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



PROLOG
  ↳ PrologToPiTRSProof

qs2(.2(X, Xs), Ys) :- part4(X, Xs, Littles, Bigs), qs2(Littles, Ls), qs2(Bigs, Bs), app3(Ls, .2(X, Bs), Ys).
qs2({}0, {}0).
part4(X, .2(Y, Xs), .2(Y, Ls), Bs) :- gt2(X, Y), part4(X, Xs, Ls, Bs).
part4(X, .2(Y, Xs), Ls, .2(Y, Bs)) :- le2(X, Y), part4(X, Xs, Ls, Bs).
part4(X, {}0, {}0, {}0).
app3(.2(X, Xs), Ys, .2(X, Zs)) :- app3(Xs, Ys, Zs).
app3({}0, Ys, Ys).
gt2(s1(X), s1(Y)) :- gt2(X, Y).
gt2(s1(00), 00).
le2(s1(X), s1(Y)) :- le2(X, Y).
le2(00, s1(00)).
le2(00, 00).


With regard to the inferred argument filtering the predicates were used in the following modes:
qs2: (b,f)
part4: (b,b,f,f)
gt2: (b,b)
le2: (b,b)
app3: (b,b,f)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:


qs_2_in_ga2(._22(X, Xs), Ys) -> if_qs_2_in_1_ga4(X, Xs, Ys, part_4_in_ggaa4(X, Xs, Littles, Bigs))
part_4_in_ggaa4(X, ._22(Y, Xs), ._22(Y, Ls), Bs) -> if_part_4_in_1_ggaa6(X, Y, Xs, Ls, Bs, 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(0_0), 0_0) -> gt_2_out_gg2(s_11(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_part_4_in_1_ggaa6(X, Y, Xs, Ls, Bs, gt_2_out_gg2(X, Y)) -> if_part_4_in_2_ggaa6(X, Y, Xs, Ls, Bs, part_4_in_ggaa4(X, Xs, Ls, Bs))
part_4_in_ggaa4(X, ._22(Y, Xs), Ls, ._22(Y, Bs)) -> if_part_4_in_3_ggaa6(X, Y, Xs, Ls, Bs, 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(0_0)) -> le_2_out_gg2(0_0, s_11(0_0))
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_part_4_in_3_ggaa6(X, Y, Xs, Ls, Bs, le_2_out_gg2(X, Y)) -> if_part_4_in_4_ggaa6(X, Y, Xs, Ls, Bs, part_4_in_ggaa4(X, Xs, Ls, Bs))
part_4_in_ggaa4(X, []_0, []_0, []_0) -> part_4_out_ggaa4(X, []_0, []_0, []_0)
if_part_4_in_4_ggaa6(X, Y, Xs, Ls, Bs, part_4_out_ggaa4(X, Xs, Ls, Bs)) -> part_4_out_ggaa4(X, ._22(Y, Xs), Ls, ._22(Y, Bs))
if_part_4_in_2_ggaa6(X, Y, Xs, Ls, Bs, part_4_out_ggaa4(X, Xs, Ls, Bs)) -> part_4_out_ggaa4(X, ._22(Y, Xs), ._22(Y, Ls), Bs)
if_qs_2_in_1_ga4(X, Xs, Ys, part_4_out_ggaa4(X, Xs, Littles, Bigs)) -> if_qs_2_in_2_ga6(X, Xs, Ys, Littles, Bigs, qs_2_in_ga2(Littles, Ls))
qs_2_in_ga2([]_0, []_0) -> qs_2_out_ga2([]_0, []_0)
if_qs_2_in_2_ga6(X, Xs, Ys, Littles, Bigs, qs_2_out_ga2(Littles, Ls)) -> if_qs_2_in_3_ga6(X, Xs, Ys, Bigs, Ls, qs_2_in_ga2(Bigs, Bs))
if_qs_2_in_3_ga6(X, Xs, Ys, Bigs, Ls, qs_2_out_ga2(Bigs, Bs)) -> if_qs_2_in_4_ga6(X, Xs, Ys, Ls, Bs, app_3_in_gga3(Ls, ._22(X, Bs), Ys))
app_3_in_gga3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_in_gga3(Xs, Ys, Zs))
app_3_in_gga3([]_0, Ys, Ys) -> app_3_out_gga3([]_0, Ys, Ys)
if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_out_gga3(Xs, Ys, Zs)) -> app_3_out_gga3(._22(X, Xs), Ys, ._22(X, Zs))
if_qs_2_in_4_ga6(X, Xs, Ys, Ls, Bs, app_3_out_gga3(Ls, ._22(X, Bs), Ys)) -> qs_2_out_ga2(._22(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_2_in_ga2(x1, x2)  =  qs_2_in_ga1(x1)
._22(x1, x2)  =  ._22(x1, x2)
[]_0  =  []_0
s_11(x1)  =  s_11(x1)
0_0  =  0_0
if_qs_2_in_1_ga4(x1, x2, x3, x4)  =  if_qs_2_in_1_ga2(x1, x4)
part_4_in_ggaa4(x1, x2, x3, x4)  =  part_4_in_ggaa2(x1, x2)
if_part_4_in_1_ggaa6(x1, x2, x3, x4, x5, x6)  =  if_part_4_in_1_ggaa4(x1, x2, x3, 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_part_4_in_2_ggaa6(x1, x2, x3, x4, x5, x6)  =  if_part_4_in_2_ggaa2(x2, x6)
if_part_4_in_3_ggaa6(x1, x2, x3, x4, x5, x6)  =  if_part_4_in_3_ggaa4(x1, x2, x3, 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_part_4_in_4_ggaa6(x1, x2, x3, x4, x5, x6)  =  if_part_4_in_4_ggaa2(x2, x6)
part_4_out_ggaa4(x1, x2, x3, x4)  =  part_4_out_ggaa2(x3, x4)
if_qs_2_in_2_ga6(x1, x2, x3, x4, x5, x6)  =  if_qs_2_in_2_ga3(x1, x5, x6)
qs_2_out_ga2(x1, x2)  =  qs_2_out_ga1(x2)
if_qs_2_in_3_ga6(x1, x2, x3, x4, x5, x6)  =  if_qs_2_in_3_ga3(x1, x5, x6)
if_qs_2_in_4_ga6(x1, x2, x3, x4, x5, x6)  =  if_qs_2_in_4_ga1(x6)
app_3_in_gga3(x1, x2, x3)  =  app_3_in_gga2(x1, x2)
if_app_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gga2(x1, x5)
app_3_out_gga3(x1, x2, x3)  =  app_3_out_gga1(x3)

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:

qs_2_in_ga2(._22(X, Xs), Ys) -> if_qs_2_in_1_ga4(X, Xs, Ys, part_4_in_ggaa4(X, Xs, Littles, Bigs))
part_4_in_ggaa4(X, ._22(Y, Xs), ._22(Y, Ls), Bs) -> if_part_4_in_1_ggaa6(X, Y, Xs, Ls, Bs, 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(0_0), 0_0) -> gt_2_out_gg2(s_11(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_part_4_in_1_ggaa6(X, Y, Xs, Ls, Bs, gt_2_out_gg2(X, Y)) -> if_part_4_in_2_ggaa6(X, Y, Xs, Ls, Bs, part_4_in_ggaa4(X, Xs, Ls, Bs))
part_4_in_ggaa4(X, ._22(Y, Xs), Ls, ._22(Y, Bs)) -> if_part_4_in_3_ggaa6(X, Y, Xs, Ls, Bs, 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(0_0)) -> le_2_out_gg2(0_0, s_11(0_0))
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_part_4_in_3_ggaa6(X, Y, Xs, Ls, Bs, le_2_out_gg2(X, Y)) -> if_part_4_in_4_ggaa6(X, Y, Xs, Ls, Bs, part_4_in_ggaa4(X, Xs, Ls, Bs))
part_4_in_ggaa4(X, []_0, []_0, []_0) -> part_4_out_ggaa4(X, []_0, []_0, []_0)
if_part_4_in_4_ggaa6(X, Y, Xs, Ls, Bs, part_4_out_ggaa4(X, Xs, Ls, Bs)) -> part_4_out_ggaa4(X, ._22(Y, Xs), Ls, ._22(Y, Bs))
if_part_4_in_2_ggaa6(X, Y, Xs, Ls, Bs, part_4_out_ggaa4(X, Xs, Ls, Bs)) -> part_4_out_ggaa4(X, ._22(Y, Xs), ._22(Y, Ls), Bs)
if_qs_2_in_1_ga4(X, Xs, Ys, part_4_out_ggaa4(X, Xs, Littles, Bigs)) -> if_qs_2_in_2_ga6(X, Xs, Ys, Littles, Bigs, qs_2_in_ga2(Littles, Ls))
qs_2_in_ga2([]_0, []_0) -> qs_2_out_ga2([]_0, []_0)
if_qs_2_in_2_ga6(X, Xs, Ys, Littles, Bigs, qs_2_out_ga2(Littles, Ls)) -> if_qs_2_in_3_ga6(X, Xs, Ys, Bigs, Ls, qs_2_in_ga2(Bigs, Bs))
if_qs_2_in_3_ga6(X, Xs, Ys, Bigs, Ls, qs_2_out_ga2(Bigs, Bs)) -> if_qs_2_in_4_ga6(X, Xs, Ys, Ls, Bs, app_3_in_gga3(Ls, ._22(X, Bs), Ys))
app_3_in_gga3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_in_gga3(Xs, Ys, Zs))
app_3_in_gga3([]_0, Ys, Ys) -> app_3_out_gga3([]_0, Ys, Ys)
if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_out_gga3(Xs, Ys, Zs)) -> app_3_out_gga3(._22(X, Xs), Ys, ._22(X, Zs))
if_qs_2_in_4_ga6(X, Xs, Ys, Ls, Bs, app_3_out_gga3(Ls, ._22(X, Bs), Ys)) -> qs_2_out_ga2(._22(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_2_in_ga2(x1, x2)  =  qs_2_in_ga1(x1)
._22(x1, x2)  =  ._22(x1, x2)
[]_0  =  []_0
s_11(x1)  =  s_11(x1)
0_0  =  0_0
if_qs_2_in_1_ga4(x1, x2, x3, x4)  =  if_qs_2_in_1_ga2(x1, x4)
part_4_in_ggaa4(x1, x2, x3, x4)  =  part_4_in_ggaa2(x1, x2)
if_part_4_in_1_ggaa6(x1, x2, x3, x4, x5, x6)  =  if_part_4_in_1_ggaa4(x1, x2, x3, 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_part_4_in_2_ggaa6(x1, x2, x3, x4, x5, x6)  =  if_part_4_in_2_ggaa2(x2, x6)
if_part_4_in_3_ggaa6(x1, x2, x3, x4, x5, x6)  =  if_part_4_in_3_ggaa4(x1, x2, x3, 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_part_4_in_4_ggaa6(x1, x2, x3, x4, x5, x6)  =  if_part_4_in_4_ggaa2(x2, x6)
part_4_out_ggaa4(x1, x2, x3, x4)  =  part_4_out_ggaa2(x3, x4)
if_qs_2_in_2_ga6(x1, x2, x3, x4, x5, x6)  =  if_qs_2_in_2_ga3(x1, x5, x6)
qs_2_out_ga2(x1, x2)  =  qs_2_out_ga1(x2)
if_qs_2_in_3_ga6(x1, x2, x3, x4, x5, x6)  =  if_qs_2_in_3_ga3(x1, x5, x6)
if_qs_2_in_4_ga6(x1, x2, x3, x4, x5, x6)  =  if_qs_2_in_4_ga1(x6)
app_3_in_gga3(x1, x2, x3)  =  app_3_in_gga2(x1, x2)
if_app_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gga2(x1, x5)
app_3_out_gga3(x1, x2, x3)  =  app_3_out_gga1(x3)


