Left Termination proof of /tmp/tmpIkpqhG/quicksort-oi.pl

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)
part_in: (f,f,f,f) (b,f,f,f) (f,b,f,f) (b,b,f,f)
gt_in: (f,f) (b,f)
le_in: (b,f) (f,f)
app_in: (b,b,f) (b,b,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_aa(X, Y))
U10_ga(X, Y, gt_out_aa(X, Y)) → gt_out_ga(s(X), s(Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
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_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))
U11_ga(X, Y, le_out_aa(X, Y)) → le_out_ga(s(X), s(Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
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))
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_agaa(X, Xs, Littles, Bigs))
part_in_agaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_agaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
U5_agaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_ga(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_ga(X, Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_ga(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)
U6_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_agaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_agaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
U7_agaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U8_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_agaa(X, [], [], []) → part_out_agaa(X, [], [], [])
U1_ga(X, Xs, Ys, part_out_agaa(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_ga(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_ggg(Ls, .(X, Bs), Ys))
app_in_ggg(.(X, Xs), Ys, .(X, Zs)) → U9_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
app_in_ggg([], Ys, Ys) → app_out_ggg([], Ys, Ys)
U9_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) → app_out_ggg(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_ggg(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → 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_aa(X, Y))
U10_ga(X, Y, gt_out_aa(X, Y)) → gt_out_ga(s(X), s(Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
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_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))
U11_ga(X, Y, le_out_aa(X, Y)) → le_out_ga(s(X), s(Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
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))
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_agaa(X, Xs, Littles, Bigs))
part_in_agaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_agaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
U5_agaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_ga(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_ga(X, Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_ga(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)
U6_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_agaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_agaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
U7_agaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U8_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_agaa(X, [], [], []) → part_out_agaa(X, [], [], [])
U1_ga(X, Xs, Ys, part_out_agaa(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_ga(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_ggg(Ls, .(X, Bs), Ys))
app_in_ggg(.(X, Xs), Ys, .(X, Zs)) → U9_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
app_in_ggg([], Ys, Ys) → app_out_ggg([], Ys, Ys)
U9_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) → app_out_ggg(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_ggg(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

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_aa(X, Y))
GT_IN_GA(s(X), s(Y)) → GT_IN_AA(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_aa(X, Y))
LE_IN_GA(s(X), s(Y)) → 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_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)
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_agaa(X, Xs, Littles, Bigs))
QS_IN_GA(.(X, Xs), Ys) → PART_IN_AGAA(X, Xs, Littles, Bigs)
PART_IN_AGAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_AGAA(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
PART_IN_AGAA(X, .(Y, Xs), .(Y, Ls), Bs) → GT_IN_AA(X, Y)
U5_AGAA(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_AGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U5_AGAA(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GGAA(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → GT_IN_GA(X, Y)
U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_ga(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_ga(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_ga(X, Y))
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → LE_IN_GA(X, Y)
U7_GGAA(X, Y, Xs, Ls, Bs, le_out_ga(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_ga(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
PART_IN_AGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_AGAA(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
PART_IN_AGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → LE_IN_AA(X, Y)
U7_AGAA(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_AGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U7_AGAA(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
U1_GA(X, Xs, Ys, part_out_agaa(X, Xs, Littles, Bigs)) → U2_GA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U1_GA(X, Xs, Ys, part_out_agaa(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_ga(Bigs, Bs))
U2_AG(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → QS_IN_GA(Bigs, Bs)
U3_AG(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_AG(X, Xs, Ys, app_in_ggg(Ls, .(X, Bs), Ys))
U3_AG(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → APP_IN_GGG(Ls, .(X, Bs), Ys)
APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) → U9_GGG(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGG(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_aa(X, Y))
U10_ga(X, Y, gt_out_aa(X, Y)) → gt_out_ga(s(X), s(Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
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_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))
U11_ga(X, Y, le_out_aa(X, Y)) → le_out_ga(s(X), s(Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
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))
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_agaa(X, Xs, Littles, Bigs))
part_in_agaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_agaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
U5_agaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_ga(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_ga(X, Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_ga(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)
U6_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_agaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_agaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
U7_agaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U8_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_agaa(X, [], [], []) → part_out_agaa(X, [], [], [])
U1_ga(X, Xs, Ys, part_out_agaa(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_ga(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_ggg(Ls, .(X, Bs), Ys))
app_in_ggg(.(X, Xs), Ys, .(X, Zs)) → U9_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
app_in_ggg([], Ys, Ys) → app_out_ggg([], Ys, Ys)
U9_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) → app_out_ggg(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_ggg(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(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
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(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(x2)
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, x2)
U8_gaaa(x1, x2, x3, x4, x5, x6) = U8_gaaa(x6)
part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa(x2, x3, x4)
part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa(x2, x3, x4)
U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6)
U8_aaaa(x1, x2, x3, x4, x5, x6) = U8_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5) = U2_ag(x2, x3, x4, x5)
qs_in_ga(x1, x2) = qs_in_ga(x1)
.(x1, x2) = .(x2)
U1_ga(x1, x2, x3, x4) = U1_ga(x4)
part_in_ggaa(x1, x2, x3, x4) = part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x3, x6)
U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x6)
U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x1, x3, x6)
U8_ggaa(x1, x2, x3, x4, x5, x6) = U8_ggaa(x6)
[] = []
part_out_ggaa(x1, x2, x3, x4) = part_out_ggaa(x3, x4)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x4, x5)
qs_out_ga(x1, x2) = qs_out_ga(x2)
U3_ga(x1, x2, x3, x4, x5) = U3_ga(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(x5)
app_out_gga(x1, x2, x3) = app_out_gga(x3)
U3_ag(x1, x2, x3, x4, x5) = U3_ag(x2, x3, x4, x5)
U4_ag(x1, x2, x3, x4) = U4_ag(x2, x4)
app_in_ggg(x1, x2, x3) = app_in_ggg(x1, x2, x3)
part_in_agaa(x1, x2, x3, x4) = part_in_agaa(x2)
U5_agaa(x1, x2, x3, x4, x5, x6) = U5_agaa(x3, x6)
U6_agaa(x1, x2, x3, x4, x5, x6) = U6_agaa(x6)
part_out_agaa(x1, x2, x3, x4) = part_out_agaa(x3, x4)
U7_agaa(x1, x2, x3, x4, x5, x6) = U7_agaa(x3, x6)
U8_agaa(x1, x2, x3, x4, x5, x6) = U8_agaa(x6)
U9_ggg(x1, x2, x3, x4, x5) = U9_ggg(x5)
app_out_ggg(x1, x2, x3) = app_out_ggg
qs_out_ag(x1, x2) = qs_out_ag(x1)
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)
U6_GGAA(x1, x2, x3, x4, x5, x6) = U6_GGAA(x6)
U6_AAAA(x1, x2, x3, x4, x5, x6) = U6_AAAA(x6)
U9_GGA(x1, x2, x3, x4, x5) = U9_GGA(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, x3, x6)
U1_GA(x1, x2, x3, x4) = U1_GA(x4)
U8_AGAA(x1, x2, x3, x4, x5, x6) = U8_AGAA(x6)
U5_AGAA(x1, x2, x3, x4, x5, x6) = U5_AGAA(x3, x6)
LE_IN_AA(x1, x2) = LE_IN_AA
PART_IN_AGAA(x1, x2, x3, x4) = PART_IN_AGAA(x2)
APP_IN_GGG(x1, x2, x3) = APP_IN_GGG(x1, x2, x3)
U6_AGAA(x1, x2, x3, x4, x5, x6) = U6_AGAA(x6)
U3_AG(x1, x2, x3, x4, x5) = U3_AG(x2, x3, x4, x5)
U7_AGAA(x1, x2, x3, x4, x5, x6) = U7_AGAA(x3, x6)
U2_GA(x1, x2, x3, x4, x5) = U2_GA(x4, x5)
U4_GA(x1, x2, x3, x4) = U4_GA(x4)
U3_GA(x1, x2, x3, x4, x5) = U3_GA(x4, x5)
PART_IN_GAAA(x1, x2, x3, x4) = PART_IN_GAAA(x1)
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)
PART_IN_GGAA(x1, x2, x3, x4) = PART_IN_GGAA(x1, x2)
QS_IN_GA(x1, x2) = QS_IN_GA(x1)
U5_GGAA(x1, x2, x3, x4, x5, x6) = U5_GGAA(x1, x3, x6)
PART_IN_AAAA(x1, x2, x3, x4) = PART_IN_AAAA
U8_GGAA(x1, x2, x3, x4, x5, x6) = U8_GGAA(x6)
U2_AG(x1, x2, x3, x4, x5) = U2_AG(x2, x3, x4, x5)
U1_AG(x1, x2, x3, x4) = U1_AG(x3, x4)
U6_GAAA(x1, x2, x3, x4, x5, x6) = U6_GAAA(x6)
GT_IN_AA(x1, x2) = GT_IN_AA
U9_GGG(x1, x2, x3, x4, x5) = U9_GGG(x5)
U11_GA(x1, x2, x3) = U11_GA(x3)
U4_AG(x1, x2, x3, x4) = U4_AG(x2, x4)
GT_IN_GA(x1, x2) = GT_IN_GA(x1)
U11_AA(x1, x2, x3) = U11_AA(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_aa(X, Y))
GT_IN_GA(s(X), s(Y)) → GT_IN_AA(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_aa(X, Y))
LE_IN_GA(s(X), s(Y)) → 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_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)
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_agaa(X, Xs, Littles, Bigs))
QS_IN_GA(.(X, Xs), Ys) → PART_IN_AGAA(X, Xs, Littles, Bigs)
PART_IN_AGAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_AGAA(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
PART_IN_AGAA(X, .(Y, Xs), .(Y, Ls), Bs) → GT_IN_AA(X, Y)
U5_AGAA(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_AGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U5_AGAA(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GGAA(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → GT_IN_GA(X, Y)
U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_ga(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_ga(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_ga(X, Y))
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → LE_IN_GA(X, Y)
U7_GGAA(X, Y, Xs, Ls, Bs, le_out_ga(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_ga(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
PART_IN_AGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_AGAA(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
PART_IN_AGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → LE_IN_AA(X, Y)
U7_AGAA(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_AGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U7_AGAA(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
U1_GA(X, Xs, Ys, part_out_agaa(X, Xs, Littles, Bigs)) → U2_GA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U1_GA(X, Xs, Ys, part_out_agaa(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_ga(Bigs, Bs))
U2_AG(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → QS_IN_GA(Bigs, Bs)
U3_AG(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_AG(X, Xs, Ys, app_in_ggg(Ls, .(X, Bs), Ys))
U3_AG(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → APP_IN_GGG(Ls, .(X, Bs), Ys)
APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) → U9_GGG(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGG(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_aa(X, Y))
U10_ga(X, Y, gt_out_aa(X, Y)) → gt_out_ga(s(X), s(Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
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_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))
U11_ga(X, Y, le_out_aa(X, Y)) → le_out_ga(s(X), s(Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
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))
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_agaa(X, Xs, Littles, Bigs))
part_in_agaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_agaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
U5_agaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_ga(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_ga(X, Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_ga(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)
U6_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_agaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_agaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
U7_agaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U8_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_agaa(X, [], [], []) → part_out_agaa(X, [], [], [])
U1_ga(X, Xs, Ys, part_out_agaa(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_ga(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_ggg(Ls, .(X, Bs), Ys))
app_in_ggg(.(X, Xs), Ys, .(X, Zs)) → U9_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
app_in_ggg([], Ys, Ys) → app_out_ggg([], Ys, Ys)
U9_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) → app_out_ggg(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_ggg(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGG(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_aa(X, Y))
U10_ga(X, Y, gt_out_aa(X, Y)) → gt_out_ga(s(X), s(Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
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_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))
U11_ga(X, Y, le_out_aa(X, Y)) → le_out_ga(s(X), s(Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
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))
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_agaa(X, Xs, Littles, Bigs))
part_in_agaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_agaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
U5_agaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_ga(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_ga(X, Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_ga(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)
U6_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_agaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_agaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
U7_agaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U8_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_agaa(X, [], [], []) → part_out_agaa(X, [], [], [])
U1_ga(X, Xs, Ys, part_out_agaa(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_ga(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_ggg(Ls, .(X, Bs), Ys))
app_in_ggg(.(X, Xs), Ys, .(X, Zs)) → U9_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
app_in_ggg([], Ys, Ys) → app_out_ggg([], Ys, Ys)
U9_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) → app_out_ggg(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_ggg(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

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

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

  ↳ PrologToPiTRSProof

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

APP_IN_GGG(.(Xs), Ys, .(Zs)) → APP_IN_GGG(Xs, Ys, 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:

APP_IN_GGG(.(Xs), Ys, .(Zs)) → APP_IN_GGG(Xs, Ys, Zs)
The graph contains the following edges 1 > 1, 2 >= 2, 3 > 3

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ 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_aa(X, Y))
U10_ga(X, Y, gt_out_aa(X, Y)) → gt_out_ga(s(X), s(Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
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_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))
U11_ga(X, Y, le_out_aa(X, Y)) → le_out_ga(s(X), s(Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
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))
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_agaa(X, Xs, Littles, Bigs))
part_in_agaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_agaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
U5_agaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_ga(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_ga(X, Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_ga(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)
U6_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_agaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_agaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
U7_agaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U8_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_agaa(X, [], [], []) → part_out_agaa(X, [], [], [])
U1_ga(X, Xs, Ys, part_out_agaa(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_ga(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_ggg(Ls, .(X, Bs), Ys))
app_in_ggg(.(X, Xs), Ys, .(X, Zs)) → U9_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
app_in_ggg([], Ys, Ys) → app_out_ggg([], Ys, Ys)
U9_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) → app_out_ggg(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_ggg(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ 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) = .(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

