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



PROLOG
  ↳ PrologToPiTRSProof

mergesort2({}0, {}0).
mergesort2(.2(E, {}0), .2(E, {}0)).
mergesort2(.2(E, .2(F, U)), V) :- split3(U, L2, L1), mergesort2(.2(E, L2), X), mergesort2(.2(F, L1), Z), merge3(X, Z, V).
merge3(X, {}0, X).
merge3({}0, X, X).
merge3(.2(A, X), .2(B, Y), .2(A, Z)) :- le2(A, B), merge3(X, .2(B, Y), Z).
merge3(.2(A, X), .2(B, Y), .2(B, Z)) :- gt2(A, B), merge3(.2(A, X), Y, Z).
split3({}0, {}0, {}0).
split3(.2(E, U), .2(E, V), W) :- split3(U, W, V).
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).


With regard to the inferred argument filtering the predicates were used in the following modes:
mergesort2: (b,f)
split3: (b,f,f)
merge3: (b,b,f)
le2: (b,b)
gt2: (b,b)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:


mergesort_2_in_ga2([]_0, []_0) -> mergesort_2_out_ga2([]_0, []_0)
mergesort_2_in_ga2(._22(E, []_0), ._22(E, []_0)) -> mergesort_2_out_ga2(._22(E, []_0), ._22(E, []_0))
mergesort_2_in_ga2(._22(E, ._22(F, U)), V) -> if_mergesort_2_in_1_ga5(E, F, U, V, split_3_in_gaa3(U, L2, L1))
split_3_in_gaa3([]_0, []_0, []_0) -> split_3_out_gaa3([]_0, []_0, []_0)
split_3_in_gaa3(._22(E, U), ._22(E, V), W) -> if_split_3_in_1_gaa5(E, U, V, W, split_3_in_gaa3(U, W, V))
if_split_3_in_1_gaa5(E, U, V, W, split_3_out_gaa3(U, W, V)) -> split_3_out_gaa3(._22(E, U), ._22(E, V), W)
if_mergesort_2_in_1_ga5(E, F, U, V, split_3_out_gaa3(U, L2, L1)) -> if_mergesort_2_in_2_ga7(E, F, U, V, L2, L1, mergesort_2_in_ga2(._22(E, L2), X))
if_mergesort_2_in_2_ga7(E, F, U, V, L2, L1, mergesort_2_out_ga2(._22(E, L2), X)) -> if_mergesort_2_in_3_ga7(E, F, U, V, L1, X, mergesort_2_in_ga2(._22(F, L1), Z))
if_mergesort_2_in_3_ga7(E, F, U, V, L1, X, mergesort_2_out_ga2(._22(F, L1), Z)) -> if_mergesort_2_in_4_ga7(E, F, U, V, X, Z, merge_3_in_gga3(X, Z, V))
merge_3_in_gga3(X, []_0, X) -> merge_3_out_gga3(X, []_0, X)
merge_3_in_gga3([]_0, X, X) -> merge_3_out_gga3([]_0, X, X)
merge_3_in_gga3(._22(A, X), ._22(B, Y), ._22(A, Z)) -> if_merge_3_in_1_gga6(A, X, B, Y, Z, le_2_in_gg2(A, B))
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_merge_3_in_1_gga6(A, X, B, Y, Z, le_2_out_gg2(A, B)) -> if_merge_3_in_2_gga6(A, X, B, Y, Z, merge_3_in_gga3(X, ._22(B, Y), Z))
merge_3_in_gga3(._22(A, X), ._22(B, Y), ._22(B, Z)) -> if_merge_3_in_3_gga6(A, X, B, Y, Z, gt_2_in_gg2(A, B))
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_merge_3_in_3_gga6(A, X, B, Y, Z, gt_2_out_gg2(A, B)) -> if_merge_3_in_4_gga6(A, X, B, Y, Z, merge_3_in_gga3(._22(A, X), Y, Z))
if_merge_3_in_4_gga6(A, X, B, Y, Z, merge_3_out_gga3(._22(A, X), Y, Z)) -> merge_3_out_gga3(._22(A, X), ._22(B, Y), ._22(B, Z))
if_merge_3_in_2_gga6(A, X, B, Y, Z, merge_3_out_gga3(X, ._22(B, Y), Z)) -> merge_3_out_gga3(._22(A, X), ._22(B, Y), ._22(A, Z))
if_mergesort_2_in_4_ga7(E, F, U, V, X, Z, merge_3_out_gga3(X, Z, V)) -> mergesort_2_out_ga2(._22(E, ._22(F, U)), V)

The argument filtering Pi contains the following mapping:
mergesort_2_in_ga2(x1, x2)  =  mergesort_2_in_ga1(x1)
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
s_11(x1)  =  s_11(x1)
0_0  =  0_0
mergesort_2_out_ga2(x1, x2)  =  mergesort_2_out_ga1(x2)
if_mergesort_2_in_1_ga5(x1, x2, x3, x4, x5)  =  if_mergesort_2_in_1_ga3(x1, x2, x5)
split_3_in_gaa3(x1, x2, x3)  =  split_3_in_gaa1(x1)
split_3_out_gaa3(x1, x2, x3)  =  split_3_out_gaa2(x2, x3)
if_split_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_split_3_in_1_gaa2(x1, x5)
if_mergesort_2_in_2_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_mergesort_2_in_2_ga3(x2, x6, x7)
if_mergesort_2_in_3_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_mergesort_2_in_3_ga2(x6, x7)
if_mergesort_2_in_4_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_mergesort_2_in_4_ga1(x7)
merge_3_in_gga3(x1, x2, x3)  =  merge_3_in_gga2(x1, x2)
merge_3_out_gga3(x1, x2, x3)  =  merge_3_out_gga1(x3)
if_merge_3_in_1_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_1_gga5(x1, x2, x3, x4, x6)
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
if_merge_3_in_2_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_2_gga2(x1, x6)
if_merge_3_in_3_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_3_gga5(x1, x2, x3, x4, x6)
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_merge_3_in_4_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_4_gga2(x3, x6)

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:

mergesort_2_in_ga2([]_0, []_0) -> mergesort_2_out_ga2([]_0, []_0)
mergesort_2_in_ga2(._22(E, []_0), ._22(E, []_0)) -> mergesort_2_out_ga2(._22(E, []_0), ._22(E, []_0))
mergesort_2_in_ga2(._22(E, ._22(F, U)), V) -> if_mergesort_2_in_1_ga5(E, F, U, V, split_3_in_gaa3(U, L2, L1))
split_3_in_gaa3([]_0, []_0, []_0) -> split_3_out_gaa3([]_0, []_0, []_0)
split_3_in_gaa3(._22(E, U), ._22(E, V), W) -> if_split_3_in_1_gaa5(E, U, V, W, split_3_in_gaa3(U, W, V))
if_split_3_in_1_gaa5(E, U, V, W, split_3_out_gaa3(U, W, V)) -> split_3_out_gaa3(._22(E, U), ._22(E, V), W)
if_mergesort_2_in_1_ga5(E, F, U, V, split_3_out_gaa3(U, L2, L1)) -> if_mergesort_2_in_2_ga7(E, F, U, V, L2, L1, mergesort_2_in_ga2(._22(E, L2), X))
if_mergesort_2_in_2_ga7(E, F, U, V, L2, L1, mergesort_2_out_ga2(._22(E, L2), X)) -> if_mergesort_2_in_3_ga7(E, F, U, V, L1, X, mergesort_2_in_ga2(._22(F, L1), Z))
if_mergesort_2_in_3_ga7(E, F, U, V, L1, X, mergesort_2_out_ga2(._22(F, L1), Z)) -> if_mergesort_2_in_4_ga7(E, F, U, V, X, Z, merge_3_in_gga3(X, Z, V))
merge_3_in_gga3(X, []_0, X) -> merge_3_out_gga3(X, []_0, X)
merge_3_in_gga3([]_0, X, X) -> merge_3_out_gga3([]_0, X, X)
merge_3_in_gga3(._22(A, X), ._22(B, Y), ._22(A, Z)) -> if_merge_3_in_1_gga6(A, X, B, Y, Z, le_2_in_gg2(A, B))
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_merge_3_in_1_gga6(A, X, B, Y, Z, le_2_out_gg2(A, B)) -> if_merge_3_in_2_gga6(A, X, B, Y, Z, merge_3_in_gga3(X, ._22(B, Y), Z))
merge_3_in_gga3(._22(A, X), ._22(B, Y), ._22(B, Z)) -> if_merge_3_in_3_gga6(A, X, B, Y, Z, gt_2_in_gg2(A, B))
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_merge_3_in_3_gga6(A, X, B, Y, Z, gt_2_out_gg2(A, B)) -> if_merge_3_in_4_gga6(A, X, B, Y, Z, merge_3_in_gga3(._22(A, X), Y, Z))
if_merge_3_in_4_gga6(A, X, B, Y, Z, merge_3_out_gga3(._22(A, X), Y, Z)) -> merge_3_out_gga3(._22(A, X), ._22(B, Y), ._22(B, Z))
if_merge_3_in_2_gga6(A, X, B, Y, Z, merge_3_out_gga3(X, ._22(B, Y), Z)) -> merge_3_out_gga3(._22(A, X), ._22(B, Y), ._22(A, Z))
if_mergesort_2_in_4_ga7(E, F, U, V, X, Z, merge_3_out_gga3(X, Z, V)) -> mergesort_2_out_ga2(._22(E, ._22(F, U)), V)

The argument filtering Pi contains the following mapping:
mergesort_2_in_ga2(x1, x2)  =  mergesort_2_in_ga1(x1)
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
s_11(x1)  =  s_11(x1)
0_0  =  0_0
mergesort_2_out_ga2(x1, x2)  =  mergesort_2_out_ga1(x2)
if_mergesort_2_in_1_ga5(x1, x2, x3, x4, x5)  =  if_mergesort_2_in_1_ga3(x1, x2, x5)
split_3_in_gaa3(x1, x2, x3)  =  split_3_in_gaa1(x1)
split_3_out_gaa3(x1, x2, x3)  =  split_3_out_gaa2(x2, x3)
if_split_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_split_3_in_1_gaa2(x1, x5)
if_mergesort_2_in_2_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_mergesort_2_in_2_ga3(x2, x6, x7)
if_mergesort_2_in_3_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_mergesort_2_in_3_ga2(x6, x7)
if_mergesort_2_in_4_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_mergesort_2_in_4_ga1(x7)
merge_3_in_gga3(x1, x2, x3)  =  merge_3_in_gga2(x1, x2)
merge_3_out_gga3(x1, x2, x3)  =  merge_3_out_gga1(x3)
if_merge_3_in_1_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_1_gga5(x1, x2, x3, x4, x6)
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
if_merge_3_in_2_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_2_gga2(x1, x6)
if_merge_3_in_3_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_3_gga5(x1, x2, x3, x4, x6)
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_merge_3_in_4_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_4_gga2(x3, x6)