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

QS_2_IN_GA2(._22(X, Xs), Ys) -> IF_QS_2_IN_1_GA4(X, Xs, Ys, part_4_in_ggaa4(X, Xs, Littles, Bigs))
QS_2_IN_GA2(._22(X, Xs), Ys) -> PART_4_IN_GGAA4(X, Xs, Littles, Bigs)
PART_4_IN_GGAA4(X, ._22(Y, Xs), ._22(Y, Ls), Bs) -> IF_PART_4_IN_1_GGAA6(X, Y, Xs, Ls, Bs, gt_2_in_gg2(X, Y))
PART_4_IN_GGAA4(X, ._22(Y, Xs), ._22(Y, Ls), Bs) -> 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_PART_4_IN_1_GGAA6(X, Y, Xs, Ls, Bs, gt_2_out_gg2(X, Y)) -> IF_PART_4_IN_2_GGAA6(X, Y, Xs, Ls, Bs, part_4_in_ggaa4(X, Xs, Ls, Bs))
IF_PART_4_IN_1_GGAA6(X, Y, Xs, Ls, Bs, gt_2_out_gg2(X, Y)) -> PART_4_IN_GGAA4(X, Xs, Ls, Bs)
PART_4_IN_GGAA4(X, ._22(Y, Xs), Ls, ._22(Y, Bs)) -> IF_PART_4_IN_3_GGAA6(X, Y, Xs, Ls, Bs, le_2_in_gg2(X, Y))
PART_4_IN_GGAA4(X, ._22(Y, Xs), Ls, ._22(Y, Bs)) -> 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)
IF_PART_4_IN_3_GGAA6(X, Y, Xs, Ls, Bs, le_2_out_gg2(X, Y)) -> IF_PART_4_IN_4_GGAA6(X, Y, Xs, Ls, Bs, part_4_in_ggaa4(X, Xs, Ls, Bs))
IF_PART_4_IN_3_GGAA6(X, Y, Xs, Ls, Bs, le_2_out_gg2(X, Y)) -> PART_4_IN_GGAA4(X, Xs, Ls, Bs)
IF_QS_2_IN_1_GA4(X, Xs, Ys, part_4_out_ggaa4(X, Xs, Littles, Bigs)) -> IF_QS_2_IN_2_GA6(X, Xs, Ys, Littles, Bigs, qs_2_in_ga2(Littles, Ls))
IF_QS_2_IN_1_GA4(X, Xs, Ys, part_4_out_ggaa4(X, Xs, Littles, Bigs)) -> QS_2_IN_GA2(Littles, Ls)
IF_QS_2_IN_2_GA6(X, Xs, Ys, Littles, Bigs, qs_2_out_ga2(Littles, Ls)) -> IF_QS_2_IN_3_GA6(X, Xs, Ys, Bigs, Ls, qs_2_in_ga2(Bigs, Bs))
IF_QS_2_IN_2_GA6(X, Xs, Ys, Littles, Bigs, qs_2_out_ga2(Littles, Ls)) -> QS_2_IN_GA2(Bigs, Bs)
IF_QS_2_IN_3_GA6(X, Xs, Ys, Bigs, Ls, qs_2_out_ga2(Bigs, Bs)) -> IF_QS_2_IN_4_GA6(X, Xs, Ys, Ls, Bs, app_3_in_gga3(Ls, ._22(X, Bs), Ys))
IF_QS_2_IN_3_GA6(X, Xs, Ys, Bigs, Ls, qs_2_out_ga2(Bigs, Bs)) -> APP_3_IN_GGA3(Ls, ._22(X, Bs), Ys)
APP_3_IN_GGA3(._22(X, Xs), Ys, ._22(X, Zs)) -> IF_APP_3_IN_1_GGA5(X, Xs, Ys, Zs, app_3_in_gga3(Xs, Ys, Zs))
APP_3_IN_GGA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APP_3_IN_GGA3(Xs, Ys, Zs)

The TRS R consists of the following rules:

qs_2_in_ga2(._22(X, Xs), Ys) -> if_qs_2_in_1_ga4(X, Xs, Ys, part_4_in_ggaa4(X, Xs, Littles, Bigs))
part_4_in_ggaa4(X, ._22(Y, Xs), ._22(Y, Ls), Bs) -> if_part_4_in_1_ggaa6(X, Y, Xs, Ls, Bs, 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(0_0), 0_0) -> gt_2_out_gg2(s_11(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_part_4_in_1_ggaa6(X, Y, Xs, Ls, Bs, gt_2_out_gg2(X, Y)) -> if_part_4_in_2_ggaa6(X, Y, Xs, Ls, Bs, part_4_in_ggaa4(X, Xs, Ls, Bs))
part_4_in_ggaa4(X, ._22(Y, Xs), Ls, ._22(Y, Bs)) -> if_part_4_in_3_ggaa6(X, Y, Xs, Ls, Bs, 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(0_0)) -> le_2_out_gg2(0_0, s_11(0_0))
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_part_4_in_3_ggaa6(X, Y, Xs, Ls, Bs, le_2_out_gg2(X, Y)) -> if_part_4_in_4_ggaa6(X, Y, Xs, Ls, Bs, part_4_in_ggaa4(X, Xs, Ls, Bs))
part_4_in_ggaa4(X, []_0, []_0, []_0) -> part_4_out_ggaa4(X, []_0, []_0, []_0)
if_part_4_in_4_ggaa6(X, Y, Xs, Ls, Bs, part_4_out_ggaa4(X, Xs, Ls, Bs)) -> part_4_out_ggaa4(X, ._22(Y, Xs), Ls, ._22(Y, Bs))
if_part_4_in_2_ggaa6(X, Y, Xs, Ls, Bs, part_4_out_ggaa4(X, Xs, Ls, Bs)) -> part_4_out_ggaa4(X, ._22(Y, Xs), ._22(Y, Ls), Bs)
if_qs_2_in_1_ga4(X, Xs, Ys, part_4_out_ggaa4(X, Xs, Littles, Bigs)) -> if_qs_2_in_2_ga6(X, Xs, Ys, Littles, Bigs, qs_2_in_ga2(Littles, Ls))
qs_2_in_ga2([]_0, []_0) -> qs_2_out_ga2([]_0, []_0)
if_qs_2_in_2_ga6(X, Xs, Ys, Littles, Bigs, qs_2_out_ga2(Littles, Ls)) -> if_qs_2_in_3_ga6(X, Xs, Ys, Bigs, Ls, qs_2_in_ga2(Bigs, Bs))
if_qs_2_in_3_ga6(X, Xs, Ys, Bigs, Ls, qs_2_out_ga2(Bigs, Bs)) -> if_qs_2_in_4_ga6(X, Xs, Ys, Ls, Bs, app_3_in_gga3(Ls, ._22(X, Bs), Ys))
app_3_in_gga3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_in_gga3(Xs, Ys, Zs))
app_3_in_gga3([]_0, Ys, Ys) -> app_3_out_gga3([]_0, Ys, Ys)
if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_out_gga3(Xs, Ys, Zs)) -> app_3_out_gga3(._22(X, Xs), Ys, ._22(X, Zs))
if_qs_2_in_4_ga6(X, Xs, Ys, Ls, Bs, app_3_out_gga3(Ls, ._22(X, Bs), Ys)) -> qs_2_out_ga2(._22(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_2_in_ga2(x1, x2)  =  qs_2_in_ga1(x1)
._22(x1, x2)  =  ._22(x1, x2)
[]_0  =  []_0
s_11(x1)  =  s_11(x1)
0_0  =  0_0
if_qs_2_in_1_ga4(x1, x2, x3, x4)  =  if_qs_2_in_1_ga2(x1, x4)
part_4_in_ggaa4(x1, x2, x3, x4)  =  part_4_in_ggaa2(x1, x2)
if_part_4_in_1_ggaa6(x1, x2, x3, x4, x5, x6)  =  if_part_4_in_1_ggaa4(x1, x2, x3, 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_part_4_in_2_ggaa6(x1, x2, x3, x4, x5, x6)  =  if_part_4_in_2_ggaa2(x2, x6)
if_part_4_in_3_ggaa6(x1, x2, x3, x4, x5, x6)  =  if_part_4_in_3_ggaa4(x1, x2, x3, 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_part_4_in_4_ggaa6(x1, x2, x3, x4, x5, x6)  =  if_part_4_in_4_ggaa2(x2, x6)
part_4_out_ggaa4(x1, x2, x3, x4)  =  part_4_out_ggaa2(x3, x4)
if_qs_2_in_2_ga6(x1, x2, x3, x4, x5, x6)  =  if_qs_2_in_2_ga3(x1, x5, x6)
qs_2_out_ga2(x1, x2)  =  qs_2_out_ga1(x2)
if_qs_2_in_3_ga6(x1, x2, x3, x4, x5, x6)  =  if_qs_2_in_3_ga3(x1, x5, x6)
if_qs_2_in_4_ga6(x1, x2, x3, x4, x5, x6)  =  if_qs_2_in_4_ga1(x6)
app_3_in_gga3(x1, x2, x3)  =  app_3_in_gga2(x1, x2)
if_app_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gga2(x1, x5)
app_3_out_gga3(x1, x2, x3)  =  app_3_out_gga1(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)
PART_4_IN_GGAA4(x1, x2, x3, x4)  =  PART_4_IN_GGAA2(x1, x2)
IF_PART_4_IN_3_GGAA6(x1, x2, x3, x4, x5, x6)  =  IF_PART_4_IN_3_GGAA4(x1, x2, x3, x6)
APP_3_IN_GGA3(x1, x2, x3)  =  APP_3_IN_GGA2(x1, x2)
IF_QS_2_IN_1_GA4(x1, x2, x3, x4)  =  IF_QS_2_IN_1_GA2(x1, x4)
IF_QS_2_IN_3_GA6(x1, x2, x3, x4, x5, x6)  =  IF_QS_2_IN_3_GA3(x1, x5, x6)
IF_LE_2_IN_1_GG3(x1, x2, x3)  =  IF_LE_2_IN_1_GG1(x3)
IF_PART_4_IN_1_GGAA6(x1, x2, x3, x4, x5, x6)  =  IF_PART_4_IN_1_GGAA4(x1, x2, x3, x6)
IF_PART_4_IN_4_GGAA6(x1, x2, x3, x4, x5, x6)  =  IF_PART_4_IN_4_GGAA2(x2, x6)
IF_QS_2_IN_2_GA6(x1, x2, x3, x4, x5, x6)  =  IF_QS_2_IN_2_GA3(x1, x5, x6)
IF_APP_3_IN_1_GGA5(x1, x2, x3, x4, x5)  =  IF_APP_3_IN_1_GGA2(x1, x5)
QS_2_IN_GA2(x1, x2)  =  QS_2_IN_GA1(x1)
IF_PART_4_IN_2_GGAA6(x1, x2, x3, x4, x5, x6)  =  IF_PART_4_IN_2_GGAA2(x2, x6)
IF_QS_2_IN_4_GA6(x1, x2, x3, x4, x5, x6)  =  IF_QS_2_IN_4_GA1(x6)
LE_2_IN_GG2(x1, x2)  =  LE_2_IN_GG2(x1, x2)