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

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

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

APP_IN_GGA(.(Xs), Ys) → APP_IN_GGA(Xs, Ys)
The graph contains the following edges 1 > 1, 2 >= 2

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ 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_aa(X, Y))
U10_ga(X, Y, gt_out_aa(X, Y)) → gt_out_ga(s(X), s(Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
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_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))
U11_ga(X, Y, le_out_aa(X, Y)) → le_out_ga(s(X), s(Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
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))
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_agaa(X, Xs, Littles, Bigs))
part_in_agaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_agaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
U5_agaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_ga(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_ga(X, Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_ga(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)
U6_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_agaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_agaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
U7_agaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U8_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_agaa(X, [], [], []) → part_out_agaa(X, [], [], [])
U1_ga(X, Xs, Ys, part_out_agaa(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_ga(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_ggg(Ls, .(X, Bs), Ys))
app_in_ggg(.(X, Xs), Ys, .(X, Zs)) → U9_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
app_in_ggg([], Ys, Ys) → app_out_ggg([], Ys, Ys)
U9_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) → app_out_ggg(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_ggg(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ 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
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

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ NonTerminationProof

              ↳ 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:

Semiunifier: [ ]
Matcher: [ ]

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

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ 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_aa(X, Y))
U10_ga(X, Y, gt_out_aa(X, Y)) → gt_out_ga(s(X), s(Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
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_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))
U11_ga(X, Y, le_out_aa(X, Y)) → le_out_ga(s(X), s(Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
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))
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_agaa(X, Xs, Littles, Bigs))
part_in_agaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_agaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
U5_agaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_ga(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_ga(X, Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_ga(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)
U6_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_agaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_agaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
U7_agaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U8_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_agaa(X, [], [], []) → part_out_agaa(X, [], [], [])
U1_ga(X, Xs, Ys, part_out_agaa(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_ga(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_ggg(Ls, .(X, Bs), Ys))
app_in_ggg(.(X, Xs), Ys, .(X, Zs)) → U9_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
app_in_ggg([], Ys, Ys) → app_out_ggg([], Ys, Ys)
U9_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) → app_out_ggg(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_ggg(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ 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
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

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ NonTerminationProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

Q DP problem:
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:

Semiunifier: [ ]
Matcher: [ ]

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

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_GGAA(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GGAA(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
U7_GGAA(X, Y, Xs, Ls, Bs, le_out_ga(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_aa(X, Y))
U10_ga(X, Y, gt_out_aa(X, Y)) → gt_out_ga(s(X), s(Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
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_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))
U11_ga(X, Y, le_out_aa(X, Y)) → le_out_ga(s(X), s(Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
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))
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_agaa(X, Xs, Littles, Bigs))
part_in_agaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_agaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
U5_agaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_ga(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_ga(X, Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_ga(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)
U6_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_agaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_agaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
U7_agaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U8_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_agaa(X, [], [], []) → part_out_agaa(X, [], [], [])
U1_ga(X, Xs, Ys, part_out_agaa(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_ga(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_ggg(Ls, .(X, Bs), Ys))
app_in_ggg(.(X, Xs), Ys, .(X, Zs)) → U9_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
app_in_ggg([], Ys, Ys) → app_out_ggg([], Ys, Ys)
U9_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) → app_out_ggg(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_ggg(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_GGAA(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GGAA(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
U7_GGAA(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)

The TRS R consists of the following rules:

le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_aa(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_aa(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U11_ga(X, Y, le_out_aa(X, Y)) → le_out_ga(s(X), s(Y))
U10_ga(X, Y, gt_out_aa(X, Y)) → gt_out_ga(s(X), s(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)
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)
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

U5_GGAA(X, Xs, gt_out_ga(Y)) → PART_IN_GGAA(X, Xs)
PART_IN_GGAA(X, .(Xs)) → U7_GGAA(X, Xs, le_in_ga(X))
PART_IN_GGAA(X, .(Xs)) → U5_GGAA(X, Xs, gt_in_ga(X))
U7_GGAA(X, Xs, le_out_ga(Y)) → PART_IN_GGAA(X, Xs)

The TRS R consists of the following rules:

le_in_ga(s) → U11_ga(le_in_aa)
le_in_ga(0) → le_out_ga(s)
le_in_ga(0) → le_out_ga(0)
gt_in_ga(s) → U10_ga(gt_in_aa)
gt_in_ga(s) → gt_out_ga(0)
U11_ga(le_out_aa(X, Y)) → le_out_ga(s)
U10_ga(gt_out_aa(X, Y)) → gt_out_ga(s)
le_in_aa → U11_aa(le_in_aa)
le_in_aa → le_out_aa(0, s)
le_in_aa → le_out_aa(0, 0)
gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s, 0)
U11_aa(le_out_aa(X, Y)) → le_out_aa(s, s)
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s, s)

The set Q consists of the following terms:

le_in_ga(x0)
gt_in_ga(x0)
U11_ga(x0)
U10_ga(x0)
le_in_aa
gt_in_aa
U11_aa(x0)
U10_aa(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:

U7_GGAA(X, Xs, le_out_ga(Y)) → PART_IN_GGAA(X, Xs)
The graph contains the following edges 1 >= 1, 2 >= 2
U5_GGAA(X, Xs, gt_out_ga(Y)) → PART_IN_GGAA(X, Xs)
The graph contains the following edges 1 >= 1, 2 >= 2
PART_IN_GGAA(X, .(Xs)) → U5_GGAA(X, Xs, gt_in_ga(X))
The graph contains the following edges 1 >= 1, 2 > 2
PART_IN_GGAA(X, .(Xs)) → U7_GGAA(X, Xs, le_in_ga(X))
The graph contains the following edges 1 >= 1, 2 > 2

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

U1_GA(X, Xs, Ys, part_out_agaa(X, Xs, Littles, Bigs)) → QS_IN_GA(Littles, Ls)
QS_IN_GA(.(X, Xs), Ys) → U1_GA(X, Xs, Ys, part_in_agaa(X, Xs, Littles, Bigs))
U1_GA(X, Xs, Ys, part_out_agaa(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_aa(X, Y))
U10_ga(X, Y, gt_out_aa(X, Y)) → gt_out_ga(s(X), s(Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
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_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))
U11_ga(X, Y, le_out_aa(X, Y)) → le_out_ga(s(X), s(Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
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))
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_agaa(X, Xs, Littles, Bigs))
part_in_agaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_agaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
U5_agaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_ga(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_ga(X, Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_ga(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)
U6_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_agaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_agaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
U7_agaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U8_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_agaa(X, [], [], []) → part_out_agaa(X, [], [], [])
U1_ga(X, Xs, Ys, part_out_agaa(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_ga(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_ggg(Ls, .(X, Bs), Ys))
app_in_ggg(.(X, Xs), Ys, .(X, Zs)) → U9_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
app_in_ggg([], Ys, Ys) → app_out_ggg([], Ys, Ys)
U9_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) → app_out_ggg(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_ggg(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

U1_GA(X, Xs, Ys, part_out_agaa(X, Xs, Littles, Bigs)) → QS_IN_GA(Littles, Ls)
QS_IN_GA(.(X, Xs), Ys) → U1_GA(X, Xs, Ys, part_in_agaa(X, Xs, Littles, Bigs))
U1_GA(X, Xs, Ys, part_out_agaa(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_agaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_agaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
part_in_agaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_agaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
part_in_agaa(X, [], [], []) → part_out_agaa(X, [], [], [])
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_agaa(X, Xs, Littles, Bigs))
qs_in_ga([], []) → qs_out_ga([], [])
U5_agaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U7_agaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U1_ga(X, Xs, Ys, part_out_agaa(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_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(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_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(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_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(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))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_ga(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_ga(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, 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_aa(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(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_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_aa(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_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
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_aa(X, Y)) → gt_out_ga(s(X), s(Y))
U11_ga(X, Y, le_out_aa(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:
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)
gt_in_ga(x1, x2) = gt_in_ga(x1)
s(x1) = s
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(x2)
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, x2)
qs_in_ga(x1, x2) = qs_in_ga(x1)
.(x1, x2) = .(x2)
U1_ga(x1, x2, x3, x4) = U1_ga(x4)
part_in_ggaa(x1, x2, x3, x4) = part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x3, x6)
U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x6)
U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x1, x3, x6)
U8_ggaa(x1, x2, x3, x4, x5, x6) = U8_ggaa(x6)
[] = []
part_out_ggaa(x1, x2, x3, x4) = part_out_ggaa(x3, x4)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x4, x5)
qs_out_ga(x1, x2) = qs_out_ga(x2)
U3_ga(x1, x2, x3, x4, x5) = U3_ga(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(x5)
app_out_gga(x1, x2, x3) = app_out_gga(x3)
part_in_agaa(x1, x2, x3, x4) = part_in_agaa(x2)
U5_agaa(x1, x2, x3, x4, x5, x6) = U5_agaa(x3, x6)
U6_agaa(x1, x2, x3, x4, x5, x6) = U6_agaa(x6)
part_out_agaa(x1, x2, x3, x4) = part_out_agaa(x3, x4)
U7_agaa(x1, x2, x3, x4, x5, x6) = U7_agaa(x3, x6)
U8_agaa(x1, x2, x3, x4, x5, x6) = U8_agaa(x6)
U1_GA(x1, x2, x3, x4) = U1_GA(x4)
U2_GA(x1, x2, x3, x4, x5) = U2_GA(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

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPOrderProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

QS_IN_GA(.(Xs)) → U1_GA(part_in_agaa(Xs))
U1_GA(part_out_agaa(Littles, Bigs)) → QS_IN_GA(Littles)
U2_GA(Bigs, qs_out_ga(Ls)) → QS_IN_GA(Bigs)
U1_GA(part_out_agaa(Littles, Bigs)) → U2_GA(Bigs, qs_in_ga(Littles))

The TRS R consists of the following rules:

part_in_agaa(.(Xs)) → U5_agaa(Xs, gt_in_aa)
part_in_agaa(.(Xs)) → U7_agaa(Xs, le_in_aa)
part_in_agaa([]) → part_out_agaa([], [])
qs_in_ga(.(Xs)) → U1_ga(part_in_agaa(Xs))
qs_in_ga([]) → qs_out_ga([])
U5_agaa(Xs, gt_out_aa(X, Y)) → U6_agaa(part_in_ggaa(X, Xs))
U7_agaa(Xs, le_out_aa(X, Y)) → U8_agaa(part_in_ggaa(X, Xs))
U1_ga(part_out_agaa(Littles, Bigs)) → U2_ga(Bigs, qs_in_ga(Littles))
gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s, 0)
U6_agaa(part_out_ggaa(Ls, Bs)) → part_out_agaa(.(Ls), Bs)
le_in_aa → U11_aa(le_in_aa)
le_in_aa → le_out_aa(0, s)
le_in_aa → le_out_aa(0, 0)
U8_agaa(part_out_ggaa(Ls, Bs)) → part_out_agaa(Ls, .(Bs))
U2_ga(Bigs, qs_out_ga(Ls)) → U3_ga(Ls, qs_in_ga(Bigs))
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s, s)
part_in_ggaa(X, .(Xs)) → U5_ggaa(X, Xs, gt_in_ga(X))
part_in_ggaa(X, .(Xs)) → U7_ggaa(X, Xs, le_in_ga(X))
part_in_ggaa(X, []) → part_out_ggaa([], [])
U11_aa(le_out_aa(X, Y)) → le_out_aa(s, s)
U3_ga(Ls, qs_out_ga(Bs)) → U4_ga(app_in_gga(Ls, .(Bs)))
U5_ggaa(X, Xs, gt_out_ga(Y)) → U6_ggaa(part_in_ggaa(X, Xs))
U7_ggaa(X, Xs, le_out_ga(Y)) → U8_ggaa(part_in_ggaa(X, Xs))
U4_ga(app_out_gga(Ys)) → qs_out_ga(Ys)
gt_in_ga(s) → U10_ga(gt_in_aa)
gt_in_ga(s) → gt_out_ga(0)
U6_ggaa(part_out_ggaa(Ls, Bs)) → part_out_ggaa(.(Ls), Bs)
le_in_ga(s) → U11_ga(le_in_aa)
le_in_ga(0) → le_out_ga(s)
le_in_ga(0) → le_out_ga(0)
U8_ggaa(part_out_ggaa(Ls, Bs)) → part_out_ggaa(Ls, .(Bs))
app_in_gga(.(Xs), Ys) → U9_gga(app_in_gga(Xs, Ys))
app_in_gga([], Ys) → app_out_gga(Ys)
U10_ga(gt_out_aa(X, Y)) → gt_out_ga(s)
U11_ga(le_out_aa(X, Y)) → le_out_ga(s)
U9_gga(app_out_gga(Zs)) → app_out_gga(.(Zs))

The set Q consists of the following terms:

part_in_agaa(x0)
qs_in_ga(x0)
U5_agaa(x0, x1)
U7_agaa(x0, x1)
U1_ga(x0)
gt_in_aa
U6_agaa(x0)
le_in_aa
U8_agaa(x0)
U2_ga(x0, x1)
U10_aa(x0)
part_in_ggaa(x0, x1)
U11_aa(x0)
U3_ga(x0, x1)
U5_ggaa(x0, x1, x2)
U7_ggaa(x0, x1, x2)
U4_ga(x0)
gt_in_ga(x0)
U6_ggaa(x0)
le_in_ga(x0)
U8_ggaa(x0)
app_in_gga(x0, x1)
U10_ga(x0)
U11_ga(x0)
U9_gga(x0)

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(.(Xs)) → U1_GA(part_in_agaa(Xs))

The remaining pairs can at least be oriented weakly.

