Left Termination proof of /tmp/tmpJNyj9y/quicksort-fb.pl

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

Clauses:

qs([], []).
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(X, [], [], []).
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
less(0, s(X)).
less(s(X), s(Y)) :- less(X, Y).

Queries:

qs(a,g).

We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
qs_in: (f,b) (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(X)) → less_out_aa(0, s(X))
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(X)) → less_out_ga(0, s(X))
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(X, [], [], []) → part_out_gaaa(X, [], [], [])
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(X, [], [], []) → part_out_aaaa(X, [], [], [])
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)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

  ↳ PrologToPiTRSProof

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

qs_in_ag([], []) → 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(X)) → less_out_aa(0, s(X))
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(X)) → less_out_ga(0, s(X))
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(X, [], [], []) → part_out_gaaa(X, [], [], [])
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(X, [], [], []) → part_out_aaaa(X, [], [], [])
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)

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(X)) → less_out_aa(0, s(X))
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(X)) → less_out_ga(0, s(X))
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(X, [], [], []) → part_out_gaaa(X, [], [], [])
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(X, [], [], []) → part_out_aaaa(X, [], [], [])
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)
U5_GAAA(x1, x2, x3, x4, x5, x6) = U5_GAAA(x1, x6)
APP_IN_AAA(x1, x2, x3) = APP_IN_AAA
U5_AAAA(x1, x2, x3, x4, x5, x6) = U5_AAAA(x6)
LESS_IN_AA(x1, x2) = LESS_IN_AA
U6_AAAA(x1, x2, x3, x4, x5, x6) = U6_AAAA(x6)
U8_AAG(x1, x2, x3, x4, x5) = U8_AAG(x1, x5)
LESS_IN_GA(x1, x2) = LESS_IN_GA(x1)
U3_AA(x1, x2, x3, x4, x5) = U3_AA(x5)
U3_AG(x1, x2, x3, x4, x5) = U3_AG(x3, x5)
U8_AAA(x1, x2, x3, x4, x5) = U8_AAA(x5)
QS_IN_AA(x1, x2) = QS_IN_AA
PART_IN_GAAA(x1, x2, x3, x4) = PART_IN_GAAA(x1)
U2_AA(x1, x2, x3, x4, x5) = U2_AA(x5)
U9_GA(x1, x2, x3) = U9_GA(x3)
U7_AAAA(x1, x2, x3, x4, x5, x6) = U7_AAAA(x6)
U1_AA(x1, x2, x3, x4) = U1_AA(x4)
QS_IN_AG(x1, x2) = QS_IN_AG(x2)
PART_IN_AAAA(x1, x2, x3, x4) = PART_IN_AAAA
U2_AG(x1, x2, x3, x4, x5) = U2_AG(x3, x5)
U1_AG(x1, x2, x3, x4) = U1_AG(x3, x4)
U4_AA(x1, x2, x3, x4) = U4_AA(x4)
U6_GAAA(x1, x2, x3, x4, x5, x6) = U6_GAAA(x6)
U4_AG(x1, x2, x3, x4) = U4_AG(x4)
APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3)
U9_AA(x1, x2, x3) = U9_AA(x3)
U7_GAAA(x1, x2, x3, x4, x5, x6) = U7_GAAA(x6)

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

  ↳ PrologToPiTRSProof

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

QS_IN_AG(.(X, Xs), Ys) → U1_AG(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
QS_IN_AG(.(X, Xs), Ys) → PART_IN_AAAA(X, Xs, Littles, Bigs)
PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_AAAA(X, Y, Xs, Ls, Bs, 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(X)) → less_out_aa(0, s(X))
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(X)) → less_out_ga(0, s(X))
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(X, [], [], []) → part_out_gaaa(X, [], [], [])
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(X, [], [], []) → part_out_aaaa(X, [], [], [])
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)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

APP_IN_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(X)) → less_out_aa(0, s(X))
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(X)) → less_out_ga(0, s(X))
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(X, [], [], []) → part_out_gaaa(X, [], [], [])
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(X, [], [], []) → part_out_aaaa(X, [], [], [])
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)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

APP_IN_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
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

APP_IN_AAG(.(X, Zs)) → APP_IN_AAG(Zs)

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

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

APP_IN_AAG(.(X, Zs)) → APP_IN_AAG(Zs)
The graph contains the following edges 1 > 1