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

↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
PiDP
          ↳ DependencyGraphProof

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

QS_2_IN_GA2(._22(X, Xs), Ys) -> IF_QS_2_IN_1_GA4(X, Xs, Ys, part_4_in_ggaa4(X, Xs, Littles, Bigs))
QS_2_IN_GA2(._22(X, Xs), Ys) -> PART_4_IN_GGAA4(X, Xs, Littles, Bigs)
PART_4_IN_GGAA4(X, ._22(Y, Xs), ._22(Y, Ls), Bs) -> IF_PART_4_IN_1_GGAA6(X, Y, Xs, Ls, Bs, gt_2_in_gg2(X, Y))
PART_4_IN_GGAA4(X, ._22(Y, Xs), ._22(Y, Ls), Bs) -> 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_PART_4_IN_1_GGAA6(X, Y, Xs, Ls, Bs, gt_2_out_gg2(X, Y)) -> IF_PART_4_IN_2_GGAA6(X, Y, Xs, Ls, Bs, part_4_in_ggaa4(X, Xs, Ls, Bs))
IF_PART_4_IN_1_GGAA6(X, Y, Xs, Ls, Bs, gt_2_out_gg2(X, Y)) -> PART_4_IN_GGAA4(X, Xs, Ls, Bs)
PART_4_IN_GGAA4(X, ._22(Y, Xs), Ls, ._22(Y, Bs)) -> IF_PART_4_IN_3_GGAA6(X, Y, Xs, Ls, Bs, le_2_in_gg2(X, Y))
PART_4_IN_GGAA4(X, ._22(Y, Xs), Ls, ._22(Y, Bs)) -> 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)
IF_PART_4_IN_3_GGAA6(X, Y, Xs, Ls, Bs, le_2_out_gg2(X, Y)) -> IF_PART_4_IN_4_GGAA6(X, Y, Xs, Ls, Bs, part_4_in_ggaa4(X, Xs, Ls, Bs))
IF_PART_4_IN_3_GGAA6(X, Y, Xs, Ls, Bs, le_2_out_gg2(X, Y)) -> PART_4_IN_GGAA4(X, Xs, Ls, Bs)
IF_QS_2_IN_1_GA4(X, Xs, Ys, part_4_out_ggaa4(X, Xs, Littles, Bigs)) -> IF_QS_2_IN_2_GA6(X, Xs, Ys, Littles, Bigs, qs_2_in_ga2(Littles, Ls))
IF_QS_2_IN_1_GA4(X, Xs, Ys, part_4_out_ggaa4(X, Xs, Littles, Bigs)) -> QS_2_IN_GA2(Littles, Ls)
IF_QS_2_IN_2_GA6(X, Xs, Ys, Littles, Bigs, qs_2_out_ga2(Littles, Ls)) -> IF_QS_2_IN_3_GA6(X, Xs, Ys, Bigs, Ls, qs_2_in_ga2(Bigs, Bs))
IF_QS_2_IN_2_GA6(X, Xs, Ys, Littles, Bigs, qs_2_out_ga2(Littles, Ls)) -> QS_2_IN_GA2(Bigs, Bs)
IF_QS_2_IN_3_GA6(X, Xs, Ys, Bigs, Ls, qs_2_out_ga2(Bigs, Bs)) -> IF_QS_2_IN_4_GA6(X, Xs, Ys, Ls, Bs, app_3_in_gga3(Ls, ._22(X, Bs), Ys))
IF_QS_2_IN_3_GA6(X, Xs, Ys, Bigs, Ls, qs_2_out_ga2(Bigs, Bs)) -> APP_3_IN_GGA3(Ls, ._22(X, Bs), Ys)
APP_3_IN_GGA3(._22(X, Xs), Ys, ._22(X, Zs)) -> IF_APP_3_IN_1_GGA5(X, Xs, Ys, Zs, app_3_in_gga3(Xs, Ys, Zs))
APP_3_IN_GGA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APP_3_IN_GGA3(Xs, Ys, Zs)

The TRS R consists of the following rules:

