Left Termination of the query pattern qs_in_2(a, g) w.r.t. the given Prolog program could not be shown:



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof

Clauses:

qs(.(X, Xs), Ys) :- ','(part(X, Xs, Littles, Bigs), ','(qs(Littles, Ls), ','(qs(Bigs, Bs), app(Ls, .(X, Bs), Ys)))).
qs([], []).
part(X, .(Y, Xs), .(Y, Ls), Bs) :- ','(gt(X, Y), part(X, Xs, Ls, Bs)).
part(X, .(Y, Xs), Ls, .(Y, Bs)) :- ','(le(X, Y), part(X, Xs, Ls, Bs)).
part(X, [], [], []).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
app([], Ys, Ys).
gt(s(X), s(Y)) :- gt(X, Y).
gt(s(0), 0).
le(s(X), s(Y)) :- le(X, Y).
le(0, s(X)).
le(0, 0).

Queries:

qs(a,g).

We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
qs_in: (f,b) (b,f) (f,f)
part_in: (f,f,f,f) (b,f,f,f) (b,b,f,f)
gt_in: (f,f) (b,f) (b,b)
le_in: (b,f) (f,f) (b,b)
app_in: (b,b,f) (b,f,f) (b,f,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

qs_in_ag(.(X, Xs), Ys) → U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
gt_in_aa(s(X), s(Y)) → U10_aa(X, Y, gt_in_aa(X, Y))
gt_in_aa(s(0), 0) → gt_out_aa(s(0), 0)
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
gt_in_ga(s(X), s(Y)) → U10_ga(X, Y, gt_in_ga(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U10_ga(X, Y, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_ga(X, Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
U11_ga(X, Y, le_out_ga(X, Y)) → le_out_ga(s(X), s(Y))
U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, [], [], []) → part_out_gaaa(X, [], [], [])
U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
le_in_aa(s(X), s(Y)) → U11_aa(X, Y, le_in_aa(X, Y))
le_in_aa(0, s(X)) → le_out_aa(0, s(X))
le_in_aa(0, 0) → le_out_aa(0, 0)
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_aaaa(X, [], [], []) → part_out_aaaa(X, [], [], [])
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa([], []) → qs_out_aa([], [])
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
app_in_gaa([], Ys, Ys) → app_out_gaa([], Ys, Ys)
U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
app_in_gag(.(X, Xs), Ys, .(X, Zs)) → U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
app_in_gag([], Ys, Ys) → app_out_gag([], Ys, Ys)
U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) → app_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt_in_aa(x1, x2)  =  gt_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)
part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
gt_in_ga(x1, x2)  =  gt_in_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
0  =  0
gt_out_ga(x1, x2)  =  gt_out_ga(x2)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x2, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
le_in_ga(x1, x2)  =  le_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x3)
le_out_ga(x1, x2)  =  le_out_ga
U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x3)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
le_in_aa(x1, x2)  =  le_in_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
le_out_aa(x1, x2)  =  le_out_aa(x1)
U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x3)
gt_out_gg(x1, x2)  =  gt_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x3)
le_out_gg(x1, x2)  =  le_out_gg
U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x2, x6)
[]  =  []
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)
qs_out_ga(x1, x2)  =  qs_out_ga(x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)
qs_out_aa(x1, x2)  =  qs_out_aa
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)
app_out_gaa(x1, x2, x3)  =  app_out_gaa
U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x5)
app_out_gag(x1, x2, x3)  =  app_out_gag(x2)
qs_out_ag(x1, x2)  =  qs_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:

qs_in_ag(.(X, Xs), Ys) → U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
gt_in_aa(s(X), s(Y)) → U10_aa(X, Y, gt_in_aa(X, Y))
gt_in_aa(s(0), 0) → gt_out_aa(s(0), 0)
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
gt_in_ga(s(X), s(Y)) → U10_ga(X, Y, gt_in_ga(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U10_ga(X, Y, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_ga(X, Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
U11_ga(X, Y, le_out_ga(X, Y)) → le_out_ga(s(X), s(Y))
U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, [], [], []) → part_out_gaaa(X, [], [], [])
U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
le_in_aa(s(X), s(Y)) → U11_aa(X, Y, le_in_aa(X, Y))
le_in_aa(0, s(X)) → le_out_aa(0, s(X))
le_in_aa(0, 0) → le_out_aa(0, 0)
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_aaaa(X, [], [], []) → part_out_aaaa(X, [], [], [])
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa([], []) → qs_out_aa([], [])
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
app_in_gaa([], Ys, Ys) → app_out_gaa([], Ys, Ys)
U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
app_in_gag(.(X, Xs), Ys, .(X, Zs)) → U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
app_in_gag([], Ys, Ys) → app_out_gag([], Ys, Ys)
U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) → app_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt_in_aa(x1, x2)  =  gt_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)
part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
gt_in_ga(x1, x2)  =  gt_in_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
0  =  0
gt_out_ga(x1, x2)  =  gt_out_ga(x2)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x2, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
le_in_ga(x1, x2)  =  le_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x3)
le_out_ga(x1, x2)  =  le_out_ga
U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x3)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
le_in_aa(x1, x2)  =  le_in_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
le_out_aa(x1, x2)  =  le_out_aa(x1)
U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x3)
gt_out_gg(x1, x2)  =  gt_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x3)
le_out_gg(x1, x2)  =  le_out_gg
U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x2, x6)
[]  =  []
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)
qs_out_ga(x1, x2)  =  qs_out_ga(x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)
qs_out_aa(x1, x2)  =  qs_out_aa
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)
app_out_gaa(x1, x2, x3)  =  app_out_gaa
U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x5)
app_out_gag(x1, x2, x3)  =  app_out_gag(x2)
qs_out_ag(x1, x2)  =  qs_out_ag


Using Dependency Pairs [1,30] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

QS_IN_AG(.(X, Xs), Ys) → U1_AG(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
QS_IN_AG(.(X, Xs), Ys) → PART_IN_AAAA(X, Xs, Littles, Bigs)
PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_AAAA(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) → GT_IN_AA(X, Y)
GT_IN_AA(s(X), s(Y)) → U10_AA(X, Y, gt_in_aa(X, Y))
GT_IN_AA(s(X), s(Y)) → GT_IN_AA(X, Y)
U5_AAAA(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_AAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U5_AAAA(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → PART_IN_GAAA(X, Xs, Ls, Bs)
PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GAAA(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) → GT_IN_GA(X, Y)
GT_IN_GA(s(X), s(Y)) → U10_GA(X, Y, gt_in_ga(X, Y))
GT_IN_GA(s(X), s(Y)) → GT_IN_GA(X, Y)
U5_GAAA(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_GAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U5_GAAA(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → PART_IN_GAAA(X, Xs, Ls, Bs)
PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_GAAA(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → LE_IN_GA(X, Y)
LE_IN_GA(s(X), s(Y)) → U11_GA(X, Y, le_in_ga(X, Y))
LE_IN_GA(s(X), s(Y)) → LE_IN_GA(X, Y)
U7_GAAA(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_GAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U7_GAAA(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → PART_IN_GAAA(X, Xs, Ls, Bs)
PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_AAAA(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → LE_IN_AA(X, Y)
LE_IN_AA(s(X), s(Y)) → U11_AA(X, Y, le_in_aa(X, Y))
LE_IN_AA(s(X), s(Y)) → LE_IN_AA(X, Y)
U7_AAAA(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_AAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U7_AAAA(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → PART_IN_GAAA(X, Xs, Ls, Bs)
U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_AG(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → QS_IN_GA(Littles, Ls)
QS_IN_GA(.(X, Xs), Ys) → U1_GA(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
QS_IN_GA(.(X, Xs), Ys) → PART_IN_GGAA(X, Xs, Littles, Bigs)
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GGAA(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → GT_IN_GG(X, Y)
GT_IN_GG(s(X), s(Y)) → U10_GG(X, Y, gt_in_gg(X, Y))
GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)
U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_GGAA(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → LE_IN_GG(X, Y)
LE_IN_GG(s(X), s(Y)) → U11_GG(X, Y, le_in_gg(X, Y))
LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)
U7_GGAA(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U7_GGAA(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_GA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → QS_IN_GA(Littles, Ls)
U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_GA(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → QS_IN_GA(Bigs, Bs)
U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_GA(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → APP_IN_GGA(Ls, .(X, Bs), Ys)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U9_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)
U2_AG(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_AG(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U2_AG(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → QS_IN_AA(Bigs, Bs)
QS_IN_AA(.(X, Xs), Ys) → U1_AA(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
QS_IN_AA(.(X, Xs), Ys) → PART_IN_AAAA(X, Xs, Littles, Bigs)
U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_AA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → QS_IN_GA(Littles, Ls)
U2_AA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_AA(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U2_AA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → QS_IN_AA(Bigs, Bs)
U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_AA(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → APP_IN_GAA(Ls, .(X, Bs), Ys)
APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) → U9_GAA(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GAA(Xs, Ys, Zs)
U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_AG(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → APP_IN_GAG(Ls, .(X, Bs), Ys)
APP_IN_GAG(.(X, Xs), Ys, .(X, Zs)) → U9_GAG(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
APP_IN_GAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

qs_in_ag(.(X, Xs), Ys) → U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
gt_in_aa(s(X), s(Y)) → U10_aa(X, Y, gt_in_aa(X, Y))
gt_in_aa(s(0), 0) → gt_out_aa(s(0), 0)
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
gt_in_ga(s(X), s(Y)) → U10_ga(X, Y, gt_in_ga(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U10_ga(X, Y, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_ga(X, Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
U11_ga(X, Y, le_out_ga(X, Y)) → le_out_ga(s(X), s(Y))
U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, [], [], []) → part_out_gaaa(X, [], [], [])
U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
le_in_aa(s(X), s(Y)) → U11_aa(X, Y, le_in_aa(X, Y))
le_in_aa(0, s(X)) → le_out_aa(0, s(X))
le_in_aa(0, 0) → le_out_aa(0, 0)
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_aaaa(X, [], [], []) → part_out_aaaa(X, [], [], [])
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa([], []) → qs_out_aa([], [])
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
app_in_gaa([], Ys, Ys) → app_out_gaa([], Ys, Ys)
U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
app_in_gag(.(X, Xs), Ys, .(X, Zs)) → U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
app_in_gag([], Ys, Ys) → app_out_gag([], Ys, Ys)
U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) → app_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt_in_aa(x1, x2)  =  gt_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)
part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
gt_in_ga(x1, x2)  =  gt_in_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
0  =  0
gt_out_ga(x1, x2)  =  gt_out_ga(x2)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x2, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
le_in_ga(x1, x2)  =  le_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x3)
le_out_ga(x1, x2)  =  le_out_ga
U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x3)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
le_in_aa(x1, x2)  =  le_in_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
le_out_aa(x1, x2)  =  le_out_aa(x1)
U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x3)
gt_out_gg(x1, x2)  =  gt_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x3)
le_out_gg(x1, x2)  =  le_out_gg
U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x2, x6)
[]  =  []
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)
qs_out_ga(x1, x2)  =  qs_out_ga(x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)
qs_out_aa(x1, x2)  =  qs_out_aa
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)
app_out_gaa(x1, x2, x3)  =  app_out_gaa
U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x5)
app_out_gag(x1, x2, x3)  =  app_out_gag(x2)
qs_out_ag(x1, x2)  =  qs_out_ag
U5_GAAA(x1, x2, x3, x4, x5, x6)  =  U5_GAAA(x1, x6)
U5_AAAA(x1, x2, x3, x4, x5, x6)  =  U5_AAAA(x6)
U8_GAAA(x1, x2, x3, x4, x5, x6)  =  U8_GAAA(x6)
U10_GA(x1, x2, x3)  =  U10_GA(x3)
LE_IN_GG(x1, x2)  =  LE_IN_GG(x1, x2)
U6_GGAA(x1, x2, x3, x4, x5, x6)  =  U6_GGAA(x2, x6)
U6_AAAA(x1, x2, x3, x4, x5, x6)  =  U6_AAAA(x2, x6)
U9_GGA(x1, x2, x3, x4, x5)  =  U9_GGA(x1, x5)
U10_AA(x1, x2, x3)  =  U10_AA(x3)
LE_IN_GA(x1, x2)  =  LE_IN_GA(x1)
U7_GGAA(x1, x2, x3, x4, x5, x6)  =  U7_GGAA(x1, x2, x3, x6)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
LE_IN_AA(x1, x2)  =  LE_IN_AA
U3_AA(x1, x2, x3, x4, x5)  =  U3_AA(x4, x5)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x3, x4, x5)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x4, x5)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x4)
U9_GAA(x1, x2, x3, x4, x5)  =  U9_GAA(x5)
QS_IN_AA(x1, x2)  =  QS_IN_AA
U2_AA(x1, x2, x3, x4, x5)  =  U2_AA(x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x4, x5)
GT_IN_GG(x1, x2)  =  GT_IN_GG(x1, x2)
PART_IN_GAAA(x1, x2, x3, x4)  =  PART_IN_GAAA(x1)
APP_IN_GAA(x1, x2, x3)  =  APP_IN_GAA(x1)
U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)
U7_AAAA(x1, x2, x3, x4, x5, x6)  =  U7_AAAA(x6)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
U8_AAAA(x1, x2, x3, x4, x5, x6)  =  U8_AAAA(x6)
QS_IN_AG(x1, x2)  =  QS_IN_AG(x2)
QS_IN_GA(x1, x2)  =  QS_IN_GA(x1)
PART_IN_GGAA(x1, x2, x3, x4)  =  PART_IN_GGAA(x1, x2)
U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)
PART_IN_AAAA(x1, x2, x3, x4)  =  PART_IN_AAAA
U8_GGAA(x1, x2, x3, x4, x5, x6)  =  U8_GGAA(x2, x6)
U2_AG(x1, x2, x3, x4, x5)  =  U2_AG(x3, x5)
U9_GAG(x1, x2, x3, x4, x5)  =  U9_GAG(x5)
U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)
U4_AA(x1, x2, x3, x4)  =  U4_AA(x4)
U6_GAAA(x1, x2, x3, x4, x5, x6)  =  U6_GAAA(x2, x6)
GT_IN_AA(x1, x2)  =  GT_IN_AA
U11_GA(x1, x2, x3)  =  U11_GA(x3)
U4_AG(x1, x2, x3, x4)  =  U4_AG(x4)
U11_GG(x1, x2, x3)  =  U11_GG(x3)
U10_GG(x1, x2, x3)  =  U10_GG(x3)
GT_IN_GA(x1, x2)  =  GT_IN_GA(x1)
U11_AA(x1, x2, x3)  =  U11_AA(x3)
APP_IN_GAG(x1, x2, x3)  =  APP_IN_GAG(x1, x3)
U7_GAAA(x1, x2, x3, x4, x5, x6)  =  U7_GAAA(x1, x6)

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:

QS_IN_AG(.(X, Xs), Ys) → U1_AG(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
QS_IN_AG(.(X, Xs), Ys) → PART_IN_AAAA(X, Xs, Littles, Bigs)
PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_AAAA(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) → GT_IN_AA(X, Y)
GT_IN_AA(s(X), s(Y)) → U10_AA(X, Y, gt_in_aa(X, Y))
GT_IN_AA(s(X), s(Y)) → GT_IN_AA(X, Y)
U5_AAAA(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_AAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U5_AAAA(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → PART_IN_GAAA(X, Xs, Ls, Bs)
PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GAAA(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) → GT_IN_GA(X, Y)
GT_IN_GA(s(X), s(Y)) → U10_GA(X, Y, gt_in_ga(X, Y))
GT_IN_GA(s(X), s(Y)) → GT_IN_GA(X, Y)
U5_GAAA(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_GAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U5_GAAA(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → PART_IN_GAAA(X, Xs, Ls, Bs)
PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_GAAA(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → LE_IN_GA(X, Y)
LE_IN_GA(s(X), s(Y)) → U11_GA(X, Y, le_in_ga(X, Y))
LE_IN_GA(s(X), s(Y)) → LE_IN_GA(X, Y)
U7_GAAA(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_GAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U7_GAAA(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → PART_IN_GAAA(X, Xs, Ls, Bs)
PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_AAAA(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → LE_IN_AA(X, Y)
LE_IN_AA(s(X), s(Y)) → U11_AA(X, Y, le_in_aa(X, Y))
LE_IN_AA(s(X), s(Y)) → LE_IN_AA(X, Y)
U7_AAAA(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_AAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U7_AAAA(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → PART_IN_GAAA(X, Xs, Ls, Bs)
U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_AG(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → QS_IN_GA(Littles, Ls)
QS_IN_GA(.(X, Xs), Ys) → U1_GA(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
QS_IN_GA(.(X, Xs), Ys) → PART_IN_GGAA(X, Xs, Littles, Bigs)
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GGAA(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → GT_IN_GG(X, Y)
GT_IN_GG(s(X), s(Y)) → U10_GG(X, Y, gt_in_gg(X, Y))
GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)
U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_GGAA(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → LE_IN_GG(X, Y)
LE_IN_GG(s(X), s(Y)) → U11_GG(X, Y, le_in_gg(X, Y))
LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)
U7_GGAA(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U7_GGAA(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_GA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → QS_IN_GA(Littles, Ls)
U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_GA(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → QS_IN_GA(Bigs, Bs)
U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_GA(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → APP_IN_GGA(Ls, .(X, Bs), Ys)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U9_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)
U2_AG(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_AG(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U2_AG(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → QS_IN_AA(Bigs, Bs)
QS_IN_AA(.(X, Xs), Ys) → U1_AA(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
QS_IN_AA(.(X, Xs), Ys) → PART_IN_AAAA(X, Xs, Littles, Bigs)
U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_AA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → QS_IN_GA(Littles, Ls)
U2_AA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_AA(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U2_AA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → QS_IN_AA(Bigs, Bs)
U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_AA(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → APP_IN_GAA(Ls, .(X, Bs), Ys)
APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) → U9_GAA(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GAA(Xs, Ys, Zs)
U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_AG(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → APP_IN_GAG(Ls, .(X, Bs), Ys)
APP_IN_GAG(.(X, Xs), Ys, .(X, Zs)) → U9_GAG(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
APP_IN_GAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

qs_in_ag(.(X, Xs), Ys) → U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
gt_in_aa(s(X), s(Y)) → U10_aa(X, Y, gt_in_aa(X, Y))
gt_in_aa(s(0), 0) → gt_out_aa(s(0), 0)
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
gt_in_ga(s(X), s(Y)) → U10_ga(X, Y, gt_in_ga(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U10_ga(X, Y, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_ga(X, Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
U11_ga(X, Y, le_out_ga(X, Y)) → le_out_ga(s(X), s(Y))
U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, [], [], []) → part_out_gaaa(X, [], [], [])
U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
le_in_aa(s(X), s(Y)) → U11_aa(X, Y, le_in_aa(X, Y))
le_in_aa(0, s(X)) → le_out_aa(0, s(X))
le_in_aa(0, 0) → le_out_aa(0, 0)
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_aaaa(X, [], [], []) → part_out_aaaa(X, [], [], [])
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa([], []) → qs_out_aa([], [])
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
app_in_gaa([], Ys, Ys) → app_out_gaa([], Ys, Ys)
U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
app_in_gag(.(X, Xs), Ys, .(X, Zs)) → U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
app_in_gag([], Ys, Ys) → app_out_gag([], Ys, Ys)
U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) → app_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt_in_aa(x1, x2)  =  gt_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)
part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
gt_in_ga(x1, x2)  =  gt_in_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
0  =  0
gt_out_ga(x1, x2)  =  gt_out_ga(x2)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x2, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
le_in_ga(x1, x2)  =  le_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x3)
le_out_ga(x1, x2)  =  le_out_ga
U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x3)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
le_in_aa(x1, x2)  =  le_in_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
le_out_aa(x1, x2)  =  le_out_aa(x1)
U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x3)
gt_out_gg(x1, x2)  =  gt_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x3)
le_out_gg(x1, x2)  =  le_out_gg
U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x2, x6)
[]  =  []
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)
qs_out_ga(x1, x2)  =  qs_out_ga(x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)
qs_out_aa(x1, x2)  =  qs_out_aa
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)
app_out_gaa(x1, x2, x3)  =  app_out_gaa
U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x5)
app_out_gag(x1, x2, x3)  =  app_out_gag(x2)
qs_out_ag(x1, x2)  =  qs_out_ag
U5_GAAA(x1, x2, x3, x4, x5, x6)  =  U5_GAAA(x1, x6)
U5_AAAA(x1, x2, x3, x4, x5, x6)  =  U5_AAAA(x6)
U8_GAAA(x1, x2, x3, x4, x5, x6)  =  U8_GAAA(x6)
U10_GA(x1, x2, x3)  =  U10_GA(x3)
LE_IN_GG(x1, x2)  =  LE_IN_GG(x1, x2)
U6_GGAA(x1, x2, x3, x4, x5, x6)  =  U6_GGAA(x2, x6)
U6_AAAA(x1, x2, x3, x4, x5, x6)  =  U6_AAAA(x2, x6)
U9_GGA(x1, x2, x3, x4, x5)  =  U9_GGA(x1, x5)
U10_AA(x1, x2, x3)  =  U10_AA(x3)
LE_IN_GA(x1, x2)  =  LE_IN_GA(x1)
U7_GGAA(x1, x2, x3, x4, x5, x6)  =  U7_GGAA(x1, x2, x3, x6)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
LE_IN_AA(x1, x2)  =  LE_IN_AA
U3_AA(x1, x2, x3, x4, x5)  =  U3_AA(x4, x5)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x3, x4, x5)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x4, x5)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x4)
U9_GAA(x1, x2, x3, x4, x5)  =  U9_GAA(x5)
QS_IN_AA(x1, x2)  =  QS_IN_AA
U2_AA(x1, x2, x3, x4, x5)  =  U2_AA(x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x4, x5)
GT_IN_GG(x1, x2)  =  GT_IN_GG(x1, x2)
PART_IN_GAAA(x1, x2, x3, x4)  =  PART_IN_GAAA(x1)
APP_IN_GAA(x1, x2, x3)  =  APP_IN_GAA(x1)
U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)
U7_AAAA(x1, x2, x3, x4, x5, x6)  =  U7_AAAA(x6)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
U8_AAAA(x1, x2, x3, x4, x5, x6)  =  U8_AAAA(x6)
QS_IN_AG(x1, x2)  =  QS_IN_AG(x2)
QS_IN_GA(x1, x2)  =  QS_IN_GA(x1)
PART_IN_GGAA(x1, x2, x3, x4)  =  PART_IN_GGAA(x1, x2)
U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)
PART_IN_AAAA(x1, x2, x3, x4)  =  PART_IN_AAAA
U8_GGAA(x1, x2, x3, x4, x5, x6)  =  U8_GGAA(x2, x6)
U2_AG(x1, x2, x3, x4, x5)  =  U2_AG(x3, x5)
U9_GAG(x1, x2, x3, x4, x5)  =  U9_GAG(x5)
U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)
U4_AA(x1, x2, x3, x4)  =  U4_AA(x4)
U6_GAAA(x1, x2, x3, x4, x5, x6)  =  U6_GAAA(x2, x6)
GT_IN_AA(x1, x2)  =  GT_IN_AA
U11_GA(x1, x2, x3)  =  U11_GA(x3)
U4_AG(x1, x2, x3, x4)  =  U4_AG(x4)
U11_GG(x1, x2, x3)  =  U11_GG(x3)
U10_GG(x1, x2, x3)  =  U10_GG(x3)
GT_IN_GA(x1, x2)  =  GT_IN_GA(x1)
U11_AA(x1, x2, x3)  =  U11_AA(x3)
APP_IN_GAG(x1, x2, x3)  =  APP_IN_GAG(x1, x3)
U7_GAAA(x1, x2, x3, x4, x5, x6)  =  U7_GAAA(x1, x6)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 13 SCCs with 42 less nodes.