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

APP_IN_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(X)) → less_out_aa(0, s(X))
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(X)) → less_out_ga(0, s(X))
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(X, [], [], []) → part_out_gaaa(X, [], [], [])
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(X, [], [], []) → part_out_aaaa(X, [], [], [])
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)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

APP_IN_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
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ NonTerminationProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

APP_IN_AAA → APP_IN_AAA

The TRS R consists of the following rules:none

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.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

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(X)) → less_out_aa(0, s(X))
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(X)) → less_out_ga(0, s(X))
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(X, [], [], []) → part_out_gaaa(X, [], [], [])
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(X, [], [], []) → part_out_aaaa(X, [], [], [])
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)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

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
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

LESS_IN_GA(s(X)) → LESS_IN_GA(X)

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, 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))
U5_GAAA(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) → 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(X)) → less_out_aa(0, s(X))
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(X)) → less_out_ga(0, s(X))
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(X, [], [], []) → part_out_gaaa(X, [], [], [])
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(X, [], [], []) → part_out_aaaa(X, [], [], [])
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)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

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

The TRS R consists of the following rules:

less_in_ga(0, s(X)) → less_out_ga(0, s(X))
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)
U5_GAAA(x1, x2, x3, x4, x5, x6) = U5_GAAA(x1, x6)
PART_IN_GAAA(x1, x2, x3, x4) = PART_IN_GAAA(x1)

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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(X) → U5_GAAA(X, less_in_ga(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.
By narrowing [15] the rule PART_IN_GAAA(X) → U5_GAAA(X, less_in_ga(X)) at position [1] we obtained the following new rules:

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Instantiation

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

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.
By instantiating [15] the rule U5_GAAA(X, less_out_ga) → PART_IN_GAAA(X) we obtained the following new rules:

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Instantiation

                              ↳ QDP

                                ↳ NonTerminationProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

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

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.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

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

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

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:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

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(X)) → less_out_aa(0, s(X))
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(X)) → less_out_ga(0, s(X))
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(X, [], [], []) → part_out_gaaa(X, [], [], [])
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(X, [], [], []) → part_out_aaaa(X, [], [], [])
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)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

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
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ NonTerminationProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

LESS_IN_AA → LESS_IN_AA

The TRS R consists of the following rules:none

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.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

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(X)) → less_out_aa(0, s(X))
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(X)) → less_out_ga(0, s(X))
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(X, [], [], []) → part_out_gaaa(X, [], [], [])
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(X, [], [], []) → part_out_aaaa(X, [], [], [])
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)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

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
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ NonTerminationProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

PART_IN_AAAA → PART_IN_AAAA

The TRS R consists of the following rules:none

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:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

  ↳ PrologToPiTRSProof

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

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(X)) → less_out_aa(0, s(X))
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(X)) → less_out_ga(0, s(X))
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(X, [], [], []) → part_out_gaaa(X, [], [], [])
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(X, [], [], []) → part_out_aaaa(X, [], [], [])
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)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

  ↳ PrologToPiTRSProof

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

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(X, [], [], []) → part_out_aaaa(X, [], [], [])
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(X)) → less_out_aa(0, s(X))
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(X, [], [], []) → part_out_gaaa(X, [], [], [])
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(X)) → less_out_ga(0, s(X))
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
U2_AA(x1, x2, x3, x4, x5) = U2_AA(x5)
U1_AA(x1, x2, x3, x4) = U1_AA(x4)

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

  ↳ PrologToPiTRSProof

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

QS_IN_AA → U1_AA(part_in_aaaa)
U1_AA(part_out_aaaa) → QS_IN_AA
U2_AA(qs_out_aa) → QS_IN_AA
U1_AA(part_out_aaaa) → U2_AA(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(X) → 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.
By narrowing [15] the rule QS_IN_AA → U1_AA(part_in_aaaa) at position [0] we obtained the following new rules:

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

  ↳ PrologToPiTRSProof

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

QS_IN_AA → U1_AA(U7_aaaa(part_in_aaaa))
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(part_out_aaaa)
U1_AA(part_out_aaaa) → U2_AA(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(X) → 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.
By narrowing [15] the rule U1_AA(part_out_aaaa) → U2_AA(qs_in_aa) at position [0] we obtained the following new rules:

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ NonTerminationProof

  ↳ PrologToPiTRSProof

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

QS_IN_AA → U1_AA(U7_aaaa(part_in_aaaa))
U1_AA(part_out_aaaa) → QS_IN_AA
U2_AA(qs_out_aa) → QS_IN_AA
U1_AA(part_out_aaaa) → U2_AA(U1_aa(part_in_aaaa))
QS_IN_AA → U1_AA(part_out_aaaa)
QS_IN_AA → U1_AA(U5_aaaa(less_in_aa))
U1_AA(part_out_aaaa) → U2_AA(qs_out_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(X) → 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.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

The TRS P consists of the following rules:

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

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:

Semiunifier: [ ]
Matcher: [ ]

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.

