(M, L), .(M, L1))
U13_gg(X, Y, notEq_out_gg(X, Y)) → notEq_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.
(55) QReductionProof (EQUIVALENT transformation)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
min1_in_ag(x0)
(56) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U1_GA(.(x0, x1), min1_out_ag(x0, .(x0, x1))) → U2_GA(.(x0, x1), x0, remove_out_gga(x0, .(x0, x1), x1))
U2_GA(L, X, remove_out_gga(X, L, L2)) → MINSORT_IN_GA(L2)
U1_GA(.(x1, x2), min1_out_ag(x0, .(x1, x2))) → U2_GA(.(x1, x2), x0, U9_gga(x0, x1, x2, notEq_in_gg(x0, x1)))
MINSORT_IN_GA(.(x0, x1)) → U1_GA(.(x0, x1), U4_ag(x0, x1, min2_in_gag(x0, x1)))
The TRS R consists of the following rules:
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))
U4_ag(X, L, min2_out_gag(X, M, L)) → min1_out_ag(M, .(X, 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))
U5_gag(X, M, L, min_out_gga(X, M, B)) → U6_gag(X, M, L, min2_in_gag(B, L))
U6_gag(X, M, L, min2_out_gag(B, A, L)) → min2_out_gag(X, A, .(M, L))
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)
U8_gga(X, Y, gt_out_gg(X, Y)) → min_out_gga(X, Y, Y)
U11_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(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)
U7_gga(X, Y, le_out_gg(X, Y)) → min_out_gga(X, Y, X)
U12_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
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))
U9_gga(N, M, L, notEq_out_gg(N, M)) → U10_gga(N, M, L, remove_in_gga(N, L))
remove_in_gga(N, []) → remove_out_gga(N, [], [])
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))
U10_gga(N, M, L, remove_out_gga(N, L, L1)) → remove_out_gga(N, .(M, L), .(M, L1))
U13_gg(X, Y, notEq_out_gg(X, Y)) → notEq_out_gg(s(X), s(Y))
The set Q consists of the following terms:
remove_in_gga(x0, x1)
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.
(57) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule
U2_GA(
L,
X,
remove_out_gga(
X,
L,
L2)) →
MINSORT_IN_GA(
L2) we obtained the following new rules [LPAR04]:
U2_GA(.(z0, z1), z0, remove_out_gga(z0, .(z0, z1), z1)) → MINSORT_IN_GA(z1)
U2_GA(.(z0, z1), z2, remove_out_gga(z2, .(z0, z1), x2)) → MINSORT_IN_GA(x2)
(58) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U1_GA(.(x0, x1), min1_out_ag(x0, .(x0, x1))) → U2_GA(.(x0, x1), x0, remove_out_gga(x0, .(x0, x1), x1))
U1_GA(.(x1, x2), min1_out_ag(x0, .(x1, x2))) → U2_GA(.(x1, x2), x0, U9_gga(x0, x1, x2, notEq_in_gg(x0, x1)))
MINSORT_IN_GA(.(x0, x1)) → U1_GA(.(x0, x1), U4_ag(x0, x1, min2_in_gag(x0, x1)))
U2_GA(.(z0, z1), z0, remove_out_gga(z0, .(z0, z1), z1)) → MINSORT_IN_GA(z1)
U2_GA(.(z0, z1), z2, remove_out_gga(z2, .(z0, z1), x2)) → MINSORT_IN_GA(x2)
The TRS R consists of the following rules:
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))
U4_ag(X, L, min2_out_gag(X, M, L)) → min1_out_ag(M, .(X, 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))
U5_gag(X, M, L, min_out_gga(X, M, B)) → U6_gag(X, M, L, min2_in_gag(B, L))
U6_gag(X, M, L, min2_out_gag(B, A, L)) → min2_out_gag(X, A, .(M, L))
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)
U8_gga(X, Y, gt_out_gg(X, Y)) → min_out_gga(X, Y, Y)
U11_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(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)
U7_gga(X, Y, le_out_gg(X, Y)) → min_out_gga(X, Y, X)
U12_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
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))
U9_gga(N, M, L, notEq_out_gg(N, M)) → U10_gga(N, M, L, remove_in_gga(N, L))
remove_in_gga(N, []) → remove_out_gga(N, [], [])
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))
U10_gga(N, M, L, remove_out_gga(N, L, L1)) → remove_out_gga(N, .(M, L), .(M, L1))
U13_gg(X, Y, notEq_out_gg(X, Y)) → notEq_out_gg(s(X), s(Y))
The set Q consists of the following terms:
remove_in_gga(x0, x1)
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.