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

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

APP_IN_GAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

qs_in_ag(.(X, Xs), Ys) → U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
gt_in_aa(s(X), s(Y)) → U10_aa(X, Y, gt_in_aa(X, Y))
gt_in_aa(s(0), 0) → gt_out_aa(s(0), 0)
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
gt_in_ga(s(X), s(Y)) → U10_ga(X, Y, gt_in_ga(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U10_ga(X, Y, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_ga(X, Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
U11_ga(X, Y, le_out_ga(X, Y)) → le_out_ga(s(X), s(Y))
U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, [], [], []) → part_out_gaaa(X, [], [], [])
U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
le_in_aa(s(X), s(Y)) → U11_aa(X, Y, le_in_aa(X, Y))
le_in_aa(0, s(X)) → le_out_aa(0, s(X))
le_in_aa(0, 0) → le_out_aa(0, 0)
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_aaaa(X, [], [], []) → part_out_aaaa(X, [], [], [])
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa([], []) → qs_out_aa([], [])
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
app_in_gaa([], Ys, Ys) → app_out_gaa([], Ys, Ys)
U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
app_in_gag(.(X, Xs), Ys, .(X, Zs)) → U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
app_in_gag([], Ys, Ys) → app_out_gag([], Ys, Ys)
U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) → app_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt_in_aa(x1, x2)  =  gt_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)
part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
gt_in_ga(x1, x2)  =  gt_in_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
0  =  0
gt_out_ga(x1, x2)  =  gt_out_ga(x2)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x2, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
le_in_ga(x1, x2)  =  le_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x3)
le_out_ga(x1, x2)  =  le_out_ga
U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x3)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
le_in_aa(x1, x2)  =  le_in_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
le_out_aa(x1, x2)  =  le_out_aa(x1)
U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x3)
gt_out_gg(x1, x2)  =  gt_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x3)
le_out_gg(x1, x2)  =  le_out_gg
U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x2, x6)
[]  =  []
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)
qs_out_ga(x1, x2)  =  qs_out_ga(x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)
qs_out_aa(x1, x2)  =  qs_out_aa
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)
app_out_gaa(x1, x2, x3)  =  app_out_gaa
U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x5)
app_out_gag(x1, x2, x3)  =  app_out_gag(x2)
qs_out_ag(x1, x2)  =  qs_out_ag
APP_IN_GAG(x1, x2, x3)  =  APP_IN_GAG(x1, x3)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

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

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

APP_IN_GAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GAG(Xs, Ys, Zs)

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

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

APP_IN_GAG(.(X, Xs), .(X, Zs)) → APP_IN_GAG(Xs, Zs)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: 



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

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

APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GAA(Xs, Ys, Zs)

The TRS R consists of the following rules:

qs_in_ag(.(X, Xs), Ys) → U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
gt_in_aa(s(X), s(Y)) → U10_aa(X, Y, gt_in_aa(X, Y))
gt_in_aa(s(0), 0) → gt_out_aa(s(0), 0)
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
gt_in_ga(s(X), s(Y)) → U10_ga(X, Y, gt_in_ga(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U10_ga(X, Y, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_ga(X, Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
U11_ga(X, Y, le_out_ga(X, Y)) → le_out_ga(s(X), s(Y))
U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, [], [], []) → part_out_gaaa(X, [], [], [])
U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
le_in_aa(s(X), s(Y)) → U11_aa(X, Y, le_in_aa(X, Y))
le_in_aa(0, s(X)) → le_out_aa(0, s(X))
le_in_aa(0, 0) → le_out_aa(0, 0)
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_aaaa(X, [], [], []) → part_out_aaaa(X, [], [], [])
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa([], []) → qs_out_aa([], [])
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
app_in_gaa([], Ys, Ys) → app_out_gaa([], Ys, Ys)
U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
app_in_gag(.(X, Xs), Ys, .(X, Zs)) → U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
app_in_gag([], Ys, Ys) → app_out_gag([], Ys, Ys)
U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) → app_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt_in_aa(x1, x2)  =  gt_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)
part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
gt_in_ga(x1, x2)  =  gt_in_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
0  =  0
gt_out_ga(x1, x2)  =  gt_out_ga(x2)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x2, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
le_in_ga(x1, x2)  =  le_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x3)
le_out_ga(x1, x2)  =  le_out_ga
U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x3)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
le_in_aa(x1, x2)  =  le_in_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
le_out_aa(x1, x2)  =  le_out_aa(x1)
U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x3)
gt_out_gg(x1, x2)  =  gt_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x3)
le_out_gg(x1, x2)  =  le_out_gg
U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x2, x6)
[]  =  []
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)
qs_out_ga(x1, x2)  =  qs_out_ga(x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)
qs_out_aa(x1, x2)  =  qs_out_aa
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)
app_out_gaa(x1, x2, x3)  =  app_out_gaa
U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x5)
app_out_gag(x1, x2, x3)  =  app_out_gag(x2)
qs_out_ag(x1, x2)  =  qs_out_ag
APP_IN_GAA(x1, x2, x3)  =  APP_IN_GAA(x1)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

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

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

APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GAA(Xs, Ys, Zs)

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

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

APP_IN_GAA(.(X, Xs)) → APP_IN_GAA(Xs)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: 



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

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

APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

qs_in_ag(.(X, Xs), Ys) → U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
gt_in_aa(s(X), s(Y)) → U10_aa(X, Y, gt_in_aa(X, Y))
gt_in_aa(s(0), 0) → gt_out_aa(s(0), 0)
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
gt_in_ga(s(X), s(Y)) → U10_ga(X, Y, gt_in_ga(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U10_ga(X, Y, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_ga(X, Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
U11_ga(X, Y, le_out_ga(X, Y)) → le_out_ga(s(X), s(Y))
U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, [], [], []) → part_out_gaaa(X, [], [], [])
U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
le_in_aa(s(X), s(Y)) → U11_aa(X, Y, le_in_aa(X, Y))
le_in_aa(0, s(X)) → le_out_aa(0, s(X))
le_in_aa(0, 0) → le_out_aa(0, 0)
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_aaaa(X, [], [], []) → part_out_aaaa(X, [], [], [])
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa([], []) → qs_out_aa([], [])
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
app_in_gaa([], Ys, Ys) → app_out_gaa([], Ys, Ys)
U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
app_in_gag(.(X, Xs), Ys, .(X, Zs)) → U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
app_in_gag([], Ys, Ys) → app_out_gag([], Ys, Ys)
U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) → app_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt_in_aa(x1, x2)  =  gt_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)
part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
gt_in_ga(x1, x2)  =  gt_in_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
0  =  0
gt_out_ga(x1, x2)  =  gt_out_ga(x2)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x2, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
le_in_ga(x1, x2)  =  le_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x3)
le_out_ga(x1, x2)  =  le_out_ga
U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x3)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
le_in_aa(x1, x2)  =  le_in_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
le_out_aa(x1, x2)  =  le_out_aa(x1)
U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x3)
gt_out_gg(x1, x2)  =  gt_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x3)
le_out_gg(x1, x2)  =  le_out_gg
U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x2, x6)
[]  =  []
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)
qs_out_ga(x1, x2)  =  qs_out_ga(x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)
qs_out_aa(x1, x2)  =  qs_out_aa
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)
app_out_gaa(x1, x2, x3)  =  app_out_gaa
U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x5)
app_out_gag(x1, x2, x3)  =  app_out_gag(x2)
qs_out_ag(x1, x2)  =  qs_out_ag
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

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

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

APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)

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

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

APP_IN_GGA(.(X, Xs), Ys) → APP_IN_GGA(Xs, Ys)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)

The TRS R consists of the following rules:

qs_in_ag(.(X, Xs), Ys) → U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
gt_in_aa(s(X), s(Y)) → U10_aa(X, Y, gt_in_aa(X, Y))
gt_in_aa(s(0), 0) → gt_out_aa(s(0), 0)
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
gt_in_ga(s(X), s(Y)) → U10_ga(X, Y, gt_in_ga(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U10_ga(X, Y, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_ga(X, Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
U11_ga(X, Y, le_out_ga(X, Y)) → le_out_ga(s(X), s(Y))
U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, [], [], []) → part_out_gaaa(X, [], [], [])
U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
le_in_aa(s(X), s(Y)) → U11_aa(X, Y, le_in_aa(X, Y))
le_in_aa(0, s(X)) → le_out_aa(0, s(X))
le_in_aa(0, 0) → le_out_aa(0, 0)
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_aaaa(X, [], [], []) → part_out_aaaa(X, [], [], [])
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa([], []) → qs_out_aa([], [])
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
app_in_gaa([], Ys, Ys) → app_out_gaa([], Ys, Ys)
U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
app_in_gag(.(X, Xs), Ys, .(X, Zs)) → U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
app_in_gag([], Ys, Ys) → app_out_gag([], Ys, Ys)
U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) → app_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt_in_aa(x1, x2)  =  gt_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)
part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
gt_in_ga(x1, x2)  =  gt_in_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
0  =  0
gt_out_ga(x1, x2)  =  gt_out_ga(x2)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x2, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
le_in_ga(x1, x2)  =  le_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x3)
le_out_ga(x1, x2)  =  le_out_ga
U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x3)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
le_in_aa(x1, x2)  =  le_in_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
le_out_aa(x1, x2)  =  le_out_aa(x1)
U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x3)
gt_out_gg(x1, x2)  =  gt_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x3)
le_out_gg(x1, x2)  =  le_out_gg
U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x2, x6)
[]  =  []
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)
qs_out_ga(x1, x2)  =  qs_out_ga(x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)
qs_out_aa(x1, x2)  =  qs_out_aa
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)
app_out_gaa(x1, x2, x3)  =  app_out_gaa
U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x5)
app_out_gag(x1, x2, x3)  =  app_out_gag(x2)
qs_out_ag(x1, x2)  =  qs_out_ag
LE_IN_GG(x1, x2)  =  LE_IN_GG(x1, x2)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

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

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

LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)

R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: 



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

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

GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)

The TRS R consists of the following rules:

qs_in_ag(.(X, Xs), Ys) → U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
gt_in_aa(s(X), s(Y)) → U10_aa(X, Y, gt_in_aa(X, Y))
gt_in_aa(s(0), 0) → gt_out_aa(s(0), 0)
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
gt_in_ga(s(X), s(Y)) → U10_ga(X, Y, gt_in_ga(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U10_ga(X, Y, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_ga(X, Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
U11_ga(X, Y, le_out_ga(X, Y)) → le_out_ga(s(X), s(Y))
U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, [], [], []) → part_out_gaaa(X, [], [], [])
U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
le_in_aa(s(X), s(Y)) → U11_aa(X, Y, le_in_aa(X, Y))
le_in_aa(0, s(X)) → le_out_aa(0, s(X))
le_in_aa(0, 0) → le_out_aa(0, 0)
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_aaaa(X, [], [], []) → part_out_aaaa(X, [], [], [])
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa([], []) → qs_out_aa([], [])
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
app_in_gaa([], Ys, Ys) → app_out_gaa([], Ys, Ys)
U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
app_in_gag(.(X, Xs), Ys, .(X, Zs)) → U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
app_in_gag([], Ys, Ys) → app_out_gag([], Ys, Ys)
U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) → app_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt_in_aa(x1, x2)  =  gt_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)
part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
gt_in_ga(x1, x2)  =  gt_in_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
0  =  0
gt_out_ga(x1, x2)  =  gt_out_ga(x2)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x2, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
le_in_ga(x1, x2)  =  le_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x3)
le_out_ga(x1, x2)  =  le_out_ga
U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x3)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
le_in_aa(x1, x2)  =  le_in_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
le_out_aa(x1, x2)  =  le_out_aa(x1)
U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x3)
gt_out_gg(x1, x2)  =  gt_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x3)
le_out_gg(x1, x2)  =  le_out_gg
U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x2, x6)
[]  =  []
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)
qs_out_ga(x1, x2)  =  qs_out_ga(x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)
qs_out_aa(x1, x2)  =  qs_out_aa
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)
app_out_gaa(x1, x2, x3)  =  app_out_gaa
U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x5)
app_out_gag(x1, x2, x3)  =  app_out_gag(x2)
qs_out_ag(x1, x2)  =  qs_out_ag
GT_IN_GG(x1, x2)  =  GT_IN_GG(x1, x2)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

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

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

GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)

R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: 



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

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

PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GGAA(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_GGAA(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
U7_GGAA(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)

The TRS R consists of the following rules:

qs_in_ag(.(X, Xs), Ys) → U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
gt_in_aa(s(X), s(Y)) → U10_aa(X, Y, gt_in_aa(X, Y))
gt_in_aa(s(0), 0) → gt_out_aa(s(0), 0)
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
gt_in_ga(s(X), s(Y)) → U10_ga(X, Y, gt_in_ga(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U10_ga(X, Y, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_ga(X, Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
U11_ga(X, Y, le_out_ga(X, Y)) → le_out_ga(s(X), s(Y))
U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, [], [], []) → part_out_gaaa(X, [], [], [])
U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
le_in_aa(s(X), s(Y)) → U11_aa(X, Y, le_in_aa(X, Y))
le_in_aa(0, s(X)) → le_out_aa(0, s(X))
le_in_aa(0, 0) → le_out_aa(0, 0)
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_aaaa(X, [], [], []) → part_out_aaaa(X, [], [], [])
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa([], []) → qs_out_aa([], [])
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
app_in_gaa([], Ys, Ys) → app_out_gaa([], Ys, Ys)
U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
app_in_gag(.(X, Xs), Ys, .(X, Zs)) → U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
app_in_gag([], Ys, Ys) → app_out_gag([], Ys, Ys)
U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) → app_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt_in_aa(x1, x2)  =  gt_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)
part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
gt_in_ga(x1, x2)  =  gt_in_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
0  =  0
gt_out_ga(x1, x2)  =  gt_out_ga(x2)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x2, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
le_in_ga(x1, x2)  =  le_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x3)
le_out_ga(x1, x2)  =  le_out_ga
U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x3)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
le_in_aa(x1, x2)  =  le_in_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
le_out_aa(x1, x2)  =  le_out_aa(x1)
U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x3)
gt_out_gg(x1, x2)  =  gt_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x3)
le_out_gg(x1, x2)  =  le_out_gg
U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x2, x6)
[]  =  []
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)
qs_out_ga(x1, x2)  =  qs_out_ga(x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)
qs_out_aa(x1, x2)  =  qs_out_aa
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)
app_out_gaa(x1, x2, x3)  =  app_out_gaa
U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x5)
app_out_gag(x1, x2, x3)  =  app_out_gag(x2)
qs_out_ag(x1, x2)  =  qs_out_ag
U7_GGAA(x1, x2, x3, x4, x5, x6)  =  U7_GGAA(x1, x2, x3, x6)
PART_IN_GGAA(x1, x2, x3, x4)  =  PART_IN_GGAA(x1, x2)
U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

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

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

PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GGAA(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_GGAA(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
U7_GGAA(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)

The TRS R consists of the following rules:

gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
0  =  0
.(x1, x2)  =  .(x1, x2)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x3)
gt_out_gg(x1, x2)  =  gt_out_gg
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x3)
le_out_gg(x1, x2)  =  le_out_gg
U7_GGAA(x1, x2, x3, x4, x5, x6)  =  U7_GGAA(x1, x2, x3, x6)
PART_IN_GGAA(x1, x2, x3, x4)  =  PART_IN_GGAA(x1, x2)
U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

U7_GGAA(X, Y, Xs, le_out_gg) → PART_IN_GGAA(X, Xs)
PART_IN_GGAA(X, .(Y, Xs)) → U5_GGAA(X, Y, Xs, gt_in_gg(X, Y))
U5_GGAA(X, Y, Xs, gt_out_gg) → PART_IN_GGAA(X, Xs)
PART_IN_GGAA(X, .(Y, Xs)) → U7_GGAA(X, Y, Xs, le_in_gg(X, Y))

The TRS R consists of the following rules:

gt_in_gg(s(X), s(Y)) → U10_gg(gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg
le_in_gg(s(X), s(Y)) → U11_gg(le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
U10_gg(gt_out_gg) → gt_out_gg
U11_gg(le_out_gg) → le_out_gg

The set Q consists of the following terms:

gt_in_gg(x0, x1)
le_in_gg(x0, x1)
U10_gg(x0)
U11_gg(x0)

We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: 



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

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

QS_IN_GA(.(X, Xs), Ys) → U1_GA(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → QS_IN_GA(Littles, Ls)
U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_GA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → QS_IN_GA(Bigs, Bs)

The TRS R consists of the following rules:

qs_in_ag(.(X, Xs), Ys) → U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
gt_in_aa(s(X), s(Y)) → U10_aa(X, Y, gt_in_aa(X, Y))
gt_in_aa(s(0), 0) → gt_out_aa(s(0), 0)
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
gt_in_ga(s(X), s(Y)) → U10_ga(X, Y, gt_in_ga(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U10_ga(X, Y, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_ga(X, Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
U11_ga(X, Y, le_out_ga(X, Y)) → le_out_ga(s(X), s(Y))
U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, [], [], []) → part_out_gaaa(X, [], [], [])
U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
le_in_aa(s(X), s(Y)) → U11_aa(X, Y, le_in_aa(X, Y))
le_in_aa(0, s(X)) → le_out_aa(0, s(X))
le_in_aa(0, 0) → le_out_aa(0, 0)
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_aaaa(X, [], [], []) → part_out_aaaa(X, [], [], [])
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa([], []) → qs_out_aa([], [])
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
app_in_gaa([], Ys, Ys) → app_out_gaa([], Ys, Ys)
U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
app_in_gag(.(X, Xs), Ys, .(X, Zs)) → U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
app_in_gag([], Ys, Ys) → app_out_gag([], Ys, Ys)
U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) → app_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt_in_aa(x1, x2)  =  gt_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)
part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
gt_in_ga(x1, x2)  =  gt_in_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
0  =  0
gt_out_ga(x1, x2)  =  gt_out_ga(x2)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x2, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
le_in_ga(x1, x2)  =  le_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x3)
le_out_ga(x1, x2)  =  le_out_ga
U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x3)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
le_in_aa(x1, x2)  =  le_in_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
le_out_aa(x1, x2)  =  le_out_aa(x1)
U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x3)
gt_out_gg(x1, x2)  =  gt_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x3)
le_out_gg(x1, x2)  =  le_out_gg
U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x2, x6)
[]  =  []
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)
qs_out_ga(x1, x2)  =  qs_out_ga(x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)
qs_out_aa(x1, x2)  =  qs_out_aa
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)
app_out_gaa(x1, x2, x3)  =  app_out_gaa
U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x5)
app_out_gag(x1, x2, x3)  =  app_out_gag(x2)
qs_out_ag(x1, x2)  =  qs_out_ag
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x4, x5)
QS_IN_GA(x1, x2)  =  QS_IN_GA(x1)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

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

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

QS_IN_GA(.(X, Xs), Ys) → U1_GA(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → QS_IN_GA(Littles, Ls)
U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_GA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → QS_IN_GA(Bigs, Bs)

The TRS R consists of the following rules:

part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
qs_in_ga([], []) → qs_out_ga([], [])
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
0  =  0
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x3)
gt_out_gg(x1, x2)  =  gt_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x3)
le_out_gg(x1, x2)  =  le_out_gg
U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x2, x6)
[]  =  []
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)
qs_out_ga(x1, x2)  =  qs_out_ga(x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x4, x5)
QS_IN_GA(x1, x2)  =  QS_IN_GA(x1)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPOrderProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

QS_IN_GA(.(X, Xs)) → U1_GA(X, part_in_ggaa(X, Xs))
U2_GA(X, Bigs, qs_out_ga(Ls)) → QS_IN_GA(Bigs)
U1_GA(X, part_out_ggaa(Littles, Bigs)) → U2_GA(X, Bigs, qs_in_ga(Littles))
U1_GA(X, part_out_ggaa(Littles, Bigs)) → QS_IN_GA(Littles)

The TRS R consists of the following rules:

part_in_ggaa(X, .(Y, Xs)) → U5_ggaa(X, Y, Xs, gt_in_gg(X, Y))
part_in_ggaa(X, .(Y, Xs)) → U7_ggaa(X, Y, Xs, le_in_gg(X, Y))
part_in_ggaa(X, []) → part_out_ggaa([], [])
qs_in_ga(.(X, Xs)) → U1_ga(X, part_in_ggaa(X, Xs))
qs_in_ga([]) → qs_out_ga([])
U5_ggaa(X, Y, Xs, gt_out_gg) → U6_ggaa(Y, part_in_ggaa(X, Xs))
U7_ggaa(X, Y, Xs, le_out_gg) → U8_ggaa(Y, part_in_ggaa(X, Xs))
U1_ga(X, part_out_ggaa(Littles, Bigs)) → U2_ga(X, Bigs, qs_in_ga(Littles))
gt_in_gg(s(X), s(Y)) → U10_gg(gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg
U6_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(.(Y, Ls), Bs)
le_in_gg(s(X), s(Y)) → U11_gg(le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
U8_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(Ls, .(Y, Bs))
U2_ga(X, Bigs, qs_out_ga(Ls)) → U3_ga(X, Ls, qs_in_ga(Bigs))
U10_gg(gt_out_gg) → gt_out_gg
U11_gg(le_out_gg) → le_out_gg
U3_ga(X, Ls, qs_out_ga(Bs)) → U4_ga(app_in_gga(Ls, .(X, Bs)))
U4_ga(app_out_gga(Ys)) → qs_out_ga(Ys)
app_in_gga(.(X, Xs), Ys) → U9_gga(X, app_in_gga(Xs, Ys))
app_in_gga([], Ys) → app_out_gga(Ys)
U9_gga(X, app_out_gga(Zs)) → app_out_gga(.(X, Zs))

The set Q consists of the following terms:

part_in_ggaa(x0, x1)
qs_in_ga(x0)
U5_ggaa(x0, x1, x2, x3)
U7_ggaa(x0, x1, x2, x3)
U1_ga(x0, x1)
gt_in_gg(x0, x1)
U6_ggaa(x0, x1)
le_in_gg(x0, x1)
U8_ggaa(x0, x1)
U2_ga(x0, x1, x2)
U10_gg(x0)
U11_gg(x0)
U3_ga(x0, x1, x2)
U4_ga(x0)
app_in_gga(x0, x1)
U9_gga(x0, x1)

We have to consider all (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


QS_IN_GA(.(X, Xs)) → U1_GA(X, part_in_ggaa(X, Xs))
The remaining pairs can at least be oriented weakly.

U2_GA(X, Bigs, qs_out_ga(Ls)) → QS_IN_GA(Bigs)
U1_GA(X, part_out_ggaa(Littles, Bigs)) → U2_GA(X, Bigs, qs_in_ga(Littles))
U1_GA(X, part_out_ggaa(Littles, Bigs)) → QS_IN_GA(Littles)
Used ordering: Polynomial interpretation [25]:

POL(.(x1, x2)) = 1 + x1 + x2   
POL(0) = 1   
POL(QS_IN_GA(x1)) = x1   
POL(U10_gg(x1)) = 1 + x1   
POL(U11_gg(x1)) = 1   
POL(U1_GA(x1, x2)) = x2   
POL(U1_ga(x1, x2)) = 0   
POL(U2_GA(x1, x2, x3)) = x2   
POL(U2_ga(x1, x2, x3)) = 0   
POL(U3_ga(x1, x2, x3)) = 0   
POL(U4_ga(x1)) = 0   
POL(U5_ggaa(x1, x2, x3, x4)) = 1 + x2 + x3   
POL(U6_ggaa(x1, x2)) = 1 + x1 + x2   
POL(U7_ggaa(x1, x2, x3, x4)) = 1 + x2 + x3   
POL(U8_ggaa(x1, x2)) = 1 + x1 + x2   
POL(U9_gga(x1, x2)) = x2   
POL([]) = 0   
POL(app_in_gga(x1, x2)) = 1 + x1   
POL(app_out_gga(x1)) = 1   
POL(gt_in_gg(x1, x2)) = x2   
POL(gt_out_gg) = 1   
POL(le_in_gg(x1, x2)) = 1 + x1   
POL(le_out_gg) = 1   
POL(part_in_ggaa(x1, x2)) = x2   
POL(part_out_ggaa(x1, x2)) = x1 + x2   
POL(qs_in_ga(x1)) = 0   
POL(qs_out_ga(x1)) = 0   
POL(s(x1)) = 1 + x1   

The following usable rules [17] were oriented:

part_in_ggaa(X, .(Y, Xs)) → U5_ggaa(X, Y, Xs, gt_in_gg(X, Y))
U8_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(Ls, .(Y, Bs))
part_in_ggaa(X, .(Y, Xs)) → U7_ggaa(X, Y, Xs, le_in_gg(X, Y))
U5_ggaa(X, Y, Xs, gt_out_gg) → U6_ggaa(Y, part_in_ggaa(X, Xs))
U7_ggaa(X, Y, Xs, le_out_gg) → U8_ggaa(Y, part_in_ggaa(X, Xs))
part_in_ggaa(X, []) → part_out_ggaa([], [])
U6_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(.(Y, Ls), Bs)



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ DependencyGraphProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

U2_GA(X, Bigs, qs_out_ga(Ls)) → QS_IN_GA(Bigs)
U1_GA(X, part_out_ggaa(Littles, Bigs)) → U2_GA(X, Bigs, qs_in_ga(Littles))
U1_GA(X, part_out_ggaa(Littles, Bigs)) → QS_IN_GA(Littles)

The TRS R consists of the following rules:

part_in_ggaa(X, .(Y, Xs)) → U5_ggaa(X, Y, Xs, gt_in_gg(X, Y))
part_in_ggaa(X, .(Y, Xs)) → U7_ggaa(X, Y, Xs, le_in_gg(X, Y))
part_in_ggaa(X, []) → part_out_ggaa([], [])
qs_in_ga(.(X, Xs)) → U1_ga(X, part_in_ggaa(X, Xs))
qs_in_ga([]) → qs_out_ga([])
U5_ggaa(X, Y, Xs, gt_out_gg) → U6_ggaa(Y, part_in_ggaa(X, Xs))
U7_ggaa(X, Y, Xs, le_out_gg) → U8_ggaa(Y, part_in_ggaa(X, Xs))
U1_ga(X, part_out_ggaa(Littles, Bigs)) → U2_ga(X, Bigs, qs_in_ga(Littles))
gt_in_gg(s(X), s(Y)) → U10_gg(gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg
U6_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(.(Y, Ls), Bs)
le_in_gg(s(X), s(Y)) → U11_gg(le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
U8_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(Ls, .(Y, Bs))
U2_ga(X, Bigs, qs_out_ga(Ls)) → U3_ga(X, Ls, qs_in_ga(Bigs))
U10_gg(gt_out_gg) → gt_out_gg
U11_gg(le_out_gg) → le_out_gg
U3_ga(X, Ls, qs_out_ga(Bs)) → U4_ga(app_in_gga(Ls, .(X, Bs)))
U4_ga(app_out_gga(Ys)) → qs_out_ga(Ys)
app_in_gga(.(X, Xs), Ys) → U9_gga(X, app_in_gga(Xs, Ys))
app_in_gga([], Ys) → app_out_gga(Ys)
U9_gga(X, app_out_gga(Zs)) → app_out_gga(.(X, Zs))

The set Q consists of the following terms:

part_in_ggaa(x0, x1)
qs_in_ga(x0)
U5_ggaa(x0, x1, x2, x3)
U7_ggaa(x0, x1, x2, x3)
U1_ga(x0, x1)
gt_in_gg(x0, x1)
U6_ggaa(x0, x1)
le_in_gg(x0, x1)
U8_ggaa(x0, x1)
U2_ga(x0, x1, x2)
U10_gg(x0)
U11_gg(x0)
U3_ga(x0, x1, x2)
U4_ga(x0)
app_in_gga(x0, x1)
U9_gga(x0, x1)

We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 3 less nodes.

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

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

LE_IN_AA(s(X), s(Y)) → LE_IN_AA(X, Y)

The TRS R consists of the following rules:

qs_in_ag(.(X, Xs), Ys) → U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
gt_in_aa(s(X), s(Y)) → U10_aa(X, Y, gt_in_aa(X, Y))
gt_in_aa(s(0), 0) → gt_out_aa(s(0), 0)
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
gt_in_ga(s(X), s(Y)) → U10_ga(X, Y, gt_in_ga(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U10_ga(X, Y, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_ga(X, Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
U11_ga(X, Y, le_out_ga(X, Y)) → le_out_ga(s(X), s(Y))
U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, [], [], []) → part_out_gaaa(X, [], [], [])
U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
le_in_aa(s(X), s(Y)) → U11_aa(X, Y, le_in_aa(X, Y))
le_in_aa(0, s(X)) → le_out_aa(0, s(X))
le_in_aa(0, 0) → le_out_aa(0, 0)
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_aaaa(X, [], [], []) → part_out_aaaa(X, [], [], [])
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa([], []) → qs_out_aa([], [])
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
app_in_gaa([], Ys, Ys) → app_out_gaa([], Ys, Ys)
U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
app_in_gag(.(X, Xs), Ys, .(X, Zs)) → U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
app_in_gag([], Ys, Ys) → app_out_gag([], Ys, Ys)
U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) → app_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt_in_aa(x1, x2)  =  gt_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)
part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
gt_in_ga(x1, x2)  =  gt_in_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
0  =  0
gt_out_ga(x1, x2)  =  gt_out_ga(x2)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x2, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
le_in_ga(x1, x2)  =  le_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x3)
le_out_ga(x1, x2)  =  le_out_ga
U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x3)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
le_in_aa(x1, x2)  =  le_in_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
le_out_aa(x1, x2)  =  le_out_aa(x1)
U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x3)
gt_out_gg(x1, x2)  =  gt_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x3)
le_out_gg(x1, x2)  =  le_out_gg
U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x2, x6)
[]  =  []
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)
qs_out_ga(x1, x2)  =  qs_out_ga(x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)
qs_out_aa(x1, x2)  =  qs_out_aa
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)
app_out_gaa(x1, x2, x3)  =  app_out_gaa
U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x5)
app_out_gag(x1, x2, x3)  =  app_out_gag(x2)
qs_out_ag(x1, x2)  =  qs_out_ag
LE_IN_AA(x1, x2)  =  LE_IN_AA

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

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

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

LE_IN_AA(s(X), s(Y)) → LE_IN_AA(X, Y)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LE_IN_AA(x1, x2)  =  LE_IN_AA

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ NonTerminationProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

LE_IN_AA → LE_IN_AA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

LE_IN_AA → LE_IN_AA

The TRS R consists of the following rules:none


s = LE_IN_AA evaluates to t =LE_IN_AA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from LE_IN_AA to LE_IN_AA.





