Left Termination proof of /tmp/tmps3Worl/pl8.3.1a.pl

Left Termination of the query pattern minsort(b,f) w.r.t. the given Prolog program could successfully be proven:

↳ PROLOG

  ↳ PrologToPiTRSProof

minsort₂({}₀, {}₀).
minsort₂(L, .₂(X, L1)) :- min1₂(X, L), remove₃(X, L, L2), minsort₂(L2, L1).
min1₂(M, .₂(X, L)) :- min2₃(X, M, L).
min2₃(X, X, {}₀).
min2₃(X, A, .₂(M, L)) :- min₃(X, M, B), min2₃(B, A, L).
min₃(X, Y, X) :- le₂(X, Y).
min₃(X, Y, Y) :- gt₂(X, Y).
remove₃(N, .₂(N, L), L).
remove₃(N, .₂(M, L), .₂(M, L1)) :- notEq₂(N, M), remove₃(N, L, L1).
gt₂(s₁(X), s₁(Y)) :- gt₂(X, Y).
gt₂(s₁(X), 0₀).
le₂(s₁(X), s₁(Y)) :- le₂(X, Y).
le₂(0₀, s₁(Y)).
le₂(0₀, 0₀).
notEq₂(s₁(X), s₁(Y)) :- notEq₂(X, Y).
notEq₂(s₁(X), 0₀).
notEq₂(0₀, s₁(X)).

With regard to the inferred argument filtering the predicates were used in the following modes:
minsort₂: (b,f)
min1₂: (f,b)
min2₃: (b,f,b)
min₃: (b,b,f)
le₂: (b,b)
gt₂: (b,b)
remove₃: (b,b,f)
notEq₂: (b,b)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

minsort_2_in_ga₂([]_0, []_0) -> minsort_2_out_ga₂([]_0, []_0)
minsort_2_in_ga₂(L, ._2₂(X, L1)) -> if_minsort_2_in_1_ga₄(L, X, L1, min1_2_in_ag₂(X, L))
min1_2_in_ag₂(M, ._2₂(X, L)) -> if_min1_2_in_1_ag₄(M, X, L, min2_3_in_gag₃(X, M, L))
min2_3_in_gag₃(X, X, []_0) -> min2_3_out_gag₃(X, X, []_0)
min2_3_in_gag₃(X, A, ._2₂(M, L)) -> if_min2_3_in_1_gag₅(X, A, M, L, min_3_in_gga₃(X, M, B))
min_3_in_gga₃(X, Y, X) -> if_min_3_in_1_gga₃(X, Y, le_2_in_gg₂(X, Y))
le_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_le_2_in_1_gg₃(X, Y, le_2_in_gg₂(X, Y))
le_2_in_gg₂(0_0, s_1₁(Y)) -> le_2_out_gg₂(0_0, s_1₁(Y))
le_2_in_gg₂(0_0, 0_0) -> le_2_out_gg₂(0_0, 0_0)
if_le_2_in_1_gg₃(X, Y, le_2_out_gg₂(X, Y)) -> le_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_min_3_in_1_gga₃(X, Y, le_2_out_gg₂(X, Y)) -> min_3_out_gga₃(X, Y, X)
min_3_in_gga₃(X, Y, Y) -> if_min_3_in_2_gga₃(X, Y, gt_2_in_gg₂(X, Y))
gt_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_gt_2_in_1_gg₃(X, Y, gt_2_in_gg₂(X, Y))
gt_2_in_gg₂(s_1₁(X), 0_0) -> gt_2_out_gg₂(s_1₁(X), 0_0)
if_gt_2_in_1_gg₃(X, Y, gt_2_out_gg₂(X, Y)) -> gt_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_min_3_in_2_gga₃(X, Y, gt_2_out_gg₂(X, Y)) -> min_3_out_gga₃(X, Y, Y)
if_min2_3_in_1_gag₅(X, A, M, L, min_3_out_gga₃(X, M, B)) -> if_min2_3_in_2_gag₆(X, A, M, L, B, min2_3_in_gag₃(B, A, L))
if_min2_3_in_2_gag₆(X, A, M, L, B, min2_3_out_gag₃(B, A, L)) -> min2_3_out_gag₃(X, A, ._2₂(M, L))
if_min1_2_in_1_ag₄(M, X, L, min2_3_out_gag₃(X, M, L)) -> min1_2_out_ag₂(M, ._2₂(X, L))
if_minsort_2_in_1_ga₄(L, X, L1, min1_2_out_ag₂(X, L)) -> if_minsort_2_in_2_ga₄(L, X, L1, remove_3_in_gga₃(X, L, L2))
remove_3_in_gga₃(N, ._2₂(N, L), L) -> remove_3_out_gga₃(N, ._2₂(N, L), L)
remove_3_in_gga₃(N, ._2₂(M, L), ._2₂(M, L1)) -> if_remove_3_in_1_gga₅(N, M, L, L1, notEq_2_in_gg₂(N, M))
notEq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_notEq_2_in_1_gg₃(X, Y, notEq_2_in_gg₂(X, Y))
notEq_2_in_gg₂(s_1₁(X), 0_0) -> notEq_2_out_gg₂(s_1₁(X), 0_0)
notEq_2_in_gg₂(0_0, s_1₁(X)) -> notEq_2_out_gg₂(0_0, s_1₁(X))
if_notEq_2_in_1_gg₃(X, Y, notEq_2_out_gg₂(X, Y)) -> notEq_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_remove_3_in_1_gga₅(N, M, L, L1, notEq_2_out_gg₂(N, M)) -> if_remove_3_in_2_gga₅(N, M, L, L1, remove_3_in_gga₃(N, L, L1))
if_remove_3_in_2_gga₅(N, M, L, L1, remove_3_out_gga₃(N, L, L1)) -> remove_3_out_gga₃(N, ._2₂(M, L), ._2₂(M, L1))
if_minsort_2_in_2_ga₄(L, X, L1, remove_3_out_gga₃(X, L, L2)) -> if_minsort_2_in_3_ga₅(L, X, L1, L2, minsort_2_in_ga₂(L2, L1))
if_minsort_2_in_3_ga₅(L, X, L1, L2, minsort_2_out_ga₂(L2, L1)) -> minsort_2_out_ga₂(L, ._2₂(X, L1))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of PROLOG

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

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

minsort_2_in_ga₂([]_0, []_0) -> minsort_2_out_ga₂([]_0, []_0)
minsort_2_in_ga₂(L, ._2₂(X, L1)) -> if_minsort_2_in_1_ga₄(L, X, L1, min1_2_in_ag₂(X, L))
min1_2_in_ag₂(M, ._2₂(X, L)) -> if_min1_2_in_1_ag₄(M, X, L, min2_3_in_gag₃(X, M, L))
min2_3_in_gag₃(X, X, []_0) -> min2_3_out_gag₃(X, X, []_0)
min2_3_in_gag₃(X, A, ._2₂(M, L)) -> if_min2_3_in_1_gag₅(X, A, M, L, min_3_in_gga₃(X, M, B))
min_3_in_gga₃(X, Y, X) -> if_min_3_in_1_gga₃(X, Y, le_2_in_gg₂(X, Y))
le_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_le_2_in_1_gg₃(X, Y, le_2_in_gg₂(X, Y))
le_2_in_gg₂(0_0, s_1₁(Y)) -> le_2_out_gg₂(0_0, s_1₁(Y))
le_2_in_gg₂(0_0, 0_0) -> le_2_out_gg₂(0_0, 0_0)
if_le_2_in_1_gg₃(X, Y, le_2_out_gg₂(X, Y)) -> le_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_min_3_in_1_gga₃(X, Y, le_2_out_gg₂(X, Y)) -> min_3_out_gga₃(X, Y, X)
min_3_in_gga₃(X, Y, Y) -> if_min_3_in_2_gga₃(X, Y, gt_2_in_gg₂(X, Y))
gt_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_gt_2_in_1_gg₃(X, Y, gt_2_in_gg₂(X, Y))
gt_2_in_gg₂(s_1₁(X), 0_0) -> gt_2_out_gg₂(s_1₁(X), 0_0)
if_gt_2_in_1_gg₃(X, Y, gt_2_out_gg₂(X, Y)) -> gt_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_min_3_in_2_gga₃(X, Y, gt_2_out_gg₂(X, Y)) -> min_3_out_gga₃(X, Y, Y)
if_min2_3_in_1_gag₅(X, A, M, L, min_3_out_gga₃(X, M, B)) -> if_min2_3_in_2_gag₆(X, A, M, L, B, min2_3_in_gag₃(B, A, L))
if_min2_3_in_2_gag₆(X, A, M, L, B, min2_3_out_gag₃(B, A, L)) -> min2_3_out_gag₃(X, A, ._2₂(M, L))
if_min1_2_in_1_ag₄(M, X, L, min2_3_out_gag₃(X, M, L)) -> min1_2_out_ag₂(M, ._2₂(X, L))
if_minsort_2_in_1_ga₄(L, X, L1, min1_2_out_ag₂(X, L)) -> if_minsort_2_in_2_ga₄(L, X, L1, remove_3_in_gga₃(X, L, L2))
remove_3_in_gga₃(N, ._2₂(N, L), L) -> remove_3_out_gga₃(N, ._2₂(N, L), L)
remove_3_in_gga₃(N, ._2₂(M, L), ._2₂(M, L1)) -> if_remove_3_in_1_gga₅(N, M, L, L1, notEq_2_in_gg₂(N, M))
notEq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_notEq_2_in_1_gg₃(X, Y, notEq_2_in_gg₂(X, Y))
notEq_2_in_gg₂(s_1₁(X), 0_0) -> notEq_2_out_gg₂(s_1₁(X), 0_0)
notEq_2_in_gg₂(0_0, s_1₁(X)) -> notEq_2_out_gg₂(0_0, s_1₁(X))
if_notEq_2_in_1_gg₃(X, Y, notEq_2_out_gg₂(X, Y)) -> notEq_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_remove_3_in_1_gga₅(N, M, L, L1, notEq_2_out_gg₂(N, M)) -> if_remove_3_in_2_gga₅(N, M, L, L1, remove_3_in_gga₃(N, L, L1))
if_remove_3_in_2_gga₅(N, M, L, L1, remove_3_out_gga₃(N, L, L1)) -> remove_3_out_gga₃(N, ._2₂(M, L), ._2₂(M, L1))
if_minsort_2_in_2_ga₄(L, X, L1, remove_3_out_gga₃(X, L, L2)) -> if_minsort_2_in_3_ga₅(L, X, L1, L2, minsort_2_in_ga₂(L2, L1))
if_minsort_2_in_3_ga₅(L, X, L1, L2, minsort_2_out_ga₂(L2, L1)) -> minsort_2_out_ga₂(L, ._2₂(X, L1))