qs_2_in_ga2(._22(X, Xs), Ys) -> if_qs_2_in_1_ga4(X, Xs, Ys, part_4_in_ggaa4(X, Xs, Littles, Bigs))
part_4_in_ggaa4(X, ._22(Y, Xs), ._22(Y, Ls), Bs) -> if_part_4_in_1_ggaa6(X, Y, Xs, Ls, Bs, 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(0_0), 0_0) -> gt_2_out_gg2(s_11(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_part_4_in_1_ggaa6(X, Y, Xs, Ls, Bs, gt_2_out_gg2(X, Y)) -> if_part_4_in_2_ggaa6(X, Y, Xs, Ls, Bs, part_4_in_ggaa4(X, Xs, Ls, Bs))
part_4_in_ggaa4(X, ._22(Y, Xs), Ls, ._22(Y, Bs)) -> if_part_4_in_3_ggaa6(X, Y, Xs, Ls, Bs, 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(0_0)) -> le_2_out_gg2(0_0, s_11(0_0))
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_part_4_in_3_ggaa6(X, Y, Xs, Ls, Bs, le_2_out_gg2(X, Y)) -> if_part_4_in_4_ggaa6(X, Y, Xs, Ls, Bs, part_4_in_ggaa4(X, Xs, Ls, Bs))
part_4_in_ggaa4(X, []_0, []_0, []_0) -> part_4_out_ggaa4(X, []_0, []_0, []_0)
if_part_4_in_4_ggaa6(X, Y, Xs, Ls, Bs, part_4_out_ggaa4(X, Xs, Ls, Bs)) -> part_4_out_ggaa4(X, ._22(Y, Xs), Ls, ._22(Y, Bs))
if_part_4_in_2_ggaa6(X, Y, Xs, Ls, Bs, part_4_out_ggaa4(X, Xs, Ls, Bs)) -> part_4_out_ggaa4(X, ._22(Y, Xs), ._22(Y, Ls), Bs)
if_qs_2_in_1_ga4(X, Xs, Ys, part_4_out_ggaa4(X, Xs, Littles, Bigs)) -> if_qs_2_in_2_ga6(X, Xs, Ys, Littles, Bigs, qs_2_in_ga2(Littles, Ls))
qs_2_in_ga2([]_0, []_0) -> qs_2_out_ga2([]_0, []_0)
if_qs_2_in_2_ga6(X, Xs, Ys, Littles, Bigs, qs_2_out_ga2(Littles, Ls)) -> if_qs_2_in_3_ga6(X, Xs, Ys, Bigs, Ls, qs_2_in_ga2(Bigs, Bs))
if_qs_2_in_3_ga6(X, Xs, Ys, Bigs, Ls, qs_2_out_ga2(Bigs, Bs)) -> if_qs_2_in_4_ga6(X, Xs, Ys, Ls, Bs, app_3_in_gga3(Ls, ._22(X, Bs), Ys))
app_3_in_gga3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_in_gga3(Xs, Ys, Zs))
app_3_in_gga3([]_0, Ys, Ys) -> app_3_out_gga3([]_0, Ys, Ys)
if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_out_gga3(Xs, Ys, Zs)) -> app_3_out_gga3(._22(X, Xs), Ys, ._22(X, Zs))
if_qs_2_in_4_ga6(X, Xs, Ys, Ls, Bs, app_3_out_gga3(Ls, ._22(X, Bs), Ys)) -> qs_2_out_ga2(._22(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_2_in_ga2(x1, x2)  =  qs_2_in_ga1(x1)
._22(x1, x2)  =  ._22(x1, x2)
[]_0  =  []_0
s_11(x1)  =  s_11(x1)
0_0  =  0_0
if_qs_2_in_1_ga4(x1, x2, x3, x4)  =  if_qs_2_in_1_ga2(x1, x4)
part_4_in_ggaa4(x1, x2, x3, x4)  =  part_4_in_ggaa2(x1, x2)
if_part_4_in_1_ggaa6(x1, x2, x3, x4, x5, x6)  =  if_part_4_in_1_ggaa4(x1, x2, x3, 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_part_4_in_2_ggaa6(x1, x2, x3, x4, x5, x6)  =  if_part_4_in_2_ggaa2(x2, x6)
if_part_4_in_3_ggaa6(x1, x2, x3, x4, x5, x6)  =  if_part_4_in_3_ggaa4(x1, x2, x3, 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_part_4_in_4_ggaa6(x1, x2, x3, x4, x5, x6)  =  if_part_4_in_4_ggaa2(x2, x6)
part_4_out_ggaa4(x1, x2, x3, x4)  =  part_4_out_ggaa2(x3, x4)
if_qs_2_in_2_ga6(x1, x2, x3, x4, x5, x6)  =  if_qs_2_in_2_ga3(x1, x5, x6)
qs_2_out_ga2(x1, x2)  =  qs_2_out_ga1(x2)
if_qs_2_in_3_ga6(x1, x2, x3, x4, x5, x6)  =  if_qs_2_in_3_ga3(x1, x5, x6)
if_qs_2_in_4_ga6(x1, x2, x3, x4, x5, x6)  =  if_qs_2_in_4_ga1(x6)
app_3_in_gga3(x1, x2, x3)  =  app_3_in_gga2(x1, x2)
if_app_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gga2(x1, x5)
app_3_out_gga3(x1, x2, x3)  =  app_3_out_gga1(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)
PART_4_IN_GGAA4(x1, x2, x3, x4)  =  PART_4_IN_GGAA2(x1, x2)
IF_PART_4_IN_3_GGAA6(x1, x2, x3, x4, x5, x6)  =  IF_PART_4_IN_3_GGAA4(x1, x2, x3, x6)
APP_3_IN_GGA3(x1, x2, x3)  =  APP_3_IN_GGA2(x1, x2)
IF_QS_2_IN_1_GA4(x1, x2, x3, x4)  =  IF_QS_2_IN_1_GA2(x1, x4)
IF_QS_2_IN_3_GA6(x1, x2, x3, x4, x5, x6)  =  IF_QS_2_IN_3_GA3(x1, x5, x6)
IF_LE_2_IN_1_GG3(x1, x2, x3)  =  IF_LE_2_IN_1_GG1(x3)
IF_PART_4_IN_1_GGAA6(x1, x2, x3, x4, x5, x6)  =  IF_PART_4_IN_1_GGAA4(x1, x2, x3, x6)
IF_PART_4_IN_4_GGAA6(x1, x2, x3, x4, x5, x6)  =  IF_PART_4_IN_4_GGAA2(x2, x6)
IF_QS_2_IN_2_GA6(x1, x2, x3, x4, x5, x6)  =  IF_QS_2_IN_2_GA3(x1, x5, x6)
IF_APP_3_IN_1_GGA5(x1, x2, x3, x4, x5)  =  IF_APP_3_IN_1_GGA2(x1, x5)
QS_2_IN_GA2(x1, x2)  =  QS_2_IN_GA1(x1)
IF_PART_4_IN_2_GGAA6(x1, x2, x3, x4, x5, x6)  =  IF_PART_4_IN_2_GGAA2(x2, x6)
IF_QS_2_IN_4_GA6(x1, x2, x3, x4, x5, x6)  =  IF_QS_2_IN_4_GA1(x6)
LE_2_IN_GG2(x1, x2)  =  LE_2_IN_GG2(x1, x2)

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:

APP_3_IN_GGA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APP_3_IN_GGA3(Xs, Ys, Zs)

The TRS R consists of the following rules:

qs_2_in_ga2(._22(X, Xs), Ys) -> if_qs_2_in_1_ga4(X, Xs, Ys, part_4_in_ggaa4(X, Xs, Littles, Bigs))
part_4_in_ggaa4(X, ._22(Y, Xs), ._22(Y, Ls), Bs) -> if_part_4_in_1_ggaa6(X, Y, Xs, Ls, Bs, 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(0_0), 0_0) -> gt_2_out_gg2(s_11(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_part_4_in_1_ggaa6(X, Y, Xs, Ls, Bs, gt_2_out_gg2(X, Y)) -> if_part_4_in_2_ggaa6(X, Y, Xs, Ls, Bs, part_4_in_ggaa4(X, Xs, Ls, Bs))
part_4_in_ggaa4(X, ._22(Y, Xs), Ls, ._22(Y, Bs)) -> if_part_4_in_3_ggaa6(X, Y, Xs, Ls, Bs, 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(0_0)) -> le_2_out_gg2(0_0, s_11(0_0))
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_part_4_in_3_ggaa6(X, Y, Xs, Ls, Bs, le_2_out_gg2(X, Y)) -> if_part_4_in_4_ggaa6(X, Y, Xs, Ls, Bs, part_4_in_ggaa4(X, Xs, Ls, Bs))
part_4_in_ggaa4(X, []_0, []_0, []_0) -> part_4_out_ggaa4(X, []_0, []_0, []_0)
if_part_4_in_4_ggaa6(X, Y, Xs, Ls, Bs, part_4_out_ggaa4(X, Xs, Ls, Bs)) -> part_4_out_ggaa4(X, ._22(Y, Xs), Ls, ._22(Y, Bs))
if_part_4_in_2_ggaa6(X, Y, Xs, Ls, Bs, part_4_out_ggaa4(X, Xs, Ls, Bs)) -> part_4_out_ggaa4(X, ._22(Y, Xs), ._22(Y, Ls), Bs)
if_qs_2_in_1_ga4(X, Xs, Ys, part_4_out_ggaa4(X, Xs, Littles, Bigs)) -> if_qs_2_in_2_ga6(X, Xs, Ys, Littles, Bigs, qs_2_in_ga2(Littles, Ls))
qs_2_in_ga2([]_0, []_0) -> qs_2_out_ga2([]_0, []_0)
if_qs_2_in_2_ga6(X, Xs, Ys, Littles, Bigs, qs_2_out_ga2(Littles, Ls)) -> if_qs_2_in_3_ga6(X, Xs, Ys, Bigs, Ls, qs_2_in_ga2(Bigs, Bs))
if_qs_2_in_3_ga6(X, Xs, Ys, Bigs, Ls, qs_2_out_ga2(Bigs, Bs)) -> if_qs_2_in_4_ga6(X, Xs, Ys, Ls, Bs, app_3_in_gga3(Ls, ._22(X, Bs), Ys))
app_3_in_gga3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_in_gga3(Xs, Ys, Zs))
app_3_in_gga3([]_0, Ys, Ys) -> app_3_out_gga3([]_0, Ys, Ys)
if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_out_gga3(Xs, Ys, Zs)) -> app_3_out_gga3(._22(X, Xs), Ys, ._22(X, Zs))
if_qs_2_in_4_ga6(X, Xs, Ys, Ls, Bs, app_3_out_gga3(Ls, ._22(X, Bs), Ys)) -> qs_2_out_ga2(._22(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_2_in_ga2(x1, x2)  =  qs_2_in_ga1(x1)
._22(x1, x2)  =  ._22(x1, x2)
[]_0  =  []_0
s_11(x1)  =  s_11(x1)
0_0  =  0_0
if_qs_2_in_1_ga4(x1, x2, x3, x4)  =  if_qs_2_in_1_ga2(x1, x4)
part_4_in_ggaa4(x1, x2, x3, x4)  =  part_4_in_ggaa2(x1, x2)
if_part_4_in_1_ggaa6(x1, x2, x3, x4, x5, x6)  =  if_part_4_in_1_ggaa4(x1, x2, x3, 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_part_4_in_2_ggaa6(x1, x2, x3, x4, x5, x6)  =  if_part_4_in_2_ggaa2(x2, x6)
if_part_4_in_3_ggaa6(x1, x2, x3, x4, x5, x6)  =  if_part_4_in_3_ggaa4(x1, x2, x3, 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_part_4_in_4_ggaa6(x1, x2, x3, x4, x5, x6)  =  if_part_4_in_4_ggaa2(x2, x6)
part_4_out_ggaa4(x1, x2, x3, x4)  =  part_4_out_ggaa2(x3, x4)
if_qs_2_in_2_ga6(x1, x2, x3, x4, x5, x6)  =  if_qs_2_in_2_ga3(x1, x5, x6)
qs_2_out_ga2(x1, x2)  =  qs_2_out_ga1(x2)
if_qs_2_in_3_ga6(x1, x2, x3, x4, x5, x6)  =  if_qs_2_in_3_ga3(x1, x5, x6)
if_qs_2_in_4_ga6(x1, x2, x3, x4, x5, x6)  =  if_qs_2_in_4_ga1(x6)
app_3_in_gga3(x1, x2, x3)  =  app_3_in_gga2(x1, x2)
if_app_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gga2(x1, x5)
app_3_out_gga3(x1, x2, x3)  =  app_3_out_gga1(x3)
APP_3_IN_GGA3(x1, x2, x3)  =  APP_3_IN_GGA2(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:

