Left Termination of the query pattern ss(f,b) w.r.t. the given Prolog program could not be shown:



PROLOG
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof

ss2(Xs, Ys) :- perm2(Xs, Ys), ordered1(Ys).
perm2({}0, {}0).
perm2(Xs, .2(X, Ys)) :- app3(X1s, .2(X, X2s), Xs), app3(X1s, X2s, Zs), perm2(Zs, Ys).
app3({}0, X, X).
app3(.2(X, Xs), Ys, .2(X, Zs)) :- app3(Xs, Ys, Zs).
ordered1({}0).
ordered1(.2(underscore, {}0)).
ordered1(.2(X, .2(Y, Xs))) :- less2(X, s1(Y)), ordered1(.2(Y, Xs)).
less2(00, s1(underscore1)).
less2(s1(X), s1(Y)) :- less2(X, Y).


With regard to the inferred argument filtering the predicates were used in the following modes:
ss2: (f,b)
perm2: (f,b)
app3: (f,f,f) (b,f,f)
ordered1: (b)
less2: (f,b) (f,f)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:


ss_2_in_ag2(Xs, Ys) -> if_ss_2_in_1_ag3(Xs, Ys, perm_2_in_ag2(Xs, Ys))
perm_2_in_ag2([]_0, []_0) -> perm_2_out_ag2([]_0, []_0)
perm_2_in_ag2(Xs, ._22(X, Ys)) -> if_perm_2_in_1_ag4(Xs, X, Ys, app_3_in_aaa3(X1s, ._22(X, X2s), Xs))
app_3_in_aaa3([]_0, X, X) -> app_3_out_aaa3([]_0, X, X)
app_3_in_aaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_aaa5(X, Xs, Ys, Zs, app_3_in_aaa3(Xs, Ys, Zs))
if_app_3_in_1_aaa5(X, Xs, Ys, Zs, app_3_out_aaa3(Xs, Ys, Zs)) -> app_3_out_aaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_perm_2_in_1_ag4(Xs, X, Ys, app_3_out_aaa3(X1s, ._22(X, X2s), Xs)) -> if_perm_2_in_2_ag6(Xs, X, Ys, X1s, X2s, app_3_in_gaa3(X1s, X2s, Zs))
app_3_in_gaa3([]_0, X, X) -> app_3_out_gaa3([]_0, X, X)
app_3_in_gaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_gaa5(X, Xs, Ys, Zs, app_3_in_gaa3(Xs, Ys, Zs))
if_app_3_in_1_gaa5(X, Xs, Ys, Zs, app_3_out_gaa3(Xs, Ys, Zs)) -> app_3_out_gaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_perm_2_in_2_ag6(Xs, X, Ys, X1s, X2s, app_3_out_gaa3(X1s, X2s, Zs)) -> if_perm_2_in_3_ag5(Xs, X, Ys, Zs, perm_2_in_ag2(Zs, Ys))
if_perm_2_in_3_ag5(Xs, X, Ys, Zs, perm_2_out_ag2(Zs, Ys)) -> perm_2_out_ag2(Xs, ._22(X, Ys))
if_ss_2_in_1_ag3(Xs, Ys, perm_2_out_ag2(Xs, Ys)) -> if_ss_2_in_2_ag3(Xs, Ys, ordered_1_in_g1(Ys))
ordered_1_in_g1([]_0) -> ordered_1_out_g1([]_0)
ordered_1_in_g1(._22(underscore, []_0)) -> ordered_1_out_g1(._22(underscore, []_0))
ordered_1_in_g1(._22(X, ._22(Y, Xs))) -> if_ordered_1_in_1_g4(X, Y, Xs, less_2_in_ag2(X, s_11(Y)))
less_2_in_ag2(0_0, s_11(underscore1)) -> less_2_out_ag2(0_0, s_11(underscore1))
less_2_in_ag2(s_11(X), s_11(Y)) -> if_less_2_in_1_ag3(X, Y, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore1)) -> less_2_out_aa2(0_0, s_11(underscore1))
less_2_in_aa2(s_11(X), s_11(Y)) -> if_less_2_in_1_aa3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_aa3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_aa2(s_11(X), s_11(Y))
if_less_2_in_1_ag3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ag2(s_11(X), s_11(Y))
if_ordered_1_in_1_g4(X, Y, Xs, less_2_out_ag2(X, s_11(Y))) -> if_ordered_1_in_2_g4(X, Y, Xs, ordered_1_in_g1(._22(Y, Xs)))
if_ordered_1_in_2_g4(X, Y, Xs, ordered_1_out_g1(._22(Y, Xs))) -> ordered_1_out_g1(._22(X, ._22(Y, Xs)))
if_ss_2_in_2_ag3(Xs, Ys, ordered_1_out_g1(Ys)) -> ss_2_out_ag2(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_2_in_ag2(x1, x2)  =  ss_2_in_ag1(x2)
[]_0  =  []_0
._22(x1, x2)  =  ._21(x2)
s_11(x1)  =  s_1
0_0  =  0_0
if_ss_2_in_1_ag3(x1, x2, x3)  =  if_ss_2_in_1_ag2(x2, x3)
perm_2_in_ag2(x1, x2)  =  perm_2_in_ag1(x2)
perm_2_out_ag2(x1, x2)  =  perm_2_out_ag
if_perm_2_in_1_ag4(x1, x2, x3, x4)  =  if_perm_2_in_1_ag2(x3, x4)
app_3_in_aaa3(x1, x2, x3)  =  app_3_in_aaa
app_3_out_aaa3(x1, x2, x3)  =  app_3_out_aaa1(x1)
if_app_3_in_1_aaa5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_aaa1(x5)
if_perm_2_in_2_ag6(x1, x2, x3, x4, x5, x6)  =  if_perm_2_in_2_ag2(x3, x6)
app_3_in_gaa3(x1, x2, x3)  =  app_3_in_gaa1(x1)
app_3_out_gaa3(x1, x2, x3)  =  app_3_out_gaa
if_app_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gaa1(x5)
if_perm_2_in_3_ag5(x1, x2, x3, x4, x5)  =  if_perm_2_in_3_ag1(x5)
if_ss_2_in_2_ag3(x1, x2, x3)  =  if_ss_2_in_2_ag1(x3)
ordered_1_in_g1(x1)  =  ordered_1_in_g1(x1)
ordered_1_out_g1(x1)  =  ordered_1_out_g
if_ordered_1_in_1_g4(x1, x2, x3, x4)  =  if_ordered_1_in_1_g2(x3, x4)
less_2_in_aa2(x1, x2)  =  less_2_in_aa
less_2_out_aa2(x1, x2)  =  less_2_out_aa2(x1, x2)
less_2_in_ag2(x1, x2)  =  less_2_in_ag1(x2)
less_2_out_ag2(x1, x2)  =  less_2_out_ag1(x1)
if_less_2_in_1_ag3(x1, x2, x3)  =  if_less_2_in_1_ag1(x3)
if_less_2_in_1_aa3(x1, x2, x3)  =  if_less_2_in_1_aa1(x3)
if_ordered_1_in_2_g4(x1, x2, x3, x4)  =  if_ordered_1_in_2_g1(x4)
ss_2_out_ag2(x1, x2)  =  ss_2_out_ag

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of PROLOG



↳ PROLOG
  ↳ PrologToPiTRSProof
PiTRS
      ↳ DependencyPairsProof
  ↳ PrologToPiTRSProof

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

ss_2_in_ag2(Xs, Ys) -> if_ss_2_in_1_ag3(Xs, Ys, perm_2_in_ag2(Xs, Ys))
perm_2_in_ag2([]_0, []_0) -> perm_2_out_ag2([]_0, []_0)
perm_2_in_ag2(Xs, ._22(X, Ys)) -> if_perm_2_in_1_ag4(Xs, X, Ys, app_3_in_aaa3(X1s, ._22(X, X2s), Xs))
app_3_in_aaa3([]_0, X, X) -> app_3_out_aaa3([]_0, X, X)
app_3_in_aaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_aaa5(X, Xs, Ys, Zs, app_3_in_aaa3(Xs, Ys, Zs))
if_app_3_in_1_aaa5(X, Xs, Ys, Zs, app_3_out_aaa3(Xs, Ys, Zs)) -> app_3_out_aaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_perm_2_in_1_ag4(Xs, X, Ys, app_3_out_aaa3(X1s, ._22(X, X2s), Xs)) -> if_perm_2_in_2_ag6(Xs, X, Ys, X1s, X2s, app_3_in_gaa3(X1s, X2s, Zs))
app_3_in_gaa3([]_0, X, X) -> app_3_out_gaa3([]_0, X, X)
app_3_in_gaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_gaa5(X, Xs, Ys, Zs, app_3_in_gaa3(Xs, Ys, Zs))
if_app_3_in_1_gaa5(X, Xs, Ys, Zs, app_3_out_gaa3(Xs, Ys, Zs)) -> app_3_out_gaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_perm_2_in_2_ag6(Xs, X, Ys, X1s, X2s, app_3_out_gaa3(X1s, X2s, Zs)) -> if_perm_2_in_3_ag5(Xs, X, Ys, Zs, perm_2_in_ag2(Zs, Ys))
if_perm_2_in_3_ag5(Xs, X, Ys, Zs, perm_2_out_ag2(Zs, Ys)) -> perm_2_out_ag2(Xs, ._22(X, Ys))
if_ss_2_in_1_ag3(Xs, Ys, perm_2_out_ag2(Xs, Ys)) -> if_ss_2_in_2_ag3(Xs, Ys, ordered_1_in_g1(Ys))
ordered_1_in_g1([]_0) -> ordered_1_out_g1([]_0)
ordered_1_in_g1(._22(underscore, []_0)) -> ordered_1_out_g1(._22(underscore, []_0))
ordered_1_in_g1(._22(X, ._22(Y, Xs))) -> if_ordered_1_in_1_g4(X, Y, Xs, less_2_in_ag2(X, s_11(Y)))
less_2_in_ag2(0_0, s_11(underscore1)) -> less_2_out_ag2(0_0, s_11(underscore1))
less_2_in_ag2(s_11(X), s_11(Y)) -> if_less_2_in_1_ag3(X, Y, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore1)) -> less_2_out_aa2(0_0, s_11(underscore1))
less_2_in_aa2(s_11(X), s_11(Y)) -> if_less_2_in_1_aa3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_aa3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_aa2(s_11(X), s_11(Y))
if_less_2_in_1_ag3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ag2(s_11(X), s_11(Y))
if_ordered_1_in_1_g4(X, Y, Xs, less_2_out_ag2(X, s_11(Y))) -> if_ordered_1_in_2_g4(X, Y, Xs, ordered_1_in_g1(._22(Y, Xs)))
if_ordered_1_in_2_g4(X, Y, Xs, ordered_1_out_g1(._22(Y, Xs))) -> ordered_1_out_g1(._22(X, ._22(Y, Xs)))
if_ss_2_in_2_ag3(Xs, Ys, ordered_1_out_g1(Ys)) -> ss_2_out_ag2(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_2_in_ag2(x1, x2)  =  ss_2_in_ag1(x2)
[]_0  =  []_0
._22(x1, x2)  =  ._21(x2)
s_11(x1)  =  s_1
0_0  =  0_0
if_ss_2_in_1_ag3(x1, x2, x3)  =  if_ss_2_in_1_ag2(x2, x3)
perm_2_in_ag2(x1, x2)  =  perm_2_in_ag1(x2)
perm_2_out_ag2(x1, x2)  =  perm_2_out_ag
if_perm_2_in_1_ag4(x1, x2, x3, x4)  =  if_perm_2_in_1_ag2(x3, x4)
app_3_in_aaa3(x1, x2, x3)  =  app_3_in_aaa
app_3_out_aaa3(x1, x2, x3)  =  app_3_out_aaa1(x1)
if_app_3_in_1_aaa5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_aaa1(x5)
if_perm_2_in_2_ag6(x1, x2, x3, x4, x5, x6)  =  if_perm_2_in_2_ag2(x3, x6)
app_3_in_gaa3(x1, x2, x3)  =  app_3_in_gaa1(x1)
app_3_out_gaa3(x1, x2, x3)  =  app_3_out_gaa
if_app_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gaa1(x5)
if_perm_2_in_3_ag5(x1, x2, x3, x4, x5)  =  if_perm_2_in_3_ag1(x5)
if_ss_2_in_2_ag3(x1, x2, x3)  =  if_ss_2_in_2_ag1(x3)
ordered_1_in_g1(x1)  =  ordered_1_in_g1(x1)
ordered_1_out_g1(x1)  =  ordered_1_out_g
if_ordered_1_in_1_g4(x1, x2, x3, x4)  =  if_ordered_1_in_1_g2(x3, x4)
less_2_in_aa2(x1, x2)  =  less_2_in_aa
less_2_out_aa2(x1, x2)  =  less_2_out_aa2(x1, x2)
less_2_in_ag2(x1, x2)  =  less_2_in_ag1(x2)
less_2_out_ag2(x1, x2)  =  less_2_out_ag1(x1)
if_less_2_in_1_ag3(x1, x2, x3)  =  if_less_2_in_1_ag1(x3)
if_less_2_in_1_aa3(x1, x2, x3)  =  if_less_2_in_1_aa1(x3)
if_ordered_1_in_2_g4(x1, x2, x3, x4)  =  if_ordered_1_in_2_g1(x4)
ss_2_out_ag2(x1, x2)  =  ss_2_out_ag


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

SS_2_IN_AG2(Xs, Ys) -> IF_SS_2_IN_1_AG3(Xs, Ys, perm_2_in_ag2(Xs, Ys))
SS_2_IN_AG2(Xs, Ys) -> PERM_2_IN_AG2(Xs, Ys)
PERM_2_IN_AG2(Xs, ._22(X, Ys)) -> IF_PERM_2_IN_1_AG4(Xs, X, Ys, app_3_in_aaa3(X1s, ._22(X, X2s), Xs))
PERM_2_IN_AG2(Xs, ._22(X, Ys)) -> APP_3_IN_AAA3(X1s, ._22(X, X2s), Xs)
APP_3_IN_AAA3(._22(X, Xs), Ys, ._22(X, Zs)) -> IF_APP_3_IN_1_AAA5(X, Xs, Ys, Zs, app_3_in_aaa3(Xs, Ys, Zs))
APP_3_IN_AAA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APP_3_IN_AAA3(Xs, Ys, Zs)
IF_PERM_2_IN_1_AG4(Xs, X, Ys, app_3_out_aaa3(X1s, ._22(X, X2s), Xs)) -> IF_PERM_2_IN_2_AG6(Xs, X, Ys, X1s, X2s, app_3_in_gaa3(X1s, X2s, Zs))
IF_PERM_2_IN_1_AG4(Xs, X, Ys, app_3_out_aaa3(X1s, ._22(X, X2s), Xs)) -> APP_3_IN_GAA3(X1s, X2s, Zs)
APP_3_IN_GAA3(._22(X, Xs), Ys, ._22(X, Zs)) -> IF_APP_3_IN_1_GAA5(X, Xs, Ys, Zs, app_3_in_gaa3(Xs, Ys, Zs))
APP_3_IN_GAA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APP_3_IN_GAA3(Xs, Ys, Zs)
IF_PERM_2_IN_2_AG6(Xs, X, Ys, X1s, X2s, app_3_out_gaa3(X1s, X2s, Zs)) -> IF_PERM_2_IN_3_AG5(Xs, X, Ys, Zs, perm_2_in_ag2(Zs, Ys))
IF_PERM_2_IN_2_AG6(Xs, X, Ys, X1s, X2s, app_3_out_gaa3(X1s, X2s, Zs)) -> PERM_2_IN_AG2(Zs, Ys)
IF_SS_2_IN_1_AG3(Xs, Ys, perm_2_out_ag2(Xs, Ys)) -> IF_SS_2_IN_2_AG3(Xs, Ys, ordered_1_in_g1(Ys))
IF_SS_2_IN_1_AG3(Xs, Ys, perm_2_out_ag2(Xs, Ys)) -> ORDERED_1_IN_G1(Ys)
ORDERED_1_IN_G1(._22(X, ._22(Y, Xs))) -> IF_ORDERED_1_IN_1_G4(X, Y, Xs, less_2_in_ag2(X, s_11(Y)))
ORDERED_1_IN_G1(._22(X, ._22(Y, Xs))) -> LESS_2_IN_AG2(X, s_11(Y))
LESS_2_IN_AG2(s_11(X), s_11(Y)) -> IF_LESS_2_IN_1_AG3(X, Y, less_2_in_aa2(X, Y))
LESS_2_IN_AG2(s_11(X), s_11(Y)) -> LESS_2_IN_AA2(X, Y)
LESS_2_IN_AA2(s_11(X), s_11(Y)) -> IF_LESS_2_IN_1_AA3(X, Y, less_2_in_aa2(X, Y))
LESS_2_IN_AA2(s_11(X), s_11(Y)) -> LESS_2_IN_AA2(X, Y)
IF_ORDERED_1_IN_1_G4(X, Y, Xs, less_2_out_ag2(X, s_11(Y))) -> IF_ORDERED_1_IN_2_G4(X, Y, Xs, ordered_1_in_g1(._22(Y, Xs)))
IF_ORDERED_1_IN_1_G4(X, Y, Xs, less_2_out_ag2(X, s_11(Y))) -> ORDERED_1_IN_G1(._22(Y, Xs))