MINSORT_2_IN_GA₂(L, ._2₂(X, L1)) -> IF_MINSORT_2_IN_1_GA₄(L, X, L1, min1_2_in_ag₂(X, L))
MINSORT_2_IN_GA₂(L, ._2₂(X, L1)) -> MIN1_2_IN_AG₂(X, L)
MIN1_2_IN_AG₂(M, ._2₂(X, L)) -> IF_MIN1_2_IN_1_AG₄(M, X, L, min2_3_in_gag₃(X, M, L))
MIN1_2_IN_AG₂(M, ._2₂(X, L)) -> MIN2_3_IN_GAG₃(X, M, L)
MIN2_3_IN_GAG₃(X, A, ._2₂(M, L)) -> IF_MIN2_3_IN_1_GAG₅(X, A, M, L, min_3_in_gga₃(X, M, B))
MIN2_3_IN_GAG₃(X, A, ._2₂(M, L)) -> MIN_3_IN_GGA₃(X, M, B)
MIN_3_IN_GGA₃(X, Y, X) -> IF_MIN_3_IN_1_GGA₃(X, Y, le_2_in_gg₂(X, Y))
MIN_3_IN_GGA₃(X, Y, X) -> LE_2_IN_GG₂(X, Y)
LE_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> IF_LE_2_IN_1_GG₃(X, Y, le_2_in_gg₂(X, Y))
LE_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> LE_2_IN_GG₂(X, Y)
MIN_3_IN_GGA₃(X, Y, Y) -> IF_MIN_3_IN_2_GGA₃(X, Y, gt_2_in_gg₂(X, Y))
MIN_3_IN_GGA₃(X, Y, Y) -> GT_2_IN_GG₂(X, Y)
GT_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> IF_GT_2_IN_1_GG₃(X, Y, gt_2_in_gg₂(X, Y))
GT_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> GT_2_IN_GG₂(X, Y)
IF_MIN2_3_IN_1_GAG₅(X, A, M, L, min_3_out_gga₃(X, M, B)) -> IF_MIN2_3_IN_2_GAG₆(X, A, M, L, B, min2_3_in_gag₃(B, A, L))
IF_MIN2_3_IN_1_GAG₅(X, A, M, L, min_3_out_gga₃(X, M, B)) -> MIN2_3_IN_GAG₃(B, A, L)
IF_MINSORT_2_IN_1_GA₄(L, X, L1, min1_2_out_ag₂(X, L)) -> IF_MINSORT_2_IN_2_GA₄(L, X, L1, remove_3_in_gga₃(X, L, L2))
IF_MINSORT_2_IN_1_GA₄(L, X, L1, min1_2_out_ag₂(X, L)) -> REMOVE_3_IN_GGA₃(X, L, L2)
REMOVE_3_IN_GGA₃(N, ._2₂(M, L), ._2₂(M, L1)) -> IF_REMOVE_3_IN_1_GGA₅(N, M, L, L1, notEq_2_in_gg₂(N, M))
REMOVE_3_IN_GGA₃(N, ._2₂(M, L), ._2₂(M, L1)) -> NOTEQ_2_IN_GG₂(N, M)
NOTEQ_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> IF_NOTEQ_2_IN_1_GG₃(X, Y, notEq_2_in_gg₂(X, Y))
NOTEQ_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> NOTEQ_2_IN_GG₂(X, Y)
IF_REMOVE_3_IN_1_GGA₅(N, M, L, L1, notEq_2_out_gg₂(N, M)) -> IF_REMOVE_3_IN_2_GGA₅(N, M, L, L1, remove_3_in_gga₃(N, L, L1))
IF_REMOVE_3_IN_1_GGA₅(N, M, L, L1, notEq_2_out_gg₂(N, M)) -> REMOVE_3_IN_GGA₃(N, L, L1)
IF_MINSORT_2_IN_2_GA₄(L, X, L1, remove_3_out_gga₃(X, L, L2)) -> IF_MINSORT_2_IN_3_GA₅(L, X, L1, L2, minsort_2_in_ga₂(L2, L1))
IF_MINSORT_2_IN_2_GA₄(L, X, L1, remove_3_out_gga₃(X, L, L2)) -> MINSORT_2_IN_GA₂(L2, L1)

The TRS R consists of the following rules:

minsort_2_in_ga₂([]_0, []_0) -> minsort_2_out_ga₂([]_0, []_0)
minsort_2_in_ga₂(L, ._2₂(X, L1)) -> if_minsort_2_in_1_ga₄(L, X, L1, min1_2_in_ag₂(X, L))
min1_2_in_ag₂(M, ._2₂(X, L)) -> if_min1_2_in_1_ag₄(M, X, L, min2_3_in_gag₃(X, M, L))
min2_3_in_gag₃(X, X, []_0) -> min2_3_out_gag₃(X, X, []_0)
min2_3_in_gag₃(X, A, ._2₂(M, L)) -> if_min2_3_in_1_gag₅(X, A, M, L, min_3_in_gga₃(X, M, B))
min_3_in_gga₃(X, Y, X) -> if_min_3_in_1_gga₃(X, Y, le_2_in_gg₂(X, Y))
le_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_le_2_in_1_gg₃(X, Y, le_2_in_gg₂(X, Y))
le_2_in_gg₂(0_0, s_1₁(Y)) -> le_2_out_gg₂(0_0, s_1₁(Y))
le_2_in_gg₂(0_0, 0_0) -> le_2_out_gg₂(0_0, 0_0)
if_le_2_in_1_gg₃(X, Y, le_2_out_gg₂(X, Y)) -> le_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_min_3_in_1_gga₃(X, Y, le_2_out_gg₂(X, Y)) -> min_3_out_gga₃(X, Y, X)
min_3_in_gga₃(X, Y, Y) -> if_min_3_in_2_gga₃(X, Y, gt_2_in_gg₂(X, Y))
gt_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_gt_2_in_1_gg₃(X, Y, gt_2_in_gg₂(X, Y))
gt_2_in_gg₂(s_1₁(X), 0_0) -> gt_2_out_gg₂(s_1₁(X), 0_0)
if_gt_2_in_1_gg₃(X, Y, gt_2_out_gg₂(X, Y)) -> gt_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_min_3_in_2_gga₃(X, Y, gt_2_out_gg₂(X, Y)) -> min_3_out_gga₃(X, Y, Y)
if_min2_3_in_1_gag₅(X, A, M, L, min_3_out_gga₃(X, M, B)) -> if_min2_3_in_2_gag₆(X, A, M, L, B, min2_3_in_gag₃(B, A, L))
if_min2_3_in_2_gag₆(X, A, M, L, B, min2_3_out_gag₃(B, A, L)) -> min2_3_out_gag₃(X, A, ._2₂(M, L))
if_min1_2_in_1_ag₄(M, X, L, min2_3_out_gag₃(X, M, L)) -> min1_2_out_ag₂(M, ._2₂(X, L))
if_minsort_2_in_1_ga₄(L, X, L1, min1_2_out_ag₂(X, L)) -> if_minsort_2_in_2_ga₄(L, X, L1, remove_3_in_gga₃(X, L, L2))
remove_3_in_gga₃(N, ._2₂(N, L), L) -> remove_3_out_gga₃(N, ._2₂(N, L), L)
remove_3_in_gga₃(N, ._2₂(M, L), ._2₂(M, L1)) -> if_remove_3_in_1_gga₅(N, M, L, L1, notEq_2_in_gg₂(N, M))
notEq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_notEq_2_in_1_gg₃(X, Y, notEq_2_in_gg₂(X, Y))
notEq_2_in_gg₂(s_1₁(X), 0_0) -> notEq_2_out_gg₂(s_1₁(X), 0_0)
notEq_2_in_gg₂(0_0, s_1₁(X)) -> notEq_2_out_gg₂(0_0, s_1₁(X))
if_notEq_2_in_1_gg₃(X, Y, notEq_2_out_gg₂(X, Y)) -> notEq_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_remove_3_in_1_gga₅(N, M, L, L1, notEq_2_out_gg₂(N, M)) -> if_remove_3_in_2_gga₅(N, M, L, L1, remove_3_in_gga₃(N, L, L1))
if_remove_3_in_2_gga₅(N, M, L, L1, remove_3_out_gga₃(N, L, L1)) -> remove_3_out_gga₃(N, ._2₂(M, L), ._2₂(M, L1))
if_minsort_2_in_2_ga₄(L, X, L1, remove_3_out_gga₃(X, L, L2)) -> if_minsort_2_in_3_ga₅(L, X, L1, L2, minsort_2_in_ga₂(L2, L1))
if_minsort_2_in_3_ga₅(L, X, L1, L2, minsort_2_out_ga₂(L2, L1)) -> minsort_2_out_ga₂(L, ._2₂(X, L1))

The argument filtering Pi contains the following mapping:
minsort_2_in_ga₂(x1, x2) = minsort_2_in_ga₁(x1)
[]_0 = []_0
._2₂(x1, x2) = ._2₂(x1, x2)
s_1₁(x1) = s_1₁(x1)
0_0 = 0_0
minsort_2_out_ga₂(x1, x2) = minsort_2_out_ga₁(x2)
if_minsort_2_in_1_ga₄(x1, x2, x3, x4) = if_minsort_2_in_1_ga₂(x1, x4)
min1_2_in_ag₂(x1, x2) = min1_2_in_ag₁(x2)
if_min1_2_in_1_ag₄(x1, x2, x3, x4) = if_min1_2_in_1_ag₁(x4)
min2_3_in_gag₃(x1, x2, x3) = min2_3_in_gag₂(x1, x3)
min2_3_out_gag₃(x1, x2, x3) = min2_3_out_gag₁(x2)
if_min2_3_in_1_gag₅(x1, x2, x3, x4, x5) = if_min2_3_in_1_gag₂(x4, x5)
min_3_in_gga₃(x1, x2, x3) = min_3_in_gga₂(x1, x2)
if_min_3_in_1_gga₃(x1, x2, x3) = if_min_3_in_1_gga₂(x1, x3)
le_2_in_gg₂(x1, x2) = le_2_in_gg₂(x1, x2)
if_le_2_in_1_gg₃(x1, x2, x3) = if_le_2_in_1_gg₁(x3)
le_2_out_gg₂(x1, x2) = le_2_out_gg
min_3_out_gga₃(x1, x2, x3) = min_3_out_gga₁(x3)
if_min_3_in_2_gga₃(x1, x2, x3) = if_min_3_in_2_gga₂(x2, x3)
gt_2_in_gg₂(x1, x2) = gt_2_in_gg₂(x1, x2)
if_gt_2_in_1_gg₃(x1, x2, x3) = if_gt_2_in_1_gg₁(x3)
gt_2_out_gg₂(x1, x2) = gt_2_out_gg
if_min2_3_in_2_gag₆(x1, x2, x3, x4, x5, x6) = if_min2_3_in_2_gag₁(x6)
min1_2_out_ag₂(x1, x2) = min1_2_out_ag₁(x1)
if_minsort_2_in_2_ga₄(x1, x2, x3, x4) = if_minsort_2_in_2_ga₂(x2, x4)
remove_3_in_gga₃(x1, x2, x3) = remove_3_in_gga₂(x1, x2)
remove_3_out_gga₃(x1, x2, x3) = remove_3_out_gga₁(x3)
if_remove_3_in_1_gga₅(x1, x2, x3, x4, x5) = if_remove_3_in_1_gga₄(x1, x2, x3, x5)
notEq_2_in_gg₂(x1, x2) = notEq_2_in_gg₂(x1, x2)
if_notEq_2_in_1_gg₃(x1, x2, x3) = if_notEq_2_in_1_gg₁(x3)
notEq_2_out_gg₂(x1, x2) = notEq_2_out_gg
if_remove_3_in_2_gga₅(x1, x2, x3, x4, x5) = if_remove_3_in_2_gga₂(x2, x5)
if_minsort_2_in_3_ga₅(x1, x2, x3, x4, x5) = if_minsort_2_in_3_ga₂(x2, x5)
GT_2_IN_GG₂(x1, x2) = GT_2_IN_GG₂(x1, x2)
IF_GT_2_IN_1_GG₃(x1, x2, x3) = IF_GT_2_IN_1_GG₁(x3)
IF_MIN2_3_IN_2_GAG₆(x1, x2, x3, x4, x5, x6) = IF_MIN2_3_IN_2_GAG₁(x6)
IF_MINSORT_2_IN_1_GA₄(x1, x2, x3, x4) = IF_MINSORT_2_IN_1_GA₂(x1, x4)
MINSORT_2_IN_GA₂(x1, x2) = MINSORT_2_IN_GA₁(x1)
MIN_3_IN_GGA₃(x1, x2, x3) = MIN_3_IN_GGA₂(x1, x2)
IF_MIN_3_IN_1_GGA₃(x1, x2, x3) = IF_MIN_3_IN_1_GGA₂(x1, x3)
IF_NOTEQ_2_IN_1_GG₃(x1, x2, x3) = IF_NOTEQ_2_IN_1_GG₁(x3)
REMOVE_3_IN_GGA₃(x1, x2, x3) = REMOVE_3_IN_GGA₂(x1, x2)
MIN1_2_IN_AG₂(x1, x2) = MIN1_2_IN_AG₁(x2)
MIN2_3_IN_GAG₃(x1, x2, x3) = MIN2_3_IN_GAG₂(x1, x3)
IF_REMOVE_3_IN_2_GGA₅(x1, x2, x3, x4, x5) = IF_REMOVE_3_IN_2_GGA₂(x2, x5)
IF_MINSORT_2_IN_2_GA₄(x1, x2, x3, x4) = IF_MINSORT_2_IN_2_GA₂(x2, x4)
IF_MIN2_3_IN_1_GAG₅(x1, x2, x3, x4, x5) = IF_MIN2_3_IN_1_GAG₂(x4, x5)
IF_LE_2_IN_1_GG₃(x1, x2, x3) = IF_LE_2_IN_1_GG₁(x3)
IF_MIN_3_IN_2_GGA₃(x1, x2, x3) = IF_MIN_3_IN_2_GGA₂(x2, x3)
NOTEQ_2_IN_GG₂(x1, x2) = NOTEQ_2_IN_GG₂(x1, x2)
IF_REMOVE_3_IN_1_GGA₅(x1, x2, x3, x4, x5) = IF_REMOVE_3_IN_1_GGA₄(x1, x2, x3, x5)
LE_2_IN_GG₂(x1, x2) = LE_2_IN_GG₂(x1, x2)
IF_MINSORT_2_IN_3_GA₅(x1, x2, x3, x4, x5) = IF_MINSORT_2_IN_3_GA₂(x2, x5)
IF_MIN1_2_IN_1_AG₄(x1, x2, x3, x4) = IF_MIN1_2_IN_1_AG₁(x4)

