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

(0) Obligation:

Clauses:

qs([], []).
qs(.(X, Xs), Ys) :- ','(part(X, Xs, Littles, Bigs), ','(qs(Littles, Ls), ','(qs(Bigs, Bs), app(Ls, .(X, Bs), Ys)))).
part(X, .(Y, Xs), .(Y, Ls), Bs) :- ','(less(X, Y), part(X, Xs, Ls, Bs)).
part(X, .(Y, Xs), Ls, .(Y, Bs)) :- part(X, Xs, Ls, Bs).
part(X1, [], [], []).
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
less(0, s(X2)).
less(s(X), s(Y)) :- less(X, Y).

Queries:

qs(a,g).

(1) PrologToPiTRSProof (SOUND transformation)

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

qs_in_ag([], []) → qs_out_ag([], [])
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, less_in_aa(X, Y))
less_in_aa(0, s(X2)) → less_out_aa(0, s(X2))
less_in_aa(s(X), s(Y)) → U9_aa(X, Y, less_in_aa(X, Y))
U9_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, less_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, less_in_ga(X, Y))
less_in_ga(0, s(X2)) → less_out_ga(0, s(X2))
less_in_ga(s(X), s(Y)) → U9_ga(X, Y, less_in_ga(X, Y))
U9_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, less_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, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X1, [], [], []) → part_out_gaaa(X1, [], [], [])
U7_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, part_in_aaaa(X, Xs, Ls, Bs))
part_in_aaaa(X1, [], [], []) → part_out_aaaa(X1, [], [], [])
U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
qs_in_aa([], []) → qs_out_aa([], [])
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
[]  =  []
qs_out_ag(x1, x2)  =  qs_out_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)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U9_aa(x1, x2, x3)  =  U9_aa(x3)
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)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1)
s(x1)  =  s(x1)
U9_ga(x1, x2, x3)  =  U9_ga(x1, x3)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
qs_out_aa(x1, x2)  =  qs_out_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
.(x1, x2)  =  .(x1, x2)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
U8_aag(x1, x2, x3, x4, x5)  =  U8_aag(x1, x4, x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

qs_in_ag([], []) → qs_out_ag([], [])
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, less_in_aa(X, Y))
less_in_aa(0, s(X2)) → less_out_aa(0, s(X2))
less_in_aa(s(X), s(Y)) → U9_aa(X, Y, less_in_aa(X, Y))
U9_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, less_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, less_in_ga(X, Y))
less_in_ga(0, s(X2)) → less_out_ga(0, s(X2))
less_in_ga(s(X), s(Y)) → U9_ga(X, Y, less_in_ga(X, Y))
U9_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, less_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, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X1, [], [], []) → part_out_gaaa(X1, [], [], [])
U7_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, part_in_aaaa(X, Xs, Ls, Bs))
part_in_aaaa(X1, [], [], []) → part_out_aaaa(X1, [], [], [])
U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
qs_in_aa([], []) → qs_out_aa([], [])
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
[]  =  []
qs_out_ag(x1, x2)  =  qs_out_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)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U9_aa(x1, x2, x3)  =  U9_aa(x3)
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)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1)
s(x1)  =  s(x1)
U9_ga(x1, x2, x3)  =  U9_ga(x1, x3)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
qs_out_aa(x1, x2)  =  qs_out_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
.(x1, x2)  =  .(x1, x2)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
U8_aag(x1, x2, x3, x4, x5)  =  U8_aag(x1, x4, x5)

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

QS_IN_AG(.(X, Xs), Ys) → U1_AG(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
QS_IN_AG(.(X, Xs), Ys) → PART_IN_AAAA(X, Xs, Littles, Bigs)
PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_AAAA(X, Y, Xs, Ls, Bs, less_in_aa(X, Y))
PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) → LESS_IN_AA(X, Y)
LESS_IN_AA(s(X), s(Y)) → U9_AA(X, Y, less_in_aa(X, Y))
LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)
U5_AAAA(X, Y, Xs, Ls, Bs, less_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, less_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, less_in_ga(X, Y))
PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) → LESS_IN_GA(X, Y)
LESS_IN_GA(s(X), s(Y)) → U9_GA(X, Y, less_in_ga(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)
U5_GAAA(X, Y, Xs, Ls, Bs, less_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, less_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, part_in_gaaa(X, Xs, Ls, Bs))
PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN_GAAA(X, Xs, Ls, Bs)
PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_AAAA(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs))
PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN_AAAA(X, Xs, Ls, Bs)
U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_AG(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → QS_IN_AA(Littles, Ls)
QS_IN_AA(.(X, Xs), Ys) → U1_AA(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
QS_IN_AA(.(X, Xs), Ys) → PART_IN_AAAA(X, Xs, Littles, Bigs)
U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_AA(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → QS_IN_AA(Littles, Ls)
U2_AA(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_AA(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U2_AA(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → QS_IN_AA(Bigs, Bs)
U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_AA(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys))
U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → APP_IN_AAA(Ls, .(X, Bs), Ys)
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → U8_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAA(Xs, Ys, Zs)
U2_AG(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_AG(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U2_AG(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → QS_IN_AA(Bigs, Bs)
U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_AG(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys))
U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → APP_IN_AAG(Ls, .(X, Bs), Ys)
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U8_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

qs_in_ag([], []) → qs_out_ag([], [])
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, less_in_aa(X, Y))
less_in_aa(0, s(X2)) → less_out_aa(0, s(X2))
less_in_aa(s(X), s(Y)) → U9_aa(X, Y, less_in_aa(X, Y))
U9_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, less_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, less_in_ga(X, Y))
less_in_ga(0, s(X2)) → less_out_ga(0, s(X2))
less_in_ga(s(X), s(Y)) → U9_ga(X, Y, less_in_ga(X, Y))
U9_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, less_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, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X1, [], [], []) → part_out_gaaa(X1, [], [], [])
U7_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, part_in_aaaa(X, Xs, Ls, Bs))
part_in_aaaa(X1, [], [], []) → part_out_aaaa(X1, [], [], [])
U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
qs_in_aa([], []) → qs_out_aa([], [])
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
[]  =  []
qs_out_ag(x1, x2)  =  qs_out_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)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U9_aa(x1, x2, x3)  =  U9_aa(x3)
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)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1)
s(x1)  =  s(x1)
U9_ga(x1, x2, x3)  =  U9_ga(x1, x3)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
qs_out_aa(x1, x2)  =  qs_out_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
.(x1, x2)  =  .(x1, x2)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
U8_aag(x1, x2, x3, x4, x5)  =  U8_aag(x1, x4, x5)
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)
LESS_IN_AA(x1, x2)  =  LESS_IN_AA
U9_AA(x1, x2, x3)  =  U9_AA(x3)
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)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
U9_GA(x1, x2, x3)  =  U9_GA(x1, x3)
U6_GAAA(x1, x2, x3, x4, x5, x6)  =  U6_GAAA(x1, x6)
U7_GAAA(x1, x2, x3, x4, x5, x6)  =  U7_GAAA(x1, x6)
U7_AAAA(x1, x2, x3, x4, x5, x6)  =  U7_AAAA(x6)
U2_AG(x1, x2, x3, x4, x5)  =  U2_AG(x3, x5)
QS_IN_AA(x1, x2)  =  QS_IN_AA
U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)
U2_AA(x1, x2, x3, x4, x5)  =  U2_AA(x5)
U3_AA(x1, x2, x3, x4, x5)  =  U3_AA(x5)
U4_AA(x1, x2, x3, x4)  =  U4_AA(x4)
APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA
U8_AAA(x1, x2, x3, x4, x5)  =  U8_AAA(x5)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x3, x5)
U4_AG(x1, x2, x3, x4)  =  U4_AG(x3, x4)
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)
U8_AAG(x1, x2, x3, x4, x5)  =  U8_AAG(x1, x4, x5)

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

(4) Obligation:

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, less_in_aa(X, Y))
PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) → LESS_IN_AA(X, Y)
LESS_IN_AA(s(X), s(Y)) → U9_AA(X, Y, less_in_aa(X, Y))
LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)
U5_AAAA(X, Y, Xs, Ls, Bs, less_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, less_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, less_in_ga(X, Y))
PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) → LESS_IN_GA(X, Y)
LESS_IN_GA(s(X), s(Y)) → U9_GA(X, Y, less_in_ga(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)
U5_GAAA(X, Y, Xs, Ls, Bs, less_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, less_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, part_in_gaaa(X, Xs, Ls, Bs))
PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN_GAAA(X, Xs, Ls, Bs)
PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_AAAA(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs))
PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN_AAAA(X, Xs, Ls, Bs)
U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_AG(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → QS_IN_AA(Littles, Ls)
QS_IN_AA(.(X, Xs), Ys) → U1_AA(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
QS_IN_AA(.(X, Xs), Ys) → PART_IN_AAAA(X, Xs, Littles, Bigs)
U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_AA(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → QS_IN_AA(Littles, Ls)
U2_AA(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_AA(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U2_AA(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → QS_IN_AA(Bigs, Bs)
U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_AA(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys))
U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → APP_IN_AAA(Ls, .(X, Bs), Ys)
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → U8_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAA(Xs, Ys, Zs)
U2_AG(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_AG(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U2_AG(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → QS_IN_AA(Bigs, Bs)
U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_AG(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys))
U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → APP_IN_AAG(Ls, .(X, Bs), Ys)
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U8_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