(59) ForwardInstantiation (EQUIVALENT transformation)
By forward instantiating [JAR06] the rule
U2_GA(
.(
z0,
z1),
z0,
remove_out_gga(
z0,
.(
z0,
z1),
z1)) →
MINSORT_IN_GA(
z1) we obtained the following new rules [LPAR04]:
U2_GA(.(x0, .(y_0, y_1)), x0, remove_out_gga(x0, .(x0, .(y_0, y_1)), .(y_0, y_1))) → MINSORT_IN_GA(.(y_0, y_1))
(60) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U1_GA(.(x0, x1), min1_out_ag(x0, .(x0, x1))) → U2_GA(.(x0, x1), x0, remove_out_gga(x0, .(x0, x1), x1))
U1_GA(.(x1, x2), min1_out_ag(x0, .(x1, x2))) → U2_GA(.(x1, x2), x0, U9_gga(x0, x1, x2, notEq_in_gg(x0, x1)))
MINSORT_IN_GA(.(x0, x1)) → U1_GA(.(x0, x1), U4_ag(x0, x1, min2_in_gag(x0, x1)))
U2_GA(.(z0, z1), z2, remove_out_gga(z2, .(z0, z1), x2)) → MINSORT_IN_GA(x2)
U2_GA(.(x0, .(y_0, y_1)), x0, remove_out_gga(x0, .(x0, .(y_0, y_1)), .(y_0, y_1))) → MINSORT_IN_GA(.(y_0, y_1))
The TRS R consists of the following rules:
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))
U4_ag(X, L, min2_out_gag(X, M, L)) → min1_out_ag(M, .(X, 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))
U5_gag(X, M, L, min_out_gga(X, M, B)) → U6_gag(X, M, L, min2_in_gag(B, L))
U6_gag(X, M, L, min2_out_gag(B, A, L)) → min2_out_gag(X, A, .(M, L))
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)
U8_gga(X, Y, gt_out_gg(X, Y)) → min_out_gga(X, Y, Y)
U11_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(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)
U7_gga(X, Y, le_out_gg(X, Y)) → min_out_gga(X, Y, X)
U12_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
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))
U9_gga(N, M, L, notEq_out_gg(N, M)) → U10_gga(N, M, L, remove_in_gga(N, L))
remove_in_gga(N, []) → remove_out_gga(N, [], [])
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))
U10_gga(N, M, L, remove_out_gga(N, L, L1)) → remove_out_gga(N, .(M, L), .(M, L1))
U13_gg(X, Y, notEq_out_gg(X, Y)) → notEq_out_gg(s(X), s(Y))
The set Q consists of the following terms:
remove_in_gga(x0, x1)
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.
(61) ForwardInstantiation (EQUIVALENT transformation)
By forward instantiating [JAR06] the rule
U2_GA(
.(
z0,
z1),
z2,
remove_out_gga(
z2,
.(
z0,
z1),
x2)) →
MINSORT_IN_GA(
x2) we obtained the following new rules [LPAR04]:
U2_GA(.(x0, x1), x2, remove_out_gga(x2, .(x0, x1), .(y_0, y_1))) → MINSORT_IN_GA(.(y_0, y_1))
(62) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U1_GA(.(x0, x1), min1_out_ag(x0, .(x0, x1))) → U2_GA(.(x0, x1), x0, remove_out_gga(x0, .(x0, x1), x1))
U1_GA(.(x1, x2), min1_out_ag(x0, .(x1, x2))) → U2_GA(.(x1, x2), x0, U9_gga(x0, x1, x2, notEq_in_gg(x0, x1)))
MINSORT_IN_GA(.(x0, x1)) → U1_GA(.(x0, x1), U4_ag(x0, x1, min2_in_gag(x0, x1)))
U2_GA(.(x0, .(y_0, y_1)), x0, remove_out_gga(x0, .(x0, .(y_0, y_1)), .(y_0, y_1))) → MINSORT_IN_GA(.(y_0, y_1))
U2_GA(.(x0, x1), x2, remove_out_gga(x2, .(x0, x1), .(y_0, y_1))) → MINSORT_IN_GA(.(y_0, y_1))
The TRS R consists of the following rules:
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))
U4_ag(X, L, min2_out_gag(X, M, L)) → min1_out_ag(M, .(X, 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))
U5_gag(X, M, L, min_out_gga(X, M, B)) → U6_gag(X, M, L, min2_in_gag(B, L))
U6_gag(X, M, L, min2_out_gag(B, A, L)) → min2_out_gag(X, A, .(M, L))
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)
U8_gga(X, Y, gt_out_gg(X, Y)) → min_out_gga(X, Y, Y)
U11_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(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)
U7_gga(X, Y, le_out_gg(X, Y)) → min_out_gga(X, Y, X)
U12_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
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))
U9_gga(N, M, L, notEq_out_gg(N, M)) → U10_gga(N, M, L, remove_in_gga(N, L))
remove_in_gga(N, []) → remove_out_gga(N, [], [])
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))
U10_gga(N, M, L, remove_out_gga(N, L, L1)) → remove_out_gga(N, .(M, L), .(M, L1))
U13_gg(X, Y, notEq_out_gg(X, Y)) → notEq_out_gg(s(X), s(Y))
The set Q consists of the following terms:
remove_in_gga(x0, x1)
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.