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

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

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

MINSORT_2_IN_GA₂(L, ._2₂(X, L1)) -> IF_MINSORT_2_IN_1_GA₄(L, X, L1, min1_2_in_ag₂(X, L))
MINSORT_2_IN_GA₂(L, ._2₂(X, L1)) -> MIN1_2_IN_AG₂(X, L)
MIN1_2_IN_AG₂(M, ._2₂(X, L)) -> IF_MIN1_2_IN_1_AG₄(M, X, L, min2_3_in_gag₃(X, M, L))
MIN1_2_IN_AG₂(M, ._2₂(X, L)) -> MIN2_3_IN_GAG₃(X, M, L)
MIN2_3_IN_GAG₃(X, A, ._2₂(M, L)) -> IF_MIN2_3_IN_1_GAG₅(X, A, M, L, min_3_in_gga₃(X, M, B))
MIN2_3_IN_GAG₃(X, A, ._2₂(M, L)) -> MIN_3_IN_GGA₃(X, M, B)
MIN_3_IN_GGA₃(X, Y, X) -> IF_MIN_3_IN_1_GGA₃(X, Y, le_2_in_gg₂(X, Y))
MIN_3_IN_GGA₃(X, Y, X) -> LE_2_IN_GG₂(X, Y)
LE_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> IF_LE_2_IN_1_GG₃(X, Y, le_2_in_gg₂(X, Y))
LE_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> LE_2_IN_GG₂(X, Y)
MIN_3_IN_GGA₃(X, Y, Y) -> IF_MIN_3_IN_2_GGA₃(X, Y, gt_2_in_gg₂(X, Y))
MIN_3_IN_GGA₃(X, Y, Y) -> GT_2_IN_GG₂(X, Y)
GT_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> IF_GT_2_IN_1_GG₃(X, Y, gt_2_in_gg₂(X, Y))
GT_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> GT_2_IN_GG₂(X, Y)
IF_MIN2_3_IN_1_GAG₅(X, A, M, L, min_3_out_gga₃(X, M, B)) -> IF_MIN2_3_IN_2_GAG₆(X, A, M, L, B, min2_3_in_gag₃(B, A, L))
IF_MIN2_3_IN_1_GAG₅(X, A, M, L, min_3_out_gga₃(X, M, B)) -> MIN2_3_IN_GAG₃(B, A, L)
IF_MINSORT_2_IN_1_GA₄(L, X, L1, min1_2_out_ag₂(X, L)) -> IF_MINSORT_2_IN_2_GA₄(L, X, L1, remove_3_in_gga₃(X, L, L2))
IF_MINSORT_2_IN_1_GA₄(L, X, L1, min1_2_out_ag₂(X, L)) -> REMOVE_3_IN_GGA₃(X, L, L2)
REMOVE_3_IN_GGA₃(N, ._2₂(M, L), ._2₂(M, L1)) -> IF_REMOVE_3_IN_1_GGA₅(N, M, L, L1, notEq_2_in_gg₂(N, M))
REMOVE_3_IN_GGA₃(N, ._2₂(M, L), ._2₂(M, L1)) -> NOTEQ_2_IN_GG₂(N, M)
NOTEQ_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> IF_NOTEQ_2_IN_1_GG₃(X, Y, notEq_2_in_gg₂(X, Y))
NOTEQ_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> NOTEQ_2_IN_GG₂(X, Y)
IF_REMOVE_3_IN_1_GGA₅(N, M, L, L1, notEq_2_out_gg₂(N, M)) -> IF_REMOVE_3_IN_2_GGA₅(N, M, L, L1, remove_3_in_gga₃(N, L, L1))
IF_REMOVE_3_IN_1_GGA₅(N, M, L, L1, notEq_2_out_gg₂(N, M)) -> REMOVE_3_IN_GGA₃(N, L, L1)
IF_MINSORT_2_IN_2_GA₄(L, X, L1, remove_3_out_gga₃(X, L, L2)) -> IF_MINSORT_2_IN_3_GA₅(L, X, L1, L2, minsort_2_in_ga₂(L2, L1))
IF_MINSORT_2_IN_2_GA₄(L, X, L1, remove_3_out_gga₃(X, L, L2)) -> MINSORT_2_IN_GA₂(L2, L1)

The TRS R consists of the following rules:

minsort_2_in_ga₂([]_0, []_0) -> minsort_2_out_ga₂([]_0, []_0)
minsort_2_in_ga₂(L, ._2₂(X, L1)) -> if_minsort_2_in_1_ga₄(L, X, L1, min1_2_in_ag₂(X, L))
min1_2_in_ag₂(M, ._2₂(X, L)) -> if_min1_2_in_1_ag₄(M, X, L, min2_3_in_gag₃(X, M, L))
min2_3_in_gag₃(X, X, []_0) -> min2_3_out_gag₃(X, X, []_0)
min2_3_in_gag₃(X, A, ._2₂(M, L)) -> if_min2_3_in_1_gag₅(X, A, M, L, min_3_in_gga₃(X, M, B))
min_3_in_gga₃(X, Y, X) -> if_min_3_in_1_gga₃(X, Y, le_2_in_gg₂(X, Y))
le_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_le_2_in_1_gg₃(X, Y, le_2_in_gg₂(X, Y))
le_2_in_gg₂(0_0, s_1₁(Y)) -> le_2_out_gg₂(0_0, s_1₁(Y))
le_2_in_gg₂(0_0, 0_0) -> le_2_out_gg₂(0_0, 0_0)
if_le_2_in_1_gg₃(X, Y, le_2_out_gg₂(X, Y)) -> le_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_min_3_in_1_gga₃(X, Y, le_2_out_gg₂(X, Y)) -> min_3_out_gga₃(X, Y, X)
min_3_in_gga₃(X, Y, Y) -> if_min_3_in_2_gga₃(X, Y, gt_2_in_gg₂(X, Y))
gt_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_gt_2_in_1_gg₃(X, Y, gt_2_in_gg₂(X, Y))
gt_2_in_gg₂(s_1₁(X), 0_0) -> gt_2_out_gg₂(s_1₁(X), 0_0)
if_gt_2_in_1_gg₃(X, Y, gt_2_out_gg₂(X, Y)) -> gt_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_min_3_in_2_gga₃(X, Y, gt_2_out_gg₂(X, Y)) -> min_3_out_gga₃(X, Y, Y)
if_min2_3_in_1_gag₅(X, A, M, L, min_3_out_gga₃(X, M, B)) -> if_min2_3_in_2_gag₆(X, A, M, L, B, min2_3_in_gag₃(B, A, L))
if_min2_3_in_2_gag₆(X, A, M, L, B, min2_3_out_gag₃(B, A, L)) -> min2_3_out_gag₃(X, A, ._2₂(M, L))
if_min1_2_in_1_ag₄(M, X, L, min2_3_out_gag₃(X, M, L)) -> min1_2_out_ag₂(M, ._2₂(X, L))
if_minsort_2_in_1_ga₄(L, X, L1, min1_2_out_ag₂(X, L)) -> if_minsort_2_in_2_ga₄(L, X, L1, remove_3_in_gga₃(X, L, L2))
remove_3_in_gga₃(N, ._2₂(N, L), L) -> remove_3_out_gga₃(N, ._2₂(N, L), L)
remove_3_in_gga₃(N, ._2₂(M, L), ._2₂(M, L1)) -> if_remove_3_in_1_gga₅(N, M, L, L1, notEq_2_in_gg₂(N, M))
notEq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_notEq_2_in_1_gg₃(X, Y, notEq_2_in_gg₂(X, Y))
notEq_2_in_gg₂(s_1₁(X), 0_0) -> notEq_2_out_gg₂(s_1₁(X), 0_0)
notEq_2_in_gg₂(0_0, s_1₁(X)) -> notEq_2_out_gg₂(0_0, s_1₁(X))
if_notEq_2_in_1_gg₃(X, Y, notEq_2_out_gg₂(X, Y)) -> notEq_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_remove_3_in_1_gga₅(N, M, L, L1, notEq_2_out_gg₂(N, M)) -> if_remove_3_in_2_gga₅(N, M, L, L1, remove_3_in_gga₃(N, L, L1))
if_remove_3_in_2_gga₅(N, M, L, L1, remove_3_out_gga₃(N, L, L1)) -> remove_3_out_gga₃(N, ._2₂(M, L), ._2₂(M, L1))
if_minsort_2_in_2_ga₄(L, X, L1, remove_3_out_gga₃(X, L, L2)) -> if_minsort_2_in_3_ga₅(L, X, L1, L2, minsort_2_in_ga₂(L2, L1))
if_minsort_2_in_3_ga₅(L, X, L1, L2, minsort_2_out_ga₂(L2, L1)) -> minsort_2_out_ga₂(L, ._2₂(X, L1))

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

NOTEQ_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> NOTEQ_2_IN_GG₂(X, Y)