qs_in_ag([], []) → qs_out_ag([], [])
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, less_in_aa(X, Y))
less_in_aa(0, s(X2)) → less_out_aa(0, s(X2))
less_in_aa(s(X), s(Y)) → U9_aa(X, Y, less_in_aa(X, Y))
U9_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, less_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, less_in_ga(X, Y))
less_in_ga(0, s(X2)) → less_out_ga(0, s(X2))
less_in_ga(s(X), s(Y)) → U9_ga(X, Y, less_in_ga(X, Y))
U9_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, less_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, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X1, [], [], []) → part_out_gaaa(X1, [], [], [])
U7_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, part_in_aaaa(X, Xs, Ls, Bs))
part_in_aaaa(X1, [], [], []) → part_out_aaaa(X1, [], [], [])
U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
qs_in_aa([], []) → qs_out_aa([], [])
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
[]  =  []
qs_out_ag(x1, x2)  =  qs_out_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)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U9_aa(x1, x2, x3)  =  U9_aa(x3)
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)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1)
s(x1)  =  s(x1)
U9_ga(x1, x2, x3)  =  U9_ga(x1, x3)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
qs_out_aa(x1, x2)  =  qs_out_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
.(x1, x2)  =  .(x1, x2)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
U8_aag(x1, x2, x3, x4, x5)  =  U8_aag(x1, x4, x5)
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)
LESS_IN_AA(x1, x2)  =  LESS_IN_AA
U9_AA(x1, x2, x3)  =  U9_AA(x3)
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)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
U9_GA(x1, x2, x3)  =  U9_GA(x1, x3)
U6_GAAA(x1, x2, x3, x4, x5, x6)  =  U6_GAAA(x1, x6)
U7_GAAA(x1, x2, x3, x4, x5, x6)  =  U7_GAAA(x1, x6)
U7_AAAA(x1, x2, x3, x4, x5, x6)  =  U7_AAAA(x6)
U2_AG(x1, x2, x3, x4, x5)  =  U2_AG(x3, x5)
QS_IN_AA(x1, x2)  =  QS_IN_AA
U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)
U2_AA(x1, x2, x3, x4, x5)  =  U2_AA(x5)
U3_AA(x1, x2, x3, x4, x5)  =  U3_AA(x5)
U4_AA(x1, x2, x3, x4)  =  U4_AA(x4)
APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA
U8_AAA(x1, x2, x3, x4, x5)  =  U8_AAA(x5)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x3, x5)
U4_AG(x1, x2, x3, x4)  =  U4_AG(x3, x4)
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)
U8_AAG(x1, x2, x3, x4, x5)  =  U8_AAG(x1, x4, x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 7 SCCs with 24 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

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

The TRS R consists of the following rules:

qs_in_ag([], []) → qs_out_ag([], [])
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, less_in_aa(X, Y))
less_in_aa(0, s(X2)) → less_out_aa(0, s(X2))
less_in_aa(s(X), s(Y)) → U9_aa(X, Y, less_in_aa(X, Y))
U9_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, less_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, less_in_ga(X, Y))
less_in_ga(0, s(X2)) → less_out_ga(0, s(X2))
less_in_ga(s(X), s(Y)) → U9_ga(X, Y, less_in_ga(X, Y))
U9_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, less_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, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X1, [], [], []) → part_out_gaaa(X1, [], [], [])
U7_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, part_in_aaaa(X, Xs, Ls, Bs))
part_in_aaaa(X1, [], [], []) → part_out_aaaa(X1, [], [], [])
U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
qs_in_aa([], []) → qs_out_aa([], [])
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
[]  =  []
qs_out_ag(x1, x2)  =  qs_out_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)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U9_aa(x1, x2, x3)  =  U9_aa(x3)
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)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1)
s(x1)  =  s(x1)
U9_ga(x1, x2, x3)  =  U9_ga(x1, x3)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
qs_out_aa(x1, x2)  =  qs_out_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
.(x1, x2)  =  .(x1, x2)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
U8_aag(x1, x2, x3, x4, x5)  =  U8_aag(x1, x4, x5)
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)

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

(8) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(9) Obligation:

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

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

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

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

(10) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(11) Obligation:

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

APP_IN_AAG(.(X, Zs)) → APP_IN_AAG(Zs)

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

(12) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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_AAG(.(X, Zs)) → APP_IN_AAG(Zs)
  The graph contains the following edges 1 > 1

(13) TRUE

(14) Obligation:

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

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

The TRS R consists of the following rules:

qs_in_ag([], []) → qs_out_ag([], [])
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, less_in_aa(X, Y))
less_in_aa(0, s(X2)) → less_out_aa(0, s(X2))
less_in_aa(s(X), s(Y)) → U9_aa(X, Y, less_in_aa(X, Y))
U9_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, less_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, less_in_ga(X, Y))
less_in_ga(0, s(X2)) → less_out_ga(0, s(X2))
less_in_ga(s(X), s(Y)) → U9_ga(X, Y, less_in_ga(X, Y))
U9_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, less_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, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X1, [], [], []) → part_out_gaaa(X1, [], [], [])
U7_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, part_in_aaaa(X, Xs, Ls, Bs))
part_in_aaaa(X1, [], [], []) → part_out_aaaa(X1, [], [], [])
U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
qs_in_aa([], []) → qs_out_aa([], [])
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
[]  =  []
qs_out_ag(x1, x2)  =  qs_out_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)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U9_aa(x1, x2, x3)  =  U9_aa(x3)
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)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1)
s(x1)  =  s(x1)
U9_ga(x1, x2, x3)  =  U9_ga(x1, x3)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
qs_out_aa(x1, x2)  =  qs_out_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
.(x1, x2)  =  .(x1, x2)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
U8_aag(x1, x2, x3, x4, x5)  =  U8_aag(x1, x4, x5)
APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA

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

(15) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(16) Obligation:

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

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

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

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

(17) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(18) Obligation:

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

APP_IN_AAA → APP_IN_AAA

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

(19) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = APP_IN_AAA evaluates to t =APP_IN_AAA

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 APP_IN_AAA to APP_IN_AAA.

(20) FALSE

(21) Obligation:

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

LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)

The TRS R consists of the following rules:

qs_in_ag([], []) → qs_out_ag([], [])
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, less_in_aa(X, Y))
less_in_aa(0, s(X2)) → less_out_aa(0, s(X2))
less_in_aa(s(X), s(Y)) → U9_aa(X, Y, less_in_aa(X, Y))
U9_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, less_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, less_in_ga(X, Y))
less_in_ga(0, s(X2)) → less_out_ga(0, s(X2))
less_in_ga(s(X), s(Y)) → U9_ga(X, Y, less_in_ga(X, Y))
U9_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, less_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, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X1, [], [], []) → part_out_gaaa(X1, [], [], [])
U7_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, part_in_aaaa(X, Xs, Ls, Bs))
part_in_aaaa(X1, [], [], []) → part_out_aaaa(X1, [], [], [])
U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
qs_in_aa([], []) → qs_out_aa([], [])
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
[]  =  []
qs_out_ag(x1, x2)  =  qs_out_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)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U9_aa(x1, x2, x3)  =  U9_aa(x3)
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)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1)
s(x1)  =  s(x1)
U9_ga(x1, x2, x3)  =  U9_ga(x1, x3)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
qs_out_aa(x1, x2)  =  qs_out_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
.(x1, x2)  =  .(x1, x2)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
U8_aag(x1, x2, x3, x4, x5)  =  U8_aag(x1, x4, x5)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)

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

(22) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(23) Obligation:

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

LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)

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

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

(24) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(25) Obligation:

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

LESS_IN_GA(s(X)) → LESS_IN_GA(X)

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

(26) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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:

• LESS_IN_GA(s(X)) → LESS_IN_GA(X)
  The graph contains the following edges 1 > 1

(27) TRUE

(28) Obligation:

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

U5_GAAA(X, Y, Xs, Ls, Bs, less_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, less_in_ga(X, Y))
PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN_GAAA(X, Xs, Ls, Bs)

The TRS R consists of the following rules:

qs_in_ag([], []) → qs_out_ag([], [])
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, less_in_aa(X, Y))
less_in_aa(0, s(X2)) → less_out_aa(0, s(X2))
less_in_aa(s(X), s(Y)) → U9_aa(X, Y, less_in_aa(X, Y))
U9_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, less_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, less_in_ga(X, Y))
less_in_ga(0, s(X2)) → less_out_ga(0, s(X2))
less_in_ga(s(X), s(Y)) → U9_ga(X, Y, less_in_ga(X, Y))
U9_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, less_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, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X1, [], [], []) → part_out_gaaa(X1, [], [], [])
U7_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, part_in_aaaa(X, Xs, Ls, Bs))
part_in_aaaa(X1, [], [], []) → part_out_aaaa(X1, [], [], [])
U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
qs_in_aa([], []) → qs_out_aa([], [])
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
[]  =  []
qs_out_ag(x1, x2)  =  qs_out_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)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U9_aa(x1, x2, x3)  =  U9_aa(x3)
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)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1)
s(x1)  =  s(x1)
U9_ga(x1, x2, x3)  =  U9_ga(x1, x3)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
qs_out_aa(x1, x2)  =  qs_out_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
.(x1, x2)  =  .(x1, x2)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
U8_aag(x1, x2, x3, x4, x5)  =  U8_aag(x1, x4, x5)
PART_IN_GAAA(x1, x2, x3, x4)  =  PART_IN_GAAA(x1)
U5_GAAA(x1, x2, x3, x4, x5, x6)  =  U5_GAAA(x1, x6)

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

(29) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(30) Obligation:

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

U5_GAAA(X, Y, Xs, Ls, Bs, less_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, less_in_ga(X, Y))
PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN_GAAA(X, Xs, Ls, Bs)