The TRS R consists of the following rules:

ss_2_in_ag2(Xs, Ys) -> if_ss_2_in_1_ag3(Xs, Ys, perm_2_in_ag2(Xs, Ys))
perm_2_in_ag2([]_0, []_0) -> perm_2_out_ag2([]_0, []_0)
perm_2_in_ag2(Xs, ._22(X, Ys)) -> if_perm_2_in_1_ag4(Xs, X, Ys, app_3_in_aaa3(X1s, ._22(X, X2s), Xs))
app_3_in_aaa3([]_0, X, X) -> app_3_out_aaa3([]_0, X, X)
app_3_in_aaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_aaa5(X, Xs, Ys, Zs, app_3_in_aaa3(Xs, Ys, Zs))
if_app_3_in_1_aaa5(X, Xs, Ys, Zs, app_3_out_aaa3(Xs, Ys, Zs)) -> app_3_out_aaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_perm_2_in_1_ag4(Xs, X, Ys, app_3_out_aaa3(X1s, ._22(X, X2s), Xs)) -> if_perm_2_in_2_ag6(Xs, X, Ys, X1s, X2s, app_3_in_gaa3(X1s, X2s, Zs))
app_3_in_gaa3([]_0, X, X) -> app_3_out_gaa3([]_0, X, X)
app_3_in_gaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_gaa5(X, Xs, Ys, Zs, app_3_in_gaa3(Xs, Ys, Zs))
if_app_3_in_1_gaa5(X, Xs, Ys, Zs, app_3_out_gaa3(Xs, Ys, Zs)) -> app_3_out_gaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_perm_2_in_2_ag6(Xs, X, Ys, X1s, X2s, app_3_out_gaa3(X1s, X2s, Zs)) -> if_perm_2_in_3_ag5(Xs, X, Ys, Zs, perm_2_in_ag2(Zs, Ys))
if_perm_2_in_3_ag5(Xs, X, Ys, Zs, perm_2_out_ag2(Zs, Ys)) -> perm_2_out_ag2(Xs, ._22(X, Ys))
if_ss_2_in_1_ag3(Xs, Ys, perm_2_out_ag2(Xs, Ys)) -> if_ss_2_in_2_ag3(Xs, Ys, ordered_1_in_g1(Ys))
ordered_1_in_g1([]_0) -> ordered_1_out_g1([]_0)
ordered_1_in_g1(._22(underscore, []_0)) -> ordered_1_out_g1(._22(underscore, []_0))
ordered_1_in_g1(._22(X, ._22(Y, Xs))) -> if_ordered_1_in_1_g4(X, Y, Xs, less_2_in_ag2(X, s_11(Y)))
less_2_in_ag2(0_0, s_11(underscore1)) -> less_2_out_ag2(0_0, s_11(underscore1))
less_2_in_ag2(s_11(X), s_11(Y)) -> if_less_2_in_1_ag3(X, Y, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore1)) -> less_2_out_aa2(0_0, s_11(underscore1))
less_2_in_aa2(s_11(X), s_11(Y)) -> if_less_2_in_1_aa3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_aa3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_aa2(s_11(X), s_11(Y))
if_less_2_in_1_ag3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ag2(s_11(X), s_11(Y))
if_ordered_1_in_1_g4(X, Y, Xs, less_2_out_ag2(X, s_11(Y))) -> if_ordered_1_in_2_g4(X, Y, Xs, ordered_1_in_g1(._22(Y, Xs)))
if_ordered_1_in_2_g4(X, Y, Xs, ordered_1_out_g1(._22(Y, Xs))) -> ordered_1_out_g1(._22(X, ._22(Y, Xs)))
if_ss_2_in_2_ag3(Xs, Ys, ordered_1_out_g1(Ys)) -> ss_2_out_ag2(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_2_in_ag2(x1, x2)  =  ss_2_in_ag1(x2)
[]_0  =  []_0
._22(x1, x2)  =  ._21(x2)
s_11(x1)  =  s_1
0_0  =  0_0
if_ss_2_in_1_ag3(x1, x2, x3)  =  if_ss_2_in_1_ag2(x2, x3)
perm_2_in_ag2(x1, x2)  =  perm_2_in_ag1(x2)
perm_2_out_ag2(x1, x2)  =  perm_2_out_ag
if_perm_2_in_1_ag4(x1, x2, x3, x4)  =  if_perm_2_in_1_ag2(x3, x4)
app_3_in_aaa3(x1, x2, x3)  =  app_3_in_aaa
app_3_out_aaa3(x1, x2, x3)  =  app_3_out_aaa1(x1)
if_app_3_in_1_aaa5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_aaa1(x5)
if_perm_2_in_2_ag6(x1, x2, x3, x4, x5, x6)  =  if_perm_2_in_2_ag2(x3, x6)
app_3_in_gaa3(x1, x2, x3)  =  app_3_in_gaa1(x1)
app_3_out_gaa3(x1, x2, x3)  =  app_3_out_gaa
if_app_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gaa1(x5)
if_perm_2_in_3_ag5(x1, x2, x3, x4, x5)  =  if_perm_2_in_3_ag1(x5)
if_ss_2_in_2_ag3(x1, x2, x3)  =  if_ss_2_in_2_ag1(x3)
ordered_1_in_g1(x1)  =  ordered_1_in_g1(x1)
ordered_1_out_g1(x1)  =  ordered_1_out_g
if_ordered_1_in_1_g4(x1, x2, x3, x4)  =  if_ordered_1_in_1_g2(x3, x4)
less_2_in_aa2(x1, x2)  =  less_2_in_aa
less_2_out_aa2(x1, x2)  =  less_2_out_aa2(x1, x2)
less_2_in_ag2(x1, x2)  =  less_2_in_ag1(x2)
less_2_out_ag2(x1, x2)  =  less_2_out_ag1(x1)
if_less_2_in_1_ag3(x1, x2, x3)  =  if_less_2_in_1_ag1(x3)
if_less_2_in_1_aa3(x1, x2, x3)  =  if_less_2_in_1_aa1(x3)
if_ordered_1_in_2_g4(x1, x2, x3, x4)  =  if_ordered_1_in_2_g1(x4)
ss_2_out_ag2(x1, x2)  =  ss_2_out_ag
LESS_2_IN_AG2(x1, x2)  =  LESS_2_IN_AG1(x2)
IF_SS_2_IN_1_AG3(x1, x2, x3)  =  IF_SS_2_IN_1_AG2(x2, x3)
IF_LESS_2_IN_1_AG3(x1, x2, x3)  =  IF_LESS_2_IN_1_AG1(x3)
IF_APP_3_IN_1_GAA5(x1, x2, x3, x4, x5)  =  IF_APP_3_IN_1_GAA1(x5)
PERM_2_IN_AG2(x1, x2)  =  PERM_2_IN_AG1(x2)
IF_PERM_2_IN_1_AG4(x1, x2, x3, x4)  =  IF_PERM_2_IN_1_AG2(x3, x4)
IF_LESS_2_IN_1_AA3(x1, x2, x3)  =  IF_LESS_2_IN_1_AA1(x3)
IF_ORDERED_1_IN_2_G4(x1, x2, x3, x4)  =  IF_ORDERED_1_IN_2_G1(x4)
SS_2_IN_AG2(x1, x2)  =  SS_2_IN_AG1(x2)
LESS_2_IN_AA2(x1, x2)  =  LESS_2_IN_AA
IF_PERM_2_IN_3_AG5(x1, x2, x3, x4, x5)  =  IF_PERM_2_IN_3_AG1(x5)
IF_ORDERED_1_IN_1_G4(x1, x2, x3, x4)  =  IF_ORDERED_1_IN_1_G2(x3, x4)
IF_PERM_2_IN_2_AG6(x1, x2, x3, x4, x5, x6)  =  IF_PERM_2_IN_2_AG2(x3, x6)
APP_3_IN_GAA3(x1, x2, x3)  =  APP_3_IN_GAA1(x1)
ORDERED_1_IN_G1(x1)  =  ORDERED_1_IN_G1(x1)
IF_SS_2_IN_2_AG3(x1, x2, x3)  =  IF_SS_2_IN_2_AG1(x3)
APP_3_IN_AAA3(x1, x2, x3)  =  APP_3_IN_AAA
IF_APP_3_IN_1_AAA5(x1, x2, x3, x4, x5)  =  IF_APP_3_IN_1_AAA1(x5)

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

↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
PiDP
          ↳ DependencyGraphProof
  ↳ PrologToPiTRSProof

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

SS_2_IN_AG2(Xs, Ys) -> IF_SS_2_IN_1_AG3(Xs, Ys, perm_2_in_ag2(Xs, Ys))
SS_2_IN_AG2(Xs, Ys) -> PERM_2_IN_AG2(Xs, Ys)
PERM_2_IN_AG2(Xs, ._22(X, Ys)) -> IF_PERM_2_IN_1_AG4(Xs, X, Ys, app_3_in_aaa3(X1s, ._22(X, X2s), Xs))
PERM_2_IN_AG2(Xs, ._22(X, Ys)) -> APP_3_IN_AAA3(X1s, ._22(X, X2s), Xs)
APP_3_IN_AAA3(._22(X, Xs), Ys, ._22(X, Zs)) -> IF_APP_3_IN_1_AAA5(X, Xs, Ys, Zs, app_3_in_aaa3(Xs, Ys, Zs))
APP_3_IN_AAA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APP_3_IN_AAA3(Xs, Ys, Zs)
IF_PERM_2_IN_1_AG4(Xs, X, Ys, app_3_out_aaa3(X1s, ._22(X, X2s), Xs)) -> IF_PERM_2_IN_2_AG6(Xs, X, Ys, X1s, X2s, app_3_in_gaa3(X1s, X2s, Zs))
IF_PERM_2_IN_1_AG4(Xs, X, Ys, app_3_out_aaa3(X1s, ._22(X, X2s), Xs)) -> APP_3_IN_GAA3(X1s, X2s, Zs)
APP_3_IN_GAA3(._22(X, Xs), Ys, ._22(X, Zs)) -> IF_APP_3_IN_1_GAA5(X, Xs, Ys, Zs, app_3_in_gaa3(Xs, Ys, Zs))
APP_3_IN_GAA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APP_3_IN_GAA3(Xs, Ys, Zs)
IF_PERM_2_IN_2_AG6(Xs, X, Ys, X1s, X2s, app_3_out_gaa3(X1s, X2s, Zs)) -> IF_PERM_2_IN_3_AG5(Xs, X, Ys, Zs, perm_2_in_ag2(Zs, Ys))
IF_PERM_2_IN_2_AG6(Xs, X, Ys, X1s, X2s, app_3_out_gaa3(X1s, X2s, Zs)) -> PERM_2_IN_AG2(Zs, Ys)
IF_SS_2_IN_1_AG3(Xs, Ys, perm_2_out_ag2(Xs, Ys)) -> IF_SS_2_IN_2_AG3(Xs, Ys, ordered_1_in_g1(Ys))
IF_SS_2_IN_1_AG3(Xs, Ys, perm_2_out_ag2(Xs, Ys)) -> ORDERED_1_IN_G1(Ys)
ORDERED_1_IN_G1(._22(X, ._22(Y, Xs))) -> IF_ORDERED_1_IN_1_G4(X, Y, Xs, less_2_in_ag2(X, s_11(Y)))
ORDERED_1_IN_G1(._22(X, ._22(Y, Xs))) -> LESS_2_IN_AG2(X, s_11(Y))
LESS_2_IN_AG2(s_11(X), s_11(Y)) -> IF_LESS_2_IN_1_AG3(X, Y, less_2_in_aa2(X, Y))
LESS_2_IN_AG2(s_11(X), s_11(Y)) -> LESS_2_IN_AA2(X, Y)
LESS_2_IN_AA2(s_11(X), s_11(Y)) -> IF_LESS_2_IN_1_AA3(X, Y, less_2_in_aa2(X, Y))
LESS_2_IN_AA2(s_11(X), s_11(Y)) -> LESS_2_IN_AA2(X, Y)
IF_ORDERED_1_IN_1_G4(X, Y, Xs, less_2_out_ag2(X, s_11(Y))) -> IF_ORDERED_1_IN_2_G4(X, Y, Xs, ordered_1_in_g1(._22(Y, Xs)))
IF_ORDERED_1_IN_1_G4(X, Y, Xs, less_2_out_ag2(X, s_11(Y))) -> ORDERED_1_IN_G1(._22(Y, Xs))

The TRS R consists of the following rules:

ss_2_in_ag2(Xs, Ys) -> if_ss_2_in_1_ag3(Xs, Ys, perm_2_in_ag2(Xs, Ys))
perm_2_in_ag2([]_0, []_0) -> perm_2_out_ag2([]_0, []_0)
perm_2_in_ag2(Xs, ._22(X, Ys)) -> if_perm_2_in_1_ag4(Xs, X, Ys, app_3_in_aaa3(X1s, ._22(X, X2s), Xs))
app_3_in_aaa3([]_0, X, X) -> app_3_out_aaa3([]_0, X, X)
app_3_in_aaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_aaa5(X, Xs, Ys, Zs, app_3_in_aaa3(Xs, Ys, Zs))
if_app_3_in_1_aaa5(X, Xs, Ys, Zs, app_3_out_aaa3(Xs, Ys, Zs)) -> app_3_out_aaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_perm_2_in_1_ag4(Xs, X, Ys, app_3_out_aaa3(X1s, ._22(X, X2s), Xs)) -> if_perm_2_in_2_ag6(Xs, X, Ys, X1s, X2s, app_3_in_gaa3(X1s, X2s, Zs))
app_3_in_gaa3([]_0, X, X) -> app_3_out_gaa3([]_0, X, X)
app_3_in_gaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_gaa5(X, Xs, Ys, Zs, app_3_in_gaa3(Xs, Ys, Zs))
if_app_3_in_1_gaa5(X, Xs, Ys, Zs, app_3_out_gaa3(Xs, Ys, Zs)) -> app_3_out_gaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_perm_2_in_2_ag6(Xs, X, Ys, X1s, X2s, app_3_out_gaa3(X1s, X2s, Zs)) -> if_perm_2_in_3_ag5(Xs, X, Ys, Zs, perm_2_in_ag2(Zs, Ys))
if_perm_2_in_3_ag5(Xs, X, Ys, Zs, perm_2_out_ag2(Zs, Ys)) -> perm_2_out_ag2(Xs, ._22(X, Ys))
if_ss_2_in_1_ag3(Xs, Ys, perm_2_out_ag2(Xs, Ys)) -> if_ss_2_in_2_ag3(Xs, Ys, ordered_1_in_g1(Ys))
ordered_1_in_g1([]_0) -> ordered_1_out_g1([]_0)
ordered_1_in_g1(._22(underscore, []_0)) -> ordered_1_out_g1(._22(underscore, []_0))
ordered_1_in_g1(._22(X, ._22(Y, Xs))) -> if_ordered_1_in_1_g4(X, Y, Xs, less_2_in_ag2(X, s_11(Y)))
less_2_in_ag2(0_0, s_11(underscore1)) -> less_2_out_ag2(0_0, s_11(underscore1))
less_2_in_ag2(s_11(X), s_11(Y)) -> if_less_2_in_1_ag3(X, Y, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore1)) -> less_2_out_aa2(0_0, s_11(underscore1))
less_2_in_aa2(s_11(X), s_11(Y)) -> if_less_2_in_1_aa3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_aa3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_aa2(s_11(X), s_11(Y))
if_less_2_in_1_ag3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ag2(s_11(X), s_11(Y))
if_ordered_1_in_1_g4(X, Y, Xs, less_2_out_ag2(X, s_11(Y))) -> if_ordered_1_in_2_g4(X, Y, Xs, ordered_1_in_g1(._22(Y, Xs)))
if_ordered_1_in_2_g4(X, Y, Xs, ordered_1_out_g1(._22(Y, Xs))) -> ordered_1_out_g1(._22(X, ._22(Y, Xs)))
if_ss_2_in_2_ag3(Xs, Ys, ordered_1_out_g1(Ys)) -> ss_2_out_ag2(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_2_in_ag2(x1, x2)  =  ss_2_in_ag1(x2)
[]_0  =  []_0
._22(x1, x2)  =  ._21(x2)
s_11(x1)  =  s_1
0_0  =  0_0
if_ss_2_in_1_ag3(x1, x2, x3)  =  if_ss_2_in_1_ag2(x2, x3)
perm_2_in_ag2(x1, x2)  =  perm_2_in_ag1(x2)
perm_2_out_ag2(x1, x2)  =  perm_2_out_ag
if_perm_2_in_1_ag4(x1, x2, x3, x4)  =  if_perm_2_in_1_ag2(x3, x4)
app_3_in_aaa3(x1, x2, x3)  =  app_3_in_aaa
app_3_out_aaa3(x1, x2, x3)  =  app_3_out_aaa1(x1)
if_app_3_in_1_aaa5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_aaa1(x5)
if_perm_2_in_2_ag6(x1, x2, x3, x4, x5, x6)  =  if_perm_2_in_2_ag2(x3, x6)
app_3_in_gaa3(x1, x2, x3)  =  app_3_in_gaa1(x1)
app_3_out_gaa3(x1, x2, x3)  =  app_3_out_gaa
if_app_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gaa1(x5)
if_perm_2_in_3_ag5(x1, x2, x3, x4, x5)  =  if_perm_2_in_3_ag1(x5)
if_ss_2_in_2_ag3(x1, x2, x3)  =  if_ss_2_in_2_ag1(x3)
ordered_1_in_g1(x1)  =  ordered_1_in_g1(x1)
ordered_1_out_g1(x1)  =  ordered_1_out_g
if_ordered_1_in_1_g4(x1, x2, x3, x4)  =  if_ordered_1_in_1_g2(x3, x4)
less_2_in_aa2(x1, x2)  =  less_2_in_aa
less_2_out_aa2(x1, x2)  =  less_2_out_aa2(x1, x2)
less_2_in_ag2(x1, x2)  =  less_2_in_ag1(x2)
less_2_out_ag2(x1, x2)  =  less_2_out_ag1(x1)
if_less_2_in_1_ag3(x1, x2, x3)  =  if_less_2_in_1_ag1(x3)
if_less_2_in_1_aa3(x1, x2, x3)  =  if_less_2_in_1_aa1(x3)
if_ordered_1_in_2_g4(x1, x2, x3, x4)  =  if_ordered_1_in_2_g1(x4)
ss_2_out_ag2(x1, x2)  =  ss_2_out_ag
LESS_2_IN_AG2(x1, x2)  =  LESS_2_IN_AG1(x2)
IF_SS_2_IN_1_AG3(x1, x2, x3)  =  IF_SS_2_IN_1_AG2(x2, x3)
IF_LESS_2_IN_1_AG3(x1, x2, x3)  =  IF_LESS_2_IN_1_AG1(x3)
IF_APP_3_IN_1_GAA5(x1, x2, x3, x4, x5)  =  IF_APP_3_IN_1_GAA1(x5)
PERM_2_IN_AG2(x1, x2)  =  PERM_2_IN_AG1(x2)
IF_PERM_2_IN_1_AG4(x1, x2, x3, x4)  =  IF_PERM_2_IN_1_AG2(x3, x4)
IF_LESS_2_IN_1_AA3(x1, x2, x3)  =  IF_LESS_2_IN_1_AA1(x3)
IF_ORDERED_1_IN_2_G4(x1, x2, x3, x4)  =  IF_ORDERED_1_IN_2_G1(x4)
SS_2_IN_AG2(x1, x2)  =  SS_2_IN_AG1(x2)
LESS_2_IN_AA2(x1, x2)  =  LESS_2_IN_AA
IF_PERM_2_IN_3_AG5(x1, x2, x3, x4, x5)  =  IF_PERM_2_IN_3_AG1(x5)
IF_ORDERED_1_IN_1_G4(x1, x2, x3, x4)  =  IF_ORDERED_1_IN_1_G2(x3, x4)
IF_PERM_2_IN_2_AG6(x1, x2, x3, x4, x5, x6)  =  IF_PERM_2_IN_2_AG2(x3, x6)
APP_3_IN_GAA3(x1, x2, x3)  =  APP_3_IN_GAA1(x1)
ORDERED_1_IN_G1(x1)  =  ORDERED_1_IN_G1(x1)
IF_SS_2_IN_2_AG3(x1, x2, x3)  =  IF_SS_2_IN_2_AG1(x3)
APP_3_IN_AAA3(x1, x2, x3)  =  APP_3_IN_AAA
IF_APP_3_IN_1_AAA5(x1, x2, x3, x4, x5)  =  IF_APP_3_IN_1_AAA1(x5)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph contains 5 SCCs with 14 less nodes.