We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
qs_in: (f,b) (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(X)) → less_out_aa(0, s(X))
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(X)) → less_out_ga(0, s(X))
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(X, [], [], []) → part_out_gaaa(X, [], [], [])
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(X, [], [], []) → part_out_aaaa(X, [], [], [])
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)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

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

qs_in_ag([], []) → 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(X)) → less_out_aa(0, s(X))
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(X)) → less_out_ga(0, s(X))
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(X, [], [], []) → part_out_gaaa(X, [], [], [])
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(X, [], [], []) → part_out_aaaa(X, [], [], [])
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)

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(X)) → less_out_aa(0, s(X))
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(X)) → less_out_ga(0, s(X))
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(X, [], [], []) → part_out_gaaa(X, [], [], [])
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(X, [], [], []) → part_out_aaaa(X, [], [], [])
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)
U5_GAAA(x1, x2, x3, x4, x5, x6) = U5_GAAA(x1, x6)
APP_IN_AAA(x1, x2, x3) = APP_IN_AAA
U5_AAAA(x1, x2, x3, x4, x5, x6) = U5_AAAA(x6)
LESS_IN_AA(x1, x2) = LESS_IN_AA
U6_AAAA(x1, x2, x3, x4, x5, x6) = U6_AAAA(x6)
U8_AAG(x1, x2, x3, x4, x5) = U8_AAG(x1, x4, x5)
LESS_IN_GA(x1, x2) = LESS_IN_GA(x1)
U3_AA(x1, x2, x3, x4, x5) = U3_AA(x5)
U3_AG(x1, x2, x3, x4, x5) = U3_AG(x3, x5)
U8_AAA(x1, x2, x3, x4, x5) = U8_AAA(x5)
QS_IN_AA(x1, x2) = QS_IN_AA
PART_IN_GAAA(x1, x2, x3, x4) = PART_IN_GAAA(x1)
U2_AA(x1, x2, x3, x4, x5) = U2_AA(x5)
U9_GA(x1, x2, x3) = U9_GA(x1, x3)
U7_AAAA(x1, x2, x3, x4, x5, x6) = U7_AAAA(x6)
U1_AA(x1, x2, x3, x4) = U1_AA(x4)
QS_IN_AG(x1, x2) = QS_IN_AG(x2)
PART_IN_AAAA(x1, x2, x3, x4) = PART_IN_AAAA
U2_AG(x1, x2, x3, x4, x5) = U2_AG(x3, x5)
U1_AG(x1, x2, x3, x4) = U1_AG(x3, x4)
U4_AA(x1, x2, x3, x4) = U4_AA(x4)
U6_GAAA(x1, x2, x3, x4, x5, x6) = U6_GAAA(x1, x6)
U4_AG(x1, x2, x3, x4) = U4_AG(x3, x4)
APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3)
U9_AA(x1, x2, x3) = U9_AA(x3)
U7_GAAA(x1, x2, x3, x4, x5, x6) = U7_GAAA(x1, x6)

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

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

QS_IN_AG(.(X, Xs), Ys) → U1_AG(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
QS_IN_AG(.(X, Xs), Ys) → PART_IN_AAAA(X, Xs, Littles, Bigs)
PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_AAAA(X, Y, Xs, Ls, Bs, 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(X)) → less_out_aa(0, s(X))
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(X)) → less_out_ga(0, s(X))
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(X, [], [], []) → part_out_gaaa(X, [], [], [])
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(X, [], [], []) → part_out_aaaa(X, [], [], [])
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)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

APP_IN_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(X)) → less_out_aa(0, s(X))
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(X)) → less_out_ga(0, s(X))
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(X, [], [], []) → part_out_gaaa(X, [], [], [])
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(X, [], [], []) → part_out_aaaa(X, [], [], [])
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)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

APP_IN_AAG(.(X, Zs)) → APP_IN_AAG(Zs)

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

APP_IN_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(X)) → less_out_aa(0, s(X))
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(X)) → less_out_ga(0, s(X))
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(X, [], [], []) → part_out_gaaa(X, [], [], [])
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(X, [], [], []) → part_out_aaaa(X, [], [], [])
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)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ NonTerminationProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

APP_IN_AAA → APP_IN_AAA

The TRS R consists of the following rules:none

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.