The TRS R consists of the following rules:

less_in_ga(0, s(X2)) → less_out_ga(0, s(X2))
less_in_ga(s(X), s(Y)) → U9_ga(X, Y, less_in_ga(X, Y))
U9_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))

The argument filtering Pi contains the following mapping:
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1)
s(x1)  =  s(x1)
U9_ga(x1, x2, x3)  =  U9_ga(x1, x3)
.(x1, x2)  =  .(x1, x2)
PART_IN_GAAA(x1, x2, x3, x4)  =  PART_IN_GAAA(x1)
U5_GAAA(x1, x2, x3, x4, x5, x6)  =  U5_GAAA(x1, x6)

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

(31) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(32) Obligation:

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

U5_GAAA(X, less_out_ga(X)) → PART_IN_GAAA(X)
PART_IN_GAAA(X) → U5_GAAA(X, less_in_ga(X))
PART_IN_GAAA(X) → PART_IN_GAAA(X)

The TRS R consists of the following rules:

less_in_ga(0) → less_out_ga(0)
less_in_ga(s(X)) → U9_ga(X, less_in_ga(X))
U9_ga(X, less_out_ga(X)) → less_out_ga(s(X))

The set Q consists of the following terms:

less_in_ga(x0)
U9_ga(x0, x1)

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

(33) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule PART_IN_GAAA(X) → U5_GAAA(X, less_in_ga(X)) at position [1] we obtained the following new rules [LPAR04]:

PART_IN_GAAA(0) → U5_GAAA(0, less_out_ga(0))
PART_IN_GAAA(s(x0)) → U5_GAAA(s(x0), U9_ga(x0, less_in_ga(x0)))

(34) Obligation:

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

U5_GAAA(X, less_out_ga(X)) → PART_IN_GAAA(X)
PART_IN_GAAA(X) → PART_IN_GAAA(X)
PART_IN_GAAA(0) → U5_GAAA(0, less_out_ga(0))
PART_IN_GAAA(s(x0)) → U5_GAAA(s(x0), U9_ga(x0, less_in_ga(x0)))

The TRS R consists of the following rules:

less_in_ga(0) → less_out_ga(0)
less_in_ga(s(X)) → U9_ga(X, less_in_ga(X))
U9_ga(X, less_out_ga(X)) → less_out_ga(s(X))

The set Q consists of the following terms:

less_in_ga(x0)
U9_ga(x0, x1)

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

(35) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U5_GAAA(X, less_out_ga(X)) → PART_IN_GAAA(X) we obtained the following new rules [LPAR04]:

U5_GAAA(0, less_out_ga(0)) → PART_IN_GAAA(0)
U5_GAAA(s(z0), less_out_ga(s(z0))) → PART_IN_GAAA(s(z0))

(36) Obligation:

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

PART_IN_GAAA(X) → PART_IN_GAAA(X)
PART_IN_GAAA(0) → U5_GAAA(0, less_out_ga(0))
PART_IN_GAAA(s(x0)) → U5_GAAA(s(x0), U9_ga(x0, less_in_ga(x0)))
U5_GAAA(0, less_out_ga(0)) → PART_IN_GAAA(0)
U5_GAAA(s(z0), less_out_ga(s(z0))) → PART_IN_GAAA(s(z0))

The TRS R consists of the following rules:

less_in_ga(0) → less_out_ga(0)
less_in_ga(s(X)) → U9_ga(X, less_in_ga(X))
U9_ga(X, less_out_ga(X)) → less_out_ga(s(X))

The set Q consists of the following terms:

less_in_ga(x0)
U9_ga(x0, x1)

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

(37) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

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

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from PART_IN_GAAA(X) to PART_IN_GAAA(X).

(38) FALSE

(39) Obligation:

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

LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)

The TRS R consists of the following rules:

qs_in_ag([], []) → qs_out_ag([], [])
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, less_in_aa(X, Y))
less_in_aa(0, s(X2)) → less_out_aa(0, s(X2))
less_in_aa(s(X), s(Y)) → U9_aa(X, Y, less_in_aa(X, Y))
U9_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, less_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, less_in_ga(X, Y))
less_in_ga(0, s(X2)) → less_out_ga(0, s(X2))
less_in_ga(s(X), s(Y)) → U9_ga(X, Y, less_in_ga(X, Y))
U9_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, less_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, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X1, [], [], []) → part_out_gaaa(X1, [], [], [])
U7_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, part_in_aaaa(X, Xs, Ls, Bs))
part_in_aaaa(X1, [], [], []) → part_out_aaaa(X1, [], [], [])
U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
qs_in_aa([], []) → qs_out_aa([], [])
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
[]  =  []
qs_out_ag(x1, x2)  =  qs_out_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)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U9_aa(x1, x2, x3)  =  U9_aa(x3)
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)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1)
s(x1)  =  s(x1)
U9_ga(x1, x2, x3)  =  U9_ga(x1, x3)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
qs_out_aa(x1, x2)  =  qs_out_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
.(x1, x2)  =  .(x1, x2)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
U8_aag(x1, x2, x3, x4, x5)  =  U8_aag(x1, x4, x5)
LESS_IN_AA(x1, x2)  =  LESS_IN_AA

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

(40) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(41) Obligation:

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

LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)

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

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

(42) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(43) Obligation:

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

LESS_IN_AA → LESS_IN_AA

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

(44) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = LESS_IN_AA evaluates to t =LESS_IN_AA

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

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

(45) FALSE

(46) Obligation:

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

PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN_AAAA(X, Xs, Ls, Bs)

The TRS R consists of the following rules:

qs_in_ag([], []) → qs_out_ag([], [])
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, less_in_aa(X, Y))
less_in_aa(0, s(X2)) → less_out_aa(0, s(X2))
less_in_aa(s(X), s(Y)) → U9_aa(X, Y, less_in_aa(X, Y))
U9_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, less_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, less_in_ga(X, Y))
less_in_ga(0, s(X2)) → less_out_ga(0, s(X2))
less_in_ga(s(X), s(Y)) → U9_ga(X, Y, less_in_ga(X, Y))
U9_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, less_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, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X1, [], [], []) → part_out_gaaa(X1, [], [], [])
U7_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, part_in_aaaa(X, Xs, Ls, Bs))
part_in_aaaa(X1, [], [], []) → part_out_aaaa(X1, [], [], [])
U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
qs_in_aa([], []) → qs_out_aa([], [])
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
[]  =  []
qs_out_ag(x1, x2)  =  qs_out_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)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U9_aa(x1, x2, x3)  =  U9_aa(x3)
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)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1)
s(x1)  =  s(x1)
U9_ga(x1, x2, x3)  =  U9_ga(x1, x3)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
qs_out_aa(x1, x2)  =  qs_out_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
.(x1, x2)  =  .(x1, x2)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
U8_aag(x1, x2, x3, x4, x5)  =  U8_aag(x1, x4, x5)
PART_IN_AAAA(x1, x2, x3, x4)  =  PART_IN_AAAA

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

(47) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(48) Obligation:

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

PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN_AAAA(X, Xs, Ls, Bs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
PART_IN_AAAA(x1, x2, x3, x4)  =  PART_IN_AAAA

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

(49) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(50) Obligation:

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

PART_IN_AAAA → PART_IN_AAAA

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

(51) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = PART_IN_AAAA evaluates to t =PART_IN_AAAA

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

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

(52) FALSE

(53) Obligation:

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

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

The TRS R consists of the following rules:

qs_in_ag([], []) → qs_out_ag([], [])
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, less_in_aa(X, Y))
less_in_aa(0, s(X2)) → less_out_aa(0, s(X2))
less_in_aa(s(X), s(Y)) → U9_aa(X, Y, less_in_aa(X, Y))
U9_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, less_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, less_in_ga(X, Y))
less_in_ga(0, s(X2)) → less_out_ga(0, s(X2))
less_in_ga(s(X), s(Y)) → U9_ga(X, Y, less_in_ga(X, Y))
U9_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, less_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, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X1, [], [], []) → part_out_gaaa(X1, [], [], [])
U7_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, part_in_aaaa(X, Xs, Ls, Bs))
part_in_aaaa(X1, [], [], []) → part_out_aaaa(X1, [], [], [])
U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
qs_in_aa([], []) → qs_out_aa([], [])
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
[]  =  []
qs_out_ag(x1, x2)  =  qs_out_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)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U9_aa(x1, x2, x3)  =  U9_aa(x3)
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)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1)
s(x1)  =  s(x1)
U9_ga(x1, x2, x3)  =  U9_ga(x1, x3)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
qs_out_aa(x1, x2)  =  qs_out_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
.(x1, x2)  =  .(x1, x2)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
U8_aag(x1, x2, x3, x4, x5)  =  U8_aag(x1, x4, x5)
QS_IN_AA(x1, x2)  =  QS_IN_AA
U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)
U2_AA(x1, x2, x3, x4, x5)  =  U2_AA(x5)

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

(54) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(55) Obligation:

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

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

The TRS R consists of the following rules:

part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y))
part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs))
part_in_aaaa(X1, [], [], []) → part_out_aaaa(X1, [], [], [])
qs_in_aa([], []) → qs_out_aa([], [])
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U5_aaaa(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) → U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
less_in_aa(0, s(X2)) → less_out_aa(0, s(X2))
less_in_aa(s(X), s(Y)) → U9_aa(X, Y, less_in_aa(X, Y))
U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U9_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y))
part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X1, [], [], []) → part_out_gaaa(X1, [], [], [])
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys))
U5_gaaa(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) → U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
less_in_ga(0, s(X2)) → less_out_ga(0, s(X2))
less_in_ga(s(X), s(Y)) → U9_ga(X, Y, less_in_ga(X, Y))
U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U9_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
[]  =  []
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U9_aa(x1, x2, x3)  =  U9_aa(x3)
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)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1)
s(x1)  =  s(x1)
U9_ga(x1, x2, x3)  =  U9_ga(x1, x3)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1)
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
qs_in_aa(x1, x2)  =  qs_in_aa
qs_out_aa(x1, x2)  =  qs_out_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
.(x1, x2)  =  .(x1, x2)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
QS_IN_AA(x1, x2)  =  QS_IN_AA
U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)
U2_AA(x1, x2, x3, x4, x5)  =  U2_AA(x5)

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

(56) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(57) Obligation:

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

QS_IN_AA → U1_AA(part_in_aaaa)
U1_AA(part_out_aaaa) → U2_AA(qs_in_aa)
U2_AA(qs_out_aa) → QS_IN_AA
U1_AA(part_out_aaaa) → QS_IN_AA

The TRS R consists of the following rules:

part_in_aaaa → U5_aaaa(less_in_aa)
part_in_aaaa → U7_aaaa(part_in_aaaa)
part_in_aaaa → part_out_aaaa
qs_in_aa → qs_out_aa
qs_in_aa → U1_aa(part_in_aaaa)
U5_aaaa(less_out_aa(X)) → U6_aaaa(part_in_gaaa(X))
U7_aaaa(part_out_aaaa) → part_out_aaaa
U1_aa(part_out_aaaa) → U2_aa(qs_in_aa)
less_in_aa → less_out_aa(0)
less_in_aa → U9_aa(less_in_aa)
U6_aaaa(part_out_gaaa(X)) → part_out_aaaa
U2_aa(qs_out_aa) → U3_aa(qs_in_aa)
U9_aa(less_out_aa(X)) → less_out_aa(s(X))
part_in_gaaa(X) → U5_gaaa(X, less_in_ga(X))
part_in_gaaa(X) → U7_gaaa(X, part_in_gaaa(X))
part_in_gaaa(X1) → part_out_gaaa(X1)
U3_aa(qs_out_aa) → U4_aa(app_in_aaa)
U5_gaaa(X, less_out_ga(X)) → U6_gaaa(X, part_in_gaaa(X))
U7_gaaa(X, part_out_gaaa(X)) → part_out_gaaa(X)
U4_aa(app_out_aaa) → qs_out_aa
less_in_ga(0) → less_out_ga(0)
less_in_ga(s(X)) → U9_ga(X, less_in_ga(X))
U6_gaaa(X, part_out_gaaa(X)) → part_out_gaaa(X)
app_in_aaa → app_out_aaa
app_in_aaa → U8_aaa(app_in_aaa)
U9_ga(X, less_out_ga(X)) → less_out_ga(s(X))
U8_aaa(app_out_aaa) → app_out_aaa

The set Q consists of the following terms:

part_in_aaaa
qs_in_aa
U5_aaaa(x0)
U7_aaaa(x0)
U1_aa(x0)
less_in_aa
U6_aaaa(x0)
U2_aa(x0)
U9_aa(x0)
part_in_gaaa(x0)
U3_aa(x0)
U5_gaaa(x0, x1)
U7_gaaa(x0, x1)
U4_aa(x0)
less_in_ga(x0)
U6_gaaa(x0, x1)
app_in_aaa
U9_ga(x0, x1)
U8_aaa(x0)

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

(58) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule QS_IN_AA → U1_AA(part_in_aaaa) at position [0] we obtained the following new rules [LPAR04]:

QS_IN_AA → U1_AA(U5_aaaa(less_in_aa))
QS_IN_AA → U1_AA(U7_aaaa(part_in_aaaa))
QS_IN_AA → U1_AA(part_out_aaaa)

(59) Obligation:

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

U1_AA(part_out_aaaa) → U2_AA(qs_in_aa)
U2_AA(qs_out_aa) → QS_IN_AA
U1_AA(part_out_aaaa) → QS_IN_AA
QS_IN_AA → U1_AA(U5_aaaa(less_in_aa))
QS_IN_AA → U1_AA(U7_aaaa(part_in_aaaa))
QS_IN_AA → U1_AA(part_out_aaaa)

The TRS R consists of the following rules:

part_in_aaaa → U5_aaaa(less_in_aa)
part_in_aaaa → U7_aaaa(part_in_aaaa)
part_in_aaaa → part_out_aaaa
qs_in_aa → qs_out_aa
qs_in_aa → U1_aa(part_in_aaaa)
U5_aaaa(less_out_aa(X)) → U6_aaaa(part_in_gaaa(X))
U7_aaaa(part_out_aaaa) → part_out_aaaa
U1_aa(part_out_aaaa) → U2_aa(qs_in_aa)
less_in_aa → less_out_aa(0)
less_in_aa → U9_aa(less_in_aa)
U6_aaaa(part_out_gaaa(X)) → part_out_aaaa
U2_aa(qs_out_aa) → U3_aa(qs_in_aa)
U9_aa(less_out_aa(X)) → less_out_aa(s(X))
part_in_gaaa(X) → U5_gaaa(X, less_in_ga(X))
part_in_gaaa(X) → U7_gaaa(X, part_in_gaaa(X))
part_in_gaaa(X1) → part_out_gaaa(X1)
U3_aa(qs_out_aa) → U4_aa(app_in_aaa)
U5_gaaa(X, less_out_ga(X)) → U6_gaaa(X, part_in_gaaa(X))
U7_gaaa(X, part_out_gaaa(X)) → part_out_gaaa(X)
U4_aa(app_out_aaa) → qs_out_aa
less_in_ga(0) → less_out_ga(0)
less_in_ga(s(X)) → U9_ga(X, less_in_ga(X))
U6_gaaa(X, part_out_gaaa(X)) → part_out_gaaa(X)
app_in_aaa → app_out_aaa
app_in_aaa → U8_aaa(app_in_aaa)
U9_ga(X, less_out_ga(X)) → less_out_ga(s(X))
U8_aaa(app_out_aaa) → app_out_aaa

The set Q consists of the following terms:

part_in_aaaa
qs_in_aa
U5_aaaa(x0)
U7_aaaa(x0)
U1_aa(x0)
less_in_aa
U6_aaaa(x0)
U2_aa(x0)
U9_aa(x0)
part_in_gaaa(x0)
U3_aa(x0)
U5_gaaa(x0, x1)
U7_gaaa(x0, x1)
U4_aa(x0)
less_in_ga(x0)
U6_gaaa(x0, x1)
app_in_aaa
U9_ga(x0, x1)
U8_aaa(x0)

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

(60) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U1_AA(part_out_aaaa) → U2_AA(qs_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U1_AA(part_out_aaaa) → U2_AA(qs_out_aa)
U1_AA(part_out_aaaa) → U2_AA(U1_aa(part_in_aaaa))

(61) Obligation:

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

U2_AA(qs_out_aa) → QS_IN_AA
U1_AA(part_out_aaaa) → QS_IN_AA
QS_IN_AA → U1_AA(U5_aaaa(less_in_aa))
QS_IN_AA → U1_AA(U7_aaaa(part_in_aaaa))
QS_IN_AA → U1_AA(part_out_aaaa)
U1_AA(part_out_aaaa) → U2_AA(qs_out_aa)
U1_AA(part_out_aaaa) → U2_AA(U1_aa(part_in_aaaa))

The TRS R consists of the following rules:

part_in_aaaa → U5_aaaa(less_in_aa)
part_in_aaaa → U7_aaaa(part_in_aaaa)
part_in_aaaa → part_out_aaaa
qs_in_aa → qs_out_aa
qs_in_aa → U1_aa(part_in_aaaa)
U5_aaaa(less_out_aa(X)) → U6_aaaa(part_in_gaaa(X))
U7_aaaa(part_out_aaaa) → part_out_aaaa
U1_aa(part_out_aaaa) → U2_aa(qs_in_aa)
less_in_aa → less_out_aa(0)
less_in_aa → U9_aa(less_in_aa)
U6_aaaa(part_out_gaaa(X)) → part_out_aaaa
U2_aa(qs_out_aa) → U3_aa(qs_in_aa)
U9_aa(less_out_aa(X)) → less_out_aa(s(X))
part_in_gaaa(X) → U5_gaaa(X, less_in_ga(X))
part_in_gaaa(X) → U7_gaaa(X, part_in_gaaa(X))
part_in_gaaa(X1) → part_out_gaaa(X1)
U3_aa(qs_out_aa) → U4_aa(app_in_aaa)
U5_gaaa(X, less_out_ga(X)) → U6_gaaa(X, part_in_gaaa(X))
U7_gaaa(X, part_out_gaaa(X)) → part_out_gaaa(X)
U4_aa(app_out_aaa) → qs_out_aa
less_in_ga(0) → less_out_ga(0)
less_in_ga(s(X)) → U9_ga(X, less_in_ga(X))
U6_gaaa(X, part_out_gaaa(X)) → part_out_gaaa(X)
app_in_aaa → app_out_aaa
app_in_aaa → U8_aaa(app_in_aaa)
U9_ga(X, less_out_ga(X)) → less_out_ga(s(X))
U8_aaa(app_out_aaa) → app_out_aaa

The set Q consists of the following terms:

part_in_aaaa
qs_in_aa
U5_aaaa(x0)
U7_aaaa(x0)
U1_aa(x0)
less_in_aa
U6_aaaa(x0)
U2_aa(x0)
U9_aa(x0)
part_in_gaaa(x0)
U3_aa(x0)
U5_gaaa(x0, x1)
U7_gaaa(x0, x1)
U4_aa(x0)
less_in_ga(x0)
U6_gaaa(x0, x1)
app_in_aaa
U9_ga(x0, x1)
U8_aaa(x0)

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

(62) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = QS_IN_AA evaluates to t =QS_IN_AA

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

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

U1_AA(part_out_aaaa) → QS_IN_AA
with rule U1_AA(part_out_aaaa) → QS_IN_AA

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.