(63) ForwardInstantiation (EQUIVALENT transformation)
By forward instantiating [JAR06] the rule
U1_GA(
.(
x0,
x1),
min1_out_ag(
x0,
.(
x0,
x1))) →
U2_GA(
.(
x0,
x1),
x0,
remove_out_gga(
x0,
.(
x0,
x1),
x1)) we obtained the following new rules [LPAR04]:
U1_GA(.(x0, .(y_1, y_2)), min1_out_ag(x0, .(x0, .(y_1, y_2)))) → U2_GA(.(x0, .(y_1, y_2)), x0, remove_out_gga(x0, .(x0, .(y_1, y_2)), .(y_1, y_2)))
(64) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U1_GA(.(x1, x2), min1_out_ag(x0, .(x1, x2))) → U2_GA(.(x1, x2), x0, U9_gga(x0, x1, x2, notEq_in_gg(x0, x1)))
MINSORT_IN_GA(.(x0, x1)) → U1_GA(.(x0, x1), U4_ag(x0, x1, min2_in_gag(x0, x1)))
U2_GA(.(x0, .(y_0, y_1)), x0, remove_out_gga(x0, .(x0, .(y_0, y_1)), .(y_0, y_1))) → MINSORT_IN_GA(.(y_0, y_1))
U2_GA(.(x0, x1), x2, remove_out_gga(x2, .(x0, x1), .(y_0, y_1))) → MINSORT_IN_GA(.(y_0, y_1))
U1_GA(.(x0, .(y_1, y_2)), min1_out_ag(x0, .(x0, .(y_1, y_2)))) → U2_GA(.(x0, .(y_1, y_2)), x0, remove_out_gga(x0, .(x0, .(y_1, y_2)), .(y_1, y_2)))
The TRS R consists of the following rules:
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))
U4_ag(X, L, min2_out_gag(X, M, L)) → min1_out_ag(M, .(X, 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))
U5_gag(X, M, L, min_out_gga(X, M, B)) → U6_gag(X, M, L, min2_in_gag(B, L))
U6_gag(X, M, L, min2_out_gag(B, A, L)) → min2_out_gag(X, A, .(M, L))
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)
U8_gga(X, Y, gt_out_gg(X, Y)) → min_out_gga(X, Y, Y)
U11_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(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)
U7_gga(X, Y, le_out_gg(X, Y)) → min_out_gga(X, Y, X)
U12_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
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))
U9_gga(N, M, L, notEq_out_gg(N, M)) → U10_gga(N, M, L, remove_in_gga(N, L))
remove_in_gga(N, []) → remove_out_gga(N, [], [])
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))
U10_gga(N, M, L, remove_out_gga(N, L, L1)) → remove_out_gga(N, .(M, L), .(M, L1))
U13_gg(X, Y, notEq_out_gg(X, Y)) → notEq_out_gg(s(X), s(Y))
The set Q consists of the following terms:
remove_in_gga(x0, x1)
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.
(65) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
U1_GA(.(x0, .(y_1, y_2)), min1_out_ag(x0, .(x0, .(y_1, y_2)))) → U2_GA(.(x0, .(y_1, y_2)), x0, remove_out_gga(x0, .(x0, .(y_1, y_2)), .(y_1, y_2)))
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)) = 1 + x1
POL(U10_gga(x1, x2, x3, x4)) = 1 + x4
POL(U11_gg(x1, x2, x3)) = 0
POL(U12_gg(x1, x2, x3)) = 0
POL(U13_gg(x1, x2, x3)) = 1
POL(U1_GA(x1, x2)) = 1 + x1
POL(U2_GA(x1, x2, x3)) = x3
POL(U4_ag(x1, x2, x3)) = 0
POL(U5_gag(x1, x2, x3, x4)) = 0
POL(U6_gag(x1, x2, x3, x4)) = 0
POL(U7_gga(x1, x2, x3)) = 0
POL(U8_gga(x1, x2, x3)) = 0
POL(U9_gga(x1, x2, x3, x4)) = 1 + x3 + x4
POL([]) = 0
POL(gt_in_gg(x1, x2)) = 0
POL(gt_out_gg(x1, x2)) = 0
POL(le_in_gg(x1, x2)) = 0
POL(le_out_gg(x1, x2)) = 0
POL(min1_out_ag(x1, x2)) = 0
POL(min2_in_gag(x1, x2)) = 0
POL(min2_out_gag(x1, x2, x3)) = 0
POL(min_in_gga(x1, x2)) = 0
POL(min_out_gga(x1, x2, x3)) = 0
POL(notEq_in_gg(x1, x2)) = 1
POL(notEq_out_gg(x1, x2)) = 1
POL(remove_in_gga(x1, x2)) = 1 + x2
POL(remove_out_gga(x1, x2, x3)) = 1 + x3
POL(s(x1)) = 0
The following usable rules [FROCOS05] were oriented:
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))
U9_gga(N, M, L, notEq_out_gg(N, M)) → U10_gga(N, M, L, remove_in_gga(N, L))
remove_in_gga(N, []) → remove_out_gga(N, [], [])
remove_in_gga(N, .(N, L)) → remove_out_gga(N, .(N, L), L)
U10_gga(N, M, L, remove_out_gga(N, L, L1)) → remove_out_gga(N, .(M, L), .(M, L1))
remove_in_gga(N, .(M, L)) → U9_gga(N, M, L, notEq_in_gg(N, M))
U13_gg(X, Y, notEq_out_gg(X, Y)) → notEq_out_gg(s(X), s(Y))
(66) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U1_GA(.(x1, x2), min1_out_ag(x0, .(x1, x2))) → U2_GA(.(x1, x2), x0, U9_gga(x0, x1, x2, notEq_in_gg(x0, x1)))
MINSORT_IN_GA(.(x0, x1)) → U1_GA(.(x0, x1), U4_ag(x0, x1, min2_in_gag(x0, x1)))
U2_GA(.(x0, .(y_0, y_1)), x0, remove_out_gga(x0, .(x0, .(y_0, y_1)), .(y_0, y_1))) → MINSORT_IN_GA(.(y_0, y_1))
U2_GA(.(x0, x1), x2, remove_out_gga(x2, .(x0, x1), .(y_0, y_1))) → MINSORT_IN_GA(.(y_0, y_1))
The TRS R consists of the following rules:
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))
U4_ag(X, L, min2_out_gag(X, M, L)) → min1_out_ag(M, .(X, 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))
U5_gag(X, M, L, min_out_gga(X, M, B)) → U6_gag(X, M, L, min2_in_gag(B, L))