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

MERGESORT_2_IN_GA2(._22(E, ._22(F, U)), V) -> IF_MERGESORT_2_IN_1_GA5(E, F, U, V, split_3_in_gaa3(U, L2, L1))
MERGESORT_2_IN_GA2(._22(E, ._22(F, U)), V) -> SPLIT_3_IN_GAA3(U, L2, L1)
SPLIT_3_IN_GAA3(._22(E, U), ._22(E, V), W) -> IF_SPLIT_3_IN_1_GAA5(E, U, V, W, split_3_in_gaa3(U, W, V))
SPLIT_3_IN_GAA3(._22(E, U), ._22(E, V), W) -> SPLIT_3_IN_GAA3(U, W, V)
IF_MERGESORT_2_IN_1_GA5(E, F, U, V, split_3_out_gaa3(U, L2, L1)) -> IF_MERGESORT_2_IN_2_GA7(E, F, U, V, L2, L1, mergesort_2_in_ga2(._22(E, L2), X))
IF_MERGESORT_2_IN_1_GA5(E, F, U, V, split_3_out_gaa3(U, L2, L1)) -> MERGESORT_2_IN_GA2(._22(E, L2), X)
IF_MERGESORT_2_IN_2_GA7(E, F, U, V, L2, L1, mergesort_2_out_ga2(._22(E, L2), X)) -> IF_MERGESORT_2_IN_3_GA7(E, F, U, V, L1, X, mergesort_2_in_ga2(._22(F, L1), Z))
IF_MERGESORT_2_IN_2_GA7(E, F, U, V, L2, L1, mergesort_2_out_ga2(._22(E, L2), X)) -> MERGESORT_2_IN_GA2(._22(F, L1), Z)
IF_MERGESORT_2_IN_3_GA7(E, F, U, V, L1, X, mergesort_2_out_ga2(._22(F, L1), Z)) -> IF_MERGESORT_2_IN_4_GA7(E, F, U, V, X, Z, merge_3_in_gga3(X, Z, V))
IF_MERGESORT_2_IN_3_GA7(E, F, U, V, L1, X, mergesort_2_out_ga2(._22(F, L1), Z)) -> MERGE_3_IN_GGA3(X, Z, V)
MERGE_3_IN_GGA3(._22(A, X), ._22(B, Y), ._22(A, Z)) -> IF_MERGE_3_IN_1_GGA6(A, X, B, Y, Z, le_2_in_gg2(A, B))
MERGE_3_IN_GGA3(._22(A, X), ._22(B, Y), ._22(A, Z)) -> LE_2_IN_GG2(A, B)
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)
IF_MERGE_3_IN_1_GGA6(A, X, B, Y, Z, le_2_out_gg2(A, B)) -> IF_MERGE_3_IN_2_GGA6(A, X, B, Y, Z, merge_3_in_gga3(X, ._22(B, Y), Z))
IF_MERGE_3_IN_1_GGA6(A, X, B, Y, Z, le_2_out_gg2(A, B)) -> MERGE_3_IN_GGA3(X, ._22(B, Y), Z)
MERGE_3_IN_GGA3(._22(A, X), ._22(B, Y), ._22(B, Z)) -> IF_MERGE_3_IN_3_GGA6(A, X, B, Y, Z, gt_2_in_gg2(A, B))
MERGE_3_IN_GGA3(._22(A, X), ._22(B, Y), ._22(B, Z)) -> GT_2_IN_GG2(A, B)
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_MERGE_3_IN_3_GGA6(A, X, B, Y, Z, gt_2_out_gg2(A, B)) -> IF_MERGE_3_IN_4_GGA6(A, X, B, Y, Z, merge_3_in_gga3(._22(A, X), Y, Z))
IF_MERGE_3_IN_3_GGA6(A, X, B, Y, Z, gt_2_out_gg2(A, B)) -> MERGE_3_IN_GGA3(._22(A, X), Y, Z)

The TRS R consists of the following rules:

mergesort_2_in_ga2([]_0, []_0) -> mergesort_2_out_ga2([]_0, []_0)
mergesort_2_in_ga2(._22(E, []_0), ._22(E, []_0)) -> mergesort_2_out_ga2(._22(E, []_0), ._22(E, []_0))
mergesort_2_in_ga2(._22(E, ._22(F, U)), V) -> if_mergesort_2_in_1_ga5(E, F, U, V, split_3_in_gaa3(U, L2, L1))
split_3_in_gaa3([]_0, []_0, []_0) -> split_3_out_gaa3([]_0, []_0, []_0)
split_3_in_gaa3(._22(E, U), ._22(E, V), W) -> if_split_3_in_1_gaa5(E, U, V, W, split_3_in_gaa3(U, W, V))
if_split_3_in_1_gaa5(E, U, V, W, split_3_out_gaa3(U, W, V)) -> split_3_out_gaa3(._22(E, U), ._22(E, V), W)
if_mergesort_2_in_1_ga5(E, F, U, V, split_3_out_gaa3(U, L2, L1)) -> if_mergesort_2_in_2_ga7(E, F, U, V, L2, L1, mergesort_2_in_ga2(._22(E, L2), X))
if_mergesort_2_in_2_ga7(E, F, U, V, L2, L1, mergesort_2_out_ga2(._22(E, L2), X)) -> if_mergesort_2_in_3_ga7(E, F, U, V, L1, X, mergesort_2_in_ga2(._22(F, L1), Z))
if_mergesort_2_in_3_ga7(E, F, U, V, L1, X, mergesort_2_out_ga2(._22(F, L1), Z)) -> if_mergesort_2_in_4_ga7(E, F, U, V, X, Z, merge_3_in_gga3(X, Z, V))
merge_3_in_gga3(X, []_0, X) -> merge_3_out_gga3(X, []_0, X)
merge_3_in_gga3([]_0, X, X) -> merge_3_out_gga3([]_0, X, X)
merge_3_in_gga3(._22(A, X), ._22(B, Y), ._22(A, Z)) -> if_merge_3_in_1_gga6(A, X, B, Y, Z, le_2_in_gg2(A, B))
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_merge_3_in_1_gga6(A, X, B, Y, Z, le_2_out_gg2(A, B)) -> if_merge_3_in_2_gga6(A, X, B, Y, Z, merge_3_in_gga3(X, ._22(B, Y), Z))
merge_3_in_gga3(._22(A, X), ._22(B, Y), ._22(B, Z)) -> if_merge_3_in_3_gga6(A, X, B, Y, Z, gt_2_in_gg2(A, B))
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_merge_3_in_3_gga6(A, X, B, Y, Z, gt_2_out_gg2(A, B)) -> if_merge_3_in_4_gga6(A, X, B, Y, Z, merge_3_in_gga3(._22(A, X), Y, Z))
if_merge_3_in_4_gga6(A, X, B, Y, Z, merge_3_out_gga3(._22(A, X), Y, Z)) -> merge_3_out_gga3(._22(A, X), ._22(B, Y), ._22(B, Z))
if_merge_3_in_2_gga6(A, X, B, Y, Z, merge_3_out_gga3(X, ._22(B, Y), Z)) -> merge_3_out_gga3(._22(A, X), ._22(B, Y), ._22(A, Z))
if_mergesort_2_in_4_ga7(E, F, U, V, X, Z, merge_3_out_gga3(X, Z, V)) -> mergesort_2_out_ga2(._22(E, ._22(F, U)), V)

The argument filtering Pi contains the following mapping:
mergesort_2_in_ga2(x1, x2)  =  mergesort_2_in_ga1(x1)
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
s_11(x1)  =  s_11(x1)
0_0  =  0_0
mergesort_2_out_ga2(x1, x2)  =  mergesort_2_out_ga1(x2)
if_mergesort_2_in_1_ga5(x1, x2, x3, x4, x5)  =  if_mergesort_2_in_1_ga3(x1, x2, x5)
split_3_in_gaa3(x1, x2, x3)  =  split_3_in_gaa1(x1)
split_3_out_gaa3(x1, x2, x3)  =  split_3_out_gaa2(x2, x3)
if_split_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_split_3_in_1_gaa2(x1, x5)
if_mergesort_2_in_2_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_mergesort_2_in_2_ga3(x2, x6, x7)
if_mergesort_2_in_3_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_mergesort_2_in_3_ga2(x6, x7)
if_mergesort_2_in_4_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_mergesort_2_in_4_ga1(x7)
merge_3_in_gga3(x1, x2, x3)  =  merge_3_in_gga2(x1, x2)
merge_3_out_gga3(x1, x2, x3)  =  merge_3_out_gga1(x3)
if_merge_3_in_1_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_1_gga5(x1, x2, x3, x4, x6)
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
if_merge_3_in_2_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_2_gga2(x1, x6)
if_merge_3_in_3_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_3_gga5(x1, x2, x3, x4, x6)
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_merge_3_in_4_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_4_gga2(x3, x6)
GT_2_IN_GG2(x1, x2)  =  GT_2_IN_GG2(x1, x2)
IF_SPLIT_3_IN_1_GAA5(x1, x2, x3, x4, x5)  =  IF_SPLIT_3_IN_1_GAA2(x1, x5)
IF_MERGE_3_IN_2_GGA6(x1, x2, x3, x4, x5, x6)  =  IF_MERGE_3_IN_2_GGA2(x1, x6)
IF_MERGE_3_IN_1_GGA6(x1, x2, x3, x4, x5, x6)  =  IF_MERGE_3_IN_1_GGA5(x1, x2, x3, x4, x6)
IF_GT_2_IN_1_GG3(x1, x2, x3)  =  IF_GT_2_IN_1_GG1(x3)
IF_MERGESORT_2_IN_4_GA7(x1, x2, x3, x4, x5, x6, x7)  =  IF_MERGESORT_2_IN_4_GA1(x7)
IF_MERGESORT_2_IN_2_GA7(x1, x2, x3, x4, x5, x6, x7)  =  IF_MERGESORT_2_IN_2_GA3(x2, x6, x7)
IF_MERGE_3_IN_4_GGA6(x1, x2, x3, x4, x5, x6)  =  IF_MERGE_3_IN_4_GGA2(x3, x6)
IF_LE_2_IN_1_GG3(x1, x2, x3)  =  IF_LE_2_IN_1_GG1(x3)
MERGE_3_IN_GGA3(x1, x2, x3)  =  MERGE_3_IN_GGA2(x1, x2)
SPLIT_3_IN_GAA3(x1, x2, x3)  =  SPLIT_3_IN_GAA1(x1)
IF_MERGE_3_IN_3_GGA6(x1, x2, x3, x4, x5, x6)  =  IF_MERGE_3_IN_3_GGA5(x1, x2, x3, x4, x6)
MERGESORT_2_IN_GA2(x1, x2)  =  MERGESORT_2_IN_GA1(x1)
IF_MERGESORT_2_IN_3_GA7(x1, x2, x3, x4, x5, x6, x7)  =  IF_MERGESORT_2_IN_3_GA2(x6, x7)
LE_2_IN_GG2(x1, x2)  =  LE_2_IN_GG2(x1, x2)
IF_MERGESORT_2_IN_1_GA5(x1, x2, x3, x4, x5)  =  IF_MERGESORT_2_IN_1_GA3(x1, x2, x5)