The TRS R consists of the following rules:

minsort_2_in_ga₂([]_0, []_0) -> minsort_2_out_ga₂([]_0, []_0)
minsort_2_in_ga₂(L, ._2₂(X, L1)) -> if_minsort_2_in_1_ga₄(L, X, L1, min1_2_in_ag₂(X, L))
min1_2_in_ag₂(M, ._2₂(X, L)) -> if_min1_2_in_1_ag₄(M, X, L, min2_3_in_gag₃(X, M, L))
min2_3_in_gag₃(X, X, []_0) -> min2_3_out_gag₃(X, X, []_0)
min2_3_in_gag₃(X, A, ._2₂(M, L)) -> if_min2_3_in_1_gag₅(X, A, M, L, min_3_in_gga₃(X, M, B))
min_3_in_gga₃(X, Y, X) -> if_min_3_in_1_gga₃(X, Y, le_2_in_gg₂(X, Y))
le_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_le_2_in_1_gg₃(X, Y, le_2_in_gg₂(X, Y))
le_2_in_gg₂(0_0, s_1₁(Y)) -> le_2_out_gg₂(0_0, s_1₁(Y))
le_2_in_gg₂(0_0, 0_0) -> le_2_out_gg₂(0_0, 0_0)
if_le_2_in_1_gg₃(X, Y, le_2_out_gg₂(X, Y)) -> le_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_min_3_in_1_gga₃(X, Y, le_2_out_gg₂(X, Y)) -> min_3_out_gga₃(X, Y, X)
min_3_in_gga₃(X, Y, Y) -> if_min_3_in_2_gga₃(X, Y, gt_2_in_gg₂(X, Y))
gt_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_gt_2_in_1_gg₃(X, Y, gt_2_in_gg₂(X, Y))
gt_2_in_gg₂(s_1₁(X), 0_0) -> gt_2_out_gg₂(s_1₁(X), 0_0)
if_gt_2_in_1_gg₃(X, Y, gt_2_out_gg₂(X, Y)) -> gt_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_min_3_in_2_gga₃(X, Y, gt_2_out_gg₂(X, Y)) -> min_3_out_gga₃(X, Y, Y)
if_min2_3_in_1_gag₅(X, A, M, L, min_3_out_gga₃(X, M, B)) -> if_min2_3_in_2_gag₆(X, A, M, L, B, min2_3_in_gag₃(B, A, L))
if_min2_3_in_2_gag₆(X, A, M, L, B, min2_3_out_gag₃(B, A, L)) -> min2_3_out_gag₃(X, A, ._2₂(M, L))
if_min1_2_in_1_ag₄(M, X, L, min2_3_out_gag₃(X, M, L)) -> min1_2_out_ag₂(M, ._2₂(X, L))
if_minsort_2_in_1_ga₄(L, X, L1, min1_2_out_ag₂(X, L)) -> if_minsort_2_in_2_ga₄(L, X, L1, remove_3_in_gga₃(X, L, L2))
remove_3_in_gga₃(N, ._2₂(N, L), L) -> remove_3_out_gga₃(N, ._2₂(N, L), L)
remove_3_in_gga₃(N, ._2₂(M, L), ._2₂(M, L1)) -> if_remove_3_in_1_gga₅(N, M, L, L1, notEq_2_in_gg₂(N, M))
notEq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_notEq_2_in_1_gg₃(X, Y, notEq_2_in_gg₂(X, Y))
notEq_2_in_gg₂(s_1₁(X), 0_0) -> notEq_2_out_gg₂(s_1₁(X), 0_0)
notEq_2_in_gg₂(0_0, s_1₁(X)) -> notEq_2_out_gg₂(0_0, s_1₁(X))
if_notEq_2_in_1_gg₃(X, Y, notEq_2_out_gg₂(X, Y)) -> notEq_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_remove_3_in_1_gga₅(N, M, L, L1, notEq_2_out_gg₂(N, M)) -> if_remove_3_in_2_gga₅(N, M, L, L1, remove_3_in_gga₃(N, L, L1))
if_remove_3_in_2_gga₅(N, M, L, L1, remove_3_out_gga₃(N, L, L1)) -> remove_3_out_gga₃(N, ._2₂(M, L), ._2₂(M, L1))
if_minsort_2_in_2_ga₄(L, X, L1, remove_3_out_gga₃(X, L, L2)) -> if_minsort_2_in_3_ga₅(L, X, L1, L2, minsort_2_in_ga₂(L2, L1))
if_minsort_2_in_3_ga₅(L, X, L1, L2, minsort_2_out_ga₂(L2, L1)) -> minsort_2_out_ga₂(L, ._2₂(X, L1))

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

NOTEQ_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> NOTEQ_2_IN_GG₂(X, Y)

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

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

NOTEQ_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> NOTEQ_2_IN_GG₂(X, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {NOTEQ_2_IN_GG₂}.
By using the subterm criterion together with the size-change analysis 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_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> NOTEQ_2_IN_GG₂(X, Y)
The graph contains the following edges 1 > 1, 2 > 2

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

IF_REMOVE_3_IN_1_GGA₅(N, M, L, L1, notEq_2_out_gg₂(N, M)) -> REMOVE_3_IN_GGA₃(N, L, L1)
REMOVE_3_IN_GGA₃(N, ._2₂(M, L), ._2₂(M, L1)) -> IF_REMOVE_3_IN_1_GGA₅(N, M, L, L1, notEq_2_in_gg₂(N, M))

The TRS R consists of the following rules:

minsort_2_in_ga₂([]_0, []_0) -> minsort_2_out_ga₂([]_0, []_0)
minsort_2_in_ga₂(L, ._2₂(X, L1)) -> if_minsort_2_in_1_ga₄(L, X, L1, min1_2_in_ag₂(X, L))
min1_2_in_ag₂(M, ._2₂(X, L)) -> if_min1_2_in_1_ag₄(M, X, L, min2_3_in_gag₃(X, M, L))
min2_3_in_gag₃(X, X, []_0) -> min2_3_out_gag₃(X, X, []_0)
min2_3_in_gag₃(X, A, ._2₂(M, L)) -> if_min2_3_in_1_gag₅(X, A, M, L, min_3_in_gga₃(X, M, B))
min_3_in_gga₃(X, Y, X) -> if_min_3_in_1_gga₃(X, Y, le_2_in_gg₂(X, Y))
le_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_le_2_in_1_gg₃(X, Y, le_2_in_gg₂(X, Y))
le_2_in_gg₂(0_0, s_1₁(Y)) -> le_2_out_gg₂(0_0, s_1₁(Y))
le_2_in_gg₂(0_0, 0_0) -> le_2_out_gg₂(0_0, 0_0)
if_le_2_in_1_gg₃(X, Y, le_2_out_gg₂(X, Y)) -> le_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_min_3_in_1_gga₃(X, Y, le_2_out_gg₂(X, Y)) -> min_3_out_gga₃(X, Y, X)
min_3_in_gga₃(X, Y, Y) -> if_min_3_in_2_gga₃(X, Y, gt_2_in_gg₂(X, Y))
gt_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_gt_2_in_1_gg₃(X, Y, gt_2_in_gg₂(X, Y))
gt_2_in_gg₂(s_1₁(X), 0_0) -> gt_2_out_gg₂(s_1₁(X), 0_0)
if_gt_2_in_1_gg₃(X, Y, gt_2_out_gg₂(X, Y)) -> gt_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_min_3_in_2_gga₃(X, Y, gt_2_out_gg₂(X, Y)) -> min_3_out_gga₃(X, Y, Y)
if_min2_3_in_1_gag₅(X, A, M, L, min_3_out_gga₃(X, M, B)) -> if_min2_3_in_2_gag₆(X, A, M, L, B, min2_3_in_gag₃(B, A, L))
if_min2_3_in_2_gag₆(X, A, M, L, B, min2_3_out_gag₃(B, A, L)) -> min2_3_out_gag₃(X, A, ._2₂(M, L))
if_min1_2_in_1_ag₄(M, X, L, min2_3_out_gag₃(X, M, L)) -> min1_2_out_ag₂(M, ._2₂(X, L))
if_minsort_2_in_1_ga₄(L, X, L1, min1_2_out_ag₂(X, L)) -> if_minsort_2_in_2_ga₄(L, X, L1, remove_3_in_gga₃(X, L, L2))
remove_3_in_gga₃(N, ._2₂(N, L), L) -> remove_3_out_gga₃(N, ._2₂(N, L), L)
remove_3_in_gga₃(N, ._2₂(M, L), ._2₂(M, L1)) -> if_remove_3_in_1_gga₅(N, M, L, L1, notEq_2_in_gg₂(N, M))
notEq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_notEq_2_in_1_gg₃(X, Y, notEq_2_in_gg₂(X, Y))
notEq_2_in_gg₂(s_1₁(X), 0_0) -> notEq_2_out_gg₂(s_1₁(X), 0_0)
notEq_2_in_gg₂(0_0, s_1₁(X)) -> notEq_2_out_gg₂(0_0, s_1₁(X))
if_notEq_2_in_1_gg₃(X, Y, notEq_2_out_gg₂(X, Y)) -> notEq_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_remove_3_in_1_gga₅(N, M, L, L1, notEq_2_out_gg₂(N, M)) -> if_remove_3_in_2_gga₅(N, M, L, L1, remove_3_in_gga₃(N, L, L1))
if_remove_3_in_2_gga₅(N, M, L, L1, remove_3_out_gga₃(N, L, L1)) -> remove_3_out_gga₃(N, ._2₂(M, L), ._2₂(M, L1))
if_minsort_2_in_2_ga₄(L, X, L1, remove_3_out_gga₃(X, L, L2)) -> if_minsort_2_in_3_ga₅(L, X, L1, L2, minsort_2_in_ga₂(L2, L1))
if_minsort_2_in_3_ga₅(L, X, L1, L2, minsort_2_out_ga₂(L2, L1)) -> minsort_2_out_ga₂(L, ._2₂(X, L1))

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

IF_REMOVE_3_IN_1_GGA₅(N, M, L, L1, notEq_2_out_gg₂(N, M)) -> REMOVE_3_IN_GGA₃(N, L, L1)
REMOVE_3_IN_GGA₃(N, ._2₂(M, L), ._2₂(M, L1)) -> IF_REMOVE_3_IN_1_GGA₅(N, M, L, L1, notEq_2_in_gg₂(N, M))

The TRS R consists of the following rules:

notEq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_notEq_2_in_1_gg₃(X, Y, notEq_2_in_gg₂(X, Y))
notEq_2_in_gg₂(s_1₁(X), 0_0) -> notEq_2_out_gg₂(s_1₁(X), 0_0)
notEq_2_in_gg₂(0_0, s_1₁(X)) -> notEq_2_out_gg₂(0_0, s_1₁(X))
if_notEq_2_in_1_gg₃(X, Y, notEq_2_out_gg₂(X, Y)) -> notEq_2_out_gg₂(s_1₁(X), s_1₁(Y))

The argument filtering Pi contains the following mapping:
._2₂(x1, x2) = ._2₂(x1, x2)
s_1₁(x1) = s_1₁(x1)
0_0 = 0_0
notEq_2_in_gg₂(x1, x2) = notEq_2_in_gg₂(x1, x2)
if_notEq_2_in_1_gg₃(x1, x2, x3) = if_notEq_2_in_1_gg₁(x3)
notEq_2_out_gg₂(x1, x2) = notEq_2_out_gg
REMOVE_3_IN_GGA₃(x1, x2, x3) = REMOVE_3_IN_GGA₂(x1, x2)
IF_REMOVE_3_IN_1_GGA₅(x1, x2, x3, x4, x5) = IF_REMOVE_3_IN_1_GGA₄(x1, x2, x3, x5)

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

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

IF_REMOVE_3_IN_1_GGA₄(N, M, L, notEq_2_out_gg) -> REMOVE_3_IN_GGA₂(N, L)
REMOVE_3_IN_GGA₂(N, ._2₂(M, L)) -> IF_REMOVE_3_IN_1_GGA₄(N, M, L, notEq_2_in_gg₂(N, M))