U6_gag(X, M, L, min2_out_gag(B, A, L)) → min2_out_gag(X, A, .(M, L))
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)
U8_gga(X, Y, gt_out_gg(X, Y)) → min_out_gga(X, Y, Y)
U11_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(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)
U7_gga(X, Y, le_out_gg(X, Y)) → min_out_gga(X, Y, X)
U12_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
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))
U9_gga(N, M, L, notEq_out_gg(N, M)) → U10_gga(N, M, L, remove_in_gga(N, L))
remove_in_gga(N, []) → remove_out_gga(N, [], [])
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))
U10_gga(N, M, L, remove_out_gga(N, L, L1)) → remove_out_gga(N, .(M, L), .(M, L1))
U13_gg(X, Y, notEq_out_gg(X, Y)) → notEq_out_gg(s(X), s(Y))
The set Q consists of the following terms:
remove_in_gga(x0, x1)
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.
(67) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
U2_GA(.(x0, .(y_0, y_1)), x0, remove_out_gga(x0, .(x0, .(y_0, y_1)), .(y_0, y_1))) → MINSORT_IN_GA(.(y_0, y_1))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
| POL(U1_GA(x1, x2)) = | -I | + | 0A | · | x1 | + | -I | · | x2 |
| POL(.(x1, x2)) = | 0A | + | 0A | · | x1 | + | 1A | · | x2 |
| POL(min1_out_ag(x1, x2)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 |
| POL(U2_GA(x1, x2, x3)) = | -I | + | 0A | · | x1 | + | -I | · | x2 | + | 0A | · | x3 |
| POL(U9_gga(x1, x2, x3, x4)) = | 0A | + | -I | · | x1 | + | 0A | · | x2 | + | 1A | · | x3 | + | -I | · | x4 |
| POL(notEq_in_gg(x1, x2)) = | 0A | + | -I | · | x1 | + | -I | · | x2 |
| POL(MINSORT_IN_GA(x1)) = | -I | + | 0A | · | x1 |
| POL(U4_ag(x1, x2, x3)) = | -I | + | 0A | · | x1 | + | -I | · | x2 | + | -I | · | x3 |
| POL(min2_in_gag(x1, x2)) = | -I | + | -I | · | x1 | + | 0A | · | x2 |
| POL(remove_out_gga(x1, x2, x3)) = | -I | + | -I | · | x1 | + | -I | · | x2 | + | 0A | · | x3 |
| POL(U13_gg(x1, x2, x3)) = | 5A | + | -I | · | x1 | + | -I | · | x2 | + | 0A | · | x3 |
| POL(notEq_out_gg(x1, x2)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 |
| POL(U10_gga(x1, x2, x3, x4)) = | 0A | + | -I | · | x1 | + | 0A | · | x2 | + | -I | · | x3 | + | 1A | · | x4 |
| POL(remove_in_gga(x1, x2)) = | -I | + | -I | · | x1 | + | 0A | · | x2 |
| POL(min2_out_gag(x1, x2, x3)) = | -I | + | 5A | · | x1 | + | 0A | · | x2 | + | 4A | · | x3 |
| POL(U5_gag(x1, x2, x3, x4)) = | -I | + | 2A | · | x1 | + | -I | · | x2 | + | 5A | · | x3 | + | -I | · | x4 |
| POL(min_in_gga(x1, x2)) = | 0A | + | -I | · | x1 | + | -I | · | x2 |
| POL(U7_gga(x1, x2, x3)) = | 5A | + | 1A | · | x1 | + | 4A | · | x2 | + | -I | · | x3 |
| POL(le_in_gg(x1, x2)) = | -I | + | -I | · | x1 | + | 0A | · | x2 |
| POL(U8_gga(x1, x2, x3)) = | 5A | + | -I | · | x1 | + | -I | · | x2 | + | -I | · | x3 |
| POL(gt_in_gg(x1, x2)) = | 4A | + | -I | · | x1 | + | -I | · | x2 |
| POL(min_out_gga(x1, x2, x3)) = | -I | + | -I | · | x1 | + | -I | · | x2 | + | 5A | · | x3 |
| POL(U6_gag(x1, x2, x3, x4)) = | -I | + | -I | · | x1 | + | 1A | · | x2 | + | -I | · | x3 | + | -I | · | x4 |
| POL(U12_gg(x1, x2, x3)) = | 0A | + | -I | · | x1 | + | -I | · | x2 | + | -I | · | x3 |
| POL(le_out_gg(x1, x2)) = | 0A | + | 0A | · | x1 | + | -I | · | x2 |
| POL(U11_gg(x1, x2, x3)) = | 4A | + | -I | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
| POL(gt_out_gg(x1, x2)) = | -I | + | 1A | · | x1 | + | 0A | · | x2 |
The following usable rules [FROCOS05] were oriented:
U9_gga(N, M, L, notEq_out_gg(N, M)) → U10_gga(N, M, L, remove_in_gga(N, L))
remove_in_gga(N, []) → remove_out_gga(N, [], [])
remove_in_gga(N, .(N, L)) → remove_out_gga(N, .(N, L), L)
U10_gga(N, M, L, remove_out_gga(N, L, L1)) → remove_out_gga(N, .(M, L), .(M, L1))
remove_in_gga(N, .(M, L)) → U9_gga(N, M, L, notEq_in_gg(N, M))
(68) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U1_GA(.(x1, x2), min1_out_ag(x0, .(x1, x2))) → U2_GA(.(x1, x2), x0, U9_gga(x0, x1, x2, notEq_in_gg(x0, x1)))
MINSORT_IN_GA(.(x0, x1)) → U1_GA(.(x0, x1), U4_ag(x0, x1, min2_in_gag(x0, x1)))
U2_GA(.(x0, x1), x2, remove_out_gga(x2, .(x0, x1), .(y_0, y_1))) → MINSORT_IN_GA(.(y_0, y_1))
The TRS R consists of the following rules:
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))
U4_ag(X, L, min2_out_gag(X, M, L)) → min1_out_ag(M, .(X, 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))
U5_gag(X, M, L, min_out_gga(X, M, B)) → U6_gag(X, M, L, min2_in_gag(B, L))
U6_gag(X, M, L, min2_out_gag(B, A, L)) → min2_out_gag(X, A, .(M, L))