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

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

LESS_2_IN_AA2(s_11(X), s_11(Y)) -> LESS_2_IN_AA2(X, Y)

The TRS R consists of the following rules:

ss_2_in_ag2(Xs, Ys) -> if_ss_2_in_1_ag3(Xs, Ys, perm_2_in_ag2(Xs, Ys))
perm_2_in_ag2([]_0, []_0) -> perm_2_out_ag2([]_0, []_0)
perm_2_in_ag2(Xs, ._22(X, Ys)) -> if_perm_2_in_1_ag4(Xs, X, Ys, app_3_in_aaa3(X1s, ._22(X, X2s), Xs))
app_3_in_aaa3([]_0, X, X) -> app_3_out_aaa3([]_0, X, X)
app_3_in_aaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_aaa5(X, Xs, Ys, Zs, app_3_in_aaa3(Xs, Ys, Zs))
if_app_3_in_1_aaa5(X, Xs, Ys, Zs, app_3_out_aaa3(Xs, Ys, Zs)) -> app_3_out_aaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_perm_2_in_1_ag4(Xs, X, Ys, app_3_out_aaa3(X1s, ._22(X, X2s), Xs)) -> if_perm_2_in_2_ag6(Xs, X, Ys, X1s, X2s, app_3_in_gaa3(X1s, X2s, Zs))
app_3_in_gaa3([]_0, X, X) -> app_3_out_gaa3([]_0, X, X)
app_3_in_gaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_gaa5(X, Xs, Ys, Zs, app_3_in_gaa3(Xs, Ys, Zs))
if_app_3_in_1_gaa5(X, Xs, Ys, Zs, app_3_out_gaa3(Xs, Ys, Zs)) -> app_3_out_gaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_perm_2_in_2_ag6(Xs, X, Ys, X1s, X2s, app_3_out_gaa3(X1s, X2s, Zs)) -> if_perm_2_in_3_ag5(Xs, X, Ys, Zs, perm_2_in_ag2(Zs, Ys))
if_perm_2_in_3_ag5(Xs, X, Ys, Zs, perm_2_out_ag2(Zs, Ys)) -> perm_2_out_ag2(Xs, ._22(X, Ys))
if_ss_2_in_1_ag3(Xs, Ys, perm_2_out_ag2(Xs, Ys)) -> if_ss_2_in_2_ag3(Xs, Ys, ordered_1_in_g1(Ys))
ordered_1_in_g1([]_0) -> ordered_1_out_g1([]_0)
ordered_1_in_g1(._22(underscore, []_0)) -> ordered_1_out_g1(._22(underscore, []_0))
ordered_1_in_g1(._22(X, ._22(Y, Xs))) -> if_ordered_1_in_1_g4(X, Y, Xs, less_2_in_ag2(X, s_11(Y)))
less_2_in_ag2(0_0, s_11(underscore1)) -> less_2_out_ag2(0_0, s_11(underscore1))
less_2_in_ag2(s_11(X), s_11(Y)) -> if_less_2_in_1_ag3(X, Y, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore1)) -> less_2_out_aa2(0_0, s_11(underscore1))
less_2_in_aa2(s_11(X), s_11(Y)) -> if_less_2_in_1_aa3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_aa3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_aa2(s_11(X), s_11(Y))
if_less_2_in_1_ag3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ag2(s_11(X), s_11(Y))
if_ordered_1_in_1_g4(X, Y, Xs, less_2_out_ag2(X, s_11(Y))) -> if_ordered_1_in_2_g4(X, Y, Xs, ordered_1_in_g1(._22(Y, Xs)))
if_ordered_1_in_2_g4(X, Y, Xs, ordered_1_out_g1(._22(Y, Xs))) -> ordered_1_out_g1(._22(X, ._22(Y, Xs)))
if_ss_2_in_2_ag3(Xs, Ys, ordered_1_out_g1(Ys)) -> ss_2_out_ag2(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_2_in_ag2(x1, x2)  =  ss_2_in_ag1(x2)
[]_0  =  []_0
._22(x1, x2)  =  ._21(x2)
s_11(x1)  =  s_1
0_0  =  0_0
if_ss_2_in_1_ag3(x1, x2, x3)  =  if_ss_2_in_1_ag2(x2, x3)
perm_2_in_ag2(x1, x2)  =  perm_2_in_ag1(x2)
perm_2_out_ag2(x1, x2)  =  perm_2_out_ag
if_perm_2_in_1_ag4(x1, x2, x3, x4)  =  if_perm_2_in_1_ag2(x3, x4)
app_3_in_aaa3(x1, x2, x3)  =  app_3_in_aaa
app_3_out_aaa3(x1, x2, x3)  =  app_3_out_aaa1(x1)
if_app_3_in_1_aaa5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_aaa1(x5)
if_perm_2_in_2_ag6(x1, x2, x3, x4, x5, x6)  =  if_perm_2_in_2_ag2(x3, x6)
app_3_in_gaa3(x1, x2, x3)  =  app_3_in_gaa1(x1)
app_3_out_gaa3(x1, x2, x3)  =  app_3_out_gaa
if_app_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gaa1(x5)
if_perm_2_in_3_ag5(x1, x2, x3, x4, x5)  =  if_perm_2_in_3_ag1(x5)
if_ss_2_in_2_ag3(x1, x2, x3)  =  if_ss_2_in_2_ag1(x3)
ordered_1_in_g1(x1)  =  ordered_1_in_g1(x1)
ordered_1_out_g1(x1)  =  ordered_1_out_g
if_ordered_1_in_1_g4(x1, x2, x3, x4)  =  if_ordered_1_in_1_g2(x3, x4)
less_2_in_aa2(x1, x2)  =  less_2_in_aa
less_2_out_aa2(x1, x2)  =  less_2_out_aa2(x1, x2)
less_2_in_ag2(x1, x2)  =  less_2_in_ag1(x2)
less_2_out_ag2(x1, x2)  =  less_2_out_ag1(x1)
if_less_2_in_1_ag3(x1, x2, x3)  =  if_less_2_in_1_ag1(x3)
if_less_2_in_1_aa3(x1, x2, x3)  =  if_less_2_in_1_aa1(x3)
if_ordered_1_in_2_g4(x1, x2, x3, x4)  =  if_ordered_1_in_2_g1(x4)
ss_2_out_ag2(x1, x2)  =  ss_2_out_ag
LESS_2_IN_AA2(x1, x2)  =  LESS_2_IN_AA

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
  ↳ PrologToPiTRSProof

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

LESS_2_IN_AA2(s_11(X), s_11(Y)) -> LESS_2_IN_AA2(X, Y)

R is empty.
The argument filtering Pi contains the following mapping:
s_11(x1)  =  s_1
LESS_2_IN_AA2(x1, x2)  =  LESS_2_IN_AA

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
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

LESS_2_IN_AA -> LESS_2_IN_AA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {LESS_2_IN_AA}.

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

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

ORDERED_1_IN_G1(._22(X, ._22(Y, Xs))) -> IF_ORDERED_1_IN_1_G4(X, Y, Xs, less_2_in_ag2(X, s_11(Y)))
IF_ORDERED_1_IN_1_G4(X, Y, Xs, less_2_out_ag2(X, s_11(Y))) -> ORDERED_1_IN_G1(._22(Y, Xs))

The TRS R consists of the following rules:

ss_2_in_ag2(Xs, Ys) -> if_ss_2_in_1_ag3(Xs, Ys, perm_2_in_ag2(Xs, Ys))
perm_2_in_ag2([]_0, []_0) -> perm_2_out_ag2([]_0, []_0)
perm_2_in_ag2(Xs, ._22(X, Ys)) -> if_perm_2_in_1_ag4(Xs, X, Ys, app_3_in_aaa3(X1s, ._22(X, X2s), Xs))
app_3_in_aaa3([]_0, X, X) -> app_3_out_aaa3([]_0, X, X)
app_3_in_aaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_aaa5(X, Xs, Ys, Zs, app_3_in_aaa3(Xs, Ys, Zs))
if_app_3_in_1_aaa5(X, Xs, Ys, Zs, app_3_out_aaa3(Xs, Ys, Zs)) -> app_3_out_aaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_perm_2_in_1_ag4(Xs, X, Ys, app_3_out_aaa3(X1s, ._22(X, X2s), Xs)) -> if_perm_2_in_2_ag6(Xs, X, Ys, X1s, X2s, app_3_in_gaa3(X1s, X2s, Zs))
app_3_in_gaa3([]_0, X, X) -> app_3_out_gaa3([]_0, X, X)
app_3_in_gaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_gaa5(X, Xs, Ys, Zs, app_3_in_gaa3(Xs, Ys, Zs))
if_app_3_in_1_gaa5(X, Xs, Ys, Zs, app_3_out_gaa3(Xs, Ys, Zs)) -> app_3_out_gaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_perm_2_in_2_ag6(Xs, X, Ys, X1s, X2s, app_3_out_gaa3(X1s, X2s, Zs)) -> if_perm_2_in_3_ag5(Xs, X, Ys, Zs, perm_2_in_ag2(Zs, Ys))
if_perm_2_in_3_ag5(Xs, X, Ys, Zs, perm_2_out_ag2(Zs, Ys)) -> perm_2_out_ag2(Xs, ._22(X, Ys))
if_ss_2_in_1_ag3(Xs, Ys, perm_2_out_ag2(Xs, Ys)) -> if_ss_2_in_2_ag3(Xs, Ys, ordered_1_in_g1(Ys))
ordered_1_in_g1([]_0) -> ordered_1_out_g1([]_0)
ordered_1_in_g1(._22(underscore, []_0)) -> ordered_1_out_g1(._22(underscore, []_0))
ordered_1_in_g1(._22(X, ._22(Y, Xs))) -> if_ordered_1_in_1_g4(X, Y, Xs, less_2_in_ag2(X, s_11(Y)))
less_2_in_ag2(0_0, s_11(underscore1)) -> less_2_out_ag2(0_0, s_11(underscore1))
less_2_in_ag2(s_11(X), s_11(Y)) -> if_less_2_in_1_ag3(X, Y, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore1)) -> less_2_out_aa2(0_0, s_11(underscore1))
less_2_in_aa2(s_11(X), s_11(Y)) -> if_less_2_in_1_aa3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_aa3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_aa2(s_11(X), s_11(Y))
if_less_2_in_1_ag3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ag2(s_11(X), s_11(Y))
if_ordered_1_in_1_g4(X, Y, Xs, less_2_out_ag2(X, s_11(Y))) -> if_ordered_1_in_2_g4(X, Y, Xs, ordered_1_in_g1(._22(Y, Xs)))
if_ordered_1_in_2_g4(X, Y, Xs, ordered_1_out_g1(._22(Y, Xs))) -> ordered_1_out_g1(._22(X, ._22(Y, Xs)))
if_ss_2_in_2_ag3(Xs, Ys, ordered_1_out_g1(Ys)) -> ss_2_out_ag2(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_2_in_ag2(x1, x2)  =  ss_2_in_ag1(x2)
[]_0  =  []_0
._22(x1, x2)  =  ._21(x2)
s_11(x1)  =  s_1
0_0  =  0_0
if_ss_2_in_1_ag3(x1, x2, x3)  =  if_ss_2_in_1_ag2(x2, x3)
perm_2_in_ag2(x1, x2)  =  perm_2_in_ag1(x2)
perm_2_out_ag2(x1, x2)  =  perm_2_out_ag
if_perm_2_in_1_ag4(x1, x2, x3, x4)  =  if_perm_2_in_1_ag2(x3, x4)
app_3_in_aaa3(x1, x2, x3)  =  app_3_in_aaa
app_3_out_aaa3(x1, x2, x3)  =  app_3_out_aaa1(x1)
if_app_3_in_1_aaa5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_aaa1(x5)
if_perm_2_in_2_ag6(x1, x2, x3, x4, x5, x6)  =  if_perm_2_in_2_ag2(x3, x6)
app_3_in_gaa3(x1, x2, x3)  =  app_3_in_gaa1(x1)
app_3_out_gaa3(x1, x2, x3)  =  app_3_out_gaa
if_app_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gaa1(x5)
if_perm_2_in_3_ag5(x1, x2, x3, x4, x5)  =  if_perm_2_in_3_ag1(x5)
if_ss_2_in_2_ag3(x1, x2, x3)  =  if_ss_2_in_2_ag1(x3)
ordered_1_in_g1(x1)  =  ordered_1_in_g1(x1)
ordered_1_out_g1(x1)  =  ordered_1_out_g
if_ordered_1_in_1_g4(x1, x2, x3, x4)  =  if_ordered_1_in_1_g2(x3, x4)
less_2_in_aa2(x1, x2)  =  less_2_in_aa
less_2_out_aa2(x1, x2)  =  less_2_out_aa2(x1, x2)
less_2_in_ag2(x1, x2)  =  less_2_in_ag1(x2)
less_2_out_ag2(x1, x2)  =  less_2_out_ag1(x1)
if_less_2_in_1_ag3(x1, x2, x3)  =  if_less_2_in_1_ag1(x3)
if_less_2_in_1_aa3(x1, x2, x3)  =  if_less_2_in_1_aa1(x3)
if_ordered_1_in_2_g4(x1, x2, x3, x4)  =  if_ordered_1_in_2_g1(x4)
ss_2_out_ag2(x1, x2)  =  ss_2_out_ag
IF_ORDERED_1_IN_1_G4(x1, x2, x3, x4)  =  IF_ORDERED_1_IN_1_G2(x3, x4)
ORDERED_1_IN_G1(x1)  =  ORDERED_1_IN_G1(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
                ↳ UsableRulesProof
PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

ORDERED_1_IN_G1(._22(X, ._22(Y, Xs))) -> IF_ORDERED_1_IN_1_G4(X, Y, Xs, less_2_in_ag2(X, s_11(Y)))
IF_ORDERED_1_IN_1_G4(X, Y, Xs, less_2_out_ag2(X, s_11(Y))) -> ORDERED_1_IN_G1(._22(Y, Xs))

The TRS R consists of the following rules:

less_2_in_ag2(0_0, s_11(underscore1)) -> less_2_out_ag2(0_0, s_11(underscore1))
less_2_in_ag2(s_11(X), s_11(Y)) -> if_less_2_in_1_ag3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_ag3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ag2(s_11(X), s_11(Y))
less_2_in_aa2(0_0, s_11(underscore1)) -> less_2_out_aa2(0_0, s_11(underscore1))
less_2_in_aa2(s_11(X), s_11(Y)) -> if_less_2_in_1_aa3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_aa3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_aa2(s_11(X), s_11(Y))

The argument filtering Pi contains the following mapping:
._22(x1, x2)  =  ._21(x2)
s_11(x1)  =  s_1
0_0  =  0_0
less_2_in_aa2(x1, x2)  =  less_2_in_aa
less_2_out_aa2(x1, x2)  =  less_2_out_aa2(x1, x2)
less_2_in_ag2(x1, x2)  =  less_2_in_ag1(x2)
less_2_out_ag2(x1, x2)  =  less_2_out_ag1(x1)
if_less_2_in_1_ag3(x1, x2, x3)  =  if_less_2_in_1_ag1(x3)
if_less_2_in_1_aa3(x1, x2, x3)  =  if_less_2_in_1_aa1(x3)
IF_ORDERED_1_IN_1_G4(x1, x2, x3, x4)  =  IF_ORDERED_1_IN_1_G2(x3, x4)
ORDERED_1_IN_G1(x1)  =  ORDERED_1_IN_G1(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
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
QDP
                        ↳ QDPPoloProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

ORDERED_1_IN_G1(._21(._21(Xs))) -> IF_ORDERED_1_IN_1_G2(Xs, less_2_in_ag1(s_1))
IF_ORDERED_1_IN_1_G2(Xs, less_2_out_ag1(X)) -> ORDERED_1_IN_G1(._21(Xs))

The TRS R consists of the following rules:

less_2_in_ag1(s_1) -> less_2_out_ag1(0_0)
less_2_in_ag1(s_1) -> if_less_2_in_1_ag1(less_2_in_aa)
if_less_2_in_1_ag1(less_2_out_aa2(X, Y)) -> less_2_out_ag1(s_1)
less_2_in_aa -> less_2_out_aa2(0_0, s_1)
less_2_in_aa -> if_less_2_in_1_aa1(less_2_in_aa)
if_less_2_in_1_aa1(less_2_out_aa2(X, Y)) -> less_2_out_aa2(s_1, s_1)

The set Q consists of the following terms:

less_2_in_ag1(x0)
if_less_2_in_1_ag1(x0)
less_2_in_aa
if_less_2_in_1_aa1(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {IF_ORDERED_1_IN_1_G2, ORDERED_1_IN_G1}.
By using a polynomial ordering, the following set of Dependency Pairs of this DP problem can be strictly oriented.

ORDERED_1_IN_G1(._21(._21(Xs))) -> IF_ORDERED_1_IN_1_G2(Xs, less_2_in_ag1(s_1))
The remaining Dependency Pairs were at least non-strictly be oriented.

IF_ORDERED_1_IN_1_G2(Xs, less_2_out_ag1(X)) -> ORDERED_1_IN_G1(._21(Xs))
With the implicit AFS there is no usable rule.

Used ordering: POLO with Polynomial interpretation:


POL(0_0) = 0   
POL(less_2_in_ag1(x1)) = 0   
POL(less_2_in_aa) = 0   
POL(ORDERED_1_IN_G1(x1)) = x1   
POL(less_2_out_aa2(x1, x2)) = 0   
POL(IF_ORDERED_1_IN_1_G2(x1, x2)) = 1 + x1   
POL(less_2_out_ag1(x1)) = 0   
POL(if_less_2_in_1_aa1(x1)) = 0   
POL(s_1) = 0   
POL(if_less_2_in_1_ag1(x1)) = 0   
POL(._21(x1)) = 1 + x1   



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

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

IF_ORDERED_1_IN_1_G2(Xs, less_2_out_ag1(X)) -> ORDERED_1_IN_G1(._21(Xs))

The TRS R consists of the following rules:

less_2_in_ag1(s_1) -> less_2_out_ag1(0_0)
less_2_in_ag1(s_1) -> if_less_2_in_1_ag1(less_2_in_aa)
if_less_2_in_1_ag1(less_2_out_aa2(X, Y)) -> less_2_out_ag1(s_1)
less_2_in_aa -> less_2_out_aa2(0_0, s_1)
less_2_in_aa -> if_less_2_in_1_aa1(less_2_in_aa)
if_less_2_in_1_aa1(less_2_out_aa2(X, Y)) -> less_2_out_aa2(s_1, s_1)

The set Q consists of the following terms:

less_2_in_ag1(x0)
if_less_2_in_1_ag1(x0)
less_2_in_aa
if_less_2_in_1_aa1(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {ORDERED_1_IN_G1, IF_ORDERED_1_IN_1_G2}.
The approximation of the Dependency Graph contains 0 SCCs with 1 less node.