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

Used ordering: Polynomial interpretation [25]:

POL(.(x₁)) = 1 + x₁
POL(0) = 0
POL(QS_IN_GA(x₁)) = 1 + x₁
POL(U10_aa(x₁)) = 0
POL(U10_ga(x₁)) = 0
POL(U11_aa(x₁)) = 0
POL(U11_ga(x₁)) = 0
POL(U1_GA(x₁)) = 1 + x₁
POL(U1_ga(x₁)) = 0
POL(U2_GA(x₁, x₂)) = 1 + x₁
POL(U2_ga(x₁, x₂)) = 0
POL(U3_ga(x₁, x₂)) = 0
POL(U4_ga(x₁)) = 0
POL(U5_agaa(x₁, x₂)) = 1 + x₁
POL(U5_ggaa(x₁, x₂, x₃)) = 1 + x₂
POL(U6_agaa(x₁)) = 1 + x₁
POL(U6_ggaa(x₁)) = 1 + x₁
POL(U7_agaa(x₁, x₂)) = 1 + x₁
POL(U7_ggaa(x₁, x₂, x₃)) = 1 + x₂
POL(U8_agaa(x₁)) = 1 + x₁
POL(U8_ggaa(x₁)) = 1 + x₁
POL(U9_gga(x₁)) = 0
POL([]) = 0
POL(app_in_gga(x₁, x₂)) = 0
POL(app_out_gga(x₁)) = 0
POL(gt_in_aa) = 0
POL(gt_in_ga(x₁)) = 0
POL(gt_out_aa(x₁, x₂)) = 0
POL(gt_out_ga(x₁)) = 0
POL(le_in_aa) = 0
POL(le_in_ga(x₁)) = 0
POL(le_out_aa(x₁, x₂)) = 0
POL(le_out_ga(x₁)) = 0
POL(part_in_agaa(x₁)) = x₁
POL(part_in_ggaa(x₁, x₂)) = x₂
POL(part_out_agaa(x₁, x₂)) = x₁ + x₂
POL(part_out_ggaa(x₁, x₂)) = x₁ + x₂
POL(qs_in_ga(x₁)) = 0
POL(qs_out_ga(x₁)) = 0
POL(s) = 0

The following usable rules [17] were oriented:

part_in_agaa(.(Xs)) → U7_agaa(Xs, le_in_aa)
U6_agaa(part_out_ggaa(Ls, Bs)) → part_out_agaa(.(Ls), Bs)
part_in_agaa(.(Xs)) → U5_agaa(Xs, gt_in_aa)
part_in_ggaa(X, []) → part_out_ggaa([], [])
U8_ggaa(part_out_ggaa(Ls, Bs)) → part_out_ggaa(Ls, .(Bs))
U8_agaa(part_out_ggaa(Ls, Bs)) → part_out_agaa(Ls, .(Bs))
U5_ggaa(X, Xs, gt_out_ga(Y)) → U6_ggaa(part_in_ggaa(X, Xs))
U6_ggaa(part_out_ggaa(Ls, Bs)) → part_out_ggaa(.(Ls), Bs)
part_in_ggaa(X, .(Xs)) → U5_ggaa(X, Xs, gt_in_ga(X))
U5_agaa(Xs, gt_out_aa(X, Y)) → U6_agaa(part_in_ggaa(X, Xs))
U7_agaa(Xs, le_out_aa(X, Y)) → U8_agaa(part_in_ggaa(X, Xs))
part_in_ggaa(X, .(Xs)) → U7_ggaa(X, Xs, le_in_ga(X))
part_in_agaa([]) → part_out_agaa([], [])
U7_ggaa(X, Xs, le_out_ga(Y)) → U8_ggaa(part_in_ggaa(X, Xs))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

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

The TRS R consists of the following rules:

part_in_agaa(.(Xs)) → U5_agaa(Xs, gt_in_aa)
part_in_agaa(.(Xs)) → U7_agaa(Xs, le_in_aa)
part_in_agaa([]) → part_out_agaa([], [])
qs_in_ga(.(Xs)) → U1_ga(part_in_agaa(Xs))
qs_in_ga([]) → qs_out_ga([])
U5_agaa(Xs, gt_out_aa(X, Y)) → U6_agaa(part_in_ggaa(X, Xs))
U7_agaa(Xs, le_out_aa(X, Y)) → U8_agaa(part_in_ggaa(X, Xs))
U1_ga(part_out_agaa(Littles, Bigs)) → U2_ga(Bigs, qs_in_ga(Littles))
gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s, 0)
U6_agaa(part_out_ggaa(Ls, Bs)) → part_out_agaa(.(Ls), Bs)
le_in_aa → U11_aa(le_in_aa)
le_in_aa → le_out_aa(0, s)
le_in_aa → le_out_aa(0, 0)
U8_agaa(part_out_ggaa(Ls, Bs)) → part_out_agaa(Ls, .(Bs))
U2_ga(Bigs, qs_out_ga(Ls)) → U3_ga(Ls, qs_in_ga(Bigs))
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s, s)
part_in_ggaa(X, .(Xs)) → U5_ggaa(X, Xs, gt_in_ga(X))
part_in_ggaa(X, .(Xs)) → U7_ggaa(X, Xs, le_in_ga(X))
part_in_ggaa(X, []) → part_out_ggaa([], [])
U11_aa(le_out_aa(X, Y)) → le_out_aa(s, s)
U3_ga(Ls, qs_out_ga(Bs)) → U4_ga(app_in_gga(Ls, .(Bs)))
U5_ggaa(X, Xs, gt_out_ga(Y)) → U6_ggaa(part_in_ggaa(X, Xs))
U7_ggaa(X, Xs, le_out_ga(Y)) → U8_ggaa(part_in_ggaa(X, Xs))
U4_ga(app_out_gga(Ys)) → qs_out_ga(Ys)
gt_in_ga(s) → U10_ga(gt_in_aa)
gt_in_ga(s) → gt_out_ga(0)
U6_ggaa(part_out_ggaa(Ls, Bs)) → part_out_ggaa(.(Ls), Bs)
le_in_ga(s) → U11_ga(le_in_aa)
le_in_ga(0) → le_out_ga(s)
le_in_ga(0) → le_out_ga(0)
U8_ggaa(part_out_ggaa(Ls, Bs)) → part_out_ggaa(Ls, .(Bs))
app_in_gga(.(Xs), Ys) → U9_gga(app_in_gga(Xs, Ys))
app_in_gga([], Ys) → app_out_gga(Ys)
U10_ga(gt_out_aa(X, Y)) → gt_out_ga(s)
U11_ga(le_out_aa(X, Y)) → le_out_ga(s)
U9_gga(app_out_gga(Zs)) → app_out_gga(.(Zs))

The set Q consists of the following terms:

part_in_agaa(x0)
qs_in_ga(x0)
U5_agaa(x0, x1)
U7_agaa(x0, x1)
U1_ga(x0)
gt_in_aa
U6_agaa(x0)
le_in_aa
U8_agaa(x0)
U2_ga(x0, x1)
U10_aa(x0)
part_in_ggaa(x0, x1)
U11_aa(x0)
U3_ga(x0, x1)
U5_ggaa(x0, x1, x2)
U7_ggaa(x0, x1, x2)
U4_ga(x0)
gt_in_ga(x0)
U6_ggaa(x0)
le_in_ga(x0)
U8_ggaa(x0)
app_in_gga(x0, x1)
U10_ga(x0)
U11_ga(x0)
U9_gga(x0)

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

                ↳ UsableRulesProof

  ↳ 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_aa(X, Y))
U10_ga(X, Y, gt_out_aa(X, Y)) → gt_out_ga(s(X), s(Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
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_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))
U11_ga(X, Y, le_out_aa(X, Y)) → le_out_ga(s(X), s(Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
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))
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_agaa(X, Xs, Littles, Bigs))
part_in_agaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_agaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
U5_agaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_ga(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_ga(X, Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_ga(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)
U6_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_agaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_agaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
U7_agaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U8_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_agaa(X, [], [], []) → part_out_agaa(X, [], [], [])
U1_ga(X, Xs, Ys, part_out_agaa(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_ga(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_ggg(Ls, .(X, Bs), Ys))
app_in_ggg(.(X, Xs), Ys, .(X, Zs)) → U9_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
app_in_ggg([], Ys, Ys) → app_out_ggg([], Ys, Ys)
U9_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) → app_out_ggg(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_ggg(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

  ↳ 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_aa(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_aa(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U11_ga(X, Y, le_out_aa(X, Y)) → le_out_ga(s(X), s(Y))
U10_ga(X, Y, gt_out_aa(X, Y)) → gt_out_ga(s(X), s(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)
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)
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

  ↳ PrologToPiTRSProof

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

U7_GAAA(X, le_out_ga(Y)) → PART_IN_GAAA(X)
PART_IN_GAAA(X) → U7_GAAA(X, le_in_ga(X))
U5_GAAA(X, gt_out_ga(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) → U11_ga(le_in_aa)
le_in_ga(0) → le_out_ga(s)
le_in_ga(0) → le_out_ga(0)
gt_in_ga(s) → U10_ga(gt_in_aa)
gt_in_ga(s) → gt_out_ga(0)
U11_ga(le_out_aa(X, Y)) → le_out_ga(s)
U10_ga(gt_out_aa(X, Y)) → gt_out_ga(s)
le_in_aa → U11_aa(le_in_aa)
le_in_aa → le_out_aa(0, s)
le_in_aa → le_out_aa(0, 0)
gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s, 0)
U11_aa(le_out_aa(X, Y)) → le_out_aa(s, s)
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s, s)

The set Q consists of the following terms:

le_in_ga(x0)
gt_in_ga(x0)
U11_ga(x0)
U10_ga(x0)
le_in_aa
gt_in_aa
U11_aa(x0)
U10_aa(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(s) → U7_GAAA(s, U11_ga(le_in_aa))
PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga(s))
PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga(0))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ 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:

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

The TRS R consists of the following rules:

le_in_ga(s) → U11_ga(le_in_aa)
le_in_ga(0) → le_out_ga(s)
le_in_ga(0) → le_out_ga(0)
gt_in_ga(s) → U10_ga(gt_in_aa)
gt_in_ga(s) → gt_out_ga(0)
U11_ga(le_out_aa(X, Y)) → le_out_ga(s)
U10_ga(gt_out_aa(X, Y)) → gt_out_ga(s)
le_in_aa → U11_aa(le_in_aa)
le_in_aa → le_out_aa(0, s)
le_in_aa → le_out_aa(0, 0)
gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s, 0)
U11_aa(le_out_aa(X, Y)) → le_out_aa(s, s)
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s, s)

The set Q consists of the following terms:

le_in_ga(x0)
gt_in_ga(x0)
U11_ga(x0)
U10_ga(x0)
le_in_aa
gt_in_aa
U11_aa(x0)
U10_aa(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

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

  ↳ PrologToPiTRSProof

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

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

The TRS R consists of the following rules:

gt_in_ga(s) → U10_ga(gt_in_aa)
gt_in_ga(s) → gt_out_ga(0)
gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s, 0)
U10_ga(gt_out_aa(X, Y)) → gt_out_ga(s)
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s, s)
le_in_aa → U11_aa(le_in_aa)
le_in_aa → le_out_aa(0, s)
le_in_aa → le_out_aa(0, 0)
U11_ga(le_out_aa(X, Y)) → le_out_ga(s)
U11_aa(le_out_aa(X, Y)) → le_out_aa(s, s)

The set Q consists of the following terms:

le_in_ga(x0)
gt_in_ga(x0)
U11_ga(x0)
U10_ga(x0)
le_in_aa
gt_in_aa
U11_aa(x0)
U10_aa(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)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ 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:

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

The TRS R consists of the following rules:

gt_in_ga(s) → U10_ga(gt_in_aa)
gt_in_ga(s) → gt_out_ga(0)
gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s, 0)
U10_ga(gt_out_aa(X, Y)) → gt_out_ga(s)
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s, s)
le_in_aa → U11_aa(le_in_aa)
le_in_aa → le_out_aa(0, s)
le_in_aa → le_out_aa(0, 0)
U11_ga(le_out_aa(X, Y)) → le_out_ga(s)
U11_aa(le_out_aa(X, Y)) → le_out_aa(s, s)

The set Q consists of the following terms:

gt_in_ga(x0)
U11_ga(x0)
U10_ga(x0)
le_in_aa
gt_in_aa
U11_aa(x0)
U10_aa(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) → U5_GAAA(s, U10_ga(gt_in_aa))
PART_IN_GAAA(s) → U5_GAAA(s, gt_out_ga(0))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

  ↳ PrologToPiTRSProof

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

U7_GAAA(X, le_out_ga(Y)) → PART_IN_GAAA(X)
PART_IN_GAAA(s) → U5_GAAA(s, gt_out_ga(0))
PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga(s))
PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga(0))
U5_GAAA(X, gt_out_ga(Y)) → PART_IN_GAAA(X)
PART_IN_GAAA(s) → U5_GAAA(s, U10_ga(gt_in_aa))
PART_IN_GAAA(s) → U7_GAAA(s, U11_ga(le_in_aa))

The TRS R consists of the following rules:

gt_in_ga(s) → U10_ga(gt_in_aa)
gt_in_ga(s) → gt_out_ga(0)
gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s, 0)
U10_ga(gt_out_aa(X, Y)) → gt_out_ga(s)
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s, s)
le_in_aa → U11_aa(le_in_aa)
le_in_aa → le_out_aa(0, s)
le_in_aa → le_out_aa(0, 0)
U11_ga(le_out_aa(X, Y)) → le_out_ga(s)
U11_aa(le_out_aa(X, Y)) → le_out_aa(s, s)