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

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

LE_IN_GA(s(X), s(Y)) → LE_IN_GA(X, Y)

The TRS R consists of the following rules:

qs_in_ag(.(X, Xs), Ys) → U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
gt_in_aa(s(X), s(Y)) → U10_aa(X, Y, gt_in_aa(X, Y))
gt_in_aa(s(0), 0) → gt_out_aa(s(0), 0)
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
gt_in_ga(s(X), s(Y)) → U10_ga(X, Y, gt_in_ga(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U10_ga(X, Y, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_ga(X, Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
U11_ga(X, Y, le_out_ga(X, Y)) → le_out_ga(s(X), s(Y))
U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, [], [], []) → part_out_gaaa(X, [], [], [])
U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
le_in_aa(s(X), s(Y)) → U11_aa(X, Y, le_in_aa(X, Y))
le_in_aa(0, s(X)) → le_out_aa(0, s(X))
le_in_aa(0, 0) → le_out_aa(0, 0)
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_aaaa(X, [], [], []) → part_out_aaaa(X, [], [], [])
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa([], []) → qs_out_aa([], [])
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
app_in_gaa([], Ys, Ys) → app_out_gaa([], Ys, Ys)
U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
app_in_gag(.(X, Xs), Ys, .(X, Zs)) → U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
app_in_gag([], Ys, Ys) → app_out_gag([], Ys, Ys)
U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) → app_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt_in_aa(x1, x2)  =  gt_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)
part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
gt_in_ga(x1, x2)  =  gt_in_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
0  =  0
gt_out_ga(x1, x2)  =  gt_out_ga(x2)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x2, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
le_in_ga(x1, x2)  =  le_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x3)
le_out_ga(x1, x2)  =  le_out_ga
U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x3)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
le_in_aa(x1, x2)  =  le_in_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
le_out_aa(x1, x2)  =  le_out_aa(x1)
U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x3)
gt_out_gg(x1, x2)  =  gt_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x3)
le_out_gg(x1, x2)  =  le_out_gg
U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x2, x6)
[]  =  []
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)
qs_out_ga(x1, x2)  =  qs_out_ga(x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)
qs_out_aa(x1, x2)  =  qs_out_aa
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)
app_out_gaa(x1, x2, x3)  =  app_out_gaa
U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x5)
app_out_gag(x1, x2, x3)  =  app_out_gag(x2)
qs_out_ag(x1, x2)  =  qs_out_ag
LE_IN_GA(x1, x2)  =  LE_IN_GA(x1)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

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

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

LE_IN_GA(s(X), s(Y)) → LE_IN_GA(X, Y)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LE_IN_GA(x1, x2)  =  LE_IN_GA(x1)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

LE_IN_GA(s(X)) → LE_IN_GA(X)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: 



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

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

GT_IN_GA(s(X), s(Y)) → GT_IN_GA(X, Y)

The TRS R consists of the following rules:

qs_in_ag(.(X, Xs), Ys) → U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
gt_in_aa(s(X), s(Y)) → U10_aa(X, Y, gt_in_aa(X, Y))
gt_in_aa(s(0), 0) → gt_out_aa(s(0), 0)
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
gt_in_ga(s(X), s(Y)) → U10_ga(X, Y, gt_in_ga(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U10_ga(X, Y, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_ga(X, Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
U11_ga(X, Y, le_out_ga(X, Y)) → le_out_ga(s(X), s(Y))
U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, [], [], []) → part_out_gaaa(X, [], [], [])
U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
le_in_aa(s(X), s(Y)) → U11_aa(X, Y, le_in_aa(X, Y))
le_in_aa(0, s(X)) → le_out_aa(0, s(X))
le_in_aa(0, 0) → le_out_aa(0, 0)
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_aaaa(X, [], [], []) → part_out_aaaa(X, [], [], [])
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa([], []) → qs_out_aa([], [])
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
app_in_gaa([], Ys, Ys) → app_out_gaa([], Ys, Ys)
U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
app_in_gag(.(X, Xs), Ys, .(X, Zs)) → U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
app_in_gag([], Ys, Ys) → app_out_gag([], Ys, Ys)
U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) → app_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt_in_aa(x1, x2)  =  gt_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)
part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
gt_in_ga(x1, x2)  =  gt_in_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
0  =  0
gt_out_ga(x1, x2)  =  gt_out_ga(x2)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x2, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
le_in_ga(x1, x2)  =  le_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x3)
le_out_ga(x1, x2)  =  le_out_ga
U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x3)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
le_in_aa(x1, x2)  =  le_in_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
le_out_aa(x1, x2)  =  le_out_aa(x1)
U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x3)
gt_out_gg(x1, x2)  =  gt_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x3)
le_out_gg(x1, x2)  =  le_out_gg
U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x2, x6)
[]  =  []
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)
qs_out_ga(x1, x2)  =  qs_out_ga(x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)
qs_out_aa(x1, x2)  =  qs_out_aa
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)
app_out_gaa(x1, x2, x3)  =  app_out_gaa
U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x5)
app_out_gag(x1, x2, x3)  =  app_out_gag(x2)
qs_out_ag(x1, x2)  =  qs_out_ag
GT_IN_GA(x1, x2)  =  GT_IN_GA(x1)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

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

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

GT_IN_GA(s(X), s(Y)) → GT_IN_GA(X, Y)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
GT_IN_GA(x1, x2)  =  GT_IN_GA(x1)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

GT_IN_GA(s(X)) → GT_IN_GA(X)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: 



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

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

PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_GAAA(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
U5_GAAA(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → PART_IN_GAAA(X, Xs, Ls, Bs)
U7_GAAA(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → PART_IN_GAAA(X, Xs, Ls, Bs)
PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GAAA(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))

The TRS R consists of the following rules:

qs_in_ag(.(X, Xs), Ys) → U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
gt_in_aa(s(X), s(Y)) → U10_aa(X, Y, gt_in_aa(X, Y))
gt_in_aa(s(0), 0) → gt_out_aa(s(0), 0)
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
gt_in_ga(s(X), s(Y)) → U10_ga(X, Y, gt_in_ga(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U10_ga(X, Y, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_ga(X, Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
U11_ga(X, Y, le_out_ga(X, Y)) → le_out_ga(s(X), s(Y))
U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, [], [], []) → part_out_gaaa(X, [], [], [])
U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
le_in_aa(s(X), s(Y)) → U11_aa(X, Y, le_in_aa(X, Y))
le_in_aa(0, s(X)) → le_out_aa(0, s(X))
le_in_aa(0, 0) → le_out_aa(0, 0)
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_aaaa(X, [], [], []) → part_out_aaaa(X, [], [], [])
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa([], []) → qs_out_aa([], [])
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
app_in_gaa([], Ys, Ys) → app_out_gaa([], Ys, Ys)
U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
app_in_gag(.(X, Xs), Ys, .(X, Zs)) → U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
app_in_gag([], Ys, Ys) → app_out_gag([], Ys, Ys)
U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) → app_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt_in_aa(x1, x2)  =  gt_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)
part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
gt_in_ga(x1, x2)  =  gt_in_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
0  =  0
gt_out_ga(x1, x2)  =  gt_out_ga(x2)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x2, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
le_in_ga(x1, x2)  =  le_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x3)
le_out_ga(x1, x2)  =  le_out_ga
U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x3)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
le_in_aa(x1, x2)  =  le_in_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
le_out_aa(x1, x2)  =  le_out_aa(x1)
U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x3)
gt_out_gg(x1, x2)  =  gt_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x3)
le_out_gg(x1, x2)  =  le_out_gg
U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x2, x6)
[]  =  []
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)
qs_out_ga(x1, x2)  =  qs_out_ga(x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)
qs_out_aa(x1, x2)  =  qs_out_aa
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)
app_out_gaa(x1, x2, x3)  =  app_out_gaa
U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x5)
app_out_gag(x1, x2, x3)  =  app_out_gag(x2)
qs_out_ag(x1, x2)  =  qs_out_ag
U5_GAAA(x1, x2, x3, x4, x5, x6)  =  U5_GAAA(x1, x6)
PART_IN_GAAA(x1, x2, x3, x4)  =  PART_IN_GAAA(x1)
U7_GAAA(x1, x2, x3, x4, x5, x6)  =  U7_GAAA(x1, x6)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

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

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

PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_GAAA(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
U5_GAAA(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → PART_IN_GAAA(X, Xs, Ls, Bs)
U7_GAAA(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → PART_IN_GAAA(X, Xs, Ls, Bs)
PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GAAA(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))

The TRS R consists of the following rules:

le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_ga(X, Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
gt_in_ga(s(X), s(Y)) → U10_ga(X, Y, gt_in_ga(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U11_ga(X, Y, le_out_ga(X, Y)) → le_out_ga(s(X), s(Y))
U10_ga(X, Y, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))

The argument filtering Pi contains the following mapping:
gt_in_ga(x1, x2)  =  gt_in_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
0  =  0
gt_out_ga(x1, x2)  =  gt_out_ga(x2)
le_in_ga(x1, x2)  =  le_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x3)
le_out_ga(x1, x2)  =  le_out_ga
.(x1, x2)  =  .(x1, x2)
U5_GAAA(x1, x2, x3, x4, x5, x6)  =  U5_GAAA(x1, x6)
PART_IN_GAAA(x1, x2, x3, x4)  =  PART_IN_GAAA(x1)
U7_GAAA(x1, x2, x3, x4, x5, x6)  =  U7_GAAA(x1, x6)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

PART_IN_GAAA(X) → U7_GAAA(X, le_in_ga(X))
U5_GAAA(X, gt_out_ga(Y)) → PART_IN_GAAA(X)
U7_GAAA(X, le_out_ga) → PART_IN_GAAA(X)
PART_IN_GAAA(X) → U5_GAAA(X, gt_in_ga(X))

The TRS R consists of the following rules:

le_in_ga(s(X)) → U11_ga(le_in_ga(X))
le_in_ga(0) → le_out_ga
gt_in_ga(s(X)) → U10_ga(gt_in_ga(X))
gt_in_ga(s(0)) → gt_out_ga(0)
U11_ga(le_out_ga) → le_out_ga
U10_ga(gt_out_ga(Y)) → gt_out_ga(s(Y))

The set Q consists of the following terms:

le_in_ga(x0)
gt_in_ga(x0)
U11_ga(x0)
U10_ga(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule PART_IN_GAAA(X) → U7_GAAA(X, le_in_ga(X)) at position [1] we obtained the following new rules:

PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga)
PART_IN_GAAA(s(x0)) → U7_GAAA(s(x0), U11_ga(le_in_ga(x0)))



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

U5_GAAA(X, gt_out_ga(Y)) → PART_IN_GAAA(X)
U7_GAAA(X, le_out_ga) → PART_IN_GAAA(X)
PART_IN_GAAA(s(x0)) → U7_GAAA(s(x0), U11_ga(le_in_ga(x0)))
PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga)
PART_IN_GAAA(X) → U5_GAAA(X, gt_in_ga(X))

The TRS R consists of the following rules:

le_in_ga(s(X)) → U11_ga(le_in_ga(X))
le_in_ga(0) → le_out_ga
gt_in_ga(s(X)) → U10_ga(gt_in_ga(X))
gt_in_ga(s(0)) → gt_out_ga(0)
U11_ga(le_out_ga) → le_out_ga
U10_ga(gt_out_ga(Y)) → gt_out_ga(s(Y))

The set Q consists of the following terms:

le_in_ga(x0)
gt_in_ga(x0)
U11_ga(x0)
U10_ga(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule PART_IN_GAAA(X) → U5_GAAA(X, gt_in_ga(X)) at position [1] we obtained the following new rules:

PART_IN_GAAA(s(x0)) → U5_GAAA(s(x0), U10_ga(gt_in_ga(x0)))
PART_IN_GAAA(s(0)) → U5_GAAA(s(0), gt_out_ga(0))



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Instantiation
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

PART_IN_GAAA(s(0)) → U5_GAAA(s(0), gt_out_ga(0))
U5_GAAA(X, gt_out_ga(Y)) → PART_IN_GAAA(X)
U7_GAAA(X, le_out_ga) → PART_IN_GAAA(X)
PART_IN_GAAA(s(x0)) → U5_GAAA(s(x0), U10_ga(gt_in_ga(x0)))
PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga)
PART_IN_GAAA(s(x0)) → U7_GAAA(s(x0), U11_ga(le_in_ga(x0)))

The TRS R consists of the following rules:

le_in_ga(s(X)) → U11_ga(le_in_ga(X))
le_in_ga(0) → le_out_ga
gt_in_ga(s(X)) → U10_ga(gt_in_ga(X))
gt_in_ga(s(0)) → gt_out_ga(0)
U11_ga(le_out_ga) → le_out_ga
U10_ga(gt_out_ga(Y)) → gt_out_ga(s(Y))

The set Q consists of the following terms:

le_in_ga(x0)
gt_in_ga(x0)
U11_ga(x0)
U10_ga(x0)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U5_GAAA(X, gt_out_ga(Y)) → PART_IN_GAAA(X) we obtained the following new rules:

U5_GAAA(s(z0), gt_out_ga(x1)) → PART_IN_GAAA(s(z0))
U5_GAAA(s(0), gt_out_ga(0)) → PART_IN_GAAA(s(0))



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Instantiation
                                  ↳ QDP
                                    ↳ Instantiation
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

PART_IN_GAAA(s(0)) → U5_GAAA(s(0), gt_out_ga(0))
U5_GAAA(s(z0), gt_out_ga(x1)) → PART_IN_GAAA(s(z0))
U5_GAAA(s(0), gt_out_ga(0)) → PART_IN_GAAA(s(0))
U7_GAAA(X, le_out_ga) → PART_IN_GAAA(X)
PART_IN_GAAA(s(x0)) → U5_GAAA(s(x0), U10_ga(gt_in_ga(x0)))
PART_IN_GAAA(s(x0)) → U7_GAAA(s(x0), U11_ga(le_in_ga(x0)))
PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga)

The TRS R consists of the following rules:

le_in_ga(s(X)) → U11_ga(le_in_ga(X))
le_in_ga(0) → le_out_ga
gt_in_ga(s(X)) → U10_ga(gt_in_ga(X))
gt_in_ga(s(0)) → gt_out_ga(0)
U11_ga(le_out_ga) → le_out_ga
U10_ga(gt_out_ga(Y)) → gt_out_ga(s(Y))

The set Q consists of the following terms:

le_in_ga(x0)
gt_in_ga(x0)
U11_ga(x0)
U10_ga(x0)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U7_GAAA(X, le_out_ga) → PART_IN_GAAA(X) we obtained the following new rules:

U7_GAAA(s(z0), le_out_ga) → PART_IN_GAAA(s(z0))
U7_GAAA(0, le_out_ga) → PART_IN_GAAA(0)



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Instantiation
                                  ↳ QDP
                                    ↳ Instantiation
                                      ↳ QDP
                                        ↳ DependencyGraphProof
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

PART_IN_GAAA(s(0)) → U5_GAAA(s(0), gt_out_ga(0))
U5_GAAA(s(z0), gt_out_ga(x1)) → PART_IN_GAAA(s(z0))
U5_GAAA(s(0), gt_out_ga(0)) → PART_IN_GAAA(s(0))
U7_GAAA(s(z0), le_out_ga) → PART_IN_GAAA(s(z0))
U7_GAAA(0, le_out_ga) → PART_IN_GAAA(0)
PART_IN_GAAA(s(x0)) → U5_GAAA(s(x0), U10_ga(gt_in_ga(x0)))
PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga)
PART_IN_GAAA(s(x0)) → U7_GAAA(s(x0), U11_ga(le_in_ga(x0)))

The TRS R consists of the following rules:

le_in_ga(s(X)) → U11_ga(le_in_ga(X))
le_in_ga(0) → le_out_ga
gt_in_ga(s(X)) → U10_ga(gt_in_ga(X))
gt_in_ga(s(0)) → gt_out_ga(0)
U11_ga(le_out_ga) → le_out_ga
U10_ga(gt_out_ga(Y)) → gt_out_ga(s(Y))

The set Q consists of the following terms:

le_in_ga(x0)
gt_in_ga(x0)
U11_ga(x0)
U10_ga(x0)

We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Instantiation
                                  ↳ QDP
                                    ↳ Instantiation
                                      ↳ QDP
                                        ↳ DependencyGraphProof
                                          ↳ AND
                                            ↳ QDP
                                              ↳ NonTerminationProof
                                            ↳ QDP
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

PART_IN_GAAA(s(0)) → U5_GAAA(s(0), gt_out_ga(0))
U5_GAAA(s(z0), gt_out_ga(x1)) → PART_IN_GAAA(s(z0))
U5_GAAA(s(0), gt_out_ga(0)) → PART_IN_GAAA(s(0))
U7_GAAA(s(z0), le_out_ga) → PART_IN_GAAA(s(z0))
PART_IN_GAAA(s(x0)) → U5_GAAA(s(x0), U10_ga(gt_in_ga(x0)))
PART_IN_GAAA(s(x0)) → U7_GAAA(s(x0), U11_ga(le_in_ga(x0)))

The TRS R consists of the following rules:

le_in_ga(s(X)) → U11_ga(le_in_ga(X))
le_in_ga(0) → le_out_ga
gt_in_ga(s(X)) → U10_ga(gt_in_ga(X))
gt_in_ga(s(0)) → gt_out_ga(0)
U11_ga(le_out_ga) → le_out_ga
U10_ga(gt_out_ga(Y)) → gt_out_ga(s(Y))

The set Q consists of the following terms:

le_in_ga(x0)
gt_in_ga(x0)
U11_ga(x0)
U10_ga(x0)

We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

The TRS P consists of the following rules:

PART_IN_GAAA(s(0)) → U5_GAAA(s(0), gt_out_ga(0))
U5_GAAA(s(z0), gt_out_ga(x1)) → PART_IN_GAAA(s(z0))
U5_GAAA(s(0), gt_out_ga(0)) → PART_IN_GAAA(s(0))
U7_GAAA(s(z0), le_out_ga) → PART_IN_GAAA(s(z0))
PART_IN_GAAA(s(x0)) → U5_GAAA(s(x0), U10_ga(gt_in_ga(x0)))
PART_IN_GAAA(s(x0)) → U7_GAAA(s(x0), U11_ga(le_in_ga(x0)))

The TRS R consists of the following rules:

le_in_ga(s(X)) → U11_ga(le_in_ga(X))
le_in_ga(0) → le_out_ga
gt_in_ga(s(X)) → U10_ga(gt_in_ga(X))
gt_in_ga(s(0)) → gt_out_ga(0)
U11_ga(le_out_ga) → le_out_ga
U10_ga(gt_out_ga(Y)) → gt_out_ga(s(Y))


s = U5_GAAA(s(0), gt_out_ga(x1)) evaluates to t =U5_GAAA(s(0), gt_out_ga(0))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:



Rewriting sequence

U5_GAAA(s(0), gt_out_ga(x1)) → PART_IN_GAAA(s(0))
with rule U5_GAAA(s(z0), gt_out_ga(x1')) → PART_IN_GAAA(s(z0)) at position [] and matcher [x1' / x1, z0 / 0]

PART_IN_GAAA(s(0)) → U5_GAAA(s(0), gt_out_ga(0))
with rule PART_IN_GAAA(s(0)) → U5_GAAA(s(0), gt_out_ga(0))

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.





↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Instantiation
                                  ↳ QDP
                                    ↳ Instantiation
                                      ↳ QDP
                                        ↳ DependencyGraphProof
                                          ↳ AND
                                            ↳ QDP
                                            ↳ QDP
                                              ↳ UsableRulesProof
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

U7_GAAA(0, le_out_ga) → PART_IN_GAAA(0)
PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga)

The TRS R consists of the following rules:

le_in_ga(s(X)) → U11_ga(le_in_ga(X))
le_in_ga(0) → le_out_ga
gt_in_ga(s(X)) → U10_ga(gt_in_ga(X))
gt_in_ga(s(0)) → gt_out_ga(0)
U11_ga(le_out_ga) → le_out_ga
U10_ga(gt_out_ga(Y)) → gt_out_ga(s(Y))

The set Q consists of the following terms:

le_in_ga(x0)
gt_in_ga(x0)
U11_ga(x0)
U10_ga(x0)

We have to consider all (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Instantiation
                                  ↳ QDP
                                    ↳ Instantiation
                                      ↳ QDP
                                        ↳ DependencyGraphProof
                                          ↳ AND
                                            ↳ QDP
                                            ↳ QDP
                                              ↳ UsableRulesProof
                                                ↳ QDP
                                                  ↳ QReductionProof
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

U7_GAAA(0, le_out_ga) → PART_IN_GAAA(0)
PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga)

R is empty.
The set Q consists of the following terms:

le_in_ga(x0)
gt_in_ga(x0)
U11_ga(x0)
U10_ga(x0)

We have to consider all (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

le_in_ga(x0)
gt_in_ga(x0)
U11_ga(x0)
U10_ga(x0)



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Instantiation
                                  ↳ QDP
                                    ↳ Instantiation
                                      ↳ QDP
                                        ↳ DependencyGraphProof
                                          ↳ AND
                                            ↳ QDP
                                            ↳ QDP
                                              ↳ UsableRulesProof
                                                ↳ QDP
                                                  ↳ QReductionProof
                                                    ↳ QDP
                                                      ↳ NonTerminationProof
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

U7_GAAA(0, le_out_ga) → PART_IN_GAAA(0)
PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

The TRS P consists of the following rules:

U7_GAAA(0, le_out_ga) → PART_IN_GAAA(0)
PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga)

The TRS R consists of the following rules:none


s = PART_IN_GAAA(0) evaluates to t =PART_IN_GAAA(0)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:



Rewriting sequence

PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga)
with rule PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga) at position [] and matcher [ ]

U7_GAAA(0, le_out_ga) → PART_IN_GAAA(0)
with rule U7_GAAA(0, le_out_ga) → PART_IN_GAAA(0)

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.





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

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

GT_IN_AA(s(X), s(Y)) → GT_IN_AA(X, Y)

The TRS R consists of the following rules:

qs_in_ag(.(X, Xs), Ys) → U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
gt_in_aa(s(X), s(Y)) → U10_aa(X, Y, gt_in_aa(X, Y))
gt_in_aa(s(0), 0) → gt_out_aa(s(0), 0)
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
gt_in_ga(s(X), s(Y)) → U10_ga(X, Y, gt_in_ga(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U10_ga(X, Y, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_ga(X, Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
U11_ga(X, Y, le_out_ga(X, Y)) → le_out_ga(s(X), s(Y))
U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, [], [], []) → part_out_gaaa(X, [], [], [])
U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
le_in_aa(s(X), s(Y)) → U11_aa(X, Y, le_in_aa(X, Y))
le_in_aa(0, s(X)) → le_out_aa(0, s(X))
le_in_aa(0, 0) → le_out_aa(0, 0)
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_aaaa(X, [], [], []) → part_out_aaaa(X, [], [], [])
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa([], []) → qs_out_aa([], [])
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
app_in_gaa([], Ys, Ys) → app_out_gaa([], Ys, Ys)
U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
app_in_gag(.(X, Xs), Ys, .(X, Zs)) → U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
app_in_gag([], Ys, Ys) → app_out_gag([], Ys, Ys)
U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) → app_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt_in_aa(x1, x2)  =  gt_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)
part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
gt_in_ga(x1, x2)  =  gt_in_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
0  =  0
gt_out_ga(x1, x2)  =  gt_out_ga(x2)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x2, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
le_in_ga(x1, x2)  =  le_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x3)
le_out_ga(x1, x2)  =  le_out_ga
U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x3)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
le_in_aa(x1, x2)  =  le_in_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
le_out_aa(x1, x2)  =  le_out_aa(x1)
U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x3)
gt_out_gg(x1, x2)  =  gt_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x3)
le_out_gg(x1, x2)  =  le_out_gg
U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x2, x6)
[]  =  []
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)
qs_out_ga(x1, x2)  =  qs_out_ga(x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)
qs_out_aa(x1, x2)  =  qs_out_aa
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)
app_out_gaa(x1, x2, x3)  =  app_out_gaa
U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x5)
app_out_gag(x1, x2, x3)  =  app_out_gag(x2)
qs_out_ag(x1, x2)  =  qs_out_ag
GT_IN_AA(x1, x2)  =  GT_IN_AA

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

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

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

GT_IN_AA(s(X), s(Y)) → GT_IN_AA(X, Y)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
GT_IN_AA(x1, x2)  =  GT_IN_AA

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ NonTerminationProof
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

GT_IN_AA → GT_IN_AA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

GT_IN_AA → GT_IN_AA

The TRS R consists of the following rules:none


s = GT_IN_AA evaluates to t =GT_IN_AA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from GT_IN_AA to GT_IN_AA.