The TRS R consists of the following rules:

notEq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_notEq_2_in_1_gg₁(notEq_2_in_gg₂(X, Y))
notEq_2_in_gg₂(s_1₁(X), 0_0) -> notEq_2_out_gg
notEq_2_in_gg₂(0_0, s_1₁(X)) -> notEq_2_out_gg
if_notEq_2_in_1_gg₁(notEq_2_out_gg) -> notEq_2_out_gg

The set Q consists of the following terms:

notEq_2_in_gg₂(x0, x1)
if_notEq_2_in_1_gg₁(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {REMOVE_3_IN_GGA₂, IF_REMOVE_3_IN_1_GGA₄}.
By using the subterm criterion together with the size-change analysis 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_3_IN_GGA₂(N, ._2₂(M, L)) -> IF_REMOVE_3_IN_1_GGA₄(N, M, L, notEq_2_in_gg₂(N, M))
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3
IF_REMOVE_3_IN_1_GGA₄(N, M, L, notEq_2_out_gg) -> REMOVE_3_IN_GGA₂(N, L)
The graph contains the following edges 1 >= 1, 3 >= 2

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

GT_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> GT_2_IN_GG₂(X, Y)

The TRS R consists of the following rules:

minsort_2_in_ga₂([]_0, []_0) -> minsort_2_out_ga₂([]_0, []_0)
minsort_2_in_ga₂(L, ._2₂(X, L1)) -> if_minsort_2_in_1_ga₄(L, X, L1, min1_2_in_ag₂(X, L))
min1_2_in_ag₂(M, ._2₂(X, L)) -> if_min1_2_in_1_ag₄(M, X, L, min2_3_in_gag₃(X, M, L))
min2_3_in_gag₃(X, X, []_0) -> min2_3_out_gag₃(X, X, []_0)
min2_3_in_gag₃(X, A, ._2₂(M, L)) -> if_min2_3_in_1_gag₅(X, A, M, L, min_3_in_gga₃(X, M, B))
min_3_in_gga₃(X, Y, X) -> if_min_3_in_1_gga₃(X, Y, le_2_in_gg₂(X, Y))
le_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_le_2_in_1_gg₃(X, Y, le_2_in_gg₂(X, Y))
le_2_in_gg₂(0_0, s_1₁(Y)) -> le_2_out_gg₂(0_0, s_1₁(Y))
le_2_in_gg₂(0_0, 0_0) -> le_2_out_gg₂(0_0, 0_0)
if_le_2_in_1_gg₃(X, Y, le_2_out_gg₂(X, Y)) -> le_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_min_3_in_1_gga₃(X, Y, le_2_out_gg₂(X, Y)) -> min_3_out_gga₃(X, Y, X)
min_3_in_gga₃(X, Y, Y) -> if_min_3_in_2_gga₃(X, Y, gt_2_in_gg₂(X, Y))
gt_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_gt_2_in_1_gg₃(X, Y, gt_2_in_gg₂(X, Y))
gt_2_in_gg₂(s_1₁(X), 0_0) -> gt_2_out_gg₂(s_1₁(X), 0_0)
if_gt_2_in_1_gg₃(X, Y, gt_2_out_gg₂(X, Y)) -> gt_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_min_3_in_2_gga₃(X, Y, gt_2_out_gg₂(X, Y)) -> min_3_out_gga₃(X, Y, Y)
if_min2_3_in_1_gag₅(X, A, M, L, min_3_out_gga₃(X, M, B)) -> if_min2_3_in_2_gag₆(X, A, M, L, B, min2_3_in_gag₃(B, A, L))
if_min2_3_in_2_gag₆(X, A, M, L, B, min2_3_out_gag₃(B, A, L)) -> min2_3_out_gag₃(X, A, ._2₂(M, L))
if_min1_2_in_1_ag₄(M, X, L, min2_3_out_gag₃(X, M, L)) -> min1_2_out_ag₂(M, ._2₂(X, L))
if_minsort_2_in_1_ga₄(L, X, L1, min1_2_out_ag₂(X, L)) -> if_minsort_2_in_2_ga₄(L, X, L1, remove_3_in_gga₃(X, L, L2))
remove_3_in_gga₃(N, ._2₂(N, L), L) -> remove_3_out_gga₃(N, ._2₂(N, L), L)
remove_3_in_gga₃(N, ._2₂(M, L), ._2₂(M, L1)) -> if_remove_3_in_1_gga₅(N, M, L, L1, notEq_2_in_gg₂(N, M))
notEq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_notEq_2_in_1_gg₃(X, Y, notEq_2_in_gg₂(X, Y))
notEq_2_in_gg₂(s_1₁(X), 0_0) -> notEq_2_out_gg₂(s_1₁(X), 0_0)
notEq_2_in_gg₂(0_0, s_1₁(X)) -> notEq_2_out_gg₂(0_0, s_1₁(X))
if_notEq_2_in_1_gg₃(X, Y, notEq_2_out_gg₂(X, Y)) -> notEq_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_remove_3_in_1_gga₅(N, M, L, L1, notEq_2_out_gg₂(N, M)) -> if_remove_3_in_2_gga₅(N, M, L, L1, remove_3_in_gga₃(N, L, L1))
if_remove_3_in_2_gga₅(N, M, L, L1, remove_3_out_gga₃(N, L, L1)) -> remove_3_out_gga₃(N, ._2₂(M, L), ._2₂(M, L1))
if_minsort_2_in_2_ga₄(L, X, L1, remove_3_out_gga₃(X, L, L2)) -> if_minsort_2_in_3_ga₅(L, X, L1, L2, minsort_2_in_ga₂(L2, L1))
if_minsort_2_in_3_ga₅(L, X, L1, L2, minsort_2_out_ga₂(L2, L1)) -> minsort_2_out_ga₂(L, ._2₂(X, L1))

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

GT_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> GT_2_IN_GG₂(X, Y)

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

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

GT_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> GT_2_IN_GG₂(X, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {GT_2_IN_GG₂}.
By using the subterm criterion together with the size-change analysis 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_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> GT_2_IN_GG₂(X, Y)
The graph contains the following edges 1 > 1, 2 > 2

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

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

LE_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> LE_2_IN_GG₂(X, Y)

The TRS R consists of the following rules:

minsort_2_in_ga₂([]_0, []_0) -> minsort_2_out_ga₂([]_0, []_0)
minsort_2_in_ga₂(L, ._2₂(X, L1)) -> if_minsort_2_in_1_ga₄(L, X, L1, min1_2_in_ag₂(X, L))
min1_2_in_ag₂(M, ._2₂(X, L)) -> if_min1_2_in_1_ag₄(M, X, L, min2_3_in_gag₃(X, M, L))
min2_3_in_gag₃(X, X, []_0) -> min2_3_out_gag₃(X, X, []_0)
min2_3_in_gag₃(X, A, ._2₂(M, L)) -> if_min2_3_in_1_gag₅(X, A, M, L, min_3_in_gga₃(X, M, B))
min_3_in_gga₃(X, Y, X) -> if_min_3_in_1_gga₃(X, Y, le_2_in_gg₂(X, Y))
le_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_le_2_in_1_gg₃(X, Y, le_2_in_gg₂(X, Y))
le_2_in_gg₂(0_0, s_1₁(Y)) -> le_2_out_gg₂(0_0, s_1₁(Y))
le_2_in_gg₂(0_0, 0_0) -> le_2_out_gg₂(0_0, 0_0)
if_le_2_in_1_gg₃(X, Y, le_2_out_gg₂(X, Y)) -> le_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_min_3_in_1_gga₃(X, Y, le_2_out_gg₂(X, Y)) -> min_3_out_gga₃(X, Y, X)
min_3_in_gga₃(X, Y, Y) -> if_min_3_in_2_gga₃(X, Y, gt_2_in_gg₂(X, Y))
gt_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_gt_2_in_1_gg₃(X, Y, gt_2_in_gg₂(X, Y))
gt_2_in_gg₂(s_1₁(X), 0_0) -> gt_2_out_gg₂(s_1₁(X), 0_0)
if_gt_2_in_1_gg₃(X, Y, gt_2_out_gg₂(X, Y)) -> gt_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_min_3_in_2_gga₃(X, Y, gt_2_out_gg₂(X, Y)) -> min_3_out_gga₃(X, Y, Y)
if_min2_3_in_1_gag₅(X, A, M, L, min_3_out_gga₃(X, M, B)) -> if_min2_3_in_2_gag₆(X, A, M, L, B, min2_3_in_gag₃(B, A, L))
if_min2_3_in_2_gag₆(X, A, M, L, B, min2_3_out_gag₃(B, A, L)) -> min2_3_out_gag₃(X, A, ._2₂(M, L))
if_min1_2_in_1_ag₄(M, X, L, min2_3_out_gag₃(X, M, L)) -> min1_2_out_ag₂(M, ._2₂(X, L))
if_minsort_2_in_1_ga₄(L, X, L1, min1_2_out_ag₂(X, L)) -> if_minsort_2_in_2_ga₄(L, X, L1, remove_3_in_gga₃(X, L, L2))
remove_3_in_gga₃(N, ._2₂(N, L), L) -> remove_3_out_gga₃(N, ._2₂(N, L), L)
remove_3_in_gga₃(N, ._2₂(M, L), ._2₂(M, L1)) -> if_remove_3_in_1_gga₅(N, M, L, L1, notEq_2_in_gg₂(N, M))
notEq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_notEq_2_in_1_gg₃(X, Y, notEq_2_in_gg₂(X, Y))
notEq_2_in_gg₂(s_1₁(X), 0_0) -> notEq_2_out_gg₂(s_1₁(X), 0_0)
notEq_2_in_gg₂(0_0, s_1₁(X)) -> notEq_2_out_gg₂(0_0, s_1₁(X))
if_notEq_2_in_1_gg₃(X, Y, notEq_2_out_gg₂(X, Y)) -> notEq_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_remove_3_in_1_gga₅(N, M, L, L1, notEq_2_out_gg₂(N, M)) -> if_remove_3_in_2_gga₅(N, M, L, L1, remove_3_in_gga₃(N, L, L1))
if_remove_3_in_2_gga₅(N, M, L, L1, remove_3_out_gga₃(N, L, L1)) -> remove_3_out_gga₃(N, ._2₂(M, L), ._2₂(M, L1))
if_minsort_2_in_2_ga₄(L, X, L1, remove_3_out_gga₃(X, L, L2)) -> if_minsort_2_in_3_ga₅(L, X, L1, L2, minsort_2_in_ga₂(L2, L1))
if_minsort_2_in_3_ga₅(L, X, L1, L2, minsort_2_out_ga₂(L2, L1)) -> minsort_2_out_ga₂(L, ._2₂(X, L1))

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

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

LE_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> LE_2_IN_GG₂(X, Y)

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

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

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

LE_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> LE_2_IN_GG₂(X, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {LE_2_IN_GG₂}.
By using the subterm criterion together with the size-change analysis 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_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> LE_2_IN_GG₂(X, Y)
The graph contains the following edges 1 > 1, 2 > 2

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

IF_MIN2_3_IN_1_GAG₅(X, A, M, L, min_3_out_gga₃(X, M, B)) -> MIN2_3_IN_GAG₃(B, A, L)
MIN2_3_IN_GAG₃(X, A, ._2₂(M, L)) -> IF_MIN2_3_IN_1_GAG₅(X, A, M, L, min_3_in_gga₃(X, M, B))