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:

MERGESORT_2_IN_GA2(._22(E, ._22(F, U)), V) -> IF_MERGESORT_2_IN_1_GA5(E, F, U, V, split_3_in_gaa3(U, L2, L1))
MERGESORT_2_IN_GA2(._22(E, ._22(F, U)), V) -> SPLIT_3_IN_GAA3(U, L2, L1)
SPLIT_3_IN_GAA3(._22(E, U), ._22(E, V), W) -> IF_SPLIT_3_IN_1_GAA5(E, U, V, W, split_3_in_gaa3(U, W, V))
SPLIT_3_IN_GAA3(._22(E, U), ._22(E, V), W) -> SPLIT_3_IN_GAA3(U, W, V)
IF_MERGESORT_2_IN_1_GA5(E, F, U, V, split_3_out_gaa3(U, L2, L1)) -> IF_MERGESORT_2_IN_2_GA7(E, F, U, V, L2, L1, mergesort_2_in_ga2(._22(E, L2), X))
IF_MERGESORT_2_IN_1_GA5(E, F, U, V, split_3_out_gaa3(U, L2, L1)) -> MERGESORT_2_IN_GA2(._22(E, L2), X)
IF_MERGESORT_2_IN_2_GA7(E, F, U, V, L2, L1, mergesort_2_out_ga2(._22(E, L2), X)) -> IF_MERGESORT_2_IN_3_GA7(E, F, U, V, L1, X, mergesort_2_in_ga2(._22(F, L1), Z))
IF_MERGESORT_2_IN_2_GA7(E, F, U, V, L2, L1, mergesort_2_out_ga2(._22(E, L2), X)) -> MERGESORT_2_IN_GA2(._22(F, L1), Z)
IF_MERGESORT_2_IN_3_GA7(E, F, U, V, L1, X, mergesort_2_out_ga2(._22(F, L1), Z)) -> IF_MERGESORT_2_IN_4_GA7(E, F, U, V, X, Z, merge_3_in_gga3(X, Z, V))
IF_MERGESORT_2_IN_3_GA7(E, F, U, V, L1, X, mergesort_2_out_ga2(._22(F, L1), Z)) -> MERGE_3_IN_GGA3(X, Z, V)
MERGE_3_IN_GGA3(._22(A, X), ._22(B, Y), ._22(A, Z)) -> IF_MERGE_3_IN_1_GGA6(A, X, B, Y, Z, le_2_in_gg2(A, B))
MERGE_3_IN_GGA3(._22(A, X), ._22(B, Y), ._22(A, Z)) -> LE_2_IN_GG2(A, B)
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)
IF_MERGE_3_IN_1_GGA6(A, X, B, Y, Z, le_2_out_gg2(A, B)) -> IF_MERGE_3_IN_2_GGA6(A, X, B, Y, Z, merge_3_in_gga3(X, ._22(B, Y), Z))
IF_MERGE_3_IN_1_GGA6(A, X, B, Y, Z, le_2_out_gg2(A, B)) -> MERGE_3_IN_GGA3(X, ._22(B, Y), Z)
MERGE_3_IN_GGA3(._22(A, X), ._22(B, Y), ._22(B, Z)) -> IF_MERGE_3_IN_3_GGA6(A, X, B, Y, Z, gt_2_in_gg2(A, B))
MERGE_3_IN_GGA3(._22(A, X), ._22(B, Y), ._22(B, Z)) -> GT_2_IN_GG2(A, B)
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_MERGE_3_IN_3_GGA6(A, X, B, Y, Z, gt_2_out_gg2(A, B)) -> IF_MERGE_3_IN_4_GGA6(A, X, B, Y, Z, merge_3_in_gga3(._22(A, X), Y, Z))
IF_MERGE_3_IN_3_GGA6(A, X, B, Y, Z, gt_2_out_gg2(A, B)) -> MERGE_3_IN_GGA3(._22(A, X), Y, Z)

The TRS R consists of the following rules:

mergesort_2_in_ga2([]_0, []_0) -> mergesort_2_out_ga2([]_0, []_0)
mergesort_2_in_ga2(._22(E, []_0), ._22(E, []_0)) -> mergesort_2_out_ga2(._22(E, []_0), ._22(E, []_0))
mergesort_2_in_ga2(._22(E, ._22(F, U)), V) -> if_mergesort_2_in_1_ga5(E, F, U, V, split_3_in_gaa3(U, L2, L1))
split_3_in_gaa3([]_0, []_0, []_0) -> split_3_out_gaa3([]_0, []_0, []_0)
split_3_in_gaa3(._22(E, U), ._22(E, V), W) -> if_split_3_in_1_gaa5(E, U, V, W, split_3_in_gaa3(U, W, V))
if_split_3_in_1_gaa5(E, U, V, W, split_3_out_gaa3(U, W, V)) -> split_3_out_gaa3(._22(E, U), ._22(E, V), W)
if_mergesort_2_in_1_ga5(E, F, U, V, split_3_out_gaa3(U, L2, L1)) -> if_mergesort_2_in_2_ga7(E, F, U, V, L2, L1, mergesort_2_in_ga2(._22(E, L2), X))
if_mergesort_2_in_2_ga7(E, F, U, V, L2, L1, mergesort_2_out_ga2(._22(E, L2), X)) -> if_mergesort_2_in_3_ga7(E, F, U, V, L1, X, mergesort_2_in_ga2(._22(F, L1), Z))
if_mergesort_2_in_3_ga7(E, F, U, V, L1, X, mergesort_2_out_ga2(._22(F, L1), Z)) -> if_mergesort_2_in_4_ga7(E, F, U, V, X, Z, merge_3_in_gga3(X, Z, V))
merge_3_in_gga3(X, []_0, X) -> merge_3_out_gga3(X, []_0, X)
merge_3_in_gga3([]_0, X, X) -> merge_3_out_gga3([]_0, X, X)
merge_3_in_gga3(._22(A, X), ._22(B, Y), ._22(A, Z)) -> if_merge_3_in_1_gga6(A, X, B, Y, Z, le_2_in_gg2(A, B))
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_merge_3_in_1_gga6(A, X, B, Y, Z, le_2_out_gg2(A, B)) -> if_merge_3_in_2_gga6(A, X, B, Y, Z, merge_3_in_gga3(X, ._22(B, Y), Z))
merge_3_in_gga3(._22(A, X), ._22(B, Y), ._22(B, Z)) -> if_merge_3_in_3_gga6(A, X, B, Y, Z, gt_2_in_gg2(A, B))
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_merge_3_in_3_gga6(A, X, B, Y, Z, gt_2_out_gg2(A, B)) -> if_merge_3_in_4_gga6(A, X, B, Y, Z, merge_3_in_gga3(._22(A, X), Y, Z))
if_merge_3_in_4_gga6(A, X, B, Y, Z, merge_3_out_gga3(._22(A, X), Y, Z)) -> merge_3_out_gga3(._22(A, X), ._22(B, Y), ._22(B, Z))
if_merge_3_in_2_gga6(A, X, B, Y, Z, merge_3_out_gga3(X, ._22(B, Y), Z)) -> merge_3_out_gga3(._22(A, X), ._22(B, Y), ._22(A, Z))
if_mergesort_2_in_4_ga7(E, F, U, V, X, Z, merge_3_out_gga3(X, Z, V)) -> mergesort_2_out_ga2(._22(E, ._22(F, U)), V)

The argument filtering Pi contains the following mapping:
mergesort_2_in_ga2(x1, x2)  =  mergesort_2_in_ga1(x1)
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
s_11(x1)  =  s_11(x1)
0_0  =  0_0
mergesort_2_out_ga2(x1, x2)  =  mergesort_2_out_ga1(x2)
if_mergesort_2_in_1_ga5(x1, x2, x3, x4, x5)  =  if_mergesort_2_in_1_ga3(x1, x2, x5)
split_3_in_gaa3(x1, x2, x3)  =  split_3_in_gaa1(x1)
split_3_out_gaa3(x1, x2, x3)  =  split_3_out_gaa2(x2, x3)
if_split_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_split_3_in_1_gaa2(x1, x5)
if_mergesort_2_in_2_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_mergesort_2_in_2_ga3(x2, x6, x7)
if_mergesort_2_in_3_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_mergesort_2_in_3_ga2(x6, x7)
if_mergesort_2_in_4_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_mergesort_2_in_4_ga1(x7)
merge_3_in_gga3(x1, x2, x3)  =  merge_3_in_gga2(x1, x2)
merge_3_out_gga3(x1, x2, x3)  =  merge_3_out_gga1(x3)
if_merge_3_in_1_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_1_gga5(x1, x2, x3, x4, x6)
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
if_merge_3_in_2_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_2_gga2(x1, x6)
if_merge_3_in_3_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_3_gga5(x1, x2, x3, x4, x6)
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_merge_3_in_4_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_4_gga2(x3, x6)
GT_2_IN_GG2(x1, x2)  =  GT_2_IN_GG2(x1, x2)
IF_SPLIT_3_IN_1_GAA5(x1, x2, x3, x4, x5)  =  IF_SPLIT_3_IN_1_GAA2(x1, x5)
IF_MERGE_3_IN_2_GGA6(x1, x2, x3, x4, x5, x6)  =  IF_MERGE_3_IN_2_GGA2(x1, x6)
IF_MERGE_3_IN_1_GGA6(x1, x2, x3, x4, x5, x6)  =  IF_MERGE_3_IN_1_GGA5(x1, x2, x3, x4, x6)
IF_GT_2_IN_1_GG3(x1, x2, x3)  =  IF_GT_2_IN_1_GG1(x3)
IF_MERGESORT_2_IN_4_GA7(x1, x2, x3, x4, x5, x6, x7)  =  IF_MERGESORT_2_IN_4_GA1(x7)
IF_MERGESORT_2_IN_2_GA7(x1, x2, x3, x4, x5, x6, x7)  =  IF_MERGESORT_2_IN_2_GA3(x2, x6, x7)
IF_MERGE_3_IN_4_GGA6(x1, x2, x3, x4, x5, x6)  =  IF_MERGE_3_IN_4_GGA2(x3, x6)
IF_LE_2_IN_1_GG3(x1, x2, x3)  =  IF_LE_2_IN_1_GG1(x3)
MERGE_3_IN_GGA3(x1, x2, x3)  =  MERGE_3_IN_GGA2(x1, x2)
SPLIT_3_IN_GAA3(x1, x2, x3)  =  SPLIT_3_IN_GAA1(x1)
IF_MERGE_3_IN_3_GGA6(x1, x2, x3, x4, x5, x6)  =  IF_MERGE_3_IN_3_GGA5(x1, x2, x3, x4, x6)
MERGESORT_2_IN_GA2(x1, x2)  =  MERGESORT_2_IN_GA1(x1)
IF_MERGESORT_2_IN_3_GA7(x1, x2, x3, x4, x5, x6, x7)  =  IF_MERGESORT_2_IN_3_GA2(x6, x7)
LE_2_IN_GG2(x1, x2)  =  LE_2_IN_GG2(x1, x2)
IF_MERGESORT_2_IN_1_GA5(x1, x2, x3, x4, x5)  =  IF_MERGESORT_2_IN_1_GA3(x1, x2, x5)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph contains 5 SCCs with 11 less nodes.

↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
PiDP
                ↳ UsableRulesProof
              ↳ PiDP
              ↳ 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:

mergesort_2_in_ga2([]_0, []_0) -> mergesort_2_out_ga2([]_0, []_0)
mergesort_2_in_ga2(._22(E, []_0), ._22(E, []_0)) -> mergesort_2_out_ga2(._22(E, []_0), ._22(E, []_0))
mergesort_2_in_ga2(._22(E, ._22(F, U)), V) -> if_mergesort_2_in_1_ga5(E, F, U, V, split_3_in_gaa3(U, L2, L1))
split_3_in_gaa3([]_0, []_0, []_0) -> split_3_out_gaa3([]_0, []_0, []_0)
split_3_in_gaa3(._22(E, U), ._22(E, V), W) -> if_split_3_in_1_gaa5(E, U, V, W, split_3_in_gaa3(U, W, V))
if_split_3_in_1_gaa5(E, U, V, W, split_3_out_gaa3(U, W, V)) -> split_3_out_gaa3(._22(E, U), ._22(E, V), W)
if_mergesort_2_in_1_ga5(E, F, U, V, split_3_out_gaa3(U, L2, L1)) -> if_mergesort_2_in_2_ga7(E, F, U, V, L2, L1, mergesort_2_in_ga2(._22(E, L2), X))
if_mergesort_2_in_2_ga7(E, F, U, V, L2, L1, mergesort_2_out_ga2(._22(E, L2), X)) -> if_mergesort_2_in_3_ga7(E, F, U, V, L1, X, mergesort_2_in_ga2(._22(F, L1), Z))
if_mergesort_2_in_3_ga7(E, F, U, V, L1, X, mergesort_2_out_ga2(._22(F, L1), Z)) -> if_mergesort_2_in_4_ga7(E, F, U, V, X, Z, merge_3_in_gga3(X, Z, V))
merge_3_in_gga3(X, []_0, X) -> merge_3_out_gga3(X, []_0, X)
merge_3_in_gga3([]_0, X, X) -> merge_3_out_gga3([]_0, X, X)
merge_3_in_gga3(._22(A, X), ._22(B, Y), ._22(A, Z)) -> if_merge_3_in_1_gga6(A, X, B, Y, Z, le_2_in_gg2(A, B))
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_merge_3_in_1_gga6(A, X, B, Y, Z, le_2_out_gg2(A, B)) -> if_merge_3_in_2_gga6(A, X, B, Y, Z, merge_3_in_gga3(X, ._22(B, Y), Z))
merge_3_in_gga3(._22(A, X), ._22(B, Y), ._22(B, Z)) -> if_merge_3_in_3_gga6(A, X, B, Y, Z, gt_2_in_gg2(A, B))
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_merge_3_in_3_gga6(A, X, B, Y, Z, gt_2_out_gg2(A, B)) -> if_merge_3_in_4_gga6(A, X, B, Y, Z, merge_3_in_gga3(._22(A, X), Y, Z))
if_merge_3_in_4_gga6(A, X, B, Y, Z, merge_3_out_gga3(._22(A, X), Y, Z)) -> merge_3_out_gga3(._22(A, X), ._22(B, Y), ._22(B, Z))
if_merge_3_in_2_gga6(A, X, B, Y, Z, merge_3_out_gga3(X, ._22(B, Y), Z)) -> merge_3_out_gga3(._22(A, X), ._22(B, Y), ._22(A, Z))
if_mergesort_2_in_4_ga7(E, F, U, V, X, Z, merge_3_out_gga3(X, Z, V)) -> mergesort_2_out_ga2(._22(E, ._22(F, U)), V)

The argument filtering Pi contains the following mapping:
mergesort_2_in_ga2(x1, x2)  =  mergesort_2_in_ga1(x1)
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
s_11(x1)  =  s_11(x1)
0_0  =  0_0
mergesort_2_out_ga2(x1, x2)  =  mergesort_2_out_ga1(x2)
if_mergesort_2_in_1_ga5(x1, x2, x3, x4, x5)  =  if_mergesort_2_in_1_ga3(x1, x2, x5)
split_3_in_gaa3(x1, x2, x3)  =  split_3_in_gaa1(x1)
split_3_out_gaa3(x1, x2, x3)  =  split_3_out_gaa2(x2, x3)
if_split_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_split_3_in_1_gaa2(x1, x5)
if_mergesort_2_in_2_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_mergesort_2_in_2_ga3(x2, x6, x7)
if_mergesort_2_in_3_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_mergesort_2_in_3_ga2(x6, x7)
if_mergesort_2_in_4_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_mergesort_2_in_4_ga1(x7)
merge_3_in_gga3(x1, x2, x3)  =  merge_3_in_gga2(x1, x2)
merge_3_out_gga3(x1, x2, x3)  =  merge_3_out_gga1(x3)
if_merge_3_in_1_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_1_gga5(x1, x2, x3, x4, x6)
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
if_merge_3_in_2_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_2_gga2(x1, x6)
if_merge_3_in_3_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_3_gga5(x1, x2, x3, x4, x6)
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_merge_3_in_4_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_4_gga2(x3, x6)
GT_2_IN_GG2(x1, x2)  =  GT_2_IN_GG2(x1, x2)

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

↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

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
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ 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:



↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
PiDP
                ↳ UsableRulesProof
              ↳ PiDP
              ↳ 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:

mergesort_2_in_ga2([]_0, []_0) -> mergesort_2_out_ga2([]_0, []_0)
mergesort_2_in_ga2(._22(E, []_0), ._22(E, []_0)) -> mergesort_2_out_ga2(._22(E, []_0), ._22(E, []_0))
mergesort_2_in_ga2(._22(E, ._22(F, U)), V) -> if_mergesort_2_in_1_ga5(E, F, U, V, split_3_in_gaa3(U, L2, L1))
split_3_in_gaa3([]_0, []_0, []_0) -> split_3_out_gaa3([]_0, []_0, []_0)
split_3_in_gaa3(._22(E, U), ._22(E, V), W) -> if_split_3_in_1_gaa5(E, U, V, W, split_3_in_gaa3(U, W, V))
if_split_3_in_1_gaa5(E, U, V, W, split_3_out_gaa3(U, W, V)) -> split_3_out_gaa3(._22(E, U), ._22(E, V), W)
if_mergesort_2_in_1_ga5(E, F, U, V, split_3_out_gaa3(U, L2, L1)) -> if_mergesort_2_in_2_ga7(E, F, U, V, L2, L1, mergesort_2_in_ga2(._22(E, L2), X))
if_mergesort_2_in_2_ga7(E, F, U, V, L2, L1, mergesort_2_out_ga2(._22(E, L2), X)) -> if_mergesort_2_in_3_ga7(E, F, U, V, L1, X, mergesort_2_in_ga2(._22(F, L1), Z))
if_mergesort_2_in_3_ga7(E, F, U, V, L1, X, mergesort_2_out_ga2(._22(F, L1), Z)) -> if_mergesort_2_in_4_ga7(E, F, U, V, X, Z, merge_3_in_gga3(X, Z, V))
merge_3_in_gga3(X, []_0, X) -> merge_3_out_gga3(X, []_0, X)
merge_3_in_gga3([]_0, X, X) -> merge_3_out_gga3([]_0, X, X)
merge_3_in_gga3(._22(A, X), ._22(B, Y), ._22(A, Z)) -> if_merge_3_in_1_gga6(A, X, B, Y, Z, le_2_in_gg2(A, B))
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_merge_3_in_1_gga6(A, X, B, Y, Z, le_2_out_gg2(A, B)) -> if_merge_3_in_2_gga6(A, X, B, Y, Z, merge_3_in_gga3(X, ._22(B, Y), Z))
merge_3_in_gga3(._22(A, X), ._22(B, Y), ._22(B, Z)) -> if_merge_3_in_3_gga6(A, X, B, Y, Z, gt_2_in_gg2(A, B))
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_merge_3_in_3_gga6(A, X, B, Y, Z, gt_2_out_gg2(A, B)) -> if_merge_3_in_4_gga6(A, X, B, Y, Z, merge_3_in_gga3(._22(A, X), Y, Z))
if_merge_3_in_4_gga6(A, X, B, Y, Z, merge_3_out_gga3(._22(A, X), Y, Z)) -> merge_3_out_gga3(._22(A, X), ._22(B, Y), ._22(B, Z))
if_merge_3_in_2_gga6(A, X, B, Y, Z, merge_3_out_gga3(X, ._22(B, Y), Z)) -> merge_3_out_gga3(._22(A, X), ._22(B, Y), ._22(A, Z))
if_mergesort_2_in_4_ga7(E, F, U, V, X, Z, merge_3_out_gga3(X, Z, V)) -> mergesort_2_out_ga2(._22(E, ._22(F, U)), V)

The argument filtering Pi contains the following mapping:
mergesort_2_in_ga2(x1, x2)  =  mergesort_2_in_ga1(x1)
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
s_11(x1)  =  s_11(x1)
0_0  =  0_0
mergesort_2_out_ga2(x1, x2)  =  mergesort_2_out_ga1(x2)
if_mergesort_2_in_1_ga5(x1, x2, x3, x4, x5)  =  if_mergesort_2_in_1_ga3(x1, x2, x5)
split_3_in_gaa3(x1, x2, x3)  =  split_3_in_gaa1(x1)
split_3_out_gaa3(x1, x2, x3)  =  split_3_out_gaa2(x2, x3)
if_split_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_split_3_in_1_gaa2(x1, x5)
if_mergesort_2_in_2_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_mergesort_2_in_2_ga3(x2, x6, x7)
if_mergesort_2_in_3_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_mergesort_2_in_3_ga2(x6, x7)
if_mergesort_2_in_4_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_mergesort_2_in_4_ga1(x7)
merge_3_in_gga3(x1, x2, x3)  =  merge_3_in_gga2(x1, x2)
merge_3_out_gga3(x1, x2, x3)  =  merge_3_out_gga1(x3)
if_merge_3_in_1_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_1_gga5(x1, x2, x3, x4, x6)
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
if_merge_3_in_2_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_2_gga2(x1, x6)
if_merge_3_in_3_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_3_gga5(x1, x2, x3, x4, x6)
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_merge_3_in_4_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_4_gga2(x3, x6)
LE_2_IN_GG2(x1, x2)  =  LE_2_IN_GG2(x1, x2)

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

↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

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
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ 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:



↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
PiDP
                ↳ UsableRulesProof
              ↳ PiDP
              ↳ PiDP

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

MERGE_3_IN_GGA3(._22(A, X), ._22(B, Y), ._22(B, Z)) -> IF_MERGE_3_IN_3_GGA6(A, X, B, Y, Z, gt_2_in_gg2(A, B))
MERGE_3_IN_GGA3(._22(A, X), ._22(B, Y), ._22(A, Z)) -> IF_MERGE_3_IN_1_GGA6(A, X, B, Y, Z, le_2_in_gg2(A, B))
IF_MERGE_3_IN_1_GGA6(A, X, B, Y, Z, le_2_out_gg2(A, B)) -> MERGE_3_IN_GGA3(X, ._22(B, Y), Z)
IF_MERGE_3_IN_3_GGA6(A, X, B, Y, Z, gt_2_out_gg2(A, B)) -> MERGE_3_IN_GGA3(._22(A, X), Y, Z)

The TRS R consists of the following rules:

mergesort_2_in_ga2([]_0, []_0) -> mergesort_2_out_ga2([]_0, []_0)
mergesort_2_in_ga2(._22(E, []_0), ._22(E, []_0)) -> mergesort_2_out_ga2(._22(E, []_0), ._22(E, []_0))
mergesort_2_in_ga2(._22(E, ._22(F, U)), V) -> if_mergesort_2_in_1_ga5(E, F, U, V, split_3_in_gaa3(U, L2, L1))
split_3_in_gaa3([]_0, []_0, []_0) -> split_3_out_gaa3([]_0, []_0, []_0)
split_3_in_gaa3(._22(E, U), ._22(E, V), W) -> if_split_3_in_1_gaa5(E, U, V, W, split_3_in_gaa3(U, W, V))
if_split_3_in_1_gaa5(E, U, V, W, split_3_out_gaa3(U, W, V)) -> split_3_out_gaa3(._22(E, U), ._22(E, V), W)
if_mergesort_2_in_1_ga5(E, F, U, V, split_3_out_gaa3(U, L2, L1)) -> if_mergesort_2_in_2_ga7(E, F, U, V, L2, L1, mergesort_2_in_ga2(._22(E, L2), X))
if_mergesort_2_in_2_ga7(E, F, U, V, L2, L1, mergesort_2_out_ga2(._22(E, L2), X)) -> if_mergesort_2_in_3_ga7(E, F, U, V, L1, X, mergesort_2_in_ga2(._22(F, L1), Z))
if_mergesort_2_in_3_ga7(E, F, U, V, L1, X, mergesort_2_out_ga2(._22(F, L1), Z)) -> if_mergesort_2_in_4_ga7(E, F, U, V, X, Z, merge_3_in_gga3(X, Z, V))
merge_3_in_gga3(X, []_0, X) -> merge_3_out_gga3(X, []_0, X)
merge_3_in_gga3([]_0, X, X) -> merge_3_out_gga3([]_0, X, X)
merge_3_in_gga3(._22(A, X), ._22(B, Y), ._22(A, Z)) -> if_merge_3_in_1_gga6(A, X, B, Y, Z, le_2_in_gg2(A, B))
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_merge_3_in_1_gga6(A, X, B, Y, Z, le_2_out_gg2(A, B)) -> if_merge_3_in_2_gga6(A, X, B, Y, Z, merge_3_in_gga3(X, ._22(B, Y), Z))
merge_3_in_gga3(._22(A, X), ._22(B, Y), ._22(B, Z)) -> if_merge_3_in_3_gga6(A, X, B, Y, Z, gt_2_in_gg2(A, B))
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_merge_3_in_3_gga6(A, X, B, Y, Z, gt_2_out_gg2(A, B)) -> if_merge_3_in_4_gga6(A, X, B, Y, Z, merge_3_in_gga3(._22(A, X), Y, Z))
if_merge_3_in_4_gga6(A, X, B, Y, Z, merge_3_out_gga3(._22(A, X), Y, Z)) -> merge_3_out_gga3(._22(A, X), ._22(B, Y), ._22(B, Z))
if_merge_3_in_2_gga6(A, X, B, Y, Z, merge_3_out_gga3(X, ._22(B, Y), Z)) -> merge_3_out_gga3(._22(A, X), ._22(B, Y), ._22(A, Z))
if_mergesort_2_in_4_ga7(E, F, U, V, X, Z, merge_3_out_gga3(X, Z, V)) -> mergesort_2_out_ga2(._22(E, ._22(F, U)), V)

The argument filtering Pi contains the following mapping:
mergesort_2_in_ga2(x1, x2)  =  mergesort_2_in_ga1(x1)
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
s_11(x1)  =  s_11(x1)
0_0  =  0_0
mergesort_2_out_ga2(x1, x2)  =  mergesort_2_out_ga1(x2)
if_mergesort_2_in_1_ga5(x1, x2, x3, x4, x5)  =  if_mergesort_2_in_1_ga3(x1, x2, x5)
split_3_in_gaa3(x1, x2, x3)  =  split_3_in_gaa1(x1)
split_3_out_gaa3(x1, x2, x3)  =  split_3_out_gaa2(x2, x3)
if_split_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_split_3_in_1_gaa2(x1, x5)
if_mergesort_2_in_2_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_mergesort_2_in_2_ga3(x2, x6, x7)
if_mergesort_2_in_3_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_mergesort_2_in_3_ga2(x6, x7)
if_mergesort_2_in_4_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_mergesort_2_in_4_ga1(x7)
merge_3_in_gga3(x1, x2, x3)  =  merge_3_in_gga2(x1, x2)
merge_3_out_gga3(x1, x2, x3)  =  merge_3_out_gga1(x3)
if_merge_3_in_1_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_1_gga5(x1, x2, x3, x4, x6)
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
if_merge_3_in_2_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_2_gga2(x1, x6)
if_merge_3_in_3_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_3_gga5(x1, x2, x3, x4, x6)
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_merge_3_in_4_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_4_gga2(x3, x6)
IF_MERGE_3_IN_1_GGA6(x1, x2, x3, x4, x5, x6)  =  IF_MERGE_3_IN_1_GGA5(x1, x2, x3, x4, x6)
MERGE_3_IN_GGA3(x1, x2, x3)  =  MERGE_3_IN_GGA2(x1, x2)
IF_MERGE_3_IN_3_GGA6(x1, x2, x3, x4, x5, x6)  =  IF_MERGE_3_IN_3_GGA5(x1, x2, x3, x4, x6)

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

↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
              ↳ PiDP

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

MERGE_3_IN_GGA3(._22(A, X), ._22(B, Y), ._22(B, Z)) -> IF_MERGE_3_IN_3_GGA6(A, X, B, Y, Z, gt_2_in_gg2(A, B))
MERGE_3_IN_GGA3(._22(A, X), ._22(B, Y), ._22(A, Z)) -> IF_MERGE_3_IN_1_GGA6(A, X, B, Y, Z, le_2_in_gg2(A, B))
IF_MERGE_3_IN_1_GGA6(A, X, B, Y, Z, le_2_out_gg2(A, B)) -> MERGE_3_IN_GGA3(X, ._22(B, Y), Z)
IF_MERGE_3_IN_3_GGA6(A, X, B, Y, Z, gt_2_out_gg2(A, B)) -> MERGE_3_IN_GGA3(._22(A, X), Y, Z)

The TRS R consists of the following rules:

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)
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_gt_2_in_1_gg3(X, Y, gt_2_out_gg2(X, Y)) -> gt_2_out_gg2(s_11(X), s_11(Y))
if_le_2_in_1_gg3(X, Y, le_2_out_gg2(X, Y)) -> le_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
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
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_MERGE_3_IN_1_GGA6(x1, x2, x3, x4, x5, x6)  =  IF_MERGE_3_IN_1_GGA5(x1, x2, x3, x4, x6)
MERGE_3_IN_GGA3(x1, x2, x3)  =  MERGE_3_IN_GGA2(x1, x2)
IF_MERGE_3_IN_3_GGA6(x1, x2, x3, x4, x5, x6)  =  IF_MERGE_3_IN_3_GGA5(x1, x2, x3, x4, x6)

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
                        ↳ QDPPoloProof
              ↳ PiDP
              ↳ PiDP

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

MERGE_3_IN_GGA2(._22(A, X), ._22(B, Y)) -> IF_MERGE_3_IN_3_GGA5(A, X, B, Y, gt_2_in_gg2(A, B))
MERGE_3_IN_GGA2(._22(A, X), ._22(B, Y)) -> IF_MERGE_3_IN_1_GGA5(A, X, B, Y, le_2_in_gg2(A, B))
IF_MERGE_3_IN_1_GGA5(A, X, B, Y, le_2_out_gg) -> MERGE_3_IN_GGA2(X, ._22(B, Y))
IF_MERGE_3_IN_3_GGA5(A, X, B, Y, gt_2_out_gg) -> MERGE_3_IN_GGA2(._22(A, X), Y)

The TRS R consists of the following rules:

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
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
if_gt_2_in_1_gg1(gt_2_out_gg) -> gt_2_out_gg
if_le_2_in_1_gg1(le_2_out_gg) -> le_2_out_gg

The set Q consists of the following terms:

gt_2_in_gg2(x0, x1)
le_2_in_gg2(x0, x1)
if_gt_2_in_1_gg1(x0)
if_le_2_in_1_gg1(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {IF_MERGE_3_IN_3_GGA5, MERGE_3_IN_GGA2, IF_MERGE_3_IN_1_GGA5}.
By using a polynomial ordering, the following set of Dependency Pairs of this DP problem can be strictly oriented.

MERGE_3_IN_GGA2(._22(A, X), ._22(B, Y)) -> IF_MERGE_3_IN_1_GGA5(A, X, B, Y, le_2_in_gg2(A, B))
The remaining Dependency Pairs were at least non-strictly be oriented.

MERGE_3_IN_GGA2(._22(A, X), ._22(B, Y)) -> IF_MERGE_3_IN_3_GGA5(A, X, B, Y, gt_2_in_gg2(A, B))
IF_MERGE_3_IN_1_GGA5(A, X, B, Y, le_2_out_gg) -> MERGE_3_IN_GGA2(X, ._22(B, Y))
IF_MERGE_3_IN_3_GGA5(A, X, B, Y, gt_2_out_gg) -> MERGE_3_IN_GGA2(._22(A, X), Y)
With the implicit AFS there is no usable rule.