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

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

QS_IN_AA(.(X, Xs), Ys) → U1_AA(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U2_AA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → QS_IN_AA(Bigs, Bs)
U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_AA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))

The TRS R consists of the following rules:

qs_in_ag(.(X, Xs), Ys) → U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
gt_in_aa(s(X), s(Y)) → U10_aa(X, Y, gt_in_aa(X, Y))
gt_in_aa(s(0), 0) → gt_out_aa(s(0), 0)
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
gt_in_ga(s(X), s(Y)) → U10_ga(X, Y, gt_in_ga(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U10_ga(X, Y, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_ga(X, Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
U11_ga(X, Y, le_out_ga(X, Y)) → le_out_ga(s(X), s(Y))
U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, [], [], []) → part_out_gaaa(X, [], [], [])
U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
le_in_aa(s(X), s(Y)) → U11_aa(X, Y, le_in_aa(X, Y))
le_in_aa(0, s(X)) → le_out_aa(0, s(X))
le_in_aa(0, 0) → le_out_aa(0, 0)
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_aaaa(X, [], [], []) → part_out_aaaa(X, [], [], [])
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa([], []) → qs_out_aa([], [])
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
app_in_gaa([], Ys, Ys) → app_out_gaa([], Ys, Ys)
U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
app_in_gag(.(X, Xs), Ys, .(X, Zs)) → U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
app_in_gag([], Ys, Ys) → app_out_gag([], Ys, Ys)
U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) → app_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt_in_aa(x1, x2)  =  gt_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)
part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
gt_in_ga(x1, x2)  =  gt_in_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
0  =  0
gt_out_ga(x1, x2)  =  gt_out_ga(x2)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x2, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
le_in_ga(x1, x2)  =  le_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x3)
le_out_ga(x1, x2)  =  le_out_ga
U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x3)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
le_in_aa(x1, x2)  =  le_in_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
le_out_aa(x1, x2)  =  le_out_aa(x1)
U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x3)
gt_out_gg(x1, x2)  =  gt_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x3)
le_out_gg(x1, x2)  =  le_out_gg
U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x2, x6)
[]  =  []
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)
qs_out_ga(x1, x2)  =  qs_out_ga(x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)
qs_out_aa(x1, x2)  =  qs_out_aa
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)
app_out_gaa(x1, x2, x3)  =  app_out_gaa
U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x5)
app_out_gag(x1, x2, x3)  =  app_out_gag(x2)
qs_out_ag(x1, x2)  =  qs_out_ag
QS_IN_AA(x1, x2)  =  QS_IN_AA
U2_AA(x1, x2, x3, x4, x5)  =  U2_AA(x5)
U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

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

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

QS_IN_AA(.(X, Xs), Ys) → U1_AA(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U2_AA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → QS_IN_AA(Bigs, Bs)
U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_AA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))

The TRS R consists of the following rules:

part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
part_in_aaaa(X, [], [], []) → part_out_aaaa(X, [], [], [])
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
qs_in_ga([], []) → qs_out_ga([], [])
U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
gt_in_aa(s(X), s(Y)) → U10_aa(X, Y, gt_in_aa(X, Y))
gt_in_aa(s(0), 0) → gt_out_aa(s(0), 0)
U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
le_in_aa(s(X), s(Y)) → U11_aa(X, Y, le_in_aa(X, Y))
le_in_aa(0, s(X)) → le_out_aa(0, s(X))
le_in_aa(0, 0) → le_out_aa(0, 0)
U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
part_in_gaaa(X, [], [], []) → part_out_gaaa(X, [], [], [])
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
gt_in_ga(s(X), s(Y)) → U10_ga(X, Y, gt_in_ga(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_ga(X, Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U10_ga(X, Y, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U11_ga(X, Y, le_out_ga(X, Y)) → le_out_ga(s(X), s(Y))
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt_in_aa(x1, x2)  =  gt_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)
part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
gt_in_ga(x1, x2)  =  gt_in_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
0  =  0
gt_out_ga(x1, x2)  =  gt_out_ga(x2)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x2, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
le_in_ga(x1, x2)  =  le_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x3)
le_out_ga(x1, x2)  =  le_out_ga
U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x3)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
le_in_aa(x1, x2)  =  le_in_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
le_out_aa(x1, x2)  =  le_out_aa(x1)
U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x3)
gt_out_gg(x1, x2)  =  gt_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x3)
le_out_gg(x1, x2)  =  le_out_gg
U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x2, x6)
[]  =  []
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)
qs_out_ga(x1, x2)  =  qs_out_ga(x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
QS_IN_AA(x1, x2)  =  QS_IN_AA
U2_AA(x1, x2, x3, x4, x5)  =  U2_AA(x5)
U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
  ↳ PrologToPiTRSProof

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

U1_AA(part_out_aaaa(Littles)) → U2_AA(qs_in_ga(Littles))
QS_IN_AA → U1_AA(part_in_aaaa)
U2_AA(qs_out_ga(Ls)) → QS_IN_AA

The TRS R consists of the following rules:

part_in_aaaa → U5_aaaa(gt_in_aa)
part_in_aaaa → U7_aaaa(le_in_aa)
part_in_aaaa → part_out_aaaa([])
qs_in_ga(.(X, Xs)) → U1_ga(X, part_in_ggaa(X, Xs))
qs_in_ga([]) → qs_out_ga([])
U5_aaaa(gt_out_aa(X, Y)) → U6_aaaa(Y, part_in_gaaa(X))
U7_aaaa(le_out_aa(X)) → U8_aaaa(part_in_gaaa(X))
U1_ga(X, part_out_ggaa(Littles, Bigs)) → U2_ga(X, Bigs, qs_in_ga(Littles))
gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s(0), 0)
U6_aaaa(Y, part_out_gaaa(Ls)) → part_out_aaaa(.(Y, Ls))
le_in_aa → U11_aa(le_in_aa)
le_in_aa → le_out_aa(0)
U8_aaaa(part_out_gaaa(Ls)) → part_out_aaaa(Ls)
part_in_ggaa(X, .(Y, Xs)) → U5_ggaa(X, Y, Xs, gt_in_gg(X, Y))
part_in_ggaa(X, .(Y, Xs)) → U7_ggaa(X, Y, Xs, le_in_gg(X, Y))
part_in_ggaa(X, []) → part_out_ggaa([], [])
U2_ga(X, Bigs, qs_out_ga(Ls)) → U3_ga(X, Ls, qs_in_ga(Bigs))
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
part_in_gaaa(X) → U5_gaaa(X, gt_in_ga(X))
part_in_gaaa(X) → U7_gaaa(X, le_in_ga(X))
part_in_gaaa(X) → part_out_gaaa([])
U11_aa(le_out_aa(X)) → le_out_aa(s(X))
U5_ggaa(X, Y, Xs, gt_out_gg) → U6_ggaa(Y, part_in_ggaa(X, Xs))
U7_ggaa(X, Y, Xs, le_out_gg) → U8_ggaa(Y, part_in_ggaa(X, Xs))
U3_ga(X, Ls, qs_out_ga(Bs)) → U4_ga(app_in_gga(Ls, .(X, Bs)))
U5_gaaa(X, gt_out_ga(Y)) → U6_gaaa(Y, part_in_gaaa(X))
U7_gaaa(X, le_out_ga) → U8_gaaa(part_in_gaaa(X))
gt_in_gg(s(X), s(Y)) → U10_gg(gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg
U6_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(.(Y, Ls), Bs)
le_in_gg(s(X), s(Y)) → U11_gg(le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
U8_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(Ls, .(Y, Bs))
U4_ga(app_out_gga(Ys)) → qs_out_ga(Ys)
gt_in_ga(s(X)) → U10_ga(gt_in_ga(X))
gt_in_ga(s(0)) → gt_out_ga(0)
U6_gaaa(Y, part_out_gaaa(Ls)) → part_out_gaaa(.(Y, Ls))
le_in_ga(s(X)) → U11_ga(le_in_ga(X))
le_in_ga(0) → le_out_ga
U8_gaaa(part_out_gaaa(Ls)) → part_out_gaaa(Ls)
U10_gg(gt_out_gg) → gt_out_gg
U11_gg(le_out_gg) → le_out_gg
app_in_gga(.(X, Xs), Ys) → U9_gga(X, app_in_gga(Xs, Ys))
app_in_gga([], Ys) → app_out_gga(Ys)
U10_ga(gt_out_ga(Y)) → gt_out_ga(s(Y))
U11_ga(le_out_ga) → le_out_ga
U9_gga(X, app_out_gga(Zs)) → app_out_gga(.(X, Zs))

The set Q consists of the following terms:

part_in_aaaa
qs_in_ga(x0)
U5_aaaa(x0)
U7_aaaa(x0)
U1_ga(x0, x1)
gt_in_aa
U6_aaaa(x0, x1)
le_in_aa
U8_aaaa(x0)
part_in_ggaa(x0, x1)
U2_ga(x0, x1, x2)
U10_aa(x0)
part_in_gaaa(x0)
U11_aa(x0)
U5_ggaa(x0, x1, x2, x3)
U7_ggaa(x0, x1, x2, x3)
U3_ga(x0, x1, x2)
U5_gaaa(x0, x1)
U7_gaaa(x0, x1)
gt_in_gg(x0, x1)
U6_ggaa(x0, x1)
le_in_gg(x0, x1)
U8_ggaa(x0, x1)
U4_ga(x0)
gt_in_ga(x0)
U6_gaaa(x0, x1)
le_in_ga(x0)
U8_gaaa(x0)
U10_gg(x0)
U11_gg(x0)
app_in_gga(x0, x1)
U10_ga(x0)
U11_ga(x0)
U9_gga(x0, x1)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule QS_IN_AA → U1_AA(part_in_aaaa) at position [0] we obtained the following new rules:

QS_IN_AA → U1_AA(U5_aaaa(gt_in_aa))
QS_IN_AA → U1_AA(part_out_aaaa([]))
QS_IN_AA → U1_AA(U7_aaaa(le_in_aa))



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ UsableRulesProof
  ↳ PrologToPiTRSProof

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

U1_AA(part_out_aaaa(Littles)) → U2_AA(qs_in_ga(Littles))
QS_IN_AA → U1_AA(part_out_aaaa([]))
U2_AA(qs_out_ga(Ls)) → QS_IN_AA
QS_IN_AA → U1_AA(U5_aaaa(gt_in_aa))
QS_IN_AA → U1_AA(U7_aaaa(le_in_aa))

The TRS R consists of the following rules:

part_in_aaaa → U5_aaaa(gt_in_aa)
part_in_aaaa → U7_aaaa(le_in_aa)
part_in_aaaa → part_out_aaaa([])
qs_in_ga(.(X, Xs)) → U1_ga(X, part_in_ggaa(X, Xs))
qs_in_ga([]) → qs_out_ga([])
U5_aaaa(gt_out_aa(X, Y)) → U6_aaaa(Y, part_in_gaaa(X))
U7_aaaa(le_out_aa(X)) → U8_aaaa(part_in_gaaa(X))
U1_ga(X, part_out_ggaa(Littles, Bigs)) → U2_ga(X, Bigs, qs_in_ga(Littles))
gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s(0), 0)
U6_aaaa(Y, part_out_gaaa(Ls)) → part_out_aaaa(.(Y, Ls))
le_in_aa → U11_aa(le_in_aa)
le_in_aa → le_out_aa(0)
U8_aaaa(part_out_gaaa(Ls)) → part_out_aaaa(Ls)
part_in_ggaa(X, .(Y, Xs)) → U5_ggaa(X, Y, Xs, gt_in_gg(X, Y))
part_in_ggaa(X, .(Y, Xs)) → U7_ggaa(X, Y, Xs, le_in_gg(X, Y))
part_in_ggaa(X, []) → part_out_ggaa([], [])
U2_ga(X, Bigs, qs_out_ga(Ls)) → U3_ga(X, Ls, qs_in_ga(Bigs))
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
part_in_gaaa(X) → U5_gaaa(X, gt_in_ga(X))
part_in_gaaa(X) → U7_gaaa(X, le_in_ga(X))
part_in_gaaa(X) → part_out_gaaa([])
U11_aa(le_out_aa(X)) → le_out_aa(s(X))
U5_ggaa(X, Y, Xs, gt_out_gg) → U6_ggaa(Y, part_in_ggaa(X, Xs))
U7_ggaa(X, Y, Xs, le_out_gg) → U8_ggaa(Y, part_in_ggaa(X, Xs))
U3_ga(X, Ls, qs_out_ga(Bs)) → U4_ga(app_in_gga(Ls, .(X, Bs)))
U5_gaaa(X, gt_out_ga(Y)) → U6_gaaa(Y, part_in_gaaa(X))
U7_gaaa(X, le_out_ga) → U8_gaaa(part_in_gaaa(X))
gt_in_gg(s(X), s(Y)) → U10_gg(gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg
U6_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(.(Y, Ls), Bs)
le_in_gg(s(X), s(Y)) → U11_gg(le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
U8_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(Ls, .(Y, Bs))
U4_ga(app_out_gga(Ys)) → qs_out_ga(Ys)
gt_in_ga(s(X)) → U10_ga(gt_in_ga(X))
gt_in_ga(s(0)) → gt_out_ga(0)
U6_gaaa(Y, part_out_gaaa(Ls)) → part_out_gaaa(.(Y, Ls))
le_in_ga(s(X)) → U11_ga(le_in_ga(X))
le_in_ga(0) → le_out_ga
U8_gaaa(part_out_gaaa(Ls)) → part_out_gaaa(Ls)
U10_gg(gt_out_gg) → gt_out_gg
U11_gg(le_out_gg) → le_out_gg
app_in_gga(.(X, Xs), Ys) → U9_gga(X, app_in_gga(Xs, Ys))
app_in_gga([], Ys) → app_out_gga(Ys)
U10_ga(gt_out_ga(Y)) → gt_out_ga(s(Y))
U11_ga(le_out_ga) → le_out_ga
U9_gga(X, app_out_gga(Zs)) → app_out_gga(.(X, Zs))

The set Q consists of the following terms:

part_in_aaaa
qs_in_ga(x0)
U5_aaaa(x0)
U7_aaaa(x0)
U1_ga(x0, x1)
gt_in_aa
U6_aaaa(x0, x1)
le_in_aa
U8_aaaa(x0)
part_in_ggaa(x0, x1)
U2_ga(x0, x1, x2)
U10_aa(x0)
part_in_gaaa(x0)
U11_aa(x0)
U5_ggaa(x0, x1, x2, x3)
U7_ggaa(x0, x1, x2, x3)
U3_ga(x0, x1, x2)
U5_gaaa(x0, x1)
U7_gaaa(x0, x1)
gt_in_gg(x0, x1)
U6_ggaa(x0, x1)
le_in_gg(x0, x1)
U8_ggaa(x0, x1)
U4_ga(x0)
gt_in_ga(x0)
U6_gaaa(x0, x1)
le_in_ga(x0)
U8_gaaa(x0)
U10_gg(x0)
U11_gg(x0)
app_in_gga(x0, x1)
U10_ga(x0)
U11_ga(x0)
U9_gga(x0, x1)

We have to consider all (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ UsableRulesProof
                              ↳ QDP
                                ↳ QReductionProof
  ↳ PrologToPiTRSProof

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

U1_AA(part_out_aaaa(Littles)) → U2_AA(qs_in_ga(Littles))
QS_IN_AA → U1_AA(part_out_aaaa([]))
U2_AA(qs_out_ga(Ls)) → QS_IN_AA
QS_IN_AA → U1_AA(U5_aaaa(gt_in_aa))
QS_IN_AA → U1_AA(U7_aaaa(le_in_aa))

The TRS R consists of the following rules:

qs_in_ga(.(X, Xs)) → U1_ga(X, part_in_ggaa(X, Xs))
qs_in_ga([]) → qs_out_ga([])
part_in_ggaa(X, .(Y, Xs)) → U5_ggaa(X, Y, Xs, gt_in_gg(X, Y))
part_in_ggaa(X, .(Y, Xs)) → U7_ggaa(X, Y, Xs, le_in_gg(X, Y))
part_in_ggaa(X, []) → part_out_ggaa([], [])
U1_ga(X, part_out_ggaa(Littles, Bigs)) → U2_ga(X, Bigs, qs_in_ga(Littles))
U2_ga(X, Bigs, qs_out_ga(Ls)) → U3_ga(X, Ls, qs_in_ga(Bigs))
U3_ga(X, Ls, qs_out_ga(Bs)) → U4_ga(app_in_gga(Ls, .(X, Bs)))
app_in_gga(.(X, Xs), Ys) → U9_gga(X, app_in_gga(Xs, Ys))
app_in_gga([], Ys) → app_out_gga(Ys)
U4_ga(app_out_gga(Ys)) → qs_out_ga(Ys)
U9_gga(X, app_out_gga(Zs)) → app_out_gga(.(X, Zs))
le_in_gg(s(X), s(Y)) → U11_gg(le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
U7_ggaa(X, Y, Xs, le_out_gg) → U8_ggaa(Y, part_in_ggaa(X, Xs))
U8_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(Ls, .(Y, Bs))
U11_gg(le_out_gg) → le_out_gg
gt_in_gg(s(X), s(Y)) → U10_gg(gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg
U5_ggaa(X, Y, Xs, gt_out_gg) → U6_ggaa(Y, part_in_ggaa(X, Xs))
U6_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(.(Y, Ls), Bs)
U10_gg(gt_out_gg) → gt_out_gg
gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s(0), 0)
U5_aaaa(gt_out_aa(X, Y)) → U6_aaaa(Y, part_in_gaaa(X))
part_in_gaaa(X) → U5_gaaa(X, gt_in_ga(X))
part_in_gaaa(X) → U7_gaaa(X, le_in_ga(X))
part_in_gaaa(X) → part_out_gaaa([])
U6_aaaa(Y, part_out_gaaa(Ls)) → part_out_aaaa(.(Y, Ls))
le_in_ga(s(X)) → U11_ga(le_in_ga(X))
le_in_ga(0) → le_out_ga
U7_gaaa(X, le_out_ga) → U8_gaaa(part_in_gaaa(X))
U8_gaaa(part_out_gaaa(Ls)) → part_out_gaaa(Ls)
U11_ga(le_out_ga) → le_out_ga
gt_in_ga(s(X)) → U10_ga(gt_in_ga(X))
gt_in_ga(s(0)) → gt_out_ga(0)
U5_gaaa(X, gt_out_ga(Y)) → U6_gaaa(Y, part_in_gaaa(X))
U6_gaaa(Y, part_out_gaaa(Ls)) → part_out_gaaa(.(Y, Ls))
U10_ga(gt_out_ga(Y)) → gt_out_ga(s(Y))
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
le_in_aa → U11_aa(le_in_aa)
le_in_aa → le_out_aa(0)
U7_aaaa(le_out_aa(X)) → U8_aaaa(part_in_gaaa(X))
U8_aaaa(part_out_gaaa(Ls)) → part_out_aaaa(Ls)
U11_aa(le_out_aa(X)) → le_out_aa(s(X))

The set Q consists of the following terms:

part_in_aaaa
qs_in_ga(x0)
U5_aaaa(x0)
U7_aaaa(x0)
U1_ga(x0, x1)
gt_in_aa
U6_aaaa(x0, x1)
le_in_aa
U8_aaaa(x0)
part_in_ggaa(x0, x1)
U2_ga(x0, x1, x2)
U10_aa(x0)
part_in_gaaa(x0)
U11_aa(x0)
U5_ggaa(x0, x1, x2, x3)
U7_ggaa(x0, x1, x2, x3)
U3_ga(x0, x1, x2)
U5_gaaa(x0, x1)
U7_gaaa(x0, x1)
gt_in_gg(x0, x1)
U6_ggaa(x0, x1)
le_in_gg(x0, x1)
U8_ggaa(x0, x1)
U4_ga(x0)
gt_in_ga(x0)
U6_gaaa(x0, x1)
le_in_ga(x0)
U8_gaaa(x0)
U10_gg(x0)
U11_gg(x0)
app_in_gga(x0, x1)
U10_ga(x0)
U11_ga(x0)
U9_gga(x0, x1)

We have to consider all (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

part_in_aaaa



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ UsableRulesProof
                              ↳ QDP
                                ↳ QReductionProof
                                  ↳ QDP
                                    ↳ Narrowing
  ↳ PrologToPiTRSProof

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

U1_AA(part_out_aaaa(Littles)) → U2_AA(qs_in_ga(Littles))
QS_IN_AA → U1_AA(part_out_aaaa([]))
U2_AA(qs_out_ga(Ls)) → QS_IN_AA
QS_IN_AA → U1_AA(U5_aaaa(gt_in_aa))
QS_IN_AA → U1_AA(U7_aaaa(le_in_aa))

The TRS R consists of the following rules:

qs_in_ga(.(X, Xs)) → U1_ga(X, part_in_ggaa(X, Xs))
qs_in_ga([]) → qs_out_ga([])
part_in_ggaa(X, .(Y, Xs)) → U5_ggaa(X, Y, Xs, gt_in_gg(X, Y))
part_in_ggaa(X, .(Y, Xs)) → U7_ggaa(X, Y, Xs, le_in_gg(X, Y))
part_in_ggaa(X, []) → part_out_ggaa([], [])
U1_ga(X, part_out_ggaa(Littles, Bigs)) → U2_ga(X, Bigs, qs_in_ga(Littles))
U2_ga(X, Bigs, qs_out_ga(Ls)) → U3_ga(X, Ls, qs_in_ga(Bigs))
U3_ga(X, Ls, qs_out_ga(Bs)) → U4_ga(app_in_gga(Ls, .(X, Bs)))
app_in_gga(.(X, Xs), Ys) → U9_gga(X, app_in_gga(Xs, Ys))
app_in_gga([], Ys) → app_out_gga(Ys)
U4_ga(app_out_gga(Ys)) → qs_out_ga(Ys)
U9_gga(X, app_out_gga(Zs)) → app_out_gga(.(X, Zs))
le_in_gg(s(X), s(Y)) → U11_gg(le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
U7_ggaa(X, Y, Xs, le_out_gg) → U8_ggaa(Y, part_in_ggaa(X, Xs))
U8_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(Ls, .(Y, Bs))
U11_gg(le_out_gg) → le_out_gg
gt_in_gg(s(X), s(Y)) → U10_gg(gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg
U5_ggaa(X, Y, Xs, gt_out_gg) → U6_ggaa(Y, part_in_ggaa(X, Xs))
U6_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(.(Y, Ls), Bs)
U10_gg(gt_out_gg) → gt_out_gg
gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s(0), 0)
U5_aaaa(gt_out_aa(X, Y)) → U6_aaaa(Y, part_in_gaaa(X))
part_in_gaaa(X) → U5_gaaa(X, gt_in_ga(X))
part_in_gaaa(X) → U7_gaaa(X, le_in_ga(X))
part_in_gaaa(X) → part_out_gaaa([])
U6_aaaa(Y, part_out_gaaa(Ls)) → part_out_aaaa(.(Y, Ls))
le_in_ga(s(X)) → U11_ga(le_in_ga(X))
le_in_ga(0) → le_out_ga
U7_gaaa(X, le_out_ga) → U8_gaaa(part_in_gaaa(X))
U8_gaaa(part_out_gaaa(Ls)) → part_out_gaaa(Ls)
U11_ga(le_out_ga) → le_out_ga
gt_in_ga(s(X)) → U10_ga(gt_in_ga(X))
gt_in_ga(s(0)) → gt_out_ga(0)
U5_gaaa(X, gt_out_ga(Y)) → U6_gaaa(Y, part_in_gaaa(X))
U6_gaaa(Y, part_out_gaaa(Ls)) → part_out_gaaa(.(Y, Ls))
U10_ga(gt_out_ga(Y)) → gt_out_ga(s(Y))
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
le_in_aa → U11_aa(le_in_aa)
le_in_aa → le_out_aa(0)
U7_aaaa(le_out_aa(X)) → U8_aaaa(part_in_gaaa(X))
U8_aaaa(part_out_gaaa(Ls)) → part_out_aaaa(Ls)
U11_aa(le_out_aa(X)) → le_out_aa(s(X))

The set Q consists of the following terms:

qs_in_ga(x0)
U5_aaaa(x0)
U7_aaaa(x0)
U1_ga(x0, x1)
gt_in_aa
U6_aaaa(x0, x1)
le_in_aa
U8_aaaa(x0)
part_in_ggaa(x0, x1)
U2_ga(x0, x1, x2)
U10_aa(x0)
part_in_gaaa(x0)
U11_aa(x0)
U5_ggaa(x0, x1, x2, x3)
U7_ggaa(x0, x1, x2, x3)
U3_ga(x0, x1, x2)
U5_gaaa(x0, x1)
U7_gaaa(x0, x1)
gt_in_gg(x0, x1)
U6_ggaa(x0, x1)
le_in_gg(x0, x1)
U8_ggaa(x0, x1)
U4_ga(x0)
gt_in_ga(x0)
U6_gaaa(x0, x1)
le_in_ga(x0)
U8_gaaa(x0)
U10_gg(x0)
U11_gg(x0)
app_in_gga(x0, x1)
U10_ga(x0)
U11_ga(x0)
U9_gga(x0, x1)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule U1_AA(part_out_aaaa(Littles)) → U2_AA(qs_in_ga(Littles)) at position [0] we obtained the following new rules:

U1_AA(part_out_aaaa([])) → U2_AA(qs_out_ga([]))
U1_AA(part_out_aaaa(.(x0, x1))) → U2_AA(U1_ga(x0, part_in_ggaa(x0, x1)))



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ UsableRulesProof
                              ↳ QDP
                                ↳ QReductionProof
                                  ↳ QDP
                                    ↳ Narrowing
                                      ↳ QDP
                                        ↳ NonTerminationProof
  ↳ PrologToPiTRSProof

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

U1_AA(part_out_aaaa([])) → U2_AA(qs_out_ga([]))
QS_IN_AA → U1_AA(part_out_aaaa([]))
U2_AA(qs_out_ga(Ls)) → QS_IN_AA
U1_AA(part_out_aaaa(.(x0, x1))) → U2_AA(U1_ga(x0, part_in_ggaa(x0, x1)))
QS_IN_AA → U1_AA(U5_aaaa(gt_in_aa))
QS_IN_AA → U1_AA(U7_aaaa(le_in_aa))

The TRS R consists of the following rules:

qs_in_ga(.(X, Xs)) → U1_ga(X, part_in_ggaa(X, Xs))
qs_in_ga([]) → qs_out_ga([])
part_in_ggaa(X, .(Y, Xs)) → U5_ggaa(X, Y, Xs, gt_in_gg(X, Y))
part_in_ggaa(X, .(Y, Xs)) → U7_ggaa(X, Y, Xs, le_in_gg(X, Y))
part_in_ggaa(X, []) → part_out_ggaa([], [])
U1_ga(X, part_out_ggaa(Littles, Bigs)) → U2_ga(X, Bigs, qs_in_ga(Littles))
U2_ga(X, Bigs, qs_out_ga(Ls)) → U3_ga(X, Ls, qs_in_ga(Bigs))
U3_ga(X, Ls, qs_out_ga(Bs)) → U4_ga(app_in_gga(Ls, .(X, Bs)))
app_in_gga(.(X, Xs), Ys) → U9_gga(X, app_in_gga(Xs, Ys))
app_in_gga([], Ys) → app_out_gga(Ys)
U4_ga(app_out_gga(Ys)) → qs_out_ga(Ys)
U9_gga(X, app_out_gga(Zs)) → app_out_gga(.(X, Zs))
le_in_gg(s(X), s(Y)) → U11_gg(le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
U7_ggaa(X, Y, Xs, le_out_gg) → U8_ggaa(Y, part_in_ggaa(X, Xs))
U8_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(Ls, .(Y, Bs))
U11_gg(le_out_gg) → le_out_gg
gt_in_gg(s(X), s(Y)) → U10_gg(gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg
U5_ggaa(X, Y, Xs, gt_out_gg) → U6_ggaa(Y, part_in_ggaa(X, Xs))
U6_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(.(Y, Ls), Bs)
U10_gg(gt_out_gg) → gt_out_gg
gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s(0), 0)
U5_aaaa(gt_out_aa(X, Y)) → U6_aaaa(Y, part_in_gaaa(X))
part_in_gaaa(X) → U5_gaaa(X, gt_in_ga(X))
part_in_gaaa(X) → U7_gaaa(X, le_in_ga(X))
part_in_gaaa(X) → part_out_gaaa([])
U6_aaaa(Y, part_out_gaaa(Ls)) → part_out_aaaa(.(Y, Ls))
le_in_ga(s(X)) → U11_ga(le_in_ga(X))
le_in_ga(0) → le_out_ga
U7_gaaa(X, le_out_ga) → U8_gaaa(part_in_gaaa(X))
U8_gaaa(part_out_gaaa(Ls)) → part_out_gaaa(Ls)
U11_ga(le_out_ga) → le_out_ga
gt_in_ga(s(X)) → U10_ga(gt_in_ga(X))
gt_in_ga(s(0)) → gt_out_ga(0)
U5_gaaa(X, gt_out_ga(Y)) → U6_gaaa(Y, part_in_gaaa(X))
U6_gaaa(Y, part_out_gaaa(Ls)) → part_out_gaaa(.(Y, Ls))
U10_ga(gt_out_ga(Y)) → gt_out_ga(s(Y))
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
le_in_aa → U11_aa(le_in_aa)
le_in_aa → le_out_aa(0)
U7_aaaa(le_out_aa(X)) → U8_aaaa(part_in_gaaa(X))
U8_aaaa(part_out_gaaa(Ls)) → part_out_aaaa(Ls)
U11_aa(le_out_aa(X)) → le_out_aa(s(X))