The TRS R consists of the following rules:

minsort_2_in_ga₂([]_0, []_0) -> minsort_2_out_ga₂([]_0, []_0)
minsort_2_in_ga₂(L, ._2₂(X, L1)) -> if_minsort_2_in_1_ga₄(L, X, L1, min1_2_in_ag₂(X, L))
min1_2_in_ag₂(M, ._2₂(X, L)) -> if_min1_2_in_1_ag₄(M, X, L, min2_3_in_gag₃(X, M, L))
min2_3_in_gag₃(X, X, []_0) -> min2_3_out_gag₃(X, X, []_0)
min2_3_in_gag₃(X, A, ._2₂(M, L)) -> if_min2_3_in_1_gag₅(X, A, M, L, min_3_in_gga₃(X, M, B))
min_3_in_gga₃(X, Y, X) -> if_min_3_in_1_gga₃(X, Y, le_2_in_gg₂(X, Y))
le_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_le_2_in_1_gg₃(X, Y, le_2_in_gg₂(X, Y))
le_2_in_gg₂(0_0, s_1₁(Y)) -> le_2_out_gg₂(0_0, s_1₁(Y))
le_2_in_gg₂(0_0, 0_0) -> le_2_out_gg₂(0_0, 0_0)
if_le_2_in_1_gg₃(X, Y, le_2_out_gg₂(X, Y)) -> le_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_min_3_in_1_gga₃(X, Y, le_2_out_gg₂(X, Y)) -> min_3_out_gga₃(X, Y, X)
min_3_in_gga₃(X, Y, Y) -> if_min_3_in_2_gga₃(X, Y, gt_2_in_gg₂(X, Y))
gt_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_gt_2_in_1_gg₃(X, Y, gt_2_in_gg₂(X, Y))
gt_2_in_gg₂(s_1₁(X), 0_0) -> gt_2_out_gg₂(s_1₁(X), 0_0)
if_gt_2_in_1_gg₃(X, Y, gt_2_out_gg₂(X, Y)) -> gt_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_min_3_in_2_gga₃(X, Y, gt_2_out_gg₂(X, Y)) -> min_3_out_gga₃(X, Y, Y)
if_min2_3_in_1_gag₅(X, A, M, L, min_3_out_gga₃(X, M, B)) -> if_min2_3_in_2_gag₆(X, A, M, L, B, min2_3_in_gag₃(B, A, L))
if_min2_3_in_2_gag₆(X, A, M, L, B, min2_3_out_gag₃(B, A, L)) -> min2_3_out_gag₃(X, A, ._2₂(M, L))
if_min1_2_in_1_ag₄(M, X, L, min2_3_out_gag₃(X, M, L)) -> min1_2_out_ag₂(M, ._2₂(X, L))
if_minsort_2_in_1_ga₄(L, X, L1, min1_2_out_ag₂(X, L)) -> if_minsort_2_in_2_ga₄(L, X, L1, remove_3_in_gga₃(X, L, L2))
remove_3_in_gga₃(N, ._2₂(N, L), L) -> remove_3_out_gga₃(N, ._2₂(N, L), L)
remove_3_in_gga₃(N, ._2₂(M, L), ._2₂(M, L1)) -> if_remove_3_in_1_gga₅(N, M, L, L1, notEq_2_in_gg₂(N, M))
notEq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_notEq_2_in_1_gg₃(X, Y, notEq_2_in_gg₂(X, Y))
notEq_2_in_gg₂(s_1₁(X), 0_0) -> notEq_2_out_gg₂(s_1₁(X), 0_0)
notEq_2_in_gg₂(0_0, s_1₁(X)) -> notEq_2_out_gg₂(0_0, s_1₁(X))
if_notEq_2_in_1_gg₃(X, Y, notEq_2_out_gg₂(X, Y)) -> notEq_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_remove_3_in_1_gga₅(N, M, L, L1, notEq_2_out_gg₂(N, M)) -> if_remove_3_in_2_gga₅(N, M, L, L1, remove_3_in_gga₃(N, L, L1))
if_remove_3_in_2_gga₅(N, M, L, L1, remove_3_out_gga₃(N, L, L1)) -> remove_3_out_gga₃(N, ._2₂(M, L), ._2₂(M, L1))
if_minsort_2_in_2_ga₄(L, X, L1, remove_3_out_gga₃(X, L, L2)) -> if_minsort_2_in_3_ga₅(L, X, L1, L2, minsort_2_in_ga₂(L2, L1))
if_minsort_2_in_3_ga₅(L, X, L1, L2, minsort_2_out_ga₂(L2, L1)) -> minsort_2_out_ga₂(L, ._2₂(X, L1))

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

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

IF_MIN2_3_IN_1_GAG₅(X, A, M, L, min_3_out_gga₃(X, M, B)) -> MIN2_3_IN_GAG₃(B, A, L)
MIN2_3_IN_GAG₃(X, A, ._2₂(M, L)) -> IF_MIN2_3_IN_1_GAG₅(X, A, M, L, min_3_in_gga₃(X, M, B))

The TRS R consists of the following rules:

min_3_in_gga₃(X, Y, X) -> if_min_3_in_1_gga₃(X, Y, le_2_in_gg₂(X, Y))
min_3_in_gga₃(X, Y, Y) -> if_min_3_in_2_gga₃(X, Y, gt_2_in_gg₂(X, Y))
if_min_3_in_1_gga₃(X, Y, le_2_out_gg₂(X, Y)) -> min_3_out_gga₃(X, Y, X)
if_min_3_in_2_gga₃(X, Y, gt_2_out_gg₂(X, Y)) -> min_3_out_gga₃(X, Y, Y)
le_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_le_2_in_1_gg₃(X, Y, le_2_in_gg₂(X, Y))
le_2_in_gg₂(0_0, s_1₁(Y)) -> le_2_out_gg₂(0_0, s_1₁(Y))
le_2_in_gg₂(0_0, 0_0) -> le_2_out_gg₂(0_0, 0_0)
gt_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_gt_2_in_1_gg₃(X, Y, gt_2_in_gg₂(X, Y))
gt_2_in_gg₂(s_1₁(X), 0_0) -> gt_2_out_gg₂(s_1₁(X), 0_0)
if_le_2_in_1_gg₃(X, Y, le_2_out_gg₂(X, Y)) -> le_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_gt_2_in_1_gg₃(X, Y, gt_2_out_gg₂(X, Y)) -> gt_2_out_gg₂(s_1₁(X), s_1₁(Y))

The argument filtering Pi contains the following mapping:
._2₂(x1, x2) = ._2₂(x1, x2)
s_1₁(x1) = s_1₁(x1)
0_0 = 0_0
min_3_in_gga₃(x1, x2, x3) = min_3_in_gga₂(x1, x2)
if_min_3_in_1_gga₃(x1, x2, x3) = if_min_3_in_1_gga₂(x1, x3)
le_2_in_gg₂(x1, x2) = le_2_in_gg₂(x1, x2)
if_le_2_in_1_gg₃(x1, x2, x3) = if_le_2_in_1_gg₁(x3)
le_2_out_gg₂(x1, x2) = le_2_out_gg
min_3_out_gga₃(x1, x2, x3) = min_3_out_gga₁(x3)
if_min_3_in_2_gga₃(x1, x2, x3) = if_min_3_in_2_gga₂(x2, x3)
gt_2_in_gg₂(x1, x2) = gt_2_in_gg₂(x1, x2)
if_gt_2_in_1_gg₃(x1, x2, x3) = if_gt_2_in_1_gg₁(x3)
gt_2_out_gg₂(x1, x2) = gt_2_out_gg
MIN2_3_IN_GAG₃(x1, x2, x3) = MIN2_3_IN_GAG₂(x1, x3)
IF_MIN2_3_IN_1_GAG₅(x1, x2, x3, x4, x5) = IF_MIN2_3_IN_1_GAG₂(x4, x5)

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

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

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

IF_MIN2_3_IN_1_GAG₂(L, min_3_out_gga₁(B)) -> MIN2_3_IN_GAG₂(B, L)
MIN2_3_IN_GAG₂(X, ._2₂(M, L)) -> IF_MIN2_3_IN_1_GAG₂(L, min_3_in_gga₂(X, M))

The TRS R consists of the following rules:

min_3_in_gga₂(X, Y) -> if_min_3_in_1_gga₂(X, le_2_in_gg₂(X, Y))
min_3_in_gga₂(X, Y) -> if_min_3_in_2_gga₂(Y, gt_2_in_gg₂(X, Y))
if_min_3_in_1_gga₂(X, le_2_out_gg) -> min_3_out_gga₁(X)
if_min_3_in_2_gga₂(Y, gt_2_out_gg) -> min_3_out_gga₁(Y)
le_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_le_2_in_1_gg₁(le_2_in_gg₂(X, Y))
le_2_in_gg₂(0_0, s_1₁(Y)) -> le_2_out_gg
le_2_in_gg₂(0_0, 0_0) -> le_2_out_gg
gt_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_gt_2_in_1_gg₁(gt_2_in_gg₂(X, Y))
gt_2_in_gg₂(s_1₁(X), 0_0) -> gt_2_out_gg
if_le_2_in_1_gg₁(le_2_out_gg) -> le_2_out_gg
if_gt_2_in_1_gg₁(gt_2_out_gg) -> gt_2_out_gg

The set Q consists of the following terms:

min_3_in_gga₂(x0, x1)
if_min_3_in_1_gga₂(x0, x1)
if_min_3_in_2_gga₂(x0, x1)
le_2_in_gg₂(x0, x1)
gt_2_in_gg₂(x0, x1)
if_le_2_in_1_gg₁(x0)
if_gt_2_in_1_gg₁(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {MIN2_3_IN_GAG₂, IF_MIN2_3_IN_1_GAG₂}.
By using the subterm criterion together with the size-change analysis 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_3_IN_GAG₂(X, ._2₂(M, L)) -> IF_MIN2_3_IN_1_GAG₂(L, min_3_in_gga₂(X, M))
The graph contains the following edges 2 > 1
IF_MIN2_3_IN_1_GAG₂(L, min_3_out_gga₁(B)) -> MIN2_3_IN_GAG₂(B, L)
The graph contains the following edges 2 > 1, 1 >= 2

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

IF_MINSORT_2_IN_2_GA₄(L, X, L1, remove_3_out_gga₃(X, L, L2)) -> MINSORT_2_IN_GA₂(L2, L1)
MINSORT_2_IN_GA₂(L, ._2₂(X, L1)) -> IF_MINSORT_2_IN_1_GA₄(L, X, L1, min1_2_in_ag₂(X, L))
IF_MINSORT_2_IN_1_GA₄(L, X, L1, min1_2_out_ag₂(X, L)) -> IF_MINSORT_2_IN_2_GA₄(L, X, L1, remove_3_in_gga₃(X, L, L2))