APP_3_IN_GGA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APP_3_IN_GGA3(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
._22(x1, x2)  =  ._22(x1, x2)
APP_3_IN_GGA3(x1, x2, x3)  =  APP_3_IN_GGA2(x1, x2)

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:

APP_3_IN_GGA2(._22(X, Xs), Ys) -> APP_3_IN_GGA2(Xs, Ys)

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

qs_2_in_ga2(._22(X, Xs), Ys) -> if_qs_2_in_1_ga4(X, Xs, Ys, part_4_in_ggaa4(X, Xs, Littles, Bigs))
part_4_in_ggaa4(X, ._22(Y, Xs), ._22(Y, Ls), Bs) -> if_part_4_in_1_ggaa6(X, Y, Xs, Ls, Bs, 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(0_0), 0_0) -> gt_2_out_gg2(s_11(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_part_4_in_1_ggaa6(X, Y, Xs, Ls, Bs, gt_2_out_gg2(X, Y)) -> if_part_4_in_2_ggaa6(X, Y, Xs, Ls, Bs, part_4_in_ggaa4(X, Xs, Ls, Bs))
part_4_in_ggaa4(X, ._22(Y, Xs), Ls, ._22(Y, Bs)) -> if_part_4_in_3_ggaa6(X, Y, Xs, Ls, Bs, 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(0_0)) -> le_2_out_gg2(0_0, s_11(0_0))
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_part_4_in_3_ggaa6(X, Y, Xs, Ls, Bs, le_2_out_gg2(X, Y)) -> if_part_4_in_4_ggaa6(X, Y, Xs, Ls, Bs, part_4_in_ggaa4(X, Xs, Ls, Bs))
part_4_in_ggaa4(X, []_0, []_0, []_0) -> part_4_out_ggaa4(X, []_0, []_0, []_0)
if_part_4_in_4_ggaa6(X, Y, Xs, Ls, Bs, part_4_out_ggaa4(X, Xs, Ls, Bs)) -> part_4_out_ggaa4(X, ._22(Y, Xs), Ls, ._22(Y, Bs))
if_part_4_in_2_ggaa6(X, Y, Xs, Ls, Bs, part_4_out_ggaa4(X, Xs, Ls, Bs)) -> part_4_out_ggaa4(X, ._22(Y, Xs), ._22(Y, Ls), Bs)
if_qs_2_in_1_ga4(X, Xs, Ys, part_4_out_ggaa4(X, Xs, Littles, Bigs)) -> if_qs_2_in_2_ga6(X, Xs, Ys, Littles, Bigs, qs_2_in_ga2(Littles, Ls))
qs_2_in_ga2([]_0, []_0) -> qs_2_out_ga2([]_0, []_0)
if_qs_2_in_2_ga6(X, Xs, Ys, Littles, Bigs, qs_2_out_ga2(Littles, Ls)) -> if_qs_2_in_3_ga6(X, Xs, Ys, Bigs, Ls, qs_2_in_ga2(Bigs, Bs))
if_qs_2_in_3_ga6(X, Xs, Ys, Bigs, Ls, qs_2_out_ga2(Bigs, Bs)) -> if_qs_2_in_4_ga6(X, Xs, Ys, Ls, Bs, app_3_in_gga3(Ls, ._22(X, Bs), Ys))
app_3_in_gga3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_in_gga3(Xs, Ys, Zs))
app_3_in_gga3([]_0, Ys, Ys) -> app_3_out_gga3([]_0, Ys, Ys)
if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_out_gga3(Xs, Ys, Zs)) -> app_3_out_gga3(._22(X, Xs), Ys, ._22(X, Zs))
if_qs_2_in_4_ga6(X, Xs, Ys, Ls, Bs, app_3_out_gga3(Ls, ._22(X, Bs), Ys)) -> qs_2_out_ga2(._22(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_2_in_ga2(x1, x2)  =  qs_2_in_ga1(x1)
._22(x1, x2)  =  ._22(x1, x2)
[]_0  =  []_0
s_11(x1)  =  s_11(x1)
0_0  =  0_0
if_qs_2_in_1_ga4(x1, x2, x3, x4)  =  if_qs_2_in_1_ga2(x1, x4)
part_4_in_ggaa4(x1, x2, x3, x4)  =  part_4_in_ggaa2(x1, x2)
if_part_4_in_1_ggaa6(x1, x2, x3, x4, x5, x6)  =  if_part_4_in_1_ggaa4(x1, x2, x3, 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_part_4_in_2_ggaa6(x1, x2, x3, x4, x5, x6)  =  if_part_4_in_2_ggaa2(x2, x6)
if_part_4_in_3_ggaa6(x1, x2, x3, x4, x5, x6)  =  if_part_4_in_3_ggaa4(x1, x2, x3, 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_part_4_in_4_ggaa6(x1, x2, x3, x4, x5, x6)  =  if_part_4_in_4_ggaa2(x2, x6)
part_4_out_ggaa4(x1, x2, x3, x4)  =  part_4_out_ggaa2(x3, x4)
if_qs_2_in_2_ga6(x1, x2, x3, x4, x5, x6)  =  if_qs_2_in_2_ga3(x1, x5, x6)
qs_2_out_ga2(x1, x2)  =  qs_2_out_ga1(x2)
if_qs_2_in_3_ga6(x1, x2, x3, x4, x5, x6)  =  if_qs_2_in_3_ga3(x1, x5, x6)
if_qs_2_in_4_ga6(x1, x2, x3, x4, x5, x6)  =  if_qs_2_in_4_ga1(x6)
app_3_in_gga3(x1, x2, x3)  =  app_3_in_gga2(x1, x2)
if_app_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gga2(x1, x5)
app_3_out_gga3(x1, x2, x3)  =  app_3_out_gga1(x3)
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:

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

The TRS R consists of the following rules:

qs_2_in_ga2(._22(X, Xs), Ys) -> if_qs_2_in_1_ga4(X, Xs, Ys, part_4_in_ggaa4(X, Xs, Littles, Bigs))
part_4_in_ggaa4(X, ._22(Y, Xs), ._22(Y, Ls), Bs) -> if_part_4_in_1_ggaa6(X, Y, Xs, Ls, Bs, 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(0_0), 0_0) -> gt_2_out_gg2(s_11(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_part_4_in_1_ggaa6(X, Y, Xs, Ls, Bs, gt_2_out_gg2(X, Y)) -> if_part_4_in_2_ggaa6(X, Y, Xs, Ls, Bs, part_4_in_ggaa4(X, Xs, Ls, Bs))
part_4_in_ggaa4(X, ._22(Y, Xs), Ls, ._22(Y, Bs)) -> if_part_4_in_3_ggaa6(X, Y, Xs, Ls, Bs, 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(0_0)) -> le_2_out_gg2(0_0, s_11(0_0))
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_part_4_in_3_ggaa6(X, Y, Xs, Ls, Bs, le_2_out_gg2(X, Y)) -> if_part_4_in_4_ggaa6(X, Y, Xs, Ls, Bs, part_4_in_ggaa4(X, Xs, Ls, Bs))
part_4_in_ggaa4(X, []_0, []_0, []_0) -> part_4_out_ggaa4(X, []_0, []_0, []_0)
if_part_4_in_4_ggaa6(X, Y, Xs, Ls, Bs, part_4_out_ggaa4(X, Xs, Ls, Bs)) -> part_4_out_ggaa4(X, ._22(Y, Xs), Ls, ._22(Y, Bs))
if_part_4_in_2_ggaa6(X, Y, Xs, Ls, Bs, part_4_out_ggaa4(X, Xs, Ls, Bs)) -> part_4_out_ggaa4(X, ._22(Y, Xs), ._22(Y, Ls), Bs)
if_qs_2_in_1_ga4(X, Xs, Ys, part_4_out_ggaa4(X, Xs, Littles, Bigs)) -> if_qs_2_in_2_ga6(X, Xs, Ys, Littles, Bigs, qs_2_in_ga2(Littles, Ls))
qs_2_in_ga2([]_0, []_0) -> qs_2_out_ga2([]_0, []_0)
if_qs_2_in_2_ga6(X, Xs, Ys, Littles, Bigs, qs_2_out_ga2(Littles, Ls)) -> if_qs_2_in_3_ga6(X, Xs, Ys, Bigs, Ls, qs_2_in_ga2(Bigs, Bs))
if_qs_2_in_3_ga6(X, Xs, Ys, Bigs, Ls, qs_2_out_ga2(Bigs, Bs)) -> if_qs_2_in_4_ga6(X, Xs, Ys, Ls, Bs, app_3_in_gga3(Ls, ._22(X, Bs), Ys))
app_3_in_gga3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_in_gga3(Xs, Ys, Zs))
app_3_in_gga3([]_0, Ys, Ys) -> app_3_out_gga3([]_0, Ys, Ys)
if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_out_gga3(Xs, Ys, Zs)) -> app_3_out_gga3(._22(X, Xs), Ys, ._22(X, Zs))
if_qs_2_in_4_ga6(X, Xs, Ys, Ls, Bs, app_3_out_gga3(Ls, ._22(X, Bs), Ys)) -> qs_2_out_ga2(._22(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_2_in_ga2(x1, x2)  =  qs_2_in_ga1(x1)
._22(x1, x2)  =  ._22(x1, x2)
[]_0  =  []_0
s_11(x1)  =  s_11(x1)
0_0  =  0_0
if_qs_2_in_1_ga4(x1, x2, x3, x4)  =  if_qs_2_in_1_ga2(x1, x4)
part_4_in_ggaa4(x1, x2, x3, x4)  =  part_4_in_ggaa2(x1, x2)
if_part_4_in_1_ggaa6(x1, x2, x3, x4, x5, x6)  =  if_part_4_in_1_ggaa4(x1, x2, x3, 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_part_4_in_2_ggaa6(x1, x2, x3, x4, x5, x6)  =  if_part_4_in_2_ggaa2(x2, x6)
if_part_4_in_3_ggaa6(x1, x2, x3, x4, x5, x6)  =  if_part_4_in_3_ggaa4(x1, x2, x3, 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_part_4_in_4_ggaa6(x1, x2, x3, x4, x5, x6)  =  if_part_4_in_4_ggaa2(x2, x6)
part_4_out_ggaa4(x1, x2, x3, x4)  =  part_4_out_ggaa2(x3, x4)
if_qs_2_in_2_ga6(x1, x2, x3, x4, x5, x6)  =  if_qs_2_in_2_ga3(x1, x5, x6)
qs_2_out_ga2(x1, x2)  =  qs_2_out_ga1(x2)
if_qs_2_in_3_ga6(x1, x2, x3, x4, x5, x6)  =  if_qs_2_in_3_ga3(x1, x5, x6)
if_qs_2_in_4_ga6(x1, x2, x3, x4, x5, x6)  =  if_qs_2_in_4_ga1(x6)
app_3_in_gga3(x1, x2, x3)  =  app_3_in_gga2(x1, x2)
if_app_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gga2(x1, x5)
app_3_out_gga3(x1, x2, x3)  =  app_3_out_gga1(x3)
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
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
PiDP
                    ↳ PiDPToQDPProof
              ↳ 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