The set Q consists of the following terms:

qs_in_ga(x0)
U5_aaaa(x0)
U7_aaaa(x0)
U1_ga(x0, x1)
gt_in_aa
U6_aaaa(x0, x1)
le_in_aa
U8_aaaa(x0)
part_in_ggaa(x0, x1)
U2_ga(x0, x1, x2)
U10_aa(x0)
part_in_gaaa(x0)
U11_aa(x0)
U5_ggaa(x0, x1, x2, x3)
U7_ggaa(x0, x1, x2, x3)
U3_ga(x0, x1, x2)
U5_gaaa(x0, x1)
U7_gaaa(x0, x1)
gt_in_gg(x0, x1)
U6_ggaa(x0, x1)
le_in_gg(x0, x1)
U8_ggaa(x0, x1)
U4_ga(x0)
gt_in_ga(x0)
U6_gaaa(x0, x1)
le_in_ga(x0)
U8_gaaa(x0)
U10_gg(x0)
U11_gg(x0)
app_in_gga(x0, x1)
U10_ga(x0)
U11_ga(x0)
U9_gga(x0, x1)

We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

The TRS P consists of the following rules:

U1_AA(part_out_aaaa([])) → U2_AA(qs_out_ga([]))
QS_IN_AA → U1_AA(part_out_aaaa([]))
U2_AA(qs_out_ga(Ls)) → QS_IN_AA
U1_AA(part_out_aaaa(.(x0, x1))) → U2_AA(U1_ga(x0, part_in_ggaa(x0, x1)))
QS_IN_AA → U1_AA(U5_aaaa(gt_in_aa))
QS_IN_AA → U1_AA(U7_aaaa(le_in_aa))

The TRS R consists of the following rules:

qs_in_ga(.(X, Xs)) → U1_ga(X, part_in_ggaa(X, Xs))
qs_in_ga([]) → qs_out_ga([])
part_in_ggaa(X, .(Y, Xs)) → U5_ggaa(X, Y, Xs, gt_in_gg(X, Y))
part_in_ggaa(X, .(Y, Xs)) → U7_ggaa(X, Y, Xs, le_in_gg(X, Y))
part_in_ggaa(X, []) → part_out_ggaa([], [])
U1_ga(X, part_out_ggaa(Littles, Bigs)) → U2_ga(X, Bigs, qs_in_ga(Littles))
U2_ga(X, Bigs, qs_out_ga(Ls)) → U3_ga(X, Ls, qs_in_ga(Bigs))
U3_ga(X, Ls, qs_out_ga(Bs)) → U4_ga(app_in_gga(Ls, .(X, Bs)))
app_in_gga(.(X, Xs), Ys) → U9_gga(X, app_in_gga(Xs, Ys))
app_in_gga([], Ys) → app_out_gga(Ys)
U4_ga(app_out_gga(Ys)) → qs_out_ga(Ys)
U9_gga(X, app_out_gga(Zs)) → app_out_gga(.(X, Zs))
le_in_gg(s(X), s(Y)) → U11_gg(le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
U7_ggaa(X, Y, Xs, le_out_gg) → U8_ggaa(Y, part_in_ggaa(X, Xs))
U8_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(Ls, .(Y, Bs))
U11_gg(le_out_gg) → le_out_gg
gt_in_gg(s(X), s(Y)) → U10_gg(gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg
U5_ggaa(X, Y, Xs, gt_out_gg) → U6_ggaa(Y, part_in_ggaa(X, Xs))
U6_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(.(Y, Ls), Bs)
U10_gg(gt_out_gg) → gt_out_gg
gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s(0), 0)
U5_aaaa(gt_out_aa(X, Y)) → U6_aaaa(Y, part_in_gaaa(X))
part_in_gaaa(X) → U5_gaaa(X, gt_in_ga(X))
part_in_gaaa(X) → U7_gaaa(X, le_in_ga(X))
part_in_gaaa(X) → part_out_gaaa([])
U6_aaaa(Y, part_out_gaaa(Ls)) → part_out_aaaa(.(Y, Ls))
le_in_ga(s(X)) → U11_ga(le_in_ga(X))
le_in_ga(0) → le_out_ga
U7_gaaa(X, le_out_ga) → U8_gaaa(part_in_gaaa(X))
U8_gaaa(part_out_gaaa(Ls)) → part_out_gaaa(Ls)
U11_ga(le_out_ga) → le_out_ga
gt_in_ga(s(X)) → U10_ga(gt_in_ga(X))
gt_in_ga(s(0)) → gt_out_ga(0)
U5_gaaa(X, gt_out_ga(Y)) → U6_gaaa(Y, part_in_gaaa(X))
U6_gaaa(Y, part_out_gaaa(Ls)) → part_out_gaaa(.(Y, Ls))
U10_ga(gt_out_ga(Y)) → gt_out_ga(s(Y))
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
le_in_aa → U11_aa(le_in_aa)
le_in_aa → le_out_aa(0)
U7_aaaa(le_out_aa(X)) → U8_aaaa(part_in_gaaa(X))
U8_aaaa(part_out_gaaa(Ls)) → part_out_aaaa(Ls)
U11_aa(le_out_aa(X)) → le_out_aa(s(X))


s = U2_AA(qs_out_ga(Ls)) evaluates to t =U2_AA(qs_out_ga([]))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:



Rewriting sequence

U2_AA(qs_out_ga(Ls)) → QS_IN_AA
with rule U2_AA(qs_out_ga(Ls')) → QS_IN_AA at position [] and matcher [Ls' / Ls]

QS_IN_AA → U1_AA(part_out_aaaa([]))
with rule QS_IN_AA → U1_AA(part_out_aaaa([])) at position [] and matcher [ ]

U1_AA(part_out_aaaa([])) → U2_AA(qs_out_ga([]))
with rule U1_AA(part_out_aaaa([])) → U2_AA(qs_out_ga([]))

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.




We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
qs_in: (f,b) (b,f) (f,f)
part_in: (f,f,f,f) (b,f,f,f) (b,b,f,f)
gt_in: (f,f) (b,f) (b,b)
le_in: (b,f) (f,f) (b,b)
app_in: (b,b,f) (b,f,f) (b,f,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

qs_in_ag(.(X, Xs), Ys) → U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
gt_in_aa(s(X), s(Y)) → U10_aa(X, Y, gt_in_aa(X, Y))
gt_in_aa(s(0), 0) → gt_out_aa(s(0), 0)
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
gt_in_ga(s(X), s(Y)) → U10_ga(X, Y, gt_in_ga(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U10_ga(X, Y, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_ga(X, Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
U11_ga(X, Y, le_out_ga(X, Y)) → le_out_ga(s(X), s(Y))
U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, [], [], []) → part_out_gaaa(X, [], [], [])
U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
le_in_aa(s(X), s(Y)) → U11_aa(X, Y, le_in_aa(X, Y))
le_in_aa(0, s(X)) → le_out_aa(0, s(X))
le_in_aa(0, 0) → le_out_aa(0, 0)
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_aaaa(X, [], [], []) → part_out_aaaa(X, [], [], [])
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa([], []) → qs_out_aa([], [])
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
app_in_gaa([], Ys, Ys) → app_out_gaa([], Ys, Ys)
U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
app_in_gag(.(X, Xs), Ys, .(X, Zs)) → U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
app_in_gag([], Ys, Ys) → app_out_gag([], Ys, Ys)
U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) → app_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt_in_aa(x1, x2)  =  gt_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)
part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
gt_in_ga(x1, x2)  =  gt_in_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
0  =  0
gt_out_ga(x1, x2)  =  gt_out_ga(x1, x2)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x2, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
le_in_ga(x1, x2)  =  le_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
le_out_ga(x1, x2)  =  le_out_ga(x1)
U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x1, x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1, x3)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
le_in_aa(x1, x2)  =  le_in_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
le_out_aa(x1, x2)  =  le_out_aa(x1)
U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)
gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x2, x3, x6)
[]  =  []
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)
qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)
qs_out_aa(x1, x2)  =  qs_out_aa
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x5)
app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x1, x2, x4, x5)
app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)
qs_out_ag(x1, x2)  =  qs_out_ag(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:

qs_in_ag(.(X, Xs), Ys) → U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
gt_in_aa(s(X), s(Y)) → U10_aa(X, Y, gt_in_aa(X, Y))
gt_in_aa(s(0), 0) → gt_out_aa(s(0), 0)
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
gt_in_ga(s(X), s(Y)) → U10_ga(X, Y, gt_in_ga(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U10_ga(X, Y, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_ga(X, Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
U11_ga(X, Y, le_out_ga(X, Y)) → le_out_ga(s(X), s(Y))
U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, [], [], []) → part_out_gaaa(X, [], [], [])
U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
le_in_aa(s(X), s(Y)) → U11_aa(X, Y, le_in_aa(X, Y))
le_in_aa(0, s(X)) → le_out_aa(0, s(X))
le_in_aa(0, 0) → le_out_aa(0, 0)
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_aaaa(X, [], [], []) → part_out_aaaa(X, [], [], [])
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa([], []) → qs_out_aa([], [])
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
app_in_gaa([], Ys, Ys) → app_out_gaa([], Ys, Ys)
U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
app_in_gag(.(X, Xs), Ys, .(X, Zs)) → U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
app_in_gag([], Ys, Ys) → app_out_gag([], Ys, Ys)
U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) → app_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt_in_aa(x1, x2)  =  gt_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)
part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
gt_in_ga(x1, x2)  =  gt_in_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
0  =  0
gt_out_ga(x1, x2)  =  gt_out_ga(x1, x2)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x2, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
le_in_ga(x1, x2)  =  le_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
le_out_ga(x1, x2)  =  le_out_ga(x1)
U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x1, x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1, x3)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
le_in_aa(x1, x2)  =  le_in_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
le_out_aa(x1, x2)  =  le_out_aa(x1)
U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)
gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x2, x3, x6)
[]  =  []
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)
qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)
qs_out_aa(x1, x2)  =  qs_out_aa
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x5)
app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x1, x2, x4, x5)
app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)
qs_out_ag(x1, x2)  =  qs_out_ag(x2)


Using Dependency Pairs [1,30] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

QS_IN_AG(.(X, Xs), Ys) → U1_AG(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
QS_IN_AG(.(X, Xs), Ys) → PART_IN_AAAA(X, Xs, Littles, Bigs)
PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_AAAA(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) → GT_IN_AA(X, Y)
GT_IN_AA(s(X), s(Y)) → U10_AA(X, Y, gt_in_aa(X, Y))
GT_IN_AA(s(X), s(Y)) → GT_IN_AA(X, Y)
U5_AAAA(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_AAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U5_AAAA(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → PART_IN_GAAA(X, Xs, Ls, Bs)
PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GAAA(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) → GT_IN_GA(X, Y)
GT_IN_GA(s(X), s(Y)) → U10_GA(X, Y, gt_in_ga(X, Y))
GT_IN_GA(s(X), s(Y)) → GT_IN_GA(X, Y)
U5_GAAA(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_GAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U5_GAAA(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → PART_IN_GAAA(X, Xs, Ls, Bs)
PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_GAAA(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → LE_IN_GA(X, Y)
LE_IN_GA(s(X), s(Y)) → U11_GA(X, Y, le_in_ga(X, Y))
LE_IN_GA(s(X), s(Y)) → LE_IN_GA(X, Y)
U7_GAAA(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_GAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U7_GAAA(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → PART_IN_GAAA(X, Xs, Ls, Bs)
PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_AAAA(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → LE_IN_AA(X, Y)
LE_IN_AA(s(X), s(Y)) → U11_AA(X, Y, le_in_aa(X, Y))
LE_IN_AA(s(X), s(Y)) → LE_IN_AA(X, Y)
U7_AAAA(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_AAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U7_AAAA(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → PART_IN_GAAA(X, Xs, Ls, Bs)
U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_AG(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → QS_IN_GA(Littles, Ls)
QS_IN_GA(.(X, Xs), Ys) → U1_GA(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
QS_IN_GA(.(X, Xs), Ys) → PART_IN_GGAA(X, Xs, Littles, Bigs)
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GGAA(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → GT_IN_GG(X, Y)
GT_IN_GG(s(X), s(Y)) → U10_GG(X, Y, gt_in_gg(X, Y))
GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)
U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_GGAA(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → LE_IN_GG(X, Y)
LE_IN_GG(s(X), s(Y)) → U11_GG(X, Y, le_in_gg(X, Y))
LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)
U7_GGAA(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U7_GGAA(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_GA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → QS_IN_GA(Littles, Ls)
U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_GA(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → QS_IN_GA(Bigs, Bs)
U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_GA(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → APP_IN_GGA(Ls, .(X, Bs), Ys)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U9_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)
U2_AG(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_AG(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U2_AG(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → QS_IN_AA(Bigs, Bs)
QS_IN_AA(.(X, Xs), Ys) → U1_AA(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
QS_IN_AA(.(X, Xs), Ys) → PART_IN_AAAA(X, Xs, Littles, Bigs)
U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_AA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → QS_IN_GA(Littles, Ls)
U2_AA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_AA(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U2_AA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → QS_IN_AA(Bigs, Bs)
U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_AA(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → APP_IN_GAA(Ls, .(X, Bs), Ys)
APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) → U9_GAA(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GAA(Xs, Ys, Zs)
U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_AG(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → APP_IN_GAG(Ls, .(X, Bs), Ys)
APP_IN_GAG(.(X, Xs), Ys, .(X, Zs)) → U9_GAG(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
APP_IN_GAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

qs_in_ag(.(X, Xs), Ys) → U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
gt_in_aa(s(X), s(Y)) → U10_aa(X, Y, gt_in_aa(X, Y))
gt_in_aa(s(0), 0) → gt_out_aa(s(0), 0)
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
gt_in_ga(s(X), s(Y)) → U10_ga(X, Y, gt_in_ga(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U10_ga(X, Y, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_ga(X, Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
U11_ga(X, Y, le_out_ga(X, Y)) → le_out_ga(s(X), s(Y))
U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, [], [], []) → part_out_gaaa(X, [], [], [])
U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
le_in_aa(s(X), s(Y)) → U11_aa(X, Y, le_in_aa(X, Y))
le_in_aa(0, s(X)) → le_out_aa(0, s(X))
le_in_aa(0, 0) → le_out_aa(0, 0)
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_aaaa(X, [], [], []) → part_out_aaaa(X, [], [], [])
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa([], []) → qs_out_aa([], [])
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
app_in_gaa([], Ys, Ys) → app_out_gaa([], Ys, Ys)
U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
app_in_gag(.(X, Xs), Ys, .(X, Zs)) → U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
app_in_gag([], Ys, Ys) → app_out_gag([], Ys, Ys)
U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) → app_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt_in_aa(x1, x2)  =  gt_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)
part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
gt_in_ga(x1, x2)  =  gt_in_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
0  =  0
gt_out_ga(x1, x2)  =  gt_out_ga(x1, x2)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x2, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
le_in_ga(x1, x2)  =  le_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
le_out_ga(x1, x2)  =  le_out_ga(x1)
U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x1, x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1, x3)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
le_in_aa(x1, x2)  =  le_in_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
le_out_aa(x1, x2)  =  le_out_aa(x1)
U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)
gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x2, x3, x6)
[]  =  []
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)
qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)
qs_out_aa(x1, x2)  =  qs_out_aa
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x5)
app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x1, x2, x4, x5)
app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)
qs_out_ag(x1, x2)  =  qs_out_ag(x2)
U5_GAAA(x1, x2, x3, x4, x5, x6)  =  U5_GAAA(x1, x6)
U5_AAAA(x1, x2, x3, x4, x5, x6)  =  U5_AAAA(x6)
U8_GAAA(x1, x2, x3, x4, x5, x6)  =  U8_GAAA(x1, x6)
U10_GA(x1, x2, x3)  =  U10_GA(x1, x3)
LE_IN_GG(x1, x2)  =  LE_IN_GG(x1, x2)
U6_GGAA(x1, x2, x3, x4, x5, x6)  =  U6_GGAA(x1, x2, x3, x6)
U6_AAAA(x1, x2, x3, x4, x5, x6)  =  U6_AAAA(x2, x6)
U9_GGA(x1, x2, x3, x4, x5)  =  U9_GGA(x1, x2, x3, x5)
U10_AA(x1, x2, x3)  =  U10_AA(x3)
LE_IN_GA(x1, x2)  =  LE_IN_GA(x1)
U7_GGAA(x1, x2, x3, x4, x5, x6)  =  U7_GGAA(x1, x2, x3, x6)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)
LE_IN_AA(x1, x2)  =  LE_IN_AA
U3_AA(x1, x2, x3, x4, x5)  =  U3_AA(x4, x5)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x3, x4, x5)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x4, x5)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x1, x2, x4)
U9_GAA(x1, x2, x3, x4, x5)  =  U9_GAA(x1, x2, x5)
QS_IN_AA(x1, x2)  =  QS_IN_AA
U2_AA(x1, x2, x3, x4, x5)  =  U2_AA(x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x4, x5)
GT_IN_GG(x1, x2)  =  GT_IN_GG(x1, x2)
PART_IN_GAAA(x1, x2, x3, x4)  =  PART_IN_GAAA(x1)
APP_IN_GAA(x1, x2, x3)  =  APP_IN_GAA(x1)
U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)
U7_AAAA(x1, x2, x3, x4, x5, x6)  =  U7_AAAA(x6)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
U8_AAAA(x1, x2, x3, x4, x5, x6)  =  U8_AAAA(x6)
QS_IN_AG(x1, x2)  =  QS_IN_AG(x2)
QS_IN_GA(x1, x2)  =  QS_IN_GA(x1)
PART_IN_GGAA(x1, x2, x3, x4)  =  PART_IN_GGAA(x1, x2)
U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)
PART_IN_AAAA(x1, x2, x3, x4)  =  PART_IN_AAAA
U8_GGAA(x1, x2, x3, x4, x5, x6)  =  U8_GGAA(x1, x2, x3, x6)
U2_AG(x1, x2, x3, x4, x5)  =  U2_AG(x3, x5)
U9_GAG(x1, x2, x3, x4, x5)  =  U9_GAG(x1, x2, x4, x5)
U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)
U4_AA(x1, x2, x3, x4)  =  U4_AA(x4)
U6_GAAA(x1, x2, x3, x4, x5, x6)  =  U6_GAAA(x1, x2, x6)
GT_IN_AA(x1, x2)  =  GT_IN_AA
U11_GA(x1, x2, x3)  =  U11_GA(x1, x3)
U4_AG(x1, x2, x3, x4)  =  U4_AG(x3, x4)
U11_GG(x1, x2, x3)  =  U11_GG(x1, x2, x3)
U10_GG(x1, x2, x3)  =  U10_GG(x1, x2, x3)
GT_IN_GA(x1, x2)  =  GT_IN_GA(x1)
U11_AA(x1, x2, x3)  =  U11_AA(x3)
APP_IN_GAG(x1, x2, x3)  =  APP_IN_GAG(x1, x3)
U7_GAAA(x1, x2, x3, x4, x5, x6)  =  U7_GAAA(x1, x6)

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:

QS_IN_AG(.(X, Xs), Ys) → U1_AG(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
QS_IN_AG(.(X, Xs), Ys) → PART_IN_AAAA(X, Xs, Littles, Bigs)
PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_AAAA(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) → GT_IN_AA(X, Y)
GT_IN_AA(s(X), s(Y)) → U10_AA(X, Y, gt_in_aa(X, Y))
GT_IN_AA(s(X), s(Y)) → GT_IN_AA(X, Y)
U5_AAAA(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_AAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U5_AAAA(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → PART_IN_GAAA(X, Xs, Ls, Bs)
PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GAAA(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) → GT_IN_GA(X, Y)
GT_IN_GA(s(X), s(Y)) → U10_GA(X, Y, gt_in_ga(X, Y))
GT_IN_GA(s(X), s(Y)) → GT_IN_GA(X, Y)
U5_GAAA(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_GAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U5_GAAA(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → PART_IN_GAAA(X, Xs, Ls, Bs)
PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_GAAA(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → LE_IN_GA(X, Y)
LE_IN_GA(s(X), s(Y)) → U11_GA(X, Y, le_in_ga(X, Y))
LE_IN_GA(s(X), s(Y)) → LE_IN_GA(X, Y)
U7_GAAA(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_GAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U7_GAAA(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → PART_IN_GAAA(X, Xs, Ls, Bs)
PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_AAAA(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → LE_IN_AA(X, Y)
LE_IN_AA(s(X), s(Y)) → U11_AA(X, Y, le_in_aa(X, Y))
LE_IN_AA(s(X), s(Y)) → LE_IN_AA(X, Y)
U7_AAAA(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_AAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U7_AAAA(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → PART_IN_GAAA(X, Xs, Ls, Bs)
U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_AG(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → QS_IN_GA(Littles, Ls)
QS_IN_GA(.(X, Xs), Ys) → U1_GA(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
QS_IN_GA(.(X, Xs), Ys) → PART_IN_GGAA(X, Xs, Littles, Bigs)
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GGAA(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → GT_IN_GG(X, Y)
GT_IN_GG(s(X), s(Y)) → U10_GG(X, Y, gt_in_gg(X, Y))
GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)
U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_GGAA(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → LE_IN_GG(X, Y)
LE_IN_GG(s(X), s(Y)) → U11_GG(X, Y, le_in_gg(X, Y))
LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)
U7_GGAA(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U7_GGAA(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_GA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → QS_IN_GA(Littles, Ls)
U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_GA(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → QS_IN_GA(Bigs, Bs)
U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_GA(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → APP_IN_GGA(Ls, .(X, Bs), Ys)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U9_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)
U2_AG(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_AG(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U2_AG(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → QS_IN_AA(Bigs, Bs)
QS_IN_AA(.(X, Xs), Ys) → U1_AA(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
QS_IN_AA(.(X, Xs), Ys) → PART_IN_AAAA(X, Xs, Littles, Bigs)
U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_AA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → QS_IN_GA(Littles, Ls)
U2_AA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_AA(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U2_AA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → QS_IN_AA(Bigs, Bs)
U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_AA(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → APP_IN_GAA(Ls, .(X, Bs), Ys)
APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) → U9_GAA(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GAA(Xs, Ys, Zs)
U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_AG(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → APP_IN_GAG(Ls, .(X, Bs), Ys)
APP_IN_GAG(.(X, Xs), Ys, .(X, Zs)) → U9_GAG(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
APP_IN_GAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

qs_in_ag(.(X, Xs), Ys) → U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
gt_in_aa(s(X), s(Y)) → U10_aa(X, Y, gt_in_aa(X, Y))
gt_in_aa(s(0), 0) → gt_out_aa(s(0), 0)
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
gt_in_ga(s(X), s(Y)) → U10_ga(X, Y, gt_in_ga(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U10_ga(X, Y, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_ga(X, Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
U11_ga(X, Y, le_out_ga(X, Y)) → le_out_ga(s(X), s(Y))
U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, [], [], []) → part_out_gaaa(X, [], [], [])
U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
le_in_aa(s(X), s(Y)) → U11_aa(X, Y, le_in_aa(X, Y))
le_in_aa(0, s(X)) → le_out_aa(0, s(X))
le_in_aa(0, 0) → le_out_aa(0, 0)
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_aaaa(X, [], [], []) → part_out_aaaa(X, [], [], [])
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa([], []) → qs_out_aa([], [])
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
app_in_gaa([], Ys, Ys) → app_out_gaa([], Ys, Ys)
U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
app_in_gag(.(X, Xs), Ys, .(X, Zs)) → U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
app_in_gag([], Ys, Ys) → app_out_gag([], Ys, Ys)
U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) → app_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt_in_aa(x1, x2)  =  gt_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)
part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
gt_in_ga(x1, x2)  =  gt_in_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
0  =  0
gt_out_ga(x1, x2)  =  gt_out_ga(x1, x2)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x2, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
le_in_ga(x1, x2)  =  le_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
le_out_ga(x1, x2)  =  le_out_ga(x1)
U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x1, x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1, x3)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
le_in_aa(x1, x2)  =  le_in_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
le_out_aa(x1, x2)  =  le_out_aa(x1)
U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)
gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x2, x3, x6)
[]  =  []
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)
qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)
qs_out_aa(x1, x2)  =  qs_out_aa
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x5)
app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x1, x2, x4, x5)
app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)
qs_out_ag(x1, x2)  =  qs_out_ag(x2)
U5_GAAA(x1, x2, x3, x4, x5, x6)  =  U5_GAAA(x1, x6)
U5_AAAA(x1, x2, x3, x4, x5, x6)  =  U5_AAAA(x6)
U8_GAAA(x1, x2, x3, x4, x5, x6)  =  U8_GAAA(x1, x6)
U10_GA(x1, x2, x3)  =  U10_GA(x1, x3)
LE_IN_GG(x1, x2)  =  LE_IN_GG(x1, x2)
U6_GGAA(x1, x2, x3, x4, x5, x6)  =  U6_GGAA(x1, x2, x3, x6)
U6_AAAA(x1, x2, x3, x4, x5, x6)  =  U6_AAAA(x2, x6)
U9_GGA(x1, x2, x3, x4, x5)  =  U9_GGA(x1, x2, x3, x5)
U10_AA(x1, x2, x3)  =  U10_AA(x3)
LE_IN_GA(x1, x2)  =  LE_IN_GA(x1)
U7_GGAA(x1, x2, x3, x4, x5, x6)  =  U7_GGAA(x1, x2, x3, x6)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)
LE_IN_AA(x1, x2)  =  LE_IN_AA
U3_AA(x1, x2, x3, x4, x5)  =  U3_AA(x4, x5)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x3, x4, x5)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x4, x5)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x1, x2, x4)
U9_GAA(x1, x2, x3, x4, x5)  =  U9_GAA(x1, x2, x5)
QS_IN_AA(x1, x2)  =  QS_IN_AA
U2_AA(x1, x2, x3, x4, x5)  =  U2_AA(x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x4, x5)
GT_IN_GG(x1, x2)  =  GT_IN_GG(x1, x2)
PART_IN_GAAA(x1, x2, x3, x4)  =  PART_IN_GAAA(x1)
APP_IN_GAA(x1, x2, x3)  =  APP_IN_GAA(x1)
U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)
U7_AAAA(x1, x2, x3, x4, x5, x6)  =  U7_AAAA(x6)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
U8_AAAA(x1, x2, x3, x4, x5, x6)  =  U8_AAAA(x6)
QS_IN_AG(x1, x2)  =  QS_IN_AG(x2)
QS_IN_GA(x1, x2)  =  QS_IN_GA(x1)
PART_IN_GGAA(x1, x2, x3, x4)  =  PART_IN_GGAA(x1, x2)
U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)
PART_IN_AAAA(x1, x2, x3, x4)  =  PART_IN_AAAA
U8_GGAA(x1, x2, x3, x4, x5, x6)  =  U8_GGAA(x1, x2, x3, x6)
U2_AG(x1, x2, x3, x4, x5)  =  U2_AG(x3, x5)
U9_GAG(x1, x2, x3, x4, x5)  =  U9_GAG(x1, x2, x4, x5)
U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)
U4_AA(x1, x2, x3, x4)  =  U4_AA(x4)
U6_GAAA(x1, x2, x3, x4, x5, x6)  =  U6_GAAA(x1, x2, x6)
GT_IN_AA(x1, x2)  =  GT_IN_AA
U11_GA(x1, x2, x3)  =  U11_GA(x1, x3)
U4_AG(x1, x2, x3, x4)  =  U4_AG(x3, x4)
U11_GG(x1, x2, x3)  =  U11_GG(x1, x2, x3)
U10_GG(x1, x2, x3)  =  U10_GG(x1, x2, x3)
GT_IN_GA(x1, x2)  =  GT_IN_GA(x1)
U11_AA(x1, x2, x3)  =  U11_AA(x3)
APP_IN_GAG(x1, x2, x3)  =  APP_IN_GAG(x1, x3)
U7_GAAA(x1, x2, x3, x4, x5, x6)  =  U7_GAAA(x1, x6)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 13 SCCs with 42 less nodes.