The TRS R consists of the following rules:

minsort_2_in_ga₂([]_0, []_0) -> minsort_2_out_ga₂([]_0, []_0)
minsort_2_in_ga₂(L, ._2₂(X, L1)) -> if_minsort_2_in_1_ga₄(L, X, L1, min1_2_in_ag₂(X, L))
min1_2_in_ag₂(M, ._2₂(X, L)) -> if_min1_2_in_1_ag₄(M, X, L, min2_3_in_gag₃(X, M, L))
min2_3_in_gag₃(X, X, []_0) -> min2_3_out_gag₃(X, X, []_0)
min2_3_in_gag₃(X, A, ._2₂(M, L)) -> if_min2_3_in_1_gag₅(X, A, M, L, min_3_in_gga₃(X, M, B))
min_3_in_gga₃(X, Y, X) -> if_min_3_in_1_gga₃(X, Y, le_2_in_gg₂(X, Y))
le_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_le_2_in_1_gg₃(X, Y, le_2_in_gg₂(X, Y))
le_2_in_gg₂(0_0, s_1₁(Y)) -> le_2_out_gg₂(0_0, s_1₁(Y))
le_2_in_gg₂(0_0, 0_0) -> le_2_out_gg₂(0_0, 0_0)
if_le_2_in_1_gg₃(X, Y, le_2_out_gg₂(X, Y)) -> le_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_min_3_in_1_gga₃(X, Y, le_2_out_gg₂(X, Y)) -> min_3_out_gga₃(X, Y, X)
min_3_in_gga₃(X, Y, Y) -> if_min_3_in_2_gga₃(X, Y, gt_2_in_gg₂(X, Y))
gt_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_gt_2_in_1_gg₃(X, Y, gt_2_in_gg₂(X, Y))
gt_2_in_gg₂(s_1₁(X), 0_0) -> gt_2_out_gg₂(s_1₁(X), 0_0)
if_gt_2_in_1_gg₃(X, Y, gt_2_out_gg₂(X, Y)) -> gt_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_min_3_in_2_gga₃(X, Y, gt_2_out_gg₂(X, Y)) -> min_3_out_gga₃(X, Y, Y)
if_min2_3_in_1_gag₅(X, A, M, L, min_3_out_gga₃(X, M, B)) -> if_min2_3_in_2_gag₆(X, A, M, L, B, min2_3_in_gag₃(B, A, L))
if_min2_3_in_2_gag₆(X, A, M, L, B, min2_3_out_gag₃(B, A, L)) -> min2_3_out_gag₃(X, A, ._2₂(M, L))
if_min1_2_in_1_ag₄(M, X, L, min2_3_out_gag₃(X, M, L)) -> min1_2_out_ag₂(M, ._2₂(X, L))
if_minsort_2_in_1_ga₄(L, X, L1, min1_2_out_ag₂(X, L)) -> if_minsort_2_in_2_ga₄(L, X, L1, remove_3_in_gga₃(X, L, L2))
remove_3_in_gga₃(N, ._2₂(N, L), L) -> remove_3_out_gga₃(N, ._2₂(N, L), L)
remove_3_in_gga₃(N, ._2₂(M, L), ._2₂(M, L1)) -> if_remove_3_in_1_gga₅(N, M, L, L1, notEq_2_in_gg₂(N, M))
notEq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_notEq_2_in_1_gg₃(X, Y, notEq_2_in_gg₂(X, Y))
notEq_2_in_gg₂(s_1₁(X), 0_0) -> notEq_2_out_gg₂(s_1₁(X), 0_0)
notEq_2_in_gg₂(0_0, s_1₁(X)) -> notEq_2_out_gg₂(0_0, s_1₁(X))
if_notEq_2_in_1_gg₃(X, Y, notEq_2_out_gg₂(X, Y)) -> notEq_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_remove_3_in_1_gga₅(N, M, L, L1, notEq_2_out_gg₂(N, M)) -> if_remove_3_in_2_gga₅(N, M, L, L1, remove_3_in_gga₃(N, L, L1))
if_remove_3_in_2_gga₅(N, M, L, L1, remove_3_out_gga₃(N, L, L1)) -> remove_3_out_gga₃(N, ._2₂(M, L), ._2₂(M, L1))
if_minsort_2_in_2_ga₄(L, X, L1, remove_3_out_gga₃(X, L, L2)) -> if_minsort_2_in_3_ga₅(L, X, L1, L2, minsort_2_in_ga₂(L2, L1))
if_minsort_2_in_3_ga₅(L, X, L1, L2, minsort_2_out_ga₂(L2, L1)) -> minsort_2_out_ga₂(L, ._2₂(X, L1))

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

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

IF_MINSORT_2_IN_2_GA₄(L, X, L1, remove_3_out_gga₃(X, L, L2)) -> MINSORT_2_IN_GA₂(L2, L1)
MINSORT_2_IN_GA₂(L, ._2₂(X, L1)) -> IF_MINSORT_2_IN_1_GA₄(L, X, L1, min1_2_in_ag₂(X, L))
IF_MINSORT_2_IN_1_GA₄(L, X, L1, min1_2_out_ag₂(X, L)) -> IF_MINSORT_2_IN_2_GA₄(L, X, L1, remove_3_in_gga₃(X, L, L2))

The TRS R consists of the following rules:

min1_2_in_ag₂(M, ._2₂(X, L)) -> if_min1_2_in_1_ag₄(M, X, L, min2_3_in_gag₃(X, M, L))
remove_3_in_gga₃(N, ._2₂(N, L), L) -> remove_3_out_gga₃(N, ._2₂(N, L), L)
remove_3_in_gga₃(N, ._2₂(M, L), ._2₂(M, L1)) -> if_remove_3_in_1_gga₅(N, M, L, L1, notEq_2_in_gg₂(N, M))
if_min1_2_in_1_ag₄(M, X, L, min2_3_out_gag₃(X, M, L)) -> min1_2_out_ag₂(M, ._2₂(X, L))
if_remove_3_in_1_gga₅(N, M, L, L1, notEq_2_out_gg₂(N, M)) -> if_remove_3_in_2_gga₅(N, M, L, L1, remove_3_in_gga₃(N, L, L1))
min2_3_in_gag₃(X, X, []_0) -> min2_3_out_gag₃(X, X, []_0)
min2_3_in_gag₃(X, A, ._2₂(M, L)) -> if_min2_3_in_1_gag₅(X, A, M, L, min_3_in_gga₃(X, M, B))
notEq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_notEq_2_in_1_gg₃(X, Y, notEq_2_in_gg₂(X, Y))
notEq_2_in_gg₂(s_1₁(X), 0_0) -> notEq_2_out_gg₂(s_1₁(X), 0_0)
notEq_2_in_gg₂(0_0, s_1₁(X)) -> notEq_2_out_gg₂(0_0, s_1₁(X))
if_remove_3_in_2_gga₅(N, M, L, L1, remove_3_out_gga₃(N, L, L1)) -> remove_3_out_gga₃(N, ._2₂(M, L), ._2₂(M, L1))
if_min2_3_in_1_gag₅(X, A, M, L, min_3_out_gga₃(X, M, B)) -> if_min2_3_in_2_gag₆(X, A, M, L, B, min2_3_in_gag₃(B, A, L))
if_notEq_2_in_1_gg₃(X, Y, notEq_2_out_gg₂(X, Y)) -> notEq_2_out_gg₂(s_1₁(X), s_1₁(Y))
min_3_in_gga₃(X, Y, X) -> if_min_3_in_1_gga₃(X, Y, le_2_in_gg₂(X, Y))
min_3_in_gga₃(X, Y, Y) -> if_min_3_in_2_gga₃(X, Y, gt_2_in_gg₂(X, Y))
if_min2_3_in_2_gag₆(X, A, M, L, B, min2_3_out_gag₃(B, A, L)) -> min2_3_out_gag₃(X, A, ._2₂(M, L))
if_min_3_in_1_gga₃(X, Y, le_2_out_gg₂(X, Y)) -> min_3_out_gga₃(X, Y, X)
if_min_3_in_2_gga₃(X, Y, gt_2_out_gg₂(X, Y)) -> min_3_out_gga₃(X, Y, Y)
le_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_le_2_in_1_gg₃(X, Y, le_2_in_gg₂(X, Y))
le_2_in_gg₂(0_0, s_1₁(Y)) -> le_2_out_gg₂(0_0, s_1₁(Y))
le_2_in_gg₂(0_0, 0_0) -> le_2_out_gg₂(0_0, 0_0)
gt_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_gt_2_in_1_gg₃(X, Y, gt_2_in_gg₂(X, Y))
gt_2_in_gg₂(s_1₁(X), 0_0) -> gt_2_out_gg₂(s_1₁(X), 0_0)
if_le_2_in_1_gg₃(X, Y, le_2_out_gg₂(X, Y)) -> le_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_gt_2_in_1_gg₃(X, Y, gt_2_out_gg₂(X, Y)) -> gt_2_out_gg₂(s_1₁(X), s_1₁(Y))

The argument filtering Pi contains the following mapping:
[]_0 = []_0
._2₂(x1, x2) = ._2₂(x1, x2)
s_1₁(x1) = s_1₁(x1)
0_0 = 0_0
min1_2_in_ag₂(x1, x2) = min1_2_in_ag₁(x2)
if_min1_2_in_1_ag₄(x1, x2, x3, x4) = if_min1_2_in_1_ag₁(x4)
min2_3_in_gag₃(x1, x2, x3) = min2_3_in_gag₂(x1, x3)
min2_3_out_gag₃(x1, x2, x3) = min2_3_out_gag₁(x2)
if_min2_3_in_1_gag₅(x1, x2, x3, x4, x5) = if_min2_3_in_1_gag₂(x4, x5)
min_3_in_gga₃(x1, x2, x3) = min_3_in_gga₂(x1, x2)
if_min_3_in_1_gga₃(x1, x2, x3) = if_min_3_in_1_gga₂(x1, x3)
le_2_in_gg₂(x1, x2) = le_2_in_gg₂(x1, x2)
if_le_2_in_1_gg₃(x1, x2, x3) = if_le_2_in_1_gg₁(x3)
le_2_out_gg₂(x1, x2) = le_2_out_gg
min_3_out_gga₃(x1, x2, x3) = min_3_out_gga₁(x3)
if_min_3_in_2_gga₃(x1, x2, x3) = if_min_3_in_2_gga₂(x2, x3)
gt_2_in_gg₂(x1, x2) = gt_2_in_gg₂(x1, x2)
if_gt_2_in_1_gg₃(x1, x2, x3) = if_gt_2_in_1_gg₁(x3)
gt_2_out_gg₂(x1, x2) = gt_2_out_gg
if_min2_3_in_2_gag₆(x1, x2, x3, x4, x5, x6) = if_min2_3_in_2_gag₁(x6)
min1_2_out_ag₂(x1, x2) = min1_2_out_ag₁(x1)
remove_3_in_gga₃(x1, x2, x3) = remove_3_in_gga₂(x1, x2)
remove_3_out_gga₃(x1, x2, x3) = remove_3_out_gga₁(x3)
if_remove_3_in_1_gga₅(x1, x2, x3, x4, x5) = if_remove_3_in_1_gga₄(x1, x2, x3, x5)
notEq_2_in_gg₂(x1, x2) = notEq_2_in_gg₂(x1, x2)
if_notEq_2_in_1_gg₃(x1, x2, x3) = if_notEq_2_in_1_gg₁(x3)
notEq_2_out_gg₂(x1, x2) = notEq_2_out_gg
if_remove_3_in_2_gga₅(x1, x2, x3, x4, x5) = if_remove_3_in_2_gga₂(x2, x5)
IF_MINSORT_2_IN_1_GA₄(x1, x2, x3, x4) = IF_MINSORT_2_IN_1_GA₂(x1, x4)
MINSORT_2_IN_GA₂(x1, x2) = MINSORT_2_IN_GA₁(x1)
IF_MINSORT_2_IN_2_GA₄(x1, x2, x3, x4) = IF_MINSORT_2_IN_2_GA₂(x2, x4)

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

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPPoloProof

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

IF_MINSORT_2_IN_2_GA₂(X, remove_3_out_gga₁(L2)) -> MINSORT_2_IN_GA₁(L2)
MINSORT_2_IN_GA₁(L) -> IF_MINSORT_2_IN_1_GA₂(L, min1_2_in_ag₁(L))
IF_MINSORT_2_IN_1_GA₂(L, min1_2_out_ag₁(X)) -> IF_MINSORT_2_IN_2_GA₂(X, remove_3_in_gga₂(X, L))