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

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

APP_3_IN_GAA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APP_3_IN_GAA3(Xs, Ys, Zs)

The TRS R consists of the following rules:

ss_2_in_ag2(Xs, Ys) -> if_ss_2_in_1_ag3(Xs, Ys, perm_2_in_ag2(Xs, Ys))
perm_2_in_ag2([]_0, []_0) -> perm_2_out_ag2([]_0, []_0)
perm_2_in_ag2(Xs, ._22(X, Ys)) -> if_perm_2_in_1_ag4(Xs, X, Ys, app_3_in_aaa3(X1s, ._22(X, X2s), Xs))
app_3_in_aaa3([]_0, X, X) -> app_3_out_aaa3([]_0, X, X)
app_3_in_aaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_aaa5(X, Xs, Ys, Zs, app_3_in_aaa3(Xs, Ys, Zs))
if_app_3_in_1_aaa5(X, Xs, Ys, Zs, app_3_out_aaa3(Xs, Ys, Zs)) -> app_3_out_aaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_perm_2_in_1_ag4(Xs, X, Ys, app_3_out_aaa3(X1s, ._22(X, X2s), Xs)) -> if_perm_2_in_2_ag6(Xs, X, Ys, X1s, X2s, app_3_in_gaa3(X1s, X2s, Zs))
app_3_in_gaa3([]_0, X, X) -> app_3_out_gaa3([]_0, X, X)
app_3_in_gaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_gaa5(X, Xs, Ys, Zs, app_3_in_gaa3(Xs, Ys, Zs))
if_app_3_in_1_gaa5(X, Xs, Ys, Zs, app_3_out_gaa3(Xs, Ys, Zs)) -> app_3_out_gaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_perm_2_in_2_ag6(Xs, X, Ys, X1s, X2s, app_3_out_gaa3(X1s, X2s, Zs)) -> if_perm_2_in_3_ag5(Xs, X, Ys, Zs, perm_2_in_ag2(Zs, Ys))
if_perm_2_in_3_ag5(Xs, X, Ys, Zs, perm_2_out_ag2(Zs, Ys)) -> perm_2_out_ag2(Xs, ._22(X, Ys))
if_ss_2_in_1_ag3(Xs, Ys, perm_2_out_ag2(Xs, Ys)) -> if_ss_2_in_2_ag3(Xs, Ys, ordered_1_in_g1(Ys))
ordered_1_in_g1([]_0) -> ordered_1_out_g1([]_0)
ordered_1_in_g1(._22(underscore, []_0)) -> ordered_1_out_g1(._22(underscore, []_0))
ordered_1_in_g1(._22(X, ._22(Y, Xs))) -> if_ordered_1_in_1_g4(X, Y, Xs, less_2_in_ag2(X, s_11(Y)))
less_2_in_ag2(0_0, s_11(underscore1)) -> less_2_out_ag2(0_0, s_11(underscore1))
less_2_in_ag2(s_11(X), s_11(Y)) -> if_less_2_in_1_ag3(X, Y, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore1)) -> less_2_out_aa2(0_0, s_11(underscore1))
less_2_in_aa2(s_11(X), s_11(Y)) -> if_less_2_in_1_aa3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_aa3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_aa2(s_11(X), s_11(Y))
if_less_2_in_1_ag3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ag2(s_11(X), s_11(Y))
if_ordered_1_in_1_g4(X, Y, Xs, less_2_out_ag2(X, s_11(Y))) -> if_ordered_1_in_2_g4(X, Y, Xs, ordered_1_in_g1(._22(Y, Xs)))
if_ordered_1_in_2_g4(X, Y, Xs, ordered_1_out_g1(._22(Y, Xs))) -> ordered_1_out_g1(._22(X, ._22(Y, Xs)))
if_ss_2_in_2_ag3(Xs, Ys, ordered_1_out_g1(Ys)) -> ss_2_out_ag2(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_2_in_ag2(x1, x2)  =  ss_2_in_ag1(x2)
[]_0  =  []_0
._22(x1, x2)  =  ._21(x2)
s_11(x1)  =  s_1
0_0  =  0_0
if_ss_2_in_1_ag3(x1, x2, x3)  =  if_ss_2_in_1_ag2(x2, x3)
perm_2_in_ag2(x1, x2)  =  perm_2_in_ag1(x2)
perm_2_out_ag2(x1, x2)  =  perm_2_out_ag
if_perm_2_in_1_ag4(x1, x2, x3, x4)  =  if_perm_2_in_1_ag2(x3, x4)
app_3_in_aaa3(x1, x2, x3)  =  app_3_in_aaa
app_3_out_aaa3(x1, x2, x3)  =  app_3_out_aaa1(x1)
if_app_3_in_1_aaa5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_aaa1(x5)
if_perm_2_in_2_ag6(x1, x2, x3, x4, x5, x6)  =  if_perm_2_in_2_ag2(x3, x6)
app_3_in_gaa3(x1, x2, x3)  =  app_3_in_gaa1(x1)
app_3_out_gaa3(x1, x2, x3)  =  app_3_out_gaa
if_app_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gaa1(x5)
if_perm_2_in_3_ag5(x1, x2, x3, x4, x5)  =  if_perm_2_in_3_ag1(x5)
if_ss_2_in_2_ag3(x1, x2, x3)  =  if_ss_2_in_2_ag1(x3)
ordered_1_in_g1(x1)  =  ordered_1_in_g1(x1)
ordered_1_out_g1(x1)  =  ordered_1_out_g
if_ordered_1_in_1_g4(x1, x2, x3, x4)  =  if_ordered_1_in_1_g2(x3, x4)
less_2_in_aa2(x1, x2)  =  less_2_in_aa
less_2_out_aa2(x1, x2)  =  less_2_out_aa2(x1, x2)
less_2_in_ag2(x1, x2)  =  less_2_in_ag1(x2)
less_2_out_ag2(x1, x2)  =  less_2_out_ag1(x1)
if_less_2_in_1_ag3(x1, x2, x3)  =  if_less_2_in_1_ag1(x3)
if_less_2_in_1_aa3(x1, x2, x3)  =  if_less_2_in_1_aa1(x3)
if_ordered_1_in_2_g4(x1, x2, x3, x4)  =  if_ordered_1_in_2_g1(x4)
ss_2_out_ag2(x1, x2)  =  ss_2_out_ag
APP_3_IN_GAA3(x1, x2, x3)  =  APP_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
                ↳ UsableRulesProof
PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

APP_3_IN_GAA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APP_3_IN_GAA3(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
._22(x1, x2)  =  ._21(x2)
APP_3_IN_GAA3(x1, x2, x3)  =  APP_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
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

APP_3_IN_GAA1(._21(Xs)) -> APP_3_IN_GAA1(Xs)

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_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
                ↳ UsableRulesProof
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

APP_3_IN_AAA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APP_3_IN_AAA3(Xs, Ys, Zs)

The TRS R consists of the following rules:

ss_2_in_ag2(Xs, Ys) -> if_ss_2_in_1_ag3(Xs, Ys, perm_2_in_ag2(Xs, Ys))
perm_2_in_ag2([]_0, []_0) -> perm_2_out_ag2([]_0, []_0)
perm_2_in_ag2(Xs, ._22(X, Ys)) -> if_perm_2_in_1_ag4(Xs, X, Ys, app_3_in_aaa3(X1s, ._22(X, X2s), Xs))
app_3_in_aaa3([]_0, X, X) -> app_3_out_aaa3([]_0, X, X)
app_3_in_aaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_aaa5(X, Xs, Ys, Zs, app_3_in_aaa3(Xs, Ys, Zs))
if_app_3_in_1_aaa5(X, Xs, Ys, Zs, app_3_out_aaa3(Xs, Ys, Zs)) -> app_3_out_aaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_perm_2_in_1_ag4(Xs, X, Ys, app_3_out_aaa3(X1s, ._22(X, X2s), Xs)) -> if_perm_2_in_2_ag6(Xs, X, Ys, X1s, X2s, app_3_in_gaa3(X1s, X2s, Zs))
app_3_in_gaa3([]_0, X, X) -> app_3_out_gaa3([]_0, X, X)
app_3_in_gaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_gaa5(X, Xs, Ys, Zs, app_3_in_gaa3(Xs, Ys, Zs))
if_app_3_in_1_gaa5(X, Xs, Ys, Zs, app_3_out_gaa3(Xs, Ys, Zs)) -> app_3_out_gaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_perm_2_in_2_ag6(Xs, X, Ys, X1s, X2s, app_3_out_gaa3(X1s, X2s, Zs)) -> if_perm_2_in_3_ag5(Xs, X, Ys, Zs, perm_2_in_ag2(Zs, Ys))
if_perm_2_in_3_ag5(Xs, X, Ys, Zs, perm_2_out_ag2(Zs, Ys)) -> perm_2_out_ag2(Xs, ._22(X, Ys))
if_ss_2_in_1_ag3(Xs, Ys, perm_2_out_ag2(Xs, Ys)) -> if_ss_2_in_2_ag3(Xs, Ys, ordered_1_in_g1(Ys))
ordered_1_in_g1([]_0) -> ordered_1_out_g1([]_0)
ordered_1_in_g1(._22(underscore, []_0)) -> ordered_1_out_g1(._22(underscore, []_0))
ordered_1_in_g1(._22(X, ._22(Y, Xs))) -> if_ordered_1_in_1_g4(X, Y, Xs, less_2_in_ag2(X, s_11(Y)))
less_2_in_ag2(0_0, s_11(underscore1)) -> less_2_out_ag2(0_0, s_11(underscore1))
less_2_in_ag2(s_11(X), s_11(Y)) -> if_less_2_in_1_ag3(X, Y, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore1)) -> less_2_out_aa2(0_0, s_11(underscore1))
less_2_in_aa2(s_11(X), s_11(Y)) -> if_less_2_in_1_aa3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_aa3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_aa2(s_11(X), s_11(Y))
if_less_2_in_1_ag3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ag2(s_11(X), s_11(Y))
if_ordered_1_in_1_g4(X, Y, Xs, less_2_out_ag2(X, s_11(Y))) -> if_ordered_1_in_2_g4(X, Y, Xs, ordered_1_in_g1(._22(Y, Xs)))
if_ordered_1_in_2_g4(X, Y, Xs, ordered_1_out_g1(._22(Y, Xs))) -> ordered_1_out_g1(._22(X, ._22(Y, Xs)))
if_ss_2_in_2_ag3(Xs, Ys, ordered_1_out_g1(Ys)) -> ss_2_out_ag2(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_2_in_ag2(x1, x2)  =  ss_2_in_ag1(x2)
[]_0  =  []_0
._22(x1, x2)  =  ._21(x2)
s_11(x1)  =  s_1
0_0  =  0_0
if_ss_2_in_1_ag3(x1, x2, x3)  =  if_ss_2_in_1_ag2(x2, x3)
perm_2_in_ag2(x1, x2)  =  perm_2_in_ag1(x2)
perm_2_out_ag2(x1, x2)  =  perm_2_out_ag
if_perm_2_in_1_ag4(x1, x2, x3, x4)  =  if_perm_2_in_1_ag2(x3, x4)
app_3_in_aaa3(x1, x2, x3)  =  app_3_in_aaa
app_3_out_aaa3(x1, x2, x3)  =  app_3_out_aaa1(x1)
if_app_3_in_1_aaa5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_aaa1(x5)
if_perm_2_in_2_ag6(x1, x2, x3, x4, x5, x6)  =  if_perm_2_in_2_ag2(x3, x6)
app_3_in_gaa3(x1, x2, x3)  =  app_3_in_gaa1(x1)
app_3_out_gaa3(x1, x2, x3)  =  app_3_out_gaa
if_app_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gaa1(x5)
if_perm_2_in_3_ag5(x1, x2, x3, x4, x5)  =  if_perm_2_in_3_ag1(x5)
if_ss_2_in_2_ag3(x1, x2, x3)  =  if_ss_2_in_2_ag1(x3)
ordered_1_in_g1(x1)  =  ordered_1_in_g1(x1)
ordered_1_out_g1(x1)  =  ordered_1_out_g
if_ordered_1_in_1_g4(x1, x2, x3, x4)  =  if_ordered_1_in_1_g2(x3, x4)
less_2_in_aa2(x1, x2)  =  less_2_in_aa
less_2_out_aa2(x1, x2)  =  less_2_out_aa2(x1, x2)
less_2_in_ag2(x1, x2)  =  less_2_in_ag1(x2)
less_2_out_ag2(x1, x2)  =  less_2_out_ag1(x1)
if_less_2_in_1_ag3(x1, x2, x3)  =  if_less_2_in_1_ag1(x3)
if_less_2_in_1_aa3(x1, x2, x3)  =  if_less_2_in_1_aa1(x3)
if_ordered_1_in_2_g4(x1, x2, x3, x4)  =  if_ordered_1_in_2_g1(x4)
ss_2_out_ag2(x1, x2)  =  ss_2_out_ag
APP_3_IN_AAA3(x1, x2, x3)  =  APP_3_IN_AAA

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
  ↳ PrologToPiTRSProof

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

APP_3_IN_AAA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APP_3_IN_AAA3(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
._22(x1, x2)  =  ._21(x2)
APP_3_IN_AAA3(x1, x2, x3)  =  APP_3_IN_AAA

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
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

APP_3_IN_AAA -> APP_3_IN_AAA

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_AAA}.

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

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

PERM_2_IN_AG2(Xs, ._22(X, Ys)) -> IF_PERM_2_IN_1_AG4(Xs, X, Ys, app_3_in_aaa3(X1s, ._22(X, X2s), Xs))
IF_PERM_2_IN_1_AG4(Xs, X, Ys, app_3_out_aaa3(X1s, ._22(X, X2s), Xs)) -> IF_PERM_2_IN_2_AG6(Xs, X, Ys, X1s, X2s, app_3_in_gaa3(X1s, X2s, Zs))
IF_PERM_2_IN_2_AG6(Xs, X, Ys, X1s, X2s, app_3_out_gaa3(X1s, X2s, Zs)) -> PERM_2_IN_AG2(Zs, Ys)

The TRS R consists of the following rules:

ss_2_in_ag2(Xs, Ys) -> if_ss_2_in_1_ag3(Xs, Ys, perm_2_in_ag2(Xs, Ys))
perm_2_in_ag2([]_0, []_0) -> perm_2_out_ag2([]_0, []_0)
perm_2_in_ag2(Xs, ._22(X, Ys)) -> if_perm_2_in_1_ag4(Xs, X, Ys, app_3_in_aaa3(X1s, ._22(X, X2s), Xs))
app_3_in_aaa3([]_0, X, X) -> app_3_out_aaa3([]_0, X, X)
app_3_in_aaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_aaa5(X, Xs, Ys, Zs, app_3_in_aaa3(Xs, Ys, Zs))
if_app_3_in_1_aaa5(X, Xs, Ys, Zs, app_3_out_aaa3(Xs, Ys, Zs)) -> app_3_out_aaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_perm_2_in_1_ag4(Xs, X, Ys, app_3_out_aaa3(X1s, ._22(X, X2s), Xs)) -> if_perm_2_in_2_ag6(Xs, X, Ys, X1s, X2s, app_3_in_gaa3(X1s, X2s, Zs))
app_3_in_gaa3([]_0, X, X) -> app_3_out_gaa3([]_0, X, X)
app_3_in_gaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_gaa5(X, Xs, Ys, Zs, app_3_in_gaa3(Xs, Ys, Zs))
if_app_3_in_1_gaa5(X, Xs, Ys, Zs, app_3_out_gaa3(Xs, Ys, Zs)) -> app_3_out_gaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_perm_2_in_2_ag6(Xs, X, Ys, X1s, X2s, app_3_out_gaa3(X1s, X2s, Zs)) -> if_perm_2_in_3_ag5(Xs, X, Ys, Zs, perm_2_in_ag2(Zs, Ys))
if_perm_2_in_3_ag5(Xs, X, Ys, Zs, perm_2_out_ag2(Zs, Ys)) -> perm_2_out_ag2(Xs, ._22(X, Ys))
if_ss_2_in_1_ag3(Xs, Ys, perm_2_out_ag2(Xs, Ys)) -> if_ss_2_in_2_ag3(Xs, Ys, ordered_1_in_g1(Ys))
ordered_1_in_g1([]_0) -> ordered_1_out_g1([]_0)
ordered_1_in_g1(._22(underscore, []_0)) -> ordered_1_out_g1(._22(underscore, []_0))
ordered_1_in_g1(._22(X, ._22(Y, Xs))) -> if_ordered_1_in_1_g4(X, Y, Xs, less_2_in_ag2(X, s_11(Y)))
less_2_in_ag2(0_0, s_11(underscore1)) -> less_2_out_ag2(0_0, s_11(underscore1))
less_2_in_ag2(s_11(X), s_11(Y)) -> if_less_2_in_1_ag3(X, Y, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore1)) -> less_2_out_aa2(0_0, s_11(underscore1))
less_2_in_aa2(s_11(X), s_11(Y)) -> if_less_2_in_1_aa3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_aa3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_aa2(s_11(X), s_11(Y))
if_less_2_in_1_ag3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ag2(s_11(X), s_11(Y))
if_ordered_1_in_1_g4(X, Y, Xs, less_2_out_ag2(X, s_11(Y))) -> if_ordered_1_in_2_g4(X, Y, Xs, ordered_1_in_g1(._22(Y, Xs)))
if_ordered_1_in_2_g4(X, Y, Xs, ordered_1_out_g1(._22(Y, Xs))) -> ordered_1_out_g1(._22(X, ._22(Y, Xs)))
if_ss_2_in_2_ag3(Xs, Ys, ordered_1_out_g1(Ys)) -> ss_2_out_ag2(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_2_in_ag2(x1, x2)  =  ss_2_in_ag1(x2)
[]_0  =  []_0
._22(x1, x2)  =  ._21(x2)
s_11(x1)  =  s_1
0_0  =  0_0
if_ss_2_in_1_ag3(x1, x2, x3)  =  if_ss_2_in_1_ag2(x2, x3)
perm_2_in_ag2(x1, x2)  =  perm_2_in_ag1(x2)
perm_2_out_ag2(x1, x2)  =  perm_2_out_ag
if_perm_2_in_1_ag4(x1, x2, x3, x4)  =  if_perm_2_in_1_ag2(x3, x4)
app_3_in_aaa3(x1, x2, x3)  =  app_3_in_aaa
app_3_out_aaa3(x1, x2, x3)  =  app_3_out_aaa1(x1)
if_app_3_in_1_aaa5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_aaa1(x5)
if_perm_2_in_2_ag6(x1, x2, x3, x4, x5, x6)  =  if_perm_2_in_2_ag2(x3, x6)
app_3_in_gaa3(x1, x2, x3)  =  app_3_in_gaa1(x1)
app_3_out_gaa3(x1, x2, x3)  =  app_3_out_gaa
if_app_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gaa1(x5)
if_perm_2_in_3_ag5(x1, x2, x3, x4, x5)  =  if_perm_2_in_3_ag1(x5)
if_ss_2_in_2_ag3(x1, x2, x3)  =  if_ss_2_in_2_ag1(x3)
ordered_1_in_g1(x1)  =  ordered_1_in_g1(x1)
ordered_1_out_g1(x1)  =  ordered_1_out_g
if_ordered_1_in_1_g4(x1, x2, x3, x4)  =  if_ordered_1_in_1_g2(x3, x4)
less_2_in_aa2(x1, x2)  =  less_2_in_aa
less_2_out_aa2(x1, x2)  =  less_2_out_aa2(x1, x2)
less_2_in_ag2(x1, x2)  =  less_2_in_ag1(x2)
less_2_out_ag2(x1, x2)  =  less_2_out_ag1(x1)
if_less_2_in_1_ag3(x1, x2, x3)  =  if_less_2_in_1_ag1(x3)
if_less_2_in_1_aa3(x1, x2, x3)  =  if_less_2_in_1_aa1(x3)
if_ordered_1_in_2_g4(x1, x2, x3, x4)  =  if_ordered_1_in_2_g1(x4)
ss_2_out_ag2(x1, x2)  =  ss_2_out_ag
PERM_2_IN_AG2(x1, x2)  =  PERM_2_IN_AG1(x2)
IF_PERM_2_IN_1_AG4(x1, x2, x3, x4)  =  IF_PERM_2_IN_1_AG2(x3, x4)
IF_PERM_2_IN_2_AG6(x1, x2, x3, x4, x5, x6)  =  IF_PERM_2_IN_2_AG2(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
              ↳ PiDP
                ↳ UsableRulesProof
PiDP
                    ↳ PiDPToQDPProof
  ↳ PrologToPiTRSProof