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

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

APP_IN_GAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

qs_in_ag(.(X, Xs), Ys) → U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
gt_in_aa(s(X), s(Y)) → U10_aa(X, Y, gt_in_aa(X, Y))
gt_in_aa(s(0), 0) → gt_out_aa(s(0), 0)
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
gt_in_ga(s(X), s(Y)) → U10_ga(X, Y, gt_in_ga(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U10_ga(X, Y, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_ga(X, Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
U11_ga(X, Y, le_out_ga(X, Y)) → le_out_ga(s(X), s(Y))
U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, [], [], []) → part_out_gaaa(X, [], [], [])
U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
le_in_aa(s(X), s(Y)) → U11_aa(X, Y, le_in_aa(X, Y))
le_in_aa(0, s(X)) → le_out_aa(0, s(X))
le_in_aa(0, 0) → le_out_aa(0, 0)
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_aaaa(X, [], [], []) → part_out_aaaa(X, [], [], [])
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa([], []) → qs_out_aa([], [])
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
app_in_gaa([], Ys, Ys) → app_out_gaa([], Ys, Ys)
U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
app_in_gag(.(X, Xs), Ys, .(X, Zs)) → U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
app_in_gag([], Ys, Ys) → app_out_gag([], Ys, Ys)
U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) → app_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt_in_aa(x1, x2)  =  gt_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)
part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
gt_in_ga(x1, x2)  =  gt_in_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
0  =  0
gt_out_ga(x1, x2)  =  gt_out_ga(x1, x2)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x2, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
le_in_ga(x1, x2)  =  le_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
le_out_ga(x1, x2)  =  le_out_ga(x1)
U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x1, x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1, x3)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
le_in_aa(x1, x2)  =  le_in_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
le_out_aa(x1, x2)  =  le_out_aa(x1)
U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)
gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x2, x3, x6)
[]  =  []
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)
qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)
qs_out_aa(x1, x2)  =  qs_out_aa
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x5)
app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x1, x2, x4, x5)
app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)
qs_out_ag(x1, x2)  =  qs_out_ag(x2)
APP_IN_GAG(x1, x2, x3)  =  APP_IN_GAG(x1, x3)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

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

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

APP_IN_GAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GAG(Xs, Ys, Zs)

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

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

APP_IN_GAG(.(X, Xs), .(X, Zs)) → APP_IN_GAG(Xs, Zs)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GAA(Xs, Ys, Zs)

The TRS R consists of the following rules:

qs_in_ag(.(X, Xs), Ys) → U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
gt_in_aa(s(X), s(Y)) → U10_aa(X, Y, gt_in_aa(X, Y))
gt_in_aa(s(0), 0) → gt_out_aa(s(0), 0)
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
gt_in_ga(s(X), s(Y)) → U10_ga(X, Y, gt_in_ga(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U10_ga(X, Y, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_ga(X, Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
U11_ga(X, Y, le_out_ga(X, Y)) → le_out_ga(s(X), s(Y))
U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, [], [], []) → part_out_gaaa(X, [], [], [])
U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
le_in_aa(s(X), s(Y)) → U11_aa(X, Y, le_in_aa(X, Y))
le_in_aa(0, s(X)) → le_out_aa(0, s(X))
le_in_aa(0, 0) → le_out_aa(0, 0)
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_aaaa(X, [], [], []) → part_out_aaaa(X, [], [], [])
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa([], []) → qs_out_aa([], [])
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
app_in_gaa([], Ys, Ys) → app_out_gaa([], Ys, Ys)
U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
app_in_gag(.(X, Xs), Ys, .(X, Zs)) → U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
app_in_gag([], Ys, Ys) → app_out_gag([], Ys, Ys)
U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) → app_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt_in_aa(x1, x2)  =  gt_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)
part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
gt_in_ga(x1, x2)  =  gt_in_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
0  =  0
gt_out_ga(x1, x2)  =  gt_out_ga(x1, x2)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x2, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
le_in_ga(x1, x2)  =  le_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
le_out_ga(x1, x2)  =  le_out_ga(x1)
U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x1, x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1, x3)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
le_in_aa(x1, x2)  =  le_in_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
le_out_aa(x1, x2)  =  le_out_aa(x1)
U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)
gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x2, x3, x6)
[]  =  []
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)
qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)
qs_out_aa(x1, x2)  =  qs_out_aa
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x5)
app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x1, x2, x4, x5)
app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)
qs_out_ag(x1, x2)  =  qs_out_ag(x2)
APP_IN_GAA(x1, x2, x3)  =  APP_IN_GAA(x1)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

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

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

APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GAA(Xs, Ys, Zs)

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

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

APP_IN_GAA(.(X, Xs)) → APP_IN_GAA(Xs)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

qs_in_ag(.(X, Xs), Ys) → U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
gt_in_aa(s(X), s(Y)) → U10_aa(X, Y, gt_in_aa(X, Y))
gt_in_aa(s(0), 0) → gt_out_aa(s(0), 0)
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
gt_in_ga(s(X), s(Y)) → U10_ga(X, Y, gt_in_ga(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U10_ga(X, Y, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_ga(X, Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
U11_ga(X, Y, le_out_ga(X, Y)) → le_out_ga(s(X), s(Y))
U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, [], [], []) → part_out_gaaa(X, [], [], [])
U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
le_in_aa(s(X), s(Y)) → U11_aa(X, Y, le_in_aa(X, Y))
le_in_aa(0, s(X)) → le_out_aa(0, s(X))
le_in_aa(0, 0) → le_out_aa(0, 0)
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_aaaa(X, [], [], []) → part_out_aaaa(X, [], [], [])
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa([], []) → qs_out_aa([], [])
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
app_in_gaa([], Ys, Ys) → app_out_gaa([], Ys, Ys)
U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
app_in_gag(.(X, Xs), Ys, .(X, Zs)) → U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
app_in_gag([], Ys, Ys) → app_out_gag([], Ys, Ys)
U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) → app_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt_in_aa(x1, x2)  =  gt_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)
part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
gt_in_ga(x1, x2)  =  gt_in_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
0  =  0
gt_out_ga(x1, x2)  =  gt_out_ga(x1, x2)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x2, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
le_in_ga(x1, x2)  =  le_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
le_out_ga(x1, x2)  =  le_out_ga(x1)
U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x1, x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1, x3)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
le_in_aa(x1, x2)  =  le_in_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
le_out_aa(x1, x2)  =  le_out_aa(x1)
U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)
gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x2, x3, x6)
[]  =  []
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)
qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)
qs_out_aa(x1, x2)  =  qs_out_aa
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x5)
app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x1, x2, x4, x5)
app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)
qs_out_ag(x1, x2)  =  qs_out_ag(x2)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

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

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

APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)

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

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

APP_IN_GGA(.(X, Xs), Ys) → APP_IN_GGA(Xs, Ys)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)

The TRS R consists of the following rules:

qs_in_ag(.(X, Xs), Ys) → U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
gt_in_aa(s(X), s(Y)) → U10_aa(X, Y, gt_in_aa(X, Y))
gt_in_aa(s(0), 0) → gt_out_aa(s(0), 0)
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
gt_in_ga(s(X), s(Y)) → U10_ga(X, Y, gt_in_ga(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U10_ga(X, Y, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_ga(X, Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
U11_ga(X, Y, le_out_ga(X, Y)) → le_out_ga(s(X), s(Y))
U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, [], [], []) → part_out_gaaa(X, [], [], [])
U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
le_in_aa(s(X), s(Y)) → U11_aa(X, Y, le_in_aa(X, Y))
le_in_aa(0, s(X)) → le_out_aa(0, s(X))
le_in_aa(0, 0) → le_out_aa(0, 0)
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_aaaa(X, [], [], []) → part_out_aaaa(X, [], [], [])
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa([], []) → qs_out_aa([], [])
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
app_in_gaa([], Ys, Ys) → app_out_gaa([], Ys, Ys)
U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
app_in_gag(.(X, Xs), Ys, .(X, Zs)) → U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
app_in_gag([], Ys, Ys) → app_out_gag([], Ys, Ys)
U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) → app_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt_in_aa(x1, x2)  =  gt_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)
part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
gt_in_ga(x1, x2)  =  gt_in_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
0  =  0
gt_out_ga(x1, x2)  =  gt_out_ga(x1, x2)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x2, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
le_in_ga(x1, x2)  =  le_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
le_out_ga(x1, x2)  =  le_out_ga(x1)
U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x1, x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1, x3)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
le_in_aa(x1, x2)  =  le_in_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
le_out_aa(x1, x2)  =  le_out_aa(x1)
U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)
gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x2, x3, x6)
[]  =  []
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)
qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)
qs_out_aa(x1, x2)  =  qs_out_aa
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x5)
app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x1, x2, x4, x5)
app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)
qs_out_ag(x1, x2)  =  qs_out_ag(x2)
LE_IN_GG(x1, x2)  =  LE_IN_GG(x1, x2)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

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

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

LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)

R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)

The TRS R consists of the following rules:

qs_in_ag(.(X, Xs), Ys) → U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
gt_in_aa(s(X), s(Y)) → U10_aa(X, Y, gt_in_aa(X, Y))
gt_in_aa(s(0), 0) → gt_out_aa(s(0), 0)
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
gt_in_ga(s(X), s(Y)) → U10_ga(X, Y, gt_in_ga(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U10_ga(X, Y, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_ga(X, Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
U11_ga(X, Y, le_out_ga(X, Y)) → le_out_ga(s(X), s(Y))
U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, [], [], []) → part_out_gaaa(X, [], [], [])
U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
le_in_aa(s(X), s(Y)) → U11_aa(X, Y, le_in_aa(X, Y))
le_in_aa(0, s(X)) → le_out_aa(0, s(X))
le_in_aa(0, 0) → le_out_aa(0, 0)
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_aaaa(X, [], [], []) → part_out_aaaa(X, [], [], [])
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa([], []) → qs_out_aa([], [])
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
app_in_gaa([], Ys, Ys) → app_out_gaa([], Ys, Ys)
U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
app_in_gag(.(X, Xs), Ys, .(X, Zs)) → U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
app_in_gag([], Ys, Ys) → app_out_gag([], Ys, Ys)
U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) → app_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt_in_aa(x1, x2)  =  gt_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)
part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
gt_in_ga(x1, x2)  =  gt_in_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
0  =  0
gt_out_ga(x1, x2)  =  gt_out_ga(x1, x2)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x2, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
le_in_ga(x1, x2)  =  le_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
le_out_ga(x1, x2)  =  le_out_ga(x1)
U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x1, x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1, x3)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
le_in_aa(x1, x2)  =  le_in_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
le_out_aa(x1, x2)  =  le_out_aa(x1)
U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)
gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x2, x3, x6)
[]  =  []
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)
qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)
qs_out_aa(x1, x2)  =  qs_out_aa
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x5)
app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x1, x2, x4, x5)
app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)
qs_out_ag(x1, x2)  =  qs_out_ag(x2)
GT_IN_GG(x1, x2)  =  GT_IN_GG(x1, x2)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

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

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

GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)

R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

GT_IN_GG(s(X), s(Y)) → GT_IN_GG(X, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GGAA(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_GGAA(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
U7_GGAA(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)

The TRS R consists of the following rules:

qs_in_ag(.(X, Xs), Ys) → U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
gt_in_aa(s(X), s(Y)) → U10_aa(X, Y, gt_in_aa(X, Y))
gt_in_aa(s(0), 0) → gt_out_aa(s(0), 0)
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
gt_in_ga(s(X), s(Y)) → U10_ga(X, Y, gt_in_ga(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U10_ga(X, Y, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_ga(X, Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
U11_ga(X, Y, le_out_ga(X, Y)) → le_out_ga(s(X), s(Y))
U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, [], [], []) → part_out_gaaa(X, [], [], [])
U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
le_in_aa(s(X), s(Y)) → U11_aa(X, Y, le_in_aa(X, Y))
le_in_aa(0, s(X)) → le_out_aa(0, s(X))
le_in_aa(0, 0) → le_out_aa(0, 0)
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_aaaa(X, [], [], []) → part_out_aaaa(X, [], [], [])
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa([], []) → qs_out_aa([], [])
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
app_in_gaa([], Ys, Ys) → app_out_gaa([], Ys, Ys)
U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
app_in_gag(.(X, Xs), Ys, .(X, Zs)) → U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
app_in_gag([], Ys, Ys) → app_out_gag([], Ys, Ys)
U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) → app_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt_in_aa(x1, x2)  =  gt_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)
part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
gt_in_ga(x1, x2)  =  gt_in_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
0  =  0
gt_out_ga(x1, x2)  =  gt_out_ga(x1, x2)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x2, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
le_in_ga(x1, x2)  =  le_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
le_out_ga(x1, x2)  =  le_out_ga(x1)
U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x1, x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1, x3)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
le_in_aa(x1, x2)  =  le_in_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
le_out_aa(x1, x2)  =  le_out_aa(x1)
U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)
gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x2, x3, x6)
[]  =  []
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)
qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)
qs_out_aa(x1, x2)  =  qs_out_aa
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x5)
app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x1, x2, x4, x5)
app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)
qs_out_ag(x1, x2)  =  qs_out_ag(x2)
U7_GGAA(x1, x2, x3, x4, x5, x6)  =  U7_GGAA(x1, x2, x3, x6)
PART_IN_GGAA(x1, x2, x3, x4)  =  PART_IN_GGAA(x1, x2)
U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

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

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

PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GGAA(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_GGAA(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
U7_GGAA(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)

The TRS R consists of the following rules:

gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
0  =  0
.(x1, x2)  =  .(x1, x2)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)
gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
U7_GGAA(x1, x2, x3, x4, x5, x6)  =  U7_GGAA(x1, x2, x3, x6)
PART_IN_GGAA(x1, x2, x3, x4)  =  PART_IN_GGAA(x1, x2)
U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

U7_GGAA(X, Y, Xs, le_out_gg(X, Y)) → PART_IN_GGAA(X, Xs)
PART_IN_GGAA(X, .(Y, Xs)) → U5_GGAA(X, Y, Xs, gt_in_gg(X, Y))
PART_IN_GGAA(X, .(Y, Xs)) → U7_GGAA(X, Y, Xs, le_in_gg(X, Y))
U5_GGAA(X, Y, Xs, gt_out_gg(X, Y)) → PART_IN_GGAA(X, Xs)

The TRS R consists of the following rules:

gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))

The set Q consists of the following terms:

gt_in_gg(x0, x1)
le_in_gg(x0, x1)
U10_gg(x0, x1, x2)
U11_gg(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

QS_IN_GA(.(X, Xs), Ys) → U1_GA(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → QS_IN_GA(Littles, Ls)
U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_GA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → QS_IN_GA(Bigs, Bs)

The TRS R consists of the following rules:

qs_in_ag(.(X, Xs), Ys) → U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
gt_in_aa(s(X), s(Y)) → U10_aa(X, Y, gt_in_aa(X, Y))
gt_in_aa(s(0), 0) → gt_out_aa(s(0), 0)
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
gt_in_ga(s(X), s(Y)) → U10_ga(X, Y, gt_in_ga(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U10_ga(X, Y, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_ga(X, Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
U11_ga(X, Y, le_out_ga(X, Y)) → le_out_ga(s(X), s(Y))
U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, [], [], []) → part_out_gaaa(X, [], [], [])
U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
le_in_aa(s(X), s(Y)) → U11_aa(X, Y, le_in_aa(X, Y))
le_in_aa(0, s(X)) → le_out_aa(0, s(X))
le_in_aa(0, 0) → le_out_aa(0, 0)
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_aaaa(X, [], [], []) → part_out_aaaa(X, [], [], [])
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa([], []) → qs_out_aa([], [])
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
app_in_gaa([], Ys, Ys) → app_out_gaa([], Ys, Ys)
U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
app_in_gag(.(X, Xs), Ys, .(X, Zs)) → U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
app_in_gag([], Ys, Ys) → app_out_gag([], Ys, Ys)
U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) → app_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt_in_aa(x1, x2)  =  gt_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)
part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
gt_in_ga(x1, x2)  =  gt_in_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
0  =  0
gt_out_ga(x1, x2)  =  gt_out_ga(x1, x2)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x2, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
le_in_ga(x1, x2)  =  le_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
le_out_ga(x1, x2)  =  le_out_ga(x1)
U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x1, x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1, x3)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
le_in_aa(x1, x2)  =  le_in_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
le_out_aa(x1, x2)  =  le_out_aa(x1)
U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)
gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x2, x3, x6)
[]  =  []
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)
qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)
qs_out_aa(x1, x2)  =  qs_out_aa
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x5)
app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x1, x2, x4, x5)
app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)
qs_out_ag(x1, x2)  =  qs_out_ag(x2)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x4, x5)
QS_IN_GA(x1, x2)  =  QS_IN_GA(x1)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

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

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

QS_IN_GA(.(X, Xs), Ys) → U1_GA(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → QS_IN_GA(Littles, Ls)
U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_GA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → QS_IN_GA(Bigs, Bs)

The TRS R consists of the following rules:

part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
qs_in_ga([], []) → qs_out_ga([], [])
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
0  =  0
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)
gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x2, x3, x6)
[]  =  []
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)
qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x4, x5)
QS_IN_GA(x1, x2)  =  QS_IN_GA(x1)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPOrderProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

QS_IN_GA(.(X, Xs)) → U1_GA(X, Xs, part_in_ggaa(X, Xs))
U1_GA(X, Xs, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_GA(X, Xs, Bigs, qs_in_ga(Littles))
U2_GA(X, Xs, Bigs, qs_out_ga(Littles, Ls)) → QS_IN_GA(Bigs)
U1_GA(X, Xs, part_out_ggaa(X, Xs, Littles, Bigs)) → QS_IN_GA(Littles)

The TRS R consists of the following rules:

part_in_ggaa(X, .(Y, Xs)) → U5_ggaa(X, Y, Xs, gt_in_gg(X, Y))
part_in_ggaa(X, .(Y, Xs)) → U7_ggaa(X, Y, Xs, le_in_gg(X, Y))
part_in_ggaa(X, []) → part_out_ggaa(X, [], [], [])
qs_in_ga(.(X, Xs)) → U1_ga(X, Xs, part_in_ggaa(X, Xs))
qs_in_ga([]) → qs_out_ga([], [])
U5_ggaa(X, Y, Xs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, part_in_ggaa(X, Xs))
U7_ggaa(X, Y, Xs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, part_in_ggaa(X, Xs))
U1_ga(X, Xs, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Bigs, qs_in_ga(Littles))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U6_ggaa(X, Y, Xs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_ggaa(X, Y, Xs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U2_ga(X, Xs, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ls, qs_in_ga(Bigs))
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U3_ga(X, Xs, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, app_in_gga(Ls, .(X, Bs)))
U4_ga(X, Xs, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
app_in_gga(.(X, Xs), Ys) → U9_gga(X, Xs, Ys, app_in_gga(Xs, Ys))
app_in_gga([], Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))

The set Q consists of the following terms:

part_in_ggaa(x0, x1)
qs_in_ga(x0)
U5_ggaa(x0, x1, x2, x3)
U7_ggaa(x0, x1, x2, x3)
U1_ga(x0, x1, x2)
gt_in_gg(x0, x1)
U6_ggaa(x0, x1, x2, x3)
le_in_gg(x0, x1)
U8_ggaa(x0, x1, x2, x3)
U2_ga(x0, x1, x2, x3)
U10_gg(x0, x1, x2)
U11_gg(x0, x1, x2)
U3_ga(x0, x1, x2, x3)
U4_ga(x0, x1, x2)
app_in_gga(x0, x1)
U9_gga(x0, x1, x2, x3)

We have to consider all (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


QS_IN_GA(.(X, Xs)) → U1_GA(X, Xs, part_in_ggaa(X, Xs))
The remaining pairs can at least be oriented weakly.

U1_GA(X, Xs, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_GA(X, Xs, Bigs, qs_in_ga(Littles))
U2_GA(X, Xs, Bigs, qs_out_ga(Littles, Ls)) → QS_IN_GA(Bigs)
U1_GA(X, Xs, part_out_ggaa(X, Xs, Littles, Bigs)) → QS_IN_GA(Littles)
Used ordering: Polynomial interpretation [25]:

POL(.(x1, x2)) = 1 + x2   
POL(0) = 0   
POL(QS_IN_GA(x1)) = x1   
POL(U10_gg(x1, x2, x3)) = 1   
POL(U11_gg(x1, x2, x3)) = 0   
POL(U1_GA(x1, x2, x3)) = x3   
POL(U1_ga(x1, x2, x3)) = 0   
POL(U2_GA(x1, x2, x3, x4)) = x3   
POL(U2_ga(x1, x2, x3, x4)) = 0   
POL(U3_ga(x1, x2, x3, x4)) = 0   
POL(U4_ga(x1, x2, x3)) = 0   
POL(U5_ggaa(x1, x2, x3, x4)) = x3 + x4   
POL(U6_ggaa(x1, x2, x3, x4)) = 1 + x2 + x4   
POL(U7_ggaa(x1, x2, x3, x4)) = 1 + x3   
POL(U8_ggaa(x1, x2, x3, x4)) = 1 + x4   
POL(U9_gga(x1, x2, x3, x4)) = 0   
POL([]) = 0   
POL(app_in_gga(x1, x2)) = 0   
POL(app_out_gga(x1, x2, x3)) = 0   
POL(gt_in_gg(x1, x2)) = 1   
POL(gt_out_gg(x1, x2)) = 1 + x1 + x2   
POL(le_in_gg(x1, x2)) = 0   
POL(le_out_gg(x1, x2)) = 0   
POL(part_in_ggaa(x1, x2)) = x2   
POL(part_out_ggaa(x1, x2, x3, x4)) = x3 + x4   
POL(qs_in_ga(x1)) = 0   
POL(qs_out_ga(x1, x2)) = 0   
POL(s(x1)) = 0   

The following usable rules [17] were oriented:

part_in_ggaa(X, .(Y, Xs)) → U5_ggaa(X, Y, Xs, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U7_ggaa(X, Y, Xs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, part_in_ggaa(X, Xs))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
part_in_ggaa(X, .(Y, Xs)) → U7_ggaa(X, Y, Xs, le_in_gg(X, Y))
U8_ggaa(X, Y, Xs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U5_ggaa(X, Y, Xs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, part_in_ggaa(X, Xs))
part_in_ggaa(X, []) → part_out_ggaa(X, [], [], [])
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U6_ggaa(X, Y, Xs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ DependencyGraphProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

U1_GA(X, Xs, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_GA(X, Xs, Bigs, qs_in_ga(Littles))
U2_GA(X, Xs, Bigs, qs_out_ga(Littles, Ls)) → QS_IN_GA(Bigs)
U1_GA(X, Xs, part_out_ggaa(X, Xs, Littles, Bigs)) → QS_IN_GA(Littles)

The TRS R consists of the following rules:

part_in_ggaa(X, .(Y, Xs)) → U5_ggaa(X, Y, Xs, gt_in_gg(X, Y))
part_in_ggaa(X, .(Y, Xs)) → U7_ggaa(X, Y, Xs, le_in_gg(X, Y))
part_in_ggaa(X, []) → part_out_ggaa(X, [], [], [])
qs_in_ga(.(X, Xs)) → U1_ga(X, Xs, part_in_ggaa(X, Xs))
qs_in_ga([]) → qs_out_ga([], [])
U5_ggaa(X, Y, Xs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, part_in_ggaa(X, Xs))
U7_ggaa(X, Y, Xs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, part_in_ggaa(X, Xs))
U1_ga(X, Xs, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Bigs, qs_in_ga(Littles))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U6_ggaa(X, Y, Xs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_ggaa(X, Y, Xs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U2_ga(X, Xs, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ls, qs_in_ga(Bigs))
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U3_ga(X, Xs, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, app_in_gga(Ls, .(X, Bs)))
U4_ga(X, Xs, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
app_in_gga(.(X, Xs), Ys) → U9_gga(X, Xs, Ys, app_in_gga(Xs, Ys))
app_in_gga([], Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))

The set Q consists of the following terms:

part_in_ggaa(x0, x1)
qs_in_ga(x0)
U5_ggaa(x0, x1, x2, x3)
U7_ggaa(x0, x1, x2, x3)
U1_ga(x0, x1, x2)
gt_in_gg(x0, x1)
U6_ggaa(x0, x1, x2, x3)
le_in_gg(x0, x1)
U8_ggaa(x0, x1, x2, x3)
U2_ga(x0, x1, x2, x3)
U10_gg(x0, x1, x2)
U11_gg(x0, x1, x2)
U3_ga(x0, x1, x2, x3)
U4_ga(x0, x1, x2)
app_in_gga(x0, x1)
U9_gga(x0, x1, x2, x3)

We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 3 less nodes.

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

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

LE_IN_AA(s(X), s(Y)) → LE_IN_AA(X, Y)

The TRS R consists of the following rules:

qs_in_ag(.(X, Xs), Ys) → U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
gt_in_aa(s(X), s(Y)) → U10_aa(X, Y, gt_in_aa(X, Y))
gt_in_aa(s(0), 0) → gt_out_aa(s(0), 0)
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
gt_in_ga(s(X), s(Y)) → U10_ga(X, Y, gt_in_ga(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U10_ga(X, Y, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_ga(X, Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
U11_ga(X, Y, le_out_ga(X, Y)) → le_out_ga(s(X), s(Y))
U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, [], [], []) → part_out_gaaa(X, [], [], [])
U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
le_in_aa(s(X), s(Y)) → U11_aa(X, Y, le_in_aa(X, Y))
le_in_aa(0, s(X)) → le_out_aa(0, s(X))
le_in_aa(0, 0) → le_out_aa(0, 0)
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_aaaa(X, [], [], []) → part_out_aaaa(X, [], [], [])
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa([], []) → qs_out_aa([], [])
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
app_in_gaa([], Ys, Ys) → app_out_gaa([], Ys, Ys)
U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
app_in_gag(.(X, Xs), Ys, .(X, Zs)) → U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
app_in_gag([], Ys, Ys) → app_out_gag([], Ys, Ys)
U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) → app_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt_in_aa(x1, x2)  =  gt_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)
part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
gt_in_ga(x1, x2)  =  gt_in_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
0  =  0
gt_out_ga(x1, x2)  =  gt_out_ga(x1, x2)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x2, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
le_in_ga(x1, x2)  =  le_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
le_out_ga(x1, x2)  =  le_out_ga(x1)
U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x1, x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1, x3)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
le_in_aa(x1, x2)  =  le_in_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
le_out_aa(x1, x2)  =  le_out_aa(x1)
U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)
gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x2, x3, x6)
[]  =  []
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)
qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)
qs_out_aa(x1, x2)  =  qs_out_aa
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x5)
app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x1, x2, x4, x5)
app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)
qs_out_ag(x1, x2)  =  qs_out_ag(x2)
LE_IN_AA(x1, x2)  =  LE_IN_AA

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

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

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

LE_IN_AA(s(X), s(Y)) → LE_IN_AA(X, Y)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LE_IN_AA(x1, x2)  =  LE_IN_AA

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ NonTerminationProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

LE_IN_AA → LE_IN_AA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

LE_IN_AA → LE_IN_AA

The TRS R consists of the following rules:none


s = LE_IN_AA evaluates to t =LE_IN_AA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from LE_IN_AA to LE_IN_AA.