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
              ↳ PiDP
PiDP
                ↳ UsableRulesProof
              ↳ PiDP

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

IF_PART_4_IN_1_GGAA6(X, Y, Xs, Ls, Bs, gt_2_out_gg2(X, Y)) -> PART_4_IN_GGAA4(X, Xs, Ls, Bs)
PART_4_IN_GGAA4(X, ._22(Y, Xs), Ls, ._22(Y, Bs)) -> IF_PART_4_IN_3_GGAA6(X, Y, Xs, Ls, Bs, le_2_in_gg2(X, Y))
IF_PART_4_IN_3_GGAA6(X, Y, Xs, Ls, Bs, le_2_out_gg2(X, Y)) -> PART_4_IN_GGAA4(X, Xs, Ls, Bs)
PART_4_IN_GGAA4(X, ._22(Y, Xs), ._22(Y, Ls), Bs) -> IF_PART_4_IN_1_GGAA6(X, Y, Xs, Ls, Bs, gt_2_in_gg2(X, Y))

The TRS R consists of the following rules:

qs_2_in_ga2(._22(X, Xs), Ys) -> if_qs_2_in_1_ga4(X, Xs, Ys, part_4_in_ggaa4(X, Xs, Littles, Bigs))
part_4_in_ggaa4(X, ._22(Y, Xs), ._22(Y, Ls), Bs) -> if_part_4_in_1_ggaa6(X, Y, Xs, Ls, Bs, 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(0_0), 0_0) -> gt_2_out_gg2(s_11(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_part_4_in_1_ggaa6(X, Y, Xs, Ls, Bs, gt_2_out_gg2(X, Y)) -> if_part_4_in_2_ggaa6(X, Y, Xs, Ls, Bs, part_4_in_ggaa4(X, Xs, Ls, Bs))
part_4_in_ggaa4(X, ._22(Y, Xs), Ls, ._22(Y, Bs)) -> if_part_4_in_3_ggaa6(X, Y, Xs, Ls, Bs, 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(0_0)) -> le_2_out_gg2(0_0, s_11(0_0))
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_part_4_in_3_ggaa6(X, Y, Xs, Ls, Bs, le_2_out_gg2(X, Y)) -> if_part_4_in_4_ggaa6(X, Y, Xs, Ls, Bs, part_4_in_ggaa4(X, Xs, Ls, Bs))
part_4_in_ggaa4(X, []_0, []_0, []_0) -> part_4_out_ggaa4(X, []_0, []_0, []_0)
if_part_4_in_4_ggaa6(X, Y, Xs, Ls, Bs, part_4_out_ggaa4(X, Xs, Ls, Bs)) -> part_4_out_ggaa4(X, ._22(Y, Xs), Ls, ._22(Y, Bs))
if_part_4_in_2_ggaa6(X, Y, Xs, Ls, Bs, part_4_out_ggaa4(X, Xs, Ls, Bs)) -> part_4_out_ggaa4(X, ._22(Y, Xs), ._22(Y, Ls), Bs)
if_qs_2_in_1_ga4(X, Xs, Ys, part_4_out_ggaa4(X, Xs, Littles, Bigs)) -> if_qs_2_in_2_ga6(X, Xs, Ys, Littles, Bigs, qs_2_in_ga2(Littles, Ls))
qs_2_in_ga2([]_0, []_0) -> qs_2_out_ga2([]_0, []_0)
if_qs_2_in_2_ga6(X, Xs, Ys, Littles, Bigs, qs_2_out_ga2(Littles, Ls)) -> if_qs_2_in_3_ga6(X, Xs, Ys, Bigs, Ls, qs_2_in_ga2(Bigs, Bs))
if_qs_2_in_3_ga6(X, Xs, Ys, Bigs, Ls, qs_2_out_ga2(Bigs, Bs)) -> if_qs_2_in_4_ga6(X, Xs, Ys, Ls, Bs, app_3_in_gga3(Ls, ._22(X, Bs), Ys))
app_3_in_gga3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_in_gga3(Xs, Ys, Zs))
app_3_in_gga3([]_0, Ys, Ys) -> app_3_out_gga3([]_0, Ys, Ys)
if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_out_gga3(Xs, Ys, Zs)) -> app_3_out_gga3(._22(X, Xs), Ys, ._22(X, Zs))
if_qs_2_in_4_ga6(X, Xs, Ys, Ls, Bs, app_3_out_gga3(Ls, ._22(X, Bs), Ys)) -> qs_2_out_ga2(._22(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_2_in_ga2(x1, x2)  =  qs_2_in_ga1(x1)
._22(x1, x2)  =  ._22(x1, x2)
[]_0  =  []_0
s_11(x1)  =  s_11(x1)
0_0  =  0_0
if_qs_2_in_1_ga4(x1, x2, x3, x4)  =  if_qs_2_in_1_ga2(x1, x4)
part_4_in_ggaa4(x1, x2, x3, x4)  =  part_4_in_ggaa2(x1, x2)
if_part_4_in_1_ggaa6(x1, x2, x3, x4, x5, x6)  =  if_part_4_in_1_ggaa4(x1, x2, x3, 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_part_4_in_2_ggaa6(x1, x2, x3, x4, x5, x6)  =  if_part_4_in_2_ggaa2(x2, x6)
if_part_4_in_3_ggaa6(x1, x2, x3, x4, x5, x6)  =  if_part_4_in_3_ggaa4(x1, x2, x3, 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_part_4_in_4_ggaa6(x1, x2, x3, x4, x5, x6)  =  if_part_4_in_4_ggaa2(x2, x6)
part_4_out_ggaa4(x1, x2, x3, x4)  =  part_4_out_ggaa2(x3, x4)
if_qs_2_in_2_ga6(x1, x2, x3, x4, x5, x6)  =  if_qs_2_in_2_ga3(x1, x5, x6)
qs_2_out_ga2(x1, x2)  =  qs_2_out_ga1(x2)
if_qs_2_in_3_ga6(x1, x2, x3, x4, x5, x6)  =  if_qs_2_in_3_ga3(x1, x5, x6)
if_qs_2_in_4_ga6(x1, x2, x3, x4, x5, x6)  =  if_qs_2_in_4_ga1(x6)
app_3_in_gga3(x1, x2, x3)  =  app_3_in_gga2(x1, x2)
if_app_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gga2(x1, x5)
app_3_out_gga3(x1, x2, x3)  =  app_3_out_gga1(x3)
PART_4_IN_GGAA4(x1, x2, x3, x4)  =  PART_4_IN_GGAA2(x1, x2)
IF_PART_4_IN_3_GGAA6(x1, x2, x3, x4, x5, x6)  =  IF_PART_4_IN_3_GGAA4(x1, x2, x3, x6)
IF_PART_4_IN_1_GGAA6(x1, x2, x3, x4, x5, x6)  =  IF_PART_4_IN_1_GGAA4(x1, x2, x3, 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
              ↳ PiDP
                ↳ UsableRulesProof
PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP

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

IF_PART_4_IN_1_GGAA6(X, Y, Xs, Ls, Bs, gt_2_out_gg2(X, Y)) -> PART_4_IN_GGAA4(X, Xs, Ls, Bs)
PART_4_IN_GGAA4(X, ._22(Y, Xs), Ls, ._22(Y, Bs)) -> IF_PART_4_IN_3_GGAA6(X, Y, Xs, Ls, Bs, le_2_in_gg2(X, Y))
IF_PART_4_IN_3_GGAA6(X, Y, Xs, Ls, Bs, le_2_out_gg2(X, Y)) -> PART_4_IN_GGAA4(X, Xs, Ls, Bs)
PART_4_IN_GGAA4(X, ._22(Y, Xs), ._22(Y, Ls), Bs) -> IF_PART_4_IN_1_GGAA6(X, Y, Xs, Ls, Bs, gt_2_in_gg2(X, Y))

The TRS R consists of the following rules:

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(0_0)) -> le_2_out_gg2(0_0, s_11(0_0))
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(0_0), 0_0) -> gt_2_out_gg2(s_11(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_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
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
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
PART_4_IN_GGAA4(x1, x2, x3, x4)  =  PART_4_IN_GGAA2(x1, x2)
IF_PART_4_IN_3_GGAA6(x1, x2, x3, x4, x5, x6)  =  IF_PART_4_IN_3_GGAA4(x1, x2, x3, x6)
IF_PART_4_IN_1_GGAA6(x1, x2, x3, x4, x5, x6)  =  IF_PART_4_IN_1_GGAA4(x1, x2, x3, 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
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP

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

IF_PART_4_IN_1_GGAA4(X, Y, Xs, gt_2_out_gg) -> PART_4_IN_GGAA2(X, Xs)
PART_4_IN_GGAA2(X, ._22(Y, Xs)) -> IF_PART_4_IN_3_GGAA4(X, Y, Xs, le_2_in_gg2(X, Y))
IF_PART_4_IN_3_GGAA4(X, Y, Xs, le_2_out_gg) -> PART_4_IN_GGAA2(X, Xs)
PART_4_IN_GGAA2(X, ._22(Y, Xs)) -> IF_PART_4_IN_1_GGAA4(X, Y, Xs, gt_2_in_gg2(X, Y))