The set Q consists of the following terms:

gt_in_ga(x0)
U11_ga(x0)
U10_ga(x0)
le_in_aa
gt_in_aa
U11_aa(x0)
U10_aa(x0)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

  ↳ PrologToPiTRSProof

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

U7_GAAA(X, le_out_ga(Y)) → PART_IN_GAAA(X)
PART_IN_GAAA(s) → U5_GAAA(s, gt_out_ga(0))
PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga(s))
U5_GAAA(X, gt_out_ga(Y)) → PART_IN_GAAA(X)
PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga(0))
PART_IN_GAAA(s) → U5_GAAA(s, U10_ga(gt_in_aa))
PART_IN_GAAA(s) → U7_GAAA(s, U11_ga(le_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, s)
le_in_aa → le_out_aa(0, 0)
U11_ga(le_out_aa(X, Y)) → le_out_ga(s)
U11_aa(le_out_aa(X, Y)) → le_out_aa(s, s)
gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s, 0)
U10_ga(gt_out_aa(X, Y)) → gt_out_ga(s)
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s, s)

The set Q consists of the following terms:

gt_in_ga(x0)
U11_ga(x0)
U10_ga(x0)
le_in_aa
gt_in_aa
U11_aa(x0)
U10_aa(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.

gt_in_ga(x0)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Instantiation

  ↳ PrologToPiTRSProof

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

U7_GAAA(X, le_out_ga(Y)) → PART_IN_GAAA(X)
PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga(s))
PART_IN_GAAA(s) → U5_GAAA(s, gt_out_ga(0))
PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga(0))
U5_GAAA(X, gt_out_ga(Y)) → PART_IN_GAAA(X)
PART_IN_GAAA(s) → U5_GAAA(s, U10_ga(gt_in_aa))
PART_IN_GAAA(s) → U7_GAAA(s, U11_ga(le_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, s)
le_in_aa → le_out_aa(0, 0)
U11_ga(le_out_aa(X, Y)) → le_out_ga(s)
U11_aa(le_out_aa(X, Y)) → le_out_aa(s, s)
gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s, 0)
U10_ga(gt_out_aa(X, Y)) → gt_out_ga(s)
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s, s)

The set Q consists of the following terms:

U11_ga(x0)
U10_ga(x0)
le_in_aa
gt_in_aa
U11_aa(x0)
U10_aa(x0)

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

U7_GAAA(0, le_out_ga(0)) → PART_IN_GAAA(0)
U7_GAAA(s, le_out_ga(x1)) → PART_IN_GAAA(s)
U7_GAAA(0, le_out_ga(s)) → PART_IN_GAAA(0)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

  ↳ PrologToPiTRSProof

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

PART_IN_GAAA(s) → U5_GAAA(s, gt_out_ga(0))
PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga(s))
U7_GAAA(0, le_out_ga(0)) → PART_IN_GAAA(0)
U5_GAAA(X, gt_out_ga(Y)) → PART_IN_GAAA(X)
PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga(0))
PART_IN_GAAA(s) → U5_GAAA(s, U10_ga(gt_in_aa))
PART_IN_GAAA(s) → U7_GAAA(s, U11_ga(le_in_aa))
U7_GAAA(0, le_out_ga(s)) → PART_IN_GAAA(0)
U7_GAAA(s, le_out_ga(x1)) → PART_IN_GAAA(s)

The TRS R consists of the following rules:

le_in_aa → U11_aa(le_in_aa)
le_in_aa → le_out_aa(0, s)
le_in_aa → le_out_aa(0, 0)
U11_ga(le_out_aa(X, Y)) → le_out_ga(s)
U11_aa(le_out_aa(X, Y)) → le_out_aa(s, s)
gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s, 0)
U10_ga(gt_out_aa(X, Y)) → gt_out_ga(s)
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s, s)

The set Q consists of the following terms:

U11_ga(x0)
U10_ga(x0)
le_in_aa
gt_in_aa
U11_aa(x0)
U10_aa(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

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ AND

                                                        ↳ QDP

                                                          ↳ UsableRulesProof

                                                        ↳ QDP

  ↳ PrologToPiTRSProof

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

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

The TRS R consists of the following rules:

le_in_aa → U11_aa(le_in_aa)
le_in_aa → le_out_aa(0, s)
le_in_aa → le_out_aa(0, 0)
U11_ga(le_out_aa(X, Y)) → le_out_ga(s)
U11_aa(le_out_aa(X, Y)) → le_out_aa(s, s)
gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s, 0)
U10_ga(gt_out_aa(X, Y)) → gt_out_ga(s)
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s, s)

The set Q consists of the following terms:

U11_ga(x0)
U10_ga(x0)
le_in_aa
gt_in_aa
U11_aa(x0)
U10_aa(x0)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ AND

                                                        ↳ QDP

                                                          ↳ UsableRulesProof

                                                            ↳ QDP

                                                              ↳ QReductionProof

                                                        ↳ QDP

  ↳ PrologToPiTRSProof

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

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

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

U11_ga(x0)
U10_ga(x0)
le_in_aa
gt_in_aa
U11_aa(x0)
U10_aa(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.

U11_ga(x0)
U10_ga(x0)
le_in_aa
gt_in_aa
U11_aa(x0)
U10_aa(x0)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ AND

                                                        ↳ QDP

                                                          ↳ UsableRulesProof

                                                            ↳ QDP

                                                              ↳ QReductionProof

                                                                ↳ QDP

                                                                  ↳ NonTerminationProof

                                                        ↳ QDP

  ↳ PrologToPiTRSProof

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

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

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

The TRS R consists of the following rules:none

s = U7_GAAA(0, le_out_ga(s)) evaluates to t =U7_GAAA(0, le_out_ga(s))

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

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

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

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

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

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ AND

                                                        ↳ QDP

                                                        ↳ QDP

                                                          ↳ Instantiation

  ↳ PrologToPiTRSProof

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

PART_IN_GAAA(s) → U5_GAAA(s, gt_out_ga(0))
U5_GAAA(X, gt_out_ga(Y)) → PART_IN_GAAA(X)
PART_IN_GAAA(s) → U5_GAAA(s, U10_ga(gt_in_aa))
PART_IN_GAAA(s) → U7_GAAA(s, U11_ga(le_in_aa))
U7_GAAA(s, le_out_ga(x1)) → PART_IN_GAAA(s)

The TRS R consists of the following rules:

le_in_aa → U11_aa(le_in_aa)
le_in_aa → le_out_aa(0, s)
le_in_aa → le_out_aa(0, 0)
U11_ga(le_out_aa(X, Y)) → le_out_ga(s)
U11_aa(le_out_aa(X, Y)) → le_out_aa(s, s)
gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s, 0)
U10_ga(gt_out_aa(X, Y)) → gt_out_ga(s)
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s, s)

The set Q consists of the following terms:

U11_ga(x0)
U10_ga(x0)
le_in_aa
gt_in_aa
U11_aa(x0)
U10_aa(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, gt_out_ga(0)) → PART_IN_GAAA(s)
U5_GAAA(s, gt_out_ga(x1)) → PART_IN_GAAA(s)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ AND

                                                        ↳ QDP

                                                        ↳ QDP

                                                          ↳ Instantiation

                                                            ↳ QDP

                                                              ↳ NonTerminationProof

  ↳ PrologToPiTRSProof

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

PART_IN_GAAA(s) → U5_GAAA(s, gt_out_ga(0))
PART_IN_GAAA(s) → U5_GAAA(s, U10_ga(gt_in_aa))
PART_IN_GAAA(s) → U7_GAAA(s, U11_ga(le_in_aa))
U5_GAAA(s, gt_out_ga(x1)) → PART_IN_GAAA(s)
U5_GAAA(s, gt_out_ga(0)) → PART_IN_GAAA(s)
U7_GAAA(s, le_out_ga(x1)) → PART_IN_GAAA(s)

The TRS R consists of the following rules:

le_in_aa → U11_aa(le_in_aa)
le_in_aa → le_out_aa(0, s)
le_in_aa → le_out_aa(0, 0)
U11_ga(le_out_aa(X, Y)) → le_out_ga(s)
U11_aa(le_out_aa(X, Y)) → le_out_aa(s, s)
gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s, 0)
U10_ga(gt_out_aa(X, Y)) → gt_out_ga(s)
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s, s)

The set Q consists of the following terms:

U11_ga(x0)
U10_ga(x0)
le_in_aa
gt_in_aa
U11_aa(x0)
U10_aa(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) → U5_GAAA(s, gt_out_ga(0))
PART_IN_GAAA(s) → U5_GAAA(s, U10_ga(gt_in_aa))
PART_IN_GAAA(s) → U7_GAAA(s, U11_ga(le_in_aa))
U5_GAAA(s, gt_out_ga(x1)) → PART_IN_GAAA(s)
U5_GAAA(s, gt_out_ga(0)) → PART_IN_GAAA(s)
U7_GAAA(s, le_out_ga(x1)) → PART_IN_GAAA(s)

The TRS R consists of the following rules:

le_in_aa → U11_aa(le_in_aa)
le_in_aa → le_out_aa(0, s)
le_in_aa → le_out_aa(0, 0)
U11_ga(le_out_aa(X, Y)) → le_out_ga(s)
U11_aa(le_out_aa(X, Y)) → le_out_aa(s, s)
gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s, 0)
U10_ga(gt_out_aa(X, Y)) → gt_out_ga(s)
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s, s)

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

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

Semiunifier: [ ]
Matcher: [x1 / 0]

Rewriting sequence

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

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

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)
part_in: (f,f,f,f) (b,f,f,f) (f,b,f,f) (b,b,f,f)
gt_in: (f,f) (b,f)
le_in: (b,f) (f,f)
app_in: (b,b,f) (b,b,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_aa(X, Y))
U10_ga(X, Y, gt_out_aa(X, Y)) → gt_out_ga(s(X), s(Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
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_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))
U11_ga(X, Y, le_out_aa(X, Y)) → le_out_ga(s(X), s(Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
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))
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_agaa(X, Xs, Littles, Bigs))
part_in_agaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_agaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
U5_agaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_ga(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_ga(X, Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_ga(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)
U6_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_agaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_agaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
U7_agaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U8_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_agaa(X, [], [], []) → part_out_agaa(X, [], [], [])
U1_ga(X, Xs, Ys, part_out_agaa(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_ga(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_ggg(Ls, .(X, Bs), Ys))
app_in_ggg(.(X, Xs), Ys, .(X, Zs)) → U9_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
app_in_ggg([], Ys, Ys) → app_out_ggg([], Ys, Ys)
U9_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) → app_out_ggg(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_ggg(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

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_aa(X, Y))
U10_ga(X, Y, gt_out_aa(X, Y)) → gt_out_ga(s(X), s(Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
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_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))
U11_ga(X, Y, le_out_aa(X, Y)) → le_out_ga(s(X), s(Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
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))
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_agaa(X, Xs, Littles, Bigs))
part_in_agaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_agaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
U5_agaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_ga(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_ga(X, Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_ga(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)
U6_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_agaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_agaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
U7_agaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U8_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_agaa(X, [], [], []) → part_out_agaa(X, [], [], [])
U1_ga(X, Xs, Ys, part_out_agaa(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_ga(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_ggg(Ls, .(X, Bs), Ys))
app_in_ggg(.(X, Xs), Ys, .(X, Zs)) → U9_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
app_in_ggg([], Ys, Ys) → app_out_ggg([], Ys, Ys)
U9_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) → app_out_ggg(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_ggg(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