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

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

LE_IN_GA(s(X), s(Y)) → LE_IN_GA(X, Y)

The TRS R consists of the following rules:

qs_in_ag(.(X, Xs), Ys) → U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
gt_in_aa(s(X), s(Y)) → U10_aa(X, Y, gt_in_aa(X, Y))
gt_in_aa(s(0), 0) → gt_out_aa(s(0), 0)
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
gt_in_ga(s(X), s(Y)) → U10_ga(X, Y, gt_in_ga(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U10_ga(X, Y, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_ga(X, Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
U11_ga(X, Y, le_out_ga(X, Y)) → le_out_ga(s(X), s(Y))
U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, [], [], []) → part_out_gaaa(X, [], [], [])
U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
le_in_aa(s(X), s(Y)) → U11_aa(X, Y, le_in_aa(X, Y))
le_in_aa(0, s(X)) → le_out_aa(0, s(X))
le_in_aa(0, 0) → le_out_aa(0, 0)
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_aaaa(X, [], [], []) → part_out_aaaa(X, [], [], [])
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa([], []) → qs_out_aa([], [])
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
app_in_gaa([], Ys, Ys) → app_out_gaa([], Ys, Ys)
U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
app_in_gag(.(X, Xs), Ys, .(X, Zs)) → U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
app_in_gag([], Ys, Ys) → app_out_gag([], Ys, Ys)
U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) → app_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt_in_aa(x1, x2)  =  gt_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)
part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
gt_in_ga(x1, x2)  =  gt_in_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
0  =  0
gt_out_ga(x1, x2)  =  gt_out_ga(x1, x2)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x2, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
le_in_ga(x1, x2)  =  le_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
le_out_ga(x1, x2)  =  le_out_ga(x1)
U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x1, x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1, x3)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
le_in_aa(x1, x2)  =  le_in_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
le_out_aa(x1, x2)  =  le_out_aa(x1)
U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)
gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x2, x3, x6)
[]  =  []
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)
qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)
qs_out_aa(x1, x2)  =  qs_out_aa
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x5)
app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x1, x2, x4, x5)
app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)
qs_out_ag(x1, x2)  =  qs_out_ag(x2)
LE_IN_GA(x1, x2)  =  LE_IN_GA(x1)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

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

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

LE_IN_GA(s(X), s(Y)) → LE_IN_GA(X, Y)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LE_IN_GA(x1, x2)  =  LE_IN_GA(x1)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

LE_IN_GA(s(X)) → LE_IN_GA(X)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

GT_IN_GA(s(X), s(Y)) → GT_IN_GA(X, Y)

The TRS R consists of the following rules:

qs_in_ag(.(X, Xs), Ys) → U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
gt_in_aa(s(X), s(Y)) → U10_aa(X, Y, gt_in_aa(X, Y))
gt_in_aa(s(0), 0) → gt_out_aa(s(0), 0)
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
gt_in_ga(s(X), s(Y)) → U10_ga(X, Y, gt_in_ga(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U10_ga(X, Y, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_ga(X, Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
U11_ga(X, Y, le_out_ga(X, Y)) → le_out_ga(s(X), s(Y))
U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, [], [], []) → part_out_gaaa(X, [], [], [])
U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
le_in_aa(s(X), s(Y)) → U11_aa(X, Y, le_in_aa(X, Y))
le_in_aa(0, s(X)) → le_out_aa(0, s(X))
le_in_aa(0, 0) → le_out_aa(0, 0)
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_aaaa(X, [], [], []) → part_out_aaaa(X, [], [], [])
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa([], []) → qs_out_aa([], [])
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
app_in_gaa([], Ys, Ys) → app_out_gaa([], Ys, Ys)
U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
app_in_gag(.(X, Xs), Ys, .(X, Zs)) → U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
app_in_gag([], Ys, Ys) → app_out_gag([], Ys, Ys)
U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) → app_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt_in_aa(x1, x2)  =  gt_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)
part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
gt_in_ga(x1, x2)  =  gt_in_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
0  =  0
gt_out_ga(x1, x2)  =  gt_out_ga(x1, x2)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x2, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
le_in_ga(x1, x2)  =  le_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
le_out_ga(x1, x2)  =  le_out_ga(x1)
U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x1, x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1, x3)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
le_in_aa(x1, x2)  =  le_in_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
le_out_aa(x1, x2)  =  le_out_aa(x1)
U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)
gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x2, x3, x6)
[]  =  []
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)
qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)
qs_out_aa(x1, x2)  =  qs_out_aa
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x5)
app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x1, x2, x4, x5)
app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)
qs_out_ag(x1, x2)  =  qs_out_ag(x2)
GT_IN_GA(x1, x2)  =  GT_IN_GA(x1)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

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

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

GT_IN_GA(s(X), s(Y)) → GT_IN_GA(X, Y)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
GT_IN_GA(x1, x2)  =  GT_IN_GA(x1)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

GT_IN_GA(s(X)) → GT_IN_GA(X)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP
              ↳ PiDP

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

PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_GAAA(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
U5_GAAA(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → PART_IN_GAAA(X, Xs, Ls, Bs)
U7_GAAA(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → PART_IN_GAAA(X, Xs, Ls, Bs)
PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GAAA(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))

The TRS R consists of the following rules:

qs_in_ag(.(X, Xs), Ys) → U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
gt_in_aa(s(X), s(Y)) → U10_aa(X, Y, gt_in_aa(X, Y))
gt_in_aa(s(0), 0) → gt_out_aa(s(0), 0)
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
gt_in_ga(s(X), s(Y)) → U10_ga(X, Y, gt_in_ga(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U10_ga(X, Y, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_ga(X, Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
U11_ga(X, Y, le_out_ga(X, Y)) → le_out_ga(s(X), s(Y))
U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, [], [], []) → part_out_gaaa(X, [], [], [])
U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
le_in_aa(s(X), s(Y)) → U11_aa(X, Y, le_in_aa(X, Y))
le_in_aa(0, s(X)) → le_out_aa(0, s(X))
le_in_aa(0, 0) → le_out_aa(0, 0)
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_aaaa(X, [], [], []) → part_out_aaaa(X, [], [], [])
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa([], []) → qs_out_aa([], [])
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
app_in_gaa([], Ys, Ys) → app_out_gaa([], Ys, Ys)
U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
app_in_gag(.(X, Xs), Ys, .(X, Zs)) → U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
app_in_gag([], Ys, Ys) → app_out_gag([], Ys, Ys)
U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) → app_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt_in_aa(x1, x2)  =  gt_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)
part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
gt_in_ga(x1, x2)  =  gt_in_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
0  =  0
gt_out_ga(x1, x2)  =  gt_out_ga(x1, x2)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x2, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
le_in_ga(x1, x2)  =  le_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
le_out_ga(x1, x2)  =  le_out_ga(x1)
U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x1, x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1, x3)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
le_in_aa(x1, x2)  =  le_in_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
le_out_aa(x1, x2)  =  le_out_aa(x1)
U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)
gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x2, x3, x6)
[]  =  []
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)
qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)
qs_out_aa(x1, x2)  =  qs_out_aa
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x5)
app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x1, x2, x4, x5)
app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)
qs_out_ag(x1, x2)  =  qs_out_ag(x2)
U5_GAAA(x1, x2, x3, x4, x5, x6)  =  U5_GAAA(x1, x6)
PART_IN_GAAA(x1, x2, x3, x4)  =  PART_IN_GAAA(x1)
U7_GAAA(x1, x2, x3, x4, x5, x6)  =  U7_GAAA(x1, x6)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

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

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

PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_GAAA(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
U5_GAAA(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → PART_IN_GAAA(X, Xs, Ls, Bs)
U7_GAAA(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → PART_IN_GAAA(X, Xs, Ls, Bs)
PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GAAA(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))

The TRS R consists of the following rules:

le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_ga(X, Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
gt_in_ga(s(X), s(Y)) → U10_ga(X, Y, gt_in_ga(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U11_ga(X, Y, le_out_ga(X, Y)) → le_out_ga(s(X), s(Y))
U10_ga(X, Y, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))

The argument filtering Pi contains the following mapping:
gt_in_ga(x1, x2)  =  gt_in_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
0  =  0
gt_out_ga(x1, x2)  =  gt_out_ga(x1, x2)
le_in_ga(x1, x2)  =  le_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
le_out_ga(x1, x2)  =  le_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U5_GAAA(x1, x2, x3, x4, x5, x6)  =  U5_GAAA(x1, x6)
PART_IN_GAAA(x1, x2, x3, x4)  =  PART_IN_GAAA(x1)
U7_GAAA(x1, x2, x3, x4, x5, x6)  =  U7_GAAA(x1, x6)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
              ↳ PiDP
              ↳ PiDP

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

PART_IN_GAAA(X) → U7_GAAA(X, le_in_ga(X))
U7_GAAA(X, le_out_ga(X)) → PART_IN_GAAA(X)
U5_GAAA(X, gt_out_ga(X, Y)) → PART_IN_GAAA(X)
PART_IN_GAAA(X) → U5_GAAA(X, gt_in_ga(X))

The TRS R consists of the following rules:

le_in_ga(s(X)) → U11_ga(X, le_in_ga(X))
le_in_ga(0) → le_out_ga(0)
gt_in_ga(s(X)) → U10_ga(X, gt_in_ga(X))
gt_in_ga(s(0)) → gt_out_ga(s(0), 0)
U11_ga(X, le_out_ga(X)) → le_out_ga(s(X))
U10_ga(X, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))

The set Q consists of the following terms:

le_in_ga(x0)
gt_in_ga(x0)
U11_ga(x0, x1)
U10_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule PART_IN_GAAA(X) → U7_GAAA(X, le_in_ga(X)) at position [1] we obtained the following new rules:

PART_IN_GAAA(s(x0)) → U7_GAAA(s(x0), U11_ga(x0, le_in_ga(x0)))
PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga(0))



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
              ↳ PiDP
              ↳ PiDP

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

PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga(0))
U7_GAAA(X, le_out_ga(X)) → PART_IN_GAAA(X)
PART_IN_GAAA(s(x0)) → U7_GAAA(s(x0), U11_ga(x0, le_in_ga(x0)))
U5_GAAA(X, gt_out_ga(X, Y)) → PART_IN_GAAA(X)
PART_IN_GAAA(X) → U5_GAAA(X, gt_in_ga(X))

The TRS R consists of the following rules:

le_in_ga(s(X)) → U11_ga(X, le_in_ga(X))
le_in_ga(0) → le_out_ga(0)
gt_in_ga(s(X)) → U10_ga(X, gt_in_ga(X))
gt_in_ga(s(0)) → gt_out_ga(s(0), 0)
U11_ga(X, le_out_ga(X)) → le_out_ga(s(X))
U10_ga(X, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))

The set Q consists of the following terms:

le_in_ga(x0)
gt_in_ga(x0)
U11_ga(x0, x1)
U10_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule PART_IN_GAAA(X) → U5_GAAA(X, gt_in_ga(X)) at position [1] we obtained the following new rules:

PART_IN_GAAA(s(x0)) → U5_GAAA(s(x0), U10_ga(x0, gt_in_ga(x0)))
PART_IN_GAAA(s(0)) → U5_GAAA(s(0), gt_out_ga(s(0), 0))



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Instantiation
              ↳ PiDP
              ↳ PiDP

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

PART_IN_GAAA(s(x0)) → U5_GAAA(s(x0), U10_ga(x0, gt_in_ga(x0)))
PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga(0))
U7_GAAA(X, le_out_ga(X)) → PART_IN_GAAA(X)
PART_IN_GAAA(s(x0)) → U7_GAAA(s(x0), U11_ga(x0, le_in_ga(x0)))
U5_GAAA(X, gt_out_ga(X, Y)) → PART_IN_GAAA(X)
PART_IN_GAAA(s(0)) → U5_GAAA(s(0), gt_out_ga(s(0), 0))

The TRS R consists of the following rules:

le_in_ga(s(X)) → U11_ga(X, le_in_ga(X))
le_in_ga(0) → le_out_ga(0)
gt_in_ga(s(X)) → U10_ga(X, gt_in_ga(X))
gt_in_ga(s(0)) → gt_out_ga(s(0), 0)
U11_ga(X, le_out_ga(X)) → le_out_ga(s(X))
U10_ga(X, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))

The set Q consists of the following terms:

le_in_ga(x0)
gt_in_ga(x0)
U11_ga(x0, x1)
U10_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U5_GAAA(X, gt_out_ga(X, Y)) → PART_IN_GAAA(X) we obtained the following new rules:

U5_GAAA(s(z0), gt_out_ga(s(z0), x1)) → PART_IN_GAAA(s(z0))
U5_GAAA(s(0), gt_out_ga(s(0), 0)) → PART_IN_GAAA(s(0))



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Instantiation
                                  ↳ QDP
                                    ↳ Instantiation
              ↳ PiDP
              ↳ PiDP

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

PART_IN_GAAA(s(x0)) → U5_GAAA(s(x0), U10_ga(x0, gt_in_ga(x0)))
U5_GAAA(s(z0), gt_out_ga(s(z0), x1)) → PART_IN_GAAA(s(z0))
PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga(0))
U7_GAAA(X, le_out_ga(X)) → PART_IN_GAAA(X)
U5_GAAA(s(0), gt_out_ga(s(0), 0)) → PART_IN_GAAA(s(0))
PART_IN_GAAA(s(x0)) → U7_GAAA(s(x0), U11_ga(x0, le_in_ga(x0)))
PART_IN_GAAA(s(0)) → U5_GAAA(s(0), gt_out_ga(s(0), 0))

The TRS R consists of the following rules:

le_in_ga(s(X)) → U11_ga(X, le_in_ga(X))
le_in_ga(0) → le_out_ga(0)
gt_in_ga(s(X)) → U10_ga(X, gt_in_ga(X))
gt_in_ga(s(0)) → gt_out_ga(s(0), 0)
U11_ga(X, le_out_ga(X)) → le_out_ga(s(X))
U10_ga(X, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))

The set Q consists of the following terms:

le_in_ga(x0)
gt_in_ga(x0)
U11_ga(x0, x1)
U10_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U7_GAAA(X, le_out_ga(X)) → PART_IN_GAAA(X) we obtained the following new rules:

U7_GAAA(s(z0), le_out_ga(s(z0))) → PART_IN_GAAA(s(z0))
U7_GAAA(0, le_out_ga(0)) → PART_IN_GAAA(0)



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Instantiation
                                  ↳ QDP
                                    ↳ Instantiation
                                      ↳ QDP
                                        ↳ DependencyGraphProof
              ↳ PiDP
              ↳ PiDP

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

U7_GAAA(s(z0), le_out_ga(s(z0))) → PART_IN_GAAA(s(z0))
PART_IN_GAAA(s(x0)) → U5_GAAA(s(x0), U10_ga(x0, gt_in_ga(x0)))
U7_GAAA(0, le_out_ga(0)) → PART_IN_GAAA(0)
PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga(0))
U5_GAAA(s(z0), gt_out_ga(s(z0), x1)) → PART_IN_GAAA(s(z0))
U5_GAAA(s(0), gt_out_ga(s(0), 0)) → PART_IN_GAAA(s(0))
PART_IN_GAAA(s(x0)) → U7_GAAA(s(x0), U11_ga(x0, le_in_ga(x0)))
PART_IN_GAAA(s(0)) → U5_GAAA(s(0), gt_out_ga(s(0), 0))

The TRS R consists of the following rules:

le_in_ga(s(X)) → U11_ga(X, le_in_ga(X))
le_in_ga(0) → le_out_ga(0)
gt_in_ga(s(X)) → U10_ga(X, gt_in_ga(X))
gt_in_ga(s(0)) → gt_out_ga(s(0), 0)
U11_ga(X, le_out_ga(X)) → le_out_ga(s(X))
U10_ga(X, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))

The set Q consists of the following terms:

le_in_ga(x0)
gt_in_ga(x0)
U11_ga(x0, x1)
U10_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Instantiation
                                  ↳ QDP
                                    ↳ Instantiation
                                      ↳ QDP
                                        ↳ DependencyGraphProof
                                          ↳ AND
                                            ↳ QDP
                                              ↳ UsableRulesProof
                                            ↳ QDP
              ↳ PiDP
              ↳ PiDP

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

PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga(0))
U7_GAAA(0, le_out_ga(0)) → PART_IN_GAAA(0)

The TRS R consists of the following rules:

le_in_ga(s(X)) → U11_ga(X, le_in_ga(X))
le_in_ga(0) → le_out_ga(0)
gt_in_ga(s(X)) → U10_ga(X, gt_in_ga(X))
gt_in_ga(s(0)) → gt_out_ga(s(0), 0)
U11_ga(X, le_out_ga(X)) → le_out_ga(s(X))
U10_ga(X, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))

The set Q consists of the following terms:

le_in_ga(x0)
gt_in_ga(x0)
U11_ga(x0, x1)
U10_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Instantiation
                                  ↳ QDP
                                    ↳ Instantiation
                                      ↳ QDP
                                        ↳ DependencyGraphProof
                                          ↳ AND
                                            ↳ QDP
                                              ↳ UsableRulesProof
                                                ↳ QDP
                                                  ↳ QReductionProof
                                            ↳ QDP
              ↳ PiDP
              ↳ PiDP

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

PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga(0))
U7_GAAA(0, le_out_ga(0)) → PART_IN_GAAA(0)

R is empty.
The set Q consists of the following terms:

le_in_ga(x0)
gt_in_ga(x0)
U11_ga(x0, x1)
U10_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

le_in_ga(x0)
gt_in_ga(x0)
U11_ga(x0, x1)
U10_ga(x0, x1)



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Instantiation
                                  ↳ QDP
                                    ↳ Instantiation
                                      ↳ QDP
                                        ↳ DependencyGraphProof
                                          ↳ AND
                                            ↳ QDP
                                              ↳ UsableRulesProof
                                                ↳ QDP
                                                  ↳ QReductionProof
                                                    ↳ QDP
                                                      ↳ NonTerminationProof
                                            ↳ QDP
              ↳ PiDP
              ↳ PiDP

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

U7_GAAA(0, le_out_ga(0)) → PART_IN_GAAA(0)
PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga(0))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

The TRS P consists of the following rules:

U7_GAAA(0, le_out_ga(0)) → PART_IN_GAAA(0)
PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga(0))

The TRS R consists of the following rules:none


s = PART_IN_GAAA(0) evaluates to t =PART_IN_GAAA(0)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:



Rewriting sequence

PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga(0))
with rule PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga(0)) at position [] and matcher [ ]

U7_GAAA(0, le_out_ga(0)) → PART_IN_GAAA(0)
with rule U7_GAAA(0, le_out_ga(0)) → PART_IN_GAAA(0)

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.





↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Instantiation
                                  ↳ QDP
                                    ↳ Instantiation
                                      ↳ QDP
                                        ↳ DependencyGraphProof
                                          ↳ AND
                                            ↳ QDP
                                            ↳ QDP
                                              ↳ NonTerminationProof
              ↳ PiDP
              ↳ PiDP

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

U7_GAAA(s(z0), le_out_ga(s(z0))) → PART_IN_GAAA(s(z0))
PART_IN_GAAA(s(x0)) → U5_GAAA(s(x0), U10_ga(x0, gt_in_ga(x0)))
U5_GAAA(s(z0), gt_out_ga(s(z0), x1)) → PART_IN_GAAA(s(z0))
U5_GAAA(s(0), gt_out_ga(s(0), 0)) → PART_IN_GAAA(s(0))
PART_IN_GAAA(s(x0)) → U7_GAAA(s(x0), U11_ga(x0, le_in_ga(x0)))
PART_IN_GAAA(s(0)) → U5_GAAA(s(0), gt_out_ga(s(0), 0))

The TRS R consists of the following rules:

le_in_ga(s(X)) → U11_ga(X, le_in_ga(X))
le_in_ga(0) → le_out_ga(0)
gt_in_ga(s(X)) → U10_ga(X, gt_in_ga(X))
gt_in_ga(s(0)) → gt_out_ga(s(0), 0)
U11_ga(X, le_out_ga(X)) → le_out_ga(s(X))
U10_ga(X, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))

The set Q consists of the following terms:

le_in_ga(x0)
gt_in_ga(x0)
U11_ga(x0, x1)
U10_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

The TRS P consists of the following rules:

U7_GAAA(s(z0), le_out_ga(s(z0))) → PART_IN_GAAA(s(z0))
PART_IN_GAAA(s(x0)) → U5_GAAA(s(x0), U10_ga(x0, gt_in_ga(x0)))
U5_GAAA(s(z0), gt_out_ga(s(z0), x1)) → PART_IN_GAAA(s(z0))
U5_GAAA(s(0), gt_out_ga(s(0), 0)) → PART_IN_GAAA(s(0))
PART_IN_GAAA(s(x0)) → U7_GAAA(s(x0), U11_ga(x0, le_in_ga(x0)))
PART_IN_GAAA(s(0)) → U5_GAAA(s(0), gt_out_ga(s(0), 0))

The TRS R consists of the following rules:

le_in_ga(s(X)) → U11_ga(X, le_in_ga(X))
le_in_ga(0) → le_out_ga(0)
gt_in_ga(s(X)) → U10_ga(X, gt_in_ga(X))
gt_in_ga(s(0)) → gt_out_ga(s(0), 0)
U11_ga(X, le_out_ga(X)) → le_out_ga(s(X))
U10_ga(X, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))


s = PART_IN_GAAA(s(0)) evaluates to t =PART_IN_GAAA(s(0))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:



Rewriting sequence

PART_IN_GAAA(s(0)) → U5_GAAA(s(0), gt_out_ga(s(0), 0))
with rule PART_IN_GAAA(s(0)) → U5_GAAA(s(0), gt_out_ga(s(0), 0)) at position [] and matcher [ ]

U5_GAAA(s(0), gt_out_ga(s(0), 0)) → PART_IN_GAAA(s(0))
with rule U5_GAAA(s(z0), gt_out_ga(s(z0), x1)) → PART_IN_GAAA(s(z0))

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.





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

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

GT_IN_AA(s(X), s(Y)) → GT_IN_AA(X, Y)

The TRS R consists of the following rules:

qs_in_ag(.(X, Xs), Ys) → U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
gt_in_aa(s(X), s(Y)) → U10_aa(X, Y, gt_in_aa(X, Y))
gt_in_aa(s(0), 0) → gt_out_aa(s(0), 0)
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
gt_in_ga(s(X), s(Y)) → U10_ga(X, Y, gt_in_ga(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U10_ga(X, Y, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_ga(X, Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
U11_ga(X, Y, le_out_ga(X, Y)) → le_out_ga(s(X), s(Y))
U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, [], [], []) → part_out_gaaa(X, [], [], [])
U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
le_in_aa(s(X), s(Y)) → U11_aa(X, Y, le_in_aa(X, Y))
le_in_aa(0, s(X)) → le_out_aa(0, s(X))
le_in_aa(0, 0) → le_out_aa(0, 0)
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_aaaa(X, [], [], []) → part_out_aaaa(X, [], [], [])
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa([], []) → qs_out_aa([], [])
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
app_in_gaa([], Ys, Ys) → app_out_gaa([], Ys, Ys)
U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
app_in_gag(.(X, Xs), Ys, .(X, Zs)) → U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
app_in_gag([], Ys, Ys) → app_out_gag([], Ys, Ys)
U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) → app_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt_in_aa(x1, x2)  =  gt_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)
part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
gt_in_ga(x1, x2)  =  gt_in_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
0  =  0
gt_out_ga(x1, x2)  =  gt_out_ga(x1, x2)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x2, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
le_in_ga(x1, x2)  =  le_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
le_out_ga(x1, x2)  =  le_out_ga(x1)
U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x1, x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1, x3)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
le_in_aa(x1, x2)  =  le_in_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
le_out_aa(x1, x2)  =  le_out_aa(x1)
U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)
gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x2, x3, x6)
[]  =  []
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)
qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)
qs_out_aa(x1, x2)  =  qs_out_aa
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x5)
app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x1, x2, x4, x5)
app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)
qs_out_ag(x1, x2)  =  qs_out_ag(x2)
GT_IN_AA(x1, x2)  =  GT_IN_AA

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

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

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

GT_IN_AA(s(X), s(Y)) → GT_IN_AA(X, Y)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
GT_IN_AA(x1, x2)  =  GT_IN_AA

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ NonTerminationProof
              ↳ PiDP

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

GT_IN_AA → GT_IN_AA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

GT_IN_AA → GT_IN_AA

The TRS R consists of the following rules:none


s = GT_IN_AA evaluates to t =GT_IN_AA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from GT_IN_AA to GT_IN_AA.