(63) FALSE

(64) PrologToPiTRSProof (SOUND transformation)

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

qs_in_ag([], []) → qs_out_ag([], [])
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, less_in_aa(X, Y))
less_in_aa(0, s(X2)) → less_out_aa(0, s(X2))
less_in_aa(s(X), s(Y)) → U9_aa(X, Y, less_in_aa(X, Y))
U9_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, less_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, less_in_ga(X, Y))
less_in_ga(0, s(X2)) → less_out_ga(0, s(X2))
less_in_ga(s(X), s(Y)) → U9_ga(X, Y, less_in_ga(X, Y))
U9_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, less_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, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X1, [], [], []) → part_out_gaaa(X1, [], [], [])
U7_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, part_in_aaaa(X, Xs, Ls, Bs))
part_in_aaaa(X1, [], [], []) → part_out_aaaa(X1, [], [], [])
U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
qs_in_aa([], []) → qs_out_aa([], [])
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
[]  =  []
qs_out_ag(x1, x2)  =  qs_out_ag
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)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U9_aa(x1, x2, x3)  =  U9_aa(x3)
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)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga
s(x1)  =  s(x1)
U9_ga(x1, x2, x3)  =  U9_ga(x3)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
qs_out_aa(x1, x2)  =  qs_out_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
.(x1, x2)  =  .(x1, x2)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U8_aag(x1, x2, x3, x4, x5)  =  U8_aag(x1, x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(65) Obligation:

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

qs_in_ag([], []) → qs_out_ag([], [])
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, less_in_aa(X, Y))
less_in_aa(0, s(X2)) → less_out_aa(0, s(X2))
less_in_aa(s(X), s(Y)) → U9_aa(X, Y, less_in_aa(X, Y))
U9_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, less_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, less_in_ga(X, Y))
less_in_ga(0, s(X2)) → less_out_ga(0, s(X2))
less_in_ga(s(X), s(Y)) → U9_ga(X, Y, less_in_ga(X, Y))
U9_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, less_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, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X1, [], [], []) → part_out_gaaa(X1, [], [], [])
U7_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, part_in_aaaa(X, Xs, Ls, Bs))
part_in_aaaa(X1, [], [], []) → part_out_aaaa(X1, [], [], [])
U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
qs_in_aa([], []) → qs_out_aa([], [])
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
[]  =  []
qs_out_ag(x1, x2)  =  qs_out_ag
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)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U9_aa(x1, x2, x3)  =  U9_aa(x3)
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)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga
s(x1)  =  s(x1)
U9_ga(x1, x2, x3)  =  U9_ga(x3)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
qs_out_aa(x1, x2)  =  qs_out_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
.(x1, x2)  =  .(x1, x2)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U8_aag(x1, x2, x3, x4, x5)  =  U8_aag(x1, x5)

(66) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

QS_IN_AG(.(X, Xs), Ys) → U1_AG(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
QS_IN_AG(.(X, Xs), Ys) → PART_IN_AAAA(X, Xs, Littles, Bigs)
PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_AAAA(X, Y, Xs, Ls, Bs, less_in_aa(X, Y))
PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) → LESS_IN_AA(X, Y)
LESS_IN_AA(s(X), s(Y)) → U9_AA(X, Y, less_in_aa(X, Y))
LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)
U5_AAAA(X, Y, Xs, Ls, Bs, less_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, less_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, less_in_ga(X, Y))
PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) → LESS_IN_GA(X, Y)
LESS_IN_GA(s(X), s(Y)) → U9_GA(X, Y, less_in_ga(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)
U5_GAAA(X, Y, Xs, Ls, Bs, less_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, less_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, part_in_gaaa(X, Xs, Ls, Bs))
PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN_GAAA(X, Xs, Ls, Bs)
PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_AAAA(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs))
PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN_AAAA(X, Xs, Ls, Bs)
U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_AG(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → QS_IN_AA(Littles, Ls)
QS_IN_AA(.(X, Xs), Ys) → U1_AA(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
QS_IN_AA(.(X, Xs), Ys) → PART_IN_AAAA(X, Xs, Littles, Bigs)
U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_AA(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → QS_IN_AA(Littles, Ls)
U2_AA(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_AA(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U2_AA(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → QS_IN_AA(Bigs, Bs)
U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_AA(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys))
U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → APP_IN_AAA(Ls, .(X, Bs), Ys)
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → U8_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAA(Xs, Ys, Zs)
U2_AG(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_AG(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U2_AG(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → QS_IN_AA(Bigs, Bs)
U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_AG(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys))
U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → APP_IN_AAG(Ls, .(X, Bs), Ys)
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U8_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

qs_in_ag([], []) → qs_out_ag([], [])
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, less_in_aa(X, Y))
less_in_aa(0, s(X2)) → less_out_aa(0, s(X2))
less_in_aa(s(X), s(Y)) → U9_aa(X, Y, less_in_aa(X, Y))
U9_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, less_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, less_in_ga(X, Y))
less_in_ga(0, s(X2)) → less_out_ga(0, s(X2))
less_in_ga(s(X), s(Y)) → U9_ga(X, Y, less_in_ga(X, Y))
U9_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, less_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, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X1, [], [], []) → part_out_gaaa(X1, [], [], [])
U7_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, part_in_aaaa(X, Xs, Ls, Bs))
part_in_aaaa(X1, [], [], []) → part_out_aaaa(X1, [], [], [])
U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
qs_in_aa([], []) → qs_out_aa([], [])
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
[]  =  []
qs_out_ag(x1, x2)  =  qs_out_ag
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)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U9_aa(x1, x2, x3)  =  U9_aa(x3)
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)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga
s(x1)  =  s(x1)
U9_ga(x1, x2, x3)  =  U9_ga(x3)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
qs_out_aa(x1, x2)  =  qs_out_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
.(x1, x2)  =  .(x1, x2)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U8_aag(x1, x2, x3, x4, x5)  =  U8_aag(x1, x5)
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)
LESS_IN_AA(x1, x2)  =  LESS_IN_AA
U9_AA(x1, x2, x3)  =  U9_AA(x3)
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)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
U9_GA(x1, x2, x3)  =  U9_GA(x3)
U6_GAAA(x1, x2, x3, x4, x5, x6)  =  U6_GAAA(x6)
U7_GAAA(x1, x2, x3, x4, x5, x6)  =  U7_GAAA(x6)
U7_AAAA(x1, x2, x3, x4, x5, x6)  =  U7_AAAA(x6)
U2_AG(x1, x2, x3, x4, x5)  =  U2_AG(x3, x5)
QS_IN_AA(x1, x2)  =  QS_IN_AA
U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)
U2_AA(x1, x2, x3, x4, x5)  =  U2_AA(x5)
U3_AA(x1, x2, x3, x4, x5)  =  U3_AA(x5)
U4_AA(x1, x2, x3, x4)  =  U4_AA(x4)
APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA
U8_AAA(x1, x2, x3, x4, x5)  =  U8_AAA(x5)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x3, x5)
U4_AG(x1, x2, x3, x4)  =  U4_AG(x4)
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)
U8_AAG(x1, x2, x3, x4, x5)  =  U8_AAG(x1, x5)

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

(67) Obligation:

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, less_in_aa(X, Y))
PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) → LESS_IN_AA(X, Y)
LESS_IN_AA(s(X), s(Y)) → U9_AA(X, Y, less_in_aa(X, Y))
LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)
U5_AAAA(X, Y, Xs, Ls, Bs, less_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, less_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, less_in_ga(X, Y))
PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) → LESS_IN_GA(X, Y)
LESS_IN_GA(s(X), s(Y)) → U9_GA(X, Y, less_in_ga(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)
U5_GAAA(X, Y, Xs, Ls, Bs, less_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, less_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, part_in_gaaa(X, Xs, Ls, Bs))
PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN_GAAA(X, Xs, Ls, Bs)
PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_AAAA(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs))
PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN_AAAA(X, Xs, Ls, Bs)
U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_AG(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → QS_IN_AA(Littles, Ls)
QS_IN_AA(.(X, Xs), Ys) → U1_AA(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
QS_IN_AA(.(X, Xs), Ys) → PART_IN_AAAA(X, Xs, Littles, Bigs)
U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_AA(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → QS_IN_AA(Littles, Ls)
U2_AA(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_AA(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U2_AA(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → QS_IN_AA(Bigs, Bs)
U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_AA(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys))
U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → APP_IN_AAA(Ls, .(X, Bs), Ys)
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → U8_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAA(Xs, Ys, Zs)
U2_AG(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_AG(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U2_AG(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → QS_IN_AA(Bigs, Bs)
U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_AG(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys))
U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → APP_IN_AAG(Ls, .(X, Bs), Ys)
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U8_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