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

PERM_2_IN_AG2(Xs, ._22(X, Ys)) -> IF_PERM_2_IN_1_AG4(Xs, X, Ys, app_3_in_aaa3(X1s, ._22(X, X2s), Xs))
IF_PERM_2_IN_1_AG4(Xs, X, Ys, app_3_out_aaa3(X1s, ._22(X, X2s), Xs)) -> IF_PERM_2_IN_2_AG6(Xs, X, Ys, X1s, X2s, app_3_in_gaa3(X1s, X2s, Zs))
IF_PERM_2_IN_2_AG6(Xs, X, Ys, X1s, X2s, app_3_out_gaa3(X1s, X2s, Zs)) -> PERM_2_IN_AG2(Zs, Ys)

The TRS R consists of the following rules:

app_3_in_aaa3([]_0, X, X) -> app_3_out_aaa3([]_0, X, X)
app_3_in_aaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_aaa5(X, Xs, Ys, Zs, app_3_in_aaa3(Xs, Ys, Zs))
app_3_in_gaa3([]_0, X, X) -> app_3_out_gaa3([]_0, X, X)
app_3_in_gaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_gaa5(X, Xs, Ys, Zs, app_3_in_gaa3(Xs, Ys, Zs))
if_app_3_in_1_aaa5(X, Xs, Ys, Zs, app_3_out_aaa3(Xs, Ys, Zs)) -> app_3_out_aaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_app_3_in_1_gaa5(X, Xs, Ys, Zs, app_3_out_gaa3(Xs, Ys, Zs)) -> app_3_out_gaa3(._22(X, Xs), Ys, ._22(X, Zs))

The argument filtering Pi contains the following mapping:
[]_0  =  []_0
._22(x1, x2)  =  ._21(x2)
app_3_in_aaa3(x1, x2, x3)  =  app_3_in_aaa
app_3_out_aaa3(x1, x2, x3)  =  app_3_out_aaa1(x1)
if_app_3_in_1_aaa5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_aaa1(x5)
app_3_in_gaa3(x1, x2, x3)  =  app_3_in_gaa1(x1)
app_3_out_gaa3(x1, x2, x3)  =  app_3_out_gaa
if_app_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gaa1(x5)
PERM_2_IN_AG2(x1, x2)  =  PERM_2_IN_AG1(x2)
IF_PERM_2_IN_1_AG4(x1, x2, x3, x4)  =  IF_PERM_2_IN_1_AG2(x3, x4)
IF_PERM_2_IN_2_AG6(x1, x2, x3, x4, x5, x6)  =  IF_PERM_2_IN_2_AG2(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
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
QDP
                        ↳ QDPSizeChangeProof
  ↳ PrologToPiTRSProof

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

PERM_2_IN_AG1(._21(Ys)) -> IF_PERM_2_IN_1_AG2(Ys, app_3_in_aaa)
IF_PERM_2_IN_1_AG2(Ys, app_3_out_aaa1(X1s)) -> IF_PERM_2_IN_2_AG2(Ys, app_3_in_gaa1(X1s))
IF_PERM_2_IN_2_AG2(Ys, app_3_out_gaa) -> PERM_2_IN_AG1(Ys)

The TRS R consists of the following rules:

app_3_in_aaa -> app_3_out_aaa1([]_0)
app_3_in_aaa -> if_app_3_in_1_aaa1(app_3_in_aaa)
app_3_in_gaa1([]_0) -> app_3_out_gaa
app_3_in_gaa1(._21(Xs)) -> if_app_3_in_1_gaa1(app_3_in_gaa1(Xs))
if_app_3_in_1_aaa1(app_3_out_aaa1(Xs)) -> app_3_out_aaa1(._21(Xs))
if_app_3_in_1_gaa1(app_3_out_gaa) -> app_3_out_gaa

The set Q consists of the following terms:

app_3_in_aaa
app_3_in_gaa1(x0)
if_app_3_in_1_aaa1(x0)
if_app_3_in_1_gaa1(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {IF_PERM_2_IN_1_AG2, PERM_2_IN_AG1, IF_PERM_2_IN_2_AG2}.
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:


With regard to the inferred argument filtering the predicates were used in the following modes:
ss2: (f,b)
perm2: (f,b)
app3: (f,f,f) (b,f,f)
ordered1: (b)
less2: (f,b) (f,f)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

ss_2_in_ag2(Xs, Ys) -> if_ss_2_in_1_ag3(Xs, Ys, perm_2_in_ag2(Xs, Ys))
perm_2_in_ag2([]_0, []_0) -> perm_2_out_ag2([]_0, []_0)
perm_2_in_ag2(Xs, ._22(X, Ys)) -> if_perm_2_in_1_ag4(Xs, X, Ys, app_3_in_aaa3(X1s, ._22(X, X2s), Xs))
app_3_in_aaa3([]_0, X, X) -> app_3_out_aaa3([]_0, X, X)
app_3_in_aaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_aaa5(X, Xs, Ys, Zs, app_3_in_aaa3(Xs, Ys, Zs))
if_app_3_in_1_aaa5(X, Xs, Ys, Zs, app_3_out_aaa3(Xs, Ys, Zs)) -> app_3_out_aaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_perm_2_in_1_ag4(Xs, X, Ys, app_3_out_aaa3(X1s, ._22(X, X2s), Xs)) -> if_perm_2_in_2_ag6(Xs, X, Ys, X1s, X2s, app_3_in_gaa3(X1s, X2s, Zs))
app_3_in_gaa3([]_0, X, X) -> app_3_out_gaa3([]_0, X, X)
app_3_in_gaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_gaa5(X, Xs, Ys, Zs, app_3_in_gaa3(Xs, Ys, Zs))
if_app_3_in_1_gaa5(X, Xs, Ys, Zs, app_3_out_gaa3(Xs, Ys, Zs)) -> app_3_out_gaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_perm_2_in_2_ag6(Xs, X, Ys, X1s, X2s, app_3_out_gaa3(X1s, X2s, Zs)) -> if_perm_2_in_3_ag5(Xs, X, Ys, Zs, perm_2_in_ag2(Zs, Ys))
if_perm_2_in_3_ag5(Xs, X, Ys, Zs, perm_2_out_ag2(Zs, Ys)) -> perm_2_out_ag2(Xs, ._22(X, Ys))
if_ss_2_in_1_ag3(Xs, Ys, perm_2_out_ag2(Xs, Ys)) -> if_ss_2_in_2_ag3(Xs, Ys, ordered_1_in_g1(Ys))
ordered_1_in_g1([]_0) -> ordered_1_out_g1([]_0)
ordered_1_in_g1(._22(underscore, []_0)) -> ordered_1_out_g1(._22(underscore, []_0))
ordered_1_in_g1(._22(X, ._22(Y, Xs))) -> if_ordered_1_in_1_g4(X, Y, Xs, less_2_in_ag2(X, s_11(Y)))
less_2_in_ag2(0_0, s_11(underscore1)) -> less_2_out_ag2(0_0, s_11(underscore1))
less_2_in_ag2(s_11(X), s_11(Y)) -> if_less_2_in_1_ag3(X, Y, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore1)) -> less_2_out_aa2(0_0, s_11(underscore1))
less_2_in_aa2(s_11(X), s_11(Y)) -> if_less_2_in_1_aa3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_aa3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_aa2(s_11(X), s_11(Y))
if_less_2_in_1_ag3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ag2(s_11(X), s_11(Y))
if_ordered_1_in_1_g4(X, Y, Xs, less_2_out_ag2(X, s_11(Y))) -> if_ordered_1_in_2_g4(X, Y, Xs, ordered_1_in_g1(._22(Y, Xs)))
if_ordered_1_in_2_g4(X, Y, Xs, ordered_1_out_g1(._22(Y, Xs))) -> ordered_1_out_g1(._22(X, ._22(Y, Xs)))
if_ss_2_in_2_ag3(Xs, Ys, ordered_1_out_g1(Ys)) -> ss_2_out_ag2(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_2_in_ag2(x1, x2)  =  ss_2_in_ag1(x2)
[]_0  =  []_0
._22(x1, x2)  =  ._21(x2)
s_11(x1)  =  s_1
0_0  =  0_0
if_ss_2_in_1_ag3(x1, x2, x3)  =  if_ss_2_in_1_ag2(x2, x3)
perm_2_in_ag2(x1, x2)  =  perm_2_in_ag1(x2)
perm_2_out_ag2(x1, x2)  =  perm_2_out_ag1(x2)
if_perm_2_in_1_ag4(x1, x2, x3, x4)  =  if_perm_2_in_1_ag2(x3, x4)
app_3_in_aaa3(x1, x2, x3)  =  app_3_in_aaa
app_3_out_aaa3(x1, x2, x3)  =  app_3_out_aaa1(x1)
if_app_3_in_1_aaa5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_aaa1(x5)
if_perm_2_in_2_ag6(x1, x2, x3, x4, x5, x6)  =  if_perm_2_in_2_ag2(x3, x6)
app_3_in_gaa3(x1, x2, x3)  =  app_3_in_gaa1(x1)
app_3_out_gaa3(x1, x2, x3)  =  app_3_out_gaa1(x1)
if_app_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gaa2(x2, x5)
if_perm_2_in_3_ag5(x1, x2, x3, x4, x5)  =  if_perm_2_in_3_ag2(x3, x5)
if_ss_2_in_2_ag3(x1, x2, x3)  =  if_ss_2_in_2_ag2(x2, x3)
ordered_1_in_g1(x1)  =  ordered_1_in_g1(x1)
ordered_1_out_g1(x1)  =  ordered_1_out_g1(x1)
if_ordered_1_in_1_g4(x1, x2, x3, x4)  =  if_ordered_1_in_1_g2(x3, x4)
less_2_in_aa2(x1, x2)  =  less_2_in_aa
less_2_out_aa2(x1, x2)  =  less_2_out_aa2(x1, x2)
less_2_in_ag2(x1, x2)  =  less_2_in_ag1(x2)
less_2_out_ag2(x1, x2)  =  less_2_out_ag2(x1, x2)
if_less_2_in_1_ag3(x1, x2, x3)  =  if_less_2_in_1_ag1(x3)
if_less_2_in_1_aa3(x1, x2, x3)  =  if_less_2_in_1_aa1(x3)
if_ordered_1_in_2_g4(x1, x2, x3, x4)  =  if_ordered_1_in_2_g2(x3, x4)
ss_2_out_ag2(x1, x2)  =  ss_2_out_ag1(x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of PROLOG



↳ PROLOG
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
PiTRS
      ↳ DependencyPairsProof

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

ss_2_in_ag2(Xs, Ys) -> if_ss_2_in_1_ag3(Xs, Ys, perm_2_in_ag2(Xs, Ys))
perm_2_in_ag2([]_0, []_0) -> perm_2_out_ag2([]_0, []_0)
perm_2_in_ag2(Xs, ._22(X, Ys)) -> if_perm_2_in_1_ag4(Xs, X, Ys, app_3_in_aaa3(X1s, ._22(X, X2s), Xs))
app_3_in_aaa3([]_0, X, X) -> app_3_out_aaa3([]_0, X, X)
app_3_in_aaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_aaa5(X, Xs, Ys, Zs, app_3_in_aaa3(Xs, Ys, Zs))
if_app_3_in_1_aaa5(X, Xs, Ys, Zs, app_3_out_aaa3(Xs, Ys, Zs)) -> app_3_out_aaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_perm_2_in_1_ag4(Xs, X, Ys, app_3_out_aaa3(X1s, ._22(X, X2s), Xs)) -> if_perm_2_in_2_ag6(Xs, X, Ys, X1s, X2s, app_3_in_gaa3(X1s, X2s, Zs))
app_3_in_gaa3([]_0, X, X) -> app_3_out_gaa3([]_0, X, X)
app_3_in_gaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_gaa5(X, Xs, Ys, Zs, app_3_in_gaa3(Xs, Ys, Zs))
if_app_3_in_1_gaa5(X, Xs, Ys, Zs, app_3_out_gaa3(Xs, Ys, Zs)) -> app_3_out_gaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_perm_2_in_2_ag6(Xs, X, Ys, X1s, X2s, app_3_out_gaa3(X1s, X2s, Zs)) -> if_perm_2_in_3_ag5(Xs, X, Ys, Zs, perm_2_in_ag2(Zs, Ys))
if_perm_2_in_3_ag5(Xs, X, Ys, Zs, perm_2_out_ag2(Zs, Ys)) -> perm_2_out_ag2(Xs, ._22(X, Ys))
if_ss_2_in_1_ag3(Xs, Ys, perm_2_out_ag2(Xs, Ys)) -> if_ss_2_in_2_ag3(Xs, Ys, ordered_1_in_g1(Ys))
ordered_1_in_g1([]_0) -> ordered_1_out_g1([]_0)
ordered_1_in_g1(._22(underscore, []_0)) -> ordered_1_out_g1(._22(underscore, []_0))
ordered_1_in_g1(._22(X, ._22(Y, Xs))) -> if_ordered_1_in_1_g4(X, Y, Xs, less_2_in_ag2(X, s_11(Y)))
less_2_in_ag2(0_0, s_11(underscore1)) -> less_2_out_ag2(0_0, s_11(underscore1))
less_2_in_ag2(s_11(X), s_11(Y)) -> if_less_2_in_1_ag3(X, Y, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore1)) -> less_2_out_aa2(0_0, s_11(underscore1))
less_2_in_aa2(s_11(X), s_11(Y)) -> if_less_2_in_1_aa3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_aa3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_aa2(s_11(X), s_11(Y))
if_less_2_in_1_ag3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ag2(s_11(X), s_11(Y))
if_ordered_1_in_1_g4(X, Y, Xs, less_2_out_ag2(X, s_11(Y))) -> if_ordered_1_in_2_g4(X, Y, Xs, ordered_1_in_g1(._22(Y, Xs)))
if_ordered_1_in_2_g4(X, Y, Xs, ordered_1_out_g1(._22(Y, Xs))) -> ordered_1_out_g1(._22(X, ._22(Y, Xs)))
if_ss_2_in_2_ag3(Xs, Ys, ordered_1_out_g1(Ys)) -> ss_2_out_ag2(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_2_in_ag2(x1, x2)  =  ss_2_in_ag1(x2)
[]_0  =  []_0
._22(x1, x2)  =  ._21(x2)
s_11(x1)  =  s_1
0_0  =  0_0
if_ss_2_in_1_ag3(x1, x2, x3)  =  if_ss_2_in_1_ag2(x2, x3)
perm_2_in_ag2(x1, x2)  =  perm_2_in_ag1(x2)
perm_2_out_ag2(x1, x2)  =  perm_2_out_ag1(x2)
if_perm_2_in_1_ag4(x1, x2, x3, x4)  =  if_perm_2_in_1_ag2(x3, x4)
app_3_in_aaa3(x1, x2, x3)  =  app_3_in_aaa
app_3_out_aaa3(x1, x2, x3)  =  app_3_out_aaa1(x1)
if_app_3_in_1_aaa5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_aaa1(x5)
if_perm_2_in_2_ag6(x1, x2, x3, x4, x5, x6)  =  if_perm_2_in_2_ag2(x3, x6)
app_3_in_gaa3(x1, x2, x3)  =  app_3_in_gaa1(x1)
app_3_out_gaa3(x1, x2, x3)  =  app_3_out_gaa1(x1)
if_app_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gaa2(x2, x5)
if_perm_2_in_3_ag5(x1, x2, x3, x4, x5)  =  if_perm_2_in_3_ag2(x3, x5)
if_ss_2_in_2_ag3(x1, x2, x3)  =  if_ss_2_in_2_ag2(x2, x3)
ordered_1_in_g1(x1)  =  ordered_1_in_g1(x1)
ordered_1_out_g1(x1)  =  ordered_1_out_g1(x1)
if_ordered_1_in_1_g4(x1, x2, x3, x4)  =  if_ordered_1_in_1_g2(x3, x4)
less_2_in_aa2(x1, x2)  =  less_2_in_aa
less_2_out_aa2(x1, x2)  =  less_2_out_aa2(x1, x2)
less_2_in_ag2(x1, x2)  =  less_2_in_ag1(x2)
less_2_out_ag2(x1, x2)  =  less_2_out_ag2(x1, x2)
if_less_2_in_1_ag3(x1, x2, x3)  =  if_less_2_in_1_ag1(x3)
if_less_2_in_1_aa3(x1, x2, x3)  =  if_less_2_in_1_aa1(x3)
if_ordered_1_in_2_g4(x1, x2, x3, x4)  =  if_ordered_1_in_2_g2(x3, x4)
ss_2_out_ag2(x1, x2)  =  ss_2_out_ag1(x2)