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)
U8_gga(X, Y, gt_out_gg(X, Y)) → min_out_gga(X, Y, Y)
U11_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(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)
U7_gga(X, Y, le_out_gg(X, Y)) → min_out_gga(X, Y, X)
U12_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
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))
U9_gga(N, M, L, notEq_out_gg(N, M)) → U10_gga(N, M, L, remove_in_gga(N, L))
remove_in_gga(N, []) → remove_out_gga(N, [], [])
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))
U10_gga(N, M, L, remove_out_gga(N, L, L1)) → remove_out_gga(N, .(M, L), .(M, L1))
U13_gg(X, Y, notEq_out_gg(X, Y)) → notEq_out_gg(s(X), s(Y))
The set Q consists of the following terms:
remove_in_gga(x0, x1)
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.
(69) 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, [], []) → remove_out_gga(N, [], [])
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
(70) 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, [], []) → remove_out_gga(N, [], [])
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)
(71) 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, [], []) → remove_out_gga(N, [], [])
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
(72) 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, [], []) → remove_out_gga(N, [], [])
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
(73) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 16 less nodes.
(74) Complex Obligation (AND)
(75) 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, [], []) → remove_out_gga(N, [], [])
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
(76) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
(77) 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
(78) PiDPToQDPProof (EQUIVALENT transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(79) 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.
(80) 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
(81) TRUE
(82) 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, [], []) → remove_out_gga(N, [], [])
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
(83) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
(84) 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
(85) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(86) 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.
(87) 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
(88) TRUE
(89) 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, [], []) → remove_out_gga(N, [], [])
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
(90) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
(91) 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
(92) PiDPToQDPProof (EQUIVALENT transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(93) 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.
(94) 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
(95) TRUE
(96) 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, [], []) → remove_out_gga(N, [], [])
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
(97) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
(98) 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
(99) PiDPToQDPProof (EQUIVALENT transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(100) 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.
(101) 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
(102) TRUE
(103) 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, [], []) → remove_out_gga(N, [], [])
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
(104) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
(105) 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
(106) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(107) 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.
(108) 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
(109) TRUE
(110) 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, [], []) → remove_out_gga(N, [], [])
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
(111) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
(112) 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, [], []) → remove_out_gga(N, [], [])
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
(113) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(114) 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, []) → remove_out_gga([])
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.