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_aa(X, Y))
GT_IN_GA(s(X), s(Y)) → GT_IN_AA(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_aa(X, Y))
LE_IN_GA(s(X), s(Y)) → 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_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)
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_agaa(X, Xs, Littles, Bigs))
QS_IN_GA(.(X, Xs), Ys) → PART_IN_AGAA(X, Xs, Littles, Bigs)
PART_IN_AGAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_AGAA(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
PART_IN_AGAA(X, .(Y, Xs), .(Y, Ls), Bs) → GT_IN_AA(X, Y)
U5_AGAA(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_AGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U5_AGAA(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GGAA(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → GT_IN_GA(X, Y)
U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_ga(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_ga(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_ga(X, Y))
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → LE_IN_GA(X, Y)
U7_GGAA(X, Y, Xs, Ls, Bs, le_out_ga(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_ga(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
PART_IN_AGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_AGAA(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
PART_IN_AGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → LE_IN_AA(X, Y)
U7_AGAA(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_AGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U7_AGAA(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
U1_GA(X, Xs, Ys, part_out_agaa(X, Xs, Littles, Bigs)) → U2_GA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U1_GA(X, Xs, Ys, part_out_agaa(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_ga(Bigs, Bs))
U2_AG(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → QS_IN_GA(Bigs, Bs)
U3_AG(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_AG(X, Xs, Ys, app_in_ggg(Ls, .(X, Bs), Ys))
U3_AG(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → APP_IN_GGG(Ls, .(X, Bs), Ys)
APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) → U9_GGG(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGG(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_aa(X, Y))
U10_ga(X, Y, gt_out_aa(X, Y)) → gt_out_ga(s(X), s(Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
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_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))
U11_ga(X, Y, le_out_aa(X, Y)) → le_out_ga(s(X), s(Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
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))
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_agaa(X, Xs, Littles, Bigs))
part_in_agaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_agaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
U5_agaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_ga(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_ga(X, Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_ga(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)
U6_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_agaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_agaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
U7_agaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U8_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_agaa(X, [], [], []) → part_out_agaa(X, [], [], [])
U1_ga(X, Xs, Ys, part_out_agaa(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_ga(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_ggg(Ls, .(X, Bs), Ys))
app_in_ggg(.(X, Xs), Ys, .(X, Zs)) → U9_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
app_in_ggg([], Ys, Ys) → app_out_ggg([], Ys, Ys)
U9_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) → app_out_ggg(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_ggg(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(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
U10_ga(x1, x2, x3) = U10_ga(x3)
0 = 0
gt_out_ga(x1, x2) = gt_out_ga(x1, x2)
U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x1, 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(x1, x2)
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, x2)
U8_gaaa(x1, x2, x3, x4, x5, x6) = U8_gaaa(x1, x6)
part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa(x1, x2, x3, x4)
part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa(x2, x3, x4)
U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6)
U8_aaaa(x1, x2, x3, x4, x5, x6) = U8_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5) = U2_ag(x2, x3, x4, x5)
qs_in_ga(x1, x2) = qs_in_ga(x1)
.(x1, x2) = .(x2)
U1_ga(x1, x2, x3, x4) = U1_ga(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, x3, x6)
U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x1, x3, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x1, x3, x6)
U8_ggaa(x1, x2, x3, x4, x5, x6) = U8_ggaa(x1, 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(x2, x4, x5)
qs_out_ga(x1, x2) = qs_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4, x5) = U3_ga(x2, x4, x5)
U4_ga(x1, x2, x3, x4) = U4_ga(x2, x4)
app_in_gga(x1, x2, x3) = app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5) = U9_gga(x2, x3, x5)
app_out_gga(x1, x2, x3) = app_out_gga(x1, x2, x3)
U3_ag(x1, x2, x3, x4, x5) = U3_ag(x2, x3, x4, x5)
U4_ag(x1, x2, x3, x4) = U4_ag(x2, x3, x4)
app_in_ggg(x1, x2, x3) = app_in_ggg(x1, x2, x3)
part_in_agaa(x1, x2, x3, x4) = part_in_agaa(x2)
U5_agaa(x1, x2, x3, x4, x5, x6) = U5_agaa(x3, x6)
U6_agaa(x1, x2, x3, x4, x5, x6) = U6_agaa(x3, x6)
part_out_agaa(x1, x2, x3, x4) = part_out_agaa(x2, x3, x4)
U7_agaa(x1, x2, x3, x4, x5, x6) = U7_agaa(x3, x6)
U8_agaa(x1, x2, x3, x4, x5, x6) = U8_agaa(x3, x6)
U9_ggg(x1, x2, x3, x4, x5) = U9_ggg(x2, x3, x4, x5)
app_out_ggg(x1, x2, x3) = app_out_ggg(x1, x2, x3)
qs_out_ag(x1, x2) = qs_out_ag(x1, 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(x3)
U6_GGAA(x1, x2, x3, x4, x5, x6) = U6_GGAA(x1, x3, x6)
U6_AAAA(x1, x2, x3, x4, x5, x6) = U6_AAAA(x6)
U9_GGA(x1, x2, x3, x4, x5) = U9_GGA(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, x3, x6)
U1_GA(x1, x2, x3, x4) = U1_GA(x2, x4)
U8_AGAA(x1, x2, x3, x4, x5, x6) = U8_AGAA(x3, x6)
U5_AGAA(x1, x2, x3, x4, x5, x6) = U5_AGAA(x3, x6)
LE_IN_AA(x1, x2) = LE_IN_AA
PART_IN_AGAA(x1, x2, x3, x4) = PART_IN_AGAA(x2)
APP_IN_GGG(x1, x2, x3) = APP_IN_GGG(x1, x2, x3)
U6_AGAA(x1, x2, x3, x4, x5, x6) = U6_AGAA(x3, x6)
U3_AG(x1, x2, x3, x4, x5) = U3_AG(x2, x3, x4, x5)
U7_AGAA(x1, x2, x3, x4, x5, x6) = U7_AGAA(x3, x6)
U2_GA(x1, x2, x3, x4, x5) = U2_GA(x2, x4, x5)
U4_GA(x1, x2, x3, x4) = U4_GA(x2, x4)
U3_GA(x1, x2, x3, x4, x5) = U3_GA(x2, x4, x5)
PART_IN_GAAA(x1, x2, x3, x4) = PART_IN_GAAA(x1)
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)
PART_IN_GGAA(x1, x2, x3, x4) = PART_IN_GGAA(x1, x2)
QS_IN_GA(x1, x2) = QS_IN_GA(x1)
U5_GGAA(x1, x2, x3, x4, x5, x6) = U5_GGAA(x1, x3, x6)
PART_IN_AAAA(x1, x2, x3, x4) = PART_IN_AAAA
U8_GGAA(x1, x2, x3, x4, x5, x6) = U8_GGAA(x1, x3, x6)
U2_AG(x1, x2, x3, x4, x5) = U2_AG(x2, x3, x4, x5)
U1_AG(x1, x2, x3, x4) = U1_AG(x3, x4)
U6_GAAA(x1, x2, x3, x4, x5, x6) = U6_GAAA(x1, x6)
GT_IN_AA(x1, x2) = GT_IN_AA
U9_GGG(x1, x2, x3, x4, x5) = U9_GGG(x2, x3, x4, x5)
U11_GA(x1, x2, x3) = U11_GA(x3)
U4_AG(x1, x2, x3, x4) = U4_AG(x2, x3, x4)
GT_IN_GA(x1, x2) = GT_IN_GA(x1)
U11_AA(x1, x2, x3) = U11_AA(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_aa(X, Y))
GT_IN_GA(s(X), s(Y)) → GT_IN_AA(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_aa(X, Y))
LE_IN_GA(s(X), s(Y)) → 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_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)
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_agaa(X, Xs, Littles, Bigs))
QS_IN_GA(.(X, Xs), Ys) → PART_IN_AGAA(X, Xs, Littles, Bigs)
PART_IN_AGAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_AGAA(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
PART_IN_AGAA(X, .(Y, Xs), .(Y, Ls), Bs) → GT_IN_AA(X, Y)
U5_AGAA(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_AGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U5_AGAA(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GGAA(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → GT_IN_GA(X, Y)
U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_ga(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_ga(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_ga(X, Y))
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → LE_IN_GA(X, Y)
U7_GGAA(X, Y, Xs, Ls, Bs, le_out_ga(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_ga(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
PART_IN_AGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_AGAA(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
PART_IN_AGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → LE_IN_AA(X, Y)
U7_AGAA(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_AGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U7_AGAA(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
U1_GA(X, Xs, Ys, part_out_agaa(X, Xs, Littles, Bigs)) → U2_GA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U1_GA(X, Xs, Ys, part_out_agaa(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_ga(Bigs, Bs))
U2_AG(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → QS_IN_GA(Bigs, Bs)
U3_AG(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_AG(X, Xs, Ys, app_in_ggg(Ls, .(X, Bs), Ys))
U3_AG(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → APP_IN_GGG(Ls, .(X, Bs), Ys)
APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) → U9_GGG(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGG(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_aa(X, Y))
U10_ga(X, Y, gt_out_aa(X, Y)) → gt_out_ga(s(X), s(Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
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_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))
U11_ga(X, Y, le_out_aa(X, Y)) → le_out_ga(s(X), s(Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
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))
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_agaa(X, Xs, Littles, Bigs))
part_in_agaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_agaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
U5_agaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_ga(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_ga(X, Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_ga(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)
U6_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_agaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_agaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
U7_agaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U8_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_agaa(X, [], [], []) → part_out_agaa(X, [], [], [])
U1_ga(X, Xs, Ys, part_out_agaa(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_ga(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_ggg(Ls, .(X, Bs), Ys))
app_in_ggg(.(X, Xs), Ys, .(X, Zs)) → U9_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
app_in_ggg([], Ys, Ys) → app_out_ggg([], Ys, Ys)
U9_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) → app_out_ggg(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_ggg(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(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
U10_ga(x1, x2, x3) = U10_ga(x3)
0 = 0
gt_out_ga(x1, x2) = gt_out_ga(x1, x2)
U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x1, 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(x1, x2)
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, x2)
U8_gaaa(x1, x2, x3, x4, x5, x6) = U8_gaaa(x1, x6)
part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa(x1, x2, x3, x4)
part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa(x2, x3, x4)
U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6)
U8_aaaa(x1, x2, x3, x4, x5, x6) = U8_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5) = U2_ag(x2, x3, x4, x5)
qs_in_ga(x1, x2) = qs_in_ga(x1)
.(x1, x2) = .(x2)
U1_ga(x1, x2, x3, x4) = U1_ga(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, x3, x6)
U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x1, x3, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x1, x3, x6)
U8_ggaa(x1, x2, x3, x4, x5, x6) = U8_ggaa(x1, 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(x2, x4, x5)
qs_out_ga(x1, x2) = qs_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4, x5) = U3_ga(x2, x4, x5)
U4_ga(x1, x2, x3, x4) = U4_ga(x2, x4)
app_in_gga(x1, x2, x3) = app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5) = U9_gga(x2, x3, x5)
app_out_gga(x1, x2, x3) = app_out_gga(x1, x2, x3)
U3_ag(x1, x2, x3, x4, x5) = U3_ag(x2, x3, x4, x5)
U4_ag(x1, x2, x3, x4) = U4_ag(x2, x3, x4)
app_in_ggg(x1, x2, x3) = app_in_ggg(x1, x2, x3)
part_in_agaa(x1, x2, x3, x4) = part_in_agaa(x2)
U5_agaa(x1, x2, x3, x4, x5, x6) = U5_agaa(x3, x6)
U6_agaa(x1, x2, x3, x4, x5, x6) = U6_agaa(x3, x6)
part_out_agaa(x1, x2, x3, x4) = part_out_agaa(x2, x3, x4)
U7_agaa(x1, x2, x3, x4, x5, x6) = U7_agaa(x3, x6)
U8_agaa(x1, x2, x3, x4, x5, x6) = U8_agaa(x3, x6)
U9_ggg(x1, x2, x3, x4, x5) = U9_ggg(x2, x3, x4, x5)
app_out_ggg(x1, x2, x3) = app_out_ggg(x1, x2, x3)
qs_out_ag(x1, x2) = qs_out_ag(x1, 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(x3)
U6_GGAA(x1, x2, x3, x4, x5, x6) = U6_GGAA(x1, x3, x6)
U6_AAAA(x1, x2, x3, x4, x5, x6) = U6_AAAA(x6)
U9_GGA(x1, x2, x3, x4, x5) = U9_GGA(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, x3, x6)
U1_GA(x1, x2, x3, x4) = U1_GA(x2, x4)
U8_AGAA(x1, x2, x3, x4, x5, x6) = U8_AGAA(x3, x6)
U5_AGAA(x1, x2, x3, x4, x5, x6) = U5_AGAA(x3, x6)
LE_IN_AA(x1, x2) = LE_IN_AA
PART_IN_AGAA(x1, x2, x3, x4) = PART_IN_AGAA(x2)
APP_IN_GGG(x1, x2, x3) = APP_IN_GGG(x1, x2, x3)
U6_AGAA(x1, x2, x3, x4, x5, x6) = U6_AGAA(x3, x6)
U3_AG(x1, x2, x3, x4, x5) = U3_AG(x2, x3, x4, x5)
U7_AGAA(x1, x2, x3, x4, x5, x6) = U7_AGAA(x3, x6)
U2_GA(x1, x2, x3, x4, x5) = U2_GA(x2, x4, x5)
U4_GA(x1, x2, x3, x4) = U4_GA(x2, x4)
U3_GA(x1, x2, x3, x4, x5) = U3_GA(x2, x4, x5)
PART_IN_GAAA(x1, x2, x3, x4) = PART_IN_GAAA(x1)
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)
PART_IN_GGAA(x1, x2, x3, x4) = PART_IN_GGAA(x1, x2)
QS_IN_GA(x1, x2) = QS_IN_GA(x1)
U5_GGAA(x1, x2, x3, x4, x5, x6) = U5_GGAA(x1, x3, x6)
PART_IN_AAAA(x1, x2, x3, x4) = PART_IN_AAAA
U8_GGAA(x1, x2, x3, x4, x5, x6) = U8_GGAA(x1, x3, x6)
U2_AG(x1, x2, x3, x4, x5) = U2_AG(x2, x3, x4, x5)
U1_AG(x1, x2, x3, x4) = U1_AG(x3, x4)
U6_GAAA(x1, x2, x3, x4, x5, x6) = U6_GAAA(x1, x6)
GT_IN_AA(x1, x2) = GT_IN_AA
U9_GGG(x1, x2, x3, x4, x5) = U9_GGG(x2, x3, x4, x5)
U11_GA(x1, x2, x3) = U11_GA(x3)
U4_AG(x1, x2, x3, x4) = U4_AG(x2, x3, x4)
GT_IN_GA(x1, x2) = GT_IN_GA(x1)
U11_AA(x1, x2, x3) = U11_AA(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 7 SCCs with 44 less nodes.