Used ordering: POLO with Polynomial interpretation:


POL(0_0) = 0   
POL(gt_2_in_gg2(x1, x2)) = 0   
POL(if_gt_2_in_1_gg1(x1)) = 0   
POL(MERGE_3_IN_GGA2(x1, x2)) = x1   
POL(._22(x1, x2)) = 1 + x2   
POL(IF_MERGE_3_IN_1_GGA5(x1, x2, x3, x4, x5)) = x2   
POL(le_2_out_gg) = 0   
POL(le_2_in_gg2(x1, x2)) = 0   
POL(s_11(x1)) = 0   
POL(gt_2_out_gg) = 0   
POL(IF_MERGE_3_IN_3_GGA5(x1, x2, x3, x4, x5)) = 1 + x2   
POL(if_le_2_in_1_gg1(x1)) = 0   



↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPPoloProof
QDP
                            ↳ DependencyGraphProof
              ↳ PiDP
              ↳ PiDP

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

MERGE_3_IN_GGA2(._22(A, X), ._22(B, Y)) -> IF_MERGE_3_IN_3_GGA5(A, X, B, Y, gt_2_in_gg2(A, B))
IF_MERGE_3_IN_1_GGA5(A, X, B, Y, le_2_out_gg) -> MERGE_3_IN_GGA2(X, ._22(B, Y))
IF_MERGE_3_IN_3_GGA5(A, X, B, Y, gt_2_out_gg) -> MERGE_3_IN_GGA2(._22(A, X), Y)

The TRS R consists of the following rules:

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
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
if_gt_2_in_1_gg1(gt_2_out_gg) -> gt_2_out_gg
if_le_2_in_1_gg1(le_2_out_gg) -> le_2_out_gg

The set Q consists of the following terms:

gt_2_in_gg2(x0, x1)
le_2_in_gg2(x0, x1)
if_gt_2_in_1_gg1(x0)
if_le_2_in_1_gg1(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {IF_MERGE_3_IN_3_GGA5, MERGE_3_IN_GGA2, IF_MERGE_3_IN_1_GGA5}.
The approximation of the Dependency Graph contains 1 SCC with 1 less node.

↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPPoloProof
                          ↳ QDP
                            ↳ DependencyGraphProof
QDP
                                ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP

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

MERGE_3_IN_GGA2(._22(A, X), ._22(B, Y)) -> IF_MERGE_3_IN_3_GGA5(A, X, B, Y, gt_2_in_gg2(A, B))
IF_MERGE_3_IN_3_GGA5(A, X, B, Y, gt_2_out_gg) -> MERGE_3_IN_GGA2(._22(A, X), Y)

The TRS R consists of the following rules:

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
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
if_gt_2_in_1_gg1(gt_2_out_gg) -> gt_2_out_gg
if_le_2_in_1_gg1(le_2_out_gg) -> le_2_out_gg

The set Q consists of the following terms:

gt_2_in_gg2(x0, x1)
le_2_in_gg2(x0, x1)
if_gt_2_in_1_gg1(x0)
if_le_2_in_1_gg1(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {IF_MERGE_3_IN_3_GGA5, MERGE_3_IN_GGA2}.
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:



↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
PiDP
                ↳ UsableRulesProof
              ↳ PiDP

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

SPLIT_3_IN_GAA3(._22(E, U), ._22(E, V), W) -> SPLIT_3_IN_GAA3(U, W, V)

The TRS R consists of the following rules:

mergesort_2_in_ga2([]_0, []_0) -> mergesort_2_out_ga2([]_0, []_0)
mergesort_2_in_ga2(._22(E, []_0), ._22(E, []_0)) -> mergesort_2_out_ga2(._22(E, []_0), ._22(E, []_0))
mergesort_2_in_ga2(._22(E, ._22(F, U)), V) -> if_mergesort_2_in_1_ga5(E, F, U, V, split_3_in_gaa3(U, L2, L1))
split_3_in_gaa3([]_0, []_0, []_0) -> split_3_out_gaa3([]_0, []_0, []_0)
split_3_in_gaa3(._22(E, U), ._22(E, V), W) -> if_split_3_in_1_gaa5(E, U, V, W, split_3_in_gaa3(U, W, V))
if_split_3_in_1_gaa5(E, U, V, W, split_3_out_gaa3(U, W, V)) -> split_3_out_gaa3(._22(E, U), ._22(E, V), W)
if_mergesort_2_in_1_ga5(E, F, U, V, split_3_out_gaa3(U, L2, L1)) -> if_mergesort_2_in_2_ga7(E, F, U, V, L2, L1, mergesort_2_in_ga2(._22(E, L2), X))
if_mergesort_2_in_2_ga7(E, F, U, V, L2, L1, mergesort_2_out_ga2(._22(E, L2), X)) -> if_mergesort_2_in_3_ga7(E, F, U, V, L1, X, mergesort_2_in_ga2(._22(F, L1), Z))
if_mergesort_2_in_3_ga7(E, F, U, V, L1, X, mergesort_2_out_ga2(._22(F, L1), Z)) -> if_mergesort_2_in_4_ga7(E, F, U, V, X, Z, merge_3_in_gga3(X, Z, V))
merge_3_in_gga3(X, []_0, X) -> merge_3_out_gga3(X, []_0, X)
merge_3_in_gga3([]_0, X, X) -> merge_3_out_gga3([]_0, X, X)
merge_3_in_gga3(._22(A, X), ._22(B, Y), ._22(A, Z)) -> if_merge_3_in_1_gga6(A, X, B, Y, Z, le_2_in_gg2(A, B))
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_merge_3_in_1_gga6(A, X, B, Y, Z, le_2_out_gg2(A, B)) -> if_merge_3_in_2_gga6(A, X, B, Y, Z, merge_3_in_gga3(X, ._22(B, Y), Z))
merge_3_in_gga3(._22(A, X), ._22(B, Y), ._22(B, Z)) -> if_merge_3_in_3_gga6(A, X, B, Y, Z, gt_2_in_gg2(A, B))
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_merge_3_in_3_gga6(A, X, B, Y, Z, gt_2_out_gg2(A, B)) -> if_merge_3_in_4_gga6(A, X, B, Y, Z, merge_3_in_gga3(._22(A, X), Y, Z))
if_merge_3_in_4_gga6(A, X, B, Y, Z, merge_3_out_gga3(._22(A, X), Y, Z)) -> merge_3_out_gga3(._22(A, X), ._22(B, Y), ._22(B, Z))
if_merge_3_in_2_gga6(A, X, B, Y, Z, merge_3_out_gga3(X, ._22(B, Y), Z)) -> merge_3_out_gga3(._22(A, X), ._22(B, Y), ._22(A, Z))
if_mergesort_2_in_4_ga7(E, F, U, V, X, Z, merge_3_out_gga3(X, Z, V)) -> mergesort_2_out_ga2(._22(E, ._22(F, U)), V)

The argument filtering Pi contains the following mapping:
mergesort_2_in_ga2(x1, x2)  =  mergesort_2_in_ga1(x1)
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
s_11(x1)  =  s_11(x1)
0_0  =  0_0
mergesort_2_out_ga2(x1, x2)  =  mergesort_2_out_ga1(x2)
if_mergesort_2_in_1_ga5(x1, x2, x3, x4, x5)  =  if_mergesort_2_in_1_ga3(x1, x2, x5)
split_3_in_gaa3(x1, x2, x3)  =  split_3_in_gaa1(x1)
split_3_out_gaa3(x1, x2, x3)  =  split_3_out_gaa2(x2, x3)
if_split_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_split_3_in_1_gaa2(x1, x5)
if_mergesort_2_in_2_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_mergesort_2_in_2_ga3(x2, x6, x7)
if_mergesort_2_in_3_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_mergesort_2_in_3_ga2(x6, x7)
if_mergesort_2_in_4_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_mergesort_2_in_4_ga1(x7)
merge_3_in_gga3(x1, x2, x3)  =  merge_3_in_gga2(x1, x2)
merge_3_out_gga3(x1, x2, x3)  =  merge_3_out_gga1(x3)
if_merge_3_in_1_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_1_gga5(x1, x2, x3, x4, x6)
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
if_merge_3_in_2_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_2_gga2(x1, x6)
if_merge_3_in_3_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_3_gga5(x1, x2, x3, x4, x6)
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_merge_3_in_4_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_4_gga2(x3, x6)
SPLIT_3_IN_GAA3(x1, x2, x3)  =  SPLIT_3_IN_GAA1(x1)

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

↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP

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

SPLIT_3_IN_GAA3(._22(E, U), ._22(E, V), W) -> SPLIT_3_IN_GAA3(U, W, V)

R is empty.
The argument filtering Pi contains the following mapping:
._22(x1, x2)  =  ._22(x1, x2)
SPLIT_3_IN_GAA3(x1, x2, x3)  =  SPLIT_3_IN_GAA1(x1)

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

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

SPLIT_3_IN_GAA1(._22(E, U)) -> SPLIT_3_IN_GAA1(U)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {SPLIT_3_IN_GAA1}.
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:



↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
PiDP
                ↳ UsableRulesProof

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

IF_MERGESORT_2_IN_1_GA5(E, F, U, V, split_3_out_gaa3(U, L2, L1)) -> MERGESORT_2_IN_GA2(._22(E, L2), X)
MERGESORT_2_IN_GA2(._22(E, ._22(F, U)), V) -> IF_MERGESORT_2_IN_1_GA5(E, F, U, V, split_3_in_gaa3(U, L2, L1))
IF_MERGESORT_2_IN_2_GA7(E, F, U, V, L2, L1, mergesort_2_out_ga2(._22(E, L2), X)) -> MERGESORT_2_IN_GA2(._22(F, L1), Z)
IF_MERGESORT_2_IN_1_GA5(E, F, U, V, split_3_out_gaa3(U, L2, L1)) -> IF_MERGESORT_2_IN_2_GA7(E, F, U, V, L2, L1, mergesort_2_in_ga2(._22(E, L2), X))