(115) Narrowing (SOUND transformation)
By narrowing [LPAR04] the rule
U1_GA(
L,
min1_out_ag(
X)) →
U2_GA(
X,
remove_in_gga(
X,
L)) at position [1] we obtained the following new rules [LPAR04]:
U1_GA([], min1_out_ag(x0)) → U2_GA(x0, remove_out_gga([]))
U1_GA(.(x0, x1), min1_out_ag(x0)) → U2_GA(x0, remove_out_gga(x1))
U1_GA(.(x1, x2), min1_out_ag(x0)) → U2_GA(x0, U9_gga(x0, x1, x2, notEq_in_gg(x0, x1)))
(116) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U2_GA(X, remove_out_gga(L2)) → MINSORT_IN_GA(L2)
MINSORT_IN_GA(L) → U1_GA(L, min1_in_ag(L))
U1_GA([], min1_out_ag(x0)) → U2_GA(x0, remove_out_gga([]))
U1_GA(.(x0, x1), min1_out_ag(x0)) → U2_GA(x0, remove_out_gga(x1))
U1_GA(.(x1, x2), min1_out_ag(x0)) → U2_GA(x0, U9_gga(x0, x1, x2, notEq_in_gg(x0, x1)))
The TRS R consists of the following rules:
remove_in_gga(N, []) → remove_out_gga([])
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.
(117) Narrowing (SOUND transformation)
By narrowing [LPAR04] the rule
MINSORT_IN_GA(
L) →
U1_GA(
L,
min1_in_ag(
L)) at position [1] we obtained the following new rules [LPAR04]:
MINSORT_IN_GA(.(x0, x1)) → U1_GA(.(x0, x1), U4_ag(min2_in_gag(x0, x1)))
(118) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U2_GA(X, remove_out_gga(L2)) → MINSORT_IN_GA(L2)
U1_GA([], min1_out_ag(x0)) → U2_GA(x0, remove_out_gga([]))
U1_GA(.(x0, x1), min1_out_ag(x0)) → U2_GA(x0, remove_out_gga(x1))
U1_GA(.(x1, x2), min1_out_ag(x0)) → U2_GA(x0, U9_gga(x0, x1, x2, notEq_in_gg(x0, x1)))
MINSORT_IN_GA(.(x0, x1)) → U1_GA(.(x0, x1), U4_ag(min2_in_gag(x0, x1)))
The TRS R consists of the following rules:
remove_in_gga(N, []) → remove_out_gga([])
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.
(119) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
(120) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MINSORT_IN_GA(.(x0, x1)) → U1_GA(.(x0, x1), U4_ag(min2_in_gag(x0, x1)))
U1_GA(.(x0, x1), min1_out_ag(x0)) → U2_GA(x0, remove_out_gga(x1))
U2_GA(X, remove_out_gga(L2)) → MINSORT_IN_GA(L2)
U1_GA(.(x1, x2), min1_out_ag(x0)) → U2_GA(x0, U9_gga(x0, x1, x2, notEq_in_gg(x0, x1)))
The TRS R consists of the following rules:
remove_in_gga(N, []) → remove_out_gga([])
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.
(121) UsableRulesProof (EQUIVALENT transformation)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
(122) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MINSORT_IN_GA(.(x0, x1)) → U1_GA(.(x0, x1), U4_ag(min2_in_gag(x0, x1)))
U1_GA(.(x0, x1), min1_out_ag(x0)) → U2_GA(x0, remove_out_gga(x1))
U2_GA(X, remove_out_gga(L2)) → MINSORT_IN_GA(L2)
U1_GA(.(x1, x2), min1_out_ag(x0)) → U2_GA(x0, U9_gga(x0, x1, x2, notEq_in_gg(x0, x1)))
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
U9_gga(N, M, L, notEq_out_gg) → U10_gga(M, remove_in_gga(N, L))
remove_in_gga(N, []) → remove_out_gga([])
remove_in_gga(N, .(N, L)) → remove_out_gga(L)
remove_in_gga(N, .(M, L)) → U9_gga(N, M, L, notEq_in_gg(N, M))
U10_gga(M, remove_out_gga(L1)) → remove_out_gga(.(M, L1))
U13_gg(notEq_out_gg) → notEq_out_gg
min2_in_gag(X, []) → min2_out_gag(X)
min2_in_gag(X, .(M, L)) → U5_gag(L, min_in_gga(X, M))
U4_ag(min2_out_gag(M)) → min1_out_ag(M)
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))
U5_gag(L, min_out_gga(B)) → U6_gag(min2_in_gag(B, L))
U6_gag(min2_out_gag(A)) → min2_out_gag(A)
gt_in_gg(s(X), s(Y)) → U11_gg(gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg
U8_gga(Y, gt_out_gg) → min_out_gga(Y)
U11_gg(gt_out_gg) → gt_out_gg
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
U7_gga(X, le_out_gg) → min_out_gga(X)
U12_gg(le_out_gg) → le_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.