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGG(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_aa(X, Y))
U10_ga(X, Y, gt_out_aa(X, Y)) → gt_out_ga(s(X), s(Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
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_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))
U11_ga(X, Y, le_out_aa(X, Y)) → le_out_ga(s(X), s(Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
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))
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_agaa(X, Xs, Littles, Bigs))
part_in_agaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_agaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
U5_agaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_ga(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_ga(X, Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_ga(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)
U6_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_agaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_agaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
U7_agaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U8_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_agaa(X, [], [], []) → part_out_agaa(X, [], [], [])
U1_ga(X, Xs, Ys, part_out_agaa(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_ga(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_ggg(Ls, .(X, Bs), Ys))
app_in_ggg(.(X, Xs), Ys, .(X, Zs)) → U9_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
app_in_ggg([], Ys, Ys) → app_out_ggg([], Ys, Ys)
U9_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) → app_out_ggg(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_ggg(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

APP_IN_GGG(.(Xs), Ys, .(Zs)) → APP_IN_GGG(Xs, Ys, Zs)

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

APP_IN_GGG(.(Xs), Ys, .(Zs)) → APP_IN_GGG(Xs, Ys, Zs)
The graph contains the following edges 1 > 1, 2 >= 2, 3 > 3

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ 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_aa(X, Y))
U10_ga(X, Y, gt_out_aa(X, Y)) → gt_out_ga(s(X), s(Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
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_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))
U11_ga(X, Y, le_out_aa(X, Y)) → le_out_ga(s(X), s(Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
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))
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_agaa(X, Xs, Littles, Bigs))
part_in_agaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_agaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
U5_agaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_ga(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_ga(X, Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_ga(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)
U6_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_agaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_agaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
U7_agaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U8_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_agaa(X, [], [], []) → part_out_agaa(X, [], [], [])
U1_ga(X, Xs, Ys, part_out_agaa(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_ga(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_ggg(Ls, .(X, Bs), Ys))
app_in_ggg(.(X, Xs), Ys, .(X, Zs)) → U9_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
app_in_ggg([], Ys, Ys) → app_out_ggg([], Ys, Ys)
U9_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) → app_out_ggg(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_ggg(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ 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)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

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

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

APP_IN_GGA(.(Xs), Ys) → APP_IN_GGA(Xs, Ys)
The graph contains the following edges 1 > 1, 2 >= 2

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ 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_aa(X, Y))
U10_ga(X, Y, gt_out_aa(X, Y)) → gt_out_ga(s(X), s(Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
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_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))
U11_ga(X, Y, le_out_aa(X, Y)) → le_out_ga(s(X), s(Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
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))
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_agaa(X, Xs, Littles, Bigs))
part_in_agaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_agaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
U5_agaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_ga(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_ga(X, Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_ga(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)
U6_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_agaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_agaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
U7_agaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U8_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_agaa(X, [], [], []) → part_out_agaa(X, [], [], [])
U1_ga(X, Xs, Ys, part_out_agaa(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_ga(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_ggg(Ls, .(X, Bs), Ys))
app_in_ggg(.(X, Xs), Ys, .(X, Zs)) → U9_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
app_in_ggg([], Ys, Ys) → app_out_ggg([], Ys, Ys)
U9_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) → app_out_ggg(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_ggg(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ 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)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ NonTerminationProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

Q DP problem:
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:

Semiunifier: [ ]
Matcher: [ ]

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

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ 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_aa(X, Y))
U10_ga(X, Y, gt_out_aa(X, Y)) → gt_out_ga(s(X), s(Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
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_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))
U11_ga(X, Y, le_out_aa(X, Y)) → le_out_ga(s(X), s(Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
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))
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_agaa(X, Xs, Littles, Bigs))
part_in_agaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_agaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
U5_agaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_ga(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_ga(X, Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_ga(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)
U6_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_agaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_agaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
U7_agaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U8_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_agaa(X, [], [], []) → part_out_agaa(X, [], [], [])
U1_ga(X, Xs, Ys, part_out_agaa(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_ga(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_ggg(Ls, .(X, Bs), Ys))
app_in_ggg(.(X, Xs), Ys, .(X, Zs)) → U9_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
app_in_ggg([], Ys, Ys) → app_out_ggg([], Ys, Ys)
U9_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) → app_out_ggg(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_ggg(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

GT_IN_AA(s(X), s(Y)) → GT_IN_AA(X, Y)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ NonTerminationProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

Q DP problem:
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:

Semiunifier: [ ]
Matcher: [ ]

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

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

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

PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_GGAA(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GGAA(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
U7_GGAA(X, Y, Xs, Ls, Bs, le_out_ga(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_aa(X, Y))
U10_ga(X, Y, gt_out_aa(X, Y)) → gt_out_ga(s(X), s(Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
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_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))
U11_ga(X, Y, le_out_aa(X, Y)) → le_out_ga(s(X), s(Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
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))
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_agaa(X, Xs, Littles, Bigs))
part_in_agaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_agaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
U5_agaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_ga(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_ga(X, Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_ga(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)
U6_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_agaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_agaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
U7_agaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U8_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_agaa(X, [], [], []) → part_out_agaa(X, [], [], [])
U1_ga(X, Xs, Ys, part_out_agaa(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_ga(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_ggg(Ls, .(X, Bs), Ys))
app_in_ggg(.(X, Xs), Ys, .(X, Zs)) → U9_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
app_in_ggg([], Ys, Ys) → app_out_ggg([], Ys, Ys)
U9_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) → app_out_ggg(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_ggg(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

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

PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_GGAA(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GGAA(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
U7_GGAA(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)

The TRS R consists of the following rules:

le_in_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_aa(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_aa(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U11_ga(X, Y, le_out_aa(X, Y)) → le_out_ga(s(X), s(Y))
U10_ga(X, Y, gt_out_aa(X, Y)) → gt_out_ga(s(X), s(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)
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)
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

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

U5_GGAA(X, Xs, gt_out_ga(X, Y)) → PART_IN_GGAA(X, Xs)
PART_IN_GGAA(X, .(Xs)) → U7_GGAA(X, Xs, le_in_ga(X))
U7_GGAA(X, Xs, le_out_ga(X, Y)) → PART_IN_GGAA(X, Xs)
PART_IN_GGAA(X, .(Xs)) → U5_GGAA(X, Xs, gt_in_ga(X))

The TRS R consists of the following rules:

le_in_ga(s) → U11_ga(le_in_aa)
le_in_ga(0) → le_out_ga(0, s)
le_in_ga(0) → le_out_ga(0, 0)
gt_in_ga(s) → U10_ga(gt_in_aa)
gt_in_ga(s) → gt_out_ga(s, 0)
U11_ga(le_out_aa(X, Y)) → le_out_ga(s, s)
U10_ga(gt_out_aa(X, Y)) → gt_out_ga(s, s)
le_in_aa → U11_aa(le_in_aa)
le_in_aa → le_out_aa(0, s)
le_in_aa → le_out_aa(0, 0)
gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s, 0)
U11_aa(le_out_aa(X, Y)) → le_out_aa(s, s)
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s, s)

The set Q consists of the following terms:

le_in_ga(x0)
gt_in_ga(x0)
U11_ga(x0)
U10_ga(x0)
le_in_aa
gt_in_aa
U11_aa(x0)
U10_aa(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:

U7_GGAA(X, Xs, le_out_ga(X, Y)) → PART_IN_GGAA(X, Xs)
The graph contains the following edges 1 >= 1, 3 > 1, 2 >= 2
U5_GGAA(X, Xs, gt_out_ga(X, Y)) → PART_IN_GGAA(X, Xs)
The graph contains the following edges 1 >= 1, 3 > 1, 2 >= 2
PART_IN_GGAA(X, .(Xs)) → U5_GGAA(X, Xs, gt_in_ga(X))
The graph contains the following edges 1 >= 1, 2 > 2
PART_IN_GGAA(X, .(Xs)) → U7_GGAA(X, Xs, le_in_ga(X))
The graph contains the following edges 1 >= 1, 2 > 2

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

U1_GA(X, Xs, Ys, part_out_agaa(X, Xs, Littles, Bigs)) → QS_IN_GA(Littles, Ls)
QS_IN_GA(.(X, Xs), Ys) → U1_GA(X, Xs, Ys, part_in_agaa(X, Xs, Littles, Bigs))
U1_GA(X, Xs, Ys, part_out_agaa(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_aa(X, Y))
U10_ga(X, Y, gt_out_aa(X, Y)) → gt_out_ga(s(X), s(Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
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_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))
U11_ga(X, Y, le_out_aa(X, Y)) → le_out_ga(s(X), s(Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
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))
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_agaa(X, Xs, Littles, Bigs))
part_in_agaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_agaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
U5_agaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_ga(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_ga(X, Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_ga(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)
U6_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_agaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_agaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
U7_agaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U8_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_agaa(X, [], [], []) → part_out_agaa(X, [], [], [])
U1_ga(X, Xs, Ys, part_out_agaa(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_ga(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_ggg(Ls, .(X, Bs), Ys))
app_in_ggg(.(X, Xs), Ys, .(X, Zs)) → U9_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
app_in_ggg([], Ys, Ys) → app_out_ggg([], Ys, Ys)
U9_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) → app_out_ggg(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_ggg(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

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

U1_GA(X, Xs, Ys, part_out_agaa(X, Xs, Littles, Bigs)) → QS_IN_GA(Littles, Ls)
QS_IN_GA(.(X, Xs), Ys) → U1_GA(X, Xs, Ys, part_in_agaa(X, Xs, Littles, Bigs))
U1_GA(X, Xs, Ys, part_out_agaa(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_agaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_agaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
part_in_agaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_agaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
part_in_agaa(X, [], [], []) → part_out_agaa(X, [], [], [])
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_agaa(X, Xs, Littles, Bigs))
qs_in_ga([], []) → qs_out_ga([], [])
U5_agaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U7_agaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U1_ga(X, Xs, Ys, part_out_agaa(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_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(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_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(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_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(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))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_ga(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_ga(X, Y)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, 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_aa(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(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_ga(s(X), s(Y)) → U11_ga(X, Y, le_in_aa(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_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
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_aa(X, Y)) → gt_out_ga(s(X), s(Y))
U11_ga(X, Y, le_out_aa(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:
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)
gt_in_ga(x1, x2) = gt_in_ga(x1)
s(x1) = s
U10_ga(x1, x2, x3) = U10_ga(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(x3)
le_out_ga(x1, x2) = le_out_ga(x1, x2)
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, x2)
qs_in_ga(x1, x2) = qs_in_ga(x1)
.(x1, x2) = .(x2)
U1_ga(x1, x2, x3, x4) = U1_ga(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, x3, x6)
U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x1, x3, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x1, x3, x6)
U8_ggaa(x1, x2, x3, x4, x5, x6) = U8_ggaa(x1, 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(x2, x4, x5)
qs_out_ga(x1, x2) = qs_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4, x5) = U3_ga(x2, x4, x5)
U4_ga(x1, x2, x3, x4) = U4_ga(x2, x4)
app_in_gga(x1, x2, x3) = app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5) = U9_gga(x2, x3, x5)
app_out_gga(x1, x2, x3) = app_out_gga(x1, x2, x3)
part_in_agaa(x1, x2, x3, x4) = part_in_agaa(x2)
U5_agaa(x1, x2, x3, x4, x5, x6) = U5_agaa(x3, x6)
U6_agaa(x1, x2, x3, x4, x5, x6) = U6_agaa(x3, x6)
part_out_agaa(x1, x2, x3, x4) = part_out_agaa(x2, x3, x4)
U7_agaa(x1, x2, x3, x4, x5, x6) = U7_agaa(x3, x6)
U8_agaa(x1, x2, x3, x4, x5, x6) = U8_agaa(x3, x6)
U1_GA(x1, x2, x3, x4) = U1_GA(x2, x4)
U2_GA(x1, x2, x3, x4, x5) = U2_GA(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

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPOrderProof

              ↳ PiDP

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

U1_GA(Xs, part_out_agaa(Xs, Littles, Bigs)) → QS_IN_GA(Littles)
U1_GA(Xs, part_out_agaa(Xs, Littles, Bigs)) → U2_GA(Xs, Bigs, qs_in_ga(Littles))
U2_GA(Xs, Bigs, qs_out_ga(Littles, Ls)) → QS_IN_GA(Bigs)
QS_IN_GA(.(Xs)) → U1_GA(Xs, part_in_agaa(Xs))

The TRS R consists of the following rules:

part_in_agaa(.(Xs)) → U5_agaa(Xs, gt_in_aa)
part_in_agaa(.(Xs)) → U7_agaa(Xs, le_in_aa)
part_in_agaa([]) → part_out_agaa([], [], [])
qs_in_ga(.(Xs)) → U1_ga(Xs, part_in_agaa(Xs))
qs_in_ga([]) → qs_out_ga([], [])
U5_agaa(Xs, gt_out_aa(X, Y)) → U6_agaa(Xs, part_in_ggaa(X, Xs))
U7_agaa(Xs, le_out_aa(X, Y)) → U8_agaa(Xs, part_in_ggaa(X, Xs))
U1_ga(Xs, part_out_agaa(Xs, Littles, Bigs)) → U2_ga(Xs, Bigs, qs_in_ga(Littles))
gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s, 0)
U6_agaa(Xs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(.(Xs), .(Ls), Bs)
le_in_aa → U11_aa(le_in_aa)
le_in_aa → le_out_aa(0, s)
le_in_aa → le_out_aa(0, 0)
U8_agaa(Xs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(.(Xs), Ls, .(Bs))
U2_ga(Xs, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(Xs, Ls, qs_in_ga(Bigs))
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s, s)
part_in_ggaa(X, .(Xs)) → U5_ggaa(X, Xs, gt_in_ga(X))
part_in_ggaa(X, .(Xs)) → U7_ggaa(X, Xs, le_in_ga(X))
part_in_ggaa(X, []) → part_out_ggaa(X, [], [], [])
U11_aa(le_out_aa(X, Y)) → le_out_aa(s, s)
U3_ga(Xs, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(Xs, app_in_gga(Ls, .(Bs)))
U5_ggaa(X, Xs, gt_out_ga(X, Y)) → U6_ggaa(X, Xs, part_in_ggaa(X, Xs))
U7_ggaa(X, Xs, le_out_ga(X, Y)) → U8_ggaa(X, Xs, part_in_ggaa(X, Xs))
U4_ga(Xs, app_out_gga(Ls, .(Bs), Ys)) → qs_out_ga(.(Xs), Ys)
gt_in_ga(s) → U10_ga(gt_in_aa)
gt_in_ga(s) → gt_out_ga(s, 0)
U6_ggaa(X, Xs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Xs), .(Ls), Bs)
le_in_ga(s) → U11_ga(le_in_aa)
le_in_ga(0) → le_out_ga(0, s)
le_in_ga(0) → le_out_ga(0, 0)
U8_ggaa(X, Xs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Xs), Ls, .(Bs))
app_in_gga(.(Xs), Ys) → U9_gga(Xs, Ys, app_in_gga(Xs, Ys))
app_in_gga([], Ys) → app_out_gga([], Ys, Ys)
U10_ga(gt_out_aa(X, Y)) → gt_out_ga(s, s)
U11_ga(le_out_aa(X, Y)) → le_out_ga(s, s)
U9_gga(Xs, Ys, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(Xs), Ys, .(Zs))

The set Q consists of the following terms:

part_in_agaa(x0)
qs_in_ga(x0)
U5_agaa(x0, x1)
U7_agaa(x0, x1)
U1_ga(x0, x1)
gt_in_aa
U6_agaa(x0, x1)
le_in_aa
U8_agaa(x0, x1)
U2_ga(x0, x1, x2)
U10_aa(x0)
part_in_ggaa(x0, x1)
U11_aa(x0)
U3_ga(x0, x1, x2)
U5_ggaa(x0, x1, x2)
U7_ggaa(x0, x1, x2)
U4_ga(x0, x1)
gt_in_ga(x0)
U6_ggaa(x0, x1, x2)
le_in_ga(x0)
U8_ggaa(x0, x1, x2)
app_in_gga(x0, x1)
U10_ga(x0)
U11_ga(x0)
U9_gga(x0, x1, x2)

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(.(Xs)) → U1_GA(Xs, part_in_agaa(Xs))

The remaining pairs can at least be oriented weakly.