The TRS R consists of the following rules:

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(0_0)) -> 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(0_0), 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:

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 {PART_4_IN_GGAA2, IF_PART_4_IN_1_GGAA4, IF_PART_4_IN_3_GGAA4}.
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
                ↳ PiDPToQDPProof

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

IF_QS_2_IN_2_GA6(X, Xs, Ys, Littles, Bigs, qs_2_out_ga2(Littles, Ls)) -> QS_2_IN_GA2(Bigs, Bs)
IF_QS_2_IN_1_GA4(X, Xs, Ys, part_4_out_ggaa4(X, Xs, Littles, Bigs)) -> QS_2_IN_GA2(Littles, Ls)
QS_2_IN_GA2(._22(X, Xs), Ys) -> IF_QS_2_IN_1_GA4(X, Xs, Ys, part_4_in_ggaa4(X, Xs, Littles, Bigs))
IF_QS_2_IN_1_GA4(X, Xs, Ys, part_4_out_ggaa4(X, Xs, Littles, Bigs)) -> IF_QS_2_IN_2_GA6(X, Xs, Ys, Littles, Bigs, qs_2_in_ga2(Littles, Ls))

The TRS R consists of the following rules:

qs_2_in_ga2(._22(X, Xs), Ys) -> if_qs_2_in_1_ga4(X, Xs, Ys, part_4_in_ggaa4(X, Xs, Littles, Bigs))
part_4_in_ggaa4(X, ._22(Y, Xs), ._22(Y, Ls), Bs) -> if_part_4_in_1_ggaa6(X, Y, Xs, Ls, Bs, 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(0_0), 0_0) -> gt_2_out_gg2(s_11(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_part_4_in_1_ggaa6(X, Y, Xs, Ls, Bs, gt_2_out_gg2(X, Y)) -> if_part_4_in_2_ggaa6(X, Y, Xs, Ls, Bs, part_4_in_ggaa4(X, Xs, Ls, Bs))
part_4_in_ggaa4(X, ._22(Y, Xs), Ls, ._22(Y, Bs)) -> if_part_4_in_3_ggaa6(X, Y, Xs, Ls, Bs, 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(0_0)) -> le_2_out_gg2(0_0, s_11(0_0))
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_part_4_in_3_ggaa6(X, Y, Xs, Ls, Bs, le_2_out_gg2(X, Y)) -> if_part_4_in_4_ggaa6(X, Y, Xs, Ls, Bs, part_4_in_ggaa4(X, Xs, Ls, Bs))
part_4_in_ggaa4(X, []_0, []_0, []_0) -> part_4_out_ggaa4(X, []_0, []_0, []_0)
if_part_4_in_4_ggaa6(X, Y, Xs, Ls, Bs, part_4_out_ggaa4(X, Xs, Ls, Bs)) -> part_4_out_ggaa4(X, ._22(Y, Xs), Ls, ._22(Y, Bs))
if_part_4_in_2_ggaa6(X, Y, Xs, Ls, Bs, part_4_out_ggaa4(X, Xs, Ls, Bs)) -> part_4_out_ggaa4(X, ._22(Y, Xs), ._22(Y, Ls), Bs)
if_qs_2_in_1_ga4(X, Xs, Ys, part_4_out_ggaa4(X, Xs, Littles, Bigs)) -> if_qs_2_in_2_ga6(X, Xs, Ys, Littles, Bigs, qs_2_in_ga2(Littles, Ls))
qs_2_in_ga2([]_0, []_0) -> qs_2_out_ga2([]_0, []_0)
if_qs_2_in_2_ga6(X, Xs, Ys, Littles, Bigs, qs_2_out_ga2(Littles, Ls)) -> if_qs_2_in_3_ga6(X, Xs, Ys, Bigs, Ls, qs_2_in_ga2(Bigs, Bs))
if_qs_2_in_3_ga6(X, Xs, Ys, Bigs, Ls, qs_2_out_ga2(Bigs, Bs)) -> if_qs_2_in_4_ga6(X, Xs, Ys, Ls, Bs, app_3_in_gga3(Ls, ._22(X, Bs), Ys))
app_3_in_gga3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_in_gga3(Xs, Ys, Zs))
app_3_in_gga3([]_0, Ys, Ys) -> app_3_out_gga3([]_0, Ys, Ys)
if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_out_gga3(Xs, Ys, Zs)) -> app_3_out_gga3(._22(X, Xs), Ys, ._22(X, Zs))
if_qs_2_in_4_ga6(X, Xs, Ys, Ls, Bs, app_3_out_gga3(Ls, ._22(X, Bs), Ys)) -> qs_2_out_ga2(._22(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_2_in_ga2(x1, x2)  =  qs_2_in_ga1(x1)
._22(x1, x2)  =  ._22(x1, x2)
[]_0  =  []_0
s_11(x1)  =  s_11(x1)
0_0  =  0_0
if_qs_2_in_1_ga4(x1, x2, x3, x4)  =  if_qs_2_in_1_ga2(x1, x4)
part_4_in_ggaa4(x1, x2, x3, x4)  =  part_4_in_ggaa2(x1, x2)
if_part_4_in_1_ggaa6(x1, x2, x3, x4, x5, x6)  =  if_part_4_in_1_ggaa4(x1, x2, x3, 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_part_4_in_2_ggaa6(x1, x2, x3, x4, x5, x6)  =  if_part_4_in_2_ggaa2(x2, x6)
if_part_4_in_3_ggaa6(x1, x2, x3, x4, x5, x6)  =  if_part_4_in_3_ggaa4(x1, x2, x3, 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_part_4_in_4_ggaa6(x1, x2, x3, x4, x5, x6)  =  if_part_4_in_4_ggaa2(x2, x6)
part_4_out_ggaa4(x1, x2, x3, x4)  =  part_4_out_ggaa2(x3, x4)
if_qs_2_in_2_ga6(x1, x2, x3, x4, x5, x6)  =  if_qs_2_in_2_ga3(x1, x5, x6)
qs_2_out_ga2(x1, x2)  =  qs_2_out_ga1(x2)
if_qs_2_in_3_ga6(x1, x2, x3, x4, x5, x6)  =  if_qs_2_in_3_ga3(x1, x5, x6)
if_qs_2_in_4_ga6(x1, x2, x3, x4, x5, x6)  =  if_qs_2_in_4_ga1(x6)
app_3_in_gga3(x1, x2, x3)  =  app_3_in_gga2(x1, x2)
if_app_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gga2(x1, x5)
app_3_out_gga3(x1, x2, x3)  =  app_3_out_gga1(x3)
IF_QS_2_IN_1_GA4(x1, x2, x3, x4)  =  IF_QS_2_IN_1_GA2(x1, x4)
IF_QS_2_IN_2_GA6(x1, x2, x3, x4, x5, x6)  =  IF_QS_2_IN_2_GA3(x1, x5, x6)
QS_2_IN_GA2(x1, x2)  =  QS_2_IN_GA1(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
              ↳ PiDP
                ↳ PiDPToQDPProof
QDP
                    ↳ QDPPoloProof

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

IF_QS_2_IN_2_GA3(X, Bigs, qs_2_out_ga1(Ls)) -> QS_2_IN_GA1(Bigs)
IF_QS_2_IN_1_GA2(X, part_4_out_ggaa2(Littles, Bigs)) -> QS_2_IN_GA1(Littles)
QS_2_IN_GA1(._22(X, Xs)) -> IF_QS_2_IN_1_GA2(X, part_4_in_ggaa2(X, Xs))
IF_QS_2_IN_1_GA2(X, part_4_out_ggaa2(Littles, Bigs)) -> IF_QS_2_IN_2_GA3(X, Bigs, qs_2_in_ga1(Littles))

The TRS R consists of the following rules:

qs_2_in_ga1(._22(X, Xs)) -> if_qs_2_in_1_ga2(X, part_4_in_ggaa2(X, Xs))
part_4_in_ggaa2(X, ._22(Y, Xs)) -> if_part_4_in_1_ggaa4(X, Y, Xs, gt_2_in_gg2(X, Y))
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(0_0), 0_0) -> gt_2_out_gg
if_gt_2_in_1_gg1(gt_2_out_gg) -> gt_2_out_gg
if_part_4_in_1_ggaa4(X, Y, Xs, gt_2_out_gg) -> if_part_4_in_2_ggaa2(Y, part_4_in_ggaa2(X, Xs))
part_4_in_ggaa2(X, ._22(Y, Xs)) -> if_part_4_in_3_ggaa4(X, Y, Xs, le_2_in_gg2(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(0_0)) -> le_2_out_gg
le_2_in_gg2(0_0, 0_0) -> le_2_out_gg
if_le_2_in_1_gg1(le_2_out_gg) -> le_2_out_gg
if_part_4_in_3_ggaa4(X, Y, Xs, le_2_out_gg) -> if_part_4_in_4_ggaa2(Y, part_4_in_ggaa2(X, Xs))
part_4_in_ggaa2(X, []_0) -> part_4_out_ggaa2([]_0, []_0)
if_part_4_in_4_ggaa2(Y, part_4_out_ggaa2(Ls, Bs)) -> part_4_out_ggaa2(Ls, ._22(Y, Bs))
if_part_4_in_2_ggaa2(Y, part_4_out_ggaa2(Ls, Bs)) -> part_4_out_ggaa2(._22(Y, Ls), Bs)
if_qs_2_in_1_ga2(X, part_4_out_ggaa2(Littles, Bigs)) -> if_qs_2_in_2_ga3(X, Bigs, qs_2_in_ga1(Littles))
qs_2_in_ga1([]_0) -> qs_2_out_ga1([]_0)
if_qs_2_in_2_ga3(X, Bigs, qs_2_out_ga1(Ls)) -> if_qs_2_in_3_ga3(X, Ls, qs_2_in_ga1(Bigs))
if_qs_2_in_3_ga3(X, Ls, qs_2_out_ga1(Bs)) -> if_qs_2_in_4_ga1(app_3_in_gga2(Ls, ._22(X, Bs)))
app_3_in_gga2(._22(X, Xs), Ys) -> if_app_3_in_1_gga2(X, app_3_in_gga2(Xs, Ys))
app_3_in_gga2([]_0, Ys) -> app_3_out_gga1(Ys)
if_app_3_in_1_gga2(X, app_3_out_gga1(Zs)) -> app_3_out_gga1(._22(X, Zs))
if_qs_2_in_4_ga1(app_3_out_gga1(Ys)) -> qs_2_out_ga1(Ys)