(123) QReductionProof (EQUIVALENT transformation)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
min1_in_ag(x0)
(124) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MINSORT_IN_GA(.(x0, x1)) → U1_GA(.(x0, x1), U4_ag(min2_in_gag(x0, x1)))
U1_GA(.(x0, x1), min1_out_ag(x0)) → U2_GA(x0, remove_out_gga(x1))
U2_GA(X, remove_out_gga(L2)) → MINSORT_IN_GA(L2)
U1_GA(.(x1, x2), min1_out_ag(x0)) → U2_GA(x0, U9_gga(x0, x1, x2, notEq_in_gg(x0, x1)))
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
U9_gga(N, M, L, notEq_out_gg) → U10_gga(M, remove_in_gga(N, L))
remove_in_gga(N, []) → remove_out_gga([])
remove_in_gga(N, .(N, L)) → remove_out_gga(L)
remove_in_gga(N, .(M, L)) → U9_gga(N, M, L, notEq_in_gg(N, M))
U10_gga(M, remove_out_gga(L1)) → remove_out_gga(.(M, L1))
U13_gg(notEq_out_gg) → notEq_out_gg
min2_in_gag(X, []) → min2_out_gag(X)
min2_in_gag(X, .(M, L)) → U5_gag(L, min_in_gga(X, M))
U4_ag(min2_out_gag(M)) → min1_out_ag(M)
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))
U5_gag(L, min_out_gga(B)) → U6_gag(min2_in_gag(B, L))
U6_gag(min2_out_gag(A)) → min2_out_gag(A)
gt_in_gg(s(X), s(Y)) → U11_gg(gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg
U8_gga(Y, gt_out_gg) → min_out_gga(Y)
U11_gg(gt_out_gg) → gt_out_gg
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
U7_gga(X, le_out_gg) → min_out_gga(X)
U12_gg(le_out_gg) → le_out_gg
The set Q consists of the following terms:
remove_in_gga(x0, x1)
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.
(125) ForwardInstantiation (EQUIVALENT transformation)
By forward instantiating [JAR06] the rule
U2_GA(
X,
remove_out_gga(
L2)) →
MINSORT_IN_GA(
L2) we obtained the following new rules [LPAR04]:
U2_GA(x0, remove_out_gga(.(y_0, y_1))) → MINSORT_IN_GA(.(y_0, y_1))
(126) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MINSORT_IN_GA(.(x0, x1)) → U1_GA(.(x0, x1), U4_ag(min2_in_gag(x0, x1)))
U1_GA(.(x0, x1), min1_out_ag(x0)) → U2_GA(x0, remove_out_gga(x1))
U1_GA(.(x1, x2), min1_out_ag(x0)) → U2_GA(x0, U9_gga(x0, x1, x2, notEq_in_gg(x0, x1)))
U2_GA(x0, remove_out_gga(.(y_0, y_1))) → MINSORT_IN_GA(.(y_0, y_1))
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
U9_gga(N, M, L, notEq_out_gg) → U10_gga(M, remove_in_gga(N, L))
remove_in_gga(N, []) → remove_out_gga([])
remove_in_gga(N, .(N, L)) → remove_out_gga(L)
remove_in_gga(N, .(M, L)) → U9_gga(N, M, L, notEq_in_gg(N, M))
U10_gga(M, remove_out_gga(L1)) → remove_out_gga(.(M, L1))
U13_gg(notEq_out_gg) → notEq_out_gg
min2_in_gag(X, []) → min2_out_gag(X)
min2_in_gag(X, .(M, L)) → U5_gag(L, min_in_gga(X, M))
U4_ag(min2_out_gag(M)) → min1_out_ag(M)
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))
U5_gag(L, min_out_gga(B)) → U6_gag(min2_in_gag(B, L))
U6_gag(min2_out_gag(A)) → min2_out_gag(A)
gt_in_gg(s(X), s(Y)) → U11_gg(gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg
U8_gga(Y, gt_out_gg) → min_out_gga(Y)
U11_gg(gt_out_gg) → gt_out_gg
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
U7_gga(X, le_out_gg) → min_out_gga(X)
U12_gg(le_out_gg) → le_out_gg
The set Q consists of the following terms:
remove_in_gga(x0, x1)
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.