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

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

QS_IN_AA(.(X, Xs), Ys) → U1_AA(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U2_AA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → QS_IN_AA(Bigs, Bs)
U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_AA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))

The TRS R consists of the following rules:

qs_in_ag(.(X, Xs), Ys) → U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
gt_in_aa(s(X), s(Y)) → U10_aa(X, Y, gt_in_aa(X, Y))
gt_in_aa(s(0), 0) → gt_out_aa(s(0), 0)
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
gt_in_ga(s(X), s(Y)) → U10_ga(X, Y, gt_in_ga(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U10_ga(X, Y, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_ga(X, Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
U11_ga(X, Y, le_out_ga(X, Y)) → le_out_ga(s(X), s(Y))
U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X, [], [], []) → part_out_gaaa(X, [], [], [])
U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
le_in_aa(s(X), s(Y)) → U11_aa(X, Y, le_in_aa(X, Y))
le_in_aa(0, s(X)) → le_out_aa(0, s(X))
le_in_aa(0, 0) → le_out_aa(0, 0)
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_aaaa(X, [], [], []) → part_out_aaaa(X, [], [], [])
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
qs_in_aa([], []) → qs_out_aa([], [])
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
app_in_gaa(.(X, Xs), Ys, .(X, Zs)) → U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
app_in_gaa([], Ys, Ys) → app_out_gaa([], Ys, Ys)
U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) → app_out_gaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
app_in_gag(.(X, Xs), Ys, .(X, Zs)) → U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
app_in_gag([], Ys, Ys) → app_out_gag([], Ys, Ys)
U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) → app_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt_in_aa(x1, x2)  =  gt_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)
part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
gt_in_ga(x1, x2)  =  gt_in_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
0  =  0
gt_out_ga(x1, x2)  =  gt_out_ga(x1, x2)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x2, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
le_in_ga(x1, x2)  =  le_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
le_out_ga(x1, x2)  =  le_out_ga(x1)
U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x1, x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1, x3)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
le_in_aa(x1, x2)  =  le_in_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
le_out_aa(x1, x2)  =  le_out_aa(x1)
U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)
gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x2, x3, x6)
[]  =  []
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)
qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)
qs_out_aa(x1, x2)  =  qs_out_aa
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x5)
app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x1, x2, x4, x5)
app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)
qs_out_ag(x1, x2)  =  qs_out_ag(x2)
QS_IN_AA(x1, x2)  =  QS_IN_AA
U2_AA(x1, x2, x3, x4, x5)  =  U2_AA(x5)
U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

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

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

QS_IN_AA(.(X, Xs), Ys) → U1_AA(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U2_AA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → QS_IN_AA(Bigs, Bs)
U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_AA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))

The TRS R consists of the following rules:

part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
part_in_aaaa(X, [], [], []) → part_out_aaaa(X, [], [], [])
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
qs_in_ga([], []) → qs_out_ga([], [])
U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
gt_in_aa(s(X), s(Y)) → U10_aa(X, Y, gt_in_aa(X, Y))
gt_in_aa(s(0), 0) → gt_out_aa(s(0), 0)
U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
le_in_aa(s(X), s(Y)) → U11_aa(X, Y, le_in_aa(X, Y))
le_in_aa(0, s(X)) → le_out_aa(0, s(X))
le_in_aa(0, 0) → le_out_aa(0, 0)
U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
part_in_gaaa(X, [], [], []) → part_out_gaaa(X, [], [], [])
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
gt_in_ga(s(X), s(Y)) → U10_ga(X, Y, gt_in_ga(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_ga(X, Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U10_ga(X, Y, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U11_ga(X, Y, le_out_ga(X, Y)) → le_out_ga(s(X), s(Y))
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt_in_aa(x1, x2)  =  gt_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)
part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)
U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)
gt_in_ga(x1, x2)  =  gt_in_ga(x1)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
0  =  0
gt_out_ga(x1, x2)  =  gt_out_ga(x1, x2)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x2, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
le_in_ga(x1, x2)  =  le_in_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
le_out_ga(x1, x2)  =  le_out_ga(x1)
U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x1, x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1, x3)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
le_in_aa(x1, x2)  =  le_in_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
le_out_aa(x1, x2)  =  le_out_aa(x1)
U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)
U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)
gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x2, x3, x6)
[]  =  []
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)
qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
QS_IN_AA(x1, x2)  =  QS_IN_AA
U2_AA(x1, x2, x3, x4, x5)  =  U2_AA(x5)
U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing

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

U1_AA(part_out_aaaa(Littles)) → U2_AA(qs_in_ga(Littles))
QS_IN_AA → U1_AA(part_in_aaaa)
U2_AA(qs_out_ga(Littles, Ls)) → QS_IN_AA

The TRS R consists of the following rules:

part_in_aaaa → U5_aaaa(gt_in_aa)
part_in_aaaa → U7_aaaa(le_in_aa)
part_in_aaaa → part_out_aaaa([])
qs_in_ga(.(X, Xs)) → U1_ga(X, Xs, part_in_ggaa(X, Xs))
qs_in_ga([]) → qs_out_ga([], [])
U5_aaaa(gt_out_aa(X, Y)) → U6_aaaa(Y, part_in_gaaa(X))
U7_aaaa(le_out_aa(X)) → U8_aaaa(part_in_gaaa(X))
U1_ga(X, Xs, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Bigs, qs_in_ga(Littles))
gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s(0), 0)
U6_aaaa(Y, part_out_gaaa(X, Ls)) → part_out_aaaa(.(Y, Ls))
le_in_aa → U11_aa(le_in_aa)
le_in_aa → le_out_aa(0)
U8_aaaa(part_out_gaaa(X, Ls)) → part_out_aaaa(Ls)
part_in_ggaa(X, .(Y, Xs)) → U5_ggaa(X, Y, Xs, gt_in_gg(X, Y))
part_in_ggaa(X, .(Y, Xs)) → U7_ggaa(X, Y, Xs, le_in_gg(X, Y))
part_in_ggaa(X, []) → part_out_ggaa(X, [], [], [])
U2_ga(X, Xs, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ls, qs_in_ga(Bigs))
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
part_in_gaaa(X) → U5_gaaa(X, gt_in_ga(X))
part_in_gaaa(X) → U7_gaaa(X, le_in_ga(X))
part_in_gaaa(X) → part_out_gaaa(X, [])
U11_aa(le_out_aa(X)) → le_out_aa(s(X))
U5_ggaa(X, Y, Xs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, part_in_ggaa(X, Xs))
U7_ggaa(X, Y, Xs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, part_in_ggaa(X, Xs))
U3_ga(X, Xs, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, app_in_gga(Ls, .(X, Bs)))
U5_gaaa(X, gt_out_ga(X, Y)) → U6_gaaa(X, Y, part_in_gaaa(X))
U7_gaaa(X, le_out_ga(X)) → U8_gaaa(X, part_in_gaaa(X))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U6_ggaa(X, Y, Xs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_ggaa(X, Y, Xs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U4_ga(X, Xs, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
gt_in_ga(s(X)) → U10_ga(X, gt_in_ga(X))
gt_in_ga(s(0)) → gt_out_ga(s(0), 0)
U6_gaaa(X, Y, part_out_gaaa(X, Ls)) → part_out_gaaa(X, .(Y, Ls))
le_in_ga(s(X)) → U11_ga(X, le_in_ga(X))
le_in_ga(0) → le_out_ga(0)
U8_gaaa(X, part_out_gaaa(X, Ls)) → part_out_gaaa(X, Ls)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
app_in_gga(.(X, Xs), Ys) → U9_gga(X, Xs, Ys, app_in_gga(Xs, Ys))
app_in_gga([], Ys) → app_out_gga([], Ys, Ys)
U10_ga(X, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U11_ga(X, le_out_ga(X)) → le_out_ga(s(X))
U9_gga(X, Xs, Ys, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))

The set Q consists of the following terms:

part_in_aaaa
qs_in_ga(x0)
U5_aaaa(x0)
U7_aaaa(x0)
U1_ga(x0, x1, x2)
gt_in_aa
U6_aaaa(x0, x1)
le_in_aa
U8_aaaa(x0)
part_in_ggaa(x0, x1)
U2_ga(x0, x1, x2, x3)
U10_aa(x0)
part_in_gaaa(x0)
U11_aa(x0)
U5_ggaa(x0, x1, x2, x3)
U7_ggaa(x0, x1, x2, x3)
U3_ga(x0, x1, x2, x3)
U5_gaaa(x0, x1)
U7_gaaa(x0, x1)
gt_in_gg(x0, x1)
U6_ggaa(x0, x1, x2, x3)
le_in_gg(x0, x1)
U8_ggaa(x0, x1, x2, x3)
U4_ga(x0, x1, x2)
gt_in_ga(x0)
U6_gaaa(x0, x1, x2)
le_in_ga(x0)
U8_gaaa(x0, x1)
U10_gg(x0, x1, x2)
U11_gg(x0, x1, x2)
app_in_gga(x0, x1)
U10_ga(x0, x1)
U11_ga(x0, x1)
U9_gga(x0, x1, x2, x3)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule QS_IN_AA → U1_AA(part_in_aaaa) at position [0] we obtained the following new rules:

QS_IN_AA → U1_AA(U5_aaaa(gt_in_aa))
QS_IN_AA → U1_AA(part_out_aaaa([]))
QS_IN_AA → U1_AA(U7_aaaa(le_in_aa))



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ UsableRulesProof

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

U1_AA(part_out_aaaa(Littles)) → U2_AA(qs_in_ga(Littles))
QS_IN_AA → U1_AA(part_out_aaaa([]))
QS_IN_AA → U1_AA(U5_aaaa(gt_in_aa))
QS_IN_AA → U1_AA(U7_aaaa(le_in_aa))
U2_AA(qs_out_ga(Littles, Ls)) → QS_IN_AA

The TRS R consists of the following rules:

part_in_aaaa → U5_aaaa(gt_in_aa)
part_in_aaaa → U7_aaaa(le_in_aa)
part_in_aaaa → part_out_aaaa([])
qs_in_ga(.(X, Xs)) → U1_ga(X, Xs, part_in_ggaa(X, Xs))
qs_in_ga([]) → qs_out_ga([], [])
U5_aaaa(gt_out_aa(X, Y)) → U6_aaaa(Y, part_in_gaaa(X))
U7_aaaa(le_out_aa(X)) → U8_aaaa(part_in_gaaa(X))
U1_ga(X, Xs, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Bigs, qs_in_ga(Littles))
gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s(0), 0)
U6_aaaa(Y, part_out_gaaa(X, Ls)) → part_out_aaaa(.(Y, Ls))
le_in_aa → U11_aa(le_in_aa)
le_in_aa → le_out_aa(0)
U8_aaaa(part_out_gaaa(X, Ls)) → part_out_aaaa(Ls)
part_in_ggaa(X, .(Y, Xs)) → U5_ggaa(X, Y, Xs, gt_in_gg(X, Y))
part_in_ggaa(X, .(Y, Xs)) → U7_ggaa(X, Y, Xs, le_in_gg(X, Y))
part_in_ggaa(X, []) → part_out_ggaa(X, [], [], [])
U2_ga(X, Xs, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ls, qs_in_ga(Bigs))
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
part_in_gaaa(X) → U5_gaaa(X, gt_in_ga(X))
part_in_gaaa(X) → U7_gaaa(X, le_in_ga(X))
part_in_gaaa(X) → part_out_gaaa(X, [])
U11_aa(le_out_aa(X)) → le_out_aa(s(X))
U5_ggaa(X, Y, Xs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, part_in_ggaa(X, Xs))
U7_ggaa(X, Y, Xs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, part_in_ggaa(X, Xs))
U3_ga(X, Xs, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, app_in_gga(Ls, .(X, Bs)))
U5_gaaa(X, gt_out_ga(X, Y)) → U6_gaaa(X, Y, part_in_gaaa(X))
U7_gaaa(X, le_out_ga(X)) → U8_gaaa(X, part_in_gaaa(X))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U6_ggaa(X, Y, Xs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_ggaa(X, Y, Xs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U4_ga(X, Xs, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
gt_in_ga(s(X)) → U10_ga(X, gt_in_ga(X))
gt_in_ga(s(0)) → gt_out_ga(s(0), 0)
U6_gaaa(X, Y, part_out_gaaa(X, Ls)) → part_out_gaaa(X, .(Y, Ls))
le_in_ga(s(X)) → U11_ga(X, le_in_ga(X))
le_in_ga(0) → le_out_ga(0)
U8_gaaa(X, part_out_gaaa(X, Ls)) → part_out_gaaa(X, Ls)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
app_in_gga(.(X, Xs), Ys) → U9_gga(X, Xs, Ys, app_in_gga(Xs, Ys))
app_in_gga([], Ys) → app_out_gga([], Ys, Ys)
U10_ga(X, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U11_ga(X, le_out_ga(X)) → le_out_ga(s(X))
U9_gga(X, Xs, Ys, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))

The set Q consists of the following terms:

part_in_aaaa
qs_in_ga(x0)
U5_aaaa(x0)
U7_aaaa(x0)
U1_ga(x0, x1, x2)
gt_in_aa
U6_aaaa(x0, x1)
le_in_aa
U8_aaaa(x0)
part_in_ggaa(x0, x1)
U2_ga(x0, x1, x2, x3)
U10_aa(x0)
part_in_gaaa(x0)
U11_aa(x0)
U5_ggaa(x0, x1, x2, x3)
U7_ggaa(x0, x1, x2, x3)
U3_ga(x0, x1, x2, x3)
U5_gaaa(x0, x1)
U7_gaaa(x0, x1)
gt_in_gg(x0, x1)
U6_ggaa(x0, x1, x2, x3)
le_in_gg(x0, x1)
U8_ggaa(x0, x1, x2, x3)
U4_ga(x0, x1, x2)
gt_in_ga(x0)
U6_gaaa(x0, x1, x2)
le_in_ga(x0)
U8_gaaa(x0, x1)
U10_gg(x0, x1, x2)
U11_gg(x0, x1, x2)
app_in_gga(x0, x1)
U10_ga(x0, x1)
U11_ga(x0, x1)
U9_gga(x0, x1, x2, x3)

We have to consider all (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ UsableRulesProof
                              ↳ QDP
                                ↳ QReductionProof

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

U1_AA(part_out_aaaa(Littles)) → U2_AA(qs_in_ga(Littles))
QS_IN_AA → U1_AA(part_out_aaaa([]))
QS_IN_AA → U1_AA(U5_aaaa(gt_in_aa))
QS_IN_AA → U1_AA(U7_aaaa(le_in_aa))
U2_AA(qs_out_ga(Littles, Ls)) → QS_IN_AA

The TRS R consists of the following rules:

le_in_aa → U11_aa(le_in_aa)
le_in_aa → le_out_aa(0)
U7_aaaa(le_out_aa(X)) → U8_aaaa(part_in_gaaa(X))
part_in_gaaa(X) → U5_gaaa(X, gt_in_ga(X))
part_in_gaaa(X) → U7_gaaa(X, le_in_ga(X))
part_in_gaaa(X) → part_out_gaaa(X, [])
U8_aaaa(part_out_gaaa(X, Ls)) → part_out_aaaa(Ls)
le_in_ga(s(X)) → U11_ga(X, le_in_ga(X))
le_in_ga(0) → le_out_ga(0)
U7_gaaa(X, le_out_ga(X)) → U8_gaaa(X, part_in_gaaa(X))
U8_gaaa(X, part_out_gaaa(X, Ls)) → part_out_gaaa(X, Ls)
U11_ga(X, le_out_ga(X)) → le_out_ga(s(X))
gt_in_ga(s(X)) → U10_ga(X, gt_in_ga(X))
gt_in_ga(s(0)) → gt_out_ga(s(0), 0)
U5_gaaa(X, gt_out_ga(X, Y)) → U6_gaaa(X, Y, part_in_gaaa(X))
U6_gaaa(X, Y, part_out_gaaa(X, Ls)) → part_out_gaaa(X, .(Y, Ls))
U10_ga(X, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U11_aa(le_out_aa(X)) → le_out_aa(s(X))
gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s(0), 0)
U5_aaaa(gt_out_aa(X, Y)) → U6_aaaa(Y, part_in_gaaa(X))
U6_aaaa(Y, part_out_gaaa(X, Ls)) → part_out_aaaa(.(Y, Ls))
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
qs_in_ga(.(X, Xs)) → U1_ga(X, Xs, part_in_ggaa(X, Xs))
qs_in_ga([]) → qs_out_ga([], [])
part_in_ggaa(X, .(Y, Xs)) → U5_ggaa(X, Y, Xs, gt_in_gg(X, Y))
part_in_ggaa(X, .(Y, Xs)) → U7_ggaa(X, Y, Xs, le_in_gg(X, Y))
part_in_ggaa(X, []) → part_out_ggaa(X, [], [], [])
U1_ga(X, Xs, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Bigs, qs_in_ga(Littles))
U2_ga(X, Xs, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ls, qs_in_ga(Bigs))
U3_ga(X, Xs, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, app_in_gga(Ls, .(X, Bs)))
app_in_gga(.(X, Xs), Ys) → U9_gga(X, Xs, Ys, app_in_gga(Xs, Ys))
app_in_gga([], Ys) → app_out_gga([], Ys, Ys)
U4_ga(X, Xs, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
U9_gga(X, Xs, Ys, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U7_ggaa(X, Y, Xs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, part_in_ggaa(X, Xs))
U8_ggaa(X, Y, Xs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U5_ggaa(X, Y, Xs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, part_in_ggaa(X, Xs))
U6_ggaa(X, Y, Xs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))

The set Q consists of the following terms:

part_in_aaaa
qs_in_ga(x0)
U5_aaaa(x0)
U7_aaaa(x0)
U1_ga(x0, x1, x2)
gt_in_aa
U6_aaaa(x0, x1)
le_in_aa
U8_aaaa(x0)
part_in_ggaa(x0, x1)
U2_ga(x0, x1, x2, x3)
U10_aa(x0)
part_in_gaaa(x0)
U11_aa(x0)
U5_ggaa(x0, x1, x2, x3)
U7_ggaa(x0, x1, x2, x3)
U3_ga(x0, x1, x2, x3)
U5_gaaa(x0, x1)
U7_gaaa(x0, x1)
gt_in_gg(x0, x1)
U6_ggaa(x0, x1, x2, x3)
le_in_gg(x0, x1)
U8_ggaa(x0, x1, x2, x3)
U4_ga(x0, x1, x2)
gt_in_ga(x0)
U6_gaaa(x0, x1, x2)
le_in_ga(x0)
U8_gaaa(x0, x1)
U10_gg(x0, x1, x2)
U11_gg(x0, x1, x2)
app_in_gga(x0, x1)
U10_ga(x0, x1)
U11_ga(x0, x1)
U9_gga(x0, x1, x2, x3)

We have to consider all (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

part_in_aaaa



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ UsableRulesProof
                              ↳ QDP
                                ↳ QReductionProof
                                  ↳ QDP
                                    ↳ Narrowing

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

U1_AA(part_out_aaaa(Littles)) → U2_AA(qs_in_ga(Littles))
QS_IN_AA → U1_AA(part_out_aaaa([]))
QS_IN_AA → U1_AA(U5_aaaa(gt_in_aa))
QS_IN_AA → U1_AA(U7_aaaa(le_in_aa))
U2_AA(qs_out_ga(Littles, Ls)) → QS_IN_AA

The TRS R consists of the following rules:

le_in_aa → U11_aa(le_in_aa)
le_in_aa → le_out_aa(0)
U7_aaaa(le_out_aa(X)) → U8_aaaa(part_in_gaaa(X))
part_in_gaaa(X) → U5_gaaa(X, gt_in_ga(X))
part_in_gaaa(X) → U7_gaaa(X, le_in_ga(X))
part_in_gaaa(X) → part_out_gaaa(X, [])
U8_aaaa(part_out_gaaa(X, Ls)) → part_out_aaaa(Ls)
le_in_ga(s(X)) → U11_ga(X, le_in_ga(X))
le_in_ga(0) → le_out_ga(0)
U7_gaaa(X, le_out_ga(X)) → U8_gaaa(X, part_in_gaaa(X))
U8_gaaa(X, part_out_gaaa(X, Ls)) → part_out_gaaa(X, Ls)
U11_ga(X, le_out_ga(X)) → le_out_ga(s(X))
gt_in_ga(s(X)) → U10_ga(X, gt_in_ga(X))
gt_in_ga(s(0)) → gt_out_ga(s(0), 0)
U5_gaaa(X, gt_out_ga(X, Y)) → U6_gaaa(X, Y, part_in_gaaa(X))
U6_gaaa(X, Y, part_out_gaaa(X, Ls)) → part_out_gaaa(X, .(Y, Ls))
U10_ga(X, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U11_aa(le_out_aa(X)) → le_out_aa(s(X))
gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s(0), 0)
U5_aaaa(gt_out_aa(X, Y)) → U6_aaaa(Y, part_in_gaaa(X))
U6_aaaa(Y, part_out_gaaa(X, Ls)) → part_out_aaaa(.(Y, Ls))
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
qs_in_ga(.(X, Xs)) → U1_ga(X, Xs, part_in_ggaa(X, Xs))
qs_in_ga([]) → qs_out_ga([], [])
part_in_ggaa(X, .(Y, Xs)) → U5_ggaa(X, Y, Xs, gt_in_gg(X, Y))
part_in_ggaa(X, .(Y, Xs)) → U7_ggaa(X, Y, Xs, le_in_gg(X, Y))
part_in_ggaa(X, []) → part_out_ggaa(X, [], [], [])
U1_ga(X, Xs, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Bigs, qs_in_ga(Littles))
U2_ga(X, Xs, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ls, qs_in_ga(Bigs))
U3_ga(X, Xs, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, app_in_gga(Ls, .(X, Bs)))
app_in_gga(.(X, Xs), Ys) → U9_gga(X, Xs, Ys, app_in_gga(Xs, Ys))
app_in_gga([], Ys) → app_out_gga([], Ys, Ys)
U4_ga(X, Xs, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
U9_gga(X, Xs, Ys, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U7_ggaa(X, Y, Xs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, part_in_ggaa(X, Xs))
U8_ggaa(X, Y, Xs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U5_ggaa(X, Y, Xs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, part_in_ggaa(X, Xs))
U6_ggaa(X, Y, Xs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))

The set Q consists of the following terms:

qs_in_ga(x0)
U5_aaaa(x0)
U7_aaaa(x0)
U1_ga(x0, x1, x2)
gt_in_aa
U6_aaaa(x0, x1)
le_in_aa
U8_aaaa(x0)
part_in_ggaa(x0, x1)
U2_ga(x0, x1, x2, x3)
U10_aa(x0)
part_in_gaaa(x0)
U11_aa(x0)
U5_ggaa(x0, x1, x2, x3)
U7_ggaa(x0, x1, x2, x3)
U3_ga(x0, x1, x2, x3)
U5_gaaa(x0, x1)
U7_gaaa(x0, x1)
gt_in_gg(x0, x1)
U6_ggaa(x0, x1, x2, x3)
le_in_gg(x0, x1)
U8_ggaa(x0, x1, x2, x3)
U4_ga(x0, x1, x2)
gt_in_ga(x0)
U6_gaaa(x0, x1, x2)
le_in_ga(x0)
U8_gaaa(x0, x1)
U10_gg(x0, x1, x2)
U11_gg(x0, x1, x2)
app_in_gga(x0, x1)
U10_ga(x0, x1)
U11_ga(x0, x1)
U9_gga(x0, x1, x2, x3)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule U1_AA(part_out_aaaa(Littles)) → U2_AA(qs_in_ga(Littles)) at position [0] we obtained the following new rules:

U1_AA(part_out_aaaa(.(x0, x1))) → U2_AA(U1_ga(x0, x1, part_in_ggaa(x0, x1)))
U1_AA(part_out_aaaa([])) → U2_AA(qs_out_ga([], []))



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ UsableRulesProof
                              ↳ QDP
                                ↳ QReductionProof
                                  ↳ QDP
                                    ↳ Narrowing
                                      ↳ QDP

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

QS_IN_AA → U1_AA(part_out_aaaa([]))
U1_AA(part_out_aaaa(.(x0, x1))) → U2_AA(U1_ga(x0, x1, part_in_ggaa(x0, x1)))
U1_AA(part_out_aaaa([])) → U2_AA(qs_out_ga([], []))
QS_IN_AA → U1_AA(U5_aaaa(gt_in_aa))
QS_IN_AA → U1_AA(U7_aaaa(le_in_aa))
U2_AA(qs_out_ga(Littles, Ls)) → QS_IN_AA

The TRS R consists of the following rules:

le_in_aa → U11_aa(le_in_aa)
le_in_aa → le_out_aa(0)
U7_aaaa(le_out_aa(X)) → U8_aaaa(part_in_gaaa(X))
part_in_gaaa(X) → U5_gaaa(X, gt_in_ga(X))
part_in_gaaa(X) → U7_gaaa(X, le_in_ga(X))
part_in_gaaa(X) → part_out_gaaa(X, [])
U8_aaaa(part_out_gaaa(X, Ls)) → part_out_aaaa(Ls)
le_in_ga(s(X)) → U11_ga(X, le_in_ga(X))
le_in_ga(0) → le_out_ga(0)
U7_gaaa(X, le_out_ga(X)) → U8_gaaa(X, part_in_gaaa(X))
U8_gaaa(X, part_out_gaaa(X, Ls)) → part_out_gaaa(X, Ls)
U11_ga(X, le_out_ga(X)) → le_out_ga(s(X))
gt_in_ga(s(X)) → U10_ga(X, gt_in_ga(X))
gt_in_ga(s(0)) → gt_out_ga(s(0), 0)
U5_gaaa(X, gt_out_ga(X, Y)) → U6_gaaa(X, Y, part_in_gaaa(X))
U6_gaaa(X, Y, part_out_gaaa(X, Ls)) → part_out_gaaa(X, .(Y, Ls))
U10_ga(X, gt_out_ga(X, Y)) → gt_out_ga(s(X), s(Y))
U11_aa(le_out_aa(X)) → le_out_aa(s(X))
gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s(0), 0)
U5_aaaa(gt_out_aa(X, Y)) → U6_aaaa(Y, part_in_gaaa(X))
U6_aaaa(Y, part_out_gaaa(X, Ls)) → part_out_aaaa(.(Y, Ls))
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
qs_in_ga(.(X, Xs)) → U1_ga(X, Xs, part_in_ggaa(X, Xs))
qs_in_ga([]) → qs_out_ga([], [])
part_in_ggaa(X, .(Y, Xs)) → U5_ggaa(X, Y, Xs, gt_in_gg(X, Y))
part_in_ggaa(X, .(Y, Xs)) → U7_ggaa(X, Y, Xs, le_in_gg(X, Y))
part_in_ggaa(X, []) → part_out_ggaa(X, [], [], [])
U1_ga(X, Xs, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Bigs, qs_in_ga(Littles))
U2_ga(X, Xs, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ls, qs_in_ga(Bigs))
U3_ga(X, Xs, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, app_in_gga(Ls, .(X, Bs)))
app_in_gga(.(X, Xs), Ys) → U9_gga(X, Xs, Ys, app_in_gga(Xs, Ys))
app_in_gga([], Ys) → app_out_gga([], Ys, Ys)
U4_ga(X, Xs, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
U9_gga(X, Xs, Ys, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
le_in_gg(s(X), s(Y)) → U11_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U7_ggaa(X, Y, Xs, le_out_gg(X, Y)) → U8_ggaa(X, Y, Xs, part_in_ggaa(X, Xs))
U8_ggaa(X, Y, Xs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U11_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
gt_in_gg(s(X), s(Y)) → U10_gg(X, Y, gt_in_gg(X, Y))
gt_in_gg(s(0), 0) → gt_out_gg(s(0), 0)
U5_ggaa(X, Y, Xs, gt_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, part_in_ggaa(X, Xs))
U6_ggaa(X, Y, Xs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U10_gg(X, Y, gt_out_gg(X, Y)) → gt_out_gg(s(X), s(Y))

The set Q consists of the following terms:

qs_in_ga(x0)
U5_aaaa(x0)
U7_aaaa(x0)
U1_ga(x0, x1, x2)
gt_in_aa
U6_aaaa(x0, x1)
le_in_aa
U8_aaaa(x0)
part_in_ggaa(x0, x1)
U2_ga(x0, x1, x2, x3)
U10_aa(x0)
part_in_gaaa(x0)
U11_aa(x0)
U5_ggaa(x0, x1, x2, x3)
U7_ggaa(x0, x1, x2, x3)
U3_ga(x0, x1, x2, x3)
U5_gaaa(x0, x1)
U7_gaaa(x0, x1)
gt_in_gg(x0, x1)
U6_ggaa(x0, x1, x2, x3)
le_in_gg(x0, x1)
U8_ggaa(x0, x1, x2, x3)
U4_ga(x0, x1, x2)
gt_in_ga(x0)
U6_gaaa(x0, x1, x2)
le_in_ga(x0)
U8_gaaa(x0, x1)
U10_gg(x0, x1, x2)
U11_gg(x0, x1, x2)
app_in_gga(x0, x1)
U10_ga(x0, x1)
U11_ga(x0, x1)
U9_gga(x0, x1, x2, x3)

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