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

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(X)) → less_out_aa(0, s(X))
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(X)) → less_out_ga(0, s(X))
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(X, [], [], []) → part_out_gaaa(X, [], [], [])
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(X, [], [], []) → part_out_aaaa(X, [], [], [])
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)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

LESS_IN_GA(s(X)) → LESS_IN_GA(X)

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, 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))
U5_GAAA(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) → 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(X)) → less_out_aa(0, s(X))
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(X)) → less_out_ga(0, s(X))
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(X, [], [], []) → part_out_gaaa(X, [], [], [])
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(X, [], [], []) → part_out_aaaa(X, [], [], [])
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)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

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

The TRS R consists of the following rules:

less_in_ga(0, s(X)) → less_out_ga(0, s(X))
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)
U5_GAAA(x1, x2, x3, x4, x5, x6) = U5_GAAA(x1, x6)
PART_IN_GAAA(x1, x2, x3, x4) = PART_IN_GAAA(x1)

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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(X) → U5_GAAA(X, less_in_ga(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.
By narrowing [15] the rule PART_IN_GAAA(X) → U5_GAAA(X, less_in_ga(X)) at position [1] we obtained the following new rules:

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Instantiation

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

PART_IN_GAAA(0) → U5_GAAA(0, less_out_ga(0))
U5_GAAA(X, less_out_ga(X)) → PART_IN_GAAA(X)
PART_IN_GAAA(X) → PART_IN_GAAA(X)
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.
By instantiating [15] the rule U5_GAAA(X, less_out_ga(X)) → PART_IN_GAAA(X) we obtained the following new rules:

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))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Instantiation

                              ↳ QDP

                                ↳ NonTerminationProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

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

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

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:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

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

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(X)) → less_out_aa(0, s(X))
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(X)) → less_out_ga(0, s(X))
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(X, [], [], []) → part_out_gaaa(X, [], [], [])
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(X, [], [], []) → part_out_aaaa(X, [], [], [])
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)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

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

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ NonTerminationProof

              ↳ PiDP

              ↳ PiDP

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

LESS_IN_AA → LESS_IN_AA

The TRS R consists of the following rules:none

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.

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

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(X)) → less_out_aa(0, s(X))
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(X)) → less_out_ga(0, s(X))
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(X, [], [], []) → part_out_gaaa(X, [], [], [])
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(X, [], [], []) → part_out_aaaa(X, [], [], [])
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)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

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

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ NonTerminationProof

              ↳ PiDP

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

PART_IN_AAAA → PART_IN_AAAA

The TRS R consists of the following rules:none

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:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

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(X)) → less_out_aa(0, s(X))
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(X)) → less_out_ga(0, s(X))
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(X, [], [], []) → part_out_gaaa(X, [], [], [])
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(X, [], [], []) → part_out_aaaa(X, [], [], [])
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)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

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

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(X, [], [], []) → part_out_aaaa(X, [], [], [])
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(X)) → less_out_aa(0, s(X))
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(X, [], [], []) → part_out_gaaa(X, [], [], [])
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(X)) → less_out_ga(0, s(X))
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
U2_AA(x1, x2, x3, x4, x5) = U2_AA(x5)
U1_AA(x1, x2, x3, x4) = U1_AA(x4)

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

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

QS_IN_AA → U1_AA(part_in_aaaa)
U1_AA(part_out_aaaa) → QS_IN_AA
U2_AA(qs_out_aa) → QS_IN_AA
U1_AA(part_out_aaaa) → U2_AA(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(X) → part_out_gaaa(X)
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.
By narrowing [15] the rule QS_IN_AA → U1_AA(part_in_aaaa) at position [0] we obtained the following new rules:

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

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

QS_IN_AA → U1_AA(U7_aaaa(part_in_aaaa))
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(part_out_aaaa)
U1_AA(part_out_aaaa) → U2_AA(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(X) → part_out_gaaa(X)
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.
By narrowing [15] the rule U1_AA(part_out_aaaa) → U2_AA(qs_in_aa) at position [0] we obtained the following new rules:

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ NonTerminationProof

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

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

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

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:

Semiunifier: [ ]
Matcher: [ ]