qs_in_ag([], []) → qs_out_ag([], [])
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, less_in_aa(X, Y))
less_in_aa(0, s(X2)) → less_out_aa(0, s(X2))
less_in_aa(s(X), s(Y)) → U9_aa(X, Y, less_in_aa(X, Y))
U9_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, less_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, less_in_ga(X, Y))
less_in_ga(0, s(X2)) → less_out_ga(0, s(X2))
less_in_ga(s(X), s(Y)) → U9_ga(X, Y, less_in_ga(X, Y))
U9_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, less_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, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X1, [], [], []) → part_out_gaaa(X1, [], [], [])
U7_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, part_in_aaaa(X, Xs, Ls, Bs))
part_in_aaaa(X1, [], [], []) → part_out_aaaa(X1, [], [], [])
U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
qs_in_aa([], []) → qs_out_aa([], [])
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
[]  =  []
qs_out_ag(x1, x2)  =  qs_out_ag
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)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U9_aa(x1, x2, x3)  =  U9_aa(x3)
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)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga
s(x1)  =  s(x1)
U9_ga(x1, x2, x3)  =  U9_ga(x3)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
qs_out_aa(x1, x2)  =  qs_out_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
.(x1, x2)  =  .(x1, x2)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U8_aag(x1, x2, x3, x4, x5)  =  U8_aag(x1, x5)
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)
LESS_IN_AA(x1, x2)  =  LESS_IN_AA
U9_AA(x1, x2, x3)  =  U9_AA(x3)
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)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
U9_GA(x1, x2, x3)  =  U9_GA(x3)
U6_GAAA(x1, x2, x3, x4, x5, x6)  =  U6_GAAA(x6)
U7_GAAA(x1, x2, x3, x4, x5, x6)  =  U7_GAAA(x6)
U7_AAAA(x1, x2, x3, x4, x5, x6)  =  U7_AAAA(x6)
U2_AG(x1, x2, x3, x4, x5)  =  U2_AG(x3, x5)
QS_IN_AA(x1, x2)  =  QS_IN_AA
U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)
U2_AA(x1, x2, x3, x4, x5)  =  U2_AA(x5)
U3_AA(x1, x2, x3, x4, x5)  =  U3_AA(x5)
U4_AA(x1, x2, x3, x4)  =  U4_AA(x4)
APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA
U8_AAA(x1, x2, x3, x4, x5)  =  U8_AAA(x5)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x3, x5)
U4_AG(x1, x2, x3, x4)  =  U4_AG(x4)
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)
U8_AAG(x1, x2, x3, x4, x5)  =  U8_AAG(x1, x5)

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

(68) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 7 SCCs with 24 less nodes.

(69) Complex Obligation (AND)

(70) Obligation:

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

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

The TRS R consists of the following rules:

qs_in_ag([], []) → qs_out_ag([], [])
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, less_in_aa(X, Y))
less_in_aa(0, s(X2)) → less_out_aa(0, s(X2))
less_in_aa(s(X), s(Y)) → U9_aa(X, Y, less_in_aa(X, Y))
U9_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, less_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, less_in_ga(X, Y))
less_in_ga(0, s(X2)) → less_out_ga(0, s(X2))
less_in_ga(s(X), s(Y)) → U9_ga(X, Y, less_in_ga(X, Y))
U9_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, less_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, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X1, [], [], []) → part_out_gaaa(X1, [], [], [])
U7_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, part_in_aaaa(X, Xs, Ls, Bs))
part_in_aaaa(X1, [], [], []) → part_out_aaaa(X1, [], [], [])
U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
qs_in_aa([], []) → qs_out_aa([], [])
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
[]  =  []
qs_out_ag(x1, x2)  =  qs_out_ag
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)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U9_aa(x1, x2, x3)  =  U9_aa(x3)
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)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga
s(x1)  =  s(x1)
U9_ga(x1, x2, x3)  =  U9_ga(x3)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
qs_out_aa(x1, x2)  =  qs_out_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
.(x1, x2)  =  .(x1, x2)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U8_aag(x1, x2, x3, x4, x5)  =  U8_aag(x1, x5)
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)

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

(71) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(72) Obligation:

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

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

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

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

(73) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(74) Obligation:

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

APP_IN_AAG(.(X, Zs)) → APP_IN_AAG(Zs)

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

(75) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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_AAG(.(X, Zs)) → APP_IN_AAG(Zs)
  The graph contains the following edges 1 > 1

(76) TRUE

(77) Obligation:

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

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

The TRS R consists of the following rules:

qs_in_ag([], []) → qs_out_ag([], [])
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, less_in_aa(X, Y))
less_in_aa(0, s(X2)) → less_out_aa(0, s(X2))
less_in_aa(s(X), s(Y)) → U9_aa(X, Y, less_in_aa(X, Y))
U9_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, less_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, less_in_ga(X, Y))
less_in_ga(0, s(X2)) → less_out_ga(0, s(X2))
less_in_ga(s(X), s(Y)) → U9_ga(X, Y, less_in_ga(X, Y))
U9_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, less_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, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X1, [], [], []) → part_out_gaaa(X1, [], [], [])
U7_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, part_in_aaaa(X, Xs, Ls, Bs))
part_in_aaaa(X1, [], [], []) → part_out_aaaa(X1, [], [], [])
U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
qs_in_aa([], []) → qs_out_aa([], [])
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
[]  =  []
qs_out_ag(x1, x2)  =  qs_out_ag
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)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U9_aa(x1, x2, x3)  =  U9_aa(x3)
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)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga
s(x1)  =  s(x1)
U9_ga(x1, x2, x3)  =  U9_ga(x3)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
qs_out_aa(x1, x2)  =  qs_out_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
.(x1, x2)  =  .(x1, x2)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U8_aag(x1, x2, x3, x4, x5)  =  U8_aag(x1, x5)
APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA

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

(78) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(79) Obligation:

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

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

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

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

(80) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(81) Obligation:

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

APP_IN_AAA → APP_IN_AAA

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

(82) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = APP_IN_AAA evaluates to t =APP_IN_AAA

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

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

(83) FALSE

(84) Obligation:

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

LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)

The TRS R consists of the following rules:

qs_in_ag([], []) → qs_out_ag([], [])
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, less_in_aa(X, Y))
less_in_aa(0, s(X2)) → less_out_aa(0, s(X2))
less_in_aa(s(X), s(Y)) → U9_aa(X, Y, less_in_aa(X, Y))
U9_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, less_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, less_in_ga(X, Y))
less_in_ga(0, s(X2)) → less_out_ga(0, s(X2))
less_in_ga(s(X), s(Y)) → U9_ga(X, Y, less_in_ga(X, Y))
U9_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, less_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, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X1, [], [], []) → part_out_gaaa(X1, [], [], [])
U7_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, part_in_aaaa(X, Xs, Ls, Bs))
part_in_aaaa(X1, [], [], []) → part_out_aaaa(X1, [], [], [])
U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
qs_in_aa([], []) → qs_out_aa([], [])
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
[]  =  []
qs_out_ag(x1, x2)  =  qs_out_ag
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)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U9_aa(x1, x2, x3)  =  U9_aa(x3)
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)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga
s(x1)  =  s(x1)
U9_ga(x1, x2, x3)  =  U9_ga(x3)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
qs_out_aa(x1, x2)  =  qs_out_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
.(x1, x2)  =  .(x1, x2)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U8_aag(x1, x2, x3, x4, x5)  =  U8_aag(x1, x5)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)

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

(85) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(86) Obligation:

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

LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)

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

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

(87) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(88) Obligation:

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

LESS_IN_GA(s(X)) → LESS_IN_GA(X)

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

(89) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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:

• LESS_IN_GA(s(X)) → LESS_IN_GA(X)
  The graph contains the following edges 1 > 1

(90) TRUE

(91) Obligation:

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

U5_GAAA(X, Y, Xs, Ls, Bs, less_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, less_in_ga(X, Y))
PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN_GAAA(X, Xs, Ls, Bs)

The TRS R consists of the following rules:

qs_in_ag([], []) → qs_out_ag([], [])
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, less_in_aa(X, Y))
less_in_aa(0, s(X2)) → less_out_aa(0, s(X2))
less_in_aa(s(X), s(Y)) → U9_aa(X, Y, less_in_aa(X, Y))
U9_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, less_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, less_in_ga(X, Y))
less_in_ga(0, s(X2)) → less_out_ga(0, s(X2))
less_in_ga(s(X), s(Y)) → U9_ga(X, Y, less_in_ga(X, Y))
U9_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, less_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, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X1, [], [], []) → part_out_gaaa(X1, [], [], [])
U7_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, part_in_aaaa(X, Xs, Ls, Bs))
part_in_aaaa(X1, [], [], []) → part_out_aaaa(X1, [], [], [])
U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
qs_in_aa([], []) → qs_out_aa([], [])
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
[]  =  []
qs_out_ag(x1, x2)  =  qs_out_ag
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)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U9_aa(x1, x2, x3)  =  U9_aa(x3)
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)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga
s(x1)  =  s(x1)
U9_ga(x1, x2, x3)  =  U9_ga(x3)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
qs_out_aa(x1, x2)  =  qs_out_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
.(x1, x2)  =  .(x1, x2)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U8_aag(x1, x2, x3, x4, x5)  =  U8_aag(x1, x5)
PART_IN_GAAA(x1, x2, x3, x4)  =  PART_IN_GAAA(x1)
U5_GAAA(x1, x2, x3, x4, x5, x6)  =  U5_GAAA(x1, x6)

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

(92) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(93) Obligation:

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

U5_GAAA(X, Y, Xs, Ls, Bs, less_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, less_in_ga(X, Y))
PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN_GAAA(X, Xs, Ls, Bs)

The TRS R consists of the following rules:

less_in_ga(0, s(X2)) → less_out_ga(0, s(X2))
less_in_ga(s(X), s(Y)) → U9_ga(X, Y, less_in_ga(X, Y))
U9_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))