The TRS R consists of the following rules:

mergesort_2_in_ga2([]_0, []_0) -> mergesort_2_out_ga2([]_0, []_0)
mergesort_2_in_ga2(._22(E, []_0), ._22(E, []_0)) -> mergesort_2_out_ga2(._22(E, []_0), ._22(E, []_0))
mergesort_2_in_ga2(._22(E, ._22(F, U)), V) -> if_mergesort_2_in_1_ga5(E, F, U, V, split_3_in_gaa3(U, L2, L1))
split_3_in_gaa3([]_0, []_0, []_0) -> split_3_out_gaa3([]_0, []_0, []_0)
split_3_in_gaa3(._22(E, U), ._22(E, V), W) -> if_split_3_in_1_gaa5(E, U, V, W, split_3_in_gaa3(U, W, V))
if_split_3_in_1_gaa5(E, U, V, W, split_3_out_gaa3(U, W, V)) -> split_3_out_gaa3(._22(E, U), ._22(E, V), W)
if_mergesort_2_in_1_ga5(E, F, U, V, split_3_out_gaa3(U, L2, L1)) -> if_mergesort_2_in_2_ga7(E, F, U, V, L2, L1, mergesort_2_in_ga2(._22(E, L2), X))
if_mergesort_2_in_2_ga7(E, F, U, V, L2, L1, mergesort_2_out_ga2(._22(E, L2), X)) -> if_mergesort_2_in_3_ga7(E, F, U, V, L1, X, mergesort_2_in_ga2(._22(F, L1), Z))
if_mergesort_2_in_3_ga7(E, F, U, V, L1, X, mergesort_2_out_ga2(._22(F, L1), Z)) -> if_mergesort_2_in_4_ga7(E, F, U, V, X, Z, merge_3_in_gga3(X, Z, V))
merge_3_in_gga3(X, []_0, X) -> merge_3_out_gga3(X, []_0, X)
merge_3_in_gga3([]_0, X, X) -> merge_3_out_gga3([]_0, X, X)
merge_3_in_gga3(._22(A, X), ._22(B, Y), ._22(A, Z)) -> if_merge_3_in_1_gga6(A, X, B, Y, Z, le_2_in_gg2(A, B))
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_merge_3_in_1_gga6(A, X, B, Y, Z, le_2_out_gg2(A, B)) -> if_merge_3_in_2_gga6(A, X, B, Y, Z, merge_3_in_gga3(X, ._22(B, Y), Z))
merge_3_in_gga3(._22(A, X), ._22(B, Y), ._22(B, Z)) -> if_merge_3_in_3_gga6(A, X, B, Y, Z, gt_2_in_gg2(A, B))
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_merge_3_in_3_gga6(A, X, B, Y, Z, gt_2_out_gg2(A, B)) -> if_merge_3_in_4_gga6(A, X, B, Y, Z, merge_3_in_gga3(._22(A, X), Y, Z))
if_merge_3_in_4_gga6(A, X, B, Y, Z, merge_3_out_gga3(._22(A, X), Y, Z)) -> merge_3_out_gga3(._22(A, X), ._22(B, Y), ._22(B, Z))
if_merge_3_in_2_gga6(A, X, B, Y, Z, merge_3_out_gga3(X, ._22(B, Y), Z)) -> merge_3_out_gga3(._22(A, X), ._22(B, Y), ._22(A, Z))
if_mergesort_2_in_4_ga7(E, F, U, V, X, Z, merge_3_out_gga3(X, Z, V)) -> mergesort_2_out_ga2(._22(E, ._22(F, U)), V)

The argument filtering Pi contains the following mapping:
mergesort_2_in_ga2(x1, x2)  =  mergesort_2_in_ga1(x1)
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
s_11(x1)  =  s_11(x1)
0_0  =  0_0
mergesort_2_out_ga2(x1, x2)  =  mergesort_2_out_ga1(x2)
if_mergesort_2_in_1_ga5(x1, x2, x3, x4, x5)  =  if_mergesort_2_in_1_ga3(x1, x2, x5)
split_3_in_gaa3(x1, x2, x3)  =  split_3_in_gaa1(x1)
split_3_out_gaa3(x1, x2, x3)  =  split_3_out_gaa2(x2, x3)
if_split_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_split_3_in_1_gaa2(x1, x5)
if_mergesort_2_in_2_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_mergesort_2_in_2_ga3(x2, x6, x7)
if_mergesort_2_in_3_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_mergesort_2_in_3_ga2(x6, x7)
if_mergesort_2_in_4_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_mergesort_2_in_4_ga1(x7)
merge_3_in_gga3(x1, x2, x3)  =  merge_3_in_gga2(x1, x2)
merge_3_out_gga3(x1, x2, x3)  =  merge_3_out_gga1(x3)
if_merge_3_in_1_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_1_gga5(x1, x2, x3, x4, x6)
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
if_merge_3_in_2_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_2_gga2(x1, x6)
if_merge_3_in_3_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_3_gga5(x1, x2, x3, x4, x6)
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_merge_3_in_4_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_4_gga2(x3, x6)
IF_MERGESORT_2_IN_2_GA7(x1, x2, x3, x4, x5, x6, x7)  =  IF_MERGESORT_2_IN_2_GA3(x2, x6, x7)
MERGESORT_2_IN_GA2(x1, x2)  =  MERGESORT_2_IN_GA1(x1)
IF_MERGESORT_2_IN_1_GA5(x1, x2, x3, x4, x5)  =  IF_MERGESORT_2_IN_1_GA3(x1, x2, x5)

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

↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
PiDP
                    ↳ PiDPToQDPProof

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

IF_MERGESORT_2_IN_1_GA5(E, F, U, V, split_3_out_gaa3(U, L2, L1)) -> MERGESORT_2_IN_GA2(._22(E, L2), X)
MERGESORT_2_IN_GA2(._22(E, ._22(F, U)), V) -> IF_MERGESORT_2_IN_1_GA5(E, F, U, V, split_3_in_gaa3(U, L2, L1))
IF_MERGESORT_2_IN_2_GA7(E, F, U, V, L2, L1, mergesort_2_out_ga2(._22(E, L2), X)) -> MERGESORT_2_IN_GA2(._22(F, L1), Z)
IF_MERGESORT_2_IN_1_GA5(E, F, U, V, split_3_out_gaa3(U, L2, L1)) -> IF_MERGESORT_2_IN_2_GA7(E, F, U, V, L2, L1, mergesort_2_in_ga2(._22(E, L2), X))

The TRS R consists of the following rules:

split_3_in_gaa3([]_0, []_0, []_0) -> split_3_out_gaa3([]_0, []_0, []_0)
split_3_in_gaa3(._22(E, U), ._22(E, V), W) -> if_split_3_in_1_gaa5(E, U, V, W, split_3_in_gaa3(U, W, V))
mergesort_2_in_ga2(._22(E, []_0), ._22(E, []_0)) -> mergesort_2_out_ga2(._22(E, []_0), ._22(E, []_0))
mergesort_2_in_ga2(._22(E, ._22(F, U)), V) -> if_mergesort_2_in_1_ga5(E, F, U, V, split_3_in_gaa3(U, L2, L1))
if_split_3_in_1_gaa5(E, U, V, W, split_3_out_gaa3(U, W, V)) -> split_3_out_gaa3(._22(E, U), ._22(E, V), W)
if_mergesort_2_in_1_ga5(E, F, U, V, split_3_out_gaa3(U, L2, L1)) -> if_mergesort_2_in_2_ga7(E, F, U, V, L2, L1, mergesort_2_in_ga2(._22(E, L2), X))
if_mergesort_2_in_2_ga7(E, F, U, V, L2, L1, mergesort_2_out_ga2(._22(E, L2), X)) -> if_mergesort_2_in_3_ga7(E, F, U, V, L1, X, mergesort_2_in_ga2(._22(F, L1), Z))
if_mergesort_2_in_3_ga7(E, F, U, V, L1, X, mergesort_2_out_ga2(._22(F, L1), Z)) -> if_mergesort_2_in_4_ga7(E, F, U, V, X, Z, merge_3_in_gga3(X, Z, V))
if_mergesort_2_in_4_ga7(E, F, U, V, X, Z, merge_3_out_gga3(X, Z, V)) -> mergesort_2_out_ga2(._22(E, ._22(F, U)), V)
merge_3_in_gga3(X, []_0, X) -> merge_3_out_gga3(X, []_0, X)
merge_3_in_gga3([]_0, X, X) -> merge_3_out_gga3([]_0, X, X)
merge_3_in_gga3(._22(A, X), ._22(B, Y), ._22(A, Z)) -> if_merge_3_in_1_gga6(A, X, B, Y, Z, le_2_in_gg2(A, B))
merge_3_in_gga3(._22(A, X), ._22(B, Y), ._22(B, Z)) -> if_merge_3_in_3_gga6(A, X, B, Y, Z, gt_2_in_gg2(A, B))
if_merge_3_in_1_gga6(A, X, B, Y, Z, le_2_out_gg2(A, B)) -> if_merge_3_in_2_gga6(A, X, B, Y, Z, merge_3_in_gga3(X, ._22(B, Y), Z))
if_merge_3_in_3_gga6(A, X, B, Y, Z, gt_2_out_gg2(A, B)) -> if_merge_3_in_4_gga6(A, X, B, Y, Z, merge_3_in_gga3(._22(A, X), Y, Z))
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_merge_3_in_2_gga6(A, X, B, Y, Z, merge_3_out_gga3(X, ._22(B, Y), Z)) -> merge_3_out_gga3(._22(A, X), ._22(B, Y), ._22(A, Z))
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_merge_3_in_4_gga6(A, X, B, Y, Z, merge_3_out_gga3(._22(A, X), Y, Z)) -> merge_3_out_gga3(._22(A, X), ._22(B, Y), ._22(B, Z))
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:
mergesort_2_in_ga2(x1, x2)  =  mergesort_2_in_ga1(x1)
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
s_11(x1)  =  s_11(x1)
0_0  =  0_0
mergesort_2_out_ga2(x1, x2)  =  mergesort_2_out_ga1(x2)
if_mergesort_2_in_1_ga5(x1, x2, x3, x4, x5)  =  if_mergesort_2_in_1_ga3(x1, x2, x5)
split_3_in_gaa3(x1, x2, x3)  =  split_3_in_gaa1(x1)
split_3_out_gaa3(x1, x2, x3)  =  split_3_out_gaa2(x2, x3)
if_split_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_split_3_in_1_gaa2(x1, x5)
if_mergesort_2_in_2_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_mergesort_2_in_2_ga3(x2, x6, x7)
if_mergesort_2_in_3_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_mergesort_2_in_3_ga2(x6, x7)
if_mergesort_2_in_4_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_mergesort_2_in_4_ga1(x7)
merge_3_in_gga3(x1, x2, x3)  =  merge_3_in_gga2(x1, x2)
merge_3_out_gga3(x1, x2, x3)  =  merge_3_out_gga1(x3)
if_merge_3_in_1_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_1_gga5(x1, x2, x3, x4, x6)
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
if_merge_3_in_2_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_2_gga2(x1, x6)
if_merge_3_in_3_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_3_gga5(x1, x2, x3, x4, x6)
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_merge_3_in_4_gga6(x1, x2, x3, x4, x5, x6)  =  if_merge_3_in_4_gga2(x3, x6)
IF_MERGESORT_2_IN_2_GA7(x1, x2, x3, x4, x5, x6, x7)  =  IF_MERGESORT_2_IN_2_GA3(x2, x6, x7)
MERGESORT_2_IN_GA2(x1, x2)  =  MERGESORT_2_IN_GA1(x1)
IF_MERGESORT_2_IN_1_GA5(x1, x2, x3, x4, x5)  =  IF_MERGESORT_2_IN_1_GA3(x1, x2, 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
                        ↳ QDPPoloProof

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

IF_MERGESORT_2_IN_1_GA3(E, F, split_3_out_gaa2(L2, L1)) -> MERGESORT_2_IN_GA1(._22(E, L2))
MERGESORT_2_IN_GA1(._22(E, ._22(F, U))) -> IF_MERGESORT_2_IN_1_GA3(E, F, split_3_in_gaa1(U))
IF_MERGESORT_2_IN_2_GA3(F, L1, mergesort_2_out_ga1(X)) -> MERGESORT_2_IN_GA1(._22(F, L1))
IF_MERGESORT_2_IN_1_GA3(E, F, split_3_out_gaa2(L2, L1)) -> IF_MERGESORT_2_IN_2_GA3(F, L1, mergesort_2_in_ga1(._22(E, L2)))