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

SS_2_IN_AG2(Xs, Ys) -> IF_SS_2_IN_1_AG3(Xs, Ys, perm_2_in_ag2(Xs, Ys))
SS_2_IN_AG2(Xs, Ys) -> PERM_2_IN_AG2(Xs, Ys)
PERM_2_IN_AG2(Xs, ._22(X, Ys)) -> IF_PERM_2_IN_1_AG4(Xs, X, Ys, app_3_in_aaa3(X1s, ._22(X, X2s), Xs))
PERM_2_IN_AG2(Xs, ._22(X, Ys)) -> APP_3_IN_AAA3(X1s, ._22(X, X2s), Xs)
APP_3_IN_AAA3(._22(X, Xs), Ys, ._22(X, Zs)) -> IF_APP_3_IN_1_AAA5(X, Xs, Ys, Zs, app_3_in_aaa3(Xs, Ys, Zs))
APP_3_IN_AAA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APP_3_IN_AAA3(Xs, Ys, Zs)
IF_PERM_2_IN_1_AG4(Xs, X, Ys, app_3_out_aaa3(X1s, ._22(X, X2s), Xs)) -> IF_PERM_2_IN_2_AG6(Xs, X, Ys, X1s, X2s, app_3_in_gaa3(X1s, X2s, Zs))
IF_PERM_2_IN_1_AG4(Xs, X, Ys, app_3_out_aaa3(X1s, ._22(X, X2s), Xs)) -> APP_3_IN_GAA3(X1s, X2s, Zs)
APP_3_IN_GAA3(._22(X, Xs), Ys, ._22(X, Zs)) -> IF_APP_3_IN_1_GAA5(X, Xs, Ys, Zs, app_3_in_gaa3(Xs, Ys, Zs))
APP_3_IN_GAA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APP_3_IN_GAA3(Xs, Ys, Zs)
IF_PERM_2_IN_2_AG6(Xs, X, Ys, X1s, X2s, app_3_out_gaa3(X1s, X2s, Zs)) -> IF_PERM_2_IN_3_AG5(Xs, X, Ys, Zs, perm_2_in_ag2(Zs, Ys))
IF_PERM_2_IN_2_AG6(Xs, X, Ys, X1s, X2s, app_3_out_gaa3(X1s, X2s, Zs)) -> PERM_2_IN_AG2(Zs, Ys)
IF_SS_2_IN_1_AG3(Xs, Ys, perm_2_out_ag2(Xs, Ys)) -> IF_SS_2_IN_2_AG3(Xs, Ys, ordered_1_in_g1(Ys))
IF_SS_2_IN_1_AG3(Xs, Ys, perm_2_out_ag2(Xs, Ys)) -> ORDERED_1_IN_G1(Ys)
ORDERED_1_IN_G1(._22(X, ._22(Y, Xs))) -> IF_ORDERED_1_IN_1_G4(X, Y, Xs, less_2_in_ag2(X, s_11(Y)))
ORDERED_1_IN_G1(._22(X, ._22(Y, Xs))) -> LESS_2_IN_AG2(X, s_11(Y))
LESS_2_IN_AG2(s_11(X), s_11(Y)) -> IF_LESS_2_IN_1_AG3(X, Y, less_2_in_aa2(X, Y))
LESS_2_IN_AG2(s_11(X), s_11(Y)) -> LESS_2_IN_AA2(X, Y)
LESS_2_IN_AA2(s_11(X), s_11(Y)) -> IF_LESS_2_IN_1_AA3(X, Y, less_2_in_aa2(X, Y))
LESS_2_IN_AA2(s_11(X), s_11(Y)) -> LESS_2_IN_AA2(X, Y)
IF_ORDERED_1_IN_1_G4(X, Y, Xs, less_2_out_ag2(X, s_11(Y))) -> IF_ORDERED_1_IN_2_G4(X, Y, Xs, ordered_1_in_g1(._22(Y, Xs)))
IF_ORDERED_1_IN_1_G4(X, Y, Xs, less_2_out_ag2(X, s_11(Y))) -> ORDERED_1_IN_G1(._22(Y, Xs))

The TRS R consists of the following rules:

ss_2_in_ag2(Xs, Ys) -> if_ss_2_in_1_ag3(Xs, Ys, perm_2_in_ag2(Xs, Ys))
perm_2_in_ag2([]_0, []_0) -> perm_2_out_ag2([]_0, []_0)
perm_2_in_ag2(Xs, ._22(X, Ys)) -> if_perm_2_in_1_ag4(Xs, X, Ys, app_3_in_aaa3(X1s, ._22(X, X2s), Xs))
app_3_in_aaa3([]_0, X, X) -> app_3_out_aaa3([]_0, X, X)
app_3_in_aaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_aaa5(X, Xs, Ys, Zs, app_3_in_aaa3(Xs, Ys, Zs))
if_app_3_in_1_aaa5(X, Xs, Ys, Zs, app_3_out_aaa3(Xs, Ys, Zs)) -> app_3_out_aaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_perm_2_in_1_ag4(Xs, X, Ys, app_3_out_aaa3(X1s, ._22(X, X2s), Xs)) -> if_perm_2_in_2_ag6(Xs, X, Ys, X1s, X2s, app_3_in_gaa3(X1s, X2s, Zs))
app_3_in_gaa3([]_0, X, X) -> app_3_out_gaa3([]_0, X, X)
app_3_in_gaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_gaa5(X, Xs, Ys, Zs, app_3_in_gaa3(Xs, Ys, Zs))
if_app_3_in_1_gaa5(X, Xs, Ys, Zs, app_3_out_gaa3(Xs, Ys, Zs)) -> app_3_out_gaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_perm_2_in_2_ag6(Xs, X, Ys, X1s, X2s, app_3_out_gaa3(X1s, X2s, Zs)) -> if_perm_2_in_3_ag5(Xs, X, Ys, Zs, perm_2_in_ag2(Zs, Ys))
if_perm_2_in_3_ag5(Xs, X, Ys, Zs, perm_2_out_ag2(Zs, Ys)) -> perm_2_out_ag2(Xs, ._22(X, Ys))
if_ss_2_in_1_ag3(Xs, Ys, perm_2_out_ag2(Xs, Ys)) -> if_ss_2_in_2_ag3(Xs, Ys, ordered_1_in_g1(Ys))
ordered_1_in_g1([]_0) -> ordered_1_out_g1([]_0)
ordered_1_in_g1(._22(underscore, []_0)) -> ordered_1_out_g1(._22(underscore, []_0))
ordered_1_in_g1(._22(X, ._22(Y, Xs))) -> if_ordered_1_in_1_g4(X, Y, Xs, less_2_in_ag2(X, s_11(Y)))
less_2_in_ag2(0_0, s_11(underscore1)) -> less_2_out_ag2(0_0, s_11(underscore1))
less_2_in_ag2(s_11(X), s_11(Y)) -> if_less_2_in_1_ag3(X, Y, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore1)) -> less_2_out_aa2(0_0, s_11(underscore1))
less_2_in_aa2(s_11(X), s_11(Y)) -> if_less_2_in_1_aa3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_aa3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_aa2(s_11(X), s_11(Y))
if_less_2_in_1_ag3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ag2(s_11(X), s_11(Y))
if_ordered_1_in_1_g4(X, Y, Xs, less_2_out_ag2(X, s_11(Y))) -> if_ordered_1_in_2_g4(X, Y, Xs, ordered_1_in_g1(._22(Y, Xs)))
if_ordered_1_in_2_g4(X, Y, Xs, ordered_1_out_g1(._22(Y, Xs))) -> ordered_1_out_g1(._22(X, ._22(Y, Xs)))
if_ss_2_in_2_ag3(Xs, Ys, ordered_1_out_g1(Ys)) -> ss_2_out_ag2(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_2_in_ag2(x1, x2)  =  ss_2_in_ag1(x2)
[]_0  =  []_0
._22(x1, x2)  =  ._21(x2)
s_11(x1)  =  s_1
0_0  =  0_0
if_ss_2_in_1_ag3(x1, x2, x3)  =  if_ss_2_in_1_ag2(x2, x3)
perm_2_in_ag2(x1, x2)  =  perm_2_in_ag1(x2)
perm_2_out_ag2(x1, x2)  =  perm_2_out_ag1(x2)
if_perm_2_in_1_ag4(x1, x2, x3, x4)  =  if_perm_2_in_1_ag2(x3, x4)
app_3_in_aaa3(x1, x2, x3)  =  app_3_in_aaa
app_3_out_aaa3(x1, x2, x3)  =  app_3_out_aaa1(x1)
if_app_3_in_1_aaa5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_aaa1(x5)
if_perm_2_in_2_ag6(x1, x2, x3, x4, x5, x6)  =  if_perm_2_in_2_ag2(x3, x6)
app_3_in_gaa3(x1, x2, x3)  =  app_3_in_gaa1(x1)
app_3_out_gaa3(x1, x2, x3)  =  app_3_out_gaa1(x1)
if_app_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gaa2(x2, x5)
if_perm_2_in_3_ag5(x1, x2, x3, x4, x5)  =  if_perm_2_in_3_ag2(x3, x5)
if_ss_2_in_2_ag3(x1, x2, x3)  =  if_ss_2_in_2_ag2(x2, x3)
ordered_1_in_g1(x1)  =  ordered_1_in_g1(x1)
ordered_1_out_g1(x1)  =  ordered_1_out_g1(x1)
if_ordered_1_in_1_g4(x1, x2, x3, x4)  =  if_ordered_1_in_1_g2(x3, x4)
less_2_in_aa2(x1, x2)  =  less_2_in_aa
less_2_out_aa2(x1, x2)  =  less_2_out_aa2(x1, x2)
less_2_in_ag2(x1, x2)  =  less_2_in_ag1(x2)
less_2_out_ag2(x1, x2)  =  less_2_out_ag2(x1, x2)
if_less_2_in_1_ag3(x1, x2, x3)  =  if_less_2_in_1_ag1(x3)
if_less_2_in_1_aa3(x1, x2, x3)  =  if_less_2_in_1_aa1(x3)
if_ordered_1_in_2_g4(x1, x2, x3, x4)  =  if_ordered_1_in_2_g2(x3, x4)
ss_2_out_ag2(x1, x2)  =  ss_2_out_ag1(x2)
LESS_2_IN_AG2(x1, x2)  =  LESS_2_IN_AG1(x2)
IF_SS_2_IN_1_AG3(x1, x2, x3)  =  IF_SS_2_IN_1_AG2(x2, x3)
IF_LESS_2_IN_1_AG3(x1, x2, x3)  =  IF_LESS_2_IN_1_AG1(x3)
IF_APP_3_IN_1_GAA5(x1, x2, x3, x4, x5)  =  IF_APP_3_IN_1_GAA2(x2, x5)
PERM_2_IN_AG2(x1, x2)  =  PERM_2_IN_AG1(x2)
IF_PERM_2_IN_1_AG4(x1, x2, x3, x4)  =  IF_PERM_2_IN_1_AG2(x3, x4)
IF_LESS_2_IN_1_AA3(x1, x2, x3)  =  IF_LESS_2_IN_1_AA1(x3)
IF_ORDERED_1_IN_2_G4(x1, x2, x3, x4)  =  IF_ORDERED_1_IN_2_G2(x3, x4)
SS_2_IN_AG2(x1, x2)  =  SS_2_IN_AG1(x2)
LESS_2_IN_AA2(x1, x2)  =  LESS_2_IN_AA
IF_PERM_2_IN_3_AG5(x1, x2, x3, x4, x5)  =  IF_PERM_2_IN_3_AG2(x3, x5)
IF_ORDERED_1_IN_1_G4(x1, x2, x3, x4)  =  IF_ORDERED_1_IN_1_G2(x3, x4)
IF_PERM_2_IN_2_AG6(x1, x2, x3, x4, x5, x6)  =  IF_PERM_2_IN_2_AG2(x3, x6)
APP_3_IN_GAA3(x1, x2, x3)  =  APP_3_IN_GAA1(x1)
ORDERED_1_IN_G1(x1)  =  ORDERED_1_IN_G1(x1)
IF_SS_2_IN_2_AG3(x1, x2, x3)  =  IF_SS_2_IN_2_AG2(x2, x3)
APP_3_IN_AAA3(x1, x2, x3)  =  APP_3_IN_AAA
IF_APP_3_IN_1_AAA5(x1, x2, x3, x4, x5)  =  IF_APP_3_IN_1_AAA1(x5)

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

↳ PROLOG
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
PiDP
          ↳ DependencyGraphProof

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

SS_2_IN_AG2(Xs, Ys) -> IF_SS_2_IN_1_AG3(Xs, Ys, perm_2_in_ag2(Xs, Ys))
SS_2_IN_AG2(Xs, Ys) -> PERM_2_IN_AG2(Xs, Ys)
PERM_2_IN_AG2(Xs, ._22(X, Ys)) -> IF_PERM_2_IN_1_AG4(Xs, X, Ys, app_3_in_aaa3(X1s, ._22(X, X2s), Xs))
PERM_2_IN_AG2(Xs, ._22(X, Ys)) -> APP_3_IN_AAA3(X1s, ._22(X, X2s), Xs)
APP_3_IN_AAA3(._22(X, Xs), Ys, ._22(X, Zs)) -> IF_APP_3_IN_1_AAA5(X, Xs, Ys, Zs, app_3_in_aaa3(Xs, Ys, Zs))
APP_3_IN_AAA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APP_3_IN_AAA3(Xs, Ys, Zs)
IF_PERM_2_IN_1_AG4(Xs, X, Ys, app_3_out_aaa3(X1s, ._22(X, X2s), Xs)) -> IF_PERM_2_IN_2_AG6(Xs, X, Ys, X1s, X2s, app_3_in_gaa3(X1s, X2s, Zs))
IF_PERM_2_IN_1_AG4(Xs, X, Ys, app_3_out_aaa3(X1s, ._22(X, X2s), Xs)) -> APP_3_IN_GAA3(X1s, X2s, Zs)
APP_3_IN_GAA3(._22(X, Xs), Ys, ._22(X, Zs)) -> IF_APP_3_IN_1_GAA5(X, Xs, Ys, Zs, app_3_in_gaa3(Xs, Ys, Zs))
APP_3_IN_GAA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APP_3_IN_GAA3(Xs, Ys, Zs)
IF_PERM_2_IN_2_AG6(Xs, X, Ys, X1s, X2s, app_3_out_gaa3(X1s, X2s, Zs)) -> IF_PERM_2_IN_3_AG5(Xs, X, Ys, Zs, perm_2_in_ag2(Zs, Ys))
IF_PERM_2_IN_2_AG6(Xs, X, Ys, X1s, X2s, app_3_out_gaa3(X1s, X2s, Zs)) -> PERM_2_IN_AG2(Zs, Ys)
IF_SS_2_IN_1_AG3(Xs, Ys, perm_2_out_ag2(Xs, Ys)) -> IF_SS_2_IN_2_AG3(Xs, Ys, ordered_1_in_g1(Ys))
IF_SS_2_IN_1_AG3(Xs, Ys, perm_2_out_ag2(Xs, Ys)) -> ORDERED_1_IN_G1(Ys)
ORDERED_1_IN_G1(._22(X, ._22(Y, Xs))) -> IF_ORDERED_1_IN_1_G4(X, Y, Xs, less_2_in_ag2(X, s_11(Y)))
ORDERED_1_IN_G1(._22(X, ._22(Y, Xs))) -> LESS_2_IN_AG2(X, s_11(Y))
LESS_2_IN_AG2(s_11(X), s_11(Y)) -> IF_LESS_2_IN_1_AG3(X, Y, less_2_in_aa2(X, Y))
LESS_2_IN_AG2(s_11(X), s_11(Y)) -> LESS_2_IN_AA2(X, Y)
LESS_2_IN_AA2(s_11(X), s_11(Y)) -> IF_LESS_2_IN_1_AA3(X, Y, less_2_in_aa2(X, Y))
LESS_2_IN_AA2(s_11(X), s_11(Y)) -> LESS_2_IN_AA2(X, Y)
IF_ORDERED_1_IN_1_G4(X, Y, Xs, less_2_out_ag2(X, s_11(Y))) -> IF_ORDERED_1_IN_2_G4(X, Y, Xs, ordered_1_in_g1(._22(Y, Xs)))
IF_ORDERED_1_IN_1_G4(X, Y, Xs, less_2_out_ag2(X, s_11(Y))) -> ORDERED_1_IN_G1(._22(Y, Xs))

The TRS R consists of the following rules:

ss_2_in_ag2(Xs, Ys) -> if_ss_2_in_1_ag3(Xs, Ys, perm_2_in_ag2(Xs, Ys))
perm_2_in_ag2([]_0, []_0) -> perm_2_out_ag2([]_0, []_0)
perm_2_in_ag2(Xs, ._22(X, Ys)) -> if_perm_2_in_1_ag4(Xs, X, Ys, app_3_in_aaa3(X1s, ._22(X, X2s), Xs))
app_3_in_aaa3([]_0, X, X) -> app_3_out_aaa3([]_0, X, X)
app_3_in_aaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_aaa5(X, Xs, Ys, Zs, app_3_in_aaa3(Xs, Ys, Zs))
if_app_3_in_1_aaa5(X, Xs, Ys, Zs, app_3_out_aaa3(Xs, Ys, Zs)) -> app_3_out_aaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_perm_2_in_1_ag4(Xs, X, Ys, app_3_out_aaa3(X1s, ._22(X, X2s), Xs)) -> if_perm_2_in_2_ag6(Xs, X, Ys, X1s, X2s, app_3_in_gaa3(X1s, X2s, Zs))
app_3_in_gaa3([]_0, X, X) -> app_3_out_gaa3([]_0, X, X)
app_3_in_gaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_gaa5(X, Xs, Ys, Zs, app_3_in_gaa3(Xs, Ys, Zs))
if_app_3_in_1_gaa5(X, Xs, Ys, Zs, app_3_out_gaa3(Xs, Ys, Zs)) -> app_3_out_gaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_perm_2_in_2_ag6(Xs, X, Ys, X1s, X2s, app_3_out_gaa3(X1s, X2s, Zs)) -> if_perm_2_in_3_ag5(Xs, X, Ys, Zs, perm_2_in_ag2(Zs, Ys))
if_perm_2_in_3_ag5(Xs, X, Ys, Zs, perm_2_out_ag2(Zs, Ys)) -> perm_2_out_ag2(Xs, ._22(X, Ys))
if_ss_2_in_1_ag3(Xs, Ys, perm_2_out_ag2(Xs, Ys)) -> if_ss_2_in_2_ag3(Xs, Ys, ordered_1_in_g1(Ys))
ordered_1_in_g1([]_0) -> ordered_1_out_g1([]_0)
ordered_1_in_g1(._22(underscore, []_0)) -> ordered_1_out_g1(._22(underscore, []_0))
ordered_1_in_g1(._22(X, ._22(Y, Xs))) -> if_ordered_1_in_1_g4(X, Y, Xs, less_2_in_ag2(X, s_11(Y)))
less_2_in_ag2(0_0, s_11(underscore1)) -> less_2_out_ag2(0_0, s_11(underscore1))
less_2_in_ag2(s_11(X), s_11(Y)) -> if_less_2_in_1_ag3(X, Y, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore1)) -> less_2_out_aa2(0_0, s_11(underscore1))
less_2_in_aa2(s_11(X), s_11(Y)) -> if_less_2_in_1_aa3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_aa3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_aa2(s_11(X), s_11(Y))
if_less_2_in_1_ag3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ag2(s_11(X), s_11(Y))
if_ordered_1_in_1_g4(X, Y, Xs, less_2_out_ag2(X, s_11(Y))) -> if_ordered_1_in_2_g4(X, Y, Xs, ordered_1_in_g1(._22(Y, Xs)))
if_ordered_1_in_2_g4(X, Y, Xs, ordered_1_out_g1(._22(Y, Xs))) -> ordered_1_out_g1(._22(X, ._22(Y, Xs)))
if_ss_2_in_2_ag3(Xs, Ys, ordered_1_out_g1(Ys)) -> ss_2_out_ag2(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_2_in_ag2(x1, x2)  =  ss_2_in_ag1(x2)
[]_0  =  []_0
._22(x1, x2)  =  ._21(x2)
s_11(x1)  =  s_1
0_0  =  0_0
if_ss_2_in_1_ag3(x1, x2, x3)  =  if_ss_2_in_1_ag2(x2, x3)
perm_2_in_ag2(x1, x2)  =  perm_2_in_ag1(x2)
perm_2_out_ag2(x1, x2)  =  perm_2_out_ag1(x2)
if_perm_2_in_1_ag4(x1, x2, x3, x4)  =  if_perm_2_in_1_ag2(x3, x4)
app_3_in_aaa3(x1, x2, x3)  =  app_3_in_aaa
app_3_out_aaa3(x1, x2, x3)  =  app_3_out_aaa1(x1)
if_app_3_in_1_aaa5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_aaa1(x5)
if_perm_2_in_2_ag6(x1, x2, x3, x4, x5, x6)  =  if_perm_2_in_2_ag2(x3, x6)
app_3_in_gaa3(x1, x2, x3)  =  app_3_in_gaa1(x1)
app_3_out_gaa3(x1, x2, x3)  =  app_3_out_gaa1(x1)
if_app_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gaa2(x2, x5)
if_perm_2_in_3_ag5(x1, x2, x3, x4, x5)  =  if_perm_2_in_3_ag2(x3, x5)
if_ss_2_in_2_ag3(x1, x2, x3)  =  if_ss_2_in_2_ag2(x2, x3)
ordered_1_in_g1(x1)  =  ordered_1_in_g1(x1)
ordered_1_out_g1(x1)  =  ordered_1_out_g1(x1)
if_ordered_1_in_1_g4(x1, x2, x3, x4)  =  if_ordered_1_in_1_g2(x3, x4)
less_2_in_aa2(x1, x2)  =  less_2_in_aa
less_2_out_aa2(x1, x2)  =  less_2_out_aa2(x1, x2)
less_2_in_ag2(x1, x2)  =  less_2_in_ag1(x2)
less_2_out_ag2(x1, x2)  =  less_2_out_ag2(x1, x2)
if_less_2_in_1_ag3(x1, x2, x3)  =  if_less_2_in_1_ag1(x3)
if_less_2_in_1_aa3(x1, x2, x3)  =  if_less_2_in_1_aa1(x3)
if_ordered_1_in_2_g4(x1, x2, x3, x4)  =  if_ordered_1_in_2_g2(x3, x4)
ss_2_out_ag2(x1, x2)  =  ss_2_out_ag1(x2)
LESS_2_IN_AG2(x1, x2)  =  LESS_2_IN_AG1(x2)
IF_SS_2_IN_1_AG3(x1, x2, x3)  =  IF_SS_2_IN_1_AG2(x2, x3)
IF_LESS_2_IN_1_AG3(x1, x2, x3)  =  IF_LESS_2_IN_1_AG1(x3)
IF_APP_3_IN_1_GAA5(x1, x2, x3, x4, x5)  =  IF_APP_3_IN_1_GAA2(x2, x5)
PERM_2_IN_AG2(x1, x2)  =  PERM_2_IN_AG1(x2)
IF_PERM_2_IN_1_AG4(x1, x2, x3, x4)  =  IF_PERM_2_IN_1_AG2(x3, x4)
IF_LESS_2_IN_1_AA3(x1, x2, x3)  =  IF_LESS_2_IN_1_AA1(x3)
IF_ORDERED_1_IN_2_G4(x1, x2, x3, x4)  =  IF_ORDERED_1_IN_2_G2(x3, x4)
SS_2_IN_AG2(x1, x2)  =  SS_2_IN_AG1(x2)
LESS_2_IN_AA2(x1, x2)  =  LESS_2_IN_AA
IF_PERM_2_IN_3_AG5(x1, x2, x3, x4, x5)  =  IF_PERM_2_IN_3_AG2(x3, x5)
IF_ORDERED_1_IN_1_G4(x1, x2, x3, x4)  =  IF_ORDERED_1_IN_1_G2(x3, x4)
IF_PERM_2_IN_2_AG6(x1, x2, x3, x4, x5, x6)  =  IF_PERM_2_IN_2_AG2(x3, x6)
APP_3_IN_GAA3(x1, x2, x3)  =  APP_3_IN_GAA1(x1)
ORDERED_1_IN_G1(x1)  =  ORDERED_1_IN_G1(x1)
IF_SS_2_IN_2_AG3(x1, x2, x3)  =  IF_SS_2_IN_2_AG2(x2, x3)
APP_3_IN_AAA3(x1, x2, x3)  =  APP_3_IN_AAA
IF_APP_3_IN_1_AAA5(x1, x2, x3, x4, x5)  =  IF_APP_3_IN_1_AAA1(x5)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph contains 5 SCCs with 14 less nodes.

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

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

LESS_2_IN_AA2(s_11(X), s_11(Y)) -> LESS_2_IN_AA2(X, Y)

The TRS R consists of the following rules:

ss_2_in_ag2(Xs, Ys) -> if_ss_2_in_1_ag3(Xs, Ys, perm_2_in_ag2(Xs, Ys))
perm_2_in_ag2([]_0, []_0) -> perm_2_out_ag2([]_0, []_0)
perm_2_in_ag2(Xs, ._22(X, Ys)) -> if_perm_2_in_1_ag4(Xs, X, Ys, app_3_in_aaa3(X1s, ._22(X, X2s), Xs))
app_3_in_aaa3([]_0, X, X) -> app_3_out_aaa3([]_0, X, X)
app_3_in_aaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_aaa5(X, Xs, Ys, Zs, app_3_in_aaa3(Xs, Ys, Zs))
if_app_3_in_1_aaa5(X, Xs, Ys, Zs, app_3_out_aaa3(Xs, Ys, Zs)) -> app_3_out_aaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_perm_2_in_1_ag4(Xs, X, Ys, app_3_out_aaa3(X1s, ._22(X, X2s), Xs)) -> if_perm_2_in_2_ag6(Xs, X, Ys, X1s, X2s, app_3_in_gaa3(X1s, X2s, Zs))
app_3_in_gaa3([]_0, X, X) -> app_3_out_gaa3([]_0, X, X)
app_3_in_gaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_gaa5(X, Xs, Ys, Zs, app_3_in_gaa3(Xs, Ys, Zs))
if_app_3_in_1_gaa5(X, Xs, Ys, Zs, app_3_out_gaa3(Xs, Ys, Zs)) -> app_3_out_gaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_perm_2_in_2_ag6(Xs, X, Ys, X1s, X2s, app_3_out_gaa3(X1s, X2s, Zs)) -> if_perm_2_in_3_ag5(Xs, X, Ys, Zs, perm_2_in_ag2(Zs, Ys))
if_perm_2_in_3_ag5(Xs, X, Ys, Zs, perm_2_out_ag2(Zs, Ys)) -> perm_2_out_ag2(Xs, ._22(X, Ys))
if_ss_2_in_1_ag3(Xs, Ys, perm_2_out_ag2(Xs, Ys)) -> if_ss_2_in_2_ag3(Xs, Ys, ordered_1_in_g1(Ys))
ordered_1_in_g1([]_0) -> ordered_1_out_g1([]_0)
ordered_1_in_g1(._22(underscore, []_0)) -> ordered_1_out_g1(._22(underscore, []_0))
ordered_1_in_g1(._22(X, ._22(Y, Xs))) -> if_ordered_1_in_1_g4(X, Y, Xs, less_2_in_ag2(X, s_11(Y)))
less_2_in_ag2(0_0, s_11(underscore1)) -> less_2_out_ag2(0_0, s_11(underscore1))
less_2_in_ag2(s_11(X), s_11(Y)) -> if_less_2_in_1_ag3(X, Y, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore1)) -> less_2_out_aa2(0_0, s_11(underscore1))
less_2_in_aa2(s_11(X), s_11(Y)) -> if_less_2_in_1_aa3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_aa3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_aa2(s_11(X), s_11(Y))
if_less_2_in_1_ag3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ag2(s_11(X), s_11(Y))
if_ordered_1_in_1_g4(X, Y, Xs, less_2_out_ag2(X, s_11(Y))) -> if_ordered_1_in_2_g4(X, Y, Xs, ordered_1_in_g1(._22(Y, Xs)))
if_ordered_1_in_2_g4(X, Y, Xs, ordered_1_out_g1(._22(Y, Xs))) -> ordered_1_out_g1(._22(X, ._22(Y, Xs)))
if_ss_2_in_2_ag3(Xs, Ys, ordered_1_out_g1(Ys)) -> ss_2_out_ag2(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_2_in_ag2(x1, x2)  =  ss_2_in_ag1(x2)
[]_0  =  []_0
._22(x1, x2)  =  ._21(x2)
s_11(x1)  =  s_1
0_0  =  0_0
if_ss_2_in_1_ag3(x1, x2, x3)  =  if_ss_2_in_1_ag2(x2, x3)
perm_2_in_ag2(x1, x2)  =  perm_2_in_ag1(x2)
perm_2_out_ag2(x1, x2)  =  perm_2_out_ag1(x2)
if_perm_2_in_1_ag4(x1, x2, x3, x4)  =  if_perm_2_in_1_ag2(x3, x4)
app_3_in_aaa3(x1, x2, x3)  =  app_3_in_aaa
app_3_out_aaa3(x1, x2, x3)  =  app_3_out_aaa1(x1)
if_app_3_in_1_aaa5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_aaa1(x5)
if_perm_2_in_2_ag6(x1, x2, x3, x4, x5, x6)  =  if_perm_2_in_2_ag2(x3, x6)
app_3_in_gaa3(x1, x2, x3)  =  app_3_in_gaa1(x1)
app_3_out_gaa3(x1, x2, x3)  =  app_3_out_gaa1(x1)
if_app_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gaa2(x2, x5)
if_perm_2_in_3_ag5(x1, x2, x3, x4, x5)  =  if_perm_2_in_3_ag2(x3, x5)
if_ss_2_in_2_ag3(x1, x2, x3)  =  if_ss_2_in_2_ag2(x2, x3)
ordered_1_in_g1(x1)  =  ordered_1_in_g1(x1)
ordered_1_out_g1(x1)  =  ordered_1_out_g1(x1)
if_ordered_1_in_1_g4(x1, x2, x3, x4)  =  if_ordered_1_in_1_g2(x3, x4)
less_2_in_aa2(x1, x2)  =  less_2_in_aa
less_2_out_aa2(x1, x2)  =  less_2_out_aa2(x1, x2)
less_2_in_ag2(x1, x2)  =  less_2_in_ag1(x2)
less_2_out_ag2(x1, x2)  =  less_2_out_ag2(x1, x2)
if_less_2_in_1_ag3(x1, x2, x3)  =  if_less_2_in_1_ag1(x3)
if_less_2_in_1_aa3(x1, x2, x3)  =  if_less_2_in_1_aa1(x3)
if_ordered_1_in_2_g4(x1, x2, x3, x4)  =  if_ordered_1_in_2_g2(x3, x4)
ss_2_out_ag2(x1, x2)  =  ss_2_out_ag1(x2)
LESS_2_IN_AA2(x1, x2)  =  LESS_2_IN_AA

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

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

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

LESS_2_IN_AA2(s_11(X), s_11(Y)) -> LESS_2_IN_AA2(X, Y)

R is empty.
The argument filtering Pi contains the following mapping:
s_11(x1)  =  s_1
LESS_2_IN_AA2(x1, x2)  =  LESS_2_IN_AA

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

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

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

ORDERED_1_IN_G1(._22(X, ._22(Y, Xs))) -> IF_ORDERED_1_IN_1_G4(X, Y, Xs, less_2_in_ag2(X, s_11(Y)))
IF_ORDERED_1_IN_1_G4(X, Y, Xs, less_2_out_ag2(X, s_11(Y))) -> ORDERED_1_IN_G1(._22(Y, Xs))

The TRS R consists of the following rules:

ss_2_in_ag2(Xs, Ys) -> if_ss_2_in_1_ag3(Xs, Ys, perm_2_in_ag2(Xs, Ys))
perm_2_in_ag2([]_0, []_0) -> perm_2_out_ag2([]_0, []_0)
perm_2_in_ag2(Xs, ._22(X, Ys)) -> if_perm_2_in_1_ag4(Xs, X, Ys, app_3_in_aaa3(X1s, ._22(X, X2s), Xs))
app_3_in_aaa3([]_0, X, X) -> app_3_out_aaa3([]_0, X, X)
app_3_in_aaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_aaa5(X, Xs, Ys, Zs, app_3_in_aaa3(Xs, Ys, Zs))
if_app_3_in_1_aaa5(X, Xs, Ys, Zs, app_3_out_aaa3(Xs, Ys, Zs)) -> app_3_out_aaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_perm_2_in_1_ag4(Xs, X, Ys, app_3_out_aaa3(X1s, ._22(X, X2s), Xs)) -> if_perm_2_in_2_ag6(Xs, X, Ys, X1s, X2s, app_3_in_gaa3(X1s, X2s, Zs))
app_3_in_gaa3([]_0, X, X) -> app_3_out_gaa3([]_0, X, X)
app_3_in_gaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_gaa5(X, Xs, Ys, Zs, app_3_in_gaa3(Xs, Ys, Zs))
if_app_3_in_1_gaa5(X, Xs, Ys, Zs, app_3_out_gaa3(Xs, Ys, Zs)) -> app_3_out_gaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_perm_2_in_2_ag6(Xs, X, Ys, X1s, X2s, app_3_out_gaa3(X1s, X2s, Zs)) -> if_perm_2_in_3_ag5(Xs, X, Ys, Zs, perm_2_in_ag2(Zs, Ys))
if_perm_2_in_3_ag5(Xs, X, Ys, Zs, perm_2_out_ag2(Zs, Ys)) -> perm_2_out_ag2(Xs, ._22(X, Ys))
if_ss_2_in_1_ag3(Xs, Ys, perm_2_out_ag2(Xs, Ys)) -> if_ss_2_in_2_ag3(Xs, Ys, ordered_1_in_g1(Ys))
ordered_1_in_g1([]_0) -> ordered_1_out_g1([]_0)
ordered_1_in_g1(._22(underscore, []_0)) -> ordered_1_out_g1(._22(underscore, []_0))
ordered_1_in_g1(._22(X, ._22(Y, Xs))) -> if_ordered_1_in_1_g4(X, Y, Xs, less_2_in_ag2(X, s_11(Y)))
less_2_in_ag2(0_0, s_11(underscore1)) -> less_2_out_ag2(0_0, s_11(underscore1))
less_2_in_ag2(s_11(X), s_11(Y)) -> if_less_2_in_1_ag3(X, Y, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore1)) -> less_2_out_aa2(0_0, s_11(underscore1))
less_2_in_aa2(s_11(X), s_11(Y)) -> if_less_2_in_1_aa3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_aa3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_aa2(s_11(X), s_11(Y))
if_less_2_in_1_ag3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ag2(s_11(X), s_11(Y))
if_ordered_1_in_1_g4(X, Y, Xs, less_2_out_ag2(X, s_11(Y))) -> if_ordered_1_in_2_g4(X, Y, Xs, ordered_1_in_g1(._22(Y, Xs)))
if_ordered_1_in_2_g4(X, Y, Xs, ordered_1_out_g1(._22(Y, Xs))) -> ordered_1_out_g1(._22(X, ._22(Y, Xs)))
if_ss_2_in_2_ag3(Xs, Ys, ordered_1_out_g1(Ys)) -> ss_2_out_ag2(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_2_in_ag2(x1, x2)  =  ss_2_in_ag1(x2)
[]_0  =  []_0
._22(x1, x2)  =  ._21(x2)
s_11(x1)  =  s_1
0_0  =  0_0
if_ss_2_in_1_ag3(x1, x2, x3)  =  if_ss_2_in_1_ag2(x2, x3)
perm_2_in_ag2(x1, x2)  =  perm_2_in_ag1(x2)
perm_2_out_ag2(x1, x2)  =  perm_2_out_ag1(x2)
if_perm_2_in_1_ag4(x1, x2, x3, x4)  =  if_perm_2_in_1_ag2(x3, x4)
app_3_in_aaa3(x1, x2, x3)  =  app_3_in_aaa
app_3_out_aaa3(x1, x2, x3)  =  app_3_out_aaa1(x1)
if_app_3_in_1_aaa5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_aaa1(x5)
if_perm_2_in_2_ag6(x1, x2, x3, x4, x5, x6)  =  if_perm_2_in_2_ag2(x3, x6)
app_3_in_gaa3(x1, x2, x3)  =  app_3_in_gaa1(x1)
app_3_out_gaa3(x1, x2, x3)  =  app_3_out_gaa1(x1)
if_app_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gaa2(x2, x5)
if_perm_2_in_3_ag5(x1, x2, x3, x4, x5)  =  if_perm_2_in_3_ag2(x3, x5)
if_ss_2_in_2_ag3(x1, x2, x3)  =  if_ss_2_in_2_ag2(x2, x3)
ordered_1_in_g1(x1)  =  ordered_1_in_g1(x1)
ordered_1_out_g1(x1)  =  ordered_1_out_g1(x1)
if_ordered_1_in_1_g4(x1, x2, x3, x4)  =  if_ordered_1_in_1_g2(x3, x4)
less_2_in_aa2(x1, x2)  =  less_2_in_aa
less_2_out_aa2(x1, x2)  =  less_2_out_aa2(x1, x2)
less_2_in_ag2(x1, x2)  =  less_2_in_ag1(x2)
less_2_out_ag2(x1, x2)  =  less_2_out_ag2(x1, x2)
if_less_2_in_1_ag3(x1, x2, x3)  =  if_less_2_in_1_ag1(x3)
if_less_2_in_1_aa3(x1, x2, x3)  =  if_less_2_in_1_aa1(x3)
if_ordered_1_in_2_g4(x1, x2, x3, x4)  =  if_ordered_1_in_2_g2(x3, x4)
ss_2_out_ag2(x1, x2)  =  ss_2_out_ag1(x2)
IF_ORDERED_1_IN_1_G4(x1, x2, x3, x4)  =  IF_ORDERED_1_IN_1_G2(x3, x4)
ORDERED_1_IN_G1(x1)  =  ORDERED_1_IN_G1(x1)

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

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

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

APP_3_IN_GAA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APP_3_IN_GAA3(Xs, Ys, Zs)