The TRS R consists of the following rules:

min1_2_in_ag₁(._2₂(X, L)) -> if_min1_2_in_1_ag₁(min2_3_in_gag₂(X, L))
remove_3_in_gga₂(N, ._2₂(N, L)) -> remove_3_out_gga₁(L)
remove_3_in_gga₂(N, ._2₂(M, L)) -> if_remove_3_in_1_gga₄(N, M, L, notEq_2_in_gg₂(N, M))
if_min1_2_in_1_ag₁(min2_3_out_gag₁(M)) -> min1_2_out_ag₁(M)
if_remove_3_in_1_gga₄(N, M, L, notEq_2_out_gg) -> if_remove_3_in_2_gga₂(M, remove_3_in_gga₂(N, L))
min2_3_in_gag₂(X, []_0) -> min2_3_out_gag₁(X)
min2_3_in_gag₂(X, ._2₂(M, L)) -> if_min2_3_in_1_gag₂(L, min_3_in_gga₂(X, M))
notEq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_notEq_2_in_1_gg₁(notEq_2_in_gg₂(X, Y))
notEq_2_in_gg₂(s_1₁(X), 0_0) -> notEq_2_out_gg
notEq_2_in_gg₂(0_0, s_1₁(X)) -> notEq_2_out_gg
if_remove_3_in_2_gga₂(M, remove_3_out_gga₁(L1)) -> remove_3_out_gga₁(._2₂(M, L1))
if_min2_3_in_1_gag₂(L, min_3_out_gga₁(B)) -> if_min2_3_in_2_gag₁(min2_3_in_gag₂(B, L))
if_notEq_2_in_1_gg₁(notEq_2_out_gg) -> notEq_2_out_gg
min_3_in_gga₂(X, Y) -> if_min_3_in_1_gga₂(X, le_2_in_gg₂(X, Y))
min_3_in_gga₂(X, Y) -> if_min_3_in_2_gga₂(Y, gt_2_in_gg₂(X, Y))
if_min2_3_in_2_gag₁(min2_3_out_gag₁(A)) -> min2_3_out_gag₁(A)
if_min_3_in_1_gga₂(X, le_2_out_gg) -> min_3_out_gga₁(X)
if_min_3_in_2_gga₂(Y, gt_2_out_gg) -> min_3_out_gga₁(Y)
le_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_le_2_in_1_gg₁(le_2_in_gg₂(X, Y))
le_2_in_gg₂(0_0, s_1₁(Y)) -> le_2_out_gg
le_2_in_gg₂(0_0, 0_0) -> le_2_out_gg
gt_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_gt_2_in_1_gg₁(gt_2_in_gg₂(X, Y))
gt_2_in_gg₂(s_1₁(X), 0_0) -> gt_2_out_gg
if_le_2_in_1_gg₁(le_2_out_gg) -> le_2_out_gg
if_gt_2_in_1_gg₁(gt_2_out_gg) -> gt_2_out_gg

The set Q consists of the following terms:

min1_2_in_ag₁(x0)
remove_3_in_gga₂(x0, x1)
if_min1_2_in_1_ag₁(x0)
if_remove_3_in_1_gga₄(x0, x1, x2, x3)
min2_3_in_gag₂(x0, x1)
notEq_2_in_gg₂(x0, x1)
if_remove_3_in_2_gga₂(x0, x1)
if_min2_3_in_1_gag₂(x0, x1)
if_notEq_2_in_1_gg₁(x0)
min_3_in_gga₂(x0, x1)
if_min2_3_in_2_gag₁(x0)
if_min_3_in_1_gga₂(x0, x1)
if_min_3_in_2_gga₂(x0, x1)
le_2_in_gg₂(x0, x1)
gt_2_in_gg₂(x0, x1)
if_le_2_in_1_gg₁(x0)
if_gt_2_in_1_gg₁(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {MINSORT_2_IN_GA₁, IF_MINSORT_2_IN_2_GA₂, IF_MINSORT_2_IN_1_GA₂}.
By using a polynomial ordering, the following set of Dependency Pairs of this DP problem can be strictly oriented.

IF_MINSORT_2_IN_2_GA₂(X, remove_3_out_gga₁(L2)) -> MINSORT_2_IN_GA₁(L2)

The remaining Dependency Pairs were at least non-strictly be oriented.

MINSORT_2_IN_GA₁(L) -> IF_MINSORT_2_IN_1_GA₂(L, min1_2_in_ag₁(L))
IF_MINSORT_2_IN_1_GA₂(L, min1_2_out_ag₁(X)) -> IF_MINSORT_2_IN_2_GA₂(X, remove_3_in_gga₂(X, L))

With the implicit AFS we had to orient the following set of usable rules non-strictly.

if_remove_3_in_2_gga₂(M, remove_3_out_gga₁(L1)) -> remove_3_out_gga₁(._2₂(M, L1))
remove_3_in_gga₂(N, ._2₂(M, L)) -> if_remove_3_in_1_gga₄(N, M, L, notEq_2_in_gg₂(N, M))
remove_3_in_gga₂(N, ._2₂(N, L)) -> remove_3_out_gga₁(L)
if_remove_3_in_1_gga₄(N, M, L, notEq_2_out_gg) -> if_remove_3_in_2_gga₂(M, remove_3_in_gga₂(N, L))

Used ordering: POLO with Polynomial interpretation:

POL(0_0) = 0
POL(gt_2_in_gg₂(x₁, x₂)) = 0
POL(if_min2_3_in_2_gag₁(x₁)) = 0
POL(if_remove_3_in_1_gga₄(x₁, x₂, x₃, x₄)) = 1 + x₃
POL(min_3_out_gga₁(x₁)) = 0
POL(if_min_3_in_1_gga₂(x₁, x₂)) = 0
POL(MINSORT_2_IN_GA₁(x₁)) = x₁
POL(min1_2_out_ag₁(x₁)) = 0
POL(if_remove_3_in_2_gga₂(x₁, x₂)) = 1 + x₂
POL(min2_3_out_gag₁(x₁)) = 0
POL(if_min2_3_in_1_gag₂(x₁, x₂)) = 0
POL(remove_3_in_gga₂(x₁, x₂)) = x₂
POL(notEq_2_out_gg) = 0
POL(IF_MINSORT_2_IN_2_GA₂(x₁, x₂)) = x₂
POL(if_gt_2_in_1_gg₁(x₁)) = 0
POL(le_2_out_gg) = 0
POL(remove_3_out_gga₁(x₁)) = 1 + x₁
POL([]_0) = 0
POL(if_min1_2_in_1_ag₁(x₁)) = 0
POL(if_le_2_in_1_gg₁(x₁)) = 0
POL(notEq_2_in_gg₂(x₁, x₂)) = 0
POL(._2₂(x₁, x₂)) = 1 + x₂
POL(min_3_in_gga₂(x₁, x₂)) = 0
POL(if_notEq_2_in_1_gg₁(x₁)) = 0
POL(min1_2_in_ag₁(x₁)) = 0
POL(IF_MINSORT_2_IN_1_GA₂(x₁, x₂)) = x₁
POL(le_2_in_gg₂(x₁, x₂)) = 0
POL(min2_3_in_gag₂(x₁, x₂)) = 0
POL(s_1₁(x₁)) = 0
POL(gt_2_out_gg) = 0
POL(if_min_3_in_2_gga₂(x₁, x₂)) = 0

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPPoloProof

                          ↳ QDP

                            ↳ DependencyGraphProof

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

MINSORT_2_IN_GA₁(L) -> IF_MINSORT_2_IN_1_GA₂(L, min1_2_in_ag₁(L))
IF_MINSORT_2_IN_1_GA₂(L, min1_2_out_ag₁(X)) -> IF_MINSORT_2_IN_2_GA₂(X, remove_3_in_gga₂(X, L))

The TRS R consists of the following rules:

min1_2_in_ag₁(._2₂(X, L)) -> if_min1_2_in_1_ag₁(min2_3_in_gag₂(X, L))
remove_3_in_gga₂(N, ._2₂(N, L)) -> remove_3_out_gga₁(L)
remove_3_in_gga₂(N, ._2₂(M, L)) -> if_remove_3_in_1_gga₄(N, M, L, notEq_2_in_gg₂(N, M))
if_min1_2_in_1_ag₁(min2_3_out_gag₁(M)) -> min1_2_out_ag₁(M)
if_remove_3_in_1_gga₄(N, M, L, notEq_2_out_gg) -> if_remove_3_in_2_gga₂(M, remove_3_in_gga₂(N, L))
min2_3_in_gag₂(X, []_0) -> min2_3_out_gag₁(X)
min2_3_in_gag₂(X, ._2₂(M, L)) -> if_min2_3_in_1_gag₂(L, min_3_in_gga₂(X, M))
notEq_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_notEq_2_in_1_gg₁(notEq_2_in_gg₂(X, Y))
notEq_2_in_gg₂(s_1₁(X), 0_0) -> notEq_2_out_gg
notEq_2_in_gg₂(0_0, s_1₁(X)) -> notEq_2_out_gg
if_remove_3_in_2_gga₂(M, remove_3_out_gga₁(L1)) -> remove_3_out_gga₁(._2₂(M, L1))
if_min2_3_in_1_gag₂(L, min_3_out_gga₁(B)) -> if_min2_3_in_2_gag₁(min2_3_in_gag₂(B, L))
if_notEq_2_in_1_gg₁(notEq_2_out_gg) -> notEq_2_out_gg
min_3_in_gga₂(X, Y) -> if_min_3_in_1_gga₂(X, le_2_in_gg₂(X, Y))
min_3_in_gga₂(X, Y) -> if_min_3_in_2_gga₂(Y, gt_2_in_gg₂(X, Y))
if_min2_3_in_2_gag₁(min2_3_out_gag₁(A)) -> min2_3_out_gag₁(A)
if_min_3_in_1_gga₂(X, le_2_out_gg) -> min_3_out_gga₁(X)
if_min_3_in_2_gga₂(Y, gt_2_out_gg) -> min_3_out_gga₁(Y)
le_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_le_2_in_1_gg₁(le_2_in_gg₂(X, Y))
le_2_in_gg₂(0_0, s_1₁(Y)) -> le_2_out_gg
le_2_in_gg₂(0_0, 0_0) -> le_2_out_gg
gt_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_gt_2_in_1_gg₁(gt_2_in_gg₂(X, Y))
gt_2_in_gg₂(s_1₁(X), 0_0) -> gt_2_out_gg
if_le_2_in_1_gg₁(le_2_out_gg) -> le_2_out_gg
if_gt_2_in_1_gg₁(gt_2_out_gg) -> gt_2_out_gg

The set Q consists of the following terms:

min1_2_in_ag₁(x0)
remove_3_in_gga₂(x0, x1)
if_min1_2_in_1_ag₁(x0)
if_remove_3_in_1_gga₄(x0, x1, x2, x3)
min2_3_in_gag₂(x0, x1)
notEq_2_in_gg₂(x0, x1)
if_remove_3_in_2_gga₂(x0, x1)
if_min2_3_in_1_gag₂(x0, x1)
if_notEq_2_in_1_gg₁(x0)
min_3_in_gga₂(x0, x1)
if_min2_3_in_2_gag₁(x0)
if_min_3_in_1_gga₂(x0, x1)
if_min_3_in_2_gga₂(x0, x1)
le_2_in_gg₂(x0, x1)
gt_2_in_gg₂(x0, x1)
if_le_2_in_1_gg₁(x0)
if_gt_2_in_1_gg₁(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {IF_MINSORT_2_IN_1_GA₂, MINSORT_2_IN_GA₁, IF_MINSORT_2_IN_2_GA₂}.
The approximation of the Dependency Graph contains 0 SCCs with 2 less nodes.