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

Used ordering: Polynomial interpretation [25]:

POL(.(x₁)) = 1 + x₁
POL(0) = 0
POL(QS_IN_GA(x₁)) = 1 + x₁
POL(U10_aa(x₁)) = x₁
POL(U10_ga(x₁)) = x₁
POL(U11_aa(x₁)) = 1
POL(U11_ga(x₁)) = 1
POL(U1_GA(x₁, x₂)) = x₂
POL(U1_ga(x₁, x₂)) = 0
POL(U2_GA(x₁, x₂, x₃)) = 1 + x₂
POL(U2_ga(x₁, x₂, x₃)) = 0
POL(U3_ga(x₁, x₂, x₃)) = 0
POL(U4_ga(x₁, x₂)) = 0
POL(U5_agaa(x₁, x₂)) = 1 + x₁ + x₂
POL(U5_ggaa(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U6_agaa(x₁, x₂)) = 1 + x₂
POL(U6_ggaa(x₁, x₂, x₃)) = 1 + x₃
POL(U7_agaa(x₁, x₂)) = 1 + x₁ + x₂
POL(U7_ggaa(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U8_agaa(x₁, x₂)) = 1 + x₂
POL(U8_ggaa(x₁, x₂, x₃)) = 1 + x₃
POL(U9_gga(x₁, x₂, x₃)) = 0
POL([]) = 0
POL(app_in_gga(x₁, x₂)) = 1 + x₂
POL(app_out_gga(x₁, x₂, x₃)) = 0
POL(gt_in_aa) = 1
POL(gt_in_ga(x₁)) = 1
POL(gt_out_aa(x₁, x₂)) = 1
POL(gt_out_ga(x₁, x₂)) = 1 + x₁
POL(le_in_aa) = 1
POL(le_in_ga(x₁)) = 1
POL(le_out_aa(x₁, x₂)) = 1
POL(le_out_ga(x₁, x₂)) = 1
POL(part_in_agaa(x₁)) = 1 + x₁
POL(part_in_ggaa(x₁, x₂)) = 1 + x₂
POL(part_out_agaa(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(part_out_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₃ + x₄
POL(qs_in_ga(x₁)) = 0
POL(qs_out_ga(x₁, x₂)) = 0
POL(s) = 0

The following usable rules [17] were oriented:

gt_in_aa → gt_out_aa(s, 0)
le_in_ga(0) → le_out_ga(0, 0)
part_in_agaa(.(Xs)) → U7_agaa(Xs, le_in_aa)
U6_agaa(Xs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(.(Xs), .(Ls), Bs)
U6_ggaa(X, Xs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Xs), .(Ls), Bs)
part_in_ggaa(X, []) → part_out_ggaa(X, [], [], [])
U5_ggaa(X, Xs, gt_out_ga(X, Y)) → U6_ggaa(X, Xs, part_in_ggaa(X, Xs))
part_in_agaa(.(Xs)) → U5_agaa(Xs, gt_in_aa)
le_in_aa → U11_aa(le_in_aa)
le_in_ga(s) → U11_ga(le_in_aa)
gt_in_aa → U10_aa(gt_in_aa)
U11_aa(le_out_aa(X, Y)) → le_out_aa(s, s)
U11_ga(le_out_aa(X, Y)) → le_out_ga(s, s)
U7_agaa(Xs, le_out_aa(X, Y)) → U8_agaa(Xs, part_in_ggaa(X, Xs))
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s, s)
U10_ga(gt_out_aa(X, Y)) → gt_out_ga(s, s)
le_in_ga(0) → le_out_ga(0, s)
part_in_ggaa(X, .(Xs)) → U5_ggaa(X, Xs, gt_in_ga(X))
U5_agaa(Xs, gt_out_aa(X, Y)) → U6_agaa(Xs, part_in_ggaa(X, Xs))
gt_in_ga(s) → gt_out_ga(s, 0)
U7_ggaa(X, Xs, le_out_ga(X, Y)) → U8_ggaa(X, Xs, part_in_ggaa(X, Xs))
le_in_aa → le_out_aa(0, 0)
U8_ggaa(X, Xs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Xs), Ls, .(Bs))
U8_agaa(Xs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(.(Xs), Ls, .(Bs))
part_in_ggaa(X, .(Xs)) → U7_ggaa(X, Xs, le_in_ga(X))
gt_in_ga(s) → U10_ga(gt_in_aa)
le_in_aa → le_out_aa(0, s)
part_in_agaa([]) → part_out_agaa([], [], [])

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

              ↳ PiDP

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

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

The TRS R consists of the following rules:

part_in_agaa(.(Xs)) → U5_agaa(Xs, gt_in_aa)
part_in_agaa(.(Xs)) → U7_agaa(Xs, le_in_aa)
part_in_agaa([]) → part_out_agaa([], [], [])
qs_in_ga(.(Xs)) → U1_ga(Xs, part_in_agaa(Xs))
qs_in_ga([]) → qs_out_ga([], [])
U5_agaa(Xs, gt_out_aa(X, Y)) → U6_agaa(Xs, part_in_ggaa(X, Xs))
U7_agaa(Xs, le_out_aa(X, Y)) → U8_agaa(Xs, part_in_ggaa(X, Xs))
U1_ga(Xs, part_out_agaa(Xs, Littles, Bigs)) → U2_ga(Xs, Bigs, qs_in_ga(Littles))
gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s, 0)
U6_agaa(Xs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(.(Xs), .(Ls), Bs)
le_in_aa → U11_aa(le_in_aa)
le_in_aa → le_out_aa(0, s)
le_in_aa → le_out_aa(0, 0)
U8_agaa(Xs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(.(Xs), Ls, .(Bs))
U2_ga(Xs, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(Xs, Ls, qs_in_ga(Bigs))
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s, s)
part_in_ggaa(X, .(Xs)) → U5_ggaa(X, Xs, gt_in_ga(X))
part_in_ggaa(X, .(Xs)) → U7_ggaa(X, Xs, le_in_ga(X))
part_in_ggaa(X, []) → part_out_ggaa(X, [], [], [])
U11_aa(le_out_aa(X, Y)) → le_out_aa(s, s)
U3_ga(Xs, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(Xs, app_in_gga(Ls, .(Bs)))
U5_ggaa(X, Xs, gt_out_ga(X, Y)) → U6_ggaa(X, Xs, part_in_ggaa(X, Xs))
U7_ggaa(X, Xs, le_out_ga(X, Y)) → U8_ggaa(X, Xs, part_in_ggaa(X, Xs))
U4_ga(Xs, app_out_gga(Ls, .(Bs), Ys)) → qs_out_ga(.(Xs), Ys)
gt_in_ga(s) → U10_ga(gt_in_aa)
gt_in_ga(s) → gt_out_ga(s, 0)
U6_ggaa(X, Xs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Xs), .(Ls), Bs)
le_in_ga(s) → U11_ga(le_in_aa)
le_in_ga(0) → le_out_ga(0, s)
le_in_ga(0) → le_out_ga(0, 0)
U8_ggaa(X, Xs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Xs), Ls, .(Bs))
app_in_gga(.(Xs), Ys) → U9_gga(Xs, Ys, app_in_gga(Xs, Ys))
app_in_gga([], Ys) → app_out_gga([], Ys, Ys)
U10_ga(gt_out_aa(X, Y)) → gt_out_ga(s, s)
U11_ga(le_out_aa(X, Y)) → le_out_ga(s, s)
U9_gga(Xs, Ys, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(Xs), Ys, .(Zs))

The set Q consists of the following terms:

part_in_agaa(x0)
qs_in_ga(x0)
U5_agaa(x0, x1)
U7_agaa(x0, x1)
U1_ga(x0, x1)
gt_in_aa
U6_agaa(x0, x1)
le_in_aa
U8_agaa(x0, x1)
U2_ga(x0, x1, x2)
U10_aa(x0)
part_in_ggaa(x0, x1)
U11_aa(x0)
U3_ga(x0, x1, x2)
U5_ggaa(x0, x1, x2)
U7_ggaa(x0, x1, x2)
U4_ga(x0, x1)
gt_in_ga(x0)
U6_ggaa(x0, x1, x2)
le_in_ga(x0)
U8_ggaa(x0, x1, x2)
app_in_gga(x0, x1)
U10_ga(x0)
U11_ga(x0)
U9_gga(x0, x1, x2)

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

                ↳ UsableRulesProof

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_aa(X, Y))
U10_ga(X, Y, gt_out_aa(X, Y)) → gt_out_ga(s(X), s(Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
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_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))
U11_ga(X, Y, le_out_aa(X, Y)) → le_out_ga(s(X), s(Y))
le_in_ga(0, s(X)) → le_out_ga(0, s(X))
le_in_ga(0, 0) → le_out_ga(0, 0)
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))
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_agaa(X, Xs, Littles, Bigs))
part_in_agaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_agaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
U5_agaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) → U6_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_ga(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_ga(X, Y))
U7_ggaa(X, Y, Xs, Ls, Bs, le_out_ga(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)
U6_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), .(Y, Ls), Bs)
part_in_agaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_agaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
U7_agaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) → U8_agaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U8_agaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_agaa(X, .(Y, Xs), Ls, .(Y, Bs))
part_in_agaa(X, [], [], []) → part_out_agaa(X, [], [], [])
U1_ga(X, Xs, Ys, part_out_agaa(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_ga(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_ggg(Ls, .(X, Bs), Ys))
app_in_ggg(.(X, Xs), Ys, .(X, Zs)) → U9_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
app_in_ggg([], Ys, Ys) → app_out_ggg([], Ys, Ys)
U9_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) → app_out_ggg(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_ggg(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)
qs_in_ag([], []) → qs_out_ag([], [])

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

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_aa(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_aa(X, Y))
gt_in_ga(s(0), 0) → gt_out_ga(s(0), 0)
U11_ga(X, Y, le_out_aa(X, Y)) → le_out_ga(s(X), s(Y))
U10_ga(X, Y, gt_out_aa(X, Y)) → gt_out_ga(s(X), s(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)
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)
U11_aa(X, Y, le_out_aa(X, Y)) → le_out_aa(s(X), s(Y))
U10_aa(X, Y, gt_out_aa(X, Y)) → gt_out_aa(s(X), s(Y))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

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, Y)) → 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) → U11_ga(le_in_aa)
le_in_ga(0) → le_out_ga(0, s)
le_in_ga(0) → le_out_ga(0, 0)
gt_in_ga(s) → U10_ga(gt_in_aa)
gt_in_ga(s) → gt_out_ga(s, 0)
U11_ga(le_out_aa(X, Y)) → le_out_ga(s, s)
U10_ga(gt_out_aa(X, Y)) → gt_out_ga(s, s)
le_in_aa → U11_aa(le_in_aa)
le_in_aa → le_out_aa(0, s)
le_in_aa → le_out_aa(0, 0)
gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s, 0)
U11_aa(le_out_aa(X, Y)) → le_out_aa(s, s)
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s, s)

The set Q consists of the following terms:

le_in_ga(x0)
gt_in_ga(x0)
U11_ga(x0)
U10_ga(x0)
le_in_aa
gt_in_aa
U11_aa(x0)
U10_aa(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(0, 0))
PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga(0, s))
PART_IN_GAAA(s) → U7_GAAA(s, U11_ga(le_in_aa))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ 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:

PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga(0, s))
U7_GAAA(X, le_out_ga(X, Y)) → PART_IN_GAAA(X)
PART_IN_GAAA(s) → U7_GAAA(s, U11_ga(le_in_aa))
PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga(0, 0))
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) → U11_ga(le_in_aa)
le_in_ga(0) → le_out_ga(0, s)
le_in_ga(0) → le_out_ga(0, 0)
gt_in_ga(s) → U10_ga(gt_in_aa)
gt_in_ga(s) → gt_out_ga(s, 0)
U11_ga(le_out_aa(X, Y)) → le_out_ga(s, s)
U10_ga(gt_out_aa(X, Y)) → gt_out_ga(s, s)
le_in_aa → U11_aa(le_in_aa)
le_in_aa → le_out_aa(0, s)
le_in_aa → le_out_aa(0, 0)
gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s, 0)
U11_aa(le_out_aa(X, Y)) → le_out_aa(s, s)
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s, s)

The set Q consists of the following terms:

le_in_ga(x0)
gt_in_ga(x0)
U11_ga(x0)
U10_ga(x0)
le_in_aa
gt_in_aa
U11_aa(x0)
U10_aa(x0)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ 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:

PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga(0, s))
U7_GAAA(X, le_out_ga(X, Y)) → PART_IN_GAAA(X)
PART_IN_GAAA(s) → U7_GAAA(s, U11_ga(le_in_aa))
PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga(0, 0))
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:

gt_in_ga(s) → U10_ga(gt_in_aa)
gt_in_ga(s) → gt_out_ga(s, 0)
gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s, 0)
U10_ga(gt_out_aa(X, Y)) → gt_out_ga(s, s)
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s, s)
le_in_aa → U11_aa(le_in_aa)
le_in_aa → le_out_aa(0, s)
le_in_aa → le_out_aa(0, 0)
U11_ga(le_out_aa(X, Y)) → le_out_ga(s, s)
U11_aa(le_out_aa(X, Y)) → le_out_aa(s, s)

The set Q consists of the following terms:

le_in_ga(x0)
gt_in_ga(x0)
U11_ga(x0)
U10_ga(x0)
le_in_aa
gt_in_aa
U11_aa(x0)
U10_aa(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)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ 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:

PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga(0, s))
U7_GAAA(X, le_out_ga(X, Y)) → PART_IN_GAAA(X)
PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga(0, 0))
PART_IN_GAAA(s) → U7_GAAA(s, U11_ga(le_in_aa))
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:

gt_in_ga(s) → U10_ga(gt_in_aa)
gt_in_ga(s) → gt_out_ga(s, 0)
gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s, 0)
U10_ga(gt_out_aa(X, Y)) → gt_out_ga(s, s)
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s, s)
le_in_aa → U11_aa(le_in_aa)
le_in_aa → le_out_aa(0, s)
le_in_aa → le_out_aa(0, 0)
U11_ga(le_out_aa(X, Y)) → le_out_ga(s, s)
U11_aa(le_out_aa(X, Y)) → le_out_aa(s, s)

The set Q consists of the following terms:

gt_in_ga(x0)
U11_ga(x0)
U10_ga(x0)
le_in_aa
gt_in_aa
U11_aa(x0)
U10_aa(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) → U5_GAAA(s, U10_ga(gt_in_aa))
PART_IN_GAAA(s) → U5_GAAA(s, gt_out_ga(s, 0))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

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

PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga(0, s))
U7_GAAA(X, le_out_ga(X, Y)) → PART_IN_GAAA(X)
PART_IN_GAAA(s) → U5_GAAA(s, U10_ga(gt_in_aa))
PART_IN_GAAA(s) → U5_GAAA(s, gt_out_ga(s, 0))
PART_IN_GAAA(s) → U7_GAAA(s, U11_ga(le_in_aa))
PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga(0, 0))
U5_GAAA(X, gt_out_ga(X, Y)) → PART_IN_GAAA(X)

The TRS R consists of the following rules:

gt_in_ga(s) → U10_ga(gt_in_aa)
gt_in_ga(s) → gt_out_ga(s, 0)
gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s, 0)
U10_ga(gt_out_aa(X, Y)) → gt_out_ga(s, s)
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s, s)
le_in_aa → U11_aa(le_in_aa)
le_in_aa → le_out_aa(0, s)
le_in_aa → le_out_aa(0, 0)
U11_ga(le_out_aa(X, Y)) → le_out_ga(s, s)
U11_aa(le_out_aa(X, Y)) → le_out_aa(s, s)

The set Q consists of the following terms:

gt_in_ga(x0)
U11_ga(x0)
U10_ga(x0)
le_in_aa
gt_in_aa
U11_aa(x0)
U10_aa(x0)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

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

PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga(0, s))
U7_GAAA(X, le_out_ga(X, Y)) → PART_IN_GAAA(X)
PART_IN_GAAA(s) → U5_GAAA(s, U10_ga(gt_in_aa))
PART_IN_GAAA(s) → U5_GAAA(s, gt_out_ga(s, 0))
PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga(0, 0))
PART_IN_GAAA(s) → U7_GAAA(s, U11_ga(le_in_aa))
U5_GAAA(X, gt_out_ga(X, Y)) → PART_IN_GAAA(X)

The TRS R consists of the following rules:

gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s, 0)
U10_ga(gt_out_aa(X, Y)) → gt_out_ga(s, s)
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s, s)
le_in_aa → U11_aa(le_in_aa)
le_in_aa → le_out_aa(0, s)
le_in_aa → le_out_aa(0, 0)
U11_ga(le_out_aa(X, Y)) → le_out_ga(s, s)
U11_aa(le_out_aa(X, Y)) → le_out_aa(s, s)

The set Q consists of the following terms:

gt_in_ga(x0)
U11_ga(x0)
U10_ga(x0)
le_in_aa
gt_in_aa
U11_aa(x0)
U10_aa(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.

gt_in_ga(x0)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Instantiation

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

PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga(0, s))
U7_GAAA(X, le_out_ga(X, Y)) → PART_IN_GAAA(X)
PART_IN_GAAA(s) → U5_GAAA(s, U10_ga(gt_in_aa))
PART_IN_GAAA(s) → U7_GAAA(s, U11_ga(le_in_aa))
PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga(0, 0))
PART_IN_GAAA(s) → U5_GAAA(s, gt_out_ga(s, 0))
U5_GAAA(X, gt_out_ga(X, Y)) → PART_IN_GAAA(X)

The TRS R consists of the following rules:

gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s, 0)
U10_ga(gt_out_aa(X, Y)) → gt_out_ga(s, s)
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s, s)
le_in_aa → U11_aa(le_in_aa)
le_in_aa → le_out_aa(0, s)
le_in_aa → le_out_aa(0, 0)
U11_ga(le_out_aa(X, Y)) → le_out_ga(s, s)
U11_aa(le_out_aa(X, Y)) → le_out_aa(s, s)

The set Q consists of the following terms:

U11_ga(x0)
U10_ga(x0)
le_in_aa
gt_in_aa
U11_aa(x0)
U10_aa(x0)

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

U7_GAAA(0, le_out_ga(0, 0)) → PART_IN_GAAA(0)
U7_GAAA(0, le_out_ga(0, s)) → PART_IN_GAAA(0)
U7_GAAA(s, le_out_ga(s, x1)) → PART_IN_GAAA(s)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

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

U7_GAAA(0, le_out_ga(0, 0)) → PART_IN_GAAA(0)
PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga(0, s))
U7_GAAA(0, le_out_ga(0, s)) → PART_IN_GAAA(0)
PART_IN_GAAA(s) → U5_GAAA(s, U10_ga(gt_in_aa))
PART_IN_GAAA(s) → U5_GAAA(s, gt_out_ga(s, 0))
PART_IN_GAAA(0) → U7_GAAA(0, le_out_ga(0, 0))
PART_IN_GAAA(s) → U7_GAAA(s, U11_ga(le_in_aa))
U5_GAAA(X, gt_out_ga(X, Y)) → PART_IN_GAAA(X)
U7_GAAA(s, le_out_ga(s, x1)) → PART_IN_GAAA(s)

The TRS R consists of the following rules:

gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s, 0)
U10_ga(gt_out_aa(X, Y)) → gt_out_ga(s, s)
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s, s)
le_in_aa → U11_aa(le_in_aa)
le_in_aa → le_out_aa(0, s)
le_in_aa → le_out_aa(0, 0)
U11_ga(le_out_aa(X, Y)) → le_out_ga(s, s)
U11_aa(le_out_aa(X, Y)) → le_out_aa(s, s)

The set Q consists of the following terms:

U11_ga(x0)
U10_ga(x0)
le_in_aa
gt_in_aa
U11_aa(x0)
U10_aa(x0)

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

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ AND

                                                        ↳ QDP

                                                          ↳ UsableRulesProof

                                                        ↳ QDP

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

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

The TRS R consists of the following rules:

gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s, 0)
U10_ga(gt_out_aa(X, Y)) → gt_out_ga(s, s)
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s, s)
le_in_aa → U11_aa(le_in_aa)
le_in_aa → le_out_aa(0, s)
le_in_aa → le_out_aa(0, 0)
U11_ga(le_out_aa(X, Y)) → le_out_ga(s, s)
U11_aa(le_out_aa(X, Y)) → le_out_aa(s, s)

The set Q consists of the following terms:

U11_ga(x0)
U10_ga(x0)
le_in_aa
gt_in_aa
U11_aa(x0)
U10_aa(x0)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ AND

                                                        ↳ QDP

                                                          ↳ UsableRulesProof

                                                            ↳ QDP

                                                              ↳ QReductionProof

                                                        ↳ QDP

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

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

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

U11_ga(x0)
U10_ga(x0)
le_in_aa
gt_in_aa
U11_aa(x0)
U10_aa(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.

U11_ga(x0)
U10_ga(x0)
le_in_aa
gt_in_aa
U11_aa(x0)
U10_aa(x0)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ AND

                                                        ↳ QDP

                                                          ↳ UsableRulesProof

                                                            ↳ QDP

                                                              ↳ QReductionProof

                                                                ↳ QDP

                                                                  ↳ NonTerminationProof

                                                        ↳ QDP

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

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

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

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

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

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ AND

                                                        ↳ QDP

                                                        ↳ QDP

                                                          ↳ Instantiation

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

PART_IN_GAAA(s) → U5_GAAA(s, U10_ga(gt_in_aa))
PART_IN_GAAA(s) → U5_GAAA(s, gt_out_ga(s, 0))
PART_IN_GAAA(s) → U7_GAAA(s, U11_ga(le_in_aa))
U5_GAAA(X, gt_out_ga(X, Y)) → PART_IN_GAAA(X)
U7_GAAA(s, le_out_ga(s, x1)) → PART_IN_GAAA(s)

The TRS R consists of the following rules:

gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s, 0)
U10_ga(gt_out_aa(X, Y)) → gt_out_ga(s, s)
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s, s)
le_in_aa → U11_aa(le_in_aa)
le_in_aa → le_out_aa(0, s)
le_in_aa → le_out_aa(0, 0)
U11_ga(le_out_aa(X, Y)) → le_out_ga(s, s)
U11_aa(le_out_aa(X, Y)) → le_out_aa(s, s)

The set Q consists of the following terms:

U11_ga(x0)
U10_ga(x0)
le_in_aa
gt_in_aa
U11_aa(x0)
U10_aa(x0)

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ AND

                                                        ↳ QDP

                                                        ↳ QDP

                                                          ↳ Instantiation

                                                            ↳ QDP

                                                              ↳ NonTerminationProof

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

U5_GAAA(s, gt_out_ga(s, x1)) → PART_IN_GAAA(s)
U5_GAAA(s, gt_out_ga(s, 0)) → PART_IN_GAAA(s)
PART_IN_GAAA(s) → U5_GAAA(s, U10_ga(gt_in_aa))
PART_IN_GAAA(s) → U7_GAAA(s, U11_ga(le_in_aa))
PART_IN_GAAA(s) → U5_GAAA(s, gt_out_ga(s, 0))
U7_GAAA(s, le_out_ga(s, x1)) → PART_IN_GAAA(s)

The TRS R consists of the following rules:

gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s, 0)
U10_ga(gt_out_aa(X, Y)) → gt_out_ga(s, s)
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s, s)
le_in_aa → U11_aa(le_in_aa)
le_in_aa → le_out_aa(0, s)
le_in_aa → le_out_aa(0, 0)
U11_ga(le_out_aa(X, Y)) → le_out_ga(s, s)
U11_aa(le_out_aa(X, Y)) → le_out_aa(s, s)

The set Q consists of the following terms:

U11_ga(x0)
U10_ga(x0)
le_in_aa
gt_in_aa
U11_aa(x0)
U10_aa(x0)

U5_GAAA(s, gt_out_ga(s, x1)) → PART_IN_GAAA(s)
U5_GAAA(s, gt_out_ga(s, 0)) → PART_IN_GAAA(s)
PART_IN_GAAA(s) → U5_GAAA(s, U10_ga(gt_in_aa))
PART_IN_GAAA(s) → U7_GAAA(s, U11_ga(le_in_aa))
PART_IN_GAAA(s) → U5_GAAA(s, gt_out_ga(s, 0))
U7_GAAA(s, le_out_ga(s, x1)) → PART_IN_GAAA(s)

The TRS R consists of the following rules:

gt_in_aa → U10_aa(gt_in_aa)
gt_in_aa → gt_out_aa(s, 0)
U10_ga(gt_out_aa(X, Y)) → gt_out_ga(s, s)
U10_aa(gt_out_aa(X, Y)) → gt_out_aa(s, s)
le_in_aa → U11_aa(le_in_aa)
le_in_aa → le_out_aa(0, s)
le_in_aa → le_out_aa(0, 0)
U11_ga(le_out_aa(X, Y)) → le_out_ga(s, s)
U11_aa(le_out_aa(X, Y)) → le_out_aa(s, s)

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

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

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

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

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

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.