The TRS R consists of the following rules:

split_3_in_gaa1([]_0) -> split_3_out_gaa2([]_0, []_0)
split_3_in_gaa1(._22(E, U)) -> if_split_3_in_1_gaa2(E, split_3_in_gaa1(U))
mergesort_2_in_ga1(._22(E, []_0)) -> mergesort_2_out_ga1(._22(E, []_0))
mergesort_2_in_ga1(._22(E, ._22(F, U))) -> if_mergesort_2_in_1_ga3(E, F, split_3_in_gaa1(U))
if_split_3_in_1_gaa2(E, split_3_out_gaa2(W, V)) -> split_3_out_gaa2(._22(E, V), W)
if_mergesort_2_in_1_ga3(E, F, split_3_out_gaa2(L2, L1)) -> if_mergesort_2_in_2_ga3(F, L1, mergesort_2_in_ga1(._22(E, L2)))
if_mergesort_2_in_2_ga3(F, L1, mergesort_2_out_ga1(X)) -> if_mergesort_2_in_3_ga2(X, mergesort_2_in_ga1(._22(F, L1)))
if_mergesort_2_in_3_ga2(X, mergesort_2_out_ga1(Z)) -> if_mergesort_2_in_4_ga1(merge_3_in_gga2(X, Z))
if_mergesort_2_in_4_ga1(merge_3_out_gga1(V)) -> mergesort_2_out_ga1(V)
merge_3_in_gga2(X, []_0) -> merge_3_out_gga1(X)
merge_3_in_gga2([]_0, X) -> merge_3_out_gga1(X)
merge_3_in_gga2(._22(A, X), ._22(B, Y)) -> if_merge_3_in_1_gga5(A, X, B, Y, le_2_in_gg2(A, B))
merge_3_in_gga2(._22(A, X), ._22(B, Y)) -> if_merge_3_in_3_gga5(A, X, B, Y, gt_2_in_gg2(A, B))
if_merge_3_in_1_gga5(A, X, B, Y, le_2_out_gg) -> if_merge_3_in_2_gga2(A, merge_3_in_gga2(X, ._22(B, Y)))
if_merge_3_in_3_gga5(A, X, B, Y, gt_2_out_gg) -> if_merge_3_in_4_gga2(B, merge_3_in_gga2(._22(A, X), 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
if_merge_3_in_2_gga2(A, merge_3_out_gga1(Z)) -> merge_3_out_gga1(._22(A, Z))
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_merge_3_in_4_gga2(B, merge_3_out_gga1(Z)) -> merge_3_out_gga1(._22(B, Z))
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:

split_3_in_gaa1(x0)
mergesort_2_in_ga1(x0)
if_split_3_in_1_gaa2(x0, x1)
if_mergesort_2_in_1_ga3(x0, x1, x2)
if_mergesort_2_in_2_ga3(x0, x1, x2)
if_mergesort_2_in_3_ga2(x0, x1)
if_mergesort_2_in_4_ga1(x0)
merge_3_in_gga2(x0, x1)
if_merge_3_in_1_gga5(x0, x1, x2, x3, x4)
if_merge_3_in_3_gga5(x0, x1, x2, x3, x4)
le_2_in_gg2(x0, x1)
if_merge_3_in_2_gga2(x0, x1)
gt_2_in_gg2(x0, x1)
if_merge_3_in_4_gga2(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 {MERGESORT_2_IN_GA1, IF_MERGESORT_2_IN_1_GA3, IF_MERGESORT_2_IN_2_GA3}.
By using a polynomial ordering, the following set of Dependency Pairs of this DP problem can be strictly oriented.

MERGESORT_2_IN_GA1(._22(E, ._22(F, U))) -> IF_MERGESORT_2_IN_1_GA3(E, F, split_3_in_gaa1(U))
The remaining Dependency Pairs were at least non-strictly be oriented.

IF_MERGESORT_2_IN_1_GA3(E, F, split_3_out_gaa2(L2, L1)) -> MERGESORT_2_IN_GA1(._22(E, L2))
IF_MERGESORT_2_IN_2_GA3(F, L1, mergesort_2_out_ga1(X)) -> MERGESORT_2_IN_GA1(._22(F, L1))
IF_MERGESORT_2_IN_1_GA3(E, F, split_3_out_gaa2(L2, L1)) -> IF_MERGESORT_2_IN_2_GA3(F, L1, mergesort_2_in_ga1(._22(E, L2)))
With the implicit AFS we had to orient the following set of usable rules non-strictly.

if_split_3_in_1_gaa2(E, split_3_out_gaa2(W, V)) -> split_3_out_gaa2(._22(E, V), W)
split_3_in_gaa1([]_0) -> split_3_out_gaa2([]_0, []_0)
split_3_in_gaa1(._22(E, U)) -> if_split_3_in_1_gaa2(E, split_3_in_gaa1(U))
Used ordering: POLO with Polynomial interpretation:

POL(0_0) = 0   
POL(merge_3_in_gga2(x1, x2)) = 0   
POL(gt_2_in_gg2(x1, x2)) = 0   
POL(split_3_out_gaa2(x1, x2)) = x1 + x2   
POL(mergesort_2_out_ga1(x1)) = 0   
POL(mergesort_2_in_ga1(x1)) = 0   
POL(if_mergesort_2_in_3_ga2(x1, x2)) = 0   
POL(MERGESORT_2_IN_GA1(x1)) = x1   
POL(merge_3_out_gga1(x1)) = 0   
POL(split_3_in_gaa1(x1)) = x1   
POL(IF_MERGESORT_2_IN_2_GA3(x1, x2, x3)) = 1 + x2   
POL(if_mergesort_2_in_1_ga3(x1, x2, x3)) = 0   
POL(if_merge_3_in_2_gga2(x1, x2)) = 0   
POL(if_gt_2_in_1_gg1(x1)) = 0   
POL(IF_MERGESORT_2_IN_1_GA3(x1, x2, x3)) = 1 + x3   
POL(if_merge_3_in_3_gga5(x1, x2, x3, x4, x5)) = 0   
POL(if_merge_3_in_4_gga2(x1, x2)) = 0   
POL(le_2_out_gg) = 0   
POL(if_mergesort_2_in_2_ga3(x1, x2, x3)) = 0   
POL(if_mergesort_2_in_4_ga1(x1)) = 0   
POL([]_0) = 0   
POL(if_le_2_in_1_gg1(x1)) = 0   
POL(._22(x1, x2)) = 1 + x2   
POL(if_merge_3_in_1_gga5(x1, x2, x3, x4, x5)) = 0   
POL(le_2_in_gg2(x1, x2)) = 0   
POL(gt_2_out_gg) = 0   
POL(s_11(x1)) = 0   
POL(if_split_3_in_1_gaa2(x1, x2)) = 1 + x2   



↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPPoloProof
QDP
                            ↳ DependencyGraphProof

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

IF_MERGESORT_2_IN_1_GA3(E, F, split_3_out_gaa2(L2, L1)) -> MERGESORT_2_IN_GA1(._22(E, L2))
IF_MERGESORT_2_IN_2_GA3(F, L1, mergesort_2_out_ga1(X)) -> MERGESORT_2_IN_GA1(._22(F, L1))
IF_MERGESORT_2_IN_1_GA3(E, F, split_3_out_gaa2(L2, L1)) -> IF_MERGESORT_2_IN_2_GA3(F, L1, mergesort_2_in_ga1(._22(E, L2)))

The TRS R consists of the following rules:

split_3_in_gaa1([]_0) -> split_3_out_gaa2([]_0, []_0)
split_3_in_gaa1(._22(E, U)) -> if_split_3_in_1_gaa2(E, split_3_in_gaa1(U))
mergesort_2_in_ga1(._22(E, []_0)) -> mergesort_2_out_ga1(._22(E, []_0))
mergesort_2_in_ga1(._22(E, ._22(F, U))) -> if_mergesort_2_in_1_ga3(E, F, split_3_in_gaa1(U))
if_split_3_in_1_gaa2(E, split_3_out_gaa2(W, V)) -> split_3_out_gaa2(._22(E, V), W)
if_mergesort_2_in_1_ga3(E, F, split_3_out_gaa2(L2, L1)) -> if_mergesort_2_in_2_ga3(F, L1, mergesort_2_in_ga1(._22(E, L2)))
if_mergesort_2_in_2_ga3(F, L1, mergesort_2_out_ga1(X)) -> if_mergesort_2_in_3_ga2(X, mergesort_2_in_ga1(._22(F, L1)))
if_mergesort_2_in_3_ga2(X, mergesort_2_out_ga1(Z)) -> if_mergesort_2_in_4_ga1(merge_3_in_gga2(X, Z))
if_mergesort_2_in_4_ga1(merge_3_out_gga1(V)) -> mergesort_2_out_ga1(V)
merge_3_in_gga2(X, []_0) -> merge_3_out_gga1(X)
merge_3_in_gga2([]_0, X) -> merge_3_out_gga1(X)
merge_3_in_gga2(._22(A, X), ._22(B, Y)) -> if_merge_3_in_1_gga5(A, X, B, Y, le_2_in_gg2(A, B))
merge_3_in_gga2(._22(A, X), ._22(B, Y)) -> if_merge_3_in_3_gga5(A, X, B, Y, gt_2_in_gg2(A, B))
if_merge_3_in_1_gga5(A, X, B, Y, le_2_out_gg) -> if_merge_3_in_2_gga2(A, merge_3_in_gga2(X, ._22(B, Y)))
if_merge_3_in_3_gga5(A, X, B, Y, gt_2_out_gg) -> if_merge_3_in_4_gga2(B, merge_3_in_gga2(._22(A, X), 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
if_merge_3_in_2_gga2(A, merge_3_out_gga1(Z)) -> merge_3_out_gga1(._22(A, Z))
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_merge_3_in_4_gga2(B, merge_3_out_gga1(Z)) -> merge_3_out_gga1(._22(B, Z))
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:

split_3_in_gaa1(x0)
mergesort_2_in_ga1(x0)
if_split_3_in_1_gaa2(x0, x1)
if_mergesort_2_in_1_ga3(x0, x1, x2)
if_mergesort_2_in_2_ga3(x0, x1, x2)
if_mergesort_2_in_3_ga2(x0, x1)
if_mergesort_2_in_4_ga1(x0)
merge_3_in_gga2(x0, x1)
if_merge_3_in_1_gga5(x0, x1, x2, x3, x4)
if_merge_3_in_3_gga5(x0, x1, x2, x3, x4)
le_2_in_gg2(x0, x1)
if_merge_3_in_2_gga2(x0, x1)
gt_2_in_gg2(x0, x1)
if_merge_3_in_4_gga2(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 {MERGESORT_2_IN_GA1, IF_MERGESORT_2_IN_1_GA3, IF_MERGESORT_2_IN_2_GA3}.
The approximation of the Dependency Graph contains 0 SCCs with 3 less nodes.