The TRS R consists of the following rules:

ss_2_in_ag2(Xs, Ys) -> if_ss_2_in_1_ag3(Xs, Ys, perm_2_in_ag2(Xs, Ys))
perm_2_in_ag2([]_0, []_0) -> perm_2_out_ag2([]_0, []_0)
perm_2_in_ag2(Xs, ._22(X, Ys)) -> if_perm_2_in_1_ag4(Xs, X, Ys, app_3_in_aaa3(X1s, ._22(X, X2s), Xs))
app_3_in_aaa3([]_0, X, X) -> app_3_out_aaa3([]_0, X, X)
app_3_in_aaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_aaa5(X, Xs, Ys, Zs, app_3_in_aaa3(Xs, Ys, Zs))
if_app_3_in_1_aaa5(X, Xs, Ys, Zs, app_3_out_aaa3(Xs, Ys, Zs)) -> app_3_out_aaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_perm_2_in_1_ag4(Xs, X, Ys, app_3_out_aaa3(X1s, ._22(X, X2s), Xs)) -> if_perm_2_in_2_ag6(Xs, X, Ys, X1s, X2s, app_3_in_gaa3(X1s, X2s, Zs))
app_3_in_gaa3([]_0, X, X) -> app_3_out_gaa3([]_0, X, X)
app_3_in_gaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_gaa5(X, Xs, Ys, Zs, app_3_in_gaa3(Xs, Ys, Zs))
if_app_3_in_1_gaa5(X, Xs, Ys, Zs, app_3_out_gaa3(Xs, Ys, Zs)) -> app_3_out_gaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_perm_2_in_2_ag6(Xs, X, Ys, X1s, X2s, app_3_out_gaa3(X1s, X2s, Zs)) -> if_perm_2_in_3_ag5(Xs, X, Ys, Zs, perm_2_in_ag2(Zs, Ys))
if_perm_2_in_3_ag5(Xs, X, Ys, Zs, perm_2_out_ag2(Zs, Ys)) -> perm_2_out_ag2(Xs, ._22(X, Ys))
if_ss_2_in_1_ag3(Xs, Ys, perm_2_out_ag2(Xs, Ys)) -> if_ss_2_in_2_ag3(Xs, Ys, ordered_1_in_g1(Ys))
ordered_1_in_g1([]_0) -> ordered_1_out_g1([]_0)
ordered_1_in_g1(._22(underscore, []_0)) -> ordered_1_out_g1(._22(underscore, []_0))
ordered_1_in_g1(._22(X, ._22(Y, Xs))) -> if_ordered_1_in_1_g4(X, Y, Xs, less_2_in_ag2(X, s_11(Y)))
less_2_in_ag2(0_0, s_11(underscore1)) -> less_2_out_ag2(0_0, s_11(underscore1))
less_2_in_ag2(s_11(X), s_11(Y)) -> if_less_2_in_1_ag3(X, Y, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore1)) -> less_2_out_aa2(0_0, s_11(underscore1))
less_2_in_aa2(s_11(X), s_11(Y)) -> if_less_2_in_1_aa3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_aa3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_aa2(s_11(X), s_11(Y))
if_less_2_in_1_ag3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ag2(s_11(X), s_11(Y))
if_ordered_1_in_1_g4(X, Y, Xs, less_2_out_ag2(X, s_11(Y))) -> if_ordered_1_in_2_g4(X, Y, Xs, ordered_1_in_g1(._22(Y, Xs)))
if_ordered_1_in_2_g4(X, Y, Xs, ordered_1_out_g1(._22(Y, Xs))) -> ordered_1_out_g1(._22(X, ._22(Y, Xs)))
if_ss_2_in_2_ag3(Xs, Ys, ordered_1_out_g1(Ys)) -> ss_2_out_ag2(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_2_in_ag2(x1, x2)  =  ss_2_in_ag1(x2)
[]_0  =  []_0
._22(x1, x2)  =  ._21(x2)
s_11(x1)  =  s_1
0_0  =  0_0
if_ss_2_in_1_ag3(x1, x2, x3)  =  if_ss_2_in_1_ag2(x2, x3)
perm_2_in_ag2(x1, x2)  =  perm_2_in_ag1(x2)
perm_2_out_ag2(x1, x2)  =  perm_2_out_ag1(x2)
if_perm_2_in_1_ag4(x1, x2, x3, x4)  =  if_perm_2_in_1_ag2(x3, x4)
app_3_in_aaa3(x1, x2, x3)  =  app_3_in_aaa
app_3_out_aaa3(x1, x2, x3)  =  app_3_out_aaa1(x1)
if_app_3_in_1_aaa5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_aaa1(x5)
if_perm_2_in_2_ag6(x1, x2, x3, x4, x5, x6)  =  if_perm_2_in_2_ag2(x3, x6)
app_3_in_gaa3(x1, x2, x3)  =  app_3_in_gaa1(x1)
app_3_out_gaa3(x1, x2, x3)  =  app_3_out_gaa1(x1)
if_app_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gaa2(x2, x5)
if_perm_2_in_3_ag5(x1, x2, x3, x4, x5)  =  if_perm_2_in_3_ag2(x3, x5)
if_ss_2_in_2_ag3(x1, x2, x3)  =  if_ss_2_in_2_ag2(x2, x3)
ordered_1_in_g1(x1)  =  ordered_1_in_g1(x1)
ordered_1_out_g1(x1)  =  ordered_1_out_g1(x1)
if_ordered_1_in_1_g4(x1, x2, x3, x4)  =  if_ordered_1_in_1_g2(x3, x4)
less_2_in_aa2(x1, x2)  =  less_2_in_aa
less_2_out_aa2(x1, x2)  =  less_2_out_aa2(x1, x2)
less_2_in_ag2(x1, x2)  =  less_2_in_ag1(x2)
less_2_out_ag2(x1, x2)  =  less_2_out_ag2(x1, x2)
if_less_2_in_1_ag3(x1, x2, x3)  =  if_less_2_in_1_ag1(x3)
if_less_2_in_1_aa3(x1, x2, x3)  =  if_less_2_in_1_aa1(x3)
if_ordered_1_in_2_g4(x1, x2, x3, x4)  =  if_ordered_1_in_2_g2(x3, x4)
ss_2_out_ag2(x1, x2)  =  ss_2_out_ag1(x2)
APP_3_IN_GAA3(x1, x2, x3)  =  APP_3_IN_GAA1(x1)

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

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

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

APP_3_IN_AAA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APP_3_IN_AAA3(Xs, Ys, Zs)

The TRS R consists of the following rules:

ss_2_in_ag2(Xs, Ys) -> if_ss_2_in_1_ag3(Xs, Ys, perm_2_in_ag2(Xs, Ys))
perm_2_in_ag2([]_0, []_0) -> perm_2_out_ag2([]_0, []_0)
perm_2_in_ag2(Xs, ._22(X, Ys)) -> if_perm_2_in_1_ag4(Xs, X, Ys, app_3_in_aaa3(X1s, ._22(X, X2s), Xs))
app_3_in_aaa3([]_0, X, X) -> app_3_out_aaa3([]_0, X, X)
app_3_in_aaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_aaa5(X, Xs, Ys, Zs, app_3_in_aaa3(Xs, Ys, Zs))
if_app_3_in_1_aaa5(X, Xs, Ys, Zs, app_3_out_aaa3(Xs, Ys, Zs)) -> app_3_out_aaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_perm_2_in_1_ag4(Xs, X, Ys, app_3_out_aaa3(X1s, ._22(X, X2s), Xs)) -> if_perm_2_in_2_ag6(Xs, X, Ys, X1s, X2s, app_3_in_gaa3(X1s, X2s, Zs))
app_3_in_gaa3([]_0, X, X) -> app_3_out_gaa3([]_0, X, X)
app_3_in_gaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_gaa5(X, Xs, Ys, Zs, app_3_in_gaa3(Xs, Ys, Zs))
if_app_3_in_1_gaa5(X, Xs, Ys, Zs, app_3_out_gaa3(Xs, Ys, Zs)) -> app_3_out_gaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_perm_2_in_2_ag6(Xs, X, Ys, X1s, X2s, app_3_out_gaa3(X1s, X2s, Zs)) -> if_perm_2_in_3_ag5(Xs, X, Ys, Zs, perm_2_in_ag2(Zs, Ys))
if_perm_2_in_3_ag5(Xs, X, Ys, Zs, perm_2_out_ag2(Zs, Ys)) -> perm_2_out_ag2(Xs, ._22(X, Ys))
if_ss_2_in_1_ag3(Xs, Ys, perm_2_out_ag2(Xs, Ys)) -> if_ss_2_in_2_ag3(Xs, Ys, ordered_1_in_g1(Ys))
ordered_1_in_g1([]_0) -> ordered_1_out_g1([]_0)
ordered_1_in_g1(._22(underscore, []_0)) -> ordered_1_out_g1(._22(underscore, []_0))
ordered_1_in_g1(._22(X, ._22(Y, Xs))) -> if_ordered_1_in_1_g4(X, Y, Xs, less_2_in_ag2(X, s_11(Y)))
less_2_in_ag2(0_0, s_11(underscore1)) -> less_2_out_ag2(0_0, s_11(underscore1))
less_2_in_ag2(s_11(X), s_11(Y)) -> if_less_2_in_1_ag3(X, Y, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore1)) -> less_2_out_aa2(0_0, s_11(underscore1))
less_2_in_aa2(s_11(X), s_11(Y)) -> if_less_2_in_1_aa3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_aa3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_aa2(s_11(X), s_11(Y))
if_less_2_in_1_ag3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ag2(s_11(X), s_11(Y))
if_ordered_1_in_1_g4(X, Y, Xs, less_2_out_ag2(X, s_11(Y))) -> if_ordered_1_in_2_g4(X, Y, Xs, ordered_1_in_g1(._22(Y, Xs)))
if_ordered_1_in_2_g4(X, Y, Xs, ordered_1_out_g1(._22(Y, Xs))) -> ordered_1_out_g1(._22(X, ._22(Y, Xs)))
if_ss_2_in_2_ag3(Xs, Ys, ordered_1_out_g1(Ys)) -> ss_2_out_ag2(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_2_in_ag2(x1, x2)  =  ss_2_in_ag1(x2)
[]_0  =  []_0
._22(x1, x2)  =  ._21(x2)
s_11(x1)  =  s_1
0_0  =  0_0
if_ss_2_in_1_ag3(x1, x2, x3)  =  if_ss_2_in_1_ag2(x2, x3)
perm_2_in_ag2(x1, x2)  =  perm_2_in_ag1(x2)
perm_2_out_ag2(x1, x2)  =  perm_2_out_ag1(x2)
if_perm_2_in_1_ag4(x1, x2, x3, x4)  =  if_perm_2_in_1_ag2(x3, x4)
app_3_in_aaa3(x1, x2, x3)  =  app_3_in_aaa
app_3_out_aaa3(x1, x2, x3)  =  app_3_out_aaa1(x1)
if_app_3_in_1_aaa5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_aaa1(x5)
if_perm_2_in_2_ag6(x1, x2, x3, x4, x5, x6)  =  if_perm_2_in_2_ag2(x3, x6)
app_3_in_gaa3(x1, x2, x3)  =  app_3_in_gaa1(x1)
app_3_out_gaa3(x1, x2, x3)  =  app_3_out_gaa1(x1)
if_app_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gaa2(x2, x5)
if_perm_2_in_3_ag5(x1, x2, x3, x4, x5)  =  if_perm_2_in_3_ag2(x3, x5)
if_ss_2_in_2_ag3(x1, x2, x3)  =  if_ss_2_in_2_ag2(x2, x3)
ordered_1_in_g1(x1)  =  ordered_1_in_g1(x1)
ordered_1_out_g1(x1)  =  ordered_1_out_g1(x1)
if_ordered_1_in_1_g4(x1, x2, x3, x4)  =  if_ordered_1_in_1_g2(x3, x4)
less_2_in_aa2(x1, x2)  =  less_2_in_aa
less_2_out_aa2(x1, x2)  =  less_2_out_aa2(x1, x2)
less_2_in_ag2(x1, x2)  =  less_2_in_ag1(x2)
less_2_out_ag2(x1, x2)  =  less_2_out_ag2(x1, x2)
if_less_2_in_1_ag3(x1, x2, x3)  =  if_less_2_in_1_ag1(x3)
if_less_2_in_1_aa3(x1, x2, x3)  =  if_less_2_in_1_aa1(x3)
if_ordered_1_in_2_g4(x1, x2, x3, x4)  =  if_ordered_1_in_2_g2(x3, x4)
ss_2_out_ag2(x1, x2)  =  ss_2_out_ag1(x2)
APP_3_IN_AAA3(x1, x2, x3)  =  APP_3_IN_AAA

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

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

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

PERM_2_IN_AG2(Xs, ._22(X, Ys)) -> IF_PERM_2_IN_1_AG4(Xs, X, Ys, app_3_in_aaa3(X1s, ._22(X, X2s), Xs))
IF_PERM_2_IN_1_AG4(Xs, X, Ys, app_3_out_aaa3(X1s, ._22(X, X2s), Xs)) -> IF_PERM_2_IN_2_AG6(Xs, X, Ys, X1s, X2s, app_3_in_gaa3(X1s, X2s, Zs))
IF_PERM_2_IN_2_AG6(Xs, X, Ys, X1s, X2s, app_3_out_gaa3(X1s, X2s, Zs)) -> PERM_2_IN_AG2(Zs, Ys)

The TRS R consists of the following rules:

ss_2_in_ag2(Xs, Ys) -> if_ss_2_in_1_ag3(Xs, Ys, perm_2_in_ag2(Xs, Ys))
perm_2_in_ag2([]_0, []_0) -> perm_2_out_ag2([]_0, []_0)
perm_2_in_ag2(Xs, ._22(X, Ys)) -> if_perm_2_in_1_ag4(Xs, X, Ys, app_3_in_aaa3(X1s, ._22(X, X2s), Xs))
app_3_in_aaa3([]_0, X, X) -> app_3_out_aaa3([]_0, X, X)
app_3_in_aaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_aaa5(X, Xs, Ys, Zs, app_3_in_aaa3(Xs, Ys, Zs))
if_app_3_in_1_aaa5(X, Xs, Ys, Zs, app_3_out_aaa3(Xs, Ys, Zs)) -> app_3_out_aaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_perm_2_in_1_ag4(Xs, X, Ys, app_3_out_aaa3(X1s, ._22(X, X2s), Xs)) -> if_perm_2_in_2_ag6(Xs, X, Ys, X1s, X2s, app_3_in_gaa3(X1s, X2s, Zs))
app_3_in_gaa3([]_0, X, X) -> app_3_out_gaa3([]_0, X, X)
app_3_in_gaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_gaa5(X, Xs, Ys, Zs, app_3_in_gaa3(Xs, Ys, Zs))
if_app_3_in_1_gaa5(X, Xs, Ys, Zs, app_3_out_gaa3(Xs, Ys, Zs)) -> app_3_out_gaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_perm_2_in_2_ag6(Xs, X, Ys, X1s, X2s, app_3_out_gaa3(X1s, X2s, Zs)) -> if_perm_2_in_3_ag5(Xs, X, Ys, Zs, perm_2_in_ag2(Zs, Ys))
if_perm_2_in_3_ag5(Xs, X, Ys, Zs, perm_2_out_ag2(Zs, Ys)) -> perm_2_out_ag2(Xs, ._22(X, Ys))
if_ss_2_in_1_ag3(Xs, Ys, perm_2_out_ag2(Xs, Ys)) -> if_ss_2_in_2_ag3(Xs, Ys, ordered_1_in_g1(Ys))
ordered_1_in_g1([]_0) -> ordered_1_out_g1([]_0)
ordered_1_in_g1(._22(underscore, []_0)) -> ordered_1_out_g1(._22(underscore, []_0))
ordered_1_in_g1(._22(X, ._22(Y, Xs))) -> if_ordered_1_in_1_g4(X, Y, Xs, less_2_in_ag2(X, s_11(Y)))
less_2_in_ag2(0_0, s_11(underscore1)) -> less_2_out_ag2(0_0, s_11(underscore1))
less_2_in_ag2(s_11(X), s_11(Y)) -> if_less_2_in_1_ag3(X, Y, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore1)) -> less_2_out_aa2(0_0, s_11(underscore1))
less_2_in_aa2(s_11(X), s_11(Y)) -> if_less_2_in_1_aa3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_aa3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_aa2(s_11(X), s_11(Y))
if_less_2_in_1_ag3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ag2(s_11(X), s_11(Y))
if_ordered_1_in_1_g4(X, Y, Xs, less_2_out_ag2(X, s_11(Y))) -> if_ordered_1_in_2_g4(X, Y, Xs, ordered_1_in_g1(._22(Y, Xs)))
if_ordered_1_in_2_g4(X, Y, Xs, ordered_1_out_g1(._22(Y, Xs))) -> ordered_1_out_g1(._22(X, ._22(Y, Xs)))
if_ss_2_in_2_ag3(Xs, Ys, ordered_1_out_g1(Ys)) -> ss_2_out_ag2(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_2_in_ag2(x1, x2)  =  ss_2_in_ag1(x2)
[]_0  =  []_0
._22(x1, x2)  =  ._21(x2)
s_11(x1)  =  s_1
0_0  =  0_0
if_ss_2_in_1_ag3(x1, x2, x3)  =  if_ss_2_in_1_ag2(x2, x3)
perm_2_in_ag2(x1, x2)  =  perm_2_in_ag1(x2)
perm_2_out_ag2(x1, x2)  =  perm_2_out_ag1(x2)
if_perm_2_in_1_ag4(x1, x2, x3, x4)  =  if_perm_2_in_1_ag2(x3, x4)
app_3_in_aaa3(x1, x2, x3)  =  app_3_in_aaa
app_3_out_aaa3(x1, x2, x3)  =  app_3_out_aaa1(x1)
if_app_3_in_1_aaa5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_aaa1(x5)
if_perm_2_in_2_ag6(x1, x2, x3, x4, x5, x6)  =  if_perm_2_in_2_ag2(x3, x6)
app_3_in_gaa3(x1, x2, x3)  =  app_3_in_gaa1(x1)
app_3_out_gaa3(x1, x2, x3)  =  app_3_out_gaa1(x1)
if_app_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gaa2(x2, x5)
if_perm_2_in_3_ag5(x1, x2, x3, x4, x5)  =  if_perm_2_in_3_ag2(x3, x5)
if_ss_2_in_2_ag3(x1, x2, x3)  =  if_ss_2_in_2_ag2(x2, x3)
ordered_1_in_g1(x1)  =  ordered_1_in_g1(x1)
ordered_1_out_g1(x1)  =  ordered_1_out_g1(x1)
if_ordered_1_in_1_g4(x1, x2, x3, x4)  =  if_ordered_1_in_1_g2(x3, x4)
less_2_in_aa2(x1, x2)  =  less_2_in_aa
less_2_out_aa2(x1, x2)  =  less_2_out_aa2(x1, x2)
less_2_in_ag2(x1, x2)  =  less_2_in_ag1(x2)
less_2_out_ag2(x1, x2)  =  less_2_out_ag2(x1, x2)
if_less_2_in_1_ag3(x1, x2, x3)  =  if_less_2_in_1_ag1(x3)
if_less_2_in_1_aa3(x1, x2, x3)  =  if_less_2_in_1_aa1(x3)
if_ordered_1_in_2_g4(x1, x2, x3, x4)  =  if_ordered_1_in_2_g2(x3, x4)
ss_2_out_ag2(x1, x2)  =  ss_2_out_ag1(x2)
PERM_2_IN_AG2(x1, x2)  =  PERM_2_IN_AG1(x2)
IF_PERM_2_IN_1_AG4(x1, x2, x3, x4)  =  IF_PERM_2_IN_1_AG2(x3, x4)
IF_PERM_2_IN_2_AG6(x1, x2, x3, x4, x5, x6)  =  IF_PERM_2_IN_2_AG2(x3, x6)

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