The argument filtering Pi contains the following mapping:
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga
s(x1)  =  s(x1)
U9_ga(x1, x2, x3)  =  U9_ga(x3)
.(x1, x2)  =  .(x1, x2)
PART_IN_GAAA(x1, x2, x3, x4)  =  PART_IN_GAAA(x1)
U5_GAAA(x1, x2, x3, x4, x5, x6)  =  U5_GAAA(x1, x6)

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

(94) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(95) Obligation:

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

U5_GAAA(X, less_out_ga) → PART_IN_GAAA(X)
PART_IN_GAAA(X) → U5_GAAA(X, less_in_ga(X))
PART_IN_GAAA(X) → PART_IN_GAAA(X)

The TRS R consists of the following rules:

less_in_ga(0) → less_out_ga
less_in_ga(s(X)) → U9_ga(less_in_ga(X))
U9_ga(less_out_ga) → less_out_ga

The set Q consists of the following terms:

less_in_ga(x0)
U9_ga(x0)

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

(96) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule PART_IN_GAAA(X) → U5_GAAA(X, less_in_ga(X)) at position [1] we obtained the following new rules [LPAR04]:

PART_IN_GAAA(0) → U5_GAAA(0, less_out_ga)
PART_IN_GAAA(s(x0)) → U5_GAAA(s(x0), U9_ga(less_in_ga(x0)))

(97) Obligation:

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

U5_GAAA(X, less_out_ga) → PART_IN_GAAA(X)
PART_IN_GAAA(X) → PART_IN_GAAA(X)
PART_IN_GAAA(0) → U5_GAAA(0, less_out_ga)
PART_IN_GAAA(s(x0)) → U5_GAAA(s(x0), U9_ga(less_in_ga(x0)))

The TRS R consists of the following rules:

less_in_ga(0) → less_out_ga
less_in_ga(s(X)) → U9_ga(less_in_ga(X))
U9_ga(less_out_ga) → less_out_ga

The set Q consists of the following terms:

less_in_ga(x0)
U9_ga(x0)

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

(98) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U5_GAAA(X, less_out_ga) → PART_IN_GAAA(X) we obtained the following new rules [LPAR04]:

U5_GAAA(0, less_out_ga) → PART_IN_GAAA(0)
U5_GAAA(s(z0), less_out_ga) → PART_IN_GAAA(s(z0))

(99) Obligation:

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

PART_IN_GAAA(X) → PART_IN_GAAA(X)
PART_IN_GAAA(0) → U5_GAAA(0, less_out_ga)
PART_IN_GAAA(s(x0)) → U5_GAAA(s(x0), U9_ga(less_in_ga(x0)))
U5_GAAA(0, less_out_ga) → PART_IN_GAAA(0)
U5_GAAA(s(z0), less_out_ga) → PART_IN_GAAA(s(z0))

The TRS R consists of the following rules:

less_in_ga(0) → less_out_ga
less_in_ga(s(X)) → U9_ga(less_in_ga(X))
U9_ga(less_out_ga) → less_out_ga

The set Q consists of the following terms:

less_in_ga(x0)
U9_ga(x0)

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

(100) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

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

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from PART_IN_GAAA(X) to PART_IN_GAAA(X).

(101) FALSE

(102) Obligation:

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

LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)

The TRS R consists of the following rules:

qs_in_ag([], []) → qs_out_ag([], [])
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, less_in_aa(X, Y))
less_in_aa(0, s(X2)) → less_out_aa(0, s(X2))
less_in_aa(s(X), s(Y)) → U9_aa(X, Y, less_in_aa(X, Y))
U9_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, less_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, less_in_ga(X, Y))
less_in_ga(0, s(X2)) → less_out_ga(0, s(X2))
less_in_ga(s(X), s(Y)) → U9_ga(X, Y, less_in_ga(X, Y))
U9_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, less_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, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X1, [], [], []) → part_out_gaaa(X1, [], [], [])
U7_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, part_in_aaaa(X, Xs, Ls, Bs))
part_in_aaaa(X1, [], [], []) → part_out_aaaa(X1, [], [], [])
U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
qs_in_aa([], []) → qs_out_aa([], [])
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
[]  =  []
qs_out_ag(x1, x2)  =  qs_out_ag
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)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U9_aa(x1, x2, x3)  =  U9_aa(x3)
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)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga
s(x1)  =  s(x1)
U9_ga(x1, x2, x3)  =  U9_ga(x3)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
qs_out_aa(x1, x2)  =  qs_out_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
.(x1, x2)  =  .(x1, x2)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U8_aag(x1, x2, x3, x4, x5)  =  U8_aag(x1, x5)
LESS_IN_AA(x1, x2)  =  LESS_IN_AA

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

(103) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(104) Obligation:

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

LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)

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

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

(105) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(106) Obligation:

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

LESS_IN_AA → LESS_IN_AA

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

(107) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = LESS_IN_AA evaluates to t =LESS_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 LESS_IN_AA to LESS_IN_AA.

(108) FALSE

(109) Obligation:

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

PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN_AAAA(X, Xs, Ls, Bs)

The TRS R consists of the following rules:

qs_in_ag([], []) → qs_out_ag([], [])
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, less_in_aa(X, Y))
less_in_aa(0, s(X2)) → less_out_aa(0, s(X2))
less_in_aa(s(X), s(Y)) → U9_aa(X, Y, less_in_aa(X, Y))
U9_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, less_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, less_in_ga(X, Y))
less_in_ga(0, s(X2)) → less_out_ga(0, s(X2))
less_in_ga(s(X), s(Y)) → U9_ga(X, Y, less_in_ga(X, Y))
U9_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, less_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, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X1, [], [], []) → part_out_gaaa(X1, [], [], [])
U7_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, part_in_aaaa(X, Xs, Ls, Bs))
part_in_aaaa(X1, [], [], []) → part_out_aaaa(X1, [], [], [])
U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
qs_in_aa([], []) → qs_out_aa([], [])
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
[]  =  []
qs_out_ag(x1, x2)  =  qs_out_ag
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)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U9_aa(x1, x2, x3)  =  U9_aa(x3)
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)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga
s(x1)  =  s(x1)
U9_ga(x1, x2, x3)  =  U9_ga(x3)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
qs_out_aa(x1, x2)  =  qs_out_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
.(x1, x2)  =  .(x1, x2)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U8_aag(x1, x2, x3, x4, x5)  =  U8_aag(x1, x5)
PART_IN_AAAA(x1, x2, x3, x4)  =  PART_IN_AAAA

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

(110) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(111) Obligation:

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

PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN_AAAA(X, Xs, Ls, Bs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
PART_IN_AAAA(x1, x2, x3, x4)  =  PART_IN_AAAA

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

(112) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(113) Obligation:

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

PART_IN_AAAA → PART_IN_AAAA

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

(114) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = PART_IN_AAAA evaluates to t =PART_IN_AAAA

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

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

(115) FALSE

(116) Obligation:

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

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

The TRS R consists of the following rules:

qs_in_ag([], []) → qs_out_ag([], [])
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, less_in_aa(X, Y))
less_in_aa(0, s(X2)) → less_out_aa(0, s(X2))
less_in_aa(s(X), s(Y)) → U9_aa(X, Y, less_in_aa(X, Y))
U9_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_aaaa(X, Y, Xs, Ls, Bs, less_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, less_in_ga(X, Y))
less_in_ga(0, s(X2)) → less_out_ga(0, s(X2))
less_in_ga(s(X), s(Y)) → U9_ga(X, Y, less_in_ga(X, Y))
U9_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U5_gaaa(X, Y, Xs, Ls, Bs, less_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, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X1, [], [], []) → part_out_gaaa(X1, [], [], [])
U7_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, part_in_aaaa(X, Xs, Ls, Bs))
part_in_aaaa(X1, [], [], []) → part_out_aaaa(X1, [], [], [])
U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
qs_in_aa([], []) → qs_out_aa([], [])
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) → qs_out_ag(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)
[]  =  []
qs_out_ag(x1, x2)  =  qs_out_ag
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)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U9_aa(x1, x2, x3)  =  U9_aa(x3)
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)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga
s(x1)  =  s(x1)
U9_ga(x1, x2, x3)  =  U9_ga(x3)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)
qs_in_aa(x1, x2)  =  qs_in_aa
qs_out_aa(x1, x2)  =  qs_out_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
.(x1, x2)  =  .(x1, x2)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U8_aag(x1, x2, x3, x4, x5)  =  U8_aag(x1, x5)
QS_IN_AA(x1, x2)  =  QS_IN_AA
U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)
U2_AA(x1, x2, x3, x4, x5)  =  U2_AA(x5)

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

(117) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(118) Obligation:

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

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

The TRS R consists of the following rules:

part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y))
part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs))
part_in_aaaa(X1, [], [], []) → part_out_aaaa(X1, [], [], [])
qs_in_aa([], []) → qs_out_aa([], [])
qs_in_aa(.(X, Xs), Ys) → U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
U5_aaaa(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) → U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) → U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls))
less_in_aa(0, s(X2)) → less_out_aa(0, s(X2))
less_in_aa(s(X), s(Y)) → U9_aa(X, Y, less_in_aa(X, Y))
U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) → U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
U9_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y))
part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
part_in_gaaa(X1, [], [], []) → part_out_gaaa(X1, [], [], [])
U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) → U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys))
U5_gaaa(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) → U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) → qs_out_aa(.(X, Xs), Ys)
less_in_ga(0, s(X2)) → less_out_ga(0, s(X2))
less_in_ga(s(X), s(Y)) → U9_ga(X, Y, less_in_ga(X, Y))
U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) → part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U9_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
[]  =  []
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1)
U9_aa(x1, x2, x3)  =  U9_aa(x3)
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)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga
s(x1)  =  s(x1)
U9_ga(x1, x2, x3)  =  U9_ga(x3)
U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x6)
U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x6)
part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa
part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa
U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)
qs_in_aa(x1, x2)  =  qs_in_aa
qs_out_aa(x1, x2)  =  qs_out_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)
U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x5)
U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)
.(x1, x2)  =  .(x1, x2)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
QS_IN_AA(x1, x2)  =  QS_IN_AA
U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)
U2_AA(x1, x2, x3, x4, x5)  =  U2_AA(x5)