The set Q consists of the following terms:

qs_2_in_ga1(x0)
part_4_in_ggaa2(x0, x1)
gt_2_in_gg2(x0, x1)
if_gt_2_in_1_gg1(x0)
if_part_4_in_1_ggaa4(x0, x1, x2, x3)
le_2_in_gg2(x0, x1)
if_le_2_in_1_gg1(x0)
if_part_4_in_3_ggaa4(x0, x1, x2, x3)
if_part_4_in_4_ggaa2(x0, x1)
if_part_4_in_2_ggaa2(x0, x1)
if_qs_2_in_1_ga2(x0, x1)
if_qs_2_in_2_ga3(x0, x1, x2)
if_qs_2_in_3_ga3(x0, x1, x2)
app_3_in_gga2(x0, x1)
if_app_3_in_1_gga2(x0, x1)
if_qs_2_in_4_ga1(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {QS_2_IN_GA1, IF_QS_2_IN_2_GA3, IF_QS_2_IN_1_GA2}.
By using a polynomial ordering, the following set of Dependency Pairs of this DP problem can be strictly oriented.

QS_2_IN_GA1(._22(X, Xs)) -> IF_QS_2_IN_1_GA2(X, part_4_in_ggaa2(X, Xs))
The remaining Dependency Pairs were at least non-strictly be oriented.

IF_QS_2_IN_2_GA3(X, Bigs, qs_2_out_ga1(Ls)) -> QS_2_IN_GA1(Bigs)
IF_QS_2_IN_1_GA2(X, part_4_out_ggaa2(Littles, Bigs)) -> QS_2_IN_GA1(Littles)
IF_QS_2_IN_1_GA2(X, part_4_out_ggaa2(Littles, Bigs)) -> IF_QS_2_IN_2_GA3(X, Bigs, qs_2_in_ga1(Littles))
With the implicit AFS we had to orient the following set of usable rules non-strictly.

if_part_4_in_2_ggaa2(Y, part_4_out_ggaa2(Ls, Bs)) -> part_4_out_ggaa2(._22(Y, Ls), Bs)
if_part_4_in_3_ggaa4(X, Y, Xs, le_2_out_gg) -> if_part_4_in_4_ggaa2(Y, part_4_in_ggaa2(X, Xs))
part_4_in_ggaa2(X, ._22(Y, Xs)) -> if_part_4_in_1_ggaa4(X, Y, Xs, gt_2_in_gg2(X, Y))
part_4_in_ggaa2(X, ._22(Y, Xs)) -> if_part_4_in_3_ggaa4(X, Y, Xs, le_2_in_gg2(X, Y))
part_4_in_ggaa2(X, []_0) -> part_4_out_ggaa2([]_0, []_0)
if_part_4_in_1_ggaa4(X, Y, Xs, gt_2_out_gg) -> if_part_4_in_2_ggaa2(Y, part_4_in_ggaa2(X, Xs))
if_part_4_in_4_ggaa2(Y, part_4_out_ggaa2(Ls, Bs)) -> part_4_out_ggaa2(Ls, ._22(Y, Bs))
Used ordering: POLO with Polynomial interpretation:

POL(if_qs_2_in_4_ga1(x1)) = 0   
POL(gt_2_in_gg2(x1, x2)) = 0   
POL(0_0) = 0   
POL(IF_QS_2_IN_1_GA2(x1, x2)) = x2   
POL(if_qs_2_in_1_ga2(x1, x2)) = 0   
POL(if_qs_2_in_2_ga3(x1, x2, x3)) = 0   
POL(if_part_4_in_4_ggaa2(x1, x2)) = 1 + x2   
POL(app_3_out_gga1(x1)) = 0   
POL(IF_QS_2_IN_2_GA3(x1, x2, x3)) = x2   
POL(QS_2_IN_GA1(x1)) = x1   
POL(if_qs_2_in_3_ga3(x1, x2, x3)) = 0   
POL(if_gt_2_in_1_gg1(x1)) = 0   
POL(qs_2_in_ga1(x1)) = 0   
POL(if_app_3_in_1_gga2(x1, x2)) = 0   
POL(part_4_in_ggaa2(x1, x2)) = x2   
POL(le_2_out_gg) = 0   
POL(qs_2_out_ga1(x1)) = 0   
POL(if_part_4_in_1_ggaa4(x1, x2, x3, x4)) = 1 + x3   
POL([]_0) = 0   
POL(if_le_2_in_1_gg1(x1)) = 0   
POL(._22(x1, x2)) = 1 + x2   
POL(part_4_out_ggaa2(x1, x2)) = x1 + x2   
POL(le_2_in_gg2(x1, x2)) = 0   
POL(if_part_4_in_2_ggaa2(x1, x2)) = 1 + x2   
POL(if_part_4_in_3_ggaa4(x1, x2, x3, x4)) = 1 + x3   
POL(s_11(x1)) = 0   
POL(gt_2_out_gg) = 0   
POL(app_3_in_gga2(x1, x2)) = 0   



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

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

IF_QS_2_IN_2_GA3(X, Bigs, qs_2_out_ga1(Ls)) -> QS_2_IN_GA1(Bigs)
IF_QS_2_IN_1_GA2(X, part_4_out_ggaa2(Littles, Bigs)) -> QS_2_IN_GA1(Littles)
IF_QS_2_IN_1_GA2(X, part_4_out_ggaa2(Littles, Bigs)) -> IF_QS_2_IN_2_GA3(X, Bigs, qs_2_in_ga1(Littles))

The TRS R consists of the following rules:

qs_2_in_ga1(._22(X, Xs)) -> if_qs_2_in_1_ga2(X, part_4_in_ggaa2(X, Xs))
part_4_in_ggaa2(X, ._22(Y, Xs)) -> if_part_4_in_1_ggaa4(X, Y, Xs, gt_2_in_gg2(X, Y))
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(0_0), 0_0) -> gt_2_out_gg
if_gt_2_in_1_gg1(gt_2_out_gg) -> gt_2_out_gg
if_part_4_in_1_ggaa4(X, Y, Xs, gt_2_out_gg) -> if_part_4_in_2_ggaa2(Y, part_4_in_ggaa2(X, Xs))
part_4_in_ggaa2(X, ._22(Y, Xs)) -> if_part_4_in_3_ggaa4(X, Y, Xs, le_2_in_gg2(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(0_0)) -> le_2_out_gg
le_2_in_gg2(0_0, 0_0) -> le_2_out_gg
if_le_2_in_1_gg1(le_2_out_gg) -> le_2_out_gg
if_part_4_in_3_ggaa4(X, Y, Xs, le_2_out_gg) -> if_part_4_in_4_ggaa2(Y, part_4_in_ggaa2(X, Xs))
part_4_in_ggaa2(X, []_0) -> part_4_out_ggaa2([]_0, []_0)
if_part_4_in_4_ggaa2(Y, part_4_out_ggaa2(Ls, Bs)) -> part_4_out_ggaa2(Ls, ._22(Y, Bs))
if_part_4_in_2_ggaa2(Y, part_4_out_ggaa2(Ls, Bs)) -> part_4_out_ggaa2(._22(Y, Ls), Bs)
if_qs_2_in_1_ga2(X, part_4_out_ggaa2(Littles, Bigs)) -> if_qs_2_in_2_ga3(X, Bigs, qs_2_in_ga1(Littles))
qs_2_in_ga1([]_0) -> qs_2_out_ga1([]_0)
if_qs_2_in_2_ga3(X, Bigs, qs_2_out_ga1(Ls)) -> if_qs_2_in_3_ga3(X, Ls, qs_2_in_ga1(Bigs))
if_qs_2_in_3_ga3(X, Ls, qs_2_out_ga1(Bs)) -> if_qs_2_in_4_ga1(app_3_in_gga2(Ls, ._22(X, Bs)))
app_3_in_gga2(._22(X, Xs), Ys) -> if_app_3_in_1_gga2(X, app_3_in_gga2(Xs, Ys))
app_3_in_gga2([]_0, Ys) -> app_3_out_gga1(Ys)
if_app_3_in_1_gga2(X, app_3_out_gga1(Zs)) -> app_3_out_gga1(._22(X, Zs))
if_qs_2_in_4_ga1(app_3_out_gga1(Ys)) -> qs_2_out_ga1(Ys)

The set Q consists of the following terms:

qs_2_in_ga1(x0)
part_4_in_ggaa2(x0, x1)
gt_2_in_gg2(x0, x1)
if_gt_2_in_1_gg1(x0)
if_part_4_in_1_ggaa4(x0, x1, x2, x3)
le_2_in_gg2(x0, x1)
if_le_2_in_1_gg1(x0)
if_part_4_in_3_ggaa4(x0, x1, x2, x3)
if_part_4_in_4_ggaa2(x0, x1)
if_part_4_in_2_ggaa2(x0, x1)
if_qs_2_in_1_ga2(x0, x1)
if_qs_2_in_2_ga3(x0, x1, x2)
if_qs_2_in_3_ga3(x0, x1, x2)
app_3_in_gga2(x0, x1)
if_app_3_in_1_gga2(x0, x1)
if_qs_2_in_4_ga1(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {QS_2_IN_GA1, IF_QS_2_IN_2_GA3, IF_QS_2_IN_1_GA2}.
The approximation of the Dependency Graph contains 0 SCCs with 3 less nodes.