(127) ForwardInstantiation (EQUIVALENT transformation)
By forward instantiating [JAR06] the rule
U1_GA(
.(
x0,
x1),
min1_out_ag(
x0)) →
U2_GA(
x0,
remove_out_gga(
x1)) we obtained the following new rules [LPAR04]:
U1_GA(.(x0, .(y_1, y_2)), min1_out_ag(x0)) → U2_GA(x0, remove_out_gga(.(y_1, y_2)))
(128) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MINSORT_IN_GA(.(x0, x1)) → U1_GA(.(x0, x1), U4_ag(min2_in_gag(x0, x1)))
U1_GA(.(x1, x2), min1_out_ag(x0)) → U2_GA(x0, U9_gga(x0, x1, x2, notEq_in_gg(x0, x1)))
U2_GA(x0, remove_out_gga(.(y_0, y_1))) → MINSORT_IN_GA(.(y_0, y_1))
U1_GA(.(x0, .(y_1, y_2)), min1_out_ag(x0)) → U2_GA(x0, remove_out_gga(.(y_1, y_2)))
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
U9_gga(N, M, L, notEq_out_gg) → U10_gga(M, remove_in_gga(N, L))
remove_in_gga(N, []) → remove_out_gga([])
remove_in_gga(N, .(N, L)) → remove_out_gga(L)
remove_in_gga(N, .(M, L)) → U9_gga(N, M, L, notEq_in_gg(N, M))
U10_gga(M, remove_out_gga(L1)) → remove_out_gga(.(M, L1))
U13_gg(notEq_out_gg) → notEq_out_gg
min2_in_gag(X, []) → min2_out_gag(X)
min2_in_gag(X, .(M, L)) → U5_gag(L, min_in_gga(X, M))
U4_ag(min2_out_gag(M)) → min1_out_ag(M)
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))
U5_gag(L, min_out_gga(B)) → U6_gag(min2_in_gag(B, L))
U6_gag(min2_out_gag(A)) → min2_out_gag(A)
gt_in_gg(s(X), s(Y)) → U11_gg(gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg
U8_gga(Y, gt_out_gg) → min_out_gga(Y)
U11_gg(gt_out_gg) → gt_out_gg
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
U7_gga(X, le_out_gg) → min_out_gga(X)
U12_gg(le_out_gg) → le_out_gg
The set Q consists of the following terms:
remove_in_gga(x0, x1)
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.
(129) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
U1_GA(.(x0, .(y_1, y_2)), min1_out_ag(x0)) → U2_GA(x0, remove_out_gga(.(y_1, y_2)))
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)) = 1
POL(U9_gga(x1, x2, x3, x4)) = 1 + x3
POL([]) = 0
POL(gt_in_gg(x1, x2)) = 0
POL(gt_out_gg) = 0
POL(le_in_gg(x1, x2)) = 0
POL(le_out_gg) = 0
POL(min1_out_ag(x1)) = 0
POL(min2_in_gag(x1, x2)) = 0
POL(min2_out_gag(x1)) = 0
POL(min_in_gga(x1, x2)) = 1 + x1
POL(min_out_gga(x1)) = 0
POL(notEq_in_gg(x1, x2)) = 0
POL(notEq_out_gg) = 0
POL(remove_in_gga(x1, x2)) = x2
POL(remove_out_gga(x1)) = x1
POL(s(x1)) = 0
The following usable rules [FROCOS05] were oriented:
U9_gga(N, M, L, notEq_out_gg) → U10_gga(M, remove_in_gga(N, L))
remove_in_gga(N, []) → remove_out_gga([])
remove_in_gga(N, .(N, L)) → remove_out_gga(L)
U10_gga(M, remove_out_gga(L1)) → remove_out_gga(.(M, L1))
remove_in_gga(N, .(M, L)) → U9_gga(N, M, L, notEq_in_gg(N, M))
(130) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MINSORT_IN_GA(.(x0, x1)) → U1_GA(.(x0, x1), U4_ag(min2_in_gag(x0, x1)))
U1_GA(.(x1, x2), min1_out_ag(x0)) → U2_GA(x0, U9_gga(x0, x1, x2, notEq_in_gg(x0, x1)))
U2_GA(x0, remove_out_gga(.(y_0, y_1))) → MINSORT_IN_GA(.(y_0, y_1))
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
U9_gga(N, M, L, notEq_out_gg) → U10_gga(M, remove_in_gga(N, L))
remove_in_gga(N, []) → remove_out_gga([])
remove_in_gga(N, .(N, L)) → remove_out_gga(L)
remove_in_gga(N, .(M, L)) → U9_gga(N, M, L, notEq_in_gg(N, M))
U10_gga(M, remove_out_gga(L1)) → remove_out_gga(.(M, L1))
U13_gg(notEq_out_gg) → notEq_out_gg
min2_in_gag(X, []) → min2_out_gag(X)
min2_in_gag(X, .(M, L)) → U5_gag(L, min_in_gga(X, M))
U4_ag(min2_out_gag(M)) → min1_out_ag(M)
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))
U5_gag(L, min_out_gga(B)) → U6_gag(min2_in_gag(B, L))
U6_gag(min2_out_gag(A)) → min2_out_gag(A)
gt_in_gg(s(X), s(Y)) → U11_gg(gt_in_gg(X, Y))
gt_in_gg(s(X), 0) → gt_out_gg
U8_gga(Y, gt_out_gg) → min_out_gga(Y)
U11_gg(gt_out_gg) → gt_out_gg
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
U7_gga(X, le_out_gg) → min_out_gga(X)
U12_gg(le_out_gg) → le_out_gg
The set Q consists of the following terms:
remove_in_gga(x0, x1)
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.