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

(119) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(120) Obligation:

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

QS_IN_AA → U1_AA(part_in_aaaa)
U1_AA(part_out_aaaa) → U2_AA(qs_in_aa)
U2_AA(qs_out_aa) → QS_IN_AA
U1_AA(part_out_aaaa) → QS_IN_AA

The TRS R consists of the following rules:

part_in_aaaa → U5_aaaa(less_in_aa)
part_in_aaaa → U7_aaaa(part_in_aaaa)
part_in_aaaa → part_out_aaaa
qs_in_aa → qs_out_aa
qs_in_aa → U1_aa(part_in_aaaa)
U5_aaaa(less_out_aa(X)) → U6_aaaa(part_in_gaaa(X))
U7_aaaa(part_out_aaaa) → part_out_aaaa
U1_aa(part_out_aaaa) → U2_aa(qs_in_aa)
less_in_aa → less_out_aa(0)
less_in_aa → U9_aa(less_in_aa)
U6_aaaa(part_out_gaaa) → part_out_aaaa
U2_aa(qs_out_aa) → U3_aa(qs_in_aa)
U9_aa(less_out_aa(X)) → less_out_aa(s(X))
part_in_gaaa(X) → U5_gaaa(X, less_in_ga(X))
part_in_gaaa(X) → U7_gaaa(part_in_gaaa(X))
part_in_gaaa(X1) → part_out_gaaa
U3_aa(qs_out_aa) → U4_aa(app_in_aaa)
U5_gaaa(X, less_out_ga) → U6_gaaa(part_in_gaaa(X))
U7_gaaa(part_out_gaaa) → part_out_gaaa
U4_aa(app_out_aaa) → qs_out_aa
less_in_ga(0) → less_out_ga
less_in_ga(s(X)) → U9_ga(less_in_ga(X))
U6_gaaa(part_out_gaaa) → part_out_gaaa
app_in_aaa → app_out_aaa
app_in_aaa → U8_aaa(app_in_aaa)
U9_ga(less_out_ga) → less_out_ga
U8_aaa(app_out_aaa) → app_out_aaa

The set Q consists of the following terms:

part_in_aaaa
qs_in_aa
U5_aaaa(x0)
U7_aaaa(x0)
U1_aa(x0)
less_in_aa
U6_aaaa(x0)
U2_aa(x0)
U9_aa(x0)
part_in_gaaa(x0)
U3_aa(x0)
U5_gaaa(x0, x1)
U7_gaaa(x0)
U4_aa(x0)
less_in_ga(x0)
U6_gaaa(x0)
app_in_aaa
U9_ga(x0)
U8_aaa(x0)

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

(121) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule QS_IN_AA → U1_AA(part_in_aaaa) at position [0] we obtained the following new rules [LPAR04]:

QS_IN_AA → U1_AA(U5_aaaa(less_in_aa))
QS_IN_AA → U1_AA(U7_aaaa(part_in_aaaa))
QS_IN_AA → U1_AA(part_out_aaaa)

(122) Obligation:

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

U1_AA(part_out_aaaa) → U2_AA(qs_in_aa)
U2_AA(qs_out_aa) → QS_IN_AA
U1_AA(part_out_aaaa) → QS_IN_AA
QS_IN_AA → U1_AA(U5_aaaa(less_in_aa))
QS_IN_AA → U1_AA(U7_aaaa(part_in_aaaa))
QS_IN_AA → U1_AA(part_out_aaaa)

The TRS R consists of the following rules:

part_in_aaaa → U5_aaaa(less_in_aa)
part_in_aaaa → U7_aaaa(part_in_aaaa)
part_in_aaaa → part_out_aaaa
qs_in_aa → qs_out_aa
qs_in_aa → U1_aa(part_in_aaaa)
U5_aaaa(less_out_aa(X)) → U6_aaaa(part_in_gaaa(X))
U7_aaaa(part_out_aaaa) → part_out_aaaa
U1_aa(part_out_aaaa) → U2_aa(qs_in_aa)
less_in_aa → less_out_aa(0)
less_in_aa → U9_aa(less_in_aa)
U6_aaaa(part_out_gaaa) → part_out_aaaa
U2_aa(qs_out_aa) → U3_aa(qs_in_aa)
U9_aa(less_out_aa(X)) → less_out_aa(s(X))
part_in_gaaa(X) → U5_gaaa(X, less_in_ga(X))
part_in_gaaa(X) → U7_gaaa(part_in_gaaa(X))
part_in_gaaa(X1) → part_out_gaaa
U3_aa(qs_out_aa) → U4_aa(app_in_aaa)
U5_gaaa(X, less_out_ga) → U6_gaaa(part_in_gaaa(X))
U7_gaaa(part_out_gaaa) → part_out_gaaa
U4_aa(app_out_aaa) → qs_out_aa
less_in_ga(0) → less_out_ga
less_in_ga(s(X)) → U9_ga(less_in_ga(X))
U6_gaaa(part_out_gaaa) → part_out_gaaa
app_in_aaa → app_out_aaa
app_in_aaa → U8_aaa(app_in_aaa)
U9_ga(less_out_ga) → less_out_ga
U8_aaa(app_out_aaa) → app_out_aaa

The set Q consists of the following terms:

part_in_aaaa
qs_in_aa
U5_aaaa(x0)
U7_aaaa(x0)
U1_aa(x0)
less_in_aa
U6_aaaa(x0)
U2_aa(x0)
U9_aa(x0)
part_in_gaaa(x0)
U3_aa(x0)
U5_gaaa(x0, x1)
U7_gaaa(x0)
U4_aa(x0)
less_in_ga(x0)
U6_gaaa(x0)
app_in_aaa
U9_ga(x0)
U8_aaa(x0)

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

(123) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U1_AA(part_out_aaaa) → U2_AA(qs_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U1_AA(part_out_aaaa) → U2_AA(qs_out_aa)
U1_AA(part_out_aaaa) → U2_AA(U1_aa(part_in_aaaa))

(124) Obligation:

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

U2_AA(qs_out_aa) → QS_IN_AA
U1_AA(part_out_aaaa) → QS_IN_AA
QS_IN_AA → U1_AA(U5_aaaa(less_in_aa))
QS_IN_AA → U1_AA(U7_aaaa(part_in_aaaa))
QS_IN_AA → U1_AA(part_out_aaaa)
U1_AA(part_out_aaaa) → U2_AA(qs_out_aa)
U1_AA(part_out_aaaa) → U2_AA(U1_aa(part_in_aaaa))

The TRS R consists of the following rules:

part_in_aaaa → U5_aaaa(less_in_aa)
part_in_aaaa → U7_aaaa(part_in_aaaa)
part_in_aaaa → part_out_aaaa
qs_in_aa → qs_out_aa
qs_in_aa → U1_aa(part_in_aaaa)
U5_aaaa(less_out_aa(X)) → U6_aaaa(part_in_gaaa(X))
U7_aaaa(part_out_aaaa) → part_out_aaaa
U1_aa(part_out_aaaa) → U2_aa(qs_in_aa)
less_in_aa → less_out_aa(0)
less_in_aa → U9_aa(less_in_aa)
U6_aaaa(part_out_gaaa) → part_out_aaaa
U2_aa(qs_out_aa) → U3_aa(qs_in_aa)
U9_aa(less_out_aa(X)) → less_out_aa(s(X))
part_in_gaaa(X) → U5_gaaa(X, less_in_ga(X))
part_in_gaaa(X) → U7_gaaa(part_in_gaaa(X))
part_in_gaaa(X1) → part_out_gaaa
U3_aa(qs_out_aa) → U4_aa(app_in_aaa)
U5_gaaa(X, less_out_ga) → U6_gaaa(part_in_gaaa(X))
U7_gaaa(part_out_gaaa) → part_out_gaaa
U4_aa(app_out_aaa) → qs_out_aa
less_in_ga(0) → less_out_ga
less_in_ga(s(X)) → U9_ga(less_in_ga(X))
U6_gaaa(part_out_gaaa) → part_out_gaaa
app_in_aaa → app_out_aaa
app_in_aaa → U8_aaa(app_in_aaa)
U9_ga(less_out_ga) → less_out_ga
U8_aaa(app_out_aaa) → app_out_aaa

The set Q consists of the following terms:

part_in_aaaa
qs_in_aa
U5_aaaa(x0)
U7_aaaa(x0)
U1_aa(x0)
less_in_aa
U6_aaaa(x0)
U2_aa(x0)
U9_aa(x0)
part_in_gaaa(x0)
U3_aa(x0)
U5_gaaa(x0, x1)
U7_gaaa(x0)
U4_aa(x0)
less_in_ga(x0)
U6_gaaa(x0)
app_in_aaa
U9_ga(x0)
U8_aaa(x0)

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

(125) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = QS_IN_AA evaluates to t =QS_IN_AA

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

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

U1_AA(part_out_aaaa) → QS_IN_AA
with rule U1_AA(part_out_aaaa) → QS_IN_AA

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.

(126) FALSE